Beginning of Section A2999
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 : setset
Hypothesis HF0 : ∀x0int, F0 x0 int
Variable G0 : setset
Hypothesis HG0 : ∀x0int, G0 x0 int
Variable H0 : setset
Hypothesis HH0 : ∀x0int, H0 x0 int
Variable U0 : setsetset
Hypothesis HU0 : ∀x0int, ∀x1int, U0 x0 x1 int
Variable V0 : setset
Hypothesis HV0 : ∀x0int, V0 x0 int
Variable SMALL : setset
Hypothesis HSMALL : ∀x0int, SMALL x0 int
Variable F1 : setset
Hypothesis HF1 : ∀x0int, F1 x0 int
Variable G1 : setset
Hypothesis HG1 : ∀x0int, G1 x0 int
Variable H1 : setset
Hypothesis HH1 : ∀x0int, H1 x0 int
Variable U1 : setsetset
Hypothesis HU1 : ∀x0int, ∀x1int, U1 x0 x1 int
Variable V1 : setset
Hypothesis HV1 : ∀x0int, V1 x0 int
Variable FAST : setset
Hypothesis HFAST : ∀x0int, FAST x0 int
Hypothesis H1 : (∀Xint, ((F0 X) = ((X + - 1) + X)))
Hypothesis H2 : (∀Xint, ((G0 X) = (X + - 2)))
Hypothesis H3 : (∀Xint, ((H0 X) = X))
Hypothesis H4 : (∀Xint, (∀Yint, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y))))))
Hypothesis H5 : (∀Xint, ((V0 X) = (U0 (G0 X) (H0 X))))
Hypothesis H6 : (∀Xint, ((SMALL X) = (((V0 X) * X) + (if (X <= 0) then 1 else X))))
Hypothesis H7 : (∀Xint, ((F1 X) = (X + X)))
Hypothesis H8 : (∀Xint, ((G1 X) = (X + - 2)))
Hypothesis H9 : (∀Xint, ((H1 X) = (X + - 1)))
Hypothesis H10 : (∀Xint, (∀Yint, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))
Hypothesis H11 : (∀Xint, ((V1 X) = (U1 (G1 X) (H1 X))))
Hypothesis H12 : (∀Xint, ((FAST X) = ((2 + (V1 X)) * (if (X <= 0) then 1 else X))))
Theorem. (A2999)
(∀Nint, ((0 <= N)((SMALL N) = (FAST N))))
Proof:
The rest of the proof is missing.

End of Section A2999