Beginning of Section A34910
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.
L9Variable F0 : set → set → set
L10Hypothesis HF0 : ∀x0 ∈ int, ∀x1 ∈ int, F0 x0 x1 ∈ int
L11Variable G0 : set → set
L12Hypothesis HG0 : ∀x0 ∈ int, G0 x0 ∈ int
L14Hypothesis HH0 : H0 ∈ int
L15Variable U0 : set → set → set
L16Hypothesis HU0 : ∀x0 ∈ int, ∀x1 ∈ int, U0 x0 x1 ∈ int
L17Variable V0 : set → set
L18Hypothesis HV0 : ∀x0 ∈ int, V0 x0 ∈ int
L19Variable SMALL : set → set
L20Hypothesis HSMALL : ∀x0 ∈ int, SMALL x0 ∈ int
L21Variable F1 : set → set
L22Hypothesis HF1 : ∀x0 ∈ int, F1 x0 ∈ int
L24Hypothesis HG1 : G1 ∈ int
L25Variable F2 : set → set
L26Hypothesis HF2 : ∀x0 ∈ int, F2 x0 ∈ int
L27Variable G2 : set → set
L28Hypothesis HG2 : ∀x0 ∈ int, G2 x0 ∈ int
L30Hypothesis HH2 : H2 ∈ int
L31Variable U2 : set → set → set
L32Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, U2 x0 x1 ∈ int
L33Variable V2 : set → set
L34Hypothesis HV2 : ∀x0 ∈ int, V2 x0 ∈ int
L35Variable H1 : set → set
L36Hypothesis HH1 : ∀x0 ∈ int, H1 x0 ∈ int
L37Variable U1 : set → set → set
L38Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, U1 x0 x1 ∈ int
L39Variable V1 : set → set
L40Hypothesis HV1 : ∀x0 ∈ int, V1 x0 ∈ int
L41Variable F3 : set → set → set
L42Hypothesis HF3 : ∀x0 ∈ int, ∀x1 ∈ int, F3 x0 x1 ∈ int
L43Variable G3 : set → set → set
L44Hypothesis HG3 : ∀x0 ∈ int, ∀x1 ∈ int, G3 x0 x1 ∈ int
L45Variable H3 : set → set
L46Hypothesis HH3 : ∀x0 ∈ int, H3 x0 ∈ int
L48Hypothesis HI3 : I3 ∈ int
L50Hypothesis HJ3 : J3 ∈ int
L51Variable U3 : set → set → set → set
L52Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U3 x0 x1 x2 ∈ int
L53Variable V3 : set → set → set → set
L54Hypothesis HV3 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V3 x0 x1 x2 ∈ int
L55Variable W3 : set → set
L56Hypothesis HW3 : ∀x0 ∈ int, W3 x0 ∈ int
L57Variable FAST : set → set
L58Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L59Hypothesis H1 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = (2 * (2 * ((2 * (X * Y)) + X))))))
L60Hypothesis H2 : (∀X ∈ int, ((G0 X) = X))
L61Hypothesis H3 : (H0 = 1)
L62Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y) X)))))
L63Hypothesis H5 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))
L64Hypothesis H6 : (∀X ∈ int, ((SMALL X) = (V0 X)))
L65Hypothesis H7 : (∀X ∈ int, ((F1 X) = (X * X)))
L66Hypothesis H8 : (G1 = 1)
L67Hypothesis H9 : (∀X ∈ int, ((F2 X) = (X + X)))
L68Hypothesis H10 : (∀X ∈ int, ((G2 X) = X))
L69Hypothesis H11 : (H2 = 1)
L70Hypothesis H12 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y))))))
L71Hypothesis H13 : (∀X ∈ int, ((V2 X) = (U2 (G2 X) H2)))
L72Hypothesis H14 : (∀X ∈ int, ((H1 X) = (V2 X)))
L73Hypothesis H15 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))
L74Hypothesis H16 : (∀X ∈ int, ((V1 X) = (U1 G1 (H1 X))))
L75Hypothesis H17 : (∀X ∈ int, (∀Y ∈ int, ((F3 X Y) = (X * Y))))
L76Hypothesis H18 : (∀X ∈ int, (∀Y ∈ int, ((G3 X Y) = (2 + Y))))
L77Hypothesis H19 : (∀X ∈ int, ((H3 X) = X))
L78Hypothesis H20 : (I3 = 1)
L79Hypothesis H21 : (J3 = (1 + 2))
L80Hypothesis H22 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U3 X Y Z) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y Z) (V3 (X + - 1) Y Z)))))))
L81Hypothesis H23 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V3 X Y Z) = (if (X <= 0) then Z else (G3 (U3 (X + - 1) Y Z) (V3 (X + - 1) Y Z)))))))
L82Hypothesis H24 : (∀X ∈ int, ((W3 X) = (U3 (H3 X) I3 J3)))
L83Hypothesis H25 : (∀X ∈ int, ((FAST X) = ((V1 X) * (W3 X))))
L84Theorem. (
A34910)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.