Beginning of Section A217471
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
L10Hypothesis HF3 : ∀x0 ∈ int, F3 x0 ∈ 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
L20Hypothesis HF2 : ∀x0 ∈ int, F2 x0 ∈ int
L22Hypothesis HG2 : G2 ∈ int
L23Variable F4 : set → set → set
L24Hypothesis HF4 : ∀x0 ∈ int, ∀x1 ∈ int, F4 x0 x1 ∈ int
L25Variable G4 : set → set
L26Hypothesis HG4 : ∀x0 ∈ int, G4 x0 ∈ int
L27Variable H4 : set → set
L28Hypothesis HH4 : ∀x0 ∈ int, H4 x0 ∈ int
L30Hypothesis HI4 : I4 ∈ int
L32Hypothesis HJ4 : J4 ∈ int
L33Variable U4 : set → set → set → set
L34Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U4 x0 x1 x2 ∈ int
L35Variable V4 : set → set → set → set
L36Hypothesis HV4 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V4 x0 x1 x2 ∈ int
L37Variable W4 : set → set
L38Hypothesis HW4 : ∀x0 ∈ int, W4 x0 ∈ int
L39Variable H2 : set → set
L40Hypothesis HH2 : ∀x0 ∈ int, H2 x0 ∈ int
L41Variable U2 : set → set → set
L42Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, U2 x0 x1 ∈ int
L43Variable V2 : set → set
L44Hypothesis HV2 : ∀x0 ∈ int, V2 x0 ∈ int
L45Variable F1 : set → set
L46Hypothesis HF1 : ∀x0 ∈ int, F1 x0 ∈ int
L48Hypothesis HG1 : G1 ∈ int
L49Variable H1 : set → set → set
L50Hypothesis HH1 : ∀x0 ∈ int, ∀x1 ∈ int, H1 x0 x1 ∈ 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
L68Hypothesis HF7 : ∀x0 ∈ int, F7 x0 ∈ int
L70Hypothesis HG7 : G7 ∈ int
L71Variable F8 : set → set → set
L72Hypothesis HF8 : ∀x0 ∈ int, ∀x1 ∈ int, F8 x0 x1 ∈ int
L73Variable G8 : set → set
L74Hypothesis HG8 : ∀x0 ∈ int, G8 x0 ∈ int
L75Variable H8 : set → set
L76Hypothesis HH8 : ∀x0 ∈ int, H8 x0 ∈ int
L78Hypothesis HI8 : I8 ∈ int
L80Hypothesis HJ8 : J8 ∈ int
L81Variable U8 : set → set → set → set
L82Hypothesis HU8 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U8 x0 x1 x2 ∈ int
L83Variable V8 : set → set → set → set
L84Hypothesis HV8 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V8 x0 x1 x2 ∈ int
L85Variable W8 : set → set
L86Hypothesis HW8 : ∀x0 ∈ int, W8 x0 ∈ int
L87Variable H7 : set → set
L88Hypothesis HH7 : ∀x0 ∈ int, H7 x0 ∈ int
L89Variable U7 : set → set → set
L90Hypothesis HU7 : ∀x0 ∈ int, ∀x1 ∈ int, U7 x0 x1 ∈ int
L91Variable V7 : set → set
L92Hypothesis HV7 : ∀x0 ∈ int, V7 x0 ∈ int
L93Variable F6 : set → set
L94Hypothesis HF6 : ∀x0 ∈ int, F6 x0 ∈ int
L96Hypothesis HG6 : G6 ∈ int
L97Variable H6 : set → set → set
L98Hypothesis HH6 : ∀x0 ∈ int, ∀x1 ∈ int, H6 x0 x1 ∈ int
L99Variable U6 : set → set → set
L100Hypothesis HU6 : ∀x0 ∈ int, ∀x1 ∈ int, U6 x0 x1 ∈ int
L101Variable V6 : set → set → set
L102Hypothesis HV6 : ∀x0 ∈ int, ∀x1 ∈ int, V6 x0 x1 ∈ int
L103Variable F5 : set → set → set
L104Hypothesis HF5 : ∀x0 ∈ int, ∀x1 ∈ int, F5 x0 x1 ∈ int
L105Variable G5 : set → set
L106Hypothesis HG5 : ∀x0 ∈ int, G5 x0 ∈ int
L108Hypothesis HH5 : H5 ∈ int
L109Variable U5 : set → set → set
L110Hypothesis HU5 : ∀x0 ∈ int, ∀x1 ∈ int, U5 x0 x1 ∈ int
L111Variable V5 : set → set
L112Hypothesis HV5 : ∀x0 ∈ int, V5 x0 ∈ int
L113Variable FAST : set → set
L114Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L115Hypothesis H1 : (∀X ∈ int, ((F3 X) = (X * X)))
L116Hypothesis H2 : (G3 = 2)
L117Hypothesis H3 : (∀X ∈ int, ((H3 X) = X))
