Beginning of Section A110224
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 F1 : set → set → set
L10Hypothesis HF1 : ∀x0 ∈ int, ∀x1 ∈ int, F1 x0 x1 ∈ int
L11Variable G1 : set → set
L12Hypothesis HG1 : ∀x0 ∈ int, G1 x0 ∈ int
L13Variable H1 : set → set → set
L14Hypothesis HH1 : ∀x0 ∈ int, ∀x1 ∈ int, H1 x0 x1 ∈ int
L16Hypothesis HI1 : I1 ∈ int
L18Hypothesis HJ1 : J1 ∈ int
L19Variable U1 : set → set → set → set
L20Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U1 x0 x1 x2 ∈ int
L21Variable V1 : set → set → set → set
L22Hypothesis HV1 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V1 x0 x1 x2 ∈ int
L23Variable W1 : set → set → set
L24Hypothesis HW1 : ∀x0 ∈ int, ∀x1 ∈ int, W1 x0 x1 ∈ int
L25Variable F0 : set → set → set
L26Hypothesis HF0 : ∀x0 ∈ int, ∀x1 ∈ int, F0 x0 x1 ∈ int
L27Variable G0 : set → set
L28Hypothesis HG0 : ∀x0 ∈ int, G0 x0 ∈ int
L30Hypothesis HH0 : H0 ∈ int
L31Variable U0 : set → set → set
L32Hypothesis HU0 : ∀x0 ∈ int, ∀x1 ∈ int, U0 x0 x1 ∈ int
L33Variable V0 : set → set
L34Hypothesis HV0 : ∀x0 ∈ int, V0 x0 ∈ int
L35Variable F2 : set → set → set
L36Hypothesis HF2 : ∀x0 ∈ int, ∀x1 ∈ int, F2 x0 x1 ∈ int
L37Variable G2 : set → set
L38Hypothesis HG2 : ∀x0 ∈ int, G2 x0 ∈ int
L39Variable H2 : set → set
L40Hypothesis HH2 : ∀x0 ∈ int, H2 x0 ∈ int
L42Hypothesis HI2 : I2 ∈ int
L44Hypothesis HJ2 : J2 ∈ int
L45Variable U2 : set → set → set → set
L46Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U2 x0 x1 x2 ∈ int
L47Variable V2 : set → set → set → set
L48Hypothesis HV2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V2 x0 x1 x2 ∈ int
L49Variable W2 : set → set
L50Hypothesis HW2 : ∀x0 ∈ int, W2 x0 ∈ int
L51Variable SMALL : set → set
L52Hypothesis HSMALL : ∀x0 ∈ int, SMALL x0 ∈ int
L53Variable F3 : set → set
L54Hypothesis HF3 : ∀x0 ∈ int, F3 x0 ∈ int
L56Hypothesis HG3 : G3 ∈ int
L57Variable F4 : set → set → set
L58Hypothesis HF4 : ∀x0 ∈ int, ∀x1 ∈ int, F4 x0 x1 ∈ int
L59Variable G4 : set → set
L60Hypothesis HG4 : ∀x0 ∈ int, G4 x0 ∈ int
L61Variable H4 : set → set
L62Hypothesis HH4 : ∀x0 ∈ int, H4 x0 ∈ int
L64Hypothesis HI4 : I4 ∈ int
L66Hypothesis HJ4 : J4 ∈ int
L67Variable U4 : set → set → set → set
L68Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U4 x0 x1 x2 ∈ int
L69Variable V4 : set → set → set → set
L70Hypothesis HV4 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V4 x0 x1 x2 ∈ int
L71Variable W4 : set → set
L72Hypothesis HW4 : ∀x0 ∈ int, W4 x0 ∈ int
L73Variable H3 : set → set
L74Hypothesis HH3 : ∀x0 ∈ int, H3 x0 ∈ int
L75Variable U3 : set → set → set
L76Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, U3 x0 x1 ∈ int
L77Variable V3 : set → set
L78Hypothesis HV3 : ∀x0 ∈ int, V3 x0 ∈ int
L79Variable F5 : set → set
L80Hypothesis HF5 : ∀x0 ∈ int, F5 x0 ∈ int
L82Hypothesis HG5 : G5 ∈ int
L83Variable F6 : set → set → set
L84Hypothesis HF6 : ∀x0 ∈ int, ∀x1 ∈ int, F6 x0 x1 ∈ int
L85Variable G6 : set → set
L86Hypothesis HG6 : ∀x0 ∈ int, G6 x0 ∈ int
L87Variable H6 : set → set
L88Hypothesis HH6 : ∀x0 ∈ int, H6 x0 ∈ int
L90Hypothesis HI6 : I6 ∈ int
L92Hypothesis HJ6 : J6 ∈ int
L93Variable U6 : set → set → set → set
L94Hypothesis HU6 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U6 x0 x1 x2 ∈ int
L95Variable V6 : set → set → set → set
L96Hypothesis HV6 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V6 x0 x1 x2 ∈ int
L97Variable W6 : set → set
L98Hypothesis HW6 : ∀x0 ∈ int, W6 x0 ∈ int
L99Variable H5 : set → set
L100Hypothesis HH5 : ∀x0 ∈ int, H5 x0 ∈ int
L101Variable U5 : set → set → set
