Beginning of Section A13873
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 F1 : set → set
L10Hypothesis HF1 : ∀x0 ∈ int, F1 x0 ∈ int
L12Hypothesis HG1 : G1 ∈ int
L14Hypothesis HH1 : H1 ∈ int
L15Variable U1 : set → set → set
L16Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, U1 x0 x1 ∈ int
L18Hypothesis HV1 : V1 ∈ int
L19Variable F0 : set → set
L20Hypothesis HF0 : ∀x0 ∈ int, F0 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 → set
L36Hypothesis HF3 : ∀x0 ∈ int, ∀x1 ∈ int, F3 x0 x1 ∈ int
L37Variable G3 : set → set → set
L38Hypothesis HG3 : ∀x0 ∈ int, ∀x1 ∈ int, G3 x0 x1 ∈ int
L39Variable H3 : set → set
L40Hypothesis HH3 : ∀x0 ∈ int, H3 x0 ∈ int
L42Hypothesis HI3 : I3 ∈ int
L44Hypothesis HJ3 : J3 ∈ int
L45Variable U3 : set → set → set → set
L46Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U3 x0 x1 x2 ∈ int
L47Variable V3 : set → set → set → set
L48Hypothesis HV3 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V3 x0 x1 x2 ∈ int
L49Variable W3 : set → set
L50Hypothesis HW3 : ∀x0 ∈ int, W3 x0 ∈ int
L51Variable H2 : set → set
L52Hypothesis HH2 : ∀x0 ∈ int, H2 x0 ∈ int
L53Variable U2 : set → set → set
L54Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, U2 x0 x1 ∈ int
L55Variable V2 : set → set
L56Hypothesis HV2 : ∀x0 ∈ int, V2 x0 ∈ int
L57Variable F4 : set → set
L58Hypothesis HF4 : ∀x0 ∈ int, F4 x0 ∈ int
L60Hypothesis HG4 : G4 ∈ int
L61Variable F5 : set → set → set
L62Hypothesis HF5 : ∀x0 ∈ int, ∀x1 ∈ int, F5 x0 x1 ∈ int
L63Variable G5 : set → set → set
L64Hypothesis HG5 : ∀x0 ∈ int, ∀x1 ∈ int, G5 x0 x1 ∈ int
L65Variable H5 : set → set
L66Hypothesis HH5 : ∀x0 ∈ int, H5 x0 ∈ int
L68Hypothesis HI5 : I5 ∈ int
L70Hypothesis HJ5 : J5 ∈ int
L71Variable U5 : set → set → set → set
L72Hypothesis HU5 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U5 x0 x1 x2 ∈ int
L73Variable V5 : set → set → set → set
L74Hypothesis HV5 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V5 x0 x1 x2 ∈ int
L75Variable W5 : set → set
L76Hypothesis HW5 : ∀x0 ∈ int, W5 x0 ∈ int
L77Variable H4 : set → set
L78Hypothesis HH4 : ∀x0 ∈ int, H4 x0 ∈ int
L79Variable U4 : set → set → set
L80Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, U4 x0 x1 ∈ int
L81Variable V4 : set → set
L82Hypothesis HV4 : ∀x0 ∈ int, V4 x0 ∈ int
L83Variable FAST : set → set
L84Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L85Hypothesis H1 : (∀X ∈ int, ((F1 X) = (X * X)))
L86Hypothesis H2 : (G1 = 2)
L87Hypothesis H3 : (H1 = 2)
L88Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))
L89Hypothesis H5 : (V1 = (U1 G1 H1))
L90Hypothesis H6 : (∀X ∈ int, ((F0 X) = ((V1 + - 2) * X)))
L91Hypothesis H7 : (∀X ∈ int, ((G0 X) = ((2 * (2 + (X + X))) + X)))
L92Hypothesis H8 : (H0 = 1)
L93Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y))))))
L94Hypothesis H10 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))
L95Hypothesis H11 : (∀X ∈ int, ((SMALL X) = (V0 X)))
L96Hypothesis H12 : (∀X ∈ int, ((F2 X) = ((X * X) * X)))
L97Hypothesis H13 : (G2 = 1)
L98Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, ((F3 X Y) = (X * Y))))
L99Hypothesis H15 : (∀X ∈ int, (∀Y ∈ int, ((G3 X Y) = Y)))
L100Hypothesis H16 : (∀X ∈ int, ((H3 X) = X))
L101Hypothesis H17 : (I3 = 1)
L102Hypothesis H18 : (J3 = (2 + (2 * (2 + (2 + 2)))))
L103Hypothesis H19 : (∀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)))))))
L104Hypothesis H20 : (∀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)))))))
L105Hypothesis H21 : (∀X ∈ int, ((W3 X) = (U3 (H3 X) I3 J3)))
L106Hypothesis H22 : (∀X ∈ int, ((H2 X) = (W3 X)))
L107Hypothesis H23 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y))))))
L108Hypothesis H24 : (∀X ∈ int, ((V2 X) = (U2 G2 (H2 X))))
L109Hypothesis H25 : (∀X ∈ int, ((F4 X) = (X * X)))
L110Hypothesis H26 : (G4 = 1)
L111Hypothesis H27 : (∀X ∈ int, (∀Y ∈ int, ((F5 X Y) = (X * Y))))
L112Hypothesis H28 : (∀X ∈ int, (∀Y ∈ int, ((G5 X Y) = Y)))
L113Hypothesis H29 : (∀X ∈ int, ((H5 X) = (2 + X)))
L114Hypothesis H30 : (I5 = 1)
L115Hypothesis H31 : (J5 = (2 + (2 * (2 + (2 + 2)))))
L116Hypothesis H32 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U5 X Y Z) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y Z) (V5 (X + - 1) Y Z)))))))
L117Hypothesis H33 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V5 X Y Z) = (if (X <= 0) then Z else (G5 (U5 (X + - 1) Y Z) (V5 (X + - 1) Y Z)))))))
L118Hypothesis H34 : (∀X ∈ int, ((W5 X) = (U5 (H5 X) I5 J5)))
L119Hypothesis H35 : (∀X ∈ int, ((H4 X) = (W5 X)))
L120Hypothesis H36 : (∀X ∈ int, (∀Y ∈ int, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y))))))
L121Hypothesis H37 : (∀X ∈ int, ((V4 X) = (U4 G4 (H4 X))))
L122Hypothesis H38 : (∀X ∈ int, ((FAST X) = ((V2 X) * (V4 X))))
L123Theorem. (
A13873)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.