Beginning of Section
QBF
Theorem.
(
conj_QBF_TMZi7mcPtvMoPPfjnE5MVYgGiYyg7Qpo7Tu
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
¬
(
(
(
(
¬
X46
)
→
(
X44
↔
X48
)
)
↔
(
X48
→
X47
→
(
X46
↔
X17
)
)
)
→
(
(
¬
(
X47
→
X25
→
False
)
)
↔
(
(
¬
(
X47
↔
(
X44
→
X9
→
X48
)
)
)
↔
(
¬
(
(
X47
→
(
X44
↔
(
X36
→
X48
)
)
)
→
(
¬
(
X48
→
(
X44
↔
(
(
¬
X25
)
↔
X36
)
)
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X45
↔
(
X42
→
(
X36
↔
X31
)
)
)
)
↔
(
(
¬
(
X42
→
(
(
¬
X36
)
↔
(
X12
→
X45
)
)
)
)
↔
(
(
¬
X42
)
↔
(
X12
→
(
¬
X6
)
→
X48
)
)
)
)
→
(
¬
(
X45
→
(
(
¬
X42
)
↔
(
X12
↔
X44
)
)
)
)
→
X44
→
X4
)
↔
(
¬
(
(
(
X29
→
(
X18
↔
(
X7
↔
False
)
)
)
↔
(
X44
↔
(
X7
→
X14
)
)
)
→
(
¬
(
(
X16
→
X7
→
X42
)
→
(
(
(
¬
X7
)
→
X45
)
↔
(
¬
(
(
¬
(
X42
↔
(
X16
↔
(
X7
→
X45
)
)
)
)
→
(
¬
(
(
¬
X42
)
→
(
X29
↔
(
X16
→
X45
)
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X45
↔
(
X39
↔
(
X38
→
X32
→
False
)
)
)
)
→
(
¬
(
X39
↔
(
X38
→
(
¬
X33
)
→
X45
)
)
)
)
)
↔
(
¬
(
(
X46
→
X43
→
X38
)
→
(
¬
(
(
¬
X46
)
↔
(
X32
↔
(
(
¬
X5
)
→
X47
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
¬
(
X47
↔
(
X41
↔
(
X30
→
X40
)
)
)
)
↔
(
(
X47
↔
(
X41
→
X40
→
X26
→
False
)
)
↔
(
X41
→
X13
)
)
)
↔
(
X40
↔
(
X26
↔
False
)
)
)
→
(
(
X47
→
(
X41
↔
(
X40
→
False
)
)
)
↔
(
¬
(
X48
→
(
X47
↔
(
X40
→
X27
→
False
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X45
)
→
(
¬
X40
)
→
(
X20
↔
X48
)
)
)
→
(
(
X41
↔
(
(
¬
X24
)
↔
(
X20
→
X45
)
)
)
↔
(
(
¬
X45
)
→
(
(
¬
X41
)
↔
(
(
¬
X24
)
↔
X48
)
)
)
)
)
)
→
(
(
X45
↔
(
X41
↔
(
X37
→
X22
→
False
)
)
)
↔
(
X45
↔
(
X40
↔
(
X37
→
X22
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
X45
)
↔
(
X34
→
(
¬
X28
)
→
X49
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X45
)
→
X34
→
(
¬
X23
)
→
X49
)
)
↔
(
X49
→
(
¬
X34
)
→
(
X8
↔
X45
)
)
)
→
(
X45
↔
(
(
¬
X34
)
↔
(
X23
↔
X47
)
)
)
)
)
)
→
(
(
¬
(
(
¬
X45
)
→
X35
→
X49
)
)
↔
(
¬
(
X49
→
(
X48
↔
(
X3
↔
False
)
)
)
)
)
→
(
(
(
(
¬
X47
)
→
X45
→
X35
→
X49
)
↔
(
¬
(
X49
→
(
X47
↔
(
(
¬
X15
)
→
X35
)
)
)
)
)
↔
(
(
¬
X47
)
↔
(
X35
→
X21
→
X49
)
)
)
)
↔
(
(
(
(
¬
X35
)
↔
(
X19
↔
(
(
¬
X1
)
↔
X45
)
)
)
↔
(
X35
↔
(
X19
→
(
X1
↔
X11
)
)
)
)
↔
(
¬
(
(
¬
(
X28
→
(
X2
↔
(
X0
→
X35
)
)
)
)
→
(
(
¬
(
X45
↔
(
(
¬
X10
)
→
X11
)
)
)
↔
(
(
(
¬
X11
)
↔
(
(
¬
X2
)
↔
X45
)
)
↔
(
(
¬
X11
)
↔
(
X10
↔
X28
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
¬
(
X43
→
(
(
¬
X33
)
↔
(
X12
↔
X47
)
)
)
)
↔
(
(
¬
X38
)
→
(
(
¬
X33
)
↔
X47
)
)
)
↔
(
¬
(
(
¬
(
X43
↔
(
X23
→
(
(
¬
X2
)
↔
X38
)
)
)
)
→
(
(
(
¬
X23
)
→
(
(
¬
X7
)
↔
(
(
¬
X6
)
↔
X38
)
)
)
↔
(
(
¬
(
(
¬
X23
)
→
X7
→
(
(
¬
X6
)
↔
X43
)
)
)
↔
(
¬
(
X1
→
X47
)
)
)
)
)
)
)
→
(
¬
(
(
X49
↔
(
X47
↔
(
X22
→
X24
)
)
)
→
(
X45
↔
(
X37
↔
X22
)
)
)
)
→
(
(
¬
(
(
¬
(
X45
↔
(
(
¬
X43
)
→
X3
→
X47
)
)
)
→
(
¬
(
(
¬
X31
)
↔
(
(
¬
X23
)
→
(
X3
↔
X45
)
)
)
)
)
)
↔
(
X45
↔
(
(
¬
X43
)
↔
(
(
¬
X31
)
↔
X49
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X45
→
X5
→
X41
)
)
↔
(
(
¬
X45
)
→
(
(
¬
X32
)
↔
X49
)
)
)
→
(
(
(
¬
X40
)
↔
(
(
¬
X20
)
→
(
X10
↔
X45
)
)
)
↔
(
(
(
¬
(
(
X45
→
(
¬
X28
)
→
(
¬
X20
)
→
X49
)
→
(
¬
(
(
¬
X45
)
→
(
X40
↔
(
(
¬
X28
)
→
X49
)
)
)
)
)
)
→
(
X40
↔
(
X32
↔
(
X28
→
X49
)
)
)
)
↔
(
¬
(
X45
→
(
X32
↔
X28
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X49
↔
(
X45
↔
(
X30
↔
X47
)
)
)
)
→
(
¬
(
X49
→
(
(
¬
X26
)
↔
X45
)
)
)
)
→
(
¬
(
X35
→
(
(
¬
X30
)
↔
(
X13
↔
X44
)
)
)
)
→
(
(
(
¬
X44
)
↔
(
X41
↔
X45
)
)
↔
(
(
X41
→
X35
→
X44
)
↔
(
X13
→
X45
)
)
)
)
)
)
↔
(
(
(
(
(
(
X29
↔
False
)
↔
(
¬
(
X46
↔
(
(
¬
X39
)
↔
(
X4
↔
X41
)
)
)
)
)
→
(
¬
(
X48
→
(
X29
↔
(
X4
→
X46
)
)
)
)
)
→
(
(
¬
X34
)
→
(
¬
X25
)
→
(
X0
↔
X41
)
)
→
(
(
(
(
¬
(
X48
→
(
X46
↔
(
X34
→
X25
→
False
)
)
)
)
↔
(
X46
↔
(
(
¬
X34
)
↔
(
(
¬
X0
)
↔
X48
)
)
)
)
↔
(
¬
(
X34
↔
(
X0
↔
X46
)
)
)
)
↔
(
X48
↔
(
X25
↔
X46
)
)
)
)
↔
(
(
(
¬
X39
)
→
(
(
¬
X16
)
↔
X46
)
)
→
(
X41
↔
(
(
¬
X36
)
→
X16
→
X39
)
)
)
)
↔
(
(
(
(
(
(
(
¬
X18
)
→
X48
)
→
(
¬
(
(
¬
(
X47
↔
(
X36
→
(
X34
↔
(
X17
↔
False
)
)
)
)
)
→
X47
→
(
X36
↔
(
X34
→
X48
)
)
)
)
)
→
(
¬
(
X48
→
(
X34
↔
(
X27
↔
X18
)
)
)
)
)
↔
(
¬
(
X48
↔
(
X34
↔
(
X27
↔
False
)
)
)
)
)
↔
(
(
¬
(
(
¬
X42
)
→
(
¬
X34
)
→
X11
→
X48
)
)
↔
(
¬
(
X42
→
(
X19
↔
X47
)
)
)
)
)
↔
(
(
(
(
¬
X48
)
↔
(
(
¬
X36
)
→
X14
→
X49
)
)
↔
(
(
¬
(
X48
→
(
X14
↔
(
X8
↔
False
)
)
)
)
↔
(
¬
(
(
¬
(
X48
→
(
X36
↔
(
X14
↔
X9
)
)
)
)
→
X48
→
(
(
¬
X14
)
↔
(
(
¬
X8
)
→
X36
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X34
→
(
X15
↔
X36
)
)
)
→
(
¬
(
(
¬
X48
)
→
X21
→
(
¬
X15
)
→
X49
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
85bcc5...
and proposition id is
91aca5...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMa2LH58us8T6dUKpxm5ESmnuZLefA658at
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
∀X51 :
prop
,
(
(
¬
(
(
(
(
(
(
¬
(
(
¬
(
X47
↔
(
X2
↔
X49
)
)
)
→
(
¬
(
X48
→
(
¬
X47
)
→
X2
→
X51
)
)
)
)
→
(
¬
(
X47
↔
(
X39
→
False
)
)
)
)
↔
(
¬
(
(
¬
X47
)
→
X39
→
X38
→
X48
)
)
)
↔
(
(
¬
(
X29
→
(
¬
X16
)
→
X40
)
)
↔
(
(
¬
(
X48
→
X32
→
X29
→
X51
)
)
↔
(
¬
(
X49
↔
(
X48
→
(
(
¬
X29
)
↔
X32
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
X36
)
→
(
¬
X31
)
→
X51
)
)
↔
(
(
X50
→
X31
→
X13
→
X0
)
→
(
¬
(
X50
↔
(
X48
↔
(
X31
→
(
X0
↔
False
)
)
)
)
)
)
)
↔
(
(
X49
↔
(
X31
↔
(
X13
↔
X36
)
)
)
↔
(
X31
→
(
(
¬
X7
)
↔
X49
)
)
)
)
↔
(
(
(
(
(
(
X44
↔
(
X23
→
(
(
¬
X21
)
↔
X51
)
)
)
↔
(
X44
→
(
X42
↔
(
X23
↔
X21
)
)
)
)
↔
(
(
¬
X44
)
→
(
(
¬
X42
)
↔
X51
)
)
)
→
(
X51
↔
(
X42
↔
(
X28
↔
False
)
)
)
)
↔
(
X51
→
X44
→
X28
)
)
↔
(
(
(
¬
(
(
X42
→
(
X23
↔
X49
)
)
→
(
¬
(
X49
→
(
¬
X21
)
→
(
(
¬
X6
)
↔
X23
)
)
)
)
)
→
(
(
(
¬
(
X42
↔
(
X11
↔
X49
)
)
)
↔
(
¬
(
X50
→
X21
→
(
X11
↔
X49
)
)
)
)
↔
(
(
¬
X11
)
↔
X50
)
)
)
↔
(
X23
↔
(
X21
→
X8
→
(
X6
↔
False
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
X49
)
→
(
¬
X46
)
→
(
¬
X4
)
→
X50
)
→
(
X50
↔
(
X45
↔
X46
)
)
→
(
¬
(
(
¬
(
X50
→
(
X49
↔
(
X46
↔
X45
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X49
)
↔
(
X27
↔
X50
)
)
)
→
(
X50
↔
(
X45
→
X49
)
)
→
(
(
¬
(
X49
↔
(
X45
→
(
(
¬
X22
)
↔
X50
)
)
)
)
↔
(
¬
(
X50
→
(
(
¬
X27
)
↔
(
X22
→
X49
)
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X46
↔
(
X41
↔
(
X15
→
X51
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X45
→
(
(
¬
X41
)
↔
(
X24
→
X51
)
)
)
)
→
(
¬
(
X46
→
(
X41
↔
(
(
¬
X5
)
↔
X51
)
)
)
)
)
)
→
(
X46
↔
(
(
¬
X24
)
→
(
X20
↔
X41
)
)
)
)
)
)
)
→
(
¬
(
(
¬
X46
)
→
(
X41
↔
X50
)
)
)
→
(
(
X46
↔
(
X41
→
(
¬
X34
)
→
X50
)
)
↔
(
¬
(
X46
↔
(
(
¬
X34
)
→
(
(
¬
X24
)
↔
X41
)
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
(
¬
(
X51
↔
(
X26
→
X37
)
)
)
→
(
(
¬
(
X26
→
(
(
¬
X25
)
↔
(
X12
↔
X51
)
)
)
)
↔
(
(
¬
X37
)
→
(
(
¬
X25
)
↔
X51
)
)
)
)
)
↔
(
¬
(
X51
↔
(
X26
↔
X12
)
)
)
)
→
(
(
X19
→
X12
→
(
(
¬
X9
)
↔
X33
)
)
→
(
¬
(
(
X51
↔
(
(
¬
X25
)
→
(
X19
↔
X33
)
)
)
→
(
X51
↔
(
(
¬
X19
)
→
(
¬
X12
)
→
X33
)
)
)
)
)
→
(
X51
↔
(
X25
↔
(
X19
→
False
)
)
)
)
↔
(
(
¬
(
(
¬
X26
)
↔
(
(
¬
X25
)
↔
(
(
¬
X10
)
↔
X45
)
)
)
)
↔
(
X26
→
X19
→
X45
)
)
)
↔
(
(
(
¬
(
X51
→
X45
→
X46
)
)
↔
(
(
¬
(
X51
→
(
X45
↔
(
X35
↔
X3
)
)
)
)
↔
(
(
¬
X46
)
→
(
¬
X3
)
→
X51
)
)
)
→
(
¬
(
(
(
X45
↔
(
X43
→
(
¬
X37
)
→
X51
)
)
↔
(
(
X51
→
X37
→
(
¬
X17
)
→
X43
)
↔
(
X43
→
X30
→
(
(
¬
X17
)
↔
X51
)
)
)
)
→
(
¬
(
(
(
¬
(
X37
→
(
¬
X1
)
→
X14
)
)
↔
(
(
X30
→
X1
→
X37
)
↔
(
¬
(
X51
→
X30
→
X18
→
X1
)
)
)
)
→
(
¬
(
X51
→
(
¬
X37
)
→
(
¬
X30
)
→
X46
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
X51
→
(
X46
↔
(
X41
→
X28
)
)
)
↔
(
(
(
¬
(
(
X41
→
X28
→
(
X24
↔
X21
)
)
→
(
¬
(
X51
→
(
¬
X24
)
→
X28
)
)
)
)
→
(
X51
↔
False
)
)
↔
(
X24
↔
(
X21
→
X51
)
)
)
)
↔
(
(
¬
(
(
X46
→
(
(
¬
X7
)
↔
X48
)
)
→
(
¬
(
(
(
¬
X28
)
↔
(
(
¬
X1
)
→
X48
)
)
→
(
X46
↔
(
(
¬
X41
)
↔
(
(
¬
X28
)
→
X48
)
)
)
)
)
)
)
↔
(
¬
(
X50
↔
(
X9
↔
(
X1
↔
X41
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
¬
X48
)
→
X47
→
(
X32
↔
X51
)
)
)
→
(
X51
↔
(
X48
↔
(
(
¬
X12
)
→
X47
)
)
)
)
)
↔
(
(
(
X51
→
X12
→
(
X6
↔
X50
)
)
→
(
¬
(
X51
→
(
(
¬
X12
)
↔
(
X6
↔
X47
)
)
)
)
)
↔
(
¬
(
X50
↔
(
X47
→
(
X4
↔
X51
)
)
)
)
)
)
→
(
X50
↔
(
(
¬
X48
)
↔
(
(
¬
X3
)
→
X51
)
)
)
)
)
→
(
(
(
(
(
X51
→
X39
→
X48
)
↔
(
X45
→
(
(
¬
X39
)
↔
(
(
¬
X29
)
→
X48
)
)
)
)
↔
(
¬
(
X51
↔
(
X39
↔
(
X11
↔
False
)
)
)
)
)
↔
(
(
(
¬
(
X51
↔
(
X48
↔
False
)
)
)
↔
(
(
X51
→
(
(
¬
X43
)
↔
X50
)
)
→
(
(
¬
(
X48
→
X43
→
X51
)
)
↔
(
¬
(
(
¬
(
(
¬
X50
)
↔
(
X43
↔
(
X30
→
X51
)
)
)
)
→
(
¬
(
X51
→
(
X48
↔
(
(
¬
X30
)
↔
X43
)
)
)
)
)
)
)
)
)
→
(
(
X45
↔
(
(
¬
X44
)
→
X8
→
X51
)
)
↔
(
¬
(
X45
→
(
X43
↔
(
X8
→
X50
)
)
)
)
)
)
)
↔
(
(
X42
↔
(
(
¬
X20
)
↔
(
(
¬
X17
)
↔
X51
)
)
)
→
(
(
¬
(
(
¬
X5
)
→
X27
)
)
↔
(
(
¬
(
X48
↔
(
(
¬
X42
)
→
X45
)
)
)
→
(
¬
(
(
¬
X42
)
→
X27
→
(
¬
X5
)
→
X48
)
)
→
(
¬
(
X45
→
(
(
¬
X27
)
↔
(
X5
→
X42
)
)
)
)
)
)
→
(
¬
(
(
(
¬
X27
)
→
(
¬
X20
)
→
X42
)
→
(
(
(
¬
X36
)
→
(
¬
X20
)
→
X42
)
↔
(
(
X36
↔
(
X20
→
(
X17
↔
X42
)
)
)
↔
(
X48
↔
(
X20
→
X27
)
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
(
(
(
X40
↔
(
(
¬
X35
)
→
(
¬
X10
)
→
X43
)
)
↔
(
X49
↔
(
(
¬
X35
)
→
(
(
¬
X10
)
↔
X40
)
)
)
)
→
(
¬
(
(
¬
X40
)
→
(
¬
X35
)
→
(
(
¬
X14
)
↔
X43
)
)
)
→
(
¬
X40
)
→
(
X26
↔
(
X14
→
X43
)
)
)
↔
(
X43
→
(
X35
↔
X40
)
)
)
→
(
¬
(
(
(
X47
↔
(
(
¬
X34
)
↔
(
(
¬
X33
)
→
X38
)
)
)
↔
(
X43
↔
(
(
¬
X35
)
→
X47
)
)
)
→
(
X47
↔
(
X40
→
(
(
¬
X25
)
↔
X33
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
X43
→
X34
→
(
X18
↔
False
)
)
→
(
¬
(
X23
↔
(
X18
→
X0
→
X43
)
)
)
)
→
(
X43
↔
(
X18
→
X34
)
)
)
)
→
(
(
¬
X34
)
↔
(
X23
→
(
¬
X18
)
→
X35
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
X48
→
(
X19
↔
(
(
¬
X16
)
↔
X38
)
)
)
)
→
(
¬
(
(
¬
X47
)
→
(
(
¬
X38
)
↔
(
X15
↔
X48
)
)
)
)
→
(
¬
(
X48
↔
(
X38
↔
(
X16
↔
X47
)
)
)
)
)
→
(
X38
↔
(
X19
↔
X48
)
)
)
)
→
(
¬
(
(
(
¬
(
X43
↔
X47
)
)
↔
(
X48
↔
(
X43
↔
(
X37
→
(
X22
↔
False
)
)
)
)
)
→
(
X48
↔
(
X47
→
(
X19
↔
X43
)
)
)
)
)
)
↔
(
(
(
¬
(
X47
→
(
X43
↔
(
X38
→
X2
)
)
)
)
↔
(
(
¬
(
X43
↔
(
(
¬
X31
)
→
X38
)
)
)
→
X38
→
(
(
¬
X31
)
↔
(
X13
→
X47
)
)
)
)
→
(
¬
X38
)
→
X31
→
X43
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
e78b44...
and proposition id is
2a8060...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMFgySSbYB9e9WAcz5pUfwcwWjD4TSD46z7
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
¬
(
(
¬
(
(
¬
(
X40
→
(
X17
↔
(
X10
→
False
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X36
)
→
(
(
¬
X21
)
↔
(
X17
→
X43
)
)
)
)
→
(
¬
(
(
¬
X40
)
↔
(
X36
→
(
X21
↔
X43
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
X38
)
↔
(
X30
↔
(
(
¬
X21
)
→
X43
)
)
)
)
↔
(
¬
(
(
¬
(
X44
→
(
(
¬
X12
)
↔
X36
)
)
)
→
(
¬
(
X44
↔
(
X43
→
(
X12
↔
X30
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
X38
)
→
(
(
¬
X35
)
↔
(
(
¬
X8
)
↔
X44
)
)
)
↔
(
(
¬
X35
)
↔
(
X8
↔
X45
)
)
)
→
(
(
¬
(
X45
→
(
(
¬
X38
)
↔
X44
)
)
)
↔
(
¬
(
(
X44
→
X32
→
X38
)
→
(
¬
(
X45
↔
(
X32
↔
(
X7
→
X38
)
)
)
)
)
)
)
)
)
↔
(
(
X40
↔
(
(
¬
X38
)
→
X35
→
X44
)
)
↔
(
¬
(
(
¬
X38
)
→
X35
→
(
(
¬
X24
)
↔
X44
)
)
)
)
)
)
↔
(
(
(
(
X44
→
X37
→
(
(
¬
X19
)
↔
X25
)
)
↔
(
¬
(
(
X40
↔
(
(
¬
X25
)
→
(
¬
X19
)
→
X44
)
)
→
(
(
(
¬
(
(
¬
(
(
¬
X40
)
↔
(
X34
→
(
X13
↔
X44
)
)
)
)
→
X44
→
(
¬
X34
)
→
(
X25
↔
X40
)
)
)
→
(
X44
↔
(
X25
↔
X34
)
)
)
↔
(
(
¬
X13
)
↔
X44
)
)
)
)
)
↔
(
(
(
(
¬
X37
)
↔
(
X26
→
X40
)
)
→
X44
→
X43
→
(
X26
↔
X40
)
)
→
(
¬
(
(
¬
X43
)
↔
(
X37
→
X44
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
(
X43
→
X41
→
(
X34
↔
X37
)
)
↔
(
¬
(
(
¬
X41
)
↔
X46
)
)
)
→
(
X46
↔
(
X43
→
(
X37
↔
(
X11
↔
False
)
)
)
)
)
)
→
(
¬
(
X41
→
X37
→
X23
→
X43
)
)
)
)
↔
(
(
X41
→
X15
→
(
¬
X2
)
→
X43
)
↔
(
¬
(
(
X43
→
(
(
¬
X42
)
↔
(
(
¬
X41
)
↔
X46
)
)
)
→
(
¬
(
X43
↔
(
(
¬
X42
)
→
(
¬
X15
)
→
X46
)
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
(
X48
↔
(
X9
↔
(
X3
→
X39
)
)
)
)
→
(
¬
X33
)
→
(
X9
↔
(
(
¬
X3
)
↔
X39
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X42
)
↔
(
X39
→
X16
→
X48
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X39
)
→
X33
→
X42
)
)
)
)
)
)
↔
(
(
¬
(
X42
→
X33
→
(
¬
X18
)
→
X47
)
)
→
(
¬
(
X47
→
(
¬
X18
)
→
(
(
¬
X4
)
↔
X31
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
¬
(
X6
↔
X39
)
)
↔
(
(
¬
X43
)
→
X29
→
X49
)
)
→
X29
→
(
X27
↔
X39
)
)
↔
(
(
(
¬
(
X49
↔
(
X45
↔
(
X42
→
X39
)
)
)
)
↔
(
¬
(
X49
↔
(
(
¬
X43
)
↔
(
(
¬
X39
)
→
X45
)
)
)
)
)
↔
(
(
¬
X43
)
↔
(
(
¬
X39
)
↔
(
X1
→
X49
)
)
)
)
)
→
(
(
(
¬
X42
)
→
(
¬
X27
)
→
X48
)
↔
(
(
(
¬
(
(
¬
X39
)
→
(
(
¬
X27
)
↔
(
X20
↔
X42
)
)
)
)
→
(
¬
(
(
¬
X39
)
→
X27
→
X5
→
X42
)
)
)
↔
(
¬
(
X42
→
X27
→
X48
)
)
)
)
)
→
(
(
(
¬
(
X49
↔
(
X45
→
X28
→
X48
)
)
)
↔
(
¬
(
(
X48
↔
(
(
¬
X28
)
↔
(
X0
→
X49
)
)
)
→
(
X48
↔
(
(
¬
X28
)
↔
(
(
¬
X22
)
↔
X49
)
)
)
)
)
)
↔
(
¬
(
(
¬
X28
)
↔
(
X14
→
X49
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
(
¬
X48
)
↔
(
X37
↔
(
X26
↔
X49
)
)
)
)
→
(
(
¬
X37
)
→
X47
)
→
(
(
¬
(
(
¬
X48
)
↔
(
X11
↔
X49
)
)
)
↔
(
(
(
X48
→
(
(
¬
X37
)
↔
X49
)
)
→
(
X47
↔
(
X37
→
(
X11
↔
False
)
)
)
)
→
(
(
¬
X48
)
↔
(
X37
→
X49
)
)
)
)
)
→
(
¬
(
(
(
(
¬
X27
)
↔
(
X25
↔
X48
)
)
↔
(
¬
(
(
¬
X45
)
↔
(
(
¬
X30
)
→
(
X25
↔
X48
)
)
)
)
)
→
(
¬
(
X37
→
(
(
¬
X33
)
↔
(
(
¬
X25
)
↔
X48
)
)
)
)
→
(
(
¬
X47
)
↔
(
X37
↔
(
X33
↔
X48
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X39
→
X3
→
X36
)
)
→
(
X49
↔
(
(
¬
X9
)
→
X3
→
X36
)
)
)
↔
(
(
(
¬
X9
)
→
X3
→
X39
)
↔
(
¬
(
X39
→
(
X9
↔
(
(
¬
X2
)
↔
X31
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X49
→
(
¬
X45
)
→
(
¬
X36
)
→
X47
)
)
↔
(
¬
(
X45
↔
(
(
¬
X39
)
→
(
¬
X36
)
→
X47
)
)
)
)
→
(
¬
(
X47
→
(
X18
↔
(
X5
→
X39
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X19
↔
X39
)
)
→
X49
→
(
¬
X21
)
→
(
(
¬
X19
)
↔
X45
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
(
X43
→
(
X35
↔
(
X22
→
False
)
)
)
→
(
X35
↔
(
X22
→
X43
)
)
→
(
X43
↔
(
X42
↔
(
(
¬
X17
)
↔
X22
)
)
)
)
)
→
(
¬
(
(
¬
(
X28
→
X12
)
)
→
(
¬
(
(
(
¬
(
(
¬
X6
)
↔
X42
)
)
↔
(
(
X42
↔
(
(
¬
X6
)
→
X14
)
)
↔
(
(
X35
↔
(
(
¬
X6
)
↔
X22
)
)
↔
(
(
X35
↔
(
(
¬
X22
)
→
(
X6
↔
X42
)
)
)
→
(
(
¬
X22
)
↔
(
X14
↔
X42
)
)
)
)
)
)
→
(
X42
↔
(
(
¬
X14
)
↔
(
(
¬
X6
)
→
X38
)
)
)
)
)
)
)
)
↔
(
(
X34
→
X22
→
X43
)
↔
(
(
X43
→
(
X35
↔
X4
)
)
→
(
¬
(
(
¬
X34
)
↔
(
X22
→
X4
→
X35
)
)
)
)
)
)
→
(
X42
→
X35
→
(
X13
↔
X32
)
)
→
(
(
(
¬
(
X46
→
(
(
¬
X40
)
↔
(
(
¬
X7
)
→
X45
)
)
)
)
↔
(
(
¬
(
X46
↔
(
X45
→
(
¬
X32
)
→
X42
)
)
)
→
(
(
¬
X45
)
↔
(
(
¬
X42
)
↔
(
X40
→
X46
)
)
)
)
)
↔
(
(
(
(
X42
↔
(
(
¬
X40
)
→
(
X34
↔
X45
)
)
)
↔
(
X45
→
X42
→
X15
→
X40
)
)
↔
(
X45
↔
(
(
¬
X40
)
↔
X42
)
)
)
→
(
¬
(
X34
→
(
(
¬
X15
)
↔
X45
)
)
)
→
(
(
(
(
X45
↔
(
(
¬
X13
)
↔
(
X8
→
X34
)
)
)
↔
(
(
¬
X34
)
↔
(
(
¬
X15
)
↔
(
(
¬
X8
)
↔
X45
)
)
)
)
→
(
(
¬
X13
)
↔
(
(
¬
X8
)
↔
X45
)
)
)
↔
(
X15
↔
(
X8
→
False
)
)
)
)
)
)
)
↔
(
(
(
(
X47
↔
(
X44
→
(
X43
↔
X48
)
)
)
→
(
X47
↔
(
X43
→
(
(
¬
X40
)
↔
X44
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X48
↔
(
X43
↔
(
(
¬
X16
)
→
X34
)
)
)
)
→
(
(
X34
↔
(
(
¬
X20
)
→
X16
→
X44
)
)
↔
(
X44
↔
(
X43
↔
(
X16
→
False
)
)
)
)
)
→
(
¬
(
(
¬
X43
)
→
X44
)
)
)
)
↔
(
X44
→
(
X20
↔
X43
)
)
)
→
(
¬
(
(
¬
X20
)
↔
(
(
¬
X16
)
→
(
(
¬
X0
)
↔
X44
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X40
→
(
¬
X23
)
→
X29
)
)
→
(
¬
(
X41
→
(
X40
↔
X23
)
)
)
→
(
¬
(
X48
→
X34
→
(
¬
X23
)
→
X41
)
)
)
→
(
¬
(
(
¬
X41
)
↔
(
(
¬
X29
)
↔
X48
)
)
)
)
)
→
(
X41
→
X34
)
→
(
(
¬
X34
)
→
(
X10
↔
(
(
¬
X1
)
→
X41
)
)
)
→
(
¬
(
X48
↔
(
(
¬
X10
)
→
X24
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
2c0643...
and proposition id is
856af8...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMUU3hfiPazPq1pJ3daHqJxDPTKXWU4ZUiJ
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
∀X51 :
prop
,
∀X52 :
prop
,
(
(
(
(
(
(
¬
(
X52
↔
(
X50
↔
(
(
¬
X18
)
↔
X49
)
)
)
)
↔
(
¬
(
(
¬
X49
)
→
(
¬
X40
)
→
X18
→
X50
)
)
)
↔
(
(
(
(
(
(
X48
→
(
¬
X16
)
→
X52
)
↔
(
¬
(
(
¬
(
X48
↔
(
X42
→
(
(
¬
X36
)
↔
X43
)
)
)
)
→
(
X49
↔
(
X42
→
(
(
¬
X36
)
↔
X43
)
)
)
)
)
)
→
(
(
¬
X22
)
↔
(
X16
→
X49
)
)
→
(
¬
(
X36
→
X16
→
X7
→
X49
)
)
)
→
(
X48
↔
(
(
¬
X43
)
↔
X49
)
)
)
↔
(
(
X49
→
(
X43
↔
(
X28
↔
X26
)
)
)
↔
(
¬
(
(
¬
X49
)
↔
(
(
¬
X43
)
→
(
(
¬
X28
)
↔
X52
)
)
)
)
)
)
↔
(
(
X50
→
(
X42
↔
(
(
¬
X3
)
→
X31
)
)
)
→
(
(
(
¬
X43
)
→
(
¬
X3
)
→
X51
)
↔
(
(
¬
X51
)
↔
(
(
¬
X42
)
→
X52
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X27
→
(
¬
(
(
¬
(
X37
→
X35
→
(
X24
↔
X27
)
)
)
→
(
¬
(
X37
↔
(
X35
→
(
X24
↔
X6
)
)
)
)
)
)
)
)
→
(
¬
(
X35
↔
(
X27
↔
(
X24
→
(
X2
↔
False
)
)
)
)
)
)
→
(
(
X43
↔
(
(
¬
X17
)
↔
(
X14
→
X27
)
)
)
↔
(
(
(
¬
(
X37
↔
(
(
¬
X27
)
→
(
X10
↔
X43
)
)
)
)
↔
(
X17
→
(
¬
X10
)
→
X43
)
)
↔
(
¬
(
X43
→
(
X35
↔
(
X27
↔
X10
)
)
)
)
)
)
)
→
(
(
(
(
X27
→
X15
→
X12
)
↔
(
X50
→
(
X43
↔
(
X15
↔
(
X12
↔
False
)
)
)
)
)
↔
(
¬
(
X50
→
(
(
¬
X33
)
↔
(
(
¬
X27
)
→
X43
)
)
)
)
)
↔
(
(
X42
↔
(
X27
→
(
¬
X0
)
→
X15
)
)
↔
(
(
X42
→
X27
→
X15
→
(
X8
↔
False
)
)
→
(
(
X43
→
(
(
¬
X15
)
↔
(
(
¬
X8
)
→
X35
)
)
)
↔
(
X35
→
X15
→
X8
→
X42
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
(
(
X46
↔
X32
)
→
(
X46
→
X29
)
→
(
(
¬
(
X46
→
X5
→
X29
)
)
↔
(
¬
(
(
¬
X39
)
↔
X45
)
)
)
)
→
(
¬
(
X45
↔
(
(
¬
X32
)
↔
(
(
¬
X20
)
→
X39
)
)
)
)
)
)
→
(
¬
(
X46
↔
(
X45
↔
(
X39
↔
False
)
)
)
)
)
→
(
(
¬
(
X50
→
X45
→
(
¬
X20
)
→
X47
)
)
↔
(
¬
(
(
¬
(
(
¬
X47
)
↔
(
X45
→
X34
→
X52
)
)
)
→
(
¬
(
X47
↔
(
X45
→
X20
→
X52
)
)
)
)
)
)
→
(
(
(
(
¬
(
X47
→
(
(
¬
X19
)
↔
(
X9
↔
X29
)
)
)
)
↔
X52
)
↔
(
X47
→
(
¬
X29
)
→
X52
)
)
↔
(
(
X45
→
(
X13
↔
X52
)
)
↔
(
(
(
X29
↔
(
(
¬
X13
)
→
X47
)
)
↔
(
¬
(
(
¬
(
X45
→
(
¬
X13
)
→
X52
)
)
→
(
¬
(
X47
→
(
¬
X29
)
→
(
(
¬
X13
)
↔
X45
)
)
)
)
)
)
→
X45
→
X47
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
(
¬
(
X46
→
(
X44
↔
(
X21
→
X1
)
)
)
)
↔
(
¬
(
(
¬
X46
)
↔
(
(
¬
X44
)
→
(
X1
↔
X50
)
)
)
)
)
→
X46
→
(
X1
↔
X21
)
)
)
→
(
(
X47
↔
(
X1
↔
X4
)
)
↔
(
(
¬
X21
)
↔
(
X4
→
(
(
¬
X1
)
↔
X50
)
)
)
)
)
)
↔
(
(
(
(
X41
↔
(
X23
→
X51
)
)
→
(
¬
(
X47
↔
(
X46
→
X41
)
)
)
)
→
(
¬
(
X51
→
(
(
¬
X30
)
↔
(
(
¬
X23
)
↔
X47
)
)
)
)
)
→
(
(
X46
↔
(
(
¬
X25
)
↔
(
(
¬
X11
)
↔
X41
)
)
)
↔
(
(
¬
(
X44
↔
(
X11
↔
X47
)
)
)
↔
(
X41
↔
(
X38
→
(
X11
↔
False
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
X33
↔
(
X22
↔
X50
)
)
→
(
¬
(
X47
↔
(
(
¬
X43
)
↔
(
X11
→
X50
)
)
)
)
→
(
¬
(
X47
→
(
X11
↔
X21
)
)
)
)
↔
(
(
(
(
¬
X45
)
→
(
(
¬
X33
)
↔
(
X32
↔
X50
)
)
)
↔
(
(
¬
X47
)
→
(
¬
X45
)
→
(
X33
↔
X50
)
)
)
↔
(
¬
(
(
¬
(
(
(
X45
→
(
(
¬
X33
)
↔
X51
)
)
↔
(
X45
↔
(
X33
↔
(
X13
→
X49
)
)
)
)
→
(
(
(
(
X45
→
(
(
¬
X41
)
↔
(
X9
↔
X51
)
)
)
↔
(
¬
(
(
¬
X49
)
→
(
¬
X45
)
→
(
X41
↔
X51
)
)
)
)
→
(
¬
(
X45
↔
(
X41
→
X49
)
)
)
)
↔
(
¬
(
X51
→
(
¬
X41
)
→
X49
)
)
)
)
)
→
(
(
¬
(
X45
→
(
X15
↔
X49
)
)
)
↔
(
X51
↔
(
(
¬
X15
)
↔
X47
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
X51
→
(
X24
↔
X39
)
)
→
(
X51
↔
(
X25
↔
X23
)
)
)
↔
(
(
(
¬
(
X25
→
(
¬
X24
)
→
(
(
¬
X7
)
↔
X51
)
)
)
↔
(
¬
(
X51
↔
(
X7
→
(
(
¬
X2
)
↔
X24
)
)
)
)
)
↔
(
¬
(
(
¬
X24
)
↔
X51
)
)
)
)
→
(
¬
(
X39
→
X24
→
(
¬
X8
)
→
X25
)
)
)
→
(
(
X49
↔
(
X39
→
(
¬
X36
)
→
X43
)
)
↔
(
(
(
(
X51
↔
(
X43
→
(
X18
↔
False
)
)
)
↔
(
(
X36
↔
(
(
¬
X18
)
→
X43
)
)
↔
(
X51
→
(
X24
↔
(
X18
→
X43
)
)
)
)
)
→
X39
→
(
¬
X24
)
→
X18
→
X51
)
→
(
¬
(
(
X36
→
(
¬
X6
)
→
X43
)
→
X39
→
(
X36
↔
(
(
¬
X6
)
↔
X12
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
(
(
(
X48
↔
(
(
¬
X19
)
↔
X30
)
)
↔
(
(
¬
X40
)
→
X38
→
(
(
¬
X19
)
↔
X48
)
)
)
→
(
¬
(
(
¬
(
(
¬
X38
)
↔
(
X30
→
X48
)
)
)
→
(
¬
(
(
¬
(
X38
→
(
X30
↔
(
(
¬
X28
)
→
X48
)
)
)
)
→
(
¬
(
X48
→
(
X38
↔
(
X0
↔
X30
)
)
)
)
)
)
)
)
)
↔
(
X48
→
(
X38
↔
X40
)
)
)
→
(
(
¬
(
(
¬
X46
)
→
(
X17
↔
X48
)
)
)
↔
(
X47
↔
(
(
¬
X28
)
↔
(
X17
→
X40
)
)
)
)
)
↔
(
¬
(
X40
↔
(
(
¬
X38
)
→
(
X4
↔
X47
)
)
)
)
)
→
(
(
¬
(
(
¬
X44
)
→
(
¬
X29
)
→
X48
)
)
↔
(
¬
(
(
¬
(
X47
→
(
X40
↔
(
X34
↔
X52
)
)
)
)
→
(
(
(
(
X29
↔
X40
)
↔
(
¬
(
(
¬
X44
)
↔
(
(
¬
X3
)
→
X47
)
)
)
)
→
(
¬
(
X52
→
X40
→
False
)
)
)
↔
(
¬
(
X47
↔
(
X44
↔
X52
)
)
)
)
)
)
)
→
(
(
(
X45
→
(
¬
X31
)
→
X48
)
↔
(
X48
→
X45
→
(
(
¬
X26
)
↔
X47
)
)
)
↔
(
(
¬
X5
)
↔
(
(
¬
(
(
¬
X48
)
↔
(
(
¬
X45
)
→
(
¬
X5
)
→
X52
)
)
)
↔
(
¬
(
(
¬
(
X48
↔
(
X45
→
X37
→
X31
→
False
)
)
)
→
(
(
¬
X45
)
↔
(
X37
→
(
X31
↔
X52
)
)
)
)
)
)
)
)
)
→
(
(
(
X50
→
X42
→
(
(
¬
X14
)
↔
X52
)
)
→
(
(
X52
→
(
X27
↔
(
(
¬
X1
)
→
X20
)
)
)
↔
(
(
¬
(
(
¬
X47
)
↔
(
X42
↔
(
(
¬
X27
)
↔
X52
)
)
)
)
↔
(
¬
(
(
(
¬
X42
)
→
(
(
¬
X1
)
↔
X52
)
)
→
(
X52
→
X50
→
X1
)
→
(
¬
(
X52
→
(
X47
↔
(
X42
→
X50
)
)
)
)
)
)
)
)
)
↔
(
(
X50
↔
(
(
¬
X35
)
↔
(
(
¬
X16
)
↔
X47
)
)
)
↔
(
¬
(
(
(
¬
(
X35
↔
(
(
¬
X16
)
↔
(
X10
↔
X48
)
)
)
)
→
(
¬
(
X35
↔
(
(
¬
X10
)
↔
X42
)
)
)
)
→
(
¬
(
X35
↔
(
(
¬
X16
)
↔
X42
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
80fe09...
and proposition id is
464e2e...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMdWPef4GhvqitTE77SaTwvqWoNXfB52cHb
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
∀X50 :
prop
,
(
¬
∀X51 :
prop
,
∀X52 :
prop
,
(
¬
∀X53 :
prop
,
∀X54 :
prop
,
(
(
(
(
¬
(
(
(
(
¬
(
(
(
¬
X50
)
↔
(
X38
→
(
X31
↔
X52
)
)
)
→
(
¬
(
(
¬
(
X53
↔
(
(
¬
X50
)
↔
X52
)
)
)
→
(
¬
(
(
X50
→
(
¬
X38
)
→
X12
→
X52
)
→
(
¬
(
(
¬
X52
)
→
(
(
¬
X50
)
↔
(
X38
↔
X53
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X52
↔
(
X0
→
False
)
)
)
→
(
¬
(
X50
→
X1
→
(
X0
↔
X53
)
)
)
)
)
)
↔
(
(
(
¬
(
(
X52
↔
(
X51
→
X50
→
False
)
)
→
(
X53
↔
(
(
¬
X50
)
↔
(
X14
↔
X51
)
)
)
)
)
↔
(
X52
→
(
(
¬
X50
)
↔
X51
)
)
)
↔
(
(
¬
(
X39
→
(
X33
↔
(
X16
↔
False
)
)
)
)
↔
(
(
¬
(
(
¬
X33
)
↔
(
(
¬
X16
)
→
X53
)
)
)
↔
(
(
¬
(
X33
↔
(
X14
↔
X53
)
)
)
↔
(
X33
→
(
(
¬
X16
)
↔
(
(
¬
X14
)
→
X39
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
X34
↔
(
X29
↔
(
X10
→
X52
)
)
)
→
(
X45
↔
(
X34
↔
(
X7
→
X52
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
X31
)
↔
(
(
¬
X29
)
↔
(
(
¬
X5
)
↔
X54
)
)
)
→
(
(
¬
X34
)
→
(
(
¬
X29
)
↔
(
X5
→
X54
)
)
)
→
(
(
¬
X45
)
↔
(
X29
→
X54
)
)
)
)
↔
(
¬
False
)
)
)
↔
(
¬
(
(
¬
(
X51
↔
(
X45
↔
(
X34
→
X29
)
)
)
)
→
(
¬
(
(
¬
X34
)
↔
(
(
¬
X29
)
↔
(
X27
→
X45
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X52
↔
(
X47
→
(
¬
X11
)
→
X40
)
)
)
↔
(
¬
(
X54
→
(
X53
↔
(
X52
→
(
X40
↔
False
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X26
↔
(
X22
↔
X54
)
)
)
→
X47
→
(
X26
↔
(
(
¬
X22
)
↔
X40
)
)
)
)
)
↔
(
(
¬
X52
)
↔
(
X11
↔
(
(
¬
X9
)
→
X54
)
)
)
)
↔
(
(
(
(
(
¬
(
X47
↔
(
(
¬
X8
)
↔
X46
)
)
)
↔
(
¬
(
(
(
¬
(
X53
→
(
¬
X20
)
→
X46
)
)
→
(
(
(
¬
X20
)
↔
(
(
¬
X2
)
↔
X53
)
)
↔
(
X47
↔
(
(
¬
X2
)
↔
X46
)
)
)
)
→
(
¬
(
(
¬
X46
)
↔
(
X20
↔
(
(
¬
X8
)
→
X47
)
)
)
)
)
)
)
→
(
¬
(
X47
→
(
(
¬
X20
)
↔
X53
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X52
)
↔
(
(
¬
X47
)
→
(
¬
X3
)
→
X53
)
)
)
→
(
¬
(
X52
↔
(
X47
↔
(
X20
↔
X3
)
)
)
)
)
→
(
¬
(
X53
↔
(
(
¬
X47
)
→
(
¬
X43
)
→
X52
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
X47
)
→
(
(
¬
X46
)
↔
(
X13
→
X53
)
)
)
)
↔
(
X53
→
(
X46
↔
(
X4
↔
X13
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
(
¬
(
X41
↔
(
X37
↔
X30
)
)
)
↔
(
¬
(
(
X49
→
(
X41
↔
(
X30
→
X37
)
)
)
→
X49
→
X30
→
(
X21
↔
False
)
)
)
)
→
(
¬
(
X49
→
(
¬
X30
)
→
X23
→
X37
)
)
)
↔
(
(
(
¬
X49
)
→
(
¬
X46
)
→
X23
→
X52
)
→
(
(
¬
X49
)
↔
(
X46
↔
(
X30
→
X52
)
)
)
)
)
↔
(
(
¬
(
X41
→
X25
→
X17
)
)
↔
(
(
¬
(
X52
→
(
(
¬
X25
)
↔
X48
)
)
)
↔
(
(
¬
(
X52
→
(
X48
↔
(
(
¬
X17
)
↔
X25
)
)
)
)
↔
(
¬
(
(
¬
X17
)
→
X23
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X47
)
↔
(
(
¬
X44
)
→
X6
→
X48
)
)
)
↔
(
(
(
¬
(
X48
↔
X46
)
)
↔
(
(
(
¬
(
X44
→
(
X6
↔
X46
)
)
)
↔
(
X52
↔
(
(
¬
X44
)
↔
(
X6
↔
X48
)
)
)
)
↔
(
X48
→
X46
→
X44
→
False
)
)
)
↔
(
X6
→
X48
)
)
)
→
(
(
¬
(
(
¬
(
X47
→
X44
→
(
(
¬
X15
)
↔
X46
)
)
)
→
X47
→
(
X46
↔
(
(
¬
X18
)
→
X44
)
)
)
)
↔
(
¬
(
X44
↔
(
X18
→
X46
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
X35
→
X19
)
↔
(
(
¬
X48
)
→
(
X42
↔
X49
)
)
)
→
(
(
¬
(
X49
→
(
(
¬
X32
)
↔
(
X28
↔
X48
)
)
)
)
↔
(
¬
(
(
¬
(
X49
→
(
(
¬
X24
)
↔
X48
)
)
)
→
(
X49
↔
(
X48
↔
(
(
¬
X32
)
→
X42
)
)
)
)
)
)
)
)
↔
(
(
X47
↔
(
X36
→
X49
)
)
↔
(
(
X47
→
(
X36
↔
X49
)
)
↔
(
¬
(
(
(
X47
→
(
(
¬
X36
)
↔
(
(
¬
X35
)
↔
X49
)
)
)
→
(
(
¬
X47
)
↔
(
(
¬
X36
)
→
(
¬
X32
)
→
X49
)
)
)
→
X49
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
(
¬
(
(
(
¬
X44
)
→
(
X32
↔
X54
)
)
→
(
(
(
¬
X39
)
↔
(
(
¬
X24
)
↔
X53
)
)
↔
(
X54
↔
(
X32
↔
(
(
¬
X24
)
→
X53
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X53
→
X42
)
)
→
(
¬
(
X53
→
X32
→
(
¬
X3
)
→
X7
)
)
)
→
(
(
(
¬
(
(
X44
→
(
(
¬
X42
)
↔
(
(
¬
X21
)
→
X53
)
)
)
→
(
¬
(
X32
→
(
¬
X21
)
→
X53
)
)
)
)
↔
(
¬
(
X53
→
X32
→
X42
)
)
)
↔
(
X53
→
(
¬
X17
)
→
X44
)
)
→
(
¬
(
(
¬
X42
)
→
(
(
¬
X32
)
↔
X44
)
)
)
)
)
)
→
(
(
(
¬
(
X44
↔
(
(
¬
X19
)
→
X15
→
X39
)
)
)
↔
(
(
(
(
¬
X11
)
↔
X39
)
↔
(
¬
(
(
(
¬
X36
)
↔
(
(
¬
X19
)
↔
(
(
¬
X11
)
→
X39
)
)
)
→
X44
→
(
X39
↔
(
X19
↔
X36
)
)
)
)
)
→
(
¬
(
X36
↔
(
X2
↔
False
)
)
)
)
)
↔
(
(
¬
(
X52
↔
(
(
¬
X44
)
→
(
(
¬
X36
)
↔
X53
)
)
)
)
↔
(
¬
(
(
¬
(
X39
↔
(
(
¬
X13
)
→
X44
)
)
)
→
(
¬
(
X53
→
X52
→
(
¬
X13
)
→
X39
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X42
)
→
X41
→
(
(
¬
X10
)
↔
X53
)
)
)
→
(
¬
(
(
¬
X53
)
↔
(
X41
↔
(
X10
↔
X54
)
)
)
)
)
)
↔
(
¬
(
X42
↔
(
X29
↔
X41
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
X43
→
X12
)
)
↔
(
¬
(
X37
→
(
¬
X12
)
→
(
¬
X6
)
→
X51
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X37
)
→
(
¬
X12
)
→
X51
)
)
→
(
¬
(
(
¬
(
(
¬
(
X51
↔
(
X38
→
X37
)
)
)
→
(
(
¬
(
(
¬
X38
)
→
X34
→
X12
→
X51
)
)
↔
(
X51
→
(
X37
↔
(
(
¬
X12
)
→
X34
)
)
)
)
)
)
→
(
¬
(
X37
→
X51
)
)
)
)
)
)
↔
(
¬
(
X52
↔
(
X12
↔
X37
)
)
)
)
)
→
X54
→
(
(
¬
X38
)
↔
(
(
¬
X6
)
↔
X43
)
)
)
↔
(
¬
(
(
X54
→
X27
→
X1
→
X38
)
→
(
(
¬
X38
)
↔
(
(
¬
X27
)
→
(
(
¬
X1
)
↔
X43
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X51
→
(
¬
X47
)
→
(
¬
X37
)
→
X52
)
)
→
(
¬
(
(
X53
→
X47
)
→
(
(
¬
(
(
¬
X48
)
→
X37
→
(
X14
↔
X52
)
)
)
↔
(
(
(
¬
X48
)
→
(
X37
↔
X53
)
)
→
(
(
X47
→
X14
)
↔
(
¬
(
X53
→
(
(
¬
X37
)
↔
(
(
¬
X14
)
↔
X48
)
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
¬
(
X52
↔
(
(
¬
X22
)
→
X45
)
)
)
→
(
¬
(
(
X45
↔
(
(
¬
X16
)
→
X22
)
)
→
(
¬
(
(
¬
X45
)
↔
(
(
¬
X0
)
↔
X52
)
)
)
)
)
)
)
→
(
(
¬
(
X52
→
(
X45
↔
(
(
¬
X16
)
↔
X28
)
)
)
)
↔
(
X52
↔
(
X45
↔
(
X9
→
X28
)
)
)
)
)
↔
(
(
(
(
X45
↔
(
X40
↔
(
(
¬
X26
)
↔
X49
)
)
)
→
(
(
(
¬
X45
)
↔
(
X25
→
X49
)
)
↔
(
¬
(
(
¬
X49
)
↔
(
(
¬
X40
)
→
(
X25
↔
X50
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
X49
)
↔
(
(
¬
X45
)
↔
(
(
¬
X4
)
↔
X53
)
)
)
)
↔
(
¬
(
X50
→
X45
→
X4
→
False
)
)
)
)
→
(
(
X50
→
X49
→
X22
→
X52
)
→
X49
→
X45
→
X22
→
False
)
→
X52
→
(
¬
X45
)
→
X50
)
)
→
(
¬
(
(
(
¬
(
X50
↔
(
X31
→
X54
)
)
)
→
(
¬
(
X54
↔
(
X46
→
X31
→
X52
)
)
)
)
→
(
(
¬
(
X50
→
(
X46
↔
(
(
¬
X8
)
↔
X54
)
)
)
)
↔
(
¬
(
X54
↔
(
X46
→
(
X30
↔
X8
)
)
)
)
)
)
)
→
(
(
(
(
(
X49
→
(
X35
↔
(
(
¬
X23
)
→
X50
)
)
)
↔
(
(
¬
(
X49
↔
(
X35
↔
X18
)
)
)
↔
(
¬
(
X49
→
(
(
¬
X35
)
↔
(
X5
↔
X50
)
)
)
)
)
)
↔
(
(
X52
↔
(
X50
→
X23
)
)
↔
(
(
X49
↔
(
(
¬
X23
)
↔
(
X20
→
X50
)
)
)
→
(
¬
X50
)
→
(
¬
X49
)
→
(
X20
↔
X52
)
)
)
)
↔
(
¬
(
X33
↔
(
X20
↔
X49
)
)
)
)
↔
(
X54
→
(
(
¬
X33
)
↔
(
(
¬
X23
)
↔
X52
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
b875cc...
and proposition id is
826589...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMTuEHy5JKhtRcVegWqmdDfxjKsgDLnZv4a
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
¬
∀X51 :
prop
,
(
(
(
¬
(
(
¬
(
(
(
¬
(
(
(
¬
(
X39
→
(
¬
X5
)
→
(
¬
X4
)
→
X28
)
)
↔
(
(
¬
X28
)
→
(
¬
X5
)
→
(
X4
↔
X44
)
)
)
→
(
¬
(
(
X44
→
(
X39
↔
(
X23
↔
False
)
)
)
→
(
¬
(
(
¬
X43
)
↔
(
(
¬
X28
)
→
X44
)
)
)
)
)
→
(
¬
X39
)
→
X44
)
)
↔
(
¬
(
(
(
¬
(
X48
↔
(
(
¬
X26
)
↔
(
X15
↔
X50
)
)
)
)
→
(
¬
(
X50
→
X48
→
(
X26
↔
X39
)
)
)
)
→
(
(
(
(
¬
(
X50
↔
(
X39
↔
(
(
¬
X28
)
→
X44
)
)
)
)
↔
(
X50
↔
(
X44
→
(
¬
X9
)
→
X39
)
)
)
↔
(
¬
(
X50
↔
(
X28
→
X21
→
X44
)
)
)
)
↔
(
(
¬
X48
)
→
(
¬
X44
)
→
X39
→
X50
)
)
)
)
)
→
(
¬
(
(
(
X50
↔
(
X18
→
X36
)
)
→
(
X50
↔
(
X48
→
X36
)
)
→
(
¬
(
X48
→
(
X36
↔
(
X18
↔
X39
)
)
)
)
)
→
(
¬
X48
)
→
(
(
¬
X39
)
↔
(
X17
→
X50
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
(
¬
X44
)
→
X43
→
(
X13
↔
X46
)
)
)
↔
(
(
(
¬
(
X46
↔
(
(
¬
X35
)
→
X44
)
)
)
→
(
(
X44
↔
(
(
¬
X34
)
→
X35
)
)
↔
(
(
¬
X44
)
↔
(
X43
→
X46
)
)
)
)
↔
(
¬
(
X34
↔
X35
)
)
)
)
→
(
(
(
¬
X44
)
↔
(
X43
↔
(
X22
↔
X48
)
)
)
↔
(
¬
(
(
X48
↔
(
X46
↔
(
(
¬
X20
)
→
X22
)
)
)
→
(
(
X48
↔
(
X44
→
X42
→
X46
)
)
↔
(
X46
→
X44
→
(
X22
↔
X48
)
)
)
)
)
)
)
)
↔
(
(
¬
(
X46
→
(
X43
↔
(
X40
→
(
X30
↔
False
)
)
)
)
)
↔
(
¬
(
X46
→
(
¬
X43
)
→
X16
→
X48
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
X24
↔
(
X19
→
(
¬
X3
)
→
X41
)
)
↔
(
¬
(
(
(
(
¬
X38
)
↔
(
(
¬
X24
)
→
(
¬
X19
)
→
X45
)
)
↔
(
¬
(
X24
↔
(
X19
→
(
¬
X3
)
→
X45
)
)
)
)
→
(
¬
(
X45
↔
(
X24
→
(
X19
↔
False
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X29
↔
(
X7
↔
False
)
)
)
→
(
(
¬
(
X47
↔
(
X37
→
(
¬
X7
)
→
X29
)
)
)
↔
(
¬
(
X37
↔
(
(
¬
X7
)
→
X44
)
)
)
)
)
→
(
(
(
(
¬
(
X45
→
(
X37
↔
False
)
)
)
↔
(
X44
→
(
¬
X29
)
→
X38
)
)
→
(
¬
(
(
¬
X29
)
→
X45
)
)
)
↔
(
X45
↔
(
X38
→
X44
)
)
)
)
)
↔
(
(
¬
X45
)
↔
(
X44
→
(
(
¬
X41
)
↔
X47
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X47
)
→
X46
→
X51
)
)
→
(
(
¬
(
(
(
¬
(
X51
↔
(
X41
↔
(
X37
↔
X47
)
)
)
)
→
X47
→
(
X41
↔
(
(
¬
X33
)
↔
X51
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X37
)
↔
(
(
¬
X33
)
→
X51
)
)
)
)
)
)
↔
(
X51
↔
(
X47
↔
(
X46
↔
X49
)
)
)
)
)
)
→
(
(
(
¬
X46
)
→
(
¬
X2
)
→
X47
)
↔
(
X33
→
(
(
¬
X25
)
↔
(
(
¬
X2
)
↔
X46
)
)
)
)
)
)
↔
(
(
(
X51
→
X49
→
(
X47
↔
(
X31
↔
False
)
)
)
→
(
(
¬
(
X51
↔
(
X27
→
X1
→
False
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X48
)
→
X1
→
X49
)
)
→
(
(
¬
X47
)
↔
(
X27
↔
(
(
¬
X1
)
→
X49
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
¬
(
(
¬
(
(
¬
X49
)
↔
(
(
¬
X14
)
→
(
¬
X12
)
→
X51
)
)
)
→
(
¬
(
X51
→
(
X49
↔
(
X47
→
X14
→
False
)
)
)
)
)
)
↔
(
X51
→
(
X47
↔
X49
)
)
)
→
(
¬
(
X51
→
X49
→
False
)
)
)
)
↔
(
(
¬
(
(
¬
X12
)
↔
(
(
¬
X11
)
→
(
¬
X8
)
→
X51
)
)
)
→
(
¬
(
X49
↔
(
X8
→
(
¬
X0
)
→
X51
)
)
)
)
)
→
(
¬
(
X32
→
(
¬
X12
)
→
(
(
¬
X6
)
↔
X14
)
)
)
→
(
¬
(
X14
↔
(
(
¬
X10
)
↔
(
(
¬
X6
)
→
X51
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
X33
↔
(
X26
↔
(
X24
↔
X32
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X24
)
↔
(
X23
→
(
X10
↔
X33
)
)
)
)
→
(
¬
(
(
¬
X24
)
↔
(
X23
↔
(
X4
→
X33
)
)
)
)
)
)
↔
(
¬
(
X32
↔
(
X10
→
X24
)
)
)
)
)
↔
(
(
¬
(
X33
↔
(
X24
↔
(
X20
↔
False
)
)
)
)
↔
(
X26
→
(
(
¬
X20
)
↔
X25
)
)
)
)
→
(
(
(
X49
→
(
(
¬
X33
)
↔
(
(
¬
X24
)
↔
X50
)
)
)
→
(
¬
(
X47
↔
(
X25
↔
(
X24
↔
False
)
)
)
)
)
↔
(
(
(
¬
X35
)
→
(
(
¬
X33
)
↔
X45
)
)
↔
(
(
¬
(
X35
→
X45
)
)
↔
(
¬
(
X49
→
X45
→
X22
→
X47
)
)
)
)
)
)
↔
(
(
(
(
X34
→
(
(
¬
X14
)
↔
X27
)
)
↔
(
¬
(
(
¬
X40
)
→
(
X34
↔
(
(
¬
X14
)
→
X45
)
)
)
)
)
→
(
(
(
(
¬
(
X45
↔
(
(
¬
X27
)
→
X36
)
)
)
↔
(
¬
(
(
¬
(
X40
↔
(
(
¬
X15
)
→
X36
)
)
)
→
(
(
¬
(
(
¬
(
X40
→
(
X36
↔
(
X15
→
X27
)
)
)
)
→
(
X45
↔
(
X40
↔
(
X27
→
X36
)
)
)
)
)
↔
(
¬
(
X45
↔
(
X27
→
X36
)
)
)
)
)
)
)
↔
(
(
¬
X40
)
↔
(
X27
↔
X45
)
)
)
↔
(
¬
(
X40
↔
(
(
¬
X21
)
↔
(
X18
↔
X35
)
)
)
)
)
)
↔
(
¬
(
(
(
X45
↔
(
X36
↔
(
X35
→
X9
→
False
)
)
)
→
X50
→
(
X36
↔
X45
)
)
→
(
(
(
(
X13
→
(
X5
↔
X44
)
)
→
(
(
(
¬
X44
)
→
(
(
¬
X13
)
↔
(
(
¬
X5
)
↔
X46
)
)
)
↔
(
¬
(
(
¬
X46
)
→
(
(
¬
X44
)
↔
(
X13
→
X50
)
)
)
)
)
)
↔
(
¬
(
X51
→
(
X44
↔
(
X5
↔
X13
)
)
)
)
)
↔
(
¬
(
(
X45
↔
(
(
¬
X35
)
→
X5
→
X46
)
)
→
(
¬
(
X51
↔
(
X5
↔
X46
)
)
)
)
)
)
)
)
)
)
→
(
(
(
(
(
(
(
X41
→
(
(
¬
X28
)
↔
(
X3
→
X31
)
)
)
↔
(
¬
(
(
¬
X41
)
↔
(
(
¬
X31
)
↔
X43
)
)
)
)
↔
(
(
(
¬
X28
)
→
(
¬
X19
)
→
(
X6
↔
X41
)
)
↔
(
¬
(
X31
↔
(
(
¬
X6
)
→
X28
)
)
)
)
)
→
X43
→
(
¬
X19
)
→
(
X12
↔
X41
)
)
↔
(
X42
→
(
¬
X19
)
→
X28
)
)
↔
(
(
(
(
X39
→
(
(
¬
X8
)
↔
(
(
¬
X7
)
↔
X31
)
)
)
↔
(
X48
↔
(
X31
→
(
X16
↔
False
)
)
)
)
↔
(
X16
→
(
¬
X7
)
→
X39
)
)
↔
(
(
(
(
¬
X17
)
→
X8
→
(
(
¬
X1
)
↔
X41
)
)
↔
(
¬
(
(
¬
X16
)
→
X17
)
)
)
→
(
¬
(
(
¬
X17
)
↔
(
(
¬
X16
)
→
(
X8
↔
X31
)
)
)
)
→
(
¬
(
(
¬
(
X31
→
(
X16
↔
False
)
)
)
→
(
¬
(
(
¬
(
X31
→
X19
→
X0
)
)
→
(
¬
X16
)
→
X8
→
(
¬
X0
)
→
X19
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
X49
↔
(
X30
↔
X29
)
)
→
(
¬
(
(
X49
→
(
X41
↔
(
X38
↔
(
X29
→
False
)
)
)
)
→
(
¬
X41
)
→
(
(
¬
X29
)
↔
X51
)
)
)
)
)
↔
(
¬
(
(
¬
X50
)
↔
(
(
¬
X49
)
↔
(
(
¬
X41
)
→
X51
)
)
)
)
)
↔
(
(
(
(
¬
(
X42
↔
(
(
¬
X37
)
↔
(
X11
↔
X51
)
)
)
)
↔
(
(
¬
X50
)
↔
(
X48
→
(
(
¬
X37
)
↔
X51
)
)
)
)
↔
(
¬
(
X51
↔
(
X48
→
X42
→
False
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X42
)
↔
(
(
¬
X38
)
→
(
(
¬
X11
)
↔
X48
)
)
)
)
→
(
(
¬
X38
)
↔
(
X11
→
(
(
¬
X2
)
↔
X48
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
d37b3c...
and proposition id is
2e8a1e...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMZY6V9hML8sw7T6PDrKkVcD7M8CCHUFZkN
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
(
(
(
¬
(
X35
↔
(
X26
↔
X41
)
)
)
→
(
X37
↔
X16
)
)
↔
(
¬
(
X41
→
(
(
¬
X26
)
↔
X37
)
)
)
)
↔
(
(
¬
(
(
¬
(
X41
→
(
X39
↔
(
X26
↔
(
X23
→
False
)
)
)
)
)
→
X39
→
X37
→
(
(
¬
X26
)
↔
X41
)
)
)
↔
(
(
(
¬
(
(
(
X41
→
(
(
¬
X19
)
↔
X21
)
)
↔
(
X41
→
(
X19
↔
(
X18
→
X37
)
)
)
)
→
X21
→
(
¬
X19
)
→
X41
)
)
↔
(
¬
(
X41
→
(
X21
↔
False
)
)
)
)
↔
(
(
X41
↔
(
X39
→
(
(
¬
X27
)
↔
X37
)
)
)
→
(
¬
(
X39
→
X37
→
(
(
¬
X19
)
↔
X41
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X34
→
(
(
¬
X13
)
↔
(
(
¬
X2
)
↔
X15
)
)
)
)
→
X37
→
X34
→
(
X13
↔
X15
)
)
→
(
¬
(
(
¬
X37
)
→
X34
→
X39
)
)
)
)
↔
(
(
(
¬
(
X49
→
(
(
¬
X15
)
↔
(
X14
↔
X28
)
)
)
)
↔
(
¬
(
X37
↔
(
X28
↔
X15
)
)
)
)
↔
(
(
¬
X37
)
→
(
¬
X28
)
→
(
¬
X14
)
→
X41
)
)
)
)
→
(
(
(
(
X48
↔
(
X29
→
X11
→
False
)
)
↔
(
¬
(
X41
↔
(
X31
→
(
X29
↔
(
X11
↔
False
)
)
)
)
)
)
↔
(
¬
(
X41
→
X37
→
X31
→
X48
)
)
)
↔
(
(
(
(
(
¬
(
X41
→
X31
→
(
X25
↔
X49
)
)
)
→
X29
→
X25
)
→
(
¬
(
X49
↔
(
X41
→
X31
→
False
)
)
)
)
→
X31
→
(
¬
X25
)
→
X49
)
→
(
¬
(
(
X41
↔
(
X4
→
(
(
¬
X3
)
↔
X31
)
)
)
→
(
(
¬
(
X49
→
X33
→
X4
)
)
↔
(
X41
↔
(
X33
↔
(
X31
↔
X49
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X46
→
(
X45
↔
(
X43
↔
(
X36
↔
False
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X45
)
→
(
(
¬
X24
)
↔
(
(
¬
X0
)
→
X46
)
)
)
)
→
(
X45
↔
(
(
¬
X44
)
→
(
¬
X0
)
→
X46
)
)
)
↔
(
(
X47
↔
(
X7
↔
(
X0
↔
X43
)
)
)
↔
(
¬
(
X46
↔
(
X43
→
(
X7
↔
X0
)
)
)
)
)
)
)
↔
(
(
X47
↔
False
)
→
(
(
¬
(
X47
↔
(
X45
→
(
¬
X36
)
→
X48
)
)
)
↔
(
¬
(
X48
→
(
¬
X17
)
→
X36
)
)
)
)
)
→
(
(
¬
(
X47
↔
(
X45
→
X32
→
False
)
)
)
↔
(
(
X47
→
(
(
¬
X42
)
↔
(
X30
↔
X45
)
)
)
↔
(
¬
(
(
(
X47
↔
(
(
¬
X42
)
↔
X46
)
)
↔
(
¬
(
X45
→
(
¬
X32
)
→
X46
)
)
)
→
(
¬
(
(
¬
X45
)
↔
(
X32
→
X1
→
X47
)
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
X12
)
→
(
¬
X8
)
→
X44
)
)
↔
(
¬
(
(
¬
X12
)
→
X9
→
X38
)
)
)
↔
(
¬
(
(
(
¬
(
(
X49
↔
(
X45
↔
X47
)
)
→
(
(
¬
(
(
¬
X45
)
↔
(
X44
→
(
¬
X41
)
→
X49
)
)
)
↔
(
X47
↔
(
(
¬
X41
)
↔
X49
)
)
)
)
)
→
(
¬
(
X49
→
X41
→
X10
)
)
)
→
(
¬
(
(
¬
(
(
¬
X38
)
↔
X45
)
)
→
X47
→
X41
→
(
¬
X5
)
→
X12
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X22
)
↔
X49
)
)
→
(
¬
(
X47
→
(
¬
X45
)
→
X40
→
X49
)
)
)
↔
(
(
X47
↔
(
(
¬
X20
)
↔
(
(
¬
X6
)
→
X49
)
)
)
↔
(
¬
(
X45
→
(
X20
↔
(
X6
↔
X47
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
(
¬
(
X47
→
(
¬
X20
)
→
(
¬
X17
)
→
X22
)
)
↔
(
X17
↔
(
X16
→
X47
)
)
)
→
(
¬
(
X20
↔
(
X16
→
False
)
)
)
)
→
(
(
X47
↔
(
X43
↔
X16
)
)
↔
(
X47
↔
(
X20
→
(
¬
X1
)
→
X43
)
)
)
)
↔
(
(
(
X49
→
X45
→
X18
)
→
(
(
¬
(
X47
→
(
X18
↔
(
(
¬
X13
)
→
X46
)
)
)
)
↔
(
¬
(
X49
↔
(
X18
↔
(
(
¬
X13
)
↔
X47
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X46
→
(
¬
X11
)
→
X49
)
)
→
(
(
¬
(
X49
→
X46
→
X23
→
X11
)
)
↔
(
X49
→
(
¬
X46
)
→
X47
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X46
)
→
X43
→
X49
)
)
→
(
¬
(
X43
→
(
(
¬
X22
)
↔
(
(
¬
X19
)
↔
X47
)
)
)
)
)
→
(
X47
↔
(
X46
↔
(
X43
↔
False
)
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X49
↔
(
X37
→
X5
→
X44
)
)
)
↔
(
X49
↔
(
X44
↔
(
X37
↔
X38
)
)
)
)
↔
(
¬
(
(
X42
↔
(
X33
↔
X44
)
)
→
(
¬
(
(
¬
(
(
¬
X44
)
→
(
¬
X38
)
→
(
(
¬
X33
)
↔
X49
)
)
)
→
(
¬
(
X38
↔
X42
)
)
→
(
¬
(
(
X38
→
X42
)
→
(
¬
(
X38
→
X33
→
(
(
¬
X25
)
↔
X44
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
X48
→
X42
→
(
¬
X12
)
→
X47
)
→
(
¬
(
(
(
(
X49
→
X48
)
→
(
¬
(
(
X47
↔
(
(
¬
X42
)
↔
(
X21
→
X48
)
)
)
→
(
¬
(
X48
→
(
X47
↔
(
(
¬
X21
)
→
X49
)
)
)
)
)
)
)
↔
(
X48
↔
(
(
¬
X47
)
↔
X49
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
(
X47
↔
(
X34
→
X3
→
X46
)
)
)
→
(
¬
X37
)
→
(
(
¬
X34
)
↔
(
(
¬
X3
)
→
X46
)
)
)
)
→
(
X37
↔
(
X34
↔
X46
)
)
)
)
↔
(
X48
→
X46
→
X37
→
X49
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
(
X46
→
(
X9
↔
X7
)
)
↔
(
¬
(
X46
↔
(
(
¬
X9
)
→
(
¬
X2
)
→
X14
)
)
)
)
→
(
(
X46
↔
(
X43
↔
X39
)
)
→
(
¬
(
(
(
¬
(
X39
↔
(
X36
↔
(
X14
↔
X9
)
)
)
)
→
(
¬
(
X36
↔
(
X14
↔
X43
)
)
)
)
→
(
(
¬
X35
)
↔
(
X14
↔
X46
)
)
)
)
)
→
(
(
¬
X36
)
↔
(
X35
↔
X39
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X41
→
(
(
¬
X39
)
↔
(
(
¬
X6
)
↔
X47
)
)
)
)
↔
(
X41
→
X39
→
X10
→
X6
→
False
)
)
→
(
¬
(
(
¬
X39
)
→
(
¬
X10
)
→
X47
)
)
)
)
↔
(
¬
(
(
(
X47
↔
(
(
¬
X29
)
↔
X41
)
)
↔
(
(
(
X47
↔
(
X46
↔
(
(
¬
X39
)
→
X41
)
)
)
↔
(
X47
→
X46
→
(
(
¬
X29
)
↔
X41
)
)
)
→
(
¬
(
X47
↔
False
)
)
)
)
→
(
¬
(
(
¬
(
X47
↔
(
X46
↔
(
X31
→
X41
)
)
)
)
→
(
(
(
¬
X46
)
→
(
X35
↔
X47
)
)
↔
(
¬
(
X35
↔
(
(
¬
X0
)
↔
X41
)
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
X39
→
(
¬
X32
)
→
(
¬
X28
)
→
X40
)
)
↔
(
(
¬
X32
)
↔
(
(
¬
X28
)
↔
(
X15
→
X40
)
)
)
)
→
(
¬
(
X46
↔
(
X32
↔
(
(
¬
X28
)
↔
X39
)
)
)
)
)
→
(
¬
(
X46
→
(
(
¬
X32
)
↔
(
(
¬
X15
)
→
X41
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
(
X40
→
(
X27
↔
(
X26
→
False
)
)
)
)
→
(
(
¬
X26
)
↔
(
X24
→
X39
)
)
)
)
→
(
¬
(
X46
→
(
(
¬
X27
)
↔
(
(
¬
X24
)
→
X39
)
)
)
)
)
)
→
(
¬
(
X40
↔
(
X39
↔
X46
)
)
)
)
→
(
¬
(
(
X46
↔
(
X26
↔
(
X4
↔
X8
)
)
)
→
X39
→
(
¬
X26
)
→
(
(
¬
X4
)
↔
X30
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
a4ea80...
and proposition id is
3354f6...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMH2H6dF39XeScZGoz87DuumUZ6cCQmghny
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
∀X50 :
prop
,
(
¬
∀X51 :
prop
,
(
¬
∀X52 :
prop
,
(
(
(
(
(
(
(
(
¬
(
X41
→
(
(
¬
X36
)
↔
(
X20
→
X52
)
)
)
)
↔
(
(
¬
(
X36
→
X20
→
False
)
)
→
(
¬
(
(
¬
X47
)
↔
(
(
¬
X31
)
↔
X52
)
)
)
)
)
↔
(
(
¬
(
X52
↔
(
(
¬
X23
)
→
X12
→
X36
)
)
)
↔
(
(
(
X52
↔
(
X12
→
(
X10
↔
False
)
)
)
↔
(
(
X52
↔
(
X31
↔
X23
)
)
↔
(
¬
(
(
¬
(
(
¬
X23
)
↔
(
(
¬
X12
)
↔
X52
)
)
)
→
(
(
X52
→
(
¬
X12
)
→
X31
)
↔
(
¬
(
(
¬
X23
)
↔
(
X12
→
X10
→
X52
)
)
)
)
)
)
)
)
→
X31
→
X12
→
X52
)
)
)
↔
(
¬
(
(
(
(
¬
X25
)
→
X11
→
X51
)
↔
(
(
X52
↔
X11
)
↔
(
¬
(
X36
→
(
(
¬
(
(
¬
X51
)
↔
(
X41
↔
(
X36
→
X52
)
)
)
)
↔
(
(
X51
↔
(
X41
↔
(
X25
↔
False
)
)
)
→
(
¬
(
X52
→
X51
→
(
X11
↔
False
)
)
)
)
)
)
)
)
)
→
(
X51
↔
(
(
¬
X41
)
→
(
X0
↔
X52
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
X46
)
↔
(
(
¬
X39
)
↔
X51
)
)
→
(
X46
→
(
(
¬
X5
)
↔
X52
)
)
→
(
¬
(
X52
↔
(
X51
↔
(
X47
→
(
X39
↔
False
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X51
→
(
X49
↔
(
X9
→
X46
)
)
)
)
→
(
¬
(
X49
→
X47
→
X9
→
(
X1
↔
False
)
)
)
)
)
→
(
¬
(
(
X47
↔
(
X40
↔
(
X29
↔
X4
)
)
)
→
(
¬
(
X40
↔
(
(
¬
X29
)
↔
(
X4
↔
X51
)
)
)
)
)
)
→
(
¬
(
(
¬
X49
)
↔
(
(
¬
X29
)
→
(
(
¬
X4
)
↔
X51
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X35
→
X21
→
X49
)
)
↔
(
(
¬
X35
)
↔
(
X21
↔
(
X15
→
X51
)
)
)
)
↔
(
(
¬
(
X21
↔
(
X15
→
(
X7
↔
X35
)
)
)
)
↔
(
¬
(
X35
↔
(
X21
→
(
X7
↔
X19
)
)
)
)
)
)
↔
(
(
¬
(
X49
↔
(
X35
→
(
X28
↔
False
)
)
)
)
↔
(
¬
(
(
X28
→
(
X18
↔
X44
)
)
→
(
¬
(
X44
→
X35
→
X49
)
)
→
(
(
X35
→
(
¬
X19
)
→
(
X18
↔
X49
)
)
↔
(
¬
(
X44
↔
(
(
¬
X19
)
→
X35
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
X35
)
→
(
X19
↔
X52
)
)
)
→
(
(
(
¬
(
X52
↔
(
X19
↔
False
)
)
)
→
(
(
¬
(
(
¬
(
X52
→
X44
→
(
¬
X8
)
→
X35
)
)
→
X35
→
(
¬
X19
)
→
(
¬
X8
)
→
X52
)
)
↔
(
¬
(
X52
→
(
(
¬
X35
)
↔
X44
)
)
)
)
)
↔
(
X52
→
(
X35
↔
X44
)
)
)
)
)
↔
(
X44
→
(
X19
↔
X52
)
)
)
)
)
↔
(
(
(
(
¬
(
(
(
¬
(
(
¬
X41
)
↔
(
X22
↔
(
(
¬
X3
)
↔
X48
)
)
)
)
→
(
(
¬
(
X45
↔
(
X3
↔
X17
)
)
)
↔
(
(
X48
→
(
X17
↔
X49
)
)
↔
(
¬
(
X45
→
(
X17
↔
(
X3
→
False
)
)
)
)
)
)
)
→
(
¬
(
X50
→
(
(
¬
X48
)
↔
(
(
¬
X46
)
↔
X49
)
)
)
)
→
(
¬
(
X48
↔
(
X46
→
X45
→
False
)
)
)
)
)
↔
(
¬
(
(
¬
(
X48
↔
(
(
¬
X13
)
↔
(
(
¬
X6
)
↔
X46
)
)
)
)
→
(
¬
(
(
X49
↔
(
X46
↔
X34
)
)
→
(
X49
↔
(
(
¬
X34
)
→
(
(
¬
X6
)
↔
X48
)
)
)
→
(
X48
↔
(
X34
→
X6
→
X46
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X48
↔
(
X38
↔
X49
)
)
)
→
(
¬
(
X49
→
(
(
¬
X45
)
↔
(
X38
→
X48
)
)
)
)
→
(
¬
(
X49
↔
(
X38
↔
(
X33
→
X45
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X49
→
(
¬
X46
)
→
(
(
¬
X14
)
↔
X50
)
)
)
→
(
X50
↔
(
X49
→
(
X42
↔
(
X2
↔
False
)
)
)
)
)
→
(
¬
(
(
¬
(
X49
↔
(
X26
↔
False
)
)
)
→
(
X49
↔
(
X42
↔
(
(
¬
X37
)
↔
X50
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X42
→
X30
→
X32
)
)
↔
(
(
¬
(
(
¬
(
X42
→
(
(
¬
X32
)
↔
X50
)
)
)
→
(
(
¬
X32
)
→
(
(
¬
X30
)
↔
(
X27
↔
X42
)
)
)
→
(
¬
(
(
¬
X42
)
→
X32
→
(
¬
X30
)
→
X50
)
)
)
)
↔
(
(
¬
X32
)
→
(
(
¬
X30
)
↔
X50
)
)
)
)
→
(
(
¬
(
X30
↔
(
(
¬
X16
)
→
X24
)
)
)
↔
(
X50
↔
(
X42
→
X24
→
X16
)
)
)
)
↔
(
(
¬
(
(
¬
X43
)
→
X42
→
X49
)
)
↔
(
X49
↔
(
(
¬
X32
)
↔
(
X30
→
X43
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
¬
(
X41
→
X35
→
(
¬
X10
)
→
X52
)
)
→
(
¬
(
(
X29
→
(
(
¬
X7
)
↔
X10
)
)
→
(
(
(
¬
(
X52
↔
(
(
¬
X29
)
→
(
(
¬
X25
)
↔
X35
)
)
)
)
→
X35
→
(
(
¬
X29
)
↔
(
(
¬
X13
)
→
X52
)
)
)
↔
(
¬
(
X52
↔
(
X29
→
X25
→
False
)
)
)
)
→
(
¬
(
X52
→
X35
→
X29
)
)
)
)
)
↔
(
¬
(
(
¬
X46
)
→
(
¬
(
X52
→
X35
)
)
→
(
¬
(
X46
→
X29
→
X51
)
)
)
)
)
→
(
(
¬
(
(
(
X37
→
(
(
¬
X23
)
↔
(
X9
→
X52
)
)
)
↔
(
X52
↔
(
X37
↔
(
X24
→
X23
→
False
)
)
)
)
→
X52
→
(
¬
X37
)
→
(
¬
X9
)
→
X46
)
)
↔
(
(
X37
↔
(
(
¬
X9
)
→
X50
)
)
↔
(
¬
(
X50
↔
(
X46
↔
(
X37
↔
(
X10
→
False
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
(
¬
X50
)
→
(
(
¬
X33
)
↔
(
(
¬
X32
)
→
X51
)
)
)
→
(
¬
(
(
X33
↔
X50
)
→
(
(
¬
(
(
¬
X51
)
↔
(
(
¬
X49
)
↔
(
(
¬
X33
)
→
X52
)
)
)
)
↔
(
X51
↔
(
(
¬
X50
)
→
X49
→
X52
)
)
)
)
)
→
(
X52
↔
(
(
¬
X49
)
↔
X50
)
)
)
)
↔
(
¬
(
(
¬
X51
)
↔
(
X44
↔
(
(
¬
X33
)
↔
X52
)
)
)
)
)
→
(
(
¬
(
(
(
(
X28
→
X3
→
X44
)
→
X50
→
(
¬
X16
)
→
X44
)
↔
(
¬
(
X44
→
X28
→
X50
)
)
)
→
(
(
X28
↔
(
X4
→
(
¬
X3
)
→
X16
)
)
↔
(
X28
↔
(
(
¬
X4
)
↔
(
(
¬
X3
)
↔
X32
)
)
)
)
)
)
↔
(
X50
↔
(
X11
↔
(
X3
→
False
)
)
)
)
)
)
→
(
(
X51
→
X45
→
X50
)
↔
(
¬
(
(
(
¬
X49
)
→
(
¬
X45
)
→
X50
)
→
(
X51
→
(
X49
↔
(
(
¬
X1
)
→
X50
)
)
)
→
(
(
¬
(
X50
→
(
¬
X49
)
→
(
X0
↔
X51
)
)
)
↔
(
X51
↔
(
X45
↔
False
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
¬
X43
)
→
(
(
¬
X39
)
↔
(
(
¬
X21
)
→
X50
)
)
)
↔
(
¬
(
X50
↔
(
(
¬
X43
)
↔
(
(
¬
X21
)
→
X48
)
)
)
)
)
→
(
(
(
¬
(
X50
↔
(
X43
→
(
X38
↔
False
)
)
)
)
↔
(
X49
↔
(
(
¬
X43
)
↔
X48
)
)
)
↔
(
¬
(
(
¬
X43
)
↔
(
(
¬
X38
)
→
(
X27
↔
X50
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
(
¬
X36
)
→
(
¬
X15
)
→
X49
)
)
→
(
¬
(
X49
→
(
X22
↔
(
X15
→
X20
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
X36
)
↔
(
(
¬
X15
)
→
X6
→
X38
)
)
)
↔
(
¬
(
(
¬
X22
)
→
(
(
¬
X15
)
↔
(
(
¬
X6
)
→
X36
)
)
)
)
)
)
↔
(
(
¬
(
X50
↔
(
X38
→
(
¬
X14
)
→
X49
)
)
)
→
(
(
¬
(
(
¬
X38
)
↔
(
X18
↔
X50
)
)
)
↔
(
¬
(
X49
→
(
¬
X20
)
→
(
X18
↔
X50
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X50
→
X49
→
X40
→
X48
)
)
↔
(
(
(
¬
X47
)
↔
(
X40
↔
(
(
¬
X12
)
↔
X50
)
)
)
→
(
¬
(
(
¬
(
(
X50
↔
(
X47
→
X49
)
)
→
(
¬
(
(
(
¬
(
X50
↔
(
X40
→
X49
)
)
)
↔
(
(
X50
→
(
X40
↔
X49
)
)
→
(
¬
(
(
X49
↔
False
)
→
(
(
¬
(
X49
↔
(
(
¬
X40
)
→
(
¬
X31
)
→
X50
)
)
)
↔
(
X50
→
(
X49
↔
(
X47
→
X40
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
X49
)
→
(
(
¬
X47
)
↔
X50
)
)
)
)
)
)
)
→
X31
→
X40
)
)
)
)
↔
(
(
¬
(
X50
↔
(
X49
→
False
)
)
)
→
(
(
X52
↔
(
X49
→
(
X5
↔
X17
)
)
)
↔
(
X48
→
(
X30
↔
(
(
¬
X5
)
→
X49
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X42
↔
(
X26
→
(
X19
↔
X2
)
)
)
)
↔
(
¬
(
(
¬
X26
)
↔
(
X19
→
X2
→
X49
)
)
)
)
↔
(
(
¬
(
X42
↔
(
(
¬
X19
)
→
(
X8
↔
X49
)
)
)
)
→
(
¬
(
(
¬
X42
)
↔
(
(
¬
X19
)
↔
X50
)
)
)
)
)
→
(
¬
(
X49
→
(
(
¬
X34
)
↔
(
X26
→
X42
)
)
)
)
→
(
¬
(
(
¬
X34
)
→
X52
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
d7eae5...
and proposition id is
0a3c53...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMPi4kqXP2WvnJhmA7PGjU1JtbcbUQsaPQi
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
∀X51 :
prop
,
(
(
(
¬
(
(
¬
(
(
(
(
X41
↔
(
(
¬
X39
)
↔
X48
)
)
↔
(
X50
→
(
X48
↔
(
(
¬
X36
)
→
X41
)
)
)
)
↔
(
(
(
X47
↔
(
X41
↔
(
X36
→
X21
)
)
)
↔
(
(
X50
↔
(
(
¬
X41
)
↔
(
X36
↔
X47
)
)
)
↔
(
X36
→
(
X24
↔
X41
)
)
)
)
↔
(
¬
(
X47
→
(
(
¬
X21
)
↔
(
X2
→
X36
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X14
→
(
X12
↔
X21
)
)
)
→
(
(
(
(
¬
X47
)
→
X43
→
X39
→
X51
)
→
(
¬
(
X43
→
(
X39
↔
(
X12
→
X51
)
)
)
)
)
↔
(
(
¬
(
X47
→
(
(
¬
X21
)
↔
(
X17
↔
X43
)
)
)
)
↔
(
¬
(
(
¬
X17
)
→
(
X16
↔
(
(
¬
X12
)
↔
X47
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X47
↔
(
X43
↔
(
X15
↔
(
X11
↔
False
)
)
)
)
)
→
(
X43
↔
X51
)
)
)
)
)
→
(
¬
(
(
(
(
¬
X43
)
→
(
X40
↔
(
X10
→
X49
)
)
)
→
(
X43
→
(
X40
↔
(
(
¬
X8
)
↔
X49
)
)
)
→
X51
→
X43
→
(
¬
X40
)
→
X49
)
→
(
(
X49
→
(
¬
X43
)
→
X34
→
X48
)
↔
(
X48
↔
(
X28
→
X34
)
)
)
→
(
(
¬
X23
)
→
(
¬
X5
)
→
X40
)
→
(
(
¬
(
X49
↔
(
X43
↔
(
X34
↔
(
X32
↔
False
)
)
)
)
)
↔
(
(
(
¬
(
X43
→
(
X34
↔
(
X23
→
X49
)
)
)
)
↔
(
X49
↔
(
X40
↔
X34
)
)
)
→
X34
→
X40
)
)
)
)
)
)
↔
(
(
(
(
(
(
(
(
¬
X47
)
→
(
(
¬
X37
)
↔
(
(
¬
X6
)
→
X51
)
)
)
→
X51
→
(
X49
↔
(
X37
→
False
)
)
)
↔
(
X51
↔
(
X49
↔
(
X47
↔
X3
)
)
)
)
↔
(
(
(
¬
X20
)
↔
X42
)
↔
(
(
¬
(
(
¬
X42
)
→
X20
→
X49
)
)
↔
(
(
¬
(
X51
→
(
X47
↔
(
X42
↔
(
X20
→
False
)
)
)
)
)
→
(
¬
(
X49
→
(
(
¬
X42
)
↔
(
X20
↔
X51
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
X37
)
→
(
X20
↔
(
(
¬
X13
)
→
X51
)
)
)
)
)
↔
(
(
(
¬
(
X46
→
(
X33
↔
(
(
¬
X22
)
→
X49
)
)
)
)
↔
(
(
¬
X42
)
→
X46
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X22
)
↔
(
X19
↔
(
X7
↔
X49
)
)
)
)
→
(
¬
(
(
¬
(
X47
→
(
(
¬
X22
)
↔
X49
)
)
)
→
(
(
¬
X47
)
↔
(
X22
↔
(
(
¬
X18
)
↔
X49
)
)
)
→
(
¬
(
X22
↔
(
X18
→
(
(
¬
X7
)
↔
X47
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
X19
↔
X7
)
→
(
¬
(
X22
→
(
(
¬
X19
)
↔
(
(
¬
X0
)
→
X46
)
)
)
)
)
↔
(
X22
→
(
X7
↔
X0
)
)
)
→
(
¬
(
X46
→
X22
→
(
X19
↔
False
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
(
¬
(
X31
→
X46
)
)
→
(
X46
↔
(
(
¬
X26
)
↔
(
X4
↔
X44
)
)
)
)
→
(
¬
(
(
¬
X44
)
→
(
(
¬
X38
)
↔
(
(
¬
X4
)
→
X46
)
)
)
)
)
)
→
(
(
(
¬
(
X38
→
X35
→
X44
)
)
↔
(
(
¬
(
(
¬
X44
)
↔
(
(
¬
X35
)
↔
X49
)
)
)
→
(
(
(
(
¬
X35
)
↔
(
X9
↔
X44
)
)
↔
(
X49
↔
(
X9
↔
X38
)
)
)
↔
(
(
¬
X38
)
↔
(
X25
→
(
(
¬
X9
)
↔
X49
)
)
)
)
)
)
↔
(
(
¬
X38
)
→
X35
→
X26
→
X44
)
)
)
)
→
(
(
¬
(
(
¬
X30
)
→
(
X27
↔
X45
)
)
)
↔
(
(
(
(
X45
→
(
(
¬
X29
)
↔
(
X1
↔
X49
)
)
)
↔
(
¬
(
X35
→
X27
→
X49
)
)
)
↔
(
¬
(
X35
↔
(
(
¬
X29
)
→
X45
)
)
)
)
→
(
¬
(
(
X44
→
X29
→
X49
)
→
(
¬
(
(
¬
(
X49
↔
(
X47
→
X30
→
X29
→
False
)
)
)
→
(
¬
(
(
¬
X35
)
↔
(
(
¬
X30
)
↔
(
(
¬
X29
)
↔
X47
)
)
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
(
¬
(
(
¬
(
X50
↔
(
X41
↔
(
X40
↔
(
X38
↔
False
)
)
)
)
)
→
(
X50
↔
(
X48
↔
(
X45
↔
(
X41
↔
False
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X38
)
↔
(
(
¬
X5
)
↔
X48
)
)
)
→
(
X48
↔
(
X38
↔
X50
)
)
→
X48
→
X38
→
X3
→
X5
)
)
)
→
(
(
(
¬
X48
)
↔
(
X45
↔
(
X27
↔
X50
)
)
)
↔
(
¬
(
X50
↔
(
X48
↔
(
X27
↔
(
X8
→
False
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
X41
)
↔
(
(
¬
X35
)
→
X2
→
X50
)
)
→
(
X49
↔
(
(
¬
X35
)
↔
(
X2
→
X50
)
)
)
)
)
→
(
(
(
¬
(
X50
→
X41
→
(
¬
X40
)
→
X49
)
)
↔
(
¬
(
X50
→
(
X40
↔
(
(
¬
X36
)
→
X41
)
)
)
)
)
↔
(
X36
↔
X50
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X50
→
(
X49
↔
(
X1
↔
False
)
)
)
)
↔
(
¬
(
X49
→
X21
→
(
X1
↔
X48
)
)
)
)
→
(
X50
↔
(
X48
↔
(
(
¬
X12
)
→
X49
)
)
)
)
)
↔
(
¬
(
(
(
X49
→
(
X48
↔
(
(
¬
X10
)
→
X24
)
)
)
↔
(
¬
(
X49
→
X48
→
(
(
¬
X10
)
↔
X47
)
)
)
)
→
(
(
(
(
¬
(
(
¬
X24
)
↔
X49
)
)
↔
(
(
X30
→
(
X24
↔
(
X19
→
False
)
)
)
↔
(
¬
(
(
¬
X30
)
→
(
X19
↔
(
(
¬
X17
)
→
X49
)
)
)
)
)
)
↔
(
(
¬
X49
)
↔
(
(
¬
X30
)
↔
(
(
¬
X24
)
→
X50
)
)
)
)
↔
(
¬
(
X48
↔
(
X24
↔
X30
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
X51
↔
(
(
¬
X33
)
→
X22
→
X43
)
)
→
(
(
X43
↔
(
(
¬
X22
)
↔
(
(
¬
X6
)
→
X47
)
)
)
↔
(
¬
(
X33
→
X22
→
(
X6
↔
X47
)
)
)
)
)
)
→
(
X47
↔
(
X42
↔
(
X31
↔
(
X15
↔
False
)
)
)
)
→
(
¬
(
X47
→
(
X43
↔
(
X37
↔
(
X31
↔
False
)
)
)
)
)
→
(
(
¬
(
X47
↔
(
X18
↔
X22
)
)
)
↔
(
(
¬
(
X43
↔
(
(
¬
X42
)
→
X47
)
)
)
↔
(
(
X47
→
(
(
¬
X42
)
↔
X43
)
)
↔
(
(
¬
(
X47
↔
(
X37
↔
(
X18
↔
False
)
)
)
)
↔
(
X43
→
(
(
¬
X37
)
↔
(
(
¬
X22
)
→
X47
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
(
¬
X34
)
→
X11
→
X42
)
→
(
(
¬
(
(
(
¬
X34
)
↔
(
(
¬
X32
)
↔
(
X11
↔
X42
)
)
)
→
X42
→
(
¬
X32
)
→
(
X11
↔
X44
)
)
)
↔
(
¬
(
X11
→
X34
)
)
)
)
)
→
(
(
¬
(
X42
→
X32
→
(
¬
X4
)
→
X47
)
)
↔
(
¬
(
(
X42
↔
(
(
¬
X0
)
→
X47
)
)
→
(
¬
(
(
¬
X42
)
↔
(
X32
→
(
(
¬
X11
)
↔
X47
)
)
)
)
)
)
)
)
)
↔
(
¬
(
X47
→
X44
→
(
X11
↔
X51
)
)
)
)
)
↔
(
(
(
(
(
¬
X47
)
→
X34
→
(
(
¬
X13
)
↔
X49
)
)
↔
(
¬
(
(
X49
↔
(
X7
→
X46
)
)
→
(
(
¬
(
(
¬
X43
)
↔
(
(
¬
X13
)
→
X49
)
)
)
↔
(
(
X49
↔
(
(
¬
X7
)
↔
X46
)
)
↔
(
X43
↔
(
(
¬
X7
)
↔
X47
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X28
→
(
(
¬
X25
)
↔
(
X14
→
X46
)
)
)
)
→
(
(
¬
(
X26
↔
(
X25
→
(
X13
↔
X14
)
)
)
)
↔
(
¬
(
(
¬
X25
)
→
(
X14
↔
X28
)
)
)
)
)
)
↔
(
¬
(
X34
→
X13
→
X28
)
)
)
)
→
(
(
(
X50
↔
(
X47
↔
X39
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
¬
X49
)
↔
(
(
¬
X47
)
↔
(
X9
→
X50
)
)
)
)
→
(
(
¬
X49
)
↔
(
X39
→
(
X9
↔
X50
)
)
)
)
)
→
(
X50
↔
(
(
¬
X20
)
→
X9
→
X49
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X47
)
→
(
X46
↔
(
X29
→
X49
)
)
)
)
→
(
¬
(
(
(
(
(
(
¬
(
X49
→
(
X46
↔
X16
)
)
)
↔
(
X46
↔
(
X23
→
(
(
¬
X20
)
↔
X49
)
)
)
)
↔
(
X49
↔
(
X23
↔
X46
)
)
)
→
(
¬
(
X23
↔
(
X20
↔
X46
)
)
)
)
→
(
X49
↔
(
X23
↔
False
)
)
)
→
X49
→
X47
→
(
X46
↔
(
X23
↔
False
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
64084b...
and proposition id is
49777d...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMGzUBAMHYSobp7963b3rYqmCprXR9i32F9
)
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
¬
(
(
(
(
X38
↔
(
(
¬
X24
)
→
X32
)
)
→
(
(
(
X32
→
(
(
¬
X24
)
↔
X38
)
)
→
(
(
¬
(
X38
→
(
(
¬
X1
)
↔
X32
)
)
)
↔
(
(
¬
(
(
¬
X32
)
→
(
¬
X24
)
→
(
¬
X22
)
→
X38
)
)
→
(
¬
X32
)
→
X22
→
(
X1
↔
X38
)
)
)
)
↔
(
¬
(
X38
→
X32
→
X23
→
(
X22
↔
False
)
)
)
)
)
↔
(
(
¬
X32
)
→
(
X24
↔
(
X4
↔
X38
)
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
X23
)
↔
X24
)
→
(
(
¬
(
X25
→
(
(
¬
X15
)
↔
(
X6
↔
X48
)
)
)
)
↔
(
¬
X24
)
)
)
)
→
(
¬
(
(
¬
(
X48
→
X31
→
(
X23
↔
False
)
)
)
→
(
X32
↔
(
(
¬
X25
)
↔
(
(
¬
X23
)
→
X48
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
X42
→
(
(
¬
X31
)
↔
(
(
¬
X30
)
↔
X43
)
)
)
↔
(
(
X30
↔
(
(
¬
X13
)
↔
X31
)
)
→
(
(
(
¬
(
(
¬
X42
)
↔
(
X29
→
(
X13
↔
X45
)
)
)
)
↔
(
X45
↔
(
X31
→
(
X29
↔
X42
)
)
)
)
↔
(
¬
(
X45
→
False
)
)
)
)
)
→
(
(
¬
(
(
(
X42
↔
(
(
¬
X41
)
↔
(
(
¬
X23
)
↔
X48
)
)
)
→
(
¬
(
(
¬
X40
)
↔
X48
)
)
)
→
(
¬
(
(
¬
X41
)
↔
(
X40
↔
X42
)
)
)
)
)
↔
(
¬
(
(
(
X48
→
(
X43
↔
(
(
¬
X41
)
↔
X42
)
)
)
↔
(
¬
(
X42
↔
(
X41
→
(
X18
↔
False
)
)
)
)
)
→
(
¬
(
X43
↔
(
X42
↔
(
X31
→
X41
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
(
¬
(
(
¬
(
(
¬
X42
)
→
(
(
¬
X26
)
↔
X46
)
)
)
→
(
(
¬
X8
)
↔
(
(
¬
X7
)
→
X42
)
)
)
)
→
(
X46
↔
(
X7
↔
X8
)
)
)
→
(
(
(
X46
↔
(
X26
→
X35
)
)
↔
(
(
¬
(
X35
↔
(
X26
→
X19
→
(
X8
↔
False
)
)
)
)
→
X46
→
X35
→
(
X19
↔
False
)
)
)
↔
(
X46
↔
(
X35
→
X8
)
)
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X19
)
→
(
¬
X8
)
→
X43
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
X33
→
X49
)
→
(
¬
X47
)
→
(
(
¬
X36
)
↔
(
X35
↔
X49
)
)
)
)
↔
(
(
X49
→
X26
→
(
¬
X21
)
→
X47
)
→
(
(
¬
X35
)
↔
(
(
¬
X26
)
↔
(
X21
→
X47
)
)
)
)
)
↔
(
(
¬
(
X20
↔
X49
)
)
↔
(
¬
(
(
¬
(
(
¬
X46
)
→
X33
→
X28
→
X47
)
)
→
(
X47
↔
(
(
¬
X28
)
→
(
(
¬
X20
)
↔
X46
)
)
)
)
)
)
)
→
(
¬
(
(
¬
X35
)
→
X26
→
(
(
¬
X11
)
↔
X47
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
X45
→
(
(
¬
X39
)
↔
(
(
¬
X2
)
↔
X47
)
)
)
↔
(
(
(
X47
→
X45
→
(
(
¬
X5
)
↔
X12
)
)
↔
(
¬
(
X39
→
(
X12
↔
(
(
¬
X5
)
→
X45
)
)
)
)
)
→
(
X45
↔
False
)
)
)
→
(
¬
(
X47
→
(
X45
↔
(
X14
→
X39
)
)
)
)
)
→
(
(
(
(
X12
↔
False
)
↔
(
¬
(
X45
↔
(
(
¬
X16
)
↔
(
(
¬
X3
)
→
X46
)
)
)
)
)
→
(
X12
↔
(
(
¬
X5
)
↔
X45
)
)
)
↔
(
(
¬
X16
)
↔
(
X12
→
X5
→
X44
)
)
)
)
→
(
(
(
(
¬
X17
)
↔
(
X0
↔
X36
)
)
↔
(
(
(
X34
→
X10
)
↔
(
¬
(
X36
→
(
X17
↔
(
X10
↔
X0
)
)
)
)
)
→
(
¬
(
X34
→
(
X17
↔
X37
)
)
)
)
)
↔
(
¬
(
(
(
(
(
X46
→
(
¬
X45
)
→
X36
→
X47
)
↔
(
¬
(
X45
↔
(
X36
↔
(
(
¬
X34
)
→
X47
)
)
)
)
)
↔
(
¬
(
X45
→
(
(
¬
X36
)
↔
X46
)
)
)
)
↔
(
(
¬
(
X47
→
(
X36
↔
(
(
¬
X17
)
↔
X27
)
)
)
)
↔
(
¬
(
(
¬
X36
)
→
(
X27
↔
(
(
¬
X17
)
↔
X46
)
)
)
)
)
)
→
(
X46
↔
(
X34
→
(
¬
X9
)
→
X36
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
(
(
(
X47
→
X46
→
X44
→
X41
)
→
(
¬
(
X47
↔
(
X46
↔
(
X20
↔
X44
)
)
)
)
)
→
(
¬
(
(
¬
X46
)
→
X41
→
X48
)
)
)
→
(
X44
↔
(
X43
↔
(
X42
↔
X46
)
)
)
)
→
(
(
X44
→
(
X22
↔
False
)
)
↔
(
X47
↔
(
X46
→
(
X37
↔
X1
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X49
↔
(
(
¬
X44
)
↔
(
(
¬
X35
)
↔
X48
)
)
)
)
→
(
¬
(
(
(
(
¬
X48
)
↔
(
X35
↔
X49
)
)
↔
(
¬
(
(
¬
X36
)
→
(
¬
X27
)
→
X48
)
)
)
→
(
(
¬
X48
)
↔
(
(
¬
X36
)
↔
X49
)
)
)
)
)
)
→
(
X49
→
(
(
¬
X36
)
↔
(
X35
↔
X43
)
)
)
→
(
¬
X38
)
)
)
→
(
(
¬
(
(
¬
(
(
(
¬
(
(
(
¬
X24
)
↔
(
X6
→
X32
)
)
→
(
¬
(
(
X34
→
(
¬
X24
)
→
X32
)
→
(
X34
↔
(
(
¬
X6
)
→
X38
)
)
)
)
)
)
↔
(
(
(
X34
→
(
(
¬
X26
)
↔
(
(
¬
X2
)
↔
X38
)
)
)
→
X38
→
(
(
¬
X32
)
↔
(
X26
↔
X34
)
)
)
↔
(
X38
↔
(
(
¬
X32
)
↔
X34
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X41
→
(
X38
↔
(
(
¬
X11
)
→
X26
)
)
)
)
→
X38
→
X32
→
X26
→
False
)
)
→
(
¬
(
(
¬
X26
)
→
(
¬
X11
)
→
X38
)
)
)
)
)
)
→
(
¬
(
X38
→
X32
→
X7
→
X26
)
)
)
)
↔
(
(
(
(
(
X34
↔
(
(
¬
X29
)
→
X14
→
X40
)
)
↔
(
X40
→
(
X34
↔
(
X15
→
X29
)
)
)
)
→
(
¬
X40
)
→
(
X29
↔
(
X15
↔
X47
)
)
)
↔
(
¬
(
(
¬
(
X47
→
(
¬
X34
)
→
X41
)
)
→
X40
→
X3
)
)
)
↔
(
(
¬
(
(
(
¬
(
X40
↔
(
X34
↔
(
X16
→
X41
)
)
)
)
↔
(
¬
(
X34
↔
(
X24
↔
(
X16
→
X41
)
)
)
)
)
→
(
¬
(
X41
→
(
X24
↔
X16
)
)
)
)
)
↔
(
(
¬
(
(
¬
X24
)
↔
X41
)
)
↔
(
(
¬
X40
)
↔
(
X19
↔
(
X5
↔
X41
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
(
(
¬
X30
)
→
X25
→
X49
)
→
(
¬
(
(
X49
↔
(
X30
→
(
X17
↔
False
)
)
)
→
(
X49
↔
(
(
¬
X18
)
↔
(
X17
↔
X31
)
)
)
→
(
X49
↔
(
X31
↔
(
X30
→
False
)
)
)
)
)
)
)
↔
(
(
¬
X31
)
↔
(
(
¬
X30
)
↔
(
(
¬
X12
)
↔
X49
)
)
)
)
→
(
¬
(
(
¬
X10
)
↔
(
(
¬
X4
)
→
X25
)
)
)
→
(
¬
(
X25
→
(
X18
↔
(
(
¬
X4
)
→
X31
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
X45
↔
(
(
¬
X39
)
→
(
¬
X0
)
→
X49
)
)
→
(
¬
(
(
¬
X45
)
→
(
X39
↔
(
(
¬
X33
)
→
X49
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X23
)
↔
(
(
¬
X21
)
↔
X45
)
)
)
→
X46
→
X45
→
X33
)
↔
(
¬
(
(
¬
(
X45
→
(
¬
X28
)
→
X33
)
)
→
(
X39
↔
(
X33
↔
(
(
¬
X21
)
→
X45
)
)
)
→
(
¬
(
(
¬
X33
)
→
(
¬
X28
)
→
X39
)
)
)
)
)
→
(
(
¬
(
X33
↔
(
X30
↔
X13
)
)
)
↔
(
¬
(
X46
↔
(
X39
↔
(
X33
↔
X30
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X48
→
X30
→
X8
→
X33
)
)
↔
(
¬
(
X46
↔
(
(
¬
X9
)
→
(
X8
↔
X30
)
)
)
)
)
→
(
X48
↔
(
X46
→
(
¬
X25
)
→
X33
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
52b189...
and proposition id is
f71ad3...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMWQqX6P83MKLhL4aRRHGqGc8KZHsxFsWLj
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
(
¬
(
(
(
¬
X42
)
→
(
(
¬
X37
)
↔
X46
)
)
→
(
(
¬
(
X42
↔
(
X37
↔
(
(
¬
X8
)
↔
X46
)
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X37
)
→
(
X8
↔
X47
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X29
→
(
¬
X25
)
→
(
(
¬
X9
)
↔
X42
)
)
)
→
(
(
¬
(
(
¬
X42
)
↔
(
X37
→
X9
→
X46
)
)
)
↔
(
(
(
¬
X42
)
→
(
(
¬
X29
)
↔
(
X20
→
X46
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X20
)
→
(
X9
↔
X29
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
(
X46
↔
(
X44
↔
X22
)
)
↔
(
X44
↔
(
X41
→
X22
→
X25
)
)
)
→
X48
→
(
(
¬
X25
)
↔
(
X22
→
X44
)
)
)
)
↔
(
X49
→
X44
→
X41
→
X46
)
)
→
(
(
¬
X44
)
↔
(
X25
↔
(
(
¬
X7
)
→
X46
)
)
)
)
)
↔
(
(
X49
↔
(
X46
→
(
X42
↔
(
X22
↔
False
)
)
)
)
↔
(
(
X49
↔
(
X22
→
(
¬
X5
)
→
X46
)
)
↔
(
¬
(
X46
↔
(
X22
↔
(
(
¬
X5
)
↔
X14
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X47
↔
(
X33
↔
(
(
¬
X27
)
↔
X43
)
)
)
)
→
(
¬
(
X43
↔
(
X27
↔
(
(
¬
X2
)
↔
X47
)
)
)
)
)
→
(
¬
(
X35
↔
(
(
¬
X27
)
→
X45
)
)
)
→
(
¬
(
X45
↔
(
X2
↔
(
(
¬
X0
)
↔
X35
)
)
)
)
)
→
(
¬
X33
)
→
(
¬
X27
)
→
(
(
¬
X2
)
↔
X40
)
)
→
(
(
¬
(
(
(
¬
(
X46
→
(
X39
↔
(
X17
↔
X15
)
)
)
)
→
(
¬
(
X48
↔
(
X46
→
X39
→
X17
)
)
)
)
→
(
¬
(
(
X48
→
(
¬
X43
)
→
X46
)
→
(
¬
(
(
(
¬
(
X48
→
(
X46
↔
(
(
¬
X15
)
↔
X45
)
)
)
)
↔
(
X46
→
(
¬
X43
)
→
(
(
¬
X15
)
↔
X45
)
)
)
→
(
¬
(
X45
↔
(
X43
↔
X46
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X39
↔
(
X26
↔
X45
)
)
)
→
X46
→
(
(
¬
X39
)
↔
(
(
¬
X18
)
→
X45
)
)
)
→
(
X46
→
(
(
¬
X35
)
↔
(
(
¬
X24
)
↔
X39
)
)
)
→
(
(
¬
X26
)
↔
(
(
¬
X24
)
→
X46
)
)
)
)
)
)
)
→
(
(
(
(
(
(
X31
↔
(
X28
→
False
)
)
↔
(
(
¬
X38
)
→
(
(
¬
X12
)
↔
(
(
¬
X10
)
↔
X48
)
)
)
)
↔
(
(
¬
X12
)
↔
(
X10
↔
X31
)
)
)
→
(
¬
(
(
X48
→
(
¬
X12
)
→
(
(
¬
X1
)
↔
X32
)
)
→
(
¬
(
X48
→
X12
→
(
(
¬
X1
)
↔
X31
)
)
)
)
)
)
↔
(
¬
(
X48
↔
(
(
¬
X32
)
↔
(
X11
↔
X38
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
X38
→
(
X23
↔
(
X6
→
X13
)
)
)
→
(
¬
(
X23
→
(
¬
X6
)
→
X48
)
)
)
↔
(
¬
(
X48
→
(
X23
↔
X13
)
)
)
)
→
(
¬
X38
)
→
(
X23
↔
X48
)
)
→
(
X38
↔
(
X13
→
(
¬
X6
)
→
X36
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X47
↔
(
X38
↔
(
X21
↔
X36
)
)
)
)
→
(
(
(
¬
X47
)
↔
(
(
¬
X36
)
→
(
¬
X32
)
→
X48
)
)
↔
(
¬
(
X38
→
(
(
¬
X32
)
↔
X48
)
)
)
)
)
)
→
(
X32
↔
X38
)
)
)
→
(
¬
(
(
X47
↔
(
(
¬
X30
)
→
X38
)
)
→
(
¬
(
(
¬
(
X47
→
X30
→
X21
→
X16
)
)
→
(
¬
(
X47
→
(
X30
↔
(
X16
↔
X38
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
X48
↔
(
X36
→
(
(
¬
X19
)
↔
X34
)
)
)
↔
(
¬
(
(
(
X48
→
(
X36
↔
(
X34
↔
X47
)
)
)
↔
(
¬
(
X48
↔
(
X36
↔
(
(
¬
X3
)
↔
X47
)
)
)
)
)
→
X49
→
(
X47
↔
(
(
¬
X34
)
→
X48
)
)
)
)
)
→
X47
→
(
¬
X34
)
→
(
(
¬
X4
)
↔
X36
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
(
(
¬
(
(
X41
↔
(
X2
↔
X49
)
)
→
X41
→
(
X30
↔
X48
)
)
)
→
(
¬
(
(
X41
→
X28
→
X22
)
→
X49
→
X28
→
(
¬
X22
)
→
X48
)
)
)
)
→
(
(
¬
X28
)
↔
(
(
¬
X18
)
↔
(
(
¬
X8
)
→
X30
)
)
)
→
(
(
(
X49
→
(
¬
X18
)
→
X8
→
X25
)
↔
(
X30
→
(
¬
X8
)
→
X18
)
)
↔
(
¬
(
X49
↔
(
X25
↔
(
X18
↔
X17
)
)
)
)
)
)
)
→
(
X49
↔
(
(
¬
X41
)
↔
(
X17
→
X47
)
)
)
)
↔
(
(
(
(
¬
(
X43
→
X36
→
X34
→
X46
)
)
↔
(
X47
↔
(
X36
→
(
X34
↔
X46
)
)
)
)
↔
(
(
(
(
¬
(
X46
↔
(
(
¬
X36
)
↔
(
X14
→
X37
)
)
)
)
↔
(
(
¬
X43
)
↔
(
X14
↔
X46
)
)
)
↔
(
¬
(
X46
→
X37
→
X36
)
)
)
↔
(
(
¬
X46
)
→
X37
→
X36
→
X48
)
)
)
→
(
¬
(
(
(
(
(
¬
X42
)
↔
(
(
¬
X13
)
→
X47
)
)
→
(
¬
(
(
¬
(
X47
↔
(
X33
↔
X7
)
)
)
→
X47
→
X43
→
(
X7
↔
X42
)
)
)
)
→
(
(
¬
(
X49
→
X43
→
X42
→
X7
)
)
↔
(
¬
(
X43
→
(
(
¬
X42
)
↔
(
X12
→
X49
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
X46
→
(
(
¬
X42
)
↔
(
(
¬
X33
)
↔
X43
)
)
)
→
(
¬
(
X46
↔
(
X43
↔
(
X42
↔
(
X5
→
False
)
)
)
)
)
)
)
↔
(
¬
(
X43
↔
(
X42
→
X33
)
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X23
)
↔
(
(
¬
X5
)
→
X33
)
)
)
)
)
↔
(
(
X42
→
X33
→
(
X4
↔
X46
)
)
→
(
X46
↔
(
X33
→
(
X10
↔
X4
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
¬
X47
)
↔
(
X46
↔
(
X32
→
X49
)
)
)
→
(
¬
(
(
¬
X46
)
→
X32
→
(
X31
↔
X49
)
)
)
)
→
(
¬
(
X49
↔
(
X47
↔
(
X26
→
X46
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
(
(
X38
→
X26
→
X3
→
X16
)
↔
(
(
¬
X16
)
↔
X32
)
)
→
(
¬
(
X38
→
(
(
¬
X3
)
↔
X26
)
)
)
)
)
→
(
¬
X38
)
)
→
X32
→
(
X27
↔
(
X16
↔
X26
)
)
)
→
(
(
¬
(
(
¬
X32
)
→
(
¬
X16
)
→
X40
)
)
↔
(
X26
→
(
X0
↔
X32
)
)
)
)
)
→
(
¬
X38
)
→
X0
→
X46
)
)
→
(
¬
(
(
(
(
X49
↔
(
(
¬
X35
)
→
X39
)
)
→
(
X46
→
X39
→
(
X24
↔
X35
)
)
→
(
¬
(
X46
→
X39
→
X24
→
X19
)
)
)
↔
(
(
¬
(
(
(
¬
X39
)
↔
(
(
¬
X35
)
↔
(
X15
→
X44
)
)
)
→
(
X44
↔
(
X39
→
(
X35
↔
(
X11
↔
False
)
)
)
)
)
)
↔
(
X44
↔
(
(
¬
X15
)
↔
(
(
¬
X9
)
→
X35
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X44
)
↔
(
(
¬
X29
)
↔
(
X6
→
X47
)
)
)
)
→
X29
→
(
X6
↔
X1
)
)
↔
(
(
(
¬
(
X46
→
X45
→
(
¬
X20
)
→
X35
)
)
→
(
X44
↔
(
X35
↔
(
X20
↔
X46
)
)
)
)
→
(
¬
(
(
(
(
¬
X46
)
→
X45
→
(
¬
X29
)
→
X47
)
→
(
¬
(
X46
↔
(
X29
↔
(
X21
↔
X45
)
)
)
)
)
→
(
¬
(
X46
→
(
X44
↔
(
(
¬
X21
)
→
X45
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
d6808e...
and proposition id is
60de36...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMTkyTAvpfS8C7xACHzee5DruWztZf3ogcR
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
(
¬
(
(
(
(
(
X50
→
(
X44
↔
(
X36
↔
(
X9
↔
False
)
)
)
)
↔
(
(
¬
(
X44
→
(
X36
↔
(
(
¬
X16
)
↔
X49
)
)
)
)
↔
(
¬
(
X50
→
X49
→
(
X43
↔
X36
)
)
)
)
)
→
(
(
¬
(
(
X49
↔
(
(
¬
X22
)
→
X36
)
)
→
(
(
X50
→
X36
→
(
X28
↔
(
X22
↔
False
)
)
)
↔
(
X50
↔
(
X36
→
X49
)
)
)
)
)
↔
(
(
¬
X49
)
↔
(
X36
→
X50
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
¬
(
(
¬
X37
)
→
(
¬
X19
)
→
(
(
¬
X2
)
↔
X49
)
)
)
↔
(
¬
(
X47
↔
(
X19
↔
(
X2
↔
X37
)
)
)
)
)
→
(
¬
(
X19
→
(
¬
X2
)
→
X49
)
)
)
)
→
(
X49
↔
(
X37
↔
(
(
¬
X10
)
↔
X19
)
)
)
)
)
→
(
(
(
(
¬
(
X47
→
X19
→
(
X7
↔
X50
)
)
)
↔
(
(
¬
X47
)
→
(
¬
X44
)
→
(
X7
↔
X50
)
)
)
→
(
¬
(
(
¬
(
X47
→
(
X45
↔
(
X44
→
X50
)
)
)
)
→
(
¬
(
X5
→
X47
)
)
)
)
)
↔
(
¬
(
X44
↔
(
X19
→
X49
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
X42
↔
(
X30
→
(
¬
X29
)
→
X45
)
)
→
(
X42
↔
(
X41
→
(
X30
↔
X50
)
)
)
→
(
(
¬
(
X50
↔
(
X45
→
X21
)
)
)
↔
(
X45
→
(
(
¬
X21
)
↔
(
X20
→
X30
)
)
)
)
)
)
↔
(
X45
→
(
X30
↔
(
X29
→
(
X4
↔
False
)
)
)
)
)
↔
(
(
¬
(
(
¬
X42
)
→
(
X11
↔
(
(
¬
X8
)
↔
X44
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X44
)
↔
(
X42
↔
X45
)
)
)
→
(
¬
(
(
¬
(
X44
→
(
(
¬
X42
)
↔
(
(
¬
X11
)
→
X45
)
)
)
)
→
(
¬
(
X45
↔
(
(
¬
X44
)
↔
(
X42
↔
X50
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
X44
)
↔
(
(
¬
X30
)
→
X50
)
)
↔
(
¬
(
X30
↔
(
(
¬
X27
)
→
(
¬
X15
)
→
X50
)
)
)
)
↔
(
X44
→
(
X15
↔
(
X12
↔
X27
)
)
)
)
↔
(
X44
→
X26
→
X50
)
)
)
)
)
↔
(
(
(
(
¬
(
X49
→
(
X48
↔
(
X46
↔
X31
)
)
)
)
→
(
X48
↔
(
X46
→
(
X41
↔
X49
)
)
)
)
↔
(
(
(
¬
(
(
X46
→
X40
→
X38
→
(
X14
↔
False
)
)
→
(
¬
(
(
(
(
¬
(
X49
→
(
X39
↔
(
X14
→
False
)
)
)
)
↔
(
X40
↔
(
X39
↔
(
X38
→
X49
)
)
)
)
↔
(
X49
↔
(
X39
↔
(
X38
→
False
)
)
)
)
→
(
(
¬
X40
)
↔
X49
)
)
)
)
)
→
(
¬
(
X40
↔
(
(
¬
X39
)
→
X14
→
X41
)
)
)
)
↔
(
(
(
¬
(
X46
→
X40
→
X49
)
)
↔
(
(
(
¬
(
(
¬
(
X46
↔
(
X35
→
X40
)
)
)
→
(
(
X49
→
X40
→
(
X35
↔
False
)
)
→
(
X49
↔
(
X46
↔
(
X40
→
X13
)
)
)
)
→
(
¬
(
(
¬
X35
)
→
(
X13
↔
X49
)
)
)
)
)
↔
(
X46
→
(
(
¬
X40
)
↔
X49
)
)
)
↔
(
¬
(
X40
↔
(
(
¬
X35
)
↔
X49
)
)
)
)
)
↔
(
(
¬
X35
)
→
(
(
¬
X25
)
↔
X40
)
)
)
)
)
→
(
(
¬
(
X34
↔
(
(
¬
X32
)
→
X1
→
X45
)
)
)
↔
(
¬
(
X46
→
(
X45
↔
X32
)
)
)
)
→
(
(
¬
(
(
¬
(
(
(
X46
↔
(
X34
→
(
X18
↔
X49
)
)
)
→
(
¬
(
X34
↔
(
(
¬
X24
)
↔
X45
)
)
)
)
→
(
¬
(
(
X45
→
X34
→
(
X24
↔
X41
)
)
→
(
X41
↔
(
X34
→
(
X17
↔
X45
)
)
)
)
)
)
)
→
(
(
X49
↔
(
X33
↔
X18
)
)
↔
(
¬
(
(
X49
↔
False
)
→
(
(
(
¬
X33
)
→
X6
→
(
X0
↔
X49
)
)
↔
(
X18
→
(
¬
X6
)
→
(
X0
↔
X45
)
)
)
)
)
)
)
)
↔
(
(
X46
↔
(
X45
↔
X41
)
)
→
X41
→
(
X3
↔
X23
)
)
)
)
)
↔
(
(
(
(
(
X10
↔
(
(
¬
X4
)
↔
X40
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X32
)
→
(
X10
↔
(
X4
→
X37
)
)
)
)
→
(
X42
↔
(
X37
→
X4
→
X32
)
)
)
)
↔
(
X10
↔
(
(
¬
X4
)
↔
X40
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X37
)
→
(
X26
↔
(
(
¬
X22
)
↔
X41
)
)
)
)
→
(
(
(
X42
↔
(
X8
↔
X37
)
)
↔
(
¬
(
X32
→
(
(
¬
X22
)
↔
(
X8
→
X37
)
)
)
)
)
↔
(
(
X32
→
(
X22
↔
(
X0
→
X26
)
)
)
↔
(
¬
(
(
¬
X26
)
↔
(
(
¬
X22
)
↔
(
(
¬
X0
)
→
X42
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
X46
↔
(
(
¬
X43
)
↔
(
X34
→
X44
)
)
)
→
(
X49
↔
(
X44
→
X43
→
X34
→
False
)
)
→
(
¬
X43
)
→
(
X34
↔
(
(
¬
X20
)
↔
X44
)
)
)
→
(
¬
(
(
X49
↔
(
X46
→
(
X42
↔
X44
)
)
)
→
X46
→
X44
→
(
¬
X40
)
→
X49
)
)
)
→
(
(
(
¬
(
X48
→
X46
→
(
¬
X38
)
→
X43
)
)
→
X44
→
(
(
¬
X38
)
↔
X43
)
)
↔
(
(
¬
(
(
(
(
(
(
¬
X41
)
→
(
¬
X38
)
→
(
(
¬
X33
)
↔
X46
)
)
→
(
¬
(
X38
↔
X43
)
)
)
↔
(
(
¬
X41
)
↔
(
X33
↔
X46
)
)
)
→
(
(
¬
(
X43
↔
X5
)
)
↔
(
¬
(
(
¬
(
X46
→
X43
→
X5
→
False
)
)
→
(
(
X46
↔
(
X41
→
X43
)
)
↔
(
(
¬
(
(
¬
X43
)
→
X5
→
X46
)
)
↔
(
¬
(
X43
→
X27
→
X41
)
)
)
)
)
)
)
)
→
(
X46
↔
(
X41
↔
(
X38
→
X19
→
False
)
)
)
)
)
→
(
(
¬
X38
)
↔
(
(
¬
X33
)
→
(
(
¬
X24
)
↔
X46
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
(
(
¬
(
X36
→
X35
→
X30
→
X43
)
)
→
(
¬
(
(
¬
X36
)
→
X35
→
(
(
¬
X7
)
↔
X43
)
)
)
)
)
→
X43
→
X7
→
False
)
)
↔
(
¬
(
(
(
¬
(
(
(
¬
X30
)
→
(
¬
X21
)
→
(
¬
X3
)
→
X43
)
→
(
X43
↔
(
(
¬
X30
)
→
X3
→
X36
)
)
)
)
↔
(
X43
→
(
¬
X21
)
→
(
X13
↔
X30
)
)
)
→
X36
→
X30
→
X21
)
)
)
↔
(
(
(
X35
↔
(
X29
→
X18
→
False
)
)
↔
(
(
¬
(
X35
→
(
¬
X29
)
→
X18
→
X43
)
)
↔
(
X43
→
X35
→
(
(
¬
X18
)
↔
X42
)
)
)
)
↔
(
X43
→
(
¬
X36
)
→
X29
→
X42
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
(
¬
X39
)
→
(
X11
↔
(
X6
→
X42
)
)
)
→
(
¬
(
X47
→
X14
)
)
)
)
↔
(
X42
→
False
)
)
→
(
¬
(
(
(
(
¬
(
X39
→
X30
→
X17
→
X48
)
)
↔
(
X47
→
(
X39
↔
X30
)
)
)
↔
(
¬
(
(
(
(
X48
→
X42
→
X39
→
(
X30
↔
False
)
)
→
(
X48
↔
(
X39
→
X30
→
X9
)
)
)
→
(
¬
(
X39
↔
(
X30
↔
(
(
¬
X1
)
↔
X48
)
)
)
)
)
→
X48
→
(
(
¬
X30
)
↔
X39
)
)
)
)
→
(
¬
X42
)
→
X39
→
X17
→
X45
)
)
)
→
(
(
(
(
X50
→
(
X35
↔
(
X25
↔
(
X16
→
False
)
)
)
)
↔
(
¬
(
X25
→
X16
→
X12
→
False
)
)
)
↔
(
(
¬
(
X42
↔
(
X35
→
False
)
)
)
→
(
¬
(
X25
→
(
¬
X2
)
→
X50
)
)
)
)
↔
(
(
¬
(
X48
↔
(
(
¬
X42
)
↔
(
X31
↔
X50
)
)
)
)
→
(
(
X45
→
(
X31
↔
(
X28
→
X15
)
)
)
↔
(
(
X45
→
(
(
¬
X31
)
↔
(
X23
→
X35
)
)
)
↔
(
X35
→
(
X28
↔
X50
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
b46d0d...
and proposition id is
fc6195...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMPgqGc1SfmxpFJ4qtXQbWwJ5GqKdRryFjp
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
(
(
(
¬
(
(
¬
X38
)
→
(
(
¬
X33
)
↔
(
X20
↔
X44
)
)
)
)
↔
(
¬
(
X38
↔
(
X33
→
(
(
¬
X17
)
↔
X44
)
)
)
)
)
→
(
X44
↔
(
X35
→
X33
→
(
X20
↔
False
)
)
)
)
→
(
(
¬
(
X35
→
(
X28
↔
(
X20
→
X17
)
)
)
)
↔
(
X17
→
X3
)
)
)
→
(
(
(
¬
(
X44
↔
(
X35
↔
(
X26
→
False
)
)
)
)
↔
(
¬
(
X44
→
(
X35
↔
(
X32
→
X4
→
False
)
)
)
)
)
↔
(
¬
(
(
(
(
(
X48
↔
(
X35
↔
(
X32
→
X44
)
)
)
↔
(
X48
→
(
(
¬
X35
)
↔
(
X32
→
X38
)
)
)
)
→
(
X44
↔
(
X42
↔
X35
)
)
)
↔
(
¬
(
(
¬
X35
)
↔
(
X32
↔
(
(
¬
X8
)
→
X42
)
)
)
)
)
→
(
X48
↔
(
X32
→
X44
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X23
↔
(
X16
→
(
X2
↔
X45
)
)
)
)
↔
(
¬
(
X39
↔
(
X16
→
(
¬
X2
)
→
X23
)
)
)
)
↔
(
(
¬
(
(
(
¬
X45
)
→
(
X27
↔
(
(
¬
X18
)
↔
X49
)
)
)
→
(
¬
(
(
¬
X45
)
→
(
(
¬
X27
)
↔
(
X19
↔
X49
)
)
)
)
)
)
→
(
(
¬
(
X49
↔
(
X39
↔
(
(
¬
X12
)
↔
X23
)
)
)
)
↔
(
(
¬
X45
)
→
(
¬
X39
)
→
(
X27
↔
X49
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
X24
)
↔
X44
)
→
(
¬
(
(
¬
X42
)
↔
(
(
¬
X24
)
→
(
¬
X23
)
→
X44
)
)
)
→
(
¬
(
(
¬
X44
)
→
(
¬
X42
)
→
(
X24
↔
X45
)
)
)
)
)
→
(
(
¬
X42
)
↔
(
(
¬
X38
)
→
X24
→
X44
)
)
)
↔
(
(
(
¬
X24
)
→
X15
→
(
¬
X10
)
→
X42
)
↔
(
(
X38
↔
(
(
¬
X27
)
→
(
¬
X15
)
→
X42
)
)
→
(
¬
(
X38
↔
(
X27
→
X24
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
(
¬
(
(
¬
X46
)
↔
(
(
¬
X45
)
↔
(
X41
↔
X48
)
)
)
)
↔
(
(
¬
X45
)
→
X44
→
X13
→
X48
)
)
↔
(
(
X48
→
X44
→
(
X21
↔
X34
)
)
↔
(
X48
↔
(
X34
→
X46
)
)
)
)
→
(
(
X46
↔
(
X44
↔
(
X29
→
X6
)
)
)
↔
(
¬
(
(
X46
→
(
(
¬
X6
)
↔
X44
)
)
→
(
X46
↔
(
X41
→
(
X6
↔
X30
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X41
→
(
¬
X9
)
→
X1
→
X11
)
)
→
(
(
X9
↔
False
)
↔
(
(
(
¬
(
(
¬
(
X45
→
X9
→
False
)
)
→
(
X44
↔
(
X25
↔
X5
)
)
)
)
→
(
X45
↔
X44
)
)
→
(
¬
(
(
X41
→
(
(
¬
X37
)
↔
(
(
¬
X9
)
→
X44
)
)
)
→
(
(
(
¬
X37
)
→
(
(
¬
X25
)
↔
(
X11
→
X41
)
)
)
→
(
¬
X44
)
)
→
(
¬
(
X44
→
(
X37
↔
X25
)
)
)
)
)
)
)
)
↔
(
X45
↔
(
X37
↔
X25
)
)
)
)
→
(
¬
(
(
(
X48
→
(
X47
↔
(
X43
→
X40
→
False
)
)
)
↔
(
¬
(
(
X47
↔
(
(
¬
X43
)
→
(
¬
X41
)
→
X48
)
)
→
(
¬
(
X47
→
(
(
¬
X41
)
↔
(
X7
↔
X48
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
(
¬
X43
)
→
X41
→
X47
)
→
(
¬
(
(
¬
X46
)
↔
(
(
¬
X43
)
→
X36
→
X47
)
)
)
)
)
→
(
X46
↔
X47
)
)
)
↔
(
(
¬
(
(
X47
↔
(
X31
→
(
X22
↔
X46
)
)
)
→
(
X22
→
X43
)
→
(
¬
(
(
(
¬
X31
)
→
(
(
¬
X22
)
↔
(
(
¬
X14
)
→
X47
)
)
)
→
(
¬
(
X47
↔
(
X22
↔
(
X14
↔
X43
)
)
)
)
)
)
)
)
→
(
¬
(
X43
→
X31
→
(
(
¬
X0
)
↔
X46
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X47
↔
(
X36
↔
(
X16
→
X41
)
)
)
)
↔
(
(
¬
(
(
¬
X45
)
↔
(
(
¬
X41
)
→
(
(
¬
X27
)
↔
X46
)
)
)
)
↔
(
(
¬
(
X46
→
(
X16
↔
X41
)
)
)
→
(
¬
(
(
X41
↔
X46
)
→
(
X41
↔
(
X36
↔
(
X8
→
X45
)
)
)
)
)
)
)
)
↔
(
(
(
X45
→
(
X26
↔
(
(
¬
X25
)
↔
X42
)
)
)
↔
(
¬
(
(
X45
↔
(
(
¬
X29
)
↔
X42
)
)
→
(
¬
(
(
X45
→
X29
→
(
(
¬
X21
)
↔
X42
)
)
→
(
¬
(
X26
↔
(
(
¬
X0
)
↔
X42
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X29
→
(
X1
↔
X46
)
)
)
↔
(
¬
(
X47
↔
(
(
¬
X3
)
↔
(
X1
↔
X29
)
)
)
)
)
↔
(
X46
↔
(
X26
→
(
¬
X1
)
→
X3
)
)
)
↔
(
¬
(
X46
→
(
X29
↔
X47
)
)
)
)
→
(
(
(
X29
→
(
X26
↔
(
X1
→
X17
)
)
)
↔
(
¬
(
X45
↔
X17
)
)
)
↔
(
¬
(
X45
↔
(
X29
↔
(
X26
↔
False
)
)
)
)
)
)
)
↔
(
(
(
(
X49
→
X48
→
(
X42
↔
(
X12
↔
False
)
)
)
↔
(
X18
→
X38
)
)
↔
(
(
(
X46
↔
(
X42
→
(
X2
↔
X7
)
)
)
→
(
¬
(
(
X7
→
(
X2
↔
X6
)
)
→
(
¬
(
X18
→
X7
→
X6
→
X42
)
)
)
)
)
↔
(
X48
↔
(
X6
→
X2
)
)
)
)
↔
(
(
(
X48
↔
(
X39
→
X49
)
)
↔
(
¬
(
(
X48
↔
(
X46
↔
X45
)
)
→
(
X45
↔
(
X39
→
X32
)
)
)
)
)
↔
(
(
¬
(
X49
↔
(
X46
→
False
)
)
)
↔
(
¬
(
(
X48
↔
(
(
¬
X44
)
→
X10
→
X49
)
)
→
(
¬
X46
)
→
(
X4
↔
X48
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
(
¬
X40
)
↔
(
(
¬
X37
)
→
X34
→
X46
)
)
)
↔
(
(
(
X46
↔
(
X40
↔
(
X15
↔
X48
)
)
)
→
X40
→
X34
→
X48
)
↔
(
¬
(
(
¬
X40
)
↔
(
X34
→
(
X15
↔
X43
)
)
)
)
)
)
→
X46
→
(
(
¬
X15
)
↔
(
X5
→
X48
)
)
)
→
(
(
(
¬
(
X43
↔
(
X30
↔
(
X20
→
X48
)
)
)
)
↔
(
(
¬
X40
)
→
(
X30
↔
(
(
¬
X23
)
→
X43
)
)
)
)
↔
(
(
¬
(
X43
↔
(
(
¬
X30
)
↔
(
X24
→
X48
)
)
)
)
↔
(
¬
(
(
¬
X24
)
↔
(
X9
↔
X43
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
(
¬
(
X43
→
(
(
¬
X14
)
↔
(
X13
→
X28
)
)
)
)
→
(
¬
(
X28
→
X19
→
(
¬
X13
)
→
X40
)
)
)
→
(
¬
(
X43
↔
(
(
¬
X28
)
→
X40
)
)
)
)
)
↔
(
(
X46
↔
(
X43
→
X19
→
X28
)
)
→
X28
→
(
X19
↔
(
X11
↔
X35
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
X40
→
X35
→
(
¬
X22
)
→
X49
)
→
(
(
¬
(
X46
→
X40
→
(
X33
↔
X49
)
)
)
↔
(
X46
↔
(
X35
→
(
¬
X33
)
→
X49
)
)
)
)
→
(
X49
↔
(
(
¬
X40
)
↔
X46
)
)
)
)
→
(
¬
(
X49
→
X33
→
X40
)
)
)
)
→
(
¬
(
X49
→
(
X46
↔
(
(
¬
X31
)
→
X33
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
0ceabf...
and proposition id is
d6b8f0...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMVpoSj79AeSTHmgBNkhNhm1Aka5eTuhrfB
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
(
¬
(
(
(
(
(
¬
(
(
(
(
¬
X34
)
→
(
X27
↔
(
(
¬
X19
)
↔
X42
)
)
)
→
(
¬
(
(
(
(
X49
→
(
X48
↔
(
X42
→
False
)
)
)
↔
(
(
(
X49
→
X48
→
X42
→
X28
)
↔
(
X49
↔
(
X48
→
(
X34
↔
X42
)
)
)
)
↔
(
(
¬
X42
)
↔
(
X34
↔
X48
)
)
)
)
↔
(
X48
↔
(
X32
→
X28
→
X27
)
)
)
→
X42
→
(
X34
↔
(
X4
↔
X49
)
)
)
)
)
→
(
¬
(
(
¬
X27
)
↔
(
(
¬
X4
)
→
X49
)
)
)
)
)
↔
(
(
(
X50
→
(
(
¬
X13
)
↔
X45
)
)
↔
(
X49
↔
(
X28
→
X13
→
X45
)
)
)
↔
(
(
¬
X28
)
↔
(
X13
→
(
X4
↔
X45
)
)
)
)
)
↔
(
(
(
¬
(
X38
↔
(
(
¬
X18
)
→
(
¬
X3
)
→
X34
)
)
)
→
(
¬
(
X18
→
(
X3
↔
X17
)
)
)
)
↔
(
(
X17
↔
X42
)
→
X50
→
X38
→
X18
→
X17
)
)
)
↔
(
(
(
(
¬
X48
)
↔
(
X21
↔
X50
)
)
↔
(
¬
(
X50
→
(
¬
X23
)
→
(
X2
↔
X48
)
)
)
)
↔
(
(
X45
↔
(
X23
→
X1
)
)
→
(
¬
(
(
¬
(
X48
↔
(
X23
↔
(
X2
↔
X45
)
)
)
)
→
X1
→
X48
)
)
)
)
)
→
(
¬
(
(
¬
(
(
X50
→
X31
→
(
¬
X0
)
→
X7
)
→
(
(
(
¬
X7
)
→
X48
)
↔
(
¬
(
(
(
¬
X49
)
→
(
X48
↔
(
X7
↔
X50
)
)
)
→
(
X50
↔
(
X49
→
X48
→
X47
→
False
)
)
→
(
¬
(
X49
↔
(
X47
↔
(
(
¬
X43
)
→
X50
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
X46
→
X20
→
X48
)
→
X48
→
(
¬
X47
)
→
(
X46
↔
X50
)
)
)
→
(
¬
(
X47
→
(
X46
↔
X20
)
)
)
)
)
↔
(
X48
→
(
X47
↔
(
(
¬
X25
)
↔
X46
)
)
)
)
→
(
¬
X46
)
→
(
(
¬
X25
)
↔
(
(
¬
X6
)
→
X47
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X9
)
→
(
(
¬
X8
)
↔
X22
)
)
)
↔
(
X40
↔
(
X22
→
(
X9
↔
X50
)
)
)
)
→
(
(
¬
(
(
(
X50
↔
False
)
↔
(
(
(
X50
↔
(
X48
→
(
¬
X5
)
→
X40
)
)
↔
(
¬
(
X5
↔
X22
)
)
)
↔
(
¬
(
X50
↔
(
X5
→
X40
)
)
)
)
)
→
X50
)
)
↔
(
(
X40
↔
(
X22
↔
(
(
¬
X16
)
↔
X50
)
)
)
↔
(
X44
↔
(
X22
→
(
¬
X16
)
→
X50
)
)
)
)
)
→
(
¬
(
(
(
(
X48
↔
(
X41
→
(
X40
↔
(
X24
→
False
)
)
)
)
↔
(
X26
↔
X48
)
)
↔
(
(
(
(
X30
↔
(
(
¬
X29
)
↔
X48
)
)
↔
(
¬
(
(
¬
(
X48
→
(
X40
↔
(
X11
↔
False
)
)
)
)
→
(
X30
↔
(
X29
↔
X48
)
)
)
)
)
↔
(
X48
→
(
X30
↔
X29
)
)
)
↔
(
(
X41
→
(
¬
X15
)
→
(
(
¬
X10
)
↔
X30
)
)
↔
(
(
X40
↔
X48
)
→
(
¬
(
(
X30
→
X41
)
→
X48
→
(
X41
↔
(
X10
→
X40
)
)
)
)
→
(
¬
(
(
¬
X41
)
→
(
X30
↔
X48
)
)
)
)
)
)
)
→
(
(
(
(
¬
X40
)
→
(
(
¬
X35
)
↔
(
X12
↔
X49
)
)
)
↔
(
¬
(
(
¬
(
X48
→
X40
→
X39
→
X50
)
)
→
(
¬
(
(
X48
→
X33
→
(
(
¬
X12
)
↔
X50
)
)
→
(
¬
(
(
¬
(
(
¬
X40
)
→
X35
→
X50
)
)
→
X48
→
X12
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X50
→
(
X48
↔
(
X14
↔
X41
)
)
)
)
→
(
X48
↔
(
X41
→
X37
→
False
)
)
)
→
(
¬
(
X48
→
(
X41
↔
(
X37
→
X49
)
)
)
)
)
→
(
¬
(
X50
→
(
¬
X48
)
→
(
¬
X36
)
→
X49
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
X48
↔
(
X41
↔
(
X24
↔
X18
)
)
)
)
→
(
(
(
¬
(
X48
↔
(
X45
→
X24
→
False
)
)
)
↔
(
¬
(
(
X41
→
(
(
¬
X12
)
↔
X45
)
)
→
(
¬
(
X45
→
(
¬
X24
)
→
(
¬
X12
)
→
X48
)
)
)
)
)
→
(
¬
(
X41
↔
(
X12
→
X45
)
)
)
)
→
X24
→
X45
)
→
(
¬
(
(
(
X49
→
(
(
¬
X42
)
↔
(
(
¬
X7
)
→
X44
)
)
)
↔
(
¬
(
X44
→
(
(
¬
X41
)
↔
(
(
¬
X7
)
↔
X42
)
)
)
)
)
→
(
¬
(
X44
↔
(
(
¬
X41
)
→
(
(
¬
X35
)
↔
X42
)
)
)
)
)
)
→
(
(
(
X49
↔
(
X42
→
X45
)
)
→
X45
→
(
(
¬
X24
)
↔
(
X3
↔
X42
)
)
)
↔
(
X49
↔
(
X44
↔
(
X24
↔
X42
)
)
)
)
)
→
(
¬
(
(
¬
(
X48
→
(
(
¬
X29
)
↔
(
X27
↔
X44
)
)
)
)
→
(
(
(
¬
X34
)
↔
(
X27
↔
(
X4
↔
X48
)
)
)
↔
(
(
¬
(
(
¬
X44
)
↔
(
(
¬
X34
)
→
X48
)
)
)
↔
(
¬
(
X48
↔
(
X46
↔
(
X27
→
X49
)
)
)
)
)
)
→
(
(
¬
(
X25
↔
(
(
¬
X20
)
→
X28
)
)
)
↔
(
¬
(
(
¬
(
X49
→
(
¬
X28
)
→
(
¬
X9
)
→
X48
)
)
→
(
X46
↔
(
X28
↔
(
X9
→
X48
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
X48
↔
(
X32
↔
(
X26
↔
X47
)
)
)
→
(
(
¬
X46
)
→
X39
→
X48
)
→
(
¬
(
(
X39
→
X32
→
(
X26
↔
(
X6
↔
False
)
)
)
→
(
X48
↔
(
X26
→
(
X6
↔
X39
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
X50
↔
(
(
¬
X37
)
→
(
¬
X30
)
→
X46
)
)
→
(
¬
(
X37
→
X30
→
X8
→
X46
)
)
)
)
→
(
¬
(
(
¬
X46
)
↔
(
(
¬
X32
)
↔
(
X8
↔
X48
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X38
→
(
X33
↔
(
(
¬
X14
)
↔
X22
)
)
)
)
→
(
¬
(
X38
↔
(
X10
→
X33
)
)
)
)
→
(
(
(
(
¬
(
X47
→
X38
→
(
X22
↔
(
X17
→
False
)
)
)
)
↔
(
X22
→
X46
)
)
↔
(
¬
(
(
¬
X38
)
→
X22
→
X2
→
X46
)
)
)
↔
(
(
¬
(
(
¬
X33
)
↔
(
(
¬
X23
)
↔
X47
)
)
)
↔
(
(
(
X47
↔
(
X39
↔
(
(
¬
X23
)
↔
X46
)
)
)
↔
(
(
¬
(
X46
↔
(
X38
→
(
X33
↔
False
)
)
)
)
→
(
(
(
¬
X39
)
→
(
¬
X33
)
→
X46
)
↔
(
¬
(
X47
→
(
X46
↔
(
X38
↔
(
X33
↔
False
)
)
)
)
)
)
)
)
↔
(
¬
(
X39
→
(
¬
X38
)
→
X47
)
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X43
→
(
(
¬
X38
)
↔
(
X0
↔
X46
)
)
)
)
↔
(
(
¬
(
(
¬
X31
)
→
(
(
¬
X1
)
↔
X46
)
)
)
→
(
¬
(
X46
↔
(
X43
↔
(
X38
→
X1
)
)
)
)
)
)
→
(
(
(
¬
(
X38
→
(
¬
X19
)
→
X46
)
)
↔
(
(
(
(
¬
X36
)
↔
(
(
¬
X19
)
↔
(
(
¬
X15
)
→
X46
)
)
)
↔
(
X46
↔
(
X38
→
(
X19
↔
X36
)
)
)
)
↔
(
X36
→
(
X19
↔
X15
)
)
)
)
→
X38
→
(
X31
↔
(
X19
→
X36
)
)
)
→
(
(
¬
(
X46
↔
(
(
¬
X38
)
↔
(
(
¬
X36
)
↔
X40
)
)
)
)
↔
(
¬
(
(
X40
→
(
X36
↔
(
X16
→
(
X13
↔
False
)
)
)
)
→
X38
→
(
(
¬
X16
)
↔
(
X13
→
X40
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
X40
)
↔
(
X38
↔
X43
)
)
)
↔
(
(
X49
↔
(
(
¬
X11
)
↔
(
(
¬
X5
)
→
X40
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
X49
→
(
¬
X21
)
→
(
X11
↔
X40
)
)
→
(
X49
↔
(
(
¬
X40
)
↔
(
X11
↔
X50
)
)
)
)
)
→
(
X50
↔
(
X11
↔
False
)
)
)
)
→
X21
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
f21f51...
and proposition id is
171771...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMbGPhScLWtcQKjurrriDosdv2i6F1B5zTV
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
∀X50 :
prop
,
(
(
(
(
¬
(
(
¬
(
(
(
(
(
¬
X49
)
↔
(
(
¬
X25
)
↔
(
X22
↔
X50
)
)
)
→
(
¬
(
X49
↔
X45
)
)
)
↔
(
(
¬
(
(
¬
X45
)
→
X49
)
)
→
(
¬
(
X50
→
X47
→
(
(
¬
X45
)
↔
X46
)
)
)
)
)
→
(
(
¬
(
(
(
X46
↔
X43
)
↔
(
(
¬
X45
)
↔
(
X43
↔
(
X30
→
X49
)
)
)
)
→
(
(
¬
X45
)
↔
(
(
¬
X43
)
→
X46
)
)
)
)
→
(
X45
↔
(
(
¬
X43
)
↔
(
(
¬
X14
)
↔
X46
)
)
)
)
→
(
¬
(
(
¬
X46
)
↔
(
X43
↔
X49
)
)
)
)
)
→
(
(
(
¬
(
X47
→
X36
→
(
(
¬
X21
)
↔
X41
)
)
)
↔
(
¬
(
X43
↔
(
(
¬
X25
)
↔
X36
)
)
)
)
→
(
¬
(
X43
→
X25
→
(
X10
↔
X47
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X18
→
X5
→
X23
)
)
↔
(
X47
↔
(
X43
↔
(
X23
↔
(
X5
↔
False
)
)
)
)
)
→
X43
→
(
¬
X13
)
→
(
X5
↔
X19
)
)
)
↔
(
(
(
¬
X41
)
→
(
X34
↔
(
X18
→
X47
)
)
)
→
X43
→
(
X34
↔
(
X18
→
False
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
X45
→
X11
→
(
X0
↔
X49
)
)
→
(
X45
↔
(
X11
↔
(
(
¬
X3
)
→
X49
)
)
)
)
→
(
X11
↔
(
X0
↔
X49
)
)
)
)
→
(
(
X11
↔
(
(
¬
X7
)
→
X45
)
)
↔
(
(
(
¬
X44
)
↔
(
(
¬
X11
)
→
(
¬
X7
)
→
X45
)
)
↔
(
X45
→
X44
→
X11
→
X3
)
)
)
)
→
(
(
¬
(
X49
↔
(
(
¬
X45
)
↔
(
X8
→
X50
)
)
)
)
↔
(
¬
(
X37
↔
(
X8
→
(
(
¬
X7
)
↔
X49
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
(
¬
X33
)
→
(
¬
X26
)
→
(
(
¬
X6
)
↔
X48
)
)
→
(
X48
↔
(
X46
↔
(
X33
→
(
X6
↔
False
)
)
)
)
)
)
↔
(
¬
(
X46
→
(
X33
↔
X48
)
)
)
)
→
(
¬
X45
)
→
(
(
¬
X20
)
↔
X48
)
)
↔
(
(
(
(
X48
→
X20
→
X16
)
→
(
(
X39
→
(
X31
↔
(
X20
→
X48
)
)
)
↔
(
¬
(
X39
↔
(
X20
→
X16
→
X31
)
)
)
)
)
↔
(
(
¬
(
X48
→
X33
→
(
X31
↔
False
)
)
)
↔
(
(
(
¬
(
X39
→
(
(
¬
X33
)
↔
(
X17
→
X48
)
)
)
)
→
(
(
¬
X39
)
↔
(
X33
↔
(
(
¬
X31
)
→
X48
)
)
)
)
↔
(
¬
(
(
¬
X33
)
↔
(
(
¬
X31
)
↔
X48
)
)
)
)
)
)
↔
(
(
¬
(
X39
→
X28
→
(
X12
↔
X48
)
)
)
↔
(
(
(
(
X12
↔
X4
)
↔
(
(
¬
(
(
¬
X28
)
↔
(
(
¬
X9
)
↔
(
(
¬
X4
)
↔
X39
)
)
)
)
↔
(
X12
↔
(
X4
→
X39
)
)
)
)
→
(
¬
(
X39
→
X9
→
X12
)
)
)
↔
(
¬
(
(
¬
(
X9
↔
(
X4
↔
(
X1
↔
False
)
)
)
)
→
(
X33
↔
(
X4
↔
(
(
¬
X1
)
↔
X39
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
X45
↔
(
(
¬
X42
)
↔
(
X29
→
X46
)
)
)
→
(
¬
(
X46
→
(
X45
↔
(
X38
↔
X29
)
)
)
)
)
→
(
¬
(
X43
→
(
(
¬
X29
)
↔
X46
)
)
)
→
(
(
X46
↔
(
(
¬
X45
)
→
(
X32
↔
X48
)
)
)
↔
(
X46
↔
(
X32
↔
(
X29
↔
X48
)
)
)
)
)
→
(
(
¬
(
(
¬
X43
)
→
(
¬
X35
)
→
X46
)
)
↔
(
¬
(
(
(
¬
(
X40
↔
(
(
¬
X38
)
↔
(
X2
→
X46
)
)
)
)
→
(
¬
(
(
¬
X43
)
→
X40
→
X24
→
X46
)
)
)
→
(
(
¬
(
(
¬
(
X38
→
(
(
¬
X27
)
↔
(
(
¬
X15
)
→
X35
)
)
)
)
→
(
X27
↔
(
X15
→
X43
)
)
)
)
↔
(
X38
↔
(
X27
→
(
(
¬
X15
)
↔
X46
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
(
¬
(
(
¬
X42
)
↔
(
(
¬
X38
)
↔
X48
)
)
)
→
(
¬
(
X48
→
(
(
¬
X38
)
↔
(
(
¬
X8
)
→
X45
)
)
)
)
)
)
↔
(
(
¬
X45
)
→
(
X42
↔
X48
)
)
)
↔
(
(
¬
(
(
¬
X8
)
→
X4
→
X29
)
)
↔
(
¬
(
(
¬
X38
)
↔
(
(
¬
X8
)
→
X42
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X47
→
X42
→
(
(
¬
X7
)
↔
X38
)
)
)
→
(
¬
(
(
(
¬
X19
)
↔
(
(
¬
X11
)
↔
(
X7
↔
X45
)
)
)
→
(
¬
(
X45
↔
(
X38
→
(
X11
↔
X7
)
)
)
)
)
)
)
)
→
X45
→
(
X38
↔
(
X17
↔
X42
)
)
)
)
)
→
(
(
(
(
(
¬
X47
)
↔
(
(
¬
X26
)
→
(
X25
↔
X50
)
)
)
→
(
¬
(
(
¬
(
X50
↔
(
X45
↔
(
X26
→
X33
)
)
)
)
→
(
¬
(
X50
→
(
X47
↔
(
X33
↔
(
X26
→
False
)
)
)
)
)
)
)
)
↔
(
(
¬
(
X5
↔
(
X2
↔
(
X1
↔
X45
)
)
)
)
↔
(
(
X47
→
X14
→
(
X1
↔
X26
)
)
↔
(
¬
(
X45
→
(
(
¬
X14
)
↔
(
(
¬
X1
)
→
X26
)
)
)
)
)
)
)
↔
(
(
(
X47
→
(
(
¬
X45
)
↔
X48
)
)
→
(
(
(
¬
X48
)
↔
(
(
¬
X45
)
→
(
¬
X18
)
→
X50
)
)
↔
(
(
¬
(
X45
→
(
X16
↔
False
)
)
)
→
(
¬
(
(
¬
X48
)
↔
(
X47
↔
X50
)
)
)
)
)
)
↔
(
(
X50
↔
(
(
¬
X47
)
→
X48
)
)
↔
(
¬
(
(
¬
(
X48
↔
(
X47
→
(
X27
↔
False
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X47
)
→
(
X27
↔
(
X12
→
X50
)
)
)
)
→
(
¬
(
X47
↔
(
X27
→
X48
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
(
(
X32
→
X15
)
↔
(
¬
(
X47
↔
(
X32
→
(
(
¬
X24
)
↔
X48
)
)
)
)
)
→
X47
→
(
¬
X32
)
→
X48
)
)
→
(
X48
↔
(
X28
→
(
X15
↔
X47
)
)
)
)
→
(
(
(
X48
→
X13
→
X47
)
↔
(
(
(
¬
X48
)
→
(
¬
X13
)
→
(
¬
X3
)
→
X49
)
↔
(
X49
↔
(
X47
↔
(
(
¬
X13
)
→
X48
)
)
)
)
)
↔
(
X49
↔
(
X47
↔
(
X13
→
X32
)
)
)
)
)
)
→
(
(
X41
↔
(
X24
→
(
X10
↔
X22
)
)
)
↔
(
(
¬
(
(
(
(
X28
→
(
X22
↔
X39
)
)
↔
(
(
X39
↔
(
X28
↔
(
(
¬
X22
)
↔
X24
)
)
)
↔
(
¬
(
(
¬
X28
)
→
X22
→
(
(
¬
X6
)
↔
X39
)
)
)
)
)
→
(
¬
(
X39
→
X28
)
)
)
→
(
X6
↔
X24
)
)
)
→
(
¬
(
(
¬
X24
)
→
X22
→
(
¬
X15
)
→
X28
)
)
→
(
¬
(
X28
→
(
(
¬
X15
)
↔
(
(
¬
X10
)
→
X22
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
X47
↔
(
X31
→
(
¬
X0
)
→
X45
)
)
→
(
¬
(
(
¬
(
(
¬
(
X47
→
(
¬
X45
)
→
X48
)
)
→
(
(
(
¬
X45
)
↔
(
(
¬
X40
)
→
X47
)
)
↔
(
X48
→
(
¬
X40
)
→
X0
→
X45
)
)
)
)
→
(
¬
(
X45
→
X0
→
X40
)
)
)
)
)
)
→
(
¬
(
(
¬
X45
)
↔
X49
)
)
)
→
(
(
¬
(
(
(
¬
(
X47
↔
(
(
¬
X37
)
→
X46
)
)
)
↔
(
¬
(
(
¬
X37
)
↔
(
(
¬
X35
)
→
(
X31
↔
X46
)
)
)
)
)
→
(
¬
(
X37
→
(
X35
↔
(
(
¬
X9
)
↔
X46
)
)
)
)
)
)
↔
(
¬
(
X45
↔
(
X44
→
X35
→
X37
)
)
)
)
)
→
(
¬
(
(
(
(
¬
X43
)
→
(
X36
↔
(
X34
↔
X47
)
)
)
↔
(
X47
↔
(
(
¬
X36
)
→
(
(
¬
X20
)
↔
X43
)
)
)
)
→
(
(
(
X43
→
(
X36
↔
(
(
¬
X23
)
↔
X47
)
)
)
↔
(
¬
(
X36
→
(
(
¬
X23
)
↔
X45
)
)
)
)
↔
(
(
¬
(
X47
→
X30
→
X21
→
X49
)
)
↔
(
X49
↔
(
X45
↔
(
(
¬
X30
)
↔
X43
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
784991...
and proposition id is
a93fbc...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMJFRbCw4u2Rm8gbXuubUukQ3zz7jaCmzNk
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
¬
(
(
(
X16
→
X8
→
(
¬
X7
)
→
X45
)
↔
(
(
(
(
¬
X33
)
↔
(
(
¬
X16
)
→
(
X8
↔
X49
)
)
)
→
(
¬
(
X16
→
(
(
¬
X7
)
↔
X33
)
)
)
)
→
(
¬
(
X49
↔
(
X46
↔
(
(
¬
X7
)
↔
X16
)
)
)
)
)
)
→
(
(
¬
(
X49
↔
(
X46
→
X33
→
X25
)
)
)
↔
(
X49
↔
(
X46
→
(
X33
↔
(
X23
→
False
)
)
)
)
)
→
(
¬
(
X46
→
X33
)
)
)
)
↔
(
(
(
¬
(
(
¬
X41
)
→
X8
→
X47
)
)
↔
(
¬
(
X47
→
(
X41
↔
(
X23
↔
(
X7
→
False
)
)
)
)
)
)
↔
(
X23
→
(
(
¬
X7
)
↔
X47
)
)
)
)
↔
(
(
(
(
(
¬
(
(
(
¬
(
(
¬
X44
)
↔
(
X27
↔
(
(
¬
X14
)
↔
X49
)
)
)
)
↔
(
X46
↔
(
X44
→
(
(
¬
X14
)
↔
X48
)
)
)
)
→
(
¬
(
(
¬
X48
)
→
(
¬
X27
)
→
X49
)
)
)
)
→
(
¬
(
X48
↔
(
X46
↔
X14
)
)
)
)
↔
(
¬
(
X48
→
(
(
¬
X27
)
↔
X46
)
)
)
)
↔
(
(
X49
→
(
X48
↔
(
(
¬
X26
)
→
X46
)
)
)
↔
(
¬
(
X46
→
(
¬
X26
)
→
X34
)
)
)
)
↔
(
¬
(
(
(
(
(
X49
→
(
X42
↔
(
X37
→
False
)
)
)
↔
(
(
X42
→
(
X30
↔
(
X20
↔
False
)
)
)
↔
(
(
(
¬
(
(
(
¬
X30
)
→
(
(
¬
X20
)
↔
(
X6
→
X48
)
)
)
→
(
¬
(
X49
→
(
(
¬
X30
)
↔
(
(
¬
X20
)
→
X48
)
)
)
)
)
)
↔
(
X49
→
X20
→
X48
)
)
→
X48
→
(
X30
↔
X37
)
)
)
)
→
X49
→
X42
→
(
¬
X22
)
→
X48
)
→
(
¬
(
(
¬
(
X22
→
(
X20
↔
(
X19
↔
X47
)
)
)
)
→
X42
→
(
¬
X19
)
→
X43
)
)
)
→
(
X43
↔
(
(
¬
X11
)
↔
(
X3
→
X46
)
)
)
→
(
(
¬
(
X46
↔
(
X43
↔
(
X3
↔
X48
)
)
)
)
↔
(
¬
(
X48
↔
(
(
¬
X46
)
↔
(
(
¬
X43
)
↔
X49
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
(
(
¬
(
(
¬
(
(
(
(
X32
→
(
¬
X0
)
→
X9
)
↔
(
¬
(
X39
→
X0
→
X49
)
)
)
↔
(
¬
(
X39
↔
(
X0
→
X49
)
)
)
)
→
X32
)
)
→
X49
→
(
X40
↔
(
(
¬
X32
)
→
X39
)
)
)
)
↔
(
¬
(
X49
→
(
X39
↔
(
X13
→
X32
)
)
)
)
)
→
(
(
¬
(
X43
→
(
(
¬
X13
)
↔
(
(
¬
X4
)
↔
X40
)
)
)
)
↔
(
¬
(
X43
↔
(
X32
↔
(
X13
→
X40
)
)
)
)
)
)
)
↔
(
(
¬
X40
)
→
X0
→
X49
)
)
→
(
(
(
X49
↔
(
(
¬
X29
)
→
(
(
¬
X15
)
↔
X31
)
)
)
→
(
(
¬
X29
)
↔
(
X15
→
X43
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
¬
(
(
¬
X31
)
→
(
X24
↔
X48
)
)
)
→
(
X31
↔
(
X29
↔
(
X18
→
X24
)
)
)
)
→
(
¬
(
X24
→
X18
→
X48
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X43
)
↔
(
X29
→
X24
→
X48
)
)
)
→
X48
→
X40
→
(
¬
X29
)
→
X43
)
)
)
→
(
¬
X40
)
→
(
X29
↔
(
X21
→
X43
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
(
¬
X39
)
→
(
(
¬
X35
)
↔
X43
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
X48
↔
(
X38
↔
(
X35
↔
X43
)
)
)
)
↔
(
¬
(
X48
↔
(
X43
↔
(
X39
→
X38
→
False
)
)
)
)
)
→
(
X39
↔
(
X38
↔
False
)
)
)
↔
(
X38
→
X35
)
)
→
(
¬
(
X39
→
(
(
¬
X38
)
↔
(
X5
→
X43
)
)
)
)
)
)
)
↔
(
(
X43
↔
X38
)
↔
(
(
X39
→
(
X38
↔
X43
)
)
↔
(
¬
(
(
¬
X36
)
↔
(
(
¬
X10
)
↔
X39
)
)
)
)
)
)
↔
(
¬
(
(
¬
X38
)
↔
(
(
¬
X35
)
→
X39
)
)
)
)
↔
(
¬
(
(
¬
X35
)
→
X39
)
)
)
↔
(
¬
(
(
(
¬
X28
)
→
X10
→
(
X1
↔
X39
)
)
→
(
(
(
(
(
X12
→
(
X10
↔
X28
)
)
↔
(
X10
→
(
X2
↔
X39
)
)
)
↔
(
X28
→
False
)
)
↔
(
X39
→
X2
→
X12
)
)
↔
(
¬
(
X12
→
X10
→
X17
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
(
(
¬
X45
)
→
(
X39
↔
(
X18
→
X48
)
)
)
↔
(
(
X42
→
(
(
¬
X41
)
↔
(
(
¬
X18
)
→
X48
)
)
)
↔
(
(
¬
X45
)
↔
(
(
¬
X41
)
↔
X49
)
)
)
)
→
(
¬
(
X42
→
(
¬
X41
)
→
(
¬
X11
)
→
X49
)
)
)
)
↔
(
(
¬
(
(
(
(
(
¬
(
X28
↔
(
X10
→
(
X0
↔
False
)
)
)
)
→
(
X40
↔
(
X30
→
X0
→
X28
)
)
)
↔
(
¬
(
X40
↔
X28
)
)
)
↔
(
¬
(
(
¬
(
X22
↔
(
(
¬
X10
)
→
X28
)
)
)
→
(
¬
(
(
X40
→
(
¬
X28
)
→
(
(
¬
X15
)
↔
X49
)
)
→
(
¬
X10
)
→
X40
)
)
)
)
)
→
(
¬
(
(
(
X40
→
X30
)
→
(
(
(
¬
(
(
X49
↔
(
X40
→
(
X30
↔
(
X19
→
False
)
)
)
)
→
X49
→
X35
→
(
X19
↔
X40
)
)
)
→
X40
)
↔
(
¬
(
X49
→
(
X35
↔
X40
)
)
)
)
)
→
(
(
X49
↔
(
X40
→
False
)
)
↔
(
X49
↔
(
X35
↔
(
(
¬
X4
)
↔
X27
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
X45
→
X29
→
(
¬
X26
)
→
X30
)
→
(
(
(
¬
(
X29
→
X26
→
X45
)
)
↔
(
¬
(
X40
↔
(
X35
↔
(
X26
↔
X45
)
)
)
)
)
↔
(
X40
↔
(
(
¬
X29
)
↔
X35
)
)
)
)
→
(
¬
(
X42
↔
(
(
¬
X26
)
↔
(
X13
→
X40
)
)
)
)
)
)
→
(
¬
(
(
¬
X35
)
↔
(
(
¬
X30
)
↔
X40
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
X33
→
(
¬
X9
)
→
X37
)
→
(
¬
(
X37
→
(
¬
X9
)
→
(
(
¬
X6
)
↔
X49
)
)
)
)
)
↔
(
¬
(
X49
→
(
(
¬
X9
)
↔
X33
)
)
)
)
↔
(
(
¬
(
X37
→
(
¬
X33
)
→
X43
)
)
→
(
(
X43
↔
(
X37
→
X33
→
False
)
)
↔
(
(
¬
X43
)
→
X37
→
(
¬
X31
)
→
X49
)
)
)
)
↔
(
¬
(
(
¬
(
X43
↔
(
X40
↔
(
X20
→
X41
)
)
)
)
→
(
¬
(
(
(
¬
(
X43
↔
(
X20
→
(
X1
↔
X40
)
)
)
)
→
(
X8
↔
(
(
¬
X1
)
↔
X40
)
)
)
→
(
(
(
¬
X43
)
→
(
¬
X38
)
→
X49
)
↔
(
(
¬
(
X38
↔
(
(
¬
X33
)
→
(
¬
X8
)
→
X43
)
)
)
→
(
¬
(
X33
→
(
(
¬
X20
)
↔
X38
)
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
X34
↔
(
X3
→
X42
)
)
→
(
(
¬
(
X46
↔
X42
)
)
↔
(
(
X42
↔
(
(
¬
X36
)
↔
X47
)
)
↔
(
(
¬
(
X47
→
(
¬
X42
)
→
(
¬
X34
)
→
X46
)
)
↔
(
¬
(
(
¬
X42
)
→
(
X36
↔
(
X34
→
X47
)
)
)
)
)
)
)
)
)
→
(
(
(
X44
↔
(
X36
↔
(
(
¬
X34
)
↔
X47
)
)
)
↔
(
(
¬
X36
)
↔
(
(
¬
X34
)
↔
(
(
¬
X32
)
→
X44
)
)
)
)
↔
(
X36
↔
(
X34
→
(
X32
↔
False
)
)
)
)
)
)
→
(
(
(
¬
(
X44
↔
(
X16
→
(
(
¬
X2
)
↔
X25
)
)
)
)
→
(
(
¬
(
X36
↔
(
(
¬
X25
)
↔
(
X24
↔
X48
)
)
)
)
↔
(
(
(
¬
X36
)
→
(
X24
↔
X48
)
)
↔
(
¬
(
X49
↔
(
(
¬
X44
)
→
(
¬
X36
)
→
X48
)
)
)
)
)
)
↔
(
(
¬
(
(
X25
↔
(
X17
→
X21
)
)
→
(
(
X23
↔
(
X21
↔
False
)
)
↔
(
(
(
X36
↔
(
X34
↔
(
X23
↔
False
)
)
)
↔
(
(
(
(
X25
↔
(
(
¬
X23
)
↔
(
(
¬
X21
)
↔
X36
)
)
)
↔
(
¬
(
X34
↔
(
X25
→
X36
)
)
)
)
→
(
X21
↔
X36
)
)
↔
(
¬
(
X34
→
(
X23
↔
False
)
)
)
)
)
↔
(
(
¬
(
X36
→
(
X34
↔
(
X23
→
(
X17
↔
False
)
)
)
)
)
↔
(
¬
(
X36
↔
(
X23
↔
(
(
¬
X14
)
↔
X17
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
X36
→
X34
→
(
X25
↔
(
X12
↔
False
)
)
)
)
↔
(
(
(
X34
↔
(
(
¬
X12
)
↔
(
(
¬
X5
)
→
X48
)
)
)
↔
(
¬
(
X48
→
(
X36
↔
(
(
¬
X7
)
→
X12
)
)
)
)
)
→
X7
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
c86875...
and proposition id is
52d9da...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMNNVeUK2GWRknJ6Ajwu7yvgWeFaM38nJTD
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
(
¬
(
X47
→
(
X1
↔
X28
)
)
)
↔
(
X49
↔
(
X5
↔
X28
)
)
)
↔
(
¬
(
(
X49
↔
(
(
¬
X16
)
→
X28
)
)
→
(
(
(
(
(
(
¬
X44
)
↔
X46
)
↔
(
X44
→
(
(
¬
X32
)
↔
(
(
¬
X28
)
→
X47
)
)
)
)
↔
(
X49
→
(
X28
↔
X32
)
)
)
↔
(
X46
↔
(
X28
↔
X44
)
)
)
↔
(
¬
(
X46
→
(
X32
↔
False
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
X47
→
(
(
¬
X4
)
↔
X38
)
)
)
↔
(
¬
(
(
(
X40
→
(
¬
X12
)
→
(
X4
↔
X38
)
)
↔
(
¬
(
X40
↔
(
X4
→
X47
)
)
)
)
→
(
¬
(
X38
↔
(
(
¬
X4
)
→
X40
)
)
)
)
)
)
→
X44
→
(
X38
↔
(
(
¬
X4
)
→
X12
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
¬
X47
)
↔
(
X26
↔
X48
)
)
→
(
¬
(
X49
→
(
X47
↔
(
X38
→
X44
)
)
)
)
)
)
→
X49
→
X47
→
(
X44
↔
(
X37
→
False
)
)
)
)
→
X47
→
(
X37
↔
X48
)
)
)
↔
(
(
¬
(
(
X38
→
(
X31
↔
X25
)
)
→
(
¬
(
(
X40
↔
(
X20
→
(
(
¬
X13
)
↔
X38
)
)
)
→
(
¬
(
(
¬
(
X40
↔
(
(
¬
X25
)
↔
X38
)
)
)
→
(
¬
(
(
¬
X31
)
↔
(
X25
→
(
(
¬
X20
)
↔
X38
)
)
)
)
)
)
)
)
)
)
→
(
(
X38
→
X20
→
(
X19
↔
False
)
)
↔
(
¬
(
(
X47
↔
(
X40
↔
X20
)
)
→
(
¬
(
X47
→
X40
→
(
¬
X14
)
→
X19
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
X47
→
(
X46
↔
X48
)
)
↔
(
X47
↔
(
X42
↔
(
X36
→
X46
)
)
)
)
↔
(
(
¬
(
(
X36
↔
(
X17
↔
X48
)
)
→
X42
→
(
X17
↔
X0
)
)
)
↔
(
(
¬
X42
)
↔
(
(
¬
X36
)
→
(
(
¬
X35
)
↔
X48
)
)
)
)
)
→
(
(
(
¬
(
X35
→
(
X33
↔
(
X2
↔
False
)
)
)
)
↔
(
¬
(
(
¬
X35
)
→
(
X29
↔
(
(
¬
X2
)
→
X47
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X47
→
(
(
¬
X35
)
↔
X42
)
)
)
→
(
¬
(
X29
↔
(
X11
→
X35
)
)
)
)
)
↔
(
X42
→
(
X29
↔
X11
)
)
)
)
→
(
¬
(
X22
↔
(
X21
→
(
¬
X6
)
→
X33
)
)
)
→
(
(
X47
→
X6
→
X33
)
→
(
X33
↔
(
(
¬
X21
)
→
(
(
¬
X6
)
↔
X29
)
)
)
)
→
(
¬
(
(
¬
X33
)
→
(
¬
X29
)
→
X10
→
X47
)
)
)
)
→
(
(
(
¬
(
(
(
(
(
¬
(
X48
→
(
X45
↔
X39
)
)
)
→
(
¬
(
X45
↔
(
(
¬
X39
)
→
X48
)
)
)
→
(
(
¬
(
X39
→
X34
→
X9
→
X45
)
)
↔
(
¬
(
(
¬
X45
)
→
(
¬
X39
)
→
(
(
¬
X34
)
↔
X48
)
)
)
)
)
→
(
¬
(
X39
↔
(
(
¬
X34
)
→
X48
)
)
)
)
↔
(
X34
↔
(
(
¬
X15
)
→
(
¬
X9
)
→
X39
)
)
)
→
(
¬
(
(
¬
(
(
X45
→
(
¬
X39
)
→
(
(
¬
X3
)
↔
X47
)
)
→
(
¬
(
(
¬
X45
)
↔
(
X15
↔
(
(
¬
X3
)
→
X46
)
)
)
)
)
)
→
(
¬
(
(
¬
X39
)
→
(
(
¬
X15
)
↔
X47
)
)
)
)
)
)
)
→
(
¬
(
X41
↔
(
X39
→
X30
→
X48
)
)
)
→
(
X48
↔
(
X41
→
(
(
¬
X9
)
↔
X39
)
)
)
)
↔
(
¬
(
(
¬
(
X48
↔
(
X46
→
(
(
¬
X27
)
↔
X47
)
)
)
)
→
(
(
¬
(
(
(
X48
→
(
X43
↔
(
X18
↔
X8
)
)
)
↔
(
¬
(
X48
→
(
X43
↔
(
X8
→
X23
)
)
)
)
)
→
(
X43
↔
(
(
¬
X23
)
→
X48
)
)
)
)
↔
(
(
X47
↔
(
X43
→
X7
→
X24
)
)
↔
(
(
¬
(
X48
↔
X43
)
)
↔
(
¬
(
X48
↔
(
X46
→
(
X18
↔
X24
)
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
X49
↔
(
X48
↔
(
(
¬
X26
)
↔
X41
)
)
)
→
(
¬
(
(
X32
↔
X41
)
→
(
(
¬
(
X46
→
(
X35
↔
(
X32
→
X41
)
)
)
)
↔
(
(
X48
↔
(
X41
↔
X49
)
)
→
(
¬
(
(
¬
(
(
¬
X48
)
↔
(
(
¬
X41
)
↔
X49
)
)
)
→
X41
→
(
¬
X35
)
→
(
¬
X32
)
→
X49
)
)
)
)
)
)
)
↔
(
(
(
(
¬
X27
)
↔
(
X16
↔
(
X2
→
X35
)
)
)
→
(
¬
(
(
(
¬
X10
)
→
(
¬
X2
)
→
X35
)
→
(
X35
→
X16
→
X10
)
→
X10
→
X4
→
(
X2
↔
X16
)
)
)
)
→
(
(
¬
(
(
¬
(
X48
→
X41
→
(
(
¬
X9
)
↔
X35
)
)
)
→
(
X27
↔
(
X10
→
X35
)
)
)
)
↔
(
X35
↔
(
X27
↔
(
X20
→
False
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X34
→
X5
→
(
(
¬
X3
)
↔
X48
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X35
)
↔
(
(
¬
X15
)
↔
(
(
¬
X5
)
↔
X48
)
)
)
)
→
(
X15
↔
(
X3
→
X5
)
)
)
)
)
→
(
(
¬
(
X35
↔
(
X17
→
(
X8
↔
X48
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X15
↔
(
X6
→
X48
)
)
)
→
(
¬
(
X35
↔
(
X17
→
X15
)
)
)
)
)
→
(
¬
(
X48
↔
(
X43
↔
X35
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
X28
)
→
X25
→
(
(
¬
X17
)
↔
X43
)
)
→
(
(
¬
(
X43
→
X34
→
X25
→
X41
)
)
↔
(
X45
↔
(
X41
↔
(
X34
↔
(
X25
↔
False
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
(
¬
(
X48
→
(
X44
↔
(
X39
↔
X42
)
)
)
)
↔
(
¬
(
X48
→
X46
→
X44
→
(
X42
↔
False
)
)
)
)
↔
(
(
¬
(
X44
↔
(
(
¬
X24
)
→
X38
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
X44
)
→
X38
→
X46
)
)
→
(
¬
(
X30
↔
(
(
¬
X24
)
↔
X42
)
)
)
)
)
↔
(
X24
↔
X38
)
)
)
)
→
(
(
¬
(
X44
→
X42
→
X7
→
X49
)
)
↔
(
¬
(
X49
→
(
X44
↔
(
X42
→
X46
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X23
)
↔
(
X19
↔
X48
)
)
)
→
(
¬
(
X39
↔
(
X19
→
X13
)
)
)
→
(
(
¬
(
(
¬
X21
)
↔
(
(
¬
X19
)
→
X13
→
X39
)
)
)
↔
(
¬
(
X39
→
(
¬
X19
)
→
X42
)
)
)
)
↔
(
¬
(
X48
↔
(
X42
→
(
¬
X21
)
→
X37
)
)
)
)
→
(
(
¬
(
X39
→
(
X37
↔
(
X33
→
(
X0
↔
False
)
)
)
)
)
↔
(
¬
(
(
¬
(
X33
→
X0
)
)
→
(
(
¬
X37
)
↔
(
(
¬
X31
)
→
X48
)
)
)
)
)
)
)
→
(
(
(
¬
(
X47
↔
(
X40
→
(
¬
X36
)
→
X46
)
)
)
→
(
(
(
(
X48
↔
(
X46
→
X40
→
(
X12
↔
False
)
)
)
→
(
¬
(
X46
→
X12
→
(
X1
↔
X48
)
)
)
)
↔
(
¬
(
X46
↔
X36
)
)
)
↔
(
(
¬
X46
)
↔
(
(
¬
X40
)
→
(
X29
↔
X48
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X40
↔
X48
)
)
→
X44
→
(
X40
↔
(
X22
→
X18
→
False
)
)
)
)
↔
(
¬
(
X48
→
(
X40
↔
(
X22
↔
(
X14
→
False
)
)
)
)
)
)
→
(
¬
(
X40
→
X36
→
X18
→
X11
)
)
→
(
¬
(
X40
→
X22
→
X11
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
0aa5de...
and proposition id is
7e90b8...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMU2BeXDrwF3ZbRiQnuVjjxoep84RHZbLYj
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
∀X50 :
prop
,
∀X51 :
prop
,
∀X52 :
prop
,
(
(
(
(
(
(
(
(
¬
X50
)
→
(
(
¬
X49
)
↔
(
X8
→
X51
)
)
)
→
(
¬
(
(
(
¬
X50
)
→
(
X49
↔
(
X8
↔
X52
)
)
)
→
(
¬
(
X52
→
(
¬
X41
)
→
X8
→
X49
)
)
)
)
)
↔
(
¬
(
(
X51
↔
(
X50
↔
(
X49
↔
(
X37
↔
False
)
)
)
)
→
(
¬
(
(
(
X51
→
(
¬
X6
)
→
(
(
¬
X4
)
↔
X50
)
)
→
X4
→
X49
)
→
X49
→
X6
→
False
)
)
)
)
)
↔
(
(
X49
→
X11
→
X8
→
X3
→
False
)
↔
(
(
¬
(
(
(
¬
X44
)
↔
(
X11
↔
(
(
¬
X8
)
↔
X49
)
)
)
→
X49
→
(
(
¬
X8
)
↔
X37
)
)
)
↔
(
X44
→
X49
)
)
)
)
→
(
(
(
X45
↔
(
X21
→
(
(
¬
X9
)
↔
X51
)
)
)
→
(
(
¬
(
(
(
(
¬
X49
)
→
(
¬
X25
)
→
(
X23
↔
X51
)
)
↔
(
X49
↔
(
X41
→
(
(
¬
X25
)
↔
X51
)
)
)
)
→
(
¬
(
X41
↔
(
X25
↔
(
X23
→
False
)
)
)
)
)
)
↔
(
(
(
¬
X45
)
↔
(
(
¬
X25
)
→
(
¬
X24
)
→
X49
)
)
↔
(
(
¬
(
(
¬
X25
)
↔
(
(
¬
X9
)
↔
X51
)
)
)
→
(
¬
(
(
¬
X25
)
↔
(
X9
→
X49
)
)
)
→
(
¬
(
(
(
¬
X45
)
↔
(
X25
↔
X51
)
)
→
(
¬
(
X51
↔
(
(
¬
X25
)
→
(
X9
↔
X49
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
X42
↔
(
(
¬
X36
)
↔
(
(
¬
X5
)
→
X51
)
)
)
↔
(
(
(
¬
X49
)
↔
(
(
¬
X25
)
↔
X51
)
)
↔
(
(
X51
↔
(
X49
↔
(
X42
↔
(
X36
↔
False
)
)
)
)
→
(
¬
(
X36
↔
(
(
¬
X25
)
↔
X42
)
)
)
)
)
)
↔
(
(
X49
→
X39
→
X36
→
(
X32
↔
False
)
)
→
(
(
(
X51
↔
(
X49
↔
(
X10
→
X36
)
)
)
↔
(
X39
→
X36
→
(
(
¬
X10
)
↔
X49
)
)
)
↔
(
¬
(
X51
↔
X39
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
(
(
(
X41
↔
(
X33
→
(
X15
↔
False
)
)
)
↔
(
(
¬
(
X41
↔
(
X22
→
X15
→
False
)
)
)
↔
(
¬
(
(
¬
X28
)
↔
(
X15
→
(
X14
↔
X41
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X46
↔
(
X43
↔
(
X33
→
X22
)
)
)
)
↔
(
X33
→
(
(
¬
X28
)
↔
(
(
¬
X22
)
→
X43
)
)
)
)
→
X33
→
(
X27
↔
X43
)
)
)
)
)
→
(
(
X46
→
X0
→
X41
)
↔
(
X46
↔
(
X43
→
(
X0
↔
X18
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X41
↔
(
X35
↔
(
(
¬
X13
)
↔
X20
)
)
)
)
↔
(
¬
(
(
¬
X38
)
→
(
¬
X20
)
→
(
X1
↔
X41
)
)
)
)
→
(
¬
(
(
(
(
¬
X38
)
→
X35
→
(
(
¬
X20
)
↔
X47
)
)
↔
(
(
¬
(
X47
→
X46
→
(
¬
X20
)
→
X48
)
)
↔
(
X48
→
(
(
¬
X20
)
↔
X38
)
)
)
)
→
(
¬
(
X38
↔
(
X20
→
X35
)
)
)
)
)
)
→
(
(
(
X47
↔
(
X43
↔
(
X17
→
False
)
)
)
↔
(
X47
↔
(
X29
↔
(
(
¬
X17
)
↔
X30
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
X47
)
↔
(
(
¬
X46
)
→
(
X17
↔
X48
)
)
)
↔
(
X47
↔
(
(
¬
X46
)
→
X29
→
X48
)
)
)
→
(
¬
(
(
¬
X46
)
→
(
X29
↔
X48
)
)
)
)
)
↔
(
X47
↔
(
X46
→
False
)
)
)
)
)
)
)
)
↔
(
(
(
X26
→
X12
→
(
¬
X2
)
→
X7
)
↔
(
¬
(
(
¬
(
X31
↔
(
X26
↔
(
(
¬
X12
)
↔
X49
)
)
)
)
→
(
¬
(
X31
↔
X7
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X49
↔
(
X48
↔
(
X47
→
(
X12
↔
False
)
)
)
)
)
↔
(
(
X49
↔
(
(
¬
X12
)
↔
(
X7
→
X48
)
)
)
↔
(
X48
↔
(
X12
→
(
¬
X7
)
→
X19
)
)
)
)
↔
(
¬
(
(
¬
(
X48
↔
(
X40
→
(
(
¬
X16
)
↔
X49
)
)
)
)
→
(
¬
(
(
¬
X16
)
→
X7
→
X49
)
)
)
)
)
→
(
(
¬
(
X49
↔
(
(
¬
X34
)
↔
(
(
¬
X12
)
→
X45
)
)
)
)
↔
(
¬
(
(
¬
(
X49
→
(
X46
↔
X34
)
)
)
→
(
(
(
¬
X45
)
→
X34
→
X46
)
↔
(
X16
→
X49
)
)
→
X49
→
(
¬
X16
)
→
X34
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
(
¬
(
(
¬
X50
)
→
(
(
¬
X49
)
↔
(
X47
↔
X52
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
X50
↔
(
X49
→
(
X14
↔
X46
)
)
)
↔
(
¬
(
X52
↔
(
X49
↔
(
X46
↔
X50
)
)
)
)
)
→
X50
→
X49
→
X52
)
)
→
X50
→
(
X15
↔
(
X14
→
X52
)
)
)
)
)
↔
(
¬
(
(
(
X50
↔
(
X49
→
X4
→
X17
)
)
↔
(
¬
X47
)
)
→
(
(
X50
↔
(
(
¬
X16
)
→
(
¬
X13
)
→
X49
)
)
↔
(
X47
↔
(
X26
↔
(
X16
↔
X13
)
)
)
)
)
)
)
↔
(
(
¬
(
X50
→
(
¬
X46
)
→
(
¬
X36
)
→
X52
)
)
↔
(
X52
↔
(
X34
↔
(
(
¬
X33
)
→
X50
)
)
)
)
)
→
(
(
¬
(
(
(
¬
X34
)
→
X46
)
→
(
¬
(
X48
→
(
X34
↔
(
X17
→
X51
)
)
)
)
→
(
¬
X48
)
→
(
X46
↔
X51
)
)
)
↔
(
X51
→
(
(
¬
X34
)
↔
X46
)
)
)
→
X48
→
(
X34
↔
X46
)
)
↔
(
(
(
(
(
¬
X43
)
→
(
¬
X27
)
→
(
(
¬
X3
)
↔
X51
)
)
↔
(
X51
→
X46
→
X3
→
False
)
)
↔
(
(
(
X46
↔
X43
)
↔
(
(
¬
(
X31
→
X46
)
)
→
(
¬
(
X51
↔
(
(
¬
X31
)
↔
(
(
¬
X19
)
→
X43
)
)
)
)
)
)
→
(
¬
(
(
¬
X43
)
→
(
X21
↔
(
X19
↔
X51
)
)
)
)
)
)
↔
(
(
(
(
(
X46
→
X38
→
X6
→
X43
)
↔
(
X46
↔
(
(
¬
X38
)
→
(
(
¬
X11
)
↔
X48
)
)
)
)
↔
(
X51
→
X11
→
(
(
¬
X6
)
↔
X48
)
)
)
→
(
¬
(
(
(
¬
X48
)
↔
(
(
¬
X46
)
→
X51
)
)
→
(
(
¬
X37
)
↔
(
(
¬
X18
)
→
X51
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X51
↔
(
X30
→
(
¬
X24
)
→
X35
)
)
)
↔
(
(
¬
X35
)
↔
(
(
¬
X30
)
→
(
¬
X0
)
→
X51
)
)
)
→
(
(
¬
(
(
¬
X35
)
↔
(
X30
→
(
(
¬
X24
)
↔
X43
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
X43
)
→
(
(
¬
X30
)
↔
X48
)
)
↔
(
(
¬
(
X43
↔
(
X30
→
X2
→
X24
)
)
)
→
X30
→
(
X24
↔
(
(
¬
X2
)
↔
X48
)
)
)
)
→
(
¬
(
X30
↔
False
)
)
)
→
X30
→
(
(
¬
X2
)
↔
X43
)
)
)
)
)
)
)
)
)
→
(
(
(
(
(
(
(
(
¬
X23
)
↔
(
(
¬
X20
)
→
X8
→
X45
)
)
↔
(
(
¬
X29
)
↔
(
(
¬
X8
)
↔
X45
)
)
)
↔
(
(
X50
→
X23
→
(
X5
↔
False
)
)
→
(
¬
X45
)
→
(
(
¬
X23
)
↔
(
X20
↔
X50
)
)
)
)
→
X45
→
(
X23
↔
(
X8
→
False
)
)
)
↔
(
(
(
(
¬
X44
)
↔
(
X29
→
X50
)
)
→
(
(
¬
X44
)
→
(
X28
↔
(
(
¬
X20
)
→
X50
)
)
)
→
X44
→
(
X29
↔
(
X28
↔
X50
)
)
)
→
(
¬
(
X50
→
X29
→
X40
)
)
)
)
↔
(
(
X50
→
X25
→
(
X22
↔
False
)
)
→
(
¬
(
(
(
X44
↔
(
(
¬
X42
)
↔
(
(
¬
X25
)
↔
X45
)
)
)
↔
(
¬
(
(
¬
X22
)
→
X45
)
)
)
→
(
X50
↔
(
X44
→
(
X25
↔
False
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
X44
↔
(
X41
→
(
X9
↔
X45
)
)
)
↔
(
¬
(
X45
↔
(
X9
→
(
(
¬
X1
)
↔
X41
)
)
)
)
)
↔
(
¬
(
X45
→
(
X9
↔
X41
)
)
)
)
→
(
X45
↔
(
(
¬
X12
)
↔
X44
)
)
→
X44
→
X41
→
(
(
¬
X32
)
↔
X45
)
)
)
→
(
(
(
X45
↔
(
X44
→
X10
→
X7
)
)
↔
(
X51
↔
(
X44
→
X10
→
X50
)
)
)
↔
(
(
¬
X50
)
↔
(
X10
↔
X51
)
)
)
→
(
(
¬
(
(
¬
X49
)
→
(
X45
↔
(
(
¬
X39
)
↔
X50
)
)
)
)
↔
(
¬
(
(
X49
→
(
X45
↔
(
(
¬
X12
)
↔
X51
)
)
)
→
(
¬
(
X51
↔
(
(
¬
X45
)
→
(
¬
X39
)
→
X49
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
c7da2b...
and proposition id is
b32a07...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMbtXJRqK7N7qGKMNbhjAcTyNeDDjpLZYxT
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
(
(
¬
(
X42
→
(
¬
X8
)
→
(
(
¬
X1
)
↔
X49
)
)
)
↔
(
X8
↔
X48
)
)
↔
(
X42
→
X40
→
X48
)
)
↔
(
(
(
¬
(
X46
→
(
X42
↔
(
(
¬
X20
)
→
X40
)
)
)
)
↔
(
¬
(
X48
↔
(
X46
↔
(
X20
→
X44
)
)
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X44
)
↔
(
(
¬
X25
)
→
X48
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X35
)
↔
(
(
¬
X32
)
→
(
¬
X26
)
→
X49
)
)
)
↔
(
(
X49
→
(
X42
↔
(
X38
↔
False
)
)
)
↔
(
¬
(
(
¬
(
X42
→
(
X26
↔
(
(
¬
X9
)
↔
X49
)
)
)
)
→
(
X42
↔
(
(
¬
X38
)
→
X49
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
X24
)
↔
(
X13
↔
X32
)
)
↔
(
(
¬
(
(
¬
X38
)
→
(
X24
↔
(
(
¬
X13
)
→
X49
)
)
)
)
↔
(
¬
(
X38
↔
(
(
¬
X13
)
↔
(
(
¬
X10
)
↔
X49
)
)
)
)
)
)
↔
(
(
¬
X38
)
↔
(
X13
→
(
X10
↔
X44
)
)
)
)
→
(
¬
(
X49
→
X44
→
X38
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
(
X39
→
(
¬
X0
)
→
X45
)
)
→
(
(
¬
(
X48
↔
(
X39
→
(
¬
X0
)
→
X27
)
)
)
↔
(
X48
↔
(
X45
↔
(
X39
↔
X27
)
)
)
)
)
)
↔
(
(
¬
(
X45
→
(
¬
X37
)
→
(
X28
↔
X48
)
)
)
↔
(
¬
(
(
(
¬
X39
)
↔
(
X28
→
(
¬
X18
)
→
X48
)
)
→
(
¬
(
X45
↔
(
(
¬
X28
)
↔
(
(
¬
X18
)
↔
X48
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
(
(
¬
X12
)
→
(
¬
X5
)
→
X28
)
)
→
X47
→
(
X37
↔
(
X12
→
False
)
)
)
)
↔
(
¬
(
(
¬
(
X47
↔
(
X45
↔
(
(
¬
X3
)
↔
X7
)
)
)
)
→
(
(
(
(
¬
(
X37
→
(
¬
X28
)
→
X47
)
)
↔
(
(
¬
(
X37
↔
(
(
¬
X28
)
↔
X47
)
)
)
→
(
X17
↔
(
(
¬
X3
)
→
X47
)
)
)
)
↔
(
¬
(
X28
↔
(
X17
→
X47
)
)
)
)
↔
(
X37
→
X28
→
X7
→
X47
)
)
)
)
)
→
(
¬
(
X45
↔
(
X28
↔
(
(
¬
X11
)
→
X37
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X41
)
→
(
(
¬
X30
)
↔
(
(
¬
X22
)
→
X44
)
)
)
)
↔
(
¬
(
X44
↔
(
(
¬
X30
)
↔
(
X14
↔
X41
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X43
)
→
X23
→
(
¬
X2
)
→
X47
)
)
→
(
¬
(
X41
→
(
¬
X30
)
→
(
¬
X23
)
→
X48
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
¬
X36
)
↔
(
X22
↔
X43
)
)
)
→
(
¬
(
X41
↔
(
X16
↔
False
)
)
)
)
→
(
(
X36
↔
X48
)
↔
(
¬
(
X41
↔
(
X22
↔
(
(
¬
X21
)
↔
X48
)
)
)
)
)
)
)
↔
(
¬
(
X43
↔
(
(
¬
X36
)
↔
X41
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
(
¬
X43
)
↔
(
X33
→
(
X6
↔
X45
)
)
)
)
→
(
¬
(
X19
→
X6
)
)
)
→
(
¬
(
X31
↔
X19
)
)
)
)
↔
(
(
X42
→
(
X34
↔
(
X31
→
X48
)
)
)
↔
(
(
(
(
¬
X34
)
↔
(
X33
→
(
(
¬
X4
)
↔
X42
)
)
)
↔
(
X48
↔
(
X34
↔
X42
)
)
)
↔
(
X43
→
X34
→
False
)
)
)
)
↔
(
(
(
X47
→
(
(
¬
X45
)
↔
(
(
¬
X29
)
→
X48
)
)
)
→
X48
→
X45
→
X43
→
X47
)
→
(
¬
X29
)
→
(
(
¬
X15
)
↔
X48
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
(
X49
→
X43
→
(
¬
X40
)
→
X47
)
→
(
¬
(
(
¬
X47
)
→
X45
→
(
(
¬
X40
)
↔
X49
)
)
)
)
)
↔
(
(
(
¬
(
X40
→
(
(
¬
X39
)
↔
(
(
¬
X22
)
→
X47
)
)
)
)
→
(
¬
(
X45
↔
(
X40
→
(
X22
↔
(
X12
→
False
)
)
)
)
)
→
(
¬
(
X45
↔
(
(
¬
X39
)
→
(
¬
X22
)
→
X40
)
)
)
)
↔
(
(
¬
(
(
¬
X40
)
↔
(
X7
↔
X45
)
)
)
↔
(
¬
(
X22
↔
(
X9
→
X40
)
)
)
)
)
)
↔
(
(
¬
(
X39
→
(
(
¬
X6
)
↔
X47
)
)
)
↔
(
X49
↔
(
X47
↔
X21
)
)
)
)
→
(
¬
(
(
(
(
X43
→
(
¬
X23
)
→
X36
)
↔
(
(
¬
(
X36
→
(
(
¬
X23
)
↔
(
X17
↔
X33
)
)
)
)
↔
(
(
(
¬
X43
)
→
(
¬
X23
)
→
X48
)
→
(
X48
↔
(
X36
→
(
X33
↔
X23
)
)
)
)
)
)
↔
(
(
¬
(
X43
→
(
X36
↔
(
X18
→
X10
)
)
)
)
↔
(
(
X43
→
(
¬
X36
)
→
X48
)
→
X43
→
(
¬
X18
)
→
X0
→
X36
)
)
)
→
(
(
(
(
¬
X38
)
→
X43
)
↔
(
(
¬
(
(
(
(
¬
(
X39
↔
X25
)
)
↔
(
X36
→
X31
→
X25
→
X38
)
)
↔
(
¬
(
X31
→
X25
→
False
)
)
)
→
X31
)
)
↔
(
¬
(
(
¬
X38
)
↔
(
X36
↔
X39
)
)
)
)
)
↔
(
(
¬
(
X43
↔
(
X34
→
(
X15
↔
X45
)
)
)
)
↔
(
¬
(
(
X48
→
X45
)
→
(
X34
↔
(
X25
↔
(
X15
→
X45
)
)
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
(
X48
→
(
¬
X11
)
→
X5
→
X27
)
→
(
¬
(
(
X48
→
X45
→
(
X5
↔
X49
)
)
→
(
¬
(
X49
→
(
X48
↔
(
X5
↔
X3
)
)
)
)
)
)
)
→
(
(
X48
→
(
X26
↔
(
X4
↔
False
)
)
)
↔
(
(
X44
→
(
X37
↔
(
(
¬
X26
)
→
X48
)
)
)
→
(
¬
(
X44
→
X37
→
(
(
¬
X4
)
↔
X26
)
)
)
)
)
→
(
(
¬
(
X45
↔
(
X26
↔
(
X1
↔
False
)
)
)
)
↔
(
X49
↔
(
X45
→
X24
→
False
)
)
)
)
)
→
(
¬
(
(
X48
↔
(
X46
↔
(
X45
↔
(
X13
→
False
)
)
)
)
→
(
(
(
X19
→
X13
→
X48
)
↔
(
¬
(
X46
→
X44
→
X13
→
False
)
)
)
↔
(
¬
(
X48
↔
X13
)
)
)
)
)
)
↔
(
(
(
(
(
(
X45
→
(
X8
↔
(
X2
→
X30
)
)
)
↔
(
(
¬
X32
)
↔
(
X30
↔
(
X2
→
X45
)
)
)
)
↔
(
(
¬
X30
)
↔
(
(
¬
X16
)
→
(
(
¬
X2
)
↔
X45
)
)
)
)
↔
(
(
(
¬
(
(
¬
X44
)
→
X30
→
X45
)
)
→
(
¬
X44
)
→
X29
→
(
¬
X2
)
→
X45
)
→
(
¬
(
(
¬
X44
)
→
X30
→
X45
)
)
)
)
→
(
X41
→
X32
→
X30
→
X48
)
→
(
¬
(
X48
↔
(
X44
↔
(
X32
→
X20
→
False
)
)
)
)
)
→
(
(
¬
(
X42
→
(
X41
↔
(
X28
↔
X44
)
)
)
)
↔
(
(
¬
(
(
¬
(
X48
→
X14
→
False
)
)
→
(
¬
(
(
¬
(
(
¬
(
X44
→
(
X35
↔
X42
)
)
)
→
X48
→
(
X35
↔
(
X28
→
(
X14
↔
False
)
)
)
)
)
→
(
¬
(
X44
→
(
(
¬
X35
)
↔
X48
)
)
)
)
)
)
)
→
(
¬
(
(
¬
X35
)
↔
(
X28
→
X48
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
9f1c23...
and proposition id is
6aaa6c...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMdPyhd1ECmcr7g7gz5nJu2yxgzcetLrWog
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
¬
(
(
X49
↔
(
X41
↔
(
X8
↔
(
X1
→
False
)
)
)
)
→
(
X4
→
X1
→
X31
)
→
X41
→
X31
→
(
X4
↔
X46
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
¬
X37
)
↔
(
X34
↔
(
X28
↔
X48
)
)
)
)
→
(
¬
(
(
¬
X48
)
↔
(
X47
↔
(
(
¬
X46
)
↔
X49
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
X41
)
↔
(
X25
→
(
(
¬
X9
)
↔
X47
)
)
)
)
→
X46
→
(
X25
↔
X41
)
)
)
↔
(
¬
(
(
¬
(
X47
↔
(
X22
→
False
)
)
)
→
(
(
¬
(
(
(
X47
↔
(
(
¬
X46
)
→
(
X13
↔
X49
)
)
)
↔
(
X49
↔
(
X47
↔
(
X46
↔
(
X25
→
False
)
)
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X22
)
↔
(
(
¬
X13
)
→
X49
)
)
)
)
)
)
↔
(
¬
(
X25
↔
(
(
¬
X13
)
→
X46
)
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
(
¬
X40
)
→
(
X35
↔
(
(
¬
X29
)
↔
X47
)
)
)
↔
(
(
(
¬
(
X35
↔
(
X16
→
X47
)
)
)
↔
(
X47
↔
(
(
¬
X21
)
↔
(
(
¬
X16
)
↔
X40
)
)
)
)
→
(
¬
(
X47
→
X35
→
X21
→
False
)
)
)
)
↔
(
¬
(
X40
→
(
X35
↔
(
X12
↔
X21
)
)
)
)
)
→
(
X47
↔
(
X29
→
False
)
)
)
→
(
(
¬
(
(
X47
→
(
¬
X20
)
→
X29
)
→
(
(
X47
→
(
X35
↔
(
X20
→
X29
)
)
)
↔
(
X48
→
(
¬
X35
)
→
(
X29
↔
X47
)
)
)
)
)
↔
(
¬
(
X35
→
(
X23
↔
(
X20
↔
X48
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X46
→
X35
→
X19
→
X0
→
False
)
)
→
(
¬
(
X46
↔
(
(
¬
X0
)
↔
X11
)
)
)
)
↔
(
¬
(
X24
↔
(
(
¬
X19
)
↔
(
(
¬
X6
)
→
X48
)
)
)
)
)
↔
(
¬
(
(
(
X18
↔
(
(
¬
X5
)
↔
X40
)
)
↔
(
¬
(
(
¬
(
X46
→
X40
→
X18
→
X35
)
)
→
(
X46
↔
(
X40
→
X5
→
X18
)
)
)
)
)
→
(
¬
(
X40
→
(
X18
↔
(
X5
↔
X48
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
(
X45
↔
X48
)
→
(
¬
(
X48
↔
(
X15
→
(
¬
X10
)
→
X43
)
)
)
)
→
X44
)
)
↔
(
(
¬
X44
)
→
(
¬
X43
)
→
(
X39
↔
X48
)
)
)
→
(
(
(
(
¬
(
(
¬
X39
)
→
(
X30
↔
(
X26
→
X45
)
)
)
)
↔
(
¬
(
X45
↔
(
X30
→
(
¬
X26
)
→
X44
)
)
)
)
↔
X39
)
↔
(
¬
(
X39
→
(
X30
↔
False
)
)
)
)
)
↔
(
¬
(
(
X49
→
(
(
¬
X44
)
↔
(
X30
↔
X47
)
)
)
→
(
¬
(
(
¬
X44
)
↔
(
(
¬
X39
)
↔
(
X30
↔
X49
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X38
↔
(
X36
↔
(
X27
→
X44
)
)
)
)
↔
(
X44
↔
(
X38
→
X36
→
X32
)
)
)
→
(
(
(
¬
(
X42
↔
(
X36
→
False
)
)
)
↔
(
(
¬
(
X42
→
(
¬
X27
)
→
(
(
¬
X2
)
↔
X36
)
)
)
↔
(
X36
↔
(
(
¬
X2
)
↔
X45
)
)
)
)
↔
(
¬
(
X45
↔
(
X27
→
X17
→
X42
)
)
)
)
)
)
→
(
(
(
X7
→
False
)
↔
(
(
X47
↔
(
X14
↔
X38
)
)
→
(
¬
(
(
X47
→
(
X14
↔
(
X7
↔
X38
)
)
)
→
(
¬
(
X38
↔
(
X33
↔
X14
)
)
)
)
)
)
)
↔
(
(
X44
↔
(
(
¬
X38
)
→
X3
→
X42
)
)
↔
(
¬
(
X47
↔
(
X44
↔
(
X33
→
X42
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
X46
↔
(
X14
↔
(
(
¬
X1
)
↔
X24
)
)
)
→
(
(
(
¬
(
(
¬
X43
)
↔
(
X24
↔
(
X2
↔
X46
)
)
)
)
→
(
¬
(
(
¬
X26
)
→
X43
)
)
)
→
(
¬
(
(
(
(
¬
X43
)
→
X26
→
(
X17
↔
X46
)
)
↔
(
¬
(
X43
↔
(
(
¬
X26
)
→
(
¬
X14
)
→
X46
)
)
)
)
→
(
X17
↔
(
X14
↔
X46
)
)
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X17
)
↔
(
X0
→
X43
)
)
)
)
)
)
↔
(
(
(
¬
(
(
X46
→
(
(
¬
X11
)
↔
X20
)
)
→
(
(
(
(
¬
X44
)
↔
(
X41
↔
(
X20
↔
X46
)
)
)
→
(
¬
(
X44
→
X11
→
False
)
)
)
↔
(
(
¬
X41
)
→
(
(
¬
X20
)
↔
X44
)
)
)
)
)
↔
(
¬
(
X20
↔
(
X11
→
X46
)
)
)
)
→
(
(
(
(
¬
X26
)
↔
(
X20
↔
(
X18
↔
X46
)
)
)
→
(
X44
↔
(
X26
↔
X20
)
)
→
(
(
¬
(
X46
→
X44
→
(
X26
↔
X20
)
)
)
↔
(
X44
↔
(
X26
→
X14
→
X46
)
)
)
)
↔
(
¬
(
X20
↔
X14
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X46
↔
(
X32
→
X21
→
(
X5
↔
False
)
)
)
)
→
(
(
(
X32
→
(
¬
X21
)
→
(
X19
↔
X41
)
)
→
(
¬
(
(
¬
X41
)
→
(
X32
↔
(
(
¬
X21
)
↔
X46
)
)
)
)
)
↔
(
X46
→
(
(
¬
X21
)
↔
X32
)
)
)
)
)
→
(
¬
(
(
¬
(
X44
→
X39
→
X21
→
X45
)
)
→
(
(
¬
(
X47
↔
(
X39
→
(
X5
↔
False
)
)
)
)
↔
(
X39
↔
(
X29
→
(
¬
X5
)
→
X45
)
)
)
→
(
(
¬
(
(
¬
X44
)
↔
(
(
¬
X19
)
→
(
¬
X5
)
→
X45
)
)
)
↔
(
X45
↔
(
X29
↔
False
)
)
)
)
)
)
)
→
(
¬
(
(
X47
→
X45
→
(
¬
X10
)
→
X44
)
→
(
(
X47
↔
(
X45
↔
(
X35
↔
False
)
)
)
↔
(
(
¬
(
X47
→
X45
→
(
¬
X35
)
→
X44
)
)
→
(
X44
↔
(
X35
↔
(
(
¬
X30
)
↔
X45
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
X44
↔
(
(
¬
X25
)
↔
(
(
¬
X13
)
→
X45
)
)
)
→
(
¬
(
(
(
¬
X44
)
↔
(
X25
↔
(
X12
→
X48
)
)
)
→
(
(
¬
(
X48
→
(
X44
↔
(
(
¬
X9
)
→
X25
)
)
)
)
↔
(
X45
↔
(
(
¬
X44
)
↔
(
X9
↔
X46
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X49
↔
(
X27
↔
(
X6
→
X48
)
)
)
)
↔
(
(
X49
→
X37
→
X3
→
X48
)
→
(
(
(
X37
↔
(
X6
→
(
X3
↔
False
)
)
)
↔
(
X48
↔
(
X46
↔
X3
)
)
)
↔
(
X37
→
X6
→
(
¬
X3
)
→
X34
)
)
)
)
↔
(
(
¬
(
(
(
(
(
(
(
(
X48
→
(
(
¬
X33
)
↔
(
X7
↔
X46
)
)
)
→
X46
→
(
X38
↔
(
X33
↔
X48
)
)
)
↔
(
X38
→
(
X7
↔
False
)
)
)
↔
(
¬
(
(
¬
X46
)
↔
(
X45
↔
(
X38
→
X48
)
)
)
)
)
↔
(
¬
(
(
¬
X45
)
↔
(
(
¬
X33
)
→
X48
)
)
)
)
↔
(
¬
(
(
¬
X45
)
→
(
¬
X33
)
→
X48
)
)
)
→
(
(
(
¬
(
X40
↔
(
X38
→
(
X7
↔
False
)
)
)
)
↔
(
X48
↔
(
(
¬
X38
)
→
(
X7
↔
X49
)
)
)
)
↔
(
¬
(
X49
→
(
X48
↔
X40
)
)
)
)
)
→
(
¬
(
X49
↔
(
X45
→
X38
→
X23
)
)
)
)
)
→
(
(
¬
X40
)
↔
(
X33
↔
(
(
¬
X8
)
→
X46
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
(
X42
↔
(
X38
↔
(
X31
→
False
)
)
)
↔
(
(
X42
→
(
¬
X36
)
→
(
X31
↔
X38
)
)
↔
(
¬
(
(
¬
X36
)
→
(
(
¬
X31
)
↔
(
(
¬
X22
)
↔
X38
)
)
)
)
)
)
→
(
¬
(
X36
↔
(
X31
→
X42
)
)
)
)
→
(
¬
(
(
X34
↔
(
(
¬
X28
)
→
X31
)
)
→
(
¬
(
X36
↔
(
X31
→
(
X28
↔
X15
)
)
)
)
)
)
)
→
(
¬
(
(
¬
X36
)
→
X28
→
(
X16
↔
X42
)
)
)
)
)
→
(
¬
(
(
¬
X34
)
→
X36
)
)
→
(
¬
(
(
¬
(
X36
↔
(
(
¬
X4
)
→
X38
)
)
)
→
X34
→
X37
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
82337f...
and proposition id is
b0a19b...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMGhzhvW6AFNFnoDGmfptKaPZfhx84jMDBr
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
∀X50 :
prop
,
(
(
(
(
¬
(
(
¬
(
(
¬
(
(
¬
(
(
¬
X46
)
→
(
¬
X22
)
→
(
(
¬
X11
)
↔
X48
)
)
)
→
(
X48
↔
(
X46
→
(
X11
↔
X43
)
)
)
)
)
→
(
(
(
¬
X34
)
↔
(
X22
↔
X49
)
)
↔
(
X49
→
X46
→
X48
)
)
)
)
→
(
(
¬
(
X48
↔
(
X47
→
(
(
¬
X46
)
↔
X49
)
)
)
)
↔
(
(
¬
(
X46
↔
(
X30
↔
(
(
¬
X13
)
↔
X49
)
)
)
)
↔
(
¬
(
(
¬
X30
)
→
X48
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
¬
(
(
¬
X27
)
→
(
¬
X20
)
→
(
X6
↔
X38
)
)
)
→
(
¬
(
X4
↔
X38
)
)
)
↔
(
¬
(
(
(
(
(
X25
→
(
X20
↔
X27
)
)
↔
(
¬
(
X38
↔
(
(
¬
X3
)
↔
X25
)
)
)
)
↔
(
X38
→
X25
)
)
→
(
¬
(
X38
→
(
X25
↔
X27
)
)
)
)
→
(
¬
(
(
¬
X3
)
→
X20
)
)
)
)
)
→
(
¬
(
X38
↔
(
X25
→
(
X24
↔
X27
)
)
)
)
)
)
↔
(
¬
(
(
(
X37
→
(
(
¬
X33
)
↔
X38
)
)
↔
(
(
¬
(
(
¬
(
(
(
X37
→
X21
→
(
X2
↔
False
)
)
↔
(
¬
(
X38
↔
(
X37
↔
(
(
¬
X2
)
→
X33
)
)
)
)
)
→
X21
→
(
(
¬
X2
)
↔
X37
)
)
)
→
X37
→
X2
)
)
↔
(
¬
(
X38
→
(
X33
↔
(
X25
→
X37
)
)
)
)
)
)
→
(
(
(
(
X33
→
(
X25
↔
(
(
¬
X24
)
→
X43
)
)
)
→
X43
→
(
(
¬
X25
)
↔
(
X24
↔
X28
)
)
)
↔
(
X43
→
(
¬
X25
)
→
X28
)
)
→
(
¬
(
X43
→
(
(
¬
X33
)
↔
(
(
¬
X28
)
↔
X38
)
)
)
)
)
→
(
(
(
X43
↔
(
X7
↔
X14
)
)
↔
(
¬
(
(
(
¬
X25
)
→
(
¬
X14
)
→
(
X7
↔
X28
)
)
→
X43
→
X25
→
(
X14
↔
(
X7
→
False
)
)
)
)
)
↔
(
X14
↔
(
X7
→
X28
)
)
)
)
)
)
→
(
(
¬
(
X39
↔
X28
)
)
↔
(
¬
(
(
X46
↔
(
X39
→
(
(
¬
X28
)
↔
X48
)
)
)
→
(
¬
(
(
(
¬
(
X46
→
(
X39
↔
(
(
¬
X28
)
→
X43
)
)
)
)
→
(
¬
X39
)
→
(
X38
↔
(
X28
↔
X46
)
)
)
→
(
¬
(
X43
→
X39
→
X28
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
¬
(
X47
↔
(
(
¬
X40
)
↔
X44
)
)
)
→
(
(
¬
(
X47
→
(
¬
X18
)
→
X40
)
)
↔
(
(
¬
(
(
¬
X44
)
↔
(
(
¬
X18
)
↔
X47
)
)
)
↔
(
¬
(
X45
→
(
(
¬
X44
)
↔
(
X40
↔
X47
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
(
X49
→
(
¬
X35
)
→
(
X10
↔
X44
)
)
)
→
(
X47
↔
(
X35
→
(
(
¬
X10
)
↔
X44
)
)
)
)
)
→
X49
→
(
¬
X44
)
→
X47
)
)
↔
(
¬
(
X40
→
X44
)
)
)
)
)
↔
(
(
(
(
X19
↔
(
X8
↔
X15
)
)
→
(
¬
X44
)
→
X19
→
(
¬
X15
)
→
X47
)
↔
X19
)
→
(
(
¬
(
(
(
¬
X16
)
→
X15
→
(
X1
↔
X45
)
)
→
(
¬
(
X47
↔
(
X16
↔
(
X15
→
X45
)
)
)
)
)
)
↔
(
X45
↔
(
X16
→
(
X15
↔
X50
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X46
↔
(
(
¬
X42
)
→
(
X36
↔
X50
)
)
)
)
↔
(
¬
(
(
¬
(
X42
→
(
¬
X23
)
→
(
¬
X17
)
→
X47
)
)
→
(
X47
↔
(
X36
↔
False
)
)
)
)
)
→
(
(
¬
(
(
X50
→
X41
)
→
(
¬
(
(
¬
X32
)
→
(
¬
X12
)
→
X47
)
)
)
)
→
(
¬
X41
)
→
(
(
¬
X12
)
↔
X47
)
)
→
(
(
X23
↔
(
X9
↔
X50
)
)
↔
(
¬
(
X50
→
(
X41
↔
X47
)
)
)
)
)
→
(
(
¬
(
(
X47
↔
(
X31
→
X29
→
X46
)
)
→
(
(
X47
↔
(
X46
↔
(
X26
↔
X50
)
)
)
↔
(
¬
(
X29
→
X50
)
)
)
)
)
↔
(
(
(
¬
X46
)
→
(
X5
↔
(
X0
→
X47
)
)
)
↔
(
(
¬
(
X46
→
(
X0
↔
X50
)
)
)
→
(
X49
↔
(
(
¬
X47
)
↔
(
(
¬
X46
)
→
X50
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
(
(
(
¬
(
X49
↔
(
X37
→
(
X11
↔
False
)
)
)
)
↔
(
(
X44
↔
(
(
¬
X11
)
↔
X37
)
)
↔
(
¬
(
(
X37
↔
(
(
¬
X9
)
↔
X49
)
)
→
(
¬
(
(
¬
X44
)
→
(
X37
↔
X45
)
)
)
)
)
)
)
↔
(
¬
(
X44
→
False
)
)
)
↔
(
(
¬
(
(
¬
X45
)
↔
(
(
¬
X37
)
↔
(
X26
→
X49
)
)
)
)
→
(
X49
↔
(
(
¬
X26
)
→
(
X22
↔
X45
)
)
)
)
)
→
(
(
X24
↔
(
(
¬
X11
)
→
(
¬
X9
)
→
X26
)
)
↔
(
X40
↔
(
X26
↔
(
X11
→
X24
)
)
)
)
)
)
↔
(
(
¬
(
X47
↔
(
(
¬
X44
)
→
X40
→
X46
)
)
)
↔
(
(
¬
(
X43
→
(
¬
X40
)
→
(
X39
↔
X49
)
)
)
→
(
(
(
¬
(
(
¬
X46
)
→
(
X36
↔
X49
)
)
)
↔
(
¬
(
X43
→
(
X32
↔
X49
)
)
)
)
↔
(
(
¬
X47
)
→
X46
→
X49
)
)
)
)
)
→
(
(
(
(
(
¬
X48
)
→
X43
→
X50
)
↔
(
¬
(
(
(
X50
↔
(
X48
→
(
(
¬
X35
)
↔
X42
)
)
)
↔
(
¬
(
X50
→
(
X43
↔
(
(
¬
X35
)
→
X42
)
)
)
)
)
→
(
¬
(
X43
↔
(
X35
→
X48
)
)
)
)
)
)
↔
(
(
(
(
X48
→
X43
→
(
X4
↔
X45
)
)
↔
(
¬
(
X48
→
(
¬
X43
)
→
(
¬
X13
)
→
X45
)
)
)
→
(
¬
(
X50
↔
(
X48
↔
(
X43
↔
(
X13
↔
False
)
)
)
)
)
)
↔
(
¬
(
(
¬
X43
)
→
X4
→
X50
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
¬
(
X8
→
X2
→
X50
)
)
→
X50
→
(
X45
↔
(
X23
→
X8
→
False
)
)
)
)
→
(
X23
↔
(
X8
→
X2
→
False
)
)
)
)
→
(
X50
→
(
X45
↔
(
X10
→
X8
→
False
)
)
)
→
(
(
X48
→
(
¬
X40
)
→
X50
)
↔
(
(
¬
X45
)
→
(
X10
↔
X50
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
X49
→
(
(
¬
X28
)
↔
X41
)
)
↔
(
(
(
X49
↔
(
(
¬
X28
)
↔
X48
)
)
↔
(
(
(
¬
X48
)
↔
(
X41
↔
X49
)
)
↔
(
(
(
¬
(
X41
↔
(
(
¬
X29
)
↔
(
(
¬
X19
)
↔
X48
)
)
)
)
↔
(
¬
(
X49
→
X48
→
X29
)
)
)
↔
(
(
¬
X48
)
→
(
(
¬
X19
)
↔
X49
)
)
)
)
)
↔
(
X49
→
X28
→
X16
→
X29
)
)
)
↔
(
X48
→
(
¬
X28
)
→
X49
)
)
→
(
(
(
(
(
X34
↔
(
(
¬
X33
)
↔
(
(
¬
X25
)
↔
X38
)
)
)
↔
(
¬
(
X38
→
(
(
¬
X33
)
↔
(
X18
→
X34
)
)
)
)
)
↔
(
(
¬
X33
)
↔
(
(
¬
X25
)
→
X38
)
)
)
↔
(
(
X38
→
X33
→
X41
)
→
(
¬
(
(
X33
↔
(
X25
→
X38
)
)
→
(
¬
(
X25
→
(
¬
X12
)
→
(
X5
↔
X38
)
)
)
)
)
)
)
↔
(
(
¬
(
(
X38
↔
(
X25
↔
(
X21
→
False
)
)
)
→
(
(
¬
(
X21
↔
(
X18
↔
(
(
¬
X0
)
↔
X25
)
)
)
)
↔
(
X38
↔
(
X25
↔
(
X18
↔
X0
)
)
)
)
)
)
↔
(
(
(
¬
X38
)
→
(
(
¬
X21
)
↔
X49
)
)
→
(
¬
(
X50
↔
(
X48
→
(
(
¬
X38
)
↔
X49
)
)
)
)
→
(
¬
(
X49
→
X38
→
X21
→
X48
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
X49
↔
(
X31
→
X14
→
False
)
)
↔
(
¬
(
X41
→
X14
→
X6
)
)
)
↔
(
(
(
(
X49
→
(
X41
↔
(
X17
↔
False
)
)
)
↔
(
¬
(
(
¬
X41
)
↔
(
X30
↔
(
X7
→
X49
)
)
)
)
)
↔
(
¬
(
X41
→
X30
→
X17
→
False
)
)
)
↔
(
(
X49
→
X17
)
→
X31
→
X30
→
(
(
¬
X17
)
↔
X20
)
)
)
)
→
(
(
X41
↔
(
X38
→
(
(
¬
X20
)
↔
X27
)
)
)
↔
(
(
X41
→
X15
→
(
X1
↔
False
)
)
↔
(
(
X15
↔
False
)
→
(
(
(
¬
X30
)
↔
(
X15
→
X41
)
)
↔
(
¬
(
(
X41
↔
X30
)
→
(
¬
(
(
¬
X15
)
↔
(
(
¬
X3
)
↔
(
(
¬
X1
)
↔
X41
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
97f7f2...
and proposition id is
975aa2...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMMphTzaus7GQR1h4iBjdcy4k238xCRtEKR
)
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
¬
(
(
(
(
(
(
(
(
¬
(
(
¬
X39
)
→
(
(
¬
X4
)
↔
X46
)
)
)
→
(
¬
(
X46
→
(
X39
↔
(
X12
↔
X40
)
)
)
)
→
(
¬
(
X39
→
(
(
¬
X12
)
↔
(
X4
↔
X46
)
)
)
)
)
↔
(
X46
→
(
¬
X39
)
→
(
¬
X25
)
→
X50
)
)
→
X46
→
X39
→
(
X2
↔
X12
)
)
→
(
¬
(
X50
↔
(
X46
↔
(
X21
→
X25
)
)
)
)
)
↔
(
(
(
X50
→
(
(
¬
X27
)
↔
(
(
¬
X5
)
↔
X34
)
)
)
↔
(
¬
(
(
¬
X40
)
→
(
(
¬
X39
)
↔
(
X27
→
X50
)
)
)
)
)
↔
(
X40
→
(
X48
↔
(
X39
→
(
X5
↔
X49
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
X39
)
→
(
X24
↔
X46
)
)
→
(
¬
(
X36
↔
(
(
¬
X32
)
→
(
¬
X24
)
→
X39
)
)
)
)
)
↔
(
(
(
¬
(
X24
→
(
X16
↔
False
)
)
)
↔
(
¬
(
(
¬
(
X39
→
(
X32
↔
False
)
)
)
→
(
¬
X24
)
→
(
(
¬
X22
)
↔
(
(
¬
X16
)
↔
X39
)
)
)
)
)
↔
(
¬
(
X39
↔
False
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
¬
(
X17
↔
X50
)
)
→
(
¬
(
(
X50
→
(
X20
↔
(
(
¬
X17
)
→
X46
)
)
)
→
X46
→
(
¬
X39
)
→
X20
→
X50
)
)
)
)
→
(
¬
(
X39
→
X20
→
X50
)
)
)
→
(
¬
(
(
¬
X46
)
→
(
¬
X20
)
→
(
X11
↔
X50
)
)
)
)
)
→
(
(
(
¬
X20
)
↔
(
X13
↔
X46
)
)
→
(
(
¬
(
X46
→
X20
→
(
(
¬
X13
)
↔
X32
)
)
)
→
(
X39
↔
X46
)
)
→
(
X39
↔
(
(
¬
X13
)
→
X46
)
)
)
→
(
X37
↔
(
X32
→
(
X20
↔
X7
)
)
)
)
)
)
→
(
(
(
(
(
(
¬
(
(
¬
X47
)
↔
(
(
¬
X44
)
↔
(
(
¬
X33
)
↔
X48
)
)
)
)
→
(
¬
(
X47
→
X44
→
(
X3
↔
False
)
)
)
)
↔
(
(
¬
(
(
¬
X47
)
↔
(
(
¬
X44
)
↔
X50
)
)
)
↔
(
¬
(
(
¬
X47
)
↔
(
(
¬
X44
)
→
(
¬
X15
)
→
X48
)
)
)
)
)
→
(
¬
(
X47
→
X33
→
(
X23
↔
(
X15
→
False
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X50
→
X43
)
)
↔
(
(
(
¬
X45
)
↔
(
X43
↔
(
(
¬
X33
)
→
X47
)
)
)
↔
(
¬
(
X50
↔
(
X47
↔
(
(
¬
X43
)
↔
X45
)
)
)
)
)
)
→
(
X47
→
(
¬
X18
)
→
(
X1
↔
X45
)
)
→
X30
→
(
¬
(
(
¬
X33
)
→
(
¬
X18
)
→
(
(
¬
X1
)
↔
X45
)
)
)
)
)
)
↔
(
(
(
¬
(
X48
↔
(
X45
→
(
X19
↔
(
X0
↔
False
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X10
)
→
(
¬
X9
)
→
(
¬
X0
)
→
X14
)
)
↔
(
(
(
¬
(
X19
↔
(
(
¬
X10
)
→
(
¬
X9
)
→
X28
)
)
)
↔
(
(
¬
X19
)
↔
(
(
¬
X9
)
↔
(
X0
→
X28
)
)
)
)
↔
(
¬
(
X10
→
(
(
¬
X9
)
↔
X19
)
)
)
)
)
↔
(
¬
(
X31
↔
(
X19
↔
(
X14
→
X28
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
X38
→
(
(
¬
X35
)
↔
(
X26
→
X42
)
)
)
→
(
¬
(
X35
→
(
(
¬
X26
)
↔
X45
)
)
)
)
↔
(
X45
→
(
X38
↔
X42
)
)
)
→
(
¬
(
X29
→
(
X8
↔
X42
)
)
)
)
→
(
¬
(
(
¬
(
(
X35
↔
(
X28
↔
(
X10
↔
False
)
)
)
→
(
¬
(
X47
→
(
X28
↔
X41
)
)
)
→
(
¬
(
X41
→
(
(
¬
X35
)
↔
(
(
¬
X10
)
↔
X47
)
)
)
)
)
)
→
X47
→
(
X42
↔
(
(
¬
X35
)
↔
X41
)
)
)
)
→
(
(
(
¬
(
X49
↔
(
(
¬
X6
)
↔
X28
)
)
)
↔
(
X49
→
(
¬
X10
)
→
X6
→
X38
)
)
↔
(
X49
→
X38
→
X6
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
X24
)
→
X4
→
X25
)
→
(
(
(
¬
(
X39
↔
(
X24
↔
(
X11
→
X49
)
)
)
)
↔
(
¬
(
X39
↔
(
(
¬
X35
)
→
(
¬
X24
)
→
X49
)
)
)
)
↔
(
X39
→
X4
→
X49
)
)
)
→
(
(
¬
(
X49
↔
(
X25
↔
(
(
¬
X7
)
↔
X21
)
)
)
)
↔
(
X50
↔
(
X49
↔
X35
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
(
(
(
¬
X43
)
→
(
(
¬
X42
)
↔
(
(
¬
X30
)
→
X50
)
)
)
→
(
¬
X42
)
→
(
X30
↔
(
X6
→
X50
)
)
)
)
→
(
¬
(
X50
→
(
¬
X42
)
→
(
X41
↔
X43
)
)
)
)
)
↔
(
(
X46
→
(
X43
↔
(
(
¬
X41
)
→
X49
)
)
)
↔
(
(
¬
(
X49
↔
(
X42
↔
X46
)
)
)
↔
(
(
X46
↔
(
X42
→
X29
→
False
)
)
↔
(
X46
→
(
X43
↔
(
X29
↔
X49
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
X46
↔
(
X36
↔
X49
)
)
→
(
(
(
X46
↔
(
X43
→
(
X36
↔
X49
)
)
)
↔
(
¬
(
X49
→
(
¬
X43
)
→
(
(
¬
X16
)
↔
X46
)
)
)
)
↔
(
¬
(
X49
→
X16
→
X43
)
)
)
)
→
(
¬
(
X46
→
(
X43
↔
X49
)
)
)
)
)
↔
(
X49
↔
(
(
¬
X43
)
↔
(
(
¬
X32
)
↔
X46
)
)
)
)
)
↔
(
(
(
(
X50
→
(
X39
↔
False
)
)
↔
(
(
(
(
¬
(
X39
→
(
¬
(
(
¬
X39
)
↔
(
(
¬
X34
)
↔
X50
)
)
)
→
(
(
¬
X49
)
↔
(
X39
→
X15
→
X50
)
)
)
)
↔
(
¬
(
X50
↔
(
X49
↔
X39
)
)
)
)
↔
(
¬
(
X34
→
(
X15
↔
X49
)
)
)
)
↔
(
¬
(
X43
→
(
(
¬
X34
)
↔
(
X15
↔
X39
)
)
)
)
)
)
→
X49
→
X34
→
X15
)
↔
(
(
¬
(
X43
→
(
X39
↔
(
X34
→
X10
)
)
)
)
↔
(
¬
(
X39
→
X34
→
X10
→
X1
)
)
)
)
)
)
→
(
(
(
(
(
¬
X40
)
↔
(
(
¬
X37
)
→
(
X0
↔
X49
)
)
)
→
X42
→
(
X20
↔
X37
)
)
↔
(
(
(
(
(
(
¬
(
(
¬
X31
)
↔
(
(
¬
X23
)
↔
(
(
¬
X2
)
↔
X45
)
)
)
)
↔
(
¬
(
X49
→
(
X45
↔
X31
)
)
)
)
↔
(
(
¬
X23
)
↔
(
X2
→
X31
)
)
)
→
(
¬
(
(
¬
X33
)
↔
(
X31
↔
(
X23
→
X45
)
)
)
)
)
↔
(
X42
→
(
X33
↔
(
(
¬
X23
)
↔
X27
)
)
)
)
→
(
(
(
X13
↔
X37
)
↔
(
X49
↔
(
X33
→
X9
→
X13
)
)
)
↔
(
¬
(
(
¬
(
X42
→
X33
→
(
¬
X9
)
→
X49
)
)
→
X49
→
(
(
¬
X33
)
↔
(
X22
↔
X42
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
(
¬
X38
)
→
(
X5
↔
(
X3
↔
X42
)
)
)
)
→
(
¬
X18
)
→
(
X5
↔
(
(
¬
X3
)
→
X38
)
)
)
→
(
¬
(
(
(
¬
X38
)
↔
(
(
¬
X18
)
↔
X42
)
)
→
(
¬
(
X44
↔
(
X42
→
X18
→
(
X12
↔
False
)
)
)
)
→
X47
→
(
X42
↔
X12
)
)
)
)
)
→
(
¬
(
(
(
X28
↔
(
(
¬
X14
)
→
X48
)
)
↔
(
¬
(
(
¬
X47
)
↔
(
(
¬
X28
)
↔
(
(
¬
X19
)
→
X49
)
)
)
)
)
→
(
(
(
X48
↔
(
X19
→
X8
→
False
)
)
↔
(
(
X48
↔
(
X19
↔
(
(
¬
X17
)
↔
X42
)
)
)
→
(
¬
(
X19
↔
(
X17
↔
(
X8
↔
X48
)
)
)
)
)
)
↔
(
¬
(
(
¬
X45
)
→
(
(
¬
X42
)
↔
(
X17
→
X47
)
)
)
)
)
)
)
→
(
(
X49
↔
(
(
¬
X43
)
↔
(
(
¬
X26
)
→
X47
)
)
)
↔
(
¬
(
X42
↔
(
(
¬
X26
)
↔
X38
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
d604c0...
and proposition id is
4fbd7a...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMScyZSSEKPpR2jYcq2SM2kVBXkA2RAbYWn
)
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
∀X50 :
prop
,
(
¬
∀X51 :
prop
,
(
(
(
(
¬
(
(
(
¬
(
X48
→
(
(
¬
X41
)
→
(
¬
X15
)
→
(
(
¬
X7
)
↔
X48
)
)
→
(
¬
(
(
¬
X48
)
↔
(
X10
↔
(
X7
↔
X51
)
)
)
)
→
X51
→
X7
→
X41
)
)
↔
(
(
X21
↔
(
X7
→
X29
)
)
↔
(
¬
(
(
¬
X48
)
↔
(
X29
↔
(
X21
→
X51
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
(
¬
(
X48
↔
(
(
¬
X32
)
→
(
(
¬
X0
)
↔
X45
)
)
)
)
↔
(
(
¬
(
(
¬
X36
)
↔
(
(
¬
X0
)
↔
X45
)
)
)
↔
(
X41
↔
(
X36
→
X32
→
X45
)
)
)
)
→
(
¬
(
X45
→
X41
→
X36
)
)
)
)
→
(
¬
(
(
¬
X45
)
↔
(
X36
→
X33
→
X48
)
)
)
)
)
→
(
(
(
(
(
(
¬
X33
)
→
X41
)
↔
(
(
(
X33
↔
(
X32
↔
(
X2
↔
False
)
)
)
↔
(
(
¬
X33
)
↔
(
X3
↔
(
(
¬
X2
)
→
X41
)
)
)
)
→
(
¬
(
X41
↔
(
X33
↔
X3
)
)
)
)
)
→
(
¬
(
X41
→
X32
→
X33
)
)
)
↔
(
(
¬
X32
)
↔
X33
)
)
↔
(
¬
(
X41
→
X33
→
(
X32
↔
(
X21
↔
False
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
X41
↔
(
X17
↔
X50
)
)
)
↔
(
(
(
¬
X41
)
→
(
¬
X37
)
→
(
¬
X25
)
→
X50
)
↔
(
¬
(
(
X50
→
(
(
¬
X41
)
↔
(
X25
↔
X45
)
)
)
→
(
¬
(
(
¬
X45
)
↔
(
X17
→
X50
)
)
)
)
)
)
)
→
(
¬
(
(
X48
→
X45
→
(
X1
↔
False
)
)
→
(
X48
↔
(
(
¬
X45
)
↔
(
X41
→
X50
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
X41
↔
X50
)
→
(
¬
(
(
(
(
¬
X47
)
→
X37
→
X12
→
X50
)
→
(
¬
(
(
¬
X12
)
↔
(
(
¬
X9
)
→
X47
)
)
)
)
→
X50
→
X12
→
X9
→
False
)
)
)
)
→
(
(
¬
(
(
¬
X47
)
→
(
¬
X41
)
→
(
¬
X37
)
→
X49
)
)
↔
(
(
¬
(
(
¬
X37
)
→
(
¬
X26
)
→
(
¬
X14
)
→
X41
)
)
↔
(
¬
(
X49
→
(
X8
↔
X37
)
)
)
)
)
)
)
↔
(
¬
(
(
X47
→
X50
)
→
X49
→
X39
→
X30
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
(
¬
(
X51
→
(
¬
X42
)
→
(
(
¬
X24
)
↔
X46
)
)
)
→
(
X51
→
(
X34
↔
False
)
)
→
(
(
¬
X46
)
→
(
(
¬
X42
)
↔
(
X34
↔
X51
)
)
)
→
(
X46
↔
(
X42
→
(
X34
↔
(
X4
→
False
)
)
)
)
)
↔
(
¬
(
X42
↔
(
X18
→
X46
)
)
)
)
↔
(
(
¬
(
(
¬
X44
)
→
X5
→
X46
)
)
↔
(
(
¬
(
(
(
(
¬
X44
)
→
(
(
¬
X42
)
↔
(
X5
↔
X51
)
)
)
→
X46
→
(
X5
↔
X51
)
)
→
(
¬
(
X51
↔
(
(
¬
X42
)
→
X44
)
)
)
)
)
→
(
X46
↔
(
X42
↔
X5
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X44
)
↔
(
(
¬
X35
)
↔
X51
)
)
)
↔
(
(
¬
(
(
(
¬
X46
)
↔
(
X35
↔
X51
)
)
→
(
¬
(
X51
→
(
(
¬
X35
)
↔
(
(
¬
X23
)
→
X44
)
)
)
)
)
)
→
(
(
¬
X44
)
↔
(
X35
→
X51
)
)
)
)
↔
(
¬
(
X44
↔
(
(
¬
X23
)
↔
(
(
¬
X11
)
↔
X51
)
)
)
)
)
↔
(
X51
↔
(
(
¬
X11
)
→
X23
)
)
)
)
→
(
(
(
¬
X28
)
→
(
(
¬
X16
)
↔
X46
)
)
↔
(
(
X46
→
X44
→
(
(
¬
X28
)
↔
X51
)
)
↔
(
¬
(
(
¬
X44
)
→
X28
→
(
X16
↔
X51
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X44
)
↔
(
(
¬
X43
)
→
(
X6
↔
X51
)
)
)
)
↔
(
¬
(
X51
↔
(
X46
↔
(
X44
↔
(
X43
↔
False
)
)
)
)
)
)
↔
(
¬
(
X46
↔
(
X44
→
(
X43
↔
False
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X44
→
(
X31
↔
(
X19
→
X13
)
)
)
)
↔
(
(
(
¬
X31
)
→
(
X27
↔
(
X19
↔
X44
)
)
)
→
(
¬
(
X48
↔
(
X38
↔
(
X19
↔
X31
)
)
)
)
)
)
↔
(
(
¬
(
(
X48
↔
(
X38
↔
X20
)
)
→
X48
→
X43
→
(
¬
X40
)
→
X50
)
)
↔
(
¬
(
(
(
¬
X40
)
→
(
¬
X22
)
→
X50
)
→
(
(
X50
→
(
¬
X35
)
→
X48
)
↔
(
X35
→
(
X31
↔
(
X22
↔
X50
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
(
(
X50
↔
(
X49
→
(
(
¬
X35
)
↔
X48
)
)
)
→
(
¬
(
X17
↔
X48
)
)
)
→
(
¬
(
X48
→
(
¬
X9
)
→
X49
)
)
)
↔
(
¬
(
X49
→
(
X48
↔
(
X17
→
False
)
)
)
)
)
↔
(
(
(
(
¬
(
X46
→
X0
→
X36
)
)
↔
(
X50
→
(
X48
↔
(
X46
→
X0
→
False
)
)
)
)
↔
(
(
¬
X48
)
→
(
¬
X46
)
→
X35
→
X50
)
)
↔
(
X50
↔
(
X46
→
X36
→
X34
)
)
)
)
↔
(
(
¬
(
X46
→
X36
)
)
↔
(
X50
↔
(
(
¬
X15
)
↔
(
(
¬
X10
)
→
X45
)
)
)
)
)
→
(
(
¬
(
(
(
(
X51
↔
(
(
¬
X18
)
↔
X48
)
)
↔
(
¬
(
(
¬
X40
)
↔
(
X28
↔
(
X18
→
X48
)
)
)
)
)
→
(
¬
(
(
¬
X40
)
→
(
(
¬
X8
)
↔
X48
)
)
)
)
→
(
¬
(
(
¬
(
(
X49
→
(
X22
↔
X28
)
)
→
(
(
¬
(
X51
→
(
X49
↔
(
X18
↔
X22
)
)
)
)
→
(
¬
(
(
¬
X28
)
↔
(
X22
↔
X51
)
)
)
)
→
(
¬
(
(
¬
X49
)
→
(
(
¬
X22
)
↔
X51
)
)
)
)
)
→
(
(
X50
→
(
X22
↔
(
X4
→
X49
)
)
)
↔
(
X49
→
(
X22
↔
X51
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X50
→
(
(
¬
X38
)
↔
(
(
¬
X25
)
↔
X47
)
)
)
)
↔
(
¬
(
(
X47
↔
X50
)
→
(
(
¬
(
(
¬
X25
)
↔
(
X19
↔
X47
)
)
)
↔
(
(
¬
X25
)
↔
(
(
¬
X24
)
↔
(
(
¬
X19
)
↔
X50
)
)
)
)
)
)
)
→
(
(
(
X51
↔
(
X12
→
(
(
¬
X2
)
↔
X50
)
)
)
→
(
(
X50
→
(
¬
X12
)
→
(
¬
X2
)
→
X49
)
↔
(
(
¬
X47
)
↔
(
X12
→
(
X2
↔
X49
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X16
→
X29
)
)
→
(
¬
(
X51
→
(
¬
X49
)
→
X50
)
)
)
)
↔
(
(
¬
(
X50
→
X49
→
(
¬
X11
)
→
X51
)
)
↔
(
¬
(
X38
↔
(
X11
↔
False
)
)
)
)
)
→
X50
→
X49
→
X51
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X43
→
(
¬
X20
)
→
(
¬
X6
)
→
X21
)
)
→
(
(
X51
↔
(
(
¬
X21
)
↔
(
(
¬
X20
)
↔
X31
)
)
)
↔
(
(
(
(
¬
(
X13
↔
(
X6
→
X51
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X31
)
↔
(
(
¬
X13
)
→
X6
→
X51
)
)
)
→
(
¬
(
X20
→
(
(
¬
X13
)
↔
(
X6
→
X31
)
)
)
)
)
)
)
↔
(
¬
(
X31
→
(
¬
X6
)
→
X51
)
)
)
→
(
¬
(
X20
→
X13
→
X51
)
)
)
)
)
↔
(
(
X37
↔
(
X31
↔
(
X21
→
False
)
)
)
→
(
¬
(
X43
↔
(
X37
→
(
(
¬
X31
)
↔
X49
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
¬
X30
)
→
(
X3
↔
X39
)
)
)
→
(
¬
(
(
¬
X42
)
→
(
X30
↔
(
X3
↔
X43
)
)
)
)
→
(
¬
(
X42
↔
(
X39
↔
X30
)
)
)
)
)
→
(
(
(
¬
(
X51
→
(
¬
X39
)
→
X30
→
X49
)
)
↔
(
X51
↔
(
(
¬
X30
)
→
X43
)
)
)
↔
(
(
¬
(
(
X30
→
X26
→
X23
→
False
)
→
(
¬
(
(
¬
X30
)
→
(
(
¬
X26
)
↔
(
(
¬
X23
)
↔
X49
)
)
)
)
→
(
X30
↔
(
(
¬
X26
)
→
X5
→
X49
)
)
)
)
→
(
¬
(
(
X51
↔
(
X49
↔
(
(
¬
X1
)
→
X23
)
)
)
→
(
(
(
X51
→
(
¬
X23
)
→
(
¬
X5
)
→
X44
)
↔
(
X44
→
X49
)
)
↔
(
¬
(
(
¬
X49
)
→
(
¬
X23
)
→
X51
)
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
(
X41
↔
(
X32
↔
X27
)
)
)
→
(
(
¬
(
X44
↔
(
X32
↔
(
X27
→
X50
)
)
)
)
↔
(
X49
→
X32
→
X27
→
X44
)
)
)
)
→
(
X44
↔
(
X27
↔
(
X14
→
False
)
)
)
)
)
↔
(
(
¬
(
(
¬
X33
)
↔
(
X27
→
(
¬
X7
)
→
X50
)
)
)
↔
(
X33
→
(
X27
↔
(
(
¬
X7
)
→
X42
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
d70d75...
and proposition id is
18dff9...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMU26oiW3YZBWzwPAZHB11VMKBoopWfYz8Q
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
¬
∀X51 :
prop
,
(
(
(
(
¬
(
(
(
¬
(
(
¬
(
X51
→
(
(
¬
X19
)
↔
X47
)
)
)
→
(
(
¬
(
X47
↔
(
X19
→
X7
→
X41
)
)
)
↔
(
¬
(
X19
↔
(
X7
↔
(
X5
→
X41
)
)
)
)
)
)
)
↔
(
X26
↔
(
X19
↔
(
X13
→
X51
)
)
)
)
→
(
¬
(
(
(
¬
(
X41
↔
(
(
¬
X4
)
↔
X31
)
)
)
↔
(
¬
(
X51
→
(
¬
X26
)
→
(
¬
X25
)
→
X41
)
)
)
→
(
(
¬
X41
)
↔
(
(
¬
X31
)
→
(
¬
X17
)
→
X51
)
)
)
)
)
)
→
(
(
¬
(
(
¬
X24
)
↔
(
(
¬
X3
)
→
X39
)
)
)
↔
(
¬
(
(
¬
(
X48
↔
(
(
¬
X24
)
↔
X47
)
)
)
→
(
(
¬
(
X47
→
X41
→
X39
→
False
)
)
↔
(
(
¬
(
(
¬
(
X48
↔
(
X47
→
X41
→
X39
→
False
)
)
)
→
(
¬
(
X47
→
(
¬
X9
)
→
X41
)
)
)
)
↔
(
¬
(
(
(
¬
(
X48
↔
(
X41
→
(
(
¬
X3
)
↔
X47
)
)
)
)
↔
(
¬
(
X47
↔
(
X24
→
X3
→
X41
)
)
)
)
→
X41
→
(
X24
↔
X48
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X42
↔
(
X24
→
X22
)
)
)
↔
(
(
¬
X42
)
→
(
(
¬
X41
)
↔
(
(
¬
X24
)
↔
X46
)
)
)
)
→
(
¬
(
(
¬
(
X46
↔
(
X41
→
X6
→
False
)
)
)
→
(
X51
↔
(
(
¬
X22
)
→
X6
→
X46
)
)
)
)
→
(
¬
X49
)
→
(
X46
↔
(
X41
↔
X51
)
)
)
↔
(
¬
(
(
¬
(
X39
→
X21
→
X12
→
X51
)
)
→
(
¬
(
X51
↔
(
X41
↔
(
(
¬
X39
)
→
X49
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
(
(
¬
(
(
¬
X43
)
↔
(
(
¬
X2
)
→
X44
)
)
)
↔
(
(
¬
X44
)
↔
(
X43
→
(
(
¬
X29
)
↔
X48
)
)
)
)
↔
(
¬
(
X44
→
(
¬
X29
)
→
(
(
¬
X14
)
↔
X48
)
)
)
)
↔
(
X48
→
(
X43
↔
(
X14
↔
False
)
)
)
)
→
(
¬
X35
)
→
(
¬
X32
)
→
(
¬
X29
)
→
X43
)
↔
(
(
(
¬
X43
)
↔
(
(
¬
X37
)
→
(
X35
↔
X49
)
)
)
↔
(
(
¬
(
X49
→
(
X45
↔
X36
)
)
)
↔
(
X49
→
(
X45
↔
(
X43
→
X44
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X37
→
X30
→
(
(
¬
X16
)
↔
X33
)
)
)
→
(
¬
(
(
¬
(
X45
→
(
X33
↔
(
(
¬
X16
)
→
X37
)
)
)
)
→
(
(
¬
X38
)
↔
(
X33
→
X16
→
X45
)
)
)
)
)
)
↔
(
(
(
(
(
X44
↔
(
X38
→
X30
→
False
)
)
↔
(
(
(
¬
(
X44
↔
X23
)
)
↔
(
(
X44
↔
(
(
¬
X23
)
→
X38
)
)
↔
(
X23
↔
(
(
¬
X18
)
→
X30
)
)
)
)
→
(
¬
(
X38
→
X23
)
)
)
)
↔
(
¬
(
X44
↔
(
X38
↔
(
X30
→
False
)
)
)
)
)
↔
(
¬
(
X38
↔
(
X33
→
(
X30
↔
(
X23
↔
False
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X27
)
↔
(
(
¬
X23
)
→
(
¬
X20
)
→
X30
)
)
)
→
(
¬
(
X38
↔
(
(
¬
X27
)
↔
(
X23
→
X30
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
(
¬
X45
)
↔
(
(
¬
X40
)
↔
(
(
¬
X10
)
→
X49
)
)
)
→
(
¬
(
X49
→
(
(
¬
X40
)
↔
(
X28
→
X45
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
X51
→
X11
→
X50
)
→
(
¬
(
X45
↔
(
(
¬
X8
)
→
X11
)
)
)
)
)
↔
(
(
¬
(
(
X51
↔
(
(
¬
X34
)
↔
X49
)
)
→
(
¬
(
X34
↔
(
X11
↔
X45
)
)
)
)
)
↔
(
¬
(
X51
→
(
¬
X45
)
→
X49
)
)
)
)
→
(
(
X45
→
X43
→
X34
→
False
)
↔
(
(
X50
→
(
(
¬
X34
)
↔
(
(
¬
X28
)
→
X43
)
)
)
→
X50
→
X45
→
(
X34
↔
False
)
)
)
)
)
)
)
→
(
(
(
(
¬
X45
)
↔
(
X1
→
(
¬
X0
)
→
X50
)
)
↔
(
(
¬
X45
)
→
(
X15
↔
(
X1
↔
X50
)
)
)
)
↔
(
¬
(
X51
→
X50
→
X45
→
(
X15
↔
False
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
(
X42
→
X28
→
(
¬
X11
)
→
X22
)
→
(
¬
(
X46
↔
(
(
¬
X28
)
↔
X42
)
)
)
)
)
→
(
X42
↔
(
X11
→
(
(
¬
X5
)
↔
X15
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
(
¬
(
X48
→
(
(
¬
X36
)
↔
X47
)
)
)
→
(
¬
(
X49
→
(
¬
X36
)
→
(
X28
↔
X47
)
)
)
)
)
↔
(
X48
↔
(
(
¬
X36
)
↔
X47
)
)
)
→
(
(
X42
↔
X47
)
↔
(
(
X49
↔
(
X47
↔
(
(
¬
X42
)
↔
X48
)
)
)
↔
(
X48
↔
(
(
¬
X47
)
↔
(
X34
→
X49
)
)
)
)
)
)
)
→
(
¬
(
(
¬
X47
)
→
(
¬
X42
)
→
(
¬
X33
)
→
X48
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X47
↔
(
X45
→
X27
→
X49
)
)
)
↔
(
¬
(
(
(
¬
X48
)
↔
(
(
¬
X47
)
→
X49
)
)
→
(
¬
(
(
¬
(
X47
↔
(
X45
↔
(
X9
↔
X48
)
)
)
)
→
(
¬
(
X48
→
(
¬
X9
)
→
X49
)
)
)
)
)
)
)
→
(
¬
(
(
X44
↔
(
X4
↔
(
X0
→
False
)
)
)
→
(
¬
(
(
(
¬
X0
)
↔
X12
)
→
(
(
¬
(
X45
↔
(
X12
↔
(
(
¬
X0
)
↔
X44
)
)
)
)
↔
(
¬
(
(
¬
X47
)
→
(
¬
X12
)
→
(
X4
↔
X49
)
)
)
)
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
(
¬
X26
)
→
(
(
¬
X1
)
↔
X49
)
)
)
→
(
¬
(
X49
→
(
X37
↔
(
X26
↔
(
X24
→
False
)
)
)
)
)
)
↔
(
(
¬
X37
)
→
(
(
¬
X1
)
↔
X49
)
)
)
→
(
¬
(
(
¬
(
(
¬
X49
)
→
(
¬
X37
)
→
(
¬
X26
)
→
X51
)
)
→
(
¬
(
X49
↔
(
X26
↔
X48
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
X49
)
→
(
X26
↔
(
X17
↔
X51
)
)
)
→
(
¬
(
X49
→
(
¬
X17
)
→
X25
)
)
)
)
)
→
(
(
(
¬
X44
)
↔
(
X18
↔
(
X7
↔
X49
)
)
)
↔
(
(
(
(
(
¬
X48
)
→
(
¬
X43
)
→
(
¬
X39
)
→
X51
)
→
(
(
(
¬
X48
)
→
(
(
¬
X43
)
↔
(
(
¬
X18
)
↔
X49
)
)
)
↔
(
¬
(
X51
→
X49
→
X18
→
X48
)
)
)
)
↔
(
X43
→
X39
→
(
X21
↔
X48
)
)
)
↔
(
(
¬
X18
)
→
X44
)
)
)
→
(
¬
(
(
¬
X48
)
→
X44
→
(
¬
X2
)
→
X51
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
(
(
¬
(
(
¬
X43
)
↔
(
(
¬
X38
)
↔
(
X31
→
X46
)
)
)
)
↔
(
X43
↔
(
X38
→
(
¬
X35
)
→
X46
)
)
)
↔
(
(
¬
X43
)
→
(
(
¬
X38
)
↔
X46
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X41
)
→
(
(
¬
X38
)
↔
(
X8
→
X50
)
)
)
)
↔
(
(
X41
↔
(
X31
↔
(
X13
→
X43
)
)
)
↔
(
(
¬
(
X43
↔
(
X31
→
X50
)
)
)
→
(
(
¬
X43
)
↔
(
(
¬
X41
)
↔
(
X31
→
X50
)
)
)
→
(
¬
(
(
¬
X43
)
→
(
X31
↔
(
(
¬
X8
)
↔
X50
)
)
)
)
)
)
)
→
(
¬
(
X50
↔
(
X41
↔
(
X38
→
False
)
)
)
)
)
)
)
)
→
(
X46
↔
(
X43
↔
(
X38
↔
False
)
)
)
)
)
↔
(
(
(
X43
↔
(
X6
→
X23
)
)
↔
(
¬
(
(
¬
X40
)
→
X32
→
(
X6
↔
X46
)
)
)
)
↔
(
¬
(
(
¬
(
X32
↔
(
(
¬
X16
)
→
X23
)
)
)
→
(
¬
(
X46
↔
(
X23
↔
X16
)
)
)
→
X32
→
(
¬
X23
)
→
(
¬
X16
)
→
X31
)
)
)
)
↔
(
(
(
(
X29
→
X20
→
(
¬
X19
)
→
X46
)
→
(
¬
(
X46
↔
(
X20
↔
(
(
¬
X10
)
→
X29
)
)
)
)
)
↔
(
¬
(
(
¬
X29
)
↔
(
X19
↔
(
X10
→
X35
)
)
)
)
)
↔
(
(
(
X29
→
X19
→
(
(
¬
X3
)
↔
X20
)
)
↔
(
(
¬
(
X30
→
X29
→
X14
)
)
→
(
¬
(
X30
↔
(
X19
→
(
X14
↔
X43
)
)
)
)
)
)
↔
(
(
¬
X43
)
→
(
¬
X19
)
→
(
¬
X14
)
→
X51
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
c16dce...
and proposition id is
23d226...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMayX7YQ8fEsoBuM1g4sA8xSz6boCwcGKMK
)
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
¬
(
(
(
¬
(
(
(
¬
(
X44
→
(
(
¬
X9
)
↔
(
X1
→
X46
)
)
)
)
↔
(
¬
(
(
¬
X44
)
→
(
¬
X9
)
→
(
(
¬
X8
)
↔
X46
)
)
)
)
→
(
(
¬
(
X49
↔
(
(
¬
X31
)
→
(
(
¬
X9
)
↔
X44
)
)
)
)
↔
(
X44
↔
(
(
¬
X9
)
→
X8
→
X49
)
)
)
)
)
↔
(
(
(
¬
(
X49
→
X10
)
)
→
(
(
(
¬
X44
)
→
X9
→
X49
)
↔
(
(
(
¬
X10
)
↔
(
X9
→
(
(
¬
X7
)
↔
X49
)
)
)
→
X44
→
(
X7
↔
False
)
)
)
)
→
X10
→
False
)
)
→
(
(
¬
(
X49
→
(
X39
↔
(
X4
↔
X6
)
)
)
)
↔
(
¬
(
(
¬
X46
)
↔
(
(
¬
X39
)
↔
(
X6
↔
X49
)
)
)
)
)
→
(
(
X44
↔
(
X6
→
(
¬
X0
)
→
X4
)
)
↔
(
(
(
¬
X34
)
→
(
X0
↔
X49
)
)
↔
(
¬
(
X49
↔
(
X48
↔
(
X44
↔
X34
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
X46
→
X35
)
→
(
¬
(
X25
↔
(
(
¬
X13
)
→
X30
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
¬
X35
)
→
X13
→
(
(
¬
X2
)
↔
X49
)
)
→
(
¬
(
(
¬
X3
)
↔
(
(
¬
X2
)
→
X48
)
)
)
)
↔
(
(
¬
X48
)
→
(
(
¬
X35
)
↔
X49
)
)
)
↔
(
¬
(
(
(
¬
X13
)
↔
(
X11
↔
(
(
¬
X2
)
↔
X48
)
)
)
→
(
¬
(
X48
→
X25
→
X11
)
)
)
)
)
→
(
¬
(
X37
→
(
¬
X35
)
→
(
X3
↔
X48
)
)
)
)
)
)
→
(
¬
(
(
¬
X48
)
↔
(
X45
↔
(
(
¬
X42
)
→
X49
)
)
)
)
→
(
¬
(
(
(
(
¬
X48
)
→
(
X46
↔
(
X42
→
X49
)
)
)
→
(
¬
(
(
¬
X42
)
↔
(
(
¬
X14
)
↔
X46
)
)
)
)
→
(
¬
False
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X36
↔
(
X33
↔
(
(
¬
X28
)
↔
X46
)
)
)
)
→
(
(
¬
(
X45
→
(
X32
↔
(
X20
→
False
)
)
)
)
→
(
X44
→
X32
→
X20
→
X19
→
False
)
→
(
¬
(
X45
↔
(
(
¬
X20
)
→
(
X19
↔
X32
)
)
)
)
)
→
(
¬
(
(
¬
(
X45
→
X32
)
)
→
X44
→
(
¬
X26
)
→
X28
)
)
)
)
→
(
(
(
(
(
(
X46
→
(
¬
X23
)
→
X22
→
X47
)
→
(
¬
(
X46
→
(
X12
↔
X47
)
)
)
)
↔
(
¬
(
(
¬
X29
)
↔
(
(
¬
X22
)
→
X12
→
X47
)
)
)
)
→
(
¬
X45
)
→
(
X29
↔
(
X12
↔
X46
)
)
)
↔
(
(
(
X40
↔
(
(
¬
X27
)
↔
(
(
¬
X15
)
→
X29
)
)
)
→
(
(
¬
(
X29
↔
(
(
¬
X15
)
→
X40
)
)
)
↔
(
(
¬
(
X29
→
(
X5
↔
X40
)
)
)
↔
(
X29
→
X23
→
X12
→
X5
)
)
)
)
→
(
X44
→
(
X23
↔
X17
)
)
→
(
(
¬
(
X17
→
(
X15
↔
(
X12
→
X23
)
)
)
)
↔
(
X44
↔
False
)
)
)
)
↔
(
(
(
X45
→
(
X41
↔
(
X38
→
X43
)
)
)
↔
(
X47
→
X41
→
X21
)
)
→
(
(
(
¬
(
(
¬
X45
)
→
X44
→
(
(
¬
X24
)
↔
X47
)
)
)
↔
(
(
¬
(
X47
→
(
X43
↔
(
(
¬
X18
)
↔
X44
)
)
)
)
↔
(
¬
(
X45
↔
(
X43
↔
(
X18
→
(
X16
↔
False
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
X45
)
↔
(
X44
→
X47
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
(
(
(
X29
↔
(
X15
↔
False
)
)
↔
(
¬
(
(
X46
→
(
X39
↔
X3
)
)
→
(
¬
(
(
¬
X29
)
→
(
(
¬
X15
)
↔
(
X3
→
X39
)
)
)
)
)
)
)
↔
(
¬
(
(
X46
↔
(
X39
→
X47
)
)
→
X39
→
X29
→
(
X18
↔
X46
)
)
)
)
→
(
¬
(
X29
→
X3
→
X46
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X22
↔
(
(
¬
X18
)
↔
X46
)
)
)
→
(
(
X46
→
(
(
¬
X18
)
↔
X43
)
)
↔
(
X46
→
(
X22
↔
(
X18
→
X39
)
)
)
)
)
)
→
(
(
¬
(
(
¬
X43
)
↔
(
(
¬
X34
)
→
(
¬
X32
)
→
X46
)
)
)
↔
(
X39
→
(
(
¬
X34
)
↔
(
X32
↔
X46
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X46
↔
(
X12
↔
X43
)
)
)
→
(
¬
(
(
(
X46
→
(
(
¬
X2
)
↔
X12
)
)
↔
(
(
¬
(
X32
→
(
X12
↔
False
)
)
)
↔
(
X46
↔
(
X30
↔
(
(
¬
X12
)
→
X43
)
)
)
)
)
→
(
¬
(
X31
→
(
(
¬
X30
)
↔
(
(
¬
X12
)
→
X32
)
)
)
)
)
)
)
)
↔
(
¬
(
X43
↔
(
X32
↔
X30
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X43
→
X28
→
(
(
¬
X1
)
↔
X20
)
)
)
↔
(
¬
(
(
¬
X43
)
→
(
¬
X28
)
→
(
¬
X6
)
→
X47
)
)
)
→
X48
→
X47
→
(
X19
↔
X43
)
)
)
→
(
(
(
(
¬
(
X34
→
(
X17
↔
(
(
¬
X5
)
→
X24
)
)
)
)
↔
(
¬
(
X49
→
(
¬
X24
)
→
(
X5
↔
X34
)
)
)
)
↔
(
¬
(
X34
→
(
X17
↔
False
)
)
)
)
↔
(
(
¬
(
(
¬
(
X42
↔
(
X11
→
(
(
¬
X7
)
↔
X49
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X34
)
↔
(
(
¬
X14
)
↔
(
X7
→
X42
)
)
)
)
→
(
¬
(
X19
↔
X14
)
)
)
)
)
)
↔
(
(
X48
→
(
X24
↔
X43
)
)
→
(
(
X24
↔
(
X14
↔
X7
)
)
↔
(
¬
(
X42
↔
(
(
¬
X7
)
→
X48
)
)
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
(
¬
(
X40
↔
(
X9
↔
False
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X40
)
↔
(
X0
↔
X48
)
)
)
)
)
)
↔
(
¬
(
X48
↔
(
X47
↔
X40
)
)
)
)
↔
(
X48
↔
(
X26
↔
(
(
¬
X9
)
→
X40
)
)
)
)
↔
(
(
(
¬
(
(
X48
→
(
X45
↔
(
X26
→
(
X10
↔
False
)
)
)
)
→
(
¬
(
X48
→
(
¬
X45
)
→
X46
)
)
)
)
→
(
X46
↔
(
X45
→
X26
)
)
)
↔
(
¬
X10
)
)
)
↔
(
(
(
(
¬
X40
)
→
(
¬
X38
)
→
(
¬
X25
)
→
X46
)
↔
(
(
¬
X40
)
→
(
(
¬
X38
)
↔
(
(
¬
X25
)
↔
X45
)
)
)
)
↔
(
(
¬
(
(
¬
(
X8
↔
X40
)
)
→
(
¬
(
(
¬
X38
)
→
X37
→
X8
→
X40
)
)
→
(
¬
(
(
¬
X38
)
↔
(
X37
→
X27
→
X40
)
)
)
)
)
↔
(
(
¬
X40
)
↔
(
(
¬
X38
)
↔
(
X37
↔
X45
)
)
)
)
)
)
→
(
(
(
¬
(
X47
↔
(
X44
→
X33
→
X46
)
)
)
↔
(
X47
→
(
X46
↔
(
X44
→
X36
)
)
)
)
→
(
¬
(
X48
↔
(
X46
→
X44
→
X47
)
)
)
)
→
(
(
¬
(
(
(
X36
↔
(
X35
↔
(
(
¬
X13
)
↔
X23
)
)
)
↔
(
¬
(
(
¬
(
X46
→
X35
→
(
X4
↔
X23
)
)
)
→
(
(
X46
↔
(
(
¬
X35
)
→
X41
)
)
↔
(
(
¬
(
X46
↔
(
X23
→
False
)
)
)
↔
(
X35
→
(
¬
X23
)
→
X13
→
X41
)
)
)
)
)
)
→
X46
→
X41
→
(
(
¬
X16
)
↔
X36
)
)
)
↔
(
(
¬
(
X46
↔
(
X41
↔
(
X21
↔
X48
)
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X23
)
↔
X48
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
b99f38...
and proposition id is
ac95ad...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMTcSr1ykKLhrLqfCraefwiu2D1wKguRSuK
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
¬
(
(
(
¬
(
X45
→
X34
→
(
X21
↔
X47
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X47
)
→
X6
→
(
(
¬
X2
)
↔
X49
)
)
)
→
(
(
(
¬
(
(
¬
(
(
¬
X45
)
→
X2
→
X47
)
)
→
(
¬
X45
)
→
(
(
¬
X21
)
↔
(
(
¬
X2
)
→
X49
)
)
)
)
→
(
¬
(
(
¬
X47
)
↔
(
X45
→
X49
)
)
)
)
↔
(
X47
↔
(
X45
↔
(
X21
→
False
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X47
↔
(
(
¬
X33
)
→
(
X9
↔
X40
)
)
)
)
↔
(
¬
(
X49
↔
X9
)
)
)
→
(
¬
(
X47
→
(
¬
X40
)
→
(
¬
X36
)
→
X49
)
)
)
)
↔
(
¬
(
(
(
¬
(
X49
→
(
X46
↔
(
X18
↔
X36
)
)
)
)
↔
(
¬
(
(
¬
(
X49
↔
(
X18
↔
X7
)
)
)
→
(
¬
X47
)
→
X46
→
X7
→
X49
)
)
)
→
(
¬
(
X49
↔
(
X47
→
X41
→
(
X36
↔
False
)
)
)
)
→
(
X45
↔
(
(
¬
X41
)
↔
(
(
¬
X4
)
→
X49
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
¬
X20
)
↔
(
(
¬
X3
)
↔
(
X1
→
X47
)
)
)
)
→
(
(
¬
(
X30
↔
False
)
)
↔
(
(
(
(
(
¬
(
(
(
¬
X30
)
→
X39
)
→
(
X30
↔
(
(
¬
X1
)
↔
X20
)
)
→
(
¬
(
(
¬
X30
)
→
(
X3
↔
(
X1
→
X39
)
)
)
)
)
)
↔
(
X30
→
X3
)
)
↔
(
X20
→
(
(
¬
X3
)
↔
X30
)
)
)
→
(
¬
(
X39
↔
(
X20
→
X3
)
)
)
)
↔
(
¬
(
X39
→
X3
→
X1
→
False
)
)
)
)
→
(
¬
(
(
¬
(
X40
→
(
X24
↔
False
)
)
)
→
X39
→
(
(
¬
X30
)
↔
(
(
¬
X1
)
↔
X40
)
)
)
)
)
→
(
¬
(
X30
↔
(
(
¬
X24
)
→
(
¬
X23
)
→
X39
)
)
)
)
)
→
(
¬
(
(
¬
(
(
X24
↔
(
(
¬
X19
)
→
(
¬
X5
)
→
X39
)
)
→
(
¬
(
X45
→
(
¬
X19
)
→
(
X5
↔
X29
)
)
)
)
)
→
(
(
¬
(
(
(
¬
X39
)
↔
(
X29
→
(
¬
X22
)
→
X40
)
)
→
(
¬
(
X40
→
(
¬
X39
)
→
(
X29
↔
X45
)
)
)
)
)
↔
(
X45
↔
(
X40
→
X29
→
X14
→
False
)
)
)
→
(
¬
X22
)
→
X14
→
(
X0
↔
X45
)
)
)
)
)
→
(
(
(
(
(
¬
(
(
¬
(
(
¬
X10
)
→
(
(
¬
X8
)
↔
X43
)
)
)
→
(
¬
(
X46
→
X28
→
(
(
¬
X10
)
↔
X43
)
)
)
)
)
↔
(
¬
(
X43
↔
(
(
¬
X8
)
→
X46
)
)
)
)
↔
(
(
¬
(
X43
→
(
¬
X8
)
→
X10
)
)
↔
(
X37
↔
(
(
¬
X8
)
↔
X17
)
)
)
)
↔
(
(
X42
↔
(
X28
→
(
(
¬
X27
)
↔
X48
)
)
)
↔
(
¬
(
(
X49
→
(
X27
↔
X43
)
)
→
(
(
(
¬
X48
)
↔
(
(
¬
X46
)
→
X49
)
)
↔
(
X42
↔
(
X27
→
X49
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
X16
)
↔
(
¬
(
(
(
¬
X43
)
↔
(
X16
↔
X48
)
)
→
(
¬
(
X43
→
(
X32
↔
(
(
¬
X13
)
→
X16
)
)
)
)
)
)
)
→
(
(
X46
→
(
(
¬
X32
)
↔
X48
)
)
↔
(
X46
↔
(
X38
→
X43
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X46
)
↔
(
(
¬
X32
)
→
(
(
¬
X25
)
↔
X48
)
)
)
)
→
(
(
¬
X46
)
↔
(
(
¬
X32
)
→
X12
→
X48
)
)
)
)
)
)
)
→
(
(
(
(
(
(
(
¬
(
X35
→
(
X15
↔
(
(
¬
X11
)
↔
X44
)
)
)
)
↔
(
(
¬
X42
)
→
X44
)
)
→
X46
→
(
(
¬
X35
)
↔
(
X15
→
X42
)
)
)
→
X44
→
(
(
¬
X11
)
↔
X35
)
)
→
(
(
(
X44
→
(
(
¬
X35
)
↔
X45
)
)
↔
(
¬
(
X35
→
(
¬
X15
)
→
X26
)
)
)
↔
(
(
¬
X44
)
↔
(
(
¬
X26
)
→
X45
)
)
)
)
↔
(
¬
(
X46
→
X42
→
X35
)
)
)
↔
(
(
X46
↔
(
X43
↔
(
X35
→
X31
)
)
)
↔
(
¬
(
X46
↔
(
X31
→
X15
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
(
X44
↔
(
(
¬
X37
)
→
X24
→
X42
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
¬
X47
)
↔
(
X40
↔
(
X4
↔
X49
)
)
)
)
→
(
(
¬
X45
)
↔
(
X40
↔
X49
)
)
)
)
→
(
(
X49
↔
(
X44
→
(
¬
X4
)
→
X42
)
)
↔
(
X44
↔
(
X40
↔
(
X4
↔
X42
)
)
)
)
)
)
)
→
(
(
(
X48
↔
(
(
¬
X45
)
→
(
¬
X42
)
→
X49
)
)
→
(
¬
X48
)
→
(
¬
X45
)
→
(
¬
X29
)
→
X49
)
↔
(
¬
(
(
¬
(
(
¬
(
(
¬
X39
)
↔
(
X36
↔
(
(
¬
X32
)
→
X49
)
)
)
)
→
(
¬
(
X45
→
(
¬
X32
)
→
X39
)
)
)
)
→
X45
→
(
¬
X39
)
→
(
(
¬
X36
)
↔
X48
)
)
)
)
)
)
↔
(
(
(
X23
→
X20
→
X47
)
↔
(
(
¬
X44
)
→
X42
→
(
(
¬
X41
)
↔
X47
)
)
)
→
X44
→
X42
→
(
X7
↔
X23
)
)
)
→
(
(
¬
(
(
(
X19
→
X14
→
False
)
↔
(
(
(
X48
↔
(
X25
→
(
¬
X19
)
→
X34
)
)
↔
(
X34
→
(
(
¬
X25
)
↔
(
X19
↔
X45
)
)
)
)
↔
(
¬
(
X48
↔
(
X34
↔
X45
)
)
)
)
)
→
X48
→
X34
→
X14
→
X13
)
)
↔
(
(
(
(
¬
(
X48
↔
(
X34
→
(
¬
X6
)
→
X38
)
)
)
↔
(
¬
(
(
X46
↔
(
(
¬
X34
)
→
(
(
¬
X6
)
↔
X38
)
)
)
→
(
X34
↔
(
X17
↔
X46
)
)
)
)
)
↔
(
(
¬
(
X48
↔
(
X17
→
X11
)
)
)
↔
(
X34
→
X11
→
X5
→
X48
)
)
)
↔
(
(
(
X48
↔
(
X45
↔
X2
)
)
↔
(
X34
→
X17
→
X48
)
)
→
(
¬
(
X34
↔
(
(
¬
X17
)
→
X45
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
X48
)
↔
(
X38
↔
(
X31
↔
X49
)
)
)
↔
(
¬
(
X49
↔
(
X31
→
X46
)
)
)
)
→
(
¬
(
(
(
¬
(
X38
↔
(
(
¬
X33
)
→
X12
→
X48
)
)
)
↔
(
(
(
¬
(
X38
→
X8
)
)
↔
(
¬
(
X49
↔
(
(
¬
X38
)
→
X12
→
X48
)
)
)
)
→
(
X49
↔
(
X12
↔
X8
)
)
)
)
→
(
(
¬
X38
)
↔
(
(
¬
X33
)
↔
X49
)
)
)
)
)
)
)
)
↔
(
(
(
(
X0
↔
X35
)
↔
(
(
X44
→
X15
→
False
)
↔
(
¬
(
(
(
X43
→
X35
→
(
X15
↔
X44
)
)
→
(
¬
(
X43
→
(
(
¬
X15
)
↔
(
(
¬
X10
)
→
X44
)
)
)
)
)
→
(
¬
(
X44
↔
(
X35
→
(
¬
X0
)
→
X15
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X35
)
→
(
¬
X26
)
→
X44
)
)
→
(
¬
(
(
X44
↔
(
X22
→
X3
→
X15
)
)
→
(
¬
(
X26
→
(
X15
↔
X3
)
)
)
)
)
)
)
)
→
(
(
(
(
X9
↔
False
)
↔
(
¬
(
X49
→
X43
→
(
(
¬
X30
)
↔
X40
)
)
)
)
→
(
(
(
¬
X40
)
→
X27
→
X43
)
↔
(
(
¬
(
X43
↔
(
X27
↔
(
X16
→
(
X1
↔
False
)
)
)
)
)
↔
(
(
(
(
¬
X30
)
→
X27
→
X16
→
X43
)
↔
(
¬
(
X43
→
(
(
¬
X30
)
↔
(
(
¬
X27
)
→
X40
)
)
)
)
)
↔
(
¬
(
(
¬
X40
)
→
(
(
¬
X30
)
↔
X43
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X44
→
(
¬
X40
)
→
X18
→
X49
)
)
→
(
(
¬
(
X40
→
(
X28
↔
X21
)
)
)
↔
(
¬
(
X49
↔
(
X43
↔
X44
)
)
)
)
)
)
↔
(
X44
→
X43
→
(
X35
↔
X40
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
05b2e3...
and proposition id is
d321d0...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMbcggCLhwEczbGhpKNbtyfNTHB5Nk3qU7N
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
∀X50 :
prop
,
(
(
(
(
(
(
(
¬
(
X28
↔
(
X4
↔
(
(
¬
X2
)
→
X46
)
)
)
)
→
X46
→
X9
→
X28
)
↔
(
(
(
(
¬
(
X39
↔
(
X33
→
(
(
¬
X28
)
↔
X47
)
)
)
)
↔
(
X47
→
(
X39
↔
(
(
¬
X1
)
→
X33
)
)
)
)
↔
(
(
¬
X28
)
→
X1
→
X39
)
)
↔
(
(
(
(
¬
(
(
¬
X9
)
↔
(
(
¬
X6
)
↔
X28
)
)
)
↔
(
X43
↔
(
(
¬
X28
)
↔
(
(
¬
X6
)
→
X46
)
)
)
)
→
(
(
¬
(
X39
↔
(
(
¬
X28
)
→
X33
)
)
)
↔
(
X43
↔
(
X33
→
X9
→
False
)
)
)
)
↔
(
X39
→
(
X6
↔
X46
)
)
)
)
)
↔
(
¬
(
(
¬
(
X46
↔
(
(
¬
X43
)
↔
(
X22
↔
X48
)
)
)
)
→
(
¬
(
X48
→
X43
→
(
X3
↔
X27
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
(
(
¬
X47
)
↔
(
(
¬
X46
)
↔
(
(
¬
X43
)
→
X48
)
)
)
)
→
(
X48
↔
(
X47
→
X3
)
)
)
)
→
(
X46
↔
(
(
¬
X43
)
↔
X47
)
)
)
)
↔
(
¬
(
X48
↔
(
X47
↔
X3
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X50
↔
(
X49
→
X31
→
(
X5
↔
False
)
)
)
)
↔
(
¬
(
X50
↔
(
(
¬
X16
)
↔
X35
)
)
)
)
↔
(
¬
(
X49
→
(
¬
X16
)
→
X8
→
X50
)
)
)
↔
(
(
(
(
¬
(
X43
↔
(
(
¬
X23
)
↔
X35
)
)
)
↔
(
¬
(
(
X23
→
(
¬
X8
)
→
X50
)
→
(
(
¬
(
X43
→
X35
→
(
X23
↔
False
)
)
)
↔
(
(
X43
→
(
X7
↔
X23
)
)
↔
(
(
X35
↔
(
(
¬
X8
)
→
X43
)
)
→
(
(
X43
→
(
X35
↔
X50
)
)
↔
(
(
¬
(
(
X43
→
(
¬
X8
)
→
X35
)
→
(
(
¬
(
X50
↔
(
X35
↔
(
X8
→
X43
)
)
)
)
↔
(
¬
(
X50
→
(
X35
↔
(
(
¬
X8
)
→
X23
)
)
)
)
)
)
)
↔
(
¬
(
X50
→
False
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
X50
→
X23
→
X43
)
)
)
→
(
X50
↔
(
X35
→
X8
)
)
)
)
↔
(
(
(
(
¬
(
X48
→
(
(
¬
X13
)
↔
(
X8
↔
X42
)
)
)
)
↔
(
(
¬
X37
)
→
(
X13
↔
(
X8
↔
X48
)
)
)
)
→
(
¬
(
(
¬
X48
)
→
(
¬
(
(
¬
(
X42
→
(
(
¬
X37
)
↔
(
(
¬
X35
)
↔
X49
)
)
)
)
→
(
¬
X48
)
→
(
X37
↔
(
X35
→
X49
)
)
)
)
)
)
)
→
(
(
¬
(
X49
→
(
(
¬
X21
)
↔
(
X18
↔
X25
)
)
)
)
↔
(
(
(
¬
X47
)
→
X48
)
↔
(
¬
(
(
¬
(
X48
↔
(
(
¬
X42
)
→
X47
)
)
)
→
(
(
¬
(
(
¬
X47
)
→
(
¬
X25
)
→
X48
)
)
↔
(
(
(
¬
(
X42
→
(
¬
X25
)
→
X48
)
)
↔
(
¬
(
(
¬
X47
)
↔
(
X42
→
(
¬
X25
)
→
X49
)
)
)
)
↔
(
¬
(
X48
↔
(
X47
↔
X49
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X47
↔
(
(
¬
X41
)
↔
X45
)
)
)
→
(
(
¬
(
X45
↔
(
X10
→
X41
)
)
)
↔
(
(
¬
(
X47
↔
(
(
¬
X10
)
↔
X36
)
)
)
→
(
X19
↔
(
X10
→
X47
)
)
→
(
¬
(
(
X47
→
(
X19
↔
X45
)
)
→
(
X36
↔
(
X19
↔
(
(
¬
X10
)
→
X47
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X47
↔
(
X26
↔
X15
)
)
)
→
(
¬
(
(
¬
X44
)
→
(
¬
X20
)
→
(
X15
↔
X47
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X44
↔
(
X0
→
X47
)
)
)
→
(
(
(
X44
↔
(
X26
→
X47
)
)
→
X44
→
(
X26
↔
(
X19
→
(
X0
↔
False
)
)
)
)
↔
(
¬
(
X47
↔
(
X44
↔
(
X26
→
False
)
)
)
)
)
)
)
→
(
¬
(
X19
→
X0
→
X26
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
(
¬
(
(
¬
X34
)
↔
(
(
¬
X24
)
↔
X46
)
)
)
→
(
¬
(
X41
↔
(
X32
↔
(
(
¬
X24
)
→
X46
)
)
)
)
)
)
→
(
X38
↔
(
X32
↔
(
(
¬
X11
)
↔
X24
)
)
)
)
)
→
(
(
(
¬
X38
)
↔
(
X34
→
X17
→
X40
)
)
↔
(
(
(
¬
(
X44
→
X40
→
(
X17
↔
X24
)
)
)
↔
(
(
¬
X34
)
→
(
X24
↔
X40
)
)
)
↔
(
(
¬
X40
)
↔
(
X34
→
(
¬
X30
)
→
X44
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X46
↔
(
X38
↔
(
X29
↔
False
)
)
)
)
↔
(
¬
(
X14
→
X12
→
False
)
)
)
→
(
(
¬
(
X47
→
(
X40
↔
(
X38
→
X50
)
)
)
)
↔
(
(
(
¬
X46
)
→
(
X40
↔
(
(
¬
X38
)
↔
X47
)
)
)
↔
(
(
¬
X14
)
↔
X40
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
(
X45
↔
(
X10
↔
False
)
)
→
(
¬
(
X42
→
(
X10
↔
(
(
¬
X2
)
↔
X40
)
)
)
)
)
↔
(
(
(
(
¬
X42
)
↔
(
X28
→
(
¬
X13
)
→
X45
)
)
↔
(
¬
(
X40
→
X13
→
X42
)
)
)
↔
(
¬
(
X40
→
(
X28
↔
(
X13
↔
X17
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X45
)
→
(
(
¬
X42
)
↔
(
X8
→
X50
)
)
)
)
→
(
X42
↔
(
X23
→
X8
)
)
)
)
→
(
¬
(
X42
↔
(
X40
→
X23
→
X12
)
)
)
)
)
→
(
¬
(
(
(
X46
→
X21
→
(
X6
↔
False
)
)
→
(
(
X46
→
(
X27
↔
X45
)
)
↔
(
X46
↔
(
X45
↔
(
(
¬
X5
)
→
X50
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
X33
)
↔
(
X27
→
(
¬
X24
)
→
X46
)
)
)
↔
(
X42
→
X24
→
False
)
)
↔
(
¬
(
X42
→
(
¬
X33
)
→
(
¬
X18
)
→
X49
)
)
)
↔
(
(
¬
(
X42
→
(
(
¬
X39
)
↔
(
X20
→
X46
)
)
)
)
↔
(
¬
(
(
(
¬
X46
)
→
(
(
¬
X42
)
↔
(
(
¬
X9
)
↔
X50
)
)
)
→
(
¬
(
(
¬
X9
)
→
X49
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
¬
(
(
¬
X25
)
→
(
X15
↔
(
(
¬
X3
)
↔
X50
)
)
)
)
→
(
¬
(
X29
↔
(
(
¬
X25
)
→
(
(
¬
X15
)
↔
X50
)
)
)
)
)
↔
(
X25
→
(
X15
↔
X29
)
)
)
→
(
(
¬
(
X46
↔
(
X4
↔
(
(
¬
X0
)
↔
X50
)
)
)
)
↔
(
X25
→
(
¬
X4
)
→
(
(
¬
X0
)
↔
X15
)
)
)
)
)
→
(
(
X31
↔
(
X4
→
X32
)
)
↔
(
(
(
¬
(
(
¬
X44
)
→
(
X31
↔
(
(
¬
X4
)
→
X47
)
)
)
)
↔
(
X44
→
(
X31
↔
(
(
¬
X4
)
↔
X30
)
)
)
)
↔
(
¬
(
(
¬
X31
)
↔
(
X30
↔
X44
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
X22
→
(
X11
↔
X49
)
)
→
(
X44
↔
(
X11
→
False
)
)
→
X49
→
(
X44
↔
(
X38
↔
X48
)
)
)
)
↔
(
(
¬
(
(
X46
→
X19
→
X38
)
→
(
¬
(
(
¬
(
X46
→
(
X38
↔
(
X7
→
X19
)
)
)
)
→
(
¬
X38
)
→
(
¬
X7
)
→
X48
)
)
)
)
↔
(
X48
→
(
X46
↔
(
(
¬
X38
)
↔
X49
)
)
)
)
)
↔
(
(
¬
(
X46
→
X41
→
(
¬
X16
)
→
X36
)
)
↔
(
(
¬
(
X44
↔
False
)
)
↔
(
X46
→
X41
→
(
X36
↔
X48
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
(
¬
X37
)
→
(
(
¬
X26
)
↔
X46
)
)
↔
(
(
¬
(
X37
→
(
X26
↔
(
X14
→
False
)
)
)
)
↔
(
¬
(
X35
↔
X34
)
)
)
)
→
(
¬
(
(
X35
→
(
(
¬
X34
)
↔
X48
)
)
→
(
¬
X37
)
→
X35
→
(
¬
X34
)
→
X44
)
)
)
)
→
(
(
X48
↔
(
X35
→
(
(
¬
X1
)
↔
X43
)
)
)
↔
(
X48
→
(
X43
↔
(
X37
↔
(
X35
↔
False
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
X47
)
↔
(
X46
→
X44
→
X48
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
736df3...
and proposition id is
a6f4f5...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMUaVM3jbW8vPQSkHiEkgVcE5NbnLrL9BrE
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
∀X50 :
prop
,
∀X51 :
prop
,
(
¬
∀X52 :
prop
,
(
¬
∀X53 :
prop
,
∀X54 :
prop
,
(
(
¬
(
(
¬
(
(
(
(
(
¬
(
(
¬
(
(
¬
(
(
¬
X48
)
→
(
¬
X40
)
→
(
X18
↔
X53
)
)
)
→
(
¬
X20
)
→
(
¬
X18
)
→
X48
)
)
→
(
¬
(
X48
↔
X53
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X20
)
↔
(
X6
↔
X41
)
)
)
→
(
¬
(
(
¬
X41
)
→
(
(
¬
X40
)
↔
(
X6
→
X48
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X18
)
→
X15
→
(
(
¬
X1
)
↔
X48
)
)
)
↔
(
(
¬
X40
)
↔
(
(
¬
X15
)
↔
(
(
¬
X7
)
→
X48
)
)
)
)
↔
(
¬
(
X48
→
(
X18
↔
(
X17
↔
X15
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
X49
)
→
(
X4
↔
X53
)
)
→
(
(
X49
→
(
X47
↔
X14
)
)
↔
(
(
(
X49
↔
(
(
¬
X47
)
↔
(
X27
→
X53
)
)
)
↔
(
X53
↔
(
X49
→
(
(
¬
X38
)
↔
X47
)
)
)
)
→
(
X53
↔
(
X38
↔
(
X14
↔
X49
)
)
)
)
)
)
)
↔
(
(
X47
↔
(
X41
↔
(
(
¬
X40
)
→
X53
)
)
)
→
X49
→
(
X40
↔
(
X21
→
False
)
)
)
)
→
(
¬
X21
)
→
(
(
¬
X14
)
↔
(
(
¬
X9
)
→
X49
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X31
→
X23
→
X3
)
)
→
(
¬
(
(
¬
(
(
¬
X42
)
↔
(
(
¬
X31
)
↔
(
(
¬
X3
)
↔
X48
)
)
)
)
→
(
¬
(
(
¬
X42
)
→
(
X3
↔
X49
)
)
)
)
)
→
(
¬
(
(
¬
(
X3
→
X42
)
)
→
(
¬
(
(
(
X42
↔
(
X26
→
X48
)
)
↔
(
(
(
¬
X42
)
↔
(
(
¬
X16
)
→
X48
)
)
→
X42
→
(
¬
X16
)
→
(
¬
X3
)
→
X26
)
)
→
(
X48
↔
(
X16
→
(
X3
↔
False
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
X48
↔
(
X42
↔
X54
)
)
→
(
¬
(
(
(
X54
↔
(
(
¬
X30
)
↔
X42
)
)
→
X49
→
X30
→
(
X24
↔
False
)
)
→
(
¬
(
X54
→
(
X24
↔
X49
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X53
→
(
X0
↔
X42
)
)
)
↔
(
(
¬
(
X54
→
(
X53
↔
(
X42
↔
X0
)
)
)
)
↔
(
¬
(
X54
↔
(
(
¬
X48
)
→
X53
)
)
)
)
)
↔
(
¬
(
X53
→
X48
→
(
X42
↔
X8
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
(
X43
↔
(
X39
→
X33
→
False
)
)
)
↔
(
(
¬
(
X43
↔
(
X39
→
X48
)
)
)
↔
(
X46
→
(
(
¬
X39
)
↔
(
(
¬
X33
)
↔
X43
)
)
)
)
)
↔
(
(
X43
↔
(
(
¬
X33
)
↔
(
(
¬
X10
)
↔
X46
)
)
)
↔
(
¬
(
(
¬
X46
)
↔
(
X37
→
(
(
¬
X10
)
↔
X51
)
)
)
)
)
)
↔
(
(
¬
X48
)
↔
(
(
¬
X43
)
↔
(
(
¬
X35
)
↔
X51
)
)
)
)
→
(
(
¬
(
(
X54
↔
X43
)
→
(
¬
(
(
¬
(
X54
↔
(
X47
↔
(
X43
↔
(
X12
→
False
)
)
)
)
)
→
(
¬
(
X51
→
(
X12
↔
X47
)
)
)
)
)
)
)
↔
(
¬
(
X43
→
(
¬
X19
)
→
X47
)
)
)
→
(
¬
(
(
(
¬
(
(
X51
↔
(
(
¬
X39
)
→
(
(
¬
X29
)
↔
X47
)
)
)
→
(
¬
(
X39
↔
(
X29
→
X25
→
X47
)
)
)
)
)
↔
(
¬
(
(
¬
X47
)
→
(
X39
↔
X51
)
)
)
)
→
(
¬
(
(
¬
X39
)
↔
(
X25
→
X43
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
(
¬
(
(
¬
(
X52
↔
(
X50
→
X48
→
X36
)
)
)
→
(
¬
(
X52
→
(
X34
↔
X48
)
)
)
)
)
↔
(
¬
(
X48
↔
(
X34
↔
X36
)
)
)
)
↔
(
(
X45
→
(
X44
↔
(
X34
↔
False
)
)
)
→
(
(
X45
→
X44
→
X36
→
(
X2
↔
False
)
)
↔
(
X36
↔
(
(
¬
X2
)
→
X34
)
)
)
)
)
→
(
¬
(
(
(
X45
↔
(
(
¬
X44
)
↔
(
(
¬
X22
)
→
X52
)
)
)
→
X44
→
(
X22
↔
(
X11
→
X52
)
)
)
→
(
X44
↔
(
(
¬
X22
)
→
X52
)
)
)
)
)
)
→
(
(
(
(
(
¬
X48
)
↔
(
(
¬
X47
)
→
(
(
¬
X32
)
↔
X51
)
)
)
→
(
X48
↔
(
(
¬
X5
)
↔
X47
)
)
)
→
(
¬
(
(
¬
X13
)
↔
(
X5
→
X48
)
)
)
)
→
(
X51
↔
(
X48
↔
X5
)
)
)
→
(
(
(
¬
X44
)
→
(
(
¬
X28
)
↔
X47
)
)
↔
(
(
¬
X32
)
→
X28
→
(
X5
↔
X45
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X47
↔
(
(
¬
X46
)
→
(
X23
↔
X50
)
)
)
)
→
(
(
(
(
¬
(
(
X52
→
X46
→
X42
→
(
X8
↔
False
)
)
→
(
X47
↔
(
X8
↔
X42
)
)
)
)
→
(
(
¬
X42
)
↔
(
(
¬
X8
)
↔
X47
)
)
)
→
X50
→
X47
→
(
X8
↔
X46
)
)
↔
(
X47
↔
(
X46
↔
(
X45
→
X50
)
)
)
)
)
↔
(
(
(
¬
(
(
X50
→
X44
→
X22
→
X46
)
→
(
X51
↔
X50
)
)
)
↔
(
¬
(
(
X50
→
(
¬
X24
)
→
(
(
¬
X6
)
↔
X40
)
)
→
(
¬
(
X51
↔
(
(
¬
X46
)
↔
(
X40
→
X50
)
)
)
)
)
)
)
→
(
(
(
¬
X52
)
→
X51
→
X54
)
↔
(
(
(
X51
↔
X54
)
↔
(
¬
(
X54
→
(
X44
↔
(
X13
→
X6
)
)
)
)
)
↔
(
(
(
¬
X13
)
↔
(
X6
→
(
X4
↔
X49
)
)
)
↔
(
¬
(
X52
↔
(
X44
→
X6
→
False
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
(
(
(
¬
X34
)
↔
(
(
¬
X28
)
↔
(
X25
↔
X50
)
)
)
→
(
¬
(
X50
↔
(
(
¬
X34
)
↔
(
(
¬
X25
)
→
X43
)
)
)
)
)
)
→
(
X50
↔
(
X28
→
X43
)
)
)
)
↔
(
X50
→
(
(
¬
X41
)
↔
(
X25
↔
X43
)
)
)
)
↔
(
(
¬
(
X34
↔
(
(
¬
X16
)
↔
X43
)
)
)
↔
(
(
¬
X43
)
↔
(
X34
→
(
X28
↔
X51
)
)
)
)
)
→
(
¬
(
(
¬
(
(
X34
→
(
(
¬
X31
)
↔
X49
)
)
→
(
(
¬
(
X41
↔
(
X34
→
X31
→
X49
)
)
)
↔
(
(
¬
X31
)
→
X25
→
X41
)
)
)
)
→
(
(
X20
↔
(
X19
↔
(
X9
→
X31
)
)
)
↔
(
¬
(
(
¬
(
X49
→
(
X2
↔
X41
)
)
)
→
(
(
(
(
X20
↔
(
(
¬
X2
)
→
X9
)
)
↔
(
(
(
X49
↔
(
X20
→
(
X9
↔
X31
)
)
)
→
(
(
¬
X41
)
↔
(
X9
→
X49
)
)
)
→
X49
→
X31
→
(
X20
↔
False
)
)
)
↔
(
¬
(
X49
→
(
¬
X9
)
→
X41
)
)
)
↔
(
¬
(
X20
→
X9
→
X49
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
¬
(
X52
→
(
¬
X32
)
→
(
(
¬
X30
)
↔
X53
)
)
)
→
(
¬
(
X54
↔
(
X53
↔
(
X52
→
False
)
)
)
)
)
→
(
(
¬
(
(
¬
X53
)
→
(
X17
↔
(
(
¬
X0
)
↔
X54
)
)
)
)
↔
(
(
X53
→
(
(
¬
X14
)
↔
(
(
¬
X0
)
→
X52
)
)
)
↔
(
¬
(
(
¬
X53
)
→
X52
→
(
¬
X14
)
→
X54
)
)
)
)
)
→
(
¬
(
X54
↔
(
X39
↔
(
X38
→
X32
→
False
)
)
)
)
→
(
¬
(
(
(
X54
→
(
(
¬
X15
)
↔
X39
)
)
↔
(
¬
(
X39
↔
(
X38
↔
(
X35
↔
X54
)
)
)
)
)
→
(
¬
(
(
X52
↔
(
X35
→
X37
)
)
→
(
X52
↔
(
X38
→
(
¬
X15
)
→
X35
)
)
→
(
X37
↔
(
X15
→
X38
)
)
)
)
)
)
)
)
→
(
(
(
X53
→
X52
→
(
¬
X18
)
→
X33
)
→
(
¬
(
(
(
¬
(
X52
↔
(
(
¬
X18
)
→
(
¬
X7
)
→
X33
)
)
)
↔
(
(
¬
X48
)
↔
(
(
¬
X18
)
↔
X52
)
)
)
→
X18
→
X7
→
False
)
)
→
X48
→
X18
)
→
(
(
¬
(
(
¬
(
(
X48
→
(
X36
↔
X52
)
)
→
(
¬
(
X54
↔
(
X36
↔
(
(
¬
X18
)
→
X27
)
)
)
)
)
)
→
(
¬
(
(
¬
X52
)
↔
(
X48
→
(
(
¬
X12
)
↔
X54
)
)
)
)
)
)
↔
(
(
X51
→
(
(
¬
X27
)
↔
(
(
¬
X21
)
→
X52
)
)
)
↔
(
(
¬
X27
)
→
(
¬
X21
)
→
X18
→
X51
)
)
)
)
→
(
(
X51
↔
(
X29
↔
(
X10
→
X26
)
)
)
→
(
X53
↔
(
X26
→
(
¬
X10
)
→
X51
)
)
)
→
(
(
(
(
¬
(
X11
↔
(
X10
↔
(
(
¬
X3
)
→
X29
)
)
)
)
↔
(
¬
(
X10
→
(
X3
↔
(
X1
↔
False
)
)
)
)
)
↔
(
X29
↔
(
(
¬
X10
)
↔
(
(
¬
X5
)
→
X11
)
)
)
)
↔
(
X33
→
(
X3
↔
(
X1
↔
X11
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
2d58c8...
and proposition id is
a71917...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMYWJacTuawBpL7vJMLrc97oXF3g2tE3DSf
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
∀X50 :
prop
,
(
¬
∀X51 :
prop
,
(
(
(
(
(
(
X45
→
X23
→
X6
→
X49
)
↔
(
(
(
¬
(
(
¬
X31
)
↔
(
(
¬
X23
)
→
X5
→
X45
)
)
)
↔
(
X6
→
X45
)
)
↔
(
(
¬
X31
)
↔
(
X23
↔
X45
)
)
)
)
↔
(
(
(
X43
→
(
X35
↔
(
(
¬
X21
)
→
X46
)
)
)
↔
(
(
¬
(
(
¬
X46
)
↔
(
(
¬
X45
)
↔
(
(
¬
X43
)
↔
X49
)
)
)
)
→
(
(
X49
↔
(
(
¬
X21
)
↔
(
(
¬
X20
)
↔
X46
)
)
)
↔
(
(
¬
(
(
¬
X45
)
↔
(
(
¬
X21
)
→
(
¬
X1
)
→
X46
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X45
)
↔
(
X21
↔
X49
)
)
)
)
)
)
)
)
→
(
X43
→
(
(
¬
X31
)
↔
(
X29
↔
X45
)
)
)
→
(
¬
(
X43
→
X35
→
(
¬
X29
)
→
X31
)
)
)
)
↔
(
(
(
(
(
¬
(
(
¬
X42
)
→
(
(
¬
X9
)
↔
(
(
¬
X4
)
→
X45
)
)
)
)
↔
(
X30
→
(
X9
↔
X4
)
)
)
↔
(
¬
(
(
¬
(
X45
↔
(
X42
→
(
X37
↔
X30
)
)
)
)
→
(
X50
↔
(
(
¬
X30
)
→
X44
)
)
)
)
)
→
(
¬
(
X44
→
X42
→
X38
)
)
→
(
(
¬
(
(
¬
(
(
¬
X42
)
↔
(
(
¬
X30
)
→
X0
→
X44
)
)
)
→
(
(
¬
X38
)
↔
(
(
¬
X0
)
↔
X44
)
)
)
)
↔
(
(
X42
→
(
(
¬
X24
)
↔
X36
)
)
↔
(
(
¬
X36
)
↔
(
(
¬
X36
)
↔
(
(
¬
(
X42
↔
(
(
¬
X36
)
→
(
¬
X24
)
→
X38
)
)
)
↔
(
¬
(
X44
↔
(
(
¬
X36
)
→
X42
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
X42
→
(
(
¬
X31
)
↔
(
(
¬
X17
)
↔
X35
)
)
)
→
(
(
X35
→
X17
→
X42
)
↔
(
¬
(
X45
→
(
¬
X17
)
→
X37
)
)
)
)
)
→
(
¬
X37
)
→
(
X35
↔
(
(
¬
X12
)
↔
X42
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
X48
)
↔
(
(
¬
X47
)
→
(
¬
X7
)
→
X49
)
)
↔
(
X49
↔
(
X48
↔
(
(
¬
X7
)
↔
X25
)
)
)
)
↔
(
(
(
(
¬
(
(
X41
↔
(
X22
↔
(
X16
↔
(
X11
↔
False
)
)
)
)
→
X22
→
X16
→
(
X11
↔
X39
)
)
)
→
(
¬
(
X41
↔
(
X22
→
(
X11
↔
X48
)
)
)
)
)
↔
(
¬
(
(
¬
X40
)
↔
(
X18
→
(
X3
↔
X41
)
)
)
)
)
→
(
(
¬
(
X49
↔
(
(
¬
X40
)
→
(
X8
↔
X41
)
)
)
)
↔
(
(
X40
↔
(
X14
↔
(
X10
↔
X41
)
)
)
↔
(
¬
(
(
¬
X14
)
→
(
X10
↔
X49
)
)
)
)
)
→
(
¬
(
(
(
¬
X47
)
↔
(
X40
→
(
X39
↔
X49
)
)
)
→
(
¬
(
X47
→
(
¬
X39
)
→
(
(
¬
X10
)
↔
X40
)
)
)
)
)
→
X39
→
X10
)
)
→
(
(
(
(
¬
(
(
¬
X48
)
→
(
(
¬
X27
)
↔
X51
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X48
↔
(
X47
→
X26
)
)
)
→
(
¬
(
(
¬
X48
)
↔
(
X47
→
(
¬
X33
)
→
X51
)
)
)
)
)
→
(
¬
(
X51
↔
(
X33
↔
X47
)
)
)
)
)
)
↔
(
¬
(
X51
→
X50
→
(
X48
↔
(
X47
→
False
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
¬
X40
)
↔
(
(
¬
X34
)
↔
(
(
¬
X15
)
↔
X46
)
)
)
→
(
¬
(
(
¬
X34
)
→
X47
)
)
)
→
(
¬
(
(
¬
X47
)
→
X46
→
X51
)
)
)
)
→
(
(
¬
(
X51
↔
(
X32
→
(
X19
↔
(
X13
↔
False
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X50
)
→
(
¬
X13
)
→
X51
)
)
↔
(
¬
(
(
¬
(
X51
→
X50
→
(
X13
↔
(
X2
↔
False
)
)
)
)
→
X51
→
X40
→
X2
)
)
)
→
(
(
(
(
(
¬
X40
)
↔
(
(
¬
X28
)
→
X51
)
)
→
(
¬
X28
)
→
(
(
X28
→
False
)
↔
(
¬
(
X50
↔
(
X40
→
X32
→
X47
)
)
)
)
)
↔
(
X47
↔
(
X32
→
X50
)
)
)
↔
(
¬
(
X28
→
X40
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X50
↔
(
(
¬
X49
)
↔
(
(
¬
X45
)
→
X51
)
)
)
)
↔
(
(
(
(
¬
(
X51
→
X50
→
(
(
¬
X39
)
↔
X49
)
)
)
↔
(
X50
↔
(
X39
↔
(
X6
↔
X51
)
)
)
)
→
(
¬
(
(
(
X51
→
(
(
¬
X35
)
↔
(
(
¬
X0
)
↔
X49
)
)
)
↔
(
X51
→
X50
→
X49
→
False
)
)
→
(
¬
(
X51
→
X49
→
(
¬
X29
)
→
X50
)
)
)
)
)
↔
(
¬
(
(
¬
(
X49
→
(
X48
↔
X51
)
)
)
→
(
(
¬
(
X49
↔
(
X48
→
(
¬
X35
)
→
X51
)
)
)
↔
(
(
X30
→
X15
)
↔
(
(
¬
(
(
¬
X49
)
↔
(
X48
↔
(
(
¬
X30
)
↔
X51
)
)
)
)
↔
(
¬
(
(
¬
X49
)
→
(
X48
↔
(
(
¬
X15
)
→
X51
)
)
)
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X48
)
→
(
(
¬
X44
)
↔
(
(
¬
X40
)
↔
X51
)
)
)
)
↔
(
(
X44
→
(
¬
X18
)
→
X40
)
↔
(
(
¬
X40
)
→
(
¬
X24
)
→
(
(
¬
X18
)
↔
X48
)
)
)
)
↔
(
(
(
X51
↔
(
X50
↔
(
X13
→
X24
)
)
)
↔
(
X44
↔
(
(
¬
X13
)
↔
X50
)
)
)
→
(
¬
(
(
¬
X50
)
↔
(
X37
→
(
X13
↔
X51
)
)
)
)
)
)
)
↔
(
¬
(
(
(
X36
→
(
(
¬
X21
)
↔
(
(
¬
X11
)
→
X38
)
)
)
→
(
(
(
X34
→
(
X21
↔
X17
)
)
→
(
(
X39
→
X21
)
↔
(
(
X46
→
(
X34
↔
X17
)
)
→
(
X21
→
X34
)
→
(
¬
(
(
¬
X39
)
→
(
X21
↔
(
X17
→
X46
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X11
)
↔
X17
)
)
↔
(
(
¬
(
X49
↔
(
X17
↔
(
X11
→
False
)
)
)
)
→
(
¬
(
X46
→
(
¬
X39
)
→
X49
)
)
)
)
↔
(
¬
(
X17
↔
(
(
¬
X11
)
↔
X46
)
)
)
)
)
)
→
(
(
(
X49
→
(
¬
X39
)
→
X51
)
→
(
(
X51
↔
(
X38
↔
X49
)
)
↔
(
(
¬
X49
)
→
(
(
¬
X43
)
↔
(
X39
↔
X51
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
X49
↔
(
X39
→
(
¬
X2
)
→
X51
)
)
)
↔
(
¬
(
(
¬
X39
)
↔
(
X33
↔
(
X2
↔
X51
)
)
)
)
)
→
(
X49
↔
False
)
)
)
↔
(
X49
↔
X33
)
)
→
(
¬
(
(
¬
X39
)
→
X32
→
X49
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
(
¬
(
X9
→
X3
→
X26
)
)
↔
(
(
¬
X41
)
→
(
(
¬
X9
)
↔
(
(
¬
X3
)
→
X48
)
)
)
)
↔
(
(
¬
(
(
¬
X26
)
→
X14
→
(
X3
↔
X49
)
)
)
↔
(
¬
(
X48
↔
(
X3
→
X14
)
)
)
)
)
↔
(
(
¬
(
(
X51
→
X14
→
(
X5
↔
False
)
)
→
(
¬
(
X51
→
(
X41
↔
(
X5
→
False
)
)
)
)
→
X26
→
X14
→
X41
)
)
↔
(
¬
(
(
¬
X26
)
↔
X41
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
X51
↔
(
(
¬
X23
)
↔
(
(
¬
X22
)
↔
X42
)
)
)
)
↔
(
¬
(
X51
→
(
(
¬
X42
)
↔
(
(
¬
X23
)
↔
X49
)
)
)
)
)
→
X49
→
X42
→
False
)
→
(
X49
↔
X22
)
)
)
↔
(
(
¬
X49
)
↔
(
(
¬
X42
)
→
(
¬
X26
)
→
X51
)
)
)
)
→
(
(
¬
(
X51
↔
(
(
¬
X12
)
→
(
¬
X4
)
→
X41
)
)
)
↔
(
(
¬
(
X41
→
X31
→
X10
)
)
↔
(
X42
→
X41
→
X31
→
X4
→
False
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X28
↔
(
(
¬
X20
)
→
X25
)
)
)
↔
(
X28
→
X25
→
X8
→
X1
→
False
)
)
→
(
(
(
¬
(
X51
↔
X25
)
)
↔
(
X28
→
(
X25
↔
(
X20
→
X19
)
)
)
)
↔
(
¬
(
(
¬
X28
)
↔
(
X19
→
X16
→
X51
)
)
)
)
)
)
→
(
¬
(
(
(
X51
→
(
X28
↔
(
X27
→
X47
)
)
)
↔
(
X47
↔
(
X41
→
(
X27
↔
X28
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X47
→
(
X27
↔
(
(
¬
X7
)
↔
X51
)
)
)
)
→
(
¬
X25
)
→
(
X7
↔
X47
)
)
)
→
X47
→
(
X27
↔
(
X23
↔
X51
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
407bc7...
and proposition id is
26058e...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMEzYFjV8YTND7fWe5fbjimRcRN4HvtAoL1
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
(
(
(
(
(
(
(
(
¬
(
X49
→
(
X48
↔
(
X34
↔
X8
)
)
)
)
→
X49
→
(
X48
↔
(
X16
→
(
X8
↔
False
)
)
)
)
↔
(
¬
(
X48
↔
(
X16
↔
(
X12
↔
X34
)
)
)
)
)
↔
(
(
¬
X36
)
↔
(
X8
→
X49
)
)
)
↔
(
¬
(
(
(
¬
X48
)
→
(
(
¬
X36
)
↔
(
X6
→
X49
)
)
)
→
X44
→
X6
→
X48
)
)
)
↔
(
(
¬
(
(
¬
(
X47
→
(
X34
↔
(
(
¬
X2
)
→
X49
)
)
)
)
→
(
X19
→
(
¬
X4
)
→
X49
)
→
(
¬
(
(
¬
X34
)
→
(
X19
↔
X47
)
)
)
)
)
→
(
X47
↔
(
X19
↔
(
X2
→
X44
)
)
)
)
)
→
(
(
(
¬
(
X44
↔
(
X41
↔
(
(
¬
X9
)
→
X48
)
)
)
)
↔
(
(
X44
→
(
(
¬
X33
)
↔
(
X25
↔
X48
)
)
)
↔
(
¬
(
X41
↔
(
X25
→
X48
)
)
)
)
)
→
(
X47
↔
(
X9
→
X25
)
)
)
→
(
¬
(
(
(
X47
→
(
X44
↔
(
(
¬
X42
)
→
X48
)
)
)
→
(
¬
(
X48
→
(
X44
↔
(
X42
↔
(
X13
↔
False
)
)
)
)
)
)
→
X42
→
X13
)
)
)
↔
(
(
(
¬
(
(
(
(
X42
→
(
(
¬
X27
)
↔
(
(
¬
X26
)
↔
X38
)
)
)
↔
(
¬
(
X42
↔
(
(
¬
X26
)
↔
X49
)
)
)
)
↔
(
X42
→
X38
)
)
→
(
¬
X38
)
→
X27
→
X11
→
X42
)
)
→
(
(
¬
(
(
¬
X42
)
→
(
X38
↔
(
(
¬
X14
)
↔
X49
)
)
)
)
↔
(
(
(
¬
X48
)
→
X38
→
X49
)
↔
(
¬
(
X14
→
X7
→
False
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
X38
↔
(
X30
↔
(
(
¬
X10
)
→
X42
)
)
)
)
→
(
(
X42
↔
X30
)
↔
(
(
¬
(
(
¬
(
X42
→
(
(
¬
X30
)
↔
(
(
¬
X0
)
↔
X32
)
)
)
)
→
(
¬
(
X30
→
X0
→
X38
)
)
)
)
↔
(
X32
→
(
(
¬
X0
)
↔
X30
)
)
)
)
)
)
→
(
(
X42
↔
(
(
¬
X32
)
→
(
¬
X11
)
→
X46
)
)
↔
(
(
¬
(
(
¬
X32
)
→
X42
)
)
↔
(
X42
↔
(
X40
→
X30
→
(
X23
↔
False
)
)
)
)
)
)
)
→
X38
→
(
X30
↔
X11
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
¬
(
X45
↔
(
X43
→
X28
)
)
)
→
(
(
¬
X43
)
↔
(
X41
↔
(
X28
↔
X50
)
)
)
)
)
↔
(
(
(
(
¬
(
X43
→
(
¬
X28
)
→
X45
)
)
→
(
(
¬
(
X22
→
(
(
¬
X17
)
↔
X45
)
)
)
↔
(
X43
↔
(
(
¬
X22
)
→
(
(
¬
X17
)
↔
X28
)
)
)
)
)
↔
(
¬
(
X45
↔
(
X28
↔
X22
)
)
)
)
↔
(
(
¬
X43
)
↔
(
(
¬
X22
)
→
X45
)
)
)
)
→
(
¬
(
(
(
X45
→
(
(
¬
X39
)
↔
(
(
¬
X3
)
→
X41
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X3
)
→
X45
)
)
)
)
→
(
¬
(
(
(
X43
↔
(
X39
→
X46
)
)
↔
(
¬
(
(
¬
X39
)
↔
(
X5
↔
X45
)
)
)
)
→
X46
→
X5
→
X45
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
X43
)
→
X29
→
(
X24
↔
X49
)
)
→
(
¬
(
(
¬
(
X37
→
(
¬
X24
)
→
(
(
¬
X1
)
↔
X49
)
)
)
→
(
¬
(
(
¬
(
(
X43
↔
(
(
¬
X24
)
→
X49
)
)
→
(
¬
(
X49
↔
(
(
¬
X18
)
↔
(
(
¬
X1
)
↔
X43
)
)
)
)
)
)
→
(
¬
(
X49
↔
(
X43
→
(
X1
↔
False
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
X41
)
→
X31
→
(
X20
↔
X46
)
)
↔
(
X43
↔
(
(
¬
X41
)
→
X46
)
)
)
→
X41
→
(
X37
↔
(
X15
→
False
)
)
)
→
(
(
¬
(
X43
→
(
X37
↔
(
X21
→
X18
)
)
)
)
↔
(
¬
(
X46
→
X35
→
X43
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
(
¬
(
X15
→
(
X13
↔
False
)
)
)
→
(
(
(
¬
X15
)
↔
(
(
¬
X13
)
→
X48
)
)
↔
(
¬
(
X50
↔
(
(
¬
X15
)
↔
X49
)
)
)
)
)
)
→
(
(
(
¬
(
X48
↔
(
(
¬
X32
)
↔
(
(
¬
X22
)
↔
X50
)
)
)
)
↔
(
(
X48
↔
(
(
¬
X8
)
↔
X32
)
)
↔
(
¬
(
X48
→
(
(
¬
X10
)
↔
(
X8
→
X50
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
(
¬
X16
)
↔
(
(
¬
X8
)
↔
X35
)
)
)
↔
(
(
¬
X49
)
→
(
(
¬
X16
)
↔
X50
)
)
)
↔
(
¬
X8
)
)
→
(
¬
(
X35
↔
(
(
¬
X16
)
↔
X50
)
)
)
)
)
→
(
¬
(
(
¬
X49
)
↔
(
X32
→
(
(
¬
X16
)
↔
X50
)
)
)
)
)
→
(
¬
(
X49
→
X35
→
(
X22
↔
X7
)
)
)
)
→
(
(
(
¬
(
X49
→
X35
→
(
X17
↔
X48
)
)
)
↔
(
X35
↔
(
(
¬
X33
)
↔
(
(
¬
X27
)
↔
X48
)
)
)
)
↔
(
(
(
¬
X0
)
→
X33
)
↔
(
¬
(
(
¬
X25
)
→
(
X17
↔
(
(
¬
X6
)
→
X27
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X48
)
→
X40
→
X28
→
X50
)
)
↔
(
(
¬
(
X50
→
(
X49
↔
(
X48
↔
(
X26
↔
False
)
)
)
)
)
↔
(
(
(
(
¬
X29
)
↔
X43
)
→
X43
→
(
¬
(
(
¬
(
X40
↔
X49
)
)
→
(
X50
↔
(
X40
↔
(
X29
↔
X43
)
)
)
)
)
)
↔
(
X50
↔
(
X49
↔
(
(
¬
X43
)
↔
X48
)
)
)
)
)
)
↔
(
X48
→
(
X47
↔
(
X29
→
X43
)
)
)
)
↔
(
(
X45
↔
(
X24
↔
(
(
¬
X21
)
↔
X47
)
)
)
→
(
X43
↔
X50
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
(
X46
→
(
(
¬
X34
)
↔
(
(
¬
X19
)
↔
X44
)
)
)
→
(
¬
(
X34
↔
(
(
¬
X19
)
↔
(
X3
→
X44
)
)
)
)
)
)
↔
(
¬
(
X46
→
(
(
¬
X42
)
↔
(
(
¬
X19
)
↔
X44
)
)
)
)
)
↔
(
¬
(
X42
↔
(
X37
→
(
(
¬
X19
)
↔
X46
)
)
)
)
)
→
(
(
(
X48
↔
(
(
¬
X44
)
↔
(
(
¬
X41
)
→
X50
)
)
)
→
X44
→
(
¬
X41
)
→
(
X30
↔
X50
)
)
↔
(
(
(
X50
→
X46
→
(
X34
↔
X2
)
)
↔
(
(
¬
(
X46
↔
(
(
¬
X2
)
→
X30
)
)
)
↔
(
X48
→
(
(
¬
X2
)
↔
X50
)
)
)
)
↔
(
X50
→
X48
→
(
¬
X23
)
→
X46
)
)
)
)
)
↔
(
(
(
(
X50
→
(
¬
X18
)
→
X11
→
X49
)
↔
(
(
X50
→
X49
→
(
(
¬
X1
)
↔
X12
)
)
↔
(
¬
(
X50
→
(
(
¬
X18
)
↔
(
X12
→
X49
)
)
)
)
)
)
↔
(
(
¬
(
X46
↔
(
(
¬
X4
)
↔
(
X1
↔
X18
)
)
)
)
→
(
¬
(
(
¬
X4
)
↔
(
(
¬
X1
)
↔
X12
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X49
↔
(
X44
↔
(
X14
→
X20
)
)
)
)
↔
(
X49
↔
(
(
¬
X44
)
→
(
X14
↔
X50
)
)
)
)
→
(
(
¬
(
X46
↔
(
X20
→
False
)
)
)
→
(
(
X20
↔
(
(
¬
X5
)
↔
X46
)
)
↔
(
(
X50
→
(
X20
↔
(
X5
→
X46
)
)
)
→
(
X50
↔
(
X46
→
(
X20
↔
X49
)
)
)
)
)
)
→
(
¬
(
X46
↔
(
X38
→
False
)
)
)
)
)
↔
(
(
(
(
¬
(
X46
↔
(
X39
↔
(
X38
↔
X31
)
)
)
)
↔
(
X46
→
(
X38
↔
(
X31
→
X36
)
)
)
)
↔
(
¬
(
X49
↔
(
X31
↔
False
)
)
)
)
↔
(
(
¬
X31
)
↔
(
(
¬
X9
)
↔
X49
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
6f50f6...
and proposition id is
b8503a...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMcpysFtzRPnRxUmsf32q6eW18hVJZAatnh
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
¬
(
(
(
(
(
(
X45
↔
(
(
¬
X25
)
→
(
X23
↔
X49
)
)
)
↔
(
(
¬
(
(
¬
(
X49
→
(
¬
X27
)
→
X23
→
X45
)
)
→
X23
→
X14
)
)
→
X49
→
(
(
¬
X27
)
↔
X45
)
)
)
↔
(
(
¬
(
(
¬
X46
)
→
(
X45
↔
(
(
¬
X39
)
→
X48
)
)
)
)
→
(
(
X25
↔
(
X6
→
X13
)
)
→
(
(
¬
(
(
¬
X45
)
→
X27
→
X20
→
X48
)
)
↔
(
(
X45
↔
(
X27
↔
(
(
¬
X13
)
↔
X20
)
)
)
↔
(
¬
(
X27
→
X20
→
X6
)
)
)
)
)
→
(
¬
(
X45
→
(
(
¬
X25
)
↔
(
(
¬
X13
)
↔
X46
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
¬
X29
)
↔
(
(
¬
X17
)
↔
(
X1
↔
X49
)
)
)
)
→
(
¬
(
X45
↔
(
X9
↔
(
X8
↔
False
)
)
)
)
)
)
→
(
(
¬
(
(
¬
X17
)
↔
(
X5
→
X39
)
)
)
↔
(
(
(
(
X45
↔
(
X29
↔
(
X17
→
False
)
)
)
↔
(
(
¬
(
X17
→
(
¬
X16
)
→
X5
→
X45
)
)
↔
(
X45
→
X16
→
X29
)
)
)
↔
(
¬
(
X45
↔
(
X5
↔
X17
)
)
)
)
↔
(
¬
(
X29
→
(
¬
X5
)
→
X16
)
)
)
)
)
)
)
→
(
(
(
X49
→
X46
→
(
X43
↔
X45
)
)
→
(
¬
(
(
(
(
(
¬
(
X45
↔
(
X34
→
(
(
¬
X30
)
↔
X44
)
)
)
)
↔
(
¬
(
X45
→
(
¬
X34
)
→
(
¬
X22
)
→
X46
)
)
)
↔
(
(
¬
X45
)
↔
(
X34
→
X46
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X30
)
↔
X34
)
)
)
)
→
(
(
¬
(
(
¬
X46
)
→
(
X45
↔
X49
)
)
)
↔
(
(
¬
X44
)
↔
(
X24
→
X45
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
¬
X40
)
↔
(
(
¬
X10
)
→
X49
)
)
)
→
(
X45
↔
(
(
¬
X26
)
→
(
¬
X10
)
→
X40
)
)
)
)
→
(
¬
(
X45
→
X43
→
X28
→
False
)
)
)
→
X43
→
X40
→
(
(
¬
X3
)
↔
X26
)
)
)
)
→
(
(
(
(
(
X45
↔
(
X32
↔
False
)
)
↔
(
(
¬
(
(
¬
(
X43
↔
(
X33
↔
(
(
¬
X18
)
→
X41
)
)
)
)
→
(
X43
↔
(
(
¬
X41
)
↔
X45
)
)
)
)
↔
(
X45
→
(
X43
↔
(
X32
→
X33
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X47
→
X41
→
(
¬
X12
)
→
X43
)
)
→
X47
→
X41
→
X12
→
X18
)
)
↔
(
(
¬
X18
)
↔
(
(
¬
X12
)
↔
X43
)
)
)
)
→
(
¬
(
X47
→
(
¬
X44
)
→
(
X21
↔
X45
)
)
)
→
(
¬
(
(
X47
→
X45
→
X42
→
X44
)
→
(
(
¬
(
(
(
X44
↔
(
X36
↔
(
X11
→
X0
)
)
)
↔
(
¬
(
(
¬
X44
)
→
(
(
¬
X0
)
↔
X45
)
)
)
)
→
(
X36
↔
(
X11
→
False
)
)
)
)
↔
(
¬
(
X44
→
X36
→
X45
)
)
)
→
(
(
¬
(
X43
→
(
(
¬
X36
)
↔
(
X11
→
X45
)
)
)
)
↔
(
¬
(
X42
↔
(
(
¬
X11
)
↔
X43
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
X43
↔
(
(
¬
X37
)
→
(
X19
↔
X45
)
)
)
)
↔
(
(
¬
(
X47
↔
(
X45
→
X43
)
)
)
↔
(
(
X47
↔
(
X45
→
(
¬
X2
)
→
X43
)
)
↔
(
X47
→
X45
→
(
X19
↔
X2
)
)
)
)
)
↔
(
¬
(
X37
↔
(
X35
↔
(
(
¬
X2
)
↔
X45
)
)
)
)
)
→
(
(
¬
X31
)
→
(
¬
X7
)
→
X45
)
→
(
(
¬
(
X47
→
(
¬
X38
)
→
(
(
¬
X31
)
↔
X43
)
)
)
↔
(
¬
(
(
¬
X45
)
→
X43
→
X47
)
)
)
)
)
↔
(
(
X35
↔
(
X15
→
(
(
¬
X4
)
↔
X38
)
)
)
↔
(
(
¬
X35
)
→
(
¬
X7
)
→
X38
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
X44
→
(
(
¬
X26
)
↔
(
(
¬
X18
)
↔
X34
)
)
)
)
↔
(
¬
(
X49
↔
(
X44
→
X32
→
(
X11
↔
False
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
X36
)
↔
(
(
¬
X32
)
→
(
(
¬
X18
)
↔
X45
)
)
)
→
(
¬
(
X45
↔
(
X30
↔
(
X14
→
X18
)
)
)
)
→
(
(
(
¬
X18
)
→
X30
)
↔
(
¬
(
(
¬
X36
)
→
(
¬
X18
)
→
(
(
¬
X16
)
↔
X45
)
)
)
)
)
)
→
(
¬
(
(
X36
↔
(
X26
↔
(
(
¬
X8
)
↔
X18
)
)
)
→
(
¬
(
(
¬
X26
)
→
(
(
¬
X18
)
↔
(
X14
↔
X36
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
(
X44
↔
(
(
¬
X35
)
→
X46
)
)
)
→
(
¬
(
X48
↔
(
X46
→
X35
→
X45
)
)
)
)
)
→
(
(
¬
X35
)
→
(
X6
↔
X44
)
)
→
(
¬
(
X35
→
(
(
¬
X20
)
↔
(
(
¬
X6
)
→
X46
)
)
)
)
)
↔
(
¬
(
(
(
X46
→
X5
→
X47
)
↔
(
X48
↔
(
X17
→
(
X4
↔
X5
)
)
)
)
→
(
(
¬
(
(
X47
→
X44
→
X39
→
X23
)
→
(
X47
↔
(
(
¬
X39
)
→
X48
)
)
)
)
↔
(
¬
(
(
(
¬
(
X47
↔
(
X44
→
X0
→
X43
)
)
)
↔
(
¬
(
(
¬
(
X0
↔
X43
)
)
→
(
¬
(
(
¬
X44
)
→
(
(
¬
X0
)
↔
X49
)
)
)
→
(
¬
(
X49
→
(
(
¬
X39
)
↔
(
(
¬
X0
)
→
X43
)
)
)
)
)
)
)
→
X44
→
(
X39
↔
X43
)
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X47
)
↔
(
X46
→
X48
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X48
)
↔
(
X46
→
(
¬
X10
)
→
X49
)
)
)
→
(
¬
(
X29
↔
(
X15
→
(
(
¬
X10
)
↔
X48
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X49
→
(
X42
↔
(
X12
↔
X40
)
)
)
)
↔
(
¬
(
(
(
(
X47
↔
(
X25
→
(
X22
↔
X40
)
)
)
→
(
¬
(
X42
→
(
(
¬
X40
)
↔
(
(
¬
X25
)
→
X47
)
)
)
)
)
↔
(
X47
↔
(
X42
→
X40
)
)
)
→
(
¬
(
X47
→
X25
→
X22
→
False
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
X37
)
↔
(
(
¬
X27
)
→
X49
)
)
→
(
¬
(
X13
↔
X40
)
)
)
)
→
(
(
¬
X42
)
→
(
¬
X3
)
→
X49
)
→
(
¬
(
X42
↔
(
(
¬
X27
)
↔
(
(
¬
X3
)
→
X37
)
)
)
)
)
)
→
(
(
(
(
¬
X24
)
↔
(
X21
→
X49
)
)
→
(
X47
↔
(
(
¬
X46
)
→
X49
)
)
)
→
(
(
¬
(
X49
↔
(
X24
→
(
X19
↔
X47
)
)
)
)
↔
(
X49
→
(
X24
↔
(
X7
↔
X47
)
)
)
)
)
→
(
¬
(
(
(
(
X49
↔
(
X46
→
X41
)
)
↔
(
(
X46
↔
(
(
¬
X41
)
→
X49
)
)
→
(
¬
(
X38
↔
(
(
¬
X2
)
→
X46
)
)
)
)
)
↔
(
¬
(
X49
→
(
¬
X41
)
→
X46
)
)
)
→
(
¬
(
(
¬
(
X49
→
(
(
¬
X38
)
↔
X42
)
)
)
→
X49
→
(
X41
↔
(
X9
→
X42
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
X33
↔
(
X1
↔
X47
)
)
)
↔
(
¬
(
(
¬
X47
)
↔
(
X46
↔
X49
)
)
)
)
↔
(
(
¬
X47
)
→
(
¬
X33
)
→
(
¬
X28
)
→
X49
)
)
↔
(
X33
→
X47
)
)
↔
(
¬
(
(
¬
(
X42
↔
(
X31
↔
(
X28
→
X49
)
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X31
)
↔
(
X28
↔
X49
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
4f3649...
and proposition id is
77d0f4...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMdNpiQpz98iJhx9PLn4kspGNvaiEHQCCw1
)
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
∀X50 :
prop
,
(
(
(
(
(
(
(
(
¬
(
(
(
¬
X46
)
→
X33
→
(
X12
↔
X47
)
)
→
X47
→
X33
→
X43
)
)
→
X47
→
X38
→
X43
)
↔
(
(
(
X38
↔
(
X28
↔
(
(
¬
X9
)
↔
X46
)
)
)
↔
(
(
¬
(
X33
↔
X38
)
)
↔
(
(
¬
(
X46
↔
(
X33
→
(
X28
↔
X38
)
)
)
)
↔
(
(
X46
→
X33
→
X13
)
↔
(
¬
(
(
(
X28
→
X13
)
↔
(
(
¬
X33
)
→
(
X13
↔
(
(
¬
X3
)
→
X46
)
)
)
)
→
(
X46
↔
(
X33
↔
X28
)
)
)
)
)
)
)
)
→
(
¬
(
X33
↔
(
X23
↔
(
(
¬
X9
)
→
X13
)
)
)
)
)
)
↔
(
(
¬
(
X46
→
(
¬
X43
)
→
X47
)
)
↔
(
¬
(
(
¬
X46
)
→
(
¬
X42
)
→
(
¬
X18
)
→
X47
)
)
)
)
↔
(
(
(
X43
↔
(
X14
→
(
(
¬
X4
)
↔
X26
)
)
)
→
(
¬
X43
)
→
(
(
¬
X4
)
↔
X47
)
)
↔
(
¬
(
(
(
X43
→
(
X26
↔
X46
)
)
→
(
¬
(
X46
→
(
X43
↔
(
(
¬
X20
)
→
X49
)
)
)
)
)
→
X49
→
(
¬
X43
)
→
X21
→
X47
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X16
)
→
(
¬
X10
)
→
X2
→
X46
)
)
↔
(
X34
↔
(
(
¬
X11
)
→
X10
→
X46
)
)
)
→
(
¬
(
(
¬
(
X34
→
X32
→
False
)
)
→
(
¬
(
X47
↔
(
(
¬
X16
)
→
X42
)
)
)
)
)
→
X34
→
X32
→
X11
→
X42
)
)
)
→
(
(
(
(
(
¬
(
(
(
¬
(
X47
↔
(
(
¬
X39
)
↔
X44
)
)
)
↔
(
¬
(
(
(
¬
X47
)
↔
(
X39
→
X48
)
)
→
(
¬
(
X47
→
(
X44
↔
(
X24
→
False
)
)
)
)
)
)
)
→
(
(
¬
(
X48
→
(
X47
↔
(
X1
→
X44
)
)
)
)
↔
(
¬
(
X44
↔
(
X1
↔
X5
)
)
)
)
)
)
↔
(
¬
(
(
(
X44
↔
(
(
¬
X24
)
→
(
¬
X19
)
→
X36
)
)
↔
(
¬
(
X39
↔
(
(
¬
X24
)
↔
X36
)
)
)
)
→
(
¬
(
X44
↔
(
X39
→
X36
→
False
)
)
)
)
)
)
↔
(
¬
(
X45
→
X39
→
(
¬
X36
)
→
X44
)
)
)
→
(
¬
(
(
(
(
(
¬
(
(
¬
X48
)
↔
(
X47
↔
(
X40
↔
X49
)
)
)
)
↔
(
(
X48
→
(
¬
X45
)
→
X49
)
↔
(
X48
→
X47
→
X45
→
X25
→
False
)
)
)
↔
(
X49
→
X45
→
(
¬
X29
)
→
X48
)
)
↔
(
¬
(
(
¬
X29
)
↔
X40
)
)
)
→
(
(
(
¬
X40
)
→
(
(
¬
X8
)
↔
X49
)
)
↔
(
(
¬
(
X27
→
X15
→
False
)
)
↔
(
X40
↔
(
(
¬
X27
)
→
(
X8
↔
X48
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
X30
)
→
X48
)
↔
(
X43
↔
(
(
¬
X41
)
↔
(
(
¬
X30
)
↔
X47
)
)
)
)
→
(
¬
(
X48
→
(
X0
↔
False
)
)
)
→
(
(
¬
X43
)
→
(
¬
X0
)
→
X48
)
→
(
¬
(
(
¬
(
(
¬
X41
)
↔
(
X17
↔
(
(
¬
X0
)
↔
X43
)
)
)
)
→
(
X43
↔
(
(
¬
X17
)
↔
(
X0
→
X48
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X50
→
(
X49
↔
(
X47
↔
X37
)
)
)
)
→
(
(
(
X50
↔
(
X47
→
(
X41
↔
X22
)
)
)
→
(
X49
↔
(
(
¬
X31
)
↔
(
(
¬
X22
)
↔
X47
)
)
)
)
↔
(
¬
(
X50
↔
(
X22
→
X31
)
)
)
)
)
→
X49
→
(
X37
↔
(
(
¬
X6
)
→
X31
)
)
)
)
↔
(
¬
(
(
X35
→
(
¬
X22
)
→
(
X7
↔
X37
)
)
→
(
¬
(
(
(
¬
(
(
¬
X43
)
→
(
(
¬
X35
)
↔
(
X22
↔
X47
)
)
)
)
↔
(
¬
(
X43
→
(
¬
X37
)
→
X35
→
X47
)
)
)
→
(
(
¬
X43
)
↔
(
(
¬
X35
)
→
X47
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X18
↔
(
(
¬
X8
)
↔
(
X3
↔
X40
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X48
→
(
(
¬
X25
)
↔
(
X8
↔
X49
)
)
)
)
→
X49
→
X40
→
X8
→
X48
)
)
→
(
(
¬
X25
)
↔
(
(
¬
X8
)
→
(
¬
X3
)
→
X48
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
¬
X48
)
→
X50
)
→
(
¬
(
(
X50
↔
(
(
¬
X44
)
→
(
¬
X16
)
→
X48
)
)
→
(
¬
(
(
¬
(
X48
↔
(
(
¬
X44
)
↔
X50
)
)
)
→
(
¬
(
(
¬
(
X48
→
(
X44
↔
(
X1
→
X50
)
)
)
)
→
X50
→
X44
→
(
(
¬
X9
)
↔
X48
)
)
)
)
)
)
)
)
→
X50
→
(
X44
↔
(
X9
↔
False
)
)
)
)
→
(
¬
(
(
¬
X40
)
→
X25
→
X7
→
X48
)
)
→
(
¬
(
(
¬
X44
)
↔
(
(
¬
X40
)
↔
(
X7
↔
X50
)
)
)
)
)
)
)
→
(
(
(
X49
→
(
X28
↔
(
(
¬
X22
)
→
X42
)
)
)
↔
(
(
¬
(
(
¬
X41
)
↔
(
(
¬
X28
)
→
X6
→
X42
)
)
)
↔
(
X34
↔
(
X28
↔
(
X22
→
X6
→
False
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
X49
→
(
X42
↔
(
X38
→
False
)
)
)
)
↔
(
¬
(
X49
→
(
X38
↔
(
(
¬
X13
)
↔
X41
)
)
)
)
)
→
(
¬
(
(
¬
(
X50
→
X42
→
(
X41
↔
X38
)
)
)
→
(
¬
(
(
X41
→
X50
)
→
(
¬
(
(
¬
X41
)
→
(
X19
↔
(
X17
↔
X42
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
X41
→
X23
→
X49
)
)
↔
(
X49
→
X43
→
(
¬
X23
)
→
X50
)
)
)
→
(
¬
(
(
(
X41
→
(
¬
X38
)
→
(
¬
X21
)
→
X48
)
↔
(
¬
(
X43
↔
X41
)
)
)
→
X32
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
¬
(
X50
→
(
(
¬
X39
)
↔
X47
)
)
)
↔
(
(
X50
↔
X39
)
↔
(
¬
(
X39
→
(
X5
↔
(
(
¬
X2
)
↔
X47
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X49
↔
(
X39
↔
(
(
¬
X33
)
→
X37
)
)
)
)
↔
(
(
¬
(
(
¬
(
X50
→
X39
→
X33
→
False
)
)
→
(
¬
(
X50
↔
(
(
¬
X37
)
→
(
(
¬
X33
)
↔
X47
)
)
)
)
)
)
→
(
¬
X37
)
→
(
¬
X33
)
→
X39
)
)
→
(
¬
(
X39
↔
(
X37
↔
(
X11
→
X49
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X49
→
(
X48
↔
(
X46
→
X33
)
)
)
)
→
(
¬
(
X39
↔
(
X36
→
X33
→
False
)
)
)
→
(
(
X46
→
X33
→
X49
)
↔
(
¬
(
(
¬
X39
)
→
(
¬
X36
)
→
X33
→
X46
)
)
)
)
)
↔
(
(
¬
X46
)
→
(
(
(
X14
↔
X20
)
↔
(
¬
(
X46
→
X36
→
(
¬
X20
)
→
X48
)
)
)
↔
(
¬
(
(
¬
X36
)
↔
(
(
¬
X20
)
↔
X46
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
X45
→
X35
→
(
¬
X26
)
→
X29
)
↔
(
¬
(
X45
→
(
(
¬
X29
)
↔
(
X26
→
X47
)
)
)
)
)
→
X29
→
(
X26
↔
False
)
)
↔
(
(
(
X47
→
X45
→
(
¬
X4
)
→
X26
)
↔
(
(
(
¬
X45
)
→
X26
→
(
(
¬
X10
)
↔
X50
)
)
↔
(
¬
(
(
¬
X47
)
↔
(
X26
↔
X50
)
)
)
)
)
↔
(
(
(
X48
→
X26
→
X0
)
→
(
¬
(
(
¬
X47
)
→
(
(
¬
X31
)
↔
(
(
¬
X26
)
↔
X48
)
)
)
)
)
↔
(
X47
→
(
¬
X31
)
→
X50
)
)
)
)
→
(
(
X30
→
False
)
↔
(
(
¬
(
X50
→
X30
→
(
X27
↔
X35
)
)
)
→
X45
→
(
(
¬
X27
)
↔
X30
)
)
)
→
(
¬
(
(
¬
(
X24
↔
(
X15
→
X45
)
)
)
→
(
¬
(
(
¬
X35
)
↔
(
(
¬
X24
)
→
(
¬
X12
)
→
X45
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
c92325...
and proposition id is
b96fcc...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMJzgqbQQZyfZMdDuSMBTqoGii2ZumPczxC
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
¬
(
(
¬
(
(
¬
(
X49
→
(
X22
↔
X45
)
)
)
→
X36
→
X21
→
False
)
)
→
(
(
(
¬
(
(
¬
X22
)
↔
(
X19
→
(
(
¬
X18
)
↔
X30
)
)
)
)
→
(
¬
(
X30
↔
(
X21
↔
(
(
¬
X19
)
↔
X22
)
)
)
)
)
↔
(
¬
(
(
¬
(
X22
↔
(
(
¬
X19
)
↔
(
X2
↔
X21
)
)
)
)
→
X49
→
(
X21
↔
(
X19
↔
X2
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X45
→
(
X32
↔
(
X13
↔
False
)
)
)
)
→
(
(
¬
(
(
¬
X32
)
→
X13
→
(
¬
X5
)
→
X45
)
)
↔
(
X45
↔
X41
)
)
)
)
↔
(
X49
→
(
(
¬
X41
)
↔
(
(
¬
X32
)
→
X45
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X45
)
→
X41
→
(
(
¬
X4
)
↔
X49
)
)
)
→
(
(
¬
(
X36
↔
X45
)
)
↔
(
(
X45
→
(
X36
↔
(
X15
↔
False
)
)
)
↔
(
X41
→
(
¬
X36
)
→
X49
)
)
)
)
)
→
(
(
¬
(
(
¬
X41
)
↔
(
X31
→
(
X23
↔
X45
)
)
)
)
↔
(
¬
(
X41
→
(
X36
↔
(
X31
→
X23
→
False
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
X48
↔
(
X44
↔
(
X9
↔
X38
)
)
)
→
(
¬
X38
)
→
(
X33
↔
X48
)
)
)
↔
(
(
X33
→
(
(
¬
X25
)
↔
X48
)
)
↔
(
(
X48
→
(
¬
X25
)
→
X44
)
↔
(
X33
→
(
(
¬
X25
)
↔
(
(
¬
X3
)
↔
X44
)
)
)
)
)
)
↔
(
(
¬
(
X44
→
X38
→
(
¬
X6
)
→
X33
)
)
→
(
(
¬
X41
)
↔
(
(
¬
X33
)
→
(
(
¬
X6
)
↔
X47
)
)
)
)
)
↔
(
(
(
¬
X48
)
→
(
(
¬
X44
)
↔
(
(
¬
X17
)
→
X49
)
)
)
→
(
X48
↔
X36
)
→
(
¬
(
(
¬
(
X41
→
X44
)
)
→
(
¬
(
X44
↔
(
X36
→
(
¬
X17
)
→
X48
)
)
)
)
)
)
)
)
→
(
(
(
(
(
(
¬
(
X48
↔
(
X45
↔
(
X20
↔
X11
)
)
)
)
↔
(
¬
(
(
¬
(
X48
→
(
(
¬
X45
)
↔
X47
)
)
)
→
(
X45
↔
(
X14
→
X20
)
)
)
)
)
↔
(
(
X48
↔
(
(
¬
X20
)
→
X10
→
X47
)
)
↔
(
¬
(
X47
↔
(
X20
↔
(
(
¬
X10
)
↔
X46
)
)
)
)
)
)
↔
(
(
(
¬
(
X45
↔
(
X37
↔
(
(
¬
X26
)
↔
X46
)
)
)
)
→
(
¬
(
X45
↔
(
(
¬
X37
)
→
(
¬
X24
)
→
X46
)
)
)
)
→
(
(
¬
(
X37
→
X8
→
False
)
)
↔
(
X48
↔
(
X45
→
(
X24
↔
False
)
)
)
)
)
)
→
(
(
X16
↔
(
X1
↔
(
(
¬
X0
)
↔
X26
)
)
)
↔
(
(
¬
(
(
¬
X46
)
↔
(
X26
↔
X47
)
)
)
↔
(
(
X46
→
(
(
¬
X16
)
↔
(
X1
↔
X26
)
)
)
↔
(
X16
→
(
(
¬
X1
)
↔
X47
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
X47
↔
(
X46
↔
(
X43
↔
X39
)
)
)
)
↔
(
¬
(
X46
↔
(
X42
↔
(
X39
→
X47
)
)
)
)
)
→
(
¬
(
X43
→
(
¬
X42
)
→
X29
→
X47
)
)
)
→
(
¬
(
(
¬
X39
)
→
(
X7
↔
X43
)
)
)
)
)
→
(
(
(
X48
→
(
(
¬
X42
)
↔
(
(
¬
X35
)
↔
X43
)
)
)
↔
(
X43
→
(
X40
↔
(
(
¬
X35
)
→
X46
)
)
)
)
↔
(
¬
(
(
X46
→
(
(
¬
X43
)
↔
(
(
¬
X28
)
→
X48
)
)
)
→
(
(
(
X48
↔
(
X46
→
X34
→
False
)
)
→
(
X48
↔
(
(
¬
X27
)
→
(
¬
X12
)
→
X34
)
)
)
↔
(
X48
→
(
X46
↔
X27
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
(
(
¬
(
X45
↔
(
X9
→
X10
)
)
)
↔
(
¬
(
X45
→
X33
→
(
¬
X18
)
→
X44
)
)
)
↔
(
X19
→
(
X18
↔
(
X9
↔
False
)
)
)
)
→
(
¬
(
(
X42
↔
(
X27
→
(
X23
↔
(
X19
↔
False
)
)
)
)
→
X42
→
(
(
¬
X23
)
↔
X29
)
)
)
)
→
(
(
(
¬
X44
)
↔
(
(
¬
X42
)
→
X46
)
)
→
(
¬
(
X41
↔
(
(
¬
X29
)
↔
(
(
¬
X21
)
→
X46
)
)
)
)
→
(
(
(
¬
X42
)
↔
(
(
¬
X41
)
↔
(
X29
↔
X44
)
)
)
↔
(
(
¬
(
X29
→
X42
)
)
↔
(
(
¬
(
(
¬
X42
)
→
(
¬
X29
)
→
X44
)
)
↔
(
¬
(
X44
→
(
X29
↔
(
X24
↔
X21
)
)
)
)
)
)
)
)
→
(
¬
(
X31
↔
False
)
)
→
X46
→
X41
→
(
X36
↔
(
X25
→
False
)
)
)
)
→
(
(
(
¬
(
(
(
(
¬
X46
)
↔
(
X1
↔
X49
)
)
→
(
(
¬
X46
)
↔
(
(
¬
X8
)
→
X48
)
)
)
→
(
(
¬
X48
)
↔
(
(
¬
X46
)
→
(
X30
↔
X49
)
)
)
)
)
→
(
(
¬
(
X49
↔
(
X48
↔
(
X46
→
X17
)
)
)
)
↔
(
¬
(
(
X48
→
(
X46
↔
(
(
¬
X45
)
↔
X49
)
)
)
→
(
X49
↔
(
X46
→
(
X45
↔
X22
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X46
↔
(
X15
→
X37
)
)
)
↔
(
X46
↔
(
(
¬
X16
)
→
(
¬
X15
)
→
X44
)
)
)
→
(
¬
(
X46
↔
(
X44
→
X37
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
X46
↔
(
(
¬
X20
)
→
X3
→
X43
)
)
→
(
¬
(
X49
→
(
X46
↔
(
X20
→
X43
)
)
)
)
)
)
→
(
¬
(
X20
→
(
¬
X3
)
→
X37
)
)
)
)
→
(
(
(
¬
(
X48
↔
(
X43
↔
(
X37
↔
False
)
)
)
)
→
(
¬
(
(
¬
X37
)
↔
(
(
¬
X16
)
↔
X46
)
)
)
)
↔
(
¬
(
X37
→
(
(
¬
X16
)
↔
X43
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
X38
↔
(
(
¬
X11
)
↔
X46
)
)
)
↔
(
(
X38
→
(
(
¬
X34
)
↔
(
(
¬
X11
)
↔
X46
)
)
)
↔
(
(
¬
X38
)
→
X47
)
)
)
↔
(
X38
→
X11
)
)
↔
(
(
(
X46
↔
(
X14
↔
(
(
¬
X0
)
↔
X34
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X14
)
↔
X47
)
)
→
(
X47
↔
(
(
¬
X46
)
↔
(
X0
→
X49
)
)
)
)
)
)
→
(
¬
X34
)
→
X14
→
(
X2
↔
X49
)
)
)
→
(
(
(
(
¬
(
(
¬
X28
)
↔
(
X26
→
(
X5
↔
X40
)
)
)
)
↔
(
¬
(
(
¬
X40
)
↔
(
X5
↔
X49
)
)
)
)
↔
(
¬
(
X40
↔
(
(
¬
X7
)
↔
(
X5
→
X35
)
)
)
)
)
↔
(
(
X47
→
(
X40
↔
(
X28
↔
(
X4
→
False
)
)
)
)
↔
(
X46
→
(
¬
X4
)
→
X5
)
)
)
)
↔
(
(
(
(
¬
X46
)
↔
(
(
¬
X32
)
↔
(
X6
↔
X47
)
)
)
↔
(
(
¬
(
X49
→
(
¬
X32
)
→
(
¬
X13
)
→
X47
)
)
↔
(
X49
↔
(
X32
↔
(
X13
↔
X44
)
)
)
)
)
↔
(
(
(
¬
(
X46
→
(
X44
↔
(
X12
→
X39
)
)
)
)
→
(
¬
(
X49
↔
(
X46
→
(
X12
↔
False
)
)
)
)
)
↔
(
¬
(
(
¬
X44
)
↔
(
(
¬
X39
)
→
X46
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
f2ba3e...
and proposition id is
3f5d95...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMVpiWDMPeDhJ1o6Az7EcGs7Yat7Y6ngRt4
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
∀X50 :
prop
,
(
(
(
(
(
(
(
¬
(
X49
→
(
X44
↔
X47
)
)
)
↔
(
(
(
X49
→
X44
)
↔
(
(
¬
X47
)
↔
(
(
¬
X39
)
↔
(
(
¬
X11
)
↔
X49
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X26
)
↔
(
(
¬
X11
)
↔
X39
)
)
)
→
(
¬
(
X50
↔
(
(
¬
X26
)
→
X49
)
)
)
)
)
)
)
→
(
¬
(
(
¬
X39
)
↔
(
(
¬
X12
)
↔
X49
)
)
)
→
(
X39
↔
(
X12
→
(
¬
X2
)
→
X50
)
)
)
→
(
¬
(
(
(
¬
(
(
(
¬
(
X48
→
X45
→
X27
→
False
)
)
↔
(
¬
(
X48
↔
(
(
¬
X47
)
↔
(
X45
↔
X49
)
)
)
)
)
→
X48
→
(
(
¬
X47
)
↔
X49
)
)
)
→
(
X47
↔
(
(
¬
X45
)
↔
X48
)
)
)
→
(
¬
(
(
(
¬
(
(
(
¬
X48
)
→
(
X43
↔
(
X10
↔
X49
)
)
)
→
(
¬
(
X50
→
(
X49
↔
(
X43
↔
X10
)
)
)
)
)
)
↔
(
X50
→
(
(
¬
X48
)
↔
(
(
¬
X46
)
↔
X49
)
)
)
)
→
(
(
¬
(
X47
→
X25
→
X31
)
)
↔
(
¬
(
(
¬
(
X50
↔
(
X47
→
X25
→
X46
)
)
)
→
(
¬
(
X48
↔
(
X47
→
X46
→
False
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X49
↔
(
(
¬
X23
)
→
X37
)
)
)
↔
(
(
¬
(
X44
→
(
X43
↔
(
(
¬
X37
)
→
X49
)
)
)
)
↔
(
¬
(
(
¬
X44
)
→
(
(
¬
X43
)
↔
(
(
¬
X35
)
→
X49
)
)
)
)
)
)
↔
(
(
(
¬
(
X43
↔
(
X41
↔
(
X37
↔
X44
)
)
)
)
↔
(
¬
(
X41
→
(
¬
X37
)
→
(
X15
↔
X43
)
)
)
)
↔
(
(
¬
(
(
¬
X13
)
→
X44
)
)
→
(
¬
(
X43
→
X41
→
(
¬
X30
)
→
X44
)
)
)
)
)
↔
(
(
X49
↔
(
X38
→
X35
→
(
X30
↔
False
)
)
)
→
(
X49
↔
(
X30
→
X1
→
X28
)
)
→
(
¬
(
X49
↔
(
X35
↔
(
X30
↔
(
X28
↔
False
)
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
X49
→
(
X40
↔
(
X36
→
X5
)
)
)
)
↔
(
(
(
(
X40
↔
(
(
¬
X36
)
↔
X49
)
)
↔
(
(
(
¬
X36
)
→
(
X8
↔
X49
)
)
↔
(
¬
(
(
X40
↔
(
X8
→
X18
)
)
→
(
¬
(
(
¬
X40
)
→
(
(
¬
X36
)
↔
(
(
¬
X18
)
↔
X49
)
)
)
)
)
)
)
)
→
X36
→
X8
)
↔
(
¬
(
(
¬
X40
)
↔
(
(
¬
X36
)
→
X33
→
X49
)
)
)
)
)
↔
(
¬
(
(
¬
(
X40
→
(
X36
↔
X50
)
)
)
→
(
(
¬
(
X50
→
X36
→
(
¬
X33
)
→
X40
)
)
↔
(
¬
(
X40
→
X36
→
(
X33
↔
(
X17
→
False
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X49
↔
(
X16
↔
X24
)
)
)
→
(
¬
(
X40
↔
(
X16
↔
X34
)
)
)
)
↔
(
(
¬
X21
)
↔
X34
)
)
→
(
¬
(
(
¬
(
(
¬
X29
)
↔
(
(
¬
X24
)
↔
X49
)
)
)
→
(
X36
↔
(
X29
↔
(
X3
→
X20
)
)
)
→
(
(
(
¬
(
X49
↔
(
X3
↔
False
)
)
)
→
X36
→
(
X24
↔
(
X3
→
X20
)
)
)
→
(
X36
↔
(
X20
↔
False
)
)
)
→
X36
→
(
¬
X20
)
→
X24
)
)
)
)
)
↔
(
(
(
(
(
¬
X42
)
↔
X48
)
↔
(
(
(
¬
(
(
¬
X22
)
↔
X48
)
)
↔
(
(
X48
↔
(
X22
↔
(
X14
→
X4
→
False
)
)
)
→
(
X42
↔
(
X22
→
(
¬
X14
)
→
X48
)
)
)
)
↔
(
¬
(
X22
→
X14
→
X42
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X14
)
↔
(
X7
→
(
¬
X6
)
→
X36
)
)
)
↔
(
¬
(
X48
→
X14
→
X36
)
)
)
→
X48
→
X36
→
(
X22
↔
X14
)
)
)
)
↔
(
(
(
(
(
(
¬
X49
)
→
(
X19
↔
(
(
¬
X9
)
↔
X50
)
)
)
→
(
¬
(
X50
↔
(
X19
→
X36
)
)
)
)
↔
(
X50
↔
(
X49
↔
(
X42
↔
X19
)
)
)
)
→
X48
→
X42
→
X0
→
X36
)
→
(
¬
(
X50
↔
(
X36
→
(
¬
X32
)
→
X48
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
(
(
(
X48
→
(
¬
X36
)
→
X27
→
X39
)
↔
(
X48
↔
X49
)
)
↔
(
(
¬
X27
)
↔
X48
)
)
↔
(
(
¬
(
X48
↔
(
X36
↔
(
X19
↔
(
X1
↔
False
)
)
)
)
)
→
(
X48
↔
(
(
¬
X1
)
→
X39
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X47
)
→
(
X39
↔
(
X36
→
X48
)
)
)
)
→
(
(
(
¬
X28
)
↔
X32
)
↔
(
¬
(
X48
↔
(
(
¬
X36
)
→
X39
)
)
)
)
)
)
)
↔
(
(
(
¬
X47
)
↔
(
(
¬
X39
)
→
(
¬
X14
)
→
X48
)
)
↔
(
(
¬
(
(
¬
X14
)
→
X10
→
(
¬
X5
)
→
X48
)
)
→
(
(
X39
→
X14
→
X10
→
False
)
↔
(
¬
(
X33
↔
(
(
¬
X10
)
↔
X48
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X46
→
(
(
¬
X35
)
↔
(
(
¬
X7
)
↔
X47
)
)
)
)
→
(
¬
(
(
¬
(
X49
↔
(
X47
→
X7
→
X3
)
)
)
→
(
(
¬
(
X49
↔
(
(
¬
X23
)
↔
(
X21
↔
X47
)
)
)
)
↔
(
¬
(
X47
→
(
X35
↔
X49
)
)
)
)
)
)
)
)
→
X49
→
(
X48
↔
(
X46
→
(
X23
↔
False
)
)
)
)
)
→
(
¬
(
(
(
X48
→
(
¬
X38
)
→
(
X17
↔
X47
)
)
↔
(
X48
↔
(
X40
↔
(
(
¬
X17
)
→
X47
)
)
)
)
→
(
¬
(
(
(
¬
(
X49
↔
(
X48
↔
(
X40
→
False
)
)
)
)
↔
(
(
¬
(
X49
→
(
¬
X30
)
→
(
X15
↔
X48
)
)
)
↔
(
¬
(
X30
→
X40
)
)
)
)
→
(
¬
(
(
X48
→
(
(
¬
X47
)
↔
X50
)
)
→
(
(
¬
(
X50
→
X49
→
X47
)
)
↔
(
X48
↔
(
(
¬
X47
)
→
(
(
¬
X40
)
↔
X50
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
(
¬
X31
)
↔
(
(
¬
X24
)
→
(
(
¬
X8
)
↔
X37
)
)
)
→
(
¬
(
X42
→
X37
→
X6
→
X24
)
)
)
)
→
(
(
(
(
¬
(
(
¬
X22
)
→
(
(
¬
X13
)
↔
X37
)
)
)
↔
(
¬
(
X41
→
(
X37
↔
(
X31
→
X13
→
False
)
)
)
)
)
↔
(
(
¬
X41
)
↔
(
(
¬
X13
)
↔
X42
)
)
)
↔
(
¬
(
(
¬
X37
)
↔
(
X31
→
(
X16
↔
X41
)
)
)
)
)
)
↔
(
¬
(
(
X48
→
X42
→
(
¬
X20
)
→
X41
)
→
(
(
(
X48
↔
(
X42
↔
(
X41
↔
False
)
)
)
↔
(
¬
(
(
¬
X41
)
→
X37
→
(
X12
↔
X42
)
)
)
)
↔
(
¬
(
(
¬
X37
)
→
(
X12
↔
X47
)
)
)
)
)
)
)
↔
(
(
(
(
(
X49
→
X45
→
(
¬
X0
)
→
X48
)
↔
(
¬
(
X45
↔
(
(
¬
X25
)
→
X47
)
)
)
)
→
(
¬
(
X49
→
(
X48
↔
(
X45
↔
X11
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X45
↔
(
X34
↔
(
X18
→
False
)
)
)
)
→
(
¬
(
X49
↔
(
X34
→
(
¬
X18
)
→
X29
)
)
)
)
)
↔
(
(
X45
→
X41
→
X34
→
X49
)
→
(
¬
(
X49
↔
(
(
¬
X34
)
↔
(
(
¬
X25
)
→
X45
)
)
)
)
)
)
)
↔
(
¬
(
(
X48
↔
(
X26
↔
(
X9
↔
X2
)
)
)
→
(
(
(
¬
(
X45
↔
(
(
¬
X44
)
↔
(
(
¬
X43
)
↔
X48
)
)
)
)
↔
(
¬
(
X48
↔
(
X45
→
(
(
¬
X9
)
↔
X44
)
)
)
)
)
↔
(
¬
(
(
X47
→
(
(
¬
X44
)
↔
(
X2
↔
X45
)
)
)
→
(
(
¬
(
(
X47
↔
(
X44
↔
(
X43
↔
(
X4
↔
False
)
)
)
)
→
(
¬
(
X47
↔
(
X45
→
(
X4
↔
X44
)
)
)
)
)
)
↔
(
X47
↔
(
X44
→
X43
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
d36622...
and proposition id is
2a998d...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMcq2cft4Rvt3F3P1Ep6mvAgDUaiyJRwACP
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
∀X50 :
prop
,
(
(
¬
(
(
(
¬
(
(
(
(
(
¬
(
X47
↔
(
X45
→
X50
)
)
)
→
(
¬
(
(
¬
X45
)
↔
(
(
¬
X12
)
→
X9
→
X47
)
)
)
)
↔
(
(
¬
X45
)
↔
X50
)
)
→
(
X47
↔
(
X9
↔
(
X0
↔
X38
)
)
)
→
(
¬
X47
)
→
X9
→
(
X0
↔
X50
)
)
→
(
¬
(
(
X49
→
(
X45
↔
(
(
¬
X3
)
↔
X26
)
)
)
→
(
(
X50
↔
(
X47
→
(
X45
↔
(
X41
↔
False
)
)
)
)
↔
(
(
¬
X47
)
→
(
X41
↔
(
(
¬
X40
)
→
X50
)
)
)
)
→
(
¬
(
X3
→
X40
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
(
X45
→
(
X44
↔
(
X31
↔
X13
)
)
)
→
(
X45
↔
(
X31
↔
(
X28
→
X44
)
)
)
)
→
(
¬
(
X44
→
(
X42
↔
(
X31
↔
X28
)
)
)
)
)
)
→
(
¬
(
X45
↔
(
(
¬
X31
)
→
(
X19
↔
X44
)
)
)
)
)
)
→
(
(
¬
(
X28
↔
(
(
¬
X23
)
↔
(
X19
↔
X27
)
)
)
)
↔
(
(
¬
X27
)
→
(
¬
X23
)
→
(
¬
X19
)
→
X42
)
)
)
→
(
(
(
(
¬
(
(
(
¬
(
X46
→
(
¬
X42
)
→
X47
)
)
↔
(
(
(
(
¬
(
(
¬
X45
)
→
X42
→
X47
)
)
↔
(
X45
→
(
(
¬
X34
)
↔
X46
)
)
)
→
(
¬
(
X42
→
X34
→
X46
)
)
)
↔
(
¬
(
X46
→
(
¬
X34
)
→
(
X14
↔
X42
)
)
)
)
)
→
(
¬
(
X46
→
X44
→
(
(
¬
X14
)
↔
X47
)
)
)
)
)
↔
(
X46
→
X45
→
(
X34
↔
False
)
)
)
↔
(
¬
(
(
¬
(
(
X45
↔
(
(
¬
X42
)
↔
(
X29
↔
X44
)
)
)
→
(
¬
(
X29
→
X16
)
)
)
)
→
(
¬
(
(
¬
X16
)
↔
(
X15
↔
(
(
¬
X8
)
→
X45
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
X44
)
→
(
¬
X29
)
→
(
(
¬
X6
)
↔
X49
)
)
)
↔
(
(
¬
X16
)
→
X10
→
X6
→
X29
)
)
)
)
→
(
(
¬
(
(
(
X49
→
X39
→
(
¬
X36
)
→
X50
)
→
(
¬
(
X49
→
(
X43
↔
False
)
)
)
)
→
(
(
(
(
X24
→
False
)
→
(
X49
→
(
X43
↔
(
X37
↔
X24
)
)
)
→
(
¬
(
X49
→
X43
→
(
X24
↔
X2
)
)
)
)
→
(
(
¬
(
(
X43
↔
(
X24
→
(
X20
↔
False
)
)
)
→
(
¬
X43
)
→
X24
→
(
X4
↔
X49
)
)
)
↔
(
¬
(
X43
→
False
)
)
)
)
↔
(
(
¬
(
(
¬
X37
)
↔
(
(
¬
X20
)
→
X39
)
)
)
↔
(
¬
(
X43
→
(
X39
↔
(
X7
→
False
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
X47
→
X17
→
(
X1
↔
X33
)
)
)
→
(
¬
(
(
X47
→
X17
→
X1
→
X22
)
→
(
¬
(
(
X48
↔
(
(
¬
X22
)
→
(
X17
↔
X47
)
)
)
→
X48
→
X22
→
(
¬
X18
)
→
X47
)
)
)
)
)
)
↔
(
(
(
¬
(
X48
↔
(
X32
↔
(
X11
↔
X49
)
)
)
)
↔
(
X49
→
(
X33
↔
X48
)
)
)
↔
(
(
(
X47
→
(
¬
X33
)
→
(
X21
↔
X49
)
)
↔
(
X49
↔
(
X33
→
(
X21
↔
X48
)
)
)
)
↔
(
¬
(
X49
↔
(
X21
↔
X48
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
(
¬
X33
)
→
(
¬
X32
)
→
(
(
¬
X5
)
↔
X37
)
)
↔
(
¬
(
(
¬
(
X47
↔
(
X37
→
X33
)
)
)
→
X37
→
X32
→
X30
)
)
)
→
(
¬
(
X47
→
(
X37
↔
(
X30
→
X43
)
)
)
)
)
)
→
(
(
¬
X33
)
↔
(
(
¬
X30
)
→
(
X25
↔
X35
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
(
X47
→
X23
→
(
X8
↔
X36
)
)
→
X47
→
(
X38
↔
(
X23
→
X43
)
)
)
)
↔
(
(
(
(
X43
→
X36
→
X7
→
X42
)
↔
(
¬
(
(
¬
X43
)
↔
(
X42
→
(
(
¬
X36
)
↔
X44
)
)
)
)
)
→
(
(
¬
X43
)
↔
(
(
¬
X36
)
↔
(
(
¬
X35
)
→
X44
)
)
)
)
↔
(
(
¬
(
X43
→
(
(
¬
X35
)
↔
(
X10
↔
X44
)
)
)
)
↔
(
X47
→
(
¬
X10
)
→
X35
)
)
)
)
→
(
(
(
X44
↔
(
X39
→
X35
)
)
↔
(
(
¬
(
X39
↔
(
X28
↔
(
X23
↔
X44
)
)
)
)
↔
(
¬
(
X44
→
(
X35
↔
X28
)
)
)
)
)
↔
(
(
¬
(
(
¬
X39
)
↔
(
(
¬
X35
)
↔
(
X2
→
X44
)
)
)
)
↔
(
¬
(
X44
→
(
¬
X35
)
→
(
(
¬
X5
)
↔
X39
)
)
)
)
)
)
)
→
(
¬
(
(
(
X39
→
X37
→
(
X29
↔
False
)
)
→
(
¬
(
(
(
¬
(
X37
→
X19
→
(
X9
↔
X39
)
)
)
↔
(
¬
(
(
¬
X39
)
→
(
(
¬
X37
)
↔
(
X19
→
X46
)
)
)
)
)
→
(
(
¬
(
X39
→
(
¬
X37
)
→
X12
→
X46
)
)
↔
(
¬
(
X37
↔
(
X29
↔
(
X12
↔
X39
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X47
↔
(
X17
→
X12
→
False
)
)
)
→
X46
→
(
(
¬
X12
)
↔
(
(
¬
X9
)
↔
X17
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
(
(
(
¬
X25
)
↔
(
X15
→
(
(
¬
X13
)
↔
X49
)
)
)
→
(
¬
(
(
¬
X25
)
↔
(
(
¬
X13
)
→
X11
→
X49
)
)
)
)
)
→
(
(
¬
(
(
¬
X15
)
↔
X47
)
)
↔
(
X49
→
X37
→
X11
→
False
)
)
→
(
(
¬
X37
)
↔
(
X15
→
X49
)
)
)
)
↔
(
¬
(
(
X47
→
X33
→
(
¬
X3
)
→
X39
)
→
(
¬
(
(
¬
(
X47
→
(
X33
↔
X39
)
)
)
→
(
¬
(
(
¬
X33
)
↔
(
(
¬
X15
)
↔
X49
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X15
)
→
X39
)
)
)
)
)
)
)
)
→
(
(
(
¬
X42
)
→
(
X21
↔
X49
)
)
↔
(
(
¬
(
X49
→
(
X33
↔
(
X24
→
X21
)
)
)
)
→
(
¬
(
X37
↔
(
X6
→
X21
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X42
→
X41
→
(
X34
↔
X45
)
)
)
→
X44
→
X1
→
False
)
)
→
(
(
¬
(
X34
→
X32
→
(
(
¬
X22
)
↔
X43
)
)
)
↔
(
¬
(
(
¬
(
X42
→
X32
→
X31
→
False
)
)
→
(
(
¬
(
(
¬
(
X32
→
X31
→
X48
)
)
→
(
(
X32
→
(
(
¬
X31
)
↔
X42
)
)
↔
(
(
(
¬
X34
)
→
(
¬
X32
)
→
(
¬
X31
)
→
X48
)
→
(
¬
(
(
¬
X42
)
↔
(
X32
→
X31
→
X48
)
)
)
)
)
)
)
↔
(
X34
→
(
X32
↔
(
X31
↔
X22
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
X42
↔
(
X32
→
(
¬
X26
)
→
X45
)
)
→
(
(
¬
(
(
¬
X31
)
→
(
X14
↔
X42
)
)
)
↔
(
X45
↔
(
X32
↔
(
X31
↔
(
X14
↔
False
)
)
)
)
)
)
→
(
¬
(
X45
→
X32
→
X26
→
False
)
)
)
)
↔
(
(
(
(
X45
→
(
X42
↔
X31
)
)
↔
(
¬
(
(
¬
(
X45
↔
(
X16
↔
X31
)
)
)
→
(
(
(
¬
X31
)
↔
(
X16
→
(
(
¬
X4
)
↔
X42
)
)
)
↔
(
X45
→
(
X42
↔
(
X31
→
X4
)
)
)
)
)
)
)
↔
(
X31
↔
(
(
¬
X16
)
↔
X45
)
)
)
↔
(
X42
→
(
¬
X16
)
→
X31
)
)
)
)
)
→
(
(
(
¬
(
X48
↔
(
(
¬
X45
)
↔
(
X27
↔
X50
)
)
)
)
↔
(
¬
(
X50
→
X48
→
X27
→
X40
)
)
)
↔
(
(
(
¬
X45
)
→
(
X43
↔
X50
)
)
→
(
(
(
(
¬
X45
)
→
X43
→
(
X18
↔
X50
)
)
↔
(
(
¬
X45
)
↔
(
X43
↔
(
X0
→
X50
)
)
)
)
↔
(
¬
(
X43
→
(
(
¬
X18
)
↔
(
(
¬
X0
)
→
X40
)
)
)
)
)
→
(
¬
(
(
¬
(
X43
→
(
(
¬
X30
)
↔
(
X20
→
X50
)
)
)
)
→
(
¬
(
X50
↔
(
(
¬
X40
)
↔
(
X20
→
X43
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
835aab...
and proposition id is
219178...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMGK6kkuMHJT63KxzLTe62RxVFV9PPdRV5K
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
(
¬
(
X19
↔
(
(
¬
X8
)
↔
X47
)
)
)
↔
(
(
¬
(
X47
↔
(
X25
↔
X21
)
)
)
→
(
X47
↔
(
X46
↔
(
X25
↔
(
X8
↔
False
)
)
)
)
)
)
↔
(
(
¬
(
X43
→
X47
)
)
↔
(
X25
↔
(
X21
→
X12
→
X1
)
)
)
)
→
(
(
(
(
¬
(
X47
↔
(
X46
↔
(
X15
↔
X41
)
)
)
)
↔
(
¬
(
X46
↔
(
X37
↔
(
(
¬
X15
)
→
X49
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
X47
↔
(
X44
↔
(
(
¬
X41
)
↔
X46
)
)
)
→
(
X47
↔
(
X46
→
(
X44
↔
(
X24
↔
False
)
)
)
)
)
)
→
X46
→
X44
→
(
(
¬
X26
)
↔
X41
)
)
)
)
↔
(
¬
(
(
X44
↔
(
X41
→
(
(
¬
X40
)
↔
X47
)
)
)
→
(
(
(
(
¬
X47
)
↔
(
X41
→
(
(
¬
X30
)
↔
X48
)
)
)
↔
(
(
(
X44
↔
(
X30
→
(
(
¬
X20
)
↔
X48
)
)
)
→
(
¬
X41
)
→
(
X30
↔
(
X20
↔
X48
)
)
)
↔
(
¬
(
(
¬
X20
)
→
X30
)
)
)
)
↔
(
(
¬
(
X48
→
(
(
¬
X41
)
↔
(
X32
→
X43
)
)
)
)
↔
(
¬
(
(
(
¬
X43
)
→
(
¬
X27
)
→
X48
)
→
(
X44
↔
(
X43
→
X41
)
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
(
(
(
¬
X42
)
→
(
X36
↔
(
(
¬
X35
)
→
X46
)
)
)
↔
(
¬
(
(
¬
X42
)
→
X36
→
(
(
¬
X17
)
↔
X46
)
)
)
)
→
(
¬
(
X46
→
X36
→
(
(
¬
X13
)
↔
X42
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
X46
→
X38
→
(
(
¬
X28
)
↔
X35
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X16
)
→
(
(
¬
X3
)
↔
X38
)
)
)
)
)
→
(
(
¬
X38
)
↔
(
X35
↔
X46
)
)
)
)
↔
(
¬
(
(
X42
→
X38
→
(
(
¬
X35
)
↔
X46
)
)
→
X42
→
(
X38
↔
(
X34
→
(
X7
↔
False
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X2
↔
X42
)
)
→
(
X38
↔
(
(
¬
X28
)
↔
X49
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
X38
)
→
X36
→
X29
→
X42
)
↔
(
X38
→
X36
→
(
(
¬
X22
)
↔
X33
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X5
)
→
X9
)
)
→
(
¬
(
(
¬
X22
)
↔
(
(
¬
X9
)
↔
(
X5
↔
X36
)
)
)
)
→
X47
→
X36
→
X22
→
X9
)
→
(
(
¬
(
X47
↔
(
(
¬
X33
)
→
X22
→
X36
)
)
)
↔
(
¬
(
X47
→
(
¬
X22
)
→
(
(
¬
X6
)
↔
X33
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
¬
X45
)
→
X39
→
(
¬
X31
)
→
X46
)
)
↔
(
X47
→
X45
→
(
X4
↔
X39
)
)
)
→
(
(
¬
X45
)
↔
(
X11
→
(
¬
X0
)
→
X47
)
)
)
↔
(
(
¬
(
X47
→
(
X46
↔
(
X23
→
(
X18
↔
False
)
)
)
)
)
↔
(
X47
→
(
(
¬
X31
)
↔
(
X18
↔
X46
)
)
)
)
)
→
(
(
¬
(
X45
↔
(
(
¬
X10
)
→
X46
)
)
)
↔
(
¬
(
(
X46
↔
(
X45
↔
(
(
¬
X10
)
→
X42
)
)
)
→
(
(
X45
→
(
¬
X42
)
→
X46
)
↔
(
¬
(
X45
→
(
¬
X14
)
→
X47
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
(
X45
→
X43
→
X46
)
↔
(
(
¬
(
(
¬
X43
)
↔
(
X38
↔
(
(
¬
X16
)
↔
X46
)
)
)
)
↔
(
¬
(
(
X45
↔
(
X43
↔
(
X38
↔
X46
)
)
)
→
(
¬
(
(
¬
X45
)
↔
(
X43
↔
(
X9
→
X46
)
)
)
)
)
)
)
)
↔
(
(
¬
(
X49
→
(
X46
↔
(
(
¬
X18
)
→
X47
)
)
)
)
→
(
¬
(
(
X47
↔
(
X46
↔
(
X9
↔
X18
)
)
)
→
(
¬
(
X37
↔
(
X18
→
X9
→
False
)
)
)
)
)
)
)
→
(
(
¬
(
X49
↔
(
X45
↔
(
X34
↔
X10
)
)
)
)
↔
(
(
¬
(
X46
→
(
X45
↔
(
X1
→
X20
)
)
)
)
↔
(
¬
(
(
(
X49
→
(
¬
X14
)
→
X46
)
↔
(
(
¬
(
(
¬
X14
)
→
(
X2
↔
X46
)
)
)
↔
(
¬
(
(
¬
X46
)
↔
(
(
¬
X22
)
↔
(
X2
→
X49
)
)
)
)
)
)
→
(
(
(
¬
(
X46
↔
(
(
¬
X14
)
→
X4
→
X49
)
)
)
→
(
X46
↔
(
X4
↔
(
X1
→
False
)
)
)
)
↔
(
¬
(
X46
↔
(
X34
→
(
X4
↔
False
)
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
(
(
¬
(
(
X38
→
(
X36
↔
(
X27
↔
X49
)
)
)
→
(
X49
↔
(
X42
→
X36
→
X38
)
)
)
)
↔
(
¬
(
X49
→
X38
→
X42
)
)
)
↔
(
X42
↔
(
X38
↔
(
(
¬
X7
)
↔
X49
)
)
)
)
→
(
¬
(
(
¬
X27
)
↔
(
X24
→
X49
)
)
)
)
)
→
(
X49
↔
(
(
¬
X42
)
→
X36
→
X45
)
)
)
)
→
(
¬
(
(
(
(
(
X21
→
X3
)
↔
(
¬
(
X45
→
(
X42
↔
(
X6
→
X21
)
)
)
)
)
↔
(
¬
(
X28
↔
(
(
¬
X21
)
→
(
X6
↔
X42
)
)
)
)
)
↔
(
X45
↔
(
X42
↔
(
X28
↔
X19
)
)
)
)
→
(
(
¬
(
X49
→
(
(
¬
X21
)
↔
(
(
¬
X5
)
→
X42
)
)
)
)
↔
(
X49
↔
(
X42
→
(
¬
X19
)
→
X21
)
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
(
(
(
(
¬
X39
)
→
(
¬
X25
)
→
X48
)
↔
(
X48
↔
(
(
¬
X43
)
→
X49
)
)
)
↔
(
¬
(
X39
↔
X48
)
)
)
→
(
¬
(
(
¬
(
(
¬
X43
)
→
(
(
¬
X33
)
↔
(
X29
→
X48
)
)
)
)
→
(
(
¬
(
(
X49
→
X43
→
X29
→
X17
→
False
)
→
(
¬
(
(
¬
X48
)
→
X29
→
X49
)
)
)
)
↔
(
X43
↔
(
X29
→
X49
)
)
)
)
)
)
→
(
¬
(
(
X48
↔
(
(
¬
X33
)
→
X41
)
)
→
(
¬
(
(
X43
→
X26
→
False
)
→
(
¬
(
(
¬
X43
)
→
X41
→
(
X33
↔
X48
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
X31
↔
(
(
¬
X26
)
↔
(
(
¬
X0
)
↔
X40
)
)
)
→
(
(
(
¬
(
(
¬
X25
)
↔
(
X11
↔
X31
)
)
)
→
(
¬
(
(
X26
→
X25
→
X11
)
→
X31
→
(
X26
↔
(
X11
→
X17
)
)
)
)
)
↔
(
(
X40
→
(
X26
↔
(
X11
↔
X25
)
)
)
→
(
X45
↔
(
X40
↔
(
X26
→
X11
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X44
)
↔
X49
)
)
↔
(
(
¬
X30
)
↔
(
X23
→
X13
→
X44
)
)
)
→
(
(
(
¬
(
(
¬
X13
)
↔
(
X8
↔
X48
)
)
)
↔
(
(
¬
(
(
¬
X35
)
↔
(
X32
→
(
X8
↔
X48
)
)
)
)
→
(
(
¬
X32
)
↔
(
X13
→
X48
)
)
)
)
→
(
(
¬
(
X35
↔
(
X32
→
(
¬
X12
)
→
X23
)
)
)
↔
(
¬
(
X35
↔
(
X23
→
(
(
¬
X15
)
↔
X48
)
)
)
)
)
)
→
(
(
X32
→
X30
)
↔
(
(
X45
↔
(
X35
→
X30
→
X32
)
)
→
(
X35
↔
(
X30
↔
(
X23
↔
X45
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
b9f43b...
