Beginning of Section A27964

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 TMRkPkWn8MwJ5gGBdDqivBTDcWDSeLQv7Y1 ***)

Variable F4 : set → set → set

L10

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

L11

Variable G4 : set → set → set

L12

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

L13

Variable H4 : set → set → set

L14

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

L15

Variable I4 : set

L16

Hypothesis HI4 : I4 ∈ int

L17

Variable J4 : set

L18

Hypothesis HJ4 : J4 ∈ int

L19

Variable U4 : set → set → set → set

L20

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

L21

Variable V4 : set → set → set → set

L22

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

L23

Variable W4 : set → set → set

L24

Hypothesis HW4 : ∀x0 ∈ int, ∀x1 ∈ int, W4 x0 x1 ∈ int

L25

Variable F3 : set → set → set

L26

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

L27

Variable G3 : set → set → set

L28

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

L29

Variable H3 : set → set

L30

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

L31

Variable U3 : set → set → set

L32

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

L33

Variable V3 : set → set → set

L34

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

L35

Variable F2 : set → set → set

L36

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

L37

Variable G2 : set → set → set

L38

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

L39

Variable H2 : set → set

L40

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

L41

Variable U2 : set → set → set

L42

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

L43

Variable V2 : set → set → set

L44

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

L45

Variable F1 : set → set → set

L46

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

L47

Variable G1 : set → set → set

L48

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

L49

Variable H1 : set → set

L50

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

L51

Variable U1 : set → set → set

L52

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

L53

Variable V1 : set → set → set

L54

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

L55

Variable F0 : set → set → set

L56

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

L57

Variable G0 : set → set

L58

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

L59

Variable H0 : set

L60

Hypothesis HH0 : H0 ∈ int

L61

Variable U0 : set → set → set

L62

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

L63

Variable V0 : set → set

L64

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

L65

Variable SMALL : set → set

L66

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

L67

Variable F9 : set → set → set

L68

Hypothesis HF9 : ∀x0 ∈ int, ∀x1 ∈ int, F9 x0 x1 ∈ int

L69

Variable G9 : set → set

L70

Hypothesis HG9 : ∀x0 ∈ int, G9 x0 ∈ int

L71

Variable H9 : set → set

L72

Hypothesis HH9 : ∀x0 ∈ int, H9 x0 ∈ int

L73

Variable I9 : set

L74

Hypothesis HI9 : I9 ∈ int

L75

Variable J9 : set

L76

Hypothesis HJ9 : J9 ∈ int

L77

Variable U9 : set → set → set → set

L78

Hypothesis HU9 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U9 x0 x1 x2 ∈ int

L79

Variable V9 : set → set → set → set

L80

Hypothesis HV9 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V9 x0 x1 x2 ∈ int

L81

Variable W9 : set → set

L82

Hypothesis HW9 : ∀x0 ∈ int, W9 x0 ∈ int

L83

Variable F8 : set → set

L84

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

L85

Variable G8 : set

L86

Hypothesis HG8 : G8 ∈ int

L87

Variable H8 : set → set → set

L88

Hypothesis HH8 : ∀x0 ∈ int, ∀x1 ∈ int, H8 x0 x1 ∈ int

L89

Variable U8 : set → set → set

L90

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

L91

Variable V8 : set → set → set

L92

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

L93

Variable F7 : set → set → set

L94

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

L95

Variable G7 : set → set

L96

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

L97

Variable H7 : set

L98

Hypothesis HH7 : H7 ∈ int

L99

Variable U7 : set → set → set

L100

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

L101

Variable V7 : set → set

L102

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

L103

Variable F6 : set → set

L104

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

L105

Variable G6 : set

L106

Hypothesis HG6 : G6 ∈ int

L107

Variable H6 : set → set → set

L108

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

L109

Variable U6 : set → set → set

L110

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

L111

Variable V6 : set → set → set

L112

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

L113

Variable F5 : set → set → set

L114

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

L115

Variable G5 : set → set

L116

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

L117

Variable H5 : set

L118

Hypothesis HH5 : H5 ∈ int

L119

Variable U5 : set → set → set

L120

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

L121

Variable V5 : set → set

L122

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

L123

Variable FAST : set → set

L124

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

L125

Hypothesis H1 : (∀X ∈ int, (∀Y ∈ int, ((F4 X Y) = Y)))

L126

Hypothesis H2 : (∀X ∈ int, (∀Y ∈ int, ((G4 X Y) = (X + Y))))

L127

Hypothesis H3 : (∀X ∈ int, (∀Y ∈ int, ((H4 X Y) = Y)))

L128

Hypothesis H4 : (I4 = 2)

L129

Hypothesis H5 : (J4 = 1)

L130

Hypothesis H6 : (∀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 H7 : (∀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 H8 : (∀X ∈ int, (∀Y ∈ int, ((W4 X Y) = (U4 (H4 X Y) I4 J4))))

L133

Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((F3 X Y) = ((W4 X Y) + X))))

L134

Hypothesis H10 : (∀X ∈ int, (∀Y ∈ int, ((G3 X Y) = Y)))

L135

Hypothesis H11 : (∀X ∈ int, ((H3 X) = X))

