Beginning of Section A34673
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
L11Variable G1 : set → set → set
L12Hypothesis HG1 : ∀x0 ∈ int, ∀x1 ∈ int, G1 x0 x1 ∈ 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 → set
L18Hypothesis HV1 : ∀x0 ∈ int, ∀x1 ∈ int, V1 x0 x1 ∈ int
L19Variable F0 : set → set → set
L20Hypothesis HF0 : ∀x0 ∈ int, ∀x1 ∈ int, F0 x0 x1 ∈ int
L21Variable G0 : set → set → set
L22Hypothesis HG0 : ∀x0 ∈ int, ∀x1 ∈ int, G0 x0 x1 ∈ int
L24Hypothesis HH0 : H0 ∈ int
L26Hypothesis HI0 : I0 ∈ int
L27Variable J0 : set → set
L28Hypothesis HJ0 : ∀x0 ∈ int, J0 x0 ∈ int
L29Variable U0 : set → set → set → set
L30Hypothesis HU0 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U0 x0 x1 x2 ∈ int
L31Variable V0 : set → set → set → set
L32Hypothesis HV0 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V0 x0 x1 x2 ∈ int
L33Variable W0 : set → set
L34Hypothesis HW0 : ∀x0 ∈ int, W0 x0 ∈ int
L35Variable F2 : set → set
L36Hypothesis HF2 : ∀x0 ∈ int, F2 x0 ∈ int
L37Variable G2 : set → set
L38Hypothesis HG2 : ∀x0 ∈ int, G2 x0 ∈ int
L40Hypothesis HH2 : H2 ∈ int
L41Variable U2 : set → set → set
L42Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, U2 x0 x1 ∈ int
L43Variable V2 : set → set
L44Hypothesis HV2 : ∀x0 ∈ int, V2 x0 ∈ int
L45Variable F3 : set → set
L46Hypothesis HF3 : ∀x0 ∈ int, F3 x0 ∈ int
L47Variable G3 : set → set
L48Hypothesis HG3 : ∀x0 ∈ int, G3 x0 ∈ int
L50Hypothesis HH3 : H3 ∈ int
L51Variable U3 : set → set → set
L52Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, U3 x0 x1 ∈ int
L53Variable V3 : set → set
L54Hypothesis HV3 : ∀x0 ∈ int, V3 x0 ∈ int
L55Variable SMALL : set → set
L56Hypothesis HSMALL : ∀x0 ∈ int, SMALL x0 ∈ int
L57Variable F4 : set → set
L58Hypothesis HF4 : ∀x0 ∈ int, F4 x0 ∈ int
L59Variable G4 : set → set
L60Hypothesis HG4 : ∀x0 ∈ int, G4 x0 ∈ int
L62Hypothesis HH4 : H4 ∈ int
L63Variable U4 : set → set → set
L64Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, U4 x0 x1 ∈ int
L65Variable V4 : set → set
L66Hypothesis HV4 : ∀x0 ∈ int, V4 x0 ∈ int
L67Variable F5 : set → set
L68Hypothesis HF5 : ∀x0 ∈ int, F5 x0 ∈ int
L69Variable G5 : set → set
L70Hypothesis HG5 : ∀x0 ∈ int, G5 x0 ∈ int
L72Hypothesis HH5 : H5 ∈ int
L73Variable U5 : set → set → set
L74Hypothesis HU5 : ∀x0 ∈ int, ∀x1 ∈ int, U5 x0 x1 ∈ int
L75Variable V5 : set → set
L76Hypothesis HV5 : ∀x0 ∈ int, V5 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 F7 : set → set → set
L88Hypothesis HF7 : ∀x0 ∈ int, ∀x1 ∈ int, F7 x0 x1 ∈ int
L89Variable G7 : set → set → set
L90Hypothesis HG7 : ∀x0 ∈ int, ∀x1 ∈ int, G7 x0 x1 ∈ int
L91Variable H7 : set → set
L92Hypothesis HH7 : ∀x0 ∈ int, H7 x0 ∈ int
L94Hypothesis HI7 : I7 ∈ int
L96Hypothesis HJ7 : J7 ∈ int
L97Variable U7 : set → set → set → set
L98Hypothesis HU7 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U7 x0 x1 x2 ∈ int
L99Variable V7 : set → set → set → set
L100Hypothesis HV7 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V7 x0 x1 x2 ∈ int
L101Variable W7 : set → set
L102Hypothesis HW7 : ∀x0 ∈ int, W7 x0 ∈ int
L103Variable FAST : set → set
L104Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L105Hypothesis H1 : (∀X ∈ int, ((F1 X) = (X + X)))