and proposition id is
22594a...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMbgCbd6azn3vbB1y3jWL3pQU3a7dpjaybA
)
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
(
(
(
(
¬
X39
)
→
(
¬
X20
)
→
(
¬
X4
)
→
X48
)
→
(
¬
(
X41
→
X32
→
X20
→
X4
→
False
)
)
)
↔
(
X37
→
(
X32
↔
X41
)
)
)
→
(
¬
(
X41
↔
(
X32
↔
(
X20
→
X22
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X48
→
X46
→
X10
→
X3
→
False
)
)
↔
(
¬
(
(
(
¬
X32
)
→
(
X30
↔
X48
)
)
→
(
(
¬
(
X48
↔
(
X10
→
X30
)
)
)
↔
(
X46
↔
(
(
¬
X10
)
↔
X32
)
)
)
)
)
)
→
(
(
X48
→
(
X47
↔
(
X12
↔
X46
)
)
)
→
(
¬
(
X48
→
(
X47
↔
(
(
¬
X8
)
→
X46
)
)
)
)
)
→
(
(
(
(
X47
↔
(
X32
↔
(
(
¬
X11
)
→
X46
)
)
)
↔
(
¬
(
X46
→
(
¬
X32
)
→
(
(
¬
X12
)
↔
X47
)
)
)
)
↔
(
X32
↔
(
X11
↔
X24
)
)
)
↔
(
X46
→
(
(
¬
X32
)
↔
(
(
¬
X12
)
→
X37
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
X35
)
↔
(
X18
↔
X37
)
)
)
→
(
(
(
(
X48
→
(
¬
X13
)
→
X42
)
↔
(
(
¬
(
X45
→
X42
→
(
X15
↔
X48
)
)
)
↔
(
(
¬
(
X45
→
X15
→
X13
→
X5
)
)
↔
(
¬
(
X48
↔
(
X45
↔
(
(
¬
X13
)
→
X15
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
X42
→
(
¬
X31
)
→
(
(
¬
X2
)
↔
X37
)
)
→
(
(
(
(
¬
X31
)
→
X18
→
(
¬
X1
)
→
X42
)
→
(
X37
↔
False
)
)
↔
(
¬
(
X37
→
(
(
¬
X31
)
↔
X42
)
)
)
)
)
→
(
¬
(
X42
→
(
(
¬
X37
)
↔
(
X2
↔
X39
)
)
)
)
)
→
(
X48
→
(
X37
↔
X42
)
)
→
(
X48
↔
(
X45
↔
(
(
¬
X18
)
→
X35
)
)
)
)
)
)
↔
(
X48
→
(
¬
X42
)
→
(
¬
X29
)
→
X45
)
)
)
)
→
(
(
¬
(
(
(
(
(
(
¬
(
X49
→
(
X40
↔
(
X25
→
False
)
)
)
)
↔
(
¬
(
X47
→
(
(
¬
X25
)
↔
(
X7
→
X49
)
)
)
)
)
→
(
¬
(
(
¬
X47
)
↔
(
X40
↔
(
(
¬
X34
)
→
X49
)
)
)
)
)
↔
(
X34
→
(
X28
↔
(
X25
→
(
X6
↔
False
)
)
)
)
)
↔
(
(
X49
↔
(
X36
→
(
¬
X26
)
→
X40
)
)
↔
(
¬
(
(
X36
↔
(
(
¬
X26
)
→
(
(
¬
X14
)
↔
X47
)
)
)
→
(
(
¬
X40
)
↔
(
(
¬
X36
)
↔
(
(
¬
X26
)
→
X47
)
)
)
)
)
)
)
→
(
(
X43
↔
(
X34
→
(
X19
↔
False
)
)
)
↔
(
¬
(
X28
↔
(
X25
↔
X19
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X44
)
→
(
¬
X9
)
→
X46
)
)
↔
(
¬
(
X47
↔
(
(
¬
X44
)
→
(
X21
↔
X46
)
)
)
)
)
↔
(
(
(
¬
(
(
X42
→
(
X38
↔
(
X0
→
X49
)
)
)
→
(
¬
(
X49
↔
(
X46
→
(
(
¬
X0
)
↔
X27
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
X44
↔
(
(
¬
X42
)
↔
(
X16
→
X43
)
)
)
→
(
X47
↔
(
X43
↔
(
X42
→
X16
→
False
)
)
)
)
)
→
(
¬
(
X47
→
(
(
¬
X44
)
↔
(
(
¬
X16
)
→
X49
)
)
)
)
)
)
↔
(
¬
(
X44
→
(
X43
↔
(
X42
↔
X46
)
)
)
)
)
)
↔
(
(
¬
(
X46
↔
(
X33
→
X17
→
X49
)
)
)
→
(
(
¬
(
(
¬
X44
)
↔
(
(
¬
X33
)
→
X23
→
X46
)
)
)
↔
(
X44
→
(
X23
↔
X17
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
(
¬
X40
)
→
X28
→
(
X3
↔
X42
)
)
↔
(
(
¬
(
X41
→
(
X40
↔
X28
)
)
)
↔
(
X41
→
(
¬
X26
)
→
(
(
¬
X3
)
↔
X40
)
)
)
)
→
(
(
(
X41
↔
(
X32
↔
(
(
¬
X3
)
↔
X46
)
)
)
↔
(
X41
↔
(
(
¬
X26
)
→
(
(
¬
X3
)
↔
X32
)
)
)
)
↔
(
¬
(
(
¬
X26
)
→
(
¬
X3
)
→
X46
)
)
)
)
→
(
(
(
(
¬
(
X47
↔
(
X44
↔
X42
)
)
)
↔
(
(
(
¬
(
X47
→
X44
→
X42
→
X7
→
False
)
)
↔
(
¬
(
(
¬
X7
)
↔
X12
)
)
)
→
(
¬
X44
)
→
(
¬
X42
)
→
X47
)
)
→
(
(
(
¬
X45
)
↔
(
X7
↔
(
X4
↔
X47
)
)
)
↔
(
(
¬
X7
)
↔
(
X4
↔
X42
)
)
)
)
↔
(
(
(
¬
(
X46
→
X45
→
X10
)
)
→
(
X41
↔
(
(
¬
X36
)
↔
(
(
¬
X23
)
↔
X42
)
)
)
)
↔
(
(
X42
↔
(
X24
→
X23
→
X47
)
)
↔
(
¬
(
(
¬
X42
)
→
(
(
¬
X23
)
↔
(
X22
↔
X46
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
(
(
(
(
X46
↔
(
X45
→
False
)
)
→
(
(
X46
→
(
(
¬
X33
)
↔
(
(
¬
X21
)
→
X47
)
)
)
↔
(
(
¬
X45
)
→
X33
→
X21
→
X46
)
)
)
→
(
X47
↔
(
X45
↔
X33
)
)
)
↔
(
(
¬
(
X46
→
(
¬
X33
)
→
X15
→
X45
)
)
↔
(
X33
↔
(
(
¬
X25
)
→
(
(
¬
X15
)
↔
X46
)
)
)
)
)
↔
(
X46
↔
(
(
¬
X45
)
↔
(
(
¬
X31
)
→
X47
)
)
)
)
↔
(
(
(
X46
→
(
(
¬
X5
)
↔
X45
)
)
↔
(
X45
↔
(
X38
→
(
¬
X13
)
→
X46
)
)
)
↔
(
¬
(
X38
→
(
X13
↔
X47
)
)
)
)
)
→
(
(
(
¬
(
X49
→
X47
)
)
→
(
¬
(
(
X47
→
X45
→
(
X38
↔
X49
)
)
→
(
X47
↔
(
(
¬
X38
)
→
(
(
¬
X29
)
↔
X49
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
¬
X38
)
↔
(
X20
↔
(
(
¬
X17
)
↔
X47
)
)
)
)
↔
(
(
X35
↔
(
(
¬
X20
)
→
(
(
¬
X17
)
↔
X38
)
)
)
↔
(
¬
(
X45
↔
(
X38
→
(
¬
X20
)
→
X35
)
)
)
)
)
↔
(
(
¬
X35
)
→
X20
→
X47
)
)
→
(
X20
↔
(
(
¬
X17
)
→
X30
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
(
¬
X16
)
→
(
(
¬
X2
)
↔
X34
)
)
↔
(
¬
(
(
(
X47
↔
(
X34
→
X16
→
False
)
)
→
(
(
(
X16
→
(
(
¬
X2
)
↔
(
(
¬
X1
)
↔
X34
)
)
)
↔
(
(
X6
→
(
X2
↔
(
X1
→
X16
)
)
)
→
(
¬
(
X16
↔
(
X6
→
X2
→
X47
)
)
)
)
)
↔
(
X34
→
X6
→
X47
)
)
)
→
(
X47
↔
(
X6
→
X16
)
)
)
)
)
→
(
¬
(
X48
→
X47
)
)
→
(
¬
(
X48
↔
(
(
¬
X11
)
→
(
X6
↔
X16
)
)
)
)
)
)
→
(
(
(
(
X27
→
(
(
¬
X11
)
↔
(
(
¬
X6
)
→
X47
)
)
)
→
(
¬
(
X43
↔
(
X27
↔
(
(
¬
X6
)
→
X11
)
)
)
)
)
↔
(
(
¬
(
X27
→
(
¬
X11
)
→
(
X0
↔
X39
)
)
)
↔
(
¬
(
(
¬
X11
)
→
(
(
¬
X9
)
↔
(
(
¬
X0
)
↔
X47
)
)
)
)
)
)
↔
(
(
¬
X43
)
↔
(
(
¬
X38
)
↔
(
(
¬
X27
)
→
X47
)
)
)
)
)
)
→
(
¬
(
(
X46
→
(
X43
↔
(
X19
↔
X39
)
)
)
→
(
¬
(
(
(
X46
→
(
¬
X38
)
→
(
(
¬
X8
)
↔
X39
)
)
↔
(
(
¬
(
X38
↔
(
(
¬
X14
)
→
X8
→
X18
)
)
)
↔
(
(
¬
X18
)
→
(
X14
↔
X39
)
)
)
)
→
X39
→
X18
→
X43
)
)
)
)
→
(
¬
(
(
¬
X38
)
→
(
¬
X37
)
→
X19
→
X39
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
d9ccc8...
and proposition id is
fd9c05...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMXvpFseuzWavXgTCNZjk7KjwYgeZ2bUCfd
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
¬
(
(
(
¬
(
(
(
¬
(
X26
→
(
¬
X25
)
→
X33
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X42
)
→
(
(
¬
X26
)
↔
(
X25
↔
X45
)
)
)
)
→
(
(
¬
(
(
¬
X37
)
→
(
(
¬
X33
)
↔
(
X25
→
X42
)
)
)
)
↔
(
¬
(
(
(
X33
↔
(
X25
→
X12
→
X26
)
)
→
(
¬
(
(
¬
X25
)
↔
X42
)
)
)
→
(
X42
↔
(
X26
↔
X33
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
X42
)
↔
(
(
¬
X37
)
↔
X45
)
)
)
)
)
→
X45
→
X26
→
X42
)
)
↔
(
(
(
(
¬
X9
)
↔
X40
)
↔
(
(
(
X37
→
X36
→
False
)
↔
(
¬
(
(
(
¬
X40
)
↔
(
(
¬
X37
)
↔
(
(
¬
X36
)
→
X42
)
)
)
→
(
(
¬
X40
)
↔
(
X37
→
(
X9
↔
X42
)
)
)
)
)
)
↔
(
(
¬
X40
)
→
(
(
¬
X9
)
↔
(
(
¬
X5
)
→
X42
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X34
)
→
(
(
¬
X31
)
↔
X42
)
)
)
↔
(
¬
(
(
(
X42
→
(
¬
X37
)
→
(
X34
↔
X40
)
)
↔
(
X42
→
X40
→
X37
→
X31
)
)
→
(
¬
(
X40
↔
(
X34
↔
X37
)
)
)
)
)
)
→
(
¬
(
X42
↔
(
X40
→
(
X8
↔
X26
)
)
)
)
→
(
¬
(
X34
→
(
X8
↔
(
X0
↔
X42
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
X47
↔
(
X40
→
X36
→
X49
)
)
)
→
(
(
¬
(
X45
→
(
¬
X36
)
→
X21
→
X47
)
)
↔
(
(
X45
↔
(
(
¬
X36
)
↔
(
(
¬
X17
)
→
X40
)
)
)
→
(
¬
(
X45
→
(
X36
↔
(
X17
↔
X21
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X47
→
(
(
¬
X45
)
↔
(
X19
→
X46
)
)
)
)
→
(
¬
(
(
X46
→
X45
→
(
X19
↔
False
)
)
→
(
(
¬
(
X46
↔
(
X45
↔
(
X23
↔
(
X7
→
False
)
)
)
)
)
↔
(
¬
(
X23
↔
(
X19
→
(
¬
X7
)
→
X45
)
)
)
)
)
)
)
↔
(
(
¬
X47
)
↔
(
(
¬
X46
)
↔
(
X45
↔
X49
)
)
)
)
↔
(
(
(
¬
X23
)
→
(
X22
↔
X29
)
)
↔
(
(
¬
(
X49
→
X22
→
False
)
)
↔
(
¬
(
(
(
X29
→
(
¬
X22
)
→
X46
)
→
(
¬
X46
)
→
(
¬
X29
)
→
(
X23
↔
X49
)
)
→
(
¬
(
(
¬
X46
)
→
(
(
¬
X29
)
↔
X49
)
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X48
↔
(
X44
→
(
X39
↔
X27
)
)
)
)
↔
(
(
(
¬
X20
)
→
(
(
¬
X15
)
↔
(
(
¬
X10
)
→
X48
)
)
)
→
(
(
(
X44
→
(
X32
↔
(
(
¬
X20
)
↔
X41
)
)
)
→
(
¬
X44
)
→
(
(
¬
X41
)
↔
X48
)
)
↔
(
¬
(
(
X48
↔
(
X27
→
(
(
¬
X13
)
↔
X41
)
)
)
→
(
¬
(
X44
↔
(
X41
→
X27
)
)
)
)
)
)
)
)
↔
(
(
¬
(
X46
→
X41
→
X11
→
X27
)
)
→
(
X27
↔
(
X15
→
X49
)
)
→
(
¬
(
X49
↔
(
X46
↔
(
(
¬
X15
)
→
X41
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
(
(
¬
(
(
X48
↔
(
(
¬
X18
)
→
(
¬
X14
)
→
X46
)
)
→
(
¬
(
X16
→
X38
)
)
)
)
→
(
¬
(
(
¬
X38
)
→
X18
→
(
(
¬
X6
)
↔
X48
)
)
)
)
)
↔
(
(
¬
(
X46
↔
(
X38
↔
(
(
¬
X35
)
→
X48
)
)
)
)
↔
(
¬
(
X48
↔
(
X46
→
X38
→
(
X3
↔
False
)
)
)
)
)
)
→
(
(
¬
(
(
¬
X44
)
↔
(
(
¬
X38
)
↔
(
(
¬
X2
)
↔
X45
)
)
)
)
↔
(
(
¬
(
(
X44
↔
(
X38
↔
(
(
¬
X35
)
→
X45
)
)
)
→
(
X45
↔
(
X44
↔
(
(
¬
X24
)
→
X35
)
)
)
)
)
↔
(
¬
(
X45
→
X44
→
(
¬
X18
)
→
X35
)
)
)
)
)
)
→
(
(
(
(
¬
(
X48
↔
(
(
¬
X30
)
→
X43
)
)
)
↔
(
¬
(
X28
→
(
X1
↔
X43
)
)
)
)
↔
(
¬
(
(
¬
X44
)
↔
(
(
¬
X43
)
→
(
X28
↔
X48
)
)
)
)
)
↔
(
(
X46
→
(
(
¬
X4
)
↔
X30
)
)
→
(
¬
(
X46
↔
(
X44
→
(
X43
↔
(
X30
↔
False
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
¬
(
X45
↔
(
X34
→
(
¬
X28
)
→
X41
)
)
)
→
(
¬
(
(
¬
X41
)
→
(
X34
↔
(
(
¬
X18
)
↔
X45
)
)
)
)
)
)
→
(
(
¬
(
X45
↔
(
X39
→
(
¬
X34
)
→
X42
)
)
)
↔
(
X47
↔
(
X39
→
X34
)
)
)
→
(
¬
(
X42
→
(
X39
↔
(
(
¬
X18
)
→
X45
)
)
)
)
)
→
(
¬
(
(
X42
↔
(
(
¬
X33
)
→
(
¬
X24
)
→
X44
)
)
→
(
(
(
(
X41
↔
(
X33
→
(
X24
↔
X42
)
)
)
→
(
¬
(
(
¬
X41
)
→
(
X33
↔
(
(
¬
X24
)
↔
X45
)
)
)
)
)
↔
(
¬
(
(
¬
X33
)
↔
(
(
¬
X24
)
↔
X45
)
)
)
)
↔
(
¬
(
(
X45
↔
(
X41
→
(
X24
↔
X46
)
)
)
→
(
¬
(
X41
↔
(
X24
↔
X19
)
)
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
(
¬
X23
)
→
X14
→
X46
)
)
↔
(
(
¬
(
(
¬
X23
)
↔
(
(
¬
X14
)
↔
(
X13
↔
X44
)
)
)
)
↔
(
X46
↔
(
X14
→
(
¬
X13
)
→
X23
)
)
)
)
↔
(
¬
(
X44
→
(
X23
↔
(
X17
→
X13
)
)
)
)
)
↔
(
X46
→
X23
→
X14
)
)
↔
(
(
(
¬
(
(
¬
X41
)
↔
(
X4
↔
(
(
¬
X1
)
→
X46
)
)
)
)
↔
(
(
¬
X44
)
↔
(
X41
→
(
X4
↔
X46
)
)
)
)
→
(
(
¬
(
X44
↔
(
X23
→
X8
→
X15
)
)
)
↔
(
¬
(
X46
→
(
(
¬
X41
)
↔
(
X15
→
X44
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X44
→
X37
→
X26
→
(
X22
↔
False
)
)
)
→
(
¬
(
(
(
X46
→
(
X44
↔
(
(
¬
X3
)
→
X26
)
)
)
↔
(
¬
(
X46
→
X44
→
(
¬
X26
)
→
X41
)
)
)
→
(
(
X46
→
(
X44
↔
(
X41
↔
(
X22
→
False
)
)
)
)
↔
(
X44
↔
(
X35
↔
(
X22
→
X41
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
X5
→
X12
)
)
→
(
¬
(
X31
↔
(
X30
↔
X44
)
)
)
)
↔
(
(
¬
(
X36
↔
(
(
¬
X30
)
↔
X44
)
)
)
↔
(
(
¬
(
(
¬
X30
)
↔
(
X11
→
X36
)
)
)
↔
(
¬
(
X36
→
(
X31
↔
X44
)
)
)
)
)
)
↔
(
(
(
(
X31
↔
(
X10
↔
False
)
)
→
(
(
¬
(
X48
↔
(
X25
→
(
X10
↔
False
)
)
)
)
↔
(
(
X25
↔
(
(
¬
X10
)
→
X31
)
)
↔
(
(
¬
X36
)
↔
(
(
¬
X25
)
↔
(
(
¬
X10
)
→
X48
)
)
)
)
)
)
↔
(
(
¬
X36
)
↔
(
(
¬
X31
)
↔
(
(
¬
X30
)
→
X48
)
)
)
)
↔
(
¬
(
(
¬
X48
)
→
(
(
¬
X30
)
↔
(
(
¬
X16
)
→
X49
)
)
)
)
)
)
→
(
(
(
(
(
X44
↔
(
(
¬
X38
)
→
(
(
¬
X20
)
↔
X43
)
)
)
↔
(
X45
↔
(
(
¬
X38
)
→
(
X20
↔
X44
)
)
)
)
↔
(
¬
(
X45
→
(
X44
↔
(
X21
→
(
X7
↔
False
)
)
)
)
)
)
→
(
(
¬
(
X40
→
X29
→
X48
)
)
↔
(
¬
(
(
(
X48
→
(
X45
↔
(
X40
→
X44
)
)
)
→
(
¬
(
X48
↔
(
X45
↔
(
X44
↔
(
X0
→
False
)
)
)
)
)
)
→
(
¬
(
X40
↔
(
X0
→
X44
)
)
)
)
)
)
→
(
(
¬
(
X45
→
X40
→
(
X20
↔
(
X9
→
False
)
)
)
)
↔
(
¬
(
(
¬
X20
)
→
X9
→
X48
)
)
)
)
↔
(
(
(
X49
↔
(
(
¬
X43
)
→
X47
)
)
↔
(
¬
(
X47
↔
(
X43
↔
(
X27
→
X2
)
)
)
)
)
→
(
¬
(
X49
→
X47
→
X45
→
X6
)
)
→
(
X47
→
(
X45
↔
(
X43
↔
X32
)
)
)
→
(
(
¬
X45
)
↔
(
(
¬
X43
)
↔
(
X40
↔
X49
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
548c03...
and proposition id is
447cab...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMa53jPSitjX9G68BryuLf1rWoQy5dYcnP8
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
¬
(
(
(
(
(
¬
(
(
(
(
X48
↔
(
(
¬
X34
)
→
(
X7
↔
X47
)
)
)
→
(
¬
(
X49
→
(
X48
↔
(
(
¬
X34
)
↔
X47
)
)
)
)
)
↔
(
¬
(
(
¬
X48
)
→
(
X47
↔
(
X14
→
X49
)
)
)
)
)
→
(
¬
(
(
(
X47
↔
(
X44
→
(
(
¬
X4
)
↔
X48
)
)
)
→
(
¬
(
X48
↔
(
X47
→
(
X44
↔
X24
)
)
)
)
)
→
(
¬
(
X44
↔
(
(
¬
X24
)
↔
X49
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X49
→
(
X44
↔
(
(
¬
X12
)
→
X39
)
)
)
)
↔
(
¬
(
(
¬
X48
)
→
(
(
¬
X44
)
↔
X49
)
)
)
)
→
(
¬
X44
)
→
X48
)
→
(
(
¬
X44
)
↔
(
X39
↔
(
(
¬
X2
)
↔
X48
)
)
)
)
)
)
↔
(
(
(
X47
→
X31
→
X13
→
False
)
→
X47
→
(
X19
↔
(
(
¬
X10
)
→
X13
)
)
)
↔
(
(
(
¬
(
(
X47
→
X31
→
(
X12
↔
False
)
)
→
(
¬
(
X47
↔
(
X12
↔
(
(
¬
X0
)
→
X49
)
)
)
)
→
(
X47
↔
(
X12
→
(
¬
X0
)
→
X31
)
)
)
)
→
(
¬
(
X49
↔
(
X31
→
X12
)
)
)
)
↔
(
(
(
X49
↔
(
X31
→
(
¬
X23
)
→
X47
)
)
→
X47
→
(
X31
↔
(
X13
↔
X49
)
)
)
↔
(
¬
(
X47
→
X31
→
(
X22
↔
X23
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X48
→
X42
→
(
X35
↔
(
X17
↔
False
)
)
)
)
→
(
X21
↔
(
X6
↔
X35
)
)
)
→
(
¬
(
(
(
¬
X47
)
→
(
(
¬
X42
)
↔
(
(
¬
X29
)
↔
X48
)
)
)
→
(
X47
↔
(
(
¬
X46
)
↔
(
(
¬
X35
)
→
X48
)
)
)
)
)
)
↔
(
(
X35
→
X17
→
X28
)
↔
(
(
¬
(
(
¬
(
(
¬
(
X49
↔
(
X28
→
(
X17
↔
X38
)
)
)
)
→
(
¬
(
X35
→
(
¬
X28
)
→
X17
→
X38
)
)
)
)
→
(
¬
(
(
¬
X35
)
↔
(
(
¬
X28
)
→
X49
)
)
)
)
)
↔
(
¬
(
X49
→
(
X17
↔
(
X8
→
False
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
(
¬
(
(
¬
X45
)
→
(
(
¬
X26
)
↔
X49
)
)
)
→
(
¬
(
X45
↔
(
(
¬
X32
)
→
X48
)
)
)
)
)
→
(
¬
(
(
(
X48
→
(
X20
↔
X49
)
)
→
(
X49
↔
(
X32
↔
X30
)
)
)
→
(
¬
(
X41
↔
(
(
¬
X30
)
↔
(
X20
↔
X48
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X45
→
(
¬
X32
)
→
(
X1
↔
X47
)
)
)
→
(
(
(
¬
X45
)
↔
(
(
¬
X32
)
↔
(
(
¬
X27
)
→
X48
)
)
)
↔
(
(
X45
↔
(
(
¬
X11
)
↔
(
X1
↔
X37
)
)
)
↔
(
(
¬
(
X45
↔
(
X27
↔
(
X1
→
X32
)
)
)
)
↔
(
¬
(
X45
↔
(
X32
↔
(
X27
→
X11
→
False
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
(
(
(
¬
X43
)
↔
(
(
¬
X33
)
→
X49
)
)
↔
(
¬
(
(
¬
X33
)
↔
(
(
¬
X5
)
→
X41
)
)
)
)
→
(
¬
X36
)
→
(
X5
↔
X41
)
)
)
↔
(
(
(
(
(
¬
(
X43
↔
(
(
¬
X36
)
↔
(
(
¬
X15
)
→
X49
)
)
)
)
→
X36
→
X16
→
(
X15
↔
X43
)
)
→
(
¬
(
X43
↔
False
)
)
)
→
(
X36
↔
(
(
¬
X16
)
→
(
¬
X9
)
→
X43
)
)
)
↔
(
X43
↔
(
X36
→
X16
→
False
)
)
)
)
↔
(
(
¬
(
X48
↔
(
X43
↔
(
X25
→
X41
)
)
)
)
↔
X49
)
)
→
(
(
¬
(
X40
↔
(
(
¬
X3
)
↔
X41
)
)
)
↔
(
(
¬
X48
)
→
X40
→
(
¬
X18
)
→
X49
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
(
¬
X26
)
→
(
(
¬
X20
)
↔
X46
)
)
)
→
(
¬
(
X44
↔
(
X33
↔
X26
)
)
)
)
↔
(
(
¬
(
X40
→
(
¬
X33
)
→
(
(
¬
X20
)
↔
X37
)
)
)
↔
(
¬
(
(
¬
X40
)
↔
(
X37
→
X33
→
X44
)
)
)
)
)
→
(
(
(
(
(
¬
X37
)
↔
(
(
¬
X16
)
→
(
(
¬
X14
)
↔
X48
)
)
)
→
(
¬
X42
)
→
(
X14
↔
X48
)
)
↔
(
(
(
X48
↔
(
X18
→
X15
→
X26
)
)
→
(
(
X48
→
(
X42
↔
(
X18
↔
X26
)
)
)
↔
(
¬
(
X48
↔
(
X37
→
X18
)
)
)
)
)
↔
(
¬
(
(
¬
(
X48
→
X6
→
X42
)
)
→
(
X42
↔
(
X37
→
X2
→
X20
)
)
)
)
)
)
↔
(
X21
↔
(
(
¬
X20
)
→
(
(
¬
X18
)
↔
X26
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X49
→
(
¬
X40
)
→
X43
)
)
→
(
(
¬
(
(
¬
(
X43
↔
(
X13
→
X1
→
X40
)
)
)
→
(
(
(
X49
→
X40
→
X13
)
→
X40
→
(
(
¬
X1
)
↔
X12
)
)
↔
(
¬
(
X49
→
(
X1
↔
X40
)
)
)
)
)
)
↔
(
¬
(
(
¬
X43
)
↔
(
(
¬
X13
)
→
X49
)
)
)
)
)
)
→
(
¬
(
X36
→
X13
→
X12
→
X25
)
)
)
→
(
(
¬
(
X49
↔
(
X29
→
X8
→
X35
)
)
)
→
X49
→
(
X29
↔
(
X8
→
X24
)
)
)
→
(
¬
(
(
¬
(
(
X43
→
X27
→
(
¬
X24
)
→
X49
)
→
(
X43
↔
(
(
¬
X29
)
↔
(
X24
↔
X49
)
)
)
)
)
→
(
(
¬
(
(
(
¬
X40
)
↔
(
(
¬
X33
)
↔
(
(
¬
X21
)
→
X43
)
)
)
→
(
¬
(
X40
↔
(
(
¬
X33
)
↔
(
X27
→
X43
)
)
)
)
)
)
↔
(
¬
(
X40
→
X21
→
X43
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
X45
↔
(
X30
↔
(
X4
↔
X41
)
)
)
↔
(
¬
(
(
(
¬
X45
)
→
X34
→
X46
)
→
(
¬
(
X46
↔
(
X30
→
(
(
¬
X4
)
↔
X45
)
)
)
)
)
)
)
↔
(
¬
(
(
(
X41
↔
(
(
¬
X32
)
↔
(
X23
→
X34
)
)
)
→
(
¬
(
X45
→
(
X34
↔
False
)
)
)
)
→
(
¬
(
X34
→
(
¬
X23
)
→
X9
→
X41
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
X32
)
→
X28
→
(
¬
X0
)
→
X34
)
)
→
X41
→
(
X34
↔
(
X32
↔
(
X0
→
False
)
)
)
)
)
↔
(
(
X41
↔
(
X34
↔
(
(
¬
X31
)
↔
X32
)
)
)
→
(
X47
↔
(
(
¬
X32
)
→
(
(
¬
X31
)
↔
X41
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
(
X45
↔
(
(
¬
X40
)
↔
(
(
¬
X11
)
→
X43
)
)
)
)
↔
(
¬
(
(
(
¬
(
X40
→
(
X17
↔
X22
)
)
)
→
(
(
X45
↔
(
(
¬
X40
)
→
(
X22
↔
X43
)
)
)
↔
(
¬
(
X45
→
(
¬
X22
)
→
(
¬
X17
)
→
X40
)
)
)
)
→
(
¬
X43
)
→
(
(
¬
X22
)
↔
X45
)
)
)
)
↔
(
X43
→
X17
→
X40
)
)
↔
(
¬
(
(
X11
→
X46
)
→
(
¬
(
X45
→
X41
→
(
X40
↔
X19
)
)
)
)
)
)
→
(
(
(
X46
↔
(
X7
↔
False
)
)
↔
(
¬
(
X41
↔
(
X39
→
(
(
¬
X10
)
↔
X22
)
)
)
)
)
↔
(
(
(
¬
X3
)
→
X39
)
↔
(
(
¬
(
X46
→
X41
→
X5
)
)
↔
(
(
X46
→
(
(
¬
X5
)
↔
X38
)
)
↔
(
(
X41
→
(
X5
↔
X46
)
)
↔
(
X38
→
(
¬
X5
)
→
X3
→
X46
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
7707f7...
and proposition id is
573aff...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMWffBBM1j9TLch6VcQWbZsiS34WRuLKcu2
)
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
¬
(
(
(
(
¬
(
(
(
(
(
¬
(
X49
→
(
X42
↔
(
X6
↔
X4
)
)
)
)
↔
(
X49
→
X45
→
(
X42
↔
False
)
)
)
→
(
X49
↔
(
(
¬
X4
)
↔
X42
)
)
)
→
(
(
¬
(
(
¬
(
X46
→
X45
→
(
X30
↔
(
X6
→
False
)
)
)
)
→
(
¬
(
X45
→
X37
→
(
X30
↔
False
)
)
)
)
)
↔
(
X46
→
X45
→
(
X30
↔
(
X4
↔
False
)
)
)
)
)
→
(
¬
(
(
X49
↔
(
X45
→
X41
→
X8
)
)
→
(
(
¬
X45
)
↔
(
X41
↔
X46
)
)
)
)
)
)
↔
(
(
(
¬
(
X46
→
X3
→
False
)
)
↔
(
(
X28
→
(
X3
↔
X20
)
)
↔
(
X46
↔
(
X40
↔
(
(
¬
X20
)
→
X28
)
)
)
)
)
↔
(
(
(
X43
↔
(
(
¬
X3
)
↔
X28
)
)
↔
(
(
¬
(
X43
→
(
X29
↔
(
X28
↔
False
)
)
)
)
↔
(
(
¬
(
X46
→
(
X43
↔
(
(
¬
X29
)
→
X49
)
)
)
)
→
X46
→
(
(
¬
X29
)
↔
(
(
¬
X28
)
↔
X49
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
X45
↔
(
(
¬
X28
)
↔
X49
)
)
→
(
¬
(
(
X49
↔
(
(
¬
X28
)
↔
(
X18
→
X34
)
)
)
→
(
¬
(
(
¬
X34
)
↔
(
X28
↔
X45
)
)
)
)
)
)
)
→
(
¬
(
(
¬
X34
)
↔
(
X28
↔
(
(
¬
X18
)
↔
X46
)
)
)
)
)
)
↔
(
¬
(
X34
→
(
(
¬
X28
)
↔
X45
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X45
→
X15
→
X11
)
)
→
(
(
(
¬
(
(
X15
↔
(
(
¬
X11
)
↔
X47
)
)
→
(
¬
(
X47
↔
(
X43
→
X11
→
X45
)
)
)
)
)
↔
(
¬
(
(
¬
X19
)
→
X11
→
X45
)
)
)
↔
(
¬
(
X47
→
(
X19
↔
(
(
¬
X0
)
↔
X43
)
)
)
)
)
)
)
→
(
¬
(
(
X45
→
X41
→
(
X19
↔
False
)
)
→
(
¬
(
(
¬
(
(
¬
(
X43
↔
(
(
¬
X41
)
↔
(
(
¬
X1
)
→
X45
)
)
)
)
→
(
(
(
¬
(
(
¬
X41
)
→
X39
→
X19
→
X45
)
)
↔
(
¬
(
(
¬
(
(
¬
X43
)
→
(
(
¬
X41
)
↔
(
X19
↔
X45
)
)
)
)
→
X45
→
(
(
¬
X41
)
↔
(
(
¬
X5
)
↔
X43
)
)
)
)
)
↔
(
X41
→
(
¬
X39
)
→
X45
)
)
)
)
→
(
¬
(
X43
→
(
X41
↔
(
X19
↔
False
)
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
(
X38
→
(
(
¬
X33
)
↔
(
(
¬
X13
)
→
X41
)
)
)
↔
(
X40
↔
(
X38
↔
(
(
¬
X33
)
↔
X41
)
)
)
)
↔
(
X41
↔
X33
)
)
↔
(
¬
(
(
¬
(
X38
↔
(
X26
→
X40
)
)
)
→
(
(
X40
→
(
X27
↔
(
(
¬
X12
)
→
X23
)
)
)
↔
(
¬
(
X41
→
X40
→
X27
→
X23
)
)
)
→
(
¬
(
X48
→
X26
→
X23
→
False
)
)
)
)
)
↔
(
(
(
¬
(
X48
↔
(
X44
↔
(
(
¬
X35
)
→
X40
)
)
)
)
↔
(
(
(
X48
→
X46
→
(
(
¬
X35
)
↔
X36
)
)
↔
(
¬
(
(
¬
X46
)
→
(
¬
X40
)
→
X36
→
X48
)
)
)
→
X48
→
X40
→
X36
→
False
)
)
→
(
¬
(
(
X32
→
X21
→
X44
)
→
(
(
¬
(
X44
→
X21
→
X7
→
X31
)
)
↔
(
X32
→
(
¬
X21
)
→
X7
→
X31
)
)
)
)
→
(
(
(
X46
→
(
¬
X44
)
→
(
(
¬
X41
)
↔
X48
)
)
→
(
¬
(
X48
↔
(
(
¬
X16
)
→
X44
)
)
)
)
↔
(
(
¬
(
X48
→
(
X21
↔
X44
)
)
)
↔
(
(
¬
X46
)
↔
(
X32
→
X21
→
X48
)
)
)
)
)
)
→
(
(
¬
(
X25
→
X22
→
(
X10
↔
X41
)
)
)
↔
(
(
(
X46
→
(
(
¬
X24
)
↔
(
(
¬
X22
)
→
X41
)
)
)
↔
(
¬
(
(
¬
(
(
(
X46
↔
(
(
¬
X25
)
↔
(
X22
↔
X41
)
)
)
→
(
¬
(
X46
→
(
(
¬
X25
)
↔
(
X9
→
X41
)
)
)
)
)
→
(
¬
(
(
¬
X41
)
→
(
(
¬
X25
)
↔
X46
)
)
)
)
)
→
(
X41
↔
(
X9
→
False
)
)
)
)
)
↔
(
X41
→
X38
→
(
(
¬
X22
)
↔
X46
)
)
)
)
→
(
(
¬
(
(
¬
X25
)
↔
(
(
¬
X17
)
↔
(
X2
→
X41
)
)
)
)
↔
(
¬
(
(
(
X38
↔
(
X25
→
(
X17
↔
X41
)
)
)
↔
(
X41
↔
(
X25
↔
(
X17
↔
(
X14
→
False
)
)
)
)
)
→
(
¬
(
X41
↔
(
X38
→
X14
→
False
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X31
↔
(
X22
→
(
¬
X11
)
→
X49
)
)
)
↔
(
(
(
(
¬
(
(
¬
X41
)
↔
(
(
¬
X31
)
→
(
(
¬
X22
)
↔
X49
)
)
)
)
→
X41
→
(
(
¬
X22
)
↔
(
X21
↔
X31
)
)
)
↔
(
¬
(
X46
↔
X12
)
)
)
↔
(
(
(
¬
X22
)
→
(
X12
↔
(
X2
→
X46
)
)
)
→
(
¬
(
X12
↔
(
(
¬
X2
)
→
X9
)
)
)
)
)
)
↔
(
(
(
X49
→
(
X42
↔
X45
)
)
↔
(
(
X42
↔
(
(
¬
X22
)
↔
(
(
¬
X4
)
→
X45
)
)
)
↔
(
(
¬
(
X49
↔
(
X42
↔
(
(
¬
X22
)
→
X45
)
)
)
)
↔
(
¬
(
X42
↔
(
(
¬
X22
)
↔
(
(
¬
X14
)
→
X49
)
)
)
)
)
)
)
↔
(
¬
(
X46
→
X45
→
(
¬
X4
)
→
X42
)
)
)
)
→
(
¬
(
(
(
(
¬
X46
)
↔
(
(
¬
X44
)
↔
X49
)
)
↔
(
(
¬
(
(
X49
→
(
X46
↔
(
(
¬
X35
)
↔
X44
)
)
)
→
(
X49
↔
(
(
¬
X44
)
↔
(
X36
↔
X46
)
)
)
)
)
→
X44
→
X35
→
X36
)
)
→
(
(
(
(
¬
(
X44
↔
(
(
¬
X35
)
→
(
(
¬
X26
)
↔
X45
)
)
)
)
↔
(
(
¬
X45
)
↔
(
(
¬
X44
)
↔
(
(
¬
X35
)
→
X46
)
)
)
)
↔
(
X45
→
X35
→
X26
)
)
↔
(
(
¬
X45
)
→
(
X44
↔
(
(
¬
X16
)
↔
X46
)
)
)
)
)
)
)
→
(
(
(
(
X43
↔
(
X38
↔
(
(
¬
X32
)
↔
X42
)
)
)
↔
(
¬
(
(
(
X43
→
(
¬
X38
)
→
(
(
¬
X30
)
↔
X46
)
)
↔
(
¬
(
(
¬
X39
)
↔
(
(
¬
X38
)
→
(
(
¬
X10
)
↔
X43
)
)
)
)
)
→
(
¬
(
(
¬
(
X46
→
(
(
¬
X42
)
↔
(
(
¬
X10
)
↔
X43
)
)
)
)
→
(
X3
↔
X46
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
(
¬
X29
)
→
X0
→
X40
)
)
↔
(
¬
(
X48
↔
(
X37
→
X29
)
)
)
)
↔
(
X48
↔
(
X37
→
(
(
¬
X29
)
↔
X32
)
)
)
)
↔
(
¬
(
(
(
X48
↔
(
X40
↔
X37
)
)
→
(
(
X37
→
(
X8
↔
X28
)
)
↔
(
¬
(
X40
↔
(
X28
→
(
¬
X8
)
→
X48
)
)
)
)
)
→
(
X28
↔
(
X8
→
(
X7
↔
X37
)
)
)
)
)
)
→
(
¬
(
(
¬
X32
)
→
X48
)
)
→
(
(
X48
↔
(
X38
↔
(
(
¬
X30
)
↔
X40
)
)
)
↔
(
¬
(
X48
↔
(
X40
↔
(
X38
↔
(
X20
↔
False
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
X46
→
(
X42
↔
(
X41
→
X17
→
False
)
)
)
→
(
¬
(
(
¬
(
X42
↔
(
(
¬
X23
)
↔
(
(
¬
X15
)
→
X41
)
)
)
)
→
(
¬
(
X46
↔
(
X41
→
X23
)
)
)
)
)
)
)
↔
(
(
¬
(
X6
→
X47
)
)
↔
(
¬
(
X41
→
X34
→
X15
→
(
X6
↔
False
)
)
)
)
)
→
(
(
(
(
(
¬
(
(
¬
X18
)
↔
(
(
¬
X5
)
→
X24
)
)
)
↔
(
(
(
¬
X25
)
↔
(
(
¬
X24
)
→
X46
)
)
↔
(
(
(
(
¬
X33
)
↔
(
(
¬
X24
)
→
X5
→
X46
)
)
↔
(
X25
→
X46
)
)
↔
(
(
¬
X33
)
↔
(
(
¬
X24
)
→
X18
→
X46
)
)
)
)
)
↔
(
X33
↔
(
X25
→
X46
)
)
)
↔
(
(
¬
(
X46
↔
(
(
¬
X25
)
→
(
¬
X1
)
→
X33
)
)
)
↔
(
(
¬
(
(
¬
X33
)
↔
(
(
¬
X25
)
→
(
(
¬
X19
)
↔
X46
)
)
)
)
↔
(
¬
(
X33
↔
(
(
¬
X25
)
→
(
¬
X13
)
→
X46
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X46
↔
(
(
¬
X33
)
↔
X34
)
)
)
→
(
X48
↔
(
X46
→
(
(
¬
X27
)
↔
X34
)
)
)
→
X48
→
X34
→
(
(
¬
X27
)
↔
X33
)
)
)
↔
(
¬
(
(
¬
(
X46
→
(
¬
X41
)
→
X48
)
)
→
(
X48
→
X41
→
X34
)
→
(
¬
(
X25
↔
X46
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
583a09...
and proposition id is
13f8ab...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMRgtrfYcgj9HqaFUHE9RmMUU9U2VTJ9mAR
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
∀X50 :
prop
,
(
¬
∀X51 :
prop
,
(
(
(
(
(
(
(
X48
↔
(
X43
↔
(
X24
↔
X35
)
)
)
↔
(
X48
→
(
X46
↔
(
X43
↔
X35
)
)
)
)
→
(
¬
X24
)
→
X43
)
→
(
(
¬
(
(
¬
(
(
X46
↔
(
X34
↔
X45
)
)
→
(
¬
(
X48
↔
(
X35
↔
(
(
¬
X7
)
→
X45
)
)
)
)
)
)
→
X34
→
X14
→
X1
→
X45
)
)
↔
(
¬
(
(
¬
X45
)
→
(
X35
↔
(
(
¬
X2
)
→
X46
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X23
→
X21
→
X8
→
False
)
)
↔
(
(
¬
X10
)
→
X4
→
X23
)
)
→
(
¬
(
(
(
¬
(
X43
↔
(
X23
↔
(
X21
→
(
X9
↔
False
)
)
)
)
)
→
(
¬
(
X43
→
(
X26
↔
(
(
¬
X9
)
↔
X23
)
)
)
)
)
→
(
¬
(
(
¬
X26
)
→
X21
→
X43
)
)
)
)
→
(
¬
(
(
(
¬
(
X48
↔
(
X43
→
(
X21
↔
(
X8
↔
False
)
)
)
)
)
↔
(
¬
(
X48
↔
(
X8
↔
X0
)
)
)
)
→
(
¬
(
X43
↔
(
X21
↔
(
X9
→
X48
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
(
X40
↔
X31
)
)
→
(
(
(
¬
X3
)
→
X16
)
↔
(
X40
↔
(
(
¬
X23
)
↔
(
X3
↔
X50
)
)
)
)
)
→
(
(
¬
X40
)
↔
(
(
¬
X31
)
→
(
X16
↔
X43
)
)
)
)
)
→
(
¬
(
(
¬
X40
)
↔
(
(
¬
X21
)
↔
(
(
¬
X19
)
↔
X43
)
)
)
)
→
(
X23
↔
(
(
¬
X21
)
→
X40
)
)
)
)
→
(
¬
(
(
(
(
¬
(
(
¬
(
(
¬
X38
)
↔
(
(
¬
X23
)
→
(
(
¬
X11
)
↔
X46
)
)
)
)
→
(
¬
X23
)
→
(
(
¬
X12
)
↔
(
(
¬
X11
)
↔
X46
)
)
)
)
↔
(
X23
→
(
¬
X11
)
→
X12
)
)
→
(
X46
↔
(
X23
→
(
X21
↔
X38
)
)
)
→
(
(
¬
(
X46
→
(
X37
↔
(
(
¬
X23
)
↔
X38
)
)
)
)
↔
(
X46
↔
(
X38
→
(
X19
↔
X37
)
)
)
)
)
→
(
¬
(
X46
→
(
¬
X23
)
→
X37
)
)
)
)
)
)
→
(
(
(
¬
(
(
(
¬
(
X51
→
(
X50
↔
(
(
¬
X15
)
↔
X47
)
)
)
)
→
(
(
X51
↔
(
X39
→
(
¬
X17
)
→
X47
)
)
↔
(
¬
(
X50
→
X47
→
X32
→
X51
)
)
)
)
→
(
¬
(
(
(
¬
(
X51
↔
(
X46
↔
(
(
¬
X29
)
→
X47
)
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X27
)
↔
X47
)
)
)
)
→
(
(
X46
↔
(
X20
↔
X51
)
)
↔
(
¬
(
(
X46
↔
X5
)
→
(
(
(
¬
X46
)
→
(
(
¬
X20
)
↔
(
X5
↔
X50
)
)
)
↔
(
X50
→
(
¬
X46
)
→
(
(
¬
X5
)
↔
X51
)
)
)
)
)
)
→
(
¬
(
X51
↔
(
X50
↔
(
(
¬
X5
)
→
X47
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
X49
↔
X36
)
→
(
¬
X28
)
→
(
X22
↔
X47
)
)
)
↔
(
¬
(
(
(
¬
X39
)
↔
(
X36
→
X49
)
)
→
(
¬
(
(
(
¬
X39
)
↔
(
(
¬
X36
)
→
X28
→
X51
)
)
→
(
¬
(
X18
↔
X28
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
X42
→
(
¬
X6
)
→
X30
)
↔
(
(
(
¬
(
(
¬
(
(
X49
→
X42
→
False
)
→
(
(
X49
→
X30
→
False
)
↔
(
¬
(
X42
↔
(
(
¬
X30
)
→
(
¬
X6
)
→
X47
)
)
)
)
)
)
→
X30
→
X6
)
)
↔
(
X49
→
X47
→
X42
→
False
)
)
↔
(
¬
(
(
¬
(
X47
↔
(
X30
→
(
¬
X25
)
→
X42
)
)
)
→
(
¬
X46
)
→
(
¬
X30
)
→
X47
)
)
)
)
→
(
(
(
¬
(
(
¬
X45
)
↔
(
X44
→
(
X33
↔
X49
)
)
)
)
→
(
¬
(
X49
↔
(
X42
→
False
)
)
)
)
↔
(
(
(
¬
X13
)
↔
X41
)
→
(
¬
(
X47
↔
(
X33
→
X13
→
False
)
)
)
→
(
(
(
X49
↔
(
X47
↔
(
X44
→
X13
→
False
)
)
)
↔
(
X49
→
X44
→
(
X41
↔
False
)
)
)
→
(
¬
X44
)
→
(
¬
X41
)
→
X47
)
→
(
¬
(
X49
↔
(
X47
↔
(
X44
↔
False
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
¬
(
(
¬
X47
)
↔
(
X44
↔
(
X5
↔
X49
)
)
)
)
→
(
(
(
X49
→
(
¬
X44
)
→
(
(
¬
X28
)
↔
X47
)
)
↔
(
(
(
X49
→
X35
)
→
(
¬
(
X49
↔
(
X35
↔
X51
)
)
)
→
X47
→
(
X42
↔
X49
)
)
↔
(
X51
↔
(
X49
→
(
¬
X28
)
→
X42
)
)
)
)
↔
(
¬
(
X51
↔
(
(
¬
X28
)
↔
X43
)
)
)
)
)
→
(
¬
(
(
(
X49
↔
(
X47
↔
X51
)
)
→
(
¬
(
(
¬
X8
)
↔
(
(
¬
X7
)
→
X49
)
)
)
)
→
(
(
¬
(
X47
↔
(
(
¬
X46
)
→
(
(
¬
X21
)
↔
X49
)
)
)
)
↔
(
(
X51
→
X49
→
(
X47
↔
(
X21
↔
False
)
)
)
↔
(
X49
→
(
¬
X21
)
→
(
X12
↔
X51
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X47
→
(
(
¬
X24
)
↔
(
(
¬
X18
)
→
X49
)
)
)
)
↔
(
(
X48
→
(
¬
X27
)
→
(
¬
X24
)
→
X49
)
→
X48
→
X27
→
X24
→
X18
)
)
→
(
(
(
(
(
¬
X41
)
→
X23
→
X51
)
↔
(
¬
(
(
¬
(
X47
→
(
¬
X29
)
→
X23
→
X51
)
)
→
(
X49
↔
(
(
¬
X47
)
→
X40
→
X51
)
)
)
)
)
→
(
¬
(
(
(
(
(
X51
→
X32
→
(
¬
X6
)
→
X47
)
↔
(
¬
(
X51
↔
(
X32
↔
(
X6
↔
X49
)
)
)
)
)
↔
False
)
→
(
¬
(
X49
↔
(
(
¬
X32
)
↔
X51
)
)
)
)
→
X49
→
X32
→
False
)
)
)
↔
(
(
¬
(
(
¬
X49
)
→
(
X48
↔
X51
)
)
)
↔
(
(
¬
(
(
X51
→
X41
→
X34
)
→
(
(
(
¬
(
X41
↔
False
)
)
↔
(
(
¬
X49
)
→
X48
→
(
¬
X34
)
→
X51
)
)
↔
(
X48
↔
(
X41
→
X49
)
)
)
)
)
↔
(
(
¬
X41
)
↔
(
(
¬
X34
)
→
X48
)
)
)
)
)
)
)
)
→
(
(
(
(
X50
↔
(
X49
↔
X45
)
)
↔
(
(
¬
(
(
X49
→
(
X47
↔
(
X45
↔
(
X22
↔
False
)
)
)
)
→
(
¬
(
X47
→
(
(
¬
X45
)
↔
(
(
¬
X2
)
→
X49
)
)
)
)
)
)
↔
(
(
¬
X22
)
→
(
(
¬
X3
)
↔
X50
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X38
)
↔
(
(
¬
X37
)
↔
(
X31
→
X49
)
)
)
)
→
(
¬
(
X51
→
(
(
¬
X37
)
↔
(
X31
↔
X38
)
)
)
)
)
↔
(
¬
(
(
¬
(
X51
↔
(
X38
→
(
(
¬
X36
)
↔
X49
)
)
)
)
→
(
X51
↔
(
X38
↔
(
X25
→
X49
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X37
↔
X38
)
)
↔
(
¬
(
(
X50
→
X38
→
X47
)
→
(
¬
(
(
¬
(
(
¬
X47
)
→
(
¬
X38
)
→
X37
→
X50
)
)
→
(
¬
(
(
¬
(
(
X47
→
X37
→
(
X10
↔
X15
)
)
→
(
¬
(
X37
↔
(
X10
↔
X50
)
)
)
)
)
→
X50
→
(
X10
↔
X15
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
X36
→
X20
→
X19
)
→
(
(
(
X47
→
(
X38
↔
(
X37
↔
(
X19
↔
False
)
)
)
)
↔
(
(
¬
X38
)
→
(
¬
X37
)
→
(
¬
X20
)
→
X47
)
)
↔
(
X38
↔
(
(
¬
X19
)
→
(
X14
↔
X37
)
)
)
)
)
)
↔
(
X38
↔
(
X19
→
(
¬
X0
)
→
X37
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
X49
↔
(
X45
→
(
X1
↔
X9
)
)
)
→
(
¬
(
X16
→
(
¬
X1
)
→
X49
)
)
)
↔
(
X49
→
X16
→
X45
)
)
↔
(
¬
(
(
¬
X17
)
→
X16
→
X49
)
)
)
↔
(
(
X47
→
X39
→
X49
)
↔
(
(
¬
(
(
¬
X39
)
↔
(
X33
→
(
¬
X9
)
→
X49
)
)
)
↔
(
X49
→
X39
→
(
X4
↔
X33
)
)
)
)
)
→
(
(
(
¬
(
X30
→
(
¬
X13
)
→
(
X11
↔
X39
)
)
)
→
(
(
¬
X45
)
↔
(
X26
→
X11
→
X49
)
)
)
↔
(
¬
(
X39
→
(
¬
X26
)
→
X30
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
d610a9...
and proposition id is
b057c4...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMMP4ZjQBaFrJTn84t1rGp1Hogc95MuQ6QR
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
¬
(
(
(
¬
(
(
(
(
(
X49
↔
(
(
¬
X46
)
↔
(
(
¬
X31
)
↔
X48
)
)
)
↔
(
¬
(
X48
↔
(
(
¬
X31
)
→
(
¬
X26
)
→
X49
)
)
)
)
→
(
¬
(
(
¬
X48
)
↔
(
(
¬
X46
)
→
X22
→
X49
)
)
)
)
→
(
¬
(
(
¬
X26
)
→
(
¬
X3
)
→
X48
)
)
→
X46
→
(
¬
X26
)
→
X15
→
X48
)
→
(
¬
(
(
¬
(
X48
↔
(
(
¬
X34
)
→
X19
→
X49
)
)
)
→
(
(
¬
(
X48
→
(
(
¬
X19
)
↔
(
(
¬
X18
)
→
X45
)
)
)
)
↔
(
(
¬
(
X45
↔
(
(
¬
X18
)
↔
X49
)
)
)
→
(
(
¬
(
X49
→
X45
→
X19
)
)
→
X49
→
X18
→
(
¬
X7
)
→
X19
)
→
X49
→
X7
→
X18
)
)
→
(
(
X38
→
(
(
¬
X19
)
↔
X46
)
)
↔
(
(
¬
(
(
¬
(
X45
→
X44
→
(
X12
↔
X19
)
)
)
→
(
(
¬
(
X44
→
False
)
)
↔
(
¬
(
(
¬
(
X46
→
X44
→
X7
→
X19
)
)
→
(
(
¬
X12
)
↔
X44
)
)
)
)
)
)
→
X46
→
(
X45
↔
(
X44
→
X38
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
(
X46
→
(
¬
X17
)
→
X45
)
↔
(
¬
(
(
¬
X17
)
↔
(
(
¬
X8
)
↔
X41
)
)
)
)
→
(
¬
(
X46
↔
(
X17
↔
X45
)
)
)
)
↔
(
(
¬
(
X45
→
X41
→
X46
)
)
↔
(
(
(
¬
(
X20
→
X5
→
X27
)
)
↔
(
(
(
¬
(
X46
→
(
X5
↔
X37
)
)
)
↔
(
(
¬
(
X46
↔
(
(
¬
X27
)
→
(
¬
X16
)
→
X37
)
)
)
↔
(
¬
(
X27
→
(
X16
↔
(
X5
↔
X37
)
)
)
)
)
)
↔
(
¬
(
X46
↔
(
X27
→
False
)
)
)
)
)
↔
(
X46
↔
(
(
¬
X27
)
→
(
(
¬
X5
)
↔
X41
)
)
)
)
)
)
→
(
X46
→
(
X45
↔
(
(
¬
X36
)
↔
X37
)
)
)
→
(
¬
(
(
(
¬
(
X46
↔
(
(
¬
X37
)
→
(
¬
X28
)
→
X49
)
)
)
→
(
¬
(
X49
→
(
(
¬
X24
)
↔
X37
)
)
)
)
→
(
¬
(
(
¬
X46
)
→
X37
→
(
¬
X36
)
→
X49
)
)
)
)
→
(
¬
(
X46
→
X37
→
X49
)
)
)
)
↔
(
¬
(
(
(
(
X13
→
(
¬
X4
)
→
X1
→
X45
)
↔
(
¬
(
X45
↔
(
X21
↔
(
X1
↔
X13
)
)
)
)
)
→
(
(
¬
X39
)
→
X37
→
(
¬
X21
)
→
X45
)
→
(
X37
↔
(
X21
↔
(
X13
↔
X45
)
)
)
)
→
(
(
¬
(
(
¬
X37
)
↔
(
(
¬
X21
)
↔
(
X13
→
X41
)
)
)
)
↔
(
¬
(
X41
↔
(
X37
→
(
(
¬
X14
)
↔
X21
)
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
(
¬
(
(
¬
(
(
¬
X38
)
↔
(
(
¬
X23
)
→
X47
)
)
)
→
(
¬
(
(
¬
X38
)
→
(
(
¬
X23
)
↔
(
(
¬
X0
)
→
X48
)
)
)
)
)
)
↔
(
(
¬
X47
)
→
X38
→
X2
→
X48
)
)
→
(
(
X47
→
X30
→
X49
)
↔
(
¬
(
X48
→
X47
→
(
X38
↔
False
)
)
)
)
)
)
↔
(
¬
(
(
X38
→
(
(
¬
X6
)
↔
X30
)
)
→
(
X49
↔
(
X11
→
(
(
¬
X10
)
↔
X38
)
)
)
→
(
¬
X6
)
)
)
)
→
(
¬
(
(
(
¬
(
(
(
X49
→
(
X40
↔
False
)
)
↔
(
¬
(
X47
→
(
¬
X40
)
→
(
¬
X32
)
→
X48
)
)
)
→
(
¬
X48
)
→
(
X47
↔
X49
)
)
)
↔
(
¬
(
(
¬
X48
)
→
X32
→
X49
)
)
)
→
(
X47
↔
(
X40
↔
X29
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X48
→
X47
→
(
¬
X42
)
→
X49
)
)
→
X48
→
(
(
¬
X43
)
↔
(
(
¬
X42
)
→
X49
)
)
)
→
(
(
X42
↔
(
X33
→
X25
→
X48
)
)
↔
(
(
(
¬
(
X48
↔
(
(
¬
X42
)
↔
X47
)
)
)
↔
(
X48
→
(
X42
↔
(
(
¬
X9
)
↔
X33
)
)
)
)
↔
(
X48
→
(
X35
↔
(
(
¬
X33
)
↔
X42
)
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
(
(
X45
→
(
(
¬
X30
)
↔
(
(
¬
X17
)
↔
X38
)
)
)
→
(
(
¬
X38
)
↔
(
X16
↔
X45
)
)
)
→
(
(
¬
(
X40
↔
(
(
¬
X38
)
↔
(
(
¬
X16
)
→
X45
)
)
)
)
↔
(
X45
→
(
¬
X38
)
→
(
¬
X8
)
→
X40
)
)
)
↔
(
X45
→
(
X30
↔
(
(
¬
X16
)
↔
X47
)
)
)
)
↔
(
¬
(
(
(
X35
↔
X40
)
↔
(
(
(
¬
(
(
¬
X40
)
→
X6
→
X45
)
)
↔
(
X40
↔
(
X38
↔
(
(
¬
X35
)
→
X45
)
)
)
)
→
X38
→
(
¬
X6
)
→
X45
)
)
→
(
(
¬
(
X40
→
X27
→
X24
→
False
)
)
↔
(
¬
(
X45
→
X24
→
X38
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X47
)
→
X49
)
)
↔
(
¬
(
X49
→
X46
→
X12
→
X40
)
)
)
↔
(
(
(
¬
(
X49
→
(
(
¬
X44
)
↔
(
(
¬
X10
)
↔
X45
)
)
)
)
→
(
(
X45
↔
(
X44
→
X49
)
)
↔
(
(
¬
(
X49
→
(
X18
↔
X45
)
)
)
↔
(
¬
(
X49
↔
(
X44
→
(
¬
X39
)
→
X45
)
)
)
)
)
)
↔
(
(
(
¬
X45
)
→
(
(
¬
X5
)
↔
X47
)
)
↔
(
¬
(
X49
→
(
¬
X46
)
→
X44
→
X47
)
)
)
)
)
)
→
(
X47
↔
(
(
¬
X29
)
→
X49
)
)
→
(
(
X36
↔
(
X29
→
X47
)
)
↔
(
(
X49
↔
(
X47
→
(
(
¬
X25
)
↔
X36
)
)
)
↔
(
(
¬
(
X47
↔
(
X25
→
(
(
¬
X4
)
↔
X36
)
)
)
)
↔
(
¬
(
X36
→
X46
)
)
)
)
)
)
)
↔
(
(
(
(
(
X21
↔
(
(
¬
X3
)
↔
(
X1
↔
X46
)
)
)
↔
(
¬
(
X3
↔
(
X2
→
(
X1
↔
False
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X43
→
X21
)
)
→
(
¬
(
X49
→
(
X43
↔
(
(
¬
X15
)
→
X32
)
)
)
)
)
)
→
(
¬
(
X49
→
X32
→
False
)
)
)
)
↔
(
(
(
¬
(
(
¬
X46
)
→
(
¬
X44
)
→
(
¬
X34
)
→
X49
)
)
↔
(
(
¬
(
X26
↔
(
X13
↔
(
(
¬
X7
)
↔
X49
)
)
)
)
↔
(
(
(
¬
(
(
¬
X34
)
→
(
X26
↔
(
X7
↔
X49
)
)
)
)
↔
(
X34
→
(
X26
↔
(
(
¬
X7
)
↔
X46
)
)
)
)
↔
(
X46
↔
X34
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X11
)
→
X22
)
)
→
(
(
(
X28
↔
False
)
→
(
¬
(
(
¬
X34
)
↔
(
X22
→
(
¬
X13
)
→
X49
)
)
)
)
↔
(
¬
(
(
¬
X32
)
→
(
¬
X22
)
→
(
(
¬
X13
)
↔
X49
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
X48
→
X23
→
X34
)
→
(
¬
(
X41
↔
(
X34
→
(
(
¬
X23
)
↔
X31
)
)
)
)
)
→
(
(
(
¬
(
X37
↔
(
X33
→
(
¬
X20
)
→
X48
)
)
)
↔
(
¬
(
X48
↔
(
X33
→
(
¬
X20
)
→
X41
)
)
)
)
↔
(
X41
↔
(
(
¬
X33
)
↔
X48
)
)
)
→
(
¬
(
(
X37
↔
(
(
¬
X34
)
↔
X41
)
)
→
(
¬
(
X34
→
X20
→
(
¬
X19
)
→
X37
)
)
→
(
¬
(
X41
→
X37
)
)
)
)
)
→
(
¬
(
(
(
X48
↔
(
X37
↔
(
(
¬
X9
)
→
X42
)
)
)
↔
(
X49
→
(
X42
↔
(
X37
→
X48
)
)
)
)
→
(
(
(
(
(
X48
↔
(
X37
↔
(
(
¬
X0
)
↔
X49
)
)
)
↔
(
¬
(
X37
→
(
¬
X32
)
→
(
(
¬
X0
)
↔
X49
)
)
)
)
→
(
¬
(
X48
↔
(
(
¬
X0
)
→
X32
)
)
)
)
↔
(
X32
→
X0
→
X48
)
)
↔
(
(
¬
(
(
X49
↔
(
X32
→
X14
)
)
→
(
(
X48
↔
(
X32
→
(
(
¬
X14
)
↔
X34
)
)
)
↔
(
¬
(
X49
→
X48
→
X14
→
False
)
)
)
)
)
→
(
¬
(
X49
→
(
X48
↔
(
X34
↔
False
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
cb54fd...
and proposition id is
5f7906...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMW3hGvJs14QT5fxo79LentkoDkAjUNYanr
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
(
(
X47
↔
(
X28
↔
(
X3
↔
X37
)
)
)
↔
(
X47
↔
(
(
¬
X28
)
→
(
¬
X17
)
→
X37
)
)
)
↔
(
(
X45
↔
X21
)
→
(
¬
(
X21
↔
(
X18
→
X17
→
X31
)
)
)
)
)
↔
(
(
¬
(
(
X47
↔
(
X37
→
(
(
¬
X21
)
↔
X34
)
)
)
→
(
¬
(
X48
↔
(
(
¬
X34
)
→
(
¬
X12
)
→
X37
)
)
)
)
)
↔
(
(
¬
(
X48
↔
(
(
¬
X46
)
↔
(
X16
↔
X47
)
)
)
)
↔
(
(
¬
(
X37
↔
(
X16
→
X0
→
X48
)
)
)
↔
(
(
¬
X37
)
→
(
¬
X16
)
→
X47
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X44
)
→
(
(
¬
X35
)
↔
X48
)
)
)
→
(
X44
↔
(
(
¬
X33
)
→
(
(
¬
X30
)
↔
X42
)
)
)
)
↔
(
(
(
X47
↔
(
X9
↔
X48
)
)
↔
(
¬
(
X35
↔
(
X9
↔
(
X7
↔
(
X4
→
False
)
)
)
)
)
)
↔
(
(
¬
X35
)
↔
(
X33
→
(
¬
X4
)
→
X47
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
(
¬
(
X48
→
(
X41
↔
(
X35
↔
X46
)
)
)
)
→
X46
→
X35
→
(
X27
↔
X48
)
)
→
(
¬
(
X35
→
X48
)
)
)
)
↔
(
X43
→
(
¬
X41
)
→
(
(
¬
X35
)
↔
X48
)
)
)
→
(
(
¬
(
X48
↔
(
(
¬
X26
)
↔
(
X10
→
X46
)
)
)
)
↔
(
X27
→
(
(
¬
X10
)
↔
X48
)
)
)
)
)
→
(
X47
↔
(
X43
→
(
X41
↔
X48
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
(
¬
X32
)
→
(
(
¬
X25
)
↔
(
X5
→
X36
)
)
)
→
(
¬
X25
)
→
(
¬
X14
)
→
X32
)
)
↔
(
(
X38
→
X36
→
X5
→
X40
)
→
(
X40
↔
(
X38
↔
(
X14
→
X36
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X32
→
(
(
¬
X19
)
↔
(
X11
→
X20
)
)
)
)
↔
(
(
¬
X20
)
→
X13
→
(
X11
↔
X32
)
)
)
↔
(
X32
↔
(
X20
→
(
X13
↔
False
)
)
)
)
→
X32
→
(
¬
X20
)
→
X22
)
↔
(
¬
(
(
¬
(
X38
↔
(
X20
↔
(
X19
↔
X40
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X38
)
→
(
X32
↔
X44
)
)
)
→
(
¬
(
(
¬
X38
)
↔
(
(
¬
X32
)
→
(
X19
↔
X40
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
X49
↔
(
(
¬
X39
)
→
(
(
¬
X8
)
↔
X43
)
)
)
↔
(
(
¬
(
(
¬
(
X49
→
X43
→
X38
→
X44
)
)
→
(
¬
(
(
(
¬
(
X49
↔
(
X43
→
(
X6
↔
X39
)
)
)
)
↔
(
(
¬
X43
)
↔
(
X39
↔
(
X6
→
X44
)
)
)
)
→
(
¬
(
X44
↔
(
X43
→
X39
)
)
)
)
)
)
)
↔
(
X49
→
(
X39
↔
X43
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
X23
)
→
X1
→
X38
)
)
→
(
(
¬
X44
)
↔
(
(
¬
X1
)
→
X47
)
)
)
)
↔
(
¬
(
X47
↔
(
X44
→
X23
→
X8
→
False
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
X29
)
↔
(
(
¬
X24
)
↔
(
(
¬
X15
)
→
X43
)
)
)
)
↔
(
X40
↔
(
X24
→
X15
→
False
)
)
)
↔
(
¬
(
X40
↔
(
X24
↔
X29
)
)
)
)
↔
(
¬
(
(
¬
(
X43
→
X47
)
)
→
(
X44
↔
(
X43
↔
(
X29
↔
X2
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
(
(
¬
X48
)
→
(
(
¬
X47
)
↔
(
X31
↔
X49
)
)
)
→
(
¬
(
(
¬
X48
)
↔
(
X46
→
X15
→
X49
)
)
)
)
↔
(
¬
(
(
(
(
¬
X46
)
↔
(
X45
↔
(
X18
↔
X49
)
)
)
→
(
(
X13
→
X6
→
X46
)
↔
(
(
(
(
X45
→
X6
→
X49
)
↔
(
¬
(
X46
↔
(
(
¬
X6
)
→
(
(
¬
X1
)
↔
X45
)
)
)
)
)
↔
(
(
¬
X46
)
→
(
¬
X45
)
→
X49
)
)
→
(
¬
(
X49
→
X45
→
X13
→
(
X6
↔
False
)
)
)
)
)
)
→
(
(
X48
↔
(
X18
→
X49
)
)
↔
(
(
¬
X45
)
↔
(
(
¬
X18
)
↔
(
(
¬
X2
)
→
X49
)
)
)
)
)
)
)
↔
(
¬
(
X18
→
(
(
¬
X1
)
↔
X46
)
)
)
)
↔
(
(
¬
(
(
¬
X29
)
↔
(
X26
→
X47
)
)
)
↔
(
(
X45
→
X26
→
X49
)
↔
(
X47
→
X45
→
(
X29
↔
(
X26
→
False
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X49
→
(
¬
X11
)
→
(
¬
X8
)
→
X35
)
)
→
(
¬
(
X35
→
X33
→
(
X27
↔
X11
)
)
)
)
→
(
(
(
X47
↔
(
(
¬
X30
)
↔
X49
)
)
↔
(
(
X49
→
X47
→
X30
→
X11
→
False
)
→
(
X49
↔
(
X27
↔
(
(
¬
X11
)
→
X30
)
)
)
)
)
→
X30
→
(
X11
↔
X47
)
)
→
X49
→
X47
)
↔
(
¬
(
(
(
¬
(
(
¬
(
(
¬
X48
)
→
(
(
¬
X33
)
↔
(
X3
→
X49
)
)
)
)
→
(
¬
(
(
¬
X38
)
↔
(
X33
→
X3
→
X48
)
)
)
)
)
→
(
X49
→
X9
→
X21
)
→
(
¬
(
(
¬
X33
)
↔
(
(
¬
X9
)
→
(
(
¬
X3
)
↔
X49
)
)
)
)
)
→
(
(
(
¬
(
X21
→
(
¬
X0
)
→
X49
)
)
↔
(
¬
(
X21
↔
(
(
¬
X20
)
↔
(
(
¬
X0
)
↔
X38
)
)
)
)
)
↔
(
(
X30
↔
(
X21
↔
X4
)
)
↔
(
(
¬
(
X30
→
(
X21
↔
(
(
¬
X4
)
→
X49
)
)
)
)
↔
(
¬
(
X49
→
(
(
¬
X30
)
↔
(
X21
→
X38
)
)
)
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
(
(
¬
(
X44
→
X43
)
)
↔
(
(
X44
↔
(
(
¬
X41
)
↔
X43
)
)
↔
(
X47
↔
(
X44
→
(
X24
↔
False
)
)
)
)
)
→
X43
→
(
¬
X41
)
→
(
¬
X36
)
→
X44
)
→
(
X24
↔
(
(
¬
X14
)
↔
X49
)
)
→
(
¬
(
X49
↔
(
X41
↔
(
X14
→
X44
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X43
↔
(
X12
↔
X47
)
)
)
→
(
X43
↔
(
(
¬
X25
)
→
X47
)
)
→
(
¬
(
(
¬
(
(
¬
(
X43
↔
(
X42
↔
(
X12
↔
X25
)
)
)
)
→
(
¬
(
X47
↔
X42
)
)
)
)
→
(
X25
↔
(
(
¬
X12
)
↔
X42
)
)
)
)
)
)
→
(
¬
(
X47
↔
False
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
(
X43
→
X22
→
X40
)
)
→
(
¬
(
(
¬
X42
)
→
(
¬
X22
)
→
X47
)
)
)
)
→
X42
→
(
X22
↔
X47
)
)
)
↔
(
X44
↔
(
X43
↔
(
X22
↔
X47
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X37
↔
(
(
¬
X32
)
→
X47
)
)
)
→
(
X37
↔
(
X32
→
X47
)
)
→
(
(
(
¬
(
(
(
¬
X42
)
↔
(
X32
↔
(
X10
→
X47
)
)
)
→
(
X32
↔
(
X16
→
X37
)
)
)
)
→
(
¬
(
X42
↔
(
X10
↔
False
)
)
)
)
↔
(
X42
↔
(
(
¬
X37
)
↔
(
(
¬
X28
)
↔
X47
)
)
)
)
)
→
(
¬
(
(
(
X39
↔
(
(
¬
X19
)
↔
X23
)
)
↔
(
¬
(
(
¬
(
X48
→
X39
→
(
X19
↔
X41
)
)
)
→
(
(
¬
(
X41
→
(
¬
X23
)
→
X48
)
)
↔
(
(
¬
(
X48
→
X47
→
(
X41
↔
X39
)
)
)
→
(
¬
(
(
¬
X47
)
→
(
(
¬
X41
)
↔
(
X23
↔
X48
)
)
)
)
)
)
)
)
)
→
(
(
X47
↔
(
X39
→
X34
→
X7
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X32
)
→
X28
→
(
X5
↔
X41
)
)
)
→
(
¬
(
(
¬
X41
)
↔
(
(
¬
X5
)
↔
X47
)
)
)
)
)
→
(
¬
(
X34
↔
(
X17
→
(
(
¬
X5
)
↔
X39
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
369f0a...
and proposition id is
a8cbe7...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMPXrNPrPhkGwP5zEsGLtR5Nfqc9Hynysaw
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
¬
(
(
(
(
(
(
X47
→
X29
→
(
¬
X1
)
→
X21
)
↔
(
X21
↔
(
X16
→
X49
)
)
)
↔
(
¬
(
X49
↔
(
X46
→
(
X21
↔
X29
)
)
)
)
)
↔
(
(
(
(
¬
(
X34
↔
(
X25
↔
(
(
¬
X14
)
→
X49
)
)
)
)
↔
(
X49
↔
(
X46
→
X25
)
)
)
→
(
¬
(
(
X49
↔
(
X46
→
(
¬
X25
)
→
X36
)
)
→
(
¬
(
X46
→
(
¬
X35
)
→
(
¬
X25
)
→
X49
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X47
→
(
X36
↔
(
X35
↔
(
X27
↔
False
)
)
)
)
)
→
(
¬
(
X6
↔
(
X5
↔
X47
)
)
)
→
(
¬
(
X47
→
(
X36
↔
(
(
¬
X27
)
→
X34
)
)
)
)
)
)
↔
(
X35
→
(
X34
↔
(
X27
↔
False
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X49
↔
(
X46
→
X26
)
)
)
→
(
(
(
¬
(
X49
↔
(
X47
↔
(
(
¬
X37
)
→
X46
)
)
)
)
↔
(
¬
(
(
¬
X46
)
→
(
¬
X26
)
→
X47
)
)
)
↔
(
¬
(
(
¬
X47
)
↔
(
(
¬
X46
)
→
X37
→
X50
)
)
)
)
)
→
X50
→
(
(
¬
X39
)
↔
(
X22
→
X46
)
)
)
↔
(
(
(
(
(
(
X47
→
(
X38
↔
(
X23
→
False
)
)
)
↔
(
¬
(
(
¬
(
X49
→
X47
→
X24
→
X38
)
)
→
(
¬
(
(
¬
X38
)
↔
(
X24
↔
(
(
¬
X23
)
→
X49
)
)
)
)
)
)
)
↔
(
¬
(
X49
↔
(
(
¬
X23
)
→
X38
)
)
)
)
→
(
¬
(
(
(
¬
X44
)
→
(
¬
X38
)
→
(
¬
X13
)
→
X47
)
→
(
X47
↔
(
(
¬
X38
)
↔
(
X13
↔
X49
)
)
)
)
)
)
↔
(
(
¬
(
X44
→
(
X17
↔
X38
)
)
)
↔
(
¬
(
X47
↔
(
X32
→
X7
)
)
)
)
)
→
(
(
X50
↔
(
X12
→
X23
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
¬
X24
)
↔
(
X23
↔
X50
)
)
)
→
(
(
¬
(
X28
→
X23
→
X24
)
)
↔
(
¬
(
X28
↔
(
(
¬
X24
)
→
(
X13
↔
X50
)
)
)
)
)
)
)
→
(
¬
(
X24
→
X23
→
(
(
¬
X12
)
↔
X50
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
¬
(
(
¬
(
X46
→
X40
→
X48
)
)
→
X40
→
(
(
¬
X33
)
↔
X41
)
)
)
↔
(
¬
(
X48
→
X46
→
X40
→
X47
)
)
)
↔
(
X48
↔
(
X47
↔
(
X45
↔
(
X11
↔
False
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
X45
→
(
X10
↔
X50
)
)
)
→
(
(
¬
(
X45
↔
(
X3
→
(
¬
X2
)
→
X10
)
)
)
↔
(
X50
→
(
X45
↔
(
X3
↔
False
)
)
)
)
)
)
↔
(
(
¬
X31
)
→
X46
)
)
↔
(
(
(
¬
(
X46
↔
(
(
¬
X41
)
→
X20
→
X45
)
)
)
↔
(
¬
(
X46
↔
(
X45
→
(
X41
↔
X50
)
)
)
)
)
→
X50
→
(
X41
↔
X46
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
X42
→
(
X41
↔
(
X8
→
X43
)
)
)
)
→
(
(
(
¬
X30
)
↔
(
X18
↔
X42
)
)
↔
(
¬
(
(
X42
→
(
X33
↔
X43
)
)
→
(
(
¬
(
(
X43
↔
(
X33
→
(
X30
↔
X42
)
)
)
→
(
¬
(
X42
↔
(
(
¬
X30
)
↔
(
X8
↔
X33
)
)
)
)
)
)
↔
(
(
¬
X33
)
↔
(
(
¬
X30
)
↔
(
X18
↔
X43
)
)
)
)
)
)
)
)
)
→
(
¬
X42
)
→
(
¬
X33
)
→
(
¬
X0
)
→
X43
)
)
↔
(
(
(
X46
↔
(
(
¬
X42
)
↔
(
X19
↔
X43
)
)
)
↔
(
¬
(
X46
→
(
(
¬
X42
)
↔
(
X15
→
X43
)
)
)
)
)
↔
(
(
(
X43
→
(
X42
↔
(
(
¬
X4
)
↔
X9
)
)
)
→
X46
→
X43
→
X42
→
False
)
↔
(
¬
(
X43
↔
(
X4
→
X41
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
(
¬
(
X26
→
X20
→
(
X4
↔
False
)
)
)
↔
(
(
X26
→
(
¬
X20
)
→
(
¬
X2
)
→
X37
)
↔
(
X37
→
(
X20
↔
(
X4
↔
False
)
)
)
)
)
→
(
¬
(
(
¬
(
(
X37
→
X20
→
X18
)
→
(
¬
(
X37
↔
(
X26
↔
(
(
¬
X3
)
↔
X18
)
)
)
)
)
)
→
(
¬
(
X37
↔
(
(
¬
X3
)
→
X43
)
)
)
)
)
)
↔
(
X37
→
X4
→
X18
)
)
→
(
(
(
X44
→
X37
→
(
(
¬
X32
)
↔
X38
)
)
↔
(
(
¬
(
X38
↔
(
X26
→
(
(
¬
X21
)
↔
X44
)
)
)
)
→
(
(
¬
(
X38
→
(
¬
X21
)
→
(
(
¬
X17
)
↔
X44
)
)
)
↔
(
X32
↔
(
(
¬
X21
)
↔
(
X17
↔
X38
)
)
)
)
)
)
↔
(
(
(
¬
X26
)
↔
(
(
¬
X20
)
→
(
¬
X12
)
→
X37
)
)
→
(
¬
(
X38
→
(
X12
↔
X20
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
X44
↔
(
(
¬
X42
)
→
X9
→
X43
)
)
↔
(
¬
(
X44
↔
(
X43
↔
(
(
¬
X37
)
↔
X42
)
)
)
)
)
→
(
(
X49
↔
(
X44
→
(
X43
↔
X29
)
)
)
↔
(
(
¬
(
X44
→
X29
→
X49
)
)
→
(
¬
X37
)
→
(
(
¬
X23
)
↔
X49
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
X28
↔
(
(
¬
X23
)
↔
(
X7
→
X42
)
)
)
)
→
(
¬
(
X31
↔
(
(
¬
X28
)
→
(
(
¬
X23
)
↔
X42
)
)
)
)
)
)
→
(
¬
(
(
¬
X23
)
↔
(
(
¬
X7
)
→
X42
)
)
)
)
)
↔
(
(
¬
X44
)
↔
(
X31
↔
(
X28
↔
X45
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X50
↔
(
X33
→
(
(
¬
X14
)
↔
X24
)
)
)
)
↔
(
¬
(
X46
→
X14
→
X50
)
)
)
↔
(
(
¬
(
(
(
¬
X46
)
↔
(
X44
↔
X49
)
)
→
(
(
X49
↔
(
(
¬
X44
)
↔
(
X33
→
X46
)
)
)
↔
(
(
¬
(
(
X49
↔
(
(
¬
X44
)
↔
(
X40
↔
X46
)
)
)
→
(
¬
(
X49
→
X44
→
X24
→
X46
)
)
)
)
↔
(
¬
(
X49
↔
(
X40
↔
False
)
)
)
)
)
)
)
↔
(
(
X46
→
X44
→
X40
)
→
(
¬
(
(
X40
↔
(
X15
→
(
¬
X10
)
→
X44
)
)
→
X15
→
X10
→
X46
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
(
¬
X46
)
→
X24
→
(
X5
↔
X48
)
)
)
→
X48
→
(
(
¬
X25
)
↔
(
(
¬
X8
)
↔
X46
)
)
)
)
↔
(
(
(
¬
(
(
¬
X46
)
↔
(
(
¬
X11
)
↔
X48
)
)
)
↔
(
(
(
¬
(
X25
→
(
X11
↔
False
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X13
)
↔
(
(
¬
X1
)
↔
X25
)
)
)
)
)
↔
(
¬
(
(
¬
X46
)
→
X25
→
(
(
¬
X13
)
↔
X48
)
)
)
)
)
→
X46
→
(
X25
↔
X48
)
)
)
↔
(
(
¬
(
X50
↔
(
X27
↔
(
X1
→
False
)
)
)
)
→
(
¬
(
X30
→
X13
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
X49
→
X47
→
X44
→
(
X39
↔
False
)
)
)
→
(
X6
↔
X47
)
)
)
↔
(
(
¬
(
(
¬
X19
)
↔
(
(
¬
X6
)
→
X49
)
)
)
→
(
¬
(
(
¬
X34
)
→
X39
)
)
)
)
→
(
(
(
X47
↔
(
X22
→
X35
)
)
→
(
¬
(
(
(
X47
↔
(
(
¬
X44
)
→
X49
)
)
↔
(
¬
(
(
¬
(
(
¬
X47
)
→
X44
→
(
X35
↔
X49
)
)
)
→
X44
→
X22
)
)
)
→
(
¬
(
X49
→
(
X35
↔
X44
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
(
¬
X36
)
→
(
X35
↔
X49
)
)
)
↔
(
(
¬
(
X42
→
X35
→
(
¬
X22
)
→
X49
)
)
↔
(
¬
(
(
¬
X36
)
→
(
(
¬
X35
)
↔
(
(
¬
X22
)
↔
X49
)
)
)
)
)
)
→
(
X42
↔
X36
)
)
)
↔
(
X49
→
X42
→
X36
)
)
)
→
(
(
(
(
X44
→
X16
→
X46
)
↔
(
(
¬
X16
)
↔
(
X0
↔
X41
)
)
)
↔
(
(
¬
X44
)
↔
(
(
¬
X16
)
↔
X46
)
)
)
↔
(
(
X49
→
(
(
¬
X41
)
↔
X44
)
)
→
(
(
¬
(
(
¬
X46
)
→
(
¬
X44
)
→
(
¬
X41
)
→
X49
)
)
↔
(
¬
(
(
¬
X44
)
↔
(
X41
↔
(
X36
→
X49
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
c0b280...
and proposition id is
a0a8e3...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMYedwspBdtCSfwZUL8xpSZeQthC5ZTR2uP
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
∀X51 :
prop
,
(
¬
∀X52 :
prop
,
∀X53 :
prop
,
(
(
¬
(
(
¬
(
(
(
(
(
(
(
¬
X51
)
→
(
(
¬
X39
)
↔
(
(
¬
X27
)
→
X52
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X46
→
(
X39
↔
(
(
¬
X18
)
↔
X51
)
)
)
)
→
X39
→
(
¬
X25
)
→
(
¬
X18
)
→
X46
)
)
→
(
¬
(
(
¬
X27
)
↔
(
(
¬
X25
)
↔
(
(
¬
X18
)
→
X46
)
)
)
)
)
)
)
↔
(
¬
(
X52
→
(
(
¬
X46
)
↔
X51
)
)
)
)
↔
(
(
X39
→
(
X37
↔
X51
)
)
↔
(
(
X51
↔
(
X37
→
(
¬
X5
)
→
X41
)
)
→
X46
→
(
X39
↔
(
X24
↔
False
)
)
)
)
)
↔
(
(
(
¬
X27
)
↔
(
(
¬
X10
)
↔
X46
)
)
↔
(
¬
(
(
X2
↔
(
¬
(
X52
→
X27
→
(
¬
X10
)
→
X46
)
)
)
→
(
¬
(
(
¬
X52
)
↔
(
X46
→
(
¬
X27
)
→
X53
)
)
)
)
)
)
)
→
(
(
(
(
¬
X40
)
→
X44
)
→
(
(
¬
(
X53
↔
(
X52
→
(
X50
↔
X51
)
)
)
)
↔
(
(
(
¬
X50
)
→
(
X33
↔
X52
)
)
→
(
(
¬
(
X53
→
(
X50
↔
X40
)
)
)
↔
(
X50
→
(
X49
↔
(
X33
→
False
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
X33
↔
(
(
¬
X17
)
↔
X21
)
)
↔
(
(
X15
↔
(
X4
→
(
¬
X0
)
→
X50
)
)
↔
(
(
¬
(
(
¬
X21
)
→
X0
→
X50
)
)
↔
(
(
¬
(
(
¬
X17
)
↔
(
(
¬
X4
)
↔
(
(
¬
X0
)
↔
X21
)
)
)
)
↔
(
X50
↔
(
X21
↔
(
X17
→
False
)
)
)
)
)
)
)
→
(
¬
(
(
¬
X21
)
→
(
¬
X17
)
→
X50
)
)
)
)
↔
(
¬
(
(
X43
→
(
X0
↔
False
)
)
→
(
¬
(
X33
↔
(
X21
→
(
¬
X0
)
→
X29
)
)
)
)
)
)
→
(
X53
↔
(
X50
→
X33
→
X29
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
(
(
¬
(
X52
↔
(
X8
↔
X1
)
)
)
↔
(
X52
→
(
X36
↔
(
X8
↔
X50
)
)
)
)
→
(
(
(
X23
→
(
X13
↔
False
)
)
↔
(
(
¬
(
(
(
X23
↔
(
X13
↔
(
X8
↔
False
)
)
)
↔
(
(
¬
X23
)
↔
(
X11
↔
(
(
¬
X8
)
↔
X50
)
)
)
)
→
(
¬
(
(
X13
↔
(
X11
↔
X12
)
)
→
(
¬
(
X23
→
(
(
¬
X11
)
↔
(
(
¬
X1
)
↔
X13
)
)
)
)
)
)
)
)
↔
(
¬
(
X23
→
(
¬
X11
)
→
X12
)
)
)
)
↔
(
(
¬
(
X16
→
(
¬
X12
)
→
(
(
¬
X11
)
↔
X50
)
)
)
↔
(
X50
→
(
¬
X12
)
→
(
(
¬
X11
)
↔
X53
)
)
)
)
)
→
(
(
(
¬
X34
)
↔
(
(
¬
X23
)
↔
(
(
¬
X6
)
→
X47
)
)
)
↔
(
(
X23
↔
(
X16
↔
X47
)
)
↔
(
(
¬
(
X23
→
(
¬
X16
)
→
X12
→
X47
)
)
↔
(
¬
(
X16
→
X34
)
)
)
)
)
)
)
↔
(
(
¬
(
(
X26
↔
(
(
¬
X7
)
→
X52
)
)
→
(
(
¬
(
X50
→
X42
→
(
X22
↔
(
X7
↔
False
)
)
)
)
↔
(
X50
→
(
X45
↔
(
X42
→
(
X7
↔
False
)
)
)
)
)
)
)
↔
(
(
¬
X47
)
↔
(
(
¬
X42
)
↔
(
(
¬
X22
)
↔
X50
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
(
¬
X50
)
↔
(
(
¬
X48
)
↔
(
(
¬
X9
)
↔
X53
)
)
)
↔
(
X53
↔
(
X20
↔
(
X9
↔
X50
)
)
)
)
↔
(
(
X53
↔
(
X35
→
X19
→
(
X9
↔
False
)
)
)
↔
(
(
¬
X35
)
↔
(
X19
→
(
¬
X9
)
→
X50
)
)
)
)
→
(
¬
X50
)
→
(
X48
↔
(
X14
↔
X53
)
)
)
→
(
(
X50
→
(
¬
X28
)
→
(
¬
X3
)
→
X48
)
↔
(
(
¬
X48
)
→
X53
)
)
)
→
(
(
¬
(
(
¬
X48
)
→
(
¬
X38
)
→
(
¬
X32
)
→
X50
)
)
→
(
¬
(
(
¬
X48
)
→
(
X30
↔
X50
)
)
)
)
→
(
¬
(
X47
↔
(
X38
→
X30
→
False
)
)
)
→
(
¬
(
X53
→
(
(
¬
X31
)
↔
X32
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
X39
↔
(
(
¬
X35
)
↔
X46
)
)
→
(
X46
→
(
(
¬
X39
)
↔
(
X35
↔
X53
)
)
)
→
(
¬
(
X53
→
(
X46
↔
(
X29
→
X39
)
)
)
)
)
→
(
¬
(
X47
↔
(
X39
↔
(
(
¬
X4
)
→
X5
)
)
)
)
→
(
¬
(
X46
→
X39
→
False
)
)
→
(
(
¬
(
X35
↔
(
X23
→
X5
→
X46
)
)
)
↔
(
(
¬
X39
)
→
(
¬
X35
)
→
(
X4
↔
X46
)
)
)
)
→
(
(
¬
(
(
X46
→
X39
→
(
X29
↔
X10
)
)
→
X39
→
(
X38
↔
(
X10
→
X29
)
)
)
)
→
(
(
X39
↔
(
X38
→
X29
)
)
↔
(
¬
(
(
¬
(
(
¬
X38
)
→
X16
→
X39
)
)
→
(
¬
(
X39
→
(
¬
X17
)
→
X29
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X46
↔
(
X39
↔
X12
)
)
)
→
(
¬
(
(
X39
↔
(
X38
→
(
(
¬
X12
)
↔
X15
)
)
)
→
X37
→
X15
→
False
)
)
)
)
→
(
¬
(
(
(
¬
(
X38
↔
(
X15
→
False
)
)
)
↔
(
¬
(
X38
↔
(
X18
↔
(
(
¬
X12
)
↔
X19
)
)
)
)
)
→
(
¬
(
(
¬
X15
)
↔
(
X12
→
X18
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
X51
→
(
X36
↔
(
(
¬
X24
)
↔
X46
)
)
)
→
(
(
X36
↔
(
(
¬
X24
)
→
X51
)
)
↔
(
¬
(
(
¬
X24
)
→
X7
→
(
(
¬
X2
)
↔
X36
)
)
)
)
)
)
↔
(
(
X47
↔
(
X46
→
(
(
¬
X6
)
↔
X51
)
)
)
→
(
(
¬
(
X46
↔
(
(
¬
X0
)
→
X47
)
)
)
↔
(
(
¬
(
(
¬
X36
)
↔
(
(
¬
X0
)
→
X46
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X36
)
→
(
X0
↔
X51
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X46
↔
(
X30
↔
(
X27
↔
False
)
)
)
)
↔
(
¬
(
(
X36
↔
(
X30
↔
(
X20
→
X46
)
)
)
→
(
¬
(
X46
↔
(
X36
→
X30
→
X27
)
)
)
)
)
)
↔
(
X36
→
X46
)
)
→
(
¬
(
X36
→
X27
→
(
¬
X14
)
→
X38
)
)
→
(
¬
(
X38
↔
(
(
¬
X30
)
→
(
X14
↔
X45
)
)
)
)
)
)
→
(
¬
(
(
(
¬
X46
)
→
(
¬
X9
)
→
(
¬
X8
)
→
X47
)
→
(
(
(
¬
X32
)
↔
(
(
¬
X20
)
→
(
¬
X14
)
→
X47
)
)
↔
(
X20
↔
(
X14
→
(
X8
↔
X9
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
(
(
(
X31
↔
(
X28
↔
False
)
)
→
(
¬
(
X51
→
X34
→
(
X1
↔
False
)
)
)
)
→
X31
→
(
X1
↔
X21
)
)
)
→
(
(
¬
(
X31
→
X51
)
)
↔
(
X48
↔
(
X13
→
X34
)
)
)
→
(
¬
(
X51
→
X13
→
False
)
)
)
→
(
(
¬
(
(
¬
X49
)
↔
(
X33
↔
X50
)
)
)
↔
(
(
¬
(
(
(
¬
X50
)
→
(
¬
X49
)
→
(
¬
X33
)
→
X52
)
→
(
¬
(
X52
↔
(
X50
→
X49
→
(
X31
↔
False
)
)
)
)
)
)
↔
(
(
¬
(
X50
→
X34
→
X33
→
X52
)
)
↔
(
X50
↔
(
X33
→
X21
→
X34
)
)
)
)
)
)
)
→
(
(
(
¬
(
X49
→
(
¬
X47
)
→
(
X40
↔
X48
)
)
)
→
(
¬
(
X48
↔
(
X47
↔
(
X40
→
X22
)
)
)
)
→
(
¬
(
X47
→
(
X26
↔
X22
)
)
)
)
↔
(
(
X41
→
X3
→
X46
)
↔
(
¬
(
(
¬
(
(
X46
→
X3
→
X47
)
→
(
¬
(
X47
→
(
(
¬
X40
)
↔
X41
)
)
)
)
)
→
(
X49
→
(
X11
↔
X46
)
)
→
(
¬
(
X49
→
(
X46
↔
(
(
¬
X40
)
↔
X47
)
)
)
)
→
(
X49
↔
(
X46
↔
X11
)
)
)
)
)
)
→
(
(
(
X51
↔
(
X49
→
(
¬
X25
)
→
X47
)
)
↔
(
(
(
¬
(
X49
↔
(
X47
↔
(
X46
→
X51
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X51
→
(
¬
X44
)
→
(
¬
X43
)
→
X49
)
)
→
X47
→
(
X44
↔
(
(
¬
X43
)
↔
X49
)
)
)
)
→
(
X51
↔
(
X47
↔
X49
)
)
)
)
)
→
(
¬
(
X47
→
(
(
¬
X44
)
↔
(
X42
↔
X49
)
)
)
)
)
)
↔
(
¬
(
X47
↔
(
(
¬
X43
)
↔
X51
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
b714c1...
and proposition id is
f8151b...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMTiXadHb9WM5RUMeiG8HrxpNTNsVYAz8Go
)
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
∀X50 :
prop
,
∀X51 :
prop
,
∀X52 :
prop
,
(
(
(
(
(
(
¬
(
(
(
¬
(
(
¬
X30
)
→
(
X24
↔
(
X7
↔
X47
)
)
)
)
→
X36
→
X7
→
X47
)
→
(
(
(
(
¬
X33
)
↔
(
X24
↔
X47
)
)
↔
(
(
¬
(
X47
↔
(
(
¬
X24
)
↔
X30
)
)
)
→
(
(
(
¬
X18
)
↔
X33
)
↔
(
¬
(
X47
↔
(
X24
↔
(
(
¬
X18
)
↔
X30
)
)
)
)
)
)
)
↔
(
X47
↔
(
X33
→
X30
→
False
)
)
)
)
)
→
(
¬
(
(
¬
(
X40
↔
(
X33
↔
X51
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X30
↔
(
X25
→
X51
)
)
)
→
(
¬
(
(
¬
X36
)
↔
(
(
¬
X30
)
→
X25
→
X40
)
)
)
)
)
→
(
X51
↔
(
X36
→
X33
→
X40
)
)
)
)
)
)
)
↔
(
(
(
X51
↔
(
(
¬
X47
)
→
X40
→
X52
)
)
↔
(
X47
→
(
X39
↔
X52
)
)
)
→
(
¬
(
X37
↔
(
(
¬
X27
)
↔
X39
)
)
)
)
)
→
(
¬
(
(
(
(
(
X50
↔
(
X48
→
X52
)
)
↔
(
X52
↔
(
X50
→
X35
→
X41
)
)
)
→
(
¬
(
(
¬
(
(
¬
X51
)
→
(
X50
↔
(
X48
→
X52
)
)
)
)
→
(
¬
(
X52
↔
(
X51
↔
(
X42
→
False
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X50
→
X39
→
(
X34
↔
X42
)
)
)
↔
(
¬
(
(
¬
(
X50
→
X42
→
(
X13
↔
False
)
)
)
→
(
(
¬
X42
)
↔
(
(
¬
X13
)
↔
(
(
¬
X9
)
↔
X51
)
)
)
→
(
¬
(
(
¬
X39
)
↔
(
X9
→
X50
)
)
)
)
)
)
→
(
¬
(
X50
→
(
X42
↔
(
(
¬
X0
)
→
X51
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X50
→
(
X29
↔
(
X21
→
X51
)
)
)
)
↔
(
(
X41
→
(
X29
↔
X51
)
)
↔
(
(
X21
→
(
X16
↔
X51
)
)
↔
(
¬
(
(
¬
X28
)
→
X41
)
)
)
)
)
→
(
¬
(
(
¬
(
X41
→
(
X26
↔
(
X10
↔
X29
)
)
)
)
→
(
¬
(
X26
↔
(
(
¬
X19
)
↔
X28
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
X48
↔
(
(
¬
X6
)
↔
(
X3
↔
X23
)
)
)
→
X48
→
(
X23
↔
(
X2
→
X1
)
)
)
→
(
¬
(
(
¬
X23
)
→
(
X14
↔
(
(
¬
X1
)
↔
X31
)
)
)
)
)
→
(
(
(
¬
X44
)
→
(
(
¬
X31
)
↔
X48
)
)
↔
(
X50
↔
(
X32
→
(
¬
X31
)
→
X48
)
)
)
→
(
(
¬
(
X48
→
(
X14
↔
(
X11
↔
X32
)
)
)
)
↔
(
X50
→
X14
→
X11
)
)
)
→
(
¬
(
(
(
(
X46
→
(
X23
↔
(
X17
→
False
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X45
)
↔
X48
)
)
)
)
→
X43
→
X17
→
X23
)
→
(
(
(
(
(
¬
X48
)
→
(
(
¬
X23
)
↔
X52
)
)
→
(
X52
↔
(
X50
↔
(
X23
→
X4
)
)
)
)
↔
(
X50
→
(
¬
X4
)
→
X45
)
)
↔
(
X52
↔
(
(
¬
X15
)
→
X48
)
)
)
)
)
→
(
¬
(
(
¬
(
(
X45
→
X5
)
→
(
(
X50
↔
(
X49
→
X38
→
(
X5
↔
False
)
)
)
↔
(
¬
(
(
¬
X45
)
→
(
(
¬
X38
)
↔
X50
)
)
)
)
)
)
→
(
(
¬
(
X49
→
(
(
¬
X45
)
↔
(
(
¬
X22
)
↔
X50
)
)
)
)
↔
(
(
¬
(
X49
↔
(
X20
→
(
(
¬
X8
)
↔
X50
)
)
)
)
→
(
(
X52
↔
(
X49
↔
(
(
¬
X12
)
↔
X50
)
)
)
↔
(
(
¬
(
(
¬
X22
)
→
(
(
¬
X20
)
↔
X50
)
)
)
↔
(
¬
(
(
¬
X20
)
↔
(
(
¬
X12
)
↔
X49
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
(
X46
→
(
¬
X40
)
→
X47
)
↔
(
(
X47
→
X40
→
(
X14
↔
False
)
)
↔
(
(
(
¬
(
X47
↔
(
X46
→
(
X6
↔
False
)
)
)
)
→
(
(
¬
(
(
¬
(
X40
↔
X46
)
)
→
(
(
(
¬
(
X47
↔
(
X40
↔
(
X6
↔
X46
)
)
)
)
→
(
¬
(
(
¬
X46
)
↔
(
X40
→
(
X14
↔
X47
)
)
)
)
)
↔
(
¬
(
X47
→
X46
)
)
)
)
)
↔
(
¬
(
X40
→
X14
→
False
)
)
)
)
↔
(
X47
↔
(
X14
→
X40
)
)
)
)
)
↔
(
X46
→
X6
→
X47
)
)
↔
(
(
¬
(
(
¬
(
(
(
(
¬
X46
)
↔
(
(
¬
X40
)
→
(
¬
X2
)
→
X48
)
)
↔
(
(
¬
X40
)
→
(
(
¬
X2
)
↔
X50
)
)
)
→
X50
→
X48
→
(
(
¬
X3
)
↔
X46
)
)
)
→
(
X50
↔
X40
)
)
)
↔
(
(
¬
(
X40
→
X19
→
X50
)
)
→
(
(
X47
→
X40
→
X3
→
X50
)
↔
(
X50
→
X40
→
X19
→
X47
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
¬
X32
)
↔
X49
)
)
→
(
(
(
(
¬
X42
)
↔
(
X24
→
X49
)
)
↔
(
X47
→
X32
→
X24
)
)
↔
(
¬
(
(
X49
↔
(
X47
↔
(
X17
↔
X46
)
)
)
→
(
X49
↔
(
X47
↔
(
(
¬
X42
)
↔
X46
)
)
)
)
)
)
)
)
→
(
(
(
¬
X47
)
→
X46
→
X23
→
X48
)
↔
(
(
X48
↔
(
X34
↔
(
X26
↔
X47
)
)
)
↔
(
X47
↔
(
X23
→
X48
)
)
)
)
→
(
(
(
¬
(
X47
→
X46
→
X29
→
X18
)
)
→
(
¬
(
X46
↔
X49
)
)
)
↔
(
¬
(
(
¬
X34
)
↔
(
X29
↔
(
(
¬
X4
)
→
X49
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
(
X51
→
(
¬
X45
)
→
X46
)
↔
(
(
¬
X45
)
→
(
¬
X35
)
→
(
X9
↔
X51
)
)
)
→
(
¬
(
X45
↔
(
X44
→
X9
→
X51
)
)
)
)
)
→
(
(
¬
X45
)
↔
(
(
¬
X35
)
↔
X46
)
)
)
)
↔
(
¬
(
(
(
¬
X49
)
→
(
X5
↔
X51
)
)
→
(
(
(
(
(
X52
→
(
(
¬
X45
)
↔
(
(
¬
X0
)
↔
X49
)
)
)
→
(
¬
(
X52
↔
(
(
¬
X49
)
↔
(
(
¬
X45
)
→
X51
)
)
)
)
)
↔
(
¬
X51
)
)
↔
(
¬
(
X49
→
X51
)
)
)
↔
(
¬
(
X52
→
X49
→
X46
→
X21
→
False
)
)
)
)
)
)
↔
(
(
¬
(
X50
→
(
(
¬
X46
)
↔
(
X44
→
X52
)
)
)
)
→
(
(
¬
(
X50
→
(
¬
X7
)
→
X10
)
)
↔
(
(
X50
↔
(
(
¬
X10
)
↔
(
X7
↔
X52
)
)
)
→
(
¬
(
(
¬
X50
)
→
(
(
¬
X46
)
↔
X52
)
)
)
)
)
→
(
¬
(
X52
↔
(
(
¬
X10
)
→
X13
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
(
¬
(
(
¬
(
X38
↔
(
(
¬
X30
)
→
(
(
¬
X1
)
↔
X50
)
)
)
)
→
(
¬
(
X50
↔
(
(
¬
X30
)
→
(
(
¬
X16
)
↔
X38
)
)
)
)
)
)
→
X50
→
(
X30
↔
X16
)
)
→
(
X41
↔
(
(
¬
X38
)
↔
(
X16
↔
X50
)
)
)
)
↔
(
(
¬
(
X43
→
X41
→
(
X38
↔
(
X20
↔
False
)
)
)
)
↔
(
(
X50
→
(
(
¬
X38
)
↔
X41
)
)
→
(
(
(
X43
↔
(
(
¬
X38
)
↔
X41
)
)
↔
(
¬
(
(
¬
X41
)
→
(
X33
↔
(
X11
↔
X43
)
)
)
)
)
↔
(
¬
(
(
¬
X41
)
→
X38
→
X50
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
X50
→
(
(
¬
X33
)
↔
(
(
¬
X12
)
↔
X47
)
)
)
)
→
X47
→
(
¬
X33
)
→
(
X30
↔
X50
)
)
)
→
(
¬
(
(
¬
X30
)
↔
X47
)
)
)
↔
(
(
¬
(
X36
↔
(
(
¬
X27
)
→
X30
)
)
)
→
(
(
X36
↔
(
X28
→
(
(
¬
X27
)
↔
X47
)
)
)
↔
(
(
X47
→
(
X30
↔
(
X27
→
X28
)
)
)
↔
(
¬
(
(
¬
X33
)
↔
(
(
¬
X27
)
→
X47
)
)
)
)
)
)
)
)
→
(
(
(
(
(
X39
→
(
¬
X37
)
→
X8
→
X47
)
↔
(
X47
↔
(
X37
→
(
¬
X8
)
→
X22
)
)
)
↔
(
¬
(
X22
↔
(
X8
→
X47
)
)
)
)
↔
(
¬
(
(
¬
X22
)
↔
(
X8
↔
X50
)
)
)
)
↔
(
(
X47
→
(
(
¬
X45
)
↔
X50
)
)
↔
(
¬
(
(
(
(
X50
→
(
(
¬
X45
)
↔
(
(
¬
X41
)
→
X47
)
)
)
↔
(
¬
(
(
¬
X41
)
→
(
(
¬
X37
)
↔
X50
)
)
)
)
↔
(
¬
(
X47
→
(
X37
↔
X45
)
)
)
)
→
(
(
(
(
¬
X31
)
→
(
(
¬
X25
)
↔
(
X15
→
X50
)
)
)
→
(
¬
(
X37
→
X31
→
X50
)
)
)
↔
(
(
¬
X31
)
↔
(
X25
↔
X47
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
a30cc1...
and proposition id is
d88a4c...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMRAZyHKtP7mMpVscuuSf2tftFL3aq5TNkG
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
∀X50 :
prop
,
(
¬
∀X51 :
prop
,
∀X52 :
prop
,
(
(
(
(
(
(
¬
(
(
X39
↔
(
(
¬
X7
)
↔
X38
)
)
→
(
(
¬
(
(
¬
X7
)
↔
X38
)
)
↔
(
(
X38
↔
(
(
¬
X29
)
→
X7
→
X52
)
)
→
(
¬
(
X38
→
(
(
¬
X29
)
↔
X39
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X49
↔
(
(
¬
X38
)
→
(
¬
X28
)
→
X41
)
)
)
↔
(
(
¬
(
(
¬
X40
)
→
(
(
¬
X34
)
↔
X49
)
)
)
→
(
X39
↔
(
X38
↔
(
(
¬
X34
)
↔
X40
)
)
)
)
)
→
(
¬
X41
)
→
(
(
¬
X40
)
↔
(
X29
→
X49
)
)
)
)
↔
(
(
(
¬
(
(
¬
X30
)
→
(
¬
X19
)
→
X52
)
)
↔
(
(
X49
→
(
X41
↔
(
X10
→
False
)
)
)
→
(
(
¬
(
(
¬
X19
)
→
X10
→
(
X1
↔
X49
)
)
)
↔
(
(
(
¬
(
X49
→
(
¬
X1
)
→
X26
)
)
↔
(
(
X52
→
(
X30
↔
(
X26
↔
False
)
)
)
→
(
¬
(
X52
→
X26
→
(
X1
↔
X49
)
)
)
)
)
↔
(
X49
↔
(
X41
↔
(
(
¬
X10
)
↔
X26
)
)
)
)
)
)
)
↔
(
X50
↔
(
(
¬
X30
)
↔
(
X26
→
X49
)
)
)
)
)
→
(
(
(
(
¬
(
(
(
¬
X52
)
↔
(
(
X50
→
(
¬
X6
)
→
(
¬
X2
)
→
X52
)
↔
(
X47
↔
(
(
¬
X2
)
→
X50
)
)
)
)
→
(
¬
(
X52
↔
(
(
¬
X33
)
↔
(
X6
↔
X43
)
)
)
)
)
)
↔
(
(
¬
(
X50
→
(
X37
↔
X47
)
)
)
↔
(
¬
(
X50
↔
(
X45
↔
(
X37
↔
X52
)
)
)
)
)
)
→
(
(
(
X52
↔
(
X45
→
X49
)
)
↔
(
(
¬
(
(
¬
X45
)
↔
(
X37
↔
X49
)
)
)
↔
(
(
¬
(
X37
↔
(
X17
↔
X52
)
)
)
↔
(
¬
(
X49
↔
(
X37
↔
(
X17
→
X45
)
)
)
)
)
)
)
↔
(
(
¬
(
X45
→
(
¬
X37
)
→
X49
)
)
↔
(
¬
(
(
X49
→
X36
→
X5
)
→
(
¬
(
X49
→
(
X37
↔
(
(
¬
X36
)
→
X45
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
X50
→
(
X46
↔
(
X14
→
False
)
)
)
↔
(
(
(
¬
X50
)
↔
(
(
¬
X46
)
↔
X52
)
)
↔
(
X14
→
(
X13
↔
X52
)
)
)
)
↔
(
X52
↔
(
(
¬
X14
)
↔
(
(
¬
X13
)
↔
X47
)
)
)
)
→
(
(
(
¬
X46
)
→
X44
→
X50
)
↔
(
¬
(
X47
→
(
X44
↔
(
X20
→
False
)
)
)
)
)
)
)
↔
(
(
(
¬
X3
)
↔
X50
)
↔
(
X52
↔
(
X47
→
X21
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
X49
→
(
¬
X42
)
→
(
X16
↔
X47
)
)
)
↔
(
(
(
¬
(
X9
→
X31
)
)
→
(
¬
(
X49
↔
(
(
¬
X42
)
↔
(
X9
↔
X47
)
)
)
)
)
↔
(
X49
↔
X42
)
)
)
→
(
¬
(
(
¬
X47
)
↔
(
X22
↔
(
(
¬
X9
)
↔
X49
)
)
)
)
)
)
→
(
¬
(
(
X48
↔
(
(
¬
X47
)
→
X49
)
)
→
(
(
(
¬
(
(
¬
X42
)
↔
X49
)
)
→
(
(
¬
(
X48
→
(
(
¬
X47
)
↔
(
(
¬
X42
)
↔
X49
)
)
)
)
↔
(
¬
(
X51
↔
(
X49
→
X42
→
X47
)
)
)
)
)
↔
(
(
X48
↔
(
(
¬
X47
)
↔
(
X32
→
X49
)
)
)
↔
(
(
X48
↔
(
(
¬
X47
)
↔
(
(
¬
X45
)
↔
X49
)
)
)
↔
(
X49
→
X48
→
(
X47
↔
X25
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
X24
↔
(
X18
→
X15
)
)
↔
(
(
¬
(
X24
→
(
X11
↔
X15
)
)
)
→
(
¬
(
(
¬
(
X15
↔
(
(
¬
X12
)
↔
(
X11
↔
X18
)
)
)
)
→
(
¬
(
(
¬
X11
)
↔
X24
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X23
→
(
¬
X4
)
→
X12
)
)
↔
(
(
X47
↔
(
X35
↔
(
X32
→
False
)
)
)
↔
(
¬
(
(
¬
(
X35
↔
(
X32
↔
(
X23
→
(
X12
↔
False
)
)
)
)
)
→
(
¬
(
X47
↔
(
X32
↔
(
X12
→
X23
)
)
)
)
)
)
)
)
→
(
(
(
(
(
(
(
(
¬
(
(
¬
X11
)
↔
(
(
¬
X8
)
↔
X47
)
)
)
→
(
(
¬
X47
)
↔
(
X8
↔
X51
)
)
→
(
(
¬
X11
)
↔
(
(
¬
X0
)
↔
X47
)
)
)
↔
(
¬
(
X51
↔
(
(
¬
X0
)
→
X47
)
)
)
)
↔
(
¬
(
X51
→
X0
)
)
)
↔
(
X51
→
(
(
¬
X0
)
↔
X47
)
)
)
→
(
¬
(
X51
→
X47
→
(
(
¬
X11
)
↔
X24
)
)
)
)
→
X51
→
(
X47
↔
(
X35
↔
X11
)
)
)
↔
(
(
¬
(
X47
↔
(
X27
↔
X4
)
)
)
↔
(
(
X51
↔
(
X47
→
X35
)
)
↔
(
X51
→
X47
→
(
X27
↔
X4
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
(
¬
(
(
¬
X14
)
→
X2
→
X44
)
)
→
(
X19
↔
(
(
¬
X2
)
→
X25
)
)
→
(
(
X12
↔
(
(
¬
X2
)
→
X25
)
)
↔
(
¬
(
X19
→
(
(
¬
X12
)
↔
(
X2
↔
X14
)
)
)
)
)
)
)
→
(
¬
(
(
(
X12
↔
(
X2
↔
X19
)
)
↔
(
X45
↔
(
X27
↔
(
X19
→
X2
→
False
)
)
)
)
→
(
¬
(
X45
↔
(
X19
↔
(
X12
↔
(
X9
↔
False
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
(
X43
↔
(
(
¬
X19
)
→
X31
)
)
↔
(
¬
(
X43
↔
(
(
¬
X42
)
→
X47
)
)
)
)
→
(
¬
(
(
¬
X31
)
→
(
X19
↔
X43
)
)
)
)
)
↔
(
X47
→
(
X43
↔
(
X31
↔
X12
)
)
)
)
↔
(
X31
→
X19
→
(
¬
X6
)
→
X47
)
)
↔
(
(
¬
X43
)
↔
(
(
¬
X19
)
↔
X47
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X47
→
(
(
¬
X29
)
↔
(
X28
→
X42
)
)
)
)
→
(
(
¬
X29
)
↔
(
X3
→
X52
)
)
→
(
¬
(
X32
↔
(
(
¬
X29
)
→
X47
)
)
)
)
→
(
(
X51
↔
(
(
¬
X48
)
↔
(
X35
↔
X52
)
)
)
→
(
(
X47
↔
(
X41
↔
X48
)
)
↔
(
X43
→
(
X42
↔
(
X41
→
(
X35
↔
False
)
)
)
)
)
)
→
X52
→
X51
→
(
X43
↔
X11
)
)
)
)
↔
(
(
(
(
¬
(
X47
→
(
X37
↔
(
X22
↔
False
)
)
)
)
→
(
(
(
¬
(
X51
↔
(
(
¬
X5
)
→
X45
)
)
)
↔
(
¬
(
X51
→
(
(
¬
X22
)
↔
(
(
¬
X5
)
↔
X50
)
)
)
)
)
↔
(
¬
(
(
(
(
X47
↔
(
X37
↔
False
)
)
↔
(
¬
(
X51
→
(
(
¬
X37
)
↔
(
(
¬
X22
)
→
X40
)
)
)
)
)
↔
(
¬
(
X50
↔
(
(
¬
X47
)
↔
(
(
¬
X37
)
↔
X51
)
)
)
)
)
→
(
X51
↔
(
(
¬
X20
)
↔
(
X4
↔
X37
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
(
¬
(
(
¬
X47
)
→
(
(
¬
X18
)
↔
X51
)
)
)
→
(
X46
↔
(
X18
→
(
(
¬
X0
)
↔
X47
)
)
)
)
)
→
(
¬
(
X47
→
(
(
¬
X46
)
↔
X51
)
)
)
)
↔
(
X51
↔
(
X47
→
(
X1
↔
X0
)
)
)
)
→
(
(
(
X51
→
(
(
¬
X24
)
↔
(
(
¬
X13
)
↔
X46
)
)
)
↔
(
(
(
(
¬
X46
)
↔
(
X21
→
X51
)
)
→
(
X46
↔
(
X39
↔
(
X24
↔
(
X21
↔
False
)
)
)
)
)
↔
(
(
¬
X39
)
↔
(
X24
↔
X46
)
)
)
)
→
(
¬
(
X46
→
X24
→
X39
)
)
)
→
(
¬
(
X51
↔
(
X47
→
X13
)
)
)
)
→
(
¬
(
X51
→
X39
→
X21
→
X50
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
(
(
(
¬
X34
)
↔
(
(
¬
X16
)
→
X46
)
)
→
X34
→
(
X30
↔
(
X16
→
X26
)
)
)
→
(
¬
(
(
¬
X30
)
↔
(
(
¬
X17
)
→
(
X16
↔
X46
)
)
)
)
)
→
(
(
¬
(
X38
↔
(
(
¬
X17
)
→
(
(
¬
X8
)
↔
X34
)
)
)
)
↔
(
X17
→
(
X8
↔
X46
)
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X30
)
→
X17
→
X46
)
)
)
)
)
→
(
(
¬
X36
)
→
(
(
¬
X34
)
↔
(
X10
→
X47
)
)
)
→
(
X47
→
(
(
¬
X16
)
↔
X33
)
)
→
(
¬
(
(
¬
X34
)
↔
(
(
¬
X16
)
↔
(
(
¬
X10
)
→
X47
)
)
)
)
)
)
↔
(
¬
(
(
X30
→
(
¬
X23
)
→
X17
→
X45
)
→
(
(
¬
(
X36
↔
(
(
¬
X17
)
↔
(
(
¬
X7
)
→
X23
)
)
)
)
↔
(
¬
(
X49
→
X45
→
X17
→
X15
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
99852f...
and proposition id is
30fd84...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMKLBN8c1hk8t2Fix6qcfH39BWnCWNNL23s
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
∀X50 :
prop
,
∀X51 :
prop
,
(
(
¬
(
(
(
(
(
(
(
(
(
¬
(
(
¬
X30
)
↔
(
(
¬
X28
)
→
(
X6
↔
X44
)
)
)
)
↔
(
X15
→
(
¬
X6
)
→
X28
)
)
→
(
X48
↔
(
(
¬
X30
)
↔
(
X28
→
X44
)
)
)
)
→
(
¬
(
(
¬
(
X48
↔
(
X25
→
(
X20
↔
X44
)
)
)
)
→
(
¬
(
X30
↔
(
X25
↔
X35
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
X30
)
→
(
X27
↔
(
(
¬
X23
)
→
X35
)
)
)
→
(
¬
(
X30
→
(
(
¬
X27
)
↔
(
(
¬
X23
)
↔
X48
)
)
)
)
)
)
)
→
(
(
(
¬
(
X44
↔
(
X43
→
X7
→
X30
)
)
)
↔
(
(
(
¬
X30
)
↔
(
(
¬
X10
)
→
X44
)
)
↔
(
X43
→
(
X23
↔
(
X10
→
X9
→
False
)
)
)
)
)
↔
(
X44
→
X21
→
(
¬
X9
)
→
X10
)
)
)
→
(
(
X50
↔
(
X48
→
(
X27
↔
X41
)
)
)
↔
(
(
¬
(
(
¬
(
X24
→
X50
)
)
→
(
(
¬
(
X48
↔
X50
)
)
↔
(
¬
(
X41
↔
(
(
¬
X35
)
↔
(
X24
→
X48
)
)
)
)
)
)
)
↔
(
¬
(
X48
→
(
¬
X35
)
→
(
¬
X0
)
→
X41
)
)
)
)
)
↔
(
(
(
(
¬
(
X29
↔
(
X1
→
X44
)
)
)
→
(
(
¬
(
X44
→
(
X29
↔
(
X12
↔
X36
)
)
)
)
↔
(
(
¬
(
(
¬
X36
)
↔
(
(
¬
X17
)
↔
X44
)
)
)
↔
(
X44
↔
(
X12
↔
(
X1
→
X17
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X44
↔
(
(
¬
X5
)
→
(
(
¬
X2
)
↔
X29
)
)
)
)
→
(
¬
(
X29
↔
(
(
¬
X5
)
→
(
(
¬
X2
)
↔
X36
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
X50
→
(
X38
↔
(
(
¬
X36
)
→
X44
)
)
)
↔
(
¬
(
X44
↔
(
(
¬
X38
)
→
X13
→
X50
)
)
)
)
↔
(
(
X46
↔
(
X44
→
X32
→
X8
)
)
↔
(
¬
(
X38
↔
(
X32
→
(
(
¬
X8
)
↔
X50
)
)
)
)
)
)
→
(
(
¬
(
(
¬
X48
)
↔
(
(
¬
X44
)
→
X50
)
)
)
↔
(
(
X48
↔
(
X17
↔
X50
)
)
↔
(
(
(
¬
X46
)
↔
(
X44
→
(
¬
X17
)
→
X50
)
)
→
(
¬
(
(
¬
X44
)
→
(
¬
X17
)
→
X48
)
)
)
)
)
)
)
)
)
→
(
(
(
X50
→
(
X45
↔
(
(
¬
X11
)
→
X14
)
)
)
→
(
(
¬
(
X47
→
(
(
¬
X45
)
↔
(
(
¬
X44
)
→
X49
)
)
)
)
↔
(
(
X47
→
(
¬
X11
)
→
X50
)
→
(
¬
(
X50
→
X49
→
X44
→
X11
)
)
)
)
)
→
(
(
X51
→
(
(
¬
X48
)
↔
(
(
¬
X16
)
→
X50
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
¬
X50
)
→
(
(
¬
X48
)
↔
(
(
¬
X44
)
↔
X51
)
)
)
)
→
(
¬
(
(
X51
→
(
¬
X44
)
→
(
(
¬
X31
)
↔
X48
)
)
→
(
¬
(
X44
→
(
X39
↔
X31
)
)
)
)
)
)
→
(
(
¬
X50
)
↔
(
X44
→
X51
)
)
)
)
→
(
(
¬
X50
)
↔
(
(
¬
X44
)
↔
X51
)
)
)
)
→
(
¬
(
(
¬
(
X44
→
X40
)
)
→
(
¬
(
X51
→
(
X47
↔
(
X44
↔
(
X33
→
False
)
)
)
)
)
)
)
→
(
¬
(
(
X51
↔
(
X39
↔
(
(
¬
X34
)
↔
X44
)
)
)
→
(
¬
(
(
(
¬
X19
)
→
X40
)
→
(
(
¬
(
(
¬
(
(
¬
X40
)
↔
(
(
¬
X34
)
→
X44
)
)
)
→
(
¬
(
(
¬
(
(
¬
X39
)
↔
(
(
¬
X34
)
→
X44
)
)
)
→
(
¬
(
(
(
¬
X39
)
↔
(
X34
→
X19
→
X44
)
)
→
(
X44
↔
(
(
¬
X39
)
→
(
(
¬
X34
)
↔
X40
)
)
)
)
)
)
)
)
)
↔
(
X44
↔
(
X39
→
X40
)
)
)
)
)
)
)
)
→
(
(
(
(
X50
↔
(
X47
→
X26
→
X4
)
)
↔
(
(
X47
↔
(
X45
→
X18
→
X50
)
)
↔
(
(
¬
X47
)
→
(
X26
↔
(
(
¬
X18
)
→
X50
)
)
)
)
)
→
(
X50
↔
(
X42
↔
(
X18
↔
(
X4
→
False
)
)
)
)
)
↔
(
(
¬
(
X22
→
(
X3
↔
X37
)
)
)
↔
(
¬
(
(
X45
→
(
X26
↔
(
X22
↔
X3
)
)
)
→
(
X50
↔
(
X45
→
(
X26
↔
X22
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
¬
X33
)
→
X15
→
X13
→
X47
)
↔
(
¬
(
(
(
X25
↔
False
)
↔
(
(
¬
X36
)
↔
(
(
¬
X35
)
→
(
¬
X33
)
→
X47
)
)
)
→
(
¬
(
(
¬
(
(
X47
↔
(
X37
→
(
X36
↔
(
X33
↔
False
)
)
)
)
→
(
¬
X33
)
→
(
¬
X15
)
→
X47
)
)
→
(
¬
(
X36
→
(
(
¬
X33
)
↔
X37
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
X49
↔
(
X45
→
(
¬
X37
)
→
X44
)
)
)
↔
(
(
(
(
(
¬
(
(
¬
X45
)
→
(
X44
↔
(
X29
→
X47
)
)
)
)
→
(
¬
(
(
¬
X45
)
↔
X49
)
)
)
→
X49
→
(
¬
X29
)
→
X47
)
→
(
X47
↔
(
(
¬
X29
)
→
X45
)
)
)
↔
(
¬
(
X47
↔
(
X44
↔
X29
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X43
↔
(
(
¬
X11
)
↔
(
(
¬
X4
)
↔
X44
)
)
)
)
→
(
(
(
X47
→
(
¬
X4
)
→
X44
)
→
(
¬
(
X49
↔
(
X11
→
X4
→
False
)
)
)
)
↔
(
X47
→
(
¬
X44
)
→
(
(
¬
X16
)
↔
X49
)
)
)
)
)
→
(
(
¬
(
X51
→
(
(
¬
X43
)
↔
X47
)
)
)
↔
(
(
(
(
¬
(
(
¬
X43
)
→
X36
→
X51
)
)
↔
(
¬
(
X47
→
(
¬
X41
)
→
X43
)
)
)
↔
(
¬
(
X51
↔
(
(
¬
X41
)
→
X43
)
)
)
)
↔
(
(
¬
X36
)
↔
X43
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
X32
→
X9
→
(
X8
↔
X50
)
)
)
→
(
¬
(
X48
→
(
(
¬
X8
)
↔
X32
)
)
)
)
↔
(
¬
(
X9
→
X50
)
)
)
↔
(
(
X48
→
X31
→
X50
)
↔
(
¬
(
(
X50
↔
(
X48
→
(
X31
↔
False
)
)
)
→
(
X45
↔
(
X42
→
(
X31
↔
X50
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
X45
)
→
(
(
¬
X38
)
↔
(
(
¬
X32
)
↔
X50
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X47
)
↔
(
X42
→
(
(
¬
X19
)
↔
X48
)
)
)
)
→
(
(
¬
(
X42
→
(
(
¬
X34
)
↔
(
(
¬
X19
)
→
X47
)
)
)
)
↔
(
¬
(
X42
↔
(
X34
→
X2
→
X19
)
)
)
)
)
↔
(
¬
(
(
(
¬
X34
)
↔
(
(
¬
X22
)
→
(
(
¬
X12
)
↔
X42
)
)
)
→
(
¬
(
X49
↔
(
X47
↔
(
X42
↔
False
)
)
)
)
)
)
)
→
(
¬
(
X48
↔
(
X47
↔
(
X22
→
X50
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X26
)
→
X21
→
(
¬
X17
)
→
X40
)
)
↔
(
(
¬
(
(
(
(
(
¬
X39
)
→
(
X6
↔
X51
)
)
↔
(
(
¬
(
(
(
(
¬
X46
)
→
X18
→
(
¬
X6
)
→
X51
)
↔
(
¬
(
X51
→
X40
)
)
)
→
X51
→
X6
→
X18
)
)
↔
(
X51
→
X46
→
X18
→
X39
)
)
)
→
(
¬
(
(
(
¬
X18
)
→
X27
)
→
(
(
(
¬
X27
)
↔
(
(
¬
X18
)
↔
X46
)
)
↔
(
X40
→
(
¬
X18
)
→
X21
)
)
)
)
)
→
X46
→
(
(
¬
X21
)
↔
X39
)
)
)
↔
(
(
X51
→
(
¬
X21
)
→
X39
)
↔
(
¬
(
(
¬
(
(
¬
X20
)
→
(
¬
X17
)
→
(
X10
↔
X39
)
)
)
→
(
X39
→
(
X20
↔
(
X10
↔
X7
)
)
)
→
X39
→
(
(
¬
X20
)
↔
(
(
¬
X7
)
↔
X21
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
X48
↔
(
X30
↔
(
(
¬
X23
)
→
X47
)
)
)
→
(
(
(
¬
X47
)
→
(
(
¬
X30
)
↔
(
(
¬
X14
)
→
X48
)
)
)
↔
(
(
(
(
¬
X47
)
↔
(
X30
→
X24
→
X48
)
)
↔
(
(
¬
X27
)
→
(
(
¬
X24
)
↔
X47
)
)
)
↔
(
X30
→
(
(
¬
X27
)
↔
X48
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X30
)
→
(
X0
↔
X48
)
)
)
→
(
¬
(
(
¬
(
X26
→
(
X3
↔
X48
)
)
)
→
(
¬
(
(
¬
(
X48
↔
(
(
¬
X0
)
↔
X3
)
)
)
→
(
(
X48
→
(
X23
↔
(
X0
→
False
)
)
)
↔
(
¬
(
X23
→
(
X3
↔
(
X0
↔
X26
)
)
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X26
↔
(
(
¬
X5
)
→
X1
→
X28
)
)
)
→
(
(
(
¬
(
X51
→
X30
→
X28
→
(
X26
↔
False
)
)
)
↔
(
¬
(
X48
→
(
X30
↔
(
(
¬
X26
)
→
X28
)
)
)
)
)
↔
(
X51
→
(
(
¬
X26
)
↔
(
(
¬
X5
)
→
X30
)
)
)
)
)
→
(
¬
(
X51
↔
(
X28
↔
(
X5
→
False
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
37600a...
and proposition id is
3d29b6...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMb3tYzb3e775brfxKDN8TF5VZai2AfjWCH
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
(
(
¬
(
(
(
(
¬
(
(
X50
→
(
X26
↔
(
(
¬
X5
)
↔
X42
)
)
)
→
X50
→
(
X42
↔
(
(
¬
X5
)
→
X28
)
)
)
)
↔
(
¬
(
(
(
(
¬
X37
)
→
X28
→
(
¬
X21
)
→
X48
)
→
(
¬
(
X50
↔
(
X48
→
X37
→
X28
)
)
)
)
→
(
¬
(
X50
→
(
X48
↔
(
X28
↔
X42
)
)
)
)
)
)
)
→
X42
→
X0
→
X21
)
→
(
(
(
¬
(
X48
→
(
X37
↔
(
X8
↔
False
)
)
)
)
↔
(
X44
→
(
X38
↔
(
X37
→
X8
)
)
)
)
↔
(
¬
(
(
(
X44
→
(
X41
↔
(
(
¬
X37
)
↔
X48
)
)
)
→
X41
→
X24
→
X48
)
→
(
¬
(
X48
→
(
(
¬
X42
)
↔
(
(
¬
X24
)
↔
X44
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
¬
(
(
¬
X47
)
↔
(
X42
↔
(
X9
↔
X48
)
)
)
)
↔
(
¬
(
X47
↔
(
(
¬
X44
)
↔
X48
)
)
)
)
→
X48
→
X42
→
X2
→
False
)
→
X47
→
X42
→
(
¬
X35
)
→
X48
)
→
(
(
X46
↔
(
(
¬
X35
)
↔
(
(
¬
X11
)
→
X48
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
¬
X35
)
↔
(
(
¬
X31
)
↔
(
(
¬
X30
)
→
X46
)
)
)
)
↔
(
(
¬
X35
)
↔
(
(
¬
X31
)
→
X11
→
X46
)
)
)
↔
(
X35
↔
(
X31
→
X46
)
)
)
→
(
X46
↔
(
X44
↔
(
X35
↔
(
X11
→
False
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
(
(
(
¬
X19
)
→
X14
→
X6
→
X46
)
↔
(
¬
(
X31
↔
(
X14
↔
X46
)
)
)
)
→
(
¬
(
X31
→
(
X20
↔
X19
)
)
)
)
)
↔
(
(
¬
X42
)
↔
(
(
¬
X20
)
→
X19
→
X46
)
)
)
→
(
X46
↔
(
(
¬
X31
)
↔
(
(
¬
X15
)
↔
X42
)
)
)
)
)
↔
(
¬
(
X46
→
X42
→
(
X20
↔
X4
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
(
X45
↔
(
X27
→
X12
)
)
)
→
(
X49
↔
(
X47
↔
(
X25
→
X12
→
False
)
)
)
)
)
→
(
¬
(
(
X49
→
X27
→
X17
)
→
(
X33
↔
(
X27
→
X13
→
X46
)
)
→
(
X46
↔
(
X25
↔
(
X13
→
False
)
)
)
)
)
)
↔
(
(
(
X36
↔
(
(
¬
X25
)
→
(
X22
↔
X40
)
)
)
→
(
¬
(
X40
↔
(
X39
↔
(
(
¬
X13
)
↔
X25
)
)
)
)
→
(
¬
(
X42
→
X40
)
)
)
↔
(
¬
(
(
(
(
X49
→
(
X42
↔
(
X36
↔
X39
)
)
)
→
(
¬
(
X42
↔
(
(
¬
X36
)
→
(
¬
X7
)
→
X39
)
)
)
)
→
(
X39
↔
(
X36
↔
X7
)
)
)
→
(
¬
(
(
¬
X36
)
→
(
X16
↔
X49
)
)
)
→
(
(
¬
X39
)
↔
(
(
¬
X29
)
↔
(
X16
↔
X49
)
)
)
)
)
)
)
↔
(
¬
(
(
(
X43
→
X40
→
(
(
¬
X23
)
↔
X39
)
)
↔
(
X43
→
X40
→
X39
→
X49
)
)
→
(
¬
(
(
(
(
¬
(
X32
→
(
X18
↔
(
X1
→
False
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X32
)
→
(
X1
↔
X34
)
)
)
↔
(
¬
(
(
¬
X32
)
↔
(
X18
→
(
¬
X1
)
→
X40
)
)
)
)
→
(
X40
↔
(
X18
↔
X32
)
)
)
)
)
↔
(
¬
(
(
(
¬
X34
)
→
(
X32
↔
(
(
¬
X10
)
↔
X39
)
)
)
→
(
¬
(
X40
→
(
¬
X32
)
→
X34
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X40
↔
(
(
¬
X34
)
→
(
¬
X3
)
→
X39
)
)
)
↔
(
¬
(
X40
→
(
X39
↔
(
(
¬
X34
)
↔
X43
)
)
)
)
)
↔
(
(
¬
X3
)
→
X34
)
)
→
(
¬
(
X43
→
X40
→
(
X34
↔
False
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
X38
↔
(
X17
↔
False
)
)
↔
(
¬
(
(
¬
(
X48
→
(
X43
↔
X38
)
)
)
→
(
(
¬
X43
)
↔
(
(
¬
X38
)
↔
(
(
¬
X18
)
→
X48
)
)
)
→
(
¬
(
X48
→
X38
→
(
¬
X5
)
→
X18
)
)
)
)
)
→
(
¬
(
(
X46
→
(
X10
↔
X48
)
)
→
(
(
(
X46
↔
(
(
¬
X5
)
↔
X48
)
)
↔
(
(
¬
X46
)
↔
(
X41
→
X10
→
X48
)
)
)
↔
(
X43
↔
(
X41
→
X10
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
(
X41
↔
X46
)
↔
(
¬
(
(
(
¬
X45
)
↔
(
X41
↔
(
X37
→
X46
)
)
)
→
(
¬
(
(
¬
X45
)
→
(
X37
↔
(
(
¬
X22
)
→
X46
)
)
)
)
)
)
)
→
(
(
¬
X41
)
↔
(
(
¬
X37
)
→
X45
)
)
)
)
↔
(
¬
(
X45
→
X37
→
X22
)
)
)
→
(
(
X48
→
(
X45
↔
X41
)
)
↔
(
(
X48
↔
(
X46
→
(
¬
X24
)
→
X45
)
)
↔
(
¬
(
X46
↔
(
X45
↔
(
X41
→
X48
)
)
)
)
)
)
)
)
→
(
¬
(
X45
→
(
X24
↔
(
X23
→
X46
)
)
)
)
→
X46
→
(
X45
↔
(
(
¬
X6
)
→
X23
)
)
)
)
→
(
(
¬
(
(
(
¬
(
(
¬
(
(
¬
X13
)
↔
(
(
¬
X7
)
→
X35
)
)
)
→
(
¬
(
(
¬
X26
)
↔
(
X12
↔
X35
)
)
)
)
)
↔
(
(
¬
(
X26
↔
(
(
¬
X12
)
→
(
¬
X3
)
→
X33
)
)
)
↔
(
(
¬
(
(
X26
↔
(
X11
↔
(
X2
→
X12
)
)
)
→
(
(
(
X35
→
X11
)
↔
(
(
¬
(
X41
↔
(
(
¬
X2
)
→
X26
)
)
)
↔
(
(
¬
(
(
¬
X35
)
→
(
(
¬
X11
)
↔
(
(
¬
X2
)
↔
X41
)
)
)
)
↔
(
(
¬
X35
)
→
(
(
¬
X26
)
↔
(
(
¬
X11
)
→
X41
)
)
)
)
)
)
↔
(
¬
(
X2
↔
False
)
)
)
)
)
↔
(
X41
→
(
¬
X12
)
→
X35
)
)
)
)
→
(
(
¬
(
X35
→
(
(
¬
X33
)
↔
(
(
¬
X1
)
↔
X41
)
)
)
)
↔
(
X35
→
(
X33
↔
(
X4
↔
(
X1
↔
False
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
(
¬
(
X45
→
(
¬
X33
)
→
(
(
¬
X8
)
↔
X41
)
)
)
↔
(
¬
(
X33
→
X16
→
(
X8
↔
False
)
)
)
)
→
(
(
¬
X41
)
↔
(
(
¬
X33
)
↔
X45
)
)
)
)
↔
(
¬
(
X41
↔
(
(
¬
X16
)
↔
X33
)
)
)
)
→
(
¬
(
X41
↔
(
X29
↔
(
(
¬
X8
)
→
X45
)
)
)
)
)
→
(
¬
(
(
¬
(
X41
↔
(
(
¬
X29
)
↔
X46
)
)
)
→
(
(
¬
(
(
¬
X41
)
↔
(
X29
→
(
¬
X13
)
→
X46
)
)
)
→
(
¬
(
X13
↔
(
X0
→
X46
)
)
)
)
→
(
¬
X16
)
→
X13
→
(
X0
↔
X46
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X46
→
X40
→
(
(
¬
X20
)
↔
X28
)
)
)
↔
(
X46
↔
(
(
¬
X20
)
→
X36
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
¬
X40
)
→
(
¬
X27
)
→
X49
)
)
→
(
(
X42
→
X27
)
↔
(
(
¬
(
X49
↔
(
X40
↔
(
X34
↔
(
X27
↔
False
)
)
)
)
)
↔
(
¬
(
X49
↔
(
X34
→
X42
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
X40
)
↔
(
(
¬
X36
)
→
(
(
¬
X25
)
↔
X46
)
)
)
)
↔
(
X46
→
(
X44
↔
(
X40
→
(
X36
↔
False
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
X46
)
→
X44
→
X40
→
X50
)
)
)
↔
(
(
(
(
(
(
¬
X48
)
↔
(
(
¬
X40
)
↔
X50
)
)
↔
(
(
¬
(
X50
↔
(
(
¬
X14
)
↔
X40
)
)
)
↔
(
(
X50
↔
(
X43
↔
(
X30
↔
X14
)
)
)
→
X50
→
(
¬
X40
)
→
X48
)
)
)
↔
(
(
(
X50
→
(
X47
↔
(
(
¬
X21
)
↔
X48
)
)
)
↔
(
¬
(
X50
→
(
X47
↔
(
(
¬
X43
)
→
X48
)
)
)
)
)
→
(
¬
(
X50
→
(
X47
↔
X48
)
)
)
)
)
↔
(
(
¬
(
X39
→
X32
→
(
X15
↔
False
)
)
)
↔
(
(
¬
(
X40
→
(
¬
X15
)
→
X43
)
)
→
(
¬
(
X47
→
(
X40
↔
(
X32
↔
X15
)
)
)
)
)
)
)
↔
(
(
(
¬
X47
)
→
(
¬
X40
)
→
(
¬
X39
)
→
X49
)
↔
(
(
¬
(
X49
→
(
X47
↔
(
X19
↔
False
)
)
)
)
→
(
(
X49
→
(
(
¬
X31
)
↔
(
X9
↔
X40
)
)
)
↔
(
(
¬
(
X47
↔
(
X40
↔
(
X19
→
X49
)
)
)
)
→
(
¬
(
X49
↔
(
X40
↔
(
(
¬
X19
)
→
X31
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
fd9dd8...
and proposition id is
d34c0b...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMdK3H8zJm3r33Bbe3wCswEZAHTXvUhq1rr
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
(
(
(
¬
X29
)
↔
(
X19
→
(
¬
X16
)
→
X40
)
)
→
X40
→
(
¬
X29
)
→
X19
→
X41
)
↔
(
¬
(
X41
→
(
X23
↔
(
X16
↔
X18
)
)
)
)
)
→
(
(
¬
(
(
(
(
¬
X33
)
↔
(
X19
→
X37
)
)
→
(
¬
(
X43
→
X37
→
(
¬
X19
)
→
X41
)
)
)
→
(
X43
↔
(
X37
↔
(
(
¬
X29
)
→
X41
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X43
↔
(
X15
→
(
(
¬
X8
)
↔
X41
)
)
)
)
→
(
¬
(
(
¬
X37
)
↔
(
X15
→
(
(
¬
X8
)
↔
X41
)
)
)
)
)
)
→
X43
→
X8
→
X27
)
↔
(
(
¬
X42
)
↔
(
(
¬
X37
)
↔
(
X27
→
X43
)
)
)
)
)
)
↔
(
(
(
(
X40
↔
(
(
¬
X26
)
↔
(
(
¬
X7
)
→
X37
)
)
)
↔
(
(
(
(
¬
(
(
¬
X22
)
→
(
(
¬
X20
)
↔
(
(
¬
X12
)
↔
X37
)
)
)
)
↔
(
X37
↔
(
X12
↔
(
X2
→
(
X1
↔
False
)
)
)
)
)
↔
(
(
¬
(
X37
↔
(
X26
↔
(
X20
↔
X40
)
)
)
)
↔
(
(
¬
X26
)
→
X22
→
(
X20
↔
X37
)
)
)
)
↔
(
¬
(
X43
↔
(
(
¬
X26
)
↔
(
X22
↔
X40
)
)
)
)
)
)
↔
(
(
¬
(
X43
→
(
(
¬
X40
)
↔
(
X34
↔
X47
)
)
)
)
↔
(
¬
(
(
X37
→
(
X5
↔
X32
)
)
→
(
¬
(
X37
↔
(
(
¬
X34
)
→
(
X32
↔
X40
)
)
)
)
)
)
)
)
→
(
(
(
X47
→
(
X40
↔
(
(
¬
X34
)
→
X49
)
)
)
↔
(
X49
↔
(
X40
→
(
(
¬
X3
)
↔
X47
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
X40
→
X49
)
↔
(
X49
→
(
(
¬
X34
)
↔
(
X14
→
X35
)
)
)
)
→
(
¬
(
X34
↔
(
(
¬
X14
)
↔
X49
)
)
)
)
)
→
(
¬
(
X35
↔
X14
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
¬
X48
)
↔
(
X28
↔
X49
)
)
)
↔
(
(
¬
(
X48
↔
(
(
¬
X28
)
→
X11
→
X47
)
)
)
↔
(
¬
(
X47
→
(
X44
↔
(
(
¬
X11
)
→
X49
)
)
)
)
)
)
↔
(
(
X31
↔
(
(
¬
X10
)
→
X47
)
)
↔
(
X48
→
(
X10
↔
X49
)
)
)
)
→
(
(
¬
(
X48
↔
(
X36
→
X24
→
X44
)
)
)
→
(
¬
(
X49
→
(
(
¬
X30
)
↔
(
X24
↔
X48
)
)
)
)
)
→
(
¬
(
(
¬
X48
)
↔
(
X44
→
(
(
¬
X39
)
↔
X49
)
)
)
)
)
→
(
(
(
(
(
¬
(
X45
→
(
X42
↔
(
X6
→
X0
)
)
)
)
↔
(
X45
↔
(
(
¬
X9
)
→
X0
→
X42
)
)
)
→
(
¬
(
X42
→
(
X9
↔
(
X6
→
X49
)
)
)
)
)
↔
(
¬
(
(
¬
X45
)
↔
(
X25
→
(
X9
↔
X49
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
(
¬
(
X46
→
(
¬
X21
)
→
(
¬
X4
)
→
X45
)
)
↔
(
¬
(
X21
↔
(
(
¬
X4
)
↔
X48
)
)
)
)
↔
(
¬
(
X45
→
X17
→
(
X4
↔
X21
)
)
)
)
→
(
¬
(
X48
→
X46
→
(
X45
↔
X13
)
)
)
)
)
→
(
(
¬
(
X48
↔
(
X46
→
(
(
¬
X21
)
↔
X38
)
)
)
)
↔
(
X47
→
(
¬
X21
)
→
X38
)
)
)
)
↔
(
¬
(
(
¬
(
X47
↔
(
X46
→
(
X38
↔
False
)
)
)
)
→
(
X49
↔
(
X46
→
X42
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
X49
→
(
(
¬
X32
)
↔
(
(
¬
X18
)
→
X43
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X32
)
→
(
¬
X30
)
→
X49
)
)
→
(
¬
(
(
(
¬
X32
)
→
(
X30
↔
(
(
¬
X12
)
→
X49
)
)
)
→
(
(
¬
(
X43
→
(
X30
↔
X42
)
)
)
↔
(
X43
→
X32
→
X30
→
X49
)
)
)
)
→
(
¬
(
(
(
¬
X42
)
↔
(
(
¬
X12
)
↔
(
X5
→
X49
)
)
)
→
(
X42
↔
(
X21
→
X12
)
)
)
)
)
)
)
↔
(
(
(
X42
↔
(
(
¬
X29
)
↔
(
(
¬
X22
)
↔
X41
)
)
)
→
(
¬
(
X42
↔
(
X22
→
X3
→
X23
)
)
)
)
↔
(
¬
(
(
(
X23
→
X9
→
(
¬
X2
)
→
X43
)
↔
(
(
¬
(
X43
↔
(
X23
→
X41
)
)
)
↔
(
¬
(
X43
→
(
X41
↔
(
X9
↔
X29
)
)
)
)
)
)
→
(
(
¬
X43
)
↔
(
(
¬
X29
)
→
(
(
¬
X9
)
↔
X49
)
)
)
)
)
)
)
→
(
(
(
¬
(
X27
→
X25
→
(
¬
X20
)
→
X32
)
)
↔
(
X45
↔
(
X44
→
(
X20
↔
X4
)
)
)
)
↔
(
(
(
(
X31
↔
(
X6
→
X49
)
)
↔
(
¬
(
(
¬
(
X49
→
X25
→
X6
→
X42
)
)
→
(
X49
↔
(
X20
→
(
X6
↔
X42
)
)
)
)
)
)
↔
(
X42
→
(
(
¬
X31
)
↔
(
X10
↔
X49
)
)
)
)
→
(
(
¬
(
(
¬
X31
)
↔
(
(
¬
X19
)
↔
X45
)
)
)
↔
(
(
X45
→
X20
→
(
(
¬
X19
)
↔
X42
)
)
→
(
¬
(
X42
↔
(
(
¬
X20
)
↔
(
(
¬
X14
)
→
X45
)
)
)
)
)
)
→
(
¬
(
X19
→
(
(
¬
X11
)
↔
X14
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
(
¬
(
X48
↔
(
X46
↔
(
X45
↔
X28
)
)
)
)
↔
(
X46
↔
(
(
¬
X28
)
↔
(
X26
→
X45
)
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X45
)
→
(
¬
X34
)
→
X48
)
)
)
)
)
↔
(
¬
(
(
¬
(
X49
→
X46
→
X35
)
)
→
(
¬
(
X49
→
(
X45
↔
(
X34
↔
X1
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
(
¬
X46
)
↔
(
X45
↔
X49
)
)
)
↔
(
(
¬
(
X49
↔
(
X46
↔
(
X45
↔
(
X13
↔
False
)
)
)
)
)
↔
(
¬
(
X49
→
(
X46
↔
(
(
¬
X39
)
→
X45
)
)
)
)
)
)
↔
(
¬
(
X46
→
X45
→
False
)
)
)
→
(
X46
↔
(
X45
→
X39
→
X7
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X45
)
→
(
(
¬
X8
)
↔
X49
)
)
)
→
(
¬
(
X45
↔
(
(
¬
X15
)
→
(
X8
↔
X46
)
)
)
)
)
)
↔
(
¬
(
(
¬
X47
)
→
(
X15
↔
(
X8
↔
X49
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X49
→
(
¬
X40
)
→
(
X38
↔
X47
)
)
)
→
(
X49
↔
(
X38
→
(
(
¬
X37
)
↔
X40
)
)
)
)
↔
(
(
¬
X40
)
↔
(
(
¬
X37
)
→
(
¬
X16
)
→
X47
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
¬
X36
)
↔
(
X17
→
X48
)
)
→
(
(
¬
(
X48
↔
(
(
¬
X40
)
→
(
X17
↔
X49
)
)
)
)
↔
(
¬
(
X48
→
(
X36
↔
(
X17
↔
X40
)
)
)
)
)
)
)
↔
(
X48
↔
(
X36
→
X40
)
)
)
→
(
¬
X48
)
→
(
(
¬
X40
)
↔
(
X38
→
X49
)
)
)
)
)
↔
(
(
(
¬
(
X48
↔
(
X33
→
X0
→
X49
)
)
)
→
(
¬
(
X43
↔
(
X33
→
X48
)
)
)
)
↔
(
(
¬
(
X48
↔
(
(
¬
X47
)
↔
(
X24
↔
X49
)
)
)
)
↔
(
¬
(
X47
↔
(
(
¬
X24
)
→
X43
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
06a71f...
and proposition id is
12b4f7...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMEg45S2Fd9xjFAvvYaFcmiXfoTAtq5a3aG
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
¬
(
(
¬
(
(
(
(
X46
→
(
(
¬
X39
)
↔
(
(
¬
X18
)
→
X41
)
)
)
↔
(
(
¬
X39
)
→
X22
→
(
¬
X18
)
→
X41
)
)
↔
(
(
¬
(
(
¬
X39
)
↔
(
(
¬
X27
)
→
(
X22
↔
X43
)
)
)
)
↔
(
(
¬
(
X41
↔
(
X39
→
X47
)
)
)
→
(
(
(
¬
X39
)
↔
(
(
¬
X27
)
→
X41
)
)
↔
(
(
(
(
¬
(
X43
→
(
(
¬
X41
)
↔
(
X39
↔
X47
)
)
)
)
↔
(
X43
→
(
(
¬
X41
)
↔
(
(
¬
X27
)
→
X47
)
)
)
)
↔
(
(
¬
X41
)
↔
(
(
¬
X27
)
→
X47
)
)
)
→
X47
→
False
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
(
X46
→
(
¬
X17
)
→
X40
)
→
(
¬
(
(
X40
↔
(
(
¬
X21
)
→
X17
→
X26
)
)
→
(
(
¬
X21
)
↔
(
(
¬
X17
)
→
X46
)
)
)
)
)
)
→
(
X46
→
X45
→
X30
→
(
X21
↔
False
)
)
→
(
¬
(
X46
↔
(
X30
→
(
X26
↔
X21
)
)
)
)
)
)
↔
(
(
(
¬
(
X46
→
(
¬
X26
)
→
(
(
¬
X9
)
↔
X30
)
)
)
↔
(
¬
(
(
¬
(
X46
→
X9
→
X4
→
False
)
)
→
(
(
¬
(
X30
↔
(
(
¬
X9
)
→
(
X4
↔
X46
)
)
)
)
↔
(
X46
↔
(
X43
→
(
(
¬
X9
)
↔
X30
)
)
)
)
)
)
)
↔
(
(
¬
(
X43
→
(
X24
↔
X26
)
)
)
↔
(
¬
(
X37
↔
(
X24
↔
X46
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X45
)
↔
(
X43
→
(
X39
↔
X46
)
)
)
)
→
(
¬
(
X46
↔
(
X45
↔
(
(
¬
X29
)
→
X43
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
X45
→
X15
→
X10
→
False
)
↔
(
¬
(
X44
→
(
X10
↔
(
(
¬
X5
)
↔
X45
)
)
)
)
)
→
(
¬
(
(
X48
↔
(
X44
↔
(
X15
↔
X45
)
)
)
→
(
¬
(
X42
→
(
(
¬
X15
)
↔
X44
)
)
)
)
)
)
)
→
(
(
¬
(
X48
→
(
X46
↔
(
X11
→
X23
)
)
)
)
→
(
¬
(
X42
→
(
X11
↔
X46
)
)
)
)
→
(
¬
(
X46
↔
(
X42
↔
(
(
¬
X11
)
↔
X45
)
)
)
)
)
)
→
(
(
(
X49
→
(
X48
↔
(
X46
↔
False
)
)
)
↔
(
X46
↔
(
X45
↔
(
X20
↔
(
X12
↔
False
)
)
)
)
)
↔
(
¬
(
X49
→
X20
→
X45
)
)
)
→
(
(
(
¬
(
(
X43
↔
(
X12
↔
(
X3
↔
False
)
)
)
→
X43
→
(
X16
↔
(
X3
→
X0
)
)
)
)
↔
(
(
¬
X3
)
→
(
X0
↔
X16
)
)
)
→
(
¬
(
(
¬
X16
)
→
(
X0
↔
X20
)
)
)
)
→
(
X46
↔
(
X43
↔
(
X16
↔
X20
)
)
)
)
)
↔
(
(
(
¬
(
(
X32
↔
(
X14
↔
(
X7
→
X8
)
)
)
→
(
X25
↔
(
X14
→
X7
→
X32
)
)
)
)
↔
(
¬
(
(
¬
(
X34
→
(
¬
X32
)
→
(
X25
↔
X36
)
)
)
→
(
(
(
X48
↔
(
X36
↔
(
(
¬
X31
)
↔
X32
)
)
)
↔
(
(
X36
↔
(
X31
↔
(
X2
↔
X48
)
)
)
→
(
(
¬
X36
)
↔
(
X34
→
(
¬
X2
)
→
X48
)
)
)
)
↔
(
¬
(
X36
→
(
(
¬
X32
)
↔
(
X8
→
X34
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
X38
)
↔
(
X33
→
(
X19
↔
X47
)
)
)
↔
(
(
¬
(
X47
→
(
¬
X38
)
→
X49
)
)
↔
(
(
X49
→
(
¬
X33
)
→
X25
→
X38
)
↔
(
(
¬
(
(
X47
→
X38
→
False
)
→
(
¬
(
X49
↔
(
X33
→
(
X28
↔
X38
)
)
)
)
)
)
↔
(
X47
→
X38
→
X33
)
)
)
)
)
→
(
X49
↔
(
(
¬
X38
)
→
(
¬
X13
)
→
X47
)
)
)
)
↔
(
(
(
X35
→
(
(
¬
X33
)
↔
X38
)
)
↔
(
(
(
(
¬
(
X33
→
(
(
¬
X32
)
↔
X35
)
)
)
↔
(
X38
→
(
¬
X32
)
→
(
¬
X31
)
→
X33
)
)
↔
(
¬
(
X35
↔
(
X33
↔
(
X6
↔
X32
)
)
)
)
)
↔
(
X35
→
(
(
¬
X33
)
↔
(
(
¬
X1
)
↔
X38
)
)
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X33
)
→
X32
→
X35
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
(
¬
(
X39
→
X31
→
(
¬
X28
)
→
X35
)
)
↔
(
(
(
(
¬
X31
)
↔
X39
)
↔
(
¬
(
X43
↔
(
X35
→
X20
→
False
)
)
)
)
↔
(
¬
(
X35
→
(
¬
X31
)
→
(
X15
↔
X43
)
)
)
)
)
↔
(
(
¬
X35
)
↔
(
(
¬
X31
)
→
(
(
¬
X30
)
↔
X43
)
)
)
)
→
(
(
¬
(
(
¬
X28
)
↔
(
X27
↔
X39
)
)
)
↔
(
X39
↔
(
(
¬
X28
)
↔
(
(
¬
X20
)
→
X49
)
)
)
)
)
↔
(
¬
(
(
X48
↔
(
(
¬
X24
)
→
X3
→
X49
)
)
→
(
(
X47
↔
(
(
¬
X16
)
↔
X31
)
)
↔
(
(
(
¬
X24
)
↔
(
(
(
¬
X31
)
↔
(
(
¬
X24
)
↔
(
X16
↔
X35
)
)
)
↔
(
¬
(
X49
↔
(
X31
→
(
¬
X16
)
→
X35
)
)
)
)
)
↔
(
¬
(
X24
→
X18
→
X48
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X48
→
(
X47
↔
(
X43
↔
X8
)
)
)
)
→
X43
→
X42
→
X48
)
→
(
X48
↔
(
(
¬
X12
)
→
(
(
¬
X8
)
↔
X43
)
)
)
)
↔
(
(
(
X48
→
(
¬
X43
)
→
(
¬
X22
)
→
X47
)
→
(
¬
(
X47
→
(
X44
↔
(
X22
↔
X48
)
)
)
)
)
↔
(
¬
(
(
¬
X44
)
↔
(
X38
→
X48
)
)
)
)
)
↔
(
(
¬
X43
)
↔
(
(
¬
X22
)
↔
(
X4
→
X44
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
¬
(
X47
↔
(
X21
↔
(
X6
→
(
X1
↔
False
)
)
)
)
)
→
(
X46
↔
(
X21
↔
(
X10
↔
X47
)
)
)
)
↔
(
¬
(
X49
↔
(
X21
→
X6
)
)
)
)
→
(
¬
(
(
X46
→
(
¬
X45
)
→
X36
→
X47
)
→
(
¬
(
(
¬
X2
)
↔
X46
)
)
→
X49
→
X47
→
X45
)
)
)
)
↔
(
(
(
¬
(
X11
→
(
X5
↔
False
)
)
)
↔
(
¬
(
X36
↔
(
(
¬
X0
)
→
X47
)
)
)
)
↔
(
(
(
¬
(
X33
↔
(
X17
→
X11
)
)
)
→
(
(
¬
(
(
X33
→
(
¬
X11
)
→
(
X5
↔
X17
)
)
→
(
¬
(
X11
→
X36
)
)
)
)
↔
(
¬
(
X36
↔
(
X11
↔
X17
)
)
)
)
)
↔
(
¬
(
(
X11
→
(
(
¬
X5
)
↔
X26
)
)
→
(
¬
(
X17
→
X46
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
(
¬
(
(
(
¬
(
(
¬
X41
)
↔
(
(
¬
X13
)
↔
X49
)
)
)
→
(
X41
↔
(
(
¬
X19
)
→
(
(
¬
X13
)
↔
X47
)
)
)
)
→
(
X49
↔
(
X47
→
X41
)
)
)
)
↔
(
¬
(
X49
→
X41
→
X47
)
)
)
↔
(
(
(
(
¬
X19
)
↔
(
(
¬
X13
)
→
X46
)
)
↔
(
¬
(
X25
→
X19
→
X47
)
)
)
↔
(
¬
(
X46
↔
(
X25
↔
X19
)
)
)
)
)
→
(
X47
↔
(
(
¬
X25
)
→
(
(
¬
X14
)
↔
X41
)
)
)
)
)
→
(
(
(
(
X48
↔
(
X41
→
X40
)
)
↔
(
(
(
¬
(
X48
→
(
X23
↔
X40
)
)
)
↔
(
(
¬
(
X41
↔
(
(
¬
X40
)
↔
(
X34
↔
X47
)
)
)
)
→
(
¬
(
X40
→
(
(
¬
X23
)
↔
X41
)
)
)
)
)
→
(
¬
(
X48
→
(
(
¬
X40
)
↔
X41
)
)
)
)
)
↔
(
(
(
¬
(
X48
↔
(
(
¬
X41
)
→
X46
)
)
)
↔
(
(
¬
(
(
¬
(
X48
↔
(
X46
→
(
¬
X32
)
→
X41
)
)
)
→
(
¬
(
X32
↔
(
(
¬
X7
)
↔
X48
)
)
)
)
)
↔
(
¬
(
X46
→
(
(
¬
X32
)
↔
X48
)
)
)
)
)
→
(
¬
(
(
¬
X41
)
→
(
(
¬
X40
)
↔
(
X9
→
X48
)
)
)
)
)
)
↔
(
(
X48
→
X47
→
(
X37
↔
False
)
)
↔
(
X46
→
X29
→
X47
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
cd8730...
and proposition id is
8040e7...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMYcLSaPRJMR3pwuF2VczDS2tMrdVCyZxBD
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
(
¬
(
(
¬
(
X48
→
(
X45
↔
(
X36
→
(
X35
↔
False
)
)
)
)
)
→
(
(
X36
↔
(
(
¬
X35
)
→
(
¬
X3
)
→
X48
)
)
↔
(
X48
→
(
X36
↔
(
(
¬
X23
)
→
X35
)
)
)
)
)
)
↔
(
¬
(
(
X48
→
X3
→
X25
)
→
(
(
¬
X23
)
→
(
¬
X19
)
→
(
¬
X3
)
→
X25
)
→
X48
→
X25
→
X3
→
X23
)
)
)
↔
(
(
¬
(
X49
→
(
¬
X36
)
→
X45
)
)
↔
(
¬
(
X46
→
(
X45
↔
(
(
¬
X36
)
→
X48
)
)
)
)
)
)
→
(
(
(
(
(
X48
↔
(
(
¬
X46
)
↔
(
X45
↔
X47
)
)
)
↔
(
(
¬
X46
)
→
(
¬
X39
)
→
X47
)
)
↔
(
(
X48
↔
(
(
¬
X28
)
↔
(
X18
→
X46
)
)
)
↔
(
(
X46
→
(
X28
↔
(
X18
↔
False
)
)
)
↔
(
(
¬
(
X46
↔
(
(
¬
X18
)
↔
(
(
¬
X15
)
→
X47
)
)
)
)
↔
(
¬
(
(
¬
X18
)
↔
(
X15
→
(
X8
↔
X47
)
)
)
)
)
)
)
)
↔
(
¬
(
(
X48
→
(
X47
↔
X46
)
)
→
(
X48
↔
(
X10
→
X4
→
X47
)
)
)
)
)
↔
(
(
¬
(
(
X47
→
(
(
¬
X46
)
↔
(
X12
→
X48
)
)
)
→
(
¬
(
(
(
X47
↔
(
X11
→
(
X7
↔
False
)
)
)
↔
(
¬
(
X47
↔
(
(
¬
X12
)
→
(
X7
↔
X48
)
)
)
)
)
→
(
(
¬
X47
)
↔
(
(
¬
X12
)
↔
X48
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X47
→
X9
→
(
¬
X2
)
→
X48
)
)
↔
(
¬
(
X42
↔
(
X9
→
False
)
)
)
)
↔
(
X42
→
(
¬
X9
)
→
X48
)
)
→
X42
→
X9
→
X47
)
↔
(
¬
(
X48
→
X47
→
X42
→
X46
)
)
)
)
)
)
→
(
(
(
(
(
X47
→
(
X34
↔
(
(
¬
X32
)
↔
X48
)
)
)
↔
(
(
(
(
X47
→
(
¬
X26
)
→
(
¬
X24
)
→
X49
)
↔
(
(
X49
↔
(
X37
→
X24
→
False
)
)
→
(
¬
(
(
¬
(
X37
→
(
X34
↔
(
X24
↔
X47
)
)
)
)
→
(
¬
(
X49
↔
(
X47
↔
(
X37
↔
X24
)
)
)
)
)
)
)
)
↔
(
X47
→
X34
→
(
X32
↔
X37
)
)
)
↔
(
¬
(
X37
→
X32
→
X30
→
X47
)
)
)
)
↔
(
¬
(
(
(
X47
↔
(
X44
↔
(
X37
→
X0
)
)
)
↔
(
¬
(
(
¬
X37
)
↔
(
(
¬
X32
)
→
(
(
¬
X0
)
↔
X44
)
)
)
)
)
→
(
X47
↔
(
X32
→
X44
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X40
)
→
X38
→
(
X20
↔
X41
)
)
)
↔
(
(
¬
X29
)
→
X40
)
)
→
(
(
¬
(
X48
→
X33
→
X41
)
)
→
(
(
¬
(
(
X38
↔
(
X33
↔
(
X22
↔
X41
)
)
)
→
(
¬
(
X48
→
X38
→
(
(
¬
X33
)
↔
X41
)
)
)
)
)
↔
(
¬
(
X47
→
X33
→
(
X22
↔
X40
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X47
↔
(
X40
→
X22
→
X6
→
False
)
)
)
→
X49
→
(
X40
↔
(
X22
→
(
X6
↔
False
)
)
)
)
)
→
(
¬
(
X49
↔
(
X6
→
X40
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
X47
)
↔
(
X27
↔
(
X13
→
X49
)
)
)
→
(
(
X47
↔
(
X27
→
False
)
)
↔
(
¬
(
(
¬
X43
)
↔
(
(
¬
X27
)
↔
(
(
¬
X13
)
→
X48
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
X31
)
↔
(
(
¬
X21
)
↔
X49
)
)
)
→
(
(
¬
X31
)
→
X21
→
(
(
¬
X1
)
↔
X48
)
)
→
X48
→
X21
→
(
¬
X1
)
→
X49
)
)
↔
(
(
(
X47
↔
(
X14
→
X49
)
)
↔
(
(
¬
(
X47
↔
(
(
¬
X21
)
↔
(
X17
→
X49
)
)
)
)
→
X49
→
(
(
¬
X21
)
↔
(
(
¬
X17
)
→
X31
)
)
)
)
↔
(
¬
(
(
¬
(
X49
↔
(
(
¬
X16
)
→
(
X5
↔
X47
)
)
)
)
→
(
¬
(
(
¬
X31
)
↔
(
X16
→
(
¬
X5
)
→
X49
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
¬
(
(
(
¬
X44
)
↔
(
(
¬
X39
)
→
(
X26
↔
X48
)
)
)
→
(
(
X44
↔
(
(
¬
X39
)
↔
(
X29
→
X47
)
)
)
↔
(
(
(
¬
(
X44
→
(
¬
X39
)
→
(
X5
↔
X47
)
)
)
↔
(
X44
↔
(
X32
↔
(
X5
→
False
)
)
)
)
→
(
X44
↔
(
X5
↔
X39
)
)
)
)
)
)
→
(
X40
↔
(
(
¬
X32
)
→
(
(
¬
X29
)
↔
X47
)
)
)
)
)
→
(
(
¬
(
(
(
(
(
(
¬
(
X39
↔
(
X30
→
(
(
¬
X27
)
↔
X28
)
)
)
)
→
(
¬
(
(
¬
X30
)
→
(
(
¬
X28
)
↔
(
X9
↔
X39
)
)
)
)
)
↔
(
¬
(
X39
↔
X27
)
)
)
↔
(
¬
(
(
¬
X30
)
→
X27
→
(
X9
↔
X48
)
)
)
)
→
(
(
¬
X30
)
↔
(
X9
→
X37
)
)
)
→
(
(
¬
(
(
(
X42
↔
(
(
¬
X36
)
↔
(
X14
↔
X39
)
)
)
→
X48
→
(
¬
X36
)
→
X42
)
→
(
¬
(
(
¬
X39
)
→
(
(
¬
X36
)
↔
(
X27
↔
X48
)
)
)
)
)
)
↔
(
X48
↔
(
(
¬
X36
)
→
X42
)
)
)
→
(
¬
(
(
¬
X42
)
↔
(
(
¬
X39
)
→
X48
)
)
)
)
)
↔
(
¬
(
(
(
¬
X27
)
↔
(
X16
↔
X47
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
X47
→
(
X42
↔
X28
)
)
→
(
¬
(
(
¬
(
(
¬
(
X42
→
X28
→
X27
→
X16
→
False
)
)
→
(
(
¬
X42
)
↔
(
(
¬
X28
)
→
(
(
¬
X27
)
↔
X47
)
)
)
)
)
→
(
¬
(
X47
→
X27
→
X28
)
)
)
)
)
)
→
(
¬
(
X28
↔
X27
)
)
)
)
→
X47
→
(
X42
↔
X28
)
)
)
)
)
)
)
)
→
(
¬
(
(
X48
→
(
X36
↔
(
X2
↔
False
)
)
)
→
(
¬
(
(
(
¬
(
X48
→
(
X30
↔
(
(
¬
X2
)
→
X46
)
)
)
)
↔
(
¬
(
X46
↔
(
X21
→
X2
)
)
)
)
→
(
¬
(
X48
↔
(
X46
↔
(
(
¬
X2
)
↔
X45
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
X48
)
→
X45
→
(
(
¬
X24
)
↔
X49
)
)
→
(
X46
↔
(
(
¬
X45
)
→
X24
→
X48
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
(
X41
→
(
X6
↔
(
X0
↔
X25
)
)
)
↔
(
X13
↔
(
(
¬
X6
)
→
(
¬
X0
)
→
X25
)
)
)
→
X15
→
(
¬
X13
)
→
X0
→
X25
)
→
(
X41
↔
(
(
¬
X15
)
↔
X25
)
)
)
)
→
(
(
¬
(
(
(
(
¬
X43
)
↔
(
(
¬
X33
)
↔
X49
)
)
→
(
(
¬
(
(
¬
X43
)
↔
(
(
¬
X22
)
↔
X49
)
)
)
↔
(
(
X43
→
(
X33
↔
(
(
¬
X22
)
→
X49
)
)
)
↔
(
X49
↔
(
X33
→
X13
→
False
)
)
)
)
)
→
(
(
¬
(
X47
→
(
X33
↔
(
X20
↔
False
)
)
)
)
↔
(
(
¬
(
(
X47
↔
(
X33
↔
X49
)
)
→
(
¬
(
X47
↔
(
X6
↔
X20
)
)
)
)
)
↔
(
¬
(
X47
→
False
)
)
)
)
→
(
X47
↔
(
X43
↔
X6
)
)
)
)
↔
(
¬
(
(
X41
↔
(
X33
↔
(
(
¬
X13
)
→
X35
)
)
)
→
(
(
(
¬
(
X41
↔
(
(
¬
X11
)
→
(
(
¬
X10
)
↔
X33
)
)
)
)
↔
(
(
X38
→
(
(
¬
X13
)
↔
(
X11
→
X33
)
)
)
↔
(
(
¬
(
X47
↔
(
X41
→
(
X33
↔
False
)
)
)
)
→
(
¬
(
X41
↔
(
X38
→
(
X13
↔
False
)
)
)
)
)
)
)
↔
(
¬
(
X33
→
(
¬
X11
)
→
X38
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
X43
↔
(
(
¬
X1
)
→
X48
)
)
↔
(
(
¬
(
X49
→
X43
→
(
X31
↔
X7
)
)
)
↔
(
X48
→
X43
→
False
)
)
)
→
(
¬
(
(
(
(
(
¬
(
(
¬
X18
)
→
X31
)
)
→
(
X49
↔
(
X43
↔
(
X34
→
(
X31
↔
False
)
)
)
)
→
X49
→
X43
→
(
(
¬
X18
)
↔
X34
)
)
↔
(
¬
(
X34
→
X18
)
)
)
↔
(
X49
→
X31
→
X18
)
)
→
(
(
(
X49
→
(
X43
↔
(
X23
→
X34
)
)
)
↔
(
(
¬
(
(
¬
X43
)
↔
(
(
¬
X23
)
↔
(
(
¬
X4
)
→
X49
)
)
)
)
→
(
X23
↔
(
(
¬
X4
)
↔
X12
)
)
)
)
↔
(
X49
→
X34
→
(
X12
↔
False
)
)
)
)
)
)
)
→
(
(
(
¬
X19
)
→
X17
→
X8
→
X49
)
↔
(
(
(
¬
(
(
¬
X34
)
→
(
X8
↔
X41
)
)
)
↔
(
(
¬
X48
)
↔
(
(
¬
X34
)
↔
(
(
¬
X8
)
↔
X49
)
)
)
)
↔
(
¬
(
(
¬
(
X48
→
X41
→
X3
)
)
→
(
X49
↔
(
(
¬
X19
)
→
X3
→
X48
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
587eca...
and proposition id is
fe64ee...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMZUZPKsCgB1sFdBu3i1dyYsnDAnyvRB3Kc
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
∀X50 :
prop
,
∀X51 :
prop
,
(
(
¬
(
(
(
(
(
(
¬
(
(
¬
(
(
X34
→
X27
)
→
(
¬
(
X39
→
(
X30
↔
X47
)
)
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X34
)
→
X25
→
X39
)
)
)
)
)
↔
(
¬
(
(
¬
(
X47
→
X39
→
(
X34
↔
False
)
)
)
→
(
X51
↔
(
(
¬
X34
)
↔
(
(
¬
X12
)
↔
X39
)
)
)
)
)
)
↔
(
¬
(
X47
→
(
(
¬
X39
)
↔
(
X2
→
X51
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
X43
)
↔
(
(
¬
X34
)
→
X49
)
)
→
(
(
(
(
¬
(
(
X48
↔
(
X29
↔
X49
)
)
→
(
¬
(
(
¬
(
X48
↔
X29
)
)
→
(
¬
(
X49
↔
(
X43
↔
(
(
¬
X29
)
→
X47
)
)
)
)
)
)
)
)
↔
(
¬
(
X48
→
X44
→
(
¬
X43
)
→
X47
)
)
)
↔
(
¬
(
X47
↔
(
(
¬
X43
)
→
X44
)
)
)
)
↔
(
X48
↔
(
X47
↔
(
(
¬
X34
)
↔
X49
)
)
)
)
)
→
(
(
(
¬
X47
)
↔
X51
)
↔
(
X48
↔
(
(
¬
X31
)
→
X44
)
)
)
)
→
(
(
¬
X48
)
↔
(
X14
→
X51
)
)
→
(
(
¬
(
X48
→
X14
)
)
↔
(
(
¬
X49
)
↔
(
X14
→
(
X4
↔
X51
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
¬
(
(
¬
X44
)
→
X20
→
X50
)
)
↔
(
(
¬
X47
)
→
(
(
¬
X19
)
↔
X50
)
)
)
↔
(
¬
(
X50
↔
False
)
)
)
↔
(
(
X48
→
X44
→
(
X19
↔
(
X9
→
False
)
)
)
↔
(
¬
(
X44
→
X19
→
(
¬
X3
)
→
X9
)
)
)
)
→
(
¬
(
X48
↔
(
X6
→
(
¬
X3
)
→
X17
)
)
)
)
→
(
(
(
(
¬
(
X48
→
(
(
¬
X28
)
↔
(
X1
→
X41
)
)
)
)
→
(
¬
(
X47
↔
(
X41
→
(
X28
↔
False
)
)
)
)
)
↔
(
¬
(
X28
→
X41
)
)
)
↔
(
¬
(
X48
→
(
X28
↔
(
X18
↔
False
)
)
)
)
)
→
(
(
(
¬
X41
)
→
(
¬
X28
)
→
(
X23
↔
X50
)
)
↔
(
(
X50
→
(
X8
↔
X48
)
)
↔
(
X28
↔
(
(
¬
X23
)
↔
X48
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
(
X46
↔
(
(
¬
X45
)
↔
X48
)
)
)
→
(
(
X46
→
(
X45
↔
(
X21
↔
False
)
)
)
↔
(
X50
↔
(
X45
→
X21
→
X48
)
)
)
)
)
→
(
(
X50
↔
(
X48
↔
(
X36
↔
False
)
)
)
↔
(
X46
→
(
(
¬
X36
)
↔
(
(
¬
X15
)
↔
X50
)
)
)
)
)
→
(
(
(
X50
→
(
(
¬
X13
)
↔
X24
)
)
↔
(
(
¬
(
X36
→
(
(
¬
X11
)
↔
X13
)
)
)
↔
(
X24
↔
(
X13
→
X50
)
)
)
)
→
(
(
¬
X36
)
↔
(
X24
→
(
¬
X0
)
→
X50
)
)
)
→
(
¬
(
(
¬
(
X50
↔
(
X45
→
X36
→
X24
)
)
)
→
(
(
¬
X38
)
↔
(
X36
→
(
(
¬
X24
)
↔
X45
)
)
)
)
)
)
↔
(
¬
(
(
(
(
X50
→
X16
→
X46
)
→
(
X50
↔
(
X46
→
(
(
¬
X26
)
↔
X48
)
)
)
)
→
(
¬
(
X49
↔
(
X48
↔
(
X46
→
X7
)
)
)
)
→
(
(
X48
↔
(
(
¬
X46
)
↔
(
(
¬
X32
)
↔
X49
)
)
)
↔
(
¬
(
X50
→
(
X46
↔
X32
)
)
)
)
)
→
(
(
X42
↔
(
X35
→
X5
→
False
)
)
↔
(
(
¬
(
X37
↔
(
(
¬
X35
)
↔
(
(
¬
X10
)
↔
X49
)
)
)
)
↔
(
¬
(
X40
↔
(
X10
→
X37
)
)
)
)
)
→
(
(
(
X48
↔
(
X42
↔
(
(
¬
X40
)
→
X50
)
)
)
↔
(
X48
→
(
(
¬
X42
)
↔
(
(
¬
X22
)
↔
X50
)
)
)
)
↔
(
(
¬
(
(
¬
X49
)
↔
(
(
¬
X48
)
→
X50
)
)
)
→
(
(
X50
↔
(
(
¬
X48
)
↔
(
(
¬
X33
)
↔
X49
)
)
)
↔
(
¬
(
X50
↔
(
X48
→
(
(
¬
X40
)
↔
X49
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X40
↔
(
X38
→
(
(
¬
X27
)
↔
X50
)
)
)
)
↔
(
(
(
X50
→
(
X33
↔
(
(
¬
X24
)
↔
X40
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X33
)
→
(
(
¬
X30
)
↔
(
X24
↔
X40
)
)
)
)
→
(
¬
(
X40
↔
(
X38
→
X30
)
)
)
)
)
)
↔
(
¬
(
X40
↔
(
X38
→
X4
→
X50
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
X33
)
↔
(
X17
↔
X50
)
)
→
(
¬
(
X50
↔
(
X40
→
X28
→
False
)
)
)
)
)
↔
(
(
¬
(
X28
↔
X5
)
)
→
X33
→
(
X18
↔
(
X17
→
False
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X39
↔
(
X23
→
(
(
¬
X14
)
↔
X19
)
)
)
)
→
(
(
X50
→
(
X48
↔
(
X10
→
X14
)
)
)
↔
(
¬
(
X50
↔
(
X39
↔
(
X19
↔
(
X14
↔
False
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
(
¬
(
(
(
¬
(
X48
→
X38
→
(
X0
↔
X11
)
)
)
↔
(
¬
(
X39
↔
(
X38
→
(
(
¬
X0
)
↔
X11
)
)
)
)
)
→
X48
→
False
)
)
↔
(
¬
(
X38
↔
(
X30
→
(
¬
X11
)
→
X48
)
)
)
)
→
(
¬
(
X38
↔
(
(
¬
X30
)
↔
X39
)
)
)
)
)
→
(
¬
(
X38
↔
(
(
¬
X0
)
→
X50
)
)
)
)
→
(
X51
↔
(
X39
↔
(
(
¬
X6
)
→
X50
)
)
)
)
)
)
)
↔
(
(
(
(
(
X42
↔
(
X41
↔
(
X32
→
X1
)
)
)
↔
(
(
X42
↔
(
(
¬
X34
)
↔
X37
)
)
→
(
(
(
¬
(
X34
→
(
X32
↔
(
X22
→
X9
)
)
)
)
↔
(
X34
→
(
X9
↔
X42
)
)
)
↔
(
¬
(
X42
↔
(
X32
→
X9
→
False
)
)
)
)
→
(
(
X34
↔
(
(
¬
X32
)
↔
X41
)
)
↔
(
¬
(
(
¬
X34
)
↔
(
X9
→
X37
)
)
)
)
)
)
↔
(
(
¬
(
X48
↔
(
X47
↔
(
X26
↔
X34
)
)
)
)
↔
(
(
(
X42
↔
(
(
¬
X34
)
→
X32
→
X46
)
)
↔
(
¬
(
X34
↔
(
X32
→
(
¬
X26
)
→
X42
)
)
)
)
↔
(
(
¬
(
X46
→
(
(
¬
X12
)
↔
(
(
¬
X8
)
→
X34
)
)
)
)
→
(
¬
(
(
¬
X34
)
↔
(
(
¬
X31
)
↔
X48
)
)
)
→
(
X46
↔
(
X32
↔
(
(
¬
X31
)
→
X48
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
X34
↔
(
X26
↔
(
(
¬
X13
)
→
X15
)
)
)
→
(
¬
(
X50
↔
X34
)
)
)
)
↔
(
X49
→
X34
→
(
X26
↔
X15
)
)
)
→
(
(
X46
→
(
X34
↔
(
(
¬
X7
)
↔
X49
)
)
)
↔
(
¬
(
(
¬
X34
)
→
(
(
¬
X20
)
↔
X49
)
)
)
)
→
(
¬
(
X49
→
X20
→
(
X15
↔
False
)
)
)
)
)
)
↔
(
(
(
¬
(
(
X49
↔
(
(
¬
X45
)
→
(
¬
X2
)
→
X50
)
)
→
(
(
¬
(
X50
→
X49
→
X45
)
)
↔
(
(
X49
→
X45
→
(
(
¬
X3
)
↔
X44
)
)
→
(
(
X50
→
X49
→
X44
→
X45
)
↔
(
¬
(
(
¬
X49
)
→
(
X45
↔
(
X21
→
X50
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
X51
→
(
X50
↔
(
X45
→
False
)
)
)
→
(
(
¬
(
(
¬
X50
)
↔
(
X45
↔
(
(
¬
X44
)
→
X51
)
)
)
)
↔
(
¬
(
(
¬
X50
)
↔
(
(
¬
X45
)
↔
(
(
¬
X25
)
↔
X51
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
X43
↔
(
X16
↔
X50
)
)
→
X43
→
(
(
¬
X29
)
↔
(
(
¬
X16
)
↔
X48
)
)
)
)
→
(
X51
↔
(
(
¬
X29
)
→
X35
)
)
)
→
(
(
X36
↔
(
X29
↔
X49
)
)
↔
(
(
X43
→
(
X29
↔
X35
)
)
↔
(
¬
(
X36
→
(
(
¬
X35
)
↔
(
(
¬
X29
)
→
X49
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
4087ff...
and proposition id is
af413b...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMKKLUScZAT1JRBopN3ejMnEWTP7Etkmz6i
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
¬
(
(
(
(
¬
(
X34
↔
(
(
¬
X31
)
→
(
(
¬
X24
)
↔
X38
)
)
)
)
→
(
(
X34
→
(
(
¬
X16
)
↔
(
X14
↔
X32
)
)
)
↔
(
¬
(
X38
↔
(
X32
→
X31
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X31
→
X6
→
X37
)
)
↔
(
(
X38
↔
(
X31
→
(
(
¬
X6
)
↔
X10
)
)
)
↔
(
(
¬
(
X10
→
(
(
¬
X6
)
↔
X38
)
)
)
→
(
(
X37
→
X10
→
(
X6
↔
X11
)
)
↔
(
X38
↔
(
(
¬
X10
)
↔
X11
)
)
)
)
)
)
↔
(
¬
(
(
¬
X38
)
→
X31
→
(
X11
↔
X44
)
)
)
)
→
(
(
¬
(
X44
→
(
X28
↔
(
X12
→
X32
)
)
)
)
↔
(
X44
→
(
X38
↔
(
X37
→
False
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X39
↔
(
(
¬
X38
)
→
X27
→
X46
)
)
)
→
(
¬
(
X39
→
(
X27
↔
(
(
¬
X9
)
↔
X46
)
)
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X38
)
↔
X39
)
)
)
)
↔
(
¬
(
(
(
(
(
(
¬
(
X49
↔
(
X34
↔
(
(
¬
X13
)
↔
X43
)
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X43
)
→
(
X13
↔
X46
)
)
)
)
)
→
(
¬
X46
)
→
(
X43
↔
(
X4
→
X49
)
)
)
↔
(
¬
(
(
¬
X13
)
→
X4
→
X43
)
)
)
↔
(
(
X43
↔
(
(
¬
X34
)
→
X49
)
)
→
(
(
(
(
¬
X43
)
↔
(
X34
→
(
¬
X22
)
→
X49
)
)
↔
(
¬
(
(
¬
X43
)
→
(
¬
X34
)
→
X45
)
)
)
↔
(
X45
↔
(
(
¬
X43
)
→
X37
→
X49
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X38
→
X46
)
)
→
X43
→
(
X27
↔
(
(
¬
X5
)
→
X46
)
)
)
)
↔
(
(
(
¬
X43
)
→
(
X21
↔
X49
)
)
→
(
¬
(
X46
→
X43
→
(
(
¬
X38
)
↔
X49
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
X33
)
→
(
(
¬
X26
)
↔
X35
)
)
→
(
(
¬
(
(
(
¬
(
(
(
(
¬
(
X46
→
(
X0
↔
X3
)
)
)
↔
(
X47
→
(
X3
↔
X45
)
)
)
↔
(
X46
↔
(
(
¬
X45
)
↔
X47
)
)
)
→
(
¬
X3
)
→
X0
→
X45
)
)
↔
(
¬
(
X35
→
(
(
¬
X3
)
↔
X46
)
)
)
)
→
(
¬
X0
)
→
X45
)
)
↔
(
¬
(
X47
→
X45
→
(
(
¬
X33
)
↔
X46
)
)
)
)
)
)
↔
(
¬
(
(
X46
→
X45
→
(
X36
↔
X19
)
)
→
(
(
(
¬
(
(
¬
(
(
¬
(
X15
↔
(
X8
→
X7
)
)
)
→
(
(
¬
(
X15
→
(
X8
↔
(
X7
→
False
)
)
)
)
↔
(
(
X46
→
(
X15
↔
(
X7
→
X2
)
)
)
↔
(
¬
(
X15
↔
(
(
¬
X8
)
↔
X46
)
)
)
)
)
)
)
→
(
¬
(
(
¬
X18
)
→
X15
→
(
¬
X7
)
→
X46
)
)
)
)
→
(
(
¬
(
X46
→
(
X36
↔
(
X15
↔
False
)
)
)
)
↔
(
¬
(
(
¬
(
X25
→
X7
→
X15
)
)
→
X46
→
X36
→
X7
→
X25
)
)
)
)
↔
(
¬
(
(
¬
(
X45
↔
(
(
¬
X36
)
↔
(
(
¬
X18
)
→
X48
)
)
)
)
→
(
(
¬
(
X48
↔
(
X35
↔
(
(
¬
X25
)
↔
X36
)
)
)
)
↔
(
¬
(
X48
→
(
¬
X25
)
→
X18
→
X35
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X47
↔
(
X46
↔
(
X44
→
X41
→
False
)
)
)
)
→
(
¬
(
(
X46
↔
(
X44
↔
(
X1
→
False
)
)
)
→
(
¬
(
X45
↔
X41
)
)
)
)
)
)
↔
(
(
X45
→
(
(
¬
X29
)
↔
(
(
¬
X20
)
↔
X40
)
)
)
↔
(
¬
(
(
¬
(
X44
→
(
X44
→
X20
)
→
(
¬
(
X45
→
X44
→
X30
→
X47
)
)
)
)
→
(
(
¬
(
(
X47
↔
(
X44
↔
(
(
¬
X17
)
↔
X42
)
)
)
→
(
X47
↔
(
(
¬
X44
)
↔
(
X17
→
X45
)
)
)
)
)
↔
(
¬
(
X47
→
X44
→
(
X42
↔
X23
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
¬
(
X48
↔
(
X46
↔
(
(
¬
X26
)
→
X37
)
)
)
)
→
(
¬
(
X48
→
X46
→
X43
)
)
)
)
↔
(
(
¬
(
(
¬
X45
)
↔
(
(
¬
X30
)
→
(
X26
↔
X46
)
)
)
)
↔
(
X45
→
(
X30
↔
(
X6
→
X46
)
)
)
)
)
→
(
(
(
(
¬
X41
)
→
(
(
¬
X36
)
↔
(
X29
→
X45
)
)
)
→
(
¬
(
X39
↔
(
(
¬
X36
)
→
X45
)
)
)
)
↔
(
(
(
¬
(
X41
↔
(
(
¬
X22
)
→
(
(
¬
X2
)
↔
X43
)
)
)
)
↔
(
(
¬
(
(
¬
X43
)
→
(
¬
X39
)
→
X17
→
X45
)
)
↔
(
X48
↔
(
X39
→
X22
→
X41
)
)
)
)
↔
(
¬
(
X43
↔
(
(
¬
X41
)
↔
(
X24
→
X45
)
)
)
)
)
)
)
↔
(
(
(
X33
↔
(
(
¬
X19
)
→
X3
→
X37
)
)
↔
(
(
¬
X37
)
→
(
(
¬
X33
)
↔
(
(
¬
X3
)
↔
X47
)
)
)
)
→
(
(
(
(
(
¬
(
X45
↔
(
(
¬
X28
)
→
(
(
¬
X19
)
↔
X30
)
)
)
)
↔
(
¬
(
X37
↔
(
X33
→
X19
→
X30
)
)
)
)
↔
(
X45
↔
(
X37
↔
(
X7
↔
X33
)
)
)
)
↔
(
(
¬
(
X33
→
False
)
)
↔
(
(
¬
(
(
¬
X21
)
↔
(
X10
↔
X28
)
)
)
↔
(
(
X37
↔
(
X33
↔
(
X21
→
X28
)
)
)
↔
(
(
¬
(
X37
→
X33
→
X28
→
X4
→
False
)
)
↔
(
¬
(
(
¬
X33
)
→
(
X10
↔
(
X4
→
X37
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
¬
X33
)
↔
(
(
¬
X30
)
→
(
(
¬
X18
)
↔
X45
)
)
)
)
↔
(
X33
→
(
(
¬
X18
)
↔
(
X11
↔
X45
)
)
)
)
↔
(
X45
↔
(
X33
↔
(
X30
↔
X20
)
)
)
)
→
(
¬
X30
)
→
(
(
¬
X20
)
↔
X33
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
(
¬
(
X46
↔
(
X40
→
X12
)
)
)
↔
(
¬
(
X46
→
X41
→
(
(
¬
X8
)
↔
X12
)
)
)
)
→
(
¬
(
X46
↔
(
X41
→
X40
→
X13
)
)
)
)
)
↔
(
(
(
(
¬
(
(
X44
↔
(
X41
↔
(
(
¬
X40
)
→
X46
)
)
)
→
(
¬
(
X46
→
(
X44
↔
(
X15
→
X41
)
)
)
)
)
)
↔
(
¬
X41
)
)
↔
(
X46
→
(
X40
↔
X15
)
)
)
↔
(
¬
(
(
¬
X46
)
↔
(
X15
↔
X48
)
)
)
)
)
↔
(
¬
(
(
(
X27
↔
(
X25
↔
X40
)
)
→
(
¬
(
(
(
X5
→
X40
)
→
(
¬
(
(
¬
(
X46
↔
(
X40
↔
(
X5
→
X27
)
)
)
)
→
(
¬
(
X27
↔
(
(
¬
X25
)
↔
(
(
¬
X5
)
↔
X40
)
)
)
)
)
)
)
→
X46
→
(
X42
↔
(
X5
→
X40
)
)
)
)
)
→
(
(
(
X42
↔
(
(
¬
X40
)
↔
X44
)
)
↔
(
¬
(
X49
↔
(
(
¬
X16
)
↔
X27
)
)
)
)
↔
(
(
X44
→
X32
)
↔
(
¬
(
(
¬
X42
)
→
(
¬
X40
)
→
(
¬
X23
)
→
X44
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X42
→
X34
→
X14
→
(
X0
↔
False
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
X44
)
→
(
X42
↔
(
X0
↔
X49
)
)
)
)
→
X42
→
(
X35
↔
(
X0
↔
X49
)
)
)
)
↔
(
¬
(
(
X44
↔
(
X42
→
(
X1
↔
False
)
)
)
→
(
¬
(
(
¬
(
X49
→
(
X34
↔
X1
)
)
)
→
(
¬
(
(
¬
X44
)
→
X42
→
(
¬
X34
)
→
X49
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
X35
)
↔
(
(
¬
X31
)
↔
(
X9
→
X38
)
)
)
→
(
¬
(
X44
↔
(
(
¬
X38
)
↔
(
X9
↔
X49
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
bb337d...
and proposition id is
ff9dbc...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMLpUraNRejJThPPFGXnhoF3sAdZ5jmfAUe
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
(
¬
(
(
¬
(
(
(
(
X38
→
X30
→
X41
)
→
(
(
(
¬
(
X47
→
(
(
¬
X41
)
↔
(
X30
↔
X50
)
)
)
)
↔
(
(
(
¬
(
(
¬
X41
)
↔
(
X33
→
X14
→
X47
)
)
)
↔
(
¬
(
X41
↔
(
X33
→
(
(
¬
X19
)
↔
X47
)
)
)
)
)
↔
(
(
¬
X19
)
↔
(
X14
↔
X50
)
)
)
)
↔
(
¬
(
(
¬
X41
)
→
(
X38
↔
(
(
¬
X33
)
↔
X47
)
)
)
)
)
)
↔
(
(
(
X33
↔
(
(
¬
X10
)
→
X31
)
)
→
(
¬
(
(
¬
X47
)
↔
(
X12
↔
(
X10
↔
X50
)
)
)
)
)
→
X50
→
(
(
¬
X24
)
↔
(
(
¬
X18
)
→
X47
)
)
)
)
→
(
(
(
¬
(
X44
↔
(
(
¬
X23
)
→
(
¬
X1
)
→
X26
)
)
)
↔
(
(
¬
(
(
(
¬
X45
)
↔
(
(
¬
X44
)
↔
(
X38
→
X46
)
)
)
→
(
(
¬
(
X44
→
X38
→
(
(
¬
X26
)
↔
X34
)
)
)
↔
(
(
(
¬
X44
)
↔
(
(
¬
X38
)
→
(
X33
↔
X45
)
)
)
↔
(
¬
(
X38
↔
(
(
¬
X33
)
→
X34
)
)
)
)
)
)
)
↔
(
¬
(
X44
→
(
X34
↔
(
X1
→
False
)
)
)
)
)
)
↔
(
¬
(
(
X45
→
(
¬
X34
)
→
(
(
¬
X20
)
↔
X44
)
)
→
(
¬
(
X45
↔
(
X44
→
(
¬
X34
)
→
X47
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
X44
→
(
(
¬
X7
)
↔
(
(
¬
X2
)
→
X46
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X44
)
↔
(
X7
↔
X48
)
)
)
)
)
→
(
(
(
(
(
¬
(
X48
→
(
(
¬
X6
)
↔
(
X3
→
X46
)
)
)
)
→
X50
→
(
(
¬
X46
)
↔
(
(
¬
X3
)
↔
X48
)
)
)
↔
(
(
¬
X46
)
→
(
¬
X44
)
→
(
(
¬
X3
)
↔
X50
)
)
)
↔
(
¬
(
X48
↔
(
(
¬
X46
)
↔
X50
)
)
)
)
↔
(
(
(
¬
X48
)
→
X40
→
X22
→
X50
)
↔
(
X22
→
X8
→
X46
)
)
)
)
)
→
(
(
¬
X44
)
→
(
¬
X15
)
→
(
(
¬
X0
)
↔
X50
)
)
→
(
X46
↔
(
X39
→
False
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
X42
↔
(
X25
→
X43
)
)
)
↔
(
¬
(
X44
↔
(
X43
→
X17
)
)
)
)
→
(
(
¬
(
(
(
(
¬
(
(
(
¬
(
(
¬
X42
)
↔
(
(
¬
X25
)
↔
(
(
¬
X5
)
→
X43
)
)
)
)
↔
(
¬
(
X36
↔
(
(
¬
X25
)
↔
X43
)
)
)
)
→
(
¬
(
(
¬
X36
)
↔
(
(
¬
X25
)
→
X43
)
)
)
)
)
↔
(
X42
→
(
X36
↔
False
)
)
)
→
(
¬
(
X43
↔
False
)
)
)
→
(
¬
(
(
¬
X36
)
→
(
¬
X25
)
→
X43
)
)
)
)
↔
(
¬
(
(
X42
→
(
¬
X36
)
→
X27
→
X43
)
→
(
¬
X36
)
→
(
X27
↔
(
X16
→
X42
)
)
)
)
)
)
→
(
¬
(
(
(
¬
X36
)
→
(
(
¬
X11
)
↔
(
(
¬
X4
)
↔
X47
)
)
)
→
(
(
X47
↔
(
X44
→
(
X29
↔
(
X11
→
False
)
)
)
)
↔
(
¬
(
(
¬
(
X50
↔
(
X47
→
X44
→
X42
)
)
)
→
(
(
¬
(
X44
↔
(
X36
↔
(
X29
→
False
)
)
)
)
↔
(
(
¬
(
X44
→
X36
)
)
↔
(
¬
(
(
¬
X47
)
→
(
¬
X36
)
→
X29
→
X50
)
)
)
)
)
)
)
)
)
)
)
→
(
(
(
(
X49
→
X47
→
X35
→
False
)
↔
(
¬
(
(
¬
(
X35
↔
(
(
¬
X28
)
↔
X37
)
)
)
→
(
(
¬
(
X37
↔
(
(
¬
X28
)
→
X35
)
)
)
↔
(
(
¬
X47
)
↔
(
(
¬
X35
)
→
(
(
¬
X28
)
↔
X49
)
)
)
)
)
)
)
↔
(
(
X47
→
X37
→
(
¬
X9
)
→
X32
)
↔
(
¬
(
(
X32
↔
X47
)
→
(
¬
(
(
¬
(
(
¬
(
X37
→
X32
→
X13
→
X35
)
)
→
X47
→
(
(
¬
X35
)
↔
(
X32
↔
X37
)
)
)
)
→
X37
→
(
¬
X32
)
→
X35
)
)
)
)
)
)
↔
(
(
(
(
¬
X44
)
↔
(
X32
→
X49
)
)
↔
(
X44
↔
(
X35
→
X21
)
)
)
↔
(
¬
(
(
¬
X42
)
↔
(
X35
→
(
X21
↔
X49
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
(
X48
↔
(
X42
↔
(
(
¬
X13
)
→
X26
)
)
)
→
(
X48
↔
(
(
¬
X13
)
↔
X45
)
)
)
)
→
(
¬
(
(
(
X50
→
(
X48
↔
(
X37
↔
X45
)
)
)
↔
(
(
(
¬
(
X37
→
X50
)
)
↔
(
¬
(
(
¬
X48
)
→
(
¬
X42
)
→
(
(
¬
X31
)
↔
X50
)
)
)
)
↔
(
X43
↔
(
(
¬
X37
)
↔
(
(
¬
X31
)
→
X42
)
)
)
)
)
→
(
¬
(
(
¬
X45
)
→
(
(
¬
X42
)
↔
(
X1
→
X50
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X50
→
X39
→
(
¬
X14
)
→
X20
)
)
↔
(
X39
→
X22
→
X20
→
X50
)
)
↔
(
(
(
¬
(
(
¬
X43
)
→
(
¬
X33
)
→
(
¬
X20
)
→
X50
)
)
↔
(
X50
↔
(
X43
→
(
(
¬
X14
)
↔
X20
)
)
)
)
→
(
¬
X33
)
→
(
(
¬
X20
)
↔
X50
)
)
)
→
(
(
(
(
X38
↔
(
X21
→
(
¬
X17
)
→
X29
)
)
→
(
¬
X38
)
→
(
X17
↔
X39
)
)
↔
(
(
¬
(
X45
↔
(
X43
↔
(
X21
→
X39
)
)
)
)
→
(
¬
(
(
¬
X45
)
↔
(
X39
→
(
¬
X38
)
→
X50
)
)
)
→
(
X45
↔
(
X43
→
(
X38
↔
False
)
)
)
)
)
↔
(
¬
(
(
¬
X39
)
→
(
X29
↔
(
(
¬
X19
)
→
X50
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X50
→
(
(
¬
X36
)
↔
(
(
¬
X9
)
→
X47
)
)
)
)
→
(
(
X47
→
X35
→
(
X6
↔
X50
)
)
↔
(
(
X50
↔
(
X44
↔
(
(
¬
X35
)
→
X36
)
)
)
↔
(
(
¬
X47
)
↔
(
(
¬
X36
)
↔
(
X35
↔
X50
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X44
→
(
(
¬
X23
)
↔
(
X11
→
X50
)
)
)
)
→
(
(
(
¬
X23
)
→
X44
)
↔
(
X50
↔
(
X47
→
X46
→
X23
→
False
)
)
)
)
)
↔
(
¬
(
X47
↔
(
X46
→
(
(
¬
X24
)
↔
X50
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
X32
)
↔
(
(
¬
X5
)
→
X46
)
)
↔
(
X47
→
X40
→
X4
→
X32
)
)
→
(
¬
(
(
¬
(
X50
↔
(
X25
→
(
(
¬
X5
)
↔
X15
)
)
)
)
→
(
(
¬
X47
)
↔
(
X32
→
X5
→
X50
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X41
↔
(
X36
→
False
)
)
)
→
(
(
¬
(
X46
↔
(
X41
↔
(
(
¬
X36
)
→
X48
)
)
)
)
↔
(
X48
↔
(
X46
↔
(
X36
↔
X49
)
)
)
)
)
→
(
¬
(
(
X46
↔
(
X41
↔
(
X12
→
X7
)
)
)
→
(
(
(
X40
→
(
X10
↔
(
X3
→
X12
)
)
)
↔
(
¬
(
X46
↔
(
X40
→
(
(
¬
X3
)
↔
X10
)
)
)
)
)
↔
(
¬
(
(
¬
(
X40
→
(
X10
↔
(
(
¬
X2
)
↔
X41
)
)
)
)
→
(
¬
(
X41
→
X12
→
X2
→
X48
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X8
↔
(
(
¬
X0
)
↔
X16
)
)
)
→
X49
→
(
X28
↔
X16
)
)
→
(
(
(
¬
X46
)
→
(
(
¬
X34
)
↔
(
X16
↔
X49
)
)
)
↔
(
X49
→
(
(
¬
X16
)
↔
X28
)
)
)
)
→
(
¬
(
(
¬
X42
)
→
(
¬
X34
)
→
X8
→
X46
)
)
)
→
(
(
¬
(
(
X49
↔
(
(
¬
X42
)
↔
(
(
¬
X41
)
↔
X46
)
)
)
→
(
¬
X46
)
→
(
¬
X41
)
→
(
¬
X30
)
→
X49
)
)
↔
(
(
(
¬
(
X46
↔
(
X42
→
(
X18
↔
False
)
)
)
)
↔
(
¬
(
(
X42
→
(
X41
↔
(
(
¬
X27
)
↔
X46
)
)
)
→
(
¬
X41
)
→
(
(
¬
X27
)
↔
(
(
¬
X18
)
→
X46
)
)
)
)
)
↔
(
¬
(
X42
→
(
(
¬
X41
)
↔
X46
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
a2eb02...
and proposition id is
1f85df...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMdXqM1tXNWYaANPHaY3HL9BurL2MosFEyA
)
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
¬
(
(
(
(
(
¬
(
(
¬
(
X20
→
(
X6
↔
X46
)
)
)
→
(
¬
(
X46
→
(
(
¬
X22
)
↔
(
X20
↔
X28
)
)
)
)
)
)
↔
(
(
¬
X20
)
→
(
¬
X6
)
→
X28
)
)
↔
(
¬
(
X28
→
(
¬
X6
)
→
X20
)
)
)
↔
(
¬
(
X20
→
(
X6
↔
False
)
)
)
)
→
(
(
X28
↔
(
X19
→
X2
)
)
↔
(
(
(
X36
↔
(
X22
↔
(
X20
→
False
)
)
)
→
(
¬
(
X35
→
(
X28
↔
(
(
¬
X20
)
→
X36
)
)
)
)
)
→
(
¬
(
X36
→
(
X22
↔
(
X2
↔
X28
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X47
→
(
¬
X44
)
→
X21
→
X46
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
X44
↔
(
X20
↔
(
X13
↔
X46
)
)
)
)
→
(
X46
↔
(
X21
↔
(
X20
→
X18
→
False
)
)
)
)
)
→
X46
→
(
(
¬
X18
)
↔
X21
)
)
)
↔
(
¬
(
X21
↔
(
(
¬
X18
)
→
(
(
¬
X11
)
↔
X20
)
)
)
)
)
)
→
(
¬
(
X18
↔
(
X13
↔
X47
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X46
)
↔
(
X5
→
X47
)
)
)
→
(
X44
↔
(
X12
→
X5
)
)
)
→
X44
→
(
X22
↔
(
(
¬
X12
)
↔
X47
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
(
¬
X44
)
→
(
X17
↔
(
X7
→
X46
)
)
)
)
↔
(
(
¬
X7
)
→
X42
)
)
→
(
X46
↔
(
(
¬
X17
)
↔
X44
)
)
)
)
↔
(
(
¬
(
X24
→
(
(
¬
X15
)
↔
X46
)
)
)
↔
(
¬
(
X42
→
(
¬
X15
)
→
X44
)
)
)
)
↔
(
(
(
(
X44
↔
(
(
¬
X25
)
→
(
X16
↔
X47
)
)
)
→
(
¬
(
(
¬
X46
)
→
X37
→
X16
→
X47
)
)
)
→
(
¬
(
(
¬
X37
)
↔
(
(
¬
X27
)
↔
(
(
¬
X25
)
↔
X44
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X46
)
↔
(
X44
→
(
(
¬
X8
)
↔
X47
)
)
)
)
→
(
X46
↔
(
X44
→
(
¬
X42
)
→
X47
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
X45
→
X14
→
X3
→
X38
)
↔
(
¬
(
X40
→
(
¬
(
X45
→
X38
)
)
→
(
¬
(
(
¬
X40
)
→
X38
→
(
X37
↔
X44
)
)
)
)
)
)
→
(
(
¬
(
(
X45
↔
(
(
¬
X29
)
→
X49
)
)
→
(
(
¬
(
X49
→
(
X45
↔
(
(
¬
X29
)
↔
X33
)
)
)
)
↔
(
X45
→
X29
→
X10
)
)
)
)
↔
(
¬
(
X45
↔
(
(
¬
X39
)
→
(
X10
↔
X49
)
)
)
)
)
→
(
¬
(
(
¬
(
(
X39
→
(
X38
↔
(
X1
↔
False
)
)
)
→
(
X49
↔
(
X44
→
(
X38
↔
False
)
)
)
)
)
→
(
¬
(
(
(
(
(
X33
↔
(
X0
→
False
)
)
↔
(
(
¬
(
X33
↔
(
(
¬
X0
)
↔
X32
)
)
)
↔
(
¬
(
X49
↔
(
X32
↔
(
X9
→
X0
→
False
)
)
)
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X0
)
→
X33
)
)
)
)
↔
(
X49
→
(
X39
↔
(
(
¬
X9
)
↔
X44
)
)
)
)
→
(
X44
↔
(
(
¬
X39
)
↔
(
X31
↔
X49
)
)
)
)
)
)
)
)
)
→
(
(
X48
→
(
X41
↔
(
(
¬
X26
)
↔
X39
)
)
)
↔
(
¬
(
(
X39
↔
(
X34
↔
(
X26
→
X48
)
)
)
→
(
¬
(
X48
→
X45
→
(
X26
↔
X23
)
)
)
)
)
)
→
(
(
(
X39
↔
(
X38
→
(
X30
↔
False
)
)
)
↔
(
¬
(
X45
→
(
(
¬
X4
)
↔
X38
)
)
)
)
→
(
¬
(
X43
↔
(
X38
→
X4
→
X45
)
)
)
)
→
(
¬
(
(
X43
→
X38
→
X26
→
X36
)
→
(
¬
(
(
¬
X39
)
→
(
X38
↔
(
X36
→
X43
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
(
X48
→
X7
→
False
)
↔
(
(
¬
(
X48
→
X46
→
(
X45
↔
False
)
)
)
↔
(
(
(
¬
X46
)
→
(
(
¬
X45
)
↔
(
X43
↔
X48
)
)
)
→
(
¬
(
X48
→
(
X19
↔
X43
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
(
(
¬
X45
)
→
(
(
¬
X24
)
↔
(
X4
↔
X48
)
)
)
)
→
(
¬
(
X4
↔
X36
)
)
)
)
↔
(
¬
(
X48
↔
X4
)
)
)
→
X43
→
X36
)
↔
(
X43
↔
(
X36
↔
X1
)
)
)
)
)
→
(
¬
(
(
¬
X45
)
→
X44
→
(
¬
X15
)
→
X47
)
)
→
(
¬
(
(
(
¬
X15
)
→
(
¬
X5
)
→
X47
)
→
(
¬
(
X44
↔
(
X43
↔
X36
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
¬
(
X40
↔
(
(
¬
X32
)
↔
(
(
¬
X2
)
→
X48
)
)
)
)
→
(
¬
(
(
(
¬
X47
)
→
X40
→
(
¬
X32
)
→
X48
)
→
(
(
¬
X40
)
↔
(
(
¬
X32
)
→
(
X20
↔
X48
)
)
)
)
)
)
)
→
(
X40
→
(
X32
↔
X46
)
)
→
(
¬
(
(
(
¬
(
(
¬
X32
)
↔
(
(
¬
X29
)
↔
X40
)
)
)
↔
(
(
X49
↔
(
(
¬
X40
)
→
X32
→
X48
)
)
↔
(
¬
(
X32
↔
(
X29
→
False
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X32
)
↔
(
X9
↔
X49
)
)
)
→
(
(
¬
X48
)
↔
(
(
¬
X46
)
→
X49
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
X47
)
↔
(
X40
↔
X49
)
)
)
)
↔
(
(
(
(
X31
→
(
X26
↔
X18
)
)
↔
(
(
X31
↔
X18
)
↔
(
X46
→
(
X31
↔
(
(
¬
X10
)
↔
X26
)
)
)
)
)
↔
(
¬
(
X46
↔
(
X40
→
(
X31
↔
False
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X31
)
↔
(
(
¬
X18
)
→
(
(
¬
X11
)
↔
X33
)
)
)
)
→
(
X33
→
(
X32
↔
(
X11
→
X46
)
)
)
→
X46
→
(
(
¬
X17
)
↔
(
X11
↔
X33
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
(
(
(
¬
X39
)
→
X12
→
(
(
¬
X3
)
↔
X42
)
)
→
(
(
(
(
¬
(
X41
→
False
)
)
↔
(
(
¬
(
(
¬
X41
)
↔
X49
)
)
→
(
(
¬
(
(
X12
↔
(
X3
→
X41
)
)
→
(
¬
(
X49
↔
(
X41
→
(
X39
↔
(
X3
→
False
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
X41
)
→
(
X3
↔
X49
)
)
)
)
→
(
¬
(
X49
→
X41
→
(
X39
↔
False
)
)
)
)
)
↔
(
¬
(
X39
→
(
(
¬
X3
)
↔
X49
)
)
)
)
↔
(
(
¬
X12
)
↔
(
(
¬
X3
)
→
X39
)
)
)
)
)
→
(
(
(
¬
(
X25
→
X22
→
X39
)
)
→
(
(
X41
↔
(
X13
↔
(
(
¬
X8
)
↔
X25
)
)
)
↔
(
(
¬
(
X25
↔
(
(
¬
X13
)
→
X42
)
)
)
↔
(
¬
(
X42
→
X41
→
(
(
¬
X22
)
↔
X25
)
)
)
)
)
)
↔
(
¬
(
X42
↔
(
X41
↔
(
X39
↔
X44
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
X49
↔
(
X0
↔
X6
)
)
)
→
(
(
X16
↔
(
(
¬
X6
)
↔
X49
)
)
↔
(
¬
(
(
¬
(
X37
→
(
X6
↔
X16
)
)
)
→
(
X37
↔
(
(
¬
X22
)
↔
(
(
¬
X6
)
→
X49
)
)
)
)
)
)
)
)
↔
(
(
X49
→
X45
→
(
¬
X30
)
→
X37
)
→
(
¬
X44
)
→
(
(
¬
X37
)
↔
(
X22
↔
X45
)
)
)
)
→
(
¬
(
X44
→
X27
→
X49
)
)
→
(
X49
↔
(
X14
↔
X30
)
)
)
)
)
→
(
(
X42
↔
(
X34
↔
(
(
¬
X23
)
→
X41
)
)
)
↔
(
X45
→
(
X42
↔
(
X41
→
X35
→
False
)
)
)
)
→
(
(
(
(
¬
X35
)
↔
(
X34
→
(
(
¬
X21
)
↔
X42
)
)
)
→
(
X42
↔
(
X38
→
X35
→
X21
→
False
)
)
)
↔
(
¬
(
(
¬
X35
)
↔
(
(
¬
X34
)
↔
(
X28
↔
X38
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
6b7b79...
and proposition id is
62259f...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMFyAAbgwiUEzF8APe515pKLeBoP9gYu4t7
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
∀X50 :
prop
,
(
(
(
(
¬
(
(
(
(
(
¬
(
X47
→
(
X36
↔
X37
)
)
)
↔
(
(
X47
↔
(
(
¬
X30
)
→
X37
)
)
→
(
¬
(
(
¬
(
(
¬
X47
)
→
X40
→
X37
→
X50
)
)
→
(
X50
↔
(
X47
↔
X30
)
)
)
)
)
)
→
(
X40
↔
X47
)
)
→
(
(
(
X40
→
(
¬
X36
)
→
(
(
¬
X1
)
↔
X37
)
)
↔
(
¬
(
X11
↔
(
(
¬
X1
)
→
X37
)
)
)
)
↔
(
X45
↔
(
X11
→
X1
→
X37
)
)
)
)
→
(
¬
(
X45
→
X40
→
X2
→
X29
)
)
→
(
(
¬
(
X50
↔
(
X45
→
X40
→
False
)
)
)
↔
(
(
¬
(
(
¬
X40
)
↔
(
(
¬
X29
)
→
(
X8
↔
X50
)
)
)
)
↔
(
¬
(
X45
→
(
¬
X29
)
→
X8
→
X40
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
(
¬
(
X48
→
(
¬
X37
)
→
X41
)
)
→
(
(
(
¬
(
(
¬
(
(
¬
X21
)
→
X37
)
)
→
(
(
¬
(
X45
→
(
(
¬
X41
)
↔
(
(
¬
X21
)
→
X48
)
)
)
)
↔
(
¬
(
X48
↔
(
X37
↔
(
X21
→
X13
)
)
)
)
)
)
)
↔
(
X45
↔
(
X41
→
(
X37
↔
False
)
)
)
)
↔
(
X45
↔
(
(
¬
X37
)
→
X48
)
)
)
)
→
X45
→
X39
)
)
↔
(
(
(
¬
(
X48
↔
(
(
¬
X16
)
→
(
¬
X14
)
→
X45
)
)
)
↔
(
(
(
X48
↔
(
X41
→
(
¬
X14
)
→
X36
)
)
→
(
¬
(
(
¬
X45
)
↔
(
(
¬
X41
)
→
X35
→
X48
)
)
)
→
(
X48
↔
(
X45
→
(
X14
↔
X41
)
)
)
)
→
(
¬
(
X35
→
X45
)
)
)
)
↔
(
X48
→
X45
→
(
X10
↔
X36
)
)
)
)
→
(
(
¬
(
(
(
X37
→
X18
→
(
¬
X5
)
→
X33
)
↔
(
¬
(
X37
→
(
X22
↔
(
(
¬
X5
)
→
X33
)
)
)
)
)
→
X37
→
X33
→
X22
→
X6
→
False
)
)
↔
(
¬
(
(
X37
→
(
X33
↔
X20
)
)
→
(
(
X45
→
(
(
¬
X20
)
↔
X22
)
)
↔
(
¬
(
X45
↔
(
X33
→
(
¬
X20
)
→
X36
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
X49
↔
(
X42
→
(
X31
↔
X23
)
)
)
→
(
(
(
(
¬
X49
)
↔
(
(
¬
X42
)
↔
(
X31
↔
X50
)
)
)
↔
(
(
¬
(
(
¬
X42
)
→
(
¬
X26
)
→
X50
)
)
↔
(
¬
(
X49
→
(
X42
↔
(
(
¬
X12
)
↔
X50
)
)
)
)
)
)
↔
(
(
¬
X42
)
↔
(
(
¬
X12
)
↔
X49
)
)
)
)
)
↔
(
(
(
¬
(
X48
→
(
X46
↔
(
X19
↔
False
)
)
)
)
↔
(
¬
(
X46
↔
(
X31
→
X19
→
X15
)
)
)
)
↔
(
(
¬
(
X48
→
(
X31
↔
(
(
¬
X26
)
→
X46
)
)
)
)
→
(
¬
(
X50
→
X48
→
(
X46
↔
(
X31
↔
False
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
X46
↔
(
X44
↔
(
X24
→
False
)
)
)
→
(
¬
(
X44
↔
(
X34
→
(
¬
X3
)
→
X32
)
)
)
→
(
¬
(
X44
→
(
(
¬
X34
)
↔
(
X3
→
X50
)
)
)
)
)
)
↔
(
(
(
(
X38
↔
(
(
¬
X27
)
→
(
X7
↔
X46
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X27
)
↔
(
X7
→
X50
)
)
)
)
)
↔
(
¬
(
X38
→
X27
→
X46
)
)
)
↔
(
(
(
¬
(
(
¬
X45
)
→
(
X32
↔
X46
)
)
)
↔
(
X45
→
X38
→
(
¬
X9
)
→
X32
)
)
→
(
¬
(
X45
→
X38
→
X32
→
False
)
)
)
)
)
↔
(
(
(
(
(
(
(
(
¬
X28
)
→
(
¬
X0
)
→
X34
)
→
(
¬
(
X48
↔
(
X28
→
X4
→
X34
)
)
)
)
→
X34
→
(
¬
X28
)
→
(
X17
↔
X48
)
)
→
(
¬
(
(
¬
X0
)
→
X48
)
)
)
↔
(
X17
↔
(
X4
↔
(
X0
→
False
)
)
)
)
↔
(
¬
(
X48
↔
(
(
¬
X17
)
↔
(
(
¬
X4
)
↔
X43
)
)
)
)
)
→
(
¬
(
(
¬
(
X43
↔
(
X34
↔
(
X25
→
X44
)
)
)
)
→
(
(
X44
→
X34
→
X32
→
X25
)
↔
(
¬
(
X44
→
X34
→
X48
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
¬
(
(
¬
X40
)
↔
(
X11
→
X47
)
)
)
↔
(
(
(
X49
→
X10
→
(
X6
↔
X11
)
)
↔
(
¬
(
(
¬
X47
)
↔
(
(
¬
X10
)
↔
X49
)
)
)
)
→
(
(
(
¬
X40
)
→
(
(
¬
X10
)
↔
(
X6
↔
X49
)
)
)
↔
(
¬
(
X49
↔
(
X4
→
False
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
X21
↔
X2
)
)
→
(
¬
X41
)
→
(
¬
X21
)
→
(
¬
X1
)
→
X49
)
↔
(
X21
↔
(
(
¬
X2
)
↔
X49
)
)
)
↔
(
¬
(
X47
↔
(
X2
↔
X41
)
)
)
)
↔
(
(
¬
(
X47
→
X41
→
X24
→
X40
)
)
→
(
(
¬
(
(
¬
X40
)
↔
(
X24
→
X49
)
)
)
↔
(
X41
↔
(
X40
→
X15
→
False
)
)
)
)
)
)
)
→
(
¬
(
(
(
X34
→
(
¬
X23
)
→
(
X8
↔
X49
)
)
↔
(
(
X49
↔
(
X47
→
X23
→
X8
)
)
↔
(
¬
(
(
¬
X41
)
↔
(
X14
→
X47
)
)
)
)
)
→
(
(
¬
(
(
(
¬
X35
)
↔
(
X32
↔
X46
)
)
→
(
(
¬
(
(
¬
X41
)
→
(
¬
X35
)
→
(
¬
X32
)
→
X47
)
)
↔
(
¬
(
X50
↔
(
X46
↔
(
X32
↔
X35
)
)
)
)
)
)
)
↔
(
X41
→
X35
→
(
(
¬
X34
)
↔
X50
)
)
)
)
)
)
→
(
(
(
(
(
(
¬
(
X31
→
X27
)
)
↔
(
(
¬
(
X41
↔
(
(
¬
X31
)
↔
(
X27
→
X48
)
)
)
)
→
(
X48
↔
(
X44
↔
(
X27
→
X31
)
)
)
)
)
↔
(
¬
(
(
X49
↔
(
X43
→
X31
→
X12
)
)
→
(
¬
(
X41
↔
(
(
¬
X36
)
→
(
¬
X27
)
→
X43
)
)
)
→
(
(
¬
X41
)
↔
(
(
¬
X36
)
↔
(
(
¬
X0
)
↔
X43
)
)
)
)
)
)
↔
(
¬
(
X44
→
(
¬
X43
)
→
(
¬
X16
)
→
X49
)
)
)
→
(
(
(
X48
→
X44
→
X19
→
False
)
→
(
¬
(
(
¬
(
X44
↔
(
X42
→
X48
)
)
)
→
(
¬
(
X48
↔
(
X29
→
(
(
¬
X19
)
↔
X44
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
X43
→
X42
→
(
X40
↔
X37
)
)
↔
(
¬
(
X42
→
(
(
¬
X28
)
↔
X40
)
)
)
)
→
(
X42
↔
(
(
¬
X28
)
↔
(
X26
→
X37
)
)
)
)
)
↔
(
X43
↔
(
X42
→
X37
)
)
)
↔
(
(
¬
(
(
¬
X37
)
↔
(
(
¬
X30
)
→
(
X3
↔
X44
)
)
)
)
↔
(
(
¬
(
X40
→
(
X28
↔
False
)
)
)
↔
(
(
¬
X37
)
↔
(
(
¬
X28
)
↔
(
X3
→
X48
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X49
→
(
(
¬
X44
)
↔
(
(
¬
X25
)
↔
X48
)
)
)
)
↔
(
(
¬
(
(
¬
X44
)
→
(
X5
↔
X49
)
)
)
↔
(
X49
↔
(
X48
↔
(
X5
→
X9
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X38
→
(
¬
X33
)
→
(
¬
X17
)
→
X45
)
)
→
X22
→
(
X17
↔
(
(
¬
X13
)
↔
X45
)
)
)
→
(
(
(
¬
(
X49
↔
(
X48
↔
(
X45
↔
X33
)
)
)
)
↔
(
(
X49
→
X48
→
(
X22
↔
X18
)
)
↔
(
(
X49
→
(
¬
X39
)
→
X22
→
X48
)
→
(
(
¬
X48
)
↔
(
(
¬
X45
)
→
X22
→
X49
)
)
)
)
)
↔
(
X49
↔
(
X39
→
X48
)
)
)
)
)
→
(
¬
(
(
¬
(
X44
→
(
¬
X39
)
→
X38
→
X49
)
)
→
(
(
(
(
¬
X20
)
→
(
¬
X7
)
→
X44
)
→
(
(
¬
(
X49
→
(
X20
↔
(
X7
↔
X44
)
)
)
)
↔
(
(
¬
X44
)
↔
(
X39
→
(
X7
↔
X49
)
)
)
)
)
↔
(
¬
(
(
¬
X44
)
→
(
(
¬
X20
)
↔
X49
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
99cc94...
and proposition id is
c79a0a...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMZkjDoUzR7GJWzd1bFUoJcwDPn1fMhvxxV
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
¬
(
(
(
(
(
X35
→
(
X30
↔
(
(
¬
X26
)
→
X41
)
)
)
↔
(
(
¬
X35
)
↔
(
X26
↔
(
(
¬
X20
)
↔
X41
)
)
)
)
→
(
(
¬
(
(
¬
X30
)
→
X26
→
(
X7
↔
X45
)
)
)
↔
(
(
¬
X30
)
↔
(
(
¬
X26
)
↔
X35
)
)
)
)
↔
(
(
¬
(
(
X45
↔
X43
)
→
(
¬
(
(
¬
X41
)
→
X30
→
X47
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X45
)
↔
(
(
¬
X41
)
↔
(
X31
↔
X47
)
)
)
)
→
(
¬
(
(
¬
(
X47
→
(
¬
X12
)
→
X45
)
)
→
(
X41
↔
(
X12
↔
(
(
¬
X1
)
→
X45
)
)
)
)
)
)
)
)
)
→
(
(
(
X47
→
X43
→
X41
)
→
X49
→
(
(
¬
X34
)
↔
(
(
¬
X3
)
→
X41
)
)
)
↔
(
¬
(
(
¬
X47
)
↔
(
(
¬
X41
)
→
X0
→
X49
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X45
)
→
(
X44
↔
(
(
¬
X22
)
↔
X46
)
)
)
)
↔
(
(
¬
(
X45
→
(
¬
X44
)
→
X39
→
X46
)
)
↔
(
(
¬
X6
)
→
X44
)
)
)
→
(
(
¬
(
(
¬
X44
)
→
(
(
¬
X43
)
↔
(
(
¬
X2
)
↔
X46
)
)
)
)
↔
(
¬
(
X46
↔
(
X43
↔
(
X27
→
X44
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X44
→
X39
→
(
X28
↔
X49
)
)
)
↔
(
¬
(
X44
→
X39
→
X28
→
X10
→
False
)
)
)
↔
(
(
¬
X45
)
→
X49
)
)
↔
(
X45
↔
(
X38
↔
(
(
¬
X10
)
→
X28
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
X46
↔
(
X42
↔
(
(
¬
X33
)
→
X47
)
)
)
→
(
(
X47
→
(
X24
↔
X29
)
)
↔
(
(
¬
(
(
(
X33
→
(
¬
X24
)
→
X42
)
↔
(
(
X47
↔
(
X42
→
X24
)
)
↔
(
(
¬
X42
)
↔
(
(
¬
X33
)
→
X4
→
X47
)
)
)
)
→
X33
→
(
X24
↔
(
X4
→
False
)
)
)
)
↔
(
X33
↔
(
X29
↔
(
(
¬
X24
)
↔
X47
)
)
)
)
)
)
)
↔
(
(
(
X42
→
(
X32
↔
(
X16
→
False
)
)
)
→
(
(
¬
(
X42
→
X33
→
(
¬
X32
)
→
X46
)
)
↔
(
X46
→
(
X42
↔
(
(
¬
X33
)
→
X48
)
)
)
)
)
→
(
(
X48
↔
(
X25
→
(
¬
X15
)
→
X42
)
)
↔
(
¬
(
(
¬
X32
)
→
(
¬
X15
)
→
X48
)
)
)
→
(
(
¬
X42
)
↔
(
(
¬
X25
)
↔
(
X19
↔
X48
)
)
)
)
)
→
(
(
(
(
(
X8
↔
(
(
¬
X5
)
→
X48
)
)
→
(
(
¬
(
X44
↔
(
(
¬
X8
)
↔
X48
)
)
)
↔
(
X48
→
X13
→
X5
→
False
)
)
)
↔
(
X46
↔
(
(
¬
X8
)
→
(
¬
X5
)
→
X44
)
)
)
↔
(
¬
(
(
¬
(
X44
→
X42
)
)
→
(
¬
(
X46
↔
(
X42
→
(
X18
↔
X48
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
X42
→
(
X14
↔
X8
)
)
→
(
¬
(
(
¬
(
X21
→
(
¬
X8
)
→
X36
)
)
→
(
X23
↔
(
X14
→
(
(
¬
X8
)
↔
X21
)
)
)
)
)
)
↔
(
X36
→
(
X23
↔
X42
)
)
)
↔
(
X23
↔
(
X21
↔
(
X9
↔
X42
)
)
)
)
→
(
¬
(
(
X17
↔
(
(
¬
X11
)
↔
(
(
¬
X9
)
→
X13
)
)
)
→
(
¬
(
(
X21
→
(
¬
X17
)
→
X13
→
X37
)
→
(
¬
(
X21
→
(
(
¬
X14
)
↔
X37
)
)
)
→
(
X40
↔
(
(
¬
X17
)
→
(
¬
X14
)
→
X21
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
(
¬
(
(
¬
X29
)
→
(
X25
↔
(
(
¬
X7
)
→
X49
)
)
)
)
↔
(
X29
→
(
¬
X15
)
→
X25
)
)
→
(
¬
(
(
¬
(
X15
→
(
X7
↔
X49
)
)
)
→
(
¬
(
(
¬
(
(
¬
X15
)
→
(
X8
↔
X34
)
)
)
→
(
¬
X34
)
→
X15
→
(
¬
X7
)
→
X39
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
X47
→
(
(
¬
X20
)
↔
(
(
¬
X19
)
↔
X29
)
)
)
→
(
X29
↔
(
X21
↔
(
(
¬
X20
)
↔
X23
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X44
↔
(
X34
→
(
(
¬
X25
)
↔
X29
)
)
)
)
→
X44
→
(
X34
↔
(
(
¬
X1
)
→
X29
)
)
)
)
↔
(
(
¬
(
X25
↔
(
(
¬
X19
)
→
(
¬
X4
)
→
X21
)
)
)
→
(
X44
↔
(
X25
→
X21
→
(
X1
↔
False
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
X28
)
→
X22
→
X2
→
X37
)
↔
(
¬
(
(
¬
X28
)
→
(
(
¬
X2
)
↔
X42
)
)
)
)
↔
(
(
(
(
(
¬
(
X43
↔
(
X17
→
X47
)
)
)
↔
(
¬
(
X38
↔
(
X17
↔
X27
)
)
)
)
↔
(
(
X44
↔
(
(
¬
X38
)
→
(
X14
↔
X43
)
)
)
↔
(
X44
→
(
¬
X42
)
→
X47
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
(
¬
X44
)
↔
(
X43
↔
(
X28
↔
X47
)
)
)
)
↔
(
¬
(
X43
→
(
X37
↔
(
(
¬
X28
)
→
X44
)
)
)
)
)
→
X47
→
(
X37
↔
X48
)
)
↔
(
¬
(
(
¬
X47
)
→
X43
→
X48
)
)
)
→
(
¬
(
(
¬
X10
)
→
X44
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X44
→
(
X43
↔
(
X5
↔
X48
)
)
)
)
→
(
(
¬
X48
)
↔
(
X43
→
X49
)
)
)
→
(
(
¬
(
(
X49
↔
(
X32
↔
X26
)
)
→
(
X32
↔
(
(
¬
X26
)
↔
X48
)
)
→
(
X32
↔
(
X16
→
X49
)
)
)
)
↔
(
X49
↔
(
X44
↔
(
X16
→
X48
)
)
)
)
)
)
)
)
)
)
→
(
(
(
X46
→
X45
→
(
X35
↔
(
X30
↔
False
)
)
)
→
(
¬
(
(
¬
(
(
X41
↔
(
X0
→
X24
)
)
→
(
¬
(
X44
↔
(
X41
→
(
¬
X30
)
→
X46
)
)
)
)
)
→
(
(
¬
X41
)
↔
(
(
¬
X35
)
↔
(
(
¬
X30
)
↔
X46
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X33
)
→
X45
)
)
→
X40
→
(
(
¬
X33
)
↔
(
X6
↔
X44
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
(
¬
X41
)
→
(
X36
↔
(
X18
↔
X44
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X41
)
↔
(
X40
→
X44
)
)
)
)
→
(
(
¬
X41
)
↔
(
X40
→
(
¬
X36
)
→
X44
)
)
)
→
X47
→
(
(
¬
X40
)
↔
X41
)
)
)
↔
(
(
(
¬
(
(
¬
X44
)
↔
(
X41
↔
X45
)
)
)
↔
(
¬
(
(
¬
X41
)
↔
(
X36
→
(
¬
X31
)
→
X45
)
)
)
)
↔
(
¬
(
X44
↔
(
(
¬
X41
)
↔
X45
)
)
)
)
)
→
(
(
(
X13
↔
(
X3
↔
X36
)
)
↔
(
¬
(
(
¬
X18
)
→
(
X13
↔
(
(
¬
X3
)
↔
X48
)
)
)
)
)
↔
(
¬
(
(
(
X47
↔
(
(
¬
X36
)
↔
(
(
¬
X11
)
↔
X48
)
)
)
→
(
¬
(
X48
↔
(
X12
↔
X47
)
)
)
)
→
(
¬
(
(
X11
↔
(
X9
↔
X18
)
)
→
(
¬
(
X47
→
(
¬
X11
)
→
X48
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
f8d5e7...
and proposition id is
d89185...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMYPT8A1rqowRLn9KHwVVtxqujmaXW8B6P5
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
¬
(
(
(
(
(
¬
(
(
(
(
X47
→
(
X37
↔
(
(
¬
X34
)
→
X46
)
)
)
→
(
X47
↔
(
(
¬
X34
)
→
(
X16
↔
X46
)
)
)
)
↔
(
(
¬
(
X46
↔
(
(
¬
X37
)
→
(
¬
X35
)
→
X49
)
)
)
→
(
X47
↔
(
X37
↔
(
(
¬
X35
)
→
X49
)
)
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X46
)
→
(
¬
X36
)
→
X49
)
)
)
)
)
↔
(
(
¬
(
X50
↔
(
X40
↔
(
X37
↔
(
X36
↔
False
)
)
)
)
)
↔
(
¬
(
(
(
¬
X49
)
→
(
¬
X37
)
→
X0
→
X50
)
→
(
¬
(
X49
→
X40
→
(
¬
X37
)
→
X50
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X32
)
↔
(
(
¬
X31
)
→
(
X18
↔
X46
)
)
)
)
↔
(
¬
(
X47
↔
(
X46
↔
(
X44
→
X31
)
)
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X45
)
→
(
¬
X44
)
→
X46
)
)
)
)
↔
(
(
X47
↔
(
X37
↔
(
X14
↔
X44
)
)
)
↔
(
¬
(
X45
→
(
X44
↔
X14
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
¬
X38
)
→
(
X7
↔
X48
)
)
→
(
X8
↔
(
X5
→
False
)
)
)
→
(
(
¬
X8
)
↔
(
(
¬
X7
)
→
X6
→
X13
)
)
→
(
¬
(
X48
↔
(
X13
↔
(
X7
↔
X6
)
)
)
)
)
)
↔
(
(
X48
↔
(
X38
→
X10
→
X45
)
)
↔
(
X48
↔
(
(
¬
X13
)
→
(
X10
↔
X45
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X40
)
↔
(
X30
↔
(
(
¬
X25
)
→
X45
)
)
)
)
→
(
(
¬
(
X48
↔
(
X38
↔
(
X19
↔
X40
)
)
)
)
↔
(
(
(
¬
(
X45
↔
(
X38
→
X21
→
X48
)
)
)
↔
(
(
¬
X45
)
↔
(
(
¬
X37
)
↔
(
X21
↔
X48
)
)
)
)
→
(
¬
(
X48
→
(
X40
↔
(
(
¬
X38
)
↔
X45
)
)
)
)
)
)
→
X48
→
(
X38
↔
(
(
¬
X30
)
→
X45
)
)
)
)
)
)
→
(
¬
(
(
(
(
X46
→
X42
→
(
X33
↔
X17
)
)
↔
(
(
(
X42
→
(
X33
↔
(
(
¬
X20
)
↔
X46
)
)
)
→
(
¬
(
X46
↔
(
X33
↔
(
X12
↔
X42
)
)
)
)
)
→
X42
→
(
(
¬
X20
)
↔
(
X9
→
X43
)
)
)
)
→
(
(
(
¬
(
(
X28
→
(
X26
↔
(
X23
↔
X42
)
)
)
→
(
(
(
X28
→
X26
→
(
¬
X23
)
→
X41
)
→
(
¬
(
X28
→
(
(
¬
X26
)
↔
X39
)
)
)
)
↔
(
¬
(
X41
↔
(
X28
↔
(
X26
→
False
)
)
)
)
)
)
)
↔
(
X46
↔
(
X41
↔
(
(
¬
X39
)
→
X42
)
)
)
)
↔
(
(
¬
(
X39
↔
(
(
¬
X2
)
→
X22
)
)
)
→
(
¬
(
(
(
(
¬
(
X39
↔
(
X28
→
(
X24
↔
(
X22
↔
False
)
)
)
)
)
↔
(
X28
↔
(
X22
→
(
(
¬
X4
)
↔
X39
)
)
)
)
↔
(
X24
↔
(
(
¬
X4
)
↔
X22
)
)
)
→
(
¬
(
(
¬
(
X24
→
X22
→
(
(
¬
X1
)
↔
X33
)
)
)
→
(
(
X41
→
X28
→
X24
→
False
)
↔
(
(
¬
X24
)
↔
(
X22
→
(
(
¬
X2
)
↔
X41
)
)
)
)
→
X28
→
(
X22
↔
(
X2
↔
X33
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
¬
(
(
¬
X33
)
↔
(
(
¬
X27
)
→
(
(
¬
X24
)
↔
X47
)
)
)
)
↔
(
¬
(
X48
↔
(
X47
↔
(
X27
↔
X24
)
)
)
)
)
↔
(
¬
(
(
¬
X33
)
→
X24
→
X48
)
)
)
→
(
¬
(
X48
→
(
X47
↔
(
X29
→
X33
)
)
)
)
)
)
↔
(
(
¬
(
X26
→
(
(
¬
X3
)
↔
X11
)
)
)
↔
(
(
¬
(
X47
↔
(
(
¬
X11
)
↔
X48
)
)
)
↔
(
X48
↔
(
X26
↔
(
(
¬
X15
)
→
X33
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
(
(
(
¬
(
X32
↔
(
X17
↔
X45
)
)
)
↔
(
X49
→
X37
→
(
X32
↔
False
)
)
)
→
(
X32
↔
(
X17
→
X37
)
)
)
→
(
¬
(
(
(
(
¬
(
(
¬
X36
)
↔
(
X17
↔
(
X9
↔
X45
)
)
)
)
→
(
X46
↔
(
(
¬
X9
)
→
X17
)
)
)
↔
(
X46
↔
X36
)
)
→
(
(
¬
X36
)
↔
(
(
¬
X17
)
↔
(
(
¬
X9
)
↔
X49
)
)
)
)
)
)
→
X45
→
(
¬
X40
)
→
X37
→
X46
)
→
(
¬
(
X46
↔
(
(
¬
X45
)
→
X22
→
X49
)
)
)
→
(
¬
(
(
¬
X22
)
→
X26
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X27
↔
X42
)
)
→
(
X43
↔
(
(
¬
X24
)
↔
(
X1
↔
X42
)
)
)
)
)
↔
(
(
¬
(
X43
→
(
X19
↔
X42
)
)
)
→
(
(
(
¬
X43
)
↔
(
X42
→
X40
→
(
X27
↔
X8
)
)
)
→
(
¬
(
X43
→
X27
→
X42
)
)
)
→
(
¬
(
(
¬
(
X27
↔
(
X19
→
X11
)
)
)
→
(
(
X27
→
X11
→
False
)
↔
(
(
¬
(
X43
↔
(
(
¬
X11
)
→
X27
)
)
)
→
(
(
X11
↔
X43
)
↔
(
X40
↔
(
X27
→
(
(
¬
X19
)
↔
X43
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
X50
→
X43
)
→
X49
→
(
¬
X45
)
→
(
X7
↔
X50
)
)
↔
(
X50
↔
(
X49
↔
X45
)
)
)
↔
(
(
¬
(
X49
↔
(
X43
→
X15
→
X50
)
)
)
↔
(
X49
↔
(
(
¬
X40
)
→
X50
)
)
)
)
→
(
¬
(
X50
→
(
¬
X45
)
→
(
(
¬
X10
)
↔
X49
)
)
)
→
(
X49
↔
(
(
¬
X42
)
→
(
X0
↔
X45
)
)
)
→
(
¬
(
X49
→
(
X45
↔
(
(
¬
X0
)
→
X50
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X50
→
X44
→
(
X12
↔
X30
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X33
→
(
X30
↔
X5
)
)
)
→
X50
→
(
X30
↔
(
X5
↔
X12
)
)
)
)
→
(
(
¬
X33
)
↔
(
X30
→
X50
)
)
)
)
)
→
(
¬
(
X50
↔
(
(
¬
X25
)
↔
(
X12
↔
X44
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
(
¬
(
X50
↔
(
X45
↔
(
(
¬
X13
)
→
X44
)
)
)
)
→
(
¬
(
X45
→
X44
→
X13
→
X18
)
)
)
→
X47
→
X44
→
(
X13
↔
X18
)
)
)
↔
(
¬
(
X50
→
(
¬
X44
)
→
(
¬
X31
)
→
X45
)
)
)
→
(
¬
(
(
(
(
¬
X35
)
→
X33
→
X45
)
→
X48
→
(
¬
X45
)
→
(
¬
X35
)
→
X50
)
→
(
¬
(
(
X50
↔
(
(
¬
X6
)
→
X35
)
)
→
(
¬
(
(
¬
(
X47
→
X45
)
)
→
X50
→
(
X6
↔
X35
)
)
)
→
(
¬
(
X45
→
(
X35
↔
(
X6
→
False
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
X33
→
(
¬
X23
)
→
(
¬
X2
)
→
X47
)
)
↔
(
X47
↔
(
X33
→
X23
→
X48
)
)
)
)
)
)
→
(
(
(
(
X47
↔
(
X41
→
X20
→
False
)
)
↔
(
(
¬
(
(
(
(
¬
(
X41
→
(
(
¬
X38
)
↔
(
(
¬
X20
)
→
X45
)
)
)
)
↔
(
X45
→
(
X38
↔
(
X29
→
X41
)
)
)
)
↔
(
¬
(
X38
↔
(
(
¬
X29
)
→
(
(
¬
X14
)
↔
X41
)
)
)
)
)
→
(
(
¬
X45
)
↔
(
X41
→
X38
→
X47
)
)
)
)
↔
(
¬
(
X45
↔
X14
)
)
)
)
↔
(
(
(
¬
X47
)
↔
(
X21
↔
X48
)
)
→
(
(
¬
X45
)
↔
(
(
¬
X41
)
→
X46
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X41
→
(
X29
↔
(
X3
↔
False
)
)
)
)
→
(
(
¬
X41
)
↔
(
(
¬
X39
)
↔
(
X3
→
X47
)
)
)
→
(
¬
(
X39
→
X29
→
(
(
¬
X3
)
↔
X41
)
)
)
)
)
↔
(
(
¬
(
X47
↔
(
X41
↔
(
X29
↔
(
X16
→
False
)
)
)
)
)
↔
(
¬
(
X47
↔
(
X29
→
(
X28
↔
X41
)
)
)
)
)
)
↔
(
X47
→
X39
→
(
X34
↔
(
X4
→
False
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
2dfa55...
and proposition id is
1f5282...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMSyfzCYLx3ZMHM4ywkEuss2qaN6juTACf7
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
¬
(
(
(
(
(
(
(
¬
X37
)
↔
(
(
¬
X28
)
↔
X46
)
)
↔
(
(
X49
→
(
X45
↔
(
(
¬
X28
)
↔
X46
)
)
)
→
(
¬
(
X37
↔
(
(
¬
X28
)
↔
X46
)
)
)
)
)
→
(
¬
(
X43
→
(
¬
X28
)
→
X23
→
X46
)
)
→
X49
→
(
(
¬
X43
)
↔
(
(
¬
X23
)
→
X46
)
)
)
↔
(
(
(
(
¬
(
(
(
(
¬
(
X45
↔
(
X39
→
(
X29
↔
X1
)
)
)
)
↔
(
X46
↔
(
(
¬
X39
)
→
(
X1
↔
X45
)
)
)
)
↔
(
X39
→
X1
)
)
→
(
X45
↔
X39
)
)
)
↔
(
X48
↔
(
(
¬
X1
)
↔
X29
)
)
)
↔
(
¬
(
X48
↔
(
X29
↔
(
(
¬
X18
)
→
X39
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X48
)
↔
(
(
¬
X39
)
↔
(
(
¬
X4
)
↔
X49
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
¬
X39
)
→
(
X32
↔
X49
)
)
)
→
(
X48
↔
(
X32
↔
(
(
¬
X14
)
→
X39
)
)
)
)
)
→
(
¬
(
(
X48
↔
(
(
¬
X32
)
↔
(
X29
→
X49
)
)
)
→
(
¬
(
X29
→
(
X11
↔
X32
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
X39
)
→
(
X29
↔
X48
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
X49
→
X20
→
X8
→
False
)
)
↔
(
¬
(
X48
↔
(
X2
→
X49
)
)
)
)
↔
(
(
¬
X48
)
↔
(
(
¬
X22
)
↔
(
(
¬
X20
)
↔
X49
)
)
)
)
↔
(
¬
(
(
¬
X46
)
→
(
X8
↔
(
(
¬
X2
)
↔
X49
)
)
)
)
)
↔
(
(
(
¬
(
X49
↔
(
(
¬
X46
)
↔
(
X26
↔
X48
)
)
)
)
↔
(
(
X48
↔
(
(
¬
X26
)
→
X0
→
X46
)
)
↔
(
¬
(
X48
→
X46
→
(
¬
X13
)
→
X26
)
)
)
)
↔
(
(
¬
(
X48
→
(
X46
↔
(
X27
↔
X49
)
)
)
)
↔
(
¬
(
(
¬
X27
)
→
(
X16
↔
(
(
¬
X7
)
↔
X49
)
)
)
)
)
)
)
↔
(
¬
(
(
(
X20
↔
(
X15
↔
(
(
¬
X6
)
→
X42
)
)
)
↔
(
¬
(
(
¬
(
X46
↔
(
X26
↔
(
X6
↔
False
)
)
)
)
→
(
¬
(
X26
↔
(
X6
↔
(
(
¬
X5
)
↔
X42
)
)
)
)
)
)
)
→
(
¬
(
(
¬
X42
)
↔
(
X20
↔
(
X10
→
X46
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
X47
↔
(
X40
↔
X46
)
)
↔
(
(
(
¬
X30
)
↔
X40
)
↔
(
(
(
(
¬
(
(
¬
X46
)
↔
(
X38
→
(
¬
X30
)
→
X47
)
)
)
→
(
¬
(
X46
↔
(
X19
↔
X30
)
)
)
)
↔
(
X38
→
(
¬
X30
)
→
X46
)
)
→
(
X38
↔
(
(
¬
X30
)
↔
X46
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X3
)
↔
X33
)
)
↔
(
X44
↔
(
X40
→
X38
→
X33
→
False
)
)
)
→
(
¬
(
X38
→
(
(
¬
X33
)
↔
(
(
¬
X3
)
→
X47
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
(
X47
→
(
(
¬
X25
)
↔
(
X17
→
X43
)
)
)
↔
(
X46
→
(
(
¬
X25
)
↔
(
(
¬
X17
)
↔
X47
)
)
)
)
↔
(
(
¬
X43
)
↔
(
X12
→
X47
)
)
)
→
(
X41
↔
(
X17
→
False
)
)
)
)
→
(
(
¬
(
X47
↔
(
X12
→
(
X9
↔
X41
)
)
)
)
↔
(
(
¬
X44
)
→
X41
→
(
(
¬
X12
)
↔
X47
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
X45
↔
(
X36
→
(
X34
↔
False
)
)
)
→
X47
→
(
X36
↔
X40
)
)
)
→
(
(
(
¬
(
(
(
¬
(
X35
→
X47
)
)
↔
(
(
¬
(
X45
→
(
(
¬
X31
)
↔
X36
)
)
)
↔
(
¬
(
(
(
¬
X45
)
↔
(
X31
→
X47
)
)
→
(
¬
(
X47
↔
(
(
¬
X35
)
↔
X36
)
)
)
)
)
)
)
→
(
X36
↔
X45
)
)
)
→
(
¬
X36
)
→
(
X35
↔
X45
)
)
↔
(
X47
→
(
¬
X35
)
→
X31
→
X43
)
)
)
↔
(
X47
→
X40
→
X43
)
)
↔
(
(
¬
(
X40
↔
(
(
¬
X36
)
→
(
(
¬
X21
)
↔
X45
)
)
)
)
↔
(
X45
↔
(
X40
→
X24
→
X36
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
(
(
¬
X46
)
↔
(
(
¬
X44
)
→
X3
→
X48
)
)
↔
(
(
(
¬
X44
)
→
(
(
¬
X33
)
↔
X48
)
)
↔
(
¬
(
(
(
¬
(
X44
↔
(
(
¬
X3
)
↔
X48
)
)
)
↔
(
¬
(
X33
→
(
X3
↔
X19
)
)
)
)
→
X44
→
(
X3
↔
X33
)
)
)
)
)
→
(
X46
→
(
(
¬
X33
)
↔
(
(
¬
X20
)
↔
X45
)
)
)
→
(
¬
(
(
¬
X45
)
↔
(
X44
↔
X46
)
)
)
)
)
↔
(
(
¬
(
X49
→
(
X46
↔
(
X39
→
False
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X45
)
↔
(
X39
→
X48
)
)
)
↔
(
(
(
(
¬
X39
)
↔
X46
)
→
X48
→
(
(
¬
X39
)
↔
(
X36
↔
X45
)
)
)
↔
(
¬
(
(
¬
X39
)
↔
(
(
¬
X36
)
→
X45
)
)
)
)
)
↔
(
¬
(
X46
→
(
X45
↔
X48
)
)
)
)
↔
(
(
¬
(
X46
→
X45
→
(
(
¬
X16
)
↔
X48
)
)
)
→
(
(
X48
↔
(
X45
↔
(
X40
↔
X41
)
)
)
↔
(
(
¬
(
X41
→
(
(
¬
X24
)
↔
X45
)
)
)
↔
(
X46
→
(
(
¬
X40
)
↔
X41
)
)
)
)
)
)
)
)
↔
(
(
(
(
X48
→
(
X35
↔
(
(
¬
X22
)
→
X34
)
)
)
↔
(
(
¬
(
(
¬
(
(
X49
→
(
¬
X34
)
→
(
(
¬
X25
)
↔
X48
)
)
→
X49
→
X48
→
(
X34
↔
X35
)
)
)
→
(
¬
X48
)
→
(
X25
↔
X49
)
)
)
→
(
¬
(
(
¬
(
X38
↔
(
X35
↔
(
X34
↔
False
)
)
)
)
→
(
(
(
¬
X35
)
↔
(
(
¬
X34
)
→
X12
→
X38
)
)
↔
(
¬
(
X25
↔
(
X12
→
X34
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
X31
)
→
(
¬
X15
)
→
X49
)
↔
(
(
¬
(
(
¬
(
(
¬
(
X46
↔
(
X15
↔
X48
)
)
)
→
X38
→
(
X31
↔
False
)
)
)
→
(
¬
(
X48
→
X38
→
X31
→
False
)
)
)
)
↔
(
(
¬
X46
)
→
(
(
¬
X38
)
↔
(
(
¬
X26
)
→
X48
)
)
)
)
)
→
(
¬
(
X49
→
X32
→
(
X31
↔
(
X11
→
False
)
)
)
)
→
(
(
¬
X32
)
↔
(
X31
↔
(
(
¬
X26
)
→
X49
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
X34
)
↔
(
X0
↔
X46
)
)
)
→
(
¬
(
X21
→
X4
)
)
)
)
↔
(
(
¬
X38
)
↔
(
X34
↔
(
X4
→
X46
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
(
¬
X46
)
↔
(
(
¬
X43
)
↔
(
X13
→
X47
)
)
)
)
↔
(
(
¬
(
X43
↔
(
(
¬
X13
)
↔
(
(
¬
X6
)
→
X45
)
)
)
)
→
(
¬
(
X47
↔
(
X18
→
(
X13
↔
X45
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
X48
→
X43
→
False
)
→
(
(
¬
X47
)
↔
(
X46
→
(
X37
↔
X48
)
)
)
)
)
↔
(
(
¬
X43
)
→
X47
)
)
→
(
¬
(
(
(
¬
(
X29
→
X8
→
X48
)
)
↔
(
¬
(
X47
→
X46
→
X8
)
)
)
→
X30
→
(
¬
X8
)
→
X46
)
)
)
)
)
→
(
(
(
(
(
¬
X10
)
→
X42
)
↔
(
(
(
X42
→
X14
→
X10
)
↔
(
(
¬
(
(
¬
X10
)
↔
X42
)
)
→
(
¬
(
(
¬
X14
)
→
(
(
¬
X10
)
↔
(
X7
→
X23
)
)
)
)
)
)
↔
(
(
X30
↔
(
X7
→
(
¬
X5
)
→
X42
)
)
↔
(
¬
(
X10
↔
(
(
¬
X7
)
→
(
¬
X5
)
→
X30
)
)
)
)
)
)
↔
(
X46
→
(
¬
X10
)
→
(
(
¬
X5
)
↔
X43
)
)
)
↔
(
(
(
¬
(
X42
↔
(
X27
→
X17
)
)
)
→
(
(
¬
(
X43
→
(
X42
↔
(
(
¬
X9
)
↔
X30
)
)
)
)
↔
(
¬
(
X43
→
(
X42
↔
(
X9
↔
X28
)
)
)
)
)
)
↔
(
¬
(
(
X27
→
(
X17
↔
(
X2
→
X9
)
)
)
→
X1
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
991d2d...
and proposition id is
4948fb...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMJjAP3pY9Yswu71Zqb8ADppAod6p9UyJfi
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
∀X51 :
prop
,
(
¬
∀X52 :
prop
,
(
¬
∀X53 :
prop
,
(
(
¬
(
(
¬
(
(
¬
(
(
(
¬
(
(
X53
↔
(
X32
↔
(
X25
↔
X35
)
)
)
→
(
¬
(
X53
→
(
X51
↔
(
X25
↔
False
)
)
)
)
→
(
¬
(
X50
↔
(
(
¬
X32
)
→
X25
→
X53
)
)
)
)
)
→
(
(
¬
(
(
¬
X2
)
↔
X32
)
)
↔
(
¬
(
(
(
¬
X51
)
→
(
X49
↔
(
X32
↔
X53
)
)
)
→
(
¬
(
X32
→
(
(
¬
X2
)
↔
X49
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X29
↔
X40
)
)
↔
(
X49
↔
(
X45
→
(
(
¬
X28
)
↔
X29
)
)
)
)
→
(
¬
(
X49
→
X40
→
X45
)
)
)
→
(
(
(
¬
(
X49
→
(
X40
↔
X51
)
)
)
↔
(
¬
(
X51
↔
(
X50
→
(
X45
↔
X49
)
)
)
)
)
→
(
(
¬
X49
)
↔
(
X45
↔
(
(
¬
X24
)
→
X50
)
)
)
)
→
(
¬
(
X51
→
(
(
¬
X26
)
↔
X40
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X19
)
→
(
¬
X3
)
→
X26
)
)
)
)
)
)
)
→
(
(
(
(
(
(
X41
↔
(
X39
↔
(
X14
→
X49
)
)
)
↔
(
¬
(
(
¬
X41
)
↔
(
(
¬
X39
)
↔
X50
)
)
)
)
→
(
X50
→
X41
→
(
¬
X40
)
→
X49
)
→
(
X49
↔
(
X40
↔
(
X6
↔
X50
)
)
)
)
↔
(
¬
(
X49
→
X41
→
X50
)
)
)
→
(
¬
(
(
¬
(
X41
↔
(
X27
→
X50
)
)
)
→
(
¬
X49
)
→
X34
→
X50
)
)
)
↔
(
(
(
(
(
¬
(
(
¬
X42
)
→
(
¬
X30
)
→
(
¬
X17
)
→
X49
)
)
↔
(
X49
→
(
X42
↔
(
X41
↔
(
X30
→
False
)
)
)
)
)
↔
(
¬
(
(
¬
(
X42
↔
(
X37
↔
X18
)
)
)
→
(
¬
(
X49
→
(
X17
↔
X37
)
)
)
)
)
)
↔
(
(
¬
X49
)
→
(
¬
X42
)
→
(
¬
X41
)
→
X50
)
)
↔
(
¬
(
(
X49
→
(
(
¬
X42
)
↔
(
X30
↔
X52
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
(
¬
(
X52
→
(
¬
X8
)
→
(
X7
↔
X42
)
)
)
↔
(
¬
(
(
¬
X42
)
→
(
¬
X8
)
→
X5
→
X52
)
)
)
→
(
¬
(
X5
→
False
)
)
)
)
→
(
¬
(
(
¬
X8
)
→
(
¬
X7
)
→
X52
)
)
)
)
→
(
¬
X8
)
→
(
X7
↔
X42
)
)
)
)
)
)
)
)
)
→
(
(
(
(
(
(
X52
→
X47
→
(
X44
↔
False
)
)
↔
(
(
¬
X44
)
↔
(
(
(
¬
(
(
¬
X47
)
→
(
X44
↔
(
(
¬
X9
)
→
X52
)
)
)
)
↔
(
X53
↔
(
(
¬
X47
)
→
X52
)
)
)
↔
(
¬
(
X53
→
(
(
¬
X47
)
↔
X52
)
)
)
)
)
)
↔
(
(
¬
(
X4
→
False
)
)
↔
(
¬
(
(
X53
→
(
¬
X44
)
→
(
(
¬
X23
)
↔
X47
)
)
→
(
¬
(
X47
↔
(
(
¬
X44
)
↔
(
(
¬
X10
)
↔
X53
)
)
)
)
)
)
)
)
↔
(
(
¬
X44
)
↔
(
(
¬
X10
)
↔
X52
)
)
)
↔
(
(
¬
(
(
¬
(
X47
↔
(
X38
↔
(
(
¬
X16
)
→
X52
)
)
)
)
→
(
¬
(
X44
↔
(
X31
↔
X16
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
X47
↔
(
(
¬
X0
)
→
X33
)
)
↔
(
¬
(
(
¬
X44
)
↔
(
X38
↔
(
X0
↔
X47
)
)
)
)
)
→
(
¬
(
X44
→
(
X38
↔
X47
)
)
)
)
)
→
(
(
(
¬
(
X38
↔
(
X21
→
X44
)
)
)
↔
(
¬
(
(
¬
(
X38
↔
(
X33
↔
(
X21
→
False
)
)
)
)
→
(
(
¬
(
(
¬
X38
)
↔
(
X33
→
X15
→
X44
)
)
)
↔
(
¬
(
X44
→
X33
→
(
(
¬
X21
)
↔
X38
)
)
)
)
)
)
)
↔
(
X38
↔
(
X33
↔
X21
)
)
)
)
→
(
(
¬
X21
)
↔
(
¬
(
(
X50
→
X15
→
X12
)
→
(
¬
(
X11
→
X21
)
)
)
)
)
→
(
¬
X21
)
→
(
X15
↔
(
X12
↔
X38
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
¬
X46
)
→
(
X13
↔
X47
)
)
)
→
(
¬
(
(
¬
X47
)
↔
(
(
¬
X1
)
→
X52
)
)
)
)
↔
(
(
(
X46
→
X43
→
X13
→
X47
)
↔
(
¬
(
X48
↔
(
X46
↔
(
X43
→
X13
→
False
)
)
)
)
)
↔
(
X47
↔
False
)
)
)
→
(
¬
(
(
¬
(
X53
↔
(
X36
→
(
X22
↔
X48
)
)
)
)
→
(
(
¬
(
(
¬
X52
)
↔
(
(
¬
X22
)
↔
(
X20
↔
X53
)
)
)
)
↔
(
(
¬
(
X36
↔
(
X20
→
False
)
)
)
→
(
¬
(
(
¬
X52
)
↔
(
(
¬
X46
)
↔
X53
)
)
)
)
)
→
(
¬
(
(
¬
X22
)
→
X52
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X53
→
(
X51
↔
(
X45
→
X52
)
)
)
)
→
(
¬
(
X53
↔
(
X52
→
(
(
¬
X32
)
↔
X44
)
)
)
)
→
(
X53
↔
(
X45
→
(
X44
↔
X52
)
)
)
)
↔
(
(
(
(
¬
(
(
(
¬
X52
)
↔
(
X40
→
X25
→
X53
)
)
→
(
X53
→
X40
→
(
(
¬
X23
)
↔
X52
)
)
→
(
¬
(
(
¬
(
(
¬
X36
)
↔
(
X23
→
X17
→
X40
)
)
)
→
(
¬
(
(
¬
X36
)
↔
(
X23
→
X52
)
)
)
)
)
)
)
↔
(
¬
(
X53
→
(
X43
↔
(
(
¬
X23
)
↔
X25
)
)
)
)
)
↔
(
X53
↔
(
X43
↔
X52
)
)
)
↔
(
(
(
(
X50
↔
(
(
¬
X40
)
→
X26
→
X53
)
)
→
(
X53
↔
(
X50
→
X40
→
X19
)
)
)
↔
(
(
X53
↔
(
X40
→
(
(
¬
X16
)
↔
X36
)
)
)
↔
(
(
¬
(
X53
↔
(
(
¬
X36
)
↔
X51
)
)
)
→
(
¬
(
X53
↔
(
X40
↔
(
X36
↔
(
X19
→
False
)
)
)
)
)
)
)
)
→
(
¬
(
X50
→
(
X40
↔
(
X36
↔
False
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
X53
→
X38
→
(
(
¬
X21
)
↔
X46
)
)
→
(
(
¬
(
(
X53
→
(
(
¬
X38
)
↔
(
X11
→
X47
)
)
)
→
(
¬
(
X52
→
(
¬
(
X52
↔
(
(
¬
X38
)
↔
(
(
¬
X5
)
↔
X49
)
)
)
)
→
X52
→
(
¬
X38
)
→
X11
→
X49
)
)
)
)
↔
(
¬
(
(
(
¬
X47
)
→
(
X30
↔
(
X21
↔
X52
)
)
)
→
(
X53
↔
(
X52
→
(
X30
↔
X21
)
)
)
)
)
)
)
)
→
(
¬
(
(
X38
↔
(
X2
↔
False
)
)
→
(
X46
↔
(
(
¬
X30
)
→
(
¬
X1
)
→
X31
)
)
)
)
)
↔
(
(
(
(
¬
X52
)
→
(
(
¬
X50
)
↔
(
X47
→
X53
)
)
)
→
X47
→
(
(
¬
X33
)
↔
X53
)
)
↔
(
(
¬
(
(
X27
↔
(
X13
↔
X53
)
)
→
(
¬
(
X50
→
(
¬
X13
)
→
X15
)
)
)
)
→
(
¬
(
(
¬
X27
)
↔
(
X15
→
(
X13
↔
X52
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X14
)
↔
(
(
¬
X12
)
↔
(
(
¬
X7
)
↔
X29
)
)
)
)
↔
(
(
(
¬
X48
)
↔
(
(
¬
X42
)
↔
X52
)
)
↔
(
(
(
X48
↔
(
(
¬
X8
)
→
(
X7
↔
X22
)
)
)
→
(
(
X52
→
(
X42
↔
(
(
¬
X22
)
→
X48
)
)
)
↔
(
¬
(
(
¬
X8
)
→
X48
)
)
)
)
↔
(
(
¬
X29
)
↔
(
X22
↔
(
X7
↔
X42
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
X42
)
↔
(
X20
→
X52
)
)
→
(
(
¬
X34
)
↔
(
X10
→
X39
)
)
)
)
→
(
(
¬
(
X52
↔
(
X39
↔
(
X9
→
X42
)
)
)
)
↔
(
X52
↔
(
X42
→
X6
→
X9
)
)
)
→
(
(
¬
(
X52
↔
(
X44
→
X29
)
)
)
↔
(
(
(
¬
(
X52
→
X29
→
(
X9
↔
False
)
)
)
↔
(
(
¬
(
X52
↔
(
X42
→
(
X9
↔
X29
)
)
)
)
↔
(
¬
(
X52
→
X29
→
X4
→
X42
)
)
)
)
→
X52
→
(
X29
↔
(
X9
→
X44
)
)
)
)
)
)
↔
(
(
(
X42
→
(
X37
↔
X48
)
)
↔
(
(
¬
(
X43
↔
(
(
¬
X41
)
↔
X52
)
)
)
↔
(
X52
→
(
¬
X41
)
→
X24
→
X48
)
)
)
↔
(
¬
(
(
(
¬
(
X42
↔
(
X3
→
False
)
)
)
↔
(
¬
(
X41
→
(
(
¬
X28
)
↔
(
X3
↔
X35
)
)
)
)
)
→
(
¬
(
(
(
X42
↔
(
X0
↔
False
)
)
↔
(
(
(
¬
X35
)
↔
(
X0
→
X42
)
)
↔
(
¬
(
X42
↔
(
X41
↔
(
X18
↔
X0
)
)
)
)
)
)
→
(
X52
↔
X18
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
dcf9ee...
and proposition id is
9610c6...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMKWYffJZ4Vk5dD82EXCQDZYL7Q4XW5199S
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
¬
(
(
¬
(
(
(
(
(
¬
(
X28
↔
(
(
¬
X27
)
↔
(
X18
→
X38
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X47
↔
(
(
¬
X28
)
→
(
X18
↔
X44
)
)
)
)
→
(
¬
(
X44
→
X28
→
X18
→
X46
)
)
)
)
↔
(
X46
→
X44
→
X28
→
(
X26
↔
False
)
)
)
→
(
X47
↔
(
X44
↔
(
(
¬
X28
)
→
X38
)
)
)
)
)
↔
(
(
(
¬
(
X37
↔
(
X5
→
X46
)
)
)
↔
(
(
(
(
¬
X37
)
↔
(
(
¬
X5
)
↔
(
(
¬
X2
)
→
X46
)
)
)
↔
(
X44
→
X2
→
False
)
)
↔
(
(
¬
X37
)
↔
(
(
¬
X5
)
↔
X44
)
)
)
)
→
(
(
¬
(
X44
↔
(
(
¬
X9
)
↔
(
(
¬
X5
)
→
X43
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X44
)
→
(
¬
X5
)
→
X47
)
)
→
(
¬
(
X43
↔
(
X5
→
(
(
¬
X2
)
↔
X47
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
X20
→
X8
)
↔
(
¬
(
X38
→
(
¬
X20
)
→
(
X7
↔
X31
)
)
)
)
↔
(
¬
(
X7
↔
(
(
¬
X0
)
↔
X20
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X43
)
→
(
X7
↔
X44
)
)
)
→
(
¬
(
(
X40
↔
X7
)
→
(
¬
(
X44
→
(
¬
X43
)
→
X7
→
X45
)
)
)
)
)
)
↔
(
(
¬
(
X45
→
(
X31
↔
(
X0
→
X44
)
)
)
)
→
X45
→
(
X44
↔
(
X0
→
X43
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
X25
)
↔
(
(
¬
X22
)
↔
X45
)
)
)
↔
(
(
(
¬
X25
)
↔
(
(
¬
X22
)
→
(
(
¬
X19
)
↔
X45
)
)
)
↔
(
¬
(
X25
→
(
X23
↔
(
X22
↔
(
X4
→
False
)
)
)
)
)
)
)
↔
(
¬
(
(
X46
→
(
X23
↔
(
X19
→
False
)
)
)
→
(
X49
↔
(
X23
↔
(
(
¬
X19
)
↔
X25
)
)
)
)
)
)
↔
(
(
(
(
(
¬
X46
)
↔
(
X43
→
(
(
¬
X36
)
↔
X49
)
)
)
→
(
X46
↔
(
X43
→
(
X34
↔
X49
)
)
)
)
→
(
¬
(
X36
↔
(
X34
↔
X43
)
)
)
)
↔
(
(
X46
→
X33
→
X10
)
↔
(
(
(
(
¬
(
X45
→
(
X33
↔
X49
)
)
)
↔
(
(
¬
(
(
¬
X45
)
→
(
X33
↔
X46
)
)
)
↔
(
(
X46
→
(
(
¬
X43
)
↔
X49
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X49
↔
(
X43
↔
(
X33
↔
False
)
)
)
)
→
(
X46
↔
(
X45
→
False
)
)
)
)
→
X46
→
(
(
¬
X45
)
↔
X49
)
)
)
)
)
)
↔
(
¬
(
X49
↔
(
X45
↔
(
X33
↔
False
)
)
)
)
)
↔
(
X46
↔
(
X45
→
(
X13
↔
X49
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X47
↔
(
X38
↔
(
X35
→
X24
)
)
)
)
→
(
(
(
X41
→
X38
→
X21
→
(
X14
↔
False
)
)
↔
(
(
X21
↔
(
X15
↔
(
X1
↔
X41
)
)
)
↔
(
X38
↔
(
X15
↔
X1
)
)
)
)
↔
(
(
(
X47
↔
(
X41
→
X35
→
False
)
)
↔
(
¬
(
X35
→
(
X15
↔
(
X11
↔
X41
)
)
)
)
)
↔
(
¬
(
X41
→
(
¬
X35
)
→
(
(
¬
X21
)
↔
X39
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
X48
↔
(
(
¬
X3
)
→
X29
)
)
→
(
(
(
¬
X41
)
↔
(
X30
↔
(
(
¬
X29
)
→
X48
)
)
)
↔
(
¬
(
(
¬
X3
)
↔
X29
)
)
)
)
)
→
(
X30
↔
(
X12
↔
(
X3
↔
X48
)
)
)
)
→
(
(
(
X39
↔
(
X30
→
X42
)
)
↔
(
¬
(
(
X42
→
(
X38
↔
X46
)
)
→
(
¬
(
(
¬
(
X46
→
(
X39
↔
(
(
¬
X38
)
↔
X42
)
)
)
)
→
X30
)
)
)
)
)
↔
(
X46
→
X42
→
(
X38
↔
X6
)
)
)
→
(
¬
(
(
¬
X41
)
→
(
(
¬
X38
)
↔
(
(
¬
X17
)
→
X42
)
)
)
)
→
(
(
(
(
¬
(
(
¬
(
(
¬
X39
)
→
X38
→
(
(
¬
X29
)
↔
X41
)
)
)
→
(
¬
(
X41
↔
(
X39
↔
(
X38
↔
(
X32
→
False
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
X39
)
→
(
¬
X38
)
→
X41
)
)
)
↔
(
¬
(
X41
→
(
X39
↔
(
X32
↔
(
X16
↔
False
)
)
)
)
)
)
↔
(
X41
↔
(
X39
↔
(
X16
→
X42
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
X48
→
(
¬
X46
)
→
X30
→
X47
)
)
↔
(
¬
(
X47
→
(
(
¬
X36
)
↔
X48
)
)
)
)
→
(
(
(
X46
→
(
X5
↔
X30
)
)
↔
(
X41
↔
(
X36
↔
X5
)
)
)
↔
(
X30
↔
(
X14
→
(
¬
X5
)
→
X45
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X45
)
↔
(
(
¬
X18
)
↔
X47
)
)
)
→
(
¬
(
X45
↔
(
X44
↔
(
X43
→
X18
)
)
)
)
)
→
(
(
X43
↔
(
X23
→
X13
→
X47
)
)
↔
(
(
(
X45
↔
(
X13
↔
(
X6
↔
X47
)
)
)
→
(
¬
X13
)
→
X6
→
X23
)
↔
(
¬
(
(
¬
X45
)
↔
(
(
¬
X23
)
↔
(
(
¬
X13
)
→
X46
)
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
(
X35
↔
(
X28
↔
(
X20
↔
X47
)
)
)
)
→
(
¬
(
X19
→
X47
)
)
)
)
↔
(
(
¬
(
X28
→
(
¬
X20
)
→
X35
)
)
→
(
¬
(
X46
↔
(
(
¬
X27
)
→
(
X20
↔
X28
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
X47
↔
(
X46
→
X19
→
X39
)
)
→
(
¬
(
(
¬
X39
)
↔
(
(
¬
X10
)
→
(
X3
↔
X47
)
)
)
)
)
)
→
(
¬
(
X48
→
X34
→
(
X10
↔
X46
)
)
)
)
)
)
↔
(
(
X47
→
X39
→
(
¬
X16
)
→
X46
)
→
(
(
¬
(
(
(
¬
(
X47
→
(
(
¬
X42
)
↔
X49
)
)
)
→
(
¬
(
(
(
¬
X47
)
↔
(
X42
→
(
¬
X39
)
→
X49
)
)
→
(
¬
(
X47
↔
(
X39
→
(
(
¬
X16
)
↔
X42
)
)
)
)
)
)
)
→
X49
→
X39
→
X15
→
X47
)
)
↔
(
¬
(
X49
↔
(
(
¬
X16
)
↔
(
(
¬
X0
)
→
X47
)
)
)
)
)
)
)
)
↔
(
(
(
(
X49
→
(
X47
↔
(
X37
↔
X17
)
)
)
→
(
(
¬
(
(
¬
X40
)
↔
(
X2
↔
X49
)
)
)
↔
(
X47
↔
(
X37
↔
(
(
¬
X2
)
↔
X40
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
X40
↔
(
X12
↔
(
X4
↔
X47
)
)
)
→
(
(
¬
(
X47
↔
(
X40
↔
(
X38
↔
(
X8
↔
False
)
)
)
)
)
↔
(
(
(
¬
(
X47
→
(
X40
↔
(
(
¬
X12
)
→
X44
)
)
)
)
↔
(
¬
(
X47
→
(
(
¬
X40
)
↔
(
(
¬
X38
)
→
X44
)
)
)
)
)
→
(
¬
X44
)
→
(
¬
X40
)
→
X47
)
)
)
)
↔
(
¬
(
(
¬
X44
)
↔
(
(
¬
X40
)
→
X47
)
)
)
)
→
(
(
¬
(
X40
→
X38
→
(
X32
↔
X9
)
)
)
↔
(
X40
→
X38
→
(
(
¬
X32
)
↔
X44
)
)
)
)
)
→
(
(
¬
(
(
¬
X47
)
→
(
(
¬
X38
)
↔
(
X32
↔
X49
)
)
)
)
↔
(
(
(
¬
X38
)
→
(
(
¬
X32
)
↔
(
(
¬
X24
)
→
X49
)
)
)
↔
(
¬
(
X32
→
(
X24
↔
(
X21
↔
X49
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
X33
→
(
X22
↔
X25
)
)
)
↔
(
¬
(
(
¬
X25
)
→
(
¬
X11
)
→
X47
)
)
)
→
(
¬
(
X26
↔
(
(
¬
X22
)
↔
(
(
¬
X11
)
↔
X25
)
)
)
)
)
)
→
(
¬
(
X47
→
(
X33
↔
(
(
¬
X7
)
→
X25
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
X29
)
↔
(
X26
→
(
(
¬
X1
)
↔
X37
)
)
)
→
(
¬
X37
)
→
X29
→
(
¬
X1
)
→
X47
)
↔
(
¬
(
(
¬
X37
)
→
(
X33
↔
(
X26
→
X47
)
)
)
)
)
→
(
(
¬
X33
)
→
X32
→
(
(
¬
X31
)
↔
X37
)
)
→
(
¬
(
(
¬
X32
)
→
X31
→
(
(
¬
X29
)
↔
X37
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
fb32ce...
and proposition id is
129978...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMd47hEbJLZUum1fX4Ss6csVYKZ7KNzn1Mp
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
¬
(
(
¬
(
(
(
(
(
¬
(
X49
↔
(
X27
→
X9
→
X41
)
)
)
↔
(
(
(
¬
(
X15
→
False
)
)
↔
(
(
¬
(
(
X29
↔
X49
)
→
(
X41
↔
(
(
¬
X10
)
↔
X15
)
)
)
)
↔
(
¬
(
X41
↔
(
X29
→
X49
)
)
)
)
)
→
(
¬
(
X41
→
X29
→
(
X27
↔
X49
)
)
)
)
)
↔
(
(
(
(
¬
(
X49
↔
(
(
¬
X34
)
→
(
X18
↔
X41
)
)
)
)
↔
(
¬
(
(
¬
X34
)
→
(
¬
X29
)
→
(
(
¬
X24
)
↔
X49
)
)
)
)
→
X43
→
X41
→
X34
→
X49
)
↔
(
X49
→
X45
→
X43
→
(
X34
↔
False
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
¬
X17
)
→
(
¬
X12
)
→
X46
)
)
→
(
¬
(
X46
↔
(
X39
→
(
¬
X17
)
→
X45
)
)
)
)
)
→
(
¬
(
(
¬
X12
)
↔
X45
)
)
)
↔
(
(
(
¬
X46
)
→
(
(
¬
X43
)
↔
X49
)
)
→
(
¬
(
(
¬
(
X49
↔
(
X46
↔
(
(
¬
X17
)
→
X43
)
)
)
)
→
(
¬
(
X45
↔
(
X17
→
X2
)
)
)
)
)
)
)
)
→
(
¬
(
X45
→
X34
→
X4
→
X26
)
)
→
(
¬
(
(
(
X43
↔
(
(
¬
X38
)
↔
(
X20
↔
X45
)
)
)
↔
(
¬
(
X38
→
(
(
¬
X20
)
↔
X48
)
)
)
)
→
X43
→
(
X38
↔
(
X4
↔
X45
)
)
)
)
)
)
→
(
X43
→
X39
→
X8
→
X16
)
→
(
(
¬
(
(
¬
(
(
¬
X41
)
↔
(
(
¬
X30
)
→
X13
→
X43
)
)
)
→
(
¬
(
(
(
¬
X39
)
→
(
X13
↔
X43
)
)
→
(
(
¬
X30
)
↔
(
(
¬
X13
)
→
(
X8
↔
X43
)
)
)
)
)
)
)
↔
(
(
¬
(
X41
→
(
X19
↔
(
(
¬
X7
)
↔
X30
)
)
)
)
↔
(
¬
(
X30
↔
(
X19
→
(
X7
↔
X43
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X48
↔
(
X11
→
(
X1
↔
X49
)
)
)
)
↔
(
(
¬
(
(
X49
→
X48
→
(
X45
↔
X14
)
)
→
(
¬
(
(
¬
X48
)
→
(
¬
X45
)
→
(
¬
X28
)
→
X49
)
)
→
(
X48
↔
(
(
¬
X11
)
→
X45
)
)
)
)
↔
(
(
¬
(
(
¬
X40
)
↔
(
X35
↔
X49
)
)
)
↔
(
X48
↔
(
X45
→
X35
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X48
↔
(
(
¬
X45
)
↔
(
X42
↔
X49
)
)
)
)
→
(
X49
↔
(
(
¬
X42
)
↔
(
(
¬
X23
)
→
X48
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
(
(
¬
(
X32
→
(
(
¬
X21
)
↔
X42
)
)
)
↔
(
(
(
¬
(
X3
↔
X42
)
)
↔
(
X48
↔
(
X42
↔
(
(
¬
X21
)
→
X32
)
)
)
)
↔
(
¬
(
X21
↔
X48
)
)
)
)
→
(
¬
X21
)
→
(
¬
X3
)
→
X42
)
)
→
X48
→
(
X37
↔
(
X3
→
X21
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X32
)
↔
X37
)
)
)
)
)
↔
(
(
¬
(
(
¬
X45
)
→
(
¬
X42
)
→
X47
)
)
→
(
(
X47
↔
(
(
¬
X32
)
→
X22
→
X45
)
)
↔
(
X42
→
(
(
¬
X32
)
↔
X45
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
X37
→
X36
→
(
(
¬
X6
)
↔
X44
)
)
→
(
X36
↔
(
X33
↔
(
X5
→
X44
)
)
)
)
)
↔
(
¬
(
X44
→
(
X37
↔
(
(
¬
X36
)
↔
X45
)
)
)
)
)
→
(
¬
(
(
(
X25
↔
False
)
→
(
X37
↔
(
X31
→
(
X25
↔
X45
)
)
)
→
(
X37
↔
(
X31
→
X0
→
X45
)
)
)
→
X45
→
X25
→
X31
)
)
→
(
(
X37
↔
X45
)
↔
(
(
X45
→
(
(
¬
X44
)
↔
X48
)
)
↔
(
(
(
¬
X37
)
→
X48
)
→
(
¬
(
X48
↔
(
X45
↔
(
X44
↔
X31
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
X44
↔
(
X39
↔
(
(
¬
X7
)
→
X23
)
)
)
)
↔
(
(
¬
(
(
¬
X47
)
↔
(
X24
↔
(
X23
↔
X48
)
)
)
)
↔
(
X23
↔
(
X14
↔
X48
)
)
)
)
↔
(
(
(
X47
↔
(
X46
↔
(
(
¬
X21
)
→
X48
)
)
)
↔
(
(
¬
(
X48
↔
(
X12
↔
(
(
¬
X8
)
↔
X46
)
)
)
)
↔
(
(
¬
X47
)
↔
(
X12
→
(
X8
↔
X48
)
)
)
)
)
→
(
¬
(
(
¬
X21
)
↔
X39
)
)
)
)
→
(
(
¬
(
X47
↔
(
(
¬
X46
)
→
X48
)
)
)
↔
(
(
X47
↔
(
X46
→
(
X38
↔
(
X19
→
False
)
)
)
)
→
(
(
¬
(
(
¬
(
X38
↔
(
X34
↔
(
X11
↔
False
)
)
)
)
→
(
(
X34
→
X2
)
↔
(
(
X38
↔
(
X34
→
X11
→
X17
)
)
→
X48
→
X34
)
)
)
)
↔
(
¬
(
X34
→
(
X17
↔
(
X11
↔
False
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
X47
)
↔
(
(
¬
X38
)
→
X33
→
X48
)
)
)
→
(
¬
(
(
X48
→
(
X20
↔
(
X16
↔
False
)
)
)
→
(
¬
(
(
¬
X47
)
→
(
¬
X33
)
→
(
X16
↔
X48
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X36
↔
(
(
¬
X5
)
→
X16
)
)
)
→
X40
→
(
(
¬
X33
)
↔
(
X5
↔
X36
)
)
)
→
(
(
X49
→
X40
→
X20
→
X3
)
↔
(
(
X49
→
X33
)
→
(
¬
(
(
¬
X36
)
→
X33
→
(
(
¬
X20
)
↔
X40
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
(
¬
(
(
X49
→
(
X32
↔
(
X6
→
X28
)
)
)
→
(
¬
(
(
¬
X32
)
→
(
(
¬
X28
)
↔
X43
)
)
)
)
)
↔
(
X49
→
X28
→
X6
)
)
→
(
(
¬
(
X49
→
(
X46
↔
False
)
)
)
↔
(
X29
→
X6
→
X32
)
)
)
)
↔
(
¬
(
X49
↔
(
(
¬
X32
)
→
X43
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
X30
↔
(
X10
→
(
X1
↔
X32
)
)
)
)
↔
(
X32
→
(
(
¬
X30
)
↔
(
(
¬
X1
)
↔
X37
)
)
)
)
↔
(
X37
↔
False
)
)
→
(
¬
(
X37
→
(
X25
↔
(
X10
→
X30
)
)
)
)
→
(
¬
(
X43
→
X37
→
X30
→
X25
→
False
)
)
)
→
(
(
(
X49
→
X41
→
X32
→
False
)
→
(
¬
(
(
¬
(
X49
↔
(
(
¬
X41
)
→
(
X32
↔
X47
)
)
)
)
→
X47
→
X41
→
X15
→
X49
)
)
)
↔
(
(
(
(
¬
X47
)
→
(
(
¬
X41
)
↔
(
X25
→
X49
)
)
)
↔
(
¬
(
(
¬
X25
)
↔
(
X10
→
X47
)
)
)
)
↔
(
X49
→
(
X25
↔
X47
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
(
¬
X45
)
→
X47
)
)
→
(
¬
(
(
X47
↔
False
)
→
(
(
¬
X42
)
↔
(
X37
↔
(
(
¬
X4
)
↔
X45
)
)
)
)
)
)
↔
(
X45
↔
(
(
¬
X42
)
↔
(
X13
↔
X47
)
)
)
)
→
(
(
(
(
¬
X27
)
↔
(
X13
↔
(
(
¬
X4
)
→
X31
)
)
)
↔
(
(
X41
→
(
X4
↔
X37
)
)
↔
(
(
¬
X31
)
↔
(
X13
↔
(
(
¬
X4
)
↔
X37
)
)
)
)
)
↔
(
(
(
¬
X37
)
→
(
(
¬
X0
)
↔
X41
)
)
↔
(
(
¬
X31
)
↔
(
X0
↔
X35
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
¬
X26
)
→
(
X18
↔
(
X9
↔
X45
)
)
)
→
X22
→
(
(
¬
X18
)
↔
(
(
¬
X9
)
→
X26
)
)
)
)
→
X45
→
(
X22
↔
X26
)
)
→
(
(
X47
→
X41
→
X9
→
X26
)
↔
(
(
¬
X45
)
→
X41
→
(
(
¬
X9
)
↔
X47
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
f282ef...
and proposition id is
d0042d...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMJGyWQPQncWsjNK7XYnmv8xW64JBNDZDca
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
∀X50 :
prop
,
(
(
(
(
(
¬
(
(
¬
(
(
¬
X40
)
↔
(
(
¬
X15
)
↔
(
X9
→
X50
)
)
)
)
→
(
(
X50
→
X40
→
(
¬
X3
)
→
X14
)
→
X50
→
(
X40
↔
(
X15
↔
X14
)
)
)
→
(
¬
X37
)
→
(
¬
X15
)
→
X50
)
)
→
(
(
(
(
¬
X41
)
↔
(
X40
→
(
X34
↔
X50
)
)
)
↔
(
(
X40
→
(
X30
↔
(
(
¬
X5
)
→
X41
)
)
)
↔
(
¬
(
X41
→
X40
→
X34
→
(
X5
↔
False
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
X46
↔
(
(
¬
X14
)
→
X5
→
X24
)
)
)
→
(
X46
↔
(
X40
→
(
¬
X5
)
→
X10
)
)
)
)
→
(
(
¬
X24
)
↔
(
(
¬
X10
)
→
(
¬
X5
)
→
X50
)
)
)
)
↔
(
X46
↔
(
X10
↔
X14
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
(
(
¬
(
(
¬
X33
)
↔
(
(
¬
X25
)
↔
X48
)
)
)
↔
(
X33
→
(
X31
↔
(
X25
↔
X2
)
)
)
)
↔
(
¬
(
X31
→
(
(
¬
X25
)
↔
(
(
¬
X22
)
→
X33
)
)
)
)
)
→
(
¬
(
X48
↔
(
X25
↔
X33
)
)
)
)
)
→
(
¬
(
X48
→
(
(
¬
X31
)
↔
(
X22
↔
X46
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X36
→
X41
)
)
↔
(
¬
(
(
¬
(
X39
→
(
¬
X36
)
→
(
¬
X22
)
→
X41
)
)
→
(
¬
(
X41
→
(
X39
↔
X25
)
)
)
)
)
)
→
(
(
(
¬
(
X40
↔
(
X22
↔
(
(
¬
X11
)
→
X48
)
)
)
)
→
(
(
¬
(
X40
→
(
X39
↔
(
X20
↔
X11
)
)
)
)
↔
(
¬
(
(
¬
X39
)
↔
(
X22
→
(
(
¬
X11
)
↔
X40
)
)
)
)
)
)
→
(
¬
(
X20
↔
(
X13
↔
(
X6
→
X40
)
)
)
)
)
→
(
(
¬
(
X38
↔
X36
)
)
↔
(
X48
→
X40
→
(
X36
↔
X39
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
X44
→
X47
)
↔
(
¬
(
X47
→
X45
→
(
¬
X42
)
→
X49
)
)
)
↔
(
(
(
X48
→
(
X45
↔
X47
)
)
↔
(
(
¬
(
X47
→
X45
→
X12
→
X48
)
)
→
(
X48
↔
(
(
¬
X45
)
↔
(
(
¬
X23
)
→
X47
)
)
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X42
)
↔
(
X23
→
X48
)
)
)
)
)
)
→
(
¬
(
(
X48
↔
(
X45
→
(
(
¬
X26
)
↔
X35
)
)
)
→
(
¬
(
(
¬
X45
)
→
X32
→
(
X27
↔
X48
)
)
)
→
(
¬
(
X48
↔
(
X35
→
(
¬
X32
)
→
X45
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X27
)
↔
(
(
¬
X26
)
↔
(
X18
↔
X47
)
)
)
)
→
(
X35
↔
(
(
¬
X26
)
↔
(
X18
→
X47
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
¬
X16
)
→
X49
)
)
→
(
(
¬
(
(
¬
(
X16
→
X8
→
X19
)
)
→
(
¬
(
(
X49
→
(
(
¬
X19
)
↔
(
(
¬
X16
)
→
X29
)
)
)
→
(
¬
(
X19
→
(
X8
↔
X29
)
)
)
)
)
)
)
↔
(
X49
↔
(
X29
→
(
X19
↔
X47
)
)
)
)
)
→
(
¬
(
X43
→
(
X8
↔
(
X7
↔
False
)
)
)
)
→
(
(
¬
(
X47
↔
(
(
¬
X16
)
↔
X43
)
)
)
→
(
X47
↔
(
X7
↔
(
(
¬
X0
)
→
X43
)
)
)
)
→
(
¬
X43
)
→
X8
→
X7
→
X47
)
→
(
(
(
¬
(
(
¬
X43
)
↔
(
(
¬
X28
)
→
(
(
¬
X21
)
↔
X49
)
)
)
)
→
(
¬
(
X29
↔
(
X21
↔
X49
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
X17
→
X46
)
)
→
X43
→
X28
→
X1
)
→
(
¬
(
X46
↔
(
X28
→
X1
→
False
)
)
)
)
)
→
X49
→
(
(
¬
X28
)
↔
(
X17
↔
X43
)
)
)
)
)
→
(
(
¬
(
X43
→
(
X4
↔
X50
)
)
)
↔
(
(
(
¬
X43
)
→
(
¬
X4
)
→
X46
)
→
(
X50
↔
(
X43
↔
(
X21
↔
X46
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
(
(
¬
(
X31
↔
(
X9
↔
X3
)
)
)
↔
(
¬
(
X42
↔
(
X34
→
(
(
¬
X9
)
↔
X28
)
)
)
)
)
→
(
(
¬
(
X31
→
(
¬
X9
)
→
(
(
¬
X3
)
↔
X21
)
)
)
↔
(
X37
→
X9
→
X31
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
(
(
¬
X23
)
↔
(
(
¬
X11
)
↔
X37
)
)
)
→
(
X28
↔
(
X11
→
(
(
¬
X4
)
↔
X42
)
)
)
)
)
↔
(
¬
(
X37
↔
(
X28
→
(
X23
↔
False
)
)
)
)
)
→
(
X37
↔
(
(
¬
X28
)
↔
X42
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
X5
)
→
X31
)
↔
(
¬
(
X31
→
(
(
¬
X22
)
↔
(
X21
↔
X45
)
)
)
)
)
→
(
(
X45
↔
(
X31
↔
(
X11
↔
X49
)
)
)
↔
(
X49
→
X45
→
X31
→
(
X21
↔
False
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X49
↔
(
X45
→
(
X42
↔
X38
)
)
)
)
↔
(
(
(
¬
X38
)
↔
(
(
¬
X30
)
→
(
(
¬
X13
)
↔
X42
)
)
)
→
(
¬
(
X42
↔
(
X38
→
(
X32
↔
X49
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
X42
→
(
¬
X26
)
→
X17
→
X46
)
→
(
X32
↔
(
(
¬
X26
)
→
(
¬
X17
)
→
X42
)
)
)
)
↔
(
¬
(
X46
→
X42
→
X17
→
False
)
)
)
→
(
¬
(
(
X32
↔
(
(
¬
X17
)
→
X26
)
)
→
(
(
X26
↔
(
(
¬
X17
)
→
(
(
¬
X0
)
↔
X49
)
)
)
↔
(
X32
→
X26
→
X17
→
X49
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
(
X39
↔
X12
)
↔
(
(
X48
→
X12
→
X39
)
↔
(
X49
→
(
¬
X39
)
→
(
(
¬
X12
)
↔
X47
)
)
)
)
→
(
(
(
(
¬
X39
)
→
X33
→
(
X12
↔
X47
)
)
↔
(
¬
(
X39
↔
(
(
¬
X33
)
→
(
X19
↔
X47
)
)
)
)
)
↔
(
X47
↔
(
(
¬
X33
)
→
X39
)
)
)
)
)
↔
(
(
(
¬
X48
)
→
X46
→
X49
)
↔
(
X50
→
(
X48
↔
(
X47
↔
(
X46
→
False
)
)
)
)
)
)
↔
(
(
X48
→
(
X47
↔
X50
)
)
↔
(
(
(
(
¬
(
X47
→
(
X8
↔
X25
)
)
)
→
(
¬
(
(
¬
X48
)
↔
(
X47
→
(
(
¬
X8
)
↔
X50
)
)
)
)
)
↔
(
¬
(
(
¬
X47
)
→
X48
)
)
)
→
X50
→
(
X25
↔
(
X10
→
False
)
)
)
)
)
↔
(
¬
(
(
(
¬
X33
)
→
(
¬
X16
)
→
(
¬
X6
)
→
X48
)
→
(
¬
(
(
X33
→
(
X24
↔
(
X1
↔
False
)
)
)
→
(
¬
X18
)
→
(
(
¬
X16
)
↔
(
(
¬
X1
)
↔
X24
)
)
)
)
→
(
(
¬
(
(
¬
X33
)
↔
(
X20
↔
(
(
¬
X16
)
↔
X46
)
)
)
)
↔
(
X46
→
X33
→
(
¬
X16
)
→
X48
)
)
)
)
)
↔
(
(
(
¬
(
(
X49
↔
(
X48
→
(
(
¬
X7
)
↔
X43
)
)
)
→
(
X48
→
X43
→
X14
→
X7
)
→
(
¬
(
(
¬
X35
)
↔
X43
)
)
)
)
→
(
(
(
¬
X47
)
→
X48
)
↔
(
(
¬
(
X49
→
(
¬
X47
)
→
(
¬
X36
)
→
X48
)
)
→
(
¬
X47
)
→
(
(
¬
X36
)
↔
(
X14
↔
X48
)
)
)
)
)
→
(
(
(
(
¬
X45
)
↔
(
(
¬
X44
)
↔
(
X43
↔
X49
)
)
)
↔
(
(
X49
→
X41
→
(
¬
X36
)
→
X44
)
↔
(
X44
→
X43
→
(
(
¬
X36
)
↔
X41
)
)
)
)
→
(
¬
X41
)
→
(
(
¬
X36
)
↔
(
X2
→
X43
)
)
)
→
(
(
¬
(
(
¬
X40
)
→
X29
→
X43
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X29
)
→
(
(
¬
X27
)
↔
X43
)
)
)
→
(
(
¬
X15
)
↔
(
¬
(
(
(
(
¬
X43
)
↔
(
X29
→
(
X15
↔
X44
)
)
)
→
(
¬
(
X44
→
(
X43
↔
(
X15
↔
X27
)
)
)
)
)
→
(
¬
(
X15
↔
X43
)
)
)
)
)
)
→
(
(
¬
(
X45
↔
(
X29
→
(
(
¬
X27
)
↔
X44
)
)
)
)
↔
(
X45
↔
(
X27
→
X40
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
c14f69...
and proposition id is
2329fb...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMZgHC6MaiHpf26y7B74DYpqVXGyEXYGoyR
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
(
(
¬
(
(
¬
(
(
¬
X40
)
↔
(
(
¬
X22
)
↔
(
X19
↔
X42
)
)
)
)
→
(
¬
(
(
¬
X42
)
↔
(
(
¬
X22
)
→
(
¬
X19
)
→
X44
)
)
)
)
)
↔
(
¬
(
X40
→
X19
)
)
)
→
(
(
(
¬
(
X44
→
(
¬
X9
)
→
(
X7
↔
X22
)
)
)
↔
(
X40
↔
(
X9
↔
X7
)
)
)
↔
(
¬
(
X9
↔
(
X0
→
X7
)
)
)
)
)
→
(
(
(
X44
↔
(
X15
↔
(
X11
→
X46
)
)
)
→
(
X40
↔
False
)
)
↔
(
(
(
X44
→
(
X40
↔
(
(
¬
X27
)
↔
X34
)
)
)
↔
(
¬
(
X44
→
(
(
¬
X40
)
↔
(
(
¬
X34
)
↔
X42
)
)
)
)
)
→
(
(
¬
(
X46
↔
(
X42
↔
(
X40
↔
X44
)
)
)
)
↔
(
X46
↔
(
(
¬
X25
)
↔
X44
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X47
→
X43
→
X20
)
)
→
(
(
¬
(
(
¬
X42
)
→
(
¬
X28
)
→
(
(
¬
X20
)
↔
X47
)
)
)
↔
(
(
¬
X45
)
→
X28
→
X20
→
X47
)
)
)
→
(
¬
(
(
(
X47
→
(
(
¬
X39
)
↔
(
(
¬
X32
)
→
X43
)
)
)
→
(
(
X47
→
(
X39
↔
(
X6
→
X48
)
)
)
↔
(
(
(
X48
↔
(
X47
→
(
(
¬
X28
)
↔
X43
)
)
)
→
(
¬
(
X48
↔
(
(
¬
X39
)
↔
(
X28
↔
X43
)
)
)
)
)
→
(
(
¬
X47
)
↔
(
X28
↔
X48
)
)
)
)
)
→
(
¬
(
(
¬
X47
)
→
X43
→
X48
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
X45
↔
(
X40
↔
X46
)
)
↔
(
X46
→
(
X45
↔
(
(
¬
X38
)
→
X48
)
)
)
)
→
(
¬
(
(
¬
(
(
X46
↔
(
X24
↔
X38
)
)
→
X48
→
X21
→
X46
)
)
→
(
¬
X46
)
→
(
(
¬
X24
)
↔
(
X12
↔
X48
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X47
↔
(
(
¬
X8
)
→
X10
)
)
)
→
(
¬
(
(
(
¬
X38
)
↔
(
X10
→
X40
)
)
→
(
¬
(
X47
↔
(
X45
→
(
(
¬
X38
)
↔
X40
)
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
X49
→
False
)
)
→
(
X48
↔
(
X43
→
(
X18
↔
X29
)
)
)
)
→
(
(
¬
(
X49
→
(
(
¬
X13
)
↔
X44
)
)
)
↔
(
¬
(
(
¬
(
X43
↔
(
X29
→
False
)
)
)
→
(
X49
↔
(
X44
↔
(
X43
↔
X13
)
)
)
)
)
)
)
↔
(
(
(
(
(
X33
↔
(
(
¬
X16
)
→
X17
)
)
↔
(
¬
(
X18
→
X17
→
(
(
¬
X16
)
↔
X49
)
)
)
)
↔
(
X29
→
X17
→
X16
→
X18
)
)
→
(
¬
(
X33
→
(
X29
↔
(
X18
↔
X5
)
)
)
)
)
↔
(
¬
(
X18
↔
(
X17
↔
(
X16
↔
False
)
)
)
)
)
)
→
(
(
(
X44
↔
(
X1
↔
X43
)
)
↔
(
(
(
¬
X44
)
→
(
(
¬
X41
)
↔
X48
)
)
→
(
(
¬
X44
)
↔
(
X43
→
(
X1
↔
X48
)
)
)
)
)
↔
(
X48
↔
(
X41
→
(
X26
↔
False
)
)
)
)
)
↔
(
(
(
(
(
X44
→
X30
→
X43
)
→
(
¬
(
X35
↔
X37
)
)
)
↔
(
(
(
¬
(
X44
↔
(
X43
→
(
X37
↔
(
X23
↔
False
)
)
)
)
)
↔
(
(
X43
↔
(
X37
→
(
¬
X23
)
→
X36
)
)
→
(
¬
(
(
¬
X37
)
↔
(
X36
↔
(
(
¬
X23
)
→
X41
)
)
)
)
)
)
↔
(
¬
(
(
¬
X41
)
↔
(
(
¬
X37
)
↔
X44
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X41
)
→
(
X14
↔
X44
)
)
)
→
(
¬
(
(
¬
(
X41
→
(
X31
↔
(
(
¬
X14
)
→
X44
)
)
)
)
→
(
X37
↔
(
(
¬
X31
)
→
X2
→
X44
)
)
)
)
)
)
↔
(
(
¬
X41
)
↔
(
(
¬
X31
)
→
X44
)
)
)
)
↔
(
(
X43
→
(
(
¬
X37
)
↔
(
(
¬
X4
)
↔
X44
)
)
)
↔
(
X44
↔
(
X43
→
(
¬
X3
)
→
X4
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
(
X47
→
(
(
¬
X42
)
↔
(
X16
↔
X43
)
)
)
↔
(
(
¬
X43
)
→
X42
→
X41
→
X47
)
)
→
(
(
X43
→
(
X42
↔
(
X29
→
False
)
)
)
↔
(
¬
(
X43
↔
(
X41
↔
(
X6
↔
X29
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
¬
X43
)
→
(
¬
X31
)
→
(
X13
↔
X49
)
)
)
→
(
¬
(
X14
↔
X31
)
)
)
)
→
(
(
¬
(
(
¬
X38
)
↔
(
X31
↔
(
(
¬
X1
)
↔
X44
)
)
)
)
↔
(
X44
↔
(
X41
↔
(
X1
→
X43
)
)
)
)
→
(
(
(
¬
(
(
¬
(
(
X49
↔
(
X36
→
(
(
¬
X3
)
↔
X42
)
)
)
→
(
¬
(
X49
→
X42
→
(
X36
↔
(
X28
↔
False
)
)
)
)
)
)
→
(
¬
X28
)
→
X3
→
X42
)
)
↔
(
¬
(
X43
↔
(
X28
→
False
)
)
)
)
↔
(
¬
(
X49
↔
(
(
¬
X42
)
↔
(
(
¬
X38
)
→
X43
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
X30
↔
(
X19
→
X11
→
X46
)
)
→
X46
→
X44
→
X19
→
X11
)
)
↔
(
(
¬
(
X47
↔
(
(
¬
X44
)
→
(
¬
X12
)
→
X46
)
)
)
↔
(
¬
(
X47
→
X44
→
(
X30
↔
X46
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X46
→
(
X42
↔
(
(
¬
X25
)
→
X47
)
)
)
)
↔
(
(
¬
(
X42
↔
(
(
¬
X22
)
↔
X25
)
)
)
→
(
(
X42
→
X15
→
X22
)
↔
(
(
¬
(
X25
↔
(
X22
↔
X46
)
)
)
↔
(
(
¬
X42
)
→
(
X22
↔
(
X15
→
X45
)
)
)
)
)
)
)
→
(
(
X46
↔
(
X44
→
X15
→
X10
)
)
↔
(
(
¬
(
X25
→
(
(
¬
X15
)
↔
(
X9
↔
X46
)
)
)
)
↔
(
X15
→
(
(
¬
X10
)
↔
(
(
¬
X9
)
→
X46
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
(
(
(
X35
↔
False
)
→
(
(
¬
X7
)
↔
(
(
¬
X5
)
→
X35
)
)
→
(
¬
X35
)
→
X21
→
(
(
¬
X5
)
↔
X46
)
)
→
(
X46
↔
(
(
¬
X2
)
→
X21
)
)
)
)
↔
(
(
¬
(
X49
→
(
X21
↔
(
(
¬
X5
)
↔
X46
)
)
)
)
↔
(
¬
(
X49
→
X21
→
(
X5
↔
X34
)
)
)
)
)
↔
(
X49
↔
(
X40
↔
(
(
¬
X34
)
→
X35
)
)
)
)
→
(
(
¬
(
X46
→
(
¬
X33
)
→
X40
)
)
↔
(
(
X41
↔
(
(
¬
X40
)
↔
(
X33
↔
X49
)
)
)
→
X49
→
(
¬
X41
)
→
X46
)
)
)
→
(
(
(
(
¬
(
(
¬
(
X32
↔
(
(
¬
X18
)
↔
X39
)
)
)
→
(
(
X48
↔
(
X32
↔
(
X18
↔
X37
)
)
)
↔
(
¬
(
(
¬
X37
)
→
(
¬
X18
)
→
X39
)
)
)
)
)
→
(
(
X39
↔
(
X32
↔
X49
)
)
↔
(
¬
(
(
X39
→
(
(
¬
X32
)
↔
(
(
¬
X18
)
↔
X49
)
)
)
→
(
¬
(
X32
↔
(
X18
↔
(
X8
→
False
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
X23
)
↔
(
(
¬
X20
)
→
X32
)
)
→
(
(
¬
X39
)
→
(
¬
X32
)
→
(
(
¬
X23
)
↔
X48
)
)
→
(
¬
(
X39
↔
(
X23
↔
(
X20
→
X32
)
)
)
)
)
↔
(
¬
(
X32
→
(
(
¬
X23
)
↔
(
X17
→
X39
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
X41
)
→
(
(
¬
X37
)
↔
(
X27
→
X48
)
)
)
→
(
¬
(
(
(
¬
X37
)
↔
(
(
¬
X34
)
↔
X41
)
)
→
(
(
¬
(
X48
↔
(
X41
↔
(
X24
→
False
)
)
)
)
↔
(
¬
(
(
¬
(
X41
→
X24
→
X34
)
)
→
(
¬
X41
)
→
(
¬
X37
)
→
X34
→
X48
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X48
↔
(
X27
↔
(
X4
↔
(
X0
→
False
)
)
)
)
)
→
(
¬
(
X48
→
(
X27
↔
(
X26
→
X49
)
)
)
)
)
)
↔
(
X41
↔
(
X0
→
False
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
6fe6ec...
and proposition id is
512042...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMNY43Rcw1obwunCPKK3pCaacaDyCx7QiBA
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
∀X50 :
prop
,
(
¬
∀X51 :
prop
,
(
¬
∀X52 :
prop
,
(
¬
∀X53 :
prop
,
(
¬
∀X54 :
prop
,
(
(
(
(
(
¬
(
(
(
(
¬
(
X46
↔
(
X36
↔
(
(
¬
X30
)
↔
X50
)
)
)
)
↔
(
(
(
(
¬
X46
)
→
(
(
¬
X30
)
↔
(
X24
→
X50
)
)
)
↔
(
¬
(
X52
→
(
X46
↔
(
X24
→
X50
)
)
)
)
)
↔
(
¬
(
X52
↔
(
X50
→
X46
)
)
)
)
)
↔
(
X52
↔
(
X46
↔
(
X45
→
X36
)
)
)
)
→
(
(
X30
→
(
¬
X24
)
→
X42
)
↔
(
¬
(
(
X45
↔
(
X30
↔
X24
)
)
→
(
(
(
¬
X30
)
→
X24
→
(
(
¬
X2
)
↔
X42
)
)
↔
(
X42
→
(
(
¬
X30
)
↔
X45
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X54
→
(
X50
↔
(
X35
→
False
)
)
)
)
↔
(
¬
(
X53
↔
(
X50
→
(
X10
↔
X35
)
)
)
)
)
↔
(
X52
→
(
X50
↔
(
(
¬
X35
)
↔
X53
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X35
)
→
(
(
¬
X14
)
↔
X50
)
)
)
→
(
¬
(
X53
↔
(
X46
↔
(
X35
↔
X14
)
)
)
)
)
→
X46
→
(
¬
X35
)
→
X53
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X53
↔
(
(
¬
X26
)
↔
(
X16
→
X50
)
)
)
)
↔
(
(
¬
X39
)
↔
(
X26
↔
(
X16
→
X53
)
)
)
)
→
(
(
(
¬
(
(
¬
X53
)
→
(
¬
X48
)
→
(
¬
X39
)
→
X54
)
)
↔
(
X53
→
(
X39
↔
(
X16
→
False
)
)
)
)
↔
(
¬
(
X53
↔
(
(
¬
X7
)
↔
X54
)
)
)
)
)
)
↔
(
(
(
(
X34
↔
(
(
¬
X22
)
→
(
(
¬
X5
)
↔
X49
)
)
)
↔
(
(
(
(
(
¬
(
X50
↔
(
X49
→
X37
)
)
)
→
(
(
X50
↔
(
X49
→
X5
)
)
↔
(
(
¬
(
X37
↔
(
(
¬
X34
)
→
X49
)
)
)
↔
(
X50
↔
(
X37
↔
(
X5
↔
X34
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
X37
)
→
(
X27
↔
X49
)
)
→
(
(
X50
→
(
(
¬
X27
)
↔
(
X15
↔
X49
)
)
)
↔
(
¬
(
X50
↔
(
X49
↔
(
X27
↔
X37
)
)
)
)
)
)
)
↔
(
¬
(
X49
↔
(
X37
→
X50
)
)
)
)
)
↔
(
X49
→
(
X27
↔
X37
)
)
)
↔
(
¬
(
X49
↔
(
X37
→
X34
→
X11
)
)
)
)
)
→
(
¬
X50
)
→
X49
→
(
¬
X37
)
→
X53
)
↔
(
(
¬
(
X53
↔
(
(
¬
X50
)
→
X54
)
)
)
↔
(
¬
(
X49
→
(
X38
↔
False
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
(
¬
(
(
¬
(
X53
↔
(
X51
↔
(
X44
↔
X25
)
)
)
)
→
(
¬
(
X53
↔
(
X44
→
(
¬
X20
)
→
X51
)
)
)
)
)
→
(
¬
(
X52
↔
(
X44
→
(
X20
↔
X25
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X40
→
(
¬
X3
)
→
X44
)
)
↔
(
¬
(
X51
→
X44
→
X19
→
X3
)
)
)
→
(
¬
X44
)
→
(
X40
↔
X51
)
)
→
X52
→
X40
→
(
X19
↔
X3
)
)
)
→
(
¬
(
X52
→
(
X9
↔
(
(
¬
X0
)
↔
X19
)
)
)
)
)
)
→
(
¬
(
X40
↔
(
(
¬
X31
)
↔
(
(
¬
X25
)
→
X53
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X44
→
(
X23
↔
(
X1
↔
X40
)
)
)
)
→
(
¬
X23
)
→
(
(
¬
X6
)
↔
(
X1
→
X40
)
)
)
)
→
(
¬
(
X23
→
X17
→
(
¬
X6
)
→
X51
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
(
(
X53
→
(
¬
X18
)
→
X12
→
X52
)
→
(
X53
↔
(
(
¬
X41
)
↔
(
X18
→
X54
)
)
)
)
)
→
(
¬
(
X47
↔
(
X41
→
X12
→
X53
)
)
)
)
)
↔
(
(
X29
→
(
X28
↔
(
X4
→
X18
)
)
)
↔
(
(
¬
(
X44
↔
(
X18
→
X28
)
)
)
↔
(
(
X47
↔
(
X44
↔
(
X12
→
X29
)
)
)
↔
(
(
¬
(
X44
↔
(
X29
→
(
(
¬
X28
)
↔
X47
)
)
)
)
→
(
¬
(
X47
↔
(
X28
→
X18
→
X44
)
)
)
)
)
)
)
)
→
(
(
¬
(
X33
→
(
X32
↔
(
(
¬
X13
)
→
X43
)
)
)
)
↔
(
¬
(
X47
→
(
(
¬
X44
)
↔
(
X13
↔
X53
)
)
)
)
)
→
(
(
¬
(
(
¬
X43
)
↔
X47
)
)
↔
(
X47
→
(
¬
X21
)
→
(
X8
↔
X53
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
(
¬
(
X50
→
(
X43
↔
(
X13
↔
X40
)
)
)
)
↔
(
¬
(
X49
↔
(
(
¬
X43
)
↔
(
(
¬
X40
)
↔
X50
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X49
)
→
(
¬
X40
)
→
X35
→
X50
)
)
↔
(
¬
(
X51
→
X49
→
X40
→
X35
→
False
)
)
)
↔
(
¬
(
X51
↔
(
X40
↔
(
(
¬
X34
)
↔
X50
)
)
)
)
)
)
→
(
(
X50
↔
(
X31
↔
False
)
)
↔
(
¬
(
X35
→
(
(
¬
X26
)
↔
X49
)
)
)
)
)
→
(
(
(
(
(
(
X51
↔
(
X50
↔
(
X41
→
(
X39
↔
False
)
)
)
)
→
X50
→
(
X43
↔
(
(
¬
X41
)
→
X51
)
)
)
↔
(
X51
→
(
X50
↔
(
X43
→
False
)
)
)
)
↔
(
X51
→
(
¬
X41
)
→
X37
→
X50
)
)
→
(
(
X48
→
(
(
¬
X9
)
↔
X39
)
)
↔
(
¬
(
(
¬
(
X51
→
(
X48
↔
(
X39
→
X50
)
)
)
)
→
(
X50
↔
(
X39
→
X9
→
X51
)
)
)
)
)
)
↔
(
(
(
(
¬
X41
)
↔
(
(
¬
X11
)
↔
(
X7
↔
X50
)
)
)
↔
(
¬
(
(
¬
(
X24
→
X50
)
)
→
(
(
¬
X24
)
↔
(
X11
↔
(
(
¬
X7
)
↔
X48
)
)
)
)
)
)
↔
(
(
¬
X41
)
↔
(
(
¬
X24
)
↔
(
(
¬
X12
)
↔
X50
)
)
)
)
)
)
↔
(
(
(
(
X45
↔
X42
)
↔
(
(
(
¬
(
X49
→
(
X45
↔
(
X42
→
X51
)
)
)
)
→
(
¬
(
(
¬
X45
)
→
(
X42
↔
(
(
¬
X21
)
↔
X49
)
)
)
)
)
→
(
X51
↔
(
X50
↔
(
(
¬
X42
)
↔
X45
)
)
)
)
)
↔
(
(
¬
(
X52
→
X51
→
X47
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
¬
X50
)
↔
(
X47
→
(
(
¬
X33
)
↔
X52
)
)
)
)
→
(
X52
↔
(
X47
→
(
¬
X33
)
→
X51
)
)
)
)
→
(
¬
(
X51
↔
(
X49
↔
X47
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
(
¬
X51
)
↔
(
(
¬
X44
)
↔
X52
)
)
↔
(
(
¬
(
(
¬
X46
)
↔
(
X38
→
X51
)
)
)
→
(
X52
→
(
(
¬
X44
)
↔
(
(
¬
X38
)
↔
X51
)
)
)
→
(
¬
(
(
¬
X51
)
↔
(
(
¬
X46
)
→
(
X38
↔
X52
)
)
)
)
)
)
→
(
X51
↔
(
(
¬
X44
)
↔
X52
)
)
)
)
→
(
X46
→
X14
→
(
(
¬
X0
)
↔
X51
)
)
→
(
¬
(
(
¬
X51
)
↔
(
(
¬
X46
)
↔
(
X14
↔
X52
)
)
)
)
)
→
(
¬
(
(
(
¬
X38
)
↔
(
(
¬
X16
)
↔
X49
)
)
→
(
¬
(
X14
↔
(
X6
→
X38
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
(
¬
X51
)
→
(
¬
X23
)
→
(
(
¬
X3
)
↔
X53
)
)
↔
(
(
X53
→
(
X49
↔
X51
)
)
↔
(
¬
(
(
¬
(
X51
→
(
X49
↔
False
)
)
)
→
(
X53
↔
(
X51
→
False
)
)
→
(
¬
X49
)
→
(
(
¬
X25
)
↔
(
(
¬
X3
)
→
X53
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X53
↔
(
(
¬
X2
)
→
X27
)
)
)
→
(
¬
(
(
(
False
→
(
¬
(
(
¬
(
X49
↔
(
X27
→
(
X1
↔
False
)
)
)
)
→
(
¬
(
X49
→
(
¬
X27
)
→
X2
→
X53
)
)
)
)
)
→
X53
→
(
X49
↔
(
X27
→
False
)
)
)
→
(
(
¬
X2
)
↔
(
X1
→
X53
)
)
)
)
)
)
↔
(
X49
→
X27
→
(
¬
X1
)
→
X30
)
)
→
(
¬
(
(
(
¬
X51
)
↔
(
X32
↔
(
X1
→
X53
)
)
)
→
X51
→
(
(
¬
X2
)
↔
(
X1
↔
X53
)
)
)
)
)
)
→
(
(
¬
(
X19
↔
(
X15
→
(
(
¬
X10
)
↔
X17
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X19
↔
(
X18
↔
(
(
¬
X4
)
↔
X53
)
)
)
)
→
(
(
¬
(
X27
→
(
X18
↔
(
X10
→
X4
→
False
)
)
)
)
↔
(
¬
(
X27
↔
(
X18
→
(
X4
↔
X53
)
)
)
)
)
)
→
(
¬
(
X27
↔
(
(
¬
X19
)
↔
(
(
¬
X10
)
↔
X47
)
)
)
)
)
)
→
(
¬
(
X51
↔
(
X47
↔
(
(
¬
X10
)
↔
X19
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
(
¬
X36
)
→
(
¬
X28
)
→
(
¬
X22
)
→
X53
)
)
→
(
X47
↔
(
X28
↔
(
(
¬
X8
)
→
X36
)
)
)
)
→
(
(
¬
(
X51
→
(
X29
↔
X22
)
)
)
↔
(
(
X51
→
(
(
¬
X22
)
↔
X47
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X5
)
→
X22
)
)
→
(
X51
↔
(
(
¬
X29
)
→
X5
→
X54
)
)
)
→
(
¬
(
(
¬
X47
)
↔
(
X36
↔
X54
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
X47
)
↔
(
(
¬
X29
)
→
X20
→
X51
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
c2da7e...
and proposition id is
1bd41b...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMcmqc2rdSWtMDR1i8BF49XTe5aK2RCTsRK
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
(
(
(
(
(
¬
(
(
¬
X33
)
→
X45
)
)
↔
(
X35
↔
(
X33
→
(
¬
X29
)
→
X48
)
)
)
↔
(
¬
(
X35
↔
(
X29
↔
(
X21
↔
False
)
)
)
)
)
↔
(
¬
(
(
¬
X45
)
→
(
(
¬
X35
)
↔
(
(
¬
X14
)
↔
X48
)
)
)
)
)
↔
(
(
¬
(
X29
↔
(
X10
→
False
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X40
)
↔
(
X35
→
(
X29
↔
X45
)
)
)
)
→
(
¬
(
X45
↔
(
X35
→
(
¬
X4
)
→
X29
)
)
)
)
)
)
)
→
(
X48
↔
(
X47
→
X45
→
X40
)
)
)
↔
(
(
(
X45
→
X35
→
(
X12
↔
X39
)
)
↔
(
X47
↔
(
X39
→
(
(
¬
X12
)
↔
X45
)
)
)
)
↔
(
(
(
X47
→
(
(
¬
X28
)
↔
(
X13
↔
X35
)
)
)
→
(
¬
(
X47
→
X28
→
(
X12
↔
X35
)
)
)
)
↔
(
X28
↔
(
(
¬
X13
)
→
X47
)
)
)
)
)
↔
(
¬
(
(
(
(
X45
↔
(
X43
→
X48
)
)
→
(
(
(
(
¬
(
X48
↔
(
X45
↔
(
(
¬
X43
)
→
X44
)
)
)
)
→
(
¬
(
X48
↔
(
X45
→
X5
→
X44
)
)
)
)
↔
(
X43
→
X44
)
)
↔
(
¬
(
X44
→
X43
)
)
)
)
→
(
(
¬
(
X48
→
X43
→
X1
→
False
)
)
↔
(
¬
(
(
X45
→
(
(
¬
X42
)
↔
(
(
¬
X31
)
→
X48
)
)
)
→
(
¬
(
(
¬
X1
)
→
X45
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
(
¬
(
(
¬
X47
)
↔
(
(
¬
X46
)
→
(
(
¬
X7
)
↔
X48
)
)
)
)
↔
(
X48
↔
(
X47
→
(
(
¬
X0
)
↔
X46
)
)
)
)
↔
(
(
X46
↔
(
(
¬
X45
)
→
X43
→
X47
)
)
→
(
(
(
¬
X47
)
→
(
(
¬
X38
)
↔
(
X19
→
X48
)
)
)
↔
(
(
X48
→
(
¬
X46
)
→
X47
)
↔
(
(
(
¬
X47
)
→
(
X46
↔
(
(
¬
X43
)
→
X48
)
)
)
↔
(
¬
(
X48
→
(
X47
↔
(
(
¬
X19
)
↔
X46
)
)
)
)
)
)
)
)
)
→
(
¬
(
X38
↔
(
X7
↔
X0
)
)
)
)
)
→
(
(
(
¬
(
X43
↔
(
X24
→
X41
)
)
)
→
(
¬
(
X48
→
X23
)
)
)
↔
(
(
X34
↔
(
(
¬
X11
)
↔
X23
)
)
↔
(
¬
(
(
X47
→
X41
→
(
X24
↔
False
)
)
→
(
(
X47
↔
(
X24
↔
(
(
¬
X11
)
→
X41
)
)
)
↔
(
(
¬
X41
)
↔
(
X23
↔
(
X11
→
X47
)
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
X49
↔
(
X46
→
X27
→
X16
)
)
→
(
¬
(
X49
↔
(
X46
↔
(
X27
→
X37
)
)
)
)
)
)
↔
(
X49
↔
(
X46
↔
(
(
¬
X32
)
→
X37
)
)
)
)
↔
(
(
¬
(
X48
→
X46
→
X32
)
)
↔
(
¬
(
(
¬
X48
)
→
X32
→
X17
→
X49
)
)
)
)
→
(
(
(
¬
(
X36
↔
(
X32
↔
(
X3
→
X46
)
)
)
)
↔
(
(
¬
X46
)
↔
(
X32
↔
X49
)
)
)
↔
(
(
¬
(
X46
→
(
X36
↔
(
X22
↔
False
)
)
)
)
↔
(
(
¬
(
X49
↔
(
(
¬
X36
)
↔
(
(
¬
X22
)
→
X46
)
)
)
)
↔
(
X46
→
(
X36
↔
(
(
¬
X18
)
↔
X22
)
)
)
)
)
)
)
↔
(
(
¬
(
(
X32
↔
(
(
¬
X25
)
→
X8
→
X36
)
)
→
X48
→
(
(
¬
X32
)
↔
(
(
¬
X8
)
→
X45
)
)
)
)
→
(
(
(
¬
(
X15
↔
X45
)
)
↔
(
(
¬
(
X45
→
X30
→
(
(
¬
X26
)
↔
X36
)
)
)
↔
(
X36
↔
(
(
¬
X9
)
↔
(
(
¬
X2
)
→
X30
)
)
)
)
)
↔
(
(
¬
(
X48
→
X36
→
X6
→
X45
)
)
↔
(
X36
↔
(
(
¬
X26
)
→
(
X20
↔
X45
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
(
X45
↔
(
X18
↔
False
)
)
↔
(
¬
(
(
(
(
¬
(
X45
→
(
X40
↔
(
X18
↔
X24
)
)
)
)
↔
(
¬
(
X48
↔
(
X40
↔
(
X18
↔
X24
)
)
)
)
)
→
X45
)
→
(
X46
↔
(
X45
↔
(
(
¬
X40
)
↔
X48
)
)
)
)
)
)
→
(
¬
X45
)
→
(
(
¬
X40
)
↔
(
(
¬
X28
)
→
X48
)
)
)
→
(
¬
(
(
(
¬
X45
)
↔
(
X34
→
X46
)
)
→
(
(
(
(
¬
X34
)
→
X40
)
↔
(
(
(
¬
(
(
¬
X45
)
→
X40
→
X46
)
)
→
(
(
(
¬
(
(
¬
X45
)
↔
(
X43
↔
(
(
¬
X40
)
→
X46
)
)
)
)
→
(
(
¬
X45
)
↔
(
X43
→
X34
→
X46
)
)
)
↔
X45
)
)
→
(
¬
(
X45
↔
X34
)
)
)
)
↔
(
(
(
¬
(
(
¬
X33
)
↔
(
X27
↔
(
(
¬
X20
)
↔
X46
)
)
)
)
↔
(
¬
(
X46
→
(
(
¬
X33
)
↔
(
(
¬
X27
)
→
X45
)
)
)
)
)
→
(
¬
(
X40
→
X27
→
X46
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X43
→
(
X38
↔
(
(
¬
X4
)
→
X5
)
)
)
)
→
(
¬
(
(
¬
(
X22
↔
(
X5
↔
X43
)
)
)
→
(
¬
(
X43
↔
(
X38
↔
(
X36
→
False
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X46
→
(
(
¬
X11
)
↔
X45
)
)
)
→
(
¬
(
X46
↔
(
X36
↔
False
)
)
)
→
(
(
¬
(
X45
→
(
X43
↔
(
X11
↔
False
)
)
)
)
↔
(
¬
(
X45
↔
(
(
¬
X43
)
↔
(
X36
↔
X46
)
)
)
)
)
)
→
(
¬
(
X46
→
(
¬
X43
)
→
(
X38
↔
X45
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
¬
(
X45
↔
(
(
¬
X7
)
→
X14
)
)
)
→
(
¬
(
X49
→
(
(
¬
X14
)
↔
(
(
¬
X7
)
→
X46
)
)
)
)
)
)
→
(
¬
X48
)
)
)
↔
(
(
¬
(
X46
→
(
X2
↔
X48
)
)
)
↔
(
¬
(
X30
↔
(
(
¬
X2
)
→
X49
)
)
)
)
)
↔
(
(
X25
→
(
¬
X2
)
→
X7
)
→
(
¬
(
X1
↔
X41
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X36
→
(
(
¬
X8
)
↔
(
(
¬
X6
)
↔
X47
)
)
)
)
↔
(
(
X10
→
(
X6
↔
X39
)
)
↔
(
¬
(
X39
→
(
X36
↔
(
(
¬
X16
)
→
X42
)
)
)
)
)
)
↔
(
(
(
¬
X45
)
→
(
(
¬
X16
)
↔
X47
)
)
↔
(
(
(
¬
X16
)
→
X15
→
X42
)
→
(
¬
(
(
¬
X43
)
↔
(
X15
↔
X45
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
X45
)
→
X42
→
X17
→
X47
)
↔
(
¬
(
(
(
¬
(
X45
→
(
X42
↔
(
X21
→
X47
)
)
)
)
↔
(
X47
→
X45
→
(
X3
↔
X42
)
)
)
→
(
¬
(
X47
↔
(
X45
↔
(
X42
↔
False
)
)
)
)
)
)
)
→
(
¬
(
(
¬
X17
)
↔
X21
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
X45
→
(
X44
↔
(
X26
→
X9
)
)
)
)
→
(
¬
(
X19
↔
(
X0
→
X44
)
)
)
→
X26
→
(
¬
X0
)
→
X9
)
)
→
(
(
¬
(
X44
→
X35
→
(
X12
↔
X43
)
)
)
↔
(
¬
(
X43
↔
(
X26
↔
(
X19
→
X44
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
X31
)
→
(
¬
X23
)
→
X43
)
↔
(
(
(
(
X43
→
X32
→
X29
→
False
)
→
(
(
X43
↔
(
X32
→
X29
→
X26
)
)
↔
(
X26
→
(
¬
X23
)
→
X32
)
)
)
→
X43
→
(
(
¬
X26
)
↔
X29
)
)
→
(
X32
↔
(
X26
→
X29
)
)
)
)
→
(
(
X42
→
X37
→
(
X26
↔
X45
)
)
↔
(
(
¬
(
X37
→
(
X26
↔
(
X23
↔
False
)
)
)
)
→
(
X45
↔
(
(
¬
X26
)
→
(
¬
X13
)
→
X37
)
)
)
)
→
(
(
¬
X42
)
↔
(
X31
↔
(
(
¬
X26
)
→
X45
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
538917...
and proposition id is
7edf77...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMcUxywjRYMeVeALGJdkowoeoQtHoaRrrAj
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
∀X50 :
prop
,
∀X51 :
prop
,
∀X52 :
prop
,
(
(
¬
(
(
(
(
(
(
X48
→
(
X45
↔
(
(
¬
X42
)
↔
X47
)
)
)
↔
(
X47
→
(
X14
↔
X42
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
X42
)
→
X29
→
X25
→
X45
)
→
(
¬
(
X45
→
(
(
¬
X29
)
↔
(
(
¬
X25
)
→
X47
)
)
)
)
)
)
→
(
(
X47
→
X25
)
↔
(
(
¬
X42
)
→
(
(
(
¬
X42
)
→
(
X25
↔
(
(
¬
X18
)
→
X47
)
)
)
↔
(
¬
(
X42
→
(
X18
↔
(
X10
→
X47
)
)
)
)
)
)
)
)
)
)
↔
(
(
X48
↔
(
(
¬
X41
)
↔
(
X25
↔
X42
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X48
↔
(
X42
→
(
(
¬
X2
)
↔
X41
)
)
)
)
→
(
X48
↔
(
X42
↔
(
X8
→
X0
)
)
)
)
)
→
(
(
(
¬
X6
)
↔
X42
)
↔
(
(
¬
X41
)
↔
(
X6
→
X0
→
X46
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
(
¬
(
X46
↔
(
X16
↔
X48
)
)
)
↔
(
(
¬
X21
)
→
(
X16
↔
X50
)
)
)
↔
(
¬
(
X48
↔
(
X45
→
False
)
)
)
)
↔
(
(
(
¬
X35
)
→
X44
)
↔
(
(
¬
X46
)
→
X45
→
(
(
¬
X35
)
↔
X48
)
)
)
)
→
(
(
¬
(
X50
↔
(
X46
↔
(
X13
↔
X4
)
)
)
)
↔
(
¬
(
X50
↔
(
(
¬
X19
)
→
X4
→
X45
)
)
)
)
)
)
→
(
(
¬
(
X48
→
(
¬
X45
)
→
(
¬
X9
)
→
X50
)
)
↔
(
(
X50
↔
(
(
¬
X45
)
↔
(
X15
↔
X48
)
)
)
→
(
¬
(
X50
↔
(
X45
→
(
(
¬
X39
)
↔
X48
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
(
X52
→
X39
→
X38
→
X45
)
→
(
¬
(
(
¬
X45
)
→
X23
→
X52
)
)
)
)
↔
(
¬
X38
)
)
→
(
X39
↔
(
X23
↔
X52
)
)
)
→
(
X52
↔
(
(
¬
X45
)
→
(
¬
X39
)
→
X50
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
(
¬
X46
)
→
(
(
¬
X36
)
↔
(
(
¬
X1
)
→
X49
)
)
)
)
↔
(
(
¬
(
(
¬
X32
)
→
(
(
¬
X7
)
↔
X43
)
)
)
→
(
(
¬
(
X43
↔
(
(
¬
X36
)
→
(
¬
X30
)
→
X46
)
)
)
↔
(
X43
→
(
¬
X36
)
→
(
¬
X1
)
→
X46
)
)
→
(
¬
(
X46
→
X32
→
(
(
¬
X30
)
↔
X36
)
)
)
)
)
→
(
(
¬
(
(
(
¬
X46
)
↔
(
X36
→
(
(
¬
X20
)
↔
X49
)
)
)
→
(
¬
(
(
(
(
(
¬
X36
)
→
X28
→
(
¬
X26
)
→
X49
)
↔
(
¬
(
X28
↔
(
X27
→
X49
)
)
)
)
→
(
¬
(
X51
↔
(
(
¬
X28
)
→
X26
→
X33
)
)
)
→
(
¬
(
X51
→
(
¬
X33
)
→
(
¬
X26
)
→
X46
)
)
)
→
(
¬
(
X36
→
(
(
¬
X27
)
↔
X46
)
)
)
)
)
)
)
↔
(
X51
→
(
X17
↔
X48
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
X31
)
→
X50
)
→
X37
→
(
X34
↔
(
(
¬
X22
)
↔
X50
)
)
)
→
(
(
(
(
¬
(
X48
→
(
(
¬
X40
)
↔
(
(
¬
X22
)
→
X43
)
)
)
)
→
(
X22
↔
(
(
¬
X11
)
↔
(
(
¬
X3
)
↔
X40
)
)
)
)
↔
(
X48
→
(
X40
↔
(
X37
→
X22
→
False
)
)
)
)
→
(
¬
(
X40
↔
(
X22
↔
X43
)
)
)
)
→
(
(
¬
(
X43
→
(
X5
↔
X37
)
)
)
↔
(
(
¬
(
X43
↔
(
X37
↔
(
X31
↔
False
)
)
)
)
→
(
(
X37
→
(
X31
↔
(
X5
↔
X11
)
)
)
↔
(
X43
→
(
(
¬
X31
)
↔
(
(
¬
X5
)
↔
X37
)
)
)
)
)
)
)
)
↔
(
¬
(
(
X50
↔
(
X33
→
X43
)
)
→
(
(
X50
→
(
X48
↔
(
X33
↔
X43
)
)
)
↔
(
¬
(
(
¬
X33
)
→
X24
→
X12
→
X43
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
X48
→
(
X38
↔
(
X9
→
X52
)
)
)
)
→
X52
→
(
X38
↔
(
X16
↔
False
)
)
)
↔
(
¬
(
(
¬
(
X52
→
(
(
¬
X16
)
↔
(
X9
→
X23
)
)
)
)
→
(
(
(
¬
(
X9
↔
False
)
)
→
(
X16
→
X9
→
(
(
¬
X6
)
↔
X52
)
)
→
X16
→
(
X9
↔
(
(
¬
X6
)
→
X50
)
)
)
↔
(
(
¬
X50
)
↔
(
X9
↔
X52
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
¬
X39
)
→
X41
)
→
(
¬
(
(
¬
X32
)
→
(
(
¬
X16
)
↔
X49
)
)
)
)
)
↔
(
X48
→
X41
→
X39
→
X16
→
False
)
)
→
(
(
X50
↔
(
X48
↔
False
)
)
↔
(
(
¬
(
X49
↔
(
(
¬
X48
)
↔
X50
)
)
)
→
(
¬
(
(
(
¬
X41
)
↔
(
X3
↔
X48
)
)
→
(
¬
(
(
¬
X49
)
→
(
¬
X41
)
→
(
(
¬
X3
)
↔
X50
)
)
)
)
)
)
)
→
(
¬
(
X50
→
(
(
¬
X41
)
↔
X49
)
)
)
)
)
)
↔
(
¬
(
(
(
(
X51
↔
X11
)
→
(
¬
(
X42
↔
(
X40
→
(
¬
X34
)
→
X51
)
)
)
)
↔
(
(
(
(
(
¬
X2
)
→
X34
)
↔
(
¬
(
(
¬
X34
)
↔
(
(
¬
X18
)
↔
(
X5
→
X40
)
)
)
)
)
↔
(
¬
(
X40
→
(
(
¬
X20
)
↔
(
X2
↔
X34
)
)
)
)
)
→
(
(
(
¬
(
X40
↔
(
X18
→
(
(
¬
X4
)
↔
X51
)
)
)
)
↔
(
X50
↔
(
(
¬
X40
)
→
(
(
¬
X4
)
↔
X51
)
)
)
)
↔
(
¬
(
(
¬
X40
)
→
(
(
¬
X20
)
↔
(
(
¬
X18
)
↔
X50
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
(
X50
↔
(
X48
→
(
X14
↔
False
)
)
)
)
→
(
(
¬
(
(
X48
↔
(
(
¬
X45
)
→
(
¬
X40
)
→
X50
)
)
→
X50
→
(
X48
↔
(
X45
↔
(
X14
→
False
)
)
)
)
)
↔
(
(
¬
X45
)
↔
(
(
¬
X40
)
→
X50
)
)
)
)
)
↔
(
X48
↔
(
X18
→
X45
)
)
)
↔
(
(
¬
(
X35
↔
(
X27
→
(
(
¬
X18
)
↔
X45
)
)
)
)
→
(
(
(
X50
↔
(
X22
→
False
)
)
↔
(
(
¬
X35
)
→
(
(
¬
X22
)
↔
(
X18
→
X52
)
)
)
)
↔
(
¬
(
(
¬
(
X45
→
(
¬
X15
)
→
X18
)
)
→
(
(
(
(
¬
(
X52
↔
(
X18
→
X45
)
)
)
↔
(
X35
→
X18
→
(
X15
↔
X45
)
)
)
↔
(
X52
→
(
X45
↔
(
X35
↔
False
)
)
)
)
↔
(
X45
↔
(
(
¬
X18
)
→
X52
)
)
)
→
X52
→
X18
→
X45
)
)
)
)
)
↔
(
¬
(
X37
→
(
X35
↔
(
X27
↔
X15
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
X49
→
X30
→
X28
→
X51
)
→
(
X51
↔
(
X49
↔
(
X30
↔
X19
)
)
)
)
)
↔
(
(
¬
(
X49
↔
(
X28
↔
(
(
¬
X7
)
↔
X51
)
)
)
)
↔
(
¬
(
X49
↔
(
X28
→
X7
→
X13
)
)
)
)
)
↔
(
(
¬
(
X51
→
(
¬
X30
)
→
(
¬
X25
)
→
X50
)
)
→
(
¬
(
X25
→
X13
)
)
)
)
↔
(
(
(
(
(
¬
X33
)
→
(
¬
X24
)
→
(
X10
↔
X51
)
)
↔
(
¬
(
X51
↔
(
X24
↔
X1
)
)
)
)
→
(
(
¬
(
(
¬
X49
)
→
(
X10
↔
X51
)
)
)
↔
(
(
(
¬
X50
)
→
X49
→
X10
→
X51
)
↔
(
¬
(
X49
→
(
X10
↔
(
(
¬
X1
)
→
X50
)
)
)
)
)
)
)
↔
(
X49
→
X44
→
(
¬
X24
)
→
X50
)
)
)
→
(
(
(
(
¬
(
X50
→
(
X36
↔
(
(
¬
X31
)
→
X43
)
)
)
)
↔
(
(
X50
↔
(
(
¬
X31
)
→
X36
)
)
→
(
¬
(
X49
↔
(
X31
→
(
(
¬
X29
)
↔
X50
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X26
)
→
(
¬
X8
)
→
X36
)
)
↔
(
¬
(
(
¬
(
(
(
X36
↔
(
X26
→
X12
→
False
)
)
→
(
X43
↔
(
X12
→
X8
→
False
)
)
)
→
(
¬
(
X36
↔
(
X12
↔
False
)
)
)
)
)
→
(
X26
↔
X12
)
)
)
)
→
(
¬
X47
)
→
(
(
¬
X43
)
↔
(
(
¬
X36
)
→
X50
)
)
)
)
↔
(
(
¬
(
(
¬
X46
)
→
(
(
¬
X31
)
↔
(
(
¬
X21
)
→
X50
)
)
)
)
↔
(
¬
(
(
(
¬
(
X17
↔
X50
)
)
↔
(
(
X50
→
X29
→
(
X17
↔
False
)
)
→
(
¬
(
X31
→
(
¬
X17
)
→
(
(
¬
X0
)
↔
X29
)
)
)
)
)
→
(
¬
(
(
¬
(
X50
↔
(
X29
→
False
)
)
)
→
(
X51
→
X50
→
(
(
¬
X17
)
↔
X29
)
)
→
(
¬
(
X51
→
(
(
¬
X46
)
↔
(
(
¬
X29
)
↔
X50
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
b367b3...
and proposition id is
3e887c...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMXeeYrvzmHEhbqjKkVRX96FYYvyZB8uvzZ
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
¬
(
(
(
(
(
(
X48
↔
(
X25
↔
(
X15
↔
X41
)
)
)
→
(
¬
(
X41
→
(
X25
↔
(
(
¬
X15
)
→
X47
)
)
)
)
)
↔
(
¬
(
X47
↔
(
X41
→
(
X25
↔
(
X4
↔
False
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
X45
)
↔
(
(
¬
X25
)
↔
X48
)
)
)
↔
(
¬
(
(
(
X45
↔
(
(
¬
X25
)
↔
(
(
¬
X23
)
→
X48
)
)
)
→
X25
→
(
¬
X22
)
→
X23
)
→
(
¬
(
X45
↔
(
X41
↔
(
X22
↔
X48
)
)
)
)
)
)
)
)
→
(
¬
(
X48
↔
(
X33
→
X35
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
¬
(
X48
→
X35
→
X37
)
)
→
(
¬
(
(
¬
X37
)
→
(
¬
X35
)
→
(
(
¬
X33
)
↔
X47
)
)
)
)
)
→
X47
→
(
X35
↔
(
X11
↔
X48
)
)
)
)
→
(
¬
(
(
¬
(
X48
→
(
¬
X37
)
→
X41
)
)
→
(
¬
(
X41
↔
(
X35
→
X48
)
)
)
→
(
¬
(
(
¬
X37
)
↔
(
X35
→
(
(
¬
X5
)
↔
X41
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X45
)
↔
(
(
¬
X19
)
→
(
(
¬
X7
)
↔
X49
)
)
)
)
→
(
¬
(
X31
→
X7
→
(
(
¬
X2
)
↔
X49
)
)
)
→
(
¬
(
(
(
X49
→
X45
→
X7
→
X31
)
→
(
(
¬
X47
)
↔
(
X45
↔
(
(
¬
X7
)
↔
X49
)
)
)
)
→
(
X47
↔
X45
)
)
)
)
→
(
¬
(
X49
→
X47
→
X45
→
X37
)
)
)
)
→
(
(
(
(
X45
↔
(
X37
↔
(
X32
→
X13
)
)
)
↔
(
¬
(
(
¬
X37
)
→
(
(
¬
X32
)
↔
(
(
¬
X16
)
↔
X45
)
)
)
)
)
↔
(
(
X37
→
(
¬
X32
)
→
X45
)
→
(
X47
→
(
¬
X37
)
→
(
¬
X28
)
→
X45
)
→
(
¬
(
X47
↔
(
X45
↔
(
X37
→
X32
)
)
)
)
)
)
↔
(
(
(
X47
↔
(
(
¬
X37
)
→
X45
)
)
↔
(
(
¬
(
X45
→
(
X34
↔
X47
)
)
)
↔
(
(
¬
(
(
¬
(
X47
↔
(
X45
→
(
¬
X14
)
→
X37
)
)
)
→
(
(
¬
X45
)
↔
(
X37
→
(
(
¬
X29
)
↔
X47
)
)
)
)
)
↔
(
X34
→
(
(
¬
X14
)
↔
X45
)
)
)
)
)
↔
(
X47
↔
(
X37
↔
(
X30
↔
(
X14
→
False
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X18
→
(
(
¬
X10
)
↔
(
(
¬
X8
)
↔
X46
)
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X10
)
→
X8
→
X49
)
)
)
)
→
(
(
(
(
¬
X46
)
↔
(
(
¬
X42
)
→
X47
)
)
→
(
¬
(
(
(
¬
(
(
¬
(
(
¬
X24
)
→
X47
)
)
→
(
(
(
X24
→
X1
→
X46
)
↔
(
(
¬
X46
)
↔
(
(
¬
X42
)
→
(
¬
X24
)
→
X47
)
)
)
↔
(
X47
↔
(
X46
↔
X1
)
)
)
→
(
¬
(
X47
↔
(
X46
↔
(
X42
→
False
)
)
)
)
)
)
↔
(
(
¬
X46
)
→
(
(
¬
X24
)
↔
X47
)
)
)
→
X47
→
(
(
¬
X24
)
↔
(
(
¬
X1
)
↔
X39
)
)
)
)
)
↔
(
(
X39
↔
X46
)
↔
(
¬
(
X49
→
X48
→
X47
→
X46
→
False
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
(
X48
→
X26
→
X6
)
)
→
(
¬
(
X48
↔
(
(
¬
X44
)
↔
(
X26
→
X47
)
)
)
)
→
X48
→
(
X44
↔
(
X40
→
X47
)
)
)
)
↔
(
(
X40
→
(
X21
↔
(
X17
→
X44
)
)
)
→
(
X48
↔
(
X44
↔
X17
)
)
)
)
→
(
(
(
¬
(
X44
↔
(
X43
→
X48
)
)
)
↔
(
¬
(
(
(
¬
(
X48
→
(
¬
X44
)
→
X20
→
X47
)
)
↔
(
(
¬
X47
)
→
(
X44
↔
(
(
¬
X12
)
→
X48
)
)
)
)
→
X47
→
X43
→
(
X12
↔
X20
)
)
)
)
↔
(
(
(
¬
(
X44
↔
X47
)
)
→
(
X47
↔
(
X46
→
(
X38
↔
X43
)
)
)
)
↔
(
¬
(
X47
→
(
X38
↔
X43
)
)
)
)
)
)
↔
(
(
¬
(
(
(
X43
→
X9
→
(
(
¬
X0
)
↔
X47
)
)
→
(
X47
↔
(
X36
↔
(
X27
→
X0
)
)
)
)
→
X36
→
(
(
¬
X9
)
↔
(
X3
→
X27
)
)
)
)
→
X48
→
(
X47
↔
(
X36
↔
X43
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
X45
↔
(
X39
→
X46
)
)
→
X35
→
(
(
¬
X17
)
↔
X49
)
)
→
(
(
¬
(
X49
↔
(
X23
→
X46
)
)
)
→
(
¬
X41
)
→
X45
)
→
(
¬
(
X49
→
(
X46
↔
(
X29
→
X45
)
)
)
)
)
↔
(
(
(
(
¬
(
X45
→
X39
→
X6
)
)
↔
(
¬
(
X45
↔
(
X32
↔
(
X6
↔
X46
)
)
)
)
)
↔
(
¬
(
(
¬
X45
)
→
X46
)
)
)
↔
(
(
¬
X32
)
→
(
¬
X6
)
→
X2
→
X39
)
)
)
→
(
(
¬
(
(
(
X46
↔
(
X41
↔
(
(
¬
X28
)
↔
X40
)
)
)
→
(
X45
↔
(
(
¬
X41
)
→
(
X40
↔
X46
)
)
)
)
→
(
¬
(
(
(
X41
→
(
X28
↔
(
X26
→
X40
)
)
)
↔
(
X40
→
(
X28
↔
(
X26
↔
(
X18
↔
False
)
)
)
)
)
→
(
X41
↔
(
X26
→
(
X11
↔
X28
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
X41
)
→
(
X36
↔
(
(
¬
X3
)
→
X44
)
)
)
→
(
(
¬
(
(
(
¬
False
)
↔
(
(
(
¬
X41
)
→
X37
→
(
¬
X36
)
→
X44
)
↔
(
¬
(
X40
↔
(
(
¬
X36
)
→
X41
)
)
)
)
)
→
(
¬
X40
)
→
(
X36
↔
(
(
¬
X31
)
↔
X44
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X37
)
↔
(
X25
↔
X46
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X16
)
→
(
X12
↔
X25
)
)
)
)
)
)
↔
(
(
(
¬
(
X46
↔
(
X31
→
(
¬
X25
)
→
X44
)
)
)
↔
(
X44
→
X37
→
False
)
)
↔
(
X44
↔
(
X25
→
False
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
X48
→
X14
→
X42
)
)
→
(
¬
(
(
¬
X33
)
→
(
(
¬
X14
)
↔
(
X8
↔
X48
)
)
)
)
)
↔
(
X48
→
(
¬
X42
)
→
(
(
¬
X34
)
↔
X43
)
)
)
↔
(
X43
→
X34
→
(
X33
↔
(
X24
↔
False
)
)
)
)
→
(
¬
(
(
(
¬
(
X46
↔
(
X22
↔
(
(
¬
X20
)
↔
X43
)
)
)
)
↔
(
X46
↔
(
X20
↔
(
(
¬
X10
)
→
X43
)
)
)
)
→
(
(
¬
(
X34
↔
(
(
¬
X10
)
→
(
(
¬
X1
)
↔
X42
)
)
)
)
↔
(
¬
(
(
¬
X42
)
↔
(
(
¬
X20
)
→
X46
)
)
)
)
)
)
→
(
¬
(
(
X46
↔
(
X38
→
X9
→
X35
)
)
→
(
(
(
¬
(
X42
→
X34
→
(
(
¬
X30
)
↔
X38
)
)
)
→
(
X46
↔
(
X38
→
X34
)
)
)
↔
(
X38
→
X35
→
(
X21
↔
X5
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X38
→
(
¬
X19
)
→
(
X15
↔
X43
)
)
)
→
(
¬
(
X43
↔
(
X38
→
(
¬
X19
)
→
X37
)
)
)
)
↔
(
(
(
(
X4
↔
X38
)
→
(
¬
(
(
¬
(
X38
↔
(
X13
↔
X37
)
)
)
→
(
¬
(
X38
↔
(
(
¬
X13
)
→
X37
)
)
)
→
(
(
¬
(
X43
↔
(
(
¬
X13
)
→
X4
→
X37
)
)
)
↔
(
¬
(
X43
↔
(
X38
→
X13
→
X4
→
False
)
)
)
)
)
)
)
↔
(
X37
→
(
¬
X4
)
→
X13
)
)
↔
(
¬
(
(
X43
→
(
(
¬
X27
)
↔
X37
)
)
→
(
X38
↔
(
(
¬
X37
)
→
(
¬
X0
)
→
X43
)
)
)
)
)
)
↔
(
(
X47
→
(
X43
↔
X42
)
)
→
(
(
¬
(
(
¬
(
X43
→
X42
→
False
)
)
→
(
¬
(
(
X42
→
(
¬
X38
)
→
(
¬
X7
)
→
X47
)
→
X43
→
(
¬
X38
)
→
(
¬
X7
)
→
X47
)
)
)
)
↔
(
(
¬
X37
)
→
X38
)
)
→
(
¬
(
X37
→
(
X7
↔
X38
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
855b8b...
and proposition id is
5c295f...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMMud9xt6C6jD8DG2YaqBA8uXZ1ssBAXTXP
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
(
¬
(
(
¬
X47
)
↔
(
X46
→
(
X16
↔
X48
)
)
)
)
↔
(
(
¬
(
(
X16
→
(
¬
X8
)
→
X47
)
→
(
(
(
¬
X20
)
↔
(
(
¬
X16
)
↔
(
(
¬
X8
)
→
X47
)
)
)
↔
(
(
¬
X47
)
↔
(
X16
→
X8
→
X48
)
)
)
)
)
↔
(
(
X47
↔
(
X16
↔
(
X1
→
X39
)
)
)
↔
(
(
¬
X39
)
→
(
¬
X20
)
→
(
X16
↔
X47
)
)
)
)
)
↔
(
(
(
X48
→
(
X46
↔
(
X43
→
False
)
)
)
↔
(
(
¬
X47
)
↔
(
X43
↔
(
X13
→
X48
)
)
)
)
↔
(
¬
(
X49
↔
(
X47
↔
(
(
¬
X46
)
→
X48
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
(
¬
X30
)
→
X21
→
X9
→
X43
)
→
(
¬
(
X43
↔
(
(
¬
X19
)
↔
(
(
¬
X9
)
↔
X47
)
)
)
)
)
)
↔
(
X47
→
(
¬
X21
)
→
X12
→
X30
)
)
↔
(
(
¬
(
X46
→
X43
→
(
¬
X38
)
→
X47
)
)
↔
(
(
(
¬
(
(
(
¬
(
(
(
¬
(
X47
↔
(
(
¬
X30
)
↔
X38
)
)
)
→
(
¬
(
(
¬
X38
)
↔
(
X37
↔
(
X10
↔
X47
)
)
)
)
)
→
(
¬
(
X38
↔
(
X37
↔
X30
)
)
)
)
)
→
(
¬
X37
)
→
X30
→
X47
)
→
(
X47
↔
(
X10
→
X30
)
)
)
)
→
(
¬
(
(
¬
X43
)
→
(
X37
↔
(
X30
↔
X47
)
)
)
)
)
↔
(
¬
(
X38
↔
(
X37
→
X30
→
False
)
)
)
)
)
)
↔
(
(
(
¬
(
X43
↔
(
X15
↔
False
)
)
)
→
(
¬
(
X43
→
X36
→
(
X35
↔
(
X29
↔
False
)
)
)
)
)
↔
(
(
(
¬
(
X43
↔
(
X36
↔
(
(
¬
X28
)
→
X47
)
)
)
)
→
(
(
X47
↔
(
X46
↔
(
X36
→
X43
)
)
)
↔
(
X47
→
(
X46
↔
(
X43
↔
(
X34
↔
False
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
(
(
¬
(
(
¬
X35
)
↔
(
X34
→
X3
→
X43
)
)
)
→
(
¬
(
X35
→
X34
→
X4
→
X43
)
)
)
↔
(
X43
→
(
(
¬
X34
)
↔
X35
)
)
)
↔
(
¬
(
X4
→
X3
→
False
)
)
)
↔
(
(
¬
X33
)
↔
(
X4
↔
(
X3
→
X35
)
)
)
)
→
(
¬
(
X43
→
(
X4
↔
False
)
)
)
)
↔
(
X43
↔
(
X33
↔
X34
)
)
)
→
(
(
(
¬
X0
)
→
X4
)
↔
(
(
¬
(
(
¬
(
(
¬
X43
)
↔
(
(
¬
X4
)
→
(
¬
X0
)
→
X46
)
)
)
→
(
¬
(
X46
→
(
X33
↔
(
X0
→
X43
)
)
)
)
)
)
↔
(
¬
(
X33
→
X4
)
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
X44
↔
(
X42
↔
(
X41
↔
X27
)
)
)
→
(
(
(
(
¬
(
X47
↔
(
(
¬
X42
)
↔
(
(
¬
X17
)
→
X44
)
)
)
)
→
(
(
¬
X44
)
↔
(
(
¬
X42
)
→
(
X5
↔
X47
)
)
)
)
→
(
¬
(
X47
→
X44
→
(
X17
↔
False
)
)
)
)
↔
(
X44
→
(
X42
↔
(
X41
↔
False
)
)
)
)
)
→
(
¬
(
(
(
X48
↔
(
(
¬
X42
)
→
(
¬
X41
)
→
X43
)
)
→
(
¬
(
X42
↔
(
(
¬
X41
)
→
X23
→
X48
)
)
)
)
→
(
(
X41
→
(
(
¬
X24
)
↔
(
(
¬
X18
)
↔
X42
)
)
)
↔
(
(
(
(
X48
→
(
¬
X27
)
→
(
¬
X2
)
→
X41
)
↔
(
¬
(
(
¬
X42
)
→
(
¬
X41
)
→
(
(
¬
X2
)
↔
X48
)
)
)
)
↔
(
¬
(
X42
→
X2
→
X27
)
)
)
↔
(
X48
→
(
X41
↔
False
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
X44
→
(
¬
X40
)
→
X7
→
X43
)
↔
(
(
X45
↔
(
X44
↔
(
X11
↔
X43
)
)
)
↔
(
¬
(
(
¬
X43
)
→
X40
→
X45
)
)
)
)
→
(
(
(
(
(
¬
(
(
¬
X32
)
→
(
(
¬
X26
)
↔
(
X6
→
X45
)
)
)
)
→
(
¬
(
(
¬
X44
)
→
X26
→
(
¬
X6
)
→
X45
)
)
)
→
(
¬
(
(
X44
→
X26
)
→
(
(
¬
(
(
¬
(
X32
↔
(
X22
→
X26
)
)
)
→
(
¬
(
(
¬
(
(
X22
↔
(
(
¬
X14
)
↔
X26
)
)
→
(
¬
(
X44
→
X32
)
)
)
)
→
(
(
¬
X22
)
↔
X44
)
)
)
)
)
↔
(
¬
(
X44
↔
(
X32
→
X26
→
False
)
)
)
)
)
)
)
→
(
¬
(
X47
→
(
X25
↔
(
X22
↔
(
X6
↔
False
)
)
)
)
)
)
↔
(
(
X45
↔
(
X31
↔
X44
)
)
↔
(
X47
↔
(
(
¬
X40
)
↔
(
(
¬
X31
)
→
X44
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
¬
(
X47
↔
(
X43
→
X30
)
)
)
→
(
(
¬
(
(
¬
X46
)
↔
(
X30
→
(
X29
↔
X47
)
)
)
)
↔
(
X25
→
False
)
)
→
(
(
¬
(
X47
→
X46
→
(
X9
↔
X43
)
)
)
↔
(
(
¬
(
X47
→
X46
→
(
X30
↔
False
)
)
)
↔
(
X46
→
X44
→
(
X43
↔
(
X30
↔
False
)
)
)
)
)
)
↔
(
(
¬
(
X46
→
(
X43
↔
(
X1
↔
X37
)
)
)
)
↔
(
(
¬
(
X46
↔
(
X43
↔
(
X37
→
X47
)
)
)
)
→
(
(
¬
(
X43
→
X37
→
X32
)
)
↔
(
¬
(
X13
→
(
(
¬
X6
)
↔
X47
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X44
↔
(
X41
→
(
(
¬
X20
)
↔
X47
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
X47
↔
X20
)
→
(
¬
(
(
¬
X41
)
↔
(
X38
↔
(
(
¬
X4
)
→
X47
)
)
)
)
)
→
X20
→
X4
→
X41
)
)
→
(
X47
↔
False
)
)
)
)
↔
(
¬
(
X47
→
X44
)
)
)
→
(
(
X47
→
(
X43
↔
(
(
¬
X41
)
↔
X44
)
)
)
↔
(
(
¬
X43
)
→
X41
→
(
¬
X19
)
→
X44
)
)
→
(
(
¬
(
X44
↔
(
X41
↔
(
X14
→
X3
)
)
)
)
↔
(
X14
↔
X47
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
(
X28
→
(
X27
↔
(
X18
↔
X46
)
)
)
)
→
(
(
¬
X36
)
↔
(
(
¬
X28
)
↔
(
X18
→
X46
)
)
)
)
)
↔
(
¬
(
X36
→
(
(
¬
X27
)
↔
X46
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X0
)
↔
X46
)
)
↔
(
(
X46
↔
(
X38
↔
(
(
¬
X0
)
↔
X28
)
)
)
↔
(
(
¬
(
(
¬
X38
)
→
X11
→
X46
)
)
→
(
¬
(
X34
↔
(
(
¬
X11
)
↔
(
(
¬
X0
)
↔
X38
)
)
)
)
)
)
)
→
(
X34
→
X18
→
(
¬
X16
)
→
X46
)
→
(
¬
(
(
¬
(
X16
↔
(
X10
↔
X38
)
)
)
→
(
X46
↔
(
X38
↔
(
X34
↔
False
)
)
)
)
)
)
)
)
↔
(
(
¬
(
X46
↔
(
(
¬
X28
)
↔
(
(
¬
X24
)
→
X34
)
)
)
)
↔
(
(
¬
(
(
¬
X46
)
↔
(
X41
→
(
¬
X34
)
→
X49
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X34
)
↔
(
(
¬
X22
)
→
(
X17
↔
X41
)
)
)
)
→
(
¬
X34
)
→
(
¬
X22
)
→
(
X17
↔
X49
)
)
→
(
¬
(
(
¬
(
(
¬
X41
)
→
(
X34
↔
X49
)
)
)
→
(
(
X41
→
X34
→
X24
)
→
(
X49
↔
(
X34
↔
(
X24
→
X41
)
)
)
→
X49
→
(
X41
↔
(
X34
→
X15
)
)
)
→
(
¬
(
X49
→
X15
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X49
↔
(
X40
↔
(
(
¬
X21
)
→
X46
)
)
)
)
↔
(
¬
(
(
X7
→
X49
)
→
(
¬
(
(
(
(
¬
X46
)
↔
(
(
¬
X45
)
↔
(
X7
↔
X49
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X45
)
→
(
X40
↔
X49
)
)
)
)
)
→
(
X45
↔
(
X40
↔
X49
)
)
)
)
→
(
(
¬
X45
)
↔
(
X40
→
X46
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X46
)
↔
(
X45
→
X49
)
)
)
↔
(
(
X49
→
X47
→
X45
→
X46
)
↔
(
¬
(
X49
↔
(
X46
→
(
(
¬
X23
)
↔
X47
)
)
)
)
)
)
↔
(
¬
(
(
¬
X47
)
↔
(
X46
↔
X49
)
)
)
)
↔
(
(
(
(
¬
X45
)
↔
(
X39
→
X48
)
)
→
(
¬
(
(
¬
X47
)
↔
(
(
¬
X45
)
↔
(
(
¬
X23
)
↔
X48
)
)
)
)
)
↔
(
¬
(
(
¬
X47
)
↔
(
(
¬
X39
)
→
X48
)
)
)
)
)
)
→
(
(
(
¬
X48
)
↔
(
(
¬
X46
)
↔
(
(
¬
X31
)
↔
X49
)
)
)
→
(
¬
(
X49
→
(
(
¬
X46
)
↔
(
X12
↔
X48
)
)
)
)
)
→
X49
→
(
¬
X31
)
→
X33
)
↔
(
(
(
¬
(
X47
→
X45
→
(
¬
X8
)
→
X42
)
)
↔
(
¬
(
X46
↔
(
(
¬
X45
)
→
(
¬
X42
)
→
X47
)
)
)
)
→
(
¬
(
(
(
¬
X46
)
↔
(
X45
→
(
¬
X26
)
→
X47
)
)
→
(
(
¬
(
X46
↔
(
(
¬
X26
)
↔
(
X5
↔
X47
)
)
)
)
↔
(
X46
↔
(
X35
↔
(
X26
→
(
X2
↔
False
)
)
)
)
)
→
(
¬
X46
)
→
(
X35
↔
X47
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
6d425c...
and proposition id is
2cf9fc...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMYj3RGC7YyCFzWmaLxk56u22FRz2tq3tci
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
¬
(
(
(
(
(
(
(
¬
X48
)
↔
(
(
¬
X46
)
→
X49
)
)
→
(
¬
(
(
(
X49
↔
(
(
¬
X46
)
↔
(
X44
↔
X48
)
)
)
↔
(
¬
(
(
¬
(
X48
→
(
X46
↔
(
X39
↔
X49
)
)
)
)
→
(
X49
↔
(
X46
↔
(
(
¬
X26
)
→
X48
)
)
)
)
)
)
→
(
¬
(
X48
→
(
X44
↔
(
(
¬
X31
)
→
X46
)
)
)
)
)
)
)
↔
(
(
X48
↔
(
X47
→
(
¬
X43
)
→
X46
)
)
↔
(
¬
(
X48
→
X46
→
(
X43
↔
X44
)
)
)
)
)
↔
(
(
(
(
¬
(
X23
→
(
¬
X12
)
→
X49
)
)
↔
(
(
X45
↔
(
(
¬
X35
)
→
(
(
¬
X23
)
↔
X49
)
)
)
↔
(
¬
(
X49
→
(
¬
X35
)
→
X14
→
X45
)
)
)
)
→
(
¬
(
X48
↔
(
X35
↔
(
X23
↔
X45
)
)
)
)
)
↔
(
(
¬
(
(
¬
X47
)
→
(
¬
X35
)
→
(
¬
X9
)
→
X49
)
)
→
(
(
¬
(
X47
↔
(
X45
→
(
X35
↔
X49
)
)
)
)
↔
(
(
¬
X35
)
↔
(
(
¬
X20
)
→
X45
)
)
)
)
)
)
→
(
(
(
(
X46
↔
(
(
¬
X37
)
↔
(
(
¬
X27
)
→
X45
)
)
)
→
(
X22
→
(
X19
↔
X37
)
)
→
(
¬
(
X46
↔
(
X45
↔
(
X19
→
X37
)
)
)
)
)
↔
(
(
(
¬
(
(
X15
↔
(
(
¬
X4
)
↔
X41
)
)
→
(
¬
(
X27
↔
(
X19
↔
(
X15
↔
X4
)
)
)
)
)
)
→
(
¬
X19
)
→
X41
)
→
(
X22
↔
X41
)
)
)
↔
(
(
(
(
(
¬
X7
)
↔
X41
)
↔
(
(
¬
(
(
¬
X41
)
→
(
(
¬
X34
)
↔
(
(
¬
X7
)
→
X45
)
)
)
)
↔
(
(
¬
X34
)
↔
(
(
¬
X27
)
→
(
(
¬
X7
)
↔
X45
)
)
)
)
)
→
(
¬
(
X45
↔
(
X36
↔
(
(
¬
X27
)
→
X41
)
)
)
)
→
(
¬
(
X41
→
X36
→
(
X27
↔
X44
)
)
)
)
→
(
¬
(
X45
↔
(
(
¬
X38
)
↔
(
(
¬
X34
)
→
X44
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
¬
(
X49
↔
(
X33
→
X50
)
)
)
↔
(
(
¬
(
X50
→
(
X47
↔
(
(
¬
X17
)
→
X21
)
)
)
)
↔
(
¬
(
X49
→
X33
→
(
(
¬
X17
)
↔
X28
)
)
)
)
)
↔
(
(
¬
(
(
¬
X49
)
→
(
(
¬
X18
)
↔
X50
)
)
)
↔
(
(
¬
(
X50
↔
(
X28
↔
X49
)
)
)
↔
(
(
(
¬
(
(
¬
X18
)
↔
(
X3
↔
X28
)
)
)
↔
(
(
¬
(
X18
↔
X49
)
)
↔
(
(
X49
↔
(
(
¬
X18
)
→
X50
)
)
↔
(
(
¬
(
(
¬
X49
)
→
(
¬
X28
)
→
X3
→
X50
)
)
↔
(
X50
→
X49
→
(
¬
X18
)
→
X28
)
)
)
)
)
↔
(
¬
(
X50
↔
(
X49
↔
(
X18
↔
False
)
)
)
)
)
)
)
)
→
(
(
X49
→
(
¬
X47
)
→
(
(
¬
X40
)
↔
X48
)
)
↔
(
(
¬
(
X49
↔
(
(
¬
X47
)
→
X50
)
)
)
→
(
¬
(
(
X49
↔
(
(
¬
X47
)
→
(
X25
↔
X50
)
)
)
→
(
¬
(
X50
↔
(
X25
→
(
¬
X2
)
→
X49
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
X30
)
→
(
X24
↔
(
X10
↔
X42
)
)
)
↔
(
¬
(
X30
↔
(
X24
↔
(
X11
↔
X10
)
)
)
)
)
↔
(
¬
(
(
¬
(
X30
↔
(
X29
↔
(
(
¬
X16
)
→
X24
)
)
)
)
→
(
¬
(
X42
→
(
X29
↔
(
X24
↔
(
X16
↔
False
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X32
→
(
¬
X29
)
→
X47
)
)
→
(
X32
↔
(
X30
→
X29
→
False
)
)
→
(
¬
(
X30
↔
(
(
¬
X29
)
↔
(
X24
↔
X47
)
)
)
)
)
→
X48
→
X29
→
X47
)
)
→
(
¬
(
(
¬
X42
)
→
(
¬
X13
)
→
(
¬
X6
)
→
X50
)
)
→
(
(
¬
(
(
X48
↔
(
(
¬
X5
)
→
(
(
¬
X1
)
↔
X8
)
)
)
→
(
¬
(
X47
→
X42
→
X5
)
)
)
)
↔
(
(
(
¬
(
(
X6
↔
(
X5
→
X13
)
)
→
(
(
¬
X47
)
↔
(
X13
→
X6
→
X48
)
)
)
)
↔
(
X48
↔
X6
)
)
↔
(
(
¬
X47
)
↔
(
(
¬
X13
)
↔
(
X0
↔
X48
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
(
X48
↔
(
X17
↔
(
(
¬
X3
)
↔
X45
)
)
)
→
(
¬
(
(
¬
(
X47
→
X45
→
(
¬
X17
)
→
X25
)
)
→
(
¬
(
X45
→
(
X17
↔
X48
)
)
)
)
)
)
→
(
¬
(
(
(
(
X45
↔
(
(
¬
X12
)
→
(
(
¬
X2
)
↔
X25
)
)
)
↔
(
X47
↔
(
X12
→
(
(
¬
X2
)
↔
X20
)
)
)
)
→
(
X20
↔
X45
)
)
→
X45
→
(
X20
↔
(
X8
↔
X25
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X45
→
X22
→
(
¬
X17
)
→
X25
)
)
→
(
¬
(
X41
↔
(
(
¬
X22
)
↔
(
X17
↔
X45
)
)
)
)
)
)
→
X17
)
)
→
(
¬
(
(
(
(
¬
(
(
¬
X33
)
→
(
(
¬
X28
)
↔
(
(
¬
X20
)
→
X49
)
)
)
)
↔
(
X49
→
(
(
¬
X28
)
↔
(
(
¬
X17
)
→
X33
)
)
)
)
↔
(
¬
(
X49
→
X33
→
(
X28
↔
X47
)
)
)
)
→
(
(
¬
X21
)
↔
(
X20
↔
(
(
¬
X13
)
↔
X28
)
)
)
→
(
¬
(
(
¬
X28
)
→
(
¬
X21
)
→
(
X20
↔
X49
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
(
X47
↔
(
(
¬
X34
)
→
X11
→
X38
)
)
)
→
(
¬
(
(
¬
X38
)
↔
(
X5
→
X47
)
)
)
)
)
→
(
¬
(
(
¬
X34
)
↔
(
(
¬
X5
)
→
X38
)
)
)
)
→
(
(
X46
→
X42
→
(
¬
X11
)
→
X34
)
↔
(
¬
(
(
¬
(
X42
→
X11
)
)
→
(
¬
(
X46
↔
(
X11
↔
(
(
¬
X4
)
↔
X38
)
)
)
)
)
)
)
→
(
(
(
¬
X42
)
→
(
¬
X9
)
→
(
(
¬
X4
)
↔
X48
)
)
↔
(
¬
(
X42
↔
(
X38
↔
(
X4
↔
False
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
X48
→
X7
→
X46
)
→
(
¬
(
X40
↔
(
(
¬
X29
)
↔
(
(
¬
X7
)
↔
X46
)
)
)
)
)
→
(
(
¬
X46
)
↔
(
(
¬
X40
)
↔
(
X0
→
X48
)
)
)
)
)
↔
(
(
¬
(
X46
↔
(
X29
→
(
¬
X27
)
→
X48
)
)
)
↔
(
(
¬
(
X48
↔
(
(
¬
X46
)
→
X27
→
X49
)
)
)
↔
(
¬
(
X49
→
X48
→
X47
)
)
)
)
)
→
(
¬
(
(
¬
X40
)
→
(
(
¬
X35
)
↔
(
X27
↔
X48
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
(
¬
(
X36
↔
(
(
¬
X32
)
→
X18
→
X48
)
)
)
→
(
(
X26
→
(
¬
X18
)
→
(
X14
↔
X36
)
)
↔
(
¬
(
X36
→
X26
→
(
(
¬
X18
)
↔
X32
)
)
)
)
)
)
↔
(
(
¬
(
X26
↔
(
X24
↔
(
X18
↔
X30
)
)
)
)
↔
(
¬
(
X36
→
X30
→
X18
→
X45
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X44
↔
(
X43
→
X15
)
)
)
↔
(
(
X49
→
(
(
¬
X43
)
↔
(
X36
→
X44
)
)
)
→
X49
→
(
X44
↔
(
X15
→
X43
)
)
)
)
↔
(
¬
(
(
(
(
¬
X36
)
↔
X45
)
→
(
(
X45
↔
(
X43
→
X36
→
(
X10
↔
False
)
)
)
↔
(
(
¬
X44
)
↔
(
X43
↔
(
X10
→
X45
)
)
)
)
)
→
(
¬
(
X45
↔
(
(
¬
X36
)
→
X43
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X43
↔
(
(
¬
X24
)
↔
(
X19
→
X31
)
)
)
)
↔
(
¬
(
X31
→
(
X24
↔
(
X16
→
False
)
)
)
)
)
→
(
(
X45
→
(
X44
↔
(
X24
↔
X31
)
)
)
↔
(
¬
(
(
¬
X44
)
→
(
(
¬
X31
)
↔
(
X16
↔
X45
)
)
)
)
)
)
→
(
X45
↔
(
X31
→
X44
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
(
¬
(
(
¬
(
(
¬
X49
)
→
(
¬
X44
)
→
(
X1
↔
X50
)
)
)
→
(
¬
(
(
¬
X49
)
→
(
(
¬
X44
)
↔
(
(
¬
X37
)
→
X50
)
)
)
)
)
)
→
(
¬
(
X50
↔
(
X49
↔
(
X44
↔
X45
)
)
)
)
)
)
→
X50
→
X37
→
(
X6
↔
X1
)
)
→
(
¬
(
(
¬
(
X39
→
(
(
¬
X37
)
↔
(
(
¬
X23
)
→
X45
)
)
)
)
→
X45
→
X39
→
X37
→
(
X1
↔
False
)
)
)
)
)
→
(
(
¬
X44
)
↔
(
(
¬
X37
)
→
X50
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
b83c16...
and proposition id is
bfb02a...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMSVUMKgBaMqyxTg2cVx9rgp5QixZ6yMLca
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
¬
(
(
(
(
¬
(
(
(
¬
(
X44
↔
(
X38
→
(
(
¬
X33
)
↔
X39
)
)
)
)
→
(
X44
↔
(
(
¬
X39
)
→
X49
)
)
→
(
(
¬
X44
)
↔
(
(
¬
X39
)
↔
(
X38
→
X49
)
)
)
→
(
(
¬
X44
)
↔
(
X38
↔
(
X9
→
X49
)
)
)
)
→
(
(
X43
→
(
X39
↔
(
X33
↔
False
)
)
)
↔
(
X47
→
(
X43
↔
(
X39
↔
X46
)
)
)
)
)
)
↔
(
(
(
(
¬
X44
)
→
(
¬
X36
)
→
(
¬
X19
)
→
X45
)
↔
(
¬
(
(
¬
X44
)
→
X36
→
(
X19
↔
X49
)
)
)
)
→
(
¬
(
X45
↔
(
X36
→
X14
→
X43
)
)
)
)
)
↔
(
(
(
¬
(
X48
→
X43
→
X45
)
)
↔
(
(
¬
(
(
X49
→
(
X45
↔
X43
)
)
→
(
¬
(
X48
→
(
X45
↔
(
X31
↔
X49
)
)
)
)
→
(
(
¬
(
X45
↔
(
(
¬
X43
)
↔
(
X31
→
X48
)
)
)
)
↔
(
X45
→
X31
→
X6
→
X43
)
)
)
)
↔
(
¬
(
X45
↔
(
X43
↔
X48
)
)
)
)
)
↔
(
(
¬
(
X49
→
X48
→
X26
→
False
)
)
↔
(
¬
(
(
¬
(
(
(
(
¬
(
X42
→
(
X41
↔
(
X16
↔
X26
)
)
)
)
→
(
¬
(
(
¬
X26
)
↔
(
X24
→
(
¬
X16
)
→
X41
)
)
)
)
↔
(
¬
(
X42
↔
(
X26
→
(
X24
↔
(
X12
→
False
)
)
)
)
)
)
→
(
¬
(
(
X45
→
(
X41
↔
(
(
¬
X16
)
→
X48
)
)
)
→
(
¬
(
X48
→
(
¬
X12
)
→
X16
)
)
)
)
)
)
→
(
(
¬
(
X45
↔
(
(
¬
X41
)
→
(
¬
X21
)
→
X49
)
)
)
↔
(
¬
(
(
X45
→
(
X26
↔
False
)
)
→
(
X49
↔
(
X45
↔
(
X41
↔
(
X4
↔
False
)
)
)
)
)
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
X44
→
(
¬
X10
)
→
(
(
¬
X5
)
↔
X37
)
)
)
↔
(
(
¬
X37
)
↔
(
X25
→
(
¬
X10
)
→
X44
)
)
)
↔
(
(
(
¬
(
(
¬
X37
)
→
X44
)
)
↔
(
X25
→
X7
→
X3
→
X22
)
)
↔
(
X22
→
(
(
¬
X3
)
↔
X7
)
)
)
)
↔
(
(
(
X44
→
(
¬
X10
)
→
X28
)
↔
(
(
¬
X40
)
↔
(
X23
→
(
(
¬
X17
)
↔
X44
)
)
)
)
→
(
(
¬
(
X37
↔
(
X28
↔
(
X10
↔
X17
)
)
)
)
↔
(
(
¬
(
(
¬
X10
)
↔
(
(
¬
X1
)
→
X28
)
)
)
→
(
¬
(
(
¬
X11
)
→
(
X1
↔
X17
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X45
↔
(
X27
→
X28
)
)
)
↔
(
(
¬
(
X47
↔
(
X27
→
X45
)
)
)
↔
(
¬
(
X37
→
(
¬
X28
)
→
X27
→
X45
)
)
)
)
→
(
(
¬
(
X45
↔
(
X37
↔
(
X29
↔
False
)
)
)
)
↔
(
(
(
¬
(
X47
↔
(
X29
↔
X8
)
)
)
↔
(
¬
(
X45
↔
(
X8
↔
X37
)
)
)
)
↔
(
X37
↔
(
X29
→
(
X8
↔
False
)
)
)
)
)
)
↔
(
X45
→
(
X28
↔
X29
)
)
)
→
(
(
¬
(
(
(
X37
↔
(
(
¬
X35
)
↔
(
(
¬
X34
)
↔
X47
)
)
)
↔
(
¬
(
(
¬
(
X37
↔
(
(
¬
X34
)
→
X45
)
)
)
→
(
(
(
¬
X37
)
→
X34
→
X18
→
X47
)
↔
(
(
¬
(
X34
→
(
(
¬
X13
)
↔
X18
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X34
)
→
(
X18
↔
X45
)
)
)
)
)
)
)
)
)
→
(
X47
→
(
X37
↔
(
X20
→
X34
)
)
)
→
X37
→
(
¬
X20
)
→
(
¬
X15
)
→
X34
)
)
↔
(
(
(
(
X45
↔
(
X44
→
X37
)
)
→
(
¬
(
X2
→
X47
)
)
→
(
¬
X44
)
→
(
X37
↔
(
(
¬
X0
)
→
X45
)
)
)
↔
(
¬
(
(
¬
X37
)
→
X2
→
X47
)
)
)
↔
(
(
¬
(
X44
→
(
(
¬
X30
)
↔
(
(
¬
X0
)
↔
X32
)
)
)
)
↔
(
(
¬
X44
)
→
(
(
¬
X32
)
↔
(
(
¬
X0
)
↔
X45
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
(
(
¬
X42
)
↔
(
X14
↔
(
X13
→
X45
)
)
)
↔
(
X40
↔
X14
)
)
→
(
X45
↔
(
(
¬
X40
)
→
X49
)
)
)
↔
(
(
(
(
¬
(
X45
↔
(
X26
→
(
X22
↔
False
)
)
)
)
→
(
¬
(
(
(
¬
(
X42
→
(
(
¬
X38
)
↔
(
(
¬
X26
)
→
X45
)
)
)
)
↔
(
¬
(
X38
→
X26
→
(
¬
X22
)
→
X42
)
)
)
→
(
(
¬
X38
)
↔
(
(
¬
X26
)
→
X45
)
)
)
)
)
↔
(
¬
(
X47
↔
(
X45
→
(
X37
↔
(
X22
→
False
)
)
)
)
)
)
↔
(
(
X47
↔
(
(
¬
X42
)
↔
(
X4
↔
X45
)
)
)
→
(
¬
(
X47
→
(
X40
↔
X42
)
)
)
→
(
¬
(
X45
→
(
¬
X42
)
→
(
¬
X31
)
→
X47
)
)
)
)
)
→
(
(
(
¬
(
X38
→
(
(
¬
X17
)
↔
(
(
¬
X15
)
→
X28
)
)
)
)
→
(
(
¬
X38
)
↔
(
(
¬
X27
)
→
X45
)
)
)
↔
(
¬
(
X28
→
(
¬
X15
)
→
X47
)
)
)
)
↔
(
(
(
¬
(
(
(
X45
→
X43
→
X3
)
↔
(
X43
→
(
(
¬
X20
)
↔
(
(
¬
X18
)
→
X45
)
)
)
)
→
(
¬
(
X45
→
(
X43
↔
(
X21
→
(
X3
↔
False
)
)
)
)
)
)
)
↔
(
(
X42
→
(
¬
X18
)
→
(
X2
↔
X20
)
)
↔
(
¬
(
(
¬
X42
)
↔
(
(
¬
X20
)
→
X43
)
)
)
)
)
→
(
(
(
X45
→
(
X41
↔
(
(
¬
X0
)
↔
X43
)
)
)
↔
(
¬
(
X42
→
(
(
¬
X41
)
↔
(
X0
↔
X43
)
)
)
)
)
↔
(
(
¬
(
X42
↔
(
X40
→
X11
→
X45
)
)
)
↔
(
¬
(
X43
→
(
X11
↔
X40
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
¬
X24
)
→
X44
)
↔
(
(
¬
X39
)
↔
(
X24
→
(
X12
↔
X47
)
)
)
)
→
(
(
¬
(
(
¬
(
X48
→
(
(
¬
X44
)
↔
(
(
¬
X30
)
↔
X49
)
)
)
)
→
X48
→
(
X5
↔
X49
)
)
)
↔
(
¬
(
(
¬
(
X48
→
X47
→
(
X39
↔
X49
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X44
)
↔
(
X39
↔
X49
)
)
)
)
)
)
)
)
)
↔
(
¬
(
X47
→
(
(
¬
X44
)
↔
(
(
¬
X36
)
→
X49
)
)
)
)
)
↔
(
(
(
X35
↔
(
(
¬
X25
)
↔
(
X8
→
X48
)
)
)
↔
(
(
(
(
¬
(
X33
↔
(
(
¬
X25
)
↔
(
(
¬
X16
)
→
X46
)
)
)
)
→
(
¬
(
X33
↔
(
X16
↔
(
(
¬
X8
)
→
X25
)
)
)
)
)
→
(
¬
(
(
X33
↔
(
(
¬
X1
)
→
X49
)
)
→
(
¬
(
(
X33
→
(
X1
↔
X16
)
)
→
(
¬
(
X49
→
(
X46
↔
(
X33
→
False
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
X49
↔
(
X46
↔
(
(
¬
X6
)
↔
X33
)
)
)
)
↔
(
(
X49
↔
(
X10
↔
X33
)
)
↔
(
¬
(
(
¬
X46
)
→
(
X19
↔
(
X10
→
X49
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X48
→
(
¬
X34
)
→
X29
→
X47
)
)
↔
(
(
¬
X7
)
→
X29
)
)
→
(
(
¬
(
(
¬
X47
)
↔
(
(
¬
X46
)
→
X48
)
)
)
↔
(
¬
(
X48
↔
(
(
¬
X29
)
→
X23
→
X46
)
)
)
)
)
→
(
¬
(
(
¬
(
X49
→
X46
→
X45
→
X47
)
)
→
(
(
¬
(
X49
↔
(
X46
→
X48
)
)
)
↔
(
(
¬
(
X46
↔
(
(
¬
X9
)
→
X48
)
)
)
↔
(
¬
(
(
¬
(
X47
→
X46
)
)
→
(
(
X49
↔
(
X47
↔
(
X32
→
X46
)
)
)
↔
(
¬
(
X46
→
X32
→
(
X9
↔
X49
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
f6fc2b...
