Beginning of Section A95807

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.

(*** $I sig/OEISPreamble.mgs ***)

(*** Bounty 1 PFG TMHLibArZa3Qx9spgD8y3DrbRJPNb4e81f6 ***)

Variable F1 : set → set → set

L10

Hypothesis HF1 : ∀x0 ∈ int, ∀x1 ∈ int, F1 x0 x1 ∈ int

L11

Variable G1 : set → set → set

L12

Hypothesis HG1 : ∀x0 ∈ int, ∀x1 ∈ int, G1 x0 x1 ∈ int

L13

Variable H1 : set → set → set

L14

Hypothesis HH1 : ∀x0 ∈ int, ∀x1 ∈ int, H1 x0 x1 ∈ int

L15

Variable I1 : set

L16

Hypothesis HI1 : I1 ∈ int

L17

Variable J1 : set

L18

Hypothesis HJ1 : J1 ∈ int

L19

Variable U1 : set → set → set → set

L20

Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U1 x0 x1 x2 ∈ int

L21

Variable V1 : set → set → set → set

L22

Hypothesis HV1 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V1 x0 x1 x2 ∈ int

L23

Variable W1 : set → set → set

L24

Hypothesis HW1 : ∀x0 ∈ int, ∀x1 ∈ int, W1 x0 x1 ∈ int

L25

Variable F2 : set → set

L26

Hypothesis HF2 : ∀x0 ∈ int, F2 x0 ∈ int

L27

Variable G2 : set

L28

Hypothesis HG2 : G2 ∈ int

L29

Variable H2 : set → set

L30

Hypothesis HH2 : ∀x0 ∈ int, H2 x0 ∈ int

L31

Variable U2 : set → set → set

L32

Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, U2 x0 x1 ∈ int

L33

Variable V2 : set → set

L34

Hypothesis HV2 : ∀x0 ∈ int, V2 x0 ∈ int

L35

Variable F0 : set → set → set

L36

Hypothesis HF0 : ∀x0 ∈ int, ∀x1 ∈ int, F0 x0 x1 ∈ int

L37

Variable G0 : set → set

L38

Hypothesis HG0 : ∀x0 ∈ int, G0 x0 ∈ int

L39

Variable H0 : set

L40

Hypothesis HH0 : H0 ∈ int

L41

Variable U0 : set → set → set

L42

Hypothesis HU0 : ∀x0 ∈ int, ∀x1 ∈ int, U0 x0 x1 ∈ int

L43

Variable V0 : set → set

L44

Hypothesis HV0 : ∀x0 ∈ int, V0 x0 ∈ int

L45

Variable SMALL : set → set

L46

Hypothesis HSMALL : ∀x0 ∈ int, SMALL x0 ∈ int

L47

Variable F4 : set → set → set

L48

Hypothesis HF4 : ∀x0 ∈ int, ∀x1 ∈ int, F4 x0 x1 ∈ int

L49

Variable G4 : set → set → set

L50

Hypothesis HG4 : ∀x0 ∈ int, ∀x1 ∈ int, G4 x0 x1 ∈ int

L51

Variable H4 : set → set

L52

Hypothesis HH4 : ∀x0 ∈ int, H4 x0 ∈ int

L53

Variable I4 : set

L54

Hypothesis HI4 : I4 ∈ int

L55

Variable J4 : set

L56

Hypothesis HJ4 : J4 ∈ int

L57

Variable U4 : set → set → set → set

L58

Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U4 x0 x1 x2 ∈ int

L59

Variable V4 : set → set → set → set

L60

Hypothesis HV4 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V4 x0 x1 x2 ∈ int

L61

Variable W4 : set → set

L62

Hypothesis HW4 : ∀x0 ∈ int, W4 x0 ∈ int

L63

Variable F5 : set → set → set

L64

Hypothesis HF5 : ∀x0 ∈ int, ∀x1 ∈ int, F5 x0 x1 ∈ int

L65

Variable G5 : set → set → set

L66

Hypothesis HG5 : ∀x0 ∈ int, ∀x1 ∈ int, G5 x0 x1 ∈ int

L67

Variable H5 : set → set

L68

Hypothesis HH5 : ∀x0 ∈ int, H5 x0 ∈ int

L69

Variable I5 : set

L70

Hypothesis HI5 : I5 ∈ int

L71

Variable J5 : set

L72

Hypothesis HJ5 : J5 ∈ int

L73

Variable U5 : set → set → set → set

L74

Hypothesis HU5 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U5 x0 x1 x2 ∈ int

L75

Variable V5 : set → set → set → set

L76

Hypothesis HV5 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V5 x0 x1 x2 ∈ int

