Beginning of Section A28155
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 → set
L10Hypothesis HF2 : ∀x0 ∈ int, ∀x1 ∈ int, F2 x0 x1 ∈ int
L11Variable G2 : set → set → set
L12Hypothesis HG2 : ∀x0 ∈ int, ∀x1 ∈ int, G2 x0 x1 ∈ int
L13Variable H2 : set → set
L14Hypothesis HH2 : ∀x0 ∈ int, H2 x0 ∈ int
L16Hypothesis HI2 : I2 ∈ int
L18Hypothesis HJ2 : J2 ∈ int
L19Variable U2 : set → set → set → set
L20Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U2 x0 x1 x2 ∈ int
L21Variable V2 : set → set → set → set
L22Hypothesis HV2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V2 x0 x1 x2 ∈ int
L23Variable W2 : set → set
L24Hypothesis HW2 : ∀x0 ∈ int, W2 x0 ∈ int
L25Variable F3 : set → set
L26Hypothesis HF3 : ∀x0 ∈ int, F3 x0 ∈ int
L27Variable G3 : set → set
L28Hypothesis HG3 : ∀x0 ∈ int, G3 x0 ∈ int
L30Hypothesis HH3 : H3 ∈ int
L31Variable U3 : set → set → set
L32Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, U3 x0 x1 ∈ int
L33Variable V3 : set → set
L34Hypothesis HV3 : ∀x0 ∈ int, V3 x0 ∈ int
L35Variable F1 : set → set
L36Hypothesis HF1 : ∀x0 ∈ int, F1 x0 ∈ int
L38Hypothesis HG1 : G1 ∈ int
L39Variable H1 : set → set → set
L40Hypothesis HH1 : ∀x0 ∈ int, ∀x1 ∈ int, H1 x0 x1 ∈ int
L41Variable U1 : set → set → set
L42Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, U1 x0 x1 ∈ int
L43Variable V1 : set → set → set
L44Hypothesis HV1 : ∀x0 ∈ int, ∀x1 ∈ int, V1 x0 x1 ∈ int
L45Variable F4 : set → set
L46Hypothesis HF4 : ∀x0 ∈ int, F4 x0 ∈ int
L48Hypothesis HG4 : G4 ∈ int
L49Variable H4 : set → set
L50Hypothesis HH4 : ∀x0 ∈ int, H4 x0 ∈ int
L51Variable U4 : set → set → set
L52Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, U4 x0 x1 ∈ int
L53Variable V4 : set → set
L54Hypothesis HV4 : ∀x0 ∈ int, V4 x0 ∈ int
L55Variable F0 : set → set → set
L56Hypothesis HF0 : ∀x0 ∈ int, ∀x1 ∈ int, F0 x0 x1 ∈ int
L57Variable G0 : set → set
L58Hypothesis HG0 : ∀x0 ∈ int, G0 x0 ∈ int
L60Hypothesis HH0 : H0 ∈ int
L61Variable U0 : set → set → set
L62Hypothesis HU0 : ∀x0 ∈ int, ∀x1 ∈ int, U0 x0 x1 ∈ int
L63Variable V0 : set → set
L64Hypothesis HV0 : ∀x0 ∈ int, V0 x0 ∈ int
L65Variable SMALL : set → set
L66Hypothesis HSMALL : ∀x0 ∈ int, SMALL x0 ∈ int
L67Variable F7 : set → set → set
L68Hypothesis HF7 : ∀x0 ∈ int, ∀x1 ∈ int, F7 x0 x1 ∈ int
L69Variable G7 : set → set → set
L70Hypothesis HG7 : ∀x0 ∈ int, ∀x1 ∈ int, G7 x0 x1 ∈ int
L71Variable H7 : set → set
L72Hypothesis HH7 : ∀x0 ∈ int, H7 x0 ∈ int
L74Hypothesis HI7 : I7 ∈ int
L76Hypothesis HJ7 : J7 ∈ int
L77Variable U7 : set → set → set → set
L78Hypothesis HU7 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U7 x0 x1 x2 ∈ int
L79Variable V7 : set → set → set → set
L80Hypothesis HV7 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V7 x0 x1 x2 ∈ int
L81Variable W7 : set → set
L82Hypothesis HW7 : ∀x0 ∈ int, W7 x0 ∈ int
L83Variable F8 : set → set
L84Hypothesis HF8 : ∀x0 ∈ int, F8 x0 ∈ int
L85Variable G8 : set → set
L86Hypothesis HG8 : ∀x0 ∈ int, G8 x0 ∈ int
L88Hypothesis HH8 : H8 ∈ int
L89Variable U8 : set → set → set
L90Hypothesis HU8 : ∀x0 ∈ int, ∀x1 ∈ int, U8 x0 x1 ∈ int
L91Variable V8 : set → set
L92Hypothesis HV8 : ∀x0 ∈ int, V8 x0 ∈ int
L93Variable F6 : set → set
L94Hypothesis HF6 : ∀x0 ∈ int, F6 x0 ∈ int
L96Hypothesis HG6 : G6 ∈ int
L97Variable H6 : set → set → set
L98Hypothesis HH6 : ∀x0 ∈ int, ∀x1 ∈ int, H6 x0 x1 ∈ int
L99Variable U6 : set → set → set
L100Hypothesis HU6 : ∀x0 ∈ int, ∀x1 ∈ int, U6 x0 x1 ∈ int
L101Variable V6 : set → set → set
L102Hypothesis HV6 : ∀x0 ∈ int, ∀x1 ∈ int, V6 x0 x1 ∈ int
L103Variable F5 : set → set → set
L104Hypothesis HF5 : ∀x0 ∈ int, ∀x1 ∈ int, F5 x0 x1 ∈ int
L105Variable G5 : set → set
L106Hypothesis HG5 : ∀x0 ∈ int, G5 x0 ∈ int
L108Hypothesis HH5 : H5 ∈ int
