Beginning of Section A20984

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 TMFGpSKVGUWyY9KM7MtjnWHsoVZgL7vXrit ***)

Variable F2 : set → set

L10

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

L11

Variable G2 : set → set → set

L12

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

L13

Variable H2 : set

L14

Hypothesis HH2 : H2 ∈ int

L15

Variable U2 : set → set → set

L16

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

L17

Variable V2 : set → set → set

L18

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

L19

Variable F3 : set → set → set

L20

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

L21

Variable G3 : set

L22

Hypothesis HG3 : G3 ∈ int

L23

Variable H3 : set → set

L24

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

L25

Variable U3 : set → set → set

L26

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

L27

Variable V3 : set → set

L28

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

L29

Variable F1 : set → set → set

L30

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

L31

Variable G1 : set → set → set

L32

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

L33

Variable H1 : set

L34

Hypothesis HH1 : H1 ∈ int

L35

Variable U1 : set → set → set

L36

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

L37

Variable V1 : set → set → set

L38

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

L39

Variable F4 : set → set → set

L40

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

L41

Variable G4 : set

L42

Hypothesis HG4 : G4 ∈ int

L43

Variable H4 : set → set

L44

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

L45

Variable U4 : set → set → set

L46

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

L47

Variable V4 : set → set

L48

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

L49

Variable F0 : set → set → set

L50

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

L51

Variable G0 : set → set

L52

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

L53

Variable H0 : set

L54

Hypothesis HH0 : H0 ∈ int

L55

Variable U0 : set → set → set

L56

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

L57

Variable V0 : set → set

L58

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

L59

Variable SMALL : set → set

L60

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

L61

Variable F7 : set → set → set

L62

Hypothesis HF7 : ∀x0 ∈ int, ∀x1 ∈ int, F7 x0 x1 ∈ int

L63

Variable G7 : set → set → set

L64

Hypothesis HG7 : ∀x0 ∈ int, ∀x1 ∈ int, G7 x0 x1 ∈ int

L65

Variable H7 : set → set

L66

Hypothesis HH7 : ∀x0 ∈ int, H7 x0 ∈ int

L67

Variable I7 : set

L68

Hypothesis HI7 : I7 ∈ int

L69

Variable J7 : set

L70

Hypothesis HJ7 : J7 ∈ int

L71

Variable U7 : set → set → set → set

L72

Hypothesis HU7 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U7 x0 x1 x2 ∈ int

L73

Variable V7 : set → set → set → set

L74

Hypothesis HV7 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V7 x0 x1 x2 ∈ int

L75

Variable W7 : set → set

L76

Hypothesis HW7 : ∀x0 ∈ int, W7 x0 ∈ int

L77

Variable F8 : set → set → set

L78

Hypothesis HF8 : ∀x0 ∈ int, ∀x1 ∈ int, F8 x0 x1 ∈ int

L79

Variable G8 : set → set → set

L80

Hypothesis HG8 : ∀x0 ∈ int, ∀x1 ∈ int, G8 x0 x1 ∈ int

L81

Variable H8 : set → set

L82

Hypothesis HH8 : ∀x0 ∈ int, H8 x0 ∈ int

L83

Variable I8 : set

L84

Hypothesis HI8 : I8 ∈ int

L85

Variable J8 : set

L86

Hypothesis HJ8 : J8 ∈ int

L87

Variable U8 : set → set → set → set

L88

Hypothesis HU8 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U8 x0 x1 x2 ∈ int

L89

Variable V8 : set → set → set → set

L90

Hypothesis HV8 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V8 x0 x1 x2 ∈ int

L91

Variable W8 : set → set

L92

Hypothesis HW8 : ∀x0 ∈ int, W8 x0 ∈ int

L93

Variable F6 : set → set

L94

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

L95

Variable G6 : set

L96

Hypothesis HG6 : G6 ∈ int

L97

Variable H6 : set → set → set

L98

Hypothesis HH6 : ∀x0 ∈ int, ∀x1 ∈ int, H6 x0 x1 ∈ int

L99

Variable U6 : set → set → set

L100

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

L101

Variable V6 : set → set → set

L102

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

L103

Variable F5 : set → set → set

L104

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

L105

Variable G5 : set → set

L106

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

L107

Variable H5 : set

L108

Hypothesis HH5 : H5 ∈ int

L109

Variable U5 : set → set → set

L110

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

L111

Variable V5 : set → set

L112

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

L113

Variable FAST : set → set

L114

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

L115

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

L116

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

L117

Hypothesis H3 : (H2 = 1)

L118

Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y))))))

L119

Hypothesis H5 : (∀X ∈ int, (∀Y ∈ int, ((V2 X Y) = (U2 (G2 X Y) H2))))

L120

Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, ((F3 X Y) = ((2 + Y) * X))))

L121

Hypothesis H7 : (G3 = 2)

L122

Hypothesis H8 : (∀X ∈ int, ((H3 X) = X))

L123

Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y) X)))))

