Beginning of Section A25985

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 TMJW1wZVbbEMbNbuTq6Jsa3qGmzGqCT7mqV ***)

Variable F3 : set → set → set

L10

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

L11

Variable G3 : set

L12

Hypothesis HG3 : G3 ∈ int

L13

Variable H3 : set → set

L14

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

L15

Variable U3 : set → set → set

L16

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

L17

Variable V3 : set → set

L18

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

L19

Variable F2 : set → set → set

L20

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

L21

Variable G2 : set → set → set

L22

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

L23

Variable H2 : set → set → set

L24

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

L25

Variable I2 : set

L26

Hypothesis HI2 : I2 ∈ int

L27

Variable J2 : set

L28

Hypothesis HJ2 : J2 ∈ int

L29

Variable U2 : set → set → set → set

L30

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

L31

Variable V2 : set → set → set → set

L32

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

L33

Variable W2 : set → set → set

L34

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

L35

Variable F4 : set → set → set

L36

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

L37

Variable G4 : set

L38

Hypothesis HG4 : G4 ∈ int

L39

Variable H4 : set → set

L40

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

L41

Variable U4 : set → set → set

L42

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

L43

Variable V4 : set → set

L44

Hypothesis HV4 : ∀x0 ∈ int, V4 x0 ∈ 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

L50

Hypothesis HH1 : H1 ∈ 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 F7 : set → set

L68

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

L69

Variable G7 : set → set

L70

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

L71

Variable H7 : set

L72

Hypothesis HH7 : H7 ∈ int

L73

Variable U7 : set → set → set

L74

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

L75

Variable V7 : set → set

L76

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

L77

Variable F8 : set → set

L78

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

L79

Variable G8 : set → set

L80

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

L81

Variable H8 : set

L82

Hypothesis HH8 : H8 ∈ int

L83

Variable U8 : set → set → set

L84

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

L85

Variable V8 : set → set

L86

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

L87

Variable F9 : set → set

L88

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

L89

Variable G9 : set → set

L90

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

L91

Variable H9 : set

L92

Hypothesis HH9 : H9 ∈ int

L93

Variable U9 : set → set → set

L94

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

L95

Variable V9 : set → set

L96

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

L97

Variable F6 : set → set

L98

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

L99

Variable G6 : set

L100

Hypothesis HG6 : G6 ∈ int

L101

Variable H6 : set → set → set

L102

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

L103

Variable U6 : set → set → set

L104

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

L105

Variable V6 : set → set → set

L106

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

L107

Variable F5 : set → set → set

L108

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

L109

Variable G5 : set → set

L110

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

L111

Variable H5 : set

L112

Hypothesis HH5 : H5 ∈ int

L113

Variable U5 : set → set → set

L114

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

L115

Variable V5 : set → set

L116

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

L117

Variable FAST : set → set

L118

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

L119

Hypothesis H1 : (∀X ∈ int, (∀Y ∈ int, ((F3 X Y) = ((2 + Y) * X))))

L120

Hypothesis H2 : (G3 = 2)

L121

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

L122

Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y) X)))))

L123

Hypothesis H5 : (∀X ∈ int, ((V3 X) = (U3 G3 (H3 X))))

L124

Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = (((V3 X) + - X) + (Y * Y)))))

L125

Hypothesis H7 : (∀X ∈ int, (∀Y ∈ int, ((G2 X Y) = (Y + Y))))

L126

Hypothesis H8 : (∀X ∈ int, (∀Y ∈ int, ((H2 X Y) = Y)))

L127

Hypothesis H9 : (I2 = 1)

L128

Hypothesis H10 : (J2 = 2)

L129

Hypothesis H11 : (∀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)))))))

L130

Hypothesis H12 : (∀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)))))))

L131

Hypothesis H13 : (∀X ∈ int, (∀Y ∈ int, ((W2 X Y) = (U2 (H2 X Y) I2 J2))))

L132

Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, ((F4 X Y) = ((2 + Y) * X))))

L133

Hypothesis H15 : (G4 = 2)

L134

Hypothesis H16 : (∀X ∈ int, ((H4 X) = X))

L135

Hypothesis H17 : (∀X ∈ int, (∀Y ∈ int, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y) X)))))

L136

Hypothesis H18 : (∀X ∈ int, ((V4 X) = (U4 G4 (H4 X))))

L137

Hypothesis H19 : (∀X ∈ int, (∀Y ∈ int, ((F1 X Y) = ((W2 X Y) + (V4 X)))))

L138

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

L139

Hypothesis H21 : (H1 = 1)

L140

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

L141

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

L142

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

L143

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

L144

Hypothesis H26 : (H0 = 1)

L145

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

L146

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

L147

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

L148

Hypothesis H30 : (∀X ∈ int, ((F7 X) = ((2 * ((X + X) + X)) + - 1)))

L149

Hypothesis H31 : (∀X ∈ int, ((G7 X) = X))

L150

Hypothesis H32 : (H7 = 2)

L151

Hypothesis H33 : (∀X ∈ int, (∀Y ∈ int, ((U7 X Y) = (if (X <= 0) then Y else (F7 (U7 (X + - 1) Y))))))

L152

Hypothesis H34 : (∀X ∈ int, ((V7 X) = (U7 (G7 X) H7)))

L153

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

L154

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

L155

Hypothesis H37 : (H8 = 1)

L156

Hypothesis H38 : (∀X ∈ int, (∀Y ∈ int, ((U8 X Y) = (if (X <= 0) then Y else (F8 (U8 (X + - 1) Y))))))

L157

Hypothesis H39 : (∀X ∈ int, ((V8 X) = (U8 (G8 X) H8)))

L158

Hypothesis H40 : (∀X ∈ int, ((F9 X) = (X + X)))

L159

Hypothesis H41 : (∀X ∈ int, ((G9 X) = X))

L160

Hypothesis H42 : (H9 = 1)

L161

Hypothesis H43 : (∀X ∈ int, (∀Y ∈ int, ((U9 X Y) = (if (X <= 0) then Y else (F9 (U9 (X + - 1) Y))))))

L162

Hypothesis H44 : (∀X ∈ int, ((V9 X) = (U9 (G9 X) H9)))

L163

Hypothesis H45 : (∀X ∈ int, ((F6 X) = (((V7 X) + - (V8 X)) * (V9 X))))

L164

Hypothesis H46 : (G6 = 1)

L165

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

L166

Hypothesis H48 : (∀X ∈ int, (∀Y ∈ int, ((U6 X Y) = (if (X <= 0) then Y else (F6 (U6 (X + - 1) Y))))))

L167

Hypothesis H49 : (∀X ∈ int, (∀Y ∈ int, ((V6 X Y) = (U6 G6 (H6 X Y)))))

L168

Hypothesis H50 : (∀X ∈ int, (∀Y ∈ int, ((F5 X Y) = (((V6 X Y) + (2 * ((2 * (X + X)) + X))) + X))))

L169

Hypothesis H51 : (∀X ∈ int, ((G5 X) = X))

L170

Hypothesis H52 : (H5 = 1)

L171

Hypothesis H53 : (∀X ∈ int, (∀Y ∈ int, ((U5 X Y) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y) X)))))

L172

Hypothesis H54 : (∀X ∈ int, ((V5 X) = (U5 (G5 X) H5)))

L173

Hypothesis H55 : (∀X ∈ int, ((FAST X) = (V5 X)))

L174

Theorem. (A25985)

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

In Proofgold the corresponding term root is 896628... and proposition id is 6e888d...

Proof:
Proof not loaded.