Beginning of Section A254
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
Hypothesis HH1 : H1 ∈ 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
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 → 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 → 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 → 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 FAST : set → set
Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
Hypothesis H1 : (∀X ∈ int, (∀Y ∈ int, ((F1 X Y) = (X * Y))))
Hypothesis H2 : (∀X ∈ int, (∀Y ∈ int, ((G1 X Y) = (Y + - 1))))
Hypothesis H3 : (H1 = 1)
Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y) X)))))
Hypothesis H5 : (∀X ∈ int, (∀Y ∈ int, ((V1 X Y) = (U1 (G1 X Y) H1))))
Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = ((X * Y) + (V1 X Y)))))
Hypothesis H7 : (∀X ∈ int, ((G0 X) = X))
Hypothesis H8 : (H0 = 0)
Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y) X)))))
Hypothesis H10 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))
Hypothesis H11 : (∀X ∈ int, ((SMALL X) = (V0 X)))
Hypothesis H12 : (∀X ∈ int, (∀Y ∈ int, ((F3 X Y) = (X * Y))))
Hypothesis H13 : (∀X ∈ int, (∀Y ∈ int, ((G3 X Y) = (Y + - 1))))
Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, ((H3 X Y) = Y)))
Hypothesis H15 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y) X)))))
Hypothesis H16 : (∀X ∈ int, (∀Y ∈ int, ((V3 X Y) = (U3 (G3 X Y) (H3 X Y)))))
Hypothesis H17 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = ((V3 X Y) + ((1 + Y) * X)))))
Hypothesis H18 : (∀X ∈ int, ((G2 X) = (X + - 1)))
Hypothesis H19 : (∀X ∈ int, ((H2 X) = (if (X <= 0) then 0 else 1)))
Hypothesis H20 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y) X)))))
Hypothesis H21 : (∀X ∈ int, ((V2 X) = (U2 (G2 X) (H2 X))))
Hypothesis H22 : (∀X ∈ int, ((FAST X) = (V2 X)))
Theorem. (
A254)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof:The rest of the proof is missing.