Beginning of Section A13742
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 F1 : set → set
L12Hypothesis HF1 : ∀x0 ∈ int, F1 x0 ∈ int
L14Hypothesis HG1 : G1 ∈ int
L15Variable H1 : set → set
L16Hypothesis HH1 : ∀x0 ∈ int, H1 x0 ∈ int
L17Variable U1 : set → set → set
L18Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, U1 x0 x1 ∈ int
L19Variable V1 : set → set
L20Hypothesis HV1 : ∀x0 ∈ int, V1 x0 ∈ int
L21Variable G0 : set → set
L22Hypothesis HG0 : ∀x0 ∈ int, G0 x0 ∈ int
L24Hypothesis HH0 : H0 ∈ int
L25Variable U0 : set → set → set
L26Hypothesis HU0 : ∀x0 ∈ int, ∀x1 ∈ int, U0 x0 x1 ∈ int
L27Variable V0 : set → set
L28Hypothesis HV0 : ∀x0 ∈ int, V0 x0 ∈ int
L29Variable SMALL : set → set
L30Hypothesis HSMALL : ∀x0 ∈ int, SMALL x0 ∈ int
L31Variable F2 : set → set
L32Hypothesis HF2 : ∀x0 ∈ int, F2 x0 ∈ int
L34Hypothesis HG2 : G2 ∈ int
L35Variable F3 : set → set
L36Hypothesis HF3 : ∀x0 ∈ int, F3 x0 ∈ int
L37Variable G3 : set → set
L38Hypothesis HG3 : ∀x0 ∈ int, G3 x0 ∈ int
L40Hypothesis HH3 : H3 ∈ int
L41Variable U3 : set → set → set
L42Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, U3 x0 x1 ∈ int
L43Variable V3 : set → set
L44Hypothesis HV3 : ∀x0 ∈ int, V3 x0 ∈ int
L45Variable H2 : set → set
L46Hypothesis HH2 : ∀x0 ∈ int, H2 x0 ∈ int
L47Variable U2 : set → set → set
L48Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, U2 x0 x1 ∈ int
L49Variable V2 : set → set
L50Hypothesis HV2 : ∀x0 ∈ int, V2 x0 ∈ int
L51Variable F4 : set → set
L52Hypothesis HF4 : ∀x0 ∈ int, F4 x0 ∈ int
L54Hypothesis HG4 : G4 ∈ int
L55Variable F5 : set → set
L56Hypothesis HF5 : ∀x0 ∈ int, F5 x0 ∈ int
L57Variable G5 : set → set
L58Hypothesis HG5 : ∀x0 ∈ int, G5 x0 ∈ int
L60Hypothesis HH5 : H5 ∈ int
L61Variable U5 : set → set → set
L62Hypothesis HU5 : ∀x0 ∈ int, ∀x1 ∈ int, U5 x0 x1 ∈ int
L63Variable V5 : set → set
L64Hypothesis HV5 : ∀x0 ∈ int, V5 x0 ∈ int
L65Variable H4 : set → set
L66Hypothesis HH4 : ∀x0 ∈ int, H4 x0 ∈ int
L67Variable U4 : set → set → set
L68Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, U4 x0 x1 ∈ int
L69Variable V4 : set → set
L70Hypothesis HV4 : ∀x0 ∈ int, V4 x0 ∈ int
L71Variable FAST : set → set
L72Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L73Hypothesis H1 : (∀X ∈ int, ((F0 X) = (X + X)))
L74Hypothesis H2 : (∀X ∈ int, ((F1 X) = ((X + X) + X)))
L75Hypothesis H3 : (G1 = 2)
L76Hypothesis H4 : (∀X ∈ int, ((H1 X) = X))
L77Hypothesis H5 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))
L78Hypothesis H6 : (∀X ∈ int, ((V1 X) = (U1 G1 (H1 X))))
L79Hypothesis H7 : (∀X ∈ int, ((G0 X) = (2 + (V1 X))))
L80Hypothesis H8 : (H0 = 2)
L81Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y))))))
L82Hypothesis H10 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))
L83Hypothesis H11 : (∀X ∈ int, ((SMALL X) = (V0 X)))
L84Hypothesis H12 : (∀X ∈ int, ((F2 X) = (2 * (X * X))))
L85Hypothesis H13 : (G2 = 2)
L86Hypothesis H14 : (∀X ∈ int, ((F3 X) = (X + X)))
L87Hypothesis H15 : (∀X ∈ int, ((G3 X) = X))
L88Hypothesis H16 : (H3 = 1)
L89Hypothesis H17 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y))))))
L90Hypothesis H18 : (∀X ∈ int, ((V3 X) = (U3 (G3 X) H3)))
L91Hypothesis H19 : (∀X ∈ int, ((H2 X) = (V3 X)))
L92Hypothesis H20 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y))))))
L93Hypothesis H21 : (∀X ∈ int, ((V2 X) = (U2 G2 (H2 X))))
L94Hypothesis H22 : (∀X ∈ int, ((F4 X) = ((((X * X) * X) * X) * X)))
L95Hypothesis H23 : (G4 = 1)
L96Hypothesis H24 : (∀X ∈ int, ((F5 X) = (X + X)))
L97Hypothesis H25 : (∀X ∈ int, ((G5 X) = X))
L98Hypothesis H26 : (H5 = 1)
L99Hypothesis H27 : (∀X ∈ int, (∀Y ∈ int, ((U5 X Y) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y))))))
L100Hypothesis H28 : (∀X ∈ int, ((V5 X) = (U5 (G5 X) H5)))
L101Hypothesis H29 : (∀X ∈ int, ((H4 X) = (V5 X)))
L102Hypothesis H30 : (∀X ∈ int, (∀Y ∈ int, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y))))))
L103Hypothesis H31 : (∀X ∈ int, ((V4 X) = (U4 G4 (H4 X))))
L104Hypothesis H32 : (∀X ∈ int, ((FAST X) = ((V2 X) * (V4 X))))
L105Theorem. (
A13742)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.