Beginning of Section A28221
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
L11Variable G3 : set → set → set
L12Hypothesis HG3 : ∀x0 ∈ int, ∀x1 ∈ int, G3 x0 x1 ∈ int
L14Hypothesis HH3 : H3 ∈ int
L15Variable U3 : set → set → set
L16Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, U3 x0 x1 ∈ int
L17Variable V3 : set → set → set
L18Hypothesis HV3 : ∀x0 ∈ int, ∀x1 ∈ int, V3 x0 x1 ∈ 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
L24Hypothesis HH2 : H2 ∈ int
L25Variable U2 : set → set → set
L26Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, U2 x0 x1 ∈ int
L27Variable V2 : set → set → set
L28Hypothesis HV2 : ∀x0 ∈ int, ∀x1 ∈ int, V2 x0 x1 ∈ int
L29Variable F4 : set → set → set
L30Hypothesis HF4 : ∀x0 ∈ int, ∀x1 ∈ int, F4 x0 x1 ∈ int
L32Hypothesis HG4 : G4 ∈ int
L33Variable H4 : set → set
L34Hypothesis HH4 : ∀x0 ∈ int, H4 x0 ∈ int
L35Variable U4 : set → set → set
L36Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, U4 x0 x1 ∈ int
L37Variable V4 : set → set
L38Hypothesis HV4 : ∀x0 ∈ int, V4 x0 ∈ int
L39Variable F1 : set → set → set
L40Hypothesis HF1 : ∀x0 ∈ int, ∀x1 ∈ int, F1 x0 x1 ∈ int
L41Variable G1 : set → set → set
L42Hypothesis HG1 : ∀x0 ∈ int, ∀x1 ∈ int, G1 x0 x1 ∈ int
L44Hypothesis HH1 : H1 ∈ int
L45Variable U1 : set → set → set
L46Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, U1 x0 x1 ∈ int
L47Variable V1 : set → set → set
L48Hypothesis HV1 : ∀x0 ∈ int, ∀x1 ∈ int, V1 x0 x1 ∈ int
L49Variable F0 : set → set → set
L50Hypothesis HF0 : ∀x0 ∈ int, ∀x1 ∈ int, F0 x0 x1 ∈ int
L51Variable G0 : set → set
L52Hypothesis HG0 : ∀x0 ∈ int, G0 x0 ∈ int
L54Hypothesis HH0 : H0 ∈ int
L55Variable U0 : set → set → set
L56Hypothesis HU0 : ∀x0 ∈ int, ∀x1 ∈ int, U0 x0 x1 ∈ int
L57Variable V0 : set → set
L58Hypothesis HV0 : ∀x0 ∈ int, V0 x0 ∈ int
L59Variable SMALL : set → set
L60Hypothesis HSMALL : ∀x0 ∈ int, SMALL x0 ∈ int
L61Variable F9 : set → set → set
L62Hypothesis HF9 : ∀x0 ∈ int, ∀x1 ∈ int, F9 x0 x1 ∈ int
L63Variable G9 : set → set → set
L64Hypothesis HG9 : ∀x0 ∈ int, ∀x1 ∈ int, G9 x0 x1 ∈ int
L65Variable H9 : set → set
L66Hypothesis HH9 : ∀x0 ∈ int, H9 x0 ∈ int
L68Hypothesis HI9 : I9 ∈ int
L70Hypothesis HJ9 : J9 ∈ int
L71Variable U9 : set → set → set → set
L72Hypothesis HU9 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U9 x0 x1 x2 ∈ int
L73Variable V9 : set → set → set → set
L74Hypothesis HV9 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V9 x0 x1 x2 ∈ int
L75Variable W9 : set → set
L76Hypothesis HW9 : ∀x0 ∈ int, W9 x0 ∈ int
L77Variable F10 : set → set
L78Hypothesis HF10 : ∀x0 ∈ int, F10 x0 ∈ int
L80Hypothesis HG10 : G10 ∈ int
L81Variable F11 : set → set
L82Hypothesis HF11 : ∀x0 ∈ int, F11 x0 ∈ int
L83Variable G11 : set → set
L84Hypothesis HG11 : ∀x0 ∈ int, G11 x0 ∈ int
L86Hypothesis HH11 : H11 ∈ int
L87Variable U11 : set → set → set
L88Hypothesis HU11 : ∀x0 ∈ int, ∀x1 ∈ int, U11 x0 x1 ∈ int
L89Variable V11 : set → set
L90Hypothesis HV11 : ∀x0 ∈ int, V11 x0 ∈ int
L91Variable H10 : set → set
L92Hypothesis HH10 : ∀x0 ∈ int, H10 x0 ∈ int
