Beginning of Section A54605
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 → set
Hypothesis HH0 : ∀x0 ∈ int, H0 x0 ∈ 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 FAST : set → set
Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
Hypothesis H1 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = (X + Y))))
Hypothesis H2 : (∀X ∈ int, ((G0 X) = (X * (X * X))))
Hypothesis H3 : (∀X ∈ int, ((H0 X) = ((X * X) + X)))
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 X))))
Hypothesis H6 : (∀X ∈ int, ((SMALL X) = (2 * (V0 X))))
Hypothesis H7 : (∀X ∈ int, ((FAST X) = ((2 + ((2 + ((((X * X) * X) * X) + X)) * X)) * X)))
Theorem. (
A54605)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof:The rest of the proof is missing.