L118Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y))))))
L119Hypothesis H5 : (∀X ∈ int, ((V3 X) = (U3 G3 (H3 X))))
L120Hypothesis H6 : (∀X ∈ int, ((F2 X) = ((V3 X) * X)))
L121Hypothesis H7 : (G2 = 1)
L122Hypothesis H8 : (∀X ∈ int, (∀Y ∈ int, ((F4 X Y) = (X + Y))))
L123Hypothesis H9 : (∀X ∈ int, ((G4 X) = X))
L124Hypothesis H10 : (∀X ∈ int, ((H4 X) = (X + X)))
L125Hypothesis H11 : (I4 = 0)
L126Hypothesis H12 : (J4 = 1)
L127Hypothesis H13 : (∀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)))))))
L128Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V4 X Y Z) = (if (X <= 0) then Z else (G4 (U4 (X + - 1) Y Z)))))))
L129Hypothesis H15 : (∀X ∈ int, ((W4 X) = (U4 (H4 X) I4 J4)))
L130Hypothesis H16 : (∀X ∈ int, ((H2 X) = (W4 X)))
L131Hypothesis H17 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y))))))
L132Hypothesis H18 : (∀X ∈ int, ((V2 X) = (U2 G2 (H2 X))))
L133Hypothesis H19 : (∀X ∈ int, ((F1 X) = (V2 X)))
L134Hypothesis H20 : (G1 = 1)
L135Hypothesis H21 : (∀X ∈ int, (∀Y ∈ int, ((H1 X Y) = Y)))
L136Hypothesis H22 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))
L137Hypothesis H23 : (∀X ∈ int, (∀Y ∈ int, ((V1 X Y) = (U1 G1 (H1 X Y)))))
L138Hypothesis H24 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = ((V1 X Y) + X))))
L139Hypothesis H25 : (∀X ∈ int, ((G0 X) = X))
L140Hypothesis H26 : (H0 = 0)
L141Hypothesis H27 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y) X)))))
L142Hypothesis H28 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))
L143Hypothesis H29 : (∀X ∈ int, ((SMALL X) = (V0 X)))
L144Hypothesis H30 : (∀X ∈ int, ((F7 X) = ((((X * X) * X) * X) * X)))
L145Hypothesis H31 : (G7 = 1)
L146Hypothesis H32 : (∀X ∈ int, (∀Y ∈ int, ((F8 X Y) = (X + Y))))
L147Hypothesis H33 : (∀X ∈ int, ((G8 X) = X))
L148Hypothesis H34 : (∀X ∈ int, ((H8 X) = ((X + - 2) + X)))
L149Hypothesis H35 : (I8 = 1)
L150Hypothesis H36 : (J8 = 1)
L151Hypothesis H37 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U8 X Y Z) = (if (X <= 0) then Y else (F8 (U8 (X + - 1) Y Z) (V8 (X + - 1) Y Z)))))))
L152Hypothesis H38 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V8 X Y Z) = (if (X <= 0) then Z else (G8 (U8 (X + - 1) Y Z)))))))
L153Hypothesis H39 : (∀X ∈ int, ((W8 X) = (U8 (H8 X) I8 J8)))
L154Hypothesis H40 : (∀X ∈ int, ((H7 X) = (W8 X)))
L155Hypothesis H41 : (∀X ∈ int, (∀Y ∈ int, ((U7 X Y) = (if (X <= 0) then Y else (F7 (U7 (X + - 1) Y))))))
L156Hypothesis H42 : (∀X ∈ int, ((V7 X) = (U7 G7 (H7 X))))
L157Hypothesis H43 : (∀X ∈ int, ((F6 X) = (V7 X)))
L158Hypothesis H44 : (G6 = 1)
L159Hypothesis H45 : (∀X ∈ int, (∀Y ∈ int, ((H6 X Y) = Y)))
L160Hypothesis H46 : (∀X ∈ int, (∀Y ∈ int, ((U6 X Y) = (if (X <= 0) then Y else (F6 (U6 (X + - 1) Y))))))
L161Hypothesis H47 : (∀X ∈ int, (∀Y ∈ int, ((V6 X Y) = (U6 G6 (H6 X Y)))))
L162Hypothesis H48 : (∀X ∈ int, (∀Y ∈ int, ((F5 X Y) = ((V6 X Y) + X))))
L163Hypothesis H49 : (∀X ∈ int, ((G5 X) = X))
L164Hypothesis H50 : (H5 = 0)
L165Hypothesis H51 : (∀X ∈ int, (∀Y ∈ int, ((U5 X Y) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y) X)))))
L166Hypothesis H52 : (∀X ∈ int, ((V5 X) = (U5 (G5 X) H5)))
L167Hypothesis H53 : (∀X ∈ int, ((FAST X) = (V5 X)))
L168Theorem. (
A217471)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.