Beginning of Section A277168

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 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 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

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

Variable I1 : set

Hypothesis HI1 : I1 ∈ int

Variable J1 : set → set

Hypothesis HJ1 : ∀x0 ∈ int, J1 x0 ∈ 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

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

Variable SMALL : set → set

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

Variable F2 : set → set → set

Hypothesis HF2 : ∀x0 ∈ int, ∀x1 ∈ int, F2 x0 x1 ∈ int

Variable G2 : set → set → set

Hypothesis HG2 : ∀x0 ∈ int, ∀x1 ∈ int, G2 x0 x1 ∈ int

Variable H2 : set → set

Hypothesis HH2 : ∀x0 ∈ int, H2 x0 ∈ int

Variable I2 : set

Hypothesis HI2 : I2 ∈ int

Variable J2 : set → set

Hypothesis HJ2 : ∀x0 ∈ int, J2 x0 ∈ int

Variable U2 : set → set → set → set

Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U2 x0 x1 x2 ∈ int

Variable V2 : set → set → set → set

Hypothesis HV2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V2 x0 x1 x2 ∈ int

Variable W2 : set → set

Hypothesis HW2 : ∀x0 ∈ int, W2 x0 ∈ int

Variable F3 : set → set → set

Hypothesis HF3 : ∀x0 ∈ int, ∀x1 ∈ int, F3 x0 x1 ∈ int

Variable G3 : set → set → set

Hypothesis HG3 : ∀x0 ∈ int, ∀x1 ∈ int, G3 x0 x1 ∈ int

Variable H3 : set → set

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

Variable I3 : set

Hypothesis HI3 : I3 ∈ int

Variable J3 : set → set

Hypothesis HJ3 : ∀x0 ∈ int, J3 x0 ∈ int

Variable U3 : set → set → set → set

Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U3 x0 x1 x2 ∈ int

Variable V3 : set → set → set → set

Hypothesis HV3 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V3 x0 x1 x2 ∈ int

Variable W3 : set → set

Hypothesis HW3 : ∀x0 ∈ int, W3 x0 ∈ int

Variable FAST : set → set

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

Hypothesis H1 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = ((2 * (X * Y)) + - X))))

Hypothesis H2 : (∀X ∈ int, ((G0 X) = X))

Hypothesis H3 : (H0 = 1)

Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y) X)))))

Hypothesis H5 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))

Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, ((F1 X Y) = (X * Y))))

Hypothesis H7 : (∀X ∈ int, (∀Y ∈ int, ((G1 X Y) = Y)))

Hypothesis H8 : (∀X ∈ int, ((H1 X) = X))

Hypothesis H9 : (I1 = 1)

Hypothesis H10 : (∀X ∈ int, ((J1 X) = (2 + (2 * (X + X)))))

Hypothesis H11 : (∀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 H12 : (∀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 H13 : (∀X ∈ int, ((W1 X) = (U1 (H1 X) I1 (J1 X))))

Hypothesis H14 : (∀X ∈ int, ((SMALL X) = ((V0 X) * (W1 X))))

Hypothesis H15 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = (X * Y))))

Hypothesis H16 : (∀X ∈ int, (∀Y ∈ int, ((G2 X Y) = Y)))

Hypothesis H17 : (∀X ∈ int, ((H2 X) = (X + - 1)))

Hypothesis H18 : (I2 = 1)

Hypothesis H19 : (∀X ∈ int, ((J2 X) = (1 + (X + X))))

Hypothesis H20 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U2 X Y Z) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y Z) (V2 (X + - 1) Y Z)))))))

Hypothesis H21 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V2 X Y Z) = (if (X <= 0) then Z else (G2 (U2 (X + - 1) Y Z) (V2 (X + - 1) Y Z)))))))

Hypothesis H22 : (∀X ∈ int, ((W2 X) = (U2 (H2 X) I2 (J2 X))))

Hypothesis H23 : (∀X ∈ int, (∀Y ∈ int, ((F3 X Y) = (X * Y))))

Hypothesis H24 : (∀X ∈ int, (∀Y ∈ int, ((G3 X Y) = (1 + Y))))

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

Hypothesis H26 : (I3 = 1)

Hypothesis H27 : (∀X ∈ int, ((J3 X) = (1 + X)))

Hypothesis H28 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U3 X Y Z) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y Z) (V3 (X + - 1) Y Z)))))))

Hypothesis H29 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V3 X Y Z) = (if (X <= 0) then Z else (G3 (U3 (X + - 1) Y Z) (V3 (X + - 1) Y Z)))))))

Hypothesis H30 : (∀X ∈ int, ((W3 X) = (U3 (H3 X) I3 (J3 X))))

Hypothesis H31 : (∀X ∈ int, ((FAST X) = (((1 + (X + X)) * (W2 X)) * (W3 X))))

Theorem. (A277168)

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

In Proofgold the corresponding term root is 94929d... and proposition id is 660b83...

Proof:

The rest of the proof is missing.

End of Section A277168