Beginning of Section A91038
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 → set
L10Hypothesis HF1 : ∀x0 ∈ int, ∀x1 ∈ int, F1 x0 x1 ∈ int
L12Hypothesis HG1 : G1 ∈ int
L13Variable H1 : set → set
L14Hypothesis HH1 : ∀x0 ∈ int, H1 x0 ∈ int
L15Variable U1 : set → set → set
L16Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, U1 x0 x1 ∈ int
L17Variable V1 : set → set
L18Hypothesis HV1 : ∀x0 ∈ int, V1 x0 ∈ int
L19Variable F0 : set → set
L20Hypothesis HF0 : ∀x0 ∈ int, F0 x0 ∈ int
L21Variable G0 : set → set
L22Hypothesis HG0 : ∀x0 ∈ int, G0 x0 ∈ int
L23Variable F2 : set → set
L24Hypothesis HF2 : ∀x0 ∈ int, F2 x0 ∈ int
L25Variable G2 : set → set
L26Hypothesis HG2 : ∀x0 ∈ int, G2 x0 ∈ int
L28Hypothesis HH2 : H2 ∈ int
L29Variable U2 : set → set → set
L30Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, U2 x0 x1 ∈ int
L31Variable V2 : set → set
L32Hypothesis HV2 : ∀x0 ∈ int, V2 x0 ∈ int
L33Variable H0 : set → set
L34Hypothesis HH0 : ∀x0 ∈ int, H0 x0 ∈ int
L35Variable U0 : set → set → set
L36Hypothesis HU0 : ∀x0 ∈ int, ∀x1 ∈ int, U0 x0 x1 ∈ int
L37Variable V0 : set → set
L38Hypothesis HV0 : ∀x0 ∈ int, V0 x0 ∈ int
L39Variable SMALL : set → set
L40Hypothesis HSMALL : ∀x0 ∈ int, SMALL x0 ∈ int
L41Variable F3 : set → set
L42Hypothesis HF3 : ∀x0 ∈ int, F3 x0 ∈ int
L44Hypothesis HG3 : G3 ∈ int
L45Variable F4 : set → set
L46Hypothesis HF4 : ∀x0 ∈ int, F4 x0 ∈ int
L47Variable G4 : set → set
L48Hypothesis HG4 : ∀x0 ∈ int, G4 x0 ∈ int
L50Hypothesis HH4 : H4 ∈ int
L51Variable U4 : set → set → set
L52Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, U4 x0 x1 ∈ int
L53Variable V4 : set → set
L54Hypothesis HV4 : ∀x0 ∈ int, V4 x0 ∈ int
L55Variable H3 : set → set
L56Hypothesis HH3 : ∀x0 ∈ int, H3 x0 ∈ int
L57Variable U3 : set → set → set
L58Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, U3 x0 x1 ∈ int
L59Variable V3 : set → set
L60Hypothesis HV3 : ∀x0 ∈ int, V3 x0 ∈ 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 F6 : set → set
L78Hypothesis HF6 : ∀x0 ∈ int, F6 x0 ∈ int
L79Variable G6 : set → set
L80Hypothesis HG6 : ∀x0 ∈ int, G6 x0 ∈ int
L82Hypothesis HH6 : H6 ∈ int
L83Variable U6 : set → set → set
L84Hypothesis HU6 : ∀x0 ∈ int, ∀x1 ∈ int, U6 x0 x1 ∈ int
L85Variable V6 : set → set
L86Hypothesis HV6 : ∀x0 ∈ int, V6 x0 ∈ int
L87Variable FAST : set → set
L88Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L89Hypothesis H1 : (∀X ∈ int, (∀Y ∈ int, ((F1 X Y) = ((X * Y) + X))))
L90Hypothesis H2 : (G1 = (2 + 2))
L91Hypothesis H3 : (∀X ∈ int, ((H1 X) = X))
L92Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y) X)))))
L93Hypothesis H5 : (∀X ∈ int, ((V1 X) = (U1 G1 (H1 X))))
L94Hypothesis H6 : (∀X ∈ int, ((F0 X) = (V1 X)))
L95Hypothesis H7 : (∀X ∈ int, ((G0 X) = X))
L96Hypothesis H8 : (∀X ∈ int, ((F2 X) = (1 + (2 * ((X + X) + X)))))
L97Hypothesis H9 : (∀X ∈ int, ((G2 X) = X))
L98Hypothesis H10 : (H2 = 1)
L99Hypothesis H11 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y))))))
L100Hypothesis H12 : (∀X ∈ int, ((V2 X) = (U2 (G2 X) H2)))
L101Hypothesis H13 : (∀X ∈ int, ((H0 X) = (V2 X)))
L102Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y))))))
L103Hypothesis H15 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) (H0 X))))
L104Hypothesis H16 : (∀X ∈ int, ((SMALL X) = (V0 X)))
L105Hypothesis H17 : (∀X ∈ int, ((F3 X) = ((X * X) * X)))
L106Hypothesis H18 : (G3 = 1)
L107Hypothesis H19 : (∀X ∈ int, ((F4 X) = (X + X)))
L108Hypothesis H20 : (∀X ∈ int, ((G4 X) = X))
L109Hypothesis H21 : (H4 = 1)
L110Hypothesis H22 : (∀X ∈ int, (∀Y ∈ int, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y))))))
L111Hypothesis H23 : (∀X ∈ int, ((V4 X) = (U4 (G4 X) H4)))
L112Hypothesis H24 : (∀X ∈ int, ((H3 X) = (V4 X)))
L113Hypothesis H25 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y))))))
L114Hypothesis H26 : (∀X ∈ int, ((V3 X) = (U3 G3 (H3 X))))
L115Hypothesis H27 : (∀X ∈ int, (∀Y ∈ int, ((F5 X Y) = (X * Y))))
L116Hypothesis H28 : (∀X ∈ int, (∀Y ∈ int, ((G5 X Y) = Y)))
L117Hypothesis H29 : (∀X ∈ int, ((H5 X) = X))
L118Hypothesis H30 : (I5 = 1)
L119Hypothesis H31 : (J5 = ((2 * (2 * (2 + 2))) + - 1))
L120Hypothesis 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)))))))
L121Hypothesis 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)))))))
L122Hypothesis H34 : (∀X ∈ int, ((W5 X) = (U5 (H5 X) I5 J5)))
L123Hypothesis H35 : (∀X ∈ int, ((F6 X) = (1 + (2 * ((X + X) + X)))))
L124Hypothesis H36 : (∀X ∈ int, ((G6 X) = X))
L125Hypothesis H37 : (H6 = 1)
L126Hypothesis H38 : (∀X ∈ int, (∀Y ∈ int, ((U6 X Y) = (if (X <= 0) then Y else (F6 (U6 (X + - 1) Y))))))
L127Hypothesis H39 : (∀X ∈ int, ((V6 X) = (U6 (G6 X) H6)))
L128Hypothesis H40 : (∀X ∈ int, ((FAST X) = (((V3 X) * (W5 X)) * (V6 X))))
L129Theorem. (
A91038)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.