Beginning of Section A16175

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 TMMLXyhnZMU1rLqZpST4WcaV7HqkCaoJehW ***)

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 H0 : set → set

L24

Hypothesis HH0 : ∀x0 ∈ int, H0 x0 ∈ 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 SMALL : set → set

L30

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

L31

Variable F2 : set → set → set

L32

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

L33

Variable G2 : set → set → set

L34

Hypothesis HG2 : ∀x0 ∈ int, ∀x1 ∈ int, G2 x0 x1 ∈ int

L35

Variable H2 : set → set

L36

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

L37

Variable F3 : set → set

L38

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

L39

Variable G3 : set → set

L40

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

L41

Variable H3 : set

L42

Hypothesis HH3 : H3 ∈ int

L43

Variable U3 : set → set → set

L44

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

L45

Variable V3 : set → set

L46

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

L47

Variable I2 : set → set

L48

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

L49

Variable J2 : set

L50

Hypothesis HJ2 : J2 ∈ int

L51

Variable U2 : set → set → set → set

L52

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

L53

Variable V2 : set → set → set → set

L54

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

L55

Variable W2 : set → set

L56

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

L57

Variable FAST : set → set

L58

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

L59

Hypothesis H1 : (∀X ∈ int, ((F0 X) = (2 * ((X + X) + X))))

L60

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

L61

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

L62

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

L63

Hypothesis H5 : (H1 = 2)

L64

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

L65

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

L66

Hypothesis H8 : (∀X ∈ int, ((H0 X) = ((V1 X) + - 1)))

L67

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

L68

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

L69

Hypothesis H11 : (∀X ∈ int, ((SMALL X) = (V0 X)))

L70

Hypothesis H12 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = (X * Y))))

L71

Hypothesis H13 : (∀X ∈ int, (∀Y ∈ int, ((G2 X Y) = Y)))

L72

Hypothesis H14 : (∀X ∈ int, ((H2 X) = X))

L73

Hypothesis H15 : (∀X ∈ int, ((F3 X) = (X + X)))

L74

Hypothesis H16 : (∀X ∈ int, ((G3 X) = X))

L75

Hypothesis H17 : (H3 = 2)

L76

Hypothesis H18 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y))))))

L77

Hypothesis H19 : (∀X ∈ int, ((V3 X) = (U3 (G3 X) H3)))

L78

Hypothesis H20 : (∀X ∈ int, ((I2 X) = ((V3 X) + - 1)))

L79

Hypothesis H21 : (J2 = (2 + (2 + 2)))

L80

Hypothesis H22 : (∀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)))))))

L81

Hypothesis H23 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V2 X Y Z) = (if (X <= 0) then Z else (G2 (U2 (X + - 1) Y Z) (V2 (X + - 1) Y Z)))))))

L82

Hypothesis H24 : (∀X ∈ int, ((W2 X) = (U2 (H2 X) (I2 X) J2)))

L83

Hypothesis H25 : (∀X ∈ int, ((FAST X) = (W2 X)))

L84

Theorem. (A16175)

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

In Proofgold the corresponding term root is 622329... and proposition id is a65fc9...

Proof:
Proof not loaded.