Beginning of Section A2571
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
L12Hypothesis HG2 : ∀x0 ∈ int, G2 x0 ∈ 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 F1 : set → set → set
L26Hypothesis HF1 : ∀x0 ∈ int, ∀x1 ∈ int, F1 x0 x1 ∈ int
L27Variable G1 : set → set → set
L28Hypothesis HG1 : ∀x0 ∈ int, ∀x1 ∈ int, G1 x0 x1 ∈ int
L30Hypothesis HH1 : H1 ∈ int
L31Variable U1 : set → set → set
L32Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, U1 x0 x1 ∈ int
L33Variable V1 : set → set → set
L34Hypothesis HV1 : ∀x0 ∈ int, ∀x1 ∈ int, V1 x0 x1 ∈ int
L35Variable F0 : set → set → set
L36Hypothesis HF0 : ∀x0 ∈ int, ∀x1 ∈ int, F0 x0 x1 ∈ int
L37Variable G0 : set → set
L38Hypothesis HG0 : ∀x0 ∈ int, G0 x0 ∈ int
L40Hypothesis HH0 : H0 ∈ int
L41Variable U0 : set → set → set
L42Hypothesis HU0 : ∀x0 ∈ int, ∀x1 ∈ int, U0 x0 x1 ∈ int
L43Variable V0 : set → set
L44Hypothesis HV0 : ∀x0 ∈ int, V0 x0 ∈ int
L45Variable SMALL : set → set
L46Hypothesis HSMALL : ∀x0 ∈ int, SMALL x0 ∈ int
L47Variable F5 : set → set → set
L48Hypothesis HF5 : ∀x0 ∈ int, ∀x1 ∈ int, F5 x0 x1 ∈ int
L49Variable G5 : set → set
L50Hypothesis HG5 : ∀x0 ∈ int, G5 x0 ∈ int
L51Variable H5 : set → set
L52Hypothesis HH5 : ∀x0 ∈ int, H5 x0 ∈ int
L54Hypothesis HI5 : I5 ∈ int
L56Hypothesis HJ5 : J5 ∈ int
L57Variable U5 : set → set → set → set
L58Hypothesis HU5 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U5 x0 x1 x2 ∈ int
L59Variable V5 : set → set → set → set
L60Hypothesis HV5 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V5 x0 x1 x2 ∈ int
L61Variable W5 : set → set
L62Hypothesis HW5 : ∀x0 ∈ int, W5 x0 ∈ int
L63Variable F6 : set → set → set
L64Hypothesis HF6 : ∀x0 ∈ int, ∀x1 ∈ int, F6 x0 x1 ∈ int
L65Variable G6 : set → set
L66Hypothesis HG6 : ∀x0 ∈ int, G6 x0 ∈ int
L67Variable H6 : set → set
L68Hypothesis HH6 : ∀x0 ∈ int, H6 x0 ∈ int
L70Hypothesis HI6 : I6 ∈ int
L72Hypothesis HJ6 : J6 ∈ int
L73Variable U6 : set → set → set → set
L74Hypothesis HU6 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U6 x0 x1 x2 ∈ int
L75Variable V6 : set → set → set → set
L76Hypothesis HV6 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V6 x0 x1 x2 ∈ int
L77Variable W6 : set → set
L78Hypothesis HW6 : ∀x0 ∈ int, W6 x0 ∈ int
L79Variable F4 : set → set
L80Hypothesis HF4 : ∀x0 ∈ int, F4 x0 ∈ int
L82Hypothesis HG4 : G4 ∈ int
L83Variable H4 : set → set → set
L84Hypothesis HH4 : ∀x0 ∈ int, ∀x1 ∈ int, H4 x0 x1 ∈ int
L85Variable U4 : set → set → set
L86Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, U4 x0 x1 ∈ int
L87Variable V4 : set → set → set
L88Hypothesis HV4 : ∀x0 ∈ int, ∀x1 ∈ int, V4 x0 x1 ∈ int
L89Variable F3 : set → set → set
L90Hypothesis HF3 : ∀x0 ∈ int, ∀x1 ∈ int, F3 x0 x1 ∈ int
L91Variable G3 : set → set
L92Hypothesis HG3 : ∀x0 ∈ int, G3 x0 ∈ int
L94Hypothesis HH3 : H3 ∈ int
L95Variable U3 : set → set → set
L96Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, U3 x0 x1 ∈ int
L97Variable V3 : set → set
L98Hypothesis HV3 : ∀x0 ∈ int, V3 x0 ∈ int
L99Variable FAST : set → set
