Beginning of Section A14941

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 TMa56oCyPgkQFi1fhnSbriUpx4cmhBX9cgJ ***)

Variable F2 : set → set → set

L10

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

L11

Variable G2 : set

L12

Hypothesis HG2 : G2 ∈ int

L13

Variable H2 : set → set

L14

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

L15

Variable U2 : set → set → set

L16

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

L17

Variable V2 : set → set

L18

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

L19

Variable F1 : set → set

L20

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

L21

Variable G1 : set → set → set

L22

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

L23

Variable H1 : set → set → set

L24

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

L25

Variable U1 : set → set → set

L26

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

L27

Variable V1 : set → set → set

L28

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

L29

Variable F0 : set → set → set

L30

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

L31

Variable G0 : set → set

L32

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

L33

Variable H0 : set

L34

Hypothesis HH0 : H0 ∈ 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 F4 : set → set → set

L42

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

L43

Variable G4 : set

L44

Hypothesis HG4 : G4 ∈ int

L45

Variable H4 : set → set

L46

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

L47

Variable U4 : set → set → set

L48

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

L49

Variable V4 : set → set

L50

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

L51

Variable F3 : set → set → set

L52

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

L53

Variable G3 : set → set → set

L54

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

L55

Variable H3 : set → set

L56

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

L57

Variable I3 : set → set

L58

Hypothesis HI3 : ∀x0 ∈ int, I3 x0 ∈ int

L59

Variable J3 : set → set

L60

Hypothesis HJ3 : ∀x0 ∈ int, J3 x0 ∈ int

L61

Variable U3 : set → set → set → set

L62

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

L63

Variable V3 : set → set → set → set

L64

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

L65

Variable W3 : set → set

L66

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

L67

Variable FAST : set → set

L68

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

L69

Hypothesis H1 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = (X * Y))))

L70

Hypothesis H2 : (G2 = (2 + 2))

L71

Hypothesis H3 : (∀X ∈ int, ((H2 X) = X))

L72

Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y) X)))))

L73

Hypothesis H5 : (∀X ∈ int, ((V2 X) = (U2 G2 (H2 X))))

L74

Hypothesis H6 : (∀X ∈ int, ((F1 X) = ((V2 X) + - X)))

L75

Hypothesis H7 : (∀X ∈ int, (∀Y ∈ int, ((G1 X Y) = Y)))

L76

Hypothesis H8 : (∀X ∈ int, (∀Y ∈ int, ((H1 X Y) = (1 + Y))))

L77

Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))

L78

Hypothesis H10 : (∀X ∈ int, (∀Y ∈ int, ((V1 X Y) = (U1 (G1 X Y) (H1 X Y)))))

L79

Hypothesis H11 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = ((V1 X Y) + X))))

L80

Hypothesis H12 : (∀X ∈ int, ((G0 X) = X))

L81

Hypothesis H13 : (H0 = 1)

L82

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

L83

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

L84

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

L85

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

L86

Hypothesis H18 : (G4 = (2 + 2))

L87

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

L88

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

L89

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

L90

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

L91

Hypothesis H23 : (∀X ∈ int, (∀Y ∈ int, ((G3 X Y) = (Y + - 1))))

L92

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

L93

Hypothesis H25 : (∀X ∈ int, ((I3 X) = (1 + X)))

L94

Hypothesis H26 : (∀X ∈ int, ((J3 X) = X))

L95

Hypothesis H27 : (∀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)))))))

L96

Hypothesis H28 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V3 X Y Z) = (if (X <= 0) then Z else (G3 (U3 (X + - 1) Y Z) (V3 (X + - 1) Y Z)))))))

L97

Hypothesis H29 : (∀X ∈ int, ((W3 X) = (U3 (H3 X) (I3 X) (J3 X))))

L98

Hypothesis H30 : (∀X ∈ int, ((FAST X) = (W3 X)))

L99

Theorem. (A14941)

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

In Proofgold the corresponding term root is 66538c... and proposition id is 7fafc5...

Proof:
Proof not loaded.