Beginning of Section A160912
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 : setsetset
Hypothesis HF0 : ∀x0int, ∀x1int, F0 x0 x1 int
Variable G0 : set
Hypothesis HG0 : G0 int
Variable H0 : setset
Hypothesis HH0 : ∀x0int, H0 x0 int
Variable I0 : set
Hypothesis HI0 : I0 int
Variable J0 : setset
Hypothesis HJ0 : ∀x0int, J0 x0 int
Variable U0 : setsetsetset
Hypothesis HU0 : ∀x0int, ∀x1int, ∀x2int, U0 x0 x1 x2 int
Variable V0 : setsetsetset
Hypothesis HV0 : ∀x0int, ∀x1int, ∀x2int, V0 x0 x1 x2 int
Variable W0 : setset
Hypothesis HW0 : ∀x0int, W0 x0 int
Variable SMALL : setset
Hypothesis HSMALL : ∀x0int, SMALL x0 int
Variable FAST : setset
Hypothesis HFAST : ∀x0int, FAST x0 int
Hypothesis H1 : (∀Xint, (∀Yint, ((F0 X Y) = ((X + Y) + Y))))
Hypothesis H2 : (G0 = 2)
Hypothesis H3 : (∀Xint, ((H0 X) = (X + X)))
Hypothesis H4 : (I0 = 1)
Hypothesis H5 : (∀Xint, ((J0 X) = X))
Hypothesis H6 : (∀Xint, (∀Yint, (∀Zint, ((U0 X Y Z) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y Z) (V0 (X + - 1) Y Z)))))))
Hypothesis H7 : (∀Xint, (∀Yint, (∀Zint, ((V0 X Y Z) = (if (X <= 0) then Z else G0)))))
Hypothesis H8 : (∀Xint, ((W0 X) = (U0 (H0 X) I0 (J0 X))))
Hypothesis H9 : (∀Xint, ((SMALL X) = (W0 X)))
Hypothesis H10 : (∀Xint, ((FAST X) = (1 + (2 * (((2 * (X + X)) + - (if (X <= 0) then 0 else 2)) + X)))))
Theorem. (A160912)
(∀Nint, ((0 <= N)((SMALL N) = (FAST N))))
Proof:
The rest of the proof is missing.

End of Section A160912