L109Variable U5 : set → set → set
L110Hypothesis HU5 : ∀x0 ∈ int, ∀x1 ∈ int, U5 x0 x1 ∈ int
L111Variable V5 : set → set
L112Hypothesis HV5 : ∀x0 ∈ int, V5 x0 ∈ int
L113Variable FAST : set → set
L114Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L115Hypothesis H1 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = (((2 * (((Y * Y) + X) + X)) + - Y) + X))))
L116Hypothesis H2 : (∀X ∈ int, (∀Y ∈ int, ((G2 X Y) = (Y + Y))))
L117Hypothesis H3 : (∀X ∈ int, ((H2 X) = X))
L118Hypothesis H4 : (I2 = 1)
L119Hypothesis H5 : (J2 = 2)
L120Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U2 X Y Z) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y Z) (V2 (X + - 1) Y Z)))))))
L121Hypothesis H7 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V2 X Y Z) = (if (X <= 0) then Z else (G2 (U2 (X + - 1) Y Z) (V2 (X + - 1) Y Z)))))))
L122Hypothesis H8 : (∀X ∈ int, ((W2 X) = (U2 (H2 X) I2 J2)))
L123Hypothesis H9 : (∀X ∈ int, ((F3 X) = (X + X)))
L124Hypothesis H10 : (∀X ∈ int, ((G3 X) = X))
L125Hypothesis H11 : (H3 = 1)
L126Hypothesis H12 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y))))))
L127Hypothesis H13 : (∀X ∈ int, ((V3 X) = (U3 (G3 X) H3)))
L128Hypothesis H14 : (∀X ∈ int, ((F1 X) = ((W2 X) * (V3 X))))
L129Hypothesis H15 : (G1 = 1)
L130Hypothesis H16 : (∀X ∈ int, (∀Y ∈ int, ((H1 X Y) = Y)))
L131Hypothesis H17 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))
L132Hypothesis H18 : (∀X ∈ int, (∀Y ∈ int, ((V1 X Y) = (U1 G1 (H1 X Y)))))
L133Hypothesis H19 : (∀X ∈ int, ((F4 X) = ((X + X) + X)))
L134Hypothesis H20 : (G4 = 2)
L135Hypothesis H21 : (∀X ∈ int, ((H4 X) = X))
L136Hypothesis H22 : (∀X ∈ int, (∀Y ∈ int, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y))))))
L137Hypothesis H23 : (∀X ∈ int, ((V4 X) = (U4 G4 (H4 X))))
L138Hypothesis H24 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = ((V1 X Y) + (V4 X)))))
L139Hypothesis H25 : (∀X ∈ int, ((G0 X) = X))
L140Hypothesis H26 : (H0 = 1)
L141Hypothesis H27 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y) X)))))
L142Hypothesis H28 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))
L143Hypothesis H29 : (∀X ∈ int, ((SMALL X) = (V0 X)))
L144Hypothesis H30 : (∀X ∈ int, (∀Y ∈ int, ((F7 X Y) = (((2 * (((Y * Y) + X) + X)) + - Y) + X))))
L145Hypothesis H31 : (∀X ∈ int, (∀Y ∈ int, ((G7 X Y) = (Y + Y))))
L146Hypothesis H32 : (∀X ∈ int, ((H7 X) = X))
L147Hypothesis H33 : (I7 = 1)
L148Hypothesis H34 : (J7 = 2)
L149Hypothesis H35 : (∀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 H36 : (∀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 H37 : (∀X ∈ int, ((W7 X) = (U7 (H7 X) I7 J7)))
L152Hypothesis H38 : (∀X ∈ int, ((F8 X) = (X + X)))
L153Hypothesis H39 : (∀X ∈ int, ((G8 X) = X))
L154Hypothesis H40 : (H8 = 1)
L155Hypothesis H41 : (∀X ∈ int, (∀Y ∈ int, ((U8 X Y) = (if (X <= 0) then Y else (F8 (U8 (X + - 1) Y))))))
L156Hypothesis H42 : (∀X ∈ int, ((V8 X) = (U8 (G8 X) H8)))
L157Hypothesis H43 : (∀X ∈ int, ((F6 X) = ((W7 X) * (V8 X))))
L158Hypothesis H44 : (G6 = 1)
L159Hypothesis H45 : (∀X ∈ int, (∀Y ∈ int, ((H6 X Y) = Y)))
L160Hypothesis H46 : (∀X ∈ int, (∀Y ∈ int, ((U6 X Y) = (if (X <= 0) then Y else (F6 (U6 (X + - 1) Y))))))
L161Hypothesis H47 : (∀X ∈ int, (∀Y ∈ int, ((V6 X Y) = (U6 G6 (H6 X Y)))))
L162Hypothesis H48 : (∀X ∈ int, (∀Y ∈ int, ((F5 X Y) = (((V6 X Y) + (2 * (2 * (X + X)))) + X))))
L163Hypothesis H49 : (∀X ∈ int, ((G5 X) = X))
L164Hypothesis H50 : (H5 = 1)
L165Hypothesis H51 : (∀X ∈ int, (∀Y ∈ int, ((U5 X Y) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y) X)))))
L166Hypothesis H52 : (∀X ∈ int, ((V5 X) = (U5 (G5 X) H5)))
L167Hypothesis H53 : (∀X ∈ int, ((FAST X) = (V5 X)))
L168Theorem. (
A28155)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.