and proposition id is
b805f3...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMTpkqvjTn2zNAHPveX2BSzsczKxAT3fWSA
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
¬
(
(
(
¬
(
X42
↔
(
X41
↔
(
X12
→
X7
→
False
)
)
)
)
↔
(
X43
→
(
X41
↔
X12
)
)
)
→
(
(
X42
→
X41
→
(
(
¬
X1
)
↔
X48
)
)
↔
(
¬
(
(
(
X48
→
X33
→
X24
→
X42
)
↔
(
¬
(
X42
↔
(
X41
↔
False
)
)
)
)
→
(
(
¬
(
X43
→
X42
→
X41
→
(
X24
↔
False
)
)
)
↔
(
¬
(
X41
↔
(
X16
→
(
(
¬
X14
)
↔
X24
)
)
)
)
)
)
)
)
)
)
→
(
(
(
X38
→
X13
→
X8
→
False
)
↔
(
¬
(
(
(
¬
(
X42
→
(
(
¬
X6
)
↔
X46
)
)
)
→
(
¬
(
(
¬
X43
)
→
(
X38
↔
(
X6
→
X46
)
)
)
)
)
→
(
¬
(
X43
→
(
X6
↔
X38
)
)
)
)
)
)
↔
(
(
¬
(
X46
↔
(
X43
↔
(
X34
↔
X41
)
)
)
)
↔
(
(
¬
(
X42
→
(
¬
X41
)
→
(
X34
↔
X43
)
)
)
→
X43
→
X42
→
(
(
¬
X29
)
↔
X41
)
)
)
)
)
↔
(
¬
(
(
(
(
X48
→
(
X22
↔
(
X9
↔
X36
)
)
)
↔
(
(
X42
→
X28
→
(
¬
X9
)
→
X22
)
→
(
(
¬
(
X28
↔
(
(
¬
X26
)
↔
(
X22
↔
X36
)
)
)
)
↔
(
(
X36
→
X9
→
(
¬
X0
)
→
X20
)
↔
(
X28
↔
(
(
¬
X9
)
→
X0
→
X26
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X48
↔
(
X22
↔
X42
)
)
)
→
(
¬
(
(
¬
(
(
¬
X22
)
↔
(
(
¬
X19
)
→
X32
)
)
)
→
(
¬
(
X48
↔
(
(
¬
X19
)
↔
X42
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
X47
→
(
¬
X10
)
→
(
¬
X2
)
→
X26
)
→
(
(
¬
(
(
(
(
X47
↔
(
X26
→
(
(
¬
X10
)
↔
X43
)
)
)
→
(
¬
(
(
¬
X43
)
→
(
X26
↔
(
X23
→
X47
)
)
)
)
)
→
(
¬
X23
)
→
(
X10
↔
X47
)
)
→
(
¬
(
X47
→
X26
→
(
X10
↔
False
)
)
)
)
)
↔
(
¬
(
X47
↔
(
X26
→
X43
)
)
)
)
)
)
→
(
¬
(
X36
↔
(
X21
→
X10
→
X32
)
)
)
→
(
(
(
X36
→
X23
→
X21
→
X3
→
False
)
↔
(
X32
↔
(
(
¬
X3
)
↔
X23
)
)
)
↔
X43
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
(
¬
(
X44
→
X17
→
X5
)
)
→
(
¬
(
X47
→
(
¬
X17
)
→
(
X5
↔
X39
)
)
)
)
↔
(
(
¬
X44
)
→
(
X39
↔
X47
)
)
)
↔
(
(
X18
→
False
)
↔
(
¬
(
(
¬
X44
)
→
(
X39
↔
(
X4
→
X47
)
)
)
)
)
)
→
(
(
¬
(
X48
↔
(
(
¬
X18
)
↔
X47
)
)
)
↔
(
¬
(
X48
→
(
X17
↔
(
(
¬
X15
)
↔
X18
)
)
)
)
)
)
)
→
(
(
¬
(
X35
→
X18
→
X5
→
X48
)
)
↔
(
(
¬
(
X18
→
X40
)
)
→
(
(
¬
(
X37
→
(
X35
↔
(
(
¬
X5
)
→
X40
)
)
)
)
↔
(
¬
(
X40
→
(
¬
X5
)
→
X18
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
¬
X40
)
→
X27
→
X45
)
)
↔
(
X40
→
(
(
¬
X27
)
↔
(
(
¬
X11
)
↔
X48
)
)
)
)
→
(
(
¬
X45
)
↔
(
X40
→
X49
)
)
→
(
X48
↔
(
(
¬
X40
)
→
X49
)
)
→
(
¬
X47
)
→
X46
→
(
¬
X40
)
→
X48
)
)
↔
(
¬
(
(
¬
(
X49
↔
(
(
¬
X31
)
→
(
(
¬
X30
)
↔
X45
)
)
)
)
→
(
(
X48
→
X31
→
(
X30
↔
False
)
)
↔
(
¬
(
X48
↔
(
X45
↔
(
(
¬
X25
)
↔
X31
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
(
(
(
X43
↔
(
X38
→
(
X20
↔
X48
)
)
)
→
(
¬
(
X43
↔
(
X20
→
X44
)
)
)
)
↔
(
¬
(
X44
→
(
(
¬
X43
)
↔
(
X25
→
X48
)
)
)
)
)
→
(
(
¬
X43
)
↔
(
X38
→
X48
)
)
)
)
→
(
¬
(
(
X45
↔
(
X44
→
(
(
¬
X20
)
↔
X25
)
)
)
→
(
X25
↔
(
(
¬
X20
)
↔
(
(
¬
X9
)
↔
X44
)
)
)
)
)
)
→
(
¬
(
(
X43
→
X20
→
X4
→
False
)
→
X38
→
(
¬
X7
)
→
(
(
¬
X4
)
↔
X20
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
(
¬
X44
)
↔
(
X29
→
X24
→
X48
)
)
)
→
(
X29
↔
(
X26
↔
(
X24
→
False
)
)
)
)
↔
(
(
X42
↔
(
X24
↔
X48
)
)
→
(
¬
(
(
(
¬
(
(
¬
X30
)
→
X24
→
(
¬
X6
)
→
X42
)
)
↔
(
¬
(
X42
↔
(
X30
↔
(
X6
→
X44
)
)
)
)
)
→
(
¬
(
X48
→
(
¬
X42
)
→
X6
→
X44
)
)
)
)
)
)
→
(
(
¬
(
X48
↔
(
X45
↔
(
X18
→
False
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
¬
X45
)
↔
(
X27
→
(
(
¬
X18
)
↔
X48
)
)
)
)
→
(
¬
(
X5
↔
X18
)
)
)
)
↔
(
X48
→
X27
→
X5
)
)
↔
(
X27
↔
(
X5
→
(
X1
↔
X18
)
)
)
)
)
→
(
(
¬
(
X44
↔
(
X42
↔
(
X33
↔
X8
)
)
)
)
↔
(
¬
(
(
¬
(
X0
→
X45
)
)
→
(
(
¬
(
(
¬
(
X45
→
(
X33
↔
X27
)
)
)
→
(
¬
(
X45
↔
(
X33
↔
X39
)
)
)
→
(
(
(
(
¬
X33
)
↔
(
(
¬
X27
)
↔
(
(
¬
X0
)
→
X45
)
)
)
↔
(
(
¬
X39
)
→
(
(
¬
X33
)
↔
X45
)
)
)
↔
(
¬
(
X27
↔
(
X0
↔
X39
)
)
)
)
)
)
↔
(
¬
(
X45
→
(
(
¬
X39
)
↔
(
(
¬
X33
)
→
X48
)
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
X42
↔
(
X19
→
(
X2
↔
X44
)
)
)
↔
(
¬
(
(
X42
↔
(
(
¬
X31
)
→
X19
→
X44
)
)
→
(
¬
(
X44
→
X35
)
)
)
)
)
→
(
(
(
(
X32
↔
(
(
¬
X19
)
↔
(
(
¬
X10
)
→
X49
)
)
)
↔
(
X49
↔
(
X42
↔
(
X19
↔
X32
)
)
)
)
↔
(
¬
(
X32
↔
X13
)
)
)
↔
(
(
(
¬
X15
)
→
X19
)
→
(
¬
(
(
¬
X42
)
↔
(
X31
↔
(
(
¬
X15
)
→
X49
)
)
)
)
)
)
)
)
↔
(
(
¬
(
X43
→
X41
→
X16
→
X44
)
)
→
(
(
X49
↔
(
X41
→
(
(
¬
X3
)
↔
X16
)
)
)
↔
(
(
(
X43
↔
(
(
¬
X16
)
→
(
X11
↔
X49
)
)
)
→
(
(
¬
(
X43
→
X42
→
X41
)
)
↔
(
X49
↔
(
X43
↔
(
X11
↔
X42
)
)
)
)
)
→
(
(
¬
X11
)
↔
X43
)
)
)
)
)
→
(
(
(
¬
X37
)
→
(
¬
X36
)
→
X40
)
↔
(
(
X40
→
X36
→
X17
→
X42
)
→
(
(
¬
(
(
¬
X36
)
→
(
¬
X21
)
→
(
(
¬
X17
)
↔
X47
)
)
)
↔
(
(
(
¬
(
X42
→
(
¬
X17
)
→
X21
)
)
↔
(
(
¬
(
X42
↔
(
(
¬
X21
)
→
X36
)
)
)
→
(
X36
↔
(
X21
↔
(
X17
↔
X37
)
)
)
)
)
↔
(
¬
(
(
¬
X36
)
→
(
X17
↔
X37
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
X49
↔
(
X37
→
(
¬
X14
)
→
X40
)
)
↔
(
(
X49
↔
(
X43
↔
(
X37
↔
X40
)
)
)
→
(
¬
X40
)
→
(
¬
X37
)
→
(
X34
↔
X43
)
)
)
→
(
¬
(
X43
↔
(
(
¬
X22
)
↔
X49
)
)
)
→
X23
→
(
¬
X22
)
→
X40
)
→
(
(
(
X49
→
(
X46
↔
X40
)
)
→
(
¬
(
(
¬
X46
)
↔
(
(
¬
X37
)
↔
(
(
¬
X28
)
→
X49
)
)
)
)
)
↔
(
(
(
¬
X36
)
→
X40
)
↔
(
¬
(
X49
→
(
¬
X12
)
→
X46
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
3c31ff...
and proposition id is
caec9f...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMVbaqfnMJ2J3inpn6NwLKPt25HZqonsqQU
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
¬
(
(
(
(
(
X47
↔
(
X11
↔
False
)
)
→
(
¬
(
(
¬
X47
)
→
X37
→
(
X12
↔
X49
)
)
)
)
↔
(
X47
→
X11
)
)
↔
(
(
(
X46
→
X47
)
↔
(
(
¬
X38
)
↔
(
(
¬
X37
)
↔
(
X20
→
X47
)
)
)
)
↔
(
¬
(
(
¬
(
X49
→
(
¬
X46
)
→
(
¬
X38
)
→
X48
)
)
→
(
¬
X48
)
→
(
X46
↔
(
X20
↔
X49
)
)
)
)
)
)
→
(
(
¬
(
X48
→
X36
→
(
X10
↔
(
X2
→
False
)
)
)
)
↔
(
X48
↔
(
X47
→
X36
→
X2
)
)
)
→
(
(
(
¬
(
X49
↔
(
X43
↔
(
(
¬
X33
)
→
X48
)
)
)
)
↔
(
¬
(
(
¬
X48
)
↔
(
X43
↔
(
(
¬
X36
)
→
X49
)
)
)
)
)
↔
(
¬
(
X36
→
X33
)
)
)
)
)
↔
(
(
(
X48
↔
(
X47
→
(
¬
X25
)
→
X45
)
)
↔
(
X47
→
(
(
¬
X45
)
↔
(
X41
↔
X48
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
X27
→
(
¬
X18
)
→
X30
)
→
(
¬
(
(
¬
(
X49
→
X30
→
False
)
)
→
(
¬
(
(
¬
(
X49
→
(
(
¬
X18
)
↔
(
(
¬
X7
)
→
X27
)
)
)
)
→
X45
→
(
(
¬
X27
)
↔
(
X7
→
X30
)
)
)
)
)
)
)
)
→
(
¬
(
X45
↔
(
X30
→
X27
→
False
)
)
)
)
→
(
¬
(
(
¬
X18
)
→
(
(
¬
X7
)
↔
X49
)
)
)
)
→
(
(
(
¬
(
X18
↔
X15
)
)
↔
(
(
(
¬
X1
)
↔
X5
)
→
(
¬
(
X45
→
(
X41
↔
(
X15
↔
(
X1
↔
False
)
)
)
)
)
)
)
↔
(
(
¬
(
X27
↔
(
X22
↔
X47
)
)
)
→
(
(
(
X45
↔
(
X22
→
(
X5
↔
False
)
)
)
↔
(
(
¬
(
(
¬
X41
)
↔
(
X22
↔
(
X5
→
X45
)
)
)
)
↔
(
X47
→
(
X45
↔
(
(
¬
X22
)
↔
X41
)
)
)
)
)
↔
(
¬
(
(
¬
X45
)
→
(
(
¬
X41
)
↔
(
X27
→
X47
)
)
)
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X47
↔
(
(
¬
X29
)
→
(
X8
↔
X34
)
)
)
)
→
(
¬
(
(
¬
X39
)
→
(
¬
X34
)
→
(
(
¬
X29
)
↔
X47
)
)
)
)
↔
(
X34
→
X39
)
)
→
(
(
(
¬
(
X47
→
(
¬
X39
)
→
X46
)
)
↔
(
(
X46
→
X35
→
(
X16
↔
False
)
)
↔
(
(
(
¬
X47
)
→
(
(
¬
X46
)
↔
(
(
¬
X39
)
↔
X48
)
)
)
↔
(
¬
(
(
X47
↔
(
X35
→
X48
)
)
→
(
¬
(
(
¬
(
(
¬
X47
)
→
(
(
¬
X46
)
↔
(
(
¬
X16
)
→
X48
)
)
)
)
→
(
¬
X47
)
→
(
(
¬
X35
)
↔
(
(
¬
X16
)
→
X48
)
)
)
)
)
)
)
)
)
↔
(
(
X47
↔
(
X34
→
X19
)
)
↔
(
(
(
(
¬
(
X34
→
X32
→
(
(
¬
X9
)
↔
X47
)
)
)
↔
(
X47
↔
(
X14
↔
X39
)
)
)
→
(
X39
↔
(
X34
→
X14
→
X23
)
)
)
→
(
(
¬
X39
)
↔
(
(
¬
X23
)
↔
(
(
¬
X19
)
→
X46
)
)
)
)
)
)
)
→
(
(
(
(
X34
↔
(
(
¬
X31
)
↔
(
(
¬
X24
)
↔
X44
)
)
)
→
X44
→
(
(
¬
X31
)
↔
(
X24
→
X46
)
)
)
↔
(
(
(
¬
(
X40
→
X31
→
X44
)
)
→
(
X44
↔
(
X40
↔
(
X31
↔
X26
)
)
)
→
(
¬
(
X48
→
(
(
¬
X40
)
↔
(
(
¬
X31
)
→
X44
)
)
)
)
)
↔
(
(
¬
(
X26
↔
(
X24
↔
X48
)
)
)
↔
(
¬
(
X48
→
(
(
¬
X31
)
↔
(
(
¬
X24
)
→
X42
)
)
)
)
)
)
)
→
(
(
X26
→
(
¬
X21
)
→
X28
)
↔
(
¬
(
X34
→
(
X28
↔
(
X21
→
X31
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
X40
→
X31
→
X17
)
→
(
¬
(
(
¬
(
X40
↔
(
X17
↔
(
(
¬
X13
)
→
X31
)
)
)
)
→
(
(
¬
(
X3
↔
X40
)
)
↔
(
(
¬
X17
)
→
(
¬
X3
)
→
(
(
¬
X0
)
↔
X31
)
)
)
)
)
)
)
↔
(
¬
(
X44
↔
(
X6
→
(
¬
X0
)
→
X17
)
)
)
)
→
(
¬
(
(
¬
X31
)
→
(
X17
↔
(
X4
↔
X44
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
X42
→
X34
→
(
(
¬
X29
)
↔
X41
)
)
↔
(
(
X45
↔
(
(
¬
X42
)
↔
X46
)
)
↔
(
(
¬
X42
)
↔
(
(
¬
X41
)
→
(
¬
X34
)
→
X45
)
)
)
)
→
(
(
(
(
¬
X45
)
→
(
X42
↔
(
(
¬
X19
)
→
X47
)
)
)
→
(
(
¬
(
X47
↔
(
X46
→
X45
→
X26
→
False
)
)
)
↔
X42
)
)
↔
(
(
¬
X46
)
→
(
X45
↔
(
X24
↔
X47
)
)
)
)
)
↔
(
¬
(
(
X45
→
(
X42
↔
(
X37
→
X47
)
)
)
→
(
(
¬
X42
)
↔
(
(
¬
X37
)
↔
(
X14
→
X47
)
)
)
)
)
)
→
(
(
(
(
¬
(
X47
↔
(
(
¬
X40
)
→
(
X8
↔
X41
)
)
)
)
↔
(
X47
↔
(
X41
→
X39
→
X0
)
)
)
↔
(
(
¬
(
X47
↔
(
X36
→
X1
→
False
)
)
)
↔
(
X41
→
X36
→
X40
)
)
)
↔
(
(
(
(
¬
(
X47
→
X41
)
)
↔
(
¬
(
X44
↔
(
X41
→
(
X11
↔
X7
)
)
)
)
)
↔
(
(
(
(
¬
(
X44
↔
(
X40
→
(
X38
↔
X47
)
)
)
)
↔
(
¬
(
(
¬
X45
)
↔
(
X40
→
X38
→
X47
)
)
)
)
↔
(
¬
(
X40
↔
(
X38
→
(
¬
X2
)
→
X45
)
)
)
)
↔
(
(
X47
↔
(
X44
→
X41
)
)
→
(
¬
(
X47
↔
(
X44
→
(
X21
↔
X45
)
)
)
)
)
)
)
↔
(
(
(
¬
X45
)
↔
(
X28
↔
X47
)
)
→
(
¬
(
(
(
(
X45
→
(
(
¬
X44
)
↔
(
X4
→
X47
)
)
)
↔
(
¬
(
X45
↔
(
X17
→
(
X4
↔
X44
)
)
)
)
)
↔
(
¬
(
(
¬
X44
)
→
(
¬
X28
)
→
(
¬
X17
)
→
X47
)
)
)
→
(
X47
↔
(
(
¬
X44
)
↔
(
(
¬
X3
)
→
X45
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
(
X49
↔
(
(
¬
X35
)
→
X43
)
)
)
→
(
(
¬
X33
)
↔
(
X25
↔
(
X15
↔
X49
)
)
)
)
)
↔
(
(
¬
X33
)
→
(
X25
↔
(
X6
↔
X43
)
)
)
)
→
(
(
(
(
X49
→
(
X48
↔
(
(
¬
X15
)
↔
X33
)
)
)
→
(
(
(
¬
(
(
¬
X48
)
↔
(
(
¬
X33
)
→
(
X18
↔
X49
)
)
)
)
↔
(
X49
→
X48
→
(
X33
↔
(
X22
↔
False
)
)
)
)
↔
(
¬
(
X48
→
(
X33
↔
(
X22
→
False
)
)
)
)
)
)
→
(
¬
(
X48
→
X18
→
X49
)
)
)
↔
(
(
¬
X33
)
↔
(
X12
↔
(
(
¬
X9
)
→
X48
)
)
)
)
)
↔
(
(
(
(
¬
(
X27
→
X46
)
)
↔
(
(
(
¬
(
X45
↔
(
X20
→
False
)
)
)
↔
(
(
¬
X46
)
→
(
X45
↔
(
(
¬
X30
)
→
X48
)
)
)
)
↔
(
X45
↔
(
X27
↔
(
(
¬
X20
)
→
X30
)
)
)
)
)
→
X46
→
X45
→
(
(
¬
X5
)
↔
X30
)
)
→
(
(
(
(
(
(
¬
(
(
(
¬
X43
)
→
(
¬
X31
)
→
X46
)
→
(
¬
(
X45
↔
(
X43
↔
False
)
)
)
→
X35
→
(
¬
X31
)
→
(
(
¬
X13
)
↔
X45
)
)
)
→
(
¬
(
X31
↔
(
(
¬
X23
)
↔
(
X10
↔
X35
)
)
)
)
→
(
¬
(
(
¬
X31
)
↔
(
(
¬
X23
)
↔
X46
)
)
)
)
↔
(
X46
→
(
¬
X13
)
→
X31
)
)
↔
(
X48
↔
(
X45
→
(
(
¬
X35
)
↔
X43
)
)
)
)
↔
(
X48
→
X46
→
(
X45
↔
X32
)
)
)
↔
(
(
¬
(
X46
↔
(
(
¬
X42
)
→
(
(
¬
X35
)
↔
X43
)
)
)
)
↔
(
X43
↔
(
(
¬
X42
)
→
X16
→
X46
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
cb5ce6...
and proposition id is
8a07fa...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMJnVxCv3x9wjxxUwfYomMw8Lq9DgNNPnVu
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
¬
(
(
(
¬
X46
)
→
(
(
¬
X38
)
↔
(
(
¬
X19
)
→
X47
)
)
)
→
(
(
¬
(
(
¬
(
X27
→
(
(
¬
X10
)
↔
X47
)
)
)
→
(
¬
(
(
¬
(
(
X47
↔
(
X38
→
X45
)
)
→
(
¬
(
X47
→
(
X38
↔
(
X22
↔
X46
)
)
)
)
→
(
¬
(
X46
→
(
X38
↔
(
X22
↔
X45
)
)
)
)
)
)
→
X47
→
X45
)
)
)
)
↔
(
(
¬
(
(
X36
→
(
(
¬
X30
)
↔
(
X18
↔
X47
)
)
)
→
(
¬
(
X30
↔
(
(
¬
X18
)
→
(
X11
↔
X24
)
)
)
)
→
(
X30
↔
(
X24
↔
(
X18
→
X47
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X47
→
X24
→
(
(
¬
X16
)
↔
X21
)
)
)
→
(
(
¬
(
X46
→
(
¬
X24
)
→
X21
→
X47
)
)
↔
(
¬
(
(
¬
X45
)
↔
(
(
¬
X24
)
→
(
X21
↔
X46
)
)
)
)
)
)
)
↔
(
X24
→
(
¬
X16
)
→
X45
)
)
↔
(
(
(
¬
X45
)
↔
X47
)
→
(
¬
(
(
¬
X46
)
→
(
(
¬
X30
)
↔
(
X12
→
X47
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
X46
→
(
X45
↔
(
X26
↔
X37
)
)
)
↔
(
(
X46
↔
(
(
¬
X25
)
↔
(
X13
↔
X26
)
)
)
↔
(
(
X46
↔
(
X26
→
(
(
¬
X8
)
↔
X13
)
)
)
↔
(
¬
(
(
¬
X45
)
→
X8
→
X46
)
)
)
)
)
→
(
(
(
¬
(
X26
→
(
¬
X23
)
→
X3
→
X47
)
)
↔
(
¬
(
X47
↔
(
X26
↔
(
(
¬
X23
)
↔
X37
)
)
)
)
)
↔
(
(
(
X47
↔
(
(
¬
X26
)
→
(
(
¬
X4
)
↔
X37
)
)
)
↔
(
¬
(
X47
→
(
X37
↔
(
X4
→
X40
)
)
)
)
)
↔
(
¬
(
(
¬
X40
)
→
(
(
¬
X37
)
↔
(
(
¬
X4
)
→
X45
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X45
↔
(
(
¬
X35
)
→
(
(
¬
X32
)
↔
X49
)
)
)
)
↔
(
(
(
X47
→
(
X28
↔
(
X1
→
X35
)
)
)
↔
(
¬
(
(
X47
↔
(
X32
→
X1
→
X28
)
)
→
X32
→
(
¬
X28
)
→
(
(
¬
X17
)
↔
X47
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X47
)
↔
(
(
¬
X35
)
→
(
X17
↔
X49
)
)
)
)
→
(
¬
(
X47
→
(
X35
↔
(
X17
↔
X0
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
X47
↔
X49
)
)
↔
(
(
X47
→
(
X28
↔
(
(
¬
X20
)
↔
X34
)
)
)
↔
(
¬
(
X49
↔
X47
)
)
)
)
↔
(
(
X48
↔
(
X47
→
(
(
¬
X7
)
↔
X49
)
)
)
→
(
¬
(
X47
↔
(
X28
→
(
X7
↔
X49
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
(
(
¬
X43
)
↔
(
(
¬
X35
)
→
(
X15
↔
X44
)
)
)
→
(
(
¬
(
(
¬
(
X43
↔
(
X33
↔
(
X15
↔
X46
)
)
)
)
→
(
X43
↔
(
(
¬
X35
)
→
X46
)
)
)
)
↔
(
X46
→
X35
→
(
X15
↔
False
)
)
)
)
→
(
(
(
X29
↔
X44
)
→
(
¬
X44
)
→
X39
→
(
X6
↔
X46
)
)
↔
(
¬
(
X46
→
X44
→
(
¬
X39
)
→
X43
)
)
)
)
)
↔
(
(
¬
(
X46
↔
X43
)
)
↔
(
(
X46
↔
(
X43
↔
X41
)
)
→
(
X39
↔
(
(
¬
X9
)
→
X43
)
)
)
)
)
→
(
¬
(
(
¬
(
(
X46
↔
(
X31
→
(
X14
↔
X45
)
)
)
→
(
¬
(
X44
↔
(
X14
→
X43
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
X43
)
→
(
X42
↔
X46
)
)
↔
(
¬
(
(
¬
(
X48
↔
(
X46
↔
X42
)
)
)
→
(
¬
X43
)
→
(
¬
X42
)
→
(
(
¬
X14
)
↔
X46
)
)
)
)
↔
(
¬
(
X48
↔
(
X46
→
(
X43
↔
(
X2
→
False
)
)
)
)
)
)
→
(
X46
↔
(
X14
→
(
¬
X5
)
→
X42
)
)
)
)
)
)
→
(
¬
(
X48
→
(
¬
X42
)
→
X45
)
)
)
)
)
)
↔
(
(
(
(
(
(
(
X49
↔
(
X48
→
(
¬
X10
)
→
X17
)
)
↔
(
X48
→
(
X10
↔
X47
)
)
)
↔
(
(
¬
X17
)
↔
(
(
¬
X10
)
→
X48
)
)
)
↔
(
(
(
X47
→
(
¬
X34
)
→
X31
→
X48
)
↔
(
(
¬
(
X49
→
(
X9
↔
X48
)
)
)
→
(
¬
(
X47
→
(
¬
X31
)
→
(
X9
↔
X49
)
)
)
)
)
↔
(
(
¬
(
X31
↔
(
(
¬
X26
)
↔
X48
)
)
)
→
(
¬
X48
)
→
(
(
¬
X29
)
↔
(
X26
↔
X49
)
)
)
)
)
↔
(
(
(
¬
(
X48
↔
(
(
¬
X41
)
↔
(
X39
→
X47
)
)
)
)
↔
(
¬
(
X47
→
X39
→
(
(
¬
X38
)
↔
X41
)
)
)
)
→
(
(
¬
(
X47
↔
(
(
¬
X24
)
→
(
(
¬
X20
)
↔
X41
)
)
)
)
↔
(
X48
↔
(
(
¬
X47
)
→
(
(
¬
X11
)
↔
X49
)
)
)
)
)
)
→
(
(
(
¬
(
(
X32
→
X48
)
→
(
(
¬
(
X48
→
(
¬
X32
)
→
X47
)
)
↔
(
X42
→
(
(
¬
X32
)
↔
(
X14
→
X47
)
)
)
)
)
)
→
(
¬
(
(
X48
→
X47
→
(
X42
↔
False
)
)
→
(
¬
(
(
(
X48
↔
(
X47
→
X5
→
X42
)
)
↔
(
¬
(
X48
→
(
(
¬
X5
)
↔
X49
)
)
)
)
→
(
¬
(
X47
→
X48
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X47
→
(
X22
↔
X2
)
)
)
→
(
¬
(
(
¬
X42
)
↔
(
(
¬
X8
)
↔
(
X2
→
X47
)
)
)
)
)
)
→
(
(
X47
→
(
X14
↔
(
(
¬
X0
)
↔
X42
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X14
)
↔
(
(
¬
X2
)
↔
X47
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
(
¬
(
X36
↔
(
(
¬
X35
)
→
(
X3
↔
X48
)
)
)
)
→
X36
→
(
(
¬
X19
)
↔
(
(
¬
X3
)
↔
X35
)
)
)
↔
(
X19
↔
(
(
¬
X3
)
→
X36
)
)
)
↔
(
¬
(
(
X37
→
(
¬
X35
)
→
(
(
¬
X33
)
↔
X36
)
)
→
(
¬
X37
)
→
X44
)
)
)
↔
(
(
¬
(
(
X48
→
(
(
¬
X28
)
↔
(
(
¬
X19
)
↔
X36
)
)
)
→
(
¬
(
X28
↔
X16
)
)
)
)
↔
(
¬
(
X28
→
(
X16
↔
X19
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
¬
X33
)
→
(
¬
X13
)
→
X36
)
)
→
(
¬
(
X43
→
X15
→
X13
)
)
)
↔
(
¬
(
(
¬
X36
)
↔
(
X33
↔
(
X28
→
X43
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X46
↔
(
(
¬
X13
)
→
X36
)
)
)
→
(
¬
(
X46
→
X36
→
(
(
¬
X12
)
↔
X43
)
)
)
)
→
X43
→
X36
→
X13
)
)
)
)
)
→
(
(
(
(
¬
(
X47
→
X40
→
(
¬
X30
)
→
X44
)
)
→
(
¬
(
X7
→
False
)
)
→
(
¬
(
(
¬
(
(
X47
↔
(
(
¬
X7
)
→
X30
)
)
→
(
X40
↔
(
X30
→
(
X7
↔
X43
)
)
)
)
)
→
(
X43
↔
(
(
¬
X30
)
→
X47
)
)
)
)
)
↔
(
(
¬
(
X40
→
X30
→
X45
)
)
→
(
(
¬
(
(
(
¬
(
X45
→
(
¬
X40
)
→
(
(
¬
X25
)
↔
X48
)
)
)
↔
(
X48
↔
(
X25
↔
(
(
¬
X4
)
↔
X40
)
)
)
)
→
(
¬
X48
)
)
)
↔
(
(
(
(
¬
X44
)
→
(
(
¬
X40
)
↔
(
(
¬
X23
)
→
X48
)
)
)
→
(
¬
(
(
X44
↔
(
X30
→
(
¬
X6
)
→
X40
)
)
→
(
(
¬
(
X45
→
(
¬
X40
)
→
X30
→
X48
)
)
↔
(
¬
(
(
¬
X45
)
→
(
X44
↔
(
X40
↔
X48
)
)
)
)
)
)
)
)
↔
(
(
¬
X44
)
↔
(
(
¬
X30
)
→
X48
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X30
)
→
(
X18
↔
X43
)
)
)
→
(
¬
(
(
¬
(
X30
↔
(
(
¬
X27
)
→
X45
)
)
)
→
(
¬
(
(
X45
→
(
(
¬
X27
)
↔
(
(
¬
X1
)
→
X30
)
)
)
→
(
¬
(
(
¬
(
X45
→
(
¬
X30
)
→
(
X21
↔
X43
)
)
)
→
X43
→
(
(
¬
X21
)
↔
(
(
¬
X1
)
→
X45
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
4dd3bd...
and proposition id is
62d4b5...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMRk8x8MsioZUcpVmb14ySfwpacjuHP2xwf
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
¬
(
(
(
(
(
(
¬
X44
)
↔
(
X18
→
X46
)
)
↔
(
(
X44
→
(
¬
X22
)
→
X45
)
↔
(
¬
(
(
¬
X46
)
→
(
¬
X45
)
→
(
(
¬
X44
)
↔
X49
)
)
)
)
)
↔
(
¬
(
(
¬
X45
)
↔
(
(
¬
X22
)
→
(
¬
X17
)
→
X49
)
)
)
)
→
(
X46
↔
(
X27
↔
(
(
¬
X15
)
↔
X44
)
)
)
→
(
¬
(
(
¬
(
X46
↔
(
(
¬
X28
)
↔
(
(
¬
X15
)
↔
X49
)
)
)
)
→
(
¬
(
X46
→
(
¬
X28
)
→
X49
)
)
→
(
(
¬
X46
)
→
(
(
¬
X44
)
↔
(
X28
→
X49
)
)
)
→
X46
→
(
(
¬
X44
)
↔
(
X0
↔
X49
)
)
)
)
)
→
(
(
(
¬
(
(
(
¬
(
(
(
X49
↔
(
(
¬
X20
)
↔
X29
)
)
↔
(
X49
→
(
¬
X40
)
→
(
(
¬
X29
)
↔
X47
)
)
)
→
(
¬
(
X47
→
(
(
¬
X40
)
↔
X49
)
)
)
)
)
↔
(
¬
(
X20
→
X49
)
)
)
→
(
(
¬
(
(
X49
↔
(
X46
↔
(
(
¬
X40
)
↔
X47
)
)
)
→
(
(
(
X47
↔
(
X25
↔
(
X11
↔
X23
)
)
)
↔
(
¬
(
X23
↔
(
X11
→
X46
)
)
)
)
↔
(
¬
(
X46
→
X23
→
(
(
¬
X11
)
↔
X49
)
)
)
)
)
)
↔
(
¬
(
X40
→
X25
→
False
)
)
)
)
)
↔
(
¬
(
(
(
X49
↔
(
(
¬
X43
)
→
(
¬
X34
)
→
X47
)
)
↔
(
X49
↔
(
(
¬
X46
)
↔
(
X43
↔
X47
)
)
)
)
→
(
¬
(
(
¬
X47
)
→
X49
)
)
)
)
)
↔
(
(
¬
(
X45
↔
(
(
¬
X43
)
↔
(
X36
↔
X49
)
)
)
)
↔
(
¬
(
(
¬
(
X45
↔
(
(
¬
X41
)
↔
(
(
¬
X12
)
→
X46
)
)
)
)
→
(
(
¬
(
(
¬
(
X49
↔
(
(
¬
X45
)
↔
(
(
¬
X33
)
↔
X46
)
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X45
)
↔
X49
)
)
)
→
(
(
X46
→
X43
→
X41
)
↔
(
¬
(
X49
→
X46
→
(
X41
↔
X45
)
)
)
)
)
)
↔
(
X46
→
(
X33
↔
X41
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
X42
→
X38
→
X8
)
↔
(
¬
(
(
¬
X8
)
→
(
X3
↔
X37
)
)
)
)
↔
(
(
(
¬
X35
)
→
(
(
¬
X10
)
↔
X42
)
)
↔
(
(
X38
↔
(
(
¬
X35
)
→
X37
)
)
↔
(
(
(
(
¬
(
(
¬
X38
)
→
X42
)
)
→
(
¬
(
X35
→
(
X10
↔
X37
)
)
)
)
↔
(
¬
(
X37
→
(
X35
↔
X10
)
)
)
)
↔
(
(
¬
X37
)
→
X10
→
X38
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X43
)
→
(
X42
↔
(
(
¬
X21
)
→
X48
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
¬
(
(
¬
X39
)
→
X37
→
X48
)
)
↔
(
(
(
¬
X42
)
→
X39
→
X43
)
↔
(
(
¬
X39
)
→
(
(
¬
X37
)
↔
(
(
¬
X16
)
↔
X43
)
)
)
)
)
→
(
¬
(
X43
→
(
X16
↔
X39
)
)
)
)
)
→
(
(
¬
X42
)
↔
(
(
¬
X39
)
↔
(
(
¬
X38
)
↔
X48
)
)
)
)
)
→
(
(
¬
X39
)
↔
(
(
¬
X38
)
→
X48
)
)
)
)
↔
(
(
X43
→
(
(
¬
X42
)
↔
(
(
¬
X19
)
→
X48
)
)
)
→
X48
→
X42
→
X24
)
)
→
(
¬
(
(
¬
(
(
¬
(
X43
→
(
¬
X26
)
→
(
¬
X24
)
→
X32
)
)
→
(
X32
↔
(
X26
↔
(
(
¬
X1
)
→
X43
)
)
)
)
)
→
X43
→
(
¬
X26
)
→
(
(
¬
X24
)
↔
X42
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X46
↔
(
X31
→
(
¬
X7
)
→
X42
)
)
)
↔
(
(
(
¬
(
(
X45
→
X13
)
→
(
¬
(
X45
↔
(
X42
→
(
(
¬
X9
)
↔
X38
)
)
)
)
)
)
→
(
X45
↔
(
X38
→
(
¬
X4
)
→
X42
)
)
→
(
¬
(
X45
↔
(
X42
→
(
(
¬
X30
)
↔
X38
)
)
)
)
)
↔
(
(
X42
→
(
(
¬
X31
)
↔
(
X14
→
X45
)
)
)
↔
(
¬
(
X46
→
(
X45
↔
(
(
¬
X30
)
→
X42
)
)
)
)
)
)
)
→
(
¬
(
(
X48
→
(
X31
↔
(
X6
↔
X42
)
)
)
→
(
¬
(
(
¬
(
X38
↔
(
(
¬
X5
)
↔
X6
)
)
)
→
(
(
X31
↔
(
X6
→
X2
)
)
↔
(
¬
(
X38
→
(
(
¬
X5
)
↔
X31
)
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X46
↔
(
X45
↔
(
(
¬
X38
)
↔
X49
)
)
)
)
↔
(
(
X38
→
X49
)
↔
(
¬
(
X46
→
X36
→
(
X14
↔
X49
)
)
)
)
)
↔
(
(
(
X45
→
X30
→
(
(
¬
X16
)
↔
X24
)
)
↔
(
(
¬
X30
)
↔
(
(
¬
X16
)
→
X49
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X16
)
→
(
¬
X11
)
→
X38
)
)
)
→
(
¬
(
X49
→
X45
→
X16
)
)
)
)
→
(
(
(
X36
↔
(
(
¬
X29
)
→
X0
→
X46
)
)
→
(
(
¬
(
X36
→
(
X22
↔
X23
)
)
)
↔
(
(
¬
(
X46
↔
(
(
¬
X23
)
→
X22
→
X36
)
)
)
→
(
¬
X29
)
→
(
¬
X23
)
→
X46
)
)
)
↔
(
(
(
(
¬
(
(
¬
(
X38
↔
X17
)
)
→
(
(
X46
→
(
X38
↔
(
X22
↔
False
)
)
)
↔
(
¬
(
(
¬
(
X48
↔
(
X38
↔
(
(
¬
X17
)
→
X22
)
)
)
)
→
(
¬
(
X46
↔
(
X17
→
False
)
)
)
)
)
)
)
)
→
(
¬
(
X48
→
X46
→
X29
→
X38
)
)
)
↔
(
¬
(
X29
↔
(
X17
→
X38
)
)
)
)
→
(
(
¬
(
(
¬
(
X46
→
X34
→
X4
)
)
→
(
¬
(
(
¬
X34
)
↔
(
(
¬
X26
)
↔
(
(
¬
X5
)
↔
X46
)
)
)
)
→
(
¬
(
X46
↔
(
X34
↔
(
X26
↔
(
X4
↔
False
)
)
)
)
)
)
)
↔
(
(
(
X46
↔
(
(
¬
X19
)
↔
(
X5
↔
X48
)
)
)
↔
(
X29
→
(
(
¬
X8
)
↔
X19
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X29
)
↔
(
X4
↔
X48
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
(
(
(
¬
(
X41
↔
X28
)
)
→
(
¬
(
X46
↔
(
X28
→
(
(
¬
X13
)
↔
X37
)
)
)
)
)
→
(
¬
(
X13
→
X37
)
)
)
↔
(
¬
(
(
¬
X46
)
↔
(
(
¬
X41
)
→
(
(
¬
X28
)
↔
X48
)
)
)
)
)
↔
(
¬
(
(
¬
X46
)
↔
(
(
¬
X41
)
→
(
¬
X18
)
→
X48
)
)
)
)
→
(
¬
(
(
(
(
(
(
¬
X46
)
↔
(
X31
→
X10
→
X48
)
)
→
(
¬
(
X48
→
X18
)
)
)
↔
(
(
¬
X46
)
→
X31
→
X48
)
)
↔
(
¬
(
(
¬
X43
)
→
(
X31
↔
X46
)
)
)
)
→
(
¬
(
X48
↔
(
(
¬
X43
)
→
X46
)
)
)
)
)
)
→
(
(
¬
(
X48
↔
(
X15
↔
X37
)
)
)
↔
(
X48
→
(
(
¬
X33
)
↔
(
X20
↔
X46
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
X37
→
X35
→
X1
→
X40
)
→
(
(
(
¬
(
X47
→
(
X40
↔
(
(
¬
X35
)
↔
X37
)
)
)
)
↔
(
X35
↔
(
(
¬
X25
)
↔
X47
)
)
)
↔
(
(
¬
X35
)
↔
X40
)
)
→
(
X47
→
X31
→
(
X27
↔
X35
)
)
→
(
¬
(
X47
↔
(
X40
↔
(
X27
↔
X31
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
(
¬
(
X45
→
(
X39
↔
(
X37
↔
X44
)
)
)
)
↔
(
(
¬
(
X44
→
(
X12
↔
X39
)
)
)
↔
(
X44
↔
(
(
¬
X32
)
↔
(
(
¬
X12
)
→
X45
)
)
)
)
)
→
(
(
¬
X39
)
↔
(
(
¬
X32
)
↔
(
(
¬
X3
)
↔
X44
)
)
)
)
)
→
(
¬
(
(
¬
(
X48
↔
(
X12
↔
(
X7
↔
False
)
)
)
)
→
(
¬
X45
)
→
(
X39
↔
(
(
¬
X7
)
→
X48
)
)
)
)
→
(
¬
(
X45
↔
(
(
¬
X12
)
→
X6
→
X48
)
)
)
)
)
↔
(
(
¬
(
(
¬
X39
)
↔
(
X35
→
(
¬
X32
)
→
X48
)
)
)
↔
(
¬
(
X48
→
X40
→
(
(
¬
X9
)
↔
X39
)
)
)
)
)
)
)
→
(
¬
(
(
¬
X47
)
→
(
X40
↔
(
X2
→
X48
)
)
)
)
→
(
(
¬
(
X47
→
(
¬
X2
)
→
X42
)
)
↔
(
(
(
X48
→
(
X42
↔
X47
)
)
↔
(
(
(
¬
X47
)
↔
(
X42
→
X48
)
)
↔
(
(
X48
↔
(
X42
↔
X2
)
)
↔
(
¬
(
X47
↔
(
X42
→
(
¬
X2
)
→
X21
)
)
)
)
)
)
↔
(
X47
→
(
X2
↔
X21
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
9b80e6...
and proposition id is
7bff28...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMN7Eb9pUtpCgsUk9HsTN9BMeWgFKrHHNH3
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
∀X51 :
prop
,
(
(
(
¬
(
(
(
(
(
(
¬
(
X49
↔
(
X22
↔
X46
)
)
)
→
(
(
¬
X49
)
↔
(
(
¬
X46
)
↔
(
X5
→
X50
)
)
)
→
(
¬
X49
)
→
(
X22
↔
(
X5
→
X50
)
)
)
↔
(
(
X46
→
X49
)
↔
(
¬
(
X50
↔
(
X44
→
(
¬
X12
)
→
X49
)
)
)
)
)
↔
(
¬
(
X50
↔
(
X49
→
(
X10
↔
X46
)
)
)
)
)
↔
(
(
(
¬
(
X51
→
X43
→
X50
)
)
↔
(
(
¬
X50
)
↔
(
X49
↔
(
(
¬
X46
)
→
X51
)
)
)
)
→
(
X42
↔
(
X40
→
(
X23
↔
X33
)
)
)
→
(
(
¬
(
(
¬
X49
)
↔
(
(
¬
X42
)
→
(
¬
X33
)
→
X51
)
)
)
↔
(
(
(
(
¬
X49
)
→
(
(
¬
X42
)
↔
X51
)
)
↔
(
¬
(
(
¬
(
X51
→
(
X46
↔
X42
)
)
)
→
(
X49
↔
(
(
¬
X46
)
→
(
X15
↔
X51
)
)
)
)
)
)
↔
(
(
¬
X46
)
→
(
¬
X15
)
→
X49
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
(
(
¬
X35
)
↔
(
X20
↔
X37
)
)
→
(
¬
(
(
(
¬
X35
)
→
(
X19
↔
(
X11
→
X37
)
)
)
→
(
¬
X20
)
→
(
(
¬
X19
)
↔
(
X11
↔
X37
)
)
)
)
)
→
(
¬
(
(
¬
X37
)
→
(
X35
↔
(
(
¬
X20
)
↔
X43
)
)
)
)
)
)
→
(
¬
(
X32
→
X31
→
(
X20
↔
X50
)
)
)
→
X50
→
X43
→
X37
→
X28
→
False
)
→
(
¬
(
(
(
¬
(
X50
→
X49
→
X37
)
)
↔
(
(
¬
(
(
X50
↔
(
X24
→
(
¬
X6
)
→
X14
)
)
→
(
X50
↔
(
X49
↔
(
(
¬
X6
)
↔
X14
)
)
)
)
)
↔
(
¬
(
(
¬
(
X24
→
(
X14
↔
(
X7
↔
X37
)
)
)
)
→
(
¬
X37
)
→
X49
)
)
)
)
→
(
¬
(
(
¬
(
X35
→
(
X24
↔
(
(
¬
X14
)
→
X36
)
)
)
)
→
(
¬
(
X49
↔
X35
)
)
)
)
)
)
→
(
(
X49
→
(
X37
↔
(
X34
↔
(
X17
→
False
)
)
)
)
↔
(
(
(
¬
(
X37
→
(
(
¬
X35
)
↔
X49
)
)
)
↔
(
¬
(
X49
↔
(
X34
→
X3
)
)
)
)
↔
(
¬
(
X35
↔
(
X34
→
(
¬
X30
)
→
X37
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
X43
→
(
X39
↔
(
X27
↔
X18
)
)
)
↔
(
(
¬
(
X39
→
X27
→
X48
)
)
↔
(
(
¬
(
X48
→
X27
→
(
(
¬
X18
)
↔
X39
)
)
)
→
X27
→
X18
→
X44
)
)
)
→
(
(
¬
(
X27
↔
(
X18
→
X2
)
)
)
↔
(
¬
(
X43
→
X18
→
X9
)
)
)
)
↔
(
X44
→
(
X43
↔
(
X39
↔
False
)
)
)
)
↔
(
(
¬
(
(
X48
→
X47
→
X26
→
X0
)
→
X44
→
X48
)
)
↔
(
(
¬
X44
)
↔
(
(
¬
X39
)
↔
(
(
¬
X26
)
↔
X47
)
)
)
)
)
→
(
(
(
¬
(
(
X48
→
(
X47
↔
(
X44
↔
(
X25
→
False
)
)
)
)
→
(
¬
(
(
¬
X45
)
→
(
X44
↔
X47
)
)
)
→
(
¬
(
X44
↔
(
(
¬
X41
)
↔
(
(
¬
X25
)
→
X45
)
)
)
)
)
)
→
(
(
¬
(
X47
↔
(
(
¬
X29
)
↔
(
X8
→
X45
)
)
)
)
↔
(
(
X47
↔
(
(
¬
X29
)
→
(
¬
X8
)
→
X41
)
)
↔
(
¬
(
X41
→
(
(
¬
X8
)
↔
(
(
¬
X4
)
↔
X47
)
)
)
)
)
)
)
↔
(
(
(
X45
→
(
X41
↔
(
X38
↔
X40
)
)
)
→
(
X48
↔
(
X40
→
(
¬
X16
)
→
X45
)
)
)
↔
(
¬
(
(
(
X40
→
(
(
¬
X38
)
↔
X45
)
)
↔
(
(
¬
(
X45
↔
(
X13
→
False
)
)
)
→
(
(
X40
↔
(
X21
→
(
(
¬
X13
)
↔
X45
)
)
)
↔
(
X45
→
(
X21
↔
(
(
¬
X1
)
↔
X40
)
)
)
)
)
)
→
(
¬
(
X45
↔
X21
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
(
X50
↔
(
(
¬
X7
)
↔
X47
)
)
↔
(
(
X50
↔
(
X45
→
X48
)
)
↔
(
¬
(
(
(
¬
X47
)
↔
(
X45
↔
(
(
¬
X7
)
↔
X48
)
)
)
→
(
¬
(
(
¬
X45
)
↔
X50
)
)
)
)
)
)
→
(
¬
(
(
¬
X46
)
↔
X50
)
)
→
(
¬
(
(
¬
X48
)
↔
(
(
¬
X7
)
→
(
X1
↔
X50
)
)
)
)
)
)
→
(
(
(
(
¬
(
X48
→
X34
→
False
)
)
→
(
(
¬
(
(
¬
X43
)
↔
(
X34
→
X50
)
)
)
↔
(
¬
(
(
¬
X48
)
→
(
¬
X45
)
→
(
X43
↔
X50
)
)
)
)
)
↔
(
¬
(
(
X48
→
X32
→
(
X17
↔
X40
)
)
→
(
X17
↔
X43
)
→
(
¬
(
(
X32
→
(
X6
↔
X17
)
)
→
(
¬
(
(
X32
↔
(
(
¬
X17
)
→
X43
)
)
→
(
¬
(
(
¬
(
X32
→
X6
→
False
)
)
→
(
¬
(
(
¬
X43
)
→
X32
→
(
¬
X17
)
→
X50
)
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
X50
↔
(
X18
↔
False
)
)
→
(
(
¬
(
X50
↔
(
X40
↔
(
X34
→
X10
→
False
)
)
)
)
↔
(
¬
(
(
¬
X40
)
↔
(
X34
→
X18
→
X50
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
(
(
¬
X34
)
↔
(
X29
→
(
¬
X25
)
→
X49
)
)
)
→
(
X34
↔
(
X29
↔
(
X25
→
X36
)
)
)
)
)
→
(
¬
(
(
¬
X32
)
→
(
¬
X29
)
→
X25
→
X34
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X45
)
→
(
¬
X34
)
→
X26
→
X49
)
)
→
(
(
(
(
¬
(
X45
→
(
¬
X34
)
→
X49
)
)
↔
(
¬
(
(
X49
→
X45
→
X31
)
→
(
(
X34
↔
(
X29
→
X31
)
)
↔
(
X49
→
(
X45
↔
(
(
¬
X29
)
↔
X31
)
)
)
)
)
)
)
→
(
¬
X34
)
)
↔
(
(
X45
→
(
X31
↔
X34
)
)
↔
(
(
X31
↔
(
X14
↔
False
)
)
↔
(
(
¬
X34
)
↔
(
(
¬
X14
)
↔
(
X0
↔
X45
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X44
→
(
X38
↔
(
X35
↔
(
X3
→
False
)
)
)
)
)
→
(
X39
↔
(
X16
↔
(
X3
↔
X30
)
)
)
→
X30
→
(
(
¬
X16
)
↔
(
X3
↔
X35
)
)
)
↔
(
(
(
(
¬
(
(
¬
X35
)
→
(
X24
↔
X51
)
)
)
↔
(
¬
(
(
¬
(
X51
↔
(
(
¬
X21
)
→
X24
)
)
)
→
X21
→
(
(
¬
X11
)
↔
X22
)
)
)
)
↔
(
¬
(
(
¬
X35
)
↔
(
(
¬
X22
)
↔
X49
)
)
)
)
→
(
(
(
¬
(
X13
→
X4
→
X39
)
)
→
(
¬
(
(
(
¬
(
X13
↔
(
X11
→
X22
)
)
)
↔
(
¬
(
X4
↔
X39
)
)
)
→
(
¬
(
X13
→
(
X11
↔
X39
)
)
)
)
)
)
↔
(
(
¬
(
X22
↔
(
X16
↔
(
(
¬
X13
)
↔
X35
)
)
)
)
→
(
¬
(
X39
→
X35
→
(
X16
↔
X22
)
)
)
)
)
)
)
→
(
(
(
¬
(
X45
→
X23
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X42
)
↔
(
(
¬
X39
)
↔
(
X28
↔
X51
)
)
)
)
↔
(
¬
(
(
¬
X39
)
↔
(
(
¬
X28
)
→
X23
→
X51
)
)
)
)
→
(
X39
↔
X42
)
)
)
)
→
(
¬
(
X39
→
X8
→
(
X5
↔
X45
)
)
)
→
(
¬
(
(
¬
(
X9
↔
(
(
¬
X2
)
↔
X8
)
)
)
→
(
¬
(
X5
↔
X2
)
)
→
X42
→
(
(
¬
X8
)
↔
(
(
¬
X5
)
↔
X39
)
)
)
)
)
→
(
¬
(
(
(
(
X45
→
(
X19
↔
(
(
¬
X12
)
→
X51
)
)
)
↔
(
X37
→
(
(
¬
X12
)
↔
X19
)
)
)
↔
(
(
X45
→
X41
→
(
X19
↔
X42
)
)
↔
(
X45
→
X41
→
X37
→
X42
)
)
)
→
(
X51
→
(
X50
↔
(
X33
→
X20
)
)
)
→
(
¬
(
(
(
¬
(
X42
↔
(
(
¬
X38
)
↔
X50
)
)
)
→
(
(
¬
(
(
¬
X42
)
↔
(
X38
→
X27
→
X50
)
)
)
↔
(
(
(
¬
X38
)
→
(
X33
↔
(
X27
↔
X50
)
)
)
→
(
X50
↔
(
X27
→
(
X15
↔
False
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X51
↔
(
X38
↔
(
(
¬
X33
)
→
X50
)
)
)
)
→
(
¬
(
X50
↔
(
X38
→
X37
→
X51
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
324aa8...
and proposition id is
2bbe64...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMZMq5RphTykVxi9ciZjocNn7KwvTPQo6zy
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
∀X50 :
prop
,
(
(
(
(
(
(
(
¬
(
(
¬
X44
)
↔
(
X42
↔
(
X38
→
X45
)
)
)
)
→
(
¬
(
(
X38
↔
(
X12
↔
X37
)
)
→
(
¬
X42
)
→
X44
)
)
)
→
(
(
(
¬
(
(
¬
X45
)
↔
(
(
¬
X0
)
→
X50
)
)
)
↔
(
¬
(
(
¬
(
X50
→
X45
→
(
X0
↔
X23
)
)
)
→
(
¬
(
(
¬
X43
)
→
(
X23
↔
X45
)
)
)
)
)
)
↔
(
(
(
X45
→
(
X29
↔
X13
)
)
↔
(
X48
↔
(
X29
↔
X42
)
)
)
→
(
(
¬
(
X50
↔
(
(
¬
X45
)
→
(
(
¬
X44
)
↔
X48
)
)
)
)
↔
(
(
(
¬
(
(
¬
X45
)
→
X44
→
X24
→
X48
)
)
↔
(
¬
(
X45
→
(
X44
↔
X42
)
)
)
)
→
(
¬
(
X45
→
(
¬
X42
)
→
X44
)
)
)
)
→
(
X50
↔
(
(
¬
X45
)
→
(
¬
X43
)
→
X48
)
)
)
)
)
→
(
(
¬
(
(
(
¬
X39
)
↔
(
(
¬
X25
)
↔
(
(
¬
X11
)
↔
X50
)
)
)
→
(
(
(
(
X34
→
X25
→
X11
)
↔
(
X42
↔
(
(
¬
X34
)
→
(
X25
↔
X50
)
)
)
)
↔
(
¬
False
)
)
↔
(
X50
→
(
X34
↔
False
)
)
)
)
)
↔
(
(
X43
→
X25
→
X10
→
X42
)
↔
(
X39
→
X10
→
X42
)
)
)
)
↔
(
¬
(
(
(
X48
→
X47
→
X18
→
X44
)
→
(
¬
(
X49
↔
(
X47
→
X7
→
X46
)
)
)
→
(
¬
(
(
¬
X47
)
→
(
X44
↔
X49
)
)
)
→
(
(
¬
(
X47
→
(
¬
X46
)
→
X49
)
)
↔
(
(
¬
(
(
¬
X47
)
↔
(
X20
↔
(
(
¬
X8
)
↔
X49
)
)
)
)
↔
(
¬
(
X47
→
(
(
¬
X44
)
↔
(
(
¬
X8
)
→
X49
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
(
X19
→
X15
→
(
(
¬
X4
)
↔
X44
)
)
)
→
(
¬
X15
)
→
(
¬
X4
)
→
(
¬
X2
)
→
X44
)
)
→
(
¬
(
X48
→
(
X19
↔
(
X15
→
X2
)
)
)
)
)
→
(
¬
(
X44
↔
(
(
¬
X19
)
→
X15
→
X27
)
)
)
)
→
(
X44
↔
(
X37
↔
(
(
¬
X2
)
↔
X15
)
)
)
)
→
(
¬
(
(
X44
→
X28
→
X37
)
→
(
(
¬
X44
)
↔
(
X28
↔
(
X27
→
X47
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
¬
(
X31
→
(
X22
↔
(
X3
↔
X30
)
)
)
)
↔
(
¬
(
X31
→
X30
→
(
X22
↔
(
X14
→
False
)
)
)
)
)
→
(
¬
(
X14
→
False
)
)
)
→
(
(
(
(
¬
X22
)
→
X31
)
→
(
(
(
¬
(
(
¬
X36
)
↔
(
X22
→
X41
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
¬
X31
)
→
X22
→
X41
)
)
→
(
¬
(
X41
↔
(
(
¬
X31
)
↔
(
(
¬
X16
)
↔
X36
)
)
)
)
)
)
→
(
¬
(
X36
↔
(
(
¬
X16
)
↔
X22
)
)
)
)
)
)
↔
(
¬
(
X31
↔
(
X22
→
X16
)
)
)
)
)
↔
(
(
¬
(
(
¬
X22
)
↔
(
X16
↔
(
X14
↔
X50
)
)
)
)
→
(
¬
(
(
¬
X31
)
→
(
X22
↔
(
(
¬
X16
)
→
X50
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X31
↔
(
X30
→
(
(
¬
X26
)
↔
X47
)
)
)
)
→
(
(
X48
→
X16
→
X30
)
↔
(
X47
↔
(
(
¬
X31
)
↔
(
X1
↔
X48
)
)
)
)
)
↔
(
¬
(
(
¬
X31
)
→
(
X9
↔
X40
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X33
)
→
X21
→
X48
)
)
↔
(
¬
(
X47
↔
(
X33
↔
(
X32
↔
X48
)
)
)
)
)
↔
(
¬
(
(
(
X48
→
(
X47
↔
(
(
¬
X35
)
→
X36
)
)
)
↔
(
X17
→
X33
)
)
→
(
¬
(
(
X36
↔
(
X6
→
(
(
¬
X5
)
↔
X33
)
)
)
→
(
¬
(
X5
↔
X47
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
X50
→
X44
→
X46
)
)
↔
(
(
¬
(
X50
↔
(
X39
↔
(
X16
→
X46
)
)
)
)
↔
(
(
¬
(
(
X46
↔
X50
)
→
(
(
(
¬
X46
)
→
(
X39
↔
X50
)
)
↔
(
¬
(
X44
↔
(
(
¬
X21
)
→
X50
)
)
)
)
)
)
↔
(
X50
↔
(
(
¬
X39
)
↔
X44
)
)
)
)
)
↔
(
¬
(
(
(
X27
→
X14
→
X25
)
↔
(
¬
(
X49
↔
(
(
¬
X25
)
→
(
(
¬
X14
)
↔
X48
)
)
)
)
)
→
(
(
(
¬
(
(
(
¬
X10
)
→
(
X8
↔
X49
)
)
→
(
X50
↔
(
(
¬
X10
)
→
X8
→
X48
)
)
)
)
↔
(
(
¬
X49
)
→
(
(
¬
X48
)
↔
X50
)
)
)
↔
(
X49
→
X48
→
(
¬
X25
)
→
X50
)
)
→
(
(
¬
(
(
¬
X48
)
↔
(
(
¬
X44
)
↔
(
X21
↔
X49
)
)
)
)
↔
(
(
¬
X49
)
↔
(
X48
↔
X50
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X31
→
(
(
(
X47
→
X31
→
X24
)
↔
(
(
¬
(
(
¬
X47
)
↔
(
X31
↔
X50
)
)
)
→
(
¬
(
(
¬
X48
)
↔
(
(
¬
X31
)
↔
(
X24
↔
X50
)
)
)
)
)
)
↔
(
X48
→
X24
→
False
)
)
)
)
↔
(
¬
(
(
(
¬
(
X48
↔
(
X31
↔
(
X0
↔
False
)
)
)
)
↔
(
X50
↔
(
X48
→
X31
→
X28
)
)
)
→
X31
→
(
¬
X28
)
→
X48
)
)
)
→
(
(
¬
(
(
(
¬
(
X47
→
(
X31
↔
X23
)
)
)
↔
(
(
X47
↔
(
X23
↔
(
X17
→
False
)
)
)
↔
(
¬
(
X47
↔
(
(
¬
X31
)
↔
X35
)
)
)
)
)
→
(
(
¬
X47
)
↔
(
X35
→
X23
→
X48
)
)
)
)
↔
(
¬
(
(
¬
X23
)
↔
X35
)
)
)
)
)
→
(
¬
(
(
¬
X31
)
↔
(
X29
→
(
¬
X3
)
→
X35
)
)
)
→
(
¬
(
(
¬
X46
)
→
(
¬
X35
)
→
X47
)
)
→
(
(
(
¬
X46
)
↔
(
X35
↔
(
(
¬
X29
)
→
X47
)
)
)
↔
(
(
¬
X46
)
→
(
X31
↔
X47
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X45
↔
(
X40
↔
X47
)
)
)
→
(
X50
↔
(
X37
↔
X45
)
)
→
(
¬
(
(
¬
X47
)
→
(
¬
X40
)
→
X37
→
X50
)
)
)
)
↔
(
X47
→
(
X40
↔
(
X32
↔
False
)
)
)
)
→
(
¬
(
(
(
¬
(
X47
→
(
¬
X43
)
→
(
(
¬
X33
)
↔
X49
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X47
)
↔
(
X15
→
X49
)
)
)
→
(
¬
(
X49
↔
(
X33
→
(
¬
X15
)
→
X47
)
)
)
→
(
X49
↔
(
X40
→
False
)
)
)
)
)
→
(
(
(
¬
(
X47
↔
(
X37
↔
(
(
¬
X4
)
→
X49
)
)
)
)
→
(
¬
(
(
¬
X43
)
↔
(
(
¬
X37
)
→
X4
→
X49
)
)
)
)
↔
(
(
X37
↔
(
(
¬
X7
)
↔
(
X5
↔
X47
)
)
)
↔
(
¬
(
X5
↔
X49
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
(
(
X36
↔
(
(
¬
X18
)
→
(
X9
↔
X41
)
)
)
→
X36
→
(
¬
X18
)
→
X20
)
)
↔
(
X41
→
(
(
¬
X9
)
↔
X18
)
)
)
↔
(
(
X41
→
(
¬
X20
)
→
X18
→
X46
)
→
(
¬
X20
)
→
(
(
¬
X18
)
↔
X30
)
)
)
→
(
(
¬
(
X38
→
X18
)
)
↔
(
¬
(
(
X42
→
X20
→
X38
)
→
(
¬
(
(
¬
X38
)
↔
(
(
¬
X18
)
→
X41
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
(
(
¬
X1
)
→
X36
)
↔
(
(
¬
(
(
¬
X13
)
→
(
(
¬
X11
)
↔
(
X1
→
X22
)
)
)
)
↔
(
(
¬
X36
)
↔
(
X11
↔
X48
)
)
)
)
→
(
¬
(
X36
→
(
X13
↔
(
(
¬
X12
)
→
X48
)
)
)
)
)
)
↔
(
¬
(
(
(
X38
↔
(
X36
↔
(
X34
↔
(
X22
→
False
)
)
)
)
↔
(
X38
↔
(
X36
↔
(
X26
↔
X34
)
)
)
)
→
(
¬
X36
)
→
(
(
¬
X26
)
↔
(
X22
→
X48
)
)
)
)
)
↔
(
(
(
(
X34
↔
(
(
¬
X19
)
→
(
(
¬
X6
)
↔
X45
)
)
)
↔
(
¬
(
X19
→
X41
)
)
)
↔
(
(
¬
X19
)
↔
(
X6
↔
(
X2
↔
X34
)
)
)
)
↔
(
(
(
¬
X45
)
→
(
X38
↔
X48
)
)
↔
(
X34
↔
(
(
¬
X19
)
→
X38
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
621fcb...
and proposition id is
8caf15...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMFoCSzoyrj6CNufG825Y6GKaFEmxtAcDmC
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
∀X51 :
prop
,
(
(
(
(
(
(
(
(
¬
(
(
¬
X46
)
→
X28
→
X50
)
)
→
(
¬
(
X50
→
(
(
¬
X44
)
↔
(
X28
↔
X47
)
)
)
)
)
↔
(
(
¬
(
X50
→
X47
→
(
¬
X24
)
→
X44
)
)
→
(
¬
(
X47
→
(
X24
↔
(
(
¬
X8
)
→
X35
)
)
)
)
)
)
↔
(
¬
(
(
¬
X35
)
→
(
X28
↔
(
(
¬
X10
)
↔
X46
)
)
)
)
)
↔
(
(
(
(
(
X47
↔
(
(
¬
X2
)
↔
X21
)
)
↔
(
(
¬
(
X46
↔
(
X21
↔
X2
)
)
)
→
(
¬
(
(
¬
(
(
¬
X47
)
↔
(
X46
→
(
¬
X21
)
→
X50
)
)
)
→
(
¬
(
(
¬
X47
)
↔
(
(
¬
X2
)
↔
X50
)
)
)
)
)
)
)
↔
(
X50
→
X46
→
(
X11
↔
X21
)
)
)
↔
(
¬
(
X50
↔
(
X47
→
(
¬
X5
)
→
X21
)
)
)
)
↔
(
(
¬
(
(
¬
X46
)
→
(
(
¬
X30
)
↔
(
(
¬
X22
)
→
X47
)
)
)
)
↔
(
(
(
¬
X47
)
↔
(
(
¬
X15
)
↔
(
(
¬
X0
)
↔
X50
)
)
)
↔
(
(
X22
→
(
X15
↔
(
X0
→
X47
)
)
)
↔
(
X46
↔
(
X0
↔
X22
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X47
→
(
¬
X31
)
→
(
X26
↔
X44
)
)
)
→
(
(
¬
(
(
¬
X38
)
→
X26
→
X47
)
)
↔
(
¬
(
X31
↔
(
(
¬
X3
)
→
X26
)
)
)
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X44
)
↔
(
(
¬
X38
)
→
X47
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
¬
X38
)
↔
(
(
¬
X29
)
↔
X48
)
)
)
→
(
(
¬
(
X48
→
(
X38
↔
(
X9
↔
X29
)
)
)
)
↔
(
¬
(
X48
↔
(
X29
↔
(
X4
↔
X38
)
)
)
)
)
)
)
↔
(
(
X38
→
(
(
¬
X4
)
↔
X19
)
)
↔
(
(
(
(
¬
X38
)
→
(
¬
X19
)
→
(
X4
↔
X46
)
)
↔
(
¬
(
X29
↔
(
(
¬
X4
)
→
X38
)
)
)
)
↔
(
X46
→
X4
)
)
)
)
→
(
(
¬
(
X48
→
(
(
¬
X40
)
↔
(
X29
↔
X46
)
)
)
)
↔
(
(
¬
X47
)
→
(
¬
X46
)
→
(
¬
X40
)
→
X48
)
)
)
)
)
↔
(
(
(
(
(
X34
↔
(
X7
↔
X41
)
)
↔
(
¬
(
X43
↔
(
X41
→
(
X18
↔
(
X7
→
False
)
)
)
)
)
)
↔
(
(
¬
(
X51
→
(
(
¬
X41
)
↔
(
(
¬
X34
)
↔
X49
)
)
)
)
↔
(
¬
(
X41
↔
(
X34
↔
(
(
¬
X33
)
↔
X51
)
)
)
)
)
)
↔
(
(
(
¬
(
(
X50
→
(
¬
X47
)
→
(
X37
↔
X49
)
)
→
(
¬
(
X49
→
(
X47
↔
(
X12
↔
X37
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X46
→
X37
→
X16
→
X50
)
)
→
(
¬
(
X49
→
(
(
¬
X37
)
↔
(
(
¬
X16
)
↔
X46
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
(
(
¬
(
(
¬
X39
)
↔
(
(
¬
X32
)
→
X47
)
)
)
→
(
¬
X46
)
→
(
¬
X41
)
→
X32
→
X47
)
)
↔
(
¬
(
(
¬
X46
)
↔
(
X43
→
(
¬
X41
)
→
X47
)
)
)
)
→
X46
→
X17
→
X47
)
)
→
(
¬
(
X49
↔
(
X39
→
(
(
¬
X32
)
↔
X46
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X23
↔
(
(
¬
X20
)
→
X47
)
)
)
→
(
¬
(
X43
↔
(
(
¬
X23
)
→
(
¬
X20
)
→
X25
)
)
)
)
)
↔
(
(
¬
(
(
(
X43
→
X25
→
X14
→
X49
)
→
(
¬
(
(
¬
X43
)
→
(
(
¬
X14
)
↔
(
X6
→
X49
)
)
)
)
)
→
(
¬
(
X49
→
(
¬
X25
)
→
X36
)
)
)
)
→
(
X14
↔
(
(
¬
X13
)
↔
X25
)
)
→
(
X49
↔
(
X25
↔
(
(
¬
X14
)
→
X20
)
)
)
)
)
→
(
(
¬
(
X25
→
X43
)
)
↔
(
(
(
¬
X27
)
↔
(
(
¬
X23
)
↔
(
X1
↔
X42
)
)
)
↔
(
(
(
¬
(
X27
↔
(
X25
↔
X45
)
)
)
→
(
X42
→
(
X27
↔
(
X1
→
X45
)
)
)
→
(
X27
↔
(
(
¬
X25
)
↔
(
(
¬
X1
)
→
X42
)
)
)
)
↔
(
X45
↔
(
X42
↔
(
(
¬
X27
)
↔
X43
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
(
(
¬
(
X47
→
(
¬
X39
)
→
(
(
¬
X25
)
↔
X49
)
)
)
→
(
X28
↔
X25
)
)
→
(
¬
(
X49
↔
(
X25
→
X28
)
)
)
)
→
(
(
¬
(
(
¬
X45
)
→
(
¬
X39
)
→
X28
→
X47
)
)
↔
(
¬
(
X39
→
X28
→
(
X5
↔
X45
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X28
)
↔
X46
)
)
↔
(
X49
↔
(
(
¬
X28
)
↔
(
X13
→
X45
)
)
)
)
→
(
X49
↔
X28
)
)
↔
(
¬
(
X46
→
(
X13
↔
X49
)
)
)
)
)
→
(
(
(
X51
↔
(
(
¬
X11
)
↔
X46
)
)
↔
(
(
¬
(
X51
↔
(
(
¬
X46
)
→
(
¬
X1
)
→
X49
)
)
)
↔
(
¬
(
X49
↔
(
X46
↔
(
X11
↔
X51
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
X47
)
↔
(
(
¬
X46
)
↔
(
(
¬
X1
)
→
X49
)
)
)
→
(
¬
(
X49
↔
(
X46
↔
(
X36
→
X1
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
X46
)
→
(
X42
↔
(
X27
→
X51
)
)
)
→
(
¬
(
X51
→
(
(
¬
X46
)
↔
(
(
¬
X42
)
→
X47
)
)
)
)
→
(
¬
(
(
¬
X47
)
↔
(
X42
→
(
¬
X12
)
→
X51
)
)
)
)
)
)
→
(
(
¬
(
(
(
X37
↔
(
(
¬
X23
)
↔
(
X14
→
X47
)
)
)
↔
(
¬
(
(
X49
↔
(
X23
↔
(
X14
→
False
)
)
)
→
(
¬
(
(
(
¬
X46
)
↔
(
X37
→
X49
)
)
→
(
¬
(
X49
↔
(
X46
↔
(
X37
↔
(
X23
↔
False
)
)
)
)
)
→
X40
→
(
¬
X23
)
→
X46
)
)
)
)
)
→
(
(
¬
(
X40
→
(
X38
↔
(
(
¬
X10
)
↔
X47
)
)
)
)
↔
(
X47
↔
(
X46
→
(
¬
X38
)
→
X40
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
X40
)
→
(
X29
↔
(
(
¬
X15
)
→
X48
)
)
)
↔
(
(
X48
↔
(
X44
→
(
X40
↔
X49
)
)
)
→
(
(
¬
(
X44
↔
(
X34
↔
X40
)
)
)
↔
(
¬
(
(
X44
→
(
(
¬
X34
)
↔
(
X15
→
X48
)
)
)
→
X40
→
X34
→
X15
)
)
)
)
)
→
(
(
¬
(
X44
↔
(
X35
↔
(
X20
→
X49
)
)
)
)
↔
(
¬
(
X49
→
X40
→
X20
→
X44
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X40
)
→
(
X33
↔
(
(
¬
X0
)
↔
X49
)
)
)
)
→
(
¬
(
(
X49
↔
(
X40
↔
(
X34
↔
(
X6
→
False
)
)
)
)
→
(
(
(
(
¬
X33
)
→
(
X17
↔
(
X8
→
X34
)
)
)
→
(
(
¬
(
X49
→
(
¬
X33
)
→
(
(
¬
X17
)
↔
X34
)
)
)
↔
(
¬
(
X33
→
X6
→
X34
)
)
)
)
↔
(
X49
↔
(
X34
→
(
X33
↔
False
)
)
)
)
→
(
¬
(
(
¬
X33
)
↔
(
X8
→
(
¬
X2
)
→
X38
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
X49
→
(
X32
↔
X45
)
)
)
↔
(
(
(
¬
(
X45
↔
(
X16
↔
X22
)
)
)
→
(
¬
(
(
¬
X32
)
↔
(
(
¬
X22
)
→
(
X16
↔
X49
)
)
)
)
)
→
(
(
¬
X22
)
↔
(
(
¬
X16
)
↔
X32
)
)
)
)
→
(
¬
(
X45
↔
(
(
¬
X43
)
↔
(
X32
↔
X49
)
)
)
)
)
↔
(
X46
→
X43
→
X32
)
)
→
(
¬
(
(
X21
↔
(
X7
↔
X43
)
)
→
(
(
¬
(
(
¬
X46
)
↔
(
X21
→
(
X3
↔
X49
)
)
)
)
↔
(
¬
(
X46
↔
(
X7
→
(
X3
↔
X21
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X31
)
→
(
¬
X4
)
→
X41
)
)
↔
(
(
X45
→
(
¬
X41
)
→
X26
→
X50
)
↔
(
¬
(
X45
→
(
X4
↔
False
)
)
)
)
)
↔
(
¬
(
X50
↔
(
X41
→
X45
)
)
)
)
↔
(
(
(
¬
(
X30
→
X24
→
X26
)
)
→
X50
→
(
X32
↔
(
X26
↔
X24
)
)
)
↔
(
(
(
¬
(
X50
↔
(
X19
→
(
X9
↔
False
)
)
)
)
→
(
¬
X26
)
→
X9
→
X49
)
↔
(
(
¬
(
(
X50
→
X41
→
X32
)
→
(
(
¬
(
X50
→
X32
→
X49
)
)
↔
(
(
¬
(
(
(
¬
(
X49
↔
(
(
¬
X32
)
→
X50
)
)
)
↔
(
¬
(
(
¬
X41
)
→
(
(
¬
X32
)
↔
(
(
¬
X18
)
↔
X49
)
)
)
)
)
→
(
¬
(
X41
↔
(
X18
↔
False
)
)
)
)
)
↔
(
X50
→
(
(
¬
X32
)
↔
X49
)
)
)
)
)
)
↔
(
X49
↔
(
X41
→
(
(
¬
X9
)
↔
X32
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
b95085...
and proposition id is
e04298...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMRZXJNppz4CxxMzZ4qHXJHpDghAqQRCiqd
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
¬
(
(
¬
(
(
(
(
(
X47
↔
(
X45
↔
(
X28
→
False
)
)
)
→
X45
→
(
(
¬
X13
)
↔
X18
)
)
↔
(
(
(
¬
(
(
X28
↔
(
(
¬
X12
)
→
(
(
¬
X7
)
↔
X13
)
)
)
→
(
¬
(
X47
→
(
X13
↔
(
X7
→
False
)
)
)
)
)
)
↔
(
X12
→
(
¬
X7
)
→
X13
)
)
↔
(
X28
↔
(
X13
↔
(
X12
→
False
)
)
)
)
)
→
(
(
¬
(
X37
↔
(
X18
↔
False
)
)
)
↔
(
(
¬
X45
)
→
X37
→
(
(
¬
X0
)
↔
X47
)
)
)
→
X37
→
(
X18
↔
X45
)
)
→
(
(
(
(
X32
↔
(
X26
↔
X45
)
)
→
(
¬
(
(
¬
X26
)
↔
(
(
¬
X18
)
→
(
(
¬
X13
)
↔
X32
)
)
)
)
→
(
X32
↔
(
(
¬
X26
)
→
(
(
¬
X13
)
↔
X45
)
)
)
)
↔
(
¬
(
X26
→
X18
→
X45
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
(
¬
X26
)
↔
(
(
¬
X18
)
→
(
¬
X8
)
→
X32
)
)
)
→
(
¬
(
X27
→
(
X26
↔
False
)
)
)
)
)
→
(
¬
(
(
¬
X27
)
→
(
(
¬
X26
)
↔
X32
)
)
)
)
)
↔
(
X32
↔
(
(
¬
X26
)
→
X27
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
(
(
(
(
(
(
¬
X43
)
↔
(
(
¬
X37
)
↔
(
X16
→
X47
)
)
)
→
X43
→
(
X37
↔
(
(
¬
X35
)
↔
X47
)
)
)
→
(
¬
(
X37
→
X35
→
X43
)
)
)
↔
(
X43
↔
(
X16
↔
X37
)
)
)
↔
(
¬
(
(
¬
X43
)
↔
(
X37
→
X47
)
)
)
)
↔
(
¬
(
X39
↔
(
X16
→
False
)
)
)
)
→
(
(
¬
(
X43
→
X15
)
)
↔
(
(
¬
(
(
¬
X37
)
→
X15
→
X43
)
)
↔
(
¬
(
(
¬
X37
)
↔
(
(
¬
X15
)
↔
(
X11
→
X39
)
)
)
)
)
)
)
→
(
X39
→
(
X23
↔
(
X15
↔
X49
)
)
)
→
X49
→
(
X35
↔
(
X15
↔
False
)
)
)
→
(
(
(
X43
→
X33
→
X39
)
↔
(
False
↔
(
¬
(
(
¬
(
X43
↔
(
(
¬
X39
)
↔
(
X33
→
X47
)
)
)
)
→
(
(
X25
→
(
X22
↔
X33
)
)
↔
(
¬
(
X47
↔
(
X25
→
(
X22
↔
X39
)
)
)
)
)
)
)
)
)
↔
(
X39
→
X33
→
X25
)
)
)
)
↔
(
(
X45
→
X39
→
(
¬
X3
)
→
X49
)
↔
(
¬
(
(
¬
X45
)
↔
(
X39
↔
(
X3
→
X47
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
X24
)
↔
(
X21
→
(
¬
X20
)
→
X42
)
)
↔
(
¬
(
X46
→
(
X42
↔
(
X21
→
X24
)
)
)
)
)
→
(
¬
(
X48
→
X20
)
)
→
(
¬
(
(
(
¬
X42
)
↔
(
X20
→
X44
)
)
→
(
X24
↔
X29
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X44
↔
(
(
¬
X42
)
→
(
(
¬
X40
)
↔
X47
)
)
)
)
→
(
¬
(
X42
→
X31
→
(
X1
↔
X47
)
)
)
)
)
→
(
(
(
X48
→
X40
→
(
X6
↔
X47
)
)
→
(
¬
(
X42
↔
(
(
¬
X6
)
↔
X48
)
)
)
)
↔
(
¬
(
X42
→
X6
)
)
)
)
)
)
→
(
(
(
(
¬
(
X49
→
(
X44
↔
(
X14
→
X2
→
False
)
)
)
)
→
(
¬
(
(
X14
↔
(
X2
→
X9
)
)
→
(
¬
(
X47
→
X46
→
X14
→
False
)
)
)
)
)
→
(
¬
(
(
(
X47
→
(
X46
↔
(
(
¬
X44
)
→
X45
)
)
)
↔
(
(
¬
X30
)
→
X45
)
)
→
(
(
X47
↔
(
X45
↔
(
X44
↔
(
X19
↔
False
)
)
)
)
↔
(
¬
(
(
¬
(
X47
↔
(
X19
→
(
X10
↔
X44
)
)
)
)
→
X47
→
(
X19
↔
(
(
¬
X5
)
↔
X44
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
X38
↔
(
X36
→
X41
)
)
↔
(
(
X47
↔
(
(
¬
X38
)
→
X36
→
X41
)
)
↔
(
X49
→
X41
→
(
¬
X36
)
→
X47
)
)
)
→
(
¬
(
X49
↔
(
X45
↔
(
X38
→
X4
)
)
)
)
→
(
¬
(
(
(
(
(
¬
(
X38
↔
(
(
¬
X34
)
→
X36
)
)
)
→
(
¬
(
X49
↔
(
X38
↔
(
X17
↔
False
)
)
)
)
)
→
(
¬
(
X45
→
(
X34
↔
(
X17
→
False
)
)
)
)
)
↔
(
X45
→
(
¬
X36
)
→
X49
)
)
→
(
¬
(
(
¬
X45
)
→
(
¬
X34
)
→
X49
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
¬
(
(
(
¬
X22
)
→
(
¬
X7
)
→
X26
)
→
(
¬
(
(
X44
↔
(
(
¬
X22
)
→
(
(
¬
X7
)
↔
X40
)
)
)
→
(
X44
↔
(
X40
→
(
X26
↔
X22
)
)
)
)
)
)
)
↔
(
(
¬
(
X40
→
X27
→
X7
→
False
)
)
→
(
¬
(
(
¬
X40
)
→
(
X7
↔
X44
)
)
)
→
(
(
¬
(
X44
→
(
¬
X40
)
→
(
¬
X37
)
→
X45
)
)
↔
(
(
¬
X40
)
↔
(
(
¬
X7
)
→
X45
)
)
)
)
)
↔
(
(
(
¬
(
X45
↔
(
X44
↔
(
(
¬
X31
)
→
X40
)
)
)
)
↔
(
(
¬
X40
)
↔
(
X31
↔
(
(
¬
X17
)
→
X44
)
)
)
)
↔
(
(
¬
(
X45
→
X38
→
X44
)
)
↔
(
(
(
¬
X44
)
→
(
X38
↔
(
(
¬
X31
)
→
X45
)
)
)
↔
(
(
¬
X44
)
↔
(
X31
↔
X48
)
)
)
)
)
)
→
(
¬
(
X47
↔
(
X46
↔
(
X44
↔
(
X39
→
False
)
)
)
)
)
→
(
(
(
(
¬
(
X44
→
(
X41
↔
(
X13
→
False
)
)
)
)
↔
(
X48
→
(
X44
↔
(
(
¬
X41
)
→
X47
)
)
)
)
→
(
¬
(
X48
→
(
¬
X33
)
→
X47
)
)
)
↔
(
(
X44
→
(
¬
X18
)
→
X33
)
↔
(
¬
(
(
¬
(
X45
↔
(
X41
↔
(
X33
↔
X46
)
)
)
)
→
(
¬
(
X46
↔
(
X41
↔
(
(
¬
X18
)
→
X45
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
X30
↔
(
X29
→
X14
)
)
)
↔
(
¬
(
(
¬
X9
)
→
X4
→
X14
)
)
)
→
(
(
¬
X9
)
↔
(
(
¬
X4
)
↔
X14
)
)
)
)
↔
(
(
X34
↔
(
X9
→
X4
→
X47
)
)
↔
(
¬
(
(
(
X47
→
(
X24
↔
(
X4
→
X46
)
)
)
↔
(
(
¬
X30
)
↔
(
(
¬
X4
)
↔
X46
)
)
)
→
X47
→
(
¬
X30
)
→
X9
→
X46
)
)
)
)
↔
(
(
X47
→
(
X30
↔
(
(
¬
X4
)
→
X43
)
)
)
↔
(
X43
↔
(
(
¬
X30
)
↔
(
(
¬
X4
)
→
X45
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
(
(
X15
↔
(
X8
→
False
)
)
→
(
¬
(
(
¬
X32
)
↔
(
(
¬
X11
)
→
(
(
¬
X8
)
↔
X36
)
)
)
)
)
→
(
¬
(
X36
→
X15
→
(
X11
↔
False
)
)
)
)
→
X11
→
(
(
¬
X8
)
↔
X15
)
)
)
↔
(
¬
(
(
(
¬
X10
)
→
X36
)
→
(
¬
(
X15
↔
(
(
¬
X11
)
↔
X47
)
)
)
)
)
)
→
(
¬
(
(
¬
X36
)
→
(
(
¬
X15
)
↔
(
X10
→
X43
)
)
)
)
)
↔
(
(
(
(
X36
↔
(
(
¬
X35
)
→
X42
)
)
↔
(
(
X43
→
(
X36
↔
(
(
¬
X35
)
→
X42
)
)
)
↔
(
(
X49
↔
(
X36
↔
(
X3
→
False
)
)
)
↔
(
¬
(
X49
→
X42
→
(
X36
↔
(
X35
→
False
)
)
)
)
)
)
)
↔
(
X43
↔
(
(
¬
X36
)
↔
X42
)
)
)
↔
(
(
¬
(
X42
→
(
(
¬
X25
)
↔
(
X23
→
X43
)
)
)
)
↔
(
¬
(
X36
→
(
X25
↔
(
(
¬
X23
)
↔
X32
)
)
)
)
)
)
)
→
(
(
(
X43
→
X12
→
X6
→
X21
)
→
(
X41
↔
(
X21
→
(
X5
↔
False
)
)
)
)
↔
(
(
¬
X41
)
↔
X43
)
)
→
(
(
(
(
X43
↔
X20
)
↔
(
X49
→
(
X21
↔
(
(
¬
X1
)
↔
X43
)
)
)
)
↔
(
(
¬
(
X20
↔
(
(
¬
X19
)
→
X21
)
)
)
↔
(
¬
(
X49
→
(
(
¬
X20
)
↔
(
X19
↔
X42
)
)
)
)
)
)
↔
(
(
(
X43
↔
(
(
¬
X41
)
→
(
¬
X20
)
→
X49
)
)
→
(
¬
(
(
X41
→
(
(
¬
X16
)
↔
(
(
¬
X0
)
↔
X43
)
)
)
→
(
X49
↔
(
X41
→
(
(
¬
X16
)
↔
X43
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
X2
)
↔
X49
)
)
↔
(
¬
(
X49
→
X43
→
(
(
¬
X28
)
↔
X41
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
6a2656...
and proposition id is
e67165...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMJUjbihU5LJSwKMvJL6cZ5qf27FGispf7Q
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
∀X50 :
prop
,
∀X51 :
prop
,
(
(
¬
(
(
(
(
(
¬
(
(
(
(
¬
X37
)
→
(
¬
X25
)
→
(
X20
↔
X46
)
)
→
(
¬
(
X25
↔
(
X0
↔
X37
)
)
)
)
→
(
¬
(
(
¬
(
X42
→
X51
)
)
→
(
¬
(
X51
↔
(
X37
↔
(
X35
↔
X20
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
(
¬
(
X50
↔
(
(
¬
X40
)
↔
X45
)
)
)
↔
(
(
¬
X42
)
↔
(
X40
→
(
¬
X23
)
→
X45
)
)
)
→
(
¬
(
X50
→
(
X45
↔
(
X42
→
X35
→
False
)
)
)
)
)
)
→
(
¬
(
X50
→
(
¬
X45
)
→
X42
→
X51
)
)
)
)
↔
(
X50
↔
(
(
¬
X42
)
→
(
¬
X22
)
→
X51
)
)
)
)
→
(
(
¬
(
(
(
(
¬
X43
)
→
(
X5
↔
X45
)
)
↔
(
(
¬
(
(
¬
X45
)
→
(
X43
↔
(
X30
↔
X46
)
)
)
)
↔
(
(
¬
X43
)
→
(
(
¬
X39
)
↔
(
X30
→
X46
)
)
)
)
)
→
(
¬
(
(
X44
→
(
(
¬
X39
)
↔
X46
)
)
→
(
X44
↔
(
(
¬
X39
)
↔
(
X6
→
X43
)
)
)
)
)
)
)
↔
(
(
¬
(
X51
→
X43
→
(
¬
X36
)
→
X46
)
)
↔
(
¬
(
(
X51
→
(
(
¬
X43
)
↔
(
X12
→
X44
)
)
)
→
(
¬
(
X51
→
X44
→
(
X43
↔
(
X19
→
False
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
(
X46
→
X45
)
)
↔
(
X46
→
X44
→
(
X38
↔
X27
)
)
)
→
(
¬
(
X46
↔
X44
)
)
)
↔
(
(
(
(
¬
(
X47
↔
(
(
¬
X27
)
→
(
¬
X17
)
→
X46
)
)
)
↔
(
(
¬
(
X17
→
(
X15
↔
X27
)
)
)
↔
(
¬
(
X47
→
(
X27
↔
(
X10
→
X17
)
)
)
)
)
)
↔
(
(
¬
X27
)
→
(
(
¬
X17
)
↔
X46
)
)
)
→
(
(
(
¬
(
(
¬
X27
)
→
(
X15
↔
X44
)
)
)
↔
(
¬
(
(
X46
↔
(
X15
↔
X27
)
)
→
X46
→
(
X27
↔
(
(
¬
X9
)
→
X44
)
)
)
)
)
↔
(
¬
(
X44
↔
(
(
¬
X27
)
↔
X46
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
(
¬
X33
)
→
(
X28
↔
(
(
¬
X4
)
↔
X34
)
)
)
)
→
(
¬
(
X34
→
(
¬
X33
)
→
(
¬
X31
)
→
X47
)
)
)
)
↔
(
¬
(
X47
→
X33
→
X34
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
(
¬
X46
)
↔
(
(
¬
X34
)
→
X47
)
)
)
→
(
(
¬
(
X47
→
(
X46
↔
(
X45
↔
(
X33
→
False
)
)
)
)
)
↔
(
X46
↔
(
X45
→
X34
→
X47
)
)
)
)
)
↔
(
¬
(
X47
→
X46
)
)
)
→
(
(
¬
X45
)
↔
(
(
¬
X34
)
→
(
(
¬
X16
)
↔
X47
)
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
(
X32
↔
X50
)
↔
(
¬
(
(
¬
X49
)
↔
(
(
¬
X46
)
→
(
¬
X14
)
→
X50
)
)
)
)
→
(
(
(
(
X49
↔
(
(
¬
X3
)
→
X32
)
)
↔
(
(
(
¬
X32
)
→
(
(
¬
X29
)
↔
(
(
¬
X3
)
↔
X49
)
)
)
→
(
X46
↔
(
(
¬
X29
)
→
X3
→
X32
)
)
)
)
↔
(
¬
(
(
¬
X32
)
→
(
¬
X29
)
→
X46
)
)
)
↔
(
X49
→
(
¬
X32
)
→
X46
)
)
)
)
→
(
¬
(
X51
↔
(
(
¬
X48
)
→
(
X46
↔
X50
)
)
)
)
)
↔
(
(
(
(
X32
→
(
(
¬
X24
)
↔
(
(
¬
X7
)
↔
X29
)
)
)
↔
(
¬
(
X24
↔
(
X8
↔
(
(
¬
X7
)
→
X48
)
)
)
)
)
↔
(
(
False
↔
(
(
X41
↔
(
X32
↔
(
X29
↔
X2
)
)
)
→
X32
→
(
¬
X29
)
→
X26
→
X41
)
)
↔
(
(
(
¬
(
X29
↔
(
X26
↔
X41
)
)
)
→
(
¬
X26
)
→
(
¬
X21
)
→
(
¬
X8
)
→
X41
)
→
(
¬
(
X41
→
X29
→
X26
)
)
)
)
)
↔
(
(
¬
(
X48
↔
(
(
¬
X1
)
↔
X41
)
)
)
↔
(
¬
(
(
(
¬
(
X18
↔
(
(
¬
X13
)
→
X11
→
X48
)
)
)
↔
(
X41
↔
(
(
¬
X18
)
→
(
(
¬
X11
)
↔
X48
)
)
)
)
→
(
(
X41
↔
(
(
¬
X29
)
↔
X48
)
)
↔
(
¬
(
(
¬
(
X48
↔
(
(
¬
X1
)
→
X29
)
)
)
→
(
¬
(
X41
→
(
(
¬
X18
)
↔
X48
)
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
(
(
¬
(
X48
→
(
X40
↔
(
X26
→
X47
)
)
)
)
→
(
(
X26
↔
(
(
¬
X19
)
↔
(
X4
↔
X39
)
)
)
↔
(
¬
(
X19
↔
X48
)
)
)
)
→
(
(
¬
(
X42
↔
(
(
¬
X40
)
→
(
(
¬
X39
)
↔
X48
)
)
)
)
↔
(
(
X50
↔
(
X42
↔
(
(
¬
X40
)
↔
X48
)
)
)
↔
(
¬
(
X50
↔
(
(
¬
X12
)
↔
X48
)
)
)
)
)
)
)
→
(
X50
→
(
(
¬
X39
)
↔
X42
)
)
→
(
X39
↔
(
(
¬
X28
)
↔
(
(
¬
X14
)
→
X42
)
)
)
→
(
¬
(
X50
↔
(
X39
↔
(
X28
↔
X42
)
)
)
)
)
)
↔
(
(
X48
→
X33
→
(
X27
↔
X47
)
)
↔
(
(
(
¬
(
X34
→
(
X27
↔
(
X7
→
X48
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X34
→
(
¬
X27
)
→
X1
→
X50
)
)
↔
(
¬
(
(
¬
X34
)
↔
(
(
¬
X1
)
→
(
X0
↔
X50
)
)
)
)
)
→
(
¬
(
X34
↔
(
X27
↔
(
X1
→
False
)
)
)
)
)
→
(
¬
X48
)
→
(
¬
X34
)
→
(
¬
X27
)
→
X50
)
)
)
↔
(
(
(
X50
→
(
X43
↔
(
(
¬
X32
)
↔
X48
)
)
)
↔
(
(
X43
→
(
¬
X32
)
→
X9
→
X48
)
↔
(
¬
(
X43
→
X32
→
(
(
¬
X9
)
↔
X33
)
)
)
)
)
→
(
¬
(
X48
↔
(
(
¬
X43
)
↔
(
X5
→
X50
)
)
)
)
→
(
¬
(
X48
↔
(
(
¬
X43
)
↔
(
(
¬
X34
)
→
X50
)
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
X38
↔
(
X10
↔
X3
)
)
)
↔
(
¬
(
(
(
(
¬
X3
)
→
X21
)
→
(
X38
↔
(
X21
→
(
X10
↔
X11
)
)
)
)
→
(
¬
(
X38
↔
(
X11
→
X21
)
)
)
)
)
)
↔
(
(
(
X47
→
(
X16
↔
(
X11
↔
False
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
X35
)
→
X16
→
(
¬
X11
)
→
X44
)
→
(
¬
(
(
¬
X44
)
→
X47
)
)
)
)
→
(
¬
(
X35
→
X16
→
X44
)
)
)
)
)
↔
(
(
¬
(
X44
→
X29
→
(
X16
↔
X35
)
)
)
↔
(
¬
(
X38
↔
(
X29
→
(
X10
↔
X44
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
X50
→
(
X47
↔
(
X35
→
X51
)
)
)
↔
(
(
¬
(
(
¬
(
X51
↔
(
X6
↔
X25
)
)
)
→
(
¬
(
(
X51
→
(
X50
↔
(
X35
↔
(
X25
→
False
)
)
)
)
→
(
X50
↔
(
(
¬
X35
)
↔
(
X6
↔
X51
)
)
)
)
)
)
)
↔
(
(
¬
X50
)
↔
(
X35
↔
X51
)
)
)
)
↔
(
(
X45
↔
(
X35
→
(
X29
↔
(
X23
↔
False
)
)
)
)
↔
(
(
X51
↔
(
X47
→
X35
→
False
)
)
→
(
X51
↔
(
X47
↔
(
X18
→
X45
)
)
)
)
)
)
→
(
¬
(
(
(
¬
X50
)
↔
(
(
¬
X46
)
→
X22
→
X51
)
)
→
(
¬
(
X47
↔
(
X8
→
X51
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X41
)
→
(
¬
X17
)
→
(
(
¬
X13
)
↔
X43
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
(
X50
→
(
X43
↔
(
X24
↔
X45
)
)
)
)
→
(
¬
(
X50
→
(
(
¬
X43
)
↔
(
X41
↔
X45
)
)
)
)
)
)
→
(
X50
↔
(
X43
↔
False
)
)
)
)
↔
(
¬
(
(
(
X45
↔
(
(
¬
X41
)
↔
(
(
¬
X17
)
↔
X47
)
)
)
↔
(
¬
(
X47
→
(
X45
↔
(
(
¬
X30
)
↔
X41
)
)
)
)
)
→
(
¬
(
X47
→
X45
→
X41
→
X24
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
X41
)
→
(
¬
X31
)
→
(
¬
X2
)
→
X49
)
→
(
¬
(
X41
↔
(
(
¬
X20
)
↔
X49
)
)
)
)
→
(
¬
(
X41
↔
(
(
¬
X31
)
↔
(
X20
→
X43
)
)
)
)
)
)
→
(
(
(
X36
→
X30
→
(
¬
X15
)
→
X43
)
→
(
(
¬
X36
)
↔
X41
)
)
↔
(
X49
↔
(
X37
→
(
X36
↔
X15
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
11decd...
and proposition id is
8fc5b6...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMGhSAptGEgqte2qMaTvRLp2wULXkENuvY7
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
¬
(
(
(
(
(
¬
(
X10
↔
(
X8
↔
(
(
¬
X6
)
→
X17
)
)
)
)
→
X49
→
(
(
¬
X10
)
↔
(
(
¬
X8
)
→
X17
)
)
)
↔
(
¬
(
(
X49
↔
(
(
¬
X42
)
→
(
X10
↔
X47
)
)
)
→
(
¬
(
X47
↔
(
X41
↔
X10
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
(
X14
↔
(
X12
→
(
¬
X4
)
→
X21
)
)
→
(
¬
(
(
¬
X21
)
→
(
¬
X4
)
→
(
¬
X3
)
→
X42
)
)
)
→
(
¬
(
X49
→
X42
→
(
(
¬
X12
)
↔
X14
)
)
)
)
)
→
(
¬
(
(
X39
↔
(
X14
↔
(
X1
→
X46
)
)
)
→
(
¬
(
(
¬
(
X49
→
(
(
¬
X14
)
↔
X46
)
)
)
→
(
(
¬
(
X49
↔
(
X46
→
X1
)
)
)
↔
(
X21
→
(
X14
↔
False
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
(
¬
X44
)
→
(
¬
X39
)
→
X46
)
→
(
(
¬
X47
)
↔
(
X46
→
(
¬
X39
)
→
X49
)
)
)
→
(
¬
(
X49
↔
(
X20
↔
X46
)
)
)
)
)
→
(
(
X47
↔
(
X46
→
(
X28
↔
X49
)
)
)
↔
(
(
X47
→
X27
→
(
(
¬
X25
)
↔
X46
)
)
↔
(
(
¬
(
X47
↔
(
X27
→
(
¬
X25
)
→
X28
)
)
)
↔
(
X28
→
X27
→
X25
→
X0
)
)
)
)
)
)
)
→
(
¬
(
(
(
X42
→
X36
→
X9
)
↔
(
¬
(
X47
↔
(
X39
↔
(
(
¬
X9
)
→
X42
)
)
)
)
)
→
(
¬
(
(
¬
X39
)
↔
(
X31
→
X42
)
)
)
→
(
¬
X42
)
→
X16
→
X47
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
X7
)
↔
(
(
¬
X5
)
↔
X40
)
)
↔
(
(
X48
→
(
¬
X47
)
→
(
(
¬
X35
)
↔
X50
)
)
↔
(
¬
(
(
X47
↔
(
X45
→
(
(
¬
X43
)
↔
X50
)
)
)
→
(
¬
(
X48
↔
(
X45
→
X40
→
X47
)
)
)
)
)
)
)
↔
(
(
(
X47
↔
(
X15
→
X5
)
)
→
(
¬
(
(
(
(
¬
X15
)
→
(
(
¬
X13
)
↔
X33
)
)
↔
(
X33
→
(
(
¬
X13
)
↔
(
X5
→
X48
)
)
)
)
→
False
)
)
)
→
(
¬
(
(
¬
X33
)
→
X15
→
X47
)
)
)
)
→
(
¬
(
(
(
¬
(
(
(
X44
→
(
(
¬
X38
)
↔
X50
)
)
↔
(
(
¬
X44
)
→
X38
→
(
(
¬
X18
)
↔
X47
)
)
)
→
(
X38
↔
(
(
¬
X18
)
→
X44
)
)
)
)
→
X47
→
(
¬
X38
)
→
X26
→
X50
)
→
(
(
(
(
¬
X44
)
↔
(
(
¬
X30
)
→
X45
)
)
→
(
(
¬
(
X30
→
(
¬
X19
)
→
X45
)
)
↔
(
X50
↔
(
X45
↔
X44
)
)
)
)
↔
(
(
¬
(
(
¬
(
X47
↔
(
X45
→
X50
)
)
)
→
(
(
¬
(
X47
↔
(
X22
→
X30
)
)
)
↔
(
(
¬
(
(
X50
→
(
X45
↔
(
X30
→
X47
)
)
)
→
(
¬
(
X50
→
X47
→
(
X45
↔
X22
)
)
)
)
)
→
X47
→
X45
→
(
X22
↔
False
)
)
)
)
)
↔
(
¬
(
X47
→
(
X45
↔
X30
)
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
X34
)
→
(
X11
↔
X49
)
)
→
(
(
¬
(
X49
→
(
X37
↔
(
X11
↔
X45
)
)
)
)
↔
(
X49
→
(
¬
X45
)
→
X50
)
)
)
↔
(
¬
(
X50
→
(
¬
X37
)
→
X2
→
X49
)
)
)
→
(
¬
(
(
¬
(
X47
↔
(
X37
↔
(
X23
→
X49
)
)
)
)
→
X49
→
(
X37
↔
(
X32
↔
X47
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
X47
)
↔
(
X37
→
X29
→
X50
)
)
)
→
(
¬
(
X37
↔
(
(
¬
X29
)
→
(
(
¬
X24
)
↔
X47
)
)
)
)
)
)
↔
(
¬
(
X50
→
(
(
¬
X32
)
↔
(
X29
→
X37
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
(
¬
(
(
¬
X44
)
↔
(
X29
↔
(
X8
↔
X48
)
)
)
)
→
(
¬
(
X44
→
X29
→
(
¬
X8
)
→
X33
)
)
)
)
↔
(
(
¬
(
(
X41
↔
(
X33
→
X43
)
)
→
(
(
X41
↔
(
X33
→
False
)
)
↔
(
(
X43
→
(
¬
X41
)
→
(
¬
X33
)
→
X48
)
→
X48
→
X43
→
X33
→
X29
)
)
)
)
↔
(
¬
(
X48
↔
(
X43
→
X33
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X41
→
(
X16
↔
(
X15
→
X49
)
)
)
)
↔
(
(
X49
↔
(
X16
↔
(
X9
→
X24
)
)
)
↔
(
X24
↔
(
(
¬
X16
)
→
X9
→
X41
)
)
)
)
→
(
¬
(
(
¬
(
X49
→
(
(
¬
X41
)
↔
(
X33
→
X47
)
)
)
)
→
(
¬
(
(
¬
X29
)
↔
X49
)
)
→
(
X47
↔
(
X24
→
X41
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
X44
→
X36
→
X47
)
↔
(
¬
(
X44
→
(
(
¬
X31
)
↔
(
X18
→
X48
)
)
)
)
)
→
(
¬
(
(
¬
(
(
X46
→
X36
→
X48
)
→
(
¬
(
X40
↔
(
(
¬
X36
)
→
X47
)
)
)
)
)
→
(
¬
(
X48
↔
(
X47
→
X46
→
False
)
)
)
)
)
→
X46
→
X40
→
X47
)
↔
(
(
(
(
X47
→
X40
→
(
(
¬
X35
)
↔
X46
)
)
↔
(
¬
(
X47
↔
(
X46
→
X40
→
X11
→
False
)
)
)
)
→
(
¬
(
X46
→
(
¬
X40
)
→
X47
)
)
)
↔
(
X47
→
X46
→
(
(
¬
X2
)
↔
X40
)
)
)
)
→
(
¬
(
(
¬
(
X44
→
(
(
¬
X39
)
↔
(
(
¬
X12
)
→
X46
)
)
)
)
→
(
(
X44
→
X48
)
↔
(
(
X48
→
(
X46
↔
(
X39
↔
False
)
)
)
↔
(
(
¬
X23
)
↔
(
X5
→
X39
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
X49
)
→
X47
→
(
(
¬
X28
)
↔
X50
)
)
→
(
(
(
¬
(
X47
↔
(
X42
→
X50
)
)
)
→
(
¬
(
(
¬
X46
)
→
X42
→
(
¬
X0
)
→
X47
)
)
)
↔
(
¬
(
(
X13
↔
(
(
¬
X7
)
↔
X45
)
)
→
(
¬
(
X50
→
(
X47
↔
(
X45
→
X7
)
)
)
)
→
(
(
¬
(
(
¬
X47
)
→
(
(
¬
X46
)
↔
(
X17
→
X50
)
)
)
)
↔
(
¬
(
X50
↔
(
X17
↔
X45
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
X47
↔
(
X26
↔
(
X14
↔
False
)
)
)
↔
(
(
¬
(
(
X50
→
X47
)
→
(
(
X50
↔
(
X14
↔
X49
)
)
↔
(
¬
(
(
X49
↔
(
(
¬
X47
)
↔
(
(
¬
X14
)
↔
X50
)
)
)
→
(
¬
(
X50
↔
(
(
¬
X26
)
→
(
X14
↔
X49
)
)
)
)
)
)
)
)
)
↔
(
X50
↔
(
(
¬
X14
)
→
X49
)
)
)
)
→
(
X50
↔
(
(
¬
X46
)
↔
(
(
¬
X26
)
→
X49
)
)
)
)
)
→
(
¬
(
(
¬
X26
)
→
(
(
¬
X14
)
↔
X46
)
)
)
)
)
→
(
¬
(
(
¬
(
X47
↔
(
(
¬
X27
)
↔
(
(
¬
X25
)
→
X50
)
)
)
)
→
(
¬
(
X49
↔
(
X27
→
(
X25
↔
X50
)
)
)
)
)
)
)
→
(
(
(
(
(
(
¬
(
(
¬
(
X49
→
(
X47
↔
(
X10
→
X38
)
)
)
)
→
(
¬
(
(
¬
X38
)
↔
(
X34
→
(
(
¬
X10
)
↔
X47
)
)
)
)
)
)
↔
(
¬
(
(
¬
X38
)
→
(
(
¬
X34
)
↔
X49
)
)
)
)
↔
(
X49
→
(
X47
↔
(
X4
→
False
)
)
)
)
↔
(
¬
(
(
(
¬
(
X50
→
(
¬
X37
)
→
(
X1
↔
X49
)
)
)
↔
(
¬
(
(
¬
X34
)
↔
(
(
¬
X1
)
↔
X49
)
)
)
)
→
(
X49
↔
(
(
¬
X37
)
↔
(
X34
↔
X47
)
)
)
)
)
)
↔
(
(
(
¬
X19
)
→
(
¬
X1
)
→
X21
)
↔
(
¬
(
X49
↔
(
(
¬
X19
)
→
X34
)
)
)
)
)
↔
(
¬
(
(
(
X49
→
(
(
¬
X32
)
↔
(
(
¬
X30
)
→
X50
)
)
)
→
(
X32
↔
(
(
¬
X22
)
↔
X49
)
)
)
→
(
(
(
¬
(
X49
↔
(
X30
→
X47
)
)
)
→
(
X47
↔
(
X30
→
(
X6
↔
X3
)
)
)
)
↔
(
(
X32
→
X47
)
↔
(
¬
(
(
¬
(
X47
↔
(
X32
↔
(
X30
↔
(
X20
↔
False
)
)
)
)
)
→
X49
→
(
X47
↔
(
X32
→
(
X20
↔
False
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
c49aa9...
and proposition id is
f91f0f...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMKXC67PW1zgEVxKpexWMt3thXirKTuJnYv
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
(
(
¬
(
X33
↔
(
(
¬
X30
)
↔
(
X27
↔
X47
)
)
)
)
→
X30
→
X27
→
(
X11
↔
X33
)
)
↔
(
(
¬
(
(
¬
X42
)
→
(
¬
X33
)
→
X30
→
X46
)
)
→
(
¬
(
X27
→
X18
)
)
→
X46
→
(
X42
↔
(
X27
↔
X33
)
)
)
)
↔
(
(
(
¬
X43
)
→
(
(
¬
X36
)
↔
(
(
¬
X28
)
→
X47
)
)
)
↔
(
(
(
¬
X43
)
↔
(
X32
↔
(
X28
↔
X47
)
)
)
↔
(
(
¬
(
X47
↔
(
(
¬
X28
)
→
(
X4
↔
X42
)
)
)
)
↔
(
¬
(
(
¬
X42
)
→
(
X28
↔
(
X18
↔
X43
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
X40
→
X31
→
(
X26
↔
X42
)
)
→
(
¬
(
(
¬
X31
)
→
(
X26
↔
(
(
¬
X15
)
↔
X40
)
)
)
)
)
)
↔
(
X42
→
(
X40
↔
(
X26
↔
(
X24
→
False
)
)
)
)
)
↔
(
(
(
X46
↔
(
X25
↔
(
X12
↔
X44
)
)
)
↔
(
¬
(
(
X13
→
(
X12
↔
(
(
¬
X7
)
→
X41
)
)
)
→
(
¬
(
X44
→
X25
→
X7
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
X47
↔
(
(
¬
X42
)
→
(
(
¬
X24
)
↔
X46
)
)
)
→
(
(
X42
→
X40
→
X19
→
False
)
↔
(
(
¬
(
X46
→
(
¬
X44
)
→
(
(
¬
X42
)
↔
X47
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X44
)
↔
(
X8
↔
X47
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X47
→
X41
→
(
(
¬
X0
)
↔
X42
)
)
)
↔
(
(
(
¬
X44
)
→
(
(
¬
X5
)
↔
X47
)
)
↔
(
(
X5
↔
X44
)
↔
(
(
¬
(
(
¬
X42
)
↔
(
(
¬
X41
)
↔
X47
)
)
)
→
(
¬
(
X47
↔
(
X44
→
X5
→
X42
)
)
)
)
)
)
)
→
X42
→
X5
→
False
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
(
(
¬
X45
)
→
(
X23
↔
(
(
¬
X21
)
↔
X46
)
)
)
↔
(
X46
→
(
X45
↔
(
X23
→
X6
→
False
)
)
)
)
↔
(
(
¬
(
(
¬
X45
)
→
(
X39
↔
(
(
¬
X29
)
→
X49
)
)
)
)
→
(
(
(
¬
(
(
¬
X23
)
↔
(
X16
→
X39
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X16
)
→
X21
)
)
)
)
↔
(
¬
(
(
¬
(
X48
→
X45
→
(
X16
↔
X23
)
)
)
→
(
(
¬
(
X21
→
(
X16
↔
X23
)
)
)
↔
(
¬
(
X48
↔
(
X1
↔
X21
)
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
X47
)
→
(
(
¬
X46
)
↔
(
X38
↔
X49
)
)
)
)
↔
(
X49
↔
(
X46
→
(
X38
↔
(
X22
→
False
)
)
)
)
)
→
(
(
¬
(
X38
↔
(
X37
↔
X22
)
)
)
↔
(
¬
(
(
¬
(
X47
→
X38
→
X22
→
False
)
)
→
(
(
X37
↔
(
X22
↔
(
X17
→
X47
)
)
)
↔
(
X47
→
(
X38
↔
(
X37
↔
False
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X49
↔
(
X47
→
X46
→
X9
)
)
)
↔
(
(
(
¬
X9
)
→
(
¬
X2
)
→
X44
)
→
(
¬
(
(
¬
X46
)
→
(
X44
↔
(
(
¬
X2
)
→
X49
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
X10
→
(
(
¬
X9
)
↔
X48
)
)
↔
(
(
(
¬
(
X49
→
(
(
¬
X9
)
↔
(
(
¬
X3
)
↔
X48
)
)
)
)
↔
(
X48
↔
(
X9
→
X3
→
X10
)
)
)
↔
(
X49
↔
(
X48
→
X3
)
)
)
)
↔
(
¬
(
X48
→
X9
)
)
)
↔
(
¬
False
)
)
→
(
¬
(
X48
→
X10
→
X49
)
)
)
)
→
(
¬
(
X48
→
(
X46
↔
X10
)
)
)
)
)
)
)
)
→
(
(
(
(
X49
↔
(
X47
↔
(
X46
↔
X34
)
)
)
↔
(
¬
(
X47
↔
(
(
¬
X20
)
→
X46
)
)
)
)
↔
(
X47
→
(
(
¬
X20
)
↔
X34
)
)
)
↔
(
(
¬
(
(
¬
X47
)
↔
(
X46
→
X14
→
X49
)
)
)
↔
(
X46
→
(
X35
↔
False
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X47
→
(
X46
↔
(
X45
→
X4
)
)
)
)
→
(
(
(
(
¬
(
X46
↔
(
X18
→
(
X4
↔
X25
)
)
)
)
↔
(
¬
(
X18
↔
(
(
¬
X0
)
↔
X25
)
)
)
)
↔
(
(
¬
X0
)
↔
X18
)
)
↔
(
(
¬
(
X45
↔
(
(
¬
X25
)
→
(
(
¬
X6
)
↔
X46
)
)
)
)
↔
(
(
¬
(
X45
→
X25
→
(
X4
↔
False
)
)
)
↔
(
(
X48
→
(
X25
↔
X4
)
)
→
(
¬
(
X46
↔
(
X45
↔
X4
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
(
¬
X32
)
→
(
¬
(
X46
↔
(
X32
→
(
X15
↔
X45
)
)
)
)
→
(
¬
(
X45
→
(
X44
↔
(
X32
↔
(
X15
↔
False
)
)
)
)
)
)
↔
(
X37
→
(
(
¬
X33
)
↔
(
(
¬
X32
)
↔
X43
)
)
)
)
→
(
(
(
X38
→
(
(
¬
X31
)
↔
X45
)
)
→
(
¬
(
X43
→
(
¬
X42
)
→
X44
)
)
)
↔
(
¬
(
X45
→
(
X42
↔
(
X10
→
X31
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X48
↔
(
X37
↔
(
(
¬
X1
)
↔
X14
)
)
)
)
↔
(
(
X14
↔
(
(
¬
X13
)
↔
(
X1
↔
X37
)
)
)
→
(
¬
(
(
¬
X37
)
→
(
¬
X14
)
→
(
¬
X1
)
→
X43
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
¬
X21
)
→
(
¬
X14
)
→
X46
)
)
→
X36
→
X21
→
X48
)
)
→
(
¬
(
(
¬
X36
)
→
(
X14
↔
(
X13
→
X48
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
X49
→
X26
→
X34
)
↔
(
(
(
(
X40
→
X28
→
(
X26
↔
X5
)
)
→
(
(
¬
X40
)
↔
(
X26
→
(
X5
↔
X49
)
)
)
)
↔
(
(
¬
X34
)
→
X28
→
(
(
¬
X26
)
↔
X40
)
)
)
↔
(
(
¬
(
X28
→
(
X23
↔
(
(
¬
X5
)
↔
X40
)
)
)
)
↔
(
¬
(
(
¬
X28
)
↔
(
X23
→
(
X3
↔
X40
)
)
)
)
)
)
)
→
X42
→
(
X28
↔
(
(
¬
X5
)
↔
X40
)
)
)
→
(
(
¬
(
(
(
¬
(
X40
↔
(
(
¬
X19
)
→
X30
)
)
)
↔
(
(
¬
(
(
¬
X34
)
↔
(
(
¬
X30
)
↔
(
X19
↔
X48
)
)
)
)
→
(
(
(
X40
↔
(
(
¬
X34
)
→
X9
→
X48
)
)
→
(
¬
(
X48
↔
(
X40
↔
(
(
¬
X28
)
↔
X34
)
)
)
)
)
↔
(
¬
(
(
¬
X40
)
↔
(
(
¬
X34
)
↔
(
(
¬
X30
)
→
X48
)
)
)
)
)
)
)
→
(
X16
↔
X2
)
→
(
¬
(
X40
↔
(
(
¬
X34
)
↔
X48
)
)
)
)
)
↔
(
(
(
¬
X22
)
↔
(
X19
→
(
X17
↔
X34
)
)
)
↔
(
¬
(
X34
→
(
(
¬
X19
)
↔
(
(
¬
X16
)
→
X22
)
)
)
)
)
)
)
)
→
(
(
(
X41
→
(
X39
↔
X48
)
)
↔
(
¬
(
(
(
X41
↔
(
X39
→
(
X27
↔
(
X12
↔
False
)
)
)
)
↔
(
(
(
X41
↔
(
X27
→
(
X26
↔
(
X24
→
False
)
)
)
)
↔
(
X41
→
(
X29
↔
False
)
)
)
↔
(
(
¬
X39
)
↔
(
(
¬
X29
)
→
(
(
¬
X26
)
↔
X41
)
)
)
)
)
→
(
¬
(
X41
↔
(
(
¬
X29
)
→
X26
→
X48
)
)
)
)
)
)
↔
(
¬
(
X47
→
(
¬
X41
)
→
X39
→
X48
)
)
)
→
(
(
X41
→
X11
→
X47
)
→
(
X47
↔
(
(
¬
X42
)
→
(
X11
↔
X48
)
)
)
)
→
(
(
(
¬
(
(
¬
(
X20
↔
(
X8
→
(
(
¬
X7
)
↔
X42
)
)
)
)
→
(
X42
↔
(
X35
→
(
(
¬
X7
)
↔
X8
)
)
)
)
)
↔
(
¬
(
X8
↔
X7
)
)
)
↔
(
¬
(
X48
→
(
¬
X42
)
→
X35
→
X47
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
2b50da...
and proposition id is
80748d...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMJyP6ebQQyTTHJyGe5FiQUwY3dxThprTfG
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
∀X50 :
prop
,
(
(
(
(
¬
(
(
¬
(
(
¬
(
(
X46
→
X45
→
X28
)
→
(
(
¬
X45
)
↔
(
(
¬
X32
)
→
X49
)
)
)
)
→
(
(
(
(
X10
↔
(
X5
→
X30
)
)
↔
(
¬
(
X39
→
(
X10
↔
(
X5
→
X17
)
)
)
)
)
↔
(
¬
(
(
¬
X39
)
→
(
X30
↔
(
X10
↔
X46
)
)
)
)
)
↔
(
(
X45
↔
(
X30
→
(
X28
↔
X39
)
)
)
→
(
¬
(
X45
↔
(
(
¬
X39
)
→
X30
→
X46
)
)
)
)
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X12
)
→
X30
)
)
)
→
(
(
X48
→
X46
→
(
X38
↔
X12
)
)
↔
(
(
¬
X30
)
↔
(
(
¬
X28
)
→
X46
)
)
)
)
)
→
(
(
X46
→
(
X41
↔
(
X13
→
X19
)
)
)
↔
(
(
(
X49
↔
(
X13
↔
False
)
)
↔
(
(
X49
→
X37
→
X13
→
X11
→
False
)
↔
(
(
X13
↔
(
X0
↔
X11
)
)
→
(
¬
(
X37
→
X19
→
X13
)
)
)
)
)
↔
(
X11
↔
(
X0
→
X48
)
)
)
)
→
(
(
(
(
¬
X37
)
↔
(
(
¬
X14
)
↔
(
X1
↔
X48
)
)
)
↔
(
¬
(
X37
↔
(
X14
→
(
(
¬
X1
)
↔
X49
)
)
)
)
)
↔
(
(
(
¬
(
X48
→
(
¬
X37
)
→
(
(
¬
X31
)
↔
X47
)
)
)
↔
(
¬
(
X48
↔
(
X37
↔
(
(
¬
X16
)
→
X47
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X47
→
(
¬
X25
)
→
X48
)
)
↔
(
(
¬
(
X47
↔
(
(
¬
X37
)
↔
(
(
¬
X25
)
→
X49
)
)
)
)
↔
(
X48
↔
X25
)
)
)
→
(
¬
(
X48
↔
(
X47
↔
(
X37
→
False
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
X45
↔
(
(
¬
X34
)
↔
X44
)
)
)
↔
(
¬
(
(
X45
↔
False
)
→
(
¬
X36
)
→
X34
→
(
(
¬
X26
)
↔
X44
)
)
)
)
↔
(
(
¬
(
X36
→
(
X34
↔
(
X27
→
X44
)
)
)
)
↔
(
¬
(
(
¬
X27
)
↔
(
X3
↔
X36
)
)
)
)
)
→
(
¬
(
X49
↔
(
X36
↔
X34
)
)
)
→
(
¬
(
(
¬
X44
)
↔
(
X36
→
X49
)
)
)
→
X44
→
(
(
¬
X20
)
↔
X34
)
)
↔
(
(
¬
(
(
(
(
¬
(
(
¬
X44
)
→
X43
→
X33
→
X49
)
)
↔
(
¬
(
(
¬
X43
)
→
(
(
¬
X33
)
↔
(
(
¬
X18
)
→
X49
)
)
)
)
)
↔
(
¬
(
X44
→
(
X41
↔
X43
)
)
)
)
→
(
¬
(
(
¬
X44
)
↔
(
(
¬
X33
)
→
(
¬
X18
)
→
X45
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X45
)
↔
(
X29
↔
(
X2
→
X50
)
)
)
)
↔
(
X45
→
(
¬
X29
)
→
(
¬
X24
)
→
X50
)
)
↔
(
¬
(
X50
↔
X29
)
)
)
↔
(
(
¬
(
(
¬
X43
)
↔
(
X29
→
X44
)
)
)
↔
(
X45
↔
(
X44
→
X50
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
¬
X42
)
→
X41
→
(
(
¬
X9
)
↔
X45
)
)
)
→
(
(
¬
(
X45
→
(
X40
↔
X41
)
)
)
↔
(
(
¬
X42
)
→
(
X41
↔
(
(
¬
X40
)
→
X50
)
)
)
)
)
)
→
(
¬
(
(
X50
→
(
X22
↔
X41
)
)
→
(
(
X41
↔
(
X22
→
X15
)
)
↔
(
¬
(
X22
↔
(
X21
→
X15
→
X50
)
)
)
)
)
)
→
(
¬
(
(
(
X42
↔
(
X15
→
X7
→
X23
)
)
→
(
¬
(
X42
↔
(
X23
→
(
(
¬
X15
)
↔
X50
)
)
)
)
)
→
(
(
¬
X23
)
↔
(
X21
↔
(
X15
→
X50
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
X40
)
→
(
(
¬
X35
)
↔
(
X8
↔
X45
)
)
)
)
↔
(
(
¬
(
(
¬
X23
)
↔
(
(
¬
X6
)
→
X4
→
X35
)
)
)
→
(
¬
(
X45
↔
(
(
¬
X35
)
↔
X40
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
X46
)
→
X39
→
X48
)
→
(
(
¬
(
X48
→
X36
→
X29
→
X43
)
)
↔
(
¬
(
(
¬
X43
)
→
(
¬
X39
)
→
(
X29
↔
X46
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X46
→
X39
→
X45
)
)
↔
(
X45
→
(
X39
↔
(
(
¬
X8
)
↔
X50
)
)
)
)
→
(
¬
(
X50
→
(
X46
↔
X39
)
)
)
)
)
↔
(
(
¬
(
X45
↔
(
X19
↔
False
)
)
)
→
(
¬
(
X45
→
(
X36
↔
(
X8
↔
X39
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
(
¬
X24
)
↔
(
X13
↔
X46
)
)
)
↔
(
¬
(
(
¬
(
X46
→
(
X24
↔
(
X4
↔
X13
)
)
)
)
→
(
(
¬
X10
)
↔
(
X0
→
X24
)
)
)
)
)
→
X15
→
(
X0
↔
X13
)
)
)
→
(
¬
(
(
X45
→
(
X43
↔
(
X25
→
X4
)
)
)
→
(
¬
(
X45
↔
(
X42
→
X43
)
)
)
→
(
(
(
X45
→
(
(
¬
X10
)
↔
X42
)
)
↔
(
(
¬
(
X45
→
(
X13
↔
X10
)
)
)
↔
(
¬
(
X45
↔
(
X43
→
X13
→
X42
)
)
)
)
)
↔
(
¬
(
X45
↔
(
X43
→
X42
→
(
X4
↔
False
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
X37
↔
(
X31
↔
(
X21
→
X12
)
)
)
↔
(
X46
↔
(
X35
→
X21
→
False
)
)
)
↔
(
¬
(
X37
→
(
(
¬
X31
)
↔
(
(
¬
X28
)
→
X46
)
)
)
)
)
↔
(
(
(
X44
→
(
X38
↔
(
X5
↔
X31
)
)
)
→
(
X35
↔
(
(
¬
X31
)
→
X44
)
)
)
↔
(
(
(
¬
(
X28
↔
(
(
¬
X3
)
→
X18
)
)
)
↔
(
(
(
¬
X35
)
↔
(
(
¬
X3
)
→
X44
)
)
→
(
(
(
X44
→
X38
→
(
X35
↔
False
)
)
↔
(
(
¬
(
X38
↔
(
X35
↔
(
X18
→
X28
)
)
)
)
→
(
(
¬
X35
)
↔
(
X18
↔
(
X3
→
X44
)
)
)
)
)
↔
(
¬
(
X38
→
X35
→
X44
)
)
)
)
)
→
(
¬
(
X28
↔
(
(
¬
X18
)
→
X38
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
X38
)
↔
(
(
¬
X34
)
→
(
¬
X30
)
→
X44
)
)
)
↔
(
¬
(
X30
↔
(
(
¬
X2
)
→
X34
)
)
)
)
↔
(
(
X34
↔
(
(
¬
X27
)
↔
X38
)
)
↔
(
¬
(
X44
↔
X37
)
)
)
)
→
(
¬
(
X37
→
(
¬
X23
)
→
(
X2
↔
X34
)
)
)
)
→
(
(
(
¬
(
(
(
(
(
¬
X17
)
→
(
(
¬
X14
)
↔
X26
)
)
↔
(
X44
→
(
X27
↔
(
X17
↔
X14
)
)
)
)
↔
(
¬
(
(
¬
X14
)
→
X27
)
)
)
→
(
X27
↔
X44
)
)
)
↔
(
¬
(
X30
→
X17
→
False
)
)
)
↔
(
X44
→
(
X30
↔
X27
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
X40
↔
(
(
¬
X39
)
↔
(
X6
→
X49
)
)
)
→
(
(
(
¬
X39
)
↔
X40
)
↔
(
(
(
X49
→
(
¬
X16
)
→
X42
)
↔
(
X40
→
(
X39
↔
X49
)
)
)
↔
(
¬
(
X40
↔
(
X16
↔
False
)
)
)
)
)
)
)
↔
(
¬
(
X49
→
X41
→
X40
→
(
X37
↔
False
)
)
)
)
↔
(
(
(
(
X39
→
(
X20
↔
False
)
)
↔
(
(
X39
→
X20
→
X41
)
→
(
¬
(
(
¬
X39
)
↔
(
X20
→
X9
→
X42
)
)
)
)
)
↔
(
¬
(
X39
↔
False
)
)
)
→
(
¬
(
X39
→
(
¬
X20
)
→
(
X9
↔
X47
)
)
)
)
)
→
(
¬
(
(
¬
(
(
X49
↔
(
X46
→
(
X42
↔
X47
)
)
)
→
(
(
¬
(
X49
→
X46
→
X1
→
X38
)
)
↔
(
(
¬
X11
)
→
X47
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X49
→
(
X47
↔
(
X42
→
(
X7
↔
False
)
)
)
)
)
↔
(
¬
(
X33
↔
(
(
¬
X32
)
→
(
X7
↔
X42
)
)
)
)
)
↔
(
¬
(
X32
→
(
(
¬
X7
)
↔
X42
)
)
)
)
→
(
¬
(
X49
→
X42
→
X22
→
X47
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
823ddb...
and proposition id is
3a9e0c...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMbRAQfZnnDdR7AnMpgCAjhxzBd7MYse1a5
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
¬
(
(
(
(
(
(
(
(
X50
→
(
X48
↔
X38
)
)
→
(
¬
(
X49
→
X48
→
(
X40
↔
False
)
)
)
)
→
X40
→
(
X38
↔
(
X7
→
X48
)
)
)
↔
(
(
(
X49
↔
(
(
¬
X13
)
→
X28
)
)
↔
(
(
¬
X48
)
↔
(
X38
↔
(
(
¬
X13
)
→
X49
)
)
)
)
↔
(
¬
(
X27
→
(
X13
↔
X49
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X40
→
(
X33
↔
(
(
¬
X15
)
↔
X50
)
)
)
)
→
X48
→
(
(
¬
X40
)
↔
(
(
¬
X33
)
↔
X50
)
)
)
)
→
(
¬
(
X33
↔
(
X18
→
X15
→
(
X5
↔
False
)
)
)
)
)
)
)
→
(
(
¬
(
X50
→
X49
→
X48
)
)
↔
(
(
¬
(
X36
↔
False
)
)
↔
(
¬
(
(
¬
(
(
X49
→
(
X48
↔
(
(
¬
X36
)
↔
X50
)
)
)
→
X48
→
X36
→
(
(
¬
X9
)
↔
X50
)
)
)
→
(
¬
(
X36
↔
X48
)
)
)
)
)
)
)
→
(
(
(
(
X48
→
X3
)
↔
(
(
X50
↔
(
X49
→
(
X47
↔
(
X3
→
False
)
)
)
)
→
(
X49
↔
(
X47
→
(
X46
↔
X3
)
)
)
)
)
→
(
(
¬
(
X47
→
(
¬
X46
)
→
X19
→
X48
)
)
↔
(
(
¬
X46
)
→
(
X44
↔
(
(
¬
X19
)
↔
X47
)
)
)
)
)
→
(
(
(
X50
→
X47
→
X49
)
↔
(
(
(
¬
(
X50
→
(
¬
X41
)
→
X39
→
X47
)
)
↔
(
X50
↔
(
X49
↔
(
(
¬
X41
)
→
X47
)
)
)
)
↔
(
X49
→
(
¬
X47
)
→
(
¬
X45
)
→
X50
)
)
)
↔
(
¬
(
(
¬
X45
)
↔
(
(
¬
X41
)
↔
(
(
¬
X10
)
↔
X47
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X44
↔
(
X42
→
(
(
¬
X4
)
↔
X45
)
)
)
)
→
(
(
X44
→
(
¬
X42
)
→
X48
)
↔
(
(
X45
→
X42
→
False
)
→
(
¬
(
(
X45
→
(
X44
↔
(
X34
→
X48
)
)
)
→
(
¬
(
(
¬
X45
)
↔
(
X44
↔
(
X42
→
X48
)
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
X44
→
(
X42
↔
(
X26
↔
X31
)
)
)
→
(
X35
↔
(
(
¬
X26
)
→
X31
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X48
↔
(
X42
→
X2
→
X35
)
)
)
↔
(
¬
(
X48
↔
(
X32
↔
(
X2
→
X35
)
)
)
)
)
→
(
¬
(
(
¬
X42
)
→
(
X35
↔
X48
)
)
)
)
)
↔
(
(
¬
(
(
¬
X42
)
↔
(
(
¬
X31
)
→
(
(
¬
X8
)
↔
X48
)
)
)
)
↔
(
X35
↔
(
X31
→
False
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X50
→
(
(
¬
X1
)
↔
X25
)
)
)
↔
(
(
(
¬
X37
)
↔
(
X29
→
(
X1
↔
X50
)
)
)
↔
(
(
¬
X47
)
↔
(
X37
→
(
(
¬
X1
)
↔
X50
)
)
)
)
)
↔
(
(
(
¬
(
X21
→
(
¬
X20
)
→
X23
)
)
→
(
(
¬
X43
)
↔
(
(
¬
X37
)
↔
X47
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
X49
↔
(
(
¬
X37
)
↔
(
(
¬
X21
)
↔
X50
)
)
)
)
→
X49
→
(
¬
X37
)
→
(
¬
X30
)
→
X50
)
)
↔
(
¬
(
X50
↔
(
X49
→
X37
)
)
)
)
→
(
¬
(
X30
→
(
¬
X23
)
→
(
(
¬
X0
)
↔
X37
)
)
)
)
)
)
)
↔
(
(
(
(
X24
→
(
(
¬
X22
)
↔
(
(
¬
X14
)
→
X43
)
)
)
→
(
¬
(
(
(
¬
X24
)
→
(
(
¬
X22
)
↔
(
(
¬
X14
)
↔
X50
)
)
)
→
X24
→
(
(
¬
X22
)
↔
(
(
¬
X11
)
↔
X50
)
)
)
)
)
↔
(
¬
(
X43
↔
(
(
¬
X16
)
↔
(
(
¬
X14
)
↔
X50
)
)
)
)
)
→
(
X17
↔
(
(
¬
X16
)
↔
(
(
¬
X12
)
→
X24
)
)
)
→
(
¬
(
(
¬
(
(
¬
X21
)
↔
(
(
¬
X12
)
↔
(
X6
↔
X43
)
)
)
)
→
(
¬
(
X43
→
(
X6
↔
X17
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X49
→
X46
→
(
(
¬
X13
)
↔
X48
)
)
)
→
(
¬
(
(
(
X49
→
(
X46
↔
(
X38
↔
X50
)
)
)
↔
(
(
(
(
¬
X37
)
↔
(
(
¬
X29
)
→
X48
)
)
↔
(
¬
(
(
¬
X37
)
→
(
X28
↔
(
(
¬
X24
)
→
X46
)
)
)
)
)
↔
(
(
X38
→
X3
)
→
(
(
X49
↔
(
X28
↔
(
(
¬
X3
)
↔
X33
)
)
)
↔
(
X28
→
(
(
¬
X19
)
↔
(
(
¬
X3
)
→
X49
)
)
)
)
→
(
¬
(
(
¬
X29
)
→
(
X19
↔
(
X3
→
X33
)
)
)
)
)
)
)
→
X24
→
X19
→
(
(
¬
X6
)
↔
X33
)
)
)
)
)
↔
(
(
¬
(
(
X49
→
(
X35
↔
(
X10
→
X50
)
)
)
→
(
¬
(
(
X49
↔
(
X45
→
(
¬
X26
)
→
X35
)
)
→
(
¬
(
X49
→
(
(
¬
X35
)
↔
(
(
¬
X26
)
→
X50
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
(
¬
X45
)
→
(
X39
↔
(
X12
→
X47
)
)
)
)
↔
(
¬
(
(
¬
(
(
X45
→
(
¬
X31
)
→
(
X11
↔
X47
)
)
→
(
(
X47
→
(
X31
↔
X45
)
)
↔
(
X47
→
X45
→
X16
→
(
X12
↔
False
)
)
)
)
)
→
(
(
¬
X12
)
↔
X45
)
)
)
)
→
(
(
X47
→
X31
)
↔
(
¬
(
(
X31
→
X47
)
→
(
¬
(
X47
↔
(
X45
→
(
¬
X15
)
→
X50
)
)
)
)
)
)
)
)
→
(
(
(
¬
X49
)
→
(
¬
X36
)
→
X8
→
X50
)
↔
(
¬
(
(
¬
(
(
X49
→
X38
→
(
X36
↔
(
X14
↔
False
)
)
)
→
(
X45
↔
(
(
¬
X38
)
↔
(
X36
↔
X49
)
)
)
)
)
→
(
¬
(
X47
↔
(
X14
→
X49
)
)
)
→
(
¬
(
(
¬
X38
)
↔
(
X14
↔
X50
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X49
↔
(
(
¬
X21
)
→
(
X20
↔
X41
)
)
)
)
→
(
¬
(
X20
↔
(
(
¬
X1
)
↔
X49
)
)
)
)
↔
(
(
¬
(
(
(
(
(
¬
X44
)
↔
(
X21
↔
(
(
¬
X2
)
↔
(
X0
→
X49
)
)
)
)
↔
(
(
¬
X41
)
→
(
X2
↔
X43
)
)
)
↔
(
X41
↔
(
X2
→
(
X0
↔
False
)
)
)
)
→
(
¬
(
(
¬
X27
)
→
X44
)
)
→
(
¬
(
(
¬
X43
)
↔
(
X42
↔
(
X27
↔
X49
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X49
↔
(
X42
↔
(
(
¬
X9
)
↔
X40
)
)
)
)
↔
(
X49
↔
(
X42
→
(
X9
↔
X44
)
)
)
)
→
(
(
(
(
¬
(
X49
→
(
X42
↔
(
X17
→
X40
)
)
)
)
↔
(
¬
(
(
¬
X47
)
↔
(
X42
↔
(
X17
→
X49
)
)
)
)
)
↔
(
¬
(
X17
↔
X42
)
)
)
↔
(
¬
(
X42
→
X40
→
X49
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
X4
↔
X34
)
→
(
X50
↔
(
X49
→
X5
)
)
)
↔
(
(
(
(
X47
→
X30
→
(
(
¬
X18
)
↔
X34
)
)
→
(
¬
(
(
¬
X47
)
↔
(
X30
↔
X49
)
)
)
)
↔
(
¬
(
X30
→
(
(
¬
X18
)
↔
X47
)
)
)
)
→
(
(
(
X50
↔
(
X30
→
X34
)
)
↔
(
(
X49
↔
(
X34
↔
(
X23
→
(
X7
↔
False
)
)
)
)
↔
(
X49
↔
(
X34
→
(
X7
↔
X30
)
)
)
)
)
↔
(
X50
↔
(
(
¬
X30
)
↔
X49
)
)
)
)
)
→
(
¬
(
(
¬
(
X44
↔
(
X34
↔
(
X32
↔
X50
)
)
)
)
→
(
¬
(
X50
→
(
X34
↔
(
X25
→
X44
)
)
)
)
)
)
→
(
(
X44
↔
(
X22
↔
X49
)
)
↔
(
(
X49
→
X34
→
X22
→
X25
)
↔
(
¬
(
X44
→
(
X25
↔
(
(
¬
X22
)
→
X34
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
8ff5c2...
and proposition id is
60d138...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMRNezK6BhbspZvsUo2iVkP7wEAzcV5La6m
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
(
(
(
(
¬
(
(
¬
X45
)
↔
(
(
¬
X43
)
↔
X48
)
)
)
→
(
¬
(
(
(
(
X35
→
X43
)
↔
(
(
X29
→
(
¬
X19
)
→
X45
)
↔
(
¬
(
X43
→
X35
→
(
X19
↔
X45
)
)
)
)
)
↔
(
X29
→
X45
)
)
→
(
(
(
¬
(
(
(
¬
(
X45
→
X48
)
)
↔
(
¬
(
X45
↔
(
X35
→
X19
→
X25
)
)
)
)
→
(
¬
(
(
¬
X25
)
→
X14
→
X35
)
)
)
)
↔
(
¬
(
X48
↔
(
X35
↔
X45
)
)
)
)
↔
(
X48
→
(
X35
↔
(
X32
↔
X45
)
)
)
)
)
)
)
→
(
(
(
X45
→
(
X44
↔
X43
)
)
→
(
¬
(
X48
↔
(
(
¬
X17
)
→
(
¬
X16
)
→
X43
)
)
)
)
↔
(
(
X44
→
X20
→
(
¬
X10
)
→
X43
)
→
X43
→
(
¬
X20
)
→
X10
→
X45
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
X22
→
X5
→
X0
→
X48
)
→
(
¬
(
X38
→
X22
→
X5
)
)
)
)
↔
(
¬
(
(
X43
↔
(
(
¬
X22
)
↔
X39
)
)
→
(
¬
(
(
¬
(
X48
↔
(
X43
↔
(
X30
→
X22
)
)
)
)
→
(
¬
(
X48
↔
(
X39
→
X0
→
X22
)
)
)
)
)
)
)
)
↔
(
¬
(
(
X45
→
X38
→
(
¬
X12
)
→
X48
)
→
(
X48
↔
(
X39
↔
False
)
)
)
)
)
→
(
(
(
(
X48
→
X45
→
X42
)
→
(
(
¬
X39
)
→
(
(
¬
X4
)
↔
X48
)
)
→
(
(
¬
X45
)
↔
(
X42
→
(
¬
X4
)
→
X48
)
)
)
↔
(
¬
(
(
¬
X39
)
↔
(
X4
→
(
¬
X2
)
→
X42
)
)
)
)
↔
(
¬
(
X48
→
(
(
¬
X42
)
↔
(
(
¬
X15
)
↔
X45
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
¬
X49
)
↔
X50
)
↔
(
¬
(
(
X47
↔
(
X38
→
X8
→
X36
)
)
→
(
¬
(
(
X38
→
X47
)
→
(
(
¬
(
X47
↔
(
(
¬
X9
)
→
X49
)
)
)
↔
(
(
X47
→
(
(
¬
X9
)
↔
(
X8
→
X49
)
)
)
→
(
¬
(
X49
→
(
¬
X38
)
→
X8
→
X47
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
X38
→
(
X28
↔
X42
)
)
↔
(
(
¬
X36
)
↔
(
X28
↔
(
X11
→
X38
)
)
)
)
→
(
¬
(
(
X42
→
(
(
¬
X36
)
↔
(
X24
↔
X49
)
)
)
→
(
¬
(
(
¬
X36
)
↔
(
X24
↔
X38
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
X21
)
→
(
¬
X18
)
→
X27
)
→
(
(
¬
X42
)
↔
(
(
¬
X21
)
↔
(
(
¬
X18
)
→
X49
)
)
)
)
→
(
¬
(
(
¬
X42
)
→
(
(
¬
X37
)
↔
(
X27
→
X49
)
)
)
)
)
↔
(
(
¬
X37
)
↔
X47
)
)
)
→
(
¬
(
(
¬
(
(
(
(
¬
X39
)
↔
(
(
¬
X34
)
↔
X47
)
)
↔
(
¬
(
X47
→
(
X46
↔
(
X6
↔
False
)
)
)
)
)
→
(
(
(
(
(
¬
(
X47
→
(
X31
↔
(
(
¬
X7
)
↔
X23
)
)
)
)
→
(
¬
X33
)
→
(
X23
↔
X47
)
)
↔
(
¬
(
(
¬
X33
)
→
(
(
¬
X7
)
↔
(
X3
→
X47
)
)
)
)
)
↔
(
¬
(
(
¬
X39
)
→
X33
→
X23
→
X47
)
)
)
↔
(
(
(
(
¬
X33
)
↔
(
X26
↔
(
(
¬
X23
)
→
X46
)
)
)
↔
(
¬
(
X46
→
(
(
¬
X23
)
↔
(
(
¬
X7
)
↔
X26
)
)
)
)
)
↔
(
¬
(
X46
↔
(
X26
→
X33
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
X42
)
↔
(
(
¬
X26
)
↔
(
(
¬
X13
)
↔
X50
)
)
)
→
(
(
X42
↔
(
(
¬
X34
)
↔
X50
)
)
↔
(
(
¬
(
X50
→
(
X34
↔
(
X13
↔
X6
)
)
)
)
↔
(
X34
↔
X42
)
)
)
)
)
→
(
(
¬
(
X42
↔
(
(
¬
X40
)
↔
(
(
¬
X26
)
→
X41
)
)
)
)
↔
(
X50
→
X26
→
X1
→
X41
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
(
¬
(
(
(
X46
→
(
X39
↔
(
(
¬
X11
)
↔
X49
)
)
)
↔
(
(
X49
↔
(
X39
↔
(
X35
→
X28
)
)
)
→
X46
→
X39
→
(
(
¬
X35
)
↔
X49
)
)
)
→
(
X39
→
X37
→
X46
)
→
(
X39
↔
(
X37
↔
(
X16
↔
X49
)
)
)
→
X49
→
(
¬
X39
)
→
X46
)
)
→
(
¬
(
(
(
(
¬
X46
)
→
X39
→
(
¬
X27
)
→
X49
)
→
(
¬
(
(
¬
X46
)
↔
(
(
¬
X39
)
→
X3
→
X49
)
)
)
)
→
(
¬
(
X46
→
(
X27
↔
(
X13
↔
X49
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
X49
↔
(
X45
↔
(
X44
↔
(
X39
↔
False
)
)
)
)
)
↔
(
(
¬
X39
)
↔
(
(
¬
X31
)
→
X44
)
)
)
↔
(
(
X24
↔
(
(
¬
X5
)
→
(
¬
X0
)
→
X37
)
)
→
(
¬
(
X37
→
(
(
¬
X24
)
↔
(
(
¬
X2
)
→
X40
)
)
)
)
→
(
¬
(
X40
↔
(
X37
→
X24
→
X0
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
(
X21
↔
False
)
→
(
(
¬
X47
)
↔
(
(
¬
X46
)
→
(
¬
X21
)
→
X49
)
)
→
(
¬
X47
)
→
(
X21
↔
(
X20
↔
X49
)
)
)
↔
(
(
(
¬
X46
)
↔
(
(
¬
X40
)
↔
(
X18
→
X47
)
)
)
↔
(
(
X49
→
X47
→
(
X46
↔
(
X33
→
False
)
)
)
↔
(
¬
(
(
(
¬
(
X49
↔
(
X47
↔
(
X46
↔
False
)
)
)
)
↔
(
(
(
(
¬
X47
)
→
(
¬
X40
)
→
(
¬
X8
)
→
X49
)
↔
(
¬
(
X49
↔
(
X46
→
(
¬
X8
)
→
X40
)
)
)
)
↔
(
X40
↔
(
X8
↔
X49
)
)
)
)
→
(
X49
↔
(
X46
→
X40
)
)
)
)
)
)
)
→
(
(
¬
(
X29
↔
(
X19
↔
X12
)
)
)
→
X49
→
X33
→
X29
)
→
(
¬
(
X12
↔
X47
)
)
)
)
→
(
¬
(
(
¬
(
X38
→
(
X17
↔
(
X4
↔
X40
)
)
)
)
→
(
¬
(
(
¬
X17
)
↔
(
(
¬
X4
)
↔
X47
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X38
→
(
X4
↔
X17
)
)
)
→
(
¬
(
(
¬
(
X47
→
(
X17
↔
X46
)
)
)
→
(
(
X46
→
(
¬
X17
)
→
(
(
¬
X14
)
↔
X47
)
)
↔
(
X47
→
(
X17
↔
(
X14
→
X4
)
)
)
)
)
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X38
)
↔
X47
)
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
(
¬
X43
)
→
(
(
¬
X25
)
↔
(
X23
→
X50
)
)
)
)
↔
(
¬
(
(
¬
(
X48
→
(
¬
X43
)
→
X25
→
X50
)
)
→
X50
→
(
(
¬
X43
)
↔
(
(
¬
X32
)
→
X48
)
)
)
)
)
↔
(
(
(
(
¬
X43
)
↔
(
X41
→
(
(
¬
X36
)
↔
X48
)
)
)
↔
(
¬
(
(
¬
X43
)
→
(
X41
↔
(
X22
→
X48
)
)
)
)
)
↔
(
(
(
(
¬
(
X50
→
(
X46
↔
(
X7
↔
X48
)
)
)
)
↔
(
X50
→
(
X46
↔
(
X7
→
X1
→
False
)
)
)
)
→
(
¬
X48
)
→
X46
→
X50
)
↔
(
(
¬
(
X50
→
(
X22
↔
X49
)
)
)
↔
(
X50
→
(
X41
↔
(
(
¬
X7
)
→
X22
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
X50
↔
(
(
¬
X34
)
→
X47
)
)
→
(
(
X50
↔
(
X47
→
(
(
¬
X6
)
↔
X30
)
)
)
↔
(
X50
→
(
X34
↔
(
X6
→
False
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
X42
)
→
(
(
¬
X30
)
↔
X50
)
)
→
(
(
¬
(
(
¬
X49
)
↔
(
(
¬
X47
)
→
X42
→
X50
)
)
)
↔
(
¬
(
X50
↔
(
(
¬
X47
)
↔
(
X30
→
X49
)
)
)
)
)
→
X42
→
(
X30
↔
X50
)
)
)
)
→
(
¬
(
(
¬
(
X26
→
(
(
¬
X10
)
↔
X48
)
)
)
→
(
(
¬
X47
)
↔
(
(
¬
X26
)
↔
(
(
¬
X10
)
↔
X49
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X48
↔
(
X47
↔
(
X15
→
False
)
)
)
)
↔
(
¬
(
(
¬
(
X48
→
(
¬
X47
)
→
(
X9
↔
X49
)
)
)
→
(
¬
(
X49
→
(
X47
↔
(
X15
→
X9
)
)
)
)
)
)
)
→
(
¬
X47
)
→
X34
→
X48
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
4af3ae...
and proposition id is
ef547e...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMM9DqN5Z2QJRF1CXFc4YN7MGEWgnnA4AiU
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
∀X50 :
prop
,
(
(
¬
(
(
(
(
(
(
(
¬
(
X48
↔
(
X36
↔
X44
)
)
)
↔
(
(
X48
↔
(
X47
→
(
X0
↔
False
)
)
)
→
X48
→
(
X44
↔
(
X36
↔
X0
)
)
)
)
→
(
(
X48
↔
(
X44
→
(
(
¬
X40
)
↔
X47
)
)
)
↔
(
¬
(
X47
→
(
X40
↔
X11
)
)
)
)
)
↔
(
(
X48
→
(
¬
X35
)
→
X38
)
↔
(
(
(
¬
(
X47
→
(
X35
↔
(
X33
↔
X44
)
)
)
)
↔
(
¬
(
(
(
(
¬
X47
)
↔
(
(
¬
X35
)
→
(
¬
X5
)
→
X48
)
)
↔
(
¬
(
X44
→
(
X35
↔
X5
)
)
)
)
→
(
¬
(
(
¬
X47
)
↔
(
(
¬
X44
)
→
X48
)
)
)
)
)
)
↔
(
¬
(
X48
↔
(
X47
→
False
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X38
→
(
(
¬
X18
)
↔
X42
)
)
)
↔
(
(
¬
(
X35
↔
(
(
¬
X11
)
↔
(
X7
↔
X42
)
)
)
)
↔
(
(
¬
(
X42
↔
(
X11
↔
X38
)
)
)
↔
(
X42
→
(
¬
X35
)
→
(
¬
X18
)
→
X38
)
)
)
)
→
X35
→
(
¬
X18
)
→
X42
)
)
)
↔
(
(
(
¬
(
(
¬
(
X38
→
X46
)
)
→
(
(
¬
(
X46
↔
(
X22
↔
(
X8
↔
X44
)
)
)
)
↔
(
X47
↔
(
X46
↔
(
(
¬
X8
)
→
X22
)
)
)
)
)
)
→
(
¬
(
(
X44
↔
(
X10
↔
False
)
)
→
(
X47
↔
(
X44
→
(
X2
↔
X1
)
)
)
)
)
→
(
¬
(
X47
↔
(
X38
↔
(
X10
→
X44
)
)
)
)
)
→
(
(
X47
↔
(
(
¬
X4
)
→
X44
)
)
↔
(
¬
(
X46
→
X44
→
(
¬
X16
)
→
X47
)
)
)
)
)
→
(
(
(
¬
(
(
(
¬
(
(
¬
X42
)
↔
(
X26
↔
X45
)
)
)
↔
(
(
(
¬
X48
)
→
(
¬
X45
)
→
(
(
¬
X24
)
↔
X50
)
)
↔
(
¬
(
(
¬
X45
)
→
(
¬
X42
)
→
X50
)
)
)
)
→
(
(
(
(
(
(
(
X20
→
X23
)
↔
(
(
¬
(
X48
↔
(
X42
→
X45
)
)
)
↔
(
(
¬
(
X42
↔
(
(
¬
X23
)
↔
(
X20
→
X48
)
)
)
)
→
(
¬
(
X45
→
(
(
¬
X42
)
↔
(
(
¬
X23
)
↔
X48
)
)
)
)
)
)
)
↔
(
X48
↔
(
X45
→
X23
→
False
)
)
)
→
(
¬
(
X48
↔
(
X42
↔
X23
)
)
)
)
↔
(
¬
(
(
¬
X42
)
↔
(
(
¬
X23
)
↔
(
(
¬
X15
)
↔
X48
)
)
)
)
)
↔
(
¬
(
(
¬
X42
)
→
(
¬
X23
)
→
X48
)
)
)
↔
(
¬
(
X41
→
(
X34
↔
(
X13
→
X20
)
)
)
)
)
)
)
→
(
(
(
X48
→
(
¬
X46
)
→
(
(
¬
X12
)
↔
X50
)
)
→
(
¬
(
X50
↔
(
(
¬
X45
)
→
(
¬
X12
)
→
X46
)
)
)
)
→
X48
→
(
(
¬
X12
)
↔
X50
)
)
→
(
(
(
(
¬
(
X50
↔
(
X46
↔
(
X31
→
X43
)
)
)
)
→
(
X31
↔
(
(
¬
X25
)
→
X46
)
)
)
↔
(
(
(
(
¬
X25
)
→
(
(
¬
X17
)
↔
X43
)
)
→
(
(
¬
(
X46
↔
(
X43
↔
(
X25
↔
X17
)
)
)
)
↔
(
X46
→
(
(
¬
X29
)
↔
(
(
¬
X25
)
↔
X43
)
)
)
)
)
↔
(
X46
→
X43
→
False
)
)
)
↔
(
(
X48
↔
(
X43
↔
X30
)
)
↔
(
X50
↔
(
(
¬
X30
)
→
X31
)
)
)
)
)
↔
(
(
(
(
X21
↔
(
X6
→
X9
)
)
↔
(
¬
(
X48
→
X21
→
(
(
¬
X9
)
↔
X37
)
)
)
)
↔
(
(
¬
(
X39
↔
(
X9
↔
X3
)
)
)
↔
(
(
¬
(
X48
→
(
¬
X43
)
→
(
¬
X39
)
→
X45
)
)
↔
(
X39
↔
(
(
¬
X37
)
→
(
¬
X3
)
→
X43
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
¬
(
X49
↔
(
X43
→
X27
)
)
)
→
(
(
¬
X27
)
→
(
¬
X14
)
→
X49
)
→
(
¬
(
X43
→
(
¬
X27
)
→
(
¬
X14
)
→
X45
)
)
)
)
→
(
X45
↔
(
X27
↔
X49
)
)
)
)
→
(
¬
(
(
(
¬
(
X48
↔
(
X32
→
(
¬
X19
)
→
X28
)
)
)
↔
(
X45
→
(
X32
↔
X43
)
)
)
→
X49
→
(
(
¬
X45
)
↔
(
X28
→
X48
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
X49
↔
(
X47
↔
(
X28
→
X37
)
)
)
↔
(
¬
(
X47
→
(
X37
↔
(
X16
↔
X49
)
)
)
)
)
↔
(
¬
(
(
X37
↔
(
X28
→
(
¬
X26
)
→
X49
)
)
→
(
¬
X37
)
→
X1
→
X49
)
)
)
→
(
(
X47
↔
(
(
¬
X43
)
↔
(
(
¬
X7
)
→
X45
)
)
)
↔
(
X45
↔
(
(
¬
X28
)
→
X14
→
X49
)
)
)
)
)
↔
(
(
(
(
X50
→
X43
→
(
X2
↔
X44
)
)
→
(
(
¬
(
X50
↔
(
(
¬
X44
)
→
(
(
¬
X12
)
↔
X48
)
)
)
)
↔
(
¬
(
(
¬
X44
)
→
X43
→
(
¬
X12
)
→
X48
)
)
)
)
↔
(
(
(
X37
↔
(
(
¬
X22
)
→
(
(
¬
X21
)
↔
X50
)
)
)
→
(
¬
(
X44
↔
(
X21
↔
X50
)
)
)
)
→
(
X22
→
X21
→
(
¬
X13
)
→
X50
)
→
(
¬
(
(
¬
X21
)
↔
(
(
¬
X13
)
→
X48
)
)
)
)
)
↔
(
(
(
¬
(
(
X50
→
(
X23
↔
(
(
¬
X9
)
↔
X44
)
)
)
→
(
(
¬
(
X23
→
X19
→
X43
)
)
↔
(
(
¬
(
(
¬
X43
)
→
(
¬
X23
)
→
(
X9
↔
X44
)
)
)
↔
(
X44
→
(
(
¬
X23
)
↔
(
X19
↔
X43
)
)
)
)
)
)
)
→
(
(
X20
↔
(
X19
→
(
¬
X9
)
→
X47
)
)
↔
(
¬
(
X20
→
(
(
¬
X15
)
↔
(
(
¬
X9
)
↔
X47
)
)
)
)
)
)
↔
(
X50
↔
(
X47
→
X43
)
)
)
)
)
↔
(
(
(
¬
(
(
(
X41
→
(
X39
↔
X24
)
)
↔
(
X29
→
(
(
¬
X24
)
↔
(
(
¬
X0
)
↔
X41
)
)
)
)
→
(
(
(
¬
(
X35
↔
(
X4
↔
X29
)
)
)
↔
(
(
¬
X39
)
↔
(
(
¬
X35
)
↔
(
X11
→
X42
)
)
)
)
↔
(
X39
→
(
¬
X24
)
→
(
¬
X4
)
→
X42
)
)
)
)
↔
(
¬
(
(
(
(
(
X8
↔
(
X6
↔
(
(
¬
X5
)
↔
X42
)
)
)
↔
(
(
¬
(
X42
→
(
(
¬
X8
)
↔
(
X6
→
X35
)
)
)
)
→
(
¬
(
X42
→
X25
)
)
)
)
↔
(
X42
→
X8
→
X6
→
X3
)
)
↔
(
¬
(
X49
↔
(
X42
↔
(
X8
→
X25
)
)
)
)
)
→
(
(
(
(
¬
(
X49
→
X31
→
X32
)
)
↔
(
¬
(
(
(
(
X49
↔
(
(
¬
X32
)
→
(
¬
X31
)
→
X35
)
)
↔
(
X35
→
(
X32
↔
(
(
¬
X18
)
→
X31
)
)
)
)
→
X35
→
(
(
¬
X31
)
↔
X49
)
)
→
X35
)
)
)
↔
(
(
¬
(
(
¬
X41
)
→
(
(
¬
X35
)
↔
(
(
¬
X18
)
→
X42
)
)
)
)
↔
(
(
¬
X42
)
→
(
(
¬
X35
)
↔
X49
)
)
)
)
↔
(
X35
→
X34
→
X32
→
X49
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X46
)
→
(
(
¬
X44
)
↔
(
X30
→
X49
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X36
)
↔
(
(
¬
X17
)
→
X46
)
)
)
↔
(
(
¬
(
(
¬
X36
)
↔
(
(
¬
X27
)
↔
X44
)
)
)
↔
(
¬
(
X46
↔
(
X36
→
X17
→
X44
)
)
)
)
)
→
X49
→
(
(
¬
X39
)
↔
(
(
¬
X36
)
→
X44
)
)
)
↔
(
¬
(
X27
↔
X49
)
)
)
)
→
(
¬
(
(
¬
(
(
X43
→
(
X42
↔
(
(
¬
X40
)
→
X44
)
)
)
→
(
¬
(
X49
↔
(
X42
→
(
¬
X40
)
→
X43
)
)
)
)
)
→
(
¬
(
(
¬
(
(
X39
→
(
X33
↔
X49
)
)
→
(
(
¬
X43
)
↔
(
X42
↔
(
(
¬
X39
)
→
X49
)
)
)
)
)
→
(
¬
(
(
¬
X43
)
↔
(
X42
→
X49
)
)
)
)
)
)
)
→
(
¬
(
(
X43
↔
(
X40
→
X44
)
)
→
(
(
¬
X38
)
↔
(
X10
↔
X43
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
3157d2...
and proposition id is
8530ee...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMWiDsu3EgvuMWD2mFNAypiesesauNCqoYb
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
¬
(
(
(
(
(
(
(
X47
→
X39
→
X9
→
False
)
↔
(
(
(
(
¬
X44
)
↔
(
(
¬
X39
)
↔
(
X9
→
X47
)
)
)
↔
(
X47
↔
(
(
¬
X39
)
↔
(
(
¬
X2
)
→
X44
)
)
)
)
→
(
¬
(
X44
↔
(
(
¬
X9
)
↔
(
X6
↔
X39
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X47
↔
(
X29
→
(
(
¬
X10
)
↔
X14
)
)
)
)
→
(
(
X47
↔
(
(
¬
X39
)
→
X7
→
X44
)
)
↔
(
X47
↔
(
X39
↔
(
X29
→
X44
)
)
)
)
→
(
X47
↔
(
X14
↔
(
(
¬
X7
)
→
X44
)
)
)
)
)
↔
(
¬
(
X44
↔
(
X39
→
X29
)
)
)
)
)
→
(
¬
(
(
X29
↔
(
X5
→
X39
)
)
→
(
(
X47
↔
(
X5
→
(
X0
↔
X27
)
)
)
↔
(
¬
(
(
¬
X29
)
→
(
(
¬
X27
)
↔
X39
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
¬
(
X45
↔
(
X26
→
X17
→
X28
)
)
)
↔
(
X28
→
X26
→
X39
)
)
→
(
¬
X39
)
→
(
X26
↔
X45
)
)
)
→
(
(
¬
X39
)
↔
(
(
¬
X33
)
↔
(
(
¬
X26
)
→
X45
)
)
)
)
)
↔
(
(
¬
(
X47
→
(
(
¬
X39
)
↔
(
(
¬
X29
)
→
X45
)
)
)
)
↔
(
¬
(
X39
→
(
(
¬
X29
)
↔
(
(
¬
X26
)
↔
X45
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
¬
X47
)
→
(
(
¬
X32
)
↔
(
(
¬
X25
)
↔
X48
)
)
)
)
→
(
(
(
¬
(
X48
↔
(
(
¬
X44
)
→
(
¬
X40
)
→
X47
)
)
)
↔
(
X23
↔
False
)
)
↔
(
¬
(
(
¬
X47
)
↔
(
(
¬
X40
)
↔
(
X32
→
X48
)
)
)
)
)
)
)
→
(
(
X44
↔
(
X38
→
(
X12
↔
X48
)
)
)
↔
(
¬
(
(
(
(
¬
(
X47
↔
(
X45
→
X12
→
X35
)
)
)
↔
(
¬
(
X47
↔
(
X38
↔
False
)
)
)
)
↔
(
¬
X45
)
)
→
(
¬
(
X48
→
(
X47
↔
(
(
¬
X3
)
↔
X38
)
)
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
X43
→
(
X41
↔
(
X22
→
X15
)
)
)
→
(
¬
(
(
¬
(
(
¬
X15
)
→
X41
)
)
→
(
X41
↔
(
X22
→
X4
→
X47
)
)
)
)
)
)
↔
(
(
¬
X43
)
→
X41
→
X19
→
X47
)
)
↔
(
(
¬
(
X1
→
X41
)
)
↔
(
(
¬
(
X41
↔
(
X34
↔
X19
)
)
)
↔
(
¬
(
(
¬
X34
)
→
X19
→
(
X1
↔
X47
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X46
)
→
(
¬
X34
)
→
X22
→
X47
)
)
→
(
(
¬
(
X47
→
(
X46
↔
(
X21
↔
(
X11
↔
False
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X34
)
↔
(
(
¬
X13
)
→
(
¬
X11
)
→
X46
)
)
)
→
(
X21
↔
(
X13
→
X34
)
)
)
)
↔
(
¬
(
(
¬
X45
)
→
(
X34
↔
(
(
¬
X11
)
→
X46
)
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X42
→
(
(
¬
X30
)
↔
(
X24
→
X36
)
)
)
)
→
(
X30
↔
(
X24
→
X49
)
)
)
→
(
(
¬
(
(
(
(
X37
↔
(
X36
↔
X45
)
)
→
(
¬
(
(
¬
(
X36
↔
X49
)
)
→
(
(
¬
X37
)
↔
(
X36
→
(
(
¬
X8
)
↔
X45
)
)
)
)
)
)
↔
(
¬
(
X45
→
(
X8
↔
X37
)
)
)
)
→
(
¬
(
X45
↔
(
X36
↔
X49
)
)
)
)
)
↔
(
(
¬
(
(
¬
X42
)
↔
(
X36
→
(
¬
X8
)
→
X47
)
)
)
↔
(
X45
↔
(
X42
→
X36
→
X47
)
)
)
)
)
↔
(
(
(
X45
↔
(
X18
→
X41
)
)
↔
(
¬
(
X48
→
(
(
¬
X16
)
↔
X42
)
)
)
)
↔
(
(
¬
(
(
X45
↔
(
(
¬
X31
)
→
(
(
¬
X20
)
↔
X41
)
)
)
→
(
¬
(
X45
→
(
X37
↔
(
(
¬
X20
)
↔
X41
)
)
)
)
)
)
↔
(
¬
(
X48
↔
(
(
¬
X41
)
↔
(
(
¬
X20
)
→
X45
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
X44
↔
(
(
¬
X38
)
↔
(
(
¬
X8
)
→
X41
)
)
)
)
↔
(
(
(
¬
(
X39
↔
(
X38
→
(
¬
X32
)
→
X41
)
)
)
→
(
¬
(
X44
↔
(
(
¬
X38
)
↔
(
X6
↔
X39
)
)
)
)
)
↔
(
X44
→
X39
→
X6
)
)
)
→
(
(
(
X39
→
(
X21
↔
False
)
)
→
(
(
¬
(
X44
→
X30
→
(
¬
X21
)
→
X23
)
)
↔
(
X39
→
(
X30
↔
(
(
¬
X23
)
↔
X44
)
)
)
)
)
↔
(
(
(
¬
(
X44
→
X30
→
(
X28
↔
X41
)
)
)
↔
(
(
(
¬
X30
)
↔
(
X28
↔
X36
)
)
↔
(
¬
(
X41
→
(
X36
↔
(
X23
↔
False
)
)
)
)
)
)
↔
(
X36
↔
(
X23
→
False
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
X41
↔
(
X35
→
(
(
¬
X3
)
↔
X36
)
)
)
↔
(
(
X41
↔
(
(
¬
X25
)
↔
(
(
¬
X13
)
→
X36
)
)
)
→
(
¬
(
X41
→
X36
→
(
(
¬
X3
)
↔
X25
)
)
)
)
)
→
(
(
¬
(
X35
↔
(
X31
→
X41
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X36
)
↔
(
(
¬
X35
)
→
X4
→
X41
)
)
)
→
(
¬
(
X41
→
(
X36
↔
(
X4
→
X31
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
X43
↔
(
(
¬
X38
)
→
X41
)
)
)
↔
(
¬
(
(
¬
(
X43
↔
(
(
¬
X36
)
↔
(
(
¬
X22
)
→
X41
)
)
)
)
→
(
¬
(
X38
→
X22
→
X4
→
False
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
X48
→
(
X44
↔
(
X11
↔
X49
)
)
)
→
(
(
¬
(
X48
↔
(
X11
→
(
X10
↔
X34
)
)
)
)
↔
(
¬
(
X11
↔
(
X0
↔
X10
)
)
)
)
)
↔
(
(
¬
(
X11
→
(
(
¬
X0
)
↔
X40
)
)
)
→
(
¬
(
(
¬
(
X49
→
X43
→
X47
)
)
→
(
(
(
¬
X47
)
→
(
(
¬
X43
)
↔
(
X0
↔
X49
)
)
)
↔
(
¬
(
X49
↔
(
X40
→
(
(
¬
X0
)
↔
X47
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
X48
→
X44
→
(
X17
↔
X42
)
)
→
(
(
¬
X42
)
→
X48
)
→
(
(
(
¬
X42
)
↔
(
(
¬
X29
)
→
X24
→
X44
)
)
↔
(
(
¬
X24
)
→
(
¬
X15
)
→
X29
)
)
)
)
↔
(
(
X47
↔
(
(
¬
X42
)
↔
(
X41
→
X44
)
)
)
↔
(
¬
(
X48
↔
(
X41
→
X44
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
X27
)
→
(
X20
↔
X47
)
)
)
↔
(
(
(
¬
(
X47
↔
(
X27
→
(
¬
X9
)
→
X42
)
)
)
→
(
¬
(
X47
→
X27
→
(
(
¬
X20
)
↔
X42
)
)
)
)
↔
(
¬
(
(
¬
X20
)
→
X9
→
X47
)
)
)
)
→
(
¬
(
X44
↔
(
(
¬
X9
)
↔
X47
)
)
)
)
↔
(
(
(
¬
(
(
¬
X27
)
↔
(
(
¬
X14
)
↔
X47
)
)
)
→
(
¬
(
X47
↔
(
X42
↔
X2
)
)
)
)
↔
(
X47
↔
(
X14
→
X42
)
)
)
)
)
)
→
(
¬
(
(
(
X46
↔
(
(
¬
X16
)
→
X34
)
)
↔
(
(
(
¬
(
(
¬
X37
)
→
(
X34
↔
(
X16
↔
X46
)
)
)
)
→
X46
→
(
X34
↔
(
X33
↔
X37
)
)
)
↔
(
¬
(
X45
↔
(
X37
→
(
(
¬
X34
)
↔
X46
)
)
)
)
)
)
→
(
¬
(
(
(
X46
→
(
X26
↔
X34
)
)
↔
(
(
¬
(
(
(
¬
X46
)
→
X18
→
X49
)
→
(
¬
(
(
¬
X34
)
→
(
¬
X18
)
→
X41
)
)
)
)
↔
(
X34
→
(
X26
↔
X18
)
)
)
)
→
(
(
(
(
¬
X46
)
→
X45
→
(
¬
X41
)
→
X49
)
↔
(
¬
(
(
¬
X45
)
↔
(
X7
→
X49
)
)
)
)
↔
(
(
(
¬
(
(
¬
X41
)
→
(
X19
↔
X46
)
)
)
→
(
X46
→
(
X19
↔
(
(
¬
X1
)
↔
X12
)
)
)
→
(
(
X41
→
(
X12
↔
(
X1
↔
X45
)
)
)
↔
(
¬
(
X46
↔
(
X41
↔
(
X12
→
X45
)
)
)
)
)
)
→
(
¬
(
X45
↔
(
X19
↔
(
X5
↔
X41
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
2d8b04...
and proposition id is
55891e...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMZiZVZ15D1frM9WXgGd84yy5rAPEvYL7g6
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
¬
(
(
(
(
(
(
(
¬
(
X42
→
X27
→
(
¬
X9
)
→
X39
)
)
↔
(
X42
↔
(
(
¬
X27
)
↔
X44
)
)
)
↔
(
¬
(
X42
→
X39
→
X44
)
)
)
→
(
(
X42
→
(
¬
X40
)
→
(
¬
X30
)
→
X44
)
↔
(
¬
(
(
(
¬
(
(
¬
X40
)
↔
(
(
¬
X39
)
→
(
X30
↔
X41
)
)
)
)
↔
(
(
¬
X40
)
↔
(
(
¬
X13
)
↔
X41
)
)
)
→
(
(
¬
(
(
X44
→
(
(
¬
X40
)
↔
(
X26
→
X41
)
)
)
→
(
¬
(
(
X40
↔
(
(
¬
X39
)
↔
(
(
¬
X26
)
→
X41
)
)
)
→
(
¬
(
X41
→
(
X40
↔
X14
)
)
)
)
)
)
)
↔
(
¬
(
X44
↔
(
(
¬
X40
)
→
X41
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X44
↔
(
X42
→
(
X12
↔
X5
)
)
)
)
↔
(
¬
(
X41
→
X42
)
)
)
→
(
(
X44
↔
(
(
¬
X41
)
↔
(
X29
↔
X42
)
)
)
↔
(
X44
→
(
(
¬
X41
)
↔
(
(
¬
X35
)
→
X42
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
X40
↔
(
X38
↔
(
X25
→
X4
)
)
)
↔
(
(
¬
(
X38
↔
(
X34
↔
(
X25
↔
(
X15
→
False
)
)
)
)
)
↔
(
X34
↔
(
(
¬
X15
)
→
(
¬
X4
)
→
X25
)
)
)
)
→
(
(
(
¬
(
X38
→
X23
→
X43
)
)
↔
(
¬
(
(
¬
X42
)
↔
(
X38
↔
(
X25
→
X50
)
)
)
)
)
↔
(
¬
(
X43
→
(
¬
X40
)
→
(
¬
X25
)
→
X50
)
)
)
)
)
→
(
(
(
X50
↔
(
X36
↔
X42
)
)
→
(
¬
X6
)
→
X38
)
↔
(
(
¬
X44
)
↔
(
X42
→
(
¬
X38
)
→
X50
)
)
)
→
(
¬
(
(
(
X21
→
(
X8
↔
(
(
¬
X2
)
↔
X50
)
)
)
↔
(
X50
↔
(
X21
↔
(
X8
→
X16
)
)
)
)
→
(
¬
(
(
¬
X42
)
↔
(
(
¬
X16
)
↔
(
X2
↔
X44
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
(
¬
(
X33
→
(
¬
X20
)
→
X47
)
)
↔
(
(
¬
(
X49
→
(
(
¬
X20
)
↔
(
(
¬
X0
)
↔
X50
)
)
)
)
↔
(
¬
(
(
X47
↔
(
X33
→
X10
→
X50
)
)
→
(
¬
(
(
¬
X33
)
↔
(
X20
↔
(
(
¬
X10
)
→
X50
)
)
)
)
)
)
)
)
↔
(
¬
(
(
X49
↔
(
(
¬
X46
)
→
(
X20
↔
X47
)
)
)
→
(
(
(
(
¬
(
X37
↔
(
(
¬
X18
)
↔
(
X1
→
X46
)
)
)
)
→
(
¬
(
X47
↔
False
)
)
)
↔
(
X46
↔
False
)
)
↔
(
¬
(
X49
→
(
¬
X37
)
→
(
¬
X18
)
→
X46
)
)
)
)
)
)
→
(
(
(
(
¬
X48
)
→
X42
→
(
¬
X3
)
→
X50
)
↔
(
(
(
¬
(
X42
↔
(
X19
→
X17
→
X50
)
)
)
↔
(
(
(
¬
X48
)
↔
(
(
¬
X45
)
↔
X50
)
)
↔
(
(
X50
→
(
X45
↔
(
(
¬
X19
)
↔
X48
)
)
)
↔
(
X50
↔
X42
)
)
)
)
→
(
X50
↔
(
X17
↔
X42
)
)
)
)
↔
(
(
(
(
¬
X46
)
→
(
¬
X17
)
→
X50
)
↔
(
¬
(
X46
↔
(
(
¬
X28
)
↔
(
(
¬
X17
)
↔
X47
)
)
)
)
)
→
(
X50
↔
(
X46
↔
(
(
¬
X45
)
→
X47
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X47
→
X44
→
(
X22
↔
X32
)
)
)
→
(
¬
(
(
(
¬
X44
)
↔
(
(
¬
X7
)
↔
X47
)
)
→
(
(
X47
→
(
X22
↔
(
X7
↔
X44
)
)
)
↔
(
¬
(
X50
→
(
(
¬
X44
)
↔
(
(
¬
X22
)
→
X47
)
)
)
)
)
)
)
)
)
→
(
(
X50
→
(
(
¬
X24
)
↔
(
X11
↔
X44
)
)
)
↔
(
¬
(
X50
↔
(
X31
↔
(
X11
→
X46
)
)
)
)
)
→
(
¬
(
(
¬
X46
)
↔
(
(
¬
X44
)
→
X24
→
X47
)
)
)
)
)
→
(
X47
↔
(
X46
↔
X50
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
(
(
¬
(
X50
→
(
(
¬
X24
)
↔
(
(
¬
X13
)
→
X39
)
)
)
)
↔
(
(
X39
↔
(
(
¬
X33
)
↔
(
(
¬
X24
)
↔
X50
)
)
)
↔
(
¬
(
X50
↔
(
X33
→
(
¬
X17
)
→
X24
)
)
)
)
)
↔
(
(
¬
(
X45
↔
(
X34
→
X16
→
X50
)
)
)
↔
(
(
¬
(
X50
↔
(
X17
↔
(
X16
→
(
X13
↔
False
)
)
)
)
)
↔
(
(
¬
X34
)
↔
X43
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X43
↔
(
(
¬
X31
)
↔
(
(
¬
X9
)
↔
X50
)
)
)
)
↔
(
¬
(
X45
↔
(
(
¬
X9
)
↔
X50
)
)
)
)
↔
(
¬
(
(
¬
X43
)
→
(
¬
X31
)
→
X50
)
)
)
→
X45
→
X31
→
False
)
)
)
↔
(
¬
(
(
(
¬
(
X50
→
X45
→
False
)
)
↔
(
¬
(
(
X45
→
(
¬
X30
)
→
(
¬
X6
)
→
X50
)
→
(
X50
↔
(
(
¬
X23
)
→
X6
→
X30
)
)
)
)
)
→
(
(
¬
(
(
X43
→
(
X12
↔
X11
)
)
→
(
(
¬
(
X12
→
X11
→
X4
→
X30
)
)
↔
(
¬
(
X43
↔
(
X11
→
X4
→
False
)
)
)
)
)
)
↔
(
(
¬
(
X14
→
(
(
¬
X12
)
↔
(
(
¬
X11
)
↔
X45
)
)
)
)
↔
(
¬
(
X43
↔
(
X14
→
(
(
¬
X11
)
↔
X12
)
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X48
↔
(
X46
→
X43
→
X35
→
False
)
)
)
→
(
(
(
¬
(
X46
→
X45
→
X35
)
)
↔
(
(
X35
↔
(
X26
→
X48
)
)
→
(
¬
(
X45
→
X35
→
X46
)
)
)
)
↔
(
X48
↔
(
(
¬
X45
)
↔
X46
)
)
)
)
↔
(
(
(
¬
(
(
(
X46
→
(
X35
↔
(
X15
→
False
)
)
)
↔
(
¬
(
(
X36
→
(
(
¬
X15
)
↔
(
X0
→
X35
)
)
)
→
(
¬
X35
)
→
(
¬
X0
)
→
X46
)
)
)
→
(
¬
(
X15
↔
(
(
¬
X0
)
→
X36
)
)
)
)
)
↔
(
(
¬
X35
)
→
X15
→
(
¬
X3
)
→
X46
)
)
↔
(
¬
(
(
(
¬
X35
)
↔
(
X15
→
X48
)
)
→
(
¬
(
X50
→
X46
)
)
)
)
)
)
↔
(
(
(
(
X45
↔
(
X28
↔
X38
)
)
↔
(
¬
(
X45
→
(
(
¬
X28
)
↔
(
X19
→
X43
)
)
)
)
)
↔
(
¬
(
X45
↔
(
X43
↔
(
X38
→
X27
→
False
)
)
)
)
)
↔
(
(
¬
(
X45
→
(
(
¬
X38
)
↔
(
X36
↔
X43
)
)
)
)
→
(
¬
(
(
¬
X36
)
↔
(
X7
→
X43
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
(
X44
↔
(
(
¬
X40
)
↔
(
X32
↔
X45
)
)
)
↔
(
(
¬
(
X49
→
X45
→
X44
→
X46
)
)
↔
(
¬
(
X46
↔
(
X18
→
X10
→
X49
)
)
)
)
)
→
(
¬
(
X42
↔
(
X18
→
(
¬
X10
)
→
X32
)
)
)
)
)
→
(
(
(
(
(
¬
X44
)
↔
(
X42
↔
(
(
¬
X29
)
↔
X47
)
)
)
↔
(
X42
↔
(
X8
→
X5
)
)
)
↔
(
(
¬
X44
)
→
(
¬
X42
)
→
X47
)
)
↔
(
¬
(
(
(
(
(
(
¬
X40
)
↔
(
(
¬
X5
)
↔
X48
)
)
↔
(
(
X40
→
(
X8
↔
X48
)
)
↔
(
¬
(
(
¬
(
(
¬
X25
)
↔
(
X5
↔
X40
)
)
)
→
(
¬
(
(
¬
X40
)
↔
(
(
¬
X25
)
→
(
(
¬
X8
)
↔
X48
)
)
)
)
)
)
)
)
↔
(
(
¬
X8
)
→
X25
)
)
↔
(
(
¬
X40
)
↔
(
X25
→
X48
)
)
)
→
(
¬
X29
)
→
(
¬
X25
)
→
X40
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
X37
↔
(
(
¬
X1
)
↔
X46
)
)
)
↔
(
X47
↔
(
X42
→
(
(
¬
X37
)
↔
X46
)
)
)
)
↔
(
¬
(
X37
→
(
(
¬
X20
)
↔
(
(
¬
X1
)
→
X47
)
)
)
)
)
→
X46
→
(
X1
↔
X37
)
)
)
→
(
(
(
¬
(
X47
→
(
(
¬
X37
)
↔
(
X21
→
X42
)
)
)
)
→
(
¬
(
(
¬
X21
)
↔
X22
)
)
)
↔
(
(
¬
(
X37
→
X41
)
)
↔
(
X47
↔
(
X22
→
(
¬
X2
)
→
X37
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
b6fc65...
and proposition id is
9ff38b...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMQ5zTVthbS2Um6z3fMhxtvinrnDXqGk5Hj
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
¬
∀X51 :
prop
,
(
¬
∀X52 :
prop
,
(
(
(
(
(
¬
(
(
(
(
(
¬
X45
)
↔
(
(
¬
X12
)
↔
X46
)
)
→
(
¬
(
X45
→
(
¬
X39
)
→
(
¬
X12
)
→
X41
)
)
)
↔
(
(
¬
(
(
¬
X12
)
↔
(
X10
↔
(
X6
↔
X46
)
)
)
)
↔
(
¬
(
X39
→
False
)
)
)
)
→
(
¬
(
(
(
¬
(
X41
↔
(
X37
→
(
¬
X32
)
→
X46
)
)
)
→
(
¬
(
(
¬
(
X41
→
(
X39
↔
(
(
¬
X32
)
→
X37
)
)
)
)
→
(
¬
(
(
(
¬
X37
)
↔
(
(
¬
X26
)
→
X39
)
)
→
(
(
¬
(
X41
→
(
X37
↔
(
X26
↔
X0
)
)
)
)
↔
(
¬
(
X41
↔
(
X26
↔
(
X0
↔
X39
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
X41
↔
(
(
¬
X37
)
→
(
¬
X3
)
→
X46
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X46
→
(
X41
↔
(
X39
↔
X38
)
)
)
)
→
(
¬
(
X41
↔
(
(
¬
X25
)
→
X39
)
)
)
→
(
¬
(
(
X38
↔
(
X16
↔
X39
)
)
→
(
(
X41
→
(
¬
X38
)
→
(
X25
↔
X39
)
)
↔
(
¬
(
X39
→
(
X38
↔
(
X16
→
X41
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
X38
)
→
X14
→
(
¬
X11
)
→
X41
)
)
↔
(
¬
(
X41
↔
(
X38
↔
(
X25
↔
(
X14
↔
False
)
)
)
)
)
)
)
)
↔
(
(
(
(
X52
↔
(
X48
→
X35
→
X45
)
)
→
(
¬
(
(
¬
X51
)
→
(
X45
↔
(
(
¬
X35
)
→
X52
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X45
)
→
(
(
¬
X35
)
↔
(
(
¬
X29
)
↔
X51
)
)
)
)
↔
(
X51
↔
(
(
¬
X35
)
→
(
X5
↔
X45
)
)
)
)
↔
(
(
¬
X45
)
↔
X51
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
¬
(
(
(
¬
X45
)
↔
(
X38
→
X9
→
X51
)
)
→
(
(
X51
→
X45
→
X24
→
X48
)
↔
(
X48
↔
(
X38
↔
(
X24
→
X51
)
)
)
)
)
)
→
(
¬
(
(
(
¬
X45
)
↔
(
X38
↔
(
X33
→
X51
)
)
)
→
(
¬
(
(
¬
X45
)
↔
(
X38
↔
(
(
¬
X21
)
↔
X51
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
X38
)
↔
(
(
¬
X19
)
↔
X45
)
)
)
↔
(
¬
(
(
¬
X28
)
→
(
¬
X19
)
→
X51
)
)
)
→
(
¬
(
X28
→
X45
)
)
)
)
→
(
(
(
¬
(
(
(
¬
(
X51
→
(
¬
X31
)
→
X30
→
X37
)
)
↔
(
¬
(
X51
↔
(
(
¬
X37
)
→
X48
)
)
)
)
→
(
¬
(
(
¬
X37
)
→
X48
)
)
)
)
↔
(
(
¬
X37
)
→
(
(
¬
X31
)
↔
(
X2
↔
X48
)
)
)
)
↔
(
X51
↔
(
X48
→
(
X30
↔
X45
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
¬
(
X48
↔
(
(
¬
X45
)
↔
(
X8
↔
X50
)
)
)
)
↔
(
(
X48
→
(
X45
↔
(
X17
↔
X40
)
)
)
↔
(
(
¬
X40
)
→
(
(
¬
X22
)
↔
(
(
¬
X17
)
→
X48
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X51
↔
(
X48
→
X42
→
X40
)
)
)
→
X40
→
(
¬
X34
)
→
X42
)
)
↔
(
¬
(
(
¬
X48
)
→
(
(
¬
X40
)
↔
(
(
¬
X22
)
↔
X51
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X50
↔
(
X48
↔
(
X15
↔
X51
)
)
)
)
→
(
(
(
¬
X50
)
↔
(
(
¬
X47
)
↔
(
X27
↔
X51
)
)
)
↔
(
(
¬
(
(
¬
X7
)
↔
X27
)
)
→
(
¬
(
X50
→
(
(
¬
X47
)
↔
(
(
¬
X7
)
→
X48
)
)
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X49
↔
(
X20
↔
(
(
¬
X1
)
→
X18
)
)
)
)
→
(
(
¬
(
X23
→
(
X18
↔
(
X1
↔
X49
)
)
)
)
↔
(
¬
(
(
¬
X23
)
→
(
X18
↔
(
(
¬
X4
)
↔
X49
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X45
→
X44
→
X43
→
False
)
)
→
(
¬
(
(
(
¬
X44
)
→
X49
)
→
(
(
(
¬
X49
)
↔
(
X45
↔
X51
)
)
↔
(
(
X51
↔
X49
)
↔
(
¬
(
X44
→
(
(
¬
X36
)
↔
X45
)
)
)
)
)
)
)
→
(
¬
(
X49
↔
(
X44
↔
False
)
)
)
)
)
↔
(
(
¬
(
X44
→
(
X18
↔
(
X13
→
X20
)
)
)
)
↔
(
(
(
¬
X44
)
↔
(
(
¬
X36
)
→
(
(
¬
X18
)
↔
X45
)
)
)
→
(
¬
(
(
¬
X44
)
→
(
¬
X36
)
→
(
¬
X20
)
→
X45
)
)
)
)
)
)
↔
(
¬
(
X51
→
(
X50
↔
(
X44
→
X43
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
(
¬
(
X48
→
(
X40
↔
(
X21
↔
X41
)
)
)
)
↔
(
X41
↔
(
X40
→
X21
→
X47
)
)
)
→
(
(
¬
(
X40
↔
(
X30
↔
X38
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X21
)
→
X52
)
)
↔
(
¬
(
X52
↔
(
X38
→
X30
→
X40
)
)
)
)
→
(
X52
↔
(
X48
→
(
¬
X9
)
→
X40
)
)
→
(
(
¬
X40
)
↔
(
(
¬
X38
)
→
X48
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X47
→
(
X10
↔
X4
)
)
)
↔
(
¬
(
X50
↔
(
X41
↔
(
(
¬
X4
)
↔
X47
)
)
)
)
)
↔
(
(
¬
X28
)
→
(
(
¬
X4
)
↔
X41
)
)
)
→
(
(
¬
(
(
(
¬
X47
)
→
(
X38
↔
X51
)
)
→
(
¬
(
(
¬
(
X31
↔
(
(
¬
X2
)
→
X38
)
)
)
→
(
(
¬
(
X47
↔
(
X31
↔
(
(
¬
X2
)
↔
X51
)
)
)
)
↔
(
¬
(
(
¬
X47
)
→
(
¬
X38
)
→
(
(
¬
X31
)
↔
X51
)
)
)
)
)
)
)
)
↔
(
(
X47
→
(
X38
↔
(
(
¬
X4
)
↔
X51
)
)
)
↔
(
¬
(
(
X38
↔
(
(
¬
X28
)
↔
(
X20
↔
X47
)
)
)
→
X47
→
X38
→
X28
→
X51
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X43
)
→
(
(
¬
X33
)
↔
(
(
¬
X24
)
↔
X48
)
)
)
)
→
(
¬
(
(
X51
→
X48
→
X43
→
X24
)
→
X48
→
X44
→
(
X43
↔
X24
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X51
↔
(
X36
↔
X19
)
)
)
→
(
(
¬
X24
)
↔
X46
)
)
→
(
¬
X46
)
→
(
¬
X36
)
→
X51
)
)
→
(
(
(
X51
→
(
(
¬
X24
)
↔
(
(
¬
X0
)
↔
X43
)
)
)
→
(
¬
(
X51
→
(
¬
X24
)
→
X14
→
X43
)
)
)
↔
(
X51
↔
(
(
¬
X24
)
→
X19
→
X43
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
(
¬
(
X48
↔
(
X24
→
(
X3
↔
X52
)
)
)
)
→
(
¬
(
(
¬
X49
)
→
(
¬
X48
)
→
(
(
¬
X24
)
↔
X52
)
)
)
)
→
(
¬
(
X49
→
X3
→
X24
)
)
)
)
→
(
¬
(
X46
→
X3
→
False
)
)
)
)
→
(
(
X52
→
X49
→
(
X48
↔
X29
)
)
↔
(
¬
(
(
¬
X49
)
↔
(
(
¬
X26
)
↔
(
X18
↔
X52
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X34
→
(
(
¬
X32
)
↔
(
X22
↔
X49
)
)
)
)
→
(
(
¬
(
X49
→
(
X32
↔
(
X22
→
(
X13
↔
False
)
)
)
)
)
→
(
¬
(
(
(
X49
→
X22
→
X16
)
↔
(
(
(
¬
(
X49
→
X42
→
X22
→
X16
)
)
↔
(
¬
(
(
¬
X39
)
↔
(
X16
→
X42
)
)
)
)
↔
(
X49
→
X42
)
)
)
→
(
¬
(
(
¬
X42
)
↔
(
X13
↔
X49
)
)
)
)
)
)
→
X48
→
X39
→
(
¬
X16
)
→
X49
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
X50
→
(
X48
↔
X1
)
)
)
→
(
(
(
(
(
¬
(
X50
↔
False
)
)
↔
(
¬
(
(
¬
X42
)
↔
(
(
¬
X7
)
↔
(
X1
→
X48
)
)
)
)
)
↔
(
X50
→
X42
→
X1
)
)
→
(
¬
(
X48
→
X42
→
X1
→
False
)
)
)
↔
(
X50
↔
(
(
¬
X42
)
↔
X48
)
)
)
)
)
→
(
X48
↔
(
X1
↔
False
)
)
)
)
↔
(
(
(
¬
X39
)
↔
(
X11
↔
X48
)
)
↔
(
¬
(
X48
→
X42
→
(
X7
↔
X11
)
)
)
)
)
)
→
(
(
¬
(
(
(
X25
↔
(
(
¬
(
X35
↔
(
(
¬
X25
)
→
(
X5
↔
X45
)
)
)
)
↔
(
¬
(
X45
↔
(
X25
→
X42
)
)
)
)
)
→
X45
→
(
X35
↔
(
X27
↔
X25
)
)
)
→
(
¬
(
(
¬
(
X39
↔
(
X15
↔
False
)
)
)
→
(
(
X45
↔
(
X35
→
X23
)
)
↔
(
X35
↔
(
(
¬
X15
)
↔
X39
)
)
)
)
)
→
(
(
X23
↔
(
(
¬
X8
)
→
X27
)
)
↔
(
(
¬
(
X37
→
(
(
¬
X8
)
↔
X39
)
)
)
↔
(
X45
→
(
X27
↔
(
X6
→
X8
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X49
↔
(
(
¬
X27
)
→
(
(
¬
X12
)
↔
X39
)
)
)
)
→
(
(
X49
↔
(
(
¬
X27
)
↔
(
X25
↔
X39
)
)
)
↔
(
X27
→
(
X25
↔
(
(
¬
X17
)
→
X49
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
46ebde...
and proposition id is
1d6184...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMWXukJRmVaGqwXSH9MHDLjjFk3YDQXphi1
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
¬
(
(
(
(
(
¬
(
(
X41
→
(
(
¬
X38
)
↔
(
X29
↔
X47
)
)
)
→
(
(
X38
→
(
(
¬
X29
)
↔
(
(
¬
X12
)
→
X41
)
)
)
↔
(
¬
(
X41
→
X12
→
(
X3
↔
X38
)
)
)
)
)
)
→
(
(
X48
→
X41
→
(
¬
X2
)
→
X38
)
↔
(
(
(
¬
(
(
¬
X38
)
→
X2
→
X46
)
)
→
(
¬
X38
)
→
X35
→
(
(
¬
X29
)
↔
X46
)
)
↔
(
X46
↔
(
X35
↔
(
X2
→
False
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
X41
→
X16
→
(
¬
X10
)
→
X31
)
)
↔
(
¬
(
X46
↔
(
X31
↔
(
X16
→
X41
)
)
)
)
)
→
(
¬
(
X48
↔
(
X46
↔
(
X41
↔
(
X31
→
False
)
)
)
)
)
)
→
(
(
¬
(
X48
→
X46
→
(
X43
↔
X41
)
)
)
↔
(
¬
(
X48
↔
(
X46
→
X43
→
(
X7
↔
False
)
)
)
)
)
)
→
(
(
¬
X45
)
↔
(
(
¬
X11
)
↔
(
X5
→
X46
)
)
)
→
(
¬
(
(
X45
↔
(
X43
→
(
X11
↔
X5
)
)
)
→
(
X48
→
(
(
¬
X11
)
↔
X15
)
)
→
(
(
(
X48
→
(
¬
X15
)
→
X45
)
↔
(
¬
(
X48
↔
(
(
¬
X15
)
→
X5
→
X42
)
)
)
)
↔
(
¬
(
(
¬
X42
)
↔
(
X15
↔
X45
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
(
¬
(
X44
↔
(
(
¬
X24
)
↔
(
X18
→
X43
)
)
)
)
→
X24
→
X33
)
→
X47
→
(
¬
X43
)
→
(
(
¬
X18
)
↔
X46
)
)
)
→
(
¬
(
X46
→
(
X44
↔
(
(
¬
X0
)
→
X47
)
)
)
)
)
→
(
X47
↔
(
X44
→
(
X39
↔
X33
)
)
)
)
↔
(
(
¬
(
X47
→
(
X44
↔
(
X43
→
False
)
)
)
)
↔
(
¬
(
(
X47
→
X43
→
(
X32
↔
(
X20
→
False
)
)
)
→
(
(
X47
↔
(
X44
→
X32
→
(
X28
↔
False
)
)
)
↔
(
¬
(
(
¬
X44
)
↔
(
X32
↔
(
X27
↔
X47
)
)
)
)
)
)
)
)
)
)
→
(
(
(
(
(
X37
→
X36
→
(
¬
X22
)
→
X40
)
↔
(
¬
(
X40
→
X36
→
X9
)
)
)
→
(
(
¬
(
(
X40
↔
(
(
¬
X34
)
→
(
X22
↔
X45
)
)
)
→
(
¬
(
(
¬
X40
)
→
X47
)
)
)
)
→
X40
→
X22
)
→
(
¬
(
(
¬
(
X34
↔
(
X30
→
X45
)
)
)
→
(
(
(
¬
(
X30
↔
X40
)
)
↔
(
X36
→
X34
→
X45
)
)
↔
(
X40
→
(
X30
↔
X34
)
)
)
)
)
)
→
(
(
(
X34
→
(
X23
↔
(
X6
→
X36
)
)
)
↔
(
(
¬
X30
)
↔
(
(
¬
X23
)
→
(
(
¬
X6
)
↔
X36
)
)
)
)
↔
(
(
¬
X34
)
↔
X36
)
)
→
X34
→
(
¬
X30
)
→
X36
)
↔
(
(
(
(
¬
(
(
X45
↔
(
X21
↔
(
X4
→
X46
)
)
)
→
(
¬
(
X46
→
X21
→
(
(
¬
X13
)
↔
X45
)
)
)
)
)
↔
(
(
¬
X21
)
→
(
X14
↔
X46
)
)
)
↔
(
¬
(
(
X37
↔
(
X14
→
(
¬
X8
)
→
X46
)
)
→
(
¬
(
(
X49
→
X4
→
X1
→
X14
)
→
(
(
¬
(
(
¬
X37
)
→
(
X4
↔
X49
)
)
)
↔
(
¬
(
X49
→
(
X46
↔
(
X37
→
X14
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
X26
)
↔
(
X25
→
(
¬
X17
)
→
X45
)
)
↔
(
¬
(
X26
↔
(
X25
↔
(
(
¬
X19
)
→
X45
)
)
)
)
)
↔
(
X26
↔
(
X19
→
X45
)
)
)
→
(
X46
→
(
¬
X30
)
→
(
¬
X26
)
→
X37
)
→
(
¬
(
(
X30
→
(
X19
↔
(
X17
↔
X37
)
)
)
→
(
¬
(
X46
↔
(
X37
↔
(
X17
↔
X30
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
(
¬
X47
)
→
X43
→
(
X41
↔
X48
)
)
↔
(
¬
(
X48
↔
(
(
¬
X43
)
↔
(
X38
→
X47
)
)
)
)
)
→
(
(
¬
(
X48
→
X46
→
(
¬
X35
)
→
X39
)
)
→
(
¬
(
X39
↔
(
(
¬
X35
)
↔
(
(
¬
X26
)
→
X48
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X46
)
→
(
X26
↔
X48
)
)
)
↔
(
X41
↔
(
X26
↔
X43
)
)
)
↔
(
X48
↔
(
(
¬
X43
)
→
X25
→
X46
)
)
)
→
(
¬
(
X25
↔
X46
)
)
)
→
(
(
X47
→
(
X46
↔
(
(
¬
X13
)
↔
X31
)
)
)
↔
(
X46
→
(
¬
X37
)
→
X0
→
X48
)
)
)
→
(
¬
(
(
(
(
¬
(
X46
→
X31
→
X3
→
X32
)
)
↔
(
(
¬
(
(
(
X37
→
X31
→
(
X29
↔
X32
)
)
↔
(
¬
(
X32
↔
(
X29
→
X27
→
X37
)
)
)
)
→
(
¬
(
X29
↔
(
X27
→
X37
)
)
)
)
)
↔
(
¬
(
(
¬
X37
)
→
(
¬
X32
)
→
(
(
¬
X31
)
↔
X46
)
)
)
)
)
→
(
X15
↔
(
¬
(
X44
→
X37
→
(
(
¬
X31
)
↔
X32
)
)
)
)
)
→
(
(
(
(
¬
X32
)
→
(
¬
X16
)
→
(
¬
X11
)
→
X37
)
→
(
(
(
¬
X32
)
→
X17
→
(
¬
X4
)
→
X37
)
↔
(
X17
→
X16
→
X4
→
X37
)
)
)
↔
(
¬
(
(
¬
(
(
X22
↔
X16
)
→
(
(
(
(
¬
(
X22
→
X17
→
X16
)
)
↔
(
X37
→
(
X22
↔
X46
)
)
)
↔
(
¬
(
X16
↔
False
)
)
)
↔
(
X37
↔
(
(
¬
X17
)
→
X22
)
)
)
)
)
→
(
¬
X37
)
→
(
¬
X22
)
→
X46
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X43
↔
(
(
¬
X42
)
→
(
X36
↔
X46
)
)
)
)
↔
(
¬
(
X43
→
(
¬
X42
)
→
(
(
¬
X7
)
↔
X46
)
)
)
)
↔
(
¬
(
(
X46
↔
(
X42
→
X36
→
False
)
)
→
(
¬
(
(
¬
(
X46
→
X18
→
X40
)
)
→
(
¬
(
(
¬
X40
)
↔
(
X36
↔
X42
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X46
→
X42
→
(
¬
X24
)
→
X47
)
)
→
(
(
¬
(
X43
→
(
¬
X42
)
→
(
(
¬
X19
)
↔
X46
)
)
)
↔
(
X24
↔
X42
)
)
→
(
X43
↔
(
X24
↔
X19
)
)
)
)
)
↔
(
(
(
(
X49
↔
(
X46
↔
(
X34
↔
(
X33
↔
False
)
)
)
)
↔
(
(
(
(
¬
(
X47
→
(
(
¬
X34
)
↔
(
X10
↔
X46
)
)
)
)
↔
(
X34
↔
(
X12
→
False
)
)
)
↔
(
(
X10
↔
X23
)
→
(
¬
(
X49
↔
(
(
¬
X12
)
↔
(
(
¬
X10
)
→
X46
)
)
)
)
)
)
↔
(
¬
(
X47
→
X34
→
X49
)
)
)
)
↔
(
(
(
X47
→
(
(
¬
X14
)
↔
X46
)
)
→
(
¬
(
X46
→
(
¬
X1
)
→
X45
)
)
)
→
X46
→
(
(
¬
X45
)
↔
(
X34
↔
X47
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
X30
→
(
X21
↔
X5
)
)
↔
(
(
¬
X21
)
→
(
¬
X20
)
→
(
(
¬
X8
)
↔
X28
)
)
)
→
(
¬
X28
)
→
(
(
¬
X21
)
↔
(
(
¬
X2
)
→
X30
)
)
)
)
→
(
(
¬
(
(
(
(
(
(
X46
→
(
X45
↔
(
X30
→
(
X28
↔
False
)
)
)
)
→
(
¬
(
(
¬
X42
)
→
X30
→
(
X28
↔
X46
)
)
)
)
↔
(
¬
(
X46
→
(
¬
X30
)
→
X42
)
)
)
↔
(
¬
(
(
¬
(
X45
↔
(
(
¬
X28
)
→
X46
)
)
)
→
X46
→
(
¬
X9
)
→
X49
)
)
)
↔
(
(
¬
X42
)
↔
(
X9
→
X45
)
)
)
→
(
¬
(
(
¬
X28
)
↔
(
X21
↔
(
X9
→
X42
)
)
)
)
)
)
↔
(
X28
↔
(
(
¬
X21
)
→
(
¬
X6
)
→
X46
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
f8d7de...
and proposition id is
cea952...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMRFg3ZYXLcPbQER5vVfhhPsvwSELotu2ob
)
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
(
(
(
(
(
(
X49
→
X42
→
X18
→
(
X4
↔
False
)
)
→
(
(
X42
→
(
X38
↔
(
(
¬
X18
)
↔
X34
)
)
)
↔
(
(
¬
X38
)
→
(
X18
↔
(
X4
↔
X42
)
)
)
)
)
↔
(
(
(
(
X49
↔
(
(
¬
X18
)
↔
(
(
¬
X0
)
→
X25
)
)
)
↔
(
¬
(
X49
↔
(
X18
→
X4
→
X25
)
)
)
)
→
(
X49
↔
(
(
¬
X18
)
↔
X25
)
)
)
↔
(
X25
→
(
X18
↔
X4
)
)
)
)
↔
(
(
(
¬
X48
)
↔
(
X25
→
(
(
¬
X15
)
↔
X50
)
)
)
→
X48
→
X41
→
X25
→
(
X15
↔
False
)
)
)
↔
(
(
¬
(
X49
↔
(
X41
→
(
¬
X27
)
→
X45
)
)
)
↔
(
(
(
¬
(
X27
↔
(
(
¬
X12
)
→
X45
)
)
)
↔
(
¬
(
(
X45
→
X11
)
→
(
¬
X25
)
→
(
X12
↔
(
X11
→
X27
)
)
)
)
)
↔
(
X49
→
X45
)
)
)
)
↔
(
(
(
(
(
(
X50
↔
X49
)
→
X47
→
(
¬
X46
)
→
(
X1
↔
X49
)
)
→
(
¬
(
X47
↔
(
X46
→
X19
→
X50
)
)
)
)
↔
(
X32
↔
(
X19
↔
(
(
¬
X1
)
→
X49
)
)
)
)
↔
(
¬
(
(
¬
(
X50
↔
(
X46
→
(
X45
↔
X49
)
)
)
)
→
(
(
(
¬
(
X49
↔
(
X45
↔
X21
)
)
)
→
(
¬
X49
)
→
(
(
¬
X21
)
↔
(
X9
→
X50
)
)
)
↔
(
X50
↔
(
X49
→
(
X45
↔
X14
)
)
)
)
)
)
)
→
(
X49
→
X45
→
X20
→
X36
)
→
(
¬
(
(
X49
→
(
X47
↔
(
(
¬
X3
)
↔
X31
)
)
)
→
(
(
¬
(
X47
→
(
¬
X20
)
→
(
X3
↔
X45
)
)
)
↔
(
¬
(
X49
→
(
X47
↔
(
X20
→
X3
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
(
X46
→
X30
)
→
(
¬
(
(
¬
X29
)
→
(
X13
↔
(
(
¬
X6
)
↔
X30
)
)
)
)
)
)
→
X46
→
(
X26
↔
(
X6
↔
X29
)
)
)
→
(
(
X43
↔
(
X30
↔
X29
)
)
↔
(
(
X16
→
False
)
↔
(
X29
↔
(
¬
(
(
¬
X16
)
→
(
¬
(
(
(
¬
X29
)
↔
X44
)
→
(
X30
↔
(
X29
↔
(
X16
↔
X44
)
)
)
→
(
¬
(
X44
→
X30
→
X13
→
X29
)
)
)
)
)
)
)
)
)
)
↔
(
(
X48
→
X26
→
(
¬
X23
)
→
X24
)
↔
(
¬
(
X46
↔
(
X43
→
(
X24
↔
X48
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
(
¬
X46
)
↔
(
(
¬
X40
)
↔
(
(
¬
X17
)
→
X48
)
)
)
)
↔
(
X46
↔
(
X45
→
(
X40
↔
X48
)
)
)
)
→
(
(
X45
→
X39
)
→
(
¬
(
(
(
¬
(
X48
→
X39
→
X46
)
)
↔
(
¬
(
X46
↔
(
X45
↔
(
(
¬
X33
)
↔
X39
)
)
)
)
)
→
(
¬
(
X46
↔
(
X39
↔
X48
)
)
)
)
)
)
→
(
¬
(
X46
↔
(
X33
↔
X45
)
)
)
)
)
→
(
¬
(
(
¬
(
X46
↔
(
X43
→
(
(
¬
X28
)
↔
X35
)
)
)
)
→
(
¬
(
(
(
X45
→
(
X22
↔
X28
)
)
↔
(
(
¬
(
(
¬
X35
)
→
(
(
¬
X28
)
↔
X45
)
)
)
↔
(
(
¬
(
X45
→
(
X35
↔
(
X28
→
(
X7
↔
False
)
)
)
)
)
↔
(
(
¬
(
X22
↔
X35
)
)
↔
(
(
¬
(
X35
→
(
X28
↔
False
)
)
)
→
(
X45
↔
(
X43
→
X22
→
False
)
)
)
)
)
)
)
→
X45
→
X40
→
(
X28
↔
(
X22
↔
False
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X10
↔
(
X5
→
X39
)
)
)
↔
(
(
X10
↔
(
X8
→
(
¬
X5
)
→
X39
)
)
↔
(
(
(
¬
(
X8
↔
(
(
¬
X5
)
→
(
¬
X2
)
→
X40
)
)
)
↔
(
X40
↔
(
(
¬
X5
)
→
(
¬
X2
)
→
X44
)
)
)
↔
(
X8
↔
(
X2
→
X39
)
)
)
)
)
→
X44
→
(
X40
↔
(
X39
→
X37
→
False
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
¬
(
(
(
¬
X42
)
↔
(
X19
→
(
X2
↔
X47
)
)
)
→
(
¬
(
X49
↔
(
X47
↔
X42
)
)
)
)
)
↔
(
(
X3
→
(
X2
↔
X26
)
)
→
(
¬
(
(
¬
(
(
¬
(
X42
→
(
¬
X26
)
→
(
¬
X2
)
→
X44
)
)
→
(
¬
(
X44
→
X42
→
X3
→
False
)
)
)
)
→
(
¬
(
X44
→
(
X42
↔
X26
)
)
)
)
)
)
)
→
(
(
(
X17
↔
(
(
¬
X12
)
→
X27
)
)
↔
(
(
X49
↔
(
X19
→
(
(
¬
X17
)
↔
X27
)
)
)
↔
(
X27
↔
(
X17
→
(
X12
↔
X49
)
)
)
)
)
↔
(
¬
(
X49
↔
(
X27
→
(
(
¬
X17
)
↔
X42
)
)
)
)
)
)
→
(
(
¬
(
(
¬
X44
)
↔
(
(
¬
X33
)
↔
X48
)
)
)
↔
(
X42
→
(
(
¬
X27
)
↔
(
X15
↔
X48
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
X40
↔
(
(
¬
X30
)
→
(
(
¬
X0
)
↔
X41
)
)
)
)
↔
(
X46
→
X30
)
)
↔
(
X48
→
(
¬
X22
)
→
X40
)
)
↔
(
X40
↔
(
(
¬
X29
)
↔
X41
)
)
)
↔
(
(
(
X44
↔
(
(
¬
X41
)
→
(
¬
X37
)
→
X47
)
)
↔
(
¬
(
X47
↔
(
X44
↔
(
(
¬
X30
)
→
X41
)
)
)
)
)
→
(
X46
↔
(
(
¬
X41
)
→
(
X30
↔
X47
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X46
↔
(
X41
→
X36
)
)
)
→
(
¬
(
(
¬
X41
)
→
X28
→
X43
)
)
→
(
¬
(
(
¬
(
(
X46
→
(
X36
↔
(
X28
↔
X25
)
)
)
→
X43
→
(
X25
↔
X28
)
)
)
→
(
¬
(
X36
↔
(
X25
→
X28
)
)
)
)
)
)
)
↔
(
(
(
¬
X41
)
↔
(
(
¬
X39
)
→
X30
→
X46
)
)
↔
(
¬
(
X39
↔
(
(
¬
X36
)
→
(
(
¬
X30
)
↔
X41
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X20
↔
(
X16
↔
X7
)
)
)
↔
(
¬
(
(
X49
↔
(
(
¬
X16
)
→
(
(
¬
X7
)
↔
X20
)
)
)
→
(
¬
(
X49
↔
(
X20
→
(
¬
X16
)
→
X50
)
)
)
)
)
)
↔
(
(
(
X50
↔
X31
)
↔
(
(
X50
↔
(
X24
→
(
X5
↔
X9
)
)
)
→
(
(
¬
X24
)
↔
(
X5
→
X49
)
)
)
)
↔
(
¬
(
X50
↔
(
X49
↔
(
X31
→
X20
)
)
)
)
)
)
↔
(
(
(
X50
→
(
¬
X31
)
→
X32
)
↔
(
¬
(
X48
↔
(
X32
↔
(
X9
→
X50
)
)
)
)
)
↔
(
¬
(
X50
→
X48
→
X11
→
False
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
X50
↔
(
X48
→
X21
→
False
)
)
→
(
¬
(
X49
→
(
X45
↔
X21
)
)
)
)
)
→
X50
→
(
¬
X45
)
→
X13
→
X49
)
)
→
(
(
(
¬
(
X38
↔
(
(
¬
X34
)
↔
X49
)
)
)
↔
(
(
(
(
¬
(
X49
→
X34
→
(
(
¬
X6
)
↔
X38
)
)
)
↔
(
(
¬
X38
)
↔
(
(
¬
X34
)
→
X50
)
)
)
↔
(
(
¬
X49
)
↔
(
(
¬
X38
)
↔
X50
)
)
)
↔
(
¬
(
X49
→
(
¬
X38
)
→
X50
)
)
)
)
↔
(
(
¬
(
(
¬
X38
)
→
X34
→
(
(
¬
X8
)
↔
X50
)
)
)
→
(
¬
(
(
(
¬
X46
)
↔
(
X38
↔
(
X8
↔
X48
)
)
)
→
(
¬
(
X34
↔
False
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
(
X35
↔
(
X23
↔
(
(
¬
X1
)
→
X46
)
)
)
)
→
(
(
¬
(
X50
↔
(
X35
→
(
(
¬
X1
)
↔
X23
)
)
)
)
↔
(
X48
↔
(
X35
↔
(
X23
↔
X50
)
)
)
)
)
→
(
¬
(
X48
→
(
¬
X46
)
→
X18
→
X50
)
)
)
)
→
(
¬
(
(
¬
(
X18
↔
(
(
¬
X10
)
↔
X46
)
)
)
→
(
¬
(
(
¬
(
X46
→
(
X14
↔
(
X4
→
X18
)
)
)
)
→
(
X14
↔
(
(
¬
X4
)
↔
X35
)
)
)
)
→
(
¬
X35
)
→
(
X18
↔
(
(
¬
X10
)
→
X46
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
93630a...
and proposition id is
881288...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMbJttPQixuQ4dJwqxiHx7KdJchiAjJjBJd
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
(
X47
↔
(
X30
↔
(
X25
→
X14
)
)
)
↔
(
¬
(
(
(
¬
(
X47
↔
(
X46
↔
X14
)
)
)
↔
(
¬
(
X47
↔
(
X41
↔
X27
)
)
)
)
→
(
¬
(
(
¬
X27
)
↔
(
X25
↔
(
(
¬
X14
)
↔
X46
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X41
→
(
X16
↔
X11
)
)
)
↔
(
X46
↔
(
X17
↔
(
X16
↔
X15
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X39
→
(
¬
X18
)
→
X17
→
X37
)
)
↔
(
¬
(
X47
↔
(
X37
↔
(
X17
↔
False
)
)
)
)
)
↔
(
(
X39
→
(
¬
X18
)
→
(
X16
↔
X46
)
)
→
(
¬
(
X39
↔
(
(
¬
X18
)
↔
X47
)
)
)
)
)
→
(
(
X37
→
(
¬
X7
)
→
(
(
¬
X6
)
↔
X17
)
)
↔
(
(
X37
↔
(
X10
↔
(
X6
→
X16
)
)
)
↔
(
(
¬
X16
)
→
(
X7
↔
(
X0
→
X37
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
(
¬
X37
)
↔
(
(
¬
X29
)
↔
X43
)
)
↔
(
¬
(
(
(
¬
(
(
(
¬
(
X43
↔
(
(
¬
X37
)
↔
X46
)
)
)
↔
(
(
¬
(
(
¬
X5
)
↔
X46
)
)
→
(
(
(
¬
X43
)
→
(
¬
X37
)
→
(
X29
↔
X46
)
)
↔
(
¬
(
X43
↔
(
X5
↔
X37
)
)
)
)
)
)
→
(
¬
X37
)
→
(
¬
X29
)
→
X46
)
)
↔
(
¬
(
X43
→
(
X37
↔
False
)
)
)
)
→
X29
→
X5
)
)
)
↔
(
¬
(
(
¬
X37
)
→
X29
→
(
X5
↔
X49
)
)
)
)
→
X43
→
X26
→
(
(
¬
X5
)
↔
X29
)
)
)
↔
(
(
(
X28
↔
(
X13
↔
(
X5
→
X29
)
)
)
↔
(
(
X38
↔
(
(
¬
X5
)
→
X45
)
)
→
(
¬
(
X45
→
(
¬
X5
)
→
X38
)
)
→
(
¬
(
(
X34
→
X28
→
(
X5
↔
X45
)
)
→
(
X45
↔
(
X38
↔
(
X28
→
X34
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
X45
)
→
X43
→
(
¬
X5
)
→
X47
)
→
(
X47
↔
(
X45
↔
(
(
¬
X4
)
→
X29
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
X48
)
↔
(
(
¬
X44
)
↔
(
(
¬
X8
)
→
X49
)
)
)
)
↔
(
¬
(
(
(
¬
(
X48
→
X33
→
(
¬
X20
)
→
X49
)
)
↔
(
X49
↔
(
X20
↔
(
(
¬
X8
)
→
X33
)
)
)
)
→
(
¬
(
X49
→
(
X48
↔
X33
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X23
→
(
X8
↔
X9
)
)
)
↔
(
¬
(
(
¬
X21
)
↔
(
(
¬
X8
)
↔
(
(
¬
X2
)
↔
X23
)
)
)
)
)
→
(
¬
(
X44
↔
(
X21
↔
(
(
¬
X8
)
→
X9
)
)
)
)
)
↔
(
(
¬
X23
)
→
(
X2
↔
X33
)
)
)
)
→
(
(
(
(
¬
(
X44
→
(
X42
↔
(
(
¬
X23
)
↔
X46
)
)
)
)
→
(
X46
↔
(
X23
↔
X12
)
)
)
↔
(
X44
↔
(
(
¬
X36
)
→
(
(
¬
X23
)
↔
X46
)
)
)
)
↔
(
(
¬
X44
)
→
X42
→
(
¬
X33
)
→
X46
)
)
)
→
(
(
(
(
¬
(
X32
→
(
¬
X22
)
→
(
(
¬
X1
)
↔
X42
)
)
)
↔
(
X32
→
X1
→
X35
)
)
↔
(
(
X48
→
X45
→
(
X35
↔
(
X22
→
False
)
)
)
↔
(
X48
→
X45
→
(
X42
↔
X35
)
)
)
)
↔
(
¬
(
(
(
X42
→
X31
→
(
X24
↔
False
)
)
↔
(
(
X42
→
(
(
¬
X31
)
↔
X36
)
)
→
(
¬
(
X42
→
(
X36
↔
(
X31
→
(
X24
↔
False
)
)
)
)
)
→
X36
→
(
X24
↔
X19
)
)
)
→
(
(
¬
(
(
X48
↔
(
X40
↔
(
(
¬
X3
)
↔
X31
)
)
)
→
(
¬
(
(
¬
X40
)
↔
(
X36
→
(
¬
X31
)
→
X42
)
)
)
)
)
↔
(
¬
(
X45
→
(
¬
X40
)
→
(
(
¬
X36
)
↔
X48
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
(
(
(
(
¬
X35
)
↔
(
(
¬
X14
)
↔
X44
)
)
→
(
¬
(
(
¬
(
X46
↔
(
X44
↔
(
(
¬
X14
)
↔
X35
)
)
)
)
→
(
¬
(
X44
→
(
X22
↔
X46
)
)
)
)
)
)
→
(
(
¬
(
X46
↔
(
X37
↔
(
(
¬
X7
)
→
X14
)
)
)
)
↔
(
X46
↔
(
X37
→
X7
→
X23
)
)
)
)
)
→
(
¬
(
(
¬
(
X46
↔
(
X36
→
(
¬
X27
)
→
X33
)
)
)
→
(
¬
(
(
¬
(
(
¬
X44
)
→
(
(
¬
X37
)
↔
(
(
¬
X32
)
→
X46
)
)
)
)
→
(
¬
(
X44
↔
(
(
¬
X32
)
→
(
¬
X27
)
→
X37
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
X46
)
→
(
(
¬
X16
)
↔
X47
)
)
↔
(
(
¬
X46
)
→
X37
→
(
X17
↔
X47
)
)
)
→
(
¬
(
(
¬
X46
)
→
X16
→
X47
)
)
)
↔
(
(
(
¬
X12
)
↔
(
¬
(
(
(
¬
X17
)
→
(
X12
↔
(
(
¬
X6
)
↔
X36
)
)
)
→
(
(
X47
↔
(
(
¬
X17
)
→
X6
→
X36
)
)
↔
(
¬
(
X47
→
X36
→
(
X17
↔
(
X3
↔
False
)
)
)
)
)
→
X36
→
(
X17
↔
X3
)
)
)
)
→
(
(
¬
X37
)
↔
(
X17
→
(
¬
X12
)
→
X47
)
)
)
)
)
→
(
(
(
X26
→
(
(
¬
(
X49
→
(
X46
↔
(
X39
→
X26
→
False
)
)
)
)
↔
(
X47
↔
(
X39
→
False
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
X49
↔
(
X47
→
(
X43
↔
X44
)
)
)
)
→
X49
→
X47
→
(
X44
↔
X46
)
)
)
→
(
X44
↔
(
X40
↔
(
X38
↔
X47
)
)
)
)
→
X47
→
(
X43
↔
(
X40
→
False
)
)
)
)
)
↔
(
(
(
¬
(
(
(
X49
→
(
(
¬
X28
)
↔
X38
)
)
→
(
(
(
¬
X38
)
→
(
X28
↔
(
X8
→
X49
)
)
)
↔
(
¬
(
(
¬
(
X39
→
X8
→
(
X0
↔
X49
)
)
)
→
(
X49
↔
(
X38
→
(
(
¬
X8
)
↔
X39
)
)
)
)
)
)
)
→
(
¬
(
X49
↔
(
X39
→
X28
→
(
X25
↔
False
)
)
)
)
)
)
↔
(
¬
(
X49
→
(
(
¬
X24
)
↔
(
X8
→
X25
)
)
)
)
)
↔
(
X49
↔
(
X38
→
(
X2
↔
X39
)
)
)
)
)
)
→
(
(
(
¬
(
X39
↔
(
X29
↔
(
(
¬
X10
)
↔
X48
)
)
)
)
↔
(
(
¬
(
(
¬
X39
)
↔
(
X13
↔
X48
)
)
)
↔
(
(
¬
(
(
¬
(
X39
↔
(
X29
↔
(
(
¬
X13
)
→
X48
)
)
)
)
→
(
¬
(
(
¬
X29
)
↔
(
X13
→
(
X11
↔
X48
)
)
)
)
)
)
→
(
¬
X29
)
→
(
X5
↔
X39
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X45
)
→
(
¬
X29
)
→
X1
→
X47
)
)
↔
(
(
¬
(
X39
↔
(
X18
↔
X29
)
)
)
↔
(
X45
↔
(
(
¬
X39
)
↔
(
(
¬
X29
)
↔
X47
)
)
)
)
)
→
(
(
(
¬
(
X47
↔
(
(
¬
X29
)
↔
(
X11
↔
X36
)
)
)
)
↔
(
¬
(
(
¬
X29
)
→
X11
→
X45
)
)
)
↔
(
¬
(
X36
→
(
(
¬
X11
)
↔
X45
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
(
(
¬
X42
)
↔
(
X31
↔
(
X30
→
X45
)
)
)
→
(
¬
(
X45
↔
(
X42
→
X31
→
X20
)
)
)
)
)
↔
(
¬
(
X42
→
X31
→
X34
)
)
)
→
(
(
X30
→
(
¬
X21
)
→
(
(
¬
X19
)
↔
X42
)
)
↔
(
¬
(
(
X48
↔
(
X42
→
X45
)
)
→
(
(
¬
(
X45
↔
(
(
¬
X30
)
→
(
(
¬
X19
)
↔
X42
)
)
)
)
↔
(
¬
(
X30
↔
(
X19
→
X48
)
)
)
)
)
)
)
→
(
(
¬
X41
)
↔
(
(
¬
X30
)
→
(
X21
↔
X42
)
)
)
)
)
↔
(
(
(
¬
(
X15
→
(
X4
↔
False
)
)
)
→
(
(
¬
X19
)
↔
(
(
¬
X15
)
→
(
X9
↔
X31
)
)
)
)
↔
(
(
¬
X45
)
↔
(
X31
↔
(
(
¬
X9
)
↔
X47
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
b715f7...
and proposition id is
c170cd...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMZNwmHBM1et4aqRWMrUWvR6epTgvGvdpob
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
∀X50 :
prop
,
(
(
(
(
(
(
(
¬
(
X46
→
(
X29
↔
X50
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X29
)
→
(
¬
X22
)
→
X50
)
)
)
→
(
(
¬
(
(
¬
X29
)
↔
X49
)
)
↔
(
¬
(
X50
↔
(
(
¬
X0
)
→
X16
)
)
)
)
)
→
(
(
¬
(
X50
→
(
¬
X28
)
→
X21
→
X46
)
)
↔
(
X50
↔
(
X49
→
(
(
¬
X11
)
↔
X28
)
)
)
)
)
↔
(
(
(
X49
→
(
(
¬
X48
)
↔
(
X46
→
X50
)
)
)
↔
(
(
X50
↔
(
X33
→
(
¬
X9
)
→
X23
)
)
↔
(
X50
→
(
X49
↔
(
(
¬
X23
)
↔
X46
)
)
)
)
)
↔
(
¬
(
(
X37
↔
(
(
¬
X4
)
→
X48
)
)
→
(
(
(
¬
(
(
¬
X37
)
→
(
(
¬
X5
)
↔
(
(
¬
X4
)
→
X46
)
)
)
)
↔
(
¬
(
(
¬
X37
)
↔
(
(
¬
X5
)
↔
(
X4
→
X49
)
)
)
)
)
→
(
¬
(
X49
↔
(
X5
→
(
X4
↔
False
)
)
)
)
)
→
(
¬
(
X49
→
(
X37
↔
X46
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
X50
→
X39
→
(
¬
X32
)
→
X41
)
↔
(
¬
(
(
¬
X39
)
↔
(
X32
→
(
¬
X13
)
→
X41
)
)
)
)
→
(
(
(
X50
→
X46
→
(
X39
↔
X41
)
)
→
(
¬
(
(
¬
X41
)
→
(
¬
X20
)
→
X46
)
)
)
↔
(
¬
(
X50
→
(
X20
↔
(
(
¬
X19
)
↔
X26
)
)
)
)
)
)
)
↔
(
¬
(
(
X39
→
X17
→
X25
)
→
(
¬
(
(
¬
X26
)
↔
(
(
¬
X25
)
→
(
(
¬
X20
)
↔
X39
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
X38
→
(
X26
↔
X32
)
)
→
(
(
(
(
¬
X46
)
↔
(
X38
↔
(
X18
→
X48
)
)
)
→
(
¬
X46
)
→
(
(
¬
X38
)
↔
(
X26
↔
X48
)
)
)
↔
(
¬
(
X38
→
(
X32
↔
(
(
¬
X18
)
→
X46
)
)
)
)
)
)
→
(
¬
(
X38
↔
(
X26
→
False
)
)
)
)
)
↔
(
X48
→
X38
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
(
¬
X30
)
↔
(
X27
→
X42
)
)
)
↔
(
(
X48
→
X45
→
X42
)
↔
(
¬
(
(
¬
(
(
¬
X44
)
↔
(
(
¬
X42
)
↔
(
(
¬
X27
)
↔
X48
)
)
)
)
→
(
X48
↔
(
X47
↔
False
)
)
)
)
)
)
↔
(
X49
↔
(
(
¬
X45
)
↔
(
(
¬
X44
)
→
X48
)
)
)
)
→
(
¬
(
(
(
X45
↔
(
X35
→
X48
)
)
↔
(
(
X50
→
(
X49
↔
(
(
¬
X45
)
↔
X48
)
)
)
→
(
¬
(
(
¬
X48
)
↔
(
(
¬
X45
)
↔
(
X40
→
X50
)
)
)
)
)
)
→
(
¬
(
(
(
X48
→
(
¬
X45
)
→
(
(
¬
X43
)
↔
X49
)
)
↔
(
X49
↔
(
(
¬
X45
)
→
(
(
¬
X7
)
↔
X48
)
)
)
)
→
(
¬
(
X48
↔
(
X43
→
X45
)
)
)
)
)
→
(
¬
(
X48
→
(
¬
X43
)
→
(
(
¬
X36
)
↔
X45
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
X31
)
→
(
(
¬
X10
)
↔
(
(
¬
X3
)
↔
X50
)
)
)
→
(
¬
(
X31
↔
(
(
¬
X12
)
↔
X50
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X12
→
(
(
¬
X6
)
↔
X14
)
)
)
→
(
¬
(
X50
↔
(
X12
↔
(
X1
↔
False
)
)
)
)
)
→
(
(
(
¬
X12
)
↔
(
X10
→
X8
→
X50
)
)
↔
(
(
¬
X12
)
→
X14
)
)
)
→
(
X14
↔
(
X12
→
(
(
¬
X1
)
↔
X2
)
)
)
)
)
)
↔
(
(
(
(
(
¬
X47
)
→
(
¬
X24
)
→
X8
→
X49
)
↔
(
X49
→
(
X34
↔
(
(
¬
X8
)
→
X15
)
)
)
)
↔
(
(
(
¬
(
X47
↔
(
X15
↔
(
X12
→
X14
)
)
)
)
↔
(
¬
(
X47
→
X34
→
(
X15
↔
(
X14
→
False
)
)
)
)
)
→
(
¬
X34
)
→
(
(
¬
X14
)
↔
X47
)
)
)
→
(
¬
(
(
¬
X47
)
→
X34
→
X49
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
X40
→
(
(
¬
X26
)
↔
(
X6
↔
X46
)
)
)
)
→
(
¬
(
(
(
¬
X37
)
→
(
X36
↔
(
(
¬
X26
)
↔
X40
)
)
)
→
X40
→
(
¬
X37
)
→
(
X26
↔
X46
)
)
)
)
↔
(
(
¬
(
X47
↔
(
X46
→
X37
→
X22
)
)
)
→
(
¬
(
(
¬
X40
)
↔
(
(
¬
X22
)
→
X47
)
)
)
)
)
→
(
(
(
(
X46
↔
(
X44
→
(
X22
↔
(
X21
↔
False
)
)
)
)
↔
(
¬
(
X44
↔
(
X21
→
X9
)
)
)
)
↔
(
X46
→
(
X21
↔
False
)
)
)
↔
(
¬
(
(
X37
→
(
¬
X21
)
→
X46
)
→
(
X44
↔
(
X31
→
(
X21
↔
X37
)
)
)
)
)
)
→
(
¬
(
(
(
X50
→
(
¬
X37
)
→
X49
)
↔
(
X31
→
(
(
¬
X2
)
↔
X50
)
)
)
→
(
X49
↔
(
(
¬
X2
)
↔
X44
)
)
)
)
)
↔
(
(
(
¬
(
(
X47
↔
(
X44
↔
(
(
¬
X3
)
→
X38
)
)
)
→
(
(
¬
(
(
¬
(
X47
→
(
X38
↔
X30
)
)
)
→
(
¬
(
X45
→
(
X30
↔
X38
)
)
)
→
(
¬
(
X44
→
(
¬
X38
)
→
X30
→
X47
)
)
)
)
↔
(
(
¬
X38
)
↔
X44
)
)
)
)
↔
(
¬
(
(
X45
→
(
X44
↔
(
X38
→
(
X17
↔
False
)
)
)
)
→
(
¬
(
X50
→
(
X45
↔
(
X17
→
X28
)
)
)
)
)
)
)
↔
(
(
¬
(
X44
→
X42
→
(
X16
↔
X45
)
)
)
↔
(
(
(
¬
(
X45
→
(
X38
↔
(
X16
→
X0
)
)
)
)
→
(
(
¬
(
X38
→
X15
→
(
¬
X4
)
→
X16
)
)
↔
(
(
¬
X38
)
→
(
(
¬
X15
)
↔
(
(
¬
X4
)
→
X45
)
)
)
)
)
↔
(
X24
→
(
X0
↔
X27
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
(
(
¬
(
X47
↔
(
(
¬
X39
)
↔
(
X32
→
X49
)
)
)
)
↔
(
(
¬
(
X49
→
(
(
¬
X13
)
↔
X47
)
)
)
↔
(
¬
(
X43
↔
(
X39
→
X47
)
)
)
)
)
→
(
(
¬
(
X49
→
(
X34
↔
(
X33
→
(
X32
↔
False
)
)
)
)
)
↔
(
¬
(
(
¬
X34
)
→
(
(
¬
X33
)
↔
(
(
¬
X32
)
→
X43
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
X43
)
↔
(
X39
→
X47
)
)
)
)
→
(
(
¬
(
X47
→
(
X7
↔
X43
)
)
)
↔
(
¬
(
X14
↔
(
X13
→
(
X11
↔
X48
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X49
→
(
X48
↔
(
X25
↔
False
)
)
)
)
↔
(
¬
(
(
(
¬
X48
)
↔
(
(
¬
X41
)
→
(
X25
↔
X49
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X41
)
→
(
X12
↔
X48
)
)
)
)
)
)
)
→
(
¬
(
(
¬
X25
)
↔
(
X12
↔
X49
)
)
)
)
)
↔
(
(
X48
↔
(
(
¬
X47
)
↔
(
X5
↔
X49
)
)
)
→
(
¬
(
(
¬
(
X47
↔
(
(
¬
X29
)
→
(
(
¬
X12
)
↔
X48
)
)
)
)
→
(
(
(
X49
↔
(
X29
→
X23
→
False
)
)
↔
(
¬
(
(
¬
X29
)
↔
(
X12
→
X48
)
)
)
)
↔
(
X29
→
(
¬
X23
)
→
(
(
¬
X10
)
↔
X48
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X50
→
(
¬
X19
)
→
(
X18
↔
X49
)
)
)
→
(
(
(
X50
↔
(
X47
↔
(
X8
↔
False
)
)
)
→
(
¬
(
X48
↔
(
X8
↔
X20
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
(
¬
X35
)
↔
(
X19
→
X48
)
)
)
→
(
(
¬
(
X50
↔
(
X35
→
X19
→
False
)
)
)
↔
(
(
¬
(
X48
→
(
¬
X35
)
→
(
X18
↔
X50
)
)
)
↔
(
¬
(
X50
→
(
X35
↔
(
X19
→
X48
)
)
)
)
)
)
)
→
(
¬
(
X48
↔
(
X35
↔
X50
)
)
)
)
)
→
(
¬
(
(
¬
(
X35
↔
(
X18
↔
(
X1
↔
X47
)
)
)
)
→
(
¬
(
(
¬
X35
)
→
(
X18
↔
(
(
¬
X1
)
↔
X48
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
508158...
and proposition id is
aad8a7...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMFpmD9gLP18dpH6k1hvnoABbt6FJxGAanb
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
(
X42
↔
(
X25
→
X4
)
)
↔
(
(
(
¬
(
(
¬
X43
)
↔
(
X42
→
X49
)
)
)
↔
(
X49
→
(
(
(
¬
X42
)
→
X49
)
↔
(
¬
(
(
¬
(
X43
↔
X15
)
)
→
(
¬
(
X49
↔
(
(
¬
X25
)
→
(
(
¬
X15
)
↔
X42
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
X38
)
→
X22
→
(
¬
X15
)
→
X43
)
↔
(
(
X49
→
X22
→
(
X9
↔
X15
)
)
↔
(
X38
↔
(
X15
↔
False
)
)
)
)
)
)
→
(
(
(
(
X49
→
X42
→
X11
→
X38
)
↔
(
¬
(
X47
→
(
X38
↔
(
X34
↔
False
)
)
)
)
)
↔
(
(
¬
(
X49
↔
(
(
¬
X40
)
↔
(
(
¬
X31
)
→
X42
)
)
)
)
↔
(
¬
(
X49
↔
(
X47
→
(
X42
↔
X40
)
)
)
)
)
)
↔
(
(
(
X47
→
X40
→
(
(
¬
X28
)
↔
X42
)
)
↔
(
¬
(
X42
↔
(
X40
↔
(
X28
↔
X16
)
)
)
)
)
↔
(
(
¬
(
(
¬
X42
)
→
(
(
¬
X26
)
↔
X47
)
)
)
↔
(
(
¬
X40
)
↔
(
(
¬
X36
)
↔
X42
)
)
)
)
)
)
↔
(
(
¬
(
(
(
X44
→
X38
→
(
X18
↔
X49
)
)
→
(
¬
X27
)
→
(
¬
X18
)
→
X38
)
→
(
(
X44
↔
(
X38
↔
(
X34
↔
X27
)
)
)
↔
(
(
¬
X44
)
↔
(
X38
↔
X46
)
)
)
)
)
↔
(
(
¬
(
X39
→
(
¬
X38
)
→
(
(
¬
X6
)
↔
X44
)
)
)
→
(
¬
(
(
X23
↔
X46
)
→
(
¬
(
(
¬
X23
)
→
X17
→
(
X6
↔
X39
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X37
)
→
(
(
¬
X33
)
↔
(
(
¬
X5
)
↔
X46
)
)
)
)
↔
(
¬
(
(
X45
↔
(
X33
→
(
X32
↔
X46
)
)
)
→
(
¬
(
X37
→
(
¬
X32
)
→
X33
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
X48
→
(
¬
X45
)
→
X46
)
↔
(
(
¬
(
X48
→
X41
→
X33
→
X46
)
)
↔
(
(
X41
↔
(
X33
↔
(
X30
→
False
)
)
)
↔
(
¬
(
(
¬
X45
)
→
X33
→
(
¬
X30
)
→
X48
)
)
)
)
)
→
(
¬
(
X41
↔
(
X30
→
X21
→
X35
)
)
)
)
)
→
(
(
¬
(
X48
↔
(
X1
→
X0
→
X33
)
)
)
↔
(
¬
(
X45
↔
(
X33
→
(
(
¬
X0
)
↔
X1
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X46
↔
(
X29
→
(
X12
↔
False
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X29
→
(
(
¬
X12
)
↔
X47
)
)
)
→
(
¬
X29
)
→
(
(
¬
X20
)
↔
X46
)
)
)
→
(
¬
X29
)
→
(
(
¬
X20
)
↔
X47
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
¬
X29
)
→
(
¬
X14
)
→
(
X8
↔
X47
)
)
)
→
(
(
¬
(
X46
→
(
(
¬
X44
)
↔
(
X14
→
X47
)
)
)
)
↔
(
¬
(
(
¬
X46
)
→
X29
→
X14
→
X47
)
)
)
)
→
(
¬
(
(
¬
X46
)
→
(
X14
↔
(
(
¬
X2
)
→
X47
)
)
)
)
)
→
(
X46
↔
(
X8
→
X47
)
)
)
)
)
↔
(
(
(
(
(
¬
X45
)
↔
(
X13
→
X47
)
)
↔
(
X47
↔
(
X13
→
(
X10
↔
X24
)
)
)
)
→
(
¬
(
(
¬
(
(
X24
↔
(
X3
→
X19
)
)
→
(
(
¬
(
X45
↔
(
(
¬
X24
)
→
(
¬
X7
)
→
X47
)
)
)
↔
(
(
X24
→
(
¬
X3
)
→
X7
)
↔
(
X19
→
(
(
¬
X7
)
↔
(
X3
→
X45
)
)
)
)
)
)
)
→
(
(
¬
X3
)
↔
X45
)
)
)
)
↔
(
¬
(
(
(
(
¬
X45
)
↔
(
X13
↔
(
(
¬
X10
)
↔
X46
)
)
)
→
(
¬
(
(
¬
X41
)
↔
(
(
¬
X13
)
↔
(
X10
→
X46
)
)
)
)
)
→
(
X45
↔
(
X44
↔
X41
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
(
¬
X46
)
↔
(
(
¬
X16
)
↔
X48
)
)
↔
(
(
(
X45
↔
(
X43
→
X48
)
)
↔
(
(
X43
↔
X4
)
→
(
¬
(
X48
↔
(
(
¬
X16
)
↔
X45
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X46
)
↔
(
X45
→
(
¬
X11
)
→
X48
)
)
)
→
(
¬
(
(
¬
X11
)
↔
(
(
¬
X4
)
→
X46
)
)
)
)
)
)
)
→
(
¬
(
(
¬
X39
)
↔
(
X35
↔
(
(
¬
X34
)
↔
X48
)
)
)
)
→
(
(
(
¬
(
(
X39
↔
(
X35
→
(
¬
X24
)
→
X30
)
)
→
(
X39
↔
(
(
¬
X35
)
↔
(
(
¬
X30
)
↔
X43
)
)
)
)
)
↔
(
X43
→
(
(
¬
X35
)
↔
(
X30
↔
X48
)
)
)
)
↔
(
(
¬
(
X45
→
X39
→
(
(
¬
X20
)
↔
X43
)
)
)
→
(
¬
(
(
¬
(
X48
→
(
X45
↔
(
X20
→
(
X8
↔
False
)
)
)
)
)
→
(
X48
↔
(
X10
↔
(
(
¬
X8
)
↔
X39
)
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
X45
↔
(
X12
↔
(
X7
↔
False
)
)
)
)
↔
(
(
¬
(
X46
→
(
(
¬
X15
)
↔
(
X7
↔
X32
)
)
)
)
↔
(
(
X32
↔
(
X12
→
X7
)
)
↔
(
X23
→
(
(
¬
X15
)
↔
(
X12
→
X32
)
)
)
)
)
)
→
(
¬
(
X32
↔
(
X15
→
X45
)
)
)
)
→
(
(
¬
(
X48
→
X32
→
X42
)
)
↔
(
¬
(
(
(
X48
→
X29
→
(
X25
↔
False
)
)
→
(
¬
X32
)
→
(
¬
X25
)
→
X45
)
→
(
¬
(
X29
↔
(
X25
↔
X42
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X32
)
↔
X45
)
)
→
(
(
¬
(
(
¬
X45
)
↔
(
X42
→
(
¬
X19
)
→
X46
)
)
)
↔
(
¬
(
X45
→
X32
→
False
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
(
¬
X37
)
↔
(
X36
→
(
X31
↔
X46
)
)
)
→
(
¬
X36
)
→
(
¬
X31
)
→
(
¬
X13
)
→
X46
)
)
↔
(
(
(
¬
(
X40
→
(
(
¬
X36
)
↔
(
(
¬
X22
)
↔
X46
)
)
)
)
↔
(
(
¬
X40
)
→
(
X36
↔
(
X13
→
X46
)
)
)
)
→
(
¬
(
(
¬
X40
)
→
(
X36
↔
(
X22
→
X44
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
X36
)
→
X22
→
X45
)
→
(
¬
(
(
X36
→
(
(
¬
X31
)
↔
X45
)
)
→
(
(
X36
→
(
¬
X31
)
→
(
(
¬
X0
)
↔
X45
)
)
→
X45
→
X31
→
X22
)
→
X22
→
X17
→
X31
)
)
)
→
(
X17
↔
(
X0
→
X47
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
X49
↔
(
X47
→
X45
→
X6
)
)
)
↔
(
¬
(
(
¬
(
X45
↔
(
(
¬
X38
)
↔
(
X26
↔
X49
)
)
)
)
→
(
X49
↔
(
X45
→
(
¬
X38
)
→
X47
)
)
)
)
)
→
(
(
¬
(
X45
↔
(
X38
→
(
(
¬
X1
)
↔
X49
)
)
)
)
↔
(
¬
(
(
X42
→
X2
→
(
¬
X1
)
→
X28
)
→
(
¬
(
X45
→
(
X38
↔
(
X28
↔
X1
)
)
)
)
→
X38
→
X28
→
(
(
¬
X1
)
↔
X42
)
)
)
)
)
)
→
(
(
(
¬
(
X49
↔
False
)
)
→
(
(
(
X49
→
(
X38
↔
X40
)
)
↔
(
X40
→
(
X38
↔
(
X33
↔
X41
)
)
)
)
↔
(
¬
(
(
¬
X40
)
→
(
X33
↔
(
(
¬
X9
)
→
X41
)
)
)
)
)
)
↔
(
(
¬
(
X49
→
X40
→
(
¬
X27
)
→
X41
)
)
↔
(
¬
(
(
X41
↔
(
X21
↔
(
X14
↔
X3
)
)
)
→
(
¬
(
(
X27
↔
(
(
¬
X21
)
↔
(
(
¬
X3
)
→
X40
)
)
)
→
(
¬
(
X41
↔
(
(
¬
X21
)
↔
X40
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
X33
→
X27
→
(
(
¬
X5
)
↔
X18
)
)
)
↔
(
(
X41
↔
(
X27
↔
X18
)
)
↔
(
¬
(
X47
→
(
X33
↔
(
(
¬
X18
)
→
X27
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
77259e...
and proposition id is
1fa9f9...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMZ93j1ToqWU8fZbHRk8hW11R9EYdWpAWvq
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
¬
(
(
¬
(
X46
↔
(
(
¬
X21
)
↔
(
(
¬
X4
)
→
X37
)
)
)
)
→
(
¬
(
(
(
¬
(
X21
↔
X49
)
)
→
(
¬
(
(
¬
X25
)
→
(
(
¬
X23
)
↔
X49
)
)
)
→
(
(
¬
(
X49
↔
(
X23
↔
X4
)
)
)
↔
(
(
¬
X25
)
↔
(
X21
↔
(
X4
→
X49
)
)
)
)
)
→
(
¬
X25
)
→
(
(
¬
X23
)
↔
X49
)
)
)
)
)
→
(
¬
(
(
(
¬
X37
)
↔
(
X12
↔
(
(
¬
X8
)
↔
X48
)
)
)
→
(
(
(
¬
X47
)
↔
(
(
¬
X38
)
→
X48
)
)
↔
(
(
¬
(
(
(
¬
(
X47
→
X4
→
False
)
)
↔
(
(
¬
X24
)
→
(
X8
↔
X48
)
)
)
→
(
¬
(
X48
↔
(
X47
→
(
(
¬
X8
)
↔
X37
)
)
)
)
)
)
↔
(
¬
(
X48
↔
(
X38
→
(
(
¬
X8
)
↔
X47
)
)
)
)
)
)
→
(
¬
(
X47
→
X38
→
X15
→
X48
)
)
→
X48
→
X47
→
(
¬
X15
)
→
X23
)
)
)
→
(
(
¬
(
(
¬
(
(
(
¬
X42
)
↔
(
(
¬
X40
)
↔
(
(
¬
X28
)
↔
X49
)
)
)
→
(
¬
X47
)
→
X42
→
(
(
¬
X40
)
↔
X49
)
)
)
→
X49
→
(
(
¬
X40
)
↔
(
(
¬
X30
)
↔
X48
)
)
)
)
↔
(
(
X29
↔
(
(
¬
X5
)
↔
X48
)
)
↔
(
¬
(
X49
→
(
X5
↔
X47
)
)
)
)
)
→
(
(
(
(
¬
(
(
(
¬
(
X49
↔
(
X36
→
X2
)
)
)
↔
(
¬
(
X45
→
(
¬
X32
)
→
(
(
¬
X2
)
↔
X49
)
)
)
)
→
(
¬
(
(
¬
X36
)
↔
(
(
¬
X32
)
→
X45
)
)
)
)
)
↔
(
¬
(
(
X49
→
X45
→
(
¬
X27
)
→
X31
)
→
X45
→
(
(
¬
X32
)
↔
(
X31
↔
X49
)
)
)
)
)
→
(
¬
(
X49
→
(
(
¬
X2
)
↔
X40
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X49
↔
(
X47
↔
(
X35
↔
(
X20
↔
False
)
)
)
)
)
→
X47
→
X45
→
(
X35
↔
X20
)
)
)
→
X47
→
(
(
¬
X35
)
↔
(
(
¬
X27
)
↔
X49
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
X44
↔
(
X41
↔
(
X26
↔
(
X11
↔
False
)
)
)
)
)
→
X44
→
X41
→
(
X11
↔
X39
)
)
↔
(
¬
(
(
X44
↔
(
X41
→
X0
)
)
→
(
X49
↔
(
X44
→
(
(
¬
X0
)
↔
X39
)
)
)
)
)
)
→
(
(
X19
↔
(
X16
→
(
(
¬
X9
)
↔
X49
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
X19
→
(
X14
↔
(
X9
→
X41
)
)
)
→
(
¬
(
(
¬
(
X44
→
(
X43
↔
(
(
¬
X14
)
↔
X41
)
)
)
)
→
(
X41
↔
(
X14
→
X9
→
X43
)
)
)
)
)
)
→
(
¬
(
X44
↔
(
(
¬
X41
)
→
X43
)
)
)
)
)
→
(
(
X44
↔
(
(
¬
X16
)
→
X43
)
)
↔
(
¬
(
(
¬
X19
)
↔
(
(
¬
X3
)
→
X44
)
)
)
)
)
→
(
¬
(
X43
↔
(
X16
↔
X19
)
)
)
)
)
)
→
(
(
(
(
¬
X34
)
→
(
X7
↔
X39
)
)
↔
(
(
(
¬
(
(
¬
X39
)
→
(
¬
X34
)
→
X46
)
)
↔
(
(
(
X34
→
(
X7
↔
X39
)
)
↔
(
¬
(
(
X46
→
(
¬
X6
)
→
X34
)
→
(
¬
(
X39
↔
(
X7
→
X6
)
)
)
)
)
)
→
X39
→
X10
→
X6
)
)
↔
(
¬
(
(
(
X46
→
X13
→
X39
)
→
(
¬
(
X44
↔
(
(
¬
X34
)
↔
(
(
¬
X13
)
↔
X39
)
)
)
)
)
→
(
(
¬
X44
)
↔
(
X34
↔
X46
)
)
)
)
)
)
↔
(
(
(
X47
↔
(
X46
→
X44
→
X1
)
)
↔
(
X47
→
(
(
¬
X44
)
↔
(
X43
↔
X46
)
)
)
)
↔
(
¬
(
(
(
X46
↔
(
(
¬
X18
)
→
X17
→
X33
)
)
→
(
¬
(
X46
↔
(
X18
↔
(
X17
→
X44
)
)
)
)
)
→
(
(
(
¬
X39
)
→
(
X22
↔
(
(
¬
X17
)
→
X46
)
)
)
↔
(
X47
↔
(
X22
→
False
)
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
(
¬
X45
)
↔
(
X41
→
(
(
¬
X40
)
↔
X47
)
)
)
→
(
¬
(
X41
→
X16
→
X0
→
X2
)
)
→
(
(
X45
↔
(
(
¬
X29
)
↔
X34
)
)
↔
(
(
(
(
X47
↔
(
X34
→
(
(
¬
X16
)
↔
X45
)
)
)
↔
(
¬
(
(
¬
X41
)
↔
(
X16
→
X45
)
)
)
)
↔
(
¬
(
X47
↔
(
X34
→
(
¬
X29
)
→
X41
)
)
)
)
→
(
¬
(
X45
↔
(
X34
↔
(
(
¬
X0
)
→
X16
)
)
)
)
)
)
)
→
(
(
X47
→
(
¬
X45
)
→
(
X41
↔
X49
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
X49
↔
(
X47
→
(
X44
↔
(
X12
→
False
)
)
)
)
)
→
(
¬
(
(
¬
X45
)
→
(
¬
X44
)
→
X49
)
)
)
)
↔
(
X47
↔
(
X45
→
X12
)
)
)
→
(
X49
↔
(
X47
→
(
¬
X6
)
→
X44
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
X35
)
↔
(
X25
↔
(
X9
↔
X37
)
)
)
→
(
¬
(
(
X37
↔
(
X35
↔
(
(
¬
X32
)
↔
X43
)
)
)
→
(
(
(
¬
(
X43
→
X35
→
X8
)
)
↔
(
¬
(
X43
↔
(
X25
→
X5
)
)
)
)
↔
(
(
¬
X35
)
↔
(
X32
↔
(
(
¬
X25
)
→
X43
)
)
)
)
)
)
→
(
¬
(
X45
→
(
(
¬
X32
)
↔
(
(
¬
X8
)
→
X35
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
X44
→
(
X38
↔
(
X20
↔
X47
)
)
)
)
→
(
X47
↔
(
(
¬
X23
)
→
(
X20
↔
X38
)
)
)
)
↔
(
X45
→
(
X44
↔
(
(
¬
X20
)
↔
X38
)
)
)
)
↔
(
(
¬
(
(
¬
X45
)
→
(
X37
↔
(
X13
→
X47
)
)
)
)
↔
(
(
¬
X45
)
↔
(
X38
→
X37
→
X47
)
)
)
)
↔
(
¬
(
X47
→
(
X45
↔
(
X44
↔
X43
)
)
)
)
)
↔
(
(
X48
→
X45
→
X14
→
X38
)
→
X38
→
(
X37
↔
False
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
X46
)
→
(
X42
↔
(
X27
↔
X48
)
)
)
→
(
¬
(
X46
↔
(
X42
↔
(
(
¬
X39
)
↔
X48
)
)
)
)
)
)
→
(
(
(
X47
→
(
¬
X39
)
→
(
(
¬
X27
)
↔
X46
)
)
↔
(
¬
(
(
(
X46
↔
X39
)
↔
(
(
¬
(
X46
↔
(
X39
→
X17
)
)
)
→
(
(
X47
↔
(
X39
↔
False
)
)
↔
(
¬
(
X46
→
(
(
¬
X17
)
↔
(
(
¬
X4
)
→
X39
)
)
)
)
)
)
)
→
X26
→
(
(
¬
X17
)
↔
(
X4
→
X46
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
X19
)
↔
(
X10
→
(
¬
X7
)
→
X46
)
)
↔
(
¬
(
X19
↔
(
X10
→
(
X3
↔
X7
)
)
)
)
)
→
X10
→
(
X3
↔
X46
)
)
)
→
(
(
X47
→
X46
→
(
X27
↔
X7
)
)
↔
(
¬
(
X46
→
(
(
¬
X7
)
↔
(
X3
→
X27
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X48
↔
(
X33
↔
(
X24
→
X28
)
)
)
)
↔
(
¬
(
(
(
(
¬
X42
)
→
X30
→
(
X28
↔
X48
)
)
↔
(
¬
(
X42
↔
(
X30
→
(
X24
↔
False
)
)
)
)
)
→
(
X42
↔
(
X28
↔
(
(
¬
X18
)
↔
X30
)
)
)
)
)
)
→
(
X42
↔
(
(
¬
X21
)
↔
X33
)
)
→
(
X48
↔
(
X42
↔
(
X21
↔
X22
)
)
)
)
)
↔
(
(
(
¬
(
X47
→
(
¬
X33
)
→
(
¬
X15
)
→
X48
)
)
→
(
(
(
¬
(
X36
→
(
(
¬
X33
)
↔
(
X1
↔
X47
)
)
)
)
→
(
¬
(
X47
↔
(
X36
↔
(
(
¬
X11
)
→
X33
)
)
)
)
)
↔
(
(
¬
X36
)
↔
(
(
¬
X33
)
→
(
X11
↔
X48
)
)
)
)
)
↔
(
(
(
X49
→
(
X42
↔
(
X36
↔
X15
)
)
)
→
(
X49
↔
(
X42
→
(
¬
X15
)
→
X31
)
)
)
→
(
¬
(
X47
↔
(
X36
→
X42
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
0f30e7...
and proposition id is
e9e38f...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMJgaxXKkxnhW1zDGz9oT3d6oEcohqZrpmu
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
∀X50 :
prop
,
(
(
(
¬
(
(
(
(
(
¬
(
X48
→
X47
→
(
X36
↔
X44
)
)
)
↔
(
¬
(
(
(
¬
(
X50
→
X48
→
X47
→
(
X44
↔
False
)
)
)
↔
(
X48
↔
(
(
¬
X47
)
↔
(
X26
↔
X50
)
)
)
)
→
(
¬
(
X48
→
False
)
)
)
)
)
→
(
X48
↔
(
X41
→
X36
→
X47
)
)
)
↔
(
(
(
(
¬
X40
)
↔
(
X16
↔
(
X3
→
X48
)
)
)
↔
(
(
X44
↔
(
X40
→
X16
→
(
X3
↔
False
)
)
)
↔
(
X40
↔
(
X16
↔
X50
)
)
)
)
↔
(
(
(
(
¬
X44
)
↔
(
(
¬
X37
)
→
(
¬
X7
)
→
X48
)
)
↔
(
X50
→
(
X37
↔
(
X28
→
X48
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X44
)
→
(
X6
↔
X50
)
)
)
→
(
(
¬
(
(
¬
X44
)
↔
(
X37
→
X50
)
)
)
↔
(
¬
(
(
¬
(
X44
↔
(
(
¬
X29
)
→
(
¬
X6
)
→
X50
)
)
)
→
(
¬
(
X50
↔
(
X37
→
False
)
)
)
)
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
(
¬
(
X50
→
(
X47
↔
(
X0
↔
X25
)
)
)
)
→
(
¬
(
X47
→
X48
)
)
)
)
↔
(
X48
↔
(
(
¬
X0
)
→
X25
)
)
)
↔
(
¬
(
(
¬
(
X50
↔
X43
)
)
→
X48
→
(
¬
X8
)
→
X47
)
)
)
↔
(
¬
(
X50
↔
(
(
¬
X43
)
↔
(
X34
→
X48
)
)
)
)
)
↔
(
(
(
X49
→
(
¬
X43
)
→
(
X4
↔
X50
)
)
↔
(
(
¬
X49
)
→
X43
→
(
X17
↔
X50
)
)
)
↔
(
¬
(
(
¬
(
X50
→
X47
→
X27
→
X43
)
)
→
(
(
X47
↔
X50
)
↔
(
(
X50
↔
(
(
¬
X47
)
→
X20
→
X49
)
)
↔
(
X49
→
(
X43
↔
(
X20
→
False
)
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
¬
(
X38
→
(
¬
X23
)
→
X11
→
X30
)
)
→
(
¬
(
X30
↔
(
X5
↔
X11
)
)
)
→
(
¬
(
(
(
¬
X30
)
↔
(
(
¬
X2
)
→
X49
)
)
→
(
(
X30
↔
(
(
¬
X23
)
↔
(
(
¬
X2
)
↔
X49
)
)
)
↔
(
(
¬
X5
)
↔
(
(
¬
X2
)
↔
X23
)
)
)
)
)
)
)
→
(
¬
(
X42
→
(
(
¬
X11
)
↔
(
(
¬
X5
)
↔
X50
)
)
)
)
)
↔
(
(
(
¬
(
X50
↔
(
X38
↔
X45
)
)
)
↔
(
(
(
X35
↔
X50
)
↔
(
(
¬
(
X50
↔
(
(
¬
X35
)
→
(
X14
↔
X45
)
)
)
)
→
(
¬
(
(
¬
X42
)
↔
(
(
¬
X14
)
↔
X45
)
)
)
)
)
↔
(
¬
(
(
¬
X35
)
→
X14
→
X42
)
)
)
)
↔
(
¬
(
X45
↔
(
(
¬
X39
)
↔
(
(
¬
X38
)
→
X42
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
X46
↔
(
X32
↔
(
(
¬
X31
)
↔
X45
)
)
)
)
↔
(
X48
↔
(
(
¬
X45
)
→
X32
→
X50
)
)
)
→
(
X48
↔
(
X46
↔
(
(
¬
X13
)
↔
X50
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X48
)
→
(
¬
X45
)
→
(
¬
X24
)
→
X50
)
)
↔
(
(
¬
X24
)
↔
(
X10
→
X45
)
)
)
↔
(
X18
↔
X48
)
)
↔
(
¬
(
X45
→
(
¬
X10
)
→
(
¬
X9
)
→
X18
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X50
→
(
¬
X38
)
→
(
¬
X33
)
→
X46
)
)
→
(
(
(
¬
(
X50
↔
(
X46
↔
(
X45
→
X12
)
)
)
)
→
(
X45
↔
(
X22
→
(
¬
X12
)
→
X46
)
)
)
↔
(
(
¬
X38
)
→
X22
→
X46
)
)
)
)
→
(
(
(
¬
X21
)
→
(
X1
↔
X33
)
)
↔
(
(
X50
→
X21
→
(
X15
↔
False
)
)
↔
(
(
¬
(
X50
↔
(
X33
→
X45
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X50
↔
(
(
¬
X33
)
↔
(
X15
→
X45
)
)
)
)
→
(
X50
↔
(
(
¬
X33
)
↔
(
(
¬
X19
)
↔
X45
)
)
)
)
)
→
(
X45
↔
(
X19
↔
(
X15
↔
False
)
)
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
(
¬
X41
)
↔
(
(
¬
X40
)
↔
(
X33
↔
X46
)
)
)
↔
(
X33
↔
X24
)
)
↔
(
(
X46
↔
(
X33
↔
(
X8
↔
X41
)
)
)
↔
(
¬
(
X41
→
(
(
¬
X8
)
↔
(
(
¬
X4
)
↔
X46
)
)
)
)
)
)
→
(
(
¬
(
X41
→
(
(
¬
X38
)
↔
(
X37
→
X40
)
)
)
)
↔
(
¬
(
(
X41
→
X40
→
X28
→
X38
)
→
X38
→
X28
→
X46
)
)
)
)
→
(
(
(
¬
(
(
X44
→
(
¬
X41
)
→
X35
→
X50
)
→
(
¬
(
(
X50
↔
(
X44
→
(
(
¬
X0
)
↔
X41
)
)
)
→
(
¬
X43
)
→
(
X0
↔
X50
)
)
)
)
)
↔
(
¬
(
(
(
X46
↔
(
X40
↔
(
X35
→
X41
)
)
)
→
(
¬
(
(
¬
X41
)
→
(
(
¬
X35
)
↔
(
X3
↔
X43
)
)
)
)
→
(
¬
(
X43
↔
(
X40
↔
X3
)
)
)
)
→
(
¬
(
X43
↔
(
(
¬
X40
)
↔
(
(
¬
X30
)
↔
X46
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X32
→
(
X10
↔
X13
)
)
)
↔
(
¬
(
X34
→
(
¬
X18
)
→
X32
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X46
↔
(
X43
↔
(
X42
↔
X34
)
)
)
)
→
(
(
(
X43
→
X42
→
X18
)
→
(
(
¬
(
X42
↔
(
X18
→
(
¬
X2
)
→
X43
)
)
)
↔
(
¬
(
X43
→
(
¬
X34
)
→
(
¬
X18
)
→
X42
)
)
)
)
→
(
¬
(
X34
→
X18
→
X2
→
False
)
)
)
→
(
X43
↔
(
X42
→
(
X34
↔
(
X32
→
False
)
)
)
)
)
)
→
(
(
¬
(
X42
→
X36
→
X29
→
False
)
)
↔
(
(
X46
↔
(
X43
↔
(
X36
↔
X41
)
)
)
↔
(
¬
(
X43
↔
(
X41
↔
(
X27
→
X29
)
)
)
)
)
)
)
)
)
)
)
→
(
(
(
(
(
(
(
X49
→
(
¬
X44
)
→
X20
→
X47
)
↔
(
(
X47
↔
(
X31
↔
X50
)
)
↔
(
¬
(
X11
→
X44
)
)
)
)
→
(
X44
↔
(
(
¬
X31
)
→
X12
→
X47
)
)
)
→
(
(
¬
X49
)
↔
(
X47
→
X50
)
)
)
↔
(
(
(
¬
(
X50
↔
(
(
¬
X48
)
→
(
X16
↔
X49
)
)
)
)
↔
(
X50
↔
(
X49
↔
(
X48
→
(
X17
↔
False
)
)
)
)
)
↔
(
(
X49
→
(
X48
↔
(
(
¬
X5
)
↔
X6
)
)
)
→
(
¬
(
(
¬
X48
)
→
(
(
¬
X44
)
↔
(
(
¬
X17
)
↔
X49
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
X44
)
→
(
¬
X25
)
→
(
X9
↔
X50
)
)
)
→
(
¬
(
X44
→
(
¬
X23
)
→
X50
)
)
)
)
↔
(
(
¬
(
(
¬
X49
)
→
(
X26
↔
(
X23
→
X50
)
)
)
)
↔
(
¬
(
X50
→
X14
→
X1
→
X9
)
)
)
)
)
↔
(
(
(
¬
(
X50
→
(
(
¬
X45
)
↔
(
X39
↔
X48
)
)
)
)
↔
(
(
¬
(
(
¬
X45
)
→
(
(
¬
X39
)
↔
(
(
¬
X22
)
↔
X48
)
)
)
)
↔
(
(
¬
X48
)
↔
(
X45
↔
X49
)
)
)
)
→
(
X48
↔
(
(
¬
X21
)
→
(
¬
X19
)
→
X50
)
)
→
(
¬
(
(
(
¬
(
X21
→
X50
)
)
→
(
(
¬
(
X50
→
X15
→
X21
)
)
↔
(
(
(
¬
X46
)
↔
(
(
¬
X21
)
↔
X50
)
)
→
(
¬
(
X49
↔
(
(
¬
X21
)
→
(
X15
↔
X46
)
)
)
)
)
)
)
→
(
(
¬
(
X50
→
X7
→
X15
)
)
↔
(
(
(
¬
(
X19
↔
(
(
¬
X15
)
→
X50
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
X21
↔
(
(
¬
X7
)
→
X15
)
)
)
→
(
¬
(
X50
→
X15
→
X19
)
)
)
→
(
¬
(
X19
→
(
¬
X7
)
→
X15
)
)
)
)
↔
(
¬
(
(
¬
X21
)
↔
(
X15
→
X50
)
)
)
)
↔
(
¬
(
X7
→
X15
)
)
)
)
↔
(
(
¬
X15
)
↔
(
X7
→
X19
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
f70772...
and proposition id is
4c7d60...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMHpwxFigSJn6fBk65ipZYXAMoeakfX3b3n
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
¬
∀X51 :
prop
,
∀X52 :
prop
,
(
(
(
(
(
(
(
¬
(
X52
↔
(
X1
↔
X31
)
)
)
↔
(
(
(
¬
(
X49
↔
(
X31
→
X1
→
X41
)
)
)
↔
(
¬
(
X49
→
(
X31
↔
(
X1
→
X21
)
)
)
)
)
↔
(
(
¬
X21
)
→
X41
)
)
)
→
(
(
(
¬
(
(
¬
X41
)
→
(
¬
X30
)
→
X29
→
X48
)
)
↔
(
(
(
X39
→
(
X29
↔
(
(
¬
X12
)
→
X41
)
)
)
→
(
¬
(
X41
→
(
(
¬
X29
)
↔
X48
)
)
)
)
→
X39
→
(
X10
↔
False
)
)
)
↔
(
¬
(
(
(
¬
(
X49
↔
(
(
¬
X41
)
↔
(
X25
↔
X52
)
)
)
)
→
(
¬
(
(
¬
X49
)
→
X41
→
X22
→
X52
)
)
)
→
(
¬
(
X25
↔
(
(
¬
X22
)
→
X49
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X48
↔
(
X18
→
X2
→
X35
)
)
)
→
(
(
¬
(
X48
→
(
¬
X18
)
→
X24
)
)
↔
(
(
¬
X35
)
→
(
X24
↔
(
(
¬
X18
)
↔
X49
)
)
)
)
)
→
(
¬
(
(
X39
→
(
¬
X18
)
→
(
¬
X4
)
→
X26
)
→
(
¬
(
X35
→
(
X35
↔
(
(
¬
X26
)
→
(
(
¬
X18
)
↔
X39
)
)
)
→
X49
→
(
X39
↔
(
X35
→
X26
)
)
)
)
)
)
)
)
)
↔
(
(
(
X23
→
X9
→
False
)
↔
(
¬
(
(
¬
(
X49
→
X20
→
X48
)
)
→
X52
→
X32
→
X20
→
X49
)
)
)
→
(
(
¬
(
(
X52
↔
(
X36
→
X48
)
)
→
(
¬
(
X49
→
(
¬
X48
)
→
X33
→
X52
)
)
)
)
↔
(
X51
↔
X48
)
)
)
)
→
(
(
¬
(
(
(
(
(
¬
(
(
¬
X45
)
↔
(
(
¬
X13
)
↔
(
(
¬
X3
)
↔
X47
)
)
)
)
↔
(
¬
(
X48
→
(
¬
X13
)
→
(
X3
↔
X47
)
)
)
)
→
(
(
¬
(
X48
↔
(
X45
↔
(
X6
↔
X34
)
)
)
)
↔
(
¬
(
(
X45
↔
(
X11
↔
(
X6
↔
False
)
)
)
→
(
(
¬
(
(
¬
X40
)
→
(
¬
X6
)
→
X45
)
)
↔
(
(
(
(
¬
X11
)
↔
X45
)
→
(
(
¬
(
X48
→
X47
→
X45
)
)
↔
(
(
X48
↔
(
X47
↔
(
X45
→
X40
)
)
)
→
(
¬
(
X47
→
X40
→
(
¬
X11
)
→
X45
)
)
)
)
)
↔
(
X40
→
(
(
¬
X11
)
↔
X48
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X52
→
(
(
¬
X48
)
↔
(
(
¬
X45
)
→
X49
)
)
)
)
↔
(
¬
(
X52
↔
(
X48
↔
(
X47
↔
X49
)
)
)
)
)
→
(
X48
↔
(
(
¬
X47
)
↔
(
(
¬
X37
)
→
X49
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X46
↔
(
X44
→
X43
→
X19
→
False
)
)
)
→
(
X46
↔
(
X44
↔
X28
)
)
)
→
(
(
(
X46
↔
(
(
¬
X28
)
→
(
¬
X27
)
→
X49
)
)
↔
(
X49
↔
(
(
¬
X27
)
↔
X43
)
)
)
↔
(
X49
→
(
(
¬
X28
)
↔
X46
)
)
)
)
)
→
(
(
¬
(
(
(
(
(
¬
(
X43
→
X28
)
)
↔
(
(
¬
(
(
¬
X43
)
→
(
¬
X42
)
→
(
¬
X28
)
→
X48
)
)
→
(
¬
X45
)
→
(
(
¬
X43
)
↔
(
X42
↔
X48
)
)
)
)
↔
(
X48
→
X43
→
(
¬
X5
)
→
X42
)
)
↔
(
¬
(
X48
↔
(
X42
→
X28
→
False
)
)
)
)
→
(
¬
(
X48
↔
(
X42
↔
False
)
)
)
)
)
↔
(
¬
(
(
¬
(
X48
→
(
X43
↔
(
X38
→
False
)
)
)
)
→
(
(
¬
X48
)
↔
(
X38
→
X5
→
X49
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
X46
)
↔
(
(
¬
X17
)
↔
(
(
¬
X15
)
↔
X48
)
)
)
↔
(
¬
(
(
¬
X17
)
↔
(
(
¬
X15
)
↔
(
X8
↔
X48
)
)
)
)
)
↔
(
X8
→
(
X0
↔
X17
)
)
)
→
(
(
X50
→
(
¬
X46
)
→
X14
→
X48
)
↔
(
(
(
(
¬
(
X48
↔
(
X46
→
X7
→
X45
)
)
)
→
(
¬
X48
)
→
X46
→
(
X45
↔
X50
)
)
↔
(
¬
(
(
¬
X46
)
↔
(
(
¬
X45
)
↔
(
(
¬
X17
)
→
X50
)
)
)
)
)
↔
(
X46
→
(
X45
↔
(
X17
→
X16
→
False
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
X50
→
X46
→
X23
→
(
X22
↔
False
)
)
→
X50
→
(
X22
↔
(
(
¬
X10
)
↔
X23
)
)
)
↔
(
(
¬
(
X49
↔
(
X32
↔
X42
)
)
)
↔
(
(
(
X32
→
(
X23
↔
(
(
¬
X19
)
↔
X49
)
)
)
↔
(
(
(
¬
(
X32
→
X22
→
X42
)
)
→
X50
→
(
¬
X23
)
→
(
(
¬
X22
)
↔
X32
)
)
→
X42
→
(
¬
X23
)
→
(
(
¬
X22
)
↔
X49
)
)
)
→
(
¬
(
X50
→
(
(
¬
X42
)
↔
X49
)
)
)
)
)
)
↔
(
¬
(
(
(
X46
↔
(
X41
→
X40
)
)
↔
(
¬
(
(
X40
↔
(
X37
→
(
X17
↔
False
)
)
)
→
(
¬
(
X46
→
X41
→
X37
→
X25
)
)
)
)
)
→
(
¬
(
(
¬
X49
)
→
(
¬
X37
)
→
(
¬
X17
)
→
X50
)
)
)
)
)
→
(
(
(
¬
X49
)
→
(
X48
↔
(
X46
→
X51
)
)
)
↔
(
(
(
(
X48
→
(
X46
↔
(
X35
→
X52
)
)
)
↔
(
(
¬
X48
)
→
(
X46
↔
(
(
¬
X21
)
↔
X52
)
)
)
)
↔
(
(
¬
(
(
(
X48
→
(
X0
↔
X52
)
)
→
(
¬
(
X52
→
(
X49
↔
(
X48
→
X46
)
)
)
)
→
(
¬
(
X52
→
(
X48
↔
(
X46
↔
(
X0
→
False
)
)
)
)
)
)
→
(
X52
↔
(
X46
↔
(
X42
↔
X48
)
)
)
)
)
→
X49
→
X46
→
(
X3
↔
X52
)
)
)
→
(
(
(
(
¬
(
X46
→
X15
→
X52
)
)
↔
(
¬
(
(
¬
X46
)
→
(
¬
X15
)
→
(
(
¬
X1
)
↔
X48
)
)
)
)
↔
(
X52
→
(
(
¬
X2
)
↔
(
X1
→
X15
)
)
)
)
↔
(
(
¬
X18
)
↔
(
X15
↔
(
X2
→
X46
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
(
¬
X43
)
→
(
(
¬
X5
)
↔
X45
)
)
)
↔
(
(
(
¬
X47
)
↔
(
X45
↔
(
X5
↔
X50
)
)
)
↔
(
(
(
X45
↔
(
X43
→
(
(
¬
X6
)
↔
X26
)
)
)
→
(
X43
↔
(
X39
→
X26
)
)
)
→
(
(
¬
(
X47
→
(
X45
↔
(
X39
↔
X50
)
)
)
)
↔
(
X45
↔
(
X43
↔
X50
)
)
)
)
)
)
→
(
(
(
(
¬
(
X45
→
X39
→
(
X33
↔
X20
)
)
)
→
(
¬
(
X13
→
X45
)
)
)
→
(
X39
→
(
¬
X33
)
→
X9
→
X43
)
→
X33
→
X9
→
X20
)
↔
(
(
¬
(
(
¬
(
(
¬
X20
)
↔
(
X16
↔
(
X4
→
X34
)
)
)
)
→
(
X45
↔
(
X20
↔
(
(
¬
X4
)
↔
X16
)
)
)
)
)
↔
(
¬
(
(
¬
X33
)
↔
(
(
¬
X20
)
↔
(
(
¬
X16
)
→
X34
)
)
)
)
)
)
)
→
(
(
(
(
X51
→
(
(
¬
X24
)
↔
X44
)
)
↔
(
X44
→
(
X31
↔
(
(
¬
X27
)
↔
X36
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X36
)
↔
(
X33
→
X38
)
)
)
→
(
(
(
¬
(
(
X44
→
X36
→
(
X8
↔
False
)
)
→
(
¬
(
(
X38
→
(
(
¬
X36
)
↔
(
X8
↔
X44
)
)
)
→
(
¬
X38
)
→
(
X8
↔
X51
)
)
)
)
)
→
(
X51
↔
(
(
¬
X36
)
→
X44
)
)
)
↔
(
(
¬
X36
)
→
X33
→
(
¬
X8
)
→
X38
)
)
)
↔
(
¬
(
X38
↔
(
(
¬
X36
)
→
X51
)
)
)
)
↔
(
(
¬
(
X30
↔
(
X29
↔
X44
)
)
)
↔
(
¬
(
(
(
X29
→
X7
→
X38
)
→
X44
→
(
¬
X36
)
→
X7
→
X38
)
→
X44
→
X29
→
X7
→
False
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X36
→
X49
)
)
↔
(
¬
(
(
X49
→
(
¬
X14
)
→
X45
)
→
(
(
¬
(
X45
↔
(
X14
↔
X11
)
)
)
→
(
X49
↔
(
(
¬
X36
)
↔
X45
)
)
)
→
(
¬
(
(
¬
X45
)
↔
(
X36
→
X49
)
)
)
)
)
)
→
(
¬
(
X45
→
(
¬
X43
)
→
(
X36
↔
X49
)
)
)
)
→
(
(
(
¬
(
(
X44
↔
False
)
→
(
(
¬
X49
)
↔
(
(
¬
X43
)
→
(
(
¬
X12
)
↔
X51
)
)
)
)
)
→
X12
)
↔
(
(
(
¬
(
X45
↔
(
X44
↔
(
(
¬
X28
)
→
X51
)
)
)
)
↔
(
(
¬
X49
)
↔
(
(
¬
X45
)
↔
(
X28
↔
X51
)
)
)
)
↔
(
X45
↔
False
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
93c53a...
and proposition id is
ef2f73...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMajDbv6SMd4HyxEvbG3KJEypQZxPSy7RGg
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
∀X50 :
prop
,
(
(
(
¬
(
(
(
(
¬
(
(
(
¬
X49
)
↔
(
X34
↔
(
(
¬
X22
)
→
X50
)
)
)
→
X49
→
(
X16
↔
(
(
¬
X14
)
→
X50
)
)
)
)
↔
(
¬
(
(
X44
→
(
(
¬
X36
)
↔
(
(
¬
X34
)
→
X43
)
)
)
→
(
¬
(
(
(
X36
→
(
X34
↔
(
X22
↔
(
X5
→
False
)
)
)
)
→
(
¬
(
(
¬
X22
)
↔
(
(
¬
X5
)
↔
(
X3
→
X34
)
)
)
)
)
→
(
(
(
X49
→
(
X34
↔
X3
)
)
↔
(
X22
↔
(
(
¬
X3
)
→
X44
)
)
)
↔
(
¬
(
(
¬
X44
)
→
(
X3
↔
X49
)
)
)
)
)
)
)
)
)
↔
(
(
(
X40
→
(
¬
X20
)
→
X44
)
↔
(
(
X44
↔
X48
)
→
(
(
¬
(
X50
↔
(
X44
↔
(
(
¬
X40
)
↔
X48
)
)
)
)
↔
(
X48
→
(
(
¬
X40
)
↔
(
(
¬
X1
)
→
X50
)
)
)
)
)
)
↔
(
(
(
¬
(
X44
→
(
X28
↔
X48
)
)
)
↔
(
¬
(
X48
→
X36
→
(
¬
X30
)
→
X50
)
)
)
↔
(
¬
(
X48
→
(
(
¬
X37
)
↔
(
(
¬
X36
)
→
X50
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
(
X37
↔
(
X29
↔
(
(
¬
X23
)
↔
X31
)
)
)
)
→
(
(
¬
(
(
¬
X37
)
↔
(
(
¬
X36
)
→
X29
→
X47
)
)
)
↔
(
(
¬
X37
)
→
X36
→
X23
→
X47
)
)
)
↔
(
(
X46
→
False
)
↔
(
(
(
¬
(
X46
→
X37
→
X31
→
(
X12
↔
False
)
)
)
→
X47
→
(
X12
↔
X37
)
)
↔
(
X13
→
(
X12
↔
False
)
)
)
)
)
↔
(
¬
(
(
¬
X31
)
↔
(
(
¬
X29
)
→
(
(
¬
X2
)
↔
X37
)
)
)
)
)
→
(
(
¬
X46
)
↔
(
X25
↔
(
(
¬
X11
)
→
X47
)
)
)
→
(
(
¬
(
(
¬
X43
)
↔
X46
)
)
↔
(
¬
(
X37
→
(
¬
X25
)
→
X47
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
¬
(
(
(
X49
→
X27
→
(
X17
↔
X19
)
)
↔
(
¬
(
X35
↔
(
X19
→
(
¬
X9
)
→
X49
)
)
)
)
→
(
X27
↔
(
X17
↔
X35
)
)
)
)
↔
(
¬
(
(
¬
X35
)
↔
(
(
¬
X27
)
↔
X49
)
)
)
)
↔
(
¬
(
X35
→
(
(
¬
X27
)
↔
(
(
¬
X0
)
→
X49
)
)
)
)
)
→
(
¬
(
(
¬
X27
)
→
(
X19
↔
X35
)
)
)
)
→
(
(
(
(
¬
X27
)
→
X10
→
X9
→
X49
)
↔
(
X49
→
(
X27
↔
(
X10
↔
(
X4
↔
False
)
)
)
)
)
↔
(
(
(
¬
(
X18
↔
X48
)
)
↔
(
¬
(
(
X49
↔
(
X27
→
X50
)
)
→
(
(
(
¬
(
X50
↔
(
X18
↔
False
)
)
)
↔
(
(
X50
→
(
X49
↔
(
X18
↔
False
)
)
)
→
X49
→
(
(
¬
X27
)
↔
X48
)
)
)
↔
(
X18
→
X49
)
)
)
)
)
→
(
¬
(
(
X48
↔
(
X18
↔
(
X9
→
X27
)
)
)
→
(
X27
↔
(
(
¬
X9
)
→
X45
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
¬
(
X41
↔
(
(
¬
X7
)
↔
(
(
¬
X6
)
↔
X50
)
)
)
)
→
(
X42
↔
(
X41
↔
X50
)
)
)
)
→
(
X38
↔
(
X26
→
(
¬
X7
)
→
X42
)
)
)
↔
(
(
¬
(
X42
↔
(
X38
→
(
¬
X33
)
→
X48
)
)
)
→
(
¬
(
(
(
X50
→
(
X33
↔
X38
)
)
↔
(
X50
↔
(
X42
→
(
(
¬
X24
)
↔
X38
)
)
)
)
→
(
¬
(
X50
→
(
(
¬
X24
)
↔
X38
)
)
)
)
)
)
)
→
(
(
(
(
X39
→
(
X32
↔
(
X26
→
X8
→
False
)
)
)
↔
(
X39
↔
(
(
¬
X21
)
→
X32
)
)
)
↔
(
¬
(
X39
→
(
X21
↔
(
(
¬
X15
)
↔
X26
)
)
)
)
)
↔
(
(
X49
→
(
X41
↔
(
X38
↔
X39
)
)
)
↔
(
X41
→
(
(
¬
X38
)
↔
(
X26
↔
X49
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
(
(
X28
→
(
¬
X25
)
→
X24
→
X41
)
↔
(
(
¬
X28
)
↔
(
(
¬
X25
)
→
(
¬
X8
)
→
X36
)
)
)
↔
(
X42
→
(
X25
↔
(
X2
↔
X28
)
)
)
)
→
(
¬
(
X36
→
(
X25
↔
X41
)
)
)
)
↔
(
(
¬
X41
)
↔
(
X36
↔
(
(
¬
X23
)
→
X42
)
)
)
)
↔
(
(
(
(
(
¬
X42
)
↔
(
X28
→
(
¬
X9
)
→
X43
)
)
↔
(
¬
(
X43
→
(
X38
↔
X42
)
)
)
)
↔
(
¬
(
X28
↔
X9
)
)
)
→
(
X38
↔
(
(
¬
X28
)
↔
(
(
¬
X25
)
→
X43
)
)
)
)
)
↔
(
¬
(
(
(
(
X42
↔
(
X41
→
(
X37
↔
X49
)
)
)
↔
(
¬
(
(
X49
→
(
(
¬
X16
)
↔
(
(
¬
X3
)
→
X48
)
)
)
→
(
(
¬
X42
)
↔
(
(
¬
X37
)
→
(
X16
↔
X49
)
)
)
)
)
)
↔
(
(
¬
(
X17
→
(
¬
X3
)
→
X48
)
)
↔
(
(
(
(
X37
↔
(
X17
↔
False
)
)
↔
(
¬
(
(
X17
→
X13
→
(
X7
↔
X48
)
)
→
(
(
¬
X37
)
↔
(
(
¬
X17
)
→
(
(
¬
X7
)
↔
X48
)
)
)
)
)
)
↔
(
X48
→
(
X37
↔
(
X17
↔
False
)
)
)
)
↔
(
X17
→
X13
→
X48
)
)
)
)
→
(
¬
(
(
(
(
(
(
¬
X48
)
→
(
¬
X43
)
→
X49
)
↔
(
¬
(
(
X48
↔
(
X43
↔
(
(
¬
X11
)
→
X40
)
)
)
→
(
¬
(
X49
↔
(
X48
→
X43
→
X40
)
)
)
)
)
)
↔
(
(
¬
X48
)
→
(
X43
↔
(
(
¬
X39
)
↔
X49
)
)
)
)
↔
(
(
¬
(
(
¬
X39
)
→
(
X29
↔
X49
)
)
)
↔
(
¬
(
X49
↔
(
(
¬
X42
)
↔
X43
)
)
)
)
)
→
(
(
(
(
¬
X40
)
→
(
¬
X32
)
→
X49
)
↔
(
(
X40
→
(
X32
↔
X0
)
)
↔
(
¬
(
(
¬
X32
)
→
(
(
¬
X6
)
↔
(
X0
→
X49
)
)
)
)
)
)
↔
(
¬
(
X48
→
(
X40
↔
X6
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
(
¬
(
X46
→
X27
→
X18
→
X48
)
)
→
(
¬
(
X27
↔
(
(
¬
X18
)
→
X5
→
X46
)
)
)
)
)
↔
(
¬
(
(
¬
X46
)
↔
(
(
¬
X5
)
↔
(
(
¬
X1
)
→
X48
)
)
)
)
)
↔
(
(
¬
(
X47
↔
(
X27
↔
X43
)
)
)
↔
(
¬
(
(
¬
X27
)
↔
(
X5
→
(
X1
↔
X43
)
)
)
)
)
)
↔
(
(
X47
→
(
¬
X12
)
→
X44
)
↔
(
(
(
(
¬
X19
)
→
X22
)
↔
(
(
¬
(
X35
↔
(
(
¬
X19
)
↔
X43
)
)
)
↔
(
(
X46
↔
(
(
¬
X35
)
→
X19
→
X43
)
)
→
(
¬
(
X43
→
(
X35
↔
(
X22
→
X46
)
)
)
)
)
)
)
↔
(
X46
→
(
(
¬
X22
)
↔
X43
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
(
¬
(
X26
↔
(
(
¬
X21
)
↔
(
X14
→
X48
)
)
)
)
↔
(
¬
(
(
¬
X33
)
→
(
¬
X21
)
→
(
¬
X14
)
→
X48
)
)
)
↔
(
¬
(
X50
→
X14
)
)
)
→
(
(
¬
(
X50
↔
(
X48
→
X10
→
X33
)
)
)
↔
(
X47
↔
(
(
¬
X46
)
↔
(
(
¬
X20
)
↔
X50
)
)
)
)
→
(
¬
(
(
X47
↔
(
(
¬
X30
)
→
X4
→
X48
)
)
→
(
X50
↔
(
(
¬
X4
)
→
X41
)
)
)
)
→
(
¬
(
X50
→
(
(
¬
X41
)
↔
X48
)
)
)
)
)
→
(
(
¬
(
(
(
(
¬
(
X47
↔
False
)
)
↔
(
¬
(
(
¬
(
(
¬
X34
)
→
X47
)
)
→
(
¬
(
(
¬
X47
)
↔
(
X31
→
(
X15
↔
X48
)
)
)
)
)
)
)
↔
(
X48
→
(
X47
↔
X34
)
)
)
→
X34
→
(
X31
↔
X48
)
)
)
↔
(
¬
(
X48
↔
(
(
¬
X45
)
↔
(
X34
→
X46
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
80123b...
and proposition id is
a9b143...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMaVJx8pUs1bsrin5tEnkL2XG5kLxVknpHZ
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
¬
(
(
(
(
X49
→
(
(
¬
X36
)
↔
(
(
¬
X29
)
→
X45
)
)
)
→
(
¬
(
X49
→
X29
→
(
¬
X24
)
→
X45
)
)
)
↔
(
(
(
X36
→
X13
→
X49
)
↔
(
X36
→
(
(
¬
X34
)
↔
(
(
¬
X32
)
→
X49
)
)
)
)
→
(
X49
↔
(
X45
↔
(
X36
↔
(
X32
→
False
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X45
)
→
(
(
¬
X34
)
↔
(
X0
→
X47
)
)
)
)
→
(
X21
↔
(
X0
↔
X36
)
)
)
)
)
)
→
(
(
(
¬
(
(
(
(
X39
→
X31
→
X44
)
↔
(
(
(
¬
(
X49
→
(
¬
X43
)
→
(
X39
↔
X47
)
)
)
↔
(
¬
(
X49
↔
(
X47
→
X43
→
X44
)
)
)
)
↔
(
X43
↔
(
X31
→
X39
)
)
)
)
↔
(
X49
↔
(
X43
→
(
X39
↔
False
)
)
)
)
→
(
(
¬
(
(
X49
→
(
(
¬
X30
)
↔
(
(
¬
X23
)
→
X44
)
)
)
→
(
¬
(
X49
→
X45
→
(
¬
X30
)
→
X47
)
)
→
X49
→
(
X45
↔
(
X44
↔
X30
)
)
)
)
↔
(
¬
(
(
¬
X30
)
↔
(
(
¬
X10
)
↔
X49
)
)
)
)
→
(
X47
↔
(
X45
↔
(
X6
↔
X49
)
)
)
)
)
↔
(
(
¬
X47
)
→
(
X45
↔
(
X41
→
X49
)
)
)
)
↔
(
(
(
¬
X47
)
→
(
¬
X28
)
→
(
(
¬
X26
)
↔
X49
)
)
→
(
(
¬
(
X48
→
X4
→
(
(
¬
X3
)
↔
X28
)
)
)
↔
(
(
X49
→
X45
→
(
X4
↔
X1
)
)
↔
(
(
X49
↔
(
(
¬
X28
)
→
X47
)
)
↔
(
¬
(
(
X49
→
(
(
¬
X28
)
↔
(
(
¬
X9
)
→
X48
)
)
)
→
(
¬
(
X49
→
(
(
¬
X47
)
↔
X48
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X49
↔
(
X42
→
X40
→
X27
→
False
)
)
)
↔
(
¬
(
X27
↔
(
(
¬
X8
)
↔
X22
)
)
)
)
↔
(
(
¬
(
(
¬
X40
)
↔
X42
)
)
↔
(
¬
(
(
¬
(
(
¬
X40
)
↔
(
(
¬
X35
)
↔
(
(
¬
X27
)
→
X42
)
)
)
)
→
X42
→
(
X40
↔
(
X27
↔
X15
)
)
)
)
)
)
↔
(
(
¬
(
X27
↔
(
X20
→
(
(
¬
X18
)
↔
X35
)
)
)
)
→
(
(
(
¬
X17
)
↔
X35
)
↔
(
(
(
¬
(
X40
→
X20
→
X37
)
)
↔
(
(
¬
X37
)
→
X35
→
(
X20
↔
X38
)
)
)
↔
(
¬
(
X40
↔
(
X37
→
(
¬
X17
)
→
X38
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
X49
↔
(
(
¬
X46
)
↔
(
X45
↔
X48
)
)
)
↔
(
¬
(
(
X49
↔
(
X48
→
(
X46
↔
X14
)
)
)
→
(
¬
(
X49
↔
(
X46
↔
(
X42
→
X48
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
X7
)
→
(
X5
↔
(
X2
→
X33
)
)
)
)
↔
(
¬
(
X40
↔
(
(
¬
X7
)
→
(
(
¬
X5
)
↔
X33
)
)
)
)
)
↔
(
X46
↔
(
X5
↔
(
X2
→
X40
)
)
)
)
↔
(
X49
↔
(
X45
→
X12
→
X5
)
)
)
)
)
↔
(
(
(
X45
→
X19
→
X49
)
↔
(
¬
(
X45
↔
(
X16
↔
X46
)
)
)
)
↔
(
X25
→
(
(
¬
X19
)
↔
(
X11
→
X49
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
(
¬
X28
)
→
X26
→
(
(
¬
X15
)
↔
X47
)
)
↔
(
(
(
¬
(
X49
→
(
X27
↔
(
X2
↔
X28
)
)
)
)
↔
(
(
¬
X47
)
↔
(
X2
→
X49
)
)
)
→
(
X28
↔
(
(
¬
X2
)
→
X15
)
)
)
)
→
(
(
X41
↔
(
X27
↔
(
(
¬
X26
)
↔
X39
)
)
)
→
(
¬
(
(
X41
↔
(
X26
↔
X24
)
)
→
(
¬
(
(
¬
X27
)
→
X24
→
(
¬
X20
)
→
X41
)
)
)
)
)
→
(
(
¬
X39
)
→
(
X28
↔
(
(
¬
X3
)
→
X41
)
)
)
→
(
X24
↔
(
X3
→
X41
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X47
)
↔
(
(
¬
X1
)
↔
X49
)
)
)
→
(
¬
(
X39
↔
(
(
¬
X18
)
→
X1
→
X47
)
)
)
)
)
↔
(
(
(
¬
(
X47
↔
(
(
¬
X23
)
↔
(
(
¬
X4
)
→
X49
)
)
)
)
↔
(
(
¬
(
(
¬
X23
)
→
X49
)
)
↔
(
X49
→
X48
→
(
X18
↔
X17
)
)
)
)
↔
(
X49
→
(
¬
X47
)
→
X48
)
)
)
)
↔
(
¬
(
(
(
¬
(
X49
↔
(
(
¬
X6
)
→
X7
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X25
)
→
X8
→
(
X6
↔
X48
)
)
)
→
(
¬
X8
)
→
(
(
¬
X7
)
↔
(
X6
↔
X47
)
)
)
)
)
→
(
¬
(
(
X47
→
X19
→
X7
→
X0
)
→
(
X35
↔
(
X19
↔
(
X0
↔
X5
)
)
)
→
X48
→
(
X35
↔
(
X5
→
X0
→
False
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
(
X47
→
X13
→
X41
)
)
→
(
¬
(
X47
↔
(
X45
→
X41
)
)
)
→
(
X46
↔
(
(
¬
X29
)
→
(
¬
X13
)
→
X47
)
)
)
)
→
(
X43
↔
(
X41
→
X13
→
X46
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
(
¬
(
X48
↔
(
X42
→
X21
→
False
)
)
)
↔
(
X48
→
(
X34
↔
(
X10
↔
False
)
)
)
)
→
(
¬
(
(
¬
(
X40
↔
(
(
¬
X37
)
↔
X42
)
)
)
→
(
(
X48
→
(
(
¬
X40
)
↔
(
X37
→
X42
)
)
)
→
X40
→
(
¬
X14
)
→
X37
)
→
(
(
¬
X42
)
↔
(
X37
↔
X48
)
)
)
)
)
)
→
(
(
¬
(
(
¬
X47
)
→
(
(
¬
X42
)
↔
(
(
¬
X30
)
→
X48
)
)
)
)
↔
(
(
X47
→
X45
→
(
(
¬
X42
)
↔
X48
)
)
↔
(
X47
↔
(
(
¬
X42
)
→
(
X40
↔
X48
)
)
)
)
)
)
)
→
(
(
(
X42
→
(
¬
X36
)
→
X44
)
↔
(
(
(
¬
X42
)
→
(
X40
↔
X47
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X40
)
→
(
¬
X9
)
→
X42
)
)
→
(
¬
(
(
¬
(
(
X42
↔
(
X36
↔
False
)
)
→
(
(
¬
X42
)
↔
(
X40
→
X9
→
X47
)
)
)
)
→
(
¬
(
X42
→
(
X40
↔
X47
)
)
)
)
)
)
)
↔
(
X44
↔
(
X40
↔
(
(
¬
X36
)
↔
X42
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X42
→
X40
→
X16
→
X46
)
)
→
(
X47
↔
(
X46
↔
(
X40
↔
False
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
X41
↔
(
X33
↔
X31
)
)
→
(
(
(
¬
(
X41
→
(
(
¬
X33
)
↔
(
(
¬
X31
)
↔
X48
)
)
)
)
→
(
(
X48
↔
(
X32
→
X31
→
X41
)
)
↔
(
¬
(
X48
→
(
X41
↔
(
X31
→
X22
)
)
)
)
)
)
↔
(
¬
(
X48
→
(
X33
↔
X41
)
)
)
)
)
)
→
(
¬
(
X38
↔
(
X31
↔
(
X12
↔
X32
)
)
)
)
)
)
→
(
(
(
(
(
¬
X41
)
→
(
¬
X11
)
→
X42
)
→
(
¬
(
X48
↔
(
X41
→
(
X38
↔
X42
)
)
)
)
→
(
¬
(
X48
↔
(
(
¬
X41
)
↔
(
(
¬
X11
)
↔
X42
)
)
)
)
)
↔
(
X38
↔
X42
)
)
↔
(
(
¬
X38
)
→
(
X11
↔
X41
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
c88c27...
and proposition id is
4cb4da...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMXa9J59NUkPvQVgfwndShCqGXLSTiRV8Lo
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
¬
(
(
X46
→
X44
→
X38
→
X36
)
→
(
(
X46
→
(
(
¬
X17
)
↔
(
(
¬
X8
)
→
X38
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
X49
↔
(
X46
→
X38
→
X44
)
)
)
↔
(
¬
(
(
¬
X43
)
→
(
¬
X38
)
→
X49
)
)
)
↔
(
X44
↔
False
)
)
→
(
X49
↔
(
X44
→
X46
)
)
)
)
↔
(
¬
(
(
¬
X43
)
→
(
¬
X38
)
→
X49
)
)
)
)
)
)
→
(
X23
↔
(
(
¬
X21
)
→
(
X1
↔
X24
)
)
)
→
(
X46
↔
(
X44
↔
(
X21
→
False
)
)
)
→
(
¬
(
(
(
X46
→
X44
→
(
X24
↔
X23
)
)
↔
(
¬
(
(
¬
X43
)
↔
(
X24
→
(
X23
↔
X44
)
)
)
)
)
→
(
¬
(
X44
↔
(
X43
→
X23
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X43
↔
(
X3
→
X38
)
)
)
→
(
(
¬
(
X43
↔
(
X38
→
(
X2
↔
X46
)
)
)
)
↔
(
¬
(
X46
→
X3
→
(
X2
↔
X43
)
)
)
)
)
↔
(
(
X43
↔
(
(
¬
X16
)
↔
X42
)
)
↔
(
(
¬
(
(
¬
(
X46
→
X43
→
(
¬
X16
)
→
X38
)
)
→
(
(
X38
↔
X18
)
↔
(
(
¬
X42
)
↔
(
(
¬
X16
)
→
X43
)
)
)
)
)
↔
(
X46
↔
False
)
)
)
)
→
(
(
¬
(
X43
↔
(
X42
↔
(
X38
↔
X37
)
)
)
)
↔
(
¬
(
X42
→
(
(
¬
X33
)
↔
X38
)
)
)
)
)
→
(
(
(
(
X46
→
X33
→
X13
→
X22
)
↔
(
X22
↔
(
X13
↔
(
(
¬
X11
)
↔
X33
)
)
)
)
↔
(
¬
(
X29
→
(
¬
X13
)
→
X11
→
X33
)
)
)
↔
(
¬
(
(
¬
(
X46
↔
(
(
¬
X38
)
↔
(
(
¬
X11
)
→
X39
)
)
)
)
→
(
(
¬
(
X39
↔
(
(
¬
X38
)
↔
(
(
¬
X33
)
↔
X46
)
)
)
)
↔
(
¬
(
(
¬
X38
)
↔
(
(
¬
X33
)
→
(
(
¬
X9
)
↔
X46
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
(
¬
X40
)
→
X6
→
X5
→
X42
)
)
→
(
¬
(
X40
↔
(
(
¬
X35
)
→
(
¬
X32
)
→
X42
)
)
)
)
→
(
(
¬
(
(
¬
X35
)
↔
(
X30
↔
(
X15
↔
X42
)
)
)
)
→
(
¬
(
(
¬
X35
)
→
(
(
¬
X30
)
↔
(
X15
→
X40
)
)
)
)
)
→
(
(
X49
→
(
X35
↔
(
(
¬
X26
)
↔
X30
)
)
)
↔
(
(
¬
X30
)
↔
(
(
¬
X26
)
→
(
(
¬
X7
)
↔
X49
)
)
)
)
→
(
¬
(
X35
↔
(
(
¬
X30
)
→
X42
)
)
)
)
↔
(
¬
(
(
(
X43
↔
(
X27
→
(
(
¬
X25
)
↔
X49
)
)
)
↔
(
(
¬
(
(
¬
X34
)
→
(
X25
↔
X35
)
)
)
↔
(
(
¬
(
X43
↔
(
(
¬
X35
)
→
X27
→
X40
)
)
)
→
(
¬
(
X34
→
(
X27
↔
False
)
)
)
)
)
)
→
(
¬
(
(
¬
X43
)
↔
(
(
¬
X27
)
↔
X49
)
)
)
)
)
)
→
(
(
(
(
X45
↔
(
X41
→
(
X31
↔
X49
)
)
)
→
X41
→
(
X40
↔
(
X31
↔
X45
)
)
)
↔
(
(
¬
(
X45
→
X43
→
X41
→
X49
)
)
→
X45
→
X43
→
(
(
¬
X20
)
↔
X41
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
X47
)
↔
(
X19
→
(
¬
X0
)
→
X49
)
)
)
→
(
(
X49
↔
(
X19
↔
X31
)
)
↔
(
(
¬
X47
)
↔
(
X31
→
(
¬
X12
)
→
X49
)
)
)
)
)
↔
(
¬
(
(
¬
(
X48
↔
(
X41
→
X31
→
(
X10
↔
False
)
)
)
)
→
(
(
¬
(
X49
→
X31
→
False
)
)
↔
(
(
X10
→
False
)
↔
(
(
¬
(
(
¬
X41
)
↔
(
(
¬
X31
)
→
X49
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X41
)
→
(
X10
↔
X47
)
)
)
→
(
(
(
¬
X31
)
→
X49
)
↔
(
X47
↔
(
X41
↔
(
X31
↔
(
X10
→
False
)
)
)
)
)
)
)
)
)
)
)
)
)
)
→
(
(
X45
→
(
¬
X28
)
→
X41
)
↔
(
(
¬
(
(
¬
X14
)
↔
(
(
¬
X4
)
↔
X28
)
)
)
↔
(
¬
(
(
¬
X41
)
→
(
X40
↔
(
(
¬
X14
)
↔
X45
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
(
(
X40
→
(
(
¬
X19
)
↔
(
X11
↔
X42
)
)
)
↔
(
¬
(
(
X35
↔
(
X19
↔
(
X11
↔
X42
)
)
)
→
(
¬
X11
)
→
X49
→
(
X42
↔
(
(
¬
X35
)
↔
X43
)
)
)
)
)
↔
(
(
¬
(
(
¬
X8
)
→
(
¬
X5
)
→
(
X0
↔
X40
)
)
)
→
(
X49
↔
(
X42
↔
False
)
)
)
)
↔
(
¬
(
(
¬
(
X40
↔
(
(
¬
X35
)
→
X43
)
)
)
→
(
¬
(
X46
→
X41
→
(
¬
X2
)
→
X40
)
)
)
)
)
→
(
X46
↔
(
X42
↔
(
X40
→
X18
→
False
)
)
)
)
→
(
X43
↔
(
X42
→
X41
→
X37
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
X41
→
(
X15
↔
(
X3
↔
X49
)
)
)
→
(
(
(
(
¬
(
X49
→
(
¬
X38
)
→
X41
)
)
↔
(
(
(
¬
X15
)
→
X7
→
X49
)
↔
(
X41
↔
(
(
¬
X38
)
↔
(
(
¬
X7
)
↔
X49
)
)
)
)
)
↔
(
¬
(
X38
↔
X7
)
)
)
↔
(
(
(
¬
X41
)
→
(
(
¬
X37
)
↔
(
(
¬
X9
)
↔
X49
)
)
)
↔
(
(
¬
X41
)
→
X38
→
X37
→
X49
)
)
)
)
)
↔
(
(
(
¬
(
X49
→
(
(
¬
X37
)
↔
(
(
¬
X22
)
→
X41
)
)
)
)
→
X45
→
X22
)
↔
(
¬
(
(
¬
(
X49
↔
(
X37
↔
(
(
¬
X32
)
↔
X45
)
)
)
)
→
(
(
¬
X37
)
↔
(
X32
↔
(
X16
→
X45
)
)
)
)
)
)
)
→
(
(
(
¬
(
X49
→
(
(
¬
X36
)
↔
(
X23
→
X45
)
)
)
)
→
(
X49
→
X41
→
(
(
¬
X27
)
↔
X34
)
)
→
(
X49
↔
(
X41
→
(
¬
X23
)
→
X27
)
)
)
↔
(
(
¬
(
X36
→
(
X31
↔
(
X27
→
X4
)
)
)
)
→
X36
→
(
X31
↔
(
X27
↔
X23
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
(
X48
↔
(
X20
↔
X14
)
)
→
(
(
¬
(
(
¬
X20
)
↔
(
X14
→
(
X13
↔
X45
)
)
)
)
↔
(
(
¬
X45
)
↔
(
X20
↔
(
(
¬
X6
)
→
X48
)
)
)
)
)
)
→
(
¬
(
(
¬
X44
)
→
(
¬
X14
)
→
X13
→
X48
)
)
)
)
↔
(
(
X48
→
X42
→
(
¬
X29
)
→
X41
)
↔
(
¬
(
(
¬
(
X45
↔
(
(
¬
X42
)
→
(
¬
X21
)
→
X48
)
)
)
→
(
¬
(
(
X42
↔
(
(
¬
X33
)
↔
X45
)
)
→
(
(
¬
(
X48
↔
(
X42
→
(
X33
↔
X29
)
)
)
)
↔
(
X48
↔
(
X42
→
(
¬
X33
)
→
X44
)
)
)
→
(
X45
↔
(
X44
↔
(
(
¬
X29
)
↔
X33
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X44
→
X30
→
(
X17
↔
X10
)
)
)
↔
(
(
X30
→
(
X28
↔
(
X24
→
(
X17
↔
False
)
)
)
)
→
(
(
X30
→
(
(
¬
X26
)
↔
(
X24
↔
X28
)
)
)
↔
(
¬
(
X26
→
(
X24
↔
(
(
¬
X10
)
↔
X30
)
)
)
)
)
)
)
→
(
¬
(
X41
↔
(
X30
→
X24
)
)
)
)
↔
(
(
(
¬
(
X48
↔
(
X39
→
(
(
¬
X17
)
↔
X24
)
)
)
)
↔
(
(
¬
(
(
(
¬
X47
)
↔
(
X39
→
(
¬
X1
)
→
X48
)
)
→
(
¬
(
X48
→
(
¬
X39
)
→
(
¬
X24
)
→
X47
)
)
)
)
↔
(
¬
(
(
¬
X39
)
→
(
(
¬
X24
)
↔
X48
)
)
)
)
)
→
(
X25
→
X24
→
X39
)
→
(
(
X39
↔
(
X24
↔
(
X12
↔
X30
)
)
)
↔
(
¬
(
(
¬
X30
)
→
X25
→
(
X24
↔
X39
)
)
)
)
→
(
¬
(
(
¬
X25
)
→
(
X17
↔
X39
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
ee3417...
and proposition id is
f5d2c2...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMP8PbZpusBS9dHcdpoF21T1UKiqzVSbBU5
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
(
X47
→
(
(
¬
X46
)
↔
(
(
¬
X32
)
↔
X48
)
)
)
↔
(
(
¬
(
(
X47
→
X31
)
→
(
(
¬
X46
)
↔
(
(
¬
X3
)
↔
X48
)
)
)
)
↔
(
¬
(
X48
→
(
¬
X44
)
→
(
X3
↔
X46
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
X49
↔
(
X46
↔
(
X44
→
False
)
)
)
→
(
(
¬
(
X48
↔
(
X46
↔
(
(
¬
X2
)
→
X49
)
)
)
)
↔
(
X49
→
X44
)
)
)
)
→
(
¬
(
X46
↔
(
X44
→
(
X8
↔
X48
)
)
)
)
)
)
→
(
X48
→
X46
→
(
¬
X1
)
→
X44
)
→
(
¬
(
(
¬
(
X48
→
X46
→
(
¬
X6
)
→
X22
)
)
→
(
¬
X44
)
→
(
X22
↔
(
(
¬
X6
)
→
X48
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
¬
(
X42
↔
(
(
¬
X25
)
→
X48
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X42
)
→
(
X25
↔
(
(
¬
X17
)
↔
X48
)
)
)
)
→
X23
→
X17
→
X25
)
)
)
→
X41
→
X35
→
X25
→
X42
)
→
(
(
¬
(
X48
→
(
¬
X36
)
→
X7
→
X42
)
)
↔
(
(
X48
↔
(
X20
→
(
(
¬
X7
)
↔
X41
)
)
)
↔
(
¬
(
X42
→
(
(
¬
X20
)
↔
(
X7
↔
X48
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
X48
↔
(
(
¬
X42
)
↔
(
X36
↔
X47
)
)
)
↔
(
(
¬
(
X48
↔
(
(
¬
X34
)
→
(
(
¬
X26
)
↔
X47
)
)
)
)
↔
(
X47
→
(
(
¬
X36
)
↔
(
X26
↔
X48
)
)
)
)
)
→
(
(
¬
(
(
¬
X42
)
↔
X47
)
)
↔
(
¬
(
(
¬
X48
)
↔
(
X42
↔
(
X12
→
X49
)
)
)
)
)
)
)
→
(
X48
↔
(
X34
↔
(
(
¬
X21
)
↔
X36
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X38
↔
(
X5
→
False
)
)
)
→
(
(
(
X42
↔
(
X40
↔
(
X5
↔
X38
)
)
)
→
(
¬
(
X38
↔
(
(
¬
X0
)
→
X40
)
)
)
)
→
(
(
¬
X13
)
↔
(
X5
↔
(
X0
↔
X42
)
)
)
)
→
(
(
X39
↔
(
X13
→
X9
)
)
↔
(
X39
→
(
¬
X5
)
→
X40
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
(
(
¬
X43
)
→
(
¬
X29
)
→
(
(
¬
X24
)
↔
X44
)
)
)
→
X45
→
X43
→
(
¬
X29
)
→
X44
)
)
→
(
(
X44
↔
(
X29
→
(
X10
↔
X33
)
)
)
↔
(
X44
→
X29
→
(
(
¬
X18
)
↔
X33
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X43
↔
(
(
¬
X38
)
→
(
X28
↔
X44
)
)
)
)
→
(
X44
↔
(
(
¬
X28
)
→
(
¬
X11
)
→
X38
)
)
)
→
(
¬
X40
)
→
X28
→
X44
)
)
↔
(
¬
(
(
X45
→
(
¬
X38
)
→
(
¬
X37
)
→
X40
)
→
(
¬
(
(
X42
→
(
¬
X40
)
→
X43
)
→
(
¬
(
(
¬
(
X42
↔
(
(
¬
X40
)
↔
X43
)
)
)
→
(
(
(
(
¬
X42
)
→
(
¬
X40
)
→
X43
)
→
(
¬
(
X43
↔
(
X37
→
X42
)
)
)
→
(
¬
(
X42
→
X40
→
(
(
¬
X37
)
↔
X38
)
)
)
)
↔
(
¬
(
X42
→
(
X38
↔
X43
)
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X19
↔
(
X14
↔
(
X4
↔
X46
)
)
)
)
→
(
¬
(
X45
→
(
¬
X19
)
→
X4
→
X46
)
)
)
↔
(
(
(
(
X46
→
(
X44
↔
(
X16
↔
X15
)
)
)
→
(
¬
(
(
¬
X19
)
↔
(
X16
↔
(
(
¬
X15
)
→
X46
)
)
)
)
)
→
X44
→
(
(
¬
X19
)
↔
(
(
¬
X4
)
→
X46
)
)
)
→
(
X44
↔
(
(
¬
X16
)
↔
X19
)
)
)
)
↔
(
(
X38
↔
(
X30
→
X27
→
X19
)
)
↔
(
X30
↔
(
X27
→
(
¬
X4
)
→
X38
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
(
(
¬
(
(
¬
X44
)
→
X15
→
X49
)
)
↔
(
(
¬
(
X44
→
(
(
¬
X36
)
↔
X49
)
)
)
↔
(
X49
↔
(
X44
→
(
¬
X15
)
→
X48
)
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X44
)
→
X27
→
X48
)
)
)
→
(
¬
(
X44
→
(
¬
X20
)
→
X48
)
)
)
)
↔
(
(
(
¬
(
X46
↔
(
X44
→
(
¬
X25
)
→
X30
)
)
)
↔
(
(
¬
(
X43
↔
(
X30
↔
X49
)
)
)
↔
(
¬
(
X44
↔
(
(
¬
X30
)
↔
X46
)
)
)
)
)
↔
(
(
¬
X36
)
→
(
X30
↔
(
X25
↔
X49
)
)
)
)
)
→
(
(
(
(
X11
→
X5
)
↔
(
¬
(
(
¬
X40
)
↔
(
X23
↔
(
(
¬
X5
)
↔
X49
)
)
)
)
)
↔
(
¬
(
X40
↔
(
(
¬
X23
)
→
X9
→
X49
)
)
)
)
↔
(
(
(
¬
(
X48
↔
(
X44
↔
X49
)
)
)
↔
(
¬
(
(
¬
(
X49
↔
(
(
¬
X23
)
↔
(
(
¬
X22
)
↔
X48
)
)
)
)
→
(
¬
(
X49
→
(
(
¬
X44
)
↔
(
X22
→
X48
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X44
)
→
X49
)
)
↔
(
(
X49
↔
(
X44
↔
(
(
¬
X6
)
→
X48
)
)
)
↔
(
¬
(
X48
→
(
X39
↔
(
X6
→
False
)
)
)
)
)
)
↔
(
X49
↔
(
X40
→
(
X39
↔
X44
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
(
(
(
¬
X31
)
→
(
(
¬
X21
)
↔
(
X10
→
X49
)
)
)
→
(
¬
(
(
X49
↔
(
X29
→
(
¬
X17
)
→
X35
)
)
→
(
¬
(
(
X35
→
X31
→
(
¬
X29
)
→
X49
)
→
(
(
¬
(
(
¬
X35
)
→
X31
→
(
(
¬
X21
)
↔
X48
)
)
)
↔
(
(
¬
X29
)
↔
(
(
¬
X21
)
↔
X35
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
X35
)
→
X31
→
(
¬
X21
)
→
X37
)
↔
(
¬
(
(
(
(
¬
X37
)
→
(
X35
↔
(
X3
→
X46
)
)
)
↔
(
¬
(
X37
↔
(
X35
↔
(
(
¬
X21
)
→
X46
)
)
)
)
)
→
X46
→
(
X21
↔
False
)
)
)
)
→
(
X31
↔
(
X21
↔
X46
)
)
)
)
)
→
(
(
(
¬
(
X48
↔
(
X47
↔
(
X24
→
X46
)
)
)
)
↔
(
X48
→
X46
→
(
¬
X8
)
→
X47
)
)
↔
(
(
(
¬
(
X46
→
X38
→
X47
)
)
↔
(
(
¬
(
X47
→
(
(
¬
X46
)
↔
X48
)
)
)
↔
(
(
(
¬
(
X48
↔
(
X32
↔
(
X1
↔
X47
)
)
)
)
↔
(
¬
(
X38
→
(
(
¬
X32
)
↔
(
(
¬
X1
)
↔
X47
)
)
)
)
)
→
(
¬
X47
)
→
(
(
¬
X32
)
↔
(
X12
↔
X48
)
)
)
)
)
↔
(
X48
↔
(
(
¬
X12
)
↔
(
X7
↔
X49
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
X4
→
(
(
¬
X0
)
↔
X33
)
)
)
↔
(
(
¬
(
X45
↔
(
(
¬
X18
)
↔
(
X0
↔
X33
)
)
)
)
↔
(
¬
(
X33
→
X4
→
X18
)
)
)
)
↔
(
X48
→
(
(
¬
X33
)
↔
(
X18
→
X45
)
)
)
)
↔
(
¬
(
X48
→
(
X33
↔
(
(
¬
X14
)
↔
X45
)
)
)
)
)
↔
(
(
(
X45
→
(
(
¬
X41
)
↔
(
X26
↔
X48
)
)
)
↔
(
¬
(
X48
↔
(
X47
→
X45
)
)
)
)
→
(
(
¬
(
(
¬
(
X48
↔
(
X26
→
(
¬
X14
)
→
X47
)
)
)
→
X26
→
(
X14
↔
(
X2
→
X47
)
)
)
)
↔
(
¬
(
X48
↔
(
X47
→
X14
)
)
)
)
)
)
→
(
(
(
(
X49
→
(
X47
↔
X28
)
)
↔
(
(
X45
→
(
X28
↔
X47
)
)
→
X47
→
(
X45
↔
(
(
¬
X16
)
→
X49
)
)
)
)
↔
(
X49
→
X28
)
)
↔
(
¬
(
(
(
¬
(
X49
→
X47
→
(
X45
↔
False
)
)
)
↔
(
X47
↔
(
(
¬
X34
)
→
(
¬
X19
)
→
X49
)
)
)
→
X45
→
X19
)
)
)
→
(
(
¬
(
X49
→
(
X48
↔
X19
)
)
)
↔
(
(
¬
(
(
¬
(
X45
↔
(
X19
→
(
(
¬
X13
)
↔
X49
)
)
)
)
→
(
X49
↔
(
X13
↔
X42
)
)
)
)
→
(
¬
(
X48
↔
(
X45
↔
X42
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
416aee...
and proposition id is
f82cc1...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMUyitGPKdtR2aDB7KhyedEbUAbYjAb9Ub3
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
(
(
(
X48
↔
(
X45
→
X6
→
X49
)
)
↔
(
¬
(
X49
↔
(
X40
→
False
)
)
)
)
↔
(
¬
(
X48
↔
(
(
¬
X6
)
→
X49
)
)
)
)
↔
(
(
¬
(
X49
→
(
X48
↔
(
X46
→
(
X45
↔
False
)
)
)
)
)
↔
(
(
¬
(
X49
↔
(
(
¬
X9
)
→
X48
)
)
)
→
(
¬
(
(
X49
→
X45
→
(
X23
↔
(
X9
→
False
)
)
)
→
X49
→
(
¬
X45
)
→
X48
)
)
)
)
)
↔
(
(
(
(
¬
X48
)
↔
(
(
¬
X45
)
↔
(
X26
↔
X49
)
)
)
↔
(
(
¬
(
(
¬
X45
)
→
(
¬
X36
)
→
(
¬
X8
)
→
X49
)
)
↔
(
¬
(
X48
→
(
X45
↔
X8
)
)
)
)
)
↔
(
¬
(
(
(
¬
X45
)
→
(
(
¬
X33
)
↔
(
X26
→
X49
)
)
)
→
X21
→
X49
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
X35
→
(
¬
X2
)
→
X46
)
↔
(
¬
(
X46
↔
(
(
¬
X17
)
↔
X49
)
)
)
)
→
(
¬
(
(
¬
X35
)
→
X11
→
X2
→
X48
)
)
)
)
→
(
(
X44
↔
X14
)
→
(
X46
↔
(
(
¬
X30
)
↔
(
(
¬
X14
)
↔
X35
)
)
)
)
→
(
¬
(
(
¬
(
X48
→
(
X3
↔
X35
)
)
)
→
(
¬
(
X46
↔
(
X44
↔
False
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X48
→
X24
→
X43
)
)
↔
(
(
(
¬
(
(
¬
X43
)
→
(
(
¬
X41
)
↔
(
(
¬
X7
)
↔
X48
)
)
)
)
↔
(
X43
↔
(
X41
↔
(
(
¬
X24
)
→
X48
)
)
)
)
↔
(
¬
(
(
¬
X41
)
→
(
X19
↔
(
(
¬
X7
)
↔
X48
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X39
)
→
(
X24
↔
X41
)
)
)
↔
(
¬
(
X41
↔
(
X39
→
(
(
¬
X31
)
↔
X48
)
)
)
)
)
→
(
(
¬
X47
)
↔
(
X31
→
(
(
¬
X24
)
↔
X48
)
)
)
)
)
↔
(
(
(
X48
↔
(
X43
↔
(
X38
→
X47
)
)
)
↔
(
(
¬
(
X48
→
(
X38
↔
(
(
¬
X22
)
↔
X47
)
)
)
)
↔
(
(
(
¬
X5
)
↔
X15
)
→
(
X48
↔
(
X47
↔
(
X38
→
X15
)
)
)
)
)
)
↔
(
(
(
X29
↔
(
(
¬
X28
)
→
X48
)
)
→
(
(
X48
→
(
X29
↔
(
X27
↔
X43
)
)
)
↔
(
¬
(
(
¬
X43
)
↔
(
X29
↔
(
(
¬
X27
)
↔
X49
)
)
)
)
)
)
↔
(
(
(
(
¬
X48
)
→
(
¬
X38
)
→
(
X28
↔
X49
)
)
→
(
X38
↔
(
(
¬
X28
)
→
(
(
¬
X20
)
↔
X49
)
)
)
)
↔
(
¬
(
X48
↔
(
X38
→
(
X28
↔
(
X12
→
False
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
(
X44
→
X43
→
(
X13
↔
X42
)
)
↔
(
¬
(
X44
↔
(
X37
→
False
)
)
)
)
→
(
X44
↔
(
X37
↔
(
X16
→
X43
)
)
)
)
)
↔
(
¬
(
(
¬
X37
)
→
X43
)
)
)
→
(
(
X43
↔
(
X37
↔
(
X34
→
False
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
¬
X43
)
↔
(
(
¬
X37
)
→
X49
)
)
)
→
(
(
X42
→
(
¬
X37
)
→
X49
)
↔
(
¬
(
X43
→
(
¬
X25
)
→
X37
)
)
)
)
→
(
¬
(
X43
↔
(
(
¬
X37
)
↔
X49
)
)
)
)
)
→
(
¬
(
(
¬
X43
)
↔
(
X42
→
(
¬
X34
)
→
X49
)
)
)
)
)
)
→
(
(
X48
↔
(
X42
↔
(
(
¬
X18
)
→
X49
)
)
)
↔
(
(
¬
(
X42
↔
X32
)
)
→
(
X48
↔
(
(
¬
X44
)
↔
X49
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X48
↔
(
X10
↔
(
X4
→
(
X0
↔
False
)
)
)
)
)
↔
(
X48
↔
(
(
¬
X4
)
↔
(
X0
↔
X44
)
)
)
)
→
(
(
¬
(
(
¬
X42
)
→
X48
)
)
↔
(
X44
↔
(
X10
→
X1
→
X48
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
¬
(
(
(
¬
(
(
¬
X44
)
↔
(
X21
↔
(
(
¬
X17
)
→
X47
)
)
)
)
↔
(
X47
→
X36
→
(
X21
↔
X44
)
)
)
→
X47
→
(
X37
↔
(
X36
→
X21
)
)
)
)
→
(
¬
(
(
¬
X44
)
→
X37
→
(
X31
↔
X47
)
)
)
→
(
¬
(
(
(
X37
↔
(
X31
↔
(
X26
↔
False
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X26
)
↔
X30
)
)
)
)
→
(
¬
(
X47
↔
(
X37
↔
False
)
)
)
)
)
)
)
→
(
(
(
¬
(
X36
↔
(
(
¬
X25
)
→
X32
)
)
)
↔
(
X36
→
(
(
¬
X10
)
↔
X37
)
)
)
↔
(
X37
↔
(
X36
→
(
X32
↔
X44
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
X49
↔
(
(
¬
X36
)
↔
(
(
¬
X29
)
→
X45
)
)
)
)
→
X49
→
(
X36
↔
(
X2
↔
X45
)
)
)
)
↔
(
X48
↔
(
(
¬
X29
)
→
(
(
¬
X9
)
↔
X49
)
)
)
)
↔
(
¬
(
(
(
¬
(
X36
↔
(
X29
→
(
X24
↔
(
X19
↔
False
)
)
)
)
)
→
(
¬
(
X48
→
(
X36
↔
(
(
¬
X8
)
↔
X24
)
)
)
)
)
→
(
¬
(
X29
→
(
X19
↔
(
X11
→
X18
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
X42
→
(
(
¬
X34
)
↔
X35
)
)
)
↔
(
(
(
¬
X35
)
→
X27
→
X49
)
↔
(
¬
(
X34
→
X32
→
False
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X49
↔
(
X42
→
X34
→
X45
)
)
)
→
(
X42
↔
(
X34
↔
(
(
¬
X3
)
→
X49
)
)
)
)
→
(
X45
↔
(
X42
↔
(
X3
→
False
)
)
)
)
)
↔
(
X3
↔
X34
)
)
)
↔
(
¬
(
(
¬
X42
)
→
X34
→
(
X6
↔
X49
)
)
)
)
↔
(
(
¬
(
(
¬
X45
)
→
X42
→
(
¬
X4
)
→
X46
)
)
↔
(
(
X46
→
(
¬
X42
)
→
X45
)
↔
(
(
(
¬
X46
)
↔
(
X45
→
(
(
¬
X23
)
↔
X47
)
)
)
↔
(
¬
(
X48
↔
(
X45
→
X35
→
False
)
)
)
)
)
)
)
)
↔
(
(
(
(
X49
↔
(
(
¬
X39
)
↔
X46
)
)
↔
(
¬
(
(
(
¬
(
X39
→
(
X28
↔
(
(
¬
X13
)
→
X49
)
)
)
)
↔
(
(
X28
→
X1
→
False
)
↔
(
(
(
X49
→
X46
→
(
X39
↔
X1
)
)
↔
(
¬
(
X39
→
(
X28
↔
X1
)
)
)
)
↔
(
¬
(
X49
↔
(
X46
→
X39
)
)
)
)
)
)
→
(
(
(
X49
→
(
(
¬
X15
)
↔
X39
)
)
↔
(
(
¬
(
X39
→
X28
→
X7
)
)
↔
(
(
(
(
¬
(
(
¬
X15
)
→
X39
)
)
→
(
(
X39
→
X7
→
X28
)
↔
(
X49
→
X15
→
X39
)
)
)
↔
(
X39
→
X28
→
X15
)
)
→
(
¬
(
X39
→
X28
→
(
X15
↔
False
)
)
)
)
)
)
↔
(
(
¬
X28
)
→
X7
→
X39
)
)
)
)
)
↔
(
(
(
X46
↔
(
X28
↔
(
X20
→
X45
)
)
)
↔
(
¬
(
(
¬
X39
)
↔
(
X28
→
X20
→
X46
)
)
)
)
↔
(
¬
(
X46
↔
(
X20
↔
(
(
¬
X0
)
→
X39
)
)
)
)
)
)
↔
(
(
(
X46
↔
(
(
¬
X45
)
→
X43
→
X49
)
)
↔
(
(
¬
(
(
¬
(
X49
↔
(
X46
→
(
¬
X12
)
→
X43
)
)
)
→
(
¬
X46
)
→
(
X43
↔
(
(
¬
X5
)
↔
X49
)
)
)
)
→
(
¬
X46
)
→
X12
→
X49
)
)
↔
(
(
(
(
¬
(
X46
↔
(
(
¬
X33
)
→
X49
)
)
)
↔
(
(
¬
(
(
¬
X46
)
→
(
(
¬
X38
)
↔
(
X22
↔
X49
)
)
)
)
↔
(
(
X46
→
(
(
¬
X33
)
↔
(
(
¬
X16
)
→
X38
)
)
)
→
(
¬
(
X49
→
(
X46
↔
False
)
)
)
)
)
)
→
(
¬
(
X46
→
X40
→
X38
→
X14
)
)
→
(
¬
(
X40
↔
(
X16
↔
(
X14
↔
X46
)
)
)
)
)
→
(
X43
→
X38
→
(
(
¬
X14
)
↔
X49
)
)
→
(
¬
(
X49
↔
(
X43
↔
(
X14
→
X41
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
271c82...
and proposition id is
963f24...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMVupbWCVGteGe9EBLTpjtfxmgvqLdCmQHs
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
(
¬
(
(
¬
(
(
(
¬
(
(
¬
(
X49
↔
(
X46
→
(
¬
X26
)
→
X31
)
)
)
→
(
¬
(
(
X35
↔
(
(
¬
X27
)
→
(
¬
X26
)
→
X49
)
)
→
(
(
X35
↔
(
X18
↔
X46
)
)
↔
(
(
X49
↔
(
X35
↔
(
X27
↔
X46
)
)
)
↔
(
¬
(
X49
↔
(
X35
↔
(
X27
↔
X18
)
)
)
)
)
)
)
)
)
)
↔
(
(
X31
→
(
¬
X21
)
→
X15
→
X34
)
↔
(
(
¬
(
(
(
¬
X40
)
→
(
X31
↔
X42
)
)
→
(
¬
(
(
¬
(
X42
→
X15
)
)
→
X46
→
X42
→
(
X40
↔
(
X31
→
False
)
)
)
)
)
)
↔
(
(
X42
↔
(
X31
→
X21
→
X26
)
)
→
(
¬
(
X40
↔
(
(
¬
X26
)
→
(
X21
↔
X31
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
X13
↔
X5
)
→
(
(
(
¬
(
(
¬
X23
)
→
X13
→
X45
)
)
↔
(
¬
(
(
¬
X42
)
↔
(
(
¬
X23
)
→
(
(
¬
X5
)
↔
X49
)
)
)
)
)
↔
(
¬
(
(
¬
X5
)
→
X49
)
)
)
)
→
(
(
(
¬
X45
)
→
(
(
¬
X42
)
↔
X46
)
)
↔
(
(
X45
→
(
X42
↔
(
X13
→
False
)
)
)
→
(
¬
(
(
¬
(
X46
↔
(
X42
↔
(
(
¬
X13
)
→
X45
)
)
)
)
→
(
¬
(
X46
→
X45
→
(
X13
↔
X1
)
)
)
)
)
)
)
→
X45
→
X43
→
X42
)
)
)
)
→
(
¬
(
(
(
(
X49
→
X11
→
X7
→
X6
→
False
)
↔
(
(
¬
X40
)
→
(
¬
X11
)
→
X6
→
X49
)
)
→
(
¬
(
(
¬
X7
)
→
(
X4
↔
(
X2
↔
X49
)
)
)
)
)
→
(
(
X46
↔
(
(
¬
X40
)
→
X49
)
)
↔
(
(
¬
(
(
¬
X46
)
→
X19
→
X49
)
)
↔
(
(
X49
↔
(
X46
↔
X40
)
)
↔
(
X49
↔
(
X46
↔
(
X19
→
(
X11
↔
False
)
)
)
)
)
)
)
→
(
(
(
X19
→
(
X11
↔
X50
)
)
↔
(
(
(
(
¬
(
X17
↔
(
(
¬
X11
)
↔
X50
)
)
)
↔
(
(
¬
(
X50
→
(
X40
↔
(
X11
↔
X19
)
)
)
)
→
(
¬
(
X17
↔
(
X11
→
X40
)
)
)
)
)
↔
(
(
¬
X17
)
↔
(
X11
↔
X40
)
)
)
→
(
X19
↔
(
X17
→
X11
)
)
)
)
↔
(
¬
(
X50
→
X40
→
X19
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
(
¬
X43
)
↔
(
(
¬
X22
)
→
X47
)
)
)
↔
(
¬
(
X16
↔
(
X9
↔
X43
)
)
)
)
→
(
(
(
(
¬
X43
)
↔
(
(
¬
X39
)
→
(
(
¬
X22
)
↔
X47
)
)
)
↔
(
(
(
¬
X47
)
→
(
(
¬
X14
)
↔
X48
)
)
→
(
¬
(
X47
↔
(
X43
→
X39
→
X14
→
False
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X43
)
→
X39
→
(
¬
X33
)
→
X47
)
)
↔
(
(
¬
X47
)
→
(
X39
↔
X50
)
)
)
↔
(
X47
↔
(
X33
→
X39
)
)
)
)
)
→
(
(
(
(
(
¬
X42
)
↔
(
(
¬
X12
)
→
(
(
¬
X0
)
↔
X48
)
)
)
↔
(
(
¬
X48
)
→
(
(
¬
X42
)
↔
X50
)
)
)
→
(
(
X42
→
(
¬
X20
)
→
(
¬
X12
)
→
X41
)
↔
(
¬
(
X29
→
(
X20
↔
(
X12
→
X41
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X47
→
(
X41
↔
X10
)
)
)
→
(
¬
(
X41
→
(
X37
↔
X42
)
)
)
)
→
(
¬
(
X50
→
X47
→
(
X42
↔
(
X8
→
False
)
)
)
)
)
)
)
)
→
(
(
¬
(
X50
→
(
(
¬
X39
)
↔
(
(
¬
X36
)
→
X44
)
)
)
)
↔
(
X44
↔
(
(
¬
X36
)
→
(
(
¬
X28
)
↔
X50
)
)
)
)
→
(
(
(
X38
→
X32
→
(
X28
↔
X25
)
)
↔
(
(
(
¬
X36
)
↔
(
X30
↔
(
X24
↔
X38
)
)
)
→
(
¬
X25
)
→
X28
)
)
↔
(
(
(
¬
X36
)
→
(
¬
X25
)
→
(
X3
↔
X48
)
)
↔
(
(
¬
(
X48
→
X44
→
(
(
¬
X36
)
↔
X38
)
)
)
→
(
¬
(
X48
↔
(
(
¬
X36
)
↔
(
X25
→
X44
)
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
X46
↔
(
X32
→
X47
)
)
↔
(
(
¬
(
(
¬
(
X42
→
(
(
¬
X3
)
↔
X47
)
)
)
→
(
¬
(
X47
→
(
¬
X10
)
→
(
(
¬
X3
)
↔
X46
)
)
)
)
)
↔
(
(
¬
X32
)
→
X42
)
)
)
↔
(
(
(
(
(
(
¬
X42
)
→
(
(
¬
X40
)
↔
X46
)
)
↔
(
(
X42
↔
(
X40
→
X19
→
(
X2
↔
False
)
)
)
→
X46
)
)
↔
(
¬
(
X42
↔
(
X19
→
(
X2
↔
X41
)
)
)
)
)
↔
(
¬
(
(
¬
X19
)
↔
(
X2
→
X46
)
)
)
)
→
(
(
¬
(
X40
↔
(
X6
→
X47
)
)
)
↔
(
(
(
¬
X40
)
↔
(
(
¬
X6
)
↔
(
(
¬
X2
)
→
X46
)
)
)
↔
(
¬
(
X47
↔
X2
)
)
)
)
)
)
↔
(
(
(
(
X41
↔
(
(
¬
X7
)
→
X48
)
)
↔
(
(
¬
(
X31
↔
X47
)
)
↔
(
¬
(
X48
→
(
X31
↔
(
(
¬
X7
)
↔
X41
)
)
)
)
)
)
↔
(
(
¬
X46
)
→
(
X30
↔
(
X7
→
X48
)
)
)
)
↔
(
(
X48
→
X46
→
(
(
¬
X13
)
↔
X47
)
)
↔
(
¬
(
X47
↔
(
(
¬
X46
)
↔
(
X27
↔
X48
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
(
¬
X42
)
↔
(
X38
→
(
X21
↔
X47
)
)
)
↔
(
¬
(
(
¬
X5
)
↔
X29
)
)
)
→
(
(
(
¬
(
X46
↔
(
X42
↔
(
X36
↔
(
X35
↔
False
)
)
)
)
)
↔
(
¬
(
X42
→
X38
→
X36
→
X35
)
)
)
↔
(
X38
→
(
X4
↔
X36
)
)
)
→
(
¬
(
X46
↔
(
X42
→
X38
→
X21
→
False
)
)
)
)
)
→
(
¬
(
(
(
¬
X36
)
→
(
(
¬
X25
)
↔
(
(
¬
X23
)
→
X47
)
)
)
→
(
¬
(
(
(
X47
↔
(
X37
→
(
X36
↔
(
X25
→
False
)
)
)
)
↔
(
(
¬
(
X47
→
X37
→
(
X36
↔
(
X24
↔
False
)
)
)
)
↔
(
X36
→
(
X24
↔
X46
)
)
)
)
→
(
¬
(
X37
→
(
(
¬
X36
)
↔
(
(
¬
X18
)
→
X46
)
)
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
(
(
(
¬
(
X50
→
(
X44
↔
(
X16
→
X17
)
)
)
)
↔
(
¬
(
X20
↔
(
(
¬
X16
)
→
X17
)
)
)
)
→
(
¬
X20
)
→
(
¬
X16
)
→
X50
)
)
↔
(
(
¬
(
X50
→
X16
→
(
X9
↔
X46
)
)
)
→
(
X50
↔
(
X20
↔
(
(
¬
X9
)
↔
X16
)
)
)
)
)
→
(
(
(
¬
(
X50
→
X48
→
(
X39
↔
X44
)
)
)
↔
(
¬
(
(
(
¬
X12
)
→
X11
→
(
X1
↔
X44
)
)
→
(
¬
(
(
¬
X44
)
→
(
(
¬
X12
)
↔
(
X11
↔
X50
)
)
)
)
)
)
)
↔
(
(
¬
X28
)
↔
(
(
¬
X15
)
↔
(
X11
↔
X46
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
X1
)
→
X9
)
↔
(
(
¬
(
X49
→
(
(
¬
X15
)
↔
(
(
¬
X9
)
→
X45
)
)
)
)
→
(
(
¬
X1
)
↔
X15
)
)
)
↔
(
¬
(
X47
→
X45
→
X15
→
X1
)
)
)
→
(
¬
(
X15
→
(
(
¬
X9
)
↔
X45
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
(
X50
↔
(
X49
→
False
)
)
)
→
(
(
¬
X43
)
↔
(
X34
↔
(
X33
↔
X49
)
)
)
)
↔
(
X50
↔
(
(
¬
X22
)
→
X49
)
)
)
→
(
(
¬
(
(
¬
X48
)
→
(
X33
↔
X49
)
)
)
↔
(
¬
(
X50
↔
(
(
¬
X48
)
↔
(
(
¬
X8
)
→
X49
)
)
)
)
)
)
→
(
¬
(
(
¬
X26
)
→
X0
→
X50
)
)
→
(
(
¬
(
X49
→
(
X34
↔
(
(
¬
X0
)
→
X14
)
)
)
)
↔
(
¬
(
X50
→
X14
→
(
X0
↔
X26
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
952c0c...
and proposition id is
562a50...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMdg5jupiKbnKJjEwR5QHkJHyHVe1XAKwEj
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
¬
(
(
(
(
(
¬
(
(
(
X40
→
X34
→
X33
→
X43
)
→
X40
→
(
X39
↔
(
X34
↔
False
)
)
)
→
(
(
(
¬
X16
)
→
X8
→
X1
→
X43
)
↔
(
(
X40
↔
(
(
¬
X39
)
→
X1
→
X43
)
)
→
(
X40
↔
(
(
¬
X8
)
↔
(
(
¬
X1
)
→
X43
)
)
)
)
)
)
)
↔
(
¬
(
(
(
X11
→
False
)
↔
(
X49
→
(
(
¬
X40
)
↔
(
(
¬
X30
)
→
X43
)
)
)
)
→
(
¬
X39
)
→
X11
→
X43
)
)
)
↔
(
(
¬
(
X39
→
(
X17
↔
X40
)
)
)
↔
(
X43
↔
(
X40
→
X17
→
X0
→
False
)
)
)
)
→
(
¬
(
(
(
(
¬
(
(
(
¬
X38
)
↔
(
(
¬
X31
)
↔
X49
)
)
→
(
X49
↔
(
(
¬
X38
)
→
(
X15
↔
X47
)
)
)
)
)
→
(
¬
(
X49
↔
(
X38
→
(
X31
↔
False
)
)
)
)
)
↔
(
(
¬
X31
)
→
(
¬
X15
)
→
X49
)
)
→
(
¬
(
(
¬
(
(
¬
(
X24
→
X10
)
)
→
(
¬
(
X47
↔
(
(
¬
X15
)
↔
(
X10
↔
X48
)
)
)
)
)
)
→
(
(
¬
(
(
¬
X24
)
↔
(
(
¬
X15
)
↔
(
(
¬
X10
)
→
X38
)
)
)
)
↔
(
X38
↔
(
X10
↔
(
(
¬
X2
)
→
X15
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X43
↔
(
X39
→
(
¬
X21
)
→
X46
)
)
)
↔
(
(
(
X43
→
X39
→
X21
→
X48
)
↔
(
X49
↔
(
X43
↔
X48
)
)
)
→
X49
→
(
(
¬
X43
)
↔
(
X9
↔
X48
)
)
)
)
→
(
(
¬
(
(
X47
↔
(
X39
→
(
¬
X36
)
→
X48
)
)
→
(
(
¬
(
(
¬
X43
)
↔
(
(
¬
X36
)
↔
X47
)
)
)
↔
(
X47
↔
(
X39
↔
(
X36
↔
(
X28
→
False
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
X39
)
↔
(
X32
→
X25
→
X47
)
)
)
→
(
¬
(
X48
→
(
¬
X32
)
→
(
X25
↔
X47
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
(
¬
(
(
¬
X44
)
→
(
(
¬
X42
)
↔
X45
)
)
)
↔
(
(
¬
(
X44
↔
(
X29
→
X22
→
X42
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X29
)
→
X45
)
)
→
(
¬
(
X29
↔
(
(
¬
X22
)
↔
X45
)
)
)
→
(
¬
(
X45
→
(
X42
↔
(
X22
→
(
X14
↔
False
)
)
)
)
)
→
(
¬
(
X42
↔
(
X29
↔
(
(
¬
X22
)
↔
X45
)
)
)
)
)
)
)
)
↔
(
(
¬
X29
)
→
(
(
¬
X22
)
↔
(
X5
↔
X45
)
)
)
)
↔
(
(
¬
(
X37
↔
(
(
¬
X29
)
↔
(
(
¬
X12
)
→
X44
)
)
)
)
↔
(
(
¬
(
(
¬
X29
)
→
(
¬
X20
)
→
(
(
¬
X19
)
↔
X45
)
)
)
↔
(
(
(
¬
X29
)
→
(
X20
↔
(
X19
↔
X44
)
)
)
↔
(
X37
↔
X19
)
)
)
)
)
↔
(
(
(
¬
(
X44
↔
(
X4
→
X3
→
False
)
)
)
↔
(
¬
(
(
¬
X29
)
↔
(
X4
→
X37
)
)
)
)
↔
(
(
X29
→
X6
→
X27
)
↔
(
¬
(
(
¬
(
X6
→
(
¬
X4
)
→
X27
)
)
→
(
¬
(
(
¬
(
X27
→
(
(
¬
X4
)
↔
(
X3
→
X29
)
)
)
)
→
(
¬
(
X48
→
(
X27
↔
(
X4
→
X29
)
)
)
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
X48
↔
(
X7
↔
False
)
)
)
↔
(
(
¬
X45
)
→
X41
→
X7
→
X46
)
)
↔
(
(
X47
→
(
X46
↔
X45
)
)
→
(
(
X45
→
X41
→
(
X26
↔
X46
)
)
↔
(
(
¬
(
(
¬
X46
)
→
(
(
¬
X35
)
↔
X47
)
)
)
↔
(
(
X47
↔
X26
)
↔
(
¬
(
X45
→
X35
→
X13
)
)
)
)
)
)
)
→
(
(
¬
X42
)
↔
(
(
¬
X23
)
↔
X46
)
)
→
(
(
¬
(
X48
→
X46
→
(
¬
X23
)
→
X47
)
)
↔
(
X46
→
(
¬
X18
)
→
X48
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
¬
(
(
X29
→
(
(
¬
X26
)
↔
X44
)
)
→
(
¬
(
(
¬
(
X44
→
(
X26
↔
X29
)
)
)
→
(
¬
(
(
¬
X29
)
→
(
X25
↔
X49
)
)
)
)
)
)
)
→
(
X49
↔
(
X26
↔
(
X8
↔
X29
)
)
)
)
)
↔
(
(
(
¬
(
X25
↔
(
X3
↔
X49
)
)
)
↔
(
(
¬
X26
)
→
(
¬
X23
)
→
(
(
¬
X19
)
↔
X49
)
)
)
↔
(
¬
(
X25
→
(
¬
X23
)
→
X49
)
)
)
)
↔
(
(
(
¬
(
X36
→
X33
→
(
¬
X26
)
→
X41
)
)
↔
(
(
(
(
X43
↔
(
X25
↔
(
X10
↔
X41
)
)
)
↔
(
(
X43
→
X41
→
(
X36
↔
(
X10
↔
False
)
)
)
↔
(
X43
→
X41
→
(
X19
↔
X36
)
)
)
)
→
(
¬
(
X25
→
X10
→
False
)
)
)
↔
(
(
(
¬
X41
)
→
(
(
¬
X36
)
↔
(
(
¬
X33
)
→
X43
)
)
)
→
(
¬
(
X43
↔
(
X41
→
(
X33
↔
X20
)
)
)
)
)
)
)
↔
(
(
¬
(
X42
→
X41
→
(
¬
X5
)
→
X26
)
)
↔
(
X43
↔
(
(
¬
X5
)
→
X26
)
)
)
)
)
↔
(
(
(
(
¬
(
X46
↔
(
X42
↔
(
X24
↔
(
X14
→
False
)
)
)
)
)
↔
(
(
(
X47
↔
(
(
¬
X42
)
↔
X43
)
)
→
(
¬
(
X46
→
X43
→
(
(
¬
X14
)
↔
X47
)
)
)
)
↔
(
X47
→
(
X14
↔
False
)
)
)
)
→
(
X46
↔
(
X43
↔
(
X42
→
X45
)
)
)
)
→
(
¬
(
(
(
(
(
¬
X44
)
↔
(
X37
↔
X46
)
)
↔
(
X43
→
X32
→
X46
)
)
↔
(
X44
↔
(
X43
↔
(
X37
↔
(
X35
→
False
)
)
)
)
)
→
(
(
¬
(
(
¬
X44
)
→
(
¬
X43
)
→
X35
→
X45
)
)
↔
(
¬
(
X45
→
(
X43
↔
(
X9
→
X35
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
(
X48
→
(
(
¬
X6
)
↔
(
(
¬
X0
)
↔
X27
)
)
)
)
→
(
¬
(
(
¬
X2
)
↔
X6
)
)
)
)
↔
(
(
¬
(
X38
↔
(
(
¬
X27
)
→
(
¬
X2
)
→
X40
)
)
)
→
(
(
(
¬
(
X40
↔
(
X38
↔
X4
)
)
)
↔
(
X27
→
X4
→
(
X0
↔
False
)
)
)
↔
(
X40
→
(
X38
↔
(
X27
↔
False
)
)
)
)
)
)
→
(
(
X48
→
X40
→
X47
)
↔
(
(
¬
(
X40
→
X27
→
X12
→
X47
)
)
↔
(
¬
(
X48
→
X27
)
)
)
)
→
(
¬
(
(
¬
(
X48
↔
(
X47
→
X27
→
X17
→
False
)
)
)
→
(
¬
(
X48
↔
(
X47
→
(
X17
↔
X15
)
)
)
)
)
)
)
↔
(
(
¬
(
(
X7
→
(
(
¬
X1
)
↔
X40
)
)
→
(
¬
(
(
(
¬
(
(
¬
(
X7
→
(
X1
↔
False
)
)
)
→
X49
→
(
X22
↔
(
(
¬
X7
)
↔
X38
)
)
)
)
↔
(
¬
(
(
¬
X38
)
→
X22
→
X49
)
)
)
→
(
¬
(
X49
↔
(
X39
↔
(
X1
→
X22
)
)
)
)
)
)
)
)
→
(
(
¬
(
X45
↔
(
X34
→
X11
)
)
)
→
(
(
(
X49
→
(
X38
↔
X18
)
)
↔
(
(
X11
→
X39
)
↔
(
¬
(
(
¬
X39
)
→
(
(
¬
X38
)
↔
(
(
¬
X34
)
→
X49
)
)
)
)
)
)
↔
(
X49
→
X38
→
X45
)
)
)
→
(
(
(
¬
X17
)
→
X13
→
X49
)
↔
(
(
X34
→
X28
→
(
(
¬
X17
)
↔
X30
)
)
↔
(
¬
(
(
¬
X34
)
↔
(
(
¬
X30
)
→
X28
→
X49
)
)
)
)
)
→
(
(
¬
(
X49
↔
(
X31
→
(
¬
X21
)
→
X34
)
)
)
↔
(
(
(
¬
X45
)
↔
(
(
¬
X28
)
↔
(
X16
↔
X49
)
)
)
↔
(
X49
→
(
X45
↔
(
X21
↔
False
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
afce0e...
and proposition id is
606c54...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMLDcqoqsoRRWLASs3qhma3H1ZTG9HUUXNr
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
¬
∀X51 :
prop
,
(
(
¬
(
(
¬
(
(
(
¬
(
(
(
¬
(
X15
↔
X43
)
)
→
(
¬
(
X41
→
(
X40
↔
(
X15
→
X49
)
)
)
)
)
→
(
(
(
¬
(
X49
→
(
(
¬
X23
)
↔
(
X12
↔
X43
)
)
)
)
↔
(
¬
(
X47
↔
(
X43
↔
(
(
¬
X23
)
↔
X49
)
)
)
)
)
↔
(
¬
(
X49
↔
(
X23
→
(
¬
X12
)
→
X50
)
)
)
)
→
(
¬
(
X47
↔
(
X41
↔
(
X12
↔
X25
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X42
→
(
(
¬
X41
)
↔
(
(
¬
X7
)
→
X51
)
)
)
)
→
(
¬
(
X47
↔
(
X42
→
X41
→
X51
)
)
)
)
)
→
(
(
X50
↔
(
X47
↔
(
X44
↔
False
)
)
)
↔
(
(
¬
(
(
¬
(
X47
→
(
(
¬
X44
)
↔
(
X41
↔
X45
)
)
)
)
→
(
¬
(
(
¬
X47
)
↔
(
(
¬
X44
)
↔
X50
)
)
)
)
)
↔
(
¬
(
(
¬
X45
)
↔
(
X44
→
X47
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
X45
)
→
X14
→
(
(
¬
X11
)
↔
X47
)
)
)
→
(
(
¬
(
X14
→
(
¬
X11
)
→
(
¬
X5
)
→
X47
)
)
↔
(
(
X11
→
X14
)
↔
(
¬
(
(
¬
X47
)
↔
(
(
¬
X14
)
↔
(
X9
→
X51
)
)
)
)
)
)
)
→
(
¬
(
(
X51
→
(
X47
↔
False
)
)
→
(
(
(
¬
X37
)
↔
(
X32
→
(
¬
X16
)
→
X47
)
)
↔
(
X47
→
(
X32
↔
(
(
¬
X27
)
→
X37
)
)
)
)
)
)
→
(
¬
(
X47
→
(
X45
↔
(
X37
↔
X51
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X27
)
↔
(
X17
→
(
X13
↔
X32
)
)
)
)
↔
(
(
¬
(
X46
↔
(
X27
↔
X18
)
)
)
↔
(
¬
(
X32
↔
(
X18
→
(
X17
↔
X46
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
X27
)
→
(
¬
X17
)
→
(
X0
↔
X32
)
)
↔
(
¬
(
X32
→
(
(
¬
X27
)
↔
(
X17
→
X45
)
)
)
)
)
→
(
X28
↔
(
X27
→
X0
→
X32
)
)
)
)
)
)
)
)
)
→
(
(
(
(
X48
→
(
¬
X46
)
→
(
(
¬
X35
)
↔
X47
)
)
↔
(
¬
(
X46
→
(
X35
↔
(
X19
↔
False
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X46
↔
(
X44
↔
(
X3
→
X31
)
)
)
)
→
(
(
¬
(
X48
↔
(
X46
↔
False
)
)
)
↔
(
X48
→
X38
→
(
¬
X3
)
→
X47
)
)
)
)
↔
(
X47
→
X44
→
X22
→
X48
)
)
)
↔
(
(
(
(
¬
(
X38
↔
(
(
¬
X34
)
↔
X48
)
)
)
↔
(
(
¬
X34
)
→
(
¬
X33
)
→
X46
)
)
→
(
¬
(
X38
↔
(
X34
↔
X46
)
)
)
)
↔
(
¬
(
(
¬
(
(
X33
↔
(
(
¬
X24
)
→
X29
)
)
→
(
¬
(
X34
↔
(
X24
↔
(
X8
→
X46
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X46
→
(
(
¬
X29
)
↔
(
X2
→
X33
)
)
)
)
→
X46
→
X21
→
X2
→
X33
)
)
)
)
)
)
→
(
¬
(
(
X47
↔
(
X30
→
False
)
)
→
(
(
¬
X47
)
→
(
X46
↔
(
(
¬
X4
)
↔
X48
)
)
)
→
(
¬
(
(
¬
(
X39
↔
(
(
¬
X4
)
→
X1
→
X48
)
)
)
→
(
X48
↔
(
X20
↔
(
X4
↔
X47
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
X26
→
(
(
¬
X10
)
↔
X47
)
)
↔
(
X10
→
X6
→
X36
)
)
→
(
¬
(
X46
→
(
X36
↔
(
X26
→
X47
)
)
)
)
)
)
↔
(
¬
(
X47
→
X46
→
X36
→
X39
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
(
¬
(
(
¬
(
X49
→
X48
→
X33
)
)
→
(
¬
(
(
¬
X8
)
→
X25
)
)
→
(
¬
(
(
¬
(
X49
→
(
(
¬
X33
)
↔
(
(
¬
X17
)
↔
X48
)
)
)
)
→
X49
→
(
¬
X33
)
→
(
X25
↔
X48
)
)
)
)
)
→
X49
→
(
¬
X35
)
→
(
X33
↔
X48
)
)
→
(
¬
(
(
(
¬
(
X50
→
(
¬
X33
)
→
(
(
¬
X22
)
↔
X49
)
)
)
↔
(
¬
(
X49
→
X33
→
X35
)
)
)
→
X22
→
False
)
)
)
→
(
(
¬
(
(
¬
(
X45
↔
(
X8
↔
(
X6
↔
X48
)
)
)
)
→
(
¬
X48
)
→
X45
→
(
¬
X8
)
→
X49
)
)
↔
(
(
¬
X6
)
↔
X45
)
)
)
→
(
(
(
X49
→
(
X43
↔
X45
)
)
↔
(
(
¬
(
(
¬
X45
)
↔
(
(
¬
X24
)
↔
X49
)
)
)
↔
(
(
X49
↔
(
X43
↔
(
X32
→
X24
)
)
)
↔
(
¬
(
X43
↔
(
(
¬
X32
)
→
(
X24
↔
X36
)
)
)
)
)
)
)
↔
(
(
X45
→
(
(
¬
X32
)
↔
X48
)
)
→
(
¬
(
X45
↔
(
X36
↔
(
(
¬
X14
)
↔
X32
)
)
)
)
)
)
→
(
(
¬
(
(
¬
X43
)
↔
(
X40
↔
(
(
¬
X36
)
↔
X45
)
)
)
)
↔
(
¬
(
(
(
¬
X43
)
↔
(
X5
→
X45
)
)
→
(
X40
↔
(
(
¬
X5
)
→
X29
)
)
)
)
)
→
(
X45
↔
(
X29
↔
(
(
¬
X11
)
→
X40
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X39
→
X37
→
(
¬
X27
)
→
X48
)
)
→
(
¬
(
X48
↔
(
X45
↔
(
X27
↔
False
)
)
)
)
)
→
(
¬
(
X48
↔
(
X37
↔
(
(
¬
X18
)
→
X39
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
X39
)
→
(
(
¬
X30
)
↔
X49
)
)
→
X48
→
(
(
¬
X39
)
↔
(
X4
→
X49
)
)
)
→
(
X48
↔
(
(
¬
X39
)
→
(
X30
↔
X45
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
¬
(
(
¬
X44
)
→
(
¬
X10
)
→
X50
)
)
↔
(
(
¬
X26
)
→
(
(
¬
X21
)
↔
(
X10
→
X44
)
)
)
)
→
X50
→
X44
→
(
X9
↔
X10
)
)
→
(
¬
(
(
(
¬
(
X26
↔
(
(
¬
X21
)
↔
(
X16
↔
X49
)
)
)
)
→
(
(
¬
X26
)
↔
(
(
¬
X16
)
↔
(
(
¬
X15
)
↔
X50
)
)
)
)
→
X21
→
(
X16
↔
(
X15
→
X7
→
False
)
)
)
)
)
)
↔
(
¬
(
(
X48
→
X28
→
X16
→
X50
)
→
(
(
(
(
¬
(
X48
→
(
X38
↔
(
(
¬
X28
)
→
X51
)
)
)
)
→
(
¬
(
(
¬
(
X50
→
(
(
¬
X38
)
↔
(
X1
↔
X48
)
)
)
)
→
(
¬
(
X38
→
(
¬
X28
)
→
X48
)
)
)
)
)
↔
(
¬
(
(
¬
X38
)
↔
(
(
¬
X28
)
→
X48
)
)
)
)
↔
(
¬
(
X51
→
(
X38
↔
(
X26
↔
X50
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
X48
)
→
X42
→
X13
→
X51
)
↔
(
¬
(
X42
→
(
¬
X2
)
→
X51
)
)
)
↔
(
¬
(
X51
→
(
X48
↔
(
X3
→
X42
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X48
→
X41
→
X23
→
X12
)
)
→
(
(
(
¬
X47
)
→
X41
→
(
(
¬
X12
)
↔
X48
)
)
↔
(
X47
↔
(
X41
↔
(
X3
↔
X48
)
)
)
)
)
↔
(
¬
(
X47
→
(
¬
X13
)
→
(
¬
X12
)
→
X41
)
)
)
→
(
X49
↔
(
X41
→
(
X23
↔
X48
)
)
)
)
)
)
→
(
(
(
¬
(
X48
↔
(
X34
↔
(
X19
↔
X49
)
)
)
)
↔
(
(
¬
(
X48
→
(
X34
↔
X49
)
)
)
↔
(
X51
→
(
(
¬
X48
)
↔
(
X20
→
X49
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
X46
)
→
X31
→
(
(
¬
X19
)
↔
X48
)
)
→
X48
→
(
X19
↔
(
X0
↔
X46
)
)
)
)
→
(
X46
↔
(
X31
→
X20
→
(
X0
↔
False
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
88bae7...
and proposition id is
2ac069...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMKsGYhPpQNcKKXqjwiJSTtyS1Pa29z3Z8N
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
¬
(
(
(
(
¬
(
(
(
¬
(
X48
↔
(
X42
→
X32
→
X45
)
)
)
→
X48
→
(
¬
X44
)
→
(
¬
X42
)
→
X45
)
→
(
¬
(
X46
→
X45
→
(
(
¬
X44
)
↔
X48
)
)
)
)
)
→
(
¬
(
(
¬
(
X48
→
(
¬
X44
)
→
X30
→
X45
)
)
→
(
¬
(
(
¬
(
X30
→
(
(
¬
X24
)
↔
(
X21
→
X34
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X45
)
→
X34
→
X7
→
X48
)
)
→
(
¬
(
(
X48
↔
(
X45
↔
(
(
¬
X24
)
↔
X34
)
)
)
→
(
¬
(
X48
→
(
X45
↔
(
(
¬
X30
)
↔
X34
)
)
)
)
)
)
)
)
)
)
)
)
)
→
(
(
(
X48
↔
(
X44
→
False
)
)
→
(
¬
(
(
¬
(
(
¬
X44
)
↔
(
(
¬
X27
)
↔
X46
)
)
)
→
(
X35
↔
(
(
¬
X27
)
→
X17
→
X46
)
)
)
)
→
X46
→
(
X44
↔
(
X27
→
False
)
)
)
↔
(
¬
(
X48
→
(
X44
↔
(
X13
↔
X46
)
)
)
)
)
)
→
(
(
(
(
(
¬
X41
)
↔
(
X29
↔
(
X2
→
X46
)
)
)
↔
(
¬
(
(
¬
X29
)
→
(
(
¬
X23
)
↔
(
X2
↔
X48
)
)
)
)
)
↔
(
(
¬
(
X48
→
(
X29
↔
X0
)
)
)
↔
(
X41
↔
(
X33
↔
(
X29
→
False
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
¬
X14
)
→
X12
→
X44
)
)
→
(
X41
↔
(
(
¬
X14
)
↔
(
X12
→
X46
)
)
)
)
→
(
(
(
¬
X41
)
→
(
(
¬
X14
)
↔
(
X5
↔
X48
)
)
)
↔
(
¬
(
X44
→
(
(
¬
X14
)
↔
(
(
¬
X5
)
→
X41
)
)
)
)
)
)
)
↔
(
(
¬
(
X44
↔
(
X39
→
X43
)
)
)
↔
(
X48
↔
(
X44
↔
X46
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X38
)
→
(
¬
X25
)
→
X22
→
X40
)
)
↔
(
(
(
(
X47
↔
(
X38
→
(
X26
↔
False
)
)
)
↔
(
¬
(
(
¬
(
X47
↔
(
X40
→
(
X26
↔
False
)
)
)
)
→
(
X47
↔
(
X40
→
X26
→
False
)
)
→
(
X47
↔
(
X38
→
(
X9
↔
False
)
)
)
)
)
)
↔
(
X38
↔
(
X26
↔
X47
)
)
)
↔
(
(
¬
(
X40
↔
(
X26
→
(
(
¬
X8
)
↔
X38
)
)
)
)
↔
(
¬
(
X40
→
X25
→
(
X8
↔
False
)
)
)
)
)
)
→
(
(
(
(
X46
↔
(
X28
↔
X37
)
)
↔
(
(
(
¬
(
X38
↔
(
(
¬
X37
)
→
X26
→
X46
)
)
)
→
X38
→
(
¬
X37
)
→
X28
→
X46
)
→
(
¬
(
(
¬
X28
)
↔
(
(
¬
X26
)
↔
X46
)
)
)
)
)
↔
(
X37
↔
(
X26
↔
X38
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
(
¬
X37
)
↔
(
X20
↔
(
X15
→
X40
)
)
)
↔
(
¬
(
X45
→
X15
→
X40
)
)
)
↔
(
¬
(
X46
↔
(
X37
↔
(
X20
→
(
X15
↔
False
)
)
)
)
)
)
→
(
¬
(
X40
↔
(
X37
→
(
X15
↔
X4
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X40
)
↔
(
(
¬
X37
)
↔
(
(
¬
X1
)
↔
X46
)
)
)
)
→
(
X37
↔
(
(
¬
X28
)
↔
(
(
¬
X1
)
→
X40
)
)
)
)
)
)
)
)
)
→
(
(
(
X47
↔
(
(
¬
X31
)
↔
X38
)
)
→
(
(
¬
(
(
¬
(
(
¬
X36
)
→
(
(
¬
X19
)
↔
X48
)
)
)
→
(
(
¬
(
(
X48
→
X36
→
X18
→
X19
)
→
(
¬
(
X36
↔
(
X19
↔
(
(
¬
X18
)
↔
X38
)
)
)
)
)
)
↔
(
¬
(
X48
→
X47
→
X36
→
(
X19
↔
False
)
)
)
)
)
)
↔
(
¬
(
(
X48
→
X31
→
(
X11
↔
X47
)
)
→
(
¬
(
X47
↔
(
X31
↔
X38
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X47
→
X36
)
)
→
(
(
(
¬
(
(
(
X49
↔
(
X47
↔
X48
)
)
↔
(
¬
(
(
¬
X36
)
↔
(
(
¬
X16
)
↔
X48
)
)
)
)
→
(
X49
↔
(
X48
↔
(
X47
↔
False
)
)
)
)
)
↔
(
X47
↔
(
(
¬
X16
)
↔
X48
)
)
)
↔
(
(
¬
X36
)
→
(
X16
↔
X48
)
)
)
)
↔
(
X48
↔
(
X47
→
(
(
¬
X10
)
↔
X49
)
)
)
)
↔
(
(
¬
(
X45
↔
(
X10
↔
(
(
¬
X6
)
↔
X48
)
)
)
)
→
(
(
¬
(
X36
↔
(
(
¬
X3
)
↔
X48
)
)
)
↔
(
X10
↔
X45
)
)
)
)
↔
(
X48
→
(
X47
↔
X36
)
)
)
)
)
↔
(
(
(
(
(
(
(
X49
→
(
¬
X45
)
→
(
X31
↔
X46
)
)
↔
(
¬
(
X49
↔
(
(
¬
X45
)
↔
(
X31
↔
X48
)
)
)
)
)
↔
(
¬
(
(
(
X49
→
(
(
¬
X19
)
↔
X45
)
)
↔
(
(
¬
X45
)
↔
(
X10
↔
(
X2
↔
X49
)
)
)
)
→
(
(
(
¬
X48
)
↔
(
(
¬
X45
)
→
X49
)
)
↔
(
¬
(
(
X45
↔
(
X19
↔
X48
)
)
→
(
¬
(
X45
↔
(
X19
↔
(
(
¬
X13
)
↔
X48
)
)
)
)
→
(
¬
(
(
¬
X45
)
→
X19
→
(
(
¬
X13
)
↔
X49
)
)
)
)
)
)
→
(
¬
(
X48
↔
(
X45
→
X49
)
)
)
)
)
)
→
(
(
X28
↔
(
X23
→
False
)
)
↔
(
(
¬
(
X49
→
X46
→
(
X45
↔
False
)
)
)
→
(
(
X49
↔
(
X45
↔
(
X28
↔
X23
)
)
)
↔
(
X49
→
(
X46
↔
X23
)
)
)
)
)
)
↔
(
(
X45
↔
(
(
¬
X36
)
↔
X43
)
)
↔
(
(
(
¬
(
X45
→
(
X9
↔
X46
)
)
)
↔
(
(
X46
↔
(
(
¬
X9
)
→
X45
)
)
→
(
(
X49
↔
(
(
¬
X45
)
→
X46
)
)
↔
(
(
¬
(
X49
↔
(
X46
→
(
X45
↔
False
)
)
)
)
→
(
(
¬
(
X46
→
X45
→
X9
)
)
↔
(
(
¬
(
X49
↔
(
X36
↔
X9
)
)
)
↔
(
¬
(
(
(
¬
X46
)
↔
(
(
¬
X45
)
↔
(
X9
↔
X49
)
)
)
→
(
X49
↔
(
X45
→
X36
→
X46
)
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
X49
→
(
X46
↔
(
(
¬
X37
)
↔
X45
)
)
)
)
↔
(
(
X46
↔
(
(
¬
X45
)
↔
(
(
¬
X17
)
↔
X49
)
)
)
→
(
¬
X46
)
→
X45
→
(
X43
↔
X49
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X27
→
(
(
X27
→
(
¬
X21
)
→
X18
→
X43
)
↔
(
(
¬
X43
)
↔
(
X27
↔
(
(
¬
X18
)
↔
X49
)
)
)
)
)
)
→
(
¬
(
(
(
(
X49
↔
(
X48
→
X39
→
(
X29
↔
False
)
)
)
↔
(
(
¬
(
(
X39
↔
(
X24
→
X21
→
X48
)
)
→
(
¬
(
X42
→
(
¬
X21
)
→
X24
)
)
)
)
↔
(
¬
(
X49
↔
(
X48
→
(
(
¬
X21
)
↔
X24
)
)
)
)
)
)
↔
(
(
X48
↔
(
X24
↔
(
X22
→
(
X6
↔
False
)
)
)
)
↔
(
X48
→
(
X29
↔
(
X24
↔
False
)
)
)
)
)
→
(
¬
(
(
X48
↔
(
X21
↔
(
X1
↔
X42
)
)
)
→
(
(
¬
(
(
¬
X42
)
→
X21
→
X18
→
X48
)
)
↔
(
¬
(
X21
↔
(
X18
↔
(
X8
→
X48
)
)
)
)
)
)
)
→
X48
→
X42
→
X21
→
X3
→
False
)
)
)
)
)
↔
(
(
(
(
¬
(
X39
↔
(
(
¬
X32
)
→
(
(
¬
X16
)
↔
X44
)
)
)
)
↔
(
(
(
(
(
¬
(
(
¬
X25
)
↔
(
X0
→
X40
)
)
)
↔
(
(
X26
↔
(
X25
↔
(
X0
→
X32
)
)
)
→
(
X34
↔
(
X32
→
X25
→
X0
)
)
)
)
↔
(
X26
↔
(
X25
→
False
)
)
)
↔
(
¬
(
(
¬
X25
)
↔
(
X4
↔
(
X0
→
X40
)
)
)
)
)
↔
(
(
X44
↔
(
(
¬
X26
)
↔
(
X0
↔
X39
)
)
)
↔
(
¬
(
X44
→
(
¬
X39
)
→
(
X0
↔
X40
)
)
)
)
)
)
↔
(
(
(
(
¬
X40
)
↔
(
(
¬
X39
)
↔
(
X33
↔
X44
)
)
)
→
(
¬
(
X39
↔
(
(
¬
X33
)
→
(
¬
X30
)
→
X44
)
)
)
)
↔
(
(
X30
↔
(
X12
↔
X40
)
)
↔
(
¬
(
(
(
(
X44
↔
(
X7
↔
X40
)
)
↔
(
¬
(
X40
→
(
X30
↔
(
(
¬
X7
)
↔
X14
)
)
)
)
)
↔
(
(
¬
X30
)
↔
X40
)
)
→
(
¬
(
X30
→
X40
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
X46
)
↔
(
(
¬
X44
)
→
(
¬
X40
)
→
X47
)
)
↔
(
¬
(
X46
→
(
X40
↔
X41
)
)
)
)
→
(
(
(
X40
→
X11
→
(
(
¬
X5
)
↔
X38
)
)
↔
(
(
(
X40
↔
(
X39
→
(
X11
↔
X41
)
)
)
↔
(
X40
→
(
X20
↔
(
X11
→
False
)
)
)
)
↔
(
X41
↔
(
(
¬
X39
)
↔
X40
)
)
)
)
↔
(
¬
(
X41
↔
(
X20
↔
(
X15
→
X39
)
)
)
)
)
→
(
¬
(
X37
↔
(
(
¬
X35
)
→
(
¬
X20
)
→
X40
)
)
)
→
(
¬
(
X38
↔
(
X37
↔
(
X20
↔
X44
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
358384...
and proposition id is
05f3b4...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMN6FWFzvyeWjinnaHfvFgnPuZXtUtDZgVi
)
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
¬
(
(
¬
(
(
(
(
¬
(
X28
→
(
(
¬
X27
)
↔
(
(
¬
X17
)
→
X46
)
)
)
)
↔
(
(
¬
(
X27
↔
(
X6
→
X28
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X27
)
→
(
X2
↔
X28
)
)
)
)
)
)
↔
(
(
(
¬
(
X28
↔
(
(
¬
X27
)
→
X37
)
)
)
→
(
(
¬
(
X45
↔
(
X28
→
(
¬
X17
)
→
X27
)
)
)
↔
(
(
¬
(
X45
↔
(
X24
↔
X28
)
)
)
↔
(
¬
(
X45
↔
(
X37
→
(
X27
↔
X24
)
)
)
)
)
)
)
↔
(
(
¬
X28
)
→
(
(
¬
X27
)
↔
(
(
¬
X0
)
→
X45
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X40
→
(
X33
↔
X16
)
)
)
↔
(
¬
(
(
¬
(
X49
↔
(
X47
↔
(
(
¬
X21
)
→
X33
)
)
)
)
→
(
(
(
¬
X40
)
↔
(
(
¬
X33
)
→
X47
)
)
↔
(
(
(
¬
(
X40
→
X27
→
X49
)
)
↔
(
(
¬
X47
)
↔
(
X27
→
X16
→
X49
)
)
)
↔
(
(
¬
X33
)
→
(
¬
X27
)
→
(
X16
↔
X40
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X47
→
(
(
¬
X33
)
↔
(
X17
↔
X40
)
)
)
)
→
(
¬
(
X47
→
(
X28
↔
(
X17
↔
X19
)
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
X48
↔
(
X47
↔
(
(
¬
X26
)
↔
X49
)
)
)
→
(
(
(
¬
(
(
¬
(
X47
↔
(
(
¬
X46
)
→
(
(
¬
X11
)
↔
X49
)
)
)
)
→
(
¬
(
X47
↔
(
X26
→
X49
)
)
)
)
)
↔
(
¬
(
X49
→
(
¬
X11
)
→
X47
)
)
)
↔
(
X49
→
(
(
¬
X26
)
↔
X47
)
)
)
)
→
(
(
(
¬
(
(
¬
X48
)
↔
(
X43
↔
(
(
¬
X41
)
→
X49
)
)
)
)
↔
(
(
¬
X43
)
↔
(
X3
↔
X47
)
)
)
↔
(
X48
→
(
X4
↔
(
X3
↔
X47
)
)
)
)
)
)
↔
(
(
X46
↔
(
X31
→
(
X23
↔
X44
)
)
)
↔
(
(
X48
↔
(
(
¬
X31
)
↔
(
(
¬
X30
)
↔
X44
)
)
)
↔
(
(
¬
X48
)
→
(
X44
↔
(
X31
↔
X49
)
)
)
)
)
)
)
)
→
(
(
(
(
(
(
¬
(
X29
→
(
X15
↔
(
X9
↔
False
)
)
)
)
↔
(
¬
(
(
¬
(
X49
→
X13
→
(
(
¬
X9
)
↔
X29
)
)
)
→
(
¬
(
X15
→
X13
→
X29
)
)
)
)
)
↔
(
¬
(
X15
↔
(
X13
↔
X9
)
)
)
)
↔
(
(
¬
(
(
¬
X42
)
↔
(
X39
↔
(
(
¬
X38
)
↔
X49
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X39
)
→
(
¬
X38
)
→
(
(
¬
X13
)
↔
X49
)
)
)
→
(
¬
(
X49
→
X39
→
(
X29
↔
(
X13
→
False
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
X42
↔
(
(
¬
X15
)
→
X1
→
X18
)
)
↔
(
¬
(
X42
↔
(
(
¬
X15
)
→
X47
)
)
)
)
↔
(
¬
(
(
¬
X42
)
→
(
¬
X18
)
→
X15
→
X48
)
)
)
→
(
¬
(
X47
→
X42
→
(
¬
X38
)
→
X48
)
)
)
)
→
(
¬
(
(
¬
X38
)
→
(
X25
↔
(
(
¬
X18
)
↔
X47
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
¬
(
(
X47
↔
(
X32
→
(
X20
↔
False
)
)
)
→
(
¬
(
(
¬
(
X49
→
(
X5
↔
X20
)
)
)
→
(
¬
(
(
¬
X47
)
→
X32
→
(
¬
X5
)
→
X49
)
)
)
)
)
)
→
(
(
¬
X5
)
↔
X32
)
)
→
(
¬
(
X32
→
(
¬
X20
)
→
X49
)
)
→
(
¬
(
X49
→
(
¬
X14
)
→
(
¬
X12
)
→
X20
)
)
)
)
↔
(
(
¬
(
X47
↔
(
X35
↔
(
(
¬
X22
)
↔
X32
)
)
)
)
↔
(
(
¬
(
X36
→
(
¬
X35
)
→
(
¬
X14
)
→
X47
)
)
↔
(
¬
(
(
¬
X35
)
↔
(
X32
↔
X47
)
)
)
)
)
)
→
(
(
X35
→
(
(
¬
X34
)
↔
(
X10
↔
X48
)
)
)
↔
(
(
X49
→
(
¬
X47
)
→
X35
→
X48
)
→
(
(
X35
→
X34
→
X8
→
X49
)
↔
(
(
¬
(
(
(
¬
(
X49
↔
(
X35
↔
(
X7
→
False
)
)
)
)
↔
(
X48
↔
(
(
¬
X34
)
→
X35
)
)
)
→
X48
→
(
(
¬
X7
)
↔
X49
)
)
)
↔
(
¬
(
X49
↔
(
X34
↔
X35
)
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
(
X22
↔
X49
)
→
(
¬
(
(
(
¬
X43
)
↔
(
X22
→
X47
)
)
→
(
¬
(
(
¬
X43
)
→
(
(
¬
X18
)
↔
(
(
¬
X16
)
→
X47
)
)
)
)
)
)
)
)
↔
(
(
X43
↔
(
(
¬
X29
)
→
(
X21
↔
X49
)
)
)
↔
(
(
¬
(
X29
↔
(
(
¬
X5
)
↔
X43
)
)
)
↔
(
¬
(
X47
↔
(
X21
↔
False
)
)
)
)
)
)
↔
(
(
X49
→
(
¬
X9
)
→
(
(
¬
X4
)
↔
X43
)
)
→
(
¬
(
X49
→
X29
→
X43
)
)
)
)
↔
(
(
(
(
(
¬
(
X41
→
(
¬
X27
)
→
(
(
¬
X14
)
↔
X42
)
)
)
↔
(
X41
↔
(
(
¬
X27
)
→
(
(
¬
X14
)
↔
X35
)
)
)
)
→
X35
→
(
X27
↔
(
X25
↔
X42
)
)
)
↔
(
(
X35
→
(
X34
↔
X25
)
)
↔
(
X43
↔
(
X35
→
(
¬
X34
)
→
X41
)
)
)
)
↔
(
(
¬
(
(
(
X41
→
X23
→
X3
→
X11
)
↔
(
¬
(
(
¬
X6
)
→
X43
)
)
)
→
(
(
(
¬
X43
)
↔
(
(
¬
X41
)
↔
X49
)
)
↔
(
(
(
X49
→
(
X47
↔
(
X43
→
X41
)
)
)
↔
(
(
X49
↔
(
(
¬
X41
)
→
(
X26
↔
X43
)
)
)
↔
X34
)
)
→
(
¬
(
(
¬
(
X43
→
(
(
¬
X23
)
↔
(
(
¬
X1
)
↔
X41
)
)
)
)
→
(
¬
(
(
¬
X23
)
↔
(
X1
→
X49
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
X33
)
↔
(
X20
↔
X47
)
)
)
↔
(
X47
↔
(
X43
↔
(
X20
↔
X49
)
)
)
)
)
)
)
→
(
(
(
(
(
(
(
¬
(
X38
→
(
X36
↔
(
(
¬
X8
)
→
X45
)
)
)
)
→
(
¬
(
X45
→
(
(
¬
X0
)
↔
X8
)
)
)
)
↔
(
(
¬
(
X38
→
X36
→
(
¬
X2
)
→
X45
)
)
↔
(
X38
↔
(
X36
→
X31
→
X45
)
)
)
)
↔
(
(
(
X48
↔
(
X45
↔
(
X38
→
(
X7
↔
False
)
)
)
)
→
(
¬
(
X48
↔
(
(
¬
X36
)
→
X38
)
)
)
→
(
(
X48
→
(
X38
↔
(
X28
↔
X45
)
)
)
↔
(
X45
→
X38
→
(
X36
↔
False
)
)
)
)
↔
(
(
¬
X28
)
→
X45
)
)
)
↔
(
(
X39
→
(
(
¬
X30
)
↔
(
(
¬
X8
)
↔
X48
)
)
)
↔
(
(
(
¬
(
X39
→
(
(
¬
X38
)
↔
(
X8
→
X48
)
)
)
)
↔
(
¬
(
X48
↔
(
(
¬
X39
)
→
(
¬
X38
)
→
X44
)
)
)
)
→
(
X48
↔
(
(
¬
X38
)
↔
X44
)
)
)
)
)
→
(
(
(
(
X49
→
X37
→
(
X13
↔
X45
)
)
↔
(
X49
↔
(
(
¬
X13
)
→
X48
)
)
)
↔
(
(
X48
↔
X39
)
↔
(
(
X49
→
(
X40
↔
(
(
¬
X39
)
↔
X45
)
)
)
↔
(
(
¬
X48
)
↔
(
X45
→
(
¬
X39
)
→
X49
)
)
)
)
)
↔
(
¬
(
X39
→
(
X37
↔
X48
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X32
)
↔
(
(
¬
X24
)
↔
(
X17
↔
X48
)
)
)
)
↔
(
(
X48
→
(
¬
X32
)
→
X24
→
X43
)
↔
(
(
¬
(
X24
→
X10
→
X32
)
)
↔
(
X43
→
(
X24
↔
(
X19
→
X32
)
)
)
)
)
)
→
(
¬
(
(
(
X46
→
(
¬
X40
)
→
X19
→
X47
)
↔
(
¬
(
(
X47
↔
(
(
¬
X43
)
↔
(
X17
↔
X44
)
)
)
→
(
X44
→
(
X40
↔
X43
)
)
→
(
X46
↔
(
X44
↔
(
X17
↔
X40
)
)
)
)
)
)
→
(
X19
→
(
(
¬
X12
)
↔
X47
)
)
→
X46
→
X19
→
X15
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
b2bb5c...
