Beginning of Section A28018
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 → set
L14Hypothesis HH2 : ∀x0 ∈ int, ∀x1 ∈ int, H2 x0 x1 ∈ 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 → set
L24Hypothesis HW2 : ∀x0 ∈ int, ∀x1 ∈ int, W2 x0 x1 ∈ int
L25Variable F3 : set → set → set
L26Hypothesis HF3 : ∀x0 ∈ int, ∀x1 ∈ int, F3 x0 x1 ∈ int
L28Hypothesis HG3 : G3 ∈ int
L29Variable H3 : set → set
L30Hypothesis HH3 : ∀x0 ∈ int, H3 x0 ∈ 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 → set
L36Hypothesis HF1 : ∀x0 ∈ int, ∀x1 ∈ int, F1 x0 x1 ∈ int
L37Variable G1 : set → set → set
L38Hypothesis HG1 : ∀x0 ∈ int, ∀x1 ∈ int, G1 x0 x1 ∈ int
L40Hypothesis HH1 : H1 ∈ 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
L74Hypothesis HF7 : ∀x0 ∈ int, F7 x0 ∈ int
L75Variable G7 : set → set
L76Hypothesis HG7 : ∀x0 ∈ int, G7 x0 ∈ int
L78Hypothesis HH7 : H7 ∈ int
L79Variable U7 : set → set → set
L80Hypothesis HU7 : ∀x0 ∈ int, ∀x1 ∈ int, U7 x0 x1 ∈ int
L81Variable V7 : set → set
L82Hypothesis HV7 : ∀x0 ∈ int, V7 x0 ∈ int
L83Variable F5 : set → set
L84Hypothesis HF5 : ∀x0 ∈ int, F5 x0 ∈ int
L86Hypothesis HG5 : G5 ∈ int
L87Variable H5 : set → set → set
L88Hypothesis HH5 : ∀x0 ∈ int, ∀x1 ∈ int, H5 x0 x1 ∈ int
L89Variable U5 : set → set → set
L90Hypothesis HU5 : ∀x0 ∈ int, ∀x1 ∈ int, U5 x0 x1 ∈ int
L91Variable V5 : set → set → set
L92Hypothesis HV5 : ∀x0 ∈ int, ∀x1 ∈ int, V5 x0 x1 ∈ int
L93Variable F4 : set → set → set
L94Hypothesis HF4 : ∀x0 ∈ int, ∀x1 ∈ int, F4 x0 x1 ∈ int
L95Variable G4 : set → set
L96Hypothesis HG4 : ∀x0 ∈ int, G4 x0 ∈ int
L98Hypothesis HH4 : H4 ∈ int
L99Variable U4 : set → set → set
L100Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, U4 x0 x1 ∈ int
L101Variable V4 : set → set
L102Hypothesis HV4 : ∀x0 ∈ int, V4 x0 ∈ int
L103Variable FAST : set → set
L104Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L105Hypothesis H1 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = ((2 * ((2 * (X + X)) + X)) + ((Y * Y) * Y)))))
L106Hypothesis H2 : (∀X ∈ int, (∀Y ∈ int, ((G2 X Y) = (Y + Y))))
L107Hypothesis H3 : (∀X ∈ int, (∀Y ∈ int, ((H2 X Y) = Y)))
L108Hypothesis H4 : (I2 = 1)
L109Hypothesis H5 : (J2 = 2)
L110Hypothesis 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)))))))
L111Hypothesis 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)))))))
L112Hypothesis H8 : (∀X ∈ int, (∀Y ∈ int, ((W2 X Y) = (U2 (H2 X Y) I2 J2))))
L113Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((F3 X Y) = ((2 + Y) * X))))
L114Hypothesis H10 : (G3 = 2)
L115Hypothesis H11 : (∀X ∈ int, ((H3 X) = X))
L116Hypothesis H12 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y) X)))))
L117Hypothesis H13 : (∀X ∈ int, ((V3 X) = (U3 G3 (H3 X))))
L118Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, ((F1 X Y) = (((W2 X Y) + - X) + (V3 X)))))
L119Hypothesis H15 : (∀X ∈ int, (∀Y ∈ int, ((G1 X Y) = Y)))
L120Hypothesis H16 : (H1 = 1)
L121Hypothesis H17 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y) X)))))
L122Hypothesis H18 : (∀X ∈ int, (∀Y ∈ int, ((V1 X Y) = (U1 (G1 X Y) H1))))
L123Hypothesis H19 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = (((V1 X Y) + X) + X))))
L124Hypothesis H20 : (∀X ∈ int, ((G0 X) = X))
L125Hypothesis H21 : (H0 = 1)
L126Hypothesis H22 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y) X)))))
L127Hypothesis H23 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))
L128Hypothesis H24 : (∀X ∈ int, ((SMALL X) = (V0 X)))
L129Hypothesis H25 : (∀X ∈ int, (∀Y ∈ int, ((F6 X Y) = (((2 * (X + X)) + X) + Y))))
L130Hypothesis H26 : (∀X ∈ int, (∀Y ∈ int, ((G6 X Y) = (1 + (2 * (Y + Y))))))
L131Hypothesis H27 : (∀X ∈ int, ((H6 X) = X))
L132Hypothesis H28 : (I6 = 1)
L133Hypothesis H29 : (J6 = (1 + (2 + 2)))
L134Hypothesis 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)))))))
L135Hypothesis 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)))))))
L136Hypothesis H32 : (∀X ∈ int, ((W6 X) = (U6 (H6 X) I6 J6)))
L137Hypothesis H33 : (∀X ∈ int, ((F7 X) = (X + X)))
L138Hypothesis H34 : (∀X ∈ int, ((G7 X) = X))
L139Hypothesis H35 : (H7 = 1)
L140Hypothesis H36 : (∀X ∈ int, (∀Y ∈ int, ((U7 X Y) = (if (X <= 0) then Y else (F7 (U7 (X + - 1) Y))))))
L141Hypothesis H37 : (∀X ∈ int, ((V7 X) = (U7 (G7 X) H7)))
L142Hypothesis H38 : (∀X ∈ int, ((F5 X) = ((W6 X) * (V7 X))))
L143Hypothesis H39 : (G5 = 1)
L144Hypothesis H40 : (∀X ∈ int, (∀Y ∈ int, ((H5 X Y) = Y)))
L145Hypothesis H41 : (∀X ∈ int, (∀Y ∈ int, ((U5 X Y) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y))))))
L146Hypothesis H42 : (∀X ∈ int, (∀Y ∈ int, ((V5 X Y) = (U5 G5 (H5 X Y)))))
L147Hypothesis H43 : (∀X ∈ int, (∀Y ∈ int, ((F4 X Y) = (((V5 X Y) + (2 * ((2 * (X + X)) + X))) + X))))
L148Hypothesis H44 : (∀X ∈ int, ((G4 X) = X))
L149Hypothesis H45 : (H4 = 1)
L150Hypothesis H46 : (∀X ∈ int, (∀Y ∈ int, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y) X)))))
L151Hypothesis H47 : (∀X ∈ int, ((V4 X) = (U4 (G4 X) H4)))
L152Hypothesis H48 : (∀X ∈ int, ((FAST X) = (V4 X)))
L153Theorem. (
A28018)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.