Beginning of Section A25957
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 → set
L10Hypothesis HF3 : ∀x0 ∈ int, ∀x1 ∈ int, F3 x0 x1 ∈ int
L12Hypothesis HG3 : G3 ∈ int
L13Variable H3 : set → set
L14Hypothesis HH3 : ∀x0 ∈ int, H3 x0 ∈ int
L15Variable U3 : set → set → set
L16Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, U3 x0 x1 ∈ int
L17Variable V3 : set → set
L18Hypothesis HV3 : ∀x0 ∈ int, V3 x0 ∈ int
L19Variable F2 : set → set → set
L20Hypothesis HF2 : ∀x0 ∈ int, ∀x1 ∈ int, F2 x0 x1 ∈ int
L21Variable G2 : set → set → set
L22Hypothesis HG2 : ∀x0 ∈ int, ∀x1 ∈ int, G2 x0 x1 ∈ int
L23Variable H2 : set → set → set
L24Hypothesis HH2 : ∀x0 ∈ int, ∀x1 ∈ int, H2 x0 x1 ∈ int
L26Hypothesis HI2 : I2 ∈ int
L28Hypothesis HJ2 : J2 ∈ int
L29Variable U2 : set → set → set → set
L30Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U2 x0 x1 x2 ∈ int
L31Variable V2 : set → set → set → set
L32Hypothesis HV2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V2 x0 x1 x2 ∈ int
L33Variable W2 : set → set → set
L34Hypothesis HW2 : ∀x0 ∈ int, ∀x1 ∈ int, W2 x0 x1 ∈ int
L35Variable F4 : set → set → set
L36Hypothesis HF4 : ∀x0 ∈ int, ∀x1 ∈ int, F4 x0 x1 ∈ int
L38Hypothesis HG4 : G4 ∈ int
L39Variable H4 : set → set
L40Hypothesis HH4 : ∀x0 ∈ int, H4 x0 ∈ int
L41Variable U4 : set → set → set
L42Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, U4 x0 x1 ∈ int
L43Variable V4 : set → set
L44Hypothesis HV4 : ∀x0 ∈ int, V4 x0 ∈ int
L45Variable F1 : set → set → set
L46Hypothesis HF1 : ∀x0 ∈ int, ∀x1 ∈ int, F1 x0 x1 ∈ int
L47Variable G1 : set → set → set
L48Hypothesis HG1 : ∀x0 ∈ int, ∀x1 ∈ int, G1 x0 x1 ∈ int
L50Hypothesis HH1 : H1 ∈ int
L51Variable U1 : set → set → set
L52Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, U1 x0 x1 ∈ int
L53Variable V1 : set → set → set
L54Hypothesis HV1 : ∀x0 ∈ int, ∀x1 ∈ int, V1 x0 x1 ∈ 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 → set
L72Hypothesis HH7 : ∀x0 ∈ int, ∀x1 ∈ int, H7 x0 x1 ∈ 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 → set
L82Hypothesis HW7 : ∀x0 ∈ int, ∀x1 ∈ int, W7 x0 x1 ∈ int
L83Variable F6 : set → set → set
L84Hypothesis HF6 : ∀x0 ∈ int, ∀x1 ∈ int, F6 x0 x1 ∈ int
L85Variable G6 : set → set → set
L86Hypothesis HG6 : ∀x0 ∈ int, ∀x1 ∈ int, G6 x0 x1 ∈ int
L88Hypothesis HH6 : H6 ∈ int
L89Variable U6 : set → set → set
L90Hypothesis HU6 : ∀x0 ∈ int, ∀x1 ∈ int, U6 x0 x1 ∈ int
L91Variable V6 : set → set → set
L92Hypothesis HV6 : ∀x0 ∈ int, ∀x1 ∈ int, V6 x0 x1 ∈ int
L93Variable F5 : set → set → set
L94Hypothesis HF5 : ∀x0 ∈ int, ∀x1 ∈ int, F5 x0 x1 ∈ int
L95Variable G5 : set → set
L96Hypothesis HG5 : ∀x0 ∈ int, G5 x0 ∈ int
L98Hypothesis HH5 : H5 ∈ int
L99Variable U5 : set → set → set
L100Hypothesis HU5 : ∀x0 ∈ int, ∀x1 ∈ int, U5 x0 x1 ∈ int
L101Variable V5 : set → set
L102Hypothesis HV5 : ∀x0 ∈ int, V5 x0 ∈ int
L103Variable FAST : set → set
L104Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L105Hypothesis H1 : (∀X ∈ int, (∀Y ∈ int, ((F3 X Y) = ((2 + Y) * X))))
L106Hypothesis H2 : (G3 = 2)
L107Hypothesis H3 : (∀X ∈ int, ((H3 X) = X))