and proposition id is
b55e33...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMNq4xWxj6AVQaBHvF2CBcmEsHbnYKZDvsd
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
(
¬
(
(
¬
(
X33
↔
(
X22
↔
(
X3
↔
False
)
)
)
)
→
(
¬
(
X42
→
(
X22
↔
(
X11
↔
False
)
)
)
)
)
)
→
(
¬
(
(
X42
→
X33
→
(
(
¬
X22
)
↔
X41
)
)
→
(
¬
(
(
¬
X22
)
↔
X49
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X42
→
X41
→
(
(
¬
X12
)
↔
X43
)
)
)
↔
(
X43
↔
(
X37
→
False
)
)
)
→
(
(
(
¬
(
X40
→
X38
→
(
X31
↔
False
)
)
)
↔
(
X42
↔
(
X38
→
X27
→
X40
)
)
)
↔
(
(
(
¬
X42
)
→
(
(
¬
X41
)
↔
(
X24
↔
X43
)
)
)
→
(
(
(
¬
(
X49
→
(
X41
↔
(
X40
↔
X38
)
)
)
)
↔
(
¬
(
X43
↔
(
X41
→
False
)
)
)
)
↔
(
¬
(
X40
→
(
X38
↔
X43
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X43
)
↔
(
X42
→
X15
→
X48
)
)
)
↔
(
¬
(
(
(
(
X45
→
(
X42
↔
X21
)
)
→
X21
→
(
X14
↔
(
X7
↔
False
)
)
)
↔
(
¬
(
X47
→
X42
→
(
¬
X14
)
→
X21
)
)
)
→
(
(
X45
→
(
X7
↔
X21
)
)
↔
(
¬
(
(
¬
(
X48
→
(
X45
↔
(
X21
↔
False
)
)
)
)
→
(
¬
(
X48
↔
(
X21
↔
(
X7
↔
X4
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X43
)
↔
(
(
¬
X42
)
→
(
¬
X23
)
→
X49
)
)
)
→
(
(
¬
(
X39
↔
X49
)
)
↔
(
(
(
¬
X43
)
↔
(
(
¬
X39
)
↔
X49
)
)
→
(
¬
(
(
¬
(
X43
→
X21
)
)
→
X49
→
(
X39
↔
(
X23
↔
(
X21
→
False
)
)
)
)
)
)
)
→
(
¬
(
X49
↔
(
X39
↔
(
X23
→
X9
)
)
)
)
)
→
(
¬
X43
)
→
(
¬
X23
)
→
(
X6
↔
X49
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
(
¬
(
(
¬
X47
)
↔
(
(
¬
X46
)
↔
(
X13
→
X49
)
)
)
)
↔
(
¬
(
(
X49
→
X45
→
(
X16
↔
X46
)
)
→
(
¬
(
X47
→
X45
)
)
)
)
)
→
(
X47
↔
(
X45
→
(
(
¬
X36
)
↔
X49
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X46
↔
(
X28
→
X19
→
False
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X45
)
↔
(
(
¬
X19
)
→
X49
)
)
)
)
)
→
(
¬
(
X45
→
X28
→
X8
→
X49
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X47
→
(
(
¬
X44
)
↔
X49
)
)
)
↔
(
(
¬
(
(
X47
→
(
X5
↔
(
(
¬
X1
)
↔
X44
)
)
)
→
(
¬
(
X5
↔
X49
)
)
)
)
↔
(
¬
(
X47
→
X46
→
(
(
¬
X5
)
↔
X49
)
)
)
)
)
→
(
X49
↔
(
X46
↔
(
(
¬
X0
)
→
X47
)
)
)
)
)
↔
(
¬
(
(
X17
↔
(
(
¬
X5
)
↔
X44
)
)
→
(
(
¬
X26
)
↔
(
(
¬
X17
)
→
(
X1
↔
X29
)
)
)
)
)
)
)
→
(
(
¬
(
X49
→
(
¬
X10
)
→
(
(
¬
X2
)
↔
X18
)
)
)
↔
(
¬
(
(
(
X35
→
X30
→
X18
)
↔
(
X47
→
(
(
¬
X30
)
↔
(
X25
→
X35
)
)
)
)
→
X25
→
(
(
¬
X18
)
↔
(
X10
↔
X35
)
)
)
)
)
→
(
¬
(
(
X49
→
X34
→
X20
→
X46
)
→
(
(
(
X20
↔
(
X2
↔
False
)
)
↔
(
(
¬
(
(
¬
X32
)
→
(
(
¬
X20
)
↔
(
(
¬
X2
)
↔
X49
)
)
)
)
→
(
(
¬
X46
)
↔
(
X20
↔
X49
)
)
)
)
↔
(
¬
(
X49
→
X46
→
False
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
X46
→
(
X35
↔
(
(
¬
X26
)
↔
X29
)
)
)
)
↔
(
¬
(
(
¬
(
X47
↔
(
(
¬
X29
)
→
X46
)
)
)
→
(
X35
↔
(
X29
→
(
(
¬
X27
)
↔
X47
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X49
→
(
X45
↔
(
X30
↔
False
)
)
)
)
→
(
(
X47
↔
(
(
¬
X27
)
↔
(
(
¬
X6
)
↔
X45
)
)
)
↔
(
¬
(
(
¬
X30
)
↔
(
(
¬
X29
)
→
X47
)
)
)
)
)
)
)
→
(
(
¬
(
X45
→
(
(
¬
X27
)
↔
X30
)
)
)
↔
(
(
¬
(
(
¬
X30
)
↔
(
(
¬
X27
)
→
X45
)
)
)
↔
(
(
X17
↔
X0
)
↔
(
(
¬
X30
)
→
(
X27
↔
(
(
¬
X17
)
↔
X45
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
X28
↔
(
(
¬
X11
)
→
X18
)
)
→
(
(
¬
(
X46
→
(
X36
↔
(
(
¬
X11
)
→
X18
)
)
)
)
↔
(
¬
(
X36
↔
(
X5
→
X46
)
)
)
)
)
→
(
(
¬
(
X18
↔
X29
)
)
↔
(
¬
(
(
X29
↔
(
(
¬
X5
)
→
X41
)
)
→
(
¬
(
X41
→
X30
→
X18
)
)
)
)
)
)
)
→
(
¬
(
(
X46
↔
(
X36
↔
(
X15
↔
False
)
)
)
→
X15
→
X13
→
(
(
¬
X4
)
↔
X36
)
)
)
→
(
(
¬
(
X14
↔
(
(
¬
X12
)
→
X30
)
)
)
↔
(
(
X30
↔
(
X13
↔
X12
)
)
↔
(
¬
(
X46
→
(
X15
↔
X13
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
X44
→
(
(
¬
X42
)
↔
X48
)
)
↔
(
(
(
(
¬
(
X49
↔
(
X42
→
X48
)
)
)
↔
(
(
¬
(
(
(
¬
X48
)
→
X49
)
→
(
¬
X42
)
→
(
(
¬
X25
)
↔
(
(
¬
X21
)
→
X49
)
)
)
)
↔
(
¬
(
X48
↔
X21
)
)
)
)
↔
(
¬
(
X42
→
(
¬
X38
)
→
(
(
¬
X25
)
↔
X48
)
)
)
)
↔
(
(
¬
X44
)
↔
(
(
¬
X38
)
→
X34
→
X48
)
)
)
)
→
(
(
(
¬
(
X45
→
(
¬
X39
)
→
X23
→
X44
)
)
→
X45
→
(
¬
X39
)
→
X38
→
X44
)
↔
(
(
X34
→
(
(
¬
X16
)
↔
X45
)
)
↔
(
X39
↔
(
(
¬
X38
)
→
X45
)
)
)
)
)
)
→
(
(
¬
(
(
(
X48
→
(
X31
↔
(
X2
→
X44
)
)
)
↔
(
X39
↔
(
X31
↔
(
(
¬
X2
)
↔
X48
)
)
)
)
→
(
¬
(
X48
↔
(
X39
→
X31
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X24
)
↔
(
X22
→
(
X9
↔
X31
)
)
)
)
↔
(
¬
(
(
¬
X24
)
↔
(
(
¬
X9
)
→
X48
)
)
)
)
→
X39
→
(
(
¬
X31
)
↔
(
X24
↔
X48
)
)
)
)
)
)
→
(
(
(
¬
(
(
(
¬
(
X45
↔
(
(
¬
X43
)
↔
(
(
¬
X20
)
→
X49
)
)
)
)
↔
(
(
(
¬
X45
)
→
X40
→
(
¬
X10
)
→
X49
)
↔
(
(
(
¬
X40
)
→
(
X10
↔
(
(
¬
X8
)
→
X49
)
)
)
↔
(
¬
(
X43
↔
(
X10
↔
X8
)
)
)
)
)
)
→
(
X49
↔
(
X47
→
X44
→
X43
)
)
→
(
¬
(
(
¬
X44
)
→
(
X8
↔
X49
)
)
)
)
)
→
(
(
(
X43
↔
(
X37
→
X33
→
X20
→
False
)
)
↔
(
(
¬
(
X43
↔
(
X33
→
(
X10
↔
X37
)
)
)
)
→
X33
→
(
X10
↔
(
X1
→
X43
)
)
)
)
↔
(
¬
(
X37
→
(
¬
X33
)
→
X43
)
)
)
)
↔
(
(
(
X7
↔
(
(
¬
X3
)
↔
X37
)
)
↔
(
(
¬
(
X37
↔
(
X32
↔
(
X19
→
(
X7
↔
False
)
)
)
)
)
↔
(
¬
(
X43
↔
(
X32
→
(
X7
↔
False
)
)
)
)
)
)
↔
(
X44
↔
(
(
¬
X19
)
→
X7
→
X32
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
3ff0a9...
and proposition id is
99f49d...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMWPMVXxFigSJoYJL8EsWzPVhEh7tYF13fd
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
(
¬
(
(
(
¬
(
(
(
¬
(
(
¬
(
X48
↔
(
X46
→
(
X45
↔
False
)
)
)
)
→
(
(
¬
(
X46
↔
(
X45
↔
(
(
¬
X41
)
→
X48
)
)
)
)
↔
(
¬
(
X41
↔
(
X38
↔
(
X17
↔
X46
)
)
)
)
)
)
)
→
(
X46
↔
(
(
¬
X45
)
→
(
(
¬
X17
)
↔
X50
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X42
→
X31
→
(
X11
↔
X50
)
)
)
→
(
¬
(
X42
→
X41
→
(
¬
X24
)
→
X31
)
)
)
)
→
(
X31
→
(
(
¬
X24
)
↔
(
X16
↔
X50
)
)
)
→
(
(
¬
X46
)
↔
(
X31
↔
(
X24
↔
X50
)
)
)
)
)
)
)
↔
(
(
(
X46
→
(
(
¬
X40
)
↔
(
X39
↔
X43
)
)
)
↔
(
(
¬
(
X50
↔
(
X46
→
(
(
¬
X39
)
↔
X43
)
)
)
)
→
(
¬
(
X50
↔
(
X43
↔
(
(
¬
X37
)
→
X46
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
(
(
¬
X45
)
↔
(
(
¬
X43
)
→
X39
→
X46
)
)
)
→
(
X45
↔
(
X43
→
X47
)
)
)
)
↔
(
(
¬
X46
)
→
(
¬
X39
)
→
X47
)
)
↔
(
¬
(
(
¬
(
X45
↔
(
(
¬
X9
)
↔
X47
)
)
)
→
(
(
X47
→
X45
→
(
X19
↔
X39
)
)
↔
(
¬
(
X47
→
(
¬
X39
)
→
X9
→
X45
)
)
)
)
)
)
↔
(
(
¬
(
X46
↔
(
X45
↔
X21
)
)
)
→
X47
→
(
(
¬
X33
)
↔
X45
)
)
)
)
)
→
(
(
(
(
(
X49
→
(
X29
↔
(
(
¬
X4
)
→
X48
)
)
)
↔
(
¬
(
X49
→
(
X29
↔
(
(
¬
X20
)
↔
X48
)
)
)
)
)
→
(
¬
(
X49
→
X48
→
(
X29
↔
False
)
)
)
)
↔
(
¬
(
X49
↔
(
X47
→
X20
→
False
)
)
)
)
↔
(
¬
(
(
(
¬
(
X36
→
X10
→
X47
)
)
↔
(
¬
(
(
¬
X47
)
↔
(
(
¬
X7
)
→
X48
)
)
)
)
→
(
¬
(
(
¬
(
X47
→
(
¬
X36
)
→
(
¬
X15
)
→
X43
)
)
→
(
X47
↔
(
(
¬
X36
)
↔
(
(
¬
X29
)
↔
X43
)
)
)
)
)
)
)
)
→
(
¬
(
(
X47
→
X34
→
False
)
→
(
¬
(
X49
→
(
X48
↔
(
X25
→
X8
)
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
X35
→
(
X27
↔
(
(
¬
X18
)
↔
X32
)
)
)
)
→
(
(
X18
↔
(
X6
→
(
¬
X2
)
→
X35
)
)
→
(
¬
(
(
¬
X27
)
↔
(
X2
↔
X35
)
)
)
)
→
(
X18
↔
(
(
¬
X6
)
↔
(
(
¬
X1
)
↔
X27
)
)
)
)
→
(
¬
(
X32
↔
(
(
¬
X18
)
→
X27
)
)
)
)
↔
(
(
X30
→
(
X6
↔
(
X5
→
X27
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
¬
(
X30
↔
(
X26
↔
(
(
¬
X5
)
→
X27
)
)
)
)
→
(
¬
(
(
¬
X30
)
↔
(
X27
↔
(
X26
→
X35
)
)
)
)
)
)
↔
(
¬
(
X27
→
(
X26
↔
X5
)
)
)
)
↔
(
¬
(
X35
↔
(
(
¬
X26
)
↔
(
(
¬
X13
)
→
X27
)
)
)
)
)
→
(
(
¬
X27
)
↔
(
(
¬
X26
)
→
X35
)
)
)
)
)
)
→
(
¬
(
(
¬
X30
)
↔
(
X27
↔
(
X23
↔
X47
)
)
)
)
→
(
¬
(
X26
→
X14
→
X27
)
)
)
→
(
(
(
(
¬
X47
)
↔
(
(
¬
X44
)
↔
X48
)
)
→
(
(
(
(
(
¬
X47
)
→
(
¬
X30
)
→
X26
→
X48
)
↔
(
X47
→
(
(
¬
X26
)
↔
(
X22
↔
X30
)
)
)
)
↔
(
X44
→
(
¬
X26
)
→
(
X22
↔
X30
)
)
)
↔
(
X44
→
(
X26
↔
X22
)
)
)
)
↔
(
¬
(
(
(
X28
↔
(
X12
→
(
(
¬
X0
)
↔
X26
)
)
)
↔
(
X50
↔
(
X26
→
(
(
¬
X12
)
↔
X28
)
)
)
)
→
(
¬
(
(
¬
(
X50
↔
(
X28
↔
(
X22
→
X26
)
)
)
)
→
(
¬
X48
)
→
X3
→
X50
)
)
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X28
)
↔
X48
)
)
)
)
↔
(
(
(
(
(
¬
(
(
¬
(
X30
↔
(
(
¬
X11
)
↔
(
X10
↔
X32
)
)
)
)
→
(
X40
↔
(
X32
↔
(
X30
→
X28
)
)
)
)
)
↔
(
(
X49
↔
(
X47
↔
(
(
¬
X32
)
↔
X40
)
)
)
→
(
¬
X40
)
→
(
(
¬
X32
)
↔
(
X30
↔
X47
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
¬
X41
)
↔
(
(
¬
X32
)
→
(
X4
↔
X47
)
)
)
)
→
X41
→
X5
→
False
)
)
→
X32
→
(
X24
↔
X40
)
)
)
)
→
(
(
(
(
X41
↔
(
X33
→
X31
→
X37
)
)
↔
(
¬
(
X41
→
X31
→
(
(
¬
X27
)
↔
X33
)
)
)
)
→
(
(
(
X47
↔
(
(
¬
X31
)
→
(
¬
X14
)
→
X33
)
)
↔
(
X47
↔
(
(
¬
X17
)
→
(
(
¬
X14
)
↔
X31
)
)
)
)
↔
(
(
(
¬
(
X47
↔
(
X27
↔
(
X21
→
X33
)
)
)
)
↔
(
¬
(
(
¬
X27
)
↔
(
X13
↔
X47
)
)
)
)
↔
(
¬
(
X33
→
X21
→
X17
)
)
)
)
)
↔
(
(
(
X41
↔
X37
)
↔
(
(
¬
(
X41
↔
X1
)
)
↔
(
X37
↔
(
(
¬
X29
)
↔
(
X1
↔
X47
)
)
)
)
)
↔
(
(
X47
→
(
¬
X29
)
→
(
X25
↔
X39
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X47
)
→
X39
→
(
(
¬
X29
)
↔
X49
)
)
)
↔
(
¬
(
X47
↔
(
(
¬
X39
)
→
(
X29
↔
X41
)
)
)
)
)
→
(
¬
(
X29
↔
X16
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
X46
↔
(
(
(
¬
X47
)
↔
(
(
¬
X46
)
→
(
X43
↔
X49
)
)
)
↔
(
(
¬
X46
)
→
X43
→
(
(
¬
X2
)
↔
X47
)
)
)
)
→
(
¬
(
(
¬
X46
)
→
(
X43
↔
(
(
¬
X18
)
→
X47
)
)
)
)
)
↔
(
(
¬
(
X49
→
(
¬
X43
)
→
X47
)
)
→
(
(
¬
(
(
¬
X43
)
→
X38
→
X49
)
)
↔
(
¬
(
(
¬
(
(
¬
X43
)
→
(
¬
X36
)
→
X7
→
X47
)
)
→
(
(
(
(
X47
↔
(
(
¬
X7
)
→
X43
)
)
↔
(
¬
(
X49
→
X43
→
X38
)
)
)
→
X47
→
X7
)
↔
(
(
¬
X47
)
↔
(
X43
↔
X49
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
(
X45
→
(
¬
X20
)
→
(
X6
↔
X22
)
)
↔
(
¬
(
(
¬
X38
)
↔
(
X20
→
X45
)
)
)
)
→
(
(
(
¬
(
X45
↔
(
X22
↔
(
(
¬
X6
)
→
X36
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X43
)
↔
(
X36
↔
X50
)
)
)
→
(
¬
(
(
¬
(
X50
↔
(
(
¬
X36
)
↔
(
(
¬
X22
)
→
X43
)
)
)
)
→
(
¬
(
X43
↔
(
X36
↔
(
X6
→
False
)
)
)
)
)
)
)
→
(
¬
(
X43
→
X22
)
)
)
)
)
→
(
¬
X45
)
→
X36
→
X50
)
→
(
¬
(
X50
↔
(
X45
→
X43
)
)
)
)
)
→
(
¬
(
X43
→
(
X20
↔
X36
)
)
)
)
)
↔
(
¬
(
X50
↔
(
X43
→
X38
→
False
)
)
)
)
)
→
(
(
(
¬
(
(
(
¬
(
X45
→
X42
→
X41
→
False
)
)
↔
(
(
¬
(
X45
↔
(
X41
↔
X42
)
)
)
→
(
X45
↔
(
X15
↔
X19
)
)
)
)
→
(
¬
(
X42
→
(
X19
↔
False
)
)
)
)
)
↔
(
¬
(
(
X44
→
X15
→
X49
)
→
(
¬
(
(
¬
(
(
¬
X45
)
↔
(
(
¬
X19
)
→
(
¬
X15
)
→
X49
)
)
)
→
(
¬
(
X48
→
False
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
X41
)
→
X23
→
(
(
¬
X12
)
↔
X49
)
)
→
(
(
¬
(
(
¬
X12
)
→
X3
→
X23
)
)
↔
(
(
¬
(
X12
↔
(
X9
→
X23
)
)
)
↔
(
¬
(
X41
↔
(
X12
→
X3
→
X9
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
X46
)
→
(
(
¬
X26
)
↔
(
X9
↔
X49
)
)
)
)
)
→
(
(
¬
(
X46
↔
(
X42
→
(
¬
X0
)
→
X49
)
)
)
↔
(
(
¬
(
(
(
(
X49
→
X46
→
(
¬
X35
)
→
X42
)
↔
(
¬
(
X49
↔
(
X35
↔
(
X26
→
X46
)
)
)
)
)
→
(
X49
↔
(
X26
→
X46
)
)
)
→
(
¬
(
X42
→
X26
→
X49
)
)
)
)
↔
(
(
¬
(
X46
↔
(
X35
→
X49
)
)
)
↔
(
(
(
¬
X35
)
→
(
¬
X34
)
→
X49
)
→
(
(
¬
(
(
¬
(
X8
↔
X49
)
)
→
(
(
X49
→
(
X35
↔
(
X34
↔
False
)
)
)
↔
(
(
X49
↔
(
X35
↔
(
X8
↔
X34
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X35
)
→
(
(
¬
X8
)
↔
X49
)
)
)
)
)
)
)
)
↔
(
X49
↔
(
X34
↔
(
X8
↔
False
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
b953ae...
and proposition id is
26f364...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMXrz4LwQUxrVWKXwxdzXYngS89fxw1STXC
)
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
(
(
(
(
X48
↔
(
(
¬
X47
)
↔
X49
)
)
→
(
¬
(
X34
↔
X46
)
)
)
→
(
(
¬
X48
)
↔
(
(
¬
X47
)
↔
X49
)
)
)
→
(
¬
(
X48
→
(
¬
X43
)
→
(
X34
↔
X49
)
)
)
)
↔
(
¬
(
X49
→
X47
→
(
X46
↔
False
)
)
)
)
↔
(
(
¬
(
(
X48
↔
(
X46
→
(
X19
↔
False
)
)
)
→
(
(
(
¬
X19
)
↔
(
X8
↔
X47
)
)
↔
(
X48
↔
(
X46
→
(
X8
↔
X47
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
(
X11
→
X48
)
)
→
(
X46
↔
(
X19
→
X17
→
False
)
)
)
→
(
¬
(
X46
→
X20
→
(
¬
X19
)
→
X48
)
)
)
)
→
(
¬
(
X48
↔
(
X17
→
X20
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
X49
→
(
X4
↔
X32
)
)
)
↔
(
X4
→
(
X2
↔
X25
)
)
)
→
(
¬
X32
)
→
(
¬
X25
)
→
X49
)
→
(
(
(
¬
(
(
¬
(
(
(
¬
X47
)
→
X29
→
X49
)
→
(
¬
(
X49
↔
(
X25
↔
X32
)
)
)
)
)
→
(
X49
↔
(
X47
↔
(
X25
↔
(
X10
→
False
)
)
)
)
)
)
↔
(
X32
→
(
(
¬
X29
)
↔
X49
)
)
)
↔
(
¬
(
X49
→
(
(
¬
X29
)
↔
(
(
¬
X28
)
→
X47
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
X31
)
↔
(
X28
↔
X48
)
)
→
(
¬
(
(
(
¬
X24
)
↔
(
(
¬
X10
)
↔
X48
)
)
→
(
(
¬
(
(
X48
↔
(
X31
→
X24
→
False
)
)
→
(
¬
(
X48
↔
(
X10
→
X28
)
)
)
)
)
↔
(
X48
↔
X31
)
)
)
)
)
↔
(
(
¬
X28
)
→
(
X24
↔
X31
)
)
)
↔
(
(
(
¬
(
X48
↔
(
X47
↔
(
X1
↔
X31
)
)
)
)
↔
(
X48
→
(
X47
↔
(
(
¬
X10
)
→
X31
)
)
)
)
→
(
¬
(
(
¬
X1
)
↔
X48
)
)
)
)
)
)
↔
(
(
(
(
(
X49
↔
(
X44
↔
(
(
¬
X23
)
↔
X30
)
)
)
↔
(
¬
(
(
¬
X23
)
→
X5
→
X3
→
X30
)
)
)
↔
(
(
¬
(
(
¬
X44
)
→
(
X39
↔
(
(
¬
X30
)
→
X48
)
)
)
)
↔
(
¬
(
X49
→
X44
)
)
)
)
↔
(
(
(
(
X48
→
(
X14
↔
(
X0
→
X44
)
)
)
↔
(
¬
(
X48
→
(
X14
↔
(
X0
↔
X13
)
)
)
)
)
↔
(
(
¬
(
(
¬
X42
)
→
X21
→
(
(
¬
X13
)
↔
X48
)
)
)
↔
(
(
X48
↔
(
(
¬
X7
)
→
X13
)
)
→
(
X44
↔
(
X13
↔
(
X7
→
X49
)
)
)
)
)
)
↔
(
(
(
¬
(
X48
↔
(
X41
↔
(
(
¬
X37
)
↔
X49
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X40
)
↔
(
(
¬
X38
)
↔
X49
)
)
)
→
(
(
X48
↔
X49
)
↔
(
¬
(
X42
→
(
¬
X38
)
→
X40
)
)
)
)
)
→
X49
→
(
X48
↔
(
(
¬
X41
)
→
X42
)
)
)
↔
(
X48
→
X42
→
X41
→
X9
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
¬
(
X36
→
(
X35
↔
(
X26
↔
X16
)
)
)
)
↔
(
X36
↔
(
X35
→
(
X26
↔
(
X15
↔
False
)
)
)
)
)
→
(
(
¬
(
X44
→
(
X16
↔
X26
)
)
)
↔
(
X45
↔
(
X44
↔
(
X16
↔
X36
)
)
)
)
)
)
→
(
¬
(
(
X44
→
(
(
¬
X15
)
↔
X16
)
)
→
(
¬
(
(
X33
↔
(
(
¬
X15
)
↔
(
(
¬
X12
)
→
X44
)
)
)
→
(
X16
↔
(
(
¬
X15
)
↔
(
X12
→
X33
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X49
→
X48
→
(
¬
X35
)
→
X45
)
)
↔
(
X48
→
(
X45
↔
(
X44
↔
X35
)
)
)
)
→
(
(
X48
→
X47
→
(
¬
X18
)
→
X27
)
↔
(
¬
(
(
¬
(
(
¬
X18
)
↔
X48
)
)
→
X49
→
(
X45
↔
(
(
¬
X27
)
↔
X48
)
)
)
)
)
)
)
↔
(
(
X49
→
X47
→
X35
)
↔
(
(
¬
(
X35
→
X22
→
X6
→
False
)
)
↔
(
¬
(
(
¬
(
X49
↔
X6
)
)
→
(
¬
(
X49
→
X47
→
(
¬
X22
)
→
X35
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
¬
(
(
¬
X29
)
↔
(
(
¬
X25
)
↔
(
X24
→
X45
)
)
)
)
↔
(
(
(
X49
→
(
(
¬
X24
)
↔
(
(
¬
X13
)
→
X29
)
)
)
↔
(
X29
→
(
¬
X24
)
→
(
(
¬
X17
)
↔
X49
)
)
)
↔
(
¬
(
(
(
¬
(
X49
→
(
¬
X26
)
→
(
X24
↔
X29
)
)
)
→
(
¬
(
X29
↔
(
(
¬
X25
)
→
(
(
¬
X24
)
↔
X26
)
)
)
)
)
→
(
¬
(
(
¬
X24
)
→
X49
)
)
)
)
)
)
→
(
(
¬
(
X48
→
X29
→
X23
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X29
)
→
(
¬
X23
)
→
X36
)
)
→
(
X49
↔
(
X48
↔
X36
)
)
)
→
(
¬
(
X48
→
(
X29
↔
(
X17
↔
X49
)
)
)
)
)
)
)
)
↔
(
(
(
X45
→
(
(
¬
X43
)
↔
(
X12
↔
X49
)
)
)
↔
(
(
¬
(
X45
→
(
¬
X30
)
→
X48
)
)
↔
(
¬
(
(
X49
→
X38
→
X30
)
→
(
(
¬
(
(
X49
↔
(
X48
→
(
¬
X30
)
→
X38
)
)
→
X12
→
False
)
)
↔
(
(
¬
X38
)
→
(
¬
X30
)
→
X48
)
)
)
)
)
)
↔
(
¬
(
(
(
(
X49
→
X40
→
(
X22
↔
False
)
)
→
(
¬
(
X45
→
X42
→
(
X40
↔
X49
)
)
)
→
(
¬
(
(
¬
(
X49
↔
(
X45
→
X42
→
X22
)
)
)
→
X49
→
(
X48
↔
(
X42
→
False
)
)
)
)
)
↔
(
¬
(
X48
→
(
(
¬
X45
)
↔
X49
)
)
)
)
→
(
¬
(
X48
↔
(
X22
↔
X49
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X40
)
↔
(
(
¬
X39
)
↔
(
X8
→
X42
)
)
)
)
↔
(
(
(
¬
X42
)
→
(
X8
↔
X48
)
)
→
(
(
¬
(
(
(
¬
X27
)
→
(
X8
↔
X48
)
)
→
(
¬
(
X48
↔
(
X39
↔
(
X16
→
X27
)
)
)
)
)
)
↔
(
¬
(
X39
→
(
¬
X27
)
→
X8
→
X42
)
)
)
)
)
↔
(
X42
↔
(
(
¬
X39
)
↔
(
X14
→
X48
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
¬
(
X48
→
X45
→
(
(
¬
X28
)
↔
X46
)
)
)
→
(
X46
↔
(
X45
↔
(
X28
→
X9
)
)
)
)
)
↔
(
(
(
¬
(
X28
↔
(
X10
→
X31
)
)
)
↔
(
(
X48
→
X45
→
X28
→
X31
)
→
(
¬
(
X45
→
(
X10
↔
X48
)
)
)
→
(
(
¬
(
(
¬
X45
)
→
(
(
¬
X31
)
↔
(
(
¬
X1
)
→
X48
)
)
)
)
↔
(
¬
(
(
¬
X45
)
↔
(
X31
↔
(
(
¬
X10
)
↔
X48
)
)
)
)
)
)
)
→
(
X41
↔
(
(
¬
X31
)
↔
(
X28
↔
X45
)
)
)
)
)
→
(
¬
(
(
(
(
X46
↔
(
(
¬
X7
)
→
X2
→
X35
)
)
↔
(
¬
(
X35
→
(
¬
X18
)
→
(
¬
X2
)
→
X46
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X46
)
→
(
¬
X34
)
→
(
(
¬
X32
)
↔
X48
)
)
)
↔
(
X46
↔
(
X35
→
False
)
)
)
→
(
¬
(
(
X48
→
(
X35
↔
X41
)
)
→
(
¬
(
X46
↔
(
X18
↔
False
)
)
)
)
)
→
(
¬
(
(
¬
X41
)
↔
(
(
¬
X35
)
↔
X46
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
(
¬
(
X37
→
(
X32
↔
(
(
¬
X19
)
↔
X41
)
)
)
)
→
(
¬
X37
)
→
(
¬
X3
)
→
X41
)
→
(
¬
(
X41
→
X37
)
)
)
)
→
(
¬
(
X47
↔
(
X37
→
(
¬
X19
)
→
X41
)
)
)
)
→
(
(
¬
(
X43
↔
(
(
¬
X37
)
→
(
X32
↔
X47
)
)
)
)
↔
(
X46
↔
False
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
X48
→
(
X47
↔
(
(
¬
X6
)
→
X43
)
)
)
→
(
¬
(
(
¬
X20
)
↔
(
X0
→
X47
)
)
)
→
(
¬
(
(
(
¬
(
X47
→
(
X20
↔
(
X6
↔
X48
)
)
)
)
↔
(
X48
↔
(
X20
→
X0
→
X47
)
)
)
→
False
)
)
)
)
↔
(
(
¬
(
(
¬
X44
)
↔
(
(
¬
X21
)
↔
(
(
¬
X6
)
↔
X47
)
)
)
)
↔
(
X44
→
(
X43
↔
(
(
¬
X6
)
↔
X21
)
)
)
)
)
↔
(
(
(
(
¬
X44
)
→
(
¬
X33
)
→
X48
)
↔
(
¬
(
X48
→
X15
→
(
(
¬
X11
)
↔
X47
)
)
)
)
↔
(
(
¬
(
X44
→
(
¬
X5
)
→
(
¬
X4
)
→
X15
)
)
→
(
¬
(
(
¬
X15
)
→
(
X5
↔
X48
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
525657...
and proposition id is
2cf66c...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMQLEodyHSas4Gp6o5fefq79CLnZjguRUaF
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
∀X50 :
prop
,
(
¬
∀X51 :
prop
,
(
(
(
(
(
(
X33
↔
X8
)
→
(
¬
(
(
¬
(
X50
→
(
X36
↔
(
X33
↔
X30
)
)
)
)
→
(
X48
↔
(
X33
↔
(
X8
→
X21
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X21
→
(
(
¬
X11
)
↔
(
(
¬
X3
)
↔
X33
)
)
)
)
→
(
(
¬
X47
)
↔
(
X39
↔
(
X34
→
X49
)
)
)
)
)
→
(
(
X46
↔
(
X39
↔
(
X8
↔
X34
)
)
)
↔
(
¬
(
X46
→
X39
→
X33
)
)
)
)
)
)
↔
(
¬
(
(
(
(
X50
↔
(
X49
↔
(
(
¬
X17
)
↔
X51
)
)
)
↔
(
(
(
(
(
¬
(
X51
→
(
¬
X49
)
→
(
X26
↔
X50
)
)
)
↔
(
X50
→
(
¬
X49
)
→
X15
→
X51
)
)
↔
(
X51
↔
(
(
¬
X26
)
↔
X49
)
)
)
↔
(
X26
↔
(
(
¬
X15
)
→
X51
)
)
)
↔
(
¬
(
(
¬
X49
)
→
(
X15
↔
X50
)
)
)
)
)
↔
(
(
(
(
¬
(
X51
↔
(
(
¬
X48
)
→
X32
→
X49
)
)
)
→
X32
→
X50
)
↔
(
(
¬
X50
)
↔
(
(
¬
X49
)
→
X51
)
)
)
↔
(
X50
↔
(
X48
→
(
(
¬
X6
)
↔
X51
)
)
)
)
)
→
(
¬
(
X15
→
(
(
¬
X6
)
↔
X51
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
¬
(
X19
↔
(
(
¬
X10
)
↔
X29
)
)
)
↔
(
X50
→
(
(
¬
X29
)
↔
X41
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X41
)
↔
(
(
¬
X25
)
→
(
¬
X18
)
→
X50
)
)
)
↔
(
¬
(
X37
→
(
(
¬
X25
)
↔
(
(
¬
X18
)
↔
X50
)
)
)
)
)
→
(
(
X49
↔
(
X25
→
(
(
¬
X19
)
↔
X41
)
)
)
↔
(
(
¬
(
X49
↔
(
(
¬
X25
)
↔
(
(
¬
X22
)
↔
X41
)
)
)
)
↔
(
X49
→
(
X37
↔
(
X25
↔
X41
)
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
(
¬
(
X40
↔
(
X22
→
X9
→
False
)
)
)
→
(
¬
(
X45
→
(
¬
X22
)
→
(
X9
↔
X50
)
)
)
)
→
(
X22
↔
X9
)
)
)
↔
(
(
(
(
X45
↔
(
X40
→
(
¬
X16
)
→
X22
)
)
↔
(
X40
→
(
¬
X22
)
→
(
¬
X16
)
→
X24
)
)
→
(
¬
(
X45
→
(
(
¬
X24
)
↔
X40
)
)
)
)
↔
(
(
¬
X22
)
→
X40
)
)
)
↔
(
¬
(
(
¬
(
X40
↔
(
X38
↔
(
X19
↔
X50
)
)
)
)
→
(
(
(
¬
(
X45
↔
(
(
¬
X40
)
↔
X50
)
)
)
↔
(
(
(
¬
(
X40
→
(
(
¬
X1
)
↔
X19
)
)
)
↔
(
¬
(
X50
↔
(
(
¬
X1
)
→
X2
)
)
)
)
↔
(
¬
(
X50
↔
(
(
¬
X40
)
→
(
¬
X1
)
→
X45
)
)
)
)
)
↔
(
¬
(
X40
→
X19
→
False
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
¬
X27
)
↔
(
X4
→
X0
→
X28
)
)
→
(
¬
(
X45
↔
(
X27
↔
(
(
¬
X0
)
→
X44
)
)
)
)
)
)
→
(
(
¬
(
(
X47
→
(
X5
↔
X20
)
)
→
(
¬
(
X35
→
X49
)
)
)
)
→
(
¬
(
X49
↔
(
X47
→
(
(
¬
X5
)
↔
X7
)
)
)
)
)
→
(
(
(
X44
↔
X47
)
→
(
(
¬
X35
)
↔
(
(
¬
X28
)
↔
X45
)
)
)
↔
(
X49
↔
(
X45
↔
(
(
¬
X44
)
→
X47
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
(
¬
X43
)
↔
(
(
¬
X31
)
↔
(
X14
→
X47
)
)
)
)
→
(
¬
(
X43
↔
(
(
¬
X14
)
↔
(
(
¬
X13
)
→
X47
)
)
)
)
)
→
(
¬
(
X31
↔
X44
)
)
)
)
→
(
¬
X44
)
→
(
¬
X31
)
→
X47
)
↔
(
¬
(
(
(
¬
X44
)
↔
(
(
¬
X43
)
↔
(
(
¬
X12
)
→
X50
)
)
)
→
(
¬
(
(
X47
→
(
X44
↔
X50
)
)
→
(
¬
(
X50
↔
(
(
¬
X23
)
↔
X43
)
)
)
)
)
→
X50
→
(
X43
↔
(
X42
→
X45
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
X51
↔
(
(
¬
X48
)
→
(
¬
X44
)
→
X50
)
)
→
(
¬
(
X50
↔
(
X44
→
(
X38
↔
(
X15
→
False
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
X23
↔
(
X8
↔
(
X2
↔
X51
)
)
)
)
→
(
X42
↔
(
(
¬
X8
)
→
(
(
¬
X2
)
↔
X51
)
)
)
)
)
→
X44
→
X42
→
(
¬
X23
)
→
X51
)
↔
(
(
(
X51
↔
X1
)
→
(
X44
↔
(
X42
↔
X37
)
)
)
↔
(
(
(
X50
↔
(
X44
↔
(
X16
→
X42
)
)
)
↔
(
(
¬
X44
)
→
(
(
¬
X16
)
↔
(
X1
→
X50
)
)
)
)
↔
(
X44
↔
(
X1
↔
X50
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X48
↔
(
X36
↔
(
(
¬
X27
)
↔
X47
)
)
)
)
→
(
¬
(
(
¬
X48
)
↔
(
(
¬
X47
)
↔
(
(
¬
X36
)
→
X49
)
)
)
)
)
↔
(
¬
(
(
¬
X49
)
↔
(
X48
↔
X51
)
)
)
)
↔
(
X49
↔
(
(
¬
X27
)
→
(
X19
↔
X36
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
(
X49
→
X30
→
(
X20
↔
False
)
)
↔
(
(
X51
↔
(
X30
→
X49
)
)
↔
(
(
X49
→
X25
→
X20
→
False
)
↔
(
X51
↔
(
X25
↔
X30
)
)
)
)
)
→
(
X30
↔
(
(
¬
X25
)
→
X49
)
)
)
)
→
(
¬
(
(
¬
(
X51
↔
(
X46
→
(
¬
X30
)
→
X49
)
)
)
→
(
(
X46
→
X48
)
↔
(
(
(
¬
X46
)
→
X14
→
X48
)
↔
(
(
¬
X49
)
→
X46
→
X51
)
)
)
)
)
)
)
→
(
(
X49
→
X51
)
↔
(
X49
↔
(
X14
↔
(
X4
↔
X50
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
(
(
(
¬
X43
)
↔
(
X17
↔
(
(
¬
X10
)
↔
X47
)
)
)
↔
(
X47
↔
(
X17
→
X3
→
X43
)
)
)
→
X29
→
X17
→
(
(
¬
X3
)
↔
X32
)
)
)
↔
(
¬
(
X43
→
X29
→
X32
)
)
)
↔
(
(
(
¬
(
X48
→
(
X35
↔
X9
)
)
)
↔
(
(
X48
→
(
X43
↔
(
X35
→
(
X9
↔
False
)
)
)
)
↔
(
X48
↔
(
X12
↔
(
(
¬
X9
)
↔
X35
)
)
)
)
)
→
(
(
¬
(
X47
↔
(
(
¬
X6
)
↔
X12
)
)
)
→
(
(
¬
(
X47
↔
(
(
¬
X29
)
→
(
¬
X9
)
→
X48
)
)
)
↔
(
¬
(
X48
↔
(
X47
→
(
X6
↔
X29
)
)
)
)
)
)
→
X29
→
(
¬
X6
)
→
X12
)
)
→
(
(
¬
(
(
¬
X47
)
↔
(
(
¬
X43
)
↔
(
(
¬
X28
)
↔
X48
)
)
)
)
↔
(
¬
(
(
X26
→
X21
→
X43
)
→
(
¬
X43
)
→
X18
→
X47
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
(
¬
X45
)
↔
(
(
¬
X40
)
↔
(
(
¬
X7
)
↔
X48
)
)
)
)
→
(
¬
(
(
(
(
X51
→
X48
→
(
¬
X41
)
→
X45
)
↔
(
(
¬
(
X45
→
X31
→
False
)
)
↔
(
¬
(
(
¬
(
(
(
¬
X41
)
→
X40
→
X31
→
X45
)
→
(
¬
(
X48
→
X45
→
(
X41
↔
X31
)
)
)
)
)
→
(
X48
↔
(
X41
↔
X40
)
)
)
)
)
)
↔
(
¬
(
X41
↔
X48
)
)
)
→
(
¬
(
X51
→
(
X31
↔
(
X22
↔
X45
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
X34
)
↔
(
(
¬
X24
)
→
(
¬
X5
)
→
X45
)
)
→
(
(
X51
→
(
X47
↔
(
(
¬
X24
)
↔
X48
)
)
)
↔
(
¬
(
(
¬
(
X48
→
(
(
¬
X45
)
↔
(
X24
→
X47
)
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X45
)
→
(
X33
↔
X48
)
)
)
)
)
)
)
)
)
↔
(
(
¬
X48
)
↔
(
X45
↔
X51
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
(
¬
(
X47
↔
(
X43
↔
(
(
¬
X13
)
→
X51
)
)
)
)
↔
(
(
¬
X43
)
→
(
(
¬
X39
)
↔
X51
)
)
)
→
(
¬
(
X47
↔
(
X43
→
X39
→
X0
)
)
)
)
)
→
(
X47
↔
(
(
¬
X13
)
↔
X51
)
)
)
)
→
(
(
(
¬
X43
)
↔
(
X39
↔
(
X11
↔
X45
)
)
)
↔
(
(
¬
X43
)
→
(
¬
X39
)
→
(
(
¬
X13
)
↔
X45
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
63adc0...
and proposition id is
c8f689...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMVWRzEDx9HQJLNwFYbFbdDDTXbkoLyMCqt
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
(
(
(
¬
(
X39
↔
(
X35
↔
(
(
¬
X3
)
→
X46
)
)
)
)
↔
(
¬
(
(
¬
X40
)
↔
(
X39
↔
(
(
¬
X35
)
↔
X46
)
)
)
)
)
↔
(
X46
→
(
X16
↔
(
(
¬
X3
)
→
X35
)
)
)
)
↔
(
(
(
¬
(
X46
↔
(
X40
→
X39
→
X34
→
False
)
)
)
→
(
¬
X39
)
→
X36
→
(
(
¬
X34
)
↔
X40
)
)
↔
(
¬
(
X46
→
X39
→
X36
)
)
)
)
↔
(
¬
(
(
(
¬
(
X40
↔
(
(
¬
X39
)
↔
X45
)
)
)
↔
(
(
¬
(
(
¬
X39
)
↔
(
(
¬
X5
)
↔
(
(
¬
X2
)
→
X45
)
)
)
)
↔
(
¬
(
X5
→
(
X2
↔
X40
)
)
)
)
)
→
(
(
(
X40
↔
(
(
¬
X35
)
→
(
(
¬
X6
)
↔
X39
)
)
)
↔
(
X39
↔
(
X35
→
X19
→
X6
→
False
)
)
)
↔
(
(
(
¬
X40
)
↔
(
X39
↔
(
(
¬
X25
)
→
X45
)
)
)
↔
(
¬
(
(
¬
X35
)
↔
(
(
¬
X25
)
→
X45
)
)
)
)
)
)
)
)
↔
(
(
(
(
X29
↔
(
(
¬
X23
)
→
X42
)
)
→
(
¬
(
X29
→
(
X20
↔
X45
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X42
)
→
(
¬
X15
)
→
(
(
¬
X10
)
↔
X46
)
)
)
↔
(
(
¬
X42
)
→
(
¬
X33
)
→
(
X15
↔
X46
)
)
)
→
(
(
¬
X33
)
↔
(
(
¬
X29
)
→
X45
)
)
)
)
)
→
(
(
(
(
X42
↔
(
X38
↔
(
(
¬
X35
)
→
X45
)
)
)
↔
(
(
X46
↔
(
X38
↔
(
X35
→
X45
)
)
)
↔
(
¬
(
X45
→
(
X35
↔
(
X0
↔
X38
)
)
)
)
)
)
↔
(
(
¬
X26
)
→
(
X13
↔
(
(
¬
X0
)
→
X46
)
)
)
)
↔
(
X42
↔
(
(
¬
X13
)
↔
(
(
¬
X4
)
↔
X38
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
¬
(
X47
↔
(
X46
↔
(
X27
→
X45
)
)
)
)
→
(
X46
↔
(
X45
↔
(
X27
→
X44
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
(
¬
X45
)
↔
(
(
¬
X44
)
→
(
(
¬
X9
)
↔
X48
)
)
)
)
→
(
¬
(
(
(
X45
↔
(
(
¬
X44
)
↔
(
X31
↔
X48
)
)
)
↔
(
X48
↔
(
X45
→
(
X37
↔
X44
)
)
)
)
→
(
¬
(
X37
→
False
)
)
)
)
)
)
→
(
¬
(
(
¬
X37
)
→
(
X9
↔
(
(
¬
X8
)
↔
X48
)
)
)
)
)
)
↔
(
X37
→
X21
→
X46
)
)
)
→
(
¬
(
(
(
X43
→
(
X24
↔
(
X22
→
X17
)
)
)
↔
(
(
X24
→
(
X22
↔
X17
)
)
→
(
¬
(
X22
↔
(
X11
→
(
¬
X1
)
→
X24
)
)
)
)
)
→
(
(
(
X37
→
(
X22
↔
(
X17
↔
X12
)
)
)
↔
(
¬
(
X43
↔
(
X22
↔
(
X17
↔
False
)
)
)
)
)
↔
(
¬
(
X37
→
(
¬
X32
)
→
(
X22
↔
X43
)
)
)
)
)
)
→
(
X32
→
(
¬
X30
)
→
X7
→
X31
)
→
(
(
X32
↔
(
(
(
¬
X31
)
↔
(
(
¬
X27
)
↔
(
X24
→
X32
)
)
)
↔
(
X24
→
(
(
¬
X22
)
↔
X32
)
)
)
)
↔
(
¬
(
X24
↔
(
X22
↔
X30
)
)
)
)
)
→
(
(
¬
(
X49
→
X45
→
X44
→
False
)
)
↔
(
(
(
¬
(
(
¬
X45
)
→
X44
→
X42
→
X49
)
)
↔
(
X42
→
(
X18
↔
(
X14
↔
False
)
)
)
)
↔
(
¬
(
(
(
(
X46
↔
(
X45
↔
X44
)
)
↔
(
¬
(
(
¬
(
X41
↔
(
X28
↔
X44
)
)
)
→
X46
→
(
X41
↔
(
X28
↔
X45
)
)
)
)
)
↔
(
¬
(
X46
↔
(
X45
→
X41
)
)
)
)
→
(
¬
X46
)
→
(
¬
X45
)
→
(
X44
↔
X49
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
(
¬
(
X49
→
X47
→
X43
→
X45
)
)
↔
(
¬
(
(
(
¬
X43
)
→
(
(
¬
X35
)
↔
(
X33
↔
X45
)
)
)
→
(
X45
↔
(
X43
↔
(
(
¬
X33
)
→
X49
)
)
)
)
)
)
↔
(
X47
↔
(
X43
↔
(
X35
→
X18
→
False
)
)
)
)
↔
(
(
¬
(
(
¬
X1
)
→
(
¬
X0
)
→
X49
)
)
↔
(
(
¬
X47
)
↔
(
(
¬
X45
)
↔
(
X0
↔
X49
)
)
)
)
)
↔
(
(
(
(
¬
X43
)
→
(
X41
↔
(
(
¬
X40
)
→
X44
)
)
)
↔
(
X47
→
(
X44
↔
(
X43
→
X41
)
)
)
)
→
(
(
¬
(
X40
↔
(
X25
↔
X49
)
)
)
→
(
¬
(
X44
→
(
¬
X43
)
→
(
X40
↔
X49
)
)
)
)
→
(
¬
(
(
X10
→
X49
)
→
(
X49
↔
(
X44
→
(
X25
↔
X27
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X43
↔
(
(
¬
X34
)
↔
(
X16
↔
X40
)
)
)
)
↔
(
¬
(
(
¬
X34
)
↔
(
X16
→
(
¬
X11
)
→
X43
)
)
)
)
→
(
(
(
¬
(
(
¬
X40
)
→
X37
→
X43
)
)
→
(
(
¬
(
(
(
(
¬
(
X47
↔
(
X46
→
X40
→
False
)
)
)
↔
(
¬
(
X47
→
X43
→
X40
→
(
X34
↔
False
)
)
)
)
↔
(
¬
(
X47
↔
(
X46
→
X43
→
False
)
)
)
)
→
(
¬
(
(
¬
X43
)
→
(
X40
↔
(
X4
→
X47
)
)
)
)
)
)
↔
(
(
¬
X37
)
→
(
¬
X34
)
→
X4
→
X43
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X6
↔
X40
)
)
→
X47
→
(
X19
↔
(
(
¬
X6
)
↔
X24
)
)
)
)
→
(
¬
(
(
¬
X40
)
↔
(
X24
→
X43
)
)
)
→
(
(
¬
(
X40
→
(
(
¬
X2
)
↔
X43
)
)
)
↔
(
¬
(
(
¬
X40
)
→
X24
→
(
¬
X6
)
→
X43
)
)
)
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
(
X21
↔
(
(
¬
X8
)
→
(
¬
X7
)
→
X26
)
)
→
(
X39
↔
(
(
¬
X21
)
→
(
¬
X8
)
→
X26
)
)
)
)
↔
(
(
(
X38
→
(
(
¬
X8
)
↔
X26
)
)
→
(
(
¬
X36
)
↔
(
X21
→
(
X8
↔
X38
)
)
)
)
→
(
X47
↔
(
X38
↔
(
X36
→
(
X26
↔
False
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X39
→
(
X31
↔
(
X20
→
(
X14
↔
False
)
)
)
)
)
↔
(
¬
(
X36
↔
(
X20
→
X31
)
)
)
)
→
(
(
X39
→
(
X38
↔
(
(
¬
X30
)
↔
X31
)
)
)
↔
(
(
(
¬
(
(
¬
X38
)
→
(
¬
X36
)
→
X3
→
X39
)
)
→
(
¬
(
X39
→
X31
→
X3
→
X36
)
)
)
↔
(
¬
(
(
¬
X38
)
↔
(
X31
→
X39
)
)
)
)
)
)
)
↔
(
¬
(
X39
↔
(
(
¬
X36
)
↔
(
X13
→
X38
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X36
)
↔
(
X31
↔
(
(
¬
X17
)
→
X43
)
)
)
)
→
(
¬
(
(
¬
X31
)
→
X17
→
X12
→
X36
)
)
)
→
(
(
¬
(
X44
↔
(
X43
→
(
X38
↔
False
)
)
)
)
↔
(
(
X44
→
X43
→
X38
)
↔
(
¬
(
X44
→
(
¬
X36
)
→
(
(
¬
X17
)
↔
X38
)
)
)
)
)
)
)
↔
(
(
(
X47
↔
(
X29
→
X28
→
X32
)
)
↔
(
(
(
¬
(
X48
→
(
X47
↔
X29
)
)
)
↔
(
(
X48
→
(
¬
X23
)
→
X29
)
→
(
¬
(
X48
→
(
X47
↔
(
X29
↔
X15
)
)
)
)
)
)
→
(
(
X48
↔
(
X47
↔
(
X39
→
False
)
)
)
↔
(
(
(
X47
→
X39
→
(
¬
X9
)
→
X48
)
↔
(
¬
(
X48
↔
(
X29
↔
False
)
)
)
)
↔
(
(
¬
X47
)
→
(
(
¬
X39
)
↔
(
(
¬
X32
)
→
X48
)
)
)
)
)
)
)
→
(
(
(
¬
(
X32
↔
(
(
¬
X22
)
→
(
X5
↔
X42
)
)
)
)
↔
(
X43
→
(
¬
X42
)
→
(
¬
X39
)
→
X47
)
)
↔
(
¬
(
X39
↔
(
X32
→
X42
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
f79c33...
and proposition id is
8f9c30...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMP6dxTG9jfJ3bKYZ4uW55r82Y8jSfQ6RzH
)
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
¬
(
(
(
(
(
(
¬
(
X45
→
(
(
¬
X41
)
↔
(
X12
→
X44
)
)
)
)
→
(
¬
(
(
¬
X44
)
↔
(
(
¬
X31
)
→
X45
)
)
)
)
→
X44
→
(
X23
↔
(
(
¬
X12
)
→
X45
)
)
)
↔
(
X45
→
(
X12
↔
False
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X45
)
→
(
¬
X44
)
→
X46
)
)
→
(
(
(
¬
(
X45
↔
(
X44
→
(
(
¬
X42
)
↔
X46
)
)
)
)
→
(
X46
→
X45
→
(
(
¬
X37
)
↔
X42
)
)
→
(
¬
X29
)
→
X46
)
↔
(
¬
(
X46
↔
(
X37
↔
(
X21
↔
X44
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X42
→
X41
→
(
X20
↔
X46
)
)
)
↔
(
¬
(
(
¬
X44
)
→
(
X42
↔
(
(
¬
X41
)
↔
X46
)
)
)
)
)
→
(
(
¬
X41
)
↔
X44
)
)
)
)
→
(
(
X31
↔
(
X29
↔
(
X27
→
X4
)
)
)
→
(
¬
(
X43
→
(
X2
↔
X4
)
)
)
→
(
¬
(
(
¬
X29
)
→
(
X4
↔
(
X2
→
X31
)
)
)
)
)
→
(
¬
(
X44
→
X43
→
X29
→
X27
→
False
)
)
)
)
↔
(
(
(
(
(
(
¬
(
X46
→
X25
→
(
X11
↔
X48
)
)
)
↔
(
(
¬
X46
)
→
(
X39
↔
(
X25
→
X48
)
)
)
)
→
(
¬
(
(
X44
→
(
¬
X26
)
→
X46
)
→
(
X48
↔
(
(
¬
X26
)
→
(
X25
↔
X44
)
)
)
)
)
)
↔
(
X44
→
(
¬
X39
)
→
X46
)
)
↔
(
¬
(
(
(
X48
↔
(
(
¬
X3
)
→
X11
)
)
→
(
¬
(
(
¬
(
(
(
¬
X11
)
→
X9
→
X46
)
→
(
¬
(
X46
↔
(
(
¬
X19
)
↔
(
X9
→
X48
)
)
)
)
)
)
→
(
X48
↔
(
X3
↔
X46
)
)
)
)
)
→
(
¬
(
X45
↔
(
X19
↔
(
X11
→
X46
)
)
)
)
)
)
)
↔
(
(
¬
(
X39
↔
(
(
¬
X19
)
→
X32
)
)
)
→
X45
→
X32
→
X13
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
(
X47
↔
(
(
¬
X17
)
→
X1
→
X28
)
)
→
(
¬
(
X17
→
(
(
¬
X1
)
↔
X10
)
)
)
)
→
(
¬
(
(
¬
(
X28
→
X0
)
)
→
(
(
X47
↔
(
(
¬
X1
)
↔
(
X0
↔
X28
)
)
)
↔
(
(
¬
X5
)
→
X1
→
X0
→
X47
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
X38
→
(
¬
X5
)
→
X8
)
↔
(
(
¬
X38
)
→
(
X6
↔
X44
)
)
)
↔
(
¬
(
X44
↔
(
X8
↔
X6
)
)
)
)
↔
(
¬
(
X8
↔
(
X5
↔
X6
)
)
)
)
→
(
¬
(
X47
→
X38
→
X6
→
X8
)
)
)
)
)
→
(
¬
(
(
X45
→
(
X44
↔
X47
)
)
→
(
¬
(
(
(
(
X47
↔
(
X44
→
X30
→
X14
)
)
→
X30
→
X28
)
↔
(
X30
↔
(
(
¬
X28
)
↔
X44
)
)
)
→
(
X30
↔
(
X28
↔
(
X14
→
X45
)
)
)
)
)
)
)
→
(
¬
(
X47
↔
(
X41
↔
X28
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X47
↔
(
(
¬
X33
)
→
X40
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
X35
)
↔
(
(
¬
X7
)
→
X41
)
)
→
(
(
¬
(
(
X47
→
X41
→
(
X35
↔
False
)
)
→
(
¬
(
X47
↔
(
X40
↔
X24
)
)
)
)
)
↔
(
¬
(
(
¬
X41
)
→
X40
→
X47
)
)
)
)
)
→
(
X47
↔
(
X41
→
(
X40
↔
X33
)
)
)
)
)
)
→
(
(
(
¬
X33
)
→
(
X18
↔
X35
)
)
↔
(
¬
(
(
X35
↔
(
X24
→
False
)
)
→
(
¬
(
X49
→
X34
→
(
(
¬
X16
)
↔
X24
)
)
)
→
(
(
¬
X34
)
↔
(
(
¬
X24
)
↔
(
(
¬
X16
)
→
X35
)
)
)
)
)
)
)
)
↔
(
(
¬
(
X47
→
(
¬
X22
)
→
(
¬
X15
)
→
X44
)
)
↔
(
(
¬
(
X49
↔
(
X40
→
X36
→
X15
)
)
)
↔
(
¬
(
X44
→
X40
→
(
X15
↔
X49
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
¬
(
(
(
(
¬
(
(
¬
(
(
(
X44
→
(
(
¬
X33
)
↔
X46
)
)
↔
(
(
¬
X27
)
↔
(
X16
↔
X33
)
)
)
→
(
X46
↔
(
X33
↔
X44
)
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X33
)
↔
X44
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X41
↔
(
X29
↔
X44
)
)
)
↔
(
X41
→
X29
→
(
X27
↔
X46
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X29
)
→
X41
)
)
)
)
)
)
↔
(
¬
(
X44
→
(
X39
↔
(
X33
↔
X41
)
)
)
)
)
→
X46
→
X16
→
X39
)
)
↔
(
(
X46
↔
X44
)
→
(
(
¬
X22
)
↔
(
X9
→
X39
)
)
)
)
→
(
¬
(
X44
↔
(
X39
→
(
(
¬
X35
)
↔
X41
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X46
→
X30
→
X36
)
)
↔
(
¬
(
X41
→
(
X28
↔
(
(
¬
X21
)
→
X30
)
)
)
)
)
↔
(
¬
(
(
¬
X36
)
→
X30
→
X46
)
)
)
↔
(
(
(
(
¬
(
(
¬
X36
)
→
X34
→
X21
→
X45
)
)
→
(
¬
(
(
¬
X42
)
↔
(
X36
→
(
X34
↔
X45
)
)
)
)
)
↔
(
(
X46
→
X34
→
(
X11
↔
X36
)
)
↔
(
X46
↔
(
X36
↔
(
(
¬
X11
)
↔
X45
)
)
)
)
)
→
(
(
¬
(
(
(
¬
X45
)
↔
(
X24
↔
X48
)
)
→
(
X39
↔
X42
)
)
)
↔
(
(
¬
(
X45
↔
(
X42
↔
(
(
¬
X2
)
→
X39
)
)
)
)
→
X46
→
X2
→
False
)
)
)
)
)
→
(
(
(
¬
(
(
(
(
(
¬
(
X49
↔
(
X40
→
X7
→
False
)
)
)
↔
(
(
X49
↔
X44
)
→
(
(
X49
↔
(
X18
→
(
¬
X7
)
→
X44
)
)
↔
(
¬
(
X40
↔
(
X18
→
False
)
)
)
)
)
)
→
(
¬
X18
)
→
X40
)
↔
(
¬
(
X44
↔
(
X40
→
X18
)
)
)
)
→
(
(
¬
(
X49
→
X40
→
X12
→
X44
)
)
↔
(
(
(
¬
(
X49
↔
(
X48
→
(
X40
↔
(
X12
→
False
)
)
)
)
)
↔
(
¬
(
(
¬
X48
)
→
(
X12
↔
(
(
¬
X1
)
↔
X49
)
)
)
)
)
↔
(
X49
→
(
¬
X40
)
→
X48
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
¬
X37
)
↔
(
X14
↔
X47
)
)
)
↔
(
¬
(
X49
↔
(
(
¬
X38
)
↔
(
X37
↔
X47
)
)
)
)
)
↔
(
X48
↔
X49
)
)
→
(
¬
(
(
¬
(
(
¬
X38
)
→
(
X25
↔
(
(
¬
X4
)
→
X48
)
)
)
)
→
(
¬
(
X48
→
(
X25
↔
(
(
¬
X4
)
↔
X49
)
)
)
)
)
)
→
(
X10
→
(
¬
X3
)
→
X38
)
→
(
¬
X48
)
→
(
¬
X10
)
→
(
¬
X3
)
→
X49
)
↔
(
(
¬
(
(
(
¬
(
X49
↔
(
X47
→
(
X19
↔
X45
)
)
)
)
→
(
¬
X48
)
→
(
X47
↔
(
(
¬
X45
)
→
X49
)
)
)
→
(
¬
(
X49
↔
(
X48
→
(
(
¬
X40
)
↔
X47
)
)
)
)
→
(
¬
(
X48
→
(
(
¬
X47
)
↔
(
(
¬
X17
)
→
X49
)
)
)
)
)
)
→
(
¬
(
(
¬
X47
)
→
X45
→
X49
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
X45
↔
(
X32
↔
(
(
¬
X0
)
↔
X15
)
)
)
)
→
X31
)
↔
(
¬
(
X15
→
False
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
¬
X26
)
↔
(
(
¬
X8
)
↔
X45
)
)
)
↔
(
¬
(
(
(
(
¬
X26
)
→
(
X8
↔
(
X5
↔
X31
)
)
)
→
(
¬
(
X45
→
(
(
¬
X5
)
↔
X8
)
)
)
)
→
(
¬
(
X45
→
X8
→
X26
)
)
)
)
)
→
(
(
¬
X31
)
↔
(
(
¬
X8
)
↔
X45
)
)
)
)
→
(
X45
↔
(
X32
↔
(
X31
→
X26
→
False
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X23
→
(
¬
X20
)
→
X31
)
)
↔
(
(
(
¬
X23
)
↔
(
X20
→
X26
)
)
→
(
¬
(
(
¬
X26
)
→
X20
→
(
(
¬
X13
)
↔
X31
)
)
)
)
)
↔
(
(
(
¬
X26
)
→
(
X23
↔
(
X20
→
X43
)
)
)
↔
(
(
¬
X43
)
→
X20
→
X6
→
X49
)
)
)
→
(
X49
↔
(
X48
↔
(
X32
↔
X43
)
)
)
→
(
¬
(
X32
→
(
X26
↔
X43
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
c62b8c...
and proposition id is
1629fe...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMVi7fWm99jG58ychS5uHQRvmn77PP75Tsc
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
(
¬
(
(
¬
(
(
¬
X44
)
→
(
X33
↔
(
X14
↔
X49
)
)
)
)
→
(
X49
↔
(
X45
↔
(
(
¬
X31
)
→
X44
)
)
)
→
X45
→
X44
→
X15
→
X31
)
)
↔
(
¬
(
(
¬
X31
)
→
(
X14
↔
(
(
¬
X7
)
↔
X48
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X44
↔
(
(
¬
X9
)
↔
X48
)
)
)
→
(
(
¬
(
X49
→
(
¬
X34
)
→
(
X9
↔
X44
)
)
)
↔
(
X48
↔
(
(
¬
X34
)
↔
(
(
¬
X9
)
→
X49
)
)
)
)
)
↔
(
¬
(
X48
↔
(
(
¬
X44
)
↔
X49
)
)
)
)
↔
(
X49
→
(
X44
↔
(
(
¬
X3
)
→
X34
)
)
)
)
→
(
¬
(
X43
↔
(
(
¬
X30
)
→
X48
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
X44
→
(
X41
↔
(
X40
→
X0
)
)
)
)
↔
(
X44
↔
(
X41
↔
(
X10
→
X45
)
)
)
)
→
(
(
¬
(
X44
→
(
¬
X28
)
→
X41
)
)
↔
(
¬
(
X45
↔
(
X44
→
X32
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
X40
)
↔
(
X36
↔
X49
)
)
)
↔
(
X40
↔
(
(
¬
X2
)
→
X16
)
)
)
↔
(
(
¬
(
X41
↔
(
X36
→
(
(
¬
X2
)
↔
X49
)
)
)
)
→
(
¬
(
X41
↔
(
X36
→
(
(
¬
X5
)
↔
X49
)
)
)
)
)
)
↔
(
¬
(
X46
→
(
(
¬
X40
)
↔
(
(
¬
X16
)
↔
X41
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
(
X49
↔
(
X46
↔
(
X44
→
False
)
)
)
)
→
(
X49
↔
(
X46
↔
(
X44
↔
X40
)
)
)
→
(
X49
↔
(
X46
→
X44
→
(
X1
↔
False
)
)
)
)
)
→
(
¬
(
X49
→
(
(
¬
X40
)
↔
(
X39
↔
X46
)
)
)
)
)
)
↔
(
X49
→
(
(
¬
X37
)
↔
(
X13
→
X46
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
(
(
¬
X24
)
→
X20
→
X27
)
→
(
(
¬
(
X48
↔
(
X27
↔
X18
)
)
)
↔
(
X47
→
(
(
¬
X20
)
↔
(
(
¬
X18
)
↔
X27
)
)
)
)
)
)
→
(
(
(
(
X47
→
(
¬
X27
)
→
X35
)
↔
(
¬
(
X45
→
X24
→
X47
)
)
)
↔
(
¬
(
X47
→
X24
→
False
)
)
)
↔
(
(
X46
→
X35
→
X11
→
X8
→
False
)
↔
(
¬
(
(
(
(
¬
X27
)
→
X8
→
X46
)
↔
(
(
X45
→
X8
→
X46
)
→
(
X46
↔
(
(
¬
X11
)
↔
(
X8
↔
X27
)
)
)
)
)
→
(
X46
↔
(
(
¬
X8
)
→
X45
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
X46
)
↔
(
(
¬
X21
)
↔
(
(
¬
X17
)
→
X48
)
)
)
)
→
X46
→
X17
→
(
X12
↔
X48
)
)
→
(
X48
↔
(
(
¬
X21
)
↔
(
(
¬
X12
)
→
X47
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X46
→
X6
→
X29
)
)
↔
(
(
¬
(
X38
↔
(
(
¬
X6
)
↔
(
X4
↔
X29
)
)
)
)
↔
(
X46
↔
(
X29
↔
(
(
¬
X6
)
→
X38
)
)
)
)
)
→
(
¬
(
(
¬
X23
)
→
(
¬
X22
)
→
(
(
¬
X6
)
↔
X46
)
)
)
→
(
X23
↔
(
(
¬
X22
)
→
X25
)
)
→
(
(
(
(
¬
X22
)
→
X4
→
X25
)
↔
(
¬
(
X46
↔
(
X25
→
(
X23
↔
False
)
)
)
)
)
↔
(
¬
(
(
¬
X25
)
→
(
¬
X23
)
→
X46
)
)
)
)
)
→
(
(
(
(
X45
↔
(
X38
↔
(
(
¬
X19
)
↔
X47
)
)
)
↔
(
X45
→
X38
→
(
X26
↔
False
)
)
)
→
(
(
X47
→
(
(
¬
X38
)
↔
(
X29
↔
X45
)
)
)
↔
(
¬
(
X45
↔
(
(
¬
X38
)
→
X29
→
X42
)
)
)
)
)
↔
(
¬
(
X45
↔
(
(
¬
X42
)
→
(
¬
X23
)
→
X47
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
¬
(
(
¬
(
X47
↔
(
X41
↔
X49
)
)
)
→
(
¬
(
(
¬
(
X44
→
X41
)
)
→
(
(
¬
(
X44
↔
(
X41
→
X9
→
X47
)
)
)
↔
(
¬
(
X44
→
(
¬
X9
)
→
X49
)
)
)
)
)
)
)
↔
(
(
X29
↔
(
(
¬
X27
)
→
(
X11
↔
X49
)
)
)
↔
(
X44
→
(
¬
X11
)
→
X27
)
)
)
↔
(
(
(
(
X47
↔
(
(
¬
X39
)
↔
(
X19
→
X49
)
)
)
↔
(
X44
→
X39
→
(
¬
X19
)
→
X49
)
)
↔
X44
)
↔
(
¬
(
X47
↔
(
X44
→
X19
)
)
)
)
)
→
(
(
(
(
(
X48
↔
(
X14
→
(
X2
↔
X47
)
)
)
↔
(
¬
(
X48
↔
(
X12
↔
False
)
)
)
)
↔
(
(
¬
X31
)
→
(
X14
↔
(
(
¬
X12
)
→
X47
)
)
)
)
→
(
¬
(
(
¬
X37
)
→
X31
→
X2
→
X49
)
)
)
↔
(
(
X47
→
(
X7
↔
(
(
¬
X1
)
→
X49
)
)
)
↔
(
(
¬
(
X23
↔
X39
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X48
↔
(
(
¬
X23
)
↔
(
(
¬
X1
)
→
X38
)
)
)
)
→
X5
→
(
(
¬
X1
)
↔
X23
)
)
)
→
X39
→
X23
→
(
(
¬
X5
)
↔
X48
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
(
¬
X46
)
→
(
(
¬
X45
)
↔
(
X17
↔
X48
)
)
)
)
→
(
¬
(
(
(
¬
X45
)
↔
(
(
¬
X36
)
↔
(
(
¬
X25
)
↔
X48
)
)
)
→
X46
→
(
X45
↔
X48
)
)
)
)
↔
(
(
¬
(
X46
→
(
X32
↔
(
X6
↔
X45
)
)
)
)
↔
(
¬
(
(
X43
↔
(
X22
→
(
(
¬
X16
)
↔
X48
)
)
)
→
(
(
X48
→
X45
→
(
X32
↔
X22
)
)
↔
(
¬
(
X46
↔
(
X45
→
X22
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
X48
→
(
(
¬
X46
)
↔
(
(
¬
X45
)
↔
X47
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X46
)
↔
(
X41
↔
(
(
¬
X30
)
↔
X47
)
)
)
)
↔
(
¬
(
X45
↔
(
X41
→
(
(
¬
X30
)
↔
X47
)
)
)
)
)
→
(
¬
X45
)
→
(
X28
↔
X47
)
)
)
)
↔
(
(
(
(
¬
(
X34
↔
(
(
¬
X30
)
→
(
X18
↔
X47
)
)
)
)
→
(
¬
(
X26
↔
(
X8
↔
(
X0
↔
X18
)
)
)
)
)
↔
(
X45
↔
(
X34
→
(
X26
↔
(
X18
→
False
)
)
)
)
)
↔
(
¬
(
X45
↔
(
X34
↔
(
X30
↔
False
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
(
(
¬
(
(
(
¬
X42
)
↔
(
(
¬
X35
)
→
X45
)
)
→
(
¬
(
(
X42
→
(
¬
X4
)
→
X24
)
→
X45
→
(
X35
↔
(
(
¬
X24
)
→
X42
)
)
)
)
)
)
↔
(
¬
(
X4
↔
X24
)
)
)
↔
(
¬
(
X48
↔
X42
)
)
)
→
(
¬
(
(
¬
(
(
¬
X42
)
→
(
X35
↔
(
(
¬
X15
)
↔
X48
)
)
)
)
→
(
(
¬
X42
)
↔
(
(
¬
X15
)
→
(
¬
X13
)
→
X48
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X42
)
→
(
(
¬
X21
)
↔
(
(
¬
X3
)
↔
X48
)
)
)
)
→
(
X48
↔
(
(
¬
X21
)
↔
(
X3
↔
X47
)
)
)
→
(
¬
(
(
¬
X21
)
↔
(
(
¬
X20
)
↔
X47
)
)
)
)
↔
(
¬
(
(
¬
(
X45
→
(
X21
↔
(
(
¬
X10
)
→
X48
)
)
)
)
→
(
X45
↔
(
X42
→
X21
)
)
)
)
)
)
→
(
¬
(
(
X48
→
(
X40
↔
(
(
¬
X35
)
↔
X47
)
)
)
→
X48
→
(
X40
↔
(
(
¬
X33
)
→
X47
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
1c00b3...
and proposition id is
51614c...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMRv42oRn5ScDDNNoiJ1BeD89N1hRpLqaTW
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
(
(
¬
(
X42
↔
(
(
¬
X37
)
→
X33
→
X40
)
)
)
→
(
(
¬
(
(
(
X40
↔
(
(
¬
X30
)
→
X38
)
)
↔
(
¬
(
(
¬
X42
)
↔
(
(
¬
X40
)
↔
(
X30
→
X47
)
)
)
)
)
→
(
¬
(
X42
↔
(
X38
→
X36
→
X5
)
)
)
→
(
¬
(
X38
↔
(
(
¬
X36
)
↔
(
(
¬
X30
)
↔
X42
)
)
)
)
)
)
↔
(
(
¬
(
X46
→
(
(
¬
X40
)
↔
(
X37
→
X47
)
)
)
)
↔
(
(
¬
X42
)
→
X40
→
X37
→
X47
)
)
)
)
↔
(
(
¬
X45
)
↔
(
X42
→
(
¬
X37
)
→
X46
)
)
)
↔
(
(
X46
↔
(
X42
↔
(
X40
↔
X8
)
)
)
↔
(
(
¬
(
X49
→
X42
→
(
X22
↔
False
)
)
)
↔
(
(
¬
(
X49
↔
(
X46
↔
(
X18
→
False
)
)
)
)
↔
(
X49
↔
(
X40
↔
(
(
¬
X22
)
↔
X42
)
)
)
)
)
)
)
→
(
(
(
(
X6
→
X38
)
↔
(
(
¬
(
X6
↔
X38
)
)
→
(
(
¬
X11
)
↔
(
X3
↔
X33
)
)
)
)
↔
(
(
(
¬
(
X33
→
X27
→
X24
→
X9
→
False
)
)
↔
(
(
(
X33
→
(
X24
↔
(
X3
↔
X25
)
)
)
↔
(
X33
→
(
(
¬
X24
)
↔
(
(
¬
X3
)
↔
X48
)
)
)
)
↔
(
¬
(
X33
↔
(
X24
↔
(
X3
↔
X27
)
)
)
)
)
)
→
X25
→
X9
→
(
(
¬
X6
)
↔
X48
)
)
)
↔
(
(
¬
(
(
¬
X42
)
↔
(
(
¬
X19
)
→
X48
)
)
)
↔
(
(
(
(
¬
X44
)
↔
(
(
¬
X42
)
↔
X48
)
)
↔
(
(
¬
X33
)
↔
X38
)
)
→
(
¬
(
X42
↔
False
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
X32
)
→
(
X21
↔
(
X13
↔
X41
)
)
)
)
→
X41
→
X32
→
X21
→
(
X16
↔
False
)
)
→
(
¬
(
(
¬
X41
)
→
(
(
¬
X39
)
↔
(
X28
→
X44
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X16
→
(
X12
↔
X34
)
)
)
→
(
(
(
X43
→
(
(
¬
X16
)
↔
(
(
¬
X0
)
↔
X34
)
)
)
↔
(
¬
(
(
¬
X32
)
→
(
¬
X10
)
→
X43
)
)
)
↔
(
(
¬
X34
)
→
(
X12
↔
(
(
¬
X10
)
→
X43
)
)
)
)
)
→
(
¬
(
X45
→
(
X44
↔
(
X32
→
X14
)
)
)
)
→
(
X45
↔
(
X44
→
X43
→
X34
→
False
)
)
)
)
)
→
(
(
(
(
(
X44
→
X32
→
(
X20
↔
(
X17
↔
False
)
)
)
↔
(
(
(
(
¬
X29
)
→
X35
)
↔
(
(
(
¬
X26
)
→
X44
)
→
(
¬
(
X17
→
X26
)
)
→
(
(
¬
(
(
¬
X29
)
↔
(
(
¬
X17
)
↔
X44
)
)
)
↔
(
(
X35
↔
(
X29
↔
X44
)
)
→
X44
→
(
X29
↔
(
X17
↔
X26
)
)
)
)
)
)
↔
(
(
X31
→
(
X29
↔
X44
)
)
→
(
X44
↔
(
X35
→
X32
)
)
)
)
)
→
(
(
¬
(
(
X41
→
(
X29
↔
X7
)
)
→
(
¬
(
(
¬
(
X17
↔
X4
)
)
→
(
¬
(
(
¬
(
X41
→
X17
→
False
)
)
→
(
¬
X17
)
→
(
¬
X7
)
→
(
X4
↔
X29
)
)
)
)
)
)
)
↔
(
X41
↔
(
X26
→
(
X7
↔
X29
)
)
)
)
)
→
(
(
¬
(
X46
→
(
¬
X15
)
→
(
X2
↔
X35
)
)
)
→
(
¬
X15
)
→
X41
)
→
(
(
¬
X41
)
↔
(
(
¬
X35
)
↔
(
X15
→
X44
)
)
)
)
↔
(
(
¬
(
X44
→
X35
→
(
X23
↔
(
X1
↔
False
)
)
)
)
↔
(
(
¬
X35
)
↔
(
X29
→
(
X23
↔
X44
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
(
X34
↔
(
X28
→
X24
→
X7
)
)
→
X24
→
X22
)
→
(
(
(
X34
→
X24
→
(
(
¬
X20
)
↔
X28
)
)
↔
(
X24
→
(
X15
↔
X20
)
)
)
↔
(
(
(
¬
X15
)
→
(
X1
↔
X24
)
)
↔
(
(
(
¬
X47
)
→
(
X24
↔
(
X15
↔
X49
)
)
)
→
(
¬
(
X47
↔
(
X20
→
False
)
)
)
)
)
)
)
)
→
(
¬
(
(
X34
→
(
¬
X24
)
→
(
(
¬
X10
)
↔
X40
)
)
→
(
(
X34
↔
(
(
¬
X24
)
↔
(
X22
↔
X40
)
)
)
↔
(
¬
(
X34
→
X22
→
(
X8
↔
False
)
)
)
)
→
(
X34
↔
(
(
¬
X24
)
↔
X40
)
)
)
)
→
(
¬
(
(
¬
X34
)
→
(
¬
X33
)
→
(
(
¬
X24
)
↔
X48
)
)
)
→
(
¬
(
X33
↔
(
(
¬
X24
)
↔
(
(
¬
X10
)
→
X34
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
¬
X39
)
→
(
(
¬
X31
)
↔
X46
)
)
↔
(
¬
(
(
¬
X42
)
→
(
(
¬
X39
)
↔
(
X13
↔
X46
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X49
→
(
X46
↔
X42
)
)
)
↔
(
¬
(
X46
↔
(
X39
→
(
(
¬
X14
)
↔
X49
)
)
)
)
)
→
(
¬
X39
)
→
(
X14
↔
X49
)
)
)
)
)
→
(
(
(
(
X43
→
(
X35
↔
(
(
¬
X18
)
→
X30
)
)
)
↔
(
¬
(
(
X43
→
(
X35
↔
(
X32
↔
(
X30
→
False
)
)
)
)
→
(
¬
(
X47
→
(
X43
↔
(
X30
↔
False
)
)
)
)
)
)
)
↔
(
¬
(
X35
↔
(
(
¬
X32
)
↔
(
X9
↔
X43
)
)
)
)
)
→
(
¬
(
(
¬
X32
)
→
(
(
¬
X30
)
↔
(
X18
↔
X39
)
)
)
)
)
→
(
(
(
¬
(
X35
→
(
¬
X18
)
→
X48
)
)
→
(
¬
(
(
¬
X18
)
↔
(
X14
↔
X43
)
)
)
)
↔
(
(
(
¬
X48
)
→
(
¬
X43
)
→
(
¬
X35
)
→
X49
)
↔
(
(
¬
(
X49
→
X43
→
X12
)
)
→
X49
→
X35
→
X12
→
X3
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
¬
(
X45
→
X41
)
)
↔
(
(
¬
X37
)
↔
(
X25
→
X5
→
X41
)
)
)
↔
(
(
(
¬
X45
)
→
(
¬
X41
)
→
X49
)
↔
(
X49
↔
(
(
¬
X25
)
↔
(
X21
↔
X41
)
)
)
)
)
→
(
(
(
X49
→
(
X43
↔
(
X17
→
False
)
)
)
↔
(
¬
(
X41
↔
(
X36
→
False
)
)
)
)
↔
(
(
X36
→
(
(
¬
X11
)
↔
X46
)
)
↔
(
(
X46
→
(
X41
↔
(
X36
↔
X43
)
)
)
↔
(
X46
↔
(
X45
↔
(
X43
↔
(
X41
↔
False
)
)
)
)
)
)
)
→
(
¬
(
X43
↔
(
(
¬
X29
)
→
X46
)
)
)
→
(
(
(
¬
X43
)
→
(
X29
↔
(
X19
↔
X46
)
)
)
↔
(
¬
(
X45
→
X29
→
(
¬
X19
)
→
X43
)
)
)
)
)
↔
(
¬
(
(
X26
→
False
)
→
(
¬
(
(
¬
(
X46
→
(
X44
↔
(
X0
↔
X26
)
)
)
)
→
(
(
¬
(
(
¬
X46
)
↔
(
(
¬
X43
)
→
(
X26
↔
X49
)
)
)
)
↔
(
¬
(
X49
↔
(
X43
→
X0
→
X26
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X27
↔
(
X6
→
X45
)
)
)
→
(
(
(
(
¬
(
X27
↔
(
(
¬
X2
)
→
X38
)
)
)
↔
(
X38
→
X23
→
(
(
¬
X6
)
↔
X27
)
)
)
↔
(
X23
→
X2
)
)
↔
X38
)
)
)
↔
(
¬
(
(
¬
(
X49
→
X46
→
(
X45
↔
X38
)
)
)
→
(
(
(
¬
(
(
¬
(
X38
→
(
(
¬
X23
)
↔
(
(
¬
X16
)
→
X49
)
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X38
)
↔
(
X23
↔
X49
)
)
)
)
)
)
↔
(
(
¬
X38
)
→
(
(
¬
X16
)
↔
X46
)
)
)
↔
(
X49
→
(
¬
X38
)
→
(
(
¬
X4
)
↔
X46
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
7b3bb7...
and proposition id is
ece9e6...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMJF1FVDAaTfWMdibPXyJusGkUB8ykvegcT
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
(
X47
↔
(
X46
↔
(
(
¬
X7
)
↔
X34
)
)
)
↔
(
(
X47
→
X34
→
(
(
¬
X33
)
↔
X46
)
)
↔
(
(
¬
(
X34
↔
(
X33
↔
X16
)
)
)
↔
(
¬
(
(
(
¬
X34
)
↔
(
(
¬
X33
)
↔
(
(
¬
X16
)
↔
X47
)
)
)
→
X47
→
(
X33
↔
(
X32
↔
X16
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
X47
→
(
(
¬
X45
)
↔
(
X34
→
X48
)
)
)
→
X48
→
X47
→
X34
→
(
X6
↔
False
)
)
→
(
¬
(
(
¬
X45
)
↔
(
(
¬
X34
)
→
(
X18
↔
X48
)
)
)
)
)
→
(
¬
(
(
(
(
X48
→
X46
→
X47
)
↔
(
¬
(
(
X47
↔
(
(
¬
X23
)
↔
X48
)
)
→
(
(
X46
↔
(
X23
→
(
¬
X21
)
→
X48
)
)
↔
(
¬
(
X48
→
(
(
¬
X23
)
↔
(
X21
→
X47
)
)
)
)
)
)
)
)
↔
(
X21
→
X47
)
)
→
(
(
(
X47
→
(
X19
↔
False
)
)
↔
(
X46
↔
(
X20
→
X48
)
)
)
↔
(
(
(
(
¬
(
X8
↔
(
(
¬
X4
)
→
X45
)
)
)
↔
(
(
X46
↔
(
(
¬
X29
)
→
(
¬
X4
)
→
X45
)
)
→
(
X47
→
(
X46
↔
X45
)
)
→
(
(
¬
X46
)
→
X45
→
(
¬
X29
)
→
X47
)
→
(
(
¬
X45
)
↔
(
(
¬
X8
)
↔
X46
)
)
)
)
↔
(
¬
(
(
¬
X29
)
↔
(
X10
↔
(
X8
→
X47
)
)
)
)
)
→
(
X28
↔
(
(
¬
X19
)
↔
X46
)
)
→
(
X47
↔
(
X29
→
(
¬
X19
)
→
X28
)
)
)
)
)
)
)
)
)
→
(
(
X48
↔
(
(
¬
X27
)
→
(
X12
↔
X31
)
)
)
→
(
(
X47
→
X46
→
X25
→
X31
)
→
(
¬
(
X31
↔
(
(
¬
X25
)
↔
(
(
¬
X12
)
↔
X47
)
)
)
)
)
→
(
¬
(
X47
↔
(
X31
→
X46
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
(
X48
→
(
X44
↔
(
X22
↔
X47
)
)
)
)
→
(
(
¬
(
(
¬
X46
)
→
(
¬
X11
)
→
X48
)
)
↔
(
¬
(
(
¬
X44
)
↔
(
(
¬
X41
)
↔
(
X11
↔
X47
)
)
)
)
)
)
)
→
(
X43
↔
(
X22
↔
(
X11
↔
X47
)
)
)
)
→
(
(
¬
X13
)
↔
(
(
¬
X0
)
↔
X44
)
)
→
(
(
(
X40
→
(
¬
X13
)
→
X44
)
→
X46
→
(
X44
↔
(
(
¬
X0
)
→
X40
)
)
)
↔
(
¬
(
(
¬
(
X47
↔
(
X40
→
X13
)
)
)
→
(
(
¬
X15
)
↔
(
(
¬
X13
)
→
X0
→
X47
)
)
→
(
¬
(
(
¬
X15
)
↔
(
X0
↔
X40
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
(
X44
↔
(
(
¬
X26
)
→
(
(
¬
X5
)
↔
X30
)
)
)
)
→
(
¬
(
X39
→
X36
→
X5
)
)
)
)
↔
(
(
(
X44
→
(
(
¬
X36
)
↔
(
X17
→
X49
)
)
)
↔
(
¬
(
X49
↔
(
X44
→
(
X42
↔
(
X36
↔
False
)
)
)
)
)
)
↔
(
(
X49
↔
(
X42
↔
(
X39
↔
X44
)
)
)
↔
(
(
¬
X44
)
→
(
(
¬
X39
)
↔
(
(
¬
X38
)
→
X49
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
X36
)
↔
(
X3
↔
X42
)
)
→
(
¬
(
(
¬
(
X49
→
X47
→
(
X3
↔
X36
)
)
)
→
(
¬
(
X49
→
X42
→
(
¬
X3
)
→
X47
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
X9
↔
(
(
¬
X2
)
↔
X24
)
)
)
↔
(
X39
→
(
X9
↔
(
X1
↔
False
)
)
)
)
→
(
¬
(
(
X42
→
X35
→
X14
→
False
)
→
(
¬
(
(
X35
↔
(
X14
↔
(
X1
↔
X9
)
)
)
→
(
¬
(
(
¬
X35
)
↔
(
X14
↔
(
X9
→
X42
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
X48
)
→
(
(
(
(
X42
↔
(
X39
↔
X44
)
)
→
(
¬
(
(
(
¬
(
X48
↔
(
X42
→
X39
→
X44
)
)
)
→
(
¬
(
X44
↔
(
X39
→
(
¬
X35
)
→
X42
)
)
)
)
→
X42
→
X48
)
)
)
→
X35
)
↔
(
¬
(
X42
→
(
X37
↔
False
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
(
(
(
¬
X45
)
↔
(
(
¬
X42
)
↔
(
X18
→
X48
)
)
)
↔
(
X48
↔
(
X18
→
(
(
¬
X0
)
↔
X42
)
)
)
)
↔
(
X45
→
(
¬
X18
)
→
(
X1
↔
X42
)
)
)
→
(
¬
(
(
(
¬
X45
)
→
X37
→
(
(
¬
X10
)
↔
X48
)
)
→
X48
→
(
X45
↔
(
(
¬
X14
)
→
X42
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
X18
)
↔
(
(
¬
X3
)
→
(
¬
X1
)
→
X28
)
)
↔
(
¬
(
(
(
X19
↔
(
(
¬
X18
)
→
(
X3
↔
X28
)
)
)
↔
(
¬
(
X18
→
(
(
¬
X3
)
↔
X48
)
)
)
)
→
(
X48
↔
(
(
¬
X18
)
↔
X19
)
)
)
)
)
→
X48
→
X3
→
False
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
X45
)
↔
(
(
¬
X43
)
↔
(
(
¬
X9
)
↔
X47
)
)
)
↔
(
¬
(
(
(
¬
X43
)
↔
(
(
¬
X9
)
↔
(
(
¬
X5
)
→
X45
)
)
)
→
(
¬
(
X46
↔
(
X43
→
False
)
)
)
)
)
)
→
X48
→
(
X46
↔
(
(
¬
X45
)
↔
X47
)
)
)
)
↔
(
(
(
X48
↔
(
X33
↔
(
X25
→
X47
)
)
)
↔
(
X48
→
(
X47
↔
X18
)
)
)
↔
(
¬
(
(
X47
↔
(
X43
→
(
(
¬
X2
)
↔
X38
)
)
)
→
(
¬
(
(
(
X18
→
X11
→
X25
)
→
(
¬
(
X43
→
(
X18
↔
(
X11
→
X38
)
)
)
)
→
X18
→
X25
)
→
(
¬
(
X43
↔
(
(
¬
X25
)
↔
(
X18
→
X47
)
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
(
(
¬
(
X44
↔
(
X30
→
(
(
¬
X27
)
↔
X35
)
)
)
)
↔
(
¬
(
X35
→
X46
)
)
)
→
(
¬
(
X44
↔
(
X30
↔
X46
)
)
)
)
↔
(
¬
(
X47
↔
(
X44
↔
(
X35
→
X27
)
)
)
)
)
→
(
¬
(
(
¬
(
(
X46
↔
(
X17
→
X7
→
X44
)
)
→
(
¬
(
X46
→
(
¬
X29
)
→
X17
→
X31
)
)
)
)
→
(
(
¬
(
(
X49
↔
(
(
¬
X44
)
↔
(
(
¬
X31
)
→
X46
)
)
)
→
(
¬
(
X47
→
(
(
¬
X46
)
↔
(
(
¬
X44
)
↔
X49
)
)
)
)
)
)
↔
(
¬
(
X49
↔
(
X44
↔
X46
)
)
)
)
)
)
)
→
(
(
(
X49
→
(
X40
↔
(
(
¬
X13
)
↔
X47
)
)
)
↔
(
¬
(
(
¬
X47
)
↔
(
(
¬
X46
)
→
(
¬
X40
)
→
X49
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
¬
X36
)
→
X24
→
X21
→
X49
)
)
→
(
(
¬
(
X49
↔
(
X24
→
X21
→
X47
)
)
)
↔
(
X47
→
(
X40
↔
(
X24
→
False
)
)
)
)
)
)
→
(
¬
(
X49
→
(
(
¬
X40
)
↔
X47
)
)
)
)
)
)
→
(
(
(
(
¬
X36
)
↔
(
X22
→
(
¬
X15
)
→
X47
)
)
↔
(
¬
(
X47
→
(
(
¬
X36
)
↔
X49
)
)
)
)
↔
(
X22
→
X15
→
X49
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X8
→
X46
)
)
→
X26
→
(
¬
X20
)
→
X12
→
X47
)
↔
(
¬
(
(
¬
(
(
¬
X43
)
↔
(
(
¬
X4
)
↔
X46
)
)
)
→
(
¬
(
X34
↔
(
X26
↔
(
X4
↔
X12
)
)
)
)
)
)
)
→
(
(
(
X49
↔
(
(
¬
X43
)
↔
X46
)
)
→
(
¬
(
(
¬
X32
)
→
(
(
¬
X16
)
↔
X46
)
)
)
→
(
(
(
¬
(
X46
↔
(
(
¬
X32
)
↔
(
X16
→
X39
)
)
)
)
→
(
¬
(
(
¬
X32
)
→
(
(
¬
X16
)
↔
(
X6
→
X46
)
)
)
)
)
→
X46
→
(
X39
↔
X32
)
)
→
(
X46
↔
(
X16
→
X39
)
)
)
↔
(
(
X43
→
X41
→
(
X32
↔
X23
)
)
↔
(
¬
(
(
¬
X46
)
↔
(
X32
→
(
(
¬
X23
)
↔
X47
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
790b1e...
and proposition id is
6b1cf0...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMN2RjrTLPqHny3LoPKqnmvEEajuRKQEuZg
)
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
¬
(
(
(
(
(
(
¬
X41
)
↔
(
X39
→
X38
→
X43
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
(
¬
X39
)
→
X26
→
(
¬
X17
)
→
X46
)
)
→
(
X39
↔
(
(
¬
X17
)
↔
X25
)
)
)
)
→
(
¬
(
X39
→
X26
→
X46
)
)
)
)
↔
(
¬
(
X46
↔
(
X39
→
(
X17
↔
False
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X5
)
→
X24
)
)
→
(
(
¬
(
X46
↔
(
X42
→
(
¬
X24
)
→
X47
)
)
)
↔
(
¬
(
X47
→
(
(
¬
X24
)
↔
(
(
¬
X5
)
→
X42
)
)
)
)
)
)
↔
(
¬
(
X46
→
False
)
)
)
↔
(
(
¬
(
(
¬
(
X46
↔
(
X33
↔
(
X31
↔
X1
)
)
)
)
→
X46
→
(
X1
↔
X41
)
)
)
→
(
(
X43
↔
(
(
¬
X31
)
→
X33
)
)
↔
(
(
X42
↔
(
X33
→
(
X12
↔
X31
)
)
)
↔
(
(
X31
→
X47
)
↔
(
¬
(
X43
↔
(
X42
→
X31
→
(
X12
↔
False
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
(
X43
↔
(
(
¬
X27
)
→
X29
)
)
→
(
¬
(
(
¬
X44
)
↔
X46
)
)
)
)
→
(
(
¬
X43
)
↔
(
X32
→
(
X29
↔
X46
)
)
)
)
)
↔
(
(
¬
(
X18
↔
(
(
¬
X8
)
↔
X29
)
)
)
→
(
¬
(
(
¬
X44
)
↔
(
X29
↔
(
X18
→
X46
)
)
)
)
)
)
→
(
(
¬
(
(
X44
→
X19
→
(
¬
X2
)
→
X46
)
→
(
¬
(
X46
→
(
(
¬
X20
)
↔
(
X19
→
X42
)
)
)
)
)
)
↔
(
¬
(
(
(
X42
↔
(
X7
↔
X19
)
)
↔
(
¬
(
X28
↔
(
(
¬
X19
)
↔
(
X7
↔
X43
)
)
)
)
)
→
X46
→
(
X42
↔
(
X7
→
X19
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
(
(
(
(
¬
(
X36
→
(
¬
X34
)
→
(
(
¬
X13
)
↔
X40
)
)
)
↔
(
¬
(
(
¬
X40
)
↔
X42
)
)
)
→
(
(
¬
X16
)
↔
(
(
¬
X13
)
↔
X42
)
)
)
→
X42
→
(
X40
↔
X36
)
)
↔
(
(
(
X36
↔
(
(
¬
X11
)
→
X44
)
)
↔
(
X38
↔
(
(
¬
X36
)
↔
X40
)
)
)
→
(
(
X42
↔
(
(
¬
X40
)
→
X44
)
)
↔
(
¬
(
(
¬
(
X44
↔
(
(
¬
X23
)
→
X40
)
)
)
→
X44
→
(
(
¬
X38
)
↔
(
(
¬
X23
)
↔
X42
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X45
↔
(
X38
↔
(
X34
↔
(
X15
↔
False
)
)
)
)
)
→
X42
→
X34
→
(
¬
X15
)
→
X45
)
)
)
)
→
(
(
(
¬
(
X30
→
X42
)
)
→
(
(
¬
(
(
(
¬
(
X36
→
X30
→
X10
)
)
↔
(
¬
(
X49
↔
(
X42
↔
X30
)
)
)
)
→
(
X49
↔
(
(
¬
X36
)
↔
X42
)
)
)
)
↔
(
(
¬
X42
)
→
X49
)
)
)
↔
(
(
¬
(
X49
→
X38
)
)
→
(
¬
(
X42
↔
(
X30
→
X3
→
X38
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X37
)
↔
(
(
¬
X9
)
↔
X48
)
)
)
→
(
(
(
¬
X42
)
→
X38
→
X37
→
X48
)
↔
(
¬
(
(
¬
(
X6
→
(
(
¬
X4
)
↔
X37
)
)
)
→
(
¬
(
X42
↔
(
(
¬
X37
)
↔
X48
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
(
¬
(
X49
→
X37
→
X48
)
)
↔
(
X49
→
(
(
¬
X21
)
↔
(
X14
→
X48
)
)
)
)
→
X21
)
↔
(
¬
X14
)
)
→
(
(
(
X49
→
X38
→
(
X0
↔
X35
)
)
↔
(
X37
↔
(
(
¬
X35
)
↔
X49
)
)
)
↔
(
X48
→
(
¬
X38
)
→
(
¬
X37
)
→
X49
)
)
)
)
↔
(
(
¬
X38
)
→
X0
→
X49
)
)
→
(
¬
(
X48
→
X38
→
(
X22
↔
X49
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
X32
↔
(
X28
→
(
(
¬
X1
)
↔
X22
)
)
)
↔
(
(
¬
(
(
¬
(
X32
↔
(
X22
↔
X28
)
)
)
→
(
(
¬
(
X29
→
(
(
¬
X9
)
↔
X18
)
)
)
→
(
¬
(
X29
→
X22
→
X9
→
X32
)
)
)
→
(
¬
(
X29
↔
(
X28
↔
(
X9
→
X22
)
)
)
)
)
)
↔
(
¬
(
X32
↔
(
X28
↔
X29
)
)
)
)
)
↔
(
(
(
¬
(
X32
→
(
X29
↔
(
X20
↔
X14
)
)
)
)
→
(
X32
↔
(
X29
↔
(
(
¬
X12
)
↔
X14
)
)
)
)
↔
(
(
(
¬
(
(
X20
↔
(
X14
→
X45
)
)
→
(
(
¬
X42
)
↔
(
X28
→
(
¬
X14
)
→
X45
)
)
→
(
¬
(
(
¬
X42
)
↔
(
X28
↔
(
X20
↔
X45
)
)
)
)
)
)
↔
(
¬
(
X45
→
(
¬
X14
)
→
X28
)
)
)
→
(
(
¬
(
(
¬
X20
)
→
X14
→
X48
)
)
↔
(
(
(
¬
X28
)
→
X20
→
X48
)
↔
(
¬
(
(
X48
→
X28
→
X5
→
False
)
→
(
(
¬
X20
)
↔
(
(
¬
X14
)
→
(
X5
↔
X48
)
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X46
→
(
X19
↔
X24
)
)
)
↔
(
¬
(
(
¬
X46
)
→
(
¬
X24
)
→
X0
→
X48
)
)
)
→
(
¬
(
X48
→
(
X24
↔
X46
)
)
)
)
)
↔
(
(
¬
(
X19
→
X0
)
)
↔
(
(
X45
→
(
X44
↔
(
X28
↔
X48
)
)
)
→
(
(
X28
↔
(
(
¬
X19
)
↔
X49
)
)
↔
(
(
¬
(
X49
→
(
¬
X44
)
→
(
¬
X19
)
→
X48
)
)
↔
(
¬
(
X44
→
(
X21
↔
X49
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
X39
↔
(
X38
↔
(
X34
↔
X40
)
)
)
)
↔
(
X38
→
X34
→
(
X33
↔
X39
)
)
)
→
(
¬
(
(
¬
X40
)
↔
(
(
¬
X39
)
→
(
¬
X34
)
→
X44
)
)
)
→
X7
→
X40
)
↔
(
¬
(
(
(
(
¬
(
X40
→
False
)
)
↔
(
(
(
¬
X40
)
→
(
(
¬
X13
)
↔
X44
)
)
↔
(
X49
↔
(
X44
↔
(
(
¬
X13
)
↔
X39
)
)
)
)
)
↔
(
(
¬
X44
)
↔
(
X40
→
X10
→
X49
)
)
)
→
X49
→
(
X39
↔
X10
)
)
)
)
↔
(
(
(
¬
X38
)
→
X49
)
↔
(
(
¬
(
X40
→
X37
→
(
(
¬
X2
)
↔
X17
)
)
)
→
(
X17
↔
(
(
¬
X15
)
→
(
¬
X2
)
→
X40
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X49
↔
(
(
¬
X44
)
↔
(
X36
↔
X47
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X44
)
↔
(
X43
↔
(
(
¬
X25
)
→
X45
)
)
)
)
→
(
(
(
(
¬
(
X49
↔
(
X41
↔
(
X6
→
False
)
)
)
)
↔
(
(
¬
X41
)
→
X30
→
X6
→
X47
)
)
↔
(
(
¬
X30
)
→
X49
)
)
↔
(
X6
→
False
)
)
)
)
)
→
(
(
(
¬
X47
)
→
(
(
¬
X44
)
↔
X49
)
)
↔
(
(
¬
(
X49
↔
(
X47
↔
(
X16
→
(
X3
↔
False
)
)
)
)
)
→
(
¬
(
X44
→
(
¬
X16
)
→
X47
)
)
)
)
→
(
X47
↔
(
X44
↔
(
(
¬
X26
)
↔
X49
)
)
)
)
→
(
¬
(
(
(
(
(
¬
(
(
(
¬
X47
)
↔
(
(
¬
X40
)
→
X31
→
X49
)
)
→
(
¬
X47
)
→
(
¬
X40
)
→
(
¬
X4
)
→
X49
)
)
↔
(
X47
↔
(
X40
→
X35
→
X23
)
)
)
→
X47
→
(
¬
X35
)
→
(
X27
↔
X49
)
)
↔
(
¬
(
(
(
¬
(
X35
↔
(
X27
↔
(
X11
↔
X40
)
)
)
)
↔
(
(
¬
X40
)
↔
(
(
¬
X27
)
↔
(
(
¬
X11
)
→
X41
)
)
)
)
→
(
X40
↔
(
X27
→
X11
)
)
)
)
)
→
(
¬
(
(
¬
X35
)
→
(
¬
X11
)
→
X8
→
X40
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
645525...
and proposition id is
40ba0d...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMZjjfEgMwieCFBCYu2FcVsbiRwGJ389Wzn
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
(
(
¬
X38
)
→
(
¬
X33
)
→
X40
)
↔
(
(
¬
(
X33
→
(
X11
↔
X40
)
)
)
→
(
(
(
(
¬
(
(
X38
→
(
X0
↔
X11
)
)
→
(
¬
(
X33
↔
(
X1
→
X38
)
)
)
)
)
→
(
(
¬
X0
)
↔
X38
)
)
↔
(
X11
→
(
¬
X1
)
→
X33
)
)
↔
(
¬
(
X33
→
X1
→
X11
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
X46
→
X37
→
(
(
¬
X2
)
↔
X32
)
)
→
(
(
¬
(
(
¬
X32
)
↔
(
(
¬
X27
)
→
(
(
¬
X16
)
↔
X40
)
)
)
)
→
(
(
¬
X32
)
→
(
¬
X20
)
→
X46
)
→
X37
→
(
(
¬
X20
)
↔
(
X16
→
X46
)
)
)
→
(
¬
(
(
¬
X37
)
→
(
(
¬
X32
)
↔
(
X11
↔
X46
)
)
)
)
)
)
→
(
(
(
¬
X37
)
→
(
X33
↔
(
(
¬
X30
)
→
X40
)
)
)
↔
(
¬
(
(
X46
→
X37
→
False
)
→
(
(
(
¬
X37
)
→
X9
→
X40
)
↔
(
X46
↔
(
(
¬
X33
)
↔
(
X9
↔
X40
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X40
↔
(
(
¬
X39
)
↔
(
X13
→
X46
)
)
)
)
↔
(
¬
(
X39
↔
(
(
¬
X38
)
→
X46
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X47
)
↔
(
(
¬
X6
)
→
X49
)
)
)
→
(
(
¬
X40
)
↔
(
(
¬
X13
)
↔
(
(
¬
X6
)
→
X49
)
)
)
→
(
X49
↔
(
(
¬
X40
)
↔
(
(
¬
X13
)
→
X47
)
)
)
)
)
↔
(
(
X49
→
X40
→
(
X34
↔
X47
)
)
→
(
¬
(
X49
→
(
(
¬
X40
)
↔
(
(
¬
X39
)
→
X47
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X31
)
↔
(
(
¬
X18
)
↔
(
X4
↔
X43
)
)
)
)
→
(
(
X43
↔
(
X40
↔
(
X17
↔
X46
)
)
)
→
(
¬
X43
)
→
X4
→
X46
)
→
(
¬
(
X46
↔
(
X43
→
(
X40
↔
False
)
)
)
)
)
↔
(
(
¬
(
(
X23
→
(
X17
↔
X14
)
)
→
(
¬
(
X40
↔
(
(
¬
X17
)
↔
X31
)
)
)
→
(
¬
(
X31
→
(
¬
X17
)
→
(
X14
↔
X23
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
X31
)
→
X26
→
X5
→
X49
)
→
(
¬
X26
)
→
(
(
¬
X17
)
↔
(
X5
↔
X49
)
)
)
)
→
(
¬
(
X28
→
X23
→
(
(
¬
X17
)
↔
X26
)
)
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
(
¬
X46
)
↔
(
X42
→
(
¬
X7
)
→
X48
)
)
)
→
X48
→
X42
→
(
(
¬
X7
)
↔
X36
)
)
↔
(
(
(
¬
X25
)
→
(
X21
↔
X48
)
)
↔
(
¬
(
(
(
¬
X46
)
↔
(
(
¬
X36
)
↔
(
X21
↔
X48
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X36
)
↔
(
X25
↔
X48
)
)
)
)
)
)
)
)
→
(
(
¬
(
X25
↔
(
X19
→
(
(
¬
X8
)
↔
X15
)
)
)
)
↔
(
(
X48
→
(
(
¬
X19
)
↔
(
(
¬
X8
)
→
X47
)
)
)
↔
(
(
(
¬
X47
)
→
(
¬
X46
)
→
(
¬
X8
)
→
X48
)
→
X48
→
X15
→
X8
→
X49
)
)
)
)
↔
(
(
¬
(
(
(
X49
↔
(
X48
↔
(
X35
↔
X22
)
)
)
→
X49
→
X44
→
(
¬
X29
)
→
X48
)
→
(
¬
(
(
¬
X48
)
↔
(
(
¬
X44
)
→
(
¬
X3
)
→
X49
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X49
↔
(
X24
↔
(
X10
↔
X41
)
)
)
)
↔
(
(
¬
(
(
¬
X45
)
↔
(
X41
↔
X48
)
)
)
→
(
¬
(
X45
↔
(
(
¬
X24
)
↔
X49
)
)
)
)
)
→
(
X45
↔
(
X41
↔
X49
)
)
)
)
→
(
¬
(
(
¬
(
X49
→
(
X35
↔
(
(
¬
X12
)
→
X47
)
)
)
)
→
(
¬
(
(
¬
(
X49
→
X48
→
(
X45
↔
False
)
)
)
→
(
¬
(
(
¬
X35
)
→
X47
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
X45
↔
(
(
¬
X4
)
↔
(
(
¬
X3
)
↔
X12
)
)
)
)
→
(
(
¬
(
X45
→
X39
→
(
X27
↔
X12
)
)
)
↔
(
¬
(
X39
→
(
X4
↔
X45
)
)
)
)
)
↔
(
(
¬
(
X47
→
(
X39
↔
(
(
¬
X10
)
↔
X49
)
)
)
)
↔
(
(
¬
(
X49
→
(
(
¬
X45
)
↔
X47
)
)
)
→
(
(
¬
(
X12
→
X47
)
)
↔
(
¬
(
X49
↔
(
X45
→
X12
→
X10
)
)
)
)
)
)
)
→
(
(
(
¬
(
X47
↔
(
(
¬
X6
)
↔
(
(
¬
X0
)
↔
X20
)
)
)
)
↔
(
X39
→
X0
→
X46
)
)
↔
(
X49
↔
(
X39
↔
(
(
¬
X20
)
→
X47
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
(
¬
(
(
¬
X45
)
↔
(
X23
→
(
¬
X13
)
→
X48
)
)
)
→
(
¬
(
X45
↔
(
X23
↔
(
X13
→
X49
)
)
)
)
)
→
(
¬
(
X49
↔
(
X38
→
(
(
¬
X13
)
↔
X23
)
)
)
)
)
→
(
¬
(
(
X49
→
X48
→
(
(
¬
X38
)
↔
X47
)
)
→
(
X49
↔
(
X38
→
(
¬
X36
)
→
X47
)
)
)
)
)
)
→
(
(
¬
(
X49
↔
(
X18
↔
(
(
¬
X8
)
↔
X29
)
)
)
)
↔
(
(
X45
↔
(
X41
→
X29
)
)
↔
(
¬
(
(
¬
(
X49
↔
(
(
¬
X41
)
→
(
X29
↔
X45
)
)
)
)
→
X49
→
(
¬
X41
)
→
(
¬
X18
)
→
X45
)
)
)
)
→
(
¬
(
X47
→
(
X29
↔
(
X18
↔
False
)
)
)
)
→
(
(
(
(
¬
(
(
¬
X29
)
↔
(
(
¬
X25
)
↔
(
X11
↔
X31
)
)
)
)
→
(
(
¬
(
X44
→
(
(
¬
X31
)
↔
(
(
¬
X11
)
↔
X47
)
)
)
)
↔
(
X31
↔
(
(
¬
X11
)
↔
X29
)
)
)
)
→
(
¬
(
X31
→
(
¬
X29
)
→
(
X18
↔
X47
)
)
)
)
↔
(
X31
→
(
X18
↔
X44
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X49
→
(
X43
↔
(
(
¬
X37
)
→
X42
)
)
)
)
→
(
¬
(
(
(
(
X47
↔
(
X17
↔
(
(
¬
X9
)
→
X35
)
)
)
→
(
¬
(
X47
↔
(
X28
↔
X35
)
)
)
)
→
(
¬
(
(
(
¬
(
X47
↔
(
X42
→
(
X40
↔
False
)
)
)
)
↔
(
X42
→
X40
→
(
¬
X17
)
→
X35
)
)
→
(
X35
↔
X47
)
)
)
)
→
(
¬
(
(
(
(
(
¬
(
X48
↔
(
X43
→
(
X42
↔
X28
)
)
)
)
↔
(
X43
↔
(
X42
↔
(
X7
→
X28
)
)
)
)
↔
(
¬
(
X42
→
(
(
¬
X28
)
↔
(
(
¬
X17
)
→
X48
)
)
)
)
)
↔
(
¬
(
X49
↔
(
(
¬
X17
)
↔
(
X7
↔
X48
)
)
)
)
)
→
(
X49
↔
(
X48
↔
X28
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
(
(
¬
(
X14
→
X30
)
)
↔
(
¬
(
(
¬
X21
)
→
(
(
¬
X14
)
↔
(
(
¬
X1
)
→
X34
)
)
)
)
)
↔
(
¬
(
X34
↔
False
)
)
)
→
(
¬
(
X30
↔
(
(
¬
X21
)
↔
(
X14
→
X24
)
)
)
)
)
)
↔
(
¬
(
(
¬
X33
)
↔
(
(
¬
X14
)
↔
(
(
¬
X1
)
↔
X34
)
)
)
)
)
→
(
¬
(
(
X34
↔
(
X30
↔
(
X24
↔
(
X5
↔
False
)
)
)
)
→
(
(
(
¬
(
X42
↔
(
X24
↔
(
(
¬
X22
)
↔
X34
)
)
)
)
↔
(
¬
(
X42
↔
(
(
¬
X22
)
→
(
X16
↔
X24
)
)
)
)
)
↔
(
¬
(
X42
↔
(
X34
→
X22
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X33
)
→
(
X32
↔
(
(
¬
X15
)
→
X42
)
)
)
)
↔
(
X42
↔
(
X34
→
(
¬
X19
)
→
X32
)
)
)
↔
(
(
(
X48
↔
(
X42
↔
(
X34
↔
X2
)
)
)
↔
(
¬
(
X42
→
(
X34
↔
(
X32
↔
X48
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X40
→
X26
→
(
X2
↔
X42
)
)
)
→
(
¬
(
X42
→
(
X32
↔
(
X2
→
X40
)
)
)
)
)
)
→
(
(
¬
X40
)
↔
(
(
¬
X26
)
↔
X42
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
6fd863...
and proposition id is
8cf490...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMSrDxTU9hcE5FmJT8MesPekS8j5nty4LPh
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
¬
(
(
(
(
(
¬
(
(
¬
(
(
X31
→
X23
→
(
(
¬
X14
)
↔
X42
)
)
→
(
(
¬
X31
)
→
(
X23
↔
(
(
¬
X14
)
→
X45
)
)
)
→
(
¬
X31
)
→
(
X23
↔
(
X5
→
X45
)
)
)
)
→
X44
→
X41
→
X20
)
)
→
(
(
¬
(
X44
→
(
¬
X28
)
→
X42
)
)
↔
(
(
¬
(
(
¬
(
X42
→
(
¬
X41
)
→
(
X39
↔
X44
)
)
)
→
(
X42
↔
(
X41
→
(
(
¬
X28
)
↔
X44
)
)
)
)
)
↔
(
X25
↔
X44
)
)
)
→
(
¬
(
(
¬
X39
)
→
X25
→
(
¬
X7
)
→
X49
)
)
→
(
¬
(
(
¬
X41
)
→
(
¬
X25
)
→
X7
→
X49
)
)
)
→
(
¬
(
(
(
¬
(
X21
↔
(
X12
↔
X49
)
)
)
↔
(
X44
→
(
¬
X21
)
→
(
(
¬
X6
)
↔
X49
)
)
)
→
(
(
(
¬
(
X13
↔
(
X6
↔
(
X1
→
False
)
)
)
)
→
(
X21
↔
(
(
¬
X6
)
→
X44
)
)
)
↔
(
¬
(
X44
↔
(
(
¬
X13
)
→
(
X6
↔
X42
)
)
)
)
)
)
)
)
→
(
¬
(
(
X49
→
(
X41
↔
X26
)
)
→
(
¬
(
(
(
¬
(
(
¬
X34
)
↔
(
(
¬
X30
)
→
X43
)
)
)
→
(
¬
X43
)
→
(
X41
↔
(
X9
↔
X44
)
)
)
→
(
¬
(
(
(
¬
(
X44
↔
(
X26
→
X43
)
)
)
↔
(
(
(
¬
X43
)
↔
(
(
¬
X26
)
↔
(
(
¬
X10
)
↔
X44
)
)
)
→
(
X49
↔
(
X26
→
X44
)
)
)
)
→
(
¬
(
(
¬
(
X44
↔
(
X43
→
X49
)
)
)
→
(
(
¬
X43
)
↔
(
X41
→
(
X2
↔
X44
)
)
)
)
)
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
(
¬
X49
)
↔
(
(
¬
X47
)
↔
(
(
¬
X16
)
↔
X50
)
)
)
)
↔
(
X50
↔
(
X40
↔
False
)
)
)
↔
(
¬
(
(
(
¬
(
X49
→
X44
→
X16
→
X48
)
)
↔
(
¬
(
X47
↔
X49
)
)
)
→
(
¬
X48
)
→
X47
→
X49
)
)
)
↔
(
(
(
X50
→
X48
→
(
X44
↔
(
X11
→
False
)
)
)
→
(
(
¬
(
X44
→
X19
→
X50
)
)
↔
(
X48
→
(
X38
↔
False
)
)
)
)
↔
(
¬
(
X48
→
(
X38
↔
(
(
¬
X37
)
→
X50
)
)
)
)
)
)
→
(
X42
→
(
¬
X33
)
→
(
¬
X24
)
→
X48
)
→
(
(
¬
(
X50
↔
(
X48
↔
(
X42
→
X33
)
)
)
)
↔
(
¬
(
X48
↔
(
(
¬
X47
)
↔
(
(
¬
X42
)
↔
X50
)
)
)
)
)
)
→
(
(
(
(
X46
→
X36
→
X18
→
X8
)
→
(
(
(
(
(
¬
X46
)
→
(
X8
↔
(
X3
→
X49
)
)
)
↔
(
¬
(
X49
↔
(
X48
↔
X8
)
)
)
)
→
(
¬
(
X49
→
(
X46
↔
(
(
¬
X35
)
↔
X48
)
)
)
)
)
↔
(
¬
(
(
¬
X8
)
↔
(
X3
↔
X18
)
)
)
)
)
↔
(
X48
→
X35
→
(
X22
↔
X46
)
)
)
↔
(
¬
(
(
(
(
¬
X46
)
→
(
X36
↔
(
X32
→
X50
)
)
)
↔
(
(
¬
(
X46
↔
(
(
¬
X36
)
→
(
(
¬
X4
)
↔
X50
)
)
)
)
→
X50
→
(
X36
↔
(
(
¬
X0
)
→
X46
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X27
)
→
(
X15
↔
X35
)
)
)
→
(
¬
(
(
(
X36
→
(
¬
X17
)
→
X29
)
↔
(
(
X50
↔
(
X36
→
X15
→
X29
)
)
↔
(
(
¬
X29
)
↔
(
X27
↔
(
(
¬
X15
)
↔
X50
)
)
)
)
)
→
(
(
¬
(
(
¬
X46
)
↔
(
X35
↔
(
X29
→
X50
)
)
)
)
↔
(
¬
(
(
¬
X35
)
↔
(
X29
→
(
¬
X17
)
→
X46
)
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
(
¬
X45
)
↔
(
X32
↔
(
(
¬
X14
)
↔
X50
)
)
)
)
→
(
(
(
¬
(
(
(
¬
X46
)
↔
(
(
¬
X45
)
↔
(
(
¬
X17
)
↔
X50
)
)
)
→
(
¬
(
(
¬
X27
)
↔
X46
)
)
)
)
→
(
X50
↔
(
X32
↔
(
X17
→
False
)
)
)
)
↔
(
(
¬
(
(
¬
X46
)
↔
(
(
¬
X45
)
→
(
¬
X0
)
→
X50
)
)
)
↔
(
X46
↔
(
X45
↔
(
X21
→
X50
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
X50
→
(
X34
↔
(
(
¬
X11
)
↔
X46
)
)
)
)
↔
(
X11
→
X4
→
X46
)
)
→
(
¬
(
(
¬
X48
)
→
(
X11
↔
(
(
¬
X4
)
→
X50
)
)
)
)
)
↔
(
X50
→
(
¬
X46
)
→
X48
)
)
↔
(
¬
(
(
X32
→
(
¬
X11
)
→
X46
)
→
(
(
¬
(
(
¬
(
(
¬
X47
)
→
X49
)
)
→
(
¬
(
(
¬
X41
)
→
X34
→
(
¬
X32
)
→
X49
)
)
)
)
↔
(
X47
→
(
¬
X41
)
→
X34
→
X46
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
(
(
(
X26
→
X15
→
(
(
¬
X9
)
↔
X24
)
)
→
(
¬
X15
)
→
(
¬
X9
)
→
(
¬
X7
)
→
X26
)
↔
(
X19
↔
(
X15
↔
(
X7
→
X9
)
)
)
)
→
(
¬
X19
)
→
(
(
¬
X9
)
↔
X26
)
)
↔
(
X15
↔
(
X7
↔
X20
)
)
)
→
(
(
X44
↔
(
X24
↔
(
(
¬
X5
)
→
X15
)
)
)
↔
(
¬
(
X47
↔
(
X44
→
(
X24
↔
(
X19
↔
False
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
X15
)
→
X20
)
→
(
¬
(
X46
↔
(
X18
→
(
X15
↔
X20
)
)
)
)
→
(
X46
↔
(
X19
→
(
(
¬
X18
)
↔
X20
)
)
)
)
)
↔
(
X43
↔
(
X20
↔
(
X19
↔
X48
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
X39
)
↔
(
X31
↔
(
(
¬
X29
)
→
X50
)
)
)
→
(
(
(
¬
(
X48
→
(
X16
↔
X25
)
)
)
↔
(
(
(
¬
(
X36
→
(
¬
X16
)
→
(
X2
↔
X31
)
)
)
↔
(
¬
(
X25
→
(
¬
X16
)
→
(
(
¬
X2
)
↔
X31
)
)
)
)
↔
(
X25
↔
(
(
¬
X16
)
→
X36
)
)
)
)
↔
(
(
(
¬
(
X49
→
(
(
¬
X29
)
↔
(
X12
↔
X36
)
)
)
)
↔
(
X50
↔
(
X29
→
X12
)
)
)
↔
(
(
(
¬
(
X50
↔
(
X36
↔
(
(
¬
X25
)
→
X49
)
)
)
)
↔
(
¬
(
(
¬
X28
)
↔
(
X25
↔
X36
)
)
)
)
↔
(
¬
(
X48
↔
(
(
¬
X36
)
↔
(
(
¬
X28
)
→
X49
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
X39
↔
(
(
¬
X13
)
→
X1
→
X38
)
)
↔
(
X39
→
X38
→
X40
)
)
↔
(
¬
(
X37
↔
(
X13
↔
(
X1
↔
False
)
)
)
)
)
→
(
(
X40
↔
(
(
¬
X37
)
↔
(
(
¬
X13
)
→
X49
)
)
)
↔
(
X40
→
(
(
¬
X37
)
↔
(
X10
↔
X49
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
¬
(
X50
→
(
X39
↔
(
X23
↔
X3
)
)
)
)
→
(
X50
↔
(
(
¬
X23
)
→
X30
)
)
)
)
→
X50
→
X39
→
X23
→
X38
)
)
→
(
(
(
(
¬
(
X49
→
X42
→
(
X35
↔
False
)
)
)
↔
(
(
¬
(
X35
→
(
X6
↔
X8
)
)
)
↔
(
X49
→
X6
→
X42
)
)
)
→
(
X50
↔
(
X38
↔
(
X33
↔
X42
)
)
)
)
↔
(
(
¬
(
X50
→
X39
→
X35
→
False
)
)
↔
(
(
X22
→
X35
)
→
(
¬
(
X49
→
(
¬
X35
)
→
(
(
¬
X22
)
↔
X39
)
)
)
→
(
¬
(
(
¬
X49
)
↔
(
(
¬
X39
)
→
(
(
¬
X22
)
↔
X50
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
74743e...
and proposition id is
ff52d4...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMG9X2ZKefgEHwfsXGavUkLPY89mSJEYgBP
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
(
(
(
(
(
X45
→
(
(
¬
X44
)
↔
(
(
¬
X35
)
↔
X47
)
)
)
→
(
¬
(
(
¬
(
(
¬
X44
)
↔
(
X36
→
X47
)
)
)
→
(
(
X47
→
(
X35
↔
(
(
¬
X26
)
↔
X36
)
)
)
↔
(
¬
(
X44
↔
(
X26
↔
X36
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X44
)
↔
(
(
¬
X39
)
→
X36
→
X47
)
)
)
→
(
¬
(
X44
→
X36
→
(
¬
X5
)
→
X47
)
)
)
)
)
→
(
(
(
X36
↔
X49
)
↔
(
X49
↔
(
(
¬
X47
)
→
(
¬
X9
)
→
X48
)
)
)
↔
(
(
(
¬
X47
)
→
(
(
¬
X39
)
↔
(
(
¬
X16
)
↔
X48
)
)
)
→
(
(
¬
(
X48
→
X39
→
X9
→
X18
)
)
↔
(
X18
→
(
(
¬
X9
)
↔
X47
)
)
)
)
)
)
→
(
(
(
(
X41
↔
(
(
¬
X30
)
↔
(
X1
↔
X48
)
)
)
→
(
¬
(
(
¬
X37
)
→
X30
→
(
X1
↔
X48
)
)
)
)
↔
(
¬
(
(
¬
X48
)
→
(
¬
X41
)
→
X30
→
X50
)
)
)
↔
(
(
(
¬
X48
)
↔
(
(
¬
X37
)
→
(
(
¬
X22
)
↔
X50
)
)
)
↔
(
¬
(
X50
→
X48
→
X8
→
X37
)
)
)
)
→
(
(
(
(
¬
(
X41
↔
(
X21
↔
(
(
¬
X14
)
→
X20
)
)
)
)
↔
(
X45
→
(
X41
↔
(
X21
→
X20
)
)
)
)
↔
(
(
(
¬
(
X48
↔
(
X45
↔
(
X13
→
X32
)
)
)
)
↔
(
¬
(
(
(
(
(
(
(
¬
(
X41
→
False
)
)
↔
(
X48
→
(
(
¬
X13
)
↔
(
(
¬
X11
)
↔
X45
)
)
)
)
↔
(
¬
(
(
¬
X21
)
→
(
X13
↔
(
(
¬
X11
)
→
X45
)
)
)
)
)
↔
(
¬
(
X41
→
(
X21
↔
X48
)
)
)
)
↔
(
X21
→
X45
)
)
↔
(
(
¬
X13
)
↔
(
(
¬
X11
)
↔
X48
)
)
)
→
X45
→
X21
→
X13
)
)
)
→
(
¬
(
X48
↔
(
X41
→
X17
→
X45
)
)
)
)
)
↔
(
¬
(
(
X48
↔
(
(
¬
X47
)
↔
(
X17
→
X50
)
)
)
→
(
¬
(
(
¬
X50
)
→
(
¬
(
(
¬
X47
)
→
X21
→
(
(
¬
X6
)
↔
X48
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X50
↔
(
X34
→
False
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X42
)
↔
(
X34
→
(
¬
X25
)
→
X43
)
)
)
→
(
¬
(
(
(
¬
X34
)
→
(
X29
↔
(
(
¬
X25
)
↔
X50
)
)
)
→
(
¬
X42
)
→
(
(
¬
X29
)
↔
(
X25
↔
X50
)
)
)
)
)
)
)
↔
(
(
¬
(
X43
→
(
¬
X27
)
→
X19
→
X34
)
)
↔
(
¬
(
X40
↔
(
(
¬
X27
)
→
(
X24
↔
X29
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X43
)
↔
X50
)
)
↔
(
X40
→
(
¬
X37
)
→
(
X23
↔
X43
)
)
)
↔
(
(
X50
↔
(
X38
↔
(
X37
→
X46
)
)
)
↔
(
X43
↔
(
X38
→
(
X35
↔
X46
)
)
)
)
)
)
↔
(
(
(
(
¬
X40
)
↔
(
X33
→
X43
)
)
↔
(
¬
(
(
(
(
¬
X40
)
↔
(
(
¬
X33
)
↔
(
X28
→
X43
)
)
)
→
(
(
¬
(
X40
→
(
X33
↔
(
X0
→
X12
)
)
)
)
↔
(
X43
↔
X12
)
)
)
→
(
(
¬
(
X33
↔
(
(
¬
X12
)
→
X10
→
X31
)
)
)
↔
(
¬
(
(
¬
X31
)
→
(
(
¬
X12
)
↔
(
(
¬
X10
)
↔
X40
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
X15
→
(
X4
↔
X40
)
)
→
(
¬
(
(
¬
(
(
X40
→
(
X31
↔
False
)
)
→
(
¬
(
(
¬
X7
)
↔
(
X2
↔
X12
)
)
)
)
)
→
(
¬
(
X12
↔
False
)
)
)
)
)
)
→
(
(
¬
(
X46
→
(
(
¬
X7
)
↔
(
X3
→
X15
)
)
)
)
↔
(
¬
(
(
¬
X15
)
→
(
¬
X7
)
→
(
X4
↔
X46
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
(
¬
(
(
¬
(
X37
→
(
X2
↔
X46
)
)
)
→
(
(
¬
(
X37
→
X2
)
)
↔
(
¬
(
(
¬
(
X46
→
X41
→
(
X39
↔
X2
)
)
)
→
(
¬
(
(
¬
X37
)
→
(
¬
X2
)
→
X46
)
)
)
)
)
)
)
↔
(
(
X37
↔
X39
)
↔
(
¬
(
X29
→
(
X18
↔
X46
)
)
)
)
)
→
(
¬
(
(
(
X46
→
(
X20
↔
X26
)
)
→
(
X26
↔
(
(
¬
X20
)
↔
X35
)
)
→
(
¬
(
(
¬
X1
)
↔
X46
)
)
)
→
(
¬
(
(
(
¬
X35
)
↔
(
(
¬
X20
)
→
(
¬
X5
)
→
X37
)
)
→
(
¬
(
(
¬
X35
)
→
(
(
¬
X29
)
↔
(
(
¬
X20
)
↔
X37
)
)
)
)
)
)
→
(
¬
(
X37
→
(
X26
↔
(
X20
↔
X29
)
)
)
)
)
)
)
)
↔
(
(
(
¬
X41
)
→
(
¬
X37
)
→
(
¬
X35
)
→
X44
)
↔
(
¬
(
X44
↔
(
X40
↔
(
X30
↔
False
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
X31
)
→
(
(
¬
X19
)
↔
X40
)
)
→
(
(
(
¬
X43
)
↔
(
X40
↔
(
X16
↔
X44
)
)
)
↔
(
¬
(
(
¬
X44
)
↔
(
X40
↔
(
(
¬
X16
)
↔
X45
)
)
)
)
)
→
(
¬
(
X43
↔
(
X22
↔
(
(
¬
X8
)
→
X16
)
)
)
)
)
)
→
(
(
(
X45
↔
(
(
¬
X37
)
→
X36
→
X43
)
)
→
(
¬
(
(
¬
X43
)
→
(
(
¬
X37
)
↔
(
(
¬
X34
)
→
X45
)
)
)
)
)
↔
(
(
(
(
¬
X44
)
→
X43
→
(
¬
X17
)
→
X45
)
↔
(
(
¬
(
X43
→
(
¬
X17
)
→
X45
)
)
↔
(
X43
→
(
¬
X6
)
→
X34
)
)
)
↔
(
¬
(
X44
→
X17
→
X25
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X49
↔
(
(
¬
X27
)
↔
X46
)
)
)
↔
(
(
¬
(
X13
→
X45
)
)
→
(
(
X48
→
(
¬
X46
)
→
X49
)
↔
(
X49
↔
(
X48
→
(
X45
↔
(
X13
→
False
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X28
↔
(
(
¬
X11
)
→
X49
)
)
)
↔
(
X48
↔
(
X28
↔
(
X11
→
X21
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
(
(
¬
(
(
¬
X0
)
→
X47
)
)
→
(
(
¬
X48
)
↔
(
(
¬
X47
)
↔
(
(
¬
X21
)
→
X50
)
)
)
)
)
→
X48
→
X47
→
(
X46
↔
X50
)
)
)
↔
X48
)
→
(
X48
↔
(
(
¬
X46
)
↔
(
(
¬
X21
)
→
X49
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
X32
→
X15
→
X14
)
→
(
¬
(
(
(
¬
X32
)
↔
(
(
¬
X15
)
↔
(
(
¬
X4
)
↔
X47
)
)
)
→
X32
→
(
¬
X15
)
→
(
(
¬
X4
)
↔
X46
)
)
)
)
↔
(
(
¬
X46
)
↔
(
X32
→
X47
)
)
)
→
(
¬
(
X48
→
(
X47
↔
(
X46
→
X15
→
False
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
X42
)
↔
(
(
¬
X38
)
→
X24
→
X44
)
)
→
(
¬
(
(
¬
(
(
X10
→
(
X9
↔
(
X3
→
X38
)
)
)
→
(
¬
(
X24
→
(
¬
X9
)
→
X10
)
)
)
)
→
(
(
X42
→
(
¬
X33
)
→
X9
→
X38
)
↔
(
X33
↔
(
(
¬
X24
)
↔
X42
)
)
)
)
)
)
)
↔
(
(
¬
(
(
X50
→
X49
→
(
X45
↔
X38
)
)
→
(
¬
(
(
(
¬
X49
)
→
(
X38
↔
X50
)
)
→
(
¬
(
X42
↔
(
X38
→
(
X7
↔
X50
)
)
)
)
)
)
)
)
↔
(
(
¬
(
X45
↔
(
X24
→
X23
→
X38
)
)
)
↔
(
(
X50
↔
(
X38
→
(
X24
↔
X12
)
)
)
↔
(
X38
→
(
¬
X24
)
→
X23
→
X50
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
a035c1...
and proposition id is
bea680...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMGByFoQW7Kc3W3oNUy6UWEmUBRMzmaiDZi
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
¬
(
(
(
(
X34
→
X31
→
X16
→
False
)
↔
(
(
¬
(
X42
↔
(
X34
→
(
(
¬
X31
)
↔
X46
)
)
)
)
→
(
X42
↔
(
(
¬
X31
)
→
(
¬
X16
)
→
X46
)
)
)
)
↔
(
X46
→
X34
→
X42
)
)
→
(
¬
(
(
¬
(
X48
→
(
X34
↔
X31
)
)
)
→
(
(
¬
X42
)
↔
(
X31
↔
X43
)
)
)
)
)
)
↔
(
(
X15
↔
(
X11
→
X46
)
)
→
(
¬
(
(
(
(
X46
↔
(
X43
↔
(
X15
↔
False
)
)
)
→
(
X46
↔
(
X15
↔
X43
)
)
→
(
¬
(
X46
↔
(
(
¬
X43
)
→
(
¬
X15
)
→
X49
)
)
)
→
(
¬
X46
)
→
X15
→
(
(
¬
X8
)
↔
X49
)
)
→
(
X49
↔
(
(
¬
X8
)
→
X43
)
)
)
→
(
(
X49
→
(
¬
X46
)
→
X8
→
X48
)
↔
(
¬
(
X49
→
X48
→
(
X46
↔
X19
)
)
)
)
)
)
→
(
(
¬
(
X46
↔
(
X43
→
X10
→
X49
)
)
)
↔
(
(
(
X49
↔
(
X43
↔
(
(
¬
X2
)
→
X11
)
)
)
→
(
¬
(
X46
→
(
X11
↔
(
X2
→
False
)
)
)
)
)
↔
(
¬
(
X46
→
(
X43
↔
X49
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
(
¬
(
X49
→
(
X37
↔
(
X5
→
False
)
)
)
)
↔
(
(
(
¬
(
X49
→
(
¬
X30
)
→
X45
)
)
↔
(
¬
(
(
X49
↔
(
(
¬
X37
)
↔
(
X5
↔
X45
)
)
)
→
(
X49
↔
(
X30
→
(
X5
↔
X37
)
)
)
)
)
)
↔
(
¬
(
X49
↔
(
X5
↔
X45
)
)
)
)
)
↔
(
(
¬
X45
)
→
X30
→
X9
→
X49
)
)
→
(
¬
(
(
X40
↔
(
(
¬
X9
)
→
X21
)
)
→
(
(
¬
(
X40
→
False
)
)
↔
(
(
(
(
¬
X40
)
↔
(
(
¬
X30
)
→
(
¬
X21
)
→
X45
)
)
↔
(
¬
(
X45
→
(
X40
↔
(
X9
↔
X30
)
)
)
)
)
→
(
¬
(
X40
→
X30
→
X9
)
)
)
)
)
)
)
)
↔
(
(
X37
↔
(
X35
↔
False
)
)
↔
(
(
X49
→
(
¬
X37
)
→
X40
)
↔
(
¬
(
X49
↔
(
X37
↔
(
X35
↔
(
X1
↔
False
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
(
X45
→
X40
→
(
X0
↔
False
)
)
↔
(
X23
↔
(
(
¬
X7
)
↔
(
(
¬
X0
)
→
X45
)
)
)
)
→
(
¬
X0
)
)
)
→
(
(
¬
(
X45
→
(
X40
↔
(
(
¬
X6
)
↔
X23
)
)
)
)
↔
(
X40
↔
(
X25
→
(
X23
↔
(
X6
→
False
)
)
)
)
)
)
↔
(
¬
(
(
(
X44
↔
(
X35
↔
(
X23
→
X48
)
)
)
↔
(
¬
(
X44
→
(
X40
↔
(
(
¬
X35
)
→
X48
)
)
)
)
)
→
(
(
¬
(
X48
→
(
X23
↔
(
X7
→
X40
)
)
)
)
↔
(
X48
→
X46
→
X13
→
False
)
)
)
)
)
)
)
)
→
(
(
(
(
X22
↔
(
X3
↔
X28
)
)
↔
(
(
(
¬
X28
)
↔
(
(
¬
X27
)
→
X3
→
X36
)
)
↔
(
X36
↔
(
X28
→
X22
→
X3
)
)
)
)
↔
(
(
(
¬
(
X38
↔
(
(
¬
X36
)
↔
X47
)
)
)
↔
(
(
¬
(
X38
→
(
¬
X36
)
→
X28
→
X47
)
)
↔
(
¬
(
X47
↔
(
(
¬
X28
)
→
(
¬
X18
)
→
X36
)
)
)
)
)
↔
(
(
¬
(
(
¬
X47
)
↔
(
(
¬
X36
)
→
(
X27
↔
X48
)
)
)
)
↔
(
X48
→
(
X38
↔
(
X27
→
X47
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
X20
)
↔
(
X14
→
X26
)
)
↔
(
(
¬
X36
)
↔
(
X20
→
X47
)
)
)
→
(
X38
→
(
X26
↔
X36
)
)
→
(
(
¬
(
X47
↔
(
(
¬
X36
)
→
X38
)
)
)
↔
(
(
¬
X46
)
↔
(
(
¬
X36
)
→
X26
→
X47
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
X39
→
(
(
¬
(
(
¬
X33
)
→
X17
→
X36
)
)
↔
(
X36
↔
(
(
¬
X24
)
→
X39
)
)
)
)
↔
(
(
(
¬
X36
)
→
(
(
¬
X17
)
↔
(
(
¬
X12
)
↔
X47
)
)
)
→
(
¬
X39
)
→
(
X36
↔
(
X17
↔
X47
)
)
)
)
→
(
¬
(
X39
↔
(
X33
↔
X24
)
)
)
)
)
→
(
(
¬
(
X33
↔
(
(
¬
X32
)
→
X24
→
X49
)
)
)
↔
(
¬
(
(
¬
X36
)
↔
(
X33
→
(
¬
X24
)
→
X49
)
)
)
)
)
)
→
(
(
¬
(
(
¬
X36
)
↔
(
X32
↔
(
X4
↔
X47
)
)
)
)
↔
(
(
¬
(
X41
→
X29
→
(
X4
↔
X36
)
)
)
→
(
¬
(
X38
→
X36
→
(
X29
↔
X41
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
¬
(
(
¬
X45
)
→
(
¬
X40
)
→
(
(
¬
X15
)
↔
X47
)
)
)
↔
(
(
¬
(
(
¬
(
X47
→
(
X40
↔
(
(
¬
X29
)
→
X41
)
)
)
)
→
(
(
¬
(
(
¬
X41
)
↔
(
(
¬
X29
)
↔
X47
)
)
)
↔
(
(
¬
(
X47
↔
(
(
¬
X25
)
→
X41
)
)
)
↔
(
¬
(
X41
→
(
X29
↔
(
X23
→
X25
)
)
)
)
)
)
)
)
↔
(
¬
(
X40
↔
(
(
¬
X29
)
→
(
X9
↔
X41
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X47
→
(
(
¬
X41
)
↔
(
(
¬
X5
)
↔
X43
)
)
)
)
→
(
¬
(
(
¬
X43
)
→
(
X41
↔
X45
)
)
)
→
(
(
¬
(
(
¬
X47
)
→
(
¬
X43
)
→
X41
→
X48
)
)
↔
(
¬
(
X48
↔
(
X47
→
X45
)
)
)
)
)
)
↔
(
(
¬
X43
)
↔
(
(
¬
X41
)
→
X45
)
)
)
)
→
(
(
(
(
(
¬
(
X39
↔
(
X33
↔
(
X2
↔
False
)
)
)
)
→
(
(
(
¬
X37
)
↔
(
X33
→
(
(
¬
X14
)
↔
X39
)
)
)
↔
(
¬
(
X39
→
(
¬
X14
)
→
(
X2
↔
X37
)
)
)
)
)
↔
(
¬
(
(
¬
(
X27
→
X14
→
X2
)
)
→
(
¬
(
X4
↔
(
(
¬
X0
)
→
X27
)
)
)
)
)
)
↔
(
(
¬
(
X39
→
(
X38
↔
(
X37
→
False
)
)
)
)
↔
(
(
X37
↔
(
X33
↔
X41
)
)
→
(
¬
X38
)
→
(
X37
↔
X39
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
¬
(
X39
→
(
X37
↔
(
(
¬
X19
)
↔
X36
)
)
)
)
↔
(
(
X47
→
(
(
¬
X37
)
↔
X39
)
)
↔
(
¬
(
X39
↔
(
X37
→
(
X22
↔
(
X19
→
False
)
)
)
)
)
)
)
→
X47
→
(
¬
X19
)
→
X37
)
)
→
(
(
¬
(
X46
→
(
¬
X37
)
→
(
¬
X35
)
→
X47
)
)
↔
(
(
X47
→
(
¬
X36
)
→
(
(
¬
X35
)
↔
X46
)
)
→
(
¬
(
(
¬
(
X47
→
(
(
¬
X36
)
↔
(
(
¬
X18
)
↔
X46
)
)
)
)
→
X46
→
(
X18
↔
(
(
¬
X7
)
↔
X47
)
)
)
)
)
)
)
→
(
(
(
¬
(
X46
→
X39
)
)
→
(
¬
(
X42
↔
(
X39
→
(
X38
↔
X32
)
)
)
)
)
↔
(
(
X42
↔
(
X40
→
X38
)
)
↔
(
(
(
X40
→
X38
→
(
X36
↔
X42
)
)
→
(
X40
↔
(
(
¬
X38
)
↔
(
X36
↔
X39
)
)
)
)
↔
(
¬
(
(
¬
X39
)
→
(
(
¬
X36
)
↔
(
X30
↔
X42
)
)
)
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
X48
→
X34
)
→
(
X49
↔
(
X12
→
X47
)
)
)
→
(
(
¬
X6
)
↔
(
(
¬
(
X48
→
(
¬
X6
)
→
(
X3
↔
X49
)
)
)
↔
(
¬
(
X49
↔
(
X48
→
X47
→
X6
→
False
)
)
)
)
)
)
→
(
(
(
X48
→
(
X28
↔
(
X20
↔
False
)
)
)
→
(
(
(
X48
→
(
X34
↔
X10
)
)
→
X28
→
(
¬
X20
)
→
(
(
¬
X10
)
↔
X48
)
)
↔
(
¬
(
X48
→
(
X10
↔
X20
)
)
)
)
)
↔
(
(
(
¬
X47
)
↔
(
(
¬
(
(
¬
X46
)
↔
X48
)
)
↔
(
(
X47
↔
(
X28
→
(
¬
X24
)
→
X34
)
)
↔
(
¬
(
X47
→
(
(
¬
X34
)
↔
(
X28
↔
X46
)
)
)
)
)
)
)
↔
(
(
¬
X47
)
↔
(
X28
↔
X48
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
¬
X26
)
↔
(
(
¬
X21
)
↔
(
(
¬
X1
)
↔
X34
)
)
)
)
→
(
(
¬
X31
)
→
(
¬
X26
)
→
X21
→
X44
)
→
(
X34
↔
(
(
¬
X26
)
↔
(
X21
→
X31
)
)
)
→
(
¬
(
X26
→
X21
→
(
X17
↔
X34
)
)
)
)
)
↔
(
X44
↔
(
X34
↔
(
X31
→
False
)
)
)
)
↔
(
(
(
¬
(
X44
↔
(
X31
→
(
(
¬
X13
)
↔
X46
)
)
)
)
→
(
(
(
¬
X31
)
↔
(
(
¬
X28
)
→
X34
)
)
↔
(
(
¬
(
X34
↔
X31
)
)
→
(
¬
(
X28
→
(
¬
X13
)
→
X11
→
X31
)
)
)
)
)
↔
(
(
¬
(
(
¬
X31
)
↔
(
(
¬
X16
)
→
(
(
¬
X8
)
↔
X46
)
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X31
)
→
(
(
¬
X16
)
↔
X34
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
e30e6f...
and proposition id is
ed13e1...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMGufNRcud81mWEGQN7LTBN5vJernfE4tru
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
(
(
(
(
(
(
(
(
¬
(
X49
→
X0
→
X5
)
)
↔
(
¬
(
X39
→
(
¬
X34
)
→
X0
→
X49
)
)
)
↔
(
¬
(
X39
→
False
)
)
)
→
(
¬
(
X49
→
X39
→
X34
)
)
)
↔
(
¬
(
X39
↔
(
X0
↔
X34
)
)
)
)
↔
(
(
¬
(
(
¬
(
X39
→
(
¬
X7
)
→
(
X0
↔
X23
)
)
)
→
(
¬
(
X39
↔
(
X7
→
(
X5
↔
X0
)
)
)
)
)
)
↔
(
¬
(
X39
→
(
¬
X7
)
→
X23
)
)
)
)
→
(
(
(
X47
↔
(
(
¬
X25
)
→
X49
)
)
↔
(
¬
(
X21
↔
(
(
¬
X6
)
→
X34
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
¬
X34
)
↔
(
(
¬
X25
)
→
X47
)
)
)
→
X35
→
X34
→
(
X23
↔
X25
)
)
)
→
(
(
(
(
¬
(
X23
→
(
X11
↔
X34
)
)
)
↔
(
X11
↔
(
(
¬
X8
)
↔
(
(
¬
X1
)
→
X34
)
)
)
)
↔
(
(
¬
X23
)
↔
(
X11
↔
X34
)
)
)
↔
(
¬
(
X47
→
(
X34
↔
(
X11
↔
X8
)
)
)
)
)
)
)
)
)
→
(
(
(
(
X47
↔
(
X42
→
(
X31
↔
X43
)
)
)
↔
(
(
¬
(
X50
→
(
X47
↔
(
X2
↔
X42
)
)
)
)
↔
(
¬
(
X47
→
(
X31
↔
(
(
¬
X2
)
↔
X50
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X49
→
X43
)
)
→
(
¬
(
(
¬
(
(
¬
X47
)
→
(
X39
↔
X49
)
)
)
→
(
(
(
¬
X43
)
→
(
¬
X42
)
→
(
(
¬
X38
)
↔
X47
)
)
↔
(
¬
(
X49
↔
(
(
¬
X39
)
→
(
¬
X38
)
→
X47
)
)
)
)
→
(
(
¬
X47
)
↔
(
X43
→
X49
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X46
)
→
(
X44
↔
(
(
¬
X20
)
→
X50
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
¬
(
(
¬
(
X49
→
(
(
¬
X44
)
↔
(
X30
→
X50
)
)
)
)
→
(
¬
X44
)
→
(
X30
↔
(
X12
↔
X50
)
)
)
)
→
(
¬
(
X50
↔
(
X44
→
X30
→
False
)
)
)
)
)
↔
(
X49
→
(
(
¬
X30
)
↔
X50
)
)
)
→
(
(
¬
X46
)
↔
(
(
¬
X30
)
↔
(
(
¬
X12
)
↔
X50
)
)
)
)
)
↔
(
(
¬
(
X50
→
(
X48
↔
(
X3
→
X49
)
)
)
)
→
X49
→
(
¬
X48
)
→
(
¬
X44
)
→
X50
)
)
)
↔
(
(
(
¬
X47
)
↔
(
(
¬
X44
)
↔
(
(
¬
X22
)
↔
X49
)
)
)
↔
(
¬
(
(
¬
(
X47
↔
(
X46
→
X24
→
X49
)
)
)
→
(
(
X49
↔
(
X47
→
(
X46
↔
X44
)
)
)
↔
(
(
¬
X47
)
↔
(
X46
↔
(
X15
↔
X49
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X47
)
↔
(
X10
↔
(
(
¬
X4
)
→
X48
)
)
)
)
→
(
(
X40
→
(
X16
↔
(
(
¬
X13
)
↔
X36
)
)
)
↔
(
¬
(
(
¬
X16
)
↔
(
X10
↔
X36
)
)
)
)
)
↔
(
(
¬
(
X37
→
(
X49
↔
(
X48
→
(
¬
X4
)
→
X41
)
)
)
)
↔
(
(
X49
→
(
¬
X37
)
→
(
(
¬
X18
)
↔
X41
)
)
→
(
(
¬
(
(
¬
X41
)
↔
(
(
¬
X18
)
↔
(
(
¬
X17
)
→
X45
)
)
)
)
↔
(
(
¬
X18
)
→
(
¬
X16
)
→
X41
)
)
→
(
¬
(
X41
→
(
X17
↔
X45
)
)
)
)
)
)
↔
(
(
(
(
(
X47
↔
(
X45
↔
X29
)
)
↔
(
(
X47
→
(
¬
X29
)
→
X28
→
X33
)
↔
(
¬
(
X47
→
(
(
¬
X33
)
↔
(
X29
↔
X45
)
)
)
)
)
)
→
(
¬
(
X33
→
X29
→
X27
→
X47
)
)
)
↔
(
(
¬
(
(
¬
X47
)
→
X48
)
)
↔
(
(
¬
(
(
¬
X47
)
→
(
(
¬
X19
)
↔
X48
)
)
)
↔
(
(
¬
(
(
¬
X28
)
↔
(
X19
↔
X48
)
)
)
↔
(
X48
→
X47
→
X33
→
(
X19
↔
False
)
)
)
)
)
)
↔
(
(
(
¬
X33
)
→
(
¬
X32
)
→
(
(
¬
X14
)
↔
X48
)
)
↔
(
(
X48
↔
(
X45
→
X26
→
X14
)
)
→
(
¬
(
(
¬
X26
)
↔
(
X14
↔
(
X9
→
X32
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
¬
(
(
¬
(
X49
↔
False
)
)
→
(
(
¬
(
(
¬
X41
)
→
(
(
¬
X35
)
↔
(
(
¬
X29
)
↔
X50
)
)
)
)
↔
(
X50
↔
(
X49
→
X41
→
X35
)
)
)
→
(
¬
(
X49
→
(
¬
X35
)
→
X50
)
)
)
)
↔
(
(
¬
(
X50
→
(
¬
X48
)
→
X23
→
X49
)
)
↔
(
¬
(
X50
→
(
(
¬
X48
)
↔
(
X41
↔
X49
)
)
)
)
)
)
→
(
X49
↔
(
(
¬
X32
)
↔
X41
)
)
→
(
(
X41
→
(
¬
X29
)
→
(
¬
X25
)
→
X49
)
↔
(
X49
→
X32
)
)
)
→
(
(
(
(
X50
↔
(
(
¬
X16
)
→
X0
→
X48
)
)
↔
(
(
¬
X43
)
→
X16
→
(
X0
↔
X48
)
)
)
→
(
¬
(
(
¬
X43
)
→
(
¬
X16
)
→
(
X4
↔
X48
)
)
)
)
↔
(
(
¬
(
(
¬
(
X50
↔
(
X47
↔
(
X39
→
X49
)
)
)
)
→
(
(
X50
→
X49
→
(
¬
X10
)
→
X43
)
↔
(
¬
(
X47
↔
(
(
¬
X43
)
↔
(
(
¬
X10
)
→
X48
)
)
)
)
)
)
)
↔
X48
)
)
)
)
→
(
(
¬
(
(
¬
(
X35
↔
(
X19
→
(
X5
↔
X42
)
)
)
)
→
(
¬
(
X42
↔
(
(
¬
X13
)
↔
(
(
¬
X5
)
↔
X33
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
(
¬
(
X35
↔
(
X28
↔
(
X1
↔
X42
)
)
)
)
↔
(
¬
(
X42
↔
(
X28
↔
(
X1
→
X19
)
)
)
)
)
→
X42
→
X28
→
(
X18
↔
(
X1
↔
False
)
)
)
)
→
X49
→
(
(
¬
X35
)
↔
(
(
¬
X1
)
→
X42
)
)
)
↔
(
(
(
¬
(
X28
→
(
(
¬
X27
)
↔
(
X14
↔
X35
)
)
)
)
↔
(
¬
(
X49
↔
(
X35
↔
(
X14
↔
X28
)
)
)
)
)
→
X49
→
(
¬
X27
)
→
X35
)
)
↔
(
¬
(
X39
→
X27
→
X19
→
X35
)
)
)
↔
(
(
¬
X42
)
↔
(
(
¬
X39
)
→
(
¬
X7
)
→
X49
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
¬
(
X47
↔
(
X44
→
(
X38
↔
(
X6
→
False
)
)
)
)
)
→
(
(
¬
(
X47
→
X45
→
(
¬
X6
)
→
X44
)
)
↔
(
X48
↔
(
X47
→
(
X44
↔
(
X6
↔
False
)
)
)
)
)
)
→
(
(
X47
→
(
X17
↔
X38
)
)
↔
(
¬
(
X47
↔
(
(
¬
X38
)
→
(
¬
X36
)
→
X45
)
)
)
)
→
(
¬
(
X47
↔
(
X45
→
X36
→
False
)
)
)
)
)
→
(
(
(
(
(
X38
→
X21
→
(
(
¬
X17
)
↔
X37
)
)
→
(
¬
(
X37
→
(
¬
X21
)
→
X15
→
X38
)
)
)
↔
(
¬
(
(
¬
X15
)
↔
X17
)
)
)
↔
(
¬
(
(
¬
X37
)
→
X21
→
(
X17
↔
X47
)
)
)
)
↔
(
¬
(
(
¬
X17
)
→
X37
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X31
→
X30
→
X40
)
)
→
(
(
X49
↔
(
X31
↔
(
(
¬
X30
)
→
X40
)
)
)
↔
(
¬
(
X40
→
(
X31
↔
(
X30
↔
X9
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X49
→
(
X40
↔
(
X20
→
False
)
)
)
)
↔
(
¬
(
X49
→
(
X40
↔
(
X30
↔
(
X26
↔
False
)
)
)
)
)
)
→
(
X34
↔
(
X30
→
(
(
¬
X3
)
↔
X11
)
)
)
→
(
¬
X11
)
→
(
X3
↔
X40
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X49
↔
(
X47
↔
(
X37
→
X12
)
)
)
)
→
(
X49
↔
(
X37
↔
(
X8
→
X24
)
)
)
)
)
↔
(
(
¬
(
(
¬
X46
)
↔
(
X2
↔
X49
)
)
)
↔
(
¬
(
(
¬
X46
)
→
(
¬
X37
)
→
X47
)
)
)
)
→
(
¬
(
(
¬
(
(
X49
↔
(
X46
↔
(
X44
↔
(
X30
↔
False
)
)
)
)
→
(
¬
(
X46
→
X30
→
X22
)
)
)
)
→
X47
→
(
(
¬
X34
)
↔
(
(
¬
X22
)
→
X49
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
527758...
and proposition id is
141ec5...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMR7sq7DmQ2Z5mmYDS91bXQxn3zxSUqfKfz
)
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
¬
∀X51 :
prop
,
∀X52 :
prop
,
(
(
¬
(
(
(
(
(
¬
(
X50
→
(
X2
↔
X28
)
)
)
→
(
¬
(
X51
↔
(
(
¬
X21
)
→
X46
)
)
)
)
→
(
(
(
(
X44
↔
(
(
¬
X7
)
→
(
(
¬
X0
)
↔
X50
)
)
)
↔
(
¬
(
X44
→
X39
→
X7
)
)
)
↔
(
(
X44
↔
(
(
¬
X39
)
→
(
(
¬
X7
)
↔
X51
)
)
)
↔
(
¬
(
(
¬
X37
)
→
(
¬
X7
)
→
X39
)
)
)
)
↔
(
(
X51
↔
(
(
¬
X37
)
↔
X39
)
)
→
(
X46
↔
(
(
¬
X35
)
↔
X39
)
)
)
)
)
→
(
(
¬
(
(
(
¬
X9
)
→
(
(
X47
→
X44
→
(
¬
X9
)
→
X36
)
↔
(
(
(
(
¬
X44
)
↔
(
(
¬
X36
)
→
X48
)
)
↔
(
(
¬
(
X44
→
(
(
¬
X36
)
↔
X48
)
)
)
↔
(
(
X48
→
(
X46
↔
(
X44
↔
X36
)
)
)
↔
(
¬
(
X44
↔
(
X36
→
(
X14
↔
X46
)
)
)
)
)
)
)
→
(
(
¬
X44
)
↔
(
X36
→
X48
)
)
)
)
)
→
(
¬
(
X50
↔
(
X49
→
(
¬
X46
)
→
X47
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X50
)
→
(
¬
X42
)
→
(
X17
↔
X52
)
)
)
→
(
¬
(
X52
→
(
¬
X42
)
→
(
(
¬
X27
)
↔
X50
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
X50
↔
(
(
¬
X47
)
→
(
(
¬
X27
)
↔
X48
)
)
)
)
→
(
¬
(
X48
→
(
X13
↔
X27
)
)
)
)
)
↔
(
(
X50
→
X48
→
X42
→
X13
)
↔
(
¬
(
X42
↔
(
(
¬
X4
)
↔
X13
)
)
)
)
)
↔
(
¬
(
X42
→
X13
→
X50
)
)
)
)
)
)
→
(
(
(
(
(
(
(
¬
(
X46
↔
(
X45
↔
(
X26
→
False
)
)
)
)
→
(
¬
(
(
X46
↔
(
X45
→
X15
)
)
→
X45
→
X26
→
(
¬
X15
)
→
X20
)
)
)
↔
(
¬
(
(
X46
→
(
(
¬
X26
)
↔
(
X16
→
X45
)
)
)
→
(
(
(
¬
(
(
¬
X45
)
↔
(
(
¬
X26
)
↔
(
X18
↔
X46
)
)
)
)
↔
(
X46
↔
(
X45
↔
(
X26
↔
X49
)
)
)
)
↔
(
(
X49
↔
(
X45
↔
X3
)
)
→
(
(
(
¬
X45
)
→
(
¬
X3
)
→
X49
)
↔
(
¬
(
X46
↔
(
(
¬
X8
)
↔
X49
)
)
)
)
)
)
)
)
)
→
(
¬
X47
)
→
(
X26
↔
(
X20
→
X49
)
)
)
↔
(
(
X49
↔
(
X41
→
(
X22
↔
X47
)
)
)
↔
(
¬
(
(
(
¬
X46
)
→
(
(
¬
X41
)
↔
X49
)
)
→
X49
→
(
X47
↔
(
(
¬
X38
)
↔
X46
)
)
)
)
)
)
↔
(
(
¬
(
X32
↔
(
X26
→
(
X12
↔
(
X1
↔
False
)
)
)
)
)
↔
(
(
X49
→
(
X26
↔
(
X12
↔
X25
)
)
)
↔
(
¬
(
(
(
(
¬
X26
)
→
(
(
¬
X25
)
↔
X41
)
)
↔
(
(
(
¬
(
X10
↔
X25
)
)
→
(
¬
(
X41
→
(
X26
↔
(
X25
↔
X23
)
)
)
)
)
↔
(
(
¬
X26
)
→
X10
→
X41
)
)
)
→
(
X26
↔
(
X23
↔
(
X1
↔
X49
)
)
)
)
)
)
)
)
↔
(
(
(
(
X40
↔
(
X34
↔
(
(
¬
X5
)
↔
X6
)
)
)
↔
(
X46
→
(
X45
↔
(
X34
↔
False
)
)
)
)
↔
(
(
(
(
(
¬
X45
)
↔
(
(
¬
X24
)
↔
X46
)
)
↔
(
(
X45
↔
(
X34
↔
(
X30
↔
X24
)
)
)
↔
(
X46
↔
(
X34
→
(
(
¬
X30
)
↔
X45
)
)
)
)
)
↔
(
(
(
¬
X33
)
→
(
¬
X30
)
→
X19
→
X46
)
↔
(
X34
↔
(
(
¬
X19
)
→
X46
)
)
)
)
→
(
(
X49
→
(
X30
↔
(
(
¬
X29
)
↔
X43
)
)
)
↔
(
¬
(
X45
↔
(
X30
↔
X43
)
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X43
)
↔
(
(
¬
X34
)
→
X45
)
)
)
)
)
)
→
(
(
¬
(
(
¬
X45
)
→
(
¬
X43
)
→
X11
→
X46
)
)
↔
(
¬
(
(
¬
X45
)
→
(
(
¬
X31
)
↔
X46
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
(
¬
(
(
X43
↔
(
(
¬
X31
)
→
(
¬
X26
)
→
X46
)
)
→
(
¬
(
(
¬
X43
)
↔
(
(
¬
X31
)
↔
(
X25
→
X46
)
)
)
)
)
)
↔
(
¬
(
X46
↔
(
X43
↔
(
X26
→
X7
)
)
)
)
)
↔
(
X31
→
(
¬
X19
)
→
X7
→
X46
)
)
→
(
(
(
¬
(
(
¬
(
X46
→
(
X33
↔
(
X26
→
False
)
)
)
)
→
(
¬
(
(
¬
X33
)
↔
(
X26
↔
(
X23
↔
X46
)
)
)
)
→
(
(
¬
X33
)
↔
(
(
¬
X31
)
→
X23
→
X46
)
)
)
)
↔
(
(
¬
X26
)
→
X46
)
)
↔
(
¬
(
X46
↔
(
(
¬
X26
)
↔
(
(
¬
X23
)
→
X48
)
)
)
)
)
)
↔
(
(
(
X2
↔
X51
)
↔
(
(
(
(
¬
X28
)
→
X46
)
↔
(
(
¬
X40
)
→
(
X30
↔
(
X2
→
X46
)
)
)
)
→
(
(
¬
X30
)
↔
(
X2
→
X31
)
)
)
)
↔
(
(
(
¬
(
X33
↔
(
(
¬
X31
)
↔
(
X20
→
X40
)
)
)
)
↔
(
(
¬
X40
)
→
(
¬
X33
)
→
(
X31
↔
X46
)
)
)
↔
(
(
¬
(
(
¬
X40
)
→
(
(
¬
X32
)
↔
(
X18
↔
X46
)
)
)
)
→
(
¬
(
X33
→
(
(
¬
X22
)
↔
(
(
¬
X18
)
→
X40
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
(
(
¬
X46
)
↔
(
X29
↔
(
X16
↔
X50
)
)
)
)
→
(
X48
↔
(
X29
→
X46
)
)
)
)
→
(
¬
(
(
¬
X48
)
↔
(
X46
→
X50
)
)
)
)
↔
(
X46
↔
(
X29
→
(
X16
↔
X51
)
)
)
)
↔
(
(
¬
(
(
¬
X50
)
→
(
¬
X48
)
→
(
¬
X14
)
→
X51
)
)
↔
(
¬
(
X50
↔
(
X34
→
X14
→
X48
)
)
)
)
)
→
(
¬
(
(
¬
(
X46
↔
(
X34
↔
(
X17
→
False
)
)
)
)
→
(
¬
(
(
X34
→
(
(
¬
X17
)
↔
X29
)
)
→
(
¬
(
X46
↔
(
X15
↔
X34
)
)
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
X50
↔
(
X48
→
X52
)
)
)
→
X45
→
(
¬
X11
)
→
X46
)
→
(
(
¬
(
(
¬
X21
)
↔
X46
)
)
↔
(
(
X52
↔
(
X21
↔
(
(
¬
X9
)
↔
X46
)
)
)
↔
(
X46
→
(
¬
X21
)
→
X48
)
)
)
)
↔
(
(
X49
→
(
(
¬
X46
)
↔
(
X0
→
X48
)
)
)
↔
(
¬
(
(
X39
↔
(
X27
→
X6
→
X0
→
False
)
)
→
(
¬
(
(
¬
X48
)
→
(
(
¬
X39
)
↔
(
X0
↔
X52
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
(
X49
↔
(
X13
→
X1
)
)
→
(
¬
(
X49
→
(
¬
X4
)
→
X44
)
)
)
)
↔
(
¬
(
X37
↔
(
(
¬
X4
)
→
X1
→
X36
)
)
)
)
→
(
¬
X41
)
→
(
¬
X37
)
→
(
(
¬
X36
)
↔
X44
)
)
)
↔
(
(
(
¬
(
X50
→
X47
→
X36
→
X37
)
)
→
(
X50
↔
(
X5
↔
X24
)
)
)
↔
(
(
X36
↔
(
X13
→
False
)
)
↔
(
X50
→
(
¬
X35
)
→
(
(
¬
X13
)
↔
X47
)
)
)
)
)
↔
(
(
(
(
¬
(
X49
→
(
X47
↔
(
(
¬
X10
)
→
X12
)
)
)
)
↔
(
(
(
¬
(
X47
→
(
X42
↔
(
(
¬
X12
)
↔
X49
)
)
)
)
↔
(
¬
(
X52
↔
(
X47
↔
False
)
)
)
)
→
(
¬
X12
)
→
X49
)
)
↔
(
(
¬
X47
)
↔
(
X10
↔
X52
)
)
)
↔
(
(
X52
→
(
X50
↔
(
X49
↔
X38
)
)
)
→
(
¬
(
(
¬
(
X47
↔
(
X8
↔
X50
)
)
)
→
(
(
(
¬
X50
)
→
(
X47
↔
(
X38
↔
X52
)
)
)
↔
(
(
(
(
X52
→
X50
→
(
X3
↔
X8
)
)
↔
(
¬
(
X52
→
X50
→
X47
)
)
)
↔
(
X8
→
X3
→
X50
)
)
↔
(
X47
→
X3
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
7b69aa...
and proposition id is
d402a4...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMbWFpLVYBFAzPHNa8gkaBgrcSA5YQ2tYan
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
(
(
(
(
(
¬
(
(
¬
X37
)
↔
(
(
¬
X33
)
→
(
X27
↔
X49
)
)
)
)
→
(
¬
(
(
¬
X37
)
→
X33
→
X40
)
)
)
↔
(
¬
(
(
X40
→
(
¬
X37
)
→
X28
→
X47
)
→
(
¬
(
X33
→
(
(
¬
X28
)
↔
X40
)
)
)
)
)
)
↔
(
(
(
¬
(
X40
→
(
¬
X37
)
→
X49
)
)
→
(
¬
(
(
¬
X29
)
→
X25
→
X40
)
)
)
↔
(
(
X49
↔
(
X37
↔
(
X30
↔
False
)
)
)
↔
(
(
¬
(
X40
↔
(
X37
→
X32
→
X48
)
)
)
↔
(
(
¬
(
X49
→
(
¬
X40
)
→
X48
)
)
↔
(
¬
(
(
X48
→
(
(
¬
X37
)
↔
(
(
¬
X9
)
↔
X49
)
)
)
→
(
¬
X48
)
→
(
¬
X40
)
→
X49
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
X49
↔
(
X39
→
(
X10
↔
X41
)
)
)
→
(
¬
(
X49
→
X48
→
(
X3
↔
X41
)
)
)
→
(
(
¬
X39
)
↔
(
(
¬
X3
)
→
X48
)
)
)
→
(
(
(
(
¬
(
X49
→
X46
→
(
X41
↔
(
X36
↔
False
)
)
)
)
↔
(
(
¬
X41
)
→
X36
→
(
(
¬
X16
)
↔
X49
)
)
)
→
(
¬
(
(
¬
X46
)
↔
(
X36
↔
(
(
¬
X34
)
↔
X49
)
)
)
)
)
↔
(
(
(
X49
↔
(
X48
→
X36
)
)
↔
(
¬
(
(
¬
(
X49
↔
(
X48
↔
(
X34
↔
X44
)
)
)
)
→
(
X49
↔
(
(
¬
X36
)
↔
(
X34
↔
X48
)
)
)
)
)
)
↔
(
¬
(
(
¬
X44
)
→
(
X36
↔
X48
)
)
)
)
)
)
)
→
(
(
(
(
¬
X35
)
↔
(
(
¬
X31
)
↔
(
X0
→
X48
)
)
)
→
(
¬
(
(
¬
X47
)
→
(
¬
X45
)
→
X48
)
)
)
↔
(
¬
(
(
X41
↔
(
X7
→
X48
)
)
→
(
¬
(
(
¬
(
(
¬
X48
)
→
X50
)
)
→
(
(
(
¬
X39
)
↔
(
X1
↔
X48
)
)
↔
(
¬
(
X48
→
X41
→
(
X1
↔
X50
)
)
)
)
)
)
→
(
(
¬
(
(
¬
X45
)
→
X41
→
(
(
¬
X35
)
↔
X50
)
)
)
↔
(
¬
(
X50
→
X45
→
X35
→
X7
)
)
)
→
(
¬
(
(
¬
X41
)
↔
(
X17
↔
X48
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
(
¬
(
X42
→
(
(
¬
X21
)
↔
(
X6
→
X46
)
)
)
)
→
(
¬
(
X46
→
X42
→
X21
→
X11
)
)
)
→
(
(
(
¬
(
X42
↔
(
(
¬
X26
)
↔
(
X6
↔
X48
)
)
)
)
→
(
(
¬
X42
)
↔
(
(
¬
X38
)
↔
(
X26
↔
X48
)
)
)
)
↔
(
¬
(
X48
→
(
¬
X38
)
→
X11
→
X42
)
)
)
)
)
↔
(
(
(
X23
↔
(
X22
↔
X14
)
)
↔
(
(
X24
→
(
¬
X23
)
→
(
(
¬
X18
)
↔
X42
)
)
↔
(
¬
(
X42
↔
(
X24
↔
(
X22
→
X18
)
)
)
)
)
)
↔
(
(
X48
↔
(
(
¬
X24
)
↔
(
(
¬
X5
)
↔
X42
)
)
)
↔
(
(
¬
(
X38
→
(
¬
X18
)
→
X5
→
X24
)
)
→
(
¬
(
X38
↔
(
X18
→
(
¬
X5
)
→
X42
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
X19
→
X15
→
X8
→
X20
)
↔
(
X43
↔
(
(
¬
X19
)
↔
X48
)
)
)
↔
(
(
X19
→
(
¬
X12
)
→
X20
)
↔
(
¬
(
X50
↔
(
X43
↔
(
X24
↔
(
X20
→
False
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X4
→
X2
)
)
→
(
(
¬
(
X26
↔
(
X13
→
(
X12
↔
X24
)
)
)
)
↔
(
X43
↔
(
X24
→
(
X12
↔
X26
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
¬
(
X50
↔
(
(
¬
X13
)
→
X18
)
)
)
↔
(
¬
(
(
¬
(
X37
→
(
X13
↔
X50
)
)
)
→
(
¬
(
X50
↔
(
X35
→
X48
)
)
)
)
)
)
→
(
(
X49
→
(
¬
X37
)
→
X45
)
↔
(
X49
↔
(
(
¬
X1
)
↔
X50
)
)
)
)
→
(
(
(
X49
↔
(
X42
↔
X45
)
)
↔
(
X49
↔
(
X45
↔
(
X29
→
X50
)
)
)
)
→
(
¬
(
X45
→
X42
→
(
(
¬
X0
)
↔
X49
)
)
)
)
→
(
¬
(
X49
→
X29
→
X50
)
)
)
)
→
(
(
(
¬
(
X28
↔
(
X14
→
(
X6
↔
X48
)
)
)
)
↔
(
¬
(
(
X34
↔
(
(
¬
X33
)
↔
(
X14
→
X36
)
)
)
→
(
(
(
X34
→
(
X28
↔
X48
)
)
→
X28
→
X14
→
X33
)
↔
(
¬
(
X33
→
(
¬
X14
)
→
X48
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X24
)
→
(
¬
X19
)
→
X49
)
)
→
(
(
(
¬
(
X49
→
(
(
¬
X24
)
↔
X25
)
)
)
↔
(
X36
↔
(
X24
↔
(
(
¬
X19
)
↔
X25
)
)
)
)
↔
(
¬
(
X36
→
(
X25
↔
(
(
¬
X24
)
↔
X34
)
)
)
)
)
)
↔
(
(
(
X25
↔
(
(
¬
X24
)
→
X4
→
X50
)
)
→
(
(
(
(
¬
(
(
¬
X25
)
→
(
¬
X10
)
→
(
X2
↔
X36
)
)
)
→
(
(
¬
(
X25
↔
(
(
¬
X2
)
↔
X50
)
)
)
↔
(
(
X50
→
(
X24
↔
(
(
¬
X2
)
↔
X25
)
)
)
↔
(
X25
↔
(
(
¬
X10
)
↔
(
X2
→
X50
)
)
)
)
)
)
↔
(
X50
→
(
X25
↔
(
(
¬
X2
)
↔
X41
)
)
)
)
↔
(
¬
(
X2
↔
X10
)
)
)
)
→
(
X50
↔
(
X48
→
(
X41
↔
X25
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
(
(
¬
X15
)
↔
(
X12
↔
X50
)
)
→
(
(
¬
(
X47
↔
(
(
¬
X40
)
→
(
X12
↔
X50
)
)
)
)
↔
(
¬
(
X50
→
(
¬
X40
)
→
(
¬
X15
)
→
X47
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X12
↔
X17
)
)
→
(
X47
↔
(
X17
→
(
¬
X15
)
→
X46
)
)
)
)
↔
(
¬
(
X46
→
(
(
¬
X38
)
↔
X47
)
)
)
)
)
↔
(
(
(
(
¬
(
X38
↔
(
(
¬
X5
)
↔
X47
)
)
)
↔
(
X47
↔
(
X23
→
X20
)
)
)
↔
(
X47
→
X38
)
)
↔
(
X50
↔
(
X23
↔
X38
)
)
)
)
→
(
(
(
X47
↔
(
X44
↔
(
X3
↔
False
)
)
)
↔
(
(
(
X47
→
X44
)
→
(
(
(
(
¬
X3
)
↔
X44
)
→
(
X50
↔
(
X48
→
(
X47
↔
(
X44
→
False
)
)
)
)
)
↔
(
¬
X3
)
)
)
↔
(
¬
(
(
¬
X47
)
→
(
X3
↔
X48
)
)
)
)
)
→
(
¬
X44
)
→
(
(
¬
X31
)
↔
(
(
¬
X21
)
↔
X47
)
)
)
→
(
¬
(
X46
→
(
(
¬
X21
)
↔
X31
)
)
)
)
→
(
(
(
(
¬
(
(
(
(
¬
X16
)
→
X39
)
↔
(
(
(
X49
→
X46
→
X39
→
False
)
↔
(
(
(
¬
(
X46
↔
(
X39
→
(
¬
X30
)
→
X49
)
)
)
↔
(
¬
(
X49
→
(
X48
↔
(
X39
→
X46
)
)
)
)
)
↔
(
X48
→
X49
)
)
)
↔
(
X49
→
(
X46
↔
(
(
¬
X16
)
→
X48
)
)
)
)
)
→
(
¬
X48
)
→
(
X46
↔
X49
)
)
)
↔
(
(
¬
(
X47
↔
(
X43
↔
(
X39
↔
(
X11
↔
False
)
)
)
)
)
↔
(
(
X49
→
(
(
¬
X39
)
↔
X48
)
)
↔
(
(
(
X49
↔
(
X47
↔
X39
)
)
↔
(
¬
(
X48
→
(
(
¬
X43
)
↔
X49
)
)
)
)
↔
(
¬
(
X43
→
X39
→
False
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X49
→
X48
→
X47
→
False
)
)
→
(
(
¬
(
(
¬
X48
)
↔
(
(
¬
X47
)
↔
(
X26
↔
X49
)
)
)
)
↔
(
¬
(
X47
→
(
X26
↔
X8
)
)
)
)
)
→
(
¬
X32
)
→
X49
)
)
)
↔
(
(
¬
(
X46
↔
(
(
¬
X22
)
↔
(
(
¬
X9
)
↔
X49
)
)
)
)
↔
(
X48
→
(
X27
↔
(
(
¬
X7
)
↔
X46
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
a17d1f...
and proposition id is
b4409c...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMNDnQNNPpPyByqU4MQyxJSqdifwSaGVAxF
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
¬
(
(
(
(
(
¬
(
X48
↔
(
(
¬
X13
)
↔
(
X11
→
X46
)
)
)
)
↔
(
¬
(
(
X18
→
(
¬
X13
)
→
X11
→
X48
)
→
(
X49
↔
(
X11
→
X13
)
)
)
)
)
↔
(
(
¬
(
(
X48
→
X13
→
(
X10
↔
X47
)
)
→
(
¬
(
(
¬
X48
)
↔
(
X13
→
X49
)
)
)
)
)
→
(
¬
(
X48
→
X47
→
(
X29
↔
X13
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X42
↔
(
X28
↔
(
X16
→
X19
)
)
)
)
→
(
(
¬
(
X42
→
(
X8
↔
(
(
¬
X3
)
→
X48
)
)
)
)
↔
(
(
(
X48
→
(
¬
X8
)
→
(
X3
↔
X19
)
)
↔
(
¬
(
X28
↔
(
X19
↔
X8
)
)
)
)
↔
X8
)
)
)
)
↔
(
(
X49
↔
(
X48
↔
(
X35
→
X42
)
)
)
↔
(
(
X49
→
(
X47
↔
X28
)
)
↔
(
(
X49
→
(
(
¬
X47
)
↔
(
(
¬
X28
)
→
X48
)
)
)
↔
(
X48
↔
(
X47
→
X35
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
(
¬
X42
)
↔
(
X15
↔
X48
)
)
→
(
¬
(
X48
↔
(
X42
↔
(
(
¬
X31
)
→
X32
)
)
)
)
)
)
→
(
(
¬
X47
)
↔
(
(
¬
X42
)
↔
(
(
¬
X32
)
↔
X48
)
)
)
)
)
↔
(
(
¬
(
X47
↔
(
X32
↔
X20
)
)
)
→
(
(
(
X42
→
X32
→
(
X20
↔
X17
)
)
→
(
¬
(
(
¬
X32
)
↔
(
X5
→
X42
)
)
)
)
↔
(
X42
↔
(
X20
→
X17
→
X30
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
(
¬
X47
)
→
(
(
¬
X41
)
↔
(
X26
↔
X48
)
)
)
→
(
¬
(
X47
→
(
X41
↔
(
X31
↔
X48
)
)
)
)
)
)
↔
(
¬
(
X48
→
(
(
¬
X44
)
↔
(
(
¬
X41
)
↔
X47
)
)
)
)
)
→
(
(
(
X44
→
X37
→
X47
)
↔
(
¬
(
X47
→
(
(
¬
X40
)
↔
(
(
¬
X37
)
→
X48
)
)
)
)
)
↔
(
(
(
¬
(
X48
↔
(
X37
↔
X33
)
)
)
→
(
¬
(
(
¬
(
(
¬
X47
)
↔
(
(
¬
X14
)
→
X48
)
)
)
→
(
¬
(
(
¬
(
X48
→
(
X47
↔
X37
)
)
)
→
(
¬
(
(
X37
→
(
X33
↔
(
X14
↔
X47
)
)
)
→
(
¬
(
X48
→
(
(
¬
X33
)
↔
X37
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
X47
)
→
X14
→
X48
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
X34
↔
(
(
¬
X22
)
→
X25
)
)
)
→
(
(
¬
(
X22
→
X1
→
X25
)
)
↔
(
¬
(
X23
↔
(
X22
→
X1
→
X34
)
)
)
)
)
↔
(
X34
→
(
(
¬
X1
)
↔
X23
)
)
)
↔
(
(
(
X49
→
(
X34
↔
(
X0
→
False
)
)
)
↔
(
X27
↔
(
(
¬
X6
)
→
X25
)
)
)
↔
(
(
X43
↔
(
X34
→
(
X12
↔
X9
)
)
)
→
(
(
X49
→
X9
→
X23
)
↔
(
(
X43
↔
(
X27
↔
(
X9
↔
X23
)
)
)
↔
(
X49
↔
(
X43
↔
(
(
¬
X23
)
↔
X27
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
X43
)
→
X39
→
(
(
¬
X27
)
↔
X49
)
)
)
→
(
¬
(
(
¬
(
X45
↔
(
X39
↔
(
X21
→
X24
)
)
)
)
→
(
¬
X27
)
→
X21
→
X39
)
)
)
→
(
(
(
X39
→
(
(
¬
X36
)
↔
(
(
¬
X4
)
↔
X38
)
)
)
→
X39
→
X36
→
(
X4
↔
X2
)
)
↔
(
(
¬
(
X49
→
X38
→
X36
)
)
→
(
(
(
¬
(
X45
→
X44
→
(
X7
↔
X42
)
)
)
→
(
¬
(
(
¬
(
X42
↔
(
(
¬
X36
)
→
X44
)
)
)
→
(
¬
(
X44
→
(
(
¬
X36
)
↔
(
X7
→
X49
)
)
)
)
)
)
)
↔
(
X49
→
X44
→
(
X39
↔
X42
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
(
¬
(
X43
→
X12
→
(
X2
↔
X40
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X40
)
↔
(
(
¬
X12
)
→
X43
)
)
)
)
)
→
(
(
X46
↔
(
X40
↔
(
(
¬
X2
)
→
X39
)
)
)
↔
(
¬
(
X40
→
(
(
¬
X39
)
↔
(
X31
→
X46
)
)
)
)
)
→
(
¬
(
X40
↔
(
(
¬
X2
)
↔
X39
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X39
↔
(
X24
→
(
X4
↔
X20
)
)
)
)
→
(
(
X39
→
(
(
¬
X24
)
↔
(
X3
↔
X43
)
)
)
↔
(
¬
(
X46
→
(
(
¬
X39
)
↔
(
X5
→
X43
)
)
)
)
)
)
)
↔
(
(
¬
(
X36
↔
(
X25
↔
(
(
¬
X3
)
↔
X39
)
)
)
)
↔
(
¬
(
X36
→
(
¬
X25
)
→
X4
→
X39
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
X45
↔
(
X40
→
(
X34
↔
X7
)
)
)
→
(
¬
(
X47
→
(
(
¬
X34
)
↔
(
X7
→
X45
)
)
)
)
)
↔
(
¬
(
X40
↔
(
X7
→
False
)
)
)
)
↔
(
¬
(
(
¬
(
X47
→
(
X43
↔
(
(
¬
X13
)
↔
X40
)
)
)
)
→
(
(
¬
X40
)
↔
(
X34
→
(
X13
↔
X47
)
)
)
)
)
)
→
(
¬
(
(
X47
↔
(
X45
↔
(
X43
→
(
X22
↔
False
)
)
)
)
→
(
(
¬
(
(
¬
(
X43
→
X41
→
X30
→
False
)
)
→
X47
→
(
¬
X30
)
→
(
¬
X27
)
→
X41
)
)
↔
(
(
¬
X43
)
↔
(
(
¬
X41
)
→
(
¬
X30
)
→
X45
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
X41
→
X16
)
→
(
X41
↔
(
X40
↔
(
X36
↔
X14
)
)
)
)
)
→
(
¬
(
(
¬
X36
)
↔
(
(
¬
X14
)
→
X40
)
)
)
)
↔
(
(
¬
(
X16
↔
(
X14
↔
X10
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X34
)
↔
(
X16
→
X10
→
X41
)
)
)
→
(
¬
(
(
¬
X34
)
→
(
¬
X14
)
→
(
(
¬
X10
)
↔
X41
)
)
)
)
)
↔
(
X14
→
(
X10
↔
(
X0
→
X34
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
X47
)
↔
(
X44
↔
(
X28
↔
X48
)
)
)
↔
(
(
¬
(
(
(
X48
→
(
X42
↔
(
(
¬
X21
)
→
X28
)
)
)
↔
(
(
¬
(
X28
↔
X48
)
)
↔
(
¬
(
(
¬
(
X42
↔
(
X32
↔
False
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X32
)
↔
(
X28
↔
X48
)
)
)
)
)
)
)
)
→
(
¬
X1
)
→
X28
)
)
→
(
X32
↔
(
X21
→
(
(
¬
X6
)
↔
X28
)
)
)
)
)
↔
(
(
(
¬
(
(
X46
→
(
X15
↔
(
X8
↔
X18
)
)
)
→
(
¬
(
(
(
¬
(
X46
→
(
X18
↔
(
X9
→
X8
)
)
)
)
→
(
¬
X46
)
→
(
¬
X18
)
→
X9
→
X49
)
→
(
¬
(
X18
→
X9
→
X46
)
)
)
)
)
)
↔
(
¬
(
(
X44
↔
(
X15
→
X11
→
X9
)
)
→
(
¬
X44
)
→
X49
)
)
)
↔
(
(
(
X48
↔
(
(
¬
X29
)
→
X44
)
)
→
(
X49
↔
(
(
¬
X44
)
↔
(
X28
→
X48
)
)
)
→
X44
→
(
X29
↔
(
X28
→
X49
)
)
)
↔
(
¬
(
X49
→
(
X46
↔
(
X44
→
X48
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X49
↔
(
(
¬
X39
)
→
X48
)
)
)
↔
(
X49
→
(
¬
X38
)
→
X35
→
X39
)
)
→
(
(
(
¬
X42
)
→
(
X23
↔
(
(
¬
X19
)
↔
X48
)
)
)
↔
(
(
(
¬
X23
)
↔
(
(
¬
X19
)
↔
X49
)
)
↔
(
¬
(
X49
→
(
X38
↔
(
(
¬
X23
)
→
X42
)
)
)
)
)
)
)
→
(
(
¬
(
X38
↔
(
X33
↔
(
(
¬
X17
)
↔
X26
)
)
)
)
↔
(
(
¬
(
(
¬
X37
)
↔
(
X26
↔
(
(
¬
X17
)
→
X49
)
)
)
)
↔
(
¬
(
X48
→
X37
→
(
X26
↔
False
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
727ea9...
and proposition id is
621cd2...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMMqQXCtRHebSbLqJ7Jt1HruVa9wWFJ1pxD
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
∀X50 :
prop
,
∀X51 :
prop
,
∀X52 :
prop
,
∀X53 :
prop
,
(
(
(
(
(
(
(
¬
(
X49
→
(
X48
↔
(
X43
→
(
X12
↔
False
)
)
)
)
)
→
(
¬
(
X51
→
(
X49
↔
(
X43
→
X12
→
False
)
)
)
)
)
→
(
X48
→
X2
→
False
)
→
(
¬
(
(
¬
X43
)
↔
(
X35
↔
(
X2
↔
X49
)
)
)
)
)
↔
(
(
(
(
¬
X5
)
↔
X8
)
↔
(
¬
(
(
(
¬
(
X33
→
X22
→
False
)
)
↔
(
(
(
X51
↔
(
(
¬
X22
)
→
X33
)
)
↔
(
¬
(
(
¬
X33
)
↔
(
(
¬
X22
)
↔
(
(
¬
X8
)
→
X52
)
)
)
)
)
↔
(
¬
(
X33
↔
X8
)
)
)
)
→
(
¬
(
X52
→
(
(
¬
X9
)
↔
(
X8
→
X33
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X48
)
↔
(
X22
↔
(
X19
↔
X52
)
)
)
)
↔
(
¬
(
X33
→
(
X19
↔
False
)
)
)
)
→
(
¬
(
(
(
¬
X33
)
→
X23
→
X43
)
→
(
¬
(
X43
↔
(
(
¬
X33
)
↔
(
(
¬
X19
)
↔
X52
)
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
X38
→
X47
)
)
↔
(
(
(
¬
(
(
¬
X10
)
→
X50
)
)
↔
(
X38
→
X30
→
X47
)
)
↔
(
¬
(
X45
↔
X30
)
)
)
)
↔
(
(
(
X50
↔
(
(
¬
X38
)
↔
(
X32
→
X45
)
)
)
↔
(
X50
↔
(
X32
→
(
¬
X0
)
→
X38
)
)
)
↔
(
(
(
X45
→
(
¬
X32
)
→
X36
)
↔
(
X38
↔
(
X32
→
X51
)
)
)
→
X51
→
X38
→
X36
)
)
)
→
(
(
(
(
X52
→
(
X44
↔
X28
)
)
↔
(
X52
→
(
X50
↔
(
X45
↔
False
)
)
)
)
→
(
(
(
X44
↔
(
(
¬
X37
)
↔
(
X28
→
X38
)
)
)
↔
(
(
¬
X44
)
↔
(
X37
↔
(
(
¬
X28
)
→
X45
)
)
)
)
↔
(
(
¬
(
X49
→
(
X24
↔
X38
)
)
)
→
(
¬
(
X45
→
(
X38
↔
(
X28
↔
False
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
X50
)
↔
(
X39
↔
(
X18
↔
X52
)
)
)
)
↔
(
¬
(
X44
→
False
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
X41
→
X53
)
)
↔
(
(
¬
(
X53
→
(
X52
↔
(
(
¬
X42
)
→
X48
)
)
)
)
→
(
¬
(
X52
→
(
(
¬
X42
)
↔
(
(
¬
X41
)
→
X53
)
)
)
)
)
)
→
(
X53
↔
(
X52
↔
(
X46
↔
X48
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X3
→
X1
)
)
↔
(
¬
(
(
¬
(
X34
↔
(
(
¬
X3
)
→
X1
→
X53
)
)
)
→
(
¬
(
X48
→
(
X3
↔
X1
)
)
)
)
)
)
→
(
(
(
(
¬
X21
)
→
X6
→
X34
)
↔
(
(
¬
X26
)
↔
(
X21
→
(
(
¬
X4
)
↔
X34
)
)
)
)
↔
(
(
X26
→
(
X21
↔
(
X15
↔
X6
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X21
)
→
X15
→
(
X6
↔
X48
)
)
)
→
(
¬
(
X48
↔
(
X21
→
(
¬
X15
)
→
X46
)
)
)
)
)
↔
(
X53
↔
(
X46
→
(
(
¬
X21
)
↔
X48
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X27
→
X25
→
X48
)
)
↔
(
¬
(
(
¬
(
X25
→
(
(
¬
X16
)
↔
X27
)
)
)
→
(
(
(
¬
X27
)
↔
(
(
¬
X25
)
↔
X48
)
)
↔
(
¬
(
(
X48
→
X27
→
(
X25
↔
False
)
)
→
(
(
¬
X25
)
↔
(
(
¬
X16
)
→
(
(
¬
X14
)
↔
X48
)
)
)
)
)
)
)
)
)
→
(
¬
(
X48
→
(
X27
↔
X50
)
)
)
→
(
¬
(
X48
↔
(
X16
↔
X53
)
)
)
)
)
→
(
(
(
(
(
(
(
¬
X25
)
↔
(
X20
↔
(
(
¬
X7
)
→
X31
)
)
)
→
(
(
X29
→
(
X25
↔
(
X7
→
X31
)
)
)
↔
(
¬
(
(
¬
X29
)
→
X7
→
X50
)
)
)
)
↔
(
X31
→
X29
→
(
(
¬
X13
)
↔
X25
)
)
)
↔
(
¬
(
(
¬
X31
)
↔
(
X25
↔
X50
)
)
)
)
↔
(
(
X50
↔
(
X40
↔
(
(
¬
X31
)
↔
X52
)
)
)
↔
(
X50
→
(
(
¬
X31
)
↔
(
(
¬
X29
)
↔
X52
)
)
)
)
)
→
(
(
¬
X20
)
↔
(
X17
↔
X27
)
)
)
→
(
¬
(
(
¬
X29
)
→
(
¬
X25
)
→
X11
→
X40
)
)
)
)
)
↔
(
(
(
¬
(
(
(
X51
↔
(
X46
→
(
X42
↔
(
X18
↔
False
)
)
)
)
↔
(
(
¬
(
X51
→
X46
→
(
X17
↔
False
)
)
)
↔
(
¬
(
X47
↔
(
(
¬
X18
)
→
X51
)
)
)
)
)
→
(
¬
(
(
¬
(
X46
↔
(
X7
→
X0
)
)
)
→
(
(
(
(
¬
(
X46
→
(
X7
↔
(
X4
→
X53
)
)
)
)
→
(
¬
(
X53
→
X7
→
X2
→
X46
)
)
)
↔
(
X53
↔
(
(
¬
X7
)
↔
X46
)
)
)
↔
(
(
¬
X46
)
↔
(
X7
↔
(
X0
→
X53
)
)
)
)
)
)
→
(
(
(
¬
X33
)
→
(
¬
X7
)
→
(
¬
X0
)
→
X51
)
↔
(
¬
(
X47
→
X33
→
False
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X47
)
↔
(
X46
↔
X51
)
)
)
↔
(
¬
(
(
(
X47
↔
(
(
¬
X46
)
→
(
X40
↔
X51
)
)
)
→
(
¬
(
X46
→
X35
→
X51
)
)
)
→
(
¬
(
(
(
¬
X47
)
→
(
(
¬
X40
)
↔
(
(
¬
X19
)
→
X51
)
)
)
→
(
¬
(
X47
↔
(
X32
→
(
X31
↔
X51
)
)
)
)
)
)
)
)
)
→
(
(
(
X46
→
(
X35
↔
X51
)
)
↔
(
(
¬
(
X46
→
X35
→
X22
)
)
↔
(
¬
(
(
X35
→
X22
)
→
(
X46
↔
(
(
¬
X35
)
↔
(
X8
↔
X51
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
X22
)
→
(
¬
X8
)
→
X35
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
X36
→
(
¬
X1
)
→
X37
)
)
→
(
¬
(
X37
→
(
X36
↔
(
X16
→
X50
)
)
)
)
)
)
→
(
(
¬
(
(
¬
X20
)
↔
(
(
¬
X5
)
→
X50
)
)
)
↔
(
(
(
¬
X28
)
↔
(
(
¬
X5
)
→
X36
)
)
↔
(
X37
→
X31
→
X26
)
)
)
)
→
(
(
¬
(
(
(
¬
X35
)
↔
(
(
¬
X20
)
↔
X50
)
)
→
X50
→
(
X37
↔
(
(
¬
X32
)
↔
X35
)
)
)
)
↔
(
(
¬
(
(
¬
X20
)
→
(
¬
X10
)
→
(
(
¬
X6
)
↔
X50
)
)
)
↔
(
¬
(
(
¬
X32
)
↔
(
(
¬
X6
)
↔
X50
)
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
(
¬
(
(
¬
(
X50
↔
(
(
¬
X39
)
↔
(
(
¬
X38
)
→
X52
)
)
)
)
→
(
¬
(
X43
→
X39
→
X52
)
)
)
)
↔
(
(
X38
↔
(
X13
↔
(
(
¬
X11
)
→
X52
)
)
)
→
(
¬
(
X52
↔
(
X43
→
False
)
)
)
)
)
→
(
¬
(
(
¬
(
(
X51
→
(
(
¬
X50
)
↔
(
(
¬
X38
)
→
X52
)
)
)
→
(
¬
(
X50
↔
(
X15
→
X52
)
)
)
)
)
→
(
¬
(
X38
↔
(
(
¬
X15
)
→
X9
→
X51
)
)
)
)
)
)
)
→
(
(
(
(
¬
X43
)
↔
(
(
¬
X34
)
→
(
¬
X29
)
→
X45
)
)
↔
(
X45
↔
(
X29
↔
(
(
¬
X24
)
→
X34
)
)
)
)
→
(
¬
(
(
¬
X43
)
→
(
¬
X24
)
→
X49
)
)
)
→
(
X49
→
(
(
¬
X3
)
↔
X51
)
)
→
(
X50
↔
(
X43
↔
(
X29
→
(
X3
↔
False
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
(
(
¬
X12
)
→
X23
)
→
(
¬
(
(
¬
(
X45
→
(
X41
↔
(
X27
→
(
X12
↔
False
)
)
)
)
)
→
X27
→
X23
→
X45
)
)
)
)
→
(
(
¬
X41
)
↔
(
(
¬
X21
)
→
(
X12
↔
X45
)
)
)
)
)
→
(
¬
(
(
X45
↔
X44
)
→
(
¬
(
X14
↔
X41
)
)
→
(
(
X45
→
(
¬
X41
)
→
(
X21
↔
X44
)
)
↔
(
¬
(
(
¬
X41
)
↔
(
(
¬
X21
)
↔
(
X14
↔
X45
)
)
)
)
)
→
X45
→
X44
→
X41
→
False
)
)
)
)
→
(
(
¬
(
(
¬
X47
)
↔
(
X45
→
(
¬
X41
)
→
X48
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
¬
X48
)
→
(
¬
X30
)
→
(
X25
↔
X50
)
)
→
(
¬
(
(
¬
X30
)
↔
(
X25
→
X51
)
)
)
)
)
→
(
¬
(
X45
↔
(
X44
↔
(
X25
→
X50
)
)
)
)
)
)
↔
(
X50
→
X47
→
X44
→
X30
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
f695dc...
and proposition id is
c6749b...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMExEtsW3i6fUwkVQSAAmT5ZjH6VCfZFgy1
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
(
(
¬
(
X48
→
(
(
¬
X40
)
↔
(
X4
↔
X47
)
)
)
)
→
(
X47
↔
(
X45
↔
(
X40
→
False
)
)
)
)
↔
(
(
(
(
(
¬
(
X47
→
(
X32
↔
X48
)
)
)
↔
(
¬
(
(
X48
↔
(
X32
↔
(
(
¬
X24
)
→
X47
)
)
)
→
(
¬
(
(
¬
X47
)
→
(
(
¬
X45
)
↔
(
(
¬
X32
)
→
X48
)
)
)
)
)
)
)
↔
(
X48
↔
(
(
¬
X24
)
↔
X32
)
)
)
↔
(
X47
→
X24
)
)
→
(
(
¬
(
(
¬
(
X47
→
(
X45
↔
(
X38
→
X32
→
False
)
)
)
)
→
X47
→
(
(
¬
X38
)
↔
(
(
¬
X2
)
↔
X45
)
)
)
)
↔
(
¬
(
X47
→
(
X45
↔
X38
)
)
)
)
)
)
→
(
(
¬
(
(
¬
X33
)
→
(
X29
↔
X40
)
)
)
↔
(
(
¬
(
X45
→
(
X39
↔
(
(
¬
X0
)
↔
X48
)
)
)
)
↔
(
¬
(
X48
↔
(
X33
↔
(
X0
↔
X42
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X27
)
→
X34
)
)
↔
(
(
(
X48
→
X45
→
(
(
¬
X36
)
↔
X49
)
)
↔
(
X48
↔
(
X45
→
(
¬
X23
)
→
X36
)
)
)
→
(
¬
(
(
¬
X45
)
→
(
¬
X36
)
→
(
(
¬
X34
)
↔
X48
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X36
)
↔
(
X31
→
X42
)
)
)
→
(
(
(
¬
(
(
¬
X36
)
↔
(
X20
→
(
(
¬
X17
)
↔
X49
)
)
)
)
↔
(
¬
(
X42
↔
(
X36
→
(
X20
↔
X49
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
X48
)
→
X36
→
X31
→
X49
)
↔
(
¬
(
X49
↔
(
X36
→
X3
→
X48
)
)
)
)
→
(
(
¬
(
(
¬
X48
)
→
(
X17
↔
(
X15
↔
X49
)
)
)
)
↔
(
X49
→
X48
→
(
(
¬
X15
)
↔
X36
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X42
)
↔
(
X36
↔
X48
)
)
)
→
(
(
¬
(
X48
↔
(
(
¬
X36
)
→
X42
)
)
)
↔
(
(
¬
(
X36
→
(
¬
X16
)
→
X48
)
)
↔
(
X16
→
X1
→
X42
)
)
)
)
)
→
X16
→
(
(
¬
X8
)
↔
(
X1
↔
X36
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X48
→
X26
→
X11
→
X6
)
)
↔
(
(
¬
X26
)
→
(
X9
↔
X28
)
)
)
↔
(
¬
(
(
(
(
¬
X26
)
→
(
(
¬
X25
)
↔
(
X11
→
X48
)
)
)
→
(
¬
(
X48
→
(
(
¬
X11
)
↔
X41
)
)
)
)
→
X25
→
X11
→
X43
)
)
)
→
(
(
(
(
(
¬
(
X48
→
X45
→
X43
→
False
)
)
→
X45
→
(
X43
↔
(
X18
↔
X37
)
)
)
↔
(
X45
↔
(
X43
→
(
X18
↔
(
X13
↔
False
)
)
)
)
)
↔
(
(
X45
→
X43
→
(
X41
↔
X18
)
)
→
(
¬
(
X43
↔
(
(
¬
X41
)
→
(
(
¬
X30
)
↔
X45
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X46
→
(
X45
↔
(
X19
→
X48
)
)
)
)
→
(
¬
(
(
¬
X43
)
↔
(
X7
→
X44
)
)
)
)
)
)
)
→
(
(
(
X44
→
(
(
¬
X21
)
↔
X22
)
)
↔
(
X45
→
(
X44
↔
(
X22
↔
X5
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X44
)
↔
(
(
¬
X12
)
→
X10
→
X45
)
)
)
→
(
(
(
¬
X44
)
→
X22
→
X12
→
X45
)
↔
(
¬
(
X45
→
(
X44
↔
(
X22
↔
X14
)
)
)
)
)
)
)
)
→
(
¬
(
X47
→
X45
→
X35
→
X46
)
)
→
(
(
¬
X22
)
↔
(
(
¬
X21
)
↔
X35
)
)
→
(
¬
(
(
¬
X45
)
↔
(
(
¬
X35
)
→
X46
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X49
↔
(
X46
→
X44
→
(
X32
↔
False
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X49
→
(
(
¬
X30
)
↔
X36
)
)
)
↔
(
¬
(
(
X49
↔
(
X43
→
X36
)
)
→
(
(
(
¬
X36
)
↔
(
X30
→
X49
)
)
↔
(
X43
→
X30
→
X26
)
)
)
)
)
→
(
¬
(
X49
→
(
X43
↔
X36
)
)
)
)
→
(
(
¬
(
(
(
X46
→
X43
→
(
¬
X30
)
→
X44
)
↔
(
(
¬
(
(
¬
X43
)
→
(
(
¬
X22
)
↔
X44
)
)
)
↔
(
(
¬
X22
)
↔
(
X7
↔
X46
)
)
)
)
→
(
¬
(
(
¬
X44
)
↔
(
(
¬
X43
)
↔
X46
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X44
)
↔
(
X43
↔
(
(
¬
X16
)
→
X46
)
)
)
)
→
(
¬
(
(
¬
X46
)
↔
(
(
¬
X44
)
↔
(
(
¬
X16
)
→
X49
)
)
)
)
)
↔
(
¬
(
X49
→
(
¬
X43
)
→
X44
)
)
)
)
)
)
)
→
(
(
(
(
(
(
¬
(
X36
→
X33
→
X42
)
)
→
(
(
¬
(
X42
→
(
X36
↔
(
X33
→
X43
)
)
)
)
↔
(
X43
→
X5
)
)
)
↔
(
¬
(
X42
↔
(
X36
→
(
X32
↔
X43
)
)
)
)
)
↔
(
X42
→
(
¬
X36
)
→
X10
→
X43
)
)
→
(
(
¬
X31
)
↔
(
(
¬
X10
)
↔
(
(
¬
X5
)
→
X42
)
)
)
)
→
(
(
X41
→
X33
→
(
(
¬
X28
)
↔
X42
)
)
→
(
¬
(
(
¬
X41
)
→
(
X33
↔
(
(
¬
X29
)
↔
X42
)
)
)
)
)
→
(
¬
(
X48
→
X36
→
X29
→
X42
)
)
)
→
(
¬
(
(
(
X43
↔
(
(
¬
X24
)
→
(
X13
↔
X25
)
)
)
↔
(
¬
(
X43
→
(
(
¬
X24
)
↔
(
X15
↔
X25
)
)
)
)
)
→
(
(
(
X41
→
(
X25
↔
(
X20
→
X19
→
False
)
)
)
→
(
(
¬
(
X20
→
X18
→
X4
→
False
)
)
↔
(
¬
(
(
¬
X20
)
→
(
¬
X19
)
→
(
¬
X18
)
→
X25
)
)
)
)
↔
(
¬
(
X42
↔
(
X20
↔
(
X19
→
(
X18
↔
False
)
)
)
)
)
)
→
(
(
¬
(
(
¬
X25
)
↔
(
X20
→
(
¬
X13
)
→
X42
)
)
)
↔
(
¬
(
(
¬
X25
)
→
(
X20
↔
(
X13
↔
X43
)
)
)
)
)
→
(
¬
(
X43
→
X42
→
X25
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
X39
)
→
(
(
¬
X3
)
↔
(
X2
→
X48
)
)
)
↔
(
¬
(
X40
↔
(
X39
→
X2
→
X48
)
)
)
)
↔
(
(
(
¬
(
X48
↔
(
(
¬
X11
)
↔
X46
)
)
)
↔
(
X46
↔
(
X40
↔
(
X14
↔
(
X11
→
False
)
)
)
)
)
↔
(
¬
(
X48
↔
(
X47
↔
(
X46
↔
X11
)
)
)
)
)
)
↔
(
(
¬
(
X46
→
(
X40
↔
(
(
¬
X1
)
↔
X27
)
)
)
)
↔
(
X27
→
(
(
¬
X11
)
↔
(
X1
→
X46
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X40
→
X21
→
X45
)
)
↔
(
(
X37
↔
(
X21
→
(
¬
X8
)
→
X40
)
)
→
(
X49
↔
(
(
¬
X37
)
→
(
X21
↔
X40
)
)
)
)
)
↔
(
X49
→
(
(
¬
X37
)
↔
(
(
¬
X17
)
↔
X45
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X46
)
→
(
(
¬
X40
)
↔
X49
)
)
)
→
(
¬
(
X6
→
X45
)
)
)
)
)
↔
(
(
(
(
X34
→
(
X0
↔
X49
)
)
→
(
(
(
¬
(
(
¬
(
X46
→
X12
→
(
X0
↔
X49
)
)
)
→
(
(
¬
X47
)
↔
(
X12
↔
(
X0
→
X49
)
)
)
)
)
↔
(
¬
(
(
¬
X47
)
→
(
X34
↔
(
(
¬
X0
)
↔
X49
)
)
)
)
)
↔
(
(
¬
X0
)
→
X49
)
)
)
→
(
(
¬
X47
)
↔
(
X46
↔
X49
)
)
)
→
(
¬
(
(
(
(
(
X40
→
X38
→
(
X23
↔
False
)
)
→
(
(
X40
↔
(
(
¬
X38
)
↔
(
X35
→
X47
)
)
)
↔
(
¬
(
X47
↔
(
X40
↔
(
X38
→
X9
)
)
)
)
)
)
↔
(
(
¬
X35
)
→
(
(
¬
X23
)
↔
X40
)
)
)
↔
(
X47
→
(
X40
↔
(
X35
→
False
)
)
)
)
→
(
¬
(
X47
↔
(
X35
→
(
X34
↔
X38
)
)
)
)
→
(
X38
↔
(
(
¬
X34
)
→
X12
→
X35
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
a535f7...
and proposition id is
f9b879...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMdD2N1kTVuPz8sBcdqAnBuZmuN7Y8dYbRd
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
∀X50 :
prop
,
∀X51 :
prop
,
(
(
(
(
(
(
¬
(
(
¬
(
X37
↔
(
X6
↔
X29
)
)
)
→
(
(
(
¬
(
(
¬
X37
)
↔
(
X29
↔
X49
)
)
)
→
X49
→
(
X37
↔
(
X24
↔
X36
)
)
)
↔
(
X49
→
X29
)
)
)
)
→
(
¬
(
(
(
¬
X28
)
→
(
¬
X24
)
→
(
X4
↔
X49
)
)
→
(
(
X49
→
X17
)
↔
(
¬
(
(
X49
↔
(
X28
→
(
¬
X17
)
→
X24
)
)
→
(
X24
↔
(
(
¬
X17
)
↔
(
X6
→
X49
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X37
→
X12
→
X36
)
)
↔
(
¬
(
(
X37
↔
(
(
¬
X36
)
↔
(
(
¬
X12
)
↔
X48
)
)
)
→
(
¬
(
(
¬
X46
)
↔
(
(
¬
X37
)
↔
(
X36
→
X48
)
)
)
)
)
)
)
→
(
(
X51
↔
(
X46
→
(
¬
X36
)
→
X37
)
)
↔
(
¬
(
X51
↔
(
X37
→
(
(
¬
X10
)
↔
X46
)
)
)
)
)
)
→
(
¬
(
(
X51
↔
(
X37
↔
(
(
¬
X9
)
↔
X48
)
)
)
→
(
(
¬
(
(
¬
X37
)
→
(
¬
X29
)
→
(
(
¬
X16
)
↔
X48
)
)
)
↔
(
¬
(
X48
→
X37
→
(
X29
↔
X51
)
)
)
)
)
)
)
)
)
↔
(
(
(
X50
→
(
X49
↔
(
(
¬
X33
)
↔
X47
)
)
)
→
(
(
(
¬
(
X49
→
(
¬
X47
)
→
(
(
¬
X45
)
↔
X50
)
)
)
→
(
¬
(
X50
↔
(
(
¬
X48
)
↔
(
X47
→
X49
)
)
)
)
)
↔
(
X45
↔
X48
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
¬
X45
)
→
X43
→
(
(
¬
X41
)
↔
X51
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X43
)
→
(
X26
↔
(
X0
→
X51
)
)
)
)
→
X51
→
(
(
¬
X0
)
↔
X45
)
)
)
)
→
(
¬
(
(
¬
X41
)
↔
X47
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
¬
X47
)
↔
(
X45
↔
X51
)
)
)
→
(
X51
↔
(
X49
→
X38
)
)
)
)
→
(
(
¬
(
X49
↔
(
X15
↔
(
X14
↔
X47
)
)
)
)
↔
(
¬
(
(
¬
X49
)
→
(
(
¬
X47
)
↔
(
(
¬
X14
)
→
X51
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X49
→
X44
→
(
(
¬
X31
)
↔
X35
)
)
)
↔
(
¬
(
(
X46
→
X44
→
(
(
¬
X31
)
↔
X50
)
)
→
(
¬
X39
)
)
)
)
↔
(
(
(
¬
(
X46
→
(
¬
X44
)
→
(
(
¬
X34
)
↔
X49
)
)
)
↔
(
¬
(
X46
→
(
X44
↔
(
X2
↔
X49
)
)
)
)
)
↔
(
¬
(
X49
→
(
X46
↔
X44
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X18
↔
X46
)
)
→
(
(
X42
↔
(
(
¬
X18
)
→
X35
)
)
↔
(
¬
(
(
¬
X35
)
↔
(
(
¬
X21
)
→
(
¬
X1
)
→
X42
)
)
)
)
)
)
→
(
(
(
(
(
(
¬
X39
)
→
(
(
¬
X23
)
↔
X46
)
)
↔
(
(
X46
→
X32
→
X23
→
False
)
↔
(
X39
→
(
(
¬
X32
)
↔
(
(
¬
X20
)
↔
X46
)
)
)
)
)
→
(
¬
X42
)
→
X39
→
(
(
¬
X32
)
↔
X46
)
)
→
(
X42
↔
(
X32
↔
(
(
¬
X7
)
↔
X39
)
)
)
)
↔
(
¬
(
(
(
¬
X32
)
→
(
X20
↔
(
(
¬
X7
)
→
X35
)
)
)
→
(
¬
(
X32
→
X20
→
X5
→
X35
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
¬
(
(
¬
(
X30
↔
(
(
¬
X19
)
↔
(
(
¬
X11
)
→
X22
)
)
)
)
→
(
¬
(
X35
→
X19
→
(
(
¬
X11
)
↔
X22
)
)
)
)
)
↔
(
(
¬
X19
)
→
(
X11
↔
X22
)
)
)
→
(
X42
↔
(
X35
↔
(
X22
↔
(
X19
→
False
)
)
)
)
)
→
(
(
(
¬
X35
)
→
(
X27
↔
(
(
¬
X22
)
→
X42
)
)
)
↔
(
X42
→
(
X22
↔
(
X13
↔
X35
)
)
)
)
)
)
→
(
(
(
¬
(
X25
→
X22
→
(
(
¬
X19
)
↔
X42
)
)
)
↔
(
X42
↔
(
X22
→
(
X19
↔
X50
)
)
)
)
↔
(
X50
→
(
X42
↔
X22
)
)
)
→
(
¬
(
(
X40
→
(
¬
X3
)
→
X25
)
→
(
¬
(
(
¬
X25
)
↔
(
X19
↔
(
(
¬
X8
)
→
X40
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
(
X47
↔
(
X19
↔
X7
)
)
↔
(
¬
(
X45
↔
(
(
¬
X22
)
↔
(
(
¬
X19
)
→
X25
)
)
)
)
)
→
(
¬
(
(
¬
X29
)
↔
(
(
¬
X25
)
→
X22
→
X45
)
)
)
)
→
(
¬
(
(
(
X45
→
X22
→
(
(
¬
X15
)
↔
X31
)
)
→
(
¬
(
(
X45
→
X31
→
False
)
→
(
¬
(
X31
→
(
¬
X22
)
→
(
X3
↔
X50
)
)
)
)
)
)
→
(
(
¬
X25
)
↔
(
(
¬
X22
)
→
X8
→
X45
)
)
)
)
)
)
↔
(
(
(
(
X48
→
X45
→
(
¬
X30
)
→
X50
)
→
(
¬
(
(
¬
X45
)
→
(
X30
↔
(
(
¬
X27
)
↔
X50
)
)
)
)
)
→
(
(
(
(
¬
(
X50
↔
(
(
¬
X48
)
↔
(
X47
↔
X49
)
)
)
)
↔
(
¬
(
X50
↔
(
X49
→
X48
→
X45
)
)
)
)
→
(
X49
↔
(
X48
→
X47
→
False
)
)
)
↔
(
¬
(
X49
↔
(
X48
→
False
)
)
)
)
→
(
¬
(
X49
↔
(
X48
↔
(
(
¬
X2
)
↔
X50
)
)
)
)
)
→
(
(
(
¬
(
(
X33
↔
(
X6
↔
X37
)
)
→
(
X50
↔
(
X37
↔
(
X26
↔
False
)
)
)
→
(
¬
(
(
¬
X37
)
↔
(
(
¬
X33
)
↔
(
(
¬
X6
)
→
X50
)
)
)
)
)
)
↔
(
¬
(
X37
↔
False
)
)
)
↔
(
(
¬
X48
)
↔
(
(
¬
X37
)
↔
(
X6
→
X49
)
)
)
)
→
(
(
X33
→
X18
→
X26
)
↔
(
(
X44
→
X26
→
(
(
¬
X18
)
↔
X49
)
)
→
(
X44
↔
(
X33
→
(
(
¬
X26
)
↔
X49
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
(
X41
→
X38
→
X34
→
X48
)
)
→
(
¬
(
(
¬
(
X48
→
(
X41
↔
(
(
¬
X4
)
→
X20
)
)
)
)
→
(
X50
↔
(
X38
↔
(
(
¬
X4
)
↔
X48
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
X42
)
↔
(
X38
→
(
(
¬
X14
)
↔
X51
)
)
)
)
→
X48
→
(
¬
X42
)
→
X50
)
)
↔
(
(
(
(
(
¬
(
X49
↔
(
X36
→
X23
)
)
)
↔
(
(
¬
(
X50
→
(
X49
↔
(
X23
↔
X9
)
)
)
)
↔
(
X50
→
(
X36
↔
(
(
¬
X9
)
↔
X23
)
)
)
)
)
↔
(
X50
↔
(
X36
↔
X49
)
)
)
→
(
¬
(
X49
↔
(
X36
↔
(
X12
→
X50
)
)
)
)
)
↔
(
(
(
(
¬
(
X39
→
(
X36
↔
(
X21
→
X49
)
)
)
)
→
(
¬
(
(
¬
X39
)
→
(
¬
X36
)
→
(
X21
↔
X48
)
)
)
)
↔
(
X49
↔
(
X39
→
X36
→
(
X13
↔
False
)
)
)
)
↔
(
¬
(
(
¬
(
X39
↔
(
(
¬
X28
)
↔
(
(
¬
X13
)
↔
X50
)
)
)
)
→
(
(
¬
(
(
¬
X48
)
↔
(
X42
→
(
(
¬
X28
)
↔
X50
)
)
)
)
↔
(
¬
X39
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X46
→
(
X16
↔
X48
)
)
)
→
(
(
¬
(
X48
→
(
(
¬
X46
)
↔
(
X32
→
X49
)
)
)
)
↔
(
¬
(
(
¬
X46
)
↔
(
X32
↔
(
X16
→
X48
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
X48
→
(
(
¬
X40
)
↔
X49
)
)
↔
(
(
¬
(
X49
↔
(
(
¬
X40
)
→
(
¬
X17
)
→
X48
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X43
)
→
X40
→
X48
)
)
)
)
)
→
(
(
X48
→
(
¬
X46
)
→
X49
)
↔
(
(
X43
↔
(
(
¬
X0
)
→
X49
)
)
↔
(
X48
↔
(
(
¬
X46
)
→
(
X0
↔
X49
)
)
)
)
)
)
)
↔
(
(
¬
(
X49
↔
(
X17
↔
(
(
¬
X5
)
→
X48
)
)
)
)
↔
(
X49
↔
(
X48
↔
(
X17
↔
X11
)
)
)
)
)
)
→
(
(
(
X35
↔
(
(
¬
X24
)
↔
X43
)
)
↔
(
¬
(
X49
↔
(
X35
→
X10
→
False
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X24
)
↔
(
X10
↔
X51
)
)
)
→
(
¬
(
(
¬
(
(
¬
X49
)
→
(
(
(
(
¬
X49
)
→
(
¬
X24
)
→
(
(
¬
X1
)
↔
X51
)
)
→
(
¬
(
X51
↔
(
X49
↔
(
X1
→
X48
)
)
)
)
)
↔
(
X51
↔
(
(
¬
X48
)
↔
X49
)
)
)
)
)
→
(
¬
(
X49
→
X10
→
(
X1
↔
False
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
4a2332...
and proposition id is
d73a19...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMdCghwGGn3c2SrmL973ivLFTSbvMphW9Zc
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
¬
(
(
(
(
(
(
¬
X45
)
→
(
¬
X27
)
→
X48
)
↔
(
¬
(
(
X48
↔
X3
)
→
(
(
X27
→
(
(
¬
X15
)
↔
X45
)
)
↔
(
¬
(
X48
→
(
X45
↔
(
X27
↔
X3
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X32
→
(
(
¬
X27
)
↔
(
(
¬
X15
)
→
X45
)
)
)
)
↔
(
X45
↔
(
(
¬
X27
)
↔
(
X21
↔
X32
)
)
)
)
→
(
¬
(
(
¬
(
X45
↔
(
X32
↔
X27
)
)
)
→
(
X45
→
(
X27
↔
X32
)
)
→
(
X32
↔
(
X27
↔
(
X10
→
X25
)
)
)
)
)
)
)
↔
(
(
X46
→
X32
→
X25
→
X41
)
↔
(
¬
(
(
(
¬
X32
)
↔
X41
)
→
(
(
¬
(
X39
→
X32
→
X21
→
X46
)
)
↔
(
(
(
X46
↔
(
(
¬
X32
)
↔
X41
)
)
→
(
(
¬
(
(
¬
X41
)
→
(
¬
X39
)
→
(
(
¬
X12
)
↔
X46
)
)
)
↔
(
¬
(
(
¬
X41
)
→
(
(
¬
X39
)
↔
(
(
¬
X32
)
↔
X46
)
)
)
)
)
)
→
(
¬
X39
)
→
(
¬
X32
)
→
X46
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
(
¬
X39
)
↔
(
(
¬
X20
)
↔
(
X16
→
X47
)
)
)
→
(
(
X47
↔
(
X2
↔
False
)
)
↔
(
¬
(
(
¬
(
X47
→
(
¬
X23
)
→
(
X20
↔
X39
)
)
)
→
X23
→
(
¬
X20
)
→
(
(
¬
X2
)
↔
X47
)
)
)
)
)
)
↔
(
(
(
¬
X30
)
↔
(
X23
↔
(
X19
→
X49
)
)
)
↔
(
¬
(
(
(
¬
X39
)
↔
(
(
¬
X30
)
→
X47
)
)
→
(
¬
(
(
¬
(
X39
↔
(
X19
↔
X23
)
)
)
→
(
¬
X39
)
→
X30
→
X19
→
X47
)
)
)
)
)
)
→
(
(
(
X47
↔
(
X32
↔
X46
)
)
→
(
(
(
X47
↔
(
X32
→
X29
)
)
↔
(
X46
→
X32
→
(
X29
↔
X39
)
)
)
↔
(
(
¬
X39
)
↔
(
X32
↔
(
(
¬
X4
)
↔
X46
)
)
)
)
)
↔
(
(
(
(
¬
X46
)
↔
(
¬
(
X49
→
X32
→
(
¬
X31
)
→
X39
)
)
)
↔
(
(
¬
(
(
¬
X31
)
↔
(
(
¬
X5
)
↔
(
X1
↔
X32
)
)
)
)
↔
(
(
¬
X1
)
→
X46
)
)
)
→
(
(
(
¬
X42
)
→
(
¬
X11
)
→
X5
→
X49
)
↔
(
X49
↔
(
(
¬
X0
)
→
X5
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
X40
→
(
(
¬
X38
)
↔
(
(
¬
X33
)
→
X44
)
)
)
↔
(
(
X49
→
(
X43
↔
(
(
¬
X28
)
→
X40
)
)
)
↔
(
¬
(
(
¬
(
X49
→
X43
→
(
¬
X33
)
→
X40
)
)
→
(
X49
↔
(
(
¬
X38
)
→
(
¬
X9
)
→
X43
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X40
→
(
X24
↔
(
X22
↔
X26
)
)
)
)
→
(
X49
↔
(
X24
↔
(
X22
↔
False
)
)
)
)
)
→
(
(
¬
(
(
¬
X47
)
→
(
¬
X44
)
→
X26
→
X49
)
)
↔
(
(
(
¬
X44
)
→
(
¬
X26
)
→
(
X24
↔
X49
)
)
↔
(
¬
(
X44
↔
(
X26
↔
(
X24
→
X18
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X47
)
→
(
X37
↔
(
X6
→
X49
)
)
)
)
→
(
¬
(
X47
→
(
X37
↔
(
X14
↔
X49
)
)
)
)
→
(
(
X49
→
X47
→
(
X37
↔
X43
)
)
↔
(
¬
(
X37
↔
(
(
¬
X8
)
↔
X47
)
)
)
)
)
)
)
↔
(
(
(
False
↔
(
(
¬
(
(
¬
X40
)
↔
(
(
¬
X38
)
↔
(
X13
↔
X46
)
)
)
)
↔
(
X40
↔
(
X38
↔
(
(
¬
X37
)
↔
X46
)
)
)
)
)
→
(
¬
(
X38
→
(
X37
↔
X46
)
)
)
)
↔
(
(
(
¬
(
(
¬
X38
)
→
(
(
¬
X35
)
↔
X46
)
)
)
↔
(
¬
(
X38
↔
(
X37
↔
(
X36
→
X46
)
)
)
)
)
→
(
(
X38
↔
(
X36
→
(
X34
↔
(
X7
↔
False
)
)
)
)
↔
(
(
X38
↔
(
X37
→
X34
→
False
)
)
↔
(
X37
→
(
X36
↔
(
X17
↔
X38
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
(
(
(
(
¬
(
(
¬
X46
)
→
(
X39
↔
(
X28
↔
X49
)
)
)
)
↔
(
¬
(
X46
↔
(
X28
↔
(
X3
→
False
)
)
)
)
)
→
(
X49
↔
(
X46
↔
X39
)
)
)
↔
(
(
¬
(
X23
→
X46
)
)
↔
(
(
¬
(
(
¬
X46
)
↔
(
X23
→
(
(
¬
X17
)
↔
X49
)
)
)
)
→
(
¬
(
X39
→
(
(
¬
X23
)
↔
X46
)
)
)
)
)
)
→
(
(
(
¬
X26
)
↔
(
(
¬
X17
)
→
X49
)
)
↔
(
X26
→
(
(
¬
X17
)
↔
(
X13
→
X23
)
)
)
)
)
↔
(
¬
(
(
¬
(
X39
↔
(
(
¬
X23
)
↔
X28
)
)
)
→
(
(
X15
↔
(
X12
→
X6
→
X48
)
)
↔
(
¬
(
(
(
(
¬
X23
)
↔
(
(
¬
X15
)
↔
(
X9
↔
X48
)
)
)
↔
(
¬
(
(
¬
X23
)
↔
(
(
¬
X15
)
→
(
X9
↔
X28
)
)
)
)
)
→
(
¬
(
X48
→
(
X9
↔
(
X6
→
X15
)
)
)
)
)
)
)
)
)
)
→
(
(
X46
↔
(
X30
→
X20
→
X13
)
)
↔
(
X15
↔
(
X13
↔
X20
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
(
¬
(
(
¬
X47
)
↔
(
(
¬
X35
)
→
(
X34
↔
X48
)
)
)
)
→
(
¬
(
(
¬
X48
)
→
(
(
¬
X47
)
↔
(
(
¬
X34
)
→
X49
)
)
)
)
)
)
→
(
¬
(
X49
→
(
X47
↔
(
X7
↔
X48
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
X40
)
→
(
X2
↔
X47
)
)
)
→
(
¬
(
X48
↔
(
(
¬
X47
)
↔
(
(
¬
X40
)
→
X49
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X48
↔
(
X42
→
X27
→
X47
)
)
)
→
(
(
X42
→
(
¬
X29
)
→
X40
)
↔
(
X48
→
(
(
¬
X42
)
↔
(
(
¬
X29
)
→
X47
)
)
)
)
)
)
↔
(
(
(
¬
(
X39
→
(
X36
↔
(
X1
↔
X48
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X1
)
↔
X36
)
)
→
(
¬
(
(
¬
X39
)
↔
(
X0
→
X47
)
)
)
)
)
)
→
(
X48
↔
(
X42
→
X39
→
X47
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X48
→
X46
→
(
(
¬
X24
)
↔
X31
)
)
)
→
(
¬
(
X49
→
X48
→
X31
→
X24
→
False
)
)
)
↔
(
(
X49
→
(
¬
X46
)
→
(
X43
↔
X48
)
)
↔
(
(
X46
→
(
X21
↔
X16
)
)
→
(
(
¬
(
X46
→
(
X16
↔
(
(
¬
X10
)
↔
X43
)
)
)
)
↔
(
X48
→
(
¬
X21
)
→
X10
→
X43
)
)
)
)
)
→
(
¬
(
(
X49
↔
(
X38
↔
(
X8
→
X48
)
)
)
→
(
(
(
¬
X46
)
↔
X48
)
↔
(
X49
↔
(
(
¬
X38
)
↔
(
X32
→
X46
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X22
)
↔
X49
)
)
↔
(
(
¬
(
X46
↔
(
(
¬
X45
)
↔
(
X22
↔
X48
)
)
)
)
→
(
¬
X46
)
→
X45
→
(
¬
X22
)
→
X49
)
)
↔
(
(
¬
(
X46
↔
(
(
¬
X37
)
↔
(
X25
→
X45
)
)
)
)
↔
(
(
(
X45
↔
(
X37
→
X4
)
)
↔
(
¬
(
X37
→
(
(
¬
X18
)
↔
(
(
¬
X4
)
↔
X25
)
)
)
)
)
↔
(
X45
→
(
X37
↔
(
X25
→
X49
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X49
↔
(
X46
→
(
X19
↔
X48
)
)
)
)
→
(
¬
(
X48
↔
(
X46
↔
(
X41
↔
(
X14
→
False
)
)
)
)
)
→
(
(
¬
X48
)
↔
(
X46
↔
(
(
¬
X41
)
→
X49
)
)
)
)
)
→
(
(
(
(
(
¬
X44
)
→
(
X41
↔
(
(
¬
X14
)
↔
X48
)
)
)
→
(
¬
(
X48
→
(
X41
↔
(
(
¬
X11
)
↔
X14
)
)
)
)
)
→
(
X44
↔
X14
)
)
↔
(
(
¬
(
X44
↔
(
(
¬
X33
)
↔
X41
)
)
)
↔
(
(
¬
(
X48
↔
(
X33
↔
False
)
)
)
↔
(
(
¬
(
X48
↔
(
X44
↔
(
(
¬
X33
)
↔
X41
)
)
)
)
↔
(
X41
→
X33
→
(
(
¬
X5
)
↔
X44
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
c9cadf...
and proposition id is
0d19bd...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMS88HP2MCVfJrFHHbstxoDqC4jWZxg1vuz
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
¬
(
(
(
¬
(
(
(
¬
(
(
¬
(
(
¬
(
X12
↔
(
X11
↔
X26
)
)
)
→
(
¬
(
X28
↔
(
(
¬
X26
)
↔
X48
)
)
)
)
)
→
(
¬
(
X48
→
(
¬
X11
)
→
X26
)
)
)
)
↔
(
¬
(
(
¬
X26
)
↔
(
(
¬
X12
)
→
(
X0
↔
X28
)
)
)
)
)
→
(
(
(
(
¬
(
X6
↔
(
X0
↔
X9
)
)
)
↔
(
¬
(
(
(
X47
↔
(
(
¬
X6
)
↔
(
(
¬
X0
)
→
X9
)
)
)
→
(
X26
↔
(
(
¬
X6
)
↔
(
X0
→
X47
)
)
)
)
→
(
¬
(
(
¬
X26
)
→
(
(
¬
X6
)
↔
X47
)
)
)
)
)
)
↔
(
(
¬
X9
)
↔
(
(
¬
X0
)
↔
X26
)
)
)
↔
(
¬
(
X26
→
(
(
¬
X9
)
↔
(
(
¬
X6
)
↔
X48
)
)
)
)
)
)
)
→
(
(
(
¬
(
X48
↔
(
X34
↔
X38
)
)
)
↔
(
(
(
¬
X38
)
↔
(
X26
→
X48
)
)
↔
(
¬
(
(
¬
X38
)
→
(
(
¬
X34
)
↔
(
X24
→
X48
)
)
)
)
)
)
↔
(
(
X48
↔
(
X24
→
(
X15
↔
X31
)
)
)
→
(
(
¬
(
(
¬
X34
)
↔
(
X15
↔
(
(
¬
X14
)
↔
X48
)
)
)
)
↔
(
(
¬
(
X48
→
(
X47
↔
(
X44
→
X14
)
)
)
)
→
(
¬
(
X24
↔
(
X15
→
X44
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
(
(
X35
↔
(
X5
↔
(
(
¬
X3
)
→
X48
)
)
)
↔
(
X43
→
(
¬
X35
)
→
(
X5
↔
X44
)
)
)
→
(
(
¬
(
X44
→
(
¬
X10
)
→
X35
)
)
↔
(
(
X44
→
X10
→
X35
)
→
(
(
¬
X29
)
↔
(
X10
→
X5
→
X44
)
)
)
)
)
)
→
(
¬
(
(
¬
X29
)
→
(
¬
X16
)
→
(
(
¬
X10
)
↔
X35
)
)
)
)
→
(
(
¬
(
(
¬
(
X46
↔
(
X20
→
(
(
¬
X7
)
↔
X48
)
)
)
)
→
(
¬
(
X29
↔
(
(
¬
X7
)
↔
X20
)
)
)
)
)
↔
(
¬
(
X44
→
(
X29
↔
(
X7
↔
X20
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X42
→
(
¬
X8
)
→
X46
)
)
→
(
¬
(
(
(
X44
↔
(
(
¬
X42
)
→
(
X8
↔
X46
)
)
)
↔
(
(
¬
X46
)
→
(
X44
↔
(
X8
↔
X47
)
)
)
)
→
(
¬
(
(
¬
X44
)
↔
(
X42
↔
(
(
¬
X37
)
↔
X46
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X42
)
→
X49
)
)
↔
(
(
¬
(
X43
↔
(
X37
→
(
(
¬
X27
)
↔
X42
)
)
)
)
↔
(
¬
(
X37
↔
(
(
¬
X27
)
↔
X49
)
)
)
)
)
↔
(
(
¬
X37
)
↔
(
X27
→
X47
)
)
)
↔
(
(
¬
X27
)
→
X43
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
¬
X39
)
↔
(
X23
↔
(
(
¬
X13
)
↔
X49
)
)
)
↔
(
¬
(
X49
↔
(
X36
↔
(
X13
↔
False
)
)
)
)
)
→
X49
→
(
¬
X33
)
→
X39
)
→
(
¬
(
(
(
(
¬
(
(
¬
X45
)
↔
(
(
¬
X39
)
↔
(
(
¬
X36
)
↔
X47
)
)
)
)
↔
(
X45
↔
(
(
¬
X36
)
↔
(
(
¬
X30
)
↔
X39
)
)
)
)
↔
(
(
¬
(
X46
→
(
(
¬
X36
)
↔
(
X22
↔
X47
)
)
)
)
↔
(
¬
(
(
X47
→
(
(
¬
X39
)
↔
(
X23
→
X46
)
)
)
→
(
¬
(
X39
→
(
(
¬
X23
)
↔
(
(
¬
X22
)
↔
X47
)
)
)
)
)
)
)
)
→
(
(
X47
→
X46
→
(
X45
↔
X17
)
)
↔
(
¬
(
X47
↔
(
(
¬
X19
)
→
X17
→
X45
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
X46
↔
(
X45
↔
(
(
¬
X4
)
→
X41
)
)
)
→
(
(
¬
(
(
¬
(
X47
→
(
X2
↔
(
(
¬
X1
)
→
X46
)
)
)
)
→
(
X47
→
X46
→
(
(
¬
X4
)
↔
X41
)
)
→
X2
)
)
↔
(
X1
→
X46
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X41
↔
(
X25
→
(
X18
↔
X45
)
)
)
)
↔
(
X41
→
(
(
¬
X25
)
↔
X40
)
)
)
→
(
¬
(
X41
↔
(
(
¬
X25
)
→
(
X18
↔
X32
)
)
)
)
)
)
↔
(
X32
↔
(
X21
↔
False
)
)
)
→
(
X46
↔
(
X45
↔
(
X21
↔
X41
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
X47
↔
(
(
¬
X46
)
→
X45
→
X49
)
)
)
→
(
¬
(
(
X46
↔
(
(
¬
X28
)
→
(
¬
X21
)
→
X41
)
)
→
(
(
¬
(
X45
↔
(
X21
↔
(
(
¬
X16
)
→
X47
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
¬
X21
)
↔
(
(
¬
X16
)
↔
(
(
¬
X8
)
↔
X46
)
)
)
)
→
X45
→
(
X16
↔
X21
)
)
)
→
(
¬
(
X8
↔
X16
)
)
)
)
)
→
X41
→
X21
→
(
¬
X8
)
→
X16
)
)
)
↔
(
(
X47
→
(
(
¬
X45
)
↔
X49
)
)
↔
(
X49
↔
(
X45
↔
(
X27
→
X31
)
)
)
)
)
→
(
(
X40
→
X38
→
X49
)
↔
(
(
X45
→
(
(
¬
X26
)
↔
(
X19
→
X38
)
)
)
↔
(
¬
(
X45
↔
(
(
¬
X38
)
↔
X47
)
)
)
)
)
→
(
(
¬
(
X49
→
(
X45
↔
False
)
)
)
↔
(
¬
(
X40
↔
(
X25
↔
(
(
¬
X11
)
→
X49
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
X33
↔
(
(
¬
X22
)
↔
(
(
¬
X17
)
↔
X44
)
)
)
)
→
X22
→
X46
)
↔
(
(
(
¬
(
X33
→
X22
→
X18
→
X43
)
)
↔
(
X42
↔
(
X22
↔
(
X18
→
X33
)
)
)
)
→
(
(
¬
(
(
¬
X40
)
→
X33
→
X22
→
X43
)
)
↔
(
X43
↔
(
(
¬
X40
)
↔
(
X33
↔
X46
)
)
)
)
)
)
↔
(
(
(
¬
(
X20
↔
(
X5
↔
X40
)
)
)
↔
(
X47
→
(
X44
↔
(
X20
→
X4
)
)
)
)
→
(
(
(
¬
(
(
(
¬
(
X39
↔
(
X15
→
X7
→
False
)
)
)
↔
(
X40
↔
(
X36
→
(
X15
↔
X7
)
)
)
)
→
(
¬
(
X40
→
X39
→
X23
→
(
X15
↔
False
)
)
)
)
)
↔
(
(
¬
X40
)
→
(
(
¬
X36
)
↔
(
X23
↔
X47
)
)
)
)
→
(
¬
(
X47
↔
(
X36
↔
(
X22
→
X39
)
)
)
)
)
→
(
(
¬
(
X44
→
X24
→
(
(
¬
X22
)
↔
X47
)
)
)
→
(
(
¬
X44
)
↔
(
X24
↔
(
(
¬
X3
)
↔
X47
)
)
)
)
→
(
¬
(
X24
↔
(
X15
↔
False
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
X34
→
X30
→
X6
→
X12
)
→
X34
→
(
X30
↔
(
(
¬
X12
)
→
X43
)
)
)
)
→
(
(
¬
(
(
¬
(
X43
→
(
(
¬
X29
)
↔
(
X12
→
X46
)
)
)
)
→
(
¬
(
X30
→
X12
→
X43
)
)
)
)
↔
(
¬
(
(
X48
→
(
X43
↔
(
X29
→
X46
)
)
)
→
(
¬
(
(
¬
X46
)
↔
(
(
¬
X43
)
→
(
(
¬
X0
)
↔
X48
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
X46
→
(
X43
↔
(
X14
→
(
X10
↔
False
)
)
)
)
↔
(
X13
→
(
X10
↔
X43
)
)
)
→
(
(
¬
(
X46
→
X43
→
(
X13
↔
X47
)
)
)
↔
(
¬
(
X47
→
X46
→
(
X43
↔
(
X1
→
False
)
)
)
)
)
)
↔
(
(
X35
↔
(
X14
↔
X2
)
)
↔
(
¬
(
X47
→
(
¬
X35
)
→
X14
→
X46
)
)
)
)
↔
(
(
(
X43
↔
(
X32
↔
X10
)
)
→
(
X48
↔
(
X37
→
X13
→
X10
)
)
→
(
¬
(
X48
↔
(
X43
↔
(
X37
→
X13
)
)
)
)
)
↔
(
X48
→
(
¬
X13
)
→
(
X9
↔
X37
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
7cb208...
and proposition id is
9907c8...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMXNqdUoK3a2n5a2XcARrGZZHsrJiCp9tsp
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
(
¬
(
(
X49
↔
(
X48
→
X23
)
)
→
X28
→
(
(
¬
X23
)
↔
(
X14
→
X48
)
)
)
)
↔
(
¬
(
(
X48
→
X28
→
(
X23
↔
X45
)
)
→
X28
→
(
(
¬
X23
)
↔
(
X12
↔
X45
)
)
)
)
)
→
(
(
¬
(
X49
→
X45
→
(
X25
↔
X44
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
¬
(
(
¬
X40
)
→
(
(
¬
X16
)
↔
X49
)
)
)
→
(
¬
X31
)
→
(
(
¬
X16
)
↔
X48
)
)
)
→
(
¬
(
X49
↔
(
X48
→
False
)
)
)
)
→
(
(
X40
↔
(
X18
↔
X45
)
)
↔
(
X48
→
(
(
¬
X40
)
↔
X49
)
)
)
)
)
↔
(
¬
(
X45
→
(
¬
X44
)
→
(
X40
↔
X48
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
X41
)
→
X39
→
X45
)
→
(
(
¬
(
(
X45
↔
(
(
¬
X42
)
→
X41
→
X49
)
)
→
(
X49
↔
(
X41
↔
X43
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X45
)
↔
(
(
¬
X43
)
↔
X49
)
)
)
→
(
¬
(
(
¬
(
(
¬
X45
)
↔
(
X43
↔
(
(
¬
X39
)
↔
X49
)
)
)
)
→
(
¬
(
X45
→
(
¬
X39
)
→
X37
→
X43
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
¬
(
(
¬
X45
)
→
(
X30
↔
(
X11
↔
X48
)
)
)
)
→
(
(
(
¬
X46
)
↔
(
(
¬
X26
)
↔
(
X11
↔
X49
)
)
)
↔
(
¬
(
X30
→
X26
→
(
X11
↔
False
)
)
)
)
)
↔
(
¬
(
X45
→
(
X30
↔
(
X24
→
X49
)
)
)
)
)
→
(
¬
(
(
(
X49
→
X43
→
(
¬
X26
)
→
X34
)
→
(
¬
(
X34
→
(
(
¬
X26
)
↔
(
X24
→
X43
)
)
)
)
)
→
(
¬
(
X49
→
X34
→
X26
→
False
)
)
)
)
)
)
↔
(
X43
↔
(
X30
↔
X49
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
X43
↔
(
X36
→
(
(
¬
X21
)
↔
X44
)
)
)
)
↔
(
(
¬
(
X43
↔
(
(
¬
X36
)
→
X44
)
)
)
→
(
(
X45
↔
(
X36
→
X44
)
)
↔
(
¬
(
X44
→
X43
→
X36
→
(
X5
↔
False
)
)
)
)
)
)
↔
(
(
X47
→
(
X43
↔
(
(
¬
X19
)
↔
X21
)
)
)
→
(
(
¬
(
(
¬
X43
)
→
(
X21
↔
(
(
¬
X15
)
→
X47
)
)
)
)
↔
(
¬
(
X43
→
X21
→
(
(
¬
X5
)
↔
X15
)
)
)
)
)
)
→
(
(
(
¬
(
X43
↔
X44
)
)
→
(
¬
(
(
(
X44
↔
(
X13
→
X45
)
)
↔
(
(
(
¬
X44
)
↔
(
X13
↔
X45
)
)
→
(
¬
(
(
¬
X13
)
→
(
(
¬
X10
)
↔
X43
)
)
)
)
)
→
(
¬
(
X45
→
X44
→
X43
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
¬
X44
)
↔
(
X43
→
X45
)
)
)
→
(
(
(
¬
(
X43
↔
(
X38
↔
(
X35
→
X45
)
)
)
)
→
X44
→
(
¬
X43
)
→
(
(
¬
X38
)
↔
X45
)
)
↔
(
X44
→
(
¬
X43
)
→
(
(
¬
X9
)
↔
X45
)
)
)
)
)
→
(
¬
(
X45
↔
(
X44
↔
(
X38
→
X3
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
X47
→
(
(
¬
X32
)
↔
(
(
¬
X17
)
↔
X45
)
)
)
→
(
¬
(
(
(
(
X45
↔
(
X22
→
X8
→
X32
)
)
↔
(
(
¬
(
X45
↔
X8
)
)
→
(
¬
(
X27
→
(
X22
↔
(
X2
→
False
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X33
→
(
X32
↔
X45
)
)
)
→
(
¬
(
(
¬
X32
)
↔
(
(
¬
X27
)
→
X45
)
)
)
→
(
X45
↔
(
(
¬
X33
)
↔
(
X27
→
X47
)
)
)
)
)
)
→
(
¬
X32
)
→
X27
→
(
¬
X1
)
→
X33
)
)
)
)
↔
(
(
X29
→
(
X27
↔
(
X22
→
X20
)
)
)
→
(
(
¬
(
(
¬
X27
)
→
(
X20
↔
(
(
¬
X17
)
→
X38
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X27
→
X20
→
(
(
¬
X0
)
↔
X29
)
)
)
↔
(
¬
(
X29
→
X20
→
(
X0
↔
X7
)
)
)
)
↔
(
¬
X27
)
)
→
(
(
¬
(
X6
→
X38
)
)
↔
(
(
X38
→
(
X27
↔
(
X4
↔
X20
)
)
)
↔
(
(
¬
X29
)
→
X20
→
(
(
¬
X4
)
↔
X38
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
(
X1
↔
False
)
→
(
X45
↔
(
(
¬
X35
)
→
X48
)
)
→
(
¬
(
X45
↔
(
(
¬
X35
)
→
X1
→
X43
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X36
→
(
X3
↔
X48
)
)
)
→
(
(
¬
X35
)
↔
(
X23
↔
(
(
¬
X3
)
↔
X36
)
)
)
)
)
↔
(
¬
(
X48
↔
(
(
¬
X36
)
↔
(
X35
↔
X43
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
¬
(
(
¬
X23
)
↔
(
X14
→
(
X0
↔
X34
)
)
)
)
↔
(
X14
→
(
(
¬
X0
)
↔
X20
)
)
)
→
(
(
¬
(
X34
→
(
(
¬
X20
)
↔
(
X1
↔
X48
)
)
)
)
↔
(
¬
(
X14
↔
(
X1
↔
False
)
)
)
)
)
)
↔
(
X34
↔
(
X23
↔
(
X1
↔
False
)
)
)
)
→
(
¬
(
X34
↔
(
X20
↔
X48
)
)
)
)
)
)
→
(
(
(
(
(
X47
→
X46
→
(
¬
X32
)
→
X48
)
→
(
¬
(
X47
→
X32
→
X17
→
X36
)
)
→
(
(
¬
(
X47
↔
(
X15
↔
False
)
)
)
↔
(
(
¬
(
X36
↔
(
(
¬
X32
)
↔
(
(
¬
X15
)
↔
X48
)
)
)
)
→
(
¬
X36
)
→
X15
→
X47
)
)
)
↔
(
¬
(
X36
→
(
(
¬
X17
)
↔
X46
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
X19
↔
(
(
¬
X8
)
→
X43
)
)
↔
(
X36
↔
(
X27
↔
(
X8
→
X47
)
)
)
)
→
(
¬
(
(
¬
X36
)
↔
(
X19
→
X10
→
X43
)
)
)
)
)
↔
(
¬
(
X36
↔
(
(
¬
X15
)
→
(
X13
↔
X27
)
)
)
)
)
→
(
¬
(
X36
↔
(
(
¬
X27
)
↔
X47
)
)
)
)
)
)
↔
(
(
(
(
¬
X43
)
→
(
(
¬
X39
)
↔
(
(
¬
X28
)
↔
X46
)
)
)
→
(
¬
(
X46
→
(
¬
X28
)
→
(
X16
↔
X47
)
)
)
)
→
(
(
X39
→
(
¬
X7
)
→
X47
)
↔
(
¬
(
(
¬
X46
)
↔
(
(
¬
X39
)
→
X4
→
X47
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
X49
↔
(
X44
→
X2
→
X43
)
)
↔
(
¬
(
X47
→
(
(
¬
X43
)
↔
X49
)
)
)
)
→
(
X47
↔
(
X43
↔
(
X42
↔
X44
)
)
)
→
(
(
¬
(
(
(
X42
↔
False
)
→
(
¬
(
X44
→
X42
→
X11
→
X40
)
)
→
X42
→
(
¬
X11
)
→
X47
)
→
(
X47
↔
(
X44
→
X42
)
)
)
)
↔
(
¬
(
X42
→
X37
→
X40
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X48
↔
(
X31
→
X25
)
)
)
→
(
X43
↔
(
(
¬
X31
)
→
(
¬
X12
)
→
X48
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
X43
)
↔
(
X42
→
X22
→
X48
)
)
)
→
(
¬
(
X38
→
(
(
¬
X22
)
↔
X42
)
)
)
)
)
↔
(
(
¬
X43
)
↔
(
X42
↔
X48
)
)
)
→
(
¬
(
(
¬
X44
)
↔
(
(
¬
X38
)
↔
(
X31
↔
X48
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
X41
)
→
(
X40
↔
(
(
¬
X26
)
→
X48
)
)
)
→
(
X40
↔
(
(
¬
X9
)
↔
X41
)
)
)
→
(
X48
↔
(
X41
↔
(
X40
→
X21
)
)
)
)
↔
(
(
(
(
(
¬
(
X48
→
X44
→
X40
)
)
→
(
(
¬
X44
)
↔
(
(
¬
X40
)
→
(
(
¬
X33
)
↔
X48
)
)
)
→
(
X40
↔
(
X33
↔
X30
)
)
)
↔
(
(
X40
→
(
(
¬
X29
)
↔
(
X5
→
X44
)
)
)
→
(
X48
↔
(
X40
→
X44
)
)
)
)
→
(
X48
↔
(
X40
→
X33
)
)
)
→
(
¬
(
(
¬
(
X48
↔
(
X41
↔
(
(
¬
X29
)
→
X33
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
(
¬
X41
)
→
(
X33
↔
(
(
¬
X6
)
↔
X48
)
)
)
→
(
¬
(
(
¬
X24
)
→
X6
→
X41
)
)
)
)
→
(
X33
↔
(
X24
↔
(
(
¬
X18
)
→
X41
)
)
)
)
)
→
(
¬
(
X41
↔
(
X33
↔
X6
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
4b838d...
and proposition id is
5368fb...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMXQ5H91LAYrjrS1sP94MPKtjfhGLmBtHwE
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
∀X50 :
prop
,
∀X51 :
prop
,
(
¬
∀X52 :
prop
,
(
¬
∀X53 :
prop
,
∀X54 :
prop
,
(
(
(
(
(
(
X37
→
X2
)
→
(
¬
(
(
¬
(
X37
↔
(
X20
→
X9
)
)
)
→
(
X53
↔
(
X51
→
(
(
¬
X9
)
↔
X20
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
X54
↔
(
X37
↔
X51
)
)
)
→
(
¬
(
(
(
¬
X51
)
→
(
¬
X50
)
→
X13
→
X54
)
→
X51
→
(
¬
X50
)
→
(
X37
↔
X54
)
)
)
)
)
→
(
¬
(
X54
→
(
(
¬
X47
)
↔
(
(
¬
X37
)
↔
X51
)
)
)
)
→
(
¬
(
X51
↔
(
X37
→
X36
→
X10
→
False
)
)
)
)
)
→
(
(
¬
(
X54
→
(
X16
↔
X51
)
)
)
↔
(
¬
(
(
(
¬
X51
)
→
(
¬
X32
)
→
(
(
¬
X16
)
↔
X54
)
)
→
(
(
(
¬
X51
)
↔
(
X49
→
(
X37
↔
X53
)
)
)
↔
(
(
¬
(
X54
↔
(
X51
↔
(
X49
→
(
X32
↔
False
)
)
)
)
)
↔
(
¬
(
X53
→
(
X51
↔
False
)
)
)
)
)
)
)
)
→
(
¬
X49
)
→
(
(
¬
X37
)
↔
(
(
¬
X21
)
↔
X54
)
)
)
)
→
(
(
(
¬
(
(
X54
↔
(
X52
↔
(
X46
↔
(
X24
→
False
)
)
)
)
→
(
¬
(
(
¬
X46
)
→
(
¬
X6
)
→
X52
)
)
)
)
→
(
(
X27
↔
(
(
¬
X6
)
↔
(
X0
↔
X54
)
)
)
→
(
¬
(
X27
↔
(
(
¬
X0
)
→
X3
)
)
)
)
→
(
¬
(
(
¬
X6
)
↔
X52
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X54
→
X52
→
(
(
¬
X1
)
↔
X46
)
)
)
→
(
(
¬
X51
)
↔
(
(
¬
X46
)
→
(
(
¬
X1
)
↔
X52
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
(
(
X52
↔
(
X51
↔
(
X31
↔
X54
)
)
)
→
(
¬
(
X52
→
(
¬
X31
)
→
(
X30
↔
X51
)
)
)
)
)
↔
(
¬
(
(
¬
X52
)
↔
(
(
¬
X51
)
→
X8
→
X54
)
)
)
)
↔
(
(
¬
X51
)
↔
(
X31
↔
X54
)
)
)
↔
(
¬
(
X54
↔
(
X51
↔
(
X49
↔
(
X30
↔
False
)
)
)
)
)
)
→
X52
→
X49
→
(
¬
X23
)
→
X54
)
)
)
)
)
)
↔
(
(
(
(
(
¬
X45
)
→
(
¬
X42
)
→
(
¬
X19
)
→
X50
)
↔
(
X45
↔
(
X42
↔
(
X19
→
(
X7
↔
False
)
)
)
)
)
↔
(
(
¬
(
X50
↔
(
(
¬
X39
)
→
(
X18
↔
X42
)
)
)
)
↔
(
¬
(
(
(
¬
(
X50
→
(
¬
X45
)
→
X53
)
)
→
(
(
(
¬
(
(
¬
X50
)
↔
(
X26
→
X53
)
)
)
↔
(
X50
→
(
X45
↔
(
X39
↔
X53
)
)
)
)
↔
(
¬
(
X26
↔
X12
)
)
)
)
→
(
(
¬
X50
)
↔
(
(
¬
X45
)
↔
(
X14
→
X53
)
)
)
→
(
¬
(
(
¬
X42
)
→
(
X14
↔
X45
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
¬
X50
)
→
(
X48
↔
(
X38
↔
X53
)
)
)
→
(
(
(
(
X53
→
(
X48
↔
(
X25
↔
X50
)
)
)
↔
(
X48
→
(
X25
↔
(
X17
↔
X53
)
)
)
)
↔
(
X53
↔
(
(
¬
X48
)
→
X50
)
)
)
↔
(
X53
→
X17
)
)
)
)
→
(
(
¬
X48
)
→
(
(
¬
X38
)
↔
(
(
¬
X35
)
→
X53
)
)
)
→
(
(
¬
(
(
X48
→
X33
→
(
X4
↔
X35
)
)
→
X38
→
(
(
¬
X33
)
↔
X48
)
)
)
↔
(
(
(
¬
X48
)
↔
(
(
¬
X35
)
→
X54
)
)
→
(
(
¬
(
X35
→
(
X4
↔
X22
)
)
)
↔
(
¬
(
X48
→
X35
→
(
(
¬
X22
)
↔
X54
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
X53
↔
(
X34
→
(
(
¬
X11
)
↔
X54
)
)
)
→
(
(
X54
→
(
(
¬
X34
)
↔
X48
)
)
↔
(
¬
(
(
¬
(
(
¬
X48
)
→
(
¬
X34
)
→
X54
)
)
→
(
¬
(
X53
↔
(
X48
→
X43
→
X54
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X48
)
→
(
(
¬
X44
)
↔
(
(
¬
X29
)
→
X54
)
)
)
)
↔
(
(
(
¬
X48
)
↔
(
(
¬
X44
)
→
(
¬
X29
)
→
X53
)
)
↔
(
(
¬
X28
)
→
X44
)
)
)
↔
(
(
¬
X41
)
↔
(
(
¬
X28
)
↔
(
X15
↔
X44
)
)
)
)
↔
(
(
X54
↔
(
X45
→
(
X40
↔
X28
)
)
)
→
(
(
X28
→
(
X15
↔
(
(
¬
X5
)
↔
X54
)
)
)
↔
(
(
¬
X40
)
→
(
¬
X15
)
→
X54
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
(
(
¬
(
(
¬
(
X47
↔
(
(
¬
X12
)
→
(
¬
X3
)
→
X35
)
)
)
→
(
¬
(
(
¬
X47
)
↔
(
X12
→
X52
)
)
)
)
)
↔
(
¬
(
(
X53
↔
(
X46
→
X35
)
)
→
(
(
X47
→
(
(
¬
X35
)
↔
X46
)
)
↔
(
(
X53
↔
(
X47
→
(
X35
↔
X41
)
)
)
→
(
X53
↔
(
(
¬
X41
)
→
(
X12
↔
X47
)
)
)
)
)
)
)
)
→
(
(
(
X53
↔
(
X41
↔
X2
)
)
↔
(
¬
(
X53
↔
(
X20
→
(
X2
↔
X32
)
)
)
)
)
↔
(
(
(
¬
X46
)
→
(
¬
X41
)
→
(
¬
X38
)
→
X53
)
→
(
X53
↔
(
X46
→
X38
→
(
X20
↔
False
)
)
)
)
)
)
)
→
(
¬
(
(
(
X38
↔
(
(
¬
X35
)
→
(
¬
X8
)
→
X37
)
)
↔
(
(
(
¬
(
(
X51
↔
(
X38
→
X5
)
)
→
(
X51
↔
(
X38
→
(
(
¬
X35
)
↔
X47
)
)
)
)
)
→
(
(
¬
X37
)
↔
(
(
¬
X35
)
↔
(
(
¬
X5
)
→
X51
)
)
)
)
↔
(
X51
→
(
(
¬
X35
)
↔
X38
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X37
)
→
(
X35
↔
(
(
¬
X26
)
→
X38
)
)
)
)
→
(
¬
(
(
¬
(
X37
→
(
¬
X29
)
→
(
X18
↔
X38
)
)
)
→
(
¬
(
X38
↔
(
X35
→
X18
)
)
)
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
X47
→
X52
)
)
↔
(
¬
(
(
X47
→
X27
→
(
(
¬
X19
)
↔
X52
)
)
→
(
¬
(
X52
→
(
X47
↔
X53
)
)
)
)
)
)
→
(
¬
(
X53
↔
(
X52
→
(
X47
↔
(
X1
→
False
)
)
)
)
)
→
(
(
(
¬
X47
)
→
(
¬
X43
)
→
(
¬
X31
)
→
X53
)
↔
(
¬
(
(
¬
(
X53
→
(
X47
↔
(
(
¬
X43
)
→
X52
)
)
)
)
→
(
¬
(
(
¬
X47
)
↔
(
X43
→
X39
→
X52
)
)
)
)
)
)
→
X43
→
(
¬
X39
)
→
X52
)
→
(
¬
(
(
(
(
X33
→
X28
→
(
(
¬
X13
)
↔
X44
)
)
→
(
X42
↔
(
(
¬
X33
)
→
X44
)
)
)
↔
(
(
¬
(
X44
↔
(
X28
↔
(
X25
↔
X52
)
)
)
)
↔
(
¬
(
X52
→
X28
→
(
X13
↔
False
)
)
)
)
)
→
(
¬
(
(
¬
(
X47
→
X23
→
(
X11
↔
X33
)
)
)
→
(
(
¬
(
(
(
(
¬
X47
)
→
(
X42
↔
(
(
¬
X11
)
→
X52
)
)
)
↔
(
¬
(
X42
→
(
X23
↔
False
)
)
)
)
→
X52
→
(
¬
X42
)
→
(
X23
↔
X53
)
)
)
↔
(
X47
↔
X52
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
(
¬
(
(
(
¬
(
(
¬
(
X40
→
(
X34
↔
(
X24
→
X6
)
)
)
)
→
X40
→
(
X24
↔
(
X6
↔
X7
)
)
)
)
↔
(
X40
↔
(
X34
↔
X24
)
)
)
→
(
¬
X40
)
→
X34
→
(
X24
↔
X52
)
)
)
↔
(
X54
→
X52
→
(
X6
↔
X34
)
)
)
↔
(
¬
(
(
¬
(
X36
→
(
X24
↔
(
(
¬
X7
)
→
X53
)
)
)
)
→
(
(
¬
X53
)
↔
(
(
¬
X36
)
→
(
(
¬
X24
)
↔
X54
)
)
)
)
)
)
↔
(
(
¬
(
X54
↔
(
(
¬
X50
)
→
X30
→
X52
)
)
)
↔
(
(
¬
X40
)
→
(
¬
X30
)
→
X50
)
)
)
↔
(
¬
(
(
X54
↔
(
X53
↔
(
X52
↔
(
X49
↔
False
)
)
)
)
→
(
X54
↔
(
X53
→
(
¬
X0
)
→
X4
)
)
→
(
¬
(
X54
↔
(
X52
↔
(
X45
→
False
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
(
X52
↔
(
X51
→
X54
)
)
)
→
(
¬
(
(
(
¬
(
X52
→
(
X51
↔
(
X36
↔
False
)
)
)
)
→
(
¬
(
(
¬
(
X54
↔
(
X52
→
X21
)
)
)
→
(
(
¬
(
X36
↔
(
(
¬
X21
)
↔
X54
)
)
)
↔
(
X54
→
(
¬
X51
)
→
(
¬
X21
)
→
X52
)
)
)
)
)
→
(
¬
(
(
¬
X51
)
→
X36
→
X10
→
X54
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
X36
→
(
X16
↔
(
X9
→
False
)
)
)
↔
(
(
X17
↔
X36
)
↔
(
(
¬
X17
)
↔
(
X16
↔
(
X9
↔
X52
)
)
)
)
)
→
(
¬
(
(
¬
X17
)
↔
(
X16
↔
X52
)
)
)
)
)
→
(
(
X51
→
X48
→
False
)
↔
(
(
¬
(
X52
↔
(
(
¬
X17
)
→
(
¬
X14
)
→
X36
)
)
)
↔
(
X48
→
(
(
¬
X17
)
↔
(
X14
↔
X52
)
)
)
)
)
)
)
)
→
(
(
¬
(
X22
↔
(
X10
↔
(
X9
→
X16
)
)
)
)
↔
(
X22
→
(
(
¬
X15
)
↔
X16
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
61e33d...
and proposition id is
3a8ac7...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMTn5bfruy6GgBZF5W3iVsUEKFXc7bKEjH7
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
∀X50 :
prop
,
(
(
(
¬
(
(
(
(
(
(
¬
(
(
(
¬
(
(
¬
(
X49
↔
(
X46
→
(
(
¬
X9
)
↔
X39
)
)
)
)
→
(
¬
(
X49
↔
(
X46
↔
(
X39
→
X12
)
)
)
)
)
)
→
(
¬
X12
)
→
X9
→
X39
)
→
(
¬
(
X43
↔
(
X12
↔
X49
)
)
)
)
)
↔
(
X50
→
(
X46
↔
X39
)
)
)
↔
(
(
X43
↔
(
X4
→
X39
)
)
↔
(
(
(
¬
X40
)
→
(
¬
X39
)
→
X43
)
↔
(
(
(
(
¬
(
X40
↔
(
(
¬
X12
)
↔
X43
)
)
)
↔
(
X43
↔
(
X39
→
(
X4
↔
X12
)
)
)
)
→
(
¬
(
X43
→
(
¬
X4
)
→
X40
)
)
)
↔
(
¬
(
X43
↔
(
X39
↔
X40
)
)
)
)
)
)
)
↔
(
(
X50
↔
(
X40
→
X11
→
X43
)
)
→
(
¬
(
X40
↔
(
X11
↔
X29
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
¬
(
(
¬
X44
)
→
(
(
¬
X28
)
↔
X45
)
)
)
→
(
(
¬
X43
)
↔
(
X28
→
X49
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X46
)
↔
(
X45
↔
(
(
¬
X43
)
→
X49
)
)
)
)
→
(
X25
↔
False
)
)
)
)
)
→
(
X45
↔
(
X44
↔
X43
)
)
)
→
(
(
X43
→
(
¬
X24
)
→
X17
→
X30
)
↔
(
(
X46
↔
(
X43
→
X49
)
)
→
(
(
(
¬
X30
)
↔
(
(
¬
X24
)
↔
X46
)
)
↔
(
¬
(
X49
↔
(
X30
↔
(
(
¬
X24
)
→
X43
)
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
X50
→
(
(
¬
X35
)
↔
(
X20
→
X45
)
)
)
)
↔
(
(
¬
X35
)
→
X47
)
)
↔
(
(
¬
(
(
X50
↔
(
X27
↔
(
X13
→
False
)
)
)
→
(
¬
(
(
¬
X32
)
↔
(
X22
↔
(
(
¬
X13
)
→
X50
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X50
↔
(
X47
↔
(
X35
↔
X32
)
)
)
)
→
(
¬
(
X47
→
(
¬
X22
)
→
X49
)
)
→
(
¬
(
(
(
¬
(
(
¬
X49
)
↔
(
X22
↔
X50
)
)
)
↔
(
(
¬
(
X49
→
X35
→
X47
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
X49
→
(
X47
↔
(
(
¬
X35
)
↔
X50
)
)
)
)
→
(
¬
(
X50
↔
(
X49
↔
(
X47
→
X22
)
)
)
)
)
)
→
X49
→
(
¬
X22
)
→
X47
)
)
→
(
¬
(
X50
↔
(
X49
→
X22
→
False
)
)
)
)
)
)
→
(
¬
(
X50
→
X35
→
(
¬
X2
)
→
X5
)
)
)
)
)
)
→
(
¬
(
X50
→
(
¬
X47
)
→
X19
→
X49
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
(
¬
X41
)
→
(
X38
↔
(
(
¬
X16
)
↔
X42
)
)
)
)
→
(
(
¬
X16
)
↔
X48
)
)
↔
(
(
(
¬
(
X16
↔
X49
)
)
↔
(
(
X42
→
(
¬
X16
)
→
X49
)
→
(
(
¬
(
(
¬
X38
)
→
(
(
¬
X16
)
↔
(
(
¬
X1
)
↔
X49
)
)
)
)
↔
(
¬
(
(
¬
X38
)
↔
(
(
¬
X16
)
→
(
(
¬
X1
)
↔
X42
)
)
)
)
)
)
)
↔
(
X38
→
X16
→
X3
→
X49
)
)
)
↔
(
(
¬
(
X41
→
(
(
¬
X18
)
↔
(
(
¬
X0
)
↔
X38
)
)
)
)
→
(
¬
(
(
¬
X42
)
↔
(
(
¬
X18
)
↔
(
(
¬
X0
)
↔
X47
)
)
)
)
)
)
↔
(
¬
(
(
(
(
X49
→
(
X37
↔
(
X26
↔
X48
)
)
)
↔
(
(
X48
↔
(
X36
↔
(
X10
↔
False
)
)
)
↔
(
(
¬
(
(
(
X48
↔
(
X36
→
(
(
¬
X7
)
↔
X37
)
)
)
↔
(
X37
↔
(
X36
→
(
(
¬
X7
)
↔
X49
)
)
)
)
→
(
X48
↔
X37
)
)
)
→
(
¬
(
(
¬
X48
)
→
(
X37
↔
(
(
¬
X10
)
↔
X49
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X47
↔
(
X37
→
(
X6
↔
False
)
)
)
)
→
(
(
¬
(
X15
↔
(
(
¬
X6
)
↔
X36
)
)
)
↔
(
(
¬
(
(
X47
→
(
¬
X6
)
→
X15
)
→
(
(
X36
→
(
(
¬
X15
)
↔
(
X6
↔
X47
)
)
)
↔
(
(
¬
X37
)
→
(
X6
↔
X47
)
)
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X36
)
↔
X37
)
)
)
)
)
)
→
(
¬
(
X6
↔
X15
)
)
)
)
↔
(
(
(
¬
X34
)
→
(
X15
↔
X37
)
)
→
(
(
¬
X47
)
↔
(
X15
→
X48
)
)
)
)
)
→
(
(
(
(
X45
↔
(
X34
→
(
X33
↔
X48
)
)
)
↔
(
X48
↔
(
X45
↔
(
X36
→
(
X33
↔
False
)
)
)
)
)
↔
(
¬
(
(
¬
X34
)
→
(
¬
X33
)
→
(
¬
X23
)
→
X45
)
)
)
↔
(
(
(
¬
(
(
¬
X31
)
→
X34
)
)
↔
(
X45
→
X36
→
X31
→
(
X14
↔
False
)
)
)
→
(
(
¬
(
(
X45
→
X36
→
(
¬
X21
)
→
X34
)
→
(
(
¬
X34
)
↔
(
X21
↔
(
X8
↔
X45
)
)
)
)
)
↔
(
¬
(
(
¬
X34
)
→
(
¬
X21
)
→
X36
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
¬
(
(
X31
→
(
¬
X29
)
→
(
X14
↔
X32
)
)
→
(
¬
X29
)
→
(
X14
↔
(
X10
→
X31
)
)
)
)
→
(
(
(
(
¬
X32
)
↔
(
X49
→
X29
→
X3
→
X10
)
)
↔
(
¬
(
X49
↔
(
(
¬
X29
)
↔
X32
)
)
)
)
↔
(
(
(
X32
↔
(
X0
↔
X41
)
)
↔
(
¬
(
(
¬
X10
)
↔
(
X3
→
X0
→
X32
)
)
)
)
↔
(
¬
(
X41
↔
(
(
¬
X10
)
↔
X32
)
)
)
)
)
→
(
X41
↔
(
X34
→
(
¬
X32
)
→
X49
)
)
)
→
(
(
(
X46
↔
(
(
¬
X9
)
↔
X41
)
)
↔
(
(
¬
X41
)
→
(
¬
X36
)
→
X49
)
)
↔
(
(
¬
(
X41
↔
(
(
¬
X36
)
→
X34
→
X49
)
)
)
↔
(
X34
↔
(
(
¬
X11
)
↔
X49
)
)
)
)
→
(
(
¬
(
(
¬
X41
)
↔
(
(
¬
X36
)
→
(
(
¬
X22
)
↔
X49
)
)
)
)
↔
(
(
¬
(
X5
↔
(
X1
→
X49
)
)
)
→
(
¬
(
(
X49
→
(
X41
↔
(
X26
→
X5
)
)
)
→
(
(
X36
→
(
¬
X5
)
→
(
¬
X1
)
→
X26
)
↔
(
¬
(
(
¬
X36
)
↔
(
(
¬
X5
)
→
X49
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
X36
)
↔
(
X32
↔
X40
)
)
→
(
(
X40
→
(
(
¬
X36
)
↔
(
X32
↔
X47
)
)
)
↔
(
X47
↔
(
X40
↔
(
(
¬
X27
)
↔
X36
)
)
)
)
)
)
↔
(
¬
(
X47
→
(
¬
X40
)
→
X27
→
X49
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
X40
↔
(
(
¬
X28
)
↔
(
X18
↔
X47
)
)
)
)
→
(
¬
(
(
(
¬
X40
)
→
(
X28
↔
(
X6
→
X47
)
)
)
→
(
X13
↔
X6
)
)
)
)
)
↔
(
(
(
¬
X40
)
↔
(
(
¬
X24
)
→
(
X18
↔
X47
)
)
)
↔
(
X47
→
(
¬
X18
)
→
(
(
¬
X15
)
↔
X40
)
)
)
)
→
(
(
(
(
X24
→
(
X8
↔
(
(
¬
X7
)
→
X47
)
)
)
↔
(
¬
(
(
¬
X24
)
↔
(
X7
→
X40
)
)
)
)
→
(
¬
(
X49
↔
(
X8
↔
(
(
¬
X7
)
→
X24
)
)
)
)
)
↔
(
X47
↔
(
(
¬
X20
)
↔
(
X7
↔
X49
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
X44
↔
(
X16
↔
X37
)
)
→
(
(
(
¬
(
(
(
(
(
¬
(
X50
↔
(
X46
↔
False
)
)
)
→
(
X44
↔
(
(
¬
X37
)
↔
(
(
¬
X23
)
→
X46
)
)
)
)
↔
(
¬
(
(
¬
X44
)
→
X46
)
)
)
→
X46
→
(
¬
X45
)
→
(
X44
↔
X50
)
)
→
(
X45
↔
(
X37
↔
X50
)
)
)
)
↔
(
¬
(
X44
→
(
X37
↔
(
X23
↔
False
)
)
)
)
)
↔
(
¬
(
(
¬
X44
)
→
X37
→
X50
)
)
)
)
→
(
(
X50
↔
(
X39
→
(
(
¬
X21
)
↔
X42
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
¬
X46
)
↔
(
(
¬
X45
)
↔
(
(
¬
X21
)
→
X49
)
)
)
)
→
X49
→
(
X21
↔
X42
)
)
→
(
X49
↔
(
(
¬
X45
)
↔
(
(
¬
X43
)
↔
X46
)
)
)
)
)
↔
(
(
(
¬
(
X50
→
(
X43
↔
(
X4
→
X21
)
)
)
)
↔
(
¬
(
X50
→
(
(
¬
X43
)
↔
(
(
¬
X4
)
→
X45
)
)
)
)
)
↔
(
¬
(
(
¬
X43
)
↔
(
X4
↔
X45
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
X49
→
(
¬
X46
)
→
X48
)
↔
(
¬
(
X48
↔
(
X46
↔
(
X45
↔
(
X25
→
False
)
)
)
)
)
)
→
(
(
(
¬
(
(
(
¬
X19
)
↔
(
(
¬
X2
)
↔
X38
)
)
→
(
(
X38
→
(
X35
↔
False
)
)
↔
(
(
(
¬
X19
)
↔
(
(
¬
X2
)
→
X46
)
)
↔
(
(
¬
(
(
¬
X46
)
→
(
¬
X38
)
→
X48
)
)
↔
(
(
X46
↔
(
(
¬
X2
)
→
X19
)
)
↔
(
¬
(
X48
↔
(
X38
→
False
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
X48
↔
(
(
¬
X38
)
↔
(
X33
→
X46
)
)
)
)
→
X46
→
(
X33
↔
(
X17
↔
X38
)
)
)
)
↔
(
(
(
¬
X46
)
↔
(
X45
↔
(
X40
→
X48
)
)
)
↔
(
(
(
¬
X45
)
→
(
(
¬
X38
)
↔
X48
)
)
↔
(
(
(
¬
(
(
¬
(
X40
↔
(
X30
→
X38
)
)
)
→
(
(
(
¬
(
(
¬
X38
)
↔
(
X30
↔
X40
)
)
)
↔
(
(
¬
(
X48
↔
(
X38
→
X30
→
X40
)
)
)
↔
(
X40
↔
(
X38
→
(
(
¬
X12
)
↔
X48
)
)
)
)
)
↔
(
X48
→
X12
→
X40
)
)
)
)
↔
(
¬
(
(
¬
X40
)
↔
(
(
¬
X38
)
↔
X48
)
)
)
)
↔
(
(
¬
X45
)
→
X38
→
(
X30
↔
X48
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
9809d6...
and proposition id is
1a6003...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMbiic2SP6D6VWqNdeScxeHXBJs4fD1rxtt
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
¬
(
(
¬
(
(
(
¬
(
X48
↔
(
X46
↔
(
X0
↔
X40
)
)
)
)
↔
(
(
¬
(
X46
↔
False
)
)
↔
(
¬
(
(
(
X49
↔
(
(
¬
X40
)
↔
X48
)
)
→
(
X48
↔
(
X46
→
X39
→
X17
→
False
)
)
→
(
(
¬
X48
)
↔
(
X40
↔
(
X39
→
X49
)
)
)
)
→
(
(
¬
X40
)
↔
(
(
¬
X39
)
→
X49
)
)
)
)
)
)
→
(
(
¬
X40
)
↔
(
X39
↔
X48
)
)
)
)
→
(
(
X49
↔
(
X39
↔
(
X12
↔
False
)
)
)
↔
(
(
¬
(
X47
↔
(
X46
↔
(
X39
↔
X12
)
)
)
)
↔
(
¬
(
X47
→
(
(
¬
X46
)
↔
(
(
¬
X39
)
→
X49
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
X47
)
↔
(
(
¬
X14
)
↔
X49
)
)
→
(
(
¬
(
X48
↔
(
(
¬
X16
)
→
X47
)
)
)
↔
(
¬
(
(
X48
↔
(
X47
↔
(
X44
↔
(
X16
→
False
)
)
)
)
→
(
¬
(
(
¬
X47
)
↔
X49
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
X35
)
→
(
(
¬
X21
)
↔
X41
)
)
↔
(
¬
(
X48
↔
(
X35
→
X30
→
X21
)
)
)
)
→
(
¬
(
(
(
(
¬
X47
)
↔
(
X41
→
(
X3
↔
X48
)
)
)
↔
(
¬
(
X22
↔
(
X3
↔
X48
)
)
)
)
→
(
(
¬
(
X48
↔
(
X41
→
(
¬
X11
)
→
X47
)
)
)
↔
(
¬
(
(
X49
↔
X41
)
→
(
¬
(
X47
→
(
(
¬
X35
)
↔
(
X11
→
X41
)
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
X46
↔
(
X43
→
X42
)
)
→
(
(
¬
(
(
X43
↔
(
X27
↔
(
X7
→
False
)
)
)
→
(
¬
(
X42
↔
(
X34
↔
False
)
)
)
)
)
↔
(
¬
(
X46
↔
(
X34
→
X42
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X9
)
↔
(
(
¬
X1
)
↔
X49
)
)
)
↔
(
X49
→
(
¬
X27
)
→
(
X1
↔
X43
)
)
)
↔
(
X49
→
(
X43
↔
(
X15
↔
(
X13
↔
False
)
)
)
)
)
↔
(
(
¬
(
(
X49
→
(
¬
X34
)
→
X13
→
X47
)
→
(
X49
↔
(
X47
↔
(
X34
→
(
X23
↔
False
)
)
)
)
)
)
↔
(
(
X47
→
X34
→
(
X9
↔
X25
)
)
→
(
¬
(
X34
→
(
X25
↔
(
(
¬
X6
)
→
X49
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
X48
↔
(
X33
→
X2
→
X49
)
)
)
↔
(
¬
(
X33
↔
(
X2
→
X38
)
)
)
)
→
(
(
¬
(
X36
→
(
X2
↔
X29
)
)
)
↔
(
¬
(
(
¬
X45
)
→
(
X36
↔
(
X2
→
X49
)
)
)
)
)
)
)
↔
(
(
X45
→
(
X38
↔
(
X24
→
X49
)
)
)
↔
(
(
(
X49
↔
(
(
¬
X38
)
→
(
¬
X10
)
→
X46
)
)
→
(
¬
(
X49
↔
(
X38
↔
(
X24
→
X10
→
False
)
)
)
)
)
↔
(
¬
(
X49
↔
(
X24
→
X46
)
)
)
)
)
)
→
(
(
(
(
X49
↔
X48
)
↔
(
¬
(
X20
↔
(
(
¬
X18
)
↔
X31
)
)
)
)
→
(
¬
(
X37
↔
(
X32
↔
(
X28
↔
False
)
)
)
)
)
→
(
(
¬
X48
)
↔
(
X28
↔
(
X19
↔
X49
)
)
)
)
→
(
(
(
¬
(
(
¬
X46
)
↔
(
(
¬
X26
)
↔
(
(
¬
X8
)
↔
X48
)
)
)
)
→
(
(
(
(
¬
(
(
¬
X8
)
→
X26
)
)
↔
(
¬
(
X32
→
(
¬
X8
)
→
(
(
¬
X4
)
↔
X46
)
)
)
)
↔
(
¬
(
X46
↔
(
X32
↔
(
X4
↔
False
)
)
)
)
)
↔
(
X32
→
X26
→
X8
→
False
)
)
)
↔
(
(
(
¬
(
X48
↔
(
X37
→
X5
→
X32
)
)
)
↔
(
¬
(
X48
→
X32
→
(
(
¬
X26
)
↔
X37
)
)
)
)
↔
(
(
¬
X5
)
↔
X26
)
)
)
)
)
↔
(
(
(
(
(
(
X46
→
X1
→
X4
)
↔
(
(
¬
(
X46
↔
(
X43
↔
(
X4
→
X44
)
)
)
)
↔
(
¬
(
X46
↔
(
X4
↔
(
(
¬
X1
)
→
X44
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
X39
)
↔
(
X12
↔
X44
)
)
)
↔
(
(
¬
X43
)
→
(
¬
X12
)
→
(
X4
↔
X44
)
)
)
)
↔
(
¬
(
(
(
¬
(
X43
→
(
(
¬
X21
)
↔
X34
)
)
)
→
(
¬
(
(
¬
X43
)
→
(
¬
X39
)
→
X44
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
¬
X44
)
→
X39
→
(
(
¬
X34
)
↔
X49
)
)
)
→
(
¬
(
(
¬
X39
)
→
X2
→
X44
)
)
)
)
→
(
¬
X44
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X44
)
↔
(
X41
↔
(
(
¬
X13
)
→
X47
)
)
)
)
↔
(
X48
→
X22
→
False
)
)
↔
(
(
(
X25
↔
X43
)
↔
(
(
(
X28
→
(
¬
X25
)
→
X48
)
↔
(
X44
↔
(
X43
↔
(
X28
→
X25
)
)
)
)
↔
(
¬
(
X48
→
X44
→
X28
)
)
)
)
↔
(
X44
→
X25
→
X22
→
(
X5
↔
False
)
)
)
)
→
(
¬
(
(
(
(
(
¬
X30
)
↔
(
X17
↔
X40
)
)
↔
(
X45
↔
(
X40
→
X37
→
X30
)
)
)
↔
(
(
(
¬
X44
)
→
(
X40
↔
(
X38
→
X47
)
)
)
↔
(
¬
(
(
(
¬
(
X38
→
(
X30
↔
(
X19
↔
X47
)
)
)
)
↔
(
¬
(
(
¬
X44
)
→
(
(
¬
X38
)
↔
(
X30
→
X47
)
)
)
)
)
→
(
¬
(
X47
→
X44
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
X44
)
→
X26
→
X45
)
)
↔
(
¬
(
(
¬
(
X44
→
(
(
¬
X0
)
↔
X45
)
)
)
→
X26
→
(
¬
X9
)
→
(
¬
X0
)
→
X45
)
)
)
↔
(
(
X45
→
X44
→
(
X33
↔
X37
)
)
↔
(
¬
(
X45
↔
(
X33
→
X26
→
X37
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
X48
↔
(
X45
→
X44
→
(
X33
↔
False
)
)
)
→
(
(
(
¬
(
X45
→
X44
→
X33
→
(
X24
↔
False
)
)
)
→
X44
→
(
(
¬
X33
)
↔
(
X20
→
X45
)
)
)
↔
(
X33
↔
(
X20
→
False
)
)
)
)
)
→
X48
→
(
¬
X44
)
→
(
(
¬
X24
)
↔
X49
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X48
↔
(
(
¬
X42
)
↔
(
X36
↔
X45
)
)
)
)
→
(
X48
↔
(
(
¬
X36
)
→
X8
→
X42
)
)
)
→
(
(
(
¬
(
(
(
¬
(
(
¬
X11
)
→
(
¬
X3
)
→
X45
)
)
↔
(
¬
(
X45
↔
(
X36
→
(
¬
X11
)
→
X47
)
)
)
)
→
X47
→
X45
→
(
X31
↔
(
X18
→
False
)
)
)
)
↔
(
X47
→
X29
→
X45
)
)
↔
(
(
(
¬
X31
)
↔
X47
)
→
(
(
¬
(
X47
→
(
(
¬
X42
)
↔
X45
)
)
)
↔
(
(
¬
X31
)
↔
(
X10
→
X47
)
)
)
)
)
)
→
(
(
(
(
¬
(
X16
↔
(
(
¬
X7
)
↔
X27
)
)
)
↔
(
¬
(
(
(
¬
X16
)
↔
(
(
¬
X7
)
↔
X42
)
)
→
(
¬
(
X48
→
(
X27
↔
(
(
¬
X16
)
↔
X42
)
)
)
)
)
)
)
→
(
(
¬
X42
)
↔
(
X35
↔
(
X16
→
X48
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X35
)
↔
(
(
¬
X32
)
→
X15
→
X45
)
)
)
↔
(
X45
→
(
¬
X32
)
→
(
X14
↔
X35
)
)
)
→
(
¬
(
(
¬
X42
)
↔
(
(
¬
X23
)
↔
(
(
¬
X6
)
↔
X48
)
)
)
)
→
(
¬
(
(
X35
→
(
X23
↔
(
X15
→
X48
)
)
)
→
(
(
X48
↔
(
X42
→
X23
)
)
↔
(
¬
(
X45
→
X42
→
(
¬
X35
)
→
X48
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
801384...
and proposition id is
1b1ca6...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMdGB7hbiR14d2sZm6Lr67V3zsoKC5YHtJU
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
¬
(
(
(
(
(
¬
X42
)
↔
(
(
¬
X30
)
→
X28
→
X47
)
)
↔
(
¬
(
X48
→
X42
)
)
)
↔
(
(
¬
(
(
¬
(
X42
→
X4
→
X15
)
)
→
(
¬
(
X42
→
(
(
¬
X17
)
↔
(
(
¬
X4
)
↔
X28
)
)
)
)
)
)
↔
(
X42
↔
(
(
¬
X15
)
↔
X48
)
)
)
)
→
(
X28
↔
(
(
¬
X24
)
→
X30
)
)
→
(
(
(
¬
X28
)
↔
(
(
¬
X24
)
→
X47
)
)
↔
(
(
X47
↔
(
X30
→
(
(
¬
X18
)
↔
X28
)
)
)
↔
(
X47
↔
(
X24
→
(
X18
↔
X28
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X42
)
↔
(
X40
↔
(
X38
→
X48
)
)
)
)
→
(
¬
(
X48
→
X39
)
)
)
↔
(
(
(
¬
(
X47
↔
(
(
¬
X35
)
→
X39
)
)
)
↔
(
(
(
¬
X47
)
→
(
X27
↔
X49
)
)
↔
(
(
(
(
(
X47
↔
(
(
¬
X35
)
→
X14
→
X49
)
)
↔
(
X49
→
(
¬
X27
)
→
X47
)
)
↔
(
¬
(
X47
↔
(
(
¬
X27
)
↔
X35
)
)
)
)
↔
(
X47
→
(
¬
X14
)
→
X35
)
)
↔
(
¬
(
X49
↔
(
X47
↔
X27
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X47
↔
(
X39
→
(
(
¬
X5
)
↔
X48
)
)
)
)
→
(
¬
(
(
¬
X48
)
↔
(
(
¬
X47
)
↔
(
(
¬
X5
)
↔
X49
)
)
)
)
)
)
→
X48
→
X38
→
X5
)
)
→
(
(
¬
(
(
X45
↔
(
X29
→
(
X21
↔
X20
)
)
)
→
X45
→
(
X29
↔
(
X21
↔
X47
)
)
)
)
↔
(
¬
(
(
¬
X45
)
→
(
¬
X29
)
→
(
(
¬
X0
)
↔
X47
)
)
)
)
→
(
¬
(
(
(
¬
X29
)
↔
(
X20
↔
X41
)
)
→
(
(
¬
(
X25
↔
(
X20
→
X45
)
)
)
↔
(
(
¬
X41
)
↔
(
(
¬
X29
)
→
X20
→
X45
)
)
)
)
)
→
(
(
(
X41
↔
(
(
¬
X20
)
↔
X48
)
)
↔
(
¬
(
X49
↔
(
X45
↔
False
)
)
)
)
↔
(
X48
↔
X45
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
(
X13
↔
(
X3
↔
X37
)
)
→
X33
→
X9
→
X36
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X43
)
→
X33
→
(
(
¬
X16
)
↔
X49
)
)
)
↔
(
¬
(
X49
→
X43
→
(
¬
X3
)
→
X33
)
)
)
→
(
X43
↔
(
X37
↔
(
(
¬
X8
)
→
X49
)
)
)
)
)
)
→
(
(
(
¬
(
X44
→
(
X43
↔
(
X31
→
X36
)
)
)
)
↔
(
¬
(
X36
→
(
(
¬
X31
)
↔
(
X10
↔
X43
)
)
)
)
)
↔
(
¬
(
(
¬
X33
)
↔
(
(
¬
X31
)
→
X26
→
X36
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
X49
↔
(
(
¬
X45
)
↔
(
(
¬
X43
)
→
X48
)
)
)
→
(
(
(
X46
↔
(
X45
↔
(
X22
→
X19
→
False
)
)
)
↔
(
¬
(
(
¬
(
X46
↔
(
X43
↔
X6
)
)
)
→
(
¬
(
(
X49
↔
(
X46
↔
(
X45
↔
(
X19
→
False
)
)
)
)
→
(
X46
↔
(
X45
→
X6
)
)
)
)
)
)
)
↔
(
X46
→
(
X34
↔
(
X22
↔
X45
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
X46
)
→
(
¬
X44
)
→
X32
→
X49
)
→
(
(
¬
(
X48
→
(
¬
X46
)
→
(
¬
X1
)
→
X49
)
)
↔
(
¬
(
(
¬
X46
)
→
(
(
¬
X32
)
↔
(
X1
→
X49
)
)
)
)
)
)
)
↔
(
(
¬
(
(
X23
↔
(
(
¬
X12
)
→
(
(
¬
X11
)
↔
X46
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X23
)
↔
(
(
¬
X12
)
→
X48
)
)
)
)
)
)
↔
(
(
¬
X49
)
↔
(
X23
→
(
(
¬
X7
)
↔
(
X2
→
X11
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
(
(
¬
X43
)
→
(
(
¬
X41
)
↔
(
(
¬
X40
)
→
X45
)
)
)
↔
(
¬
(
X45
↔
(
X40
→
X8
→
X41
)
)
)
)
→
(
(
(
(
X43
↔
(
(
¬
X40
)
→
(
(
¬
X31
)
↔
X45
)
)
)
↔
(
X33
↔
(
X31
→
X45
)
)
)
↔
(
¬
(
X45
↔
(
X43
↔
X31
)
)
)
)
↔
(
(
(
¬
X40
)
↔
(
X31
→
X49
)
)
↔
(
(
X45
→
(
(
¬
X15
)
↔
X31
)
)
↔
(
X49
→
(
X40
↔
(
X15
↔
X45
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X45
→
(
¬
X10
)
→
X49
)
)
→
X12
→
(
X11
↔
(
X10
↔
X45
)
)
)
→
(
(
(
X27
→
(
X12
↔
(
(
¬
X11
)
→
X19
)
)
)
↔
(
¬
(
X45
↔
(
X12
→
X11
)
)
)
)
↔
(
X27
→
(
X19
↔
(
X12
→
False
)
)
)
)
)
)
→
(
¬
(
X49
→
X45
→
X43
→
X12
)
)
)
→
(
(
(
(
¬
(
X49
↔
(
(
¬
X30
)
→
X41
)
)
)
→
(
¬
(
X49
→
X30
→
X18
→
X45
)
)
)
→
(
¬
(
X44
→
(
(
¬
X30
)
↔
(
(
¬
X18
)
→
X49
)
)
)
)
)
↔
(
(
¬
(
(
X36
↔
(
X18
↔
X4
)
)
→
(
X30
↔
(
(
¬
X4
)
↔
X41
)
)
→
(
(
X18
→
(
(
¬
X4
)
↔
X41
)
)
↔
(
X30
↔
(
X4
→
X36
)
)
)
)
)
↔
(
X41
→
X36
→
(
X18
↔
False
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X42
↔
(
(
¬
X24
)
→
(
X5
↔
X49
)
)
)
)
→
(
(
(
X49
↔
(
X45
→
X48
)
)
→
(
(
(
¬
X36
)
↔
(
(
¬
X17
)
↔
X49
)
)
↔
(
¬
(
X49
→
(
(
¬
X45
)
↔
(
X17
↔
X48
)
)
)
)
)
)
↔
(
(
¬
(
X48
→
(
(
¬
X46
)
↔
(
(
¬
X24
)
↔
X49
)
)
)
)
↔
(
X48
→
X45
→
(
(
¬
X16
)
↔
X46
)
)
)
)
)
)
↔
(
(
¬
(
X49
↔
(
X25
↔
(
(
¬
X0
)
↔
X23
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X49
↔
(
X45
→
X42
→
False
)
)
)
↔
(
(
(
¬
X45
)
→
(
X23
↔
X49
)
)
↔
(
¬
(
(
¬
X42
)
→
X23
→
(
¬
X14
)
→
X45
)
)
)
)
↔
(
(
¬
X45
)
↔
(
(
¬
X42
)
↔
X49
)
)
)
→
(
X42
↔
(
(
¬
X25
)
↔
(
(
¬
X14
)
↔
X45
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X44
→
(
¬
X38
)
→
X47
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
X37
→
(
(
¬
X13
)
↔
X47
)
)
)
→
(
¬
(
X47
→
(
¬
X38
)
→
X37
→
X39
)
)
)
)
↔
(
X37
→
X47
)
)
→
(
¬
(
(
(
X44
→
(
(
¬
X3
)
↔
(
(
¬
X1
)
↔
X38
)
)
)
↔
(
X39
↔
(
X38
↔
(
X3
→
X44
)
)
)
)
→
(
¬
(
(
¬
X39
)
→
(
¬
X38
)
→
(
¬
X1
)
→
X47
)
)
)
)
)
)
)
↔
(
(
(
(
X35
→
(
X28
↔
(
(
¬
X22
)
→
X39
)
)
)
↔
(
¬
(
(
X32
↔
X44
)
→
(
X39
↔
(
(
¬
X28
)
↔
(
(
¬
X26
)
→
X35
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X32
↔
(
X21
→
X6
→
False
)
)
)
→
(
(
(
X40
→
X35
→
(
X32
↔
X6
)
)
→
(
X32
↔
(
X6
↔
X21
)
)
)
↔
(
(
X35
→
(
¬
X32
)
→
(
(
¬
X28
)
↔
X38
)
)
↔
(
X38
→
(
X35
↔
(
X32
→
X21
→
False
)
)
)
)
)
)
)
)
→
(
(
(
X29
↔
(
(
¬
X9
)
↔
X20
)
)
↔
(
(
¬
(
(
¬
X9
)
↔
(
X2
→
X20
)
)
)
↔
(
(
(
¬
X29
)
→
(
(
¬
X22
)
↔
(
(
¬
X2
)
↔
X35
)
)
)
↔
(
¬
(
(
¬
X29
)
→
X22
→
X20
→
X35
)
)
)
)
)
↔
(
X22
→
X9
→
(
X2
↔
False
)
)
)
→
(
(
X35
↔
(
X34
→
X20
)
)
↔
(
(
¬
(
X48
↔
(
X35
→
X22
→
X21
→
False
)
)
)
↔
(
¬
(
(
X22
↔
(
(
¬
X20
)
→
X21
)
)
→
(
¬
(
(
¬
(
X48
→
(
X20
↔
(
X7
↔
False
)
)
)
)
→
X48
→
X22
→
(
X21
↔
X7
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
428e49...
and proposition id is
fd755e...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMTTvFgieHsqYsHYiSMUz8mwfP64Le59yi3
)
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
(
¬
(
(
(
(
(
¬
(
(
X50
→
X48
→
X32
)
→
(
(
X50
↔
(
X32
↔
(
X31
↔
X48
)
)
)
↔
(
¬
(
(
¬
(
X50
↔
(
(
¬
X11
)
↔
(
(
¬
X4
)
↔
X48
)
)
)
)
→
(
X48
↔
(
X32
↔
(
X11
→
X50
)
)
)
)
)
)
)
)
→
(
X48
↔
(
(
¬
X31
)
→
X14
→
X50
)
)
)
↔
(
(
(
¬
(
(
¬
X46
)
↔
(
(
¬
X32
)
↔
X50
)
)
)
↔
(
X50
↔
(
X48
↔
X3
)
)
)
→
(
X50
↔
(
X46
↔
(
X29
→
X48
)
)
)
)
)
↔
(
(
¬
(
(
X50
↔
(
X36
↔
(
(
¬
X21
)
↔
X46
)
)
)
→
(
(
(
¬
(
X50
↔
(
X45
→
(
¬
X39
)
→
X48
)
)
)
↔
(
X50
↔
(
(
¬
X21
)
→
X48
)
)
)
↔
(
(
¬
X21
)
↔
X48
)
)
)
)
→
(
¬
(
X36
→
(
¬
X21
)
→
(
X15
↔
X39
)
)
)
)
)
→
(
(
(
(
¬
X37
)
↔
(
(
¬
X35
)
↔
X48
)
)
↔
(
(
X43
↔
(
X35
↔
(
(
¬
X12
)
→
X24
)
)
)
→
(
¬
X28
)
→
(
¬
X24
)
→
(
X12
↔
X43
)
)
)
↔
(
¬
(
(
(
(
¬
(
X50
→
X48
→
(
X44
↔
(
X40
↔
False
)
)
)
)
↔
(
¬
(
(
¬
X33
)
↔
(
X19
↔
X44
)
)
)
)
↔
(
X48
↔
(
X45
↔
(
(
¬
X40
)
→
X44
)
)
)
)
→
(
(
(
¬
(
X43
↔
(
X1
→
X40
)
)
)
↔
(
¬
(
X50
↔
(
X48
→
X43
→
(
X1
↔
False
)
)
)
)
)
↔
(
¬
(
(
(
¬
X43
)
→
(
¬
X40
)
→
(
X18
↔
X50
)
)
→
(
X50
↔
(
(
¬
X37
)
→
(
¬
X18
)
→
X43
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
(
¬
(
X50
↔
(
X26
↔
(
(
¬
X8
)
→
X34
)
)
)
)
→
(
¬
(
(
¬
X26
)
→
X8
→
(
(
¬
X2
)
↔
X45
)
)
)
)
→
(
(
(
(
X17
↔
X45
)
↔
(
(
X45
↔
(
X34
→
(
X17
↔
(
X9
→
False
)
)
)
)
→
(
¬
(
X38
↔
(
(
¬
X9
)
→
X45
)
)
)
)
)
↔
(
X34
↔
(
X17
→
X10
→
X45
)
)
)
↔
(
X38
→
X34
→
X10
→
X50
)
)
)
↔
(
(
¬
(
X50
↔
(
X45
→
(
(
¬
X7
)
↔
X41
)
)
)
)
↔
(
(
¬
(
(
X50
↔
(
X45
→
(
X34
↔
(
X16
→
False
)
)
)
)
→
X34
→
X16
→
X6
→
X50
)
)
→
(
¬
(
X41
→
(
(
¬
X16
)
↔
(
X6
↔
X45
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X49
→
(
¬
X47
)
→
(
X38
↔
X50
)
)
)
↔
(
(
¬
(
X50
↔
(
X45
→
(
X25
↔
X47
)
)
)
)
↔
(
(
¬
X47
)
→
(
X45
↔
(
X38
↔
X50
)
)
)
)
)
→
(
X50
↔
X47
)
)
↔
(
X38
→
(
¬
X25
)
→
X50
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
X47
)
↔
(
X42
↔
(
X27
↔
X49
)
)
)
)
→
(
(
(
(
¬
X5
)
→
(
(
¬
X0
)
↔
X27
)
)
→
(
¬
(
X20
→
(
¬
X0
)
→
X23
)
)
)
↔
(
(
(
X47
↔
(
(
¬
X27
)
↔
(
X20
↔
X49
)
)
)
↔
(
(
¬
X0
)
→
X27
)
)
→
(
¬
(
X47
→
(
(
¬
X27
)
↔
(
(
¬
X0
)
↔
X30
)
)
)
)
)
)
)
)
↔
(
(
(
X49
→
X42
→
(
¬
X23
)
→
X50
)
↔
(
¬
(
X23
↔
(
(
¬
X13
)
↔
X49
)
)
)
)
↔
(
(
¬
(
(
¬
X48
)
→
(
X42
↔
X50
)
)
)
↔
(
(
X50
↔
(
(
¬
X23
)
↔
X48
)
)
→
(
(
X42
→
(
(
¬
X23
)
↔
X48
)
)
↔
(
¬
(
X48
↔
(
X23
↔
(
(
¬
X22
)
↔
X50
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
X49
↔
(
X44
↔
False
)
)
↔
(
¬
(
X50
→
X45
→
X44
→
X25
)
)
)
↔
(
¬
(
(
¬
(
X49
→
(
X48
↔
(
X44
↔
(
X39
↔
False
)
)
)
)
)
→
(
(
X44
↔
(
(
¬
X9
)
↔
X50
)
)
↔
(
(
¬
X48
)
↔
(
(
¬
X44
)
↔
(
(
¬
X39
)
→
X50
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X32
↔
(
(
¬
X0
)
↔
X49
)
)
)
↔
(
(
(
¬
(
(
¬
X49
)
→
X46
→
X0
→
X50
)
)
↔
(
(
¬
(
X48
↔
(
X46
→
(
X32
↔
X50
)
)
)
)
↔
(
X49
↔
(
(
¬
X32
)
→
(
(
¬
X27
)
↔
X46
)
)
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X35
)
→
(
(
¬
X32
)
↔
X49
)
)
)
)
)
)
→
(
(
(
X48
↔
(
(
¬
X11
)
→
X41
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X50
→
(
¬
X48
)
→
(
X41
↔
X49
)
)
)
→
(
X49
↔
(
X41
↔
X11
)
)
)
)
→
(
(
¬
X48
)
↔
(
(
¬
X41
)
→
X49
)
)
)
)
)
↔
(
X50
↔
(
X49
→
X41
→
False
)
)
)
)
)
)
)
→
(
(
(
(
X50
→
(
X30
↔
(
X13
↔
X12
)
)
)
→
(
¬
(
X43
↔
(
(
¬
X30
)
↔
(
(
¬
X13
)
↔
X50
)
)
)
)
)
↔
(
X50
↔
(
(
¬
X30
)
↔
(
X28
↔
X43
)
)
)
)
↔
(
(
(
¬
(
(
¬
X49
)
→
(
(
¬
X47
)
↔
(
(
¬
X16
)
↔
X50
)
)
)
)
→
(
¬
(
X49
→
X47
→
(
X45
↔
(
X16
→
False
)
)
)
)
)
→
(
(
(
¬
X49
)
→
X50
)
↔
(
¬
(
(
¬
X47
)
↔
(
X43
↔
(
(
¬
X28
)
→
X49
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
(
¬
X49
)
→
(
(
¬
X46
)
↔
(
X23
→
X50
)
)
)
)
↔
(
X33
→
(
¬
X23
)
→
X50
)
)
↔
(
(
¬
(
(
¬
(
X50
→
X38
→
(
X37
↔
(
X5
↔
False
)
)
)
)
→
(
(
(
¬
X38
)
→
(
¬
X33
)
→
X5
→
X49
)
↔
(
¬
(
(
¬
X33
)
→
(
(
¬
X15
)
↔
(
X5
↔
X50
)
)
)
)
)
)
)
↔
(
¬
(
(
(
X31
↔
(
(
¬
X19
)
↔
X49
)
)
↔
(
¬
(
X50
→
(
¬
X31
)
→
(
X14
↔
X49
)
)
)
)
→
(
(
¬
X49
)
↔
(
X37
↔
X50
)
)
→
(
(
(
¬
(
X37
↔
(
X19
→
(
X18
↔
X10
)
)
)
)
→
X49
)
↔
(
X50
↔
(
X19
↔
(
X10
→
X18
)
)
)
)
)
)
)
)
→
(
(
(
X7
→
(
(
¬
X4
)
↔
(
(
¬
X3
)
→
X36
)
)
)
→
X36
→
X3
→
X2
→
X7
)
↔
(
(
(
(
(
(
¬
X40
)
→
X50
)
↔
(
(
¬
(
X40
↔
(
X38
→
(
X37
↔
X50
)
)
)
)
↔
(
X38
↔
(
X37
↔
(
X4
↔
False
)
)
)
)
)
↔
(
¬
(
X37
↔
(
(
¬
X4
)
→
X50
)
)
)
)
↔
(
¬
(
X38
→
(
(
¬
X4
)
↔
X40
)
)
)
)
↔
(
(
¬
(
X40
↔
(
(
¬
X4
)
→
X7
)
)
)
↔
(
(
¬
(
X50
→
(
¬
X36
)
→
X4
→
X40
)
)
→
(
¬
(
X36
↔
(
X7
↔
False
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X38
↔
(
(
¬
X24
)
↔
X49
)
)
)
→
(
(
¬
X38
)
↔
(
X24
→
X6
→
X42
)
)
)
)
→
(
¬
(
X49
→
X42
→
X24
)
)
)
→
(
(
(
¬
(
(
¬
(
(
¬
(
(
¬
(
X42
→
(
¬
X34
)
→
X26
→
X38
)
)
→
(
¬
(
X38
↔
(
X26
→
(
¬
X20
)
→
X34
)
)
)
)
)
→
(
¬
X26
)
→
(
¬
X20
)
→
X38
)
)
→
(
¬
(
(
¬
X34
)
→
(
(
¬
X26
)
↔
X38
)
)
)
)
)
↔
(
(
¬
(
X42
→
(
¬
X38
)
→
(
(
¬
X17
)
↔
X49
)
)
)
↔
(
(
(
¬
X34
)
↔
(
X17
→
X49
)
)
→
(
X49
↔
(
X42
↔
(
X34
→
False
)
)
)
→
(
¬
(
(
¬
X34
)
↔
(
X17
→
(
X8
↔
X42
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X38
)
↔
(
(
¬
X29
)
→
X49
)
)
)
→
(
(
X29
→
X49
)
↔
(
X50
↔
(
X49
↔
(
X1
↔
X42
)
)
)
)
)
→
(
¬
(
X22
↔
(
(
¬
X21
)
→
(
X1
↔
X50
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
eb8761...
and proposition id is
1e0fe2...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMY2BcT6wqKMqUtKPwRmkwmKaokMQjophd1
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
¬
∀X51 :
prop
,
∀X52 :
prop
,
(
¬
∀X53 :
prop
,
(
¬
∀X54 :
prop
,
(
(
(
(
(
(
(
(
(
¬
X53
)
↔
(
X52
↔
(
(
¬
X40
)
→
X54
)
)
)
→
(
X54
↔
(
(
¬
X40
)
↔
(
(
¬
X19
)
→
X52
)
)
)
)
↔
(
¬
(
(
¬
X53
)
→
X52
→
(
X45
↔
X54
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
¬
X11
)
→
X52
)
)
→
(
X30
↔
(
X27
↔
(
X11
→
X10
)
)
)
)
)
→
(
(
(
¬
(
X45
↔
(
X27
→
X30
)
)
)
→
(
¬
(
X52
↔
(
X27
↔
(
X21
→
X30
)
)
)
)
)
↔
(
¬
(
X45
↔
(
X40
↔
(
X30
↔
X52
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
X54
→
X53
→
(
X29
↔
X52
)
)
↔
(
(
X53
↔
(
X41
↔
(
X29
↔
X12
)
)
)
↔
(
X41
→
X54
)
)
)
→
(
(
X29
↔
(
X22
→
X14
→
X52
)
)
↔
(
(
¬
(
X43
→
(
X22
↔
False
)
)
)
→
(
(
(
¬
X43
)
↔
(
X41
→
(
¬
X14
)
→
X52
)
)
→
X41
→
(
X22
↔
(
X14
→
X52
)
)
)
→
(
¬
(
X14
→
False
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
(
(
X52
→
(
¬
X7
)
→
X33
)
→
X52
→
(
(
¬
X43
)
↔
(
X7
↔
X50
)
)
)
↔
(
¬
(
X43
→
X33
→
False
)
)
)
→
(
¬
(
X52
↔
(
(
¬
X32
)
→
X50
)
)
)
)
↔
(
¬
(
(
¬
(
X42
→
(
(
¬
X32
)
↔
(
X26
→
X33
)
)
)
)
→
X42
→
(
X33
↔
(
(
¬
X32
)
↔
X43
)
)
)
)
)
↔
(
X54
↔
(
X43
→
(
X42
↔
X50
)
)
)
)
→
(
(
X50
↔
(
(
¬
X17
)
→
(
(
¬
X1
)
↔
X52
)
)
)
↔
(
X42
→
(
¬
X17
)
→
X35
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
¬
(
(
(
¬
(
X54
↔
(
X51
↔
(
(
¬
X18
)
→
X38
)
)
)
)
↔
(
(
¬
X51
)
↔
(
(
¬
X38
)
↔
(
X31
→
X54
)
)
)
)
→
(
¬
(
X51
→
(
X49
↔
(
X38
↔
X31
)
)
)
)
)
)
→
(
X49
↔
(
X31
→
X4
→
X38
)
)
)
→
X49
→
(
(
¬
X48
)
↔
(
(
¬
X38
)
→
X51
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X48
)
↔
(
X15
→
(
¬
X3
)
→
X51
)
)
)
→
(
(
(
X48
→
(
(
¬
X47
)
↔
(
X2
↔
X51
)
)
)
↔
(
¬
(
X51
→
(
¬
X47
)
→
(
X8
↔
X48
)
)
)
)
↔
(
¬
(
X47
↔
(
(
¬
X15
)
↔
X51
)
)
)
)
→
(
X48
↔
(
X5
→
False
)
)
)
→
(
¬
(
(
X38
→
(
(
¬
X20
)
↔
X47
)
)
→
(
(
(
(
(
¬
X20
)
↔
(
(
¬
X16
)
↔
(
X3
↔
X38
)
)
)
→
X20
→
X16
→
X47
)
→
(
¬
X3
)
→
X16
)
↔
(
¬
(
X38
→
(
¬
X16
)
→
X5
→
X47
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
X52
↔
(
X34
↔
False
)
)
→
(
(
(
X46
→
X34
)
↔
(
X53
→
X52
→
(
(
¬
X34
)
↔
X51
)
)
)
↔
(
¬
(
X53
↔
X51
)
)
)
)
↔
(
(
X53
→
X51
→
X28
→
X46
)
→
(
¬
(
X48
↔
(
(
¬
X46
)
→
X51
)
)
)
)
)
↔
(
¬
(
(
(
¬
X48
)
↔
(
(
¬
X46
)
→
X28
→
X54
)
)
→
(
¬
(
(
¬
X46
)
↔
(
(
¬
X28
)
↔
(
X0
→
X48
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
X53
→
(
X44
↔
(
X24
→
(
X9
↔
False
)
)
)
)
→
X9
→
X54
)
)
→
(
¬
(
X53
↔
(
(
¬
X24
)
→
X54
)
)
)
)
)
→
(
¬
(
X53
↔
(
X9
→
X46
)
)
)
)
↔
(
(
(
X25
→
X13
→
X6
→
X44
)
→
(
¬
(
(
X46
→
(
X25
↔
(
X6
↔
False
)
)
)
→
(
¬
(
X36
→
(
(
¬
X13
)
↔
X25
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
X37
)
↔
(
X23
↔
(
X6
↔
X39
)
)
)
→
(
X39
↔
(
X25
→
X37
)
)
)
→
(
¬
(
X54
→
(
¬
X25
)
→
X39
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
False
→
(
¬
(
X40
→
X16
→
X48
)
)
→
(
¬
(
(
¬
X37
)
↔
(
(
¬
X16
)
↔
X41
)
)
)
)
)
→
(
(
(
(
¬
X41
)
→
(
(
¬
X13
)
↔
X48
)
)
→
(
X37
↔
(
X22
↔
(
(
¬
X13
)
↔
X48
)
)
)
)
↔
(
X48
→
(
(
¬
X22
)
↔
X37
)
)
)
)
↔
(
(
(
X40
→
(
(
¬
X39
)
↔
X48
)
)
↔
(
¬
(
(
¬
X39
)
→
(
(
¬
X27
)
↔
(
(
¬
X2
)
↔
X48
)
)
)
)
)
→
(
(
X41
→
(
¬
X27
)
→
(
X23
↔
X38
)
)
↔
(
(
¬
(
X41
↔
(
(
¬
X38
)
↔
X40
)
)
)
↔
(
X40
→
(
X27
↔
(
(
¬
X2
)
↔
X38
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
X53
↔
(
(
¬
X31
)
→
X46
)
)
→
(
(
¬
(
X53
↔
(
X46
→
(
(
¬
X31
)
↔
X52
)
)
)
)
↔
(
(
¬
X52
)
→
(
¬
X46
)
→
(
¬
X17
)
→
X53
)
)
)
)
↔
(
(
¬
(
(
¬
X47
)
→
X31
→
(
X11
↔
X52
)
)
)
→
(
¬
(
X52
→
(
X47
↔
(
X11
↔
X1
)
)
)
)
→
(
¬
(
X52
→
X17
→
(
X11
↔
(
X1
→
False
)
)
)
)
)
)
↔
(
(
¬
(
X52
→
X47
→
X0
)
)
↔
(
¬
(
X53
→
X52
→
X51
)
)
)
)
↔
(
¬
(
(
¬
X47
)
→
(
(
¬
X46
)
↔
(
(
¬
X19
)
↔
X52
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
X52
→
(
X48
↔
X4
)
)
)
→
(
(
¬
(
X52
↔
(
(
¬
X48
)
→
(
X21
↔
X51
)
)
)
)
↔
(
(
¬
X52
)
↔
(
(
¬
X51
)
↔
X53
)
)
)
)
)
↔
(
(
¬
(
X51
→
X52
)
)
↔
(
¬
(
X53
↔
(
X41
→
(
(
¬
X40
)
↔
X51
)
)
)
)
)
)
↔
(
¬
(
X51
→
X40
→
X4
→
X5
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
X51
→
(
(
¬
X50
)
↔
(
(
¬
X8
)
↔
X54
)
)
)
)
→
(
X50
↔
(
X8
↔
X52
)
)
)
→
(
¬
(
(
¬
X52
)
→
(
X51
↔
X54
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X24
→
(
¬
X12
)
→
X54
)
)
→
(
X52
↔
(
X26
↔
(
X12
→
X54
)
)
)
)
)
→
(
¬
(
(
¬
X45
)
↔
(
X26
↔
X51
)
)
)
→
(
(
X45
→
X26
)
↔
(
(
¬
(
X26
↔
(
(
¬
X12
)
→
X45
)
)
)
↔
(
X51
→
X24
→
(
X12
↔
False
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X52
→
(
X34
↔
(
X32
→
X43
)
)
)
)
↔
(
¬
(
(
¬
X43
)
↔
(
X34
→
X49
)
)
)
)
↔
(
(
¬
X44
)
↔
(
(
¬
X43
)
→
(
(
¬
X34
)
↔
X52
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
¬
(
X52
↔
(
X7
→
(
X3
↔
X44
)
)
)
)
→
X44
→
(
X32
↔
(
(
¬
X3
)
→
X7
)
)
)
)
↔
(
X52
↔
(
(
¬
X32
)
→
X44
)
)
)
→
X52
→
X3
→
X44
)
→
X32
→
(
(
¬
X3
)
↔
X7
)
)
)
)
↔
(
(
¬
(
X52
↔
(
(
¬
X49
)
↔
(
(
¬
X44
)
→
X50
)
)
)
)
↔
(
X52
↔
(
(
¬
X44
)
→
(
X18
↔
X51
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
¬
(
(
¬
X53
)
↔
(
X36
↔
X54
)
)
)
↔
(
X52
→
X36
→
(
¬
X15
)
→
X54
)
)
↔
(
¬
(
(
¬
X52
)
→
X36
→
(
X10
↔
X54
)
)
)
)
↔
(
(
X54
→
X50
→
(
X36
↔
False
)
)
↔
(
¬
(
(
¬
(
X53
→
X50
→
X29
→
X36
)
)
→
(
X54
↔
(
X50
↔
(
X36
→
X29
→
False
)
)
)
)
)
)
)
↔
(
¬
(
X54
↔
(
X53
→
(
X35
↔
X36
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
(
¬
X52
)
→
X42
→
X53
)
↔
(
X33
→
(
(
¬
X6
)
↔
X52
)
)
)
→
(
X52
↔
(
X33
→
False
)
)
)
)
→
(
(
(
X53
→
(
(
¬
X36
)
↔
(
(
¬
X30
)
↔
X52
)
)
)
↔
(
(
(
¬
(
(
¬
X36
)
↔
(
X25
↔
X52
)
)
)
→
(
(
(
¬
X36
)
→
(
¬
X25
)
→
(
(
¬
X9
)
↔
X52
)
)
↔
(
X53
↔
(
X36
→
X25
)
)
)
)
↔
(
¬
(
X36
↔
(
X9
→
X53
)
)
)
)
)
↔
(
(
X53
↔
(
X52
→
(
X28
↔
X20
)
)
)
→
(
¬
(
X53
→
X28
→
X52
)
)
→
(
(
(
¬
(
X53
→
X52
→
X28
→
False
)
)
→
(
(
(
¬
(
X52
→
X36
→
(
X28
↔
(
X14
→
False
)
)
)
)
↔
(
X53
→
(
X52
↔
(
X28
↔
False
)
)
)
)
→
X53
→
(
(
¬
X28
)
↔
X36
)
)
→
(
X53
↔
(
X52
→
(
X14
↔
False
)
)
)
)
↔
(
X53
↔
(
X42
↔
(
X14
↔
X52
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
4d5433...
and proposition id is
7ee897...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMK7G3Zp5Hyw71WTAkaXCednq4g8mj2gcnp
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
¬
(
(
(
(
(
¬
X32
)
→
X31
→
(
¬
X4
)
→
X46
)
↔
(
(
¬
(
(
¬
(
X31
↔
(
X30
→
(
X27
↔
X46
)
)
)
)
→
(
¬
(
X38
↔
(
(
¬
X30
)
↔
X46
)
)
)
)
)
↔
(
(
¬
X38
)
↔
(
(
¬
X32
)
↔
(
(
¬
X30
)
↔
X46
)
)
)
)
)
↔
(
¬
(
X46
→
X31
→
X30
→
False
)
)
)
→
(
(
(
(
(
(
¬
X30
)
↔
(
(
¬
X10
)
→
X43
)
)
→
(
¬
(
(
¬
X30
)
↔
(
X27
↔
(
(
¬
X18
)
↔
X43
)
)
)
)
)
→
(
¬
(
X48
↔
(
X30
→
X10
→
X18
)
)
)
)
→
(
X43
↔
(
(
¬
X10
)
↔
X18
)
)
)
↔
(
¬
(
(
¬
X31
)
↔
(
X30
↔
(
(
¬
X27
)
→
X43
)
)
)
)
)
)
)
→
(
¬
(
(
X38
→
X13
→
X8
)
→
(
¬
(
X30
→
(
¬
X19
)
→
(
¬
X15
)
→
X48
)
)
→
(
¬
(
(
¬
X19
)
→
(
X15
↔
X41
)
)
)
)
)
)
→
(
(
(
X46
↔
(
(
¬
X45
)
↔
(
X38
→
X48
)
)
)
↔
(
X48
→
(
(
¬
X6
)
↔
X45
)
)
)
↔
(
(
(
¬
X36
)
↔
(
X9
→
(
X7
↔
X45
)
)
)
→
(
¬
(
(
X48
→
X46
→
(
X42
↔
X9
)
)
→
(
¬
X9
)
→
(
¬
X7
)
→
X42
)
)
)
)
)
↔
(
(
(
¬
(
(
(
X0
→
X16
)
↔
(
(
X20
↔
X17
)
→
(
(
¬
(
X41
→
X23
→
(
X17
↔
(
X0
↔
False
)
)
)
)
↔
(
X17
→
X16
)
)
)
)
→
(
(
¬
(
(
X35
→
X11
→
X3
)
→
(
¬
(
(
¬
X11
)
↔
(
X3
→
X23
)
)
)
→
(
(
¬
(
(
¬
X23
)
→
(
¬
X11
)
→
(
(
¬
X3
)
↔
X35
)
)
)
↔
(
X20
→
(
(
¬
X11
)
↔
(
(
¬
X3
)
→
X35
)
)
)
)
)
)
↔
(
¬
(
X3
→
False
)
)
)
→
(
¬
(
(
¬
X23
)
→
(
¬
X17
)
→
(
X3
↔
X40
)
)
)
)
)
→
(
(
(
(
¬
(
X44
↔
(
X40
→
X34
→
X24
)
)
)
→
(
(
¬
X24
)
↔
(
X12
→
X34
)
)
→
(
¬
(
X40
→
X47
)
)
)
↔
(
(
(
(
X22
↔
(
(
¬
X21
)
↔
X44
)
)
↔
(
¬
(
X44
↔
(
X35
↔
(
X22
→
X1
)
)
)
)
)
↔
(
(
¬
X21
)
↔
(
X1
→
X35
)
)
)
→
(
¬
(
X47
→
(
¬
X35
)
→
X22
→
X40
)
)
)
)
↔
(
(
X49
→
(
X44
↔
(
X35
→
False
)
)
)
↔
(
(
¬
X41
)
→
(
¬
X35
)
→
X47
)
)
)
)
→
(
(
X49
→
X43
→
X25
→
X46
)
↔
(
(
¬
(
(
(
(
¬
X37
)
→
X28
→
(
(
¬
X5
)
↔
X46
)
)
↔
(
X37
→
(
¬
X29
)
→
(
X5
↔
X46
)
)
)
→
X44
→
(
X28
↔
(
X26
↔
X47
)
)
)
)
↔
(
(
(
¬
(
(
¬
X43
)
→
(
(
¬
X39
)
↔
(
(
¬
X14
)
↔
X49
)
)
)
)
↔
(
(
X47
→
X37
→
X14
→
X39
)
↔
(
X47
↔
(
X39
↔
(
X37
→
X43
)
)
)
)
)
↔
(
(
X49
↔
(
X47
↔
(
X44
↔
X33
)
)
)
↔
(
(
(
¬
X47
)
↔
(
X44
↔
(
(
¬
X2
)
→
X49
)
)
)
↔
(
(
¬
X47
)
↔
(
X44
→
X37
→
X49
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
(
(
(
X45
↔
(
X41
↔
(
(
¬
X27
)
↔
X32
)
)
)
↔
(
¬
(
(
¬
X45
)
↔
(
X41
↔
X46
)
)
)
)
→
X45
→
(
X32
↔
(
X27
→
False
)
)
)
→
(
¬
(
X45
→
X41
→
X32
→
False
)
)
)
↔
(
¬
(
X46
↔
(
X45
↔
X32
)
)
)
)
↔
(
(
X46
→
(
X41
↔
X48
)
)
↔
(
(
¬
(
(
¬
X25
)
→
(
(
¬
X6
)
↔
X48
)
)
)
↔
(
(
X46
→
X32
→
(
¬
X6
)
→
X25
)
↔
(
X25
↔
(
X11
→
(
X6
↔
X48
)
)
)
)
)
)
)
→
(
(
(
¬
(
X45
↔
(
X41
→
(
(
¬
X16
)
↔
X39
)
)
)
)
→
(
¬
(
X39
↔
(
X32
↔
(
X15
↔
X45
)
)
)
)
→
(
(
¬
(
X39
↔
(
X32
↔
(
X2
↔
False
)
)
)
)
↔
(
¬
(
X32
→
X23
→
X16
)
)
)
→
(
¬
(
(
¬
X23
)
→
(
(
¬
X16
)
↔
(
X2
→
X45
)
)
)
)
)
↔
(
X41
↔
(
X32
↔
(
X30
↔
X5
)
)
)
)
)
↔
(
(
(
(
X37
↔
(
X29
→
(
X28
↔
X45
)
)
)
↔
(
X40
↔
(
(
¬
X29
)
→
(
¬
X28
)
→
X42
)
)
)
↔
(
¬
(
(
X45
→
X34
→
X40
)
→
(
X42
↔
(
X41
→
(
¬
X14
)
→
X45
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
X49
→
X46
→
(
X45
↔
X37
)
)
→
(
¬
(
X49
→
(
¬
X40
)
→
X45
)
)
)
)
→
(
¬
(
(
X45
↔
(
X21
→
(
¬
X18
)
→
X40
)
)
→
(
X37
↔
X45
)
→
(
X21
→
X9
→
False
)
→
(
¬
X37
)
→
(
¬
X18
)
→
X45
)
)
)
→
(
¬
(
X49
↔
(
X45
↔
(
X37
→
False
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X43
→
(
(
¬
X35
)
↔
(
(
¬
X4
)
↔
X38
)
)
)
)
→
(
(
¬
(
X44
↔
(
X4
↔
False
)
)
)
↔
(
(
¬
(
(
(
¬
X44
)
→
X35
→
(
(
¬
X17
)
↔
X48
)
)
→
(
¬
(
X48
→
(
X35
↔
(
X17
→
X4
→
False
)
)
)
)
)
)
→
(
¬
(
X48
→
X35
→
X24
→
X44
)
)
)
)
)
)
↔
(
(
(
(
X38
→
(
X31
↔
(
X20
↔
False
)
)
)
↔
(
¬
(
(
¬
X35
)
→
X3
→
X44
)
)
)
→
(
¬
(
X44
→
(
¬
X31
)
→
(
(
¬
X1
)
↔
X38
)
)
)
)
↔
(
¬
(
(
X45
↔
(
X13
↔
X35
)
)
→
(
¬
(
X13
→
(
X12
↔
X38
)
)
)
→
(
¬
(
X45
↔
(
(
¬
X12
)
↔
X20
)
)
)
)
)
)
)
↔
(
(
(
(
(
X48
→
X44
→
X26
→
X46
)
↔
(
¬
(
(
¬
(
(
¬
X33
)
↔
(
X26
↔
(
(
¬
X19
)
↔
X48
)
)
)
)
→
(
¬
X44
)
→
(
(
¬
X33
)
↔
(
X26
↔
X48
)
)
)
)
)
↔
(
(
¬
X46
)
→
(
(
¬
X44
)
↔
(
(
¬
X36
)
↔
X48
)
)
)
)
↔
(
(
(
¬
(
X33
↔
(
(
¬
X19
)
↔
(
X8
↔
X46
)
)
)
)
↔
(
X46
↔
(
X7
↔
X33
)
)
)
↔
(
¬
(
X8
→
X46
)
)
)
)
→
(
(
X33
↔
(
X10
↔
X36
)
)
→
(
¬
(
X36
→
(
(
¬
X22
)
↔
X47
)
)
)
)
→
(
¬
(
(
¬
X33
)
→
X22
→
X48
)
)
→
(
¬
(
(
(
¬
(
(
¬
X45
)
↔
(
X22
↔
(
(
¬
X0
)
→
X47
)
)
)
)
→
(
¬
(
X45
↔
(
(
¬
X22
)
→
(
(
¬
X0
)
↔
X48
)
)
)
)
)
→
(
X48
↔
(
X47
↔
(
X45
↔
X33
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
1e64e6...
and proposition id is
3f9404...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMSCQYNUv8H99Yt66mu2Kjg2fvoZCqteXqt
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
(
¬
(
X48
→
(
X47
↔
(
X42
↔
X34
)
)
)
)
→
(
(
¬
(
X48
→
(
X47
↔
(
X39
↔
(
X24
↔
False
)
)
)
)
)
↔
(
(
(
¬
X47
)
↔
(
(
¬
X39
)
↔
(
(
¬
X26
)
→
X48
)
)
)
→
(
¬
(
(
(
(
X36
→
X2
)
→
(
¬
(
X48
↔
(
X34
→
X2
→
X16
)
)
)
)
↔
(
(
(
¬
X39
)
→
(
¬
X36
)
→
X48
)
↔
(
(
¬
X39
)
→
(
¬
X36
)
→
X2
→
X47
)
)
)
→
(
¬
(
(
X48
↔
(
X43
→
(
X16
↔
X11
)
)
)
→
(
¬
(
X39
→
(
¬
X16
)
→
(
(
¬
X11
)
↔
X43
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
X48
↔
(
X41
↔
(
(
¬
X10
)
→
X22
)
)
)
↔
(
(
X47
↔
(
(
¬
X22
)
→
X48
)
)
→
(
(
X48
↔
(
(
¬
X17
)
→
(
¬
X10
)
→
X22
)
)
↔
(
X47
→
(
X17
↔
(
X10
→
False
)
)
)
)
)
)
→
(
X22
↔
X47
)
)
→
(
(
X47
→
X42
→
X48
)
↔
(
(
(
X48
↔
(
X42
↔
(
X15
→
X22
)
)
)
→
(
X42
↔
(
(
¬
X22
)
→
X30
)
)
→
(
(
X47
↔
(
X42
↔
(
X22
↔
False
)
)
)
↔
(
(
X42
↔
(
X30
→
X47
)
)
→
(
(
(
X42
→
X22
→
(
¬
X15
)
→
X47
)
→
(
¬
(
(
¬
X30
)
↔
(
X22
↔
(
X15
→
X42
)
)
)
)
)
↔
(
(
¬
X42
)
→
(
¬
X15
)
→
X47
)
)
)
)
)
↔
(
¬
(
X22
→
X15
→
X48
)
)
)
)
)
)
→
(
(
(
¬
(
X39
↔
(
X32
↔
(
X1
→
X48
)
)
)
)
→
(
¬
(
X48
→
(
¬
X1
)
→
X42
)
)
)
↔
(
X48
↔
(
(
¬
X32
)
↔
X39
)
)
)
→
(
(
¬
(
X39
↔
(
(
¬
X28
)
↔
(
X20
↔
X37
)
)
)
)
↔
(
(
¬
(
X28
↔
X37
)
)
↔
(
(
X48
↔
(
X37
↔
(
(
¬
X18
)
↔
X39
)
)
)
→
(
¬
(
(
¬
X37
)
↔
(
(
¬
X28
)
↔
(
X18
→
X48
)
)
)
)
)
)
)
→
X48
→
(
X42
↔
(
(
¬
X28
)
↔
X39
)
)
)
→
(
(
¬
(
(
(
¬
(
(
X48
↔
(
X46
↔
X27
)
)
→
(
(
¬
X45
)
↔
(
X38
→
(
X27
↔
X46
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
X41
↔
(
X31
→
X44
)
)
→
(
(
¬
X45
)
↔
(
X44
↔
X48
)
)
)
)
→
(
¬
(
X45
↔
(
X44
→
(
X41
↔
X46
)
)
)
)
)
)
→
(
X45
↔
(
X41
→
(
X31
↔
False
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
(
¬
X29
)
→
X25
→
(
X19
↔
X46
)
)
→
X44
→
X19
→
X25
)
)
→
(
¬
(
(
¬
(
X48
→
(
¬
X33
)
→
X44
)
)
→
(
(
¬
(
X48
→
X33
→
(
X29
↔
X44
)
)
)
↔
(
¬
(
(
¬
X29
)
→
X46
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X44
↔
X12
)
)
↔
(
(
¬
(
X48
→
(
¬
X12
)
→
(
¬
X9
)
→
X44
)
)
→
(
¬
(
X46
↔
X39
)
)
)
)
→
X48
→
(
X44
↔
(
X21
↔
X39
)
)
)
)
)
)
↔
(
(
(
(
(
(
X48
→
(
X40
↔
(
X14
→
False
)
)
)
↔
(
¬
(
(
¬
(
X48
↔
(
X40
→
(
¬
X6
)
→
X35
)
)
)
→
(
¬
(
(
¬
X35
)
↔
(
(
¬
X14
)
↔
(
X6
↔
X40
)
)
)
)
)
)
)
→
(
¬
(
X48
→
X40
→
(
(
¬
X23
)
↔
X35
)
)
)
)
↔
(
¬
(
X35
↔
(
X6
↔
X40
)
)
)
)
→
(
¬
X40
)
→
(
(
¬
X35
)
↔
(
X13
→
X48
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
X7
↔
X48
)
↔
(
(
¬
X44
)
→
(
¬
X40
)
→
(
(
¬
X8
)
↔
X48
)
)
)
→
(
¬
(
X44
↔
(
X3
→
False
)
)
)
)
)
↔
(
(
¬
X41
)
→
(
X3
↔
X44
)
)
)
→
(
(
(
X41
↔
(
X4
→
(
X0
↔
X49
)
)
)
→
(
(
¬
(
(
(
¬
(
X44
↔
(
(
¬
X4
)
→
X8
)
)
)
↔
(
¬
(
X41
↔
(
X14
↔
(
X8
↔
False
)
)
)
)
)
→
(
¬
(
X44
↔
X4
)
)
)
)
↔
(
¬
(
X49
→
(
X14
↔
X41
)
)
)
)
)
→
X44
→
(
¬
X8
)
→
X14
)
→
(
¬
(
X49
→
(
X44
↔
(
X8
↔
(
X5
→
False
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
¬
(
(
¬
X37
)
↔
(
X36
↔
(
X26
→
X40
)
)
)
)
→
(
¬
(
(
¬
(
X40
↔
(
X2
↔
X26
)
)
)
→
(
(
¬
X39
)
↔
(
X37
↔
(
X26
↔
X40
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
X47
→
(
¬
X33
)
→
(
X32
↔
X49
)
)
)
↔
(
X49
→
(
(
¬
X32
)
↔
(
X26
→
X47
)
)
)
)
→
(
(
(
¬
X13
)
↔
(
(
¬
X5
)
↔
X26
)
)
↔
(
(
(
¬
X26
)
↔
(
X13
→
(
¬
X5
)
→
X47
)
)
↔
(
X26
→
(
(
¬
X13
)
↔
X33
)
)
)
)
)
↔
(
X49
↔
(
X47
↔
(
X36
→
X32
)
)
)
)
↔
(
(
(
¬
(
(
¬
X48
)
↔
(
(
¬
X23
)
↔
X49
)
)
)
↔
(
(
¬
X37
)
↔
(
(
¬
X33
)
→
X23
→
X48
)
)
)
→
(
¬
(
(
¬
(
X49
↔
(
X48
→
X24
→
False
)
)
)
→
(
¬
(
X48
→
(
X36
↔
(
(
¬
X24
)
→
X37
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X49
→
X45
→
X41
→
X7
→
False
)
)
↔
(
(
X47
→
(
X28
↔
X41
)
)
↔
(
¬
(
X48
→
(
X47
↔
(
X28
↔
(
X7
→
False
)
)
)
)
)
)
)
↔
(
(
¬
(
X48
→
(
X47
↔
(
X12
↔
X27
)
)
)
)
↔
(
(
¬
X48
)
↔
(
(
¬
X27
)
↔
(
X12
↔
X49
)
)
)
)
)
→
(
(
X49
↔
(
X43
↔
(
X10
→
False
)
)
)
↔
(
X45
↔
(
X43
→
(
(
¬
X10
)
↔
X35
)
)
)
)
→
(
(
(
(
¬
X45
)
↔
(
(
¬
X43
)
→
X20
→
X48
)
)
↔
(
X48
→
(
(
¬
X20
)
↔
X49
)
)
)
↔
(
(
¬
X20
)
→
X18
→
X45
)
)
)
↔
(
(
(
(
¬
X1
)
↔
X48
)
↔
(
(
¬
X47
)
→
(
X31
↔
(
X1
→
X49
)
)
)
)
↔
(
(
¬
(
X49
→
(
(
¬
X22
)
↔
(
X21
→
X47
)
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X47
)
→
X48
)
)
)
)
)
)
)
→
(
(
(
X45
↔
(
X30
→
X4
→
X38
)
)
↔
(
(
X45
→
(
¬
X42
)
→
(
¬
X30
)
→
X47
)
↔
(
¬
(
(
¬
X4
)
↔
X30
)
)
)
)
↔
(
(
(
(
(
¬
X17
)
→
X25
)
↔
(
X42
→
X38
→
X25
→
X47
)
)
↔
(
(
(
¬
(
X17
↔
(
X15
→
(
(
¬
X14
)
↔
X25
)
)
)
)
↔
(
X19
↔
(
X14
↔
X17
)
)
)
↔
(
(
¬
X19
)
↔
(
X17
→
(
(
¬
X14
)
↔
X47
)
)
)
)
)
↔
(
(
¬
(
(
X46
→
(
X42
↔
(
(
¬
X29
)
↔
X48
)
)
)
→
(
(
X47
→
(
¬
X3
)
→
X48
)
↔
(
¬
(
X48
→
(
X47
↔
(
(
¬
X42
)
→
X46
)
)
)
)
)
)
)
↔
(
X48
→
(
X38
↔
X42
)
)
)
)
)
→
(
(
(
¬
(
(
(
(
¬
(
(
¬
X16
)
→
(
(
¬
X6
)
↔
X48
)
)
)
→
(
¬
(
X47
↔
(
X34
↔
(
X6
↔
X48
)
)
)
)
)
→
(
¬
(
X34
→
False
)
)
)
→
X48
→
(
¬
X38
)
→
(
¬
X34
)
→
X47
)
)
→
(
¬
(
(
¬
X34
)
↔
(
(
¬
X6
)
↔
X47
)
)
)
)
↔
(
(
(
¬
(
(
¬
X38
)
↔
(
X8
→
X48
)
)
)
→
(
¬
(
(
¬
(
X16
↔
(
X11
→
X38
)
)
)
→
(
¬
(
X48
→
X16
→
X8
)
)
)
)
)
↔
(
(
(
X44
→
(
¬
X11
)
→
X0
→
X16
)
→
(
¬
(
X44
→
(
X9
↔
X16
)
)
)
)
→
(
¬
(
X38
↔
(
(
¬
X11
)
→
(
(
¬
X9
)
↔
X16
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
b545a9...
and proposition id is
59f630...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMGw5Le5jaXKYYPKAFKf9YDEEeEdrjG8Wx4
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
¬
(
(
¬
(
(
¬
(
(
¬
(
(
¬
(
(
(
¬
X41
)
↔
(
X39
↔
X48
)
)
→
(
(
X48
↔
(
X41
↔
X32
)
)
↔
(
X41
→
X39
→
X21
)
)
)
)
→
(
¬
(
(
(
¬
X41
)
→
X0
→
X48
)
→
X48
→
(
X39
↔
(
(
¬
X0
)
↔
X23
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
(
(
X47
↔
False
)
→
(
(
¬
(
X32
↔
(
X28
↔
(
X21
→
False
)
)
)
)
↔
(
X47
↔
(
X32
↔
(
(
¬
X28
)
→
X48
)
)
)
)
)
)
→
(
¬
(
X47
↔
X21
)
)
)
)
↔
(
¬
(
X48
→
(
X30
↔
(
(
¬
X21
)
↔
X32
)
)
)
)
)
→
(
(
¬
X48
)
→
(
¬
X17
)
→
(
X7
↔
X49
)
)
→
(
(
¬
(
X49
↔
(
X48
↔
X28
)
)
)
↔
(
(
X48
↔
(
X6
→
X49
)
)
↔
(
X48
→
X47
→
(
X6
↔
X28
)
)
)
)
→
(
X28
↔
(
(
¬
X17
)
↔
(
X6
↔
X48
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X39
↔
(
X33
→
X24
→
(
X12
↔
False
)
)
)
)
→
X31
→
X12
)
↔
(
(
¬
X27
)
→
(
¬
X12
)
→
X2
→
X39
)
)
↔
(
(
¬
(
(
(
X44
↔
(
X41
↔
(
(
¬
X27
)
↔
X39
)
)
)
↔
(
¬
(
X41
→
(
(
¬
X39
)
↔
(
X4
→
X44
)
)
)
)
)
→
(
¬
(
X44
↔
(
X31
↔
(
X8
↔
False
)
)
)
)
)
)
↔
(
(
¬
(
X48
→
(
X31
↔
(
X22
↔
X39
)
)
)
)
→
(
(
¬
(
X31
↔
(
X18
↔
False
)
)
)
→
(
¬
(
X39
→
(
(
¬
X18
)
↔
(
(
¬
X8
)
→
X48
)
)
)
)
)
→
X48
→
(
X39
↔
X18
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
X43
→
(
X11
↔
(
(
¬
X3
)
↔
X9
)
)
)
)
↔
(
¬
(
X43
→
(
(
¬
X20
)
↔
X46
)
)
)
)
→
(
¬
(
(
X45
↔
(
X43
→
(
X1
↔
X48
)
)
)
→
(
(
¬
(
(
¬
X11
)
↔
(
X1
↔
X48
)
)
)
↔
(
(
¬
(
X48
→
X43
→
X1
)
)
↔
(
¬
(
X43
→
(
(
¬
X3
)
↔
(
(
¬
X1
)
↔
X11
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
X45
↔
(
X36
→
(
(
¬
X16
)
↔
X46
)
)
)
)
↔
(
¬
(
(
¬
(
(
X48
↔
(
X46
↔
(
X45
↔
False
)
)
)
→
(
(
X48
→
X46
→
(
X36
↔
X34
)
)
↔
(
X48
↔
(
X45
→
(
(
¬
X25
)
↔
X36
)
)
)
)
)
)
→
(
¬
(
(
¬
X34
)
↔
X46
)
)
)
)
)
)
↔
(
¬
(
(
(
(
X15
→
X13
→
X19
)
↔
(
(
(
¬
(
X49
→
(
(
¬
X40
)
↔
(
X19
↔
X48
)
)
)
)
↔
(
X47
→
(
X40
↔
(
(
¬
X19
)
↔
X48
)
)
)
)
↔
(
(
X19
↔
(
X15
→
(
X13
↔
(
X10
↔
False
)
)
)
)
↔
(
X40
↔
(
(
¬
X10
)
↔
X13
)
)
)
)
)
↔
(
(
¬
(
X35
→
X15
→
X10
→
X46
)
)
→
(
¬
(
(
¬
X35
)
↔
(
X10
↔
X49
)
)
)
)
)
→
(
¬
(
(
¬
(
(
X45
↔
(
(
¬
X37
)
↔
(
X5
↔
X48
)
)
)
→
(
¬
(
X48
↔
(
(
¬
X42
)
→
X29
→
X45
)
)
)
→
(
¬
(
X45
→
(
X42
↔
(
(
¬
X37
)
↔
X48
)
)
)
)
)
)
→
(
X38
↔
(
X35
↔
(
(
¬
X26
)
↔
X49
)
)
)
→
(
(
(
(
(
X45
→
X38
→
X49
)
→
(
¬
(
(
(
(
¬
(
(
¬
(
X38
↔
X45
)
)
→
(
¬
X38
)
→
(
(
¬
X35
)
↔
(
(
¬
X14
)
↔
X49
)
)
)
)
↔
(
X45
→
X14
)
)
→
(
X49
↔
(
(
¬
X37
)
↔
(
(
¬
X14
)
↔
X45
)
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X37
)
→
X38
)
)
)
)
)
)
→
(
¬
(
(
¬
X38
)
→
(
X37
↔
X49
)
)
)
)
→
(
¬
(
X45
→
(
¬
X14
)
→
X37
)
)
)
↔
(
¬
(
X38
→
(
X37
↔
X48
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
(
(
¬
(
X44
↔
(
X28
→
(
(
¬
X21
)
↔
X29
)
)
)
)
↔
(
¬
(
X44
↔
(
X30
→
X29
→
(
X28
↔
False
)
)
)
)
)
→
X44
→
(
(
¬
X29
)
↔
(
(
¬
X28
)
→
X37
)
)
)
)
↔
(
(
¬
(
(
¬
X8
)
↔
X37
)
)
↔
(
(
X37
→
(
X21
↔
(
X8
↔
X44
)
)
)
↔
(
¬
(
X42
→
(
X37
↔
(
X8
→
X44
)
)
)
)
)
)
)
↔
(
(
X36
→
(
X32
↔
(
X7
→
X39
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
X42
)
↔
(
(
¬
X32
)
→
(
X29
↔
X49
)
)
)
)
→
(
(
X42
↔
(
X17
↔
X30
)
)
↔
(
¬
(
X32
→
(
¬
X30
)
→
X49
)
)
)
)
)
↔
(
(
¬
X32
)
↔
(
(
¬
X30
)
→
(
(
¬
X17
)
↔
X36
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
(
X45
→
(
¬
X33
)
→
(
¬
X31
)
→
X39
)
→
(
¬
(
X45
↔
(
X33
↔
(
(
¬
X19
)
↔
X39
)
)
)
)
)
)
↔
(
¬
(
X39
↔
(
(
¬
X31
)
→
X19
→
X49
)
)
)
)
↔
(
X45
→
X33
→
X49
)
)
↔
(
X33
→
(
¬
X19
)
→
X31
)
)
→
(
X49
↔
(
X44
→
(
X33
↔
X45
)
)
)
)
)
)
→
(
¬
(
(
(
(
X18
↔
(
X12
↔
(
(
¬
X0
)
→
X22
)
)
)
↔
(
(
¬
(
(
X48
→
X35
→
X22
→
False
)
→
(
(
¬
(
(
X48
→
X35
→
X24
→
X22
)
→
(
(
¬
X35
)
↔
(
(
¬
X22
)
↔
(
(
¬
X18
)
↔
X48
)
)
)
)
)
↔
(
(
¬
X35
)
↔
(
X24
↔
X48
)
)
)
)
)
↔
(
X35
→
X24
→
X22
→
False
)
)
)
↔
(
¬
(
(
X49
↔
(
X23
↔
X4
)
)
→
(
(
(
¬
(
(
¬
X36
)
→
(
X35
↔
(
(
¬
X23
)
→
X48
)
)
)
)
↔
(
¬
(
(
X49
↔
(
(
¬
X23
)
→
(
(
¬
X4
)
↔
X48
)
)
)
→
(
(
¬
X48
)
↔
(
X36
↔
(
(
¬
X4
)
↔
X49
)
)
)
)
)
)
↔
(
¬
(
X36
↔
(
X4
↔
X23
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X25
↔
X22
)
)
→
(
(
¬
(
X39
→
(
X36
↔
(
X35
↔
False
)
)
)
)
↔
(
(
¬
(
X35
↔
(
X22
↔
False
)
)
)
↔
(
¬
(
X39
↔
(
X36
↔
(
X11
→
X35
)
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
X46
↔
(
(
¬
X40
)
↔
(
X27
→
X44
)
)
)
→
(
¬
(
X44
↔
(
(
¬
X40
)
→
(
X34
↔
X46
)
)
)
)
)
→
(
(
(
X46
→
X2
→
X40
)
→
(
¬
(
X40
→
False
)
)
→
(
X47
↔
(
X6
↔
(
(
¬
X2
)
→
X34
)
)
)
)
↔
(
X46
→
(
¬
X6
)
→
X34
)
)
)
)
→
(
(
¬
(
(
¬
(
X40
↔
(
X3
↔
X9
)
)
)
→
(
¬
X45
)
→
(
X40
↔
(
(
¬
X34
)
↔
X46
)
)
)
)
↔
(
(
¬
(
X46
↔
(
(
¬
X40
)
→
X26
→
X45
)
)
)
↔
(
¬
(
X45
→
(
¬
X40
)
→
(
X1
↔
X46
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X48
→
X20
→
(
X15
↔
(
X14
↔
False
)
)
)
)
↔
(
¬
(
(
(
(
(
X48
↔
(
X45
↔
X43
)
)
→
X48
→
X15
→
X20
)
↔
(
¬
(
X45
↔
(
X20
→
X15
)
)
)
)
↔
(
(
¬
(
X45
→
(
X41
↔
(
(
¬
X20
)
↔
X43
)
)
)
)
→
(
¬
(
X45
→
X43
→
X5
→
X41
)
)
)
)
→
(
¬
(
X20
→
X43
)
)
)
)
)
→
(
¬
(
X48
↔
(
(
¬
X41
)
↔
(
(
¬
X13
)
→
X43
)
)
)
)
)
)
↔
(
(
¬
(
(
X47
↔
X16
)
→
(
¬
(
X46
→
(
¬
X45
)
→
(
(
¬
X38
)
↔
X47
)
)
)
)
)
→
(
X45
↔
(
X20
↔
X16
)
)
→
(
(
(
¬
(
X46
↔
(
(
¬
X45
)
↔
(
(
¬
X10
)
↔
X48
)
)
)
)
→
X46
→
(
(
¬
X45
)
↔
(
X16
↔
X48
)
)
)
↔
(
(
¬
X16
)
↔
X45
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
c33c51...
and proposition id is
532b5a...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMZidEnFKTQqHxbwotgLjPLiwBy2ESBpNSg
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
∀X50 :
prop
,
∀X51 :
prop
,
(
(
¬
(
(
(
(
(
(
X46
→
(
X28
↔
X47
)
)
→
(
(
¬
(
X16
↔
(
(
¬
X5
)
↔
X28
)
)
)
→
(
¬
(
(
¬
(
X46
→
(
X16
↔
(
X8
→
X28
)
)
)
)
→
(
¬
(
X46
↔
(
X28
→
X5
→
X8
)
)
)
)
)
)
→
(
¬
(
X28
↔
(
X16
↔
X8
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X28
)
↔
(
X16
↔
(
(
¬
X5
)
→
X51
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X16
)
→
(
¬
X2
)
→
X49
)
)
→
(
¬
X28
)
→
X5
→
X49
)
→
X5
→
(
(
¬
X2
)
↔
X16
)
)
)
)
↔
(
X49
↔
(
X16
↔
X2
)
)
)
↔
(
X51
↔
(
X37
↔
(
X28
→
X49
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
(
X42
↔
(
X10
→
X15
)
)
)
↔
(
X42
↔
(
(
¬
X16
)
↔
(
X15
↔
X43
)
)
)
)
→
(
X15
↔
X10
)
)
↔
(
(
(
¬
(
X46
→
(
(
¬
X10
)
↔
X15
)
)
)
→
(
¬
(
X46
→
X16
→
(
(
¬
X15
)
↔
X47
)
)
)
)
↔
(
(
¬
X46
)
↔
(
(
¬
X44
)
↔
(
(
¬
X16
)
↔
X47
)
)
)
)
)
→
(
(
(
¬
(
X25
↔
(
X20
→
X13
→
X42
)
)
)
↔
(
X25
→
X20
→
X13
→
X16
)
)
↔
(
¬
(
X47
→
(
¬
X20
)
→
(
X16
↔
X25
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
¬
X48
)
→
(
¬
X32
)
→
X49
)
)
→
(
(
(
X32
↔
(
X4
↔
False
)
)
↔
(
¬
(
X48
↔
(
(
¬
X35
)
→
(
X4
↔
X51
)
)
)
)
)
↔
(
¬
(
X48
↔
(
(
¬
X35
)
→
(
¬
X32
)
→
X49
)
)
)
)
→
(
¬
X48
)
→
(
¬
X35
)
→
X17
→
X51
)
↔
(
(
¬
X32
)
↔
(
(
¬
X4
)
↔
(
X0
→
X48
)
)
)
)
↔
(
¬
(
(
(
¬
(
X49
→
X19
→
X11
→
X33
)
)
↔
(
¬
(
X49
→
(
¬
X19
)
→
X11
→
X30
)
)
)
→
(
(
(
¬
X30
)
→
X11
→
X40
)
↔
(
¬
(
X51
→
(
X32
↔
(
(
¬
X11
)
→
X40
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X49
→
(
X39
↔
(
X18
→
X51
)
)
)
)
↔
(
¬
(
(
¬
(
X41
→
X39
→
(
¬
X18
)
→
X48
)
)
→
X41
→
(
X34
↔
X18
)
)
)
)
↔
(
(
¬
(
(
¬
X49
)
→
(
X48
↔
(
X47
↔
X51
)
)
)
)
↔
(
X51
↔
(
X48
→
X27
→
X47
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
(
¬
X14
)
→
(
(
¬
X1
)
↔
X24
)
)
)
→
(
(
¬
(
(
¬
(
X50
↔
X45
)
)
→
(
¬
(
X45
↔
(
(
¬
X22
)
↔
(
(
¬
X1
)
→
X51
)
)
)
)
)
)
↔
(
X29
→
(
(
¬
X22
)
↔
X51
)
)
)
)
→
(
(
(
(
¬
X50
)
→
X6
→
X51
)
↔
(
(
¬
(
(
¬
X6
)
→
X3
→
X51
)
)
→
X36
→
(
(
¬
X31
)
↔
(
(
¬
X6
)
→
X51
)
)
)
)
↔
(
(
(
¬
(
X45
↔
(
(
¬
X36
)
→
X49
)
)
)
→
(
¬
(
X48
→
X6
→
X36
)
)
)
↔
(
X48
→
X36
→
(
¬
X31
)
→
X45
)
)
)
)
)
↔
(
(
¬
(
(
(
X23
↔
(
X12
→
False
)
)
↔
(
¬
(
X51
↔
(
X38
↔
(
X12
↔
X50
)
)
)
)
)
→
X38
→
X21
→
(
(
¬
X9
)
↔
X12
)
)
)
↔
(
¬
(
(
(
¬
(
X49
↔
(
X26
→
(
¬
X9
)
→
X23
)
)
)
↔
(
¬
(
X23
→
(
X9
↔
(
(
¬
X7
)
↔
X49
)
)
)
)
)
→
(
¬
(
(
¬
X9
)
↔
(
(
¬
X7
)
→
X50
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
(
¬
X18
)
↔
(
(
¬
X13
)
↔
(
(
¬
X1
)
→
X36
)
)
)
↔
(
X44
→
(
X18
↔
(
X1
→
X41
)
)
)
)
→
(
(
¬
(
X51
↔
(
X24
→
X18
→
False
)
)
)
↔
(
¬
(
X44
↔
(
X39
→
(
¬
X18
)
→
X51
)
)
)
)
→
(
¬
(
(
¬
(
X51
↔
(
X50
→
(
X18
↔
(
X11
→
False
)
)
)
)
)
→
(
(
¬
(
(
¬
X50
)
↔
(
X36
↔
(
X11
↔
X51
)
)
)
)
↔
(
¬
(
X50
↔
(
X44
→
False
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
X48
↔
(
X44
→
X22
→
X16
→
False
)
)
→
(
X22
↔
(
X21
→
(
¬
X16
)
→
X50
)
)
)
)
↔
(
X50
→
(
X44
↔
(
X7
→
X21
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
X51
↔
(
X48
→
X40
→
False
)
)
→
X48
→
(
X40
↔
(
(
¬
X26
)
↔
X44
)
)
)
)
→
(
¬
(
X44
→
X51
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X44
)
↔
(
X30
→
(
X4
↔
X51
)
)
)
)
→
(
¬
X50
)
→
X30
→
(
¬
X4
)
→
X51
)
→
(
X51
↔
(
X50
↔
X4
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
X49
→
(
¬
X37
)
→
X2
→
X41
)
↔
(
(
¬
(
X5
→
(
X3
↔
(
X2
↔
X49
)
)
)
)
↔
(
¬
(
X49
→
(
¬
X5
)
→
X3
→
X37
)
)
)
)
→
(
(
X38
↔
(
X20
→
X37
)
)
↔
(
(
(
¬
(
(
¬
X38
)
↔
(
X26
→
X41
)
)
)
→
(
¬
(
X39
→
X26
→
X3
→
X41
)
)
)
↔
(
¬
(
X41
↔
(
X37
↔
(
(
¬
X26
)
↔
X39
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X46
↔
(
X34
↔
(
X27
→
X45
)
)
)
)
→
(
¬
(
(
¬
(
X46
→
X34
→
False
)
)
→
(
¬
(
(
¬
X34
)
→
(
X33
↔
(
X0
→
X45
)
)
)
)
)
)
)
→
(
(
X51
↔
(
(
¬
X34
)
↔
(
(
¬
X17
)
↔
X42
)
)
)
↔
(
(
¬
(
(
X42
↔
(
X34
↔
(
X19
↔
X17
)
)
)
→
X42
→
X34
→
X19
→
(
X12
↔
False
)
)
)
↔
(
¬
(
X45
→
(
(
¬
X34
)
↔
(
(
¬
X12
)
→
X48
)
)
)
)
)
)
→
(
¬
(
(
X49
↔
(
X43
→
X45
)
)
→
(
(
¬
(
(
X51
↔
(
X48
↔
X45
)
)
→
(
(
(
¬
(
X51
↔
(
X43
↔
X48
)
)
)
↔
(
(
¬
(
X48
↔
(
(
¬
X43
)
↔
(
(
¬
X33
)
→
X51
)
)
)
)
→
(
¬
(
X51
→
X45
→
(
(
¬
X33
)
↔
X48
)
)
)
)
)
↔
(
(
¬
X45
)
↔
(
X43
↔
X48
)
)
)
)
)
↔
X51
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
X45
→
(
¬
X29
)
→
X25
→
X32
)
)
↔
(
(
¬
(
(
X45
→
(
¬
X25
)
→
(
(
¬
X6
)
↔
X29
)
)
→
(
¬
(
X45
→
X25
→
(
X9
↔
False
)
)
)
)
)
↔
(
(
¬
X25
)
↔
(
X6
↔
X29
)
)
)
)
→
(
X25
↔
(
(
¬
X9
)
→
X29
)
)
)
)
→
(
(
(
(
¬
X45
)
↔
(
(
¬
X25
)
↔
X47
)
)
→
(
¬
(
(
(
¬
(
(
¬
X31
)
→
(
X25
↔
(
X10
→
X45
)
)
)
)
↔
(
X45
→
(
X31
↔
X47
)
)
)
→
X25
)
)
→
(
¬
(
X31
→
X10
→
False
)
)
)
↔
(
X47
→
(
X29
↔
(
(
¬
X10
)
↔
X25
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
X47
→
(
(
¬
X45
)
↔
(
X35
↔
X46
)
)
)
→
(
(
¬
(
X45
→
(
X23
↔
(
X15
↔
False
)
)
)
)
↔
(
¬
(
X46
↔
(
X15
→
X35
)
)
)
)
)
)
→
(
(
¬
(
X46
→
X35
→
X8
→
False
)
)
↔
(
¬
(
(
(
¬
X45
)
↔
(
X35
→
X46
)
)
→
(
¬
(
X46
→
X35
→
(
X14
↔
(
X8
↔
False
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
(
(
¬
X45
)
↔
(
(
¬
X43
)
→
(
¬
X28
)
→
X46
)
)
↔
(
¬
(
X46
→
(
X45
↔
(
X43
↔
X35
)
)
)
)
)
→
(
¬
(
(
¬
X28
)
→
X43
)
)
)
↔
(
X46
→
X45
→
X43
→
False
)
)
→
(
¬
(
(
¬
X43
)
↔
(
X35
↔
(
X32
↔
X45
)
)
)
)
)
)
↔
(
¬
(
X45
→
(
X35
↔
(
X32
→
False
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
16b0eb...
and proposition id is
588b16...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMXvNRBpgZYqREUqntoHJUDt4jvz7HXhiyp
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
(
¬
(
X28
→
X13
→
(
¬
X9
)
→
X42
)
)
↔
(
¬
(
(
¬
X26
)
↔
(
X13
→
(
¬
X9
)
→
X42
)
)
)
)
↔
(
¬
(
(
X13
↔
(
X12
→
(
(
¬
X6
)
↔
X24
)
)
)
→
(
¬
(
X6
→
X44
)
)
)
)
)
→
(
(
(
(
¬
(
X49
↔
(
X44
→
X39
)
)
)
↔
(
(
¬
(
X44
→
X20
→
X42
)
)
↔
(
X49
→
X43
→
(
¬
X20
)
→
X41
)
)
)
↔
(
X44
↔
(
X43
↔
(
(
¬
X8
)
→
X49
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X45
↔
(
X44
→
X39
→
False
)
)
)
↔
(
(
(
X44
→
(
¬
X39
)
→
X41
)
↔
(
¬
(
X44
→
(
X39
↔
(
(
¬
X0
)
→
X45
)
)
)
)
)
↔
(
X41
→
(
X25
↔
X45
)
)
)
)
→
X45
→
X44
→
(
X39
↔
X16
)
)
)
↔
(
¬
(
(
(
(
X37
→
(
X34
↔
X45
)
)
↔
(
(
¬
(
X37
→
X34
→
(
X22
↔
X26
)
)
)
↔
(
¬
(
X45
↔
(
X34
↔
(
X26
↔
X37
)
)
)
)
)
)
↔
(
¬
(
X37
→
(
¬
X26
)
→
(
¬
X21
)
→
X45
)
)
)
→
(
(
¬
(
X22
↔
X37
)
)
↔
(
(
X44
↔
(
X37
↔
X22
)
)
→
(
(
¬
X41
)
↔
X45
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
(
X31
→
(
¬
X18
)
→
(
(
¬
X14
)
↔
X32
)
)
)
→
(
¬
(
X18
→
(
X14
↔
(
(
¬
X1
)
→
X32
)
)
)
)
)
)
→
(
(
¬
(
X31
→
(
(
¬
X19
)
↔
(
X1
→
X41
)
)
)
)
↔
(
(
(
X31
→
(
¬
X19
)
→
X41
)
↔
(
¬
(
X38
↔
(
X19
→
X18
→
False
)
)
)
)
↔
(
¬
(
X18
→
X41
)
)
)
)
)
↔
(
(
(
(
X43
↔
(
X40
→
(
X30
↔
X46
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X30
)
↔
(
X29
→
X43
)
)
)
)
)
→
(
(
¬
(
X43
→
(
X41
↔
(
X40
→
X11
→
False
)
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X11
)
↔
X43
)
)
)
)
)
→
(
(
(
¬
(
X36
↔
(
X30
↔
(
X17
→
X46
)
)
)
)
↔
(
(
¬
X36
)
↔
(
(
¬
X30
)
→
(
X18
↔
X46
)
)
)
)
↔
(
¬
(
(
¬
(
X29
→
(
X23
↔
X36
)
)
)
→
(
(
X46
↔
(
X45
→
X29
→
X30
)
)
↔
(
(
¬
(
(
¬
X29
)
→
X23
→
X30
)
)
↔
(
(
¬
(
X45
↔
(
(
¬
X30
)
↔
(
(
¬
X14
)
↔
X36
)
)
)
)
↔
(
¬
(
(
X23
→
(
(
¬
X14
)
↔
X45
)
)
→
(
¬
(
X14
→
(
X5
↔
False
)
)
)
→
(
¬
(
X45
→
(
X36
↔
(
(
¬
X5
)
↔
X29
)
)
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
X46
↔
(
X33
↔
(
(
¬
X3
)
→
X48
)
)
)
)
→
(
(
¬
(
X48
→
X46
→
X43
→
X35
)
)
→
(
X43
↔
(
X35
→
(
(
¬
X33
)
↔
X46
)
)
)
)
→
(
¬
(
X48
↔
(
X33
↔
X35
)
)
)
)
→
(
¬
X46
)
→
(
X35
↔
X48
)
)
→
(
¬
(
X46
↔
(
(
¬
X43
)
↔
(
(
¬
X4
)
↔
X48
)
)
)
)
)
↔
(
(
(
¬
X43
)
→
(
(
¬
X35
)
↔
(
(
¬
X2
)
→
X46
)
)
)
↔
(
(
(
(
¬
(
X46
↔
(
(
¬
X35
)
↔
X43
)
)
)
→
(
¬
(
X43
↔
(
X10
↔
X15
)
)
)
)
↔
(
X46
↔
(
X43
↔
(
(
¬
X35
)
→
X47
)
)
)
)
→
(
¬
X43
)
→
X35
→
X46
)
)
)
↔
(
(
(
(
¬
(
X48
→
(
X46
↔
(
(
¬
X7
)
↔
X47
)
)
)
)
↔
(
X46
→
X27
→
X48
)
)
↔
(
¬
(
X47
↔
(
X7
→
X46
)
)
)
)
↔
(
¬
(
X48
→
(
X45
↔
X46
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
(
X44
↔
(
X32
→
X0
→
X48
)
)
↔
(
¬
(
X48
↔
(
(
¬
X32
)
↔
(
(
¬
X23
)
→
X44
)
)
)
)
)
↔
(
X48
↔
(
X23
→
X32
)
)
)
↔
(
¬
(
(
(
(
X34
→
(
(
¬
X27
)
↔
(
X15
→
X32
)
)
)
→
(
¬
(
(
¬
X36
)
→
(
¬
X34
)
→
X15
→
X38
)
)
→
(
X36
↔
(
(
¬
X34
)
↔
(
X32
↔
X38
)
)
)
)
→
X38
→
(
X36
↔
(
X27
↔
(
X23
↔
False
)
)
)
)
→
(
(
¬
(
X32
↔
X36
)
)
↔
(
¬
(
X44
↔
(
(
¬
X34
)
→
X28
→
X36
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X38
→
(
(
¬
X32
)
↔
X43
)
)
)
↔
(
¬
(
(
X47
↔
(
X38
↔
X43
)
)
→
(
(
¬
(
X47
↔
(
X46
↔
X43
)
)
)
↔
(
¬
(
X47
→
X27
→
X32
)
)
)
)
)
)
→
(
(
X43
→
X24
→
(
¬
X8
)
→
X47
)
↔
(
(
¬
(
(
¬
X46
)
↔
(
(
¬
X24
)
→
X47
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X46
)
→
X43
→
X49
)
)
→
(
(
X46
↔
(
(
¬
X43
)
↔
(
(
¬
X24
)
↔
X49
)
)
)
↔
(
X49
↔
(
X47
↔
(
X46
↔
False
)
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
¬
(
X44
↔
(
(
¬
X5
)
↔
(
X4
↔
X22
)
)
)
)
→
(
¬
(
X44
↔
(
(
¬
X17
)
↔
(
X5
→
X22
)
)
)
)
)
↔
(
(
(
¬
X17
)
→
X44
)
↔
(
(
X36
↔
(
X4
↔
False
)
)
↔
(
(
X44
→
X17
→
(
X14
↔
False
)
)
↔
(
¬
(
(
¬
(
X44
→
X36
→
X4
→
X17
)
)
→
(
¬
(
X44
→
X17
→
(
¬
X14
)
→
X36
)
)
)
)
)
)
)
)
→
(
X44
→
X17
→
X10
→
X47
)
→
(
(
¬
X17
)
↔
(
(
¬
X10
)
→
X36
)
)
)
↔
(
¬
(
X36
↔
(
X30
↔
(
(
¬
X22
)
↔
X47
)
)
)
)
)
→
(
(
X25
↔
X44
)
↔
(
(
¬
(
(
(
X36
→
X11
→
X7
)
↔
(
¬
(
(
¬
X17
)
↔
(
(
¬
X12
)
↔
(
(
¬
X11
)
↔
X36
)
)
)
)
)
→
(
¬
(
X17
→
(
¬
X11
)
→
X36
)
)
)
)
→
X36
→
X17
→
X12
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
(
X9
→
(
X2
↔
X41
)
)
)
→
(
X13
↔
(
(
¬
X9
)
↔
X47
)
)
)
)
↔
(
(
¬
(
X41
↔
(
X31
→
X2
)
)
)
↔
(
¬
(
X48
→
X37
→
X2
→
X41
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
¬
(
X47
↔
(
X41
→
(
(
¬
X19
)
↔
X42
)
)
)
)
→
(
(
¬
X6
)
↔
X41
)
)
)
→
(
¬
(
X6
→
X47
)
)
)
)
→
(
¬
(
(
(
¬
(
X47
↔
(
X20
↔
(
X3
↔
X41
)
)
)
)
↔
(
(
¬
X37
)
↔
(
(
¬
X20
)
→
(
(
¬
X3
)
↔
X41
)
)
)
)
→
(
¬
(
X47
→
(
(
¬
X19
)
↔
(
(
¬
X3
)
↔
X41
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
X49
↔
(
X40
→
(
X18
↔
X16
)
)
)
↔
(
(
(
¬
X47
)
→
(
(
¬
X16
)
↔
(
(
¬
X1
)
↔
X49
)
)
)
↔
(
(
¬
X40
)
↔
(
(
¬
X16
)
↔
X47
)
)
)
)
→
(
(
(
X47
→
(
X21
↔
X45
)
)
↔
(
X47
→
(
X35
↔
(
(
¬
X21
)
→
X49
)
)
)
)
↔
(
X29
↔
(
(
(
¬
(
(
¬
X47
)
→
(
(
¬
X29
)
↔
X48
)
)
)
→
(
(
¬
(
X48
↔
(
X47
↔
(
X40
↔
(
X29
→
False
)
)
)
)
)
↔
(
¬
(
X48
↔
(
X40
↔
(
(
¬
X35
)
→
X49
)
)
)
)
)
)
↔
(
X49
↔
(
X47
→
X40
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
X45
↔
(
X42
→
(
¬
X33
)
→
X39
)
)
→
(
¬
(
X26
↔
X42
)
)
)
)
→
(
¬
(
X42
→
X39
→
X33
)
)
)
→
(
¬
(
(
¬
X45
)
↔
(
(
¬
X39
)
→
(
(
¬
X33
)
↔
X47
)
)
)
)
)
)
↔
(
¬
(
X48
→
X47
→
(
¬
X39
)
→
X45
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
e52794...
and proposition id is
168f58...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMQv54sYaGc8qvqEXqrP78ApNPJz7jwKdLv
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
¬
(
(
(
(
¬
(
(
(
X42
→
(
¬
X13
)
→
X7
→
X16
)
→
(
(
(
¬
X42
)
↔
(
X13
↔
(
X2
→
X47
)
)
)
↔
(
¬
(
X16
↔
X2
)
)
)
)
→
(
(
¬
(
(
¬
X46
)
↔
(
X45
↔
(
(
¬
X20
)
→
X47
)
)
)
)
↔
(
(
(
¬
X45
)
→
(
¬
X34
)
→
X47
)
↔
(
¬
(
(
(
(
¬
(
(
¬
X34
)
↔
(
(
¬
X2
)
→
X46
)
)
)
↔
(
(
¬
X2
)
↔
(
(
(
X46
↔
(
X34
→
(
X2
↔
X45
)
)
)
↔
(
¬
(
X46
→
X45
→
X25
→
X34
)
)
)
↔
(
¬
(
(
¬
X34
)
→
(
¬
X25
)
→
X46
)
)
)
)
)
↔
(
(
¬
X45
)
↔
(
X2
↔
X47
)
)
)
→
(
¬
(
(
¬
(
X46
↔
(
X26
↔
X45
)
)
)
→
(
¬
(
X46
↔
(
X26
→
(
¬
X13
)
→
X45
)
)
)
→
(
¬
(
X45
→
X26
→
X47
)
)
)
)
)
)
)
)
)
)
→
(
(
¬
X43
)
↔
(
X42
→
(
X34
↔
X46
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
X9
)
→
(
¬
X0
)
→
X46
)
)
→
(
(
¬
(
X43
↔
(
(
¬
X42
)
↔
X46
)
)
)
↔
(
(
¬
(
X42
→
X9
→
X46
)
)
↔
(
X43
→
(
(
¬
X42
)
↔
(
(
¬
X0
)
→
X46
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
X43
)
↔
(
(
¬
X9
)
↔
X46
)
)
)
)
)
→
(
(
(
¬
(
(
(
X49
→
X36
→
(
X34
↔
(
X3
↔
False
)
)
)
→
(
¬
(
X46
→
X36
→
(
X34
↔
X3
)
)
)
)
→
(
(
¬
(
(
¬
X15
)
→
X34
)
)
↔
(
¬
(
(
X34
↔
(
(
¬
X15
)
→
X32
)
)
→
(
¬
(
(
¬
(
X36
→
(
¬
X34
)
→
(
(
¬
X15
)
↔
X46
)
)
)
→
X36
→
(
X34
↔
(
X32
↔
False
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
X32
)
→
(
X23
↔
(
X22
↔
X34
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
X34
)
→
(
¬
X23
)
→
X47
)
)
→
X34
→
X32
→
(
(
¬
X15
)
↔
X23
)
)
)
↔
(
¬
(
X32
↔
(
X23
→
X15
)
)
)
)
)
)
)
↔
(
¬
(
(
(
X47
→
X41
→
(
(
¬
X30
)
↔
X36
)
)
↔
(
X47
↔
(
X44
↔
(
X30
↔
False
)
)
)
)
→
(
(
(
¬
X47
)
↔
(
X44
→
(
(
¬
X28
)
↔
X49
)
)
)
→
(
¬
(
X47
→
(
X36
↔
(
X28
→
False
)
)
)
)
)
→
(
X49
↔
(
X44
↔
X47
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
(
¬
(
X37
→
(
¬
(
(
X47
→
X8
→
X27
)
→
(
¬
(
(
¬
(
X47
→
(
X37
↔
(
(
¬
X27
)
→
X39
)
)
)
)
→
X39
→
(
(
¬
X27
)
↔
(
X8
↔
X37
)
)
)
)
)
)
)
)
→
(
¬
(
X47
↔
(
X4
→
X39
)
)
)
→
(
(
(
¬
(
X27
↔
(
X24
→
X4
)
)
)
↔
(
¬
(
X47
→
(
X39
↔
(
X4
→
X24
)
)
)
)
)
↔
(
X4
→
X24
)
)
)
→
(
¬
(
(
(
¬
X39
)
↔
(
(
¬
X37
)
↔
X47
)
)
→
X49
→
X46
→
(
(
¬
X37
)
↔
X39
)
)
)
)
→
(
X49
↔
(
X39
↔
(
X14
→
X46
)
)
)
)
→
(
(
(
(
(
(
¬
(
(
¬
(
X17
→
(
X1
↔
False
)
)
)
→
(
¬
(
X37
→
(
X6
↔
(
(
¬
X1
)
↔
X48
)
)
)
)
)
)
→
(
X49
↔
(
X48
→
(
¬
X17
)
→
X37
)
)
)
→
(
X48
↔
(
X39
↔
(
X1
↔
X6
)
)
)
)
↔
(
¬
(
X48
↔
(
(
¬
X37
)
↔
X39
)
)
)
)
↔
(
¬
(
X48
→
(
¬
X1
)
→
X37
)
)
)
↔
(
(
¬
(
(
¬
X48
)
↔
(
X37
→
X10
→
X49
)
)
)
↔
(
X49
→
X37
→
X11
→
X10
)
)
)
)
→
(
(
¬
(
(
(
X47
↔
(
(
¬
X19
)
↔
X35
)
)
→
(
¬
(
(
¬
(
X38
→
(
X35
↔
(
X33
↔
(
X19
↔
False
)
)
)
)
)
→
(
X33
→
(
¬
X19
)
→
X38
)
→
(
¬
(
X47
↔
(
(
¬
X33
)
↔
(
X19
→
X40
)
)
)
)
)
)
)
→
(
(
(
X48
→
X40
→
(
X31
↔
False
)
)
→
(
¬
(
(
¬
(
X48
↔
(
X31
→
(
X29
↔
False
)
)
)
)
→
(
¬
(
X33
→
(
(
¬
X31
)
↔
(
(
¬
X29
)
↔
X40
)
)
)
)
)
)
)
↔
(
X48
↔
(
(
¬
X38
)
↔
(
(
¬
X33
)
↔
X40
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X21
↔
(
(
¬
X12
)
↔
X18
)
)
)
→
(
(
(
¬
(
X18
→
X49
)
)
↔
(
¬
(
X48
↔
(
(
¬
X35
)
↔
(
(
¬
X18
)
→
X40
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X49
→
(
X12
↔
X48
)
)
)
↔
(
¬
(
(
¬
(
X48
↔
(
(
¬
X12
)
→
X49
)
)
)
→
(
(
(
(
X48
→
X12
→
(
¬
X5
)
→
X49
)
↔
(
¬
(
X48
↔
(
X18
→
(
X5
↔
False
)
)
)
)
)
→
(
¬
(
(
¬
X5
)
↔
X49
)
)
)
↔
(
X48
↔
(
X12
↔
X18
)
)
)
)
)
)
↔
(
¬
(
X49
↔
(
X18
↔
X48
)
)
)
)
→
(
¬
(
(
¬
X18
)
↔
(
(
¬
X12
)
→
X49
)
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
X45
↔
(
(
¬
X18
)
↔
X48
)
)
→
(
(
¬
(
X44
→
(
X18
↔
X45
)
)
)
↔
(
¬
(
(
¬
(
X48
→
X45
→
False
)
)
→
(
¬
(
X48
↔
(
X47
→
X44
→
X18
)
)
)
)
)
)
)
→
(
(
X48
↔
(
X45
→
X44
→
X33
)
)
↔
(
¬
(
X48
↔
(
(
¬
X33
)
→
(
X31
↔
X45
)
)
)
)
)
→
(
X45
↔
(
X33
↔
False
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
X48
→
(
¬
X15
)
→
X32
)
)
→
(
¬
X32
)
→
(
(
¬
X21
)
↔
X44
)
)
↔
(
X48
→
X21
→
X44
)
)
→
(
(
(
¬
(
X48
↔
(
X44
→
X33
→
(
X15
↔
False
)
)
)
)
→
(
X44
↔
(
(
¬
X38
)
↔
(
X33
→
X48
)
)
)
)
↔
(
¬
(
(
¬
X44
)
→
(
¬
X38
)
→
(
(
¬
X25
)
↔
X48
)
)
)
)
)
→
(
¬
(
(
¬
X33
)
→
X15
→
X1
→
X44
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
¬
X33
)
→
(
(
¬
X7
)
↔
X45
)
)
)
→
(
¬
(
X43
↔
(
X17
↔
(
X7
↔
False
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
X45
↔
(
X43
→
(
X23
↔
(
X6
↔
False
)
)
)
)
)
→
(
¬
(
(
¬
X43
)
↔
(
X23
↔
(
(
¬
X8
)
→
X45
)
)
)
)
)
)
→
X33
→
(
(
¬
X23
)
↔
(
(
¬
X6
)
↔
X43
)
)
)
→
(
¬
(
(
(
X45
↔
(
X33
↔
(
(
¬
X8
)
→
X27
)
)
)
→
(
¬
(
X45
↔
(
(
¬
X27
)
↔
(
(
¬
X19
)
↔
X33
)
)
)
)
)
→
(
X33
↔
X45
)
)
)
)
)
)
→
(
¬
(
X39
→
(
X33
↔
(
X27
↔
X43
)
)
)
)
)
→
(
(
¬
(
(
X39
↔
(
(
¬
X33
)
→
(
X12
↔
X43
)
)
)
→
(
¬
(
(
¬
(
(
¬
X33
)
↔
(
X24
→
(
¬
X9
)
→
X39
)
)
)
→
(
(
(
¬
X33
)
→
(
¬
X24
)
→
(
(
¬
X9
)
↔
X43
)
)
↔
(
¬
(
X43
↔
(
(
¬
X9
)
↔
(
X4
↔
X24
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
X39
→
X24
→
(
X22
↔
(
X16
↔
False
)
)
)
→
(
¬
(
(
¬
(
X47
→
(
¬
X24
)
→
X39
)
)
→
(
(
X39
→
X47
)
↔
(
¬
(
X43
↔
(
(
¬
X22
)
→
X24
)
)
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
(
¬
(
X43
→
(
(
¬
X28
)
↔
(
X0
→
X42
)
)
)
)
↔
(
¬
X14
)
)
↔
(
¬
(
(
¬
(
X37
↔
(
(
¬
X28
)
↔
X35
)
)
)
→
(
(
¬
(
X37
→
(
X35
↔
X43
)
)
)
↔
(
X37
→
X28
→
(
¬
X13
)
→
X42
)
)
)
)
)
→
(
(
¬
(
(
¬
X42
)
→
(
(
¬
X29
)
↔
X43
)
)
)
↔
(
¬
(
(
¬
X42
)
→
(
X37
↔
(
(
¬
X20
)
→
X43
)
)
)
)
)
→
(
X29
↔
X42
)
)
→
(
(
(
¬
(
X10
↔
X28
)
)
↔
(
(
¬
(
(
X42
→
X37
→
(
X11
↔
X44
)
)
→
X44
→
(
X42
↔
(
X37
→
(
X5
↔
False
)
)
)
)
)
↔
(
X37
↔
(
X11
→
X42
)
)
)
)
↔
(
¬
(
X44
→
X28
→
(
X11
↔
False
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X48
→
X30
→
(
¬
X3
)
→
X43
)
)
→
(
(
¬
(
X42
→
(
¬
X30
)
→
(
X3
↔
X43
)
)
)
↔
(
X43
→
(
X3
↔
(
(
¬
X2
)
→
X42
)
)
)
)
)
)
→
X46
→
X42
→
(
X26
↔
(
X2
→
False
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
X42
)
→
X44
)
→
(
¬
(
X49
→
(
X46
↔
X48
)
)
)
)
)
→
(
¬
(
(
X48
↔
(
X44
↔
(
X40
→
X42
)
)
)
→
(
¬
(
X42
→
(
¬
X41
)
→
X44
)
)
)
)
)
)
→
(
(
¬
(
X49
→
(
(
¬
X42
)
↔
(
(
¬
X36
)
↔
X46
)
)
)
)
↔
(
¬
(
(
¬
X46
)
↔
(
X42
→
(
(
¬
X34
)
↔
X49
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
8004b7...
and proposition id is
f9eb2b...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMLuWcoBg4gRswejPLw7bPtRGYy1dNn9kJY
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
¬
(
(
(
(
(
¬
(
X38
→
X5
)
)
↔
(
(
¬
(
X46
↔
(
X15
↔
X43
)
)
)
→
(
(
X38
↔
(
(
¬
X15
)
↔
(
X10
→
X43
)
)
)
↔
(
(
X46
↔
(
X38
↔
(
(
¬
X5
)
↔
X43
)
)
)
↔
(
¬
(
X43
↔
(
(
¬
X38
)
→
(
(
¬
X10
)
↔
X46
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
X47
→
X46
→
(
X44
↔
(
X35
↔
False
)
)
)
→
X46
→
X43
→
X35
→
False
)
)
↔
(
¬
(
X47
↔
(
X13
→
X10
→
X44
)
)
)
)
)
↔
(
¬
(
(
¬
(
X49
↔
(
X47
→
(
(
¬
X46
)
↔
X48
)
)
)
)
→
(
(
¬
(
(
¬
(
(
(
X48
→
(
X47
↔
(
X44
→
False
)
)
)
↔
(
(
¬
X47
)
→
X45
→
X7
→
X48
)
)
→
(
¬
(
X47
→
(
¬
X44
)
→
X45
)
)
)
)
→
(
¬
(
X47
↔
(
X45
→
X48
)
)
)
)
)
↔
(
(
(
¬
X45
)
↔
(
(
¬
X23
)
→
X48
)
)
→
(
¬
(
X49
→
(
X48
↔
(
X23
↔
X47
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
X44
→
X24
→
(
(
¬
X20
)
↔
X43
)
)
→
(
(
¬
(
X44
→
(
X24
↔
(
X1
→
X43
)
)
)
)
→
(
X43
↔
(
X24
→
(
X1
↔
X8
)
)
)
)
→
(
X43
↔
(
X24
→
X44
)
)
)
→
(
¬
(
(
¬
X24
)
↔
(
(
¬
X11
)
→
X8
→
X43
)
)
)
)
→
(
(
¬
(
X43
↔
(
X41
→
X33
→
X22
)
)
)
↔
(
(
(
¬
(
X41
↔
(
X36
→
(
(
¬
X33
)
↔
X47
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X33
)
→
(
(
¬
X22
)
↔
X41
)
)
)
→
(
X47
↔
(
X24
→
X22
)
)
)
)
)
↔
(
¬
(
(
¬
(
X47
↔
(
X24
↔
(
X11
→
X41
)
)
)
)
→
X41
→
(
X24
↔
(
(
¬
X2
)
→
X11
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
X49
↔
(
X39
↔
(
X31
→
False
)
)
)
↔
(
X42
→
(
(
¬
X18
)
↔
(
(
¬
X4
)
→
X31
)
)
)
)
↔
(
¬
(
(
¬
X42
)
→
(
(
¬
X39
)
↔
(
(
¬
X32
)
↔
X49
)
)
)
)
)
↔
(
(
(
(
(
¬
X9
)
↔
(
(
¬
X3
)
→
X30
)
)
↔
(
¬
(
X42
↔
(
X32
→
(
(
¬
X9
)
↔
X30
)
)
)
)
)
↔
(
X49
↔
(
X32
↔
(
(
¬
X3
)
→
X9
)
)
)
)
→
(
X9
↔
(
(
¬
X3
)
→
X30
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
¬
(
(
¬
(
X30
↔
(
(
¬
X26
)
↔
(
(
¬
X14
)
↔
X46
)
)
)
)
→
(
X46
↔
(
X30
→
(
X12
↔
X26
)
)
)
)
)
↔
(
X49
↔
(
X30
→
(
(
¬
X12
)
↔
X40
)
)
)
)
→
(
¬
(
X26
↔
(
X12
→
X30
)
)
)
)
)
→
(
¬
(
X32
↔
(
X30
↔
(
(
¬
X26
)
→
X46
)
)
)
)
)
)
→
(
(
X32
→
(
X30
↔
(
X6
→
False
)
)
)
↔
(
(
X49
→
X32
→
X30
→
X6
→
False
)
↔
(
(
¬
(
X49
↔
(
X32
↔
(
(
¬
X27
)
→
X28
)
)
)
)
↔
(
¬
(
X49
↔
(
X32
↔
(
X30
→
X28
→
False
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X47
↔
(
X16
↔
(
(
¬
X0
)
↔
X21
)
)
)
)
↔
(
(
(
¬
(
X37
↔
(
X17
↔
(
(
¬
X0
)
→
X42
)
)
)
)
→
(
¬
(
X42
→
X39
→
X17
→
X37
)
)
)
→
(
(
¬
X29
)
↔
(
(
¬
X0
)
↔
X39
)
)
)
)
↔
(
(
(
¬
(
X47
→
X25
→
X19
)
)
↔
(
X47
→
(
X34
↔
(
X29
↔
X19
)
)
)
)
→
(
X47
↔
(
X39
→
X19
→
X46
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
X47
→
X41
→
(
X35
↔
X42
)
)
→
(
¬
(
(
¬
(
X49
↔
(
X6
→
(
(
¬
X4
)
↔
X5
)
)
)
)
→
(
¬
(
X49
↔
(
X47
→
(
X35
↔
(
X4
→
False
)
)
)
)
)
)
)
)
→
(
X47
↔
(
X42
→
(
X41
↔
X7
)
)
)
→
(
(
¬
X41
)
→
(
X40
↔
(
(
¬
X14
)
→
X42
)
)
)
→
(
¬
(
(
(
¬
(
X47
↔
(
X42
↔
(
X33
→
X41
)
)
)
)
→
X47
→
X42
→
X41
→
(
X40
↔
False
)
)
→
(
¬
(
X41
↔
X40
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
X46
→
(
(
¬
X31
)
↔
(
(
¬
X10
)
→
X49
)
)
)
)
→
(
¬
(
(
¬
X31
)
→
(
(
¬
X21
)
↔
(
(
¬
X10
)
→
X49
)
)
)
)
)
)
→
(
(
X42
→
(
¬
X21
)
→
(
¬
X9
)
→
X46
)
↔
(
X10
→
(
X9
↔
X45
)
)
)
)
→
(
(
X42
→
(
X19
↔
X28
)
)
↔
(
¬
(
X49
↔
(
X41
→
(
X19
↔
X45
)
)
)
)
)
)
)
)
→
(
(
(
(
X48
→
X47
→
(
X39
↔
False
)
)
→
(
(
X48
→
(
(
¬
X2
)
↔
X39
)
)
↔
(
¬
(
(
¬
X47
)
↔
(
X46
→
(
¬
X39
)
→
X48
)
)
)
)
→
(
(
(
(
¬
X47
)
↔
(
X46
→
(
¬
X37
)
→
X48
)
)
↔
(
¬
(
X48
→
X47
→
X34
→
False
)
)
)
↔
(
X48
↔
(
X47
↔
(
X46
→
False
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X48
→
(
¬
X46
)
→
(
(
¬
X43
)
↔
X47
)
)
)
→
(
¬
(
(
¬
(
X48
↔
(
X47
→
(
X44
↔
X3
)
)
)
)
→
(
¬
(
X47
↔
(
X46
→
(
(
¬
X3
)
↔
X48
)
)
)
)
)
)
)
)
↔
(
(
¬
(
X48
→
(
¬
X26
)
→
(
(
¬
X22
)
↔
X47
)
)
)
↔
(
(
X43
→
X13
→
X22
)
→
(
¬
(
(
¬
X26
)
↔
(
(
¬
X22
)
↔
X47
)
)
)
)
)
)
)
→
(
X44
↔
(
X16
→
False
)
)
→
(
¬
(
(
¬
(
X48
↔
(
X46
↔
X44
)
)
)
→
(
(
¬
(
X48
→
X46
→
X16
→
X25
)
)
↔
(
¬
(
(
¬
X24
)
↔
X46
)
)
)
→
X48
→
X25
→
X24
→
False
)
)
)
→
(
(
(
(
X47
↔
(
(
¬
X8
)
→
X0
→
X44
)
)
↔
(
¬
(
X44
→
(
X39
↔
(
X8
↔
X30
)
)
)
)
)
→
(
¬
(
(
X49
→
(
(
¬
X39
)
↔
(
X18
→
X47
)
)
)
→
(
(
(
X49
↔
(
X39
↔
(
X30
↔
X47
)
)
)
↔
(
X39
↔
(
X32
→
False
)
)
)
↔
(
X49
↔
(
(
¬
X39
)
↔
X47
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
X30
)
↔
(
(
¬
X15
)
→
X44
)
)
)
↔
(
¬
(
X39
→
X36
→
(
¬
X30
)
→
X44
)
)
)
)
→
(
(
(
¬
(
X44
↔
(
(
¬
X38
)
→
X39
)
)
)
→
(
(
(
(
(
¬
(
X44
→
(
(
¬
X29
)
↔
X38
)
)
)
↔
(
(
X44
→
(
(
¬
X1
)
↔
X29
)
)
↔
(
X39
↔
(
X38
↔
(
X1
↔
X29
)
)
)
)
)
↔
(
(
¬
X38
)
↔
(
(
¬
X29
)
↔
X39
)
)
)
↔
(
¬
(
X29
↔
(
(
¬
X12
)
↔
(
(
¬
X1
)
↔
X39
)
)
)
)
)
↔
(
X29
→
False
)
)
)
↔
(
(
(
X39
↔
(
(
¬
X38
)
→
(
¬
X27
)
→
X44
)
)
↔
(
¬
(
(
X44
↔
(
X39
↔
(
(
¬
X23
)
→
X38
)
)
)
→
(
¬
X23
)
→
(
(
¬
X17
)
↔
X44
)
)
)
)
↔
(
(
X38
↔
(
X20
→
(
(
¬
X11
)
↔
X47
)
)
)
↔
(
(
(
(
¬
X38
)
→
(
X27
↔
(
X20
↔
X47
)
)
)
→
(
¬
(
X38
→
X27
→
X23
→
False
)
)
)
↔
(
¬
(
X47
→
(
(
¬
X27
)
↔
X38
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
bfcd05...
