Beginning of Section A13897

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 TMUA7CupvYSfmBRNGqsxJ491AYr9K2XP98A ***)

Variable F1 : set → set

L10

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

L11

Variable G1 : set

L12

Hypothesis HG1 : G1 ∈ int

L13

Variable H1 : set

L14

Hypothesis HH1 : H1 ∈ int

L15

Variable U1 : set → set → set

L16

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

L17

Variable V1 : set

L18

Hypothesis HV1 : V1 ∈ int

L19

Variable F0 : set → set

L20

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

L21

Variable G0 : set → set

L22

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

L23

Variable H0 : set

L24

Hypothesis HH0 : H0 ∈ int

L25

Variable U0 : set → set → set

L26

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

L27

Variable V0 : set → set

L28

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

L29

Variable SMALL : set → set

L30

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

L31

Variable F2 : set → set

L32

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

L33

Variable G2 : set

L34

Hypothesis HG2 : G2 ∈ int

L35

Variable F3 : set → set → set

L36

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

L37

Variable G3 : set → set → set

L38

Hypothesis HG3 : ∀x0 ∈ int, ∀x1 ∈ int, G3 x0 x1 ∈ int

L39

Variable H3 : set → set

L40

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

L41

Variable I3 : set

L42

Hypothesis HI3 : I3 ∈ int

L43

Variable J3 : set

L44

Hypothesis HJ3 : J3 ∈ int

L45

Variable U3 : set → set → set → set

L46

Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U3 x0 x1 x2 ∈ int

L47

Variable V3 : set → set → set → set

L48

Hypothesis HV3 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V3 x0 x1 x2 ∈ int

L49

Variable W3 : set → set

L50

Hypothesis HW3 : ∀x0 ∈ int, W3 x0 ∈ int

L51

Variable H2 : set → set

L52

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

L53

Variable U2 : set → set → set

L54

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

L55

Variable V2 : set → set

L56

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

L57

Variable F4 : set → set → set

L58

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

L59

Variable G4 : set → set → set

L60

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

L61

Variable H4 : set → set

L62

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

L63

Variable I4 : set

L64

Hypothesis HI4 : I4 ∈ int

L65

Variable J4 : set

L66

Hypothesis HJ4 : J4 ∈ int

L67

Variable U4 : set → set → set → set

L68

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

L69

Variable V4 : set → set → set → set

L70

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

L71

Variable W4 : set → set

L72

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

L73

Variable F5 : set → set

L74

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

L75

Variable G5 : set

L76

Hypothesis HG5 : G5 ∈ int

L77

Variable F6 : set → set → set

L78

Hypothesis HF6 : ∀x0 ∈ int, ∀x1 ∈ int, F6 x0 x1 ∈ int

L79

Variable G6 : set → set → set

L80

Hypothesis HG6 : ∀x0 ∈ int, ∀x1 ∈ int, G6 x0 x1 ∈ int

L81

Variable H6 : set → set

L82

Hypothesis HH6 : ∀x0 ∈ int, H6 x0 ∈ int

L83

Variable I6 : set

L84

Hypothesis HI6 : I6 ∈ int

L85

Variable J6 : set

L86

Hypothesis HJ6 : J6 ∈ int

L87

Variable U6 : set → set → set → set

L88

Hypothesis HU6 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U6 x0 x1 x2 ∈ int

L89

Variable V6 : set → set → set → set

L90

Hypothesis HV6 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V6 x0 x1 x2 ∈ int

L91

Variable W6 : set → set

L92

Hypothesis HW6 : ∀x0 ∈ int, W6 x0 ∈ int

L93

Variable H5 : set → set

L94

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

L95

Variable U5 : set → set → set

L96

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

L97

Variable V5 : set → set

L98

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

L99

Variable FAST : set → set

L100

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

L101

Hypothesis H1 : (∀X ∈ int, ((F1 X) = (2 * (2 + X))))

L102

Hypothesis H2 : (G1 = 2)

L103

Hypothesis H3 : (H1 = 2)

L104

Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))

L105

Hypothesis H5 : (V1 = (U1 G1 H1))

L106

Hypothesis H6 : (∀X ∈ int, ((F0 X) = (V1 * X)))

