Beginning of Section A13745
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
L10Hypothesis HF0 : ∀x0 ∈ int, F0 x0 ∈ 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 → set
L26Hypothesis HF2 : ∀x0 ∈ int, ∀x1 ∈ int, F2 x0 x1 ∈ int
L27Variable G2 : set → set → set
L28Hypothesis HG2 : ∀x0 ∈ int, ∀x1 ∈ int, G2 x0 x1 ∈ int
L29Variable H2 : set → set
L30Hypothesis HH2 : ∀x0 ∈ int, H2 x0 ∈ int
L32Hypothesis HI2 : I2 ∈ int
L34Hypothesis HJ2 : J2 ∈ int
L35Variable U2 : set → set → set → set
L36Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U2 x0 x1 x2 ∈ int
L37Variable V2 : set → set → set → set
L38Hypothesis HV2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V2 x0 x1 x2 ∈ int
L39Variable W2 : set → set
L40Hypothesis HW2 : ∀x0 ∈ int, W2 x0 ∈ int
L41Variable H1 : set → set
L42Hypothesis HH1 : ∀x0 ∈ int, H1 x0 ∈ int
L43Variable U1 : set → set → set
L44Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, U1 x0 x1 ∈ int
L45Variable V1 : set → set
L46Hypothesis HV1 : ∀x0 ∈ int, V1 x0 ∈ int
L47Variable F3 : set → set
L48Hypothesis HF3 : ∀x0 ∈ int, F3 x0 ∈ int
L50Hypothesis HG3 : G3 ∈ int
L51Variable F4 : set → set → set
L52Hypothesis HF4 : ∀x0 ∈ int, ∀x1 ∈ int, F4 x0 x1 ∈ int
L53Variable G4 : set → set → set
L54Hypothesis HG4 : ∀x0 ∈ int, ∀x1 ∈ int, G4 x0 x1 ∈ int
L55Variable H4 : set → set
L56Hypothesis HH4 : ∀x0 ∈ int, H4 x0 ∈ int
L58Hypothesis HI4 : I4 ∈ int
L60Hypothesis HJ4 : J4 ∈ int
L61Variable U4 : set → set → set → set
L62Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U4 x0 x1 x2 ∈ int
L63Variable V4 : set → set → set → set
L64Hypothesis HV4 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V4 x0 x1 x2 ∈ int
L65Variable W4 : set → set
L66Hypothesis HW4 : ∀x0 ∈ int, W4 x0 ∈ int
L67Variable H3 : set → set
L68Hypothesis HH3 : ∀x0 ∈ int, H3 x0 ∈ int
L69Variable U3 : set → set → set
L70Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, U3 x0 x1 ∈ int
L71Variable V3 : set → set
L72Hypothesis HV3 : ∀x0 ∈ int, V3 x0 ∈ int
L73Variable FAST : set → set
L74Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L75Hypothesis H1 : (∀X ∈ int, ((F0 X) = ((X + X) + X)))
L76Hypothesis H2 : (∀X ∈ int, ((G0 X) = (2 * (2 + ((X + X) + X)))))
L77Hypothesis H3 : (H0 = 1)
L78Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y))))))
L79Hypothesis H5 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))
L80Hypothesis H6 : (∀X ∈ int, ((SMALL X) = (V0 X)))
L81Hypothesis H7 : (∀X ∈ int, ((F1 X) = (X * X)))
L82Hypothesis H8 : (G1 = 1)
L83Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = (X * Y))))
L84Hypothesis H10 : (∀X ∈ int, (∀Y ∈ int, ((G2 X Y) = Y)))
L85Hypothesis H11 : (∀X ∈ int, ((H2 X) = X))
L86Hypothesis H12 : (I2 = (1 + 2))
L87Hypothesis H13 : (J2 = (1 + 2))
L88Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U2 X Y Z) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y Z) (V2 (X + - 1) Y Z)))))))
L89Hypothesis H15 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V2 X Y Z) = (if (X <= 0) then Z else (G2 (U2 (X + - 1) Y Z) (V2 (X + - 1) Y Z)))))))
L90Hypothesis H16 : (∀X ∈ int, ((W2 X) = (U2 (H2 X) I2 J2)))
L91Hypothesis H17 : (∀X ∈ int, ((H1 X) = (W2 X)))
L92Hypothesis H18 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))
L93Hypothesis H19 : (∀X ∈ int, ((V1 X) = (U1 G1 (H1 X))))
L94Hypothesis H20 : (∀X ∈ int, ((F3 X) = (X * X)))
L95Hypothesis H21 : (G3 = 1)
L96Hypothesis H22 : (∀X ∈ int, (∀Y ∈ int, ((F4 X Y) = (X * Y))))
L97Hypothesis H23 : (∀X ∈ int, (∀Y ∈ int, ((G4 X Y) = Y)))
L98Hypothesis H24 : (∀X ∈ int, ((H4 X) = X))
L99Hypothesis H25 : (I4 = (1 + 2))
L100Hypothesis H26 : (J4 = (1 + (2 * (2 + 2))))
L101Hypothesis H27 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U4 X Y Z) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y Z) (V4 (X + - 1) Y Z)))))))
L102Hypothesis H28 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V4 X Y Z) = (if (X <= 0) then Z else (G4 (U4 (X + - 1) Y Z) (V4 (X + - 1) Y Z)))))))
L103Hypothesis H29 : (∀X ∈ int, ((W4 X) = (U4 (H4 X) I4 J4)))
L104Hypothesis H30 : (∀X ∈ int, ((H3 X) = (W4 X)))
L105Hypothesis H31 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y))))))
L106Hypothesis H32 : (∀X ∈ int, ((V3 X) = (U3 G3 (H3 X))))
L107Hypothesis H33 : (∀X ∈ int, ((FAST X) = ((V1 X) * (V3 X))))
L108Theorem. (
A13745)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.