Beginning of Section A28218
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 F0 : set → set → set
L46Hypothesis HF0 : ∀x0 ∈ int, ∀x1 ∈ int, F0 x0 x1 ∈ int
L47Variable G0 : set → set
L48Hypothesis HG0 : ∀x0 ∈ int, G0 x0 ∈ int
L50Hypothesis HH0 : H0 ∈ int
L51Variable U0 : set → set → set
L52Hypothesis HU0 : ∀x0 ∈ int, ∀x1 ∈ int, U0 x0 x1 ∈ int
L53Variable V0 : set → set
L54Hypothesis HV0 : ∀x0 ∈ int, V0 x0 ∈ int
L55Variable SMALL : set → set
L56Hypothesis HSMALL : ∀x0 ∈ int, SMALL x0 ∈ int
L57Variable F6 : set → set → set
L58Hypothesis HF6 : ∀x0 ∈ int, ∀x1 ∈ int, F6 x0 x1 ∈ int
L59Variable G6 : set → set → set
L60Hypothesis HG6 : ∀x0 ∈ int, ∀x1 ∈ int, G6 x0 x1 ∈ int
L61Variable H6 : set → set
L62Hypothesis HH6 : ∀x0 ∈ int, H6 x0 ∈ int
L64Hypothesis HI6 : I6 ∈ int
L66Hypothesis HJ6 : J6 ∈ int
L67Variable U6 : set → set → set → set
L68Hypothesis HU6 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U6 x0 x1 x2 ∈ int
L69Variable V6 : set → set → set → set
L70Hypothesis HV6 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V6 x0 x1 x2 ∈ int
L71Variable W6 : set → set
L72Hypothesis HW6 : ∀x0 ∈ int, W6 x0 ∈ int
L73Variable F7 : set → set → set
L74Hypothesis HF7 : ∀x0 ∈ int, ∀x1 ∈ int, F7 x0 x1 ∈ int
L75Variable G7 : set → set → set
L76Hypothesis HG7 : ∀x0 ∈ int, ∀x1 ∈ int, G7 x0 x1 ∈ int
L77Variable H7 : set → set
L78Hypothesis HH7 : ∀x0 ∈ int, H7 x0 ∈ int
L80Hypothesis HI7 : I7 ∈ int
L82Hypothesis HJ7 : J7 ∈ int
L83Variable U7 : set → set → set → set
L84Hypothesis HU7 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U7 x0 x1 x2 ∈ int
L85Variable V7 : set → set → set → set
L86Hypothesis HV7 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V7 x0 x1 x2 ∈ int
L87Variable W7 : set → set
L88Hypothesis HW7 : ∀x0 ∈ int, W7 x0 ∈ int
L89Variable F5 : set → set
L90Hypothesis HF5 : ∀x0 ∈ int, F5 x0 ∈ int
L92Hypothesis HG5 : G5 ∈ int
L93Variable H5 : set → set → set
L94Hypothesis HH5 : ∀x0 ∈ int, ∀x1 ∈ int, H5 x0 x1 ∈ int
L95Variable U5 : set → set → set
L96Hypothesis HU5 : ∀x0 ∈ int, ∀x1 ∈ int, U5 x0 x1 ∈ int
L97Variable V5 : set → set → set
L98Hypothesis HV5 : ∀x0 ∈ int, ∀x1 ∈ int, V5 x0 x1 ∈ int
L99Variable F4 : set → set → set
L100Hypothesis HF4 : ∀x0 ∈ int, ∀x1 ∈ int, F4 x0 x1 ∈ int
L101Variable G4 : set → set
L102Hypothesis HG4 : ∀x0 ∈ int, G4 x0 ∈ int
L104Hypothesis HH4 : H4 ∈ int
L105Variable U4 : set → set → set
L106Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, U4 x0 x1 ∈ int
L107Variable V4 : set → set
L108Hypothesis HV4 : ∀x0 ∈ int, V4 x0 ∈ int
L109Variable FAST : set → set
L110Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L111Hypothesis H1 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = (((2 * ((Y * Y) + X)) + - Y) + X))))
L112Hypothesis H2 : (∀X ∈ int, (∀Y ∈ int, ((G2 X Y) = (Y + Y))))
L113Hypothesis H3 : (∀X ∈ int, ((H2 X) = X))
L114Hypothesis H4 : (I2 = 1)
L115Hypothesis H5 : (J2 = 2)
L116Hypothesis 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)))))))
