Beginning of Section A19040
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 : set → set → set
Hypothesis HF2 : ∀x0 ∈ int, ∀x1 ∈ int, F2 x0 x1 ∈ int
Variable G2 : set
Hypothesis HG2 : G2 ∈ int
Variable H2 : set → set
Hypothesis HH2 : ∀x0 ∈ int, H2 x0 ∈ 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 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
Hypothesis HJ1 : J1 ∈ 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 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
Hypothesis HI4 : I4 ∈ int
Variable J4 : set
Hypothesis HJ4 : J4 ∈ 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 F3 : set → set → set
Hypothesis HF3 : ∀x0 ∈ int, ∀x1 ∈ int, F3 x0 x1 ∈ 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 FAST : set → set
Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
Hypothesis H1 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = ((2 + Y) * X))))
Hypothesis H2 : (G2 = 2)
Hypothesis H3 : (∀X ∈ int, ((H2 X) = X))
Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y) X)))))
Hypothesis H5 : (∀X ∈ int, ((V2 X) = (U2 G2 (H2 X))))
Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, ((F1 X Y) = (((V2 X) + - X) + (Y * Y)))))
Hypothesis H7 : (∀X ∈ int, (∀Y ∈ int, ((G1 X Y) = (Y + Y))))
Hypothesis H8 : (∀X ∈ int, (∀Y ∈ int, ((H1 X Y) = Y)))
Hypothesis H9 : (I1 = 1)
Hypothesis H10 : (J1 = 2)
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, (∀Y ∈ int, ((W1 X Y) = (U1 (H1 X Y) I1 J1))))
Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = (((W1 X Y) + (2 * (X + X))) + X))))
Hypothesis H15 : (∀X ∈ int, ((G0 X) = X))
Hypothesis H16 : (H0 = 1)
Hypothesis H17 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y) X)))))
Hypothesis H18 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))
Hypothesis H19 : (∀X ∈ int, ((SMALL X) = (V0 X)))
Hypothesis H20 : (∀X ∈ int, (∀Y ∈ int, ((F4 X Y) = (((2 * (X + X)) + (Y * Y)) + X))))
Hypothesis H21 : (∀X ∈ int, (∀Y ∈ int, ((G4 X Y) = (Y + Y))))
Hypothesis H22 : (∀X ∈ int, (∀Y ∈ int, ((H4 X Y) = Y)))
Hypothesis H23 : (I4 = 1)
Hypothesis H24 : (J4 = 2)
Hypothesis H25 : (∀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 H26 : (∀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 H27 : (∀X ∈ int, (∀Y ∈ int, ((W4 X Y) = (U4 (H4 X Y) I4 J4))))
Hypothesis H28 : (∀X ∈ int, (∀Y ∈ int, ((F3 X Y) = (((W4 X Y) + (2 * ((2 * (X + X)) + X))) + X))))
Hypothesis H29 : (∀X ∈ int, ((G3 X) = X))
Hypothesis H30 : (H3 = 1)
Hypothesis H31 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y) X)))))
Hypothesis H32 : (∀X ∈ int, ((V3 X) = (U3 (G3 X) H3)))
Hypothesis H33 : (∀X ∈ int, ((FAST X) = (V3 X)))
Theorem. (
A19040)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof:The rest of the proof is missing.