Beginning of Section A95917

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 TMHRBGAJSorkHLf5Jd7RJkYRAPW38JWoin5 ***)

Variable F1 : set → set

L10

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

L11

Variable G1 : set

L12

Hypothesis HG1 : G1 ∈ int

L13

Variable F2 : set → set → set

L14

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

L15

Variable G2 : set → set

L16

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

L17

Variable H2 : set → set → set

L18

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

L19

Variable I2 : set

L20

Hypothesis HI2 : I2 ∈ int

L21

Variable J2 : set

L22

Hypothesis HJ2 : J2 ∈ int

L23

Variable U2 : set → set → set → set

L24

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

L25

Variable V2 : set → set → set → set

L26

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

L27

Variable W2 : set → set → set

L28

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

L29

Variable H1 : set → set → set

L30

Hypothesis HH1 : ∀x0 ∈ int, ∀x1 ∈ int, H1 x0 x1 ∈ int

L31

Variable U1 : set → set → set

L32

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

L33

Variable V1 : set → set → set

L34

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

L35

Variable F0 : set → set → set

L36

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

L37

Variable G0 : set → set

L38

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

L39

Variable H0 : set

L40

Hypothesis HH0 : H0 ∈ int

L41

Variable U0 : set → set → set

L42

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

L43

Variable V0 : set → set

L44

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

L45

Variable F3 : set → set → set

L46

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

L47

Variable G3 : set → set

L48

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

L49

Variable H3 : set → set

L50

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

L51

Variable I3 : set

L52

Hypothesis HI3 : I3 ∈ int

L53

Variable J3 : set

L54

Hypothesis HJ3 : J3 ∈ int

L55

Variable U3 : set → set → set → set

L56

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

L57

Variable V3 : set → set → set → set

L58

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

L59

Variable W3 : set → set

L60

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

L61

Variable SMALL : set → set

L62

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

L63

Variable F5 : set → set

L64

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

L65

Variable G5 : set

L66

Hypothesis HG5 : G5 ∈ int

L67

Variable F6 : set → set → set

L68

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

L69

Variable G6 : set → set

L70

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

L71

Variable H6 : set → set → set

L72

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

L73

Variable I6 : set

L74

Hypothesis HI6 : I6 ∈ int

L75

Variable J6 : set

L76

Hypothesis HJ6 : J6 ∈ int

L77

Variable U6 : set → set → set → set

L78

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

L79

Variable V6 : set → set → set → set

L80

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

L81

Variable W6 : set → set → set

L82

Hypothesis HW6 : ∀x0 ∈ int, ∀x1 ∈ int, W6 x0 x1 ∈ int

L83

Variable H5 : set → set → set

L84

Hypothesis HH5 : ∀x0 ∈ int, ∀x1 ∈ int, H5 x0 x1 ∈ int

L85

Variable U5 : set → set → set

L86

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

L87

Variable V5 : set → set → set

L88

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

L89

Variable F4 : set → set → set

L90

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

L91

Variable G4 : set → set

L92

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

L93

Variable H4 : set

L94

Hypothesis HH4 : H4 ∈ int

L95

Variable U4 : set → set → set

L96

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

L97

Variable V4 : set → set

L98

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

L99

Variable F7 : set → set → set

L100

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

L101

Variable G7 : set → set

L102

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

L103

Variable H7 : set → set

L104

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

L105

Variable I7 : set

L106

Hypothesis HI7 : I7 ∈ int

L107

Variable J7 : set

L108

Hypothesis HJ7 : J7 ∈ int

L109

Variable U7 : set → set → set → set

L110

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

L111

Variable V7 : set → set → set → set

L112

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

L113

Variable W7 : set → set

L114

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

L115

Variable FAST : set → set

L116

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

L117

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

L118

Hypothesis H2 : (G1 = 1)

L119

Hypothesis H3 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = (X + Y))))

L120

Hypothesis H4 : (∀X ∈ int, ((G2 X) = X))

L121

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

L122

Hypothesis H6 : (I2 = 1)

L123

Hypothesis H7 : (J2 = 0)

L124

Hypothesis H8 : (∀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)))))))

L125

Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V2 X Y Z) = (if (X <= 0) then Z else (G2 (U2 (X + - 1) Y Z)))))))

L126

Hypothesis H10 : (∀X ∈ int, (∀Y ∈ int, ((W2 X Y) = (U2 (H2 X Y) I2 J2))))