and proposition id is
c57da0...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMNtey15xztbrqh5yKtnjCoVEw46PR7z1UL
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
¬
(
(
(
(
(
(
¬
(
X47
↔
(
X40
↔
(
(
¬
X24
)
→
X32
)
)
)
)
↔
(
X47
↔
(
X24
→
(
X15
↔
X40
)
)
)
)
↔
(
(
¬
X32
)
→
X15
→
X40
)
)
↔
(
X40
↔
(
X32
↔
X47
)
)
)
↔
(
(
¬
X24
)
→
X19
→
X15
→
X44
)
)
→
(
(
(
¬
(
X40
↔
(
X35
↔
(
X34
↔
(
X9
→
False
)
)
)
)
)
→
(
¬
(
X49
→
(
X40
↔
X34
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
¬
X34
)
↔
(
X32
↔
(
(
¬
X27
)
→
X49
)
)
)
)
→
(
X47
↔
(
(
¬
X40
)
↔
(
(
¬
X27
)
→
X42
)
)
)
)
)
→
(
¬
(
(
¬
(
X49
→
X42
→
False
)
)
→
(
(
¬
X40
)
→
X42
)
→
(
¬
(
(
¬
X46
)
↔
(
X40
→
X39
→
X49
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X28
↔
(
(
¬
X21
)
↔
(
(
¬
X8
)
↔
X49
)
)
)
)
→
(
(
X49
↔
(
X46
→
(
¬
X21
)
→
X39
)
)
↔
(
¬
(
X28
→
X21
→
X49
)
)
)
→
(
¬
(
X28
→
(
X21
↔
X46
)
)
)
)
↔
(
¬
(
(
¬
(
(
X46
↔
(
(
¬
X39
)
↔
(
X31
→
X49
)
)
)
→
(
¬
(
X46
→
(
X39
↔
(
X31
↔
(
X18
↔
False
)
)
)
)
)
)
)
→
(
X39
↔
(
(
¬
X31
)
↔
X49
)
)
)
)
)
→
(
¬
(
(
(
(
¬
X12
)
↔
(
(
¬
X7
)
↔
X49
)
)
↔
(
¬
(
(
(
X31
↔
(
X25
→
X49
)
)
↔
(
(
X31
→
X14
→
(
X7
↔
X25
)
)
↔
(
¬
(
(
¬
X31
)
↔
X49
)
)
)
)
→
X25
→
X49
)
)
)
→
(
(
¬
X39
)
↔
(
¬
(
(
¬
(
(
¬
X12
)
↔
(
X6
↔
X39
)
)
)
→
(
¬
(
(
X25
→
(
¬
X12
)
→
X39
)
→
(
¬
(
(
(
¬
X12
)
↔
(
(
¬
X6
)
↔
X25
)
)
→
(
¬
(
X49
↔
(
(
¬
X12
)
↔
(
(
¬
X6
)
→
X39
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
X41
↔
(
X37
→
X36
→
False
)
)
)
→
(
X47
↔
(
X41
→
X38
)
)
→
(
¬
(
(
X49
→
X47
→
X38
→
False
)
→
(
¬
(
X49
↔
(
X47
↔
(
X36
→
X41
)
)
)
)
)
)
)
→
(
(
(
(
(
(
¬
(
(
¬
X43
)
→
X23
→
(
X0
↔
X45
)
)
)
↔
(
(
¬
X43
)
↔
(
X23
→
(
(
¬
X0
)
↔
X49
)
)
)
)
↔
(
¬
(
X49
↔
(
X45
→
X23
)
)
)
)
↔
(
X45
↔
(
(
¬
X23
)
↔
X43
)
)
)
↔
(
(
(
¬
X43
)
↔
(
X36
→
(
X5
↔
X45
)
)
)
↔
(
¬
(
X43
↔
(
X5
↔
(
X0
↔
False
)
)
)
)
)
)
↔
(
(
(
X49
→
(
X41
↔
(
X4
→
X30
)
)
)
→
(
(
(
X41
↔
(
(
¬
X30
)
↔
(
(
¬
X17
)
→
X49
)
)
)
→
(
¬
(
(
¬
X17
)
↔
X43
)
)
)
↔
(
¬
(
(
¬
X30
)
↔
(
X17
→
X41
)
)
)
)
)
↔
(
X49
↔
(
X43
→
(
¬
X30
)
→
X45
)
)
)
)
)
)
→
(
(
¬
(
(
(
X20
↔
(
(
¬
X16
)
↔
(
(
¬
X10
)
↔
X47
)
)
)
→
(
(
(
X29
→
X16
→
(
(
¬
X10
)
↔
X47
)
)
→
X47
→
X33
→
X10
→
X16
)
↔
(
¬
(
X33
↔
(
X16
→
(
X10
↔
False
)
)
)
)
)
)
→
(
(
X47
→
(
(
¬
X10
)
↔
(
(
¬
X3
)
→
X11
)
)
)
↔
(
¬
(
X47
→
(
X11
↔
(
X2
↔
X10
)
)
)
)
)
→
(
(
¬
(
(
¬
X47
)
→
(
¬
X16
)
→
(
¬
X11
)
→
X48
)
)
↔
(
X16
→
(
X11
↔
(
(
¬
X10
)
→
X47
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
X49
↔
(
X47
↔
(
X46
↔
(
X45
→
False
)
)
)
)
↔
(
(
¬
(
X49
↔
X48
)
)
→
(
(
¬
(
X49
→
(
X45
↔
(
(
¬
X13
)
→
X46
)
)
)
)
↔
(
X49
→
X45
→
(
(
¬
X13
)
↔
X48
)
)
)
)
)
→
(
¬
(
(
¬
X48
)
→
(
X26
↔
X49
)
)
)
→
(
¬
(
X49
→
(
¬
X33
)
→
X47
)
)
)
)
↔
(
¬
(
X48
↔
(
X26
→
X1
→
X46
)
)
)
)
→
(
¬
(
X49
→
(
X48
↔
(
X26
→
(
X22
↔
False
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
(
(
¬
X48
)
↔
(
(
¬
X45
)
→
X31
→
X49
)
)
→
(
X14
↔
(
X9
↔
X48
)
)
)
→
(
¬
X14
)
→
(
X9
↔
X48
)
)
↔
(
X49
→
X45
→
X10
→
X48
)
)
↔
(
(
¬
(
(
X31
↔
(
(
¬
X13
)
↔
(
(
¬
X0
)
→
X49
)
)
)
→
X49
→
X2
→
(
(
¬
X0
)
↔
X45
)
)
)
→
(
¬
(
X49
→
(
X45
↔
(
X31
→
X47
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
¬
X48
)
→
(
X43
↔
(
X38
→
X49
)
)
)
)
↔
(
(
(
(
¬
(
X43
↔
(
(
¬
X38
)
→
(
(
¬
X28
)
↔
X46
)
)
)
)
→
(
¬
X46
)
→
(
(
¬
X43
)
↔
(
X28
↔
X48
)
)
)
↔
(
X48
→
(
X3
↔
X38
)
)
)
↔
(
(
¬
(
X42
↔
(
(
¬
X38
)
→
X43
)
)
)
↔
(
(
¬
X42
)
↔
(
X19
↔
X48
)
)
)
)
)
↔
(
(
(
¬
X46
)
→
(
¬
X43
)
→
(
¬
X32
)
→
X47
)
↔
(
(
¬
(
(
X40
→
X29
→
X47
)
→
(
(
X40
→
(
¬
X32
)
→
X29
→
X47
)
↔
(
¬
(
X46
↔
(
(
¬
X32
)
→
X40
)
)
)
)
)
)
↔
(
¬
(
(
X47
→
X28
→
False
)
→
(
¬
(
X46
↔
(
(
¬
X40
)
→
(
(
¬
X23
)
↔
X47
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
X47
)
↔
(
(
¬
X39
)
↔
X48
)
)
↔
(
X48
↔
(
(
¬
X12
)
↔
X49
)
)
)
→
X47
→
(
(
¬
X39
)
↔
(
X37
↔
X48
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
X44
↔
(
(
¬
X34
)
→
(
¬
X11
)
→
X45
)
)
)
→
(
(
¬
X34
)
↔
(
(
¬
X18
)
→
X44
)
)
→
(
(
X47
↔
(
(
¬
X18
)
→
X45
)
)
↔
(
X44
→
(
X34
↔
False
)
)
)
)
→
(
X47
↔
(
X44
→
X15
→
X45
)
)
)
↔
(
(
(
¬
(
X45
↔
(
X8
→
X47
)
)
)
↔
(
(
(
¬
(
X41
→
(
X30
↔
(
(
¬
X8
)
↔
X47
)
)
)
)
↔
(
¬
(
X30
↔
(
(
¬
X8
)
↔
X45
)
)
)
)
↔
(
(
¬
X8
)
↔
X45
)
)
)
→
X45
→
X44
→
X36
→
X47
)
)
→
(
¬
(
(
(
(
X20
→
(
¬
X7
)
→
(
(
¬
X1
)
↔
X24
)
)
→
(
(
(
¬
(
X46
↔
False
)
)
↔
(
(
X46
↔
(
X36
→
X35
→
X7
)
)
→
(
¬
(
X45
↔
(
(
¬
X36
)
→
(
(
¬
X35
)
↔
X46
)
)
)
)
)
)
↔
(
X45
↔
(
(
¬
X36
)
↔
(
X25
↔
X46
)
)
)
)
)
↔
(
X46
→
(
X24
↔
X25
)
)
)
→
(
(
(
X21
↔
X27
)
→
(
¬
(
(
¬
X36
)
↔
(
X35
→
(
(
¬
X21
)
↔
X46
)
)
)
)
)
↔
(
¬
(
X21
→
X16
→
X46
)
)
)
)
)
→
(
¬
(
(
(
(
(
X33
→
(
(
¬
X17
)
↔
(
X6
→
X35
)
)
)
↔
(
(
X35
↔
(
(
¬
X33
)
→
(
(
¬
X27
)
↔
X44
)
)
)
↔
(
¬
(
X35
↔
(
X27
↔
(
X6
↔
X44
)
)
)
)
)
)
↔
(
X35
↔
(
X33
↔
(
(
¬
X22
)
↔
X44
)
)
)
)
↔
(
¬
(
(
¬
X35
)
↔
(
X33
↔
(
(
¬
X4
)
↔
X44
)
)
)
)
)
→
(
(
(
¬
X44
)
→
X33
→
(
¬
X5
)
→
X46
)
↔
(
(
(
¬
X33
)
↔
(
X27
→
X44
)
)
↔
(
(
X46
↔
(
(
¬
X27
)
↔
(
X26
→
X44
)
)
)
↔
(
¬
(
X33
↔
(
X26
→
False
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
c28ee4...
and proposition id is
b218fa...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMGYj9A322G6pVJXgZjdAZjqruEyR7YvtdS
)
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
∀X50 :
prop
,
∀X51 :
prop
,
(
(
¬
(
(
(
(
(
¬
(
(
(
¬
X27
)
↔
X51
)
→
(
(
¬
(
X48
↔
(
(
¬
X2
)
↔
X44
)
)
)
↔
(
X51
→
(
X48
↔
(
X40
→
(
X2
↔
False
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X44
↔
(
X40
↔
(
(
¬
X25
)
↔
X48
)
)
)
)
↔
(
¬
(
X40
→
(
¬
X25
)
→
X4
→
X48
)
)
)
→
(
¬
(
X48
↔
(
X25
→
(
¬
X4
)
→
X49
)
)
)
)
)
→
(
(
(
(
(
¬
X48
)
↔
(
X36
↔
(
(
¬
X30
)
→
X51
)
)
)
↔
(
(
X51
↔
(
X40
→
X30
→
X48
)
)
↔
(
¬
(
(
¬
X40
)
→
(
¬
X30
)
→
X5
→
X51
)
)
)
)
→
X48
→
X40
→
(
¬
X11
)
→
X51
)
↔
(
(
¬
(
X51
↔
(
X15
→
X9
→
X48
)
)
)
↔
(
X51
↔
(
X15
→
(
(
¬
X9
)
↔
X40
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X51
↔
(
(
¬
X32
)
→
(
¬
X12
)
→
X47
)
)
)
→
(
(
X14
↔
(
(
¬
X12
)
→
X46
)
)
↔
(
X45
↔
(
X14
→
X12
→
X49
)
)
)
)
)
↔
(
(
(
¬
(
X48
→
(
¬
X47
)
→
X51
)
)
↔
(
X7
↔
(
X6
→
X48
)
)
)
↔
(
(
¬
(
X48
↔
(
X42
↔
(
(
¬
X8
)
↔
X47
)
)
)
)
→
(
(
X51
→
(
¬
X43
)
→
(
(
¬
X24
)
↔
X48
)
)
↔
(
¬
(
(
X51
↔
(
X43
↔
(
X42
↔
False
)
)
)
→
(
¬
(
X49
↔
(
X48
→
X42
)
)
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
(
X49
→
(
¬
X41
)
→
(
¬
X39
)
→
X48
)
)
→
(
¬
(
(
¬
(
X41
↔
(
X35
→
X49
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X35
↔
(
X1
↔
X41
)
)
)
→
(
¬
(
X49
→
(
¬
X41
)
→
X1
→
X51
)
)
)
)
→
(
X41
↔
(
(
¬
X1
)
→
X35
)
)
)
)
)
)
→
X51
→
(
X41
↔
(
(
¬
X39
)
→
X49
)
)
)
)
↔
(
(
(
X51
→
(
X38
↔
(
X17
↔
X48
)
)
)
↔
(
(
¬
X49
)
→
X38
→
(
X21
↔
X51
)
)
)
→
(
¬
(
X51
→
(
¬
X38
)
→
X48
)
)
)
)
↔
(
(
(
X33
→
(
¬
X31
)
→
X0
→
X49
)
↔
(
X49
→
X33
→
(
X28
↔
X0
)
)
)
↔
(
(
¬
(
(
X48
→
X31
→
X51
)
→
(
¬
(
(
¬
(
X48
↔
(
(
¬
X26
)
→
(
X23
↔
X31
)
)
)
)
→
(
X31
↔
(
X23
↔
X3
)
)
)
)
→
(
¬
(
X51
→
(
X26
↔
(
(
¬
X3
)
↔
X31
)
)
)
)
)
)
↔
(
(
(
¬
X31
)
↔
(
(
¬
X28
)
↔
(
(
¬
X26
)
→
X49
)
)
)
↔
(
¬
(
X31
↔
(
X28
→
(
X26
↔
X23
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
X20
→
X19
→
X16
→
(
X10
↔
False
)
)
↔
(
¬
(
X51
↔
(
X38
↔
(
X20
→
False
)
)
)
)
)
↔
(
X38
↔
(
X10
→
False
)
)
)
→
(
(
¬
(
X22
↔
(
X20
↔
(
X19
↔
X39
)
)
)
)
↔
(
¬
(
(
X16
→
X34
)
→
(
¬
(
X22
→
X20
→
X39
)
)
)
)
)
)
)
→
(
(
X22
→
(
(
¬
X18
)
↔
(
(
¬
X13
)
→
X19
)
)
)
↔
(
¬
(
X37
→
(
X19
↔
(
X18
→
X13
)
)
)
)
)
→
(
¬
(
(
X29
→
X19
→
X13
)
→
(
¬
(
X37
↔
(
(
¬
X34
)
↔
(
(
¬
X29
)
→
X50
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
(
(
(
¬
(
X36
↔
(
(
¬
X3
)
↔
X47
)
)
)
↔
(
¬
(
(
(
¬
X36
)
↔
(
(
¬
X3
)
↔
X49
)
)
→
(
¬
(
X13
↔
(
(
¬
X3
)
↔
X36
)
)
)
→
X47
→
X36
→
(
X23
↔
(
X3
→
False
)
)
)
)
)
→
(
(
¬
(
X49
→
(
(
¬
X18
)
↔
(
X16
↔
X47
)
)
)
)
↔
(
X36
→
X16
)
)
)
↔
(
(
¬
(
X36
↔
(
X18
↔
(
X2
↔
X47
)
)
)
)
↔
(
¬
(
(
¬
X47
)
→
(
X18
↔
(
X12
↔
X48
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X49
→
(
(
¬
X37
)
↔
X50
)
)
)
→
(
X50
→
X37
→
X23
→
False
)
→
(
(
X23
↔
(
X16
→
X10
)
)
↔
(
¬
(
X50
↔
(
(
¬
X10
)
↔
X37
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
X47
)
→
(
X37
↔
(
(
¬
X35
)
→
X48
)
)
)
↔
(
¬
(
X48
↔
(
X47
→
(
(
¬
X32
)
↔
X35
)
)
)
)
)
↔
(
X47
↔
(
(
¬
X37
)
→
X48
)
)
)
→
X48
→
(
(
¬
X37
)
↔
(
(
¬
X30
)
→
X47
)
)
)
)
↔
(
(
(
(
¬
(
X49
↔
(
(
¬
X44
)
→
X45
)
)
)
↔
(
(
X46
→
(
(
¬
X29
)
↔
(
X27
→
X49
)
)
)
→
(
¬
(
(
¬
(
X49
→
X46
→
X45
)
)
→
(
¬
(
(
¬
(
X46
↔
(
X45
↔
(
X44
→
(
X29
↔
False
)
)
)
)
)
→
(
X49
↔
X29
)
)
)
)
)
)
)
→
(
X44
→
(
(
¬
X27
)
↔
(
X1
→
X29
)
)
)
→
(
¬
(
(
¬
(
X0
↔
X25
)
)
→
(
(
¬
(
X29
↔
X25
)
)
↔
(
(
¬
(
X27
→
X0
→
X29
)
)
↔
(
¬
(
(
¬
(
(
X29
↔
(
(
¬
X0
)
→
X44
)
)
→
(
¬
(
X27
→
(
¬
X0
)
→
X39
)
)
)
)
→
X29
→
(
¬
X25
)
→
X27
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
X47
)
↔
(
(
¬
X8
)
↔
(
X5
→
X48
)
)
)
→
(
¬
(
X46
→
X47
)
)
)
)
↔
(
X47
↔
(
X46
→
(
(
¬
X43
)
↔
X48
)
)
)
)
→
(
¬
(
(
¬
X29
)
↔
(
(
¬
X6
)
→
X39
)
)
)
→
(
¬
(
(
X48
→
X47
→
(
X6
↔
False
)
)
→
(
¬
(
X43
↔
(
X39
↔
X6
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
X42
↔
(
X38
→
X4
→
False
)
)
→
(
(
¬
(
X50
↔
(
X49
↔
(
X38
↔
X51
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
X49
→
(
X15
↔
(
X4
→
False
)
)
)
)
↔
(
(
¬
X49
)
↔
(
X24
↔
(
(
¬
X15
)
↔
X51
)
)
)
)
→
(
(
¬
X49
)
↔
(
(
¬
X24
)
↔
X51
)
)
)
)
→
X49
→
X38
→
X24
→
X51
)
)
)
)
→
(
(
(
(
¬
(
(
¬
X42
)
→
(
¬
X38
)
→
(
¬
X31
)
→
X51
)
)
→
(
(
X51
→
(
X42
↔
(
X31
→
X11
→
False
)
)
)
↔
(
(
¬
X42
)
→
X31
→
X28
→
X51
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X34
↔
(
(
¬
X21
)
↔
(
X20
→
X42
)
)
)
)
→
(
(
X51
↔
(
X38
↔
(
X34
↔
False
)
)
)
↔
(
X42
↔
(
X38
↔
(
(
¬
X20
)
→
X51
)
)
)
)
)
)
→
(
(
(
¬
(
X38
↔
(
X22
↔
X20
)
)
)
→
(
¬
(
(
¬
(
X34
→
X22
→
(
X20
↔
False
)
)
)
→
(
X40
↔
(
X38
↔
(
X22
↔
(
X20
→
False
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
X38
)
↔
(
X22
→
X40
)
)
)
)
)
)
)
↔
(
X41
→
(
¬
X40
)
→
(
¬
X38
)
→
X51
)
)
)
)
↔
(
(
(
(
¬
(
X7
→
False
)
)
→
(
(
¬
X50
)
↔
(
X49
↔
(
X47
↔
X51
)
)
)
)
→
(
¬
(
(
(
¬
X33
)
↔
(
X26
↔
(
X14
↔
X50
)
)
)
→
(
X49
↔
(
(
¬
X33
)
↔
(
X14
↔
X51
)
)
)
)
)
)
↔
(
(
¬
(
X47
↔
(
X17
→
(
(
¬
X9
)
↔
X19
)
)
)
)
↔
(
¬
(
(
(
¬
(
X51
↔
(
X47
→
(
X9
↔
False
)
)
)
)
↔
(
(
¬
X17
)
→
X50
)
)
→
X50
→
(
¬
X9
)
→
X17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
d895e5...
and proposition id is
be8a38...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMHfpQ81Mpsj82p8ywkKBQqtwKWk3hpdHQA
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
(
(
¬
(
X18
↔
(
(
¬
X7
)
→
(
¬
X5
)
→
X46
)
)
)
→
(
X5
↔
(
X4
→
X30
)
)
)
↔
(
¬
(
(
¬
X7
)
→
(
¬
X5
)
→
X4
→
X16
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
¬
X29
)
→
X4
→
(
X0
↔
X36
)
)
)
→
(
¬
(
X40
→
(
¬
X29
)
→
(
¬
X0
)
→
X39
)
)
)
)
→
(
(
X36
→
X18
→
X46
)
↔
(
X24
→
(
¬
X18
)
→
(
¬
X0
)
→
X29
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
(
¬
(
X30
↔
(
X13
↔
X28
)
)
)
↔
(
¬
(
X47
↔
(
X30
↔
(
(
¬
X13
)
→
X29
)
)
)
)
)
↔
(
¬
(
X28
→
(
X13
↔
X30
)
)
)
)
→
(
(
X47
↔
(
(
¬
X30
)
→
(
¬
X10
)
→
X46
)
)
↔
(
X30
↔
(
X28
→
(
X10
↔
X46
)
)
)
)
)
→
(
¬
(
(
¬
X30
)
↔
(
X10
→
X46
)
)
)
)
)
↔
(
(
X44
→
(
X28
↔
(
X15
↔
False
)
)
)
→
(
¬
(
(
X45
↔
(
(
¬
X26
)
→
(
X15
↔
X47
)
)
)
→
(
¬
(
X47
↔
(
X44
→
X26
→
X45
)
)
)
→
(
¬
(
X46
→
X44
→
X47
)
)
)
)
)
)
→
(
(
¬
(
X44
↔
(
X41
↔
(
(
¬
X21
)
→
X45
)
)
)
)
↔
(
X41
→
(
(
¬
X21
)
↔
X46
)
)
)
→
(
(
¬
(
(
X45
↔
(
(
¬
X42
)
↔
(
X41
→
X44
)
)
)
→
(
(
¬
X25
)
↔
X41
)
)
)
↔
(
(
¬
X41
)
→
(
X25
↔
X45
)
)
)
)
)
↔
(
¬
(
(
(
(
(
X31
→
X22
→
X12
→
X33
)
↔
(
X40
↔
(
X35
→
X3
)
)
)
↔
(
(
(
X35
↔
(
X33
↔
(
(
¬
X1
)
→
X48
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X20
)
→
(
¬
X6
)
→
X35
)
)
→
(
¬
(
(
¬
X35
)
→
(
(
¬
X33
)
↔
(
X31
→
X48
)
)
)
)
)
)
)
↔
(
X35
↔
(
X33
→
X31
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
X46
↔
(
(
¬
X44
)
↔
(
X35
→
X48
)
)
)
)
↔
(
¬
(
X48
→
X11
)
)
)
↔
(
(
¬
X44
)
→
(
X33
↔
X46
)
)
)
→
(
¬
X46
)
→
(
(
¬
X33
)
↔
X48
)
)
)
↔
(
¬
(
(
¬
(
X44
↔
False
)
)
→
(
¬
(
X46
↔
(
(
¬
X34
)
→
X9
→
X44
)
)
)
→
(
¬
(
X9
↔
X48
)
)
)
)
)
)
→
(
¬
(
(
(
(
X44
→
(
X38
↔
(
X37
↔
X46
)
)
)
→
(
¬
(
X40
↔
(
(
¬
X37
)
↔
(
X14
→
X44
)
)
)
)
)
↔
(
(
¬
(
X49
→
X40
→
(
¬
X19
)
→
X43
)
)
→
(
(
¬
(
X38
↔
(
X37
→
X19
→
X40
)
)
)
↔
(
¬
(
(
¬
X37
)
→
(
¬
X8
)
→
X43
)
)
)
)
)
→
(
(
(
¬
(
X43
→
X32
→
X27
→
(
X2
↔
False
)
)
)
↔
(
(
¬
(
X40
→
(
X27
↔
(
X17
↔
X32
)
)
)
)
↔
(
X43
→
(
X32
↔
(
(
¬
X27
)
↔
X40
)
)
)
)
)
↔
(
(
X23
→
X27
)
→
(
¬
(
X43
↔
(
X40
↔
(
X27
→
X44
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
(
(
(
(
¬
X42
)
↔
(
(
¬
X22
)
→
(
¬
X2
)
→
X47
)
)
↔
(
¬
(
X42
↔
(
X2
↔
X6
)
)
)
)
↔
(
¬
(
X47
↔
(
(
¬
X22
)
↔
(
(
¬
X16
)
→
X42
)
)
)
)
)
→
(
(
¬
X22
)
↔
(
X19
↔
(
X16
↔
X47
)
)
)
)
→
(
¬
(
(
(
(
X46
↔
(
(
¬
X44
)
↔
(
(
¬
X22
)
→
X47
)
)
)
↔
(
¬
(
X46
→
X27
)
)
)
↔
(
X46
→
(
(
¬
X44
)
↔
X47
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X44
)
→
(
(
¬
X42
)
↔
(
X27
→
X47
)
)
)
)
→
(
X47
↔
(
X44
→
X27
→
X7
→
False
)
)
)
→
(
¬
(
X44
→
X7
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
X47
↔
(
(
¬
X42
)
→
X44
)
)
→
(
¬
(
(
(
¬
X42
)
↔
(
(
¬
X33
)
→
X9
→
X44
)
)
→
(
¬
(
X47
↔
(
X44
→
(
X33
↔
(
X9
→
False
)
)
)
)
)
)
)
)
→
(
X33
↔
(
X9
→
False
)
)
)
)
→
(
¬
(
X42
↔
(
X9
→
X33
)
)
)
)
)
→
(
X42
↔
(
(
¬
X18
)
→
(
(
¬
X13
)
↔
X47
)
)
)
→
X33
→
(
(
¬
X13
)
↔
(
(
¬
X8
)
→
X47
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
¬
X12
)
↔
(
X4
→
X41
)
)
)
→
(
(
¬
(
(
¬
X41
)
↔
(
(
¬
X12
)
→
X4
→
X49
)
)
)
↔
(
¬
(
(
¬
X47
)
↔
X49
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X12
)
→
(
(
¬
X4
)
↔
X48
)
)
)
)
)
)
→
(
(
X44
↔
(
(
¬
X43
)
→
(
¬
X5
)
→
X48
)
)
↔
(
(
¬
(
(
¬
X43
)
→
(
(
¬
X12
)
↔
X47
)
)
)
↔
(
(
¬
(
X47
→
X43
→
X5
)
)
→
(
¬
(
(
¬
X12
)
↔
X43
)
)
→
(
(
(
¬
X43
)
↔
(
(
¬
X12
)
→
X5
→
X47
)
)
↔
(
X48
↔
(
X12
↔
(
(
¬
X5
)
↔
X47
)
)
)
)
)
)
)
)
↔
(
(
(
(
X48
↔
(
(
¬
X28
)
→
(
¬
X15
)
→
X49
)
)
↔
(
¬
(
(
¬
X48
)
→
(
¬
X26
)
→
X49
)
)
)
→
(
(
(
¬
X20
)
→
X47
)
↔
(
¬
(
(
¬
X47
)
→
X48
→
(
(
¬
X26
)
↔
(
X20
↔
X28
)
)
)
)
)
)
↔
(
(
X46
↔
(
(
¬
X24
)
↔
(
(
¬
X11
)
↔
X49
)
)
)
↔
(
¬
(
(
¬
X29
)
↔
(
(
¬
X20
)
↔
X46
)
)
)
)
)
)
)
→
(
(
(
(
(
(
X31
→
(
(
¬
X30
)
↔
(
X10
→
X37
)
)
)
↔
(
¬
(
(
¬
X30
)
↔
(
X25
→
X37
)
)
)
)
→
(
(
¬
X37
)
↔
(
X30
→
(
¬
X10
)
→
X46
)
)
)
→
(
¬
(
(
¬
(
(
¬
X39
)
→
(
(
¬
X37
)
↔
(
(
¬
X31
)
↔
X49
)
)
)
)
→
(
X49
→
(
X39
↔
(
(
¬
X34
)
↔
X37
)
)
)
→
(
(
X39
↔
(
X34
↔
(
(
¬
X17
)
→
X37
)
)
)
→
(
¬
(
X34
→
(
¬
X17
)
→
X23
)
)
)
→
(
X39
↔
(
X37
→
(
X34
↔
X30
)
)
)
)
)
)
↔
(
(
¬
(
(
X43
↔
(
X34
→
(
X3
↔
X21
)
)
)
→
(
¬
(
(
¬
(
X43
↔
(
X1
→
X21
)
)
)
→
(
¬
X36
)
→
(
¬
X3
)
→
(
¬
X1
)
→
X45
)
)
)
)
↔
(
(
X46
→
(
(
¬
X3
)
↔
(
(
¬
X0
)
→
X45
)
)
)
→
X46
→
(
X37
↔
X14
)
)
)
)
↔
(
(
(
¬
X43
)
↔
(
X38
↔
(
(
¬
X35
)
→
X45
)
)
)
↔
(
(
¬
(
X49
↔
(
X40
→
(
X32
↔
X45
)
)
)
)
→
(
X38
↔
(
(
¬
X32
)
→
X35
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
1d4f0e...
and proposition id is
8f95ad...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMQpJGHoMcNb1rDqTAUiCv9XvrPEta3hYyD
)
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
(
(
(
(
X48
→
X44
→
(
X30
↔
(
X1
→
False
)
)
)
↔
(
(
(
X48
→
X44
)
→
(
X48
→
X44
→
(
X1
↔
X18
)
)
→
(
¬
X41
)
)
↔
(
X48
↔
(
X44
↔
X41
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
X44
→
(
X36
↔
(
(
¬
X20
)
↔
X41
)
)
)
)
↔
(
¬
(
X36
→
(
X6
↔
False
)
)
)
)
↔
(
(
¬
(
X41
↔
X6
)
)
↔
(
¬
(
(
¬
X44
)
→
X41
→
(
¬
X30
)
→
X47
)
)
)
)
→
(
(
¬
X41
)
↔
(
X36
→
(
(
¬
X14
)
↔
X44
)
)
)
)
→
(
¬
X36
)
→
(
¬
X32
)
→
(
(
¬
X20
)
↔
X47
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
¬
(
X44
↔
(
(
¬
X21
)
↔
X25
)
)
)
→
(
(
¬
X27
)
↔
(
(
¬
X25
)
→
(
¬
X21
)
→
X34
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X35
)
→
(
X27
↔
(
(
¬
X2
)
↔
X36
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X35
)
→
(
¬
X34
)
→
X27
→
X36
)
)
→
(
(
X48
↔
(
X35
↔
(
X27
↔
False
)
)
)
→
(
¬
(
X35
↔
(
X34
↔
(
(
¬
X25
)
↔
X27
)
)
)
)
)
→
(
¬
(
X27
→
(
X25
↔
False
)
)
)
)
)
)
)
)
↔
(
¬
(
X48
↔
(
X27
↔
(
(
¬
X11
)
→
X44
)
)
)
)
)
→
(
¬
(
(
(
¬
X34
)
↔
(
X21
→
X44
)
)
→
(
(
X44
↔
(
X34
↔
(
(
¬
X21
)
→
X50
)
)
)
↔
(
(
¬
X21
)
→
(
X17
↔
X34
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
(
X50
↔
(
(
¬
X22
)
↔
X33
)
)
)
→
(
(
X42
→
(
(
¬
X22
)
↔
(
(
¬
X16
)
↔
X33
)
)
)
↔
(
X42
→
(
X39
↔
(
X0
↔
X22
)
)
)
)
→
X33
→
(
X16
↔
(
X0
↔
False
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
(
¬
(
X40
↔
(
(
¬
X28
)
↔
X49
)
)
)
↔
(
(
X50
→
(
(
¬
X40
)
↔
(
(
¬
X13
)
↔
X49
)
)
)
↔
(
¬
(
X50
→
(
X28
↔
(
X13
→
False
)
)
)
)
)
)
→
(
X50
↔
(
X28
→
X13
)
)
)
↔
(
¬
(
X49
↔
(
X42
→
False
)
)
)
)
→
(
¬
(
X50
→
(
¬
X39
)
→
X28
→
X42
)
)
)
)
→
(
¬
(
(
¬
X29
)
↔
(
X28
↔
X40
)
)
)
)
)
)
↔
(
(
X45
→
(
¬
X42
)
→
X12
→
X47
)
→
(
(
(
(
¬
(
X47
↔
(
X39
↔
(
X3
→
False
)
)
)
)
↔
(
X47
↔
(
X45
→
(
X42
↔
(
X39
→
False
)
)
)
)
)
→
(
¬
(
X47
↔
(
X45
→
(
X24
↔
X42
)
)
)
)
)
↔
(
(
X47
→
(
¬
X39
)
→
X15
→
X45
)
↔
(
¬
(
(
¬
X47
)
→
(
(
¬
X39
)
↔
(
X15
↔
X49
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
X26
→
(
X8
↔
(
X7
↔
X38
)
)
)
↔
(
X48
↔
(
X8
↔
(
X7
→
X5
→
False
)
)
)
)
→
(
¬
(
(
¬
(
X26
↔
(
(
¬
X7
)
→
(
(
¬
X4
)
↔
X45
)
)
)
)
→
(
¬
(
X8
→
(
X4
↔
X26
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
(
X48
→
X43
→
(
X7
↔
X31
)
)
)
↔
(
¬
(
X26
→
(
(
¬
X7
)
↔
X43
)
)
)
)
→
(
¬
(
X48
↔
(
X31
↔
False
)
)
)
)
)
→
(
¬
(
X48
→
(
(
¬
X31
)
↔
X43
)
)
)
)
)
)
)
↔
(
(
(
X50
→
(
X43
↔
(
(
¬
X23
)
→
X47
)
)
)
↔
(
X50
→
(
¬
X38
)
→
X19
→
X47
)
)
↔
(
(
(
(
X43
→
(
X37
↔
(
X9
→
X46
)
)
)
↔
(
¬
(
(
X50
→
(
X43
↔
X46
)
)
→
(
(
¬
(
X46
↔
(
(
¬
X37
)
↔
(
X10
→
X43
)
)
)
)
↔
(
X50
↔
(
(
¬
X43
)
→
(
(
¬
X10
)
↔
X46
)
)
)
)
)
)
)
→
X50
→
(
(
¬
X38
)
↔
(
X9
↔
X46
)
)
)
→
(
¬
(
X46
↔
(
X31
↔
(
X9
→
X43
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
(
¬
X43
)
↔
(
X42
→
(
X40
↔
X50
)
)
)
)
↔
(
¬
(
(
(
¬
(
X43
↔
(
(
¬
X42
)
↔
(
X14
↔
X50
)
)
)
)
→
(
¬
(
(
¬
X43
)
→
(
¬
X42
)
→
X46
)
)
)
→
(
¬
(
X42
→
X46
)
)
)
)
)
↔
(
(
(
¬
X40
)
↔
(
(
¬
X26
)
↔
(
(
¬
X25
)
↔
X46
)
)
)
↔
(
(
¬
(
X43
→
X25
→
False
)
)
→
(
¬
(
(
¬
(
X48
→
(
(
¬
X25
)
↔
(
X19
→
X43
)
)
)
)
→
(
¬
X46
)
→
X43
→
(
(
¬
X19
)
↔
X48
)
)
)
→
(
¬
(
X48
→
X43
→
X46
)
)
)
)
)
↔
(
(
¬
(
(
X50
→
(
X49
↔
(
X46
↔
False
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
X46
)
↔
(
X43
↔
(
X13
→
X50
)
)
)
)
→
(
(
¬
X46
)
↔
(
X43
↔
X49
)
)
)
)
↔
(
¬
(
X50
→
(
¬
X43
)
→
(
X6
↔
X49
)
)
)
)
)
)
↔
(
(
¬
(
X49
→
X41
→
(
(
¬
X39
)
↔
X43
)
)
)
↔
(
(
¬
(
X46
↔
(
(
¬
X41
)
→
(
¬
X12
)
→
X49
)
)
)
↔
(
X49
↔
(
X43
↔
(
X41
→
X12
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X10
→
X9
→
(
X0
↔
X31
)
)
)
→
(
(
X31
↔
(
(
¬
X3
)
↔
(
(
¬
X0
)
→
X37
)
)
)
↔
(
(
(
X31
→
X9
)
→
(
(
¬
(
X37
→
X31
→
(
X9
↔
(
X3
↔
False
)
)
)
)
↔
(
X37
→
X36
→
X9
→
X31
)
)
)
↔
(
X36
→
(
X31
↔
X37
)
)
)
)
)
→
(
(
(
X48
→
X42
→
X29
→
X36
)
→
(
(
(
¬
(
X20
→
X37
)
)
→
(
¬
(
X48
↔
(
X20
↔
(
(
¬
X1
)
↔
X22
)
)
)
)
)
↔
(
(
¬
X36
)
→
(
(
¬
X20
)
↔
(
(
¬
X1
)
→
X42
)
)
)
)
)
↔
(
(
¬
(
X42
→
(
X41
↔
(
(
¬
X37
)
→
X48
)
)
)
)
↔
(
X41
↔
(
X31
↔
X48
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
X38
↔
(
X17
→
X21
)
)
↔
(
X49
↔
(
X17
↔
(
X2
→
False
)
)
)
)
→
(
(
¬
(
X17
↔
(
(
¬
X2
)
→
X33
)
)
)
↔
(
X49
→
(
X33
↔
(
(
¬
X11
)
→
X17
)
)
)
)
)
→
(
¬
(
(
(
X47
↔
(
X30
↔
(
(
¬
X7
)
↔
X44
)
)
)
↔
(
(
¬
(
X44
→
X38
)
)
→
(
(
(
(
¬
X44
)
→
(
(
¬
X38
)
↔
X47
)
)
→
(
(
(
X44
→
X30
→
(
¬
X17
)
→
X47
)
→
(
(
¬
X38
)
↔
(
X30
↔
(
(
¬
X17
)
↔
X44
)
)
)
)
↔
(
¬
(
X47
→
X38
)
)
)
→
(
¬
(
X38
→
(
¬
X30
)
→
X47
)
)
)
↔
(
X47
↔
(
(
¬
X17
)
→
X30
)
)
)
)
)
→
(
X49
↔
(
(
¬
X38
)
→
(
X17
↔
X47
)
)
)
)
)
)
)
→
(
(
(
(
(
(
¬
(
X35
→
(
(
¬
X28
)
↔
(
(
¬
X8
)
→
X44
)
)
)
)
→
(
X8
↔
(
X4
→
X44
)
)
)
↔
(
¬
(
X45
↔
(
(
¬
X28
)
→
X8
→
X44
)
)
)
)
↔
(
¬
(
(
¬
X44
)
→
X35
→
X15
→
X45
)
)
)
↔
(
(
(
¬
(
(
¬
X38
)
→
X34
→
(
X23
↔
X44
)
)
)
↔
(
¬
(
X45
→
(
(
¬
X35
)
↔
(
X23
→
X44
)
)
)
)
)
↔
(
(
X50
→
X48
→
(
X44
↔
X34
)
)
↔
(
X44
→
(
¬
X34
)
→
X23
→
X48
)
)
)
)
↔
(
(
(
(
¬
X47
)
↔
(
X35
↔
(
X32
→
X50
)
)
)
↔
(
(
¬
X32
)
→
X18
→
X50
)
)
→
(
(
(
(
¬
(
X47
→
X32
→
(
(
¬
X5
)
↔
X45
)
)
)
↔
(
(
¬
X45
)
→
(
(
¬
X35
)
↔
(
X5
→
X47
)
)
)
)
→
(
(
¬
X35
)
↔
(
(
¬
X32
)
↔
(
(
¬
X16
)
↔
X47
)
)
)
)
↔
(
(
¬
(
(
X32
→
X27
→
X24
)
→
(
¬
(
X35
→
(
X5
↔
False
)
)
)
)
)
→
(
¬
(
X35
↔
X27
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
9cb5ad...
and proposition id is
05ae7c...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMQCpuwsgmeVkFKzPW5WTFz5ws3ocjEhB25
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
∀X50 :
prop
,
(
(
(
(
¬
(
(
(
(
¬
(
X43
→
(
(
¬
X35
)
↔
X47
)
)
)
↔
(
(
¬
X47
)
↔
(
(
¬
X35
)
↔
(
X22
↔
X50
)
)
)
)
↔
(
(
¬
(
(
¬
X22
)
→
(
(
¬
X12
)
↔
(
X8
→
X35
)
)
)
)
↔
(
¬
(
X43
→
X35
→
(
(
¬
X12
)
↔
X22
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X43
↔
(
X42
↔
(
(
¬
X22
)
→
X45
)
)
)
)
→
(
¬
(
X45
→
X43
→
(
(
¬
X12
)
↔
X42
)
)
)
)
→
(
¬
(
(
¬
(
(
(
X45
↔
(
(
¬
X27
)
→
X43
)
)
↔
(
¬
(
X32
↔
(
X27
↔
(
(
¬
X22
)
→
X45
)
)
)
)
)
→
(
¬
(
X45
↔
(
X43
↔
(
X27
↔
False
)
)
)
)
→
(
X43
→
X27
→
(
X9
↔
X3
)
)
→
(
¬
(
X45
↔
(
X43
→
(
¬
X3
)
→
X9
)
)
)
)
)
→
(
(
¬
X32
)
↔
(
X22
↔
(
(
¬
X19
)
↔
X45
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
X44
)
↔
(
(
¬
X40
)
→
(
X7
↔
X47
)
)
)
↔
(
¬
(
X37
→
X47
)
)
)
↔
(
(
(
¬
X49
)
→
X44
→
(
(
¬
X28
)
↔
X50
)
)
↔
(
¬
(
(
¬
(
X47
→
(
(
¬
X44
)
↔
(
(
¬
X34
)
→
X49
)
)
)
)
→
(
¬
(
X44
→
(
(
¬
X40
)
↔
(
(
¬
X21
)
↔
X49
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
(
(
¬
X47
)
→
(
X41
↔
(
(
¬
X36
)
↔
X50
)
)
)
)
→
(
(
X41
→
X36
→
X17
)
↔
(
X47
→
(
X36
↔
X23
)
)
)
)
)
→
(
(
¬
(
X41
→
(
X36
↔
(
X23
→
False
)
)
)
)
↔
(
X36
→
(
(
¬
X11
)
↔
X45
)
)
)
)
↔
(
(
¬
(
X17
↔
(
(
¬
X11
)
↔
X39
)
)
)
↔
(
(
X50
↔
(
X17
→
(
¬
X11
)
→
X29
)
)
→
X29
→
X6
→
X50
)
)
)
↔
(
(
(
X50
↔
X47
)
↔
(
¬
(
(
¬
(
(
¬
X47
)
→
(
X45
↔
X50
)
)
)
→
(
¬
(
X47
→
X44
→
X10
→
False
)
)
→
X50
→
X44
→
X45
)
)
)
↔
(
¬
(
(
¬
X10
)
↔
X45
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X47
↔
(
X46
→
(
X26
↔
(
X15
→
False
)
)
)
)
)
↔
(
¬
(
(
(
¬
X26
)
→
(
X18
↔
(
(
¬
X15
)
→
X47
)
)
)
→
X47
→
(
X15
↔
X4
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X38
→
(
X20
↔
(
(
¬
X15
)
→
X47
)
)
)
)
→
(
X47
↔
(
X15
→
(
(
¬
X2
)
↔
X38
)
)
)
)
)
→
(
¬
(
X38
↔
(
X20
→
(
X15
↔
X33
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
X33
↔
(
X31
↔
(
(
¬
X1
)
→
X38
)
)
)
→
(
¬
(
(
¬
X33
)
→
(
¬
X31
)
→
(
(
¬
X5
)
↔
X38
)
)
)
)
↔
(
X45
↔
(
X33
→
(
X31
↔
X50
)
)
)
)
→
(
¬
(
(
X50
↔
(
(
¬
X38
)
→
(
¬
X30
)
→
X45
)
)
→
(
¬
(
X50
↔
(
(
¬
X24
)
→
X48
)
)
)
)
)
)
↔
(
(
(
(
(
¬
X25
)
↔
(
(
¬
X14
)
→
X46
)
)
↔
(
(
¬
(
(
¬
X46
)
→
(
¬
X31
)
→
(
(
¬
X14
)
↔
X48
)
)
)
↔
(
¬
(
X48
→
X46
→
X25
→
X31
)
)
)
)
↔
(
¬
(
X48
↔
(
X46
→
(
X25
↔
(
X13
↔
False
)
)
)
)
)
)
→
(
¬
(
(
X46
↔
(
(
¬
X33
)
→
X31
→
X48
)
)
→
(
X48
↔
(
(
¬
X31
)
→
(
¬
X16
)
→
X33
)
)
)
)
→
(
¬
(
(
¬
X0
)
↔
X48
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X21
↔
(
(
¬
X5
)
↔
(
(
¬
X3
)
↔
X48
)
)
)
)
↔
(
¬
(
X50
↔
(
X48
↔
(
X42
↔
(
X21
↔
False
)
)
)
)
)
)
→
(
(
¬
(
X50
→
X26
→
X48
)
)
↔
(
(
¬
X46
)
↔
(
X41
→
X50
)
)
)
→
(
(
X48
→
(
(
¬
X42
)
↔
(
(
¬
X20
)
→
X49
)
)
)
↔
(
(
(
(
¬
X48
)
↔
(
X42
↔
(
(
¬
X34
)
↔
X49
)
)
)
→
X50
→
(
X48
↔
(
(
¬
X34
)
↔
X46
)
)
)
→
(
¬
(
X50
↔
(
(
¬
X48
)
↔
(
X40
↔
X49
)
)
)
)
)
)
)
→
(
(
(
¬
(
X27
↔
False
)
)
↔
(
(
¬
(
(
¬
X42
)
↔
(
(
¬
X27
)
↔
X45
)
)
)
→
(
(
¬
(
X46
↔
(
X42
↔
(
(
¬
X27
)
↔
X45
)
)
)
)
↔
(
X46
→
(
(
¬
X42
)
↔
(
X6
→
X45
)
)
)
)
)
)
↔
(
¬
(
(
(
X50
↔
(
X46
→
(
X45
↔
X39
)
)
)
→
X46
→
X42
)
→
(
(
(
¬
(
X46
↔
(
X24
↔
(
X11
↔
False
)
)
)
)
↔
(
(
¬
X24
)
↔
(
(
¬
X15
)
↔
(
X11
→
X45
)
)
)
)
↔
(
(
(
¬
X24
)
↔
(
X22
↔
(
X15
↔
X50
)
)
)
↔
(
¬
(
X50
→
(
X45
↔
(
X24
↔
(
X22
↔
False
)
)
)
)
)
)
)
)
)
)
→
(
X22
↔
(
(
¬
X8
)
↔
(
X7
↔
X39
)
)
)
→
(
¬
(
X8
↔
X27
)
)
)
↔
(
¬
(
(
(
(
¬
(
X50
→
(
¬
X48
)
→
X49
)
)
↔
(
(
¬
X47
)
→
(
X37
↔
(
X19
→
X50
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X49
)
↔
(
(
¬
X47
)
↔
(
(
¬
X14
)
↔
X50
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X32
)
↔
(
X17
→
(
X10
↔
X50
)
)
)
)
↔
(
X38
↔
(
X17
→
X50
)
)
)
→
X32
→
X17
→
X10
)
)
→
(
(
X50
→
X49
→
X38
→
X16
)
↔
(
X49
↔
(
(
¬
X38
)
→
X32
→
X50
)
)
)
)
)
)
→
(
(
(
¬
(
(
X48
→
X43
→
(
¬
X28
)
→
X44
)
→
(
(
¬
(
X43
→
X29
)
)
↔
(
(
(
¬
X2
)
↔
X29
)
↔
(
¬
(
(
¬
(
X43
↔
(
X36
↔
(
(
¬
X29
)
↔
X48
)
)
)
)
→
(
X48
↔
(
(
¬
X36
)
→
(
(
¬
X2
)
↔
X43
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X48
↔
(
X43
→
X18
→
X36
)
)
)
→
(
¬
(
X18
↔
(
(
¬
X0
)
↔
X36
)
)
)
)
)
↔
(
¬
(
X48
→
(
¬
X36
)
→
X28
→
X43
)
)
)
)
)
↔
(
¬
(
(
¬
(
X43
→
X28
→
(
X4
↔
X35
)
)
)
→
(
(
(
¬
(
(
¬
X35
)
↔
(
X28
↔
X49
)
)
)
↔
(
¬
(
(
¬
X43
)
↔
(
X28
→
X44
)
)
)
)
↔
(
X43
→
X35
→
(
(
¬
X25
)
↔
X44
)
)
)
)
)
)
↔
(
(
¬
(
(
X44
↔
(
X31
→
X33
)
)
→
(
(
(
¬
(
X49
→
(
X44
↔
(
(
¬
X23
)
→
X30
)
)
)
)
↔
(
¬
(
X48
→
X44
→
(
X30
↔
X50
)
)
)
)
↔
(
(
X50
→
X33
→
False
)
↔
(
(
X48
→
X44
→
X1
)
→
(
X33
↔
(
X31
↔
(
X12
→
False
)
)
)
)
)
)
)
)
→
(
(
¬
(
X49
↔
(
(
¬
X13
)
↔
(
(
¬
X9
)
→
X48
)
)
)
)
↔
(
X50
↔
(
X48
↔
False
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
3a840b...
and proposition id is
3f9143...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMEuvmXzyyCjnTGauQhXWoq6TtupxUGubp8
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
∀X51 :
prop
,
(
(
¬
(
(
(
(
(
¬
(
(
(
¬
X49
)
↔
(
X19
→
X50
)
)
→
(
(
¬
X33
)
↔
(
(
¬
X28
)
↔
(
X19
→
X50
)
)
)
)
)
→
(
¬
(
(
¬
(
(
X50
→
X49
→
(
(
¬
X32
)
↔
X40
)
)
→
(
¬
(
X50
↔
(
X49
→
X40
→
X51
)
)
)
)
)
→
(
¬
(
(
¬
X50
)
↔
(
(
¬
X49
)
↔
(
X28
↔
X51
)
)
)
)
)
)
)
→
(
¬
(
(
(
X49
→
X45
→
X11
→
X43
)
→
(
X49
→
(
(
¬
X45
)
↔
X50
)
)
→
(
(
¬
(
(
¬
X49
)
↔
(
(
¬
X43
)
↔
(
(
¬
X15
)
→
X50
)
)
)
)
↔
(
(
¬
X45
)
↔
(
(
¬
X43
)
→
X49
)
)
)
)
→
(
¬
X45
)
→
X11
→
X49
)
)
→
(
(
(
X51
→
X45
→
X8
)
↔
(
X43
↔
(
(
¬
X7
)
→
X8
)
)
)
↔
(
X50
↔
(
X8
→
X7
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
X50
↔
(
X42
↔
(
X41
→
X31
→
False
)
)
)
)
→
(
(
(
¬
X9
)
↔
(
(
X9
→
False
)
↔
(
(
(
¬
(
X42
↔
(
X41
→
X36
→
X50
)
)
)
↔
(
X50
→
X42
→
X9
→
X41
)
)
↔
(
¬
(
X50
↔
(
X41
→
X42
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
X41
)
↔
(
(
¬
X36
)
→
X3
→
X50
)
)
→
(
¬
(
X50
→
X41
→
(
¬
X2
)
→
X36
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X48
)
→
(
¬
X21
)
→
(
¬
X18
)
→
X50
)
)
→
(
X50
↔
(
X48
↔
(
X21
↔
X44
)
)
)
)
)
↔
(
¬
(
(
(
¬
X48
)
→
X44
→
(
X42
↔
X50
)
)
→
(
¬
(
(
¬
X44
)
↔
(
(
¬
X42
)
↔
(
X31
→
X50
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
(
¬
(
(
¬
X46
)
↔
(
(
¬
X18
)
→
(
X9
↔
X49
)
)
)
)
↔
(
¬
(
X50
↔
(
X49
→
X9
→
False
)
)
)
)
→
(
¬
(
X18
→
(
X9
↔
X50
)
)
)
)
)
→
(
(
¬
X46
)
↔
(
X29
→
(
X18
↔
X50
)
)
)
)
)
→
(
¬
(
X49
→
(
X46
↔
X17
)
)
)
→
(
¬
(
X50
↔
(
X5
→
X46
)
)
)
)
)
)
→
(
(
(
(
¬
X42
)
→
X37
→
(
¬
X14
)
→
X51
)
↔
(
¬
(
X42
↔
(
X37
→
(
X14
↔
(
X4
→
False
)
)
)
)
)
)
↔
(
(
X50
→
(
X48
↔
(
X37
→
False
)
)
)
↔
(
(
¬
(
X48
↔
(
X37
→
X50
)
)
)
↔
(
¬
(
(
¬
X37
)
→
(
X22
↔
(
X14
→
X50
)
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
X48
→
(
¬
X39
)
→
(
X13
↔
X51
)
)
↔
(
(
¬
(
(
¬
X48
)
→
(
(
¬
X35
)
↔
(
(
¬
X13
)
↔
X50
)
)
)
)
→
(
¬
(
X51
→
(
¬
X48
)
→
(
X35
↔
X50
)
)
)
)
)
↔
(
(
¬
(
X34
↔
(
(
¬
X30
)
↔
(
X10
→
X51
)
)
)
)
↔
(
¬
(
X39
↔
(
(
¬
X30
)
→
X50
)
)
)
)
)
→
(
(
(
¬
(
X24
↔
(
(
¬
X23
)
→
X39
)
)
)
↔
(
¬
(
X51
↔
(
(
¬
X35
)
→
(
X1
↔
X39
)
)
)
)
)
↔
(
¬
(
X39
→
(
¬
X35
)
→
(
X24
↔
X38
)
)
)
)
)
)
↔
(
(
(
(
X35
→
X27
)
↔
(
¬
(
X50
↔
(
X47
↔
(
(
¬
X0
)
→
X27
)
)
)
)
)
→
X50
→
X35
→
X25
→
X47
)
→
(
X51
↔
(
(
¬
X25
)
↔
X47
)
)
→
(
¬
(
(
(
(
X26
↔
X35
)
↔
(
(
(
¬
X12
)
→
X35
)
→
(
(
X26
↔
(
(
¬
X12
)
↔
(
(
¬
X6
)
↔
X35
)
)
)
↔
(
X26
↔
(
(
¬
X6
)
→
X47
)
)
)
)
)
→
(
¬
(
(
(
X35
↔
(
X20
→
(
(
¬
X12
)
↔
X26
)
)
)
→
X51
→
(
X26
↔
(
X20
↔
X12
)
)
)
→
(
¬
(
X26
↔
False
)
)
)
)
)
→
(
¬
(
X16
↔
(
(
¬
X6
)
↔
X25
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
(
¬
(
X38
→
X39
)
)
→
(
(
¬
(
X47
→
(
¬
X39
)
→
(
¬
X26
)
→
X48
)
)
↔
(
X47
↔
(
(
¬
X39
)
→
(
¬
X38
)
→
X48
)
)
)
)
)
↔
(
X48
→
(
(
¬
X26
)
↔
X47
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
X24
→
(
¬
X21
)
→
(
¬
X3
)
→
X47
)
)
→
(
X26
↔
(
(
¬
X21
)
→
(
¬
X3
)
→
X47
)
)
)
→
(
¬
(
X26
→
(
¬
X7
)
→
(
¬
X3
)
→
X24
)
)
)
↔
(
X47
↔
X3
)
)
→
(
¬
(
X51
→
X21
→
X38
)
)
→
X26
→
X24
→
(
¬
X21
)
→
X38
)
)
)
↔
(
(
¬
(
X42
→
(
X38
↔
(
(
¬
X31
)
↔
X46
)
)
)
)
↔
(
(
(
¬
X38
)
↔
(
X24
↔
X46
)
)
↔
(
X46
→
X39
→
(
¬
X20
)
→
X42
)
)
)
)
↔
(
(
(
(
X34
→
(
¬
X28
)
→
(
(
¬
X16
)
↔
X35
)
)
↔
(
(
¬
X28
)
→
(
(
¬
X16
)
↔
X51
)
)
)
↔
(
(
X37
↔
(
X10
→
False
)
)
↔
(
(
X51
→
X37
→
X35
→
X34
→
False
)
→
X51
→
(
X37
↔
X25
)
)
)
)
→
(
(
¬
(
(
¬
(
X49
→
(
X47
↔
X43
)
)
)
→
(
¬
(
(
¬
X48
)
↔
(
X47
→
X37
→
X49
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X37
)
→
(
¬
X16
)
→
X49
)
)
→
(
¬
(
(
¬
X37
)
→
(
(
¬
X35
)
↔
(
(
¬
X1
)
→
X47
)
)
)
)
→
(
¬
(
X1
↔
(
(
¬
X0
)
↔
X47
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
X50
↔
(
X41
→
X22
→
(
X17
↔
False
)
)
)
)
↔
(
¬
(
(
¬
(
X29
→
(
(
¬
X22
)
↔
(
X17
→
X44
)
)
)
)
→
(
(
¬
(
(
¬
X44
)
→
(
X22
↔
X50
)
)
)
↔
(
¬
(
X50
↔
(
X44
→
(
X29
↔
X5
)
)
)
)
)
)
)
)
→
(
¬
(
X50
↔
(
(
¬
X36
)
→
(
X33
↔
X44
)
)
)
)
→
(
¬
(
(
(
X46
→
(
(
¬
X44
)
↔
(
X33
→
X50
)
)
)
→
(
(
¬
X44
)
↔
(
(
¬
X33
)
↔
(
X29
↔
X50
)
)
)
)
→
(
(
¬
X44
)
↔
X46
)
)
)
)
)
↔
(
(
(
(
¬
(
X8
↔
X48
)
)
↔
(
¬
(
X44
↔
(
X11
→
(
X6
↔
X49
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X41
)
→
(
X40
↔
(
X36
↔
X44
)
)
)
)
↔
(
¬
(
X48
↔
(
X36
↔
X40
)
)
)
)
↔
(
(
¬
X41
)
↔
(
X40
→
(
¬
X36
)
→
X49
)
)
)
)
↔
(
(
¬
(
(
¬
X45
)
→
X44
→
(
(
¬
X41
)
↔
X49
)
)
)
↔
(
¬
(
(
¬
X45
)
↔
(
X44
↔
(
(
¬
X13
)
↔
X49
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
(
¬
X18
)
→
X48
)
)
→
(
¬
(
(
¬
X19
)
→
X15
→
X32
)
)
)
↔
(
¬
(
(
¬
X32
)
→
(
(
¬
X19
)
↔
(
X4
→
X48
)
)
)
)
)
→
(
¬
(
(
X27
→
X46
)
→
(
¬
(
(
¬
X46
)
↔
(
(
¬
X30
)
→
(
X2
↔
X50
)
)
)
)
→
(
X32
↔
(
X27
→
(
X2
↔
X50
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
X23
)
↔
(
(
¬
X9
)
↔
X48
)
)
)
→
(
¬
(
(
¬
(
X30
↔
(
(
¬
X23
)
→
(
(
¬
X9
)
↔
X48
)
)
)
)
→
X50
→
X23
→
X12
→
X9
)
)
)
)
↔
(
X23
↔
(
X14
→
X12
→
X50
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
dd3260...
and proposition id is
73282e...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMbWCEm5VCdkHXj1eS3vdHiy36kX8hMzgzW
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
(
(
(
¬
X42
)
→
X11
→
X8
→
X45
)
→
X46
→
(
X27
↔
(
(
¬
X8
)
↔
X45
)
)
)
↔
(
¬
(
(
¬
(
X46
↔
(
X45
↔
(
X42
↔
(
X37
↔
False
)
)
)
)
)
→
(
¬
(
(
X46
→
(
X42
↔
(
X1
↔
X37
)
)
)
→
X46
→
(
X37
↔
X48
)
)
)
)
)
)
→
(
¬
(
X48
↔
(
X42
→
X24
→
X37
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X46
→
X44
→
X19
→
X33
)
)
→
(
(
¬
(
X46
↔
(
(
¬
X44
)
→
X18
→
X47
)
)
)
↔
(
(
(
X47
↔
(
X44
→
X33
)
)
↔
(
¬
(
X47
→
(
X46
↔
(
(
¬
X28
)
→
X33
)
)
)
)
)
↔
(
(
¬
X44
)
→
(
¬
X28
)
→
X47
)
)
)
)
→
(
(
(
¬
(
X47
→
X44
→
X46
)
)
↔
(
(
(
(
¬
X44
)
→
X36
→
(
(
¬
X13
)
↔
X46
)
)
↔
(
¬
(
(
(
¬
X46
)
→
(
(
¬
X13
)
↔
X47
)
)
→
(
¬
(
X46
→
(
(
¬
X30
)
↔
X36
)
)
)
)
)
)
→
(
¬
X46
)
→
(
¬
X44
)
→
X47
)
)
↔
(
¬
(
X46
↔
(
X36
↔
X47
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X44
→
(
(
¬
X32
)
↔
X48
)
)
)
→
(
(
¬
X16
)
→
X48
)
→
(
¬
(
X48
↔
(
X44
→
(
¬
X32
)
→
X36
)
)
)
)
↔
(
¬
(
X47
→
X44
→
(
X36
↔
X48
)
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
X44
)
→
X48
)
→
(
(
¬
(
(
¬
X37
)
↔
(
(
¬
X26
)
↔
X44
)
)
)
↔
(
¬
(
(
(
¬
X44
)
→
(
(
¬
X23
)
↔
X48
)
)
→
(
¬
(
(
¬
(
X44
↔
(
X37
↔
(
(
¬
X26
)
→
X48
)
)
)
)
→
(
X26
↔
(
X23
↔
False
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
X36
↔
(
X26
↔
(
X25
↔
False
)
)
)
→
X37
→
(
X36
↔
(
X25
↔
X44
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
X38
→
X20
→
(
X4
↔
X43
)
)
→
(
X49
↔
(
X43
↔
(
X38
↔
X20
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X7
)
↔
(
X6
→
(
¬
X2
)
→
X39
)
)
)
→
(
¬
(
(
¬
X7
)
↔
(
X6
↔
(
X5
→
X39
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X43
→
(
X38
↔
(
X20
→
X7
)
)
)
)
↔
(
(
¬
X20
)
↔
(
(
¬
X7
)
→
(
(
¬
X6
)
↔
X38
)
)
)
)
↔
(
¬
(
X43
↔
(
(
¬
X20
)
↔
X38
)
)
)
)
↔
(
X20
↔
(
X7
→
False
)
)
)
↔
(
X20
↔
X7
)
)
)
)
↔
(
(
(
(
(
¬
X43
)
→
X22
→
(
(
¬
X3
)
↔
X49
)
)
→
X43
→
(
X22
↔
(
(
¬
X21
)
→
X49
)
)
)
↔
(
X49
↔
(
X44
→
(
(
¬
X22
)
↔
X43
)
)
)
)
↔
(
(
X39
↔
(
(
¬
X15
)
→
X21
)
)
→
(
¬
(
(
¬
(
X39
→
(
X14
↔
(
X3
→
X21
)
)
)
)
→
X22
→
X3
)
)
)
)
)
↔
(
(
(
¬
(
(
X40
↔
(
X35
↔
(
X31
→
X45
)
)
)
→
(
¬
(
X40
→
X9
→
X45
)
)
)
)
↔
(
(
¬
(
X46
↔
(
X12
↔
X40
)
)
)
↔
(
(
(
¬
X31
)
↔
(
(
¬
X17
)
→
X44
)
)
→
(
(
¬
(
(
(
¬
(
(
X39
→
False
)
→
(
¬
(
(
¬
(
(
¬
X39
)
→
(
(
¬
X35
)
↔
X44
)
)
)
→
(
¬
(
(
¬
X39
)
↔
(
X17
→
X40
)
)
)
)
)
)
)
→
(
¬
(
X40
→
(
¬
X35
)
→
(
¬
X31
)
→
X44
)
)
)
→
(
¬
(
(
¬
X35
)
→
X31
→
X12
→
X40
)
)
)
)
↔
(
¬
(
(
¬
X35
)
→
X31
→
X39
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
(
X45
↔
(
(
¬
X10
)
↔
X46
)
)
)
↔
(
¬
(
X41
↔
(
X34
→
X46
)
)
)
)
→
(
X44
↔
(
(
¬
X41
)
↔
X46
)
)
)
)
→
(
¬
(
(
(
¬
X44
)
↔
(
X0
↔
X46
)
)
→
(
(
X44
→
X0
→
X34
)
↔
(
(
¬
X43
)
↔
(
X34
→
X29
→
X44
)
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
(
(
¬
X37
)
↔
(
X33
→
X5
→
X41
)
)
↔
(
(
¬
X37
)
→
(
¬
X33
)
→
(
¬
X8
)
→
X41
)
)
→
(
(
¬
X33
)
↔
X44
)
)
→
(
(
(
(
X44
↔
(
X41
↔
(
X19
→
False
)
)
)
↔
(
(
(
¬
X41
)
↔
(
X2
→
X45
)
)
→
(
¬
(
X44
↔
(
X41
↔
(
(
¬
X19
)
→
X45
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X41
→
(
X40
↔
(
(
¬
X37
)
→
X45
)
)
)
)
↔
(
(
¬
X41
)
↔
(
(
¬
X40
)
↔
X44
)
)
)
→
(
¬
(
(
¬
X41
)
↔
(
X37
→
X45
)
)
)
)
)
)
↔
(
X22
↔
(
X19
→
X41
)
)
)
)
↔
(
(
¬
(
(
¬
(
X44
↔
(
(
¬
X27
)
↔
(
X12
↔
X41
)
)
)
)
→
(
¬
(
(
¬
X27
)
→
(
(
¬
X17
)
↔
(
(
¬
X12
)
↔
X41
)
)
)
)
)
)
↔
(
(
X44
→
(
(
¬
X41
)
↔
(
(
¬
X27
)
→
X43
)
)
)
↔
(
X43
→
X41
→
X33
→
X44
)
)
)
)
→
(
(
(
¬
(
(
X49
→
(
(
¬
X6
)
↔
X10
)
)
→
(
(
¬
(
X48
→
(
¬
X42
)
→
X49
)
)
↔
(
(
¬
(
X49
→
(
¬
X7
)
→
(
X6
↔
X42
)
)
)
↔
(
¬
(
(
¬
X48
)
→
X42
→
(
(
¬
X7
)
↔
X49
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
X45
)
↔
(
(
¬
X44
)
↔
(
X41
↔
X48
)
)
)
→
X45
→
(
X41
↔
X49
)
)
)
↔
(
(
¬
(
X44
→
(
X42
↔
(
X41
↔
False
)
)
)
)
↔
(
(
(
X44
↔
(
(
¬
X41
)
↔
(
(
¬
X16
)
→
X42
)
)
)
↔
(
¬
(
X49
↔
(
(
¬
X42
)
→
X44
)
)
)
)
→
X49
→
(
X42
↔
False
)
)
)
)
↔
(
¬
(
X44
→
(
X41
↔
(
(
¬
X29
)
→
X49
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
¬
X35
)
↔
X36
)
)
↔
(
(
¬
(
X36
↔
(
X14
→
X44
)
)
)
↔
(
X45
→
X35
→
(
¬
X14
)
→
X36
)
)
)
→
(
¬
(
X45
→
(
X15
↔
(
(
¬
X0
)
↔
X35
)
)
)
)
→
(
X45
↔
(
X36
→
(
X11
↔
X15
)
)
)
)
)
↔
(
¬
(
(
¬
(
X44
→
(
X41
↔
(
(
¬
X14
)
↔
X39
)
)
)
)
→
(
¬
(
(
¬
(
(
X41
↔
(
(
¬
X15
)
→
(
¬
X4
)
→
X44
)
)
→
(
¬
(
(
¬
X18
)
→
(
X15
↔
(
X4
↔
X41
)
)
)
)
)
)
→
(
(
X44
↔
(
X39
↔
X15
)
)
↔
(
X15
↔
(
X13
→
(
¬
X4
)
→
X44
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
X46
→
(
(
¬
(
(
(
X30
→
X20
)
↔
(
(
(
¬
X30
)
→
(
X24
↔
X32
)
)
↔
(
(
¬
(
(
¬
X24
)
↔
(
X20
↔
(
(
¬
X9
)
↔
X32
)
)
)
)
↔
(
X30
↔
(
X24
↔
X20
)
)
)
)
)
→
(
¬
(
X25
→
X20
→
(
(
¬
X9
)
↔
X32
)
)
)
)
)
↔
(
X32
→
(
(
¬
X30
)
↔
(
(
¬
X25
)
→
X41
)
)
)
)
)
↔
(
(
(
(
¬
X42
)
→
(
(
¬
X26
)
↔
(
(
¬
X1
)
→
X46
)
)
)
↔
(
¬
(
X46
↔
(
X32
↔
(
X1
→
False
)
)
)
)
)
↔
(
(
(
X46
→
(
¬
X32
)
→
X30
→
X42
)
↔
(
X46
→
(
¬
X32
)
→
(
¬
X21
)
→
X42
)
)
↔
(
X42
↔
(
(
¬
X32
)
↔
X46
)
)
)
)
)
↔
(
(
(
(
X47
↔
(
X34
→
(
¬
X31
)
→
X38
)
)
↔
(
X47
→
(
(
¬
X38
)
↔
(
(
¬
X34
)
↔
X39
)
)
)
)
→
X47
→
(
X38
↔
(
X34
→
X3
)
)
)
→
(
¬
(
(
¬
(
(
¬
X41
)
→
X31
→
(
¬
X28
)
→
X42
)
)
→
(
(
(
¬
(
X41
→
(
¬
X28
)
→
(
¬
X23
)
→
X47
)
)
↔
(
X46
→
(
X41
↔
X28
)
)
)
↔
(
¬
(
X41
↔
(
(
¬
X23
)
↔
X47
)
)
)
)
→
(
¬
(
(
X47
↔
(
X46
→
X41
→
False
)
)
→
(
(
¬
(
X41
↔
(
(
¬
X31
)
→
X34
)
)
)
↔
(
(
¬
(
(
¬
X31
)
→
X34
)
)
→
(
X46
↔
(
X41
→
X31
)
)
→
(
X34
↔
(
(
¬
X31
)
↔
X47
)
)
)
)
→
(
¬
(
X47
→
(
X31
↔
X34
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
775c22...
and proposition id is
9eeede...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMXyAQ6FKWZQQPGEdJN8FueHPNrjyGxbDr1
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
¬
(
(
(
(
(
(
(
X45
↔
(
X8
→
X3
→
X37
)
)
↔
(
¬
(
X8
→
(
X3
↔
(
(
¬
X0
)
→
X45
)
)
)
)
)
↔
(
(
¬
X30
)
↔
(
(
¬
X8
)
→
(
(
¬
X3
)
↔
X45
)
)
)
)
↔
(
(
(
X48
→
X8
→
(
X3
↔
False
)
)
↔
(
¬
(
X17
↔
(
(
¬
X8
)
→
X3
→
X11
)
)
)
)
→
(
(
X37
→
False
)
↔
(
¬
(
X48
→
(
X8
↔
(
(
¬
X1
)
↔
X11
)
)
)
)
)
)
)
→
(
(
(
(
¬
X2
)
→
X39
)
↔
(
(
X41
↔
(
X39
↔
(
X37
→
X45
)
)
)
↔
(
¬
(
(
¬
X41
)
↔
(
(
¬
X39
)
↔
(
(
¬
X2
)
↔
X45
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
X37
)
→
(
X21
↔
(
X0
↔
X39
)
)
)
)
→
(
¬
(
X41
↔
(
X39
→
X21
→
X0
)
)
)
)
)
)
→
(
(
X48
↔
(
(
¬
X44
)
→
(
X27
↔
X47
)
)
)
↔
(
(
¬
(
X44
↔
(
X40
↔
X30
)
)
)
↔
(
X48
↔
(
X40
→
(
X30
↔
X47
)
)
)
)
)
→
(
(
(
(
(
X46
→
X11
)
↔
(
¬
(
X49
→
X30
→
(
(
¬
X11
)
↔
X20
)
)
)
)
↔
(
¬
(
(
¬
X30
)
↔
(
X20
→
X46
)
)
)
)
→
(
¬
X30
)
→
X20
→
X46
)
→
(
X46
↔
(
X30
→
(
¬
X20
)
→
X47
)
)
)
→
(
(
¬
(
X46
→
(
X13
↔
X30
)
)
)
↔
(
¬
(
(
¬
(
X49
↔
(
X46
↔
(
X44
→
X30
→
False
)
)
)
)
→
(
¬
(
X44
↔
(
X36
↔
X49
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
(
(
(
¬
(
X48
↔
(
X46
→
(
(
¬
X28
)
↔
X35
)
)
)
)
→
(
X46
↔
(
(
¬
X35
)
→
(
¬
X28
)
→
X47
)
)
)
→
(
¬
(
X47
→
(
¬
X28
)
→
(
X15
↔
X35
)
)
)
)
→
X48
→
(
X46
↔
(
X35
→
(
X31
↔
False
)
)
)
)
)
→
X31
→
(
(
¬
X28
)
↔
X48
)
)
→
(
(
¬
X46
)
↔
X47
)
)
→
(
(
¬
(
(
(
X46
↔
(
X35
→
(
X32
↔
X29
)
)
)
→
(
X35
↔
(
(
¬
X32
)
→
(
¬
X22
)
→
X46
)
)
)
→
(
(
¬
X32
)
→
(
¬
X26
)
→
X48
)
→
(
¬
(
X46
↔
(
X29
→
X32
)
)
)
)
)
↔
(
(
X34
↔
(
(
¬
X23
)
→
(
¬
X6
)
→
X29
)
)
↔
(
(
(
X35
↔
(
X23
↔
X32
)
)
↔
(
(
¬
(
X32
→
X23
→
X16
)
)
→
X35
→
X32
→
X16
→
X34
)
)
↔
(
¬
(
X29
↔
(
(
¬
X23
)
↔
X35
)
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
(
¬
(
(
¬
X42
)
→
(
¬
X38
)
→
X12
→
X46
)
)
→
(
X46
↔
(
X38
→
(
X12
↔
X43
)
)
)
)
)
↔
(
(
(
(
¬
X47
)
→
(
¬
X43
)
→
(
¬
X38
)
→
X48
)
↔
(
¬
(
(
¬
(
X48
→
(
(
¬
X19
)
↔
(
(
¬
X12
)
→
X47
)
)
)
)
→
X48
→
X43
→
(
(
¬
X19
)
↔
X47
)
)
)
)
↔
(
X43
↔
X19
)
)
)
↔
(
(
¬
X42
)
→
(
X38
↔
X43
)
)
)
→
(
(
¬
(
X48
→
X46
→
(
X33
↔
(
X4
↔
False
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X47
)
→
(
¬
X46
)
→
(
¬
X25
)
→
X48
)
)
→
X48
→
X46
→
(
X4
↔
X25
)
)
)
)
)
↔
(
(
(
(
X33
→
(
X14
↔
(
(
¬
X5
)
→
X24
)
)
)
↔
(
¬
(
(
(
¬
(
X14
→
(
¬
X9
)
→
(
(
¬
X5
)
↔
X43
)
)
)
↔
(
¬
(
X9
↔
(
X5
→
X24
)
)
)
)
→
X24
→
(
X14
↔
X9
)
)
)
)
→
(
(
¬
X33
)
↔
(
(
¬
X9
)
→
(
¬
X7
)
→
X43
)
)
)
→
(
¬
(
X38
→
(
X10
↔
(
(
¬
X9
)
→
X18
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
X21
→
(
¬
X5
)
→
X24
)
↔
(
(
¬
(
X24
→
X21
→
(
X5
↔
X27
)
)
)
↔
(
(
¬
X24
)
→
(
X21
↔
(
X10
↔
X27
)
)
)
)
)
→
(
(
¬
(
X46
→
(
X44
↔
X49
)
)
)
→
(
¬
(
(
X44
→
(
(
¬
X10
)
↔
X27
)
)
→
X46
→
(
(
¬
X44
)
↔
X49
)
)
)
)
→
(
X46
↔
(
(
¬
X27
)
→
(
(
¬
X5
)
↔
X49
)
)
)
)
→
(
¬
(
(
(
¬
(
(
(
(
¬
(
X49
→
X47
→
X45
→
(
X43
↔
False
)
)
)
↔
(
¬
(
X49
↔
(
X47
→
(
(
¬
X18
)
↔
X43
)
)
)
)
)
↔
(
X47
→
(
X45
↔
False
)
)
)
→
(
¬
(
X43
→
X49
)
)
)
)
→
(
¬
(
(
¬
X45
)
→
(
¬
X43
)
→
X8
→
X47
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
X45
)
→
(
¬
X44
)
→
X49
)
)
→
(
¬
(
X47
↔
(
X46
→
X45
)
)
)
)
)
↔
(
X45
→
(
(
¬
X44
)
↔
X46
)
)
)
)
)
)
→
(
(
(
(
¬
(
X49
→
X45
→
(
X33
↔
False
)
)
)
↔
(
(
(
¬
X45
)
↔
(
(
¬
X33
)
→
(
X23
↔
X47
)
)
)
↔
(
X47
→
X45
→
X33
→
(
X2
↔
False
)
)
)
)
→
(
(
¬
X47
)
↔
(
X45
↔
(
(
¬
X36
)
↔
X49
)
)
)
→
(
¬
X36
)
→
(
¬
X34
)
→
X45
)
↔
(
¬
(
(
(
¬
(
X38
↔
(
X20
→
X9
→
X49
)
)
)
↔
(
(
¬
(
X26
→
X20
→
X38
)
)
→
(
(
(
(
¬
X39
)
→
X26
→
(
¬
X20
)
→
X49
)
→
(
X49
↔
(
X39
→
(
X6
↔
X20
)
)
)
)
↔
(
¬
(
X39
→
(
X26
↔
(
(
¬
X12
)
↔
X49
)
)
)
)
)
)
)
→
(
(
X47
↔
(
X38
↔
(
(
¬
X4
)
↔
X26
)
)
)
↔
(
¬
(
X38
↔
(
(
¬
X20
)
↔
X45
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
X48
↔
(
(
¬
X41
)
↔
(
(
¬
X22
)
↔
X43
)
)
)
↔
(
X49
→
X43
→
(
(
¬
X22
)
↔
X42
)
)
)
↔
(
¬
(
(
(
(
X42
→
X37
→
X35
)
↔
(
X41
↔
(
X37
↔
(
X35
→
X7
)
)
)
)
↔
(
X42
→
(
¬
X22
)
→
(
X7
↔
X37
)
)
)
→
X42
→
(
(
¬
X37
)
↔
(
X14
→
X41
)
)
)
)
)
→
(
¬
(
(
X43
↔
(
(
¬
X25
)
→
X19
→
X42
)
)
→
(
(
(
¬
(
(
(
X48
→
X43
→
X3
)
↔
(
¬
(
X48
→
X19
→
(
¬
X3
)
→
X25
)
)
)
→
(
X48
↔
(
X19
→
X43
)
)
)
)
↔
(
X25
→
(
X19
↔
False
)
)
)
↔
(
¬
(
X43
→
X3
)
)
)
)
)
)
↔
(
(
(
X49
→
(
X43
↔
(
X41
→
(
X29
↔
False
)
)
)
)
→
(
(
(
(
¬
(
X41
↔
X16
)
)
→
(
¬
(
(
¬
(
X49
→
X43
→
(
(
¬
X16
)
↔
X41
)
)
)
→
(
(
¬
X43
)
↔
(
(
¬
X16
)
→
(
X15
↔
X49
)
)
)
)
)
)
→
(
X49
↔
(
(
¬
X41
)
↔
(
(
¬
X13
)
↔
X43
)
)
)
)
↔
(
¬
(
X43
↔
(
(
¬
X16
)
↔
(
X0
→
X41
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
(
(
¬
(
(
¬
X28
)
↔
(
(
¬
X17
)
↔
(
(
¬
X11
)
→
X31
)
)
)
)
↔
(
X49
→
X17
→
X28
)
)
→
(
X31
↔
(
X28
→
X49
)
)
)
)
→
(
(
¬
X32
)
↔
(
X28
↔
(
(
¬
X11
)
→
X49
)
)
)
)
)
→
(
¬
(
(
X49
↔
(
X28
↔
X43
)
)
→
(
X30
↔
(
(
¬
X28
)
→
(
¬
X11
)
→
X43
)
)
)
)
)
→
(
(
(
¬
(
X49
→
(
X41
↔
(
X40
↔
False
)
)
)
)
↔
(
X41
↔
(
X40
→
(
X31
↔
(
X1
↔
False
)
)
)
)
)
↔
(
¬
(
X49
→
(
(
¬
X43
)
↔
(
X31
↔
X46
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
def27f...
and proposition id is
dbda10...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMMTXQR3PYwALcoNB6ZePRr4VbRKvCnmDks
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
(
(
(
X47
→
(
(
¬
X35
)
↔
(
(
¬
X32
)
→
X46
)
)
)
↔
(
X32
→
X4
→
False
)
)
↔
(
¬
(
(
¬
(
(
(
¬
X39
)
↔
(
(
¬
X35
)
↔
X47
)
)
→
(
(
¬
(
X46
→
(
(
¬
X39
)
↔
X48
)
)
)
↔
(
¬
(
X48
→
(
(
¬
X39
)
↔
(
(
¬
X35
)
→
X47
)
)
)
)
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X39
)
→
(
¬
X9
)
→
X48
)
)
)
)
)
)
↔
(
(
(
¬
X44
)
→
X48
)
↔
(
¬
(
(
(
(
¬
X47
)
→
(
¬
X46
)
→
X48
)
↔
(
¬
(
X33
→
X44
)
)
)
→
(
¬
(
X47
→
X33
→
X48
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X44
→
(
¬
X29
)
→
(
X28
↔
X48
)
)
)
→
(
¬
X25
)
)
→
(
¬
(
(
¬
(
X49
↔
(
X47
↔
(
X29
→
X48
)
)
)
)
→
X47
→
(
X29
↔
(
(
¬
X25
)
↔
X49
)
)
)
)
)
)
→
(
X48
↔
(
(
¬
X12
)
↔
X29
)
)
→
(
(
¬
(
(
X48
↔
(
X29
→
(
¬
X18
)
→
X44
)
)
→
(
¬
(
X29
→
(
X0
↔
X48
)
)
)
)
)
↔
(
(
X18
↔
(
X3
→
X1
→
False
)
)
→
(
¬
(
(
¬
X12
)
→
(
¬
X3
)
→
(
(
¬
X1
)
↔
X18
)
)
)
→
(
¬
(
X12
↔
(
(
¬
X1
)
↔
X44
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
X43
↔
(
(
¬
X42
)
→
(
X20
↔
X44
)
)
)
→
(
(
(
X43
↔
(
X20
↔
False
)
)
→
(
X43
↔
(
(
¬
X15
)
→
X8
→
X42
)
)
→
X20
→
(
(
¬
X7
)
↔
X8
)
)
↔
(
¬
(
X43
↔
(
X15
↔
X20
)
)
)
)
)
)
→
(
(
¬
(
X44
↔
(
X42
↔
(
X37
↔
False
)
)
)
)
↔
(
¬
(
(
¬
(
X48
→
X43
→
X44
)
)
→
(
(
X43
→
X48
)
↔
(
(
¬
(
X44
→
(
X43
↔
(
X42
→
X48
)
)
)
)
→
(
(
¬
X43
)
↔
(
(
¬
X42
)
→
(
(
¬
X37
)
↔
X48
)
)
)
)
)
)
)
)
)
)
→
(
(
(
X42
↔
(
(
¬
X16
)
→
(
X2
↔
X39
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X43
)
→
(
¬
X41
)
→
(
¬
X34
)
→
X44
)
)
→
(
¬
(
(
(
¬
(
(
¬
X16
)
→
X39
)
)
→
(
(
(
¬
(
(
¬
X16
)
→
(
¬
X10
)
→
X39
)
)
↔
(
¬
(
X42
→
X38
→
(
¬
X16
)
→
X39
)
)
)
↔
(
¬
(
(
¬
X39
)
↔
(
(
¬
X38
)
↔
X42
)
)
)
)
)
→
(
(
¬
(
(
X44
↔
(
X43
↔
(
X41
↔
(
X39
→
False
)
)
)
)
→
X42
→
(
X41
↔
(
(
¬
X39
)
→
X43
)
)
)
)
↔
(
¬
(
X44
↔
(
X39
↔
X42
)
)
)
)
)
)
)
)
)
↔
(
¬
(
X44
→
(
¬
X38
)
→
(
¬
X6
)
→
X41
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
X30
)
↔
(
X19
→
X43
)
)
→
(
¬
(
(
(
¬
X43
)
→
X40
→
X19
→
X45
)
→
(
¬
(
X49
↔
(
X45
↔
(
X19
↔
X40
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X24
→
(
¬
X17
)
→
X40
)
)
↔
(
X48
↔
(
X43
↔
(
X24
↔
X40
)
)
)
)
↔
(
(
X48
→
(
X36
↔
(
X27
→
X49
)
)
)
↔
(
(
X43
→
(
(
¬
X27
)
↔
X36
)
)
↔
(
(
¬
(
X48
→
X43
→
X40
→
X36
)
)
→
(
¬
(
(
¬
(
X48
→
(
X40
↔
(
X14
→
X27
)
)
)
)
→
(
X43
↔
(
(
¬
X40
)
↔
(
(
¬
X27
)
→
X48
)
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
X43
↔
(
X36
→
X31
)
)
)
↔
(
(
(
(
¬
(
X27
→
(
X22
↔
False
)
)
)
↔
(
(
(
X36
→
X27
→
(
(
¬
X22
)
↔
X31
)
)
↔
(
X43
↔
(
X31
↔
(
X22
→
X27
)
)
)
)
↔
(
X43
→
(
(
¬
X31
)
↔
X36
)
)
)
)
↔
(
X43
↔
(
(
¬
X22
)
→
X27
)
)
)
↔
(
X31
↔
(
(
¬
X27
)
↔
(
(
¬
X23
)
↔
X43
)
)
)
)
)
→
(
(
(
¬
X26
)
↔
(
X21
→
(
(
¬
X5
)
↔
X36
)
)
)
↔
(
¬
(
(
(
(
¬
(
X36
↔
(
X31
↔
(
X21
↔
X43
)
)
)
)
↔
(
¬
(
X43
→
(
¬
X21
)
→
(
¬
X11
)
→
X31
)
)
)
↔
(
¬
(
X43
↔
(
X21
→
X13
→
X31
)
)
)
)
→
(
X26
↔
(
(
¬
X21
)
→
X13
→
X31
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
¬
(
(
(
X5
→
X45
)
→
(
(
¬
X45
)
↔
(
X40
→
X35
→
X48
)
)
)
→
(
(
(
¬
X45
)
→
(
X35
↔
(
(
¬
X33
)
↔
X48
)
)
)
↔
(
X45
→
(
¬
X33
)
→
(
(
¬
X32
)
↔
X35
)
)
)
)
)
→
(
¬
(
X48
→
(
X32
↔
(
X29
↔
X45
)
)
)
)
)
)
→
(
(
¬
(
X48
→
(
X46
↔
(
(
¬
X35
)
↔
X43
)
)
)
)
↔
(
¬
(
(
X46
↔
(
(
¬
X40
)
→
X44
)
)
→
(
¬
(
(
¬
(
X46
↔
(
X44
↔
X35
)
)
)
→
(
¬
(
(
¬
(
X44
↔
(
X40
↔
(
(
¬
X35
)
↔
X46
)
)
)
)
→
(
¬
(
X44
↔
(
(
¬
X35
)
↔
(
(
¬
X17
)
→
X46
)
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
X42
↔
X22
)
)
↔
(
(
¬
(
(
¬
(
X42
→
X19
→
(
¬
X10
)
→
X22
)
)
→
(
X18
↔
False
)
)
)
→
(
¬
(
X18
→
(
X10
↔
X42
)
)
)
)
)
→
(
¬
(
(
¬
(
X18
→
False
)
)
→
(
(
¬
(
X27
↔
(
X22
→
(
(
¬
X19
)
↔
X48
)
)
)
)
↔
(
¬
(
(
¬
X46
)
↔
(
(
¬
X27
)
↔
X48
)
)
)
)
)
)
)
↔
(
(
¬
X46
)
↔
(
X42
→
X48
)
)
)
→
(
¬
(
(
¬
(
(
(
X49
↔
(
(
¬
X42
)
→
X30
→
X48
)
)
→
(
¬
(
(
(
X49
↔
(
(
¬
X45
)
↔
(
X39
→
X48
)
)
)
↔
(
¬
(
X49
↔
(
X48
→
X45
→
(
X42
↔
False
)
)
)
)
)
→
(
¬
X45
)
→
X49
)
)
)
→
(
X49
↔
(
X45
→
(
X42
↔
(
X9
→
False
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X36
)
→
(
(
¬
X34
)
↔
(
X12
→
X42
)
)
)
)
↔
(
(
(
¬
X46
)
→
(
(
¬
X42
)
↔
X49
)
)
↔
(
¬
(
(
(
¬
X42
)
↔
(
X36
↔
X46
)
)
→
(
(
X36
→
(
(
¬
X21
)
↔
X46
)
)
↔
(
(
(
¬
X21
)
→
(
X12
↔
X42
)
)
→
(
X36
↔
(
X21
↔
(
X12
→
X46
)
)
)
)
)
)
)
)
)
↔
(
X45
↔
(
(
¬
X36
)
→
X21
→
X46
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X48
)
→
(
¬
X46
)
→
X49
)
)
↔
(
(
¬
(
X49
↔
(
X48
→
(
X14
↔
False
)
)
)
)
↔
(
¬
(
(
¬
X46
)
→
X31
→
(
¬
X14
)
→
X48
)
)
)
)
→
X49
→
(
¬
X31
)
→
(
(
¬
X23
)
↔
X46
)
)
)
)
)
)
→
(
(
(
(
(
(
¬
X47
)
→
(
(
¬
X13
)
↔
(
(
¬
X8
)
→
X49
)
)
)
↔
(
(
(
(
(
X7
→
X8
)
→
(
¬
(
(
¬
X47
)
↔
(
(
¬
X28
)
↔
(
(
¬
X8
)
↔
X49
)
)
)
)
)
→
(
¬
(
X28
↔
(
X8
→
X49
)
)
)
)
→
X47
→
(
(
¬
X7
)
↔
X8
)
)
↔
(
(
¬
X28
)
↔
(
(
¬
X7
)
↔
X47
)
)
)
)
→
(
(
¬
(
X49
↔
(
X47
↔
(
X37
→
X28
)
)
)
)
↔
(
(
(
(
¬
(
X47
↔
(
X37
↔
(
X11
↔
False
)
)
)
)
↔
(
¬
(
X47
→
X37
→
(
(
¬
X0
)
↔
X26
)
)
)
)
↔
(
X26
↔
X20
)
)
↔
(
(
(
X47
→
(
X37
↔
(
X26
↔
(
X1
↔
False
)
)
)
)
↔
X25
)
↔
(
X47
→
X28
→
X26
→
X25
)
)
)
)
)
→
(
X47
↔
(
(
¬
X24
)
→
(
X16
↔
X49
)
)
)
→
(
(
(
¬
X43
)
→
(
X28
↔
X47
)
)
→
(
X47
↔
(
(
¬
X28
)
↔
(
X24
↔
X49
)
)
)
)
→
(
(
¬
(
X6
↔
X28
)
)
↔
(
(
X49
↔
(
X6
→
(
X4
↔
False
)
)
)
↔
(
X28
→
(
(
¬
X6
)
↔
(
X4
↔
X43
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
X49
↔
(
X48
→
X41
)
)
↔
(
¬
(
(
¬
(
(
X2
→
False
)
→
(
X41
↔
(
(
¬
X15
)
↔
X48
)
)
→
(
¬
X2
)
→
X49
)
)
→
(
¬
(
X49
↔
(
X48
↔
X41
)
)
)
)
)
)
→
(
(
¬
X41
)
↔
(
(
¬
X2
)
→
X48
)
)
)
)
↔
(
X49
↔
(
X48
→
(
X2
↔
X38
)
)
)
)
→
(
¬
(
(
(
X47
↔
(
X38
↔
(
X15
↔
X41
)
)
)
↔
(
(
¬
X38
)
↔
(
(
¬
X28
)
→
(
(
¬
X15
)
↔
X47
)
)
)
)
→
X28
→
X15
→
(
(
¬
X3
)
↔
X38
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
3e86d7...
and proposition id is
bf2526...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TML7StfMLt6Zq5PjvYy9STw2wpffTc1NQ4Y
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
∀X51 :
prop
,
(
(
(
(
(
(
(
¬
(
(
¬
X36
)
→
(
¬
X25
)
→
X49
)
)
→
X36
→
(
¬
X25
)
→
(
X24
↔
X50
)
)
→
(
X47
↔
(
X36
↔
(
(
¬
X13
)
→
X49
)
)
)
→
X47
→
X25
→
X5
)
↔
(
(
¬
(
(
(
(
¬
(
X50
↔
(
X49
→
(
(
¬
X36
)
↔
X48
)
)
)
)
→
(
¬
(
X50
→
(
¬
X3
)
→
X49
)
)
)
↔
(
X48
→
(
X36
↔
(
X0
→
False
)
)
)
)
→
(
¬
(
X48
→
X36
→
(
¬
X27
)
→
X49
)
)
)
)
↔
(
¬
(
X50
→
X48
→
X36
→
(
X34
↔
False
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
X48
)
↔
(
X43
↔
(
(
¬
X33
)
→
X50
)
)
)
→
(
X50
↔
(
(
¬
X47
)
→
(
¬
X33
)
→
X48
)
)
)
→
(
(
(
X48
↔
(
X42
→
(
¬
X9
)
→
X50
)
)
↔
(
¬
(
X50
→
(
X48
↔
(
X43
↔
(
X42
→
False
)
)
)
)
)
)
↔
(
(
¬
(
X50
→
(
(
¬
X20
)
↔
X48
)
)
)
↔
(
(
¬
(
X48
↔
(
(
¬
X20
)
↔
X43
)
)
)
↔
(
(
X50
↔
(
X43
→
(
(
¬
X7
)
↔
X20
)
)
)
→
(
¬
(
(
¬
X48
)
↔
(
(
¬
X43
)
↔
(
(
¬
X7
)
→
X50
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
X45
)
↔
(
(
¬
X40
)
↔
(
(
¬
X11
)
→
X48
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X45
)
↔
(
(
¬
X40
)
↔
X46
)
)
)
↔
(
(
¬
(
X46
↔
X50
)
)
↔
(
(
¬
(
X48
→
X46
→
X45
→
X40
)
)
↔
(
(
¬
X46
)
↔
(
(
¬
X45
)
→
(
¬
X40
)
→
X50
)
)
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X46
)
↔
(
(
¬
X45
)
→
X48
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
X47
)
→
(
¬
X46
)
→
X48
)
)
)
↔
(
(
¬
(
(
¬
X48
)
→
X46
→
X39
→
X50
)
)
↔
(
(
(
¬
X39
)
→
(
X35
↔
X46
)
)
↔
(
(
X50
→
(
X35
↔
X15
)
)
↔
(
¬
(
X50
→
(
X39
↔
X46
)
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X47
↔
(
X38
→
(
¬
X31
)
→
X32
)
)
)
↔
(
(
¬
(
(
¬
X38
)
↔
(
X32
→
(
¬
X22
)
→
X46
)
)
)
↔
(
X46
→
(
X38
↔
(
X32
↔
(
X31
↔
False
)
)
)
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X16
)
↔
X21
)
)
)
→
(
(
(
¬
(
(
(
¬
X37
)
→
X46
)
→
(
(
¬
(
(
X46
↔
(
(
¬
X18
)
→
(
X1
↔
X47
)
)
)
→
(
¬
(
X47
→
(
X38
↔
(
(
¬
X1
)
↔
X46
)
)
)
)
)
)
↔
(
¬
(
X47
↔
(
X38
→
(
X37
↔
X18
)
)
)
)
)
)
)
→
(
¬
(
(
¬
X38
)
↔
(
X37
↔
(
(
¬
X21
)
↔
X46
)
)
)
)
→
(
X46
↔
(
X37
↔
(
(
¬
X19
)
→
X38
)
)
)
)
↔
(
¬
(
(
(
¬
X21
)
↔
(
X18
↔
(
X16
→
X38
)
)
)
→
X38
→
(
(
¬
X16
)
↔
X29
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
X47
)
↔
(
(
¬
X46
)
↔
(
X2
→
X51
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X4
)
→
(
X2
↔
X47
)
)
)
)
)
↔
(
X51
→
(
¬
X46
)
→
(
X37
↔
X47
)
)
)
→
(
(
(
(
¬
(
(
¬
X30
)
→
(
¬
X23
)
→
X47
)
)
↔
(
¬
(
X47
→
(
(
¬
X30
)
↔
(
X28
↔
X44
)
)
)
)
)
↔
(
(
X51
→
(
X17
↔
(
X6
↔
False
)
)
)
↔
(
(
X47
→
(
(
¬
X30
)
↔
(
(
¬
X17
)
→
X51
)
)
)
↔
(
¬
(
X51
→
X49
→
X30
→
False
)
)
)
)
)
↔
(
(
(
(
¬
(
X51
↔
(
X44
↔
(
X26
↔
X41
)
)
)
)
→
(
X51
↔
(
X46
→
X41
→
(
X8
↔
False
)
)
)
→
(
(
¬
X46
)
↔
(
X44
↔
(
X8
↔
X51
)
)
)
)
→
(
¬
(
X51
↔
(
X44
→
(
(
¬
X12
)
↔
X46
)
)
)
)
)
↔
(
(
(
¬
X46
)
↔
(
(
¬
X14
)
→
X47
)
)
↔
(
(
¬
(
X46
→
(
X41
↔
(
(
¬
X10
)
→
X26
)
)
)
)
→
(
¬
(
X46
→
(
X26
↔
(
X10
↔
X14
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
(
¬
(
(
(
(
(
¬
(
X45
→
(
X37
↔
(
X26
↔
(
X6
→
False
)
)
)
)
)
↔
(
X45
→
(
(
¬
X37
)
↔
(
(
¬
X26
)
↔
X43
)
)
)
)
↔
(
X45
→
(
X26
↔
X6
)
)
)
↔
(
¬
(
X37
↔
(
(
¬
X26
)
↔
(
X6
↔
X41
)
)
)
)
)
→
(
(
X43
→
(
¬
X36
)
→
X41
)
↔
(
(
¬
X41
)
↔
(
X36
↔
(
X1
↔
X45
)
)
)
)
)
)
↔
(
(
(
¬
(
X18
→
(
¬
X10
)
→
X36
)
)
→
(
¬
(
(
X36
↔
(
(
¬
X10
)
↔
X18
)
)
→
(
¬
(
X37
→
X18
→
X41
)
)
)
)
)
↔
(
(
¬
(
X41
→
(
X37
↔
(
X0
↔
False
)
)
)
)
↔
(
(
(
X41
→
(
X18
↔
(
X0
↔
X5
)
)
)
↔
(
¬
(
X41
→
(
X37
↔
(
X5
→
X18
)
)
)
)
)
↔
(
(
¬
X37
)
→
X18
→
X41
)
)
)
)
)
→
(
(
(
¬
(
X43
↔
(
(
¬
X30
)
↔
(
(
¬
X17
)
→
X46
)
)
)
)
↔
(
¬
(
X46
→
X43
→
X41
→
X30
)
)
)
↔
(
(
(
X48
↔
(
X45
→
(
(
¬
X24
)
↔
X41
)
)
)
→
(
(
¬
X45
)
↔
(
X41
↔
(
X24
↔
X46
)
)
)
)
↔
(
(
X30
↔
X46
)
↔
(
(
¬
(
(
¬
X30
)
→
X45
)
)
↔
(
X45
→
(
¬
X42
)
→
(
X7
↔
X48
)
)
)
)
)
)
)
)
→
(
(
(
(
(
X35
→
X28
→
X3
→
X45
)
↔
(
(
¬
(
X44
↔
(
X19
↔
X28
)
)
)
→
(
X35
↔
(
X28
→
(
(
¬
X3
)
↔
X19
)
)
)
)
)
↔
(
¬
(
X44
→
X43
→
X16
)
)
)
↔
(
¬
(
X45
↔
(
X43
↔
(
(
¬
X15
)
↔
X44
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X43
→
X41
→
(
X20
↔
X44
)
)
)
↔
(
(
X45
→
(
X44
↔
(
X20
↔
(
X9
↔
False
)
)
)
)
→
(
¬
(
X45
↔
(
X44
→
(
X41
↔
(
X20
→
False
)
)
)
)
)
)
)
→
(
(
(
¬
X21
)
↔
(
(
¬
X12
)
↔
(
X2
→
X44
)
)
)
↔
(
(
¬
(
(
X44
→
(
(
¬
X27
)
↔
(
(
¬
X12
)
↔
X45
)
)
)
→
X27
→
(
X21
↔
(
X12
↔
False
)
)
)
)
↔
(
X27
→
X21
→
(
X12
↔
False
)
)
)
)
→
(
¬
X43
)
→
(
¬
X27
)
→
(
X21
↔
X44
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
X47
)
→
(
X33
↔
(
(
¬
X13
)
↔
X48
)
)
)
→
(
(
¬
(
(
¬
X33
)
→
(
¬
X31
)
→
(
(
¬
X22
)
↔
X47
)
)
)
↔
(
(
(
¬
(
X48
↔
(
X22
→
(
(
¬
X8
)
↔
X33
)
)
)
)
→
(
¬
(
(
¬
X33
)
→
(
X31
↔
X48
)
)
)
)
↔
(
¬
(
(
¬
X31
)
↔
X48
)
)
)
)
)
→
(
¬
(
(
(
X44
↔
(
(
¬
X43
)
→
(
(
¬
X33
)
↔
X47
)
)
)
↔
(
¬
(
(
¬
(
X47
↔
(
(
¬
X44
)
→
(
X43
↔
X48
)
)
)
)
→
X47
→
(
¬
X14
)
→
X40
)
)
)
→
(
X48
↔
(
X44
→
(
¬
X4
)
→
X33
)
)
)
)
)
)
→
(
(
(
(
(
¬
X44
)
↔
(
(
¬
X39
)
→
(
X29
↔
X50
)
)
)
→
(
(
X44
↔
(
(
¬
X38
)
↔
(
X25
→
X50
)
)
)
↔
(
X44
→
X39
→
(
(
¬
X38
)
↔
X50
)
)
)
)
→
(
¬
(
(
X49
→
(
¬
X39
)
→
(
X34
↔
X44
)
)
→
(
X49
↔
(
X44
→
X39
→
X50
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X47
↔
(
(
¬
X38
)
↔
(
(
¬
X32
)
↔
X50
)
)
)
)
↔
(
X50
→
(
¬
X23
)
→
X48
)
)
→
(
¬
(
X50
→
X32
→
X23
)
)
)
)
→
(
¬
(
(
¬
(
X49
↔
(
X47
→
(
X11
↔
X51
)
)
)
)
→
X51
→
X11
→
X50
)
)
→
X49
→
(
X47
↔
(
X11
↔
X48
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
ec90fb...
and proposition id is
9bbe0a...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMYfhLfv7rNRjxmsbuoW5XsC4VJf5a77qVY
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
∀X51 :
prop
,
(
¬
∀X52 :
prop
,
(
(
(
(
¬
(
(
¬
(
X51
→
(
(
¬
X49
)
↔
(
(
¬
X40
)
→
X50
)
)
)
)
→
(
(
(
¬
(
X49
↔
(
X48
↔
(
X2
→
X34
)
)
)
)
↔
(
(
(
X49
↔
(
(
¬
X34
)
→
(
(
¬
X2
)
↔
X44
)
)
)
↔
(
¬
(
X51
→
(
X49
↔
(
X2
↔
X34
)
)
)
)
)
↔
(
X51
↔
(
(
¬
X34
)
↔
X49
)
)
)
)
↔
(
(
X50
↔
(
X29
↔
(
X20
→
X34
)
)
)
↔
(
(
(
¬
(
(
X50
↔
(
X48
→
(
X41
↔
False
)
)
)
→
(
(
¬
(
X34
↔
(
X20
→
False
)
)
)
→
(
¬
X48
)
→
(
(
¬
X41
)
↔
X50
)
)
→
(
(
¬
X48
)
↔
(
(
¬
X41
)
↔
X50
)
)
)
)
↔
(
(
¬
(
(
¬
X8
)
↔
X44
)
)
↔
(
(
X48
↔
(
(
¬
X44
)
↔
(
X41
→
X50
)
)
)
↔
(
¬
(
X50
→
(
X44
↔
(
(
¬
X8
)
↔
X41
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
X48
)
→
(
(
¬
X44
)
↔
(
X28
→
X50
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
X47
↔
(
X36
→
(
(
¬
X10
)
↔
X38
)
)
)
→
(
(
X47
↔
(
X36
↔
(
X22
↔
X38
)
)
)
↔
(
¬
(
X16
→
X22
)
)
)
)
)
→
X47
→
(
(
¬
X45
)
↔
(
X31
→
X51
)
)
)
)
→
(
(
(
(
¬
X38
)
↔
(
X26
↔
X47
)
)
→
(
(
X47
→
X25
→
X48
)
↔
(
(
¬
(
X48
→
X47
→
X36
→
X49
)
)
→
(
(
¬
(
X49
↔
(
X48
↔
(
(
¬
X25
)
↔
X26
)
)
)
)
↔
(
¬
(
X47
→
(
(
¬
X26
)
↔
X48
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
X38
→
(
X35
↔
X49
)
)
↔
(
(
¬
(
X38
↔
(
X36
↔
(
X35
↔
X7
)
)
)
)
→
X49
→
X36
→
(
¬
X35
)
→
X38
)
)
↔
(
X36
→
(
X35
↔
X49
)
)
)
→
(
¬
(
(
¬
X36
)
→
X35
→
(
(
¬
X27
)
↔
X49
)
)
)
)
)
↔
(
(
(
¬
(
X36
→
(
¬
X35
)
→
(
¬
X6
)
→
X49
)
)
↔
(
X36
→
X35
→
X6
→
X3
)
)
→
(
¬
(
X35
↔
X3
)
)
)
)
↔
(
(
¬
(
X38
→
X27
→
X13
→
X0
→
False
)
)
→
(
X38
↔
(
X35
↔
(
X27
→
(
X13
↔
False
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
(
(
(
¬
X43
)
↔
(
(
¬
X21
)
↔
(
(
¬
X1
)
↔
X46
)
)
)
↔
(
X46
→
(
X43
↔
(
(
¬
X1
)
→
X49
)
)
)
)
↔
(
(
¬
(
X49
↔
(
X43
↔
X46
)
)
)
→
X46
→
X24
→
X12
→
X49
)
)
→
(
X24
↔
(
(
¬
X21
)
→
(
¬
X12
)
→
X23
)
)
)
)
↔
(
X43
→
(
X23
↔
X49
)
)
)
↔
(
¬
(
(
¬
(
(
X49
↔
(
(
¬
X21
)
→
X9
→
X33
)
)
→
(
(
¬
X43
)
↔
(
X33
→
X49
)
)
)
)
→
(
¬
(
(
¬
X33
)
→
(
¬
X17
)
→
X43
)
)
)
)
)
→
(
¬
(
X51
→
(
X23
↔
X46
)
)
)
→
(
(
(
¬
X37
)
→
(
¬
X23
)
→
X46
)
↔
(
¬
(
X49
↔
(
(
¬
X46
)
→
X37
→
X51
)
)
)
)
)
↔
(
(
(
(
X52
→
X39
→
(
X15
↔
X42
)
)
↔
(
¬
(
(
¬
X42
)
→
(
¬
X39
)
→
(
X4
↔
X52
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
¬
X15
)
→
(
X5
↔
X39
)
)
)
↔
(
¬
(
X43
→
X30
→
X5
→
X11
)
)
)
↔
(
(
¬
(
X43
↔
(
(
¬
X15
)
→
X51
)
)
)
↔
(
X50
→
(
X43
↔
(
(
¬
X15
)
→
X30
)
)
)
)
)
→
(
¬
(
(
¬
X51
)
↔
(
X50
↔
(
(
¬
X43
)
↔
X52
)
)
)
)
→
X52
→
(
X50
↔
(
X43
→
X18
)
)
)
)
)
→
(
(
X51
→
(
(
¬
X43
)
↔
(
(
¬
X19
)
→
X52
)
)
)
↔
(
¬
(
(
¬
(
X46
→
(
(
¬
X32
)
↔
(
X14
↔
X43
)
)
)
)
→
(
¬
(
X46
↔
(
X32
→
(
(
¬
X19
)
↔
X43
)
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
¬
(
(
¬
(
(
X48
→
X40
→
(
(
¬
X2
)
↔
X39
)
)
→
(
(
¬
X48
)
↔
(
(
¬
X40
)
→
(
¬
X39
)
→
X52
)
)
)
)
→
X48
→
(
(
¬
X39
)
↔
X52
)
)
)
→
(
(
¬
(
X39
↔
(
(
¬
X20
)
↔
(
X2
→
X40
)
)
)
)
↔
(
X40
↔
(
X39
→
X0
→
False
)
)
)
)
)
→
(
(
(
(
¬
X48
)
↔
(
(
¬
X22
)
↔
(
X4
↔
X52
)
)
)
→
(
(
¬
(
X48
↔
(
X40
→
(
(
¬
X37
)
↔
X52
)
)
)
)
↔
(
¬
(
X52
→
(
(
¬
X40
)
↔
(
(
¬
X4
)
→
X48
)
)
)
)
)
)
↔
(
¬
(
X48
↔
(
(
¬
X37
)
→
(
¬
X16
)
→
X40
)
)
)
)
)
→
(
¬
(
(
(
(
¬
X39
)
→
(
X13
↔
(
X11
↔
X52
)
)
)
↔
(
(
(
¬
(
X52
→
X39
→
(
¬
X13
)
→
X26
)
)
↔
(
¬
(
X52
↔
(
(
¬
X39
)
↔
(
(
¬
X26
)
↔
X49
)
)
)
)
)
→
(
X52
↔
(
X39
↔
(
(
¬
X21
)
→
X49
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
(
(
¬
(
(
¬
X47
)
↔
(
(
¬
X34
)
↔
(
(
¬
X17
)
→
X52
)
)
)
)
→
(
(
¬
X47
)
↔
(
(
¬
X37
)
↔
(
(
¬
X17
)
↔
X52
)
)
)
)
)
→
(
¬
(
X52
↔
(
X37
↔
X47
)
)
)
)
→
(
¬
(
(
¬
X47
)
→
X37
→
(
¬
X34
)
→
X51
)
)
)
)
→
(
(
X47
↔
(
(
¬
X34
)
→
(
X21
↔
X51
)
)
)
↔
(
¬
(
X51
↔
(
X47
↔
(
X34
↔
X19
)
)
)
)
)
→
(
¬
(
X19
→
(
(
¬
X9
)
↔
X26
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
(
¬
X28
)
↔
X52
)
→
(
¬
(
X48
↔
(
(
¬
X45
)
→
(
X32
↔
X46
)
)
)
)
)
)
↔
(
X48
→
X46
→
(
X32
↔
False
)
)
)
→
(
(
(
(
¬
(
X52
→
X46
→
(
¬
X38
)
→
X48
)
)
↔
(
X52
→
X46
→
X43
→
X48
)
)
↔
(
X52
→
(
X46
↔
X43
)
)
)
↔
(
¬
(
X48
↔
(
X46
↔
X52
)
)
)
)
)
→
(
(
¬
(
X52
→
(
(
¬
X45
)
↔
(
(
¬
X35
)
→
X46
)
)
)
)
↔
(
(
(
¬
(
X46
→
X45
→
(
X31
↔
X18
)
)
)
→
X46
→
(
(
¬
X45
)
↔
(
X31
→
X52
)
)
)
→
(
¬
(
(
(
(
¬
(
X30
→
X12
→
X31
)
)
→
(
¬
X25
)
→
X12
→
X45
)
↔
(
X23
↔
X12
)
)
→
(
¬
(
(
¬
X30
)
↔
(
X14
→
(
(
¬
X12
)
↔
X52
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
¬
(
(
(
X50
→
X29
→
X33
)
→
(
¬
X50
)
→
(
(
¬
X46
)
↔
X52
)
)
→
(
¬
(
(
¬
X46
)
↔
(
X29
↔
X50
)
)
)
)
)
→
(
¬
(
X46
→
(
¬
X24
)
→
X50
)
)
)
↔
(
(
X33
↔
(
X29
↔
X50
)
)
↔
(
¬
(
X44
→
(
X41
↔
(
X8
↔
False
)
)
)
)
)
)
→
(
(
¬
(
X42
→
X7
→
(
X3
↔
X46
)
)
)
↔
(
(
(
(
¬
X7
)
↔
(
(
¬
X1
)
→
X33
)
)
→
X42
→
(
(
¬
X7
)
↔
(
(
¬
X3
)
→
X33
)
)
)
↔
(
¬
(
(
¬
X7
)
→
(
X1
↔
X33
)
)
)
)
)
→
(
¬
X44
)
→
X3
→
X46
)
)
↔
(
¬
(
(
(
(
¬
(
X52
↔
(
X49
↔
(
(
¬
X5
)
→
X46
)
)
)
)
→
(
¬
(
X49
→
(
(
¬
X6
)
↔
(
(
¬
X5
)
→
X52
)
)
)
)
)
↔
(
(
¬
(
X42
→
(
X15
↔
X5
)
)
)
↔
(
(
¬
(
X42
↔
(
X15
↔
(
X5
↔
X45
)
)
)
)
→
(
¬
(
(
¬
X45
)
→
(
¬
X42
)
→
X15
→
X52
)
)
)
)
)
→
(
¬
(
X52
↔
(
X45
→
X36
→
X27
→
False
)
)
)
→
(
¬
(
(
¬
(
X46
↔
X27
)
)
→
X45
→
(
X36
↔
(
X27
↔
X10
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
83bd63...
and proposition id is
5c948b...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMFmtHLvFNJy6tcNWnKksA59CC8pFooN8Ws
)
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
¬
(
(
(
(
(
(
¬
X46
)
→
X34
→
(
¬
X23
)
→
X49
)
↔
(
¬
(
(
¬
(
X49
↔
(
X34
→
(
X19
↔
False
)
)
)
)
→
(
¬
(
(
X49
→
(
X23
↔
(
(
¬
X19
)
↔
X45
)
)
)
→
(
(
¬
(
X49
→
X34
→
(
X1
↔
X45
)
)
)
↔
(
X34
→
(
X23
↔
X1
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X45
↔
(
X18
→
(
X2
↔
X4
)
)
)
)
↔
(
¬
(
X46
↔
(
X18
→
(
¬
X4
)
→
X47
)
)
)
)
↔
(
¬
(
X35
→
(
¬
X18
)
→
X4
→
X47
)
)
)
↔
(
(
(
(
(
¬
X46
)
↔
(
(
¬
X35
)
→
(
(
¬
X9
)
↔
X47
)
)
)
↔
(
¬
(
X49
↔
(
X46
↔
X47
)
)
)
)
→
(
X49
↔
(
X47
→
X46
)
)
)
→
(
¬
(
X47
↔
(
X46
↔
(
X9
→
(
X3
↔
False
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X47
→
X30
→
(
X12
↔
X42
)
)
)
→
(
(
¬
(
(
¬
X46
)
↔
(
X42
↔
X49
)
)
)
↔
(
X48
↔
(
(
¬
X30
)
→
X8
→
X49
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
X44
↔
(
X37
→
(
X36
↔
X31
)
)
)
→
(
(
¬
(
(
¬
X37
)
↔
(
X31
↔
X40
)
)
)
↔
(
¬
(
X37
→
X31
→
(
(
¬
X6
)
↔
X36
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X36
↔
(
X27
→
X26
)
)
)
↔
(
¬
(
(
¬
(
X36
↔
(
(
¬
X26
)
↔
(
(
¬
X5
)
→
X31
)
)
)
)
→
X31
→
(
X26
↔
(
X5
↔
X27
)
)
)
)
)
→
(
X31
→
(
(
¬
X25
)
↔
(
(
¬
X5
)
↔
X44
)
)
)
→
(
¬
X26
)
→
X5
→
X44
)
)
)
↔
(
(
(
(
¬
(
X48
→
(
X44
↔
(
X39
→
(
X36
↔
False
)
)
)
)
)
→
(
X44
↔
(
(
¬
X0
)
↔
X16
)
)
→
(
¬
(
(
¬
X39
)
↔
(
(
¬
X16
)
↔
(
(
¬
X0
)
↔
X48
)
)
)
)
)
↔
(
X48
↔
(
(
¬
X39
)
→
X44
)
)
)
→
(
¬
(
X40
↔
(
X39
↔
(
X36
↔
X48
)
)
)
)
→
(
¬
(
(
¬
(
X39
↔
(
(
¬
X32
)
↔
(
X14
↔
X36
)
)
)
)
→
X40
→
(
X32
↔
(
X14
→
X29
)
)
)
)
)
)
↔
(
(
(
(
(
X43
→
X39
→
(
X22
↔
(
X20
↔
False
)
)
)
→
(
(
(
¬
X43
)
↔
(
X39
↔
X47
)
)
↔
(
(
¬
(
X39
↔
(
X24
→
X22
→
X47
)
)
)
→
(
¬
(
X47
→
(
X43
↔
(
X24
↔
False
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
X11
→
X21
)
↔
(
(
(
X47
→
(
X44
↔
X20
)
)
→
(
¬
(
X39
↔
(
(
¬
X20
)
↔
(
(
¬
X11
)
→
X47
)
)
)
)
)
→
X39
→
(
X20
↔
(
X11
↔
False
)
)
)
)
↔
(
X39
↔
(
X20
↔
X21
)
)
)
→
(
¬
(
X44
→
(
¬
X39
)
→
X47
)
)
)
→
X47
→
(
X39
↔
X44
)
)
↔
(
¬
(
(
¬
X39
)
↔
(
(
¬
X21
)
↔
X44
)
)
)
)
)
→
(
(
¬
(
X49
→
X48
→
X39
)
)
↔
(
¬
(
(
(
¬
(
X39
→
X13
→
X33
)
)
↔
(
¬
(
X48
↔
(
(
¬
X39
)
→
(
¬
X33
)
→
X49
)
)
)
)
→
(
¬
(
X48
→
(
(
¬
X33
)
↔
(
(
¬
X17
)
→
X49
)
)
)
)
)
)
)
→
(
X33
↔
X39
)
→
(
¬
(
(
¬
X44
)
↔
(
X33
→
X21
→
X49
)
)
)
)
↔
(
¬
(
(
¬
(
X44
↔
(
X43
→
(
X10
↔
X15
)
)
)
)
→
(
¬
(
(
(
X28
→
X10
→
X15
)
↔
(
(
(
X47
→
X28
)
↔
(
¬
(
X47
→
(
¬
X15
)
→
(
(
¬
X7
)
↔
X38
)
)
)
)
↔
(
X47
→
(
¬
X28
)
→
X38
)
)
)
→
X41
→
(
X28
↔
(
X15
→
X47
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
(
(
¬
(
X30
→
(
¬
X0
)
→
X43
)
)
↔
(
(
(
X31
↔
(
X30
↔
(
X24
→
False
)
)
)
↔
(
(
¬
(
(
(
¬
(
X43
→
(
(
¬
X21
)
↔
X24
)
)
)
↔
(
(
¬
X31
)
↔
(
X21
↔
(
(
¬
X0
)
↔
X43
)
)
)
)
→
(
¬
(
X30
→
(
X21
↔
(
(
¬
X0
)
↔
X17
)
)
)
)
→
(
¬
(
(
¬
X17
)
→
(
(
¬
X0
)
↔
X43
)
)
)
)
)
→
(
X31
↔
(
(
¬
X30
)
↔
X43
)
)
)
)
↔
(
¬
(
X21
→
X24
)
)
)
)
→
(
¬
(
X30
↔
(
(
¬
X29
)
→
X0
→
X31
)
)
)
)
)
→
(
(
(
(
X48
↔
(
X31
↔
(
X23
→
X15
)
)
)
↔
(
X23
→
(
(
¬
X15
)
↔
(
X2
↔
X31
)
)
)
)
↔
(
¬
(
X49
→
(
X23
↔
(
(
¬
X2
)
↔
X43
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X43
)
↔
(
X31
→
X49
)
)
)
↔
(
¬
(
(
¬
(
X49
↔
(
X43
→
X30
→
X28
)
)
)
→
(
¬
(
X49
↔
(
X30
↔
X28
)
)
)
→
X31
→
X30
→
(
X27
↔
False
)
)
)
)
↔
(
¬
(
(
(
¬
X26
)
↔
(
(
¬
X6
)
↔
X43
)
)
→
(
(
¬
(
X44
↔
(
X28
→
(
X26
↔
(
X6
↔
False
)
)
)
)
)
↔
(
X44
↔
(
X30
→
(
X6
↔
X28
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X16
)
↔
(
(
¬
X5
)
↔
X40
)
)
)
→
X42
→
X35
)
↔
(
¬
(
(
X49
→
(
X47
↔
(
(
¬
X3
)
↔
X40
)
)
)
→
(
¬
X40
)
→
(
X31
↔
(
(
¬
X3
)
↔
X47
)
)
)
)
)
→
(
(
¬
(
(
X40
→
False
)
→
(
¬
(
(
¬
(
X40
→
X35
→
(
X20
↔
X18
)
)
)
→
(
(
¬
(
X20
↔
X35
)
)
↔
(
¬
(
X48
↔
(
X40
→
X20
→
X9
→
False
)
)
)
)
)
)
)
)
↔
(
(
¬
(
X48
↔
(
X47
↔
False
)
)
)
↔
(
¬
(
(
¬
X46
)
↔
(
(
¬
X40
)
→
(
(
¬
X9
)
↔
X48
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
(
X49
→
X46
→
(
X41
↔
(
X38
↔
False
)
)
)
↔
(
¬
(
(
(
¬
(
X46
→
(
¬
X41
)
→
X49
)
)
↔
(
(
X49
↔
(
(
¬
X36
)
→
X46
)
)
↔
(
¬
(
(
¬
(
X36
↔
(
(
¬
X19
)
↔
X41
)
)
)
→
(
X46
↔
(
X36
↔
(
(
¬
X19
)
↔
X49
)
)
)
)
)
)
)
→
X46
→
X19
→
X49
)
)
)
→
(
(
¬
(
(
X41
→
(
X32
↔
X46
)
)
→
(
X41
↔
(
(
¬
X32
)
→
X46
)
)
→
(
¬
(
X49
↔
(
X41
→
(
(
¬
X13
)
↔
X46
)
)
)
)
)
)
↔
(
(
(
¬
X45
)
→
(
X41
↔
(
(
¬
X33
)
↔
X46
)
)
)
↔
(
¬
(
(
X45
→
(
¬
X33
)
→
X46
)
→
(
(
¬
(
X45
↔
(
(
¬
X33
)
↔
(
(
¬
X13
)
↔
X46
)
)
)
)
↔
(
¬
(
(
¬
X33
)
→
X11
→
X46
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
X34
→
X25
→
X49
)
)
↔
(
(
(
¬
X36
)
→
(
¬
X34
)
→
X25
→
X49
)
↔
(
X49
→
X41
→
(
X34
↔
(
X25
↔
False
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
X44
→
X22
→
X41
)
)
↔
(
X45
→
(
X7
↔
False
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X37
→
(
X12
↔
X41
)
)
)
→
(
¬
(
X44
↔
(
(
¬
X41
)
↔
(
(
¬
X12
)
→
X45
)
)
)
)
)
)
→
(
¬
(
(
¬
X41
)
↔
(
X37
→
(
(
¬
X10
)
↔
X45
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X37
)
→
(
(
¬
X1
)
↔
X49
)
)
)
→
(
¬
(
(
¬
(
X49
↔
(
X44
→
(
¬
X1
)
→
X37
)
)
)
→
(
¬
X41
)
→
(
X37
↔
(
(
¬
X1
)
↔
X44
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X14
→
(
(
¬
X12
)
↔
(
(
¬
X8
)
↔
X39
)
)
)
)
→
(
(
¬
(
(
¬
(
(
X39
→
X22
→
(
X14
↔
X12
)
)
→
X39
→
(
X37
↔
(
(
¬
X12
)
→
X14
)
)
)
)
→
X39
→
(
X37
↔
(
X12
↔
False
)
)
)
)
↔
(
X37
→
X22
→
X14
→
X4
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
e7d55e...
and proposition id is
7ffa5c...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMRMQxTG6e2Ge8gyBN2HQsEnXXdbBdSnuSn
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
(
(
(
¬
(
X49
→
X15
→
X46
)
)
→
(
¬
(
X49
→
X31
→
(
X1
↔
X46
)
)
)
)
→
(
(
X31
↔
(
(
¬
X29
)
→
X46
)
)
→
X48
→
(
(
¬
X46
)
↔
(
X31
↔
X49
)
)
)
→
(
¬
(
X49
→
X46
→
X29
)
)
)
↔
(
¬
(
X49
→
(
X48
↔
(
X46
→
X25
)
)
)
)
)
↔
(
(
(
(
X41
↔
(
X37
↔
X43
)
)
↔
(
¬
(
(
¬
(
X41
→
(
X24
↔
(
X13
↔
False
)
)
)
)
→
(
X37
↔
(
(
¬
X24
)
↔
(
X13
→
X43
)
)
)
)
)
)
↔
(
¬
(
X48
→
X41
→
X37
→
(
X24
↔
False
)
)
)
)
↔
(
(
X49
↔
(
X48
→
X46
)
)
→
(
X49
↔
(
(
¬
X43
)
↔
X48
)
)
→
(
(
¬
(
X49
→
(
X46
↔
(
X43
↔
False
)
)
)
)
↔
(
(
X48
↔
(
X46
↔
(
X3
↔
False
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X43
)
→
X49
)
)
→
(
(
¬
(
X46
→
(
X43
↔
(
X41
↔
False
)
)
)
)
↔
(
¬
(
X49
→
X48
→
(
X46
↔
(
X43
→
False
)
)
)
)
)
)
)
↔
(
X48
→
(
X46
↔
(
X41
↔
False
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X49
→
(
¬
X27
)
→
(
X20
↔
X37
)
)
)
→
(
(
¬
X32
)
→
X16
→
(
X10
↔
X37
)
)
→
(
¬
(
X32
→
(
X16
↔
(
X10
→
X27
)
)
)
)
)
→
(
(
¬
(
(
(
(
¬
X32
)
→
(
¬
X17
)
→
X33
)
→
(
¬
(
(
¬
X32
)
↔
(
(
¬
X20
)
→
(
X17
↔
X49
)
)
)
)
)
→
(
¬
(
X33
→
(
X20
↔
X49
)
)
)
)
)
↔
(
(
¬
X33
)
↔
(
X17
→
(
¬
X0
)
→
X49
)
)
)
)
)
↔
(
(
¬
(
X43
↔
(
X34
→
(
(
¬
X27
)
↔
X37
)
)
)
)
→
(
¬
(
(
¬
X27
)
→
X20
→
X37
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
(
¬
X47
)
↔
(
(
¬
X46
)
→
X5
→
X48
)
)
)
→
(
¬
(
(
¬
X48
)
↔
(
(
¬
X47
)
↔
(
(
¬
X46
)
↔
X49
)
)
)
)
)
↔
(
(
(
(
(
¬
X38
)
→
(
X28
↔
(
(
¬
X14
)
→
X47
)
)
)
↔
(
(
¬
(
X49
→
(
X47
↔
(
X14
↔
X38
)
)
)
)
↔
(
X48
→
(
X14
↔
X38
)
)
)
)
↔
(
(
¬
(
X48
↔
(
X47
↔
(
X11
↔
False
)
)
)
)
↔
(
X36
→
(
X11
↔
X49
)
)
)
)
↔
(
(
¬
(
X48
↔
(
X36
↔
(
X30
↔
(
X28
→
False
)
)
)
)
)
↔
(
X48
→
X36
→
(
X30
↔
X21
)
)
)
)
)
→
(
¬
(
(
(
(
¬
X38
)
↔
(
(
¬
X9
)
→
X40
)
)
↔
(
(
(
(
¬
(
(
¬
X38
)
→
(
(
¬
X22
)
↔
(
(
¬
X2
)
→
X40
)
)
)
)
→
(
X49
↔
(
(
¬
X38
)
→
(
¬
X22
)
→
X40
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X9
)
→
(
¬
X2
)
→
X22
)
)
)
)
→
X40
→
X38
→
False
)
)
→
(
(
¬
(
X47
↔
(
X18
↔
X45
)
)
)
↔
(
¬
(
(
¬
(
(
X49
↔
(
X18
→
(
(
¬
X9
)
↔
X45
)
)
)
→
(
¬
(
X49
→
(
X22
↔
(
(
¬
X18
)
↔
X47
)
)
)
)
)
)
→
(
¬
(
(
¬
X22
)
↔
(
X18
↔
X47
)
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
¬
(
(
X38
↔
(
(
¬
X6
)
↔
X42
)
)
→
(
¬
X42
)
→
(
X39
↔
(
X38
↔
X44
)
)
)
)
↔
(
(
(
¬
(
(
(
X45
→
X36
→
X42
)
↔
(
X45
↔
(
X44
→
X39
→
X36
)
)
)
→
(
¬
(
(
¬
X36
)
→
X39
)
)
)
)
→
(
¬
X45
)
)
↔
(
X44
↔
(
X42
↔
X45
)
)
)
)
↔
(
¬
(
X44
→
(
(
¬
X39
)
↔
(
(
¬
X8
)
→
X45
)
)
)
)
)
→
X45
→
(
X44
↔
(
X19
↔
X42
)
)
)
)
↔
(
(
(
(
X35
↔
(
(
¬
X7
)
↔
(
(
¬
X4
)
↔
X44
)
)
)
↔
(
(
¬
X44
)
↔
(
X26
→
X45
)
)
)
↔
(
¬
(
(
¬
(
X45
↔
(
(
¬
X26
)
↔
(
(
¬
X23
)
↔
X35
)
)
)
)
→
(
¬
(
(
¬
X35
)
↔
(
(
¬
X26
)
↔
(
(
¬
X12
)
↔
X45
)
)
)
)
)
)
)
→
(
(
¬
X39
)
↔
(
X35
↔
(
(
¬
X26
)
↔
X45
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
(
(
X42
→
(
X31
↔
X39
)
)
↔
(
(
(
¬
(
X42
→
X39
→
(
(
¬
X38
)
↔
X40
)
)
)
→
X42
→
(
(
¬
X38
)
↔
(
(
¬
X31
)
→
X40
)
)
)
↔
(
(
¬
X38
)
↔
(
(
¬
X31
)
↔
(
X21
→
X40
)
)
)
)
)
→
(
¬
(
(
(
(
¬
X42
)
→
(
X39
↔
(
(
¬
X17
)
↔
X48
)
)
)
↔
(
¬
(
X48
↔
(
X39
→
X17
→
X11
)
)
)
)
→
(
(
¬
X40
)
↔
(
X11
↔
X48
)
)
)
)
)
)
↔
(
(
¬
(
X42
→
(
X38
↔
(
(
¬
X13
)
↔
X22
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X39
→
(
¬
X7
)
→
X13
)
)
↔
(
¬
(
(
¬
X39
)
↔
(
X38
↔
(
(
¬
X7
)
↔
X42
)
)
)
)
)
↔
(
¬
(
X42
→
False
)
)
)
→
(
¬
(
X42
→
X39
→
X13
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
X40
)
↔
(
(
¬
X24
)
↔
(
X9
→
X42
)
)
)
)
↔
(
(
¬
(
X42
→
(
X39
↔
(
X9
→
(
X6
↔
False
)
)
)
)
)
→
X39
→
(
¬
X24
)
→
X15
→
X42
)
)
)
↔
(
¬
(
(
(
(
X48
→
(
¬
X39
)
→
(
¬
X16
)
→
X42
)
↔
(
(
¬
(
X42
↔
(
X10
↔
(
(
¬
X2
)
↔
X39
)
)
)
)
→
(
(
¬
X37
)
↔
(
(
¬
X2
)
↔
X48
)
)
→
(
¬
(
(
¬
X42
)
↔
(
X37
↔
(
(
¬
X10
)
→
X48
)
)
)
)
)
)
→
(
¬
X39
)
→
(
¬
X37
)
→
X48
)
→
(
(
X48
↔
(
X30
→
(
¬
X27
)
→
X36
)
)
→
(
(
(
¬
X48
)
↔
(
¬
(
(
¬
(
X48
→
(
(
¬
X30
)
↔
(
(
¬
X27
)
→
X42
)
)
)
)
→
(
¬
(
(
¬
X34
)
↔
(
X30
↔
(
X27
↔
X48
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
X30
)
↔
(
X27
↔
X34
)
)
)
)
→
(
¬
(
X34
↔
(
(
¬
X30
)
→
X48
)
)
)
)
→
(
(
(
¬
X39
)
↔
(
(
¬
X34
)
→
(
(
¬
X23
)
↔
X42
)
)
)
↔
(
(
¬
X39
)
→
X36
→
(
(
¬
X34
)
↔
X42
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
X49
→
X43
→
X33
→
X4
)
→
(
(
X44
→
(
(
¬
X33
)
↔
(
X3
→
X43
)
)
)
↔
(
X49
→
(
¬
X43
)
→
(
¬
X33
)
→
X44
)
)
)
)
↔
(
(
(
(
(
¬
X42
)
↔
(
(
¬
(
X42
→
(
(
¬
X18
)
↔
(
(
¬
X3
)
↔
X49
)
)
)
)
→
(
¬
X42
)
→
(
¬
X20
)
→
(
(
¬
X18
)
↔
X49
)
)
)
↔
(
X49
↔
(
X20
→
X18
)
)
)
↔
(
¬
(
X49
→
X42
→
X3
)
)
)
↔
(
(
¬
X18
)
↔
(
X3
↔
X20
)
)
)
)
→
(
(
(
(
X43
↔
(
X42
→
X26
→
X14
)
)
↔
(
¬
(
(
(
¬
(
X43
↔
(
X42
↔
(
X5
↔
False
)
)
)
)
↔
(
X43
→
(
¬
X42
)
→
(
X41
↔
X46
)
)
)
→
(
X43
↔
(
X41
→
X26
→
X46
)
)
)
)
)
↔
(
X43
↔
(
(
¬
X26
)
↔
(
(
¬
X14
)
→
X37
)
)
)
)
↔
(
(
(
X47
↔
(
X42
↔
(
(
¬
X12
)
→
X44
)
)
)
↔
(
¬
(
X43
↔
(
(
¬
X42
)
→
(
¬
X12
)
→
X44
)
)
)
)
↔
(
X47
↔
(
X46
→
(
(
¬
X42
)
↔
X44
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X46
↔
(
X45
→
X42
→
False
)
)
)
→
(
¬
(
(
¬
X44
)
↔
(
(
¬
X25
)
↔
X45
)
)
)
)
→
(
X46
→
X32
→
X19
→
(
X8
↔
False
)
)
→
(
¬
(
(
¬
(
X44
↔
(
X40
→
X35
→
X19
)
)
)
→
(
(
(
¬
X44
)
→
(
X35
↔
(
(
¬
X19
)
→
X45
)
)
)
↔
(
X45
↔
(
X32
→
(
X19
↔
X44
)
)
)
)
)
)
)
)
↔
(
(
¬
(
X46
↔
(
(
¬
X29
)
→
(
(
¬
X0
)
↔
X44
)
)
)
)
↔
(
(
(
¬
X28
)
↔
(
X1
→
(
X0
↔
X45
)
)
)
↔
(
¬
(
X46
→
(
X28
↔
(
X1
→
X0
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
0a0ce6...
and proposition id is
44c9b7...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMWuscECQKHcufswEMqUpEHzrNUcUDM5a5W
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
∀X50 :
prop
,
(
(
(
(
(
(
(
(
(
X30
↔
(
X15
↔
(
(
¬
X6
)
↔
X48
)
)
)
↔
(
(
¬
X40
)
↔
(
(
¬
X30
)
→
(
(
¬
X15
)
↔
X48
)
)
)
)
↔
(
¬
(
X10
↔
(
(
¬
X6
)
→
X15
)
)
)
)
↔
(
X48
→
(
(
¬
X27
)
↔
X31
)
)
)
↔
(
¬
(
(
(
¬
X31
)
→
(
X13
↔
(
X1
→
X48
)
)
)
→
(
X48
↔
(
X31
→
(
X27
↔
False
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
X31
)
→
(
(
¬
X26
)
↔
(
X17
↔
X32
)
)
)
)
→
(
(
¬
(
(
(
(
¬
(
X47
↔
(
X3
→
X32
)
)
)
↔
(
¬
(
(
(
¬
X26
)
→
(
(
¬
X4
)
↔
(
X3
↔
X47
)
)
)
→
(
X32
↔
X3
)
)
)
)
↔
(
X47
↔
(
X4
↔
(
(
¬
X3
)
↔
X40
)
)
)
)
→
(
(
¬
(
X26
↔
X24
)
)
↔
(
(
¬
(
X26
→
(
X17
↔
X40
)
)
)
↔
(
(
¬
X32
)
→
X24
→
(
¬
X17
)
→
X40
)
)
)
)
)
↔
(
¬
(
X47
↔
(
X32
↔
(
X30
↔
(
X17
↔
False
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X47
↔
(
X42
→
X38
→
X39
)
)
)
↔
(
(
X39
↔
(
(
¬
X28
)
↔
(
(
¬
X22
)
→
X38
)
)
)
↔
(
¬
(
X42
↔
(
X28
↔
X22
)
)
)
)
)
↔
(
(
(
X42
↔
(
(
¬
X25
)
→
X32
)
)
↔
(
(
X39
→
X38
→
(
X12
↔
X32
)
)
↔
(
¬
(
X39
→
(
X38
↔
(
X32
↔
X25
)
)
)
)
)
)
→
(
¬
(
X32
↔
(
X12
→
X38
)
)
)
)
)
↔
(
(
X49
↔
(
(
¬
X39
)
↔
X50
)
)
→
(
¬
(
(
¬
X48
)
→
(
¬
X47
)
→
(
(
¬
X42
)
↔
X49
)
)
)
→
(
X48
↔
(
(
¬
X47
)
↔
(
X19
↔
X49
)
)
)
)
)
)
↔
(
(
(
(
(
X20
→
X49
)
↔
(
(
(
¬
X45
)
↔
(
(
¬
X37
)
↔
X49
)
)
→
(
X49
→
X37
→
X33
→
X45
)
→
(
(
¬
X45
)
↔
(
(
¬
X33
)
↔
(
X20
↔
X49
)
)
)
)
)
→
(
(
(
¬
(
(
(
(
¬
(
X23
↔
(
X18
→
X11
→
False
)
)
)
↔
(
(
¬
X18
)
↔
(
(
¬
X5
)
↔
X33
)
)
)
↔
(
X18
↔
(
X5
→
False
)
)
)
→
(
¬
X33
)
)
)
→
(
¬
(
X37
→
X23
→
X8
)
)
→
(
¬
(
(
(
X37
↔
(
X5
↔
X45
)
)
↔
(
X37
↔
(
X23
↔
(
X8
↔
X5
)
)
)
)
→
(
¬
(
X45
→
(
X23
↔
X37
)
)
)
)
)
)
↔
(
(
¬
X23
)
↔
(
(
¬
X8
)
→
X37
)
)
)
)
→
(
¬
(
(
¬
(
X49
→
X45
→
X35
→
X33
→
False
)
)
→
(
(
¬
(
X49
↔
(
X33
→
X16
→
X35
)
)
)
↔
(
(
X47
→
(
(
¬
X9
)
↔
(
X0
→
X35
)
)
)
→
X49
→
(
X47
↔
(
X9
↔
X0
)
)
)
)
)
)
)
↔
(
(
(
X46
↔
(
(
¬
X7
)
→
X14
)
)
↔
(
(
X44
→
X7
→
False
)
↔
(
¬
(
(
¬
X46
)
→
(
X21
↔
X47
)
)
)
)
)
↔
(
(
(
(
¬
(
X47
↔
(
X44
↔
X46
)
)
)
↔
(
(
(
X47
↔
(
(
¬
X41
)
→
X36
→
X46
)
)
→
(
¬
(
X48
↔
(
X44
↔
X36
)
)
)
)
↔
(
¬
(
X46
→
(
X36
↔
X48
)
)
)
)
)
→
(
(
(
¬
(
X47
→
(
X44
↔
(
X34
→
X48
)
)
)
)
↔
(
(
X45
↔
(
X34
→
X48
)
)
→
(
(
X44
↔
(
(
¬
X37
)
→
(
¬
X34
)
→
X45
)
)
↔
(
X45
↔
(
X34
→
(
X2
↔
False
)
)
)
)
)
)
↔
(
¬
(
X48
↔
(
X47
↔
X43
)
)
)
)
)
↔
(
X45
↔
(
X43
→
(
X29
↔
X47
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
(
¬
(
(
(
X49
↔
(
X46
↔
(
X36
↔
X45
)
)
)
↔
(
¬
(
X49
→
X45
→
(
X8
↔
False
)
)
)
)
→
X45
→
False
)
)
→
(
(
¬
X45
)
↔
(
(
¬
X36
)
↔
X46
)
)
)
→
(
¬
X36
)
→
X3
→
X45
)
)
↔
(
¬
(
(
X42
↔
(
(
¬
X19
)
→
X45
)
)
→
(
(
(
¬
(
(
¬
(
X46
↔
(
X42
→
(
X31
↔
False
)
)
)
)
→
(
(
¬
(
X42
↔
(
X10
→
X46
)
)
)
↔
(
¬
(
X36
→
(
(
¬
X10
)
↔
X31
)
)
)
)
→
X46
→
(
X36
↔
X42
)
)
)
↔
(
¬
(
X31
↔
(
(
¬
X10
)
↔
X46
)
)
)
)
↔
(
X45
↔
(
(
¬
X10
)
→
X31
)
)
)
)
)
)
→
(
(
¬
(
(
X45
↔
(
(
¬
X15
)
→
(
(
¬
X11
)
↔
X39
)
)
)
→
(
¬
(
X45
↔
(
X15
→
X39
)
)
)
→
(
¬
(
X15
↔
(
(
¬
X9
)
↔
(
X5
→
X45
)
)
)
)
)
)
↔
(
(
(
¬
(
X45
↔
(
(
¬
X39
)
↔
X49
)
)
)
↔
(
¬
(
(
¬
X39
)
↔
(
X30
↔
(
X11
↔
X49
)
)
)
)
)
↔
(
(
¬
(
(
¬
X45
)
↔
(
X42
↔
X49
)
)
)
↔
(
¬
(
(
¬
X45
)
→
(
¬
X39
)
→
X4
→
X49
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X46
↔
(
X32
→
(
X2
↔
X48
)
)
)
)
↔
(
(
¬
(
(
¬
X40
)
↔
(
X2
→
X48
)
)
)
↔
(
¬
(
(
X49
→
X48
→
X32
→
X40
)
→
X48
→
X40
→
(
¬
X2
)
→
X32
)
)
)
)
→
(
(
X50
→
X48
)
↔
(
¬
(
X49
→
(
(
¬
X24
)
↔
X40
)
)
)
)
→
(
(
(
¬
X49
)
↔
(
X48
↔
(
(
¬
X13
)
→
X50
)
)
)
↔
(
¬
(
(
¬
X49
)
↔
(
X13
→
(
X1
↔
X50
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X48
)
↔
(
X46
→
(
X21
↔
X49
)
)
)
)
→
(
(
¬
X46
)
↔
(
(
¬
X42
)
→
(
(
¬
X21
)
↔
X48
)
)
)
)
↔
(
¬
(
X49
→
(
X48
↔
(
(
¬
X12
)
→
X42
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
X49
↔
(
X48
↔
(
X28
→
(
X26
↔
False
)
)
)
)
↔
(
¬
(
(
¬
(
X49
↔
(
X48
↔
X28
)
)
)
→
X48
→
(
¬
X28
)
→
X18
→
X44
)
)
)
→
(
¬
(
(
(
(
¬
(
X49
↔
(
X48
→
(
X46
↔
(
X44
→
False
)
)
)
)
)
↔
(
¬
(
X48
→
X47
→
X44
→
X34
→
False
)
)
)
→
(
(
(
X48
↔
(
X47
↔
(
X35
→
X6
)
)
)
→
(
(
¬
X6
)
↔
X49
)
)
↔
(
¬
(
(
(
X48
↔
(
(
¬
X29
)
→
X7
→
X35
)
)
↔
(
X48
↔
(
X29
→
X49
)
)
)
→
X48
→
X25
→
(
X7
↔
X49
)
)
)
)
)
→
(
(
¬
(
(
(
X49
↔
(
X33
→
(
X27
↔
X37
)
)
)
↔
(
(
¬
(
(
¬
X41
)
↔
(
(
¬
X37
)
↔
(
X27
↔
X44
)
)
)
)
→
(
¬
(
X41
↔
(
(
¬
X27
)
→
X49
)
)
)
)
)
→
X49
→
X37
→
X41
)
)
↔
(
¬
(
X49
→
X37
→
X33
)
)
)
→
(
X49
↔
(
X47
↔
(
X41
↔
X44
)
)
)
)
)
)
)
→
(
(
(
¬
(
X43
↔
(
(
¬
X41
)
↔
X46
)
)
)
↔
(
¬
(
X48
↔
(
X41
→
X23
)
)
)
)
↔
(
X48
→
X46
→
X43
→
X20
→
False
)
)
→
(
(
X46
↔
(
X17
→
X14
)
)
→
(
¬
(
X46
↔
(
(
¬
X22
)
→
X38
)
)
)
)
→
(
(
(
¬
X43
)
→
X16
→
(
¬
X0
)
→
X48
)
↔
(
(
X38
→
(
(
¬
X14
)
↔
(
X0
↔
X46
)
)
)
→
(
¬
(
X48
↔
(
(
¬
X38
)
↔
(
(
¬
X14
)
↔
X46
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
8d97b5...
and proposition id is
f8f03a...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMLY6HJd8XQT11FEjySE111WTgL6xrYCek4
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
∀X50 :
prop
,
∀X51 :
prop
,
(
(
(
¬
(
(
(
(
(
X50
↔
(
X47
→
(
¬
X19
)
→
X48
)
)
→
(
¬
(
X48
→
(
¬
X47
)
→
(
X31
↔
X50
)
)
)
)
→
(
¬
(
X50
↔
(
X48
↔
(
(
¬
X46
)
↔
X47
)
)
)
)
)
↔
(
(
¬
(
X47
→
(
X46
↔
(
X40
↔
X14
)
)
)
)
↔
(
X50
↔
(
(
¬
X46
)
→
X40
→
X47
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X37
→
(
¬
X16
)
→
X51
)
)
↔
(
(
¬
(
X51
→
(
X50
↔
(
X29
→
X16
)
)
)
)
↔
(
¬
(
(
(
¬
(
X29
↔
(
(
¬
X27
)
→
X28
)
)
)
→
(
¬
(
(
¬
X50
)
→
(
¬
X28
)
→
X27
→
X51
)
)
→
(
¬
(
(
¬
X29
)
→
(
X28
↔
X50
)
)
)
)
→
X27
→
X50
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
X42
)
→
X29
→
(
¬
X27
)
→
X51
)
→
(
X51
↔
(
X29
↔
(
(
¬
X1
)
→
X27
)
)
)
)
)
→
(
(
X27
↔
(
(
¬
X6
)
↔
X20
)
)
↔
(
(
X42
↔
(
X29
→
(
X27
↔
(
X6
→
False
)
)
)
)
↔
(
(
¬
X27
)
↔
(
(
¬
X20
)
→
(
¬
X6
)
→
X42
)
)
)
)
)
)
)
→
(
(
(
(
X51
↔
(
X48
→
X47
→
X29
)
)
→
(
¬
(
(
¬
X29
)
↔
(
X4
→
X48
)
)
)
)
↔
(
¬
(
X48
↔
(
X29
→
X51
)
)
)
)
↔
(
(
¬
(
(
¬
X29
)
↔
(
(
¬
X21
)
→
X5
→
X42
)
)
)
→
(
¬
(
(
¬
(
X48
↔
(
X21
↔
(
(
¬
X7
)
→
X42
)
)
)
)
→
(
¬
(
X42
→
(
(
¬
X4
)
↔
X29
)
)
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
(
¬
(
X30
→
(
X13
↔
X8
)
)
)
→
(
¬
(
(
¬
X43
)
→
(
(
¬
X30
)
↔
(
(
¬
X8
)
→
X49
)
)
)
)
)
↔
(
(
X50
↔
(
X49
↔
(
X30
→
X51
)
)
)
↔
(
(
¬
X49
)
↔
(
X43
↔
(
X30
↔
X51
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X51
↔
(
X50
↔
(
X24
→
X38
)
)
)
)
↔
(
X39
→
False
)
)
→
(
(
X51
↔
(
X49
↔
False
)
)
→
(
X39
↔
(
X24
↔
X26
)
)
→
(
X49
↔
(
(
¬
X39
)
↔
X51
)
)
)
→
(
¬
(
X36
↔
X26
)
)
)
)
)
→
(
(
(
(
(
¬
(
X49
→
(
(
¬
X41
)
↔
X44
)
)
)
↔
(
(
¬
(
(
¬
(
X41
↔
(
X12
↔
False
)
)
)
→
(
¬
(
X44
↔
(
X41
↔
(
X15
→
X49
)
)
)
)
)
)
↔
(
(
¬
X44
)
→
X49
)
)
)
↔
(
¬
(
X44
↔
(
X15
→
X41
)
)
)
)
↔
(
¬
(
(
X48
↔
(
X12
↔
X49
)
)
→
(
(
¬
(
X15
→
(
X12
↔
X22
)
)
)
↔
(
X49
↔
(
X15
→
(
¬
X12
)
→
X48
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X51
→
X48
→
X33
→
X50
)
)
→
(
¬
(
(
X49
→
X48
→
(
(
¬
X33
)
↔
X51
)
)
→
(
(
¬
(
(
¬
X33
)
→
(
X18
↔
(
(
¬
X15
)
→
X48
)
)
)
)
↔
(
X18
↔
(
(
¬
X11
)
→
(
(
¬
X10
)
↔
X51
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
X51
→
(
¬
X9
)
→
(
¬
X0
)
→
X49
)
→
(
¬
(
X49
↔
(
(
¬
X48
)
↔
(
X0
↔
X51
)
)
)
)
)
)
→
(
X51
→
X25
→
(
¬
X23
)
→
X48
)
→
(
¬
(
X25
↔
(
(
¬
X0
)
↔
X51
)
)
)
)
→
(
(
¬
(
(
(
¬
X35
)
↔
(
(
¬
X3
)
→
X2
→
X48
)
)
→
(
¬
X35
)
→
(
X25
↔
(
(
¬
X3
)
↔
X48
)
)
)
)
↔
(
X35
→
X25
→
(
(
¬
X23
)
↔
X48
)
)
)
)
)
→
(
X49
→
X32
→
(
¬
X17
)
→
X48
)
→
(
¬
(
(
(
¬
X48
)
→
(
¬
X45
)
→
X51
)
→
(
(
¬
(
X45
→
(
¬
X34
)
→
X17
→
X51
)
)
↔
(
(
¬
X32
)
→
X45
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X46
↔
(
X35
→
X12
)
)
)
→
(
¬
(
(
(
(
(
(
(
X47
→
(
X46
↔
(
X20
↔
X14
)
)
)
↔
(
X46
↔
(
(
¬
X20
)
↔
(
X0
↔
X47
)
)
)
)
↔
(
(
¬
X46
)
→
(
¬
X20
)
→
X47
)
)
↔
(
X47
→
X46
→
(
¬
X20
)
→
X43
)
)
→
(
¬
(
X47
↔
(
(
¬
X0
)
→
X46
)
)
)
)
↔
(
X47
↔
(
X43
↔
(
X14
↔
False
)
)
)
)
→
(
¬
(
(
(
¬
(
X46
↔
(
(
¬
X35
)
↔
(
X12
→
X43
)
)
)
)
↔
(
¬
(
X35
↔
(
X20
↔
X43
)
)
)
)
→
(
¬
(
X46
→
X35
→
X43
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X32
→
X14
→
(
X11
↔
X48
)
)
)
↔
(
X48
→
(
X32
↔
(
X11
↔
X43
)
)
)
)
→
(
¬
(
(
¬
X43
)
→
X11
→
X48
)
)
)
→
(
(
¬
(
(
X50
↔
(
X45
→
X43
→
X48
)
)
→
(
(
¬
(
X14
↔
X2
)
)
↔
(
(
X48
→
X14
→
(
(
¬
X2
)
↔
X45
)
)
↔
(
X50
→
(
X14
↔
X45
)
)
)
)
)
)
↔
(
X50
→
(
X43
↔
(
X0
↔
X48
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
(
¬
(
X26
↔
(
X22
→
(
X16
↔
(
X7
→
False
)
)
)
)
)
→
(
¬
(
(
¬
X26
)
↔
(
X16
→
X7
→
X30
)
)
)
)
→
X26
→
X22
→
X16
)
)
↔
(
¬
(
(
X19
→
(
X16
↔
(
(
¬
X7
)
→
X31
)
)
)
→
(
¬
(
X31
→
(
X26
↔
(
(
¬
X16
)
↔
X19
)
)
)
)
)
)
)
→
(
(
X48
→
X41
)
↔
(
¬
(
X46
→
X30
→
X26
)
)
)
→
(
(
¬
X46
)
→
X41
→
X5
→
X48
)
→
(
X46
↔
X22
)
)
↔
(
(
¬
(
(
(
(
¬
X29
)
→
X4
→
X31
)
↔
(
(
X47
↔
(
(
¬
X29
)
→
X8
→
X41
)
)
→
(
¬
(
X41
↔
(
X31
→
(
X29
↔
False
)
)
)
)
)
)
→
(
¬
(
X29
→
(
¬
X9
)
→
X47
)
)
)
)
↔
(
(
X28
→
(
¬
X6
)
→
X48
)
↔
(
¬
(
(
X48
↔
(
(
¬
X28
)
→
X31
)
)
→
(
X31
↔
(
(
¬
X8
)
→
(
X4
↔
X28
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
(
¬
(
X44
↔
(
(
¬
X42
)
↔
(
(
¬
X23
)
→
X48
)
)
)
)
→
(
¬
(
X48
→
(
X42
↔
(
X17
↔
X23
)
)
)
)
)
→
(
(
(
¬
(
X51
↔
(
(
¬
X44
)
→
(
X37
↔
X47
)
)
)
)
↔
(
¬
(
(
¬
X47
)
→
(
¬
X17
)
→
X51
)
)
)
↔
(
¬
(
X51
↔
(
X47
→
(
X44
↔
False
)
)
)
)
)
)
)
↔
(
X51
→
(
¬
X44
)
→
X48
)
)
↔
(
(
¬
(
(
¬
(
(
(
¬
(
(
¬
X47
)
→
X39
→
(
¬
X25
)
→
X51
)
)
↔
(
X51
↔
(
X47
↔
(
(
¬
X34
)
→
X39
)
)
)
)
→
(
¬
(
(
¬
X34
)
→
(
X25
↔
X47
)
)
)
)
)
→
(
¬
X39
)
→
X34
→
(
¬
X21
)
→
X47
)
)
→
(
(
(
¬
X47
)
↔
(
(
¬
X36
)
↔
X51
)
)
↔
(
¬
(
(
¬
(
X47
↔
(
X44
→
(
¬
X24
)
→
X36
)
)
)
→
(
¬
(
(
¬
X44
)
↔
(
X24
→
X51
)
)
)
)
)
)
→
X51
→
(
X39
↔
X24
)
)
)
→
(
¬
(
(
(
(
(
(
X40
↔
(
X10
↔
X50
)
)
↔
(
¬
(
X44
↔
(
(
¬
X38
)
→
X10
→
X40
)
)
)
)
→
X44
→
(
X38
↔
(
X10
↔
False
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X38
)
→
(
X33
↔
X50
)
)
)
→
(
(
¬
(
(
¬
X40
)
→
X38
→
(
(
¬
X33
)
↔
X50
)
)
)
↔
(
¬
(
(
¬
X40
)
→
X33
→
(
(
¬
X27
)
↔
X50
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X47
→
X44
→
X40
→
X18
→
False
)
)
→
(
¬
(
(
¬
X44
)
→
(
X40
↔
(
X18
↔
X49
)
)
)
)
)
↔
(
(
(
X49
↔
(
(
¬
X38
)
→
X44
)
)
→
(
X47
↔
(
X15
↔
False
)
)
)
↔
(
(
(
¬
X13
)
↔
X38
)
↔
(
X47
→
X44
→
X1
)
)
)
)
)
→
(
¬
(
(
(
X50
↔
(
X49
↔
(
X3
↔
False
)
)
)
→
(
¬
(
(
(
(
(
X50
↔
(
(
¬
X3
)
↔
X49
)
)
↔
(
X49
→
(
(
¬
X42
)
↔
(
X3
→
X47
)
)
)
)
→
(
¬
(
X49
↔
(
X47
→
X3
)
)
)
)
↔
(
X50
↔
(
X47
→
X3
)
)
)
→
(
¬
(
X42
→
X49
)
)
)
)
)
→
(
X47
↔
(
X42
↔
X3
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
cd03af...
and proposition id is
59aeef...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMMuVQE5LcShGiLVADQKfkGz1iVFiB43aDj
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
¬
(
(
(
(
(
(
(
¬
(
(
X45
→
(
X2
↔
(
X0
→
False
)
)
)
→
(
X45
↔
(
(
¬
X38
)
→
(
X2
↔
X47
)
)
)
)
)
→
(
¬
(
X47
↔
(
X45
→
(
X38
↔
False
)
)
)
)
)
↔
(
(
(
¬
(
X47
→
X29
→
X42
)
)
→
(
¬
(
(
(
¬
X38
)
↔
(
(
¬
X29
)
→
X47
)
)
→
(
(
X46
→
(
¬
X17
)
→
X38
)
↔
(
(
¬
X46
)
→
(
X42
↔
(
X29
→
X47
)
)
)
)
)
)
)
↔
(
¬
(
X46
↔
(
X42
→
X2
→
X47
)
)
)
)
)
↔
(
(
X46
→
(
¬
X38
)
→
(
X20
↔
X39
)
)
→
(
¬
(
(
¬
(
X39
↔
(
(
¬
X38
)
↔
X48
)
)
)
→
(
(
¬
X20
)
↔
(
(
¬
X10
)
→
X38
)
)
)
)
→
(
¬
(
X38
→
X10
→
X39
)
)
)
)
→
(
¬
(
(
(
¬
(
X48
→
(
¬
X47
)
→
X49
)
)
↔
(
(
¬
(
(
¬
(
(
(
¬
(
X48
↔
(
(
¬
X46
)
→
(
X35
↔
X47
)
)
)
)
↔
(
X46
→
X35
→
(
(
¬
X7
)
↔
X48
)
)
)
→
(
X46
↔
(
X37
↔
(
(
¬
X7
)
↔
X35
)
)
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X37
)
↔
X48
)
)
)
)
)
→
(
¬
(
X49
→
X35
→
X48
)
)
)
)
→
(
¬
(
X48
→
X47
→
X34
→
X37
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
¬
X30
)
→
(
¬
X27
)
→
(
(
¬
X19
)
↔
X47
)
)
↔
(
¬
(
X27
→
(
¬
X4
)
→
X30
)
)
)
↔
(
X47
→
(
X27
↔
(
X19
↔
False
)
)
)
)
→
(
X47
↔
(
X19
↔
X30
)
)
)
)
↔
(
(
¬
(
X47
↔
(
X39
↔
(
(
¬
X12
)
↔
X49
)
)
)
)
↔
(
¬
(
(
(
(
¬
X39
)
↔
(
(
¬
X36
)
→
(
¬
X6
)
→
X49
)
)
↔
(
¬
(
(
¬
(
X49
↔
(
X47
→
(
¬
X36
)
→
X39
)
)
)
→
(
¬
X39
)
→
(
(
¬
X36
)
↔
(
(
¬
X27
)
→
X47
)
)
)
)
)
→
(
¬
X47
)
→
(
¬
X39
)
→
(
¬
X30
)
→
X49
)
)
)
)
)
→
(
(
¬
(
(
(
(
(
¬
(
(
¬
X32
)
↔
(
X31
→
X46
)
)
)
↔
(
¬
(
(
(
¬
X31
)
→
(
X9
↔
X44
)
)
→
(
¬
X44
)
→
X28
→
(
¬
X9
)
→
X46
)
)
)
↔
(
¬
(
X33
↔
(
(
¬
X31
)
→
X24
→
X44
)
)
)
)
↔
(
(
X48
↔
(
(
¬
X44
)
→
X46
)
)
↔
(
(
¬
(
X46
→
(
¬
X33
)
→
(
(
¬
X18
)
↔
X48
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X44
)
↔
X48
)
)
)
)
)
)
→
(
(
¬
(
(
¬
X26
)
→
(
X23
↔
(
(
¬
X21
)
→
X32
)
)
)
)
↔
(
(
(
¬
(
X32
↔
(
X23
↔
(
X5
→
X33
)
)
)
)
→
(
X26
↔
(
X23
→
X33
)
)
)
↔
(
X32
→
(
¬
X23
)
→
X44
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
X47
↔
(
(
¬
X40
)
↔
X48
)
)
↔
(
(
X47
↔
(
X44
→
X40
→
X48
)
)
→
(
¬
(
(
¬
X14
)
↔
X44
)
)
)
)
→
(
(
¬
(
X48
↔
(
X47
↔
(
X25
→
X3
→
False
)
)
)
)
↔
(
X48
↔
(
X44
→
X25
→
X3
→
False
)
)
)
)
↔
(
(
X47
↔
(
X44
↔
X40
)
)
↔
(
¬
(
X47
→
(
X46
↔
(
X43
↔
X40
)
)
)
)
)
)
→
X48
→
X25
→
(
¬
X14
)
→
X15
)
)
→
(
(
(
X44
↔
(
X41
↔
(
X11
→
X16
)
)
)
↔
(
(
(
¬
(
X48
→
(
X47
↔
(
X44
↔
(
X16
↔
False
)
)
)
)
)
↔
(
(
¬
X47
)
→
(
X44
↔
(
X41
↔
X48
)
)
)
)
↔
(
(
¬
(
(
¬
X41
)
→
(
X13
↔
(
(
¬
X1
)
↔
X47
)
)
)
)
↔
(
¬
(
X44
↔
(
X41
→
X13
)
)
)
)
)
)
↔
(
(
(
X48
→
X22
→
X44
)
↔
(
(
(
¬
X44
)
↔
(
X41
→
(
¬
X22
)
→
X48
)
)
→
(
¬
(
X48
→
(
¬
X22
)
→
X8
→
X44
)
)
)
)
↔
(
¬
(
X48
→
X44
→
X41
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
(
¬
X42
)
→
X31
→
(
¬
X30
)
→
X46
)
↔
(
¬
(
X31
→
(
¬
X17
)
→
X42
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X40
)
↔
(
X37
→
(
¬
X1
)
→
X49
)
)
)
→
(
(
¬
(
X40
→
(
X33
↔
X37
)
)
)
↔
(
¬
(
(
(
¬
(
X31
↔
X33
)
)
→
(
X49
↔
(
X37
→
(
¬
X31
)
→
X40
)
)
)
→
(
X40
↔
(
(
¬
X31
)
→
X37
)
)
)
)
)
)
)
→
(
(
(
¬
(
X42
↔
(
(
¬
X33
)
→
(
¬
X1
)
→
X46
)
)
)
↔
(
X48
→
X33
→
(
(
¬
X1
)
↔
X46
)
)
)
↔
(
¬
(
X49
↔
(
X42
→
X1
→
False
)
)
)
)
)
)
↔
(
(
(
X35
↔
(
(
¬
X6
)
→
X40
)
)
↔
(
¬
(
X42
→
(
(
¬
X24
)
↔
(
(
¬
X6
)
↔
X35
)
)
)
)
)
→
(
(
¬
(
X42
↔
(
X35
↔
(
X24
↔
(
X12
→
False
)
)
)
)
)
↔
X49
)
)
)
↔
(
(
(
(
(
¬
(
(
¬
X14
)
→
(
¬
X11
)
→
X42
)
)
↔
(
(
(
X42
→
(
X19
↔
X11
)
)
↔
(
(
¬
(
X42
↔
(
X19
↔
(
X14
→
X11
)
)
)
)
↔
(
(
¬
X42
)
→
(
X19
↔
X49
)
)
)
)
→
(
¬
(
(
X49
→
X42
→
(
X8
↔
False
)
)
→
(
¬
(
X49
→
(
X8
↔
(
X3
→
X11
)
)
)
)
)
)
)
)
↔
(
(
¬
(
X48
↔
(
X42
→
X8
→
X20
)
)
)
→
X29
→
(
X20
↔
(
X19
→
X42
)
)
)
)
↔
(
(
X47
→
(
X25
↔
(
(
¬
X11
)
→
X49
)
)
)
→
(
¬
(
X47
↔
(
X26
↔
(
(
¬
X25
)
→
X49
)
)
)
)
)
)
→
(
(
X25
→
(
X20
↔
(
X18
→
X29
)
)
)
↔
(
(
¬
(
X29
→
(
X20
↔
(
(
¬
X2
)
→
X4
)
)
)
)
→
X29
→
(
X2
↔
X25
)
)
)
)
)
→
(
¬
(
(
(
(
(
X49
↔
(
X41
↔
(
(
¬
X9
)
→
X47
)
)
)
↔
(
¬
(
(
¬
(
X48
→
(
X41
↔
(
(
¬
X36
)
→
X38
)
)
)
)
→
(
(
¬
X36
)
↔
(
(
¬
X9
)
↔
X48
)
)
)
)
)
↔
(
¬
(
(
X22
→
(
¬
X7
)
→
X45
)
→
(
(
¬
(
(
X45
↔
(
(
¬
X22
)
↔
X41
)
)
→
(
(
¬
X41
)
↔
(
X36
→
X22
→
X49
)
)
)
)
↔
(
¬
(
X49
→
(
(
¬
X41
)
↔
(
X7
→
X45
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X47
↔
(
X44
→
X5
→
X45
)
)
)
↔
(
¬
(
(
¬
(
X47
↔
(
(
¬
X44
)
↔
(
X10
→
X45
)
)
)
)
→
(
¬
(
X45
→
(
X44
↔
(
(
¬
X23
)
↔
X47
)
)
)
)
)
)
)
→
(
(
¬
X47
)
↔
(
X23
↔
(
X16
→
X49
)
)
)
→
(
¬
(
(
(
¬
X45
)
↔
(
(
¬
X15
)
→
X47
)
)
→
X49
→
(
X23
↔
X15
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
(
X44
→
(
¬
X39
)
→
(
(
¬
X27
)
↔
X47
)
)
↔
(
(
¬
(
(
¬
X13
)
→
(
¬
X0
)
→
X47
)
)
↔
(
¬
(
X47
→
(
¬
X27
)
→
X13
→
X39
)
)
)
)
↔
(
¬
(
(
¬
X28
)
↔
(
(
¬
X0
)
↔
X49
)
)
)
)
→
(
¬
(
X49
↔
(
X43
↔
(
X39
↔
X44
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
X47
)
↔
(
(
¬
X44
)
↔
(
(
¬
X21
)
→
X49
)
)
)
→
(
¬
X34
)
→
(
¬
X21
)
→
X44
)
)
↔
(
¬
(
X49
↔
(
(
¬
X44
)
→
(
(
¬
X32
)
↔
X47
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
1d17bc...
and proposition id is
e71ed8...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMMxoBdCwV3aFiVzXUjU5sgJiDn6wgWKLpp
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
¬
(
(
¬
(
(
(
¬
(
(
¬
(
(
X47
↔
(
(
¬
X43
)
→
(
¬
X7
)
→
X48
)
)
→
(
¬
(
(
¬
X43
)
→
(
(
¬
X20
)
↔
(
X7
→
X48
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X19
)
↔
(
X13
↔
X48
)
)
)
↔
(
X49
↔
(
X13
→
(
(
¬
X10
)
↔
X19
)
)
)
)
↔
(
¬
(
(
(
¬
(
X47
↔
(
X32
↔
(
(
¬
X13
)
↔
X49
)
)
)
)
→
(
¬
(
(
¬
X19
)
↔
X32
)
)
)
→
(
¬
(
(
¬
X43
)
→
X19
→
X49
)
)
)
)
)
)
)
↔
(
(
(
X49
↔
(
(
¬
X25
)
→
X47
)
)
↔
(
(
X47
↔
(
X42
→
X20
→
X49
)
)
↔
(
¬
(
X42
→
(
X25
↔
(
(
¬
X20
)
↔
X47
)
)
)
)
)
)
↔
(
(
¬
(
X25
→
(
(
¬
X17
)
↔
X49
)
)
)
↔
(
(
(
X45
→
(
¬
X25
)
→
X49
)
→
X47
→
(
X17
↔
X45
)
)
↔
(
¬
(
X49
→
(
X45
↔
X47
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
X35
)
↔
(
(
¬
X26
)
↔
(
(
¬
X8
)
→
X47
)
)
)
)
→
(
(
X35
→
X8
→
X43
)
↔
(
X48
→
X8
→
(
¬
X0
)
→
X43
)
)
)
)
↔
(
(
X39
↔
(
(
¬
X23
)
→
(
¬
X1
)
→
X26
)
)
↔
(
(
¬
(
(
¬
(
X39
↔
(
X11
↔
X22
)
)
)
→
(
¬
(
X39
↔
(
(
¬
X31
)
↔
X48
)
)
)
)
)
→
(
(
¬
(
X39
→
X8
→
False
)
)
↔
(
(
(
¬
(
X22
→
(
X8
↔
False
)
)
)
↔
(
¬
(
(
(
X39
→
(
(
¬
X8
)
↔
X23
)
)
↔
(
X48
→
X22
→
(
(
¬
X8
)
↔
X23
)
)
)
→
(
(
¬
X8
)
↔
X48
)
)
)
)
↔
(
¬
(
X39
↔
(
X23
↔
(
(
¬
X8
)
→
X26
)
)
)
)
)
)
)
)
)
)
)
→
(
(
(
(
X47
↔
(
X21
↔
X44
)
)
→
(
¬
(
X44
↔
(
X34
↔
(
(
¬
X16
)
→
X47
)
)
)
)
)
↔
(
¬
(
(
X40
→
(
X16
↔
(
X15
→
X44
)
)
)
→
(
(
(
¬
X44
)
↔
(
(
¬
X12
)
→
X47
)
)
↔
(
(
(
¬
X44
)
→
(
¬
X40
)
→
(
¬
X3
)
→
X47
)
↔
(
¬
(
X12
→
X44
)
)
)
)
→
(
(
(
X44
→
(
¬
X16
)
→
(
(
¬
X12
)
↔
X41
)
)
↔
(
X16
↔
(
(
¬
X12
)
↔
X40
)
)
)
↔
(
X44
↔
(
(
¬
X16
)
→
X41
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
¬
(
(
¬
X9
)
→
X38
)
)
↔
(
¬
(
X41
↔
(
X38
↔
(
(
¬
X29
)
↔
X47
)
)
)
)
)
↔
(
X47
↔
(
(
¬
X33
)
↔
(
X29
↔
X41
)
)
)
)
→
(
¬
(
X47
→
(
X38
↔
(
X14
↔
X41
)
)
)
)
)
)
↔
(
¬
(
X44
↔
(
X41
↔
(
X38
→
X47
)
)
)
)
)
)
→
(
¬
(
(
(
X40
↔
(
X5
↔
X4
)
)
↔
(
(
(
¬
(
X40
↔
(
X28
→
X49
)
)
)
→
(
¬
X38
)
→
(
(
¬
X30
)
↔
(
(
¬
X28
)
→
X41
)
)
)
↔
(
¬
(
X49
→
(
¬
X38
)
→
(
X4
↔
X41
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
(
X46
→
(
X24
↔
(
(
¬
X18
)
→
X36
)
)
)
→
(
(
¬
(
(
¬
(
X24
↔
(
(
¬
X18
)
↔
X46
)
)
)
→
(
¬
(
(
X49
↔
(
X27
↔
(
X24
↔
X46
)
)
)
→
(
¬
(
X49
↔
(
X27
↔
(
X18
↔
X46
)
)
)
)
)
)
)
)
↔
(
¬
(
X27
→
(
X24
↔
X46
)
)
)
)
)
)
↔
(
(
¬
(
X49
→
(
X48
↔
(
X46
→
(
X36
↔
False
)
)
)
)
)
↔
(
¬
(
(
¬
X46
)
↔
(
X41
↔
(
(
¬
X36
)
↔
X49
)
)
)
)
)
)
→
(
¬
(
(
(
¬
X30
)
→
X6
→
X49
)
→
(
¬
(
(
(
¬
(
X48
↔
(
X41
↔
(
(
¬
X6
)
↔
X30
)
)
)
)
↔
(
(
¬
X30
)
↔
(
(
¬
X6
)
→
X2
→
X48
)
)
)
→
(
X48
↔
(
X41
↔
X6
)
)
)
)
→
(
X6
↔
X48
)
)
)
)
→
(
¬
(
(
¬
X48
)
→
(
X46
↔
(
(
¬
X37
)
↔
X49
)
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
¬
(
(
(
(
¬
X43
)
↔
(
X17
↔
(
X13
↔
X46
)
)
)
↔
(
(
¬
X41
)
↔
X46
)
)
→
(
¬
(
(
¬
X43
)
↔
(
X41
↔
(
X26
↔
X46
)
)
)
)
)
)
→
(
(
(
(
¬
(
X46
→
(
X41
↔
X43
)
)
)
→
(
X40
↔
(
X24
→
False
)
)
→
(
¬
(
(
¬
X43
)
↔
(
(
¬
X41
)
→
X46
)
)
)
)
↔
(
X46
↔
(
X43
→
X41
)
)
)
↔
(
¬
(
(
¬
(
X43
→
(
X41
↔
(
X40
↔
(
X20
↔
False
)
)
)
)
)
→
(
¬
(
X40
↔
(
X20
↔
X11
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X46
→
X37
→
False
)
)
↔
(
(
¬
X47
)
↔
(
(
¬
X40
)
↔
(
X37
↔
X49
)
)
)
)
→
(
(
¬
(
X47
↔
(
X44
→
(
¬
X37
)
→
X49
)
)
)
↔
(
(
X49
→
X44
→
X43
)
↔
(
¬
(
(
¬
(
(
(
X37
→
X49
)
↔
(
X37
↔
(
(
¬
X21
)
↔
(
(
¬
X14
)
→
X33
)
)
)
)
→
X44
→
(
X37
↔
(
X21
↔
X33
)
)
)
)
→
X37
→
X33
→
(
X14
↔
X43
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
(
X25
↔
(
(
¬
X10
)
→
X37
)
)
)
→
(
¬
X29
)
→
(
(
¬
X25
)
↔
(
(
¬
X6
)
→
X33
)
)
)
→
(
(
(
X10
→
(
X6
↔
(
X5
→
X7
)
)
)
↔
(
X41
↔
(
X7
→
X6
→
X48
)
)
)
↔
(
X48
→
X25
→
(
¬
X7
)
→
X41
)
)
)
)
→
(
X37
↔
(
X33
→
(
X31
↔
X41
)
)
)
)
)
)
)
↔
(
(
(
(
(
X38
→
(
(
¬
X15
)
↔
(
(
¬
X12
)
↔
X39
)
)
)
↔
(
¬
(
X44
→
(
X39
↔
(
X36
↔
False
)
)
)
)
)
↔
(
(
¬
(
X38
↔
(
(
¬
X16
)
→
X44
)
)
)
→
(
¬
(
(
X38
↔
(
X16
↔
(
X4
→
(
X1
↔
False
)
)
)
)
→
(
¬
X38
)
→
(
(
¬
X16
)
↔
X39
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
X22
)
↔
X42
)
→
(
¬
(
X39
→
(
X38
↔
(
X34
↔
X22
)
)
)
)
)
↔
(
¬
(
(
¬
(
X23
→
(
(
¬
X0
)
↔
X42
)
)
)
→
(
(
¬
(
X42
↔
(
X0
↔
X39
)
)
)
↔
(
X38
↔
(
X30
↔
(
X0
↔
X39
)
)
)
)
)
)
)
→
(
(
(
¬
X30
)
→
(
X28
↔
(
(
¬
X23
)
→
X42
)
)
)
↔
(
¬
(
(
X42
↔
(
X36
↔
(
X28
↔
X2
)
)
)
→
(
¬
(
(
¬
(
(
¬
X42
)
↔
(
(
¬
X30
)
↔
X47
)
)
)
→
X42
→
(
(
¬
X2
)
↔
X28
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X39
)
↔
(
X19
↔
X47
)
)
)
↔
(
(
¬
(
X47
→
X18
→
X19
)
)
↔
(
¬
(
(
¬
X45
)
→
(
(
¬
X41
)
↔
(
(
¬
X19
)
→
X47
)
)
)
)
)
)
→
(
(
X41
→
X39
→
X35
→
X38
)
↔
(
(
(
X41
→
(
X9
↔
(
(
¬
X8
)
↔
X39
)
)
)
→
X41
→
X9
→
(
¬
X8
)
→
X32
)
→
(
X41
↔
(
X32
↔
X39
)
)
)
)
→
(
(
¬
(
(
¬
X27
)
↔
(
(
¬
X9
)
→
(
(
¬
X3
)
↔
X38
)
)
)
)
↔
(
¬
(
X47
↔
(
X38
↔
X27
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
9ed835...
and proposition id is
ab988f...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMSdFLDtALJqiZJV8rjsodnVyd4Ap7Z6bkT
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
(
(
(
(
¬
(
(
¬
X48
)
↔
(
X45
↔
(
(
¬
X41
)
↔
X49
)
)
)
)
↔
(
(
¬
X45
)
↔
(
X42
↔
(
(
¬
X41
)
→
X48
)
)
)
)
→
(
X48
↔
X49
)
)
↔
(
X49
→
X48
→
X42
→
False
)
)
↔
(
X48
↔
(
X42
↔
(
X41
↔
(
X33
↔
False
)
)
)
)
)
↔
(
(
(
X48
↔
(
X39
↔
(
(
¬
X36
)
→
X42
)
)
)
→
(
X49
↔
(
X44
↔
(
(
¬
X36
)
↔
X42
)
)
)
)
↔
(
(
¬
(
(
¬
(
X48
→
(
(
¬
X44
)
↔
(
(
¬
X16
)
→
X47
)
)
)
)
→
(
X48
↔
(
X47
→
X44
→
(
X42
↔
False
)
)
)
)
)
→
X42
→
(
X16
↔
X47
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
¬
(
X46
↔
(
(
¬
X17
)
↔
X35
)
)
)
→
(
X44
↔
(
X22
↔
(
X17
→
X35
)
)
)
)
)
→
X46
→
(
X37
↔
(
X17
↔
X22
)
)
)
)
↔
(
(
(
X47
↔
(
(
¬
X37
)
→
(
(
¬
X23
)
↔
X45
)
)
)
↔
(
(
¬
(
X47
→
(
X24
↔
X45
)
)
)
→
(
¬
(
X47
↔
(
X44
→
X37
→
False
)
)
)
)
)
↔
(
(
¬
(
(
¬
X13
)
↔
X46
)
)
→
(
(
X47
↔
(
X44
↔
(
(
¬
X24
)
↔
X43
)
)
)
↔
(
(
¬
(
(
(
¬
X46
)
↔
(
X44
↔
X47
)
)
→
(
X44
↔
(
(
¬
X13
)
↔
X43
)
)
)
)
↔
(
X47
→
X46
→
X43
→
False
)
)
)
)
)
)
↔
(
(
(
¬
(
X34
→
(
¬
X30
)
→
X9
→
X46
)
)
↔
(
¬
(
X34
→
(
X30
↔
X47
)
)
)
)
→
(
(
(
¬
(
(
¬
(
X38
↔
(
(
¬
X28
)
↔
(
X11
→
X44
)
)
)
)
→
(
¬
(
X28
→
(
¬
X11
)
→
X14
)
)
)
)
→
(
(
(
¬
(
X44
↔
(
X34
↔
(
X28
→
X26
)
)
)
)
→
(
¬
(
(
X30
→
(
(
¬
X28
)
↔
(
X26
→
X34
)
)
)
→
(
(
(
¬
X30
)
↔
(
X28
→
(
¬
X14
)
→
X34
)
)
↔
(
X34
↔
(
X28
↔
(
X14
↔
(
X4
→
False
)
)
)
)
)
)
)
)
↔
(
¬
(
X14
→
(
¬
X4
)
→
X44
)
)
)
)
→
(
(
¬
(
(
¬
(
X38
↔
(
(
¬
X30
)
↔
(
X21
↔
X46
)
)
)
)
→
(
X46
↔
(
X44
↔
(
X21
→
X30
)
)
)
)
)
↔
(
X46
↔
(
(
¬
X38
)
↔
(
X5
→
X44
)
)
)
)
)
→
(
¬
(
(
¬
X29
)
→
(
¬
X28
)
→
X26
→
X34
)
)
)
)
)
→
(
(
¬
(
(
X43
↔
(
X40
→
X20
→
X19
)
)
→
(
¬
(
(
¬
X40
)
→
(
(
¬
X20
)
↔
(
(
¬
X18
)
↔
X43
)
)
)
)
)
)
→
(
(
¬
(
X27
↔
(
(
¬
X18
)
↔
(
X12
↔
X31
)
)
)
)
↔
(
(
(
¬
(
X27
↔
(
(
¬
X20
)
→
X31
)
)
)
↔
(
(
¬
(
X31
→
(
X19
↔
X10
)
)
)
→
X20
→
X10
→
X27
)
)
↔
(
¬
(
X27
↔
(
(
¬
X19
)
↔
(
(
¬
X12
)
↔
X20
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X32
)
↔
(
X27
↔
(
X8
↔
X40
)
)
)
)
↔
(
¬
(
X45
→
(
X40
↔
(
(
¬
X0
)
↔
X46
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
(
¬
X45
)
↔
(
X7
↔
(
(
¬
X6
)
↔
X46
)
)
)
→
(
¬
(
(
¬
X7
)
↔
(
(
¬
X6
)
→
X2
→
X46
)
)
)
)
→
(
X46
↔
(
X7
↔
X2
)
)
)
→
(
¬
(
X40
→
(
¬
X7
)
→
X2
→
X46
)
)
)
)
→
(
(
X46
→
(
(
¬
X25
)
↔
(
(
¬
X1
)
→
X40
)
)
)
↔
(
¬
(
X25
↔
(
X7
→
(
X3
↔
X46
)
)
)
)
)
)
↔
(
(
¬
(
X45
→
X19
→
X15
→
X46
)
)
↔
(
¬
(
X46
↔
(
X25
→
False
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
¬
(
(
¬
(
X49
↔
(
(
¬
X31
)
↔
(
X30
→
X45
)
)
)
)
→
(
X46
↔
(
(
¬
X10
)
↔
X45
)
)
)
)
→
(
¬
(
(
(
¬
(
(
(
¬
X47
)
↔
(
X30
↔
(
X9
→
X49
)
)
)
→
X47
→
X30
→
(
X9
↔
X34
)
)
)
→
(
¬
(
X34
→
(
(
¬
X9
)
↔
X30
)
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
(
X36
↔
(
X34
→
X30
)
)
)
↔
(
¬
(
X36
↔
(
X4
↔
X49
)
)
)
)
→
(
¬
(
(
¬
X36
)
→
X34
→
X49
)
)
)
)
→
(
¬
(
X36
↔
(
(
¬
X17
)
→
(
(
¬
X4
)
↔
X49
)
)
)
)
)
)
→
X34
→
(
¬
X30
)
→
(
(
¬
X10
)
↔
X49
)
)
)
)
→
(
(
(
X46
↔
(
X16
↔
(
(
¬
X13
)
↔
X42
)
)
)
↔
(
(
(
(
X12
→
X16
)
→
(
X47
↔
(
X32
→
(
X16
↔
X5
)
)
)
)
↔
(
X42
↔
(
(
¬
X16
)
↔
(
X5
→
X47
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X16
↔
(
(
¬
X6
)
→
X12
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X12
)
↔
(
(
¬
X6
)
↔
X13
)
)
)
)
)
)
→
(
¬
(
(
¬
X12
)
→
(
(
¬
X6
)
↔
X16
)
)
)
)
)
)
)
↔
(
(
(
(
X32
↔
(
(
¬
X23
)
→
X45
)
)
→
X32
→
(
X8
↔
X46
)
)
→
(
¬
X41
)
→
(
¬
X32
)
→
X23
→
X45
)
↔
(
¬
(
X46
→
X41
→
(
(
¬
X32
)
↔
X42
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
X41
↔
(
X33
↔
(
X28
↔
X48
)
)
)
→
(
(
¬
(
X48
↔
(
X45
→
X41
→
(
X33
↔
False
)
)
)
)
→
(
(
¬
(
X48
→
(
X45
↔
(
X25
→
False
)
)
)
)
↔
(
X48
→
(
X41
↔
(
X21
↔
False
)
)
)
)
)
→
(
¬
(
(
¬
X33
)
↔
(
X25
→
X41
)
)
)
)
)
→
(
¬
(
X48
↔
(
(
¬
X28
)
↔
(
X3
↔
X33
)
)
)
)
)
)
→
(
(
(
¬
X33
)
↔
(
(
¬
X25
)
↔
(
X14
→
X36
)
)
)
↔
(
¬
(
X36
↔
(
X14
↔
X48
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
(
(
¬
(
(
X48
→
X43
→
X39
→
X45
)
→
(
¬
(
X48
↔
(
X47
↔
(
X45
↔
X39
)
)
)
)
)
)
→
(
¬
(
(
¬
X47
)
↔
(
(
¬
X45
)
↔
X48
)
)
)
)
)
→
(
(
¬
(
(
¬
X27
)
↔
(
(
¬
X20
)
→
X45
)
)
)
↔
(
¬
(
(
¬
X43
)
↔
(
X38
→
(
¬
X19
)
→
X48
)
)
)
)
)
)
→
(
(
(
X45
→
X43
→
(
X40
↔
(
X35
↔
False
)
)
)
→
(
(
¬
(
X35
→
(
(
¬
X1
)
↔
X40
)
)
)
↔
(
¬
(
(
¬
X7
)
→
X43
)
)
)
)
↔
(
¬
(
(
X43
→
X24
)
→
(
(
¬
X45
)
→
X43
→
(
X7
↔
X47
)
)
→
(
X45
↔
(
(
¬
X43
)
↔
(
X18
↔
X47
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
(
¬
X45
)
↔
(
(
¬
X44
)
→
(
(
¬
X38
)
↔
X48
)
)
)
)
↔
(
¬
(
X45
↔
(
X38
↔
(
X2
→
X44
)
)
)
)
)
→
(
(
(
X29
→
(
X22
↔
X44
)
)
↔
(
(
(
X45
↔
(
X29
→
X22
→
False
)
)
→
(
X48
↔
(
X29
↔
(
X22
↔
X44
)
)
)
)
↔
(
¬
(
(
¬
X44
)
↔
(
X37
↔
(
(
¬
X29
)
→
X48
)
)
)
)
)
)
↔
(
¬
(
(
¬
X37
)
↔
(
X29
→
X44
)
)
)
)
)
)
→
(
¬
(
X37
↔
(
(
¬
X26
)
↔
(
(
¬
X15
)
↔
X38
)
)
)
)
→
(
X38
↔
(
X11
→
X0
→
X44
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
090ffb...
and proposition id is
575d3f...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMP3Hw7yib4j8b5X1XbKuNS8apGPuxSme57
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
∀X51 :
prop
,
∀X52 :
prop
,
(
¬
∀X53 :
prop
,
∀X54 :
prop
,
(
(
¬
(
(
¬
(
(
(
¬
(
(
(
¬
(
X47
→
X44
→
(
X17
↔
X5
)
)
)
↔
(
¬
(
X54
→
X44
→
(
X5
↔
(
X4
↔
False
)
)
)
)
)
→
(
¬
(
X47
→
X17
→
(
X5
↔
X53
)
)
)
→
(
(
¬
(
X17
→
X53
)
)
↔
(
¬
(
X51
↔
(
(
¬
X47
)
↔
(
X0
↔
X53
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X51
↔
(
X49
→
X37
→
X8
→
False
)
)
)
→
(
¬
(
(
¬
(
X51
→
(
X48
↔
(
(
¬
X8
)
→
X49
)
)
)
)
→
(
X49
↔
(
(
¬
X45
)
→
X8
→
X48
)
)
)
)
)
)
→
(
¬
(
(
¬
X51
)
↔
(
(
¬
X49
)
→
(
X24
↔
X54
)
)
)
)
→
(
(
¬
(
(
¬
(
X51
→
(
¬
X49
)
→
(
¬
X15
)
→
X54
)
)
→
(
(
X54
→
X49
→
X48
→
X51
)
↔
(
(
X53
↔
(
(
¬
X49
)
↔
(
X36
↔
X51
)
)
)
↔
(
X29
→
X49
)
)
)
)
)
↔
(
(
¬
X53
)
↔
(
(
¬
X49
)
↔
(
(
¬
X26
)
→
X54
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X41
)
→
(
X34
↔
(
X20
→
X53
)
)
)
)
↔
(
(
X53
→
X34
→
(
X31
↔
X41
)
)
↔
(
(
¬
(
X31
↔
(
(
¬
X12
)
→
X53
)
)
)
↔
(
¬
(
X34
→
(
(
¬
X31
)
↔
(
(
¬
X10
)
→
X53
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
X49
↔
(
(
¬
X25
)
↔
(
X3
→
X53
)
)
)
↔
(
(
¬
X49
)
→
X22
→
(
X3
↔
X53
)
)
)
→
(
(
¬
X51
)
↔
(
X49
↔
(
(
¬
X25
)
→
X53
)
)
)
)
→
X51
→
(
(
¬
X49
)
↔
X53
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
X53
↔
(
(
¬
X28
)
↔
(
X20
↔
X31
)
)
)
)
→
(
¬
(
(
¬
X51
)
↔
(
(
¬
X28
)
↔
X53
)
)
)
)
→
(
X51
↔
(
X31
→
X25
→
X53
)
)
)
)
→
(
¬
(
X31
↔
(
X28
↔
(
X2
→
X51
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
(
¬
(
X52
↔
(
X42
→
X40
)
)
)
↔
(
X52
↔
(
X35
↔
(
(
¬
X14
)
↔
X53
)
)
)
)
→
(
(
(
¬
X42
)
↔
(
(
¬
X27
)
↔
(
X23
↔
X52
)
)
)
↔
(
¬
(
X50
→
(
(
¬
X35
)
↔
(
(
¬
X23
)
→
X52
)
)
)
)
)
)
)
→
(
(
(
X49
↔
(
(
¬
X7
)
↔
X53
)
)
↔
(
(
(
X49
↔
(
(
¬
X46
)
→
(
(
¬
X1
)
↔
X53
)
)
)
→
(
¬
(
X53
→
(
X7
↔
(
X1
→
X46
)
)
)
)
)
↔
(
¬
(
X49
↔
(
X46
→
X53
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X52
↔
(
X49
→
(
X46
↔
X53
)
)
)
)
→
(
X53
↔
(
X49
↔
(
(
¬
X18
)
↔
X52
)
)
)
)
)
↔
(
(
X53
→
(
X52
↔
X39
)
)
↔
(
X49
→
(
¬
X38
)
→
X53
)
)
)
)
→
(
(
(
¬
X50
)
↔
(
X49
↔
(
(
¬
X9
)
↔
X52
)
)
)
↔
(
(
¬
(
X50
→
(
¬
X30
)
→
(
X19
↔
X46
)
)
)
↔
(
(
(
¬
X19
)
↔
X50
)
↔
(
¬
(
(
¬
(
X49
↔
X52
)
)
→
(
¬
(
X50
↔
(
X30
→
X19
→
X52
)
)
)
)
)
)
)
)
)
)
→
(
(
(
X33
↔
(
X13
→
X6
→
X32
)
)
↔
(
(
¬
(
X11
↔
X13
)
)
↔
(
(
X46
↔
(
(
¬
X13
)
↔
(
(
¬
X6
)
↔
X32
)
)
)
↔
(
¬
(
(
¬
X32
)
→
X13
→
(
X11
↔
X46
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
X33
)
→
(
¬
X16
)
→
X43
)
)
→
(
¬
(
(
¬
(
(
(
(
X54
→
X11
→
X43
)
↔
(
¬
(
(
X54
↔
(
X43
↔
X11
)
)
→
(
¬
(
X16
→
X21
)
)
)
)
)
→
(
¬
(
X54
→
X43
→
X21
)
)
)
→
(
¬
(
(
X50
→
X21
→
(
(
¬
X16
)
↔
X46
)
)
→
(
¬
(
X54
↔
(
X49
↔
(
X46
↔
X16
)
)
)
)
)
)
)
)
→
(
¬
(
X49
↔
(
X33
↔
(
X21
→
X50
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
(
¬
(
X48
↔
(
X32
→
(
(
¬
X14
)
↔
X45
)
)
)
)
↔
(
¬
(
X20
→
X14
→
X48
)
)
)
↔
(
(
¬
(
X46
→
(
¬
X40
)
→
X48
)
)
↔
(
(
X53
→
(
X46
↔
(
X32
→
X48
)
)
)
↔
(
(
¬
(
(
¬
X40
)
→
(
¬
X2
)
→
X48
)
)
→
(
(
(
¬
(
X53
→
X46
→
False
)
)
↔
(
(
¬
(
X46
→
(
(
¬
X20
)
↔
(
X2
→
X53
)
)
)
)
↔
(
¬
(
X53
↔
(
X40
↔
(
(
¬
X20
)
→
X46
)
)
)
)
)
)
↔
(
X53
↔
(
X48
↔
(
(
¬
X40
)
→
X46
)
)
)
)
)
)
)
)
↔
(
(
X46
→
X45
→
(
X32
↔
X11
)
)
→
(
¬
(
(
(
(
¬
X46
)
↔
(
X45
↔
X51
)
)
↔
(
¬
(
X45
→
(
¬
X7
)
→
X52
)
)
)
→
(
¬
(
(
¬
X51
)
↔
(
(
¬
X32
)
↔
(
X7
↔
X52
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
¬
(
(
¬
X31
)
→
(
¬
X30
)
→
(
(
¬
X16
)
↔
X40
)
)
)
→
(
¬
(
X40
↔
(
X36
→
(
X31
↔
X30
)
)
)
)
)
)
↔
(
¬
(
X40
→
X36
→
X31
→
X5
→
False
)
)
)
↔
(
(
X36
↔
(
X22
→
(
X1
↔
False
)
)
)
↔
(
(
(
¬
X31
)
→
(
X30
↔
(
X1
→
X36
)
)
)
↔
(
X51
↔
(
(
¬
X31
)
→
(
X30
↔
X36
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
¬
(
X51
↔
(
X27
→
(
X5
↔
X48
)
)
)
)
↔
(
X48
↔
(
X35
↔
(
X27
→
X5
→
False
)
)
)
)
↔
(
(
¬
X36
)
↔
(
X35
↔
(
X27
→
X51
)
)
)
)
→
(
¬
(
X35
→
(
(
¬
X5
)
↔
X27
)
)
)
)
)
↔
(
¬
(
X51
→
X36
→
False
)
)
)
)
)
)
↔
(
(
¬
(
(
X52
→
(
X51
↔
(
X29
↔
X37
)
)
)
→
(
(
¬
(
(
¬
X29
)
→
X28
→
(
(
¬
X25
)
↔
X52
)
)
)
↔
(
(
¬
(
X28
→
(
(
¬
X25
)
↔
X37
)
)
)
↔
(
¬
(
(
¬
X46
)
→
(
X25
↔
X52
)
)
)
)
)
→
(
(
¬
X46
)
↔
(
X29
→
(
(
¬
X23
)
↔
X52
)
)
)
)
)
↔
(
(
X46
→
(
¬
X26
)
→
(
¬
X0
)
→
X52
)
→
(
X51
↔
(
(
¬
X0
)
→
X26
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
(
(
X47
→
(
X10
↔
(
(
¬
X9
)
→
X24
)
)
)
↔
(
¬
(
(
¬
X24
)
↔
(
(
¬
X10
)
→
X54
)
)
)
)
→
(
¬
(
X24
→
X10
→
X47
)
)
)
)
↔
(
(
(
X38
↔
(
(
¬
X24
)
↔
(
(
¬
X10
)
→
X52
)
)
)
↔
(
¬
(
X33
→
X24
→
X10
)
)
)
→
X10
→
(
¬
X3
)
→
X47
)
)
↔
(
¬
(
(
(
(
X52
↔
(
X21
→
X17
→
X6
→
False
)
)
→
(
(
¬
X48
)
↔
(
(
¬
X21
)
→
X6
→
X54
)
)
→
X54
→
X6
→
X39
)
→
(
(
(
¬
(
X50
↔
X48
)
)
↔
(
(
¬
(
X52
→
X50
→
(
¬
X43
)
→
X54
)
)
→
(
(
¬
(
(
¬
X48
)
→
X50
)
)
↔
(
(
¬
(
X54
↔
(
(
¬
X48
)
→
X47
→
X52
)
)
)
→
X54
→
(
(
¬
X48
)
↔
X50
)
)
)
)
)
↔
(
X48
→
(
(
¬
X47
)
↔
X54
)
)
)
)
→
(
¬
(
(
(
¬
(
X50
↔
(
X15
↔
(
(
¬
X13
)
→
X54
)
)
)
)
↔
(
¬
(
X54
→
(
(
¬
X13
)
↔
X52
)
)
)
)
→
(
¬
(
X41
→
(
(
¬
X15
)
↔
(
X13
↔
X54
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
(
¬
X52
)
→
X51
→
(
(
¬
X42
)
↔
X54
)
)
→
(
X52
↔
(
X51
↔
(
X42
→
(
X12
↔
False
)
)
)
)
)
↔
(
¬
(
(
X51
↔
(
X42
→
X34
→
X4
→
False
)
)
→
(
¬
X51
)
→
(
(
¬
X42
)
↔
X54
)
)
)
)
→
(
¬
(
(
X42
→
X12
)
→
(
¬
(
(
¬
X18
)
↔
(
X12
↔
(
(
¬
X8
)
↔
X52
)
)
)
)
)
)
)
→
(
(
(
¬
X44
)
→
X52
)
↔
(
(
(
¬
(
X54
→
(
(
¬
X44
)
↔
(
(
¬
X42
)
↔
X50
)
)
)
)
↔
(
¬
(
X54
↔
X49
)
)
)
↔
(
(
¬
(
X54
→
(
¬
X49
)
→
X50
)
)
↔
(
¬
(
X52
→
X19
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
eeb444...
and proposition id is
062f05...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMX2XRQX9pr6WEfCUhwCmPPtguPYQRcQkse
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
¬
(
(
(
(
¬
(
(
(
X40
→
(
X30
↔
(
(
¬
X7
)
→
X45
)
)
)
↔
(
(
(
X48
→
(
¬
X45
)
→
(
¬
X9
)
→
X49
)
↔
(
¬
(
(
¬
X45
)
↔
(
(
¬
X40
)
→
X49
)
)
)
)
→
(
(
¬
(
(
¬
(
X48
↔
(
X44
→
(
X7
↔
X28
)
)
)
)
→
(
¬
(
X48
→
X45
→
False
)
)
)
)
↔
(
¬
(
X45
↔
(
X44
→
(
¬
X7
)
→
X49
)
)
)
)
→
(
¬
(
X45
→
X48
)
)
)
)
→
(
(
(
X24
↔
(
X22
↔
(
X9
↔
False
)
)
)
→
(
(
¬
X9
)
↔
(
X1
↔
X46
)
)
)
↔
(
(
¬
(
X46
→
X40
→
(
X22
↔
X45
)
)
)
↔
(
¬
(
(
X40
↔
(
X24
→
X22
)
)
→
(
(
¬
X40
)
↔
(
(
¬
X24
)
→
(
(
¬
X0
)
↔
X45
)
)
)
)
)
)
)
)
)
↔
(
¬
(
X44
→
(
X30
↔
(
X2
→
X46
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
(
¬
(
X45
↔
(
X44
↔
(
(
¬
X27
)
↔
X34
)
)
)
)
↔
(
¬
(
(
¬
X44
)
↔
(
X36
→
(
¬
X27
)
→
X45
)
)
)
)
→
(
X44
↔
(
X34
→
X21
→
X45
)
)
)
)
→
(
(
(
(
¬
X27
)
→
X44
)
→
(
¬
(
(
¬
(
(
¬
X47
)
↔
(
X36
↔
(
X3
↔
X49
)
)
)
)
→
X49
→
X44
→
X3
)
)
)
→
X49
→
(
X27
↔
(
X3
→
False
)
)
)
→
(
¬
(
X49
↔
(
X47
→
X44
→
False
)
)
)
)
→
(
(
X45
↔
(
X29
→
X25
→
X14
)
)
↔
(
¬
(
(
¬
X44
)
↔
(
X43
→
X13
→
X47
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
(
¬
X43
)
→
X19
→
(
(
¬
X5
)
↔
X44
)
)
→
(
¬
(
X43
↔
(
X18
→
(
X15
↔
X19
)
)
)
)
)
)
↔
(
(
(
X32
→
(
¬
X19
)
→
X43
)
→
(
¬
(
(
¬
X32
)
→
(
X19
↔
X44
)
)
)
→
(
¬
X32
)
→
(
¬
X19
)
→
(
X4
↔
X38
)
)
↔
(
X32
→
X4
→
False
)
)
)
↔
(
(
(
(
¬
(
X38
→
(
X23
↔
X31
)
)
)
↔
(
¬
(
X20
↔
X42
)
)
)
↔
(
(
(
¬
(
X42
→
(
X31
↔
(
(
¬
X16
)
→
X38
)
)
)
)
↔
(
¬
(
X32
↔
(
(
¬
X16
)
↔
X38
)
)
)
)
↔
(
X38
↔
(
(
¬
X16
)
↔
X32
)
)
)
)
↔
(
(
¬
(
False
→
(
(
(
(
(
¬
X23
)
→
X26
)
→
(
(
(
¬
(
X26
↔
(
X23
↔
X44
)
)
)
→
(
(
¬
(
(
X26
→
(
X23
↔
(
(
¬
X10
)
→
X44
)
)
)
→
(
X41
↔
(
(
¬
X10
)
→
X23
)
)
)
)
↔
(
¬
(
(
¬
X41
)
↔
(
(
¬
X26
)
↔
X44
)
)
)
)
)
↔
(
¬
(
X41
↔
(
X26
↔
X23
)
)
)
)
)
→
(
¬
(
X44
→
(
X23
↔
False
)
)
)
)
↔
(
(
¬
X41
)
↔
(
X23
→
X42
)
)
)
)
)
↔
(
(
¬
(
X44
↔
(
X42
↔
(
(
¬
X12
)
↔
X31
)
)
)
)
↔
(
¬
(
X42
→
X41
→
X12
→
X44
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
X48
↔
(
X44
↔
(
(
¬
X37
)
→
X42
)
)
)
)
↔
(
¬
(
(
¬
X44
)
↔
(
(
¬
X38
)
↔
(
X37
→
X48
)
)
)
)
)
↔
(
(
¬
X37
)
→
(
(
¬
X6
)
↔
X48
)
)
)
→
(
X44
↔
(
X33
↔
(
(
¬
X11
)
↔
X38
)
)
)
→
(
(
¬
X44
)
↔
(
(
¬
X33
)
→
(
X11
↔
X48
)
)
)
)
)
→
(
(
¬
(
(
X42
↔
(
(
¬
X35
)
→
(
(
¬
X17
)
↔
X39
)
)
)
→
X48
→
(
X42
↔
(
X35
↔
(
X17
↔
False
)
)
)
)
)
↔
(
(
X38
↔
(
X8
→
X42
)
)
↔
(
¬
(
(
¬
X35
)
→
(
X17
↔
(
X8
↔
X38
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
(
(
¬
(
(
¬
(
(
¬
X29
)
↔
(
X26
→
(
(
¬
X25
)
↔
X46
)
)
)
)
→
(
(
(
(
(
(
X46
↔
(
X29
→
X34
)
)
↔
(
(
X29
↔
(
X26
→
X46
)
)
→
(
(
¬
(
X29
↔
(
X26
→
X46
)
)
)
↔
(
(
¬
(
X34
↔
(
X29
↔
(
(
¬
X24
)
↔
X46
)
)
)
)
↔
(
X34
↔
(
(
¬
X29
)
→
X26
→
X46
)
)
)
)
)
)
↔
(
X34
→
X26
→
False
)
)
↔
(
X24
↔
X34
)
)
↔
(
X46
→
(
X34
↔
(
X24
→
False
)
)
)
)
↔
(
¬
(
(
¬
X29
)
↔
X46
)
)
)
)
)
↔
(
(
¬
X43
)
↔
(
X34
→
X29
→
X46
)
)
)
→
(
¬
(
X43
→
X34
→
X26
→
X49
)
)
)
)
↔
(
¬
(
(
(
¬
(
X46
→
(
(
¬
X24
)
↔
(
(
¬
X17
)
→
X25
)
)
)
)
↔
(
¬
(
X30
↔
(
(
¬
X17
)
↔
X24
)
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X28
)
→
(
(
¬
X3
)
↔
X45
)
)
)
)
→
(
X17
↔
X30
)
)
)
)
→
(
¬
(
X36
→
X34
→
(
¬
X20
)
→
X33
)
)
→
(
(
(
¬
(
X34
↔
(
X33
↔
(
X20
→
False
)
)
)
)
→
(
¬
X43
)
→
(
X34
↔
(
X20
→
X49
)
)
)
↔
(
X34
↔
(
(
¬
X33
)
↔
X49
)
)
)
)
)
→
(
(
(
(
X45
→
X19
→
X1
→
X47
)
↔
(
¬
(
(
X48
→
(
¬
X45
)
→
(
X14
↔
X47
)
)
→
(
¬
(
(
¬
(
X48
→
X1
)
)
→
(
¬
(
(
¬
X48
)
→
(
(
¬
X45
)
↔
(
X14
→
X49
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
X48
)
→
(
X45
↔
(
(
¬
X2
)
→
X49
)
)
)
)
)
↔
(
¬
(
(
X45
↔
(
X43
→
X13
→
X47
)
)
→
(
(
(
¬
(
X5
↔
X19
)
)
→
(
¬
(
X45
↔
(
(
¬
X21
)
→
(
(
¬
X5
)
↔
X43
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X47
↔
(
X19
→
X4
→
False
)
)
)
→
(
¬
(
X47
↔
(
X19
↔
(
(
¬
X4
)
↔
X21
)
)
)
)
→
(
¬
(
(
¬
X21
)
↔
(
X19
→
(
X6
↔
X47
)
)
)
)
)
→
(
¬
(
X43
↔
(
X21
↔
(
X6
↔
(
X4
↔
False
)
)
)
)
)
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
X49
↔
(
X16
↔
(
(
¬
X11
)
→
X38
)
)
)
)
↔
(
X38
→
X47
)
)
↔
(
(
¬
(
X49
→
X48
→
X47
→
False
)
)
↔
(
¬
(
(
(
(
¬
X48
)
↔
(
X47
↔
(
X38
→
X49
)
)
)
↔
(
¬
(
X49
↔
(
X47
→
(
(
¬
X18
)
↔
X38
)
)
)
)
)
→
X38
→
(
(
¬
X18
)
↔
(
(
¬
X15
)
↔
X47
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
¬
(
(
(
X47
→
X10
→
X40
)
↔
(
¬
(
(
¬
X27
)
→
X7
→
X40
)
)
)
→
(
(
X40
↔
(
X12
→
False
)
)
↔
(
X48
↔
(
X47
→
(
¬
X10
)
→
X12
)
)
)
)
)
↔
(
¬
(
X48
↔
(
(
¬
X27
)
↔
(
X22
→
X47
)
)
)
)
)
↔
(
(
¬
X22
)
→
(
(
¬
X12
)
↔
X40
)
)
)
→
(
¬
(
(
¬
(
X47
→
X40
→
(
(
¬
X8
)
↔
X27
)
)
)
→
(
(
¬
X40
)
↔
(
(
¬
X35
)
→
(
¬
X27
)
→
X47
)
)
)
)
)
)
↔
(
(
X49
↔
(
X12
↔
(
(
¬
X0
)
→
X47
)
)
)
↔
(
¬
(
(
¬
X47
)
→
(
¬
X22
)
→
X49
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
(
X47
→
(
(
¬
X41
)
↔
X42
)
)
)
→
(
¬
(
X41
→
X39
→
(
¬
X31
)
→
X42
)
)
)
)
↔
(
¬
(
X47
→
X39
)
)
)
↔
(
(
(
(
X44
→
(
X39
↔
(
X32
→
X47
)
)
)
↔
(
¬
(
X47
↔
(
(
¬
X39
)
→
(
¬
X9
)
→
X44
)
)
)
)
→
(
X47
↔
(
(
¬
X39
)
→
X44
)
)
)
↔
(
(
¬
X44
)
↔
(
(
¬
X42
)
↔
(
(
¬
X32
)
→
X47
)
)
)
)
)
↔
(
(
(
(
¬
X48
)
↔
(
X47
↔
(
(
¬
X37
)
↔
X49
)
)
)
↔
(
¬
(
(
¬
X48
)
↔
(
X47
→
X41
→
X49
)
)
)
)
↔
(
¬
(
(
X48
→
X37
→
X23
→
X49
)
→
(
(
¬
X42
)
↔
(
(
¬
X37
)
→
(
X23
↔
X48
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
5e7667...
and proposition id is
45ff06...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMU8cEQGzKXfDVLqzDaoLAcAaL6sqatkmCF
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
¬
(
(
(
(
(
X49
→
X48
→
X41
→
X9
→
False
)
→
(
X49
↔
(
X41
→
(
X9
↔
X33
)
)
)
)
↔
(
(
(
X49
↔
X33
)
→
(
(
(
¬
X45
)
→
X30
→
X49
)
↔
(
¬
(
(
¬
X45
)
→
(
(
¬
X33
)
↔
(
(
¬
X30
)
→
X48
)
)
)
)
)
)
↔
X45
)
)
→
(
¬
(
(
¬
X48
)
↔
(
X46
↔
(
(
¬
X12
)
↔
X49
)
)
)
)
→
(
(
¬
(
(
¬
X48
)
→
(
¬
X45
)
→
(
(
¬
X12
)
↔
X49
)
)
)
↔
(
(
¬
X45
)
↔
(
(
¬
X18
)
→
X12
→
X48
)
)
)
)
→
(
(
¬
(
(
(
(
X33
↔
(
X28
→
X16
→
X44
)
)
↔
(
¬
(
X44
→
(
¬
X19
)
→
(
¬
X16
)
→
X33
)
)
)
→
(
(
X44
↔
(
X16
→
(
¬
X1
)
→
X26
)
)
↔
(
¬
(
X16
→
(
(
¬
X1
)
↔
X28
)
)
)
)
)
→
(
(
(
¬
(
X8
↔
X28
)
)
→
(
¬
(
(
X44
→
(
¬
X42
)
→
X45
)
→
(
¬
(
(
¬
X28
)
→
X16
→
X45
)
)
)
)
)
↔
(
¬
(
X44
↔
(
X37
↔
(
X28
↔
X42
)
)
)
)
)
)
)
↔
(
(
X15
↔
X11
)
↔
(
(
X45
→
(
X41
↔
X11
)
)
↔
(
(
X11
↔
(
(
¬
X6
)
→
X45
)
)
→
(
¬
(
X44
↔
(
X42
↔
(
(
¬
X6
)
↔
X11
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
(
¬
X45
)
↔
X49
)
→
(
¬
(
(
(
¬
(
X20
→
X38
)
)
↔
(
(
X48
↔
(
X39
↔
(
X20
→
X49
)
)
)
→
(
¬
(
X49
→
(
X36
↔
False
)
)
)
)
)
→
(
¬
(
(
X45
→
X39
→
X48
)
→
(
¬
(
X45
↔
(
X38
→
(
(
¬
X32
)
↔
X48
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X48
↔
(
X47
↔
(
X34
↔
(
X31
→
False
)
)
)
)
)
→
(
¬
(
X48
→
X45
→
X38
→
X47
)
)
)
)
)
↔
(
(
(
X48
↔
(
(
¬
X38
)
→
(
X24
↔
X43
)
)
)
↔
(
¬
(
X45
↔
(
(
¬
X38
)
→
(
(
¬
X29
)
↔
X43
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X45
↔
(
X40
↔
(
X7
↔
X43
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X43
)
↔
(
X21
↔
(
X4
↔
X45
)
)
)
)
→
(
(
X48
↔
(
X43
↔
(
(
¬
X40
)
→
X45
)
)
)
↔
(
(
¬
X45
)
↔
(
X43
↔
(
X4
↔
X48
)
)
)
)
)
)
)
)
↔
(
(
¬
X43
)
→
X7
→
(
X5
↔
X48
)
)
)
)
)
↔
(
¬
(
(
(
(
X45
→
(
X40
↔
(
(
¬
X35
)
→
X48
)
)
)
↔
(
¬
(
(
(
X35
↔
(
X14
→
X23
)
)
↔
(
X14
→
(
¬
X10
)
→
X45
)
)
→
X48
→
(
X45
↔
(
(
¬
X10
)
↔
X35
)
)
)
)
)
→
(
¬
(
(
X48
→
(
(
¬
X2
)
↔
X13
)
)
→
(
(
X48
↔
(
(
¬
X35
)
→
X2
→
X40
)
)
↔
(
¬
(
X48
↔
(
X13
↔
(
X2
→
False
)
)
)
)
)
)
)
)
→
(
(
¬
(
X48
→
(
X40
↔
(
X27
↔
X45
)
)
)
)
↔
(
¬
(
(
X48
→
(
(
¬
X40
)
↔
(
X25
↔
X45
)
)
)
→
(
(
(
¬
X17
)
→
(
¬
X3
)
→
(
¬
X0
)
→
X25
)
↔
(
(
(
¬
X17
)
→
X40
)
↔
(
(
¬
X22
)
→
(
¬
X17
)
→
X3
→
X48
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
(
¬
X29
)
→
(
X6
↔
(
X0
→
X37
)
)
)
)
↔
(
¬
(
X29
→
(
X22
↔
X46
)
)
)
)
↔
(
(
¬
(
X21
→
X29
)
)
→
(
X37
↔
(
X11
↔
X46
)
)
)
)
→
(
(
¬
(
X47
↔
(
(
¬
X45
)
↔
(
X21
↔
X46
)
)
)
)
↔
(
(
(
(
¬
X45
)
↔
(
X32
↔
(
(
¬
X21
)
↔
X46
)
)
)
↔
(
¬
(
X46
→
X45
→
(
(
¬
X10
)
↔
X32
)
)
)
)
→
(
¬
(
(
X45
→
(
(
¬
X14
)
↔
X37
)
)
→
(
¬
(
(
X46
→
(
X45
↔
(
X32
→
False
)
)
)
→
(
¬
(
X45
↔
(
X37
→
X7
→
X14
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
X46
→
(
(
¬
X34
)
↔
X39
)
)
↔
(
¬
(
X47
→
(
X46
↔
(
(
¬
X15
)
↔
X39
)
)
)
)
)
→
(
(
¬
(
X39
→
(
X16
↔
(
(
¬
X9
)
↔
X15
)
)
)
)
↔
(
¬
(
X47
↔
(
X15
→
X9
)
)
)
)
)
)
→
(
¬
(
(
(
¬
X45
)
↔
(
X39
→
(
¬
X34
)
→
X46
)
)
→
(
¬
(
(
(
(
¬
X43
)
→
(
X39
↔
(
X34
↔
X44
)
)
)
→
(
¬
(
(
¬
X43
)
→
(
X35
↔
(
X34
→
X44
)
)
)
)
)
→
(
(
X39
↔
(
(
¬
X35
)
↔
(
(
¬
X17
)
→
X46
)
)
)
↔
(
X39
→
(
¬
X17
)
→
(
X2
↔
X44
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X46
↔
(
X41
↔
(
(
¬
X20
)
→
X33
)
)
)
)
→
(
¬
(
X42
↔
(
X41
→
X19
→
X4
)
)
)
)
)
↔
(
(
(
(
¬
(
X46
↔
(
(
¬
X41
)
↔
(
X27
↔
X48
)
)
)
)
↔
(
X49
→
(
X48
↔
(
X46
↔
False
)
)
)
)
→
(
(
(
X46
↔
(
X27
→
X33
)
)
↔
(
¬
(
X49
↔
(
X46
→
X8
→
X27
)
)
)
)
↔
(
X46
→
X25
→
(
X8
↔
X33
)
)
)
)
→
(
X49
↔
(
X46
→
(
X44
↔
(
X26
↔
False
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X46
)
↔
(
(
¬
X44
)
↔
X49
)
)
)
→
(
¬
(
X49
↔
(
X48
→
X44
→
(
X23
↔
False
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
X36
↔
(
(
¬
X18
)
→
(
¬
X3
)
→
X46
)
)
↔
(
X36
→
X18
→
(
X1
↔
X3
)
)
)
↔
(
(
¬
(
X12
→
(
X5
↔
X13
)
)
)
↔
(
(
(
¬
(
(
¬
X13
)
→
(
(
¬
X5
)
↔
X46
)
)
)
↔
(
(
¬
(
X47
→
X46
→
(
X5
↔
X36
)
)
)
↔
(
X46
↔
(
(
¬
X36
)
↔
(
(
¬
X18
)
→
X47
)
)
)
)
)
↔
(
X47
→
X18
→
X36
)
)
)
)
→
(
¬
(
X40
↔
(
(
¬
X13
)
→
(
X5
↔
X18
)
)
)
)
)
→
(
(
(
(
X44
↔
(
X41
↔
(
X38
→
X47
)
)
)
↔
(
¬
(
X47
→
X41
→
X38
→
X36
→
False
)
)
)
↔
(
¬
(
X47
→
X41
→
X38
)
)
)
↔
(
(
¬
(
(
¬
(
X41
↔
(
X36
→
X30
→
X28
→
False
)
)
)
→
(
(
X41
→
(
¬
X36
)
→
X30
→
X47
)
↔
(
¬
(
(
(
X24
↔
X36
)
↔
(
(
¬
(
X36
↔
(
X24
→
(
(
¬
X18
)
↔
X41
)
)
)
)
↔
(
X47
↔
X36
)
)
)
→
(
¬
(
X36
→
(
(
¬
X18
)
↔
X47
)
)
)
)
)
)
)
)
↔
(
(
¬
X41
)
→
(
(
¬
X31
)
↔
(
X24
→
X47
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
cd4821...
and proposition id is
64c615...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMRAWCe1wuNoxhxjbgdJBYwhFdpPQBJVcKL
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
(
(
¬
(
(
(
¬
(
X44
→
(
(
¬
X16
)
↔
(
X6
→
X48
)
)
)
)
↔
(
¬
(
(
¬
(
X23
→
(
(
¬
X22
)
↔
(
(
¬
X6
)
↔
X43
)
)
)
)
→
X48
→
(
(
¬
X22
)
↔
X43
)
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
X41
)
↔
(
(
¬
X34
)
→
(
(
¬
X4
)
↔
X43
)
)
)
→
(
¬
(
(
¬
X43
)
↔
(
X23
→
(
(
¬
X4
)
↔
X44
)
)
)
)
)
)
→
(
¬
(
(
¬
X43
)
↔
(
X41
↔
(
(
¬
X17
)
↔
X44
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X49
↔
(
X44
↔
(
(
¬
X8
)
→
X31
)
)
)
)
↔
(
(
(
X44
↔
X25
)
→
(
(
¬
(
(
¬
X31
)
↔
(
(
¬
X25
)
↔
(
(
¬
X20
)
→
X42
)
)
)
)
↔
(
¬
(
X49
→
(
¬
X25
)
→
(
X20
↔
X42
)
)
)
)
)
→
(
(
¬
(
X25
↔
(
X3
↔
(
X2
→
X44
)
)
)
)
↔
(
¬
(
X49
→
(
¬
X25
)
→
X2
→
X44
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X43
→
X33
→
X27
)
)
↔
(
¬
(
(
¬
X42
)
↔
(
X33
↔
(
X27
↔
X49
)
)
)
)
)
↔
(
X49
→
(
(
¬
X27
)
↔
(
X0
→
X33
)
)
)
)
→
(
(
(
X49
↔
(
X48
→
(
¬
X21
)
→
X44
)
)
↔
(
¬
(
(
(
¬
X21
)
↔
(
(
¬
X15
)
→
X49
)
)
→
(
¬
X33
)
→
(
X21
↔
X44
)
)
)
)
↔
(
¬
(
(
¬
(
X48
→
X27
→
X10
→
X21
)
)
→
(
X48
↔
(
X33
↔
(
(
¬
X10
)
→
X21
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
(
(
¬
(
(
¬
(
X47
↔
(
X26
↔
(
X7
↔
False
)
)
)
)
→
(
¬
(
X47
→
(
(
¬
X44
)
↔
X45
)
)
)
)
)
→
(
(
¬
X45
)
↔
(
(
¬
X26
)
↔
(
(
¬
X11
)
→
X47
)
)
)
)
)
→
(
X45
↔
(
(
¬
X44
)
↔
(
(
¬
X14
)
→
X47
)
)
)
)
)
↔
(
¬
(
(
¬
(
X47
↔
(
(
¬
X45
)
↔
(
X44
→
X46
)
)
)
)
→
(
¬
(
(
¬
X46
)
→
(
¬
X45
)
→
(
¬
X44
)
→
X48
)
)
)
)
)
↔
(
(
(
X50
↔
(
X44
→
X48
)
)
↔
(
(
¬
(
(
X50
↔
(
X44
→
X48
)
)
→
(
(
X48
↔
(
X39
↔
(
(
¬
X18
)
→
X50
)
)
)
↔
(
¬
(
X44
→
X39
→
X48
)
)
)
)
)
↔
(
X47
↔
(
(
¬
X39
)
→
(
(
¬
X18
)
↔
X50
)
)
)
)
)
→
(
(
(
(
¬
(
X44
↔
(
X30
→
X48
)
)
)
↔
(
(
X44
→
X32
→
X29
)
↔
(
(
(
¬
X32
)
↔
(
X5
↔
X48
)
)
→
(
(
¬
(
X32
↔
(
X30
→
X5
→
False
)
)
)
↔
(
¬
(
X32
↔
(
X30
↔
(
(
¬
X29
)
↔
X48
)
)
)
)
)
)
)
)
↔
(
X30
↔
X32
)
)
↔
(
¬
(
(
¬
(
(
X36
→
(
X29
↔
(
(
¬
X24
)
↔
X32
)
)
)
→
(
¬
X36
)
→
X32
→
(
X24
↔
X47
)
)
)
→
(
X36
↔
(
X32
↔
(
X29
→
X44
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
X48
→
(
X46
↔
(
X40
→
False
)
)
)
↔
(
¬
(
X48
↔
(
X40
→
X19
→
X37
)
)
)
)
↔
(
¬
(
(
¬
(
X49
→
(
(
¬
X48
)
↔
(
X46
→
X50
)
)
)
)
→
(
¬
(
(
¬
(
X50
↔
(
X40
↔
(
X38
↔
False
)
)
)
)
→
(
¬
(
(
¬
X46
)
↔
(
(
¬
X40
)
→
X49
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
X49
)
↔
(
(
¬
X46
)
→
(
(
¬
X13
)
↔
X50
)
)
)
→
(
¬
(
X49
↔
(
X46
↔
(
X13
→
X12
→
False
)
)
)
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X13
)
→
X9
→
X50
)
)
)
)
)
)
↔
(
(
(
(
¬
X35
)
→
X49
)
↔
(
(
¬
X38
)
↔
(
X35
→
X28
→
X46
)
)
)
→
X49
→
(
X46
↔
(
X35
↔
X1
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
(
¬
(
X48
↔
(
(
¬
X43
)
→
X32
→
X44
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X44
)
→
(
X32
↔
X46
)
)
)
→
(
¬
(
(
¬
(
X13
↔
X48
)
)
→
(
X46
↔
(
(
¬
X32
)
→
(
(
¬
X13
)
↔
X44
)
)
)
)
)
)
)
)
→
(
X48
↔
(
X46
→
(
X13
↔
X23
)
)
)
)
)
→
(
(
X23
→
X19
→
X7
→
False
)
↔
(
¬
(
X45
↔
(
X23
↔
(
(
¬
X7
)
→
X9
)
)
)
)
)
→
(
(
¬
(
(
(
¬
X43
)
→
(
X9
↔
(
(
¬
X8
)
→
X44
)
)
)
→
(
¬
(
(
¬
X44
)
↔
(
(
¬
X43
)
→
(
X9
↔
X45
)
)
)
)
)
)
↔
(
X44
↔
(
(
¬
X7
)
↔
X45
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
X46
→
(
(
¬
X42
)
↔
(
(
¬
X38
)
→
X48
)
)
)
)
↔
(
¬
(
(
(
X26
↔
X2
)
↔
(
¬
(
X46
→
X42
→
X2
→
X48
)
)
)
→
X34
→
(
X26
↔
(
(
¬
X2
)
→
X48
)
)
)
)
)
→
(
(
¬
X48
)
↔
(
(
¬
X46
)
→
X50
)
)
→
(
¬
(
(
¬
X46
)
↔
(
(
¬
X42
)
↔
(
X33
↔
X50
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
(
¬
X46
)
↔
(
(
¬
X31
)
→
(
(
¬
X20
)
↔
X50
)
)
)
)
↔
(
(
(
X42
↔
(
X31
→
X20
→
X50
)
)
↔
(
X45
↔
(
(
¬
X31
)
↔
(
X20
↔
X50
)
)
)
)
↔
(
¬
(
(
¬
X45
)
→
(
X42
↔
X50
)
)
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X31
)
↔
X45
)
)
)
)
)
→
(
¬
(
X20
↔
(
(
¬
X3
)
↔
X31
)
)
)
)
)
)
↔
(
(
(
X48
↔
(
(
¬
X40
)
↔
(
(
¬
X10
)
→
X46
)
)
)
↔
(
(
¬
X40
)
↔
(
X10
→
X5
→
X46
)
)
)
↔
(
(
¬
X45
)
↔
(
X10
↔
(
X4
→
X48
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
(
X41
→
(
(
¬
X36
)
↔
(
X12
↔
X46
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X36
)
↔
(
X25
↔
X41
)
)
)
)
)
↔
(
¬
(
X46
→
(
¬
X36
)
→
X41
)
)
)
↔
(
¬
(
X49
→
X12
→
(
¬
X1
)
→
X41
)
)
)
↔
(
(
¬
(
(
¬
(
(
X16
↔
(
¬
(
X49
→
(
X27
↔
(
X21
↔
X46
)
)
)
)
)
→
(
¬
(
(
¬
X46
)
→
X27
→
(
¬
X15
)
→
X49
)
)
)
)
→
(
¬
(
X49
→
X46
→
X21
→
X47
)
)
)
)
↔
(
X47
↔
(
(
¬
X46
)
↔
(
X41
→
X49
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
¬
X30
)
↔
(
X24
↔
X46
)
)
)
→
(
X30
↔
(
(
¬
X25
)
→
X49
)
)
)
)
→
(
X46
→
(
X35
↔
(
(
¬
X25
)
→
X30
)
)
)
→
(
¬
(
(
¬
X35
)
→
(
X30
↔
(
(
¬
X14
)
→
X46
)
)
)
)
)
)
→
X24
→
(
(
¬
X14
)
↔
(
(
¬
X0
)
↔
X46
)
)
)
)
↔
(
(
¬
(
(
¬
(
X44
↔
(
X29
↔
(
(
¬
X11
)
→
X37
)
)
)
)
→
(
(
(
¬
(
(
¬
X37
)
↔
(
X29
→
(
¬
X18
)
→
X45
)
)
)
↔
(
¬
(
X44
↔
(
(
¬
X29
)
→
(
X18
↔
X37
)
)
)
)
)
↔
(
(
¬
(
(
¬
X18
)
→
(
X6
↔
X44
)
)
)
↔
(
X30
→
(
(
¬
X29
)
↔
(
(
¬
X6
)
↔
X44
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
X45
→
(
X39
↔
(
X17
↔
X30
)
)
)
)
→
(
¬
(
(
(
¬
X28
)
→
X39
)
→
X45
→
X39
→
(
¬
X22
)
→
X30
)
)
)
↔
(
¬
(
X39
→
(
(
¬
X35
)
↔
(
(
¬
X30
)
↔
X45
)
)
)
)
)
↔
(
¬
(
X45
↔
(
(
¬
X35
)
→
X39
)
)
)
)
↔
(
(
¬
(
(
(
¬
X28
)
→
X22
→
X49
)
→
(
¬
(
(
(
¬
X37
)
→
(
X30
↔
(
X28
→
X49
)
)
)
→
(
¬
(
X49
→
X28
→
(
(
¬
X22
)
↔
X37
)
)
)
)
)
)
)
↔
(
X28
→
X37
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
a9273e...
and proposition id is
0410a3...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMWSCmvmiJg8cXnPDL1hvqbDf3ZVRSvuLK1
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
¬
(
(
(
(
(
¬
(
(
¬
X46
)
→
(
(
¬
X45
)
↔
(
(
¬
X26
)
↔
X48
)
)
)
)
→
(
¬
(
X45
↔
(
(
¬
X33
)
↔
(
(
¬
X26
)
→
X46
)
)
)
)
)
↔
(
¬
(
(
X48
→
(
X31
↔
(
(
¬
X12
)
↔
X43
)
)
)
→
(
¬
(
(
¬
(
(
(
(
X48
↔
(
X46
→
(
(
¬
X6
)
↔
X45
)
)
)
↔
(
X46
→
X45
→
(
X6
↔
(
X4
→
False
)
)
)
)
↔
(
X45
↔
(
X6
↔
X48
)
)
)
→
(
X46
↔
False
)
)
)
→
(
(
¬
(
X48
→
(
¬
X31
)
→
(
¬
X25
)
→
X46
)
)
↔
(
¬
(
(
¬
(
(
¬
X45
)
↔
(
(
¬
X43
)
→
(
(
¬
X31
)
↔
X48
)
)
)
)
→
(
¬
(
X48
→
(
X46
↔
(
X45
→
X31
→
False
)
)
)
)
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
X43
)
↔
(
X33
→
X31
→
X44
)
)
)
→
(
¬
(
(
¬
X43
)
↔
(
X33
→
(
¬
X2
)
→
X44
)
)
)
)
→
(
¬
(
X33
↔
(
(
¬
X31
)
↔
(
(
¬
X1
)
↔
X43
)
)
)
)
)
↔
(
(
(
¬
X43
)
→
(
X33
↔
X49
)
)
↔
(
¬
(
X43
→
X31
→
X15
→
False
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
(
X41
↔
(
X27
→
(
X10
↔
X47
)
)
)
)
→
(
(
(
¬
(
(
(
X46
↔
(
X17
↔
X47
)
)
↔
(
¬
(
(
¬
(
X47
→
(
(
¬
X17
)
↔
X28
)
)
)
→
(
(
¬
X46
)
↔
(
(
¬
X17
)
→
(
¬
X10
)
→
X47
)
)
)
)
)
→
(
X46
↔
(
X10
↔
X41
)
)
)
)
↔
(
(
¬
X46
)
→
(
¬
X41
)
→
X47
)
)
↔
(
(
(
¬
(
X46
↔
(
X41
→
X7
→
X17
)
)
)
↔
(
(
¬
X41
)
↔
(
X9
→
(
¬
X7
)
→
X46
)
)
)
↔
(
¬
(
X41
→
X7
→
X9
)
)
)
)
)
→
(
(
(
(
X34
→
(
X30
↔
X27
)
)
↔
(
¬
(
X49
↔
(
X30
↔
(
X19
→
X27
)
)
)
)
)
↔
(
(
¬
X30
)
→
X27
→
X49
)
)
↔
(
¬
(
(
(
¬
X46
)
→
(
(
¬
X29
)
↔
X47
)
)
→
(
(
¬
(
X47
→
(
X46
↔
(
X19
→
X34
)
)
)
)
↔
(
¬
(
X47
↔
(
(
¬
X34
)
↔
(
X29
→
X46
)
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
(
X45
↔
(
X40
→
X39
→
False
)
)
)
→
X49
→
X40
→
(
(
¬
X39
)
↔
X47
)
)
)
→
(
¬
(
X49
→
(
X47
↔
(
X45
→
False
)
)
)
)
)
)
↔
(
(
¬
X47
)
↔
(
(
¬
X41
)
↔
(
X40
↔
(
X13
↔
X45
)
)
)
)
)
→
(
X45
↔
(
X39
↔
(
X35
→
X41
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X23
→
(
¬
X18
)
→
X32
)
)
↔
(
X38
↔
(
X36
↔
(
X32
→
False
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
¬
(
(
X42
→
(
(
¬
X23
)
↔
(
(
¬
X8
)
↔
X45
)
)
)
→
X45
→
(
X23
↔
X38
)
)
)
→
(
(
(
X45
↔
(
X42
↔
(
X24
↔
(
X8
→
False
)
)
)
)
→
(
¬
(
X45
↔
(
X24
↔
(
X11
→
X8
)
)
)
)
)
↔
(
X45
↔
(
X24
↔
(
X8
↔
False
)
)
)
)
)
)
→
(
¬
(
(
X38
→
X36
→
X37
)
→
(
¬
(
(
¬
(
X37
↔
(
X36
→
X23
)
)
)
→
(
¬
(
X42
↔
(
(
¬
X36
)
↔
(
X23
→
X38
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
X46
↔
(
X45
→
(
(
¬
X36
)
↔
X42
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X49
→
(
(
¬
X21
)
↔
(
(
¬
X3
)
↔
X46
)
)
)
)
↔
(
X38
→
(
X22
↔
(
X3
↔
False
)
)
)
)
↔
(
(
(
(
¬
(
X21
→
X16
)
)
↔
(
¬
(
X46
→
(
¬
X14
)
→
(
(
¬
X0
)
↔
X21
)
)
)
)
↔
(
X14
→
X5
→
X0
→
X46
)
)
↔
(
(
X49
↔
(
X42
↔
(
X20
→
X14
→
False
)
)
)
→
(
(
¬
X21
)
↔
(
X20
→
X5
→
X49
)
)
→
(
¬
X42
)
→
(
¬
X21
)
→
X20
→
X49
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
X47
↔
(
(
¬
X20
)
↔
(
X5
→
X37
)
)
)
→
(
¬
(
(
¬
X47
)
↔
(
X37
→
X20
→
X49
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X37
)
→
(
¬
X36
)
→
X19
→
X49
)
)
↔
(
(
(
(
¬
X34
)
↔
(
X19
↔
X37
)
)
→
(
¬
(
X31
↔
(
X19
→
(
X12
↔
X34
)
)
)
)
)
↔
(
¬
(
X31
↔
X36
)
)
)
)
→
(
¬
(
X47
↔
(
X34
↔
(
X12
→
X19
)
)
)
)
)
)
)
↔
(
(
¬
(
X45
→
(
(
¬
X31
)
↔
(
(
¬
X20
)
↔
X36
)
)
)
)
→
X45
→
X36
→
X4
→
X20
)
)
→
(
¬
(
(
(
(
(
¬
(
X45
→
X11
→
False
)
)
↔
(
¬
(
X24
↔
(
X23
↔
X9
)
)
)
)
→
(
¬
(
X49
↔
(
X45
→
(
X9
↔
X23
)
)
)
)
)
↔
(
(
X49
↔
(
X45
→
X27
→
(
X6
↔
False
)
)
)
→
(
¬
X27
)
→
(
¬
X24
)
→
X6
→
X45
)
)
→
(
(
(
¬
(
X49
↔
(
X37
→
X25
)
)
)
↔
(
(
X49
→
X23
→
(
¬
X21
)
→
X25
)
↔
(
¬
(
X49
→
(
(
¬
X25
)
↔
(
(
¬
X23
)
→
X37
)
)
)
)
)
)
↔
(
(
(
¬
X25
)
↔
(
X23
→
(
X6
↔
X37
)
)
)
↔
(
X37
↔
(
X23
↔
(
(
¬
X18
)
→
X25
)
)
)
)
)
)
)
)
↔
(
(
(
(
X47
→
(
X14
↔
(
X0
↔
X17
)
)
)
↔
(
(
¬
(
X17
↔
(
X0
↔
False
)
)
)
↔
(
¬
(
X43
→
(
¬
X3
)
→
X49
)
)
)
)
↔
(
(
(
(
¬
X48
)
↔
(
(
¬
X43
)
↔
(
X38
↔
X49
)
)
)
→
(
(
(
¬
(
X32
↔
(
X30
→
X16
→
X43
)
)
)
→
(
¬
(
(
¬
(
X49
↔
(
(
¬
X30
)
↔
(
(
¬
X7
)
→
X32
)
)
)
)
→
(
(
¬
(
(
¬
X43
)
↔
(
X32
↔
(
X13
↔
X49
)
)
)
)
↔
(
X49
→
X7
→
(
X2
↔
False
)
)
)
)
)
)
↔
(
¬
(
X43
↔
(
X38
↔
(
X30
↔
X49
)
)
)
)
)
)
↔
(
(
X49
↔
(
X47
↔
(
X29
→
X48
)
)
)
↔
(
(
¬
(
X47
↔
(
X45
↔
X49
)
)
)
↔
(
(
(
X48
↔
(
X47
→
X49
)
)
↔
(
(
(
¬
X48
)
↔
(
X47
↔
(
X38
→
X49
)
)
)
↔
(
¬
(
X44
→
X38
→
False
)
)
)
)
↔
(
¬
(
(
X49
→
(
X48
↔
(
X26
→
X47
)
)
)
→
(
¬
(
X49
→
(
X48
↔
(
(
¬
X26
)
↔
X45
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X46
→
(
X22
↔
X39
)
)
)
→
(
(
(
(
¬
(
X47
→
(
¬
X8
)
→
X22
)
)
→
(
¬
(
X46
→
(
(
¬
X22
)
↔
(
(
¬
X8
)
→
X35
)
)
)
)
→
(
X47
↔
(
X46
↔
(
X35
→
X8
)
)
)
)
→
(
¬
X47
)
→
(
(
¬
X8
)
↔
X48
)
)
↔
(
X35
↔
(
X8
↔
X15
)
)
)
→
(
(
¬
(
X35
→
X8
→
X48
)
)
↔
(
¬
(
X39
→
X8
→
X1
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
¬
X47
)
↔
(
(
¬
X46
)
→
(
(
¬
X41
)
↔
X49
)
)
)
→
(
¬
(
X49
→
X46
→
(
(
¬
X28
)
↔
X41
)
)
)
)
)
↔
(
¬
(
X47
↔
(
(
¬
X46
)
→
(
X40
↔
X49
)
)
)
)
)
→
(
¬
(
(
¬
(
X47
↔
(
(
¬
X46
)
→
(
X42
↔
X48
)
)
)
)
→
(
(
(
(
¬
(
X47
↔
(
X41
↔
(
(
¬
X33
)
↔
X42
)
)
)
)
↔
(
¬
(
X48
↔
(
(
¬
X41
)
→
X47
)
)
)
)
↔
(
¬
(
(
¬
X41
)
→
(
¬
X10
)
→
X47
)
)
)
↔
(
¬
(
(
¬
X47
)
↔
(
(
¬
X33
)
↔
X48
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
81a55f...
and proposition id is
4cb68f...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMEmtr6DPVbHedWm1sHvyAtW5kyFRCXViZo
)
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
(
X47
↔
(
X44
→
X41
→
X36
)
)
→
(
¬
(
X41
↔
X49
)
)
→
(
¬
(
X49
→
False
)
)
→
(
(
¬
(
(
¬
(
X49
↔
(
(
¬
X41
)
↔
(
X19
↔
X48
)
)
)
)
→
(
X48
↔
(
X47
→
X19
)
)
)
)
↔
(
¬
(
(
¬
X48
)
↔
(
X19
↔
X49
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X48
↔
(
X0
→
X46
)
)
)
→
(
¬
(
X49
→
(
X48
↔
(
X3
→
X6
)
)
)
)
→
(
(
X49
↔
(
(
¬
X46
)
↔
(
X6
→
X48
)
)
)
↔
(
X49
→
(
¬
X0
)
→
X46
)
)
)
→
(
(
¬
(
(
(
¬
X44
)
↔
(
(
¬
X9
)
↔
(
X6
→
X48
)
)
)
→
(
(
(
X17
→
X9
→
X6
→
X0
)
↔
(
(
¬
X6
)
↔
X44
)
)
↔
(
¬
(
X17
→
X9
→
X6
→
X11
)
)
)
)
)
↔
(
¬
(
X44
→
X0
)
)
)
)
)
)
→
(
(
¬
(
X49
↔
(
X45
↔
X18
)
)
)
↔
(
(
(
¬
X45
)
↔
(
X18
↔
(
(
¬
X5
)
→
X49
)
)
)
↔
(
¬
(
(
¬
X40
)
→
(
(
¬
X18
)
↔
(
(
¬
X5
)
→
X49
)
)
)
)
)
)
→
(
(
¬
(
X48
→
(
X43
↔
(
X28
→
False
)
)
)
)
↔
(
(
¬
(
(
¬
X47
)
→
(
(
¬
X43
)
↔
X48
)
)
)
→
(
(
(
¬
(
X47
↔
(
(
¬
X43
)
↔
(
(
¬
X40
)
→
X48
)
)
)
)
↔
(
X43
→
(
¬
X40
)
→
(
¬
X23
)
→
X48
)
)
↔
(
X47
→
(
¬
X40
)
→
X48
)
)
)
)
→
(
(
(
¬
(
(
(
¬
(
X45
→
X31
)
)
→
(
X47
↔
(
X34
↔
(
(
¬
X31
)
→
X49
)
)
)
→
(
¬
(
X49
→
X16
→
X45
)
)
)
→
(
¬
(
(
¬
X45
)
→
X31
→
(
¬
X22
)
→
X49
)
)
)
)
→
(
(
X47
↔
(
X45
↔
(
X40
↔
X43
)
)
)
↔
X49
)
)
→
(
X49
↔
(
(
¬
X40
)
→
(
(
¬
X35
)
↔
X45
)
)
)
)
→
(
(
¬
(
X49
→
(
X31
↔
(
(
¬
X13
)
→
X35
)
)
)
)
↔
(
X35
↔
(
X30
→
X13
)
)
)
)
→
(
(
¬
(
(
(
(
(
(
¬
(
(
¬
(
(
¬
X42
)
→
(
¬
X29
)
→
X26
→
X48
)
)
→
(
¬
(
X48
→
X47
→
(
¬
X29
)
→
X42
)
)
)
)
↔
(
X47
↔
(
(
¬
X42
)
↔
X48
)
)
)
→
(
¬
(
X42
↔
(
(
¬
X26
)
↔
X47
)
)
)
)
↔
(
(
¬
X29
)
→
(
(
¬
X26
)
↔
X47
)
)
)
↔
(
(
X48
→
X47
→
X42
→
(
X33
↔
False
)
)
→
X33
→
(
X20
↔
X42
)
)
)
→
(
(
(
(
X48
→
X47
→
(
¬
X24
)
→
X45
)
→
X48
→
(
X47
↔
(
X45
→
X27
)
)
)
↔
(
¬
(
X48
→
X45
→
(
X27
↔
X21
)
)
)
)
↔
(
(
X47
↔
(
X42
→
(
¬
X12
)
→
X37
)
)
→
(
¬
(
(
(
¬
X42
)
→
(
X37
↔
(
X27
→
X48
)
)
)
→
(
¬
(
X42
↔
(
(
¬
X27
)
↔
(
(
¬
X8
)
→
X47
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
X38
→
(
X37
↔
(
(
¬
X1
)
↔
X4
)
)
)
→
(
¬
(
(
¬
X37
)
→
(
(
¬
X4
)
↔
X45
)
)
)
)
)
↔
(
(
X4
↔
(
X1
↔
X38
)
)
↔
(
¬
(
X33
→
X2
→
(
(
¬
X1
)
↔
X38
)
)
)
)
)
→
(
(
¬
(
(
X45
→
(
(
¬
X33
)
↔
(
X14
→
X39
)
)
)
→
(
¬
(
(
¬
(
(
X37
↔
(
X33
→
(
X14
↔
X15
)
)
)
→
(
(
¬
X15
)
↔
(
X10
↔
X33
)
)
)
)
→
X37
→
X33
→
X10
)
)
)
)
↔
(
(
¬
(
X45
↔
(
X15
↔
(
X10
↔
X32
)
)
)
)
→
(
¬
(
X25
↔
(
(
¬
X7
)
↔
X45
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
(
¬
X36
)
→
X22
→
(
X11
↔
X37
)
)
→
(
(
X46
→
(
(
¬
X37
)
↔
(
(
¬
X36
)
→
X45
)
)
)
↔
(
¬
(
(
X45
→
(
(
¬
X36
)
↔
(
X33
→
X37
)
)
)
→
(
X36
↔
(
X22
↔
X37
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
X36
→
X46
)
↔
(
X36
→
X29
→
(
(
¬
X8
)
↔
X33
)
)
)
↔
(
¬
(
(
¬
X36
)
↔
(
X33
↔
(
X29
→
X39
)
)
)
)
)
→
(
X36
↔
(
X35
↔
False
)
)
→
(
X29
↔
(
(
¬
X14
)
→
(
X8
↔
X20
)
)
)
)
)
)
)
→
(
(
¬
(
X45
↔
(
X19
→
(
(
¬
X16
)
↔
X48
)
)
)
)
→
(
¬
(
(
¬
(
X45
↔
(
X38
↔
(
(
¬
X16
)
↔
X27
)
)
)
)
→
(
¬
(
X48
→
X38
→
(
X36
↔
(
X16
→
False
)
)
)
)
)
)
)
→
(
(
X48
↔
(
X36
↔
(
X23
↔
X5
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X36
)
↔
(
X33
→
X27
→
X40
)
)
)
→
(
¬
(
(
¬
X36
)
→
(
(
¬
X33
)
↔
(
(
¬
X23
)
↔
X40
)
)
)
)
)
)
→
(
¬
(
X23
↔
X33
)
)
)
)
)
↔
(
(
(
(
(
(
X49
→
(
X10
↔
(
(
¬
X3
)
→
X32
)
)
)
→
X32
→
(
(
¬
X21
)
↔
(
(
¬
X10
)
↔
X46
)
)
)
→
(
¬
(
X48
→
(
X21
↔
(
X10
→
X32
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X46
↔
(
X43
↔
X0
)
)
)
→
(
X43
↔
(
X9
→
(
(
¬
X0
)
↔
X7
)
)
)
)
)
↔
(
(
X43
→
(
X7
↔
(
(
¬
X0
)
↔
X32
)
)
)
↔
(
X43
↔
(
X10
↔
X12
)
)
)
)
)
↔
(
¬
(
(
(
(
(
X47
→
X46
→
(
(
¬
X18
)
↔
X25
)
)
→
X48
→
X47
→
(
X45
↔
(
X18
↔
False
)
)
)
↔
(
(
¬
X25
)
↔
(
X1
→
X46
)
)
)
→
(
X48
↔
(
X46
→
X47
)
)
)
→
(
(
¬
(
(
(
¬
(
X43
→
(
X15
↔
X46
)
)
)
↔
(
X47
↔
(
X6
↔
X43
)
)
)
→
(
¬
(
X45
↔
(
(
¬
X6
)
→
X46
)
)
)
)
)
↔
(
X46
↔
(
(
¬
X43
)
→
X17
→
X45
)
)
)
→
(
X43
↔
(
(
¬
X6
)
↔
(
X4
→
X46
)
)
)
)
)
)
→
(
(
(
(
X42
↔
X46
)
→
(
¬
(
X44
↔
X13
)
)
→
(
¬
(
(
¬
X42
)
↔
(
X30
→
X46
)
)
)
)
↔
(
(
X46
→
X44
→
X34
→
X42
)
→
(
(
¬
X34
)
↔
(
X2
↔
X44
)
)
→
X44
→
(
(
¬
X43
)
↔
(
(
¬
X24
)
↔
X46
)
)
)
)
↔
(
(
¬
(
X46
↔
(
(
¬
X41
)
→
X31
→
X44
)
)
)
↔
(
¬
(
(
X48
→
(
¬
X31
)
→
X30
→
X41
)
→
(
(
(
¬
(
X30
→
(
X28
↔
(
X26
↔
X31
)
)
)
)
→
(
¬
(
X41
↔
(
X30
↔
X44
)
)
)
)
↔
(
(
¬
X26
)
↔
X41
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
914e95...
and proposition id is
f9dd8a...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMT22digjE5WsUmkY8H59HJNCSBBtNsv617
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
(
¬
(
(
(
(
¬
(
X45
↔
(
X24
↔
(
(
¬
X14
)
→
X49
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
X14
)
→
X9
→
X49
)
→
(
X49
↔
(
X45
→
(
X14
↔
(
X5
↔
False
)
)
)
)
)
)
↔
(
¬
(
X45
→
(
X14
↔
False
)
)
)
)
→
(
X14
↔
(
X5
→
X45
)
)
)
)
→
(
¬
(
X49
↔
(
X45
→
(
X9
↔
X44
)
)
)
)
)
→
X49
→
(
X44
↔
(
X39
↔
X45
)
)
)
)
↔
(
(
X39
→
X14
→
(
X2
↔
X7
)
)
↔
(
(
¬
(
X44
↔
(
X39
↔
False
)
)
)
↔
(
(
¬
(
X44
↔
X7
)
)
↔
(
X44
↔
(
X41
↔
(
X14
→
X39
)
)
)
)
)
)
)
↔
(
(
(
X29
→
(
X25
↔
(
(
¬
X14
)
↔
X41
)
)
)
→
(
¬
(
X41
→
(
X14
↔
X19
)
)
)
)
↔
(
(
(
¬
(
(
¬
X38
)
→
(
(
¬
X25
)
↔
(
X13
↔
X41
)
)
)
)
↔
(
¬
(
(
¬
X38
)
↔
(
X24
↔
(
(
¬
X13
)
→
X41
)
)
)
)
)
→
(
¬
(
X41
↔
(
X38
↔
(
X19
↔
X24
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
X48
↔
(
(
¬
X33
)
↔
X46
)
)
)
↔
(
¬
(
(
X48
→
(
¬
X33
)
→
(
(
¬
X1
)
↔
X46
)
)
→
(
¬
(
X33
↔
(
(
¬
X21
)
↔
(
X1
↔
X48
)
)
)
)
)
)
)
↔
(
(
¬
X46
)
→
(
¬
X33
)
→
(
¬
X31
)
→
X48
)
)
↔
(
¬
(
X48
→
(
¬
X41
)
→
X31
→
X46
)
)
)
→
(
¬
(
X46
→
(
¬
X31
)
→
(
X1
↔
X39
)
)
)
)
→
(
(
¬
(
(
¬
X39
)
↔
(
(
¬
X15
)
↔
X46
)
)
)
↔
(
(
¬
(
X39
↔
(
(
¬
X6
)
↔
X46
)
)
)
↔
(
X45
↔
(
(
¬
X37
)
↔
(
X6
↔
X39
)
)
)
)
)
→
(
(
(
X48
→
X37
→
(
(
¬
X15
)
↔
X36
)
)
↔
(
¬
(
X45
→
(
(
¬
X15
)
↔
X37
)
)
)
)
↔
(
¬
(
(
¬
X36
)
↔
(
(
¬
X15
)
→
X48
)
)
)
)
)
)
→
(
(
(
(
¬
X42
)
↔
(
X20
↔
X48
)
)
↔
(
¬
(
(
¬
(
X47
↔
(
(
¬
X46
)
→
(
¬
X27
)
→
X48
)
)
)
→
(
X47
↔
(
(
¬
X27
)
→
(
¬
X20
)
→
X46
)
)
)
)
)
↔
(
(
¬
(
X42
↔
(
X16
→
X47
)
)
)
↔
(
(
(
¬
(
X48
→
(
X47
↔
(
X16
→
X11
)
)
)
)
→
(
¬
(
(
¬
X16
)
↔
(
X11
↔
(
X0
↔
X48
)
)
)
)
)
↔
(
¬
(
(
¬
X32
)
→
(
X16
↔
(
(
¬
X11
)
↔
X42
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
X47
→
(
X34
↔
(
X28
↔
X46
)
)
)
↔
(
¬
(
X46
↔
(
X10
→
X47
)
)
)
)
↔
(
(
X48
→
(
¬
X30
)
→
(
(
¬
X23
)
↔
X46
)
)
→
(
¬
(
X48
↔
(
X47
→
(
X34
↔
X23
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
X26
→
(
(
¬
X22
)
↔
X48
)
)
→
(
(
(
¬
X46
)
→
(
(
¬
X26
)
↔
(
(
¬
X18
)
→
X47
)
)
)
↔
(
(
(
¬
X46
)
↔
(
X26
→
X12
→
X48
)
)
↔
(
¬
(
X48
→
X46
→
(
(
¬
X12
)
↔
X18
)
)
)
)
)
)
)
→
X47
→
(
¬
X22
)
→
X26
)
)
)
→
(
¬
(
(
(
X44
→
(
X40
↔
(
(
¬
X35
)
↔
X48
)
)
)
↔
(
(
X48
↔
(
X40
→
X44
)
)
↔
(
¬
(
X47
→
(
¬
X43
)
→
(
(
¬
X17
)
↔
X48
)
)
)
)
)
→
(
(
(
X46
→
X35
→
X43
)
↔
(
(
¬
(
(
¬
(
X48
↔
(
X43
→
X8
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X43
)
→
(
¬
X35
)
→
X48
)
)
)
)
)
→
(
X43
↔
(
X35
→
X48
)
)
)
)
↔
(
(
(
(
¬
X46
)
↔
(
X4
↔
X48
)
)
↔
(
X35
→
(
¬
X4
)
→
(
¬
X3
)
→
X46
)
)
↔
(
X4
→
(
X3
↔
X35
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
(
¬
(
X49
↔
(
X48
↔
(
(
¬
X33
)
→
X44
)
)
)
)
↔
(
¬
(
X44
→
X33
→
X34
)
)
)
→
(
¬
(
X49
↔
(
X48
↔
(
X44
↔
X29
)
)
)
)
)
)
↔
(
(
X44
→
(
X33
↔
(
X2
↔
X29
)
)
)
→
(
¬
(
(
¬
(
X33
→
(
X29
↔
X20
)
)
)
→
(
(
(
¬
(
X33
→
(
(
¬
X29
)
↔
(
X20
→
X44
)
)
)
)
→
(
X29
↔
(
(
¬
X20
)
→
(
¬
X8
)
→
X44
)
)
)
↔
(
X33
↔
(
X20
→
X44
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X33
↔
(
(
¬
X22
)
→
(
X21
↔
X38
)
)
)
)
→
X38
→
X21
→
X44
)
)
→
(
(
¬
X22
)
↔
(
X11
→
X38
)
)
)
→
(
(
X49
↔
(
(
¬
X38
)
↔
(
X25
→
X44
)
)
)
↔
(
¬
(
(
¬
X38
)
→
(
(
¬
X25
)
↔
(
(
¬
X4
)
↔
X49
)
)
)
)
)
→
(
(
(
¬
X38
)
↔
(
X33
→
X49
)
)
↔
(
(
X49
↔
(
X25
↔
(
(
¬
X23
)
↔
X38
)
)
)
→
(
¬
(
(
¬
X38
)
→
(
(
¬
X33
)
↔
(
X25
→
X49
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X49
→
(
¬
X19
)
→
(
¬
X3
)
→
X38
)
)
→
(
¬
(
(
¬
X48
)
→
(
(
¬
X19
)
↔
X49
)
)
)
)
↔
(
X48
→
X38
→
X49
)
)
↔
(
¬
(
(
¬
(
(
¬
X19
)
→
(
X12
↔
(
X3
→
X45
)
)
)
)
→
X19
→
(
X12
↔
X48
)
)
)
)
↔
(
(
(
(
¬
X37
)
→
X9
→
X38
)
↔
(
¬
(
(
¬
X37
)
↔
(
(
¬
X30
)
→
X9
→
X45
)
)
)
)
→
(
(
(
X49
→
X32
→
(
X27
↔
(
X9
↔
False
)
)
)
→
(
¬
(
(
¬
X37
)
→
(
X32
↔
(
X27
↔
X45
)
)
)
)
)
↔
(
X37
↔
(
(
¬
X32
)
→
X49
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
¬
(
(
¬
X40
)
→
(
(
¬
X17
)
↔
X48
)
)
)
→
(
(
(
¬
X46
)
→
(
¬
X40
)
→
(
¬
X17
)
→
X48
)
↔
(
¬
(
(
¬
X40
)
↔
(
X17
→
(
X15
↔
X48
)
)
)
)
)
)
)
→
X46
→
X43
→
(
X40
↔
X17
)
)
↔
(
(
¬
(
X43
↔
(
(
¬
X39
)
↔
(
(
¬
X18
)
→
X40
)
)
)
)
↔
(
¬
(
X39
↔
(
X18
↔
X48
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X49
↔
(
X42
↔
X46
)
)
)
→
(
¬
(
X49
→
(
X48
↔
(
(
¬
X44
)
↔
X46
)
)
)
)
)
↔
(
X46
→
X42
→
(
X5
↔
X49
)
)
)
↔
(
(
¬
(
(
¬
X13
)
→
(
X7
↔
X16
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
X16
)
→
(
X13
↔
(
(
¬
X5
)
→
X42
)
)
)
)
→
(
¬
(
(
¬
X42
)
→
(
X16
↔
(
(
¬
X13
)
↔
X46
)
)
)
)
)
)
↔
(
X42
→
(
X13
↔
X16
)
)
)
)
)
→
(
X48
↔
(
(
¬
X44
)
→
(
(
¬
X18
)
↔
X49
)
)
)
)
)
→
(
(
(
(
(
(
¬
(
(
¬
X46
)
→
(
(
¬
X41
)
↔
(
X31
↔
X49
)
)
)
)
↔
(
X49
↔
(
X46
↔
(
X31
↔
X6
)
)
)
)
→
(
¬
(
X46
→
X41
→
X31
)
)
)
↔
(
(
X48
↔
(
(
¬
X41
)
↔
(
X31
→
X46
)
)
)
↔
(
¬
(
X48
→
(
(
¬
X41
)
↔
(
X10
→
X46
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X47
↔
(
X35
→
X28
)
)
)
→
(
(
¬
(
(
¬
X24
)
→
X14
→
X35
)
)
↔
(
(
¬
(
X28
↔
(
X14
↔
X47
)
)
)
→
(
¬
(
X47
→
(
X35
↔
X24
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X35
→
X31
→
(
(
¬
X24
)
↔
X28
)
)
)
→
(
¬
(
X49
↔
(
X24
→
X35
)
)
)
→
(
(
(
¬
(
X46
↔
(
X35
→
(
¬
X0
)
→
X28
)
)
)
→
(
¬
(
X35
→
(
(
¬
X24
)
↔
X46
)
)
)
)
↔
(
X46
↔
X0
)
)
)
)
↔
(
X49
→
(
X46
↔
(
(
¬
X1
)
↔
X31
)
)
)
)
)
)
→
(
¬
(
(
¬
X47
)
→
(
¬
X46
)
→
X49
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X47
)
↔
(
(
¬
X46
)
→
X48
)
)
)
→
(
¬
(
(
¬
(
X48
↔
(
X47
↔
(
X36
→
X26
)
)
)
)
→
X46
→
(
(
¬
X36
)
↔
(
(
¬
X26
)
→
X48
)
)
)
)
→
(
¬
(
X48
→
(
X47
↔
(
X36
→
False
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
630b0b...
and proposition id is
5538e7...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMc42k6Uuk6FzNnhgFEuyT7SFCa7vS5CYV8
)
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
¬
(
(
(
(
(
¬
(
X47
→
(
X46
↔
X49
)
)
)
→
(
(
¬
X46
)
→
X20
→
X47
)
→
(
(
¬
(
X46
↔
(
X20
→
(
¬
X6
)
→
X49
)
)
)
↔
(
(
X47
→
(
X5
↔
X6
)
)
→
(
¬
(
X46
→
X20
→
X6
→
X47
)
)
)
)
)
↔
(
(
X20
↔
(
X11
↔
X46
)
)
→
(
¬
(
(
¬
(
X46
→
X41
→
X11
→
False
)
)
→
(
X49
↔
(
X11
→
X20
)
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X41
)
↔
X49
)
)
)
)
)
↔
(
(
(
X39
→
X17
)
→
(
¬
(
X46
↔
(
X33
↔
(
(
¬
X22
)
↔
X49
)
)
)
)
→
X33
→
(
X17
↔
X49
)
)
↔
(
(
(
¬
(
X1
↔
X34
)
)
↔
(
(
¬
(
X49
↔
(
X34
↔
(
X1
↔
X39
)
)
)
)
→
(
¬
(
(
¬
X39
)
→
X34
→
(
¬
X1
)
→
X46
)
)
)
)
↔
(
¬
(
X47
→
False
)
)
)
)
)
→
(
(
(
(
¬
(
X28
↔
(
(
¬
X27
)
→
X47
)
)
)
→
(
¬
(
(
¬
(
X28
↔
(
X24
→
X27
)
)
)
→
(
¬
(
X48
→
X7
→
X27
)
)
)
)
→
(
¬
X24
)
→
X28
)
→
(
(
¬
(
(
(
¬
(
(
¬
(
(
X48
↔
X7
)
→
X19
→
X16
→
(
(
¬
X7
)
↔
X37
)
)
)
→
X37
→
(
¬
X16
)
→
X48
)
)
↔
(
¬
(
X48
→
(
¬
X28
)
→
X37
)
)
)
→
(
¬
(
X37
↔
(
X16
↔
False
)
)
)
)
)
↔
(
¬
(
(
¬
X19
)
↔
(
(
¬
X16
)
↔
(
X3
↔
X37
)
)
)
)
)
)
↔
(
(
(
(
X47
↔
(
(
¬
X14
)
→
X48
)
)
↔
(
¬
(
X38
→
(
(
¬
X19
)
↔
X48
)
)
)
)
→
(
¬
(
(
X49
↔
(
X38
↔
(
(
¬
X19
)
↔
X47
)
)
)
→
(
(
X49
↔
(
X47
→
(
¬
X9
)
→
X46
)
)
↔
(
¬
(
X49
→
(
¬
X46
)
→
(
X38
↔
X47
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X49
↔
(
X47
↔
(
X38
↔
(
X15
→
False
)
)
)
)
)
→
(
¬
(
(
¬
X45
)
↔
(
(
¬
X15
)
→
X49
)
)
)
)
↔
(
(
¬
X38
)
→
X15
→
X47
)
)
)
)
)
)
↔
(
(
(
(
(
X45
→
(
X42
↔
X40
)
)
→
(
X40
↔
(
X35
→
X26
→
X42
)
)
→
X45
→
X35
→
X42
)
↔
(
X47
↔
(
X26
↔
X45
)
)
)
↔
(
¬
(
(
X36
↔
(
X2
→
X26
)
)
→
(
X49
↔
(
X47
→
(
X40
↔
False
)
)
)
)
)
)
↔
(
¬
(
(
(
X43
↔
(
X23
→
(
X12
↔
False
)
)
)
→
(
(
(
X47
→
(
X31
↔
X45
)
)
→
(
(
¬
(
(
¬
X31
)
↔
(
X30
→
X47
)
)
)
↔
(
X45
↔
(
X31
→
X29
→
X47
)
)
)
)
↔
(
¬
(
X45
→
(
X31
↔
(
X30
↔
(
X13
→
False
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X44
→
False
)
)
→
(
¬
(
X45
↔
(
X42
↔
(
(
¬
X25
)
↔
X47
)
)
)
)
→
(
(
¬
X42
)
↔
(
(
¬
X32
)
↔
(
X25
→
X44
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X31
)
↔
(
X21
↔
(
X10
→
X44
)
)
)
)
→
X31
→
(
(
¬
X8
)
↔
X10
)
)
)
→
(
(
(
(
¬
(
(
¬
X31
)
↔
(
(
¬
X4
)
↔
X47
)
)
)
↔
(
¬
(
X47
↔
(
(
¬
X4
)
→
X0
→
X45
)
)
)
)
↔
(
(
¬
(
X45
→
X44
→
X31
→
False
)
)
→
(
¬
(
(
¬
(
(
¬
X44
)
→
(
¬
X31
)
→
(
X4
↔
X45
)
)
)
→
(
X44
↔
(
X31
↔
(
(
¬
X4
)
→
X10
)
)
)
)
)
)
)
↔
(
X45
→
(
X44
↔
(
X31
↔
(
X18
↔
False
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
(
¬
(
(
(
¬
X45
)
→
X19
→
(
¬
X7
)
→
X48
)
→
(
(
¬
X45
)
↔
(
X42
→
X7
→
X48
)
)
)
)
↔
(
¬
(
(
(
(
X42
→
X34
→
(
¬
X11
)
→
X13
)
→
(
X45
↔
(
X34
↔
(
X13
→
X42
)
)
)
)
↔
(
X13
↔
(
X11
↔
X34
)
)
)
→
(
¬
(
X48
→
X45
→
X42
→
X13
)
)
)
)
)
→
(
(
(
X48
→
(
¬
X42
)
→
X30
→
X45
)
↔
(
(
¬
X45
)
↔
(
(
¬
X42
)
↔
(
X31
→
X48
)
)
)
)
↔
(
(
X42
→
(
X25
↔
X45
)
)
↔
(
(
X42
↔
(
X36
→
(
X25
↔
False
)
)
)
→
(
¬
(
X48
→
X42
→
X21
→
X25
)
)
)
)
)
)
)
→
(
¬
(
X45
↔
(
X41
↔
(
X12
↔
X47
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
X41
)
→
(
X25
↔
(
X18
→
X47
)
)
)
)
→
X45
→
(
X41
↔
(
X25
↔
X47
)
)
)
)
↔
(
(
(
¬
(
(
¬
X41
)
→
X3
→
(
(
¬
X0
)
↔
X47
)
)
)
↔
(
¬
(
X41
→
(
(
¬
X0
)
↔
X2
)
)
)
)
↔
(
X30
→
(
X18
↔
(
X0
↔
False
)
)
)
)
)
)
↔
(
(
(
(
(
¬
X44
)
↔
(
X37
→
X45
)
)
↔
(
(
(
¬
(
X45
→
X44
→
X43
)
)
↔
(
(
X45
→
X43
→
X37
→
X46
)
→
(
¬
(
(
¬
X44
)
↔
(
(
¬
X37
)
→
X45
)
)
)
)
)
→
X46
→
(
X37
↔
(
(
¬
X6
)
→
X45
)
)
)
)
↔
(
(
X48
→
X37
→
(
X10
↔
X45
)
)
↔
(
¬
(
(
X45
→
(
(
¬
X43
)
↔
(
(
¬
X37
)
↔
X47
)
)
)
→
(
¬
(
(
¬
X45
)
→
(
(
¬
X37
)
↔
(
(
¬
X14
)
→
X47
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X43
)
→
(
(
¬
X28
)
↔
X45
)
)
)
→
(
¬
X43
)
→
(
¬
X17
)
→
X48
)
)
↔
(
X45
→
(
X43
↔
(
(
¬
X15
)
→
X28
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X47
→
X39
→
(
X38
↔
X40
)
)
)
↔
(
X47
→
(
X4
↔
X38
)
)
)
→
(
X48
↔
(
(
¬
X38
)
→
(
X22
↔
X40
)
)
)
→
(
¬
(
(
¬
(
X38
↔
(
X16
↔
(
X8
→
X48
)
)
)
)
→
X38
→
X20
→
(
X16
↔
(
X8
→
False
)
)
)
)
→
(
X8
→
(
X1
↔
X20
)
)
→
(
(
X48
↔
(
X8
→
(
X1
↔
X20
)
)
)
↔
(
X40
→
(
X20
↔
X4
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X39
)
→
(
X33
↔
(
(
¬
X27
)
→
X48
)
)
)
)
→
(
X48
↔
(
X44
→
(
¬
X26
)
→
X33
)
)
→
(
¬
(
(
¬
X44
)
→
(
X39
↔
X48
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
X49
↔
(
(
¬
X23
)
↔
X48
)
)
)
→
(
X48
↔
(
X44
↔
X32
)
)
)
)
→
(
¬
(
(
(
¬
X39
)
↔
(
X32
↔
(
(
¬
X9
)
↔
X44
)
)
)
→
(
¬
(
X39
↔
X49
)
)
)
)
→
X49
→
X32
→
(
X9
↔
X29
)
)
→
(
(
(
¬
(
(
(
X24
→
X5
)
↔
(
X44
→
(
X39
↔
(
X38
→
X5
→
False
)
)
)
)
→
X39
→
False
)
)
→
(
¬
(
X39
↔
(
X38
↔
(
X35
→
X44
)
)
)
)
)
↔
(
X49
→
(
X5
↔
False
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
172808...
and proposition id is
c7f9a7...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMZfYqBBcQ3pRNiMPKKkTyhbnAWJNWgefyx
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
(
(
¬
(
(
¬
(
(
¬
X43
)
↔
(
X40
↔
X49
)
)
)
→
(
¬
(
(
¬
X40
)
→
X24
→
(
(
¬
X2
)
↔
X49
)
)
)
)
)
→
(
(
X40
→
(
X36
↔
(
(
¬
X32
)
→
X43
)
)
)
→
(
¬
(
(
¬
X40
)
→
(
¬
X32
)
→
(
(
¬
X29
)
↔
X43
)
)
)
)
→
(
¬
(
X40
→
(
X32
↔
(
(
¬
X29
)
↔
X49
)
)
)
)
)
↔
(
¬
(
(
¬
(
X49
→
X41
→
X36
)
)
→
(
¬
(
(
(
¬
(
(
¬
X36
)
↔
(
X32
→
(
(
¬
X19
)
↔
X41
)
)
)
)
↔
(
¬
(
X41
↔
(
X32
→
(
(
¬
X6
)
↔
X19
)
)
)
)
)
→
(
¬
(
(
¬
X32
)
→
X19
→
(
X15
↔
X49
)
)
)
→
(
(
(
¬
X19
)
→
(
(
¬
X9
)
↔
(
(
¬
X7
)
→
X32
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X36
)
↔
(
(
¬
X32
)
↔
(
X9
↔
X49
)
)
)
)
→
(
¬
(
(
¬
X32
)
→
(
(
¬
X7
)
↔
X36
)
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X49
↔
(
X43
→
(
¬
X18
)
→
X41
)
)
)
→
(
X43
↔
(
(
¬
X41
)
→
(
¬
X33
)
→
X46
)
)
→
(
¬
(
(
¬
X46
)
→
(
X43
↔
(
X20
↔
X49
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
X46
→
(
X37
↔
(
(
¬
X30
)
→
X43
)
)
)
→
(
X37
↔
(
(
¬
X30
)
→
(
(
¬
X25
)
↔
X43
)
)
)
)
)
→
(
¬
X43
)
→
(
¬
X41
)
→
(
¬
X39
)
→
X46
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
(
(
(
¬
(
X48
→
X46
→
X22
)
)
↔
(
¬
(
(
¬
X46
)
→
(
X41
↔
(
X16
→
X48
)
)
)
)
)
↔
(
X48
↔
(
(
¬
X17
)
↔
X22
)
)
)
→
(
¬
(
(
¬
X41
)
↔
(
X22
↔
X48
)
)
)
)
)
→
(
X46
↔
(
X17
→
X41
)
)
)
)
→
(
¬
(
X41
→
(
¬
X34
)
→
(
¬
X22
)
→
X48
)
)
)
)
↔
(
¬
(
X48
↔
(
(
¬
X41
)
↔
(
(
¬
X39
)
→
X46
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
(
¬
(
(
¬
X31
)
→
(
¬
X26
)
→
(
(
¬
X23
)
↔
X47
)
)
)
↔
(
(
X47
→
X48
→
X31
→
(
X8
↔
X26
)
)
↔
(
¬
(
(
¬
X26
)
↔
(
(
¬
X8
)
↔
X48
)
)
)
)
)
→
(
¬
(
(
¬
X8
)
→
X26
)
)
→
(
X48
↔
(
X13
↔
(
X8
→
X45
)
)
)
)
)
→
(
(
X26
→
(
X23
↔
X48
)
)
↔
(
(
X49
→
(
(
¬
X26
)
↔
(
(
¬
X23
)
↔
X46
)
)
)
↔
(
X46
→
(
(
¬
X26
)
↔
(
(
¬
X23
)
↔
X48
)
)
)
)
)
→
(
¬
(
X46
↔
(
X35
→
(
(
¬
X26
)
↔
X49
)
)
)
)
)
)
→
(
(
(
X49
↔
(
X48
↔
(
X46
→
False
)
)
)
→
(
¬
(
X49
↔
(
X46
↔
(
X0
→
X28
)
)
)
)
)
↔
(
¬
(
(
¬
X48
)
↔
(
X28
→
(
(
¬
X27
)
↔
X49
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X48
→
(
(
¬
X21
)
↔
(
(
¬
X1
)
→
X45
)
)
)
)
↔
(
X45
↔
(
X44
↔
(
(
¬
X21
)
→
X48
)
)
)
)
↔
(
X44
↔
(
(
¬
X21
)
→
(
X14
↔
X48
)
)
)
)
→
(
(
(
¬
(
X44
→
False
)
)
→
X45
→
X38
→
(
¬
X21
)
→
X46
)
↔
(
¬
(
X21
→
(
(
¬
X12
)
↔
X38
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X48
→
(
X21
↔
(
X11
↔
(
X4
↔
False
)
)
)
)
)
→
(
¬
X44
)
→
(
(
¬
(
(
¬
(
(
¬
X21
)
→
(
(
¬
X5
)
↔
X48
)
)
)
→
(
X44
↔
(
X42
↔
X48
)
)
)
)
↔
(
¬
(
(
¬
X42
)
↔
(
(
¬
X21
)
→
(
¬
X11
)
→
X44
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X44
↔
(
(
¬
X21
)
→
X48
)
)
)
→
(
(
¬
(
X44
↔
(
(
¬
X3
)
↔
X21
)
)
)
↔
(
X21
→
(
X10
↔
X48
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
(
¬
(
X48
↔
(
(
¬
X47
)
→
(
(
¬
X11
)
↔
X49
)
)
)
)
→
X48
→
(
X41
↔
X49
)
)
↔
(
¬
(
X47
→
(
¬
X41
)
→
(
X17
↔
X49
)
)
)
)
→
(
¬
(
(
(
X41
→
(
X25
↔
(
(
¬
X22
)
→
X49
)
)
)
→
(
(
(
(
X22
→
X25
)
↔
(
(
¬
(
X41
↔
(
X36
↔
X25
)
)
)
↔
(
¬
(
(
¬
X22
)
→
(
¬
X19
)
→
X25
)
)
)
)
↔
(
¬
(
(
¬
X22
)
→
(
(
¬
X19
)
↔
X36
)
)
)
)
↔
(
¬
(
X41
↔
(
(
¬
X22
)
→
X25
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X47
)
↔
(
X25
→
X19
→
X49
)
)
)
→
(
¬
(
X49
→
X13
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
X27
↔
(
(
¬
X24
)
↔
X46
)
)
→
(
(
¬
X43
)
↔
(
X24
↔
(
X6
→
X46
)
)
)
)
→
(
¬
(
X46
→
X43
→
(
(
¬
X16
)
↔
X27
)
)
)
)
)
→
(
(
(
X48
→
X47
→
X27
→
(
X0
↔
False
)
)
→
(
¬
(
X47
↔
(
(
¬
X16
)
↔
(
X0
↔
X27
)
)
)
)
)
↔
(
(
X27
→
(
X16
↔
(
(
¬
X9
)
↔
X43
)
)
)
→
(
(
¬
X27
)
↔
(
X16
↔
X49
)
)
)
)
)
)
→
(
(
(
¬
(
X49
↔
(
X46
→
(
¬
X1
)
→
X21
)
)
)
→
(
X46
↔
(
X39
↔
(
X1
↔
False
)
)
)
→
(
¬
(
X33
↔
(
X32
→
(
(
¬
X1
)
↔
X39
)
)
)
)
)
↔
(
(
¬
X46
)
→
X39
→
(
¬
X33
)
→
X47
)
)
)
)
↔
(
(
(
¬
(
(
(
(
(
X48
↔
(
X34
→
X12
)
)
↔
(
¬
(
X48
↔
(
X20
→
X7
→
X34
)
)
)
)
→
(
¬
(
(
¬
X44
)
↔
(
(
¬
X34
)
→
(
¬
X20
)
→
X48
)
)
)
)
→
(
¬
(
(
¬
X34
)
→
(
(
¬
X20
)
↔
(
X12
↔
X49
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X38
)
↔
(
(
¬
X15
)
↔
(
X14
↔
X48
)
)
)
)
→
(
(
¬
(
X49
→
X15
)
)
↔
(
(
¬
(
X49
→
X35
)
)
→
(
¬
(
X49
↔
(
(
¬
X38
)
↔
(
X14
→
X44
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
¬
(
X44
→
(
(
¬
X20
)
↔
X29
)
)
)
↔
(
(
X20
→
(
(
¬
X10
)
↔
X29
)
)
↔
(
(
¬
X44
)
↔
(
X20
→
(
¬
X10
)
→
X45
)
)
)
)
→
(
X26
↔
(
X10
→
(
X4
↔
X20
)
)
)
)
)
↔
(
X44
→
(
¬
X20
)
→
X26
)
)
→
(
¬
(
(
X35
↔
(
(
¬
X20
)
→
(
X8
↔
X40
)
)
)
→
(
(
(
(
¬
(
X44
→
(
X3
↔
X45
)
)
)
→
(
(
X40
→
X8
→
X3
)
↔
(
X44
↔
(
(
¬
X8
)
→
(
¬
X3
)
→
X45
)
)
)
)
→
(
¬
(
X45
→
(
X44
↔
(
X40
→
False
)
)
)
)
)
↔
(
(
¬
X8
)
→
X44
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
X48
→
(
X42
↔
(
X37
↔
(
X23
↔
False
)
)
)
)
→
(
¬
(
(
(
¬
X37
)
↔
(
(
¬
X31
)
↔
X49
)
)
→
(
¬
(
(
¬
(
(
¬
(
X48
→
(
X37
↔
(
(
¬
X31
)
→
X49
)
)
)
)
→
(
¬
(
(
¬
X31
)
↔
(
X23
↔
X48
)
)
)
)
)
→
(
¬
(
X37
→
(
(
¬
X18
)
↔
X48
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X49
→
(
X48
↔
(
X42
↔
X46
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X2
)
↔
X28
)
)
↔
(
¬
(
X49
↔
(
X40
→
(
X28
↔
X48
)
)
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X30
)
↔
(
(
¬
X5
)
→
X48
)
)
)
)
→
(
(
¬
X46
)
→
(
¬
X30
)
→
(
¬
X28
)
→
X48
)
→
(
¬
(
X48
→
(
(
¬
X46
)
↔
(
(
¬
X30
)
→
X49
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
49a7c4...
and proposition id is
5e905e...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMQn1HNEEnsjoxViYUHacfsAeXao7cGiVVd
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
¬
(
(
(
¬
(
(
(
(
X40
→
(
¬
X36
)
→
X46
)
↔
(
¬
(
(
¬
(
X46
→
X36
→
(
(
¬
X14
)
↔
X40
)
)
)
→
X46
→
X36
→
(
X33
↔
X40
)
)
)
)
↔
(
X46
↔
(
(
¬
X36
)
↔
(
X14
↔
X37
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
X33
)
↔
(
X27
→
(
(
¬
X18
)
↔
X41
)
)
)
)
→
(
¬
(
(
¬
(
X41
→
(
¬
X6
)
→
X27
)
)
→
(
X46
↔
(
(
¬
X33
)
↔
X41
)
)
)
)
)
)
↔
(
¬
(
X41
↔
(
X40
↔
(
(
¬
X18
)
↔
X27
)
)
)
)
)
)
)
→
(
(
(
X40
↔
(
X25
↔
(
(
¬
X16
)
→
X37
)
)
)
→
(
¬
(
X40
→
X30
→
(
X28
↔
False
)
)
)
→
(
¬
(
(
(
X31
↔
(
X28
→
False
)
)
→
(
X37
→
(
(
¬
X31
)
↔
(
X16
↔
X40
)
)
)
→
X28
→
(
¬
X16
)
→
X37
)
→
(
¬
(
X37
→
X31
→
X40
)
)
)
)
)
↔
(
(
X40
→
X4
→
X30
)
↔
(
(
¬
X35
)
↔
(
X31
↔
(
X4
→
X41
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
(
X26
→
X17
→
False
)
)
→
(
¬
(
(
X49
→
(
X46
↔
(
X45
→
X40
)
)
)
→
(
(
¬
(
X40
→
(
(
¬
X38
)
↔
X49
)
)
)
↔
(
¬
(
(
¬
(
X49
↔
(
X40
↔
(
X17
↔
False
)
)
)
)
→
(
X49
↔
(
X46
↔
(
X38
→
(
X17
↔
False
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
X46
)
↔
(
(
¬
X45
)
↔
(
(
¬
X43
)
→
X49
)
)
)
→
(
¬
(
(
X43
↔
(
X13
→
(
(
¬
X12
)
↔
X49
)
)
)
→
(
¬
(
X13
→
(
(
¬
X12
)
↔
X45
)
)
)
)
)
)
)
→
X45
→
X43
)
↔
(
¬
(
X49
↔
(
X45
→
(
X10
↔
X43
)
)
)
)
)
)
→
(
(
(
X49
→
(
X46
↔
(
X37
→
False
)
)
)
↔
(
(
¬
(
X46
↔
(
X29
→
(
¬
X19
)
→
X49
)
)
)
→
(
¬
(
X48
↔
(
(
¬
X46
)
→
(
¬
X37
)
→
X49
)
)
)
)
)
↔
(
¬
(
X48
↔
(
X46
↔
X49
)
)
)
)
→
(
¬
(
(
X45
↔
(
X19
→
(
X2
↔
X49
)
)
)
→
X49
→
(
X29
↔
False
)
)
)
→
(
X45
↔
(
(
¬
X19
)
→
(
X5
↔
X49
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
¬
(
(
¬
X32
)
→
X24
→
(
(
¬
X20
)
↔
X42
)
)
)
→
X39
→
X20
)
→
(
(
¬
X32
)
↔
(
(
¬
X8
)
↔
(
(
¬
X7
)
↔
X44
)
)
)
)
)
→
(
(
¬
(
X44
→
(
X22
↔
(
X0
→
X32
)
)
)
)
↔
(
(
¬
(
(
¬
(
X47
↔
(
X32
↔
(
X3
↔
X21
)
)
)
)
→
X21
→
(
(
¬
X3
)
↔
X23
)
)
)
↔
(
¬
(
(
(
(
¬
(
X44
→
X32
→
(
X22
↔
(
X21
↔
False
)
)
)
)
↔
(
¬
(
(
¬
X32
)
→
X22
→
(
¬
X3
)
→
X44
)
)
)
↔
(
¬
(
(
¬
X32
)
↔
(
X3
↔
X44
)
)
)
)
→
(
¬
(
X44
↔
(
X39
→
False
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
¬
X46
)
→
(
¬
X45
)
→
(
¬
X15
)
→
X47
)
)
→
(
¬
(
(
¬
X46
)
→
(
(
¬
X45
)
↔
(
X42
↔
X47
)
)
)
)
)
→
(
¬
(
X39
→
(
X15
↔
X48
)
)
)
)
)
↔
(
(
(
X9
↔
(
(
¬
X1
)
→
X34
)
)
↔
(
(
¬
(
X34
↔
(
(
¬
X1
)
↔
X42
)
)
)
→
(
(
¬
(
X42
↔
(
X34
→
X11
)
)
)
↔
(
X45
→
(
(
¬
X11
)
↔
(
X9
→
X42
)
)
)
)
)
)
↔
(
¬
(
X47
↔
(
X42
→
(
¬
X11
)
→
X45
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
¬
(
(
X43
→
X41
→
(
X20
↔
X22
)
)
→
(
¬
(
X42
→
(
(
¬
X22
)
↔
(
(
¬
X4
)
↔
X43
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
(
X37
↔
(
X22
→
X14
→
X28
)
)
)
→
(
(
¬
(
(
¬
X28
)
↔
X43
)
)
↔
(
(
¬
(
(
¬
X42
)
→
X43
)
)
↔
(
¬
(
X42
→
(
X28
↔
(
(
¬
X22
)
↔
X37
)
)
)
)
)
)
)
)
→
(
(
¬
(
X42
→
(
X37
↔
(
X10
→
X21
)
)
)
)
↔
(
¬
(
X37
↔
(
X28
→
(
X10
↔
X21
)
)
)
)
)
)
)
↔
(
(
¬
X28
)
↔
(
(
¬
X22
)
→
X42
)
)
)
)
↔
(
¬
(
(
¬
(
X43
↔
(
(
¬
X30
)
→
(
X21
↔
X41
)
)
)
)
→
(
(
X41
↔
X43
)
↔
(
(
¬
(
X41
→
(
X30
↔
X41
)
→
(
¬
(
(
¬
X43
)
↔
(
X5
↔
X46
)
)
)
)
)
↔
(
¬
(
(
¬
X43
)
→
(
X41
↔
X46
)
)
)
)
)
)
)
)
↔
(
(
X42
↔
(
(
¬
X30
)
↔
(
X22
↔
X35
)
)
)
↔
(
X35
→
(
X30
↔
(
X22
→
X25
)
)
)
)
)
→
(
(
¬
(
(
(
¬
X47
)
→
(
¬
X45
)
→
(
(
¬
X34
)
↔
X49
)
)
→
(
(
X49
↔
(
X37
→
X45
)
)
↔
(
¬
(
X47
↔
(
X45
→
(
(
¬
X26
)
↔
X37
)
)
)
)
)
→
(
¬
(
(
¬
X47
)
↔
(
X37
→
X49
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X45
↔
(
X24
↔
(
X12
→
X34
)
)
)
)
→
(
¬
(
(
¬
(
X43
↔
(
X24
↔
(
(
¬
X23
)
↔
X47
)
)
)
)
→
(
¬
(
X34
→
(
¬
X24
)
→
X23
→
X47
)
)
)
)
)
)
↔
(
(
(
X45
→
X41
→
(
X29
↔
(
X17
→
False
)
)
)
↔
(
¬
(
X45
→
(
X41
↔
(
X29
→
X19
→
False
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
X47
→
X41
→
(
¬
X29
)
→
X34
)
→
(
¬
X45
)
→
X47
)
)
→
X47
→
X41
→
X29
)
)
)
)
)
)
→
(
¬
(
(
(
(
X48
→
(
¬
X27
)
→
(
X16
↔
X31
)
)
→
(
X45
↔
(
(
¬
X38
)
→
(
¬
X33
)
→
X48
)
)
)
→
(
¬
(
(
(
(
¬
X2
)
↔
X45
)
→
(
¬
(
(
(
¬
X33
)
↔
(
X15
→
(
(
¬
X2
)
↔
X45
)
)
)
→
(
¬
(
(
X33
→
X15
→
X41
)
→
(
X33
↔
(
X18
↔
(
(
¬
X2
)
→
X41
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X18
→
X8
→
(
(
¬
X0
)
↔
X41
)
)
)
↔
(
X0
↔
X15
)
)
→
(
(
X41
↔
(
X38
→
X18
→
X40
)
)
↔
(
(
(
¬
X40
)
→
(
(
¬
X18
)
↔
X46
)
)
↔
(
(
X41
→
(
X40
↔
X18
)
)
↔
(
(
¬
X41
)
↔
(
X40
→
(
¬
X0
)
→
X46
)
)
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
X6
→
(
¬
X3
)
→
X13
)
→
(
(
X36
↔
(
X13
↔
(
(
¬
X11
)
→
X46
)
)
)
↔
(
X46
→
(
X11
↔
(
(
¬
X6
)
→
X36
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
X46
→
X36
→
(
¬
X1
)
→
X39
)
→
(
X36
↔
(
(
¬
X13
)
↔
X46
)
)
)
↔
(
X46
→
(
X13
↔
X39
)
)
)
→
(
X46
↔
(
X39
→
X36
)
)
)
→
(
¬
(
(
¬
(
X48
↔
(
X46
→
X39
→
False
)
)
)
→
X46
→
(
X45
↔
(
(
¬
X13
)
→
X48
)
)
)
)
)
)
)
↔
(
(
(
(
X36
→
(
¬
X32
)
→
(
(
¬
X13
)
↔
X39
)
)
↔
(
(
(
¬
X39
)
↔
(
X32
→
X44
)
)
↔
(
(
¬
(
X39
↔
(
X32
→
(
¬
X7
)
→
X44
)
)
)
→
(
X44
↔
(
(
¬
X13
)
↔
(
(
¬
X7
)
↔
X32
)
)
)
)
)
)
↔
(
(
¬
X39
)
↔
(
(
¬
X36
)
→
(
(
¬
X9
)
↔
X44
)
)
)
)
↔
(
¬
(
X36
→
(
X7
↔
X13
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
5adf44...
and proposition id is
9d5b3a...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMce4PwaRVc1DZGRfbXGBacqV4gTEHPkbFt
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
(
¬
(
(
(
(
¬
(
(
X39
↔
(
(
¬
X31
)
→
X46
)
)
→
X45
→
X41
→
(
(
¬
X39
)
↔
X46
)
)
)
↔
(
X45
→
X43
→
(
X41
↔
X46
)
)
)
→
(
(
(
(
(
¬
(
(
¬
X41
)
→
(
(
¬
X36
)
↔
(
X32
↔
X50
)
)
)
)
↔
(
X45
↔
(
X32
→
X50
)
)
)
↔
(
¬
(
X41
→
(
(
¬
X36
)
↔
(
(
¬
X20
)
→
X45
)
)
)
)
)
→
(
X50
↔
(
X43
↔
(
X41
→
X45
)
)
)
)
↔
(
(
(
¬
X36
)
↔
(
(
¬
X10
)
↔
X45
)
)
↔
(
X20
↔
(
X10
→
X23
)
)
)
)
)
→
(
(
(
(
(
X45
→
(
X25
↔
(
X19
→
X9
)
)
)
→
(
¬
(
X46
→
X45
→
(
X19
↔
X9
)
)
)
)
↔
(
¬
(
X19
→
(
(
¬
X9
)
↔
X25
)
)
)
)
↔
(
¬
(
(
¬
(
X35
↔
(
(
¬
X19
)
↔
(
(
¬
X15
)
↔
X46
)
)
)
)
→
(
(
(
¬
(
(
¬
X45
)
↔
(
(
¬
X35
)
↔
(
(
¬
X30
)
→
X46
)
)
)
)
→
(
X35
↔
(
(
¬
X8
)
→
X30
)
)
)
↔
(
¬
(
X45
→
X38
→
X35
→
X46
)
)
)
)
)
)
↔
(
(
(
X50
↔
(
X40
↔
X5
)
)
↔
(
¬
(
(
(
(
(
(
¬
X25
)
→
X19
→
(
¬
X5
)
→
X40
)
↔
(
X50
→
(
X40
↔
(
X5
→
False
)
)
)
)
↔
(
¬
(
X40
→
(
X25
↔
X50
)
)
)
)
→
(
X50
↔
(
X40
↔
X25
)
)
)
→
(
¬
(
X50
→
X5
→
X40
)
)
)
)
)
↔
(
(
(
X50
↔
(
X19
↔
X2
)
)
↔
(
(
(
¬
X2
)
→
X19
)
→
(
X50
→
(
X19
↔
(
(
¬
X5
)
→
X35
)
)
)
→
(
¬
(
(
¬
X35
)
→
(
X19
↔
(
(
¬
X2
)
↔
X50
)
)
)
)
)
)
→
(
¬
(
X19
→
(
X2
↔
False
)
)
)
)
)
)
→
(
(
¬
(
X13
↔
(
(
¬
X1
)
→
X38
)
)
)
↔
(
(
(
¬
(
X38
↔
(
(
¬
X14
)
↔
(
X4
→
X46
)
)
)
)
↔
(
¬
(
(
(
¬
(
X49
→
X14
→
(
X13
↔
X38
)
)
)
→
(
¬
(
X49
↔
(
X46
→
(
X13
↔
X14
)
)
)
)
)
→
(
¬
(
(
¬
X38
)
↔
(
(
¬
X13
)
→
X46
)
)
)
)
)
)
→
(
¬
(
(
X46
↔
X40
)
→
(
(
(
¬
X40
)
↔
(
X38
↔
(
(
¬
X1
)
→
X46
)
)
)
↔
(
¬
(
X49
↔
(
X46
↔
(
X38
↔
X1
)
)
)
)
)
)
)
)
)
)
)
→
(
(
(
(
(
(
¬
(
X42
→
(
X37
↔
(
X33
→
X45
)
)
)
)
↔
(
¬
(
X45
↔
(
X42
→
X37
→
X24
→
False
)
)
)
)
↔
(
(
¬
(
(
¬
X46
)
→
(
¬
X18
)
→
X48
)
)
↔
(
(
X46
→
(
(
¬
X37
)
↔
X48
)
)
→
(
(
(
¬
X37
)
↔
(
X18
↔
X48
)
)
↔
(
X48
↔
(
X46
↔
(
X37
→
X24
→
False
)
)
)
)
)
)
)
↔
(
X48
↔
(
X46
↔
(
X42
→
False
)
)
)
)
→
(
¬
(
(
(
(
X33
→
(
¬
X11
)
→
(
X6
↔
X12
)
)
↔
(
X18
→
(
X11
↔
(
(
¬
X6
)
→
X12
)
)
)
)
→
(
¬
(
X42
→
(
X16
↔
(
X12
↔
X6
)
)
)
)
)
→
(
(
¬
(
(
X42
↔
(
(
¬
X16
)
→
X12
→
X44
)
)
→
(
¬
(
(
¬
X42
)
↔
(
X12
→
X3
→
X44
)
)
)
)
)
↔
(
¬
(
X16
↔
(
X12
↔
X44
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X26
)
→
(
X0
↔
X42
)
)
)
↔
(
¬
(
X46
→
(
¬
X29
)
→
(
¬
X28
)
→
X34
)
)
)
→
(
(
(
(
¬
X34
)
↔
(
(
¬
X29
)
→
(
X22
↔
X44
)
)
)
↔
(
¬
(
(
(
(
X47
↔
(
X29
↔
(
X22
↔
X27
)
)
)
→
X27
→
(
X22
↔
(
X7
↔
False
)
)
)
↔
(
(
¬
X27
)
→
(
X22
↔
X29
)
)
)
→
(
X47
↔
(
(
¬
X27
)
→
(
¬
X22
)
→
X44
)
)
)
)
)
↔
(
¬
(
(
(
X21
↔
X29
)
↔
(
¬
(
(
¬
X44
)
→
(
¬
X22
)
→
(
X21
↔
X50
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X29
)
→
(
(
¬
X28
)
↔
(
(
¬
X17
)
→
X44
)
)
)
)
→
(
¬
(
(
¬
X44
)
→
(
(
¬
X29
)
↔
(
(
¬
X28
)
↔
X50
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
X47
→
(
X25
↔
(
X24
↔
X1
)
)
)
→
(
(
¬
(
(
X45
↔
(
(
¬
X42
)
↔
X44
)
)
→
(
(
X44
↔
(
(
¬
X42
)
→
X25
→
X45
)
)
→
(
¬
(
X45
→
X44
→
X42
→
(
X33
↔
False
)
)
)
)
→
(
¬
X33
)
→
X25
→
(
(
¬
X1
)
↔
X45
)
)
)
↔
(
¬
(
X25
↔
(
X1
→
False
)
)
)
)
)
)
↔
(
(
X33
→
(
¬
X25
)
→
X11
→
X49
)
↔
(
(
¬
X42
)
→
(
¬
X33
)
→
(
X25
↔
X49
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
X49
↔
(
(
¬
X39
)
↔
(
X6
→
X50
)
)
)
→
(
¬
(
X49
→
(
(
¬
X39
)
↔
(
(
¬
X29
)
→
X50
)
)
)
)
)
)
→
(
¬
(
(
X19
↔
(
X13
→
(
X6
↔
X41
)
)
)
→
X50
→
(
X41
↔
(
X19
→
X13
→
False
)
)
)
)
)
)
→
(
(
(
¬
(
X35
→
X23
→
X7
→
False
)
)
→
(
(
¬
(
X49
↔
(
(
¬
X35
)
→
(
¬
X19
)
→
X45
)
)
)
↔
(
(
X23
→
(
¬
X7
)
→
X49
)
↔
(
¬
(
(
(
X45
↔
(
X35
↔
(
(
¬
X7
)
↔
X23
)
)
)
→
X45
→
X49
)
→
(
X49
↔
(
(
¬
X7
)
→
X45
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X29
)
→
(
¬
X17
)
→
X35
)
)
↔
(
X42
→
(
(
¬
X29
)
↔
(
X17
↔
X49
)
)
)
)
→
(
(
(
¬
X23
)
→
(
¬
X2
)
→
X29
)
↔
(
(
(
¬
X35
)
↔
(
(
¬
X23
)
→
(
X14
↔
X42
)
)
)
↔
(
X29
↔
(
X23
→
(
X14
↔
False
)
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
(
(
X48
↔
(
(
¬
X37
)
→
X43
)
)
↔
(
(
¬
X43
)
→
(
(
¬
X37
)
↔
(
(
¬
X12
)
↔
X46
)
)
)
)
→
(
(
¬
X46
)
↔
(
(
¬
X43
)
→
X48
)
)
)
↔
(
¬
(
X48
→
X46
→
(
X43
↔
(
X34
→
False
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X50
↔
(
(
¬
X40
)
↔
X46
)
)
)
↔
(
(
¬
(
(
¬
X48
)
→
X40
→
X50
)
)
↔
(
(
¬
(
X46
↔
(
X40
→
X18
→
False
)
)
)
↔
(
¬
(
X50
↔
X48
)
)
)
)
)
→
(
(
¬
X48
)
→
X46
→
(
(
¬
X26
)
↔
X50
)
)
→
(
¬
(
(
(
¬
X43
)
↔
(
(
¬
X20
)
↔
X46
)
)
→
(
¬
(
X50
↔
(
X46
→
(
(
¬
X43
)
↔
X48
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
¬
(
X50
→
X32
→
X8
→
X46
)
)
→
(
¬
(
(
¬
X49
)
↔
(
(
¬
X46
)
↔
X50
)
)
)
)
↔
(
¬
(
X49
↔
(
(
¬
X32
)
→
X46
)
)
)
)
↔
(
X46
→
(
X32
↔
(
X15
↔
X50
)
)
)
)
↔
(
(
(
(
X49
→
(
¬
X28
)
→
X32
)
→
(
¬
(
(
¬
X48
)
↔
(
(
¬
X32
)
↔
(
X28
↔
X49
)
)
)
)
→
(
¬
(
X36
→
X28
→
X32
)
)
)
→
(
X49
↔
(
X36
→
X48
)
)
)
↔
(
¬
(
X50
↔
(
(
¬
X32
)
↔
(
(
¬
X28
)
→
X36
)
)
)
)
)
)
→
(
(
(
¬
(
X49
→
X45
→
(
X9
↔
X48
)
)
)
↔
(
X49
↔
(
X48
↔
(
X45
→
X38
)
)
)
)
↔
(
¬
(
(
¬
X46
)
↔
(
(
¬
X45
)
→
(
X9
↔
X48
)
)
)
)
)
)
)
)
→
(
(
X22
↔
(
X21
↔
(
(
¬
X4
)
→
X5
)
)
)
↔
(
(
(
X30
↔
(
(
¬
X22
)
→
(
¬
X10
)
→
X27
)
)
→
(
¬
(
(
¬
X30
)
↔
(
X10
↔
X37
)
)
)
)
↔
(
¬
(
X30
↔
(
(
¬
X27
)
↔
(
(
¬
X21
)
↔
X37
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X49
↔
(
X38
↔
(
X21
↔
False
)
)
)
)
→
(
¬
(
X49
↔
(
X3
↔
X31
)
)
)
)
→
(
(
¬
(
(
¬
X38
)
→
(
X0
↔
X49
)
)
)
↔
(
(
X49
↔
(
(
¬
X16
)
↔
(
X0
↔
X31
)
)
)
↔
(
(
¬
X16
)
↔
(
X0
→
X38
)
)
)
)
)
→
(
X38
↔
(
X21
→
(
X16
↔
X27
)
)
)
)
)
→
(
¬
(
(
¬
X38
)
↔
(
X31
↔
(
(
¬
X30
)
↔
X49
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
af28de...
and proposition id is
da594b...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMYXRryBFqURy5evp3ExYPiaJqqhvzHmcF8
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
(
(
¬
(
(
¬
(
(
X32
→
(
X27
↔
(
(
¬
X6
)
→
X41
)
)
)
→
X41
→
X43
)
)
→
(
X32
↔
(
(
¬
X27
)
↔
X41
)
)
)
)
→
X41
→
X43
)
→
(
(
¬
(
(
(
(
¬
X32
)
↔
(
X24
↔
X43
)
)
↔
(
¬
(
(
(
(
¬
X41
)
→
X32
→
(
X24
↔
X43
)
)
↔
(
X41
↔
X21
)
)
→
(
¬
(
X41
→
(
(
¬
X32
)
↔
X43
)
)
)
)
)
)
→
X43
→
(
X41
↔
X32
)
)
)
↔
(
(
¬
(
X41
→
(
X14
↔
(
X10
→
X23
)
)
)
)
→
(
(
X41
→
(
(
¬
X24
)
↔
(
(
¬
X14
)
↔
X43
)
)
)
↔
(
X43
↔
(
X41
↔
(
X10
↔
False
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
X44
)
→
(
¬
X21
)
→
(
X2
↔
X49
)
)
)
↔
(
(
¬
X44
)
↔
(
X33
↔
(
X2
→
X49
)
)
)
)
→
(
X44
↔
(
(
¬
X33
)
↔
(
(
¬
X27
)
↔
X49
)
)
)
)
↔
(
¬
(
X33
↔
(
(
¬
X2
)
→
(
¬
X1
)
→
X49
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X30
)
→
X11
→
X32
)
)
↔
(
(
(
¬
X33
)
↔
(
X32
→
(
X11
↔
X49
)
)
)
→
X49
→
(
X32
↔
(
X30
↔
X11
)
)
)
)
→
(
¬
(
(
¬
X44
)
↔
(
(
¬
X30
)
→
X11
→
X49
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
X49
→
X48
→
(
X5
↔
X29
)
)
)
↔
(
X29
↔
(
(
¬
X3
)
→
X17
)
)
)
→
(
X49
↔
(
X20
→
X17
→
X48
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X39
↔
(
X5
→
False
)
)
)
→
X26
→
X22
→
X5
→
X40
)
)
→
(
¬
(
(
(
(
¬
(
X43
↔
(
(
¬
X26
)
→
X22
→
X46
)
)
)
↔
(
X39
↔
(
(
¬
X26
)
↔
X46
)
)
)
↔
(
(
¬
(
X46
→
(
X44
↔
X22
)
)
)
→
(
¬
(
X44
↔
(
(
¬
X26
)
→
X45
)
)
)
)
)
→
(
(
X45
→
(
(
¬
X3
)
↔
X46
)
)
↔
(
(
(
¬
(
X46
→
(
X20
↔
(
(
¬
X3
)
→
X26
)
)
)
)
→
(
¬
(
(
¬
X26
)
→
(
X0
↔
X46
)
)
)
)
→
(
¬
(
X0
↔
False
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
X47
↔
(
X15
→
X45
)
)
↔
(
¬
(
X45
↔
(
X35
→
(
¬
X13
)
→
X47
)
)
)
)
↔
(
X47
↔
(
(
¬
X15
)
↔
X45
)
)
)
↔
(
(
(
¬
(
(
¬
X42
)
↔
(
(
¬
X35
)
→
X45
)
)
)
→
(
X45
→
X35
→
X34
→
X8
→
False
)
→
(
¬
(
(
¬
X35
)
→
(
X8
↔
X42
)
)
)
)
↔
(
X47
→
(
X36
↔
(
X18
→
X42
)
)
)
)
)
↔
(
(
(
X38
↔
(
X19
→
X35
)
)
↔
(
(
X45
↔
(
X35
↔
(
(
¬
X19
)
↔
X38
)
)
)
↔
(
X38
→
(
¬
X19
)
→
X15
→
X35
)
)
)
↔
(
¬
(
(
¬
X38
)
↔
(
(
¬
X35
)
→
(
(
¬
X28
)
↔
X45
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
(
¬
(
X43
→
X42
→
(
X31
↔
X46
)
)
)
↔
(
X43
→
X9
→
False
)
)
→
(
¬
(
X46
→
(
¬
X43
)
→
(
¬
X31
)
→
X47
)
)
)
)
↔
(
(
¬
(
(
¬
X43
)
→
X38
→
X4
→
X46
)
)
→
(
¬
(
X46
→
(
X7
↔
(
X4
↔
X42
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X47
↔
(
X46
→
(
X42
↔
(
X37
↔
False
)
)
)
)
)
→
(
¬
(
(
¬
(
X46
↔
(
X25
↔
(
X16
→
X42
)
)
)
)
→
(
¬
(
X46
↔
(
X42
↔
(
X37
→
(
X16
↔
False
)
)
)
)
)
)
)
)
)
→
X47
→
(
X46
↔
(
(
¬
X12
)
→
X25
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
¬
X45
)
→
(
¬
X41
)
→
(
¬
X30
)
→
X47
)
)
→
(
X44
↔
(
(
¬
X30
)
↔
(
X13
→
X41
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
¬
(
(
X36
↔
(
X5
→
X24
)
)
→
(
(
X45
↔
(
X5
→
(
¬
X2
)
→
X24
)
)
↔
(
X45
↔
(
X36
↔
(
X5
→
(
X2
↔
False
)
)
)
)
)
→
(
(
¬
X36
)
↔
(
(
¬
X24
)
↔
X45
)
)
)
)
→
(
(
¬
(
X40
→
X36
→
(
X0
↔
X45
)
)
)
↔
(
X36
↔
(
(
¬
X5
)
→
(
(
¬
X0
)
↔
X40
)
)
)
)
)
)
↔
(
¬
(
(
¬
X40
)
→
X36
→
(
(
¬
X19
)
↔
X45
)
)
)
)
↔
(
¬
(
(
X41
↔
(
X24
↔
False
)
)
→
(
¬
(
(
¬
(
(
¬
(
X47
→
(
X45
↔
(
X41
↔
X24
)
)
)
)
→
(
X41
↔
(
X36
↔
X47
)
)
)
)
→
(
¬
(
X36
↔
(
(
¬
X24
)
↔
X41
)
)
)
)
)
)
)
)
→
(
¬
(
X41
↔
(
X24
↔
(
X5
→
False
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X36
↔
(
(
¬
X29
)
↔
(
X25
→
X40
)
)
)
)
→
(
(
(
¬
X34
)
→
(
(
¬
X29
)
↔
(
(
¬
X25
)
↔
X36
)
)
)
↔
(
(
¬
X36
)
↔
(
(
¬
X34
)
↔
(
(
¬
X29
)
→
X37
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
X37
→
X36
→
(
(
¬
X10
)
↔
X31
)
)
→
(
(
¬
(
X31
→
(
(
¬
X10
)
↔
X37
)
)
)
↔
(
X32
↔
(
X10
→
(
X7
↔
X31
)
)
)
)
)
)
→
X37
→
(
¬
X32
)
→
X36
)
→
(
¬
(
X37
↔
(
X34
↔
(
X32
→
X36
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
(
¬
X27
)
↔
(
(
¬
X8
)
↔
X37
)
)
)
→
(
¬
(
(
¬
X40
)
↔
(
(
¬
X37
)
→
(
(
¬
X36
)
↔
X45
)
)
)
)
)
→
(
¬
(
(
¬
X35
)
→
(
¬
X17
)
→
X49
)
)
→
(
(
¬
(
X37
↔
(
X35
→
(
(
¬
X17
)
↔
X36
)
)
)
)
↔
(
X37
→
X36
→
(
¬
X35
)
→
X40
)
)
)
)
→
(
¬
X36
)
→
X35
→
(
X4
↔
X37
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
(
(
¬
(
X26
→
X11
→
X42
)
)
↔
(
X48
↔
(
X45
↔
(
X26
↔
X44
)
)
)
)
→
(
X45
↔
(
X44
↔
(
X39
↔
X48
)
)
)
)
↔
(
¬
(
X48
→
(
¬
X39
)
→
X12
→
X42
)
)
)
→
(
¬
(
(
(
¬
(
X48
→
X43
→
(
X42
↔
X3
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X48
→
(
¬
X28
)
→
X43
)
)
→
(
¬
(
X39
→
(
¬
X28
)
→
X3
→
X48
)
)
)
)
→
(
X28
↔
(
(
¬
X3
)
→
X39
)
)
)
)
)
→
(
¬
(
(
¬
X42
)
→
(
X39
↔
X43
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
X45
)
↔
(
X43
↔
(
X20
→
X49
)
)
)
→
(
(
(
(
¬
X39
)
↔
(
(
¬
X33
)
↔
X49
)
)
↔
(
¬
(
X45
↔
(
X43
↔
(
X39
↔
X33
)
)
)
)
)
↔
(
(
X45
↔
(
X39
→
(
(
¬
X1
)
↔
X49
)
)
)
→
(
X39
↔
(
X1
→
X38
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
X23
↔
(
X16
→
X15
)
)
↔
(
(
¬
(
(
¬
X23
)
↔
(
X16
↔
(
(
¬
X6
)
→
X45
)
)
)
)
→
(
¬
(
(
¬
X16
)
→
X15
→
X6
→
X23
)
)
)
)
↔
X15
)
↔
(
(
¬
(
(
¬
X21
)
↔
(
X16
→
X45
)
)
)
↔
(
(
(
(
¬
X14
)
↔
(
(
¬
X9
)
→
X23
)
)
↔
(
¬
(
(
X16
→
(
¬
X14
)
→
X9
→
X23
)
→
(
X46
↔
(
(
¬
X16
)
→
(
¬
X14
)
→
X23
)
)
)
)
)
↔
(
(
¬
X21
)
↔
(
(
¬
X16
)
↔
(
X14
↔
X46
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
X22
)
↔
(
X14
→
X44
)
)
→
(
¬
(
X44
→
(
(
¬
X21
)
↔
X45
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
X23
)
↔
(
(
¬
X22
)
→
X18
→
X44
)
)
→
X44
→
X42
→
False
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
140d38...
and proposition id is
f6c0b2...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMaMWP8JynPbXZw9FnM6XtkTS5t3NgCbpbK
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
¬
(
(
(
(
(
(
¬
X21
)
→
(
¬
X14
)
→
(
X1
↔
X40
)
)
↔
(
(
¬
X28
)
→
(
(
¬
X1
)
↔
X40
)
)
)
↔
(
(
¬
(
(
¬
X43
)
→
(
(
¬
X40
)
↔
X44
)
)
)
↔
(
(
X44
↔
(
X43
→
(
X21
↔
X28
)
)
)
↔
(
(
¬
X40
)
↔
(
X28
→
(
X21
↔
X44
)
)
)
)
)
)
↔
(
(
(
¬
(
X47
→
X35
→
(
¬
X34
)
→
X43
)
)
→
(
(
(
¬
(
(
(
¬
X44
)
→
X43
→
(
¬
X19
)
→
X47
)
→
(
¬
(
X43
↔
(
(
¬
X29
)
↔
X47
)
)
)
)
)
→
(
¬
(
(
¬
X44
)
↔
(
(
¬
X43
)
→
X34
→
X49
)
)
)
→
(
¬
(
X49
→
(
X47
↔
(
(
¬
X40
)
→
X44
)
)
)
)
)
↔
(
X29
→
(
¬
X19
)
→
X43
)
)
)
↔
(
X47
→
(
(
¬
X35
)
↔
(
(
¬
X8
)
→
X49
)
)
)
)
)
→
(
(
(
X49
→
(
¬
X36
)
→
(
(
¬
X31
)
↔
X44
)
)
↔
(
X49
→
X36
→
X31
→
X47
)
)
↔
(
(
(
¬
(
X47
↔
(
X32
→
(
X0
↔
X13
)
)
)
)
↔
(
(
¬
(
X26
→
X30
)
)
→
(
¬
(
X32
↔
(
(
¬
X0
)
→
X47
)
)
)
)
)
↔
(
¬
(
(
¬
(
X47
↔
(
X44
→
(
X30
↔
X36
)
)
)
)
→
(
X22
↔
(
(
¬
X20
)
→
(
X12
↔
X47
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X39
→
X38
→
(
¬
X4
)
→
X48
)
)
↔
(
(
(
¬
X48
)
→
(
(
¬
X5
)
↔
(
X4
↔
X49
)
)
)
→
(
X46
↔
(
(
¬
X5
)
↔
(
X4
↔
X49
)
)
)
)
)
→
(
¬
(
X46
↔
(
X4
↔
(
X2
→
X48
)
)
)
)
→
(
¬
X46
)
→
(
X45
↔
(
(
¬
X2
)
↔
X48
)
)
)
→
(
¬
(
(
¬
(
(
(
X49
↔
(
X47
→
(
X42
↔
X44
)
)
)
→
(
(
X47
→
(
(
¬
X45
)
↔
(
X15
→
X49
)
)
)
↔
(
(
X41
↔
(
(
¬
X10
)
→
X47
)
)
→
(
¬
(
X49
→
(
X15
↔
X47
)
)
)
)
)
→
(
(
¬
(
X44
→
X15
→
X7
→
X45
)
)
↔
(
X45
→
X44
→
X15
→
X49
)
)
)
→
(
(
(
X47
→
(
(
¬
X41
)
↔
(
(
¬
X17
)
→
X44
)
)
)
↔
(
X44
↔
(
X41
→
(
X27
↔
False
)
)
)
)
↔
(
(
(
(
(
(
¬
(
X47
→
(
X25
↔
(
X23
→
X27
)
)
)
)
→
X47
→
(
¬
X25
)
→
(
X23
↔
X41
)
)
↔
(
(
¬
X27
)
→
(
X25
↔
X41
)
)
)
↔
(
¬
X27
)
)
→
(
¬
(
X47
→
(
X27
↔
(
(
¬
X24
)
↔
X41
)
)
)
)
)
→
(
¬
X41
)
→
X27
→
X47
)
)
→
(
¬
(
X44
↔
(
(
¬
X41
)
→
(
(
¬
X16
)
↔
X47
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
X37
↔
(
X11
→
X6
)
)
→
X48
→
(
X11
↔
X45
)
)
)
↔
(
¬
(
(
¬
X37
)
↔
(
X33
↔
(
X11
→
X47
)
)
)
)
)
→
(
¬
(
(
X18
→
(
X9
↔
(
X3
→
X47
)
)
)
→
(
(
X47
→
(
X37
↔
(
X18
↔
(
X6
→
False
)
)
)
)
↔
(
X49
→
X37
→
X18
)
)
→
(
¬
(
X47
→
X18
→
X6
→
False
)
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
(
(
¬
(
X45
→
X39
→
X47
)
)
↔
(
(
¬
X47
)
↔
(
(
¬
X39
)
↔
(
X7
↔
X49
)
)
)
)
→
(
¬
X45
)
→
(
X39
↔
(
(
¬
X35
)
→
X47
)
)
)
→
(
(
(
(
(
(
¬
X40
)
↔
X46
)
↔
(
X47
↔
(
(
¬
X45
)
→
(
(
¬
X22
)
↔
X46
)
)
)
)
↔
(
¬
(
(
¬
X40
)
→
X45
)
)
)
↔
(
¬
(
X47
→
(
X46
↔
X45
)
)
)
)
↔
(
X49
↔
(
X47
→
(
¬
X8
)
→
X40
)
)
)
)
→
(
¬
(
(
(
(
X38
↔
(
(
¬
X28
)
↔
X44
)
)
↔
(
X44
→
X38
→
X3
→
X48
)
)
↔
(
(
(
X48
↔
(
X38
↔
X18
)
)
→
(
(
¬
(
X48
↔
(
(
¬
X38
)
↔
X49
)
)
)
↔
(
(
¬
X38
)
→
(
X18
↔
(
(
¬
X1
)
↔
X49
)
)
)
)
)
→
(
X49
↔
(
X28
→
X18
→
X19
)
)
)
)
→
(
(
(
¬
X48
)
→
(
X47
↔
(
(
¬
X46
)
↔
X49
)
)
)
↔
(
¬
(
X47
→
(
(
¬
X38
)
↔
(
(
¬
X16
)
→
X46
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
X48
↔
(
X37
→
X31
→
X11
→
False
)
)
→
(
¬
(
X17
→
(
X11
↔
False
)
)
)
)
↔
(
(
X37
→
(
X31
↔
X30
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X37
)
↔
(
(
¬
X30
)
↔
(
(
¬
X6
)
↔
X48
)
)
)
)
↔
(
¬
(
(
¬
X31
)
↔
(
(
¬
X30
)
↔
X48
)
)
)
)
→
(
¬
(
X6
→
X31
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X47
)
↔
(
(
¬
X37
)
↔
(
X14
↔
X48
)
)
)
)
↔
(
(
(
X45
↔
(
(
¬
X4
)
↔
X48
)
)
↔
(
¬
(
X37
↔
X21
)
)
)
↔
(
(
¬
X37
)
→
(
(
¬
X21
)
↔
(
X14
→
X48
)
)
)
)
)
→
(
(
X48
↔
(
(
¬
X43
)
↔
X45
)
)
↔
(
¬
(
X47
↔
(
X45
↔
(
X43
↔
(
X30
→
False
)
)
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
X27
)
↔
(
X25
↔
(
(
¬
X10
)
↔
X41
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
X41
↔
(
X27
↔
X47
)
)
)
→
(
¬
(
(
¬
(
X29
→
(
(
¬
X27
)
↔
(
X10
↔
X41
)
)
)
)
→
(
X32
↔
(
X29
→
(
(
¬
X10
)
↔
X27
)
)
)
)
)
)
)
→
(
¬
(
(
¬
X29
)
↔
(
X9
→
X32
)
)
)
)
)
→
(
(
(
(
¬
X43
)
↔
(
X32
↔
X47
)
)
→
(
¬
(
(
(
X47
→
(
X27
↔
X24
)
)
↔
(
(
¬
(
X32
→
(
X27
↔
False
)
)
)
↔
(
¬
(
X47
↔
(
X27
→
(
(
¬
X24
)
↔
X43
)
)
)
)
)
)
→
(
X43
↔
(
X32
→
X47
)
)
)
)
)
↔
(
(
¬
X32
)
→
(
¬
X27
)
→
(
X0
↔
X43
)
)
)
)
)
↔
(
(
X36
→
X32
→
(
X20
↔
(
X5
→
False
)
)
)
→
(
¬
(
(
X47
→
(
¬
X5
)
→
X43
)
→
(
(
¬
(
X43
↔
(
X5
→
X42
)
)
)
↔
(
¬
(
X47
↔
(
(
¬
X43
)
↔
(
(
¬
X42
)
↔
X48
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
X43
→
(
(
¬
X33
)
↔
X34
)
)
↔
(
(
¬
(
X12
→
X33
)
)
↔
(
(
X13
→
X12
→
X34
)
→
(
¬
X33
)
→
(
X13
↔
X43
)
)
)
)
→
(
X47
↔
(
X43
↔
(
X42
↔
(
X33
→
False
)
)
)
)
→
(
X43
↔
(
(
¬
X42
)
→
(
(
¬
X34
)
↔
X47
)
)
)
)
)
→
(
(
¬
(
X43
→
(
X26
↔
(
X23
→
(
X2
↔
False
)
)
)
)
)
↔
(
¬
(
X43
→
X26
→
X23
→
X15
→
False
)
)
)
→
(
(
¬
(
X13
↔
X47
)
)
↔
(
X43
→
(
¬
X2
)
→
X15
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
923985...
and proposition id is
e9689f...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMaU7W3cEjNDHYsrGvGY3wqD1yhKJDzGBdA
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
¬
(
(
(
(
(
(
(
(
¬
X33
)
↔
(
(
¬
X32
)
↔
(
X1
↔
X40
)
)
)
↔
(
¬
(
(
¬
X40
)
→
(
X32
↔
(
X1
→
X45
)
)
)
)
)
↔
(
(
X44
→
(
X40
↔
(
(
¬
X19
)
→
X45
)
)
)
→
(
(
X44
↔
(
X32
→
X27
)
)
↔
(
X45
↔
(
X38
→
(
(
¬
X27
)
↔
X40
)
)
)
)
)
)
↔
(
(
¬
X32
)
↔
(
(
¬
X13
)
↔
(
X1
↔
X33
)
)
)
)
→
(
(
¬
(
(
¬
(
(
(
¬
(
X44
↔
(
(
¬
X34
)
↔
(
X17
↔
X45
)
)
)
)
→
(
X44
↔
(
(
¬
X34
)
→
X26
→
X45
)
)
)
→
(
¬
(
X45
↔
(
(
¬
X35
)
→
(
X34
↔
X44
)
)
)
)
)
)
→
(
(
X45
↔
(
(
¬
X17
)
↔
(
(
¬
X7
)
→
X35
)
)
)
↔
(
(
¬
X38
)
→
(
X17
↔
X45
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X26
)
↔
(
(
¬
X23
)
↔
(
(
¬
X14
)
→
X35
)
)
)
)
↔
(
¬
(
X23
↔
(
X17
→
(
X14
↔
X35
)
)
)
)
)
→
X26
→
(
X14
↔
(
(
¬
X5
)
↔
X35
)
)
)
)
)
→
(
(
(
¬
(
(
(
¬
(
(
X45
→
(
¬
X40
)
→
X16
→
X46
)
→
(
¬
(
X45
→
X43
→
(
¬
X16
)
→
X46
)
)
)
)
↔
(
X45
↔
(
X44
↔
(
(
¬
X16
)
↔
X40
)
)
)
)
→
(
¬
(
(
¬
(
X45
↔
(
X44
→
(
X40
↔
X2
)
)
)
)
→
(
¬
(
X47
↔
(
X45
↔
(
X3
→
X2
)
)
)
)
)
)
→
X47
→
(
X45
↔
(
X40
↔
False
)
)
)
)
→
(
¬
(
(
(
¬
X45
)
→
(
(
¬
X44
)
↔
(
(
¬
X28
)
→
X47
)
)
)
→
(
¬
(
X47
→
(
X45
↔
(
(
¬
X6
)
→
X28
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
X43
)
↔
(
(
¬
X4
)
→
X44
)
)
)
→
(
(
¬
(
(
¬
(
X44
↔
(
X21
↔
(
X9
↔
X43
)
)
)
)
→
(
¬
(
X43
↔
(
(
¬
X4
)
↔
X9
)
)
)
→
X21
→
(
(
¬
X9
)
↔
X40
)
)
)
↔
(
X44
↔
(
X40
→
X43
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
(
X49
↔
(
X42
↔
(
X30
↔
X39
)
)
)
)
↔
(
¬
(
(
(
¬
(
X39
↔
X8
)
)
↔
(
(
X48
↔
(
X42
↔
(
X39
↔
X49
)
)
)
↔
(
X49
↔
(
X48
↔
(
(
¬
X8
)
↔
X39
)
)
)
)
)
→
X48
→
X42
→
(
X39
↔
False
)
)
)
)
↔
(
(
¬
X47
)
↔
(
X45
→
X8
→
X48
)
)
)
→
(
¬
(
(
¬
(
(
X48
→
X22
→
X10
→
False
)
→
X49
→
X48
→
X41
→
X22
)
)
→
(
(
¬
(
X49
↔
(
X29
→
X20
→
X45
)
)
)
↔
(
(
(
¬
X45
)
↔
(
X10
↔
X48
)
)
→
(
¬
(
X11
↔
X49
)
)
)
)
)
)
)
→
(
(
(
X49
↔
(
X45
→
(
¬
X12
)
→
X46
)
)
→
X49
→
X12
→
X15
)
→
(
¬
(
(
X49
→
(
X37
↔
(
X24
↔
X45
)
)
)
→
(
¬
(
X49
↔
(
X46
↔
(
X36
→
X37
)
)
)
)
)
)
)
→
(
(
(
¬
X37
)
↔
(
(
¬
X25
)
↔
(
(
¬
X0
)
↔
X45
)
)
)
→
(
X47
↔
(
X37
↔
(
X25
→
(
X0
↔
False
)
)
)
)
)
→
(
(
(
¬
(
X47
↔
(
X37
→
(
(
¬
X18
)
↔
X31
)
)
)
)
↔
(
X37
↔
(
X31
→
X18
→
X45
)
)
)
↔
(
X47
→
X45
→
(
X31
↔
X36
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
¬
(
(
¬
(
X49
→
X47
→
X39
→
X26
→
False
)
)
→
(
¬
(
X47
→
X39
→
(
X8
↔
X26
)
)
)
)
)
→
(
¬
(
(
(
¬
X39
)
→
X30
→
(
¬
X20
)
→
X47
)
→
(
X49
↔
False
)
)
)
)
↔
(
(
¬
(
(
(
X48
→
(
X30
↔
X35
)
)
→
(
X46
↔
(
X7
→
False
)
)
)
→
(
(
¬
(
(
¬
X47
)
→
(
¬
X7
)
→
(
X6
↔
X48
)
)
)
↔
(
(
¬
(
X7
→
False
)
)
↔
(
¬
(
(
¬
(
X48
→
X47
→
X39
→
X7
)
)
→
(
X48
↔
(
X47
↔
(
(
¬
X39
)
↔
X46
)
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
X46
↔
(
X10
↔
X12
)
)
)
↔
(
X46
↔
(
X28
↔
(
X10
→
X37
)
)
)
)
↔
(
(
(
¬
X47
)
→
(
(
¬
X46
)
↔
(
(
¬
X37
)
↔
X48
)
)
)
↔
(
¬
(
X47
→
X46
→
(
X10
↔
X48
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X48
→
(
X19
↔
(
(
¬
X5
)
↔
X41
)
)
)
)
→
(
¬
(
(
X49
↔
(
X41
↔
(
(
¬
X5
)
→
X48
)
)
)
→
(
(
¬
X48
)
↔
(
(
¬
X41
)
→
(
¬
X3
)
→
X49
)
)
)
)
)
→
(
X19
↔
(
X0
↔
X29
)
)
→
(
(
(
¬
X48
)
↔
(
X29
↔
X49
)
)
→
(
¬
(
(
¬
(
X49
→
X48
→
X0
)
)
→
(
¬
(
(
X33
↔
(
X29
→
False
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
¬
X48
)
→
(
¬
X33
)
→
(
¬
X29
)
→
X49
)
)
→
(
¬
(
X29
→
X0
)
)
)
)
→
(
¬
(
X49
→
(
X0
↔
X48
)
)
)
)
)
)
)
)
)
)
→
(
¬
X0
)
)
→
(
¬
(
(
(
¬
(
(
¬
X45
)
↔
(
X34
→
X48
)
)
)
→
(
X48
↔
(
X41
→
X34
→
X45
)
)
→
(
¬
(
(
¬
X45
)
↔
(
X41
↔
(
X23
↔
X48
)
)
)
)
)
→
(
¬
(
X34
↔
(
X27
↔
(
X14
↔
X48
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
(
X49
↔
(
(
¬
X21
)
→
X23
)
)
)
→
(
(
¬
(
X23
→
(
X13
↔
(
(
¬
X9
)
↔
X21
)
)
)
)
↔
(
¬
(
X49
↔
(
X23
→
(
¬
X9
)
→
X13
)
)
)
)
)
)
↔
(
¬
(
X40
→
(
¬
X21
)
→
(
¬
X13
)
→
X49
)
)
)
→
(
(
(
¬
(
X34
→
X27
→
(
X21
↔
X13
)
)
)
→
(
¬
(
X47
→
X27
→
(
X21
↔
X34
)
)
)
)
↔
(
¬
(
X34
↔
(
X27
↔
(
X23
→
X47
)
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X44
→
(
¬
X17
)
→
X22
)
)
↔
(
¬
(
(
(
(
(
(
¬
X24
)
↔
(
(
¬
X17
)
→
(
X15
↔
X44
)
)
)
↔
(
X49
↔
X44
)
)
↔
(
X49
↔
(
(
¬
X17
)
→
X44
)
)
)
→
X17
→
X24
)
→
(
¬
(
(
¬
X24
)
→
X44
)
)
)
)
)
↔
(
(
(
(
¬
X31
)
→
(
(
¬
X18
)
↔
(
(
¬
X17
)
↔
X47
)
)
)
↔
(
¬
(
(
¬
X22
)
→
X18
→
(
X15
↔
X31
)
)
)
)
↔
(
X17
↔
(
(
¬
X15
)
↔
(
(
¬
X11
)
→
X31
)
)
)
)
)
→
(
(
(
¬
(
X48
↔
(
(
¬
X4
)
→
(
(
¬
X1
)
↔
X43
)
)
)
)
↔
(
¬
(
X43
→
(
(
¬
X1
)
↔
X47
)
)
)
)
↔
(
(
¬
(
(
¬
(
X48
→
(
¬
X47
)
→
(
X32
↔
X49
)
)
)
→
(
(
X49
↔
(
X48
↔
(
X47
↔
X43
)
)
)
↔
(
¬
(
X47
↔
X44
)
)
)
)
)
→
(
(
X49
→
X47
→
(
X42
↔
X48
)
)
↔
(
¬
(
X47
↔
(
(
¬
X42
)
→
(
X38
↔
X49
)
)
)
)
)
)
)
)
)
→
(
(
X42
↔
(
X25
→
(
(
¬
X2
)
↔
X48
)
)
)
↔
(
(
¬
(
(
¬
X47
)
→
X36
→
(
X25
↔
X48
)
)
)
↔
(
X48
→
(
X36
↔
(
(
¬
X25
)
↔
X42
)
)
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X42
)
→
(
X16
↔
X48
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
764584...
and proposition id is
3c7277...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMPFKMiJwnYUNrfBap37MinhFqPTyFeeJbh
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
(
(
X33
↔
(
X29
→
X22
→
X11
→
False
)
)
↔
(
¬
(
(
¬
X29
)
↔
X37
)
)
)
→
(
¬
(
X33
→
(
X22
↔
(
X17
↔
(
X11
→
False
)
)
)
)
)
)
↔
(
(
(
(
¬
X37
)
→
(
X33
↔
(
X29
→
X47
)
)
)
→
(
¬
(
X41
↔
(
X37
→
X29
→
X33
)
)
)
)
↔
(
X47
↔
(
X37
→
X33
)
)
)
)
→
(
(
(
¬
(
(
(
¬
(
X48
→
(
X39
↔
(
X31
↔
X38
)
)
)
)
↔
(
¬
(
X48
→
X38
→
(
X31
↔
(
X21
→
False
)
)
)
)
)
→
(
¬
(
(
¬
(
X19
→
X1
→
X41
)
)
→
(
X38
→
(
(
¬
X31
)
↔
(
X19
→
X41
)
)
)
→
(
¬
(
X48
↔
(
(
¬
X19
)
↔
X31
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X41
↔
(
X30
→
(
X3
↔
X48
)
)
)
)
↔
(
(
¬
X41
)
→
X3
→
X49
)
)
↔
(
(
¬
(
(
X49
→
(
X48
↔
X6
)
)
→
(
¬
(
X49
↔
(
(
¬
X30
)
→
X6
→
X39
)
)
)
)
)
↔
(
X48
→
(
X39
↔
(
X30
→
False
)
)
)
)
)
→
(
¬
(
X41
→
(
X30
↔
X49
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
X48
→
(
X43
↔
(
(
¬
X35
)
→
X45
)
)
)
)
→
(
¬
(
(
¬
X45
)
↔
(
(
¬
X43
)
→
(
(
¬
X33
)
↔
X48
)
)
)
)
)
)
→
(
¬
(
X35
↔
(
X33
→
X31
→
X45
)
)
)
)
)
↔
(
(
X45
↔
(
(
¬
X33
)
↔
(
(
¬
X30
)
↔
X47
)
)
)
↔
(
¬
(
X48
→
(
¬
X30
)
→
X2
→
X45
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
(
(
¬
(
(
¬
X46
)
↔
(
X28
↔
X48
)
)
)
↔
(
(
(
X18
↔
(
(
¬
X4
)
↔
X46
)
)
↔
(
(
¬
X18
)
→
(
(
¬
X8
)
↔
(
(
¬
X4
)
→
X28
)
)
)
)
↔
(
¬
(
X48
→
X18
→
(
¬
X4
)
→
X8
)
)
)
)
↔
(
¬
(
X28
→
(
(
¬
X12
)
↔
(
(
¬
X4
)
↔
X46
)
)
)
)
)
↔
(
¬
(
(
X49
→
(
X18
↔
(
(
¬
X9
)
↔
X46
)
)
)
→
X49
→
(
¬
X18
)
→
(
¬
X12
)
→
X46
)
)
)
→
(
(
¬
(
(
¬
(
X40
↔
(
(
¬
X16
)
→
X5
→
X34
)
)
)
→
(
(
(
¬
(
X46
→
(
X34
↔
(
(
¬
X20
)
↔
X40
)
)
)
)
→
(
¬
(
(
¬
X40
)
↔
(
(
¬
X34
)
→
(
¬
X5
)
→
X46
)
)
)
)
↔
(
¬
(
X46
→
X20
→
X40
)
)
)
)
)
↔
(
(
(
¬
X48
)
→
(
X34
↔
(
X20
→
X49
)
)
)
↔
(
¬
(
X34
↔
(
X20
→
X46
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X47
↔
(
X40
↔
(
(
¬
X7
)
→
X23
)
)
)
)
→
(
(
¬
(
X23
→
X20
→
(
(
¬
X18
)
↔
X40
)
)
)
↔
(
(
¬
X40
)
→
(
(
¬
X23
)
↔
(
X20
→
X47
)
)
)
)
)
→
(
¬
(
(
¬
(
X42
↔
(
(
¬
X25
)
↔
(
(
¬
X20
)
→
X36
)
)
)
)
→
(
(
¬
(
X42
→
X40
→
(
X36
↔
X25
)
)
)
↔
(
¬
(
X40
↔
(
X36
→
X25
→
(
X23
↔
False
)
)
)
)
)
)
)
)
)
)
)
→
(
(
(
(
X49
↔
(
X48
→
(
(
¬
X14
)
↔
X15
)
)
)
↔
(
¬
(
(
X47
↔
(
X15
↔
(
X13
↔
X14
)
)
)
→
(
X15
↔
(
X13
→
X48
)
)
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X47
)
→
(
¬
X24
)
→
X48
)
)
)
)
↔
(
(
¬
(
(
¬
X27
)
↔
(
X14
↔
(
(
¬
X0
)
→
X32
)
)
)
)
↔
(
(
X49
↔
(
(
¬
X32
)
↔
(
(
¬
X26
)
↔
X44
)
)
)
↔
(
¬
(
(
(
¬
X44
)
→
X26
→
(
(
¬
X13
)
↔
X49
)
)
→
(
¬
(
(
(
¬
(
X49
→
X10
→
X14
)
)
↔
(
X44
→
X14
→
(
X13
↔
X10
)
)
)
→
(
¬
(
X44
↔
(
X14
↔
(
X13
→
False
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
X49
↔
(
X43
↔
(
X23
↔
X7
)
)
)
→
(
¬
(
(
(
(
¬
X23
)
→
X17
→
X42
)
↔
(
¬
(
(
(
¬
X42
)
↔
(
(
¬
X17
)
→
X49
)
)
→
(
¬
(
X49
→
(
X23
↔
X43
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X43
→
(
X23
↔
X42
)
)
)
→
(
¬
X23
)
→
X12
→
X6
→
X42
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
X48
→
X23
→
X30
)
↔
(
(
(
¬
X42
)
→
(
(
¬
X30
)
↔
(
X23
↔
X48
)
)
)
→
(
¬
(
X48
→
X38
→
(
(
¬
X30
)
↔
X42
)
)
)
)
)
→
(
¬
(
X48
↔
(
(
¬
X41
)
→
(
¬
X23
)
→
X49
)
)
)
)
)
→
X48
→
(
X30
↔
X42
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X42
)
→
X34
→
(
X19
↔
X43
)
)
)
↔
(
¬
(
X35
→
X34
→
(
X19
↔
X43
)
)
)
)
→
(
(
¬
(
X48
↔
(
(
¬
X35
)
→
(
X32
↔
X43
)
)
)
)
↔
(
¬
(
(
¬
(
(
X42
→
(
(
¬
X32
)
↔
(
X31
↔
X48
)
)
)
→
X42
→
(
X35
↔
(
X16
↔
X18
)
)
)
)
→
(
¬
(
X48
↔
(
X32
→
False
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
X46
↔
(
X28
↔
(
(
¬
X13
)
↔
X22
)
)
)
→
(
¬
X39
)
→
(
X28
↔
X49
)
)
↔
(
(
¬
(
(
(
X39
→
X27
→
(
(
¬
X21
)
↔
X49
)
)
→
(
(
¬
(
X46
→
(
¬
X36
)
→
(
¬
X27
)
→
X39
)
)
↔
(
¬
(
X49
↔
(
X27
→
(
(
¬
X24
)
↔
X39
)
)
)
)
)
)
→
(
¬
(
X36
↔
(
(
¬
X21
)
↔
(
(
¬
X2
)
→
X24
)
)
)
)
)
)
↔
(
¬
(
X49
↔
(
(
¬
X21
)
↔
(
(
¬
X20
)
→
X24
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
X42
↔
(
X28
↔
X48
)
)
→
(
¬
(
(
(
¬
X42
)
→
(
¬
X28
)
→
X48
)
→
(
¬
(
X48
↔
(
X27
→
(
(
¬
X10
)
↔
X42
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
X25
)
↔
(
(
¬
X4
)
→
(
X3
↔
X28
)
)
)
→
(
(
¬
(
(
¬
X44
)
→
(
X4
↔
X48
)
)
)
↔
(
¬
(
(
¬
X42
)
↔
(
X28
↔
(
X0
→
X48
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
X44
→
(
(
¬
X25
)
↔
(
X24
→
X42
)
)
)
)
↔
(
X25
↔
(
X24
→
(
(
¬
X1
)
↔
X42
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
(
X40
↔
(
X29
→
(
X5
↔
X49
)
)
)
)
→
(
(
(
¬
(
X26
→
X14
)
)
↔
(
(
¬
(
X26
→
(
(
¬
(
(
¬
X26
)
→
(
¬
X15
)
→
X14
→
X49
)
)
↔
(
¬
(
X15
→
(
¬
X14
)
→
X5
→
X49
)
)
)
)
)
↔
(
X48
→
(
(
¬
X15
)
↔
X49
)
)
)
)
↔
(
X49
↔
(
X33
↔
(
X15
→
X14
→
False
)
)
)
)
)
)
→
(
¬
(
X49
→
(
(
¬
X45
)
↔
(
(
¬
X14
)
↔
X48
)
)
)
)
→
(
¬
(
X49
→
(
X8
↔
X37
)
)
)
→
(
¬
(
X49
→
(
X45
↔
X48
)
)
)
)
)
↔
(
(
¬
(
X47
↔
(
(
¬
X45
)
→
(
X9
↔
X49
)
)
)
)
↔
(
¬
(
(
X48
↔
(
X46
↔
(
X11
→
X9
)
)
)
→
(
¬
(
X47
↔
(
X11
→
(
(
¬
X9
)
↔
X48
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
2b04dd...
and proposition id is
e352a1...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMc3PMLA2fzt8LNVeQo7aEPva6qHvDKHos9
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
∀X50 :
prop
,
(
¬
∀X51 :
prop
,
(
(
¬
(
(
(
(
¬
(
(
¬
(
X51
→
(
(
¬
X16
)
↔
X48
)
)
)
→
(
(
X37
↔
(
(
¬
X11
)
↔
X32
)
)
↔
(
¬
(
X48
→
(
¬
X32
)
→
(
X16
↔
X37
)
)
)
)
→
(
¬
(
X48
↔
(
(
¬
X11
)
→
X37
)
)
)
)
)
↔
(
¬
(
X48
→
X32
→
X16
→
X49
)
)
)
→
(
(
(
(
(
X46
→
(
X41
↔
X48
)
)
↔
(
X41
↔
(
(
¬
X34
)
↔
(
(
¬
X25
)
↔
X46
)
)
)
)
→
(
(
(
(
¬
(
(
(
X46
↔
(
X41
→
(
(
¬
X37
)
↔
X50
)
)
)
↔
(
X48
→
(
X46
↔
False
)
)
)
→
(
¬
(
X46
→
(
X37
↔
False
)
)
)
)
)
↔
(
¬
(
X46
↔
(
X37
→
(
X20
↔
X50
)
)
)
)
)
↔
(
¬
(
X48
→
(
X46
↔
(
X32
↔
X41
)
)
)
)
)
↔
(
(
¬
X48
)
→
(
¬
X46
)
→
(
X39
↔
X50
)
)
)
)
↔
(
¬
(
(
(
X46
↔
(
X12
↔
(
X10
→
X13
)
)
)
↔
(
X50
↔
(
X46
→
X13
→
X12
)
)
)
→
(
¬
(
X48
→
(
(
¬
X13
)
↔
(
X12
→
X46
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X48
↔
(
(
¬
X46
)
↔
(
X24
→
X50
)
)
)
)
→
(
¬
(
(
¬
(
(
X48
↔
(
(
¬
X2
)
→
X6
)
)
→
(
X50
↔
(
X46
↔
(
(
¬
X5
)
↔
X48
)
)
)
)
)
→
X48
→
(
¬
X46
)
→
X50
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
(
(
X47
↔
(
X29
↔
(
X18
↔
False
)
)
)
↔
(
(
(
¬
(
X42
→
(
¬
X29
)
→
X33
)
)
↔
(
(
¬
X33
)
→
X29
→
(
X14
↔
X47
)
)
)
↔
(
X42
↔
(
X14
→
X18
)
)
)
)
→
(
¬
(
X47
↔
(
X42
→
X33
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
X29
)
→
(
(
¬
X18
)
↔
(
X14
→
X45
)
)
)
→
(
¬
(
X29
→
(
(
¬
X14
)
↔
(
(
¬
X7
)
→
X45
)
)
)
)
)
)
↔
(
(
X18
↔
(
X7
↔
(
X0
↔
X40
)
)
)
↔
(
X40
→
(
X7
↔
X45
)
)
)
)
↔
(
X45
↔
X3
)
)
)
↔
(
(
(
¬
X40
)
→
(
X17
↔
(
(
¬
X8
)
↔
X42
)
)
)
↔
(
(
(
¬
(
(
¬
(
X48
↔
(
X45
↔
(
X36
↔
False
)
)
)
)
→
(
(
X48
↔
(
X45
→
X42
→
X40
)
)
↔
(
¬
(
(
¬
X40
)
→
(
(
¬
X36
)
↔
X42
)
)
)
)
)
)
→
(
X48
↔
(
X45
↔
(
X17
↔
X35
)
)
)
)
↔
(
¬
(
X40
→
(
X35
↔
X36
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
(
(
¬
(
(
¬
(
X43
→
(
X38
↔
(
X9
↔
(
X1
→
False
)
)
)
)
)
→
X44
→
X43
→
(
X1
↔
X38
)
)
)
↔
(
¬
(
X43
→
X38
→
False
)
)
)
→
(
¬
(
X47
→
X44
→
(
X43
↔
X38
)
)
)
)
)
↔
(
(
¬
(
(
X50
↔
(
(
¬
X31
)
→
X28
→
X43
)
)
→
X47
→
(
¬
X43
)
→
X31
→
X50
)
)
→
(
¬
(
(
¬
X38
)
↔
(
(
¬
X31
)
→
(
X28
↔
X43
)
)
)
)
)
)
↔
(
¬
(
(
(
X47
→
X44
→
(
X22
↔
(
X21
↔
False
)
)
)
↔
(
¬
(
(
¬
X44
)
→
(
¬
X43
)
→
(
X22
↔
X47
)
)
)
)
→
(
¬
(
(
X44
→
(
¬
X19
)
→
X15
→
X45
)
→
(
¬
(
(
¬
X21
)
↔
(
(
¬
X19
)
→
X15
→
X44
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X30
↔
(
X28
→
X44
)
)
)
↔
(
¬
(
(
X30
↔
(
X28
↔
(
X26
→
False
)
)
)
→
(
¬
(
(
¬
X28
)
↔
(
(
¬
X23
)
→
X44
)
)
)
)
)
)
→
(
¬
(
(
(
¬
X48
)
→
(
¬
X38
)
→
(
(
¬
X26
)
↔
X49
)
)
→
(
¬
(
(
¬
X48
)
↔
(
(
¬
X44
)
→
X49
)
)
)
)
)
→
(
(
¬
(
(
¬
X28
)
→
X4
→
X48
)
)
↔
(
(
X44
↔
(
(
¬
X4
)
↔
X27
)
)
↔
(
(
¬
(
(
¬
X27
)
→
(
(
¬
X4
)
↔
X44
)
)
)
→
(
¬
X28
)
→
X4
→
X49
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
X36
→
(
X18
↔
(
X16
↔
X42
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X41
)
→
X36
→
(
¬
X20
)
→
X42
)
)
↔
(
¬
(
X42
→
(
X41
↔
(
X15
↔
X36
)
)
)
)
)
→
(
(
¬
(
X20
↔
False
)
)
↔
(
X42
→
X36
→
(
¬
X9
)
→
X18
)
)
)
)
)
↔
(
(
(
(
¬
(
X34
↔
(
X33
↔
(
X20
↔
X36
)
)
)
)
→
(
¬
(
X33
↔
(
(
¬
X20
)
→
X10
→
X34
)
)
)
)
↔
(
¬
(
(
¬
(
X38
↔
(
(
¬
X33
)
→
X4
→
X34
)
)
)
→
X38
→
(
X34
↔
(
(
¬
X20
)
→
X33
)
)
)
)
)
→
(
X41
→
(
¬
X33
)
→
X3
→
X38
)
→
X38
→
X33
→
(
X7
↔
X36
)
)
)
↔
(
(
X38
↔
(
X29
→
(
¬
X8
)
→
X33
)
)
→
(
¬
(
(
X33
↔
(
X29
→
False
)
)
→
(
(
¬
(
X38
→
X33
→
X6
)
)
↔
(
(
(
¬
(
X32
↔
(
(
¬
X29
)
→
X38
)
)
)
↔
(
¬
(
(
¬
X34
)
→
(
¬
X29
)
→
(
(
¬
X6
)
↔
X38
)
)
)
)
→
X34
→
(
(
¬
X32
)
↔
(
(
¬
X29
)
↔
X33
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
X43
↔
(
X38
↔
X34
)
)
)
↔
(
(
¬
(
(
¬
X42
)
→
(
(
¬
X38
)
↔
(
(
¬
X31
)
↔
X43
)
)
)
)
→
(
(
¬
X38
)
↔
(
(
¬
X34
)
→
(
(
¬
X31
)
↔
X42
)
)
)
)
)
→
(
X31
↔
(
(
¬
X26
)
→
X42
)
)
)
→
(
¬
(
(
(
(
X48
→
(
X26
↔
(
X22
→
X23
)
)
)
↔
(
X48
↔
(
X34
→
(
X23
↔
X26
)
)
)
)
↔
(
(
X47
→
(
X34
↔
(
X23
↔
X26
)
)
)
↔
(
X31
→
X34
)
)
)
→
(
(
¬
(
X34
↔
(
X26
→
X1
→
X47
)
)
)
↔
(
X47
↔
(
X26
→
X12
→
X34
)
)
)
)
)
)
)
↔
(
(
¬
(
X43
↔
(
(
¬
X42
)
↔
(
(
¬
X2
)
↔
X47
)
)
)
)
→
(
¬
(
(
(
¬
(
X42
→
(
¬
X24
)
→
(
X19
↔
X43
)
)
)
↔
(
X47
→
(
¬
X42
)
→
(
(
¬
X24
)
↔
X43
)
)
)
→
(
¬
(
X47
↔
(
X19
↔
X24
)
)
)
)
)
)
)
)
→
(
(
(
(
X44
→
(
X42
↔
(
(
¬
X11
)
→
X39
)
)
)
↔
(
¬
(
X42
↔
(
X11
↔
(
X0
→
False
)
)
)
)
)
↔
(
X44
↔
(
X41
↔
(
X39
↔
False
)
)
)
)
→
(
(
(
(
¬
(
X44
→
(
X30
↔
False
)
)
)
↔
(
¬
(
X51
→
X28
→
(
¬
X14
)
→
X30
)
)
)
→
(
(
¬
X28
)
→
(
¬
X11
)
→
X51
)
→
(
¬
(
X39
↔
(
X11
↔
(
(
¬
X5
)
→
X44
)
)
)
)
)
↔
(
¬
(
(
¬
X45
)
↔
(
X39
→
(
¬
X28
)
→
X51
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
(
¬
(
X48
↔
(
X46
↔
(
X42
↔
X41
)
)
)
)
→
(
(
(
¬
(
X46
↔
(
(
¬
X27
)
→
X48
)
)
)
↔
(
¬
(
(
¬
(
X48
→
X21
→
X42
)
)
→
(
X27
↔
(
(
¬
X21
)
→
X46
)
)
)
)
)
↔
(
¬
X48
)
)
)
→
(
¬
(
(
X50
↔
(
(
¬
X46
)
→
(
X42
↔
X48
)
)
)
→
X48
→
X35
→
X17
→
X50
)
)
)
)
↔
(
(
(
¬
(
X48
↔
(
X46
→
(
X45
↔
(
X37
→
False
)
)
)
)
)
→
(
¬
(
X48
↔
(
X46
↔
(
X45
→
X50
)
)
)
)
)
→
(
¬
(
X48
↔
(
X46
→
X13
→
X50
)
)
)
)
)
→
(
(
¬
(
(
X48
→
(
X46
↔
X50
)
)
→
X50
→
(
X40
↔
(
(
¬
X25
)
→
X49
)
)
)
)
↔
(
¬
(
X49
→
(
(
¬
X46
)
↔
X48
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
890374...
and proposition id is
ccf76e...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMQcXb67t146t8ybDrvKbZrnRXDw1KSz3QE
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
(
¬
(
(
¬
(
(
¬
X4
)
→
X47
)
)
→
(
(
(
¬
(
(
¬
X25
)
↔
X45
)
)
↔
(
(
¬
(
X49
↔
(
(
¬
X45
)
→
X46
)
)
)
→
(
¬
(
(
(
¬
(
X49
→
(
X46
↔
(
(
¬
X4
)
→
X25
)
)
)
)
↔
(
X25
↔
X45
)
)
→
(
¬
(
X46
→
(
¬
X4
)
→
X45
)
)
)
)
)
)
↔
(
¬
(
X25
→
X4
→
X45
)
)
)
)
)
↔
(
¬
(
X47
↔
(
(
¬
X46
)
→
(
¬
X0
)
→
X49
)
)
)
)
↔
(
(
¬
(
X42
→
(
¬
X16
)
→
(
X13
↔
X26
)
)
)
↔
(
(
(
X49
→
(
X42
↔
False
)
)
→
(
(
¬
X26
)
↔
(
X16
↔
X45
)
)
)
↔
(
X46
↔
(
X45
→
(
(
¬
X26
)
↔
X49
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X45
)
↔
(
(
¬
X33
)
↔
(
X10
→
X49
)
)
)
)
→
(
X9
↔
(
(
¬
X5
)
→
X10
)
)
→
(
¬
(
(
X10
↔
(
(
¬
X9
)
→
(
¬
X5
)
→
X33
)
)
→
(
X49
↔
(
X33
↔
(
X9
→
X5
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X48
)
↔
(
(
¬
X47
)
→
(
¬
X45
)
→
X49
)
)
)
↔
(
¬
(
(
(
(
¬
X18
)
→
X48
)
↔
(
(
X49
→
(
¬
X18
)
→
X48
)
→
(
(
X49
→
(
(
¬
X18
)
↔
X24
)
)
↔
(
X49
→
(
X48
↔
(
X47
→
(
X18
↔
False
)
)
)
)
)
)
)
→
(
(
X49
→
X48
→
X32
)
↔
(
(
X49
→
(
¬
X32
)
→
(
¬
X23
)
→
X47
)
↔
(
(
¬
X48
)
↔
(
(
¬
X32
)
→
(
¬
X23
)
→
X49
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
X49
↔
X47
)
↔
(
(
¬
X33
)
→
(
X21
↔
(
X6
→
X49
)
)
)
)
→
(
¬
(
X47
→
(
(
¬
X33
)
↔
X49
)
)
)
)
)
→
X47
→
X48
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X22
)
→
(
¬
(
(
¬
(
X48
↔
(
X46
↔
X22
)
)
)
→
(
¬
(
(
¬
(
X48
→
(
(
¬
X22
)
↔
X41
)
)
)
→
(
(
¬
(
X46
→
(
X22
↔
X41
)
)
)
↔
(
(
¬
(
(
¬
X46
)
↔
(
(
¬
X1
)
→
X48
)
)
)
↔
(
¬
(
X46
↔
(
X41
→
(
(
¬
X1
)
↔
X22
)
)
)
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
X46
↔
(
X40
→
False
)
)
)
↔
(
¬
(
X41
↔
(
X40
→
(
X22
↔
X12
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
(
(
X48
↔
(
X40
→
(
¬
X19
)
→
X47
)
)
→
X48
→
(
X47
↔
(
X41
→
X40
)
)
)
↔
(
¬
(
X48
↔
(
(
¬
X19
)
→
X40
)
)
)
)
→
(
¬
(
X41
↔
(
X40
↔
(
X28
→
False
)
)
)
)
)
)
→
(
¬
(
X41
→
(
¬
X7
)
→
X47
)
)
→
(
(
¬
X47
)
↔
(
(
¬
X31
)
↔
X48
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
X40
)
→
(
¬
X37
)
→
X41
)
→
(
(
¬
X40
)
↔
(
X39
→
(
¬
X15
)
→
X41
)
)
)
)
→
(
¬
(
X41
↔
(
X39
↔
(
X38
→
False
)
)
)
)
→
(
¬
(
(
¬
(
X41
↔
(
X40
→
X39
)
)
)
→
X39
→
(
X38
↔
(
(
¬
X34
)
↔
X40
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X48
↔
(
X41
↔
(
X40
→
X38
)
)
)
)
→
(
¬
(
(
¬
X41
)
→
(
X40
↔
(
(
¬
X17
)
↔
X48
)
)
)
)
)
)
↔
(
(
(
(
X40
→
(
(
¬
X38
)
↔
(
(
¬
X2
)
↔
X41
)
)
)
→
(
¬
(
X40
→
X14
→
(
¬
X2
)
→
X41
)
)
)
→
(
¬
(
X38
→
X14
→
(
X2
↔
False
)
)
)
)
→
(
¬
(
X40
↔
(
X38
→
X14
→
False
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
X43
↔
(
X30
↔
(
(
¬
X3
)
↔
X46
)
)
)
→
(
X46
↔
(
X43
↔
(
X3
↔
X47
)
)
)
)
↔
(
¬
(
(
¬
(
X47
↔
(
X30
→
X20
)
)
)
→
(
¬
(
X30
→
(
X29
↔
X43
)
)
)
)
)
)
→
X47
→
(
¬
X46
)
→
(
¬
X43
)
→
X48
)
)
→
(
(
(
(
(
¬
(
(
¬
(
X46
→
(
X11
↔
X35
)
)
)
→
(
X36
↔
X44
)
→
(
(
¬
X46
)
→
(
(
¬
X36
)
↔
(
(
¬
X11
)
→
X47
)
)
)
→
X47
→
X44
)
)
→
(
¬
(
(
¬
X36
)
→
(
¬
X11
)
→
X46
)
)
)
→
(
X47
↔
(
(
¬
X36
)
↔
X46
)
)
)
→
(
(
¬
X36
)
↔
(
X27
→
(
(
¬
X11
)
↔
X46
)
)
)
)
↔
(
(
(
(
¬
X47
)
→
(
(
¬
X36
)
↔
(
(
¬
X8
)
↔
X48
)
)
)
↔
(
X47
→
X46
→
(
¬
X8
)
→
X48
)
)
↔
(
X46
→
(
(
¬
X36
)
↔
X47
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
(
¬
(
(
¬
(
(
(
X35
↔
(
X9
↔
X25
)
)
↔
(
¬
(
X36
→
(
X35
↔
(
(
¬
X25
)
↔
X43
)
)
)
)
)
→
X36
→
X9
→
False
)
)
→
(
¬
(
(
¬
X35
)
→
(
X25
↔
(
X7
↔
X43
)
)
)
)
)
)
→
(
¬
(
(
¬
X9
)
↔
(
X7
↔
X36
)
)
)
)
)
→
(
(
(
(
(
¬
X35
)
↔
(
(
¬
X30
)
↔
(
(
¬
X14
)
→
X44
)
)
)
↔
(
¬
(
X43
↔
(
X35
→
X30
)
)
)
)
→
(
X35
↔
X43
)
)
→
(
X48
↔
(
X43
→
(
(
¬
X35
)
↔
X44
)
)
)
)
→
(
X48
↔
(
X35
↔
X14
)
)
)
→
(
(
¬
(
X42
→
X36
→
X15
→
X44
)
)
↔
(
(
¬
(
X36
→
X20
→
(
X19
↔
(
X15
→
False
)
)
)
)
↔
(
X30
→
(
¬
X15
)
→
(
X4
↔
X19
)
)
)
)
)
→
(
¬
(
(
(
(
X49
→
(
(
¬
X3
)
↔
X37
)
)
↔
(
¬
(
(
¬
(
(
¬
X37
)
→
(
X11
↔
(
X3
↔
X49
)
)
)
)
→
(
¬
X37
)
→
(
(
¬
X32
)
↔
(
X11
→
X49
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X43
→
(
(
¬
X28
)
↔
(
X0
→
X37
)
)
)
)
→
(
X43
↔
(
(
¬
X37
)
↔
(
(
¬
X16
)
↔
X49
)
)
)
)
)
→
(
¬
(
X37
↔
(
(
¬
X32
)
↔
(
X10
→
X49
)
)
)
)
→
(
¬
(
X49
↔
(
X37
↔
(
X8
→
False
)
)
)
)
)
)
)
→
(
(
X49
→
(
X46
↔
(
(
¬
X10
)
↔
X37
)
)
)
↔
(
¬
(
X46
↔
(
X44
↔
(
X37
↔
False
)
)
)
)
)
)
)
→
(
(
(
¬
(
X48
↔
(
X28
→
X23
→
X37
)
)
)
↔
(
¬
X32
)
)
↔
(
X37
→
X32
→
(
(
¬
X18
)
↔
X48
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
¬
X46
)
→
(
X27
↔
(
X2
↔
X48
)
)
)
→
(
X46
↔
(
X45
↔
False
)
)
)
)
→
(
(
¬
(
(
¬
X27
)
↔
(
X21
↔
X45
)
)
)
↔
(
¬
(
(
¬
(
X26
↔
(
X13
↔
(
X12
→
X21
)
)
)
)
→
(
(
¬
(
(
(
X26
↔
(
X21
↔
(
X12
↔
X46
)
)
)
↔
(
(
¬
X46
)
↔
(
(
¬
X21
)
↔
(
(
¬
X12
)
→
X47
)
)
)
)
→
(
X47
↔
(
X46
↔
X26
)
)
)
)
↔
(
X47
↔
(
(
¬
X12
)
→
X46
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
X1
)
↔
X38
)
→
(
¬
(
X41
→
(
¬
X39
)
→
X38
→
X46
)
)
→
(
¬
(
(
X41
→
(
(
¬
X5
)
↔
(
(
¬
X1
)
↔
X39
)
)
)
→
X46
→
(
X41
↔
(
X39
→
X5
)
)
)
)
)
)
→
(
(
(
¬
X38
)
→
(
(
¬
X29
)
↔
(
(
¬
X1
)
↔
X39
)
)
)
↔
(
¬
(
X38
→
(
¬
X1
)
→
X34
)
)
)
)
→
(
X47
↔
(
(
¬
X40
)
↔
(
(
¬
X24
)
→
X44
)
)
)
→
(
(
(
(
X47
↔
(
X40
→
(
(
¬
X22
)
↔
X48
)
)
)
→
X47
→
X40
→
X33
→
False
)
↔
(
(
¬
(
(
¬
(
(
¬
X47
)
→
(
(
¬
X34
)
↔
(
X22
→
X48
)
)
)
)
→
X39
→
X22
→
X47
)
)
↔
(
¬
(
X48
↔
(
X44
↔
(
X39
→
X34
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
X44
)
↔
(
(
¬
X31
)
→
X6
→
X47
)
)
)
↔
(
(
¬
(
(
X48
↔
(
(
¬
X17
)
↔
X44
)
)
→
(
(
¬
(
X47
↔
(
(
¬
X44
)
↔
(
(
¬
X31
)
↔
X48
)
)
)
)
↔
(
¬
(
X44
↔
(
X29
↔
False
)
)
)
)
)
)
↔
(
¬
(
X48
↔
(
X47
↔
X29
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
eaa0ee...
and proposition id is
d0e12a...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMMZnFtazWYThv9JsQZHKwmj87n4M4reUpY
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
¬
∀X51 :
prop
,
(
¬
∀X52 :
prop
,
(
(
(
(
¬
(
(
¬
(
(
¬
(
(
(
X48
↔
(
X29
↔
X37
)
)
↔
(
¬
(
(
¬
(
(
¬
X46
)
→
(
¬
X29
)
→
X48
)
)
→
X40
→
X37
→
X46
)
)
)
→
(
X48
↔
(
X40
↔
(
X29
↔
X7
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X48
↔
(
X44
↔
(
X40
→
(
X9
↔
False
)
)
)
)
)
→
(
¬
(
(
¬
X44
)
↔
(
X41
→
(
X40
↔
X46
)
)
)
)
)
→
X39
→
False
)
)
)
)
→
(
(
(
(
X46
→
X40
→
(
¬
X20
)
→
X47
)
→
(
¬
(
X48
↔
(
(
¬
X46
)
→
(
¬
X20
)
→
X47
)
)
)
)
↔
(
X47
↔
(
X46
↔
X20
)
)
)
↔
(
¬
(
(
¬
(
X40
→
X27
→
X0
)
)
→
(
¬
(
(
(
¬
X27
)
↔
(
(
¬
X0
)
↔
X48
)
)
→
X40
→
X27
→
X46
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
X26
↔
(
X17
→
X4
→
X45
)
)
)
↔
(
¬
(
(
¬
X26
)
↔
(
X17
→
X5
→
X45
)
)
)
)
↔
(
¬
(
X26
→
X4
→
X5
)
)
)
↔
(
¬
(
X45
→
X26
→
X17
)
)
)
→
(
(
(
¬
X49
)
→
(
X46
↔
(
(
¬
X31
)
↔
X51
)
)
)
↔
(
(
¬
(
X49
↔
(
X26
→
(
X5
↔
False
)
)
)
)
↔
(
(
¬
X46
)
→
(
(
¬
X26
)
↔
(
X12
↔
X51
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X40
↔
(
X3
→
X51
)
)
)
→
(
(
¬
(
X51
↔
(
X40
→
X50
)
)
)
↔
(
X40
↔
(
X3
↔
X30
)
)
)
)
→
(
X40
↔
(
X3
↔
X30
)
)
)
)
↔
(
(
(
(
X24
→
(
¬
X15
)
→
X50
)
→
(
¬
(
X51
→
(
X50
↔
(
X46
↔
(
X15
↔
False
)
)
)
)
)
→
(
(
X42
↔
(
(
¬
X24
)
→
X46
)
)
↔
(
(
X50
↔
(
X42
→
X24
→
X46
)
)
↔
(
X51
↔
(
(
¬
X46
)
→
(
¬
X24
)
→
X50
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X50
)
↔
(
(
¬
X46
)
→
(
(
¬
X45
)
↔
X51
)
)
)
)
↔
(
(
¬
X50
)
→
X46
→
(
¬
X38
)
→
X51
)
)
→
(
(
X50
↔
(
X45
↔
X40
)
)
→
X51
→
X50
→
(
¬
X19
)
→
X45
)
→
(
¬
(
X51
→
(
X50
↔
(
X40
↔
False
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X44
)
↔
(
(
¬
X24
)
→
(
¬
X14
)
→
X50
)
)
)
→
(
¬
(
X45
↔
(
(
¬
X24
)
→
X50
)
)
)
)
)
→
(
¬
(
X45
↔
(
X44
↔
(
X24
→
(
X16
↔
False
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X34
→
(
(
¬
X28
)
↔
(
X6
↔
X49
)
)
)
)
→
(
¬
(
X49
↔
(
X34
↔
X48
)
)
)
)
↔
(
(
¬
(
X23
↔
(
(
¬
X22
)
↔
X25
)
)
)
↔
(
¬
(
(
(
(
¬
X28
)
↔
(
X23
↔
(
X8
↔
X34
)
)
)
→
(
¬
(
(
¬
X49
)
→
(
(
¬
X28
)
↔
(
X8
↔
X50
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X50
)
↔
(
X25
↔
(
X23
→
X52
)
)
)
)
↔
(
X50
→
(
¬
X23
)
→
(
¬
X22
)
→
X25
)
)
↔
(
(
(
¬
X23
)
→
(
X13
↔
X49
)
)
↔
(
(
X50
→
X49
→
X22
)
↔
(
(
¬
(
(
¬
X49
)
↔
(
X23
↔
(
(
¬
X13
)
→
X50
)
)
)
)
↔
(
¬
(
X50
↔
(
X22
↔
X13
)
)
)
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X32
↔
X33
)
)
↔
(
(
¬
(
X52
↔
(
X32
→
(
¬
X21
)
→
X33
)
)
)
↔
(
X52
→
(
X33
↔
(
X21
↔
(
X11
→
False
)
)
)
)
)
)
↔
(
(
X43
→
(
(
¬
X36
)
↔
X52
)
)
↔
(
(
¬
(
X52
↔
(
(
¬
X36
)
→
X11
→
X43
)
)
)
→
(
¬
(
(
(
X43
→
X11
→
X33
)
↔
(
(
X43
→
(
X36
↔
(
X11
↔
X33
)
)
)
↔
(
¬
(
X36
↔
(
(
¬
X33
)
↔
(
(
¬
X2
)
→
X43
)
)
)
)
)
)
→
(
¬
(
X33
↔
X36
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
(
¬
X18
)
↔
X50
)
→
(
¬
(
X43
↔
(
X1
→
X49
)
)
)
)
)
→
(
(
¬
(
X43
↔
(
(
¬
X33
)
→
X10
→
X36
)
)
)
↔
(
X36
→
X18
→
X10
→
X33
)
)
→
(
¬
(
(
¬
X35
)
↔
(
(
¬
X33
)
→
X18
→
X50
)
)
)
)
)
→
(
¬
X18
)
→
(
X10
↔
X35
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
(
X39
↔
(
X36
↔
(
X35
→
X33
)
)
)
→
(
X39
→
(
(
¬
X35
)
↔
(
(
¬
X13
)
↔
X36
)
)
)
→
(
¬
(
X39
→
(
¬
X36
)
→
(
X35
↔
X49
)
)
)
)
)
↔
(
(
¬
(
(
¬
X27
)
→
(
X24
↔
X51
)
)
)
↔
(
(
X33
↔
(
X24
→
X51
)
)
↔
(
¬
(
(
¬
X36
)
→
(
(
¬
X27
)
↔
(
(
¬
X24
)
→
X49
)
)
)
)
)
)
)
↔
(
(
X51
→
(
¬
X12
)
→
X11
→
X23
)
→
(
X23
→
(
¬
X11
)
→
X5
→
X36
)
→
(
¬
(
X51
→
(
(
¬
X33
)
↔
X36
)
)
)
)
)
→
(
(
(
(
¬
(
X39
↔
(
(
¬
X15
)
↔
X45
)
)
)
→
(
(
(
¬
(
X39
↔
(
X19
→
(
X15
↔
X31
)
)
)
)
↔
(
¬
(
(
¬
X45
)
↔
(
X39
→
(
(
¬
X19
)
↔
X48
)
)
)
)
)
↔
(
X48
→
X45
→
X31
)
)
)
↔
(
(
(
¬
(
X28
↔
(
X18
→
X16
)
)
)
→
(
X45
↔
(
X28
→
X16
→
False
)
)
→
(
¬
(
X45
→
(
X39
↔
(
X28
↔
(
X14
→
False
)
)
)
)
)
→
(
¬
(
X45
→
(
¬
X28
)
→
X16
→
X39
)
)
)
↔
(
(
¬
(
X31
→
(
X28
↔
X39
)
)
)
↔
(
X29
↔
(
(
¬
X28
)
→
(
(
¬
X18
)
↔
X31
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
X29
→
X36
)
→
(
X36
↔
False
)
→
(
(
X39
↔
(
X29
↔
False
)
)
↔
(
¬
(
X39
↔
(
(
¬
X20
)
↔
(
X1
→
X36
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
X29
)
↔
(
(
¬
X21
)
↔
(
(
¬
X1
)
→
X45
)
)
)
)
)
↔
(
(
(
¬
(
X20
↔
X49
)
)
↔
(
X52
→
(
¬
X39
)
→
X29
→
X49
)
)
↔
(
X39
↔
False
)
)
)
↔
(
¬
(
X39
→
(
X36
↔
(
X21
↔
False
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
(
(
¬
(
X50
↔
(
(
¬
X26
)
↔
X46
)
)
)
↔
(
(
X46
↔
(
(
¬
X26
)
→
X50
)
)
→
(
(
X46
→
(
X37
↔
(
X32
↔
X50
)
)
)
↔
(
(
¬
X46
)
→
(
(
¬
X32
)
↔
(
X26
↔
X50
)
)
)
)
)
)
→
(
¬
X46
)
→
(
(
¬
X32
)
↔
X50
)
)
)
↔
(
(
¬
X37
)
→
X46
)
)
→
(
¬
(
(
¬
(
(
¬
X32
)
→
X26
→
(
(
¬
X3
)
↔
X51
)
)
)
→
(
¬
(
X42
↔
(
(
¬
X32
)
↔
(
X3
→
X37
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X50
)
→
X41
→
(
(
¬
X8
)
↔
X51
)
)
)
↔
(
(
(
X42
↔
(
(
¬
X41
)
↔
(
(
¬
X8
)
↔
X48
)
)
)
↔
(
¬
(
(
¬
X42
)
↔
(
X34
↔
X48
)
)
)
)
→
(
¬
(
X42
↔
(
X6
→
False
)
)
)
→
(
(
¬
X48
)
↔
(
X41
↔
X51
)
)
)
)
↔
(
(
(
(
(
(
¬
(
X51
↔
(
(
¬
X30
)
→
X48
)
)
)
↔
(
¬
(
(
¬
X48
)
→
(
(
¬
X47
)
↔
(
(
¬
X30
)
↔
X50
)
)
)
)
)
↔
(
(
¬
X47
)
↔
X51
)
)
↔
(
¬
(
X51
↔
(
X30
↔
X47
)
)
)
)
↔
(
¬
(
X51
↔
(
X50
→
X48
→
False
)
)
)
)
↔
(
(
¬
(
(
¬
X50
)
→
(
¬
X47
)
→
(
¬
X9
)
→
X51
)
)
↔
(
¬
(
(
¬
X50
)
→
(
(
¬
X9
)
↔
(
X4
↔
X51
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
X52
→
(
(
¬
X48
)
↔
X51
)
)
↔
(
¬
(
(
¬
(
X51
→
(
X50
↔
(
X2
→
X48
)
)
)
)
→
(
(
X48
↔
(
X10
→
X50
)
)
↔
(
(
¬
X51
)
→
(
¬
X50
)
→
(
X48
↔
X52
)
)
)
)
)
)
↔
(
(
X52
→
X48
→
False
)
↔
(
(
¬
(
(
(
¬
X50
)
↔
(
(
¬
X48
)
→
(
¬
X43
)
→
X52
)
)
→
(
¬
(
(
¬
X43
)
↔
X44
)
)
)
)
→
(
¬
(
X52
↔
(
X48
↔
(
(
¬
X38
)
→
X50
)
)
)
)
)
)
)
→
(
¬
(
(
X51
→
(
(
¬
X25
)
↔
(
(
¬
X7
)
↔
X44
)
)
)
→
(
X25
↔
(
X7
→
(
X0
↔
False
)
)
)
)
)
→
(
(
(
(
X44
↔
(
X22
→
X17
→
X25
)
)
→
(
(
X44
↔
False
)
↔
(
(
X50
→
(
X40
↔
(
(
¬
X22
)
↔
X51
)
)
)
↔
(
¬
(
(
¬
X50
)
→
X44
→
(
(
¬
X40
)
↔
X51
)
)
)
)
)
→
(
¬
(
X44
↔
(
(
¬
X40
)
→
(
(
¬
X25
)
↔
X50
)
)
)
)
)
↔
(
(
¬
X44
)
↔
(
X40
↔
X50
)
)
)
↔
(
X40
↔
(
(
¬
X25
)
↔
X51
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
ab797e...
