Beginning of Section A16092

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 TMJgMPKj6bzgy1hTvMWacV9zN86AQ1dF2zJ ***)

Variable F3 : set → set

L10

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

L11

Variable G3 : set → set → set

L12

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

L13

Variable H3 : set

L14

Hypothesis HH3 : H3 ∈ int

L15

Variable U3 : set → set → set

L16

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

L17

Variable V3 : set → set → set

L18

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

L24

Hypothesis HH2 : H2 ∈ int

L25

Variable U2 : set → set → set

L26

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

L27

Variable V2 : set → set → set

L28

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

L29

Variable F4 : set → set → set

L30

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

L31

Variable G4 : set

L32

Hypothesis HG4 : G4 ∈ int

L33

Variable H4 : set → set

L34

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

L35

Variable U4 : set → set → set

L36

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

L37

Variable V4 : set → set

L38

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

L39

Variable F1 : set → set → set

L40

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

L41

Variable G1 : set → set → set

L42

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

L43

Variable H1 : set

L44

Hypothesis HH1 : H1 ∈ int

L45

Variable U1 : set → set → set

L46

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

L47

Variable V1 : set → set → set

L48

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

L49

Variable F5 : set → set → set

L50

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

L51

Variable G5 : set

L52

Hypothesis HG5 : G5 ∈ int

L53

Variable H5 : set → set

L54

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

L55

Variable U5 : set → set → set

L56

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

L57

Variable V5 : set → set

L58

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

L59

Variable F0 : set → set → set

L60

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

L61

Variable G0 : set → set

L62

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

L63

Variable H0 : set

L64

Hypothesis HH0 : H0 ∈ int

L65

Variable U0 : set → set → set

L66

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

L67

Variable V0 : set → set

L68

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

L69

Variable SMALL : set → set

L70

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

L71

Variable F10 : set → set → set

L72

Hypothesis HF10 : ∀x0 ∈ int, ∀x1 ∈ int, F10 x0 x1 ∈ int

L73

Variable G10 : set → set → set

L74

Hypothesis HG10 : ∀x0 ∈ int, ∀x1 ∈ int, G10 x0 x1 ∈ int

L75

Variable H10 : set → set

L76

Hypothesis HH10 : ∀x0 ∈ int, H10 x0 ∈ int

L77

Variable I10 : set

L78

Hypothesis HI10 : I10 ∈ int

L79

Variable J10 : set

L80

Hypothesis HJ10 : J10 ∈ int

L81

Variable U10 : set → set → set → set

L82

Hypothesis HU10 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U10 x0 x1 x2 ∈ int

L83

Variable V10 : set → set → set → set

L84

Hypothesis HV10 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V10 x0 x1 x2 ∈ int

