Beginning of Section A25991
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 F3 : set → set
L10Hypothesis HF3 : ∀x0 ∈ int, F3 x0 ∈ int
L11Variable G3 : set → set
L12Hypothesis HG3 : ∀x0 ∈ int, G3 x0 ∈ int
L13Variable F4 : set → set
L14Hypothesis HF4 : ∀x0 ∈ int, F4 x0 ∈ int
L15Variable G4 : set → set
L16Hypothesis HG4 : ∀x0 ∈ int, G4 x0 ∈ int
L18Hypothesis HH4 : H4 ∈ int
L19Variable U4 : set → set → set
L20Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, U4 x0 x1 ∈ int
L21Variable V4 : set → set
L22Hypothesis HV4 : ∀x0 ∈ int, V4 x0 ∈ int
L23Variable H3 : set → set
L24Hypothesis HH3 : ∀x0 ∈ int, H3 x0 ∈ int
L25Variable U3 : set → set → set
L26Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, U3 x0 x1 ∈ int
L27Variable V3 : set → set
L28Hypothesis HV3 : ∀x0 ∈ int, V3 x0 ∈ int
L29Variable F2 : set → set
L30Hypothesis HF2 : ∀x0 ∈ int, F2 x0 ∈ int
L32Hypothesis HG2 : G2 ∈ int
L33Variable H2 : set → set → set
L34Hypothesis HH2 : ∀x0 ∈ int, ∀x1 ∈ int, H2 x0 x1 ∈ int
L35Variable U2 : set → set → set
L36Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, U2 x0 x1 ∈ int
L37Variable V2 : set → set → set
L38Hypothesis HV2 : ∀x0 ∈ int, ∀x1 ∈ int, V2 x0 x1 ∈ int
L39Variable F1 : set → set → set
L40Hypothesis HF1 : ∀x0 ∈ int, ∀x1 ∈ int, F1 x0 x1 ∈ int
L41Variable G1 : set → set → set
L42Hypothesis HG1 : ∀x0 ∈ int, ∀x1 ∈ int, G1 x0 x1 ∈ int
L44Hypothesis HH1 : H1 ∈ int
L45Variable U1 : set → set → set
L46Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, U1 x0 x1 ∈ int
L47Variable V1 : set → set → set
L48Hypothesis HV1 : ∀x0 ∈ int, ∀x1 ∈ int, V1 x0 x1 ∈ int
L49Variable F0 : set → set → set
L50Hypothesis HF0 : ∀x0 ∈ int, ∀x1 ∈ int, F0 x0 x1 ∈ int
L51Variable G0 : set → set
L52Hypothesis HG0 : ∀x0 ∈ int, G0 x0 ∈ int
L54Hypothesis HH0 : H0 ∈ int
L55Variable U0 : set → set → set
L56Hypothesis HU0 : ∀x0 ∈ int, ∀x1 ∈ int, U0 x0 x1 ∈ int
L57Variable V0 : set → set
L58Hypothesis HV0 : ∀x0 ∈ int, V0 x0 ∈ int
L59Variable SMALL : set → set
L60Hypothesis HSMALL : ∀x0 ∈ int, SMALL 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 F8 : set → set
L78Hypothesis HF8 : ∀x0 ∈ int, F8 x0 ∈ int
L79Variable G8 : set → set
L80Hypothesis HG8 : ∀x0 ∈ int, G8 x0 ∈ int
L82Hypothesis HH8 : H8 ∈ int
L83Variable U8 : set → set → set
L84Hypothesis HU8 : ∀x0 ∈ int, ∀x1 ∈ int, U8 x0 x1 ∈ int
L85Variable V8 : set → set
L86Hypothesis HV8 : ∀x0 ∈ int, V8 x0 ∈ int
L87Variable F6 : set → set
L88Hypothesis HF6 : ∀x0 ∈ int, F6 x0 ∈ int
L90Hypothesis HG6 : G6 ∈ int
L91Variable H6 : set → set → set
L92Hypothesis HH6 : ∀x0 ∈ int, ∀x1 ∈ int, H6 x0 x1 ∈ int
L93Variable U6 : set → set → set
L94Hypothesis HU6 : ∀x0 ∈ int, ∀x1 ∈ int, U6 x0 x1 ∈ int
L95Variable V6 : set → set → set
L96Hypothesis HV6 : ∀x0 ∈ int, ∀x1 ∈ int, V6 x0 x1 ∈ int
L97Variable F5 : set → set → set
L98Hypothesis HF5 : ∀x0 ∈ int, ∀x1 ∈ int, F5 x0 x1 ∈ int
L99Variable G5 : set → set
L100Hypothesis HG5 : ∀x0 ∈ int, G5 x0 ∈ int
L102Hypothesis HH5 : H5 ∈ int
L103Variable U5 : set → set → set
L104Hypothesis HU5 : ∀x0 ∈ int, ∀x1 ∈ int, U5 x0 x1 ∈ int
L105Variable V5 : set → set