L117Hypothesis 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)))))))
L118Hypothesis H8 : (∀X ∈ int, ((W2 X) = (U2 (H2 X) I2 J2)))
L119Hypothesis H9 : (∀X ∈ int, ((F3 X) = ((X + X) + X)))
L120Hypothesis H10 : (∀X ∈ int, ((G3 X) = X))
L121Hypothesis H11 : (H3 = 1)
L122Hypothesis H12 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y))))))
L123Hypothesis H13 : (∀X ∈ int, ((V3 X) = (U3 (G3 X) H3)))
L124Hypothesis H14 : (∀X ∈ int, ((F1 X) = ((W2 X) * (V3 X))))
L125Hypothesis H15 : (G1 = 1)
L126Hypothesis H16 : (∀X ∈ int, (∀Y ∈ int, ((H1 X Y) = Y)))
L127Hypothesis H17 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))
L128Hypothesis H18 : (∀X ∈ int, (∀Y ∈ int, ((V1 X Y) = (U1 G1 (H1 X Y)))))
L129Hypothesis H19 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = (((V1 X Y) + (2 * ((2 * (X + X)) + X))) + X))))
L130Hypothesis H20 : (∀X ∈ int, ((G0 X) = X))
L131Hypothesis H21 : (H0 = 1)
L132Hypothesis H22 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y) X)))))
L133Hypothesis H23 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))
L134Hypothesis H24 : (∀X ∈ int, ((SMALL X) = (V0 X)))
L135Hypothesis H25 : (∀X ∈ int, (∀Y ∈ int, ((F6 X Y) = (((2 * ((Y * Y) + X)) + - Y) + X))))
L136Hypothesis H26 : (∀X ∈ int, (∀Y ∈ int, ((G6 X Y) = (Y + Y))))
L137Hypothesis H27 : (∀X ∈ int, ((H6 X) = X))
L138Hypothesis H28 : (I6 = 1)
L139Hypothesis H29 : (J6 = 2)
L140Hypothesis H30 : (∀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)))))))
L141Hypothesis H31 : (∀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)))))))
L142Hypothesis H32 : (∀X ∈ int, ((W6 X) = (U6 (H6 X) I6 J6)))
L143Hypothesis H33 : (∀X ∈ int, (∀Y ∈ int, ((F7 X Y) = (X * Y))))
L144Hypothesis H34 : (∀X ∈ int, (∀Y ∈ int, ((G7 X Y) = Y)))
L145Hypothesis H35 : (∀X ∈ int, ((H7 X) = X))
L146Hypothesis H36 : (I7 = 1)
L147Hypothesis H37 : (J7 = (1 + 2))
L148Hypothesis H38 : (∀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)))))))
L149Hypothesis H39 : (∀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)))))))
L150Hypothesis H40 : (∀X ∈ int, ((W7 X) = (U7 (H7 X) I7 J7)))
L151Hypothesis H41 : (∀X ∈ int, ((F5 X) = ((W6 X) * (W7 X))))
L152Hypothesis H42 : (G5 = 1)
L153Hypothesis H43 : (∀X ∈ int, (∀Y ∈ int, ((H5 X Y) = Y)))
L154Hypothesis H44 : (∀X ∈ int, (∀Y ∈ int, ((U5 X Y) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y))))))
L155Hypothesis H45 : (∀X ∈ int, (∀Y ∈ int, ((V5 X Y) = (U5 G5 (H5 X Y)))))
L156Hypothesis H46 : (∀X ∈ int, (∀Y ∈ int, ((F4 X Y) = (((V5 X Y) + (2 * ((2 * (X + X)) + X))) + X))))
L157Hypothesis H47 : (∀X ∈ int, ((G4 X) = X))
L158Hypothesis H48 : (H4 = 1)
L159Hypothesis H49 : (∀X ∈ int, (∀Y ∈ int, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y) X)))))
L160Hypothesis H50 : (∀X ∈ int, ((V4 X) = (U4 (G4 X) H4)))
L161Hypothesis H51 : (∀X ∈ int, ((FAST X) = (V4 X)))
L162Theorem. (
A28218)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.