L93Variable U10 : set → set → set
L94Hypothesis HU10 : ∀x0 ∈ int, ∀x1 ∈ int, U10 x0 x1 ∈ int
L95Variable V10 : set → set
L96Hypothesis HV10 : ∀x0 ∈ int, V10 x0 ∈ int
L97Variable F8 : set → set
L98Hypothesis HF8 : ∀x0 ∈ int, F8 x0 ∈ int
L100Hypothesis HG8 : G8 ∈ int
L101Variable H8 : set → set → set
L102Hypothesis HH8 : ∀x0 ∈ int, ∀x1 ∈ int, H8 x0 x1 ∈ int
L103Variable U8 : set → set → set
L104Hypothesis HU8 : ∀x0 ∈ int, ∀x1 ∈ int, U8 x0 x1 ∈ int
L105Variable V8 : set → set → set
L106Hypothesis HV8 : ∀x0 ∈ int, ∀x1 ∈ int, V8 x0 x1 ∈ int
L107Variable F7 : set → set → set
L108Hypothesis HF7 : ∀x0 ∈ int, ∀x1 ∈ int, F7 x0 x1 ∈ int
L109Variable G7 : set → set
L110Hypothesis HG7 : ∀x0 ∈ int, G7 x0 ∈ int
L112Hypothesis HH7 : H7 ∈ int
L113Variable U7 : set → set → set
L114Hypothesis HU7 : ∀x0 ∈ int, ∀x1 ∈ int, U7 x0 x1 ∈ int
L115Variable V7 : set → set
L116Hypothesis HV7 : ∀x0 ∈ int, V7 x0 ∈ int
L117Variable F6 : set → set
L118Hypothesis HF6 : ∀x0 ∈ int, F6 x0 ∈ int
L120Hypothesis HG6 : G6 ∈ int
L121Variable H6 : set → set → set
L122Hypothesis HH6 : ∀x0 ∈ int, ∀x1 ∈ int, H6 x0 x1 ∈ int
L123Variable U6 : set → set → set
L124Hypothesis HU6 : ∀x0 ∈ int, ∀x1 ∈ int, U6 x0 x1 ∈ int
L125Variable V6 : set → set → set
L126Hypothesis HV6 : ∀x0 ∈ int, ∀x1 ∈ int, V6 x0 x1 ∈ int
L127Variable F5 : set → set → set
L128Hypothesis HF5 : ∀x0 ∈ int, ∀x1 ∈ int, F5 x0 x1 ∈ int
L129Variable G5 : set → set
L130Hypothesis HG5 : ∀x0 ∈ int, G5 x0 ∈ int
L132Hypothesis HH5 : H5 ∈ int
L133Variable U5 : set → set → set
L134Hypothesis HU5 : ∀x0 ∈ int, ∀x1 ∈ int, U5 x0 x1 ∈ int
L135Variable V5 : set → set
L136Hypothesis HV5 : ∀x0 ∈ int, V5 x0 ∈ int
L137Variable FAST : set → set
L138Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L139Hypothesis H1 : (∀X ∈ int, ((F3 X) = ((X + X) + X)))
L140Hypothesis H2 : (∀X ∈ int, (∀Y ∈ int, ((G3 X Y) = (Y + Y))))
L141Hypothesis H3 : (H3 = 1)
L142Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y))))))
L143Hypothesis H5 : (∀X ∈ int, (∀Y ∈ int, ((V3 X Y) = (U3 (G3 X Y) H3))))
L144Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = ((V3 X Y) + (2 * (2 * (X + X)))))))
L145Hypothesis H7 : (∀X ∈ int, (∀Y ∈ int, ((G2 X Y) = Y)))
L146Hypothesis H8 : (H2 = 1)
L147Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y) X)))))
L148Hypothesis H10 : (∀X ∈ int, (∀Y ∈ int, ((V2 X Y) = (U2 (G2 X Y) H2))))
L149Hypothesis H11 : (∀X ∈ int, (∀Y ∈ int, ((F4 X Y) = ((2 + Y) * X))))
L150Hypothesis H12 : (G4 = 2)
L151Hypothesis H13 : (∀X ∈ int, ((H4 X) = X))
L152Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y) X)))))
L153Hypothesis H15 : (∀X ∈ int, ((V4 X) = (U4 G4 (H4 X))))
L154Hypothesis H16 : (∀X ∈ int, (∀Y ∈ int, ((F1 X Y) = ((V2 X Y) + (V4 X)))))
L155Hypothesis H17 : (∀X ∈ int, (∀Y ∈ int, ((G1 X Y) = Y)))
L156Hypothesis H18 : (H1 = 1)
L157Hypothesis H19 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y) X)))))
L158Hypothesis H20 : (∀X ∈ int, (∀Y ∈ int, ((V1 X Y) = (U1 (G1 X Y) H1))))
L159Hypothesis H21 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = (((V1 X Y) + (2 * ((X + X) + X))) + X))))
