Beginning of Section A45756

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.

Variable F1 : set → set

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

Variable G1 : set

Hypothesis HG1 : G1 ∈ int

Variable H1 : set → set → set

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

Variable U1 : set → set → set

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

Variable V1 : set → set → set

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

Variable F0 : set → set → set

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

Variable G0 : set → set

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

Variable H0 : set

Hypothesis HH0 : H0 ∈ int

Variable U0 : set → set → set

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

Variable V0 : set → set

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

Variable SMALL : set → set

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

Variable F2 : set → set → set

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

Variable G2 : set → set

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

Variable H2 : set

Hypothesis HH2 : H2 ∈ int

Variable U2 : set → set → set

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

Variable V2 : set → set

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

Variable FAST : set → set

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

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

Hypothesis H2 : (G1 = 2)

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

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

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

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

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

Hypothesis H8 : (H0 = 1)

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

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

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

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

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

Hypothesis H14 : (H2 = 1)

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

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

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

Theorem. (A45756)

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

In Proofgold the corresponding term root is 18cac3... and proposition id is 49b1a5...

Proof:

The rest of the proof is missing.

End of Section A45756