Beginning of Section A182543

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 TMEyVfSwd8NNBKJFz36o1kbKhtroFdt39HD ***)

Variable F1 : set → set → set

L10

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

L11

Variable G1 : set → set → set

L12

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

L13

Variable H1 : set

L14

Hypothesis HH1 : H1 ∈ int

L15

Variable U1 : set → set → set

L16

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

L17

Variable V1 : set → set → set

L18

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

L19

Variable F0 : set → set → set

L20

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

L21

Variable G0 : set → set

L22

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

L23

Variable H0 : set

L24

Hypothesis HH0 : H0 ∈ int

L25

Variable U0 : set → set → set

L26

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

L27

Variable V0 : set → set

L28

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

L29

Variable F2 : set → set → set

L30

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

L31

Variable G2 : set → set

L32

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

L33

Variable H2 : set

L34

Hypothesis HH2 : H2 ∈ int

L35

Variable U2 : set → set → set

L36

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

L37

Variable V2 : set → set

L38

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

L39

Variable SMALL : set → set

L40

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

L41

Variable F4 : set → set → set

L42

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

L43

Variable G4 : set → set → set

L44

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

L45

Variable H4 : set → set → set

L46

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

L47

Variable U4 : set → set → set

L48

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

L49

Variable V4 : set → set → set

L50

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

L51

Variable F3 : set → set → set

L52

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

L53

Variable G3 : set → set

L54

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

L55

Variable H3 : set

L56

Hypothesis HH3 : H3 ∈ 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

L64

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

L65

Variable H5 : set

L66

Hypothesis HH5 : H5 ∈ int

L67

Variable U5 : set → set → set

L68

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

L69

Variable V5 : set → set

L70

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

L71

Variable FAST : set → set

L72

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

L73

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

L74

Hypothesis H2 : (∀X ∈ int, (∀Y ∈ int, ((G1 X Y) = Y)))

L75

Hypothesis H3 : (H1 = 1)

L76

Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y) X)))))

L77

Hypothesis H5 : (∀X ∈ int, (∀Y ∈ int, ((V1 X Y) = (U1 (G1 X Y) H1))))

L78

Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = (((X * Y) + (V1 X Y)) + X))))

L79

Hypothesis H7 : (∀X ∈ int, ((G0 X) = X))

L80

Hypothesis H8 : (H0 = 1)

L81

Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y) X)))))

L82

Hypothesis H10 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))

L83

Hypothesis H11 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = (X * Y))))

L84

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

L85

Hypothesis H13 : (H2 = 1)

L86

Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y) X)))))

L87

Hypothesis H15 : (∀X ∈ int, ((V2 X) = (U2 (G2 X) H2)))

L88

Hypothesis H16 : (∀X ∈ int, ((SMALL X) = (((((V0 X) + (V2 X)) * 2) + - (if (X <= 0) then 0 else 1)) + 1)))

L89

Hypothesis H17 : (∀X ∈ int, (∀Y ∈ int, ((F4 X Y) = (X * Y))))

L90

Hypothesis H18 : (∀X ∈ int, (∀Y ∈ int, ((G4 X Y) = (Y + - 1))))

L91

Hypothesis H19 : (∀X ∈ int, (∀Y ∈ int, ((H4 X Y) = Y)))

L92

Hypothesis H20 : (∀X ∈ int, (∀Y ∈ int, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y) X)))))

L93

Hypothesis H21 : (∀X ∈ int, (∀Y ∈ int, ((V4 X Y) = (U4 (G4 X Y) (H4 X Y)))))

L94

Hypothesis H22 : (∀X ∈ int, (∀Y ∈ int, ((F3 X Y) = (((V4 X Y) + (X * Y)) + X))))

L95

Hypothesis H23 : (∀X ∈ int, ((G3 X) = X))

L96

Hypothesis H24 : (H3 = 1)

L97

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

L98

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

L99

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

L100

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

L101

Hypothesis H29 : (H5 = 1)

L102

Hypothesis H30 : (∀X ∈ int, (∀Y ∈ int, ((U5 X Y) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y) X)))))

L103

Hypothesis H31 : (∀X ∈ int, ((V5 X) = (U5 (G5 X) H5)))

L104

Hypothesis H32 : (∀X ∈ int, ((FAST X) = (((((V3 X) + (V5 X)) * 2) + 2) + - (if (X <= 0) then 1 else 2))))

L105

Theorem. (A182543)

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

In Proofgold the corresponding term root is 9c91fa... and proposition id is 11609a...

Proof:
Proof not loaded.