Beginning of Section A25940

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 TMKw8iU4Q1iBQt6MT2pTnQNHt7S1sknkMba ***)

Variable F2 : set → set → set

L10

Hypothesis HF2 : ∀x0 ∈ int, ∀x1 ∈ int, F2 x0 x1 ∈ int

L11

Variable G2 : set → set → set

L12

Hypothesis HG2 : ∀x0 ∈ int, ∀x1 ∈ int, G2 x0 x1 ∈ int

L13

Variable H2 : set → set → set

L14

Hypothesis HH2 : ∀x0 ∈ int, ∀x1 ∈ int, H2 x0 x1 ∈ int

L15

Variable I2 : set

L16

Hypothesis HI2 : I2 ∈ int

L17

Variable J2 : set

L18

Hypothesis HJ2 : J2 ∈ int

L19

Variable U2 : set → set → set → set

L20

Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U2 x0 x1 x2 ∈ int

L21

Variable V2 : set → set → set → set

L22

Hypothesis HV2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V2 x0 x1 x2 ∈ int

L23

Variable W2 : set → set → set

L24

Hypothesis HW2 : ∀x0 ∈ int, ∀x1 ∈ int, W2 x0 x1 ∈ int

L25

Variable F1 : set → set → set

L26

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

L27

Variable G1 : set → set → set

L28

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

L29

Variable H1 : set

L30

Hypothesis HH1 : H1 ∈ int

L31

Variable U1 : set → set → set

L32

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

L33

Variable V1 : set → set → set

L34

Hypothesis HV1 : ∀x0 ∈ int, ∀x1 ∈ int, V1 x0 x1 ∈ 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 F5 : set → set → set

L48

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

L49

Variable G5 : set → set → set

L50

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

L51

Variable H5 : set → set

L52

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

L53

Variable I5 : set

L54

Hypothesis HI5 : I5 ∈ int

L55

Variable J5 : set

L56

Hypothesis HJ5 : J5 ∈ int

L57

Variable U5 : set → set → set → set

L58

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

L59

Variable V5 : set → set → set → set

L60

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

L61

Variable W5 : set → set

L62

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

L63

Variable F6 : set → set

L64

Hypothesis HF6 : ∀x0 ∈ int, F6 x0 ∈ int

L65

Variable G6 : set → set

L66

Hypothesis HG6 : ∀x0 ∈ int, G6 x0 ∈ int

L67

Variable H6 : set

L68

Hypothesis HH6 : H6 ∈ int

L69

Variable U6 : set → set → set

L70

Hypothesis HU6 : ∀x0 ∈ int, ∀x1 ∈ int, U6 x0 x1 ∈ int

L71

Variable V6 : set → set

L72

Hypothesis HV6 : ∀x0 ∈ int, V6 x0 ∈ int

L73

Variable F4 : set → set

L74

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

L75

Variable G4 : set

L76

Hypothesis HG4 : G4 ∈ int

L77

Variable H4 : set → set → set

L78

Hypothesis HH4 : ∀x0 ∈ int, ∀x1 ∈ int, H4 x0 x1 ∈ int

L79

Variable U4 : set → set → set

L80

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

L81

Variable V4 : set → set → set

L82

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

L83

Variable F3 : set → set → set

L84

Hypothesis HF3 : ∀x0 ∈ int, ∀x1 ∈ int, F3 x0 x1 ∈ int

L85

Variable G3 : set → set

L86

Hypothesis HG3 : ∀x0 ∈ int, G3 x0 ∈ int

L87

Variable H3 : set

L88

Hypothesis HH3 : H3 ∈ int

L89

Variable U3 : set → set → set

L90

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

L91

Variable V3 : set → set

L92

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

L93

Variable FAST : set → set

L94

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

L95

Hypothesis H1 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = ((2 * ((X + X) + X)) + Y))))

L96

Hypothesis H2 : (∀X ∈ int, (∀Y ∈ int, ((G2 X Y) = (Y + Y))))

