Beginning of Section A28071
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
minus_SNo.
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add_SNo.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul_SNo.
Notation. We use
< as an infix operator with priority 490 and no associativity corresponding to applying term
SNoLt.
Notation. We use
<= as an infix operator with priority 490 and no associativity corresponding to applying term
SNoLe.
Variable F3 : set → set
Hypothesis HF3 : ∀x0 ∈ int, F3 x0 ∈ int
Variable G3 : set → set
Hypothesis HG3 : ∀x0 ∈ int, G3 x0 ∈ int
Variable F4 : set → set
Hypothesis HF4 : ∀x0 ∈ int, F4 x0 ∈ int
Variable G4 : set → set
Hypothesis HG4 : ∀x0 ∈ int, G4 x0 ∈ int
Variable H4 : set
Hypothesis HH4 : H4 ∈ int
Variable U4 : set → set → set
Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, U4 x0 x1 ∈ int
Variable V4 : set → set
Hypothesis HV4 : ∀x0 ∈ int, V4 x0 ∈ int
Variable H3 : set → set
Hypothesis HH3 : ∀x0 ∈ int, H3 x0 ∈ int
Variable U3 : set → set → set
Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, U3 x0 x1 ∈ int
Variable V3 : set → set
Hypothesis HV3 : ∀x0 ∈ int, V3 x0 ∈ int
Variable F2 : set → set
Hypothesis HF2 : ∀x0 ∈ int, F2 x0 ∈ int
Variable G2 : set
Hypothesis HG2 : G2 ∈ int
Variable H2 : set → set → set
Hypothesis HH2 : ∀x0 ∈ int, ∀x1 ∈ int, H2 x0 x1 ∈ int
Variable U2 : set → set → set
Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, U2 x0 x1 ∈ int
Variable V2 : set → set → set
Hypothesis HV2 : ∀x0 ∈ int, ∀x1 ∈ int, V2 x0 x1 ∈ int
Variable F5 : set → set → set
Hypothesis HF5 : ∀x0 ∈ int, ∀x1 ∈ int, F5 x0 x1 ∈ int
Variable G5 : set
Hypothesis HG5 : G5 ∈ int
Variable H5 : set → set
Hypothesis HH5 : ∀x0 ∈ int, H5 x0 ∈ int
Variable U5 : set → set → set
Hypothesis HU5 : ∀x0 ∈ int, ∀x1 ∈ int, U5 x0 x1 ∈ int
Variable V5 : set → set
Hypothesis HV5 : ∀x0 ∈ int, V5 x0 ∈ int
Variable F1 : set → set → set
Hypothesis HF1 : ∀x0 ∈ int, ∀x1 ∈ int, F1 x0 x1 ∈ int
Variable G1 : set → set → set
Hypothesis HG1 : ∀x0 ∈ int, ∀x1 ∈ int, G1 x0 x1 ∈ int
Variable H1 : set
Hypothesis HH1 : H1 ∈ int
Variable U1 : set → set → set
Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, U1 x0 x1 ∈ int
Variable V1 : set → set → set
Hypothesis HV1 : ∀x0 ∈ int, ∀x1 ∈ int, V1 x0 x1 ∈ int
Variable F0 : set → set → set
Hypothesis HF0 : ∀x0 ∈ int, ∀x1 ∈ int, F0 x0 x1 ∈ int
Variable G0 : set → set
Hypothesis HG0 : ∀x0 ∈ int, G0 x0 ∈ int
Variable H0 : set
Hypothesis HH0 : H0 ∈ int
Variable U0 : set → set → set
Hypothesis HU0 : ∀x0 ∈ int, ∀x1 ∈ int, U0 x0 x1 ∈ int
Variable V0 : set → set
Hypothesis HV0 : ∀x0 ∈ int, V0 x0 ∈ int
Variable SMALL : set → set
Hypothesis HSMALL : ∀x0 ∈ int, SMALL x0 ∈ int
Variable F10 : set → set
Hypothesis HF10 : ∀x0 ∈ int, F10 x0 ∈ int
Variable G10 : set → set
Hypothesis HG10 : ∀x0 ∈ int, G10 x0 ∈ int
Variable H10 : set
Hypothesis HH10 : H10 ∈ int
Variable U10 : set → set → set
Hypothesis HU10 : ∀x0 ∈ int, ∀x1 ∈ int, U10 x0 x1 ∈ int
Variable V10 : set → set
Hypothesis HV10 : ∀x0 ∈ int, V10 x0 ∈ int
Variable F11 : set → set → set
