Beginning of Section A334991
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 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
L18Hypothesis HV1 : ∀x0 ∈ int, V1 x0 ∈ 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 F2 : set → set
L30Hypothesis HF2 : ∀x0 ∈ int, F2 x0 ∈ int
L31Variable G2 : set → set
L32Hypothesis HG2 : ∀x0 ∈ int, G2 x0 ∈ int
L34Hypothesis HH2 : H2 ∈ int
L35Variable U2 : set → set → set
L36Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, U2 x0 x1 ∈ int
L37Variable V2 : set → set
L38Hypothesis HV2 : ∀x0 ∈ int, V2 x0 ∈ int
L39Variable SMALL : set → set
L40Hypothesis HSMALL : ∀x0 ∈ int, SMALL x0 ∈ int
L41Variable F3 : set → set
L42Hypothesis HF3 : ∀x0 ∈ int, F3 x0 ∈ int
L43Variable G3 : set → set
L44Hypothesis HG3 : ∀x0 ∈ int, G3 x0 ∈ int
L46Hypothesis HH3 : H3 ∈ int
L47Variable U3 : set → set → set
L48Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, U3 x0 x1 ∈ int
L49Variable V3 : set → set
L50Hypothesis HV3 : ∀x0 ∈ int, V3 x0 ∈ int
L51Variable F4 : set → set → set
L52Hypothesis HF4 : ∀x0 ∈ int, ∀x1 ∈ int, F4 x0 x1 ∈ int
L53Variable G4 : set → set → set
L54Hypothesis HG4 : ∀x0 ∈ int, ∀x1 ∈ int, G4 x0 x1 ∈ int
L55Variable H4 : set → set
L56Hypothesis HH4 : ∀x0 ∈ int, H4 x0 ∈ int
L58Hypothesis HI4 : I4 ∈ int
L60Hypothesis HJ4 : J4 ∈ int
L61Variable U4 : set → set → set → set
L62Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U4 x0 x1 x2 ∈ int
L63Variable V4 : set → set → set → set
L64Hypothesis HV4 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V4 x0 x1 x2 ∈ int
L65Variable W4 : set → set
L66Hypothesis HW4 : ∀x0 ∈ int, W4 x0 ∈ int
L67Variable F5 : set → set
L68Hypothesis HF5 : ∀x0 ∈ int, F5 x0 ∈ int
L69Variable G5 : set → set
L70Hypothesis HG5 : ∀x0 ∈ int, G5 x0 ∈ int
L72Hypothesis HH5 : H5 ∈ int
L73Variable U5 : set → set → set
L74Hypothesis HU5 : ∀x0 ∈ int, ∀x1 ∈ int, U5 x0 x1 ∈ int
L75Variable V5 : set → set
L76Hypothesis HV5 : ∀x0 ∈ int, V5 x0 ∈ int
L77Variable FAST : set → set
L78Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L79Hypothesis H1 : (∀X ∈ int, ((F1 X) = ((X + X) + X)))
L80Hypothesis H2 : (G1 = 2)
L81Hypothesis H3 : (∀X ∈ int, ((H1 X) = X))
L82Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))
L83Hypothesis H5 : (∀X ∈ int, ((V1 X) = (U1 G1 (H1 X))))
L84Hypothesis H6 : (∀X ∈ int, ((F0 X) = (2 * (V1 X))))
L85Hypothesis H7 : (∀X ∈ int, ((G0 X) = X))
L86Hypothesis H8 : (H0 = (1 + 2))
L87Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y))))))
L88Hypothesis H10 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))
L89Hypothesis H11 : (∀X ∈ int, ((F2 X) = (X + X)))
L90Hypothesis H12 : (∀X ∈ int, ((G2 X) = (X + X)))
L91Hypothesis H13 : (H2 = 1)
L92Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y))))))
L93Hypothesis H15 : (∀X ∈ int, ((V2 X) = (U2 (G2 X) H2)))
L94Hypothesis H16 : (∀X ∈ int, ((SMALL X) = ((V0 X) + (V2 X))))
L95Hypothesis H17 : (∀X ∈ int, ((F3 X) = (X + X)))
L96Hypothesis H18 : (∀X ∈ int, ((G3 X) = X))
L97Hypothesis H19 : (H3 = 1)
L98Hypothesis H20 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y))))))
L99Hypothesis H21 : (∀X ∈ int, ((V3 X) = (U3 (G3 X) H3)))
L100Hypothesis H22 : (∀X ∈ int, (∀Y ∈ int, ((F4 X Y) = (X * Y))))
L101Hypothesis H23 : (∀X ∈ int, (∀Y ∈ int, ((G4 X Y) = Y)))
L102Hypothesis H24 : (∀X ∈ int, ((H4 X) = X))
L103Hypothesis H25 : (I4 = (1 + 2))
L104Hypothesis H26 : (J4 = (1 + (2 * (2 + 2))))
L105Hypothesis H27 : (∀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)))))))
L106Hypothesis H28 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V4 X Y Z) = (if (X <= 0) then Z else (G4 (U4 (X + - 1) Y Z) (V4 (X + - 1) Y Z)))))))
L107Hypothesis H29 : (∀X ∈ int, ((W4 X) = (U4 (H4 X) I4 J4)))
L108Hypothesis H30 : (∀X ∈ int, ((F5 X) = (X + X)))
L109Hypothesis H31 : (∀X ∈ int, ((G5 X) = X))
L110Hypothesis H32 : (H5 = 1)
L111Hypothesis H33 : (∀X ∈ int, (∀Y ∈ int, ((U5 X Y) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y))))))
L112Hypothesis H34 : (∀X ∈ int, ((V5 X) = (U5 (G5 X) H5)))
L113Hypothesis H35 : (∀X ∈ int, ((FAST X) = (((V3 X) + (W4 X)) * (V5 X))))
L114Theorem. (
A334991)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.