Beginning of Section A1553
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 F0 : set → set
L10Hypothesis HF0 : ∀x0 ∈ int, F0 x0 ∈ int
L11Variable G0 : set → set
L12Hypothesis HG0 : ∀x0 ∈ int, G0 x0 ∈ int
L14Hypothesis HH0 : H0 ∈ int
L15Variable U0 : set → set → set
L16Hypothesis HU0 : ∀x0 ∈ int, ∀x1 ∈ int, U0 x0 x1 ∈ int
L17Variable V0 : set → set
L18Hypothesis HV0 : ∀x0 ∈ int, V0 x0 ∈ int
L19Variable F1 : set → set
L20Hypothesis HF1 : ∀x0 ∈ int, F1 x0 ∈ int
L21Variable G1 : set → set
L22Hypothesis HG1 : ∀x0 ∈ int, G1 x0 ∈ int
L24Hypothesis HH1 : H1 ∈ int
L25Variable U1 : set → set → set
L26Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, U1 x0 x1 ∈ int
L27Variable V1 : set → set
L28Hypothesis HV1 : ∀x0 ∈ int, V1 x0 ∈ int
L29Variable F2 : set → set
L30Hypothesis HF2 : ∀x0 ∈ int, F2 x0 ∈ int
L31Variable G2 : set → set
L32Hypothesis HG2 : ∀x0 ∈ int, G2 x0 ∈ int
L34Hypothesis HH2 : H2 ∈ int
L35Variable U2 : set → set → set
L36Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, U2 x0 x1 ∈ int
L37Variable V2 : set → set
L38Hypothesis HV2 : ∀x0 ∈ int, V2 x0 ∈ int
L39Variable F3 : set → set
L40Hypothesis HF3 : ∀x0 ∈ int, F3 x0 ∈ int
L41Variable G3 : set → set
L42Hypothesis HG3 : ∀x0 ∈ int, G3 x0 ∈ int
L44Hypothesis HH3 : H3 ∈ int
L45Variable U3 : set → set → set
L46Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, U3 x0 x1 ∈ int
L47Variable V3 : set → set
L48Hypothesis HV3 : ∀x0 ∈ int, V3 x0 ∈ int
L49Variable SMALL : set → set
L50Hypothesis HSMALL : ∀x0 ∈ int, SMALL x0 ∈ int
L51Variable F4 : set → set
L52Hypothesis HF4 : ∀x0 ∈ int, F4 x0 ∈ int
L54Hypothesis HG4 : G4 ∈ int
L55Variable F5 : set → set
L56Hypothesis HF5 : ∀x0 ∈ int, F5 x0 ∈ int
L57Variable G5 : set → set
L58Hypothesis HG5 : ∀x0 ∈ int, G5 x0 ∈ int
L60Hypothesis HH5 : H5 ∈ int
L61Variable U5 : set → set → set
L62Hypothesis HU5 : ∀x0 ∈ int, ∀x1 ∈ int, U5 x0 x1 ∈ int
L63Variable V5 : set → set
L64Hypothesis HV5 : ∀x0 ∈ int, V5 x0 ∈ int
L65Variable H4 : set → set
L66Hypothesis HH4 : ∀x0 ∈ int, H4 x0 ∈ int
L67Variable U4 : set → set → set
L68Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, U4 x0 x1 ∈ int
L69Variable V4 : set → set
L70Hypothesis HV4 : ∀x0 ∈ int, V4 x0 ∈ 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
L77Variable F7 : set → set
L78Hypothesis HF7 : ∀x0 ∈ int, F7 x0 ∈ int
L79Variable G7 : set → set
L80Hypothesis HG7 : ∀x0 ∈ int, G7 x0 ∈ int
L82Hypothesis HH7 : H7 ∈ int
L83Variable U7 : set → set → set
L84Hypothesis HU7 : ∀x0 ∈ int, ∀x1 ∈ int, U7 x0 x1 ∈ int
L85Variable V7 : set → set
L86Hypothesis HV7 : ∀x0 ∈ int, V7 x0 ∈ int
L87Variable I6 : set → set
L88Hypothesis HI6 : ∀x0 ∈ int, I6 x0 ∈ int
L90Hypothesis HJ6 : J6 ∈ int
L91Variable U6 : set → set → set → set
L92Hypothesis HU6 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U6 x0 x1 x2 ∈ int
L93Variable V6 : set → set → set → set
L94Hypothesis HV6 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V6 x0 x1 x2 ∈ int
L95Variable W6 : set → set
L96Hypothesis HW6 : ∀x0 ∈ int, W6 x0 ∈ int
L97Variable F8 : set → set → set
L98Hypothesis HF8 : ∀x0 ∈ int, ∀x1 ∈ int, F8 x0 x1 ∈ int
L99Variable G8 : set → set → set
L100Hypothesis HG8 : ∀x0 ∈ int, ∀x1 ∈ int, G8 x0 x1 ∈ int
L101Variable H8 : set → set
L102Hypothesis HH8 : ∀x0 ∈ int, H8 x0 ∈ int
L104Hypothesis HI8 : I8 ∈ int
L106Hypothesis HJ8 : J8 ∈ int
L107Variable U8 : set → set → set → set
L108Hypothesis HU8 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U8 x0 x1 x2 ∈ int
L109Variable V8 : set → set → set → set