L102Hypothesis HU5 : ∀x0 ∈ int, ∀x1 ∈ int, U5 x0 x1 ∈ int
L103Variable V5 : set → set
L104Hypothesis HV5 : ∀x0 ∈ int, V5 x0 ∈ int
L105Variable FAST : set → set
L106Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L107Hypothesis H1 : (∀X ∈ int, (∀Y ∈ int, ((F1 X Y) = (Y + - X))))
L108Hypothesis H2 : (∀X ∈ int, ((G1 X) = X))
L109Hypothesis H3 : (∀X ∈ int, (∀Y ∈ int, ((H1 X Y) = (Y + Y))))
L110Hypothesis H4 : (I1 = 2)
L111Hypothesis H5 : (J1 = 2)
L112Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U1 X Y Z) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y Z) (V1 (X + - 1) Y Z)))))))
L113Hypothesis H7 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V1 X Y Z) = (if (X <= 0) then Z else (G1 (U1 (X + - 1) Y Z)))))))
L114Hypothesis H8 : (∀X ∈ int, (∀Y ∈ int, ((W1 X Y) = (U1 (H1 X Y) I1 J1))))
L115Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = ((W1 X Y) + - X))))
L116Hypothesis H10 : (∀X ∈ int, ((G0 X) = X))
L117Hypothesis H11 : (H0 = 1)
L118Hypothesis H12 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y) X)))))
L119Hypothesis H13 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))
L120Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = (X + Y))))
L121Hypothesis H15 : (∀X ∈ int, ((G2 X) = X))
L122Hypothesis H16 : (∀X ∈ int, ((H2 X) = X))
L123Hypothesis H17 : (I2 = 1)
L124Hypothesis H18 : (J2 = 1)
L125Hypothesis H19 : (∀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)))))))
L126Hypothesis H20 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V2 X Y Z) = (if (X <= 0) then Z else (G2 (U2 (X + - 1) Y Z)))))))
L127Hypothesis H21 : (∀X ∈ int, ((W2 X) = (U2 (H2 X) I2 J2)))
L128Hypothesis H22 : (∀X ∈ int, ((SMALL X) = ((V0 X) * (W2 X))))
L129Hypothesis H23 : (∀X ∈ int, ((F3 X) = ((X * X) * X)))
L130Hypothesis H24 : (G3 = 1)
L131Hypothesis H25 : (∀X ∈ int, (∀Y ∈ int, ((F4 X Y) = (X + Y))))
L132Hypothesis H26 : (∀X ∈ int, ((G4 X) = X))
L133Hypothesis H27 : (∀X ∈ int, ((H4 X) = (X + - 1)))
L134Hypothesis H28 : (I4 = 1)
L135Hypothesis H29 : (J4 = 1)
L136Hypothesis H30 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U4 X Y Z) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y Z) (V4 (X + - 1) Y Z)))))))
L137Hypothesis H31 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V4 X Y Z) = (if (X <= 0) then Z else (G4 (U4 (X + - 1) Y Z)))))))
L138Hypothesis H32 : (∀X ∈ int, ((W4 X) = (U4 (H4 X) I4 J4)))
L139Hypothesis H33 : (∀X ∈ int, ((H3 X) = (W4 X)))
L140Hypothesis H34 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y))))))
L141Hypothesis H35 : (∀X ∈ int, ((V3 X) = (U3 G3 (H3 X))))
L142Hypothesis H36 : (∀X ∈ int, ((F5 X) = ((X * X) * X)))
L143Hypothesis H37 : (G5 = 1)
L144Hypothesis H38 : (∀X ∈ int, (∀Y ∈ int, ((F6 X Y) = (X + Y))))
L145Hypothesis H39 : (∀X ∈ int, ((G6 X) = X))
L146Hypothesis H40 : (∀X ∈ int, ((H6 X) = X))
L147Hypothesis H41 : (I6 = 0)
L148Hypothesis H42 : (J6 = 1)
L149Hypothesis H43 : (∀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)))))))
L150Hypothesis H44 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V6 X Y Z) = (if (X <= 0) then Z else (G6 (U6 (X + - 1) Y Z)))))))
L151Hypothesis H45 : (∀X ∈ int, ((W6 X) = (U6 (H6 X) I6 J6)))
L152Hypothesis H46 : (∀X ∈ int, ((H5 X) = (W6 X)))
L153Hypothesis H47 : (∀X ∈ int, (∀Y ∈ int, ((U5 X Y) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y))))))
L154Hypothesis H48 : (∀X ∈ int, ((V5 X) = (U5 G5 (H5 X))))
L155Hypothesis H49 : (∀X ∈ int, ((FAST X) = ((V3 X) + (V5 X))))
L156Theorem. (
A110224)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.