Beginning of Section A227137
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
L14Hypothesis HF2 : ∀x0 ∈ int, F2 x0 ∈ int
L16Hypothesis HG2 : G2 ∈ int
L18Hypothesis HH2 : H2 ∈ int
L19Variable U2 : set → set → set
L20Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, U2 x0 x1 ∈ int
L22Hypothesis HV2 : V2 ∈ int
L24Hypothesis HH1 : H1 ∈ int
L25Variable U1 : set → set → set
L26Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, U1 x0 x1 ∈ int
L28Hypothesis HV1 : V1 ∈ int
L29Variable F0 : set → set → set
L30Hypothesis HF0 : ∀x0 ∈ int, ∀x1 ∈ int, F0 x0 x1 ∈ int
L31Variable G0 : set → set
L32Hypothesis HG0 : ∀x0 ∈ int, G0 x0 ∈ int
L33Variable H0 : set → set
L34Hypothesis HH0 : ∀x0 ∈ int, H0 x0 ∈ int
L36Hypothesis HI0 : I0 ∈ int
L38Hypothesis HJ0 : J0 ∈ int
L39Variable U0 : set → set → set → set
L40Hypothesis HU0 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U0 x0 x1 x2 ∈ int
L41Variable V0 : set → set → set → set
L42Hypothesis HV0 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V0 x0 x1 x2 ∈ int
L43Variable W0 : set → set
L44Hypothesis HW0 : ∀x0 ∈ int, W0 x0 ∈ int
L45Variable SMALL : set → set
L46Hypothesis HSMALL : ∀x0 ∈ int, SMALL x0 ∈ int
L47Variable F4 : set → set
L48Hypothesis HF4 : ∀x0 ∈ int, F4 x0 ∈ int
L50Hypothesis HG4 : G4 ∈ int
L51Variable F5 : set → set
L52Hypothesis HF5 : ∀x0 ∈ int, F5 x0 ∈ int
L54Hypothesis HG5 : G5 ∈ int
L56Hypothesis HH5 : H5 ∈ int
L57Variable U5 : set → set → set
L58Hypothesis HU5 : ∀x0 ∈ int, ∀x1 ∈ int, U5 x0 x1 ∈ int
L60Hypothesis HV5 : V5 ∈ int
L62Hypothesis HH4 : H4 ∈ int
L63Variable U4 : set → set → set
L64Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, U4 x0 x1 ∈ int
L66Hypothesis HV4 : V4 ∈ int
L67Variable F3 : set → set → set
L68Hypothesis HF3 : ∀x0 ∈ int, ∀x1 ∈ int, F3 x0 x1 ∈ int
L69Variable G3 : set → set
L70Hypothesis HG3 : ∀x0 ∈ int, G3 x0 ∈ int
L71Variable H3 : set → set
L72Hypothesis HH3 : ∀x0 ∈ int, H3 x0 ∈ int
L74Hypothesis HI3 : I3 ∈ int
L76Hypothesis HJ3 : J3 ∈ int
L77Variable U3 : set → set → set → set
L78Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U3 x0 x1 x2 ∈ int
L79Variable V3 : set → set → set → set
L80Hypothesis HV3 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V3 x0 x1 x2 ∈ int
L81Variable W3 : set → set
L82Hypothesis HW3 : ∀x0 ∈ int, W3 x0 ∈ int
L83Variable FAST : set → set
L84Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L85Hypothesis H1 : (∀X ∈ int, ((F1 X) = ((X * X) * X)))
L86Hypothesis H2 : (G1 = 1)
L87Hypothesis H3 : (∀X ∈ int, ((F2 X) = (1 + ((2 + X) * X))))
L88Hypothesis H4 : (G2 = 2)
L89Hypothesis H5 : (H2 = 2)
L90Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y))))))
L91Hypothesis H7 : (V2 = (U2 G2 H2))
L92Hypothesis H8 : (H1 = V2)
L93Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))
L94Hypothesis H10 : (V1 = (U1 G1 H1))
L95Hypothesis H11 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = (((2 + V1) * X) + - Y))))
L96Hypothesis H12 : (∀X ∈ int, ((G0 X) = X))
L97Hypothesis H13 : (∀X ∈ int, ((H0 X) = X))
L98Hypothesis H14 : (I0 = 1)
L99Hypothesis H15 : (J0 = (0 + - 1))
L100Hypothesis H16 : (∀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)))))))
L101Hypothesis H17 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V0 X Y Z) = (if (X <= 0) then Z else (G0 (U0 (X + - 1) Y Z)))))))
L102Hypothesis H18 : (∀X ∈ int, ((W0 X) = (U0 (H0 X) I0 J0)))
L103Hypothesis H19 : (∀X ∈ int, ((SMALL X) = (W0 X)))
L104Hypothesis H20 : (∀X ∈ int, ((F4 X) = ((X * X) * X)))
L105Hypothesis H21 : (G4 = 1)
L106Hypothesis H22 : (∀X ∈ int, ((F5 X) = (X * X)))
L107Hypothesis H23 : (G5 = 1)
L108Hypothesis H24 : (H5 = (2 + (2 * (2 + 2))))
L109Hypothesis H25 : (∀X ∈ int, (∀Y ∈ int, ((U5 X Y) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y))))))
L110Hypothesis H26 : (V5 = (U5 G5 H5))
L111Hypothesis H27 : (H4 = V5)
L112Hypothesis H28 : (∀X ∈ int, (∀Y ∈ int, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y))))))
L113Hypothesis H29 : (V4 = (U4 G4 H4))
L114Hypothesis H30 : (∀X ∈ int, (∀Y ∈ int, ((F3 X Y) = (((2 + V4) * X) + - Y))))
L115Hypothesis H31 : (∀X ∈ int, ((G3 X) = X))
L116Hypothesis H32 : (∀X ∈ int, ((H3 X) = X))
L117Hypothesis H33 : (I3 = 1)
L118Hypothesis H34 : (J3 = (0 + - 1))
L119Hypothesis H35 : (∀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)))))))
L120Hypothesis H36 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V3 X Y Z) = (if (X <= 0) then Z else (G3 (U3 (X + - 1) Y Z)))))))
L121Hypothesis H37 : (∀X ∈ int, ((W3 X) = (U3 (H3 X) I3 J3)))
L122Hypothesis H38 : (∀X ∈ int, ((FAST X) = (W3 X)))
L123Theorem. (
A227137)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.