Beginning of Section A138199

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 TMZ9Hg3Yq86SQCjJt1R6TAfHmbDzAvm67U6 ***)

Variable F1 : set → set

L10

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

L11

Variable G1 : set

L12

Hypothesis HG1 : G1 ∈ 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

L18

Hypothesis HV1 : V1 ∈ int

L19

Variable F0 : set → set

L20

Hypothesis HF0 : ∀x0 ∈ int, F0 x0 ∈ 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

L30

Hypothesis HF2 : ∀x0 ∈ int, F2 x0 ∈ 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 F3 : set → set → set

L42

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

L43

Variable G3 : set → set → set

L44

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

L45

Variable H3 : set → set

L46

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

L47

Variable I3 : set

L48

Hypothesis HI3 : I3 ∈ int

L49

Variable J3 : set

L50

Hypothesis HJ3 : J3 ∈ int

L51

Variable U3 : set → set → set → set

L52

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

L53

Variable V3 : set → set → set → set

L54

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

L55

Variable W3 : set → set

L56

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

L57

Variable F4 : set → set

L58

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

L59

Variable G4 : set

L60

Hypothesis HG4 : G4 ∈ int

L61

Variable F5 : set → set

L62

Hypothesis HF5 : ∀x0 ∈ int, F5 x0 ∈ 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 H4 : set → set

L72

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

L73

Variable U4 : set → set → set

L74

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

L75

Variable V4 : set → set

L76

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

L77

Variable FAST : set → set

L78

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

L79

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

L80

Hypothesis H2 : (G1 = 2)

L81

Hypothesis H3 : (H1 = 2)

L82

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

L83

Hypothesis H5 : (V1 = (U1 G1 H1))

L84

Hypothesis H6 : (∀X ∈ int, ((F0 X) = ((V1 + - 2) * X)))

L85

Hypothesis H7 : (∀X ∈ int, ((G0 X) = (1 + (X + X))))

L86

Hypothesis H8 : (H0 = 1)

L87

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

L88

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

L89

Hypothesis H11 : (∀X ∈ int, ((F2 X) = ((2 * X) + X)))

L90

Hypothesis H12 : (∀X ∈ int, ((G2 X) = ((X + 1) + X)))

L91

Hypothesis H13 : (H2 = 1)

L92

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

L93

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

L94

Hypothesis H16 : (∀X ∈ int, ((SMALL X) = ((V0 X) + (V2 X))))

L95

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

L96

Hypothesis H18 : (∀X ∈ int, (∀Y ∈ int, ((G3 X Y) = Y)))

L97

Hypothesis H19 : (∀X ∈ int, ((H3 X) = (1 + (X + X))))

L98

Hypothesis H20 : (I3 = 1)

L99

Hypothesis H21 : (J3 = (2 + (2 * (2 + (2 + 2)))))

L100

Hypothesis H22 : (∀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)))))))

L101

Hypothesis H23 : (∀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)))))))

L102

Hypothesis H24 : (∀X ∈ int, ((W3 X) = (U3 (H3 X) I3 J3)))

L103

Hypothesis H25 : (∀X ∈ int, ((F4 X) = (X * (X * (1 + 2)))))

L104

Hypothesis H26 : (G4 = 1)

L105

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

L106

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

L107

Hypothesis H29 : (H5 = 1)

L108

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

L109

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

L110

Hypothesis H32 : (∀X ∈ int, ((H4 X) = (V5 X)))

L111

Hypothesis H33 : (∀X ∈ int, (∀Y ∈ int, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y))))))

L112

Hypothesis H34 : (∀X ∈ int, ((V4 X) = (U4 G4 (H4 X))))

L113

Hypothesis H35 : (∀X ∈ int, ((FAST X) = ((W3 X) + (V4 X))))

L114

Theorem. (A138199)

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

In Proofgold the corresponding term root is da5aff... and proposition id is 28df44...

Proof:
Proof not loaded.