Hypothesis HF11 : ∀x0 ∈ int, ∀x1 ∈ int, F11 x0 x1 ∈ int
Variable G11 : set → set → set
Hypothesis HG11 : ∀x0 ∈ int, ∀x1 ∈ int, G11 x0 x1 ∈ int
Variable H11 : set → set
Hypothesis HH11 : ∀x0 ∈ int, H11 x0 ∈ int
Variable I11 : set
Hypothesis HI11 : I11 ∈ int
Variable J11 : set
Hypothesis HJ11 : J11 ∈ int
Variable U11 : set → set → set → set
Hypothesis HU11 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U11 x0 x1 x2 ∈ int
Variable V11 : set → set → set → set
Hypothesis HV11 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V11 x0 x1 x2 ∈ int
Variable W11 : set → set
Hypothesis HW11 : ∀x0 ∈ int, W11 x0 ∈ int
Variable F9 : set → set
Hypothesis HF9 : ∀x0 ∈ int, F9 x0 ∈ int
Variable G9 : set
Hypothesis HG9 : G9 ∈ int
Variable H9 : set → set → set
Hypothesis HH9 : ∀x0 ∈ int, ∀x1 ∈ int, H9 x0 x1 ∈ int
Variable U9 : set → set → set
Hypothesis HU9 : ∀x0 ∈ int, ∀x1 ∈ int, U9 x0 x1 ∈ int
Variable V9 : set → set → set
Hypothesis HV9 : ∀x0 ∈ int, ∀x1 ∈ int, V9 x0 x1 ∈ int
Variable F8 : set → set → set
Hypothesis HF8 : ∀x0 ∈ int, ∀x1 ∈ int, F8 x0 x1 ∈ int
Variable G8 : set → set
Hypothesis HG8 : ∀x0 ∈ int, G8 x0 ∈ int
Variable H8 : set
Hypothesis HH8 : H8 ∈ int
Variable U8 : set → set → set
Hypothesis HU8 : ∀x0 ∈ int, ∀x1 ∈ int, U8 x0 x1 ∈ int
Variable V8 : set → set
Hypothesis HV8 : ∀x0 ∈ int, V8 x0 ∈ int
Variable F7 : set → set
Hypothesis HF7 : ∀x0 ∈ int, F7 x0 ∈ int
Variable G7 : set
Hypothesis HG7 : G7 ∈ int
Variable H7 : set → set → set
Hypothesis HH7 : ∀x0 ∈ int, ∀x1 ∈ int, H7 x0 x1 ∈ int
Variable U7 : set → set → set
Hypothesis HU7 : ∀x0 ∈ int, ∀x1 ∈ int, U7 x0 x1 ∈ int
Variable V7 : set → set → set
Hypothesis HV7 : ∀x0 ∈ int, ∀x1 ∈ int, V7 x0 x1 ∈ int
Variable F6 : set → set → set
Hypothesis HF6 : ∀x0 ∈ int, ∀x1 ∈ int, F6 x0 x1 ∈ int
Variable G6 : set → set
Hypothesis HG6 : ∀x0 ∈ int, G6 x0 ∈ int
Variable H6 : set
Hypothesis HH6 : H6 ∈ int
Variable U6 : set → set → set
Hypothesis HU6 : ∀x0 ∈ int, ∀x1 ∈ int, U6 x0 x1 ∈ int
Variable V6 : set → set
Hypothesis HV6 : ∀x0 ∈ int, V6 x0 ∈ int
Variable FAST : set → set
Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
Hypothesis H1 : (∀X ∈ int, ((F3 X) = ((2 * (X + X)) + X)))
Hypothesis H2 : (∀X ∈ int, ((G3 X) = X))
Hypothesis H3 : (∀X ∈ int, ((F4 X) = (X + X)))
Hypothesis H4 : (∀X ∈ int, ((G4 X) = X))
Hypothesis H5 : (H4 = 2)
Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y))))))
Hypothesis H7 : (∀X ∈ int, ((V4 X) = (U4 (G4 X) H4)))
Hypothesis H8 : (∀X ∈ int, ((H3 X) = ((V4 X) + - 1)))
Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y))))))
Hypothesis H10 : (∀X ∈ int, ((V3 X) = (U3 (G3 X) (H3 X))))
Hypothesis H11 : (∀X ∈ int, ((F2 X) = (V3 X)))
Hypothesis H12 : (G2 = 1)
Hypothesis H13 : (∀X ∈ int, (∀Y ∈ int, ((H2 X Y) = Y)))
Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y))))))
Hypothesis H15 : (∀X ∈ int, (∀Y ∈ int, ((V2 X Y) = (U2 G2 (H2 X Y)))))
Hypothesis H16 : (∀X ∈ int, (∀Y ∈ int, ((F5 X Y) = ((2 + Y) * X))))
Hypothesis H17 : (G5 = 2)
Hypothesis H18 : (∀X ∈ int, ((H5 X) = X))
Hypothesis H19 : (∀X ∈ int, (∀Y ∈ int, ((U5 X Y) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y) X)))))
