Beginning of Section A241031
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
L11Variable G1 : set → set → set
L12Hypothesis HG1 : ∀x0 ∈ int, ∀x1 ∈ int, G1 x0 x1 ∈ int
L13Variable H1 : set → set
L14Hypothesis HH1 : ∀x0 ∈ int, H1 x0 ∈ int
L15Variable U1 : set → set → set
L16Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, U1 x0 x1 ∈ int
L17Variable V1 : set → set → set
L18Hypothesis HV1 : ∀x0 ∈ int, ∀x1 ∈ int, V1 x0 x1 ∈ int
L19Variable F0 : set → set → set
L20Hypothesis HF0 : ∀x0 ∈ int, ∀x1 ∈ int, F0 x0 x1 ∈ int
L21Variable G0 : set → set → set
L22Hypothesis HG0 : ∀x0 ∈ int, ∀x1 ∈ int, G0 x0 x1 ∈ int
L24Hypothesis HH0 : H0 ∈ int
L26Hypothesis HI0 : I0 ∈ int
L27Variable J0 : set → set
L28Hypothesis HJ0 : ∀x0 ∈ int, J0 x0 ∈ 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 F2 : set → set
L36Hypothesis HF2 : ∀x0 ∈ int, F2 x0 ∈ int
L37Variable G2 : set → set
L38Hypothesis HG2 : ∀x0 ∈ int, G2 x0 ∈ int
L40Hypothesis HH2 : H2 ∈ int
L41Variable U2 : set → set → set
L42Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, U2 x0 x1 ∈ int
L43Variable V2 : set → set
L44Hypothesis HV2 : ∀x0 ∈ int, V2 x0 ∈ int
L45Variable SMALL : set → set
L46Hypothesis HSMALL : ∀x0 ∈ int, SMALL x0 ∈ int
L47Variable F3 : set → set
L48Hypothesis HF3 : ∀x0 ∈ int, F3 x0 ∈ int
L50Hypothesis HG3 : G3 ∈ int
L51Variable F4 : set → set
L52Hypothesis HF4 : ∀x0 ∈ int, F4 x0 ∈ int
L53Variable G4 : set → set
L54Hypothesis HG4 : ∀x0 ∈ int, G4 x0 ∈ int
L56Hypothesis HH4 : H4 ∈ int
L57Variable U4 : set → set → set
L58Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, U4 x0 x1 ∈ int
L59Variable V4 : set → set
L60Hypothesis HV4 : ∀x0 ∈ int, V4 x0 ∈ int
L61Variable H3 : set → set
L62Hypothesis HH3 : ∀x0 ∈ int, H3 x0 ∈ int
L63Variable U3 : set → set → set
L64Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, U3 x0 x1 ∈ int
L65Variable V3 : set → set
L66Hypothesis HV3 : ∀x0 ∈ int, V3 x0 ∈ 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 FAST : set → set
L84Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L85Hypothesis H1 : (∀X ∈ int, ((F1 X) = (X + X)))
L86Hypothesis H2 : (∀X ∈ int, (∀Y ∈ int, ((G1 X Y) = Y)))
L87Hypothesis H3 : (∀X ∈ int, ((H1 X) = X))
L88Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))
L89Hypothesis H5 : (∀X ∈ int, (∀Y ∈ int, ((V1 X Y) = (U1 (G1 X Y) (H1 X)))))
L90Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = (1 + (V1 X Y)))))
L91Hypothesis H7 : (∀X ∈ int, (∀Y ∈ int, ((G0 X Y) = Y)))
L92Hypothesis H8 : (H0 = 2)
L93Hypothesis H9 : (I0 = 1)
L94Hypothesis H10 : (∀X ∈ int, ((J0 X) = X))
L95Hypothesis 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)))))))
L96Hypothesis H12 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V0 X Y Z) = (if (X <= 0) then Z else (G0 (U0 (X + - 1) Y Z) (V0 (X + - 1) Y Z)))))))
L97Hypothesis H13 : (∀X ∈ int, ((W0 X) = (U0 H0 I0 (J0 X))))
L98Hypothesis H14 : (∀X ∈ int, ((F2 X) = ((2 * ((X + X) + X)) + X)))
L99Hypothesis H15 : (∀X ∈ int, ((G2 X) = X))
L100Hypothesis H16 : (H2 = 1)
L101Hypothesis H17 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y))))))
L102Hypothesis H18 : (∀X ∈ int, ((V2 X) = (U2 (G2 X) H2)))
L103Hypothesis H19 : (∀X ∈ int, ((SMALL X) = ((W0 X) * (1 + (V2 X)))))
L104Hypothesis H20 : (∀X ∈ int, ((F3 X) = ((X * X) + X)))
L105Hypothesis H21 : (G3 = 1)
L106Hypothesis H22 : (∀X ∈ int, ((F4 X) = (X + X)))
L107Hypothesis H23 : (∀X ∈ int, ((G4 X) = X))
L108Hypothesis H24 : (H4 = 1)
L109Hypothesis H25 : (∀X ∈ int, (∀Y ∈ int, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y))))))
L110Hypothesis H26 : (∀X ∈ int, ((V4 X) = (U4 (G4 X) H4)))
L111Hypothesis H27 : (∀X ∈ int, ((H3 X) = (V4 X)))
L112Hypothesis H28 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y))))))
L113Hypothesis H29 : (∀X ∈ int, ((V3 X) = (U3 G3 (H3 X))))
L114Hypothesis H30 : (∀X ∈ int, (∀Y ∈ int, ((F5 X Y) = (X * Y))))
L115Hypothesis H31 : (∀X ∈ int, (∀Y ∈ int, ((G5 X Y) = Y)))
L116Hypothesis H32 : (∀X ∈ int, ((H5 X) = X))
L117Hypothesis H33 : (I5 = 1)
L118Hypothesis H34 : (J5 = (1 + (2 + (2 + 2))))
L119Hypothesis 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)))))))
L120Hypothesis 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)))))))
L121Hypothesis H37 : (∀X ∈ int, ((W5 X) = (U5 (H5 X) I5 J5)))
L122Hypothesis H38 : (∀X ∈ int, ((FAST X) = ((1 + (V3 X)) * (1 + (W5 X)))))
L123Theorem. (
A241031)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.