and proposition id is
4c7efe...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMbMvnZn7hhV8ktjHnkwrJRXK94hRxkrTc7
)
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
(
¬
(
(
¬
X41
)
↔
(
(
¬
X32
)
→
(
¬
X28
)
→
X44
)
)
)
↔
(
(
(
¬
(
(
¬
X41
)
→
(
X19
↔
(
X13
↔
X42
)
)
)
)
↔
(
¬
(
X41
→
X19
→
(
X17
↔
X13
)
)
)
)
→
(
(
X33
→
(
¬
X28
)
→
(
X9
↔
X32
)
)
↔
(
(
X41
↔
(
X32
↔
(
X28
→
X17
→
False
)
)
)
→
(
X28
↔
(
X19
↔
X41
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X44
→
X6
)
)
→
(
(
¬
(
X16
→
(
(
¬
X6
)
↔
X41
)
)
)
↔
(
¬
(
(
¬
X21
)
→
X44
)
)
)
)
→
(
¬
(
X41
→
(
(
¬
X38
)
↔
(
(
¬
X4
)
↔
X44
)
)
)
)
→
(
¬
(
X41
↔
(
(
¬
X26
)
↔
(
(
¬
X21
)
→
X44
)
)
)
)
)
)
→
(
¬
(
X28
↔
(
(
¬
X21
)
→
X16
→
X41
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
(
X39
↔
(
X35
→
X48
)
)
)
→
(
(
¬
(
(
¬
X41
)
→
(
(
¬
X39
)
↔
X44
)
)
)
↔
(
X42
↔
(
X39
→
(
¬
X35
)
→
X41
)
)
)
→
(
X49
↔
(
X42
→
(
¬
X35
)
→
X39
)
)
)
)
↔
(
(
(
¬
(
X11
↔
(
X10
↔
(
X1
→
X35
)
)
)
)
↔
(
¬
(
X11
→
(
X10
↔
X14
)
)
)
)
↔
(
¬
(
(
(
X39
→
(
X14
↔
(
(
¬
X5
)
↔
X48
)
)
)
↔
(
(
X35
↔
(
X10
→
False
)
)
↔
(
X48
↔
(
(
¬
X35
)
→
(
¬
X14
)
→
X39
)
)
)
)
→
(
¬
(
X39
→
(
¬
X14
)
→
X35
)
)
)
)
)
)
→
(
(
(
¬
X31
)
↔
(
X30
↔
(
(
¬
X15
)
→
X49
)
)
)
↔
(
¬
(
X48
↔
(
X15
↔
(
(
¬
X12
)
↔
X31
)
)
)
)
)
→
(
(
¬
(
X48
→
X3
→
X31
)
)
↔
(
(
¬
(
X49
↔
(
X44
↔
False
)
)
)
→
(
¬
(
(
¬
(
X49
→
(
(
¬
X31
)
↔
X44
)
)
)
→
(
¬
(
(
¬
X18
)
→
(
(
¬
X3
)
↔
X31
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
(
¬
X36
)
↔
(
X20
↔
(
(
¬
X2
)
↔
X40
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
X42
↔
(
X0
→
False
)
)
↔
(
¬
(
(
X45
↔
(
X42
→
X36
→
False
)
)
→
(
¬
(
(
(
¬
X36
)
↔
(
X34
↔
(
X0
↔
X45
)
)
)
→
(
¬
(
(
¬
X42
)
↔
(
X34
→
(
(
¬
X0
)
↔
X45
)
)
)
)
)
)
)
)
)
→
(
¬
(
X45
↔
(
X42
→
X34
→
X40
)
)
)
)
)
↔
(
(
¬
X42
)
→
(
X36
↔
X45
)
)
)
→
(
¬
(
X36
→
X20
→
X0
)
)
)
)
)
→
(
X43
↔
(
(
¬
X36
)
↔
X44
)
)
→
(
¬
(
(
X45
→
X44
→
(
(
¬
X40
)
↔
X46
)
)
→
(
(
(
¬
X44
)
↔
(
(
¬
X43
)
→
X29
→
X46
)
)
↔
(
(
¬
X43
)
→
X45
)
)
)
)
)
)
→
(
(
(
¬
(
(
X43
→
(
(
¬
X27
)
↔
X47
)
)
→
(
¬
(
X43
→
(
X25
↔
(
X24
↔
X27
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
X41
)
→
(
(
¬
X27
)
↔
X43
)
)
→
(
¬
(
(
¬
(
(
¬
X43
)
→
(
(
¬
X37
)
↔
(
X27
→
X47
)
)
)
)
→
(
(
X43
→
X27
)
↔
(
X47
→
(
(
¬
X27
)
↔
(
X22
→
X41
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X42
↔
(
X23
→
(
X8
↔
(
X7
↔
False
)
)
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X7
)
→
X23
)
)
)
)
)
↔
(
¬
(
X46
↔
(
X43
→
(
¬
X27
)
→
X47
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
(
¬
X9
)
→
X41
)
)
→
(
(
¬
(
X40
→
(
X23
↔
(
(
¬
X9
)
→
X43
)
)
)
)
↔
(
¬
(
X41
↔
(
(
¬
X40
)
→
(
X9
↔
X43
)
)
)
)
)
)
↔
(
¬
(
X26
↔
(
X9
→
X22
)
)
)
)
→
(
(
X47
→
(
X40
↔
(
(
¬
X5
)
↔
X19
)
)
)
↔
(
(
¬
(
X41
→
(
(
¬
X5
)
↔
X42
)
)
)
→
(
¬
(
(
X42
↔
(
X5
→
X47
)
)
→
(
(
¬
(
X43
↔
(
X41
↔
(
X40
→
False
)
)
)
)
↔
(
(
X43
→
X41
→
(
X5
↔
X42
)
)
→
(
¬
(
X43
↔
(
X41
→
X40
)
)
)
)
)
)
)
→
(
X41
↔
(
X40
↔
X42
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
¬
(
(
¬
X41
)
↔
(
(
¬
X16
)
↔
X42
)
)
)
→
(
(
¬
(
X4
↔
X41
)
)
↔
(
(
¬
(
(
¬
X35
)
↔
(
(
¬
X16
)
↔
X42
)
)
)
→
(
¬
(
X42
→
(
X35
↔
(
(
¬
X4
)
↔
X41
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
(
¬
(
X42
↔
(
X35
↔
X48
)
)
)
→
X35
→
(
(
¬
X31
)
↔
(
X24
↔
X48
)
)
)
)
→
(
(
¬
(
X31
↔
(
X21
→
X48
)
)
)
↔
(
(
¬
(
X48
↔
(
X41
→
(
(
¬
X21
)
↔
X31
)
)
)
)
→
X48
→
X35
→
X21
→
X41
)
)
)
↔
(
¬
(
X31
→
X24
→
X41
)
)
)
→
(
¬
(
(
X48
↔
(
(
¬
X31
)
↔
(
(
¬
X29
)
↔
X40
)
)
)
→
(
¬
(
(
¬
(
X42
→
(
¬
X40
)
→
(
(
¬
X20
)
↔
X48
)
)
)
→
(
¬
X42
)
→
(
¬
X40
)
→
(
X31
↔
X48
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
X48
)
→
(
(
¬
X41
)
↔
(
X40
↔
X49
)
)
)
)
→
(
¬
(
X49
→
X46
→
X11
→
X41
)
)
)
)
↔
(
(
X46
→
X18
→
(
X12
↔
X40
)
)
→
X46
→
(
(
¬
X40
)
↔
(
(
¬
X18
)
→
X41
)
)
)
)
)
)
)
↔
(
(
(
(
(
X49
↔
(
X39
→
(
¬
X33
)
→
X45
)
)
→
(
¬
(
X49
↔
(
X45
→
X39
→
(
X36
↔
False
)
)
)
)
)
↔
(
(
¬
(
X49
→
(
(
¬
X39
)
↔
(
X6
→
X45
)
)
)
)
↔
(
¬
(
(
(
¬
(
X17
↔
(
X6
↔
(
(
¬
X0
)
→
X49
)
)
)
)
↔
(
¬
(
X39
→
(
¬
X6
)
→
(
(
¬
X0
)
↔
X49
)
)
)
)
→
(
¬
X39
)
)
)
)
)
↔
(
(
(
X47
→
(
X39
↔
(
(
¬
X15
)
→
X33
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X39
→
X32
→
X28
→
X33
)
)
→
(
¬
X39
)
→
(
(
¬
X33
)
↔
(
(
¬
X32
)
↔
X47
)
)
)
)
→
(
¬
(
(
¬
X39
)
→
(
¬
X33
)
→
(
X2
↔
X47
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X47
)
→
(
X25
↔
(
X3
→
X49
)
)
)
)
↔
(
¬
X33
)
)
↔
(
(
X49
→
(
¬
X25
)
→
(
X7
↔
X28
)
)
↔
(
(
¬
(
X47
↔
(
(
¬
X1
)
→
X49
)
)
)
↔
(
(
¬
(
(
¬
X28
)
↔
(
(
¬
X25
)
→
X49
)
)
)
↔
(
X47
↔
(
X10
↔
X28
)
)
)
)
)
)
)
)
→
(
(
(
(
(
(
(
¬
(
X45
→
(
¬
X44
)
→
X49
)
)
↔
(
¬
(
(
¬
X47
)
↔
(
X44
→
X34
→
X49
)
)
)
)
→
(
¬
(
(
¬
X45
)
↔
(
X44
→
X49
)
)
)
)
→
X49
→
(
X47
↔
False
)
)
↔
(
¬
(
(
¬
X45
)
↔
(
X44
→
(
¬
X8
)
→
X49
)
)
)
)
→
(
(
(
¬
X44
)
→
X37
→
X49
)
↔
(
(
¬
X45
)
→
X27
→
X49
)
)
)
↔
(
(
(
¬
X34
)
→
(
¬
X13
)
→
X46
)
↔
(
¬
(
(
X46
↔
(
(
¬
X37
)
↔
X38
)
)
→
(
(
X38
↔
(
(
¬
X37
)
→
X34
→
X46
)
)
↔
(
(
¬
(
X37
→
(
X34
↔
(
X30
↔
X45
)
)
)
)
↔
(
¬
(
X37
→
(
X30
↔
(
(
¬
X14
)
→
X34
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
6e1b5b...
and proposition id is
6e81dd...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMHoccTfGP4sopar8CfEa4Q58EGCWbiXssW
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
¬
(
(
¬
(
(
¬
(
X46
→
(
(
¬
X31
)
↔
(
(
¬
X26
)
→
X44
)
)
)
)
→
(
(
¬
X39
)
→
(
(
¬
X37
)
↔
(
X31
↔
X44
)
)
)
→
(
¬
(
(
¬
(
X49
↔
(
X46
↔
(
(
¬
X31
)
↔
X39
)
)
)
)
→
(
X39
→
(
X37
↔
X46
)
)
→
(
¬
(
(
¬
(
(
X49
→
X37
→
(
¬
X4
)
→
X46
)
→
X46
→
X39
→
(
X4
↔
X49
)
)
)
→
(
X49
↔
(
X46
→
X39
)
)
)
)
)
)
)
)
→
(
(
¬
(
X26
↔
(
(
¬
X9
)
→
X44
)
)
)
↔
(
¬
(
(
(
¬
X44
)
↔
(
X9
→
X48
)
)
→
(
¬
(
X48
→
X46
→
X26
→
X9
)
)
)
)
)
)
)
→
(
(
X45
→
(
¬
X39
)
→
(
(
¬
X28
)
↔
X49
)
)
→
(
¬
(
(
(
¬
X28
)
→
X45
)
→
X28
→
X26
→
(
(
¬
X16
)
↔
X39
)
)
)
)
→
(
(
(
(
(
(
¬
X39
)
→
X5
→
X45
)
→
(
¬
(
X45
→
(
¬
X14
)
→
X49
)
)
)
↔
(
¬
(
X45
→
X39
→
X5
)
)
)
↔
(
X49
→
(
X14
↔
X5
)
)
)
↔
(
¬
(
(
¬
X45
)
→
(
X15
↔
(
X14
→
X49
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
X49
↔
(
X48
↔
(
X32
↔
False
)
)
)
→
(
¬
(
(
¬
(
X48
↔
(
X32
→
(
(
¬
X6
)
↔
X33
)
)
)
)
→
(
¬
(
X49
→
X48
→
X32
→
X6
)
)
)
)
)
)
↔
(
(
(
X33
↔
(
X19
→
X32
)
)
→
(
X49
↔
(
X45
→
False
)
)
)
↔
(
¬
(
(
¬
X33
)
↔
X49
)
)
)
)
↔
(
¬
(
X43
→
X32
→
(
X19
↔
X33
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
(
¬
X0
)
↔
X3
)
)
↔
(
X48
→
X27
→
(
(
¬
X3
)
↔
X36
)
)
)
→
(
¬
(
X45
→
(
X36
↔
(
(
¬
X0
)
↔
X27
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
¬
(
X49
↔
(
(
¬
X27
)
↔
(
(
¬
X20
)
→
X48
)
)
)
)
→
(
(
¬
X48
)
↔
(
(
¬
X36
)
→
X27
→
X49
)
)
)
)
→
X49
→
(
¬
X27
)
→
X36
)
)
→
X36
→
(
X27
↔
(
X24
→
X48
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
X45
↔
(
X42
→
X38
→
X30
)
)
)
↔
(
(
(
¬
X42
)
→
(
(
¬
X34
)
↔
(
(
¬
X22
)
↔
X49
)
)
)
→
(
¬
(
X18
↔
X42
)
)
)
)
→
(
(
¬
(
(
¬
X38
)
↔
(
(
¬
X25
)
↔
X41
)
)
)
↔
(
(
(
¬
X42
)
↔
(
(
¬
X41
)
→
(
X23
↔
X49
)
)
)
→
(
¬
(
(
¬
X41
)
↔
(
(
¬
X38
)
→
(
¬
X23
)
→
X49
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
X45
→
X40
→
X13
→
False
)
)
→
(
¬
(
X49
↔
(
X7
→
X45
)
)
)
)
)
→
(
¬
(
(
¬
(
(
X42
→
(
(
¬
X40
)
↔
(
X1
↔
X45
)
)
)
→
(
¬
(
X45
↔
X13
)
)
)
)
→
(
X17
↔
(
(
¬
X11
)
→
(
(
¬
X1
)
↔
X45
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
X47
↔
(
X45
↔
(
X10
→
X2
)
)
)
↔
(
(
¬
(
(
¬
(
X47
→
(
X40
↔
(
X10
↔
X45
)
)
)
)
→
(
¬
(
X45
→
(
(
¬
X40
)
↔
(
(
¬
X8
)
→
X47
)
)
)
)
)
)
→
X45
→
(
X40
↔
(
X10
↔
False
)
)
)
)
↔
(
¬
(
X49
↔
(
X45
→
X10
→
X2
)
)
)
)
↔
(
¬
(
X47
↔
(
X10
↔
(
X2
→
X46
)
)
)
)
)
→
(
(
(
X46
→
(
X35
↔
(
X29
→
X45
)
)
)
↔
(
(
¬
(
X46
→
X29
)
)
↔
(
¬
(
X45
→
X12
→
X40
)
)
)
)
↔
(
X40
↔
(
X21
→
X12
→
False
)
)
)
→
(
¬
X40
)
→
(
X35
↔
X46
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
¬
(
X43
→
X42
→
X40
→
X2
)
)
→
(
¬
(
(
(
¬
X42
)
→
(
(
¬
X40
)
↔
(
X4
→
X47
)
)
)
→
X47
→
X43
→
(
(
¬
X4
)
↔
X31
)
)
)
)
)
↔
(
X43
→
X42
→
(
(
¬
X31
)
↔
X49
)
)
)
↔
(
(
(
¬
(
X21
↔
X33
)
)
→
(
(
¬
(
X43
→
(
X33
↔
X21
)
)
)
↔
(
(
(
(
¬
X47
)
↔
(
X43
→
(
(
¬
X33
)
↔
X49
)
)
)
→
(
X49
↔
(
X43
↔
(
(
¬
X21
)
→
X33
)
)
)
)
↔
(
X49
→
(
¬
X21
)
→
X43
)
)
)
)
→
(
¬
(
(
¬
(
X43
→
X14
→
(
X1
↔
False
)
)
)
→
(
¬
(
X47
↔
(
X43
↔
(
(
¬
X39
)
↔
X49
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
(
¬
X41
)
↔
(
(
¬
X28
)
→
X18
→
X47
)
)
)
↔
(
¬
(
(
¬
X41
)
→
(
X18
↔
(
X7
↔
X47
)
)
)
)
)
→
(
¬
(
X41
→
X28
→
X15
→
X47
)
)
→
(
(
¬
(
(
¬
(
(
(
¬
X19
)
→
(
¬
X13
)
→
X25
)
→
(
(
¬
(
(
¬
X25
)
→
(
X19
↔
(
X13
↔
X28
)
)
)
)
↔
(
¬
(
X41
→
(
X19
↔
X25
)
)
)
)
)
)
→
X28
→
X19
)
)
↔
(
¬
(
(
¬
X28
)
→
(
(
¬
X25
)
↔
X41
)
)
)
)
)
→
(
¬
(
X47
↔
(
X28
→
(
X25
↔
X6
)
)
)
)
→
(
¬
(
(
X44
↔
(
(
¬
X17
)
↔
(
(
¬
X6
)
↔
X28
)
)
)
→
X25
→
X6
→
X28
)
)
)
)
↔
(
(
(
X49
↔
(
X29
↔
(
(
¬
X8
)
→
X43
)
)
)
↔
(
(
(
X43
↔
(
(
¬
X38
)
↔
(
(
¬
X28
)
↔
X49
)
)
)
↔
(
¬
(
X49
→
X43
→
(
X38
↔
X12
)
)
)
)
↔
(
¬
(
X43
↔
(
X29
↔
(
(
¬
X28
)
→
X38
)
)
)
)
)
)
↔
(
(
¬
(
X47
↔
(
(
¬
X44
)
↔
(
(
¬
X43
)
→
X49
)
)
)
)
→
(
X43
↔
X38
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
¬
X47
)
↔
X48
)
↔
(
(
(
¬
X46
)
↔
(
X22
→
X3
→
X48
)
)
↔
(
X45
↔
(
(
¬
X3
)
→
X48
)
)
)
)
→
(
(
(
¬
(
X43
→
X11
→
X45
)
)
↔
(
(
(
(
(
¬
(
X46
↔
(
X43
→
X11
→
X48
)
)
)
→
(
X43
↔
(
X11
→
X45
)
)
)
→
(
X46
↔
(
X45
→
X48
)
)
)
↔
(
X48
↔
(
X11
↔
(
X0
→
X43
)
)
)
)
→
(
X46
↔
(
X45
↔
X11
)
)
)
)
↔
(
(
(
X27
→
(
¬
X11
)
→
X45
)
↔
(
(
(
X46
↔
(
X45
↔
X11
)
)
↔
(
X46
→
X10
→
X27
)
)
→
(
¬
(
X45
↔
False
)
)
)
)
↔
(
X45
→
X46
)
)
)
)
)
→
(
(
(
(
X48
↔
(
X35
↔
X9
)
)
↔
(
¬
(
X46
↔
(
(
¬
X36
)
→
(
(
¬
X9
)
↔
X45
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X9
)
→
(
X5
↔
X45
)
)
)
↔
(
(
¬
(
(
¬
X9
)
→
X35
)
)
→
(
X45
↔
(
(
¬
X30
)
↔
(
X5
↔
X35
)
)
)
)
)
↔
(
X44
↔
(
(
¬
X35
)
→
(
(
¬
X30
)
↔
X45
)
)
)
)
)
↔
(
(
¬
(
X46
→
(
X24
↔
(
(
¬
X16
)
→
X48
)
)
)
)
↔
(
(
¬
(
(
X48
↔
(
X34
→
(
X24
↔
X16
)
)
)
→
(
¬
X45
)
→
X24
→
X48
)
)
↔
(
¬
(
X34
↔
X26
)
)
)
)
)
→
(
(
(
¬
X45
)
↔
(
(
¬
X37
)
→
(
X20
↔
X48
)
)
)
↔
(
¬
(
(
¬
(
X37
→
(
¬
X35
)
→
(
¬
X32
)
→
X48
)
)
→
(
¬
(
X45
→
(
X23
↔
X48
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
845bb7...
and proposition id is
3c70e1...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMa5XRrXJtrRVXMnb2StpY2MtNA63SW3xcQ
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
(
¬
(
(
¬
X45
)
→
(
¬
X43
)
→
(
¬
X8
)
→
X48
)
)
↔
(
(
¬
(
(
X48
→
(
X0
↔
X8
)
)
→
(
(
(
¬
X42
)
→
(
¬
X22
)
→
X48
)
↔
(
(
(
¬
X42
)
↔
X48
)
→
X22
→
(
X8
↔
(
X0
→
False
)
)
)
)
)
)
↔
(
X45
↔
(
X22
→
(
¬
X0
)
→
X42
)
)
)
)
↔
(
(
¬
(
(
¬
X21
)
↔
(
X12
→
X10
→
X48
)
)
)
↔
(
(
X48
↔
X43
)
↔
(
(
¬
X12
)
→
(
¬
X10
)
→
X42
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
X43
→
X30
)
)
↔
(
¬
(
(
X36
↔
(
(
¬
X32
)
↔
X45
)
)
→
(
¬
(
X43
→
(
X36
↔
(
X32
→
False
)
)
)
)
→
(
(
¬
(
X36
↔
(
(
¬
X30
)
→
X45
)
)
)
↔
(
¬
(
(
X45
→
(
X43
↔
X36
)
)
→
(
¬
(
X43
↔
(
X36
→
(
(
¬
X30
)
↔
X32
)
)
)
)
)
)
)
)
)
)
→
(
X43
→
X31
→
X17
→
False
)
→
(
X45
↔
(
X43
↔
(
X32
→
False
)
)
)
)
)
→
(
(
(
(
¬
(
X43
↔
(
X28
↔
(
(
¬
X2
)
→
X36
)
)
)
)
→
X43
→
(
(
¬
X28
)
↔
(
X27
→
X36
)
)
)
↔
(
(
(
(
(
¬
(
X35
→
(
¬
X33
)
→
X36
)
)
↔
(
¬
(
X43
↔
(
X35
↔
(
X33
↔
(
X27
↔
False
)
)
)
)
)
)
↔
(
¬
(
(
¬
X35
)
→
(
(
¬
X27
)
↔
X43
)
)
)
)
↔
(
¬
(
X43
→
X36
→
X35
)
)
)
↔
(
¬
(
X43
↔
(
X35
↔
X36
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
X43
→
X42
→
X36
→
X35
)
)
↔
(
X43
↔
(
(
¬
X35
)
↔
(
(
¬
X23
)
↔
X36
)
)
)
)
→
(
¬
(
(
¬
(
X36
↔
(
X26
↔
X23
)
)
)
→
(
¬
(
X36
→
(
X26
↔
(
X11
↔
X43
)
)
)
)
)
)
)
)
→
(
¬
X43
)
→
(
(
¬
X42
)
↔
(
(
¬
X35
)
→
X45
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
X47
↔
(
X45
↔
(
(
¬
X6
)
→
X42
)
)
)
→
(
(
X47
↔
(
X42
→
X4
→
False
)
)
↔
(
¬
(
X45
→
X42
→
X41
→
False
)
)
)
)
↔
(
(
(
X47
↔
(
X41
↔
(
(
¬
X25
)
→
X46
)
)
)
↔
(
X25
↔
(
(
¬
X15
)
↔
X47
)
)
)
↔
(
(
¬
(
X47
↔
(
X41
→
X34
→
X25
)
)
)
↔
(
X47
→
X42
→
X34
→
X41
)
)
)
)
→
(
(
(
(
X45
→
X20
)
↔
(
(
¬
X45
)
→
(
¬
X19
)
→
(
¬
X14
)
→
X47
)
)
→
(
¬
(
X45
↔
X19
)
)
)
↔
(
¬
(
(
¬
X37
)
→
(
¬
X19
)
→
(
X14
↔
X47
)
)
)
)
→
(
¬
(
(
(
¬
(
(
(
¬
(
X47
→
X45
→
X41
→
False
)
)
→
(
¬
(
(
(
X45
↔
(
(
¬
X41
)
↔
X47
)
)
↔
(
X47
→
(
X39
↔
X16
)
)
)
→
(
X47
↔
(
(
¬
X16
)
↔
X41
)
)
)
)
)
→
X41
→
(
X16
↔
False
)
)
)
↔
(
X47
↔
(
X41
↔
(
X39
↔
False
)
)
)
)
→
(
¬
(
X45
↔
(
(
¬
X39
)
→
(
¬
X37
)
→
X41
)
)
)
)
)
→
(
X39
↔
(
(
¬
X5
)
↔
X41
)
)
)
)
→
(
(
(
(
¬
(
X47
↔
(
X46
→
X45
→
(
X29
↔
False
)
)
)
)
→
(
(
¬
(
X49
→
(
X46
↔
(
(
¬
X24
)
↔
X47
)
)
)
)
↔
(
X49
↔
(
X45
→
X47
)
)
)
)
↔
(
¬
(
(
(
¬
(
X3
↔
X47
)
)
↔
(
¬
(
X49
→
(
¬
X44
)
→
(
(
¬
X3
)
↔
X45
)
)
)
)
→
X44
→
(
X38
↔
(
X3
→
X13
)
)
)
)
)
↔
(
(
(
¬
(
X43
→
(
X7
↔
(
(
¬
X1
)
→
X40
)
)
)
)
↔
(
¬
(
X47
→
(
(
¬
X40
)
↔
(
X7
↔
X43
)
)
)
)
)
→
(
(
(
(
X47
→
(
(
¬
X29
)
↔
(
X18
↔
X40
)
)
)
↔
(
(
¬
X24
)
↔
(
(
¬
X18
)
↔
(
(
¬
X9
)
→
X47
)
)
)
)
↔
(
X29
→
X18
→
X40
)
)
↔
(
X47
→
(
X43
↔
(
X29
→
X24
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
(
(
X47
↔
(
X37
↔
(
X31
↔
X20
)
)
)
→
(
¬
(
(
X47
→
X20
→
X15
→
(
X11
↔
False
)
)
→
(
¬
(
X32
→
X11
)
)
)
)
→
(
¬
(
(
¬
X37
)
↔
(
(
¬
X32
)
↔
(
(
¬
X5
)
→
X47
)
)
)
)
)
)
→
(
¬
(
X40
→
(
X37
↔
(
(
¬
X20
)
↔
X47
)
)
)
)
)
)
↔
(
(
(
(
(
¬
X39
)
↔
(
X37
↔
(
X21
→
X49
)
)
)
→
(
¬
(
X48
↔
(
X37
→
X45
)
)
)
)
↔
(
X49
→
X47
→
(
X37
↔
X45
)
)
)
↔
(
(
¬
(
X47
→
(
X40
↔
(
X0
→
X49
)
)
)
)
↔
(
(
¬
(
X47
→
(
X40
↔
(
(
¬
X4
)
→
X49
)
)
)
)
→
(
X47
↔
X48
)
)
)
)
)
↔
(
¬
(
(
(
X38
→
(
(
¬
X33
)
↔
X47
)
)
↔
(
X49
↔
X17
)
)
→
(
(
¬
(
(
X37
↔
(
X7
↔
(
(
¬
X2
)
→
X47
)
)
)
→
(
(
(
X37
↔
(
X7
↔
(
X2
↔
X49
)
)
)
↔
(
(
¬
X18
)
↔
(
X7
↔
(
X2
→
X49
)
)
)
)
↔
(
¬
(
(
¬
X37
)
→
(
X2
↔
X49
)
)
)
)
)
)
↔
(
(
¬
(
X47
→
X38
→
(
(
¬
X22
)
↔
X49
)
)
)
↔
(
(
(
¬
(
(
¬
(
X47
→
(
(
¬
X22
)
↔
(
X10
↔
X37
)
)
)
)
→
(
X37
↔
(
(
¬
X10
)
↔
(
(
¬
X1
)
→
X47
)
)
)
)
)
↔
(
(
¬
X22
)
→
X10
→
X47
)
)
→
(
¬
(
X49
↔
(
X47
↔
(
(
¬
X22
)
↔
X37
)
)
)
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
X43
)
→
X42
→
(
(
¬
X6
)
↔
X46
)
)
↔
(
(
¬
X45
)
→
(
¬
X42
)
→
(
X6
↔
X46
)
)
)
→
(
(
¬
(
X46
↔
(
X45
→
(
X23
↔
X25
)
)
)
)
↔
(
X46
→
X45
→
(
X23
↔
X43
)
)
)
)
↔
(
(
(
¬
(
(
¬
X43
)
↔
(
(
¬
X3
)
↔
X44
)
)
)
↔
(
X45
↔
(
X43
→
(
X16
↔
False
)
)
)
)
↔
(
(
(
¬
X44
)
→
(
(
X45
→
(
(
¬
X41
)
↔
(
(
¬
X26
)
→
X43
)
)
)
↔
(
X43
↔
X44
)
)
)
↔
(
¬
(
(
X30
↔
False
)
→
(
(
(
¬
X44
)
↔
(
(
¬
X42
)
→
X30
→
X45
)
)
↔
(
¬
(
X44
→
(
X43
↔
(
X30
→
False
)
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
(
X49
↔
(
X47
↔
(
X45
↔
False
)
)
)
→
(
¬
(
(
(
¬
(
X49
↔
(
X35
→
X47
)
)
)
→
(
X45
↔
(
(
¬
X34
)
→
X35
)
)
)
→
(
¬
(
X49
→
(
X47
↔
(
X34
↔
False
)
)
)
)
)
)
)
→
(
¬
(
(
X49
↔
(
X47
↔
(
X46
↔
(
X45
→
False
)
)
)
)
→
(
X49
↔
(
X47
↔
(
(
¬
X36
)
↔
X46
)
)
)
)
)
→
X49
→
X47
→
X36
)
)
→
(
(
¬
(
(
¬
(
X45
↔
(
X27
↔
(
X13
↔
(
X12
→
False
)
)
)
)
)
→
(
X49
↔
X12
)
)
)
↔
(
(
(
X28
→
X27
→
(
(
¬
X24
)
↔
X46
)
)
↔
(
¬
(
X45
↔
(
X27
↔
(
X24
↔
X46
)
)
)
)
)
↔
(
X28
→
X27
→
(
(
¬
X8
)
↔
X49
)
)
)
)
)
)
↔
(
(
(
¬
(
X49
→
(
(
¬
X14
)
↔
X43
)
)
)
↔
(
X49
↔
(
X45
↔
(
X43
→
(
X9
↔
False
)
)
)
)
)
→
(
(
(
X43
→
(
X14
↔
X29
)
)
↔
(
¬
(
(
¬
X45
)
↔
(
(
¬
X43
)
→
(
¬
X19
)
→
X46
)
)
)
)
↔
(
(
¬
X45
)
→
X14
→
X46
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
95b229...
and proposition id is
532ab5...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMN8nReVRWcoSt13aWVcogkiSA4EppzYxxE
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
(
¬
(
X47
→
(
X42
↔
(
X21
→
X38
)
)
)
)
↔
(
(
(
¬
X45
)
→
X2
→
X47
)
→
(
(
¬
X38
)
↔
(
X21
↔
X39
)
)
→
(
(
¬
(
(
¬
(
(
¬
X21
)
→
(
X12
↔
X38
)
)
)
→
(
¬
(
(
¬
X39
)
↔
(
(
¬
X12
)
↔
X47
)
)
)
)
)
↔
(
¬
(
X21
↔
False
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X45
→
(
X42
↔
X26
)
)
)
↔
(
X47
→
X42
→
(
¬
X0
)
→
X45
)
)
→
(
X46
↔
(
X42
→
X26
→
X1
)
)
→
X26
→
(
¬
X1
)
→
X0
→
X46
)
)
)
↔
(
¬
(
(
(
(
(
¬
X44
)
↔
(
X24
↔
X46
)
)
↔
(
¬
(
X46
→
X40
→
X5
→
False
)
)
)
→
(
(
X44
↔
(
X6
↔
X17
)
)
↔
(
(
¬
(
(
¬
X46
)
↔
(
(
¬
X17
)
↔
(
X6
→
X47
)
)
)
)
→
(
¬
(
(
¬
(
X47
→
(
¬
X24
)
→
(
X6
↔
X46
)
)
)
→
X46
→
(
(
¬
X24
)
↔
(
(
¬
X6
)
→
X44
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
(
¬
X36
)
↔
(
(
¬
X33
)
↔
(
X13
→
X41
)
)
)
→
(
¬
(
(
¬
(
X36
→
(
¬
X8
)
→
X41
)
)
→
(
(
¬
(
X41
→
X33
→
X8
)
)
↔
(
¬
(
(
¬
(
(
X36
↔
(
(
¬
X8
)
→
X46
)
)
→
(
¬
(
X36
↔
(
(
¬
X33
)
→
(
X8
↔
X41
)
)
)
)
)
)
→
X41
→
X36
→
X33
)
)
)
)
)
)
)
↔
(
(
X45
→
(
X41
↔
(
X33
→
False
)
)
)
↔
(
X46
→
(
X41
↔
(
(
¬
X11
)
↔
X45
)
)
)
)
)
→
(
(
(
¬
(
X44
↔
(
X11
↔
(
(
¬
X3
)
↔
X7
)
)
)
)
→
(
¬
(
X11
↔
(
X7
→
X33
)
)
)
)
↔
(
¬
(
(
X44
↔
(
X7
→
X47
)
)
→
(
¬
(
X11
→
(
¬
X7
)
→
X41
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
(
¬
X32
)
↔
(
(
¬
X10
)
↔
X48
)
)
)
↔
(
X48
↔
(
X20
↔
(
(
¬
X10
)
→
X15
)
)
)
)
→
(
X45
↔
(
X15
↔
X46
)
)
)
)
→
(
¬
(
(
¬
(
X46
→
(
X43
↔
(
X18
→
X32
)
)
)
)
→
(
¬
(
(
(
¬
(
X46
→
X43
→
(
X34
↔
False
)
)
)
↔
(
X46
→
(
¬
X18
)
→
X20
)
)
→
(
¬
(
X34
↔
(
(
¬
X9
)
↔
X18
)
)
)
→
(
(
X34
↔
(
X18
↔
X46
)
)
↔
(
(
X34
↔
(
(
¬
X18
)
↔
X46
)
)
→
(
¬
(
(
¬
(
X46
↔
(
X4
↔
X34
)
)
)
→
X46
→
X18
→
(
X9
↔
X4
)
)
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X45
↔
(
X37
→
X30
→
X31
)
)
)
↔
(
(
¬
(
X30
→
(
X28
↔
X37
)
)
)
↔
(
¬
(
(
X37
→
(
X30
↔
(
(
¬
X28
)
→
X43
)
)
)
→
X37
→
(
X30
↔
(
(
¬
X28
)
→
X47
)
)
)
)
)
)
→
(
(
(
¬
(
(
(
X47
→
X43
→
(
(
¬
X14
)
↔
X23
)
)
↔
(
X49
→
(
X37
↔
(
(
¬
X23
)
↔
X43
)
)
)
)
→
(
¬
(
X43
↔
(
X14
→
X47
)
)
)
)
)
↔
(
X47
→
(
(
¬
X37
)
↔
(
(
¬
X22
)
↔
X43
)
)
)
)
↔
(
¬
(
X49
↔
(
X43
↔
(
X37
→
False
)
)
)
)
)
→
(
(
X43
↔
(
(
¬
X37
)
↔
X49
)
)
↔
(
¬
(
(
X49
↔
(
X43
↔
(
X37
→
(
X27
↔
False
)
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X16
)
↔
X27
)
)
)
)
)
)
)
↔
(
¬
(
(
X45
→
(
(
¬
X43
)
↔
(
X29
→
X46
)
)
)
→
(
(
(
X46
↔
(
X45
↔
X37
)
)
→
(
(
¬
(
X46
→
(
X19
↔
False
)
)
)
↔
(
¬
(
(
¬
X45
)
→
(
¬
X43
)
→
X37
→
X46
)
)
)
)
↔
(
(
X45
↔
(
X43
↔
(
(
¬
X35
)
↔
X37
)
)
)
↔
(
¬
(
(
¬
X43
)
→
X37
→
X25
→
X45
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
X48
↔
(
(
¬
X44
)
→
X18
→
X45
)
)
)
↔
(
(
X49
↔
(
X48
↔
X5
)
)
→
(
(
¬
(
(
¬
X47
)
→
(
(
¬
X46
)
↔
(
(
¬
X45
)
→
X49
)
)
)
)
↔
(
¬
(
(
¬
X46
)
→
(
X45
↔
(
(
¬
X35
)
→
X47
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
X26
)
↔
(
X12
↔
(
(
¬
X1
)
↔
X48
)
)
)
)
↔
(
(
X45
→
X38
→
X26
)
→
(
(
¬
X45
)
↔
(
(
¬
X26
)
↔
(
X12
→
X48
)
)
)
)
)
→
(
(
(
¬
X26
)
↔
(
X21
→
X45
)
)
↔
(
(
(
(
X46
↔
(
X45
→
(
X26
↔
X3
)
)
)
↔
(
¬
(
X38
→
X3
→
X46
)
)
)
↔
(
X46
↔
(
X26
↔
(
(
¬
X25
)
→
X45
)
)
)
)
↔
(
X26
→
(
X25
↔
(
X21
→
X46
)
)
)
)
)
)
→
(
(
(
¬
(
X42
↔
(
X34
→
(
¬
X30
)
→
X41
)
)
)
↔
(
(
¬
(
(
¬
X34
)
↔
(
X23
→
(
(
¬
X20
)
↔
X42
)
)
)
)
↔
(
X41
→
X23
→
(
(
¬
X20
)
↔
X42
)
)
)
)
↔
(
(
¬
(
X49
↔
(
(
¬
X34
)
→
X43
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
X34
→
(
X30
↔
X42
)
)
→
(
X49
↔
(
X43
↔
(
X42
↔
False
)
)
)
)
)
→
(
¬
(
X49
→
(
¬
X30
)
→
X42
)
)
)
)
→
(
¬
(
(
¬
X34
)
↔
(
(
¬
X30
)
↔
X49
)
)
)
)
)
)
)
→
(
(
X45
↔
(
X43
→
X24
→
X41
)
)
↔
(
(
(
¬
X47
)
→
(
(
¬
X41
)
↔
(
X24
→
X49
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X24
)
↔
(
(
¬
X19
)
→
(
¬
X14
)
→
X45
)
)
)
→
X47
→
(
X45
↔
(
(
¬
X14
)
↔
X19
)
)
)
)
)
)
)
→
(
(
(
(
(
(
¬
X39
)
→
X11
→
X46
)
↔
(
(
(
X48
↔
X39
)
→
(
¬
(
X48
→
(
(
¬
X11
)
↔
(
X0
↔
X46
)
)
)
)
)
↔
(
(
(
X39
↔
(
X13
→
X46
)
)
→
(
¬
(
X48
→
X39
→
X32
)
)
)
→
(
X46
↔
(
X39
→
(
¬
X4
)
→
X13
)
)
)
)
)
↔
(
¬
(
X48
↔
(
(
¬
X33
)
→
(
¬
X32
)
→
X39
)
)
)
)
↔
(
¬
(
(
(
X46
→
(
X22
↔
X33
)
)
↔
(
(
¬
(
X46
↔
(
X33
↔
(
(
¬
X22
)
→
X32
)
)
)
)
↔
(
X46
→
X33
→
(
X7
↔
X22
)
)
)
)
→
(
¬
(
X37
↔
(
X7
↔
(
X6
↔
X32
)
)
)
)
→
(
¬
(
X31
↔
(
(
¬
X22
)
↔
X37
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
(
(
X49
↔
(
(
¬
X37
)
→
(
¬
X10
)
→
X48
)
)
→
(
¬
(
(
¬
X48
)
→
(
X37
↔
(
X29
↔
X49
)
)
)
)
)
)
→
(
(
X48
↔
(
X37
↔
X28
)
)
↔
(
(
X37
↔
(
X28
→
False
)
)
↔
(
X48
→
(
(
¬
X28
)
↔
(
(
¬
X15
)
→
X49
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
X37
)
→
(
(
¬
X36
)
↔
(
X28
↔
X48
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
(
¬
X27
)
↔
(
(
¬
X9
)
↔
(
X8
→
X29
)
)
)
)
↔
(
(
X29
→
(
¬
X16
)
→
(
(
¬
X9
)
↔
X27
)
)
↔
(
X37
↔
(
(
¬
X16
)
↔
(
(
¬
X9
)
↔
X27
)
)
)
)
)
↔
(
¬
(
(
¬
X27
)
↔
(
X8
→
X37
)
)
)
)
→
(
X40
↔
(
X28
→
(
X27
↔
(
X8
↔
False
)
)
)
)
)
↔
(
¬
(
(
¬
X27
)
↔
(
(
¬
X16
)
→
X29
)
)
)
)
↔
(
X40
↔
(
(
¬
X27
)
↔
(
(
¬
X17
)
→
X28
)
)
)
)
)
→
(
(
(
¬
X29
)
→
X2
→
X33
)
↔
(
X48
→
X16
→
X2
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
5ba3aa...
and proposition id is
8a5873...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMakDrbEaJGfDKLMyRQVQ74xgSGnqoq1gYN
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
¬
(
(
(
¬
(
X48
↔
(
X42
→
X45
)
)
)
↔
(
(
(
¬
(
(
¬
(
X44
→
X5
→
X38
)
)
→
X45
→
X44
→
X40
→
X38
)
)
→
(
¬
(
X48
↔
X38
)
)
)
↔
(
(
¬
(
X48
→
X44
→
X38
)
)
→
(
(
X48
↔
(
X42
↔
(
X22
↔
False
)
)
)
↔
(
(
X44
→
(
X42
↔
(
X22
↔
X48
)
)
)
↔
(
¬
(
(
¬
X22
)
↔
X38
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X45
)
→
(
(
¬
X32
)
↔
X48
)
)
)
→
(
(
¬
(
(
¬
X46
)
↔
(
(
¬
X45
)
→
(
¬
X32
)
→
X48
)
)
)
↔
(
(
¬
(
(
¬
X44
)
↔
(
(
¬
X31
)
↔
X48
)
)
)
→
(
¬
X32
)
→
(
(
¬
X31
)
↔
X46
)
)
)
)
)
)
)
↔
(
(
(
(
X48
↔
(
(
¬
X45
)
↔
(
(
¬
X17
)
→
X47
)
)
)
↔
(
X45
→
(
¬
X21
)
→
(
¬
X17
)
→
X47
)
)
→
(
¬
(
(
¬
(
X21
→
(
(
¬
X10
)
↔
X47
)
)
)
→
(
¬
(
X48
↔
(
X21
↔
X7
)
)
)
)
)
)
↔
(
¬
(
(
X35
→
(
(
¬
X21
)
↔
X47
)
)
→
(
(
¬
(
X21
↔
X1
)
)
↔
(
(
¬
(
X21
→
X1
→
X35
)
)
↔
(
¬
(
X45
→
X46
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X47
→
X38
→
X24
→
X11
→
False
)
)
↔
(
¬
(
X38
→
X24
→
(
(
¬
X11
)
↔
X49
)
)
)
)
↔
(
¬
(
(
(
X49
↔
(
X44
→
X30
→
X38
)
)
↔
(
(
¬
(
X49
↔
(
(
¬
X30
)
→
(
X0
↔
X44
)
)
)
)
→
(
¬
(
(
¬
X44
)
↔
(
X30
→
X26
→
X49
)
)
)
)
)
→
(
(
(
(
¬
(
X38
→
X26
→
(
¬
X0
)
→
X47
)
)
→
(
¬
(
X47
→
(
(
¬
X34
)
↔
(
X26
↔
X38
)
)
)
)
)
↔
(
X47
→
X38
→
False
)
)
↔
(
(
¬
X34
)
↔
(
X26
→
(
(
¬
X20
)
↔
X47
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
(
(
¬
X25
)
→
(
¬
X15
)
→
(
¬
X4
)
→
X39
)
)
↔
(
(
(
(
¬
X33
)
→
X15
→
X39
)
→
(
¬
X36
)
→
(
¬
X4
)
→
X39
)
↔
(
X36
↔
(
X15
→
X13
→
False
)
)
)
)
↔
(
¬
(
(
X39
→
(
X16
↔
(
X3
↔
X37
)
)
)
→
(
¬
(
(
¬
X13
)
↔
(
X4
→
(
¬
X3
)
→
X16
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X47
→
X37
→
(
X36
↔
X19
)
)
)
→
(
¬
(
X25
↔
(
X19
↔
X47
)
)
)
)
)
↔
(
(
¬
(
X47
→
(
X43
↔
(
X14
↔
X41
)
)
)
)
→
(
X47
↔
(
(
¬
X37
)
↔
(
X36
→
X43
)
)
)
)
)
)
→
(
(
(
¬
(
(
X47
→
X45
→
(
X33
↔
X44
)
)
→
(
¬
(
X47
→
X44
→
(
X39
↔
False
)
)
)
)
)
↔
(
(
¬
(
X33
→
(
(
¬
X12
)
↔
X47
)
)
)
→
(
(
¬
(
(
¬
X33
)
↔
(
(
¬
X27
)
→
X47
)
)
)
↔
(
(
¬
(
X33
↔
(
X27
↔
(
X9
↔
X12
)
)
)
)
↔
(
(
¬
(
X47
→
X27
)
)
↔
(
¬
(
(
¬
(
X39
↔
(
(
¬
X12
)
→
X33
)
)
)
→
(
X39
↔
(
(
¬
X27
)
→
(
¬
X12
)
→
X47
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
X47
→
(
X6
↔
X39
)
)
↔
(
X29
↔
(
X28
↔
False
)
)
)
→
(
(
(
¬
(
(
¬
X39
)
↔
(
(
¬
X23
)
↔
X45
)
)
)
↔
(
(
¬
(
X23
↔
X29
)
)
→
(
X45
↔
(
X39
→
(
¬
X23
)
→
X47
)
)
)
)
↔
(
(
¬
(
(
¬
X18
)
↔
(
(
¬
X8
)
↔
X39
)
)
)
→
(
¬
(
X29
→
(
¬
X18
)
→
X8
→
X47
)
)
)
)
)
)
↔
(
X39
→
(
X18
↔
(
X2
↔
X8
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
X43
↔
(
(
¬
X40
)
↔
(
(
¬
X4
)
↔
X46
)
)
)
↔
(
¬
(
(
X47
↔
(
X46
→
X43
→
False
)
)
→
(
(
(
X48
→
(
¬
X23
)
→
X40
)
→
(
(
X48
→
(
¬
X23
)
→
X47
)
↔
(
¬
(
X40
↔
(
(
¬
X23
)
→
X46
)
)
)
)
)
↔
(
X48
↔
(
(
¬
X43
)
→
(
¬
X23
)
→
X47
)
)
)
)
)
)
↔
(
(
¬
(
X43
→
(
X23
↔
(
X14
→
X47
)
)
)
)
↔
(
X23
↔
(
(
¬
X14
)
↔
X31
)
)
)
)
↔
(
¬
(
(
(
X28
→
(
¬
X24
)
→
X47
)
→
(
(
(
¬
(
X43
↔
(
X24
→
X5
→
X47
)
)
)
↔
(
¬
(
X24
→
(
X20
↔
X47
)
)
)
)
↔
(
(
¬
(
X48
→
(
¬
X28
)
→
(
(
¬
X20
)
↔
X47
)
)
)
→
(
X47
↔
(
X24
↔
False
)
)
)
)
)
→
(
(
¬
(
X28
→
X24
→
X12
→
(
X6
↔
False
)
)
)
↔
(
¬
(
(
¬
(
X48
↔
(
X44
→
(
X28
↔
(
X12
→
False
)
)
)
)
)
→
(
¬
X19
)
→
(
¬
(
X28
→
X48
)
)
→
(
¬
(
X19
→
(
X12
↔
(
(
¬
X6
)
↔
X28
)
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
X43
)
→
(
¬
X16
)
→
X1
→
X47
)
→
(
¬
(
(
¬
(
X43
↔
(
X42
↔
(
(
¬
X9
)
↔
X46
)
)
)
)
→
(
¬
(
X47
→
X16
→
X7
→
X42
)
)
)
)
)
)
↔
(
(
(
¬
(
X42
↔
(
X22
↔
(
(
¬
X11
)
↔
X38
)
)
)
)
→
(
¬
(
X22
↔
(
X21
↔
False
)
)
)
→
(
(
(
¬
X21
)
→
(
X10
↔
X22
)
)
↔
(
¬
(
X46
↔
(
(
¬
X21
)
↔
(
(
¬
X10
)
→
X43
)
)
)
)
)
)
→
(
(
¬
(
X35
↔
(
X25
→
X17
→
X46
)
)
)
↔
(
(
X38
↔
(
X35
↔
(
X25
→
X46
)
)
)
→
(
¬
(
(
¬
(
X46
↔
(
X35
→
X42
)
)
)
→
(
(
¬
(
X46
↔
(
(
¬
X26
)
↔
(
(
¬
X25
)
↔
X35
)
)
)
)
↔
(
X42
↔
(
X26
→
X25
)
)
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
X39
↔
(
(
¬
X34
)
↔
(
X2
↔
X45
)
)
)
→
(
¬
(
(
(
(
(
(
¬
X39
)
→
(
(
¬
X34
)
↔
X49
)
)
↔
(
¬
(
X39
→
(
(
¬
X36
)
↔
X45
)
)
)
)
↔
(
X49
↔
(
X39
↔
X36
)
)
)
→
(
¬
X36
)
)
→
(
X49
↔
(
(
¬
X36
)
→
X30
→
X39
)
)
)
)
)
)
→
(
¬
(
X47
→
X45
→
(
X34
↔
X36
)
)
)
)
↔
(
(
¬
X39
)
↔
(
(
¬
X34
)
↔
(
(
¬
X29
)
↔
X47
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X47
→
X45
→
(
(
¬
X27
)
↔
X32
)
)
)
→
(
(
¬
(
X44
→
X32
→
X49
)
)
↔
(
(
¬
(
X44
→
(
(
¬
X32
)
↔
X49
)
)
)
↔
(
(
¬
(
X47
↔
(
(
¬
X45
)
↔
(
(
¬
X44
)
↔
X49
)
)
)
)
→
(
¬
(
X44
↔
(
(
¬
X32
)
→
X45
)
)
)
)
)
)
)
→
(
¬
(
(
¬
X44
)
→
(
X42
↔
X49
)
)
)
)
)
↔
(
(
X42
↔
(
X41
↔
(
X13
→
X45
)
)
)
↔
(
(
(
¬
X44
)
→
(
X42
↔
X45
)
)
→
(
X45
→
(
X41
↔
X44
)
)
→
(
¬
(
(
¬
(
X45
↔
(
X42
→
(
(
¬
X41
)
↔
X44
)
)
)
)
→
(
¬
(
X45
→
(
X44
↔
(
X42
↔
X33
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X49
→
(
X47
↔
(
X15
→
X45
)
)
)
)
→
(
¬
(
(
X49
↔
(
X47
↔
(
X37
↔
X45
)
)
)
→
(
¬
X45
)
→
(
X37
↔
(
(
¬
X3
)
↔
X49
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X37
→
X0
→
X8
)
)
→
(
¬
(
X18
↔
(
(
¬
X15
)
↔
(
X8
→
X47
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
fd1fec...
and proposition id is
65b8f8...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMXpTYrtASm4NCEy9G5DEyDH4Vmm9urWNzQ
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
¬
(
(
¬
(
(
¬
(
(
¬
(
X38
↔
(
X32
↔
(
X30
→
X28
→
False
)
)
)
)
→
(
¬
(
(
¬
(
X32
→
(
X30
↔
(
(
¬
X23
)
↔
X28
)
)
)
)
→
X32
→
(
X28
↔
(
X9
↔
X30
)
)
)
)
→
X32
→
(
X30
↔
X28
)
)
)
→
(
X46
↔
(
X38
→
(
X32
↔
(
X30
→
False
)
)
)
)
)
)
→
(
¬
(
X45
→
X9
→
X24
)
)
→
(
(
X24
↔
(
X1
↔
X28
)
)
↔
(
¬
(
(
X46
↔
(
X45
↔
(
X28
→
X9
→
False
)
)
)
→
(
X45
↔
(
X28
→
(
¬
X1
)
→
X46
)
)
)
)
)
)
)
↔
(
(
¬
X38
)
↔
(
(
¬
X30
)
↔
(
(
¬
X4
)
↔
X45
)
)
)
)
→
(
(
¬
(
(
(
X45
→
X42
→
(
X37
↔
X44
)
)
↔
(
(
¬
X42
)
↔
(
X37
→
X10
→
X45
)
)
)
→
(
¬
X42
)
→
(
¬
X37
)
→
(
¬
X6
)
→
X45
)
)
↔
(
(
(
(
(
¬
X27
)
→
(
(
¬
X2
)
↔
X28
)
)
↔
(
(
¬
X28
)
↔
(
X14
→
(
(
¬
X2
)
↔
X37
)
)
)
)
↔
(
¬
(
X29
↔
(
X28
→
X14
→
X2
)
)
)
)
↔
(
(
¬
(
(
¬
(
X44
→
X12
→
X42
)
)
→
(
(
¬
(
X42
→
(
X29
↔
(
X15
↔
X44
)
)
)
)
↔
(
¬
(
X44
↔
(
X42
→
X27
→
False
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X35
)
↔
(
X28
↔
X44
)
)
)
→
(
X35
→
(
(
¬
X28
)
↔
(
(
¬
X15
)
→
X32
)
)
)
→
X44
→
(
X35
↔
(
X32
↔
(
X15
↔
False
)
)
)
)
→
X32
→
(
¬
X29
)
→
X15
→
X35
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
(
¬
(
(
(
X46
→
(
X18
↔
(
X7
↔
False
)
)
)
↔
(
¬
(
(
¬
X41
)
↔
(
(
¬
X25
)
↔
(
X7
↔
X46
)
)
)
)
)
→
(
¬
(
X41
↔
(
X7
→
X25
)
)
)
)
)
↔
(
(
(
(
¬
X41
)
↔
(
X8
↔
X46
)
)
↔
(
¬
(
(
¬
X33
)
→
(
(
X46
→
(
X33
↔
(
X25
→
X8
)
)
)
↔
(
¬
(
X46
→
(
X41
↔
(
X25
→
False
)
)
)
)
)
)
)
)
→
(
X25
↔
X46
)
)
)
→
X26
→
X8
→
(
(
¬
X7
)
↔
X46
)
)
)
→
(
¬
X41
)
→
(
(
¬
X26
)
↔
(
(
¬
X5
)
↔
X46
)
)
)
→
(
(
¬
(
(
¬
(
X41
↔
(
X33
→
(
X25
↔
False
)
)
)
)
→
(
X41
↔
(
X39
→
(
X20
↔
X25
)
)
)
)
)
↔
(
(
(
¬
(
X43
→
(
¬
X13
)
→
X49
)
)
→
(
¬
(
(
X11
↔
X49
)
→
(
(
¬
(
X49
→
(
(
¬
X11
)
↔
X43
)
)
)
↔
(
X33
↔
(
X13
↔
X43
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X25
)
↔
(
X13
→
X43
)
)
)
→
(
¬
(
(
(
¬
X33
)
→
(
¬
X25
)
→
(
¬
X13
)
→
X43
)
→
(
¬
(
X43
↔
(
(
¬
X25
)
→
(
¬
X20
)
→
X33
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
X46
↔
(
X45
↔
(
X34
→
X36
)
)
)
↔
(
¬
(
X45
→
(
¬
X44
)
→
X36
→
X46
)
)
)
→
(
¬
(
(
X45
↔
(
(
¬
X41
)
→
X21
→
X43
)
)
→
(
(
(
(
(
X43
↔
(
(
¬
X19
)
↔
X45
)
)
↔
(
¬
(
X45
→
(
X44
↔
X17
)
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X44
)
↔
(
(
¬
X43
)
→
X45
)
)
)
)
)
↔
(
¬
(
X45
→
X19
→
X46
)
)
)
↔
(
X44
↔
(
X43
↔
(
(
¬
X41
)
↔
X46
)
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
X49
→
X47
→
X46
→
X41
)
)
↔
(
X49
→
(
¬
X41
)
→
X3
→
X47
)
)
↔
(
(
(
(
(
X49
↔
(
X47
↔
(
X0
→
X40
)
)
)
↔
(
¬
(
X47
↔
(
X41
→
X40
)
)
)
)
↔
(
¬
(
X49
→
(
X47
↔
X40
)
)
)
)
→
(
(
X48
↔
(
X47
→
X16
→
X0
→
False
)
)
↔
(
¬
(
X48
→
X47
→
X16
→
X40
)
)
)
)
↔
(
(
(
(
(
(
¬
(
X47
↔
(
X31
↔
(
(
¬
X22
)
↔
X49
)
)
)
)
↔
(
¬
(
(
¬
X48
)
↔
(
X31
↔
(
X22
→
X49
)
)
)
)
)
↔
(
¬
(
X49
↔
(
X48
↔
X22
)
)
)
)
↔
(
X47
↔
X31
)
)
→
(
¬
(
(
¬
X48
)
→
(
¬
X47
)
→
X49
)
)
)
↔
(
X49
↔
(
X31
↔
(
X22
↔
False
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
X46
)
↔
(
X45
↔
(
(
¬
X30
)
→
X49
)
)
)
→
(
(
¬
(
X45
→
X30
→
X17
)
)
↔
(
(
(
¬
(
(
¬
X47
)
→
X30
→
X49
)
)
→
(
(
X47
→
(
X25
↔
X9
)
)
↔
(
¬
(
X47
↔
(
(
¬
X30
)
→
X49
)
)
)
)
)
↔
(
(
¬
(
(
(
(
X17
→
(
X9
↔
(
X1
→
False
)
)
)
↔
(
¬
(
X25
→
X17
→
(
(
¬
X9
)
↔
X49
)
)
)
)
↔
(
X25
↔
(
(
¬
X9
)
↔
X17
)
)
)
→
X49
→
(
X1
↔
X9
)
)
)
→
(
¬
(
(
¬
X25
)
→
(
¬
X17
)
→
X49
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
(
¬
X40
)
↔
(
(
¬
X33
)
→
(
(
¬
X0
)
↔
X46
)
)
)
)
↔
(
X40
→
(
(
¬
X11
)
↔
(
X0
↔
X26
)
)
)
)
→
(
¬
(
(
¬
(
(
(
X46
→
X44
→
(
X40
↔
X3
)
)
→
(
(
(
¬
(
X44
→
(
¬
X3
)
→
X47
)
)
→
(
¬
(
X40
↔
(
X32
→
X44
)
)
)
)
↔
(
(
¬
X44
)
↔
(
X3
→
X47
)
)
)
→
(
¬
(
X32
→
X47
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X44
)
→
(
¬
X26
)
→
X46
)
)
)
)
)
→
(
¬
(
X44
↔
(
(
¬
X40
)
↔
X47
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
X46
)
→
(
X45
↔
X47
)
)
↔
(
(
X15
→
X47
)
↔
(
¬
(
X46
→
X45
→
X40
→
X47
)
)
)
)
→
(
¬
(
(
¬
X45
)
→
X40
→
(
(
¬
X10
)
↔
X47
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
X45
)
→
(
(
¬
X42
)
↔
X46
)
)
↔
(
¬
(
(
(
X47
→
(
(
¬
X38
)
↔
(
X18
→
X39
)
)
)
↔
(
¬
(
(
¬
X39
)
↔
X42
)
)
)
→
(
(
¬
(
(
¬
X45
)
→
(
¬
X38
)
→
X47
)
)
↔
(
(
¬
(
(
¬
X45
)
↔
(
(
¬
X38
)
↔
(
X18
→
X47
)
)
)
)
↔
(
X46
↔
(
(
¬
X45
)
→
X47
)
)
)
)
)
)
)
→
(
¬
(
(
X47
→
(
X23
↔
X45
)
)
→
(
¬
X46
)
→
(
(
¬
X45
)
↔
(
X35
→
X47
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
¬
X47
)
↔
(
(
¬
X34
)
↔
X48
)
)
)
↔
(
¬
(
X48
↔
(
X47
↔
(
X14
↔
X45
)
)
)
)
)
↔
(
(
(
¬
X47
)
↔
(
(
¬
X28
)
→
X14
→
X48
)
)
↔
(
(
X48
→
X14
→
(
¬
X5
)
→
X47
)
→
(
¬
(
X5
→
X20
)
)
)
)
)
→
(
(
(
(
¬
(
X16
↔
X48
)
)
→
(
¬
(
X41
↔
(
(
¬
X12
)
↔
(
(
¬
X4
)
→
X16
)
)
)
)
)
↔
(
(
(
¬
X36
)
→
(
(
¬
X34
)
↔
X47
)
)
↔
(
(
¬
(
(
¬
X34
)
→
(
X16
↔
X36
)
)
)
→
(
(
¬
(
(
¬
X36
)
→
X16
→
X12
→
X47
)
)
↔
(
¬
(
(
¬
X41
)
↔
(
X36
↔
(
(
¬
X16
)
↔
X47
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X41
↔
(
X19
↔
(
X7
↔
X31
)
)
)
)
↔
(
(
¬
X31
)
↔
(
(
¬
X21
)
↔
(
(
¬
X7
)
↔
X48
)
)
)
)
→
(
¬
(
X48
→
(
¬
X19
)
→
(
X7
↔
X36
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
X49
→
(
(
¬
X41
)
↔
(
(
¬
X22
)
↔
X48
)
)
)
→
(
(
¬
(
(
X49
↔
(
(
¬
X29
)
→
X24
→
X48
)
)
→
(
¬
(
X49
↔
(
X48
↔
(
X24
→
X22
)
)
)
)
)
)
↔
(
¬
(
X29
↔
(
(
¬
X24
)
→
X48
)
)
)
)
)
→
(
(
X48
↔
(
X6
→
(
(
¬
X2
)
↔
X41
)
)
)
↔
(
(
¬
(
(
¬
X13
)
→
(
(
¬
X6
)
↔
(
(
¬
X2
)
↔
X49
)
)
)
)
→
(
¬
(
X48
↔
(
(
¬
X6
)
→
(
X2
↔
X49
)
)
)
)
)
)
→
(
¬
(
X48
→
(
¬
X6
)
→
X13
)
)
)
)
→
(
¬
(
(
X37
↔
(
X36
→
(
X13
↔
X27
)
)
)
→
X43
→
X37
→
(
X27
↔
X48
)
)
)
→
(
X49
↔
X13
)
→
(
¬
(
X43
↔
(
X27
↔
X8
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
94ae49...
and proposition id is
890547...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMX28i3cgVpFnUExujCLSbzbNMRncLTLwoG
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
¬
(
(
(
¬
(
(
(
¬
(
(
X49
↔
(
X42
→
X43
)
)
→
X46
→
(
¬
X42
)
→
(
¬
X41
)
→
X43
)
)
→
(
(
(
(
¬
X41
)
↔
(
X13
→
X43
)
)
↔
(
¬
(
X41
→
(
X32
↔
X42
)
)
)
)
↔
(
¬
(
X43
→
(
X41
↔
(
X32
→
X47
)
)
)
)
)
)
→
(
(
(
X47
↔
(
X36
↔
False
)
)
↔
(
X47
↔
(
X46
↔
(
(
¬
X14
)
→
X49
)
)
)
)
↔
(
¬
(
(
(
X47
↔
(
X46
→
(
(
¬
X21
)
↔
X24
)
)
)
↔
(
¬
(
X47
↔
X36
)
)
)
→
(
¬
(
X47
→
(
X46
↔
X24
)
)
)
)
)
)
→
(
(
(
(
¬
X46
)
↔
(
X29
↔
X47
)
)
↔
(
(
(
¬
X47
)
→
(
(
¬
X46
)
↔
(
X29
↔
X49
)
)
)
↔
(
X49
→
(
X47
↔
(
(
¬
X18
)
→
X46
)
)
)
)
)
↔
(
(
X49
→
(
X29
↔
False
)
)
↔
(
¬
(
X47
→
(
(
¬
X34
)
↔
(
X1
→
X49
)
)
)
)
)
)
)
)
↔
(
(
(
(
X42
↔
(
X8
↔
(
X0
↔
X27
)
)
)
↔
(
¬
(
(
¬
X42
)
→
(
X10
↔
(
(
¬
X0
)
↔
X44
)
)
)
)
)
↔
(
(
X44
→
X10
→
X4
→
X9
)
↔
(
(
(
¬
X9
)
↔
(
(
¬
X4
)
↔
X38
)
)
↔
(
¬
(
X27
→
(
X10
↔
(
X7
↔
X9
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X44
)
↔
(
(
¬
X43
)
→
X49
)
)
)
↔
(
(
¬
(
X44
→
X28
→
(
X16
↔
X38
)
)
)
→
(
(
(
(
¬
(
X44
↔
(
(
¬
X16
)
→
X43
)
)
)
↔
(
¬
(
X44
→
(
X28
↔
X49
)
)
)
)
→
(
X43
↔
(
X28
→
(
X16
↔
False
)
)
)
)
↔
(
¬
(
(
¬
X43
)
→
X28
→
X44
)
)
)
)
)
→
(
¬
(
(
X43
↔
(
X42
↔
(
X38
→
X30
→
False
)
)
)
→
(
(
¬
X38
)
↔
(
X16
↔
X42
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
(
X40
↔
(
X22
↔
(
X11
→
X48
)
)
)
↔
(
¬
(
X40
→
(
(
¬
X31
)
↔
(
X11
↔
X48
)
)
)
)
)
→
(
(
¬
(
(
¬
X31
)
→
(
(
¬
X15
)
↔
(
X6
→
X40
)
)
)
)
→
X48
→
X15
→
X40
)
→
(
(
¬
X40
)
↔
(
(
¬
X31
)
→
X25
→
X48
)
)
)
)
↔
(
(
¬
(
X48
↔
(
X35
→
(
X2
↔
X25
)
)
)
)
↔
(
X25
→
(
(
¬
X6
)
↔
X35
)
)
)
)
↔
(
(
(
(
(
(
¬
X47
)
→
(
X45
↔
(
X44
→
X48
)
)
)
↔
(
¬
(
(
¬
X47
)
→
(
X44
↔
(
X33
↔
X48
)
)
)
)
)
↔
(
(
¬
X33
)
→
X45
)
)
→
(
(
X40
→
(
¬
X23
)
→
(
(
¬
X12
)
↔
X48
)
)
↔
(
(
¬
(
X40
→
X37
→
X12
)
)
→
(
¬
(
(
X47
→
(
X44
↔
X40
)
)
→
(
¬
(
(
(
¬
(
X47
→
(
(
¬
X17
)
↔
(
X12
→
X23
)
)
)
)
→
(
¬
(
X47
↔
(
X44
↔
(
X17
↔
X23
)
)
)
)
)
→
(
¬
(
(
¬
X40
)
→
(
X17
↔
(
(
¬
X12
)
→
X47
)
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
X39
→
(
X20
↔
(
(
¬
X3
)
→
X35
)
)
)
↔
(
False
↔
(
(
¬
(
X47
↔
(
(
¬
X35
)
↔
(
X20
↔
X45
)
)
)
)
→
(
X48
↔
(
X20
↔
X45
)
)
)
)
)
↔
(
¬
(
X41
→
(
X39
↔
(
(
¬
X35
)
↔
X48
)
)
)
)
)
→
(
(
(
X47
→
(
X45
↔
(
X26
→
False
)
)
)
↔
(
¬
(
X45
→
(
¬
X26
)
→
X19
→
X48
)
)
)
↔
(
(
¬
X47
)
↔
(
X19
→
(
(
¬
X5
)
↔
X48
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
(
X48
→
(
X45
↔
(
X28
↔
X41
)
)
)
)
→
(
¬
(
(
(
X49
→
X48
→
X47
→
X43
→
False
)
↔
(
X48
→
(
X47
↔
(
X45
→
X49
)
)
)
)
→
(
X47
↔
(
X45
→
(
X41
↔
X43
)
)
)
)
)
→
(
(
(
¬
X43
)
→
(
X33
↔
(
(
¬
X22
)
→
X47
)
)
)
↔
(
¬
(
X47
↔
(
X41
→
(
(
¬
X22
)
↔
X43
)
)
)
)
)
)
)
↔
(
¬
(
(
(
X38
↔
(
(
¬
X16
)
↔
X48
)
)
→
(
X36
↔
(
X29
↔
False
)
)
)
→
(
¬
(
(
(
X14
↔
(
X9
↔
X44
)
)
↔
(
¬
(
(
¬
(
X47
↔
(
(
¬
X9
)
→
X44
)
)
)
→
(
¬
(
(
¬
X14
)
→
(
¬
X8
)
→
X44
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
(
¬
(
X36
↔
(
(
¬
X29
)
↔
X47
)
)
)
↔
(
¬
(
X38
↔
(
X29
→
(
(
¬
X17
)
↔
X36
)
)
)
)
)
→
(
X29
↔
X47
)
)
)
↔
(
¬
X47
)
)
→
(
(
(
¬
(
X47
↔
(
X5
→
(
¬
X3
)
→
X38
)
)
)
↔
(
¬
(
X38
→
X3
→
X36
)
)
)
↔
(
¬
X3
)
)
→
(
¬
X5
)
→
X38
)
)
→
(
(
¬
(
(
(
(
¬
X36
)
↔
X49
)
↔
(
(
¬
(
X49
↔
(
X44
↔
(
X36
→
False
)
)
)
)
↔
(
(
(
(
¬
X47
)
→
(
(
¬
X36
)
↔
X49
)
)
↔
(
(
(
X36
→
X47
)
↔
(
¬
(
X44
→
X36
→
(
¬
X14
)
→
X49
)
)
)
↔
(
X44
↔
(
X36
→
X47
)
)
)
)
↔
(
¬
(
X49
↔
(
X47
↔
(
X44
→
False
)
)
)
)
)
)
)
→
(
¬
X44
)
)
)
↔
(
¬
(
X38
→
(
X36
↔
(
(
¬
X14
)
↔
X49
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
(
¬
X34
)
↔
(
X30
↔
X46
)
)
)
→
(
¬
(
X36
→
(
¬
X1
)
→
X34
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X34
)
→
(
¬
X20
)
→
(
(
¬
X1
)
↔
X38
)
)
)
→
(
X34
↔
(
X20
→
(
¬
X1
)
→
X36
)
)
)
)
→
(
¬
(
(
¬
X20
)
→
X34
)
)
)
→
(
¬
(
(
¬
(
X49
↔
(
X46
→
(
¬
X2
)
→
X20
)
)
)
→
(
(
X46
↔
(
(
¬
X36
)
→
(
¬
X12
)
→
X49
)
)
↔
(
X46
↔
(
X36
→
(
X12
↔
X20
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
¬
(
(
(
(
X39
→
(
X21
↔
(
(
¬
X18
)
→
X23
)
)
)
↔
(
X25
↔
X23
)
)
↔
(
¬
(
(
¬
X25
)
↔
(
X23
→
X39
)
)
)
)
→
X39
→
(
X25
↔
X18
)
)
)
↔
(
¬
(
(
X46
↔
(
X25
→
(
¬
X21
)
→
X39
)
)
→
(
¬
(
X39
↔
(
(
¬
X25
)
↔
(
X0
→
X46
)
)
)
)
)
)
)
→
X37
→
(
¬
X23
)
→
X19
→
X46
)
)
→
(
(
(
X48
↔
(
X10
↔
(
X4
→
X39
)
)
)
↔
(
X42
→
(
(
¬
X18
)
↔
(
(
¬
X10
)
→
X48
)
)
)
)
→
(
(
X48
→
X42
→
X21
→
X39
)
↔
(
¬
(
X42
→
(
¬
X21
)
→
X11
→
X39
)
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X10
)
↔
X42
)
)
)
)
→
(
¬
(
(
(
(
(
(
(
¬
X31
)
↔
(
(
¬
X27
)
↔
(
(
¬
X6
)
↔
X39
)
)
)
→
(
¬
(
X39
→
X37
→
(
X31
↔
(
X24
↔
False
)
)
)
)
)
→
X39
→
(
X37
↔
X31
)
)
↔
(
¬
(
X39
→
(
X31
↔
(
X6
↔
X40
)
)
)
)
)
↔
(
(
¬
(
X49
→
(
X39
↔
(
(
¬
X13
)
↔
X15
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
X24
→
(
X7
↔
X40
)
)
→
(
¬
(
(
¬
X13
)
↔
(
X7
↔
X15
)
)
)
)
)
→
(
¬
(
X40
↔
(
(
¬
X15
)
↔
X24
)
)
)
)
)
→
(
¬
(
X24
↔
(
X13
↔
X49
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X49
↔
(
X40
↔
(
X32
↔
X39
)
)
)
)
→
(
(
X35
↔
(
(
¬
X32
)
→
X40
)
)
↔
(
(
X39
↔
(
(
¬
X35
)
→
(
(
¬
X32
)
↔
X40
)
)
)
↔
(
X40
→
X32
→
X26
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
0e200f...
and proposition id is
f47011...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMUvLR61CXmUhxzWEoemdAbPMB2vuHCkZh8
)
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
(
(
(
¬
(
X48
↔
(
X47
↔
X49
)
)
)
↔
(
(
¬
(
(
(
X48
↔
(
X45
→
X47
)
)
↔
(
(
¬
X44
)
→
X49
)
)
→
(
¬
(
X44
→
(
X22
↔
False
)
)
)
)
)
↔
(
(
¬
X48
)
↔
(
X47
→
(
¬
X38
)
→
X49
)
)
)
)
→
(
(
¬
X48
)
↔
(
X44
→
X49
)
)
)
↔
(
(
(
(
X49
→
X22
→
(
X1
↔
False
)
)
↔
(
¬
(
(
X24
↔
(
(
¬
X22
)
→
(
X1
↔
X38
)
)
)
→
(
X49
↔
(
X24
→
X1
→
X22
)
)
)
)
)
↔
(
X38
→
(
¬
X22
)
→
X49
)
)
↔
(
(
X38
↔
(
X14
→
X24
)
)
↔
(
¬
(
X38
↔
(
X22
→
X14
→
X17
)
)
)
)
)
)
↔
(
(
X44
↔
(
X35
↔
(
X29
↔
False
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X47
→
(
X18
↔
X21
)
)
)
→
(
X44
↔
(
X21
→
X18
)
)
→
(
¬
X47
)
→
(
(
¬
X44
)
↔
(
(
¬
X18
)
↔
X49
)
)
)
)
→
(
¬
(
(
¬
(
X35
→
(
X29
↔
(
(
¬
X21
)
↔
X42
)
)
)
)
→
(
X44
↔
(
X29
→
(
¬
X18
)
→
X42
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
(
(
(
X46
→
(
¬
X5
)
→
X2
→
X47
)
→
(
¬
(
X5
↔
X26
)
)
)
→
X49
→
(
(
¬
X5
)
↔
(
(
¬
X2
)
→
X46
)
)
)
)
→
(
(
X26
→
(
(
¬
X2
)
↔
X13
)
)
↔
(
¬
(
(
¬
X13
)
→
(
X10
↔
X46
)
)
)
)
)
)
↔
(
¬
(
X49
↔
(
X47
↔
(
X46
↔
(
X30
→
False
)
)
)
)
)
)
→
(
(
(
(
¬
X30
)
→
(
(
¬
X26
)
↔
X48
)
)
→
(
¬
X31
)
→
X30
→
X12
→
X48
)
↔
(
(
(
X6
→
(
X4
↔
False
)
)
↔
(
(
¬
(
(
¬
X32
)
↔
(
X12
↔
X48
)
)
)
↔
(
(
(
X32
↔
(
(
¬
X4
)
↔
X6
)
)
↔
(
(
¬
(
X48
↔
(
X6
→
(
¬
X4
)
→
X32
)
)
)
↔
(
X32
↔
(
X6
↔
X47
)
)
)
)
↔
(
(
¬
X6
)
↔
X32
)
)
)
)
→
X48
→
(
X32
↔
(
X26
→
X12
→
False
)
)
)
)
)
)
→
(
¬
(
(
(
X47
→
(
X32
↔
(
(
¬
X15
)
→
X46
)
)
)
↔
(
(
(
¬
X46
)
↔
(
X32
↔
(
(
¬
X23
)
↔
X47
)
)
)
↔
(
(
¬
X32
)
↔
(
(
¬
X23
)
→
X7
→
X47
)
)
)
)
→
(
¬
(
(
(
¬
X46
)
→
(
(
¬
X0
)
↔
X47
)
)
→
(
¬
(
(
(
(
¬
(
X46
→
X32
→
X25
→
X16
)
)
↔
(
(
¬
X46
)
↔
(
X25
→
X47
)
)
)
→
(
¬
(
X47
→
(
X32
↔
X16
)
)
)
)
→
(
X25
↔
False
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X43
↔
(
(
¬
X36
)
→
X40
)
)
)
→
(
(
(
X39
↔
(
X36
→
(
(
¬
X8
)
↔
X43
)
)
)
↔
(
(
¬
X36
)
↔
(
(
¬
X8
)
→
X40
)
)
)
↔
(
¬
(
(
¬
X37
)
↔
(
X36
→
(
X8
↔
X43
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X48
→
(
¬
X37
)
→
X34
→
X40
)
)
↔
(
X37
→
X11
)
)
↔
(
X40
↔
(
X39
→
(
X37
↔
X34
)
)
)
)
↔
(
X46
↔
(
X40
↔
(
X3
→
False
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X46
↔
(
X33
↔
X36
)
)
)
↔
(
X41
→
(
(
¬
X33
)
↔
X47
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
¬
X41
)
→
(
¬
X20
)
→
(
¬
X19
)
→
X47
)
↔
(
(
¬
X41
)
↔
(
(
¬
X19
)
→
X46
)
)
)
→
X41
→
X20
→
False
)
)
→
(
¬
(
(
(
¬
(
(
¬
X46
)
→
(
(
¬
X41
)
↔
(
X9
↔
X47
)
)
)
)
↔
(
¬
(
X47
↔
(
(
¬
X9
)
↔
X37
)
)
)
)
→
(
¬
(
X47
→
(
X41
↔
X37
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
X46
)
→
(
X27
↔
X47
)
)
→
(
¬
(
(
¬
X41
)
→
(
¬
X28
)
→
(
X27
↔
X47
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
(
(
(
¬
X22
)
↔
(
X6
↔
X32
)
)
↔
(
(
X44
→
X22
→
(
X6
↔
X32
)
)
↔
(
X22
↔
(
(
¬
X14
)
↔
(
X6
→
X32
)
)
)
)
)
↔
(
X22
→
(
(
¬
X6
)
↔
X14
)
)
)
→
X47
→
X45
→
(
X14
↔
(
X6
→
False
)
)
)
)
↔
(
(
¬
(
(
¬
(
X44
→
(
X24
↔
(
X12
→
X21
)
)
)
)
→
X44
→
(
¬
X24
)
→
(
¬
X12
)
→
X45
)
)
↔
(
(
¬
(
X44
→
X32
→
(
(
¬
X20
)
↔
X42
)
)
)
↔
(
(
X44
↔
(
X32
→
(
¬
X24
)
→
X42
)
)
↔
(
(
X45
↔
(
(
¬
(
X48
→
(
X45
↔
(
X15
→
False
)
)
)
)
→
(
(
¬
(
X48
→
X32
→
X15
→
X44
)
)
→
(
¬
(
X45
↔
(
X44
↔
X32
)
)
)
)
→
X32
→
(
X15
↔
X44
)
)
)
↔
(
X45
↔
(
(
¬
X42
)
→
X44
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X47
→
(
X31
↔
(
X18
→
X48
)
)
)
)
→
X31
→
(
¬
X13
)
→
X36
)
↔
(
(
¬
(
(
¬
X35
)
→
(
X18
↔
X49
)
)
)
→
(
(
(
¬
X48
)
↔
(
(
¬
X9
)
→
X49
)
)
↔
(
¬
(
(
X49
↔
(
X35
→
X9
→
False
)
)
→
(
¬
(
(
¬
X35
)
→
(
X18
↔
(
(
¬
X13
)
↔
X49
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
X47
)
↔
(
(
¬
X35
)
↔
(
X2
→
X49
)
)
)
)
→
(
¬
(
(
X47
→
X35
→
X0
)
→
(
¬
(
X49
→
X34
→
(
X0
↔
X35
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
(
(
¬
X40
)
→
(
(
¬
X27
)
↔
X48
)
)
)
→
(
¬
(
(
¬
(
(
¬
X42
)
→
(
(
¬
X40
)
↔
(
(
¬
X3
)
→
X48
)
)
)
)
→
(
¬
(
X48
→
(
¬
X27
)
→
(
¬
X3
)
→
X40
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X48
→
X42
→
(
(
¬
X40
)
↔
X49
)
)
)
→
(
X49
↔
(
X42
→
X40
→
(
X38
↔
False
)
)
)
)
)
)
→
(
¬
(
(
¬
X38
)
→
(
X23
↔
X48
)
)
)
→
(
X40
↔
(
(
¬
X38
)
↔
(
X25
→
X48
)
)
)
)
↔
(
(
¬
X47
)
↔
(
X42
→
(
X40
↔
X48
)
)
)
)
)
)
↔
(
(
(
(
(
¬
X39
)
→
(
X11
↔
(
(
¬
X7
)
→
X49
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
X41
→
(
¬
X7
)
→
X45
)
)
→
(
X49
↔
(
X43
→
X41
→
X7
→
False
)
)
)
)
→
X45
→
(
X43
↔
X41
)
)
→
(
X49
↔
(
(
¬
X43
)
↔
X45
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
X45
↔
(
X33
→
(
(
¬
X10
)
↔
X30
)
)
)
↔
(
X45
→
X33
→
(
X10
↔
X41
)
)
)
↔
(
(
¬
X30
)
→
(
¬
X10
)
→
(
X5
↔
X33
)
)
)
↔
(
(
¬
(
(
X17
→
X45
)
→
(
X45
↔
(
(
¬
X33
)
→
(
(
¬
X26
)
↔
X41
)
)
)
)
)
↔
(
¬
(
X41
↔
(
X26
→
X33
)
)
)
)
)
→
X43
→
X30
→
X33
)
)
↔
(
(
¬
(
X41
→
X29
→
X45
)
)
→
(
¬
(
X45
→
(
X29
↔
(
X8
↔
X43
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X48
)
↔
(
(
¬
X46
)
↔
(
X43
↔
X49
)
)
)
)
↔
(
X41
↔
X46
)
)
↔
(
(
(
X19
→
(
(
¬
(
X49
↔
(
X28
↔
(
(
¬
X19
)
→
X48
)
)
)
)
↔
(
X37
→
X28
→
X19
)
)
)
↔
(
X49
↔
(
X46
↔
(
X37
→
(
X19
↔
False
)
)
)
)
)
↔
(
(
(
(
¬
(
X49
↔
(
X16
↔
(
X4
↔
X48
)
)
)
)
↔
(
¬
(
X48
→
(
X16
↔
(
X4
→
X1
)
)
)
)
)
→
(
¬
(
X49
→
(
(
¬
X37
)
↔
(
X16
→
X48
)
)
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X37
)
→
X48
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
561c55...
and proposition id is
613067...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMbjbPnHoo7MXhaRP78YeaG42xWS6YhFYgV
)
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
(
(
(
X49
→
X38
→
X4
→
X46
)
↔
(
(
X46
→
(
¬
X41
)
→
X49
)
→
(
X49
↔
(
X41
↔
(
X38
→
X29
)
)
)
)
)
→
X49
→
(
X38
↔
(
X13
→
X4
→
False
)
)
)
↔
(
(
¬
X48
)
→
(
(
¬
X41
)
↔
(
X38
→
X49
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
(
(
¬
X20
)
→
X18
→
(
X9
↔
X46
)
)
)
→
X49
→
X46
→
(
X9
↔
False
)
)
)
→
(
¬
(
(
¬
X46
)
↔
(
(
¬
X27
)
→
(
X18
↔
X49
)
)
)
)
)
)
↔
(
¬
(
(
(
X46
↔
(
X24
→
X41
)
)
→
(
X46
↔
(
X20
↔
(
X9
→
False
)
)
)
)
→
(
¬
(
X41
↔
(
(
¬
X24
)
→
(
X23
↔
X46
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
X49
→
X36
→
(
¬
X15
)
→
X33
)
→
X37
→
(
(
¬
X36
)
↔
(
(
¬
X15
)
↔
X43
)
)
)
)
→
(
¬
(
X49
↔
(
X36
→
X43
)
)
)
)
→
(
¬
(
(
¬
(
X43
↔
(
(
¬
X33
)
↔
X48
)
)
)
→
(
X48
↔
(
X37
→
X26
→
False
)
)
)
)
)
)
↔
(
¬
(
(
(
X49
→
X40
→
(
¬
X30
)
→
X34
)
→
(
¬
(
(
(
X34
↔
(
(
¬
X8
)
→
X33
)
)
→
(
¬
(
X40
↔
(
X31
→
(
X8
↔
False
)
)
)
)
)
→
(
¬
(
(
X33
→
X40
)
→
(
(
¬
(
X47
→
X40
→
X34
→
X33
)
)
↔
(
¬
(
(
¬
X1
)
↔
X47
)
)
)
)
)
→
(
(
¬
(
(
X49
↔
(
X47
↔
X34
)
)
→
(
¬
(
(
¬
X34
)
→
X33
→
X17
→
X49
)
)
)
)
↔
(
¬
(
X49
↔
(
X33
↔
False
)
)
)
)
)
)
)
→
(
(
X47
↔
(
X30
↔
X46
)
)
↔
(
(
(
¬
X47
)
→
(
(
¬
X46
)
↔
(
(
¬
X30
)
↔
X49
)
)
)
→
(
¬
(
X49
→
X47
→
(
X36
↔
X46
)
)
)
)
)
)
)
)
)
→
(
(
(
(
X44
→
(
(
¬
X14
)
↔
(
X2
→
X28
)
)
)
↔
(
X48
↔
(
X14
→
(
¬
X12
)
→
X28
)
)
)
↔
(
(
¬
(
X48
↔
(
X22
↔
(
(
¬
X14
)
↔
X19
)
)
)
)
→
(
(
¬
(
(
(
(
¬
X28
)
→
(
¬
X0
)
→
X32
)
↔
(
X45
↔
(
X28
↔
(
X11
→
X0
)
)
)
)
→
(
¬
X28
)
→
(
(
¬
X22
)
↔
(
(
¬
X11
)
↔
X45
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X32
)
→
(
X28
↔
(
(
¬
X14
)
→
X45
)
)
)
)
→
(
¬
(
X19
↔
(
(
¬
X14
)
↔
X32
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
X48
↔
(
X47
↔
(
(
¬
X39
)
→
X45
)
)
)
↔
(
(
(
¬
(
X48
↔
(
(
¬
X42
)
↔
(
X10
↔
X47
)
)
)
)
→
(
X47
↔
(
X45
↔
(
X10
→
X42
)
)
)
)
↔
(
X48
↔
(
X42
→
(
¬
X25
)
→
X45
)
)
)
)
→
(
(
(
¬
(
(
X48
↔
(
X3
→
X47
)
)
→
(
¬
X48
)
→
(
¬
X47
)
→
(
X45
↔
X49
)
)
)
↔
(
¬
False
)
)
↔
(
X47
↔
(
(
¬
X3
)
→
X48
)
)
)
)
)
→
(
(
¬
(
(
X44
↔
(
(
¬
X21
)
→
X42
)
)
→
(
¬
(
X48
→
(
X21
↔
(
(
¬
X6
)
→
X35
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
X7
)
→
X16
)
↔
(
(
X21
→
(
¬
X16
)
→
X35
)
↔
(
X35
↔
(
(
¬
X21
)
→
(
X7
↔
X44
)
)
)
)
)
→
(
¬
(
(
¬
X35
)
↔
(
(
¬
X21
)
↔
(
X5
→
X44
)
)
)
)
)
)
)
→
X48
→
(
(
¬
X44
)
↔
(
(
¬
X42
)
↔
X47
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
(
(
¬
(
(
¬
(
X44
↔
(
X33
↔
(
X32
↔
X22
)
)
)
)
→
(
(
¬
X32
)
↔
(
(
¬
X22
)
↔
(
X5
→
X33
)
)
)
)
)
↔
(
X44
↔
(
X22
→
X32
)
)
)
↔
(
¬
(
X32
→
X14
→
(
X5
↔
X44
)
)
)
)
↔
(
(
¬
X22
)
↔
(
X7
→
(
X5
↔
X44
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X32
)
↔
(
X14
→
X5
→
X42
)
)
)
↔
(
¬
(
X42
↔
(
X14
→
(
X4
↔
False
)
)
)
)
)
→
(
¬
(
(
X42
→
(
X14
↔
(
X0
↔
X32
)
)
)
→
(
X48
↔
(
X32
↔
(
(
¬
X14
)
↔
X42
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X46
→
(
X25
↔
X48
)
)
)
→
(
¬
(
X48
→
X46
→
X42
→
X25
)
)
→
X48
→
(
X44
↔
(
(
¬
X42
)
→
X46
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
(
(
¬
(
X48
↔
(
X30
↔
(
X15
↔
X40
)
)
)
)
→
X48
→
X40
→
(
(
¬
X30
)
↔
X44
)
)
→
(
X48
↔
(
(
¬
X30
)
↔
X40
)
)
)
↔
(
(
(
(
¬
(
X30
↔
(
(
¬
X15
)
→
(
¬
X13
)
→
X46
)
)
)
→
(
¬
(
(
¬
X30
)
↔
(
X19
→
X13
→
X48
)
)
)
)
↔
(
X30
→
X15
→
X19
)
)
↔
(
¬
(
X48
→
(
¬
X15
)
→
X3
→
X19
)
)
)
)
→
(
(
(
X46
→
(
X44
↔
(
(
¬
X38
)
→
X40
)
)
)
↔
(
¬
(
X40
↔
(
X10
↔
False
)
)
)
)
→
(
¬
(
X40
↔
(
(
¬
X38
)
→
X46
)
)
)
)
→
(
X46
↔
(
X44
↔
False
)
)
)
)
→
(
(
¬
(
X13
→
(
(
¬
X2
)
↔
X48
)
)
)
↔
(
(
(
X47
↔
(
X45
↔
(
X1
→
X24
)
)
)
↔
(
X45
→
(
(
¬
X15
)
↔
(
(
¬
X1
)
↔
X24
)
)
)
)
↔
(
(
¬
(
X13
↔
(
(
¬
X2
)
↔
X47
)
)
)
→
(
¬
(
(
¬
X47
)
→
(
¬
X45
)
→
(
¬
X13
)
→
X48
)
)
→
(
¬
(
X47
→
X15
→
(
X13
↔
X45
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
(
¬
(
X41
↔
(
X39
→
(
X28
↔
X20
)
)
)
)
↔
(
¬
(
X44
→
X39
→
(
X20
↔
X28
)
)
)
)
→
(
(
(
¬
(
(
(
X34
↔
(
(
¬
X28
)
→
X41
)
)
↔
(
(
X43
↔
(
X41
→
X34
)
)
↔
(
(
X34
↔
X41
)
→
(
X49
↔
(
X41
↔
(
(
¬
X28
)
→
X43
)
)
)
)
)
)
→
(
¬
(
X49
→
X28
)
)
)
)
→
(
¬
(
(
¬
X43
)
→
(
(
¬
X28
)
↔
X49
)
)
)
)
↔
(
¬
(
X39
→
X34
→
(
¬
X28
)
→
X43
)
)
)
)
)
↔
(
(
¬
(
X44
↔
(
(
¬
X39
)
↔
X49
)
)
)
→
(
¬
X44
)
→
(
¬
X27
)
→
(
(
¬
X18
)
↔
X49
)
)
)
→
(
(
X46
→
(
X39
↔
(
X12
→
X43
)
)
)
↔
(
X44
↔
(
X43
↔
(
X35
→
X46
)
)
)
)
→
(
(
(
X44
↔
(
X8
↔
X39
)
)
↔
(
X43
↔
(
X21
→
X8
)
)
)
↔
(
X44
↔
(
(
¬
X39
)
↔
(
X23
→
X43
)
)
)
)
)
↔
(
(
(
¬
(
X31
→
(
(
¬
X17
)
↔
(
X6
↔
X26
)
)
)
)
↔
(
¬
(
X17
→
(
(
¬
X11
)
↔
X42
)
)
)
)
↔
(
(
(
(
X42
↔
(
X26
→
X16
→
False
)
)
→
(
X29
↔
(
X26
↔
X37
)
)
)
→
(
(
X29
↔
X49
)
↔
(
(
X49
↔
(
X29
→
False
)
)
→
(
¬
(
(
(
¬
X36
)
↔
(
(
¬
X29
)
↔
(
X26
↔
X37
)
)
)
→
(
¬
(
(
¬
X37
)
→
(
¬
X36
)
→
(
¬
X26
)
→
X49
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
X49
↔
(
X31
↔
(
X26
↔
X16
)
)
)
)
↔
(
X43
→
(
¬
X16
)
→
X49
)
)
→
(
X43
↔
(
X31
→
X26
→
X44
)
)
)
↔
(
¬
(
(
¬
X31
)
↔
(
(
¬
X16
)
↔
X44
)
)
)
)
↔
(
X49
→
X44
→
(
X43
↔
(
X9
→
False
)
)
)
)
↔
(
¬
(
X49
→
X43
→
X44
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
7fc23a...
and proposition id is
5bfb86...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMbkPCEgmbwquHqLA3gJFr3Rwnimcq5awy3
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
¬
(
(
¬
(
(
(
(
(
¬
(
X50
↔
(
X31
↔
(
X26
↔
X41
)
)
)
)
↔
(
(
X26
↔
(
(
¬
X17
)
↔
X50
)
)
↔
(
(
¬
(
X50
↔
(
X26
→
(
¬
X10
)
→
X17
)
)
)
↔
(
(
¬
(
X50
↔
(
X10
↔
X8
)
)
)
↔
(
¬
(
(
(
(
¬
X41
)
↔
(
(
¬
X17
)
→
X8
→
X50
)
)
→
X50
→
X41
→
X10
→
False
)
→
(
¬
X10
)
→
(
¬
X8
)
→
X50
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X41
)
→
(
X19
↔
(
(
¬
X9
)
→
X42
)
)
)
)
↔
(
¬
(
X31
↔
(
(
¬
X19
)
↔
X42
)
)
)
)
→
(
¬
(
(
X3
→
(
X1
↔
False
)
)
→
(
(
¬
(
(
¬
X31
)
→
(
(
¬
X3
)
↔
X50
)
)
)
↔
(
¬
(
X50
→
X19
→
X9
→
X31
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
X42
)
↔
(
(
¬
X31
)
→
(
(
¬
X0
)
↔
X49
)
)
)
↔
(
X49
↔
(
X42
↔
(
X39
→
X31
)
)
)
)
→
(
¬
(
(
X39
→
(
¬
X15
)
→
(
¬
X11
)
→
X42
)
→
(
¬
(
X42
→
(
X15
↔
(
X13
→
False
)
)
)
)
)
)
→
(
¬
(
X42
↔
(
(
¬
X26
)
→
X13
→
X31
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X48
→
(
¬
X34
)
→
X22
→
X50
)
)
→
(
¬
(
X49
↔
(
X40
→
(
X22
↔
X46
)
)
)
)
)
)
↔
(
(
(
(
X39
↔
(
X38
→
(
X36
↔
X48
)
)
)
→
(
(
¬
(
X39
→
(
¬
X29
)
→
(
(
¬
X20
)
↔
X48
)
)
)
↔
(
¬
(
(
¬
X14
)
↔
X48
)
)
)
→
(
¬
(
X38
↔
(
(
¬
X20
)
↔
(
X14
↔
X48
)
)
)
)
)
↔
(
X48
↔
(
(
¬
X39
)
↔
(
X20
→
X46
)
)
)
)
→
(
(
(
¬
(
(
¬
X29
)
→
X27
→
X46
)
)
→
(
X27
↔
(
X16
↔
(
X7
→
X29
)
)
)
)
↔
(
(
X50
→
(
(
¬
X14
)
↔
(
X7
↔
X16
)
)
)
↔
(
¬
(
X29
↔
(
X7
↔
X14
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
(
X42
↔
(
X32
↔
X24
)
)
↔
(
¬
(
X45
→
(
X44
↔
(
(
¬
X24
)
↔
X42
)
)
)
)
)
↔
(
X47
↔
(
(
¬
X44
)
→
(
X24
↔
X45
)
)
)
)
→
(
(
X42
→
(
X12
↔
X47
)
)
↔
(
(
(
¬
(
X50
↔
(
X42
→
X45
)
)
)
↔
(
¬
(
(
¬
X47
)
↔
(
X12
↔
X50
)
)
)
)
↔
(
(
¬
X45
)
↔
(
(
¬
X42
)
↔
(
X21
↔
X47
)
)
)
)
)
)
↔
(
(
(
(
X50
→
(
¬
X25
)
→
X47
)
↔
(
(
¬
(
X37
→
(
X33
↔
(
X25
→
X47
)
)
)
)
↔
(
X50
↔
(
X25
↔
False
)
)
)
)
↔
(
(
¬
X47
)
→
(
¬
X23
)
→
X50
)
)
↔
(
X47
↔
(
X42
↔
(
X33
→
X50
)
)
)
)
)
→
(
(
(
(
¬
X43
)
↔
(
X42
→
(
¬
X18
)
→
X50
)
)
↔
(
¬
(
(
¬
(
X44
→
X35
→
(
X5
↔
X43
)
)
)
→
(
X44
↔
(
(
¬
X18
)
→
X43
)
)
)
)
)
↔
(
(
(
¬
(
X49
→
X28
→
(
X4
↔
X44
)
)
)
→
(
¬
(
X35
→
X28
→
X4
→
X43
)
)
)
↔
(
(
(
¬
(
(
¬
X43
)
→
(
X30
↔
X49
)
)
)
↔
(
(
X43
→
(
(
¬
X2
)
↔
X44
)
)
↔
(
¬
(
X49
→
(
X30
↔
(
X2
↔
False
)
)
)
)
)
)
→
(
(
¬
X6
)
↔
(
X2
↔
X49
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
X41
↔
(
X38
→
X10
→
X13
)
)
↔
(
(
¬
(
X41
→
(
X40
↔
X44
)
)
)
→
(
¬
(
X44
↔
(
X41
↔
(
(
¬
X10
)
↔
X38
)
)
)
)
)
)
↔
(
¬
(
X44
↔
(
X38
→
X34
→
X40
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X41
→
X1
)
)
→
(
¬
X38
)
→
X17
→
X15
→
X41
)
↔
(
(
(
¬
(
(
¬
X16
)
→
(
(
¬
X15
)
↔
(
(
¬
X1
)
→
X22
)
)
)
)
↔
(
X22
→
(
X17
↔
(
X16
↔
(
X1
↔
False
)
)
)
)
)
→
(
¬
(
X48
→
(
(
¬
X17
)
↔
(
X15
↔
X22
)
)
)
)
)
)
→
(
¬
(
X40
↔
(
X17
↔
(
(
¬
X1
)
→
X22
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
(
¬
X47
)
↔
(
(
¬
X44
)
↔
(
(
¬
X32
)
↔
X49
)
)
)
→
(
¬
(
X32
↔
X37
)
)
)
→
(
X47
→
(
X34
↔
(
X14
↔
False
)
)
)
→
(
¬
(
(
¬
(
X47
→
(
¬
X14
)
→
X34
)
)
→
(
¬
(
X49
→
(
X44
↔
(
X34
→
X47
)
)
)
)
)
)
)
↔
(
(
X40
↔
(
(
¬
X32
)
→
(
(
¬
X21
)
↔
X33
)
)
)
↔
(
¬
(
X40
→
(
(
¬
X21
)
↔
(
(
¬
X14
)
↔
X37
)
)
)
)
)
)
→
(
¬
(
(
X37
→
X32
→
(
¬
X23
)
→
X44
)
→
(
¬
(
X44
→
(
¬
X12
)
→
X32
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
(
X50
→
(
¬
X11
)
→
X46
)
→
X50
→
(
X48
↔
X30
)
)
)
↔
(
(
(
X50
→
(
X48
↔
(
X29
→
X30
)
)
)
→
X30
→
(
(
¬
X29
)
↔
(
(
¬
X3
)
↔
X48
)
)
)
↔
(
(
(
¬
X30
)
↔
(
X18
→
X48
)
)
↔
(
¬
(
X50
→
X30
→
(
(
¬
X18
)
↔
X19
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
X49
)
→
(
¬
X8
)
→
X0
→
X50
)
→
(
¬
(
(
¬
(
(
¬
X29
)
→
X25
→
(
X8
↔
X50
)
)
)
→
(
¬
(
X25
→
X8
→
(
X0
↔
X29
)
)
)
)
)
)
)
→
X50
→
(
X49
↔
(
X46
→
X25
→
False
)
)
)
→
(
¬
(
X49
→
X46
)
)
→
(
¬
(
X47
↔
(
(
¬
X29
)
→
(
X5
↔
X46
)
)
)
)
)
)
↔
(
(
(
(
X39
→
X29
→
(
¬
X9
)
→
X19
)
→
(
X19
↔
X35
)
)
↔
(
¬
(
(
X39
↔
(
(
¬
X20
)
↔
(
(
¬
X18
)
→
X35
)
)
)
→
(
(
¬
(
X35
→
(
(
¬
X7
)
↔
X29
)
)
)
↔
(
(
X29
↔
(
X7
↔
False
)
)
↔
(
(
X39
↔
(
X29
→
(
X20
↔
X7
)
)
)
→
(
¬
(
X35
→
(
(
¬
X7
)
↔
X20
)
)
)
)
)
)
)
)
)
↔
(
¬
(
X39
↔
(
X36
→
X35
→
X20
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
X48
↔
(
X45
↔
(
X43
→
X49
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X50
↔
False
)
)
→
(
(
X50
→
(
X48
↔
X43
)
)
↔
(
(
(
¬
X49
)
→
(
X48
↔
X50
)
)
↔
(
¬
(
(
(
X48
↔
(
X43
↔
(
X6
↔
False
)
)
)
↔
(
¬
(
(
¬
X49
)
↔
(
X43
→
(
X6
↔
X50
)
)
)
)
)
→
(
¬
(
X50
→
(
(
¬
X48
)
↔
X49
)
)
)
)
)
)
)
)
)
→
(
¬
(
X50
→
(
X48
↔
X49
)
)
)
)
)
)
)
→
(
(
(
X50
↔
(
X49
↔
(
X42
→
X4
→
False
)
)
)
↔
(
¬
(
X50
→
(
X49
↔
(
X48
↔
(
X4
↔
False
)
)
)
)
)
)
↔
(
¬
(
X42
→
X26
)
)
)
)
)
→
(
(
X50
↔
(
X49
→
(
(
¬
X24
)
↔
X28
)
)
)
↔
(
¬
(
(
X48
→
(
(
¬
X31
)
↔
X49
)
)
→
(
(
X48
→
(
¬
X27
)
→
X31
)
↔
(
¬
(
(
X49
→
(
¬
X27
)
→
X2
→
X31
)
→
(
¬
(
(
¬
X27
)
↔
(
X24
→
X49
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
37ce87...
and proposition id is
daebf1...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMMvnFYZsnqQtgvoQcThWBngFwKM89uu9sC
)
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
∀X51 :
prop
,
(
(
(
¬
(
(
(
(
(
X22
↔
X48
)
↔
(
(
¬
X48
)
↔
(
(
¬
X5
)
↔
(
(
¬
X1
)
↔
X49
)
)
)
)
→
(
¬
(
(
X49
→
(
X47
↔
False
)
)
→
(
¬
(
(
¬
X49
)
→
(
X48
↔
(
X47
↔
X50
)
)
)
)
→
(
(
(
¬
X22
)
↔
X50
)
↔
(
¬
(
X48
→
X47
→
(
(
¬
X22
)
↔
X25
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X3
)
→
(
¬
X0
)
→
X49
)
)
→
(
¬
(
(
X50
↔
(
X49
↔
(
X22
→
X0
)
)
)
→
(
(
¬
X22
)
↔
(
(
¬
X3
)
→
(
X0
↔
X50
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
X50
↔
(
X49
↔
(
(
¬
X17
)
↔
X43
)
)
)
↔
(
¬
(
X49
↔
(
(
¬
X43
)
↔
(
X36
→
X50
)
)
)
)
)
→
(
(
X50
→
(
X49
↔
(
X48
→
X43
)
)
)
↔
(
¬
(
(
(
¬
(
X30
↔
(
X10
→
(
¬
X9
)
→
X50
)
)
)
→
X49
→
X10
→
(
X9
↔
X30
)
)
→
(
(
X49
→
X26
→
X7
)
↔
(
¬
(
(
¬
(
(
¬
(
(
X50
→
(
(
¬
X48
)
↔
(
(
¬
X7
)
→
X49
)
)
)
→
(
¬
(
(
¬
X49
)
→
(
(
¬
X48
)
↔
(
X10
↔
X50
)
)
)
)
)
)
→
(
¬
(
X50
↔
(
X49
↔
(
X26
→
X48
)
)
)
)
)
)
→
(
(
¬
X7
)
↔
X48
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
(
¬
X37
)
→
(
¬
X29
)
→
(
(
¬
X23
)
↔
X50
)
)
→
(
(
X50
→
(
X48
↔
(
(
¬
X4
)
↔
X27
)
)
)
↔
(
(
¬
(
(
¬
X48
)
↔
(
X37
↔
X50
)
)
)
→
(
(
¬
X37
)
→
(
¬
X27
)
→
(
¬
X14
)
→
X50
)
→
(
(
¬
X48
)
↔
(
X37
↔
(
X29
→
X50
)
)
)
→
(
¬
(
X48
→
X27
→
False
)
)
)
)
)
→
(
¬
(
(
X29
↔
(
X27
↔
(
X12
↔
X8
)
)
)
→
(
¬
(
(
¬
X27
)
→
(
(
¬
X12
)
↔
X37
)
)
)
)
)
)
)
→
(
(
¬
(
X50
↔
(
(
¬
X44
)
↔
(
X23
↔
X49
)
)
)
)
↔
(
X50
↔
(
X23
→
X29
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
X47
→
(
X28
↔
(
X2
↔
X51
)
)
)
→
X45
→
(
(
¬
X28
)
↔
(
(
¬
X2
)
↔
X41
)
)
)
↔
(
(
(
(
X48
→
(
¬
X35
)
→
(
(
¬
X21
)
↔
X45
)
)
→
(
¬
(
X45
→
X35
→
X16
→
X21
)
)
)
↔
(
(
(
¬
(
X45
↔
(
X42
→
(
X21
↔
X35
)
)
)
)
↔
(
(
¬
(
X51
→
(
X42
↔
(
X38
↔
X6
)
)
)
)
↔
(
¬
(
X42
↔
(
X38
→
(
X21
↔
X51
)
)
)
)
)
)
↔
(
¬
(
X51
↔
(
X6
↔
X35
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
X48
→
(
X33
↔
False
)
)
→
(
¬
X47
)
→
(
¬
X35
)
→
(
X20
↔
X48
)
)
)
→
(
¬
(
(
X38
↔
(
X35
↔
X33
)
)
→
(
¬
(
(
¬
(
(
¬
X35
)
↔
(
(
¬
X33
)
↔
(
(
¬
X15
)
↔
X38
)
)
)
)
→
X48
→
(
¬
X33
)
→
X38
)
)
)
)
)
)
↔
(
¬
(
(
¬
X33
)
↔
(
X20
↔
X38
)
)
)
)
)
)
→
(
(
(
¬
(
X40
↔
(
(
¬
X31
)
→
(
X24
↔
X51
)
)
)
)
→
(
¬
(
(
¬
X39
)
→
(
(
¬
X32
)
↔
X45
)
)
)
→
(
(
¬
(
(
(
¬
(
(
¬
X34
)
→
X24
→
X19
→
X51
)
)
↔
(
X51
→
X32
→
False
)
)
→
(
¬
X39
)
→
(
(
¬
X19
)
↔
X51
)
)
)
↔
(
¬
(
X51
→
(
¬
X39
)
→
X19
→
X45
)
)
)
)
↔
(
(
¬
(
X48
↔
(
X46
→
X45
→
X39
→
False
)
)
)
↔
(
¬
(
(
(
X24
→
(
(
¬
X18
)
↔
(
X13
↔
X43
)
)
)
→
(
X43
↔
(
X24
→
(
X18
↔
X48
)
)
)
)
→
(
(
(
X45
↔
X43
)
↔
(
X48
↔
(
X43
→
X24
→
X11
→
False
)
)
)
↔
(
X48
↔
(
X39
→
X24
→
X43
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
(
¬
(
(
¬
(
(
X51
→
X47
→
X50
)
→
(
(
¬
(
X50
↔
(
X0
→
X51
)
)
)
↔
(
¬
(
(
X51
↔
(
(
¬
X13
)
→
(
¬
X0
)
→
X50
)
)
→
(
¬
(
X47
→
(
(
¬
X0
)
↔
X50
)
)
)
)
)
)
→
X0
→
False
)
)
→
(
¬
(
(
¬
X47
)
↔
(
(
¬
X31
)
→
X13
→
X51
)
)
)
)
)
↔
(
¬
(
X51
→
(
¬
X31
)
→
X50
)
)
)
→
(
¬
(
X50
→
(
¬
(
(
X51
↔
(
X45
→
X7
→
False
)
)
→
(
¬
(
(
(
X50
↔
(
(
¬
X7
)
→
(
X5
↔
X45
)
)
)
→
(
¬
X31
)
→
(
¬
X7
)
→
X45
)
→
(
¬
(
(
¬
X7
)
→
X5
→
X45
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X45
→
(
(
¬
X7
)
↔
X47
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X7
)
↔
(
X4
↔
X48
)
)
)
)
)
)
↔
(
¬
(
X48
↔
(
X45
↔
(
X4
→
False
)
)
)
)
)
)
→
(
(
(
(
X30
→
(
¬
X9
)
→
X28
)
↔
(
(
¬
(
X28
↔
(
(
¬
X14
)
→
(
¬
X9
)
→
X30
)
)
)
→
(
¬
X26
)
→
(
X14
↔
(
X9
↔
X30
)
)
)
)
↔
(
(
¬
(
(
X39
→
X31
→
(
X30
↔
(
X28
↔
False
)
)
)
→
(
¬
(
X39
→
(
X31
↔
(
X22
↔
False
)
)
)
)
)
)
↔
(
(
(
¬
(
X31
↔
(
(
¬
X28
)
→
X39
)
)
)
↔
(
¬
(
X51
↔
(
X39
↔
(
X31
↔
(
X21
↔
False
)
)
)
)
)
)
→
X28
→
X26
→
X51
)
)
)
↔
(
¬
(
X51
→
(
(
¬
X39
)
↔
(
X30
→
X48
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
X47
↔
(
X43
↔
(
(
¬
X35
)
→
X48
)
)
)
)
→
(
X47
↔
(
X43
→
X19
→
False
)
)
)
→
(
¬
(
(
¬
X43
)
↔
(
(
¬
X19
)
↔
X46
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X48
)
→
(
¬
X46
)
→
X50
)
)
→
(
(
X43
↔
(
(
¬
X18
)
↔
X48
)
)
↔
(
¬
(
X50
↔
(
X46
↔
(
X18
→
X48
)
)
)
)
)
→
X48
→
(
X46
↔
X43
)
)
)
)
→
(
(
¬
(
(
(
(
(
¬
(
X33
→
(
(
¬
X23
)
↔
(
X16
→
X46
)
)
)
)
↔
(
X33
↔
(
X23
↔
(
X12
↔
False
)
)
)
)
→
(
X33
↔
(
(
¬
X12
)
↔
X50
)
)
)
→
(
¬
X23
)
→
(
¬
X16
)
→
X46
)
→
(
¬
(
(
X33
↔
(
(
¬
X20
)
→
(
X6
↔
X50
)
)
)
→
(
¬
(
X46
→
(
X33
↔
(
(
¬
X20
)
→
X50
)
)
)
)
)
)
)
)
↔
(
(
X48
↔
(
X46
→
(
(
¬
X20
)
↔
X33
)
)
)
→
(
¬
(
(
¬
X46
)
→
(
¬
X33
)
→
(
¬
X17
)
→
X48
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
X44
→
X42
→
(
(
¬
X15
)
↔
X49
)
)
→
(
¬
(
(
¬
X15
)
→
X47
)
)
)
)
↔
(
(
¬
(
X49
↔
(
(
¬
X42
)
→
(
¬
X34
)
→
X44
)
)
)
→
(
(
¬
X44
)
↔
(
(
¬
X25
)
↔
(
X24
↔
X49
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X44
)
→
(
X41
↔
X51
)
)
)
↔
(
(
(
¬
X40
)
↔
(
X32
→
(
(
¬
X11
)
↔
X41
)
)
)
↔
(
¬
(
X51
→
X41
→
X40
→
X44
)
)
)
)
→
(
(
(
X44
→
(
¬
X24
)
→
X49
)
↔
(
(
¬
(
X51
↔
(
X49
↔
(
X44
↔
X24
)
)
)
)
→
X44
→
(
X40
↔
(
X24
↔
X49
)
)
)
)
↔
(
¬
(
(
¬
X49
)
→
(
X44
↔
(
X2
→
X51
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
X40
↔
(
(
¬
X29
)
↔
X48
)
)
→
(
X40
↔
(
X38
↔
(
X27
↔
X48
)
)
)
)
→
(
¬
(
X48
→
X38
→
X40
)
)
)
)
↔
(
X37
→
(
(
¬
X36
)
↔
(
X27
↔
X38
)
)
)
)
→
(
(
(
¬
(
X40
→
X37
→
X42
)
)
↔
(
¬
(
(
(
¬
X40
)
→
X38
→
X37
→
X51
)
→
X42
→
(
¬
X37
)
→
X51
)
)
)
↔
(
(
(
X51
↔
(
X38
↔
(
(
¬
X8
)
↔
X40
)
)
)
→
(
(
X51
→
(
(
¬
X38
)
↔
(
(
¬
X1
)
↔
X40
)
)
)
↔
(
¬
(
X40
→
X38
→
(
X1
↔
X10
)
)
)
)
)
↔
(
X40
→
(
¬
X38
)
→
(
X3
↔
X51
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
ca2031...
and proposition id is
ea4834...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMY4nD1F6V82k4UiFzsiuhuR69YYzSTbNFx
)
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
¬
∀X51 :
prop
,
∀X52 :
prop
,
∀X53 :
prop
,
(
¬
∀X54 :
prop
,
(
(
¬
(
(
¬
(
(
(
(
X53
↔
(
X45
↔
(
X38
→
False
)
)
)
↔
(
(
¬
(
X53
→
(
(
¬
X38
)
↔
(
X29
↔
X50
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X45
)
→
(
X29
↔
(
(
¬
X12
)
→
X47
)
)
)
)
→
X53
→
(
¬
X45
)
→
X12
→
X47
)
)
↔
(
¬
(
X50
↔
(
(
¬
X45
)
→
(
¬
X23
)
→
X53
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
X50
)
↔
(
X40
→
(
¬
X9
)
→
X53
)
)
)
→
(
¬
(
X53
↔
(
X50
→
(
(
¬
X9
)
↔
X30
)
)
)
)
)
)
↔
(
(
X54
↔
(
(
¬
X47
)
↔
(
X40
↔
X53
)
)
)
↔
(
X54
↔
(
X53
→
(
X47
↔
X50
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X46
)
→
(
(
¬
X45
)
↔
(
(
¬
X27
)
↔
X48
)
)
)
)
→
(
(
¬
(
(
¬
(
X49
↔
(
X45
↔
(
X27
↔
(
X16
↔
False
)
)
)
)
)
→
(
(
(
¬
X16
)
↔
X50
)
↔
(
X49
↔
(
X48
→
X16
→
X27
)
)
)
)
)
↔
(
¬
(
X27
→
(
(
¬
X16
)
↔
X50
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
(
¬
(
X24
→
(
¬
X3
)
→
(
¬
X2
)
→
X28
)
)
→
(
¬
(
X50
↔
(
X28
↔
(
X24
↔
False
)
)
)
)
)
)
→
X50
→
(
X20
↔
(
X3
↔
X28
)
)
)
)
→
(
X24
↔
(
X20
↔
(
(
¬
X8
)
→
X28
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X39
)
↔
(
(
¬
X3
)
→
X49
)
)
)
→
(
X39
↔
(
X24
→
X3
→
X44
)
)
)
→
(
(
¬
(
X49
↔
(
(
¬
X47
)
→
(
X44
↔
X48
)
)
)
)
↔
(
¬
(
(
¬
X48
)
↔
(
X47
↔
(
X39
↔
X49
)
)
)
)
)
)
)
→
(
¬
(
X44
→
(
X28
↔
(
X24
↔
X47
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
(
(
X51
→
(
X35
↔
(
X10
↔
False
)
)
)
→
(
¬
(
(
X53
↔
(
X51
→
X35
→
X15
→
False
)
)
→
(
X51
↔
(
(
¬
X15
)
↔
(
X10
↔
X53
)
)
)
)
)
)
)
↔
(
(
¬
(
X53
→
(
(
¬
X41
)
↔
(
X10
↔
X54
)
)
)
)
→
(
¬
X51
)
)
)
→
(
¬
(
(
¬
X53
)
↔
(
(
¬
X52
)
→
X14
→
X54
)
)
)
→
(
(
¬
(
(
¬
X51
)
↔
(
(
¬
X14
)
→
X4
→
X54
)
)
)
↔
(
X54
→
(
X52
↔
(
X51
→
X14
)
)
)
)
)
)
→
(
(
(
(
(
(
¬
(
X33
→
X17
)
)
↔
(
¬
(
X53
→
X51
→
X37
→
X17
→
False
)
)
)
↔
(
¬
(
(
¬
X33
)
↔
X37
)
)
)
→
(
¬
(
X51
↔
(
X49
→
(
X37
↔
X53
)
)
)
)
)
→
(
¬
(
(
X53
→
(
X37
↔
(
X5
→
X26
)
)
)
→
(
(
(
¬
X19
)
↔
(
X5
→
(
X0
↔
X53
)
)
)
↔
(
(
X36
↔
(
(
¬
X26
)
↔
(
X0
→
X53
)
)
)
↔
(
X36
↔
(
X17
→
False
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X54
→
X6
→
X5
→
False
)
)
→
(
¬
(
X53
→
(
X51
↔
(
X6
→
X5
)
)
)
)
)
)
↔
(
X53
↔
(
X51
→
False
)
)
)
)
)
)
→
(
(
(
X53
↔
(
X32
→
(
X18
↔
X21
)
)
)
↔
(
¬
(
(
¬
(
(
(
X31
→
(
(
¬
X25
)
↔
(
X22
→
X42
)
)
)
↔
(
X32
→
(
(
¬
X25
)
↔
(
X22
→
X42
)
)
)
)
→
(
¬
(
X42
↔
(
X22
→
X34
)
)
)
)
)
→
(
(
X42
↔
(
X32
↔
(
X1
↔
False
)
)
)
↔
(
(
¬
(
(
¬
(
X32
↔
(
(
¬
X18
)
↔
X42
)
)
)
→
(
¬
X34
)
→
X32
→
X53
)
)
→
(
¬
(
X42
→
(
X18
↔
(
(
¬
X13
)
→
X34
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
X51
→
X42
→
(
X11
↔
X53
)
)
→
(
X51
↔
(
(
¬
X43
)
↔
(
(
¬
X31
)
→
X53
)
)
)
→
X51
→
(
¬
X42
)
→
X31
→
X43
)
→
(
¬
(
X52
→
(
(
¬
X31
)
↔
X53
)
)
)
→
(
¬
(
(
¬
X42
)
↔
(
(
¬
X31
)
→
X52
)
)
)
→
(
¬
(
(
¬
(
X52
→
(
¬
X42
)
→
X31
→
X53
)
)
→
(
¬
(
X52
→
X42
→
(
¬
X7
)
→
X31
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
(
¬
X12
)
↔
(
(
¬
X10
)
→
X54
)
)
↔
(
¬
(
X50
→
X48
→
X10
→
X12
)
)
)
→
(
(
X50
↔
(
X48
→
(
X1
↔
X17
)
)
)
↔
(
(
¬
X10
)
↔
(
X1
→
X17
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X48
)
→
X47
→
X34
→
X50
)
)
→
(
(
(
(
(
¬
(
(
¬
X39
)
→
(
(
¬
X3
)
↔
X43
)
)
)
→
(
(
(
¬
X43
)
↔
(
X39
↔
X54
)
)
↔
(
(
X47
→
(
(
¬
X39
)
↔
(
(
¬
X3
)
↔
X54
)
)
)
↔
(
X54
→
(
X47
↔
(
(
¬
X32
)
↔
X39
)
)
)
)
)
)
↔
(
(
¬
X43
)
↔
(
X32
→
(
(
¬
X27
)
↔
X47
)
)
)
)
→
(
X43
↔
(
(
¬
X39
)
→
(
X18
↔
X54
)
)
)
)
↔
(
(
X18
→
(
X9
↔
X48
)
)
→
(
¬
(
(
¬
X48
)
→
(
¬
X47
)
→
(
X27
↔
X50
)
)
)
→
(
¬
(
(
X47
↔
(
(
¬
X27
)
→
(
(
¬
X18
)
↔
X48
)
)
)
→
(
X48
↔
(
(
¬
X27
)
↔
(
X9
→
X47
)
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
(
X21
↔
(
(
¬
(
(
¬
X46
)
→
X43
→
X52
)
)
→
X52
→
(
(
¬
X43
)
↔
(
(
¬
X21
)
→
X50
)
)
)
)
→
(
¬
X46
)
→
(
(
¬
X43
)
↔
(
X23
↔
X52
)
)
)
→
(
(
(
¬
(
X52
→
X35
→
X16
)
)
→
(
¬
X35
)
→
(
¬
X25
)
→
X43
)
↔
(
(
(
X43
↔
(
(
¬
X35
)
→
(
(
¬
X31
)
↔
X50
)
)
)
↔
(
¬
(
X52
↔
(
X50
↔
(
X43
→
(
X35
↔
False
)
)
)
)
)
)
↔
(
X50
→
(
X43
↔
(
X35
→
False
)
)
)
)
)
)
)
→
(
¬
(
(
¬
X50
)
↔
(
X46
↔
(
(
¬
X19
)
↔
X52
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X50
)
→
(
(
¬
X46
)
↔
(
X40
→
X54
)
)
)
)
↔
(
¬
(
(
¬
(
X54
↔
(
X50
↔
(
X35
↔
(
X6
↔
False
)
)
)
)
)
→
(
(
¬
X40
)
↔
(
X6
→
X50
)
)
)
)
)
↔
(
X54
↔
(
X50
→
(
¬
X5
)
→
X46
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X53
→
X52
→
(
(
¬
X45
)
↔
X51
)
)
)
→
(
(
(
(
X51
↔
(
X49
↔
(
(
¬
X13
)
↔
X37
)
)
)
→
(
¬
(
X51
↔
(
X49
↔
(
(
¬
X37
)
↔
X45
)
)
)
)
)
→
(
¬
(
(
¬
X51
)
→
(
¬
X49
)
→
(
¬
X37
)
→
X53
)
)
)
↔
(
(
X44
→
(
¬
X15
)
→
(
X14
↔
X53
)
)
→
(
X51
↔
(
X49
→
(
X38
↔
False
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
X45
)
→
(
(
¬
X42
)
↔
(
X36
↔
X51
)
)
)
↔
(
(
¬
(
(
¬
(
X42
→
X24
→
(
X7
↔
False
)
)
)
→
(
(
¬
X45
)
↔
(
(
¬
X42
)
↔
(
X7
↔
X51
)
)
)
)
)
↔
(
(
¬
X24
)
→
(
X7
↔
(
(
¬
X2
)
↔
X51
)
)
)
)
)
↔
(
(
X42
→
X36
→
X45
)
↔
(
(
¬
(
X45
→
X8
→
X36
)
)
↔
(
¬
(
(
¬
(
X42
↔
(
(
¬
X36
)
↔
(
(
¬
X29
)
→
X45
)
)
)
)
→
(
X49
↔
(
(
¬
X36
)
↔
(
X29
↔
X42
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
X11
↔
(
(
¬
X8
)
→
(
(
¬
X0
)
↔
X29
)
)
)
→
(
(
¬
X22
)
↔
(
X0
↔
X29
)
)
)
→
(
¬
(
(
¬
X8
)
↔
(
X0
→
X22
)
)
)
→
(
(
¬
(
X45
→
X8
→
(
¬
X0
)
→
X41
)
)
↔
(
¬
(
X45
↔
(
X22
→
X8
→
X0
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
X53
→
X47
→
X46
→
X20
)
)
↔
(
(
(
(
X47
↔
(
X33
↔
(
(
¬
X28
)
↔
X49
)
)
)
↔
(
(
X53
↔
(
X28
→
X47
)
)
↔
(
¬
(
X46
→
X33
→
False
)
)
)
)
↔
(
¬
(
(
¬
X49
)
→
X47
→
X53
)
)
)
↔
(
¬
(
(
¬
(
X33
↔
(
X30
↔
(
X26
↔
X51
)
)
)
)
→
(
(
¬
X49
)
→
(
¬
X46
)
→
(
X4
↔
X51
)
)
→
(
¬
(
X51
↔
(
(
¬
X33
)
↔
(
(
¬
X26
)
↔
X49
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
13fb1a...
and proposition id is
b082bc...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMLcXnf5mGnvLqsmpk4g7j54vsKg6oHnwNf
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
∀X50 :
prop
,
(
(
¬
(
(
(
(
(
¬
(
(
¬
X41
)
↔
(
(
¬
X40
)
↔
(
(
¬
X23
)
↔
X49
)
)
)
)
↔
(
¬
(
(
¬
(
X49
↔
(
X41
→
X47
)
)
)
→
(
(
X49
→
(
X41
↔
X10
)
)
↔
(
X49
→
X47
→
(
¬
X6
)
→
X10
)
)
→
(
¬
(
X49
→
(
X40
↔
(
X6
→
X10
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X40
→
X2
→
X4
)
)
↔
(
(
X7
→
X2
→
X1
)
↔
(
X49
↔
(
X1
→
X29
)
)
)
)
→
(
(
(
X44
→
(
X4
↔
(
X2
→
X1
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
X16
↔
(
X4
→
X44
)
)
)
→
(
¬
(
X44
→
X41
→
(
(
¬
X1
)
↔
X4
)
)
)
)
)
↔
(
¬
(
X41
↔
(
(
¬
X16
)
→
X44
)
)
)
)
→
X44
→
(
¬
X1
)
→
X4
)
)
)
↔
(
X44
↔
(
X4
↔
(
X2
↔
False
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X49
→
(
¬
X41
)
→
(
X21
↔
X48
)
)
)
→
(
¬
(
X48
→
(
(
¬
X41
)
↔
(
X40
↔
X49
)
)
)
)
)
↔
(
(
X48
→
X41
→
(
X31
↔
False
)
)
↔
(
¬
(
X49
↔
(
X41
↔
(
X31
↔
(
X25
→
False
)
)
)
)
)
)
)
→
(
X49
↔
(
X41
↔
X48
)
)
)
→
(
(
(
X49
→
X41
→
(
X39
↔
X0
)
)
→
(
(
¬
(
X49
→
(
(
¬
X11
)
↔
X39
)
)
)
↔
(
(
¬
X48
)
↔
(
(
¬
X12
)
↔
(
(
¬
X11
)
→
X49
)
)
)
)
)
→
X49
→
(
(
¬
X24
)
↔
(
(
¬
X0
)
→
X48
)
)
)
→
(
(
¬
(
(
¬
X39
)
→
(
¬
X20
)
→
X41
)
)
↔
(
(
¬
(
X49
→
(
(
¬
X35
)
↔
(
(
¬
X20
)
↔
X41
)
)
)
)
↔
(
¬
(
(
X49
→
X38
)
→
(
(
(
(
X38
↔
(
(
¬
X26
)
→
(
¬
X20
)
→
X49
)
)
↔
(
¬
(
X49
↔
(
X35
→
(
(
¬
X20
)
↔
X26
)
)
)
)
)
↔
(
¬
(
X38
→
(
¬
X35
)
→
X49
)
)
)
↔
(
¬
(
X38
→
(
X26
↔
X35
)
)
)
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
X44
→
(
(
¬
X37
)
↔
(
X3
↔
X45
)
)
)
)
→
(
(
X50
↔
(
(
¬
X45
)
→
(
¬
X28
)
→
X49
)
)
↔
(
¬
(
X49
↔
(
X37
→
(
X28
↔
X45
)
)
)
)
)
)
→
X50
→
(
X49
↔
(
X44
↔
False
)
)
)
→
(
(
¬
(
(
(
¬
(
X43
↔
(
(
¬
X42
)
→
(
(
¬
X8
)
↔
X45
)
)
)
)
→
(
(
¬
(
X46
→
X45
→
X42
)
)
↔
(
(
¬
(
(
X46
→
X45
→
X5
→
X32
)
→
(
¬
(
X46
→
(
X45
↔
(
X32
→
X42
)
)
)
)
)
)
↔
(
¬
(
X46
↔
(
X45
↔
(
X32
↔
(
X13
→
False
)
)
)
)
)
)
)
→
(
¬
(
X43
→
(
X13
↔
(
X5
↔
X42
)
)
)
)
)
→
(
X45
↔
(
X43
↔
(
(
¬
X18
)
↔
X32
)
)
)
)
)
↔
(
(
(
(
X46
→
X43
)
→
(
¬
(
X50
→
(
X44
↔
(
X34
→
False
)
)
)
)
)
↔
(
¬
(
X45
↔
(
(
¬
X44
)
→
X46
)
)
)
)
→
(
¬
(
X45
↔
(
X43
→
(
X30
↔
X46
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
¬
(
X15
→
(
(
¬
X9
)
↔
X45
)
)
)
↔
(
(
¬
(
(
X36
→
X15
→
False
)
→
(
X46
↔
(
X45
↔
(
(
¬
X9
)
→
X15
)
)
)
)
)
→
(
X45
↔
(
X36
→
X46
)
)
)
)
→
(
X46
↔
(
X45
→
X36
→
X22
)
)
)
↔
(
¬
(
X36
→
X22
→
X46
)
)
)
→
(
¬
(
X36
→
(
X15
↔
X17
)
)
)
)
→
(
¬
(
(
(
¬
(
X43
→
X36
→
(
¬
X14
)
→
X19
)
)
↔
(
(
¬
X36
)
→
(
¬
X33
)
→
(
¬
X19
)
→
X47
)
)
→
(
X33
↔
(
(
¬
X27
)
↔
(
(
¬
X22
)
→
X36
)
)
)
→
(
¬
(
X45
↔
(
X43
↔
(
(
¬
X22
)
→
X36
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
(
X48
↔
(
X24
→
X2
→
X49
)
)
→
(
¬
(
X39
↔
X49
)
)
)
)
↔
(
(
¬
(
(
¬
X24
)
↔
(
X2
→
X32
)
)
)
↔
(
¬
(
X49
→
(
¬
X24
)
→
(
(
¬
X1
)
↔
X32
)
)
)
)
)
→
X16
→
(
X2
↔
X32
)
)
↔
(
(
(
¬
(
X44
→
X41
→
X5
)
)
→
(
X46
→
(
(
¬
X5
)
↔
(
X2
↔
X41
)
)
)
→
(
(
¬
X44
)
↔
(
X41
↔
(
X27
→
X46
)
)
)
)
↔
(
¬
(
X44
↔
(
X5
↔
X46
)
)
)
)
)
↔
(
(
(
¬
(
(
X17
↔
(
¬
(
(
¬
(
X44
↔
(
X17
→
X43
)
)
)
→
(
¬
(
X43
↔
(
(
¬
X34
)
↔
(
(
¬
X8
)
↔
X44
)
)
)
)
)
)
)
→
(
¬
X44
)
→
X41
→
X34
→
X50
)
)
↔
(
¬
(
(
(
¬
(
X43
↔
(
X21
↔
X44
)
)
)
↔
(
(
X43
→
(
X9
↔
(
X3
→
False
)
)
)
→
(
(
X44
↔
(
X9
→
(
X3
↔
False
)
)
)
↔
(
(
¬
X43
)
→
(
¬
X21
)
→
(
¬
X3
)
→
X44
)
)
)
)
→
(
¬
(
X44
↔
(
(
¬
X43
)
→
X21
→
X50
)
)
)
)
)
)
↔
(
(
(
¬
(
X40
→
(
¬
X31
)
→
(
X23
↔
X48
)
)
)
→
(
(
X31
→
(
(
¬
X12
)
↔
X40
)
)
↔
(
(
(
¬
(
X48
↔
(
X44
→
X31
)
)
)
↔
(
¬
(
X44
↔
(
X40
→
(
(
¬
X12
)
↔
X48
)
)
)
)
)
→
(
¬
(
X40
→
(
(
¬
X12
)
↔
(
(
¬
X0
)
→
X48
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X31
↔
(
X23
→
X10
→
X50
)
)
)
↔
(
¬
(
(
¬
X31
)
→
(
¬
X10
)
→
X48
)
)
)
↔
(
(
¬
(
X50
↔
X41
)
)
↔
(
X48
→
X35
→
(
X29
↔
False
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
X22
→
(
X18
↔
(
X7
→
X49
)
)
)
↔
(
¬
(
X22
↔
(
(
¬
X18
)
↔
(
X7
↔
X46
)
)
)
)
)
↔
X46
)
↔
(
X25
↔
(
(
¬
X18
)
→
X49
)
)
)
↔
(
(
(
(
¬
(
X45
↔
(
X25
→
X19
→
X14
)
)
)
↔
(
¬
(
X47
→
(
X19
↔
False
)
)
)
)
→
(
¬
(
X45
→
(
X20
↔
(
(
¬
X14
)
↔
X25
)
)
)
)
)
→
X49
→
X46
→
(
X25
↔
X20
)
)
)
→
(
(
(
¬
(
(
¬
(
X47
↔
(
X42
→
(
(
¬
X36
)
↔
X45
)
)
)
)
→
(
(
(
(
¬
(
X47
→
X45
)
)
↔
(
¬
(
(
(
¬
X4
)
→
X11
)
→
(
X45
→
(
(
¬
X4
)
↔
X11
)
)
→
(
¬
(
X38
↔
(
X11
↔
(
X4
→
X37
)
)
)
)
)
)
)
↔
(
(
¬
X37
)
↔
(
(
¬
X11
)
↔
(
X4
→
X42
)
)
)
)
↔
(
X37
→
(
(
¬
X33
)
↔
X47
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X46
↔
(
X45
→
(
(
¬
X30
)
↔
X37
)
)
)
)
↔
(
¬
(
X48
↔
X45
)
)
)
↔
(
(
¬
(
(
¬
(
X45
→
(
X37
↔
(
X28
↔
X15
)
)
)
)
→
(
¬
(
X45
↔
(
X37
→
(
¬
X28
)
→
X46
)
)
)
)
)
→
(
¬
(
(
¬
X36
)
↔
(
(
¬
X15
)
↔
X45
)
)
)
)
)
→
(
¬
(
X48
↔
(
X37
→
(
X28
↔
X13
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
X46
)
↔
(
X26
↔
X47
)
)
→
(
(
¬
(
X48
↔
(
X47
→
(
X26
↔
X46
)
)
)
)
↔
(
(
¬
X46
)
↔
(
(
¬
X26
)
→
(
X6
↔
X47
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
5a902f...
and proposition id is
602112...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMLP78dAuuL4fa3Zd16vfi6S3nT7sRMHdCW
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
¬
(
(
(
(
(
(
¬
X35
)
↔
(
(
¬
X31
)
→
X30
→
X45
)
)
→
(
X46
→
X45
→
(
X10
↔
X35
)
)
→
(
¬
(
X46
↔
(
X35
→
(
X28
↔
False
)
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X35
)
↔
(
X31
→
X45
)
)
)
)
)
→
(
(
¬
(
X46
→
(
(
¬
X35
)
↔
(
X34
→
X48
)
)
)
)
↔
(
X48
↔
(
X46
→
X30
→
X35
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X45
)
→
(
¬
X35
)
→
(
(
¬
X7
)
↔
X46
)
)
)
→
(
¬
(
X45
→
(
X26
↔
X35
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
(
¬
X44
)
↔
(
(
¬
X15
)
↔
(
(
¬
X3
)
→
X46
)
)
)
)
→
(
X47
→
(
(
¬
X29
)
↔
(
(
¬
X15
)
↔
X44
)
)
)
→
(
X44
↔
(
X3
↔
X47
)
)
)
)
→
(
(
(
X46
→
(
¬
X17
)
→
X15
→
X44
)
→
(
¬
(
X2
↔
X17
)
)
)
↔
(
X49
↔
(
(
¬
X44
)
→
X15
→
X46
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
(
¬
(
(
(
¬
X24
)
↔
X46
)
→
X48
→
X46
→
(
X23
↔
X25
)
)
)
→
(
¬
(
(
¬
X45
)
↔
(
X25
→
(
(
¬
X23
)
↔
X48
)
)
)
)
)
)
→
X46
)
)
↔
(
(
(
(
X47
↔
(
X25
→
(
¬
X24
)
→
X46
)
)
→
X33
→
(
¬
X25
)
→
X47
)
→
X33
→
X25
→
X47
)
→
(
(
¬
X41
)
↔
(
X33
→
X25
→
X47
)
)
)
)
→
(
(
(
(
¬
(
X42
→
(
¬
X22
)
→
X24
)
)
↔
(
(
X33
→
X19
→
X22
)
↔
(
¬
(
X33
↔
(
X6
↔
(
X4
→
X22
)
)
)
)
)
)
↔
(
¬
(
X24
→
(
X22
↔
X33
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X48
)
↔
(
X42
↔
(
(
¬
X24
)
↔
X49
)
)
)
)
→
(
X24
↔
(
X6
↔
(
(
¬
X0
)
→
X18
)
)
)
→
(
(
¬
X42
)
→
(
X18
↔
(
(
¬
X6
)
↔
X49
)
)
)
→
(
¬
(
X49
→
(
¬
X18
)
→
(
¬
X0
)
→
X42
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
(
¬
X38
)
→
(
X32
↔
(
X27
→
X39
)
)
)
)
↔
(
¬
(
(
X39
↔
(
X38
→
(
X32
↔
(
X8
↔
False
)
)
)
)
→
(
X43
↔
(
X39
→
(
¬
X8
)
→
X38
)
)
)
)
)
→
(
(
¬
(
X39
↔
(
(
¬
X27
)
→
(
(
¬
X8
)
↔
X48
)
)
)
)
↔
(
¬
(
(
¬
X39
)
→
(
(
¬
X37
)
↔
(
(
¬
X27
)
↔
X48
)
)
)
)
)
)
)
↔
(
(
(
(
X48
↔
(
(
¬
X39
)
↔
(
(
¬
X38
)
→
X49
)
)
)
↔
(
¬
(
(
¬
X38
)
→
(
(
¬
X36
)
↔
(
(
¬
X21
)
→
X48
)
)
)
)
)
↔
(
¬
(
X39
→
(
X38
↔
(
X5
↔
X48
)
)
)
)
)
↔
(
(
(
X39
→
(
(
¬
X11
)
↔
(
(
¬
X9
)
↔
X49
)
)
)
↔
(
¬
(
X38
→
(
X11
↔
X39
)
)
)
)
↔
(
(
¬
(
X11
↔
(
X9
↔
X14
)
)
)
→
(
(
(
¬
X14
)
↔
(
(
¬
X13
)
→
X9
→
X38
)
)
↔
(
¬
(
(
¬
X14
)
↔
(
X11
→
X38
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X43
→
X40
→
(
(
¬
X12
)
↔
X49
)
)
)
→
(
(
(
¬
X20
)
↔
(
(
¬
X12
)
→
X43
)
)
↔
(
¬
(
(
X40
↔
(
(
¬
X20
)
↔
(
X1
→
X43
)
)
)
→
(
¬
(
X12
→
(
(
¬
X1
)
↔
X40
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
X43
)
→
(
¬
X40
)
→
(
¬
X16
)
→
X45
)
↔
(
¬
(
X45
↔
X48
)
)
)
→
X45
→
(
¬
X43
)
→
(
X20
↔
X48
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X48
↔
(
X41
↔
X43
)
)
)
→
(
¬
(
(
(
(
(
¬
X31
)
↔
(
X28
→
X6
→
X48
)
)
↔
(
X48
→
X7
→
(
¬
X6
)
→
X31
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X31
)
→
X28
→
(
(
¬
X7
)
↔
X41
)
)
)
→
(
(
¬
X31
)
↔
(
X28
→
X7
→
X43
)
)
)
)
↔
(
¬
(
(
¬
X41
)
→
(
(
¬
X28
)
↔
(
X15
↔
X43
)
)
)
)
)
)
→
(
(
X48
↔
(
(
¬
X41
)
↔
X43
)
)
↔
(
(
¬
(
(
¬
X43
)
↔
(
X21
↔
X48
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X48
→
(
(
¬
X21
)
↔
(
X19
↔
X43
)
)
)
)
→
(
¬
(
(
¬
X43
)
↔
(
X41
→
(
¬
X21
)
→
X48
)
)
)
)
)
→
(
¬
(
X48
↔
(
X19
↔
X21
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
¬
(
X49
→
X39
)
)
→
(
(
¬
X46
)
↔
(
(
¬
X39
)
↔
(
(
¬
X10
)
↔
X47
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X39
↔
(
(
¬
X29
)
→
(
¬
X26
)
→
X47
)
)
)
↔
(
(
¬
X39
)
↔
(
(
¬
X26
)
↔
(
(
¬
X2
)
→
X47
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X47
→
(
¬
X46
)
→
X48
)
)
→
(
(
¬
X47
)
↔
(
X30
↔
X48
)
)
→
(
(
¬
(
(
¬
X46
)
→
(
(
¬
X30
)
↔
(
X29
→
X48
)
)
)
)
↔
(
¬
(
X48
↔
(
X47
→
(
X46
↔
(
X30
↔
False
)
)
)
)
)
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X30
)
↔
(
(
¬
X26
)
→
X46
)
)
)
)
)
)
)
→
(
¬
(
X48
↔
(
X47
→
(
(
¬
X33
)
↔
X46
)
)
)
)
)
)
)
)
→
(
(
X49
→
(
X48
↔
(
X34
↔
False
)
)
)
↔
(
(
¬
(
X49
→
X48
→
X36
)
)
→
(
¬
(
(
¬
(
X36
↔
(
(
¬
X8
)
↔
X49
)
)
)
→
(
X49
↔
(
X34
→
X48
)
)
)
)
)
)
→
(
¬
(
(
X49
↔
(
X41
→
X36
)
)
→
(
(
X46
↔
(
X31
→
X36
)
)
↔
(
¬
(
X36
→
(
(
¬
X34
)
↔
(
X31
→
X49
)
)
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X45
↔
(
X32
→
(
¬
X14
)
→
X38
)
)
)
→
(
¬
(
(
(
(
¬
(
(
¬
X14
)
→
(
¬
X9
)
→
X47
)
)
↔
(
(
X32
→
(
X14
↔
False
)
)
↔
(
(
¬
X40
)
↔
(
(
¬
X32
)
→
(
¬
X9
)
→
X47
)
)
)
)
→
(
X45
↔
(
X40
↔
(
X32
↔
X14
)
)
)
)
→
(
¬
(
X42
→
X40
→
(
X14
↔
X47
)
)
)
)
)
)
↔
(
(
(
X49
→
(
¬
X42
)
→
(
(
¬
X16
)
↔
X47
)
)
↔
(
¬
(
(
¬
X47
)
↔
(
X40
↔
X49
)
)
)
)
→
(
¬
(
(
¬
X40
)
→
(
¬
X16
)
→
X49
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X20
↔
(
(
¬
X12
)
→
X45
)
)
)
→
(
(
¬
X45
)
↔
(
X20
→
X11
→
X47
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X12
)
↔
(
(
¬
X0
)
↔
X47
)
)
)
)
)
)
↔
(
(
(
X23
→
X11
→
(
(
¬
X3
)
↔
X44
)
)
↔
(
(
¬
X12
)
→
X11
→
(
X3
↔
X49
)
)
)
↔
(
¬
(
(
¬
X23
)
↔
(
(
¬
X11
)
→
(
¬
X5
)
→
X49
)
)
)
)
)
)
→
(
(
(
(
¬
(
X35
→
X4
→
(
X1
↔
X17
)
)
)
↔
(
X24
→
(
(
¬
X13
)
↔
(
X1
→
X17
)
)
)
)
→
(
¬
(
X48
→
(
X24
↔
(
(
¬
X13
)
→
X17
)
)
)
)
)
↔
(
(
X48
→
X37
→
(
X25
↔
X47
)
)
→
(
¬
(
(
(
¬
(
X49
→
X45
→
X47
)
)
↔
(
(
¬
(
X37
→
(
X25
↔
(
X18
→
X47
)
)
)
)
↔
(
¬
(
X49
→
X45
→
X37
→
X25
→
False
)
)
)
)
→
(
¬
(
(
(
X37
↔
(
X27
↔
(
X17
↔
(
X13
↔
False
)
)
)
)
↔
(
¬
(
X37
→
X17
→
(
¬
X13
)
→
X47
)
)
)
→
(
¬
(
(
¬
X17
)
↔
(
X13
→
X37
)
)
)
→
(
(
¬
X37
)
↔
(
(
¬
X22
)
→
X45
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
f18de9...
and proposition id is
45f363...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMR1jnzdhW6J3AszC6fP8FR7Tzby3u3oa3C
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
¬
(
(
(
(
(
¬
(
X29
→
(
¬
X15
)
→
(
(
¬
X8
)
↔
X39
)
)
)
→
X47
→
X8
)
→
(
¬
(
X47
→
(
(
¬
X29
)
↔
X39
)
)
)
)
→
(
¬
(
(
(
(
¬
(
(
¬
X29
)
→
X34
)
)
↔
(
(
(
¬
X29
)
→
X43
)
→
(
¬
(
X37
↔
(
X35
↔
(
X34
↔
(
X29
→
False
)
)
)
)
)
)
)
→
(
(
(
¬
(
X47
→
X29
→
X39
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
(
¬
(
(
¬
X39
)
↔
(
(
¬
X29
)
↔
(
(
¬
X21
)
→
X47
)
)
)
)
↔
(
¬
(
X24
→
(
X21
↔
False
)
)
)
)
→
(
¬
(
X34
→
X29
→
(
X21
↔
X47
)
)
)
)
)
→
(
¬
(
X29
→
(
X21
↔
X47
)
)
)
)
)
→
(
(
¬
X39
)
↔
(
X34
→
X47
)
)
)
)
)
↔
(
¬
(
X47
→
X39
→
(
¬
X34
)
→
X43
)
)
)
→
(
¬
(
(
¬
X43
)
↔
(
(
¬
X39
)
↔
(
(
¬
X16
)
→
X47
)
)
)
)
)
→
(
(
¬
(
(
¬
X35
)
→
(
¬
X22
)
→
(
¬
X6
)
→
X43
)
)
↔
(
¬
(
(
¬
X23
)
→
(
X22
↔
X24
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
(
¬
X44
)
→
X43
→
(
X18
↔
X45
)
)
)
→
(
¬
(
(
X45
↔
(
(
¬
X34
)
→
(
¬
X10
)
→
X44
)
)
→
(
X43
↔
(
(
¬
X34
)
→
(
¬
X0
)
→
X44
)
)
→
(
X44
↔
(
X43
→
X45
)
)
)
)
)
)
→
(
(
X43
↔
(
(
¬
X25
)
↔
(
(
¬
X10
)
→
X32
)
)
)
↔
(
¬
(
X45
↔
(
(
¬
X25
)
→
(
(
¬
X10
)
↔
X43
)
)
)
)
)
)
↔
(
(
(
(
X18
↔
(
X9
→
(
(
¬
X3
)
↔
X37
)
)
)
↔
(
¬
(
X45
↔
(
X37
→
X3
)
)
)
)
↔
(
(
¬
X37
)
↔
(
X9
↔
X45
)
)
)
↔
(
¬
(
(
¬
(
(
X37
→
X32
→
X1
→
X13
)
→
(
¬
(
X37
↔
(
(
¬
X18
)
→
(
X13
↔
X32
)
)
)
)
)
)
→
X37
→
(
X32
↔
(
X1
→
False
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
(
¬
X7
)
↔
X28
)
)
→
(
(
X41
→
(
(
¬
X19
)
↔
X28
)
)
↔
(
X48
→
X19
→
X7
)
)
)
→
(
¬
(
(
X28
→
(
¬
X7
)
→
X30
)
→
(
¬
(
X47
→
(
X30
↔
(
(
¬
X19
)
↔
X28
)
)
)
)
)
)
)
→
(
¬
(
X40
↔
(
(
¬
X28
)
↔
(
(
¬
X5
)
↔
X49
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X44
→
(
X7
↔
(
(
¬
X4
)
↔
X40
)
)
)
)
↔
(
¬
(
X7
↔
(
(
¬
X4
)
→
X47
)
)
)
)
→
(
¬
(
X47
→
X27
→
X7
→
False
)
)
)
)
↔
(
X44
→
(
X40
↔
X47
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X33
↔
(
X14
↔
X49
)
)
)
→
X36
→
(
¬
(
X42
↔
(
(
¬
X33
)
→
(
X14
↔
X45
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X42
↔
(
X26
↔
X45
)
)
)
→
(
X45
↔
(
(
¬
X42
)
↔
(
X38
→
X49
)
)
)
)
→
X49
→
(
X41
↔
(
X26
→
False
)
)
)
)
)
)
→
(
(
(
¬
(
(
(
¬
X31
)
↔
(
(
¬
X17
)
↔
(
(
¬
X2
)
→
X45
)
)
)
→
(
¬
(
X49
→
(
¬
X31
)
→
X45
)
)
)
)
→
X46
→
X31
→
X2
→
X45
)
↔
(
(
¬
(
X49
→
(
X47
↔
(
X31
→
X20
)
)
)
)
→
(
(
(
¬
X47
)
→
(
(
¬
X46
)
↔
(
(
¬
X42
)
→
X49
)
)
)
↔
(
X49
↔
(
X42
↔
X31
)
)
)
→
(
X42
↔
(
(
¬
X12
)
↔
(
(
¬
X11
)
→
X46
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
X43
→
X35
)
)
↔
(
¬
(
X43
↔
(
X17
→
X6
→
X1
→
False
)
)
)
)
→
(
(
(
X35
→
X48
)
↔
(
¬
(
X49
→
X46
→
X0
→
X48
)
)
)
↔
(
(
(
¬
X46
)
→
(
(
¬
X17
)
↔
X49
)
)
↔
(
¬
(
X48
↔
(
X46
→
X6
)
)
)
)
)
→
(
¬
(
(
¬
X35
)
→
X16
→
(
X5
↔
X48
)
)
)
→
(
¬
(
X48
↔
(
X35
→
(
(
¬
X16
)
↔
X17
)
)
)
)
)
)
↔
(
(
(
(
X39
→
X33
→
(
¬
X8
)
→
X22
)
→
(
(
¬
X27
)
↔
(
(
¬
X8
)
→
X38
)
)
)
↔
(
(
(
(
¬
(
(
X43
↔
False
)
→
(
¬
(
(
¬
(
(
¬
X43
)
↔
(
(
¬
X38
)
↔
(
X28
→
X49
)
)
)
)
→
(
¬
(
(
¬
X43
)
↔
(
(
¬
X38
)
↔
(
(
¬
X14
)
↔
X49
)
)
)
)
)
)
)
)
↔
(
¬
(
X47
→
(
¬
X28
)
→
X14
→
X43
)
)
)
→
(
(
(
(
X49
→
(
X39
↔
(
X38
↔
X43
)
)
)
→
(
X44
↔
(
X39
↔
X38
)
)
)
↔
(
X44
↔
(
X38
↔
False
)
)
)
↔
(
X44
↔
(
X43
→
(
(
¬
X19
)
↔
X38
)
)
)
)
)
→
(
X47
→
X43
→
(
¬
X33
)
→
X44
)
→
(
¬
(
(
¬
(
X44
↔
(
X43
↔
(
(
¬
X25
)
→
X45
)
)
)
)
→
(
¬
(
(
X43
↔
(
X33
→
X25
→
X31
)
)
→
(
¬
(
X44
→
X43
→
(
X25
↔
X31
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X34
→
X26
)
)
→
(
(
¬
(
X34
↔
(
X28
↔
(
X7
↔
X49
)
)
)
)
↔
(
X49
↔
(
(
¬
X28
)
↔
X46
)
)
)
)
)
↔
(
¬
(
X49
↔
(
X34
→
X21
→
X28
)
)
)
)
↔
(
(
(
(
X49
→
(
X34
↔
(
X12
→
(
X11
↔
False
)
)
)
)
→
(
¬
(
X49
→
(
X43
↔
(
X12
→
X11
)
)
)
)
)
→
(
(
¬
(
X49
→
X26
→
(
¬
X15
)
→
X43
)
)
↔
(
¬
(
(
¬
X43
)
↔
(
X26
↔
(
(
¬
X11
)
↔
X49
)
)
)
)
)
)
↔
(
¬
(
(
¬
X43
)
→
(
¬
X15
)
→
(
X9
↔
X49
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
X49
↔
(
X47
→
X41
→
X2
)
)
↔
(
(
X49
→
X42
)
↔
(
X47
→
X42
→
X41
→
(
X32
↔
False
)
)
)
)
↔
(
(
¬
(
X41
↔
(
X32
↔
X37
)
)
)
↔
(
¬
(
X37
→
(
(
¬
X18
)
↔
X49
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X29
)
→
X20
→
X18
→
X42
)
)
→
(
(
¬
X42
)
↔
(
X29
↔
(
X18
↔
X49
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
(
¬
(
X36
↔
(
(
¬
X29
)
→
X42
)
)
)
↔
(
(
¬
(
(
¬
X40
)
→
X36
→
X24
→
X42
)
)
↔
(
(
X40
→
(
(
¬
X24
)
↔
(
X10
↔
X29
)
)
)
→
(
X40
↔
(
X36
→
(
X29
↔
(
X10
↔
False
)
)
)
)
)
)
)
→
X42
→
(
X24
↔
(
X4
→
X40
)
)
)
)
→
(
(
¬
(
(
X42
→
X30
→
(
X23
↔
X36
)
)
→
(
X40
↔
(
X30
→
X23
)
)
)
)
↔
(
X42
↔
(
X40
↔
(
(
¬
X13
)
→
X30
)
)
)
)
)
)
→
(
¬
(
X43
↔
(
X32
↔
(
X3
↔
X42
)
)
)
)
→
(
(
¬
X40
)
↔
(
X36
→
X42
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
c19a8e...
and proposition id is
a12e50...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMKeSLta436SeFHWHuXfmEDGVnmyc6679YJ
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
¬
(
(
¬
(
(
¬
(
X44
→
X38
→
(
(
¬
X37
)
↔
X47
)
)
)
→
(
¬
(
(
¬
(
(
¬
X39
)
↔
(
(
¬
X38
)
→
(
X22
↔
X47
)
)
)
)
→
(
(
¬
X47
)
↔
(
(
¬
X22
)
↔
X48
)
)
)
)
)
)
→
(
(
¬
(
X23
→
X9
→
False
)
)
↔
(
(
¬
(
X48
→
X34
→
X25
)
)
↔
(
(
¬
(
X37
→
X23
→
X47
)
)
↔
(
¬
(
X48
→
X47
→
X37
→
X25
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X44
→
(
X20
↔
X39
)
)
)
→
(
(
(
¬
(
(
(
¬
(
(
¬
X15
)
→
X36
)
)
→
(
¬
(
(
¬
X31
)
→
X20
→
X49
)
)
)
→
(
X49
↔
(
(
¬
X31
)
↔
X36
)
)
)
)
↔
(
¬
(
X36
↔
(
(
¬
X20
)
→
X49
)
)
)
)
↔
(
X39
↔
X49
)
)
)
)
↔
(
¬
(
(
(
¬
(
X35
→
(
(
¬
X31
)
↔
X36
)
)
)
↔
(
(
¬
(
X49
↔
(
X35
→
(
¬
X31
)
→
X33
)
)
)
↔
(
¬
(
(
¬
(
X49
→
(
X31
↔
(
X14
↔
X28
)
)
)
)
→
X49
→
X35
→
X14
)
)
)
)
→
(
¬
(
X49
↔
(
X33
→
(
(
¬
X19
)
↔
X31
)
)
)
)
)
)
)
→
(
(
¬
(
X47
→
(
X39
↔
(
X18
↔
X10
)
)
)
)
↔
(
¬
(
(
(
¬
X33
)
↔
(
X29
→
X35
)
)
→
(
¬
(
(
(
¬
(
X39
↔
(
X33
↔
X29
)
)
)
↔
(
¬
(
X35
↔
(
X10
→
(
(
¬
X8
)
↔
X29
)
)
)
)
)
→
X47
→
(
X35
↔
(
X33
→
X10
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
(
(
(
¬
(
X43
→
(
(
¬
X21
)
↔
(
(
¬
X16
)
→
X41
)
)
)
)
↔
(
¬
(
X43
→
X41
→
(
(
¬
X11
)
↔
X16
)
)
)
)
→
(
¬
(
X43
↔
(
X41
↔
X16
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
(
¬
(
X21
↔
(
X11
↔
(
X2
↔
X47
)
)
)
)
↔
(
¬
(
(
¬
X21
)
→
(
¬
X11
)
→
X4
→
X47
)
)
)
→
X21
→
False
)
)
↔
X4
)
↔
(
¬
(
X21
↔
(
X16
↔
X11
)
)
)
)
→
(
¬
(
(
¬
X21
)
→
(
X11
↔
X47
)
)
)
)
)
)
→
X47
→
X41
→
(
X16
↔
False
)
)
)
→
(
¬
X43
)
→
X40
→
(
(
¬
X16
)
↔
X47
)
)
)
→
(
¬
(
(
(
(
X43
↔
(
X42
→
X41
→
X17
)
)
↔
(
X45
↔
(
X44
→
X43
→
X17
)
)
)
↔
(
(
¬
X41
)
→
X44
)
)
→
(
¬
(
(
¬
(
X42
→
(
(
¬
X41
)
↔
(
(
¬
X39
)
→
X43
)
)
)
)
→
(
(
¬
(
(
X47
↔
(
X43
→
X42
→
X41
)
)
→
(
¬
(
(
¬
X43
)
↔
(
X41
↔
(
(
¬
X30
)
→
X47
)
)
)
)
)
)
↔
(
¬
(
(
¬
X43
)
↔
(
(
¬
X41
)
→
X47
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
X26
)
↔
(
X13
→
(
¬
X0
)
→
X27
)
)
↔
(
¬
(
(
¬
(
X43
→
X27
→
X13
→
X45
)
)
→
(
¬
(
(
¬
X43
)
↔
(
(
¬
X13
)
↔
(
(
¬
X0
)
↔
X45
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X26
)
↔
(
(
¬
X24
)
→
(
X12
↔
X46
)
)
)
)
→
(
(
X45
↔
(
X44
→
(
X12
↔
X3
)
)
)
↔
(
X45
→
(
X24
↔
(
X7
→
X46
)
)
)
)
)
↔
(
(
(
(
¬
(
X44
→
(
X43
↔
False
)
)
)
↔
(
X46
→
(
(
¬
X44
)
↔
(
X5
↔
X45
)
)
)
)
↔
(
¬
(
(
¬
X32
)
→
(
¬
X6
)
→
X5
→
X46
)
)
)
→
(
¬
(
X43
→
X1
→
X45
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
(
¬
(
(
¬
X38
)
→
X31
→
(
¬
X8
)
→
X47
)
)
↔
(
¬
(
X31
→
X8
→
(
X3
↔
X38
)
)
)
)
→
X38
→
(
X31
↔
(
X8
↔
(
X1
→
False
)
)
)
)
)
→
(
¬
(
(
X38
→
(
X31
↔
(
(
¬
X10
)
↔
X30
)
)
)
→
(
(
X10
→
X49
)
↔
(
¬
(
X38
↔
(
X31
→
(
X10
↔
(
X8
↔
False
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X49
→
False
)
)
→
(
¬
(
(
¬
X38
)
→
(
X31
↔
X46
)
)
)
→
(
X49
↔
(
X46
→
X2
→
X31
)
)
)
)
)
→
(
(
(
(
¬
X42
)
↔
(
X38
↔
(
(
¬
X33
)
↔
X46
)
)
)
→
(
(
¬
X33
)
↔
(
X27
↔
(
(
¬
X24
)
↔
X46
)
)
)
)
↔
(
(
(
¬
(
X17
→
(
¬
X5
)
→
X38
)
)
↔
(
¬
(
(
X41
↔
(
X32
↔
(
X15
→
X42
)
)
)
→
(
X42
↔
(
X32
↔
(
(
¬
X5
)
→
X41
)
)
)
)
)
)
→
(
(
¬
X38
)
→
(
X23
↔
(
(
¬
X9
)
→
X41
)
)
)
→
(
¬
(
X46
→
(
¬
X23
)
→
(
(
¬
X9
)
↔
X32
)
)
)
)
)
→
(
(
(
¬
(
(
(
X48
→
(
¬
X46
)
→
(
(
¬
X26
)
↔
X47
)
)
→
(
¬
(
X48
→
(
(
¬
X42
)
↔
X46
)
)
)
)
→
(
¬
(
X48
→
X47
→
(
X41
↔
X46
)
)
)
)
)
↔
(
X46
→
X41
)
)
↔
(
(
¬
(
X42
↔
(
X18
→
X48
)
)
)
→
(
¬
(
X47
→
(
¬
X20
)
→
X18
→
X42
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
(
¬
(
X45
→
(
X21
↔
(
X19
↔
False
)
)
)
)
↔
(
(
(
¬
X44
)
↔
(
(
¬
X39
)
↔
X45
)
)
↔
(
(
(
¬
X44
)
↔
(
X39
↔
(
X19
↔
X45
)
)
)
→
X39
→
(
(
¬
X21
)
↔
(
X19
↔
X44
)
)
)
)
)
→
(
(
X45
↔
(
X29
→
X35
)
)
→
(
¬
(
(
X45
→
X22
→
X29
)
→
(
(
¬
X35
)
↔
(
X29
↔
(
(
¬
X22
)
↔
X44
)
)
)
)
)
)
→
(
(
X45
↔
(
(
¬
X21
)
→
(
(
¬
X0
)
↔
X35
)
)
)
↔
(
¬
(
(
¬
(
X45
↔
(
X34
↔
(
X21
→
X35
)
)
)
)
→
(
¬
(
X45
→
X44
)
)
)
)
)
)
)
→
(
(
(
¬
X44
)
→
(
(
¬
X39
)
↔
(
X34
→
X49
)
)
)
↔
(
(
(
¬
X40
)
→
X34
→
X49
)
→
(
¬
X34
)
→
(
¬
X11
)
→
X44
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X47
)
→
X40
→
(
(
¬
X6
)
↔
X48
)
)
)
↔
(
X48
→
X47
→
(
¬
X6
)
→
X28
)
)
↔
(
(
¬
(
X48
↔
(
X40
↔
(
(
¬
X4
)
↔
X47
)
)
)
)
↔
(
(
(
¬
(
X48
→
X47
→
X40
)
)
→
(
(
X40
→
X48
)
↔
(
¬
(
X47
↔
(
(
¬
X45
)
→
X16
→
X48
)
)
)
)
)
→
(
¬
(
(
¬
X40
)
→
(
X16
↔
X48
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
(
(
¬
(
X44
↔
(
(
¬
X40
)
→
(
(
¬
X14
)
↔
X45
)
)
)
)
→
(
¬
(
(
(
¬
(
X37
↔
(
X7
↔
X43
)
)
)
↔
(
¬
(
X45
↔
X44
)
)
)
→
X43
→
(
X40
↔
X37
)
)
)
)
)
→
(
X45
↔
X43
)
)
→
(
(
(
X44
→
(
(
¬
X37
)
↔
X47
)
)
↔
(
¬
(
(
X45
→
X37
→
(
(
¬
X36
)
↔
X44
)
)
→
(
¬
(
(
¬
(
(
¬
X45
)
↔
(
X37
↔
(
X25
→
X47
)
)
)
)
→
(
X47
↔
(
(
¬
X44
)
→
X25
→
X45
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
X37
)
→
X36
→
(
(
¬
X12
)
↔
X47
)
)
)
)
)
)
→
(
¬
(
X44
→
(
X40
↔
(
X13
→
X47
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
2b353b...
and proposition id is
9de08a...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMPxJTFNtdBqfTFdzF47Vo9fD4TBUKCHkWe
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
(
(
(
(
¬
(
X45
→
(
¬
X19
)
→
(
X9
↔
X42
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
X42
)
↔
(
(
¬
X33
)
↔
(
(
¬
X19
)
→
X45
)
)
)
→
(
(
¬
X44
)
↔
(
(
¬
X42
)
→
X33
→
X45
)
)
)
)
→
(
X42
↔
(
(
¬
X19
)
→
X3
→
X33
)
)
)
)
)
→
(
(
X44
↔
(
X42
→
X25
→
X38
)
)
↔
(
¬
(
(
X44
→
(
X42
↔
(
X25
→
False
)
)
)
→
(
X42
↔
(
X15
→
X45
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
(
¬
(
X47
↔
(
X42
↔
X44
)
)
)
↔
(
(
(
¬
X42
)
→
(
(
¬
X6
)
↔
X49
)
)
↔
(
X49
↔
(
X47
→
X44
→
X42
)
)
)
)
→
(
¬
(
X44
↔
(
X42
↔
X16
)
)
)
)
)
↔
(
(
¬
X44
)
↔
(
(
¬
X42
)
→
X47
)
)
)
↔
(
(
(
X14
↔
X47
)
↔
(
¬
(
X30
↔
(
X28
→
X42
)
)
)
)
↔
(
¬
(
(
X50
→
X42
→
(
(
¬
X11
)
↔
X47
)
)
→
(
¬
(
X50
↔
(
X47
→
(
(
¬
X28
)
↔
X49
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
X45
→
(
X44
↔
(
X4
↔
False
)
)
)
→
X50
→
(
X45
↔
X0
)
)
→
(
¬
(
X23
→
(
X0
↔
X27
)
)
)
)
↔
(
¬
(
X45
→
(
¬
X27
)
→
X0
→
X47
)
)
)
→
(
¬
(
X47
→
(
(
¬
X23
)
↔
(
X12
↔
X45
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
X48
↔
(
(
¬
X44
)
→
X50
)
)
↔
(
(
(
X48
↔
(
X29
↔
(
X18
→
X50
)
)
)
→
(
¬
(
X50
→
(
X48
↔
(
(
¬
X29
)
↔
X44
)
)
)
)
)
↔
(
X50
→
(
X29
↔
(
X18
→
False
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X48
↔
(
(
¬
X29
)
→
X24
→
X46
)
)
)
↔
(
X48
↔
(
X46
↔
(
X18
↔
False
)
)
)
)
→
(
¬
(
X48
↔
(
X29
↔
(
X18
↔
False
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
¬
(
X48
↔
(
X44
→
(
X26
↔
False
)
)
)
)
↔
(
X26
↔
(
(
¬
X10
)
↔
X17
)
)
)
↔
(
¬
(
X44
→
(
X26
↔
(
X8
↔
X10
)
)
)
)
)
→
(
(
X50
↔
(
X43
→
(
¬
X26
)
→
X48
)
)
↔
(
¬
(
(
(
X50
↔
(
X48
→
X43
)
)
↔
(
(
¬
(
(
¬
(
X43
↔
(
(
¬
X21
)
→
X36
)
)
)
→
(
¬
(
(
(
¬
(
X43
↔
(
X21
→
(
X10
↔
X48
)
)
)
)
↔
(
(
¬
X21
)
→
X36
)
)
→
(
¬
(
X48
↔
(
X36
→
(
X10
↔
False
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
X21
)
→
X50
)
)
)
)
→
X48
→
X21
→
(
X1
↔
False
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X48
)
→
(
X37
↔
(
(
¬
X34
)
↔
X50
)
)
)
)
→
(
X22
↔
(
(
¬
X5
)
→
X37
)
)
)
)
→
(
X50
↔
(
(
¬
X43
)
→
(
(
¬
X41
)
↔
X48
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X42
→
(
X36
↔
(
(
¬
X35
)
↔
X43
)
)
)
)
→
(
¬
(
(
(
(
X43
→
X36
→
(
(
¬
X32
)
↔
X42
)
)
→
(
¬
(
X43
→
(
X42
↔
(
X32
↔
(
X20
→
False
)
)
)
)
)
)
→
(
¬
(
X20
→
X42
)
)
)
→
(
¬
(
(
¬
X42
)
↔
(
(
¬
X20
)
↔
X43
)
)
)
)
)
)
↔
(
(
¬
(
(
X42
↔
(
X40
→
(
(
¬
X36
)
↔
X39
)
)
)
→
(
¬
X39
)
→
X36
→
X32
→
X42
)
)
↔
(
¬
(
(
¬
X32
)
↔
X40
)
)
)
)
↔
(
(
(
X31
→
(
(
¬
X7
)
↔
X43
)
)
↔
(
X50
↔
(
X43
→
(
¬
X7
)
→
X13
)
)
)
↔
(
(
(
X43
↔
(
X40
↔
(
(
¬
X2
)
↔
X39
)
)
)
→
(
¬
(
X40
→
(
¬
X39
)
→
(
¬
X31
)
→
X43
)
)
)
↔
(
X50
→
(
X40
↔
(
(
¬
X39
)
→
X43
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
(
(
X46
↔
(
(
¬
X12
)
→
(
X5
↔
X49
)
)
)
↔
(
¬
(
(
¬
X45
)
→
(
(
¬
X12
)
↔
(
(
¬
X5
)
↔
X46
)
)
)
)
)
→
(
¬
(
(
¬
X46
)
→
(
¬
X45
)
→
(
X14
↔
X49
)
)
)
)
→
(
X45
↔
(
X14
→
(
(
¬
X1
)
↔
X49
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
X45
→
(
¬
X41
)
→
(
¬
X35
)
→
X48
)
)
→
(
¬
X46
)
)
)
↔
(
(
¬
X46
)
→
X49
)
)
↔
(
(
¬
(
(
X21
↔
(
(
¬
X9
)
→
X43
)
)
→
(
¬
(
(
(
¬
(
(
¬
X27
)
↔
(
X21
↔
X43
)
)
)
↔
(
¬
(
(
¬
X46
)
↔
(
X43
→
X27
→
X49
)
)
)
)
→
(
¬
(
X43
→
(
(
¬
X21
)
↔
(
X9
→
X49
)
)
)
)
)
)
)
)
↔
(
¬
(
X46
→
(
(
¬
X45
)
↔
(
X21
↔
X49
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
(
(
¬
X42
)
→
(
¬
X30
)
→
X49
)
)
→
(
¬
(
(
¬
(
X50
→
(
¬
X43
)
→
X49
)
)
→
(
¬
X43
)
→
(
(
¬
(
X30
↔
(
(
¬
X28
)
↔
X49
)
)
)
↔
(
¬
(
(
X43
↔
(
(
¬
X42
)
→
X50
)
)
→
X49
→
(
X43
↔
(
(
¬
X28
)
→
X30
)
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X42
→
(
X6
↔
(
X4
↔
X45
)
)
)
)
↔
(
¬
(
(
¬
X30
)
→
(
X6
↔
X45
)
)
)
)
↔
(
¬
(
(
¬
X45
)
→
X30
→
X50
)
)
)
↔
(
(
¬
X30
)
→
(
X13
↔
(
X4
→
X45
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X45
→
X17
→
(
X2
↔
X49
)
)
)
→
(
¬
(
(
¬
(
X30
↔
(
(
¬
X8
)
↔
(
X2
→
X49
)
)
)
)
→
X45
→
X20
→
X49
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
(
¬
(
(
(
¬
(
(
¬
X45
)
↔
(
(
¬
X44
)
→
X33
→
X50
)
)
)
↔
(
¬
(
(
¬
X45
)
→
X44
→
(
(
¬
X36
)
↔
X50
)
)
)
)
→
(
X45
↔
X44
)
)
)
↔
(
(
¬
(
X44
→
(
(
¬
X39
)
↔
(
X31
→
X50
)
)
)
)
↔
(
¬
(
(
X46
→
X31
→
X36
)
→
(
(
¬
(
(
¬
X44
)
↔
(
X39
→
X46
)
)
)
↔
(
(
(
¬
(
(
¬
X39
)
↔
(
X31
↔
X44
)
)
)
↔
(
¬
(
(
¬
X44
)
→
(
X36
↔
(
(
¬
X31
)
↔
X46
)
)
)
)
)
→
(
¬
(
(
¬
X44
)
→
X46
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
X47
)
→
X45
→
(
¬
X44
)
→
X50
)
)
)
)
↔
(
(
(
X45
↔
(
X40
↔
(
(
¬
X39
)
↔
X47
)
)
)
↔
(
X45
↔
(
X40
↔
(
X38
↔
X39
)
)
)
)
↔
(
X47
↔
(
X38
→
X45
)
)
)
)
→
(
(
(
(
¬
(
(
¬
X46
)
→
(
¬
X32
)
→
(
X29
↔
X50
)
)
)
↔
(
(
¬
X32
)
→
(
¬
X29
)
→
(
¬
X7
)
→
X46
)
)
→
X32
→
(
X29
↔
(
X7
→
X44
)
)
)
↔
(
X50
→
(
X46
↔
(
(
¬
X32
)
→
X37
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X34
)
↔
(
X22
→
X10
→
X40
)
)
)
↔
(
(
(
¬
(
X45
→
(
X16
↔
False
)
)
)
↔
(
(
X34
→
(
X25
↔
(
X15
→
X16
)
)
)
↔
(
¬
(
X45
→
X25
→
X16
→
X34
)
)
)
)
↔
(
X34
→
(
¬
X25
)
→
(
(
¬
X22
)
↔
X40
)
)
)
)
↔
(
(
(
¬
(
X45
→
X34
→
(
X24
↔
X40
)
)
)
↔
(
(
¬
(
(
¬
X40
)
↔
(
X23
↔
(
X3
↔
X45
)
)
)
)
↔
(
(
(
¬
X24
)
→
(
(
¬
X23
)
↔
(
(
¬
X3
)
↔
X40
)
)
)
↔
(
¬
(
X40
↔
(
X24
↔
(
X23
↔
(
X18
↔
False
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X34
→
(
X24
↔
X47
)
)
)
→
(
¬
(
(
X24
→
(
X19
↔
(
X0
→
X34
)
)
)
→
(
¬
(
(
¬
(
X45
→
X34
→
(
X26
↔
False
)
)
)
→
(
¬
(
(
(
¬
(
X47
→
X34
→
X26
→
(
X11
↔
False
)
)
)
↔
(
¬
(
X26
→
(
X24
↔
(
(
¬
X11
)
↔
X47
)
)
)
)
)
→
X34
→
(
¬
X24
)
→
(
X19
↔
X47
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
a2bdcd...
and proposition id is
fd5496...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMRoSfbhqz8ArzERE873DEequ7vBBTLXtwC
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
(
¬
(
X44
↔
(
(
¬
X32
)
→
X10
→
X40
)
)
)
→
X37
→
(
¬
X10
)
→
X32
)
↔
(
(
¬
(
(
¬
(
X49
↔
(
X40
→
(
X11
↔
X25
)
)
)
)
→
(
X40
→
X37
→
(
¬
X25
)
→
X35
)
→
(
¬
(
X49
→
(
X40
↔
(
X37
→
X25
→
False
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
X49
→
(
X32
↔
X40
)
)
)
→
(
¬
(
X49
↔
(
X31
→
X21
→
X40
)
)
)
)
)
→
(
X31
↔
(
X21
↔
(
X6
↔
X49
)
)
)
)
)
↔
(
(
(
¬
X31
)
↔
(
X21
↔
X40
)
)
↔
(
X37
→
(
(
¬
X21
)
↔
(
X16
↔
X31
)
)
)
)
)
)
)
↔
(
(
(
(
X48
↔
(
(
¬
X5
)
↔
X15
)
)
↔
(
(
(
X48
→
(
X36
↔
(
X15
→
X49
)
)
)
→
X36
→
X19
→
X48
)
↔
(
¬
(
X49
→
(
X48
↔
X19
)
)
)
)
)
→
(
(
(
(
¬
X1
)
↔
X48
)
↔
(
¬
(
(
¬
(
X47
↔
(
(
¬
X36
)
→
X49
)
)
)
→
(
¬
(
X48
↔
(
X47
→
X1
→
X36
)
)
)
)
)
)
↔
(
¬
(
X48
→
(
(
¬
X1
)
↔
X49
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X47
)
→
(
X44
↔
(
(
¬
X42
)
→
X48
)
)
)
)
→
(
¬
(
X47
↔
(
X44
→
(
¬
X38
)
→
X42
)
)
)
)
)
→
(
(
(
X49
→
(
¬
X40
)
→
(
(
¬
X13
)
↔
X48
)
)
↔
(
(
¬
(
(
¬
X40
)
→
(
(
¬
X39
)
↔
(
X38
→
X49
)
)
)
)
→
(
¬
(
X40
↔
(
X39
↔
X48
)
)
)
)
)
↔
(
(
X49
↔
(
X39
↔
X33
)
)
↔
(
(
¬
(
X49
→
X42
→
X40
→
X39
→
False
)
)
↔
(
¬
(
X40
↔
(
(
¬
X39
)
↔
(
(
¬
X33
)
↔
X49
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
X47
→
(
X45
↔
(
X22
↔
X34
)
)
)
→
(
(
X34
↔
(
X27
→
X14
)
)
↔
(
¬
(
(
¬
(
(
X20
→
X14
→
X45
)
→
(
¬
(
X34
→
(
X20
↔
(
X14
→
X47
)
)
)
)
)
)
→
X47
→
(
¬
X20
)
→
X45
)
)
)
)
)
→
(
¬
(
(
(
¬
X27
)
→
(
¬
X23
)
→
(
X2
↔
X45
)
)
→
(
¬
(
X34
↔
(
X27
→
(
(
¬
X2
)
↔
X23
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
X46
→
(
¬
X34
)
→
X49
)
)
→
(
¬
(
(
X49
→
X46
→
(
X8
↔
X34
)
)
→
X49
→
X34
→
(
X12
↔
X46
)
)
)
)
)
→
(
(
(
X49
→
X17
→
X46
)
↔
(
X49
↔
(
X17
→
X4
→
X34
)
)
)
↔
(
X49
→
(
X46
↔
(
X34
→
False
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X45
↔
(
X34
↔
(
(
¬
X27
)
↔
X46
)
)
)
)
→
(
(
(
¬
X18
)
→
X49
)
↔
(
¬
(
(
(
(
¬
X46
)
↔
(
(
¬
X45
)
↔
X49
)
)
↔
(
(
X46
↔
(
X18
→
X49
)
)
↔
(
¬
(
X45
↔
(
X34
→
X49
)
)
)
)
)
→
X45
→
(
X34
↔
X49
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X49
↔
(
X41
↔
(
X30
↔
X43
)
)
)
)
→
(
(
¬
(
X46
→
X7
→
X48
)
)
↔
(
(
¬
(
X48
↔
(
X46
→
(
¬
X30
)
→
X49
)
)
)
↔
(
¬
(
X49
→
X46
→
(
X7
↔
X48
)
)
)
)
)
→
X7
→
X30
)
↔
(
¬
(
X48
↔
(
(
¬
X46
)
↔
(
X45
↔
X49
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X45
→
(
¬
X26
)
→
X46
)
)
→
(
¬
(
(
X46
↔
(
X45
→
(
X0
↔
False
)
)
)
→
(
¬
(
X45
↔
(
X26
↔
(
X24
→
X0
→
False
)
)
)
)
)
)
→
(
¬
X45
)
→
(
¬
X26
)
→
X46
)
)
→
X46
→
X24
→
(
¬
X3
)
→
X45
)
↔
(
(
(
(
¬
X29
)
→
(
¬
X28
)
→
(
¬
X26
)
→
X46
)
→
(
¬
(
(
(
X46
↔
(
X29
↔
(
X24
↔
X9
)
)
)
↔
(
¬
(
(
¬
X28
)
↔
(
(
¬
X24
)
→
X46
)
)
)
)
→
X46
→
X29
→
False
)
)
)
↔
(
¬
(
(
¬
X46
)
↔
(
(
¬
X28
)
→
X26
→
X48
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
¬
(
X33
→
X23
→
X29
)
)
→
(
¬
(
X41
↔
(
(
¬
X33
)
→
(
X23
↔
X36
)
)
)
)
)
)
↔
(
(
(
¬
X36
)
→
(
(
¬
X23
)
↔
(
(
¬
X16
)
→
X41
)
)
)
↔
(
¬
(
X41
↔
(
X37
→
X7
)
)
)
)
)
→
(
(
(
¬
(
(
(
¬
X23
)
→
X14
→
(
X1
↔
X36
)
)
→
X36
→
(
¬
X14
)
→
(
¬
X10
)
→
X23
)
)
↔
(
¬
(
X36
→
(
X23
↔
X14
)
)
)
)
↔
(
(
X14
→
(
(
¬
X12
)
↔
(
(
¬
X10
)
↔
X23
)
)
)
↔
(
(
(
¬
X37
)
→
(
¬
X12
)
→
(
(
¬
X10
)
↔
X45
)
)
→
(
¬
(
(
¬
X37
)
→
(
(
¬
X23
)
↔
(
X12
↔
X45
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
X44
↔
(
X43
→
(
X42
↔
X18
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
X44
)
↔
(
X39
↔
(
X36
→
X46
)
)
)
→
(
X43
↔
(
(
¬
X39
)
→
X44
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X43
)
→
X42
→
X5
→
X46
)
)
→
(
¬
(
X5
→
X45
)
)
)
)
)
)
)
↔
(
X46
↔
(
(
¬
X42
)
→
(
(
¬
X41
)
↔
X44
)
)
)
)
→
(
(
¬
(
X48
↔
(
X46
→
X44
→
X17
)
)
)
↔
(
(
¬
(
X46
↔
(
(
¬
X44
)
→
X32
→
X48
)
)
)
→
(
¬
(
X32
→
(
(
¬
X25
)
↔
X46
)
)
)
→
(
¬
(
(
¬
(
(
¬
X39
)
↔
(
X32
↔
(
X25
↔
X48
)
)
)
)
→
X46
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
X49
→
X31
→
(
X0
↔
X3
)
)
↔
(
(
(
(
¬
(
(
¬
X15
)
↔
(
X3
↔
(
X0
↔
X35
)
)
)
)
↔
(
¬
(
(
¬
X35
)
→
(
(
¬
X3
)
↔
X47
)
)
)
)
↔
(
¬
(
(
¬
X47
)
→
X35
→
(
(
¬
X15
)
↔
X49
)
)
)
)
→
(
X49
↔
(
(
¬
X35
)
→
(
X15
↔
X38
)
)
)
)
)
→
(
(
¬
(
X49
↔
(
X31
↔
(
(
¬
X21
)
↔
X35
)
)
)
)
↔
(
¬
(
X49
↔
(
(
¬
X21
)
↔
(
X4
→
X35
)
)
)
)
)
→
(
X31
↔
(
X21
→
X19
→
X35
)
)
)
→
(
¬
(
(
(
(
¬
(
(
¬
X43
)
→
(
(
¬
X27
)
↔
(
X6
→
X48
)
)
)
)
↔
(
X48
→
(
¬
X20
)
→
X27
)
)
↔
(
(
(
X28
→
X20
→
X35
)
↔
(
¬
(
X49
↔
(
(
¬
X35
)
↔
(
(
¬
X28
)
→
X43
)
)
)
)
)
→
(
¬
(
X49
→
(
X35
↔
(
X27
→
X28
)
)
)
)
)
)
→
(
¬
(
(
(
(
X48
↔
(
(
¬
X42
)
↔
X43
)
)
↔
(
X49
→
(
(
¬
X42
)
↔
(
X22
→
X48
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
X24
→
X13
)
)
→
(
¬
(
X30
↔
(
X13
→
X9
→
X43
)
)
)
)
)
→
(
¬
(
X49
↔
(
X48
→
X13
→
X30
)
)
)
)
)
↔
(
¬
(
(
¬
X48
)
→
(
X43
↔
(
(
¬
X2
)
→
X49
)
)
)
)
)
)
→
(
(
(
X40
→
(
X38
↔
(
X26
→
X11
→
False
)
)
)
↔
(
¬
(
(
¬
X40
)
→
(
¬
X38
)
→
(
X11
↔
X48
)
)
)
)
↔
(
X48
→
(
(
¬
X8
)
↔
X43
)
)
)
→
(
(
(
X49
↔
(
(
¬
X34
)
→
X40
)
)
↔
(
X40
→
(
¬
X38
)
→
X48
)
)
↔
(
¬
X49
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
9be796...
and proposition id is
a52617...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMJtp5w9SCxhQgpY7qrGYKza5Ej1Ci8A4uD
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
¬
(
(
(
(
¬
(
(
(
(
X43
→
(
X42
↔
X3
)
)
↔
(
¬
(
X47
↔
(
X43
↔
(
X38
↔
X3
)
)
)
)
)
→
(
¬
(
X43
→
(
(
¬
X38
)
↔
X47
)
)
)
)
→
(
¬
(
X45
↔
(
(
¬
X42
)
→
(
¬
X3
)
→
X43
)
)
)
)
)
→
(
¬
(
(
¬
(
X43
↔
(
(
¬
X16
)
→
(
(
¬
X1
)
↔
X45
)
)
)
)
→
(
(
(
¬
(
(
¬
(
X43
→
(
¬
X37
)
→
(
(
¬
X30
)
↔
X40
)
)
)
→
(
¬
(
(
¬
X42
)
↔
(
X40
→
(
(
¬
X37
)
↔
X43
)
)
)
)
)
)
→
(
¬
X40
)
→
X16
→
X45
)
↔
(
¬
(
(
¬
X43
)
↔
(
(
¬
X42
)
↔
(
(
¬
X6
)
→
X45
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X45
↔
(
X34
→
(
X27
↔
(
X11
→
False
)
)
)
)
)
→
(
(
(
¬
(
X45
→
(
X27
↔
X42
)
)
)
→
(
X45
↔
(
(
¬
X27
)
↔
(
X2
→
X34
)
)
)
)
↔
(
X42
↔
(
X34
↔
X2
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X37
)
↔
(
(
¬
X8
)
↔
X45
)
)
)
→
(
¬
(
X45
→
X37
→
(
X27
↔
X42
)
)
)
)
)
)
→
(
(
X42
→
(
(
¬
X27
)
↔
(
(
¬
X21
)
↔
X45
)
)
)
↔
(
(
X45
→
(
¬
X27
)
→
(
X4
↔
X42
)
)
↔
(
¬
(
(
¬
X42
)
→
(
¬
X19
)
→
(
¬
X4
)
→
X45
)
)
)
)
→
(
¬
(
(
¬
(
X45
↔
(
X42
→
X19
→
X35
)
)
)
→
(
¬
(
(
¬
(
X19
↔
X46
)
)
→
(
(
X35
→
(
(
¬
X19
)
↔
(
(
¬
X9
)
↔
X45
)
)
)
↔
(
(
¬
X45
)
→
(
¬
X35
)
→
X46
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
(
(
X31
→
X24
→
(
X14
↔
X32
)
)
↔
(
¬
(
X31
→
X24
→
(
X18
↔
False
)
)
)
)
→
(
(
(
X31
↔
(
X24
↔
X23
)
)
→
(
¬
(
(
(
(
X32
↔
(
(
¬
X23
)
→
X31
)
)
↔
(
X32
→
(
¬
X24
)
→
(
X23
↔
X29
)
)
)
→
(
¬
X29
)
→
X32
)
→
(
(
¬
X31
)
↔
(
(
¬
X23
)
↔
X32
)
)
)
)
)
↔
(
¬
(
X32
↔
(
(
¬
X24
)
→
X29
)
)
)
)
)
)
↔
(
(
(
(
¬
X29
)
→
(
¬
X28
)
→
(
X17
↔
X45
)
)
↔
(
¬
(
(
¬
X32
)
→
(
X29
↔
(
(
¬
X28
)
↔
X45
)
)
)
)
)
↔
(
(
X45
→
False
)
↔
(
(
X29
↔
(
X28
→
(
X24
↔
X45
)
)
)
↔
(
X29
→
(
X24
↔
(
(
¬
X0
)
↔
X45
)
)
)
)
)
)
)
→
(
(
(
¬
(
X44
→
(
¬
X32
)
→
(
¬
X29
)
→
X49
)
)
→
(
(
X31
↔
(
X7
↔
X32
)
)
↔
(
¬
(
(
¬
(
(
(
¬
X36
)
↔
(
X31
→
(
X7
↔
X49
)
)
)
→
(
X49
↔
(
X31
→
X7
→
X32
)
)
)
)
→
(
(
X49
↔
(
(
¬
X29
)
→
(
(
¬
X13
)
↔
X36
)
)
)
↔
(
X36
↔
(
(
¬
X29
)
→
(
¬
X13
)
→
X31
)
)
)
)
)
)
)
↔
(
(
X32
→
(
X10
↔
False
)
)
→
(
¬
(
(
X44
↔
(
X31
→
False
)
)
→
(
(
X44
→
X32
→
X10
→
False
)
→
(
¬
(
(
¬
(
(
¬
X33
)
→
X32
→
(
(
¬
X10
)
↔
X44
)
)
)
→
X33
→
(
X10
↔
(
(
¬
X5
)
↔
X32
)
)
)
)
)
→
(
¬
(
X44
→
(
¬
X32
)
→
(
X31
↔
X33
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
X42
→
(
X39
↔
(
X31
→
X32
)
)
)
→
(
¬
(
X42
↔
(
X31
↔
(
(
¬
X12
)
→
X32
)
)
)
)
)
→
(
(
(
¬
X32
)
↔
(
X31
→
(
(
¬
X26
)
↔
X41
)
)
)
↔
(
¬
(
X41
↔
(
X32
↔
(
X12
↔
False
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
X33
)
↔
(
(
¬
X31
)
↔
(
(
¬
X20
)
↔
X41
)
)
)
)
↔
(
¬
(
(
¬
X12
)
↔
X31
)
)
)
)
↔
(
(
(
(
¬
(
X39
↔
(
(
¬
X25
)
→
(
X15
↔
X46
)
)
)
)
→
(
X48
↔
(
X46
→
X39
→
False
)
)
)
↔
(
(
X48
→
X46
→
X39
→
X31
)
→
(
¬
(
X39
→
(
(
¬
X31
)
↔
(
(
¬
X22
)
↔
X46
)
)
)
)
)
)
→
(
¬
(
X48
→
X41
→
(
¬
X31
)
→
X46
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
(
(
¬
(
X34
→
X21
→
(
X0
↔
False
)
)
)
↔
(
(
X34
↔
(
X0
↔
X44
)
)
↔
(
X49
↔
(
X21
→
X44
)
)
)
)
→
(
(
X49
↔
(
X47
↔
(
X21
↔
X25
)
)
)
↔
(
X49
↔
(
X44
↔
(
X25
↔
X17
)
)
)
)
)
↔
(
¬
(
X21
↔
(
X16
↔
(
X0
→
X44
)
)
)
)
)
→
(
(
(
¬
X12
)
↔
(
(
(
¬
X19
)
→
X12
→
X47
)
↔
(
¬
(
X25
→
X12
→
(
X6
↔
X47
)
)
)
)
)
↔
(
(
¬
(
(
¬
X46
)
↔
(
X44
→
(
(
¬
X25
)
↔
X47
)
)
)
)
↔
(
¬
(
(
¬
(
X46
↔
(
(
¬
X2
)
→
X47
)
)
)
→
(
X47
↔
(
X46
↔
(
X44
→
X19
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
X46
)
→
(
¬
X32
)
→
(
X14
↔
X48
)
)
→
(
(
¬
X48
)
↔
(
X32
→
X14
→
X49
)
)
)
)
→
(
(
X49
↔
(
X48
↔
(
(
¬
X22
)
→
X46
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
¬
X44
)
↔
(
X15
↔
(
X13
↔
X49
)
)
)
→
X49
→
(
X48
↔
X13
)
)
→
(
¬
X48
)
)
)
→
(
X44
↔
(
(
¬
X22
)
→
(
¬
X13
)
→
X49
)
)
)
)
)
)
→
(
(
X43
→
X40
→
X36
)
↔
(
¬
(
(
(
(
¬
X37
)
↔
(
X1
↔
X41
)
)
↔
(
X43
↔
(
X36
→
(
(
¬
X1
)
↔
X41
)
)
)
)
→
(
¬
(
X40
↔
(
(
¬
X36
)
↔
X44
)
)
)
)
)
)
→
(
(
¬
(
X44
→
X43
→
X46
)
)
↔
(
(
X48
↔
(
X43
↔
(
(
¬
X40
)
→
X44
)
)
)
→
(
(
¬
(
X43
↔
(
X40
↔
X44
)
)
)
↔
(
(
¬
(
X44
→
(
X40
↔
(
X11
↔
X43
)
)
)
)
→
(
X44
↔
(
X43
↔
(
(
¬
X24
)
→
X40
)
)
)
)
)
)
)
)
)
→
(
(
(
(
X49
↔
(
X48
↔
(
X45
→
X39
→
False
)
)
)
↔
(
X45
↔
(
(
¬
X39
)
↔
(
(
¬
X5
)
↔
X49
)
)
)
)
↔
(
(
(
(
X49
→
X47
→
X45
→
False
)
→
(
¬
(
(
X42
→
X10
→
X45
)
→
(
¬
(
(
(
X49
↔
(
X45
↔
(
(
¬
X10
)
→
X42
)
)
)
↔
(
¬
(
X49
↔
(
X45
→
X23
→
X42
)
)
)
)
→
(
¬
(
X45
→
(
¬
X42
)
→
(
(
¬
X23
)
↔
X47
)
)
)
)
)
)
)
)
↔
(
X18
↔
(
(
¬
X10
)
→
X42
)
)
)
↔
(
(
X42
↔
X7
)
↔
(
(
X49
↔
(
X45
↔
(
X7
↔
X48
)
)
)
→
(
X42
↔
X49
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
X49
→
X29
→
(
X26
↔
(
X20
↔
False
)
)
)
)
↔
(
¬
(
X38
↔
(
X29
↔
(
X20
→
X26
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X45
)
→
(
(
¬
X42
)
↔
(
(
¬
X27
)
→
X49
)
)
)
)
→
(
(
¬
(
(
¬
(
X49
→
(
X42
↔
(
X29
→
X28
)
)
)
)
→
(
¬
(
X42
↔
(
X29
→
X26
)
)
)
)
)
↔
(
(
X49
↔
(
(
¬
X26
)
→
X8
→
X42
)
)
→
(
¬
(
(
¬
X45
)
↔
(
X42
↔
(
(
¬
X8
)
→
X49
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
(
X42
→
(
X38
↔
(
X30
↔
False
)
)
)
↔
(
¬
(
X45
→
X38
→
(
X33
↔
X42
)
)
)
)
→
(
¬
(
X45
→
(
X33
↔
(
(
¬
X4
)
→
X42
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X38
↔
(
X35
↔
(
(
¬
X30
)
→
X45
)
)
)
)
→
X45
→
X35
→
(
¬
X31
)
→
X38
)
)
)
→
(
¬
(
(
¬
(
X31
→
X9
→
(
¬
X3
)
→
X33
)
)
→
(
X33
↔
(
(
¬
X3
)
→
X38
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
7180af...
and proposition id is
6b2b89...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMLboB79kDmqnVjEwMJsJx56cSLqsoEK5fd
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
∀X50 :
prop
,
(
(
(
(
(
(
(
¬
(
X35
→
(
X15
↔
(
(
¬
X12
)
↔
X31
)
)
)
)
↔
(
X35
↔
(
(
¬
X31
)
→
(
X12
↔
X48
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X27
)
→
X16
→
(
¬
X15
)
→
X35
)
)
→
(
X16
↔
X48
)
)
)
→
(
X35
↔
(
X15
→
False
)
)
)
)
↔
(
(
(
(
¬
(
X50
→
(
¬
X41
)
→
X47
)
)
↔
(
X47
↔
(
X41
→
(
X35
↔
X48
)
)
)
)
↔
(
¬
(
(
(
¬
X26
)
↔
X50
)
→
(
(
¬
(
X47
↔
(
(
¬
X26
)
↔
X48
)
)
)
↔
(
X48
→
(
(
¬
X47
)
↔
(
(
¬
X31
)
↔
X50
)
)
)
)
)
)
)
→
(
(
¬
(
X47
↔
(
(
¬
X28
)
↔
X41
)
)
)
↔
(
(
¬
(
X50
↔
(
X41
↔
X44
)
)
)
↔
(
X44
→
(
X41
↔
(
X28
→
X47
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X47
→
(
X45
↔
(
(
¬
X44
)
→
X50
)
)
)
)
↔
(
¬
(
(
(
¬
(
X50
↔
(
X45
→
X13
→
X32
)
)
)
↔
(
X50
↔
(
(
¬
X17
)
↔
(
X13
→
X32
)
)
)
)
→
(
¬
(
(
¬
X44
)
↔
(
(
¬
X17
)
→
X50
)
)
)
)
)
)
↔
(
¬
(
X47
→
X45
→
(
X43
↔
X38
)
)
)
)
↔
(
(
(
¬
(
(
X38
→
(
X30
↔
(
X14
→
False
)
)
)
→
(
X30
↔
(
(
¬
X8
)
→
X22
)
)
)
)
↔
(
(
¬
(
X44
↔
(
X38
→
(
¬
X21
)
→
X30
)
)
)
↔
(
X44
↔
(
X30
→
X22
→
X38
)
)
)
)
→
(
(
¬
(
(
¬
(
X32
→
(
(
¬
X30
)
↔
(
(
¬
X22
)
↔
X44
)
)
)
)
→
X22
→
X7
→
X30
)
)
↔
(
X44
→
X30
→
X22
→
X3
→
False
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X49
↔
(
X24
→
(
X20
↔
X40
)
)
)
)
↔
(
(
(
¬
(
(
¬
X49
)
→
(
(
¬
X42
)
↔
(
(
¬
X6
)
↔
X50
)
)
)
)
↔
(
¬
(
(
¬
X49
)
→
X42
→
X33
→
X50
)
)
)
↔
(
(
(
¬
X42
)
→
X24
→
X49
)
→
(
¬
(
(
¬
X49
)
↔
(
X42
↔
(
(
¬
X20
)
→
X50
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X49
↔
(
X37
↔
(
X24
→
X42
)
)
)
)
→
(
¬
(
(
(
¬
(
X49
↔
(
X25
→
X37
)
)
)
↔
(
(
¬
(
X47
→
X42
)
)
↔
(
(
(
X49
→
X25
→
(
X10
↔
X42
)
)
↔
(
(
¬
X25
)
↔
(
X10
→
X37
)
)
)
↔
(
¬
(
X37
→
(
X10
↔
False
)
)
)
)
)
)
→
(
(
¬
X47
)
↔
(
X42
↔
X49
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
(
¬
X46
)
→
(
X37
↔
(
(
¬
X2
)
↔
X49
)
)
)
)
→
(
(
X49
↔
(
X46
↔
(
X4
↔
X19
)
)
)
↔
(
¬
(
X49
→
X19
→
X37
)
)
)
)
→
(
(
X4
→
X49
)
↔
(
(
¬
(
X46
→
X45
→
(
X18
↔
X49
)
)
)
↔
(
¬
(
(
¬
X46
)
↔
(
X45
↔
(
X34
→
X49
)
)
)
)
)
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X45
)
→
(
¬
X40
)
→
X46
)
)
)
)
→
(
(
(
¬
(
(
¬
(
(
¬
X40
)
→
X39
→
X23
→
X45
)
)
→
(
¬
(
(
¬
X37
)
→
(
X23
↔
X40
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
(
¬
(
X40
↔
(
X39
→
(
X9
↔
X45
)
)
)
)
↔
(
¬
(
X45
↔
(
X39
↔
(
X5
↔
X40
)
)
)
)
)
↔
(
X39
↔
(
X5
→
X9
)
)
)
→
(
X45
↔
X5
)
)
)
→
(
¬
(
X40
→
(
¬
X5
)
→
X45
)
)
)
→
(
¬
(
(
¬
X40
)
→
(
(
¬
X39
)
↔
X45
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X29
→
X11
→
X40
)
)
↔
(
(
¬
X37
)
→
X36
→
(
¬
X29
)
→
X45
)
)
↔
(
X37
↔
(
X36
↔
(
X29
→
X1
)
)
)
)
↔
(
¬
(
(
¬
X37
)
↔
(
(
¬
X36
)
↔
(
(
¬
X0
)
→
X40
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
¬
(
(
¬
(
X49
↔
(
X45
↔
(
X40
↔
(
X39
↔
False
)
)
)
)
)
→
(
(
¬
(
(
X50
↔
(
X23
↔
(
X3
↔
X24
)
)
)
→
(
(
¬
(
(
¬
(
X49
↔
(
X3
→
X23
)
)
)
→
(
¬
(
X50
→
X39
)
)
)
)
↔
(
X50
↔
(
X39
↔
X49
)
)
)
)
)
↔
(
¬
(
X24
→
(
(
¬
X23
)
↔
X39
)
)
)
)
)
)
→
(
¬
(
X49
→
(
(
¬
X40
)
↔
(
(
¬
X38
)
→
X45
)
)
)
)
)
)
→
(
(
¬
X47
)
→
X24
→
X49
)
→
(
X47
→
(
X16
↔
False
)
)
→
(
¬
X45
)
→
X24
→
(
X21
↔
X49
)
)
)
→
(
(
(
(
¬
(
X32
↔
X30
)
)
→
(
(
False
→
(
(
X42
↔
(
(
¬
X37
)
→
X47
)
)
↔
(
¬
(
(
¬
(
X42
→
(
X37
↔
(
(
¬
X15
)
↔
X47
)
)
)
)
→
(
¬
(
X42
→
(
¬
X37
)
→
(
(
¬
X32
)
↔
X47
)
)
)
)
)
)
)
↔
(
X30
→
(
X15
↔
X32
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X42
→
X31
→
X49
)
)
↔
(
¬
(
X50
↔
(
X49
→
X9
→
X31
)
)
)
)
↔
(
¬
(
(
(
¬
X32
)
↔
(
(
¬
X31
)
→
X47
)
)
→
(
¬
(
X50
↔
(
X47
→
X9
→
X32
)
)
)
)
)
)
→
(
¬
(
X47
→
(
(
¬
X32
)
↔
X42
)
)
)
)
)
)
↔
(
(
(
(
(
¬
X46
)
→
(
¬
X29
)
→
(
(
¬
X13
)
↔
X49
)
)
→
(
¬
(
X49
↔
(
X29
→
(
(
¬
X7
)
↔
X46
)
)
)
)
)
→
X4
→
X29
)
↔
(
¬
(
(
¬
(
(
¬
X47
)
↔
(
(
¬
X46
)
→
(
X13
↔
X49
)
)
)
)
→
(
¬
(
X47
→
X46
→
(
¬
X13
)
→
X50
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
X49
↔
(
X19
↔
(
(
¬
X0
)
↔
X8
)
)
)
)
→
(
¬
(
(
X34
→
X0
→
X49
)
→
(
¬
(
(
¬
X34
)
↔
(
(
¬
X8
)
↔
(
(
¬
X1
)
→
X49
)
)
)
)
)
)
)
→
(
¬
(
X48
→
(
X35
↔
(
(
¬
X5
)
→
X8
)
)
)
)
)
→
(
(
¬
(
X48
→
(
(
¬
X18
)
↔
X45
)
)
)
↔
(
(
(
X49
↔
(
(
¬
X45
)
→
(
(
¬
X17
)
↔
X48
)
)
)
↔
(
(
¬
(
X35
↔
(
X18
→
(
X17
↔
X48
)
)
)
)
↔
(
X45
→
(
(
¬
X35
)
↔
(
X18
→
X48
)
)
)
)
)
↔
(
¬
(
X49
↔
(
(
¬
X35
)
↔
X45
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
X50
↔
(
X45
→
X43
→
X44
)
)
)
→
(
¬
(
X43
↔
X2
)
)
)
)
↔
(
(
(
¬
(
X43
→
(
X36
↔
X44
)
)
)
↔
(
(
¬
(
(
¬
X43
)
↔
(
(
¬
X41
)
→
X45
)
)
)
↔
(
(
X44
→
(
¬
X41
)
→
(
¬
X36
)
→
X45
)
↔
(
(
¬
(
X44
↔
(
X43
→
(
X36
↔
X41
)
)
)
)
↔
(
(
¬
X43
)
↔
(
X41
↔
(
X6
→
X44
)
)
)
)
)
)
)
↔
(
¬
(
(
X44
↔
(
X33
↔
(
(
¬
X12
)
→
X20
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X12
↔
X22
)
)
→
(
¬
(
(
¬
X41
)
↔
(
X33
→
X12
→
X44
)
)
)
)
)
→
(
¬
X44
)
→
(
X33
↔
(
(
¬
X26
)
→
X45
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
X43
)
↔
(
X27
→
(
(
¬
X14
)
↔
X49
)
)
)
)
↔
(
(
¬
X14
)
↔
X45
)
)
→
(
(
(
(
X45
↔
(
(
¬
X43
)
→
X27
→
X50
)
)
↔
(
(
(
X50
↔
(
X43
→
(
X11
↔
X45
)
)
)
↔
(
¬
(
(
¬
X45
)
↔
(
X43
↔
(
(
¬
X25
)
↔
X50
)
)
)
)
)
↔
(
¬
(
X50
→
(
X25
↔
X43
)
)
)
)
)
↔
(
¬
(
X28
→
(
X27
↔
(
(
¬
X11
)
↔
X50
)
)
)
)
)
↔
(
¬
(
X48
→
X11
→
(
(
¬
X10
)
↔
X28
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
d3992c...
and proposition id is
423873...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMMfWJLQJe9FEc4aWYJ3SXsMwDLudgzavP3
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
¬
(
(
(
(
¬
(
(
¬
X42
)
→
X40
→
(
(
¬
X6
)
↔
X45
)
)
)
↔
(
(
(
X45
→
X42
→
(
X40
↔
X32
)
)
↔
(
X45
↔
(
X42
↔
X24
)
)
)
→
(
¬
(
(
¬
X40
)
↔
(
X32
↔
(
X31
↔
X45
)
)
)
)
)
)
↔
(
(
¬
(
(
(
X10
→
X42
)
→
X45
→
X44
→
(
X42
↔
(
X40
→
False
)
)
)
→
(
(
¬
(
(
¬
X40
)
→
(
X15
↔
(
X10
→
X44
)
)
)
)
↔
(
¬
(
X40
↔
(
(
¬
X15
)
↔
(
X5
→
X44
)
)
)
)
)
)
)
↔
(
X40
↔
(
(
¬
X5
)
→
X10
)
)
)
)
→
(
(
¬
(
(
(
(
X48
↔
(
X44
→
(
(
¬
X9
)
↔
X43
)
)
)
→
(
X45
↔
X9
)
)
→
X9
→
X48
)
→
(
(
(
(
¬
(
X43
→
(
(
¬
X42
)
↔
X45
)
)
)
↔
(
(
(
¬
X44
)
↔
(
(
¬
X42
)
↔
(
X21
↔
X45
)
)
)
↔
(
X45
↔
(
X43
→
(
(
¬
X42
)
↔
X44
)
)
)
)
)
↔
(
¬
(
X45
↔
(
X44
↔
(
X42
↔
X30
)
)
)
)
)
↔
(
X43
↔
(
(
¬
X30
)
↔
X45
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X41
)
↔
(
X16
↔
X45
)
)
)
↔
(
(
X16
↔
X41
)
↔
(
(
¬
X45
)
→
(
¬
X44
)
→
X48
)
)
)
→
X41
→
(
X16
↔
X44
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
X41
→
(
X40
↔
(
X26
→
X12
)
)
)
→
X40
→
(
X19
↔
(
X12
↔
X27
)
)
)
)
→
(
¬
(
(
¬
X26
)
→
X41
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X2
)
→
(
¬
X1
)
→
X12
)
)
↔
(
X47
↔
(
(
¬
X12
)
→
(
X2
↔
X42
)
)
)
)
↔
(
¬
(
X40
→
X2
→
X12
)
)
)
↔
(
¬
(
X41
↔
(
X2
→
(
X1
↔
False
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
¬
(
X40
↔
(
(
¬
X28
)
↔
(
(
¬
X17
)
→
X34
)
)
)
)
→
(
¬
(
X40
→
(
X34
↔
(
X3
→
X28
)
)
)
)
)
↔
(
(
X40
↔
(
X34
→
(
X13
↔
X28
)
)
)
→
(
¬
(
X44
→
(
X40
↔
X28
)
)
)
)
)
→
(
(
¬
X34
)
↔
(
X28
↔
(
X8
↔
X40
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
X34
↔
(
X4
→
X46
)
)
→
(
¬
(
X46
↔
(
X25
↔
(
X4
↔
X37
)
)
)
)
)
)
→
(
¬
(
X46
↔
(
X25
→
X4
)
)
)
)
)
↔
(
(
¬
(
(
¬
X37
)
↔
X46
)
)
↔
(
¬
(
(
¬
X38
)
→
X37
→
(
X34
↔
X39
)
)
)
)
)
↔
(
¬
(
(
(
X44
↔
(
X40
↔
(
X33
→
X39
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X33
)
→
X39
)
)
)
)
→
(
(
¬
(
X39
→
(
(
¬
X38
)
↔
(
(
¬
X23
)
↔
X46
)
)
)
)
↔
(
¬
(
X49
→
(
X46
↔
(
(
¬
X23
)
↔
X44
)
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
X29
)
→
(
(
¬
X22
)
↔
(
(
¬
X14
)
↔
X44
)
)
)
↔
(
(
(
¬
(
(
X41
→
X18
→
X44
)
→
(
(
X44
→
(
¬
X40
)
→
X18
→
X41
)
↔
(
¬
(
X44
→
X40
→
X22
)
)
)
)
)
↔
(
¬
(
(
¬
X40
)
→
X22
→
X44
)
)
)
→
(
¬
(
(
X41
↔
(
(
¬
X36
)
↔
(
X18
↔
X44
)
)
)
→
(
¬
(
X44
↔
(
(
¬
X18
)
↔
(
(
¬
X14
)
↔
X36
)
)
)
)
)
)
→
(
¬
(
X36
↔
False
)
)
)
)
↔
(
(
¬
(
X7
→
(
(
¬
(
X44
→
(
X40
↔
(
X20
↔
X36
)
)
)
)
↔
(
X40
→
(
X36
↔
(
(
¬
X7
)
↔
X20
)
)
)
)
)
)
↔
(
(
(
X36
↔
(
X7
→
X49
)
)
→
(
(
¬
(
(
X36
→
(
X35
↔
(
X7
→
X49
)
)
)
→
(
X36
↔
(
(
¬
X11
)
→
X7
→
X35
)
)
)
)
↔
(
¬
(
X36
→
X35
→
X49
)
)
)
→
(
¬
(
X49
→
X36
→
X35
)
)
)
→
(
¬
(
(
¬
(
X49
↔
(
X36
→
X44
)
)
)
→
(
(
X49
→
(
X35
↔
X44
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X44
→
(
(
¬
X35
)
↔
(
(
¬
X0
)
↔
X49
)
)
)
)
→
(
X49
↔
(
X44
↔
(
X36
→
(
X0
↔
False
)
)
)
)
)
)
→
(
¬
(
X49
→
X35
→
False
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X49
→
X45
→
(
X29
↔
X40
)
)
)
↔
(
(
(
¬
(
X29
↔
(
(
¬
X20
)
→
(
¬
X18
)
→
X45
)
)
)
→
(
¬
(
X45
↔
(
X40
↔
X29
)
)
)
→
(
(
X29
↔
(
(
¬
X20
)
→
X26
)
)
↔
(
(
¬
(
X45
↔
(
X29
↔
(
(
¬
X21
)
↔
X40
)
)
)
)
→
(
X29
↔
(
X21
→
(
X20
↔
X45
)
)
)
)
)
)
↔
(
(
X40
↔
(
X38
→
(
X17
↔
X26
)
)
)
↔
(
X29
↔
(
(
¬
X17
)
→
X26
)
)
)
)
)
↔
(
(
¬
(
X47
↔
(
(
¬
X45
)
↔
(
X40
→
X49
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X49
→
X32
→
X45
)
)
→
(
¬
(
(
(
¬
X45
)
→
(
X32
↔
(
(
¬
X8
)
↔
X49
)
)
)
→
(
(
X47
→
(
(
¬
X32
)
↔
(
X30
↔
X45
)
)
)
→
(
¬
(
(
¬
X3
)
↔
X49
)
)
→
(
¬
(
(
¬
(
(
¬
X45
)
↔
(
X30
↔
(
X3
→
X47
)
)
)
)
→
X47
→
(
¬
X30
)
→
X49
)
)
→
X30
→
X3
)
→
(
¬
(
X49
↔
(
X47
↔
(
X45
→
(
X5
↔
False
)
)
)
)
)
)
)
)
)
→
(
¬
(
X49
↔
(
X13
→
(
X8
↔
X1
)
)
)
)
→
(
¬
X13
)
→
X1
→
X32
)
)
)
)
↔
(
¬
(
(
(
X47
↔
(
X42
↔
(
X11
→
X19
)
)
)
→
(
¬
(
X49
→
X45
→
(
X19
↔
False
)
)
)
)
→
(
(
¬
(
X42
↔
X44
)
)
↔
(
¬
(
(
X45
↔
(
X42
↔
(
X25
↔
(
X4
→
False
)
)
)
)
→
(
¬
(
(
¬
X42
)
↔
(
X7
→
X49
)
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
(
(
(
(
¬
X27
)
↔
(
X24
↔
(
X9
→
X31
)
)
)
↔
(
(
¬
X31
)
↔
(
X27
↔
(
X24
↔
X37
)
)
)
)
↔
(
X44
→
(
(
¬
X27
)
↔
(
X24
→
X31
)
)
)
)
→
(
¬
(
X31
→
(
(
¬
X24
)
↔
(
X0
→
X37
)
)
)
)
)
)
→
(
X23
→
X27
)
→
(
(
(
¬
X23
)
→
(
(
¬
X16
)
↔
X33
)
)
↔
(
X33
↔
(
(
¬
X24
)
↔
(
(
¬
X23
)
↔
X27
)
)
)
)
→
(
¬
(
X33
→
(
(
¬
X23
)
↔
(
X16
↔
X31
)
)
)
)
)
)
↔
(
(
¬
(
(
X34
↔
X48
)
→
(
(
¬
X28
)
↔
(
(
¬
X27
)
→
(
¬
X10
)
→
X34
)
)
→
(
(
¬
(
X34
→
(
X10
↔
(
(
¬
X6
)
↔
X37
)
)
)
)
↔
(
¬
(
X48
↔
(
X34
↔
(
(
¬
X10
)
→
X37
)
)
)
)
)
)
)
↔
(
(
¬
(
X45
↔
(
(
¬
X23
)
↔
(
(
¬
X22
)
→
X48
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
X27
)
↔
(
(
¬
X22
)
→
X34
)
)
→
(
X41
↔
(
(
¬
X23
)
→
X22
→
X27
)
)
)
)
→
(
¬
(
X41
→
(
(
¬
X27
)
↔
X34
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
(
X38
→
(
¬
X14
)
→
X2
→
X45
)
)
→
(
¬
(
X45
→
X38
→
X15
→
X14
→
False
)
)
)
)
↔
(
X45
↔
(
X15
↔
X38
)
)
)
↔
(
(
¬
(
(
(
¬
(
X46
↔
(
X45
→
X39
→
X12
)
)
)
↔
(
X48
→
X46
→
X39
→
X12
→
False
)
)
→
(
¬
(
X48
→
(
¬
X39
)
→
X46
)
)
)
)
→
X39
→
(
X12
↔
X38
)
)
)
→
(
(
(
(
¬
(
X43
→
(
X35
↔
(
X12
↔
X14
)
)
)
)
→
(
X41
↔
(
X35
→
(
X12
↔
X43
)
)
)
)
↔
(
¬
(
X36
→
(
X35
↔
(
X14
→
X12
→
False
)
)
)
)
)
↔
(
¬
(
(
¬
X41
)
↔
(
(
¬
X38
)
↔
(
X36
↔
X43
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
6e3ed4...
and proposition id is
33f52e...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMHXdsYbtt9ivBHUkkD6yyWmcgGGz5irHm8
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
∀X50 :
prop
,
(
¬
∀X51 :
prop
,
(
(
(
¬
(
(
¬
(
(
(
(
(
X49
→
X48
→
(
X4
↔
X19
)
)
↔
(
X42
↔
(
(
¬
X19
)
↔
X48
)
)
)
→
(
X42
↔
(
X19
↔
X48
)
)
)
→
(
¬
X48
)
→
X31
→
(
(
¬
X19
)
↔
X49
)
)
→
(
(
¬
(
(
(
¬
X42
)
→
(
¬
X15
)
→
(
(
¬
X11
)
↔
X49
)
)
→
(
(
¬
(
X49
↔
(
X40
↔
X26
)
)
)
↔
(
(
¬
(
X49
→
(
(
¬
X26
)
↔
(
(
¬
X11
)
→
X40
)
)
)
)
↔
(
¬
(
X49
→
X42
→
(
X11
↔
X40
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
X49
→
X40
→
X48
)
→
(
¬
(
X49
→
(
X48
↔
(
(
¬
X37
)
→
X42
)
)
)
)
)
)
→
(
(
X51
↔
(
X49
→
(
(
¬
X40
)
↔
X48
)
)
)
↔
(
¬
(
(
¬
X49
)
↔
(
(
¬
X48
)
→
(
¬
X30
)
→
X51
)
)
)
)
)
→
(
¬
(
(
¬
X49
)
↔
(
X37
↔
(
(
¬
X24
)
→
X51
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
X48
)
→
(
(
¬
X39
)
↔
(
X5
↔
X51
)
)
)
↔
(
(
¬
(
X50
↔
X46
)
)
→
(
¬
(
X48
→
X39
→
False
)
)
→
(
¬
(
X51
→
(
X50
↔
(
X9
→
X48
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X49
→
X46
→
X0
→
X47
)
)
↔
(
¬
(
(
¬
X47
)
↔
(
X25
↔
(
(
¬
X20
)
↔
X50
)
)
)
)
)
→
(
(
(
¬
X47
)
↔
(
(
¬
X39
)
→
(
¬
X25
)
→
X50
)
)
↔
(
¬
(
X50
→
X39
→
X18
→
False
)
)
)
)
→
(
¬
(
X49
→
(
X46
↔
(
(
¬
X3
)
→
X39
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
(
(
(
¬
X45
)
↔
(
X35
→
X47
)
)
↔
(
¬
(
(
¬
(
X35
→
X12
→
(
(
¬
X1
)
↔
X45
)
)
)
→
(
¬
(
X47
↔
(
X35
↔
(
(
¬
X12
)
↔
X45
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X29
↔
False
)
)
→
X35
→
(
(
¬
X12
)
↔
(
(
¬
X1
)
↔
X13
)
)
)
)
→
X35
→
(
X1
↔
X12
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X49
→
X45
→
(
(
¬
X10
)
↔
X48
)
)
)
↔
(
X48
↔
(
X45
→
X28
→
X13
)
)
)
→
(
(
¬
(
X49
→
(
(
¬
X38
)
↔
(
(
¬
X7
)
→
X48
)
)
)
)
↔
(
(
(
X38
↔
(
X2
↔
X28
)
)
→
X48
→
X38
→
X28
→
X49
)
↔
(
¬
(
(
¬
X35
)
↔
(
(
¬
X2
)
↔
X49
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X32
)
↔
(
(
¬
X22
)
→
X45
)
)
)
→
(
(
¬
(
X47
↔
(
(
¬
X16
)
↔
X36
)
)
)
↔
(
(
¬
(
X47
↔
(
X45
↔
(
X38
→
X32
)
)
)
)
→
(
¬
(
(
X47
→
(
(
¬
X22
)
↔
(
(
¬
X16
)
→
X27
)
)
)
→
(
X47
↔
(
X45
→
X16
→
X38
)
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
X45
→
X6
→
X17
)
)
↔
(
¬
(
X41
↔
(
(
¬
X6
)
→
X21
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X43
↔
(
X23
→
X49
)
)
)
→
(
(
¬
(
(
¬
X43
)
↔
(
(
¬
X23
)
→
(
(
¬
X17
)
↔
X49
)
)
)
)
↔
(
X43
→
X41
→
(
X23
↔
X49
)
)
)
)
)
↔
(
(
¬
X23
)
→
(
¬
X17
)
→
X43
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
(
¬
(
X45
→
X43
→
(
¬
X14
)
→
X17
)
)
→
(
¬
(
X23
→
(
X14
↔
X43
)
)
)
)
)
→
(
¬
X43
)
→
(
X23
↔
(
X8
↔
X45
)
)
)
)
↔
(
(
¬
X43
)
↔
(
(
¬
X23
)
↔
X45
)
)
)
→
(
¬
X23
)
)
)
)
)
→
(
(
(
X45
→
X23
)
↔
(
(
X45
↔
(
(
¬
X34
)
↔
X47
)
)
↔
(
¬
(
(
¬
(
X44
→
(
X34
↔
X45
)
)
)
→
(
X47
↔
(
(
¬
X44
)
↔
(
X34
→
X45
)
)
)
→
(
¬
(
X45
↔
(
(
¬
X44
)
↔
(
(
¬
X33
)
↔
X47
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
X43
)
→
(
(
¬
X34
)
↔
(
X33
→
X45
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
(
¬
(
(
(
¬
X38
)
↔
(
X20
→
X51
)
)
→
(
(
¬
X46
)
→
(
¬
X15
)
→
X51
)
→
(
¬
(
(
¬
(
X46
→
(
¬
X37
)
→
(
¬
X20
)
→
X51
)
)
→
(
¬
(
X51
↔
(
X15
↔
X46
)
)
)
)
)
)
)
↔
(
X51
→
(
¬
X37
)
→
X38
)
)
↔
(
X46
↔
(
(
¬
X37
)
↔
X51
)
)
)
↔
(
(
X40
↔
(
X38
↔
(
(
¬
X22
)
↔
X41
)
)
)
↔
(
(
(
X41
→
(
¬
X37
)
→
X40
)
→
(
(
¬
(
(
¬
X41
)
→
(
X40
↔
(
X38
→
X51
)
)
)
)
↔
(
(
¬
X38
)
→
(
¬
X37
)
→
X51
)
)
)
↔
(
(
¬
(
(
(
¬
X43
)
→
(
X38
↔
X51
)
)
→
(
(
X43
↔
X51
)
↔
(
(
¬
(
X51
→
(
¬
X4
)
→
X43
)
)
→
(
(
X51
↔
(
X38
→
X4
→
False
)
)
↔
(
¬
(
(
¬
(
X43
↔
(
X40
↔
(
X4
→
False
)
)
)
)
→
(
¬
(
(
¬
X38
)
→
X51
)
)
)
)
)
)
)
)
)
→
(
X51
↔
(
X38
→
(
X18
↔
X40
)
)
)
)
)
)
)
→
(
(
(
(
(
(
X41
→
(
X29
↔
False
)
)
↔
(
(
¬
(
X50
↔
(
(
¬
X9
)
↔
X41
)
)
)
↔
(
(
X29
→
X9
→
(
X7
↔
False
)
)
↔
(
¬
(
X41
↔
(
(
¬
X29
)
→
X50
)
)
)
)
)
)
↔
(
¬
(
(
¬
X9
)
↔
(
X7
↔
X50
)
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X9
)
→
(
X7
↔
X41
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X46
)
→
(
X29
↔
X50
)
)
)
↔
(
¬
(
X48
→
X46
→
(
X39
↔
X23
)
)
)
)
→
(
¬
(
(
¬
X48
)
↔
(
(
¬
X29
)
↔
(
(
¬
X12
)
↔
X50
)
)
)
)
→
(
¬
(
(
¬
(
X50
→
X48
→
(
(
¬
X26
)
↔
X29
)
)
)
→
(
X29
↔
(
X26
↔
(
X13
→
(
X2
↔
False
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
X50
→
(
X43
↔
(
X35
↔
X31
)
)
)
→
(
X43
↔
(
X31
→
(
X0
↔
False
)
)
)
)
)
→
(
X49
↔
(
(
¬
X48
)
→
(
¬
X43
)
→
X50
)
)
→
(
¬
(
(
(
¬
(
X49
↔
(
X43
→
(
¬
X31
)
→
X46
)
)
)
→
X50
→
(
X49
↔
(
X43
→
X46
)
)
)
→
(
¬
(
X46
↔
(
X43
→
X31
→
X1
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X51
↔
(
X50
→
X48
)
)
)
→
(
(
¬
(
(
¬
(
X51
↔
(
(
¬
X21
)
↔
(
(
¬
X10
)
↔
X50
)
)
)
)
→
(
(
¬
(
(
¬
(
X50
→
(
¬
X6
)
→
(
¬
X5
)
→
X48
)
)
→
(
¬
(
X21
↔
False
)
)
)
)
↔
(
X50
↔
X5
)
)
→
(
¬
(
X51
↔
(
X21
→
(
¬
X5
)
→
X6
)
)
)
)
)
↔
(
X51
↔
(
X21
→
X6
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X51
↔
(
(
¬
X32
)
↔
(
(
¬
X11
)
→
X33
)
)
)
)
→
X42
→
(
X33
↔
(
(
¬
X28
)
→
X32
)
)
)
→
(
X51
↔
(
(
¬
X42
)
↔
(
(
¬
X11
)
→
X48
)
)
)
)
→
(
(
¬
X50
)
↔
(
(
¬
X48
)
→
(
¬
X27
)
→
X51
)
)
→
(
(
(
X51
→
(
(
¬
X42
)
↔
(
(
¬
X27
)
↔
X50
)
)
)
→
X50
→
(
(
¬
X27
)
↔
(
X25
→
X51
)
)
)
↔
(
¬
(
(
¬
X50
)
→
(
X42
↔
X51
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X51
↔
(
X50
→
X48
→
(
X16
↔
False
)
)
)
)
→
(
¬
(
X51
→
(
X50
↔
(
X48
↔
(
X19
↔
False
)
)
)
)
)
)
)
↔
(
(
(
(
X45
↔
(
X36
↔
(
X24
→
X51
)
)
)
→
(
¬
(
(
¬
X3
)
↔
X45
)
)
)
↔
(
¬
(
(
X51
↔
(
X36
↔
(
X24
→
False
)
)
)
→
(
¬
(
(
X51
↔
(
X47
↔
X44
)
)
→
(
¬
(
(
(
X44
↔
(
X24
→
X47
)
)
→
(
¬
X24
)
→
X17
→
X51
)
→
(
X44
↔
(
(
¬
X36
)
→
(
(
¬
X17
)
↔
X51
)
)
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X50
→
(
X45
↔
X48
)
)
)
↔
(
(
¬
X50
)
→
(
(
¬
X45
)
↔
(
(
¬
X30
)
↔
X51
)
)
)
)
↔
(
¬
(
(
¬
(
X51
→
(
(
¬
X45
)
↔
X48
)
)
)
→
(
(
X47
→
(
¬
X45
)
→
(
(
¬
X34
)
↔
X51
)
)
↔
(
(
¬
X47
)
↔
(
X45
→
(
¬
X34
)
→
X48
)
)
)
)
)
)
↔
(
(
X51
→
X47
→
X14
)
→
(
(
X47
↔
(
X14
→
(
(
¬
X8
)
↔
X48
)
)
)
↔
(
X51
→
(
X48
↔
(
X47
↔
(
X14
↔
False
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
770a2c...
and proposition id is
a094e9...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMWbq79Gz7JNLFvkTKe437itFjjcWb9Y2jP
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
(
(
X47
↔
False
)
→
(
¬
(
(
¬
X45
)
→
(
(
¬
X28
)
↔
(
(
¬
X24
)
→
X46
)
)
)
)
)
→
(
(
¬
(
X46
↔
(
X45
→
X40
→
X28
→
False
)
)
)
↔
(
¬
(
(
¬
(
X43
→
(
(
¬
X40
)
↔
(
(
¬
X30
)
→
X46
)
)
)
)
→
(
¬
X45
)
→
(
(
¬
X40
)
↔
(
(
¬
X30
)
→
X46
)
)
)
)
)
)
↔
(
(
¬
(
X48
↔
(
X28
→
(
¬
X4
)
→
X30
)
)
)
↔
(
(
(
¬
X45
)
→
(
(
¬
X30
)
↔
(
(
¬
X27
)
→
X48
)
)
)
↔
(
(
(
X28
↔
(
(
¬
X27
)
↔
(
(
¬
X20
)
↔
X30
)
)
)
↔
(
X30
↔
(
X27
↔
(
X20
→
X48
)
)
)
)
↔
(
(
¬
X27
)
→
(
(
¬
X20
)
↔
X48
)
)
)
)
)
)
↔
(
(
(
¬
(
X47
↔
(
(
¬
X42
)
↔
X45
)
)
)
→
(
¬
(
(
(
X47
→
(
X45
↔
(
X42
↔
X7
)
)
)
↔
(
(
¬
(
X47
→
X45
→
(
X35
↔
(
X15
↔
False
)
)
)
)
↔
(
¬
(
X47
↔
(
X45
↔
(
X42
→
X15
→
False
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X45
→
(
X42
↔
(
(
¬
X13
)
→
X36
)
)
)
)
→
(
¬
X36
)
→
X13
→
(
¬
X2
)
→
X45
)
→
(
(
¬
X45
)
↔
(
X13
↔
(
X2
→
X47
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
X45
→
X40
→
X16
→
X11
→
False
)
→
(
¬
(
X45
↔
(
(
¬
X8
)
→
X16
)
)
)
)
)
↔
(
X1
→
X8
)
)
→
(
(
¬
(
X40
→
X13
→
X45
)
)
↔
(
(
X40
↔
(
X13
↔
(
X8
→
X45
)
)
)
↔
(
¬
(
X47
→
X40
→
X8
→
X45
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
(
¬
(
X33
↔
(
(
¬
X32
)
→
X42
)
)
)
↔
(
X23
↔
(
(
(
¬
(
X33
↔
(
X23
↔
(
X22
→
False
)
)
)
)
↔
(
X42
↔
(
X33
↔
(
X23
↔
X32
)
)
)
)
→
(
¬
X32
)
→
X42
)
)
)
→
(
(
¬
(
X22
→
X44
)
)
↔
(
(
¬
(
(
¬
X33
)
→
(
(
¬
X23
)
↔
(
(
¬
X14
)
→
X44
)
)
)
)
↔
(
X22
→
X14
→
X33
)
)
)
)
↔
(
X44
↔
(
(
¬
X33
)
↔
(
(
¬
X22
)
↔
X47
)
)
)
)
↔
(
(
(
¬
(
X26
↔
(
X25
↔
(
(
¬
X12
)
↔
X33
)
)
)
)
↔
(
(
X41
↔
(
(
¬
X12
)
→
X33
)
)
↔
(
X44
→
(
X26
↔
X33
)
)
)
)
→
X41
→
(
X33
↔
(
X26
↔
X23
)
)
)
)
↔
(
(
(
X47
→
(
(
¬
X3
)
↔
X38
)
)
↔
(
(
X49
↔
(
(
¬
X44
)
↔
(
(
¬
X41
)
↔
X47
)
)
)
↔
(
¬
(
X49
→
(
X44
↔
(
X41
↔
(
X38
↔
False
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X9
)
↔
(
X3
→
X44
)
)
)
→
X44
→
(
¬
X6
)
→
(
¬
X3
)
→
X43
)
)
)
)
→
(
(
¬
(
(
(
(
X37
→
X34
→
(
(
¬
X17
)
↔
X19
)
)
→
(
¬
(
X42
→
X37
)
)
)
→
(
(
¬
X19
)
↔
(
(
¬
X5
)
→
X37
)
)
→
(
¬
X18
)
→
X42
)
→
(
(
X49
→
(
X42
↔
(
X37
↔
X34
)
)
)
↔
(
X42
↔
(
(
¬
X34
)
↔
(
X10
↔
X37
)
)
)
)
)
)
↔
(
(
X42
→
X37
→
(
X31
↔
X49
)
)
↔
(
(
¬
(
(
¬
(
X49
↔
(
X42
→
(
X39
↔
X37
)
)
)
)
→
(
¬
(
X42
↔
(
X39
→
(
¬
X29
)
→
X37
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
X37
)
↔
(
(
¬
X29
)
↔
(
X0
↔
X44
)
)
)
↔
(
¬
(
X42
↔
(
(
¬
X37
)
↔
(
X21
↔
X44
)
)
)
)
)
→
(
¬
(
X44
↔
X37
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
¬
(
(
¬
(
X48
↔
(
X33
→
X44
)
)
)
→
(
¬
(
X48
→
(
(
¬
X10
)
↔
X17
)
)
)
)
)
→
(
¬
(
(
¬
X33
)
↔
(
(
¬
X10
)
↔
X48
)
)
)
)
→
(
(
X33
↔
(
X9
↔
X39
)
)
↔
(
¬
(
X48
→
(
X39
↔
(
(
¬
X9
)
→
X44
)
)
)
)
)
)
↔
(
(
(
X49
→
X44
→
(
X11
↔
X47
)
)
↔
(
(
(
¬
(
(
(
¬
X47
)
↔
(
(
¬
X42
)
→
(
X38
↔
X48
)
)
)
→
(
X49
↔
(
(
¬
X47
)
→
(
X38
↔
X48
)
)
)
)
)
↔
(
¬
X48
)
)
↔
(
X48
→
(
¬
X47
)
→
(
(
¬
X44
)
↔
X49
)
)
)
)
↔
(
¬
(
(
X49
↔
(
X47
→
(
¬
X20
)
→
X39
)
)
→
(
¬
(
(
X49
↔
(
X44
→
(
X40
↔
X47
)
)
)
→
X47
→
X44
→
(
¬
X39
)
→
X49
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
¬
X46
)
→
(
X27
↔
(
(
¬
X16
)
→
X49
)
)
)
)
↔
(
¬
(
X49
→
X46
→
(
X28
↔
(
X27
→
False
)
)
)
)
)
→
(
¬
(
(
(
(
¬
X35
)
↔
(
(
¬
X28
)
↔
(
(
¬
X22
)
↔
X46
)
)
)
↔
(
(
X46
→
X31
→
(
X27
↔
False
)
)
↔
(
(
¬
(
X31
→
(
(
¬
X13
)
↔
X27
)
)
)
→
X35
→
(
X31
↔
(
X22
↔
X27
)
)
)
)
)
→
(
¬
(
X27
→
X46
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X49
→
X28
→
X31
)
)
→
(
(
(
¬
X47
)
↔
(
(
¬
X23
)
→
(
¬
X4
)
→
X48
)
)
↔
(
X28
→
X23
→
X12
→
X48
)
)
)
)
→
(
(
(
¬
(
X47
→
(
¬
X23
)
→
X31
)
)
→
(
¬
(
X19
↔
X3
)
)
)
↔
(
X31
↔
(
X23
↔
False
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
(
(
¬
X41
)
→
(
¬
X1
)
→
X48
)
→
(
X48
↔
(
X47
→
(
X0
↔
X1
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
X48
→
X46
→
(
X0
↔
X47
)
)
→
(
X46
↔
(
X45
→
(
(
¬
X0
)
↔
X48
)
)
)
)
)
↔
(
¬
(
X48
→
(
X47
↔
X46
)
)
)
)
→
(
¬
(
X45
→
X0
→
X47
)
)
)
)
↔
(
¬
(
X48
↔
(
X46
↔
(
X34
→
X47
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X48
↔
(
X47
↔
X46
)
)
)
↔
(
(
¬
(
X47
→
(
(
¬
X46
)
↔
(
X14
↔
X48
)
)
)
)
↔
(
¬
(
X21
↔
(
X14
→
False
)
)
)
)
)
→
(
(
(
¬
(
X47
→
(
X21
↔
X43
)
)
)
↔
(
¬
(
X47
↔
(
X32
→
(
X21
↔
X48
)
)
)
)
)
↔
(
X47
↔
(
X32
↔
(
(
¬
X2
)
→
X43
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X46
)
↔
(
(
¬
X37
)
→
(
X8
↔
X47
)
)
)
)
↔
(
¬
(
X46
→
(
X41
↔
(
(
¬
X30
)
→
X36
)
)
)
)
)
↔
(
(
(
¬
(
(
(
X41
↔
(
X18
↔
False
)
)
→
(
¬
(
(
(
X18
→
(
¬
X15
)
→
X24
)
↔
(
X41
→
X29
→
X15
)
)
→
(
(
¬
X29
)
↔
(
X25
↔
(
(
¬
X18
)
↔
X41
)
)
)
)
)
)
→
(
¬
(
X41
→
(
X29
↔
(
(
¬
X5
)
→
X25
)
)
)
)
)
)
→
(
X46
↔
(
¬
(
(
X29
↔
(
X24
→
X7
)
)
→
(
¬
(
X46
↔
(
(
¬
X24
)
↔
(
(
¬
X18
)
↔
X29
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
X41
↔
X15
)
→
(
X26
↔
(
X24
↔
(
X15
↔
(
X6
→
False
)
)
)
)
)
)
↔
(
¬
(
X24
→
(
(
¬
X6
)
↔
X41
)
)
)
)
→
(
(
¬
X41
)
↔
(
(
¬
X15
)
↔
X47
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
8e838b...
and proposition id is
796609...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMQn216BpoXVb8XorVSjrsfJ7c5LGifQq6n
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
(
(
¬
(
(
(
(
(
¬
X46
)
↔
(
X45
↔
(
X31
→
X49
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
¬
(
X36
→
(
¬
X31
)
→
(
(
¬
X17
)
↔
X39
)
)
)
→
(
X28
↔
X39
)
)
)
→
X49
→
(
X39
↔
(
X31
↔
X17
)
)
)
)
→
(
(
X28
→
(
X17
↔
(
X14
→
X41
)
)
)
↔
(
(
¬
X39
)
↔
(
(
¬
X28
)
↔
X46
)
)
)
)
)
)
→
(
(
¬
X36
)
→
(
(
¬
X8
)
↔
X49
)
)
→
X41
→
X36
→
X8
→
X48
)
→
(
(
(
¬
(
X39
↔
X23
)
)
↔
(
(
(
¬
(
X44
→
X27
→
X23
→
X36
)
)
↔
(
X44
↔
(
X36
↔
(
X27
↔
(
X12
↔
False
)
)
)
)
)
↔
(
¬
(
X36
↔
(
(
¬
X23
)
↔
X44
)
)
)
)
)
↔
(
(
¬
(
X38
↔
(
(
¬
X23
)
→
(
(
¬
X12
)
↔
X39
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X45
→
X39
→
X38
→
False
)
)
↔
(
(
¬
(
X39
→
(
(
¬
X38
)
↔
(
X0
→
X45
)
)
)
)
↔
(
X39
→
(
(
¬
X12
)
↔
(
X0
→
X45
)
)
)
)
)
→
(
¬
(
X39
↔
(
X38
→
False
)
)
)
)
)
→
(
¬
(
X45
→
(
¬
X0
)
→
X39
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X47
↔
(
X30
↔
(
X24
↔
X45
)
)
)
)
↔
(
(
(
¬
X26
)
↔
(
X18
↔
X45
)
)
↔
(
¬
(
X26
→
(
(
¬
X24
)
↔
(
(
¬
X18
)
→
X30
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X26
→
(
¬
X18
)
→
(
X4
↔
X30
)
)
)
→
(
X30
↔
(
(
¬
X26
)
→
(
¬
X4
)
→
X49
)
)
)
)
)
↔
(
(
(
X47
↔
(
X45
→
(
X22
↔
X6
)
)
)
↔
(
(
¬
(
X45
↔
(
X21
→
X6
→
X43
)
)
)
↔
(
(
¬
X43
)
→
(
(
¬
X41
)
↔
(
X6
→
X45
)
)
)
)
)
↔
(
¬
(
X47
↔
(
X43
↔
(
X21
→
X30
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
X29
→
(
¬
X13
)
→
X10
→
X49
)
→
(
¬
(
X25
↔
(
(
¬
X13
)
↔
(
(
¬
X10
)
↔
X49
)
)
)
)
)
↔
(
X29
→
(
¬
X19
)
→
(
¬
X13
)
→
X49
)
)
→
(
(
¬
(
X49
↔
(
(
¬
X25
)
↔
(
X15
→
X40
)
)
)
)
↔
(
(
¬
X19
)
→
(
¬
X15
)
→
(
¬
X1
)
→
X40
)
)
)
↔
(
(
(
¬
(
(
¬
(
X50
↔
(
X20
→
X47
)
)
)
→
(
(
¬
X49
)
↔
(
(
¬
X47
)
↔
(
X15
↔
X50
)
)
)
)
)
→
(
¬
(
X20
→
X15
→
(
¬
X7
)
→
X47
)
)
→
(
(
¬
(
(
X20
→
(
X15
↔
(
X13
↔
X9
)
)
)
→
(
¬
(
X47
↔
(
X13
↔
(
X9
→
X20
)
)
)
)
)
)
↔
(
¬
(
X47
↔
(
X15
→
X13
→
False
)
)
)
)
)
→
(
¬
(
X50
→
X47
→
X33
→
X13
→
False
)
)
)
)
→
(
(
¬
(
(
(
(
(
X49
↔
(
X42
→
(
¬
X3
)
→
X16
)
)
↔
(
¬
(
X42
↔
(
X16
→
X3
→
X32
)
)
)
)
↔
(
¬
(
(
¬
X32
)
↔
(
X11
↔
(
X3
→
X47
)
)
)
)
)
↔
(
¬
(
(
¬
X42
)
→
(
¬
X16
)
→
(
X2
↔
X49
)
)
)
)
→
(
(
¬
(
X50
→
(
X35
↔
(
(
¬
X34
)
↔
X45
)
)
)
)
↔
(
X47
→
(
¬
X45
)
→
(
(
¬
X35
)
↔
X50
)
)
)
→
(
(
(
¬
(
X50
→
(
X35
↔
(
X32
↔
(
X2
→
False
)
)
)
)
)
↔
(
¬
(
X45
→
(
X35
↔
(
X32
→
(
X2
↔
False
)
)
)
)
)
)
↔
(
¬
(
X45
→
X2
)
)
)
→
(
¬
(
(
¬
X45
)
↔
(
(
¬
X35
)
↔
X50
)
)
)
)
)
↔
(
(
¬
(
(
¬
X47
)
↔
(
(
¬
X37
)
→
X5
→
X50
)
)
)
→
(
(
¬
X37
)
↔
X49
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
(
¬
(
(
¬
X36
)
→
(
X20
↔
X50
)
)
)
→
(
(
(
¬
X43
)
↔
(
X36
→
(
¬
X20
)
→
X46
)
)
↔
X49
)
)
↔
(
¬
(
X50
→
X46
→
X43
)
)
)
↔
(
(
¬
X46
)
↔
(
(
X49
→
X46
→
(
¬
X25
)
→
X48
)
↔
(
(
¬
X48
)
↔
(
(
¬
X25
)
→
(
¬
X20
)
→
X49
)
)
)
)
)
↔
(
(
(
(
X50
↔
(
(
¬
X47
)
→
X48
)
)
↔
(
(
X50
↔
(
X37
→
(
(
¬
X24
)
↔
X27
)
)
)
→
(
¬
(
(
X48
→
(
X37
↔
(
X27
→
X47
)
)
)
→
(
X48
↔
(
X37
↔
(
(
¬
X24
)
↔
X47
)
)
)
)
)
)
)
↔
(
¬
(
(
X50
↔
False
)
→
(
¬
(
(
X50
→
X49
→
(
X47
↔
X48
)
)
→
(
X49
↔
(
X48
↔
(
X47
↔
X4
)
)
)
)
)
)
)
)
↔
(
¬
(
(
X28
→
X27
→
(
(
¬
X24
)
↔
X48
)
)
→
X48
→
X27
→
(
¬
X9
)
→
X24
)
)
)
)
→
(
¬
(
(
(
¬
(
X39
→
(
X23
↔
(
X19
→
(
X6
↔
False
)
)
)
)
)
→
(
X42
↔
(
X39
→
X6
)
)
)
→
(
X48
↔
(
X42
↔
(
X14
↔
X23
)
)
)
→
(
¬
(
X42
→
(
(
¬
X23
)
↔
(
X12
↔
X39
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
(
¬
X45
)
→
(
¬
X39
)
→
X46
)
)
→
(
¬
(
X47
↔
X32
)
)
)
)
→
(
(
(
(
¬
X42
)
→
(
(
¬
X39
)
↔
(
X35
→
X48
)
)
)
↔
(
¬
(
X30
→
X42
)
)
)
↔
(
(
¬
(
X48
→
(
X47
↔
(
X39
↔
(
X30
↔
False
)
)
)
)
)
→
(
¬
(
X48
↔
(
X47
↔
(
(
¬
X39
)
↔
X46
)
)
)
)
)
)
)
↔
(
(
X46
↔
(
X3
→
X1
→
X48
)
)
↔
(
(
¬
(
(
(
¬
X45
)
→
(
X29
↔
(
X1
↔
X48
)
)
)
→
(
¬
(
X45
→
(
X29
↔
(
X13
↔
False
)
)
)
)
)
)
↔
(
¬
(
X45
↔
(
X29
↔
(
X13
→
X3
)
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X43
↔
(
X42
↔
X41
)
)
)
→
(
X44
↔
(
(
¬
X43
)
↔
(
(
¬
X21
)
→
X45
)
)
)
→
(
¬
(
(
¬
X42
)
→
(
¬
X41
)
→
(
¬
X8
)
→
X44
)
)
)
→
(
¬
(
(
¬
X10
)
↔
(
X7
→
X43
)
)
)
→
(
¬
X42
)
→
(
(
¬
X10
)
↔
(
(
¬
X7
)
↔
X45
)
)
)
→
(
(
(
X38
→
X15
)
→
(
¬
(
X43
↔
(
(
¬
X38
)
→
(
(
¬
X31
)
↔
X42
)
)
)
)
)
↔
(
(
¬
(
X45
↔
(
X42
↔
(
(
¬
X38
)
↔
X43
)
)
)
)
→
(
(
(
(
¬
(
(
¬
X42
)
→
X43
)
)
→
(
(
(
¬
X42
)
→
(
(
¬
X34
)
↔
(
(
¬
X0
)
→
X45
)
)
)
↔
(
X43
→
X42
→
X34
→
X45
)
)
)
↔
(
¬
(
(
¬
X43
)
→
(
(
¬
X34
)
↔
X45
)
)
)
)
↔
(
¬
(
X43
→
(
(
¬
X0
)
↔
X42
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
(
X45
→
(
X22
↔
(
(
¬
X5
)
↔
X17
)
)
)
→
X22
→
(
(
¬
X11
)
↔
(
(
¬
X5
)
→
X45
)
)
)
)
↔
(
(
¬
(
(
(
¬
X22
)
↔
(
X17
→
X45
)
)
→
(
¬
(
X33
↔
(
X17
→
(
X2
↔
X45
)
)
)
)
)
)
→
(
¬
(
X33
↔
(
X22
↔
(
X2
↔
False
)
)
)
)
)
)
→
(
X43
→
(
X22
↔
(
(
¬
X11
)
↔
X45
)
)
)
→
(
X45
↔
(
X43
↔
(
X40
↔
X22
)
)
)
)
)
↔
(
(
X40
↔
(
X18
→
X16
→
X45
)
)
↔
(
(
X33
→
(
¬
X26
)
→
(
X16
↔
X42
)
)
↔
(
X42
→
X40
→
(
¬
X16
)
→
X45
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
a4b328...
and proposition id is
eb4a75...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMJzNq1iiYmRQR2WDeqp59CfVHWp163tBpL
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
(
(
X49
→
(
¬
X40
)
→
X38
→
X45
)
↔
(
¬
(
X49
↔
(
(
¬
X38
)
→
(
X36
↔
X40
)
)
)
)
)
↔
(
¬
(
(
(
(
(
X49
↔
(
X40
↔
(
(
¬
X4
)
→
X43
)
)
)
→
(
¬
(
X34
↔
X43
)
)
)
↔
(
X49
↔
(
X38
→
(
(
¬
X4
)
↔
X34
)
)
)
)
↔
(
X40
↔
(
(
¬
X4
)
↔
X43
)
)
)
→
(
¬
(
(
(
¬
X40
)
→
(
(
¬
X39
)
↔
(
X38
↔
X43
)
)
)
→
(
¬
(
(
X40
→
(
¬
X11
)
→
(
(
¬
X3
)
↔
X43
)
)
→
(
¬
(
X38
↔
X11
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
X43
)
→
(
(
¬
X27
)
↔
X49
)
)
)
↔
(
¬
(
X45
→
X27
→
(
X26
↔
X43
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X49
→
(
X44
↔
(
(
¬
X13
)
↔
X40
)
)
)
)
↔
(
(
(
(
¬
X48
)
→
(
(
¬
X40
)
↔
X49
)
)
↔
(
¬
(
(
(
X48
↔
(
X40
→
(
¬
X6
)
→
X33
)
)
↔
(
¬
(
X13
→
X6
→
X40
)
)
)
→
(
¬
(
X49
→
(
(
¬
X40
)
↔
(
X33
↔
X48
)
)
)
)
)
)
)
→
(
X49
↔
(
X13
↔
X48
)
)
)
)
→
(
(
¬
(
X48
↔
(
(
¬
X43
)
↔
(
X19
→
X44
)
)
)
)
↔
(
(
¬
X44
)
→
(
(
¬
X43
)
↔
(
(
¬
X40
)
→
X48
)
)
)
)
)
→
(
¬
(
(
(
(
¬
X28
)
→
(
(
¬
X22
)
↔
X49
)
)
→
(
¬
(
X43
↔
(
X29
↔
(
X12
↔
False
)
)
)
)
)
→
(
(
¬
(
X49
→
X48
→
(
X29
↔
(
X12
→
False
)
)
)
)
↔
(
X48
↔
(
X41
→
X20
→
X12
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
X5
)
→
(
¬
X0
)
→
X15
)
↔
(
¬
(
(
¬
X30
)
→
(
¬
X2
)
→
(
(
¬
X0
)
↔
X35
)
)
)
)
→
(
(
¬
(
X24
↔
(
X8
↔
(
X5
→
X37
)
)
)
)
↔
(
(
X35
↔
(
X17
→
(
¬
X8
)
→
X24
)
)
→
(
(
¬
(
(
¬
X35
)
↔
(
(
¬
X7
)
→
X49
)
)
)
↔
(
¬
(
X17
↔
(
(
¬
X7
)
↔
X24
)
)
)
)
)
)
)
)
→
(
(
(
(
(
X45
→
False
)
↔
(
¬
(
X49
↔
(
(
¬
X31
)
→
(
X16
↔
X32
)
)
)
)
)
↔
(
¬
(
X47
→
(
X42
↔
(
X32
↔
X16
)
)
)
)
)
↔
(
(
¬
(
(
X46
→
X23
→
X14
)
→
(
¬
(
(
¬
False
)
→
(
¬
(
(
X46
→
X21
→
X14
→
X23
)
→
(
¬
(
(
¬
X46
)
→
(
(
¬
X23
)
↔
(
(
¬
X21
)
→
X47
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X32
↔
(
(
¬
X23
)
↔
(
X14
→
X42
)
)
)
)
→
(
¬
(
X32
↔
(
X23
↔
(
X18
↔
X42
)
)
)
)
)
→
(
(
¬
X23
)
↔
X32
)
)
→
(
¬
(
X21
↔
(
X14
→
X23
)
)
)
)
)
)
)
↔
(
(
(
X46
↔
(
X32
→
X1
→
X10
)
)
↔
(
¬
(
(
¬
X46
)
→
(
¬
X42
)
→
(
(
¬
X10
)
↔
X48
)
)
)
)
↔
(
(
(
¬
(
(
X48
→
X46
→
(
X42
↔
(
X25
↔
False
)
)
)
→
X46
→
(
¬
X25
)
→
(
X9
↔
X42
)
)
)
↔
(
X42
↔
X25
)
)
→
X48
→
(
(
¬
X37
)
↔
(
X35
→
X42
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
¬
(
(
(
X47
↔
(
(
¬
X28
)
↔
(
X17
↔
X34
)
)
)
→
(
¬
(
X47
↔
(
X23
↔
X34
)
)
)
→
X34
→
(
¬
X26
)
→
X23
→
X28
)
→
(
(
(
¬
(
X42
→
(
X26
↔
(
X5
→
X39
)
)
)
)
→
(
¬
(
(
¬
(
X48
→
(
X39
↔
(
X34
→
False
)
)
)
)
→
(
¬
(
X47
→
(
(
¬
X34
)
↔
(
X5
→
X39
)
)
)
)
)
)
)
↔
(
X48
→
(
X34
↔
X42
)
)
)
)
)
→
(
¬
(
(
¬
(
X26
↔
X49
)
)
→
(
(
(
¬
(
(
¬
(
(
¬
X34
)
→
(
(
¬
X28
)
↔
(
(
¬
X10
)
↔
X36
)
)
)
)
→
(
¬
(
X34
↔
(
X28
↔
(
X1
→
X36
)
)
)
)
)
)
↔
(
¬
(
X34
→
(
X28
↔
(
X26
↔
X1
)
)
)
)
)
↔
(
¬
(
X34
→
(
(
¬
X26
)
↔
X36
)
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
X41
)
→
X8
→
(
X3
↔
X47
)
)
↔
(
¬
(
X45
↔
(
X41
→
X8
→
X3
→
False
)
)
)
)
↔
(
¬
(
X45
→
(
X40
↔
X42
)
)
)
)
→
(
X45
↔
(
(
¬
X42
)
→
X7
→
X47
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
¬
X32
)
↔
(
(
¬
X22
)
→
X18
→
X40
)
)
)
→
(
(
¬
(
X41
↔
(
(
¬
X32
)
↔
X40
)
)
)
↔
(
¬
(
X40
↔
(
(
¬
X18
)
↔
X21
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X49
→
(
X41
↔
(
(
¬
X2
)
↔
X32
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X32
)
↔
(
X19
↔
(
X15
→
X41
)
)
)
)
→
(
¬
(
(
X49
↔
(
X41
→
X32
→
X19
→
False
)
)
→
(
¬
(
(
¬
X19
)
→
X18
→
X41
)
)
)
)
)
)
)
)
↔
(
(
(
¬
X47
)
→
(
¬
X32
)
→
(
¬
X29
)
→
X49
)
↔
(
¬
(
X49
→
X41
→
(
(
¬
X32
)
↔
X47
)
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
X43
↔
(
X37
→
(
¬
X33
)
→
X48
)
)
→
(
(
¬
(
X44
→
(
X6
↔
X47
)
)
)
↔
(
(
¬
(
X43
↔
(
X37
↔
X35
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X44
)
→
(
X37
↔
(
X6
↔
X48
)
)
)
)
→
(
¬
(
X6
↔
X43
)
)
)
→
(
¬
(
X46
→
(
X43
↔
(
X6
↔
X44
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
X48
↔
(
X16
→
(
(
¬
X11
)
↔
X37
)
)
)
↔
(
X37
↔
(
(
¬
X11
)
→
(
¬
X0
)
→
X48
)
)
)
↔
(
X44
→
(
(
¬
X16
)
↔
(
X11
→
X48
)
)
)
)
↔
(
¬
(
X16
→
(
¬
X11
)
→
X9
→
X48
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
¬
(
X38
↔
(
(
¬
X13
)
↔
X37
)
)
)
→
(
¬
(
(
¬
X38
)
↔
(
X37
→
X27
→
X44
)
)
)
)
)
→
X38
→
(
(
¬
X37
)
↔
X44
)
)
)
→
(
¬
(
(
(
(
¬
(
X49
→
(
X37
↔
(
X31
→
X30
)
)
)
)
→
(
¬
(
X49
↔
(
X20
→
X30
)
)
)
)
↔
(
¬
(
(
¬
(
X49
↔
(
X31
↔
(
(
¬
X25
)
↔
X48
)
)
)
)
→
(
¬
(
X49
↔
(
X30
↔
(
X12
→
X31
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X48
→
(
(
¬
X14
)
↔
X25
)
)
)
→
(
(
X44
↔
(
X24
↔
(
X14
↔
X25
)
)
)
↔
(
¬
(
X48
↔
(
X44
↔
(
X25
↔
(
X14
↔
False
)
)
)
)
)
)
)
)
→
(
(
X44
→
X38
→
(
(
¬
X4
)
↔
X48
)
)
↔
(
(
X44
↔
(
(
¬
X4
)
↔
X30
)
)
↔
(
(
¬
X24
)
↔
(
(
¬
X4
)
→
X48
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
967b6a...
and proposition id is
8b68c1...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMLBc316Kx37tDx5NWSXTA6DzV6r78AgtBQ
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
¬
(
(
X43
→
(
¬
X41
)
→
(
¬
X11
)
→
X49
)
→
(
(
¬
(
X12
→
X49
)
)
→
(
¬
(
X49
→
(
X13
↔
(
X5
↔
X29
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
¬
X26
)
↔
(
X25
↔
(
(
¬
X13
)
↔
X43
)
)
)
)
→
(
¬
(
X25
↔
(
X13
↔
X29
)
)
)
)
)
→
(
(
¬
(
X41
↔
(
X29
↔
(
(
¬
X25
)
↔
X26
)
)
)
)
↔
(
¬
(
X29
↔
(
X25
→
(
¬
X12
)
→
X41
)
)
)
)
)
)
)
)
→
(
(
(
¬
X48
)
↔
(
(
¬
X45
)
↔
X49
)
)
↔
(
(
¬
(
X45
→
(
(
¬
X29
)
↔
X43
)
)
)
↔
(
¬
(
(
¬
X33
)
→
(
X28
↔
X45
)
)
)
)
)
)
↔
(
(
¬
(
(
(
X6
→
X2
→
X45
)
↔
(
(
X43
↔
(
X27
↔
(
X6
→
X2
)
)
)
→
(
¬
(
X14
↔
X43
)
)
)
)
→
(
¬
(
(
¬
X27
)
↔
(
X14
→
(
¬
X6
)
→
X38
)
)
)
)
)
→
(
(
(
(
(
¬
(
(
¬
X39
)
↔
(
(
¬
X38
)
↔
(
(
¬
X8
)
↔
X45
)
)
)
)
↔
(
(
X38
↔
(
X36
↔
X45
)
)
→
(
¬
(
(
X45
↔
(
X36
→
(
(
¬
X8
)
↔
X38
)
)
)
→
X48
→
(
¬
X38
)
→
(
X36
↔
X45
)
)
)
)
)
→
(
X43
↔
(
(
¬
X36
)
→
X8
→
X48
)
)
)
↔
(
¬
(
X39
→
(
X38
↔
(
(
¬
X35
)
↔
X43
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X43
→
(
(
¬
X36
)
↔
(
X9
→
X45
)
)
)
)
↔
(
X45
→
X43
→
X17
→
X36
)
)
→
(
¬
(
X45
→
X36
→
X17
)
)
)
↔
(
¬
(
X45
→
(
X43
↔
(
X36
→
X0
→
False
)
)
)
)
)
→
(
¬
(
X9
→
X36
)
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
X45
)
→
(
X44
↔
(
(
¬
X43
)
→
X48
)
)
)
→
(
X48
↔
(
(
¬
X43
)
↔
(
(
¬
X23
)
→
X44
)
)
)
)
)
→
(
¬
(
X48
↔
X44
)
)
)
)
→
X45
→
X44
→
(
¬
X32
)
→
X43
)
)
↔
(
(
(
(
(
X40
→
X24
→
X49
)
↔
(
(
X41
↔
(
X24
→
(
¬
X21
)
→
X46
)
)
→
(
¬
X24
)
→
(
X21
↔
X49
)
)
)
→
(
(
X40
→
(
¬
X22
)
→
(
¬
X18
)
→
X24
)
↔
(
¬
(
X49
→
(
X40
↔
(
X24
↔
X22
)
)
)
)
)
)
↔
(
(
(
¬
(
X41
→
(
X16
↔
X40
)
)
)
→
(
¬
(
(
¬
X37
)
↔
(
X31
→
X47
)
)
)
)
→
(
(
¬
(
X46
→
(
(
¬
X37
)
↔
X40
)
)
)
↔
(
(
¬
(
(
¬
X37
)
→
X46
)
)
↔
(
¬
(
X40
→
(
¬
X37
)
→
(
X34
↔
X41
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
(
¬
X15
)
→
X19
)
)
→
(
¬
X47
)
→
(
X42
↔
(
(
¬
X15
)
↔
X48
)
)
)
→
(
¬
(
(
¬
X30
)
↔
(
X19
→
X15
→
X48
)
)
)
)
)
→
(
(
(
X37
↔
(
(
¬
X4
)
→
(
¬
X1
)
→
X40
)
)
→
(
¬
(
(
¬
X40
)
↔
(
(
¬
X37
)
→
(
X4
↔
X47
)
)
)
)
)
↔
(
¬
(
X47
↔
X42
)
)
)
→
(
(
(
X48
↔
(
X42
↔
(
X40
↔
X47
)
)
)
↔
(
X47
↔
(
(
¬
X40
)
↔
(
X7
→
X48
)
)
)
)
↔
(
(
X48
→
(
X40
↔
(
(
¬
X10
)
→
X42
)
)
)
↔
(
(
(
¬
X20
)
↔
(
X10
→
X40
)
)
↔
(
(
¬
(
X48
↔
(
X20
→
False
)
)
)
↔
(
(
¬
(
X40
→
X20
→
X3
→
False
)
)
↔
(
(
¬
(
X48
↔
(
(
¬
X10
)
↔
(
(
¬
X3
)
↔
X40
)
)
)
)
↔
(
X20
→
X10
)
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
(
¬
X21
)
↔
(
(
¬
X19
)
↔
(
X6
↔
X31
)
)
)
↔
(
(
¬
(
(
¬
X31
)
→
(
¬
X26
)
→
(
X21
↔
X47
)
)
)
↔
(
X31
→
(
¬
X26
)
→
(
¬
X19
)
→
X47
)
)
)
↔
(
(
¬
(
(
¬
X26
)
→
(
(
¬
X16
)
↔
X39
)
)
)
↔
(
¬
(
X47
→
(
X19
↔
X39
)
)
)
)
)
→
(
¬
(
(
(
(
X39
→
(
(
¬
X32
)
↔
(
X31
↔
X33
)
)
)
↔
(
¬
(
(
(
(
X48
↔
(
X39
↔
X31
)
)
↔
(
¬
(
(
¬
X36
)
→
(
¬
X31
)
→
(
X21
↔
X48
)
)
)
)
↔
(
X36
↔
X48
)
)
→
(
¬
(
(
¬
X33
)
→
X31
→
X36
)
)
)
)
)
→
(
X48
↔
(
X33
↔
X32
)
)
)
→
(
(
X48
↔
(
(
¬
X32
)
↔
(
(
¬
X5
)
→
X33
)
)
)
↔
(
X48
→
X33
→
(
¬
X8
)
→
X32
)
)
)
)
)
↔
(
(
(
(
¬
(
X49
↔
(
(
¬
X31
)
↔
(
(
¬
X24
)
↔
X38
)
)
)
)
→
(
¬
(
X38
↔
(
(
¬
X12
)
↔
X49
)
)
)
→
(
¬
(
X49
↔
(
X32
↔
(
X31
→
X12
)
)
)
)
)
↔
(
(
¬
(
X49
↔
(
X29
→
X24
→
X18
→
False
)
)
)
↔
(
¬
(
(
¬
(
X29
→
(
¬
X18
)
→
X24
)
)
→
(
¬
(
X29
↔
(
(
¬
X24
)
↔
(
(
¬
X4
)
→
X31
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X28
↔
(
X0
→
False
)
)
)
↔
(
¬
(
(
¬
(
X38
→
(
X32
↔
(
X28
↔
X49
)
)
)
)
→
(
X32
↔
(
(
¬
X26
)
→
X28
)
)
)
)
)
→
(
(
¬
X32
)
↔
(
(
¬
X10
)
↔
(
X0
↔
X49
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
(
(
X49
↔
(
X7
↔
X48
)
)
↔
(
(
¬
X44
)
↔
(
X43
↔
X49
)
)
)
→
(
(
(
¬
(
(
¬
X44
)
→
X41
→
X49
)
)
↔
(
(
¬
X43
)
→
(
¬
X2
)
→
X44
)
)
↔
(
X49
→
(
X41
↔
X43
)
)
)
)
)
→
(
¬
(
X48
→
(
X13
↔
(
(
¬
X7
)
→
X42
)
)
)
)
)
)
↔
(
(
(
X49
↔
(
X30
→
X15
)
)
↔
(
¬
(
(
(
¬
X44
)
→
X23
→
(
¬
X1
)
→
X49
)
→
(
¬
X30
)
→
(
¬
X23
)
→
X44
)
)
)
↔
(
(
X44
→
(
¬
X30
)
→
X35
)
→
(
¬
(
(
¬
X44
)
→
(
X41
↔
X49
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X34
→
(
¬
X20
)
→
(
X9
↔
X27
)
)
)
↔
(
¬
(
X27
→
X25
→
X20
→
X9
→
False
)
)
)
↔
(
(
(
¬
X47
)
↔
(
(
¬
X37
)
↔
(
X20
↔
X48
)
)
)
↔
(
(
¬
(
(
¬
X37
)
↔
(
X20
→
(
¬
X3
)
→
X47
)
)
)
↔
(
¬
(
(
¬
(
X9
→
X27
)
)
→
(
(
¬
X27
)
→
(
¬
X20
)
→
(
(
¬
X9
)
↔
X47
)
)
→
(
¬
(
(
¬
X20
)
↔
(
X9
→
(
¬
X3
)
→
X47
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
X44
)
→
(
(
¬
X34
)
↔
(
X11
↔
X48
)
)
)
→
(
(
X46
→
(
X44
↔
(
X27
→
X11
→
False
)
)
)
↔
(
¬
(
X48
→
(
X44
↔
X46
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
X44
)
↔
(
(
¬
X42
)
→
(
(
¬
X34
)
↔
X46
)
)
)
→
(
¬
(
X46
→
(
X42
↔
(
(
¬
X22
)
↔
X44
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X48
→
(
X14
↔
False
)
)
)
→
(
¬
(
(
X47
↔
(
X45
↔
(
X44
→
X40
)
)
)
→
(
(
X47
↔
(
X40
↔
(
(
¬
X14
)
→
X45
)
)
)
↔
(
¬
(
X47
↔
(
X45
→
X40
→
X17
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
732c86...
and proposition id is
b7446c...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMUHBbGUzobMPnUHz3JvKYhjiWhnzqfLoCw
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
¬
∀X51 :
prop
,
(
(
(
¬
(
(
(
(
(
(
¬
(
X50
→
X16
→
X7
)
)
↔
(
¬
(
(
¬
X46
)
→
(
X43
↔
(
X7
→
X50
)
)
)
)
)
↔
(
(
X49
→
(
¬
X27
)
→
(
X15
↔
X43
)
)
→
X46
→
X43
→
X15
)
)
→
(
(
(
(
¬
(
(
(
¬
(
X47
→
(
(
¬
X43
)
↔
(
X24
↔
X50
)
)
)
)
↔
(
(
¬
X41
)
→
(
¬
X24
)
→
X47
)
)
→
(
¬
(
X47
↔
X43
)
)
)
)
→
(
¬
(
X43
↔
(
X24
↔
X47
)
)
)
)
↔
(
¬
(
X50
↔
X24
)
)
)
↔
(
(
¬
X47
)
→
X50
)
)
→
X51
→
(
X50
↔
(
X24
→
X49
)
)
)
→
(
(
(
(
¬
(
(
¬
X47
)
→
(
(
¬
X40
)
↔
(
X27
→
X49
)
)
)
)
↔
(
X49
→
X27
→
(
¬
X19
)
→
X47
)
)
↔
(
¬
(
X40
↔
(
(
¬
X29
)
↔
(
(
¬
X19
)
↔
X47
)
)
)
)
)
↔
(
(
¬
(
X49
↔
(
X40
→
(
(
¬
X29
)
↔
X42
)
)
)
)
→
(
¬
(
(
(
X40
→
X2
→
(
(
¬
X1
)
↔
X42
)
)
↔
(
¬
(
(
¬
X40
)
→
(
X31
↔
(
X2
↔
X42
)
)
)
)
)
→
(
¬
X40
)
→
(
X31
↔
(
X1
→
X49
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
(
(
(
¬
X45
)
→
(
¬
X28
)
→
(
¬
X21
)
→
X51
)
→
(
¬
(
X49
↔
(
(
¬
X45
)
↔
(
X28
↔
X50
)
)
)
)
)
→
(
¬
(
X45
→
(
X21
↔
X50
)
)
)
)
)
↔
(
¬
(
X50
↔
(
(
¬
X45
)
→
(
¬
X4
)
→
X51
)
)
)
)
↔
(
¬
(
(
(
(
X49
→
(
X25
↔
False
)
)
↔
(
¬
(
(
X51
→
(
X50
↔
X14
)
)
→
(
¬
(
X49
↔
(
(
¬
X25
)
→
(
X14
↔
X50
)
)
)
)
)
)
)
↔
(
(
¬
X50
)
→
X51
)
)
→
(
¬
(
(
¬
X50
)
↔
(
(
¬
X25
)
↔
X51
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X23
→
(
X13
↔
(
X11
→
X51
)
)
)
)
→
(
(
¬
X48
)
↔
X51
)
)
↔
(
(
(
X38
↔
(
X23
↔
(
X13
→
X51
)
)
)
↔
(
X51
→
(
¬
X23
)
→
(
¬
X12
)
→
X38
)
)
↔
(
X51
→
X13
→
False
)
)
)
→
(
(
¬
(
X49
↔
X30
)
)
↔
(
(
¬
(
(
¬
X50
)
→
(
¬
X49
)
→
X51
)
)
→
(
X49
↔
(
(
¬
X9
)
↔
X10
)
)
)
)
→
(
(
¬
(
(
X49
→
X38
→
(
¬
X10
)
→
X48
)
→
(
¬
(
X38
↔
(
X10
→
(
¬
X6
)
→
X49
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X49
)
↔
(
(
¬
X48
)
↔
X50
)
)
)
↔
(
(
¬
X38
)
↔
(
(
¬
X18
)
→
X50
)
)
)
↔
(
X48
↔
X49
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
(
(
(
¬
X39
)
↔
(
(
¬
X35
)
→
(
¬
X26
)
→
X50
)
)
→
(
¬
(
(
¬
X32
)
↔
(
X26
↔
X35
)
)
)
)
)
→
(
(
¬
(
X37
↔
(
X5
↔
False
)
)
)
↔
(
¬
(
X50
→
X32
→
X39
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X35
)
→
(
¬
X33
)
→
(
(
¬
X26
)
↔
X44
)
)
)
→
(
X37
↔
(
(
¬
X34
)
→
(
¬
X33
)
→
X35
)
)
→
(
X34
↔
(
(
¬
X26
)
→
(
X3
↔
X37
)
)
)
→
(
¬
(
X35
→
X34
→
(
(
¬
X3
)
↔
X26
)
)
)
)
)
)
→
(
(
¬
(
X34
→
(
¬
X20
)
→
X49
)
)
↔
(
¬
(
X49
→
X35
→
X33
→
X20
→
False
)
)
)
)
→
(
(
X49
↔
(
(
¬
X36
)
↔
(
(
¬
X0
)
→
X48
)
)
)
↔
(
¬
(
(
¬
X39
)
↔
(
X36
→
(
X0
↔
X49
)
)
)
)
)
→
(
(
(
X44
↔
X8
)
↔
(
(
X49
↔
(
X36
↔
(
(
¬
X17
)
→
X44
)
)
)
↔
(
X49
→
(
X48
↔
(
X44
→
(
X36
↔
False
)
)
)
)
)
)
↔
(
X48
↔
(
(
¬
X44
)
→
(
¬
X22
)
→
X49
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
¬
(
(
X30
→
X1
→
(
¬
X0
)
→
X11
)
→
(
¬
(
X1
→
X0
→
X30
)
)
→
(
X48
↔
(
X26
↔
(
(
¬
X0
)
→
X30
)
)
)
)
)
↔
(
¬
(
X50
→
X26
→
(
X11
↔
X48
)
)
)
)
→
(
(
(
¬
X48
)
↔
(
(
¬
X11
)
→
X7
→
X50
)
)
↔
(
(
X50
↔
(
X48
→
X11
→
X3
)
)
↔
(
X49
→
(
X11
↔
(
X3
→
False
)
)
)
)
)
)
↔
(
(
¬
(
(
X50
→
(
X48
↔
X49
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
X29
→
(
¬
X16
)
→
X50
)
→
(
¬
(
X50
↔
(
X22
↔
X48
)
)
)
)
)
→
(
¬
(
X50
↔
(
X22
→
False
)
)
)
)
)
→
(
(
¬
(
X49
→
(
(
¬
X29
)
↔
X48
)
)
)
↔
(
¬
(
(
¬
(
X48
↔
(
X29
↔
(
(
¬
X17
)
↔
X49
)
)
)
)
→
X50
→
(
X49
↔
X29
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
X22
)
↔
(
X17
→
(
¬
X13
)
→
X47
)
)
)
↔
(
¬
(
(
¬
X47
)
→
(
X22
↔
(
(
¬
X17
)
→
X50
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
X46
)
→
(
X41
↔
(
X10
↔
X48
)
)
)
→
(
(
¬
(
X50
→
X49
→
X48
→
X41
→
False
)
)
↔
(
(
(
¬
X49
)
↔
(
X48
↔
(
X18
↔
X50
)
)
)
↔
(
X50
→
(
(
¬
X46
)
↔
(
(
¬
X18
)
↔
X48
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X42
)
↔
(
X28
↔
(
X19
→
X45
)
)
)
)
→
(
(
¬
X45
)
↔
(
(
¬
X28
)
→
X50
)
)
)
↔
(
¬
(
(
(
¬
X39
)
↔
X42
)
→
(
X50
↔
(
X49
↔
(
X28
→
X42
)
)
)
)
)
)
→
(
(
(
X49
→
X46
→
(
¬
X23
)
→
X45
)
↔
(
¬
(
(
¬
X49
)
↔
(
(
¬
X46
)
↔
(
X45
→
X50
)
)
)
)
)
↔
(
¬
(
(
¬
X46
)
→
(
X45
↔
(
(
¬
X12
)
↔
X50
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
X8
↔
X24
)
↔
(
(
¬
(
X46
↔
(
X36
→
(
(
¬
X24
)
↔
X34
)
)
)
)
↔
(
¬
(
X43
→
(
X34
↔
(
(
¬
X24
)
↔
X46
)
)
)
)
)
)
→
(
(
¬
(
X44
↔
(
X34
↔
X31
)
)
)
↔
(
(
X36
↔
(
X34
↔
X6
)
)
↔
(
(
¬
(
X34
↔
(
X6
→
False
)
)
)
→
(
X36
↔
(
X31
→
(
(
¬
X4
)
↔
X6
)
)
)
)
)
)
→
(
(
X6
↔
(
(
¬
X2
)
→
X46
)
)
↔
(
X46
→
(
X36
↔
(
X31
→
X2
)
)
)
)
→
(
¬
X31
)
→
(
(
¬
X6
)
↔
X46
)
)
)
↔
(
(
(
¬
(
X44
↔
(
(
¬
X25
)
→
X51
)
)
)
↔
(
¬
(
X44
→
(
X36
↔
(
(
¬
X25
)
→
X50
)
)
)
)
)
→
(
(
X51
→
X36
→
(
X31
↔
(
X21
↔
False
)
)
)
↔
(
X51
↔
(
(
¬
X36
)
→
(
(
¬
X31
)
↔
X50
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
(
X45
↔
(
X44
↔
(
(
¬
X20
)
↔
X51
)
)
)
)
↔
(
(
(
¬
X15
)
↔
(
(
¬
X9
)
→
X44
)
)
→
(
¬
X45
)
→
X44
→
(
(
¬
X9
)
↔
X51
)
)
)
→
(
(
¬
(
X51
→
X40
→
(
(
¬
X38
)
↔
X50
)
)
)
→
(
¬
(
X51
→
X40
→
(
X38
↔
X45
)
)
)
)
→
X40
→
X5
→
X44
)
)
→
(
¬
(
(
(
(
¬
(
X51
↔
(
X45
→
(
X35
↔
X32
)
)
)
)
↔
(
¬
(
(
¬
X45
)
↔
(
X35
↔
(
X14
↔
X51
)
)
)
)
)
↔
(
(
¬
(
X33
→
X27
→
X45
)
)
→
(
(
(
(
(
(
¬
(
X33
↔
(
X27
↔
X45
)
)
)
→
(
(
X45
→
X33
→
(
¬
X27
)
→
X51
)
↔
(
X44
→
(
(
¬
X27
)
↔
X45
)
)
)
)
↔
(
¬
(
X51
↔
(
(
¬
X35
)
↔
(
X33
↔
X44
)
)
)
)
)
↔
(
¬
(
(
¬
X33
)
↔
(
(
¬
X27
)
→
X35
)
)
)
)
↔
(
¬
(
(
¬
X35
)
→
(
¬
X27
)
→
X51
)
)
)
↔
(
X45
→
(
(
¬
X35
)
↔
X51
)
)
)
)
)
→
(
¬
(
X45
→
X37
→
(
(
¬
X33
)
↔
X44
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
473318...
and proposition id is
8e8da1...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMFNHWEECi3ruJosjPMaGkJeWnqciMb27E4
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
∀X50 :
prop
,
(
(
(
(
(
(
¬
(
(
¬
(
X49
↔
(
(
¬
X45
)
↔
(
X32
→
X50
)
)
)
)
→
(
¬
(
X49
→
X45
→
X34
→
X50
)
)
)
)
↔
(
(
X49
→
(
X34
↔
(
X32
↔
(
X23
→
False
)
)
)
)
→
(
¬
(
(
(
¬
X32
)
→
(
¬
X23
)
→
(
¬
X2
)
→
X49
)
→
(
X49
↔
(
X23
↔
(
X2
↔
X28
)
)
)
)
)
)
)
→
(
(
X45
↔
(
(
¬
X34
)
→
(
X17
↔
X48
)
)
)
↔
(
(
X48
↔
(
(
¬
X32
)
→
(
(
¬
X19
)
↔
X45
)
)
)
↔
(
X45
↔
(
X32
↔
(
(
¬
X18
)
↔
X34
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
X45
→
(
X42
↔
(
(
¬
X33
)
↔
X41
)
)
)
↔
(
(
X42
→
(
¬
X41
)
→
(
X26
↔
X45
)
)
↔
(
(
¬
X42
)
↔
(
X41
→
X37
→
X45
)
)
)
)
↔
(
X26
→
X37
)
)
→
(
X42
→
(
X16
↔
X45
)
)
→
(
(
¬
(
(
(
X45
↔
(
X44
→
(
¬
X16
)
→
X26
)
)
→
(
¬
(
(
¬
X44
)
→
(
X42
↔
(
X26
→
X45
)
)
)
)
)
→
(
¬
X45
)
→
(
(
¬
X44
)
↔
(
(
¬
X26
)
↔
X49
)
)
)
)
↔
(
X44
→
X42
→
X45
)
)
)
)
↔
(
(
(
(
¬
(
(
(
¬
X42
)
→
(
¬
X27
)
→
(
(
¬
X6
)
↔
X44
)
)
→
(
X48
↔
(
X44
↔
(
X6
→
X27
)
)
)
)
)
→
(
(
¬
(
(
(
(
¬
(
X42
↔
(
(
¬
X6
)
→
X44
)
)
)
↔
(
¬
(
X33
↔
(
X10
→
(
¬
X6
)
→
X44
)
)
)
)
→
(
¬
(
X44
↔
(
(
¬
X10
)
↔
X42
)
)
)
)
→
(
X44
↔
(
X33
→
False
)
)
)
)
↔
(
¬
(
X42
↔
(
(
¬
X33
)
→
X44
)
)
)
)
)
↔
(
X42
→
(
¬
X33
)
→
X29
→
X48
)
)
↔
(
X42
↔
(
X29
→
X1
→
False
)
)
)
)
)
→
(
(
(
(
(
(
¬
(
X50
→
X47
→
False
)
)
↔
(
(
¬
(
X50
→
X38
→
(
(
¬
X12
)
↔
X36
)
)
)
↔
(
X47
↔
(
(
¬
X36
)
→
(
(
¬
X12
)
↔
X38
)
)
)
)
)
→
(
(
(
X30
→
(
¬
X5
)
→
X47
)
↔
(
¬
(
X47
↔
(
(
¬
X22
)
↔
X50
)
)
)
)
↔
(
(
¬
X5
)
→
X38
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X22
)
↔
(
(
¬
X4
)
→
(
¬
X0
)
→
X50
)
)
)
→
(
(
X46
↔
(
(
¬
X4
)
→
(
¬
X0
)
→
X22
)
)
↔
(
(
¬
X30
)
↔
(
X22
↔
X46
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
X47
)
↔
(
X38
↔
X48
)
)
→
(
(
¬
(
(
¬
X47
)
↔
(
(
¬
X38
)
→
X48
)
)
)
↔
(
(
¬
X43
)
→
(
¬
X38
)
→
(
¬
X14
)
→
X48
)
)
)
)
↔
(
(
(
¬
(
X49
↔
(
X48
↔
(
X47
↔
(
X11
→
False
)
)
)
)
)
↔
(
X49
↔
(
X47
↔
(
X43
→
False
)
)
)
)
→
X48
→
(
(
¬
X11
)
↔
X47
)
)
)
↔
(
(
(
¬
(
X47
→
(
(
¬
X7
)
↔
X15
)
)
)
↔
(
X49
↔
(
(
¬
X35
)
→
(
(
¬
X15
)
↔
X47
)
)
)
)
↔
(
(
¬
(
X49
↔
(
(
¬
X36
)
→
(
(
¬
X7
)
↔
X43
)
)
)
)
↔
(
¬
(
X43
↔
(
X15
↔
X7
)
)
)
)
)
)
)
↔
(
(
(
(
X47
↔
X9
)
↔
(
¬
(
(
(
X47
↔
(
X9
→
X39
)
)
↔
(
¬
(
X49
→
(
X47
↔
(
(
¬
X24
)
→
X39
)
)
)
)
)
→
(
¬
X47
)
→
(
(
¬
X39
)
↔
(
X25
↔
X49
)
)
)
)
)
↔
(
¬
(
(
X49
→
X40
→
X39
)
→
(
¬
(
X43
↔
(
X40
↔
X47
)
)
)
)
)
)
→
(
(
¬
(
X49
↔
(
X47
↔
(
X21
→
X31
)
)
)
)
↔
(
¬
(
(
(
¬
(
X31
↔
X3
)
)
↔
(
(
(
¬
(
X39
→
X49
)
)
→
(
¬
(
(
¬
X31
)
→
(
¬
X13
)
→
(
¬
X8
)
→
X49
)
)
)
→
(
X49
↔
(
X39
→
X31
→
X3
)
)
)
)
→
(
¬
(
X31
↔
(
(
¬
X20
)
→
(
¬
X13
)
→
X47
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
(
¬
(
X47
→
(
X45
↔
(
X6
→
X40
)
)
)
)
↔
(
(
¬
(
(
¬
X45
)
→
(
¬
X40
)
→
(
(
¬
X30
)
↔
X47
)
)
)
↔
(
X40
↔
(
X30
→
X8
→
False
)
)
)
)
→
(
¬
(
(
¬
(
X35
↔
(
(
¬
X9
)
↔
(
(
¬
X8
)
→
X48
)
)
)
)
→
(
(
(
X40
↔
(
X9
→
(
X6
↔
X48
)
)
)
↔
(
(
¬
X48
)
→
(
¬
X40
)
→
(
X9
↔
X49
)
)
)
↔
(
¬
(
X48
↔
(
X35
→
(
(
¬
X9
)
↔
X49
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
X45
)
→
X46
)
↔
(
¬
(
X45
↔
(
X32
→
(
X30
↔
(
X24
→
False
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X35
→
(
X32
↔
X45
)
)
)
↔
(
(
¬
(
(
¬
X23
)
→
(
(
¬
X3
)
↔
X32
)
)
)
↔
(
¬
(
X45
↔
(
(
¬
X23
)
→
X35
)
)
)
)
)
→
(
¬
(
(
¬
X35
)
↔
(
(
¬
X4
)
↔
X46
)
)
)
)
)
)
)
)
→
(
(
(
X26
↔
(
X25
↔
(
(
¬
X10
)
→
X47
)
)
)
↔
(
¬
(
(
X33
→
X31
→
X26
)
→
(
¬
(
X47
↔
(
X26
↔
X36
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
¬
(
X47
→
(
(
¬
X38
)
↔
X49
)
)
)
→
(
(
¬
(
X40
↔
(
X38
↔
(
(
¬
X33
)
↔
X47
)
)
)
)
↔
(
¬
(
(
¬
X47
)
↔
(
X38
↔
(
(
¬
X1
)
↔
X49
)
)
)
)
)
)
)
→
(
¬
(
(
¬
X38
)
↔
(
(
¬
X25
)
→
X1
→
X49
)
)
)
)
)
→
X49
→
(
X40
↔
(
X38
↔
False
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
(
¬
(
X5
↔
(
(
¬
X2
)
→
X39
)
)
)
→
(
¬
(
X50
↔
(
(
¬
X5
)
↔
X42
)
)
)
)
)
→
(
¬
(
(
X50
↔
(
X41
↔
(
(
¬
X39
)
↔
X48
)
)
)
→
(
(
(
¬
(
X50
↔
(
(
¬
X34
)
→
(
¬
X18
)
→
X41
)
)
)
→
(
¬
(
(
¬
X48
)
↔
(
X41
→
X34
→
X50
)
)
)
)
↔
(
(
¬
(
X50
→
(
X48
↔
(
X37
→
X5
)
)
)
)
→
(
¬
(
X48
↔
(
(
¬
X34
)
↔
(
(
¬
X5
)
→
X50
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
X48
→
X44
→
False
)
↔
(
(
(
X50
↔
(
X48
↔
(
X44
↔
(
X20
→
False
)
)
)
)
→
(
¬
(
X50
→
(
¬
X43
)
→
X44
)
)
→
(
(
¬
(
(
¬
X48
)
→
(
X44
↔
(
(
¬
X43
)
→
X50
)
)
)
)
↔
(
(
¬
X44
)
→
X43
→
(
X22
↔
X50
)
)
)
)
↔
(
¬
(
X50
→
(
X43
↔
X44
)
)
)
)
)
→
X50
→
X22
→
X12
→
X44
)
→
(
¬
(
(
¬
(
X44
↔
(
X22
↔
(
X11
↔
False
)
)
)
)
→
(
X50
→
X42
)
→
(
X42
→
(
(
¬
X11
)
↔
(
X0
↔
X50
)
)
)
→
(
¬
(
(
¬
X44
)
↔
(
X42
↔
(
X11
↔
X50
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X43
↔
(
X41
→
X14
→
X15
)
)
)
↔
(
(
¬
(
X14
↔
X41
)
)
↔
(
X49
→
X41
→
(
(
¬
X15
)
↔
X48
)
)
)
)
→
(
X43
→
(
X7
↔
X13
)
)
→
(
¬
(
X49
↔
(
X43
↔
(
X7
→
X41
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X28
↔
(
X27
→
X48
)
)
)
→
(
¬
(
X44
→
(
¬
X29
)
→
X28
→
X48
)
)
)
)
↔
(
(
¬
(
(
X42
→
(
X21
↔
(
(
¬
X17
)
↔
X44
)
)
)
→
(
(
X47
→
(
¬
X29
)
→
X42
)
↔
(
(
¬
X29
)
↔
(
(
¬
X21
)
↔
(
X17
→
X47
)
)
)
)
)
)
↔
(
(
(
¬
X47
)
↔
(
X29
↔
(
X19
↔
X48
)
)
)
→
(
X47
↔
(
X42
→
(
X29
↔
X48
)
)
)
→
X48
→
X29
→
X16
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
9e93f1...
and proposition id is
79a896...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMFmaTXipDQj2xVShiL8JNBi9sDs5XcG8fb
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
∀X51 :
prop
,
∀X52 :
prop
,
∀X53 :
prop
,
(
(
(
(
(
(
(
(
(
¬
(
(
¬
(
X48
→
(
¬
X40
)
→
(
¬
X14
)
→
X49
)
)
→
X49
→
(
X48
↔
(
X40
→
X51
)
)
)
)
↔
(
X40
↔
(
X14
↔
X51
)
)
)
→
(
¬
(
X51
↔
(
X48
↔
(
(
¬
X24
)
↔
X49
)
)
)
)
)
↔
(
(
(
X51
→
(
(
¬
X45
)
↔
(
(
¬
X44
)
→
X46
)
)
)
↔
(
(
¬
X46
)
↔
(
X44
→
X49
)
)
)
↔
(
(
¬
(
X49
→
(
(
¬
X46
)
↔
(
X45
↔
X48
)
)
)
)
↔
(
¬
(
X48
↔
(
(
¬
X45
)
↔
(
(
¬
X39
)
→
X46
)
)
)
)
)
)
)
→
(
(
¬
X48
)
↔
(
X46
↔
(
X45
↔
X50
)
)
)
)
→
(
(
(
(
(
X48
→
(
(
¬
X43
)
↔
(
X2
→
X47
)
)
)
↔
(
(
¬
(
X50
↔
(
(
¬
X47
)
↔
(
(
¬
X37
)
→
X48
)
)
)
)
↔
(
(
¬
X47
)
↔
(
(
¬
X37
)
→
(
(
¬
X2
)
↔
X50
)
)
)
)
)
→
X37
→
(
X23
↔
(
X2
↔
False
)
)
)
↔
(
¬
(
X2
→
False
)
)
)
→
(
¬
(
(
(
¬
X13
)
↔
(
(
¬
X6
)
→
X50
)
)
→
(
¬
(
(
¬
(
(
¬
X47
)
↔
(
(
¬
X43
)
→
X50
)
)
)
→
(
(
¬
(
X47
→
(
X43
↔
(
(
¬
X13
)
→
X50
)
)
)
)
↔
(
X50
→
X6
→
False
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X37
)
↔
(
X31
↔
(
X1
↔
X43
)
)
)
)
↔
(
X49
→
X21
→
X1
→
X43
)
)
→
(
(
(
¬
(
(
¬
X43
)
→
X37
→
(
X31
↔
X47
)
)
)
↔
(
X47
→
(
X43
↔
(
X5
↔
False
)
)
)
)
↔
(
¬
(
X43
↔
(
X5
↔
X47
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
X50
↔
(
X33
↔
X42
)
)
→
(
¬
(
(
(
X50
↔
(
X28
↔
X15
)
)
↔
(
(
X42
↔
(
(
¬
X28
)
↔
(
(
¬
X15
)
↔
X33
)
)
)
→
(
¬
(
X15
→
X50
)
)
)
)
→
(
¬
(
X51
→
(
X50
↔
(
(
¬
X15
)
→
X18
)
)
)
)
)
)
)
→
(
(
(
X51
↔
(
X35
→
X33
→
X30
)
)
→
X28
→
X3
)
↔
(
X33
→
X3
→
X51
)
)
)
→
(
(
(
(
¬
(
X47
↔
(
(
¬
X35
)
↔
(
(
¬
X7
)
↔
X53
)
)
)
)
↔
(
¬
(
X53
↔
(
(
¬
X47
)
→
(
X7
↔
X50
)
)
)
)
)
↔
(
(
(
(
¬
(
X50
↔
(
X29
↔
(
(
¬
X10
)
→
X35
)
)
)
)
↔
(
X50
↔
(
(
¬
X35
)
↔
(
X29
→
X42
)
)
)
)
↔
(
(
¬
X47
)
→
X42
→
X35
→
X50
)
)
↔
(
¬
(
X35
↔
X47
)
)
)
)
↔
(
¬
(
X53
↔
(
X47
→
(
X42
↔
X17
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
X47
)
→
(
¬
X38
)
→
(
(
¬
X16
)
↔
X52
)
)
↔
(
(
X47
↔
(
X12
→
X51
)
)
→
(
(
¬
(
(
¬
X51
)
↔
(
X4
↔
X52
)
)
)
↔
(
¬
(
X47
↔
(
X36
↔
X4
)
)
)
)
→
(
¬
(
(
X51
→
(
(
¬
X38
)
↔
(
(
¬
X12
)
→
X52
)
)
)
→
(
¬
(
X52
↔
(
X51
→
X47
)
)
)
)
)
)
)
↔
(
(
¬
(
(
X47
→
X11
→
(
X8
↔
False
)
)
→
(
X22
↔
(
X11
→
(
(
¬
X8
)
↔
X36
)
)
)
→
(
(
¬
X36
)
↔
(
(
¬
X32
)
↔
(
X11
→
X47
)
)
)
)
)
→
(
¬
(
(
¬
X36
)
↔
(
X16
↔
(
X11
→
X47
)
)
)
)
)
)
→
(
(
(
X38
↔
(
X20
↔
(
X19
→
X25
)
)
)
→
(
(
(
¬
(
X51
→
(
X41
↔
X38
)
)
)
→
(
(
¬
(
(
¬
X44
)
↔
(
(
¬
X41
)
→
(
(
¬
X38
)
↔
X51
)
)
)
)
↔
(
¬
(
(
¬
X44
)
↔
(
X41
↔
(
X20
→
X51
)
)
)
)
)
)
↔
(
(
¬
X41
)
→
(
X38
↔
(
X19
↔
X44
)
)
)
)
)
↔
(
(
(
(
¬
(
X51
↔
X9
)
)
↔
(
(
(
¬
X44
)
↔
(
(
¬
X27
)
→
X9
→
X51
)
)
→
(
X51
↔
X41
)
)
)
↔
(
(
(
¬
(
X41
↔
(
X34
↔
(
(
¬
X0
)
→
X26
)
)
)
)
↔
(
¬
(
X47
↔
(
X26
↔
X0
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X51
→
X44
→
(
X41
↔
(
X34
↔
False
)
)
)
)
→
(
X44
↔
(
X41
↔
X47
)
)
)
)
→
(
¬
(
X44
→
X41
)
)
)
)
)
)
↔
(
¬
(
X47
↔
(
X44
↔
X51
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
(
¬
(
(
(
¬
X20
)
→
X9
→
X7
→
X37
)
→
(
X37
→
(
¬
X7
)
→
(
¬
X3
)
→
X9
)
→
X48
)
)
→
(
X48
↔
(
X9
→
(
(
¬
X7
)
↔
X28
)
)
)
)
↔
(
X36
→
(
X20
↔
(
X9
→
X7
)
)
)
)
→
(
(
(
(
(
¬
X2
)
↔
X49
)
→
X28
→
X2
→
(
¬
X0
)
→
X20
)
↔
(
(
¬
(
X26
→
(
¬
X20
)
→
X0
→
X49
)
)
→
(
¬
X49
)
→
(
(
¬
X26
)
↔
(
(
¬
X20
)
→
X53
)
)
)
)
↔
(
¬
(
(
(
¬
(
X28
↔
(
(
¬
X20
)
→
X29
)
)
)
↔
(
X49
↔
(
X48
↔
(
(
¬
X20
)
↔
X29
)
)
)
)
→
(
¬
X48
)
→
(
¬
X29
)
→
(
(
¬
X11
)
↔
X49
)
)
)
)
→
(
¬
(
X53
→
(
X48
↔
(
(
¬
X29
)
↔
X45
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
X43
)
→
X15
→
X13
→
X49
)
)
↔
(
¬
(
X49
→
X1
)
)
)
↔
(
¬
(
(
(
(
¬
X37
)
↔
(
X22
→
(
¬
X13
)
→
X51
)
)
↔
(
¬
(
(
¬
(
X30
↔
(
(
¬
X22
)
↔
(
(
¬
X18
)
↔
X51
)
)
)
)
→
(
¬
(
X51
↔
(
(
¬
X30
)
↔
(
(
¬
X22
)
→
X37
)
)
)
)
)
)
)
→
(
(
(
¬
X30
)
↔
(
X22
→
X43
)
)
↔
(
(
¬
(
X51
→
X30
→
X49
)
)
↔
(
¬
(
X49
↔
(
X30
→
(
¬
X22
)
→
X43
)
)
)
)
)
)
)
)
↔
(
¬
(
(
X23
↔
(
(
¬
X18
)
→
X37
)
)
→
(
¬
(
(
X43
→
(
(
¬
X18
)
↔
X23
)
)
→
(
¬
(
X48
→
(
¬
X37
)
→
(
¬
X23
)
→
X43
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
X39
↔
(
(
¬
X24
)
↔
(
X14
→
X51
)
)
)
→
(
(
¬
(
X39
→
X24
→
X16
)
)
↔
(
X52
↔
(
X44
↔
(
(
¬
X8
)
→
X51
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
X50
)
↔
(
X27
↔
(
(
¬
X6
)
→
X53
)
)
)
→
X50
→
(
X31
↔
False
)
)
↔
(
¬
(
(
¬
X42
)
→
(
(
¬
X6
)
↔
X51
)
)
)
)
↔
(
(
¬
(
X50
↔
(
(
¬
X38
)
↔
(
(
¬
X31
)
↔
X52
)
)
)
)
↔
(
X52
→
(
X38
↔
(
X31
↔
(
X27
↔
False
)
)
)
)
)
)
↔
(
¬
(
(
X52
↔
(
X33
→
X39
)
)
→
(
(
(
X50
→
X32
→
X52
)
↔
(
(
¬
X39
)
↔
(
X33
→
X50
)
)
)
↔
(
X53
→
X52
→
X33
→
X39
)
)
)
)
)
)
→
(
(
(
(
X53
↔
(
X49
↔
X46
)
)
↔
(
X53
↔
(
(
¬
X44
)
↔
(
X41
↔
X46
)
)
)
)
↔
(
¬
(
(
(
X40
↔
(
X25
→
False
)
)
→
(
¬
(
(
¬
(
(
X53
↔
(
X46
→
X49
)
)
→
(
(
X53
→
(
¬
X40
)
→
(
¬
X25
)
→
X46
)
↔
(
¬
(
X49
→
(
X46
↔
X53
)
)
)
)
)
)
→
(
X53
↔
(
X40
↔
X49
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X49
)
→
(
¬
X46
)
→
(
¬
X35
)
→
X53
)
)
→
X46
→
(
X35
↔
(
X12
→
X5
)
)
)
)
)
)
)
→
(
(
(
¬
(
X46
→
X21
→
X4
→
False
)
)
→
(
(
X46
↔
(
X10
↔
(
X4
→
False
)
)
)
↔
(
X53
→
(
(
¬
X44
)
↔
(
(
¬
X4
)
↔
X46
)
)
)
)
)
↔
(
¬
(
(
X53
↔
(
(
¬
X34
)
↔
(
(
¬
X10
)
→
X52
)
)
)
→
(
¬
X52
)
→
(
X44
↔
(
(
¬
X34
)
↔
X53
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X53
↔
(
X47
→
X17
→
X49
)
)
)
↔
(
¬
(
X49
→
(
(
¬
X19
)
↔
(
X17
→
X47
)
)
)
)
)
→
(
¬
(
X53
→
X49
→
(
X47
↔
False
)
)
)
)
)
↔
(
¬
(
(
¬
X52
)
→
X49
→
(
X47
↔
X53
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
7decf4...
and proposition id is
b6d4b9...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMJY2s6bgo4YCVCEbP2tbWroxDGruxTGWW4
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
(
(
¬
(
X46
↔
(
X41
→
X40
→
X42
)
)
)
↔
(
(
(
X42
↔
(
(
¬
X41
)
↔
(
(
¬
X29
)
↔
X43
)
)
)
↔
(
¬
(
X43
↔
(
(
¬
X29
)
→
(
X11
↔
X41
)
)
)
)
)
↔
(
X43
↔
(
(
¬
X41
)
↔
(
(
¬
X40
)
→
X42
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
(
¬
(
X46
→
(
X42
↔
(
(
¬
X24
)
→
X39
)
)
)
)
→
(
¬
(
(
X46
→
(
X12
↔
X42
)
)
→
(
¬
X39
)
→
(
X12
↔
(
(
¬
X8
)
↔
X46
)
)
)
)
)
→
(
(
¬
X42
)
↔
(
(
¬
X39
)
→
(
¬
X37
)
→
X46
)
)
)
)
→
(
¬
(
X37
↔
(
X24
→
(
X22
↔
X8
)
)
)
)
)
)
→
(
X43
↔
(
X42
→
(
(
¬
X39
)
↔
X46
)
)
)
)
)
)
↔
(
(
(
(
X43
→
(
(
¬
X31
)
↔
X46
)
)
↔
(
¬
(
(
¬
X33
)
↔
(
X26
↔
X46
)
)
)
)
↔
(
(
¬
X42
)
→
(
¬
X33
)
→
(
(
¬
X31
)
↔
X46
)
)
)
→
(
(
¬
X42
)
↔
(
(
¬
X31
)
→
(
(
¬
X23
)
↔
X43
)
)
)
)
)
↔
(
(
(
(
¬
X44
)
→
X42
→
(
(
¬
X14
)
↔
X46
)
)
→
X46
→
X44
→
(
¬
X36
)
→
X42
)
→
(
¬
(
(
(
¬
(
(
(
(
¬
(
X36
↔
(
X19
→
(
¬
X15
)
→
X43
)
)
)
↔
(
¬
(
X43
→
X36
→
X19
→
(
X4
↔
False
)
)
)
)
↔
(
X43
↔
(
X15
↔
X46
)
)
)
→
(
¬
(
X43
↔
(
X36
↔
X19
)
)
)
)
)
↔
(
¬
(
X43
→
(
X15
↔
X4
)
)
)
)
→
(
¬
(
(
(
¬
(
X44
↔
(
X28
↔
X46
)
)
)
↔
(
(
¬
(
X46
→
(
X44
↔
(
(
¬
X19
)
↔
X28
)
)
)
)
↔
(
¬
(
(
¬
X20
)
→
(
(
¬
X7
)
↔
X46
)
)
)
)
)
→
(
(
¬
(
(
¬
X43
)
↔
(
(
¬
X28
)
↔
(
X20
↔
X44
)
)
)
)
↔
(
¬
(
X44
→
X20
→
X18
→
X43
)
)
)
)
)
)
)
)
)
→
(
(
(
(
(
X47
→
X32
→
(
X10
↔
X48
)
)
↔
(
X48
↔
(
X47
↔
(
X32
→
X17
)
)
)
)
→
(
(
¬
(
(
¬
X17
)
→
(
X0
↔
X45
)
)
)
↔
(
(
¬
(
(
¬
(
(
X47
→
(
¬
X17
)
→
(
(
¬
X0
)
↔
X27
)
)
→
(
¬
(
X30
↔
(
X0
→
X47
)
)
)
)
)
→
(
X47
↔
(
X30
→
X17
)
)
)
)
↔
(
¬
(
X27
→
(
X0
↔
X48
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
X38
)
→
(
(
¬
X16
)
↔
(
X5
↔
X47
)
)
)
→
(
¬
(
X40
↔
(
(
¬
X5
)
↔
X16
)
)
)
)
)
→
(
¬
(
X16
→
(
X6
↔
X40
)
)
)
→
(
¬
(
(
¬
X40
)
→
(
¬
X38
)
→
(
(
¬
X6
)
↔
X45
)
)
)
)
↔
(
(
(
¬
(
X45
→
(
X40
↔
False
)
)
)
↔
(
(
(
¬
(
(
¬
X47
)
↔
(
(
¬
X45
)
↔
(
(
¬
X40
)
→
X48
)
)
)
)
↔
(
(
¬
X45
)
↔
(
(
¬
X42
)
→
X40
→
X48
)
)
)
↔
(
X42
↔
(
(
¬
X40
)
↔
X48
)
)
)
)
→
(
¬
(
(
¬
X45
)
→
(
(
¬
X9
)
↔
X48
)
)
)
→
(
¬
X45
)
→
(
(
¬
X9
)
↔
(
X2
↔
X47
)
)
)
)
)
↔
(
(
(
(
(
¬
X48
)
↔
(
X43
↔
(
(
¬
X42
)
↔
X49
)
)
)
↔
(
X42
↔
(
(
¬
X35
)
↔
X49
)
)
)
→
(
¬
(
(
¬
(
(
X48
↔
(
(
¬
X3
)
→
X43
)
)
→
X43
→
(
X34
↔
X13
)
)
)
→
(
X49
↔
(
X13
↔
(
X3
→
X34
)
)
)
)
)
)
↔
(
¬
(
(
(
X25
↔
(
(
¬
X21
)
↔
(
(
¬
X13
)
↔
X34
)
)
)
↔
(
¬
(
(
¬
X25
)
→
(
¬
X1
)
→
X34
)
)
)
→
(
¬
X47
)
→
(
(
¬
X34
)
↔
(
(
¬
X21
)
→
X49
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
(
X48
↔
(
X47
→
X33
→
X41
)
)
→
X40
→
(
¬
X33
)
→
(
(
¬
X11
)
↔
X41
)
)
)
→
(
(
¬
(
(
(
X40
→
False
)
↔
(
(
(
X42
↔
(
(
¬
X41
)
↔
(
(
¬
X5
)
↔
X48
)
)
)
→
(
(
X41
→
(
(
¬
X38
)
↔
X42
)
)
↔
(
¬
(
X42
↔
(
(
¬
X40
)
→
X48
)
)
)
)
)
↔
(
X48
→
(
X42
↔
(
X40
↔
False
)
)
)
)
)
→
(
¬
(
(
X48
→
(
X40
↔
X31
)
)
→
(
(
¬
X31
)
↔
(
(
¬
X0
)
↔
X42
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
¬
(
(
X48
↔
(
(
¬
X41
)
→
X49
)
)
→
(
¬
(
(
¬
X24
)
↔
(
(
¬
X19
)
→
X48
)
)
)
)
)
→
(
¬
(
X49
→
X41
→
X24
→
False
)
)
)
)
→
(
(
(
¬
(
X49
↔
(
(
¬
X45
)
↔
X48
)
)
)
↔
(
(
(
¬
X45
)
↔
X49
)
↔
(
¬
(
(
¬
(
(
X49
→
(
¬
X45
)
→
(
(
¬
X24
)
↔
X48
)
)
→
(
¬
X45
)
→
(
(
¬
X42
)
↔
(
(
¬
X24
)
→
X48
)
)
)
)
→
(
¬
(
X49
↔
X45
)
)
)
)
)
)
↔
(
(
¬
X45
)
↔
(
(
¬
X24
)
↔
X49
)
)
)
)
)
→
(
¬
X48
)
→
(
(
¬
X45
)
↔
(
X2
↔
X49
)
)
)
)
)
)
)
→
(
(
(
(
(
(
¬
(
(
¬
X36
)
→
(
(
¬
X28
)
↔
X41
)
)
)
↔
(
(
¬
X36
)
→
(
(
¬
X28
)
↔
(
X23
→
X46
)
)
)
)
→
(
¬
(
X41
→
(
X28
↔
(
(
¬
X23
)
→
X40
)
)
)
)
)
↔
(
X41
↔
(
(
¬
X23
)
↔
X36
)
)
)
↔
(
(
¬
(
X41
↔
(
(
¬
X34
)
↔
X46
)
)
)
↔
(
¬
(
(
(
¬
X34
)
↔
(
(
¬
X28
)
→
X41
)
)
→
(
(
¬
X34
)
↔
(
X15
→
X46
)
)
)
)
)
)
↔
(
(
(
(
¬
X42
)
→
(
(
¬
X36
)
↔
(
(
¬
X1
)
↔
X46
)
)
)
↔
(
(
¬
(
X42
→
X32
→
X18
→
X8
→
False
)
)
↔
(
¬
(
X42
→
(
¬
X25
)
→
(
¬
X18
)
→
X32
)
)
)
)
↔
(
(
(
¬
(
(
X41
→
X36
→
X16
→
X22
)
→
(
X41
↔
(
X16
↔
(
X10
↔
X36
)
)
)
)
)
↔
(
(
(
(
¬
(
X32
↔
(
X10
↔
X3
)
)
)
→
(
¬
(
(
¬
X32
)
→
(
¬
(
(
¬
(
X36
↔
(
X25
→
X22
→
False
)
)
)
→
(
¬
(
(
¬
X32
)
→
(
X25
↔
(
X10
↔
X36
)
)
)
)
)
)
)
)
)
↔
(
X36
↔
(
X32
→
X25
)
)
)
↔
(
¬
(
X22
→
(
X10
↔
X32
)
)
)
)
)
↔
(
¬
(
(
X36
→
(
(
¬
X17
)
↔
(
(
¬
X10
)
→
X22
)
)
)
→
(
¬
(
X36
→
(
¬
X6
)
→
X22
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
X46
)
→
X45
→
(
X37
↔
X48
)
)
→
(
¬
(
X46
→
(
(
¬
X45
)
↔
(
(
¬
X43
)
→
X48
)
)
)
)
)
)
→
(
(
¬
(
X45
↔
(
X35
→
(
(
¬
X29
)
↔
X43
)
)
)
)
↔
(
¬
(
(
(
¬
(
X45
→
X37
→
X20
→
X12
→
False
)
)
↔
(
¬
(
X35
↔
(
(
¬
X20
)
↔
(
(
¬
X12
)
↔
X37
)
)
)
)
)
→
(
X37
→
(
¬
X27
)
→
X43
)
→
(
¬
(
X37
→
(
X35
↔
(
X20
→
X43
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
X30
↔
(
X7
↔
False
)
)
↔
(
X44
→
(
X9
↔
(
(
¬
X7
)
→
X46
)
)
)
)
↔
(
¬
(
X44
→
(
X39
↔
(
X9
↔
X30
)
)
)
)
)
→
(
(
X44
→
(
¬
X13
)
→
X45
)
↔
(
¬
(
X48
→
X44
→
X30
→
(
X7
↔
False
)
)
)
)
)
)
→
(
¬
(
(
X21
↔
X48
)
→
(
X45
↔
(
X26
↔
(
X21
→
X4
→
False
)
)
)
)
)
→
(
¬
(
X45
→
(
X39
↔
(
(
¬
X14
)
→
X26
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
f9e947...
and proposition id is
d9ace3...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMdrvzHFspYrdawGsTWAd9613Z7cnD12oFe
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
¬
(
(
¬
(
(
¬
(
(
¬
(
(
¬
(
(
(
(
¬
(
(
¬
X31
)
↔
(
(
¬
X19
)
→
X37
)
)
)
↔
(
¬
(
(
¬
(
X41
↔
(
(
¬
X37
)
↔
(
(
¬
X19
)
→
X39
)
)
)
)
→
(
¬
(
X41
→
(
(
¬
X31
)
↔
X37
)
)
)
)
)
)
→
(
¬
(
X41
→
(
X39
↔
(
X31
→
False
)
)
)
)
)
→
(
¬
(
X37
↔
(
X19
↔
(
X17
→
X31
)
)
)
)
)
)
→
(
X41
→
(
X39
↔
(
(
¬
X37
)
→
X48
)
)
)
→
(
¬
(
X39
→
X30
→
X27
)
)
→
(
¬
(
X48
↔
(
X39
↔
(
(
¬
X30
)
↔
X41
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
(
¬
X41
)
→
(
X28
↔
X43
)
)
→
(
X44
↔
(
X41
↔
(
X5
↔
X43
)
)
)
)
)
↔
(
¬
(
X41
→
(
X28
↔
(
(
¬
X22
)
→
X43
)
)
)
)
)
→
(
(
X44
→
(
¬
X41
)
→
X25
→
X43
)
↔
(
¬
(
X48
↔
False
)
)
)
→
(
¬
(
(
X44
→
(
X14
↔
X10
)
)
→
(
(
(
¬
(
X39
↔
(
X3
→
X48
)
)
)
→
(
¬
(
X44
→
X41
→
(
X39
↔
X3
)
)
)
)
↔
(
¬
(
(
X48
↔
(
X44
→
X41
→
X2
→
False
)
)
→
(
¬
(
(
¬
X44
)
→
(
(
¬
X41
)
↔
(
(
¬
X14
)
↔
X48
)
)
)
)
)
)
)
)
)
)
)
)
)
→
(
(
(
(
X48
↔
(
X46
↔
X44
)
)
↔
(
(
¬
(
X46
↔
(
(
¬
X44
)
↔
(
X7
→
X48
)
)
)
)
↔
(
¬
(
(
¬
X46
)
↔
(
X41
↔
(
(
¬
X7
)
→
X48
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X41
↔
(
X13
↔
X15
)
)
)
→
X44
→
(
X41
↔
(
(
¬
X13
)
→
X34
)
)
)
)
↔
(
X44
→
(
¬
X41
)
→
(
(
¬
X34
)
↔
X48
)
)
)
)
→
(
(
(
X41
→
X13
→
(
¬
X0
)
→
X44
)
↔
(
¬
(
X42
→
(
¬
X41
)
→
(
(
¬
X13
)
↔
X44
)
)
)
)
↔
(
X42
↔
(
X0
→
X44
)
)
)
)
→
(
(
(
(
X38
→
X44
)
↔
(
¬
(
X44
→
(
X23
↔
(
X16
→
X24
)
)
)
)
)
→
X24
→
X16
→
X41
)
↔
(
(
¬
(
X38
↔
(
(
¬
X23
)
↔
(
(
¬
X16
)
↔
X46
)
)
)
)
↔
(
(
X46
→
X23
→
(
X4
↔
False
)
)
↔
(
¬
(
(
X46
→
X4
→
X1
→
X38
)
→
(
¬
(
X46
↔
(
X4
↔
X23
)
)
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
X48
↔
(
X47
→
(
X46
↔
(
X20
↔
False
)
)
)
)
→
(
¬
(
(
(
¬
(
X48
↔
(
X47
→
(
¬
X12
)
→
X32
)
)
)
→
(
(
¬
(
X47
↔
(
X45
→
X11
)
)
)
↔
(
¬
(
(
X47
→
(
¬
X32
)
→
X11
→
X45
)
→
(
¬
(
X48
↔
(
X45
→
X32
→
X11
→
False
)
)
)
)
)
)
)
→
(
(
¬
(
X46
→
(
(
¬
X32
)
↔
(
X29
→
X48
)
)
)
)
↔
(
¬
(
(
¬
X32
)
→
(
¬
X29
)
→
(
(
¬
X11
)
↔
X48
)
)
)
)
)
)
)
)
↔
(
¬
(
(
X48
→
X46
→
X33
)
→
(
¬
(
X46
↔
(
(
¬
X45
)
↔
(
(
¬
X44
)
→
X48
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X49
↔
(
(
¬
X26
)
↔
(
(
¬
X18
)
→
X35
)
)
)
)
→
(
¬
X26
)
→
(
¬
X18
)
→
(
¬
X8
)
→
X35
)
↔
(
¬
(
X49
→
X36
→
(
X26
↔
X18
)
)
)
)
↔
(
X48
↔
(
X21
↔
(
(
¬
X18
)
→
X49
)
)
)
)
→
(
(
(
X47
↔
(
X9
↔
False
)
)
↔
(
(
¬
(
(
X26
→
(
(
¬
X18
)
↔
(
X9
→
X36
)
)
)
→
(
¬
X46
)
→
(
¬
X26
)
→
(
¬
X18
)
→
X47
)
)
↔
(
¬
(
(
¬
X36
)
→
(
(
¬
X6
)
↔
X46
)
)
)
)
)
↔
(
X47
→
X46
→
X26
→
X40
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
(
X47
→
X39
→
X25
→
X37
)
→
(
¬
(
(
¬
X47
)
↔
(
(
¬
X25
)
→
X48
)
)
)
)
→
X45
→
(
¬
X39
)
→
(
¬
X25
)
→
X47
)
)
→
(
(
¬
X39
)
→
X13
→
(
X2
↔
X48
)
)
→
(
¬
(
X48
→
X47
→
(
X2
↔
False
)
)
)
)
↔
(
¬
(
(
X48
→
(
X37
↔
(
(
¬
X20
)
→
X44
)
)
)
→
(
¬
(
(
(
(
¬
X37
)
↔
(
(
¬
X21
)
→
X48
)
)
↔
(
¬
(
(
¬
X37
)
↔
(
(
¬
X3
)
↔
X45
)
)
)
)
→
(
¬
(
(
(
¬
(
X48
↔
(
X45
→
X21
→
X1
)
)
)
↔
(
(
¬
X45
)
↔
(
(
¬
X21
)
↔
(
X20
↔
X48
)
)
)
)
→
X48
→
(
¬
X45
)
→
(
¬
X20
)
→
X46
)
)
→
(
¬
(
X46
↔
(
X45
↔
(
X1
→
False
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X41
)
↔
(
(
¬
X40
)
→
X42
)
)
)
↔
(
(
¬
(
X47
↔
(
X46
↔
(
X41
↔
(
X24
→
False
)
)
)
)
)
↔
(
(
X47
↔
(
(
¬
X23
)
↔
X41
)
)
→
(
¬
(
(
(
X42
→
X29
→
False
)
↔
(
(
¬
(
X46
↔
(
(
¬
X29
)
↔
(
X23
→
X40
)
)
)
)
→
(
(
¬
X42
)
↔
(
(
¬
X40
)
↔
(
(
¬
X29
)
→
X46
)
)
)
)
)
→
(
¬
(
X40
↔
(
X23
↔
X47
)
)
)
→
(
(
¬
(
X46
→
X40
→
X23
→
X41
)
)
↔
(
X40
→
X15
→
X46
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
X48
↔
(
X47
↔
(
X17
→
False
)
)
)
→
(
X41
↔
(
X36
→
X17
→
X42
)
)
→
(
¬
(
X48
↔
(
X36
→
X42
)
)
)
)
)
→
(
(
¬
(
X47
↔
(
(
¬
X6
)
→
X41
)
)
)
↔
(
(
X47
→
X34
→
X17
)
↔
(
X47
→
X41
→
(
(
¬
X6
)
↔
X34
)
)
)
)
→
(
¬
(
X41
→
(
X34
↔
(
(
¬
X6
)
↔
X11
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
X49
↔
(
X43
↔
(
(
¬
X38
)
↔
X44
)
)
)
↔
(
X38
→
X30
→
X43
)
)
↔
(
¬
(
(
¬
(
(
(
¬
X44
)
→
(
X27
↔
X49
)
)
→
(
(
X44
↔
(
X27
→
(
¬
X14
)
→
X28
)
)
↔
(
¬
(
X44
↔
(
X28
→
X27
→
X26
)
)
)
)
)
)
→
(
(
(
X49
↔
(
X44
→
X18
→
X0
→
False
)
)
→
(
¬
(
X44
→
(
(
¬
X28
)
↔
X49
)
)
)
→
(
¬
(
X38
↔
X18
)
)
)
↔
(
¬
(
X44
→
(
X0
↔
X18
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
X33
→
(
X9
↔
X32
)
)
)
↔
(
X10
→
(
(
¬
X9
)
↔
(
(
¬
X7
)
→
X33
)
)
)
)
↔
(
(
(
(
¬
X12
)
↔
(
(
¬
X10
)
→
(
(
¬
X9
)
↔
X22
)
)
)
→
(
X22
↔
(
X9
↔
(
(
¬
X7
)
↔
X10
)
)
)
)
→
(
¬
(
X22
↔
(
X12
→
X9
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
X44
)
↔
X49
)
↔
(
¬
(
(
¬
X47
)
→
X8
→
(
X4
↔
X49
)
)
)
)
→
(
(
(
(
X44
↔
(
X33
↔
(
X5
↔
X46
)
)
)
→
(
(
¬
X44
)
↔
(
(
¬
X33
)
→
(
X12
↔
X46
)
)
)
)
→
(
(
¬
X46
)
↔
(
(
¬
X33
)
↔
(
(
¬
X5
)
↔
X49
)
)
)
)
↔
(
¬
(
X47
↔
(
(
¬
X46
)
→
(
¬
X12
)
→
X49
)
)
)
)
→
(
(
X44
↔
(
X33
→
X19
→
X22
)
)
↔
(
X33
→
(
X22
↔
X49
)
)
)
)
)
)
↔
(
(
(
¬
X47
)
↔
(
X44
↔
(
X35
↔
X49
)
)
)
↔
(
(
X47
→
X43
→
X16
)
→
(
X47
↔
(
X44
↔
(
X31
→
X16
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
a041b5...
and proposition id is
a0331b...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMY21YdUt6G2VS4d9z2p39qZ1HYH5KUcwpE
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
¬
(
(
(
¬
(
(
(
(
(
X40
↔
(
X31
→
X22
→
(
X10
↔
False
)
)
)
↔
(
¬
(
X35
→
(
X31
↔
(
X22
→
X40
)
)
)
)
)
↔
(
X35
↔
(
X31
↔
(
(
¬
X22
)
→
X39
)
)
)
)
↔
(
¬
(
(
(
X31
↔
(
(
¬
X21
)
↔
X35
)
)
↔
(
¬
(
(
(
X40
→
(
X31
↔
X21
)
)
↔
(
X35
→
(
X31
↔
(
X3
→
X21
)
)
)
)
→
(
¬
(
X35
→
(
X31
↔
X40
)
)
)
)
)
)
→
(
(
¬
X40
)
↔
(
X31
↔
(
(
¬
X3
)
↔
X48
)
)
)
)
)
)
→
(
(
(
X48
↔
(
X40
↔
(
X25
↔
X8
)
)
)
↔
(
(
(
(
¬
(
X40
→
(
X23
↔
X50
)
)
)
→
(
¬
(
X50
↔
(
X46
↔
(
X0
↔
False
)
)
)
)
)
↔
(
X50
→
X48
→
X46
→
(
X40
↔
False
)
)
)
↔
(
¬
(
X50
↔
(
X48
→
(
X25
↔
(
X23
→
False
)
)
)
)
)
)
)
↔
(
(
X46
↔
(
(
¬
X9
)
→
X48
)
)
→
(
(
¬
X48
)
↔
(
(
¬
X46
)
→
(
(
¬
X31
)
↔
X50
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
X38
→
X12
→
X48
)
)
→
(
¬
X38
)
→
X33
→
(
(
¬
X24
)
↔
X48
)
)
→
(
¬
(
(
X26
↔
(
(
¬
X14
)
→
X48
)
)
→
(
¬
(
(
X48
↔
(
X38
↔
(
(
¬
X14
)
↔
X26
)
)
)
→
(
(
¬
X38
)
↔
(
X33
→
(
¬
X26
)
→
X48
)
)
)
)
)
)
)
↔
(
(
¬
(
(
X33
↔
(
(
¬
X30
)
→
(
(
¬
X16
)
↔
X39
)
)
)
→
(
¬
(
(
X46
↔
(
X39
→
X33
)
)
→
(
(
X46
→
(
X39
↔
(
(
¬
X16
)
↔
X41
)
)
)
↔
(
X46
↔
(
X41
→
(
(
¬
X39
)
↔
X48
)
)
)
)
)
)
)
)
↔
(
(
X48
→
X46
→
(
X38
↔
False
)
)
↔
(
(
¬
(
X48
↔
(
X46
↔
X38
)
)
)
→
(
¬
(
X48
↔
(
(
¬
X33
)
↔
X38
)
)
)
→
(
(
¬
(
X38
↔
(
(
¬
X33
)
→
X48
)
)
)
↔
(
¬
(
X48
→
X46
→
(
X2
↔
False
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
X45
↔
(
X38
→
X24
→
X11
→
False
)
)
→
(
¬
(
X43
→
(
(
¬
X24
)
↔
(
X11
↔
X45
)
)
)
)
)
)
↔
(
X41
→
(
¬
X18
)
→
X11
→
X38
)
)
)
)
→
(
(
(
¬
(
(
(
(
(
¬
X45
)
↔
(
X28
↔
X46
)
)
→
X46
→
(
X45
↔
(
(
¬
X32
)
↔
X37
)
)
)
↔
(
¬
(
X46
↔
(
X28
↔
False
)
)
)
)
→
(
X46
↔
(
(
¬
X28
)
↔
X47
)
)
)
)
→
(
¬
(
(
¬
(
X47
→
(
¬
X1
)
→
X15
)
)
→
(
¬
(
(
X46
↔
(
X17
→
(
X15
↔
(
X1
↔
False
)
)
)
)
→
(
¬
(
X45
→
(
(
¬
X1
)
↔
X17
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
X49
→
(
¬
X46
)
→
X47
)
)
→
(
¬
(
X49
↔
(
(
¬
X19
)
↔
(
(
¬
X5
)
→
X46
)
)
)
)
)
)
→
(
(
X47
→
(
(
¬
X20
)
↔
(
(
¬
X4
)
↔
X49
)
)
)
↔
(
¬
(
(
¬
(
X47
→
(
X46
↔
(
X4
↔
X20
)
)
)
)
→
(
X46
↔
(
(
¬
X37
)
→
X47
)
)
)
)
)
)
↔
(
¬
(
X49
→
(
X47
↔
(
X46
→
(
X42
↔
False
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
X44
)
→
(
¬
X34
)
→
(
(
¬
X29
)
↔
X49
)
)
)
↔
(
X49
↔
(
(
¬
X44
)
↔
(
X34
↔
X50
)
)
)
)
→
(
¬
(
(
¬
X34
)
→
(
X29
↔
X49
)
)
)
)
→
(
(
(
¬
(
X49
↔
(
X45
↔
(
(
¬
X36
)
→
X50
)
)
)
)
↔
(
¬
(
X50
→
X49
→
X45
→
X34
→
False
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
X49
↔
(
(
¬
X45
)
→
(
X13
↔
X50
)
)
)
)
→
(
¬
(
X49
↔
(
X45
↔
(
X7
→
X13
)
)
)
)
)
)
→
(
¬
(
(
¬
X27
)
→
(
¬
X13
)
→
(
¬
X7
)
→
X50
)
)
)
)
↔
(
¬
(
X50
↔
(
X27
↔
X13
)
)
)
)
)
)
→
(
(
¬
(
X44
↔
False
)
)
→
X50
→
(
(
¬
X46
)
↔
(
X6
↔
X49
)
)
)
→
X50
→
(
(
¬
X44
)
↔
X46
)
)
)
↔
(
(
¬
(
(
(
(
(
(
(
¬
X24
)
→
X27
)
→
(
¬
(
(
X48
→
(
¬
X24
)
→
(
(
¬
X21
)
↔
X27
)
)
→
(
¬
(
X27
→
(
X24
↔
(
(
¬
X5
)
→
X48
)
)
)
)
)
)
)
↔
(
(
¬
(
X24
↔
(
(
¬
X18
)
↔
X48
)
)
)
→
(
X48
↔
(
X27
→
(
(
¬
X18
)
↔
X19
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X24
→
X18
→
(
X5
↔
X22
)
)
)
↔
(
X24
↔
(
(
¬
X22
)
→
(
X18
↔
X23
)
)
)
)
→
(
¬
(
X29
↔
X5
)
)
)
)
)
↔
(
(
(
X16
→
X27
)
→
(
(
(
¬
X47
)
↔
(
(
¬
X16
)
↔
X50
)
)
↔
(
¬
(
(
¬
X27
)
↔
(
X24
→
X16
→
X50
)
)
)
)
)
↔
(
(
¬
(
X48
↔
(
(
¬
X47
)
→
(
X27
↔
X50
)
)
)
)
↔
(
(
(
X50
↔
X2
)
↔
(
¬
(
(
(
¬
X48
)
→
(
¬
X47
)
→
(
X34
↔
X50
)
)
→
(
X50
↔
(
X47
↔
(
X2
↔
X34
)
)
)
)
)
)
↔
(
(
¬
X48
)
↔
(
(
¬
X47
)
↔
(
X25
→
X50
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
X45
→
(
¬
X32
)
→
(
X14
↔
X50
)
)
→
(
¬
(
(
¬
X0
)
↔
X50
)
)
)
→
(
¬
(
X14
→
(
(
¬
X1
)
↔
(
X0
→
X47
)
)
)
)
)
↔
(
(
X50
↔
(
X32
↔
(
(
¬
X26
)
→
X45
)
)
)
→
(
¬
(
(
¬
(
(
¬
X45
)
→
(
¬
X32
)
→
(
(
¬
X28
)
↔
X50
)
)
)
→
(
¬
(
X50
↔
(
(
¬
X44
)
→
X28
→
X45
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
(
(
¬
X36
)
↔
(
(
¬
X33
)
↔
X48
)
)
)
→
(
¬
(
X46
↔
(
X45
↔
False
)
)
)
)
)
↔
(
(
¬
(
X46
↔
(
X44
↔
(
(
¬
X37
)
↔
X50
)
)
)
)
↔
(
¬
(
(
¬
(
X50
→
(
X48
↔
(
X44
↔
X45
)
)
)
)
→
(
¬
(
X48
→
(
¬
X46
)
→
(
(
¬
X45
)
↔
X50
)
)
)
)
)
)
)
→
(
(
¬
(
(
X44
↔
(
(
¬
X30
)
↔
(
(
¬
X3
)
→
X48
)
)
)
→
(
¬
(
X30
↔
X50
)
)
)
)
↔
(
¬
(
X48
↔
X17
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
X42
→
X7
→
X43
)
↔
(
(
(
¬
(
(
¬
X43
)
↔
(
X42
→
X7
→
X46
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X39
)
→
(
¬
X7
)
→
X42
)
)
)
)
↔
(
(
¬
X7
)
↔
X42
)
)
)
→
(
(
¬
(
X46
→
(
X43
↔
(
(
¬
X38
)
→
X42
)
)
)
)
↔
(
¬
(
(
X43
↔
(
X42
→
(
X15
↔
False
)
)
)
→
(
¬
(
X42
→
(
X38
↔
(
(
¬
X15
)
↔
X20
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X49
↔
(
X46
→
X39
→
X10
)
)
)
→
X49
→
(
X42
↔
(
X39
→
X13
→
False
)
)
)
↔
(
(
¬
(
X49
→
X40
→
X42
)
)
↔
(
¬
(
(
¬
X42
)
→
(
X40
↔
(
X35
→
X46
)
)
)
)
)
)
)
→
(
(
¬
(
X42
↔
(
X12
↔
X49
)
)
)
↔
(
X47
↔
(
X41
↔
(
X4
↔
False
)
)
)
)
→
(
(
(
(
¬
X41
)
→
(
(
¬
X31
)
↔
(
X6
→
X48
)
)
)
→
(
¬
(
X43
→
X31
→
X48
)
)
)
→
(
¬
(
X43
↔
(
X41
↔
(
X31
→
False
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
X8
)
↔
X43
)
→
(
¬
(
(
X47
↔
(
X41
→
X43
)
)
→
(
(
¬
(
(
¬
X43
)
↔
(
(
¬
X8
)
↔
X49
)
)
)
↔
(
(
(
X49
↔
(
X8
↔
X31
)
)
→
(
¬
(
X43
↔
(
X41
↔
(
(
¬
X8
)
↔
X49
)
)
)
)
)
↔
(
¬
(
X49
↔
(
(
¬
X31
)
→
X41
)
)
)
)
)
→
(
X43
↔
(
(
¬
X8
)
↔
X31
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
X41
)
↔
(
X11
↔
X47
)
)
)
→
(
¬
(
(
¬
X11
)
→
X9
→
X43
)
)
)
)
↔
(
¬
(
X47
→
(
(
¬
X41
)
↔
X43
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
c8c75d...
and proposition id is
e8ebf2...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMKhCCPnRBP283vFGRxB9vEJC5XUQEBsihv
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
(
(
¬
(
X33
→
(
¬
X1
)
→
X45
)
)
↔
(
¬
(
(
¬
(
(
(
X47
→
(
X42
↔
X33
)
)
↔
(
(
¬
X42
)
→
X30
→
X48
)
)
→
(
¬
(
X47
→
(
(
¬
X1
)
↔
X30
)
)
)
)
)
→
X47
→
X42
→
X30
→
False
)
)
)
↔
(
(
(
¬
(
X47
→
(
¬
X45
)
→
(
(
¬
X32
)
↔
X48
)
)
)
↔
(
(
¬
(
(
(
X48
↔
(
(
¬
X45
)
→
(
¬
X7
)
→
X47
)
)
↔
(
¬
(
X48
↔
(
X47
→
X44
)
)
)
)
→
(
X48
↔
(
X47
→
(
X7
↔
False
)
)
)
)
)
→
(
¬
(
X16
→
X7
)
)
)
)
↔
(
(
¬
(
X45
→
X44
→
(
X20
↔
X32
)
)
)
↔
(
¬
(
X20
→
(
(
¬
X10
)
↔
X45
)
)
)
)
)
)
↔
(
¬
(
(
X42
↔
(
X40
↔
X45
)
)
→
(
¬
X42
)
→
(
(
¬
X40
)
↔
(
(
¬
X27
)
→
X44
)
)
)
)
)
↔
(
(
(
(
X45
→
X43
→
X22
→
X3
)
↔
(
(
(
X37
→
(
X28
↔
(
X22
↔
X42
)
)
)
↔
(
(
¬
X37
)
↔
(
X28
↔
(
X3
↔
X42
)
)
)
)
↔
(
X28
→
(
X3
↔
X22
)
)
)
)
→
(
¬
(
(
¬
(
X44
↔
(
X42
→
(
X25
↔
X4
)
)
)
)
→
(
X42
↔
(
X22
→
(
(
¬
X4
)
↔
X25
)
)
)
)
)
→
(
¬
(
X44
→
(
X4
↔
(
X3
→
X42
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
¬
X43
)
↔
(
X42
↔
(
(
¬
X19
)
→
X49
)
)
)
)
↔
(
¬
(
X49
↔
(
(
¬
X19
)
↔
X45
)
)
)
)
→
(
¬
(
(
¬
X42
)
↔
(
X41
→
(
¬
X19
)
→
X45
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X45
)
→
X43
→
X8
→
X49
)
)
→
(
¬
(
(
(
¬
X43
)
→
X39
→
(
(
¬
X6
)
↔
X45
)
)
→
X45
→
(
(
¬
X43
)
↔
(
(
¬
X39
)
→
X49
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
X47
↔
(
X45
↔
(
(
¬
X38
)
↔
X42
)
)
)
↔
(
¬
(
X45
→
(
X38
↔
(
(
¬
X21
)
↔
X42
)
)
)
)
)
↔
(
(
(
¬
(
X38
↔
(
X17
↔
(
(
¬
X14
)
↔
X29
)
)
)
)
↔
(
(
¬
(
X45
↔
(
X29
↔
(
(
¬
X17
)
→
X35
)
)
)
)
→
(
(
(
¬
X35
)
↔
(
(
¬
X29
)
↔
(
(
¬
X0
)
↔
X38
)
)
)
↔
(
¬
(
X35
→
X17
→
X0
)
)
)
)
)
↔
(
X47
↔
(
X45
↔
(
(
¬
X0
)
→
X29
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
X36
↔
(
X31
↔
X34
)
)
)
→
(
¬
(
X34
↔
(
(
¬
X31
)
→
(
¬
X24
)
→
X46
)
)
)
→
(
X36
↔
(
X24
→
X34
)
)
)
→
(
¬
(
X34
↔
(
(
¬
X26
)
→
X24
→
X31
)
)
)
)
↔
(
¬
(
(
¬
X36
)
→
(
(
¬
X26
)
↔
(
(
¬
X24
)
→
X47
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
(
X47
→
(
¬
X23
)
→
X42
)
)
→
(
(
¬
(
X42
→
(
X26
↔
(
X23
→
False
)
)
)
)
↔
(
¬
(
X23
→
(
X12
↔
X24
)
)
)
)
)
)
↔
(
(
X24
↔
(
X23
→
(
X2
↔
X11
)
)
)
↔
(
(
¬
X24
)
→
(
X23
↔
(
(
¬
X2
)
↔
X26
)
)
)
)
)
→
(
(
(
¬
X26
)
→
(
(
¬
X15
)
↔
(
(
¬
X9
)
↔
X42
)
)
)
→
(
(
¬
X26
)
→
X45
)
→
(
¬
(
X18
→
X15
→
(
¬
X9
)
→
X45
)
)
)
→
(
(
¬
(
X45
↔
(
X18
→
(
(
¬
X13
)
↔
X26
)
)
)
)
→
(
¬
(
X45
→
(
¬
X18
)
→
(
(
¬
X5
)
↔
X26
)
)
)
)
→
(
X45
↔
(
(
¬
X13
)
→
(
(
¬
X11
)
↔
X26
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
(
(
X48
→
X42
→
X32
→
X12
)
↔
(
X48
↔
(
X42
↔
(
X32
↔
(
X29
→
False
)
)
)
)
)
→
(
X37
↔
(
X29
↔
(
X12
↔
X39
)
)
)
)
)
→
(
¬
(
X48
↔
(
X42
→
(
¬
X22
)
→
X37
)
)
)
)
↔
(
(
(
¬
(
X34
↔
(
X29
→
X37
)
)
)
→
(
X34
↔
(
(
¬
X20
)
↔
(
X19
↔
X29
)
)
)
→
(
¬
(
X34
↔
(
X29
→
X42
)
)
)
)
↔
(
¬
(
X29
↔
X20
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
X48
↔
(
X10
→
X4
→
X8
)
)
)
→
(
¬
(
(
¬
X8
)
→
X4
→
X39
)
)
)
)
→
(
¬
X4
)
→
X39
)
)
↔
(
¬
(
(
(
¬
X10
)
↔
X42
)
→
(
(
(
¬
(
X48
↔
(
X42
→
X21
→
X10
)
)
)
↔
(
X48
↔
(
X42
↔
(
X10
→
X8
→
False
)
)
)
)
↔
(
¬
(
(
¬
X8
)
↔
X48
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
¬
X33
)
→
X40
)
)
↔
(
¬
(
(
(
¬
X33
)
→
(
¬
X31
)
→
X45
)
→
(
X37
↔
(
X33
↔
(
X31
↔
(
X7
→
False
)
)
)
)
)
)
)
→
(
(
(
(
¬
X37
)
↔
(
(
¬
X30
)
→
(
(
¬
X28
)
↔
X45
)
)
)
↔
(
¬
(
(
¬
X40
)
→
X30
→
(
(
¬
X28
)
↔
X45
)
)
)
)
↔
(
X45
↔
(
X30
→
(
X28
↔
False
)
)
)
)
)
↔
(
(
¬
(
X45
→
X31
)
)
↔
(
¬
(
(
X31
→
(
X26
↔
X45
)
)
→
(
(
¬
(
(
¬
X31
)
↔
X42
)
)
↔
(
(
¬
X47
)
↔
(
X33
→
(
X31
↔
(
(
¬
X26
)
→
X42
)
)
)
)
)
→
X47
→
(
X42
↔
(
(
¬
X31
)
↔
X45
)
)
)
)
)
)
↔
(
(
(
(
¬
X31
)
↔
(
(
¬
X14
)
→
X40
)
)
→
(
¬
(
X40
↔
(
(
¬
X34
)
→
X46
)
)
)
)
→
(
(
(
X37
→
X1
→
X14
)
→
(
¬
(
(
¬
X31
)
→
(
(
¬
X14
)
↔
(
(
¬
X6
)
→
X34
)
)
)
)
)
↔
(
X34
→
(
¬
X6
)
→
X3
→
X14
)
)
)
)
)
→
(
(
(
X49
↔
(
X47
↔
(
(
¬
X44
)
↔
X45
)
)
)
↔
(
(
(
¬
(
X49
→
(
X44
↔
(
X24
→
(
X2
↔
False
)
)
)
)
)
↔
(
X24
↔
(
X2
→
(
¬
X0
)
→
X49
)
)
)
↔
(
X49
↔
(
(
¬
X0
)
→
X44
)
)
)
)
↔
(
(
¬
(
(
X44
↔
(
X41
→
(
X18
↔
X9
)
)
)
→
(
¬
(
(
¬
X41
)
→
X27
→
(
X18
↔
X43
)
)
)
)
)
↔
(
(
¬
(
(
¬
X44
)
→
(
(
¬
X43
)
↔
X49
)
)
)
→
X47
→
(
¬
X44
)
→
(
(
¬
X27
)
↔
X49
)
)
)
)
→
(
(
¬
(
(
(
(
¬
(
X49
→
(
(
¬
X16
)
↔
X25
)
)
)
↔
(
(
(
¬
X47
)
→
X25
→
X16
→
X48
)
↔
(
¬
(
X47
→
X25
→
(
¬
X5
)
→
X48
)
)
)
)
↔
(
X49
→
X47
→
X48
)
)
→
(
¬
(
(
X48
→
(
X47
↔
(
(
¬
X13
)
↔
X45
)
)
)
→
X16
→
X13
)
)
)
)
↔
(
¬
(
(
(
¬
(
X48
→
X38
→
(
X15
↔
X17
)
)
)
↔
(
(
(
¬
(
X48
↔
(
(
¬
X38
)
↔
X47
)
)
)
↔
(
¬
(
(
¬
(
X47
→
X23
)
)
→
(
(
(
¬
X47
)
↔
(
X38
→
(
(
¬
X23
)
↔
X48
)
)
)
↔
(
X47
→
X38
→
(
(
¬
X11
)
↔
X48
)
)
)
)
)
)
→
(
¬
(
X48
→
(
(
¬
X11
)
↔
X17
)
)
)
)
)
→
(
¬
(
X47
→
X36
→
(
¬
X35
)
→
X49
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
1988a1...
and proposition id is
c6b5f2...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMUP7vVeqZrv52HHK58EEcZa7TKJ6grBBan
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
∀X50 :
prop
,
(
¬
∀X51 :
prop
,
(
(
¬
(
(
(
¬
(
(
(
X50
↔
(
X32
↔
(
X11
↔
(
X8
→
False
)
)
)
)
↔
(
(
(
¬
X18
)
→
(
(
(
¬
X38
)
↔
(
X32
↔
X50
)
)
↔
(
(
¬
(
(
¬
X32
)
→
X11
→
X50
)
)
↔
(
(
¬
(
X50
↔
(
X38
→
X11
)
)
)
↔
(
¬
(
X38
↔
(
X32
↔
(
X0
→
False
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
X32
)
→
(
¬
X29
)
→
X50
)
)
↔
(
(
X29
→
X11
→
X32
)
↔
(
(
¬
(
(
¬
X32
)
→
(
X29
↔
X50
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
X50
→
(
X29
↔
X32
)
)
→
(
¬
(
X32
↔
(
X11
↔
(
X7
↔
False
)
)
)
)
)
→
(
X32
↔
(
X7
↔
False
)
)
)
→
X50
→
(
(
¬
X7
)
↔
X29
)
)
)
→
X50
→
(
X32
↔
X7
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
(
¬
(
X24
→
(
¬
X18
)
→
(
X13
↔
X39
)
)
)
→
X39
→
(
¬
X18
)
→
X42
)
)
↔
(
X46
→
(
(
¬
X24
)
↔
(
(
¬
X18
)
↔
X39
)
)
)
)
→
(
(
X39
↔
(
X16
→
False
)
)
↔
(
X25
→
X24
→
X16
→
X42
)
)
)
)
↔
(
(
¬
(
X44
→
(
¬
X39
)
→
(
¬
X29
)
→
X42
)
)
→
(
¬
(
X51
↔
(
X46
↔
(
X39
→
X38
)
)
)
)
)
)
)
)
↔
(
(
(
(
X49
→
X4
)
↔
(
¬
(
(
¬
(
(
(
¬
(
(
¬
X44
)
→
(
X35
↔
(
X15
→
X49
)
)
)
)
↔
(
¬
(
X35
→
X4
→
False
)
)
)
→
(
¬
(
(
¬
X15
)
→
X44
)
)
)
)
→
(
¬
(
X49
↔
(
X44
→
(
X15
↔
False
)
)
)
)
)
)
)
↔
(
¬
(
X44
→
X4
→
X2
→
X51
)
)
)
→
(
(
¬
(
(
¬
X46
)
→
(
(
¬
X44
)
↔
(
(
¬
X33
)
→
X51
)
)
)
)
→
(
(
(
¬
(
(
¬
(
X51
→
X44
→
X46
)
)
→
(
(
¬
(
X51
↔
(
X46
→
(
X5
↔
False
)
)
)
)
↔
(
X51
↔
(
X44
→
(
X5
↔
X49
)
)
)
)
)
)
↔
(
¬
(
(
¬
X5
)
→
X46
)
)
)
↔
(
¬
(
X51
→
(
X49
↔
(
X44
↔
False
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
X41
→
(
X27
↔
X1
)
)
)
→
(
(
¬
(
X33
↔
(
X27
→
(
¬
X1
)
→
X51
)
)
)
↔
(
¬
(
(
¬
X33
)
→
(
¬
X1
)
→
X41
)
)
)
)
→
X33
→
(
¬
X27
)
→
X41
)
→
(
¬
X41
)
→
(
¬
X33
)
→
X51
)
↔
(
¬
(
X44
→
(
X41
↔
(
X27
→
False
)
)
)
)
)
)
)
→
(
(
(
(
X50
→
(
(
¬
X36
)
↔
(
(
¬
X34
)
→
X48
)
)
)
↔
(
(
X36
→
X34
→
X10
→
X48
)
→
(
¬
(
(
X23
→
(
(
¬
X10
)
↔
X48
)
)
→
(
¬
(
(
(
X48
↔
(
X23
→
X6
→
X10
)
)
→
(
¬
(
(
¬
X10
)
→
X6
→
X34
)
)
)
→
(
¬
(
X48
→
(
X23
↔
X34
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
X9
)
→
X50
)
→
(
(
X50
→
X48
→
(
(
¬
X9
)
↔
X22
)
)
↔
(
¬
(
(
¬
X48
)
→
X43
→
(
¬
X22
)
→
X50
)
)
)
)
↔
(
¬
(
X49
↔
(
(
¬
X43
)
↔
(
(
¬
X22
)
↔
X50
)
)
)
)
)
→
(
(
(
X48
↔
(
(
¬
X45
)
→
(
X21
↔
X50
)
)
)
↔
(
¬
(
X21
↔
X49
)
)
)
↔
(
(
(
¬
X48
)
→
(
¬
X45
)
→
(
(
¬
X34
)
↔
X50
)
)
↔
(
X50
↔
(
X45
↔
(
(
¬
X34
)
→
X37
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
(
¬
X28
)
→
X31
)
)
↔
(
(
¬
(
X31
↔
(
(
¬
X30
)
↔
X36
)
)
)
↔
(
¬
(
(
¬
(
X40
↔
(
(
¬
X31
)
↔
(
(
¬
X30
)
↔
X36
)
)
)
)
→
(
¬
X36
)
→
(
¬
X31
)
→
(
¬
X28
)
→
X40
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X47
↔
(
X40
→
X37
)
)
)
↔
(
(
¬
X40
)
→
(
(
¬
X31
)
↔
(
X30
↔
X47
)
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X31
)
→
X37
)
)
)
)
)
↔
(
¬
(
(
¬
X37
)
↔
(
X31
→
(
X17
↔
X47
)
)
)
)
)
)
)
→
(
(
(
X51
↔
(
(
¬
X20
)
→
(
X12
↔
X26
)
)
)
→
(
(
¬
(
(
(
(
¬
(
(
¬
X37
)
↔
(
(
¬
X36
)
↔
(
X26
↔
X51
)
)
)
)
↔
(
X51
↔
(
X37
↔
(
(
¬
X12
)
→
X36
)
)
)
)
↔
(
(
¬
X36
)
↔
(
(
¬
X26
)
→
X37
)
)
)
→
(
¬
(
X51
→
X37
→
X36
→
False
)
)
)
)
↔
(
¬
(
(
X51
→
(
X30
↔
(
(
¬
X14
)
→
X37
)
)
)
→
(
¬
(
(
¬
X26
)
↔
(
(
¬
X14
)
↔
X30
)
)
)
)
)
)
)
↔
(
¬
(
(
X50
↔
(
(
¬
X26
)
→
(
X19
↔
X51
)
)
)
→
(
¬
(
X50
↔
(
(
¬
X37
)
→
(
(
¬
X3
)
↔
X51
)
)
)
)
→
(
¬
(
X51
↔
(
(
¬
X3
)
↔
X26
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
(
(
¬
X40
)
↔
(
X1
↔
X46
)
)
→
X44
→
X23
)
↔
(
(
(
X44
→
X38
→
X32
→
X1
→
False
)
↔
(
¬
(
X44
→
X38
→
X23
→
X32
)
)
)
↔
(
¬
(
X44
→
X38
→
X23
)
)
)
)
↔
(
¬
(
(
¬
(
X51
→
X32
→
(
X8
↔
False
)
)
)
→
(
¬
(
X44
↔
(
X40
↔
X32
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X46
→
(
X38
↔
(
X9
→
X32
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X41
)
↔
(
(
¬
X38
)
↔
X43
)
)
)
→
(
(
¬
(
(
¬
X38
)
↔
(
(
¬
X32
)
↔
X43
)
)
)
↔
(
(
X41
↔
(
(
¬
X22
)
↔
(
X9
↔
X32
)
)
)
→
(
¬
(
X43
→
X38
→
(
X32
↔
X22
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
¬
(
X47
→
X46
→
X45
→
X24
)
)
→
(
¬
(
(
¬
X45
)
→
(
X30
↔
(
(
¬
X24
)
↔
X46
)
)
)
)
)
↔
(
(
¬
(
X41
↔
(
X14
↔
False
)
)
)
↔
(
X47
↔
(
X41
↔
(
X30
↔
X46
)
)
)
)
)
→
X46
→
(
X35
↔
X45
)
)
)
↔
(
X45
↔
(
X41
↔
(
X35
→
False
)
)
)
)
↔
(
(
(
¬
(
(
(
(
¬
X29
)
→
X18
→
X36
)
↔
(
¬
(
X45
↔
(
(
¬
X36
)
↔
(
(
¬
X18
)
↔
X46
)
)
)
)
)
→
(
¬
(
(
X47
→
(
(
¬
X20
)
↔
X29
)
)
→
(
¬
(
(
¬
X36
)
↔
(
X29
↔
X46
)
)
)
)
)
)
)
→
(
(
(
¬
X45
)
↔
(
(
¬
X29
)
↔
(
X7
↔
X47
)
)
)
→
(
¬
(
X51
→
(
X45
↔
(
X0
→
X7
)
)
)
)
)
→
X47
→
(
¬
X45
)
→
X10
→
X51
)
↔
(
¬
(
(
(
¬
(
(
(
X33
↔
(
(
¬
X2
)
→
X37
)
)
↔
(
(
¬
(
X47
→
X37
→
X33
→
X2
→
False
)
)
↔
(
¬
(
X46
↔
(
X2
↔
X33
)
)
)
)
)
→
X46
→
(
X33
↔
X37
)
)
)
↔
(
X46
↔
(
(
¬
X37
)
↔
X47
)
)
)
→
(
(
¬
(
X47
→
(
X31
↔
(
X6
↔
X2
)
)
)
)
↔
(
(
¬
X47
)
↔
(
X6
→
X51
)
)
)
)
)
)
)
)
↔
(
(
(
(
X49
→
(
X25
↔
(
(
¬
X4
)
→
X34
)
)
)
↔
(
X50
↔
(
X25
→
X4
→
X34
)
)
)
↔
(
(
¬
(
(
(
¬
X39
)
↔
(
(
¬
X34
)
↔
(
X25
↔
X51
)
)
)
→
(
¬
(
(
¬
(
X51
↔
(
X39
↔
(
X5
↔
False
)
)
)
)
→
(
X47
→
(
(
¬
X21
)
↔
(
(
¬
X5
)
↔
X48
)
)
)
→
(
¬
(
X50
→
(
¬
X47
)
→
(
¬
X21
)
→
X48
)
)
)
)
)
)
→
(
X50
↔
(
(
¬
X48
)
↔
(
X19
→
X51
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X46
)
↔
(
X28
↔
(
(
¬
X3
)
↔
X49
)
)
)
)
→
(
¬
(
(
(
(
(
¬
X42
)
→
(
(
¬
X27
)
↔
X50
)
)
↔
(
(
¬
(
(
¬
X49
)
→
(
¬
X42
)
→
X50
)
)
→
(
(
¬
(
X42
↔
(
(
¬
X27
)
↔
X49
)
)
)
↔
(
¬
(
(
(
¬
X42
)
→
(
¬
X27
)
→
(
(
¬
X11
)
↔
X49
)
)
→
(
¬
(
X50
↔
(
X42
↔
(
X27
→
X49
)
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X50
↔
(
(
¬
X46
)
→
X42
→
X48
)
)
)
↔
(
¬
(
(
¬
X48
)
↔
(
X46
→
(
(
¬
X28
)
↔
X50
)
)
)
)
)
→
(
¬
(
(
¬
X48
)
→
X46
→
X50
)
)
)
↔
(
¬
(
X48
→
(
(
¬
X42
)
↔
X46
)
)
)
)
)
→
(
¬
(
X42
→
(
¬
X28
)
→
X46
)
)
)
)
)
)
→
(
¬
(
(
(
(
X49
→
(
(
¬
X17
)
↔
X28
)
)
↔
(
(
¬
(
X26
↔
(
X17
↔
(
(
¬
X13
)
→
X16
)
)
)
)
→
(
(
¬
X15
)
↔
(
X13
↔
X17
)
)
)
)
→
(
¬
(
X26
↔
(
(
¬
X13
)
→
X49
)
)
)
)
→
(
(
¬
(
X42
↔
(
X17
↔
X12
)
)
)
↔
(
(
(
¬
(
(
(
¬
X42
)
↔
(
X17
→
X12
→
X49
)
)
→
(
¬
(
X42
→
(
¬
X26
)
→
(
X12
↔
X49
)
)
)
)
)
→
(
¬
(
X49
→
X26
→
(
X17
↔
False
)
)
)
)
↔
(
¬
(
X42
→
(
¬
X12
)
→
X26
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
329b3e...
and proposition id is
9a37dc...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMHaXMKEBF8NvBGqqyNHMPavREz3Kevj1pD
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
(
¬
(
(
(
(
(
¬
(
(
¬
X32
)
→
(
(
¬
X18
)
↔
X50
)
)
)
→
(
¬
X30
)
→
(
(
¬
X18
)
↔
(
X13
→
X32
)
)
)
↔
(
(
¬
X44
)
→
(
X32
↔
(
X18
↔
X50
)
)
)
)
→
(
(
¬
(
(
(
X32
↔
(
(
¬
X24
)
→
(
X15
↔
X50
)
)
)
↔
(
¬
(
(
¬
X40
)
↔
(
X18
↔
X50
)
)
)
)
→
(
(
¬
(
X50
→
(
X46
↔
X32
)
)
)
↔
(
¬
(
(
¬
X32
)
→
(
¬
X31
)
→
X40
)
)
)
)
)
↔
(
¬
(
X46
↔
(
X15
↔
(
(
¬
X6
)
→
X18
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
X22
→
(
¬
X20
)
→
(
¬
X3
)
→
X46
)
→
(
(
X49
↔
(
X34
↔
(
X10
↔
X19
)
)
)
↔
(
(
(
¬
(
X20
↔
(
(
¬
X10
)
↔
(
(
¬
X3
)
→
X22
)
)
)
)
↔
(
(
(
(
¬
X3
)
↔
(
¬
(
(
¬
(
X22
↔
(
(
¬
X20
)
→
(
(
¬
X3
)
↔
X49
)
)
)
)
→
X49
→
(
X34
↔
(
X22
→
X20
→
False
)
)
)
)
)
↔
(
¬
(
X22
↔
X20
)
)
)
→
(
(
¬
X34
)
↔
(
(
¬
X20
)
↔
X49
)
)
)
)
↔
(
(
X36
↔
(
(
¬
X34
)
→
X20
→
X49
)
)
↔
(
¬
(
X49
↔
(
X34
↔
X37
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
X45
→
X44
→
X23
→
X46
)
)
↔
(
¬
(
X44
↔
(
(
¬
X17
)
→
X46
)
)
)
)
→
(
¬
(
(
X50
→
(
¬
X23
)
→
(
¬
X7
)
→
X44
)
→
(
(
X50
↔
(
X49
→
(
¬
X44
)
→
X46
)
)
↔
(
(
¬
(
X49
→
(
(
¬
X7
)
↔
X50
)
)
)
→
(
X46
→
(
X7
↔
X50
)
)
→
(
X50
↔
(
(
¬
X21
)
↔
X49
)
)
)
)
)
)
)
→
(
(
X47
→
X37
→
X46
)
↔
(
(
¬
(
X47
→
(
¬
X37
)
→
X27
→
X49
)
)
↔
(
X47
→
X46
→
False
)
)
)
→
(
X49
↔
(
(
¬
X27
)
↔
(
(
¬
X17
)
→
X37
)
)
)
)
)
)
)
↔
(
(
(
(
X45
↔
(
X42
→
(
(
¬
X0
)
↔
X38
)
)
)
→
(
¬
(
(
(
¬
(
X49
↔
(
X42
↔
(
X38
↔
(
X26
→
False
)
)
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X0
)
→
X42
)
)
)
)
→
(
¬
(
(
¬
X46
)
↔
(
X38
↔
(
(
¬
X26
)
→
X49
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
X9
)
→
(
X1
↔
X29
)
)
→
(
¬
(
X45
↔
(
X29
↔
(
(
¬
X1
)
→
X49
)
)
)
)
)
)
→
(
¬
(
(
¬
X45
)
↔
(
(
¬
X42
)
→
X29
→
X49
)
)
)
)
↔
(
¬
(
X49
→
X45
→
X42
)
)
)
)
↔
(
(
¬
(
(
(
X43
↔
(
X33
↔
(
(
¬
X25
)
→
X41
)
)
)
↔
(
(
¬
(
(
¬
X43
)
↔
(
X41
→
(
X39
↔
X45
)
)
)
)
↔
(
¬
(
(
¬
X41
)
↔
(
(
¬
X39
)
→
(
(
¬
X33
)
↔
X45
)
)
)
)
)
)
→
(
(
(
¬
(
X49
↔
(
X43
→
X39
→
X35
)
)
)
↔
(
(
¬
(
X35
↔
(
X28
→
(
X11
↔
X49
)
)
)
)
→
(
(
(
¬
X43
)
→
(
¬
X35
)
→
(
X2
↔
X49
)
)
↔
(
X35
↔
(
(
¬
X28
)
→
X43
)
)
)
)
)
↔
(
X39
↔
(
(
¬
X33
)
→
(
X28
↔
X49
)
)
)
)
)
)
↔
(
(
(
¬
(
X46
↔
(
(
¬
X41
)
→
(
(
¬
X12
)
↔
X43
)
)
)
)
↔
(
X46
↔
(
(
¬
X41
)
→
X39
→
X43
)
)
)
↔
(
(
(
(
¬
(
(
¬
X46
)
→
(
(
¬
X4
)
↔
X48
)
)
)
↔
(
(
¬
(
X46
↔
False
)
)
→
(
(
(
(
¬
(
(
¬
(
X48
→
X5
→
X4
→
False
)
)
→
(
¬
X46
)
→
(
X16
↔
(
X5
→
X48
)
)
)
)
→
X46
→
(
X5
↔
X16
)
)
↔
(
¬
(
X16
↔
(
X5
→
X48
)
)
)
)
↔
(
¬
(
(
¬
X16
)
→
(
X4
↔
X46
)
)
)
)
)
)
→
(
(
¬
(
(
¬
X41
)
→
(
(
¬
X16
)
↔
(
X14
→
X46
)
)
)
)
↔
(
(
¬
X41
)
↔
(
(
¬
X14
)
→
(
¬
X5
)
→
X46
)
)
)
)
→
(
¬
(
X46
→
X41
→
(
X39
↔
(
X8
→
False
)
)
)
)
→
(
X48
↔
(
X8
→
X41
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
(
(
¬
(
X47
↔
(
(
¬
X39
)
→
(
¬
X38
)
→
X41
)
)
)
↔
(
(
¬
(
X41
↔
(
X39
→
X37
→
X23
)
)
)
↔
(
(
¬
(
X41
→
False
)
)
↔
(
(
¬
(
X47
→
(
X39
↔
X23
)
)
)
↔
(
¬
(
(
¬
X41
)
→
(
(
¬
X23
)
↔
(
(
¬
X9
)
→
X47
)
)
)
)
)
)
)
)
→
(
(
¬
X39
)
↔
(
X23
→
(
¬
X8
)
→
X41
)
)
)
↔
(
¬
(
(
¬
X39
)
↔
(
(
¬
X38
)
↔
(
(
¬
X4
)
↔
X47
)
)
)
)
)
↔
(
(
(
X39
→
(
(
¬
X16
)
↔
X38
)
)
→
(
(
¬
(
(
X39
→
(
X38
↔
(
(
¬
X26
)
↔
X48
)
)
)
→
(
¬
(
X48
↔
(
X38
→
X16
→
X6
)
)
)
)
)
↔
(
¬
(
X16
→
X6
)
)
)
)
→
(
¬
(
X41
→
(
X39
↔
(
(
¬
X16
)
→
X32
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X2
)
↔
X38
)
)
↔
(
(
¬
(
(
¬
X18
)
→
(
X13
↔
(
X7
↔
X22
)
)
)
)
→
(
X22
↔
(
X13
↔
(
X2
↔
False
)
)
)
)
)
→
(
¬
(
(
X46
→
X38
→
(
¬
X18
)
→
X22
)
→
(
¬
(
X41
↔
(
(
¬
X38
)
→
(
(
¬
X22
)
↔
X46
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
X27
)
↔
(
(
¬
X21
)
→
(
(
¬
X14
)
↔
X39
)
)
)
↔
(
¬
(
(
X46
↔
(
(
¬
X27
)
→
(
(
¬
X21
)
↔
X38
)
)
)
→
(
¬
(
(
¬
X39
)
→
(
X25
↔
(
X12
↔
X46
)
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X27
)
→
(
¬
X25
)
→
X38
)
)
)
)
)
)
→
(
(
¬
X41
)
↔
(
(
¬
X39
)
↔
(
X25
↔
X46
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
X45
→
(
X44
↔
(
X43
↔
X46
)
)
)
)
→
(
¬
(
(
(
X50
↔
(
(
¬
X42
)
→
(
¬
X3
)
→
X45
)
)
↔
(
¬
(
(
¬
X44
)
→
(
X42
↔
(
(
¬
X3
)
↔
X45
)
)
)
)
)
→
X50
→
X45
→
False
)
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X36
)
→
X28
→
X50
)
)
)
)
↔
(
(
X44
→
(
X43
↔
X50
)
)
→
(
(
X31
→
X30
→
(
¬
X29
)
→
X50
)
↔
(
X44
→
(
X43
↔
(
X29
→
X31
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
X34
↔
(
(
¬
X20
)
→
(
X0
↔
X43
)
)
)
)
↔
(
(
¬
(
(
¬
X49
)
→
(
¬
X43
)
→
X34
→
X50
)
)
→
(
¬
(
X43
→
(
(
¬
X0
)
↔
X50
)
)
)
)
)
↔
(
(
(
X35
→
(
X34
↔
X17
)
)
↔
(
¬
(
X49
↔
(
X35
↔
(
(
¬
X34
)
↔
X43
)
)
)
)
)
↔
(
¬
(
X35
→
(
(
¬
X34
)
↔
X43
)
)
)
)
)
→
(
(
¬
X44
)
↔
(
X34
→
(
X10
↔
X50
)
)
)
→
(
¬
(
(
¬
(
X50
→
X44
→
X5
→
X43
)
)
→
(
¬
X34
)
→
(
(
¬
X5
)
↔
X43
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
(
X47
→
(
(
¬
X43
)
↔
(
(
¬
X19
)
↔
X49
)
)
)
↔
(
¬
(
X48
→
X43
)
)
)
→
(
X48
↔
(
(
¬
X33
)
→
(
X19
↔
X43
)
)
)
)
)
→
X48
→
X47
)
)
→
(
(
¬
(
(
¬
X40
)
→
X49
)
)
→
(
¬
(
X48
↔
(
X44
↔
(
(
¬
X40
)
→
X43
)
)
)
)
)
→
(
X40
→
(
¬
X11
)
→
(
¬
X1
)
→
X43
)
→
(
(
¬
(
X43
→
(
X40
↔
(
X24
↔
X48
)
)
)
)
↔
(
X48
→
(
X40
↔
(
X15
→
X24
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
fcefa6...
and proposition id is
fddcd4...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMFTngJ6KfXz785Nimdxw19sxquRmrpKQaC
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
¬
(
(
(
(
¬
(
(
(
(
X36
→
X0
→
X45
)
→
(
¬
(
(
¬
(
(
¬
X36
)
→
X31
→
(
¬
X25
)
→
X38
)
)
→
(
X38
↔
(
(
¬
X31
)
↔
X37
)
)
)
)
)
↔
(
¬
(
X45
↔
(
X42
→
X37
→
X25
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X38
↔
(
X37
→
(
X22
↔
X42
)
)
)
)
→
(
X42
↔
(
X38
→
(
X37
↔
(
X14
↔
False
)
)
)
)
)
↔
(
¬
(
X38
↔
False
)
)
)
→
(
¬
(
X42
→
(
X37
↔
X22
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X40
→
(
X37
↔
(
(
¬
X7
)
→
X42
)
)
)
)
→
(
¬
(
(
¬
X40
)
↔
(
X37
↔
(
X27
↔
X42
)
)
)
)
)
→
X42
→
X37
→
(
X1
↔
X40
)
)
)
→
(
¬
(
(
(
(
X27
↔
(
X9
↔
False
)
)
↔
(
X42
↔
(
(
¬
X27
)
↔
(
(
¬
X2
)
↔
X37
)
)
)
)
→
(
X27
↔
(
(
¬
X9
)
↔
X42
)
)
)
→
(
X45
↔
(
X3
↔
(
(
¬
X2
)
↔
X37
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X36
)
→
(
(
¬
X34
)
↔
(
X29
→
X40
)
)
)
)
→
(
(
¬
X20
)
→
X40
)
→
X36
→
(
X34
↔
(
(
¬
X21
)
→
X37
)
)
)
↔
(
(
(
¬
X37
)
→
(
X21
↔
(
(
¬
X10
)
→
X41
)
)
)
↔
(
(
X37
↔
(
X35
↔
(
(
¬
X5
)
↔
X10
)
)
)
↔
(
X37
→
(
X21
↔
False
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
X49
→
X26
→
(
¬
X19
)
→
X24
)
→
(
¬
(
(
¬
X26
)
↔
(
(
¬
X23
)
→
X49
)
)
)
→
(
¬
(
(
¬
(
X32
→
(
¬
X23
)
→
X49
)
)
→
(
¬
(
(
¬
X23
)
→
(
X19
↔
X26
)
)
)
)
)
)
↔
(
(
¬
(
(
X47
↔
(
(
¬
X33
)
→
(
X24
↔
X49
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
¬
X47
)
↔
(
X33
→
X49
)
)
)
→
X49
→
(
X17
↔
X16
)
)
)
→
X49
→
(
X33
↔
(
X17
→
(
X15
↔
False
)
)
)
)
)
→
(
X47
↔
(
X33
→
X15
)
)
)
)
↔
(
(
¬
X47
)
↔
(
(
¬
X33
)
→
(
¬
X19
)
→
X49
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X48
↔
(
(
¬
X40
)
↔
(
X30
→
X47
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X45
)
↔
(
(
¬
X40
)
↔
(
X4
↔
X47
)
)
)
)
→
(
¬
X46
)
→
X45
→
X47
)
)
→
(
¬
X45
)
)
)
→
(
¬
(
X47
↔
(
X40
↔
False
)
)
)
)
)
↔
(
(
X47
→
X39
→
(
(
¬
X32
)
↔
X48
)
)
↔
(
¬
(
(
(
¬
X39
)
↔
X46
)
→
(
¬
(
X46
→
(
¬
X33
)
→
(
¬
X32
)
→
X47
)
)
)
)
)
)
→
(
(
¬
X45
)
→
(
X13
↔
(
X4
→
X46
)
)
)
→
(
¬
(
(
¬
(
X46
↔
(
X28
↔
(
(
¬
X13
)
↔
X45
)
)
)
)
→
(
¬
(
(
¬
X45
)
→
(
¬
X28
)
→
(
¬
X6
)
→
X46
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X47
↔
(
(
¬
X18
)
↔
(
X12
→
X49
)
)
)
)
→
(
¬
(
(
X49
↔
(
X46
↔
X18
)
)
→
(
¬
(
(
X49
↔
(
X47
→
X37
→
X46
)
)
→
X47
→
X46
→
X18
→
X49
)
)
)
)
→
(
¬
(
X47
↔
(
X46
↔
(
(
¬
X11
)
↔
X49
)
)
)
)
)
)
→
(
X49
↔
(
X43
→
X44
)
)
→
(
(
(
X49
→
(
¬
X44
)
→
(
(
¬
X43
)
↔
X46
)
)
↔
(
¬
(
(
¬
X47
)
→
(
(
¬
X8
)
↔
X49
)
)
)
)
↔
(
¬
(
X43
→
X47
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
(
(
(
¬
X40
)
→
(
¬
X35
)
→
(
(
¬
X25
)
↔
X47
)
)
↔
(
X5
→
False
)
)
↔
(
(
X35
→
(
X15
↔
(
X5
→
X49
)
)
)
→
(
¬
(
(
¬
X47
)
↔
(
X35
→
(
X5
↔
X48
)
)
)
)
→
(
(
¬
X47
)
↔
(
X35
→
X15
→
X48
)
)
)
)
→
(
(
¬
X48
)
↔
(
(
¬
X47
)
↔
X49
)
)
)
)
↔
(
(
(
(
(
¬
(
(
¬
X43
)
↔
(
X40
↔
(
X20
→
X49
)
)
)
)
↔
(
¬
(
X43
↔
(
(
¬
X20
)
→
X47
)
)
)
)
↔
(
¬
(
(
¬
X43
)
↔
(
(
¬
X40
)
↔
(
(
¬
X13
)
→
X49
)
)
)
)
)
↔
(
X49
→
X13
→
(
¬
X4
)
→
X20
)
)
↔
(
(
X34
→
X26
→
X13
)
→
(
¬
(
(
(
X40
↔
(
X34
↔
(
(
¬
X19
)
→
X28
)
)
)
↔
(
X40
↔
(
(
¬
X28
)
→
(
¬
X13
)
→
X34
)
)
)
→
(
X43
↔
(
X34
→
(
X28
↔
X40
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X43
↔
(
X32
→
(
(
¬
X22
)
↔
X47
)
)
)
)
→
(
(
¬
(
(
¬
X22
)
↔
(
X10
↔
X43
)
)
)
↔
(
X31
→
X22
→
(
X10
↔
X32
)
)
)
)
→
(
(
(
¬
X32
)
→
(
X12
↔
X48
)
)
↔
(
¬
(
(
X17
↔
(
X10
→
X48
)
)
→
(
(
¬
(
X48
↔
(
X43
↔
(
(
¬
X17
)
→
X32
)
)
)
)
↔
(
X48
↔
(
X43
↔
(
X32
↔
X10
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X43
↔
(
(
¬
X30
)
↔
(
X3
↔
X44
)
)
)
)
→
(
¬
(
X43
↔
(
X32
↔
(
X30
→
False
)
)
)
)
→
(
¬
(
X45
↔
(
X43
→
(
X30
↔
X44
)
)
)
)
)
↔
(
¬
(
X48
→
X44
→
(
X8
↔
X3
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
¬
(
X46
→
(
¬
X41
)
→
(
X33
↔
X49
)
)
)
↔
(
¬
(
X41
↔
(
(
¬
X9
)
→
X46
)
)
)
)
→
(
(
¬
(
X41
↔
(
X39
→
X18
→
X46
)
)
)
↔
(
(
¬
(
X49
↔
(
X47
→
X41
→
X39
→
False
)
)
)
↔
(
X46
→
(
(
¬
X41
)
↔
(
X39
↔
X47
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X24
)
↔
(
(
¬
X23
)
→
X42
)
)
)
→
(
(
(
X24
↔
(
X23
→
False
)
)
↔
(
(
¬
(
X49
↔
(
X47
→
(
X24
↔
False
)
)
)
)
↔
(
¬
(
X42
↔
(
X24
→
X23
)
)
)
)
)
↔
(
X24
↔
False
)
)
→
(
¬
(
(
¬
(
(
¬
X42
)
↔
(
(
¬
X24
)
→
(
(
¬
X21
)
↔
X49
)
)
)
)
→
(
X42
↔
(
(
¬
X21
)
→
X11
→
X49
)
)
)
)
)
)
→
(
(
X42
↔
(
X7
↔
(
(
¬
X2
)
↔
X49
)
)
)
↔
(
(
(
¬
X36
)
↔
(
(
¬
X24
)
↔
(
(
¬
X7
)
→
X42
)
)
)
→
(
(
¬
(
X36
→
X7
→
False
)
)
↔
(
¬
(
X41
→
(
X7
↔
(
(
¬
X2
)
↔
X42
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
X27
→
(
(
(
(
¬
X38
)
→
(
X37
↔
(
X29
→
X47
)
)
)
↔
(
(
¬
X38
)
→
(
¬
X29
)
→
(
¬
X14
)
→
X47
)
)
↔
(
(
X27
↔
(
X6
→
(
¬
X0
)
→
X47
)
)
↔
(
¬
(
X47
→
X38
→
(
X0
↔
X37
)
)
)
)
)
)
)
↔
(
(
(
¬
X27
)
↔
(
(
(
¬
X16
)
→
(
(
¬
X1
)
↔
X49
)
)
↔
(
(
¬
X47
)
→
(
¬
X38
)
→
(
(
¬
X16
)
↔
X49
)
)
)
)
→
(
(
¬
(
X47
↔
(
X43
→
(
X39
↔
X49
)
)
)
)
↔
(
(
¬
X43
)
↔
(
X39
↔
(
X38
↔
X49
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
634b35...
and proposition id is
3c9827...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMbZeCLNFmLVf33FrD2e1YaspLLMEQm7fGU
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
∀X50 :
prop
,
(
(
(
(
(
¬
(
(
¬
(
(
¬
(
X25
↔
(
X18
↔
X27
)
)
)
→
X44
→
(
¬
X25
)
→
X12
→
X27
)
)
→
(
¬
(
(
¬
X43
)
→
(
¬
X25
)
→
X18
→
X44
)
)
)
)
↔
(
¬
(
(
¬
(
X49
→
X43
→
(
X37
↔
X47
)
)
)
→
(
¬
(
(
X50
→
(
X48
↔
(
X47
↔
X44
)
)
)
→
(
(
X48
→
X37
→
X43
)
↔
(
(
(
¬
(
(
¬
X47
)
↔
(
(
¬
X43
)
↔
(
X18
→
X48
)
)
)
)
↔
(
X47
→
(
X18
↔
X50
)
)
)
↔
(
X47
→
X50
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
X50
↔
(
X35
↔
(
(
¬
X1
)
↔
X48
)
)
)
)
→
(
¬
(
(
¬
X35
)
→
(
¬
X14
)
→
(
¬
X1
)
→
X45
)
)
)
)
↔
(
¬
(
(
(
¬
(
X42
↔
(
X33
↔
(
(
¬
X6
)
→
X14
)
)
)
)
↔
(
¬
(
(
(
¬
X42
)
→
(
(
¬
X33
)
↔
(
(
¬
X26
)
→
X50
)
)
)
→
X50
→
X33
→
(
¬
X6
)
→
X42
)
)
)
→
(
¬
(
(
X45
→
X42
→
X33
→
False
)
→
(
(
X33
→
X26
→
X30
)
↔
(
¬
(
X45
↔
(
(
¬
X30
)
→
X42
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
X49
)
↔
(
(
¬
X42
)
↔
(
(
¬
X23
)
→
X50
)
)
)
↔
(
¬
(
(
¬
(
X48
↔
(
X21
↔
X42
)
)
)
→
(
¬
(
(
X42
→
(
X17
↔
(
X8
→
X21
)
)
)
→
(
¬
(
X50
↔
(
(
¬
X23
)
↔
(
X17
→
X42
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
X46
)
→
(
X42
↔
(
(
¬
X24
)
↔
X50
)
)
)
→
(
¬
(
X48
↔
(
X42
→
X50
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
(
X40
→
(
¬
X36
)
→
X48
)
→
(
(
X41
→
X40
→
X15
)
↔
(
¬
(
X48
↔
(
X15
→
(
X13
↔
X36
)
)
)
)
)
)
)
↔
(
¬
(
(
X48
→
X41
→
X36
→
(
X0
↔
False
)
)
→
(
¬
(
(
¬
X36
)
↔
(
(
¬
X34
)
→
(
(
¬
X0
)
↔
X48
)
)
)
)
)
)
)
→
(
(
(
¬
(
X41
↔
(
X7
↔
X11
)
)
)
↔
(
(
¬
(
X39
↔
(
(
¬
X11
)
→
(
(
¬
X7
)
↔
X48
)
)
)
)
↔
(
¬
(
(
¬
X39
)
→
(
¬
X11
)
→
(
(
¬
X5
)
↔
X48
)
)
)
)
)
↔
(
(
(
(
X39
↔
(
X22
↔
X40
)
)
→
(
(
X39
↔
(
X31
↔
(
X22
→
False
)
)
)
→
X40
→
X31
→
(
¬
X3
)
→
X22
)
→
(
X40
↔
(
X31
↔
False
)
)
)
→
(
¬
X4
)
→
(
¬
X3
)
→
X40
)
↔
(
(
¬
(
(
¬
X22
)
→
(
¬
X4
)
→
(
X3
↔
X48
)
)
)
↔
(
¬
(
X22
→
(
(
¬
X11
)
↔
(
X4
→
X48
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
(
X41
↔
(
(
¬
X32
)
↔
(
(
¬
X28
)
→
X36
)
)
)
↔
(
(
¬
(
(
¬
X36
)
↔
(
(
¬
X32
)
→
X44
)
)
)
↔
(
(
¬
(
X44
↔
(
X32
→
X41
)
)
)
↔
(
¬
(
(
¬
X41
)
→
(
X36
↔
(
(
¬
X19
)
↔
X44
)
)
)
)
)
)
)
↔
(
¬
(
X41
→
(
(
¬
X36
)
↔
(
(
¬
X32
)
↔
X47
)
)
)
)
)
↔
(
¬
(
(
¬
X44
)
→
(
¬
X41
)
→
(
¬
X40
)
→
X47
)
)
)
↔
(
(
X40
↔
(
(
¬
X32
)
→
(
¬
X2
)
→
X41
)
)
↔
(
X41
↔
(
X38
↔
(
X32
↔
X40
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
X20
→
(
X16
↔
(
X9
→
X38
)
)
)
→
(
¬
(
X20
→
(
¬
X10
)
→
X16
)
)
)
→
(
¬
(
X10
↔
(
X9
→
X29
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X47
↔
(
X38
→
X29
→
X20
→
False
)
)
)
→
(
(
¬
X38
)
↔
(
(
¬
X29
)
↔
(
X20
→
X44
)
)
)
)
)
→
X47
→
(
X29
↔
False
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
(
¬
(
X44
→
(
X32
↔
(
X30
↔
X47
)
)
)
)
→
(
(
X49
→
(
(
¬
X8
)
↔
X47
)
)
↔
(
(
(
¬
(
(
¬
X32
)
→
(
X8
↔
(
X3
↔
X49
)
)
)
)
→
(
¬
(
X32
→
X8
→
X47
)
)
)
↔
(
¬
(
X47
→
X44
→
(
(
¬
X8
)
↔
X32
)
)
)
)
)
)
)
↔
(
(
¬
X43
)
→
(
X35
↔
(
X3
→
X49
)
)
)
)
→
(
(
(
¬
(
(
¬
X47
)
↔
(
(
¬
X46
)
→
(
(
¬
X38
)
↔
X49
)
)
)
)
↔
(
(
(
(
¬
(
X46
→
(
X44
↔
(
(
¬
X26
)
↔
X49
)
)
)
)
↔
(
¬
(
(
¬
X47
)
↔
(
X46
↔
(
(
¬
X44
)
↔
X49
)
)
)
)
)
↔
(
¬
(
X46
→
X26
→
X49
)
)
)
↔
(
¬
(
X47
↔
(
X44
↔
X46
)
)
)
)
)
↔
(
(
(
(
X48
↔
(
X44
→
X22
→
False
)
)
↔
(
X44
→
(
X38
↔
(
X22
↔
X33
)
)
)
)
↔
(
(
(
(
X48
↔
(
(
¬
X38
)
→
X44
)
)
↔
(
¬
(
X48
→
X46
→
(
X38
↔
(
X17
→
False
)
)
)
)
)
↔
(
(
¬
X44
)
↔
(
(
¬
X38
)
↔
X48
)
)
)
↔
(
¬
(
X48
↔
(
X44
→
False
)
)
)
)
)
↔
(
X46
↔
(
X33
→
(
¬
X5
)
→
X48
)
)
)
)
)
)
↔
(
(
(
(
X41
→
(
X36
↔
(
X6
→
X37
)
)
)
→
(
(
¬
(
X37
↔
(
(
¬
X36
)
→
X2
→
X41
)
)
)
↔
(
¬
(
X48
→
(
X36
↔
(
X2
→
X37
)
)
)
)
)
)
↔
(
(
¬
X12
)
↔
(
(
¬
X2
)
↔
X37
)
)
)
↔
(
(
(
X48
↔
(
X37
→
(
¬
X36
)
→
X49
)
)
↔
(
(
(
¬
(
(
¬
X48
)
↔
(
X39
↔
(
X37
↔
X49
)
)
)
)
→
(
X39
↔
(
X37
↔
(
X10
→
X48
)
)
)
)
→
(
¬
(
X48
→
X10
→
X37
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X48
→
(
X46
↔
(
X39
→
False
)
)
)
)
→
(
¬
(
(
¬
X48
)
→
(
¬
X39
)
→
(
X27
↔
X49
)
)
)
)
↔
(
X46
→
(
X39
↔
(
X9
→
X48
)
)
)
)
→
(
(
¬
X39
)
↔
(
(
¬
X9
)
↔
X48
)
)
)
)
)
)
)
↔
(
(
(
(
(
X40
↔
(
X20
↔
(
X19
→
X25
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
X34
)
→
(
¬
X25
)
→
(
X7
↔
X40
)
)
→
X40
→
X34
→
(
X25
↔
(
X19
→
False
)
)
)
)
→
(
(
¬
X25
)
↔
(
X7
↔
X34
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X34
→
(
¬
X25
)
→
(
X24
↔
X49
)
)
)
↔
(
X48
↔
(
X34
→
(
X25
↔
X49
)
)
)
)
↔
(
X49
↔
X24
)
)
→
(
(
¬
(
(
¬
(
(
¬
X49
)
→
(
¬
X48
)
→
(
(
¬
X25
)
↔
X50
)
)
)
→
(
X49
↔
(
X48
→
X40
→
X50
)
)
)
)
↔
(
X49
→
X48
→
(
(
¬
X21
)
↔
X50
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
(
¬
(
X48
↔
(
X42
→
(
X15
↔
(
X14
↔
False
)
)
)
)
)
→
(
(
¬
X15
)
↔
(
(
¬
X13
)
→
X42
)
)
)
→
(
¬
(
X48
→
False
)
)
)
→
(
¬
(
X48
↔
(
X15
→
X42
)
)
)
)
↔
(
¬
(
(
¬
X15
)
→
(
¬
X13
)
→
X48
)
)
)
→
(
¬
(
X31
↔
(
X15
↔
X42
)
)
)
)
→
(
¬
(
(
¬
(
X42
↔
(
(
¬
X25
)
→
X48
)
)
)
→
(
(
X42
→
X40
→
(
¬
X25
)
→
X31
)
↔
(
X42
↔
(
X40
→
X23
→
False
)
)
)
)
)
)
)
)
→
(
(
¬
(
X45
↔
(
X31
→
X18
→
X11
→
False
)
)
)
↔
(
(
(
¬
X16
)
→
X46
)
↔
(
(
¬
(
(
(
¬
X46
)
→
(
¬
X18
)
→
(
X16
↔
X49
)
)
→
(
¬
(
X28
↔
(
X16
→
X46
)
)
)
)
)
→
(
X31
↔
(
X18
↔
False
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X29
→
X16
→
(
X1
↔
X0
)
)
)
→
(
(
¬
X16
)
↔
X18
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X49
)
↔
(
X29
→
(
¬
X4
)
→
X50
)
)
)
↔
(
X4
↔
(
(
¬
X0
)
↔
X50
)
)
)
→
False
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
a42512...
