Beginning of Section A5970
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
L10Hypothesis HF1 : ∀x0 ∈ int, F1 x0 ∈ int
L12Hypothesis HG1 : G1 ∈ int
L13Variable F2 : set → set → set
L14Hypothesis HF2 : ∀x0 ∈ int, ∀x1 ∈ int, F2 x0 x1 ∈ int
L15Variable G2 : set → set
L16Hypothesis HG2 : ∀x0 ∈ int, G2 x0 ∈ int
L17Variable H2 : set → set → set
L18Hypothesis HH2 : ∀x0 ∈ int, ∀x1 ∈ int, H2 x0 x1 ∈ int
L20Hypothesis HI2 : I2 ∈ int
L22Hypothesis HJ2 : J2 ∈ int
L23Variable U2 : set → set → set → set
L24Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U2 x0 x1 x2 ∈ int
L25Variable V2 : set → set → set → set
L26Hypothesis HV2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V2 x0 x1 x2 ∈ int
L27Variable W2 : set → set → set
L28Hypothesis HW2 : ∀x0 ∈ int, ∀x1 ∈ int, W2 x0 x1 ∈ int
L29Variable H1 : set → set → set
L30Hypothesis HH1 : ∀x0 ∈ int, ∀x1 ∈ int, H1 x0 x1 ∈ int
L31Variable U1 : set → set → set
L32Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, U1 x0 x1 ∈ int
L33Variable V1 : set → set → set
L34Hypothesis HV1 : ∀x0 ∈ int, ∀x1 ∈ int, V1 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 F3 : set → set → set
L48Hypothesis HF3 : ∀x0 ∈ int, ∀x1 ∈ int, F3 x0 x1 ∈ int
L49Variable G3 : set → set
L50Hypothesis HG3 : ∀x0 ∈ int, G3 x0 ∈ int
L51Variable H3 : set → set
L52Hypothesis HH3 : ∀x0 ∈ int, H3 x0 ∈ int
L54Hypothesis HI3 : I3 ∈ int
L56Hypothesis HJ3 : J3 ∈ int
L57Variable U3 : set → set → set → set
L58Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U3 x0 x1 x2 ∈ int
L59Variable V3 : set → set → set → set
L60Hypothesis HV3 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V3 x0 x1 x2 ∈ int
L61Variable W3 : set → set
L62Hypothesis HW3 : ∀x0 ∈ int, W3 x0 ∈ int
L63Variable F4 : set → set → set
L64Hypothesis HF4 : ∀x0 ∈ int, ∀x1 ∈ int, F4 x0 x1 ∈ int
L65Variable G4 : set → set
L66Hypothesis HG4 : ∀x0 ∈ int, G4 x0 ∈ int
L67Variable H4 : set → set
L68Hypothesis HH4 : ∀x0 ∈ int, H4 x0 ∈ int
L70Hypothesis HI4 : I4 ∈ int
L72Hypothesis HJ4 : J4 ∈ int
L73Variable U4 : set → set → set → set
L74Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U4 x0 x1 x2 ∈ int
L75Variable V4 : set → set → set → set
L76Hypothesis HV4 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V4 x0 x1 x2 ∈ int
L77Variable W4 : set → set
L78Hypothesis HW4 : ∀x0 ∈ int, W4 x0 ∈ int
L79Variable FAST : set → set
L80Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L81Hypothesis H1 : (∀X ∈ int, ((F1 X) = (X * X)))
L82Hypothesis H2 : (G1 = 1)
L83Hypothesis H3 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = (X + Y))))
L84Hypothesis H4 : (∀X ∈ int, ((G2 X) = X))
L85Hypothesis H5 : (∀X ∈ int, (∀Y ∈ int, ((H2 X Y) = Y)))
L86Hypothesis H6 : (I2 = 1)
L87Hypothesis H7 : (J2 = 2)
L88Hypothesis H8 : (∀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)))))))
L89Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V2 X Y Z) = (if (X <= 0) then Z else (G2 (U2 (X + - 1) Y Z)))))))
L90Hypothesis H10 : (∀X ∈ int, (∀Y ∈ int, ((W2 X Y) = (U2 (H2 X Y) I2 J2))))
L91Hypothesis H11 : (∀X ∈ int, (∀Y ∈ int, ((H1 X Y) = (W2 X Y))))
L92Hypothesis H12 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))
L93Hypothesis H13 : (∀X ∈ int, (∀Y ∈ int, ((V1 X Y) = (U1 G1 (H1 X Y)))))
L94Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = ((V1 X Y) + X))))
L95Hypothesis H15 : (∀X ∈ int, ((G0 X) = X))
L96Hypothesis H16 : (H0 = 1)
L97Hypothesis H17 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y) X)))))
L98Hypothesis H18 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))
L99Hypothesis H19 : (∀X ∈ int, ((SMALL X) = (V0 X)))
L100Hypothesis H20 : (∀X ∈ int, (∀Y ∈ int, ((F3 X Y) = (X + Y))))
L101Hypothesis H21 : (∀X ∈ int, ((G3 X) = X))
L102Hypothesis H22 : (∀X ∈ int, ((H3 X) = X))
L103Hypothesis H23 : (I3 = 1)
L104Hypothesis H24 : (J3 = 2)
L105Hypothesis H25 : (∀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)))))))
L106Hypothesis H26 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V3 X Y Z) = (if (X <= 0) then Z else (G3 (U3 (X + - 1) Y Z)))))))
L107Hypothesis H27 : (∀X ∈ int, ((W3 X) = (U3 (H3 X) I3 J3)))
L108Hypothesis H28 : (∀X ∈ int, (∀Y ∈ int, ((F4 X Y) = (X + Y))))
L109Hypothesis H29 : (∀X ∈ int, ((G4 X) = X))
L110Hypothesis H30 : (∀X ∈ int, ((H4 X) = X))
L111Hypothesis H31 : (I4 = (1 + 2))
L112Hypothesis H32 : (J4 = 1)
L113Hypothesis H33 : (∀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)))))))
L114Hypothesis H34 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V4 X Y Z) = (if (X <= 0) then Z else (G4 (U4 (X + - 1) Y Z)))))))
L115Hypothesis H35 : (∀X ∈ int, ((W4 X) = (U4 (H4 X) I4 J4)))
L116Hypothesis H36 : (∀X ∈ int, ((FAST X) = (((W3 X) * (W4 X)) + - 2)))
L117Theorem. (
A5970)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.