Beginning of Section A431
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 → set
Hypothesis HG0 : ∀x0 ∈ int, ∀x1 ∈ int, G0 x0 x1 ∈ int
Variable H0 : set → set
Hypothesis HH0 : ∀x0 ∈ int, H0 x0 ∈ int
Variable I0 : set
Hypothesis HI0 : I0 ∈ int
Variable J0 : set
Hypothesis HJ0 : J0 ∈ int
Variable U0 : set → set → set → set
Hypothesis HU0 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U0 x0 x1 x2 ∈ int
Variable V0 : set → set → set → set
Hypothesis HV0 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V0 x0 x1 x2 ∈ int
Variable W0 : set → set
Hypothesis HW0 : ∀x0 ∈ int, W0 x0 ∈ int
Variable SMALL : set → set
Hypothesis HSMALL : ∀x0 ∈ int, SMALL x0 ∈ int
Variable F1 : set → set
Hypothesis HF1 : ∀x0 ∈ int, F1 x0 ∈ int
Variable G1 : set → set
Hypothesis HG1 : ∀x0 ∈ int, G1 x0 ∈ 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
Hypothesis HV1 : ∀x0 ∈ int, V1 x0 ∈ int
Variable F2 : set → set
Hypothesis HF2 : ∀x0 ∈ int, F2 x0 ∈ 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, ((F0 X Y) = ((((Y * Y) + - Y) + X) + X))))
Hypothesis H2 : (∀X ∈ int, (∀Y ∈ int, ((G0 X Y) = (Y + Y))))
Hypothesis H3 : (∀X ∈ int, ((H0 X) = (X + - 1)))
Hypothesis H4 : (I0 = 0)
Hypothesis H5 : (J0 = 1)
Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U0 X Y Z) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y Z) (V0 (X + - 1) Y Z)))))))
Hypothesis H7 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V0 X Y Z) = (if (X <= 0) then Z else (G0 (U0 (X + - 1) Y Z) (V0 (X + - 1) Y Z)))))))
Hypothesis H8 : (∀X ∈ int, ((W0 X) = (U0 (H0 X) I0 J0)))
Hypothesis H9 : (∀X ∈ int, ((SMALL X) = (W0 X)))
Hypothesis H10 : (∀X ∈ int, ((F1 X) = (X + X)))
Hypothesis H11 : (∀X ∈ int, ((G1 X) = (X + - 1)))
Hypothesis H12 : (H1 = 1)
Hypothesis H13 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))
Hypothesis H14 : (∀X ∈ int, ((V1 X) = (U1 (G1 X) H1)))
Hypothesis H15 : (∀X ∈ int, ((F2 X) = (2 * X)))
Hypothesis H16 : (∀X ∈ int, ((G2 X) = (X + - 2)))
Hypothesis H17 : (H2 = 1)
Hypothesis H18 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y))))))
Hypothesis H19 : (∀X ∈ int, ((V2 X) = (U2 (G2 X) H2)))
Hypothesis H20 : (∀X ∈ int, ((FAST X) = (((V1 X) + - (if ((X + - 1) <= 0) then 1 else X)) * (V2 X))))
Theorem. (
A431)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof:The rest of the proof is missing.