Beginning of Section A13896
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
L10Hypothesis HF1 : ∀x0 ∈ int, F1 x0 ∈ int
L12Hypothesis HG1 : G1 ∈ int
L14Hypothesis HH1 : H1 ∈ int
L15Variable U1 : set → set → set
L16Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, U1 x0 x1 ∈ int
L18Hypothesis HV1 : V1 ∈ int
L19Variable F0 : set → set
L20Hypothesis HF0 : ∀x0 ∈ int, F0 x0 ∈ int
L21Variable F2 : set → set
L22Hypothesis HF2 : ∀x0 ∈ int, F2 x0 ∈ int
L24Hypothesis HG2 : G2 ∈ int
L25Variable H2 : set → set
L26Hypothesis HH2 : ∀x0 ∈ int, H2 x0 ∈ int
L27Variable U2 : set → set → set
L28Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, U2 x0 x1 ∈ int
L29Variable V2 : set → set
L30Hypothesis HV2 : ∀x0 ∈ int, V2 x0 ∈ int
L31Variable G0 : set → set
L32Hypothesis HG0 : ∀x0 ∈ int, G0 x0 ∈ int
L34Hypothesis HH0 : H0 ∈ int
L35Variable U0 : set → set → set
L36Hypothesis HU0 : ∀x0 ∈ int, ∀x1 ∈ int, U0 x0 x1 ∈ int
L37Variable V0 : set → set
L38Hypothesis HV0 : ∀x0 ∈ int, V0 x0 ∈ int
L39Variable SMALL : set → set
L40Hypothesis HSMALL : ∀x0 ∈ int, SMALL x0 ∈ int
L41Variable F3 : set → set
L42Hypothesis HF3 : ∀x0 ∈ int, F3 x0 ∈ int
L44Hypothesis HG3 : G3 ∈ int
L45Variable F4 : set → set → set
L46Hypothesis HF4 : ∀x0 ∈ int, ∀x1 ∈ int, F4 x0 x1 ∈ int
L47Variable G4 : set → set → set
L48Hypothesis HG4 : ∀x0 ∈ int, ∀x1 ∈ int, G4 x0 x1 ∈ int
L49Variable H4 : set → set
L50Hypothesis HH4 : ∀x0 ∈ int, H4 x0 ∈ int
L52Hypothesis HI4 : I4 ∈ int
L54Hypothesis HJ4 : J4 ∈ int
L55Variable U4 : set → set → set → set
L56Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U4 x0 x1 x2 ∈ int
L57Variable V4 : set → set → set → set
L58Hypothesis HV4 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V4 x0 x1 x2 ∈ int
L59Variable W4 : set → set
L60Hypothesis HW4 : ∀x0 ∈ int, W4 x0 ∈ int
L61Variable H3 : set → set
L62Hypothesis HH3 : ∀x0 ∈ int, H3 x0 ∈ int
L63Variable U3 : set → set → set
L64Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, U3 x0 x1 ∈ int
L65Variable V3 : set → set
L66Hypothesis HV3 : ∀x0 ∈ int, V3 x0 ∈ int
L67Variable F5 : set → set → set
L68Hypothesis HF5 : ∀x0 ∈ int, ∀x1 ∈ int, F5 x0 x1 ∈ int
L69Variable G5 : set → set → set
L70Hypothesis HG5 : ∀x0 ∈ int, ∀x1 ∈ int, G5 x0 x1 ∈ int
L71Variable H5 : set → set
L72Hypothesis HH5 : ∀x0 ∈ int, H5 x0 ∈ int
L74Hypothesis HI5 : I5 ∈ int
L76Hypothesis HJ5 : J5 ∈ int
L77Variable U5 : set → set → set → set
L78Hypothesis HU5 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U5 x0 x1 x2 ∈ int
L79Variable V5 : set → set → set → set
L80Hypothesis HV5 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V5 x0 x1 x2 ∈ int
L81Variable W5 : set → set
L82Hypothesis HW5 : ∀x0 ∈ int, W5 x0 ∈ int
L83Variable F6 : set → set
L84Hypothesis HF6 : ∀x0 ∈ int, F6 x0 ∈ int
L86Hypothesis HG6 : G6 ∈ int
L87Variable F7 : set → set → set
L88Hypothesis HF7 : ∀x0 ∈ int, ∀x1 ∈ int, F7 x0 x1 ∈ int
L89Variable G7 : set → set → set
L90Hypothesis HG7 : ∀x0 ∈ int, ∀x1 ∈ int, G7 x0 x1 ∈ int
L91Variable H7 : set → set
L92Hypothesis HH7 : ∀x0 ∈ int, H7 x0 ∈ int
L94Hypothesis HI7 : I7 ∈ int
L96Hypothesis HJ7 : J7 ∈ int
L97Variable U7 : set → set → set → set
L98Hypothesis HU7 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U7 x0 x1 x2 ∈ int
L99Variable V7 : set → set → set → set
L100Hypothesis HV7 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V7 x0 x1 x2 ∈ int
L101Variable W7 : set → set
L102Hypothesis HW7 : ∀x0 ∈ int, W7 x0 ∈ int
L103Variable H6 : set → set
L104Hypothesis HH6 : ∀x0 ∈ int, H6 x0 ∈ int
