Beginning of Section A227138
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 → set
L20Hypothesis HF0 : ∀x0 ∈ int, ∀x1 ∈ int, F0 x0 x1 ∈ int
L21Variable G0 : set → set
L22Hypothesis HG0 : ∀x0 ∈ int, G0 x0 ∈ int
L23Variable H0 : set → set
L24Hypothesis HH0 : ∀x0 ∈ int, H0 x0 ∈ int
L26Hypothesis HI0 : I0 ∈ int
L28Hypothesis HJ0 : J0 ∈ int
L29Variable U0 : set → set → set → set
L30Hypothesis HU0 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U0 x0 x1 x2 ∈ int
L31Variable V0 : set → set → set → set
L32Hypothesis HV0 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V0 x0 x1 x2 ∈ int
L33Variable W0 : set → set
L34Hypothesis HW0 : ∀x0 ∈ int, W0 x0 ∈ int
L35Variable SMALL : set → set
L36Hypothesis HSMALL : ∀x0 ∈ int, SMALL x0 ∈ int
L37Variable F3 : set → set
L38Hypothesis HF3 : ∀x0 ∈ int, F3 x0 ∈ int
L40Hypothesis HG3 : G3 ∈ int
L41Variable F4 : set → set
L42Hypothesis HF4 : ∀x0 ∈ int, F4 x0 ∈ int
L44Hypothesis HG4 : G4 ∈ int
L46Hypothesis HH4 : H4 ∈ int
L47Variable U4 : set → set → set
L48Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, U4 x0 x1 ∈ int
L50Hypothesis HV4 : V4 ∈ int
L52Hypothesis HH3 : H3 ∈ int
L53Variable U3 : set → set → set
L54Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, U3 x0 x1 ∈ int
L56Hypothesis HV3 : V3 ∈ int
L57Variable F2 : set → set → set
L58Hypothesis HF2 : ∀x0 ∈ int, ∀x1 ∈ int, F2 x0 x1 ∈ int
L59Variable G2 : set → set
L60Hypothesis HG2 : ∀x0 ∈ int, G2 x0 ∈ int
L61Variable H2 : set → set
L62Hypothesis HH2 : ∀x0 ∈ int, H2 x0 ∈ int
L64Hypothesis HI2 : I2 ∈ int
L66Hypothesis HJ2 : J2 ∈ int
L67Variable U2 : set → set → set → set
L68Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U2 x0 x1 x2 ∈ int
L69Variable V2 : set → set → set → set
L70Hypothesis HV2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V2 x0 x1 x2 ∈ int
L71Variable W2 : set → set
L72Hypothesis HW2 : ∀x0 ∈ int, W2 x0 ∈ int
L73Variable FAST : set → set
L74Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L75Hypothesis H1 : (∀X ∈ int, ((F1 X) = (2 + ((X * X) * X))))
L76Hypothesis H2 : (G1 = 2)
L77Hypothesis H3 : (H1 = 2)
L78Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))
L79Hypothesis H5 : (V1 = (U1 G1 H1))
L80Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = (((V1 + - 2) * X) + Y))))
L81Hypothesis H7 : (∀X ∈ int, ((G0 X) = X))
L82Hypothesis H8 : (∀X ∈ int, ((H0 X) = (X + X)))
L83Hypothesis H9 : (I0 = 1)
L84Hypothesis H10 : (J0 = 0)
L85Hypothesis H11 : (∀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)))))))
L86Hypothesis H12 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V0 X Y Z) = (if (X <= 0) then Z else (G0 (U0 (X + - 1) Y Z)))))))
L87Hypothesis H13 : (∀X ∈ int, ((W0 X) = (U0 (H0 X) I0 J0)))
L88Hypothesis H14 : (∀X ∈ int, ((SMALL X) = (W0 X)))
L89Hypothesis H15 : (∀X ∈ int, ((F3 X) = ((X * X) * X)))
L90Hypothesis H16 : (G3 = 1)
L91Hypothesis H17 : (∀X ∈ int, ((F4 X) = (X * X)))
L92Hypothesis H18 : (G4 = 1)
L93Hypothesis H19 : (H4 = (2 + (2 * (2 + 2))))
L94Hypothesis H20 : (∀X ∈ int, (∀Y ∈ int, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y))))))
L95Hypothesis H21 : (V4 = (U4 G4 H4))
L96Hypothesis H22 : (H3 = V4)
L97Hypothesis H23 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y))))))
L98Hypothesis H24 : (V3 = (U3 G3 H3))
L99Hypothesis H25 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = (((2 + V3) * X) + - Y))))
L100Hypothesis H26 : (∀X ∈ int, ((G2 X) = X))
L101Hypothesis H27 : (∀X ∈ int, ((H2 X) = X))
L102Hypothesis H28 : (I2 = 1)
L103Hypothesis H29 : (J2 = 1)
L104Hypothesis H30 : (∀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)))))))
L105Hypothesis H31 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V2 X Y Z) = (if (X <= 0) then Z else (G2 (U2 (X + - 1) Y Z)))))))
L106Hypothesis H32 : (∀X ∈ int, ((W2 X) = (U2 (H2 X) I2 J2)))
L107Hypothesis H33 : (∀X ∈ int, ((FAST X) = (W2 X)))
L108Theorem. (
A227138)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.