Beginning of Section A28130
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 F0 : set → set → set
L26Hypothesis HF0 : ∀x0 ∈ int, ∀x1 ∈ int, F0 x0 x1 ∈ int
L27Variable G0 : set → set
L28Hypothesis HG0 : ∀x0 ∈ int, G0 x0 ∈ int
L30Hypothesis HH0 : H0 ∈ int
L31Variable U0 : set → set → set
L32Hypothesis HU0 : ∀x0 ∈ int, ∀x1 ∈ int, U0 x0 x1 ∈ int
L33Variable V0 : set → set
L34Hypothesis HV0 : ∀x0 ∈ int, V0 x0 ∈ int
L35Variable SMALL : set → set
L36Hypothesis HSMALL : ∀x0 ∈ int, SMALL x0 ∈ int
L37Variable F4 : set → set → set
L38Hypothesis HF4 : ∀x0 ∈ int, ∀x1 ∈ int, F4 x0 x1 ∈ int
L39Variable G4 : set → set → set
L40Hypothesis HG4 : ∀x0 ∈ int, ∀x1 ∈ int, G4 x0 x1 ∈ int
L41Variable H4 : set → set
L42Hypothesis HH4 : ∀x0 ∈ int, H4 x0 ∈ int
L44Hypothesis HI4 : I4 ∈ int
L46Hypothesis HJ4 : J4 ∈ int
L47Variable U4 : set → set → set → set
L48Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U4 x0 x1 x2 ∈ int
L49Variable V4 : set → set → set → set
L50Hypothesis HV4 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V4 x0 x1 x2 ∈ int
L51Variable W4 : set → set
L52Hypothesis HW4 : ∀x0 ∈ int, W4 x0 ∈ int
L53Variable F5 : set → set
L54Hypothesis HF5 : ∀x0 ∈ int, F5 x0 ∈ int
L55Variable G5 : set → set
L56Hypothesis HG5 : ∀x0 ∈ int, G5 x0 ∈ int
L58Hypothesis HH5 : H5 ∈ int
L59Variable U5 : set → set → set
L60Hypothesis HU5 : ∀x0 ∈ int, ∀x1 ∈ int, U5 x0 x1 ∈ int
L61Variable V5 : set → set
L62Hypothesis HV5 : ∀x0 ∈ int, V5 x0 ∈ int
L63Variable F3 : set → set
L64Hypothesis HF3 : ∀x0 ∈ int, F3 x0 ∈ int
L66Hypothesis HG3 : G3 ∈ int
L67Variable H3 : set → set → set
L68Hypothesis HH3 : ∀x0 ∈ int, ∀x1 ∈ int, H3 x0 x1 ∈ int
L69Variable U3 : set → set → set
L70Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, U3 x0 x1 ∈ int
L71Variable V3 : set → set → set
L72Hypothesis HV3 : ∀x0 ∈ int, ∀x1 ∈ int, V3 x0 x1 ∈ int
L73Variable F2 : set → set → set
L74Hypothesis HF2 : ∀x0 ∈ int, ∀x1 ∈ int, F2 x0 x1 ∈ int
L75Variable G2 : set → set
L76Hypothesis HG2 : ∀x0 ∈ int, G2 x0 ∈ int
L78Hypothesis HH2 : H2 ∈ int
L79Variable U2 : set → set → set
L80Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, U2 x0 x1 ∈ int
L81Variable V2 : set → set
L82Hypothesis HV2 : ∀x0 ∈ int, V2 x0 ∈ int
L83Variable FAST : set → set
L84Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L85Hypothesis H1 : (∀X ∈ int, (∀Y ∈ int, ((F1 X Y) = (((((Y + - 1) + Y) * (Y * Y)) + X) + (2 * ((X + X) + X))))))
L86Hypothesis H2 : (∀X ∈ int, (∀Y ∈ int, ((G1 X Y) = (Y + Y))))
L87Hypothesis H3 : (∀X ∈ int, (∀Y ∈ int, ((H1 X Y) = Y)))
L88Hypothesis H4 : (I1 = 1)
L89Hypothesis H5 : (J1 = 2)
L90Hypothesis 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)))))))
L91Hypothesis 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)))))))
L92Hypothesis H8 : (∀X ∈ int, (∀Y ∈ int, ((W1 X Y) = (U1 (H1 X Y) I1 J1))))
L93Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = ((W1 X Y) + (2 * ((X + X) + X))))))
L94Hypothesis H10 : (∀X ∈ int, ((G0 X) = X))
L95Hypothesis H11 : (H0 = 1)
L96Hypothesis H12 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y) X)))))
L97Hypothesis H13 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))
L98Hypothesis H14 : (∀X ∈ int, ((SMALL X) = (V0 X)))
L99Hypothesis H15 : (∀X ∈ int, (∀Y ∈ int, ((F4 X Y) = (((2 * ((Y * Y) + X)) + - Y) + X))))
L100Hypothesis H16 : (∀X ∈ int, (∀Y ∈ int, ((G4 X Y) = (Y + Y))))
L101Hypothesis H17 : (∀X ∈ int, ((H4 X) = X))
L102Hypothesis H18 : (I4 = 1)
L103Hypothesis H19 : (J4 = 2)
L104Hypothesis H20 : (∀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)))))))
L105Hypothesis H21 : (∀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)))))))
L106Hypothesis H22 : (∀X ∈ int, ((W4 X) = (U4 (H4 X) I4 J4)))
L107Hypothesis H23 : (∀X ∈ int, ((F5 X) = (X + X)))
L108Hypothesis H24 : (∀X ∈ int, ((G5 X) = X))
L109Hypothesis H25 : (H5 = 1)
L110Hypothesis H26 : (∀X ∈ int, (∀Y ∈ int, ((U5 X Y) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y))))))
L111Hypothesis H27 : (∀X ∈ int, ((V5 X) = (U5 (G5 X) H5)))
L112Hypothesis H28 : (∀X ∈ int, ((F3 X) = ((W4 X) * (V5 X))))
L113Hypothesis H29 : (G3 = 1)
L114Hypothesis H30 : (∀X ∈ int, (∀Y ∈ int, ((H3 X Y) = Y)))
L115Hypothesis H31 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y))))))
L116Hypothesis H32 : (∀X ∈ int, (∀Y ∈ int, ((V3 X Y) = (U3 G3 (H3 X Y)))))
L117Hypothesis H33 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = (((V3 X Y) + (2 * ((X + X) + X))) + X))))
L118Hypothesis H34 : (∀X ∈ int, ((G2 X) = X))
L119Hypothesis H35 : (H2 = 1)
L120Hypothesis H36 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y) X)))))
L121Hypothesis H37 : (∀X ∈ int, ((V2 X) = (U2 (G2 X) H2)))
L122Hypothesis H38 : (∀X ∈ int, ((FAST X) = (V2 X)))
L123Theorem. (
A28130)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.