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 : set → set → set

Hypothesis HF1 : ∀x0 ∈ int, ∀x1 ∈ int, F1 x0 x1 ∈ int

Variable G1 : set → set → set

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

Variable H1 : set → set → set

Hypothesis HH1 : ∀x0 ∈ int, ∀x1 ∈ int, H1 x0 x1 ∈ int

Variable I1 : set

Hypothesis HI1 : I1 ∈ int

Variable J1 : set → set → set

Hypothesis HJ1 : ∀x0 ∈ int, ∀x1 ∈ int, J1 x0 x1 ∈ int

Variable U1 : set → set → set → set

Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U1 x0 x1 x2 ∈ int

Variable V1 : set → set → set → set

Hypothesis HV1 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V1 x0 x1 x2 ∈ int

Variable W1 : set → set → set

Hypothesis HW1 : ∀x0 ∈ int, ∀x1 ∈ int, W1 x0 x1 ∈ int

Variable F0 : set → set → set

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

Variable G0 : set → set

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

Variable H0 : set

Hypothesis HH0 : H0 ∈ int

Variable U0 : set → set → set

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

Variable V0 : set → set

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

Variable SMALL : set → set

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

Variable F3 : set → set

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

Variable G3 : set

Hypothesis HG3 : G3 ∈ int

Variable F4 : set → set → set

Hypothesis HF4 : ∀x0 ∈ int, ∀x1 ∈ int, F4 x0 x1 ∈ int

Variable G4 : set → set → set

Hypothesis HG4 : ∀x0 ∈ int, ∀x1 ∈ int, G4 x0 x1 ∈ int

Variable H4 : set → set → set

Hypothesis HH4 : ∀x0 ∈ int, ∀x1 ∈ int, H4 x0 x1 ∈ int

Variable I4 : set → set → set

Hypothesis HI4 : ∀x0 ∈ int, ∀x1 ∈ int, I4 x0 x1 ∈ int

Variable J4 : set → set → set

Hypothesis HJ4 : ∀x0 ∈ int, ∀x1 ∈ int, J4 x0 x1 ∈ int

Variable U4 : set → set → set → set

Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U4 x0 x1 x2 ∈ int

Variable V4 : set → set → set → set

Hypothesis HV4 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V4 x0 x1 x2 ∈ int

Variable W4 : set → set → set

Hypothesis HW4 : ∀x0 ∈ int, ∀x1 ∈ int, W4 x0 x1 ∈ int

Variable H3 : set → set → set

Hypothesis HH3 : ∀x0 ∈ int, ∀x1 ∈ int, H3 x0 x1 ∈ int

Variable U3 : set → set → set

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

Variable V3 : set → set → set

Hypothesis HV3 : ∀x0 ∈ int, ∀x1 ∈ int, V3 x0 x1 ∈ int

Variable F2 : set → set → set

Hypothesis HF2 : ∀x0 ∈ int, ∀x1 ∈ int, F2 x0 x1 ∈ 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 FAST : set → set

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

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

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

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

Hypothesis H4 : (I1 = 1)

Hypothesis H5 : (∀X ∈ int, (∀Y ∈ int, ((J1 X Y) = Y)))

Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U1 X Y Z) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y Z) (V1 (X + - 1) Y Z)))))))

Hypothesis H7 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V1 X Y Z) = (if (X <= 0) then Z else (G1 (U1 (X + - 1) Y Z) (V1 (X + - 1) Y Z)))))))

Hypothesis H8 : (∀X ∈ int, (∀Y ∈ int, ((W1 X Y) = (U1 (H1 X Y) I1 (J1 X Y)))))

Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = (X + (W1 X Y)))))

Hypothesis H10 : (∀X ∈ int, ((G0 X) = X))

Hypothesis H11 : (H0 = 1)

Hypothesis H12 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y) X)))))

Hypothesis H13 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))

Hypothesis H14 : (∀X ∈ int, ((SMALL X) = (V0 X)))

Hypothesis H15 : (∀X ∈ int, ((F3 X) = (X * X)))

Hypothesis H16 : (G3 = 1)

Hypothesis H17 : (∀X ∈ int, (∀Y ∈ int, ((F4 X Y) = (X * Y))))

Hypothesis H18 : (∀X ∈ int, (∀Y ∈ int, ((G4 X Y) = Y)))

Hypothesis H19 : (∀X ∈ int, (∀Y ∈ int, ((H4 X Y) = (Y + - 1))))

Hypothesis H20 : (∀X ∈ int, (∀Y ∈ int, ((I4 X Y) = Y)))

Hypothesis H21 : (∀X ∈ int, (∀Y ∈ int, ((J4 X Y) = Y)))

Hypothesis H22 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U4 X Y Z) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y Z) (V4 (X + - 1) Y Z)))))))

Hypothesis H23 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V4 X Y Z) = (if (X <= 0) then Z else (G4 (U4 (X + - 1) Y Z) (V4 (X + - 1) Y Z)))))))

Hypothesis H24 : (∀X ∈ int, (∀Y ∈ int, ((W4 X Y) = (U4 (H4 X Y) (I4 X Y) (J4 X Y)))))

Hypothesis H25 : (∀X ∈ int, (∀Y ∈ int, ((H3 X Y) = (W4 X Y))))

Hypothesis H26 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y))))))

Hypothesis H27 : (∀X ∈ int, (∀Y ∈ int, ((V3 X Y) = (U3 G3 (H3 X Y)))))

Hypothesis H28 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = ((V3 X Y) + X))))

Hypothesis H29 : (∀X ∈ int, ((G2 X) = X))

Hypothesis H30 : (H2 = 1)

Hypothesis H31 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y) X)))))

Hypothesis H32 : (∀X ∈ int, ((V2 X) = (U2 (G2 X) H2)))

Hypothesis H33 : (∀X ∈ int, ((FAST X) = (V2 X)))

Theorem. (A350008)

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

In Proofgold the corresponding term root is 001b0a... and proposition id is 4de6e3...

Proof:

The rest of the proof is missing.

End of Section A350008