Beginning of Section A350008
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 F1 : setsetset
Hypothesis HF1 : ∀x0int, ∀x1int, F1 x0 x1 int
Variable G1 : setsetset
Hypothesis HG1 : ∀x0int, ∀x1int, G1 x0 x1 int
Variable H1 : setsetset
Hypothesis HH1 : ∀x0int, ∀x1int, H1 x0 x1 int
Variable I1 : set
Hypothesis HI1 : I1 int
Variable J1 : setsetset
Hypothesis HJ1 : ∀x0int, ∀x1int, J1 x0 x1 int
Variable U1 : setsetsetset
Hypothesis HU1 : ∀x0int, ∀x1int, ∀x2int, U1 x0 x1 x2 int
Variable V1 : setsetsetset
Hypothesis HV1 : ∀x0int, ∀x1int, ∀x2int, V1 x0 x1 x2 int
Variable W1 : setsetset
Hypothesis HW1 : ∀x0int, ∀x1int, W1 x0 x1 int
Variable F0 : setsetset
Hypothesis HF0 : ∀x0int, ∀x1int, F0 x0 x1 int
Variable G0 : setset
Hypothesis HG0 : ∀x0int, G0 x0 int
Variable H0 : set
Hypothesis HH0 : H0 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 F3 : setset
Hypothesis HF3 : ∀x0int, F3 x0 int
Variable G3 : set
Hypothesis HG3 : G3 int
Variable F4 : setsetset
Hypothesis HF4 : ∀x0int, ∀x1int, F4 x0 x1 int
Variable G4 : setsetset
Hypothesis HG4 : ∀x0int, ∀x1int, G4 x0 x1 int
Variable H4 : setsetset
Hypothesis HH4 : ∀x0int, ∀x1int, H4 x0 x1 int
Variable I4 : setsetset
Hypothesis HI4 : ∀x0int, ∀x1int, I4 x0 x1 int
Variable J4 : setsetset
Hypothesis HJ4 : ∀x0int, ∀x1int, J4 x0 x1 int
Variable U4 : setsetsetset
Hypothesis HU4 : ∀x0int, ∀x1int, ∀x2int, U4 x0 x1 x2 int
Variable V4 : setsetsetset
Hypothesis HV4 : ∀x0int, ∀x1int, ∀x2int, V4 x0 x1 x2 int
Variable W4 : setsetset
Hypothesis HW4 : ∀x0int, ∀x1int, W4 x0 x1 int
Variable H3 : setsetset
Hypothesis HH3 : ∀x0int, ∀x1int, H3 x0 x1 int
Variable U3 : setsetset
Hypothesis HU3 : ∀x0int, ∀x1int, U3 x0 x1 int
Variable V3 : setsetset
Hypothesis HV3 : ∀x0int, ∀x1int, V3 x0 x1 int
Variable F2 : setsetset
Hypothesis HF2 : ∀x0int, ∀x1int, F2 x0 x1 int
Variable G2 : setset
Hypothesis HG2 : ∀x0int, G2 x0 int
Variable H2 : set
Hypothesis HH2 : H2 int
Variable U2 : setsetset
Hypothesis HU2 : ∀x0int, ∀x1int, U2 x0 x1 int
Variable V2 : setset
Hypothesis HV2 : ∀x0int, V2 x0 int
Variable FAST : setset
Hypothesis HFAST : ∀x0int, FAST x0 int
Hypothesis H1 : (∀Xint, (∀Yint, ((F1 X Y) = ((X * Y) * Y))))
Hypothesis H2 : (∀Xint, (∀Yint, ((G1 X Y) = Y)))
Hypothesis H3 : (∀Xint, (∀Yint, ((H1 X Y) = Y)))
Hypothesis H4 : (I1 = 1)
Hypothesis H5 : (∀Xint, (∀Yint, ((J1 X Y) = Y)))
Hypothesis H6 : (∀Xint, (∀Yint, (∀Zint, ((U1 X Y Z) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y Z) (V1 (X + - 1) Y Z)))))))
Hypothesis H7 : (∀Xint, (∀Yint, (∀Zint, ((V1 X Y Z) = (if (X <= 0) then Z else (G1 (U1 (X + - 1) Y Z) (V1 (X + - 1) Y Z)))))))
Hypothesis H8 : (∀Xint, (∀Yint, ((W1 X Y) = (U1 (H1 X Y) I1 (J1 X Y)))))
Hypothesis H9 : (∀Xint, (∀Yint, ((F0 X Y) = (X + (W1 X Y)))))
Hypothesis H10 : (∀Xint, ((G0 X) = X))
Hypothesis H11 : (H0 = 1)
Hypothesis H12 : (∀Xint, (∀Yint, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y) X)))))
Hypothesis H13 : (∀Xint, ((V0 X) = (U0 (G0 X) H0)))
Hypothesis H14 : (∀Xint, ((SMALL X) = (V0 X)))
Hypothesis H15 : (∀Xint, ((F3 X) = (X * X)))
Hypothesis H16 : (G3 = 1)
Hypothesis H17 : (∀Xint, (∀Yint, ((F4 X Y) = (X * Y))))
Hypothesis H18 : (∀Xint, (∀Yint, ((G4 X Y) = Y)))
Hypothesis H19 : (∀Xint, (∀Yint, ((H4 X Y) = (Y + - 1))))
Hypothesis H20 : (∀Xint, (∀Yint, ((I4 X Y) = Y)))
Hypothesis H21 : (∀Xint, (∀Yint, ((J4 X Y) = Y)))
Hypothesis H22 : (∀Xint, (∀Yint, (∀Zint, ((U4 X Y Z) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y Z) (V4 (X + - 1) Y Z)))))))
Hypothesis H23 : (∀Xint, (∀Yint, (∀Zint, ((V4 X Y Z) = (if (X <= 0) then Z else (G4 (U4 (X + - 1) Y Z) (V4 (X + - 1) Y Z)))))))
Hypothesis H24 : (∀Xint, (∀Yint, ((W4 X Y) = (U4 (H4 X Y) (I4 X Y) (J4 X Y)))))
Hypothesis H25 : (∀Xint, (∀Yint, ((H3 X Y) = (W4 X Y))))
Hypothesis H26 : (∀Xint, (∀Yint, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y))))))
Hypothesis H27 : (∀Xint, (∀Yint, ((V3 X Y) = (U3 G3 (H3 X Y)))))
Hypothesis H28 : (∀Xint, (∀Yint, ((F2 X Y) = ((V3 X Y) + X))))
Hypothesis H29 : (∀Xint, ((G2 X) = X))
Hypothesis H30 : (H2 = 1)
Hypothesis H31 : (∀Xint, (∀Yint, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y) X)))))
Hypothesis H32 : (∀Xint, ((V2 X) = (U2 (G2 X) H2)))
Hypothesis H33 : (∀Xint, ((FAST X) = (V2 X)))
Theorem. (A350008)
(∀Nint, ((0 <= N)((SMALL N) = (FAST N))))
Proof:
The rest of the proof is missing.

End of Section A350008