Beginning of Section A20594
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 F2 : setset
Hypothesis HF2 : ∀x0int, F2 x0 int
Variable G2 : setset
Hypothesis HG2 : ∀x0int, G2 x0 int
Variable F3 : setset
Hypothesis HF3 : ∀x0int, F3 x0 int
Variable G3 : setset
Hypothesis HG3 : ∀x0int, G3 x0 int
Variable H3 : set
Hypothesis HH3 : H3 int
Variable U3 : setsetset
Hypothesis HU3 : ∀x0int, ∀x1int, U3 x0 x1 int
Variable V3 : setset
Hypothesis HV3 : ∀x0int, V3 x0 int
Variable H2 : setset
Hypothesis HH2 : ∀x0int, H2 x0 int
Variable U2 : setsetset
Hypothesis HU2 : ∀x0int, ∀x1int, U2 x0 x1 int
Variable V2 : setset
Hypothesis HV2 : ∀x0int, V2 x0 int
Variable F1 : setset
Hypothesis HF1 : ∀x0int, F1 x0 int
Variable G1 : set
Hypothesis HG1 : G1 int
Variable H1 : setsetset
Hypothesis HH1 : ∀x0int, ∀x1int, H1 x0 x1 int
Variable U1 : setsetset
Hypothesis HU1 : ∀x0int, ∀x1int, U1 x0 x1 int
Variable V1 : setsetset
Hypothesis HV1 : ∀x0int, ∀x1int, V1 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 F6 : setset
Hypothesis HF6 : ∀x0int, F6 x0 int
Variable G6 : setset
Hypothesis HG6 : ∀x0int, G6 x0 int
Variable H6 : set
Hypothesis HH6 : H6 int
Variable U6 : setsetset
Hypothesis HU6 : ∀x0int, ∀x1int, U6 x0 x1 int
Variable V6 : setset
Hypothesis HV6 : ∀x0int, V6 x0 int
Variable F7 : setsetset
Hypothesis HF7 : ∀x0int, ∀x1int, F7 x0 x1 int
Variable G7 : setsetset
Hypothesis HG7 : ∀x0int, ∀x1int, G7 x0 x1 int
Variable H7 : setset
Hypothesis HH7 : ∀x0int, H7 x0 int
Variable I7 : set
Hypothesis HI7 : I7 int
Variable J7 : set
Hypothesis HJ7 : J7 int
Variable U7 : setsetsetset
Hypothesis HU7 : ∀x0int, ∀x1int, ∀x2int, U7 x0 x1 x2 int
Variable V7 : setsetsetset
Hypothesis HV7 : ∀x0int, ∀x1int, ∀x2int, V7 x0 x1 x2 int
Variable W7 : setset
Hypothesis HW7 : ∀x0int, W7 x0 int
Variable F5 : setset
Hypothesis HF5 : ∀x0int, F5 x0 int
Variable G5 : set
Hypothesis HG5 : G5 int
Variable H5 : setsetset
Hypothesis HH5 : ∀x0int, ∀x1int, H5 x0 x1 int
Variable U5 : setsetset
Hypothesis HU5 : ∀x0int, ∀x1int, U5 x0 x1 int
Variable V5 : setsetset
Hypothesis HV5 : ∀x0int, ∀x1int, V5 x0 x1 int
Variable F4 : setsetset
Hypothesis HF4 : ∀x0int, ∀x1int, F4 x0 x1 int
Variable G4 : setset
Hypothesis HG4 : ∀x0int, G4 x0 int
Variable H4 : set
Hypothesis HH4 : H4 int
Variable U4 : setsetset
Hypothesis HU4 : ∀x0int, ∀x1int, U4 x0 x1 int
Variable V4 : setset
Hypothesis HV4 : ∀x0int, V4 x0 int
Variable F8 : setset
Hypothesis HF8 : ∀x0int, F8 x0 int
Variable G8 : setset
Hypothesis HG8 : ∀x0int, G8 x0 int
Variable H8 : set
Hypothesis HH8 : H8 int
Variable U8 : setsetset
Hypothesis HU8 : ∀x0int, ∀x1int, U8 x0 x1 int
Variable V8 : setset
Hypothesis HV8 : ∀x0int, V8 x0 int
Variable FAST : setset
Hypothesis HFAST : ∀x0int, FAST x0 int
Hypothesis H1 : (∀Xint, ((F2 X) = (2 * ((X + X) + X))))
Hypothesis H2 : (∀Xint, ((G2 X) = X))
Hypothesis H3 : (∀Xint, ((F3 X) = (X + X)))