Hypothesis H20 : (∀X ∈ int, ((V5 X) = (U5 G5 (H5 X))))
Hypothesis H21 : (∀X ∈ int, (∀Y ∈ int, ((F1 X Y) = ((V2 X Y) + ((V5 X) + - X)))))
Hypothesis H22 : (∀X ∈ int, (∀Y ∈ int, ((G1 X Y) = Y)))
Hypothesis H23 : (H1 = 1)
Hypothesis H24 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y) X)))))
Hypothesis H25 : (∀X ∈ int, (∀Y ∈ int, ((V1 X Y) = (U1 (G1 X Y) H1))))
Hypothesis H26 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = ((((V1 X Y) + X) + X) + X))))
Hypothesis H27 : (∀X ∈ int, ((G0 X) = X))
Hypothesis H28 : (H0 = 1)
Hypothesis H29 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y) X)))))
Hypothesis H30 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))
Hypothesis H31 : (∀X ∈ int, ((SMALL X) = (V0 X)))
Hypothesis H32 : (∀X ∈ int, ((F10 X) = (X + X)))
Hypothesis H33 : (∀X ∈ int, ((G10 X) = X))
Hypothesis H34 : (H10 = 2)
Hypothesis H35 : (∀X ∈ int, (∀Y ∈ int, ((U10 X Y) = (if (X <= 0) then Y else (F10 (U10 (X + - 1) Y))))))
Hypothesis H36 : (∀X ∈ int, ((V10 X) = (U10 (G10 X) H10)))
Hypothesis H37 : (∀X ∈ int, (∀Y ∈ int, ((F11 X Y) = (X * Y))))
Hypothesis H38 : (∀X ∈ int, (∀Y ∈ int, ((G11 X Y) = Y)))
Hypothesis H39 : (∀X ∈ int, ((H11 X) = X))
Hypothesis H40 : (I11 = 1)
Hypothesis H41 : (J11 = (1 + (2 + 2)))
Hypothesis H42 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U11 X Y Z) = (if (X <= 0) then Y else (F11 (U11 (X + - 1) Y Z) (V11 (X + - 1) Y Z)))))))
Hypothesis H43 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V11 X Y Z) = (if (X <= 0) then Z else (G11 (U11 (X + - 1) Y Z) (V11 (X + - 1) Y Z)))))))
Hypothesis H44 : (∀X ∈ int, ((W11 X) = (U11 (H11 X) I11 J11)))
Hypothesis H45 : (∀X ∈ int, ((F9 X) = (((V10 X) + - 1) * (W11 X))))
Hypothesis H46 : (G9 = 1)
Hypothesis H47 : (∀X ∈ int, (∀Y ∈ int, ((H9 X Y) = Y)))
Hypothesis H48 : (∀X ∈ int, (∀Y ∈ int, ((U9 X Y) = (if (X <= 0) then Y else (F9 (U9 (X + - 1) Y))))))
Hypothesis H49 : (∀X ∈ int, (∀Y ∈ int, ((V9 X Y) = (U9 G9 (H9 X Y)))))
Hypothesis H50 : (∀X ∈ int, (∀Y ∈ int, ((F8 X Y) = ((((V9 X Y) + X) + X) + X))))
Hypothesis H51 : (∀X ∈ int, ((G8 X) = X))
Hypothesis H52 : (H8 = 1)
Hypothesis H53 : (∀X ∈ int, (∀Y ∈ int, ((U8 X Y) = (if (X <= 0) then Y else (F8 (U8 (X + - 1) Y) X)))))
Hypothesis H54 : (∀X ∈ int, ((V8 X) = (U8 (G8 X) H8)))
Hypothesis H55 : (∀X ∈ int, ((F7 X) = (V8 X)))
Hypothesis H56 : (G7 = 1)
Hypothesis H57 : (∀X ∈ int, (∀Y ∈ int, ((H7 X Y) = Y)))
Hypothesis H58 : (∀X ∈ int, (∀Y ∈ int, ((U7 X Y) = (if (X <= 0) then Y else (F7 (U7 (X + - 1) Y))))))
Hypothesis H59 : (∀X ∈ int, (∀Y ∈ int, ((V7 X Y) = (U7 G7 (H7 X Y)))))
Hypothesis H60 : (∀X ∈ int, (∀Y ∈ int, ((F6 X Y) = (((V7 X Y) + ((((X + X) * 2) + X) * 2)) + X))))
Hypothesis H61 : (∀X ∈ int, ((G6 X) = X))
Hypothesis H62 : (H6 = 1)
Hypothesis H63 : (∀X ∈ int, (∀Y ∈ int, ((U6 X Y) = (if (X <= 0) then Y else (F6 (U6 (X + - 1) Y) X)))))
Hypothesis H64 : (∀X ∈ int, ((V6 X) = (U6 (G6 X) H6)))
Hypothesis H65 : (∀X ∈ int, ((FAST X) = (V6 X)))
Theorem. (
A28071)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof:The rest of the proof is missing.