and proposition id is
ada11e...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMQnfq5t5pnqypurrq287QxQGnQHheU2arY
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
∀X50 :
prop
,
∀X51 :
prop
,
∀X52 :
prop
,
(
¬
∀X53 :
prop
,
(
¬
∀X54 :
prop
,
(
(
(
(
¬
(
(
(
(
X38
↔
(
X18
↔
(
X13
→
False
)
)
)
→
(
¬
(
X47
↔
X18
)
)
→
(
¬
(
X51
↔
(
(
¬
X38
)
↔
(
(
¬
X18
)
→
X49
)
)
)
)
)
↔
(
(
¬
X49
)
↔
(
X47
→
(
¬
X40
)
→
X51
)
)
)
→
(
(
(
(
¬
(
(
¬
(
X51
↔
(
X18
→
False
)
)
)
→
(
¬
X52
)
→
(
(
¬
X51
)
↔
(
(
¬
X49
)
↔
X54
)
)
)
)
↔
(
X54
↔
(
X52
↔
(
X6
→
X49
)
)
)
)
↔
(
¬
(
X54
→
X51
→
X49
→
X53
)
)
)
↔
(
(
¬
(
X54
→
(
X52
↔
(
X49
→
(
X1
↔
False
)
)
)
)
)
→
(
¬
(
X52
↔
(
X43
→
X1
→
X49
)
)
)
)
)
)
)
↔
(
(
(
X35
↔
(
X22
↔
X8
)
)
↔
(
(
(
X54
↔
(
X35
↔
X25
)
)
→
(
X52
↔
(
(
¬
X35
)
↔
(
(
¬
X22
)
↔
X54
)
)
)
)
↔
(
¬
(
(
X35
→
(
¬
X8
)
→
X53
)
→
(
¬
(
X54
↔
(
X53
→
(
X52
↔
(
X35
→
False
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
X53
)
↔
(
(
¬
X39
)
→
(
(
¬
X27
)
↔
X54
)
)
)
↔
(
(
¬
(
X54
→
X53
→
(
X41
↔
X52
)
)
)
↔
(
(
X41
→
(
(
¬
X17
)
↔
X54
)
)
↔
(
¬
(
(
¬
X53
)
→
(
X41
↔
(
X24
→
X54
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
(
¬
X28
)
→
(
¬
X23
)
→
X12
→
X52
)
)
↔
(
¬
(
X44
↔
(
(
¬
X28
)
↔
(
X12
→
X52
)
)
)
)
)
↔
(
X44
→
X23
→
X28
)
)
↔
(
(
¬
(
X52
→
(
X45
↔
(
X28
→
X23
→
False
)
)
)
)
→
(
(
(
¬
(
X45
↔
(
X37
↔
False
)
)
)
↔
(
(
X50
→
X45
→
(
(
¬
X37
)
↔
X54
)
)
→
(
¬
(
(
X52
→
(
(
¬
X28
)
↔
X54
)
)
→
(
X54
↔
(
X50
↔
(
X45
→
X52
)
)
)
)
)
)
)
↔
(
X52
→
(
¬
X45
)
→
X54
)
)
)
)
→
(
(
(
¬
(
X50
→
X48
→
(
(
¬
X42
)
↔
X52
)
)
)
↔
(
(
(
X54
→
(
X48
↔
X5
)
)
→
(
X54
↔
(
(
¬
X42
)
→
X52
)
)
)
→
(
(
X52
↔
(
(
¬
X42
)
→
(
(
¬
X21
)
↔
X54
)
)
)
↔
(
(
¬
(
(
X42
→
X21
)
→
(
(
¬
X21
)
↔
(
(
¬
X3
)
→
(
X2
↔
X52
)
)
)
)
)
→
X42
→
(
¬
X21
)
→
X52
)
)
)
)
→
(
(
(
(
¬
(
X52
→
(
(
¬
X26
)
↔
(
(
¬
X16
)
↔
X54
)
)
)
)
→
(
(
X52
↔
X16
)
↔
(
¬
(
(
¬
X52
)
→
(
X33
↔
(
X32
→
X54
)
)
)
)
)
)
→
X52
→
X42
→
(
¬
X16
)
→
X32
)
↔
(
¬
(
X33
↔
(
X32
→
(
¬
X0
)
→
X54
)
)
)
)
)
→
(
(
(
X52
↔
(
X50
→
(
X9
↔
False
)
)
)
↔
(
(
¬
X31
)
↔
(
(
¬
X14
)
→
(
(
¬
X9
)
↔
X50
)
)
)
)
↔
(
X50
→
(
X33
↔
False
)
)
)
)
↔
(
(
(
(
(
¬
(
(
(
(
¬
X50
)
→
(
X15
↔
(
(
¬
X11
)
↔
X52
)
)
)
→
(
X52
↔
(
(
¬
X50
)
→
(
(
¬
X15
)
↔
X51
)
)
)
)
→
(
X51
↔
(
X50
↔
False
)
)
)
)
→
X15
→
(
¬
X11
)
→
X50
)
↔
(
X52
↔
(
X50
→
X36
→
X15
→
False
)
)
)
↔
(
(
(
¬
(
X30
↔
(
X29
→
False
)
)
)
↔
(
X52
↔
(
X36
→
(
X30
↔
(
X4
→
False
)
)
)
)
)
↔
(
(
X30
→
(
¬
X29
)
→
(
¬
X7
)
→
X52
)
↔
(
(
(
(
¬
(
(
¬
X19
)
↔
(
X10
→
X30
)
)
)
→
(
¬
(
(
¬
X30
)
↔
(
(
¬
X29
)
↔
(
(
¬
X10
)
→
X52
)
)
)
)
)
↔
(
¬
(
X19
→
X10
→
X52
)
)
)
↔
(
¬
(
X50
↔
(
(
¬
X19
)
↔
(
(
¬
X10
)
→
X52
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
X51
)
→
(
X46
↔
(
(
¬
X34
)
↔
X54
)
)
)
↔
(
¬
(
X20
↔
X54
)
)
)
→
(
¬
(
(
¬
(
(
¬
X51
)
→
X50
→
X54
)
)
→
(
¬
(
(
¬
(
X51
→
(
¬
X36
)
→
X54
)
)
→
(
¬
(
X54
→
(
(
¬
X50
)
↔
(
(
¬
X46
)
↔
X51
)
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
X52
↔
(
(
¬
X41
)
↔
(
(
¬
X18
)
↔
X51
)
)
)
)
↔
(
X54
→
(
(
¬
X41
)
↔
X52
)
)
)
↔
(
(
(
¬
(
X34
↔
(
X4
↔
False
)
)
)
↔
(
X36
→
X34
→
(
X18
↔
X54
)
)
)
↔
(
X36
↔
(
X4
→
X18
)
)
)
)
→
(
(
¬
(
(
(
¬
X50
)
↔
(
X46
→
X54
)
)
→
(
¬
(
(
X54
→
(
¬
X36
)
→
(
(
¬
X21
)
↔
X51
)
)
→
(
(
¬
(
X46
↔
X21
)
)
↔
(
(
¬
(
X51
↔
(
(
¬
X21
)
↔
X54
)
)
)
↔
(
¬
(
(
¬
X46
)
→
X50
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
X46
→
(
(
¬
X12
)
↔
X54
)
)
)
↔
(
¬
(
(
¬
(
X54
↔
(
X11
↔
X12
)
)
)
→
(
(
X54
↔
(
X50
↔
(
X46
→
X11
)
)
)
↔
(
(
¬
X50
)
↔
(
X46
→
X12
→
X54
)
)
)
)
)
)
→
(
¬
(
(
¬
X46
)
→
X50
)
)
)
→
(
¬
(
(
¬
X50
)
→
(
X11
↔
X54
)
)
)
)
↔
(
X54
→
(
X46
↔
X11
)
)
)
↔
(
(
¬
X12
)
→
(
X11
↔
X50
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
(
X52
↔
(
X43
↔
(
X42
↔
X54
)
)
)
)
→
(
¬
(
X52
↔
(
X43
↔
(
(
¬
X29
)
↔
X54
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
X54
↔
(
X53
→
(
¬
X2
)
→
X43
)
)
→
(
¬
X42
)
→
(
(
¬
X2
)
↔
X43
)
)
)
→
(
¬
(
X44
↔
(
X43
↔
(
(
¬
X42
)
↔
X54
)
)
)
)
)
)
)
→
(
X51
↔
(
X47
↔
(
(
¬
X43
)
↔
X44
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
¬
(
X37
→
(
(
¬
X28
)
↔
(
(
¬
X19
)
→
X52
)
)
)
)
→
(
¬
X51
)
)
)
→
X28
→
(
X19
↔
X52
)
)
)
→
(
¬
(
(
(
¬
X20
)
↔
X53
)
→
X52
→
(
¬
X28
)
→
(
(
¬
X20
)
↔
X37
)
)
)
→
(
¬
(
(
¬
X52
)
→
(
(
¬
X28
)
↔
X53
)
)
)
)
→
(
(
(
(
(
¬
(
X51
→
X28
→
X10
)
)
↔
(
X28
↔
(
(
¬
X27
)
↔
(
(
¬
X20
)
→
X39
)
)
)
)
→
X27
→
(
¬
X20
)
→
(
X7
↔
X39
)
)
↔
(
X51
→
(
¬
X27
)
→
X28
)
)
↔
(
¬
(
X51
→
(
¬
X27
)
→
(
X15
↔
X39
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
(
X53
↔
(
X51
→
(
X45
↔
False
)
)
)
→
(
(
¬
(
(
X9
→
X45
)
→
(
¬
(
X26
↔
(
(
¬
X16
)
↔
(
(
¬
X9
)
→
X53
)
)
)
)
)
)
↔
(
X51
→
X26
→
X16
→
(
X9
↔
False
)
)
)
)
→
X45
→
(
¬
X26
)
→
(
(
¬
X6
)
↔
X53
)
)
→
(
¬
(
X45
↔
(
X26
→
(
¬
X9
)
→
X49
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
¬
X53
)
↔
(
X23
↔
(
X14
↔
X54
)
)
)
→
(
(
(
X26
→
(
(
¬
X14
)
↔
X53
)
)
→
(
(
¬
(
(
¬
(
(
¬
X16
)
→
X54
)
)
→
(
¬
(
X26
→
(
¬
X16
)
→
(
¬
X14
)
→
X53
)
)
)
)
↔
(
¬
(
X53
↔
(
X16
↔
(
X14
→
False
)
)
)
)
)
)
↔
(
X54
↔
(
X26
→
X16
→
False
)
)
)
)
)
↔
(
¬
(
X54
↔
(
(
¬
X23
)
→
X26
)
)
)
)
→
(
¬
(
X54
→
(
(
¬
X23
)
↔
X53
)
)
)
)
)
)
→
(
(
(
(
(
X32
→
(
(
¬
X31
)
↔
(
(
¬
X0
)
→
X40
)
)
)
↔
(
(
¬
X32
)
→
(
¬
X31
)
→
(
X8
↔
X40
)
)
)
↔
(
(
¬
X48
)
→
X40
→
(
X8
↔
X49
)
)
)
→
(
(
X49
→
(
X31
↔
X32
)
)
↔
(
¬
(
(
(
¬
X49
)
→
(
(
¬
X31
)
↔
(
(
¬
X17
)
→
X51
)
)
)
→
(
¬
(
X51
→
(
X49
↔
(
X31
↔
X32
)
)
)
)
)
)
)
)
↔
(
¬
(
X53
↔
(
X51
↔
(
X32
↔
(
X17
↔
False
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X54
↔
(
(
¬
X38
)
↔
(
(
¬
X35
)
↔
X51
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X51
↔
(
X38
↔
(
(
¬
X13
)
↔
X35
)
)
)
)
→
X38
→
(
X35
↔
(
X13
→
X1
→
False
)
)
)
→
(
¬
(
(
¬
X13
)
↔
(
(
¬
X1
)
↔
X51
)
)
)
)
→
(
(
¬
X38
)
↔
(
X35
↔
X51
)
)
)
)
)
→
(
(
(
(
¬
X48
)
↔
(
(
¬
X33
)
↔
X51
)
)
↔
(
¬
(
(
¬
X22
)
→
(
¬
X3
)
→
X38
)
)
)
↔
(
X51
↔
(
X35
→
(
(
¬
X33
)
↔
X38
)
)
)
)
)
→
(
(
(
¬
(
X51
↔
(
X25
↔
(
X24
↔
X38
)
)
)
)
↔
(
¬
(
X38
→
X25
→
(
X5
↔
X24
)
)
)
)
↔
(
¬
(
X51
↔
(
X5
↔
X25
)
)
)
)
→
(
(
¬
X38
)
↔
(
X30
↔
(
X24
↔
X48
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
e43fe3...
and proposition id is
f5ffd8...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMayf3VtXXSwwh311zQufRhDQp4qGCZ8sWu
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
(
(
(
(
X48
→
(
(
¬
X39
)
↔
(
(
¬
X11
)
↔
X45
)
)
)
↔
(
(
¬
(
X48
→
X39
→
(
(
¬
X38
)
↔
X45
)
)
)
→
(
X45
→
(
X15
↔
(
(
¬
X13
)
→
X49
)
)
)
→
(
¬
(
(
¬
(
(
¬
X38
)
↔
(
X15
↔
(
X13
→
X39
)
)
)
)
→
(
¬
(
X49
→
(
X38
↔
(
X15
↔
(
X13
↔
False
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X50
↔
(
X38
↔
(
X22
↔
False
)
)
)
)
→
(
(
X36
→
(
¬
X22
)
→
(
(
¬
X9
)
↔
X50
)
)
↔
(
X50
→
X38
→
X36
→
X9
→
False
)
)
)
)
↔
(
(
¬
(
X36
→
(
X9
↔
X49
)
)
)
→
(
X49
↔
(
X45
↔
X50
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
¬
X29
)
↔
X49
)
→
(
¬
(
(
¬
X17
)
→
(
¬
X2
)
→
X49
)
)
→
(
X49
↔
(
(
¬
X24
)
→
(
(
¬
X17
)
↔
X39
)
)
)
)
↔
(
X40
↔
(
X29
↔
(
X17
→
X39
)
)
)
)
↔
(
(
¬
(
(
(
X47
→
(
¬
X39
)
→
(
X33
↔
X41
)
)
↔
(
¬
(
X47
↔
(
X41
↔
(
X39
→
X40
)
)
)
)
)
→
X39
→
X32
→
(
¬
X4
)
→
X40
)
)
→
(
(
(
(
¬
(
(
¬
(
X40
↔
(
(
¬
X33
)
→
(
X28
↔
X47
)
)
)
)
→
X47
→
X33
→
(
X29
↔
X40
)
)
)
↔
(
¬
(
(
¬
X33
)
↔
(
X28
↔
X40
)
)
)
)
↔
(
(
¬
X33
)
↔
(
(
¬
X28
)
→
X40
)
)
)
↔
(
¬
(
X40
↔
(
X33
→
X29
)
)
)
)
)
)
→
(
X39
→
(
(
¬
X31
)
↔
(
X10
↔
X38
)
)
)
→
(
(
(
¬
(
X47
→
(
(
¬
X26
)
↔
(
(
¬
X25
)
→
X40
)
)
)
)
→
(
¬
(
X39
→
(
(
¬
X31
)
↔
(
X3
→
X40
)
)
)
)
)
↔
(
(
X47
↔
(
(
¬
X39
)
→
(
¬
X38
)
→
X40
)
)
→
(
X40
↔
(
X39
↔
(
X6
↔
X47
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
(
(
X27
→
(
(
¬
X7
)
↔
X18
)
)
↔
(
(
X44
→
(
X27
↔
(
X7
↔
(
X1
↔
False
)
)
)
)
↔
(
¬
(
X44
→
X18
→
X7
)
)
)
)
↔
(
¬
(
(
¬
X27
)
↔
(
(
¬
X7
)
→
X44
)
)
)
)
→
(
X27
↔
(
(
¬
X18
)
→
(
(
¬
X7
)
↔
X30
)
)
)
)
→
(
¬
(
X44
↔
(
X7
→
False
)
)
)
)
→
(
X46
↔
(
(
¬
X27
)
→
(
X23
↔
X44
)
)
)
→
(
(
¬
(
X30
→
X27
→
X46
)
)
↔
(
(
X46
↔
(
X27
→
(
¬
X14
)
→
X44
)
)
→
(
(
(
¬
X44
)
→
X30
→
(
X27
↔
X49
)
)
↔
(
¬
(
(
¬
X46
)
↔
(
(
¬
X44
)
↔
(
(
¬
X30
)
↔
X49
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
¬
X47
)
→
(
X19
↔
(
(
¬
X0
)
→
X48
)
)
)
)
→
(
(
(
¬
(
X47
→
(
(
¬
X46
)
↔
(
X19
↔
X48
)
)
)
)
→
(
¬
(
X48
↔
(
X47
↔
(
X46
↔
(
X18
→
False
)
)
)
)
)
)
↔
(
X48
→
(
¬
X46
)
→
X47
)
)
)
)
→
(
(
¬
(
X48
↔
(
X47
↔
(
X23
↔
X46
)
)
)
)
→
X46
→
X43
→
X23
→
False
)
→
(
¬
(
(
¬
X43
)
↔
(
X23
↔
X46
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
X12
↔
X47
)
→
(
¬
(
(
¬
(
X49
↔
(
X37
↔
X16
)
)
)
→
X21
→
(
X8
↔
X37
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
(
¬
X47
)
→
X46
→
(
¬
X42
)
→
X49
)
)
→
(
(
¬
X46
)
↔
(
X42
→
(
X5
↔
X49
)
)
)
)
→
X37
→
(
¬
X5
)
→
X49
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X37
→
(
X34
↔
(
(
¬
X20
)
→
X46
)
)
)
)
→
X37
→
(
(
¬
X34
)
↔
(
(
¬
X20
)
→
X35
)
)
)
)
→
(
X42
↔
(
X37
↔
False
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
(
(
¬
(
X44
↔
(
X33
→
X46
)
)
)
↔
(
¬
(
(
¬
(
X46
→
X33
→
X44
)
)
→
X44
→
X43
→
X49
)
)
)
→
(
¬
(
X46
→
(
¬
X33
)
→
X49
)
)
)
)
→
(
¬
(
(
X49
↔
(
X33
↔
(
(
¬
X27
)
→
X47
)
)
)
→
(
(
¬
(
X50
→
(
X49
↔
(
(
¬
X27
)
↔
X33
)
)
)
)
↔
(
(
¬
X43
)
→
(
¬
X27
)
→
X49
)
)
)
)
)
→
(
(
(
¬
(
X50
↔
(
(
¬
X48
)
↔
(
X8
↔
X49
)
)
)
)
↔
(
(
(
¬
X8
)
↔
(
X7
→
X49
)
)
↔
(
¬
(
(
¬
X48
)
→
(
(
¬
X37
)
↔
(
X7
→
X49
)
)
)
)
)
)
↔
(
¬
(
X48
↔
(
X47
↔
(
X8
↔
X37
)
)
)
)
)
)
↔
(
(
(
¬
(
X49
↔
(
X44
↔
(
X2
→
X29
)
)
)
)
↔
(
(
X49
→
(
X44
↔
(
X6
↔
X33
)
)
)
→
(
¬
(
X6
→
X2
)
)
)
)
↔
(
(
(
¬
X33
)
↔
(
X29
→
X5
→
X44
)
)
→
X18
→
X5
)
)
)
↔
(
(
¬
(
(
¬
(
X40
→
X38
→
X9
→
X35
)
)
→
(
(
(
¬
(
(
X25
→
X19
→
(
(
¬
X10
)
↔
X23
)
)
→
(
¬
(
X25
→
(
X23
↔
(
(
¬
X19
)
→
X35
)
)
)
)
)
)
→
(
(
(
¬
X25
)
→
X13
→
(
¬
X10
)
→
X45
)
↔
(
(
X45
↔
(
X23
↔
(
X13
↔
X38
)
)
)
↔
(
X25
→
(
X23
↔
X38
)
)
)
)
)
↔
(
(
X40
↔
(
X38
↔
(
X31
↔
(
X23
→
False
)
)
)
)
↔
(
¬
(
X31
↔
(
X19
↔
X40
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
(
X45
→
(
¬
X35
)
→
(
(
¬
X4
)
↔
X41
)
)
↔
(
X46
→
X41
→
(
(
¬
X35
)
↔
X40
)
)
)
→
(
¬
(
(
¬
(
(
¬
X45
)
↔
(
X41
↔
X46
)
)
)
→
(
¬
(
X50
→
(
X49
↔
(
X41
↔
X45
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
X46
)
→
(
(
¬
X41
)
↔
X50
)
)
)
)
)
↔
(
(
X40
↔
(
(
¬
X22
)
→
X41
)
)
↔
(
¬
(
X50
↔
(
X22
↔
(
X0
→
X41
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
X36
↔
(
X32
↔
X49
)
)
→
(
(
¬
X36
)
→
(
¬
X34
)
→
X15
→
X49
)
→
X36
→
(
(
¬
X34
)
↔
(
X32
↔
X42
)
)
)
)
↔
(
(
¬
(
X42
→
X39
→
X36
)
)
→
(
¬
(
(
¬
(
X42
→
(
X34
↔
(
(
¬
X28
)
↔
X39
)
)
)
)
→
(
¬
(
(
¬
X39
)
↔
(
(
¬
X36
)
↔
(
(
¬
X34
)
↔
X42
)
)
)
)
)
)
)
)
→
(
(
¬
X49
)
→
(
¬
X42
)
→
(
X32
↔
X50
)
)
→
(
¬
(
(
¬
(
(
(
¬
X32
)
→
X24
→
(
(
¬
X20
)
↔
X49
)
)
→
(
(
¬
X24
)
↔
X26
)
)
)
→
(
¬
X20
)
→
X42
)
)
)
↔
(
(
¬
(
X49
↔
(
(
¬
X39
)
→
X46
)
)
)
↔
(
¬
(
(
X49
↔
(
(
¬
X30
)
→
X11
→
X36
)
)
→
(
(
X49
↔
(
X39
↔
X46
)
)
↔
(
¬
(
X46
→
(
X30
↔
X11
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
X50
↔
(
(
¬
X17
)
↔
(
X16
→
X21
)
)
)
↔
(
¬
(
X50
↔
(
X1
→
X16
)
)
)
)
↔
(
(
¬
X16
)
↔
(
(
¬
X3
)
↔
(
(
¬
X1
)
→
X17
)
)
)
)
→
(
(
¬
(
(
¬
X49
)
↔
(
X42
↔
(
(
¬
X14
)
↔
X50
)
)
)
)
↔
(
¬
(
(
¬
X42
)
↔
(
(
¬
X14
)
→
(
(
¬
X12
)
↔
X49
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
6fcb8b...
and proposition id is
4df964...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMJL1PhRnUFgG1im3svgxVLtgnmtWUduzMN
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
¬
∀X51 :
prop
,
(
(
¬
(
(
(
(
¬
(
(
(
¬
(
(
¬
X34
)
↔
(
(
¬
X17
)
↔
(
(
¬
X6
)
↔
X51
)
)
)
)
↔
(
X39
→
(
¬
X12
)
→
X31
)
)
→
(
¬
(
(
¬
X39
)
→
(
¬
X31
)
→
(
X6
↔
X47
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X49
)
↔
(
X46
↔
(
X30
↔
X50
)
)
)
)
↔
(
(
(
X46
→
X1
→
X49
)
↔
(
X30
→
X18
→
(
¬
X1
)
→
X31
)
)
→
(
¬
(
X49
→
(
X46
↔
X18
)
)
)
)
)
↔
(
(
¬
(
(
(
(
X34
→
(
¬
X23
)
→
X42
)
→
(
(
(
¬
X34
)
→
(
(
¬
X23
)
↔
X42
)
)
↔
(
(
¬
(
X34
→
(
X1
↔
X23
)
)
)
→
(
¬
(
X42
↔
(
(
¬
X34
)
→
(
X23
↔
X49
)
)
)
)
)
)
)
→
(
¬
(
X42
→
(
¬
X31
)
→
(
(
¬
X23
)
↔
X34
)
)
)
)
→
(
¬
(
X49
↔
X42
)
)
)
)
↔
(
(
X12
↔
(
X1
→
X0
→
X23
)
)
↔
(
¬
(
X23
↔
(
X1
→
(
X0
↔
X31
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
X45
→
X32
→
(
(
¬
X21
)
↔
X48
)
)
→
(
¬
(
X51
↔
(
(
¬
X4
)
→
X47
)
)
)
)
↔
(
¬
(
(
¬
X32
)
→
(
¬
X21
)
→
(
X11
↔
X51
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
¬
(
(
¬
X21
)
↔
X45
)
)
→
(
X48
↔
(
(
¬
X20
)
↔
(
(
¬
X3
)
→
X21
)
)
)
)
)
↔
(
X45
↔
(
X3
→
X21
)
)
)
→
(
¬
(
X45
→
(
(
¬
X3
)
↔
X21
)
)
)
)
)
→
(
¬
(
(
¬
X21
)
↔
(
(
¬
X20
)
↔
X48
)
)
)
)
)
→
(
¬
(
(
(
X51
→
(
X44
↔
(
X42
→
X35
)
)
)
↔
(
(
(
¬
(
X44
→
(
¬
X35
)
→
X27
→
X51
)
)
↔
(
(
¬
X44
)
→
(
¬
X41
)
→
X35
→
X51
)
)
↔
(
¬
(
(
¬
X41
)
↔
(
(
¬
X35
)
↔
X44
)
)
)
)
)
→
(
(
¬
(
(
¬
X48
)
↔
(
(
¬
X44
)
↔
(
(
¬
X22
)
↔
X51
)
)
)
)
↔
(
(
¬
X48
)
→
(
¬
X47
)
→
(
(
¬
X22
)
↔
X51
)
)
)
→
(
(
X48
↔
(
X47
→
(
(
¬
X38
)
↔
X40
)
)
)
↔
(
(
(
¬
(
(
(
¬
X42
)
↔
(
(
¬
X38
)
↔
(
X9
→
X44
)
)
)
→
(
¬
(
X44
↔
(
X42
→
(
(
¬
X38
)
↔
X48
)
)
)
)
)
)
→
(
¬
(
X44
→
(
(
¬
X38
)
↔
X48
)
)
)
)
↔
(
(
¬
(
X42
↔
(
(
¬
X29
)
↔
X48
)
)
)
↔
(
(
X47
↔
(
X29
→
X9
→
False
)
)
→
(
(
¬
(
X48
→
(
X47
↔
(
X42
→
False
)
)
)
)
↔
(
¬
(
X48
→
(
(
¬
X29
)
↔
(
X9
↔
X47
)
)
)
)
)
)
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
X33
)
→
(
X24
↔
(
(
¬
X8
)
→
X45
)
)
)
)
→
(
¬
(
X33
→
X24
→
(
¬
X8
)
→
X43
)
)
)
↔
(
(
¬
(
(
(
¬
(
X45
→
(
X37
↔
(
X8
↔
X36
)
)
)
)
↔
(
(
(
(
(
¬
(
X37
↔
(
X36
↔
(
X8
→
X15
)
)
)
)
↔
(
¬
(
X37
→
(
(
¬
X15
)
↔
(
X8
↔
X48
)
)
)
)
)
→
(
¬
X15
)
→
X48
)
→
X48
→
X37
→
(
(
¬
X19
)
↔
X36
)
)
↔
(
X48
↔
(
(
¬
X19
)
→
X37
)
)
)
)
→
(
¬
(
(
¬
(
X36
→
False
)
)
→
(
(
¬
(
(
¬
X43
)
→
X19
→
(
X5
↔
X48
)
)
)
↔
(
X48
↔
(
X36
↔
False
)
)
)
→
X19
→
(
¬
X5
)
→
X48
)
)
)
)
→
(
¬
(
(
¬
(
X43
↔
(
(
¬
X16
)
→
(
X2
↔
X36
)
)
)
)
→
(
¬
(
X48
→
(
X43
↔
(
X36
↔
False
)
)
)
)
)
)
→
(
X43
↔
(
(
¬
X2
)
↔
X36
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X45
↔
(
X43
→
(
X37
↔
X14
)
)
)
)
↔
(
(
¬
X43
)
↔
(
(
¬
X37
)
→
(
X28
↔
X45
)
)
)
)
→
(
¬
(
(
X43
↔
(
X37
→
(
X10
↔
X25
)
)
)
→
(
¬
(
X43
↔
(
X25
↔
X45
)
)
)
)
)
)
)
→
(
X43
↔
(
X28
→
(
X7
↔
False
)
)
)
→
(
(
X43
↔
(
X13
↔
X26
)
)
↔
(
¬
(
(
¬
X28
)
↔
(
X26
↔
(
X14
↔
X43
)
)
)
)
)
→
(
X46
↔
(
X14
↔
X28
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
(
(
¬
(
X46
↔
(
X42
↔
(
(
¬
X39
)
↔
X40
)
)
)
)
↔
(
¬
(
X46
→
X45
→
X42
→
(
X40
↔
False
)
)
)
)
→
(
¬
(
(
(
¬
X29
)
→
X39
)
→
(
¬
(
X42
→
(
X39
↔
(
(
¬
X1
)
→
X45
)
)
)
)
)
)
)
→
(
(
X45
→
X43
→
X49
)
↔
(
(
¬
(
X42
↔
(
X17
→
(
(
¬
X5
)
↔
X43
)
)
)
)
↔
(
(
¬
(
(
¬
(
X45
→
X43
→
(
X17
↔
False
)
)
)
→
(
X42
→
X40
→
X45
)
→
X45
→
(
(
¬
X42
)
↔
(
X17
↔
X43
)
)
)
)
→
X43
→
(
(
¬
X40
)
↔
X45
)
)
)
)
)
)
↔
(
(
X47
→
(
¬
X45
)
→
(
¬
X39
)
→
X46
)
↔
(
(
¬
(
X39
→
(
(
¬
X38
)
↔
(
X19
↔
X46
)
)
)
)
↔
(
(
(
¬
(
X45
↔
(
X39
↔
(
X33
→
X22
→
False
)
)
)
)
↔
(
¬
(
(
¬
X39
)
↔
(
(
¬
X38
)
↔
X45
)
)
)
)
↔
(
X46
↔
(
X45
→
X38
→
X22
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
X45
→
(
¬
X20
)
→
(
X11
↔
X44
)
)
→
(
¬
(
X49
↔
(
X45
↔
False
)
)
)
)
→
(
(
(
(
(
¬
(
(
(
¬
X34
)
→
(
X9
↔
X45
)
)
→
X49
→
(
X45
↔
(
X34
→
X44
)
)
)
)
→
X49
→
X44
→
X9
→
False
)
↔
(
(
¬
X47
)
↔
X49
)
)
↔
(
¬
(
(
¬
X47
)
↔
(
X45
↔
(
(
¬
X41
)
↔
X49
)
)
)
)
)
↔
(
X49
→
(
(
¬
X47
)
↔
(
(
¬
X45
)
→
X48
)
)
)
)
)
)
→
(
(
X37
↔
(
X15
→
X47
)
)
↔
(
¬
(
(
¬
(
(
¬
X15
)
→
X16
)
)
→
(
X48
↔
(
X44
→
(
(
¬
X15
)
↔
X37
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
X51
→
(
¬
X26
)
→
X49
)
↔
(
(
(
(
(
¬
X45
)
↔
(
X27
↔
(
(
¬
X2
)
→
X51
)
)
)
↔
(
¬
(
X45
↔
(
(
¬
X27
)
→
(
(
¬
X14
)
↔
X51
)
)
)
)
)
↔
(
¬
(
X51
↔
(
X45
→
X2
→
False
)
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X27
)
↔
(
X2
→
X51
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X45
)
↔
(
X32
↔
(
X0
↔
X47
)
)
)
)
→
(
(
¬
(
X49
→
X47
→
(
¬
X32
)
→
X51
)
)
↔
(
(
¬
X49
)
→
(
X45
↔
(
(
¬
X32
)
↔
X51
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
(
¬
X30
)
↔
(
X25
↔
(
X21
→
X36
)
)
)
)
↔
(
(
(
¬
(
X23
→
X6
→
X30
)
)
→
X49
→
(
¬
X23
)
→
(
(
¬
X4
)
↔
X36
)
)
↔
(
¬
(
(
¬
X30
)
↔
(
(
¬
X25
)
→
(
¬
X23
)
→
X36
)
)
)
)
)
→
(
(
¬
(
(
(
(
X31
↔
(
(
¬
X24
)
→
(
X7
↔
X46
)
)
)
→
(
¬
(
X31
→
X24
→
X23
→
X46
)
)
)
→
(
¬
X24
)
→
(
X23
↔
X46
)
)
→
(
¬
(
X49
↔
(
X31
↔
(
X23
→
X24
)
)
)
)
)
)
↔
(
(
X35
↔
(
(
¬
X24
)
↔
(
X7
↔
X46
)
)
)
↔
(
X24
→
(
(
¬
X21
)
↔
X35
)
)
)
)
)
)
↔
(
(
(
¬
(
(
X50
↔
(
X31
↔
(
X12
↔
X36
)
)
)
→
(
¬
(
X50
↔
(
X13
↔
X3
)
)
)
)
)
↔
(
¬
(
X31
→
X18
→
(
¬
X3
)
→
X36
)
)
)
↔
(
(
X47
→
X10
→
(
¬
X8
)
→
X36
)
↔
(
(
(
¬
(
X31
↔
(
X10
→
False
)
)
)
↔
(
(
(
¬
(
X31
↔
(
X28
→
X47
)
)
)
↔
(
¬
(
(
¬
(
X47
→
X28
)
)
→
(
¬
(
(
¬
X31
)
→
(
(
¬
X10
)
↔
(
X8
↔
X47
)
)
)
)
)
)
)
→
(
X50
↔
(
X31
→
(
X8
↔
X10
)
)
)
)
)
↔
(
X46
↔
(
X28
→
X8
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
3908ae...
and proposition id is
74a8c7...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMXNwUbXLyqTXXEzEhGyjuXAZYq6MSx6o9i
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
∀X50 :
prop
,
(
¬
∀X51 :
prop
,
(
¬
∀X52 :
prop
,
(
(
(
(
(
(
(
¬
(
X48
→
(
X42
↔
X41
)
)
)
↔
(
(
¬
(
X48
↔
(
X42
↔
(
X41
→
X50
)
)
)
)
↔
(
(
(
X50
→
(
(
¬
X41
)
↔
(
(
¬
X30
)
→
X42
)
)
)
→
(
¬
(
X42
→
(
¬
X41
)
→
(
X24
↔
X50
)
)
)
)
→
(
¬
(
X50
→
X42
→
X41
)
)
)
)
)
→
(
(
¬
X48
)
↔
(
X22
↔
X50
)
)
)
↔
(
(
¬
(
(
¬
X24
)
↔
(
X22
→
(
¬
X10
)
→
X39
)
)
)
↔
(
¬
(
(
(
¬
(
X22
→
(
(
¬
X10
)
↔
(
X2
→
X41
)
)
)
)
↔
(
(
¬
X19
)
→
X2
→
X39
)
)
→
(
X41
↔
(
X22
↔
(
X19
↔
False
)
)
)
)
)
)
)
↔
(
(
(
(
X52
↔
(
X48
→
(
X45
↔
X6
)
)
)
↔
(
¬
(
(
¬
X45
)
↔
(
(
¬
X36
)
→
(
(
¬
X6
)
↔
X52
)
)
)
)
)
→
(
¬
(
X48
↔
(
X45
↔
(
(
¬
X17
)
↔
X50
)
)
)
)
→
(
¬
(
(
¬
X50
)
↔
(
X45
→
(
X17
↔
X52
)
)
)
)
)
→
(
(
(
(
¬
(
(
X52
↔
(
X50
→
X47
→
X48
)
)
→
(
¬
(
(
¬
X50
)
↔
(
X48
↔
(
X41
↔
X52
)
)
)
)
)
)
↔
(
¬
(
(
X50
→
(
X35
↔
(
X9
↔
X48
)
)
)
→
(
X50
↔
(
X48
→
(
X35
↔
X52
)
)
)
)
)
)
→
(
¬
(
(
¬
X48
)
→
X41
→
(
(
¬
X8
)
↔
X52
)
)
)
→
(
(
(
¬
(
X48
↔
(
X21
↔
(
(
¬
X8
)
↔
X41
)
)
)
)
↔
(
(
¬
X41
)
→
X28
→
(
(
¬
X21
)
↔
X48
)
)
)
↔
(
X41
↔
(
X21
↔
False
)
)
)
)
↔
(
X52
→
(
X48
↔
(
X41
→
X34
→
False
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
¬
(
(
¬
(
X16
→
X46
)
)
→
(
(
X40
↔
(
(
¬
X33
)
↔
(
(
¬
X15
)
↔
X50
)
)
)
→
X33
→
X16
)
→
(
¬
(
X50
→
X40
→
X16
→
False
)
)
)
)
↔
(
X46
↔
(
(
¬
X33
)
→
X40
)
)
)
→
(
(
(
¬
X33
)
→
(
¬
X26
)
→
X40
)
↔
(
(
¬
(
X50
→
(
¬
X33
)
→
X26
→
X40
)
)
↔
(
X40
↔
(
(
¬
X26
)
→
X25
→
X50
)
)
)
)
)
→
(
(
(
(
¬
(
(
(
(
¬
(
(
(
X50
↔
(
X40
→
X1
→
X52
)
)
↔
(
¬
(
X1
↔
X51
)
)
)
→
(
¬
(
(
¬
X51
)
→
(
(
¬
X40
)
↔
X52
)
)
)
)
)
→
(
X51
↔
(
X1
↔
X50
)
)
)
↔
(
X51
↔
(
X40
↔
X50
)
)
)
→
(
X50
↔
X1
)
)
)
→
(
X50
↔
(
X4
↔
(
X1
→
X40
)
)
)
)
↔
(
X52
↔
(
X7
↔
(
(
¬
X1
)
↔
X4
)
)
)
)
↔
(
(
¬
(
X51
↔
(
X46
→
(
X31
↔
X27
)
)
)
)
→
(
(
X51
↔
(
X7
↔
False
)
)
↔
(
(
¬
(
X51
↔
(
(
¬
X31
)
↔
(
(
¬
X7
)
↔
X46
)
)
)
)
↔
(
(
¬
X1
)
↔
X7
)
)
)
)
)
)
→
(
(
(
¬
(
X50
↔
(
X43
→
(
(
¬
X29
)
↔
X52
)
)
)
)
↔
(
¬
(
(
X52
→
X29
→
X40
)
→
(
¬
(
X52
→
(
X43
↔
(
X29
→
X13
→
False
)
)
)
)
→
(
X52
↔
(
X43
↔
(
X40
↔
False
)
)
)
)
)
)
↔
(
(
¬
(
X43
→
X29
→
(
X3
↔
(
X0
→
False
)
)
)
)
↔
(
¬
(
(
¬
X18
)
↔
(
X0
↔
X50
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
(
¬
X20
)
→
(
X14
↔
X49
)
)
→
(
(
X49
↔
(
(
¬
X20
)
↔
(
(
¬
X5
)
↔
X51
)
)
)
↔
(
X51
↔
(
X20
→
X14
→
X49
)
)
)
)
)
↔
(
(
¬
(
(
¬
X50
)
↔
(
(
¬
X46
)
→
X23
→
X51
)
)
)
↔
(
(
¬
(
X38
↔
(
X14
↔
X51
)
)
)
↔
(
(
¬
(
X50
↔
(
X38
↔
X46
)
)
)
↔
(
¬
(
X51
↔
(
(
¬
X46
)
↔
(
X14
→
X50
)
)
)
)
)
)
)
)
↔
(
(
(
(
X49
↔
(
X48
↔
(
X32
↔
(
X12
→
False
)
)
)
)
→
(
(
¬
(
X48
↔
(
X12
↔
X44
)
)
)
↔
(
¬
(
X50
↔
(
X44
↔
(
X12
↔
X49
)
)
)
)
)
→
(
¬
(
X50
→
(
X49
↔
(
X44
↔
False
)
)
)
)
)
↔
(
(
X51
↔
(
(
¬
X49
)
→
(
(
¬
X11
)
↔
X50
)
)
)
↔
(
X51
↔
(
X49
↔
(
(
¬
X48
)
↔
X50
)
)
)
)
)
↔
(
X44
→
(
X37
↔
(
(
¬
X32
)
→
X51
)
)
)
)
)
)
)
↔
(
(
(
(
X51
→
(
¬
X36
)
→
(
(
¬
X14
)
↔
X48
)
)
↔
(
(
(
(
(
¬
X36
)
→
(
X28
↔
(
(
¬
X7
)
→
X40
)
)
)
→
(
X40
↔
(
X31
↔
(
(
¬
X7
)
→
X28
)
)
)
)
↔
(
(
(
¬
X31
)
↔
(
(
¬
X28
)
↔
(
X10
→
X48
)
)
)
↔
(
(
¬
(
(
¬
X40
)
↔
(
X34
↔
(
X28
→
X48
)
)
)
)
↔
(
¬
(
(
¬
(
X40
→
X28
→
X27
→
X31
)
)
→
(
(
¬
X34
)
↔
(
X31
↔
(
(
¬
X27
)
→
X40
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X48
→
(
¬
X36
)
→
(
¬
X29
)
→
X46
)
)
↔
(
¬
(
(
¬
X40
)
↔
(
X36
→
X48
)
)
)
)
→
(
(
¬
X37
)
↔
(
X29
↔
X40
)
)
)
)
↔
(
¬
(
(
X51
→
X40
)
→
(
¬
(
X52
↔
(
(
¬
X48
)
→
X18
→
X51
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X50
↔
(
X9
→
X4
)
)
)
↔
(
(
(
¬
X47
)
↔
(
(
¬
X26
)
↔
(
X1
↔
X50
)
)
)
↔
(
(
¬
(
X52
→
X50
→
(
X4
↔
X1
)
)
)
↔
(
¬
(
X52
→
(
(
¬
X26
)
↔
(
(
¬
X1
)
↔
X50
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X50
→
(
¬
X43
)
→
(
(
¬
X30
)
↔
X47
)
)
)
→
(
¬
(
(
¬
X47
)
↔
(
(
¬
X43
)
→
(
(
¬
X42
)
↔
X50
)
)
)
)
)
)
→
(
¬
(
(
¬
X30
)
↔
X47
)
)
)
↔
(
(
X52
→
X43
→
X35
→
X50
)
↔
(
¬
(
X52
↔
(
X50
→
(
X47
↔
X43
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X46
)
→
(
¬
X40
)
→
(
¬
X11
)
→
X50
)
)
↔
(
(
¬
(
(
¬
X13
)
↔
(
(
¬
X8
)
→
(
X6
↔
X50
)
)
)
)
↔
(
(
¬
(
(
¬
X40
)
→
(
(
¬
X13
)
↔
(
X8
↔
X50
)
)
)
)
↔
(
X46
→
X40
→
(
X13
↔
(
X8
↔
False
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X46
→
(
X43
↔
False
)
)
)
↔
(
X50
↔
(
X47
↔
(
X38
↔
False
)
)
)
)
→
X47
→
(
(
¬
X46
)
↔
X50
)
)
)
→
(
(
¬
(
X33
↔
(
X17
→
X40
)
)
)
↔
(
X38
↔
(
X17
↔
X47
)
)
)
→
(
¬
X40
)
→
(
X38
↔
(
(
¬
X33
)
→
X43
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
(
X39
→
(
¬
X25
)
→
X46
)
↔
(
(
¬
(
X41
→
X2
→
X19
)
)
↔
(
¬
(
X39
→
(
X25
↔
(
(
¬
X19
)
↔
X46
)
)
)
)
)
)
→
(
¬
(
X39
↔
(
(
¬
X19
)
↔
X46
)
)
)
)
↔
(
(
X44
↔
(
X39
↔
(
X23
↔
(
X21
→
False
)
)
)
)
↔
(
¬
(
(
X51
→
(
X46
↔
(
X25
↔
X3
)
)
)
→
(
(
(
¬
(
X3
→
X44
)
)
↔
(
X51
→
(
¬
X5
)
→
(
(
¬
X3
)
↔
X25
)
)
)
↔
(
¬
(
(
¬
X44
)
→
(
¬
X25
)
→
X51
)
)
)
)
)
)
)
→
(
(
X15
↔
X20
)
↔
(
(
¬
(
X21
↔
(
(
¬
X20
)
↔
(
X15
↔
X41
)
)
)
)
↔
(
¬
(
X44
↔
(
(
¬
X20
)
→
X41
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
X45
↔
(
X32
→
(
(
¬
X24
)
↔
X46
)
)
)
)
↔
(
(
¬
X32
)
↔
(
(
¬
X24
)
→
(
¬
X0
)
→
X45
)
)
)
→
(
(
X39
↔
(
(
¬
X24
)
↔
(
(
¬
X16
)
↔
X46
)
)
)
↔
(
X39
→
X24
→
(
(
¬
X16
)
↔
X32
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X46
)
↔
(
X32
→
(
¬
X22
)
→
X48
)
)
)
→
X48
→
(
(
¬
X39
)
↔
(
X32
↔
X46
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
(
¬
X23
)
→
X49
)
)
→
(
X45
↔
(
X12
↔
X32
)
)
)
)
↔
(
X45
↔
(
(
¬
X32
)
↔
X49
)
)
)
→
(
¬
(
X46
→
(
X32
↔
(
X23
↔
X49
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
959c88...
and proposition id is
a77e76...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMYnCA1K5NRGjXLiyJpTSLored7ATHbErRn
)
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
¬
∀X51 :
prop
,
(
¬
∀X52 :
prop
,
∀X53 :
prop
,
(
¬
∀X54 :
prop
,
(
(
(
(
(
¬
(
(
¬
(
X49
↔
(
X48
→
(
X9
↔
X52
)
)
)
)
→
(
¬
(
(
¬
(
X53
↔
(
X37
→
X9
→
X48
)
)
)
→
(
¬
(
(
X53
↔
(
X37
↔
(
X26
→
X39
)
)
)
→
(
¬
(
(
¬
(
X52
→
(
(
¬
X39
)
↔
(
X37
↔
X53
)
)
)
)
→
(
¬
(
X53
↔
(
X39
↔
(
(
¬
X9
)
↔
X52
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X38
)
→
(
(
¬
X11
)
↔
X51
)
)
)
↔
(
(
¬
(
(
X48
→
X2
→
X11
)
→
(
X51
↔
(
X48
→
(
X18
↔
X2
)
)
)
)
)
↔
(
(
¬
X18
)
→
(
(
¬
X15
)
↔
(
X2
→
X48
)
)
)
)
)
→
(
¬
(
X51
↔
(
(
¬
X38
)
→
(
(
¬
X28
)
↔
X48
)
)
)
)
)
↔
(
(
X39
→
(
(
¬
X38
)
↔
(
(
¬
X28
)
↔
X50
)
)
)
↔
(
(
(
(
(
X50
↔
(
(
¬
X38
)
→
X5
→
X49
)
)
↔
(
(
¬
X49
)
→
X38
→
(
¬
X28
)
→
X50
)
)
→
(
¬
(
X50
↔
(
X38
→
(
X23
↔
X49
)
)
)
)
)
↔
(
X28
↔
(
X5
→
X23
)
)
)
↔
(
(
¬
(
X34
↔
(
(
¬
X18
)
→
X23
)
)
)
→
(
(
¬
X28
)
↔
(
(
¬
X18
)
↔
(
(
¬
X5
)
→
X34
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
X42
→
X51
)
→
(
¬
X50
)
→
(
X43
↔
X52
)
)
)
↔
(
(
¬
(
X52
→
(
(
¬
X50
)
↔
X51
)
)
)
↔
(
¬
(
(
¬
(
X51
↔
(
X50
→
X47
→
False
)
)
)
→
(
¬
(
X51
↔
(
(
¬
X47
)
→
(
¬
X8
)
→
X52
)
)
)
)
)
)
)
→
(
(
X51
→
(
X43
↔
(
(
¬
X7
)
↔
X47
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
X51
↔
(
(
¬
X45
)
↔
(
X41
↔
X47
)
)
)
↔
(
¬
(
X51
↔
(
X47
→
X45
→
X30
)
)
)
)
→
(
¬
(
X45
→
X30
→
X17
→
X51
)
)
)
)
→
(
¬
(
X47
↔
(
X17
→
(
¬
X6
)
→
X30
)
)
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X45
)
→
(
X43
↔
X51
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
¬
(
(
(
X40
↔
(
X32
→
X46
)
)
→
(
¬
(
(
¬
X46
)
↔
(
X32
→
(
(
¬
X20
)
↔
X48
)
)
)
)
)
→
(
¬
(
X40
↔
False
)
)
)
)
→
X46
→
(
X40
↔
X48
)
)
)
↔
(
¬
(
(
¬
(
X49
→
(
X46
↔
X27
)
)
)
→
(
X40
↔
(
(
¬
X33
)
→
(
X32
↔
X46
)
)
)
)
)
)
→
(
(
¬
(
(
X46
→
(
(
¬
X31
)
↔
(
(
¬
X19
)
→
X40
)
)
)
→
(
X40
↔
(
(
¬
X31
)
↔
(
X12
→
X46
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X54
→
(
X48
↔
(
X35
→
(
X31
↔
False
)
)
)
)
)
→
(
¬
(
(
X46
↔
(
(
¬
X31
)
→
X35
)
)
→
(
¬
(
X48
→
(
¬
X31
)
→
(
X19
↔
X35
)
)
)
)
)
)
)
→
(
¬
(
X46
↔
X31
)
)
)
)
)
↔
(
(
¬
(
X54
→
(
X3
↔
X14
)
)
)
→
(
¬
(
(
X22
→
X16
)
→
(
X54
↔
(
(
¬
X48
)
↔
(
X16
→
X49
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
¬
(
(
¬
X36
)
↔
(
X1
↔
X54
)
)
)
↔
(
¬
(
(
(
¬
(
X48
↔
(
(
¬
X36
)
↔
X54
)
)
)
↔
(
¬
(
(
¬
X36
)
↔
(
X25
↔
(
X1
↔
X54
)
)
)
)
)
→
(
¬
X36
)
→
(
¬
X25
)
→
X48
)
)
)
↔
(
X53
→
(
X48
↔
(
X25
→
X36
)
)
)
)
→
(
(
¬
(
X53
→
(
¬
X48
)
→
X4
→
X54
)
)
↔
(
(
X54
↔
(
(
¬
X48
)
↔
(
X24
↔
X53
)
)
)
↔
(
X54
↔
(
(
¬
X48
)
↔
(
X10
↔
X53
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
X46
)
↔
(
(
¬
X44
)
→
X29
→
X54
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
(
¬
X48
)
↔
(
X13
→
X53
)
)
)
→
(
(
X53
↔
(
X44
→
X13
)
)
↔
(
X54
→
(
X44
↔
(
X13
↔
X48
)
)
)
)
)
)
→
(
¬
(
(
¬
X29
)
→
(
¬
X13
)
→
X48
)
)
)
)
↔
(
(
¬
(
X54
→
X29
→
X0
)
)
↔
(
X44
↔
(
X29
→
(
(
¬
X21
)
↔
X54
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
X52
→
(
(
¬
X48
)
↔
(
(
¬
X46
)
→
X53
)
)
)
↔
(
X53
→
(
(
¬
X48
)
↔
(
(
¬
X46
)
↔
X51
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
X46
)
↔
(
X39
→
(
¬
X31
)
→
X48
)
)
→
(
¬
X46
)
→
X31
→
X52
)
)
↔
(
¬
(
X48
→
(
¬
X39
)
→
(
(
¬
X18
)
↔
X46
)
)
)
)
↔
(
¬
(
(
(
(
¬
X46
)
→
(
X39
↔
(
X24
↔
X48
)
)
)
↔
(
(
(
(
¬
X46
)
→
X39
→
(
X38
↔
X48
)
)
↔
(
(
¬
X46
)
→
(
(
¬
X38
)
↔
(
(
¬
X7
)
↔
X48
)
)
)
)
→
(
(
¬
X7
)
↔
X46
)
)
)
→
X34
→
(
(
¬
X24
)
↔
(
(
¬
X7
)
→
X39
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
(
(
X37
↔
(
X9
↔
(
X4
↔
False
)
)
)
↔
(
X41
→
X37
→
(
(
¬
X2
)
↔
X9
)
)
)
→
(
¬
(
X53
→
(
X37
↔
(
X2
↔
X9
)
)
)
)
)
)
↔
(
(
(
¬
X37
)
→
X2
→
X50
)
↔
(
¬
(
(
¬
(
X37
→
(
X36
↔
(
(
¬
X2
)
→
X9
)
)
)
)
→
(
¬
(
X50
→
(
X37
↔
(
X36
↔
(
X9
↔
False
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
X52
↔
(
X50
↔
(
X36
↔
False
)
)
)
→
(
¬
(
(
¬
X51
)
↔
(
(
¬
X50
)
↔
(
(
¬
X37
)
↔
X52
)
)
)
)
)
)
↔
(
X45
→
(
¬
X36
)
→
X52
)
)
)
→
(
¬
(
(
X50
→
(
(
¬
X45
)
↔
(
X28
↔
X51
)
)
)
→
(
¬
(
X54
→
(
X28
↔
X25
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
(
¬
(
(
¬
(
(
X54
↔
(
X51
↔
(
X33
↔
False
)
)
)
→
(
¬
X21
)
→
(
(
¬
X12
)
↔
X33
)
)
)
→
(
¬
X21
)
)
)
↔
(
X54
↔
(
X51
→
X21
)
)
)
→
(
X54
↔
(
X33
→
X21
)
)
)
)
↔
(
¬
(
X12
↔
False
)
)
)
→
(
¬
(
(
(
¬
(
X51
↔
(
(
¬
X43
)
→
(
¬
X19
)
→
X54
)
)
)
→
(
X51
→
(
(
¬
X19
)
↔
X33
)
)
→
(
¬
X33
)
→
(
¬
X30
)
→
X43
)
→
(
(
(
¬
X20
)
→
(
X0
↔
X43
)
)
↔
(
¬
(
(
¬
X43
)
→
(
X30
↔
(
(
¬
X20
)
↔
X51
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
X51
→
X11
→
(
¬
X8
)
→
X32
)
→
X49
→
(
X11
↔
(
X8
↔
X32
)
)
)
)
→
(
X49
↔
(
X32
↔
(
X11
↔
X5
)
)
)
)
→
(
¬
X49
)
→
(
(
¬
X33
)
↔
(
X8
→
X51
)
)
)
→
(
X30
↔
(
(
¬
X10
)
→
X33
)
)
→
(
¬
(
X51
→
(
X33
↔
(
X32
↔
(
X30
→
False
)
)
)
)
)
)
)
→
(
(
(
(
(
(
¬
(
X51
→
(
(
¬
X15
)
↔
X54
)
)
)
↔
(
(
X16
↔
(
X15
↔
X51
)
)
↔
(
(
¬
X35
)
↔
(
X16
↔
X54
)
)
)
)
→
(
¬
(
X35
↔
(
X6
→
X15
)
)
)
)
↔
(
¬
(
X51
→
(
(
¬
X35
)
↔
(
(
¬
X16
)
↔
X49
)
)
)
)
)
→
(
¬
(
(
(
(
¬
X27
)
↔
(
X22
↔
(
X17
→
X49
)
)
)
→
(
¬
(
X54
↔
(
(
¬
X44
)
→
X22
→
X51
)
)
)
→
(
¬
(
X51
→
(
¬
X27
)
→
X54
)
)
)
→
(
¬
X44
)
→
X27
→
(
(
¬
X13
)
↔
X54
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X49
↔
(
X29
↔
(
(
¬
X26
)
→
X42
)
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X29
)
↔
(
(
¬
X23
)
→
X42
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X52
)
→
(
X49
↔
(
X29
→
X54
)
)
)
)
↔
(
¬
(
(
¬
X52
)
→
(
X49
↔
(
(
¬
X14
)
→
X54
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X54
↔
(
(
¬
X49
)
↔
X52
)
)
)
→
(
¬
(
X47
↔
(
X40
↔
X54
)
)
)
)
)
↔
(
¬
(
(
¬
X52
)
↔
(
X49
→
(
(
¬
X26
)
↔
X54
)
)
)
)
)
)
)
)
↔
(
(
¬
(
X47
↔
(
(
¬
X1
)
↔
X49
)
)
)
→
(
(
(
(
¬
(
(
¬
X47
)
→
(
(
¬
X23
)
↔
(
X3
→
X51
)
)
)
)
↔
(
¬
(
X51
↔
(
(
¬
X47
)
↔
X49
)
)
)
)
↔
(
X49
→
(
(
¬
X3
)
↔
X23
)
)
)
↔
(
¬
(
(
¬
X23
)
↔
(
(
¬
X3
)
→
(
(
¬
X1
)
↔
X49
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
e2d5fa...
and proposition id is
f61e27...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMdxyrnoYAMQJdwjqjcqfmaWaqSV4kJSr8g
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
∀X50 :
prop
,
(
¬
∀X51 :
prop
,
∀X52 :
prop
,
(
(
(
(
(
(
(
(
(
¬
X49
)
→
(
(
¬
X35
)
↔
(
(
¬
X16
)
↔
X52
)
)
)
↔
(
X49
→
(
¬
X16
)
→
X25
)
)
→
(
¬
(
X52
→
X34
→
(
X24
↔
X49
)
)
)
)
→
(
(
¬
X49
)
↔
(
(
¬
(
X49
↔
(
(
¬
X36
)
→
(
¬
X0
)
→
X52
)
)
)
↔
(
¬
(
(
¬
X49
)
→
(
X36
↔
(
(
¬
X34
)
↔
X52
)
)
)
)
)
)
)
→
(
(
(
(
X52
→
(
¬
X18
)
→
X28
)
↔
(
¬
(
(
X52
→
(
X11
↔
X18
)
)
→
(
(
(
¬
X11
)
↔
X48
)
↔
(
¬
(
X52
↔
(
X48
↔
(
(
¬
X18
)
→
X28
)
)
)
)
)
)
)
)
↔
(
X48
→
(
X34
↔
(
(
¬
X18
)
↔
X28
)
)
)
)
↔
(
(
(
¬
X48
)
↔
(
(
¬
X36
)
↔
(
(
¬
X23
)
→
X52
)
)
)
↔
(
X52
→
(
¬
X36
)
→
(
X34
↔
X48
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
X41
↔
(
X22
↔
(
X2
↔
X51
)
)
)
)
→
(
¬
(
(
¬
X48
)
→
(
X41
↔
(
(
¬
X22
)
→
X51
)
)
)
)
)
)
↔
(
(
¬
(
X49
→
(
X30
↔
X51
)
)
)
→
(
¬
(
(
¬
(
(
¬
X49
)
→
(
¬
X41
)
→
(
(
¬
X30
)
↔
X51
)
)
)
→
(
¬
(
(
¬
(
(
X49
→
X41
→
(
X3
↔
X2
)
)
→
(
¬
(
X51
↔
(
(
¬
X41
)
↔
(
(
¬
X2
)
→
X49
)
)
)
)
)
)
→
(
X51
↔
(
X3
→
X2
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
X52
↔
(
(
¬
X41
)
↔
(
(
¬
X12
)
→
X46
)
)
)
↔
(
(
¬
X49
)
→
(
X12
↔
X52
)
)
)
→
(
¬
(
X46
→
(
X12
↔
X41
)
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X41
)
→
X49
)
)
)
)
→
(
X49
↔
(
(
¬
X46
)
→
(
¬
X7
)
→
X52
)
)
)
↔
(
(
(
¬
(
X49
↔
(
X43
↔
(
X13
↔
X42
)
)
)
)
↔
(
X46
→
(
X42
↔
X52
)
)
)
→
(
X52
↔
(
(
¬
X46
)
↔
(
(
¬
X30
)
→
X49
)
)
)
)
)
)
)
→
(
(
(
X47
→
X44
→
(
(
¬
X20
)
↔
X40
)
)
↔
(
(
(
X40
↔
(
X10
↔
X19
)
)
↔
(
(
(
X32
→
X29
→
X19
→
X40
)
→
X40
→
(
X29
↔
(
(
¬
X5
)
→
X19
)
)
)
↔
(
¬
(
X40
↔
(
X29
→
X32
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
X44
)
→
(
(
¬
X32
)
↔
(
X5
↔
X47
)
)
)
→
(
¬
(
(
¬
X5
)
↔
X17
)
)
)
)
→
X47
→
(
¬
X32
)
→
X44
)
)
)
↔
(
(
(
(
X45
↔
(
X40
→
(
X29
↔
False
)
)
)
↔
(
(
¬
(
X45
→
(
¬
X15
)
→
X44
)
)
→
X40
→
(
(
¬
X29
)
↔
(
X15
→
X44
)
)
)
)
→
(
¬
(
(
¬
X40
)
→
X29
→
X45
)
)
)
↔
(
¬
(
X44
↔
(
(
¬
X15
)
→
X45
)
)
)
)
)
→
(
(
(
(
¬
(
X33
↔
(
(
¬
X27
)
↔
X37
)
)
)
→
(
¬
(
(
X33
↔
(
X27
→
X14
→
X48
)
)
→
(
(
X48
→
(
(
¬
X14
)
↔
X31
)
)
↔
(
(
(
¬
(
(
¬
X31
)
↔
(
X27
→
X1
→
X37
)
)
)
↔
(
¬
(
(
X48
↔
(
X27
→
X14
→
X31
)
)
→
(
(
¬
(
X37
→
(
¬
X27
)
→
(
¬
X26
)
→
X31
)
)
↔
(
X37
↔
(
X31
↔
(
(
¬
X14
)
↔
X26
)
)
)
)
)
)
)
→
(
¬
(
(
¬
X27
)
↔
(
(
¬
X14
)
↔
(
(
¬
X4
)
→
X31
)
)
)
)
)
)
)
)
)
↔
(
¬
(
X37
→
(
¬
X33
)
→
(
X6
↔
X48
)
)
)
)
↔
(
¬
(
(
(
¬
(
X48
↔
(
X44
→
(
X31
↔
X40
)
)
)
)
↔
(
(
(
X51
↔
(
X44
↔
X38
)
)
→
(
¬
(
X51
↔
(
X50
↔
(
(
¬
X21
)
↔
X44
)
)
)
)
)
↔
(
(
(
¬
(
X50
→
(
¬
X44
)
→
(
X9
↔
X46
)
)
)
↔
(
¬
(
X50
↔
(
X46
↔
(
(
¬
X40
)
→
X44
)
)
)
)
)
↔
(
(
¬
(
X51
→
X50
→
(
X33
↔
X44
)
)
)
↔
(
¬
(
(
¬
(
X50
↔
X31
)
)
→
(
X51
↔
(
X46
↔
(
X31
→
X33
)
)
)
)
)
)
)
)
)
→
(
X51
→
(
X48
↔
(
X8
→
X40
)
)
)
→
(
(
X51
↔
(
X48
→
X40
→
X39
)
)
→
(
¬
(
X51
↔
(
(
¬
X40
)
↔
(
X39
→
X47
)
)
)
)
)
→
(
¬
(
X48
↔
(
X47
↔
X51
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
(
X49
→
(
(
¬
X35
)
↔
X52
)
)
→
(
¬
(
(
(
X49
↔
(
(
¬
X39
)
↔
(
(
¬
X35
)
→
X52
)
)
)
↔
(
(
¬
(
X52
→
(
X49
↔
X21
)
)
)
↔
(
(
(
X52
↔
(
X49
→
X20
)
)
→
(
¬
(
X52
→
X49
→
(
(
¬
X21
)
↔
X39
)
)
)
)
↔
(
X20
→
X52
)
)
)
)
→
(
X35
↔
(
X20
→
(
(
¬
X15
)
↔
X52
)
)
)
)
)
)
)
↔
(
¬
(
(
X49
↔
(
X39
↔
(
X35
→
X32
)
)
)
→
(
¬
(
X39
→
X28
→
X16
→
X35
)
)
)
)
)
→
(
¬
(
X39
↔
(
(
¬
X35
)
→
X52
)
)
)
)
↔
(
(
(
X18
↔
(
X9
→
X51
)
)
↔
(
(
¬
(
X42
↔
(
(
¬
X31
)
→
(
¬
X18
)
→
X35
)
)
)
↔
(
(
X35
→
(
X18
↔
(
(
¬
X9
)
→
X42
)
)
)
↔
(
¬
(
(
¬
X42
)
→
(
¬
X35
)
→
(
(
¬
X18
)
↔
X51
)
)
)
)
)
)
→
(
(
X51
↔
(
(
¬
X9
)
↔
(
X3
↔
X14
)
)
)
↔
(
(
(
¬
X18
)
↔
(
(
¬
X14
)
→
(
X9
↔
X49
)
)
)
↔
(
(
¬
X49
)
→
(
X14
↔
X52
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
X49
↔
(
(
¬
X37
)
↔
(
(
¬
X26
)
↔
X45
)
)
)
)
→
(
¬
(
(
X45
↔
(
X17
→
False
)
)
→
(
¬
(
X49
→
X26
→
X25
)
)
)
)
)
)
→
(
¬
(
X49
↔
(
X45
→
X37
→
X27
→
False
)
)
)
)
)
→
(
¬
(
(
(
(
X46
→
(
X45
↔
X40
)
)
↔
(
¬
(
(
¬
(
X52
↔
(
X40
→
(
(
¬
X23
)
↔
X46
)
)
)
)
→
(
¬
(
X46
↔
(
X45
↔
X52
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X46
)
→
(
¬
X45
)
→
X49
)
)
→
(
¬
(
(
¬
X45
)
→
(
X40
↔
(
(
¬
X7
)
→
X46
)
)
)
)
)
)
)
→
(
(
X52
→
X40
→
X37
→
X46
)
↔
(
¬
(
X51
↔
(
X46
→
(
X37
↔
X40
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
X52
→
(
X11
↔
X38
)
)
↔
(
(
(
¬
(
X2
→
X50
)
)
↔
(
¬
(
(
¬
(
X11
→
(
X2
↔
X52
)
)
)
→
(
(
X50
↔
X38
)
↔
(
(
(
¬
X2
)
↔
X11
)
↔
(
¬
(
X50
→
(
¬
X38
)
→
(
¬
X2
)
→
X52
)
)
)
)
)
)
)
→
(
X52
↔
(
X11
→
(
X2
↔
False
)
)
)
)
)
↔
(
X47
→
(
X38
↔
(
X2
↔
False
)
)
)
)
↔
(
(
¬
(
(
X47
→
(
(
¬
X34
)
↔
(
(
¬
X8
)
→
X52
)
)
)
→
(
¬
(
X50
→
(
X34
↔
(
X12
↔
X52
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X34
→
X29
→
(
X12
↔
X4
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X12
)
→
(
X4
↔
X51
)
)
)
)
)
)
)
)
↔
(
(
(
X48
↔
X10
)
↔
(
(
¬
(
(
¬
X47
)
↔
(
X10
↔
(
(
¬
X0
)
→
X48
)
)
)
)
→
(
X50
↔
(
X10
↔
X48
)
)
)
)
↔
(
(
(
¬
(
X50
↔
(
X47
↔
(
(
¬
X41
)
→
X52
)
)
)
)
→
(
(
¬
(
(
¬
X41
)
→
(
X22
↔
X52
)
)
)
↔
(
(
¬
(
(
¬
(
X52
→
(
X48
↔
(
X47
→
X22
)
)
)
)
→
(
¬
X48
)
→
(
(
¬
X47
)
↔
(
X41
→
X52
)
)
)
)
→
X47
→
X41
)
)
→
(
X52
↔
(
X47
↔
(
X24
↔
X41
)
)
)
)
↔
(
(
¬
X41
)
→
(
X24
↔
X48
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
(
¬
(
X44
→
(
(
¬
X43
)
↔
(
(
¬
X30
)
↔
X51
)
)
)
)
→
(
(
X52
↔
(
X44
→
(
(
¬
X6
)
↔
X51
)
)
)
↔
(
X44
↔
(
X43
→
X6
→
X52
)
)
)
)
→
(
¬
(
X6
↔
X30
)
)
)
)
→
(
¬
(
(
¬
(
X51
↔
(
X5
→
X43
)
)
)
→
(
X51
→
(
X50
↔
(
X43
→
X13
→
False
)
)
)
→
(
¬
(
(
(
¬
X50
)
↔
(
(
¬
X13
)
→
(
(
¬
X5
)
↔
X51
)
)
)
→
X52
→
X51
→
(
X13
↔
(
X5
↔
False
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
X45
)
→
X36
→
(
X1
↔
X51
)
)
)
→
(
X45
↔
(
X36
→
X33
)
)
)
→
(
(
X50
↔
(
X45
→
X19
→
X43
)
)
↔
(
¬
(
(
¬
(
X50
↔
(
X36
↔
(
X19
↔
False
)
)
)
)
→
(
¬
(
(
¬
X45
)
↔
(
X33
↔
(
X19
→
X50
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
f00aec...
and proposition id is
51c016...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMSJXvL4RW3oWD2nJLatBBKhod3DKch7rxS
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
∀X50 :
prop
,
(
¬
∀X51 :
prop
,
(
(
¬
(
(
¬
(
(
(
(
(
(
(
¬
X3
)
→
X15
)
↔
(
(
X48
→
(
X15
↔
X3
)
)
↔
(
¬
(
X49
↔
(
(
¬
X15
)
→
(
(
¬
X3
)
↔
X50
)
)
)
)
)
)
↔
(
¬
(
X47
↔
(
X30
→
(
(
¬
X15
)
↔
X36
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
X46
)
↔
(
X40
→
(
X31
↔
X50
)
)
)
)
→
(
(
¬
(
(
¬
X46
)
↔
(
(
¬
X31
)
→
X48
)
)
)
↔
(
¬
(
X50
↔
X33
)
)
)
)
)
↔
(
¬
(
X49
→
(
X31
↔
X27
)
)
)
)
)
↔
(
(
X30
→
(
X17
↔
(
X11
↔
False
)
)
)
→
X46
→
(
X33
↔
False
)
)
)
→
(
(
(
(
X46
→
(
X39
↔
(
(
¬
X1
)
↔
X25
)
)
)
↔
(
X43
→
(
¬
X1
)
→
X46
)
)
↔
(
(
(
¬
(
(
¬
X44
)
→
X39
→
(
¬
X25
)
→
X47
)
)
→
(
(
X47
→
(
X39
↔
(
X22
↔
X14
)
)
)
↔
(
¬
(
(
¬
X44
)
↔
(
X39
↔
(
X14
→
X47
)
)
)
)
)
)
↔
(
¬
(
X46
→
(
X44
↔
(
X39
↔
(
X22
→
False
)
)
)
)
)
)
)
→
X46
→
(
X44
↔
(
X32
↔
X47
)
)
)
→
(
¬
(
(
(
¬
(
X51
↔
(
(
¬
X39
)
→
(
¬
X23
)
→
X47
)
)
)
↔
(
¬
(
(
(
X51
→
(
¬
X39
)
→
X35
→
X50
)
→
(
(
¬
(
(
¬
X47
)
→
(
X6
↔
X51
)
)
)
↔
(
¬
(
X50
→
(
X35
↔
(
X6
↔
X51
)
)
)
)
)
)
→
(
¬
(
X47
↔
(
X6
↔
X16
)
)
)
)
)
)
→
(
(
X50
↔
(
X24
↔
(
X9
→
X47
)
)
)
↔
(
(
¬
(
(
¬
X50
)
↔
(
(
¬
X43
)
→
X24
→
X51
)
)
)
↔
(
(
¬
(
X50
→
(
¬
X4
)
→
X51
)
)
→
(
X51
→
(
X50
↔
(
X47
→
(
X4
↔
False
)
)
)
)
→
(
¬
(
X47
→
X24
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
(
X48
↔
(
X37
→
(
X28
↔
(
X26
→
False
)
)
)
)
)
→
(
(
(
X48
↔
X38
)
↔
(
(
X45
↔
(
X38
↔
False
)
)
↔
(
¬
(
(
¬
(
X48
↔
(
X45
→
(
¬
X20
)
→
X38
)
)
)
→
X38
→
X37
→
X20
→
False
)
)
)
)
↔
(
¬
(
(
¬
X38
)
↔
(
(
¬
X37
)
↔
(
(
¬
X28
)
→
X48
)
)
)
)
)
)
)
→
(
(
¬
X28
)
↔
(
(
¬
X26
)
↔
(
(
¬
X12
)
↔
X42
)
)
)
→
(
X45
↔
(
X42
↔
(
X26
→
X7
→
False
)
)
)
)
→
(
¬
(
(
(
(
(
¬
(
(
¬
X34
)
↔
(
X13
→
X49
)
)
)
↔
(
(
X47
↔
(
X34
→
X19
→
X13
)
)
→
X47
→
(
(
¬
X19
)
↔
(
X13
↔
X45
)
)
)
)
→
(
¬
(
X49
↔
(
X0
→
False
)
)
)
)
↔
(
(
X49
→
X8
→
X5
→
X47
)
→
(
¬
X45
)
→
(
X5
↔
X49
)
)
)
→
(
(
(
(
(
(
(
¬
(
(
¬
X48
)
→
(
¬
X45
)
→
(
(
¬
X41
)
↔
X49
)
)
)
↔
(
¬
(
X29
↔
X48
)
)
)
↔
(
(
¬
X41
)
↔
X48
)
)
↔
(
(
¬
X41
)
↔
(
(
¬
X29
)
→
X45
)
)
)
↔
(
X41
→
X29
→
(
(
¬
X21
)
↔
X48
)
)
)
→
(
X49
↔
(
X10
↔
X41
)
)
→
(
¬
(
(
¬
(
X41
↔
(
X10
→
X49
)
)
)
→
(
¬
(
(
¬
X48
)
↔
(
X41
→
(
(
¬
X18
)
↔
X49
)
)
)
)
)
)
)
↔
(
(
¬
(
X45
↔
(
X42
↔
(
X2
→
X48
)
)
)
)
→
(
(
X42
↔
(
(
¬
X29
)
→
X47
)
)
↔
(
(
¬
(
X48
→
(
X42
↔
(
X2
→
False
)
)
)
)
↔
(
(
¬
X47
)
↔
(
(
¬
X29
)
↔
X48
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
(
(
(
X45
↔
(
X42
↔
(
(
¬
X39
)
↔
X50
)
)
)
→
(
(
(
¬
(
X45
→
(
¬
X39
)
→
(
X19
↔
X50
)
)
)
→
(
¬
(
(
¬
X39
)
→
(
X19
↔
(
X6
↔
X45
)
)
)
)
)
↔
(
¬
(
(
¬
X34
)
↔
(
(
¬
X6
)
↔
X39
)
)
)
)
)
↔
(
X45
↔
(
(
¬
X42
)
→
(
(
¬
X11
)
↔
X50
)
)
)
)
↔
(
(
¬
(
X39
↔
(
X29
→
X42
)
)
)
↔
(
(
(
(
¬
(
(
¬
X29
)
↔
(
(
¬
X8
)
↔
X42
)
)
)
→
(
(
¬
(
X50
→
X39
→
False
)
)
↔
(
¬
(
X39
↔
(
X29
→
X8
→
X42
)
)
)
)
)
↔
(
¬
(
X42
→
(
X39
↔
X50
)
)
)
)
↔
X29
)
)
)
→
(
(
(
(
¬
(
(
¬
X50
)
↔
(
(
¬
X49
)
↔
X51
)
)
)
→
(
(
¬
X49
)
→
(
X45
↔
(
(
¬
X25
)
→
X50
)
)
)
→
(
X50
↔
(
X45
↔
X51
)
)
)
↔
(
X51
→
(
X49
↔
(
X9
→
False
)
)
)
)
↔
(
(
¬
(
(
¬
X42
)
→
(
(
¬
X38
)
↔
X49
)
)
)
→
(
¬
(
(
¬
X45
)
→
X50
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
(
X48
↔
(
X28
→
(
(
¬
X22
)
↔
X40
)
)
)
)
↔
(
¬
(
(
¬
X48
)
→
X40
→
X50
)
)
)
→
(
(
¬
(
X48
↔
(
X40
→
(
X1
↔
X50
)
)
)
)
↔
(
X50
↔
(
X40
↔
(
X1
↔
X49
)
)
)
)
)
→
(
(
¬
(
X50
→
(
X14
↔
False
)
)
)
↔
(
¬
(
(
¬
(
(
X17
→
(
X14
↔
(
X4
↔
X28
)
)
)
→
(
¬
(
X28
→
(
X22
↔
(
(
¬
X14
)
→
X17
)
)
)
)
)
)
→
(
X22
↔
(
(
¬
X17
)
→
X28
)
)
)
)
)
)
→
(
(
¬
X28
)
→
X45
)
→
(
¬
(
(
X28
↔
(
X23
→
X50
)
)
→
(
¬
(
(
¬
(
X23
→
X22
→
X50
)
)
→
(
(
¬
(
X50
↔
(
X28
↔
False
)
)
)
↔
(
¬
(
X28
↔
(
X22
↔
(
X21
↔
X23
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X45
↔
X48
)
)
↔
(
¬
(
(
¬
X48
)
↔
(
X41
→
X31
→
X50
)
)
)
)
→
(
¬
(
(
(
X50
→
X48
→
X45
)
↔
(
¬
(
(
¬
(
(
¬
X48
)
→
(
X44
↔
(
(
¬
X36
)
↔
X50
)
)
)
)
→
(
¬
(
(
¬
X45
)
↔
(
(
¬
X44
)
→
X50
)
)
)
)
)
)
→
(
¬
(
X45
↔
(
(
¬
X7
)
→
X48
)
)
)
)
)
)
↔
(
X48
→
(
¬
X43
)
→
(
X41
↔
X50
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
X50
→
X48
→
X47
→
X33
→
False
)
↔
(
(
X47
→
(
¬
X18
)
→
X33
)
→
(
¬
(
(
¬
(
X50
↔
X26
)
)
→
(
X32
↔
(
(
¬
X18
)
↔
X33
)
)
)
)
)
)
↔
(
(
¬
(
X50
→
X37
→
X27
→
False
)
)
→
(
(
(
(
(
¬
(
(
¬
X48
)
→
(
X27
↔
(
X12
↔
X51
)
)
)
)
↔
(
¬
(
X48
→
(
X37
↔
(
X27
→
X12
→
False
)
)
)
)
)
↔
(
X51
↔
False
)
)
↔
(
X48
→
(
X27
↔
(
X12
↔
False
)
)
)
)
↔
(
X48
→
(
X37
↔
(
(
¬
X32
)
↔
X51
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
X46
)
↔
(
X35
↔
X48
)
)
)
↔
(
¬
(
X2
↔
X37
)
)
)
↔
(
(
¬
(
X37
↔
(
X32
↔
(
(
¬
X15
)
→
X46
)
)
)
)
↔
(
(
(
¬
X35
)
→
(
(
¬
X15
)
↔
(
(
¬
X0
)
→
X37
)
)
)
→
X37
→
(
X35
↔
(
(
¬
X15
)
→
X32
)
)
)
)
)
↔
(
X37
→
X35
→
(
¬
X13
)
→
X48
)
)
)
)
→
(
(
(
¬
(
X37
→
X10
→
(
¬
X3
)
→
X20
)
)
→
(
¬
(
(
¬
X16
)
↔
(
X10
→
X37
)
)
)
)
↔
(
¬
(
(
(
X50
→
(
X37
↔
(
X10
→
X5
)
)
)
↔
(
(
¬
(
(
¬
X37
)
↔
(
(
¬
X30
)
→
(
(
¬
X5
)
↔
X46
)
)
)
)
↔
(
X46
↔
(
X37
↔
(
(
¬
X10
)
↔
X30
)
)
)
)
)
→
(
¬
(
X50
↔
(
(
¬
X16
)
→
(
(
¬
X10
)
↔
X24
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
80d8d1...
and proposition id is
2376d8...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMJ85Jf5gPaSyRScYqinHveoSz2U9RsEHHx
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
¬
(
(
(
¬
(
(
(
X46
→
X23
→
(
¬
X7
)
→
X43
)
↔
(
(
¬
(
(
¬
X46
)
↔
(
X42
→
X7
→
X47
)
)
)
→
X47
→
(
X42
↔
(
X38
→
X7
)
)
)
)
→
(
¬
(
(
¬
(
(
X38
→
(
X30
↔
(
X21
↔
X9
)
)
)
→
(
¬
(
X8
↔
X30
)
)
)
)
→
(
X42
→
(
X38
↔
(
(
¬
X21
)
→
X23
)
)
)
→
(
X23
→
(
¬
X21
)
→
(
¬
X9
)
→
X48
)
→
(
¬
X42
)
→
(
¬
X23
)
→
(
(
¬
X21
)
↔
X48
)
)
)
)
)
→
(
(
(
¬
(
X48
→
X32
→
X46
)
)
→
(
¬
(
(
X47
↔
(
X32
↔
X46
)
)
→
(
(
¬
(
X47
↔
(
X46
→
X45
→
(
X14
↔
False
)
)
)
)
↔
(
¬
(
X47
↔
(
X45
→
(
X32
↔
X46
)
)
)
)
)
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X32
)
↔
(
(
¬
X14
)
→
X35
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X33
)
↔
(
X1
→
X47
)
)
)
→
(
X47
↔
(
X20
↔
(
(
¬
X1
)
↔
X29
)
)
)
)
↔
(
¬
(
(
(
(
¬
X16
)
→
(
X15
↔
X47
)
)
↔
(
X29
↔
(
X15
↔
(
X2
↔
X47
)
)
)
)
→
(
(
(
¬
(
(
¬
(
X45
→
X33
→
X29
→
(
X4
↔
False
)
)
)
→
X35
→
(
¬
X33
)
→
(
X4
↔
X45
)
)
)
↔
(
X47
→
(
X35
↔
(
X29
→
False
)
)
)
)
↔
(
¬
(
X35
→
(
¬
X33
)
→
(
X15
↔
X47
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
(
(
X49
→
(
X46
↔
(
X31
↔
(
X11
→
False
)
)
)
)
→
(
(
¬
(
(
¬
(
X46
→
X31
→
X36
)
)
→
X46
→
(
X31
↔
(
(
¬
X22
)
↔
X49
)
)
)
)
↔
(
¬
(
X22
↔
False
)
)
)
)
)
↔
(
¬
(
(
(
¬
X24
)
↔
(
X22
→
(
X3
↔
X31
)
)
)
→
(
¬
(
X24
→
(
¬
X22
)
→
(
(
¬
X11
)
↔
X31
)
)
)
)
)
)
→
(
(
(
¬
(
(
(
(
¬
(
X48
→
(
(
¬
X46
)
↔
(
(
¬
X40
)
↔
X49
)
)
)
)
↔
(
¬
(
(
¬
X46
)
→
(
X40
↔
(
X36
↔
X48
)
)
)
)
)
↔
(
(
¬
X46
)
↔
(
X40
→
(
(
¬
X25
)
↔
X49
)
)
)
)
→
(
X25
↔
X36
)
)
)
↔
(
¬
(
X49
↔
(
X48
↔
(
X12
→
X46
)
)
)
)
)
↔
(
(
(
(
X46
→
(
X0
↔
X49
)
)
↔
(
¬
(
X46
→
X37
→
(
X13
↔
False
)
)
)
)
→
(
¬
(
X49
→
(
¬
X13
)
→
X37
)
)
)
↔
(
¬
(
(
¬
X48
)
→
X49
)
)
)
)
)
)
→
(
(
(
(
X34
↔
(
(
¬
X19
)
↔
(
X17
↔
X39
)
)
)
↔
(
X26
→
X19
)
)
→
(
(
¬
(
X41
→
(
(
¬
(
(
¬
X47
)
→
(
X27
↔
X49
)
)
)
↔
(
¬
(
X49
↔
(
X47
↔
(
X17
↔
X41
)
)
)
)
)
)
)
↔
(
(
(
X41
→
(
X40
↔
(
X26
→
X47
)
)
)
↔
(
¬
(
(
¬
X40
)
↔
(
(
¬
X39
)
→
X47
)
)
)
)
↔
(
¬
(
X49
→
X41
→
(
X39
↔
(
X26
↔
False
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X49
→
(
(
¬
X28
)
↔
(
(
¬
X18
)
→
X40
)
)
)
)
↔
(
(
(
(
¬
X46
)
↔
(
(
¬
X6
)
→
X49
)
)
↔
(
¬
(
(
¬
(
X49
→
(
X28
↔
X40
)
)
)
→
(
(
X44
↔
(
(
¬
X40
)
→
(
(
¬
X28
)
↔
X49
)
)
)
↔
(
¬
(
X44
→
(
¬
X40
)
→
X6
→
X49
)
)
)
)
)
)
→
(
X28
↔
False
)
)
)
→
(
¬
X46
)
→
(
X44
↔
X49
)
)
↔
(
(
(
X49
→
(
X46
↔
(
(
¬
X10
)
↔
X39
)
)
)
→
(
X49
↔
(
X40
↔
(
(
¬
X10
)
↔
X46
)
)
)
)
↔
(
¬
(
X49
↔
(
X46
→
(
X39
↔
X5
)
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
¬
(
(
(
(
X34
→
(
X29
↔
(
(
¬
X4
)
→
X13
)
)
)
↔
(
X29
↔
(
(
¬
X13
)
↔
(
X9
↔
X43
)
)
)
)
↔
(
X43
→
(
X9
↔
(
X4
→
False
)
)
)
)
→
(
¬
(
(
¬
X41
)
→
(
(
¬
X13
)
↔
(
X4
↔
X49
)
)
)
)
)
)
→
(
(
¬
(
X34
→
(
X20
↔
X41
)
)
)
↔
(
(
X43
→
X42
→
(
X34
↔
X41
)
)
↔
(
¬
(
X43
→
X34
→
X30
→
X41
)
)
)
)
)
↔
(
(
X49
→
(
X45
↔
(
X8
→
X43
)
)
)
→
(
¬
(
(
¬
(
(
(
(
¬
(
X42
→
(
X36
↔
(
X16
→
False
)
)
)
)
→
(
¬
(
(
¬
X43
)
↔
(
X42
→
(
(
¬
X36
)
↔
X46
)
)
)
)
)
→
(
¬
X43
)
→
X16
→
X46
)
→
(
¬
(
X46
↔
(
X43
→
(
X2
↔
False
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
X46
)
↔
(
(
¬
X45
)
→
X49
)
)
↔
(
¬
(
(
¬
(
X46
↔
(
(
¬
X45
)
→
(
(
¬
X43
)
↔
X49
)
)
)
)
→
(
¬
(
X49
→
(
X46
↔
(
(
¬
X36
)
↔
X45
)
)
)
)
)
)
)
↔
(
X45
↔
(
(
¬
X43
)
→
X49
)
)
)
→
(
¬
(
X49
↔
(
X46
↔
(
X43
↔
(
X0
→
False
)
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X46
→
X42
→
(
X5
↔
False
)
)
)
→
(
¬
(
X48
↔
(
X46
→
(
X42
↔
X49
)
)
)
)
)
)
→
(
(
(
¬
(
X49
↔
(
X42
↔
(
X15
→
X1
)
)
)
)
↔
(
(
X37
→
(
¬
X15
)
→
X49
)
→
(
¬
(
(
¬
X42
)
→
X15
→
X48
)
)
)
)
↔
(
¬
(
(
X42
↔
(
X37
↔
(
X33
↔
X3
)
)
)
→
(
(
¬
X42
)
↔
(
(
¬
X3
)
↔
X48
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
(
X40
→
X25
→
(
¬
X19
)
→
X35
)
)
↔
(
¬
(
(
¬
X35
)
→
(
¬
X25
)
→
(
X14
↔
X40
)
)
)
)
→
(
¬
(
X6
↔
X35
)
)
→
(
X32
↔
(
X19
→
X17
→
False
)
)
)
↔
(
(
(
¬
(
X48
↔
(
X26
↔
(
X25
↔
False
)
)
)
)
→
(
X48
→
(
X45
↔
X25
)
)
→
(
(
¬
X45
)
↔
(
X44
↔
(
X26
→
X48
)
)
)
)
↔
(
(
X39
↔
(
X32
↔
(
(
¬
X10
)
↔
X23
)
)
)
↔
(
(
¬
(
X45
→
X32
→
(
¬
X10
)
→
X40
)
)
↔
(
¬
(
(
X44
→
(
¬
X40
)
→
(
(
¬
X32
)
↔
X45
)
)
→
(
¬
(
(
(
¬
(
X45
↔
(
(
¬
X32
)
↔
(
X23
→
X40
)
)
)
)
↔
(
(
¬
X40
)
→
(
¬
X23
)
→
X48
)
)
→
(
(
¬
X40
)
↔
X48
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
(
X47
↔
(
X38
→
(
X11
↔
X31
)
)
)
↔
(
X47
→
(
X31
↔
(
X21
→
X38
)
)
)
)
→
(
¬
(
(
¬
X38
)
→
(
¬
X31
)
→
X47
)
)
)
↔
(
X40
↔
(
X38
↔
(
X31
→
X11
→
False
)
)
)
)
↔
(
¬
(
X40
↔
(
(
¬
X38
)
↔
(
X28
↔
X48
)
)
)
)
)
→
(
X48
↔
X12
)
→
(
(
(
X47
→
X28
→
X18
→
X42
)
↔
(
(
¬
(
X47
→
(
(
¬
X28
)
↔
X42
)
)
)
↔
(
(
¬
(
(
¬
X42
)
→
(
X28
↔
(
X22
→
X47
)
)
)
)
→
(
¬
(
(
¬
X28
)
↔
(
X27
↔
X42
)
)
)
)
)
)
↔
(
(
(
X28
→
(
¬
X27
)
→
(
(
¬
X24
)
↔
X42
)
)
→
(
¬
(
X42
↔
(
X28
→
(
X7
↔
X27
)
)
)
)
)
→
(
X28
↔
(
X7
↔
X32
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
c615d8...
and proposition id is
942a57...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMShesJJCNe2f2LGXwLxeXwxy2sLw4ZDdp9
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
(
(
(
¬
(
X41
↔
(
(
¬
X33
)
→
(
¬
X27
)
→
X47
)
)
)
→
(
¬
(
X41
↔
(
X39
→
False
)
)
)
)
↔
(
¬
(
(
X41
→
(
(
¬
X38
)
↔
(
X33
↔
X39
)
)
)
→
(
¬
X39
)
→
(
(
¬
X38
)
↔
(
X31
↔
X41
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X39
)
→
(
X8
↔
(
(
¬
X7
)
↔
X48
)
)
)
)
→
(
X47
↔
(
X19
↔
(
X8
↔
(
X7
→
False
)
)
)
)
)
)
↔
(
(
¬
(
X41
→
X39
→
X19
→
(
X9
↔
False
)
)
)
→
(
¬
X41
)
→
(
¬
X19
)
→
(
(
¬
X9
)
↔
X47
)
)
)
)
↔
(
(
(
X48
→
(
(
¬
X41
)
↔
X45
)
)
↔
(
¬
(
X45
→
X41
→
(
X29
↔
(
X11
↔
False
)
)
)
)
)
↔
(
(
(
¬
(
X45
→
(
(
¬
X13
)
↔
X14
)
)
)
→
(
(
¬
(
X47
→
(
(
¬
X13
)
↔
X29
)
)
)
↔
(
X47
↔
(
X45
→
X14
→
(
X13
↔
False
)
)
)
)
)
→
(
¬
(
X45
→
(
(
¬
X41
)
↔
X47
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X48
)
↔
(
X47
↔
(
(
¬
X28
)
↔
X49
)
)
)
)
↔
(
¬
(
(
¬
(
X49
→
X32
→
X3
→
X48
)
)
→
(
¬
X47
)
→
(
(
¬
X36
)
↔
(
X32
→
X49
)
)
)
)
)
→
(
(
X48
→
(
¬
X18
)
→
X3
→
X45
)
↔
(
(
X32
→
X18
→
X37
)
→
(
¬
(
(
¬
(
X32
→
(
X18
↔
(
X3
↔
False
)
)
)
)
→
(
(
X18
↔
(
X2
↔
X37
)
)
↔
(
(
¬
(
X45
↔
(
X37
→
X18
→
X32
)
)
)
↔
(
X32
↔
X2
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X49
↔
(
X48
→
X17
→
X44
)
)
)
↔
(
(
(
(
X49
→
X0
→
X48
)
→
X49
→
(
(
¬
X25
)
↔
(
(
¬
X24
)
→
X48
)
)
)
↔
(
X48
→
X25
→
X49
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X25
)
→
X24
→
X44
)
)
→
(
(
X49
→
(
X44
↔
(
X25
↔
False
)
)
)
↔
(
(
¬
(
(
¬
(
X24
→
(
X21
↔
X44
)
)
)
→
(
¬
(
X49
→
X25
→
X24
)
)
)
)
→
(
(
X49
→
(
X44
↔
(
X15
↔
X22
)
)
)
↔
(
¬
(
X49
→
(
¬
X22
)
→
X24
)
)
)
)
)
)
)
↔
(
X49
→
(
¬
X24
)
→
(
¬
X16
)
→
X44
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
¬
X44
)
↔
(
X42
→
(
(
¬
X5
)
↔
X48
)
)
)
)
↔
(
(
¬
X12
)
↔
(
(
¬
X5
)
↔
X44
)
)
)
→
(
(
X48
→
(
(
¬
X42
)
↔
(
X26
→
X44
)
)
)
↔
(
X48
→
(
X26
↔
(
(
¬
X25
)
↔
X44
)
)
)
)
)
)
↔
(
¬
(
(
X48
↔
(
X42
→
(
X20
↔
(
X6
↔
False
)
)
)
)
→
(
¬
(
X48
→
(
¬
X42
)
→
(
X20
↔
X44
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
(
¬
(
(
¬
X34
)
↔
(
(
¬
X30
)
↔
(
X1
↔
X47
)
)
)
)
→
(
¬
X43
)
→
(
(
¬
X34
)
↔
(
X30
→
X47
)
)
)
)
↔
(
¬
(
(
¬
X43
)
↔
(
(
¬
X4
)
↔
(
X1
↔
X47
)
)
)
)
)
↔
(
(
¬
(
X46
→
X43
→
X34
→
X30
)
)
↔
(
¬
(
(
¬
X46
)
→
X43
→
(
¬
X34
)
→
X48
)
)
)
)
→
(
¬
(
X47
→
(
X46
↔
(
X40
→
X48
)
)
)
)
→
(
¬
(
X47
↔
(
X46
↔
(
(
¬
X40
)
→
X42
)
)
)
)
)
)
→
(
(
¬
(
(
(
X40
↔
(
(
¬
X35
)
↔
(
X34
↔
X44
)
)
)
↔
(
(
¬
X40
)
→
(
X35
↔
(
X34
↔
X42
)
)
)
)
→
(
¬
(
(
¬
X42
)
↔
(
(
¬
X34
)
↔
(
X10
↔
X44
)
)
)
)
)
)
↔
(
¬
(
X42
↔
(
X40
↔
(
(
¬
X10
)
→
X23
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
X49
↔
(
X48
→
X2
→
False
)
)
↔
(
X49
→
X44
→
(
(
¬
X2
)
↔
X17
)
)
)
→
(
¬
(
X49
↔
(
X46
→
X44
)
)
)
)
↔
(
(
X44
→
(
(
¬
X38
)
↔
X46
)
)
→
(
(
(
¬
(
(
¬
X20
)
↔
(
X10
→
X48
)
)
)
↔
(
X46
↔
(
X38
→
X10
→
X48
)
)
)
↔
(
(
¬
(
X44
↔
(
X23
→
X10
→
X38
)
)
)
↔
(
(
¬
X20
)
→
(
(
¬
X10
)
↔
X23
)
)
)
)
)
)
→
(
¬
(
(
(
X49
→
X46
→
(
¬
X13
)
→
X24
)
→
(
(
(
¬
X46
)
↔
(
X45
→
(
¬
X13
)
→
X49
)
)
↔
(
(
X15
→
(
¬
X4
)
→
X48
)
→
(
¬
(
X48
→
(
X13
↔
(
X4
↔
X49
)
)
)
)
)
)
)
→
(
X46
→
(
(
¬
X12
)
↔
X24
)
)
→
(
¬
(
X12
→
X48
)
)
→
(
(
¬
(
X48
↔
(
X46
↔
(
X12
→
X3
)
)
)
)
↔
(
¬
(
X21
↔
(
X12
↔
(
X3
↔
X48
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
(
(
¬
(
(
¬
X43
)
↔
(
X35
→
(
(
¬
X33
)
↔
X47
)
)
)
)
↔
(
X40
↔
(
(
¬
X35
)
→
(
(
¬
X33
)
↔
X43
)
)
)
)
↔
(
¬
(
X47
→
False
)
)
)
↔
(
¬
(
X47
→
(
X40
↔
(
X35
→
X19
→
False
)
)
)
)
)
→
X46
→
X40
→
X35
→
X43
)
↔
(
(
¬
(
X49
↔
(
X32
→
X40
)
)
)
→
(
(
(
¬
X37
)
↔
(
X19
↔
X40
)
)
↔
(
(
(
¬
X37
)
↔
(
X32
↔
(
X19
↔
X49
)
)
)
↔
(
¬
(
X40
→
(
(
¬
X32
)
↔
X49
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
(
¬
X41
)
↔
(
(
¬
X16
)
↔
(
(
¬
X9
)
→
X49
)
)
)
)
↔
(
X43
→
(
(
¬
X41
)
↔
(
(
¬
X16
)
→
X49
)
)
)
)
↔
(
¬
(
(
¬
X31
)
↔
(
X9
→
X5
→
X41
)
)
)
)
→
(
(
¬
X36
)
↔
(
X18
↔
(
X16
→
X41
)
)
)
→
X43
→
(
X16
↔
False
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
¬
X46
)
↔
(
X27
↔
(
(
¬
X14
)
↔
X49
)
)
)
→
X42
→
(
X27
↔
X14
)
)
)
→
(
¬
(
X49
↔
(
X46
→
X43
→
X22
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
¬
(
(
¬
X43
)
→
X25
→
X49
)
)
→
(
¬
(
(
¬
X42
)
→
X25
→
(
¬
X0
)
→
X43
)
)
)
)
→
(
¬
(
X46
↔
(
X43
↔
(
X42
→
X49
)
)
)
)
)
)
→
X46
→
(
X42
↔
X43
)
)
)
)
→
(
(
¬
(
X49
↔
(
X43
→
(
¬
X25
)
→
X41
)
)
)
↔
(
¬
(
(
X49
→
(
X41
↔
(
X39
→
X7
)
)
)
→
(
¬
(
(
¬
X41
)
↔
(
(
¬
X7
)
→
X43
)
)
)
)
)
)
→
(
¬
(
X43
↔
(
(
¬
X25
)
→
X6
→
X41
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
(
X34
→
(
¬
X26
)
→
X28
)
)
→
(
X26
↔
(
X11
→
X47
)
)
)
)
↔
(
(
(
X47
→
(
¬
X11
)
→
X34
)
→
(
X49
↔
(
X34
→
(
(
¬
X11
)
↔
X29
)
)
)
)
↔
(
X49
→
(
X48
↔
(
(
¬
X30
)
↔
X47
)
)
)
)
)
→
(
(
¬
(
X48
↔
(
X46
→
X8
→
(
X1
↔
False
)
)
)
)
↔
(
(
¬
X26
)
↔
(
X1
↔
X49
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
a9f8f4...
and proposition id is
23395e...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMYk1nSGVKSFNwRTpVM8iDK6u237hummgpV
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
(
¬
(
X49
↔
(
X45
→
X27
→
X36
)
)
)
↔
(
X36
↔
X43
)
)
↔
(
(
¬
(
(
¬
(
(
(
¬
(
X44
↔
X28
)
)
↔
(
X48
↔
(
X43
↔
(
(
¬
X11
)
↔
X28
)
)
)
)
→
(
¬
(
X28
→
(
¬
X11
)
→
X44
)
)
)
)
→
(
(
¬
(
X44
↔
(
(
¬
X43
)
↔
X48
)
)
)
↔
(
X48
↔
(
X45
→
X13
→
X44
)
)
)
)
)
↔
(
(
(
¬
(
X45
→
(
(
¬
X33
)
↔
(
(
¬
X28
)
↔
X49
)
)
)
)
↔
(
¬
(
X49
→
(
(
¬
X28
)
↔
(
X8
→
X33
)
)
)
)
)
↔
(
¬
(
X48
↔
(
X24
→
X18
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X25
↔
(
X10
→
False
)
)
)
↔
(
(
¬
(
(
X43
↔
(
X42
→
(
(
¬
X22
)
↔
X25
)
)
)
→
(
¬
(
(
X42
↔
(
X25
→
X10
→
X44
)
)
→
(
¬
X43
)
→
(
¬
X42
)
→
(
X10
↔
X44
)
)
)
)
)
→
(
¬
X22
)
→
X44
)
)
→
(
(
¬
(
X44
→
(
X21
↔
X25
)
)
)
↔
(
¬
(
X42
↔
(
X25
↔
(
(
¬
X0
)
↔
X44
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
¬
X36
)
↔
(
X26
→
(
¬
X23
)
→
X44
)
)
)
→
(
(
¬
(
X44
→
(
(
¬
X36
)
↔
X42
)
)
)
↔
(
¬
(
X44
↔
(
X26
→
X21
)
)
)
)
)
)
→
(
¬
(
(
¬
X44
)
→
X26
→
(
X5
↔
X47
)
)
)
→
(
¬
(
X36
→
(
X26
↔
X44
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X41
↔
(
X38
↔
(
X19
↔
X48
)
)
)
)
→
(
¬
(
X48
→
(
(
¬
X38
)
↔
(
X35
→
X41
)
)
)
)
)
)
→
(
(
¬
(
X32
→
X14
→
(
¬
X9
)
→
X35
)
)
→
(
(
¬
(
(
¬
X38
)
→
(
(
¬
X35
)
↔
(
(
¬
X32
)
↔
X46
)
)
)
)
↔
(
X38
→
(
X35
↔
(
(
¬
X32
)
→
X49
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
X35
)
→
(
X30
↔
(
(
¬
X6
)
↔
X41
)
)
)
→
(
¬
(
X35
→
(
(
¬
X30
)
↔
(
X15
→
X32
)
)
)
)
→
(
¬
(
(
¬
X35
)
→
(
(
¬
X30
)
↔
(
X15
→
X41
)
)
)
)
)
)
→
(
¬
(
X49
→
(
X35
↔
(
X32
→
X15
→
False
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X46
↔
(
(
¬
X37
)
→
(
¬
X17
)
→
X48
)
)
)
↔
(
(
¬
(
X47
↔
(
X17
→
(
(
¬
X2
)
↔
X46
)
)
)
)
↔
(
¬
(
X34
→
(
X17
↔
False
)
)
)
)
)
↔
(
(
¬
(
X46
→
X39
→
(
X29
↔
X16
)
)
)
→
(
¬
(
(
(
¬
X46
)
↔
(
X39
→
X47
)
)
→
(
(
¬
(
(
¬
X29
)
↔
(
X1
↔
X48
)
)
)
↔
(
(
X47
↔
(
X46
↔
X48
)
)
↔
(
(
¬
(
X46
↔
(
X39
→
(
X1
↔
X48
)
)
)
)
↔
(
¬
(
X47
↔
X1
)
)
)
)
)
)
)
)
)
→
(
(
(
X49
↔
X4
)
↔
(
¬
(
(
¬
(
(
(
¬
(
X49
→
X20
→
X12
→
X40
)
)
↔
(
¬
(
X46
→
X40
→
X49
)
)
)
→
(
¬
(
X40
↔
(
(
¬
X12
)
↔
X20
)
)
)
)
)
→
(
(
¬
X46
)
↔
(
X12
→
X49
)
)
)
)
)
↔
(
(
X47
↔
(
X40
→
(
¬
X31
)
→
X46
)
)
↔
(
(
X48
→
(
(
¬
X40
)
↔
(
(
¬
X7
)
→
X49
)
)
)
↔
(
(
¬
(
X48
→
(
X7
↔
(
(
¬
X3
)
↔
X31
)
)
)
)
↔
(
¬
(
X47
→
(
¬
X7
)
→
(
X3
↔
X31
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
X46
↔
(
X43
↔
(
X25
↔
(
X14
↔
False
)
)
)
)
→
(
¬
(
(
X31
↔
(
X25
→
(
X14
↔
(
X10
↔
False
)
)
)
)
→
(
X43
↔
(
X31
↔
(
(
¬
X14
)
↔
X25
)
)
)
)
)
)
→
(
X31
↔
(
X25
↔
X44
)
)
)
↔
(
(
(
(
(
X32
→
(
X31
↔
(
X24
↔
X18
)
)
)
↔
(
¬
(
X16
→
X18
)
)
)
↔
(
X32
↔
(
X18
↔
X24
)
)
)
→
(
¬
(
(
(
X18
→
X13
→
(
X12
↔
(
X9
→
False
)
)
)
→
X13
→
X12
→
(
¬
X1
)
→
X9
)
→
(
¬
(
(
¬
(
X25
→
X24
→
X12
→
X31
)
)
→
(
(
¬
X24
)
↔
(
(
¬
X12
)
→
X9
→
X31
)
)
)
)
)
)
)
→
X44
→
(
(
¬
X25
)
↔
(
X18
→
X31
)
)
)
)
↔
(
(
(
(
¬
(
(
X44
→
(
X40
↔
(
X19
↔
X27
)
)
)
→
(
X46
↔
(
X44
→
(
X27
↔
X40
)
)
)
)
)
→
(
(
¬
(
X44
↔
(
(
¬
X4
)
→
(
¬
X2
)
→
X40
)
)
)
↔
(
¬
(
X46
↔
(
X44
↔
(
X40
→
X2
)
)
)
)
)
)
↔
(
(
X40
→
(
(
¬
X26
)
↔
(
(
¬
X5
)
→
X28
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X27
)
→
X28
)
)
→
(
(
(
¬
X27
)
↔
X43
)
↔
(
¬
(
X28
→
(
(
¬
X11
)
↔
(
X5
→
X27
)
)
)
)
)
)
→
(
X11
↔
X46
)
→
(
X46
↔
(
X28
→
(
(
¬
X5
)
↔
X26
)
)
)
→
(
X11
↔
(
X5
→
False
)
)
)
)
)
)
→
(
(
¬
(
X6
→
X8
)
)
→
X46
→
X43
→
(
X8
↔
X49
)
)
→
(
¬
(
(
(
¬
(
X44
↔
(
X33
↔
X49
)
)
)
↔
(
¬
(
X46
↔
(
X42
→
X44
)
)
)
)
→
(
¬
(
(
(
(
¬
X46
)
→
(
(
¬
X44
)
↔
(
(
¬
X15
)
→
X49
)
)
)
→
(
X46
↔
(
X15
↔
(
X8
→
X44
)
)
)
)
→
(
X46
↔
(
X44
↔
(
X15
↔
False
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
(
X49
↔
(
(
¬
X22
)
→
(
¬
X3
)
→
X38
)
)
↔
(
(
(
¬
(
(
¬
X38
)
↔
(
X36
↔
X49
)
)
)
↔
(
(
¬
(
X49
↔
(
(
¬
X38
)
↔
(
(
¬
X36
)
→
X45
)
)
)
)
↔
(
¬
(
(
¬
X41
)
↔
(
X38
→
(
(
¬
X36
)
↔
X49
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X45
→
X38
→
False
)
)
→
(
X41
↔
(
X38
→
(
X22
↔
X21
)
)
)
)
)
)
)
↔
(
(
¬
(
X47
→
X45
→
X17
→
X49
)
)
→
(
(
¬
(
(
¬
(
(
¬
X47
)
→
(
(
¬
X29
)
↔
(
X22
→
X49
)
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X22
)
↔
(
X17
↔
X29
)
)
)
)
)
)
↔
(
¬
(
X49
↔
(
X47
→
X29
→
(
X7
↔
False
)
)
)
)
)
)
)
→
(
(
(
¬
(
X44
↔
(
X34
→
(
(
¬
X23
)
↔
X41
)
)
)
)
↔
(
X41
→
(
X34
↔
(
X23
↔
(
X20
→
False
)
)
)
)
)
↔
(
(
(
¬
X38
)
→
(
(
¬
X34
)
↔
(
(
¬
X23
)
↔
X44
)
)
)
↔
(
(
¬
X44
)
→
(
X23
↔
X45
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
X37
↔
(
(
¬
X0
)
↔
X48
)
)
)
→
(
¬
(
X39
↔
(
X35
→
(
X0
↔
False
)
)
)
)
)
)
↔
(
(
(
¬
X48
)
→
(
X44
↔
(
X43
→
X49
)
)
)
↔
(
(
(
X49
↔
(
X47
→
X37
)
)
↔
(
(
¬
(
X47
→
(
X39
↔
X30
)
)
)
→
(
¬
(
X48
↔
(
X43
↔
(
X37
→
X30
→
False
)
)
)
)
)
)
↔
(
¬
(
X49
→
(
X48
↔
(
X47
↔
False
)
)
)
)
)
)
)
↔
(
¬
(
X44
→
(
(
¬
X39
)
↔
X47
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
b95cc9...
and proposition id is
53e4da...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMGEbJrrvpDkCekxXhfq5PYn86nuQBZnLy9
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
¬
(
(
(
¬
(
X41
↔
(
X31
↔
(
X2
↔
False
)
)
)
)
↔
(
(
(
X43
↔
(
X31
→
(
X28
↔
X2
)
)
)
→
(
X44
↔
(
X31
↔
(
X28
→
False
)
)
)
)
↔
(
X43
→
X31
→
X44
)
)
)
→
(
¬
(
(
(
(
¬
(
X40
→
(
X31
↔
(
X30
↔
X23
)
)
)
)
→
X40
→
X33
→
X30
→
X23
)
↔
(
(
(
¬
(
X47
→
(
(
¬
X31
)
↔
(
(
¬
X23
)
↔
X43
)
)
)
)
↔
(
(
¬
X33
)
↔
(
(
¬
X31
)
↔
(
(
¬
X23
)
→
X43
)
)
)
)
↔
(
¬
(
X43
↔
(
(
¬
X31
)
→
X25
→
X47
)
)
)
)
)
→
(
(
¬
(
X44
→
(
X36
↔
(
X31
↔
X25
)
)
)
)
↔
(
¬
(
(
(
(
¬
X30
)
↔
(
X20
↔
(
(
¬
X4
)
→
X31
)
)
)
↔
(
X31
↔
(
X14
↔
(
X12
→
False
)
)
)
)
→
(
(
¬
(
X36
↔
(
(
¬
X20
)
↔
X44
)
)
)
↔
(
X30
→
(
¬
X20
)
→
(
¬
X14
)
→
X44
)
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
X48
)
→
(
¬
X43
)
→
X5
→
X49
)
)
→
(
X43
↔
(
X5
↔
X34
)
)
)
→
(
(
¬
(
(
(
¬
(
X43
↔
(
X0
→
False
)
)
)
↔
(
¬
(
X43
→
X40
→
X37
→
X36
)
)
)
→
(
¬
(
X40
→
(
(
¬
X37
)
↔
(
(
¬
X36
)
↔
X48
)
)
)
)
)
)
↔
(
¬
(
X48
→
(
(
¬
X37
)
↔
(
X36
→
X49
)
)
)
)
)
)
↔
(
(
(
¬
(
X49
→
X43
→
X13
→
X16
)
)
↔
(
(
¬
X36
)
↔
(
(
¬
X16
)
→
(
(
¬
X13
)
↔
X43
)
)
)
)
↔
(
(
(
X43
↔
(
(
¬
X45
)
→
(
X24
↔
(
(
¬
X8
)
↔
X49
)
)
)
)
↔
(
¬
(
X45
→
(
(
¬
X24
)
↔
(
X6
↔
X49
)
)
)
)
)
↔
(
(
X36
↔
(
(
¬
X24
)
→
(
¬
X15
)
→
X43
)
)
↔
(
X36
↔
(
(
¬
X15
)
↔
X49
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
¬
(
X48
↔
X35
)
)
→
(
(
(
¬
X42
)
→
X38
→
X10
→
X48
)
↔
(
(
X48
→
(
X38
↔
(
X35
↔
(
X32
→
False
)
)
)
)
↔
(
¬
(
X42
→
(
¬
X38
)
→
X35
→
X48
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
¬
(
(
X48
→
X42
→
X38
→
False
)
→
(
¬
(
(
¬
(
X48
↔
(
X42
↔
(
X38
↔
(
X22
↔
False
)
)
)
)
)
→
(
¬
(
(
¬
X39
)
↔
(
X22
→
X42
)
)
)
)
)
→
(
X48
↔
(
X42
→
X39
)
)
)
)
→
(
¬
(
(
¬
X39
)
↔
(
(
¬
X38
)
↔
(
(
¬
X9
)
→
X42
)
)
)
)
)
)
→
X39
→
(
X9
↔
X22
)
)
)
↔
(
¬
(
(
¬
X39
)
→
(
(
¬
X38
)
↔
X42
)
)
)
)
→
X48
→
(
X42
↔
X38
)
)
)
↔
(
¬
(
(
¬
X38
)
→
(
(
¬
X32
)
↔
(
(
¬
X7
)
↔
X39
)
)
)
)
)
→
(
(
(
(
¬
(
X38
↔
(
X27
→
X19
)
)
)
↔
(
¬
(
X42
→
X38
→
(
X19
↔
X48
)
)
)
)
→
(
(
(
(
¬
X19
)
→
X1
→
X42
)
↔
(
(
¬
X27
)
→
(
(
¬
X19
)
↔
(
(
¬
X1
)
→
X29
)
)
)
)
↔
(
X42
↔
(
X1
↔
False
)
)
)
)
↔
(
(
(
¬
(
(
¬
X42
)
↔
(
X38
↔
(
(
¬
X27
)
↔
X47
)
)
)
)
↔
(
¬
(
(
(
¬
X38
)
↔
(
X26
↔
X47
)
)
→
(
¬
(
X47
↔
(
(
¬
X27
)
→
X26
→
X38
)
)
)
→
(
X26
↔
(
X11
↔
X27
)
)
)
)
)
↔
(
¬
(
X27
↔
(
X26
→
X47
)
)
)
)
)
)
↔
(
(
(
¬
(
X49
↔
(
X46
↔
(
X38
↔
X18
)
)
)
)
↔
(
¬
(
X46
→
(
¬
X38
)
→
(
¬
X17
)
→
X49
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
¬
X21
)
→
X18
→
X3
→
X39
)
)
→
(
¬
(
X46
↔
(
(
¬
X18
)
↔
(
(
¬
X3
)
→
X39
)
)
)
)
)
)
→
(
¬
(
X21
↔
(
X3
→
X42
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
(
¬
(
X46
↔
(
X33
↔
X31
)
)
)
↔
(
(
X47
↔
(
X45
→
X42
)
)
↔
(
¬
(
(
X45
→
(
(
¬
X42
)
↔
(
X36
→
X46
)
)
)
→
(
X47
↔
(
X46
→
(
X45
↔
(
X33
↔
False
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X36
)
→
(
¬
X24
)
→
X45
)
)
→
(
X46
↔
(
X36
→
(
X24
↔
X33
)
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
X38
→
(
X4
↔
X46
)
)
)
↔
(
(
¬
(
X42
→
(
¬
X38
)
→
X22
→
X46
)
)
→
(
¬
(
X38
↔
(
X22
↔
X4
)
)
)
)
)
↔
(
X46
→
X22
→
(
(
¬
X4
)
↔
X45
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X45
→
(
X39
↔
(
X32
→
(
X26
↔
False
)
)
)
)
)
→
(
(
¬
(
X42
→
X32
→
X39
)
)
↔
(
¬
(
X39
↔
(
(
¬
X32
)
↔
(
(
¬
X18
)
↔
X45
)
)
)
)
)
)
→
(
(
(
¬
X39
)
→
(
(
¬
X21
)
↔
(
(
¬
X0
)
↔
X45
)
)
)
↔
(
¬
(
X46
↔
(
X13
↔
(
X0
↔
X42
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
X35
→
X19
)
→
(
¬
(
(
¬
X34
)
↔
(
(
¬
X19
)
→
X17
→
X44
)
)
)
)
)
→
(
(
¬
(
X47
→
(
X46
↔
(
X44
→
X45
)
)
)
)
↔
(
(
¬
(
(
¬
(
X44
→
(
¬
X35
)
→
(
(
¬
X10
)
↔
X45
)
)
)
→
X45
→
(
¬
X35
)
→
(
¬
X34
)
→
X44
)
)
→
(
¬
(
(
X46
↔
(
X45
↔
(
X3
→
False
)
)
)
→
(
¬
(
X34
↔
(
X8
↔
X3
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X47
↔
(
X29
→
(
¬
X27
)
→
X44
)
)
)
→
(
¬
(
X47
→
(
X46
↔
X27
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
(
¬
X40
)
↔
(
X28
↔
(
X2
→
X42
)
)
)
↔
(
X47
↔
(
X40
→
(
¬
X2
)
→
X42
)
)
)
↔
(
¬
(
X47
→
X42
→
X40
)
)
)
↔
(
(
(
X49
↔
(
X40
→
(
¬
X2
)
→
X47
)
)
↔
(
X49
↔
(
X48
→
(
X47
↔
(
X40
→
False
)
)
)
)
)
↔
(
X49
→
X48
→
X47
→
X1
)
)
)
↔
(
(
¬
(
(
X40
↔
(
(
¬
X15
)
→
X23
)
)
→
(
(
X47
↔
(
X5
→
X23
)
)
↔
(
X43
→
(
X40
↔
(
(
¬
X23
)
→
X47
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
(
¬
(
X43
→
(
¬
X28
)
→
X41
)
)
→
(
(
(
¬
X40
)
→
(
(
¬
X28
)
↔
X43
)
)
↔
(
X41
→
(
X7
↔
X40
)
)
)
)
)
→
(
¬
(
X43
↔
(
X28
→
X41
)
)
)
)
)
→
X41
→
(
X28
↔
(
X7
→
X23
)
)
)
)
→
(
¬
(
X28
→
(
X7
↔
X41
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
X37
↔
(
X30
→
X45
)
)
)
→
(
¬
(
(
¬
X42
)
↔
(
X37
→
(
(
¬
X30
)
↔
X43
)
)
)
)
)
↔
(
¬
(
X30
↔
X45
)
)
)
↔
(
X43
→
X42
→
(
¬
X11
)
→
X45
)
)
↔
(
¬
(
(
¬
(
(
¬
X20
)
↔
(
(
¬
X9
)
→
X45
)
)
)
→
(
X49
↔
(
X11
↔
(
(
¬
X6
)
→
X43
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X25
)
→
(
¬
X16
)
→
(
X14
↔
X43
)
)
)
→
(
¬
(
(
¬
X25
)
→
(
¬
X16
)
→
(
(
¬
X12
)
↔
X43
)
)
)
)
↔
(
(
¬
X42
)
→
(
(
¬
X25
)
↔
(
X16
↔
X49
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
e368a5...
and proposition id is
19ffef...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMEgfP8TzNRbFeczrRbvm29D3vMvKWgD5TZ
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
¬
(
(
(
(
¬
(
(
X48
↔
(
X46
→
(
X25
↔
X9
)
)
)
→
(
¬
(
(
¬
(
(
¬
X25
)
↔
(
X6
↔
X46
)
)
)
→
(
¬
(
X46
→
X25
→
(
X9
↔
X49
)
)
)
)
)
)
)
↔
(
¬
(
X25
→
(
X9
↔
X46
)
)
)
)
→
(
(
X49
→
X44
→
(
¬
X24
)
→
X48
)
↔
(
¬
(
(
X48
→
X44
→
X49
)
→
(
(
¬
(
X49
↔
(
(
¬
X46
)
→
(
X34
↔
X48
)
)
)
)
↔
(
X49
→
X48
→
(
X46
↔
X27
)
)
)
→
(
¬
(
X46
↔
(
X44
↔
X27
)
)
)
)
)
)
)
→
(
(
X32
→
X21
→
X43
)
→
(
(
¬
(
X48
→
(
¬
X32
)
→
(
X22
↔
X43
)
)
)
↔
(
X37
↔
(
(
¬
X32
)
↔
(
(
¬
X22
)
↔
X48
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
(
¬
X37
)
↔
(
(
¬
X10
)
↔
X49
)
)
)
→
(
(
¬
(
X46
→
(
¬
X14
)
→
X49
)
)
↔
(
(
¬
(
(
¬
X46
)
→
(
¬
X14
)
→
(
(
¬
X10
)
↔
X49
)
)
)
→
X46
→
(
(
¬
X14
)
↔
(
(
¬
X10
)
↔
X37
)
)
)
)
)
↔
(
¬
(
X37
→
(
X10
↔
(
X5
↔
X46
)
)
)
)
)
→
(
¬
(
(
¬
(
X49
↔
(
(
¬
X32
)
→
X43
)
)
)
→
(
X46
↔
(
(
¬
X32
)
↔
(
X14
→
X43
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
X8
→
X30
)
↔
(
(
¬
(
(
(
¬
X30
)
↔
(
(
¬
X13
)
→
X8
→
X48
)
)
→
(
¬
(
X30
→
(
(
¬
X13
)
↔
(
(
¬
X11
)
↔
X48
)
)
)
)
)
)
→
(
¬
(
X48
→
(
¬
X11
)
→
X30
)
)
)
)
→
X46
→
(
(
¬
X30
)
↔
(
(
¬
X13
)
↔
X48
)
)
)
)
↔
(
(
¬
(
X49
→
(
X46
↔
(
X13
↔
X2
)
)
)
)
→
(
¬
(
(
¬
X13
)
→
X8
→
X46
)
)
)
)
↔
(
(
(
X49
→
(
¬
X13
)
→
(
X8
↔
X31
)
)
↔
(
(
(
¬
(
X41
↔
(
X31
→
(
¬
X1
)
→
X8
)
)
)
→
(
¬
(
(
¬
X41
)
→
(
¬
X31
)
→
X49
)
)
)
↔
(
X49
↔
(
X41
→
(
X31
↔
False
)
)
)
)
)
↔
(
(
(
¬
X13
)
↔
(
(
¬
X8
)
↔
X20
)
)
↔
(
X41
↔
(
X33
→
X13
→
False
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X44
↔
(
X43
↔
(
X17
↔
X36
)
)
)
)
↔
(
(
(
X43
→
(
(
¬
X42
)
↔
(
X36
→
X45
)
)
)
↔
(
(
¬
X44
)
→
X42
→
X45
)
)
↔
(
¬
(
X36
↔
(
(
¬
X18
)
→
(
¬
X3
)
→
X45
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X43
)
→
(
¬
X38
)
→
(
¬
X29
)
→
X48
)
)
↔
(
(
¬
X38
)
↔
(
X36
↔
(
X29
→
X43
)
)
)
)
↔
(
(
(
¬
X36
)
→
(
X15
↔
X45
)
)
→
(
¬
X43
)
→
(
(
¬
X38
)
↔
X45
)
)
)
)
→
(
(
¬
(
(
(
(
(
¬
X35
)
↔
(
(
¬
X28
)
→
X40
)
)
↔
(
(
¬
(
X40
→
(
X28
↔
(
X19
↔
X35
)
)
)
)
↔
(
(
¬
(
X40
→
(
(
¬
X23
)
↔
X35
)
)
)
→
(
(
¬
(
X23
↔
(
X16
↔
(
(
¬
X4
)
→
X40
)
)
)
)
↔
(
¬
(
X40
↔
(
X28
↔
X35
)
)
)
)
)
)
)
↔
(
X47
↔
(
X35
→
(
X23
↔
X40
)
)
)
)
→
(
(
¬
(
X35
↔
(
(
¬
X12
)
→
X40
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X28
)
→
X12
→
X35
)
)
↔
(
X35
↔
(
X28
↔
(
X0
→
X40
)
)
)
)
→
(
¬
X12
)
→
(
X0
↔
X35
)
)
)
)
)
)
↔
(
(
(
(
(
X45
→
(
(
¬
X40
)
↔
(
X28
↔
X44
)
)
)
→
(
¬
(
(
¬
X44
)
→
(
¬
X40
)
→
(
¬
X39
)
→
X45
)
)
)
↔
(
¬
(
X44
↔
(
X40
↔
False
)
)
)
)
↔
(
¬
(
(
¬
X44
)
↔
(
X43
→
X28
→
X45
)
)
)
)
↔
(
¬
(
(
¬
(
X47
→
(
(
¬
X40
)
↔
X45
)
)
)
→
(
(
(
¬
X45
)
↔
(
X43
↔
X47
)
)
↔
(
X43
→
(
¬
X26
)
→
X7
→
X45
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
(
X46
→
(
X42
↔
(
X2
→
X37
)
)
)
↔
(
¬
(
(
¬
(
X45
→
(
X37
↔
X46
)
)
)
→
(
¬
(
X45
→
X37
→
(
¬
X36
)
→
X42
)
)
)
)
)
→
(
X42
↔
(
X41
↔
(
X37
↔
False
)
)
)
)
↔
(
(
X46
→
X36
→
X15
→
X45
)
→
(
(
(
(
¬
X44
)
→
(
X36
↔
(
X20
→
X46
)
)
)
→
(
(
¬
X20
)
↔
(
X9
→
X45
)
)
)
↔
(
(
¬
(
X46
↔
(
X45
→
(
¬
X21
)
→
X36
)
)
)
↔
(
¬
(
X46
↔
(
X45
↔
(
X28
↔
X36
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
X44
)
↔
(
(
¬
X25
)
↔
(
(
¬
X23
)
↔
X45
)
)
)
→
(
¬
(
X44
→
X25
→
X1
→
X23
)
)
)
→
(
¬
(
(
¬
(
X46
↔
(
X14
→
(
¬
X1
)
→
X44
)
)
)
→
(
(
¬
X44
)
→
X23
→
(
¬
X13
)
→
X46
)
→
(
¬
(
(
¬
X23
)
→
X14
→
(
(
¬
X13
)
↔
X44
)
)
)
)
)
)
→
(
X45
↔
(
X38
↔
(
X25
→
(
X23
↔
False
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
X38
→
(
X37
↔
(
(
¬
X29
)
↔
X34
)
)
)
→
(
(
X40
→
(
¬
X38
)
→
(
X0
↔
X47
)
)
↔
(
(
(
¬
(
X34
↔
(
(
¬
X29
)
→
(
X0
↔
X47
)
)
)
)
↔
(
(
¬
X18
)
→
X34
)
)
↔
(
(
¬
X29
)
↔
(
X18
↔
X47
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X48
→
X37
→
X32
→
False
)
)
→
(
¬
(
(
¬
X37
)
→
(
(
¬
X32
)
↔
(
(
¬
X19
)
→
X49
)
)
)
)
)
)
→
(
(
¬
(
X48
↔
(
X38
→
(
X22
↔
X40
)
)
)
)
↔
(
(
(
¬
X32
)
→
X22
→
(
¬
X16
)
→
X48
)
↔
(
(
¬
X38
)
→
X32
→
(
X22
↔
X48
)
)
)
)
→
(
(
¬
X44
)
→
(
X38
↔
X49
)
)
→
X49
→
X44
→
X22
→
X32
)
)
)
)
)
→
(
(
(
(
(
X48
↔
(
X31
↔
X4
)
)
↔
(
¬
(
(
(
(
¬
X45
)
→
(
(
¬
X39
)
↔
X48
)
)
→
(
¬
(
X45
↔
(
X39
→
(
X4
↔
X31
)
)
)
)
)
→
(
¬
(
X39
→
False
)
)
)
)
)
↔
(
(
¬
X27
)
↔
(
X6
↔
(
(
¬
X4
)
→
X45
)
)
)
)
↔
(
(
(
¬
(
X45
→
(
(
¬
X33
)
↔
(
(
¬
X31
)
→
X48
)
)
)
)
↔
(
(
X48
→
(
(
¬
X10
)
↔
(
(
¬
X7
)
→
X45
)
)
)
↔
(
(
(
X45
↔
(
X33
→
(
¬
X10
)
→
X48
)
)
↔
(
X48
↔
(
X33
↔
(
X5
→
X45
)
)
)
)
↔
(
¬
(
X45
→
(
¬
X5
)
→
X33
)
)
)
)
)
→
X48
→
(
(
¬
X45
)
↔
(
(
¬
X33
)
↔
X47
)
)
)
)
↔
(
(
(
¬
(
X48
↔
(
X39
→
(
X3
↔
X11
)
)
)
)
↔
(
X48
↔
(
X43
↔
(
(
¬
X12
)
↔
X39
)
)
)
)
↔
(
(
¬
(
(
¬
(
X30
↔
(
X24
↔
X26
)
)
)
→
X48
→
X39
→
X30
→
X35
)
)
↔
(
(
(
¬
(
X47
→
X8
→
X35
)
)
→
X43
→
X35
→
(
X8
↔
X17
)
)
↔
(
(
X47
↔
False
)
→
(
(
¬
(
X47
↔
(
X24
→
X17
→
X30
)
)
)
↔
(
X47
→
(
(
¬
X24
)
↔
(
(
¬
X17
)
→
X35
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
905557...
and proposition id is
0e0d7f...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMQmhVRS8teYrccVUKxezSQN6G5zYNe86Vc
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
¬
(
(
(
(
(
X35
↔
(
X33
→
(
X11
↔
X3
)
)
)
↔
(
(
¬
(
(
X35
→
(
(
¬
X11
)
↔
(
(
¬
X3
)
↔
X45
)
)
)
→
(
¬
(
X45
↔
(
X3
→
(
X1
↔
X11
)
)
)
)
)
)
↔
(
¬
(
X45
→
(
(
¬
X3
)
↔
X35
)
)
)
)
)
↔
(
¬
(
(
X45
↔
(
X37
→
X15
→
X47
)
)
→
(
(
¬
(
X35
↔
(
(
¬
X33
)
↔
(
(
¬
X15
)
↔
X46
)
)
)
)
↔
(
¬
(
X37
↔
(
(
¬
X15
)
→
X46
)
)
)
)
)
)
)
↔
(
(
(
X47
↔
(
(
¬
X27
)
→
X45
)
)
↔
(
(
X45
→
(
X29
↔
(
X27
→
X6
→
False
)
)
)
→
X47
→
(
X45
↔
(
X29
↔
(
X27
→
False
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X46
→
X45
→
(
¬
X8
)
→
X23
)
)
↔
(
X45
→
X38
→
(
(
¬
X8
)
↔
X46
)
)
)
→
(
X45
↔
X8
)
)
→
(
¬
(
(
X26
→
(
X17
↔
X46
)
)
→
(
¬
(
X42
↔
(
X27
→
X17
→
X29
)
)
)
)
)
→
X38
→
X27
→
(
¬
X26
)
→
X45
)
)
)
)
→
(
(
(
(
¬
X45
)
→
(
(
¬
X44
)
↔
(
X4
↔
X47
)
)
)
↔
(
(
¬
(
(
¬
X44
)
↔
(
(
¬
X43
)
→
(
X30
↔
X47
)
)
)
)
↔
(
(
(
¬
(
X44
→
(
(
¬
X28
)
↔
X47
)
)
)
→
(
(
¬
(
(
X47
→
X28
→
X44
)
→
(
¬
(
(
¬
X43
)
→
(
X28
↔
X45
)
)
)
)
)
↔
(
X28
↔
False
)
)
)
→
(
X43
↔
(
(
¬
X28
)
↔
(
(
¬
X21
)
→
X47
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X44
→
X41
→
X39
)
)
↔
(
X47
↔
(
(
¬
X41
)
→
(
(
¬
X36
)
↔
X44
)
)
)
)
→
(
(
¬
(
X44
↔
(
X7
→
X46
)
)
)
↔
(
(
(
X44
→
X41
→
X19
→
X5
)
↔
(
¬
(
(
¬
False
)
→
(
¬
(
(
X45
→
(
X41
↔
(
X5
→
False
)
)
)
→
(
¬
(
X44
→
(
X39
↔
X5
)
)
)
)
)
)
)
)
→
(
¬
(
X45
→
(
X44
↔
(
X7
→
X41
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
(
¬
(
X46
↔
(
(
¬
X31
)
→
(
(
¬
X9
)
↔
X34
)
)
)
)
→
(
¬
(
X46
↔
(
X13
→
X9
→
False
)
)
)
)
→
(
X49
↔
(
X31
→
X9
)
)
)
↔
(
(
¬
(
X25
→
(
X24
↔
X49
)
)
)
→
(
(
¬
(
(
¬
X25
)
↔
(
(
¬
X24
)
→
X13
→
X49
)
)
)
↔
(
¬
(
(
¬
X24
)
→
(
X13
↔
(
X9
→
X25
)
)
)
)
)
)
)
↔
(
(
(
¬
X34
)
→
X24
→
(
X12
↔
X46
)
)
→
(
¬
(
X34
→
X20
→
(
¬
X12
)
→
X32
)
)
)
)
↔
(
¬
(
(
X32
↔
(
X2
↔
(
(
¬
X0
)
↔
X16
)
)
)
→
(
¬
(
(
¬
(
(
¬
X31
)
↔
(
(
¬
X2
)
↔
(
(
¬
X0
)
↔
X34
)
)
)
)
→
X32
→
(
(
¬
X2
)
↔
(
X0
→
X34
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
X40
↔
(
(
¬
X10
)
→
X32
)
)
)
→
(
¬
(
X49
→
(
(
¬
X40
)
↔
(
X10
↔
X48
)
)
)
)
)
→
X40
→
(
(
¬
X18
)
↔
(
(
¬
X10
)
↔
X48
)
)
)
→
(
X40
↔
(
X18
↔
X43
)
)
)
↔
(
(
X43
↔
(
X40
→
(
X32
↔
(
X22
→
False
)
)
)
)
↔
(
¬
(
(
¬
(
X43
↔
(
X32
→
X24
)
)
)
→
(
¬
(
(
¬
(
(
¬
X24
)
↔
(
X14
↔
X43
)
)
)
→
(
¬
(
X32
↔
(
(
¬
X14
)
→
X22
)
)
)
)
)
)
)
)
)
→
X48
→
X40
→
(
(
¬
X32
)
↔
X42
)
)
)
↔
(
(
(
(
(
¬
(
(
(
(
¬
X40
)
→
X34
→
(
¬
X23
)
→
X48
)
↔
(
X48
→
(
X23
↔
X7
)
)
)
→
(
¬
(
(
¬
(
X48
↔
(
X45
↔
(
(
¬
X23
)
→
X43
)
)
)
)
→
(
¬
(
X48
→
X45
→
(
X5
↔
False
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X45
↔
(
(
¬
X38
)
→
X41
)
)
)
→
(
(
¬
(
X45
↔
(
X36
→
(
¬
X0
)
→
X38
)
)
)
↔
(
¬
(
X43
↔
(
X36
→
X45
)
)
)
)
)
)
↔
(
(
¬
(
X41
↔
(
X40
↔
(
(
¬
X38
)
→
X42
)
)
)
)
↔
(
(
(
X40
↔
(
(
¬
X36
)
→
X15
→
X41
)
)
→
X42
→
X15
)
↔
(
X36
→
X40
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
X40
→
X37
→
X17
→
False
)
→
(
X48
→
X47
→
(
X37
↔
X32
)
)
→
(
X48
↔
(
X32
→
X40
)
)
)
)
→
(
(
(
X32
↔
(
(
¬
X8
)
↔
X49
)
)
↔
(
¬
(
(
(
(
(
¬
(
(
¬
X40
)
↔
(
(
¬
X38
)
→
(
X32
↔
X49
)
)
)
)
→
X49
→
(
(
¬
X38
)
↔
(
(
¬
X10
)
→
X40
)
)
)
→
(
X49
↔
(
(
¬
X32
)
↔
X38
)
)
)
↔
(
¬
(
X49
→
X32
→
X8
→
X38
)
)
)
→
(
¬
(
(
¬
X38
)
→
(
X10
↔
X40
)
)
)
)
)
)
↔
(
¬
(
(
¬
X17
)
↔
(
X10
↔
(
X8
→
X38
)
)
)
)
)
)
)
)
→
(
(
(
X31
→
(
¬
X14
)
→
X12
→
X47
)
↔
(
(
¬
(
X4
↔
(
(
¬
X3
)
↔
X49
)
)
)
↔
(
(
X49
→
(
X3
↔
False
)
)
↔
(
¬
(
X49
→
(
X14
↔
(
X4
↔
X31
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X49
↔
(
X31
↔
(
X27
↔
X45
)
)
)
)
→
(
¬
(
(
¬
X45
)
→
(
¬
X31
)
→
(
X12
↔
X49
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
(
(
¬
(
X28
→
X24
→
X35
)
)
↔
(
¬
(
(
X35
→
X28
→
(
(
¬
X24
)
↔
X46
)
)
→
(
¬
(
X46
→
(
(
¬
X28
)
↔
(
(
¬
X21
)
→
X35
)
)
)
)
)
)
)
↔
(
¬
(
X28
→
X21
→
(
¬
X6
)
→
X24
)
)
)
→
(
¬
(
X44
→
(
(
¬
X6
)
↔
X48
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X25
)
→
X18
→
(
X1
↔
X30
)
)
)
↔
(
(
(
¬
(
(
X48
↔
(
X30
→
(
¬
X25
)
→
X46
)
)
→
(
¬
X30
)
→
(
X25
↔
(
(
¬
X1
)
→
X48
)
)
)
)
↔
(
(
¬
X30
)
↔
(
X25
↔
X46
)
)
)
↔
(
¬
False
)
)
)
→
(
¬
(
(
X30
↔
(
X25
↔
X44
)
)
→
(
(
¬
(
X44
→
X18
→
X13
→
X25
)
)
↔
(
¬
(
(
¬
X25
)
→
(
¬
X13
)
→
X30
)
)
)
)
)
)
)
)
↔
(
¬
(
(
X48
→
(
X47
↔
(
X2
↔
False
)
)
)
→
(
(
¬
(
X48
↔
(
X47
↔
X46
)
)
)
↔
(
¬
(
X49
→
X30
→
False
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
¬
(
X47
↔
(
X19
↔
X33
)
)
)
→
(
¬
(
X26
↔
(
(
¬
X9
)
→
X33
)
)
)
)
)
→
(
(
(
¬
X47
)
→
(
(
¬
X33
)
↔
(
X22
↔
X49
)
)
)
↔
(
¬
(
X49
→
(
¬
X33
)
→
(
¬
X26
)
→
X47
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
(
X47
→
X33
→
(
¬
X11
)
→
X48
)
)
→
(
X47
↔
(
(
¬
X33
)
→
X16
→
X48
)
)
)
)
→
(
X47
↔
(
X33
↔
(
X26
↔
X16
)
)
)
)
)
↔
(
X48
↔
(
X47
→
(
X33
↔
False
)
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X29
)
→
(
X20
↔
X33
)
)
)
)
→
(
¬
(
(
X48
→
X47
→
(
(
¬
X29
)
↔
X39
)
)
→
(
X47
↔
(
(
¬
X39
)
→
(
X20
↔
X48
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
43915c...
and proposition id is
e8ea2f...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMZABx3ZsAWgb1Waxt5mmxb1Rh2J1kJvupe
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
¬
(
(
¬
(
(
(
X8
↔
X18
)
↔
(
¬
(
(
¬
(
X43
→
X18
→
(
¬
X8
)
→
X34
)
)
→
X46
→
X43
→
X8
)
)
)
→
(
(
X43
↔
(
(
¬
X39
)
→
X46
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
¬
(
(
¬
X43
)
→
(
(
¬
X39
)
↔
(
(
¬
X8
)
↔
X46
)
)
)
)
→
(
X46
↔
(
(
¬
X39
)
→
(
X31
↔
X43
)
)
)
)
)
→
X46
→
X39
→
(
X31
↔
False
)
)
)
→
(
¬
(
X46
↔
(
X39
→
X31
→
False
)
)
)
)
)
)
)
)
→
(
(
¬
(
X31
↔
(
X16
↔
(
X7
↔
False
)
)
)
)
↔
(
(
X31
→
(
X16
↔
X7
)
)
→
(
(
X43
↔
(
X31
↔
(
X18
→
X7
)
)
)
↔
(
¬
(
X18
↔
(
X16
→
X7
→
X43
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X47
→
X46
→
(
X11
↔
X10
)
)
)
→
(
(
(
¬
X31
)
→
(
¬
X11
)
→
(
X2
↔
X46
)
)
↔
(
(
¬
(
X31
→
(
¬
X11
)
→
X1
→
X34
)
)
→
(
¬
(
(
¬
X34
)
→
X10
→
(
(
¬
X1
)
↔
X46
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
X37
↔
(
(
¬
X3
)
→
X47
)
)
→
(
(
¬
(
(
X37
↔
(
X14
→
X43
)
)
→
(
¬
(
X33
↔
(
X29
→
X3
→
False
)
)
)
)
)
↔
(
X43
→
(
X33
↔
X14
)
)
)
)
)
↔
(
X37
→
(
(
¬
X33
)
↔
X47
)
)
)
→
(
(
¬
(
X47
→
(
X43
↔
(
X37
↔
False
)
)
)
)
→
(
(
¬
(
X48
→
(
¬
X21
)
→
X43
)
)
↔
(
¬
(
(
¬
X37
)
→
(
¬
X21
)
→
X47
)
)
)
)
→
X48
→
(
¬
X21
)
→
(
(
¬
X0
)
↔
X43
)
)
↔
(
(
(
(
¬
(
(
(
(
¬
(
X46
→
(
X43
↔
(
X37
→
X12
)
)
)
)
→
(
¬
(
X43
↔
(
X27
→
(
X12
↔
False
)
)
)
)
)
→
(
¬
(
(
¬
X37
)
↔
(
X12
→
X43
)
)
)
)
→
(
¬
(
X27
→
X12
→
X46
)
)
)
)
↔
(
(
¬
X43
)
→
(
X37
↔
X46
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X46
)
→
(
(
¬
X43
)
↔
(
(
¬
X38
)
→
X47
)
)
)
)
→
X47
→
(
X38
↔
(
X27
→
X43
)
)
)
)
)
↔
(
(
X46
↔
(
X38
→
X20
→
X32
)
)
→
X43
→
(
(
¬
X32
)
↔
(
X20
→
X38
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
¬
(
X48
↔
(
X47
→
(
X40
↔
(
X22
→
False
)
)
)
)
)
→
(
¬
(
X25
→
X22
→
X48
)
)
)
)
↔
(
(
(
¬
(
X25
→
(
¬
X22
)
→
(
X19
↔
X26
)
)
)
↔
(
X40
↔
(
X26
↔
(
X19
→
False
)
)
)
)
↔
(
X40
→
(
X22
↔
X25
)
)
)
)
→
(
(
(
(
¬
X40
)
↔
(
X13
↔
(
X9
→
X45
)
)
)
↔
(
(
(
¬
X28
)
→
(
(
¬
X26
)
↔
(
(
¬
X15
)
→
X40
)
)
)
↔
(
X46
↔
(
X28
→
(
¬
X15
)
→
X26
)
)
)
)
↔
(
¬
(
(
¬
(
X47
→
(
X28
↔
X41
)
)
)
→
(
¬
(
X45
→
(
X41
↔
(
X40
↔
X28
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
X44
↔
(
X42
→
(
X17
↔
X48
)
)
)
→
(
¬
(
X48
↔
(
X44
→
(
(
¬
X23
)
↔
X42
)
)
)
)
)
↔
(
X44
→
(
(
¬
X42
)
↔
X48
)
)
)
→
(
¬
(
(
(
¬
(
X49
↔
(
X48
↔
(
X44
→
X24
)
)
)
)
↔
(
¬
(
X49
→
(
(
¬
X24
)
↔
(
(
¬
X23
)
↔
X48
)
)
)
)
)
→
X44
→
X24
→
(
X4
↔
False
)
)
)
)
↔
(
(
(
¬
(
(
X46
→
(
X45
↔
(
(
¬
X42
)
→
X44
)
)
)
→
(
(
(
¬
X45
)
↔
(
(
¬
X44
)
→
X5
→
X46
)
)
↔
(
X46
↔
(
X30
→
(
¬
X5
)
→
X45
)
)
)
)
)
↔
(
X30
→
False
)
)
↔
(
(
¬
(
X48
→
(
(
¬
X30
)
↔
(
(
¬
X6
)
→
X46
)
)
)
)
↔
(
¬
(
(
(
(
X48
→
X35
→
X42
)
↔
(
¬
(
X46
↔
(
X42
→
X30
)
)
)
)
↔
(
X46
↔
(
X42
↔
(
X30
→
False
)
)
)
)
→
X46
→
(
X42
↔
(
(
¬
X36
)
↔
X48
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
¬
(
X49
↔
(
X43
→
X29
→
X27
→
False
)
)
)
↔
(
(
(
¬
X18
)
→
X43
)
↔
(
(
X29
↔
X18
)
↔
(
¬
(
X43
→
X27
→
X18
→
X44
)
)
)
)
)
→
(
(
(
X32
↔
(
X28
→
(
X13
↔
X49
)
)
)
↔
(
X16
→
X5
→
(
X1
↔
X49
)
)
)
→
(
¬
(
X49
→
(
X32
↔
(
(
¬
X28
)
→
X44
)
)
)
)
)
→
(
¬
(
(
X44
→
X34
→
(
(
¬
X14
)
↔
X49
)
)
→
(
(
¬
(
(
¬
(
X43
→
(
X34
↔
(
X14
→
False
)
)
)
)
→
(
¬
(
(
¬
X34
)
↔
(
(
¬
X16
)
→
X14
→
X44
)
)
)
)
)
↔
(
¬
(
X44
↔
(
X34
→
(
X16
↔
False
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
(
(
¬
X48
)
↔
(
X44
↔
(
(
¬
X43
)
↔
X49
)
)
)
)
→
(
¬
(
X49
↔
(
X44
↔
(
X43
↔
X35
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X49
↔
(
X44
↔
X48
)
)
)
↔
(
(
¬
X37
)
→
(
X9
↔
X44
)
)
)
→
(
¬
(
X48
↔
(
(
¬
X37
)
→
X44
)
)
)
)
)
)
↔
(
(
¬
(
(
X48
→
(
¬
X43
)
→
(
X8
↔
X49
)
)
→
(
(
(
¬
(
(
¬
X43
)
→
X2
→
X48
)
)
↔
(
(
¬
(
X49
→
(
¬
X43
)
→
(
X22
↔
X48
)
)
)
↔
(
X49
↔
(
X48
↔
(
X2
↔
False
)
)
)
)
)
↔
(
X49
→
X43
→
(
¬
X22
)
→
X36
)
)
)
)
↔
(
¬
(
(
¬
X22
)
↔
(
X8
↔
(
X6
↔
X49
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
X49
↔
(
X47
↔
(
X41
→
X44
)
)
)
)
→
(
X47
↔
(
X44
→
X7
)
)
)
)
↔
(
(
¬
(
X41
↔
(
(
¬
X40
)
→
(
(
¬
X33
)
↔
X47
)
)
)
)
→
(
¬
(
X47
→
(
¬
X41
)
→
(
¬
X40
)
→
X44
)
)
→
(
¬
(
X47
↔
(
X41
→
X40
→
X48
)
)
)
)
)
↔
(
(
¬
(
X48
→
(
X44
↔
(
X19
↔
False
)
)
)
)
→
X48
→
(
X41
↔
(
(
¬
X4
)
→
X44
)
)
)
)
)
)
)
)
→
(
(
(
(
X20
↔
(
(
¬
X10
)
↔
X45
)
)
↔
(
¬
(
(
¬
(
X38
↔
(
(
¬
X20
)
→
(
¬
X10
)
→
X48
)
)
)
→
(
(
¬
X38
)
↔
(
X20
→
X45
)
)
)
)
)
↔
(
¬
(
(
¬
(
X48
↔
(
(
¬
X38
)
→
(
X20
↔
X39
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
(
(
¬
X38
)
↔
(
(
¬
X24
)
↔
X48
)
)
)
→
(
¬
(
(
¬
X39
)
→
X24
→
X3
→
X48
)
)
)
)
↔
(
X38
→
(
X24
↔
(
X3
↔
False
)
)
)
)
→
(
¬
(
(
¬
X38
)
→
X48
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
X48
↔
(
X15
→
False
)
)
↔
(
X48
→
(
(
¬
X39
)
↔
(
X26
↔
X47
)
)
)
)
→
(
¬
(
(
¬
(
X47
↔
(
X15
↔
X45
)
)
)
→
(
¬
(
X48
→
(
(
¬
X23
)
↔
(
X15
↔
X45
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X47
↔
(
X46
→
X45
→
(
X30
↔
False
)
)
)
)
↔
(
(
(
(
(
¬
X11
)
↔
X30
)
↔
(
X47
↔
(
X46
→
(
X11
↔
X17
)
)
)
)
↔
(
X30
↔
(
X17
→
X11
→
False
)
)
)
↔
(
X46
↔
X30
)
)
)
↔
(
¬
(
X45
↔
(
(
¬
X30
)
→
(
X21
↔
X46
)
)
)
)
)
↔
(
¬
(
X48
→
(
X46
↔
(
X45
→
X47
)
)
)
)
)
)
)
→
(
(
(
X46
↔
(
X45
→
X31
→
X42
)
)
↔
(
¬
(
(
(
(
¬
(
(
¬
X48
)
→
(
¬
X46
)
→
(
¬
X0
)
→
X49
)
)
→
X48
→
X46
→
X31
→
X0
→
False
)
→
(
¬
X0
)
→
X31
)
→
(
X42
↔
(
(
¬
X31
)
→
X46
)
)
)
)
)
↔
(
(
(
¬
X46
)
→
(
X45
↔
(
(
¬
X25
)
↔
X49
)
)
)
↔
(
X45
→
(
(
¬
X25
)
↔
(
(
¬
X12
)
↔
X46
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
c7a86d...
and proposition id is
4a6b02...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMNoEq971VTtvkKyNRUrDeNVRm7xW6tANzo
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
¬
(
(
(
(
(
(
X44
→
X29
→
(
X8
↔
X39
)
)
→
(
¬
(
X48
→
X29
→
X14
→
X8
)
)
)
↔
(
(
(
X49
→
X44
→
X48
)
↔
(
¬
(
X49
↔
(
X39
→
(
¬
X3
)
→
X35
)
)
)
)
↔
(
(
X44
↔
X5
)
↔
(
¬
(
(
¬
X12
)
→
(
(
¬
X5
)
↔
(
X3
→
X50
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X0
)
→
X44
)
)
↔
(
(
¬
(
(
¬
(
(
X44
↔
(
X9
↔
X48
)
)
→
X48
→
(
X9
↔
(
(
¬
X0
)
→
X43
)
)
)
)
→
(
¬
(
X44
→
X43
→
X48
)
)
)
)
↔
(
¬
(
(
¬
X25
)
↔
(
X9
→
(
(
¬
X0
)
↔
X44
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X43
→
(
X42
↔
(
X40
→
False
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X43
)
↔
(
X42
→
(
¬
X40
)
→
X44
)
)
)
→
(
¬
(
(
¬
X44
)
↔
(
(
¬
X43
)
→
(
X42
↔
X49
)
)
)
)
)
)
)
→
(
¬
(
(
¬
X48
)
↔
X49
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
(
(
¬
(
X43
↔
(
X41
↔
(
X34
→
X2
)
)
)
)
→
(
(
¬
X38
)
↔
(
(
¬
X34
)
→
X41
)
)
)
→
(
X41
↔
(
(
¬
X34
)
→
X17
→
X43
)
)
)
)
→
(
¬
(
X41
↔
(
X17
→
X43
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X26
)
↔
X38
)
)
→
(
X43
↔
(
X41
↔
(
X26
↔
(
X22
↔
False
)
)
)
)
)
)
)
→
(
(
¬
(
X7
→
(
(
¬
X6
)
↔
X44
)
)
)
↔
(
(
¬
X42
)
↔
(
(
¬
X34
)
→
(
¬
X6
)
→
X44
)
)
)
→
(
(
¬
(
(
(
(
¬
X41
)
→
X38
→
(
(
¬
X34
)
↔
X42
)
)
↔
(
X42
→
(
X38
↔
X7
)
)
)
→
X38
→
(
(
¬
X34
)
↔
X42
)
)
)
↔
(
(
¬
X41
)
↔
(
X34
→
X42
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
X27
)
↔
(
(
¬
X18
)
→
X13
→
X47
)
)
)
↔
(
X33
↔
(
X18
↔
(
(
¬
X13
)
↔
X47
)
)
)
)
→
(
(
X50
→
(
X48
↔
(
X47
→
X27
)
)
)
↔
(
(
¬
(
(
(
(
X50
↔
(
(
¬
X27
)
↔
(
X19
↔
X31
)
)
)
↔
(
¬
(
(
¬
X31
)
→
(
X27
↔
X47
)
)
)
)
↔
(
(
¬
X47
)
→
(
(
¬
X19
)
↔
X50
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
(
X50
↔
(
X30
↔
False
)
)
)
→
X31
→
(
(
¬
X18
)
↔
X47
)
)
)
→
(
¬
(
X47
↔
(
X31
↔
X18
)
)
)
)
→
(
(
¬
X31
)
↔
(
(
¬
X30
)
→
X50
)
)
)
)
)
)
↔
(
(
¬
X47
)
↔
(
X30
→
X50
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X50
↔
(
X48
↔
(
X15
→
False
)
)
)
)
→
(
(
¬
(
X48
→
(
X10
↔
X15
)
)
)
↔
(
(
X21
→
X31
)
↔
(
X48
→
X21
→
(
(
¬
X15
)
↔
X50
)
)
)
)
)
)
→
(
¬
(
X50
↔
(
X31
→
(
(
¬
X1
)
↔
X48
)
)
)
)
→
(
¬
(
(
¬
X24
)
→
(
X1
↔
X31
)
)
)
)
)
→
(
(
(
(
X50
↔
(
X47
↔
(
(
¬
X23
)
→
X49
)
)
)
→
(
(
(
(
X50
→
(
(
¬
X23
)
↔
X49
)
)
↔
(
X36
↔
(
X20
↔
X49
)
)
)
→
(
¬
(
X36
↔
(
(
¬
X20
)
↔
X23
)
)
)
)
↔
(
(
¬
X49
)
↔
(
X23
↔
(
(
¬
X16
)
→
X50
)
)
)
)
)
↔
(
(
(
(
¬
(
(
X50
→
X47
→
X46
→
X28
→
False
)
→
(
¬
(
X49
→
X47
→
X46
→
X50
)
)
)
)
↔
(
X49
→
(
X46
↔
False
)
)
)
↔
(
(
¬
X49
)
↔
(
X47
↔
X50
)
)
)
→
(
¬
(
(
¬
X47
)
↔
(
(
¬
X28
)
→
X49
)
)
)
)
)
↔
(
(
(
¬
(
X46
→
(
X45
↔
(
X37
→
X32
→
False
)
)
)
)
↔
(
¬
(
X50
→
(
X32
↔
False
)
)
)
)
↔
(
(
(
¬
(
X48
↔
(
X37
→
(
(
¬
X32
)
↔
X49
)
)
)
)
↔
(
¬
(
X48
→
X46
→
X49
)
)
)
→
(
¬
(
X37
→
X11
→
(
X4
↔
X49
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
(
¬
(
(
X35
→
(
¬
X25
)
→
(
(
¬
X9
)
↔
X30
)
)
→
(
¬
(
X42
→
(
X25
↔
X9
)
)
)
→
(
(
¬
(
(
¬
X35
)
→
(
(
¬
X25
)
↔
X42
)
)
)
↔
(
X49
↔
(
X42
↔
(
(
¬
X9
)
↔
X35
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X35
↔
(
X26
↔
(
X1
→
False
)
)
)
)
↔
(
¬
(
X49
↔
(
X28
→
(
X26
↔
X30
)
)
)
)
)
↔
(
¬
(
X30
↔
(
X28
→
X49
)
)
)
)
→
(
¬
(
X35
↔
(
X26
→
X30
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
(
¬
X35
)
→
(
(
¬
X32
)
↔
(
(
¬
X8
)
→
X49
)
)
)
)
→
(
¬
(
(
¬
(
X49
→
(
(
¬
X19
)
↔
(
X8
→
X50
)
)
)
)
→
(
¬
(
X35
↔
(
X19
↔
(
(
¬
X8
)
↔
X49
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
X42
)
→
(
X22
↔
X49
)
)
)
↔
(
¬
(
X50
→
(
X35
↔
False
)
)
)
)
)
→
(
¬
(
(
¬
X23
)
→
X20
→
X49
)
)
→
(
¬
(
(
¬
X42
)
→
(
(
¬
X35
)
↔
(
(
¬
X23
)
→
X49
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
X50
↔
(
X47
↔
False
)
)
→
(
(
¬
(
X32
→
(
(
¬
X15
)
↔
X35
)
)
)
↔
(
¬
(
(
¬
X47
)
→
X35
→
(
X15
↔
X50
)
)
)
)
)
)
↔
(
¬
(
X47
→
(
X35
↔
X50
)
)
)
)
→
X35
→
(
X15
↔
X47
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
X34
)
→
(
X14
↔
X38
)
)
→
(
¬
(
(
X34
→
X27
→
X46
)
→
(
(
X50
→
X47
→
(
¬
X27
)
→
X38
)
↔
(
(
X50
→
(
X38
↔
(
X36
→
X34
)
)
)
→
X50
→
(
X47
↔
(
X36
↔
X38
)
)
)
)
)
)
→
(
(
(
X49
↔
(
X38
↔
(
(
¬
X14
)
↔
X36
)
)
)
↔
(
¬
(
X49
→
X38
→
(
¬
X36
)
→
X47
)
)
)
↔
(
X49
↔
(
X47
→
X38
→
False
)
)
)
)
)
↔
(
(
X39
→
X0
→
X13
)
→
(
¬
(
(
(
¬
(
X47
↔
(
X13
↔
(
(
¬
X3
)
→
X50
)
)
)
)
↔
(
(
(
¬
(
X3
→
X0
→
X49
)
)
→
(
(
¬
(
(
¬
X3
)
→
(
X0
↔
X6
)
)
)
↔
(
(
¬
(
(
¬
(
X49
↔
(
X6
→
(
(
¬
X0
)
↔
X3
)
)
)
)
→
(
¬
(
X49
↔
(
X13
→
False
)
)
)
)
)
↔
(
X6
↔
X49
)
)
)
)
→
(
¬
X3
)
→
(
¬
X0
)
→
X6
)
)
→
(
X50
↔
(
X39
→
(
X3
↔
X49
)
)
)
)
)
)
)
)
→
(
(
(
(
(
(
¬
X47
)
→
(
(
¬
X24
)
↔
(
(
¬
X17
)
→
X48
)
)
)
→
(
(
(
¬
(
X48
→
X24
→
X17
→
X4
)
)
↔
(
¬
X50
)
)
↔
X50
)
)
→
(
X50
↔
(
X49
↔
(
X24
↔
X48
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X40
)
→
(
X17
↔
(
(
¬
X2
)
→
X49
)
)
)
)
→
(
(
¬
(
X40
↔
(
X17
↔
(
X5
→
False
)
)
)
)
↔
(
X47
↔
(
X17
↔
(
(
¬
X5
)
→
X49
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
X48
)
→
(
(
¬
X47
)
↔
(
(
¬
X43
)
→
X50
)
)
)
→
(
¬
(
(
¬
(
X50
→
X48
→
(
(
¬
X21
)
↔
X31
)
)
)
→
(
¬
(
X45
↔
(
X31
↔
(
(
¬
X21
)
↔
X48
)
)
)
)
)
)
→
(
(
(
(
X47
↔
(
(
¬
X41
)
↔
(
X16
↔
X50
)
)
)
↔
(
(
¬
X41
)
→
X48
)
)
↔
(
(
¬
X47
)
↔
(
(
¬
X16
)
↔
X48
)
)
)
↔
(
X48
→
(
(
¬
X31
)
↔
(
X16
↔
X41
)
)
)
)
)
→
(
¬
(
X41
→
(
X37
↔
(
(
¬
X10
)
→
X47
)
)
)
)
→
(
¬
(
(
¬
(
X45
↔
(
(
¬
X37
)
↔
(
(
¬
X10
)
→
X47
)
)
)
)
→
X47
→
(
X45
↔
(
X12
↔
X37
)
)
)
)
)
)
↔
(
(
¬
(
X50
↔
(
X29
→
(
¬
X18
)
→
X33
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X47
)
↔
(
X44
→
X11
→
X48
)
)
)
→
(
(
¬
(
X44
↔
(
X33
→
(
X7
↔
X48
)
)
)
)
↔
(
¬
(
X48
→
(
(
¬
X11
)
↔
(
X7
↔
X44
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X50
↔
(
X48
→
(
X18
↔
X33
)
)
)
)
→
(
¬
(
X50
↔
(
(
¬
X18
)
↔
X44
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
fd4f43...
and proposition id is
b8b325...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMRvN7gTmv9t3cN8TWWKyFq1LTWuL5RgYB1
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
∀X50 :
prop
,
(
(
(
(
¬
(
(
(
(
(
X49
→
(
(
¬
X44
)
↔
(
X40
↔
X48
)
)
)
→
X49
→
X40
→
(
X15
↔
X44
)
)
→
(
¬
(
(
(
¬
X15
)
↔
(
X9
→
(
(
¬
X5
)
↔
X35
)
)
)
→
(
¬
(
(
(
¬
(
X42
→
(
(
¬
X9
)
↔
(
X5
→
X49
)
)
)
)
↔
(
(
¬
X15
)
→
X9
→
X42
)
)
→
(
¬
(
X49
↔
(
X48
↔
(
X42
→
X15
)
)
)
)
)
)
)
)
)
→
(
(
X44
→
(
(
¬
X26
)
↔
X49
)
)
↔
(
¬
(
X44
→
(
(
¬
X43
)
↔
(
(
¬
X26
)
↔
X48
)
)
)
)
)
)
→
(
(
¬
(
X49
→
(
¬
X18
)
→
X50
)
)
→
(
(
(
(
(
(
¬
X49
)
→
X44
→
(
(
¬
X18
)
↔
X50
)
)
→
X48
→
(
(
¬
X18
)
↔
X49
)
)
→
(
¬
(
X18
→
X49
)
)
)
↔
(
¬
(
X48
→
X49
)
)
)
↔
(
X48
↔
(
(
¬
X44
)
→
X49
)
)
)
)
→
(
¬
(
(
(
(
X49
→
(
(
¬
X36
)
↔
(
X16
↔
X50
)
)
)
↔
(
(
¬
X49
)
→
(
(
¬
X48
)
↔
X50
)
)
)
↔
(
X50
↔
(
X49
↔
(
X16
↔
False
)
)
)
)
→
(
¬
(
(
¬
X49
)
→
X48
→
(
¬
X29
)
→
X50
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
X48
→
X37
→
X11
→
False
)
↔
(
¬
(
X47
→
(
(
¬
X37
)
↔
(
X11
↔
X49
)
)
)
)
)
→
(
(
(
(
(
X48
↔
(
(
¬
X37
)
↔
(
X3
↔
X49
)
)
)
↔
(
¬
(
X48
↔
(
X37
↔
(
(
¬
X3
)
↔
X50
)
)
)
)
)
↔
(
(
¬
X37
)
→
(
(
¬
X3
)
↔
X50
)
)
)
→
(
¬
(
X49
→
(
¬
X3
)
→
X50
)
)
)
↔
(
X50
→
(
X48
↔
X3
)
)
)
→
X49
→
X37
→
(
X20
↔
X50
)
)
)
→
(
(
X50
→
X1
→
X47
)
↔
(
(
(
X50
↔
(
(
¬
X7
)
→
X1
→
X47
)
)
→
X49
→
X47
→
X10
→
(
X7
↔
False
)
)
↔
(
X50
↔
(
(
¬
X1
)
↔
X10
)
)
)
)
→
(
¬
(
X50
→
X47
→
(
(
¬
X6
)
↔
X49
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
¬
X28
)
↔
(
X13
→
X47
)
)
)
↔
(
(
(
(
¬
X13
)
↔
(
X0
↔
X28
)
)
↔
(
X49
→
X28
→
X13
→
X38
)
)
↔
(
X49
→
(
X47
↔
(
X0
↔
False
)
)
)
)
)
↔
(
(
X22
→
(
¬
X13
)
→
X47
)
↔
(
¬
(
(
¬
(
X49
↔
(
(
¬
X41
)
→
X47
)
)
)
→
(
¬
(
(
¬
(
X47
→
X41
→
X13
)
)
→
X41
→
(
X25
↔
(
(
¬
X13
)
↔
X49
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X48
→
X38
→
X19
→
X25
)
)
↔
(
¬
(
X19
→
(
(
¬
X17
)
↔
X38
)
)
)
)
→
(
¬
(
(
(
X34
↔
(
X28
↔
X25
)
)
→
(
X34
↔
(
X25
↔
(
(
¬
X17
)
↔
X48
)
)
)
)
→
(
¬
(
(
¬
X28
)
↔
(
(
¬
X25
)
→
X34
)
)
)
)
)
)
↔
(
¬
(
X48
↔
(
X28
→
X24
→
X38
)
)
)
)
)
↔
(
(
(
(
X49
↔
(
X47
→
X45
)
)
→
(
(
(
(
¬
(
(
¬
X41
)
→
(
¬
X33
)
→
X47
)
)
→
(
¬
(
X47
→
X45
→
(
X33
↔
X49
)
)
)
)
↔
(
¬
(
X45
→
False
)
)
)
↔
(
¬
(
(
¬
X41
)
↔
X47
)
)
)
→
(
¬
(
X49
→
(
(
¬
X33
)
↔
(
X27
↔
X41
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
X33
)
→
(
(
¬
X30
)
↔
X45
)
)
↔
(
X45
↔
(
X31
↔
(
(
¬
X8
)
→
X33
)
)
)
)
→
(
¬
(
(
(
¬
X39
)
→
(
¬
X33
)
→
X45
)
→
X45
→
(
(
¬
X39
)
↔
(
X31
→
X47
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
X45
)
→
X41
→
(
X12
↔
X46
)
)
↔
(
¬
(
(
X45
→
(
X41
↔
(
X23
↔
X46
)
)
)
→
(
X41
↔
(
(
¬
X23
)
→
X21
→
X32
)
)
)
)
)
→
(
¬
(
X47
↔
(
X21
↔
(
(
¬
X12
)
↔
X45
)
)
)
)
)
↔
(
¬
(
(
¬
(
X47
↔
(
(
¬
X14
)
↔
(
(
¬
X2
)
↔
X45
)
)
)
)
→
(
¬
(
X47
→
X45
)
)
→
(
¬
(
X46
↔
(
X14
↔
(
X4
↔
X45
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
¬
(
X46
→
(
X30
↔
(
X1
→
False
)
)
)
)
→
(
(
(
X30
→
(
¬
X22
)
→
X0
→
X48
)
↔
(
¬
(
(
¬
X22
)
↔
(
X1
→
X48
)
)
)
)
↔
(
¬
(
X22
→
(
(
¬
X1
)
↔
X30
)
)
)
)
)
)
↔
(
(
¬
(
X34
→
(
(
¬
X27
)
↔
(
X0
→
X46
)
)
)
)
↔
(
¬
(
X27
→
X6
→
(
(
¬
X1
)
↔
X34
)
)
)
)
)
↔
(
(
¬
(
X47
↔
(
X46
↔
(
(
¬
X30
)
→
X48
)
)
)
)
→
(
¬
(
X48
→
(
X47
↔
(
X46
→
(
X43
↔
False
)
)
)
)
)
)
)
→
(
(
¬
(
X47
↔
(
X41
→
(
X40
↔
X46
)
)
)
)
↔
(
(
(
X41
→
(
(
¬
X13
)
↔
X46
)
)
↔
(
(
¬
(
X40
→
(
(
¬
X13
)
↔
(
(
¬
X12
)
↔
X46
)
)
)
)
→
X46
→
X41
→
X13
→
False
)
)
↔
(
X46
→
X13
→
X12
)
)
)
→
(
(
(
(
¬
X43
)
→
(
X40
↔
X49
)
)
↔
(
¬
(
(
(
(
X47
→
(
X46
↔
(
X3
↔
X40
)
)
)
→
(
¬
(
X49
↔
X40
)
)
)
↔
(
X49
↔
(
X43
↔
(
(
¬
X3
)
↔
X40
)
)
)
)
→
X49
→
X43
→
X5
→
X47
)
)
)
↔
(
¬
(
(
¬
(
(
X47
→
(
X36
↔
(
X35
→
False
)
)
)
→
(
(
¬
(
(
X47
→
(
(
¬
X46
)
↔
(
X2
↔
X49
)
)
)
→
(
¬
(
X35
↔
X49
)
)
)
)
↔
(
¬
(
X49
→
X47
→
X35
)
)
)
→
(
¬
(
X49
↔
(
X47
→
X35
)
)
)
)
)
→
(
¬
(
X46
↔
(
X35
→
(
¬
X20
)
→
X36
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
X47
→
(
¬
X15
)
→
X19
)
↔
(
¬
(
(
¬
(
X31
→
X46
)
)
→
X47
→
(
X28
↔
(
(
¬
X19
)
↔
X31
)
)
)
)
)
↔
(
X19
→
(
(
¬
X15
)
↔
X47
)
)
)
↔
(
(
X31
→
(
X28
↔
(
X26
→
X10
)
)
)
→
(
X32
↔
(
X31
↔
(
X19
↔
(
X10
↔
False
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
¬
(
(
¬
X38
)
↔
(
(
¬
X29
)
↔
(
X25
→
X44
)
)
)
)
↔
(
X44
→
X29
→
X25
→
X17
)
)
→
(
¬
(
(
X46
→
X44
→
(
X38
↔
False
)
)
→
(
(
(
¬
X38
)
→
(
(
¬
X25
)
↔
(
X9
↔
X46
)
)
)
↔
(
X44
↔
(
(
¬
X25
)
→
(
(
¬
X19
)
↔
X46
)
)
)
)
)
)
)
)
→
X50
→
(
X46
↔
(
X17
→
X44
)
)
)
)
→
(
X50
↔
(
X44
↔
(
(
¬
X28
)
→
X38
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
(
(
¬
X42
)
→
X37
→
(
X4
↔
X44
)
)
→
(
(
¬
(
(
¬
X42
)
→
(
¬
X37
)
→
(
X4
↔
X45
)
)
)
↔
(
¬
(
X16
→
(
X4
↔
X37
)
)
)
)
)
)
→
(
X45
↔
(
X42
→
(
(
¬
X16
)
↔
X33
)
)
)
)
→
(
¬
(
X46
↔
(
X45
→
X42
→
X44
)
)
)
)
→
(
(
¬
(
X39
→
X18
→
X7
)
)
↔
(
¬
(
(
¬
(
(
(
¬
X37
)
→
X21
→
(
¬
X7
)
→
X44
)
→
(
¬
X18
)
→
(
(
¬
X16
)
↔
(
X7
↔
X44
)
)
)
)
→
(
¬
(
X16
→
X37
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
X50
→
X45
→
(
¬
X24
)
→
X40
)
→
(
¬
(
X11
↔
X39
)
)
)
)
→
(
(
¬
(
X40
→
(
X14
↔
(
X11
→
X23
)
)
)
)
↔
(
¬
(
X39
→
(
¬
X23
)
→
(
¬
X14
)
→
X40
)
)
)
)
)
↔
(
(
X39
↔
(
(
¬
X24
)
↔
(
(
¬
X8
)
↔
X37
)
)
)
→
(
¬
(
(
¬
X37
)
↔
(
X8
→
X50
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
f82649...
and proposition id is
87b90c...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMW4AJScpozZyKj8BRMDsNrpQfP1ybLwivA
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
∀X50 :
prop
,
(
¬
∀X51 :
prop
,
(
¬
∀X52 :
prop
,
(
(
¬
(
(
(
(
(
(
(
¬
(
(
¬
X46
)
→
(
X44
↔
X49
)
)
)
→
(
(
(
X49
↔
(
X45
→
(
X41
↔
(
X15
↔
False
)
)
)
)
→
X46
→
X41
→
X15
→
False
)
↔
(
X49
↔
(
X15
↔
X45
)
)
)
→
(
(
¬
X46
)
↔
(
(
¬
X44
)
→
X41
→
X49
)
)
)
↔
(
X49
→
X45
→
X44
→
(
X27
↔
False
)
)
)
→
(
(
¬
X46
)
↔
(
(
¬
X41
)
↔
X49
)
)
)
↔
(
(
¬
(
(
(
¬
(
X49
↔
(
X33
↔
(
X19
↔
X12
)
)
)
)
↔
(
¬
(
(
¬
X41
)
↔
(
X33
→
(
X19
↔
X45
)
)
)
)
)
→
(
X45
↔
(
(
¬
X5
)
→
X41
)
)
→
(
X49
↔
(
X45
↔
X32
)
)
)
)
→
(
¬
(
X41
→
X33
→
X49
)
)
)
)
↔
(
(
¬
(
(
(
X33
↔
(
X24
↔
(
X20
→
X46
)
)
)
→
(
(
¬
X24
)
↔
(
X20
↔
(
X7
↔
X33
)
)
)
→
(
¬
(
X33
↔
(
X24
↔
(
X7
→
(
X1
↔
False
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
X48
)
↔
(
(
¬
X33
)
→
(
¬
X24
)
→
X52
)
)
→
(
¬
(
(
¬
X46
)
→
(
X24
↔
X52
)
)
)
)
↔
(
¬
(
(
¬
X17
)
↔
X40
)
)
)
→
(
¬
(
(
¬
X40
)
↔
(
X33
→
(
¬
X29
)
→
X46
)
)
)
)
)
)
)
↔
(
(
(
(
X51
→
(
(
¬
X37
)
↔
(
X33
↔
X44
)
)
)
↔
(
¬
(
(
¬
X46
)
→
(
(
¬
X44
)
↔
(
(
¬
X37
)
→
X51
)
)
)
)
)
↔
(
¬
(
X46
↔
(
X44
→
X37
)
)
)
)
→
(
(
X48
↔
(
(
¬
X44
)
↔
X51
)
)
↔
(
(
(
¬
(
X51
↔
(
X48
↔
(
X46
→
(
X22
↔
False
)
)
)
)
)
↔
(
X51
→
(
X46
↔
(
(
¬
X22
)
→
X44
)
)
)
)
↔
(
¬
(
(
¬
X46
)
↔
(
X22
↔
X51
)
)
)
)
)
)
)
)
→
(
(
(
(
(
(
¬
X28
)
↔
(
X26
→
(
(
¬
X11
)
↔
X47
)
)
)
↔
(
(
(
¬
(
(
¬
X46
)
→
(
(
¬
X28
)
↔
(
(
¬
X26
)
→
X47
)
)
)
)
↔
(
X42
→
(
¬
X39
)
→
X47
)
)
↔
(
X42
↔
(
(
¬
X39
)
→
X46
)
)
)
)
↔
(
(
(
(
¬
X23
)
→
X25
)
↔
(
X42
→
X34
→
(
X14
↔
X25
)
)
)
→
(
(
(
¬
(
X46
↔
(
X13
↔
(
X8
↔
False
)
)
)
)
→
(
X42
↔
(
X34
→
X8
→
X13
)
)
)
↔
(
¬
(
X46
↔
(
X42
→
X14
→
X13
→
False
)
)
)
)
→
(
¬
(
(
¬
X11
)
↔
(
X8
↔
X14
)
)
)
)
)
↔
(
(
¬
(
(
¬
X38
)
→
(
(
¬
X6
)
↔
(
X2
→
X42
)
)
)
)
↔
(
¬
(
X47
→
(
(
¬
X34
)
↔
(
(
¬
X2
)
→
X38
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
(
¬
(
X51
→
(
(
¬
X39
)
↔
(
X18
↔
X50
)
)
)
)
↔
(
X51
→
(
X47
↔
(
(
¬
X18
)
↔
X50
)
)
)
)
↔
(
X50
→
X18
)
)
↔
(
(
(
(
¬
(
(
¬
X39
)
↔
(
(
¬
X16
)
→
(
(
¬
X3
)
↔
X50
)
)
)
)
↔
(
X50
↔
(
X16
↔
(
X3
→
X46
)
)
)
)
↔
(
X16
↔
(
X3
→
(
X0
↔
X50
)
)
)
)
↔
(
¬
(
X46
→
(
¬
X9
)
→
X3
→
X36
)
)
)
)
→
(
¬
(
(
¬
X50
)
↔
(
(
¬
X47
)
↔
X51
)
)
)
)
)
→
(
(
(
(
(
X31
↔
(
X10
→
(
X4
↔
X52
)
)
)
↔
(
¬
(
X51
↔
(
X10
↔
False
)
)
)
)
↔
(
X51
↔
(
X31
→
False
)
)
)
↔
(
¬
(
X51
→
X50
→
X43
)
)
)
↔
(
X35
↔
(
(
¬
(
(
¬
X47
)
→
(
¬
X35
)
→
(
(
¬
X30
)
↔
X51
)
)
)
→
(
¬
(
(
¬
(
X52
↔
(
X30
→
X21
→
X47
)
)
)
→
(
¬
(
(
¬
X50
)
↔
(
X47
↔
(
X35
→
X52
)
)
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
X49
→
(
X39
↔
(
X3
↔
X48
)
)
)
→
(
(
¬
(
(
¬
(
X49
→
(
¬
X39
)
→
X51
)
)
→
(
¬
(
(
¬
X39
)
↔
(
(
¬
X5
)
→
(
(
¬
X3
)
↔
X49
)
)
)
)
)
)
↔
(
¬
(
X49
→
X5
→
X3
)
)
)
)
↔
(
(
¬
(
X14
↔
(
(
¬
X0
)
→
X39
)
)
)
→
(
(
¬
(
(
(
¬
X49
)
↔
(
(
¬
X14
)
→
X9
→
X51
)
)
→
(
X51
↔
(
X39
↔
(
X14
↔
X9
)
)
)
)
)
↔
(
¬
(
X51
↔
(
X49
↔
(
X39
→
X0
→
False
)
)
)
)
)
)
)
↔
(
(
X51
→
(
X39
↔
(
X1
→
False
)
)
)
→
X49
→
X39
→
X2
)
)
→
(
(
¬
(
X47
↔
(
X24
↔
(
X8
↔
X45
)
)
)
)
↔
(
¬
(
(
¬
(
X49
→
(
X47
↔
(
(
¬
X11
)
→
X45
)
)
)
)
→
(
(
¬
(
(
(
(
¬
X47
)
→
X45
→
(
(
¬
X10
)
↔
X51
)
)
→
(
¬
(
(
(
¬
X49
)
→
X45
→
(
¬
X10
)
→
X52
)
→
(
¬
(
X52
↔
(
X51
↔
(
X45
→
X10
)
)
)
)
)
)
)
→
(
X51
↔
(
X47
↔
(
(
¬
X10
)
→
X24
)
)
)
)
)
↔
(
X51
→
(
¬
X24
)
→
(
¬
X21
)
→
X47
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X50
)
→
X49
→
(
¬
X30
)
→
X52
)
)
↔
(
¬
(
(
¬
X49
)
→
(
X48
↔
X52
)
)
)
)
→
(
¬
(
(
X52
→
X40
→
(
(
¬
X31
)
↔
X50
)
)
→
X51
→
X49
→
(
X20
↔
False
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
¬
X44
)
→
X52
→
(
X46
↔
(
(
¬
X12
)
↔
X42
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X32
↔
(
(
¬
X23
)
↔
(
(
¬
X22
)
→
X52
)
)
)
)
→
(
¬
(
(
(
¬
X38
)
↔
(
X19
→
X42
)
)
→
(
(
¬
(
X42
↔
(
(
¬
X23
)
→
X19
→
X32
)
)
)
↔
(
(
¬
X32
)
↔
X38
)
)
)
)
)
→
(
¬
(
(
¬
(
X32
→
(
X23
↔
(
(
¬
X15
)
→
X44
)
)
)
)
→
(
X52
↔
(
X44
→
(
(
¬
X23
)
↔
X32
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X44
→
(
¬
X34
)
→
X38
)
)
↔
(
¬
(
X44
↔
(
(
¬
X34
)
↔
(
(
¬
X17
)
→
X52
)
)
)
)
)
↔
(
¬
(
X44
→
(
X38
↔
(
X17
→
X35
)
)
)
)
)
→
(
X52
↔
(
X44
↔
X35
)
)
)
↔
(
X50
↔
(
(
¬
X38
)
↔
(
(
¬
X35
)
↔
X44
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X51
↔
(
X44
→
X13
)
)
)
↔
(
X50
↔
(
X41
→
X13
→
X46
)
)
)
→
(
(
¬
X46
)
↔
(
(
¬
X41
)
→
(
¬
X28
)
→
X51
)
)
→
(
(
¬
(
(
¬
X46
)
↔
(
(
¬
X41
)
↔
(
X7
↔
X51
)
)
)
)
↔
(
¬
(
X43
↔
(
X27
→
X46
)
)
)
)
→
(
¬
(
X43
↔
(
(
¬
X41
)
↔
X51
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X41
→
(
X18
↔
(
X4
↔
False
)
)
)
)
→
(
(
¬
(
X44
→
(
(
¬
X41
)
↔
(
(
¬
X18
)
→
X52
)
)
)
)
↔
(
X52
↔
(
X41
↔
(
X36
↔
X18
)
)
)
)
)
↔
(
(
¬
(
X52
→
(
X41
↔
(
X37
↔
X6
)
)
)
)
↔
(
X41
→
X44
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
¬
X33
)
↔
(
X26
↔
(
(
¬
X16
)
↔
X52
)
)
)
)
↔
(
(
¬
X33
)
→
(
X26
↔
(
X16
→
X44
)
)
)
)
↔
(
X52
→
False
)
)
→
(
(
X29
→
(
¬
X25
)
→
(
(
¬
X16
)
↔
X52
)
)
↔
(
(
(
(
¬
X26
)
→
(
X16
↔
X52
)
)
↔
(
¬
(
X52
→
(
¬
X29
)
→
(
X26
↔
X41
)
)
)
)
↔
(
X29
→
X52
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
ac01bb...
and proposition id is
cf52db...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMSyMWmTN8HamzgUSTvz3PeRzXBvandS16Y
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
¬
(
(
(
(
¬
(
(
(
¬
X41
)
↔
(
X39
→
X17
→
X42
)
)
→
(
(
(
X49
→
(
(
¬
X41
)
↔
(
(
¬
X17
)
→
X42
)
)
)
↔
(
¬
(
X49
→
(
¬
X40
)
→
X41
)
)
)
↔
(
(
¬
X42
)
→
(
X14
↔
(
X6
→
X49
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X27
)
→
(
¬
X14
)
→
(
X11
↔
X39
)
)
)
→
(
¬
(
X14
→
(
X11
↔
(
(
¬
X2
)
↔
X39
)
)
)
)
)
→
(
¬
(
X39
→
(
(
¬
X27
)
↔
(
(
¬
X4
)
→
X41
)
)
)
)
)
)
)
→
(
(
(
(
(
(
X49
→
(
X46
↔
(
X26
↔
X3
)
)
)
↔
(
(
¬
(
(
(
¬
(
X46
→
(
X26
↔
(
X16
→
X49
)
)
)
)
→
(
X46
↔
(
(
¬
X26
)
↔
(
X25
↔
X49
)
)
)
)
→
(
¬
X16
)
→
X49
)
)
↔
(
¬
(
(
¬
X46
)
↔
(
X25
→
X49
)
)
)
)
)
↔
(
¬
(
(
¬
X26
)
→
(
X16
↔
X46
)
)
)
)
↔
(
(
¬
(
X41
↔
(
X25
→
(
X20
↔
False
)
)
)
)
↔
(
X49
↔
(
X46
→
(
(
¬
X20
)
↔
X41
)
)
)
)
)
↔
(
(
(
¬
X26
)
↔
(
(
¬
X25
)
↔
X46
)
)
↔
(
¬
(
(
¬
X34
)
→
(
X26
↔
(
X24
↔
X46
)
)
)
)
)
)
↔
(
(
(
¬
(
X48
→
(
X47
↔
(
X12
↔
X45
)
)
)
)
→
(
¬
(
(
¬
X47
)
→
X40
→
X12
→
X48
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
X47
→
X1
→
X36
)
)
↔
(
(
(
¬
X46
)
↔
(
(
¬
X30
)
↔
(
(
¬
X1
)
↔
X47
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X30
)
↔
(
(
¬
X1
)
↔
X48
)
)
)
)
)
)
→
(
¬
(
X48
→
X46
→
False
)
)
)
)
↔
(
(
¬
X32
)
→
(
X1
↔
X45
)
)
)
→
(
¬
(
(
(
¬
(
X48
→
(
(
¬
X24
)
↔
(
X19
→
X45
)
)
)
)
→
(
¬
X30
)
→
(
¬
X24
)
→
X19
→
X48
)
→
(
(
¬
X30
)
↔
(
(
¬
X20
)
↔
(
(
¬
X19
)
→
X48
)
)
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
X46
↔
(
X10
→
X29
)
)
)
↔
(
¬
(
(
¬
X43
)
↔
(
(
¬
X38
)
→
(
X29
↔
X46
)
)
)
)
)
↔
(
¬
(
(
¬
(
X44
↔
(
X31
→
(
(
¬
X22
)
↔
X43
)
)
)
)
→
(
X46
↔
(
X31
↔
X38
)
)
)
)
)
→
X45
→
X33
→
X10
)
↔
(
(
(
¬
(
X45
↔
(
(
¬
X31
)
→
(
¬
X23
)
→
X33
)
)
)
→
(
¬
(
X45
→
(
(
¬
X31
)
↔
X48
)
)
)
)
↔
(
(
¬
(
X18
↔
(
X13
↔
X33
)
)
)
→
(
X18
↔
(
X13
↔
X48
)
)
→
(
¬
(
X31
→
(
¬
X18
)
→
(
¬
X8
)
→
X48
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X43
→
(
¬
X37
)
→
X35
→
X48
)
)
↔
(
¬
(
X48
↔
(
X46
↔
(
X35
↔
False
)
)
)
)
)
→
(
(
¬
(
(
X0
↔
False
)
→
(
¬
(
X43
→
(
(
¬
X9
)
↔
(
(
¬
X0
)
↔
X37
)
)
)
)
→
(
¬
(
X35
↔
(
(
¬
X21
)
→
X0
→
X43
)
)
)
)
)
↔
(
X35
↔
(
X28
→
X9
→
False
)
)
)
)
)
↔
(
(
(
¬
(
X48
→
X46
→
(
X7
↔
(
X5
→
False
)
)
)
)
↔
(
(
¬
(
X48
↔
(
(
¬
X15
)
↔
X37
)
)
)
↔
(
X48
→
(
X46
↔
(
X5
↔
X15
)
)
)
)
)
→
(
¬
(
X7
↔
(
(
¬
X5
)
↔
X15
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
(
(
¬
(
X47
↔
(
X32
→
(
X3
↔
False
)
)
)
)
↔
(
X36
↔
(
(
¬
X31
)
↔
(
(
¬
X3
)
↔
X47
)
)
)
)
→
(
(
¬
(
X47
→
X44
→
X36
→
False
)
)
↔
(
(
X44
↔
(
X19
↔
X47
)
)
↔
(
¬
(
X44
→
(
(
¬
X32
)
↔
X36
)
)
)
)
)
)
)
↔
(
(
(
X49
→
(
¬
X32
)
→
(
X28
↔
X47
)
)
↔
(
X47
→
X31
→
(
¬
X28
)
→
X49
)
)
→
(
X31
↔
(
X27
↔
(
X16
↔
X32
)
)
)
)
)
→
(
(
(
¬
(
X49
↔
(
X48
→
(
X46
↔
X47
)
)
)
)
↔
(
X49
→
(
(
¬
X38
)
↔
X48
)
)
)
→
(
¬
(
(
(
(
¬
(
X47
↔
(
(
¬
X43
)
↔
(
(
¬
X11
)
→
X48
)
)
)
)
→
(
¬
(
(
(
¬
(
X43
↔
(
(
¬
X2
)
↔
X47
)
)
)
↔
(
X43
→
(
(
¬
X12
)
↔
(
X2
→
X48
)
)
)
)
→
X43
→
(
¬
X12
)
→
X48
)
)
)
→
(
X47
↔
(
(
¬
X12
)
→
X48
)
)
)
→
(
¬
(
(
¬
X46
)
↔
(
(
¬
X12
)
→
(
X11
↔
X48
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X38
)
→
(
(
¬
X12
)
↔
X40
)
)
)
→
(
¬
(
X44
→
(
X39
↔
(
X12
→
X40
)
)
)
)
)
→
X48
→
(
X44
↔
(
(
¬
X38
)
↔
X40
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
X22
→
(
X15
↔
X47
)
)
↔
(
(
X45
↔
(
X39
↔
(
X24
↔
X47
)
)
)
↔
(
X47
→
(
X45
↔
X22
)
)
)
)
↔
(
(
¬
X24
)
↔
(
(
¬
X15
)
→
X47
)
)
)
→
(
(
(
¬
X26
)
↔
(
X24
→
(
X22
↔
X45
)
)
)
↔
(
X45
→
X35
→
(
¬
X22
)
→
X24
)
)
)
)
→
(
¬
(
(
(
X37
→
X33
→
(
X7
↔
False
)
)
↔
(
(
¬
(
X33
→
(
(
¬
X7
)
↔
X37
)
)
)
↔
(
¬
(
(
¬
(
X33
↔
(
(
¬
X15
)
→
X7
→
X45
)
)
)
→
(
X37
↔
(
X33
↔
X7
)
)
)
)
)
)
→
(
¬
(
X45
→
X35
→
X33
→
X37
)
)
→
(
¬
(
X45
→
X38
→
X33
→
X37
)
)
)
)
)
→
(
(
(
¬
(
X38
↔
(
X21
↔
(
X14
↔
X34
)
)
)
)
→
(
¬
(
X34
↔
(
X14
→
(
¬
X10
)
→
X25
)
)
)
)
↔
(
(
(
¬
X39
)
↔
(
(
¬
X34
)
↔
(
X1
→
X47
)
)
)
↔
(
¬
(
X39
↔
(
(
¬
X14
)
→
X34
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
X41
↔
(
X20
→
X29
)
)
↔
(
X41
↔
(
(
¬
X29
)
→
(
(
¬
X6
)
↔
X35
)
)
)
)
→
(
¬
(
(
(
¬
X35
)
→
X41
)
→
(
(
(
(
(
X37
↔
(
(
¬
X29
)
→
(
¬
X4
)
→
X35
)
)
↔
(
X41
→
(
¬
X29
)
→
(
(
¬
X18
)
↔
X37
)
)
)
↔
(
¬
(
X37
→
(
X35
↔
X41
)
)
)
)
↔
(
X41
→
(
¬
X29
)
→
X35
)
)
↔
(
(
¬
(
X37
↔
(
(
¬
X9
)
↔
(
(
¬
X4
)
↔
X17
)
)
)
)
↔
(
¬
(
(
¬
X9
)
↔
X18
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
(
¬
X23
)
→
(
(
¬
X13
)
↔
(
X0
→
X35
)
)
)
↔
(
¬
(
X42
↔
(
(
¬
X13
)
↔
(
(
¬
X0
)
↔
X23
)
)
)
)
)
→
(
¬
X0
)
→
X23
)
)
→
(
(
(
¬
(
(
¬
X35
)
↔
(
(
¬
X30
)
↔
(
(
¬
X23
)
→
X42
)
)
)
)
↔
(
¬
(
X42
→
(
X35
↔
(
X30
→
X8
)
)
)
)
)
↔
(
(
(
X39
↔
(
X30
→
False
)
)
↔
(
(
¬
X35
)
→
(
(
¬
X20
)
↔
(
X5
→
X39
)
)
)
)
→
(
(
¬
(
(
¬
X30
)
↔
(
(
¬
X23
)
→
(
¬
X5
)
→
X35
)
)
)
↔
(
X30
→
X29
→
(
X5
↔
False
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
b69d6b...
and proposition id is
e74f17...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMJJTeSzCrSn258SXSkeek12cAyubizQhPc
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
¬
∀X51 :
prop
,
(
¬
∀X52 :
prop
,
∀X53 :
prop
,
(
(
¬
(
(
(
(
(
(
¬
(
(
¬
(
(
¬
(
X53
↔
(
X43
→
X47
)
)
)
→
(
(
X43
→
(
(
¬
X35
)
↔
(
(
¬
X24
)
→
X53
)
)
)
↔
(
(
¬
X43
)
→
X35
→
X47
)
)
)
)
→
(
¬
(
X53
↔
(
(
¬
X24
)
→
X47
)
)
)
)
)
↔
(
(
(
X53
→
(
(
¬
X3
)
↔
X43
)
)
↔
(
(
X53
↔
(
X50
↔
(
X3
→
False
)
)
)
↔
(
X43
↔
(
(
¬
X8
)
↔
(
(
¬
X3
)
↔
X53
)
)
)
)
)
↔
(
(
(
X53
→
(
X43
↔
(
X42
→
X50
)
)
)
↔
(
¬
(
X50
↔
(
X47
↔
(
X43
→
False
)
)
)
)
)
→
X53
→
X47
)
)
)
↔
(
¬
(
(
¬
(
(
X53
→
(
X32
↔
(
X23
→
X42
)
)
)
→
(
(
¬
(
X53
↔
(
(
¬
X32
)
→
X1
→
X38
)
)
)
↔
(
(
¬
X32
)
→
(
X14
↔
(
(
¬
X1
)
→
X38
)
)
)
)
→
(
(
(
¬
X32
)
→
(
¬
X23
)
→
(
(
¬
X6
)
↔
X38
)
)
↔
(
X23
→
(
¬
X14
)
→
(
¬
X6
)
→
X38
)
)
)
)
→
(
(
¬
(
X43
↔
X53
)
)
↔
(
(
X53
↔
(
X43
→
X38
→
X5
)
)
↔
(
X43
↔
(
(
¬
X38
)
→
(
X32
↔
X53
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
X50
)
↔
(
X47
→
(
(
¬
X12
)
↔
X53
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X33
)
↔
X50
)
)
)
→
(
(
¬
(
X47
↔
X44
)
)
↔
(
(
(
X47
→
X34
→
X22
→
X33
)
↔
(
X50
↔
(
(
¬
X33
)
↔
(
X22
↔
X47
)
)
)
)
→
(
¬
(
X50
↔
(
X34
↔
X47
)
)
)
)
)
)
)
)
→
(
(
¬
(
X45
→
X40
→
(
(
¬
X13
)
↔
X18
)
)
)
↔
(
¬
(
(
(
¬
(
X53
↔
(
X50
→
X29
)
)
)
↔
(
X40
↔
(
X29
→
(
X18
↔
False
)
)
)
)
→
(
¬
(
(
(
¬
(
(
X48
↔
X13
)
→
(
(
(
X47
→
X13
)
↔
(
¬
(
(
¬
(
(
X28
→
(
(
¬
X13
)
↔
X48
)
)
→
(
¬
(
(
X47
→
(
(
¬
X28
)
↔
(
X13
↔
X40
)
)
)
→
(
¬
(
(
¬
X47
)
→
X40
→
(
X28
↔
X48
)
)
)
)
)
)
)
→
X48
→
X47
→
X28
)
)
)
↔
(
¬
(
(
¬
X28
)
↔
X40
)
)
)
)
)
↔
(
X47
↔
(
(
¬
X13
)
→
X50
)
)
)
→
(
¬
(
X53
→
X48
→
(
X40
↔
X28
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
(
(
¬
(
X50
→
(
X48
↔
(
X36
→
False
)
)
)
)
→
(
X50
↔
(
(
¬
X27
)
↔
(
X7
→
X36
)
)
)
)
)
↔
(
(
¬
(
X48
→
X47
→
X36
)
)
↔
(
X48
↔
(
(
¬
X47
)
→
X30
→
X50
)
)
)
)
↔
(
¬
(
(
X36
→
X31
→
(
X26
↔
X50
)
)
→
(
(
¬
(
(
¬
(
(
¬
X36
)
→
(
¬
X31
)
→
(
X15
↔
X50
)
)
)
→
X48
→
(
¬
X36
)
→
(
X15
↔
X50
)
)
)
↔
(
X48
→
X31
→
X21
→
X36
)
)
)
)
)
→
(
¬
(
(
¬
(
X50
↔
(
(
¬
X36
)
↔
(
X11
→
X52
)
)
)
)
→
(
(
X50
↔
(
(
¬
X48
)
↔
(
(
¬
X2
)
→
X52
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X48
)
↔
(
X11
→
X52
)
)
)
→
(
¬
(
X48
→
(
X0
↔
X52
)
)
)
→
(
¬
(
X11
→
(
(
¬
X0
)
↔
X52
)
)
)
→
(
¬
(
(
¬
X50
)
→
(
X48
↔
(
(
¬
X11
)
→
X52
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X52
↔
(
X50
↔
(
(
¬
X19
)
↔
X51
)
)
)
)
↔
(
(
¬
(
X52
→
(
X51
↔
X25
)
)
)
→
(
¬
(
(
(
X52
→
X51
→
(
X39
↔
False
)
)
↔
(
(
X51
↔
(
X50
→
(
¬
X39
)
→
X52
)
)
→
X52
→
X51
→
(
(
¬
X37
)
↔
X50
)
)
)
→
(
¬
(
X51
↔
(
(
¬
X39
)
↔
(
(
¬
X25
)
→
X52
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
¬
X51
)
→
(
(
¬
X41
)
↔
(
X20
↔
X52
)
)
)
)
→
(
¬
(
(
¬
(
X52
→
(
(
¬
X41
)
↔
(
X39
→
X51
)
)
)
)
→
(
¬
X51
)
→
(
(
¬
X41
)
↔
(
(
¬
X16
)
→
X52
)
)
)
)
)
)
→
(
¬
(
X52
↔
(
(
¬
X20
)
↔
(
X9
↔
X51
)
)
)
)
)
)
)
→
(
¬
(
(
X51
↔
(
X50
→
X17
→
X52
)
)
→
(
(
(
X46
→
(
X10
↔
X49
)
)
↔
(
(
(
(
¬
X49
)
→
(
¬
X46
)
→
(
¬
X10
)
→
X50
)
→
(
¬
(
X49
→
(
¬
X46
)
→
(
X4
↔
X50
)
)
)
)
↔
(
¬
(
X50
→
X49
→
X10
→
X47
)
)
)
)
↔
(
¬
(
X51
→
X47
→
(
(
¬
X4
)
↔
X52
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
(
X52
→
X50
→
(
X47
↔
(
X20
↔
False
)
)
)
↔
(
(
¬
X50
)
↔
(
(
¬
X47
)
↔
(
X30
↔
X52
)
)
)
)
↔
(
(
¬
(
(
¬
X50
)
→
X35
→
(
(
¬
X30
)
↔
X52
)
)
)
→
(
¬
(
(
¬
X18
)
→
X10
→
X35
)
)
→
(
¬
(
(
X52
↔
(
X30
→
(
X18
↔
(
X4
↔
False
)
)
)
)
→
(
¬
(
X10
→
X30
)
)
)
)
)
)
→
(
(
X47
→
X42
→
X28
)
→
(
(
¬
X28
)
↔
(
(
¬
X0
)
↔
X52
)
)
)
→
(
¬
(
(
(
(
¬
X44
)
↔
(
X42
↔
(
(
¬
X5
)
↔
X50
)
)
)
→
(
(
X42
↔
X5
)
↔
(
X50
→
(
X42
↔
(
X11
↔
X26
)
)
)
)
)
→
(
¬
(
(
X42
↔
False
)
→
(
(
(
¬
X50
)
↔
(
X47
→
X42
→
X52
)
)
↔
(
¬
(
X50
→
X47
→
X26
→
X42
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
X52
↔
(
X50
↔
(
X43
↔
X47
)
)
)
↔
(
¬
(
(
¬
X47
)
↔
(
X43
↔
(
(
¬
X6
)
→
X52
)
)
)
)
)
↔
(
¬
(
X52
→
X47
)
)
)
↔
(
¬
(
(
¬
(
(
X51
→
(
¬
X43
)
→
(
(
¬
X9
)
↔
X52
)
)
→
X51
→
(
¬
X47
)
→
X43
→
X52
)
)
→
X51
→
(
¬
X43
)
→
X52
)
)
)
↔
(
(
¬
(
X46
↔
X50
)
)
↔
(
(
X46
→
(
¬
X43
)
→
X33
→
X50
)
→
(
¬
X50
)
→
(
(
¬
X43
)
↔
(
X33
→
X52
)
)
)
)
)
)
↔
(
(
(
(
(
X22
→
(
(
¬
X8
)
↔
X45
)
)
↔
(
¬
(
X50
→
(
(
¬
X7
)
↔
X8
)
)
)
)
↔
(
(
¬
X45
)
→
(
(
¬
X22
)
↔
(
X8
→
X52
)
)
)
)
→
(
¬
(
(
¬
(
X49
↔
(
X45
→
X22
)
)
)
→
X50
→
(
X49
↔
(
X12
→
False
)
)
)
)
)
↔
(
(
¬
(
(
X51
↔
X49
)
→
(
X52
↔
(
X49
→
(
(
¬
X21
)
↔
X31
)
)
)
)
)
↔
(
(
X49
↔
(
X45
→
X39
→
False
)
)
↔
(
(
¬
X51
)
↔
(
(
¬
X39
)
→
X52
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
¬
X25
)
↔
(
X17
↔
X40
)
)
)
→
(
(
X25
→
X24
→
(
X17
↔
X34
)
)
↔
(
X40
↔
(
X24
→
X25
)
)
)
)
→
(
(
¬
(
(
¬
X25
)
→
X16
→
X29
)
)
↔
(
(
X53
↔
X47
)
→
(
X53
↔
(
X40
→
X29
→
X25
→
False
)
)
)
)
→
(
(
X47
→
(
(
¬
X25
)
↔
X40
)
)
↔
(
(
X40
→
X47
)
↔
(
X40
→
X25
→
X17
→
X19
)
)
)
)
→
(
(
(
(
¬
(
X49
↔
(
X41
→
(
X14
↔
X1
)
)
)
)
↔
(
(
¬
X49
)
↔
(
X13
↔
(
X1
→
X52
)
)
)
)
↔
(
(
(
X52
→
X23
→
(
X13
↔
False
)
)
↔
(
¬
(
X48
↔
(
(
¬
X14
)
→
(
¬
X13
)
→
X40
)
)
)
)
→
(
¬
X48
)
→
(
¬
X13
)
→
X52
)
)
↔
(
(
(
¬
(
(
(
(
¬
(
X53
↔
(
X52
→
(
X49
↔
X15
)
)
)
)
↔
(
X52
→
X49
→
X3
→
X53
)
)
↔
(
(
¬
X52
)
→
(
¬
X49
)
→
X53
)
)
→
(
¬
(
X47
→
(
¬
X15
)
→
(
¬
X3
)
→
X49
)
)
)
)
→
(
(
(
¬
(
X53
→
(
¬
X36
)
→
(
(
¬
X27
)
↔
X38
)
)
)
→
(
(
X53
→
X38
→
X49
)
↔
(
(
¬
(
X53
↔
(
X49
→
(
¬
X36
)
→
X38
)
)
)
↔
(
¬
(
(
¬
X49
)
↔
(
(
¬
X38
)
↔
(
(
¬
X2
)
→
X53
)
)
)
)
)
)
)
↔
(
X53
↔
(
X48
↔
(
X47
→
X2
)
)
)
)
)
↔
(
(
¬
(
X52
↔
(
(
¬
X37
)
→
X47
)
)
)
→
(
¬
(
(
X48
↔
(
(
¬
X47
)
→
(
(
¬
X37
)
↔
X52
)
)
)
→
(
¬
(
X32
→
X37
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
b63446...
and proposition id is
5f25dc...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMNnRTmPCNCPFHGPEsKRn6Y8LjXcJguxMPU
)
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
(
¬
(
(
¬
(
(
¬
X36
)
→
(
(
¬
X34
)
↔
(
X14
→
X45
)
)
)
)
→
(
¬
(
(
¬
X45
)
↔
X49
)
)
→
(
¬
(
X48
↔
(
X8
→
X36
)
)
)
)
)
↔
(
(
¬
X36
)
↔
(
(
¬
X34
)
↔
(
X24
↔
X48
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
(
¬
X46
)
→
(
(
¬
X36
)
↔
(
(
¬
X6
)
↔
X49
)
)
)
)
→
(
X39
↔
X36
)
)
)
→
(
¬
(
(
¬
(
X34
→
(
X16
↔
(
X10
↔
X46
)
)
)
)
→
(
(
¬
X34
)
↔
(
X10
→
X6
→
X36
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X39
↔
(
X36
→
(
(
¬
X20
)
↔
X34
)
)
)
)
→
(
¬
(
(
¬
X34
)
→
X20
→
X7
→
X39
)
)
)
→
X36
→
(
(
¬
X34
)
↔
(
(
¬
X5
)
→
X39
)
)
)
)
)
)
→
(
(
(
(
¬
(
X1
↔
X49
)
)
↔
(
¬
(
(
(
¬
(
X43
→
X30
→
(
(
¬
X1
)
↔
X21
)
)
)
→
(
¬
(
X44
↔
(
X30
→
X21
→
(
X1
↔
False
)
)
)
)
)
→
(
(
¬
X44
)
↔
(
(
¬
X1
)
↔
X49
)
)
)
)
)
↔
(
(
¬
(
X48
↔
(
X45
↔
(
(
¬
X43
)
→
X44
)
)
)
)
→
(
X48
↔
(
X44
→
(
X30
↔
X43
)
)
)
)
)
↔
(
(
(
X43
↔
(
(
¬
X23
)
→
X11
→
X31
)
)
↔
(
X30
↔
X31
)
)
↔
(
(
X45
→
X23
→
X28
)
↔
(
¬
(
(
¬
(
(
X45
↔
(
X18
↔
X23
)
)
→
X31
→
(
(
¬
X0
)
↔
X23
)
)
)
→
(
¬
(
X28
→
(
¬
X23
)
→
(
¬
X11
)
→
X31
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
(
(
¬
(
X47
↔
(
X46
↔
(
X22
↔
X42
)
)
)
)
→
(
¬
(
X33
↔
False
)
)
→
(
(
¬
X46
)
↔
(
(
¬
X22
)
↔
(
X12
↔
X47
)
)
)
)
)
→
(
¬
(
X22
→
(
¬
X12
)
→
X33
)
)
)
)
→
(
(
(
¬
X19
)
→
X40
)
↔
(
(
(
(
X46
→
X42
→
(
(
¬
X32
)
↔
X47
)
)
↔
(
¬
(
(
¬
X46
)
↔
(
(
¬
X42
)
→
(
X3
↔
X47
)
)
)
)
)
↔
(
X42
→
(
¬
X32
)
→
X46
)
)
↔
(
¬
(
X42
→
X3
→
False
)
)
)
)
)
↔
(
¬
(
(
X33
↔
(
(
¬
X32
)
→
(
X13
↔
X47
)
)
)
→
(
¬
(
X27
→
X32
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
¬
X46
)
→
(
X45
↔
(
(
¬
X9
)
→
X49
)
)
)
→
(
¬
(
X49
↔
(
X41
→
X9
)
)
)
)
→
(
(
(
(
X46
→
(
X45
↔
(
X9
→
False
)
)
)
↔
(
(
X46
→
(
¬
X45
)
→
(
X15
↔
X47
)
)
↔
(
¬
(
X47
→
(
X15
↔
(
(
¬
X9
)
→
X46
)
)
)
)
)
)
↔
(
¬
(
(
¬
X15
)
→
X47
)
)
)
↔
(
X47
↔
(
(
¬
X15
)
↔
X45
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
(
¬
(
X29
↔
(
(
¬
X4
)
→
X37
)
)
)
→
(
¬
(
X49
→
X38
→
X4
)
)
)
↔
(
X38
→
(
¬
X29
)
→
X37
)
)
→
(
X42
↔
(
X37
↔
(
X29
→
X4
)
)
)
)
)
→
(
(
(
¬
(
X49
→
(
X38
↔
(
X2
↔
False
)
)
)
)
↔
(
(
¬
(
X35
→
(
(
¬
X26
)
↔
(
X17
↔
X49
)
)
)
)
↔
(
X26
↔
(
(
¬
X2
)
→
X35
)
)
)
)
↔
(
¬
(
X17
→
(
¬
X2
)
→
X35
)
)
)
→
(
¬
(
(
¬
(
(
¬
X46
)
↔
(
(
¬
X42
)
↔
(
X38
→
X47
)
)
)
)
→
(
¬
(
(
¬
(
X46
↔
(
(
¬
X38
)
↔
X47
)
)
)
→
X47
→
(
X35
↔
(
X25
→
X46
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
¬
(
(
(
X43
→
(
X34
↔
(
X7
→
X44
)
)
)
↔
(
¬
(
X44
→
(
X43
↔
(
X34
↔
X2
)
)
)
)
)
→
(
¬
(
X2
→
X34
)
)
)
)
→
X47
→
(
¬
X34
)
→
X7
→
X43
)
)
→
(
(
X12
↔
(
(
¬
X7
)
↔
(
(
¬
X2
)
→
X31
)
)
)
↔
(
X31
↔
(
(
¬
X12
)
↔
(
(
¬
X2
)
→
X34
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X43
↔
(
X40
→
(
X23
↔
X44
)
)
)
)
→
(
¬
(
X44
↔
(
(
¬
X40
)
↔
X47
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
(
¬
(
X47
→
X32
→
X22
→
X45
)
)
→
(
¬
(
(
¬
X40
)
→
(
X32
↔
X47
)
)
)
)
)
→
(
X45
↔
(
(
¬
X32
)
→
X47
)
)
)
)
↔
(
(
X44
↔
(
X40
→
X47
)
)
↔
(
(
(
¬
X44
)
↔
X47
)
→
(
¬
(
X40
↔
(
(
¬
X19
)
↔
X45
)
)
)
)
)
)
→
(
X47
↔
(
X44
→
(
X38
↔
X45
)
)
)
)
)
)
→
(
¬
(
(
(
X45
↔
(
X35
↔
(
X30
↔
X15
)
)
)
→
X45
→
X44
→
(
¬
X15
)
→
X30
)
→
(
(
(
(
X46
↔
(
(
¬
X43
)
↔
(
X35
↔
X44
)
)
)
→
X3
→
X43
)
↔
(
(
X44
↔
(
X35
↔
X21
)
)
↔
(
X44
↔
(
(
¬
X43
)
↔
X45
)
)
)
)
↔
(
¬
(
(
¬
X44
)
↔
(
(
¬
X21
)
↔
(
X5
↔
X46
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X28
↔
(
(
¬
X8
)
↔
(
(
¬
X1
)
→
X47
)
)
)
)
→
X28
→
(
X24
↔
(
(
¬
X8
)
→
X44
)
)
)
→
(
¬
(
(
(
¬
(
X45
↔
(
X28
↔
False
)
)
)
↔
(
(
X45
→
X10
→
X47
)
→
(
(
¬
(
(
¬
X45
)
→
(
¬
X44
)
→
(
¬
X28
)
→
X49
)
)
↔
(
(
X47
→
X49
)
→
(
X49
↔
(
(
¬
X44
)
↔
(
(
¬
X10
)
↔
X45
)
)
)
)
)
)
)
→
(
(
¬
(
X44
↔
(
(
¬
X10
)
↔
(
(
¬
X0
)
↔
X39
)
)
)
)
↔
(
X39
↔
(
(
¬
X10
)
→
X49
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
X45
→
X6
→
False
)
)
→
(
¬
(
(
X6
→
X47
)
→
(
¬
(
(
¬
X45
)
↔
(
(
¬
X37
)
→
(
(
¬
X16
)
↔
X47
)
)
)
)
→
(
X47
↔
(
X6
↔
X37
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
X47
↔
(
(
¬
X33
)
↔
(
(
¬
X29
)
→
X37
)
)
)
)
→
(
X47
↔
(
X45
↔
(
X29
→
False
)
)
)
)
→
(
X45
↔
(
(
¬
X29
)
→
X37
)
)
)
)
→
(
¬
(
X45
↔
(
(
¬
X29
)
→
X47
)
)
)
)
)
)
↔
(
(
¬
(
X48
→
(
X38
↔
(
(
¬
X29
)
→
X37
)
)
)
)
↔
(
(
(
(
X37
→
X26
→
(
X14
↔
X18
)
)
→
(
X36
↔
(
(
¬
X14
)
→
(
(
¬
X13
)
↔
X26
)
)
)
)
↔
(
¬
(
X14
↔
X26
)
)
)
↔
(
(
X37
→
(
(
¬
X36
)
↔
(
(
¬
X33
)
↔
X48
)
)
)
↔
(
(
¬
(
(
X36
→
(
X18
↔
X33
)
)
→
(
(
¬
(
X38
→
X33
→
(
¬
X18
)
→
X36
)
)
↔
(
X48
↔
(
(
¬
X36
)
→
(
X33
↔
X38
)
)
)
)
)
)
↔
(
¬
(
(
¬
X36
)
→
(
(
¬
X33
)
↔
X48
)
)
)
)
)
)
)
)
→
(
(
(
(
X27
↔
(
X20
→
False
)
)
↔
(
¬
(
(
(
X47
→
X27
→
(
X20
↔
X45
)
)
↔
(
¬
(
(
¬
X27
)
↔
(
X20
↔
X42
)
)
)
)
→
(
¬
(
X42
→
(
¬
X27
)
→
X47
)
)
)
)
)
↔
(
¬
(
(
(
¬
X17
)
↔
X45
)
→
(
¬
X42
)
→
(
(
¬
X11
)
↔
(
(
¬
X9
)
→
X48
)
)
)
)
)
↔
(
(
(
(
¬
X45
)
↔
(
X42
↔
(
(
¬
X41
)
↔
X47
)
)
)
↔
(
¬
(
X47
→
X41
→
(
(
¬
X25
)
↔
X42
)
)
)
)
→
(
(
¬
X45
)
↔
(
(
¬
X42
)
↔
(
X4
↔
X47
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
3b68c6...
and proposition id is
a6b02f...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMUTH5QKVrDoB3kKm49BDku4b75RgGw6QeS
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
(
¬
(
(
¬
(
X37
↔
(
X30
↔
(
(
¬
X17
)
→
X48
)
)
)
)
→
(
(
(
¬
(
(
(
¬
X32
)
→
X27
→
(
¬
X17
)
→
X37
)
→
(
¬
(
(
¬
X30
)
→
X27
→
(
(
¬
X17
)
↔
X32
)
)
)
)
)
↔
(
X30
→
X27
→
X32
)
)
↔
(
¬
(
X37
→
(
X27
↔
(
(
¬
X8
)
→
X17
)
)
)
)
)
)
)
→
(
(
X47
↔
(
X30
→
X17
→
X32
)
)
↔
(
(
¬
X17
)
↔
X45
)
)
)
→
(
(
(
(
X26
↔
(
X5
→
X48
)
)
↔
(
X19
↔
(
X18
→
(
(
¬
X5
)
↔
X26
)
)
)
)
↔
(
(
¬
(
X48
→
(
(
¬
X26
)
↔
(
(
¬
X19
)
→
X41
)
)
)
)
→
(
¬
(
(
¬
X28
)
→
(
(
¬
X26
)
↔
X42
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X43
→
(
X28
↔
X45
)
)
)
↔
(
(
¬
X28
)
→
(
¬
X15
)
→
X14
→
X42
)
)
→
(
¬
(
(
X26
→
(
(
¬
X22
)
↔
X28
)
)
→
(
(
X45
→
(
X28
↔
X22
)
)
↔
(
(
¬
(
X28
→
X26
→
(
¬
X6
)
→
X45
)
)
↔
(
¬
(
(
¬
X26
)
↔
(
X22
↔
X45
)
)
)
)
)
→
(
(
¬
(
(
¬
X45
)
→
(
X42
↔
(
X26
→
X47
)
)
)
)
↔
(
¬
(
X47
↔
(
X42
↔
(
X6
→
X26
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
X36
↔
(
(
¬
X25
)
→
(
(
¬
X13
)
↔
X46
)
)
)
→
(
¬
(
(
¬
X36
)
→
(
¬
X29
)
→
X44
)
)
)
)
↔
(
(
(
(
¬
X34
)
↔
(
X25
→
X7
→
X46
)
)
↔
(
¬
(
X46
↔
(
X7
→
X48
)
)
)
)
↔
(
X46
↔
(
X29
↔
(
(
¬
X7
)
→
X25
)
)
)
)
)
→
(
(
X45
↔
(
(
¬
X0
)
↔
X48
)
)
→
(
¬
X45
)
→
(
(
¬
X44
)
↔
(
X31
→
X48
)
)
)
→
(
(
(
¬
(
(
¬
X44
)
↔
(
X20
↔
X46
)
)
)
↔
(
¬
(
(
X48
↔
(
(
¬
X44
)
→
(
X38
↔
X46
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X38
)
↔
(
(
¬
X33
)
→
X48
)
)
)
)
)
)
)
↔
(
¬
(
(
X43
↔
(
X33
→
X44
)
)
→
(
X45
↔
(
X20
↔
False
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
¬
X45
)
↔
(
X24
→
(
X23
↔
X48
)
)
)
↔
(
(
X45
→
X24
→
X23
)
→
(
(
X45
→
(
X10
↔
X23
)
)
↔
(
¬
(
X24
→
(
¬
X12
)
→
X10
→
X23
)
)
)
→
(
X24
↔
(
X12
→
X45
)
)
)
)
→
(
(
X49
↔
(
X48
→
(
(
¬
X2
)
↔
X24
)
)
)
↔
(
(
¬
X24
)
↔
(
(
¬
X12
)
→
(
X2
↔
X43
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X48
→
X39
→
(
(
¬
X3
)
↔
X43
)
)
)
→
(
¬
(
X49
↔
(
X39
↔
(
(
¬
X4
)
↔
X21
)
)
)
)
)
)
)
→
(
(
(
(
¬
X40
)
→
X16
→
X45
)
↔
(
X49
↔
(
X45
→
(
¬
X9
)
→
X11
)
)
)
↔
(
(
¬
(
(
(
X48
↔
(
(
¬
X40
)
→
X1
→
X49
)
)
↔
(
¬
(
X49
↔
(
X40
→
(
(
¬
X1
)
↔
X45
)
)
)
)
)
→
(
¬
(
X49
→
(
X40
↔
X1
)
)
)
)
)
↔
(
X48
↔
(
X40
→
(
¬
X1
)
→
X35
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
(
¬
(
(
(
X47
↔
X48
)
↔
(
¬
(
X48
↔
(
(
¬
X38
)
→
(
X36
↔
X45
)
)
)
)
)
→
X48
→
(
X47
↔
(
X21
↔
X26
)
)
)
)
→
(
(
¬
(
(
¬
X38
)
↔
(
(
¬
X37
)
→
(
X23
↔
X48
)
)
)
)
↔
(
(
X48
→
(
X47
↔
(
X11
→
X38
)
)
)
↔
(
¬
(
(
¬
X38
)
→
(
(
¬
X37
)
↔
X48
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
X49
↔
(
(
¬
X15
)
→
(
X13
↔
X35
)
)
)
→
(
X35
↔
(
X32
→
X3
→
X15
)
)
)
)
→
(
¬
(
(
¬
(
X45
→
(
X25
↔
X49
)
)
)
→
(
¬
(
X49
→
(
X46
↔
(
X32
↔
(
X13
→
False
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X49
↔
(
X36
→
X38
)
)
)
→
(
¬
(
X46
→
(
(
¬
X45
)
↔
(
(
¬
X4
)
↔
X49
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X28
→
(
X27
↔
False
)
)
)
↔
(
¬
(
X44
↔
(
X28
→
X17
→
X7
)
)
)
)
→
(
(
¬
(
X46
→
X7
→
X16
)
)
↔
(
¬
(
(
¬
X44
)
↔
(
(
¬
X40
)
↔
(
(
¬
X16
)
↔
X49
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X49
↔
(
X47
→
(
X46
↔
X6
)
)
)
)
→
(
¬
(
X47
↔
(
X24
→
(
X6
↔
X46
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X47
↔
(
X24
↔
(
(
¬
X22
)
↔
X49
)
)
)
)
↔
(
X47
↔
(
X22
→
X40
)
)
)
→
(
(
X24
↔
(
(
¬
X22
)
→
X33
)
)
↔
(
(
¬
(
X47
↔
(
X24
↔
(
X22
↔
X19
)
)
)
)
→
(
¬
X33
)
→
(
¬
X24
)
→
X47
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
(
¬
(
(
X44
↔
(
X31
↔
(
X14
↔
(
X9
→
False
)
)
)
)
→
X44
→
(
X10
↔
(
X9
↔
X14
)
)
)
)
↔
(
¬
(
X44
↔
(
X14
↔
X31
)
)
)
)
↔
(
(
X8
→
X0
→
X10
)
↔
(
¬
(
X44
→
(
X14
↔
(
X10
↔
False
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X34
↔
(
(
¬
X31
)
→
(
(
¬
X14
)
↔
X44
)
)
)
)
→
(
¬
(
X44
↔
(
X34
↔
(
X31
→
X20
)
)
)
)
)
)
)
→
(
¬
(
(
X43
↔
(
X14
→
X1
→
X20
)
)
→
(
¬
(
X46
→
(
(
¬
X20
)
↔
X34
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
X44
↔
(
X39
↔
(
(
¬
X12
)
↔
X41
)
)
)
→
(
¬
(
X41
→
(
(
¬
X18
)
↔
(
(
¬
X12
)
↔
X39
)
)
)
)
)
→
(
¬
(
X41
↔
X18
)
)
)
→
(
X46
↔
(
X18
→
(
X12
↔
(
X2
→
False
)
)
)
)
→
(
¬
(
(
¬
X39
)
→
X18
→
(
¬
X2
)
→
X46
)
)
)
)
↔
(
(
(
(
¬
(
(
X48
→
(
¬
X30
)
→
(
¬
X5
)
→
X41
)
→
(
X41
↔
(
(
¬
X30
)
↔
(
X5
↔
X43
)
)
)
)
)
→
(
¬
(
(
¬
X41
)
↔
(
(
¬
X30
)
→
(
X29
↔
X43
)
)
)
)
)
→
(
¬
(
X44
→
(
X43
↔
(
X41
↔
X48
)
)
)
)
)
↔
(
(
¬
(
X43
↔
(
X42
↔
X39
)
)
)
↔
(
(
X41
→
(
(
¬
X39
)
↔
X43
)
)
↔
(
¬
(
X44
→
(
X41
↔
(
(
¬
X39
)
→
X42
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
3055a0...
and proposition id is
2021f3...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMJM6VCwJHBwcMPxJCNJQ5cAGDMuYpbo813
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
∀X51 :
prop
,
(
(
¬
(
(
(
(
(
(
X50
→
(
X40
↔
(
X24
↔
X12
)
)
)
→
(
¬
(
X40
↔
(
X18
→
(
X12
↔
False
)
)
)
)
)
↔
(
¬
(
(
(
X46
↔
(
X18
↔
(
X12
↔
X8
)
)
)
↔
(
X46
→
(
X18
↔
(
X11
↔
X12
)
)
)
)
→
X50
→
(
¬
X18
)
→
(
(
¬
X8
)
↔
X44
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
(
¬
X48
)
↔
(
X19
→
(
X9
↔
X50
)
)
)
)
↔
(
X46
↔
(
X9
→
X48
)
)
)
→
(
¬
(
X48
→
(
X34
↔
X46
)
)
)
)
)
↔
(
(
¬
(
X50
↔
(
(
¬
X34
)
↔
(
(
¬
X27
)
↔
X46
)
)
)
)
↔
(
¬
(
(
¬
X46
)
→
(
X38
↔
(
X27
→
X50
)
)
)
)
)
)
→
(
¬
(
(
(
¬
X46
)
→
X17
→
(
¬
X14
)
→
X48
)
→
(
¬
(
(
¬
X44
)
→
(
X17
↔
X50
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
(
¬
(
X20
→
X44
)
)
↔
(
¬
(
(
¬
X45
)
→
(
(
¬
X32
)
↔
X48
)
)
)
)
→
(
(
¬
(
X48
→
X44
→
X5
→
X32
)
)
↔
(
X44
↔
(
X32
→
X10
→
X48
)
)
)
)
)
↔
(
¬
(
X48
↔
(
X45
→
(
X44
↔
(
X43
↔
False
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X50
↔
(
X44
↔
(
X22
→
X16
)
)
)
)
→
(
¬
(
X48
↔
(
X44
→
(
X22
↔
X50
)
)
)
)
)
)
↔
(
(
(
¬
X42
)
↔
(
X22
→
(
(
¬
X16
)
↔
X43
)
)
)
↔
(
X22
→
(
(
¬
X16
)
↔
X50
)
)
)
)
)
→
(
¬
(
(
(
X37
→
(
X36
↔
(
(
¬
X15
)
↔
X20
)
)
)
→
X37
→
(
X20
↔
X49
)
)
→
(
(
¬
X32
)
→
(
X20
↔
(
(
¬
X7
)
↔
X49
)
)
)
→
(
¬
(
(
(
¬
(
X32
→
X39
)
)
↔
(
¬
(
(
¬
X49
)
↔
(
X37
↔
(
X32
↔
X50
)
)
)
)
)
→
(
X50
↔
(
X32
→
X20
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
X44
)
↔
(
(
¬
X23
)
→
(
X2
↔
X51
)
)
)
)
→
(
(
(
¬
(
X51
→
(
X33
↔
(
X6
↔
X28
)
)
)
)
↔
(
(
¬
(
X51
↔
(
X44
→
X28
)
)
)
→
(
(
X44
→
X33
→
False
)
↔
(
¬
(
X51
↔
(
(
¬
X33
)
→
(
¬
X28
)
→
X49
)
)
)
)
→
X44
→
X33
→
X51
)
)
↔
(
¬
(
X51
↔
(
X44
↔
(
X33
↔
(
X23
↔
False
)
)
)
)
)
)
)
→
(
(
(
¬
X28
)
↔
(
(
¬
X26
)
→
(
X6
↔
X44
)
)
)
↔
(
(
¬
(
X44
→
X30
→
(
X28
↔
False
)
)
)
→
(
(
¬
(
X30
→
X28
→
X3
→
X44
)
)
↔
(
¬
(
X30
→
(
¬
X26
)
→
X44
)
)
)
)
)
)
↔
(
(
(
X51
→
(
X44
↔
(
(
¬
X40
)
↔
X49
)
)
)
↔
(
X49
→
X35
→
(
¬
X29
)
→
X40
)
)
↔
(
(
¬
(
(
¬
(
X31
↔
(
(
¬
X21
)
→
(
X0
↔
X51
)
)
)
)
→
(
¬
(
(
(
(
X21
→
(
X13
↔
(
X4
↔
False
)
)
)
↔
(
X44
↔
(
(
¬
X13
)
→
(
(
¬
X4
)
↔
X31
)
)
)
)
→
(
¬
(
X44
↔
(
X21
→
(
X13
↔
X1
)
)
)
)
)
→
(
¬
(
(
X51
↔
(
(
¬
X25
)
→
X31
)
)
→
(
(
(
¬
(
(
¬
X31
)
↔
(
X25
→
X21
→
X51
)
)
)
↔
(
X51
↔
(
X31
→
(
¬
X13
)
→
X21
)
)
)
↔
(
X21
→
X31
)
)
)
)
)
)
)
)
↔
(
(
X47
→
(
X44
↔
(
X41
→
X40
→
False
)
)
)
→
(
(
X51
↔
X47
)
↔
(
¬
(
(
(
(
(
X51
→
(
X44
↔
(
X41
↔
X31
)
)
)
↔
(
X47
→
(
X41
↔
(
(
¬
X31
)
↔
X44
)
)
)
)
→
X51
→
(
X47
↔
X41
)
)
↔
(
X41
↔
(
X31
↔
X47
)
)
)
→
X47
→
(
(
¬
X41
)
↔
X44
)
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
X49
↔
(
(
¬
X47
)
↔
(
(
¬
X9
)
↔
X48
)
)
)
→
(
¬
(
(
(
¬
X48
)
↔
(
(
¬
X45
)
→
(
X39
↔
X49
)
)
)
→
(
(
¬
(
X40
→
X39
→
X45
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X47
)
↔
(
(
¬
X3
)
↔
X48
)
)
)
→
(
¬
(
X40
↔
(
X6
↔
(
(
¬
X3
)
↔
X48
)
)
)
)
)
)
↔
(
X48
↔
(
(
¬
X3
)
→
X45
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
X49
→
(
X48
↔
(
X47
↔
(
X32
↔
False
)
)
)
)
→
(
(
¬
(
X48
→
(
X47
↔
(
(
¬
X26
)
↔
X44
)
)
)
)
↔
(
¬
(
X49
↔
(
X47
→
X44
→
X48
)
)
)
)
)
)
↔
(
(
(
X49
→
(
(
¬
X44
)
↔
(
(
¬
X14
)
↔
X47
)
)
)
↔
(
(
¬
(
(
¬
X44
)
→
X11
→
X49
)
)
↔
(
X49
↔
(
X47
↔
(
X38
→
X44
)
)
)
)
)
→
(
X44
↔
(
(
¬
X14
)
→
X49
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
(
X51
↔
(
X30
↔
X49
)
)
)
→
(
(
(
(
X2
→
X51
)
↔
(
(
¬
(
(
¬
X47
)
→
X51
)
)
→
(
¬
(
X51
→
(
X49
↔
(
X48
↔
X2
)
)
)
)
)
)
↔
(
¬
(
(
¬
X49
)
↔
(
(
¬
X48
)
→
X30
→
X51
)
)
)
)
↔
(
X51
→
(
(
¬
X48
)
↔
X49
)
)
)
)
)
→
(
(
¬
(
(
¬
X48
)
↔
(
X10
→
(
X7
↔
X49
)
)
)
)
↔
(
¬
(
X47
→
X10
→
X7
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X43
)
→
X34
→
X49
)
)
→
(
(
X46
→
(
X43
↔
(
X28
↔
X49
)
)
)
↔
(
¬
(
X43
→
X34
)
)
)
)
)
→
(
¬
(
(
¬
(
X48
→
(
(
¬
X46
)
↔
(
X17
→
X47
)
)
)
)
→
(
¬
(
(
(
(
(
¬
(
(
¬
X22
)
↔
(
X17
↔
X47
)
)
)
↔
(
X49
→
X17
→
X48
)
)
→
(
X49
↔
(
(
¬
X22
)
↔
X47
)
)
)
↔
(
X48
→
False
)
)
→
(
¬
(
X49
→
(
(
¬
X43
)
↔
(
(
¬
X22
)
↔
X47
)
)
)
)
)
)
)
)
)
)
)
)
→
(
(
(
(
(
(
(
(
¬
X48
)
↔
(
X13
↔
X50
)
)
↔
(
(
¬
(
X51
↔
(
(
¬
X13
)
↔
(
X1
↔
X50
)
)
)
)
→
(
¬
X48
)
→
(
¬
X13
)
→
X51
)
)
→
(
¬
(
(
¬
X13
)
→
(
X12
↔
(
(
¬
X1
)
↔
X51
)
)
)
)
)
→
(
¬
(
X51
→
X1
)
)
)
↔
(
(
(
¬
(
X51
→
(
(
¬
X41
)
↔
(
(
¬
X31
)
→
X42
)
)
)
)
→
(
¬
(
(
¬
X50
)
→
(
(
¬
X42
)
↔
X51
)
)
)
→
(
(
¬
(
X51
→
(
X42
↔
False
)
)
)
↔
(
X50
↔
(
X31
→
(
X29
↔
X42
)
)
)
)
)
↔
(
(
¬
(
X41
→
(
¬
X21
)
→
X42
)
)
↔
(
(
(
X51
↔
(
X48
↔
False
)
)
→
(
(
X51
→
(
X48
↔
(
X42
↔
(
X41
↔
False
)
)
)
)
↔
(
(
¬
X41
)
↔
(
X15
↔
X51
)
)
)
)
↔
(
¬
(
X41
→
X21
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X51
→
(
X41
↔
(
X27
→
False
)
)
)
)
→
X51
→
(
X50
↔
(
X19
↔
X27
)
)
)
↔
(
¬
(
(
¬
(
X41
↔
(
X35
↔
(
(
¬
X0
)
→
X51
)
)
)
)
→
(
¬
(
X51
→
(
(
¬
X0
)
↔
X27
)
)
)
)
)
)
→
(
(
¬
X35
)
↔
(
(
¬
X27
)
→
(
(
¬
X5
)
↔
X41
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X25
)
→
X23
→
(
¬
X16
)
→
X42
)
)
→
(
(
¬
(
X37
↔
(
(
¬
X25
)
↔
X33
)
)
)
↔
(
(
¬
(
X37
→
(
¬
X16
)
→
X25
)
)
↔
(
¬
(
X37
→
(
X33
↔
(
X23
→
False
)
)
)
)
)
)
→
(
(
(
¬
(
(
X35
↔
(
X25
↔
X8
)
)
→
X23
→
(
¬
X8
)
→
(
X4
↔
X25
)
)
)
↔
(
¬
(
(
¬
X25
)
↔
(
X23
↔
X50
)
)
)
)
↔
(
¬
(
(
¬
X33
)
→
(
(
¬
X25
)
↔
(
X23
↔
X35
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X49
→
X20
→
X36
)
)
→
(
(
X33
↔
(
X25
↔
(
(
¬
X20
)
↔
X37
)
)
)
↔
(
X49
↔
(
X33
→
False
)
)
)
)
→
(
(
¬
(
(
¬
X24
)
→
X37
)
)
↔
(
(
X49
→
(
¬
X18
)
→
X24
)
↔
(
X33
→
(
X18
↔
X37
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
7495db...
and proposition id is
46c511...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMZhMcjNX5MAUBKpPmftYRHMvNR8Pbw7hsH
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
(
¬
(
X44
→
(
X29
↔
(
X1
↔
X36
)
)
)
)
↔
(
(
(
X44
→
X25
→
X29
)
↔
(
¬
(
X44
→
X29
→
(
X0
↔
X36
)
)
)
)
↔
(
(
(
¬
X44
)
→
X25
→
(
¬
X15
)
→
X49
)
↔
(
(
X49
↔
(
X44
→
X25
)
)
↔
(
¬
(
(
¬
(
(
X49
↔
(
X44
↔
X25
)
)
→
(
¬
(
X25
→
(
(
¬
X0
)
↔
X7
)
)
)
)
)
→
(
X44
↔
(
(
¬
X0
)
↔
X25
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X44
→
(
X43
↔
(
X22
→
X27
)
)
)
)
↔
(
¬
(
X44
→
(
X27
↔
X25
)
)
)
)
↔
(
(
(
X43
→
X27
→
X36
)
→
(
¬
(
X36
↔
(
X19
→
X24
)
)
)
)
↔
(
(
X21
↔
(
(
¬
X10
)
↔
X44
)
)
↔
(
¬
(
X21
→
(
X19
↔
(
X10
→
X24
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
X36
→
X33
→
X23
→
X35
)
↔
(
¬
(
(
X42
→
(
X35
↔
(
X33
→
False
)
)
)
→
(
¬
(
(
¬
X36
)
↔
(
(
¬
X31
)
↔
X42
)
)
)
)
)
)
→
(
(
¬
(
X42
→
(
¬
X35
)
→
(
(
¬
X20
)
↔
X44
)
)
)
↔
(
¬
(
X44
→
(
X33
↔
X20
)
)
)
)
)
)
→
(
(
X41
↔
(
X33
→
(
X16
↔
(
X8
→
False
)
)
)
)
→
(
(
X43
→
(
X36
↔
X41
)
)
↔
(
¬
(
X33
↔
(
X16
→
X36
)
)
)
)
)
→
(
(
¬
(
X43
↔
(
X33
→
(
¬
X2
)
→
X44
)
)
)
↔
(
¬
(
X33
↔
(
X2
↔
X16
)
)
)
)
→
(
¬
(
X43
→
X33
→
(
(
¬
X26
)
↔
X44
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
X43
→
X17
→
(
¬
X11
)
→
X46
)
↔
(
(
X34
↔
(
(
¬
X11
)
↔
(
X3
↔
X46
)
)
)
→
(
¬
(
(
¬
X34
)
↔
(
X17
↔
(
(
¬
X3
)
→
X46
)
)
)
)
)
)
→
(
(
X46
→
(
(
¬
X3
)
↔
X43
)
)
↔
(
X47
↔
(
X37
↔
X48
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X49
→
X48
→
X47
→
X12
→
False
)
)
↔
(
(
¬
(
(
¬
X48
)
↔
(
X47
→
(
X18
↔
X49
)
)
)
)
→
(
¬
(
(
¬
X47
)
→
(
(
¬
X40
)
↔
(
(
¬
X18
)
↔
X49
)
)
)
)
)
)
↔
(
¬
(
(
(
X47
↔
(
X18
↔
(
X13
↔
X9
)
)
)
→
X40
→
X13
)
→
(
¬
(
X40
→
(
(
¬
X9
)
↔
X47
)
)
)
)
)
)
→
(
(
X48
→
(
X43
↔
(
(
¬
X4
)
↔
X49
)
)
)
↔
(
(
¬
(
X49
→
(
X48
↔
(
X18
↔
(
X4
→
False
)
)
)
)
)
↔
(
¬
(
X41
↔
(
(
¬
X4
)
↔
X49
)
)
)
)
)
)
)
)
)
↔
(
(
(
X48
↔
(
X39
↔
(
X28
→
X6
)
)
)
↔
(
¬
(
(
¬
(
X45
↔
(
(
¬
X43
)
→
(
X32
↔
X48
)
)
)
)
→
X48
→
(
X28
↔
X45
)
)
)
)
↔
(
¬
(
(
(
(
X45
→
(
¬
X30
)
→
(
X5
↔
X46
)
)
↔
(
X48
→
(
X46
↔
(
X30
↔
False
)
)
)
)
→
(
¬
(
X46
↔
(
X45
↔
X48
)
)
)
)
→
(
¬
(
(
¬
(
X46
→
(
¬
X43
)
→
(
X14
↔
X48
)
)
)
→
X45
→
(
X38
↔
X46
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
(
¬
(
X49
↔
(
(
¬
X15
)
→
(
X3
↔
X32
)
)
)
)
→
(
¬
(
X31
→
(
¬
X15
)
→
X32
)
)
)
)
↔
(
¬
(
X33
↔
(
(
¬
X31
)
→
(
¬
X3
)
→
X48
)
)
)
)
↔
(
(
X33
↔
(
(
¬
X32
)
↔
X45
)
)
→
(
X47
→
X32
→
(
(
¬
X5
)
↔
X49
)
)
→
(
(
X49
↔
(
X33
↔
(
X32
↔
False
)
)
)
↔
(
(
(
X45
→
(
X32
↔
X5
)
)
↔
(
¬
(
X45
↔
(
(
¬
X23
)
↔
(
(
¬
X17
)
↔
X49
)
)
)
)
)
↔
(
X45
→
X32
→
(
¬
X17
)
→
X33
)
)
)
)
)
↔
(
(
(
(
¬
X48
)
↔
(
X47
→
X30
→
X49
)
)
→
(
(
¬
(
X14
→
X48
)
)
↔
(
(
X30
→
(
X26
↔
X49
)
)
→
X26
→
(
(
¬
X14
)
↔
X48
)
)
)
)
↔
(
¬
(
(
¬
X48
)
↔
(
(
¬
X30
)
→
X49
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X45
→
(
¬
(
X27
↔
(
X25
→
(
X20
↔
X43
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
X42
→
(
X27
↔
(
X10
→
X20
)
)
)
)
→
(
¬
(
(
¬
X27
)
→
(
X20
↔
(
X10
↔
X41
)
)
)
)
)
)
→
(
¬
(
X42
↔
(
X20
→
X10
)
)
)
)
)
↔
(
¬
(
X36
↔
(
X27
→
(
X20
↔
X41
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X42
)
→
(
X41
↔
(
(
¬
X9
)
↔
X45
)
)
)
)
→
(
¬
(
X42
↔
(
X41
→
(
¬
X9
)
→
X28
)
)
)
)
)
)
→
(
(
¬
(
X49
→
X41
→
X27
→
X12
→
False
)
)
↔
(
(
(
¬
(
X48
→
X41
→
(
X37
↔
X49
)
)
)
↔
(
X49
↔
(
(
¬
X41
)
↔
(
X27
↔
X48
)
)
)
)
→
X48
→
(
X45
↔
(
(
¬
X41
)
↔
X49
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
X34
→
(
X21
↔
X22
)
)
↔
(
¬
(
X22
→
(
(
¬
X21
)
↔
(
X19
↔
X38
)
)
)
)
)
→
(
(
¬
(
X46
↔
(
(
¬
X19
)
↔
(
X13
↔
X35
)
)
)
)
→
(
¬
(
X46
→
(
X21
↔
(
(
¬
X19
)
→
X35
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X35
↔
(
X0
↔
False
)
)
)
→
(
¬
(
(
¬
X19
)
→
(
¬
X16
)
→
(
¬
X0
)
→
X38
)
)
)
)
↔
(
(
(
¬
(
(
¬
X35
)
↔
(
(
¬
X16
)
→
X38
)
)
)
↔
(
(
(
¬
X38
)
↔
(
(
¬
X35
)
↔
(
X16
→
X46
)
)
)
→
(
¬
(
X46
→
X6
→
False
)
)
)
)
→
(
X35
↔
(
X16
↔
(
X1
↔
X46
)
)
)
)
)
)
→
(
(
(
(
¬
(
X49
↔
(
X47
↔
(
X40
↔
False
)
)
)
)
↔
(
¬
(
(
¬
X47
)
→
X8
→
(
X7
↔
X49
)
)
)
)
↔
(
(
¬
(
X44
→
(
(
¬
X24
)
↔
(
X4
↔
X46
)
)
)
)
↔
(
(
¬
(
X47
↔
(
X24
→
X8
→
X48
)
)
)
→
(
(
X7
↔
(
(
¬
X4
)
↔
X24
)
)
↔
(
(
¬
(
(
¬
X47
)
→
(
X7
↔
(
(
¬
X4
)
↔
X48
)
)
)
)
↔
(
¬
(
(
X47
→
X24
→
(
¬
X7
)
→
X44
)
→
(
¬
(
(
¬
X47
)
→
(
¬
X7
)
→
X48
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
X48
↔
(
(
¬
X47
)
→
(
¬
X18
)
→
X49
)
)
↔
(
X49
↔
(
X48
→
(
X18
↔
X39
)
)
)
)
→
(
¬
(
X49
→
X39
→
X35
→
X18
)
)
)
↔
(
¬
(
X49
→
X47
→
(
X34
↔
X18
)
)
)
)
→
(
¬
(
(
¬
X35
)
→
(
X34
↔
(
(
¬
X2
)
↔
X39
)
)
)
)
→
X34
→
(
X29
↔
(
X11
↔
X2
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
40a178...
and proposition id is
3d72e8...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMLUYxZJ5gBLcRBAuEc9a7uopzZqp66nmWt
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
¬
(
(
(
¬
(
(
(
¬
(
X48
↔
(
X47
↔
(
(
¬
X3
)
↔
X44
)
)
)
)
↔
(
X48
→
X43
→
(
(
¬
X3
)
↔
X4
)
)
)
→
(
¬
(
(
(
X44
→
X47
)
↔
(
¬
(
(
¬
X33
)
↔
(
X25
↔
(
(
¬
X22
)
↔
X47
)
)
)
)
)
→
(
X43
↔
(
X25
↔
(
(
¬
X7
)
↔
X44
)
)
)
)
)
)
)
→
(
(
X47
↔
(
X44
↔
(
X26
↔
X18
)
)
)
↔
(
(
(
¬
(
(
¬
X36
)
↔
(
(
¬
X12
)
↔
(
(
¬
X1
)
↔
X44
)
)
)
)
↔
(
(
(
¬
X36
)
→
(
(
¬
X33
)
↔
X44
)
)
→
(
¬
(
(
¬
X41
)
→
X33
→
(
(
¬
X18
)
↔
X44
)
)
)
)
)
→
(
¬
(
X41
→
X36
→
(
¬
X26
)
→
X33
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
(
¬
(
X47
→
(
(
¬
X39
)
↔
(
(
¬
X16
)
↔
X43
)
)
)
)
↔
(
¬
(
X39
↔
(
X35
↔
(
X16
→
X47
)
)
)
)
)
→
(
(
¬
X35
)
↔
X47
)
→
X43
→
(
¬
X9
)
→
X0
→
X47
)
)
→
(
¬
(
(
¬
(
(
¬
X43
)
→
(
X39
↔
(
(
¬
X34
)
→
X47
)
)
)
)
→
(
¬
(
X47
↔
(
X39
↔
(
X32
→
X43
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X34
→
(
(
¬
X28
)
↔
(
X15
→
X32
)
)
)
)
→
(
¬
(
X47
→
X34
→
X28
)
)
)
)
↔
(
¬
(
(
X48
→
(
X17
↔
(
(
¬
X14
)
→
X34
)
)
)
→
(
X34
→
(
(
¬
X17
)
↔
X48
)
)
→
(
X48
↔
(
X32
↔
X47
)
)
)
)
)
)
)
)
→
(
(
(
(
X37
↔
(
(
¬
X10
)
↔
X45
)
)
↔
(
(
(
¬
(
X49
↔
(
X42
↔
(
X13
↔
(
X10
↔
False
)
)
)
)
)
↔
(
(
¬
X42
)
↔
(
X37
→
X10
→
X49
)
)
)
↔
(
¬
(
(
X42
↔
(
(
¬
X13
)
↔
(
(
¬
X6
)
→
X37
)
)
)
→
(
(
¬
X42
)
↔
(
(
¬
X6
)
→
X45
)
)
)
)
)
)
→
(
(
¬
X37
)
↔
(
X10
↔
(
X8
↔
X44
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
X23
↔
(
X20
→
X29
)
)
)
↔
(
(
¬
(
X46
→
(
(
¬
X23
)
↔
(
X20
→
X30
)
)
)
)
→
(
¬
(
X46
↔
(
X30
↔
(
X29
↔
(
X23
↔
False
)
)
)
)
)
)
)
→
(
X29
↔
(
(
¬
X20
)
→
X45
)
)
)
)
→
(
X46
↔
(
X45
↔
False
)
)
)
↔
(
(
(
¬
(
X42
→
X38
→
False
)
)
↔
(
(
¬
(
(
¬
(
X44
↔
(
X38
↔
X30
)
)
)
→
(
¬
(
(
¬
(
X44
→
(
¬
X19
)
→
X30
)
)
→
(
¬
(
(
(
¬
(
X30
↔
(
(
¬
X19
)
↔
(
X2
→
X44
)
)
)
)
→
(
¬
(
X2
→
X38
)
)
)
→
(
X38
↔
(
X19
→
X30
)
)
)
)
→
(
¬
(
X38
↔
(
(
¬
X30
)
→
X44
)
)
)
)
)
)
)
→
(
¬
(
X44
↔
(
X38
→
False
)
)
)
)
)
→
(
(
(
X45
→
X44
→
(
(
¬
X21
)
↔
X38
)
)
→
X38
→
X29
→
(
¬
X21
)
→
X45
)
↔
(
¬
(
X45
↔
(
X44
→
(
(
¬
X38
)
↔
X42
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
X48
↔
(
X45
→
X11
→
X40
)
)
→
(
¬
(
(
¬
(
(
¬
X40
)
→
(
(
¬
X11
)
↔
X45
)
)
)
→
(
¬
(
X45
→
X11
→
(
¬
X5
)
→
X49
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
X40
)
→
(
¬
X24
)
→
X46
)
→
(
(
(
X48
→
(
X39
↔
(
(
¬
X27
)
↔
X49
)
)
)
↔
(
¬
(
(
¬
X31
)
→
(
X24
↔
X48
)
)
)
)
↔
(
X46
→
(
X39
↔
X49
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
(
¬
(
(
(
¬
(
(
X32
↔
(
(
¬
X24
)
→
X43
)
)
→
(
¬
(
(
¬
(
X32
→
(
X2
↔
False
)
)
)
→
(
(
¬
(
(
¬
X24
)
↔
(
X2
↔
X43
)
)
)
↔
(
¬
(
(
¬
X39
)
↔
(
(
¬
X32
)
→
(
X2
↔
X43
)
)
)
)
)
)
)
)
)
↔
(
¬
(
X32
→
(
(
¬
X22
)
↔
(
X2
↔
X39
)
)
)
)
)
→
(
X45
→
(
X39
↔
(
(
¬
X24
)
↔
X43
)
)
)
→
(
¬
(
(
¬
X24
)
→
(
X9
↔
(
X5
↔
X43
)
)
)
)
)
)
→
(
(
(
¬
(
X47
↔
(
X39
↔
(
X8
→
X31
)
)
)
)
↔
(
¬
(
X47
↔
(
(
¬
X42
)
↔
(
(
¬
X39
)
→
X43
)
)
)
)
)
↔
(
¬
(
(
¬
X31
)
↔
(
X24
→
(
(
¬
X8
)
↔
X43
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
(
(
X45
→
(
¬
X23
)
→
X30
)
→
(
¬
(
(
(
¬
(
X43
→
(
X23
↔
X45
)
)
)
↔
(
¬
(
X43
↔
(
(
¬
X1
)
↔
X30
)
)
)
)
→
(
¬
(
(
¬
X43
)
→
X30
→
X45
)
)
)
)
)
)
↔
(
(
¬
X43
)
↔
(
(
¬
X30
)
→
(
¬
X10
)
→
X45
)
)
)
→
(
¬
(
X30
↔
(
X1
↔
X43
)
)
)
)
)
→
(
(
¬
(
(
¬
X32
)
↔
(
X7
→
X45
)
)
)
↔
(
(
X29
→
(
(
¬
X7
)
↔
X45
)
)
→
X29
→
X23
→
X7
→
X32
)
)
→
(
¬
(
X43
→
X28
→
(
X23
↔
X32
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X14
↔
X40
)
)
↔
(
(
(
¬
X34
)
↔
(
(
¬
X14
)
↔
X45
)
)
→
X45
→
X42
→
(
¬
X14
)
→
X40
)
)
↔
(
X42
↔
(
X37
↔
(
X18
↔
X49
)
)
)
)
↔
(
(
¬
(
(
(
X42
→
(
¬
X35
)
→
X33
→
X40
)
↔
(
¬
(
(
X45
↔
(
(
¬
X35
)
↔
(
X33
↔
X40
)
)
)
→
(
¬
(
X45
↔
(
X40
↔
(
X33
→
(
X11
↔
False
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
X42
)
↔
(
X40
→
(
X17
↔
X45
)
)
)
)
↔
(
X45
↔
(
X42
→
X36
)
)
)
)
)
↔
(
¬
(
(
¬
X43
)
↔
(
(
¬
X35
)
→
(
(
¬
X33
)
↔
X45
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
(
(
¬
X46
)
↔
(
(
¬
X43
)
↔
(
X27
↔
X47
)
)
)
)
↔
(
(
X47
↔
(
X42
↔
X46
)
)
→
(
(
X47
↔
(
(
¬
X41
)
↔
(
X27
→
X46
)
)
)
↔
(
X46
→
X27
→
(
(
¬
X26
)
↔
X41
)
)
)
)
)
↔
(
(
¬
(
X47
↔
(
X38
→
(
(
¬
X13
)
↔
X46
)
)
)
)
↔
(
(
(
¬
X43
)
↔
(
(
¬
X38
)
→
(
(
¬
X19
)
↔
X46
)
)
)
→
(
¬
(
X47
↔
(
X43
↔
(
X19
↔
False
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X25
↔
False
)
)
→
(
(
X38
→
(
(
¬
X12
)
↔
X42
)
)
↔
(
¬
(
(
¬
X42
)
↔
(
(
¬
X38
)
→
(
(
¬
X25
)
↔
X46
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
X21
)
→
(
¬
X20
)
→
(
¬
X4
)
→
X48
)
)
→
(
¬
(
(
¬
X20
)
→
(
¬
X3
)
→
X21
)
)
)
)
→
X48
→
(
(
¬
X16
)
↔
(
(
¬
X4
)
→
X21
)
)
)
→
(
(
(
(
¬
X44
)
→
(
¬
X26
)
→
(
X21
↔
X48
)
)
↔
(
(
¬
(
X26
→
(
(
¬
X6
)
↔
X21
)
)
)
↔
(
(
¬
(
X26
↔
(
X21
→
X15
→
False
)
)
)
→
(
(
¬
(
X26
↔
(
X21
→
X44
)
)
)
↔
(
(
¬
(
(
¬
X21
)
↔
(
(
¬
X15
)
→
(
(
¬
X6
)
↔
X44
)
)
)
)
↔
(
X26
→
(
X15
↔
(
(
¬
X6
)
→
X21
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X26
↔
(
X20
→
False
)
)
)
↔
(
(
¬
(
X26
↔
(
X25
↔
X15
)
)
)
→
(
¬
(
X26
↔
(
X20
→
(
¬
X6
)
→
X15
)
)
)
)
)
↔
(
¬
(
(
¬
(
X47
↔
(
X26
→
X15
→
(
X0
↔
False
)
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X15
)
↔
(
X6
→
X26
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
836247...
and proposition id is
46f3d6...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMF2Lv6cFwkJuZ7enxaULTfSyagshtuMWjk
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
(
¬
(
(
¬
(
(
¬
(
(
(
¬
(
(
¬
X42
)
→
X47
)
)
↔
(
¬
(
X46
↔
(
X18
→
X43
)
)
)
)
→
(
¬
(
(
X42
→
(
X31
↔
(
(
¬
X18
)
→
X47
)
)
)
→
(
¬
(
X42
↔
(
X18
↔
(
(
¬
X12
)
→
X31
)
)
)
)
)
)
)
)
→
(
(
X43
→
(
¬
X32
)
→
(
¬
X31
)
→
X42
)
↔
(
X32
→
(
(
¬
X31
)
↔
(
X21
→
X42
)
)
)
)
→
(
¬
(
X42
→
(
¬
X31
)
→
(
(
¬
X0
)
↔
X50
)
)
)
)
)
→
(
(
¬
(
(
(
(
¬
(
(
X2
→
X26
)
→
(
¬
(
X45
↔
(
(
¬
X8
)
↔
(
X2
→
X50
)
)
)
)
)
)
→
(
¬
(
X50
↔
(
X29
↔
False
)
)
)
→
X50
→
X48
→
X46
→
X8
→
False
)
→
X48
→
(
¬
X26
)
→
X46
)
→
(
(
¬
(
(
(
¬
(
(
¬
X41
)
→
(
(
¬
X20
)
↔
(
X1
→
X45
)
)
)
)
→
X46
→
X45
→
X1
→
X20
)
→
(
¬
(
X20
↔
X1
)
)
)
)
↔
(
(
X41
↔
(
X27
→
(
(
¬
X6
)
↔
X46
)
)
)
↔
(
¬
(
X45
→
(
X29
↔
(
X6
→
X27
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
X46
)
→
(
¬
X40
)
→
(
X14
↔
X47
)
)
→
(
¬
(
X45
↔
(
X44
→
(
(
¬
X14
)
↔
X46
)
)
)
)
)
↔
(
¬
(
X45
↔
(
X44
↔
(
X24
→
(
X14
↔
False
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
¬
X45
)
→
(
X38
↔
(
X13
↔
X47
)
)
)
)
→
(
(
¬
X38
)
→
(
X22
↔
X48
)
)
→
(
(
X48
→
(
¬
X38
)
→
(
(
¬
X22
)
↔
X47
)
)
↔
(
X45
→
(
¬
X22
)
→
X47
)
)
)
↔
(
¬
(
(
(
X45
↔
(
X38
→
X10
→
X46
)
)
↔
(
(
¬
X45
)
↔
(
X10
↔
(
(
¬
X4
)
↔
X46
)
)
)
)
→
(
¬
(
(
¬
X38
)
↔
(
X22
↔
(
(
¬
X10
)
↔
X46
)
)
)
)
)
)
)
→
(
¬
(
X47
↔
(
X38
↔
X45
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
(
X46
→
X36
→
(
X28
↔
(
X17
→
False
)
)
)
→
(
¬
(
X50
↔
(
X46
↔
(
X36
→
(
X28
↔
False
)
)
)
)
)
)
)
→
(
X46
↔
(
X28
↔
(
X17
→
False
)
)
)
)
→
(
¬
(
(
X39
↔
(
X36
↔
(
(
¬
X30
)
→
X50
)
)
)
→
(
¬
(
X50
↔
(
X46
↔
(
(
¬
X36
)
→
X39
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
X36
)
↔
(
X34
→
(
X16
↔
X50
)
)
)
)
→
X39
)
)
→
(
(
(
(
X39
↔
(
(
¬
X37
)
→
(
¬
X23
)
→
X50
)
)
↔
(
(
(
¬
(
X50
↔
(
X47
→
X19
→
X46
)
)
)
↔
(
X50
↔
(
(
¬
X46
)
→
(
¬
X37
)
→
X47
)
)
)
↔
(
¬
(
X50
→
(
X46
↔
(
X39
→
(
X19
↔
False
)
)
)
)
)
)
)
↔
(
(
X49
→
X35
→
X23
→
X9
)
↔
(
(
(
(
¬
(
X49
→
X33
→
X15
→
X50
)
)
→
(
X50
↔
(
X37
↔
(
X33
↔
X49
)
)
)
)
↔
(
¬
(
(
¬
(
X50
↔
(
(
¬
X37
)
→
(
(
¬
X35
)
↔
X49
)
)
)
)
→
(
¬
(
X50
→
(
X35
↔
X23
)
)
)
)
)
)
↔
(
X49
↔
(
X33
→
X35
)
)
)
)
)
↔
(
(
(
¬
(
X25
→
(
¬
X11
)
→
X49
)
)
↔
(
(
¬
X49
)
→
(
(
¬
X46
)
↔
(
X3
↔
X50
)
)
)
)
↔
(
¬
(
(
X49
↔
(
X46
→
(
X7
↔
False
)
)
)
→
(
¬
(
X50
↔
(
X46
→
X7
→
X5
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
(
X38
→
(
X34
↔
(
X24
↔
False
)
)
)
↔
(
(
¬
(
X34
↔
X24
)
)
↔
(
(
¬
(
(
¬
(
X44
↔
(
X38
→
X34
→
X10
→
False
)
)
)
→
(
(
¬
X38
)
↔
(
(
¬
X34
)
→
X24
→
X44
)
)
)
)
↔
(
(
¬
X34
)
→
(
¬
X10
)
→
X44
)
)
)
)
↔
(
X34
→
(
(
¬
X10
)
↔
X24
)
)
)
↔
(
(
(
¬
X7
)
→
X20
)
↔
(
(
¬
(
(
¬
X38
)
↔
(
(
¬
X34
)
→
(
X7
↔
X44
)
)
)
)
↔
(
X44
↔
(
X38
→
X34
→
X20
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X38
)
↔
(
(
¬
X21
)
→
(
X10
↔
X40
)
)
)
)
↔
(
¬
(
(
¬
X38
)
→
(
(
¬
X21
)
↔
(
X10
↔
X45
)
)
)
)
)
→
(
¬
(
X38
→
(
X8
↔
False
)
)
)
)
)
→
(
¬
(
(
(
¬
X38
)
↔
(
X24
→
X40
)
)
→
(
(
(
¬
(
X40
→
(
¬
X38
)
→
(
X1
↔
X49
)
)
)
↔
(
X38
↔
(
X8
→
X40
)
)
)
↔
(
(
¬
X8
)
↔
(
X1
→
X40
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
(
X47
↔
(
X44
→
X5
→
X50
)
)
↔
(
¬
(
X47
↔
(
(
¬
X13
)
↔
(
(
¬
X5
)
→
X50
)
)
)
)
)
→
(
(
¬
(
X48
↔
(
X13
→
X47
)
)
)
↔
(
(
X31
↔
(
(
¬
X13
)
→
X44
)
)
→
(
¬
(
X47
↔
(
(
¬
X44
)
→
(
(
¬
X13
)
↔
X48
)
)
)
)
)
)
)
)
→
(
(
(
X39
→
(
X37
↔
(
X31
→
False
)
)
)
↔
(
(
¬
(
X44
↔
(
(
¬
X37
)
→
(
¬
X31
)
→
X39
)
)
)
→
(
X44
→
False
)
→
(
¬
(
X37
→
(
¬
X13
)
→
(
X4
↔
X39
)
)
)
)
)
→
(
¬
(
X39
↔
(
X13
→
X31
)
)
)
)
→
(
¬
(
X45
→
(
¬
X37
)
→
(
X31
↔
X39
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
X46
↔
X35
)
)
→
(
(
¬
(
X46
↔
(
X45
↔
(
X3
↔
X35
)
)
)
)
↔
(
X46
→
(
X45
↔
(
(
¬
X27
)
→
X35
)
)
)
)
)
)
↔
(
¬
(
(
¬
X45
)
↔
(
(
¬
X35
)
↔
(
(
¬
X23
)
↔
X46
)
)
)
)
)
→
(
¬
(
X46
↔
(
X40
↔
(
(
¬
X35
)
↔
X45
)
)
)
)
)
→
(
(
¬
(
(
¬
X35
)
→
X28
→
(
(
¬
X23
)
↔
X50
)
)
)
↔
(
¬
(
X44
↔
(
X28
→
(
X23
↔
False
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
(
¬
(
X41
↔
(
X36
→
(
¬
X32
)
→
X43
)
)
)
↔
(
X50
↔
(
X36
→
(
(
¬
X32
)
↔
X41
)
)
)
)
→
(
(
X41
→
(
X30
↔
(
X17
↔
X12
)
)
)
↔
(
¬
(
X43
↔
(
X30
↔
(
(
¬
X17
)
↔
X50
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
X50
↔
(
(
¬
X29
)
↔
X36
)
)
↔
(
(
¬
X43
)
→
(
X29
↔
(
(
¬
X18
)
↔
X50
)
)
)
)
→
X43
→
(
(
¬
X29
)
↔
(
X2
↔
X36
)
)
)
)
→
(
¬
(
X47
↔
(
X18
→
X43
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
¬
(
X42
↔
(
(
¬
X14
)
→
X43
)
)
)
→
(
(
¬
(
X42
↔
(
X19
→
X9
)
)
)
↔
(
(
¬
X19
)
→
(
¬
X14
)
→
X43
)
)
)
→
(
¬
(
X14
↔
(
X9
↔
False
)
)
)
)
→
X47
→
X43
→
X42
→
False
)
)
↔
(
X48
↔
(
X47
↔
(
X42
→
False
)
)
)
)
→
(
(
(
¬
X43
)
→
(
X33
↔
(
X16
↔
X50
)
)
)
→
(
(
¬
(
X47
↔
(
X16
→
(
X6
↔
False
)
)
)
)
↔
(
(
¬
(
X33
→
(
(
¬
X6
)
↔
X43
)
)
)
↔
(
¬
(
X47
↔
(
X33
→
(
X16
↔
X43
)
)
)
)
)
)
)
→
(
(
X48
↔
(
X6
↔
X16
)
)
↔
(
(
X16
→
(
(
¬
X0
)
↔
X6
)
)
↔
(
¬
(
(
¬
X43
)
→
X16
→
(
¬
X0
)
→
X48
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
X26
↔
(
X25
→
X42
)
)
↔
(
(
¬
(
X26
→
X25
→
X15
→
False
)
)
↔
(
¬
(
X50
↔
(
X42
→
X15
)
)
)
)
)
↔
(
¬
(
(
¬
X42
)
→
X25
→
(
¬
X15
)
→
X43
)
)
)
→
(
¬
X42
)
→
X26
→
X43
)
)
→
(
(
(
¬
(
X50
↔
(
X26
↔
(
X11
↔
X22
)
)
)
)
↔
(
X25
↔
X50
)
)
↔
(
¬
(
X26
→
X22
→
X25
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
28da5e...
and proposition id is
074248...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMGCBw2xaMGkZ7QbVuw4bzsGK4QXaKimWGL
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
¬
(
(
(
(
¬
(
(
(
X48
↔
(
X40
→
X41
)
)
↔
(
(
(
(
¬
(
X48
→
X41
→
(
X40
↔
False
)
)
)
→
X22
→
(
¬
X10
)
→
X41
)
→
(
¬
(
X48
↔
(
X41
↔
False
)
)
)
)
↔
(
X41
→
(
X40
↔
(
X10
↔
False
)
)
)
)
)
→
(
¬
(
(
(
X48
↔
(
X43
↔
(
(
¬
X6
)
↔
X40
)
)
)
↔
(
X40
→
(
¬
X15
)
→
(
¬
X6
)
→
X48
)
)
→
(
X15
↔
(
X6
↔
False
)
)
)
)
→
(
¬
(
X48
↔
(
X40
↔
(
X22
→
X43
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
X28
↔
(
(
¬
X3
)
↔
X8
)
)
)
→
(
¬
(
X48
→
(
X24
↔
(
X8
→
(
X3
↔
False
)
)
)
)
)
)
↔
(
(
¬
(
X28
↔
(
X18
→
X24
)
)
)
↔
(
(
X24
↔
(
(
¬
X11
)
↔
X48
)
)
↔
(
¬
(
(
¬
X28
)
→
X24
→
(
(
¬
X18
)
↔
X48
)
)
)
)
)
)
→
(
¬
(
X48
↔
(
(
¬
X18
)
→
(
X17
↔
X28
)
)
)
)
)
)
↔
(
(
(
¬
(
X40
→
X28
→
(
(
¬
X7
)
↔
X43
)
)
)
↔
(
X48
→
(
(
¬
X40
)
↔
(
X28
→
X43
)
)
)
)
↔
(
¬
(
(
X48
↔
(
X43
↔
X0
)
)
→
(
(
(
¬
(
(
¬
X40
)
↔
(
X37
↔
X43
)
)
)
→
(
X40
↔
(
(
¬
X21
)
↔
(
(
¬
X18
)
→
X37
)
)
)
)
↔
(
X37
→
X21
→
(
(
¬
X0
)
↔
X43
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
¬
(
X45
→
(
(
¬
X42
)
↔
(
(
¬
X12
)
→
X43
)
)
)
)
→
X12
→
(
¬
(
X45
→
X37
→
(
X16
↔
X43
)
)
)
)
)
→
X43
→
(
¬
X42
)
→
(
¬
X39
)
→
X45
)
)
→
(
(
(
(
¬
(
(
¬
X45
)
→
(
X37
↔
X46
)
)
)
↔
(
X45
↔
(
X42
→
(
¬
X26
)
→
X37
)
)
)
↔
(
X46
↔
(
X45
↔
X42
)
)
)
↔
(
(
¬
X45
)
↔
(
(
¬
X42
)
→
X46
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X45
→
(
(
¬
X43
)
↔
(
X14
↔
X46
)
)
)
)
→
X46
→
(
¬
X22
)
→
(
X14
↔
X43
)
)
)
→
(
¬
(
X22
→
(
X14
↔
X45
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
(
¬
(
X47
↔
(
X46
↔
(
(
¬
X2
)
→
X29
)
)
)
)
↔
(
X46
↔
(
X19
↔
(
X2
→
X47
)
)
)
)
↔
(
(
¬
X46
)
↔
(
X29
↔
(
X1
→
X47
)
)
)
)
→
(
(
¬
X33
)
↔
X46
)
)
→
(
(
¬
(
(
¬
X33
)
↔
(
(
¬
X19
)
↔
(
(
¬
X2
)
↔
X38
)
)
)
)
↔
(
X38
→
X33
→
(
X2
↔
X45
)
)
)
)
)
↔
(
(
¬
(
(
X31
→
(
(
¬
X5
)
↔
X35
)
)
→
(
X35
↔
(
X31
→
(
X13
↔
X46
)
)
)
)
)
→
(
¬
(
X46
→
(
¬
X31
)
→
(
(
¬
X9
)
↔
X47
)
)
)
→
(
¬
(
(
¬
X45
)
→
X9
→
(
X5
↔
X47
)
)
)
)
)
→
(
¬
(
(
(
X38
→
(
¬
X35
)
→
(
X34
↔
X49
)
)
↔
(
X38
→
(
X35
↔
(
X34
→
(
X4
↔
False
)
)
)
)
)
→
(
(
¬
(
X36
↔
(
X35
↔
(
X34
→
X23
→
False
)
)
)
)
↔
(
¬
(
X47
→
X34
→
(
¬
X23
)
→
X38
)
)
)
)
)
→
(
(
¬
(
(
X42
→
(
X36
↔
(
X35
↔
X38
)
)
)
→
(
X44
↔
(
X38
↔
(
X20
↔
X35
)
)
)
)
)
↔
(
¬
(
(
(
¬
X38
)
↔
(
X32
→
(
X27
↔
X42
)
)
)
→
(
¬
(
(
(
¬
X25
)
↔
X32
)
→
(
¬
(
(
¬
(
(
(
¬
(
X44
→
(
¬
X32
)
→
(
¬
X30
)
→
X42
)
)
→
X42
→
(
X38
↔
(
X32
↔
(
X30
↔
False
)
)
)
)
→
(
¬
(
X42
↔
(
X32
→
X30
→
False
)
)
)
)
)
→
(
X42
↔
(
X38
→
(
X25
↔
X44
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
(
X46
↔
(
X43
↔
(
X10
→
X38
)
)
)
↔
(
X38
↔
X48
)
)
↔
(
(
X43
↔
(
X38
→
(
X35
↔
X48
)
)
)
→
(
¬
(
(
¬
X38
)
→
(
¬
X35
)
→
(
(
¬
X2
)
↔
X48
)
)
)
)
)
↔
(
¬
(
X35
↔
(
X30
↔
(
X2
↔
X43
)
)
)
)
)
→
(
¬
(
(
X37
↔
(
X35
→
X3
→
False
)
)
→
(
¬
(
X40
↔
(
X37
→
(
X5
↔
(
X3
→
False
)
)
)
)
)
)
)
→
(
X35
↔
(
X18
↔
X46
)
)
→
(
(
(
¬
(
X43
→
(
(
¬
X37
)
↔
X46
)
)
)
↔
(
¬
(
(
¬
X37
)
→
(
(
¬
X18
)
↔
(
(
¬
X5
)
→
X43
)
)
)
)
)
↔
(
¬
(
X46
→
(
X37
↔
(
X35
→
False
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X47
→
(
X46
↔
(
X43
↔
X8
)
)
)
)
↔
(
(
X45
↔
(
X43
→
(
¬
X8
)
→
X47
)
)
↔
(
¬
(
X47
↔
(
X45
↔
(
(
¬
X7
)
↔
X43
)
)
)
)
)
)
→
(
(
X45
↔
(
X43
→
X34
→
False
)
)
↔
(
(
¬
(
X47
→
X20
→
X15
→
False
)
)
↔
(
(
X47
→
X43
→
X11
→
X45
)
↔
(
(
¬
X43
)
↔
(
(
¬
X15
)
↔
(
(
¬
X11
)
↔
X47
)
)
)
)
)
)
)
→
(
(
(
(
(
X46
→
X45
→
X32
→
False
)
↔
(
(
(
X46
↔
(
X28
→
(
(
¬
X16
)
↔
X45
)
)
)
↔
(
¬
(
X45
↔
(
(
¬
X32
)
↔
(
X16
→
X46
)
)
)
)
)
↔
(
¬
(
X46
→
(
(
¬
X28
)
↔
X45
)
)
)
)
)
↔
(
¬
(
X46
→
(
X32
↔
X45
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
(
X41
→
False
)
)
→
(
¬
(
(
X46
↔
(
(
¬
X16
)
↔
(
X14
↔
X28
)
)
)
→
X41
→
X14
→
False
)
)
)
)
→
(
X46
↔
(
X41
↔
(
X16
→
X1
→
False
)
)
)
)
)
↔
(
X28
→
X14
→
X46
)
)
)
↔
(
(
(
¬
(
X46
→
(
X41
↔
X6
)
)
)
↔
(
¬
(
(
(
¬
(
X46
↔
(
X6
→
(
X4
↔
X32
)
)
)
)
→
(
¬
X41
)
)
→
(
¬
X6
)
→
(
X4
↔
X32
)
)
)
)
↔
(
(
(
¬
X47
)
↔
X48
)
→
(
¬
(
X47
→
(
X46
↔
(
(
¬
X4
)
→
X6
)
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
X48
↔
(
X44
→
X33
→
(
X25
↔
False
)
)
)
→
X48
→
(
(
¬
X33
)
↔
(
X9
↔
X44
)
)
)
→
(
(
(
¬
X44
)
→
(
¬
X36
)
→
(
(
¬
X33
)
↔
X45
)
)
→
(
¬
(
(
(
¬
X45
)
→
(
¬
X33
)
→
X48
)
→
(
¬
(
X48
→
X36
→
(
X19
↔
X17
)
)
)
)
)
)
→
(
(
(
X44
↔
(
X12
→
X17
)
)
↔
(
¬
(
(
(
¬
X29
)
↔
(
(
¬
X22
)
→
(
¬
X12
)
→
X44
)
)
→
(
(
¬
(
(
¬
X44
)
→
(
¬
X17
)
→
X48
)
)
↔
(
¬
(
X29
→
(
X22
↔
(
(
¬
X17
)
→
X48
)
)
)
)
)
)
)
)
↔
(
X26
↔
(
X12
↔
X22
)
)
)
→
X29
→
(
(
¬
X22
)
↔
X26
)
)
)
→
(
(
(
¬
(
(
(
(
¬
X39
)
→
X13
→
X45
)
↔
(
X45
→
(
¬
X23
)
→
(
X13
↔
X48
)
)
)
→
(
¬
(
(
(
¬
(
X42
→
X23
)
)
→
(
(
X48
↔
(
(
¬
X42
)
→
(
¬
X23
)
→
X45
)
)
↔
(
¬
(
X46
→
(
¬
X45
)
→
X48
)
)
)
)
→
(
¬
(
(
¬
X23
)
→
X46
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X44
→
(
X39
↔
(
(
¬
X27
)
↔
X48
)
)
)
)
→
X49
→
X45
→
X39
→
False
)
)
)
↔
(
(
X49
→
(
X24
↔
(
(
¬
X0
)
↔
X48
)
)
)
↔
(
¬
(
(
(
X45
→
(
X31
↔
(
(
¬
X24
)
→
X49
)
)
)
↔
(
¬
(
X49
↔
X48
)
)
)
→
(
¬
(
(
¬
X45
)
↔
(
X24
↔
(
X21
→
X49
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
b01c0b...
and proposition id is
192ecf...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMU4bB8oS1SneG4C67wtjPuYQTexiwVBHVV
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
¬
(
(
(
(
(
(
(
¬
X47
)
→
X0
→
X48
)
→
(
¬
(
X46
→
X42
→
(
X8
↔
X48
)
)
)
)
↔
(
X46
↔
(
X42
↔
X48
)
)
)
↔
(
¬
(
(
(
¬
X37
)
↔
(
(
¬
X25
)
↔
(
(
¬
X18
)
↔
X47
)
)
)
→
(
(
X47
→
(
¬
X42
)
→
X46
)
↔
(
(
¬
(
(
¬
X42
)
→
(
¬
X25
)
→
X47
)
)
↔
(
¬
(
X47
→
X42
→
(
(
¬
X37
)
↔
X46
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X45
→
X38
→
X19
→
X48
)
)
↔
(
¬
(
X32
↔
(
(
¬
X19
)
→
X48
)
)
)
)
→
(
¬
X45
)
→
(
¬
X38
)
→
X48
)
)
↔
(
X46
→
(
X45
↔
(
X19
↔
X20
)
)
)
)
→
(
¬
(
(
(
X47
↔
(
X46
→
(
¬
X20
)
→
X48
)
)
→
X47
→
(
X20
↔
(
(
¬
X6
)
↔
X48
)
)
)
→
(
¬
(
X48
→
X47
→
(
X46
↔
False
)
)
)
)
)
)
→
(
(
(
¬
(
X44
↔
(
X36
↔
X12
)
)
)
↔
(
¬
(
X41
↔
(
(
¬
X36
)
→
X12
→
X48
)
)
)
)
↔
(
¬
(
(
X47
↔
(
X31
→
X17
→
X12
)
)
→
(
(
(
¬
(
(
X48
→
(
X17
↔
(
(
¬
X12
)
→
X35
)
)
)
→
(
X48
→
(
X10
↔
X41
)
)
→
X35
→
(
(
¬
X12
)
↔
(
(
¬
X10
)
→
X41
)
)
)
)
→
(
¬
(
X48
↔
(
(
¬
X35
)
→
X17
→
X47
)
)
)
)
↔
(
X12
↔
(
X10
→
(
¬
X3
)
→
X35
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
(
¬
X31
)
→
(
¬
X23
)
→
X45
)
→
X36
→
X30
→
(
(
¬
X23
)
↔
X31
)
)
)
→
(
X41
↔
(
(
¬
X31
)
↔
X36
)
)
)
)
↔
(
(
¬
(
(
¬
X31
)
→
(
X23
↔
(
X21
→
X41
)
)
)
)
↔
(
¬
(
X41
↔
(
X23
↔
(
X11
↔
X31
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
X45
)
↔
(
X43
↔
(
(
¬
X34
)
↔
X48
)
)
)
→
(
¬
(
(
¬
(
X34
→
(
X29
↔
X13
)
)
)
→
(
¬
X45
)
→
(
(
¬
X29
)
↔
(
X16
→
X48
)
)
)
)
)
)
→
(
(
(
¬
(
(
(
X49
→
(
¬
X33
)
→
X46
)
↔
(
¬
(
X49
→
(
X46
↔
(
(
¬
X22
)
→
X34
)
)
)
)
)
→
(
(
X48
→
(
(
¬
X34
)
↔
X49
)
)
↔
(
¬
(
(
(
¬
(
X48
↔
(
X34
↔
(
X33
→
X9
→
False
)
)
)
)
↔
(
¬
(
(
¬
X48
)
→
(
(
¬
X34
)
↔
(
X9
→
X49
)
)
)
)
)
→
(
X48
↔
(
X33
→
X9
→
False
)
)
)
)
)
)
)
→
X49
→
(
¬
X46
)
→
(
(
¬
X24
)
↔
X48
)
)
↔
(
(
(
¬
(
X48
→
False
)
)
↔
(
(
¬
(
X49
↔
(
(
¬
X33
)
→
X5
→
X48
)
)
)
↔
(
(
¬
X48
)
↔
(
(
¬
X45
)
→
(
¬
X5
)
→
X49
)
)
)
)
↔
(
¬
(
(
¬
X45
)
↔
(
(
¬
X33
)
→
(
(
¬
X2
)
↔
X48
)
)
)
)
)
)
)
)
→
(
(
(
X43
↔
(
(
¬
X14
)
↔
(
(
¬
X7
)
↔
X40
)
)
)
→
(
¬
(
X28
→
(
X1
↔
False
)
)
)
→
(
(
X14
↔
(
X1
↔
X40
)
)
↔
(
(
(
(
¬
(
(
¬
X39
)
→
(
(
¬
X7
)
↔
(
(
¬
X1
)
→
X40
)
)
)
)
↔
(
¬
(
(
¬
X14
)
↔
X39
)
)
)
↔
(
¬
(
(
¬
X39
)
→
X14
→
X40
)
)
)
↔
(
X7
→
X1
→
X14
)
)
)
)
↔
(
¬
(
(
¬
(
X15
→
X28
)
)
→
(
¬
(
(
¬
(
X40
↔
(
X28
→
(
X15
↔
X27
)
)
)
)
→
(
¬
(
X43
→
X28
)
)
)
)
)
)
)
→
(
(
(
(
¬
X40
)
↔
(
(
¬
X4
)
→
X46
)
)
→
(
X48
→
(
X4
↔
False
)
)
→
(
¬
(
(
X40
↔
(
(
¬
X26
)
↔
X48
)
)
→
(
(
¬
(
X48
→
(
(
¬
X26
)
↔
(
X4
→
X40
)
)
)
)
↔
(
¬
(
X48
↔
(
(
¬
X40
)
→
(
X26
↔
X46
)
)
)
)
)
)
)
)
↔
(
(
X46
→
(
¬
X43
)
→
X48
)
→
(
(
¬
(
(
¬
X40
)
↔
(
(
¬
X39
)
→
X46
)
)
)
↔
(
¬
(
X46
→
(
X43
↔
(
X40
→
X48
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
(
(
X44
→
X42
→
(
(
¬
X37
)
↔
X43
)
)
↔
(
¬
(
X43
↔
(
(
¬
X14
)
→
X42
)
)
)
)
→
(
¬
(
X47
→
(
X44
↔
(
(
¬
X37
)
→
X45
)
)
)
)
)
)
→
(
(
¬
X43
)
→
(
¬
X38
)
→
X44
)
→
(
(
¬
X38
)
↔
(
X2
→
X43
)
)
→
(
(
(
¬
(
(
(
¬
(
X44
↔
(
X38
↔
(
(
¬
X2
)
↔
X43
)
)
)
)
↔
(
X45
→
(
X44
↔
(
X2
→
X43
)
)
)
)
→
(
¬
(
(
¬
X38
)
→
(
(
¬
X2
)
↔
X44
)
)
)
)
)
→
X43
→
X38
)
↔
(
(
¬
X38
)
↔
X45
)
)
)
↔
(
¬
(
(
¬
(
(
X47
↔
(
X21
↔
(
(
¬
X18
)
→
X26
)
)
)
→
(
X47
↔
(
(
¬
X26
)
→
(
X21
↔
X37
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
(
X49
↔
(
X38
↔
(
X13
→
X47
)
)
)
)
→
(
¬
(
X38
→
(
X26
↔
(
(
¬
X21
)
→
X47
)
)
)
)
)
)
↔
(
¬
(
(
¬
X38
)
→
(
(
¬
X30
)
↔
(
X26
→
X47
)
)
)
)
)
→
(
¬
(
X38
→
X26
)
)
)
)
)
)
)
↔
(
(
(
(
(
X45
↔
(
(
¬
X32
)
→
(
¬
X20
)
→
X44
)
)
↔
(
(
¬
X44
)
→
(
X20
↔
(
(
¬
X19
)
→
X45
)
)
)
)
↔
(
X20
↔
(
X19
↔
False
)
)
)
↔
(
(
¬
(
(
¬
(
X49
→
(
¬
X39
)
→
X32
→
X46
)
)
→
(
X49
↔
(
X39
→
X35
→
X32
)
)
)
)
↔
(
¬
(
X49
→
(
¬
X39
)
→
(
¬
X20
)
→
X46
)
)
)
)
↔
(
(
(
¬
X40
)
↔
(
(
¬
X39
)
→
(
¬
X10
)
→
X46
)
)
↔
(
(
X40
→
X15
→
X10
)
↔
(
(
¬
(
X40
→
(
X39
↔
(
X15
→
X0
→
False
)
)
)
)
↔
(
X40
→
X45
)
)
)
)
)
)
→
(
(
(
(
(
X33
↔
(
(
¬
X29
)
→
X43
)
)
→
X43
→
(
(
(
¬
X33
)
↔
(
X29
→
X43
)
)
↔
(
¬
(
X36
↔
(
(
¬
X5
)
→
X43
)
)
)
)
)
→
(
(
X43
→
X41
→
(
X33
↔
(
X29
↔
False
)
)
)
↔
(
(
(
¬
X33
)
↔
(
(
¬
X29
)
↔
X41
)
)
↔
(
(
(
¬
X29
)
→
(
(
¬
X23
)
↔
(
X16
↔
X41
)
)
)
↔
(
X41
→
(
X29
↔
(
X23
↔
X33
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X36
→
X27
→
X45
)
)
↔
(
X43
↔
(
X27
→
X1
→
X36
)
)
)
↔
(
(
(
(
¬
(
X36
→
X27
→
(
X11
↔
(
X6
↔
False
)
)
)
)
→
(
(
¬
(
X43
↔
(
X33
↔
(
X11
→
False
)
)
)
)
↔
(
X33
↔
(
(
¬
X27
)
↔
(
(
¬
X11
)
↔
X36
)
)
)
)
)
↔
(
X36
↔
(
(
¬
X33
)
↔
(
X3
↔
X43
)
)
)
)
→
(
X36
↔
(
X27
→
(
X22
↔
False
)
)
)
→
(
(
X36
→
(
X27
↔
False
)
)
↔
(
¬
(
(
¬
X22
)
→
X12
→
X43
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
(
¬
X45
)
→
(
¬
X34
)
→
X49
)
→
(
¬
(
(
¬
X34
)
→
(
(
¬
X4
)
↔
X36
)
)
)
)
)
→
(
¬
(
X49
→
(
X41
↔
(
X4
→
X34
)
)
)
)
)
→
(
¬
(
X41
↔
X34
)
)
)
↔
(
¬
(
(
X49
↔
(
X48
→
(
X41
↔
X4
)
)
)
→
(
X49
↔
(
(
¬
X4
)
↔
X17
)
)
)
)
)
→
(
¬
(
(
X49
↔
(
X45
↔
X48
)
)
→
(
(
¬
(
X49
↔
(
(
¬
X28
)
↔
(
X7
→
X45
)
)
)
)
↔
(
(
X49
↔
(
X48
→
(
¬
X28
)
→
X45
)
)
↔
(
X48
↔
(
X45
→
(
X9
↔
X49
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X48
)
→
(
¬
X45
)
→
(
¬
X8
)
→
X49
)
)
→
(
(
¬
(
X49
→
(
X8
↔
X31
)
)
)
↔
(
X49
→
(
X45
↔
(
X24
→
X25
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
111ce4...
and proposition id is
66ae0d...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMUfV1Y5rLD5pzgkZLm571NDcefJdd63BUf
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
¬
(
(
¬
(
(
(
¬
(
(
¬
(
X46
↔
(
(
¬
X30
)
→
(
¬
X26
)
→
X49
)
)
)
→
(
(
X30
→
(
(
¬
X26
)
↔
(
X4
→
X46
)
)
)
↔
(
¬
(
(
¬
X26
)
→
(
¬
X4
)
→
X44
)
)
)
→
(
(
X30
→
(
(
¬
X26
)
↔
(
X15
→
X48
)
)
)
↔
(
¬
(
X48
→
X30
→
(
(
¬
X4
)
↔
X15
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
(
¬
(
X48
↔
(
X43
→
False
)
)
)
↔
(
(
X48
→
(
X47
↔
X6
)
)
→
(
X48
↔
(
(
¬
X46
)
↔
(
(
¬
X43
)
↔
X47
)
)
)
)
)
→
(
X47
↔
(
X46
→
X24
→
False
)
)
)
)
→
(
X48
↔
(
(
¬
X46
)
↔
(
X24
→
X49
)
)
)
)
→
(
¬
(
X48
↔
(
X46
↔
(
X43
↔
(
X7
→
False
)
)
)
)
)
)
)
↔
(
(
(
¬
X46
)
↔
(
X20
↔
(
X1
→
X49
)
)
)
↔
(
(
(
¬
X46
)
→
X41
→
X36
→
X47
)
↔
(
¬
(
X46
→
(
X41
↔
(
X5
↔
X47
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X44
)
↔
(
X39
→
(
¬
X23
)
→
X47
)
)
)
↔
(
X44
↔
(
(
¬
X23
)
→
X38
)
)
)
↔
(
X47
→
(
X44
↔
(
(
¬
X32
)
→
X39
)
)
)
)
→
(
X44
↔
(
X21
→
(
X10
↔
X49
)
)
)
)
)
→
(
(
(
(
¬
(
(
(
¬
(
(
¬
X42
)
→
(
(
¬
X34
)
↔
(
(
¬
X33
)
↔
X46
)
)
)
)
→
(
X42
→
(
¬
X25
)
→
(
(
¬
X13
)
↔
X34
)
)
→
(
X34
↔
(
(
¬
X13
)
→
X33
)
)
)
→
X42
→
X40
→
(
(
¬
X33
)
↔
X34
)
)
)
↔
(
(
¬
(
(
¬
(
X37
↔
(
X8
→
X43
)
)
)
→
X46
→
X43
→
X42
→
(
X8
↔
False
)
)
)
↔
(
¬
(
X37
→
(
(
¬
X25
)
↔
(
(
¬
X8
)
↔
X43
)
)
)
)
)
)
↔
(
(
(
X43
→
(
X42
↔
(
X28
↔
X46
)
)
)
↔
(
(
X43
→
X42
→
X28
→
X22
)
→
(
¬
(
X45
→
(
X42
↔
(
X12
→
X22
)
)
)
)
)
)
→
(
(
¬
(
X42
→
(
¬
X35
)
→
(
(
¬
X2
)
↔
X43
)
)
)
↔
(
¬
(
X45
↔
(
X43
→
(
X2
↔
X35
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
X19
↔
(
X11
↔
(
X9
↔
X0
)
)
)
→
(
X19
↔
(
(
¬
X9
)
↔
(
X0
↔
X42
)
)
)
)
)
↔
(
¬
(
X42
→
X11
→
X19
)
)
)
→
(
¬
(
(
¬
(
X46
↔
(
X9
↔
(
(
¬
X3
)
↔
X48
)
)
)
)
→
(
X42
↔
(
(
¬
X40
)
→
(
(
¬
X11
)
↔
X48
)
)
)
)
)
)
)
→
(
(
¬
(
X40
↔
(
X27
→
X18
→
False
)
)
)
↔
(
(
(
(
(
¬
(
X42
↔
(
X18
↔
X48
)
)
)
↔
(
(
¬
(
(
¬
X42
)
→
(
X40
↔
(
(
¬
X17
)
→
X48
)
)
)
)
↔
(
¬
(
X48
→
(
X42
↔
(
X18
↔
False
)
)
)
)
)
)
↔
(
X42
↔
(
X18
↔
(
X17
→
False
)
)
)
)
↔
(
¬
(
X42
↔
(
(
¬
X18
)
↔
(
(
¬
X16
)
↔
X31
)
)
)
)
)
→
(
(
(
¬
(
X27
↔
(
X14
↔
X31
)
)
)
↔
(
¬
(
(
X40
↔
(
(
¬
X27
)
↔
(
X14
→
X29
)
)
)
→
(
X31
↔
(
(
¬
X27
)
→
(
X14
↔
X40
)
)
)
)
)
)
↔
(
(
¬
X31
)
↔
(
(
¬
X29
)
→
X42
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
(
(
¬
(
(
X44
↔
(
X12
↔
(
(
¬
X6
)
↔
X47
)
)
)
→
(
¬
(
(
(
¬
(
X46
→
X41
)
)
↔
(
(
¬
(
X44
↔
(
(
¬
X41
)
→
(
X6
↔
X46
)
)
)
)
↔
(
¬
(
X47
↔
(
X44
↔
X46
)
)
)
)
)
→
(
¬
(
X41
→
X47
)
)
)
)
)
)
→
(
¬
(
X46
→
X44
→
X41
)
)
)
→
(
X44
→
(
X41
↔
(
(
¬
X25
)
→
X46
)
)
)
→
X41
→
(
X23
↔
False
)
)
)
↔
(
(
(
¬
(
(
¬
X44
)
→
(
(
¬
X41
)
↔
X46
)
)
)
↔
(
(
X38
↔
False
)
↔
(
(
¬
(
X44
↔
(
X38
→
X30
)
)
)
↔
(
X46
↔
(
(
¬
X41
)
↔
(
X38
↔
X44
)
)
)
)
)
)
↔
(
X44
↔
(
(
¬
X32
)
→
X41
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
X45
)
↔
(
(
¬
X41
)
↔
X47
)
)
)
→
(
¬
(
X45
↔
(
X41
→
(
X11
↔
X46
)
)
)
)
)
)
↔
(
(
¬
(
X41
↔
(
(
¬
X36
)
↔
(
(
¬
X31
)
↔
X45
)
)
)
)
→
(
¬
(
X47
↔
(
X36
↔
(
X31
↔
X45
)
)
)
)
)
)
)
)
↔
(
(
(
X45
→
(
X17
↔
X22
)
)
→
(
(
(
¬
(
(
¬
X40
)
→
(
(
¬
X22
)
↔
X43
)
)
)
↔
(
¬
(
(
¬
(
X40
↔
(
X35
↔
(
X22
→
False
)
)
)
)
→
(
¬
(
X43
→
(
X35
↔
X2
)
)
)
)
)
)
↔
(
(
¬
(
X40
↔
(
X17
↔
X43
)
)
)
→
(
X27
↔
(
(
¬
X22
)
↔
(
X17
→
X45
)
)
)
)
)
)
→
(
(
(
¬
X45
)
→
(
X44
↔
(
(
¬
X28
)
↔
X46
)
)
)
↔
(
(
¬
(
(
(
¬
X44
)
→
(
X43
↔
(
X13
→
X45
)
)
)
→
(
¬
(
X45
↔
(
X15
↔
X9
)
)
)
)
)
↔
(
(
¬
(
X43
→
X40
→
X15
→
X46
)
)
→
(
¬
(
(
¬
X44
)
→
X48
)
)
)
)
)
→
(
(
(
(
X34
→
(
(
¬
X26
)
↔
(
X4
↔
X44
)
)
)
↔
(
(
¬
X1
)
↔
X44
)
)
→
(
¬
(
X26
↔
X44
)
)
)
↔
(
¬
(
(
(
¬
X44
)
↔
(
(
¬
X4
)
↔
X46
)
)
→
(
(
X46
↔
(
X7
→
X44
)
)
↔
(
(
¬
(
(
¬
X44
)
↔
(
(
¬
X34
)
↔
(
X7
↔
X46
)
)
)
)
↔
(
¬
(
X34
→
X4
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
X44
)
→
(
X29
↔
(
X8
→
X45
)
)
)
→
(
(
¬
(
X44
→
(
X29
↔
(
X7
↔
X8
)
)
)
)
↔
(
¬
(
X46
→
X8
→
X44
)
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X19
↔
X48
)
)
→
(
¬
(
(
¬
(
X42
↔
(
(
¬
X37
)
↔
(
(
¬
X19
)
↔
X47
)
)
)
)
→
(
¬
(
X42
→
X20
→
X19
→
X48
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X47
↔
(
(
¬
X45
)
→
X18
→
X49
)
)
)
→
(
(
X47
↔
(
(
¬
X37
)
↔
X49
)
)
↔
(
¬
(
(
¬
(
(
¬
X37
)
↔
(
X0
→
X49
)
)
)
→
(
¬
(
X37
↔
(
(
¬
X18
)
↔
(
(
¬
X0
)
→
X47
)
)
)
)
)
)
)
)
)
↔
(
(
(
X49
→
(
X45
↔
(
X20
→
X21
)
)
)
↔
(
¬
(
X49
↔
(
X47
→
(
(
¬
X21
)
↔
X45
)
)
)
)
)
↔
(
¬
(
X47
→
X45
→
(
¬
X3
)
→
X49
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X47
↔
(
X33
→
(
(
¬
X16
)
↔
X48
)
)
)
)
↔
(
(
(
¬
(
X47
↔
(
X33
→
(
X24
↔
False
)
)
)
)
↔
(
¬
(
X47
↔
(
X24
↔
(
X14
↔
X39
)
)
)
)
)
→
(
¬
(
X48
→
(
(
¬
X39
)
↔
(
(
¬
X24
)
↔
X47
)
)
)
)
)
)
→
(
¬
(
X47
↔
(
X5
→
X24
)
)
)
→
(
X45
↔
(
X39
→
(
X24
↔
False
)
)
)
)
)
→
X48
→
(
X47
↔
(
X45
→
X10
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
5f0bc7...
and proposition id is
d4c3c3...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMdH3JC4jWmhB9c5MmQR2MnFzVa3MsincGk
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
∀X51 :
prop
,
(
(
(
(
(
¬
(
(
(
X45
↔
(
(
¬
X41
)
↔
(
(
¬
X32
)
→
X51
)
)
)
↔
(
¬
(
(
X41
↔
(
(
¬
X32
)
↔
(
X29
↔
X49
)
)
)
→
(
X45
↔
(
X41
↔
(
(
¬
X29
)
→
X32
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X51
↔
False
)
)
↔
(
(
¬
(
X31
→
X13
→
X22
)
)
→
(
(
(
¬
X45
)
→
(
¬
X31
)
→
(
(
¬
X22
)
↔
X51
)
)
↔
(
¬
(
X31
→
X13
→
X22
)
)
)
)
)
→
(
¬
(
X45
↔
(
X32
→
(
X31
↔
(
X22
→
False
)
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
X51
↔
(
(
¬
X42
)
→
X17
→
X45
)
)
)
↔
(
X51
→
(
¬
X42
)
→
(
(
¬
X17
)
↔
X49
)
)
)
↔
(
(
¬
X49
)
→
(
(
(
X47
→
X45
)
→
(
¬
(
X51
↔
(
X47
→
X49
)
)
)
→
(
X51
↔
(
(
¬
X9
)
↔
X45
)
)
)
→
(
¬
(
X45
↔
False
)
)
)
→
(
¬
(
(
¬
X45
)
→
(
¬
X9
)
→
X47
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
(
(
(
¬
(
X39
→
(
X21
↔
(
X5
→
X42
)
)
)
)
↔
(
(
¬
X42
)
↔
(
(
¬
X39
)
→
X21
→
X51
)
)
)
↔
(
X42
↔
(
X41
↔
X39
)
)
)
→
(
¬
(
(
¬
X42
)
↔
(
X39
↔
(
(
¬
X21
)
↔
X50
)
)
)
)
)
)
↔
(
¬
(
X51
→
(
X42
↔
(
(
¬
X36
)
↔
X50
)
)
)
)
)
→
(
(
(
(
¬
X28
)
→
(
(
¬
X12
)
↔
X41
)
)
↔
(
X51
→
(
(
¬
X28
)
↔
(
(
¬
X12
)
→
X42
)
)
)
)
↔
(
(
X51
↔
(
X30
↔
(
X24
↔
X28
)
)
)
↔
(
¬
(
X42
→
X30
→
(
¬
X28
)
→
X51
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X39
↔
(
(
¬
X20
)
↔
(
X15
↔
X51
)
)
)
)
→
(
¬
(
X20
↔
(
(
¬
X19
)
↔
(
(
¬
X15
)
↔
X25
)
)
)
)
)
)
↔
(
(
X51
→
(
(
¬
X34
)
↔
(
(
¬
X20
)
↔
X41
)
)
)
↔
(
X51
→
(
X34
↔
(
X19
→
X41
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X51
→
(
(
¬
X18
)
↔
(
X16
→
X39
)
)
)
)
→
(
¬
(
X25
↔
(
(
¬
X16
)
→
X39
)
)
)
)
)
→
(
¬
X39
)
→
(
(
¬
X18
)
↔
(
X7
↔
X51
)
)
)
)
)
)
)
↔
(
(
(
(
(
X48
↔
(
X40
→
(
X1
↔
X44
)
)
)
↔
(
¬
(
(
X51
↔
(
X48
↔
(
(
¬
X14
)
↔
X44
)
)
)
→
(
X48
→
(
¬
X40
)
→
X2
→
X46
)
→
(
(
X51
→
(
¬
X46
)
→
(
(
¬
X44
)
↔
X48
)
)
↔
(
X46
→
X2
→
False
)
)
)
)
)
↔
(
(
¬
(
(
¬
X46
)
→
(
¬
X40
)
→
X51
)
)
↔
(
(
¬
(
X37
→
X4
→
X40
)
)
→
(
¬
(
(
(
X51
↔
(
(
¬
X37
)
↔
(
X4
→
X46
)
)
)
↔
(
¬
(
(
¬
X40
)
→
X37
→
X4
→
X46
)
)
)
→
X4
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X48
↔
(
X38
↔
(
X8
↔
False
)
)
)
)
↔
(
(
(
X46
↔
(
X38
→
X23
)
)
↔
(
(
¬
(
(
¬
X46
)
→
(
(
¬
X38
)
↔
(
(
¬
X8
)
→
X48
)
)
)
)
→
(
¬
(
X38
→
(
X23
↔
(
(
¬
X11
)
→
X46
)
)
)
)
)
)
→
(
¬
(
X48
→
X46
)
)
)
)
→
(
(
¬
(
(
¬
X44
)
↔
(
X33
↔
(
(
¬
X10
)
↔
X50
)
)
)
)
↔
(
¬
(
X44
↔
(
X38
↔
(
(
¬
X33
)
→
X50
)
)
)
)
)
→
(
¬
(
X44
↔
(
X38
↔
(
X3
↔
X50
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X35
)
↔
(
X27
↔
(
(
¬
X26
)
→
X50
)
)
)
)
→
(
¬
(
X50
→
X49
)
)
)
↔
(
(
¬
(
X49
↔
(
(
¬
X27
)
→
(
(
¬
X6
)
↔
X50
)
)
)
)
→
(
(
X49
↔
(
(
¬
X43
)
↔
(
X0
↔
X50
)
)
)
↔
(
¬
(
(
¬
(
X51
→
(
¬
X46
)
→
X49
)
)
→
X51
→
(
X43
↔
(
(
¬
X0
)
→
X49
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
¬
(
X30
→
(
X22
↔
X43
)
)
)
↔
(
¬
(
(
(
(
(
X43
→
(
¬
X30
)
→
(
¬
X21
)
→
X40
)
↔
(
¬
(
X40
→
(
(
¬
X30
)
↔
(
(
¬
X14
)
→
X43
)
)
)
)
)
↔
(
¬
(
X43
→
(
X40
↔
False
)
)
)
)
→
(
¬
(
X40
↔
(
(
¬
X21
)
↔
X30
)
)
)
)
→
(
X43
↔
(
X40
↔
(
X21
→
X22
)
)
)
)
)
)
→
(
¬
(
(
¬
X40
)
→
(
X35
↔
X48
)
)
)
→
(
(
(
(
¬
X35
)
↔
(
(
¬
X30
)
↔
(
X22
→
X48
)
)
)
↔
(
(
¬
(
(
¬
(
X47
→
(
X40
↔
(
X30
↔
False
)
)
)
)
→
(
(
¬
X9
)
↔
(
¬
(
(
¬
(
(
X47
→
(
¬
X9
)
→
X35
)
→
(
¬
(
X40
↔
X22
)
)
)
)
→
(
X40
↔
(
X35
↔
(
X30
→
X47
)
)
)
)
)
)
)
)
↔
(
X35
↔
(
X30
→
(
X22
↔
False
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X48
→
(
(
¬
X40
)
↔
(
(
¬
X18
)
↔
X47
)
)
)
)
→
(
¬
(
X40
↔
(
X18
→
X11
→
X48
)
)
)
)
)
→
(
(
¬
X40
)
↔
(
X11
→
X47
)
)
)
↔
(
X48
→
(
¬
X18
)
→
X40
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X35
)
↔
(
X22
→
(
X15
↔
X46
)
)
)
)
→
X50
→
(
(
¬
X15
)
↔
X40
)
)
)
→
X40
→
(
(
¬
X7
)
↔
X22
)
)
)
→
(
(
(
¬
(
(
¬
(
(
¬
(
(
¬
X34
)
→
(
X33
↔
X45
)
)
)
→
(
¬
(
(
¬
X34
)
↔
(
(
¬
X20
)
↔
(
(
¬
X13
)
→
X45
)
)
)
)
)
)
→
(
(
X47
↔
(
X3
↔
X26
)
)
↔
(
X34
→
(
X13
↔
X3
)
)
)
)
)
→
(
(
(
¬
(
X47
↔
(
X45
↔
(
X36
→
X34
)
)
)
)
↔
(
(
(
X47
↔
(
X36
→
(
(
¬
X19
)
↔
X45
)
)
)
↔
(
¬
(
X50
↔
(
(
¬
X19
)
→
X47
)
)
)
)
→
(
X45
↔
(
X19
→
False
)
)
)
)
↔
(
¬
(
X50
→
(
X47
↔
(
X27
→
X36
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
¬
X23
)
→
X36
)
)
→
(
(
X48
↔
(
(
¬
X36
)
→
(
¬
X28
)
→
X50
)
)
↔
(
¬
(
X50
↔
(
X48
→
X29
)
)
)
)
)
)
→
(
¬
(
X50
→
X48
→
X36
→
X47
)
)
)
↔
(
(
¬
(
(
¬
X48
)
↔
(
(
¬
X36
)
↔
X50
)
)
)
↔
(
(
X48
↔
(
X16
→
(
X8
↔
X36
)
)
)
→
X48
→
(
X16
↔
(
X8
↔
X50
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
(
(
¬
(
(
¬
X43
)
↔
(
(
¬
X25
)
↔
(
(
¬
X24
)
→
X49
)
)
)
)
↔
(
X43
↔
(
X25
→
(
¬
X12
)
→
X49
)
)
)
↔
(
¬
(
X49
→
(
X24
↔
False
)
)
)
)
↔
(
¬
(
X49
↔
(
X48
↔
(
X43
↔
(
X25
→
False
)
)
)
)
)
)
↔
(
(
¬
(
X47
↔
(
X5
→
X49
)
)
)
↔
(
X25
↔
(
X12
↔
X5
)
)
)
)
→
(
¬
(
X37
→
(
(
¬
X25
)
↔
(
X12
↔
X49
)
)
)
)
)
)
→
(
(
(
(
¬
(
X49
→
X47
→
X48
)
)
↔
(
(
¬
X47
)
↔
(
¬
(
(
¬
(
X48
↔
(
X47
↔
(
X41
→
False
)
)
)
)
→
(
X48
↔
(
(
¬
X31
)
↔
X49
)
)
)
)
)
)
↔
(
¬
(
X49
→
X48
→
X47
→
False
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X41
)
→
X48
)
)
)
)
→
(
¬
(
X48
→
(
X47
↔
X1
)
)
)
→
(
¬
X47
)
→
(
¬
X4
)
→
(
¬
X1
)
→
X49
)
→
(
(
¬
(
(
(
(
¬
(
X42
→
X6
→
X39
)
)
↔
(
¬
(
X48
→
(
(
¬
X39
)
↔
X47
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
X39
↔
X48
)
→
(
¬
(
X49
↔
(
X10
↔
(
X0
↔
X48
)
)
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X39
)
→
X42
)
)
)
)
)
→
(
(
¬
X10
)
↔
(
(
¬
X0
)
↔
X49
)
)
)
)
)
→
(
(
X47
→
(
X39
↔
(
X38
↔
X49
)
)
)
↔
(
¬
(
X47
→
(
X38
↔
X32
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X48
↔
(
X39
↔
(
(
¬
X17
)
→
X49
)
)
)
)
→
(
X39
↔
X51
)
)
)
→
X49
→
(
¬
X44
)
→
(
(
¬
X39
)
↔
X48
)
)
↔
(
¬
(
X51
→
(
X44
↔
(
X17
→
X2
→
False
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
03c7cf...
and proposition id is
b7fcf5...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMVCC9kJrBXpS6yJ8EtbsvefppH6pkQwJGi
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
¬
∀X51 :
prop
,
(
¬
∀X52 :
prop
,
∀X53 :
prop
,
(
(
(
(
(
¬
(
(
(
(
(
(
¬
X10
)
→
(
(
¬
X2
)
↔
X45
)
)
→
(
¬
(
(
¬
(
X52
→
(
X45
↔
(
X10
→
(
X2
↔
False
)
)
)
)
)
→
X52
→
X42
→
(
(
¬
X10
)
↔
X45
)
)
)
)
↔
(
(
(
¬
X50
)
↔
(
X29
→
X10
→
X52
)
)
↔
(
X42
→
(
X10
↔
X50
)
)
)
)
↔
(
(
¬
(
X45
→
X42
→
(
¬
X36
)
→
X50
)
)
→
(
(
¬
X43
)
↔
(
(
¬
X36
)
→
X50
)
)
)
)
→
(
(
¬
(
(
¬
X45
)
→
(
(
¬
X38
)
↔
(
X35
↔
X52
)
)
)
)
↔
(
(
(
(
X43
↔
(
X5
→
X45
)
)
→
(
(
¬
X43
)
↔
(
(
¬
X38
)
→
(
¬
X5
)
→
X52
)
)
)
↔
(
¬
(
X45
↔
(
X38
→
X52
)
)
)
)
↔
(
X45
→
(
X43
↔
(
X32
→
X52
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
X48
→
(
X43
↔
(
X33
↔
X49
)
)
)
→
(
¬
(
(
¬
X48
)
→
(
¬
X43
)
→
(
¬
X22
)
→
X49
)
)
)
→
(
¬
(
X49
↔
(
X48
→
(
(
¬
X43
)
↔
X52
)
)
)
)
)
)
↔
(
(
X52
→
(
(
¬
X48
)
↔
(
(
¬
X23
)
↔
X49
)
)
)
↔
(
(
(
¬
X49
)
↔
(
(
¬
X48
)
↔
(
(
¬
X3
)
↔
X52
)
)
)
↔
(
¬
(
(
¬
(
X52
↔
X48
)
)
→
(
¬
(
(
¬
X45
)
→
(
X26
↔
X49
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X52
→
(
X49
↔
(
X30
→
X27
→
False
)
)
)
)
→
(
¬
X27
)
→
(
X12
↔
X30
)
)
→
(
X53
↔
(
X52
→
(
X50
↔
X30
)
)
)
→
(
¬
(
X50
→
(
X49
↔
False
)
)
)
→
(
¬
(
X52
→
X46
→
X53
)
)
)
↔
(
(
(
(
¬
X46
)
→
(
¬
X41
)
→
(
X7
↔
X53
)
)
↔
(
(
¬
(
X49
→
X41
→
X25
→
X46
)
)
→
(
¬
(
X53
↔
(
X49
→
(
X46
↔
False
)
)
)
)
)
)
→
(
¬
(
X53
↔
(
X47
↔
(
X46
→
(
X7
↔
False
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
¬
(
X49
↔
(
X6
→
(
(
¬
X1
)
↔
X51
)
)
)
)
→
(
X53
↔
(
(
¬
X6
)
↔
(
X1
→
X21
)
)
)
)
)
→
(
¬
(
X53
→
X49
→
X21
)
)
)
)
→
(
¬
(
(
(
¬
X49
)
↔
(
(
¬
X41
)
↔
(
(
¬
X20
)
↔
X53
)
)
)
→
(
X49
↔
(
X41
→
(
¬
X6
)
→
X53
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X44
→
(
(
¬
X37
)
↔
(
(
¬
X15
)
↔
X51
)
)
)
)
↔
(
(
(
¬
X47
)
→
(
(
¬
X15
)
↔
(
(
¬
X8
)
↔
X51
)
)
)
↔
(
(
¬
(
X51
→
X47
→
(
X15
↔
False
)
)
)
↔
(
(
(
¬
X44
)
↔
(
X18
→
X47
)
)
↔
(
(
(
¬
X47
)
↔
(
X18
→
X51
)
)
↔
(
(
¬
(
X47
→
(
X44
↔
(
X15
↔
X18
)
)
)
)
↔
(
¬
(
X51
↔
(
X44
→
(
(
¬
X15
)
↔
X18
)
)
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
X13
→
(
X0
↔
X51
)
)
)
→
(
(
X44
→
(
X13
↔
(
X0
→
X51
)
)
)
↔
(
¬
(
X51
↔
(
X37
↔
(
X13
→
X0
)
)
)
)
)
)
↔
(
(
X44
→
(
X13
↔
X34
)
)
→
X44
→
(
X37
↔
(
(
¬
X9
)
→
X34
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
(
(
¬
(
X47
→
X39
→
X4
→
False
)
)
→
(
¬
(
X28
→
X51
)
)
)
)
↔
(
X39
↔
(
X28
→
X51
)
)
)
→
(
X47
↔
(
X28
↔
(
(
¬
X24
)
→
X51
)
)
)
)
↔
(
(
X40
↔
(
(
¬
X31
)
↔
(
X19
↔
X47
)
)
)
↔
(
¬
(
(
X40
↔
(
(
¬
X19
)
↔
(
X11
→
X24
)
)
)
→
(
(
(
¬
(
X24
↔
X11
)
)
→
(
X40
↔
(
(
¬
X28
)
↔
X31
)
)
)
↔
(
X40
→
(
X31
↔
(
X28
→
False
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
X48
→
(
X40
↔
(
(
¬
X16
)
→
X47
)
)
)
→
(
X48
→
X47
→
X40
→
X17
)
→
X40
→
(
X14
↔
X17
)
)
→
X48
→
X47
→
(
¬
X37
)
→
X40
)
)
↔
(
X51
↔
(
(
¬
X40
)
→
(
(
¬
X17
)
↔
X47
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
(
X50
↔
(
X49
→
(
¬
X11
)
→
X46
)
)
→
(
X49
↔
(
(
¬
X19
)
↔
(
(
¬
X11
)
↔
X46
)
)
)
)
)
↔
(
(
(
¬
(
X50
→
(
(
¬
X44
)
↔
(
(
¬
X27
)
→
X46
)
)
)
)
↔
(
X50
→
X46
→
X19
→
False
)
)
↔
(
¬
(
(
¬
X27
)
→
(
¬
X19
)
→
X44
)
)
)
)
↔
(
¬
(
(
¬
(
X50
→
(
X44
↔
(
X39
→
False
)
)
)
)
→
(
¬
(
(
(
¬
X27
)
→
(
X12
↔
X44
)
)
→
(
X50
↔
(
(
¬
X12
)
→
X15
)
)
)
)
)
)
)
→
(
(
(
¬
X44
)
↔
(
(
¬
X25
)
↔
(
X21
→
X45
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X46
)
→
(
¬
X45
)
→
(
(
¬
X44
)
↔
X50
)
)
)
↔
(
¬
(
X50
↔
(
(
¬
X45
)
↔
(
X26
↔
X46
)
)
)
)
)
→
(
¬
(
(
¬
X44
)
↔
(
X26
→
(
(
¬
X25
)
↔
X50
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
X26
↔
(
X6
↔
(
X2
↔
X40
)
)
)
→
(
¬
(
X40
↔
(
X6
→
X4
)
)
)
)
)
↔
(
X40
→
(
¬
X4
)
→
X6
)
)
→
(
¬
(
X26
↔
(
X2
↔
X45
)
)
)
)
↔
(
¬
(
X40
→
(
(
¬
X4
)
↔
(
X2
→
X44
)
)
)
)
)
)
↔
(
(
(
(
(
¬
X35
)
→
X34
→
(
(
¬
X9
)
↔
X53
)
)
↔
(
¬
(
(
¬
(
(
X53
→
X34
→
X23
→
X13
)
→
X53
→
(
X31
↔
(
X23
→
X34
)
)
)
)
→
(
X34
↔
(
(
¬
X31
)
→
X13
→
X35
)
)
)
)
)
↔
(
(
(
¬
X16
)
→
X35
)
↔
(
(
¬
(
(
¬
X1
)
→
X23
)
)
→
(
¬
(
X49
→
(
X16
↔
X31
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X46
)
↔
(
X45
→
(
¬
X35
)
→
X53
)
)
)
→
(
X53
↔
(
(
¬
X45
)
↔
(
(
¬
X35
)
↔
X49
)
)
)
)
)
→
(
¬
(
(
(
¬
X20
)
↔
(
(
¬
X10
)
→
(
¬
X7
)
→
X24
)
)
→
(
¬
(
(
(
(
(
X10
→
(
(
¬
X0
)
↔
X53
)
)
↔
(
¬
(
X53
→
(
¬
X24
)
→
X45
)
)
)
↔
(
¬
(
X45
↔
(
X24
↔
X53
)
)
)
)
↔
(
X45
→
(
(
¬
X10
)
↔
(
(
¬
X0
)
→
X46
)
)
)
)
→
(
¬
(
X46
↔
(
X45
↔
X24
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
(
(
X43
↔
(
(
¬
X37
)
↔
(
(
¬
X18
)
→
X51
)
)
)
→
(
¬
(
(
¬
X37
)
↔
(
(
¬
X28
)
→
(
X8
↔
X51
)
)
)
)
→
X51
→
(
¬
X28
)
→
X18
→
X37
)
→
X43
→
X37
→
X8
→
X30
)
)
↔
(
(
(
¬
X43
)
↔
(
(
¬
X28
)
↔
(
X14
↔
X49
)
)
)
→
(
X52
→
(
X49
↔
(
(
¬
X14
)
→
X50
)
)
)
→
(
¬
(
X50
→
X37
→
X14
)
)
)
)
→
(
(
(
(
¬
X42
)
→
X36
→
(
¬
X32
)
→
X50
)
→
(
¬
(
X51
→
(
X42
↔
(
X32
→
False
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
X41
↔
(
(
¬
X17
)
→
X51
)
)
→
(
¬
(
X42
↔
(
(
¬
X41
)
→
X17
→
X51
)
)
)
→
(
¬
(
X51
↔
(
X50
↔
False
)
)
)
)
)
↔
(
(
¬
(
(
¬
X38
)
↔
X51
)
)
↔
(
(
¬
X42
)
→
(
(
X51
↔
(
X38
↔
X42
)
)
↔
(
¬
(
X53
→
(
(
¬
X42
)
↔
(
(
¬
X38
)
↔
X50
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X38
↔
(
(
¬
X3
)
→
X52
)
)
)
→
(
¬
(
X52
↔
(
X51
→
(
X41
↔
X3
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
X52
)
↔
(
(
¬
X48
)
↔
(
X47
→
X53
)
)
)
↔
(
(
¬
X52
)
→
(
X47
↔
(
(
¬
X29
)
↔
X53
)
)
)
)
↔
(
X29
→
(
X22
↔
X47
)
)
)
↔
(
(
(
X52
→
(
X47
↔
(
X33
→
X48
)
)
)
↔
(
X51
→
(
X33
↔
X48
)
)
)
↔
(
X52
↔
(
X47
→
X33
→
X5
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
827e76...
and proposition id is
0f1085...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMNQd6LHJfqixbpvzfsNxyNcRfTb1LxvaA9
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
¬
(
(
(
(
(
(
¬
(
X46
→
(
¬
X38
)
→
(
(
¬
X8
)
↔
X47
)
)
)
↔
(
¬
(
(
¬
X26
)
→
X8
→
X47
)
)
)
↔
(
¬
(
X49
↔
(
(
¬
X46
)
→
(
X38
↔
X47
)
)
)
)
)
→
(
(
(
¬
X46
)
↔
(
X45
→
(
(
¬
X35
)
↔
X49
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X35
)
→
(
(
¬
X25
)
↔
X45
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X40
)
↔
(
(
¬
X35
)
↔
(
(
¬
X34
)
→
X47
)
)
)
)
↔
(
(
(
X47
↔
(
X46
→
X35
→
(
X30
↔
False
)
)
)
↔
(
¬
(
(
¬
X35
)
↔
(
(
¬
X30
)
↔
(
X2
→
X47
)
)
)
)
)
↔
(
X46
↔
(
X35
→
(
X2
↔
X40
)
)
)
)
)
→
(
(
(
¬
X43
)
→
(
¬
X7
)
→
X2
→
X49
)
↔
(
¬
(
X49
→
(
X43
↔
X40
)
)
)
)
)
)
)
→
(
(
(
(
X43
↔
(
X37
→
(
X29
↔
X49
)
)
)
↔
(
(
X49
↔
(
X37
↔
(
X0
→
X43
)
)
)
↔
(
¬
(
X49
→
(
¬
X37
)
→
(
X10
↔
X43
)
)
)
)
)
↔
(
(
¬
(
X13
→
(
X3
↔
X32
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X34
→
(
X32
↔
X43
)
)
)
→
(
(
¬
(
(
¬
X37
)
↔
(
X32
↔
X49
)
)
)
↔
(
¬
(
(
¬
X34
)
→
(
(
¬
X32
)
↔
X43
)
)
)
)
)
)
→
(
¬
X37
)
→
X43
)
)
)
)
↔
(
(
¬
(
X34
↔
(
(
¬
X32
)
↔
(
X24
→
X37
)
)
)
)
→
(
(
(
¬
X32
)
↔
(
(
¬
X27
)
↔
(
(
¬
X23
)
→
X37
)
)
)
→
X37
→
(
(
¬
X32
)
↔
(
X27
↔
X36
)
)
)
→
(
¬
X36
)
→
X34
→
(
(
¬
X32
)
↔
X37
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X42
↔
(
X17
→
X1
→
X28
)
)
)
→
(
(
X48
→
(
(
¬
X22
)
↔
(
(
¬
X1
)
↔
X42
)
)
)
↔
(
(
X22
↔
(
(
¬
X17
)
↔
X42
)
)
↔
(
¬
(
X48
→
(
X17
↔
(
(
¬
X6
)
↔
X22
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X44
↔
(
(
¬
X39
)
→
(
X28
↔
X42
)
)
)
)
↔
(
(
¬
(
X44
↔
(
(
¬
X41
)
→
X39
→
X42
)
)
)
↔
(
(
X44
↔
(
X22
→
(
X21
↔
X41
)
)
)
→
(
X44
↔
(
X42
→
(
¬
X22
)
→
X41
)
)
)
)
)
→
X41
→
(
¬
X21
)
→
X44
)
)
→
(
(
¬
(
(
(
(
¬
(
(
¬
X33
)
→
X20
→
(
¬
X19
)
→
X44
)
)
↔
(
¬
(
X33
→
X19
→
X42
)
)
)
↔
(
(
¬
(
X44
↔
(
(
¬
X33
)
↔
(
(
¬
X9
)
→
X46
)
)
)
)
↔
(
(
¬
(
X41
↔
(
(
¬
X33
)
→
X46
)
)
)
→
(
(
¬
(
X42
→
X11
)
)
↔
(
(
X33
↔
(
X9
↔
False
)
)
→
(
¬
(
(
¬
(
X41
→
(
¬
X11
)
→
(
(
¬
X9
)
↔
X42
)
)
)
→
(
¬
(
(
¬
X33
)
→
(
X11
↔
(
X9
↔
X42
)
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X14
)
↔
X15
)
)
↔
(
(
(
¬
(
X42
→
X18
→
(
X15
↔
(
X14
→
False
)
)
)
)
↔
(
(
¬
X18
)
↔
(
X14
↔
(
X5
→
X41
)
)
)
)
→
X42
→
(
¬
X15
)
→
X18
)
)
→
(
¬
(
X46
↔
(
X41
↔
(
X14
→
X5
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
X41
→
(
X16
↔
(
X12
↔
(
X4
↔
False
)
)
)
)
↔
(
¬
(
X31
↔
(
X16
→
(
¬
X12
)
→
X41
)
)
)
)
→
(
¬
(
X41
↔
(
X31
→
X16
→
False
)
)
)
)
)
→
(
¬
(
X42
↔
X46
)
)
)
↔
(
(
(
¬
(
(
¬
X31
)
↔
(
(
¬
X16
)
↔
(
(
¬
X12
)
↔
X45
)
)
)
)
↔
(
(
¬
X16
)
↔
(
(
¬
X12
)
→
X49
)
)
)
→
X49
→
(
(
¬
X16
)
↔
X31
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
¬
(
(
¬
(
X48
↔
(
X43
→
(
X41
↔
False
)
)
)
)
→
(
¬
X41
)
→
X32
→
X17
→
X48
)
)
→
(
(
(
¬
X40
)
→
X32
→
X43
)
↔
(
(
(
¬
(
(
¬
X32
)
→
(
¬
X17
)
→
X40
)
)
→
(
¬
(
X40
↔
(
X32
→
X12
→
X43
)
)
)
)
→
(
¬
(
X12
↔
X32
)
)
)
)
)
)
→
(
(
X41
→
(
¬
X18
)
→
(
¬
X4
)
→
X40
)
↔
(
¬
(
(
¬
X40
)
↔
(
(
¬
X18
)
↔
(
X4
→
X48
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X41
→
(
¬
X13
)
→
(
¬
X9
)
→
X43
)
)
→
(
¬
(
X43
→
(
X41
↔
(
X30
↔
(
X13
→
False
)
)
)
)
)
)
→
(
(
(
X43
↔
(
X41
→
(
¬
X30
)
→
X44
)
)
→
(
¬
X44
)
→
(
¬
X43
)
→
(
X30
↔
X48
)
)
↔
(
(
¬
(
(
¬
X43
)
→
(
¬
X24
)
→
(
(
¬
X15
)
↔
X44
)
)
)
↔
(
¬
(
X30
↔
X44
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
(
X41
→
(
X28
↔
X43
)
)
→
(
¬
(
(
¬
X23
)
→
X28
)
)
→
(
¬
(
X41
→
X23
→
(
¬
X7
)
→
X43
)
)
)
)
→
(
X43
↔
(
X23
↔
(
X10
↔
False
)
)
)
)
↔
(
X43
↔
(
X33
↔
(
X28
→
X41
)
)
)
)
→
(
¬
(
X48
→
(
¬
X28
)
→
(
X10
↔
X41
)
)
)
→
(
(
¬
X41
)
↔
(
(
¬
X21
)
↔
(
(
¬
X10
)
↔
X48
)
)
)
)
)
→
(
(
X41
→
(
¬
X14
)
→
X46
)
↔
(
¬
(
(
¬
(
(
¬
(
X26
↔
(
X6
↔
X33
)
)
)
→
(
¬
(
(
¬
X33
)
→
X26
→
(
X6
↔
X46
)
)
)
→
(
(
¬
X43
)
↔
(
X33
→
X26
→
X46
)
)
)
)
→
X46
→
(
X33
↔
(
(
¬
X25
)
↔
X26
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
X49
→
X48
→
(
¬
X41
)
→
X45
)
)
↔
(
X49
↔
(
(
¬
X41
)
→
(
(
¬
X36
)
↔
X48
)
)
)
)
→
(
(
(
¬
X42
)
↔
(
X41
↔
X45
)
)
↔
(
(
¬
X47
)
→
(
X45
↔
X49
)
)
)
)
→
(
(
(
(
¬
X27
)
→
(
¬
X20
)
→
(
¬
X3
)
→
X35
)
↔
(
(
(
¬
(
X44
→
(
X41
↔
(
X19
→
X3
)
)
)
)
↔
(
X20
↔
(
X3
↔
X44
)
)
)
→
(
¬
(
X20
↔
(
(
¬
X19
)
↔
X41
)
)
)
)
)
↔
(
(
(
X20
↔
(
X5
↔
X48
)
)
→
(
¬
(
(
(
¬
X38
)
→
X27
→
(
X5
↔
X48
)
)
→
(
¬
(
(
¬
X42
)
→
X20
→
X48
)
)
→
(
¬
(
(
¬
X38
)
↔
(
X27
↔
X42
)
)
)
)
)
)
↔
(
(
¬
(
X27
→
(
(
¬
X20
)
↔
(
X3
→
X48
)
)
)
)
↔
(
¬
(
X27
→
X3
→
(
(
¬
X1
)
↔
X20
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X47
→
(
X39
↔
X37
)
)
)
↔
(
(
(
X48
→
X39
→
X37
)
↔
(
¬
(
(
X48
↔
(
X39
→
(
X29
↔
X37
)
)
)
→
(
(
¬
(
X48
→
(
X37
↔
X45
)
)
)
↔
(
¬
(
(
¬
(
X39
→
(
X37
↔
(
X2
→
X45
)
)
)
)
→
(
X48
↔
(
X39
→
(
X37
↔
X2
)
)
)
)
)
)
→
(
¬
(
X37
↔
(
(
¬
X2
)
→
X48
)
)
)
)
)
)
→
(
X45
↔
(
X37
↔
X29
)
)
)
)
↔
(
(
¬
(
(
(
(
X47
↔
(
(
¬
X22
)
→
(
X16
↔
X39
)
)
)
↔
(
¬
(
X47
→
(
¬
X37
)
→
X39
)
)
)
→
(
¬
(
X47
→
X37
→
X22
)
)
)
→
(
(
X39
→
(
¬
X31
)
→
(
(
¬
X8
)
↔
X37
)
)
↔
(
¬
(
X47
↔
(
(
¬
X8
)
↔
X31
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X47
↔
(
X0
↔
X37
)
)
)
→
(
¬
(
X39
↔
(
X34
→
(
(
¬
X0
)
↔
X37
)
)
)
)
)
)
↔
(
X39
→
X34
→
(
¬
X11
)
→
X37
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
7ed82c...
and proposition id is
5ad3da...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMFDh6Wzh6fV9A7HrQWyzaKgjhckSwknWW5
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
(
(
(
¬
(
X46
→
(
X42
↔
(
(
¬
X30
)
↔
X38
)
)
)
)
→
(
(
¬
X42
)
↔
(
X40
↔
(
X30
↔
X46
)
)
)
)
→
(
¬
(
(
¬
X40
)
↔
(
(
¬
X38
)
↔
(
X16
→
X42
)
)
)
)
)
↔
(
(
(
(
X37
→
(
X8
↔
(
X1
→
False
)
)
)
↔
(
(
¬
(
(
¬
X40
)
→
X8
→
(
¬
X1
)
→
X42
)
)
↔
(
¬
(
X40
↔
(
X37
↔
(
(
¬
X1
)
→
X8
)
)
)
)
)
)
→
X40
→
(
(
¬
X8
)
↔
X42
)
)
↔
(
X42
→
(
X38
↔
(
(
¬
X8
)
→
X37
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X41
)
↔
(
(
¬
X36
)
↔
X43
)
)
)
→
(
(
¬
(
X42
↔
(
(
¬
X41
)
↔
(
X38
→
X43
)
)
)
)
↔
(
(
X43
→
(
X40
↔
(
X4
↔
X41
)
)
)
↔
(
X43
↔
(
X38
↔
X4
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
X41
→
X25
→
(
¬
X13
)
→
X39
)
→
X39
→
X17
→
X43
)
)
→
(
X13
→
(
¬
X2
)
→
X17
)
→
(
X41
↔
(
(
¬
X17
)
→
(
¬
X13
)
→
X46
)
)
)
→
(
¬
(
(
(
¬
(
X43
↔
(
X39
↔
(
X22
↔
X34
)
)
)
)
↔
(
¬
(
(
(
X41
→
X34
→
X43
)
→
(
(
X34
↔
(
X27
→
X22
)
)
↔
(
(
¬
(
(
¬
X27
)
↔
(
(
¬
X22
)
→
X41
)
)
)
↔
(
(
X41
↔
(
X34
→
X22
→
X27
)
)
↔
(
¬
(
X43
↔
(
X34
↔
(
X22
→
False
)
)
)
)
)
)
)
)
→
(
¬
(
X43
→
(
X34
↔
X27
)
)
)
)
)
)
→
(
¬
(
(
¬
X41
)
↔
(
X39
↔
(
X0
→
X43
)
)
)
)
)
)
→
(
(
(
(
¬
(
X24
↔
(
X23
→
(
X18
↔
X19
)
)
)
)
→
(
¬
(
X39
→
(
(
¬
X18
)
↔
X24
)
)
)
)
↔
(
(
X43
→
X34
→
X10
→
False
)
→
(
¬
(
X23
→
(
(
¬
X19
)
↔
(
(
¬
X18
)
↔
X43
)
)
)
)
)
)
↔
(
¬
(
X43
↔
(
X39
→
X26
→
X34
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
(
X49
↔
(
X47
↔
(
X41
→
False
)
)
)
→
(
(
¬
(
(
¬
X11
)
↔
X44
)
)
↔
(
X47
↔
(
X41
↔
(
X28
↔
(
X11
→
False
)
)
)
)
)
)
)
↔
(
(
¬
(
X41
↔
(
(
¬
X29
)
→
(
¬
X12
)
→
X44
)
)
)
→
X44
→
(
X28
↔
(
(
¬
X12
)
→
X29
)
)
)
)
→
(
(
(
¬
(
(
¬
(
X45
↔
(
(
¬
X41
)
↔
(
(
¬
X33
)
↔
X49
)
)
)
)
→
(
(
¬
X41
)
↔
(
X33
→
(
(
¬
X20
)
↔
X49
)
)
)
)
)
↔
(
¬
(
X49
→
(
¬
X20
)
→
X41
)
)
)
↔
(
(
(
¬
X33
)
→
(
X20
↔
(
X15
↔
X44
)
)
)
↔
(
X44
→
(
X33
↔
(
X32
↔
X20
)
)
)
)
)
)
↔
(
(
(
¬
(
X48
→
(
¬
X41
)
→
(
¬
X32
)
→
X47
)
)
→
X47
→
X42
)
↔
False
)
)
↔
(
(
(
(
¬
(
X35
→
(
¬
X9
)
→
(
X3
↔
X45
)
)
)
↔
(
(
X49
→
(
X45
↔
(
X3
→
False
)
)
)
→
X9
→
(
X5
↔
(
(
¬
X3
)
→
X45
)
)
)
)
↔
(
¬
(
(
¬
(
X21
→
X6
→
X45
)
)
→
(
¬
(
(
¬
X21
)
↔
X35
)
)
→
(
¬
(
(
¬
X42
)
↔
(
X14
↔
(
(
¬
X6
)
→
X49
)
)
)
)
)
)
)
↔
(
(
X42
→
X35
→
X45
)
↔
(
(
(
X45
↔
(
X35
↔
(
(
¬
X7
)
↔
X42
)
)
)
↔
(
X43
→
X42
→
(
(
¬
X35
)
↔
X45
)
)
)
↔
(
¬
(
X43
↔
(
X42
→
(
¬
X31
)
→
X45
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
¬
(
(
¬
(
X44
→
X38
→
(
X37
↔
X49
)
)
)
→
(
X44
↔
(
X38
↔
(
X23
→
X49
)
)
)
)
)
→
(
¬
(
X44
↔
X49
)
)
)
)
↔
(
X43
↔
(
X39
→
(
X14
↔
X30
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X24
)
↔
(
(
¬
X13
)
↔
X35
)
)
)
↔
(
(
X35
↔
(
(
¬
X24
)
↔
(
(
¬
X13
)
↔
X34
)
)
)
↔
(
X49
→
X35
)
)
)
→
(
(
¬
X34
)
↔
(
X13
↔
X35
)
)
)
↔
(
(
X43
↔
(
X27
→
X16
→
X35
)
)
↔
(
(
X49
↔
(
(
¬
X43
)
→
(
¬
X16
)
→
X44
)
)
↔
(
¬
(
(
(
X44
→
(
X5
↔
False
)
)
→
X49
→
(
¬
X8
)
→
(
X5
↔
X34
)
)
→
(
¬
(
X5
→
X35
)
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
X45
)
→
X41
→
X28
→
X47
)
↔
(
(
¬
(
X41
↔
(
(
¬
X25
)
↔
X28
)
)
)
↔
(
(
(
X45
↔
(
(
¬
X28
)
↔
X41
)
)
↔
(
(
X49
→
(
¬
(
X45
↔
(
(
¬
X28
)
↔
(
(
¬
X25
)
↔
X41
)
)
)
)
)
↔
(
¬
(
(
¬
X41
)
→
(
(
¬
X28
)
↔
X45
)
)
)
)
)
↔
(
¬
(
X49
↔
(
(
¬
X25
)
↔
X45
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
X47
→
X26
→
(
X12
↔
X45
)
)
)
↔
(
X45
↔
(
X43
→
(
(
¬
X12
)
↔
X26
)
)
)
)
→
(
¬
X43
)
)
→
X43
→
(
X26
↔
X45
)
)
)
↔
(
(
X49
→
(
X47
↔
(
(
¬
X6
)
↔
X43
)
)
)
→
(
(
¬
X47
)
↔
(
(
¬
X45
)
→
(
(
¬
X6
)
↔
X49
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
(
X48
↔
(
X44
→
(
(
¬
X43
)
↔
X49
)
)
)
↔
(
¬
(
X48
↔
(
(
¬
X43
)
↔
(
X31
→
X49
)
)
)
)
)
→
X44
→
X43
→
X31
)
)
→
(
¬
(
X49
↔
(
(
¬
X31
)
→
X44
)
)
)
)
→
(
¬
(
(
X43
↔
(
X41
→
X44
)
)
→
(
(
¬
(
X48
↔
(
X43
↔
X11
)
)
)
↔
(
¬
(
(
X48
→
(
(
¬
X43
)
↔
(
X41
↔
X44
)
)
)
→
(
¬
(
X43
→
(
¬
X41
)
→
(
¬
X11
)
→
X44
)
)
)
)
)
)
)
)
)
↔
(
(
(
X49
↔
(
(
¬
X1
)
↔
X31
)
)
→
(
¬
(
X49
↔
(
X1
↔
(
(
¬
X0
)
↔
X44
)
)
)
)
)
↔
(
¬
(
X44
→
X9
→
X0
→
X29
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
X33
→
X32
→
X17
→
X42
)
)
→
(
¬
(
(
(
¬
(
X46
→
(
X33
↔
X42
)
)
)
↔
(
¬
(
X48
→
(
X33
↔
(
X22
→
X42
)
)
)
)
)
→
(
(
¬
X46
)
↔
(
(
¬
X32
)
↔
(
X22
→
X48
)
)
)
)
)
)
)
→
(
X48
↔
(
X33
↔
X42
)
)
)
)
↔
(
(
(
(
X46
↔
(
X43
↔
(
X42
→
X10
)
)
)
↔
(
(
¬
(
(
¬
X42
)
→
X40
→
(
¬
X15
)
→
X46
)
)
↔
(
(
¬
X43
)
↔
(
(
¬
X42
)
→
(
X15
↔
X46
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X15
↔
(
(
¬
X10
)
↔
X40
)
)
)
↔
(
X48
↔
(
X40
↔
X20
)
)
)
↔
(
X48
→
(
X20
↔
(
X19
↔
X15
)
)
)
)
→
X48
→
(
X20
↔
X10
)
)
↔
(
X10
→
X19
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
X41
↔
(
(
¬
X7
)
→
(
X2
↔
X36
)
)
)
→
(
¬
(
X41
↔
(
(
¬
X7
)
→
(
X2
↔
X48
)
)
)
)
)
)
↔
(
(
¬
X36
)
↔
(
X7
→
X41
)
)
)
↔
(
X43
→
X36
→
(
¬
X7
)
→
X48
)
)
→
(
¬
(
(
(
¬
(
X43
→
(
¬
X21
)
→
X46
)
)
↔
(
(
¬
(
(
¬
X21
)
→
(
X18
↔
(
(
¬
X4
)
↔
X43
)
)
)
)
↔
(
(
¬
X18
)
→
X46
)
)
)
→
(
¬
(
(
(
X3
↔
X21
)
↔
(
¬
(
(
¬
(
X36
→
X21
→
(
¬
X18
)
→
X43
)
)
→
(
¬
(
(
¬
X3
)
→
X43
)
)
)
)
)
→
(
X41
↔
(
X21
↔
(
(
¬
X18
)
→
X43
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
139dc9...
and proposition id is
a04df8...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMa4ABQLjMz2gE7HjPZ9rRbq5du5yqBmKEJ
)
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
(
¬
(
X48
→
(
X44
↔
(
X41
→
X36
→
False
)
)
)
)
↔
(
(
(
X41
→
(
X33
↔
(
(
¬
X22
)
↔
X44
)
)
)
→
(
(
X49
→
(
X44
↔
(
X33
→
X20
)
)
)
↔
(
(
(
¬
X48
)
→
(
(
¬
X33
)
↔
X49
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X44
)
↔
(
(
¬
X41
)
↔
(
X33
→
X49
)
)
)
)
→
(
X48
↔
(
(
¬
X41
)
→
X33
→
X49
)
)
)
→
(
¬
(
(
¬
X41
)
→
(
X33
↔
X44
)
)
)
)
)
)
)
)
→
(
(
¬
X48
)
↔
(
X44
↔
(
(
¬
X43
)
↔
X49
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
X48
→
(
X19
↔
(
X17
↔
X3
)
)
)
)
→
(
¬
(
X31
→
(
¬
X17
)
→
(
¬
X3
)
→
X19
)
)
)
)
↔
(
¬
(
(
(
¬
(
X39
↔
(
(
¬
X19
)
→
X31
)
)
)
→
(
(
X39
→
X17
→
(
(
¬
X5
)
↔
X31
)
)
↔
(
¬
(
X17
→
X19
)
)
)
→
(
¬
(
X31
→
X19
→
False
)
)
)
→
(
¬
(
(
¬
X31
)
↔
(
(
¬
X19
)
↔
(
(
¬
X1
)
→
X39
)
)
)
)
)
)
)
→
(
(
¬
(
(
X44
→
X33
→
X31
→
False
)
→
(
¬
(
(
¬
(
(
X48
→
(
(
¬
X39
)
↔
(
X31
↔
X44
)
)
)
→
(
¬
(
X44
↔
(
(
¬
X39
)
→
(
X33
↔
X48
)
)
)
)
)
)
→
(
¬
(
X39
→
X33
→
False
)
)
)
)
)
)
↔
(
¬
(
(
¬
X44
)
↔
(
X33
→
X48
)
)
)
)
→
(
¬
(
X44
→
(
(
¬
X31
)
↔
(
X15
→
X33
)
)
)
)
)
)
)
↔
(
(
X43
→
X32
→
(
(
¬
X23
)
↔
X30
)
)
↔
(
¬
(
(
¬
(
(
X8
↔
(
X4
→
X23
)
)
→
(
¬
(
(
(
X37
↔
(
X32
→
(
¬
X8
)
→
X47
)
)
→
X47
→
(
¬
X34
)
→
X8
→
X37
)
→
(
(
X34
↔
(
(
¬
X23
)
↔
(
X4
→
X32
)
)
)
↔
(
X34
↔
(
(
¬
X32
)
↔
(
(
¬
X23
)
→
X47
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
X49
↔
(
(
¬
X37
)
→
X47
)
)
)
→
(
(
X49
↔
(
(
¬
X41
)
↔
X47
)
)
↔
(
(
X49
↔
(
(
¬
X37
)
↔
(
X6
↔
X41
)
)
)
↔
(
X47
↔
(
(
¬
X41
)
↔
X49
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
X41
↔
(
X37
→
(
(
¬
X7
)
↔
X47
)
)
)
→
(
(
¬
X37
)
↔
(
(
¬
X27
)
↔
(
(
¬
X7
)
→
X41
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X47
↔
(
X37
↔
(
X11
↔
X27
)
)
)
)
↔
(
¬
(
X43
→
X27
→
X37
)
)
)
→
(
¬
X43
)
→
(
¬
X34
)
→
(
(
¬
X27
)
↔
X47
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
X25
)
→
(
(
¬
X13
)
↔
(
X9
→
X35
)
)
)
↔
(
¬
(
X45
↔
(
X35
→
X25
→
(
X13
↔
False
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X47
→
(
¬
X46
)
→
X45
→
X48
)
)
→
(
X48
↔
(
X46
→
(
X28
↔
X47
)
)
)
)
)
→
(
(
(
¬
(
X48
→
X45
→
X13
)
)
↔
(
(
(
¬
(
X13
↔
(
(
¬
X0
)
→
X48
)
)
)
→
(
¬
X45
)
→
(
(
¬
X13
)
↔
(
(
¬
X0
)
→
X46
)
)
)
↔
(
¬
(
X45
→
(
X13
↔
X0
)
)
)
)
)
↔
(
¬
(
X48
↔
(
(
¬
X9
)
↔
X46
)
)
)
)
)
)
)
↔
(
X47
→
(
¬
X46
)
→
X12
→
X48
)
)
↔
(
¬
(
(
(
(
(
¬
(
(
X49
→
X40
→
X45
)
→
(
¬
X47
)
→
(
(
¬
X18
)
↔
X49
)
)
)
↔
(
¬
(
X45
↔
(
(
¬
X40
)
↔
X49
)
)
)
)
↔
(
X47
→
X45
→
X16
→
X49
)
)
→
(
¬
(
(
X46
↔
(
X38
→
X42
)
)
→
X42
→
(
X40
↔
(
X10
↔
X21
)
)
)
)
→
X40
→
(
X21
↔
(
(
¬
X16
)
→
X18
)
)
)
→
(
(
(
X47
→
(
(
¬
X24
)
↔
(
(
¬
X2
)
→
X29
)
)
)
→
(
¬
(
X42
↔
(
X26
→
X24
→
False
)
)
)
)
↔
(
X49
→
(
X14
↔
X2
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
(
(
X43
↔
(
X42
↔
(
X7
→
X40
)
)
)
→
(
¬
(
X35
→
(
(
¬
X7
)
↔
X47
)
)
)
)
↔
(
(
(
X20
↔
(
(
¬
X16
)
→
(
(
¬
X8
)
↔
X47
)
)
)
↔
(
(
¬
(
X43
↔
(
X20
→
X26
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X43
)
→
X26
→
(
¬
X20
)
→
X47
)
)
→
(
¬
(
X43
↔
(
(
¬
X20
)
→
(
X8
↔
X47
)
)
)
)
)
)
)
)
↔
(
(
X4
↔
X20
)
→
(
(
¬
X42
)
→
X29
→
X43
)
→
X43
→
X42
→
(
¬
X16
)
→
X20
)
)
)
→
(
(
¬
(
X35
→
(
¬
X30
)
→
(
(
¬
X25
)
↔
X36
)
)
)
↔
(
(
X30
↔
(
X17
↔
X40
)
)
→
(
(
(
X36
→
(
X35
↔
(
(
¬
X17
)
↔
X40
)
)
)
↔
(
X36
→
X17
→
X26
)
)
↔
(
(
¬
X30
)
→
(
¬
X26
)
→
X40
)
)
)
)
)
)
→
(
(
(
(
¬
X42
)
→
(
(
¬
X41
)
↔
(
(
¬
X32
)
→
X48
)
)
)
↔
(
X45
↔
(
X41
↔
X43
)
)
)
↔
(
X48
→
(
X42
↔
X47
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X45
)
↔
(
X22
→
X48
)
)
)
↔
(
(
¬
(
X22
↔
X48
)
)
↔
(
(
X34
→
X22
→
(
X14
↔
False
)
)
→
(
(
(
X41
↔
(
X34
→
(
X22
↔
X48
)
)
)
↔
(
¬
(
X48
→
(
X34
↔
(
X22
→
X45
)
)
)
)
)
↔
(
¬
(
(
X14
↔
X48
)
→
(
¬
(
(
¬
X22
)
↔
(
(
¬
X14
)
→
X5
→
X41
)
)
)
)
)
)
)
)
)
→
(
¬
(
X45
↔
(
X41
→
(
¬
X21
)
→
X48
)
)
)
→
(
(
(
(
¬
X45
)
→
X13
→
X48
)
↔
(
(
(
¬
X46
)
→
X45
→
(
¬
X13
)
→
X48
)
→
X48
→
(
X13
↔
(
X12
↔
X45
)
)
)
)
↔
(
¬
(
X48
→
X46
→
X44
→
X41
)
)
)
)
↔
(
¬
(
(
¬
(
X46
→
X41
→
(
X33
↔
X38
)
)
)
→
(
¬
(
(
(
X23
→
X46
)
↔
(
¬
(
X44
→
X2
→
X41
)
)
)
→
X46
→
(
X41
↔
(
X38
→
X44
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X46
→
(
(
¬
X18
)
↔
X44
)
)
)
↔
(
(
(
¬
X44
)
→
(
¬
X18
)
→
(
X11
↔
X46
)
)
→
X49
→
(
X27
↔
(
X18
↔
(
X3
↔
False
)
)
)
)
)
↔
(
X49
→
X19
→
X3
→
X27
)
)
↔
(
X37
↔
(
(
¬
X19
)
↔
(
X18
→
X27
)
)
)
)
→
(
¬
(
(
(
(
¬
(
(
¬
X37
)
↔
(
X6
↔
X49
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X37
)
→
X31
→
(
(
¬
X1
)
↔
X49
)
)
)
→
(
¬
(
X49
→
X31
→
(
¬
X0
)
→
X1
)
)
)
)
)
→
(
(
¬
X37
)
↔
(
X31
→
X49
)
)
)
→
(
X48
↔
(
X39
→
(
(
¬
X6
)
↔
X31
)
)
)
→
(
¬
(
(
¬
X31
)
→
(
¬
X0
)
→
X39
)
)
)
)
→
(
(
(
¬
(
X49
→
(
¬
X10
)
→
X44
)
)
↔
(
(
¬
X48
)
↔
(
(
¬
X28
)
↔
(
X10
↔
X49
)
)
)
)
↔
(
(
(
(
X49
↔
(
(
¬
X39
)
↔
(
(
¬
X24
)
↔
X44
)
)
)
→
(
¬
(
X44
↔
(
X39
↔
(
X9
↔
X49
)
)
)
)
)
↔
(
¬
(
X49
↔
(
X39
↔
X24
)
)
)
)
↔
(
(
¬
(
X49
→
(
X48
↔
(
X44
↔
(
X15
↔
False
)
)
)
)
)
↔
(
X49
↔
(
(
¬
X44
)
→
X24
→
X48
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
50bc71...
and proposition id is
b53d4f...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMYLQqbMUZmdwiM6mYnJTsEPNsLyFrrVH4i
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
¬
(
(
(
(
(
X27
→
X18
→
X19
)
→
(
X49
→
(
(
¬
X18
)
↔
(
X16
↔
X19
)
)
)
→
(
¬
(
(
¬
X27
)
↔
(
(
¬
X19
)
↔
(
X18
→
X49
)
)
)
)
)
↔
(
X32
→
(
(
¬
X19
)
↔
(
(
¬
X18
)
↔
X49
)
)
)
)
↔
(
(
(
(
(
¬
(
(
X27
→
X18
→
(
X0
↔
X36
)
)
→
(
¬
X36
)
→
(
(
¬
X27
)
↔
(
(
¬
X0
)
→
X49
)
)
)
)
→
(
¬
(
X49
↔
(
X0
→
False
)
)
)
)
↔
(
¬
(
X49
↔
(
X0
↔
X36
)
)
)
)
→
(
¬
(
X18
↔
(
X0
→
X49
)
)
)
)
↔
(
¬
(
(
¬
(
X49
↔
(
X18
→
(
X10
↔
X17
)
)
)
)
→
(
(
¬
X27
)
↔
(
(
¬
X10
)
↔
(
X1
→
X49
)
)
)
→
(
¬
(
X49
→
(
X27
↔
(
X3
↔
X17
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
(
X47
↔
(
X45
→
(
X41
↔
(
X27
↔
False
)
)
)
)
)
→
(
¬
(
(
¬
X45
)
→
X48
)
)
)
↔
(
(
(
X45
↔
(
X31
↔
(
X24
↔
False
)
)
)
↔
(
¬
(
(
¬
(
X32
↔
(
X24
↔
X6
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X24
)
→
X6
→
X31
)
)
)
)
)
)
↔
(
¬
(
(
¬
X47
)
→
(
X31
↔
(
X24
→
(
¬
X9
)
→
X41
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
X27
↔
(
X24
→
X12
→
X42
)
)
→
(
X45
↔
(
X27
→
(
(
¬
X12
)
↔
X24
)
)
)
)
)
↔
(
(
¬
X42
)
→
X27
→
(
(
¬
X5
)
↔
X45
)
)
)
→
(
X27
↔
(
(
¬
X5
)
→
X12
)
)
)
)
↔
(
(
¬
X27
)
→
X24
→
(
X22
↔
X42
)
)
)
)
→
(
(
(
X48
→
(
¬
X45
)
→
X23
→
X49
)
→
(
X45
↔
(
X20
↔
X49
)
)
)
↔
(
¬
(
(
¬
(
X49
↔
(
X47
→
X45
→
X48
)
)
)
→
(
X48
↔
(
X45
→
(
(
¬
X26
)
↔
X47
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
¬
(
(
¬
X39
)
↔
(
(
¬
X37
)
↔
(
X15
↔
X46
)
)
)
)
→
(
(
(
X38
↔
(
(
¬
X8
)
↔
(
(
¬
X7
)
→
X37
)
)
)
→
X46
→
(
X38
↔
(
X13
↔
X37
)
)
)
↔
(
(
X13
↔
X8
)
↔
(
¬
(
X46
↔
(
X15
↔
(
X8
→
X11
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X35
→
(
¬
X29
)
→
(
(
¬
X15
)
↔
X39
)
)
)
↔
(
(
¬
(
(
¬
X35
)
→
X28
→
(
X15
↔
X38
)
)
)
↔
(
X38
→
X28
→
(
X15
↔
X39
)
)
)
)
→
(
X28
↔
False
)
)
)
)
↔
(
¬
(
(
X46
↔
(
X39
→
X4
→
X25
)
)
→
(
(
(
(
¬
X39
)
→
(
(
¬
X30
)
↔
(
(
¬
X25
)
↔
X46
)
)
)
→
(
¬
(
X46
→
(
¬
X30
)
→
(
¬
X21
)
→
X35
)
)
)
↔
(
¬
(
X48
↔
(
X46
→
X25
→
X21
→
False
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X46
→
(
¬
X40
)
→
(
(
¬
X35
)
↔
X44
)
)
)
→
(
(
¬
(
X46
↔
(
X40
↔
(
(
¬
X35
)
↔
X43
)
)
)
)
↔
(
¬
(
X46
→
(
X40
↔
(
X35
↔
X33
)
)
)
)
)
)
↔
(
(
X43
→
(
(
¬
X35
)
↔
X41
)
)
↔
(
(
X41
↔
X34
)
→
(
¬
(
X49
→
X41
→
(
(
¬
X14
)
↔
X35
)
)
)
)
)
)
→
(
¬
(
(
X46
↔
(
X44
↔
(
X37
→
X49
)
)
)
→
(
¬
(
(
¬
X46
)
→
(
X44
↔
(
(
¬
X2
)
↔
X49
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
(
¬
(
X46
→
X43
→
(
X25
↔
X47
)
)
)
↔
(
(
X43
→
X47
)
↔
(
(
¬
(
(
(
X4
→
X43
)
→
(
¬
(
X47
→
(
X46
↔
(
X43
→
X27
)
)
)
)
)
→
(
¬
(
(
¬
X27
)
↔
(
(
¬
X4
)
→
X43
)
)
)
)
)
↔
(
¬
(
X47
→
(
X43
↔
X46
)
)
)
)
)
)
↔
(
X47
→
(
¬
X4
)
→
X27
)
)
↔
(
X46
↔
(
(
¬
X22
)
↔
(
(
¬
X10
)
→
X27
)
)
)
)
↔
(
(
(
(
X46
→
X30
→
(
(
¬
X19
)
↔
X43
)
)
→
(
(
(
(
¬
X43
)
→
(
¬
X30
)
→
X16
→
X46
)
↔
(
X43
→
(
X36
↔
(
X30
↔
X46
)
)
)
)
↔
(
¬
(
X43
→
(
X36
↔
(
(
¬
X13
)
↔
X16
)
)
)
)
)
)
↔
(
¬
(
X43
→
(
(
¬
X36
)
↔
(
X25
→
X46
)
)
)
)
)
↔
(
(
(
X36
↔
(
X25
→
X44
)
)
→
X25
→
(
(
¬
X19
)
↔
(
(
¬
X11
)
→
X45
)
)
)
→
(
¬
(
X45
→
X44
→
False
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
X46
↔
(
X40
→
(
X34
↔
X45
)
)
)
)
→
(
¬
X45
)
→
(
(
¬
X40
)
↔
(
X20
→
X46
)
)
)
)
↔
(
X46
→
X34
→
X17
→
X40
)
)
↔
(
(
¬
(
(
¬
(
(
(
X33
→
(
(
¬
X7
)
↔
X24
)
)
↔
(
X33
↔
(
X26
→
(
X6
↔
X24
)
)
)
)
→
(
X24
↔
(
(
¬
X6
)
↔
X26
)
)
)
)
→
(
¬
X33
)
→
(
(
¬
X7
)
↔
X47
)
)
)
→
(
(
¬
(
X46
↔
(
X45
↔
(
X40
→
X26
→
False
)
)
)
)
↔
(
(
¬
(
X46
↔
(
(
¬
X33
)
↔
X40
)
)
)
↔
(
(
X46
↔
(
X40
↔
(
X33
↔
False
)
)
)
↔
(
(
X46
→
False
)
↔
(
¬
(
(
¬
(
(
¬
X45
)
→
(
¬
X40
)
→
(
X24
↔
X46
)
)
)
→
(
¬
(
X40
→
X24
→
X33
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
(
¬
X44
)
→
(
(
¬
X31
)
↔
X45
)
)
→
X44
→
(
¬
X15
)
→
X32
)
)
↔
(
(
¬
(
(
¬
X45
)
↔
(
X44
→
(
X41
↔
X48
)
)
)
)
→
(
(
¬
(
X44
↔
(
(
¬
X38
)
→
(
(
¬
X32
)
↔
X41
)
)
)
)
→
(
X45
↔
(
X41
↔
X44
)
)
)
→
X45
→
(
¬
X32
)
→
X38
)
)
↔
(
¬
(
X48
→
(
¬
X31
)
→
X38
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
X44
↔
(
X37
→
(
(
¬
X21
)
↔
X42
)
)
)
)
→
(
¬
(
X45
↔
(
X37
→
X21
→
X42
)
)
)
)
)
→
(
X42
↔
(
(
¬
X21
)
→
(
(
¬
X1
)
↔
X44
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X42
)
↔
(
(
¬
X32
)
↔
(
X12
→
X44
)
)
)
)
→
(
¬
(
(
(
¬
X32
)
→
(
X21
↔
(
(
¬
X12
)
→
X42
)
)
)
→
(
¬
(
X32
→
(
(
¬
X3
)
↔
X42
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
(
X41
→
(
X9
↔
(
X5
↔
X14
)
)
)
)
→
(
¬
(
X41
↔
(
X9
→
(
X5
↔
X47
)
)
)
)
)
)
↔
(
(
¬
(
X47
→
X14
→
(
(
¬
X5
)
↔
X49
)
)
)
↔
(
X49
↔
(
X47
→
(
X14
↔
X48
)
)
)
)
)
↔
(
(
¬
(
X41
→
(
X18
↔
(
X8
↔
X28
)
)
)
)
↔
(
¬
(
(
(
¬
X18
)
↔
X47
)
→
(
X48
↔
(
X28
↔
(
X18
→
X41
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X35
↔
(
X29
↔
X47
)
)
)
→
(
(
X47
→
X42
→
(
¬
X29
)
→
X35
)
↔
(
(
X47
↔
(
X42
↔
X2
)
)
→
(
(
X28
↔
(
X23
→
X29
)
)
↔
(
¬
(
(
¬
(
(
¬
X28
)
→
(
¬
X23
)
→
(
¬
X2
)
→
X42
)
)
→
X42
→
(
X23
↔
(
(
¬
X2
)
↔
X29
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
X49
→
X39
)
)
→
(
¬
(
X47
→
(
¬
X39
)
→
(
¬
X0
)
→
X42
)
)
)
)
↔
(
¬
(
X49
→
(
X42
↔
(
X39
↔
False
)
)
)
)
)
↔
(
¬
(
(
¬
X47
)
↔
(
X42
→
X49
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
4f9134...
and proposition id is
a72cc1...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMcChS6guUyvyZ7uNQY4KUtEYy9KEP3gKcn
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
∀X50 :
prop
,
(
(
(
(
(
(
(
¬
(
X50
→
X11
→
False
)
)
→
(
X11
↔
(
¬
(
(
X47
↔
(
X11
→
X33
)
)
→
X50
→
X44
→
X11
)
)
)
)
↔
(
¬
(
X47
→
(
X44
↔
(
X3
→
X33
)
)
)
)
)
↔
(
(
(
¬
X48
)
→
(
X47
↔
(
X30
↔
X50
)
)
)
↔
(
¬
(
(
(
X50
↔
(
(
¬
X38
)
↔
(
X30
→
X45
)
)
)
↔
(
(
X47
↔
(
X33
↔
(
X30
→
X38
)
)
)
↔
(
¬
(
X38
→
(
¬
X33
)
→
X45
)
)
)
)
→
(
(
¬
X38
)
↔
(
(
¬
X30
)
↔
(
X2
↔
X45
)
)
)
)
)
)
)
↔
(
(
(
X47
↔
(
X45
→
(
X17
↔
X36
)
)
)
↔
(
(
(
(
(
X45
↔
(
(
¬
X31
)
↔
(
X16
↔
X36
)
)
)
↔
(
(
¬
X36
)
→
X31
→
X47
)
)
↔
(
X47
↔
(
X45
↔
X36
)
)
)
↔
(
¬
(
X47
↔
(
(
¬
X28
)
↔
X45
)
)
)
)
→
(
¬
(
(
¬
(
X47
→
(
X23
↔
(
X18
↔
False
)
)
)
)
→
(
¬
(
X45
→
(
X36
↔
(
X8
↔
X18
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X37
↔
(
(
¬
X6
)
↔
(
X5
→
X28
)
)
)
)
↔
(
X6
↔
(
X5
→
X12
)
)
)
↔
(
¬
(
(
¬
(
X37
→
(
¬
X12
)
→
(
¬
X9
)
→
X50
)
)
→
(
(
¬
(
X47
→
(
¬
X31
)
→
(
X6
↔
X50
)
)
)
↔
(
X50
↔
(
(
¬
X9
)
→
(
(
¬
X6
)
↔
X31
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
(
¬
(
(
X50
→
(
(
¬
X42
)
↔
(
X22
→
X48
)
)
)
→
X48
→
(
(
¬
X4
)
↔
X42
)
)
)
→
(
(
¬
(
X46
↔
(
X43
→
(
(
¬
X42
)
↔
X48
)
)
)
)
↔
(
¬
(
X48
↔
(
X42
→
X50
)
)
)
)
)
↔
(
¬
(
(
X50
↔
(
X42
→
X35
→
X10
→
False
)
)
→
(
X46
↔
(
(
¬
X42
)
↔
X50
)
)
)
)
)
→
(
¬
(
(
¬
X46
)
↔
(
X42
↔
(
X25
↔
X48
)
)
)
)
)
)
→
(
(
¬
(
X43
→
(
(
¬
X42
)
↔
(
(
¬
X0
)
↔
X46
)
)
)
)
↔
(
(
¬
(
(
¬
(
X46
↔
(
X42
↔
False
)
)
)
→
X42
→
(
(
¬
X32
)
↔
(
X13
↔
X40
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
X42
)
→
(
(
¬
X26
)
↔
(
(
¬
X19
)
↔
X43
)
)
)
→
(
¬
(
(
¬
X42
)
↔
(
(
¬
X26
)
→
X15
→
X43
)
)
)
)
↔
(
(
¬
X19
)
→
(
X15
↔
X32
)
)
)
→
(
X42
↔
(
(
¬
X19
)
↔
(
(
¬
X15
)
↔
X40
)
)
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
X39
)
→
X29
→
X14
→
X49
)
→
X14
→
X48
)
↔
(
(
X49
→
(
X47
↔
(
X29
→
X24
)
)
)
→
(
(
¬
(
(
¬
(
(
X49
→
X47
→
(
X39
↔
False
)
)
→
(
(
X49
↔
(
X47
↔
(
X35
→
(
X29
↔
False
)
)
)
)
↔
(
X39
→
(
X35
↔
(
X29
→
False
)
)
)
)
)
)
→
(
¬
(
(
¬
X39
)
↔
(
X35
↔
X47
)
)
)
)
)
↔
(
¬
(
X47
→
(
¬
X29
)
→
X49
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X49
↔
(
X47
→
X34
)
)
)
→
(
X48
↔
(
X32
↔
(
X7
→
X20
)
)
)
)
)
↔
(
(
(
(
(
¬
X27
)
→
X1
→
X47
)
↔
(
(
X47
↔
(
X21
↔
(
X1
↔
X41
)
)
)
→
(
¬
(
X27
→
(
(
¬
X1
)
↔
X21
)
)
)
)
)
↔
(
X41
→
(
X21
↔
False
)
)
)
↔
(
(
¬
X7
)
↔
(
(
¬
(
X47
→
X41
→
(
X32
↔
X27
)
)
)
↔
(
X41
↔
(
(
¬
X32
)
→
(
¬
X7
)
→
X47
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
¬
(
X50
↔
(
(
¬
X48
)
→
(
¬
X21
)
→
X49
)
)
)
↔
(
(
¬
(
X50
↔
(
(
¬
X25
)
↔
(
X17
→
X49
)
)
)
)
↔
(
¬
(
X49
→
(
X25
↔
X21
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
¬
X31
)
↔
(
(
¬
X30
)
↔
(
(
¬
X1
)
→
X48
)
)
)
)
→
(
¬
(
X50
→
X31
→
False
)
)
)
)
→
(
(
(
X46
→
(
¬
X20
)
→
X22
)
↔
(
(
¬
(
X49
→
X30
→
(
X20
↔
False
)
)
)
→
X46
→
(
¬
X20
)
→
X22
)
)
↔
(
¬
(
X30
→
(
¬
X22
)
→
X20
→
X48
)
)
)
→
(
X46
↔
(
X31
↔
(
X30
↔
X48
)
)
)
)
)
)
→
(
X46
↔
(
X42
→
(
X36
↔
X23
)
)
)
→
(
¬
(
(
¬
X48
)
↔
(
X36
↔
X49
)
)
)
)
→
(
(
(
¬
(
(
¬
X47
)
→
(
¬
X46
)
→
(
(
¬
X8
)
↔
X49
)
)
)
↔
(
(
¬
X16
)
↔
(
(
¬
X15
)
↔
(
(
¬
X8
)
↔
X49
)
)
)
)
↔
(
(
(
¬
(
X50
→
X27
→
(
X7
↔
X41
)
)
)
↔
(
¬
(
(
¬
(
(
X47
→
(
(
¬
X41
)
↔
(
(
¬
X27
)
↔
X49
)
)
)
→
(
X50
↔
(
X49
→
X41
→
False
)
)
)
)
→
X50
→
X47
)
)
)
→
(
(
¬
(
(
(
X49
→
X45
→
X28
→
False
)
→
(
¬
(
(
¬
X45
)
↔
(
X28
→
X46
)
)
)
→
(
(
¬
X24
)
↔
X49
)
)
→
(
X49
↔
(
X46
↔
(
(
¬
X28
)
→
X41
)
)
)
)
)
→
(
¬
(
X46
→
X24
)
)
)
→
(
(
(
¬
(
X50
↔
(
(
¬
X41
)
↔
(
X28
→
X45
)
)
)
)
↔
(
(
¬
X4
)
→
X28
)
)
↔
(
¬
X4
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
(
(
(
¬
X40
)
↔
(
X33
→
X44
)
)
↔
(
(
¬
(
X44
↔
(
X33
↔
(
X29
→
False
)
)
)
)
↔
(
¬
(
(
(
¬
X44
)
→
(
X40
↔
(
(
¬
X29
)
→
X48
)
)
)
→
(
¬
(
X44
→
(
X33
↔
(
X29
↔
False
)
)
)
)
)
)
)
)
↔
(
¬
(
(
X40
→
(
(
¬
X13
)
↔
(
X12
↔
X48
)
)
)
→
(
¬
(
X13
↔
(
(
¬
X12
)
↔
X44
)
)
)
)
)
)
↔
(
(
(
¬
(
X44
→
(
X5
↔
False
)
)
)
→
(
¬
(
(
¬
(
X50
↔
(
X48
↔
(
X9
↔
X10
)
)
)
)
→
(
¬
(
(
¬
X10
)
↔
(
(
¬
X9
)
→
(
X5
↔
X50
)
)
)
)
)
)
)
↔
(
(
¬
X44
)
→
(
X40
↔
(
(
¬
X10
)
→
X50
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X40
)
↔
(
(
¬
X34
)
↔
(
(
¬
X19
)
→
X50
)
)
)
)
→
(
¬
(
(
¬
X48
)
→
(
(
¬
X43
)
↔
(
X19
→
X50
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
¬
(
X43
→
(
X38
↔
(
X18
→
X11
→
False
)
)
)
)
↔
(
(
X50
→
(
X38
↔
(
X18
↔
X32
)
)
)
↔
(
¬
(
X50
↔
(
X38
↔
(
X18
↔
(
X11
↔
False
)
)
)
)
)
)
)
↔
(
X50
↔
(
(
¬
X38
)
↔
X43
)
)
)
→
(
(
¬
(
X43
→
(
X32
↔
(
X11
→
X0
)
)
)
)
↔
(
X43
→
X32
→
X39
)
)
)
)
↔
(
(
X48
↔
(
X39
↔
(
X38
↔
X35
)
)
)
→
(
X39
↔
(
X18
↔
X35
)
)
)
)
)
→
(
(
(
¬
(
(
(
(
X44
→
(
¬
X3
)
→
(
(
¬
X2
)
↔
X43
)
)
→
(
¬
(
(
¬
X6
)
↔
(
X3
↔
(
X2
→
X26
)
)
)
)
)
↔
(
¬
(
X44
→
(
X26
↔
(
(
¬
X14
)
→
X43
)
)
)
)
)
→
(
¬
(
(
¬
X6
)
→
(
(
¬
X2
)
↔
X14
)
)
)
)
)
↔
(
X26
→
(
X14
↔
(
X2
→
False
)
)
)
)
↔
(
(
¬
(
X40
→
(
X37
↔
(
(
¬
X14
)
↔
X44
)
)
)
)
→
X49
→
X44
→
(
¬
X6
)
→
X37
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
f51b38...
and proposition id is
e89632...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMNQAAaU5uLzWu8JGmsESDZUftbrqSMeyDV
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
¬
(
(
(
X49
→
X47
→
X12
→
X27
)
↔
(
X48
↔
(
X46
→
X12
)
)
)
→
(
(
(
(
(
(
¬
X31
)
→
X13
→
X47
)
↔
(
X46
→
(
(
¬
X23
)
↔
X47
)
)
)
→
(
X47
↔
(
X23
→
False
)
)
)
↔
(
¬
(
X48
→
X47
→
(
(
¬
X23
)
↔
X31
)
)
)
)
↔
(
(
¬
X13
)
↔
(
(
¬
X47
)
↔
(
(
¬
X44
)
↔
(
X20
↔
X48
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X49
→
(
(
¬
X42
)
↔
(
X40
→
X48
)
)
)
)
→
(
(
(
(
(
¬
X30
)
↔
(
(
¬
X24
)
→
X42
)
)
↔
(
¬
(
X30
→
X24
→
(
(
¬
X21
)
↔
X40
)
)
)
)
↔
(
¬
(
X48
↔
(
(
¬
X30
)
↔
(
(
¬
X24
)
↔
X40
)
)
)
)
)
↔
(
¬
(
(
¬
X42
)
↔
(
(
¬
X30
)
↔
(
X2
↔
X48
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
X24
)
→
(
¬
X6
)
→
X30
)
→
X30
→
(
¬
X16
)
→
X48
)
)
→
(
X30
↔
(
X24
↔
X48
)
)
)
)
↔
(
(
(
X36
↔
(
(
¬
X1
)
→
X46
)
)
↔
(
¬
(
(
X46
→
(
X40
↔
X42
)
)
→
(
(
¬
(
X42
↔
(
X40
↔
X46
)
)
)
↔
(
X46
↔
(
X34
↔
(
(
¬
X1
)
→
X42
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X48
→
(
X42
↔
(
(
¬
X0
)
→
X46
)
)
)
)
→
(
¬
(
X42
↔
(
X40
↔
(
X4
↔
X46
)
)
)
)
→
(
X48
↔
(
X42
↔
(
X4
→
False
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
(
¬
(
(
X46
↔
(
X45
→
(
X29
↔
X35
)
)
)
→
X35
→
(
(
¬
X28
)
↔
X29
)
)
)
→
(
(
¬
(
X45
↔
(
X29
→
X49
)
)
)
↔
(
X46
→
(
¬
X45
)
→
(
¬
X43
)
→
X49
)
)
)
)
→
(
(
X45
→
(
X11
↔
X43
)
)
↔
(
X35
→
(
X22
↔
False
)
)
)
)
)
↔
(
(
¬
(
(
¬
X43
)
↔
(
(
¬
X41
)
→
(
(
¬
X33
)
↔
X46
)
)
)
)
↔
(
(
¬
(
(
¬
(
X39
→
X8
→
X43
)
)
→
(
¬
(
X45
↔
X46
)
)
)
)
↔
(
(
¬
X41
)
→
X45
)
)
)
)
↔
(
(
¬
(
(
(
(
X38
↔
(
(
¬
X15
)
↔
(
X14
→
X45
)
)
)
↔
(
(
¬
(
(
¬
X25
)
→
X15
→
(
X9
↔
X38
)
)
)
↔
(
X38
↔
(
X7
↔
X45
)
)
)
)
↔
(
(
¬
(
(
X38
→
(
¬
X25
)
→
(
X17
↔
X46
)
)
→
(
¬
(
X46
↔
(
(
¬
X25
)
↔
(
X17
↔
X45
)
)
)
)
)
)
→
(
X38
↔
(
(
¬
X25
)
→
X45
)
)
)
)
→
(
(
(
¬
X45
)
→
(
(
¬
X38
)
↔
(
(
¬
X37
)
↔
X46
)
)
)
↔
(
¬
(
(
¬
X45
)
↔
(
X37
↔
(
(
¬
X32
)
→
X46
)
)
)
)
)
)
)
→
(
(
(
(
X49
→
(
X46
↔
(
X18
→
X45
)
)
)
↔
(
(
¬
X45
)
↔
(
(
¬
X19
)
→
(
¬
X18
)
→
X46
)
)
)
→
X46
→
X45
→
X44
)
→
X46
→
(
X45
↔
X44
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
X49
↔
(
X5
→
X45
)
)
→
(
(
¬
(
X49
→
(
X46
↔
(
(
¬
X26
)
→
X45
)
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X26
)
↔
(
(
¬
X5
)
↔
X45
)
)
)
)
)
→
(
X49
↔
(
X26
↔
(
X3
→
X46
)
)
)
)
)
→
(
¬
(
X49
→
(
X46
↔
(
X37
↔
(
X26
↔
False
)
)
)
)
)
)
)
→
X49
→
(
X26
↔
(
X10
→
X37
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
¬
(
(
X28
→
X22
→
(
X1
↔
False
)
)
→
X45
→
(
X28
↔
(
(
¬
X1
)
↔
X41
)
)
)
)
↔
(
X22
→
X1
)
)
↔
(
(
X41
↔
(
(
¬
X28
)
→
X45
)
)
↔
(
(
¬
(
X45
↔
(
X41
↔
(
(
¬
X16
)
→
X36
)
)
)
)
↔
(
¬
(
X28
→
(
¬
X16
)
→
X36
)
)
)
)
)
↔
(
(
(
(
X45
↔
(
X41
↔
(
(
¬
X40
)
↔
X49
)
)
)
↔
(
(
¬
(
X49
↔
(
X45
↔
(
X13
→
X33
)
)
)
)
→
X49
→
(
X45
↔
(
X40
↔
(
X33
↔
False
)
)
)
)
)
↔
(
¬
(
X49
→
(
(
¬
X41
)
↔
(
(
¬
X29
)
→
X45
)
)
)
)
)
↔
(
(
(
X48
→
(
(
¬
X45
)
↔
(
X6
↔
X49
)
)
)
↔
(
¬
(
(
X49
↔
(
X11
↔
X48
)
)
→
(
(
¬
(
X36
→
(
X11
↔
X49
)
)
)
↔
(
¬
(
X36
↔
(
(
¬
X31
)
→
X45
)
)
)
)
)
)
)
↔
(
¬
(
X49
↔
(
(
¬
X45
)
↔
(
X41
↔
X48
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
(
X48
→
X20
→
X40
)
)
↔
(
¬
(
X40
↔
(
X30
→
X20
→
X46
)
)
)
)
→
(
X37
↔
(
X23
→
X30
)
)
)
↔
(
(
(
(
X41
→
(
X40
↔
(
(
¬
X8
)
→
X30
)
)
)
↔
(
(
¬
X9
)
↔
(
X8
↔
X30
)
)
)
↔
(
X30
→
X23
→
False
)
)
↔
(
X40
→
(
X30
↔
False
)
)
)
)
→
(
(
¬
X44
)
→
(
¬
X26
)
→
X24
→
X46
)
→
(
¬
(
(
X44
↔
(
X40
↔
(
X18
→
X48
)
)
)
→
(
¬
(
(
¬
X40
)
→
X48
)
)
→
(
¬
(
X46
↔
(
X44
↔
X26
)
)
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
X47
↔
(
(
¬
X12
)
→
X35
)
)
)
→
(
(
X43
↔
(
X39
→
X35
→
X10
)
)
↔
(
(
(
X47
↔
(
X43
↔
(
X35
↔
(
X12
↔
False
)
)
)
)
→
(
¬
(
X39
↔
(
(
¬
X12
)
→
X35
)
)
)
)
→
(
X39
↔
(
X35
↔
X47
)
)
)
)
)
↔
(
¬
(
(
¬
X47
)
↔
(
X43
→
(
¬
X12
)
→
X48
)
)
)
)
→
(
¬
(
(
(
(
X48
↔
(
X35
↔
(
X21
↔
X46
)
)
)
→
(
¬
(
X48
→
X46
→
(
X2
↔
X35
)
)
)
)
↔
(
(
X48
→
(
X46
↔
(
X25
→
(
X17
↔
False
)
)
)
)
→
X46
→
(
(
¬
X35
)
↔
(
X25
↔
X48
)
)
)
)
→
(
(
X46
↔
(
(
¬
X25
)
→
(
¬
X19
)
→
X35
)
)
↔
(
¬
(
X46
→
X35
→
(
¬
X25
)
→
X47
)
)
)
)
)
)
↔
(
(
(
(
X39
→
(
(
¬
X34
)
↔
(
(
¬
X4
)
↔
X48
)
)
)
↔
(
X39
→
X34
→
X32
→
X48
)
)
↔
(
(
(
(
¬
X32
)
↔
(
X27
↔
(
X5
→
X38
)
)
)
↔
(
(
X38
↔
(
X5
→
X0
→
X27
)
)
↔
(
(
¬
X27
)
→
(
(
¬
X5
)
↔
X34
)
)
)
)
↔
(
(
¬
(
X38
↔
X3
)
)
→
(
(
¬
(
X38
↔
(
(
¬
X5
)
↔
(
(
¬
X3
)
↔
X39
)
)
)
)
↔
(
¬
(
X39
↔
(
X38
→
X0
→
X5
)
)
)
)
)
)
)
↔
(
(
(
X42
↔
(
X39
→
X38
→
(
X14
↔
False
)
)
)
↔
(
(
¬
(
X42
↔
(
X32
↔
X39
)
)
)
↔
(
(
¬
(
X32
→
(
X15
↔
X42
)
)
)
↔
(
X38
→
(
(
¬
X32
)
↔
(
(
¬
X15
)
→
X39
)
)
)
)
)
)
→
(
(
(
¬
(
X39
↔
X42
)
)
↔
(
(
X47
↔
(
X7
↔
X45
)
)
↔
(
¬
(
X45
→
(
¬
X39
)
→
(
(
¬
X7
)
↔
X42
)
)
)
)
)
↔
(
¬
(
(
¬
X39
)
↔
(
(
¬
X15
)
↔
X42
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
1b920a...
