Beginning of Section A21194
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 → set
L12Hypothesis HG1 : ∀x0 ∈ int, ∀x1 ∈ int, G1 x0 x1 ∈ 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 F2 : set → set
L26Hypothesis HF2 : ∀x0 ∈ int, F2 x0 ∈ int
L28Hypothesis HG2 : G2 ∈ int
L29Variable H2 : set → set
L30Hypothesis HH2 : ∀x0 ∈ int, H2 x0 ∈ int
L31Variable U2 : set → set → set
L32Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, U2 x0 x1 ∈ int
L33Variable V2 : set → set
L34Hypothesis HV2 : ∀x0 ∈ int, V2 x0 ∈ 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
L48Hypothesis HF5 : ∀x0 ∈ int, F5 x0 ∈ int
L49Variable G5 : set → set
L50Hypothesis HG5 : ∀x0 ∈ int, G5 x0 ∈ int
L52Hypothesis HH5 : H5 ∈ int
L53Variable U5 : set → set → set
L54Hypothesis HU5 : ∀x0 ∈ int, ∀x1 ∈ int, U5 x0 x1 ∈ int
L55Variable V5 : set → set
L56Hypothesis HV5 : ∀x0 ∈ int, V5 x0 ∈ int
L57Variable F6 : set → set
L58Hypothesis HF6 : ∀x0 ∈ int, F6 x0 ∈ int
L59Variable G6 : set → set
L60Hypothesis HG6 : ∀x0 ∈ int, G6 x0 ∈ int
L62Hypothesis HH6 : H6 ∈ int
L63Variable U6 : set → set → set
L64Hypothesis HU6 : ∀x0 ∈ int, ∀x1 ∈ int, U6 x0 x1 ∈ int
L65Variable V6 : set → set
L66Hypothesis HV6 : ∀x0 ∈ int, V6 x0 ∈ int
L67Variable F4 : set → set
L68Hypothesis HF4 : ∀x0 ∈ int, F4 x0 ∈ int
L70Hypothesis HG4 : G4 ∈ int
L71Variable H4 : set → set → set
L72Hypothesis HH4 : ∀x0 ∈ int, ∀x1 ∈ int, H4 x0 x1 ∈ int
L73Variable U4 : set → set → set
L74Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, U4 x0 x1 ∈ int
L75Variable V4 : set → set → set
L76Hypothesis HV4 : ∀x0 ∈ int, ∀x1 ∈ int, V4 x0 x1 ∈ int
L77Variable F3 : set → set → set
L78Hypothesis HF3 : ∀x0 ∈ int, ∀x1 ∈ int, F3 x0 x1 ∈ int
L79Variable G3 : set → set
L80Hypothesis HG3 : ∀x0 ∈ int, G3 x0 ∈ int
L82Hypothesis HH3 : H3 ∈ int
L83Variable U3 : set → set → set
L84Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, U3 x0 x1 ∈ int
L85Variable V3 : set → set
L86Hypothesis HV3 : ∀x0 ∈ int, V3 x0 ∈ int
L87Variable FAST : set → set
L88Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L89Hypothesis H1 : (∀X ∈ int, (∀Y ∈ int, ((F1 X Y) = ((2 * (((X + X) + X) + Y)) + - 1))))
L90Hypothesis H2 : (∀X ∈ int, (∀Y ∈ int, ((G1 X Y) = (Y + Y))))
L91Hypothesis H3 : (∀X ∈ int, (∀Y ∈ int, ((H1 X Y) = Y)))
L92Hypothesis H4 : (I1 = 1)
L93Hypothesis H5 : (J1 = 2)
L94Hypothesis 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)))))))
L95Hypothesis H7 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V1 X Y Z) = (if (X <= 0) then Z else (G1 (U1 (X + - 1) Y Z) (V1 (X + - 1) Y Z)))))))
L96Hypothesis H8 : (∀X ∈ int, (∀Y ∈ int, ((W1 X Y) = (U1 (H1 X Y) I1 J1))))
L97Hypothesis H9 : (∀X ∈ int, ((F2 X) = ((X + X) + X)))
L98Hypothesis H10 : (G2 = 2)
L99Hypothesis H11 : (∀X ∈ int, ((H2 X) = X))
L100Hypothesis H12 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y))))))
L101Hypothesis H13 : (∀X ∈ int, ((V2 X) = (U2 G2 (H2 X))))
L102Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = ((W1 X Y) + (V2 X)))))
L103Hypothesis H15 : (∀X ∈ int, ((G0 X) = X))
L104Hypothesis H16 : (H0 = 1)
L105Hypothesis H17 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y) X)))))
L106Hypothesis H18 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))
L107Hypothesis H19 : (∀X ∈ int, ((SMALL X) = (V0 X)))
L108Hypothesis H20 : (∀X ∈ int, ((F5 X) = ((2 * ((X + X) + X)) + - 1)))
L109Hypothesis H21 : (∀X ∈ int, ((G5 X) = X))
L110Hypothesis H22 : (H5 = 2)
L111Hypothesis H23 : (∀X ∈ int, (∀Y ∈ int, ((U5 X Y) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y))))))
L112Hypothesis H24 : (∀X ∈ int, ((V5 X) = (U5 (G5 X) H5)))
L113Hypothesis H25 : (∀X ∈ int, ((F6 X) = (X + X)))
L114Hypothesis H26 : (∀X ∈ int, ((G6 X) = X))
L115Hypothesis H27 : (H6 = 1)
L116Hypothesis H28 : (∀X ∈ int, (∀Y ∈ int, ((U6 X Y) = (if (X <= 0) then Y else (F6 (U6 (X + - 1) Y))))))
L117Hypothesis H29 : (∀X ∈ int, ((V6 X) = (U6 (G6 X) H6)))
L118Hypothesis H30 : (∀X ∈ int, ((F4 X) = ((V5 X) + - (V6 X))))
L119Hypothesis H31 : (G4 = 1)
L120Hypothesis H32 : (∀X ∈ int, (∀Y ∈ int, ((H4 X Y) = Y)))
L121Hypothesis H33 : (∀X ∈ int, (∀Y ∈ int, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y))))))
L122Hypothesis H34 : (∀X ∈ int, (∀Y ∈ int, ((V4 X Y) = (U4 G4 (H4 X Y)))))
L123Hypothesis H35 : (∀X ∈ int, (∀Y ∈ int, ((F3 X Y) = (((V4 X Y) + (2 * (2 * (X + X)))) + X))))
L124Hypothesis H36 : (∀X ∈ int, ((G3 X) = X))
L125Hypothesis H37 : (H3 = 1)
L126Hypothesis H38 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y) X)))))
L127Hypothesis H39 : (∀X ∈ int, ((V3 X) = (U3 (G3 X) H3)))
L128Hypothesis H40 : (∀X ∈ int, ((FAST X) = (V3 X)))
L129Theorem. (
A21194)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.