L110Hypothesis HV8 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V8 x0 x1 x2 ∈ int
L111Variable W8 : set → set
L112Hypothesis HW8 : ∀x0 ∈ int, W8 x0 ∈ int
L113Variable FAST : set → set
L114Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L115Hypothesis H1 : (∀X ∈ int, ((F0 X) = ((X + X) + X)))
L116Hypothesis H2 : (∀X ∈ int, ((G0 X) = X))
L117Hypothesis H3 : (H0 = 1)
L118Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y))))))
L119Hypothesis H5 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))
L120Hypothesis H6 : (∀X ∈ int, ((F1 X) = (X + X)))
L121Hypothesis H7 : (∀X ∈ int, ((G1 X) = X))
L122Hypothesis H8 : (H1 = 1)
L123Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))
L124Hypothesis H10 : (∀X ∈ int, ((V1 X) = (U1 (G1 X) H1)))
L125Hypothesis H11 : (∀X ∈ int, ((F2 X) = (X + X)))
L126Hypothesis H12 : (∀X ∈ int, ((G2 X) = X))
L127Hypothesis H13 : (H2 = 1)
L128Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y))))))
L129Hypothesis H15 : (∀X ∈ int, ((V2 X) = (U2 (G2 X) H2)))
L130Hypothesis H16 : (∀X ∈ int, ((F3 X) = ((2 * (X + X)) + X)))
L131Hypothesis H17 : (∀X ∈ int, ((G3 X) = X))
L132Hypothesis H18 : (H3 = 1)
L133Hypothesis H19 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y))))))
L134Hypothesis H20 : (∀X ∈ int, ((V3 X) = (U3 (G3 X) H3)))
L135Hypothesis H21 : (∀X ∈ int, ((SMALL X) = ((((V0 X) + (V1 X)) * (1 + (V2 X))) + (1 + (V3 X)))))
L136Hypothesis H22 : (∀X ∈ int, ((F4 X) = ((X * X) + X)))
L137Hypothesis H23 : (G4 = 1)
L138Hypothesis H24 : (∀X ∈ int, ((F5 X) = (X + X)))
L139Hypothesis H25 : (∀X ∈ int, ((G5 X) = X))
L140Hypothesis H26 : (H5 = 1)
L141Hypothesis H27 : (∀X ∈ int, (∀Y ∈ int, ((U5 X Y) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y))))))
L142Hypothesis H28 : (∀X ∈ int, ((V5 X) = (U5 (G5 X) H5)))
L143Hypothesis H29 : (∀X ∈ int, ((H4 X) = (V5 X)))
L144Hypothesis H30 : (∀X ∈ int, (∀Y ∈ int, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y))))))
L145Hypothesis H31 : (∀X ∈ int, ((V4 X) = (U4 G4 (H4 X))))
L146Hypothesis H32 : (∀X ∈ int, (∀Y ∈ int, ((F6 X Y) = (X * Y))))
L147Hypothesis H33 : (∀X ∈ int, (∀Y ∈ int, ((G6 X Y) = Y)))
L148Hypothesis H34 : (∀X ∈ int, ((H6 X) = X))
L149Hypothesis H35 : (∀X ∈ int, ((F7 X) = (X + X)))
L150Hypothesis H36 : (∀X ∈ int, ((G7 X) = X))
L151Hypothesis H37 : (H7 = 1)
L152Hypothesis H38 : (∀X ∈ int, (∀Y ∈ int, ((U7 X Y) = (if (X <= 0) then Y else (F7 (U7 (X + - 1) Y))))))
L153Hypothesis H39 : (∀X ∈ int, ((V7 X) = (U7 (G7 X) H7)))
L154Hypothesis H40 : (∀X ∈ int, ((I6 X) = (1 + (V7 X))))
L155Hypothesis H41 : (J6 = (1 + 2))
L156Hypothesis H42 : (∀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)))))))
L157Hypothesis H43 : (∀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)))))))
L158Hypothesis H44 : (∀X ∈ int, ((W6 X) = (U6 (H6 X) (I6 X) J6)))
L159Hypothesis H45 : (∀X ∈ int, (∀Y ∈ int, ((F8 X Y) = (X * Y))))
L160Hypothesis H46 : (∀X ∈ int, (∀Y ∈ int, ((G8 X Y) = Y)))
L161Hypothesis H47 : (∀X ∈ int, ((H8 X) = X))
L162Hypothesis H48 : (I8 = 1)
L163Hypothesis H49 : (J8 = (1 + (2 + 2)))
L164Hypothesis H50 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U8 X Y Z) = (if (X <= 0) then Y else (F8 (U8 (X + - 1) Y Z) (V8 (X + - 1) Y Z)))))))
L165Hypothesis H51 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V8 X Y Z) = (if (X <= 0) then Z else (G8 (U8 (X + - 1) Y Z) (V8 (X + - 1) Y Z)))))))
L166Hypothesis H52 : (∀X ∈ int, ((W8 X) = (U8 (H8 X) I8 J8)))
L167Hypothesis H53 : (∀X ∈ int, ((FAST X) = (((1 + (V4 X)) + (W6 X)) + (W8 X))))
L168Theorem. (
A1553)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.