L160Hypothesis H22 : (∀X ∈ int, ((G0 X) = X))
L161Hypothesis H23 : (H0 = 1)
L162Hypothesis H24 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y) X)))))
L163Hypothesis H25 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))
L164Hypothesis H26 : (∀X ∈ int, ((SMALL X) = (V0 X)))
L165Hypothesis H27 : (∀X ∈ int, (∀Y ∈ int, ((F9 X Y) = (((X + X) + X) + Y))))
L166Hypothesis H28 : (∀X ∈ int, (∀Y ∈ int, ((G9 X Y) = (Y + Y))))
L167Hypothesis H29 : (∀X ∈ int, ((H9 X) = X))
L168Hypothesis H30 : (I9 = 1)
L169Hypothesis H31 : (J9 = 2)
L170Hypothesis H32 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U9 X Y Z) = (if (X <= 0) then Y else (F9 (U9 (X + - 1) Y Z) (V9 (X + - 1) Y Z)))))))
L171Hypothesis H33 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V9 X Y Z) = (if (X <= 0) then Z else (G9 (U9 (X + - 1) Y Z) (V9 (X + - 1) Y Z)))))))
L172Hypothesis H34 : (∀X ∈ int, ((W9 X) = (U9 (H9 X) I9 J9)))
L173Hypothesis H35 : (∀X ∈ int, ((F10 X) = (X * X)))
L174Hypothesis H36 : (G10 = 1)
L175Hypothesis H37 : (∀X ∈ int, ((F11 X) = (X + X)))
L176Hypothesis H38 : (∀X ∈ int, ((G11 X) = X))
L177Hypothesis H39 : (H11 = 1)
L178Hypothesis H40 : (∀X ∈ int, (∀Y ∈ int, ((U11 X Y) = (if (X <= 0) then Y else (F11 (U11 (X + - 1) Y))))))
L179Hypothesis H41 : (∀X ∈ int, ((V11 X) = (U11 (G11 X) H11)))
L180Hypothesis H42 : (∀X ∈ int, ((H10 X) = (V11 X)))
L181Hypothesis H43 : (∀X ∈ int, (∀Y ∈ int, ((U10 X Y) = (if (X <= 0) then Y else (F10 (U10 (X + - 1) Y))))))
L182Hypothesis H44 : (∀X ∈ int, ((V10 X) = (U10 G10 (H10 X))))
L183Hypothesis H45 : (∀X ∈ int, ((F8 X) = ((W9 X) * (V10 X))))
L184Hypothesis H46 : (G8 = 1)
L185Hypothesis H47 : (∀X ∈ int, (∀Y ∈ int, ((H8 X Y) = Y)))
L186Hypothesis H48 : (∀X ∈ int, (∀Y ∈ int, ((U8 X Y) = (if (X <= 0) then Y else (F8 (U8 (X + - 1) Y))))))
L187Hypothesis H49 : (∀X ∈ int, (∀Y ∈ int, ((V8 X Y) = (U8 G8 (H8 X Y)))))
L188Hypothesis H50 : (∀X ∈ int, (∀Y ∈ int, ((F7 X Y) = (((V8 X Y) + (2 * ((X + X) + X))) + X))))
L189Hypothesis H51 : (∀X ∈ int, ((G7 X) = X))
L190Hypothesis H52 : (H7 = 1)
L191Hypothesis H53 : (∀X ∈ int, (∀Y ∈ int, ((U7 X Y) = (if (X <= 0) then Y else (F7 (U7 (X + - 1) Y) X)))))
L192Hypothesis H54 : (∀X ∈ int, ((V7 X) = (U7 (G7 X) H7)))
L193Hypothesis H55 : (∀X ∈ int, ((F6 X) = (V7 X)))
L194Hypothesis H56 : (G6 = 1)
L195Hypothesis H57 : (∀X ∈ int, (∀Y ∈ int, ((H6 X Y) = Y)))
L196Hypothesis H58 : (∀X ∈ int, (∀Y ∈ int, ((U6 X Y) = (if (X <= 0) then Y else (F6 (U6 (X + - 1) Y))))))
L197Hypothesis H59 : (∀X ∈ int, (∀Y ∈ int, ((V6 X Y) = (U6 G6 (H6 X Y)))))
L198Hypothesis H60 : (∀X ∈ int, (∀Y ∈ int, ((F5 X Y) = (((V6 X Y) + X) + (((X + X) * 2) * 2)))))
L199Hypothesis H61 : (∀X ∈ int, ((G5 X) = X))
L200Hypothesis H62 : (H5 = 1)
L201Hypothesis H63 : (∀X ∈ int, (∀Y ∈ int, ((U5 X Y) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y) X)))))
L202Hypothesis H64 : (∀X ∈ int, ((V5 X) = (U5 (G5 X) H5)))
L203Hypothesis H65 : (∀X ∈ int, ((FAST X) = (V5 X)))
L204Theorem. (
A28221)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.