Beginning of Section A114182

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 TMZWQTkaujAQR3EdbS8msnbrzQ9jwQk23CQ ***)

Variable F1 : set → set → set

L10

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

L11

Variable G1 : set → set

L12

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

L13

Variable H1 : set → set → set

L14

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

L15

Variable I1 : set

L16

Hypothesis HI1 : I1 ∈ int

L17

Variable J1 : set

L18

Hypothesis HJ1 : J1 ∈ int

L19

Variable U1 : set → set → set → set

L20

Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U1 x0 x1 x2 ∈ int

L21

Variable V1 : set → set → set → set

L22

Hypothesis HV1 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V1 x0 x1 x2 ∈ int

L23

Variable W1 : set → set → set

L24

Hypothesis HW1 : ∀x0 ∈ int, ∀x1 ∈ int, W1 x0 x1 ∈ int

L25

Variable F0 : set → set → set

L26

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

L27

Variable G0 : set → set

L28

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

L29

Variable H0 : set

L30

Hypothesis HH0 : H0 ∈ int

L31

Variable U0 : set → set → set

L32

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

L33

Variable V0 : set → set

L34

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

L35

Variable SMALL : set → set

L36

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

L37

Variable F2 : set → set → set

L38

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

L39

Variable G2 : set → set

L40

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

L41

Variable H2 : set → set

L42

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

L43

Variable I2 : set → set

L44

Hypothesis HI2 : ∀x0 ∈ int, I2 x0 ∈ int

L45

Variable J2 : set

L46

Hypothesis HJ2 : J2 ∈ int

L47

Variable U2 : set → set → set → set

L48

Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U2 x0 x1 x2 ∈ int

L49

Variable V2 : set → set → set → set

L50

Hypothesis HV2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V2 x0 x1 x2 ∈ int

L51

Variable W2 : set → set

L52

Hypothesis HW2 : ∀x0 ∈ int, W2 x0 ∈ int

L53

Variable FAST : set → set

L54

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

L55

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

L56

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

L57

Hypothesis H3 : (∀X ∈ int, (∀Y ∈ int, ((H1 X Y) = (Y + Y))))

L58

Hypothesis H4 : (I1 = 2)

L59

Hypothesis H5 : (J1 = 1)

L60

Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U1 X Y Z) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y Z) (V1 (X + - 1) Y Z)))))))

L61

Hypothesis H7 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V1 X Y Z) = (if (X <= 0) then Z else (G1 (U1 (X + - 1) Y Z)))))))

L62

Hypothesis H8 : (∀X ∈ int, (∀Y ∈ int, ((W1 X Y) = (U1 (H1 X Y) I1 J1))))

L63

Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = (((W1 X Y) + - 1) + X))))

L64

Hypothesis H10 : (∀X ∈ int, ((G0 X) = (X + X)))

L65

Hypothesis H11 : (H0 = 0)

L66

Hypothesis H12 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y) X)))))

L67

Hypothesis H13 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))

L68

Hypothesis H14 : (∀X ∈ int, ((SMALL X) = (V0 X)))

L69

Hypothesis H15 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = (((2 * ((X + X) + X)) + - Y) + X))))

L70

Hypothesis H16 : (∀X ∈ int, ((G2 X) = X))

L71

Hypothesis H17 : (∀X ∈ int, ((H2 X) = (X + - 1)))

L72

Hypothesis H18 : (∀X ∈ int, ((I2 X) = (if (X <= 0) then 1 else (1 + (2 + (2 + 2))))))

L73

Hypothesis H19 : (J2 = 1)

L74

Hypothesis H20 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U2 X Y Z) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y Z) (V2 (X + - 1) Y Z)))))))

L75

Hypothesis H21 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V2 X Y Z) = (if (X <= 0) then Z else (G2 (U2 (X + - 1) Y Z)))))))

L76

Hypothesis H22 : (∀X ∈ int, ((W2 X) = (U2 (H2 X) (I2 X) J2)))

L77

Hypothesis H23 : (∀X ∈ int, ((FAST X) = (((((W2 X) + - 1) + - X) * (2 + 1)) + X)))

L78

Theorem. (A114182)

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

In Proofgold the corresponding term root is 69f0e2... and proposition id is e3e14a...

Proof:
Proof not loaded.