Beginning of Section A2999

Variable G0 : set → set

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

Variable H0 : set → set

Hypothesis HH0 : ∀x0 ∈ int, H0 x0 ∈ 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 F1 : set → set

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

Variable G1 : set → set

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

Variable H1 : set → set

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

Variable U1 : set → set → set

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

Variable V1 : set → set

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

Variable FAST : set → set

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

Hypothesis H1 : (∀X ∈ int, ((F0 X) = ((X + - 1) + X)))

Hypothesis H2 : (∀X ∈ int, ((G0 X) = (X + - 2)))

Hypothesis H3 : (∀X ∈ int, ((H0 X) = X))

Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y))))))

Hypothesis H5 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) (H0 X))))

Hypothesis H6 : (∀X ∈ int, ((SMALL X) = (((V0 X) * X) + (if (X <= 0) then 1 else X))))

Hypothesis H7 : (∀X ∈ int, ((F1 X) = (X + X)))

Hypothesis H8 : (∀X ∈ int, ((G1 X) = (X + - 2)))

Hypothesis H9 : (∀X ∈ int, ((H1 X) = (X + - 1)))

Hypothesis H10 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))

Hypothesis H11 : (∀X ∈ int, ((V1 X) = (U1 (G1 X) (H1 X))))

Hypothesis H12 : (∀X ∈ int, ((FAST X) = ((2 + (V1 X)) * (if (X <= 0) then 1 else X))))

Theorem. (A2999)

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

In Proofgold the corresponding term root is b6e0cc... and proposition id is a561ba...

Proof:

The rest of the proof is missing.

End of Section A2999