Beginning of Section A74533

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 TMbuoYsQSjAh3hjv8KAeBh7Ku4uvoDPfy4Z ***)

Variable F0 : set → set

L10

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

L11

Variable G0 : set → set

L12

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

L13

Variable F1 : set → set

L14

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

L15

Variable G1 : set → set

L16

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

L17

Variable H1 : set

L18

Hypothesis HH1 : H1 ∈ int

L19

Variable U1 : set → set → set

L20

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

L21

Variable V1 : set → set

L22

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

L23

Variable F2 : set → set

L24

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

L25

Variable G2 : set → set

L26

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

L27

Variable H2 : set

L28

Hypothesis HH2 : H2 ∈ int

L29

Variable U2 : set → set → set

L30

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

L31

Variable V2 : set → set

L32

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

L33

Variable H0 : set → set

L34

Hypothesis HH0 : ∀x0 ∈ int, H0 x0 ∈ int

L35

Variable U0 : set → set → set

L36

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

L37

Variable V0 : set → set

L38

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

L39

Variable SMALL : set → set

L40

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

L41

Variable F3 : set → set

L42

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

L43

Variable G3 : set

L44

Hypothesis HG3 : G3 ∈ int

L45

Variable F4 : set → set

L46

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

L47

Variable G4 : set → set

L48

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

L49

Variable H4 : set

L50

Hypothesis HH4 : H4 ∈ int

L51

Variable U4 : set → set → set

L52

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

L53

Variable V4 : set → set

L54

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

L55

Variable H3 : set → set

L56

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

L57

Variable U3 : set → set → set

L58

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

L59

Variable V3 : set → set

L60

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

L61

Variable F5 : set → set → set

L62

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

L63

Variable G5 : set → set → set

L64

Hypothesis HG5 : ∀x0 ∈ int, ∀x1 ∈ int, G5 x0 x1 ∈ int

L65

Variable H5 : set → set

L66

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

L67

Variable I5 : set

L68

Hypothesis HI5 : I5 ∈ int

L69

Variable J5 : set

L70

Hypothesis HJ5 : J5 ∈ int

L71

Variable U5 : set → set → set → set

L72

Hypothesis HU5 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U5 x0 x1 x2 ∈ int

L73

Variable V5 : set → set → set → set

L74

Hypothesis HV5 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V5 x0 x1 x2 ∈ int

L75

Variable W5 : set → set

L76

Hypothesis HW5 : ∀x0 ∈ int, W5 x0 ∈ int

L77

Variable FAST : set → set

L78

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

L79

Hypothesis H1 : (∀X ∈ int, ((F0 X) = (X + X)))

L80

Hypothesis H2 : (∀X ∈ int, ((G0 X) = X))

L81

Hypothesis H3 : (∀X ∈ int, ((F1 X) = ((X + X) + X)))

L82

Hypothesis H4 : (∀X ∈ int, ((G1 X) = X))

L83

Hypothesis H5 : (H1 = 1)

L84

Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))

L85

Hypothesis H7 : (∀X ∈ int, ((V1 X) = (U1 (G1 X) H1)))

L86

Hypothesis H8 : (∀X ∈ int, ((F2 X) = (X + X)))

L87

Hypothesis H9 : (∀X ∈ int, ((G2 X) = X))

L88

Hypothesis H10 : (H2 = 1)

L89

Hypothesis H11 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y))))))

L90

Hypothesis H12 : (∀X ∈ int, ((V2 X) = (U2 (G2 X) H2)))

L91

Hypothesis H13 : (∀X ∈ int, ((H0 X) = (1 + ((V1 X) + (V2 X)))))

L92

Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y))))))

L93

Hypothesis H15 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) (H0 X))))

L94

Hypothesis H16 : (∀X ∈ int, ((SMALL X) = (V0 X)))

L95

Hypothesis H17 : (∀X ∈ int, ((F3 X) = ((X * X) + X)))

L96

Hypothesis H18 : (G3 = 1)

L97

Hypothesis H19 : (∀X ∈ int, ((F4 X) = (X + X)))

L98

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

L99

Hypothesis H21 : (H4 = 1)

L100

Hypothesis H22 : (∀X ∈ int, (∀Y ∈ int, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y))))))

L101

Hypothesis H23 : (∀X ∈ int, ((V4 X) = (U4 (G4 X) H4)))

L102

Hypothesis H24 : (∀X ∈ int, ((H3 X) = (V4 X)))

L103

Hypothesis H25 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y))))))

L104

Hypothesis H26 : (∀X ∈ int, ((V3 X) = (U3 G3 (H3 X))))

L105

Hypothesis H27 : (∀X ∈ int, (∀Y ∈ int, ((F5 X Y) = (X * Y))))

L106

Hypothesis H28 : (∀X ∈ int, (∀Y ∈ int, ((G5 X Y) = Y)))

L107

Hypothesis H29 : (∀X ∈ int, ((H5 X) = X))

L108

Hypothesis H30 : (I5 = 1)

L109

Hypothesis H31 : (J5 = (2 + (2 + 2)))

L110

Hypothesis H32 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U5 X Y Z) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y Z) (V5 (X + - 1) Y Z)))))))

L111

Hypothesis H33 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V5 X Y Z) = (if (X <= 0) then Z else (G5 (U5 (X + - 1) Y Z) (V5 (X + - 1) Y Z)))))))

L112

Hypothesis H34 : (∀X ∈ int, ((W5 X) = (U5 (H5 X) I5 J5)))

L113

Hypothesis H35 : (∀X ∈ int, ((FAST X) = ((V3 X) + (W5 X))))

L114

Theorem. (A74533)

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

In Proofgold the corresponding term root is 95eb5c... and proposition id is 25fd17...

Proof:
Proof not loaded.