Beginning of Section A20594
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
L12Hypothesis HG2 : ∀x0 ∈ int, G2 x0 ∈ int
L13Variable F3 : set → set
L14Hypothesis HF3 : ∀x0 ∈ int, F3 x0 ∈ int
L15Variable G3 : set → set
L16Hypothesis HG3 : ∀x0 ∈ int, G3 x0 ∈ int
L18Hypothesis HH3 : H3 ∈ int
L19Variable U3 : set → set → set
L20Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, U3 x0 x1 ∈ int
L21Variable V3 : set → set
L22Hypothesis HV3 : ∀x0 ∈ int, V3 x0 ∈ int
L23Variable H2 : set → set
L24Hypothesis HH2 : ∀x0 ∈ int, H2 x0 ∈ int
L25Variable U2 : set → set → set
L26Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, U2 x0 x1 ∈ int
L27Variable V2 : set → set
L28Hypothesis HV2 : ∀x0 ∈ int, V2 x0 ∈ int
L29Variable F1 : set → set
L30Hypothesis HF1 : ∀x0 ∈ int, F1 x0 ∈ int
L32Hypothesis HG1 : G1 ∈ int
L33Variable H1 : set → set → set
L34Hypothesis HH1 : ∀x0 ∈ int, ∀x1 ∈ int, H1 x0 x1 ∈ 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
L52Hypothesis HF6 : ∀x0 ∈ int, F6 x0 ∈ int
L53Variable G6 : set → set
L54Hypothesis HG6 : ∀x0 ∈ int, G6 x0 ∈ int
L56Hypothesis HH6 : H6 ∈ int
L57Variable U6 : set → set → set
L58Hypothesis HU6 : ∀x0 ∈ int, ∀x1 ∈ int, U6 x0 x1 ∈ int
L59Variable V6 : set → set
L60Hypothesis HV6 : ∀x0 ∈ int, V6 x0 ∈ int
L61Variable F7 : set → set → set
L62Hypothesis HF7 : ∀x0 ∈ int, ∀x1 ∈ int, F7 x0 x1 ∈ int
L63Variable G7 : set → set → set
L64Hypothesis HG7 : ∀x0 ∈ int, ∀x1 ∈ int, G7 x0 x1 ∈ int
L65Variable H7 : set → set
L66Hypothesis HH7 : ∀x0 ∈ int, H7 x0 ∈ int
L68Hypothesis HI7 : I7 ∈ int
L70Hypothesis HJ7 : J7 ∈ int
L71Variable U7 : set → set → set → set
L72Hypothesis HU7 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U7 x0 x1 x2 ∈ int
L73Variable V7 : set → set → set → set
L74Hypothesis HV7 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V7 x0 x1 x2 ∈ int
L75Variable W7 : set → set
L76Hypothesis HW7 : ∀x0 ∈ int, W7 x0 ∈ int
L77Variable F5 : set → set
L78Hypothesis HF5 : ∀x0 ∈ int, F5 x0 ∈ int
L80Hypothesis HG5 : G5 ∈ int
L81Variable H5 : set → set → set
L82Hypothesis HH5 : ∀x0 ∈ int, ∀x1 ∈ int, H5 x0 x1 ∈ int
L83Variable U5 : set → set → set
L84Hypothesis HU5 : ∀x0 ∈ int, ∀x1 ∈ int, U5 x0 x1 ∈ int
L85Variable V5 : set → set → set
L86Hypothesis HV5 : ∀x0 ∈ int, ∀x1 ∈ int, V5 x0 x1 ∈ int
L87Variable F4 : set → set → set
L88Hypothesis HF4 : ∀x0 ∈ int, ∀x1 ∈ int, F4 x0 x1 ∈ int
L89Variable G4 : set → set
L90Hypothesis HG4 : ∀x0 ∈ int, G4 x0 ∈ int
L92Hypothesis HH4 : H4 ∈ int
L93Variable U4 : set → set → set
L94Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, U4 x0 x1 ∈ int
L95Variable V4 : set → set
L96Hypothesis HV4 : ∀x0 ∈ int, V4 x0 ∈ int
L97Variable F8 : set → set
L98Hypothesis HF8 : ∀x0 ∈ int, F8 x0 ∈ int
L99Variable G8 : set → set
L100Hypothesis HG8 : ∀x0 ∈ int, G8 x0 ∈ int
L102Hypothesis HH8 : H8 ∈ int
L103Variable U8 : set → set → set
L104Hypothesis HU8 : ∀x0 ∈ int, ∀x1 ∈ int, U8 x0 x1 ∈ int
L105Variable V8 : set → set
L106Hypothesis HV8 : ∀x0 ∈ int, V8 x0 ∈ int
L107Variable FAST : set → set
L108Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L109Hypothesis H1 : (∀X ∈ int, ((F2 X) = (2 * ((X + X) + X))))
