Beginning of Section A13802
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
L14Hypothesis HH1 : H1 ∈ int
L15Variable U1 : set → set → set
L16Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, U1 x0 x1 ∈ int
L18Hypothesis HV1 : V1 ∈ int
L19Variable F0 : set → set
L20Hypothesis HF0 : ∀x0 ∈ int, F0 x0 ∈ int
L21Variable G0 : set → set
L22Hypothesis HG0 : ∀x0 ∈ int, G0 x0 ∈ int
L24Hypothesis HH0 : H0 ∈ int
L25Variable U0 : set → set → set
L26Hypothesis HU0 : ∀x0 ∈ int, ∀x1 ∈ int, U0 x0 x1 ∈ int
L27Variable V0 : set → set
L28Hypothesis HV0 : ∀x0 ∈ int, V0 x0 ∈ int
L29Variable SMALL : set → set
L30Hypothesis HSMALL : ∀x0 ∈ int, SMALL x0 ∈ int
L31Variable F2 : set → set → set
L32Hypothesis HF2 : ∀x0 ∈ int, ∀x1 ∈ int, F2 x0 x1 ∈ int
L33Variable G2 : set → set → set
L34Hypothesis HG2 : ∀x0 ∈ int, ∀x1 ∈ int, G2 x0 x1 ∈ int
L35Variable H2 : set → set
L36Hypothesis HH2 : ∀x0 ∈ int, H2 x0 ∈ int
L38Hypothesis HI2 : I2 ∈ int
L40Hypothesis HJ2 : J2 ∈ int
L41Variable U2 : set → set → set → set
L42Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U2 x0 x1 x2 ∈ int
L43Variable V2 : set → set → set → set
L44Hypothesis HV2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V2 x0 x1 x2 ∈ int
L45Variable W2 : set → set
L46Hypothesis HW2 : ∀x0 ∈ int, W2 x0 ∈ int
L47Variable F3 : set → set → set
L48Hypothesis HF3 : ∀x0 ∈ int, ∀x1 ∈ int, F3 x0 x1 ∈ int
L49Variable G3 : set → set → set
L50Hypothesis HG3 : ∀x0 ∈ int, ∀x1 ∈ int, G3 x0 x1 ∈ 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
L64Hypothesis HF4 : ∀x0 ∈ int, F4 x0 ∈ int
L66Hypothesis HG4 : G4 ∈ int
L67Variable F5 : set → set → set
L68Hypothesis HF5 : ∀x0 ∈ int, ∀x1 ∈ int, F5 x0 x1 ∈ int
L69Variable G5 : set → set → set
L70Hypothesis HG5 : ∀x0 ∈ int, ∀x1 ∈ int, G5 x0 x1 ∈ int
L71Variable H5 : set → set
L72Hypothesis HH5 : ∀x0 ∈ int, H5 x0 ∈ int
L74Hypothesis HI5 : I5 ∈ int
L76Hypothesis HJ5 : J5 ∈ int
L77Variable U5 : set → set → set → set
L78Hypothesis HU5 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U5 x0 x1 x2 ∈ int
L79Variable V5 : set → set → set → set
L80Hypothesis HV5 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V5 x0 x1 x2 ∈ int
L81Variable W5 : set → set
L82Hypothesis HW5 : ∀x0 ∈ int, W5 x0 ∈ int
L83Variable H4 : set → set
L84Hypothesis HH4 : ∀x0 ∈ int, H4 x0 ∈ int
L85Variable U4 : set → set → set
L86Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, U4 x0 x1 ∈ int
L87Variable V4 : set → set
L88Hypothesis HV4 : ∀x0 ∈ int, V4 x0 ∈ int
L89Variable FAST : set → set
L90Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L91Hypothesis H1 : (∀X ∈ int, ((F1 X) = (X * X)))
L92Hypothesis H2 : (G1 = 2)
L93Hypothesis H3 : (H1 = 2)
L94Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))
L95Hypothesis H5 : (V1 = (U1 G1 H1))
L96Hypothesis H6 : (∀X ∈ int, ((F0 X) = ((V1 * X) + - X)))
L97Hypothesis H7 : (∀X ∈ int, ((G0 X) = (1 + (2 * (X + X)))))
L98Hypothesis H8 : (H0 = 1)
L99Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y))))))
L100Hypothesis H10 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))
L101Hypothesis H11 : (∀X ∈ int, ((SMALL X) = (V0 X)))
L102Hypothesis H12 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = (X * Y))))
L103Hypothesis H13 : (∀X ∈ int, (∀Y ∈ int, ((G2 X Y) = Y)))
L104Hypothesis H14 : (∀X ∈ int, ((H2 X) = (1 + X)))
L105Hypothesis H15 : (I2 = 1)
L106Hypothesis H16 : (J2 = ((2 * (2 * (2 + 2))) + - 1))
L107Hypothesis H17 : (∀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 H18 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V2 X Y Z) = (if (X <= 0) then Z else (G2 (U2 (X + - 1) Y Z) (V2 (X + - 1) Y Z)))))))
L109Hypothesis H19 : (∀X ∈ int, ((W2 X) = (U2 (H2 X) I2 J2)))
L110Hypothesis H20 : (∀X ∈ int, (∀Y ∈ int, ((F3 X Y) = (X * Y))))
L111Hypothesis H21 : (∀X ∈ int, (∀Y ∈ int, ((G3 X Y) = Y)))
L112Hypothesis H22 : (∀X ∈ int, ((H3 X) = X))
L113Hypothesis H23 : (I3 = 1)
L114Hypothesis H24 : (J3 = ((2 * (2 * (2 + 2))) + - 1))
L115Hypothesis 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)))))))
L116Hypothesis H26 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V3 X Y Z) = (if (X <= 0) then Z else (G3 (U3 (X + - 1) Y Z) (V3 (X + - 1) Y Z)))))))
L117Hypothesis H27 : (∀X ∈ int, ((W3 X) = (U3 (H3 X) I3 J3)))
L118Hypothesis H28 : (∀X ∈ int, ((F4 X) = (X * X)))
L119Hypothesis H29 : (G4 = 1)
L120Hypothesis H30 : (∀X ∈ int, (∀Y ∈ int, ((F5 X Y) = (X * Y))))
L121Hypothesis H31 : (∀X ∈ int, (∀Y ∈ int, ((G5 X Y) = Y)))
L122Hypothesis H32 : (∀X ∈ int, ((H5 X) = X))
L123Hypothesis H33 : (I5 = 1)
L124Hypothesis H34 : (J5 = ((2 * (2 * (2 + 2))) + - 1))
L125Hypothesis H35 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U5 X Y Z) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y Z) (V5 (X + - 1) Y Z)))))))
L126Hypothesis H36 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V5 X Y Z) = (if (X <= 0) then Z else (G5 (U5 (X + - 1) Y Z) (V5 (X + - 1) Y Z)))))))
L127Hypothesis H37 : (∀X ∈ int, ((W5 X) = (U5 (H5 X) I5 J5)))
L128Hypothesis H38 : (∀X ∈ int, ((H4 X) = (W5 X)))
L129Hypothesis H39 : (∀X ∈ int, (∀Y ∈ int, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y))))))
L130Hypothesis H40 : (∀X ∈ int, ((V4 X) = (U4 G4 (H4 X))))
L131Hypothesis H41 : (∀X ∈ int, ((FAST X) = (((W2 X) * (W3 X)) * (V4 X))))
L132Theorem. (
A13802)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.