L136

Hypothesis H12 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y) X)))))

L137

Hypothesis H13 : (∀X ∈ int, (∀Y ∈ int, ((V3 X Y) = (U3 (G3 X Y) (H3 X)))))

L138

Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = (V3 X Y))))

L139

Hypothesis H15 : (∀X ∈ int, (∀Y ∈ int, ((G2 X Y) = Y)))

L140

Hypothesis H16 : (∀X ∈ int, ((H2 X) = X))

L141

Hypothesis H17 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y) X)))))

L142

Hypothesis H18 : (∀X ∈ int, (∀Y ∈ int, ((V2 X Y) = (U2 (G2 X Y) (H2 X)))))

L143

Hypothesis H19 : (∀X ∈ int, (∀Y ∈ int, ((F1 X Y) = (V2 X Y))))

L144

Hypothesis H20 : (∀X ∈ int, (∀Y ∈ int, ((G1 X Y) = (1 + Y))))

L145

Hypothesis H21 : (∀X ∈ int, ((H1 X) = X))

L146

Hypothesis H22 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y) X)))))

L147

Hypothesis H23 : (∀X ∈ int, (∀Y ∈ int, ((V1 X Y) = (U1 (G1 X Y) (H1 X)))))

L148

Hypothesis H24 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = (V1 X Y))))

L149

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

L150

Hypothesis H26 : (H0 = 1)

L151

Hypothesis H27 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y) X)))))

L152

Hypothesis H28 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))

L153

Hypothesis H29 : (∀X ∈ int, ((SMALL X) = (V0 X)))

L154

Hypothesis H30 : (∀X ∈ int, (∀Y ∈ int, ((F9 X Y) = (X + Y))))

L155

Hypothesis H31 : (∀X ∈ int, ((G9 X) = X))

L156

Hypothesis H32 : (∀X ∈ int, ((H9 X) = X))

L157

Hypothesis H33 : (I9 = (2 + 2))

L158

Hypothesis H34 : (J9 = (1 + 2))

L159

Hypothesis H35 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U9 X Y Z) = (if (X <= 0) then Y else (F9 (U9 (X + - 1) Y Z) (V9 (X + - 1) Y Z)))))))

L160

Hypothesis H36 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V9 X Y Z) = (if (X <= 0) then Z else (G9 (U9 (X + - 1) Y Z)))))))

L161

Hypothesis H37 : (∀X ∈ int, ((W9 X) = (U9 (H9 X) I9 J9)))

L162

Hypothesis H38 : (∀X ∈ int, ((F8 X) = (((((W9 X) + - 2) + - X) + - X) + - X)))

L163

Hypothesis H39 : (G8 = 1)

L164

Hypothesis H40 : (∀X ∈ int, (∀Y ∈ int, ((H8 X Y) = (2 + Y))))

L165

Hypothesis H41 : (∀X ∈ int, (∀Y ∈ int, ((U8 X Y) = (if (X <= 0) then Y else (F8 (U8 (X + - 1) Y))))))

L166

Hypothesis H42 : (∀X ∈ int, (∀Y ∈ int, ((V8 X Y) = (U8 G8 (H8 X Y)))))

L167

Hypothesis H43 : (∀X ∈ int, (∀Y ∈ int, ((F7 X Y) = ((V8 X Y) + X))))

L168

Hypothesis H44 : (∀X ∈ int, ((G7 X) = X))

L169

Hypothesis H45 : (H7 = 0)

L170

Hypothesis H46 : (∀X ∈ int, (∀Y ∈ int, ((U7 X Y) = (if (X <= 0) then Y else (F7 (U7 (X + - 1) Y) X)))))

L171

Hypothesis H47 : (∀X ∈ int, ((V7 X) = (U7 (G7 X) H7)))

L172

Hypothesis H48 : (∀X ∈ int, ((F6 X) = (V7 X)))

L173

Hypothesis H49 : (G6 = 1)

L174

Hypothesis H50 : (∀X ∈ int, (∀Y ∈ int, ((H6 X Y) = Y)))

L175

Hypothesis H51 : (∀X ∈ int, (∀Y ∈ int, ((U6 X Y) = (if (X <= 0) then Y else (F6 (U6 (X + - 1) Y))))))

L176

Hypothesis H52 : (∀X ∈ int, (∀Y ∈ int, ((V6 X Y) = (U6 G6 (H6 X Y)))))

L177

Hypothesis H53 : (∀X ∈ int, (∀Y ∈ int, ((F5 X Y) = ((V6 X Y) + X))))

L178

Hypothesis H54 : (∀X ∈ int, ((G5 X) = X))

L179

Hypothesis H55 : (H5 = 1)

L180

Hypothesis H56 : (∀X ∈ int, (∀Y ∈ int, ((U5 X Y) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y) X)))))

L181

Hypothesis H57 : (∀X ∈ int, ((V5 X) = (U5 (G5 X) H5)))

L182

Hypothesis H58 : (∀X ∈ int, ((FAST X) = ((V5 X) + - (X * X))))

L183

Theorem. (A27964)

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

In Proofgold the corresponding term root is 029cde... and proposition id is 3d2d61...

Proof:
Proof not loaded.