L127

Hypothesis H11 : (∀X ∈ int, (∀Y ∈ int, ((H1 X Y) = (W2 X Y))))

L128

Hypothesis H12 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))

L129

Hypothesis H13 : (∀X ∈ int, (∀Y ∈ int, ((V1 X Y) = (U1 G1 (H1 X Y)))))

L130

Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = ((V1 X Y) * X))))

L131

Hypothesis H15 : (∀X ∈ int, ((G0 X) = X))

L132

Hypothesis H16 : (H0 = 1)

L133

Hypothesis H17 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y) X)))))

L134

Hypothesis H18 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))

L135

Hypothesis H19 : (∀X ∈ int, (∀Y ∈ int, ((F3 X Y) = (X + Y))))

L136

Hypothesis H20 : (∀X ∈ int, ((G3 X) = X))

L137

Hypothesis H21 : (∀X ∈ int, ((H3 X) = X))

L138

Hypothesis H22 : (I3 = 1)

L139

Hypothesis H23 : (J3 = 0)

L140

Hypothesis H24 : (∀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)))))))

L141

Hypothesis H25 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V3 X Y Z) = (if (X <= 0) then Z else (G3 (U3 (X + - 1) Y Z)))))))

L142

Hypothesis H26 : (∀X ∈ int, ((W3 X) = (U3 (H3 X) I3 J3)))

L143

Hypothesis H27 : (∀X ∈ int, ((SMALL X) = ((V0 X) * (W3 X))))

L144

Hypothesis H28 : (∀X ∈ int, ((F5 X) = (X * X)))

L145

Hypothesis H29 : (G5 = 1)

L146

Hypothesis H30 : (∀X ∈ int, (∀Y ∈ int, ((F6 X Y) = (X + Y))))

L147

Hypothesis H31 : (∀X ∈ int, ((G6 X) = X))

L148

Hypothesis H32 : (∀X ∈ int, (∀Y ∈ int, ((H6 X Y) = Y)))

L149

Hypothesis H33 : (I6 = 1)

L150

Hypothesis H34 : (J6 = 1)

L151

Hypothesis H35 : (∀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)))))))

L152

Hypothesis H36 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V6 X Y Z) = (if (X <= 0) then Z else (G6 (U6 (X + - 1) Y Z)))))))

L153

Hypothesis H37 : (∀X ∈ int, (∀Y ∈ int, ((W6 X Y) = (U6 (H6 X Y) I6 J6))))

L154

Hypothesis H38 : (∀X ∈ int, (∀Y ∈ int, ((H5 X Y) = (W6 X Y))))

L155

Hypothesis H39 : (∀X ∈ int, (∀Y ∈ int, ((U5 X Y) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y))))))

L156

Hypothesis H40 : (∀X ∈ int, (∀Y ∈ int, ((V5 X Y) = (U5 G5 (H5 X Y)))))

L157

Hypothesis H41 : (∀X ∈ int, (∀Y ∈ int, ((F4 X Y) = ((V5 X Y) * X))))

L158

Hypothesis H42 : (∀X ∈ int, ((G4 X) = (X + - 1)))

L159

Hypothesis H43 : (H4 = 1)

L160

Hypothesis H44 : (∀X ∈ int, (∀Y ∈ int, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y) X)))))

L161

Hypothesis H45 : (∀X ∈ int, ((V4 X) = (U4 (G4 X) H4)))

L162

Hypothesis H46 : (∀X ∈ int, (∀Y ∈ int, ((F7 X Y) = (X + Y))))

L163

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

L164

Hypothesis H48 : (∀X ∈ int, ((H7 X) = (X + - 1)))

L165

Hypothesis H49 : (I7 = 1)

L166

Hypothesis H50 : (J7 = 1)

L167

Hypothesis H51 : (∀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)))))))

L168

Hypothesis H52 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V7 X Y Z) = (if (X <= 0) then Z else (G7 (U7 (X + - 1) Y Z)))))))

L169

Hypothesis H53 : (∀X ∈ int, ((W7 X) = (U7 (H7 X) I7 J7)))

L170

Hypothesis H54 : (∀X ∈ int, ((FAST X) = ((V4 X) * (W7 X))))

L171

Theorem. (A95917)

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

In Proofgold the corresponding term root is 8fca73... and proposition id is ca0c46...

Proof:
Proof not loaded.