L77

Variable W5 : set → set

L78

Hypothesis HW5 : ∀x0 ∈ int, W5 x0 ∈ int

L79

Variable F3 : set → set

L80

Hypothesis HF3 : ∀x0 ∈ int, F3 x0 ∈ int

L81

Variable G3 : set

L82

Hypothesis HG3 : G3 ∈ int

L83

Variable H3 : set → set

L84

Hypothesis HH3 : ∀x0 ∈ int, H3 x0 ∈ int

L85

Variable U3 : set → set → set

L86

Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, U3 x0 x1 ∈ int

L87

Variable V3 : set → set

L88

Hypothesis HV3 : ∀x0 ∈ int, V3 x0 ∈ int

L89

Variable FAST : set → set

L90

Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int

L91

Hypothesis H1 : (∀X ∈ int, (∀Y ∈ int, ((F1 X Y) = (X * Y))))

L92

Hypothesis H2 : (∀X ∈ int, (∀Y ∈ int, ((G1 X Y) = Y)))

L93

Hypothesis H3 : (∀X ∈ int, (∀Y ∈ int, ((H1 X Y) = Y)))

L94

Hypothesis H4 : (I1 = 1)

L95

Hypothesis H5 : (J1 = (2 + (2 * (2 + 2))))

L96

Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U1 X Y Z) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y Z) (V1 (X + - 1) Y Z)))))))

L97

Hypothesis H7 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V1 X Y Z) = (if (X <= 0) then Z else (G1 (U1 (X + - 1) Y Z) (V1 (X + - 1) Y Z)))))))

L98

Hypothesis H8 : (∀X ∈ int, (∀Y ∈ int, ((W1 X Y) = (U1 (H1 X Y) I1 J1))))

L99

Hypothesis H9 : (∀X ∈ int, ((F2 X) = ((X + X) + X)))

L100

Hypothesis H10 : (G2 = 2)

L101

Hypothesis H11 : (∀X ∈ int, ((H2 X) = (X + - 1)))

L102

Hypothesis H12 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y))))))

L103

Hypothesis H13 : (∀X ∈ int, ((V2 X) = (U2 G2 (H2 X))))

L104

Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = ((W1 X Y) + (V2 X)))))

L105

Hypothesis H15 : (∀X ∈ int, ((G0 X) = X))

L106

Hypothesis H16 : (H0 = 1)

L107

Hypothesis H17 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y) X)))))

L108

Hypothesis H18 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))

L109

Hypothesis H19 : (∀X ∈ int, ((SMALL X) = (V0 X)))

L110

Hypothesis H20 : (∀X ∈ int, (∀Y ∈ int, ((F4 X Y) = (X * Y))))

L111

Hypothesis H21 : (∀X ∈ int, (∀Y ∈ int, ((G4 X Y) = Y)))

L112

Hypothesis H22 : (∀X ∈ int, ((H4 X) = (X + - 1)))

L113

Hypothesis H23 : (I4 = 1)

L114

Hypothesis H24 : (J4 = (2 + (2 * (2 + 2))))

L115

Hypothesis H25 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U4 X Y Z) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y Z) (V4 (X + - 1) Y Z)))))))

L116

Hypothesis H26 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V4 X Y Z) = (if (X <= 0) then Z else (G4 (U4 (X + - 1) Y Z) (V4 (X + - 1) Y Z)))))))

L117

Hypothesis H27 : (∀X ∈ int, ((W4 X) = (U4 (H4 X) I4 J4)))

L118

Hypothesis H28 : (∀X ∈ int, (∀Y ∈ int, ((F5 X Y) = ((X + - 1) * Y))))

L119

Hypothesis H29 : (∀X ∈ int, (∀Y ∈ int, ((G5 X Y) = Y)))

L120

Hypothesis H30 : (∀X ∈ int, ((H5 X) = X))

L121

Hypothesis H31 : (I5 = 1)

L122

Hypothesis H32 : (J5 = (1 + (2 * (2 + 2))))

L123

Hypothesis H33 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U5 X Y Z) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y Z) (V5 (X + - 1) Y Z)))))))

L124

Hypothesis H34 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V5 X Y Z) = (if (X <= 0) then Z else (G5 (U5 (X + - 1) Y Z) (V5 (X + - 1) Y Z)))))))

L125

Hypothesis H35 : (∀X ∈ int, ((W5 X) = (U5 (H5 X) I5 J5)))

L126

Hypothesis H36 : (∀X ∈ int, ((F3 X) = ((W4 X) + (W5 X))))

L127

Hypothesis H37 : (G3 = 1)

L128

Hypothesis H38 : (∀X ∈ int, ((H3 X) = (2 + X)))

L129

Hypothesis H39 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y))))))

L130

Hypothesis H40 : (∀X ∈ int, ((V3 X) = (U3 G3 (H3 X))))

L131

Hypothesis H41 : (∀X ∈ int, ((FAST X) = (V3 X)))

L132

Theorem. (A95807)

(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))

In Proofgold the corresponding term root is 19c9bd... and proposition id is 169c6b...

Proof:
Proof not loaded.