Beginning of Section A272263
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 : setsetset
Hypothesis HG0 : ∀x0int, ∀x1int, G0 x0 x1 int
Variable H0 : setset
Hypothesis HH0 : ∀x0int, H0 x0 int
Variable I0 : set
Hypothesis HI0 : I0 int
Variable J0 : set
Hypothesis HJ0 : J0 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 F1 : setset
Hypothesis HF1 : ∀x0int, F1 x0 int
Variable G1 : setset
Hypothesis HG1 : ∀x0int, G1 x0 int
Variable F2 : setsetset
Hypothesis HF2 : ∀x0int, ∀x1int, F2 x0 x1 int
Variable G2 : setset
Hypothesis HG2 : ∀x0int, G2 x0 int
Variable H2 : setset
Hypothesis HH2 : ∀x0int, H2 x0 int
Variable I2 : set
Hypothesis HI2 : I2 int
Variable J2 : set
Hypothesis HJ2 : J2 int
Variable U2 : setsetsetset
Hypothesis HU2 : ∀x0int, ∀x1int, ∀x2int, U2 x0 x1 x2 int
Variable V2 : setsetsetset
Hypothesis HV2 : ∀x0int, ∀x1int, ∀x2int, V2 x0 x1 x2 int
Variable W2 : setset
Hypothesis HW2 : ∀x0int, W2 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, (∀Yint, ((F0 X Y) = (Y + Y))))
Hypothesis H2 : (∀Xint, (∀Yint, ((G0 X Y) = (2 * (X + Y)))))
Hypothesis H3 : (∀Xint, ((H0 X) = X))
Hypothesis H4 : (I0 = 2)
Hypothesis H5 : (J0 = 1)
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 (U0 (X + - 1) Y Z) (V0 (X + - 1) Y Z)))))))
Hypothesis H8 : (∀Xint, ((W0 X) = (U0 (H0 X) I0 J0)))
Hypothesis H9 : (∀Xint, ((SMALL X) = ((W0 X) + - 1)))
Hypothesis H10 : (∀Xint, ((F1 X) = (X + X)))
Hypothesis H11 : (∀Xint, ((G1 X) = (X + - 1)))
Hypothesis H12 : (∀Xint, (∀Yint, ((F2 X Y) = (X + Y))))
Hypothesis H13 : (∀Xint, ((G2 X) = X))
Hypothesis H14 : (∀Xint, ((H2 X) = (X + - 1)))
Hypothesis H15 : (I2 = 1)
Hypothesis H16 : (J2 = 2)
Hypothesis H17 : (∀Xint, (∀Yint, (∀Zint, ((U2 X Y Z) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y Z) (V2 (X + - 1) Y Z)))))))
Hypothesis H18 : (∀Xint, (∀Yint, (∀Zint, ((V2 X Y Z) = (if (X <= 0) then Z else (G2 (U2 (X + - 1) Y Z)))))))
Hypothesis H19 : (∀Xint, ((W2 X) = (U2 (H2 X) I2 J2)))
Hypothesis H20 : (∀Xint, ((H1 X) = (W2 X)))
Hypothesis H21 : (∀Xint, (∀Yint, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))
Hypothesis H22 : (∀Xint, ((V1 X) = (U1 (G1 X) (H1 X))))
Hypothesis H23 : (∀Xint, ((FAST X) = ((2 * (V1 X)) + - 1)))
Theorem. (A272263)
(∀Nint, ((0 <= N)((SMALL N) = (FAST N))))
Proof:
The rest of the proof is missing.

End of Section A272263