Beginning of Section A61788
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 : 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 F4 : setset
Hypothesis HF4 : ∀x0int, F4 x0 int
Variable G4 : set
Hypothesis HG4 : G4 int
Variable F5 : setset
Hypothesis HF5 : ∀x0int, F5 x0 int
Variable G5 : setset
Hypothesis HG5 : ∀x0int, G5 x0 int
Variable H5 : set
Hypothesis HH5 : H5 int
Variable U5 : setsetset
Hypothesis HU5 : ∀x0int, ∀x1int, U5 x0 x1 int
Variable V5 : setset
Hypothesis HV5 : ∀x0int, V5 x0 int
Variable H4 : setset
Hypothesis HH4 : ∀x0int, H4 x0 int
Variable U4 : setsetset
Hypothesis HU4 : ∀x0int, ∀x1int, U4 x0 x1 int
Variable V4 : setset
Hypothesis HV4 : ∀x0int, V4 x0 int
Variable F6 : setsetset
Hypothesis HF6 : ∀x0int, ∀x1int, F6 x0 x1 int
Variable G6 : setsetset
Hypothesis HG6 : ∀x0int, ∀x1int, G6 x0 x1 int
Variable H6 : setset
Hypothesis HH6 : ∀x0int, H6 x0 int
Variable I6 : setset
Hypothesis HI6 : ∀x0int, I6 x0 int
Variable J6 : setset
Hypothesis HJ6 : ∀x0int, J6 x0 int
Variable U6 : setsetsetset
Hypothesis HU6 : ∀x0int, ∀x1int, ∀x2int, U6 x0 x1 x2 int
Variable V6 : setsetsetset
Hypothesis HV6 : ∀x0int, ∀x1int, ∀x2int, V6 x0 x1 x2 int
Variable W6 : setset
Hypothesis HW6 : ∀x0int, W6 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 : setset
Hypothesis HI7 : ∀x0int, I7 x0 int
Variable J7 : setset
Hypothesis HJ7 : ∀x0int, J7 x0 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 F3 : setset
Hypothesis HF3 : ∀x0int, F3 x0 int
Variable G3 : set
Hypothesis HG3 : G3 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))))
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) = ((W1 X Y) + X))))
Hypothesis H10 : (∀Xint, (∀Yint, ((G0 X Y) = (2 + Y))))
Hypothesis H11 : (∀Xint, ((H0 X) = (X + 1)))
Hypothesis H12 : (I0 = 0)
Hypothesis H13 : (J0 = 2)
Hypothesis H14 : (∀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 H15 : (∀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 H16 : (∀Xint, ((W0 X) = (U0 (H0 X) I0 J0)))
Hypothesis H17 : (∀Xint, ((SMALL X) = (W0 X)))
Hypothesis H18 : (∀Xint, ((F4 X) = (X * X)))
Hypothesis H19 : (G4 = 1)
Hypothesis H20 : (∀Xint, ((F5 X) = (X + X)))
Hypothesis H21 : (∀Xint, ((G5 X) = X))
Hypothesis H22 : (H5 = 1)
Hypothesis H23 : (∀Xint, (∀Yint, ((U5 X Y) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y))))))
Hypothesis H24 : (∀Xint, ((V5 X) = (U5 (G5 X) H5)))
Hypothesis H25 : (∀Xint, ((H4 X) = (V5 X)))
Hypothesis H26 : (∀Xint, (∀Yint, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y))))))
Hypothesis H27 : (∀Xint, ((V4 X) = (U4 G4 (H4 X))))
Hypothesis H28 : (∀Xint, (∀Yint, ((F6 X Y) = (X * Y))))
Hypothesis H29 : (∀Xint, (∀Yint, ((G6 X Y) = Y)))
Hypothesis H30 : (∀Xint, ((H6 X) = X))
Hypothesis H31 : (∀Xint, ((I6 X) = (1 + X)))
Hypothesis H32 : (∀Xint, ((J6 X) = (1 + X)))
Hypothesis H33 : (∀Xint, (∀Yint, (∀Zint, ((U6 X Y Z) = (if (X <= 0) then Y else (F6 (U6 (X + - 1) Y Z) (V6 (X + - 1) Y Z)))))))
Hypothesis H34 : (∀Xint, (∀Yint, (∀Zint, ((V6 X Y Z) = (if (X <= 0) then Z else (G6 (U6 (X + - 1) Y Z) (V6 (X + - 1) Y Z)))))))
Hypothesis H35 : (∀Xint, ((W6 X) = (U6 (H6 X) (I6 X) (J6 X))))
Hypothesis H36 : (∀Xint, (∀Yint, ((F7 X Y) = (X * Y))))
Hypothesis H37 : (∀Xint, (∀Yint, ((G7 X Y) = Y)))
Hypothesis H38 : (∀Xint, ((H7 X) = X))
Hypothesis H39 : (∀Xint, ((I7 X) = (1 + X)))
Hypothesis H40 : (∀Xint, ((J7 X) = (1 + X)))
Hypothesis H41 : (∀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 H42 : (∀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 H43 : (∀Xint, ((W7 X) = (U7 (H7 X) (I7 X) (J7 X))))
Hypothesis H44 : (∀Xint, ((F3 X) = (((2 * (V4 X)) * (W6 X)) * (W7 X))))
Hypothesis H45 : (G3 = 1)
Hypothesis H46 : (∀Xint, (∀Yint, ((H3 X Y) = Y)))
Hypothesis H47 : (∀Xint, (∀Yint, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y))))))
Hypothesis H48 : (∀Xint, (∀Yint, ((V3 X Y) = (U3 G3 (H3 X Y)))))
Hypothesis H49 : (∀Xint, (∀Yint, ((F2 X Y) = ((V3 X Y) + X))))
Hypothesis H50 : (∀Xint, ((G2 X) = X))
Hypothesis H51 : (H2 = 1)
Hypothesis H52 : (∀Xint, (∀Yint, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y) X)))))
Hypothesis H53 : (∀Xint, ((V2 X) = (U2 (G2 X) H2)))
Hypothesis H54 : (∀Xint, ((FAST X) = (((V2 X) * 2) + 2)))
Theorem. (A61788)
(∀Nint, ((0 <= N)((SMALL N) = (FAST N))))
Proof:
The rest of the proof is missing.

End of Section A61788