L106Hypothesis HV5 : ∀x0 ∈ int, V5 x0 ∈ int
L107Variable FAST : set → set
L108Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L109Hypothesis H1 : (∀X ∈ int, ((F3 X) = (2 * ((X + X) + X))))
L110Hypothesis H2 : (∀X ∈ int, ((G3 X) = X))
L111Hypothesis H3 : (∀X ∈ int, ((F4 X) = (X + X)))
L112Hypothesis H4 : (∀X ∈ int, ((G4 X) = X))
L113Hypothesis H5 : (H4 = 2)
L114Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y))))))
L115Hypothesis H7 : (∀X ∈ int, ((V4 X) = (U4 (G4 X) H4)))
L116Hypothesis H8 : (∀X ∈ int, ((H3 X) = ((V4 X) + - 1)))
L117Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y))))))
L118Hypothesis H10 : (∀X ∈ int, ((V3 X) = (U3 (G3 X) (H3 X))))
L119Hypothesis H11 : (∀X ∈ int, ((F2 X) = (V3 X)))
L120Hypothesis H12 : (G2 = 1)
L121Hypothesis H13 : (∀X ∈ int, (∀Y ∈ int, ((H2 X Y) = Y)))
L122Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y))))))
L123Hypothesis H15 : (∀X ∈ int, (∀Y ∈ int, ((V2 X Y) = (U2 G2 (H2 X Y)))))
L124Hypothesis H16 : (∀X ∈ int, (∀Y ∈ int, ((F1 X Y) = (((V2 X Y) + (2 * (X + X))) + X))))
L125Hypothesis H17 : (∀X ∈ int, (∀Y ∈ int, ((G1 X Y) = Y)))
L126Hypothesis H18 : (H1 = 1)
L127Hypothesis H19 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y) X)))))
L128Hypothesis H20 : (∀X ∈ int, (∀Y ∈ int, ((V1 X Y) = (U1 (G1 X Y) H1))))
L129Hypothesis H21 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = (((V1 X Y) + X) + X))))
L130Hypothesis H22 : (∀X ∈ int, ((G0 X) = X))
L131Hypothesis H23 : (H0 = 1)
L132Hypothesis H24 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y) X)))))
L133Hypothesis H25 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))
L134Hypothesis H26 : (∀X ∈ int, ((SMALL X) = (V0 X)))
L135Hypothesis H27 : (∀X ∈ int, (∀Y ∈ int, ((F7 X Y) = ((2 * ((X + X) + X)) + Y))))
L136Hypothesis H28 : (∀X ∈ int, (∀Y ∈ int, ((G7 X Y) = (1 + ((Y + Y) + Y)))))
L137Hypothesis H29 : (∀X ∈ int, ((H7 X) = X))
L138Hypothesis H30 : (I7 = 1)
L139Hypothesis H31 : (J7 = (2 + 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, ((F8 X) = (X + X)))
L144Hypothesis H36 : (∀X ∈ int, ((G8 X) = X))
L145Hypothesis H37 : (H8 = 1)
L146Hypothesis H38 : (∀X ∈ int, (∀Y ∈ int, ((U8 X Y) = (if (X <= 0) then Y else (F8 (U8 (X + - 1) Y))))))
L147Hypothesis H39 : (∀X ∈ int, ((V8 X) = (U8 (G8 X) H8)))
L148Hypothesis H40 : (∀X ∈ int, ((F6 X) = ((W7 X) * (V8 X))))
L149Hypothesis H41 : (G6 = 1)
L150Hypothesis H42 : (∀X ∈ int, (∀Y ∈ int, ((H6 X Y) = Y)))
L151Hypothesis H43 : (∀X ∈ int, (∀Y ∈ int, ((U6 X Y) = (if (X <= 0) then Y else (F6 (U6 (X + - 1) Y))))))
L152Hypothesis H44 : (∀X ∈ int, (∀Y ∈ int, ((V6 X Y) = (U6 G6 (H6 X Y)))))
L153Hypothesis H45 : (∀X ∈ int, (∀Y ∈ int, ((F5 X Y) = (((V6 X Y) + (2 * (X + X))) + X))))
L154Hypothesis H46 : (∀X ∈ int, ((G5 X) = X))
L155Hypothesis H47 : (H5 = 1)
L156Hypothesis H48 : (∀X ∈ int, (∀Y ∈ int, ((U5 X Y) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y) X)))))
L157Hypothesis H49 : (∀X ∈ int, ((V5 X) = (U5 (G5 X) H5)))
L158Hypothesis H50 : (∀X ∈ int, ((FAST X) = (V5 X)))
L159Theorem. (
A25991)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.