and proposition id is
f6f585...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMdxRompRFMxt7P1oN9TxNSNxYtVQmqXNPJ
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
(
(
¬
(
(
¬
(
(
X45
→
(
¬
X36
)
→
(
X16
↔
X50
)
)
→
(
¬
(
(
¬
X18
)
↔
(
(
¬
X15
)
↔
X25
)
)
)
→
(
(
X36
↔
(
(
¬
X18
)
↔
X45
)
)
↔
(
(
X45
→
X36
→
(
(
¬
X18
)
↔
X25
)
)
→
(
¬
(
X18
↔
(
(
¬
X15
)
→
X45
)
)
)
)
)
)
)
→
(
(
(
X48
→
(
(
¬
X26
)
↔
X34
)
)
↔
(
X35
↔
(
X34
↔
X36
)
)
)
↔
(
(
(
X36
→
(
(
¬
X35
)
↔
X45
)
)
→
(
X36
↔
(
X35
↔
(
X9
↔
X50
)
)
)
→
(
(
(
¬
X36
)
→
(
(
¬
X35
)
↔
(
(
¬
X34
)
↔
X45
)
)
)
↔
(
(
¬
X36
)
→
(
(
¬
X35
)
↔
(
(
¬
X34
)
↔
X50
)
)
)
)
)
↔
(
¬
(
X50
↔
(
X45
→
False
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X50
→
(
(
¬
X17
)
↔
X48
)
)
)
↔
(
(
¬
(
(
(
¬
(
X48
→
(
X19
↔
(
(
¬
X17
)
↔
X45
)
)
)
)
→
(
X38
↔
(
X19
→
(
¬
X17
)
→
X48
)
)
)
→
(
X45
↔
(
X19
→
X38
)
)
)
)
↔
(
(
X45
↔
(
(
¬
X0
)
→
X33
)
)
↔
(
¬
(
(
X45
→
(
(
¬
X33
)
↔
X48
)
)
→
(
¬
(
(
¬
X38
)
→
(
(
¬
X33
)
↔
(
X0
↔
X45
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
X50
→
(
X48
↔
(
(
¬
X38
)
↔
X41
)
)
)
)
↔
(
¬
(
X48
↔
(
X38
→
(
X2
↔
X41
)
)
)
)
)
→
(
(
(
X50
↔
(
X38
↔
X48
)
)
↔
(
X43
→
X6
)
)
↔
(
¬
(
X48
→
(
X6
↔
False
)
)
)
)
)
)
→
(
(
X49
→
(
X43
↔
(
X31
↔
(
X10
→
False
)
)
)
)
↔
(
(
¬
(
X50
→
(
X48
↔
(
X10
→
False
)
)
)
)
↔
(
X50
↔
(
X49
↔
(
X10
→
X43
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
¬
X24
)
→
(
X20
↔
(
X1
→
X40
)
)
)
↔
(
(
(
X40
↔
(
X22
↔
(
X11
↔
False
)
)
)
↔
(
¬
(
(
X22
→
(
(
¬
X11
)
↔
X44
)
)
→
(
¬
(
X44
→
(
X40
↔
(
(
¬
X11
)
↔
X24
)
)
)
)
)
)
)
↔
(
X24
↔
(
X1
→
X44
)
)
)
)
→
(
¬
(
(
¬
X30
)
→
(
X29
↔
(
(
¬
X22
)
→
X44
)
)
)
)
→
(
(
¬
(
(
¬
X44
)
→
(
¬
X22
)
→
X46
)
)
↔
(
¬
(
(
(
¬
X29
)
→
(
(
¬
X13
)
↔
X40
)
)
→
(
¬
(
X44
→
(
¬
X29
)
→
X13
→
X46
)
)
)
)
)
)
)
→
(
(
(
(
(
(
(
¬
(
X39
→
(
X30
↔
X3
)
)
)
↔
(
(
¬
(
X39
→
X30
→
(
¬
X3
)
→
X46
)
)
↔
(
¬
(
X46
→
(
(
¬
X30
)
↔
(
X8
→
X39
)
)
)
)
)
)
↔
(
¬
(
X46
↔
(
X39
→
X30
)
)
)
)
↔
(
¬
(
X39
→
X30
→
X8
→
False
)
)
)
→
(
X46
↔
(
X39
↔
X8
)
)
)
→
(
¬
(
X46
↔
(
X39
↔
(
X30
↔
X49
)
)
)
)
)
↔
(
(
X30
→
X21
→
(
X4
↔
X46
)
)
↔
(
¬
(
X46
→
(
(
¬
X21
)
↔
X39
)
)
)
)
)
→
(
(
(
¬
X46
)
↔
(
(
¬
X28
)
↔
X49
)
)
↔
(
(
(
¬
(
X49
→
X46
→
(
¬
X12
)
→
X39
)
)
→
(
X22
↔
X12
)
)
→
(
¬
(
(
¬
X39
)
→
X22
→
X49
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X50
↔
(
X46
↔
(
X44
→
X47
)
)
)
)
→
(
(
X47
↔
(
X44
→
(
X32
↔
(
X14
→
False
)
)
)
)
↔
(
¬
(
X47
→
X46
→
X44
→
X32
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
(
¬
X42
)
↔
(
X39
↔
(
X5
→
X44
)
)
)
)
→
(
(
¬
(
X46
→
(
(
¬
X39
)
↔
X50
)
)
)
↔
(
X44
→
(
X42
↔
(
(
¬
X39
)
↔
X46
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X50
↔
(
X44
→
(
X27
↔
X39
)
)
)
)
→
(
¬
(
(
¬
X44
)
→
(
(
¬
X39
)
↔
(
(
¬
X7
)
↔
X50
)
)
)
)
)
)
)
→
X50
→
(
(
¬
X27
)
↔
X39
)
)
)
)
→
(
(
(
¬
(
X48
↔
(
(
¬
X39
)
→
(
¬
X23
)
→
X42
)
)
)
→
(
¬
(
(
¬
X37
)
→
X23
→
X48
)
)
)
↔
(
¬
(
X48
→
(
X37
↔
X44
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
X46
↔
(
X34
↔
X14
)
)
→
(
(
(
(
¬
(
X28
↔
(
(
¬
X19
)
→
X34
)
)
)
↔
(
X46
↔
(
X43
→
X28
→
X34
)
)
)
→
(
(
(
¬
(
(
¬
X46
)
↔
(
(
¬
X43
)
→
(
(
¬
X34
)
↔
X49
)
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X34
)
→
(
X14
↔
X49
)
)
)
)
)
↔
(
X49
↔
(
X14
→
X46
)
)
)
)
↔
(
¬
(
X49
↔
(
X47
→
(
X46
↔
(
X43
→
False
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
X7
→
(
X2
↔
X16
)
)
→
(
(
(
(
¬
(
(
¬
X30
)
→
(
¬
X16
)
→
(
X7
↔
X32
)
)
)
↔
(
¬
(
X30
→
(
X16
↔
X49
)
)
)
)
↔
(
X32
→
(
¬
X7
)
→
X30
)
)
↔
(
¬
(
X50
→
X7
→
(
¬
X2
)
→
X49
)
)
)
)
)
→
(
(
(
X47
→
(
X46
↔
(
(
¬
X30
)
↔
X32
)
)
)
→
X31
→
X32
)
↔
(
¬
(
X46
↔
(
(
¬
X31
)
→
X47
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X47
↔
(
X1
→
X24
)
)
)
→
(
¬
(
(
(
(
X49
→
X38
→
(
(
¬
X24
)
↔
X25
)
)
→
(
¬
(
(
(
X46
↔
(
X24
→
X25
)
)
→
(
¬
(
X49
↔
(
X25
→
(
X1
↔
X24
)
)
)
)
)
→
X49
→
X25
→
X46
)
)
)
↔
(
¬
(
X46
→
(
(
¬
X25
)
↔
X49
)
)
)
)
→
(
(
¬
X46
)
↔
(
(
¬
X38
)
↔
(
X8
↔
X47
)
)
)
→
(
¬
(
(
X38
→
X36
→
X24
→
X8
→
False
)
→
(
X47
↔
(
X24
↔
(
X8
→
X38
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
X46
↔
(
(
¬
X38
)
↔
(
(
¬
X20
)
↔
X44
)
)
)
)
↔
(
¬
(
(
X49
↔
(
X44
→
X38
)
)
→
(
¬
(
(
(
¬
(
(
¬
X38
)
↔
X46
)
)
↔
(
X49
→
(
X38
↔
(
(
¬
X4
)
↔
X44
)
)
)
)
→
X49
→
(
(
¬
X44
)
↔
(
(
¬
X9
)
→
X46
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X50
↔
(
(
¬
X33
)
↔
(
(
¬
X22
)
↔
X48
)
)
)
)
→
(
(
(
¬
X48
)
→
(
(
¬
X33
)
↔
(
X23
→
X50
)
)
)
↔
(
X33
↔
(
(
¬
X29
)
↔
X50
)
)
)
)
↔
(
(
¬
(
(
¬
X48
)
↔
(
(
¬
X33
)
→
X49
)
)
)
↔
(
(
X49
↔
(
X45
→
(
(
¬
X13
)
↔
X48
)
)
)
↔
(
(
¬
X48
)
↔
(
(
¬
X23
)
↔
X49
)
)
)
)
)
→
(
¬
(
(
(
(
X15
→
X11
→
X29
)
→
(
(
¬
(
X11
→
X6
→
X29
)
)
↔
(
X29
↔
(
X15
→
(
¬
X11
)
→
X48
)
)
)
)
↔
(
¬
(
X29
↔
(
X11
↔
X15
)
)
)
)
→
(
(
X47
→
(
X17
↔
(
(
¬
X0
)
→
X11
)
)
)
↔
(
(
X47
→
(
X22
↔
X17
)
)
↔
(
(
¬
X11
)
↔
(
(
¬
(
(
¬
X11
)
↔
X17
)
)
↔
(
X47
↔
(
X22
→
(
X15
↔
X17
)
)
)
)
)
)
)
→
(
¬
(
X29
↔
(
X0
↔
X11
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
X50
↔
(
X37
↔
(
X10
→
X49
)
)
)
→
(
X26
↔
(
(
¬
X10
)
↔
X49
)
)
)
)
↔
(
¬
(
(
¬
(
(
X40
↔
(
(
¬
X39
)
↔
(
X27
↔
X41
)
)
)
→
(
¬
(
(
(
¬
(
X50
↔
(
X40
↔
(
X39
→
X5
)
)
)
)
↔
(
¬
(
X40
→
(
X39
↔
(
X27
↔
X50
)
)
)
)
)
→
(
X50
↔
(
(
¬
X40
)
→
X42
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
(
¬
(
(
¬
X40
)
↔
(
X37
→
(
(
¬
X21
)
↔
X41
)
)
)
)
↔
(
X41
↔
(
(
¬
X37
)
↔
(
X10
↔
X40
)
)
)
)
↔
(
¬
(
X41
→
(
¬
X37
)
→
X40
)
)
)
↔
(
¬
(
(
¬
X21
)
→
X10
→
X40
)
)
)
→
X50
→
(
(
¬
X40
)
↔
(
(
¬
X37
)
→
X41
)
)
)
→
(
X40
↔
(
X37
↔
(
(
¬
X18
)
→
X50
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
X49
→
X40
→
(
¬
X12
)
→
X35
)
→
(
¬
(
X49
→
(
¬
X35
)
→
X26
→
X40
)
)
)
)
↔
(
X49
→
(
(
¬
X26
)
↔
X35
)
)
)
↔
(
(
¬
(
X48
↔
(
X3
→
X35
)
)
)
→
(
¬
(
X40
→
X26
→
X35
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
8a551a...
and proposition id is
aaa88d...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMWDt9GSN1jFaVGz3P7ENY2oSsybNryDNhT
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
∀X50 :
prop
,
(
(
(
(
¬
(
(
¬
(
(
(
X48
↔
(
X38
↔
(
X35
↔
X13
)
)
)
→
(
¬
(
X48
→
(
¬
X38
)
→
(
(
¬
X35
)
↔
X46
)
)
)
)
→
(
(
(
(
¬
X38
)
↔
(
(
¬
X37
)
↔
(
X18
→
X46
)
)
)
→
X46
→
(
¬
X18
)
→
X48
)
↔
(
X46
→
X38
→
X37
)
)
→
(
¬
(
(
¬
(
X50
→
(
X18
↔
(
(
¬
X17
)
→
X37
)
)
)
)
→
(
X46
↔
(
(
¬
X37
)
↔
X50
)
)
)
)
)
)
→
(
(
(
¬
(
X39
→
(
X38
↔
(
X29
→
X4
)
)
)
)
↔
(
(
¬
(
X39
↔
(
X38
→
False
)
)
)
↔
(
X37
↔
(
X29
↔
X46
)
)
)
)
↔
(
¬
(
(
¬
(
X45
↔
(
X38
↔
(
(
¬
X2
)
↔
X29
)
)
)
)
→
(
¬
(
(
¬
(
X45
↔
(
X25
↔
X38
)
)
)
→
(
(
(
¬
X29
)
↔
(
X25
→
X37
)
)
↔
(
(
¬
X38
)
↔
(
(
¬
X25
)
↔
X45
)
)
)
)
)
)
)
)
)
)
→
(
(
(
(
X23
↔
(
X14
→
X9
→
False
)
)
↔
(
(
¬
(
X14
→
X9
)
)
→
(
¬
X45
)
→
(
¬
X23
)
→
(
¬
X14
)
→
X46
)
)
→
(
(
¬
X14
)
↔
(
(
¬
X9
)
↔
(
(
¬
X7
)
→
X23
)
)
)
)
↔
(
(
(
¬
(
X36
↔
(
X15
→
(
¬
X5
)
→
X49
)
)
)
→
(
¬
X23
)
→
(
(
¬
X15
)
↔
(
(
¬
X5
)
→
X36
)
)
)
→
(
(
¬
(
(
X48
→
(
X31
↔
(
X14
→
X46
)
)
)
→
X49
→
(
X31
↔
(
X14
↔
False
)
)
)
)
↔
(
X49
→
X48
→
X27
→
X46
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
X49
→
(
¬
X41
)
→
(
(
¬
X19
)
↔
X47
)
)
)
↔
(
X47
→
(
¬
X24
)
→
(
¬
X8
)
→
X49
)
)
↔
(
(
(
¬
(
(
¬
(
X47
→
X42
→
(
(
¬
X24
)
↔
X44
)
)
)
→
X47
→
(
X44
↔
(
X42
→
X22
)
)
)
)
→
(
(
¬
X44
)
↔
(
X42
↔
X47
)
)
)
↔
(
(
¬
X24
)
↔
(
(
¬
X22
)
↔
X42
)
)
)
)
→
(
¬
(
(
¬
(
X40
→
(
X26
↔
(
(
¬
X20
)
→
X41
)
)
)
)
→
(
(
¬
(
(
¬
X42
)
↔
(
(
¬
X41
)
→
(
X40
↔
X47
)
)
)
)
↔
(
X41
↔
(
X40
↔
(
(
¬
X1
)
→
X26
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
X49
→
False
)
→
(
X43
↔
(
X12
→
(
(
¬
X10
)
↔
X50
)
)
)
)
↔
(
(
X49
↔
(
X44
→
(
X34
↔
X48
)
)
)
↔
(
(
¬
(
X49
→
X44
→
X10
→
X48
)
)
↔
(
X49
→
(
¬
X44
)
→
(
X43
↔
X48
)
)
)
)
)
→
(
(
X43
↔
(
X28
→
X21
)
)
→
(
¬
(
X44
↔
(
X43
→
(
(
¬
X28
)
↔
X32
)
)
)
)
)
→
(
¬
(
X49
→
X28
→
X21
)
)
)
→
(
(
(
(
(
¬
(
X50
↔
(
X33
↔
(
X6
↔
(
X3
→
False
)
)
)
)
)
↔
(
X50
→
(
(
¬
X3
)
↔
X44
)
)
)
↔
(
X50
↔
(
X44
↔
(
X33
→
False
)
)
)
)
→
(
X44
↔
(
(
¬
X6
)
→
X33
)
)
)
→
(
¬
(
(
¬
(
X49
↔
(
X42
→
(
¬
X16
)
→
X50
)
)
)
→
(
¬
(
(
¬
X44
)
→
(
X42
↔
X50
)
)
)
)
)
→
(
¬
(
X49
→
X44
→
X42
→
X11
)
)
→
(
¬
(
X49
→
X44
→
(
X33
↔
X42
)
)
)
)
→
(
(
X42
→
(
X33
↔
X50
)
)
↔
(
¬
(
(
¬
(
X33
→
(
X30
↔
(
(
¬
X0
)
→
X42
)
)
)
)
→
(
¬
X30
)
→
(
X0
↔
X50
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X50
→
(
X46
↔
(
X28
↔
X36
)
)
)
)
→
(
X47
↔
(
X32
→
(
¬
X31
)
→
X49
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
X49
→
X37
→
(
(
¬
X29
)
↔
X47
)
)
)
→
(
(
¬
(
(
¬
X47
)
↔
(
X46
→
X49
)
)
)
↔
(
(
X49
↔
(
(
¬
X46
)
→
(
(
¬
X13
)
↔
X47
)
)
)
↔
(
X46
↔
(
(
¬
X37
)
→
(
¬
X13
)
→
X47
)
)
)
)
)
)
→
(
¬
(
X46
↔
(
X37
→
(
¬
X20
)
→
X47
)
)
)
)
)
↔
(
X49
→
(
X29
↔
(
(
¬
X8
)
↔
X47
)
)
)
)
)
→
(
(
(
X48
→
X46
→
(
X40
↔
X1
)
)
↔
(
(
¬
X2
)
→
(
¬
X1
)
→
X40
)
)
→
(
(
(
X48
→
X18
→
X5
)
→
(
¬
(
(
¬
(
X50
↔
(
X48
→
(
¬
X17
)
→
X40
)
)
)
→
(
¬
(
X48
↔
(
X40
→
X5
→
X17
)
)
)
)
)
)
↔
(
X46
→
(
¬
X40
)
→
X18
→
X50
)
)
)
→
(
(
(
(
¬
(
X15
→
X45
)
)
↔
(
X47
↔
(
(
¬
X45
)
↔
(
X35
↔
X48
)
)
)
)
→
(
¬
(
(
¬
X46
)
→
(
X45
↔
(
X35
→
X47
)
)
)
)
)
↔
(
(
¬
(
X33
→
X47
)
)
→
(
(
¬
(
X50
→
(
(
¬
X46
)
↔
X47
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X33
)
→
X50
)
)
→
(
¬
(
(
(
¬
X47
)
↔
(
(
¬
X46
)
↔
(
(
¬
X4
)
→
X48
)
)
)
→
(
¬
(
(
¬
(
X50
↔
(
X47
→
(
X46
↔
X48
)
)
)
)
→
X50
→
(
¬
X46
)
→
(
¬
X33
)
→
X47
)
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
(
X48
↔
(
X44
↔
(
X7
↔
False
)
)
)
→
(
X26
→
X9
→
X25
)
→
X48
→
(
X26
↔
(
X9
→
X7
)
)
)
→
(
¬
(
(
¬
X9
)
↔
X48
)
)
)
↔
(
X26
↔
(
X10
↔
X25
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X49
)
↔
(
X44
→
(
X23
↔
X50
)
)
)
)
→
(
(
¬
(
X26
→
X25
→
(
¬
X6
)
→
X44
)
)
↔
(
(
¬
X49
)
↔
(
(
¬
X26
)
↔
(
(
¬
X6
)
↔
X50
)
)
)
)
)
)
↔
(
(
X44
↔
(
X21
↔
False
)
)
→
(
(
(
(
(
X26
→
X25
→
(
X3
↔
X21
)
)
↔
(
¬
(
X26
↔
(
(
¬
X25
)
→
(
X21
↔
X44
)
)
)
)
)
↔
(
X44
↔
(
X26
↔
X21
)
)
)
↔
(
¬
(
(
¬
X21
)
→
X44
)
)
)
↔
(
¬
(
X44
→
(
X25
↔
(
X21
→
False
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
X44
)
→
(
X27
↔
(
X16
↔
X48
)
)
)
↔
(
X48
→
(
¬
X27
)
→
(
X12
↔
X42
)
)
)
↔
(
(
¬
X12
)
→
(
¬
X11
)
→
X48
)
)
→
(
¬
(
(
X42
↔
(
(
¬
X16
)
↔
(
(
¬
X14
)
→
X48
)
)
)
→
(
¬
X14
)
→
X49
)
)
)
)
)
)
↔
(
(
(
(
X24
→
(
X19
↔
X48
)
)
↔
(
(
¬
(
(
¬
X19
)
→
X24
)
)
→
(
¬
(
(
X43
↔
(
(
¬
X24
)
→
X48
)
)
→
(
(
X19
→
X48
)
↔
(
¬
(
(
¬
(
X48
→
(
X30
↔
(
X24
→
(
X19
↔
False
)
)
)
)
)
→
(
¬
(
(
¬
X43
)
→
(
¬
X19
)
→
X48
)
)
)
)
)
)
)
)
)
↔
(
(
X50
→
(
X38
↔
False
)
)
→
(
¬
(
(
(
X38
→
(
X22
↔
X50
)
)
↔
(
¬
(
X22
↔
(
X19
↔
X39
)
)
)
)
→
(
¬
(
X50
→
(
X19
↔
X22
)
)
)
)
)
)
)
→
(
(
(
¬
(
X39
→
(
(
¬
X38
)
↔
(
X34
↔
X48
)
)
)
)
↔
(
¬
(
X38
→
X0
→
X48
)
)
)
↔
(
(
¬
(
(
X48
↔
(
X41
↔
X34
)
)
→
(
X49
↔
(
X48
↔
(
X41
↔
(
X39
↔
False
)
)
)
)
→
(
X48
↔
(
X39
→
(
¬
X34
)
→
X49
)
)
)
)
↔
(
¬
(
X48
→
(
X34
↔
X49
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
dd27ed...
and proposition id is
39f03e...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMQtDmKwAq6wiewQF6HM3EQCEneAGFsXfub
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
∀X51 :
prop
,
(
(
(
¬
(
(
(
(
(
¬
(
(
X43
↔
(
X35
→
X21
→
X26
)
)
→
X26
→
(
X10
↔
False
)
)
)
↔
(
X35
→
X43
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
¬
X35
)
↔
(
(
¬
X29
)
→
(
¬
X15
)
→
X47
)
)
)
→
(
(
(
(
¬
X29
)
↔
(
(
¬
X21
)
↔
(
X20
→
X43
)
)
)
→
(
¬
(
X47
→
X29
)
)
)
↔
(
X29
→
(
X24
↔
(
(
¬
X20
)
↔
X35
)
)
)
)
)
)
→
X47
→
(
X43
↔
X35
)
)
)
)
→
(
¬
(
(
(
¬
(
X42
↔
(
X38
↔
(
X35
→
X21
→
False
)
)
)
)
→
(
X42
↔
(
X35
→
X5
→
X21
)
)
)
→
(
(
X47
→
X38
→
X42
)
↔
(
(
X38
↔
(
X21
→
X25
)
)
↔
(
¬
(
(
¬
(
X47
→
(
X38
↔
X6
)
)
)
→
(
X42
↔
(
X38
↔
(
X25
→
X21
)
)
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
(
X51
→
X48
→
X12
)
→
(
X48
↔
(
X44
↔
(
(
¬
X12
)
↔
X51
)
)
)
→
(
¬
X44
)
→
X50
)
)
→
(
(
X51
↔
(
X50
↔
(
X48
→
X22
)
)
)
↔
(
(
¬
X50
)
→
(
(
¬
X48
)
↔
(
X28
↔
X51
)
)
)
)
)
)
↔
(
(
¬
(
(
X50
→
(
X28
↔
(
(
¬
X7
)
→
X17
)
)
)
→
(
¬
X28
)
→
X17
→
(
(
¬
X7
)
↔
X44
)
)
)
↔
(
X50
→
(
(
¬
X7
)
↔
X28
)
)
)
)
↔
(
(
(
¬
(
(
X51
↔
(
(
¬
X32
)
→
(
¬
X4
)
→
X50
)
)
→
(
X51
↔
(
X38
↔
(
X4
↔
X41
)
)
)
)
)
↔
(
¬
(
X50
↔
(
X32
→
(
(
¬
X27
)
↔
X51
)
)
)
)
)
↔
(
(
(
¬
(
X50
→
X48
→
X41
→
X51
)
)
↔
(
¬
(
X51
→
(
¬
X48
)
→
X42
→
X50
)
)
)
↔
(
¬
(
X48
→
(
X42
↔
(
X35
↔
X50
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
(
(
¬
X45
)
→
(
¬
X34
)
→
X46
)
→
(
(
X40
↔
X46
)
↔
(
X49
↔
(
X40
↔
(
X34
→
X23
)
)
)
)
)
→
(
(
¬
X46
)
↔
(
X45
↔
X49
)
)
)
)
→
X34
→
X30
→
X49
)
↔
(
¬
(
X44
→
(
¬
X30
)
→
(
X9
↔
X45
)
)
)
)
→
(
(
¬
(
(
(
(
(
X33
→
X42
)
→
(
¬
(
(
¬
(
X46
→
(
(
¬
X33
)
↔
X49
)
)
)
→
(
¬
(
X49
→
X33
→
X39
)
)
→
(
¬
(
(
¬
(
X49
↔
(
X42
↔
(
X39
→
X46
)
)
)
)
→
(
X39
↔
(
X33
→
False
)
)
)
)
)
)
)
→
(
X49
↔
(
X46
↔
X42
)
)
)
↔
(
¬
(
X49
↔
(
X45
↔
(
X33
↔
X46
)
)
)
)
)
→
(
(
¬
(
X46
↔
(
(
¬
X31
)
↔
X49
)
)
)
↔
(
(
X46
↔
(
(
¬
X45
)
↔
(
X31
↔
X49
)
)
)
↔
(
¬
(
X45
→
X16
)
)
)
)
)
)
↔
(
¬
(
(
X49
↔
(
X46
↔
(
X37
↔
X45
)
)
)
→
X46
→
(
(
¬
X37
)
↔
(
X13
→
X45
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
(
(
(
(
¬
X45
)
↔
X49
)
→
(
(
X19
↔
(
(
¬
X8
)
→
(
(
¬
X1
)
↔
X49
)
)
)
↔
(
(
¬
X45
)
↔
(
(
¬
X19
)
↔
(
(
¬
X1
)
→
X49
)
)
)
)
)
→
X49
→
(
X19
↔
(
(
¬
X18
)
→
X45
)
)
)
)
↔
(
¬
(
X49
↔
(
X46
↔
(
X45
↔
X19
)
)
)
)
)
→
(
(
¬
(
X19
→
(
¬
X14
)
→
(
(
¬
X2
)
↔
X49
)
)
)
↔
(
¬
(
X49
→
(
(
¬
X14
)
↔
(
X2
↔
X45
)
)
)
)
)
)
↔
(
X46
→
(
¬
X45
)
→
X43
→
X49
)
)
→
(
(
¬
(
X45
→
(
(
¬
X19
)
↔
(
X11
↔
X46
)
)
)
)
↔
(
(
X19
↔
(
X11
→
X0
→
X36
)
)
↔
(
(
X11
↔
(
X3
→
X45
)
)
↔
(
(
(
¬
X36
)
→
(
(
¬
X11
)
↔
X45
)
)
↔
(
¬
(
X3
→
(
X0
↔
X45
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
¬
(
(
¬
X22
)
→
(
¬
X2
)
→
X49
)
)
→
(
¬
(
X14
↔
(
(
¬
X2
)
→
X37
)
)
)
)
→
(
(
¬
X14
)
↔
(
(
¬
X2
)
↔
X22
)
)
)
→
(
¬
(
X50
↔
(
(
¬
X9
)
↔
X14
)
)
)
→
(
(
¬
(
X50
↔
(
X37
→
(
X32
↔
False
)
)
)
)
↔
(
(
(
¬
(
(
¬
X32
)
↔
(
X22
→
(
(
¬
X9
)
↔
X50
)
)
)
)
↔
(
X37
→
(
¬
X22
)
→
(
X9
↔
X50
)
)
)
→
(
¬
(
X9
→
X32
)
)
)
)
)
→
(
(
(
(
¬
(
(
(
(
X37
→
(
(
¬
X29
)
↔
(
X3
↔
X51
)
)
)
→
X37
→
(
¬
X29
)
→
X34
)
→
(
¬
(
X34
↔
(
(
¬
X3
)
↔
X29
)
)
)
)
→
(
¬
(
X34
→
X3
→
X37
)
)
)
)
↔
(
¬
(
X51
→
(
¬
X34
)
→
X33
→
X37
)
)
)
↔
(
(
¬
(
X51
↔
(
X32
→
(
X7
↔
False
)
)
)
)
→
(
¬
(
(
¬
(
(
X40
↔
(
(
¬
X22
)
↔
(
(
¬
X7
)
→
X29
)
)
)
→
(
¬
(
X40
→
(
X29
↔
(
X22
→
X37
)
)
)
)
)
)
→
(
(
¬
X29
)
↔
(
X22
→
X37
)
)
)
)
)
)
↔
(
(
X51
↔
(
(
¬
X35
)
↔
X36
)
)
→
(
(
(
¬
(
(
X36
↔
(
(
¬
X32
)
→
X35
)
)
→
(
¬
(
X34
→
(
X30
↔
False
)
)
)
)
)
→
(
¬
(
X36
↔
(
X34
→
X30
)
)
)
)
↔
(
¬
(
(
X51
→
(
X36
↔
(
X34
→
(
X32
↔
False
)
)
)
)
→
X51
→
(
¬
X32
)
→
X23
→
X36
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
(
¬
X46
)
↔
(
X28
↔
(
(
¬
X16
)
↔
X50
)
)
)
→
(
X48
→
(
(
¬
X38
)
↔
(
X28
→
X50
)
)
)
→
(
(
¬
X46
)
↔
(
X38
→
X48
)
)
→
(
(
¬
X46
)
→
(
X38
↔
X48
)
)
→
(
¬
(
(
¬
(
X46
→
(
¬
X38
)
→
X48
)
)
→
(
(
¬
X38
)
↔
(
(
¬
X28
)
→
(
¬
X25
)
→
X46
)
)
)
)
)
↔
(
(
X38
↔
(
(
¬
X17
)
↔
X46
)
)
→
(
¬
(
X38
→
(
(
¬
X19
)
↔
X46
)
)
)
→
(
(
¬
(
X46
↔
(
X25
↔
X38
)
)
)
↔
(
(
(
(
¬
(
X46
↔
(
X25
↔
X17
)
)
)
↔
(
(
¬
X25
)
→
(
¬
X19
)
→
(
X17
↔
X38
)
)
)
→
(
X46
↔
(
(
¬
X19
)
↔
X25
)
)
)
→
(
¬
(
X46
→
(
X38
↔
X25
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X42
→
(
X38
↔
(
X15
→
X12
)
)
)
)
→
(
¬
(
X46
→
(
(
¬
X42
)
↔
(
X38
↔
X48
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
(
¬
(
(
(
¬
X43
)
→
X20
→
(
X1
↔
X46
)
)
→
X48
→
X46
→
X1
→
X20
)
)
→
(
¬
(
X48
↔
X46
)
)
)
)
→
(
(
¬
(
X27
↔
(
X21
→
X1
)
)
)
↔
(
¬
(
X43
→
(
X20
↔
(
(
¬
X1
)
↔
X27
)
)
)
)
)
)
)
→
(
¬
(
(
X45
↔
(
(
¬
X43
)
↔
(
(
¬
X27
)
↔
X50
)
)
)
→
(
¬
(
(
¬
X21
)
↔
(
X8
↔
X45
)
)
)
→
X50
→
(
(
¬
X8
)
↔
X43
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X41
→
X27
→
(
(
¬
X11
)
↔
X44
)
)
)
→
(
¬
(
(
¬
X44
)
↔
(
(
¬
X41
)
→
(
X13
↔
X50
)
)
)
)
)
)
↔
(
(
¬
X13
)
→
(
X6
↔
X50
)
)
)
→
(
(
¬
(
X45
→
X41
→
(
¬
X24
)
→
X27
)
)
↔
(
(
(
¬
(
X47
→
(
(
¬
X41
)
↔
(
X27
→
X50
)
)
)
)
→
X47
→
(
X45
↔
(
X41
→
False
)
)
)
↔
(
X47
→
(
X41
↔
(
X27
↔
False
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X42
↔
(
X39
↔
(
X18
→
X49
)
)
)
)
↔
(
¬
(
X51
→
X26
→
X49
)
)
)
→
(
¬
(
(
(
(
(
¬
(
(
(
¬
X39
)
↔
(
X5
↔
(
X0
↔
X49
)
)
)
→
(
¬
(
(
¬
X39
)
→
(
X5
↔
(
(
¬
X0
)
↔
X47
)
)
)
)
)
)
↔
(
(
(
¬
X39
)
↔
(
(
¬
X10
)
→
X5
→
X49
)
)
↔
(
(
X46
→
(
(
¬
X31
)
↔
(
X10
↔
X49
)
)
)
→
X46
→
X39
)
)
)
↔
(
¬
(
X39
→
X31
→
(
(
¬
X26
)
↔
X49
)
)
)
)
→
(
¬
(
X46
→
(
X31
↔
X26
)
)
)
)
→
(
¬
(
X49
↔
(
X46
→
(
X39
↔
X4
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
d5c285...
and proposition id is
83c248...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMQYNHuXAMxQL5DSpmyjSbSydbDpSjgF7k7
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
∀X50 :
prop
,
∀X51 :
prop
,
(
(
¬
(
(
(
(
¬
(
(
(
X51
↔
(
X11
↔
X37
)
)
↔
(
¬
(
(
¬
(
X20
↔
(
X11
→
(
(
¬
X8
)
↔
X21
)
)
)
)
→
(
(
¬
X21
)
↔
(
X20
↔
(
(
¬
X8
)
→
X37
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
(
(
¬
X46
)
↔
(
X31
→
X51
)
)
)
↔
(
¬
(
(
X46
→
X31
→
X10
)
→
(
¬
(
(
(
¬
X46
)
↔
(
(
¬
X31
)
→
X51
)
)
→
(
(
¬
(
(
X51
↔
(
(
¬
X34
)
↔
(
X31
→
X46
)
)
)
→
(
(
¬
X46
)
↔
(
X34
→
(
X10
↔
X51
)
)
)
)
)
↔
(
X51
↔
(
X10
→
False
)
)
)
)
)
)
)
)
↔
(
¬
(
X51
→
(
X31
↔
X46
)
)
)
)
→
(
¬
X46
)
→
(
X31
↔
X51
)
)
→
(
(
¬
X37
)
↔
(
X21
→
X10
→
X46
)
)
)
)
)
)
↔
(
(
X48
↔
(
X31
↔
(
X20
↔
X43
)
)
)
↔
(
¬
(
(
¬
X43
)
↔
(
X20
→
X46
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
¬
X43
)
→
X46
)
)
→
(
¬
(
X51
↔
(
X40
→
(
¬
X19
)
→
X47
)
)
)
)
)
→
(
¬
(
(
¬
(
X51
↔
X47
)
)
→
(
(
¬
(
(
¬
(
X51
→
(
X47
↔
(
X43
→
X32
)
)
)
)
→
X51
→
(
(
¬
X43
)
↔
(
(
¬
X29
)
↔
X47
)
)
)
)
↔
False
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X46
→
X28
→
X9
→
X29
)
)
↔
(
(
¬
X28
)
→
X9
→
(
(
¬
X4
)
↔
X46
)
)
)
↔
(
¬
(
X39
→
X28
→
(
X22
↔
(
X9
↔
False
)
)
)
)
)
→
(
(
¬
(
X36
→
X29
→
(
¬
X23
)
→
X51
)
)
↔
(
¬
(
(
(
¬
(
X51
↔
(
(
¬
X16
)
→
X36
)
)
)
↔
(
X36
→
(
X23
↔
(
X3
↔
X16
)
)
)
)
→
X46
→
(
¬
X23
)
→
X36
)
)
)
→
(
¬
(
X39
→
X23
→
(
¬
X16
)
→
X29
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
(
¬
(
(
¬
X45
)
→
(
X42
↔
(
X14
→
X51
)
)
)
)
→
(
(
(
(
¬
(
X24
↔
X45
)
)
↔
(
(
¬
(
X42
→
X24
→
X17
)
)
↔
(
X45
↔
(
(
¬
X42
)
↔
(
(
¬
X24
)
→
X51
)
)
)
)
)
→
(
¬
(
X51
→
X42
)
)
)
↔
(
X51
→
(
X25
↔
X42
)
)
)
)
)
↔
(
(
¬
(
(
¬
X45
)
↔
(
X24
→
X49
)
)
)
→
(
¬
(
(
X49
↔
(
X45
→
(
X0
↔
X51
)
)
)
→
(
X51
↔
(
X24
→
X45
)
)
)
)
)
)
→
(
(
(
(
¬
(
X45
↔
(
X2
↔
False
)
)
)
↔
(
¬
(
(
(
(
¬
(
X41
↔
X49
)
)
↔
(
¬
(
(
(
¬
(
X49
↔
(
X45
↔
(
X18
↔
False
)
)
)
)
↔
(
¬
(
X49
→
(
(
¬
X18
)
↔
(
X2
↔
X41
)
)
)
)
)
→
(
¬
(
(
¬
X45
)
→
X41
→
X49
)
)
)
)
)
↔
(
X45
↔
X49
)
)
→
(
X49
↔
(
X41
↔
X18
)
)
)
)
)
↔
(
(
(
X18
→
X26
)
↔
(
X45
→
X41
→
(
X7
↔
False
)
)
)
↔
(
¬
(
(
¬
X18
)
↔
(
(
¬
X7
)
↔
X45
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X51
→
False
)
)
→
(
¬
X44
)
→
(
(
¬
X26
)
↔
(
(
¬
X15
)
↔
X45
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X26
)
→
(
(
¬
X15
)
↔
X44
)
)
)
↔
(
¬
(
X26
↔
(
X15
↔
(
(
¬
X1
)
→
X49
)
)
)
)
)
↔
(
¬
(
(
¬
X26
)
→
(
X1
↔
X49
)
)
)
)
→
(
¬
(
(
¬
X44
)
↔
(
X26
→
X49
)
)
)
)
)
→
(
X45
↔
(
X15
→
X33
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
(
X43
↔
(
X30
↔
(
X6
→
X27
)
)
)
)
→
(
¬
(
(
¬
(
X50
↔
(
X49
↔
(
(
¬
X6
)
→
X30
)
)
)
)
→
(
¬
X30
)
→
(
(
¬
X27
)
↔
(
X6
↔
X49
)
)
)
)
)
)
→
(
(
(
¬
(
X50
→
X38
→
(
X30
↔
X12
)
)
)
↔
(
¬
(
X42
→
(
(
¬
X38
)
↔
(
(
¬
X30
)
→
X50
)
)
)
)
)
↔
(
X43
→
(
X42
↔
(
X38
↔
X50
)
)
)
)
→
(
(
¬
X43
)
↔
(
(
¬
X42
)
↔
(
(
¬
X5
)
↔
X50
)
)
)
)
→
(
¬
(
(
(
¬
(
(
(
¬
(
(
¬
X30
)
↔
(
X27
→
X13
→
X35
)
)
)
→
(
¬
(
X46
↔
(
X30
→
X27
→
X13
)
)
)
)
→
(
(
¬
X35
)
↔
(
X30
→
X46
)
)
)
)
↔
(
(
¬
X38
)
→
X35
→
(
X30
↔
X46
)
)
)
→
(
¬
X41
)
→
(
(
¬
X30
)
↔
(
(
¬
X27
)
→
X46
)
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
(
(
¬
(
X47
↔
(
(
¬
X43
)
→
(
¬
X32
)
→
X48
)
)
)
↔
(
X48
→
(
(
¬
X43
)
↔
(
(
¬
X20
)
→
X47
)
)
)
)
↔
(
(
¬
(
X48
↔
(
X43
→
X42
→
False
)
)
)
→
(
(
¬
(
(
(
¬
(
X47
↔
(
(
¬
X43
)
↔
(
X42
↔
X48
)
)
)
)
↔
(
¬
(
X48
↔
(
X47
↔
(
X43
↔
(
X39
→
False
)
)
)
)
)
)
→
(
¬
(
(
¬
X47
)
→
(
(
¬
X42
)
↔
X48
)
)
)
)
)
↔
(
¬
(
X48
↔
(
X47
↔
(
X43
↔
False
)
)
)
)
)
)
)
→
(
¬
(
(
(
X48
↔
(
(
¬
X15
)
↔
(
(
¬
X1
)
↔
X39
)
)
)
→
(
X48
↔
(
X43
↔
(
X15
↔
(
X1
↔
False
)
)
)
)
)
→
(
¬
(
X43
↔
(
(
¬
X39
)
↔
X48
)
)
)
)
)
)
)
↔
(
(
(
(
X42
↔
(
X48
→
X30
→
(
(
¬
X7
)
↔
X45
)
)
)
→
(
¬
(
X48
↔
(
X45
→
X42
)
)
)
)
→
(
(
(
(
¬
X34
)
→
X7
→
X45
)
→
(
X51
↔
(
(
¬
X34
)
↔
(
(
¬
X7
)
→
X48
)
)
)
)
↔
(
¬
(
X7
↔
False
)
)
)
→
(
X51
↔
(
X7
↔
X48
)
)
)
↔
(
(
(
(
(
¬
(
(
¬
X0
)
↔
X37
)
)
↔
(
(
X37
→
(
X34
↔
(
(
¬
X24
)
→
X43
)
)
)
→
(
X48
↔
X34
)
)
)
↔
(
(
X48
→
(
(
¬
X14
)
↔
X43
)
)
↔
(
(
X48
↔
(
X43
→
(
X14
↔
X30
)
)
)
→
(
¬
(
X43
→
(
X37
↔
X30
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
X30
)
↔
(
(
¬
X5
)
→
X48
)
)
)
↔
(
¬
(
X43
→
(
(
¬
X30
)
↔
(
(
¬
X27
)
↔
X48
)
)
)
)
)
)
→
(
¬
(
X37
→
(
(
¬
X27
)
↔
(
X19
→
X48
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
X51
→
(
(
¬
X23
)
↔
(
(
¬
X16
)
→
X47
)
)
)
→
(
(
¬
(
(
¬
X16
)
→
X47
)
)
↔
(
(
X47
→
X23
→
(
X17
↔
(
X16
→
False
)
)
)
↔
(
X40
→
(
¬
X17
)
→
X23
)
)
)
)
→
(
(
¬
(
(
¬
X49
)
↔
(
(
¬
X40
)
↔
(
(
¬
X9
)
→
X51
)
)
)
)
↔
(
X51
→
(
X49
↔
(
X40
↔
(
X23
↔
False
)
)
)
)
)
)
)
→
(
¬
(
(
X51
↔
(
X40
↔
(
X29
→
X48
)
)
)
→
(
¬
(
(
(
¬
(
X40
→
(
¬
X22
)
→
X51
)
)
→
(
(
(
¬
(
(
¬
X40
)
→
X29
→
X22
→
X49
)
)
↔
(
¬
(
X49
→
X40
→
X29
→
X51
)
)
)
↔
(
¬
(
X51
↔
X49
)
)
)
)
→
(
¬
(
X51
→
(
¬
X29
)
→
X49
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
(
¬
(
X50
→
X47
→
(
¬
X39
)
→
X44
)
)
→
X46
→
(
(
¬
X44
)
↔
(
X39
↔
X50
)
)
)
→
(
¬
(
X46
→
X50
)
)
)
)
↔
(
¬
(
X47
↔
(
X46
→
(
X44
↔
X28
)
)
)
)
)
→
(
(
¬
(
(
¬
X46
)
↔
(
X36
↔
(
(
¬
X12
)
↔
X50
)
)
)
)
↔
(
(
¬
(
X50
↔
(
X13
→
(
(
¬
X12
)
↔
X36
)
)
)
)
↔
(
(
(
X50
↔
(
X44
→
(
X11
↔
X36
)
)
)
↔
(
¬
(
X50
→
X44
→
X13
)
)
)
↔
(
¬
(
X44
↔
(
X36
↔
X13
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X31
)
↔
(
X25
→
X46
)
)
)
↔
(
(
X46
→
(
X31
↔
(
(
¬
X25
)
→
X50
)
)
)
↔
(
X50
→
(
X31
↔
(
(
¬
X6
)
→
X46
)
)
)
)
)
↔
(
(
(
¬
X18
)
→
(
¬
X8
)
→
X50
)
↔
(
¬
(
(
X50
↔
(
X25
↔
X8
)
)
→
(
¬
(
(
X46
↔
(
(
¬
X25
)
↔
X50
)
)
→
(
¬
(
(
¬
X25
)
↔
(
(
¬
X10
)
→
(
(
¬
X8
)
↔
X50
)
)
)
)
)
)
→
(
X50
↔
(
X8
↔
X25
)
)
)
)
)
)
→
(
(
X50
→
(
X44
↔
(
(
¬
X31
)
→
X46
)
)
)
↔
(
¬
(
X50
→
X21
→
(
X3
↔
X31
)
)
)
)
)
)
↔
(
(
(
¬
(
X50
→
(
X46
↔
X42
)
)
)
→
(
(
¬
(
X44
→
X35
)
)
↔
(
(
¬
(
X50
→
X46
→
(
X42
↔
X44
)
)
)
↔
(
(
¬
(
X50
↔
(
X46
↔
(
(
¬
X35
)
→
X42
)
)
)
)
↔
(
¬
(
X50
→
(
X46
↔
(
X4
↔
X42
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
X44
→
(
X42
↔
(
X26
→
X2
)
)
)
)
→
(
¬
(
(
¬
(
X41
↔
(
(
¬
X2
)
→
X44
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
X41
)
→
(
(
¬
X33
)
↔
(
X2
↔
X44
)
)
)
)
→
(
X44
↔
(
X42
→
(
X33
↔
(
X2
→
False
)
)
)
)
)
)
↔
(
(
¬
X41
)
→
X42
)
)
)
)
)
→
X44
→
(
¬
X33
)
→
X26
→
X38
)
→
(
¬
(
X42
→
X33
→
X2
→
X46
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
dba9c2...
and proposition id is
70d263...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMSviGY23oj4iy9BceMibkT5JBHTVcik1pT
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
∀X50 :
prop
,
∀X51 :
prop
,
(
(
(
(
(
(
¬
(
(
¬
(
(
X51
→
X48
→
(
X39
↔
(
X26
↔
False
)
)
)
→
(
(
¬
X48
)
↔
(
(
¬
X39
)
→
X49
)
)
)
)
→
(
¬
(
(
¬
X49
)
→
X48
→
(
¬
X24
)
→
X51
)
)
)
)
→
(
¬
(
X48
→
(
X8
↔
(
X2
↔
X51
)
)
)
)
)
↔
(
(
(
(
X37
→
(
X32
↔
X47
)
)
→
(
(
¬
(
X49
→
(
X47
↔
(
X11
→
X37
)
)
)
)
↔
(
(
¬
(
(
¬
X32
)
→
(
(
¬
X18
)
↔
X47
)
)
)
→
(
(
¬
X47
)
↔
(
(
¬
X37
)
↔
(
(
¬
X18
)
→
X49
)
)
)
)
)
)
→
(
¬
(
(
X32
↔
(
(
¬
X18
)
↔
(
(
¬
X4
)
↔
X39
)
)
)
→
(
¬
(
(
¬
X39
)
→
(
(
¬
X18
)
↔
(
X4
→
X47
)
)
)
)
)
)
)
→
(
(
¬
(
X47
↔
(
X46
↔
(
X38
↔
X48
)
)
)
)
↔
(
(
X46
↔
(
X13
→
X38
)
)
↔
(
X48
↔
(
(
¬
X38
)
→
(
¬
X13
)
→
X49
)
)
)
)
→
(
¬
(
(
X48
→
X39
→
(
¬
X7
)
→
X46
)
→
(
¬
(
X46
↔
(
X39
↔
(
X13
↔
X7
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
X51
↔
(
(
¬
X47
)
↔
(
(
¬
X34
)
↔
X48
)
)
)
↔
(
X51
→
(
(
¬
X34
)
↔
(
(
¬
X19
)
→
X48
)
)
)
)
↔
(
X48
↔
(
X36
↔
(
(
¬
X27
)
↔
X42
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X25
↔
X42
)
)
↔
(
¬
(
X43
↔
(
(
¬
X42
)
↔
(
(
¬
X23
)
↔
X48
)
)
)
)
)
↔
(
¬
(
X51
↔
(
(
¬
X25
)
↔
(
X23
→
X48
)
)
)
)
)
→
(
¬
(
(
X42
→
(
X23
↔
(
(
¬
X0
)
↔
X15
)
)
)
→
X25
)
)
)
)
→
(
(
(
X51
→
(
X23
↔
(
X10
→
X0
→
False
)
)
)
↔
(
(
¬
(
X40
→
X10
→
X5
→
False
)
)
↔
(
X51
↔
(
X40
↔
(
X23
→
X10
→
False
)
)
)
)
)
→
(
¬
X10
)
→
(
(
¬
X5
)
↔
X23
)
)
→
(
(
¬
X40
)
↔
(
(
¬
X10
)
↔
X51
)
)
)
)
)
↔
(
(
(
(
¬
(
X41
→
(
¬
X30
)
→
(
X9
↔
X44
)
)
)
→
(
(
(
¬
(
(
¬
X41
)
→
(
(
¬
(
X50
↔
(
(
¬
X30
)
↔
(
X28
↔
X48
)
)
)
)
↔
(
¬
(
X48
→
X41
→
(
(
¬
X28
)
↔
X30
)
)
)
)
)
)
↔
(
(
¬
X41
)
↔
(
X28
↔
X48
)
)
)
↔
(
¬
(
(
¬
(
X44
→
X30
→
(
¬
X3
)
→
X41
)
)
→
(
¬
(
X50
→
(
X30
↔
(
(
¬
X3
)
→
X41
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X50
↔
(
X48
↔
(
X16
→
X44
)
)
)
)
↔
(
¬
(
(
¬
X16
)
↔
X49
)
)
)
↔
(
(
(
¬
X29
)
→
X16
→
(
(
¬
X1
)
↔
X44
)
)
→
(
(
¬
X45
)
↔
(
X44
→
(
¬
X21
)
→
X50
)
)
→
(
X47
↔
(
X29
→
(
X21
↔
X16
)
)
)
→
X47
→
X45
→
(
¬
X16
)
→
X44
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
¬
(
(
¬
X31
)
→
(
¬
X20
)
→
(
¬
X6
)
→
X35
)
)
↔
(
¬
(
X31
→
(
¬
X6
)
→
X22
)
)
)
→
(
(
(
¬
(
X48
→
(
X14
↔
(
(
¬
X6
)
↔
X22
)
)
)
)
↔
(
¬
(
X49
↔
(
(
¬
X6
)
→
X14
)
)
)
)
↔
(
X49
→
(
(
¬
X22
)
↔
X48
)
)
)
)
)
↔
(
(
¬
X45
)
→
(
X35
↔
(
(
¬
X31
)
↔
X49
)
)
)
)
→
(
(
X45
↔
(
X33
→
(
X17
↔
X48
)
)
)
↔
(
¬
(
(
X35
→
X48
)
→
(
X48
↔
(
X45
→
X33
→
X12
→
False
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
¬
(
(
¬
(
(
¬
X50
)
↔
(
X4
→
(
(
¬
X1
)
↔
X51
)
)
)
)
→
(
¬
(
X37
↔
(
(
¬
X1
)
→
X50
)
)
)
)
)
→
X49
→
(
¬
X37
)
→
(
(
¬
X1
)
↔
X50
)
)
)
↔
(
(
(
¬
(
X46
→
(
X36
↔
(
X14
→
X51
)
)
)
)
↔
(
(
(
¬
(
X50
→
X49
→
X36
→
X22
)
)
↔
(
X49
↔
(
(
¬
X14
)
↔
X50
)
)
)
↔
(
¬
(
X50
↔
(
X46
↔
X22
)
)
)
)
)
→
(
(
¬
(
X36
↔
(
(
¬
X21
)
↔
X51
)
)
)
↔
(
(
(
¬
X50
)
→
(
¬
X21
)
→
(
¬
X18
)
→
X51
)
↔
(
¬
(
X36
↔
(
X18
→
X51
)
)
)
)
)
)
)
→
(
(
(
¬
(
X49
→
(
X38
↔
(
(
¬
X10
)
→
X46
)
)
)
)
↔
(
¬
(
(
¬
X38
)
↔
(
(
¬
X10
)
→
X51
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X46
)
→
(
X38
↔
(
X33
→
X48
)
)
)
)
→
(
X49
↔
(
X48
↔
(
(
¬
X38
)
→
X46
)
)
)
)
)
)
→
(
¬
(
(
(
¬
X46
)
→
(
¬
X45
)
→
X25
→
X48
)
→
(
(
X48
↔
(
X46
→
X51
)
)
↔
(
¬
(
(
X6
↔
X46
)
→
(
(
¬
(
(
(
¬
X46
)
↔
(
X45
↔
(
(
¬
X6
)
↔
X51
)
)
)
→
(
¬
(
X51
→
X46
→
(
X41
↔
X45
)
)
)
)
)
↔
(
¬
(
(
¬
X48
)
↔
(
(
¬
X45
)
↔
(
(
¬
X41
)
→
X51
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
X50
→
(
X26
↔
(
X12
↔
X46
)
)
)
↔
(
¬
(
X50
→
(
X39
↔
(
(
¬
X12
)
↔
X46
)
)
)
)
)
→
(
(
¬
X39
)
↔
(
X26
↔
(
(
¬
X12
)
↔
X43
)
)
)
)
)
↔
(
(
¬
(
X50
→
X46
→
(
X43
↔
(
X9
↔
False
)
)
)
)
→
(
¬
(
X48
↔
(
(
¬
X43
)
→
(
¬
X34
)
→
X46
)
)
)
→
(
¬
(
X43
→
(
¬
X34
)
→
X9
→
X48
)
)
→
(
¬
(
X48
↔
(
(
¬
X34
)
↔
(
X2
→
X43
)
)
)
)
)
)
↔
(
(
(
X48
→
X34
→
X28
→
(
X17
↔
False
)
)
→
(
¬
(
X48
↔
(
(
¬
X3
)
→
X17
)
)
)
)
↔
(
(
(
¬
(
X43
→
(
(
¬
X34
)
↔
X48
)
)
)
↔
(
(
X43
↔
(
(
¬
X27
)
↔
X32
)
)
→
(
¬
(
X48
→
(
¬
X34
)
→
X43
)
)
)
)
→
(
X34
↔
(
(
¬
X29
)
→
(
X28
↔
X43
)
)
)
→
X48
→
(
X43
↔
(
X29
↔
X34
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
X16
)
↔
(
(
¬
X5
)
↔
X42
)
)
→
(
¬
(
X35
↔
X47
)
)
)
)
↔
(
(
(
(
X47
→
(
X46
↔
X49
)
)
→
(
X49
→
X46
→
X35
→
False
)
→
(
(
X46
→
X35
→
(
¬
X30
)
→
X47
)
↔
(
X49
→
X35
→
X30
)
)
)
↔
(
¬
(
X49
↔
(
X35
→
(
X7
↔
X46
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
X31
↔
(
X30
→
X46
)
)
)
→
(
X49
↔
(
(
¬
X46
)
↔
X51
)
)
)
)
↔
(
X51
↔
(
(
¬
X31
)
→
X49
)
)
)
→
(
¬
(
(
¬
X42
)
→
(
¬
X24
)
→
(
X19
↔
X46
)
)
)
→
(
¬
(
(
¬
(
X51
→
(
(
¬
X42
)
↔
(
(
¬
X24
)
↔
X49
)
)
)
)
→
(
¬
X49
)
→
(
X46
↔
(
(
¬
X24
)
→
X51
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X51
↔
(
X42
↔
(
X11
→
X44
)
)
)
)
→
(
¬
(
(
¬
X46
)
↔
(
X42
→
X11
→
X51
)
)
)
)
→
X44
→
X20
→
(
X11
↔
X42
)
)
)
→
(
(
X46
↔
(
(
¬
X42
)
→
X13
→
X44
)
)
↔
(
(
¬
(
X44
→
(
X42
↔
(
X23
→
X46
)
)
)
)
→
(
(
¬
(
(
¬
X15
)
→
(
¬
X8
)
→
(
X0
↔
X42
)
)
)
↔
(
¬
(
(
¬
(
X44
→
X8
)
)
→
(
¬
(
X44
↔
(
X15
↔
(
X0
→
X46
)
)
)
)
)
)
)
)
)
→
(
¬
(
X46
↔
(
X42
↔
(
X40
↔
X44
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
e542b1...
and proposition id is
2973b0...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMa97Lsjs2uZ7avQKbzVsP7KuFFANxg8Dv5
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
(
(
¬
(
(
(
(
(
X43
↔
(
X41
↔
(
(
¬
X22
)
↔
X49
)
)
)
→
(
(
(
¬
X41
)
→
X2
→
X49
)
↔
(
(
(
¬
X47
)
→
(
¬
X41
)
→
(
¬
X0
)
→
X49
)
↔
(
¬
(
X2
→
X0
→
X49
)
)
)
)
)
↔
(
X48
→
X47
→
X43
→
(
X22
↔
False
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
X41
↔
(
X1
↔
False
)
)
)
→
(
¬
(
(
¬
(
(
¬
X11
)
→
X41
)
)
→
(
¬
(
X47
↔
(
X14
↔
(
X11
↔
X1
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
X41
)
↔
X48
)
)
)
→
(
¬
X11
)
)
)
)
→
(
(
(
X50
→
X41
→
X39
)
↔
(
X50
→
(
¬
X39
)
→
(
X38
↔
X40
)
)
)
→
(
(
(
X41
↔
(
X25
→
X39
)
)
↔
(
¬
(
X41
→
X38
→
X49
)
)
)
↔
(
(
¬
(
X41
↔
(
X25
↔
(
X20
→
X38
)
)
)
)
→
(
¬
(
X25
→
(
¬
X20
)
→
(
¬
X3
)
→
X38
)
)
)
)
)
→
(
¬
X41
)
→
(
X40
↔
(
(
¬
X3
)
↔
X47
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
X41
)
↔
(
X39
→
(
¬
X18
)
→
X48
)
)
↔
(
(
(
¬
(
(
(
(
¬
X41
)
↔
(
(
¬
X39
)
↔
X48
)
)
↔
(
¬
(
(
¬
X33
)
↔
(
(
¬
X5
)
→
X41
)
)
)
)
→
(
(
X41
→
X5
→
False
)
↔
(
(
(
¬
X5
)
↔
X30
)
↔
(
¬
(
(
¬
X41
)
→
(
¬
X39
)
→
X44
)
)
)
)
)
)
↔
(
¬
(
X44
↔
(
X30
→
X48
)
)
)
)
↔
(
¬
(
X44
↔
(
(
¬
X33
)
→
X41
)
)
)
)
)
→
(
¬
(
(
¬
X44
)
↔
(
(
¬
X41
)
↔
(
X24
↔
X48
)
)
)
)
)
)
→
(
(
¬
(
(
X48
→
(
X36
↔
(
X35
↔
X21
)
)
)
→
(
¬
(
(
¬
(
(
¬
X44
)
→
(
¬
X36
)
→
X48
)
)
→
(
¬
(
(
¬
X35
)
↔
X41
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X31
→
(
¬
X13
)
→
X44
)
)
→
(
¬
(
(
(
¬
(
(
¬
(
(
¬
X28
)
→
X31
)
)
→
(
¬
(
(
¬
X31
)
↔
(
X13
→
X4
→
X48
)
)
)
)
)
↔
(
¬
(
X28
→
(
¬
X13
)
→
X31
)
)
)
→
(
¬
(
(
¬
X44
)
↔
(
(
¬
X28
)
→
(
(
¬
X13
)
↔
X48
)
)
)
)
)
)
)
)
→
(
¬
(
X44
↔
(
X31
→
(
¬
X28
)
→
X36
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
X26
→
(
(
¬
X16
)
↔
(
(
¬
X7
)
↔
X19
)
)
)
↔
(
X46
↔
(
(
¬
X16
)
↔
X26
)
)
)
→
X46
→
(
(
¬
X9
)
↔
X26
)
)
→
(
¬
(
(
(
X50
→
(
X46
↔
(
X32
↔
(
X29
↔
False
)
)
)
)
↔
(
(
(
(
(
¬
X29
)
→
(
(
¬
X23
)
↔
(
(
¬
X19
)
→
X46
)
)
)
→
(
X29
↔
(
X23
↔
(
X19
→
X16
→
False
)
)
)
)
↔
(
¬
(
X29
↔
(
(
¬
X19
)
→
X46
)
)
)
)
→
(
¬
X23
)
→
(
(
¬
X19
)
↔
X46
)
)
)
→
(
¬
(
X46
↔
(
X29
↔
X23
)
)
)
)
)
)
→
(
(
(
¬
(
(
X50
→
(
(
¬
X32
)
↔
(
X17
↔
X48
)
)
)
→
(
X48
↔
(
(
¬
X32
)
↔
(
X29
↔
X50
)
)
)
)
)
→
(
¬
(
X48
↔
(
X37
→
(
¬
X29
)
→
X32
)
)
)
)
↔
(
(
¬
X37
)
↔
(
(
¬
X27
)
→
X50
)
)
)
→
X37
→
X32
→
(
X29
↔
False
)
)
↔
(
¬
(
(
(
X49
↔
(
X45
↔
(
X8
↔
X32
)
)
)
↔
(
(
¬
(
X49
↔
(
X29
→
(
X26
↔
X32
)
)
)
)
↔
(
(
¬
X32
)
↔
(
(
¬
X29
)
↔
(
(
¬
X26
)
→
X45
)
)
)
)
)
→
(
(
(
¬
X45
)
↔
(
X42
→
X34
→
X49
)
)
↔
(
(
¬
(
(
(
¬
(
(
¬
X34
)
↔
(
X29
↔
(
X6
↔
X45
)
)
)
)
→
(
X42
↔
(
(
¬
X29
)
→
(
(
¬
X10
)
↔
X34
)
)
)
)
→
(
(
¬
X42
)
↔
(
(
¬
X34
)
↔
(
(
¬
X15
)
→
X45
)
)
)
)
)
↔
(
X45
↔
(
X29
→
(
(
¬
X12
)
↔
X42
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
(
¬
(
X35
→
(
X34
↔
X48
)
)
)
→
(
(
¬
(
X35
→
X32
→
(
X25
↔
X34
)
)
)
↔
(
(
¬
X35
)
↔
(
(
¬
X34
)
↔
(
X25
→
X48
)
)
)
)
)
→
(
X49
↔
(
X48
↔
(
X35
↔
(
X34
→
False
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X26
)
↔
(
X22
↔
X34
)
)
)
→
(
¬
(
X35
→
(
X28
↔
(
X22
→
X34
)
)
)
)
)
)
↔
(
(
X34
→
(
(
¬
X28
)
↔
(
X26
↔
X47
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X34
)
↔
(
X28
→
(
¬
X26
)
→
X49
)
)
)
→
X34
→
(
(
¬
X28
)
↔
(
X24
↔
X49
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
(
¬
(
X48
→
(
¬
X18
)
→
X50
)
)
→
(
¬
(
(
¬
X18
)
↔
(
(
¬
X6
)
↔
X48
)
)
)
→
(
¬
(
X50
→
(
X21
↔
(
X18
↔
X6
)
)
)
)
)
)
↔
(
X31
↔
(
X18
→
False
)
)
)
↔
(
(
(
(
¬
(
X48
→
(
X44
↔
(
X13
→
False
)
)
)
)
↔
(
(
X31
→
(
(
¬
X13
)
↔
(
X0
→
X44
)
)
)
↔
(
¬
(
X48
↔
(
(
¬
X13
)
↔
(
X0
↔
X44
)
)
)
)
)
)
↔
(
¬
(
X18
↔
(
X13
→
X48
)
)
)
)
↔
(
¬
(
X48
↔
(
X31
→
(
¬
X11
)
→
X44
)
)
)
)
)
→
(
¬
(
X50
→
(
¬
X23
)
→
X48
)
)
→
(
X49
↔
(
X48
→
X31
)
)
)
→
(
(
(
(
¬
(
X47
→
(
X19
↔
X50
)
)
)
→
(
(
X48
→
X45
→
False
)
↔
(
(
(
X48
↔
(
(
¬
X47
)
→
X42
→
X50
)
)
→
(
¬
(
X50
↔
(
(
¬
X42
)
↔
(
X19
→
X48
)
)
)
)
)
→
X48
→
X42
→
X50
)
)
)
→
(
X50
↔
(
X45
↔
(
X37
→
X47
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
¬
X42
)
→
X37
→
X48
)
→
(
¬
(
X42
↔
(
X37
↔
(
(
¬
X14
)
↔
X49
)
)
)
)
)
)
→
(
¬
(
(
¬
X37
)
→
(
X14
↔
X42
)
)
)
)
)
→
X48
→
(
X3
↔
X14
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
(
¬
(
X45
→
X40
→
X39
)
)
↔
(
X45
↔
(
(
¬
X15
)
→
X2
→
X41
)
)
)
→
(
¬
(
(
(
¬
(
X48
→
(
(
¬
X15
)
↔
(
X4
↔
X41
)
)
)
)
→
(
(
¬
X40
)
↔
(
(
¬
X15
)
→
(
X4
↔
X48
)
)
)
)
→
(
¬
X41
)
→
(
¬
X4
)
→
X48
)
)
)
)
→
(
(
(
(
¬
X46
)
→
X38
→
(
(
¬
X7
)
↔
X48
)
)
↔
(
(
(
¬
(
X38
↔
X46
)
)
→
(
¬
(
X7
→
(
(
¬
X1
)
↔
X2
)
)
)
)
→
(
¬
(
(
¬
X2
)
↔
(
(
¬
X1
)
→
X46
)
)
)
)
)
↔
(
(
¬
(
X39
→
(
X38
↔
X46
)
)
)
↔
(
(
X48
↔
(
X46
↔
False
)
)
↔
(
¬
(
X46
↔
(
X38
↔
(
X9
→
X39
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
(
(
X43
→
X40
→
(
(
¬
X29
)
↔
X46
)
)
→
(
(
¬
X43
)
↔
(
X17
→
X10
→
X50
)
)
)
)
→
(
¬
(
X46
↔
(
X40
↔
X29
)
)
)
)
)
↔
(
(
¬
(
X12
→
X27
)
)
↔
(
¬
(
X40
→
(
¬
X12
)
→
(
(
¬
X10
)
↔
X17
)
)
)
)
)
↔
(
(
(
¬
X30
)
↔
(
(
¬
X27
)
→
(
¬
X12
)
→
X46
)
)
↔
(
¬
(
X46
→
X33
→
(
(
¬
X12
)
↔
X30
)
)
)
)
)
→
(
¬
(
(
¬
(
X46
↔
(
(
¬
X20
)
↔
(
X5
→
X36
)
)
)
)
→
(
(
(
¬
(
X43
↔
(
(
¬
X40
)
→
(
X5
↔
X46
)
)
)
)
↔
(
X43
→
X8
→
X46
)
)
↔
(
¬
(
(
¬
X43
)
→
(
(
¬
X36
)
↔
(
(
¬
X8
)
→
X46
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
X46
)
↔
(
X40
→
X48
)
)
→
(
¬
(
X46
↔
(
X36
→
(
(
¬
X16
)
↔
X48
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
bb271e...
and proposition id is
84b83f...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMNxvs2Re6qYXT7oHRTSZCKRqqXT9nYnNm1
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
(
(
X47
→
(
¬
X7
)
→
X0
→
X46
)
↔
(
¬
(
X36
→
(
(
¬
X19
)
↔
(
(
¬
X0
)
↔
X47
)
)
)
)
)
↔
(
(
(
¬
X36
)
↔
(
(
¬
X34
)
↔
(
X5
→
X46
)
)
)
→
(
X34
↔
(
(
¬
X19
)
↔
X31
)
)
→
(
¬
(
X46
→
(
X31
↔
(
X19
→
X34
)
)
)
)
→
(
X36
↔
(
X34
→
False
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
X46
→
X36
→
False
)
)
→
(
(
X46
→
(
(
¬
X8
)
↔
X10
)
)
↔
(
¬
(
X38
→
X36
→
X10
→
X8
)
)
)
)
↔
(
X49
↔
(
X34
↔
(
X4
↔
X46
)
)
)
)
↔
(
¬
(
X46
→
X38
→
(
¬
X29
)
→
X36
)
)
)
→
(
(
¬
X36
)
↔
(
X26
→
X38
)
)
→
(
X46
↔
(
X36
→
X20
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
X44
)
→
(
¬
X32
)
→
X27
→
X49
)
)
→
(
¬
(
X49
→
(
X21
↔
X33
)
)
)
)
)
↔
(
(
(
(
X44
→
(
X32
↔
(
(
¬
X31
)
↔
X36
)
)
)
↔
(
X44
↔
(
X36
→
(
X30
↔
X31
)
)
)
)
→
(
X37
↔
(
X36
→
X32
→
X14
)
)
)
→
(
(
(
X49
→
(
(
¬
X15
)
↔
(
X9
→
X31
)
)
)
↔
(
¬
(
X49
→
(
X36
↔
(
X9
↔
X31
)
)
)
)
)
↔
(
¬
(
X15
↔
(
X9
→
X31
)
)
)
)
→
(
X32
↔
(
X31
↔
(
X9
↔
X49
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
¬
(
X41
→
X40
→
(
X23
↔
X39
)
)
)
→
X42
→
X41
→
False
)
→
(
¬
X39
)
→
(
X23
↔
X40
)
)
↔
(
(
(
¬
X45
)
→
(
X41
↔
X46
)
)
↔
(
¬
(
(
¬
(
X41
→
(
(
¬
X40
)
↔
X42
)
)
)
→
(
¬
(
X46
↔
(
X42
↔
(
X41
↔
X45
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
X25
→
(
X22
↔
X28
)
)
)
↔
(
¬
(
(
(
X41
↔
(
(
¬
X40
)
↔
X42
)
)
↔
(
(
¬
(
X40
→
X25
→
(
X22
↔
X42
)
)
)
→
(
X41
↔
(
X40
↔
X25
)
)
)
)
→
(
¬
(
X41
→
(
X40
↔
(
X28
→
(
X22
↔
False
)
)
)
)
)
)
)
)
↔
(
(
¬
(
X28
→
(
X3
↔
X16
)
)
)
↔
(
¬
(
(
¬
X22
)
↔
(
(
¬
X12
)
→
X25
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X17
↔
(
X6
↔
(
(
¬
X1
)
→
X41
)
)
)
)
→
(
(
¬
(
(
¬
(
X49
↔
(
(
¬
X47
)
↔
(
(
¬
X41
)
↔
X48
)
)
)
)
→
(
¬
(
X48
↔
(
X47
→
(
¬
X6
)
→
X41
)
)
)
)
)
↔
(
(
¬
X47
)
→
(
(
¬
X41
)
↔
X48
)
)
)
)
↔
(
(
¬
(
(
X48
→
X46
→
X43
→
False
)
→
(
(
(
(
¬
X24
)
→
X11
→
X48
)
↔
(
X49
→
X24
→
(
¬
X11
)
→
X43
)
)
↔
(
X48
↔
(
X13
↔
X46
)
)
)
)
)
↔
(
(
¬
(
X49
↔
(
(
¬
X47
)
↔
(
X35
↔
X48
)
)
)
)
→
(
(
(
¬
(
X48
→
(
¬
X45
)
→
X2
→
X46
)
)
↔
(
(
X49
↔
(
X45
→
X18
)
)
→
(
¬
(
X46
↔
(
X18
↔
(
X2
↔
False
)
)
)
)
)
)
↔
(
X49
→
(
X47
↔
(
X45
→
(
X2
↔
False
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
(
X22
→
(
X10
↔
X0
)
)
→
(
(
¬
(
X47
↔
(
X27
→
X46
)
)
)
↔
(
¬
(
(
¬
(
(
X47
→
X12
)
→
(
¬
(
(
¬
X45
)
→
(
¬
X27
)
→
(
X12
↔
X46
)
)
)
)
)
→
X45
→
X12
→
X27
)
)
)
)
)
→
(
(
(
¬
X46
)
→
(
¬
X45
)
→
(
(
¬
X44
)
↔
X47
)
)
↔
(
(
¬
X45
)
→
(
X44
↔
(
(
¬
X2
)
→
X46
)
)
)
)
)
→
(
X25
↔
(
X20
→
X46
)
)
→
(
(
¬
(
(
¬
(
X47
→
X46
→
(
X45
↔
False
)
)
)
→
X46
→
(
X45
↔
(
X25
↔
(
X20
→
False
)
)
)
)
)
→
X47
→
X46
→
X20
→
False
)
→
(
¬
(
X46
→
(
X45
↔
(
(
¬
X15
)
↔
X47
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
(
X48
→
(
X45
↔
(
X26
↔
X46
)
)
)
→
(
¬
(
(
¬
X45
)
→
(
X31
↔
(
X26
↔
X46
)
)
)
)
)
→
(
(
¬
(
X45
→
(
X33
↔
X48
)
)
)
↔
(
(
¬
(
(
¬
X36
)
→
(
¬
(
(
X46
↔
(
X45
→
X33
→
False
)
)
→
(
¬
(
X45
↔
(
(
¬
X36
)
↔
X48
)
)
)
)
)
)
)
→
(
(
¬
X46
)
↔
(
X33
→
X48
)
)
)
)
)
)
→
(
(
X26
→
(
¬
X14
)
→
(
(
¬
X9
)
↔
X34
)
)
↔
(
(
¬
(
X46
→
(
¬
X14
)
→
(
¬
X6
)
→
X26
)
)
↔
(
(
¬
(
X45
→
(
X34
↔
(
X6
→
X46
)
)
)
)
→
X34
→
(
¬
X14
)
→
(
(
¬
X6
)
↔
X45
)
)
)
)
)
)
→
(
¬
(
(
(
X48
↔
(
X44
↔
(
X32
↔
(
X23
↔
False
)
)
)
)
→
(
¬
(
(
(
¬
X45
)
↔
(
X28
→
(
¬
X23
)
→
X48
)
)
→
(
¬
X44
)
→
(
¬
X23
)
→
X45
)
)
)
→
(
(
¬
(
(
¬
X44
)
→
(
¬
X24
)
→
X46
)
)
↔
(
(
¬
(
X44
→
(
X23
↔
X28
)
)
)
↔
(
X46
→
(
(
¬
X28
)
↔
(
X24
→
X44
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
X47
↔
(
X45
→
(
¬
X35
)
→
X49
)
)
)
↔
(
¬
(
X45
→
(
¬
X35
)
→
(
¬
X11
)
→
X49
)
)
)
→
(
¬
(
(
(
¬
X8
)
↔
X49
)
→
(
(
(
X45
→
(
(
¬
X39
)
↔
(
X8
→
X49
)
)
)
↔
(
X49
→
(
(
¬
X17
)
↔
X45
)
)
)
↔
(
X47
↔
(
(
¬
X39
)
↔
(
(
¬
X17
)
→
X49
)
)
)
)
)
)
→
(
(
¬
(
X47
↔
(
X39
↔
(
X37
↔
X42
)
)
)
)
↔
(
X49
→
X47
→
X42
→
X39
)
)
→
(
¬
(
X42
↔
(
(
¬
X37
)
↔
X49
)
)
)
)
)
↔
(
(
¬
(
(
X44
↔
(
X42
→
(
¬
X5
)
→
X19
)
)
→
(
¬
X45
)
→
(
(
¬
X42
)
↔
(
X5
↔
X47
)
)
)
)
→
(
¬
(
(
¬
(
X29
↔
(
X19
→
(
X5
↔
X42
)
)
)
)
→
(
¬
X42
)
→
X19
→
(
X5
↔
X43
)
)
)
→
(
X42
↔
(
X29
↔
(
X16
→
X43
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
¬
(
(
¬
(
(
¬
X21
)
→
(
X3
↔
X40
)
)
)
→
(
¬
(
X30
↔
(
X7
→
X18
)
)
)
)
)
→
X30
→
(
¬
X18
)
→
X13
→
X40
)
)
↔
(
X38
→
(
X13
↔
(
X7
→
X30
)
)
)
)
→
(
(
(
¬
(
X21
↔
(
X18
↔
(
X1
→
X13
)
)
)
)
↔
(
¬
(
(
¬
X21
)
↔
(
(
¬
X18
)
↔
X49
)
)
)
)
↔
(
¬
(
X21
↔
(
(
¬
X1
)
→
X18
)
)
)
)
→
(
(
¬
X18
)
↔
(
(
¬
X13
)
↔
(
(
¬
X4
)
→
X21
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X44
)
→
X43
→
(
(
¬
X18
)
↔
X49
)
)
)
↔
(
¬
(
X49
↔
(
(
¬
X21
)
→
X43
)
)
)
)
→
X18
)
)
)
→
(
¬
(
(
¬
X44
)
→
(
¬
X43
)
→
(
¬
X41
)
→
X49
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
d940db...
and proposition id is
b5a0c5...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMbojk6FZ7vqY4gd3Cb9N62xnDfP1zWYbiQ
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
(
¬
(
X42
→
X38
→
X1
→
X49
)
)
↔
(
¬
(
(
¬
(
X48
→
X45
→
(
X37
↔
X38
)
)
)
→
(
(
X49
→
(
(
¬
X42
)
↔
X45
)
)
↔
(
(
X42
↔
(
(
¬
X32
)
↔
X37
)
)
↔
(
¬
(
(
(
¬
(
X38
↔
(
X37
↔
X45
)
)
)
→
(
¬
(
X38
↔
(
(
¬
X37
)
↔
(
(
¬
X32
)
↔
X49
)
)
)
)
)
→
(
¬
(
(
¬
X42
)
→
(
¬
X37
)
→
X45
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X42
↔
(
(
¬
X37
)
↔
(
X32
→
X46
)
)
)
)
→
(
¬
(
X42
↔
(
X37
→
X18
→
X46
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
X48
→
(
¬
X16
)
→
(
¬
X6
)
→
X46
)
↔
(
X46
→
X16
→
X6
→
X49
)
)
↔
(
¬
(
X46
→
X16
→
(
X2
↔
X48
)
)
)
)
→
(
X46
↔
(
X34
→
X16
→
False
)
)
→
(
(
(
¬
(
X49
↔
(
X23
↔
False
)
)
)
↔
(
¬
(
X49
→
(
X46
↔
(
X16
↔
X34
)
)
)
)
)
↔
(
X46
→
(
¬
X23
)
→
X11
→
X49
)
)
)
)
↔
(
¬
(
X48
↔
(
X23
↔
X34
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X48
↔
(
X30
→
X10
)
)
)
→
(
¬
(
X48
→
(
X21
↔
X30
)
)
)
→
(
X43
↔
X49
)
)
)
→
(
¬
(
(
¬
(
X49
↔
(
X39
↔
(
(
¬
X25
)
↔
X30
)
)
)
)
→
(
X48
↔
(
X39
→
X25
→
X30
)
)
)
)
)
↔
(
(
X46
→
X41
→
X48
)
↔
(
(
(
¬
(
X41
→
(
(
¬
X20
)
↔
X49
)
)
)
↔
(
¬
(
X46
→
(
¬
X20
)
→
(
(
¬
X17
)
↔
X41
)
)
)
)
→
(
X46
↔
(
X43
↔
(
(
¬
X20
)
→
X41
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X42
→
(
¬
X36
)
→
X41
)
)
↔
(
(
(
¬
X43
)
↔
(
X42
↔
(
X41
↔
X47
)
)
)
↔
(
X41
→
(
(
¬
X36
)
↔
X47
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
(
X47
↔
(
X42
→
X43
)
)
)
→
(
(
(
¬
(
X43
→
False
)
)
↔
(
(
(
¬
X43
)
↔
(
X35
→
(
X31
↔
X47
)
)
)
→
(
¬
X43
)
→
(
X35
↔
(
(
¬
X22
)
↔
X47
)
)
)
)
↔
(
(
¬
X42
)
↔
(
X31
↔
X47
)
)
)
→
(
¬
(
X47
→
(
X35
↔
(
(
¬
X33
)
→
X43
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X43
→
(
(
¬
X42
)
↔
(
(
¬
X19
)
→
X47
)
)
)
)
→
X42
→
(
X8
↔
X47
)
)
)
↔
(
X47
↔
(
X19
↔
X43
)
)
)
)
→
(
¬
(
X44
↔
(
X43
→
(
X22
↔
X47
)
)
)
)
)
)
)
)
↔
(
(
¬
(
X47
↔
(
X42
→
X35
→
(
X7
↔
False
)
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X35
)
↔
(
(
¬
X4
)
→
X42
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
¬
(
(
X29
→
X28
→
X9
)
→
X33
→
(
¬
X28
)
→
(
(
¬
X5
)
↔
X29
)
)
)
↔
(
¬
(
X29
→
X9
→
(
X5
↔
False
)
)
)
)
↔
(
¬
(
(
¬
X27
)
↔
(
(
¬
X9
)
↔
X33
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X27
)
→
X28
)
)
→
(
¬
(
X40
↔
(
X33
↔
(
(
¬
X24
)
→
X28
)
)
)
)
)
)
)
)
→
(
(
X28
↔
(
X26
↔
(
(
¬
X13
)
→
X27
)
)
)
↔
(
¬
(
(
¬
X27
)
↔
(
X13
↔
(
(
¬
X5
)
→
X28
)
)
)
)
)
)
→
(
(
(
¬
(
X42
↔
(
X29
↔
(
X26
→
X12
→
False
)
)
)
)
↔
(
(
X44
→
X42
→
(
X29
↔
False
)
)
→
(
¬
(
X44
→
X42
→
(
X29
↔
False
)
)
)
→
(
¬
(
(
¬
X42
)
↔
(
X26
↔
(
(
¬
X0
)
↔
X44
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X44
→
X42
→
X28
→
False
)
)
↔
(
¬
(
X44
↔
(
X14
→
X3
→
False
)
)
)
)
→
(
¬
X29
)
→
(
X15
↔
(
X14
→
X42
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
¬
(
(
X44
↔
(
(
¬
X14
)
→
X27
)
)
→
(
(
¬
(
X37
→
X14
)
)
↔
(
X27
↔
(
X14
↔
(
X11
↔
X44
)
)
)
)
)
)
↔
(
(
¬
(
X44
↔
(
(
¬
X37
)
→
X0
→
X41
)
)
)
↔
(
¬
(
X44
↔
(
X28
→
(
X27
↔
X37
)
)
)
)
)
)
→
(
(
(
¬
(
(
X44
↔
(
(
¬
X30
)
→
X47
)
)
→
(
¬
(
(
¬
(
X47
↔
(
X37
↔
(
X30
↔
False
)
)
)
)
→
X47
→
(
(
¬
X10
)
↔
X44
)
)
)
)
)
→
(
¬
(
X42
→
X30
→
X4
→
X44
)
)
→
(
(
(
¬
X37
)
→
(
(
¬
X30
)
↔
X42
)
)
↔
(
(
X42
→
(
¬
X37
)
→
(
¬
X2
)
→
X44
)
↔
(
¬
(
X37
→
X30
→
False
)
)
)
)
→
(
¬
(
X42
→
(
X2
↔
X37
)
)
)
)
↔
(
(
(
¬
X30
)
→
(
X27
↔
(
X18
↔
X42
)
)
)
↔
(
¬
(
X27
→
(
¬
X18
)
→
(
¬
X4
)
→
X42
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
(
X44
→
(
¬
X12
)
→
(
X7
↔
X47
)
)
)
→
(
(
¬
(
X47
→
(
X38
↔
X25
)
)
)
↔
(
(
X44
→
X7
→
False
)
→
(
X44
↔
(
X38
↔
(
X25
↔
X47
)
)
)
)
)
)
)
↔
(
(
¬
X12
)
↔
(
X8
↔
X38
)
)
)
↔
(
(
(
X25
→
X13
→
False
)
→
(
(
¬
(
X49
↔
(
X47
→
X44
→
X41
→
False
)
)
)
↔
(
(
¬
X47
)
→
(
¬
X41
)
→
X13
→
X49
)
)
)
↔
(
X47
↔
(
X41
→
(
¬
X5
)
→
X25
)
)
)
)
↔
(
(
¬
(
X47
↔
(
(
¬
X26
)
→
X49
)
)
)
→
(
¬
X44
)
→
(
¬
X36
)
→
X26
→
X47
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
(
(
¬
X40
)
→
X3
→
X42
)
)
→
(
¬
(
(
(
(
¬
X40
)
↔
(
(
¬
X35
)
↔
(
X3
→
X43
)
)
)
↔
(
(
¬
X43
)
↔
(
X40
→
X46
)
)
)
→
(
(
X35
↔
(
(
¬
X34
)
↔
X46
)
)
↔
(
(
¬
(
(
¬
X43
)
↔
(
X35
↔
(
X34
→
X46
)
)
)
)
↔
(
(
¬
X42
)
→
X35
→
(
(
¬
X34
)
↔
X46
)
)
)
)
)
)
)
)
→
(
¬
(
X46
→
(
X43
↔
(
(
¬
X32
)
↔
X40
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X43
→
(
(
¬
X39
)
↔
X47
)
)
)
↔
(
¬
(
(
¬
(
X47
↔
(
(
¬
X31
)
→
X43
)
)
)
→
(
¬
(
X45
→
X43
→
X31
→
X39
)
)
)
)
)
→
(
¬
(
(
(
X45
↔
(
X39
→
X33
→
X47
)
)
↔
(
¬
(
X46
→
False
)
)
)
→
(
¬
(
X45
→
X33
)
)
)
)
)
)
↔
(
¬
(
X47
↔
(
X46
→
X43
→
X42
→
False
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
X44
↔
(
X17
↔
False
)
)
→
(
¬
(
X45
↔
(
X44
↔
(
(
¬
X9
)
↔
X48
)
)
)
)
)
)
↔
(
(
(
(
¬
X46
)
↔
(
X24
→
(
X21
↔
X48
)
)
)
↔
(
¬
(
(
¬
(
(
X44
→
X24
→
X29
)
→
(
X24
↔
(
(
¬
X21
)
↔
(
(
¬
X19
)
→
X44
)
)
)
→
(
(
¬
X29
)
↔
(
X21
→
X44
)
)
)
)
→
(
¬
(
X44
↔
(
X24
↔
(
(
¬
X21
)
↔
X46
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X46
)
→
(
(
¬
X6
)
↔
X47
)
)
)
↔
(
(
X46
↔
(
(
¬
X45
)
↔
(
X44
→
X48
)
)
)
↔
(
(
X47
→
X45
→
X6
→
False
)
↔
(
X46
→
(
X45
↔
X48
)
)
)
)
)
↔
(
X47
↔
(
X46
→
X23
→
X45
)
)
)
)
)
→
(
(
¬
(
(
(
X48
→
X22
→
X44
)
↔
(
(
X45
→
(
X22
↔
(
(
¬
X20
)
↔
X44
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X22
)
↔
X44
)
)
→
(
(
(
¬
(
X45
→
X44
→
X16
→
X22
)
)
→
(
¬
(
(
¬
X45
)
↔
(
(
¬
X22
)
↔
(
X16
↔
X48
)
)
)
)
)
↔
(
X48
→
(
X45
↔
X16
)
)
)
)
)
)
)
→
(
¬
(
X44
↔
(
X20
↔
(
X15
→
X16
)
)
)
)
)
)
↔
(
¬
(
X48
↔
(
X16
↔
(
X1
↔
X15
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
81fe0e...
and proposition id is
d1d6ec...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMVpVwT2aop9HqbDZ7zPT7pEbmVaEgz3CbE
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
¬
(
(
(
(
¬
(
(
(
(
¬
(
X48
↔
(
X2
→
X0
→
X35
)
)
)
↔
(
(
X2
↔
X48
)
↔
(
X48
→
(
(
¬
X22
)
↔
(
(
¬
X15
)
↔
X35
)
)
)
)
)
↔
(
¬
(
X15
→
(
(
¬
X2
)
↔
X48
)
)
)
)
→
(
¬
(
(
¬
(
X35
↔
(
X6
↔
X48
)
)
)
→
(
¬
(
X42
↔
(
X39
↔
(
X35
→
X6
)
)
)
)
)
)
→
(
¬
(
X48
↔
(
X42
↔
(
(
¬
X15
)
→
X39
)
)
)
)
)
)
↔
(
(
(
¬
X30
)
↔
(
X26
↔
(
X16
↔
X40
)
)
)
↔
(
(
(
X48
↔
(
(
¬
X39
)
↔
(
(
¬
X16
)
↔
X44
)
)
)
↔
(
(
¬
(
(
¬
X39
)
↔
(
(
¬
X16
)
↔
X40
)
)
)
↔
(
(
X40
↔
(
X35
↔
(
X33
↔
False
)
)
)
↔
(
(
X35
→
(
X33
↔
(
(
¬
X14
)
↔
X39
)
)
)
↔
(
¬
(
X44
↔
(
(
¬
X33
)
↔
(
X14
→
X35
)
)
)
)
)
)
)
)
→
(
¬
(
X48
→
(
¬
X39
)
→
(
X26
↔
X44
)
)
)
)
)
)
→
(
(
¬
(
X40
→
X28
→
X8
→
X42
)
)
↔
(
¬
(
(
X40
→
(
X8
↔
X48
)
)
→
(
X48
→
X32
→
(
X28
↔
(
X8
→
False
)
)
)
→
(
(
¬
X40
)
↔
(
X28
↔
(
X8
→
X48
)
)
)
)
)
)
→
(
(
(
X48
↔
(
X45
→
(
(
¬
X29
)
↔
X40
)
)
)
↔
(
(
(
¬
(
X40
→
(
¬
X34
)
→
X45
)
)
→
(
X48
↔
(
X42
→
X40
→
X45
)
)
)
→
(
¬
(
X48
→
X45
→
(
X40
↔
False
)
)
)
)
)
↔
(
X40
→
X34
→
X37
)
)
)
→
(
(
¬
(
(
(
(
(
(
(
¬
X19
)
↔
(
(
¬
X4
)
↔
X21
)
)
↔
(
¬
(
(
¬
(
X4
↔
X21
)
)
→
(
¬
(
(
¬
X44
)
↔
(
X12
↔
(
(
¬
X4
)
↔
X46
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
X21
)
→
(
(
¬
X12
)
↔
(
X4
↔
X47
)
)
)
)
)
↔
(
¬
(
X44
→
(
X25
↔
(
(
¬
X19
)
→
X47
)
)
)
)
)
↔
(
¬
(
(
X47
→
(
¬
X46
)
→
X49
)
→
(
(
(
¬
(
X49
→
X47
→
X38
→
X21
)
)
↔
(
X47
↔
(
X46
→
X38
→
X49
)
)
)
↔
(
(
(
¬
(
(
¬
X38
)
→
X27
→
X41
)
)
↔
(
¬
(
(
¬
(
(
¬
X46
)
→
X41
→
(
(
¬
X38
)
↔
X49
)
)
)
→
(
X49
↔
(
(
¬
X41
)
↔
(
X3
→
X46
)
)
)
)
)
)
↔
(
X27
↔
False
)
)
)
)
)
)
→
(
¬
(
(
X47
↔
(
X36
↔
(
X1
↔
X49
)
)
)
→
(
(
¬
X46
)
↔
(
(
¬
X36
)
↔
X47
)
)
→
(
¬
(
(
¬
X47
)
↔
(
(
¬
X36
)
→
(
X10
↔
X48
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
X46
↔
(
X42
→
X49
)
)
↔
(
¬
(
X46
→
(
X43
↔
(
X42
→
X11
)
)
)
)
)
→
(
(
¬
X46
)
↔
(
X17
↔
(
(
¬
X7
)
↔
X49
)
)
)
→
(
¬
(
X9
→
(
(
¬
X7
)
↔
X43
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X43
→
(
(
¬
X31
)
↔
(
(
¬
X24
)
↔
X49
)
)
)
)
↔
(
¬
(
(
¬
X42
)
→
X43
)
)
)
→
(
¬
(
(
¬
(
X49
↔
(
X48
↔
(
X31
→
False
)
)
)
)
→
X49
→
(
X48
↔
(
X5
→
X43
)
)
)
)
)
→
(
(
¬
(
X49
→
X20
→
(
(
¬
X18
)
↔
X48
)
)
)
↔
(
(
(
X48
↔
(
X43
↔
(
X13
→
X49
)
)
)
↔
(
¬
(
X49
→
X20
→
(
¬
X13
)
→
X43
)
)
)
↔
(
¬
(
(
¬
X43
)
→
(
¬
X23
)
→
(
X13
↔
X48
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
(
X44
→
X43
→
(
X37
↔
False
)
)
↔
(
(
¬
(
X46
→
(
¬
X7
)
→
X37
)
)
↔
(
¬
(
(
¬
(
(
¬
X43
)
↔
(
X42
→
X7
→
X44
)
)
)
→
(
X46
↔
(
X44
↔
(
X42
↔
X7
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
X13
)
↔
(
(
¬
X7
)
↔
X39
)
)
↔
(
¬
(
(
¬
(
X39
↔
(
(
¬
X7
)
↔
X13
)
)
)
→
(
¬
(
X42
↔
(
X39
↔
(
X37
↔
X7
)
)
)
)
)
)
)
→
(
¬
X39
)
)
→
(
¬
(
X39
↔
(
X37
↔
(
X7
↔
False
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
X44
↔
(
X20
→
X47
)
)
→
(
¬
(
(
(
¬
(
X44
↔
(
X20
↔
(
X9
↔
X47
)
)
)
)
→
(
¬
(
(
(
¬
X46
)
↔
(
X20
→
X47
)
)
→
(
X44
→
(
X20
↔
(
X5
→
X46
)
)
)
→
(
¬
(
(
¬
X46
)
↔
(
(
¬
X44
)
→
X47
)
)
)
)
)
)
→
(
¬
(
(
¬
X44
)
↔
(
X20
→
X46
)
)
)
)
)
)
→
X46
→
X20
→
(
X9
↔
False
)
)
→
X47
→
(
¬
X44
)
→
(
X34
↔
X46
)
)
→
(
¬
(
X46
→
(
¬
X43
)
→
(
¬
X20
)
→
X44
)
)
)
)
)
)
↔
(
(
(
(
X46
↔
(
X32
↔
(
X18
→
X38
)
)
)
↔
(
X46
↔
(
X32
↔
(
X29
→
X38
)
)
)
)
→
(
(
(
(
¬
(
(
(
¬
(
(
¬
X32
)
→
X15
→
(
X11
↔
X46
)
)
)
→
X15
→
X11
→
(
(
¬
X8
)
↔
X46
)
)
→
(
X46
↔
(
X8
→
False
)
)
)
)
↔
(
X46
→
X15
)
)
↔
(
¬
(
(
¬
(
X37
↔
(
(
¬
X11
)
→
X28
)
)
)
→
(
¬
(
(
¬
X28
)
→
(
(
¬
X11
)
↔
(
X1
→
X46
)
)
)
)
)
)
)
↔
(
¬
(
(
(
X37
→
(
X33
↔
(
(
¬
X29
)
↔
X46
)
)
)
↔
(
(
¬
X33
)
→
(
¬
X32
)
→
(
X29
↔
X46
)
)
)
→
(
¬
(
X37
→
(
¬
X29
)
→
X46
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
X48
↔
(
X38
→
X37
)
)
)
→
(
¬
(
(
X47
→
X38
→
(
X35
↔
False
)
)
→
(
¬
(
X48
↔
(
X47
↔
(
X35
↔
X37
)
)
)
)
)
)
)
)
→
(
¬
(
X47
→
(
(
¬
X46
)
↔
(
X38
→
X48
)
)
)
)
)
↔
(
(
¬
X46
)
↔
(
X37
↔
(
(
¬
X10
)
↔
X47
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
(
X49
↔
(
X6
↔
X48
)
)
)
→
(
(
¬
(
X49
↔
(
X42
↔
X22
)
)
)
↔
(
¬
(
X49
→
X48
→
(
X6
↔
X22
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X48
→
X23
→
False
)
)
→
(
¬
(
(
X41
↔
X48
)
→
(
¬
(
X48
→
(
(
¬
X16
)
↔
(
X14
→
X23
)
)
)
)
)
)
)
→
(
¬
(
(
¬
X41
)
↔
(
X16
↔
X49
)
)
)
)
)
)
↔
(
(
(
X49
↔
(
X41
↔
(
X23
→
X30
)
)
)
→
(
(
¬
(
X41
→
(
(
¬
X40
)
↔
(
(
¬
X30
)
→
X47
)
)
)
)
↔
(
(
¬
X47
)
→
(
X41
↔
(
(
¬
X40
)
↔
X49
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X40
→
(
X3
↔
X31
)
)
)
→
(
(
¬
(
X40
↔
(
X36
→
X30
→
X21
)
)
)
↔
(
(
¬
(
(
¬
X21
)
→
(
¬
X12
)
→
X31
)
)
→
(
¬
(
(
¬
(
X36
↔
(
X30
→
(
X12
↔
X31
)
)
)
)
→
(
X31
↔
(
(
¬
X30
)
↔
(
X0
↔
X36
)
)
)
)
)
)
)
)
)
→
(
¬
(
X47
↔
(
X40
→
(
X36
↔
X31
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
X47
)
↔
(
(
¬
X27
)
→
(
(
¬
X24
)
↔
X48
)
)
)
↔
(
(
X48
↔
(
X24
↔
(
X2
↔
X4
)
)
)
↔
(
¬
(
(
¬
(
X47
→
X24
→
X48
)
)
→
(
¬
(
X48
↔
(
X47
↔
(
X42
↔
(
X24
↔
False
)
)
)
)
)
→
(
¬
X47
)
→
(
¬
X24
)
→
(
(
¬
X4
)
↔
X48
)
)
)
)
)
↔
(
X47
→
(
(
¬
(
(
¬
(
X48
↔
(
X45
↔
(
X27
↔
(
X26
→
False
)
)
)
)
)
→
(
(
X48
→
(
X19
↔
(
(
¬
X17
)
→
X27
)
)
)
→
(
¬
(
X45
↔
(
X27
↔
False
)
)
)
)
→
(
¬
(
X45
↔
(
(
¬
X27
)
→
X48
)
)
)
)
)
↔
(
X48
→
X27
→
X45
)
)
→
(
(
¬
(
(
¬
(
X48
↔
(
X45
→
(
¬
X25
)
→
X47
)
)
)
→
(
X45
↔
(
X25
→
X27
)
)
)
)
→
(
X48
↔
(
(
¬
X45
)
→
X47
)
)
)
→
(
¬
(
X27
↔
(
X25
→
False
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
13bf3d...
and proposition id is
ee5aa4...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMHb5KnAUpZiRqQ6cLSgeyZ6UifqgJP88DJ
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
(
¬
(
(
¬
(
X48
↔
(
(
¬
X30
)
↔
X40
)
)
)
→
(
(
¬
(
X48
→
(
(
¬
X40
)
↔
(
(
¬
X39
)
→
X49
)
)
)
)
↔
(
X48
↔
(
X40
→
(
¬
X22
)
→
X49
)
)
)
)
)
↔
(
(
¬
(
(
¬
X35
)
↔
X49
)
)
→
(
¬
(
(
(
¬
(
X39
↔
(
X37
→
(
X8
↔
False
)
)
)
)
↔
(
¬
(
(
X49
→
X39
→
(
X8
↔
False
)
)
→
(
¬
(
X39
→
(
¬
X30
)
→
(
¬
X8
)
→
X37
)
)
)
)
)
→
X39
→
X35
→
(
X8
↔
X30
)
)
)
)
)
↔
(
X48
↔
(
(
¬
X40
)
→
(
(
¬
X37
)
↔
X42
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
(
¬
X45
)
↔
(
X43
↔
(
X10
↔
X48
)
)
)
)
→
X45
→
(
X28
↔
(
X10
↔
False
)
)
)
↔
(
X48
↔
(
(
¬
X43
)
↔
X45
)
)
)
→
(
¬
(
(
¬
(
X48
↔
(
(
¬
X25
)
→
X7
→
X43
)
)
)
→
(
(
(
¬
(
(
¬
(
(
(
X5
↔
X47
)
↔
(
(
¬
X47
)
→
(
¬
X18
)
→
X10
→
X48
)
)
→
(
X43
↔
(
X18
↔
(
X10
↔
X47
)
)
)
)
)
→
X43
→
(
X10
↔
X47
)
)
)
↔
(
¬
(
X48
↔
(
X43
→
X47
)
)
)
)
↔
(
¬
(
X47
→
(
X10
↔
(
X5
→
X25
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X46
→
(
¬
X40
)
→
(
(
¬
X27
)
↔
X42
)
)
)
↔
(
(
(
(
¬
(
X42
↔
(
X27
→
(
(
¬
X4
)
↔
X6
)
)
)
)
↔
(
¬
(
X42
→
(
X31
↔
(
X27
→
X4
)
)
)
)
)
↔
(
¬
(
X42
→
X40
→
X4
→
X6
)
)
)
↔
(
(
(
¬
X42
)
→
X2
→
X45
)
↔
(
X45
↔
(
X31
→
(
(
¬
X2
)
↔
X27
)
)
)
)
)
)
↔
(
(
(
¬
(
X47
→
X36
→
X32
→
X48
)
)
↔
(
(
X47
↔
(
X32
↔
X42
)
)
→
X48
→
X45
)
)
↔
(
(
¬
(
X48
↔
(
(
¬
X36
)
↔
(
(
¬
X32
)
↔
X46
)
)
)
)
↔
(
(
¬
(
X32
↔
(
X20
→
X16
)
)
)
↔
(
¬
(
X46
→
(
X32
↔
(
X16
↔
X36
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
X45
→
(
¬
X15
)
→
X14
→
X19
)
↔
(
(
(
¬
X24
)
→
(
¬
X19
)
→
X48
)
↔
(
¬
(
(
¬
(
X48
↔
(
X24
↔
(
X19
↔
X41
)
)
)
)
→
(
(
X24
→
(
X15
↔
(
X14
→
X48
)
)
)
↔
(
(
¬
X24
)
→
(
X19
↔
X48
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X45
→
(
X12
↔
(
X3
→
False
)
)
)
)
→
(
(
¬
(
(
¬
X26
)
↔
(
(
¬
X12
)
→
(
X1
↔
X48
)
)
)
)
↔
(
¬
(
(
(
¬
X34
)
→
(
(
¬
X3
)
↔
X48
)
)
→
(
X34
↔
(
X12
↔
(
(
¬
X3
)
↔
X26
)
)
)
→
X48
→
(
(
¬
X12
)
↔
(
(
¬
X3
)
→
X34
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X11
→
X26
)
)
→
(
¬
(
X48
→
(
(
¬
X26
)
↔
(
X11
→
X45
)
)
)
)
→
X48
→
(
¬
X26
)
→
(
¬
X11
)
→
X41
)
)
)
)
)
↔
(
(
(
(
(
X33
→
(
X29
↔
(
X9
↔
X48
)
)
)
↔
(
X38
↔
(
X33
↔
False
)
)
)
↔
(
¬
(
X38
→
X33
→
X48
)
)
)
↔
(
X33
→
(
¬
X29
)
→
X38
)
)
↔
(
¬
(
(
(
(
(
¬
X39
)
↔
X48
)
↔
(
¬
(
X39
↔
(
X38
→
(
¬
X13
)
→
X44
)
)
)
)
→
(
¬
(
X33
↔
X21
)
)
)
→
(
¬
(
(
(
(
X41
→
X39
→
(
X17
↔
X45
)
)
↔
(
¬
(
X45
↔
(
X39
→
(
X0
↔
X41
)
)
)
)
)
↔
(
¬
(
X45
→
X41
→
X39
→
X23
)
)
)
→
(
(
¬
(
X48
↔
(
X44
→
X39
→
X45
)
)
)
↔
(
X48
→
X44
→
(
¬
X23
)
→
X45
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
¬
(
(
X43
→
(
X33
↔
X24
)
)
→
X45
→
(
X24
↔
(
X16
→
X22
)
)
)
)
→
(
¬
(
(
¬
(
X31
↔
(
X24
→
X32
)
)
)
→
(
(
¬
X33
)
↔
(
X32
↔
(
(
¬
X24
)
↔
X45
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X39
→
(
X33
↔
(
X27
→
X45
)
)
)
)
↔
(
¬
(
X45
↔
(
X39
↔
(
X33
→
X26
→
False
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X42
)
→
(
¬
X39
)
→
(
X26
↔
X47
)
)
)
↔
(
X47
↔
(
X39
→
(
¬
X33
)
→
X42
)
)
)
↔
(
¬
(
X42
→
(
X33
↔
(
X0
↔
X47
)
)
)
)
)
)
↔
(
(
¬
(
X48
→
(
(
¬
X43
)
↔
(
X18
→
X47
)
)
)
)
→
(
¬
(
(
X43
↔
(
(
¬
X42
)
→
X47
)
)
→
(
(
¬
(
X39
→
X5
→
X1
)
)
↔
(
¬
(
X48
→
X39
→
(
¬
X1
)
→
X43
)
)
)
)
)
)
)
)
→
(
(
(
(
X45
↔
(
X36
↔
False
)
)
↔
(
¬
(
(
X36
→
(
(
¬
X10
)
↔
X43
)
)
→
(
(
¬
(
(
¬
X43
)
↔
(
X36
→
X45
)
)
)
↔
(
¬
(
X43
→
(
¬
X28
)
→
X10
→
X36
)
)
)
)
)
)
↔
(
(
X40
→
(
¬
X36
)
→
(
X12
↔
X43
)
)
→
(
(
¬
X40
)
↔
(
X12
↔
X45
)
)
)
)
↔
(
(
(
(
X47
↔
(
(
¬
X15
)
↔
X49
)
)
→
(
¬
(
(
¬
(
(
¬
X40
)
→
(
¬
X15
)
→
(
¬
X11
)
→
X49
)
)
→
(
¬
X47
)
→
(
¬
X15
)
→
(
¬
X11
)
→
X49
)
)
)
→
(
¬
(
X47
↔
(
X45
→
X21
→
False
)
)
)
)
↔
(
¬
(
(
(
¬
X8
)
↔
(
¬
(
X47
↔
(
X40
↔
(
(
¬
X4
)
↔
X7
)
)
)
)
)
→
(
¬
(
X49
↔
(
X47
↔
X36
)
)
)
)
)
)
)
)
→
(
(
(
(
(
X47
↔
(
(
¬
X25
)
→
(
X17
↔
X37
)
)
)
↔
(
(
(
(
(
¬
X41
)
↔
(
(
¬
X29
)
↔
(
(
¬
X25
)
↔
X47
)
)
)
↔
(
(
X47
↔
X41
)
→
(
¬
(
(
(
X47
↔
(
X29
↔
(
X20
→
X37
)
)
)
↔
(
¬
(
X47
→
(
(
¬
X29
)
↔
X41
)
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X37
)
↔
X41
)
)
)
)
)
)
)
→
(
¬
X29
)
→
(
X25
↔
X47
)
)
→
(
¬
(
(
¬
X25
)
→
(
(
¬
X20
)
↔
X47
)
)
)
)
)
→
X29
→
(
X25
↔
X17
)
)
↔
(
¬
(
(
(
(
¬
(
(
¬
X41
)
↔
(
X38
→
(
¬
X6
)
→
X46
)
)
)
↔
(
¬
(
(
¬
X38
)
→
(
¬
X37
)
→
(
¬
X6
)
→
X46
)
)
)
→
(
(
¬
X41
)
↔
(
X37
↔
(
(
¬
X35
)
↔
X46
)
)
)
)
→
(
(
¬
(
(
¬
X35
)
→
(
X29
↔
X37
)
)
)
↔
(
(
(
¬
X37
)
→
(
(
¬
X35
)
↔
X38
)
)
↔
(
¬
(
X48
↔
(
X38
↔
(
X29
→
False
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
X30
)
→
(
X13
↔
X34
)
)
↔
(
(
¬
X44
)
→
(
(
¬
X13
)
↔
(
X9
↔
X49
)
)
)
)
→
(
(
X49
↔
(
X41
→
X30
→
X2
)
)
→
(
(
¬
(
(
¬
(
X30
→
X49
)
)
→
(
¬
(
X49
↔
(
X23
↔
(
X19
→
X34
)
)
)
)
)
)
↔
(
(
(
¬
X34
)
↔
(
X23
→
(
¬
X19
)
→
X41
)
)
↔
(
¬
(
X23
↔
(
(
¬
X3
)
↔
X14
)
)
)
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X41
)
↔
(
X34
→
X46
)
)
)
)
→
X46
→
(
X38
↔
(
(
¬
X34
)
↔
X41
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
3f37ef...
and proposition id is
37ae1b...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMHFQF9M3LX7Dhx9z5TWj9JZnZcg3HxHDjv
)
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
(
(
(
¬
(
(
(
¬
(
X44
→
(
(
¬
X26
)
↔
X27
)
)
)
→
(
¬
(
(
¬
(
X29
→
(
¬
X27
)
→
(
X23
↔
X47
)
)
)
→
(
(
X44
→
X27
→
(
¬
X23
)
→
X29
)
↔
(
¬
(
X27
→
(
X26
↔
X44
)
)
)
)
)
)
)
→
(
(
X44
→
X27
→
X47
)
↔
(
X48
→
X24
→
False
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X42
)
→
(
X26
↔
(
X0
↔
X47
)
)
)
)
→
X49
→
(
X45
↔
(
X0
→
False
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
¬
(
(
(
X47
↔
(
X43
↔
(
X25
→
X22
→
False
)
)
)
↔
(
¬
(
X45
↔
(
X25
↔
X43
)
)
)
)
→
X45
)
)
→
(
¬
X45
)
→
X47
)
)
→
(
X47
↔
(
(
¬
X43
)
↔
(
X14
→
X45
)
)
)
)
)
↔
(
¬
(
X47
→
X25
→
(
X22
↔
X49
)
)
)
)
→
(
(
(
(
X49
↔
X13
)
→
(
(
¬
(
X43
→
(
¬
X36
)
→
X13
→
X49
)
)
↔
(
X36
→
(
(
¬
X13
)
↔
(
X7
↔
X49
)
)
)
)
)
↔
(
¬
(
X47
→
(
(
¬
X43
)
↔
(
X36
→
X49
)
)
)
)
)
↔
(
X49
→
(
X36
↔
(
(
¬
X27
)
↔
X43
)
)
)
)
)
)
)
↔
(
(
(
(
X45
→
X41
→
X38
)
↔
(
(
(
¬
(
(
X49
↔
(
X43
→
X9
→
X38
)
)
→
(
¬
(
X43
↔
(
(
¬
X38
)
↔
(
X9
→
X45
)
)
)
)
)
)
→
X38
→
X49
)
↔
(
¬
(
X41
→
X9
→
False
)
)
)
)
↔
(
¬
(
(
¬
X43
)
→
(
X41
↔
(
X3
→
X45
)
)
)
)
)
↔
(
(
X49
↔
(
X32
→
(
¬
X1
)
→
X43
)
)
→
(
¬
(
(
¬
X16
)
↔
(
(
¬
X1
)
↔
X38
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
(
X39
→
(
(
¬
X4
)
↔
(
X2
→
X50
)
)
)
↔
(
(
(
¬
X42
)
→
(
X39
↔
(
X4
↔
X50
)
)
)
→
(
X50
↔
(
X42
↔
(
X39
↔
X33
)
)
)
)
)
↔
(
X42
↔
(
(
¬
X39
)
↔
(
(
¬
X34
)
→
X50
)
)
)
)
→
(
¬
(
(
¬
X20
)
→
(
(
¬
X15
)
↔
(
X11
→
X34
)
)
)
)
→
(
(
¬
(
(
¬
X44
)
→
X39
→
X20
→
X50
)
)
↔
(
¬
(
(
¬
(
(
¬
X20
)
→
X15
→
(
¬
X12
)
→
X30
)
)
→
(
(
¬
X30
)
↔
(
(
¬
X20
)
↔
(
X15
→
X44
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
X37
)
↔
(
X35
→
X42
)
)
↔
(
(
X37
→
X35
→
(
¬
X18
)
→
X42
)
→
(
(
(
¬
(
(
¬
(
X42
↔
(
(
¬
X31
)
↔
(
X5
↔
X35
)
)
)
)
→
(
¬
(
(
¬
X31
)
→
(
X18
↔
(
(
¬
X5
)
↔
X35
)
)
)
)
)
)
→
X31
→
X5
→
X18
)
↔
(
¬
(
X31
→
(
X5
↔
X35
)
)
)
)
)
)
↔
(
(
(
(
X42
↔
(
X37
↔
(
X17
→
X10
→
False
)
)
)
→
(
(
(
(
¬
(
X42
→
(
¬
X18
)
→
X37
)
)
↔
(
(
¬
X28
)
↔
(
X18
↔
(
(
¬
X10
)
→
X37
)
)
)
)
↔
(
X40
↔
(
(
¬
X37
)
→
(
(
¬
X18
)
↔
X42
)
)
)
)
↔
X18
)
)
↔
(
¬
(
(
(
¬
X18
)
→
(
(
¬
X8
)
↔
X37
)
)
→
(
(
¬
X28
)
↔
(
X8
↔
(
(
¬
X6
)
↔
X37
)
)
)
)
)
)
→
(
¬
(
X40
→
X28
→
X18
)
)
)
)
→
(
¬
(
(
(
(
(
¬
X39
)
→
(
(
¬
X37
)
↔
(
X19
↔
X46
)
)
)
↔
(
¬
(
(
¬
X40
)
→
(
X39
↔
(
X19
→
X46
)
)
)
)
)
↔
(
¬
(
X46
↔
(
X45
→
X42
→
X40
)
)
)
)
→
(
¬
(
(
(
¬
(
X44
→
(
X42
↔
(
X19
→
False
)
)
)
)
↔
(
X44
→
(
X21
↔
(
X19
↔
X46
)
)
)
)
→
X46
→
X21
)
)
)
)
→
(
¬
(
X39
↔
(
X19
→
X44
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
(
(
(
¬
(
X28
↔
(
X20
↔
(
X10
↔
X45
)
)
)
)
↔
(
¬
(
X48
→
(
X28
↔
(
X10
↔
X45
)
)
)
)
)
↔
(
(
¬
X45
)
↔
(
(
¬
X34
)
→
(
¬
X20
)
→
X48
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
(
X34
→
X10
)
)
↔
(
(
(
¬
X40
)
↔
(
(
¬
X34
)
→
(
X10
↔
X50
)
)
)
↔
(
¬
(
(
¬
X34
)
↔
(
X10
→
X36
)
)
)
)
)
→
(
¬
(
X36
↔
(
(
¬
X34
)
↔
X40
)
)
)
)
)
→
(
¬
(
X34
↔
(
(
¬
X28
)
→
X40
)
)
)
)
)
)
)
↔
(
(
(
X45
→
(
¬
X17
)
→
X48
)
→
(
¬
(
(
(
(
¬
(
X7
→
X45
)
)
↔
(
¬
(
X47
↔
(
(
¬
X45
)
↔
(
(
¬
X14
)
→
X48
)
)
)
)
)
↔
(
¬
(
(
¬
X45
)
↔
(
X40
→
X48
)
)
)
)
→
X45
→
(
X17
↔
(
X14
↔
X40
)
)
)
)
)
↔
(
(
¬
X45
)
↔
(
X40
↔
X48
)
)
)
)
↔
(
(
(
(
(
X41
→
(
X15
↔
(
(
¬
X2
)
→
X50
)
)
)
↔
(
¬
(
(
¬
X15
)
↔
X45
)
)
)
→
(
(
¬
X41
)
↔
(
(
¬
X19
)
→
(
¬
X15
)
→
X45
)
)
)
↔
(
(
X47
→
(
X19
↔
X41
)
)
→
(
(
¬
X47
)
↔
(
(
¬
X44
)
→
X50
)
)
)
)
↔
(
¬
(
X50
↔
(
X44
→
(
X19
↔
X0
)
)
)
)
)
)
↔
(
(
(
¬
X48
)
→
X41
→
X1
→
X49
)
→
(
(
¬
(
X44
↔
(
(
¬
X41
)
→
(
(
¬
X37
)
↔
X49
)
)
)
)
↔
(
¬
(
X42
↔
(
(
¬
X6
)
→
X48
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
(
¬
(
(
¬
(
(
¬
X43
)
→
(
¬
X30
)
→
(
X26
↔
X46
)
)
)
→
(
¬
(
X47
→
X46
→
X43
→
X30
)
)
)
)
↔
(
(
¬
X30
)
↔
(
(
¬
X26
)
↔
X47
)
)
)
→
(
¬
(
X46
→
(
(
¬
X43
)
↔
(
(
¬
X29
)
↔
X47
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X35
→
X24
→
(
(
¬
X11
)
↔
X31
)
)
)
→
(
(
¬
(
X47
↔
(
X46
↔
(
X35
↔
X43
)
)
)
)
↔
(
(
(
¬
X43
)
↔
(
(
¬
X35
)
→
X47
)
)
→
(
(
¬
(
X35
↔
(
X24
→
False
)
)
)
↔
(
¬
(
X47
↔
(
X35
↔
(
X31
→
X43
)
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
X35
→
X21
→
X43
)
→
(
¬
(
X43
→
X21
→
X9
→
X33
)
)
)
)
↔
(
(
(
X43
→
(
X35
↔
(
(
¬
X18
)
→
X46
)
)
)
→
(
¬
(
X43
→
X35
→
(
(
¬
X33
)
↔
X46
)
)
)
)
↔
(
¬
(
(
¬
X18
)
→
X35
)
)
)
)
)
↔
(
(
(
(
(
(
(
(
X39
→
X13
→
(
¬
X12
)
→
X50
)
↔
(
¬
(
X50
→
X47
→
(
X39
↔
False
)
)
)
)
↔
(
X50
↔
(
X39
→
X47
)
)
)
↔
(
¬
(
(
¬
X47
)
↔
(
(
¬
X13
)
→
X50
)
)
)
)
↔
(
¬
(
X50
↔
X5
)
)
)
↔
(
(
X44
→
X39
→
(
X38
↔
(
X23
↔
False
)
)
)
↔
(
(
¬
(
(
¬
X47
)
↔
(
(
¬
X23
)
↔
X50
)
)
)
↔
(
¬
(
X50
→
X47
→
(
X39
↔
(
X38
→
False
)
)
)
)
)
)
)
↔
(
(
¬
(
X50
→
X47
→
(
X39
↔
(
X4
↔
False
)
)
)
)
↔
(
(
¬
(
(
X39
↔
(
(
¬
X22
)
→
X35
)
)
→
(
¬
(
X39
→
(
X35
↔
(
X32
↔
X4
)
)
)
)
→
(
¬
(
(
¬
X35
)
↔
(
X32
→
(
(
¬
X22
)
↔
X39
)
)
)
)
)
)
→
(
¬
(
(
¬
X32
)
→
X22
→
X35
)
)
)
)
)
↔
(
(
(
(
X32
↔
(
X27
↔
(
X25
↔
(
X3
↔
False
)
)
)
)
↔
(
(
X32
→
(
(
¬
X16
)
↔
(
(
¬
X8
)
→
X25
)
)
)
→
(
(
¬
X27
)
↔
(
X25
↔
(
X8
→
X32
)
)
)
)
)
↔
(
¬
(
X43
↔
(
X27
↔
(
(
¬
X25
)
↔
X32
)
)
)
)
)
→
(
(
¬
(
X39
↔
X27
)
)
↔
(
(
¬
(
X43
→
(
(
¬
X27
)
↔
(
X8
→
X35
)
)
)
)
↔
(
(
¬
X39
)
→
(
(
¬
X35
)
↔
(
(
¬
X27
)
→
X43
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
0ddfb8...
and proposition id is
2d7f3b...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMb5upGT9XknMzEJ9mwFzHi7GhsoGJjcF2G
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
¬
(
(
X46
↔
(
(
¬
X43
)
↔
X45
)
)
→
(
¬
(
X45
→
(
X43
↔
X46
)
)
)
→
(
X44
↔
(
X31
↔
X43
)
)
→
(
¬
(
X43
→
X22
→
X46
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
(
X48
↔
(
(
¬
X44
)
↔
(
(
¬
X43
)
→
X46
)
)
)
)
→
(
¬
(
X48
↔
(
X44
→
(
X42
↔
False
)
)
)
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X42
)
↔
X48
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X46
↔
(
X44
→
(
X38
↔
False
)
)
)
)
→
(
¬
(
(
¬
(
X46
→
X44
→
(
X38
↔
False
)
)
)
→
(
¬
(
X44
↔
(
X32
↔
X26
)
)
)
)
)
)
↔
(
(
¬
(
X48
→
(
X46
↔
(
X12
↔
X32
)
)
)
)
↔
(
(
¬
X32
)
↔
(
X26
↔
(
(
¬
X12
)
↔
X48
)
)
)
)
)
→
(
X48
↔
(
X46
→
X26
)
)
)
↔
(
(
(
(
¬
(
(
¬
X38
)
↔
(
X24
↔
(
(
¬
X18
)
↔
X44
)
)
)
)
↔
(
¬
(
(
¬
X44
)
↔
(
(
¬
X39
)
→
X48
)
)
)
)
↔
(
X48
↔
(
X38
↔
(
(
¬
X17
)
↔
X39
)
)
)
)
↔
(
X44
↔
(
(
¬
X38
)
→
X48
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
¬
(
X49
→
X40
→
(
X7
↔
X15
)
)
)
↔
(
X15
↔
(
X10
→
(
(
¬
X7
)
↔
X49
)
)
)
)
↔
(
(
(
¬
(
X42
↔
False
)
)
↔
(
(
¬
(
X49
↔
(
X40
→
(
X5
↔
False
)
)
)
)
→
(
(
¬
X40
)
↔
(
(
¬
X15
)
↔
X44
)
)
)
)
→
(
(
X40
→
(
X36
↔
(
X4
↔
X42
)
)
)
↔
(
X44
→
(
X36
↔
(
(
¬
X4
)
↔
X42
)
)
)
)
)
)
→
(
¬
X40
)
→
(
X36
↔
(
(
¬
X1
)
→
X49
)
)
)
)
→
(
(
(
(
¬
(
X43
↔
(
X27
→
X47
)
)
)
→
X36
→
(
X27
↔
X23
)
)
→
(
(
¬
X41
)
↔
(
(
¬
X23
)
→
X47
)
)
)
↔
(
(
X47
→
(
¬
X40
)
→
X27
→
X49
)
↔
(
X49
↔
(
X47
↔
(
X36
→
(
X25
↔
False
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
(
¬
(
(
¬
X45
)
↔
(
X44
↔
X46
)
)
)
→
(
(
¬
(
(
¬
X44
)
↔
(
X13
→
X46
)
)
)
↔
(
(
(
X13
↔
(
X11
↔
X45
)
)
↔
(
X45
↔
(
(
¬
X11
)
→
(
¬
X9
)
→
X44
)
)
)
↔
(
(
¬
X44
)
↔
(
X13
↔
X45
)
)
)
)
)
)
→
(
¬
(
(
X44
→
(
X20
↔
(
X8
→
X45
)
)
)
→
(
¬
(
(
¬
X44
)
→
(
¬
X20
)
→
(
(
¬
X8
)
↔
X46
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X46
↔
(
X29
↔
(
(
¬
X2
)
→
X20
)
)
)
)
→
(
¬
(
(
¬
X37
)
→
(
X29
↔
X46
)
)
)
)
)
→
(
¬
(
(
¬
X37
)
→
X29
→
(
¬
X2
)
→
X44
)
)
)
)
)
)
↔
(
¬
(
(
(
(
X33
→
(
¬
X6
)
→
X30
)
↔
(
(
X35
↔
(
(
¬
X30
)
↔
(
(
¬
X6
)
→
X33
)
)
)
↔
(
¬
(
(
¬
X30
)
→
(
¬
X16
)
→
X6
→
X33
)
)
)
)
↔
(
(
X21
→
(
X16
↔
(
(
¬
X14
)
↔
X33
)
)
)
↔
(
¬
(
X33
↔
(
X30
↔
(
X16
→
X21
)
)
)
)
)
)
→
(
(
(
X45
↔
(
(
¬
X33
)
↔
(
X0
↔
X46
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X45
)
→
X33
→
X30
→
X46
)
)
↔
(
¬
(
X45
→
X33
→
(
¬
X28
)
→
X46
)
)
)
→
(
(
X45
↔
(
(
¬
X30
)
↔
(
(
¬
X3
)
→
X44
)
)
)
↔
(
¬
(
X44
↔
(
(
¬
X3
)
↔
X46
)
)
)
)
)
)
)
↔
(
(
(
¬
X45
)
→
(
¬
X44
)
→
(
(
¬
X34
)
↔
X46
)
)
↔
(
(
¬
(
X35
→
(
X34
↔
X45
)
)
)
↔
(
(
¬
(
X35
↔
(
(
¬
X19
)
→
X45
)
)
)
↔
(
X46
→
X34
→
X19
→
False
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
(
X42
→
X30
→
(
¬
X26
)
→
X41
)
↔
(
¬
(
X43
→
X42
→
(
X30
↔
(
X26
→
False
)
)
)
)
)
↔
(
X42
→
(
X41
↔
(
X29
→
X43
)
)
)
)
↔
(
(
X48
→
(
X43
↔
(
(
¬
X4
)
↔
X42
)
)
)
↔
(
¬
(
X48
→
(
X42
↔
(
X41
↔
False
)
)
)
)
)
)
→
(
(
(
X44
→
(
X42
↔
(
X11
→
X48
)
)
)
→
(
¬
X42
)
→
(
¬
X20
)
→
(
(
¬
X11
)
↔
X43
)
)
↔
(
¬
(
(
¬
(
X48
↔
(
X18
↔
(
X12
↔
X42
)
)
)
)
→
(
(
X20
→
(
¬
X16
)
→
(
X12
↔
X48
)
)
↔
(
(
¬
X20
)
→
(
¬
X12
)
→
X42
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X34
→
(
¬
X2
)
→
X15
)
)
→
(
X25
↔
(
X2
→
(
(
¬
X0
)
↔
X15
)
)
)
)
→
(
(
(
¬
(
(
¬
X34
)
→
X2
→
X35
)
)
→
(
X36
↔
(
X35
→
X2
→
X31
)
)
)
↔
(
X36
↔
(
X35
↔
X25
)
)
)
)
→
(
(
(
(
X31
→
X7
→
X42
)
↔
(
¬
(
X48
↔
(
X36
→
(
X23
↔
False
)
)
)
)
)
→
(
(
(
¬
(
X36
→
X21
→
(
¬
X10
)
→
X42
)
)
↔
(
X48
→
X42
)
)
↔
(
¬
False
)
)
)
↔
(
(
(
(
¬
X32
)
↔
(
X13
↔
X44
)
)
↔
(
X48
↔
(
(
¬
X32
)
↔
(
(
¬
X14
)
↔
X44
)
)
)
)
→
(
(
¬
(
(
¬
(
X32
↔
(
(
¬
X23
)
↔
(
(
¬
X13
)
↔
X35
)
)
)
)
→
X31
→
X23
→
X13
→
X35
)
)
↔
(
¬
(
X32
↔
(
X23
→
X31
)
)
)
)
→
(
¬
(
X35
↔
(
X32
→
(
X23
↔
(
X14
→
False
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X48
→
(
(
¬
X24
)
↔
(
X3
→
X45
)
)
)
)
→
(
¬
(
X38
→
X24
)
)
)
↔
(
¬
(
(
¬
(
(
X48
→
(
(
¬
X28
)
↔
(
(
¬
X9
)
→
X49
)
)
)
→
X48
→
(
(
¬
X38
)
↔
(
X9
→
X49
)
)
)
)
→
(
(
(
(
¬
X28
)
↔
(
X9
↔
(
X8
↔
X47
)
)
)
→
(
¬
(
X38
↔
(
(
¬
X28
)
↔
(
(
¬
X9
)
→
X47
)
)
)
)
)
↔
(
(
¬
X9
)
↔
(
(
¬
X8
)
↔
X47
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X47
)
→
(
X45
↔
(
(
¬
X27
)
→
X49
)
)
)
)
↔
(
¬
(
(
¬
X45
)
↔
(
(
¬
X27
)
→
(
X5
↔
X47
)
)
)
)
)
→
(
¬
(
X45
→
X43
→
X33
→
X47
)
)
)
)
→
(
(
(
(
¬
X48
)
→
(
X46
↔
(
X39
↔
X49
)
)
)
→
(
(
X46
→
(
¬
X39
)
→
(
(
¬
X17
)
↔
X40
)
)
↔
(
(
¬
(
(
(
X1
→
X39
)
↔
(
¬
(
(
X42
→
X39
→
(
(
¬
X1
)
↔
X46
)
)
→
(
¬
(
(
¬
X42
)
→
X49
)
)
)
)
)
→
(
¬
(
(
¬
X46
)
↔
(
X42
↔
X49
)
)
)
)
)
↔
(
X42
→
X17
→
X46
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X37
)
→
X6
→
X42
)
)
→
(
(
X48
→
X40
→
(
X6
↔
False
)
)
↔
(
X47
→
X42
→
X6
→
X40
)
)
)
)
→
(
X48
→
X47
→
(
(
¬
X19
)
↔
X40
)
)
→
(
(
¬
X40
)
↔
(
X22
↔
(
(
¬
X19
)
↔
X47
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
3e25ff...
and proposition id is
8ba317...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMQpQEsePNr7vtbNf21cMK3KvGwNW12axKv
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
∀X51 :
prop
,
(
¬
∀X52 :
prop
,
(
(
¬
(
(
(
(
(
(
(
X48
→
(
X44
↔
(
X41
→
X32
)
)
)
↔
(
¬
(
X48
→
(
(
¬
X44
)
↔
(
X41
↔
X51
)
)
)
)
)
↔
(
¬
(
(
X44
↔
X48
)
→
(
(
(
X46
→
X5
→
False
)
↔
(
X51
→
(
X48
↔
(
X43
→
X5
)
)
)
)
↔
(
(
¬
X46
)
↔
(
(
¬
X43
)
→
X48
)
)
)
)
)
)
→
(
¬
(
X51
↔
(
X44
↔
X46
)
)
)
)
↔
(
(
(
(
¬
X21
)
↔
X48
)
↔
(
(
X48
↔
(
X47
→
X24
→
X49
)
)
→
(
(
¬
X48
)
↔
(
X47
→
(
¬
X21
)
→
X49
)
)
)
)
↔
(
(
(
(
X21
→
X18
→
(
(
¬
X9
)
↔
X47
)
)
→
X21
→
X18
→
X10
→
(
X9
↔
False
)
)
↔
(
¬
(
X47
→
(
(
¬
X44
)
↔
(
X24
→
X51
)
)
)
)
)
→
(
¬
(
X18
↔
(
(
¬
X10
)
→
X47
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X39
→
(
¬
X33
)
→
X47
)
)
→
(
¬
(
X33
↔
(
(
¬
X12
)
↔
X37
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X40
↔
(
(
¬
X30
)
↔
(
(
¬
X27
)
↔
X51
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X39
→
(
X20
↔
(
(
¬
X16
)
→
X40
)
)
)
)
→
X40
→
X37
→
(
¬
X20
)
→
X39
)
)
↔
(
¬
(
(
¬
X20
)
↔
(
(
¬
X16
)
↔
X40
)
)
)
)
↔
(
(
(
¬
X30
)
↔
(
(
¬
X20
)
↔
(
X8
↔
X39
)
)
)
→
(
X51
↔
(
X39
↔
(
X37
→
X8
)
)
)
)
)
)
→
(
(
(
¬
(
X50
↔
X11
)
)
→
(
¬
(
X50
↔
(
X47
↔
(
X40
→
False
)
)
)
)
→
(
(
¬
(
X40
↔
(
X11
↔
X47
)
)
)
↔
(
(
¬
X11
)
→
(
X4
↔
X50
)
)
)
)
↔
(
(
(
¬
X39
)
↔
(
(
¬
X27
)
↔
(
X4
↔
X47
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X40
↔
(
X15
↔
X48
)
)
)
→
X47
→
X27
→
X15
→
X40
)
)
→
(
¬
(
(
¬
X27
)
↔
(
X15
→
X47
)
)
)
)
)
)
)
)
)
)
)
→
(
(
(
(
(
(
¬
(
X38
→
X34
→
(
(
¬
X3
)
↔
X23
)
)
)
↔
(
¬
(
X52
→
X23
→
(
¬
X3
)
→
X34
)
)
)
↔
(
¬
(
(
¬
X38
)
→
(
X34
↔
(
X25
↔
X52
)
)
)
)
)
→
(
X45
↔
(
X38
↔
(
(
¬
X34
)
→
X52
)
)
)
)
→
(
¬
(
(
(
X52
↔
(
(
¬
X42
)
↔
(
X26
→
X49
)
)
)
↔
(
¬
(
X38
→
(
X7
↔
X49
)
)
)
)
→
(
X42
↔
(
(
¬
X38
)
↔
X52
)
)
→
(
¬
(
(
¬
(
(
¬
X34
)
↔
X51
)
)
→
(
¬
(
X51
↔
(
(
¬
X38
)
↔
(
X2
↔
X52
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
X36
→
(
X6
↔
X19
)
)
↔
(
X35
→
(
X19
↔
(
(
¬
X17
)
↔
X36
)
)
)
)
→
(
(
¬
(
X14
→
(
X6
↔
X22
)
)
)
↔
(
(
¬
(
X22
↔
(
X19
→
X6
→
X35
)
)
)
↔
(
X19
→
X14
→
(
(
¬
X1
)
↔
X35
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
X36
→
X28
)
→
(
¬
(
X50
↔
(
(
¬
X29
)
→
(
¬
X28
)
→
X52
)
)
)
)
)
↔
(
(
¬
(
X29
→
X22
→
X50
)
)
↔
(
X50
↔
(
X36
↔
(
X13
↔
X22
)
)
)
)
)
→
(
(
¬
(
X50
→
X36
→
(
X0
↔
X35
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
X38
)
↔
(
X36
→
X42
)
)
→
(
X50
↔
(
X35
↔
(
X31
→
X42
)
)
)
)
)
→
(
X42
↔
(
X38
↔
(
X31
↔
(
X28
→
False
)
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
(
X49
↔
(
X29
↔
(
X14
→
X48
)
)
)
)
→
(
(
(
¬
(
(
X29
↔
X47
)
→
(
(
(
¬
X47
)
↔
(
(
¬
X30
)
↔
X48
)
)
↔
(
(
¬
X30
)
↔
(
(
¬
X14
)
→
(
X11
↔
X48
)
)
)
)
)
)
↔
(
¬
(
X30
↔
(
(
¬
X29
)
→
X48
)
)
)
)
↔
(
X14
→
X30
)
)
)
)
↔
(
(
(
X47
→
X6
)
↔
(
(
(
(
X47
↔
(
X9
→
X6
→
False
)
)
↔
(
(
X49
↔
(
X6
→
X48
)
)
→
(
¬
(
X49
↔
(
X47
↔
(
(
¬
X9
)
→
X48
)
)
)
)
)
)
↔
(
¬
(
(
¬
X47
)
→
X24
→
(
X6
↔
X48
)
)
)
)
→
X49
→
X24
→
X9
→
False
)
)
→
(
¬
(
X49
→
(
(
¬
X24
)
↔
(
(
¬
X2
)
↔
X46
)
)
)
)
→
X24
→
(
X12
↔
(
X2
↔
X48
)
)
)
)
→
(
(
¬
(
(
(
¬
(
(
¬
X8
)
↔
(
X3
↔
X45
)
)
)
→
(
¬
(
X47
→
(
¬
X3
)
→
X33
)
)
)
→
(
(
X44
↔
X48
)
↔
(
(
X35
→
X8
→
(
¬
X3
)
→
X49
)
↔
(
(
¬
X35
)
↔
(
(
¬
X3
)
→
X47
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
(
¬
X38
)
→
X32
→
X20
→
X49
)
)
↔
(
X49
→
(
X48
↔
(
X38
→
False
)
)
)
)
↔
(
(
¬
(
X49
↔
(
X38
→
X20
→
X5
)
)
)
↔
(
¬
(
X49
→
(
X38
↔
(
X16
→
X20
)
)
)
)
)
)
→
(
X48
↔
(
X38
→
X32
)
)
)
)
→
(
¬
(
X49
→
(
X16
↔
(
X10
↔
X47
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
(
(
(
(
¬
X51
)
→
(
¬
X17
)
→
X0
→
X52
)
↔
(
¬
(
(
¬
X51
)
↔
(
X25
→
(
X0
↔
X52
)
)
)
)
)
↔
(
¬
(
X45
→
(
¬
X25
)
→
X0
→
X51
)
)
)
→
(
¬
X42
)
→
(
¬
X25
)
→
X17
→
X45
)
)
↔
(
¬
(
(
(
(
¬
X42
)
↔
(
(
¬
X25
)
↔
X51
)
)
↔
(
¬
(
X51
↔
(
X40
↔
X39
)
)
)
)
→
(
¬
(
X42
↔
(
X40
→
X39
)
)
)
)
)
)
→
(
(
¬
X42
)
→
(
(
¬
X40
)
↔
(
X37
→
X51
)
)
)
→
(
X42
↔
(
X37
→
(
X19
↔
X40
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
X52
→
X36
→
(
X26
↔
(
X22
→
False
)
)
)
→
(
(
X47
→
(
¬
X36
)
→
(
¬
X22
)
→
X42
)
↔
(
¬
(
X36
→
(
(
¬
X31
)
↔
(
X22
↔
X42
)
)
)
)
)
)
)
↔
(
¬
(
X47
↔
(
X36
→
X26
)
)
)
)
→
(
(
¬
(
X52
→
(
(
¬
X36
)
↔
(
(
¬
X26
)
↔
X45
)
)
)
)
↔
(
(
X52
↔
(
X36
↔
(
(
¬
X23
)
→
X45
)
)
)
→
(
X36
↔
(
X23
↔
X27
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
¬
(
X52
↔
(
X51
↔
(
X15
→
X21
)
)
)
)
↔
(
¬
(
X52
↔
(
X28
→
X51
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X43
)
→
(
X15
↔
(
X13
→
X51
)
)
)
)
↔
(
(
X51
↔
(
X43
↔
X15
)
)
↔
(
¬
(
X43
→
(
X21
↔
X4
)
)
)
)
)
→
(
¬
(
X51
→
(
X43
↔
X15
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X51
→
(
X44
↔
X41
)
)
)
→
(
(
¬
X50
)
↔
(
(
¬
X44
)
↔
(
(
¬
X41
)
→
X52
)
)
)
)
)
↔
(
(
¬
(
X51
↔
(
X50
→
(
(
¬
X43
)
↔
X52
)
)
)
)
↔
(
¬
(
X52
↔
(
X50
↔
(
(
¬
X1
)
↔
X51
)
)
)
)
)
)
)
)
↔
(
(
(
X52
↔
(
X51
↔
X50
)
)
↔
(
(
X52
→
X51
→
(
X50
↔
(
X18
→
False
)
)
)
→
(
X50
↔
(
(
¬
X47
)
↔
(
(
¬
X18
)
→
X51
)
)
)
)
)
↔
(
(
¬
X51
)
→
(
¬
X50
)
→
(
(
¬
X7
)
↔
X52
)
)
)
)
↔
(
¬
(
(
¬
X50
)
↔
(
(
¬
X44
)
→
X34
→
X51
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
1425b7...
and proposition id is
fcbdc2...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMRJentna7jRMQ523D5DkcKspKhccPjPY11
)
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
¬
(
(
(
(
(
(
¬
(
(
¬
X41
)
↔
(
(
¬
X34
)
→
X31
→
X45
)
)
)
→
(
¬
X45
)
→
(
(
¬
X41
)
↔
X49
)
)
↔
(
(
¬
(
X49
→
(
X42
↔
X41
)
)
)
→
(
X49
↔
(
X30
↔
X42
)
)
→
(
(
¬
(
X49
↔
(
X41
↔
(
X31
↔
X30
)
)
)
)
↔
(
¬
(
X49
→
(
(
¬
X31
)
↔
(
(
¬
X30
)
↔
X42
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
X45
)
↔
(
X41
→
X39
→
X49
)
)
)
)
↔
(
(
(
¬
(
X45
→
X42
→
(
X38
↔
X24
)
)
)
→
(
¬
(
(
¬
(
X48
↔
(
(
¬
X15
)
↔
X42
)
)
)
→
X48
→
X45
→
(
X38
↔
X42
)
)
)
)
↔
(
¬
(
(
¬
X48
)
↔
(
(
¬
X45
)
↔
(
X42
→
X49
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
(
¬
(
(
¬
X29
)
→
(
(
¬
X11
)
↔
X48
)
)
)
↔
(
¬
(
X49
→
(
X42
↔
(
X11
↔
X48
)
)
)
)
)
→
(
X42
↔
(
(
¬
X33
)
→
X48
)
)
)
)
→
(
¬
X11
)
→
X6
→
X49
)
)
↔
(
X49
↔
(
(
(
¬
X42
)
→
(
(
¬
X37
)
↔
(
(
¬
X19
)
→
X49
)
)
)
→
(
X37
↔
(
(
¬
X19
)
→
X48
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
(
X43
↔
(
(
¬
X2
)
↔
X37
)
)
→
(
X43
↔
(
X41
↔
(
X2
↔
X47
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
(
¬
X43
)
↔
(
(
¬
X16
)
↔
(
(
¬
X1
)
→
X47
)
)
)
)
→
(
¬
X43
)
→
(
¬
X41
)
→
(
X16
↔
X47
)
)
)
→
(
¬
(
X47
↔
(
X41
↔
X43
)
)
)
)
)
↔
(
X49
↔
(
X16
→
X1
)
)
)
)
↔
(
(
X43
↔
(
(
¬
X20
)
↔
(
X2
→
X49
)
)
)
↔
(
X3
↔
(
(
¬
X2
)
→
X49
)
)
)
)
→
(
(
¬
(
(
(
(
(
¬
X35
)
↔
(
X25
↔
(
X5
→
X37
)
)
)
→
(
¬
(
(
¬
X37
)
→
(
X35
↔
(
(
¬
X25
)
↔
X43
)
)
)
)
)
↔
(
X43
→
(
X37
↔
X35
)
)
)
→
X37
→
False
)
)
↔
(
(
(
X37
↔
(
X35
→
(
(
¬
X18
)
↔
X49
)
)
)
↔
(
(
¬
(
(
¬
X37
)
→
(
(
¬
X35
)
↔
X43
)
)
)
↔
(
X43
→
X37
→
X9
→
X49
)
)
)
↔
(
X49
→
(
¬
X18
)
→
X37
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
X44
↔
(
(
¬
X40
)
→
X36
→
X47
)
)
)
↔
(
X23
→
X12
→
(
(
¬
X8
)
↔
X47
)
)
)
→
(
¬
(
X40
↔
(
X12
↔
(
(
¬
X8
)
→
X14
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
(
¬
X32
)
↔
X44
)
↔
(
¬
(
X46
↔
(
X44
↔
(
(
¬
X40
)
→
X47
)
)
)
)
)
→
X46
→
(
X32
↔
X44
)
)
)
→
(
(
(
¬
X32
)
→
(
X26
↔
X47
)
)
↔
(
(
X47
↔
X26
)
→
(
(
¬
(
(
¬
X44
)
→
X32
→
X45
)
)
→
(
¬
(
X47
↔
(
(
¬
X44
)
→
(
¬
X26
)
→
X45
)
)
)
)
→
(
¬
(
(
¬
X32
)
→
(
X26
↔
X47
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
X40
↔
(
X36
→
(
X28
↔
X46
)
)
)
)
→
(
¬
(
X46
→
(
X36
↔
(
(
¬
X17
)
→
X40
)
)
)
)
)
→
(
¬
(
X36
→
X46
)
)
)
)
↔
(
¬
(
(
¬
(
X28
→
(
¬
X13
)
→
X0
→
X46
)
)
→
X13
→
(
¬
X0
)
→
X36
)
)
)
→
(
¬
(
(
(
¬
(
(
(
X10
↔
(
(
¬
X4
)
↔
X27
)
)
→
(
¬
(
(
¬
X40
)
↔
(
X27
→
X46
)
)
)
)
→
X22
→
X10
→
(
(
¬
X4
)
↔
X40
)
)
)
↔
(
(
¬
X27
)
↔
(
X22
↔
(
X7
↔
X46
)
)
)
)
→
(
¬
(
(
(
¬
(
X27
→
(
X21
↔
False
)
)
)
↔
(
(
¬
(
X27
↔
(
(
¬
X21
)
↔
X40
)
)
)
↔
(
¬
(
(
X36
↔
(
(
¬
X27
)
↔
X43
)
)
→
(
¬
(
X43
↔
(
X40
→
(
X21
↔
False
)
)
)
)
)
)
)
)
→
(
(
¬
X43
)
↔
(
X40
→
X27
→
X46
)
)
)
)
→
(
¬
(
X36
→
(
X21
↔
X46
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
(
¬
X44
)
↔
(
X41
→
(
¬
X33
)
→
X47
)
)
)
↔
(
¬
(
(
¬
X46
)
↔
(
(
¬
X4
)
↔
X47
)
)
)
)
↔
(
¬
(
(
(
(
¬
X46
)
→
(
X41
↔
(
(
¬
X26
)
↔
X47
)
)
)
↔
(
¬
(
(
X49
→
(
X47
↔
(
X41
↔
X45
)
)
)
→
(
¬
X11
)
)
)
)
→
(
(
X39
→
(
(
¬
X13
)
↔
(
(
¬
X1
)
↔
X49
)
)
)
↔
(
(
¬
(
X39
↔
(
(
¬
X29
)
→
X19
→
X46
)
)
)
→
(
(
¬
(
X41
↔
(
X39
→
(
X29
↔
X1
)
)
)
)
↔
(
(
¬
X46
)
↔
(
X41
→
(
(
¬
X29
)
↔
X49
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X46
↔
(
X17
↔
X20
)
)
)
→
(
(
X17
→
X9
→
X42
)
↔
(
(
¬
(
(
¬
X9
)
→
X49
)
)
→
(
(
¬
(
X49
↔
(
X46
↔
False
)
)
)
↔
(
¬
(
X46
↔
(
X17
↔
(
(
¬
X9
)
↔
X20
)
)
)
)
)
)
)
)
↔
(
(
¬
(
X18
↔
(
X9
↔
X42
)
)
)
↔
(
(
¬
(
(
(
¬
X36
)
↔
(
(
¬
X22
)
→
X42
)
)
→
(
¬
(
X2
→
X0
→
X42
)
)
)
)
↔
(
(
(
¬
X20
)
↔
(
X0
→
X36
)
)
↔
(
¬
(
X38
↔
(
(
¬
X36
)
→
(
(
¬
X20
)
↔
X42
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
(
¬
X46
)
→
(
(
¬
X38
)
↔
(
X10
→
X47
)
)
)
)
→
(
(
¬
(
X47
↔
(
(
¬
X24
)
→
(
¬
X21
)
→
X38
)
)
)
↔
(
(
¬
(
X47
→
X46
→
(
X21
↔
X23
)
)
)
↔
(
X47
↔
(
(
¬
X38
)
→
(
(
¬
X23
)
↔
X46
)
)
)
)
)
)
)
→
(
(
¬
X46
)
↔
(
X21
→
X47
)
)
)
→
(
(
(
(
X48
→
(
¬
X42
)
→
(
X38
↔
X46
)
)
↔
(
X5
→
False
)
)
↔
(
¬
(
X38
↔
(
(
¬
X24
)
↔
(
(
¬
X5
)
→
X42
)
)
)
)
)
↔
(
¬
(
(
¬
X42
)
↔
(
X5
→
X46
)
)
)
)
)
)
)
)
→
(
(
(
(
X43
↔
(
(
¬
X37
)
↔
(
X31
↔
X47
)
)
)
↔
(
X46
↔
(
X43
↔
(
(
¬
X25
)
↔
X47
)
)
)
)
↔
(
(
X43
↔
(
X32
→
(
(
¬
X3
)
↔
X47
)
)
)
↔
X37
)
)
→
(
(
(
¬
(
X32
→
(
X16
↔
X47
)
)
)
↔
(
¬
(
(
¬
(
X15
↔
(
X7
→
X35
)
)
)
→
(
(
¬
(
X47
↔
(
X35
→
X15
→
X7
)
)
)
↔
(
X35
→
(
¬
X32
)
→
(
X15
↔
X47
)
)
)
)
)
)
↔
(
(
¬
(
X47
↔
(
X45
→
(
X35
↔
X46
)
)
)
)
↔
(
(
X47
↔
(
X35
↔
(
X15
↔
(
X14
↔
False
)
)
)
)
↔
(
¬
(
(
(
¬
X35
)
↔
(
(
¬
X15
)
↔
X47
)
)
→
(
X46
↔
(
X8
↔
X15
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
X47
→
X45
→
X27
→
X46
)
↔
(
(
X46
↔
(
(
¬
X45
)
↔
(
(
¬
X30
)
↔
X47
)
)
)
↔
(
(
¬
X45
)
↔
(
X40
→
(
(
¬
X30
)
↔
X46
)
)
)
)
)
→
(
¬
(
X46
↔
(
X45
↔
(
(
¬
X6
)
↔
X28
)
)
)
)
→
(
(
¬
(
(
¬
(
X47
↔
(
X46
↔
(
X12
↔
X34
)
)
)
)
→
(
¬
X12
)
→
(
(
¬
X6
)
↔
X46
)
)
)
↔
(
¬
(
(
¬
X12
)
↔
X34
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
5fba09...
and proposition id is
c920ac...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMUDq3DwYncuJ4UFY2JDfsggfz1Ux15LmPW
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
¬
(
(
(
(
(
¬
(
X49
↔
(
X46
→
(
(
¬
X1
)
↔
X48
)
)
)
)
↔
(
(
(
¬
(
(
¬
X38
)
→
X49
)
)
→
(
X49
→
(
X46
↔
(
X13
→
X48
)
)
)
→
(
¬
(
(
X46
→
X38
)
→
(
¬
(
(
¬
X46
)
→
(
(
¬
X10
)
↔
X48
)
)
)
→
(
¬
(
(
¬
X46
)
→
(
¬
X38
)
→
(
(
¬
X10
)
↔
X49
)
)
)
)
)
→
(
¬
(
X48
→
(
X38
↔
X46
)
)
)
)
→
(
¬
(
X49
→
(
X46
↔
X48
)
)
)
)
)
↔
(
(
¬
(
X48
↔
(
X47
→
X49
)
)
)
↔
(
(
¬
(
(
¬
X0
)
↔
X47
)
)
→
(
(
¬
(
X49
→
(
X38
↔
(
X0
↔
X47
)
)
)
)
↔
(
(
¬
X47
)
↔
(
(
¬
X38
)
→
(
X0
↔
X48
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
X35
)
↔
(
(
¬
X32
)
→
(
¬
X3
)
→
X49
)
)
→
(
¬
(
(
¬
(
X48
→
(
(
¬
X38
)
↔
(
(
¬
X3
)
→
X49
)
)
)
)
→
(
(
(
¬
(
X38
→
(
¬
X32
)
→
(
¬
X4
)
→
X49
)
)
↔
(
X38
↔
(
X32
→
X4
→
X48
)
)
)
↔
(
¬
(
(
¬
X38
)
↔
(
(
¬
X32
)
↔
X48
)
)
)
)
)
)
)
)
)
→
(
(
(
X48
↔
(
X25
↔
(
X20
↔
X2
)
)
)
↔
(
(
(
X46
→
(
¬
X43
)
→
(
¬
X25
)
→
X48
)
↔
(
(
¬
(
(
¬
X43
)
↔
(
(
¬
X25
)
↔
(
(
¬
X5
)
↔
X48
)
)
)
)
→
X43
→
(
¬
X5
)
→
X23
)
)
↔
(
X48
→
(
X46
↔
(
X2
↔
X23
)
)
)
)
)
↔
(
(
(
X43
→
(
X27
↔
X47
)
)
↔
(
¬
(
X43
→
(
¬
X30
)
→
X27
→
X46
)
)
)
→
(
(
(
X47
→
(
X6
↔
False
)
)
↔
(
¬
(
(
¬
(
X30
→
(
X8
↔
X6
)
)
)
→
(
X47
↔
(
X8
↔
(
X6
↔
X46
)
)
)
)
)
)
↔
(
(
X46
↔
(
X12
→
X6
→
X48
)
)
→
(
¬
(
X46
↔
X47
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
(
¬
(
(
(
¬
X44
)
↔
(
(
¬
X22
)
→
X47
)
)
→
(
X31
↔
(
X19
↔
X22
)
)
)
)
→
(
¬
(
(
¬
(
(
(
(
¬
X42
)
↔
(
(
¬
X34
)
↔
(
X19
→
X44
)
)
)
→
(
¬
X34
)
→
(
X31
↔
(
X19
↔
X42
)
)
)
→
X34
→
X42
)
)
→
(
¬
(
X44
→
(
X42
↔
(
X34
→
False
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X44
↔
(
X22
→
X9
→
X49
)
)
)
→
(
¬
(
X31
↔
(
X22
↔
(
X9
↔
X44
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X44
)
→
(
¬
X42
)
→
X40
→
X49
)
)
↔
(
¬
(
(
¬
X44
)
→
(
¬
X42
)
→
(
X36
↔
X49
)
)
)
)
→
(
¬
(
X16
↔
X14
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X49
↔
(
(
¬
X39
)
↔
(
X11
→
X40
)
)
)
)
→
(
(
¬
(
(
¬
X40
)
→
X34
→
(
X7
↔
X49
)
)
)
↔
(
(
¬
(
X39
↔
(
(
¬
X28
)
→
(
¬
X7
)
→
X40
)
)
)
→
X40
→
X49
)
)
)
→
(
(
X34
→
(
¬
X24
)
→
X37
)
↔
(
(
X39
→
(
(
¬
X34
)
↔
X37
)
)
↔
(
(
(
¬
X34
)
→
(
X28
↔
(
X24
→
X39
)
)
)
→
(
¬
(
(
¬
X34
)
→
(
X28
↔
(
X24
→
X37
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
(
¬
X48
)
→
(
(
¬
X45
)
↔
(
X44
→
X49
)
)
)
→
(
(
¬
(
(
¬
X45
)
↔
(
(
¬
X33
)
→
(
X17
↔
X49
)
)
)
)
↔
(
X49
→
(
(
¬
X45
)
↔
(
X33
→
X48
)
)
)
)
)
)
→
(
¬
(
(
¬
X45
)
↔
(
(
¬
X17
)
→
X49
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
X47
↔
(
X29
→
(
(
¬
X17
)
↔
X45
)
)
)
→
(
¬
(
X47
↔
(
X33
→
X29
→
X45
)
)
)
)
)
→
(
X45
↔
(
X29
→
X33
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
X49
→
(
(
¬
X41
)
↔
X48
)
)
)
→
(
X48
↔
(
X47
↔
(
(
¬
X41
)
→
X49
)
)
)
→
(
X48
↔
(
X47
→
(
(
¬
X18
)
↔
X49
)
)
)
)
)
↔
(
¬
(
X49
→
(
(
¬
X18
)
↔
(
X15
→
X48
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
X48
↔
(
X47
→
X26
→
X21
→
False
)
)
→
(
X47
↔
(
(
¬
X18
)
↔
X48
)
)
)
)
→
(
¬
(
(
¬
X18
)
→
X48
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
(
(
¬
(
X48
→
(
X46
↔
(
X26
→
False
)
)
)
)
→
(
(
¬
X42
)
↔
(
(
¬
X26
)
↔
(
(
¬
X15
)
→
X46
)
)
)
)
↔
(
¬
(
(
¬
(
X26
→
(
(
¬
X11
)
↔
X37
)
)
)
→
(
¬
(
X48
↔
(
X26
→
X15
→
(
X11
↔
False
)
)
)
)
)
)
)
↔
(
(
X49
→
X47
→
X46
)
↔
(
X48
→
X26
→
X46
)
)
)
↔
(
(
(
¬
(
(
X49
↔
(
X26
↔
False
)
)
→
(
¬
(
X26
↔
(
X7
→
X5
→
X37
)
)
)
→
(
¬
(
X49
→
(
(
¬
X26
)
↔
(
(
¬
X7
)
↔
X37
)
)
)
)
)
)
↔
(
X49
↔
(
X7
↔
X5
)
)
)
→
X26
→
(
¬
X7
)
→
X5
→
X36
)
)
→
(
(
¬
(
(
¬
(
(
¬
(
X47
↔
(
(
¬
X17
)
→
(
X1
↔
X29
)
)
)
)
→
(
X47
↔
(
X29
↔
(
X17
→
X4
)
)
)
)
)
→
(
¬
(
X49
↔
(
X29
→
X17
→
X47
)
)
)
)
)
↔
(
¬
(
X4
→
X1
→
X17
)
)
)
→
(
(
X47
↔
(
X29
↔
(
X14
→
X49
)
)
)
↔
(
¬
(
X49
↔
(
X46
↔
(
X14
↔
X47
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X47
→
X41
→
(
X21
↔
False
)
)
)
→
(
¬
(
X47
↔
(
X41
↔
(
X31
→
X33
)
)
)
)
)
)
→
(
(
(
X49
→
(
(
¬
X22
)
↔
(
X16
→
X41
)
)
)
↔
(
(
¬
(
X22
→
X16
)
)
↔
(
(
¬
(
X47
↔
(
(
¬
X33
)
→
(
(
¬
X22
)
↔
X49
)
)
)
)
→
(
¬
(
(
X49
→
X22
→
X33
)
→
(
(
(
¬
X22
)
→
(
(
¬
X16
)
↔
X46
)
)
↔
(
¬
(
X49
→
(
X46
↔
(
X16
→
X33
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
X49
↔
(
X46
→
(
X41
↔
(
X27
↔
False
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
X47
→
(
X34
↔
(
X28
→
X24
→
False
)
)
)
↔
(
X28
→
X23
→
X47
)
)
→
(
¬
(
X47
→
(
X43
↔
(
X34
↔
X23
)
)
)
)
)
↔
(
(
X47
↔
(
X40
→
X13
→
X43
)
)
→
(
(
(
¬
X43
)
→
X40
→
X47
)
↔
(
(
¬
(
X47
→
(
X40
↔
(
(
¬
X28
)
↔
X43
)
)
)
)
→
(
¬
(
X43
→
X18
→
False
)
)
)
)
)
)
→
(
(
(
¬
(
(
(
¬
(
X46
→
X12
→
X3
→
X0
)
)
↔
(
X34
↔
X46
)
)
→
(
X46
↔
(
X32
→
(
X12
↔
X34
)
)
)
)
)
→
X3
→
X0
→
X48
)
↔
(
(
¬
(
X34
→
(
¬
X10
)
→
(
(
¬
X3
)
↔
X19
)
)
)
→
(
¬
(
X40
↔
(
X34
↔
(
(
¬
X10
)
↔
X28
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
(
X46
↔
(
X44
→
(
X8
↔
X35
)
)
)
)
↔
(
¬
(
(
¬
X44
)
→
(
(
¬
X35
)
↔
(
X25
→
X46
)
)
)
)
)
→
(
¬
(
X35
→
X20
→
False
)
)
)
↔
(
(
¬
(
X45
→
X44
→
X8
→
X47
)
)
↔
(
X45
→
(
(
¬
X20
)
↔
(
(
¬
X8
)
→
X47
)
)
)
)
)
→
(
(
X48
↔
(
X44
→
(
X6
↔
X45
)
)
)
↔
(
(
¬
(
(
¬
X47
)
↔
(
X39
↔
X48
)
)
)
→
(
(
¬
(
X48
→
X45
→
X39
)
)
↔
(
X47
→
(
¬
X45
)
→
(
X44
↔
X48
)
)
)
)
)
)
)
→
(
(
(
(
X39
↔
(
(
¬
X2
)
↔
X38
)
)
↔
(
X30
↔
(
X9
↔
X39
)
)
)
↔
(
X45
↔
(
X38
→
(
¬
X2
)
→
X30
)
)
)
↔
(
X44
→
(
X38
↔
X30
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
3d8477...
and proposition id is
4d752e...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMZL4LPCSsint8PUjHLVi8c5SdgyhY4miep
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
∀X51 :
prop
,
∀X52 :
prop
,
(
(
(
(
(
(
¬
(
(
¬
X43
)
↔
(
X24
↔
(
(
¬
X21
)
↔
X51
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X42
)
↔
(
X39
→
(
(
¬
X36
)
↔
X51
)
)
)
)
→
(
(
¬
(
X52
↔
(
(
¬
X39
)
→
(
X24
↔
X51
)
)
)
)
↔
(
(
(
¬
(
X51
→
X36
→
(
(
¬
X21
)
↔
X24
)
)
)
→
X51
→
(
X36
↔
(
(
¬
X21
)
↔
X52
)
)
)
↔
(
X52
↔
(
X21
↔
X36
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X52
→
X3
→
X32
)
)
→
(
(
(
¬
X32
)
→
(
(
¬
X17
)
↔
(
(
¬
X3
)
→
X47
)
)
)
↔
(
¬
(
X51
↔
(
X17
↔
(
(
¬
X3
)
→
X47
)
)
)
)
)
→
(
(
¬
(
(
(
X52
↔
(
(
¬
X17
)
→
(
¬
X1
)
→
X32
)
)
→
(
¬
X32
)
→
(
(
¬
X17
)
↔
X47
)
)
→
(
¬
(
X52
↔
(
(
¬
X17
)
↔
X32
)
)
)
)
)
↔
(
¬
(
X17
↔
X1
)
)
)
)
)
→
(
(
X52
→
(
X34
↔
(
X22
→
(
X19
↔
False
)
)
)
)
↔
(
(
(
(
X42
↔
False
)
↔
(
¬
(
(
(
X47
→
X42
→
(
X22
↔
(
X14
↔
False
)
)
)
↔
(
X51
↔
(
(
¬
X22
)
↔
(
X14
↔
X47
)
)
)
)
→
(
X47
↔
False
)
)
)
)
↔
(
X42
→
(
X14
↔
X51
)
)
)
↔
(
X52
→
(
X51
↔
(
X42
↔
X14
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X41
→
(
X25
↔
(
(
¬
X13
)
→
X46
)
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X41
)
→
(
¬
X13
)
→
X48
)
)
)
)
↔
(
¬
(
(
(
¬
X46
)
↔
(
(
¬
X41
)
↔
(
X4
↔
X48
)
)
)
→
(
¬
(
X46
↔
(
X41
↔
(
(
¬
X4
)
↔
X6
)
)
)
)
)
)
)
↔
(
(
X15
↔
(
(
¬
X13
)
→
(
¬
X7
)
→
X25
)
)
→
(
(
¬
X15
)
↔
(
(
¬
X4
)
→
X35
)
)
)
)
→
(
¬
(
(
(
¬
X42
)
→
X35
→
X18
→
X44
)
→
(
¬
(
X42
↔
(
X35
→
(
¬
X16
)
→
X52
)
)
)
)
)
→
(
(
¬
(
X47
↔
(
(
¬
X44
)
→
(
(
¬
X18
)
↔
X48
)
)
)
)
↔
(
(
(
(
¬
X47
)
↔
(
(
¬
X41
)
↔
(
(
¬
X27
)
↔
X48
)
)
)
→
(
(
¬
X47
)
↔
(
(
¬
X44
)
↔
X48
)
)
)
→
X48
→
(
X44
↔
(
X16
↔
X47
)
)
)
)
)
)
↔
(
(
(
(
(
(
X12
→
X33
)
↔
(
(
¬
(
(
X44
↔
(
X33
→
(
X12
↔
False
)
)
)
→
X30
→
(
(
¬
X23
)
↔
X33
)
)
)
↔
(
X33
↔
(
X12
↔
False
)
)
)
)
↔
(
(
(
(
¬
(
X45
↔
(
X30
↔
(
(
¬
X10
)
↔
X52
)
)
)
)
→
(
X45
↔
(
X44
→
(
X10
↔
False
)
)
)
)
↔
(
¬
(
X52
↔
(
X30
→
(
¬
X10
)
→
X50
)
)
)
)
↔
(
¬
(
X50
↔
(
X30
→
(
X10
↔
X33
)
)
)
)
)
)
↔
(
(
¬
(
(
X50
↔
(
(
¬
X9
)
↔
X30
)
)
→
(
¬
(
X50
→
(
¬
X9
)
→
X45
)
)
→
(
X9
↔
(
X8
→
X30
)
)
)
)
→
(
¬
(
(
¬
X45
)
→
(
¬
X30
)
→
(
¬
X9
)
→
X46
)
)
)
)
↔
(
(
(
(
¬
X26
)
→
X52
)
→
(
(
¬
(
X45
→
(
X26
↔
(
(
¬
X11
)
↔
X38
)
)
)
)
↔
(
¬
(
(
¬
X49
)
→
(
(
¬
X45
)
↔
(
X38
↔
X52
)
)
)
)
)
)
↔
(
(
(
X45
↔
(
X33
↔
(
X26
→
X0
)
)
)
→
(
¬
(
X45
→
(
X33
↔
X52
)
)
)
)
↔
(
X52
↔
(
X20
→
X0
)
)
)
)
)
↔
(
(
(
(
X49
↔
(
(
¬
X40
)
→
(
X29
↔
X50
)
)
)
→
X50
→
(
X49
↔
(
X28
↔
False
)
)
)
↔
(
¬
(
(
(
¬
X29
)
↔
(
X28
↔
X37
)
)
→
(
(
X37
↔
(
(
¬
X29
)
↔
(
X5
→
X31
)
)
)
↔
(
(
¬
(
X37
→
(
¬
X31
)
→
(
X29
↔
X49
)
)
)
→
(
¬
(
X37
↔
(
(
¬
X28
)
↔
X49
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
X49
)
↔
(
(
¬
X29
)
↔
(
X2
→
X52
)
)
)
)
↔
(
(
¬
X49
)
→
(
X40
↔
(
(
¬
X29
)
→
X52
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
X50
→
X24
→
(
(
¬
X20
)
↔
X41
)
)
→
(
(
X43
→
(
(
¬
X41
)
↔
(
(
¬
X28
)
↔
X50
)
)
)
→
(
¬
(
(
X43
↔
(
(
¬
X24
)
↔
X28
)
)
→
(
¬
(
X43
↔
(
X28
→
X17
→
X50
)
)
)
→
(
¬
(
X24
↔
False
)
)
)
)
)
→
(
¬
(
X50
→
(
X43
↔
(
X41
↔
False
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X43
↔
(
(
¬
X15
)
↔
X21
)
)
)
→
X43
→
X24
→
X50
)
↔
(
X43
↔
(
X24
↔
(
X21
↔
False
)
)
)
)
↔
(
¬
(
(
¬
X21
)
→
X15
→
(
(
¬
X11
)
↔
X50
)
)
)
)
↔
(
(
¬
X24
)
→
X21
→
X2
→
X50
)
)
)
→
(
(
(
(
X52
→
X35
)
↔
(
¬
(
(
(
¬
X41
)
↔
(
X22
↔
(
(
¬
X12
)
↔
X50
)
)
)
→
(
X52
↔
(
X50
↔
(
X41
↔
(
X22
→
False
)
)
)
)
)
)
)
↔
(
(
X52
→
(
X43
↔
X35
)
)
↔
(
(
(
¬
X43
)
→
(
¬
X35
)
→
(
¬
X4
)
→
X52
)
↔
(
(
¬
(
X52
↔
(
X43
→
X41
→
False
)
)
)
↔
(
¬
(
X52
→
X41
→
(
X8
↔
X35
)
)
)
)
)
)
)
↔
(
¬
(
X50
→
X48
→
(
X43
↔
X35
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
¬
X34
)
→
X25
→
(
¬
X10
)
→
X37
)
↔
(
¬
(
(
¬
X34
)
→
(
(
¬
X10
)
↔
(
(
¬
X5
)
→
X36
)
)
)
)
)
→
(
¬
(
(
¬
X34
)
→
(
¬
(
X48
↔
(
X37
↔
(
X36
→
X52
)
)
)
)
)
)
→
(
¬
(
(
¬
X37
)
→
X36
→
X48
)
)
)
)
→
(
(
X48
↔
(
(
¬
X34
)
↔
(
(
¬
X30
)
↔
X37
)
)
)
↔
(
¬
(
X48
→
(
X45
↔
(
X37
→
(
X30
↔
False
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
(
¬
X1
)
→
X52
)
)
→
(
X52
→
X48
→
X18
)
→
(
¬
(
(
X48
→
X9
→
(
¬
X1
)
→
X52
)
→
X18
→
X52
)
)
)
→
(
¬
X48
)
→
(
(
¬
X9
)
↔
X52
)
)
→
(
¬
(
(
¬
(
X52
↔
(
X48
↔
(
(
¬
X9
)
↔
X14
)
)
)
)
→
(
¬
(
X14
↔
X42
)
)
)
)
)
)
→
(
(
X50
↔
(
(
¬
X48
)
↔
(
X18
→
X52
)
)
)
↔
(
(
¬
(
X52
→
X38
→
(
X36
↔
X23
)
)
)
↔
(
¬
(
(
X52
→
(
X50
↔
(
X38
→
X36
→
False
)
)
)
→
(
¬
(
X48
→
X38
→
False
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
X33
↔
(
X19
↔
X7
)
)
→
(
¬
(
(
X46
↔
(
X45
↔
(
(
¬
X27
)
↔
X33
)
)
)
→
(
(
(
(
¬
X33
)
→
(
(
¬
X27
)
↔
(
(
¬
X7
)
↔
X45
)
)
)
→
(
X27
↔
(
X7
→
(
(
¬
X3
)
↔
X45
)
)
)
)
↔
(
X51
→
X45
→
(
X33
↔
X27
)
)
)
)
)
)
)
↔
(
(
(
X45
→
(
X33
↔
X6
)
)
↔
(
¬
(
(
¬
(
(
X49
↔
(
(
¬
X33
)
↔
X51
)
)
→
(
¬
(
X48
↔
(
X47
→
X33
)
)
)
)
)
→
(
(
¬
(
X49
→
(
(
¬
X29
)
↔
X48
)
)
)
→
(
¬
(
(
¬
X45
)
→
(
X6
↔
X48
)
)
)
→
(
¬
(
X48
↔
(
X47
↔
X29
)
)
)
)
→
(
X48
↔
(
(
¬
X6
)
↔
X49
)
)
)
)
)
↔
(
X45
↔
(
(
¬
X39
)
↔
(
(
¬
X33
)
→
X51
)
)
)
)
)
↔
(
(
(
(
¬
(
X50
↔
(
(
¬
X48
)
↔
(
(
¬
X40
)
→
X49
)
)
)
)
↔
(
(
¬
(
(
X44
↔
(
(
¬
(
(
¬
X48
)
↔
(
X26
↔
X49
)
)
)
↔
(
X50
→
X48
→
X44
→
(
X26
↔
False
)
)
)
)
→
(
¬
(
X44
→
(
(
¬
X32
)
↔
(
(
¬
X26
)
→
X48
)
)
)
)
)
)
↔
(
(
¬
X48
)
↔
(
(
¬
X44
)
↔
X49
)
)
)
)
↔
(
¬
(
X47
↔
(
X44
↔
(
X40
→
X50
)
)
)
)
)
↔
(
¬
(
(
X31
↔
(
(
¬
X16
)
↔
(
X0
↔
X44
)
)
)
→
(
¬
(
(
(
X50
→
(
X47
↔
X31
)
)
→
(
X47
↔
(
(
¬
X44
)
→
(
¬
X16
)
→
X50
)
)
)
→
X50
→
(
(
¬
X16
)
↔
(
(
¬
X13
)
→
X31
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
745e1a...
and proposition id is
b49b5c...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMayBakyAWLWVmdsuFBGgBSf4NuBNNp7Cse
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
∀X50 :
prop
,
(
¬
∀X51 :
prop
,
(
(
(
(
¬
(
(
¬
(
(
(
(
(
¬
(
X31
→
X30
→
X48
)
)
↔
(
¬
(
(
¬
X49
)
↔
(
(
¬
X48
)
↔
(
(
¬
X31
)
↔
X50
)
)
)
)
)
↔
(
(
¬
X49
)
→
X48
→
(
X47
↔
X50
)
)
)
→
(
¬
(
X51
↔
(
X49
→
X47
→
X31
→
False
)
)
)
)
→
(
(
(
¬
X50
)
↔
(
X49
↔
X51
)
)
↔
(
(
¬
(
X49
↔
(
(
¬
X45
)
→
X51
)
)
)
↔
(
(
(
¬
X50
)
→
(
¬
X49
)
→
(
¬
X45
)
→
X51
)
↔
(
(
¬
X50
)
↔
(
X49
↔
(
X41
↔
X51
)
)
)
)
)
)
)
)
→
(
¬
(
(
X47
→
(
X39
↔
(
X32
↔
X48
)
)
)
→
(
(
¬
(
(
¬
(
X48
↔
(
X12
→
(
(
¬
X11
)
↔
X51
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X2
)
↔
X51
)
)
→
(
¬
X14
)
→
(
(
¬
X12
)
↔
(
X2
→
X48
)
)
)
→
(
¬
(
X14
→
(
(
¬
X12
)
↔
X48
)
)
)
)
)
)
)
↔
(
(
¬
(
X14
↔
(
X2
→
X48
)
)
)
↔
(
¬
(
X47
↔
(
X39
→
(
X14
↔
(
X2
↔
False
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X22
→
X47
)
)
↔
(
(
¬
(
(
¬
X22
)
↔
X51
)
)
↔
(
(
(
¬
(
X50
↔
(
X33
→
(
X24
↔
X22
)
)
)
)
→
(
(
¬
X50
)
↔
(
(
¬
X33
)
→
(
¬
X22
)
→
X51
)
)
)
↔
(
(
¬
X50
)
→
(
X22
↔
X51
)
)
)
)
)
→
(
(
X47
↔
(
X27
→
False
)
)
→
(
(
¬
(
X51
→
(
X27
↔
X47
)
)
)
↔
(
X51
→
(
X50
↔
(
(
¬
X27
)
↔
X36
)
)
)
)
)
→
(
¬
(
X51
→
(
X47
↔
(
X27
→
False
)
)
)
)
)
↔
(
(
(
¬
(
X23
↔
(
(
¬
X20
)
↔
(
(
¬
X4
)
↔
X29
)
)
)
)
→
(
¬
(
X29
↔
(
(
¬
X28
)
↔
(
X20
↔
X51
)
)
)
)
)
↔
(
(
(
X51
↔
(
X50
→
X25
→
X38
)
)
→
(
¬
(
(
¬
X44
)
→
X50
)
)
)
↔
(
(
¬
(
X29
↔
(
X28
↔
X33
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X38
→
X29
→
(
X18
↔
X28
)
)
)
→
(
¬
(
X51
→
(
X38
↔
(
X29
→
False
)
)
)
)
)
)
→
(
¬
(
X33
→
(
X28
↔
(
X18
↔
False
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
X46
↔
(
X42
↔
(
(
¬
X7
)
↔
X15
)
)
)
)
→
(
(
¬
(
(
¬
X42
)
↔
(
X26
↔
X46
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X47
)
↔
(
(
¬
X42
)
→
(
¬
X26
)
→
X50
)
)
)
→
X50
→
X46
→
(
(
¬
X42
)
↔
X47
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X19
)
→
X10
→
X42
)
)
→
(
(
¬
X42
)
↔
(
(
¬
X10
)
↔
(
(
¬
X1
)
→
X49
)
)
)
)
→
(
¬
(
X46
→
X43
)
)
)
)
)
→
(
(
(
(
(
X42
→
(
(
¬
X21
)
↔
(
X17
↔
X35
)
)
)
↔
(
(
¬
(
X42
↔
(
(
¬
X21
)
↔
(
(
¬
X5
)
→
X35
)
)
)
)
→
(
(
¬
X34
)
↔
(
X5
→
X35
)
)
)
)
↔
(
¬
(
(
(
(
(
X34
↔
(
(
¬
X21
)
↔
(
X6
↔
X40
)
)
)
→
(
¬
(
X49
→
(
X40
↔
(
(
¬
X6
)
↔
X34
)
)
)
)
)
↔
(
(
¬
X40
)
→
(
(
¬
X35
)
↔
X49
)
)
)
→
(
¬
(
X35
→
(
X21
↔
(
X6
↔
False
)
)
)
)
)
→
(
X49
↔
(
X34
→
(
¬
X21
)
→
X50
)
)
)
)
)
↔
(
(
X35
→
(
(
¬
X16
)
↔
X50
)
)
→
(
¬
(
X21
→
X37
)
)
)
)
↔
(
(
(
¬
(
X37
↔
(
X8
↔
(
X3
↔
X40
)
)
)
)
↔
(
¬
(
X13
→
(
¬
X8
)
→
X40
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
X8
↔
(
(
¬
X3
)
↔
X13
)
)
)
→
(
X9
→
(
(
¬
X0
)
↔
X13
)
)
→
X8
→
(
(
¬
X3
)
↔
(
(
¬
X0
)
↔
X9
)
)
)
→
(
¬
(
(
¬
X8
)
↔
(
X3
↔
X9
)
)
)
)
→
(
X9
↔
(
(
¬
X8
)
↔
X13
)
)
)
)
↔
(
¬
(
(
¬
X13
)
↔
(
(
¬
X3
)
↔
(
(
¬
X0
)
→
X49
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
(
(
¬
(
(
X48
↔
(
X8
→
X10
)
)
→
(
(
X49
↔
(
(
¬
X10
)
↔
(
X8
↔
X48
)
)
)
↔
(
(
¬
X10
)
↔
(
X4
↔
X49
)
)
)
)
)
↔
(
X8
↔
X10
)
)
→
(
¬
X4
)
→
X48
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X49
→
(
X13
↔
(
X10
→
X8
)
)
)
)
→
(
¬
(
X51
↔
(
(
¬
X10
)
↔
X13
)
)
)
→
(
(
X51
↔
(
X10
→
(
(
¬
X6
)
↔
X49
)
)
)
↔
(
¬
(
X49
↔
(
(
¬
X13
)
↔
X51
)
)
)
)
)
)
→
(
¬
(
X49
→
(
X10
↔
X8
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
X51
→
(
X32
↔
(
(
¬
X16
)
→
X27
)
)
)
→
(
¬
(
X48
→
(
(
¬
X16
)
↔
X51
)
)
)
)
→
(
(
(
X49
↔
(
X48
↔
(
X32
↔
(
X27
→
False
)
)
)
)
→
(
(
¬
(
(
X49
→
(
¬
X27
)
→
(
X14
↔
X32
)
)
→
(
¬
(
X49
→
(
X21
↔
X32
)
)
)
)
)
↔
(
¬
(
X32
→
(
¬
X21
)
→
X49
)
)
)
)
↔
(
¬
(
X48
↔
(
X32
↔
X49
)
)
)
)
)
)
→
(
(
¬
(
X47
↔
(
X32
→
(
¬
X28
)
→
X34
)
)
)
↔
(
(
(
¬
(
X32
↔
(
X28
↔
(
X27
→
X21
)
)
)
)
↔
(
¬
(
(
¬
X32
)
↔
(
(
¬
X28
)
→
(
(
¬
X27
)
↔
X34
)
)
)
)
)
↔
(
¬
(
X34
→
X32
)
)
)
)
)
)
)
)
→
(
(
(
(
(
¬
(
X51
↔
(
X46
↔
(
X20
↔
False
)
)
)
)
↔
(
(
(
¬
(
X51
↔
(
X49
↔
(
X46
→
X20
→
False
)
)
)
)
→
X49
→
X46
→
X3
)
↔
(
X46
→
X49
)
)
)
↔
(
X51
→
X46
→
X3
)
)
→
(
(
(
¬
X49
)
↔
(
(
¬
X31
)
↔
X51
)
)
↔
(
(
(
¬
(
X49
↔
(
(
¬
X31
)
↔
X51
)
)
)
↔
(
¬
(
(
¬
(
X20
↔
X49
)
)
→
(
(
¬
X31
)
↔
(
X30
↔
X51
)
)
)
)
)
↔
(
¬
(
(
¬
X31
)
→
X30
→
X51
)
)
)
)
)
↔
(
¬
(
X49
→
(
(
¬
X30
)
↔
X31
)
)
)
)
→
(
¬
(
(
(
(
¬
X47
)
↔
(
(
¬
X23
)
↔
(
X2
↔
X51
)
)
)
→
(
X51
↔
(
X23
↔
(
(
¬
X22
)
↔
X47
)
)
)
)
→
(
(
(
(
(
¬
X45
)
↔
(
(
¬
X30
)
↔
(
X12
↔
X46
)
)
)
→
(
¬
(
X47
↔
(
X45
↔
(
(
¬
X12
)
↔
X46
)
)
)
)
)
↔
(
¬
(
(
¬
X45
)
→
(
(
¬
X12
)
↔
X47
)
)
)
)
↔
(
(
(
¬
X30
)
→
(
¬
X2
)
→
X47
)
↔
(
X45
↔
(
(
¬
X23
)
→
X2
→
X30
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
(
¬
X43
)
↔
(
(
¬
X39
)
↔
(
(
¬
X7
)
→
X48
)
)
)
↔
(
X39
→
(
X11
↔
X48
)
)
)
→
(
¬
(
X43
→
(
X11
↔
(
X7
↔
X24
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X42
↔
(
(
¬
X40
)
→
(
¬
X0
)
→
X47
)
)
)
→
(
¬
(
(
¬
(
X48
→
X43
→
X40
)
)
→
(
¬
(
X43
↔
(
(
¬
X24
)
→
(
¬
X0
)
→
X40
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
X44
↔
(
X38
↔
(
(
¬
X35
)
→
X37
)
)
)
)
↔
(
(
(
X15
→
(
X5
↔
X38
)
)
↔
(
¬
(
X50
→
(
X5
↔
X37
)
)
)
)
↔
(
¬
(
(
(
X38
↔
(
X35
→
(
X33
↔
X37
)
)
)
→
(
¬
(
X50
↔
(
X35
↔
False
)
)
)
)
→
(
¬
(
(
¬
X42
)
→
(
¬
X38
)
→
(
(
¬
X9
)
↔
X50
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
X44
↔
(
(
¬
X42
)
↔
(
(
¬
X26
)
→
X51
)
)
)
→
(
(
X26
→
(
X17
↔
X36
)
)
↔
(
¬
(
(
¬
(
X51
↔
(
(
¬
X36
)
→
X26
→
X44
)
)
)
→
(
¬
(
X51
→
X44
→
X17
→
X36
)
)
)
)
)
)
↔
(
(
¬
(
X41
↔
(
X37
↔
(
(
¬
X18
)
↔
X43
)
)
)
)
→
X44
→
X43
→
X42
→
X18
→
False
)
)
→
(
¬
(
(
(
¬
(
(
¬
X19
)
↔
X41
)
)
↔
(
(
(
(
(
(
¬
(
X25
→
(
X1
↔
X42
)
)
)
↔
(
¬
(
(
¬
X25
)
↔
(
(
¬
X19
)
↔
X41
)
)
)
)
↔
(
(
¬
X41
)
↔
(
X25
→
X42
)
)
)
↔
(
¬
(
X42
↔
(
X41
→
X25
)
)
)
)
→
(
¬
(
X42
↔
(
X19
↔
(
X1
→
False
)
)
)
)
)
↔
(
X41
↔
(
X19
→
X25
)
)
)
)
→
(
¬
(
(
¬
(
X51
→
(
X48
↔
(
(
¬
X29
)
→
X42
)
)
)
)
→
X51
→
(
¬
X29
)
→
(
(
¬
X1
)
↔
X48
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
58368a...
and proposition id is
dd49c7...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMUd6NyLx2ZseMp8bLFAuiVtZW17dpfvPwq
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
¬
(
(
¬
(
(
(
¬
(
(
(
¬
(
X48
→
(
X46
↔
(
X38
→
X45
)
)
)
)
→
(
(
¬
(
X48
↔
(
X9
↔
X41
)
)
)
↔
(
¬
(
X31
→
X28
→
X9
)
)
)
→
(
(
¬
(
X46
↔
(
X45
→
X9
→
X48
)
)
)
↔
(
(
¬
X41
)
→
(
(
¬
X9
)
↔
X45
)
)
)
)
→
(
(
¬
X24
)
↔
(
(
¬
X13
)
→
X46
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
X45
↔
X41
)
)
→
(
X49
↔
(
(
¬
X25
)
→
(
¬
X19
)
→
X34
)
)
)
→
(
¬
(
(
¬
(
X49
→
(
X25
↔
X48
)
)
)
→
X48
→
(
X34
↔
(
X14
→
False
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X35
→
(
(
¬
X33
)
↔
(
X27
→
X41
)
)
)
)
→
(
¬
(
(
¬
X33
)
↔
(
X27
↔
(
X0
↔
X41
)
)
)
)
)
)
→
(
(
X41
↔
(
X27
↔
(
(
¬
X6
)
→
X33
)
)
)
↔
(
¬
(
X49
↔
(
X48
→
(
(
¬
X6
)
↔
X27
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X37
↔
(
X8
↔
(
(
¬
X5
)
→
X46
)
)
)
)
↔
(
X42
→
(
¬
X8
)
→
X46
)
)
→
X37
→
X32
→
(
¬
X5
)
→
X39
)
↔
(
(
(
(
¬
(
X49
↔
(
(
¬
X22
)
↔
X37
)
)
)
→
(
¬
(
(
¬
X39
)
↔
(
X37
↔
(
(
¬
X20
)
↔
X49
)
)
)
)
)
→
(
¬
X22
)
→
(
X20
↔
X49
)
)
↔
(
(
¬
(
X22
→
X20
→
X46
)
)
→
(
¬
(
X22
→
(
(
¬
X21
)
↔
(
X20
→
X37
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X44
↔
(
X42
→
X39
→
False
)
)
)
→
(
(
¬
(
(
¬
X44
)
↔
(
X18
→
X46
)
)
)
↔
(
(
(
¬
(
(
¬
X44
)
→
X39
→
X2
→
X48
)
)
↔
(
¬
(
(
¬
X39
)
→
(
¬
X18
)
→
X44
)
)
)
→
(
(
(
¬
(
X48
↔
(
X42
→
X26
)
)
)
↔
(
¬
(
X44
↔
(
(
¬
X39
)
→
(
¬
X26
)
→
X42
)
)
)
)
↔
(
¬
(
X44
→
X42
→
X26
)
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X43
↔
(
X30
→
X35
)
)
)
→
(
¬
(
X43
↔
(
(
¬
X7
)
→
X1
→
X30
)
)
)
)
→
(
¬
(
X43
→
X35
→
(
¬
X23
)
→
X30
)
)
)
→
(
(
¬
(
X49
↔
(
(
¬
X29
)
↔
(
X3
→
X47
)
)
)
)
↔
(
(
(
(
¬
X43
)
→
(
¬
X35
)
→
X11
→
X47
)
→
(
¬
(
X49
↔
(
(
¬
X11
)
↔
X47
)
)
)
)
→
(
X47
↔
X35
)
→
(
¬
(
X49
→
X47
→
(
X30
↔
X29
)
)
)
)
)
)
→
(
(
(
¬
(
(
(
¬
X17
)
↔
(
X12
↔
(
X4
↔
X49
)
)
)
→
(
(
¬
(
X17
→
(
(
¬
X16
)
↔
(
X4
→
X34
)
)
)
)
↔
(
(
¬
X34
)
↔
(
X4
↔
X49
)
)
)
)
)
↔
(
(
X34
→
(
(
¬
X16
)
↔
(
(
¬
X10
)
↔
X35
)
)
)
→
(
¬
(
X35
→
(
X34
↔
(
X12
→
X16
)
)
)
)
)
)
↔
(
(
X49
↔
(
(
¬
X40
)
→
(
(
¬
X34
)
↔
X43
)
)
)
↔
(
(
¬
(
X40
↔
(
X34
↔
X49
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X36
)
↔
(
(
¬
X12
)
→
X40
)
)
)
→
X36
→
(
¬
X34
)
→
(
(
¬
X12
)
↔
X49
)
)
)
→
(
¬
(
(
¬
X36
)
↔
(
(
¬
X34
)
↔
(
(
¬
X15
)
↔
X40
)
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
(
(
(
X47
→
(
X43
↔
(
X20
↔
X22
)
)
)
↔
(
X47
→
(
(
¬
X32
)
↔
(
(
¬
X20
)
↔
X43
)
)
)
)
→
(
(
¬
(
(
¬
X45
)
↔
(
(
¬
X32
)
→
(
X22
↔
X47
)
)
)
)
↔
(
¬
(
X47
↔
(
(
¬
X32
)
→
(
X6
↔
X45
)
)
)
)
)
)
→
(
(
¬
(
X32
→
X22
→
(
(
¬
X4
)
↔
X23
)
)
)
↔
(
(
¬
X23
)
→
(
¬
X4
)
→
(
(
¬
X0
)
↔
X47
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
¬
(
X46
→
(
¬
X19
)
→
X43
)
)
→
(
¬
(
(
¬
(
(
¬
X17
)
↔
(
(
¬
X12
)
→
X46
)
)
)
→
(
¬
(
X46
↔
(
X43
↔
(
X19
→
False
)
)
)
)
)
)
)
)
↔
(
(
¬
X17
)
→
(
¬
X12
)
→
X19
)
)
→
(
¬
(
X43
↔
(
X19
↔
X46
)
)
)
→
(
(
X45
↔
(
X43
→
(
X19
↔
False
)
)
)
↔
(
¬
(
X47
↔
(
X43
↔
(
(
¬
X19
)
↔
X46
)
)
)
)
)
)
)
↔
(
¬
(
(
X47
↔
(
X17
→
X3
)
)
→
(
¬
(
X12
→
X3
→
X7
)
)
)
)
)
)
↔
(
(
¬
(
X38
↔
(
(
¬
X22
)
→
(
¬
X16
)
→
X28
)
)
)
↔
(
¬
(
X45
↔
(
(
¬
X23
)
↔
(
(
¬
X16
)
→
X28
)
)
)
)
)
)
→
(
(
(
X45
↔
(
X35
↔
(
X13
↔
X24
)
)
)
↔
(
¬
(
(
(
¬
X43
)
↔
(
(
¬
X27
)
↔
(
X13
↔
X45
)
)
)
→
(
X45
↔
(
X43
↔
(
(
¬
X25
)
→
X27
)
)
)
)
)
)
↔
(
(
¬
(
(
X43
→
(
(
¬
X35
)
↔
X46
)
)
→
(
¬
(
(
¬
(
(
¬
X11
)
↔
X43
)
)
→
(
(
¬
X46
)
↔
(
(
¬
X45
)
↔
(
X21
→
X49
)
)
)
)
)
)
)
↔
(
(
¬
X42
)
↔
(
(
¬
X21
)
↔
(
(
¬
X18
)
→
X43
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
(
¬
(
(
(
¬
(
(
¬
X44
)
↔
(
(
¬
X30
)
↔
X46
)
)
)
↔
(
(
(
X49
→
(
X46
↔
(
X15
↔
X44
)
)
)
→
X49
→
(
X46
↔
(
X30
→
X15
)
)
)
↔
(
¬
(
X49
↔
(
X46
↔
(
X44
↔
False
)
)
)
)
)
)
→
(
X48
→
(
(
¬
X46
)
↔
(
X30
↔
X49
)
)
)
→
(
¬
(
X48
→
(
X30
↔
(
(
¬
X10
)
↔
X46
)
)
)
)
)
)
↔
(
X46
→
(
(
¬
X44
)
↔
(
(
¬
X14
)
↔
X49
)
)
)
)
→
(
(
¬
(
X44
↔
(
(
¬
X37
)
↔
X48
)
)
)
↔
(
X49
→
X48
→
X33
)
)
→
(
¬
(
(
¬
X33
)
→
(
X31
↔
X49
)
)
)
)
)
↔
(
¬
(
(
(
(
(
X47
↔
(
X44
→
(
¬
X26
)
→
X48
)
)
↔
(
¬
(
(
¬
X47
)
→
(
X44
↔
(
X36
↔
X48
)
)
)
)
)
↔
(
¬
(
X48
→
(
¬
X47
)
→
X26
→
X49
)
)
)
→
X49
→
(
X47
↔
(
(
¬
X39
)
→
X48
)
)
)
→
X48
→
(
X44
↔
(
X1
↔
X49
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
(
X46
↔
(
X44
↔
(
(
¬
X2
)
→
X40
)
)
)
)
→
(
(
(
(
¬
(
X46
↔
(
(
¬
X41
)
→
(
(
¬
X40
)
↔
X44
)
)
)
)
↔
(
(
¬
X41
)
↔
(
(
¬
X40
)
↔
(
(
¬
X9
)
↔
X46
)
)
)
)
↔
(
X44
→
(
(
¬
X41
)
↔
X46
)
)
)
↔
(
X46
→
(
(
¬
X40
)
↔
(
(
¬
X37
)
→
X41
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
X41
)
↔
(
X29
↔
X46
)
)
)
→
(
¬
(
X41
↔
(
(
¬
X29
)
→
(
X8
↔
X40
)
)
)
)
)
)
↔
(
X46
↔
(
(
¬
X8
)
↔
X29
)
)
)
↔
(
(
X34
→
(
¬
X29
)
→
X40
)
↔
(
(
¬
(
X34
→
X37
)
)
↔
(
X37
↔
(
(
¬
X29
)
↔
(
(
¬
X5
)
→
X40
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
3ab2fb...
and proposition id is
cee3f7...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMbS9jGJ7iXJtV6wJwr2ZATn4pAa4cBCRwq
)
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
(
(
(
¬
X46
)
↔
(
X38
↔
(
(
¬
X6
)
↔
X48
)
)
)
→
(
¬
(
(
(
¬
(
X48
→
(
X36
↔
(
X23
→
X38
)
)
)
)
↔
(
¬
(
(
X46
→
(
¬
X38
)
→
X48
)
→
(
¬
(
X41
↔
(
(
¬
X38
)
↔
(
X23
→
X48
)
)
)
)
)
)
)
→
(
¬
(
X41
→
X36
→
(
X23
↔
False
)
)
)
)
)
)
↔
(
(
(
X48
→
(
X47
↔
(
X41
↔
False
)
)
)
↔
(
¬
(
(
¬
X48
)
→
X41
→
X3
→
X49
)
)
)
↔
(
X47
→
X23
→
False
)
)
)
→
(
(
¬
(
(
¬
(
(
(
¬
(
(
¬
(
X49
→
(
¬
X14
)
→
X39
)
)
→
(
¬
(
X29
↔
(
X9
↔
X14
)
)
)
)
)
↔
(
X49
↔
(
X29
→
X14
)
)
)
→
(
X39
↔
(
X29
→
X9
→
X38
)
)
)
)
→
(
(
¬
X38
)
↔
(
X14
→
X40
)
)
)
)
↔
(
(
¬
(
X49
→
X38
→
X40
)
)
→
(
(
¬
(
(
¬
X45
)
→
X40
→
X38
→
X49
)
)
↔
(
X49
→
(
¬
X40
)
→
(
X19
↔
X45
)
)
)
)
)
→
(
¬
(
(
(
(
¬
X45
)
→
(
¬
X33
)
→
(
X1
↔
X49
)
)
↔
(
X49
→
X46
→
X1
)
)
→
(
¬
(
(
¬
(
(
¬
(
X49
↔
X45
)
)
→
(
X46
↔
(
X45
↔
(
(
¬
X25
)
↔
X40
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X45
↔
(
X40
→
X5
→
False
)
)
)
→
(
¬
(
X46
↔
(
X45
↔
(
X5
↔
(
X2
↔
False
)
)
)
)
)
)
→
X5
→
X45
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
(
(
¬
(
X46
↔
(
(
¬
X42
)
↔
(
X15
↔
X48
)
)
)
)
→
(
¬
(
X48
↔
(
X42
→
(
(
¬
X21
)
↔
X46
)
)
)
)
)
)
→
(
¬
(
X42
→
X22
→
(
¬
X21
)
→
X48
)
)
)
)
↔
(
(
(
(
¬
X37
)
→
(
¬
X21
)
→
(
¬
X10
)
→
X46
)
↔
(
¬
(
X46
→
(
(
¬
X22
)
↔
(
X21
→
X37
)
)
)
)
)
↔
(
(
(
¬
X43
)
→
(
(
¬
X22
)
↔
X46
)
)
↔
(
¬
(
(
X46
→
X21
→
False
)
→
(
¬
(
(
¬
X44
)
↔
(
X43
→
(
X22
↔
X46
)
)
)
)
)
)
)
)
)
→
(
(
(
(
X42
→
X26
)
↔
(
¬
(
X44
↔
(
X28
→
(
X4
↔
X42
)
)
)
)
)
↔
(
X44
→
(
(
¬
X31
)
↔
(
X28
→
X42
)
)
)
)
↔
(
(
¬
(
(
(
¬
X28
)
→
(
¬
X17
)
→
X44
)
→
(
¬
(
X46
↔
(
X44
→
(
X42
↔
(
X28
→
False
)
)
)
)
)
)
)
↔
(
¬
(
X46
→
(
(
¬
X28
)
↔
X42
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
(
X44
↔
(
(
¬
X30
)
→
(
¬
X18
)
→
X47
)
)
→
(
¬
(
(
¬
(
X44
↔
(
X24
↔
(
X7
→
X30
)
)
)
)
→
(
¬
(
X47
→
(
(
¬
X7
)
↔
X24
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
X47
↔
(
X24
↔
(
(
¬
X8
)
→
X44
)
)
)
)
→
(
¬
(
X44
↔
(
(
¬
X43
)
↔
(
X24
→
X47
)
)
)
)
)
)
)
↔
(
(
(
X46
↔
(
(
¬
X43
)
↔
(
(
¬
X35
)
↔
X47
)
)
)
↔
(
¬
(
X46
→
X43
→
(
X11
↔
X47
)
)
)
)
→
(
¬
(
(
¬
(
X35
↔
(
X30
→
False
)
)
)
→
(
(
¬
(
(
¬
X43
)
↔
(
X35
→
(
X30
↔
X47
)
)
)
)
↔
(
(
¬
X43
)
→
(
X35
↔
(
(
¬
X34
)
↔
X47
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
X35
→
(
X27
↔
X47
)
)
)
↔
(
(
(
¬
(
X47
→
(
X35
↔
(
X27
↔
(
X12
→
False
)
)
)
)
)
↔
(
X47
↔
(
X35
↔
(
X20
→
X27
)
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X27
)
↔
(
X16
→
X35
)
)
)
)
)
)
↔
(
(
¬
(
(
X32
→
(
(
¬
X16
)
↔
(
X0
→
X35
)
)
)
→
(
(
(
¬
X27
)
↔
(
X16
→
(
¬
X0
)
→
X35
)
)
↔
(
¬
(
X35
↔
(
(
¬
X27
)
↔
X44
)
)
)
)
)
)
→
(
¬
(
X35
→
(
X27
↔
(
X13
→
X0
)
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
(
(
(
(
(
¬
(
X32
→
(
(
¬
X16
)
↔
(
X2
↔
X36
)
)
)
)
↔
(
¬
(
(
¬
X36
)
↔
(
X16
↔
X49
)
)
)
)
↔
(
¬
(
X16
→
(
X2
↔
X49
)
)
)
)
→
(
(
¬
(
X49
→
(
¬
X37
)
→
(
X32
↔
X45
)
)
)
↔
(
X36
↔
(
(
¬
X35
)
↔
X49
)
)
)
)
↔
(
(
¬
(
(
¬
X37
)
→
X36
→
X45
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X48
→
(
X45
↔
(
(
¬
X3
)
↔
X37
)
)
)
)
→
(
¬
(
X45
→
(
(
¬
X37
)
↔
(
X36
→
X48
)
)
)
)
)
)
→
(
¬
(
X37
→
(
¬
X36
)
→
X45
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X38
)
↔
(
(
¬
X37
)
→
(
(
¬
X36
)
↔
X48
)
)
)
)
→
(
(
¬
(
(
¬
X38
)
↔
(
(
¬
X37
)
↔
(
X22
→
X48
)
)
)
)
↔
(
(
¬
(
X31
→
X37
)
)
→
(
¬
(
X38
→
X37
→
(
¬
X33
)
→
X48
)
)
)
)
)
)
→
X36
→
False
)
→
(
(
X39
↔
(
X24
↔
(
(
¬
X0
)
↔
X49
)
)
)
↔
(
(
X39
↔
(
X24
→
X36
)
)
→
(
(
X39
→
(
(
¬
X36
)
↔
X49
)
)
↔
(
(
(
(
¬
(
X36
↔
X24
)
)
↔
(
X39
↔
(
X35
→
(
X24
↔
X49
)
)
)
)
↔
(
¬
(
X49
→
(
X39
↔
(
X24
→
X20
)
)
)
)
)
↔
(
¬
(
X36
→
(
(
¬
X20
)
↔
X49
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
X46
→
(
¬
X26
)
→
(
(
¬
X14
)
↔
X49
)
)
→
(
¬
(
(
(
¬
X26
)
↔
(
(
¬
X14
)
→
X43
)
)
→
X46
→
(
X45
↔
(
X43
↔
(
X26
→
False
)
)
)
)
)
)
→
(
(
X46
→
(
X27
↔
(
X23
→
X18
)
)
)
↔
(
(
(
(
¬
(
(
¬
X45
)
→
(
¬
X27
)
→
X49
)
)
→
(
(
¬
(
X45
↔
(
(
¬
X27
)
→
X23
→
X49
)
)
)
↔
(
¬
(
(
¬
X45
)
↔
(
X30
↔
(
X23
→
X49
)
)
)
)
)
)
→
(
¬
X45
)
→
(
X27
↔
X49
)
)
→
(
X23
↔
X27
)
→
(
(
¬
X30
)
↔
(
(
¬
X23
)
↔
(
(
¬
X18
)
→
X39
)
)
)
)
)
→
(
¬
(
X49
→
X46
→
X17
→
X39
)
)
→
(
(
¬
X39
)
→
X17
→
(
X9
↔
X44
)
)
→
(
¬
(
X49
↔
(
X39
→
(
X9
↔
X44
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X29
↔
(
X8
↔
(
(
¬
X6
)
→
X40
)
)
)
)
↔
(
¬
(
(
¬
(
X44
→
(
¬
X40
)
→
(
(
¬
X29
)
↔
X45
)
)
)
→
(
(
¬
(
(
¬
X45
)
→
(
(
¬
X40
)
↔
(
X8
→
X49
)
)
)
)
↔
(
(
¬
X44
)
→
(
X12
↔
(
(
¬
X8
)
↔
X45
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
(
(
¬
X42
)
→
(
¬
X10
)
→
X49
)
)
→
(
¬
(
(
¬
X10
)
↔
(
X5
↔
X40
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X10
)
↔
(
X5
↔
X42
)
)
)
)
)
→
(
X40
↔
(
(
¬
X10
)
↔
(
X5
↔
X19
)
)
)
)
)
→
(
¬
X42
)
→
X29
→
(
(
¬
X19
)
↔
X49
)
)
)
)
→
(
(
(
(
X7
↔
X47
)
↔
(
(
¬
(
(
¬
X45
)
↔
(
X28
→
(
X7
↔
X47
)
)
)
)
↔
(
X47
→
(
(
¬
X45
)
↔
(
(
¬
X28
)
→
X49
)
)
)
)
)
↔
(
(
(
¬
X45
)
↔
(
(
¬
X34
)
↔
X49
)
)
↔
(
(
(
(
¬
(
X49
→
(
X47
↔
X34
)
)
)
→
(
¬
(
X45
→
(
X44
↔
(
(
¬
X34
)
→
X47
)
)
)
)
)
↔
(
X49
→
X45
)
)
→
X49
→
X45
→
(
X21
↔
X44
)
)
)
)
↔
(
(
(
¬
(
X47
↔
(
(
¬
X41
)
↔
(
(
¬
X13
)
→
X45
)
)
)
)
↔
(
(
¬
(
(
¬
X41
)
↔
(
(
¬
X1
)
→
X45
)
)
)
→
(
¬
(
X25
→
(
X13
↔
(
(
¬
X1
)
→
X41
)
)
)
)
)
)
↔
(
(
(
¬
(
X49
↔
(
(
¬
X41
)
→
(
¬
X13
)
→
X47
)
)
)
↔
(
(
¬
(
X13
→
X11
→
X4
→
False
)
)
↔
(
(
¬
X41
)
→
(
(
¬
X13
)
↔
(
X11
↔
X47
)
)
)
)
)
↔
(
¬
(
X49
→
(
X15
↔
(
X13
→
X11
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
a1bf6f...
and proposition id is
236382...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMHDh2osbm63sLVi1ZcSYigSQSBovACoowE
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
¬
(
(
(
(
(
¬
(
(
¬
X39
)
↔
(
X38
→
(
(
¬
X2
)
↔
X41
)
)
)
)
↔
(
¬
(
(
(
¬
X39
)
↔
(
(
¬
X38
)
→
(
¬
X37
)
→
X40
)
)
→
(
¬
(
X46
↔
(
(
¬
X37
)
↔
(
(
¬
X1
)
→
X41
)
)
)
)
)
)
)
↔
(
(
¬
(
X29
→
(
X1
↔
X2
)
)
)
↔
(
(
¬
(
(
¬
(
X29
→
(
(
¬
X28
)
↔
(
(
¬
X2
)
↔
X37
)
)
)
)
→
(
¬
(
(
¬
X29
)
↔
(
(
¬
X28
)
→
(
¬
X8
)
→
X37
)
)
)
)
)
→
X29
→
X28
→
(
X2
↔
False
)
)
)
)
→
(
(
X46
↔
(
(
¬
X40
)
↔
X48
)
)
↔
(
(
¬
(
(
¬
X40
)
↔
(
X39
→
(
¬
X27
)
→
X46
)
)
)
↔
(
(
(
(
¬
(
X39
→
X40
)
)
→
(
(
¬
(
X46
↔
(
X30
→
X21
→
X39
)
)
)
↔
(
¬
(
X40
↔
(
(
¬
X39
)
↔
X46
)
)
)
)
)
→
(
X40
↔
False
)
)
↔
(
(
X48
→
X46
→
(
X39
↔
X26
)
)
↔
(
¬
(
X46
↔
(
X39
→
(
(
¬
X30
)
↔
X48
)
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X38
→
(
X33
↔
(
(
¬
X5
)
↔
X44
)
)
)
)
↔
(
¬
(
(
¬
X38
)
↔
(
(
¬
X32
)
→
(
¬
X5
)
→
X44
)
)
)
)
↔
(
¬
(
(
¬
(
X44
↔
(
X38
↔
(
X37
↔
X33
)
)
)
)
→
(
(
¬
(
X49
→
(
X33
↔
(
X0
↔
X37
)
)
)
)
↔
(
(
(
¬
(
(
¬
X37
)
→
(
X33
↔
(
(
¬
X15
)
→
X38
)
)
)
)
→
(
¬
(
X37
→
X15
→
X11
→
False
)
)
)
↔
(
¬
(
(
¬
X38
)
↔
(
X37
→
(
¬
X33
)
→
X49
)
)
)
)
)
)
)
)
↔
(
(
(
X32
↔
False
)
→
(
(
¬
(
(
¬
X10
)
↔
X32
)
)
↔
(
¬
(
X38
↔
(
X35
→
(
X32
↔
X41
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
(
(
X38
↔
(
X33
↔
(
X19
↔
False
)
)
)
→
(
(
¬
(
X22
↔
(
(
¬
X19
)
↔
X38
)
)
)
↔
(
¬
(
X41
→
(
(
¬
X33
)
↔
(
X22
→
X38
)
)
)
)
)
)
↔
(
X38
→
(
(
¬
X19
)
↔
X33
)
)
)
→
(
¬
(
(
¬
X38
)
↔
(
X33
→
X41
)
)
)
)
)
↔
False
)
→
X33
→
(
X22
↔
X19
)
)
)
↔
(
(
¬
(
X33
→
(
¬
X32
)
→
(
(
¬
X19
)
↔
X45
)
)
)
→
(
¬
(
X45
→
(
¬
X14
)
→
(
(
¬
X6
)
↔
X38
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
¬
(
X48
↔
(
X16
↔
X7
)
)
)
↔
(
¬
(
(
¬
X16
)
→
(
X13
↔
X25
)
)
)
)
↔
(
¬
(
X48
↔
(
X25
→
X7
→
X43
)
)
)
)
→
(
¬
(
X49
→
(
X25
↔
(
(
¬
X16
)
→
X48
)
)
)
)
)
)
→
(
(
¬
(
X48
→
(
¬
X43
)
→
(
(
¬
X34
)
↔
X49
)
)
)
↔
(
(
(
(
(
X49
→
X43
→
X17
)
↔
(
(
¬
X48
)
→
(
¬
X47
)
→
X43
→
X49
)
)
↔
(
X49
→
X47
→
(
(
¬
X24
)
↔
X43
)
)
)
↔
(
(
¬
X43
)
↔
(
(
¬
X24
)
→
X49
)
)
)
↔
(
¬
(
X43
↔
(
X31
→
False
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X42
↔
(
(
¬
X12
)
→
X47
)
)
)
→
(
X49
→
False
)
→
(
(
¬
(
X42
↔
X12
)
)
↔
(
X49
→
(
¬
X20
)
→
X47
)
)
)
→
(
(
X47
→
(
X20
↔
(
X18
→
(
X3
↔
False
)
)
)
)
↔
(
(
¬
X20
)
↔
(
(
¬
X3
)
↔
X49
)
)
)
)
↔
(
X20
↔
(
X3
↔
X47
)
)
)
↔
(
(
(
(
(
X46
→
(
X36
↔
(
X9
→
False
)
)
)
↔
(
¬
(
X46
↔
(
(
¬
X20
)
↔
(
(
¬
X9
)
→
X49
)
)
)
)
)
↔
(
¬
(
(
¬
X47
)
↔
(
X46
→
(
X36
↔
X49
)
)
)
)
)
↔
(
X23
→
X49
)
)
↔
(
(
¬
(
X48
↔
(
X47
↔
X49
)
)
)
↔
(
¬
(
X48
↔
(
X47
→
(
X46
↔
(
X4
→
False
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
¬
(
X46
↔
(
X20
↔
(
(
¬
X18
)
→
X25
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
X35
)
→
(
(
¬
X25
)
↔
(
(
¬
X18
)
→
X45
)
)
)
→
(
¬
X35
)
→
(
(
¬
X20
)
↔
(
X4
↔
X47
)
)
)
)
→
(
X20
↔
(
X18
→
False
)
)
)
)
)
→
(
(
(
(
¬
X42
)
→
X32
→
(
(
¬
X24
)
↔
X43
)
)
↔
(
X42
↔
(
(
¬
X21
)
→
(
(
¬
X7
)
↔
X24
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
X42
→
(
¬
X35
)
→
X47
)
)
→
(
X47
↔
(
X43
↔
(
X24
→
X42
)
)
)
)
↔
(
X47
→
(
(
¬
X42
)
↔
(
(
¬
X31
)
→
X43
)
)
)
)
↔
(
X47
↔
(
(
¬
X42
)
↔
X43
)
)
)
→
(
(
X46
→
X43
→
(
X35
↔
X42
)
)
↔
(
(
¬
X43
)
↔
(
(
¬
X42
)
↔
(
X29
↔
X46
)
)
)
)
)
)
)
)
→
(
(
(
X23
→
X16
→
(
X12
↔
X45
)
)
↔
(
(
¬
(
X47
↔
(
X13
↔
(
(
¬
X12
)
→
X16
)
)
)
)
→
(
¬
(
(
¬
X13
)
→
X12
→
X23
)
)
)
)
↔
(
(
(
¬
X43
)
→
(
X16
↔
(
X12
→
X49
)
)
)
↔
(
(
X49
↔
(
X43
↔
(
(
¬
X13
)
↔
X16
)
)
)
↔
(
(
¬
X43
)
→
(
X16
↔
(
(
¬
X15
)
↔
X49
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X48
↔
(
X40
→
X11
→
False
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X45
)
→
(
¬
X40
)
→
X47
)
)
→
(
¬
(
X48
↔
(
X45
↔
(
(
¬
X1
)
→
X47
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X46
→
(
¬
X45
)
→
(
¬
X14
)
→
X47
)
)
↔
(
¬
(
X40
↔
False
)
)
)
→
(
(
X14
↔
(
X8
↔
False
)
)
↔
(
(
¬
(
X46
↔
(
X8
→
X34
)
)
)
→
(
¬
(
X46
↔
(
X40
↔
(
X34
↔
X14
)
)
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
(
¬
(
(
¬
(
X38
↔
(
X30
→
X5
)
)
)
→
(
(
X38
↔
(
(
¬
X5
)
→
(
X0
↔
X30
)
)
)
↔
(
X22
↔
(
(
¬
X5
)
→
X30
)
)
)
)
)
→
X22
→
(
X5
↔
(
X0
→
X36
)
)
)
)
→
(
(
¬
(
X38
↔
(
X30
↔
X22
)
)
)
↔
(
(
¬
X33
)
→
(
¬
X30
)
→
(
X28
↔
X38
)
)
)
→
(
(
X48
↔
(
X41
→
X30
→
(
X6
↔
False
)
)
)
→
(
(
¬
(
(
¬
X41
)
↔
(
(
¬
X33
)
↔
X48
)
)
)
↔
(
(
¬
(
X33
→
X38
)
)
↔
(
¬
(
(
X48
→
(
¬
X30
)
→
X38
)
→
(
¬
(
(
¬
(
X48
→
(
(
¬
X38
)
↔
(
(
¬
X33
)
→
X41
)
)
)
)
→
(
¬
(
X48
→
X41
→
(
X33
↔
(
X30
↔
False
)
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
X48
↔
(
X41
→
(
(
¬
X17
)
↔
X30
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
X36
)
→
(
(
¬
X33
)
↔
(
X2
↔
X38
)
)
)
)
↔
(
(
¬
(
X46
↔
(
X37
→
X36
→
X19
)
)
)
↔
(
¬
(
(
(
¬
X36
)
→
(
X33
↔
X37
)
)
→
X37
→
(
X36
↔
(
(
¬
X19
)
↔
X38
)
)
)
)
)
)
)
↔
(
(
(
(
¬
X40
)
→
(
¬
X26
)
→
X10
→
X44
)
→
(
(
¬
(
X40
↔
(
X39
↔
(
X26
↔
X37
)
)
)
)
→
(
¬
(
(
¬
X39
)
↔
(
X37
↔
(
X26
↔
X44
)
)
)
)
)
→
(
¬
(
(
¬
X40
)
→
(
X39
↔
(
X3
↔
X44
)
)
)
)
)
→
(
(
X40
→
(
X10
↔
X41
)
)
↔
(
X38
↔
(
X10
↔
(
X9
→
X27
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
aa3a82...
and proposition id is
e95ac4...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMXg79YBC6cnxNsh8FdVCezvrYERaWQYrrg
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
(
¬
(
(
¬
(
(
(
X10
→
(
X6
↔
(
(
¬
X1
)
↔
X19
)
)
)
→
(
¬
(
X6
→
(
¬
X4
)
→
X0
→
X10
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
(
(
X31
→
X16
→
(
¬
X6
)
→
X48
)
→
X48
→
(
X6
↔
X1
)
)
)
→
(
X31
↔
(
X16
→
(
X1
↔
False
)
)
)
)
)
↔
(
¬
(
(
¬
X6
)
↔
X48
)
)
)
→
(
¬
(
X50
→
(
(
¬
X41
)
↔
(
X16
↔
X48
)
)
)
)
→
(
¬
(
(
¬
X48
)
↔
(
(
¬
X41
)
→
(
¬
X31
)
→
X50
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
X39
)
→
X30
→
(
(
¬
X5
)
↔
X41
)
)
)
↔
(
(
¬
(
(
¬
False
)
→
(
(
X39
→
(
(
¬
X5
)
↔
X48
)
)
↔
(
¬
(
X42
↔
(
(
¬
X5
)
↔
X26
)
)
)
)
)
)
↔
(
¬
(
X48
↔
(
X42
→
(
X5
↔
False
)
)
)
)
)
)
↔
(
(
(
(
¬
X41
)
↔
(
(
¬
X12
)
→
X49
)
)
↔
(
(
¬
(
X49
→
X42
→
(
X41
↔
False
)
)
)
→
(
(
X48
↔
(
X42
↔
X41
)
)
↔
(
X49
→
(
X48
↔
X12
)
)
)
)
)
↔
(
(
¬
(
X49
→
(
(
¬
X42
)
↔
(
(
¬
X14
)
→
X48
)
)
)
)
↔
(
(
¬
X48
)
↔
(
X14
→
(
¬
X8
)
→
X49
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X50
↔
(
X42
→
X40
→
(
X37
↔
False
)
)
)
)
→
(
(
¬
(
X42
↔
(
X40
↔
(
(
¬
X21
)
↔
X50
)
)
)
)
↔
(
X40
↔
(
X23
↔
X42
)
)
)
)
)
→
(
(
¬
(
X48
↔
(
(
¬
X41
)
→
(
(
¬
X23
)
↔
X50
)
)
)
)
↔
(
¬
(
X50
→
(
X48
↔
(
X41
↔
X21
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
X35
)
↔
(
(
¬
X13
)
→
X36
)
)
)
→
(
(
(
¬
(
(
¬
(
X45
→
(
¬
X35
)
→
X50
)
)
→
(
¬
(
X49
→
X45
→
X9
→
X50
)
)
)
)
→
(
¬
(
X45
→
(
(
¬
X35
)
↔
X49
)
)
)
)
↔
(
¬
(
X49
↔
(
X45
→
(
¬
X36
)
→
X50
)
)
)
)
)
→
(
(
X48
↔
(
X35
→
X2
→
X45
)
)
↔
(
(
X48
→
X44
→
(
X13
↔
X33
)
)
↔
(
(
¬
(
X45
→
(
¬
X33
)
→
X15
→
X35
)
)
↔
(
X45
→
(
¬
X33
)
→
(
(
¬
X13
)
↔
X44
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
X27
↔
(
X22
↔
(
(
¬
X18
)
→
X29
)
)
)
→
(
X29
↔
(
(
¬
X20
)
↔
X44
)
)
→
X46
→
(
X20
↔
X22
)
)
)
→
(
¬
(
(
(
X50
↔
(
X34
→
(
(
¬
X22
)
↔
X44
)
)
)
→
X50
→
X34
→
(
¬
X27
)
→
X44
)
→
X50
→
(
(
¬
X27
)
↔
(
(
¬
X11
)
→
X44
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
(
X32
→
(
(
¬
X24
)
↔
(
X17
↔
X44
)
)
)
→
(
¬
(
(
¬
(
(
¬
X24
)
↔
(
(
¬
X17
)
↔
X46
)
)
)
→
(
¬
(
X32
↔
(
X24
↔
(
X17
→
(
X7
↔
False
)
)
)
)
)
)
)
)
→
(
¬
(
X25
↔
(
X24
→
X44
)
)
)
→
(
(
¬
X38
)
↔
(
X25
→
X24
→
X46
)
)
)
)
↔
(
X44
↔
(
X36
↔
(
X25
↔
X17
)
)
)
)
→
(
(
(
¬
X47
)
→
X44
→
(
X43
↔
X50
)
)
↔
(
¬
(
(
(
(
¬
X44
)
↔
(
(
¬
X28
)
↔
(
X3
→
X47
)
)
)
↔
(
(
¬
(
X32
↔
(
(
¬
X28
)
→
X44
)
)
)
→
(
(
(
¬
X44
)
↔
(
X28
→
X47
)
)
↔
(
(
¬
X44
)
↔
(
X43
→
(
(
¬
X32
)
↔
X47
)
)
)
)
)
)
→
(
X47
↔
(
(
¬
X32
)
↔
X43
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
(
¬
(
X44
↔
X4
)
)
↔
(
¬
(
(
¬
(
X48
→
X46
)
)
→
(
(
¬
(
X44
↔
(
X37
↔
(
X4
↔
X48
)
)
)
)
↔
(
¬
(
X46
→
(
(
¬
X4
)
↔
X44
)
)
)
)
)
)
)
↔
(
(
(
¬
X46
)
↔
(
X44
↔
(
X41
→
X48
)
)
)
↔
(
X46
↔
(
X44
↔
(
(
¬
X13
)
↔
X25
)
)
)
)
)
↔
(
(
¬
(
(
¬
X44
)
→
(
¬
X19
)
→
(
(
¬
X2
)
↔
X46
)
)
)
↔
(
(
¬
(
X20
↔
(
X2
→
X19
)
)
)
↔
(
(
¬
(
X20
→
X19
→
(
(
¬
X2
)
↔
X41
)
)
)
↔
(
X46
→
(
X41
↔
(
(
¬
X19
)
↔
X20
)
)
)
)
)
)
)
→
(
¬
(
X48
→
(
(
¬
X28
)
↔
(
X14
→
X29
)
)
)
)
→
(
(
X48
↔
(
X44
↔
(
(
¬
X27
)
↔
X49
)
)
)
↔
(
(
X48
→
(
X46
↔
X29
)
)
↔
(
(
X48
↔
(
X46
→
(
X44
↔
X49
)
)
)
↔
(
(
¬
(
X48
↔
(
(
¬
X28
)
→
X49
)
)
)
↔
(
¬
(
X49
→
(
X48
↔
(
X29
↔
X44
)
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
X50
→
(
¬
X26
)
→
(
¬
X5
)
→
X47
)
)
→
(
¬
(
(
X50
↔
(
X49
→
(
X48
↔
(
X38
→
False
)
)
)
)
→
(
(
(
¬
(
X49
→
(
X33
↔
(
X5
→
False
)
)
)
)
↔
(
¬
(
X50
→
(
X49
↔
(
X44
→
X33
)
)
)
)
)
↔
(
¬
(
X47
↔
(
X33
↔
(
(
¬
X5
)
→
X48
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
X42
→
(
(
¬
X21
)
↔
X31
)
)
↔
(
X31
↔
(
(
¬
X24
)
↔
(
(
¬
X21
)
↔
X50
)
)
)
)
↔
(
X50
↔
(
(
¬
X31
)
→
X21
→
X49
)
)
)
→
(
(
(
(
¬
X48
)
→
(
X12
↔
X49
)
)
↔
(
X48
↔
(
(
¬
X21
)
↔
X24
)
)
)
↔
(
(
¬
(
X48
↔
(
(
¬
X9
)
→
X10
)
)
)
↔
(
¬
(
(
¬
X21
)
→
X10
→
(
X9
↔
X49
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
X49
→
X22
→
(
(
¬
X16
)
↔
X48
)
)
)
↔
(
(
(
¬
(
X40
↔
(
(
¬
X35
)
↔
(
X11
↔
X49
)
)
)
)
↔
(
X35
→
(
X11
↔
X43
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
(
X1
→
X34
)
→
(
X34
↔
(
X22
↔
(
X16
→
X40
)
)
)
)
)
→
(
¬
(
(
¬
X40
)
→
(
X16
↔
(
X1
→
X43
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X43
↔
(
X22
↔
X3
)
)
)
↔
(
X43
→
(
(
¬
X34
)
↔
(
(
¬
X22
)
↔
X39
)
)
)
)
→
X39
→
X3
→
X22
)
)
↔
(
X39
↔
(
X22
→
X43
)
)
)
)
)
↔
(
X49
→
(
X43
↔
(
X34
↔
(
X22
→
False
)
)
)
)
)
)
)
↔
(
(
(
X39
→
X8
→
X23
)
↔
(
(
¬
(
(
¬
X43
)
→
(
¬
X40
)
→
X6
→
X45
)
)
↔
(
(
¬
(
X50
→
X40
→
(
(
¬
X6
)
↔
X23
)
)
)
→
(
X40
↔
(
X23
→
X45
)
)
)
)
)
→
(
¬
(
(
¬
(
(
X17
↔
(
X15
↔
(
X7
→
X50
)
)
)
→
(
¬
(
X32
↔
(
X15
↔
X17
)
)
)
)
)
→
X40
→
X17
)
)
→
(
(
¬
(
(
(
(
¬
(
X45
↔
(
X30
↔
(
X0
→
X7
)
)
)
)
↔
(
X30
→
(
X17
↔
X0
)
)
)
↔
(
¬
(
(
¬
X0
)
↔
X45
)
)
)
→
(
¬
(
X40
↔
(
X36
→
(
X17
↔
X18
)
)
)
)
)
)
↔
(
(
¬
X40
)
↔
(
X36
↔
X45
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
aa6d0a...
and proposition id is
15397f...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMVGKy1gVyMtbrLBAJrkQ1s6mwQJvvFNBfu
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
∀X51 :
prop
,
∀X52 :
prop
,
(
¬
∀X53 :
prop
,
(
(
¬
(
(
(
(
(
(
¬
(
(
(
¬
X45
)
↔
(
(
¬
X29
)
→
X28
→
X49
)
)
→
(
¬
(
(
X53
↔
(
X49
↔
(
(
¬
X9
)
↔
X45
)
)
)
→
(
¬
(
X53
→
(
X49
↔
(
X29
→
X45
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
X49
)
→
(
(
¬
X45
)
↔
X53
)
)
)
)
↔
(
¬
(
X53
→
(
X49
↔
(
(
¬
X11
)
→
X29
)
)
)
)
)
↔
(
(
¬
(
(
¬
X9
)
→
(
¬
X8
)
→
(
¬
X4
)
→
X53
)
)
→
(
(
(
X51
↔
(
(
¬
X23
)
→
X53
)
)
↔
(
(
X53
↔
(
(
¬
X9
)
→
X8
→
X23
)
)
→
(
¬
(
X53
→
(
X51
↔
X9
)
)
)
)
)
↔
(
X53
↔
(
X51
↔
(
X8
→
False
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
X52
↔
(
X51
↔
(
(
¬
X16
)
→
X53
)
)
)
→
(
(
X41
↔
(
(
¬
(
(
¬
X49
)
→
(
(
¬
X48
)
↔
(
X18
↔
X52
)
)
)
)
↔
(
X52
→
(
X41
↔
X49
)
)
)
)
↔
(
(
(
X49
↔
(
(
¬
X19
)
↔
X41
)
)
↔
(
X53
→
(
(
¬
X41
)
↔
(
X19
→
X52
)
)
)
)
→
(
¬
X51
)
→
(
¬
X49
)
→
X19
→
X53
)
)
)
)
↔
(
(
(
¬
(
X44
→
X6
→
X53
)
)
→
(
¬
(
X53
↔
(
(
¬
X35
)
→
(
¬
X10
)
→
X44
)
)
)
)
↔
(
(
(
X44
→
(
X35
↔
(
X10
↔
False
)
)
)
↔
(
(
¬
(
X52
→
X51
→
X30
→
X10
)
)
↔
(
¬
(
(
¬
X44
)
↔
(
X30
↔
X52
)
)
)
)
)
→
(
¬
(
X51
→
(
X35
↔
X44
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
(
¬
X47
)
→
(
X37
↔
(
(
¬
X2
)
→
X53
)
)
)
)
→
(
¬
(
(
¬
X37
)
↔
(
(
¬
X32
)
↔
(
(
¬
X2
)
→
X47
)
)
)
)
)
)
→
(
(
(
(
¬
(
X53
↔
(
X13
→
X7
→
False
)
)
)
↔
(
¬
(
(
¬
(
X47
→
(
(
¬
X13
)
↔
(
X7
→
X53
)
)
)
)
→
X37
→
X13
→
(
(
¬
X7
)
↔
X47
)
)
)
)
↔
(
¬
(
(
¬
X37
)
↔
X47
)
)
)
↔
(
¬
(
(
¬
X37
)
→
(
X7
↔
X47
)
)
)
)
)
↔
(
(
(
¬
(
(
X47
→
X37
→
(
¬
X12
)
→
X27
)
→
X37
→
X7
→
X47
)
)
↔
(
X47
→
(
X27
↔
X49
)
)
)
↔
(
(
(
¬
X12
)
→
(
X7
↔
(
X5
→
X41
)
)
)
↔
(
(
¬
X41
)
↔
(
X12
↔
X47
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
(
¬
X45
)
↔
(
X40
↔
X48
)
)
→
(
(
¬
(
(
¬
X38
)
→
(
X3
↔
X40
)
)
)
↔
(
¬
(
(
¬
X40
)
↔
(
X21
→
X3
→
X48
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
X42
)
→
(
(
¬
X40
)
↔
X45
)
)
→
(
¬
(
X42
→
(
(
¬
X40
)
↔
(
(
¬
X33
)
→
X46
)
)
)
)
)
)
↔
(
¬
(
(
¬
X46
)
→
(
¬
X42
)
→
X40
→
X48
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X46
↔
(
X20
↔
(
X0
↔
X42
)
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X25
)
→
X0
→
X42
)
)
)
)
)
→
(
X25
→
(
(
¬
X20
)
↔
(
X0
↔
X26
)
)
)
→
(
¬
(
(
¬
X40
)
↔
(
(
¬
X20
)
↔
X43
)
)
)
)
)
)
)
↔
(
(
(
(
X49
→
(
X40
↔
(
(
¬
X15
)
→
X50
)
)
)
→
(
¬
(
(
¬
X49
)
→
(
¬
X48
)
→
X50
)
)
)
↔
(
X50
↔
(
X49
→
False
)
)
)
→
(
¬
(
(
(
¬
(
X48
→
(
¬
X46
)
→
X39
→
X50
)
)
↔
(
¬
(
(
(
X48
↔
(
(
¬
X39
)
↔
(
X22
↔
X46
)
)
)
↔
(
X49
→
X48
→
(
X22
↔
X46
)
)
)
→
X49
→
X36
)
)
)
→
(
(
(
¬
(
X48
→
X1
)
)
→
(
X50
→
X39
→
(
X1
↔
False
)
)
→
X50
→
X48
→
(
X14
↔
(
X1
→
False
)
)
)
↔
(
(
¬
X39
)
→
(
(
¬
X14
)
↔
X50
)
)
)
)
)
→
(
(
(
X49
↔
(
X31
→
(
¬
X24
)
→
X46
)
)
↔
(
¬
(
X36
→
(
X24
↔
(
X17
→
X46
)
)
)
)
)
↔
(
¬
(
(
¬
X46
)
↔
(
X34
→
(
¬
X17
)
→
X49
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
(
¬
(
X49
→
(
¬
X46
)
→
(
¬
X30
)
→
X51
)
)
↔
(
(
¬
X19
)
→
X14
→
(
¬
X0
)
→
X30
)
)
→
(
(
(
(
(
(
X49
→
(
(
¬
X23
)
↔
(
(
¬
X19
)
→
X52
)
)
)
↔
(
¬
(
X52
↔
(
X19
→
X46
)
)
)
)
↔
(
X23
→
(
¬
X19
)
→
X52
)
)
→
(
¬
(
(
¬
X46
)
↔
X49
)
)
)
↔
(
¬
(
X49
↔
(
X46
→
(
(
¬
X42
)
↔
X52
)
)
)
)
)
↔
(
(
¬
(
(
¬
X19
)
↔
(
(
¬
X11
)
↔
(
X5
↔
X52
)
)
)
)
↔
(
¬
(
X52
↔
(
(
¬
X19
)
→
X23
)
)
)
)
)
)
)
↔
(
(
(
(
X48
→
(
¬
X12
)
→
X42
)
↔
(
(
(
¬
X42
)
↔
(
X34
→
(
(
¬
X12
)
↔
X48
)
)
)
→
(
X49
↔
(
X34
↔
(
(
¬
X12
)
→
X48
)
)
)
)
)
↔
(
¬
(
X53
→
X42
→
(
X12
↔
X49
)
)
)
)
→
(
(
(
¬
(
X48
→
(
¬
X42
)
→
X1
→
X52
)
)
→
(
¬
(
X52
↔
(
X33
→
(
¬
X17
)
→
X53
)
)
)
)
↔
(
(
¬
X48
)
→
(
(
¬
X1
)
↔
X52
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
X44
)
↔
(
(
¬
X24
)
→
(
X22
↔
X51
)
)
)
↔
(
¬
(
(
X22
→
X18
→
(
X6
↔
X9
)
)
→
(
(
¬
(
X47
↔
(
(
¬
X9
)
↔
X22
)
)
)
↔
(
X24
→
X18
→
X9
→
X44
)
)
)
)
)
↔
(
(
(
¬
(
X53
↔
(
(
¬
X47
)
↔
(
(
¬
X18
)
↔
X51
)
)
)
)
→
X53
→
(
(
¬
X18
)
↔
(
(
¬
X10
)
→
X47
)
)
)
↔
(
X53
→
(
X51
↔
(
(
¬
X7
)
→
X47
)
)
)
)
)
→
(
¬
(
(
¬
(
(
X24
→
(
X4
↔
X47
)
)
→
(
¬
(
(
¬
X49
)
↔
(
(
¬
X15
)
↔
(
X4
→
X51
)
)
)
)
)
)
→
(
(
¬
X47
)
↔
(
(
¬
X15
)
→
(
¬
X3
)
→
X51
)
)
)
)
)
)
)
)
↔
(
(
(
(
X53
↔
(
X52
↔
(
X8
↔
X37
)
)
)
→
(
(
(
X53
↔
(
X40
↔
X51
)
)
↔
(
¬
(
(
X51
→
(
¬
X40
)
→
X16
→
X53
)
→
(
¬
(
(
X53
→
X37
→
False
)
→
(
(
¬
(
X37
↔
X8
)
)
↔
(
X53
↔
(
X40
→
(
¬
X37
)
→
X51
)
)
)
)
)
)
)
)
↔
(
X53
→
(
(
¬
X37
)
↔
X51
)
)
)
)
↔
(
(
(
(
¬
(
X53
→
X45
→
(
X38
↔
X41
)
)
)
↔
(
¬
(
X51
↔
(
X38
→
X52
)
)
)
)
↔
(
¬
(
X51
→
(
¬
X41
)
→
(
X39
↔
X52
)
)
)
)
→
(
¬
(
X52
→
(
¬
X29
)
→
X39
)
)
→
(
(
(
¬
X31
)
↔
(
(
¬
X29
)
→
X39
)
)
↔
(
(
X51
→
X50
→
(
X39
↔
False
)
)
↔
(
¬
(
(
¬
X51
)
→
(
X31
↔
X52
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
(
X50
→
(
¬
X43
)
→
(
X35
↔
X45
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X35
)
↔
(
(
¬
X32
)
→
X50
)
)
)
→
(
¬
X35
)
→
(
(
¬
X21
)
↔
X43
)
)
)
)
→
(
X43
↔
(
X21
↔
X32
)
)
)
↔
(
(
¬
(
X50
↔
(
X35
↔
(
X26
↔
(
X2
→
False
)
)
)
)
)
↔
(
¬
(
X45
↔
(
(
¬
X35
)
↔
(
X28
↔
X50
)
)
)
)
)
)
↔
(
(
X53
↔
(
(
¬
X20
)
↔
X36
)
)
↔
(
(
(
X53
↔
(
X35
↔
X45
)
)
↔
(
(
¬
(
(
¬
X35
)
↔
X53
)
)
→
(
X53
↔
(
X45
↔
(
X28
↔
X36
)
)
)
)
)
→
(
¬
(
X53
↔
(
X36
→
(
X20
↔
X45
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
X45
↔
(
X27
→
(
X25
↔
False
)
)
)
)
↔
(
¬
(
(
(
¬
X45
)
↔
(
(
¬
X43
)
→
X53
)
)
→
(
¬
(
X53
↔
(
X27
→
X25
)
)
)
)
)
)
→
(
¬
(
X53
↔
(
X25
↔
X43
)
)
)
)
↔
(
X45
↔
(
(
¬
X25
)
↔
(
(
¬
X13
)
→
X53
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
299791...
and proposition id is
cf5ed5...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMRRfht5ftgMHsPXAYt6p4zePygFef8aKxK
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
¬
(
(
(
X46
→
(
X43
↔
(
(
¬
X26
)
↔
X44
)
)
)
→
(
¬
(
(
(
X46
↔
(
(
¬
X5
)
↔
X25
)
)
↔
(
X46
→
X26
→
X5
→
X44
)
)
→
(
¬
(
X46
↔
(
X44
↔
False
)
)
)
)
)
)
→
(
(
X46
↔
(
X7
→
X20
)
)
↔
(
¬
(
(
¬
X16
)
↔
(
(
¬
X13
)
→
X46
)
)
)
)
→
(
(
(
(
¬
(
X42
→
(
X26
↔
(
X0
→
X43
)
)
)
)
↔
(
¬
(
X42
↔
(
X26
↔
(
X0
→
X46
)
)
)
)
)
↔
(
¬
(
(
¬
X43
)
↔
(
X26
↔
(
X16
→
X46
)
)
)
)
)
↔
(
¬
(
(
¬
X43
)
→
(
X42
↔
(
(
¬
X20
)
↔
X46
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
X37
↔
(
X34
→
(
¬
X12
)
→
X35
)
)
↔
(
¬
(
(
(
¬
(
X37
→
(
X35
↔
(
(
¬
X22
)
↔
X34
)
)
)
)
↔
(
X47
→
X34
→
X22
→
False
)
)
→
(
(
(
¬
(
X37
→
X36
→
(
(
¬
X14
)
↔
X48
)
)
)
↔
(
¬
(
(
¬
X36
)
→
(
¬
X34
)
→
(
¬
X14
)
→
X37
)
)
)
↔
(
¬
(
X34
→
X36
)
)
)
→
(
¬
(
(
X48
↔
(
X46
↔
(
X37
→
(
X35
↔
False
)
)
)
)
→
(
¬
(
(
¬
X46
)
→
X36
→
(
X35
↔
X48
)
)
)
)
)
)
)
)
→
(
(
(
X43
↔
False
)
→
(
¬
(
(
(
(
¬
X47
)
→
(
X46
↔
(
X43
↔
X48
)
)
)
↔
(
X48
↔
(
X46
↔
(
(
¬
X4
)
→
X47
)
)
)
)
→
(
¬
X46
)
→
(
¬
X43
)
→
X48
)
)
)
↔
(
X48
→
(
X46
↔
(
(
¬
X18
)
↔
X43
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
X46
↔
(
(
¬
X19
)
→
(
X2
↔
X41
)
)
)
→
(
¬
X42
)
→
X2
→
X46
)
)
→
(
¬
(
X46
↔
(
X41
↔
X2
)
)
)
)
→
(
¬
(
X47
↔
(
X46
↔
(
X6
↔
X19
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
¬
(
(
¬
X40
)
→
(
¬
X39
)
→
(
¬
X11
)
→
X47
)
)
→
X24
→
X11
)
)
↔
(
¬
(
X39
↔
(
X24
→
X11
→
X43
)
)
)
)
→
(
¬
(
X47
↔
(
X39
→
X40
)
)
)
)
)
↔
(
(
¬
(
X42
↔
(
X39
→
X40
)
)
)
↔
(
X41
→
X24
→
X42
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
X49
↔
(
X43
↔
False
)
)
→
(
¬
X48
)
→
(
¬
X46
)
→
X17
→
X49
)
↔
(
¬
(
(
¬
(
X48
→
(
¬
X15
)
→
(
(
¬
X3
)
↔
X49
)
)
)
→
(
X46
↔
(
X3
↔
X49
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X31
)
↔
X49
)
)
→
(
¬
(
(
X49
↔
(
X33
↔
X1
)
)
→
(
¬
(
X49
↔
(
(
¬
X31
)
→
X48
)
)
)
)
)
)
→
(
¬
(
X48
↔
(
X21
→
X1
→
False
)
)
)
)
)
→
(
¬
(
(
¬
X46
)
↔
(
X31
↔
(
(
¬
X21
)
↔
X48
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X32
↔
(
(
¬
X8
)
→
X38
)
)
)
↔
(
¬
(
X43
↔
(
X32
↔
(
X29
↔
(
X27
↔
False
)
)
)
)
)
)
→
(
(
¬
(
(
¬
X43
)
↔
(
X38
↔
X45
)
)
)
↔
(
¬
(
X32
→
(
X10
↔
False
)
)
)
)
)
↔
(
(
X46
↔
(
X43
→
(
X30
↔
X38
)
)
)
↔
(
(
¬
(
(
¬
X38
)
↔
(
X30
↔
(
(
¬
X28
)
→
X46
)
)
)
)
↔
(
X38
→
X28
→
X45
)
)
)
)
→
(
(
¬
(
X33
→
X23
→
X9
→
X43
)
)
↔
(
(
¬
(
(
¬
X38
)
↔
(
(
¬
X33
)
↔
(
(
¬
X9
)
↔
X45
)
)
)
)
→
(
X38
↔
(
X23
→
X9
→
X45
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
X48
→
(
X27
↔
(
X8
↔
False
)
)
)
→
(
(
¬
(
X48
→
(
¬
X8
)
→
X34
)
)
↔
(
(
(
¬
X34
)
↔
(
X8
→
X48
)
)
→
(
¬
(
X27
→
X2
→
False
)
)
)
)
)
↔
(
(
X33
→
(
X15
↔
X8
)
)
→
(
(
¬
(
X45
→
(
(
¬
X21
)
↔
(
(
¬
X8
)
↔
X48
)
)
)
)
↔
(
¬
(
X21
→
(
¬
X15
)
→
X45
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X48
↔
(
(
¬
X45
)
→
(
¬
X44
)
→
X46
)
)
)
↔
(
¬
(
(
¬
X44
)
↔
(
X34
↔
(
(
¬
X20
)
↔
X46
)
)
)
)
)
→
(
¬
(
X34
↔
X20
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X44
↔
(
X11
↔
(
X3
→
X29
)
)
)
)
↔
(
¬
(
X44
↔
(
X41
↔
(
X3
→
X11
)
)
)
)
)
→
(
X33
→
(
X5
↔
(
(
¬
X4
)
↔
X29
)
)
)
→
(
(
(
¬
X29
)
↔
(
(
¬
X5
)
→
(
X4
↔
X44
)
)
)
↔
(
X41
↔
(
X4
→
X5
)
)
)
)
)
↔
(
(
(
¬
(
X44
↔
(
X30
→
X1
→
X23
)
)
)
↔
(
¬
(
X41
→
X33
→
X23
)
)
)
→
(
(
(
¬
X30
)
→
(
(
¬
X26
)
↔
(
X10
↔
X48
)
)
)
↔
(
X44
↔
(
X41
↔
(
X30
↔
X48
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
(
¬
X45
)
↔
(
X43
↔
X48
)
)
↔
(
(
¬
(
X48
↔
(
X42
↔
(
(
¬
X39
)
↔
X45
)
)
)
)
→
(
¬
(
(
¬
(
X48
↔
(
(
¬
X42
)
↔
(
(
¬
X19
)
↔
X45
)
)
)
)
→
X48
→
(
X43
↔
(
X25
↔
X22
)
)
)
)
)
)
→
(
¬
(
(
(
X46
↔
(
(
¬
X22
)
→
X47
)
)
↔
(
(
¬
(
X46
→
(
X18
↔
(
X12
→
X38
)
)
)
)
↔
(
X46
↔
(
(
¬
X38
)
↔
(
(
¬
X18
)
↔
X48
)
)
)
)
)
→
(
(
X38
→
X25
→
(
(
¬
X9
)
↔
X48
)
)
↔
(
¬
(
X47
↔
(
X38
↔
(
(
¬
X9
)
↔
X48
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
(
X46
→
(
¬
X28
)
→
X45
)
)
↔
(
(
¬
(
X45
→
X28
→
(
¬
X13
)
→
X46
)
)
↔
(
¬
(
(
X46
→
X28
→
X6
→
X45
)
→
(
¬
(
X46
↔
(
X37
↔
(
(
¬
X28
)
→
X45
)
)
)
)
)
)
)
)
→
(
X37
→
(
(
¬
X17
)
↔
(
X6
→
X32
)
)
)
→
(
¬
(
X37
→
X32
→
X49
)
)
→
(
(
X37
→
(
¬
X32
)
→
(
¬
X17
)
→
X49
)
↔
(
¬
(
(
¬
X32
)
→
(
X28
↔
(
(
¬
X17
)
→
X37
)
)
)
)
)
)
→
(
¬
(
(
X40
↔
(
X35
→
(
(
¬
X32
)
↔
X46
)
)
)
→
X49
→
X35
→
(
¬
X16
)
→
X46
)
)
)
→
(
(
(
¬
(
(
¬
(
X45
↔
(
(
¬
X40
)
→
X43
)
)
)
→
X36
→
(
¬
X24
)
→
X45
)
)
→
(
(
(
¬
(
X40
↔
X0
)
)
↔
(
(
¬
(
X45
↔
(
X43
→
X0
→
X31
)
)
)
↔
(
¬
(
X40
↔
(
(
¬
X0
)
→
X43
)
)
)
)
)
↔
(
¬
(
X45
↔
(
X31
→
X0
→
X49
)
)
)
)
)
↔
(
(
¬
(
(
¬
X7
)
→
X45
)
)
→
(
¬
(
X43
→
(
(
¬
X40
)
↔
(
X14
→
X45
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
45eeaf...
and proposition id is
32fd58...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMbZ8mUehkzUQubgL2w9dxgYWRWfXQVcpko
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
∀X50 :
prop
,
(
(
(
¬
(
(
¬
(
(
(
(
(
X46
→
X38
→
(
X12
↔
X39
)
)
↔
(
(
(
(
¬
X39
)
→
(
(
¬
X12
)
↔
(
X5
↔
X46
)
)
)
→
(
X39
↔
(
X20
↔
X46
)
)
)
→
X20
→
(
X12
↔
X39
)
)
)
↔
(
(
¬
(
X38
↔
(
X12
→
(
¬
X10
)
→
X49
)
)
)
↔
(
¬
(
X39
→
X12
)
)
)
)
↔
(
(
(
¬
(
(
¬
X34
)
↔
(
(
¬
X31
)
↔
(
(
¬
X12
)
↔
X38
)
)
)
)
↔
(
X39
→
(
X34
↔
X38
)
)
)
↔
(
¬
(
X31
↔
(
X12
↔
X39
)
)
)
)
)
→
(
(
X49
→
(
X47
↔
(
(
¬
X34
)
→
X39
)
)
)
↔
(
X49
→
(
X39
↔
(
(
¬
X16
)
↔
X34
)
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X34
)
↔
(
X11
→
(
(
¬
X1
)
↔
X41
)
)
)
)
↔
(
(
(
¬
X27
)
↔
(
X11
→
X41
)
)
→
(
X41
↔
(
X27
↔
(
X14
→
X46
)
)
)
→
(
¬
(
X27
↔
(
X13
↔
(
X11
→
False
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
(
(
¬
X25
)
→
X38
)
)
→
(
¬
(
X34
↔
(
X25
↔
(
(
¬
X24
)
→
X46
)
)
)
)
)
)
→
(
(
X41
→
X38
→
(
X7
↔
X46
)
)
↔
(
(
X46
→
(
X41
↔
(
X38
↔
X28
)
)
)
→
(
¬
X38
)
→
X34
→
X41
)
)
)
)
↔
(
(
(
X50
→
X38
→
X37
→
(
X28
↔
False
)
)
→
(
¬
(
X50
→
X34
→
(
X28
↔
X44
)
)
)
→
(
¬
(
X50
→
X44
→
(
X34
↔
X37
)
)
)
)
↔
(
X44
→
(
¬
X38
)
→
(
X17
↔
X50
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
X48
→
X46
→
X35
)
↔
(
(
X48
→
(
(
¬
X35
)
↔
(
X9
↔
X46
)
)
)
↔
(
X46
→
(
X32
↔
(
X9
→
False
)
)
)
)
)
→
(
(
(
¬
(
X35
→
X48
)
)
→
X42
→
(
(
¬
X35
)
↔
(
(
¬
X23
)
↔
X46
)
)
)
↔
(
X48
→
(
¬
X33
)
→
(
X23
↔
X42
)
)
)
)
)
→
(
(
(
¬
(
(
X46
→
X32
→
(
X29
↔
X50
)
)
→
(
¬
(
X35
↔
(
X29
↔
X32
)
)
)
→
(
¬
(
(
¬
(
(
¬
X35
)
→
X29
→
X46
)
)
→
(
(
X35
→
(
X32
↔
(
(
¬
X21
)
↔
X29
)
)
)
↔
(
(
¬
(
(
¬
X32
)
↔
(
(
¬
X29
)
→
(
¬
X19
)
→
X46
)
)
)
↔
(
X35
↔
(
X32
↔
X46
)
)
)
)
)
)
)
)
↔
(
X46
↔
(
(
¬
X40
)
↔
(
(
¬
X35
)
↔
X50
)
)
)
)
↔
(
(
(
¬
X46
)
→
(
X45
↔
X50
)
)
↔
(
(
X50
↔
(
X45
↔
(
X15
↔
False
)
)
)
↔
(
(
¬
(
(
X50
→
(
¬
X32
)
→
(
X15
↔
X45
)
)
→
X46
→
(
X15
↔
X32
)
)
)
↔
(
X50
↔
X32
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
X50
→
(
(
¬
X0
)
↔
X40
)
)
↔
(
(
¬
(
X36
→
(
X8
↔
X18
)
)
)
→
(
(
¬
(
X18
↔
(
X0
→
X36
)
)
)
↔
(
¬
(
X47
↔
(
X36
↔
(
X0
→
X8
)
)
)
)
)
)
)
→
(
(
(
¬
X18
)
→
(
(
¬
X4
)
↔
(
X3
↔
X48
)
)
)
↔
(
X36
→
X8
→
False
)
)
)
)
→
(
¬
(
(
(
¬
X26
)
→
(
¬
X2
)
→
X48
)
→
(
¬
(
(
¬
(
X6
↔
X30
)
)
→
(
¬
(
X30
→
(
X26
↔
(
X22
→
X48
)
)
)
)
)
)
)
)
→
(
(
¬
(
X50
↔
(
X48
↔
(
(
¬
X6
)
→
X43
)
)
)
)
↔
(
X48
↔
(
X43
↔
(
(
¬
X26
)
↔
X40
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
¬
(
(
¬
(
(
¬
(
(
¬
X17
)
→
X24
)
)
→
(
¬
(
(
¬
(
(
¬
X40
)
↔
(
X37
↔
(
(
¬
X3
)
↔
X50
)
)
)
)
→
X50
→
(
(
¬
X17
)
↔
(
X3
↔
X40
)
)
)
)
)
)
→
(
(
X44
↔
(
X8
↔
X40
)
)
↔
(
¬
(
X50
↔
(
X44
→
X37
→
X40
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X46
→
(
(
¬
X40
)
↔
(
(
¬
X29
)
→
X50
)
)
)
)
→
(
(
(
(
(
¬
(
(
¬
X44
)
→
X21
→
X46
)
)
↔
(
(
X44
↔
(
(
¬
X40
)
→
(
(
¬
X21
)
↔
X46
)
)
)
→
(
¬
(
X46
→
(
(
¬
X40
)
↔
(
X7
↔
X44
)
)
)
)
)
)
↔
(
X46
↔
(
X40
↔
X44
)
)
)
→
(
X46
↔
(
X7
→
X4
)
)
)
↔
(
¬
(
(
¬
X46
)
→
(
X44
↔
(
X4
→
X50
)
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X49
)
↔
(
(
¬
X46
)
↔
(
X16
→
X50
)
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X1
)
→
X46
)
)
)
)
↔
(
(
¬
(
X49
→
X47
→
(
¬
X16
)
→
X50
)
)
↔
(
¬
(
X50
↔
(
(
¬
X47
)
→
X14
→
X49
)
)
)
)
)
→
(
(
(
¬
(
X50
→
(
X13
↔
X29
)
)
)
↔
(
¬
(
X45
↔
(
X29
→
X27
)
)
)
)
↔
(
(
X47
→
(
X45
↔
(
X27
↔
(
X13
↔
False
)
)
)
)
↔
(
(
¬
X46
)
↔
(
(
¬
X45
)
↔
(
X27
→
X47
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X9
)
→
(
(
¬
X6
)
↔
(
(
¬
X2
)
→
X11
)
)
)
)
→
(
¬
X48
)
→
(
X9
↔
(
X2
↔
X50
)
)
)
→
(
(
¬
(
(
(
¬
X18
)
→
(
X11
↔
X47
)
)
→
(
¬
(
X50
→
(
(
¬
X11
)
↔
(
(
¬
X6
)
↔
X45
)
)
)
)
)
)
↔
(
(
¬
X45
)
→
X18
→
(
(
¬
X0
)
↔
X50
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
X49
→
(
X30
↔
X34
)
)
↔
(
(
¬
(
X49
↔
(
X30
→
(
(
¬
X12
)
↔
X43
)
)
)
)
↔
(
¬
(
(
¬
X43
)
↔
(
X34
→
(
¬
X12
)
→
X49
)
)
)
)
)
↔
(
(
¬
(
X47
→
X43
→
(
X19
↔
X41
)
)
)
↔
(
(
(
¬
(
X43
↔
(
X34
→
X33
→
X41
)
)
)
↔
(
(
(
¬
X41
)
→
(
X34
↔
(
X30
↔
X43
)
)
)
→
(
¬
(
X47
↔
X41
)
)
)
)
→
(
¬
(
(
¬
X33
)
→
X30
→
(
¬
X28
)
→
X34
)
)
)
)
)
→
(
(
(
(
(
X43
→
X15
)
→
(
(
X43
→
(
X36
↔
(
X15
↔
X47
)
)
)
↔
(
X43
→
(
(
¬
X36
)
↔
(
X26
→
X47
)
)
)
)
)
↔
(
¬
(
(
X43
→
(
X20
↔
X36
)
)
→
(
X43
↔
(
(
¬
X41
)
↔
X47
)
)
)
)
)
↔
(
¬
(
X47
↔
(
X36
→
(
(
¬
X5
)
↔
X15
)
)
)
)
)
↔
(
¬
(
X47
↔
(
X41
→
X36
→
X32
)
)
)
)
→
(
¬
(
X41
↔
(
X32
→
X15
→
False
)
)
)
)
)
→
(
(
(
(
¬
X43
)
→
X42
→
(
X22
↔
X48
)
)
↔
(
(
(
¬
(
(
¬
(
X49
→
X43
→
X35
)
)
→
(
¬
(
X43
→
(
X42
↔
(
X35
↔
False
)
)
)
)
→
(
¬
(
(
(
X49
→
(
X42
↔
X35
)
)
↔
(
¬
(
(
¬
X31
)
↔
(
(
¬
X23
)
→
X35
)
)
)
)
→
(
¬
(
X49
→
(
X43
↔
(
X35
↔
X42
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
X42
)
→
X35
→
(
¬
X25
)
→
X43
)
)
)
↔
(
¬
(
(
¬
X48
)
→
(
¬
X43
)
→
(
¬
X35
)
→
X49
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
X42
)
↔
(
(
¬
X39
)
→
(
¬
X38
)
→
X47
)
)
→
(
X47
↔
(
X39
↔
(
(
¬
X10
)
→
X38
)
)
)
)
)
→
(
¬
(
(
X39
↔
(
X38
→
X49
)
)
→
(
¬
(
(
¬
X48
)
↔
(
X47
↔
(
(
¬
X38
)
↔
X49
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
08aa8a...
and proposition id is
3f3f66...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMSEk21Qwqpya64gYaP1fVrryEW39NNtEP6
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
(
(
(
(
X46
↔
(
(
¬
X32
)
↔
(
X30
→
X43
)
)
)
↔
(
¬
(
X43
→
(
X32
↔
(
X30
→
X12
→
False
)
)
)
)
)
↔
(
¬
(
X30
→
X12
→
X32
)
)
)
↔
(
(
¬
(
X46
↔
(
X43
↔
(
X34
→
X9
)
)
)
)
→
(
X46
↔
(
X43
↔
(
(
¬
X9
)
↔
X30
)
)
)
)
)
↔
(
¬
(
X38
→
(
(
¬
X34
)
↔
(
(
¬
X32
)
→
X46
)
)
)
)
)
→
(
(
(
(
¬
X45
)
→
(
(
¬
X6
)
↔
(
(
¬
X2
)
→
X48
)
)
)
↔
(
X48
↔
(
X43
→
X2
→
X45
)
)
)
→
(
(
(
X38
→
(
X6
↔
X46
)
)
→
(
(
¬
X6
)
→
(
(
¬
X5
)
↔
X45
)
)
→
(
¬
(
X43
↔
(
X38
→
X5
→
X45
)
)
)
)
↔
(
¬
(
X46
→
(
(
¬
X43
)
↔
X45
)
)
)
)
)
→
(
X48
↔
(
(
¬
X43
)
→
(
(
¬
X20
)
↔
X45
)
)
)
)
→
(
(
(
(
(
¬
(
X46
→
(
¬
X33
)
→
X25
→
X38
)
)
↔
(
(
¬
(
X33
↔
(
X13
↔
X38
)
)
)
↔
(
X38
↔
(
X25
↔
(
(
¬
X13
)
→
X48
)
)
)
)
)
→
(
X48
↔
(
X38
→
(
X14
↔
X46
)
)
)
)
→
(
¬
(
X38
→
(
(
¬
X25
)
↔
X33
)
)
)
)
↔
(
(
X33
→
(
¬
X25
)
→
X42
)
↔
(
(
(
¬
(
(
¬
X38
)
↔
(
X33
↔
X42
)
)
)
↔
(
¬
(
X47
↔
(
(
¬
X21
)
→
X42
)
)
)
)
↔
(
X25
→
X47
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
X44
↔
(
X28
↔
(
X26
↔
X42
)
)
)
→
X48
→
(
¬
X24
)
→
X28
)
↔
(
¬
(
(
¬
(
(
¬
(
(
¬
X47
)
↔
(
X17
→
(
X15
↔
X48
)
)
)
)
→
(
¬
(
(
¬
X47
)
→
(
¬
X44
)
→
X17
→
X48
)
)
)
)
→
(
¬
(
X44
↔
(
(
¬
X23
)
↔
X48
)
)
)
→
(
¬
(
(
¬
X47
)
↔
(
X44
→
(
X26
↔
X48
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X42
)
→
(
(
¬
X40
)
↔
(
X22
→
X47
)
)
)
)
→
(
(
¬
(
X42
→
X49
)
)
↔
(
(
(
¬
X44
)
→
X42
→
(
(
¬
X40
)
↔
X49
)
)
↔
(
¬
(
X49
→
(
X44
↔
(
X42
↔
X47
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
(
¬
(
(
(
¬
(
X47
→
(
X36
↔
X40
)
)
)
↔
(
¬
(
X47
→
(
¬
X40
)
→
(
¬
X4
)
→
X42
)
)
)
→
(
X47
↔
(
X40
→
X42
)
)
)
)
↔
(
X42
→
X40
→
X36
)
)
↔
(
¬
(
X4
↔
X40
)
)
)
→
(
¬
(
X47
↔
(
X4
→
X36
)
)
)
)
→
X47
)
→
X47
→
X36
)
)
)
)
↔
(
(
(
¬
(
X47
→
(
¬
X37
)
→
X42
)
)
→
(
¬
(
X42
↔
(
X37
→
X8
)
)
)
→
(
¬
(
X47
→
X42
→
X27
→
(
X8
↔
False
)
)
)
)
→
(
¬
(
X47
→
(
X37
↔
(
(
¬
X27
)
↔
X49
)
)
)
)
)
)
→
(
(
X49
↔
(
X44
→
(
X31
↔
(
X11
↔
False
)
)
)
)
↔
(
(
(
¬
(
X39
→
X29
→
(
(
¬
X18
)
↔
X49
)
)
)
↔
(
X44
↔
(
X41
→
(
(
¬
X29
)
↔
X49
)
)
)
)
→
(
(
¬
(
(
¬
X44
)
→
X35
→
(
X31
↔
X49
)
)
)
→
(
¬
(
X49
→
(
X44
↔
(
X35
↔
(
X7
↔
False
)
)
)
)
)
)
→
(
(
¬
X41
)
↔
(
(
¬
X3
)
→
X44
)
)
)
)
→
(
(
¬
(
X41
↔
(
X16
→
(
X0
↔
False
)
)
)
)
↔
(
(
(
X47
↔
X0
)
→
(
¬
(
(
¬
(
X47
↔
(
X1
↔
(
X0
↔
False
)
)
)
)
→
X49
→
X47
→
(
(
¬
X1
)
↔
X44
)
)
)
)
↔
(
(
(
¬
(
X19
↔
(
X16
↔
(
X10
↔
X49
)
)
)
)
↔
(
(
¬
X44
)
↔
(
X10
↔
X49
)
)
)
↔
(
¬
(
(
¬
X44
)
→
X19
→
(
(
¬
X10
)
↔
X47
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
(
X31
↔
(
¬
(
X39
→
(
¬
X38
)
→
(
X1
↔
X47
)
)
)
)
↔
(
X39
↔
(
X31
↔
X1
)
)
)
↔
(
¬
(
X49
→
(
(
¬
X1
)
↔
X38
)
)
)
)
→
(
¬
(
(
(
¬
(
X49
→
X46
→
(
(
¬
X29
)
↔
X38
)
)
)
→
(
¬
(
(
¬
X46
)
→
(
X38
↔
(
(
¬
X7
)
→
X49
)
)
)
)
)
→
(
X46
↔
(
X29
↔
(
(
¬
X9
)
↔
X38
)
)
)
)
)
)
↔
(
(
(
X38
→
(
X28
↔
(
X17
↔
False
)
)
)
↔
(
(
¬
(
(
¬
X17
)
→
(
¬
X0
)
→
X46
)
)
↔
(
¬
(
X28
→
(
(
¬
X17
)
↔
(
(
¬
X0
)
↔
X38
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
X46
↔
(
X28
→
X17
)
)
→
(
(
(
X46
→
X35
→
X34
→
False
)
→
(
(
¬
(
X46
↔
(
(
¬
X17
)
→
(
¬
X6
)
→
X34
)
)
)
↔
(
(
¬
X34
)
→
(
(
¬
X6
)
↔
X35
)
)
)
)
↔
(
X46
→
(
(
¬
X28
)
↔
(
(
¬
X17
)
↔
X35
)
)
)
)
)
)
→
(
¬
(
X31
↔
(
X17
↔
(
X6
→
False
)
)
)
)
→
(
¬
(
X28
↔
(
X6
→
(
X4
↔
False
)
)
)
)
)
)
→
(
¬
(
X17
→
(
¬
X4
)
→
(
X2
↔
X6
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X36
↔
(
(
¬
X24
)
↔
(
X16
→
X42
)
)
)
)
→
(
¬
(
(
¬
X22
)
→
(
(
¬
X20
)
↔
(
X16
↔
X47
)
)
)
)
)
↔
(
¬
(
X39
→
(
(
¬
X16
)
↔
X24
)
)
)
)
↔
(
¬
(
(
X42
→
X39
→
X27
→
X46
)
→
(
¬
(
(
¬
X39
)
↔
(
(
¬
X36
)
→
(
X27
↔
X42
)
)
)
)
)
)
)
→
(
(
(
¬
X37
)
→
(
¬
X36
)
→
(
X25
↔
X42
)
)
↔
(
¬
(
(
¬
X42
)
→
(
(
¬
X36
)
↔
(
(
¬
X8
)
↔
X47
)
)
)
)
)
)
)
↔
(
(
(
(
(
(
¬
X48
)
→
(
(
¬
X23
)
↔
(
(
¬
X3
)
→
X49
)
)
)
↔
(
¬
(
(
(
X43
↔
(
X10
↔
X5
)
)
↔
(
(
X49
→
X23
→
(
(
¬
X18
)
↔
X43
)
)
↔
(
X49
→
(
X23
↔
(
X10
→
X43
)
)
)
)
)
→
X49
→
X48
→
X5
→
X43
)
)
)
↔
(
(
(
(
X49
↔
(
X46
→
(
(
¬
X14
)
↔
X43
)
)
)
→
(
X41
↔
(
(
¬
X14
)
→
X46
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X43
)
→
(
X14
↔
X48
)
)
)
)
)
↔
(
¬
(
X48
→
X46
→
(
X41
↔
False
)
)
)
)
)
↔
(
(
(
¬
(
X44
↔
(
(
¬
X42
)
→
(
¬
X40
)
→
X46
)
)
)
↔
(
(
(
¬
(
X46
→
(
(
¬
X32
)
↔
X44
)
)
)
↔
(
¬
(
(
X46
↔
(
X44
↔
(
(
¬
X32
)
↔
X41
)
)
)
→
(
¬
(
X46
↔
(
X21
→
X32
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X42
↔
(
(
¬
X32
)
→
(
X21
↔
X48
)
)
)
)
↔
(
(
¬
X46
)
→
(
X42
↔
(
(
¬
X32
)
→
X48
)
)
)
)
↔
(
X48
→
(
X46
↔
X32
)
)
)
↔
(
X32
↔
(
X21
↔
X42
)
)
)
→
(
(
¬
X46
)
↔
(
(
¬
X42
)
↔
X48
)
)
)
)
)
↔
(
(
(
¬
X46
)
↔
(
(
¬
X45
)
↔
(
(
¬
X12
)
↔
X48
)
)
)
↔
(
X48
↔
(
X46
→
(
X44
↔
False
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X49
→
(
X46
↔
X19
)
)
)
↔
(
(
X46
→
X13
)
→
X49
→
(
X19
↔
X33
)
)
)
→
(
(
X49
→
X46
→
X33
)
↔
(
(
(
¬
(
X46
↔
(
X30
↔
X49
)
)
)
↔
(
¬
(
(
¬
X48
)
→
X33
→
X30
→
X49
)
)
)
↔
(
(
(
¬
(
X26
→
(
X11
↔
X48
)
)
)
↔
(
X48
→
(
X46
↔
X49
)
)
)
→
(
¬
(
X26
↔
(
X15
↔
(
X11
→
X48
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
16653b...
and proposition id is
02b651...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMYMMagNtyfQANkbeZQaLsJaTGbpaXHJyPT
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
¬
(
(
(
(
¬
(
(
X46
→
X32
→
(
X30
↔
(
X20
↔
False
)
)
)
→
(
(
X42
→
(
X32
↔
(
(
¬
X19
)
↔
X46
)
)
)
↔
(
¬
(
X42
→
X19
→
X20
)
)
)
)
)
→
(
(
(
X50
↔
(
X5
↔
False
)
)
↔
(
(
¬
(
X42
→
X30
→
X50
)
)
→
(
¬
(
(
¬
(
X50
→
X42
→
(
X30
↔
(
X10
↔
False
)
)
)
)
→
(
¬
(
X50
→
X10
→
(
X5
↔
X42
)
)
)
)
)
)
)
↔
(
(
(
(
¬
X47
)
→
(
¬
X5
)
→
X1
→
X50
)
↔
(
(
(
(
¬
X47
)
↔
(
(
¬
X30
)
↔
(
X5
↔
X50
)
)
)
→
(
¬
(
X50
→
X49
→
X5
→
False
)
)
)
→
X49
→
(
X30
↔
X47
)
)
)
↔
(
X32
→
(
X30
↔
(
X5
↔
X49
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
(
(
¬
(
X47
→
(
(
¬
X16
)
↔
X23
)
)
)
→
(
(
(
(
¬
X46
)
↔
(
X42
→
(
X16
↔
X47
)
)
)
↔
(
¬
(
(
¬
X46
)
↔
(
X42
↔
(
(
¬
X23
)
↔
X47
)
)
)
)
)
↔
(
(
¬
X23
)
→
X42
)
)
)
→
(
¬
(
X42
↔
(
X16
→
False
)
)
)
)
↔
(
X47
↔
(
X46
→
X41
→
X42
)
)
)
↔
(
(
¬
(
(
¬
X31
)
↔
(
(
¬
X26
)
→
X47
)
)
)
↔
(
(
¬
(
X31
→
(
¬
X26
)
→
X47
)
)
→
(
¬
(
X47
↔
(
(
¬
X42
)
↔
X46
)
)
)
→
(
¬
(
(
¬
(
X42
→
(
¬
X33
)
→
X46
)
)
→
(
¬
(
X47
↔
(
X42
↔
(
X33
↔
X31
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
X33
→
False
)
)
↔
(
(
(
X41
→
(
X33
↔
(
X31
↔
X27
)
)
)
↔
(
(
¬
X34
)
→
X33
→
X31
→
X41
)
)
↔
(
(
¬
X34
)
↔
(
(
¬
X33
)
→
X41
)
)
)
)
)
→
(
¬
(
(
X47
→
(
X46
↔
(
(
¬
X3
)
↔
X33
)
)
)
→
(
(
(
X47
→
X29
→
(
(
¬
X8
)
↔
X48
)
)
↔
(
X47
↔
(
(
¬
X29
)
→
(
¬
X8
)
→
X46
)
)
)
↔
(
(
¬
X46
)
→
(
(
¬
X29
)
↔
(
(
¬
X3
)
↔
X48
)
)
)
)
)
)
)
)
↔
(
(
¬
(
X47
→
X25
→
X21
→
(
X11
↔
False
)
)
)
↔
(
¬
(
X46
↔
(
X42
→
X28
)
)
)
)
)
)
→
(
¬
(
(
(
(
X50
→
(
X49
↔
(
X44
↔
(
X14
→
False
)
)
)
)
↔
(
¬
(
(
¬
(
(
X49
→
(
X43
↔
(
(
¬
X4
)
→
X44
)
)
)
→
(
X44
↔
(
X43
↔
(
(
¬
X39
)
→
X49
)
)
)
)
)
→
X50
→
(
X43
↔
False
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
¬
X43
)
→
(
¬
X38
)
→
(
X9
↔
X44
)
)
)
→
(
(
X43
↔
(
(
¬
X38
)
→
(
¬
X36
)
→
X44
)
)
↔
(
¬
(
X43
↔
X17
)
)
)
)
→
(
(
X22
↔
(
X9
→
(
X2
↔
X43
)
)
)
↔
(
(
¬
X24
)
↔
(
(
¬
X7
)
→
(
¬
X2
)
→
X44
)
)
)
)
)
↔
(
(
(
(
X49
→
(
(
¬
X18
)
↔
X38
)
)
→
(
¬
(
X40
↔
(
(
¬
X38
)
→
(
X18
↔
X48
)
)
)
)
→
(
X49
↔
(
X18
↔
X38
)
)
)
↔
(
¬
(
X48
→
(
¬
X43
)
→
(
¬
X40
)
→
X49
)
)
)
↔
(
¬
(
(
¬
X48
)
→
(
¬
X43
)
→
(
(
¬
X24
)
↔
X49
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
X43
)
↔
(
(
¬
X12
)
↔
X45
)
)
→
(
¬
(
X45
↔
(
X43
↔
(
(
¬
X6
)
→
X15
)
)
)
)
→
(
(
(
¬
X40
)
↔
(
X15
↔
(
(
¬
X12
)
↔
X43
)
)
)
→
(
X45
↔
(
(
¬
X15
)
↔
X43
)
)
)
→
(
¬
(
X45
↔
(
X43
→
X40
→
False
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X50
↔
(
(
¬
X13
)
↔
(
(
¬
X0
)
→
X40
)
)
)
)
→
(
(
X50
↔
(
X40
→
(
X35
↔
False
)
)
)
↔
(
¬
(
(
¬
X13
)
→
(
¬
(
(
¬
X37
)
↔
(
X35
→
(
¬
X13
)
→
X40
)
)
)
→
X50
→
X37
→
X35
→
X13
)
)
)
)
→
(
(
X43
→
(
X35
↔
X40
)
)
↔
(
(
X45
↔
(
(
¬
X35
)
↔
(
X13
→
X40
)
)
)
↔
(
X43
↔
(
(
¬
X40
)
↔
(
(
¬
X35
)
↔
X45
)
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
X25
→
X39
)
→
(
¬
(
(
X25
↔
(
X23
→
(
X6
↔
X48
)
)
)
→
(
¬
(
X25
↔
(
(
¬
X23
)
↔
X26
)
)
)
)
)
)
↔
(
X26
↔
(
X11
↔
(
(
¬
X6
)
→
X39
)
)
)
)
↔
(
(
¬
(
(
(
¬
X43
)
↔
(
X39
→
X25
→
X49
)
)
→
(
¬
(
X49
→
X44
→
X43
→
X25
→
False
)
)
)
)
→
(
(
¬
(
(
(
¬
X48
)
↔
(
(
¬
X43
)
→
(
(
¬
X24
)
↔
X49
)
)
)
→
(
¬
(
(
¬
X48
)
→
(
¬
X43
)
→
(
X25
↔
X49
)
)
)
)
)
↔
(
¬
(
(
(
¬
X13
)
↔
(
X10
→
X48
)
)
→
(
¬
(
X49
↔
(
X48
→
X24
→
X10
)
)
)
)
)
)
)
)
→
(
(
(
(
X36
↔
(
(
¬
X35
)
↔
X38
)
)
↔
(
(
X36
→
X35
→
X17
→
X28
)
↔
(
(
(
(
¬
(
(
¬
X28
)
↔
(
X17
→
(
X12
↔
X44
)
)
)
)
↔
(
¬
(
X44
→
X36
)
)
)
↔
(
X44
→
X48
)
)
→
(
X38
↔
(
(
¬
X12
)
↔
X48
)
)
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X28
)
↔
X35
)
)
)
)
↔
(
(
(
¬
(
(
¬
X27
)
↔
(
X16
↔
(
(
¬
X0
)
→
X48
)
)
)
)
↔
(
(
X49
→
(
X35
↔
X27
)
)
→
(
¬
(
(
(
¬
X35
)
↔
(
X16
→
X49
)
)
→
(
¬
(
(
(
¬
X35
)
→
(
X16
↔
(
(
¬
X0
)
↔
X49
)
)
)
→
(
¬
(
X27
→
X49
)
)
)
)
)
)
)
)
↔
(
(
¬
(
X42
→
(
¬
X27
)
→
(
¬
X20
)
→
X49
)
)
↔
(
¬
(
X35
→
(
X27
↔
X42
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
(
¬
(
X42
↔
(
(
¬
X22
)
→
(
(
¬
X2
)
↔
X46
)
)
)
)
↔
(
¬
(
(
¬
(
X46
→
(
(
¬
X41
)
↔
(
X22
↔
X42
)
)
)
)
→
(
(
¬
(
X44
→
(
¬
X42
)
→
(
¬
X41
)
→
X45
)
)
↔
(
X45
→
(
(
¬
X44
)
↔
(
(
¬
X41
)
→
X47
)
)
)
)
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X45
)
→
(
X14
↔
X46
)
)
)
)
)
)
↔
(
(
¬
(
X29
↔
(
X22
↔
False
)
)
)
→
(
(
X47
↔
(
(
¬
X14
)
→
X29
)
)
↔
(
X31
↔
(
X14
↔
X22
)
)
)
)
)
↔
(
(
(
X43
→
(
¬
X9
)
→
X33
)
↔
(
(
X43
→
X42
→
(
(
¬
X33
)
↔
X45
)
)
→
(
¬
X9
)
→
X45
)
)
↔
(
(
(
X45
→
(
(
¬
X19
)
↔
(
X9
→
X42
)
)
)
↔
(
¬
(
X50
↔
(
(
¬
X42
)
→
X9
→
X44
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
X34
)
↔
(
X5
↔
X45
)
)
)
→
(
(
X45
→
(
X34
↔
False
)
)
↔
(
¬
(
X34
↔
(
(
¬
X9
)
↔
(
(
¬
X5
)
↔
X42
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
(
X40
→
X30
→
X7
→
X8
)
→
(
¬
(
X50
→
X30
→
X7
→
X8
)
)
)
↔
(
X8
→
(
X7
↔
X50
)
)
)
→
(
¬
(
X32
↔
(
X8
→
X7
)
)
)
)
↔
(
(
(
¬
(
X21
↔
X7
)
)
→
(
(
(
¬
X21
)
→
(
(
¬
X18
)
↔
(
X8
→
X43
)
)
)
↔
(
X43
→
(
X8
↔
(
(
¬
X7
)
→
X21
)
)
)
)
)
→
X8
→
X21
)
)
→
(
(
(
¬
(
(
¬
X3
)
↔
(
X1
↔
X4
)
)
)
→
(
¬
(
X45
→
(
X4
↔
(
(
¬
X1
)
→
X30
)
)
)
)
→
(
X15
↔
(
X3
→
X30
)
)
)
↔
(
(
(
(
(
¬
X30
)
↔
(
X3
→
X37
)
)
↔
(
(
X37
→
(
¬
X3
)
→
X32
)
↔
(
(
¬
(
X37
→
(
X30
↔
X21
)
)
)
↔
(
(
¬
(
(
¬
X32
)
↔
(
(
¬
X30
)
↔
X37
)
)
)
↔
(
X21
→
X3
→
X37
)
)
)
)
)
→
(
¬
(
X37
→
X30
→
X21
)
)
)
↔
(
(
¬
X30
)
→
X3
→
X1
→
X37
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
a41de4...
and proposition id is
cf85eb...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMFYH17XHZUseNEPQgKBCsqaJFFLKvor2Ao
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
∀X51 :
prop
,
(
¬
∀X52 :
prop
,
∀X53 :
prop
,
(
¬
∀X54 :
prop
,
(
(
(
(
(
(
¬
(
(
(
X43
↔
(
X41
↔
(
X39
→
X22
)
)
)
↔
(
¬
(
X41
→
(
(
¬
X39
)
↔
(
(
¬
X17
)
→
X43
)
)
)
)
)
→
(
¬
(
X54
→
X17
→
X12
)
)
→
(
¬
(
(
(
¬
X39
)
→
(
X22
↔
(
(
¬
X17
)
↔
X54
)
)
)
→
(
X8
↔
X54
)
)
)
→
(
(
¬
X12
)
↔
(
X8
↔
X17
)
)
)
)
↔
(
¬
(
X25
→
(
X17
↔
X43
)
)
)
)
↔
(
¬
(
(
(
¬
X10
)
→
X49
)
→
(
¬
(
(
(
¬
(
X54
↔
(
X46
↔
(
X38
→
X12
)
)
)
)
↔
(
X54
→
X38
→
X10
→
False
)
)
→
(
(
(
¬
(
(
¬
X10
)
→
(
¬
X1
)
→
X25
)
)
→
(
¬
(
(
X46
→
(
¬
X12
)
→
X10
→
X25
)
→
(
¬
(
(
¬
X25
)
→
(
X10
↔
(
X1
↔
X46
)
)
)
)
)
)
)
↔
(
¬
(
X46
→
(
X25
↔
X12
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X50
↔
(
X46
→
(
(
¬
X45
)
↔
X48
)
)
)
)
↔
(
¬
(
(
(
¬
(
X50
→
(
¬
X45
)
→
X48
)
)
→
(
¬
(
X50
↔
(
X49
→
(
(
¬
X2
)
↔
X48
)
)
)
)
)
→
(
(
¬
X48
)
↔
(
(
¬
X2
)
↔
X49
)
)
)
)
)
→
(
(
X33
→
X32
→
(
(
¬
X26
)
↔
X50
)
)
↔
(
(
(
X33
→
(
¬
X32
)
→
X26
→
X49
)
↔
(
(
¬
X48
)
→
X33
→
(
(
¬
X32
)
↔
X49
)
)
)
↔
(
X48
↔
(
(
¬
X33
)
↔
X49
)
)
)
)
)
↔
(
(
(
(
¬
X50
)
↔
(
X45
↔
X54
)
)
↔
(
(
¬
(
X50
→
(
(
¬
X40
)
↔
(
(
¬
X28
)
→
X54
)
)
)
)
→
(
(
X50
→
X40
→
(
(
¬
X6
)
↔
X28
)
)
↔
(
¬
(
(
¬
X45
)
→
(
X40
↔
(
X28
→
X50
)
)
)
)
)
→
(
¬
(
X50
→
X46
→
(
X45
↔
(
X40
→
False
)
)
)
)
)
)
↔
(
(
X45
↔
(
X6
→
X5
)
)
↔
(
X50
↔
(
X45
→
(
X6
↔
X34
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X51
→
(
(
¬
X9
)
↔
X50
)
)
)
↔
(
(
¬
(
(
¬
X51
)
→
X53
)
)
↔
(
X50
→
(
X13
↔
(
X9
↔
False
)
)
)
)
)
↔
(
¬
(
(
¬
X50
)
→
(
¬
X37
)
→
(
(
¬
X13
)
↔
X53
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
X47
↔
(
(
¬
X31
)
↔
X51
)
)
)
→
(
¬
(
X42
→
X50
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X50
)
↔
(
(
¬
X44
)
↔
(
X16
→
X54
)
)
)
)
↔
(
(
X53
↔
(
X50
→
X44
→
X16
)
)
↔
(
¬
(
X44
→
(
(
¬
X19
)
↔
(
X16
↔
X50
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X50
)
↔
(
X47
↔
(
X3
↔
X54
)
)
)
)
↔
(
¬
(
X50
→
(
X47
↔
(
X7
↔
X54
)
)
)
)
)
↔
(
(
¬
(
(
X53
↔
(
(
¬
X47
)
↔
(
(
¬
X31
)
→
X50
)
)
)
→
(
¬
(
X53
→
(
¬
X31
)
→
X0
→
X50
)
)
)
)
↔
(
¬
(
(
¬
X50
)
→
(
(
¬
X31
)
↔
(
X23
↔
X53
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
X51
↔
(
X36
→
X27
→
(
X21
↔
False
)
)
)
→
(
¬
(
(
¬
(
X35
↔
(
(
¬
X27
)
↔
X36
)
)
)
→
(
¬
(
(
¬
(
X51
↔
(
X36
→
X35
→
X27
)
)
)
→
X36
→
(
X35
↔
(
(
¬
X4
)
↔
X51
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
¬
X51
)
↔
(
(
¬
X36
)
↔
(
(
¬
X35
)
↔
X52
)
)
)
)
→
X52
→
X20
)
)
→
(
(
¬
(
X50
↔
(
(
¬
X49
)
↔
(
(
¬
X30
)
→
X52
)
)
)
)
↔
(
(
¬
X35
)
→
(
¬
X30
)
→
X51
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X53
→
(
X29
↔
(
X24
↔
X52
)
)
)
)
↔
(
X49
→
X29
→
X11
→
False
)
)
→
(
(
¬
(
X15
↔
False
)
)
↔
(
(
(
¬
(
(
X51
↔
(
X49
↔
(
X24
→
X53
)
)
)
→
(
X53
↔
(
X51
→
X14
)
)
)
)
↔
(
¬
(
X49
→
X24
→
(
(
¬
X14
)
↔
X18
)
)
)
)
↔
(
¬
(
(
¬
X24
)
→
(
¬
X18
)
→
X53
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
(
¬
X20
)
→
(
(
¬
X13
)
↔
X51
)
)
↔
(
¬
(
(
(
¬
X47
)
↔
(
X13
→
X51
)
)
→
(
(
X50
→
(
¬
X20
)
→
(
X13
↔
X47
)
)
↔
(
¬
(
X51
→
X20
)
)
)
)
)
)
→
(
¬
(
(
(
X43
→
(
¬
X39
)
→
(
(
¬
X24
)
↔
X50
)
)
→
(
¬
(
X43
↔
X47
)
)
)
→
(
¬
(
X50
↔
(
X43
↔
X39
)
)
)
)
)
→
(
¬
(
X47
↔
X51
)
)
)
↔
(
(
¬
(
(
(
X53
↔
(
X49
↔
(
(
¬
X26
)
↔
X51
)
)
)
→
(
¬
(
(
¬
X49
)
→
X43
→
(
¬
X29
)
→
X51
)
)
)
→
(
(
(
(
X51
→
X19
→
(
(
¬
X10
)
↔
X31
)
)
→
(
X51
↔
(
X47
↔
(
X31
→
(
X29
↔
False
)
)
)
)
)
↔
(
¬
(
(
¬
X19
)
→
(
(
¬
X10
)
↔
X51
)
)
)
)
↔
(
X53
↔
(
X51
→
(
¬
X19
)
→
X31
)
)
)
)
)
↔
(
(
(
X31
↔
X18
)
↔
(
¬
(
(
¬
(
X47
↔
(
(
¬
X29
)
↔
(
(
¬
X6
)
↔
X30
)
)
)
)
→
(
¬
(
X30
→
(
(
¬
X29
)
↔
(
X12
→
X47
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
X30
)
↔
(
(
¬
X18
)
↔
X31
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X53
→
(
X51
↔
(
(
¬
X38
)
→
X52
)
)
)
)
→
(
¬
(
X38
↔
(
(
¬
X32
)
→
X52
)
)
)
)
↔
(
(
(
(
¬
(
X53
→
(
¬
X51
)
→
(
X50
↔
X52
)
)
)
↔
(
(
X50
→
X42
→
(
X14
↔
X51
)
)
→
(
¬
(
(
¬
X51
)
→
(
X50
↔
(
X42
→
X53
)
)
)
)
)
)
↔
(
X52
→
(
X33
↔
(
X14
→
X50
)
)
)
)
→
X52
→
(
X42
↔
X14
)
)
)
→
(
(
(
¬
(
(
¬
X37
)
↔
(
(
¬
X8
)
→
(
(
¬
X0
)
↔
X51
)
)
)
)
→
(
¬
(
X37
↔
(
X11
↔
X0
)
)
)
)
↔
(
(
(
(
¬
(
X51
→
X53
)
)
→
(
¬
(
(
¬
X8
)
→
(
X7
↔
X52
)
)
)
)
↔
(
¬
(
(
¬
X22
)
→
X8
→
X51
)
)
)
↔
(
(
X53
↔
(
(
¬
X11
)
→
(
X3
↔
X52
)
)
)
↔
(
¬
(
(
¬
X52
)
↔
(
(
¬
X51
)
↔
(
(
¬
X11
)
↔
X53
)
)
)
)
)
)
)
→
(
¬
(
(
X53
↔
(
(
¬
X17
)
→
X11
→
X51
)
)
→
X51
→
X11
→
(
¬
X9
)
→
X17
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
(
X53
→
(
X52
↔
(
(
¬
X16
)
↔
X46
)
)
)
→
(
(
X53
→
(
(
¬
X35
)
↔
(
(
¬
X28
)
↔
X46
)
)
)
↔
(
(
¬
(
X28
↔
(
X1
↔
False
)
)
)
↔
(
¬
(
(
¬
X46
)
↔
(
(
¬
X35
)
→
X28
→
X52
)
)
)
)
)
)
)
↔
(
¬
(
X44
→
(
(
¬
X1
)
↔
X52
)
)
)
)
→
(
(
¬
(
(
X45
↔
(
(
¬
X5
)
→
X35
)
)
→
(
(
X44
↔
(
X35
↔
(
X21
↔
X45
)
)
)
↔
(
X45
→
X35
→
(
X5
↔
X44
)
)
)
)
)
↔
(
(
¬
(
X44
↔
(
(
¬
X28
)
→
(
(
¬
X27
)
↔
X35
)
)
)
)
→
X28
→
X45
)
)
)
)
↔
(
(
(
(
(
X48
→
X45
→
(
X2
↔
X52
)
)
↔
(
X53
→
(
¬
X45
)
→
X48
)
)
↔
(
(
¬
(
X53
→
(
¬
X41
)
→
X2
→
X48
)
)
→
(
(
X15
→
(
(
¬
X2
)
↔
X53
)
)
↔
(
(
(
X53
↔
(
X41
↔
False
)
)
↔
(
(
¬
(
X53
→
(
(
¬
X2
)
↔
X41
)
)
)
↔
(
¬
(
X36
→
X15
→
False
)
)
)
)
→
X36
→
X41
)
)
)
)
→
(
(
(
¬
(
X54
→
X53
→
(
X51
↔
(
X4
→
False
)
)
)
)
↔
(
¬
(
X53
→
X48
→
(
¬
X4
)
→
X51
)
)
)
↔
(
¬
(
X54
→
X48
→
(
(
¬
X41
)
↔
X51
)
)
)
)
)
→
(
(
¬
X45
)
→
(
¬
X40
)
→
(
¬
X34
)
→
X54
)
→
(
¬
(
(
¬
(
(
¬
(
X52
↔
(
X45
→
X23
→
X40
)
)
)
→
(
¬
(
(
¬
(
X52
→
X45
→
X25
→
X34
)
)
→
(
¬
(
(
¬
X40
)
→
(
X25
↔
X52
)
)
)
)
)
)
)
→
(
¬
(
X40
↔
(
(
¬
X34
)
↔
X45
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
fc9e5e...
and proposition id is
78b10c...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMWSQp9bQfej29TNXhfCcLccVGYDTa4ieXV
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
¬
(
(
(
(
(
¬
(
(
(
¬
X36
)
↔
(
X35
→
X18
→
X43
)
)
→
(
¬
(
(
X44
↔
(
(
¬
X37
)
→
(
(
¬
X35
)
↔
X43
)
)
)
→
(
X43
↔
(
(
¬
X37
)
→
(
(
¬
X18
)
↔
X44
)
)
)
)
)
)
)
↔
(
(
X43
→
X35
→
X33
→
(
X19
↔
False
)
)
↔
(
¬
(
X43
↔
(
X37
↔
(
X33
↔
X19
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X43
)
→
X36
→
(
(
¬
X3
)
↔
X44
)
)
)
→
(
(
(
¬
(
X48
→
(
(
¬
X16
)
↔
(
(
¬
X3
)
↔
X36
)
)
)
)
↔
(
X43
↔
(
(
¬
X36
)
↔
(
(
¬
X16
)
→
X48
)
)
)
)
↔
(
¬
(
X16
↔
X36
)
)
)
)
→
X48
→
(
¬
X27
)
→
(
¬
X3
)
→
X43
)
)
↔
(
(
(
(
(
¬
(
X34
↔
(
X8
→
X45
)
)
)
↔
(
X44
→
(
X34
↔
X7
)
)
)
→
(
(
X47
↔
(
X45
→
X43
→
X8
→
False
)
)
→
(
¬
(
(
¬
X40
)
→
X8
→
X45
)
)
)
→
X47
→
(
X8
↔
(
X7
→
False
)
)
)
→
(
(
X47
→
False
)
↔
(
(
(
X47
↔
(
X45
↔
(
X43
→
(
X42
↔
False
)
)
)
)
→
(
(
(
X47
↔
(
X45
→
(
(
¬
X6
)
↔
X43
)
)
)
↔
(
¬
(
X45
→
X43
→
(
X6
↔
X20
)
)
)
)
↔
(
¬
(
X45
→
X47
)
)
)
)
↔
(
¬
(
X47
↔
(
X43
↔
False
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X47
→
(
¬
X24
)
→
X48
)
)
→
X48
→
(
(
¬
X39
)
↔
(
(
¬
X24
)
→
X43
)
)
)
↔
(
(
¬
(
X48
↔
(
X47
→
X43
→
(
X26
↔
False
)
)
)
)
→
(
X48
↔
(
(
¬
X28
)
→
(
X26
↔
X47
)
)
)
)
)
↔
(
¬
(
(
(
X47
↔
(
(
¬
X2
)
↔
X25
)
)
↔
(
(
¬
(
(
¬
X45
)
↔
(
(
¬
X39
)
↔
X47
)
)
)
→
(
(
¬
(
X47
↔
(
X25
↔
(
X2
→
X45
)
)
)
)
↔
(
¬
(
(
¬
X39
)
→
X47
)
)
)
)
)
→
(
¬
(
(
X43
→
(
X39
↔
(
X9
→
X25
)
)
)
→
(
(
(
¬
X43
)
→
(
¬
X39
)
→
(
X25
↔
X47
)
)
↔
(
¬
(
X25
↔
(
X21
↔
X39
)
)
)
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
(
¬
(
(
¬
(
X48
→
X47
→
X42
→
X12
→
False
)
)
→
(
¬
(
X30
→
(
X22
↔
(
X12
→
X47
)
)
)
)
)
)
→
(
(
(
¬
X31
)
→
X47
)
↔
(
¬
(
X42
↔
(
(
¬
X30
)
→
(
X29
↔
X47
)
)
)
)
)
)
)
→
(
(
¬
(
X42
↔
(
(
¬
X5
)
→
(
(
¬
X1
)
↔
X31
)
)
)
)
↔
(
X38
↔
(
X5
→
(
(
¬
X1
)
↔
X47
)
)
)
)
)
)
↔
(
(
(
X47
↔
(
X38
→
X22
→
False
)
)
→
(
¬
(
X38
→
X10
→
X17
)
)
)
→
(
¬
(
X38
→
X17
→
(
(
¬
X4
)
↔
X22
)
)
)
)
)
↔
(
(
¬
(
(
X46
↔
(
X32
→
X14
→
X48
)
)
→
(
¬
(
(
(
¬
X47
)
→
X32
→
X49
)
→
(
X49
↔
(
(
¬
X41
)
↔
(
X32
→
X46
)
)
)
→
(
¬
(
(
¬
X46
)
→
X48
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
X42
)
↔
(
X11
↔
(
(
¬
X0
)
→
X47
)
)
)
→
(
(
¬
X47
)
↔
(
(
¬
X42
)
→
(
X11
↔
X49
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
X42
→
(
X23
↔
(
X15
→
X13
)
)
)
↔
(
¬
(
X41
↔
(
(
¬
X15
)
↔
(
(
¬
X13
)
↔
X23
)
)
)
)
)
→
X23
→
(
X15
↔
(
X13
→
X47
)
)
)
)
→
(
¬
X42
)
→
(
X41
↔
(
X15
→
X49
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
¬
(
(
X46
↔
(
X38
↔
X49
)
)
→
(
¬
(
X42
→
X48
)
)
)
)
↔
(
¬
(
X49
→
(
X38
↔
(
(
¬
X15
)
→
X27
)
)
)
)
)
↔
(
(
X40
↔
(
X19
→
(
¬
X10
)
→
X38
)
)
↔
(
¬
(
(
X40
→
(
X36
↔
(
X28
→
X13
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X41
)
→
(
X40
↔
X48
)
)
)
→
(
¬
(
(
¬
X19
)
→
(
(
¬
X10
)
↔
X41
)
)
)
)
→
(
(
(
¬
(
X48
→
(
X41
↔
(
X36
↔
X40
)
)
)
)
↔
(
(
¬
X48
)
↔
(
(
¬
X41
)
→
(
(
¬
X40
)
↔
X49
)
)
)
)
↔
(
¬
(
X48
↔
(
(
¬
X41
)
→
X49
)
)
)
)
)
)
)
)
)
)
→
(
(
(
X49
↔
(
X46
↔
(
X37
→
X4
→
False
)
)
)
↔
(
X49
→
(
X5
↔
(
X4
↔
False
)
)
)
)
↔
(
(
¬
(
X49
↔
(
X31
↔
X40
)
)
)
→
X37
→
X5
)
)
→
(
¬
(
(
¬
X21
)
→
(
(
¬
X5
)
↔
(
(
¬
X4
)
→
X49
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
¬
(
(
¬
(
X47
→
(
X40
↔
(
(
¬
X32
)
↔
X48
)
)
)
)
→
(
¬
(
(
¬
X47
)
→
(
X20
↔
X48
)
)
)
)
)
↔
(
¬
(
(
¬
X42
)
↔
(
(
¬
X40
)
↔
(
(
¬
X20
)
→
X47
)
)
)
)
)
↔
(
(
¬
(
X48
→
X40
→
X17
→
X42
)
)
↔
(
¬
(
X42
→
(
X18
↔
(
X17
→
X40
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X47
↔
(
X20
→
X14
→
X12
)
)
)
→
X14
→
(
(
¬
X12
)
↔
(
(
¬
X0
)
→
X47
)
)
)
)
→
(
X42
↔
(
X12
↔
(
X0
↔
X47
)
)
)
)
→
(
X40
↔
(
(
¬
X35
)
↔
X39
)
)
→
X42
→
(
(
¬
X40
)
↔
(
(
¬
X7
)
↔
X48
)
)
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
(
X29
↔
(
(
¬
X9
)
↔
X23
)
)
)
→
(
(
¬
(
X45
↔
(
X29
→
X23
→
X3
)
)
)
↔
(
¬
(
(
¬
X22
)
→
(
(
¬
X9
)
↔
(
X3
→
X45
)
)
)
)
)
)
)
↔
(
(
¬
(
X47
→
(
¬
X22
)
→
(
X1
↔
X34
)
)
)
↔
(
X47
↔
(
X45
→
(
¬
X1
)
→
X29
)
)
)
)
→
(
(
(
¬
X45
)
↔
(
X6
→
(
(
¬
X2
)
↔
X47
)
)
)
↔
(
(
X34
↔
(
X30
↔
X2
)
)
↔
(
(
¬
(
X49
→
(
X24
↔
False
)
)
)
↔
(
¬
(
(
¬
X45
)
→
(
X30
↔
(
X2
→
X49
)
)
)
)
)
)
)
)
)
→
(
(
(
(
(
(
(
¬
X47
)
→
(
(
¬
X45
)
↔
(
(
¬
X16
)
↔
X49
)
)
)
→
(
¬
X48
)
→
(
¬
X47
)
→
(
X45
↔
X49
)
)
→
(
¬
(
X48
→
(
¬
X16
)
→
X45
)
)
)
↔
(
¬
(
X48
↔
(
X47
↔
(
X8
→
X16
)
)
)
)
)
↔
(
(
¬
(
X45
↔
(
X25
↔
X49
)
)
)
→
X49
→
X43
→
X48
)
)
↔
(
¬
(
(
X49
→
X26
→
X44
)
→
(
(
¬
(
X47
→
(
¬
X43
)
→
X33
→
X49
)
)
↔
(
¬
(
(
(
X44
→
X33
→
(
X11
↔
X43
)
)
↔
(
¬
(
(
¬
(
X49
→
(
(
¬
X33
)
↔
(
(
¬
X26
)
↔
X43
)
)
)
)
→
(
¬
X33
)
→
X44
)
)
)
→
(
X49
↔
(
X43
↔
X44
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
66766c...
and proposition id is
3ea257...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMTN4nxEZQpbq8Ac2Ryirn1XswC1FSX9CQZ
)
(
¬
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
¬
(
(
(
(
(
(
¬
(
(
¬
(
X46
→
X27
→
X19
→
X39
)
)
→
(
X46
↔
X50
)
)
)
→
X49
→
(
¬
X19
)
→
X46
)
→
(
(
¬
(
X50
→
X46
→
X12
→
False
)
)
↔
(
(
(
¬
X46
)
↔
(
X32
↔
(
X12
↔
X49
)
)
)
↔
(
¬
(
X50
↔
(
X46
↔
(
(
¬
X12
)
→
X32
)
)
)
)
)
)
)
→
(
¬
(
(
(
X50
→
(
(
¬
X46
)
↔
(
(
¬
X43
)
→
X49
)
)
)
→
(
¬
(
X23
↔
(
(
¬
X7
)
→
X49
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X46
)
↔
(
(
¬
X7
)
→
X50
)
)
)
)
)
→
(
(
(
¬
(
X23
↔
(
(
¬
X14
)
→
X43
)
)
)
↔
(
X14
→
X6
→
X15
)
)
↔
(
¬
(
(
X23
→
X18
→
(
(
¬
X6
)
↔
X14
)
)
→
(
(
¬
(
(
¬
(
X49
↔
(
(
¬
X23
)
→
(
¬
X14
)
→
X45
)
)
)
→
X45
→
(
¬
X14
)
→
X6
→
X49
)
)
↔
(
(
¬
X14
)
→
X45
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
¬
(
X49
→
(
X44
↔
(
X37
↔
X50
)
)
)
)
→
(
(
(
(
(
X46
↔
(
(
¬
X1
)
→
(
(
¬
X0
)
↔
X37
)
)
)
↔
(
¬
(
X46
→
X5
)
)
)
↔
(
¬
(
(
¬
X1
)
→
X5
)
)
)
↔
(
(
¬
(
X50
↔
(
(
¬
X46
)
↔
X49
)
)
)
↔
(
X37
↔
(
(
¬
X8
)
→
(
X5
↔
X46
)
)
)
)
)
↔
(
¬
(
X37
↔
(
X5
↔
X50
)
)
)
)
)
→
(
¬
(
(
X44
→
X28
)
→
(
¬
(
X46
→
(
¬
X28
)
→
X3
→
X49
)
)
)
)
)
)
→
(
(
X49
→
(
X29
↔
X38
)
)
↔
(
(
X50
→
(
X44
↔
(
X31
↔
X29
)
)
)
↔
(
¬
(
(
(
¬
X44
)
→
X38
→
X49
)
→
(
¬
(
X31
→
X38
)
)
→
(
X44
→
X20
)
→
(
(
(
X31
→
(
(
¬
X20
)
↔
X29
)
)
↔
(
¬
(
(
¬
(
X44
→
X31
→
(
(
¬
X29
)
↔
X49
)
)
)
→
(
X44
↔
(
X29
→
(
¬
X20
)
→
X49
)
)
)
)
)
↔
(
¬
(
X49
→
X44
→
X31
)
)
)
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
X41
)
→
X30
→
(
(
¬
X13
)
↔
X42
)
)
)
↔
(
¬
(
X41
↔
(
X36
→
X30
→
X42
)
)
)
)
↔
(
(
X47
→
(
(
¬
X42
)
↔
(
(
¬
X41
)
↔
X44
)
)
)
→
(
¬
(
(
(
¬
(
(
¬
X44
)
→
(
(
¬
X34
)
↔
(
X26
→
X47
)
)
)
)
→
(
¬
(
X47
→
X44
→
X42
→
X26
)
)
)
→
X44
→
(
(
¬
X30
)
↔
(
X26
→
X47
)
)
)
)
)
)
↔
(
(
(
¬
(
X41
↔
(
X21
↔
X25
)
)
)
↔
(
(
¬
(
X36
↔
(
(
¬
X30
)
↔
(
X21
→
X49
)
)
)
)
→
X41
→
(
X9
↔
X36
)
)
)
→
(
(
(
(
¬
(
(
¬
(
(
¬
(
(
¬
X41
)
→
X49
)
)
→
(
X30
↔
(
X24
→
(
¬
X4
)
→
X41
)
)
)
)
→
(
¬
(
(
¬
X41
)
↔
(
(
¬
X30
)
→
X49
)
)
)
)
)
→
(
¬
(
(
¬
X41
)
→
(
(
¬
X30
)
↔
X49
)
)
)
)
↔
(
¬
(
X49
↔
(
X41
→
(
X24
↔
False
)
)
)
)
)
↔
(
¬
(
X49
→
(
X41
↔
(
X24
→
False
)
)
)
)
)
)
)
→
(
(
(
X49
→
X48
→
(
X44
↔
(
X22
→
False
)
)
)
→
X45
→
(
X44
↔
X48
)
)
→
(
(
¬
(
(
¬
X44
)
↔
(
(
¬
X10
)
↔
X48
)
)
)
↔
(
¬
(
(
X49
→
False
)
→
(
¬
(
X48
→
(
¬
X36
)
→
X10
→
X40
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
(
(
¬
(
X49
→
X33
)
)
↔
(
(
¬
(
X44
↔
(
X11
↔
X49
)
)
)
↔
(
X44
↔
(
(
¬
X17
)
→
X33
)
)
)
)
↔
X11
)
→
(
¬
(
X33
→
X17
→
X11
)
)
)
)
→
(
¬
(
X33
↔
(
(
¬
X11
)
↔
X17
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X16
)
↔
(
X11
→
X44
)
)
)
↔
(
(
¬
X44
)
↔
(
X2
↔
X48
)
)
)
↔
(
X48
↔
(
X44
→
X35
→
(
X17
↔
False
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
(
X48
→
(
(
¬
X35
)
↔
(
X0
→
X42
)
)
)
↔
(
¬
(
(
¬
X47
)
→
(
(
¬
X35
)
↔
(
X0
↔
X48
)
)
)
)
)
→
(
¬
(
(
¬
X35
)
↔
(
(
¬
X0
)
↔
X42
)
)
)
)
)
↔
(
(
¬
(
X50
→
X42
→
(
X35
↔
X47
)
)
)
↔
(
X47
↔
(
(
¬
X35
)
→
(
X31
↔
X42
)
)
)
)
)
↔
(
(
(
(
¬
(
X32
↔
(
(
¬
X16
)
→
(
X15
↔
X46
)
)
)
)
↔
(
(
(
X46
→
(
¬
X25
)
→
(
(
¬
X16
)
↔
X40
)
)
↔
(
¬
(
X46
↔
(
(
¬
X25
)
→
(
X9
↔
X40
)
)
)
)
)
↔
(
¬
(
X40
→
(
(
¬
X9
)
↔
X32
)
)
)
)
)
↔
(
¬
(
(
X50
↔
(
X37
→
X19
→
X48
)
)
→
(
(
(
(
¬
(
X50
↔
(
X40
↔
X48
)
)
)
↔
(
¬
(
(
¬
(
X46
↔
(
X40
→
(
X16
↔
X48
)
)
)
)
→
(
(
¬
(
X50
↔
X46
)
)
↔
(
(
¬
(
X50
↔
(
X46
↔
(
X40
↔
X37
)
)
)
)
→
(
¬
X40
)
→
X37
→
(
¬
X16
)
→
X46
)
)
)
)
)
↔
(
X46
↔
(
X37
→
(
X33
↔
X50
)
)
)
)
↔
(
X46
→
(
X33
↔
X37
)
)
)
)
)
)
→
(
X47
→
X40
→
(
X22
↔
X48
)
)
→
(
¬
(
(
¬
(
X46
→
(
(
¬
(
X47
↔
(
(
¬
X44
)
↔
(
(
¬
X22
)
↔
X46
)
)
)
)
↔
(
¬
(
X47
→
(
X46
↔
(
(
¬
X44
)
→
X48
)
)
)
)
)
)
)
→
(
(
¬
X44
)
↔
(
X22
↔
(
(
¬
X13
)
↔
X48
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
(
X38
↔
(
X36
→
X8
→
X42
)
)
→
(
(
¬
X38
)
↔
(
X36
→
(
(
¬
X8
)
↔
X49
)
)
)
)
)
↔
(
(
¬
(
(
¬
X45
)
→
X42
→
(
X30
↔
X49
)
)
)
↔
(
X45
↔
(
(
¬
X30
)
→
X36
)
)
)
)
→
(
(
¬
(
(
¬
(
X45
→
(
X7
↔
X5
)
)
)
→
(
¬
(
(
¬
(
X49
↔
(
X4
→
X7
)
)
)
→
(
¬
(
X49
→
X7
→
(
(
¬
X5
)
↔
X20
)
)
)
)
)
)
)
↔
(
(
¬
(
X47
↔
(
X42
→
X7
→
False
)
)
)
↔
(
¬
(
(
X7
↔
X12
)
→
(
X49
↔
(
X42
↔
(
X12
→
X5
)
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
¬
(
(
¬
X36
)
→
(
(
¬
X2
)
↔
X38
)
)
)
→
(
¬
(
X36
→
(
(
¬
X26
)
↔
(
(
¬
X21
)
→
X38
)
)
)
)
)
)
↔
(
¬
(
X36
→
X2
→
X26
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
(
¬
(
(
¬
X38
)
↔
(
(
¬
X10
)
→
X6
→
X49
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X6
)
↔
X46
)
)
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X38
)
↔
X49
)
)
)
)
)
→
(
(
¬
(
X49
↔
(
X21
→
X38
)
)
)
↔
(
X21
→
(
X10
↔
(
X2
↔
X49
)
)
)
)
)
)
→
(
¬
(
X46
↔
(
(
¬
X21
)
↔
X49
)
)
)
)
)
→
X38
→
(
(
¬
X24
)
↔
(
(
¬
X10
)
↔
X46
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
X23
→
(
¬
X18
)
→
X14
→
X45
)
)
→
(
¬
(
X47
→
X43
→
(
X11
↔
X14
)
)
)
→
(
(
(
¬
(
X23
↔
(
X18
↔
(
X3
↔
False
)
)
)
)
→
(
¬
(
X47
→
(
(
¬
X18
)
↔
X43
)
)
)
)
↔
(
¬
(
X3
↔
False
)
)
)
)
→
(
(
¬
(
X47
↔
(
(
¬
X18
)
→
(
(
¬
X14
)
↔
X27
)
)
)
)
↔
(
(
¬
X27
)
→
(
X18
↔
(
(
¬
X14
)
→
X34
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X49
↔
(
(
¬
X17
)
→
X43
)
)
)
→
(
¬
(
X47
↔
(
X43
→
X34
)
)
)
)
↔
(
X49
→
(
X43
↔
(
X17
→
False
)
)
)
)
→
(
¬
(
(
¬
X47
)
→
X34
→
X49
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X45
→
X41
→
(
X39
↔
(
X1
→
False
)
)
)
)
↔
(
X45
↔
(
X28
↔
X39
)
)
)
→
(
(
(
¬
(
X28
→
X49
)
)
→
(
(
¬
X45
)
↔
(
(
¬
X29
)
↔
(
(
¬
X28
)
↔
X47
)
)
)
→
(
¬
(
X49
↔
(
X45
↔
(
X29
↔
X28
)
)
)
)
)
↔
(
X47
→
(
X29
↔
(
X28
↔
X39
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
456e12...
and proposition id is
bd0b18...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMRBntxENxNoBU7GScdiezfMvMiMVZPRGqN
)
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
∀X42 :
prop
,
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
∀X48 :
prop
,
∀X49 :
prop
,
(
(
¬
(
(
¬
(
(
(
(
(
(
(
¬
X21
)
↔
(
X10
→
X33
)
)
↔
(
¬
(
X36
↔
(
X33
→
(
X21
↔
X1
)
)
)
)
)
↔
(
¬
(
X21
→
X10
→
X36
)
)
)
→
(
¬
(
(
¬
(
X41
↔
(
X33
↔
(
(
¬
X10
)
↔
X34
)
)
)
)
→
(
(
¬
X1
)
↔
X33
)
)
)
)
↔
(
(
(
(
¬
X45
)
→
(
(
¬
X36
)
↔
(
(
¬
X34
)
→
X48
)
)
)
↔
(
(
(
¬
X29
)
↔
(
X13
→
(
¬
X2
)
→
X35
)
)
↔
(
¬
(
X45
↔
(
X36
↔
X34
)
)
)
)
)
↔
(
¬
(
(
¬
(
X48
→
X36
→
(
¬
X25
)
→
X35
)
)
→
(
¬
(
X48
→
X35
→
(
(
¬
X21
)
↔
X25
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
X41
→
(
(
¬
X27
)
↔
(
(
¬
X20
)
→
X47
)
)
)
→
(
¬
(
(
¬
X20
)
→
(
¬
X16
)
→
X41
)
)
)
↔
(
(
(
¬
(
(
¬
X34
)
↔
(
X24
↔
(
(
¬
X16
)
↔
X45
)
)
)
)
↔
(
¬
(
X47
↔
(
X45
→
X24
→
X34
)
)
)
)
→
(
(
¬
(
X34
↔
(
(
¬
X12
)
↔
X24
)
)
)
↔
(
¬
(
(
¬
(
X41
→
(
X24
↔
(
X14
→
X16
)
)
)
)
→
(
¬
(
X24
↔
(
X16
↔
(
X12
→
False
)
)
)
)
)
)
)
)
)
→
(
(
¬
(
X34
↔
(
X31
→
(
X23
↔
X24
)
)
)
)
↔
(
¬
(
(
(
X40
↔
(
X23
↔
(
X6
→
X5
)
)
)
↔
(
(
¬
X31
)
↔
(
X6
↔
(
X5
↔
X40
)
)
)
)
→
(
¬
(
X34
↔
(
X31
→
(
X23
↔
(
X6
→
False
)
)
)
)
)
)
)
)
→
(
(
¬
X34
)
↔
(
X24
↔
(
(
¬
X19
)
↔
X47
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X46
→
X44
→
(
(
¬
X18
)
↔
X30
)
)
)
→
(
(
(
X44
→
(
(
¬
X28
)
↔
X46
)
)
↔
(
¬
(
X38
→
X28
→
(
X18
↔
False
)
)
)
)
↔
(
¬
(
X18
↔
False
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X44
→
(
(
¬
X32
)
↔
(
X9
→
X38
)
)
)
)
↔
(
X44
→
X38
→
(
X9
↔
X28
)
)
)
→
(
¬
(
X38
→
X28
→
X44
)
)
)
→
(
¬
(
(
¬
X38
)
↔
(
(
¬
X32
)
→
X17
→
X44
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
¬
(
(
¬
X47
)
→
(
¬
X38
)
→
X26
→
X48
)
)
→
(
¬
(
X48
→
(
¬
X28
)
→
X38
)
)
)
→
(
¬
X28
)
→
(
X26
↔
X38
)
)
)
↔
(
¬
(
(
¬
(
X48
↔
(
X28
↔
(
X11
→
X46
)
)
)
)
→
(
¬
(
X38
↔
False
)
)
)
)
)
→
(
(
¬
(
X43
↔
(
X42
↔
(
X8
→
X7
)
)
)
)
↔
(
(
¬
X15
)
→
(
¬
X8
)
→
(
¬
X7
)
→
X48
)
)
→
(
(
¬
(
(
¬
(
X48
↔
(
X47
↔
(
X46
→
X15
→
False
)
)
)
)
→
(
¬
(
X46
↔
(
X43
↔
(
(
¬
X15
)
→
X47
)
)
)
)
)
)
↔
(
(
¬
X46
)
→
X15
→
X47
)
)
)
)
→
(
(
¬
(
X46
↔
(
X37
→
(
¬
X3
)
→
X49
)
)
)
→
(
X49
↔
(
X48
→
(
X46
↔
X3
)
)
)
→
(
X49
↔
(
X0
→
X39
)
)
→
X49
→
(
X48
↔
(
X46
→
X39
→
False
)
)
)
→
(
¬
(
(
(
¬
(
X22
↔
(
(
¬
X4
)
↔
X49
)
)
)
→
X45
→
X22
→
X48
)
→
(
¬
(
(
X48
↔
(
X46
↔
(
X45
↔
False
)
)
)
→
(
(
¬
X45
)
→
X46
)
→
(
¬
(
X48
↔
(
(
¬
X43
)
→
(
(
¬
X4
)
↔
X45
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
X45
→
(
¬
X19
)
→
X47
)
↔
(
(
X45
↔
(
X19
↔
(
X3
→
X17
)
)
)
↔
(
(
X45
→
(
X11
↔
X19
)
)
→
(
¬
(
(
¬
(
X47
↔
(
X45
→
(
X19
↔
False
)
)
)
)
→
(
¬
(
(
¬
(
X47
→
(
X19
↔
(
X11
→
X3
→
False
)
)
)
)
→
(
X47
↔
X45
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X47
↔
(
X46
↔
(
(
¬
X9
)
↔
X28
)
)
)
)
→
(
(
(
¬
(
(
X47
↔
(
(
¬
X28
)
↔
(
X17
↔
X45
)
)
)
→
(
¬
(
X47
↔
(
X28
→
(
X17
↔
X46
)
)
)
)
)
)
→
X45
→
(
¬
X28
)
→
X47
)
↔
(
X45
→
(
(
¬
X17
)
↔
X47
)
)
)
)
)
)
)
→
(
(
(
(
¬
X37
)
↔
(
(
¬
X34
)
↔
(
(
¬
X26
)
→
X40
)
)
)
↔
(
(
¬
(
X47
→
X45
→
X39
→
X34
→
False
)
)
→
(
¬
(
(
¬
(
(
¬
X37
)
↔
(
X33
↔
X39
)
)
)
→
(
¬
(
(
¬
X45
)
↔
(
(
¬
X39
)
↔
(
(
¬
X6
)
→
X47
)
)
)
)
)
)
→
(
¬
X39
)
→
X37
→
X45
)
)
↔
(
¬
(
(
(
(
¬
(
(
¬
(
X46
→
(
¬
X37
)
→
(
¬
X4
)
→
X47
)
)
→
X37
→
(
X14
↔
(
X4
↔
False
)
)
)
)
→
(
¬
(
X46
→
(
¬
X14
)
→
X37
)
)
)
→
(
¬
(
X14
→
X5
→
X37
)
)
)
→
(
X48
↔
(
X46
→
(
X4
↔
X32
)
)
)
→
(
(
¬
(
X46
↔
(
(
¬
X32
)
→
X14
→
X34
)
)
)
↔
(
¬
(
X48
→
(
X46
↔
(
X14
↔
X32
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
(
(
X44
↔
(
X27
→
X24
→
X49
)
)
↔
(
(
(
¬
X44
)
↔
(
X38
↔
X49
)
)
↔
(
¬
(
X48
→
X44
→
X38
→
X27
)
)
)
)
→
(
(
¬
(
X44
↔
(
X42
↔
(
(
¬
X38
)
↔
X49
)
)
)
)
↔
(
(
¬
X44
)
→
(
(
¬
X38
)
↔
(
X36
→
X49
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
¬
(
X48
→
(
¬
X23
)
→
(
X20
↔
X30
)
)
)
→
(
¬
(
(
¬
X30
)
↔
(
X23
→
(
¬
X16
)
→
X48
)
)
)
)
)
→
(
¬
(
(
¬
(
(
X30
→
(
X22
↔
(
X15
→
X20
)
)
)
→
(
¬
(
(
¬
X22
)
↔
(
(
¬
X20
)
→
X16
→
X30
)
)
)
)
)
→
(
X44
↔
(
X22
→
X20
→
X30
)
)
)
)
)
)
↔
(
(
¬
(
X42
↔
(
(
¬
X16
)
↔
X48
)
)
)
→
(
(
¬
(
(
¬
X15
)
↔
X48
)
)
↔
(
¬
(
X47
↔
(
X42
→
X16
→
False
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
(
(
(
¬
X40
)
→
(
¬
X25
)
→
(
X12
↔
X43
)
)
↔
(
(
¬
X41
)
→
(
¬
X40
)
→
(
X12
↔
X43
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X40
)
→
(
X25
↔
X43
)
)
)
)
)
→
(
¬
(
(
X43
→
X41
→
(
X31
↔
False
)
)
→
(
¬
(
(
(
¬
X31
)
↔
X43
)
→
(
¬
(
X47
→
X41
→
X21
→
False
)
)
)
)
)
)
)
)
↔
(
(
(
(
X41
↔
X0
)
↔
(
¬
(
X47
→
(
X35
↔
(
X10
→
X43
)
)
)
)
)
↔
(
(
¬
X43
)
↔
(
X35
→
X47
)
)
)
↔
(
X35
→
(
X10
↔
(
(
¬
X0
)
↔
X13
)
)
)
)
)
↔
(
(
(
(
¬
X29
)
↔
(
X18
→
(
¬
X2
)
→
X48
)
)
↔
(
¬
(
(
(
(
¬
(
(
¬
X18
)
→
(
(
¬
X8
)
↔
X49
)
)
)
→
(
¬
(
X49
→
(
¬
X1
)
→
X29
)
)
)
→
(
X8
↔
(
X1
→
X49
)
)
)
→
(
¬
(
(
¬
X18
)
↔
(
(
¬
X8
)
→
(
¬
X2
)
→
X29
)
)
)
)
)
)
↔
(
(
X47
↔
(
X7
↔
X43
)
)
→
(
¬
(
(
¬
(
X48
→
(
X47
↔
(
X41
↔
X8
)
)
)
)
→
(
(
¬
(
(
¬
X41
)
↔
(
X18
→
(
(
¬
X8
)
↔
X43
)
)
)
)
↔
(
X47
↔
(
(
¬
X18
)
↔
(
(
¬
X8
)
→
X41
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
54249a...
and proposition id is
ff2d68...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMNySy1Uc2gTE2ke7VQGcLA42PXZFFAZysQ
)
∀X0 :
prop
,
(
¬
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
¬
∀X50 :
prop
,
(
(
(
(
(
¬
(
(
(
(
¬
(
X45
→
(
(
¬
X3
)
↔
X49
)
)
)
↔
(
(
¬
(
(
¬
X45
)
↔
(
(
¬
X42
)
→
(
(
¬
X7
)
↔
X49
)
)
)
)
→
(
¬
(
(
¬
X42
)
→
(
X32
↔
(
(
¬
X7
)
→
X49
)
)
)
)
)
)
→
(
¬
X45
)
→
(
X42
↔
(
X2
→
X49
)
)
)
→
(
(
(
¬
X32
)
→
(
¬
X21
)
→
(
¬
X9
)
→
X46
)
↔
(
X13
→
X9
→
X46
)
)
→
(
(
X50
↔
(
X46
↔
(
X42
↔
False
)
)
)
↔
(
(
¬
(
(
¬
X46
)
→
X45
→
(
¬
X9
)
→
X50
)
)
↔
(
¬
(
X45
↔
(
(
¬
X42
)
↔
(
(
¬
X9
)
→
X46
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
X49
↔
(
(
¬
X44
)
→
X41
→
X46
)
)
↔
(
¬
(
X44
↔
(
X12
↔
X49
)
)
)
)
→
(
(
(
(
X45
↔
(
(
¬
X38
)
↔
X49
)
)
↔
(
¬
(
(
¬
(
X45
→
(
X38
↔
X30
)
)
)
→
X49
→
(
¬
X30
)
→
(
(
¬
X12
)
↔
X38
)
)
)
)
→
(
(
¬
X45
)
↔
(
X12
↔
X46
)
)
)
↔
(
X30
↔
(
X12
→
X5
→
X49
)
)
)
)
)
↔
(
(
¬
(
X46
↔
(
(
¬
X38
)
→
(
(
¬
X37
)
↔
X41
)
)
)
)
↔
(
(
(
(
¬
X18
)
→
X37
)
↔
(
X41
↔
(
(
¬
X16
)
→
X27
)
)
)
↔
(
¬
(
X41
↔
(
X38
↔
(
(
¬
X16
)
→
X27
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
X41
)
→
(
(
¬
X39
)
↔
(
X29
→
X49
)
)
)
↔
(
¬
(
(
X49
↔
(
X41
↔
(
X38
→
X26
→
False
)
)
)
→
(
(
¬
X39
)
↔
(
X26
→
X41
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X41
↔
(
(
¬
X39
)
↔
(
(
¬
X15
)
→
X47
)
)
)
)
→
(
¬
(
X47
→
(
X39
↔
(
X29
→
X15
)
)
)
)
)
→
(
¬
(
(
¬
X29
)
↔
(
(
¬
X15
)
↔
X41
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
¬
(
(
¬
X47
)
↔
(
(
¬
X40
)
→
X48
)
)
)
↔
(
(
¬
X36
)
↔
(
(
¬
X22
)
↔
(
(
¬
X20
)
↔
X48
)
)
)
)
↔
(
¬
(
X47
↔
(
X40
↔
(
X36
→
False
)
)
)
)
)
→
(
(
(
¬
X10
)
↔
(
X0
↔
X40
)
)
↔
(
¬
(
X48
↔
(
X22
→
(
X10
↔
(
X0
→
False
)
)
)
)
)
)
→
X48
→
(
X40
↔
X10
)
)
→
(
¬
(
(
¬
(
(
¬
(
X50
→
X47
→
X31
→
X35
)
)
→
(
¬
(
X35
↔
(
(
¬
X28
)
→
(
X14
↔
X31
)
)
)
)
)
)
→
(
(
¬
(
(
(
¬
(
X50
↔
(
X48
↔
X40
)
)
)
↔
(
X48
↔
(
X31
↔
X23
)
)
)
→
(
¬
(
X24
→
X50
)
)
)
)
↔
(
(
(
¬
X48
)
→
(
(
¬
X46
)
↔
X50
)
)
↔
(
(
¬
(
X50
↔
(
(
¬
X46
)
→
X48
)
)
)
↔
(
(
¬
X40
)
↔
(
(
¬
X35
)
→
X46
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
¬
(
X43
↔
(
X34
↔
(
X33
→
X1
)
)
)
)
↔
(
(
¬
(
X50
↔
(
X43
↔
X33
)
)
)
→
(
¬
(
X50
→
X8
→
X1
)
)
)
)
→
(
¬
(
(
¬
(
X50
↔
(
X34
→
(
¬
X4
)
→
X19
)
)
)
→
(
¬
X34
)
→
(
X33
↔
(
X19
→
X50
)
)
)
)
)
)
→
(
(
¬
(
X34
→
(
(
¬
X11
)
↔
(
(
¬
X4
)
→
X47
)
)
)
)
↔
(
(
(
(
X34
→
(
¬
X11
)
→
(
¬
X6
)
→
X19
)
→
(
¬
(
X47
↔
(
X34
↔
(
X11
↔
X19
)
)
)
)
)
→
(
¬
(
X34
↔
(
(
¬
X6
)
→
X19
)
)
)
)
↔
(
X11
↔
(
X6
↔
X34
)
)
)
)
)
→
(
(
(
X43
→
(
X34
↔
(
X25
↔
False
)
)
)
→
(
(
¬
X46
)
↔
(
(
¬
X25
)
↔
(
X17
→
X47
)
)
)
)
↔
(
(
¬
X25
)
↔
(
X17
→
X47
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
(
(
¬
(
X43
→
X16
)
)
→
(
¬
(
(
¬
X49
)
→
(
¬
X16
)
→
X50
)
)
→
X50
→
X44
→
X43
→
False
)
→
(
(
(
¬
(
X26
↔
(
(
¬
X25
)
→
(
X18
↔
X50
)
)
)
)
↔
(
¬
(
X49
↔
(
(
¬
X26
)
→
(
¬
X25
)
→
X44
)
)
)
)
↔
(
(
(
¬
X29
)
→
X27
→
(
(
¬
X2
)
↔
X49
)
)
↔
(
X49
↔
(
(
¬
X44
)
↔
(
X30
↔
X50
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X8
→
(
X6
↔
False
)
)
)
→
(
¬
(
X45
↔
(
X43
→
(
(
¬
X8
)
↔
X30
)
)
)
)
)
)
↔
(
(
X43
↔
(
X30
↔
(
X5
→
False
)
)
)
↔
(
¬
(
X43
→
(
¬
(
(
X43
→
(
X30
↔
(
(
¬
X5
)
↔
X49
)
)
)
→
(
X30
↔
(
X5
↔
X45
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
X24
↔
X49
)
)
→
(
¬
(
X50
→
(
X45
↔
(
(
¬
X9
)
→
X49
)
)
)
)
)
)
↔
(
¬
(
X21
→
(
(
¬
X9
)
↔
X45
)
)
)
)
→
(
¬
(
X49
→
X45
→
X9
→
X35
)
)
)
)
)
→
(
(
(
¬
X36
)
→
(
¬
X4
)
→
X49
)
↔
(
(
¬
(
X49
↔
(
X45
→
X44
)
)
)
→
(
(
¬
X41
)
↔
(
(
¬
X20
)
↔
X44
)
)
)
)
→
(
¬
(
(
X13
→
(
X7
↔
X49
)
)
→
(
¬
(
X17
↔
(
(
¬
X13
)
→
X7
→
X49
)
)
)
→
(
¬
(
X17
→
(
X13
↔
(
X7
→
X3
→
False
)
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
(
¬
(
X49
→
(
X48
↔
(
X47
→
False
)
)
)
)
→
(
(
¬
(
X49
→
(
X48
↔
False
)
)
)
↔
(
¬
(
X48
→
(
(
¬
X37
)
↔
(
X12
→
X47
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X49
↔
(
X47
→
X44
)
)
)
→
X49
→
X48
→
(
(
¬
X42
)
↔
X44
)
)
)
)
)
→
(
(
¬
(
(
¬
X47
)
→
(
(
¬
X40
)
↔
(
X39
↔
X48
)
)
)
)
↔
(
(
¬
X40
)
→
(
X39
↔
(
(
¬
X22
)
↔
X47
)
)
)
)
→
(
(
(
¬
X39
)
↔
(
X19
→
(
¬
X10
)
→
X46
)
)
↔
(
(
¬
(
X49
→
X19
→
X28
)
)
↔
(
(
¬
X47
)
→
X48
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
(
¬
X11
)
→
X0
→
X23
)
)
↔
(
X38
→
(
¬
X23
)
→
(
(
¬
X0
)
↔
X47
)
)
)
→
(
(
(
¬
X34
)
→
(
¬
X33
)
→
(
¬
X23
)
→
X38
)
↔
(
¬
(
X45
↔
(
(
¬
X34
)
↔
(
(
¬
X33
)
→
X38
)
)
)
)
)
)
→
(
(
¬
(
(
X46
→
(
X34
↔
(
X31
→
X32
)
)
)
→
(
X38
→
(
X34
↔
(
X14
→
X1
→
False
)
)
)
→
(
¬
(
X38
↔
(
X32
→
(
¬
X14
)
→
X34
)
)
)
)
)
↔
(
(
X48
↔
(
(
¬
X15
)
→
(
X14
↔
X47
)
)
)
↔
(
¬
(
(
¬
X15
)
↔
(
X14
↔
X34
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
242dfa...
and proposition id is
a0ebfe...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMQEiTQrB3nZyHzmcAbyM5eS1TAWcBiHu6W
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
∀X38 :
prop
,
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
∀X50 :
prop
,
(
(
(
(
¬
(
(
¬
(
(
(
(
X43
→
X35
→
X46
)
↔
(
¬
(
X35
→
(
¬
X1
)
→
X19
)
)
)
↔
(
X35
→
X19
→
(
X17
↔
X43
)
)
)
→
(
(
(
X43
→
X17
→
(
¬
X14
)
→
X50
)
↔
(
¬
(
(
¬
X42
)
↔
(
X7
→
X43
)
)
)
)
↔
(
¬
(
(
¬
(
X43
↔
(
X17
→
(
¬
X13
)
→
X19
)
)
)
→
(
¬
(
X42
↔
(
(
¬
X20
)
↔
(
(
¬
X15
)
↔
X43
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X46
→
(
X22
↔
(
(
¬
X4
)
→
X48
)
)
)
)
→
(
(
X48
→
(
¬
X4
)
→
X35
)
↔
(
X46
→
(
X35
↔
(
(
¬
X4
)
→
X44
)
)
)
)
)
)
→
(
(
(
(
¬
(
X46
↔
(
(
¬
X3
)
↔
X33
)
)
)
↔
(
(
X48
↔
(
X33
→
(
¬
X3
)
→
X46
)
)
↔
(
¬
(
X42
→
(
(
¬
X3
)
↔
X46
)
)
)
)
)
↔
(
¬
(
X48
→
(
X33
↔
(
(
¬
X0
)
→
X42
)
)
)
)
)
↔
(
(
(
X42
↔
(
X10
↔
X46
)
)
↔
(
X46
↔
(
(
¬
X42
)
↔
(
(
¬
X22
)
↔
X48
)
)
)
)
↔
(
¬
False
)
)
)
)
)
)
)
→
(
(
(
(
X49
↔
X11
)
→
(
X44
↔
(
(
¬
X23
)
↔
(
X16
↔
X25
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X25
)
↔
(
X23
→
X29
)
)
)
→
(
(
(
X30
→
(
X16
↔
X29
)
)
↔
(
¬
(
X29
↔
(
(
¬
X23
)
→
(
X16
↔
X25
)
)
)
)
)
↔
(
X25
↔
(
X16
↔
X9
)
)
)
)
)
)
↔
(
(
(
(
¬
(
(
(
¬
(
(
¬
X44
)
↔
(
X23
↔
(
(
¬
X2
)
→
X48
)
)
)
)
↔
(
X44
→
(
(
¬
X27
)
↔
(
X23
→
X48
)
)
)
)
→
(
¬
X23
)
→
(
¬
X2
)
→
X27
)
)
→
(
¬
(
X44
→
X27
→
X2
)
)
)
↔
(
¬
(
(
¬
X23
)
↔
X27
)
)
)
↔
(
(
¬
(
X48
→
(
X44
↔
X25
)
)
)
↔
(
(
X48
↔
(
X34
↔
(
X25
↔
X44
)
)
)
↔
(
(
¬
X46
)
↔
(
X44
↔
(
X34
→
X48
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
(
(
¬
(
X47
→
X45
→
X41
→
X49
)
)
↔
(
¬
(
X47
→
X45
→
(
X31
↔
X49
)
)
)
)
↔
(
(
(
¬
X47
)
↔
(
(
¬
X41
)
↔
(
(
¬
X26
)
→
X49
)
)
)
→
(
¬
(
X49
→
X47
→
(
X18
↔
X26
)
)
)
)
)
→
(
¬
(
(
(
¬
X47
)
→
(
¬
X12
)
→
X49
)
→
(
(
¬
X40
)
↔
(
(
¬
X18
)
→
(
X12
↔
X47
)
)
)
)
)
)
)
↔
(
(
X47
→
(
X32
↔
X40
)
)
↔
(
(
X47
↔
(
X40
→
(
X24
↔
X39
)
)
)
↔
(
(
X50
↔
(
X39
→
(
X32
↔
X40
)
)
)
↔
(
(
X47
→
X39
→
X32
→
False
)
↔
(
¬
(
(
¬
(
X40
→
X32
→
(
X31
↔
False
)
)
)
→
(
(
¬
(
X47
→
(
X40
↔
(
(
¬
X21
)
↔
X31
)
)
)
)
→
(
¬
X40
)
→
X32
→
(
¬
X31
)
→
X47
)
→
(
¬
(
(
¬
X39
)
→
(
X32
↔
(
(
¬
X21
)
↔
X40
)
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
X49
↔
(
X48
↔
(
X8
→
X5
→
False
)
)
)
↔
(
¬
(
X49
→
(
X8
↔
(
X5
→
X50
)
)
)
)
)
→
(
(
(
X47
→
X36
→
X6
→
X38
)
→
(
(
(
¬
X48
)
→
X38
→
X49
)
↔
(
¬
(
X47
→
(
¬
X6
)
→
X48
)
)
)
)
↔
(
¬
(
X47
↔
(
X36
↔
(
X28
→
X48
)
)
)
)
)
→
(
(
¬
(
(
X50
↔
(
X47
→
X48
)
)
→
(
¬
(
X47
→
(
(
¬
X37
)
↔
(
(
¬
X36
)
→
X50
)
)
)
)
→
(
¬
(
X48
↔
(
(
¬
X47
)
→
(
(
¬
X37
)
↔
X50
)
)
)
)
)
)
↔
(
¬
(
X50
↔
(
(
¬
X37
)
→
X48
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X47
→
X6
→
(
X4
↔
(
X2
→
False
)
)
)
)
↔
(
(
¬
X31
)
↔
(
X4
↔
(
(
¬
X2
)
↔
X47
)
)
)
)
→
(
(
(
¬
(
X47
↔
(
X27
↔
X42
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X42
)
→
(
¬
X27
)
→
X47
)
)
→
(
¬
(
X43
→
(
X13
↔
X47
)
)
)
)
)
)
↔
(
(
¬
(
X42
↔
(
X27
→
(
X2
↔
X43
)
)
)
)
↔
(
X44
→
(
(
¬
X42
)
↔
(
X27
↔
X43
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
X41
→
(
X35
↔
(
X0
↔
X7
)
)
)
→
X0
→
X42
)
)
→
(
X42
↔
(
(
¬
X7
)
→
X35
)
)
)
)
→
(
¬
(
X34
→
(
¬
X25
)
→
(
¬
X0
)
→
X48
)
)
)
↔
(
(
(
¬
(
(
¬
X33
)
→
(
(
¬
X26
)
↔
(
X18
↔
X38
)
)
)
)
→
(
(
X43
→
X26
→
(
(
¬
X24
)
↔
X38
)
)
↔
(
¬
(
X43
→
X33
→
(
¬
X24
)
→
X38
)
)
)
)
↔
(
(
(
¬
(
X44
→
X38
→
(
(
¬
X33
)
↔
X42
)
)
)
↔
(
X44
↔
(
X42
↔
(
(
¬
X38
)
→
X48
)
)
)
)
↔
(
¬
(
(
¬
(
X48
↔
(
X44
↔
(
X42
↔
X10
)
)
)
)
→
(
X48
↔
(
X3
→
X35
)
)
→
(
(
¬
X44
)
↔
(
X42
→
(
X35
↔
X48
)
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
(
(
¬
(
X49
→
X43
→
X8
→
X36
)
)
↔
(
¬
(
X50
↔
(
X49
↔
(
X8
↔
X36
)
)
)
)
)
→
(
(
X50
→
(
X15
↔
(
X9
→
X36
)
)
)
↔
(
X50
↔
(
X43
→
(
X36
↔
(
X9
→
False
)
)
)
)
)
→
(
¬
(
X15
↔
(
X11
→
X9
→
X36
)
)
)
)
)
→
(
¬
(
(
¬
(
X41
↔
(
X20
↔
X43
)
)
)
→
(
¬
(
X46
↔
(
X20
→
X15
→
False
)
)
)
)
)
)
→
(
¬
(
(
X41
→
(
X39
↔
X46
)
)
→
(
X49
→
X46
→
X17
)
→
(
(
X42
→
False
)
↔
(
¬
(
(
(
¬
X46
)
↔
(
X42
↔
(
(
¬
X39
)
↔
X49
)
)
)
→
(
¬
(
(
¬
X39
)
→
(
(
¬
X17
)
↔
X49
)
)
)
)
)
)
)
)
→
(
(
(
X41
↔
(
X40
↔
(
(
¬
X23
)
↔
X32
)
)
)
↔
(
X40
↔
(
(
¬
X32
)
→
X42
)
)
)
→
(
¬
(
(
¬
X32
)
↔
(
(
¬
X23
)
→
X41
)
)
)
)
→
(
¬
(
(
X19
↔
False
)
→
(
(
(
¬
X40
)
→
(
(
¬
X32
)
↔
(
(
¬
X14
)
↔
X41
)
)
)
↔
(
¬
(
(
¬
X40
)
↔
(
(
¬
X32
)
↔
(
(
¬
X5
)
↔
X41
)
)
)
)
)
→
(
¬
X41
)
→
(
(
¬
X40
)
↔
(
X32
→
X49
)
)
)
)
)
)
→
(
(
¬
(
X45
↔
(
(
¬
X39
)
→
(
(
¬
X12
)
↔
X42
)
)
)
)
↔
(
¬
(
(
(
(
X40
→
(
X39
↔
(
X28
→
X29
)
)
)
↔
(
(
X40
→
(
(
¬
X39
)
↔
(
X30
↔
X42
)
)
)
↔
(
X39
→
(
X30
↔
(
(
¬
X28
)
↔
X40
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
X37
→
(
¬
X21
)
→
(
¬
X1
)
→
X40
)
→
(
¬
(
(
¬
X39
)
↔
(
X21
↔
X40
)
)
)
)
)
→
(
¬
(
X40
→
X21
→
X1
)
)
)
)
↔
(
¬
X29
)
)
)
→
(
(
¬
(
X30
↔
(
(
¬
X16
)
→
X40
)
)
)
↔
(
(
(
¬
(
X45
→
X30
)
)
↔
(
¬
(
X45
↔
(
X39
→
X16
→
X40
)
)
)
)
→
X40
→
X39
)
)
→
(
X45
↔
(
X39
→
(
¬
X22
)
→
X40
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
8641de...
and proposition id is
f7a807...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMNPdm1YBoJTcdo6Cp2BJkcZKK2wjSNAxLr
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
(
¬
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
(
¬
∀X28 :
prop
,
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
(
¬
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
∀X50 :
prop
,
(
(
(
(
(
(
(
¬
(
(
¬
X42
)
↔
(
(
¬
X40
)
→
(
¬
X14
)
→
X49
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X42
)
→
(
X39
↔
(
X0
↔
X49
)
)
)
)
→
X49
→
(
X40
↔
(
X0
↔
X39
)
)
)
→
(
¬
(
X49
↔
False
)
)
)
)
)
→
(
(
¬
(
X40
→
(
¬
X17
)
→
(
X2
↔
X39
)
)
)
↔
(
(
X40
↔
(
(
¬
X17
)
→
X49
)
)
↔
(
¬
(
(
(
¬
X17
)
↔
(
(
¬
X15
)
→
X40
)
)
→
X39
→
X15
→
X49
)
)
)
)
→
(
¬
(
X40
→
X39
→
(
X27
↔
X49
)
)
)
→
(
X39
↔
(
(
¬
X27
)
↔
(
(
¬
X11
)
↔
X40
)
)
)
)
→
(
(
¬
X43
)
→
(
X42
↔
X47
)
)
→
(
¬
(
(
(
X43
↔
(
X42
→
X31
→
X47
)
)
↔
(
¬
(
(
¬
(
X47
→
(
(
¬
X40
)
↔
X42
)
)
)
→
X43
→
X15
→
False
)
)
)
→
(
X42
↔
(
X15
→
X47
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
X47
↔
(
X45
→
(
(
¬
X7
)
↔
X25
)
)
)
→
(
¬
X25
)
→
(
(
¬
X24
)
↔
(
(
¬
X7
)
↔
X47
)
)
)
→
(
X47
↔
(
X45
↔
(
X38
↔
(
X25
↔
False
)
)
)
)
)
→
(
¬
(
X45
→
(
(
¬
X24
)
↔
X47
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
(
X46
→
False
)
)
→
(
¬
(
X47
→
(
X38
↔
(
(
¬
X8
)
→
X29
)
)
)
)
)
)
→
(
¬
(
X47
↔
(
(
¬
X38
)
↔
X46
)
)
)
)
)
↔
(
(
(
¬
X38
)
→
(
(
¬
X32
)
↔
X47
)
)
↔
(
(
(
(
X38
→
(
¬
X32
)
→
X19
→
X45
)
→
(
¬
(
X19
→
X1
)
)
)
↔
(
X45
↔
(
(
¬
X32
)
↔
(
(
¬
X29
)
→
X38
)
)
)
)
→
(
(
X45
↔
X38
)
↔
(
X38
→
(
¬
X32
)
→
X21
→
X47
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
X45
)
↔
X47
)
→
(
(
¬
(
X33
↔
False
)
)
↔
(
(
(
¬
X47
)
→
(
(
¬
X45
)
↔
(
X33
→
X49
)
)
)
↔
(
(
¬
X47
)
↔
(
(
¬
X45
)
→
(
(
¬
X43
)
↔
X49
)
)
)
)
)
)
→
(
(
¬
(
X49
↔
(
X46
→
(
¬
X16
)
→
X33
)
)
)
↔
(
¬
(
(
(
¬
X46
)
↔
(
(
¬
X43
)
→
(
(
¬
X33
)
↔
X49
)
)
)
→
X49
→
(
X46
↔
(
X26
→
X43
)
)
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
(
¬
(
X36
→
X34
→
X50
)
)
→
(
(
(
¬
X36
)
→
(
(
¬
X34
)
↔
(
(
¬
X5
)
↔
X50
)
)
)
↔
(
X50
↔
(
(
¬
X36
)
→
(
(
¬
X34
)
↔
X37
)
)
)
)
)
→
(
(
¬
(
X37
↔
(
X36
→
X50
)
)
)
↔
(
X47
→
(
X37
↔
(
X18
→
False
)
)
)
)
→
(
¬
X47
)
→
(
¬
X18
)
→
(
¬
X3
)
→
X50
)
)
→
(
(
¬
(
(
(
¬
X46
)
↔
(
X36
↔
(
(
¬
X18
)
↔
X48
)
)
)
→
X48
→
X46
→
X36
→
X5
)
)
↔
(
X36
→
(
X5
↔
X18
)
)
)
)
↔
(
¬
(
(
(
X50
→
(
X47
↔
(
X30
→
X44
)
)
)
↔
(
(
X47
→
(
¬
X41
)
→
(
(
¬
X4
)
↔
X44
)
)
↔
(
¬
(
X50
→
X41
→
(
(
¬
X35
)
↔
X44
)
)
)
)
)
→
(
(
¬
(
(
(
(
(
¬
X44
)
↔
(
(
¬
X12
)
→
(
¬
X6
)
→
X48
)
)
→
(
¬
X22
)
→
(
X12
↔
(
(
¬
X6
)
→
X44
)
)
)
→
(
X47
↔
(
X23
→
False
)
)
)
→
(
X45
→
(
¬
X22
)
→
(
X9
↔
X44
)
)
→
(
¬
(
(
¬
(
X22
→
(
¬
X20
)
→
X45
)
)
→
(
(
¬
(
X22
↔
(
X20
→
X48
)
)
)
↔
(
X44
↔
(
X22
→
(
¬
X20
)
→
X45
)
)
)
)
)
)
)
↔
(
(
X23
↔
(
(
¬
X13
)
↔
(
X10
→
X28
)
)
)
↔
(
(
X28
↔
(
X23
→
X20
→
X50
)
)
↔
(
¬
(
X48
↔
(
(
¬
X28
)
→
X20
→
X50
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
¬
(
(
X50
→
(
X38
↔
(
X31
→
X16
)
)
)
→
(
X38
↔
(
X17
↔
(
X16
→
X1
→
False
)
)
)
→
(
¬
X38
)
→
(
X17
↔
(
(
¬
X16
)
↔
X50
)
)
)
)
→
(
(
¬
(
(
¬
X17
)
↔
(
X16
→
(
X14
↔
X43
)
)
)
)
↔
(
¬
(
X43
↔
(
X38
↔
(
(
¬
X14
)
→
X17
)
)
)
)
)
)
↔
(
(
(
¬
(
X13
→
X47
)
)
→
(
¬
(
(
¬
(
X47
↔
(
X13
↔
X43
)
)
)
→
(
¬
(
X47
→
(
X13
↔
(
(
¬
X3
)
↔
X17
)
)
)
)
)
)
)
↔
(
(
¬
(
X27
→
X2
→
False
)
)
→
(
(
X27
↔
(
X13
→
(
¬
X2
)
→
X31
)
)
↔
(
¬
(
X47
↔
X27
)
)
)
)
)
)
→
(
(
(
¬
(
X47
→
(
X39
↔
False
)
)
)
↔
(
¬
(
(
¬
X43
)
→
(
X42
↔
(
(
¬
X10
)
↔
X46
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
X46
→
(
¬
X43
)
→
(
X7
↔
X50
)
)
)
↔
(
¬
(
(
¬
X43
)
↔
(
X42
→
(
(
¬
X7
)
↔
X50
)
)
)
)
)
→
X46
→
(
(
¬
X42
)
↔
X50
)
)
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X44
↔
(
(
¬
X40
)
↔
(
(
¬
X26
)
→
X49
)
)
)
)
↔
(
(
¬
(
X49
→
X34
→
(
(
¬
X20
)
↔
X46
)
)
)
→
(
¬
(
X44
↔
(
X20
→
False
)
)
)
→
(
X34
↔
(
X24
↔
X40
)
)
)
)
↔
(
X26
→
(
¬
X20
)
→
X49
)
)
→
(
¬
(
(
(
¬
(
(
X44
↔
(
(
¬
X6
)
→
X46
)
)
→
(
(
¬
(
X49
→
X46
→
(
¬
X6
)
→
X21
)
)
↔
(
¬
(
(
¬
X46
)
→
X44
→
(
¬
X21
)
→
X49
)
)
)
)
)
→
(
¬
(
(
¬
X6
)
→
X46
)
)
)
→
(
(
(
X44
↔
(
X23
→
(
¬
X6
)
→
X21
)
)
↔
(
(
¬
X21
)
↔
X42
)
)
↔
(
(
¬
(
X42
↔
(
X23
↔
(
X21
↔
(
X4
→
False
)
)
)
)
)
→
(
¬
(
(
¬
X23
)
↔
(
(
¬
X5
)
→
(
¬
X4
)
→
X43
)
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
(
¬
(
X45
→
X36
→
(
¬
X35
)
→
X46
)
)
→
(
¬
(
(
¬
X36
)
↔
(
(
¬
X35
)
→
X32
→
X46
)
)
)
)
→
X46
→
X19
→
X11
→
X35
)
→
(
(
¬
(
X36
→
(
X33
↔
(
(
¬
X28
)
↔
X35
)
)
)
)
↔
(
X45
↔
(
X36
↔
(
X35
↔
X33
)
)
)
)
)
→
(
X41
→
(
(
¬
X33
)
↔
X46
)
)
→
X37
→
X32
→
False
)
→
(
X41
→
(
¬
X25
)
→
(
(
¬
X0
)
↔
X46
)
)
→
(
¬
(
(
¬
X45
)
↔
(
X41
↔
(
X25
→
X46
)
)
)
)
)
)
↔
(
(
(
X49
↔
(
X30
→
X9
→
X50
)
)
↔
(
¬
(
(
¬
X46
)
→
X50
)
)
)
→
(
(
(
¬
(
(
(
X50
↔
(
(
¬
X37
)
↔
(
X15
↔
X48
)
)
)
↔
(
¬
(
X50
→
X48
→
X46
)
)
)
→
(
¬
(
(
¬
X46
)
→
(
X45
↔
(
(
¬
X37
)
↔
X48
)
)
)
)
)
)
↔
(
(
¬
(
X49
↔
(
X46
→
X48
)
)
)
↔
(
¬
(
(
¬
(
X50
→
X49
→
(
X8
↔
X46
)
)
)
→
(
¬
(
(
¬
X48
)
↔
(
(
¬
X46
)
↔
X49
)
)
)
)
)
)
)
↔
(
(
(
(
¬
(
X48
→
X45
→
(
¬
X12
)
→
X29
)
)
→
(
X45
↔
(
X29
→
(
(
¬
X22
)
↔
X48
)
)
)
)
→
X48
→
(
X45
↔
(
X37
→
(
X22
↔
False
)
)
)
)
↔
(
¬
(
X48
→
X45
→
X29
→
X18
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
4bbf22...
and proposition id is
8159e5...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMLKnYVetG9bva7WtMDTcordo48TaddRNwa
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
∀X3 :
prop
,
∀X4 :
prop
,
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
(
¬
∀X8 :
prop
,
∀X9 :
prop
,
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
(
¬
∀X15 :
prop
,
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
(
¬
∀X21 :
prop
,
(
¬
∀X22 :
prop
,
∀X23 :
prop
,
(
¬
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
(
¬
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
∀X44 :
prop
,
(
¬
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
(
¬
∀X49 :
prop
,
(
(
(
(
(
(
(
(
¬
(
X48
↔
(
X47
↔
X5
)
)
)
↔
(
(
¬
(
X48
→
(
¬
X38
)
→
X5
→
X45
)
)
→
X48
→
(
X38
↔
(
X13
↔
X45
)
)
)
)
→
(
(
(
¬
X45
)
↔
(
(
¬
X19
)
→
X48
)
)
↔
(
¬
(
(
¬
(
(
¬
X47
)
→
X45
→
X27
→
X48
)
)
→
(
¬
(
X47
→
(
X27
↔
(
(
¬
X19
)
↔
X45
)
)
)
)
)
)
)
)
→
(
(
¬
(
(
X19
↔
(
X13
→
(
X12
↔
X45
)
)
)
→
(
¬
(
X20
→
(
X19
↔
X12
)
)
)
)
)
↔
(
(
¬
X19
)
→
X13
→
(
¬
X12
)
→
X40
)
)
)
↔
(
(
X43
→
(
¬
X27
)
→
X45
)
↔
(
(
(
¬
(
(
¬
X27
)
↔
(
(
¬
X14
)
↔
(
(
¬
X3
)
→
X45
)
)
)
)
↔
(
¬
(
X45
↔
(
X43
→
(
X14
↔
X27
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X40
→
(
X22
↔
(
(
¬
X14
)
→
X43
)
)
)
)
→
(
X40
↔
(
(
¬
X22
)
↔
(
X2
↔
X43
)
)
)
)
)
↔
(
¬
(
X43
↔
(
X40
↔
(
X22
→
X3
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
¬
(
(
¬
(
X49
↔
(
(
¬
X34
)
→
X43
)
)
)
→
(
(
X43
↔
(
X34
→
(
¬
X7
)
→
X49
)
)
↔
(
X43
→
(
X34
↔
(
X7
↔
X24
)
)
)
)
)
)
↔
(
X38
→
X34
→
X24
)
)
→
(
¬
(
(
(
¬
(
(
¬
X38
)
→
(
X34
↔
X49
)
)
)
↔
(
X46
↔
(
X38
→
(
X34
↔
X36
)
)
)
)
→
(
¬
(
(
¬
X36
)
↔
(
X34
↔
(
X10
↔
X49
)
)
)
)
)
)
)
)
↔
(
(
(
(
¬
X47
)
↔
(
(
¬
X11
)
↔
X49
)
)
↔
(
X49
→
X47
→
X46
→
(
X43
↔
False
)
)
)
↔
(
¬
(
(
¬
X43
)
↔
(
(
¬
X11
)
→
X47
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
¬
(
X45
→
X37
→
X44
)
)
↔
(
(
¬
X37
)
↔
(
X31
↔
(
(
¬
X21
)
→
X45
)
)
)
)
→
(
(
(
¬
X45
)
→
(
¬
X31
)
→
(
X4
↔
X48
)
)
↔
(
(
¬
(
X37
→
X15
→
(
X4
↔
X48
)
)
)
↔
(
¬
(
X45
↔
(
X15
→
False
)
)
)
)
)
→
(
¬
(
(
(
X30
↔
(
(
¬
X29
)
↔
(
X15
↔
X37
)
)
)
↔
(
¬
(
X37
↔
(
(
¬
X30
)
→
(
(
¬
X15
)
↔
X48
)
)
)
)
)
→
(
¬
(
(
¬
X37
)
→
(
¬
X29
)
→
(
(
¬
X15
)
↔
X42
)
)
)
)
)
)
→
(
¬
(
(
(
X47
→
(
(
¬
X41
)
↔
X44
)
)
↔
(
(
(
(
X48
↔
(
X33
↔
(
X28
↔
False
)
)
)
→
(
¬
(
(
¬
(
X48
↔
(
X33
→
(
(
¬
X6
)
↔
X28
)
)
)
)
→
X28
→
(
(
¬
X0
)
↔
X6
)
)
)
)
↔
(
¬
(
(
X28
↔
X48
)
→
(
(
(
¬
X41
)
↔
(
X33
↔
(
(
¬
X28
)
↔
X48
)
)
)
↔
(
¬
(
X44
→
(
X41
↔
X28
)
)
)
)
)
)
)
↔
(
(
¬
X41
)
↔
(
X28
↔
(
X6
→
X47
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
(
¬
X44
)
↔
(
X39
→
(
(
¬
X18
)
↔
X45
)
)
)
)
↔
(
X44
→
X39
→
X26
→
X45
)
)
→
(
¬
(
X48
↔
(
(
¬
X44
)
→
X45
)
)
)
)
→
(
¬
(
X48
→
(
X45
↔
(
X41
↔
X39
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
X17
)
→
X25
)
↔
(
¬
(
(
¬
(
X23
↔
(
(
¬
X17
)
↔
X25
)
)
)
→
(
(
¬
X25
)
↔
(
X23
→
(
X16
↔
X47
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
X25
)
→
(
¬
X23
)
→
X8
→
X47
)
→
X47
→
X44
→
(
X23
↔
False
)
)
→
(
X44
↔
(
X8
→
False
)
)
)
)
)
)
→
(
(
(
¬
(
X47
→
X9
→
X32
)
)
↔
(
¬
(
(
¬
X45
)
↔
(
X9
↔
(
X1
→
X47
)
)
)
)
)
↔
(
X47
→
(
X1
↔
X45
)
)
)
→
(
¬
(
(
X45
→
(
X32
↔
(
X25
↔
X47
)
)
)
→
(
¬
(
X45
→
X44
→
(
X25
↔
X35
)
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
(
(
¬
(
X22
→
X45
)
)
↔
(
(
¬
(
X39
↔
(
X3
→
False
)
)
)
↔
(
X45
→
X22
→
X21
)
)
)
↔
(
X22
↔
X40
)
)
→
(
(
X21
↔
(
X10
→
X17
)
)
↔
(
(
¬
(
(
(
¬
(
X3
↔
X8
)
)
↔
(
(
¬
(
(
X47
↔
(
X21
→
(
X8
↔
False
)
)
)
→
(
X46
↔
(
X10
→
X8
)
)
)
)
→
(
¬
(
X46
→
(
(
¬
X21
)
↔
(
X3
↔
X47
)
)
)
)
)
)
→
(
¬
(
(
¬
X21
)
→
(
(
¬
X17
)
↔
X46
)
)
)
→
(
¬
(
X47
→
(
X21
↔
(
X17
↔
(
X11
↔
False
)
)
)
)
)
)
)
↔
(
¬
(
X21
↔
(
(
¬
X17
)
→
X47
)
)
)
)
)
)
)
↔
(
¬
(
(
(
X34
↔
(
X20
↔
(
X6
↔
X12
)
)
)
→
(
(
X20
→
(
¬
X6
)
→
X24
)
↔
(
¬
(
(
¬
X34
)
→
(
(
¬
X15
)
↔
X40
)
)
)
)
)
→
(
(
¬
X39
)
→
X24
→
(
(
¬
X12
)
↔
X49
)
)
→
(
¬
(
(
X49
↔
(
(
¬
X39
)
↔
(
(
¬
X24
)
↔
X47
)
)
)
→
(
X42
↔
(
X39
→
(
¬
X24
)
→
X49
)
)
)
)
)
)
)
→
(
(
(
(
¬
(
(
¬
(
(
¬
X23
)
→
X14
→
X13
→
X33
)
)
→
(
(
¬
(
X35
↔
(
X32
→
(
(
¬
X13
)
↔
X23
)
)
)
)
↔
(
(
X32
→
(
X27
↔
(
X25
↔
X35
)
)
)
↔
(
(
(
X35
↔
(
X27
→
(
X14
↔
False
)
)
)
↔
(
(
¬
(
X27
↔
(
X23
→
X14
)
)
)
→
(
(
¬
(
X23
↔
(
X14
↔
X32
)
)
)
↔
(
(
¬
X27
)
↔
X35
)
)
)
)
→
(
¬
(
(
¬
X32
)
↔
(
X14
↔
X35
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
(
X33
↔
(
X28
↔
(
X27
→
X48
)
)
)
→
(
(
X38
→
(
(
¬
X35
)
↔
(
(
¬
X33
)
→
X48
)
)
)
↔
(
(
(
¬
X35
)
→
(
¬
X33
)
→
X48
)
↔
(
(
(
¬
X33
)
→
X28
→
X48
)
↔
(
X35
↔
(
X33
↔
(
X28
↔
X7
)
)
)
)
)
)
)
)
→
(
¬
(
X35
↔
(
(
¬
X27
)
↔
(
(
¬
X1
)
→
X48
)
)
)
)
)
)
)
→
(
(
(
¬
(
(
¬
X47
)
→
X46
→
(
X29
↔
X48
)
)
)
↔
(
(
X47
→
(
¬
X46
)
→
X48
)
↔
(
¬
(
X46
→
X38
→
(
(
¬
X32
)
↔
X47
)
)
)
)
)
↔
(
(
(
(
X46
↔
(
(
¬
X4
)
↔
X19
)
)
→
(
X46
↔
(
X44
↔
(
X29
→
X19
)
)
)
)
↔
(
¬
(
X46
→
X29
→
(
X4
↔
False
)
)
)
)
↔
(
¬
(
(
(
(
¬
(
X29
→
(
(
¬
X5
)
↔
(
X2
→
X38
)
)
)
)
→
X38
→
(
X19
↔
(
X4
→
X2
→
False
)
)
)
→
(
¬
(
X29
→
X4
→
X2
→
False
)
)
)
→
(
¬
(
X19
↔
(
(
¬
X4
)
↔
X29
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
¬
(
X36
↔
(
X31
↔
X37
)
)
)
→
(
¬
(
(
¬
X37
)
↔
(
X31
→
(
¬
X30
)
→
X44
)
)
)
)
↔
(
¬
(
(
¬
X30
)
↔
X44
)
)
)
→
(
(
¬
(
X31
↔
(
X26
↔
(
X0
↔
X37
)
)
)
)
↔
(
¬
(
(
¬
X37
)
→
(
(
¬
X0
)
↔
X43
)
)
)
)
→
(
X43
↔
(
X36
→
(
(
¬
X31
)
↔
X37
)
)
)
)
)
→
(
(
(
(
¬
(
X41
→
(
¬
X30
)
→
X16
→
X49
)
)
↔
(
¬
(
X49
→
(
(
¬
X30
)
↔
X37
)
)
)
)
↔
(
¬
(
X37
↔
(
X16
→
False
)
)
)
)
↔
(
(
¬
(
X44
↔
(
X30
→
(
X16
↔
X9
)
)
)
)
↔
(
¬
(
X44
↔
X49
)
)
)
)
→
(
(
X46
→
(
(
¬
X30
)
↔
(
X18
→
X44
)
)
)
↔
(
(
(
¬
X46
)
→
(
(
¬
X44
)
↔
X47
)
)
↔
(
(
¬
X46
)
↔
(
(
¬
X30
)
↔
X49
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
39cfc6...
and proposition id is
62785b...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMZkMSKu1qNTNMdDpiKTC2PQaRdDze7qgRQ
)
∀X0 :
prop
,
∀X1 :
prop
,
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
∀X4 :
prop
,
(
¬
∀X5 :
prop
,
(
¬
∀X6 :
prop
,
(
¬
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
(
¬
∀X10 :
prop
,
∀X11 :
prop
,
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
(
¬
∀X17 :
prop
,
∀X18 :
prop
,
(
¬
∀X19 :
prop
,
(
¬
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
∀X25 :
prop
,
∀X26 :
prop
,
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
(
¬
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
∀X32 :
prop
,
(
¬
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
(
¬
∀X36 :
prop
,
(
¬
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
(
¬
∀X39 :
prop
,
(
¬
∀X40 :
prop
,
(
¬
∀X41 :
prop
,
∀X42 :
prop
,
(
¬
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
(
¬
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
(
(
(
(
¬
(
(
(
(
X46
↔
(
(
¬
X16
)
→
X12
→
X28
)
)
↔
(
¬
(
(
¬
X28
)
↔
(
X16
→
X42
)
)
)
)
↔
(
(
¬
(
(
¬
(
(
¬
X42
)
→
(
X41
↔
X46
)
)
)
→
(
(
(
X41
↔
(
(
¬
X20
)
↔
X28
)
)
↔
(
¬
(
(
¬
X42
)
↔
(
X20
↔
X46
)
)
)
)
↔
(
¬
False
)
)
)
)
↔
(
¬
(
X46
→
(
X28
↔
X41
)
)
)
)
)
→
(
¬
(
(
¬
(
(
¬
(
X48
→
X47
→
X46
)
)
→
(
¬
(
(
(
¬
(
X48
↔
(
X47
→
(
X46
↔
False
)
)
)
)
↔
(
(
X46
↔
(
(
¬
X38
)
→
X6
→
X47
)
)
↔
(
(
¬
X47
)
↔
(
X46
→
(
¬
X38
)
→
X48
)
)
)
)
→
(
¬
(
X47
→
(
X6
↔
X38
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
(
X47
↔
(
X45
→
(
¬
X12
)
→
X48
)
)
→
(
¬
(
(
¬
X20
)
↔
X48
)
)
)
↔
(
X47
↔
X12
)
)
↔
(
X48
→
X47
→
(
X20
↔
X46
)
)
)
→
(
¬
(
X46
→
(
¬
X20
)
→
X48
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
(
(
(
¬
X45
)
↔
(
(
¬
X31
)
↔
(
X18
↔
X47
)
)
)
→
(
¬
(
X31
↔
(
(
¬
X19
)
→
(
X18
↔
X47
)
)
)
)
)
)
→
(
(
¬
(
(
¬
(
(
X47
↔
(
X19
↔
(
(
¬
X9
)
↔
X35
)
)
)
→
(
¬
(
(
¬
X33
)
→
(
(
¬
X19
)
↔
X35
)
)
)
→
(
(
¬
(
X33
→
(
¬
X18
)
→
X19
)
)
↔
(
¬
(
X47
→
X33
→
X35
)
)
)
)
)
→
X47
→
(
(
¬
X18
)
↔
X19
)
)
)
↔
(
(
¬
(
(
X35
↔
(
X30
↔
(
X7
↔
X47
)
)
)
→
(
¬
(
X33
→
X30
→
(
¬
X7
)
→
X47
)
)
)
)
↔
(
¬
(
(
¬
X35
)
↔
(
(
¬
X33
)
↔
X47
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
X29
↔
(
X15
→
X33
)
)
↔
(
¬
(
(
¬
(
X47
→
(
X33
↔
False
)
)
)
→
(
¬
(
(
¬
X26
)
→
(
X15
↔
X43
)
)
)
)
)
)
↔
(
(
¬
(
(
¬
(
X46
↔
X35
)
)
→
(
X35
↔
(
(
¬
X26
)
→
(
(
¬
X15
)
↔
X43
)
)
)
)
)
↔
(
(
¬
(
(
¬
X26
)
→
(
X14
↔
(
(
¬
X10
)
↔
X43
)
)
)
)
↔
(
X26
↔
(
(
¬
X15
)
→
(
X14
↔
X43
)
)
)
)
)
)
→
(
¬
X29
)
→
(
¬
X26
)
→
(
¬
X13
)
→
X46
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
(
X23
→
(
¬
X42
)
→
X40
→
(
¬
X1
)
→
X44
)
→
(
¬
(
X40
→
False
)
)
)
→
(
¬
(
X42
↔
(
X41
↔
(
X24
↔
X40
)
)
)
)
→
(
(
X41
↔
(
X36
→
X42
)
)
↔
(
(
X41
↔
(
(
¬
X24
)
↔
X44
)
)
↔
(
(
¬
X42
)
→
(
¬
X36
)
→
X24
→
X44
)
)
)
)
)
↔
(
¬
(
(
¬
(
X44
↔
(
(
¬
X40
)
→
X22
→
X42
)
)
)
→
(
¬
(
(
¬
X39
)
→
(
(
¬
X21
)
↔
(
(
¬
X0
)
↔
X44
)
)
)
)
)
)
)
→
(
(
(
(
X44
↔
(
X25
→
(
X8
↔
False
)
)
)
↔
(
(
(
(
¬
(
X44
↔
(
X25
→
X41
)
)
)
↔
(
¬
(
(
¬
X11
)
→
X8
→
X41
)
)
)
↔
(
¬
(
(
¬
X11
)
↔
(
X8
↔
X44
)
)
)
)
↔
(
X47
→
(
X41
↔
(
X25
↔
False
)
)
)
)
)
↔
(
(
¬
(
(
X47
↔
(
(
¬
X42
)
↔
(
X25
→
X49
)
)
)
→
(
¬
(
X49
→
(
(
¬
X42
)
↔
(
X34
↔
X47
)
)
)
)
)
)
↔
(
¬
(
(
X49
→
(
¬
X8
)
→
X17
)
→
X49
→
(
X34
↔
(
X17
↔
X47
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
X37
)
→
X32
→
(
(
¬
X5
)
↔
X47
)
)
)
→
(
X47
↔
(
X42
↔
(
X32
→
False
)
)
)
)
↔
(
(
(
X37
→
X42
)
→
(
(
(
X37
↔
(
X4
→
X2
)
)
↔
(
X3
→
(
X2
↔
X42
)
)
)
↔
(
X42
↔
(
(
¬
X4
)
↔
X37
)
)
)
)
→
(
¬
(
(
X42
↔
(
X27
↔
(
X4
→
X49
)
)
)
→
(
(
¬
(
X49
→
(
X47
↔
X42
)
)
)
↔
(
(
(
¬
X47
)
↔
(
X42
↔
(
(
¬
X27
)
↔
X49
)
)
)
↔
(
X47
↔
(
(
¬
X27
)
↔
(
X3
→
X42
)
)
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
¬
(
(
(
(
(
¬
X46
)
→
(
¬
X27
)
→
(
(
¬
X13
)
↔
X47
)
)
↔
(
X47
→
X27
→
(
(
¬
X13
)
↔
X48
)
)
)
→
(
(
¬
(
X47
→
(
(
¬
X27
)
↔
(
(
¬
X20
)
↔
X48
)
)
)
)
↔
(
(
(
¬
(
X47
→
(
¬
X27
)
→
(
X26
↔
X29
)
)
)
↔
(
(
X47
↔
X29
)
→
(
¬
X29
)
→
(
(
¬
X20
)
↔
(
(
¬
X18
)
→
X37
)
)
)
)
→
(
X45
↔
(
X37
↔
(
(
¬
X27
)
→
X47
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X24
→
(
¬
X6
)
→
(
(
¬
X0
)
↔
X47
)
)
)
→
(
¬
X23
)
→
X0
→
X46
)
)
→
(
(
(
¬
X42
)
→
X31
→
X24
→
X46
)
→
(
X46
↔
(
X42
→
(
X24
↔
X48
)
)
)
)
→
(
¬
(
(
(
X46
→
(
X42
↔
X7
)
)
↔
(
¬
(
(
(
¬
(
(
¬
X7
)
↔
X42
)
)
→
X46
→
(
¬
X42
)
→
X36
→
X47
)
→
(
X46
↔
(
X7
↔
False
)
)
)
)
)
→
(
(
¬
(
(
¬
(
X48
↔
(
X46
→
(
(
¬
X1
)
↔
X36
)
)
)
)
→
(
X48
↔
(
(
¬
X36
)
→
(
(
¬
X7
)
↔
X46
)
)
)
)
)
↔
(
(
¬
X7
)
↔
(
X5
→
X46
)
)
)
)
)
)
)
↔
(
(
(
¬
(
(
¬
(
(
(
¬
(
(
¬
X46
)
→
(
X35
↔
(
(
¬
X12
)
→
X48
)
)
)
)
↔
(
(
¬
X46
)
↔
(
X43
→
X12
→
X48
)
)
)
→
(
(
¬
(
X46
↔
(
X15
→
X43
)
)
)
↔
(
¬
(
X43
→
X10
→
X48
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
(
(
¬
(
X48
→
(
X44
↔
(
(
¬
X15
)
→
X35
)
)
)
)
↔
(
X48
↔
(
X44
↔
(
X3
→
X35
)
)
)
)
→
(
¬
(
X35
↔
(
X8
→
(
(
¬
X3
)
↔
X44
)
)
)
)
)
)
→
X8
→
X48
)
)
↔
(
X8
→
(
X3
↔
X44
)
)
)
↔
(
X48
↔
X8
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
(
X48
→
(
¬
X40
)
→
(
¬
X16
)
→
X43
)
)
↔
(
¬
(
X48
↔
(
X16
↔
(
X11
→
X47
)
)
)
)
)
→
(
X11
↔
X43
)
)
)
→
(
X46
↔
(
(
¬
X40
)
↔
X43
)
)
→
(
(
X48
→
(
X43
↔
(
X40
↔
X46
)
)
)
↔
(
(
¬
(
(
¬
X43
)
→
(
(
¬
X2
)
↔
X48
)
)
)
→
X43
→
X19
→
X2
→
X40
)
)
)
)
)
→
(
(
(
(
(
(
X48
↔
(
X46
↔
(
X22
→
X28
)
)
)
→
(
(
(
¬
(
X48
→
(
X34
↔
(
X4
→
X46
)
)
)
)
↔
(
¬
(
X34
↔
(
X28
↔
False
)
)
)
)
↔
(
¬
(
X48
↔
(
(
¬
X41
)
↔
(
(
¬
X34
)
→
X46
)
)
)
)
)
)
→
(
¬
(
X48
↔
(
(
¬
X41
)
↔
(
X14
→
X46
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X46
)
→
(
(
¬
X41
)
↔
(
(
¬
X30
)
↔
X48
)
)
)
)
→
(
¬
(
X48
→
(
¬
X41
)
→
(
(
¬
X32
)
↔
X46
)
)
)
)
)
)
→
(
(
(
¬
(
X46
↔
(
X17
↔
(
X9
↔
X32
)
)
)
)
→
(
¬
X38
)
→
(
(
¬
X17
)
↔
(
X9
→
X46
)
)
)
↔
(
(
¬
(
X38
↔
(
(
¬
X17
)
↔
(
(
¬
X9
)
↔
X34
)
)
)
)
→
(
¬
(
X49
→
(
X34
↔
X38
)
)
)
)
)
)
↔
(
(
(
¬
(
X47
↔
(
X41
→
X39
→
False
)
)
)
→
(
¬
(
X47
↔
(
X46
→
X41
→
X48
)
)
)
)
↔
(
(
¬
(
(
¬
X33
)
↔
(
X21
→
X47
)
)
)
↔
(
(
X46
↔
(
X33
→
(
(
¬
X25
)
↔
X47
)
)
)
↔
(
¬
(
X46
↔
(
X33
→
X48
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
1566e4...
and proposition id is
864cbb...
Proof:
The rest of this subproof is missing.
∎
Theorem.
(
conj_QBF_TMW4w9MMRFWwtyhhjKVgYLs2LxPT4SNCmzg
)
(
¬
∀X0 :
prop
,
∀X1 :
prop
,
(
¬
∀X2 :
prop
,
(
¬
∀X3 :
prop
,
(
¬
∀X4 :
prop
,
∀X5 :
prop
,
∀X6 :
prop
,
∀X7 :
prop
,
∀X8 :
prop
,
(
¬
∀X9 :
prop
,
∀X10 :
prop
,
(
¬
∀X11 :
prop
,
(
¬
∀X12 :
prop
,
(
¬
∀X13 :
prop
,
(
¬
∀X14 :
prop
,
∀X15 :
prop
,
(
¬
∀X16 :
prop
,
∀X17 :
prop
,
(
¬
∀X18 :
prop
,
∀X19 :
prop
,
∀X20 :
prop
,
∀X21 :
prop
,
∀X22 :
prop
,
∀X23 :
prop
,
∀X24 :
prop
,
(
¬
∀X25 :
prop
,
(
¬
∀X26 :
prop
,
(
¬
∀X27 :
prop
,
∀X28 :
prop
,
(
¬
∀X29 :
prop
,
∀X30 :
prop
,
(
¬
∀X31 :
prop
,
(
¬
∀X32 :
prop
,
∀X33 :
prop
,
∀X34 :
prop
,
∀X35 :
prop
,
∀X36 :
prop
,
∀X37 :
prop
,
(
¬
∀X38 :
prop
,
∀X39 :
prop
,
∀X40 :
prop
,
∀X41 :
prop
,
(
¬
∀X42 :
prop
,
∀X43 :
prop
,
(
¬
∀X44 :
prop
,
∀X45 :
prop
,
∀X46 :
prop
,
(
¬
∀X47 :
prop
,
(
¬
∀X48 :
prop
,
∀X49 :
prop
,
∀X50 :
prop
,
(
¬
∀X51 :
prop
,
(
(
(
(
(
(
¬
(
(
X31
↔
(
(
¬
X22
)
↔
X29
)
)
→
(
¬
(
X39
↔
(
(
¬
X29
)
→
(
¬
X22
)
→
X31
)
)
)
→
(
¬
(
(
¬
X31
)
→
X29
→
(
¬
X11
)
→
X39
)
)
)
)
↔
(
¬
(
(
(
X31
→
X22
→
(
X6
↔
X49
)
)
↔
(
¬
(
X49
→
X31
→
(
X6
↔
X29
)
)
)
)
→
(
¬
(
X49
↔
(
X31
→
X22
→
X27
)
)
)
)
)
)
↔
(
(
¬
(
(
X50
→
X48
→
(
¬
X39
)
→
X51
)
→
(
¬
(
X39
↔
X25
)
)
)
)
→
(
¬
(
(
(
X32
→
(
X29
↔
(
X12
→
X39
)
)
)
↔
(
¬
(
X31
↔
X32
)
)
)
→
(
(
(
¬
(
X29
→
(
¬
X0
)
→
X50
)
)
↔
(
¬
(
X48
→
X32
→
(
X13
↔
False
)
)
)
)
↔
(
¬
(
(
¬
(
X19
↔
(
X13
↔
(
X0
→
X39
)
)
)
)
→
(
¬
(
X50
↔
(
X39
↔
(
X13
→
False
)
)
)
)
)
)
)
)
)
)
)
↔
(
(
(
X50
↔
(
X38
↔
(
X1
↔
X16
)
)
)
↔
(
(
X47
↔
(
X34
↔
(
(
¬
X16
)
↔
X51
)
)
)
↔
(
(
(
(
¬
(
(
¬
X38
)
→
(
(
¬
X33
)
↔
(
X16
→
X47
)
)
)
)
↔
(
¬
(
X47
→
X38
→
X1
→
False
)
)
)
↔
(
(
¬
X38
)
↔
(
X16
→
X47
)
)
)
↔
(
(
¬
X34
)
↔
(
(
¬
X33
)
→
(
X16
↔
X38
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
(
(
¬
(
X49
→
(
X42
↔
(
X30
↔
X28
)
)
)
)
→
(
(
¬
X37
)
↔
(
X30
→
X42
)
)
)
)
↔
(
X51
→
X49
→
X38
→
X37
→
False
)
)
→
(
(
¬
(
(
¬
X34
)
→
(
¬
X26
)
→
(
¬
X14
)
→
X42
)
)
↔
(
¬
(
(
X14
↔
X34
)
→
(
¬
(
(
X34
↔
(
(
¬
X17
)
↔
(
X14
↔
X49
)
)
)
→
(
¬
(
X38
→
(
X34
↔
(
(
¬
X14
)
→
X49
)
)
)
)
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
(
(
¬
(
X49
→
X46
→
(
X43
↔
(
X35
↔
False
)
)
)
)
→
(
(
(
(
¬
(
X49
→
X45
→
X23
)
)
↔
(
(
¬
(
X45
→
(
¬
X23
)
→
(
X10
↔
X49
)
)
)
↔
(
(
¬
X21
)
→
(
¬
X10
)
→
X23
)
)
)
↔
(
(
¬
X10
)
→
X23
)
)
↔
(
¬
(
(
(
¬
X35
)
↔
(
X21
→
X10
→
X43
)
)
→
(
¬
(
X49
↔
(
X35
→
(
(
¬
X10
)
↔
X43
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
¬
X46
)
↔
(
X8
→
(
(
¬
X7
)
↔
X49
)
)
)
→
(
(
¬
(
(
¬
(
X21
↔
(
X9
→
(
X8
↔
X49
)
)
)
)
→
(
¬
(
X21
→
(
X9
↔
(
X8
→
X46
)
)
)
)
)
)
↔
(
¬
(
X21
↔
X49
)
)
)
)
)
)
→
(
¬
(
X45
↔
(
X43
→
X41
→
X5
)
)
)
→
(
X45
↔
(
(
¬
X18
)
→
(
(
¬
X5
)
↔
X49
)
)
)
)
)
→
(
¬
(
(
(
¬
(
X51
↔
(
X40
→
(
X36
↔
False
)
)
)
)
→
(
¬
(
X49
↔
(
(
¬
X46
)
→
(
(
¬
X40
)
↔
X51
)
)
)
)
)
→
(
(
X36
→
(
X15
↔
(
X2
↔
X46
)
)
)
↔
(
(
(
¬
X46
)
↔
(
(
¬
X36
)
↔
X51
)
)
↔
(
(
(
(
¬
(
(
¬
X44
)
→
(
(
¬
X36
)
↔
(
X15
↔
X46
)
)
)
)
→
(
¬
(
X51
↔
(
X36
↔
X46
)
)
)
)
↔
(
X44
↔
False
)
)
↔
(
¬
(
X46
→
False
)
)
)
)
)
→
X46
→
X36
→
X24
→
(
X15
↔
False
)
)
)
→
(
(
X44
↔
(
(
¬
X20
)
→
(
X3
↔
X45
)
)
)
↔
(
(
¬
(
(
(
¬
X3
)
↔
X4
)
→
(
¬
(
X49
↔
(
X20
↔
X46
)
)
)
)
)
↔
(
(
(
¬
(
X49
→
(
(
¬
X45
)
↔
X46
)
)
)
↔
(
¬
(
(
(
¬
X45
)
↔
(
(
¬
X3
)
→
X46
)
)
→
(
¬
(
(
X49
↔
(
(
¬
X45
)
↔
X50
)
)
→
(
¬
(
(
¬
X46
)
↔
(
(
¬
X45
)
→
X3
→
X50
)
)
)
)
)
)
)
)
→
X46
→
(
(
¬
X45
)
↔
X49
)
)
)
)
)
↔
(
(
(
(
¬
(
X48
→
(
¬
X46
)
→
(
X32
↔
X47
)
)
)
→
(
(
¬
(
(
(
¬
(
X50
→
(
X21
↔
(
X16
↔
X48
)
)
)
)
↔
(
X47
→
X21
)
)
→
(
¬
(
(
¬
(
(
¬
(
(
¬
(
X50
↔
(
X44
→
X47
)
)
)
→
(
¬
(
X50
↔
(
X46
→
(
X29
↔
False
)
)
)
)
)
)
→
(
X46
↔
(
(
¬
X29
)
↔
X44
)
)
)
)
→
(
¬
(
(
¬
X48
)
↔
(
(
¬
X46
)
↔
(
X29
→
X50
)
)
)
)
)
)
)
)
↔
(
¬
(
(
¬
(
(
(
¬
X47
)
→
(
X27
↔
X48
)
)
→
X48
→
(
X38
↔
X50
)
)
)
→
(
¬
(
X50
→
(
X48
↔
(
X36
→
X47
)
)
)
)
)
)
)
)
↔
(
¬
(
(
(
(
(
(
¬
(
X47
↔
(
X22
↔
X18
)
)
)
→
X48
→
(
X18
↔
X51
)
)
→
(
¬
(
X48
→
(
X47
↔
(
(
¬
X22
)
↔
X50
)
)
)
)
)
↔
(
(
X51
↔
(
X35
→
X18
→
False
)
)
→
(
X31
↔
(
X22
↔
(
(
¬
X18
)
↔
X35
)
)
)
)
)
↔
(
¬
(
(
¬
(
X48
↔
(
X47
↔
(
X2
→
X51
)
)
)
)
→
(
¬
(
(
(
(
X19
→
(
(
¬
X0
)
↔
X47
)
)
↔
(
¬
(
X48
→
X19
→
(
(
¬
X0
)
↔
X1
)
)
)
)
↔
(
¬
(
X48
↔
(
X1
↔
X19
)
)
)
)
→
(
¬
(
X2
↔
(
(
¬
X1
)
↔
(
(
¬
X0
)
→
X48
)
)
)
)
)
)
)
)
)
→
(
¬
(
(
¬
(
X48
→
X35
→
X4
)
)
→
(
¬
(
(
¬
(
(
¬
X48
)
→
(
¬
X24
)
→
X13
→
X50
)
)
→
X50
→
(
¬
X8
)
→
X13
)
)
)
)
)
)
)
↔
(
(
(
(
(
¬
(
X49
↔
(
X43
→
(
X11
↔
(
X7
↔
False
)
)
)
)
)
↔
(
¬
(
X49
→
(
X45
↔
(
X11
→
False
)
)
)
)
)
→
(
(
(
¬
(
X49
↔
(
X43
↔
X33
)
)
)
→
(
¬
(
(
(
¬
X43
)
↔
(
(
¬
X33
)
↔
(
X15
↔
X45
)
)
)
→
(
¬
X45
)
→
(
(
¬
X43
)
↔
(
(
¬
X15
)
→
X49
)
)
)
)
)
↔
(
¬
(
X47
→
(
X17
↔
(
X15
↔
X49
)
)
)
)
)
)
↔
(
(
(
X46
↔
(
X43
↔
(
X20
↔
X45
)
)
)
→
(
¬
(
(
¬
X28
)
→
X43
)
)
)
↔
(
(
(
¬
(
X45
↔
(
X43
↔
(
X42
→
False
)
)
)
)
↔
(
(
X45
↔
(
X43
→
X42
→
(
X37
↔
False
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X42
)
↔
(
(
¬
X10
)
↔
X49
)
)
)
→
(
¬
(
X37
↔
X45
)
)
)
)
)
)
↔
(
(
(
X42
→
(
(
¬
X14
)
↔
(
(
¬
X6
)
↔
X45
)
)
)
↔
(
¬
(
(
¬
(
(
¬
X42
)
→
(
¬
X28
)
→
X14
→
X45
)
)
→
(
X28
↔
(
(
¬
X14
)
→
X49
)
)
)
)
)
↔
(
X49
→
X42
→
X45
)
)
)
)
)
↔
(
¬
(
(
(
(
X46
→
(
¬
X34
)
→
(
X30
↔
X41
)
)
→
X34
→
(
(
¬
X30
)
↔
(
X26
↔
X40
)
)
)
↔
(
(
X46
↔
(
(
¬
X39
)
↔
(
(
¬
X26
)
→
X41
)
)
)
↔
(
(
¬
(
(
¬
X41
)
↔
(
(
¬
X26
)
↔
(
(
¬
X12
)
↔
X46
)
)
)
)
→
X46
→
(
X26
↔
(
X3
→
X12
)
)
)
)
)
→
(
(
(
¬
(
(
¬
(
X49
→
X45
→
X5
→
False
)
)
→
(
X34
↔
(
X30
↔
X49
)
)
)
)
↔
(
(
(
¬
X46
)
→
(
¬
X30
)
→
X23
→
X49
)
↔
(
¬
(
X45
↔
(
X30
↔
X23
)
)
)
)
)
↔
(
(
(
(
X46
↔
(
X45
↔
(
(
¬
X9
)
↔
X44
)
)
)
↔
(
¬
(
X46
↔
(
X45
→
X44
→
X25
→
False
)
)
)
)
→
X44
→
(
X9
↔
X25
)
)
↔
(
¬
(
X49
→
(
X46
↔
(
X45
↔
(
X44
→
False
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
In Proofgold the corresponding term root is
3b6042...
and proposition id is
7d6a07...
Proof:
The rest of this subproof is missing.
∎
End of Section
QBF