Beginning of Section A61787

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 TMMar6ZUwELFRiHZmLjPQtGyLrpUL9trav6 ***)

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 → set → set

L18

Hypothesis HJ1 : ∀x0 ∈ int, ∀x1 ∈ int, J1 x0 x1 ∈ 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 F0 : set → set → set

L26

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

L27

Variable G0 : set → set → set

L28

Hypothesis HG0 : ∀x0 ∈ int, ∀x1 ∈ int, G0 x0 x1 ∈ int

L29

Variable H0 : set → set

L30

Hypothesis HH0 : ∀x0 ∈ int, H0 x0 ∈ int

L31

Variable I0 : set

L32

Hypothesis HI0 : I0 ∈ int

L33

Variable J0 : set

L34

Hypothesis HJ0 : J0 ∈ int

L35

Variable U0 : set → set → set → set

L36

Hypothesis HU0 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U0 x0 x1 x2 ∈ int

L37

Variable V0 : set → set → set → set

L38

Hypothesis HV0 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V0 x0 x1 x2 ∈ int

L39

Variable W0 : set → set

L40

Hypothesis HW0 : ∀x0 ∈ int, W0 x0 ∈ int

L41

Variable SMALL : set → set

L42

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

L43

Variable F4 : set → set → set

L44

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

L45

Variable G4 : set → set → set

L46

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

L47

Variable H4 : set → set

L48

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

L49

Variable I4 : set

L50

Hypothesis HI4 : I4 ∈ int

L51

Variable J4 : set → set

L52

Hypothesis HJ4 : ∀x0 ∈ int, J4 x0 ∈ int

L53

Variable U4 : set → set → set → set

L54

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

L55

Variable V4 : set → set → set → set

L56

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

L57

Variable W4 : set → set

L58

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

L59

Variable F5 : set → set → set

L60

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

L61

Variable G5 : set → set → set

L62

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

L63

Variable H5 : set → set

L64

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

L65

Variable I5 : set

L66

Hypothesis HI5 : I5 ∈ int

L67

Variable J5 : set → set

L68

Hypothesis HJ5 : ∀x0 ∈ int, J5 x0 ∈ int

L69

Variable U5 : set → set → set → set

L70

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

L71

Variable V5 : set → set → set → set

L72

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

L73

Variable W5 : set → set

L74

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

L75

Variable F3 : set → set

L76

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

L77

Variable G3 : set

L78

Hypothesis HG3 : G3 ∈ int

L79

Variable H3 : set → set → set

L80

Hypothesis HH3 : ∀x0 ∈ int, ∀x1 ∈ int, H3 x0 x1 ∈ int

L81

Variable U3 : set → set → set

L82

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

L83

Variable V3 : set → set → set

L84

Hypothesis HV3 : ∀x0 ∈ int, ∀x1 ∈ int, V3 x0 x1 ∈ int

L85

Variable F2 : set → set → set

L86

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

L87

Variable G2 : set → set

L88

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

L89

Variable H2 : set

L90

Hypothesis HH2 : H2 ∈ int

L91

Variable U2 : set → set → set

L92

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

L93

Variable V2 : set → set

L94

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

L95

Variable FAST : set → set

L96

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

L97

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

L98

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

L99

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

L100

Hypothesis H4 : (I1 = 1)

L101

Hypothesis H5 : (∀X ∈ int, (∀Y ∈ int, ((J1 X Y) = Y)))

L102

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)))))))

L103

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)))))))

L104

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

L105

Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = ((W1 X Y) + X))))

L106

Hypothesis H10 : (∀X ∈ int, (∀Y ∈ int, ((G0 X Y) = (2 + Y))))

L107

Hypothesis H11 : (∀X ∈ int, ((H0 X) = X))

L108

Hypothesis H12 : (I0 = 1)

L109

Hypothesis H13 : (J0 = (1 + 2))

L110

Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U0 X Y Z) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y Z) (V0 (X + - 1) Y Z)))))))

L111

Hypothesis H15 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V0 X Y Z) = (if (X <= 0) then Z else (G0 (U0 (X + - 1) Y Z) (V0 (X + - 1) Y Z)))))))

L112

Hypothesis H16 : (∀X ∈ int, ((W0 X) = (U0 (H0 X) I0 J0)))

L113

Hypothesis H17 : (∀X ∈ int, ((SMALL X) = (W0 X)))

L114

Hypothesis H18 : (∀X ∈ int, (∀Y ∈ int, ((F4 X Y) = (X * Y))))

L115

Hypothesis H19 : (∀X ∈ int, (∀Y ∈ int, ((G4 X Y) = Y)))

L116

Hypothesis H20 : (∀X ∈ int, ((H4 X) = (1 + X)))

L117

Hypothesis H21 : (I4 = 1)

L118

Hypothesis H22 : (∀X ∈ int, ((J4 X) = (1 + (X + X))))

L119

Hypothesis H23 : (∀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)))))))

L120

Hypothesis H24 : (∀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)))))))

L121

Hypothesis H25 : (∀X ∈ int, ((W4 X) = (U4 (H4 X) I4 (J4 X))))

L122

Hypothesis H26 : (∀X ∈ int, (∀Y ∈ int, ((F5 X Y) = (X * Y))))

L123

Hypothesis H27 : (∀X ∈ int, (∀Y ∈ int, ((G5 X Y) = Y)))

L124

Hypothesis H28 : (∀X ∈ int, ((H5 X) = X))

L125

Hypothesis H29 : (I5 = 1)

L126

Hypothesis H30 : (∀X ∈ int, ((J5 X) = (1 + (X + X))))

L127

Hypothesis H31 : (∀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)))))))

L128

Hypothesis H32 : (∀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)))))))

L129

Hypothesis H33 : (∀X ∈ int, ((W5 X) = (U5 (H5 X) I5 (J5 X))))

L130

Hypothesis H34 : (∀X ∈ int, ((F3 X) = ((W4 X) * (W5 X))))

L131

Hypothesis H35 : (G3 = 1)

L132

Hypothesis H36 : (∀X ∈ int, (∀Y ∈ int, ((H3 X Y) = Y)))

L133

Hypothesis H37 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y))))))

L134

Hypothesis H38 : (∀X ∈ int, (∀Y ∈ int, ((V3 X Y) = (U3 G3 (H3 X Y)))))

L135

Hypothesis H39 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = ((V3 X Y) + X))))

L136

Hypothesis H40 : (∀X ∈ int, ((G2 X) = X))

L137

Hypothesis H41 : (H2 = 1)

L138

Hypothesis H42 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y) X)))))

L139

Hypothesis H43 : (∀X ∈ int, ((V2 X) = (U2 (G2 X) H2)))

L140

Hypothesis H44 : (∀X ∈ int, ((FAST X) = (V2 X)))

L141

Theorem. (A61787)

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

In Proofgold the corresponding term root is 642cd2... and proposition id is 6584ab...

Proof:
Proof not loaded.