L107

Hypothesis H7 : (∀X ∈ int, ((G0 X) = ((2 * (2 + (X + X))) + X)))

L108

Hypothesis H8 : (H0 = 1)

L109

Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y))))))

L110

Hypothesis H10 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))

L111

Hypothesis H11 : (∀X ∈ int, ((SMALL X) = (V0 X)))

L112

Hypothesis H12 : (∀X ∈ int, ((F2 X) = (X * X)))

L113

Hypothesis H13 : (G2 = 1)

L114

Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, ((F3 X Y) = (X * Y))))

L115

Hypothesis H15 : (∀X ∈ int, (∀Y ∈ int, ((G3 X Y) = Y)))

L116

Hypothesis H16 : (∀X ∈ int, ((H3 X) = X))

L117

Hypothesis H17 : (I3 = 1)

L118

Hypothesis H18 : (J3 = (2 * (2 + (2 * (2 + 2)))))

L119

Hypothesis H19 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U3 X Y Z) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y Z) (V3 (X + - 1) Y Z)))))))

L120

Hypothesis H20 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V3 X Y Z) = (if (X <= 0) then Z else (G3 (U3 (X + - 1) Y Z) (V3 (X + - 1) Y Z)))))))

L121

Hypothesis H21 : (∀X ∈ int, ((W3 X) = (U3 (H3 X) I3 J3)))

L122

Hypothesis H22 : (∀X ∈ int, ((H2 X) = (W3 X)))

L123

Hypothesis H23 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y))))))

L124

Hypothesis H24 : (∀X ∈ int, ((V2 X) = (U2 G2 (H2 X))))

L125

Hypothesis H25 : (∀X ∈ int, (∀Y ∈ int, ((F4 X Y) = (X * Y))))

L126

Hypothesis H26 : (∀X ∈ int, (∀Y ∈ int, ((G4 X Y) = Y)))

L127

Hypothesis H27 : (∀X ∈ int, ((H4 X) = (2 + X)))

L128

Hypothesis H28 : (I4 = 1)

L129

Hypothesis H29 : (J4 = (2 * (2 + (2 * (2 + 2)))))

L130

Hypothesis H30 : (∀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)))))))

L131

Hypothesis H31 : (∀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)))))))

L132

Hypothesis H32 : (∀X ∈ int, ((W4 X) = (U4 (H4 X) I4 J4)))

L133

Hypothesis H33 : (∀X ∈ int, ((F5 X) = (X * X)))

L134

Hypothesis H34 : (G5 = 1)

L135

Hypothesis H35 : (∀X ∈ int, (∀Y ∈ int, ((F6 X Y) = (X * Y))))

L136

Hypothesis H36 : (∀X ∈ int, (∀Y ∈ int, ((G6 X Y) = Y)))

L137

Hypothesis H37 : (∀X ∈ int, ((H6 X) = (1 + X)))

L138

Hypothesis H38 : (I6 = 1)

L139

Hypothesis H39 : (J6 = (2 * (2 + (2 * (2 + 2)))))

L140

Hypothesis H40 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U6 X Y Z) = (if (X <= 0) then Y else (F6 (U6 (X + - 1) Y Z) (V6 (X + - 1) Y Z)))))))

L141

Hypothesis H41 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V6 X Y Z) = (if (X <= 0) then Z else (G6 (U6 (X + - 1) Y Z) (V6 (X + - 1) Y Z)))))))

L142

Hypothesis H42 : (∀X ∈ int, ((W6 X) = (U6 (H6 X) I6 J6)))

L143

Hypothesis H43 : (∀X ∈ int, ((H5 X) = (W6 X)))

L144

Hypothesis H44 : (∀X ∈ int, (∀Y ∈ int, ((U5 X Y) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y))))))

L145

Hypothesis H45 : (∀X ∈ int, ((V5 X) = (U5 G5 (H5 X))))

L146

Hypothesis H46 : (∀X ∈ int, ((FAST X) = (((V2 X) * (W4 X)) * (V5 X))))

L147

Theorem. (A13897)

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

In Proofgold the corresponding term root is 3659a4... and proposition id is c77d47...

Proof:
Proof not loaded.