L85

Variable W10 : set → set

L86

Hypothesis HW10 : ∀x0 ∈ int, W10 x0 ∈ int

L87

Variable F11 : set → set

L88

Hypothesis HF11 : ∀x0 ∈ int, F11 x0 ∈ int

L89

Variable G11 : set

L90

Hypothesis HG11 : G11 ∈ int

L91

Variable F12 : set → set

L92

Hypothesis HF12 : ∀x0 ∈ int, F12 x0 ∈ int

L93

Variable G12 : set → set

L94

Hypothesis HG12 : ∀x0 ∈ int, G12 x0 ∈ int

L95

Variable H12 : set

L96

Hypothesis HH12 : H12 ∈ int

L97

Variable U12 : set → set → set

L98

Hypothesis HU12 : ∀x0 ∈ int, ∀x1 ∈ int, U12 x0 x1 ∈ int

L99

Variable V12 : set → set

L100

Hypothesis HV12 : ∀x0 ∈ int, V12 x0 ∈ int

L101

Variable H11 : set → set

L102

Hypothesis HH11 : ∀x0 ∈ int, H11 x0 ∈ int

L103

Variable U11 : set → set → set

L104

Hypothesis HU11 : ∀x0 ∈ int, ∀x1 ∈ int, U11 x0 x1 ∈ int

L105

Variable V11 : set → set

L106

Hypothesis HV11 : ∀x0 ∈ int, V11 x0 ∈ int

L107

Variable F9 : set → set

L108

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

L109

Variable G9 : set

L110

Hypothesis HG9 : G9 ∈ int

L111

Variable H9 : set → set → set

L112

Hypothesis HH9 : ∀x0 ∈ int, ∀x1 ∈ int, H9 x0 x1 ∈ int

L113

Variable U9 : set → set → set

L114

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

L115

Variable V9 : set → set → set

L116

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

L117

Variable F8 : set → set → set

L118

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

L119

Variable G8 : set → set

L120

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

L121

Variable H8 : set

L122

Hypothesis HH8 : H8 ∈ int

L123

Variable U8 : set → set → set

L124

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

L125

Variable V8 : set → set

L126

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

L127

Variable F7 : set → set

L128

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

L129

Variable G7 : set

L130

Hypothesis HG7 : G7 ∈ int

L131

Variable H7 : set → set → set

L132

Hypothesis HH7 : ∀x0 ∈ int, ∀x1 ∈ int, H7 x0 x1 ∈ int

L133

Variable U7 : set → set → set

L134

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

L135

Variable V7 : set → set → set

L136

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

L137

Variable F6 : set → set → set

L138

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

L139

Variable G6 : set → set

L140

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

L141

Variable H6 : set

L142

Hypothesis HH6 : H6 ∈ int

L143

Variable U6 : set → set → set

L144

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

L145

Variable V6 : set → set

L146

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

L147

Variable FAST : set → set

L148

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

L149

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

L150

Hypothesis H2 : (∀X ∈ int, (∀Y ∈ int, ((G3 X Y) = (Y + Y))))

L151

Hypothesis H3 : (H3 = 1)

L152

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

L153

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

L154

Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = ((V3 X Y) + (2 * (2 * (X + X)))))))

L155

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

L156

Hypothesis H8 : (H2 = 1)

L157

Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y) X)))))

L158

Hypothesis H10 : (∀X ∈ int, (∀Y ∈ int, ((V2 X Y) = (U2 (G2 X Y) H2))))

L159

Hypothesis H11 : (∀X ∈ int, (∀Y ∈ int, ((F4 X Y) = ((2 + Y) * X))))

L160

Hypothesis H12 : (G4 = 2)

L161

Hypothesis H13 : (∀X ∈ int, ((H4 X) = X))

L162

Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y) X)))))

L163

Hypothesis H15 : (∀X ∈ int, ((V4 X) = (U4 G4 (H4 X))))

L164

Hypothesis H16 : (∀X ∈ int, (∀Y ∈ int, ((F1 X Y) = ((V2 X Y) + (V4 X)))))

L165

Hypothesis H17 : (∀X ∈ int, (∀Y ∈ int, ((G1 X Y) = Y)))

L166

Hypothesis H18 : (H1 = 1)

L167

Hypothesis H19 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y) X)))))

L168

Hypothesis H20 : (∀X ∈ int, (∀Y ∈ int, ((V1 X Y) = (U1 (G1 X Y) H1))))

L169

Hypothesis H21 : (∀X ∈ int, (∀Y ∈ int, ((F5 X Y) = ((2 + Y) * X))))

L170

Hypothesis H22 : (G5 = 2)

L171

Hypothesis H23 : (∀X ∈ int, ((H5 X) = X))

L172

Hypothesis H24 : (∀X ∈ int, (∀Y ∈ int, ((U5 X Y) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y) X)))))

L173

Hypothesis H25 : (∀X ∈ int, ((V5 X) = (U5 G5 (H5 X))))

L174

Hypothesis H26 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = (((V1 X Y) + - X) + (V5 X)))))

L175

Hypothesis H27 : (∀X ∈ int, ((G0 X) = X))

L176

Hypothesis H28 : (H0 = 1)

L177

Hypothesis H29 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y) X)))))

L178

Hypothesis H30 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))

L179

Hypothesis H31 : (∀X ∈ int, ((SMALL X) = (V0 X)))

L180

Hypothesis H32 : (∀X ∈ int, (∀Y ∈ int, ((F10 X Y) = (((X + X) + X) + Y))))

L181

Hypothesis H33 : (∀X ∈ int, (∀Y ∈ int, ((G10 X Y) = (Y + Y))))

L182

Hypothesis H34 : (∀X ∈ int, ((H10 X) = X))

L183

Hypothesis H35 : (I10 = 1)

L184

Hypothesis H36 : (J10 = 2)

L185

Hypothesis H37 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U10 X Y Z) = (if (X <= 0) then Y else (F10 (U10 (X + - 1) Y Z) (V10 (X + - 1) Y Z)))))))