L97

Hypothesis H3 : (∀X ∈ int, (∀Y ∈ int, ((H2 X Y) = Y)))

L98

Hypothesis H4 : (I2 = 1)

L99

Hypothesis H5 : (J2 = 2)

L100

Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U2 X Y Z) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y Z) (V2 (X + - 1) Y Z)))))))

L101

Hypothesis H7 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V2 X Y Z) = (if (X <= 0) then Z else (G2 (U2 (X + - 1) Y Z) (V2 (X + - 1) Y Z)))))))

L102

Hypothesis H8 : (∀X ∈ int, (∀Y ∈ int, ((W2 X Y) = (U2 (H2 X Y) I2 J2))))

L103

Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((F1 X Y) = ((W2 X Y) + (2 * ((2 * (X + X)) + X))))))

L104

Hypothesis H10 : (∀X ∈ int, (∀Y ∈ int, ((G1 X Y) = Y)))

L105

Hypothesis H11 : (H1 = 1)

L106

Hypothesis H12 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y) X)))))

L107

Hypothesis H13 : (∀X ∈ int, (∀Y ∈ int, ((V1 X Y) = (U1 (G1 X Y) H1))))

L108

Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = ((((V1 X Y) + X) + X) + X))))

L109

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

L110

Hypothesis H16 : (H0 = 1)

L111

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

L112

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

L113

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

L114

Hypothesis H20 : (∀X ∈ int, (∀Y ∈ int, ((F5 X Y) = (((2 * (X + X)) + X) + Y))))

L115

Hypothesis H21 : (∀X ∈ int, (∀Y ∈ int, ((G5 X Y) = (1 + ((Y + Y) + Y)))))

L116

Hypothesis H22 : (∀X ∈ int, ((H5 X) = X))

L117

Hypothesis H23 : (I5 = 1)

L118

Hypothesis H24 : (J5 = (2 + 2))

L119

Hypothesis H25 : (∀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)))))))

L120

Hypothesis H26 : (∀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)))))))

L121

Hypothesis H27 : (∀X ∈ int, ((W5 X) = (U5 (H5 X) I5 J5)))

L122

Hypothesis H28 : (∀X ∈ int, ((F6 X) = (X + X)))

L123

Hypothesis H29 : (∀X ∈ int, ((G6 X) = X))

L124

Hypothesis H30 : (H6 = 1)

L125

Hypothesis H31 : (∀X ∈ int, (∀Y ∈ int, ((U6 X Y) = (if (X <= 0) then Y else (F6 (U6 (X + - 1) Y))))))

L126

Hypothesis H32 : (∀X ∈ int, ((V6 X) = (U6 (G6 X) H6)))

L127

Hypothesis H33 : (∀X ∈ int, ((F4 X) = ((W5 X) * (V6 X))))

L128

Hypothesis H34 : (G4 = 1)

L129

Hypothesis H35 : (∀X ∈ int, (∀Y ∈ int, ((H4 X Y) = Y)))

L130

Hypothesis H36 : (∀X ∈ int, (∀Y ∈ int, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y))))))

L131

Hypothesis H37 : (∀X ∈ int, (∀Y ∈ int, ((V4 X Y) = (U4 G4 (H4 X Y)))))

L132

Hypothesis H38 : (∀X ∈ int, (∀Y ∈ int, ((F3 X Y) = ((((V4 X Y) + X) + X) + X))))

L133

Hypothesis H39 : (∀X ∈ int, ((G3 X) = X))

L134

Hypothesis H40 : (H3 = 1)

L135

Hypothesis H41 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y) X)))))

L136

Hypothesis H42 : (∀X ∈ int, ((V3 X) = (U3 (G3 X) H3)))

L137

Hypothesis H43 : (∀X ∈ int, ((FAST X) = (V3 X)))

L138

Theorem. (A25940)

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

In Proofgold the corresponding term root is 3b9419... and proposition id is b62694...

Proof:
Proof not loaded.