L124

Hypothesis H10 : (∀X ∈ int, ((V3 X) = (U3 G3 (H3 X))))

L125

Hypothesis H11 : (∀X ∈ int, (∀Y ∈ int, ((F1 X Y) = ((V2 X Y) + (V3 X)))))

L126

Hypothesis H12 : (∀X ∈ int, (∀Y ∈ int, ((G1 X Y) = Y)))

L127

Hypothesis H13 : (H1 = 1)

L128

Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y) X)))))

L129

Hypothesis H15 : (∀X ∈ int, (∀Y ∈ int, ((V1 X Y) = (U1 (G1 X Y) H1))))

L130

Hypothesis H16 : (∀X ∈ int, (∀Y ∈ int, ((F4 X Y) = ((2 + Y) * X))))

L131

Hypothesis H17 : (G4 = 2)

L132

Hypothesis H18 : (∀X ∈ int, ((H4 X) = X))

L133

Hypothesis H19 : (∀X ∈ int, (∀Y ∈ int, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y) X)))))

L134

Hypothesis H20 : (∀X ∈ int, ((V4 X) = (U4 G4 (H4 X))))

L135

Hypothesis H21 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = (((V1 X Y) + - X) + (V4 X)))))

L136

Hypothesis H22 : (∀X ∈ int, ((G0 X) = X))

L137

Hypothesis H23 : (H0 = 1)

L138

Hypothesis H24 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y) X)))))

L139

Hypothesis H25 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))

L140

Hypothesis H26 : (∀X ∈ int, ((SMALL X) = (V0 X)))

L141

Hypothesis H27 : (∀X ∈ int, (∀Y ∈ int, ((F7 X Y) = ((((Y * Y) + X) + X) + X))))

L142

Hypothesis H28 : (∀X ∈ int, (∀Y ∈ int, ((G7 X Y) = (Y + Y))))

L143

Hypothesis H29 : (∀X ∈ int, ((H7 X) = X))

L144

Hypothesis H30 : (I7 = 1)

L145

Hypothesis H31 : (J7 = 2)

L146

Hypothesis H32 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U7 X Y Z) = (if (X <= 0) then Y else (F7 (U7 (X + - 1) Y Z) (V7 (X + - 1) Y Z)))))))

L147

Hypothesis H33 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V7 X Y Z) = (if (X <= 0) then Z else (G7 (U7 (X + - 1) Y Z) (V7 (X + - 1) Y Z)))))))

L148

Hypothesis H34 : (∀X ∈ int, ((W7 X) = (U7 (H7 X) I7 J7)))

L149

Hypothesis H35 : (∀X ∈ int, (∀Y ∈ int, ((F8 X Y) = (X * Y))))

L150

Hypothesis H36 : (∀X ∈ int, (∀Y ∈ int, ((G8 X Y) = Y)))

L151

Hypothesis H37 : (∀X ∈ int, ((H8 X) = X))

L152

Hypothesis H38 : (I8 = 1)

L153

Hypothesis H39 : (J8 = (1 + 2))

L154

Hypothesis H40 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U8 X Y Z) = (if (X <= 0) then Y else (F8 (U8 (X + - 1) Y Z) (V8 (X + - 1) Y Z)))))))

L155

Hypothesis H41 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V8 X Y Z) = (if (X <= 0) then Z else (G8 (U8 (X + - 1) Y Z) (V8 (X + - 1) Y Z)))))))

L156

Hypothesis H42 : (∀X ∈ int, ((W8 X) = (U8 (H8 X) I8 J8)))

L157

Hypothesis H43 : (∀X ∈ int, ((F6 X) = ((W7 X) * (W8 X))))

L158

Hypothesis H44 : (G6 = 1)

L159

Hypothesis H45 : (∀X ∈ int, (∀Y ∈ int, ((H6 X Y) = Y)))

L160

Hypothesis H46 : (∀X ∈ int, (∀Y ∈ int, ((U6 X Y) = (if (X <= 0) then Y else (F6 (U6 (X + - 1) Y))))))

L161

Hypothesis H47 : (∀X ∈ int, (∀Y ∈ int, ((V6 X Y) = (U6 G6 (H6 X Y)))))

L162

Hypothesis H48 : (∀X ∈ int, (∀Y ∈ int, ((F5 X Y) = (((V6 X Y) + X) + ((((X * 2) * 2) + X) * 2)))))

L163

Hypothesis H49 : (∀X ∈ int, ((G5 X) = X))

L164

Hypothesis H50 : (H5 = 1)

L165

Hypothesis H51 : (∀X ∈ int, (∀Y ∈ int, ((U5 X Y) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y) X)))))

L166

Hypothesis H52 : (∀X ∈ int, ((V5 X) = (U5 (G5 X) H5)))

L167

Hypothesis H53 : (∀X ∈ int, ((FAST X) = (V5 X)))

L168

Theorem. (A20984)

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

In Proofgold the corresponding term root is 581ef5... and proposition id is 7441d2...

Proof:
Proof not loaded.