L186

Hypothesis H38 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V10 X Y Z) = (if (X <= 0) then Z else (G10 (U10 (X + - 1) Y Z) (V10 (X + - 1) Y Z)))))))

L187

Hypothesis H39 : (∀X ∈ int, ((W10 X) = (U10 (H10 X) I10 J10)))

L188

Hypothesis H40 : (∀X ∈ int, ((F11 X) = (X * X)))

L189

Hypothesis H41 : (G11 = 1)

L190

Hypothesis H42 : (∀X ∈ int, ((F12 X) = (X + X)))

L191

Hypothesis H43 : (∀X ∈ int, ((G12 X) = X))

L192

Hypothesis H44 : (H12 = 1)

L193

Hypothesis H45 : (∀X ∈ int, (∀Y ∈ int, ((U12 X Y) = (if (X <= 0) then Y else (F12 (U12 (X + - 1) Y))))))

L194

Hypothesis H46 : (∀X ∈ int, ((V12 X) = (U12 (G12 X) H12)))

L195

Hypothesis H47 : (∀X ∈ int, ((H11 X) = (V12 X)))

L196

Hypothesis H48 : (∀X ∈ int, (∀Y ∈ int, ((U11 X Y) = (if (X <= 0) then Y else (F11 (U11 (X + - 1) Y))))))

L197

Hypothesis H49 : (∀X ∈ int, ((V11 X) = (U11 G11 (H11 X))))

L198

Hypothesis H50 : (∀X ∈ int, ((F9 X) = ((W10 X) * (V11 X))))

L199

Hypothesis H51 : (G9 = 1)

L200

Hypothesis H52 : (∀X ∈ int, (∀Y ∈ int, ((H9 X Y) = Y)))

L201

Hypothesis H53 : (∀X ∈ int, (∀Y ∈ int, ((U9 X Y) = (if (X <= 0) then Y else (F9 (U9 (X + - 1) Y))))))

L202

Hypothesis H54 : (∀X ∈ int, (∀Y ∈ int, ((V9 X Y) = (U9 G9 (H9 X Y)))))

L203

Hypothesis H55 : (∀X ∈ int, (∀Y ∈ int, ((F8 X Y) = (((V9 X Y) + X) + (((X * 2) * 2) * 2)))))

L204

Hypothesis H56 : (∀X ∈ int, ((G8 X) = X))

L205

Hypothesis H57 : (H8 = 1)

L206

Hypothesis H58 : (∀X ∈ int, (∀Y ∈ int, ((U8 X Y) = (if (X <= 0) then Y else (F8 (U8 (X + - 1) Y) X)))))

L207

Hypothesis H59 : (∀X ∈ int, ((V8 X) = (U8 (G8 X) H8)))

L208

Hypothesis H60 : (∀X ∈ int, ((F7 X) = (V8 X)))

L209

Hypothesis H61 : (G7 = 1)

L210

Hypothesis H62 : (∀X ∈ int, (∀Y ∈ int, ((H7 X Y) = Y)))

L211

Hypothesis H63 : (∀X ∈ int, (∀Y ∈ int, ((U7 X Y) = (if (X <= 0) then Y else (F7 (U7 (X + - 1) Y))))))

L212

Hypothesis H64 : (∀X ∈ int, (∀Y ∈ int, ((V7 X Y) = (U7 G7 (H7 X Y)))))

L213

Hypothesis H65 : (∀X ∈ int, (∀Y ∈ int, ((F6 X Y) = (((V7 X Y) + X) + ((((X * 2) * 2) + X) * 2)))))

L214

Hypothesis H66 : (∀X ∈ int, ((G6 X) = X))

L215

Hypothesis H67 : (H6 = 1)

L216

Hypothesis H68 : (∀X ∈ int, (∀Y ∈ int, ((U6 X Y) = (if (X <= 0) then Y else (F6 (U6 (X + - 1) Y) X)))))

L217

Hypothesis H69 : (∀X ∈ int, ((V6 X) = (U6 (G6 X) H6)))

L218

Hypothesis H70 : (∀X ∈ int, ((FAST X) = (V6 X)))

L219

Theorem. (A16092)

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

In Proofgold the corresponding term root is 0d6b32... and proposition id is 631417...

Proof:
Proof not loaded.