L108Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y) X)))))
L109Hypothesis H5 : (∀X ∈ int, ((V3 X) = (U3 G3 (H3 X))))
L110Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = (((V3 X) + - X) + Y))))
L111Hypothesis H7 : (∀X ∈ int, (∀Y ∈ int, ((G2 X Y) = (Y + Y))))
L112Hypothesis H8 : (∀X ∈ int, (∀Y ∈ int, ((H2 X Y) = Y)))
L113Hypothesis H9 : (I2 = 1)
L114Hypothesis H10 : (J2 = 2)
L115Hypothesis H11 : (∀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)))))))
L116Hypothesis H12 : (∀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)))))))
L117Hypothesis H13 : (∀X ∈ int, (∀Y ∈ int, ((W2 X Y) = (U2 (H2 X Y) I2 J2))))
L118Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, ((F4 X Y) = ((2 + Y) * X))))
L119Hypothesis H15 : (G4 = 2)
L120Hypothesis H16 : (∀X ∈ int, ((H4 X) = X))
L121Hypothesis H17 : (∀X ∈ int, (∀Y ∈ int, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y) X)))))
L122Hypothesis H18 : (∀X ∈ int, ((V4 X) = (U4 G4 (H4 X))))
L123Hypothesis H19 : (∀X ∈ int, (∀Y ∈ int, ((F1 X Y) = ((W2 X Y) + (V4 X)))))
L124Hypothesis H20 : (∀X ∈ int, (∀Y ∈ int, ((G1 X Y) = Y)))
L125Hypothesis H21 : (H1 = 1)
L126Hypothesis H22 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y) X)))))
L127Hypothesis H23 : (∀X ∈ int, (∀Y ∈ int, ((V1 X Y) = (U1 (G1 X Y) H1))))
L128Hypothesis H24 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = ((((V1 X Y) + X) + X) + X))))
L129Hypothesis H25 : (∀X ∈ int, ((G0 X) = X))
L130Hypothesis H26 : (H0 = 1)
L131Hypothesis H27 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y) X)))))
L132Hypothesis H28 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))
L133Hypothesis H29 : (∀X ∈ int, ((SMALL X) = (V0 X)))
L134Hypothesis H30 : (∀X ∈ int, (∀Y ∈ int, ((F7 X Y) = (((X + X) + X) + Y))))
L135Hypothesis H31 : (∀X ∈ int, (∀Y ∈ int, ((G7 X Y) = (Y + Y))))
L136Hypothesis H32 : (∀X ∈ int, (∀Y ∈ int, ((H7 X Y) = Y)))
L137Hypothesis H33 : (I7 = 1)
L138Hypothesis H34 : (J7 = 2)
L139Hypothesis 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)))))))
L140Hypothesis 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)))))))
L141Hypothesis H37 : (∀X ∈ int, (∀Y ∈ int, ((W7 X Y) = (U7 (H7 X Y) I7 J7))))
L142Hypothesis H38 : (∀X ∈ int, (∀Y ∈ int, ((F6 X Y) = ((W7 X Y) + (2 * (2 * ((X + X) + X)))))))
L143Hypothesis H39 : (∀X ∈ int, (∀Y ∈ int, ((G6 X Y) = Y)))
L144Hypothesis H40 : (H6 = 1)
L145Hypothesis H41 : (∀X ∈ int, (∀Y ∈ int, ((U6 X Y) = (if (X <= 0) then Y else (F6 (U6 (X + - 1) Y) X)))))
L146Hypothesis H42 : (∀X ∈ int, (∀Y ∈ int, ((V6 X Y) = (U6 (G6 X Y) H6))))
L147Hypothesis H43 : (∀X ∈ int, (∀Y ∈ int, ((F5 X Y) = (((V6 X Y) + (2 * ((2 * (X + X)) + X))) + X))))
L148Hypothesis H44 : (∀X ∈ int, ((G5 X) = X))
L149Hypothesis H45 : (H5 = 1)
L150Hypothesis H46 : (∀X ∈ int, (∀Y ∈ int, ((U5 X Y) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y) X)))))
L151Hypothesis H47 : (∀X ∈ int, ((V5 X) = (U5 (G5 X) H5)))
L152Hypothesis H48 : (∀X ∈ int, ((FAST X) = (V5 X)))
L153Theorem. (
A25957)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.