L106Hypothesis H2 : (∀X ∈ int, (∀Y ∈ int, ((G1 X Y) = Y)))
L107Hypothesis H3 : (∀X ∈ int, ((H1 X) = X))
L108Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))
L109Hypothesis H5 : (∀X ∈ int, (∀Y ∈ int, ((V1 X Y) = (U1 (G1 X Y) (H1 X)))))
L110Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = (1 + (V1 X Y)))))
L111Hypothesis H7 : (∀X ∈ int, (∀Y ∈ int, ((G0 X Y) = Y)))
L112Hypothesis H8 : (H0 = (1 + 2))
L113Hypothesis H9 : (I0 = 1)
L114Hypothesis H10 : (∀X ∈ int, ((J0 X) = X))
L115Hypothesis H11 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U0 X Y Z) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y Z) (V0 (X + - 1) Y Z)))))))
L116Hypothesis H12 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V0 X Y Z) = (if (X <= 0) then Z else (G0 (U0 (X + - 1) Y Z) (V0 (X + - 1) Y Z)))))))
L117Hypothesis H13 : (∀X ∈ int, ((W0 X) = (U0 H0 I0 (J0 X))))
L118Hypothesis H14 : (∀X ∈ int, ((F2 X) = ((2 * (X + X)) + X)))
L119Hypothesis H15 : (∀X ∈ int, ((G2 X) = X))
L120Hypothesis H16 : (H2 = 1)
L121Hypothesis H17 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y))))))
L122Hypothesis H18 : (∀X ∈ int, ((V2 X) = (U2 (G2 X) H2)))
L123Hypothesis H19 : (∀X ∈ int, ((F3 X) = ((X + X) + X)))
L124Hypothesis H20 : (∀X ∈ int, ((G3 X) = X))
L125Hypothesis H21 : (H3 = 1)
L126Hypothesis H22 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y))))))
L127Hypothesis H23 : (∀X ∈ int, ((V3 X) = (U3 (G3 X) H3)))
L128Hypothesis H24 : (∀X ∈ int, ((SMALL X) = (((W0 X) * (1 + (V2 X))) * (1 + (V3 X)))))
L129Hypothesis H25 : (∀X ∈ int, ((F4 X) = ((2 * (X + X)) + X)))
L130Hypothesis H26 : (∀X ∈ int, ((G4 X) = X))
L131Hypothesis H27 : (H4 = 1)
L132Hypothesis H28 : (∀X ∈ int, (∀Y ∈ int, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y))))))
L133Hypothesis H29 : (∀X ∈ int, ((V4 X) = (U4 (G4 X) H4)))
L134Hypothesis H30 : (∀X ∈ int, ((F5 X) = (X + X)))
L135Hypothesis H31 : (∀X ∈ int, ((G5 X) = X))
L136Hypothesis H32 : (H5 = 1)
L137Hypothesis H33 : (∀X ∈ int, (∀Y ∈ int, ((U5 X Y) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y))))))
L138Hypothesis H34 : (∀X ∈ int, ((V5 X) = (U5 (G5 X) H5)))
L139Hypothesis H35 : (∀X ∈ int, ((F6 X) = ((X + X) + X)))
L140Hypothesis H36 : (∀X ∈ int, ((G6 X) = X))
L141Hypothesis H37 : (H6 = 1)
L142Hypothesis H38 : (∀X ∈ int, (∀Y ∈ int, ((U6 X Y) = (if (X <= 0) then Y else (F6 (U6 (X + - 1) Y))))))
L143Hypothesis H39 : (∀X ∈ int, ((V6 X) = (U6 (G6 X) H6)))
L144Hypothesis H40 : (∀X ∈ int, (∀Y ∈ int, ((F7 X Y) = (X * Y))))
L145Hypothesis H41 : (∀X ∈ int, (∀Y ∈ int, ((G7 X Y) = Y)))
L146Hypothesis H42 : (∀X ∈ int, ((H7 X) = X))
L147Hypothesis H43 : (I7 = 1)
L148Hypothesis H44 : (J7 = (2 + 2))
L149Hypothesis H45 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U7 X Y Z) = (if (X <= 0) then Y else (F7 (U7 (X + - 1) Y Z) (V7 (X + - 1) Y Z)))))))
L150Hypothesis H46 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V7 X Y Z) = (if (X <= 0) then Z else (G7 (U7 (X + - 1) Y Z) (V7 (X + - 1) Y Z)))))))
L151Hypothesis H47 : (∀X ∈ int, ((W7 X) = (U7 (H7 X) I7 J7)))
L152Hypothesis H48 : (∀X ∈ int, ((FAST X) = ((((1 + (V4 X)) * (1 + (V5 X))) * (1 + (V6 X))) * (1 + (W7 X)))))
L153Theorem. (
A34673)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.