Beginning of Section A13797
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 F2 : set → set
L22Hypothesis HF2 : ∀x0 ∈ int, F2 x0 ∈ int
L24Hypothesis HG2 : G2 ∈ int
L25Variable H2 : set → set
L26Hypothesis HH2 : ∀x0 ∈ int, H2 x0 ∈ int
L27Variable U2 : set → set → set
L28Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, U2 x0 x1 ∈ int
L29Variable V2 : set → set
L30Hypothesis HV2 : ∀x0 ∈ int, V2 x0 ∈ int
L31Variable G0 : set → set
L32Hypothesis HG0 : ∀x0 ∈ int, G0 x0 ∈ int
L34Hypothesis HH0 : H0 ∈ 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 → set
L46Hypothesis HF4 : ∀x0 ∈ int, ∀x1 ∈ int, F4 x0 x1 ∈ int
L47Variable G4 : set → set → set
L48Hypothesis HG4 : ∀x0 ∈ int, ∀x1 ∈ int, G4 x0 x1 ∈ int
L49Variable H4 : set → set
L50Hypothesis HH4 : ∀x0 ∈ int, H4 x0 ∈ int
L52Hypothesis HI4 : I4 ∈ int
L54Hypothesis HJ4 : J4 ∈ int
L55Variable U4 : set → set → set → set
L56Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U4 x0 x1 x2 ∈ int
L57Variable V4 : set → set → set → set
L58Hypothesis HV4 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V4 x0 x1 x2 ∈ int
L59Variable W4 : set → set
L60Hypothesis HW4 : ∀x0 ∈ int, W4 x0 ∈ int
L61Variable H3 : set → set
L62Hypothesis HH3 : ∀x0 ∈ int, H3 x0 ∈ int
L63Variable U3 : set → set → set
L64Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, U3 x0 x1 ∈ int
L65Variable V3 : set → set
L66Hypothesis HV3 : ∀x0 ∈ int, V3 x0 ∈ int
L67Variable F5 : set → set
L68Hypothesis HF5 : ∀x0 ∈ int, F5 x0 ∈ int
L70Hypothesis HG5 : G5 ∈ int
L71Variable F6 : set → set → set
L72Hypothesis HF6 : ∀x0 ∈ int, ∀x1 ∈ int, F6 x0 x1 ∈ int
L73Variable G6 : set → set → set
L74Hypothesis HG6 : ∀x0 ∈ int, ∀x1 ∈ int, G6 x0 x1 ∈ int
L75Variable H6 : set → set
L76Hypothesis HH6 : ∀x0 ∈ int, H6 x0 ∈ int
L78Hypothesis HI6 : I6 ∈ int
L80Hypothesis HJ6 : J6 ∈ int
L81Variable U6 : set → set → set → set
L82Hypothesis HU6 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U6 x0 x1 x2 ∈ int
L83Variable V6 : set → set → set → set
L84Hypothesis HV6 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V6 x0 x1 x2 ∈ int
L85Variable W6 : set → set
L86Hypothesis HW6 : ∀x0 ∈ int, W6 x0 ∈ int
L87Variable H5 : set → set
L88Hypothesis HH5 : ∀x0 ∈ int, H5 x0 ∈ int
L89Variable U5 : set → set → set
L90Hypothesis HU5 : ∀x0 ∈ int, ∀x1 ∈ int, U5 x0 x1 ∈ int
L91Variable V5 : set → set
L92Hypothesis HV5 : ∀x0 ∈ int, V5 x0 ∈ int
L93Variable FAST : set → set
L94Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L95Hypothesis H1 : (∀X ∈ int, (∀Y ∈ int, ((F1 X Y) = ((2 + Y) * X))))
L96Hypothesis H2 : (G1 = 2)
L97Hypothesis H3 : (∀X ∈ int, ((H1 X) = X))
L98Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y) X)))))
L99Hypothesis H5 : (∀X ∈ int, ((V1 X) = (U1 G1 (H1 X))))
L100Hypothesis H6 : (∀X ∈ int, ((F0 X) = (V1 X)))
L101Hypothesis H7 : (∀X ∈ int, ((F2 X) = (1 + (X + X))))
L102Hypothesis H8 : (G2 = 2)
L103Hypothesis H9 : (∀X ∈ int, ((H2 X) = X))
L104Hypothesis H10 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y))))))
L105Hypothesis H11 : (∀X ∈ int, ((V2 X) = (U2 G2 (H2 X))))
L106Hypothesis H12 : (∀X ∈ int, ((G0 X) = (V2 X)))
L107Hypothesis H13 : (H0 = 1)
L108Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y))))))
L109Hypothesis H15 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))
L110Hypothesis H16 : (∀X ∈ int, ((SMALL X) = (V0 X)))
L111Hypothesis H17 : (∀X ∈ int, ((F3 X) = (X * X)))
L112Hypothesis H18 : (G3 = 1)
L113Hypothesis H19 : (∀X ∈ int, (∀Y ∈ int, ((F4 X Y) = (X * Y))))
L114Hypothesis H20 : (∀X ∈ int, (∀Y ∈ int, ((G4 X Y) = Y)))
L115Hypothesis H21 : (∀X ∈ int, ((H4 X) = X))
L116Hypothesis H22 : (I4 = 1)
L117Hypothesis H23 : (J4 = (2 * (2 + (2 + 2))))
L118Hypothesis H24 : (∀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)))))))
L119Hypothesis H25 : (∀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)))))))
L120Hypothesis H26 : (∀X ∈ int, ((W4 X) = (U4 (H4 X) I4 J4)))
L121Hypothesis H27 : (∀X ∈ int, ((H3 X) = (W4 X)))
L122Hypothesis H28 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y))))))
L123Hypothesis H29 : (∀X ∈ int, ((V3 X) = (U3 G3 (H3 X))))
L124Hypothesis H30 : (∀X ∈ int, ((F5 X) = (X * X)))
L125Hypothesis H31 : (G5 = 1)
L126Hypothesis H32 : (∀X ∈ int, (∀Y ∈ int, ((F6 X Y) = (X * Y))))
L127Hypothesis H33 : (∀X ∈ int, (∀Y ∈ int, ((G6 X Y) = Y)))
L128Hypothesis H34 : (∀X ∈ int, ((H6 X) = (1 + X)))
L129Hypothesis H35 : (I6 = 1)
L130Hypothesis H36 : (J6 = (2 * (2 + (2 + 2))))
L131Hypothesis H37 : (∀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)))))))
L132Hypothesis H38 : (∀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)))))))
L133Hypothesis H39 : (∀X ∈ int, ((W6 X) = (U6 (H6 X) I6 J6)))
L134Hypothesis H40 : (∀X ∈ int, ((H5 X) = (W6 X)))
L135Hypothesis H41 : (∀X ∈ int, (∀Y ∈ int, ((U5 X Y) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y))))))
L136Hypothesis H42 : (∀X ∈ int, ((V5 X) = (U5 G5 (H5 X))))
L137Hypothesis H43 : (∀X ∈ int, ((FAST X) = (((V3 X) * (2 * (2 + (2 + 2)))) * (V5 X))))
L138Theorem. (
A13797)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.