Beginning of Section A34673

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 : set → set

Hypothesis HF1 : ∀x0 ∈ int, F1 x0 ∈ int

Variable G1 : set → set → set

Hypothesis HG1 : ∀x0 ∈ int, ∀x1 ∈ int, G1 x0 x1 ∈ int

Variable H1 : set → set

Hypothesis HH1 : ∀x0 ∈ int, H1 x0 ∈ int

Variable U1 : set → set → set

Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, U1 x0 x1 ∈ int

Variable V1 : set → set → set

Hypothesis HV1 : ∀x0 ∈ int, ∀x1 ∈ int, V1 x0 x1 ∈ int

Variable F0 : set → set → set

Hypothesis HF0 : ∀x0 ∈ int, ∀x1 ∈ int, F0 x0 x1 ∈ int

Variable G0 : set → set → set

Hypothesis HG0 : ∀x0 ∈ int, ∀x1 ∈ int, G0 x0 x1 ∈ int

Variable H0 : set

Hypothesis HH0 : H0 ∈ int

Variable I0 : set

Hypothesis HI0 : I0 ∈ int

Variable J0 : set → set

Hypothesis HJ0 : ∀x0 ∈ int, J0 x0 ∈ int

Variable U0 : set → set → set → set

Hypothesis HU0 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U0 x0 x1 x2 ∈ int

Variable V0 : set → set → set → set

Hypothesis HV0 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V0 x0 x1 x2 ∈ int

Variable W0 : set → set

Hypothesis HW0 : ∀x0 ∈ int, W0 x0 ∈ int

Variable F2 : set → set

Hypothesis HF2 : ∀x0 ∈ int, F2 x0 ∈ int

Variable G2 : set → set

Hypothesis HG2 : ∀x0 ∈ int, G2 x0 ∈ int

Variable H2 : set

Hypothesis HH2 : H2 ∈ int

Variable U2 : set → set → set

Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, U2 x0 x1 ∈ int

Variable V2 : set → set

Hypothesis HV2 : ∀x0 ∈ int, V2 x0 ∈ int

Variable F3 : set → set

Hypothesis HF3 : ∀x0 ∈ int, F3 x0 ∈ int

Variable G3 : set → set

Hypothesis HG3 : ∀x0 ∈ int, G3 x0 ∈ int

Variable H3 : set

Hypothesis HH3 : H3 ∈ int

Variable U3 : set → set → set

Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, U3 x0 x1 ∈ int

Variable V3 : set → set

Hypothesis HV3 : ∀x0 ∈ int, V3 x0 ∈ int

Variable SMALL : set → set

Hypothesis HSMALL : ∀x0 ∈ int, SMALL x0 ∈ int

Variable F4 : set → set

Hypothesis HF4 : ∀x0 ∈ int, F4 x0 ∈ int

Variable G4 : set → set

Hypothesis HG4 : ∀x0 ∈ int, G4 x0 ∈ int

Variable H4 : set

Hypothesis HH4 : H4 ∈ int

Variable U4 : set → set → set

Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, U4 x0 x1 ∈ int

Variable V4 : set → set

Hypothesis HV4 : ∀x0 ∈ int, V4 x0 ∈ int

Variable F5 : set → set

Hypothesis HF5 : ∀x0 ∈ int, F5 x0 ∈ int

Variable G5 : set → set

Hypothesis HG5 : ∀x0 ∈ int, G5 x0 ∈ int

Variable H5 : set

Hypothesis HH5 : H5 ∈ int

Variable U5 : set → set → set

Hypothesis HU5 : ∀x0 ∈ int, ∀x1 ∈ int, U5 x0 x1 ∈ int

Variable V5 : set → set

Hypothesis HV5 : ∀x0 ∈ int, V5 x0 ∈ int

Variable F6 : set → set

Hypothesis HF6 : ∀x0 ∈ int, F6 x0 ∈ int

Variable G6 : set → set

Hypothesis HG6 : ∀x0 ∈ int, G6 x0 ∈ int

Variable H6 : set

Hypothesis HH6 : H6 ∈ int

Variable U6 : set → set → set

Hypothesis HU6 : ∀x0 ∈ int, ∀x1 ∈ int, U6 x0 x1 ∈ int

Variable V6 : set → set

Hypothesis HV6 : ∀x0 ∈ int, V6 x0 ∈ int

Variable F7 : set → set → set

Hypothesis HF7 : ∀x0 ∈ int, ∀x1 ∈ int, F7 x0 x1 ∈ int

Variable G7 : set → set → set

Hypothesis HG7 : ∀x0 ∈ int, ∀x1 ∈ int, G7 x0 x1 ∈ int

Variable H7 : set → set

Hypothesis HH7 : ∀x0 ∈ int, H7 x0 ∈ int

Variable I7 : set

Hypothesis HI7 : I7 ∈ int

Variable J7 : set

Hypothesis HJ7 : J7 ∈ int

Variable U7 : set → set → set → set

Hypothesis HU7 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U7 x0 x1 x2 ∈ int

