Beginning of Section A5571

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 TMKmg78VG8AeLyz6anive2zkYNx2qVfBeGC ***)

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 → set

L14

Hypothesis HH1 : ∀x0 ∈ int, H1 x0 ∈ 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 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

L34

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

L35

Variable H2 : set

L36

Hypothesis HH2 : H2 ∈ int

L37

Variable U2 : set → set → set

L38

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

L39

Variable V2 : set → set

L40

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

L41

Variable FAST : set → set

L42

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

L43

Hypothesis H1 : (∀X ∈ int, (∀Y ∈ int, ((F1 X Y) = (2 + (X + Y)))))

L44

Hypothesis H2 : (∀X ∈ int, (∀Y ∈ int, ((G1 X Y) = (2 * (Y + Y)))))

L45

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

L46

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

L47

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

L48

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

L49

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

L50

Hypothesis H8 : (H0 = 1)

L51

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

L52

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

L53

Hypothesis H11 : (∀X ∈ int, ((SMALL X) = ((((V0 X) + X) * 2) * 2)))

L54

Hypothesis H12 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = (((((1 + Y) * (2 + 2)) + 1) * (2 * Y)) + X))))

L55

Hypothesis H13 : (∀X ∈ int, ((G2 X) = (X + 1)))

L56

Hypothesis H14 : (H2 = 1)

L57

Hypothesis H15 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y) X)))))

L58

Hypothesis H16 : (∀X ∈ int, ((V2 X) = (U2 (G2 X) H2)))

L59

Hypothesis H17 : (∀X ∈ int, ((FAST X) = ((((V2 X) + X) * 2) * 2)))

L60

Theorem. (A5571)

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

In Proofgold the corresponding term root is 8c1f14... and proposition id is 4e9651...

Proof:
Proof not loaded.