Beginning of Section A24167
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
Hypothesis HH3 : H3 ∈ 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 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 FAST : set → set
Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
Hypothesis H1 : (∀X ∈ int, (∀Y ∈ int, ((F1 X Y) = (0 + - (X * Y)))))
Hypothesis H2 : (∀X ∈ int, (∀Y ∈ int, ((G1 X Y) = Y)))
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)) + X))))
Hypothesis H7 : (∀X ∈ int, ((G0 X) = X))
Hypothesis H8 : (H0 = 1)
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)))
Hypothesis H14 : (H3 = 1)
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))))
Hypothesis H17 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = ((V3 X Y) + - ((1 + Y) * X)))))
Hypothesis H18 : (∀X ∈ int, ((G2 X) = X))
Hypothesis H19 : (H2 = 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)))
Hypothesis H22 : (∀X ∈ int, ((F4 X) = (0 + - X)))
Hypothesis H23 : (∀X ∈ int, ((G4 X) = X))
Hypothesis H24 : (H4 = 1)
Hypothesis H25 : (∀X ∈ int, (∀Y ∈ int, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y))))))
Hypothesis H26 : (∀X ∈ int, ((V4 X) = (U4 (G4 X) H4)))
Hypothesis H27 : (∀X ∈ int, ((FAST X) = ((V2 X) * (V4 X))))
Theorem. (
A24167)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof:The rest of the proof is missing.