Beginning of Section A20527

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 F0 : set → set

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

Variable G0 : set → set

Hypothesis HG0 : ∀x0 ∈ int, G0 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

Hypothesis HH1 : H1 ∈ 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 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 F2 : set → set

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

Variable G2 : set

Hypothesis HG2 : G2 ∈ int

Variable F3 : set → set

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

Variable G3 : set → set

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

Variable H3 : set

Hypothesis HH3 : H3 ∈ int

Variable U3 : set → set → set

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

Variable V3 : set → set

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

Variable H2 : set → set

Hypothesis HH2 : ∀x0 ∈ int, H2 x0 ∈ 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, ((F0 X) = (X + X)))

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

Hypothesis H3 : (∀X ∈ int, ((F1 X) = (2 * (2 + X))))

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

Hypothesis H5 : (H1 = 2)

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

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

Hypothesis H8 : (∀X ∈ int, ((H0 X) = ((V1 X) + - 2)))

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

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

Hypothesis H11 : (∀X ∈ int, ((SMALL X) = ((V0 X) + 1)))

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

Hypothesis H13 : (G2 = 1)

Hypothesis H14 : (∀X ∈ int, ((F3 X) = (X + X)))

Hypothesis H15 : (∀X ∈ int, ((G3 X) = X))

Hypothesis H16 : (H3 = 1)

Hypothesis H17 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y))))))

Hypothesis H18 : (∀X ∈ int, ((V3 X) = (U3 (G3 X) H3)))

Hypothesis H19 : (∀X ∈ int, ((H2 X) = (V3 X)))

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

Hypothesis H21 : (∀X ∈ int, ((V2 X) = (U2 G2 (H2 X))))

Hypothesis H22 : (∀X ∈ int, ((FAST X) = (1 + (2 * ((1 + 2) * (V2 X))))))

Theorem. (A20527)

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

In Proofgold the corresponding term root is 8d79db... and proposition id is e5f1a7...

Proof:

The rest of the proof is missing.

End of Section A20527