Hypothesis H4 : (∀Xint, ((G3 X) = X))
Hypothesis H5 : (H3 = 2)
Hypothesis H6 : (∀Xint, (∀Yint, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y))))))
Hypothesis H7 : (∀Xint, ((V3 X) = (U3 (G3 X) H3)))
Hypothesis H8 : (∀Xint, ((H2 X) = ((V3 X) + - 1)))
Hypothesis H9 : (∀Xint, (∀Yint, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y))))))
Hypothesis H10 : (∀Xint, ((V2 X) = (U2 (G2 X) (H2 X))))
Hypothesis H11 : (∀Xint, ((F1 X) = (V2 X)))
Hypothesis H12 : (G1 = 1)
Hypothesis H13 : (∀Xint, (∀Yint, ((H1 X Y) = Y)))
Hypothesis H14 : (∀Xint, (∀Yint, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))
Hypothesis H15 : (∀Xint, (∀Yint, ((V1 X Y) = (U1 G1 (H1 X Y)))))
Hypothesis H16 : (∀Xint, (∀Yint, ((F0 X Y) = ((V1 X Y) + (2 * (2 * (X + X)))))))
Hypothesis H17 : (∀Xint, ((G0 X) = X))
Hypothesis H18 : (H0 = 1)
Hypothesis H19 : (∀Xint, (∀Yint, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y) X)))))
Hypothesis H20 : (∀Xint, ((V0 X) = (U0 (G0 X) H0)))
Hypothesis H21 : (∀Xint, ((SMALL X) = (V0 X)))
Hypothesis H22 : (∀Xint, ((F6 X) = (X + X)))
Hypothesis H23 : (∀Xint, ((G6 X) = X))
Hypothesis H24 : (H6 = 2)
Hypothesis H25 : (∀Xint, (∀Yint, ((U6 X Y) = (if (X <= 0) then Y else (F6 (U6 (X + - 1) Y))))))
Hypothesis H26 : (∀Xint, ((V6 X) = (U6 (G6 X) H6)))
Hypothesis H27 : (∀Xint, (∀Yint, ((F7 X Y) = (X * Y))))
Hypothesis H28 : (∀Xint, (∀Yint, ((G7 X Y) = Y)))
Hypothesis H29 : (∀Xint, ((H7 X) = X))
Hypothesis H30 : (I7 = 1)
Hypothesis H31 : (J7 = (1 + 2))
Hypothesis H32 : (∀Xint, (∀Yint, (∀Zint, ((U7 X Y Z) = (if (X <= 0) then Y else (F7 (U7 (X + - 1) Y Z) (V7 (X + - 1) Y Z)))))))
Hypothesis H33 : (∀Xint, (∀Yint, (∀Zint, ((V7 X Y Z) = (if (X <= 0) then Z else (G7 (U7 (X + - 1) Y Z) (V7 (X + - 1) Y Z)))))))
Hypothesis H34 : (∀Xint, ((W7 X) = (U7 (H7 X) I7 J7)))
Hypothesis H35 : (∀Xint, ((F5 X) = (((V6 X) + - 1) * (W7 X))))
Hypothesis H36 : (G5 = 1)
Hypothesis H37 : (∀Xint, (∀Yint, ((H5 X Y) = Y)))
Hypothesis H38 : (∀Xint, (∀Yint, ((U5 X Y) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y))))))
Hypothesis H39 : (∀Xint, (∀Yint, ((V5 X Y) = (U5 G5 (H5 X Y)))))
Hypothesis H40 : (∀Xint, (∀Yint, ((F4 X Y) = ((V5 X Y) + (2 * (X + X))))))
Hypothesis H41 : (∀Xint, ((G4 X) = X))
Hypothesis H42 : (H4 = 1)
Hypothesis H43 : (∀Xint, (∀Yint, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y) X)))))
Hypothesis H44 : (∀Xint, ((V4 X) = (U4 (G4 X) H4)))
Hypothesis H45 : (∀Xint, ((F8 X) = (X + X)))
Hypothesis H46 : (∀Xint, ((G8 X) = X))
Hypothesis H47 : (H8 = 1)
Hypothesis H48 : (∀Xint, (∀Yint, ((U8 X Y) = (if (X <= 0) then Y else (F8 (U8 (X + - 1) Y))))))
Hypothesis H49 : (∀Xint, ((V8 X) = (U8 (G8 X) H8)))
Hypothesis H50 : (∀Xint, ((FAST X) = ((V4 X) * (V8 X))))
Theorem. (A20594)
(∀Nint, ((0 <= N)((SMALL N) = (FAST N))))
Proof:
The rest of the proof is missing.

End of Section A20594