Variable V7 : set → set → set → set

Hypothesis HV7 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V7 x0 x1 x2 ∈ int

Variable W7 : set → set

Hypothesis HW7 : ∀x0 ∈ int, W7 x0 ∈ int

Variable FAST : set → set

Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int

Hypothesis H1 : (∀X ∈ int, ((F1 X) = (X + X)))

Hypothesis H2 : (∀X ∈ int, (∀Y ∈ int, ((G1 X Y) = Y)))

Hypothesis H3 : (∀X ∈ int, ((H1 X) = X))

Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))

Hypothesis H5 : (∀X ∈ int, (∀Y ∈ int, ((V1 X Y) = (U1 (G1 X Y) (H1 X)))))

Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = (1 + (V1 X Y)))))

Hypothesis H7 : (∀X ∈ int, (∀Y ∈ int, ((G0 X Y) = Y)))

Hypothesis H8 : (H0 = (1 + 2))

Hypothesis H9 : (I0 = 1)

Hypothesis H10 : (∀X ∈ int, ((J0 X) = X))

Hypothesis H11 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U0 X Y Z) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y Z) (V0 (X + - 1) Y Z)))))))

Hypothesis H12 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V0 X Y Z) = (if (X <= 0) then Z else (G0 (U0 (X + - 1) Y Z) (V0 (X + - 1) Y Z)))))))

Hypothesis H13 : (∀X ∈ int, ((W0 X) = (U0 H0 I0 (J0 X))))

Hypothesis H14 : (∀X ∈ int, ((F2 X) = ((2 * (X + X)) + X)))

Hypothesis H15 : (∀X ∈ int, ((G2 X) = X))

Hypothesis H16 : (H2 = 1)

Hypothesis H17 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y))))))

Hypothesis H18 : (∀X ∈ int, ((V2 X) = (U2 (G2 X) H2)))

Hypothesis H19 : (∀X ∈ int, ((F3 X) = ((X + X) + X)))

Hypothesis H20 : (∀X ∈ int, ((G3 X) = X))

Hypothesis H21 : (H3 = 1)

Hypothesis H22 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y))))))

Hypothesis H23 : (∀X ∈ int, ((V3 X) = (U3 (G3 X) H3)))

Hypothesis H24 : (∀X ∈ int, ((SMALL X) = (((W0 X) * (1 + (V2 X))) * (1 + (V3 X)))))

Hypothesis H25 : (∀X ∈ int, ((F4 X) = ((2 * (X + X)) + X)))

Hypothesis H26 : (∀X ∈ int, ((G4 X) = X))

Hypothesis H27 : (H4 = 1)

Hypothesis H28 : (∀X ∈ int, (∀Y ∈ int, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y))))))

Hypothesis H29 : (∀X ∈ int, ((V4 X) = (U4 (G4 X) H4)))

Hypothesis H30 : (∀X ∈ int, ((F5 X) = (X + X)))

Hypothesis H31 : (∀X ∈ int, ((G5 X) = X))

Hypothesis H32 : (H5 = 1)

Hypothesis H33 : (∀X ∈ int, (∀Y ∈ int, ((U5 X Y) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y))))))

Hypothesis H34 : (∀X ∈ int, ((V5 X) = (U5 (G5 X) H5)))

Hypothesis H35 : (∀X ∈ int, ((F6 X) = ((X + X) + X)))

Hypothesis H36 : (∀X ∈ int, ((G6 X) = X))

Hypothesis H37 : (H6 = 1)

Hypothesis H38 : (∀X ∈ int, (∀Y ∈ int, ((U6 X Y) = (if (X <= 0) then Y else (F6 (U6 (X + - 1) Y))))))

Hypothesis H39 : (∀X ∈ int, ((V6 X) = (U6 (G6 X) H6)))

Hypothesis H40 : (∀X ∈ int, (∀Y ∈ int, ((F7 X Y) = (X * Y))))

Hypothesis H41 : (∀X ∈ int, (∀Y ∈ int, ((G7 X Y) = Y)))

Hypothesis H42 : (∀X ∈ int, ((H7 X) = X))

Hypothesis H43 : (I7 = 1)

Hypothesis H44 : (J7 = (2 + 2))

Hypothesis H45 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U7 X Y Z) = (if (X <= 0) then Y else (F7 (U7 (X + - 1) Y Z) (V7 (X + - 1) Y Z)))))))

Hypothesis H46 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V7 X Y Z) = (if (X <= 0) then Z else (G7 (U7 (X + - 1) Y Z) (V7 (X + - 1) Y Z)))))))

Hypothesis H47 : (∀X ∈ int, ((W7 X) = (U7 (H7 X) I7 J7)))

Hypothesis H48 : (∀X ∈ int, ((FAST X) = ((((1 + (V4 X)) * (1 + (V5 X))) * (1 + (V6 X))) * (1 + (W7 X)))))

Theorem. (A34673)

(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))

In Proofgold the corresponding term root is 00a873... and proposition id is 2b20f5...

Proof:

The rest of the proof is missing.

End of Section A34673