Beginning of Section A1582
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 F0 : set → set → set
L10Hypothesis HF0 : ∀x0 ∈ int, ∀x1 ∈ int, F0 x0 x1 ∈ int
L11Variable G0 : set → set
L12Hypothesis HG0 : ∀x0 ∈ int, G0 x0 ∈ int
L13Variable H0 : set → set
L14Hypothesis HH0 : ∀x0 ∈ int, H0 x0 ∈ int
L15Variable F1 : set → set → set
L16Hypothesis HF1 : ∀x0 ∈ int, ∀x1 ∈ int, F1 x0 x1 ∈ int
L17Variable G1 : set → set
L18Hypothesis HG1 : ∀x0 ∈ int, G1 x0 ∈ int
L19Variable H1 : set → set
L20Hypothesis HH1 : ∀x0 ∈ int, H1 x0 ∈ int
L22Hypothesis HI1 : I1 ∈ int
L24Hypothesis HJ1 : J1 ∈ int
L25Variable U1 : set → set → set → set
L26Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U1 x0 x1 x2 ∈ int
L27Variable V1 : set → set → set → set
L28Hypothesis HV1 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V1 x0 x1 x2 ∈ int
L29Variable W1 : set → set
L30Hypothesis HW1 : ∀x0 ∈ int, W1 x0 ∈ int
L31Variable I0 : set → set
L32Hypothesis HI0 : ∀x0 ∈ int, I0 x0 ∈ int
L34Hypothesis HJ0 : J0 ∈ int
L35Variable U0 : set → set → set → set
L36Hypothesis HU0 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U0 x0 x1 x2 ∈ int
L37Variable V0 : set → set → set → set
L38Hypothesis HV0 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V0 x0 x1 x2 ∈ int
L39Variable W0 : set → set
L40Hypothesis HW0 : ∀x0 ∈ int, W0 x0 ∈ int
L41Variable SMALL : set → set
L42Hypothesis HSMALL : ∀x0 ∈ int, SMALL x0 ∈ int
L43Variable F2 : set → set → set
L44Hypothesis HF2 : ∀x0 ∈ int, ∀x1 ∈ int, F2 x0 x1 ∈ int
L45Variable G2 : set → set
L46Hypothesis HG2 : ∀x0 ∈ int, G2 x0 ∈ int
L47Variable H2 : set → set
L48Hypothesis HH2 : ∀x0 ∈ int, H2 x0 ∈ int
L49Variable F3 : set → set → set
L50Hypothesis HF3 : ∀x0 ∈ int, ∀x1 ∈ int, F3 x0 x1 ∈ int
L51Variable G3 : set → set
L52Hypothesis HG3 : ∀x0 ∈ int, G3 x0 ∈ int
L53Variable H3 : set → set
L54Hypothesis HH3 : ∀x0 ∈ int, H3 x0 ∈ int
L56Hypothesis HI3 : I3 ∈ int
L58Hypothesis HJ3 : J3 ∈ int
L59Variable U3 : set → set → set → set
L60Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U3 x0 x1 x2 ∈ int
L61Variable V3 : set → set → set → set
L62Hypothesis HV3 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V3 x0 x1 x2 ∈ int
L63Variable W3 : set → set
L64Hypothesis HW3 : ∀x0 ∈ int, W3 x0 ∈ int
L65Variable I2 : set → set
L66Hypothesis HI2 : ∀x0 ∈ int, I2 x0 ∈ int
L68Hypothesis HJ2 : J2 ∈ int
L69Variable U2 : set → set → set → set
L70Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U2 x0 x1 x2 ∈ int
L71Variable V2 : set → set → set → set
L72Hypothesis HV2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V2 x0 x1 x2 ∈ int
L73Variable W2 : set → set
L74Hypothesis HW2 : ∀x0 ∈ int, W2 x0 ∈ int
L75Variable FAST : set → set
L76Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L77Hypothesis H1 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = ((X + X) + Y))))
L78Hypothesis H2 : (∀X ∈ int, ((G0 X) = X))
L79Hypothesis H3 : (∀X ∈ int, ((H0 X) = X))
L80Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((F1 X Y) = (X + Y))))
L81Hypothesis H5 : (∀X ∈ int, ((G1 X) = X))
L82Hypothesis H6 : (∀X ∈ int, ((H1 X) = X))
L83Hypothesis H7 : (I1 = 1)
L84Hypothesis H8 : (J1 = 0)
L85Hypothesis H9 : (∀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 H10 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V1 X Y Z) = (if (X <= 0) then Z else (G1 (U1 (X + - 1) Y Z)))))))
L87Hypothesis H11 : (∀X ∈ int, ((W1 X) = (U1 (H1 X) I1 J1)))
L88Hypothesis H12 : (∀X ∈ int, ((I0 X) = (W1 X)))
L89Hypothesis H13 : (J0 = 0)
L90Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U0 X Y Z) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y Z) (V0 (X + - 1) Y Z)))))))
L91Hypothesis H15 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V0 X Y Z) = (if (X <= 0) then Z else (G0 (U0 (X + - 1) Y Z)))))))
L92Hypothesis H16 : (∀X ∈ int, ((W0 X) = (U0 (H0 X) (I0 X) J0)))
L93Hypothesis H17 : (∀X ∈ int, ((SMALL X) = (W0 X)))
L94Hypothesis H18 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = ((X + X) + Y))))
L95Hypothesis H19 : (∀X ∈ int, ((G2 X) = X))
L96Hypothesis H20 : (∀X ∈ int, ((H2 X) = X))
L97Hypothesis H21 : (∀X ∈ int, (∀Y ∈ int, ((F3 X Y) = (X + Y))))
L98Hypothesis H22 : (∀X ∈ int, ((G3 X) = X))
L99Hypothesis H23 : (∀X ∈ int, ((H3 X) = (X + - 1)))
L100Hypothesis H24 : (I3 = 1)
L101Hypothesis H25 : (J3 = 1)
L102Hypothesis H26 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U3 X Y Z) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y Z) (V3 (X + - 1) Y Z)))))))
L103Hypothesis H27 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V3 X Y Z) = (if (X <= 0) then Z else (G3 (U3 (X + - 1) Y Z)))))))
L104Hypothesis H28 : (∀X ∈ int, ((W3 X) = (U3 (H3 X) I3 J3)))
L105Hypothesis H29 : (∀X ∈ int, ((I2 X) = (W3 X)))
L106Hypothesis H30 : (J2 = 0)
L107Hypothesis H31 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U2 X Y Z) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y Z) (V2 (X + - 1) Y Z)))))))
L108Hypothesis H32 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V2 X Y Z) = (if (X <= 0) then Z else (G2 (U2 (X + - 1) Y Z)))))))
L109Hypothesis H33 : (∀X ∈ int, ((W2 X) = (U2 (H2 X) (I2 X) J2)))
L110Hypothesis H34 : (∀X ∈ int, ((FAST X) = (W2 X)))
L111Theorem. (
A1582)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.