Beginning of Section A20499
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 F2 : set → set
L10Hypothesis HF2 : ∀x0 ∈ int, F2 x0 ∈ int
L11Variable G2 : set → set → set
L12Hypothesis HG2 : ∀x0 ∈ int, ∀x1 ∈ int, G2 x0 x1 ∈ int
L14Hypothesis HH2 : H2 ∈ int
L15Variable U2 : set → set → set
L16Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, U2 x0 x1 ∈ int
L17Variable V2 : set → set → set
L18Hypothesis HV2 : ∀x0 ∈ int, ∀x1 ∈ int, V2 x0 x1 ∈ int
L19Variable F3 : set → set → set
L20Hypothesis HF3 : ∀x0 ∈ int, ∀x1 ∈ int, F3 x0 x1 ∈ int
L22Hypothesis HG3 : G3 ∈ int
L23Variable H3 : set → set
L24Hypothesis HH3 : ∀x0 ∈ int, H3 x0 ∈ int
L25Variable U3 : set → set → set
L26Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, U3 x0 x1 ∈ int
L27Variable V3 : set → set
L28Hypothesis HV3 : ∀x0 ∈ int, V3 x0 ∈ int
L29Variable F1 : set → set → set
L30Hypothesis HF1 : ∀x0 ∈ int, ∀x1 ∈ int, F1 x0 x1 ∈ int
L31Variable G1 : set → set → set
L32Hypothesis HG1 : ∀x0 ∈ int, ∀x1 ∈ int, G1 x0 x1 ∈ int
L34Hypothesis HH1 : H1 ∈ int
L35Variable U1 : set → set → set
L36Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, U1 x0 x1 ∈ int
L37Variable V1 : set → set → set
L38Hypothesis HV1 : ∀x0 ∈ int, ∀x1 ∈ int, V1 x0 x1 ∈ int
L39Variable F0 : set → set → set
L40Hypothesis HF0 : ∀x0 ∈ int, ∀x1 ∈ int, F0 x0 x1 ∈ int
L41Variable G0 : set → set
L42Hypothesis HG0 : ∀x0 ∈ int, G0 x0 ∈ int
L44Hypothesis HH0 : H0 ∈ int
L45Variable U0 : set → set → set
L46Hypothesis HU0 : ∀x0 ∈ int, ∀x1 ∈ int, U0 x0 x1 ∈ int
L47Variable V0 : set → set
L48Hypothesis HV0 : ∀x0 ∈ int, V0 x0 ∈ int
L49Variable SMALL : set → set
L50Hypothesis HSMALL : ∀x0 ∈ int, SMALL x0 ∈ int
L51Variable F6 : set → set → set
L52Hypothesis HF6 : ∀x0 ∈ int, ∀x1 ∈ int, F6 x0 x1 ∈ int
L53Variable G6 : set → set → set
L54Hypothesis HG6 : ∀x0 ∈ int, ∀x1 ∈ int, G6 x0 x1 ∈ int
L55Variable H6 : set → set
L56Hypothesis HH6 : ∀x0 ∈ int, H6 x0 ∈ int
L58Hypothesis HI6 : I6 ∈ int
L60Hypothesis HJ6 : J6 ∈ int
L61Variable U6 : set → set → set → set
L62Hypothesis HU6 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U6 x0 x1 x2 ∈ int
L63Variable V6 : set → set → set → set
L64Hypothesis HV6 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V6 x0 x1 x2 ∈ int
L65Variable W6 : set → set
L66Hypothesis HW6 : ∀x0 ∈ int, W6 x0 ∈ int
L67Variable F5 : set → set
L68Hypothesis HF5 : ∀x0 ∈ int, F5 x0 ∈ int
L70Hypothesis HG5 : G5 ∈ int
L71Variable H5 : set → set → set
L72Hypothesis HH5 : ∀x0 ∈ int, ∀x1 ∈ int, H5 x0 x1 ∈ int
L73Variable U5 : set → set → set
L74Hypothesis HU5 : ∀x0 ∈ int, ∀x1 ∈ int, U5 x0 x1 ∈ int
L75Variable V5 : set → set → set
L76Hypothesis HV5 : ∀x0 ∈ int, ∀x1 ∈ int, V5 x0 x1 ∈ int
L77Variable F4 : set → set → set
L78Hypothesis HF4 : ∀x0 ∈ int, ∀x1 ∈ int, F4 x0 x1 ∈ int
L79Variable G4 : set → set
L80Hypothesis HG4 : ∀x0 ∈ int, G4 x0 ∈ int
L82Hypothesis HH4 : H4 ∈ int
L83Variable U4 : set → set → set
L84Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, U4 x0 x1 ∈ int