L110Hypothesis H2 : (∀X ∈ int, ((G2 X) = X))
L111Hypothesis H3 : (∀X ∈ int, ((F3 X) = (X + X)))
L112Hypothesis H4 : (∀X ∈ int, ((G3 X) = X))
L113Hypothesis H5 : (H3 = 2)
L114Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y))))))
L115Hypothesis H7 : (∀X ∈ int, ((V3 X) = (U3 (G3 X) H3)))
L116Hypothesis H8 : (∀X ∈ int, ((H2 X) = ((V3 X) + - 1)))
L117Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y))))))
L118Hypothesis H10 : (∀X ∈ int, ((V2 X) = (U2 (G2 X) (H2 X))))
L119Hypothesis H11 : (∀X ∈ int, ((F1 X) = (V2 X)))
L120Hypothesis H12 : (G1 = 1)
L121Hypothesis H13 : (∀X ∈ int, (∀Y ∈ int, ((H1 X Y) = Y)))
L122Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))
L123Hypothesis H15 : (∀X ∈ int, (∀Y ∈ int, ((V1 X Y) = (U1 G1 (H1 X Y)))))
L124Hypothesis H16 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = ((V1 X Y) + (2 * (2 * (X + X)))))))
L125Hypothesis H17 : (∀X ∈ int, ((G0 X) = X))
L126Hypothesis H18 : (H0 = 1)
L127Hypothesis H19 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y) X)))))
L128Hypothesis H20 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))
L129Hypothesis H21 : (∀X ∈ int, ((SMALL X) = (V0 X)))
L130Hypothesis H22 : (∀X ∈ int, ((F6 X) = (X + X)))
L131Hypothesis H23 : (∀X ∈ int, ((G6 X) = X))
L132Hypothesis H24 : (H6 = 2)
L133Hypothesis H25 : (∀X ∈ int, (∀Y ∈ int, ((U6 X Y) = (if (X <= 0) then Y else (F6 (U6 (X + - 1) Y))))))
L134Hypothesis H26 : (∀X ∈ int, ((V6 X) = (U6 (G6 X) H6)))
L135Hypothesis H27 : (∀X ∈ int, (∀Y ∈ int, ((F7 X Y) = (X * Y))))
L136Hypothesis H28 : (∀X ∈ int, (∀Y ∈ int, ((G7 X Y) = Y)))
L137Hypothesis H29 : (∀X ∈ int, ((H7 X) = X))
L138Hypothesis H30 : (I7 = 1)
L139Hypothesis H31 : (J7 = (1 + 2))
L140Hypothesis H32 : (∀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)))))))
L141Hypothesis H33 : (∀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)))))))
L142Hypothesis H34 : (∀X ∈ int, ((W7 X) = (U7 (H7 X) I7 J7)))
L143Hypothesis H35 : (∀X ∈ int, ((F5 X) = (((V6 X) + - 1) * (W7 X))))
L144Hypothesis H36 : (G5 = 1)
L145Hypothesis H37 : (∀X ∈ int, (∀Y ∈ int, ((H5 X Y) = Y)))
L146Hypothesis H38 : (∀X ∈ int, (∀Y ∈ int, ((U5 X Y) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y))))))
L147Hypothesis H39 : (∀X ∈ int, (∀Y ∈ int, ((V5 X Y) = (U5 G5 (H5 X Y)))))
L148Hypothesis H40 : (∀X ∈ int, (∀Y ∈ int, ((F4 X Y) = ((V5 X Y) + (2 * (X + X))))))
L149Hypothesis H41 : (∀X ∈ int, ((G4 X) = X))
L150Hypothesis H42 : (H4 = 1)
L151Hypothesis H43 : (∀X ∈ int, (∀Y ∈ int, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y) X)))))
L152Hypothesis H44 : (∀X ∈ int, ((V4 X) = (U4 (G4 X) H4)))
L153Hypothesis H45 : (∀X ∈ int, ((F8 X) = (X + X)))
L154Hypothesis H46 : (∀X ∈ int, ((G8 X) = X))
L155Hypothesis H47 : (H8 = 1)
L156Hypothesis H48 : (∀X ∈ int, (∀Y ∈ int, ((U8 X Y) = (if (X <= 0) then Y else (F8 (U8 (X + - 1) Y))))))
L157Hypothesis H49 : (∀X ∈ int, ((V8 X) = (U8 (G8 X) H8)))
L158Hypothesis H50 : (∀X ∈ int, ((FAST X) = ((V4 X) * (V8 X))))
L159Theorem. (
A20594)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.