Beginning of Section A19040
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
L12Hypothesis HG2 : G2 ∈ int
L13Variable H2 : set → set
L14Hypothesis HH2 : ∀x0 ∈ int, H2 x0 ∈ int
L15Variable U2 : set → set → set
L16Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, U2 x0 x1 ∈ int
L17Variable V2 : set → set
L18Hypothesis HV2 : ∀x0 ∈ int, V2 x0 ∈ int
L19Variable F1 : set → set → set
L20Hypothesis HF1 : ∀x0 ∈ int, ∀x1 ∈ int, F1 x0 x1 ∈ int
L21Variable G1 : set → set → set
L22Hypothesis HG1 : ∀x0 ∈ int, ∀x1 ∈ int, G1 x0 x1 ∈ int
L23Variable H1 : set → set → set
L24Hypothesis HH1 : ∀x0 ∈ int, ∀x1 ∈ int, H1 x0 x1 ∈ int
L26Hypothesis HI1 : I1 ∈ int
L28Hypothesis HJ1 : J1 ∈ int
L29Variable U1 : set → set → set → set
L30Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U1 x0 x1 x2 ∈ int
L31Variable V1 : set → set → set → set
L32Hypothesis HV1 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V1 x0 x1 x2 ∈ int
L33Variable W1 : set → set → set
L34Hypothesis HW1 : ∀x0 ∈ int, ∀x1 ∈ int, W1 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 F4 : set → set → set
L48Hypothesis HF4 : ∀x0 ∈ int, ∀x1 ∈ int, F4 x0 x1 ∈ int
L49Variable G4 : set → set → set
L50Hypothesis HG4 : ∀x0 ∈ int, ∀x1 ∈ int, G4 x0 x1 ∈ int
L51Variable H4 : set → set → set
L52Hypothesis HH4 : ∀x0 ∈ int, ∀x1 ∈ int, H4 x0 x1 ∈ int
L54Hypothesis HI4 : I4 ∈ int
L56Hypothesis HJ4 : J4 ∈ int
L57Variable U4 : set → set → set → set
L58Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U4 x0 x1 x2 ∈ int
L59Variable V4 : set → set → set → set
L60Hypothesis HV4 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V4 x0 x1 x2 ∈ int
L61Variable W4 : set → set → set
L62Hypothesis HW4 : ∀x0 ∈ int, ∀x1 ∈ int, W4 x0 x1 ∈ int
L63Variable F3 : set → set → set
L64Hypothesis HF3 : ∀x0 ∈ int, ∀x1 ∈ int, F3 x0 x1 ∈ int
L65Variable G3 : set → set
L66Hypothesis HG3 : ∀x0 ∈ int, G3 x0 ∈ int
L68Hypothesis HH3 : H3 ∈ int
L69Variable U3 : set → set → set
L70Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, U3 x0 x1 ∈ int
L71Variable V3 : set → set
L72Hypothesis HV3 : ∀x0 ∈ int, V3 x0 ∈ int
L73Variable FAST : set → set
L74Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L75Hypothesis H1 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = ((2 + Y) * X))))
L76Hypothesis H2 : (G2 = 2)
L77Hypothesis H3 : (∀X ∈ int, ((H2 X) = X))
L78Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y) X)))))
L79Hypothesis H5 : (∀X ∈ int, ((V2 X) = (U2 G2 (H2 X))))
L80Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, ((F1 X Y) = (((V2 X) + - X) + (Y * Y)))))
L81Hypothesis H7 : (∀X ∈ int, (∀Y ∈ int, ((G1 X Y) = (Y + Y))))
L82Hypothesis H8 : (∀X ∈ int, (∀Y ∈ int, ((H1 X Y) = Y)))
L83Hypothesis H9 : (I1 = 1)
L84Hypothesis H10 : (J1 = 2)
L85Hypothesis H11 : (∀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)))))))
L86Hypothesis H12 : (∀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)))))))
L87Hypothesis H13 : (∀X ∈ int, (∀Y ∈ int, ((W1 X Y) = (U1 (H1 X Y) I1 J1))))
L88Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = (((W1 X Y) + (2 * (X + X))) + X))))
L89Hypothesis H15 : (∀X ∈ int, ((G0 X) = X))
L90Hypothesis H16 : (H0 = 1)
L91Hypothesis H17 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y) X)))))
L92Hypothesis H18 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))
L93Hypothesis H19 : (∀X ∈ int, ((SMALL X) = (V0 X)))
L94Hypothesis H20 : (∀X ∈ int, (∀Y ∈ int, ((F4 X Y) = (((2 * (X + X)) + (Y * Y)) + X))))
L95Hypothesis H21 : (∀X ∈ int, (∀Y ∈ int, ((G4 X Y) = (Y + Y))))
L96Hypothesis H22 : (∀X ∈ int, (∀Y ∈ int, ((H4 X Y) = Y)))
L97Hypothesis H23 : (I4 = 1)
L98Hypothesis H24 : (J4 = 2)
L99Hypothesis H25 : (∀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)))))))
L100Hypothesis H26 : (∀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)))))))
L101Hypothesis H27 : (∀X ∈ int, (∀Y ∈ int, ((W4 X Y) = (U4 (H4 X Y) I4 J4))))
L102Hypothesis H28 : (∀X ∈ int, (∀Y ∈ int, ((F3 X Y) = (((W4 X Y) + (2 * ((2 * (X + X)) + X))) + X))))
L103Hypothesis H29 : (∀X ∈ int, ((G3 X) = X))
L104Hypothesis H30 : (H3 = 1)
L105Hypothesis H31 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y) X)))))
L106Hypothesis H32 : (∀X ∈ int, ((V3 X) = (U3 (G3 X) H3)))
L107Hypothesis H33 : (∀X ∈ int, ((FAST X) = (V3 X)))
L108Theorem. (
A19040)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.