L85Variable V4 : set → set
L86Hypothesis HV4 : ∀x0 ∈ int, V4 x0 ∈ int
L87Variable FAST : set → set
L88Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L89Hypothesis H1 : (∀X ∈ int, ((F2 X) = ((X + X) + X)))
L90Hypothesis H2 : (∀X ∈ int, (∀Y ∈ int, ((G2 X Y) = (Y + Y))))
L91Hypothesis H3 : (H2 = 1)
L92Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y))))))
L93Hypothesis H5 : (∀X ∈ int, (∀Y ∈ int, ((V2 X Y) = (U2 (G2 X Y) H2))))
L94Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, ((F3 X Y) = ((2 + Y) * X))))
L95Hypothesis H7 : (G3 = 2)
L96Hypothesis H8 : (∀X ∈ int, ((H3 X) = X))
L97Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y) X)))))
L98Hypothesis H10 : (∀X ∈ int, ((V3 X) = (U3 G3 (H3 X))))
L99Hypothesis H11 : (∀X ∈ int, (∀Y ∈ int, ((F1 X Y) = ((V2 X Y) + ((V3 X) + - X)))))
L100Hypothesis H12 : (∀X ∈ int, (∀Y ∈ int, ((G1 X Y) = Y)))
L101Hypothesis H13 : (H1 = 1)
L102Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y) X)))))
L103Hypothesis H15 : (∀X ∈ int, (∀Y ∈ int, ((V1 X Y) = (U1 (G1 X Y) H1))))
L104Hypothesis H16 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = (((V1 X Y) + (2 * (X + X))) + X))))
L105Hypothesis H17 : (∀X ∈ int, ((G0 X) = X))
L106Hypothesis H18 : (H0 = 1)
L107Hypothesis H19 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y) X)))))
L108Hypothesis H20 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))
L109Hypothesis H21 : (∀X ∈ int, ((SMALL X) = (V0 X)))
L110Hypothesis H22 : (∀X ∈ int, (∀Y ∈ int, ((F6 X Y) = (((2 * (X + X)) + (Y * Y)) + X))))
L111Hypothesis H23 : (∀X ∈ int, (∀Y ∈ int, ((G6 X Y) = ((Y + Y) + Y))))
L112Hypothesis H24 : (∀X ∈ int, ((H6 X) = X))
L113Hypothesis H25 : (I6 = 1)
L114Hypothesis H26 : (J6 = (1 + 2))
L115Hypothesis H27 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U6 X Y Z) = (if (X <= 0) then Y else (F6 (U6 (X + - 1) Y Z) (V6 (X + - 1) Y Z)))))))
L116Hypothesis H28 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V6 X Y Z) = (if (X <= 0) then Z else (G6 (U6 (X + - 1) Y Z) (V6 (X + - 1) Y Z)))))))
L117Hypothesis H29 : (∀X ∈ int, ((W6 X) = (U6 (H6 X) I6 J6)))
L118Hypothesis H30 : (∀X ∈ int, ((F5 X) = (W6 X)))
L119Hypothesis H31 : (G5 = 1)
L120Hypothesis H32 : (∀X ∈ int, (∀Y ∈ int, ((H5 X Y) = Y)))
L121Hypothesis H33 : (∀X ∈ int, (∀Y ∈ int, ((U5 X Y) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y))))))
L122Hypothesis H34 : (∀X ∈ int, (∀Y ∈ int, ((V5 X Y) = (U5 G5 (H5 X Y)))))
L123Hypothesis H35 : (∀X ∈ int, (∀Y ∈ int, ((F4 X Y) = (((V5 X Y) + (2 * ((2 * (X + X)) + X))) + X))))
L124Hypothesis H36 : (∀X ∈ int, ((G4 X) = X))
L125Hypothesis H37 : (H4 = 1)
L126Hypothesis H38 : (∀X ∈ int, (∀Y ∈ int, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y) X)))))
L127Hypothesis H39 : (∀X ∈ int, ((V4 X) = (U4 (G4 X) H4)))
L128Hypothesis H40 : (∀X ∈ int, ((FAST X) = (V4 X)))
L129Theorem. (
A20499)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.