L100Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L101Hypothesis H1 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = (X + Y))))
L102Hypothesis H2 : (∀X ∈ int, ((G2 X) = X))
L103Hypothesis H3 : (∀X ∈ int, (∀Y ∈ int, ((H2 X Y) = (Y + Y))))
L104Hypothesis H4 : (I2 = 1)
L105Hypothesis H5 : (J2 = 1)
L106Hypothesis 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)))))))
L107Hypothesis H7 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V2 X Y Z) = (if (X <= 0) then Z else (G2 (U2 (X + - 1) Y Z)))))))
L108Hypothesis H8 : (∀X ∈ int, (∀Y ∈ int, ((W2 X Y) = (U2 (H2 X Y) I2 J2))))
L109Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((F1 X Y) = ((W2 X Y) + - X))))
L110Hypothesis H10 : (∀X ∈ int, (∀Y ∈ int, ((G1 X Y) = (1 + Y))))
L111Hypothesis H11 : (H1 = 1)
L112Hypothesis H12 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y) X)))))
L113Hypothesis H13 : (∀X ∈ int, (∀Y ∈ int, ((V1 X Y) = (U1 (G1 X Y) H1))))
L114Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = ((V1 X Y) + - X))))
L115Hypothesis H15 : (∀X ∈ int, ((G0 X) = X))
L116Hypothesis H16 : (H0 = 1)
L117Hypothesis H17 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y) X)))))
L118Hypothesis H18 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))
L119Hypothesis H19 : (∀X ∈ int, ((SMALL X) = (V0 X)))
L120Hypothesis H20 : (∀X ∈ int, (∀Y ∈ int, ((F5 X Y) = (X + Y))))
L121Hypothesis H21 : (∀X ∈ int, ((G5 X) = X))
L122Hypothesis H22 : (∀X ∈ int, ((H5 X) = (X + - 1)))
L123Hypothesis H23 : (I5 = 2)
L124Hypothesis H24 : (J5 = 1)
L125Hypothesis H25 : (∀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)))))))
L126Hypothesis H26 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V5 X Y Z) = (if (X <= 0) then Z else (G5 (U5 (X + - 1) Y Z)))))))
L127Hypothesis H27 : (∀X ∈ int, ((W5 X) = (U5 (H5 X) I5 J5)))
L128Hypothesis H28 : (∀X ∈ int, (∀Y ∈ int, ((F6 X Y) = (X + Y))))
L129Hypothesis H29 : (∀X ∈ int, ((G6 X) = X))
L130Hypothesis H30 : (∀X ∈ int, ((H6 X) = X))
L131Hypothesis H31 : (I6 = 2)
L132Hypothesis H32 : (J6 = 1)
L133Hypothesis H33 : (∀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)))))))
L134Hypothesis H34 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V6 X Y Z) = (if (X <= 0) then Z else (G6 (U6 (X + - 1) Y Z)))))))
L135Hypothesis H35 : (∀X ∈ int, ((W6 X) = (U6 (H6 X) I6 J6)))
L136Hypothesis H36 : (∀X ∈ int, ((F4 X) = ((W5 X) * (W6 X))))
L137Hypothesis H37 : (G4 = 1)
L138Hypothesis H38 : (∀X ∈ int, (∀Y ∈ int, ((H4 X Y) = Y)))
L139Hypothesis H39 : (∀X ∈ int, (∀Y ∈ int, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y))))))
L140Hypothesis H40 : (∀X ∈ int, (∀Y ∈ int, ((V4 X Y) = (U4 G4 (H4 X Y)))))
L141Hypothesis H41 : (∀X ∈ int, (∀Y ∈ int, ((F3 X Y) = ((V4 X Y) + - X))))
L142Hypothesis H42 : (∀X ∈ int, ((G3 X) = X))
L143Hypothesis H43 : (H3 = 1)
L144Hypothesis H44 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y) X)))))
L145Hypothesis H45 : (∀X ∈ int, ((V3 X) = (U3 (G3 X) H3)))
L146Hypothesis H46 : (∀X ∈ int, ((FAST X) = (V3 X)))
L147Theorem. (
A2571)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.