Beginning of Section A61705

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 → 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 → set

Hypothesis HH0 : ∀x0 ∈ int, H0 x0 ∈ int

Variable I0 : set

Hypothesis HI0 : I0 ∈ int

Variable J0 : set → set

Hypothesis HJ0 : ∀x0 ∈ int, J0 x0 ∈ int

Variable U0 : set → set → set → set

Hypothesis HU0 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U0 x0 x1 x2 ∈ int

Variable V0 : set → set → set → set

Hypothesis HV0 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V0 x0 x1 x2 ∈ int

Variable W0 : set → set

Hypothesis HW0 : ∀x0 ∈ int, W0 x0 ∈ int

Variable SMALL : set → set

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

Variable F1 : set → set → set

Hypothesis HF1 : ∀x0 ∈ int, ∀x1 ∈ int, F1 x0 x1 ∈ int

Variable G1 : set → set

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

Variable H1 : set → set

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

Variable I1 : set → set

Hypothesis HI1 : ∀x0 ∈ int, I1 x0 ∈ int

Variable J1 : set

Hypothesis HJ1 : J1 ∈ int

Variable U1 : set → set → set → set

Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U1 x0 x1 x2 ∈ int

Variable V1 : set → set → set → set

Hypothesis HV1 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V1 x0 x1 x2 ∈ int

Variable W1 : set → set

Hypothesis HW1 : ∀x0 ∈ int, W1 x0 ∈ int

Variable FAST : set → set

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

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

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

Hypothesis H3 : (∀X ∈ int, ((H0 X) = (1 + X)))

Hypothesis H4 : (I0 = 2)

Hypothesis H5 : (∀X ∈ int, ((J0 X) = X))

Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U0 X Y Z) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y Z) (V0 (X + - 1) Y Z)))))))

Hypothesis H7 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V0 X Y Z) = (if (X <= 0) then Z else (G0 (U0 (X + - 1) Y Z)))))))

Hypothesis H8 : (∀X ∈ int, ((W0 X) = (U0 (H0 X) I0 (J0 X))))

Hypothesis H9 : (∀X ∈ int, ((SMALL X) = (W0 X)))

Hypothesis H10 : (∀X ∈ int, (∀Y ∈ int, ((F1 X Y) = (X + Y))))

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

Hypothesis H12 : (∀X ∈ int, ((H1 X) = X))

Hypothesis H13 : (∀X ∈ int, ((I1 X) = (2 + X)))

Hypothesis H14 : (J1 = 2)

Hypothesis H15 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U1 X Y Z) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y Z) (V1 (X + - 1) Y Z)))))))

Hypothesis H16 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V1 X Y Z) = (if (X <= 0) then Z else (G1 (U1 (X + - 1) Y Z)))))))

Hypothesis H17 : (∀X ∈ int, ((W1 X) = (U1 (H1 X) (I1 X) J1)))

Hypothesis H18 : (∀X ∈ int, ((FAST X) = (W1 X)))

Theorem. (A61705)

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

In Proofgold the corresponding term root is 925880... and proposition id is 22bc79...

Proof:

The rest of the proof is missing.

End of Section A61705