L105Variable U6 : set → set → set
L106Hypothesis HU6 : ∀x0 ∈ int, ∀x1 ∈ int, U6 x0 x1 ∈ int
L107Variable V6 : set → set
L108Hypothesis HV6 : ∀x0 ∈ int, V6 x0 ∈ int
L109Variable FAST : set → set
L110Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L111Hypothesis H1 : (∀X ∈ int, ((F1 X) = (2 * (2 + X))))
L112Hypothesis H2 : (G1 = 2)
L113Hypothesis H3 : (H1 = 2)
L114Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))
L115Hypothesis H5 : (V1 = (U1 G1 H1))
L116Hypothesis H6 : (∀X ∈ int, ((F0 X) = (V1 * X)))
L117Hypothesis H7 : (∀X ∈ int, ((F2 X) = (1 + (X + X))))
L118Hypothesis H8 : (G2 = 2)
L119Hypothesis H9 : (∀X ∈ int, ((H2 X) = X))
L120Hypothesis H10 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y))))))
L121Hypothesis H11 : (∀X ∈ int, ((V2 X) = (U2 G2 (H2 X))))
L122Hypothesis H12 : (∀X ∈ int, ((G0 X) = ((V2 X) + X)))
L123Hypothesis H13 : (H0 = 1)
L124Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y))))))
L125Hypothesis H15 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))
L126Hypothesis H16 : (∀X ∈ int, ((SMALL X) = (V0 X)))
L127Hypothesis H17 : (∀X ∈ int, ((F3 X) = (X * X)))
L128Hypothesis H18 : (G3 = 1)
L129Hypothesis H19 : (∀X ∈ int, (∀Y ∈ int, ((F4 X Y) = (X * Y))))
L130Hypothesis H20 : (∀X ∈ int, (∀Y ∈ int, ((G4 X Y) = Y)))
L131Hypothesis H21 : (∀X ∈ int, ((H4 X) = X))
L132Hypothesis H22 : (I4 = 1)
L133Hypothesis H23 : (J4 = (2 * (2 + (2 * (2 + 2)))))
L134Hypothesis H24 : (∀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)))))))
L135Hypothesis H25 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V4 X Y Z) = (if (X <= 0) then Z else (G4 (U4 (X + - 1) Y Z) (V4 (X + - 1) Y Z)))))))
L136Hypothesis H26 : (∀X ∈ int, ((W4 X) = (U4 (H4 X) I4 J4)))
L137Hypothesis H27 : (∀X ∈ int, ((H3 X) = (W4 X)))
L138Hypothesis H28 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y))))))
L139Hypothesis H29 : (∀X ∈ int, ((V3 X) = (U3 G3 (H3 X))))
L140Hypothesis H30 : (∀X ∈ int, (∀Y ∈ int, ((F5 X Y) = (X * Y))))
L141Hypothesis H31 : (∀X ∈ int, (∀Y ∈ int, ((G5 X Y) = Y)))
L142Hypothesis H32 : (∀X ∈ int, ((H5 X) = (1 + X)))
L143Hypothesis H33 : (I5 = 1)
L144Hypothesis H34 : (J5 = (2 * (2 + (2 * (2 + 2)))))
L145Hypothesis H35 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U5 X Y Z) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y Z) (V5 (X + - 1) Y Z)))))))
L146Hypothesis H36 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V5 X Y Z) = (if (X <= 0) then Z else (G5 (U5 (X + - 1) Y Z) (V5 (X + - 1) Y Z)))))))
L147Hypothesis H37 : (∀X ∈ int, ((W5 X) = (U5 (H5 X) I5 J5)))
L148Hypothesis H38 : (∀X ∈ int, ((F6 X) = (X * X)))
L149Hypothesis H39 : (G6 = 1)
L150Hypothesis H40 : (∀X ∈ int, (∀Y ∈ int, ((F7 X Y) = (X * Y))))
L151Hypothesis H41 : (∀X ∈ int, (∀Y ∈ int, ((G7 X Y) = Y)))
L152Hypothesis H42 : (∀X ∈ int, ((H7 X) = (1 + X)))
L153Hypothesis H43 : (I7 = 1)
L154Hypothesis H44 : (J7 = (2 * (2 + (2 * (2 + 2)))))
L155Hypothesis H45 : (∀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)))))))
L156Hypothesis H46 : (∀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)))))))
L157Hypothesis H47 : (∀X ∈ int, ((W7 X) = (U7 (H7 X) I7 J7)))
L158Hypothesis H48 : (∀X ∈ int, ((H6 X) = (W7 X)))
L159Hypothesis H49 : (∀X ∈ int, (∀Y ∈ int, ((U6 X Y) = (if (X <= 0) then Y else (F6 (U6 (X + - 1) Y))))))
L160Hypothesis H50 : (∀X ∈ int, ((V6 X) = (U6 G6 (H6 X))))
L161Hypothesis H51 : (∀X ∈ int, ((FAST X) = (((V3 X) * (W5 X)) * (V6 X))))
L162Theorem. (
A13896)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.