Beginning of Section A306472
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 F0 : set → set
L10Hypothesis HF0 : ∀x0 ∈ int, F0 x0 ∈ int
L11Variable G0 : set → set
L12Hypothesis HG0 : ∀x0 ∈ int, G0 x0 ∈ int
L13Variable F1 : set → set
L14Hypothesis HF1 : ∀x0 ∈ int, F1 x0 ∈ int
L16Hypothesis HG1 : G1 ∈ int
L18Hypothesis HH1 : H1 ∈ int
L19Variable U1 : set → set → set
L20Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, U1 x0 x1 ∈ int
L22Hypothesis HV1 : V1 ∈ 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 F3 : set → set
L32Hypothesis HF3 : ∀x0 ∈ int, F3 x0 ∈ int
L34Hypothesis HG3 : G3 ∈ int
L36Hypothesis HH3 : H3 ∈ int
L37Variable U3 : set → set → set
L38Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, U3 x0 x1 ∈ int
L40Hypothesis HV3 : V3 ∈ int
L41Variable F2 : set → set
L42Hypothesis HF2 : ∀x0 ∈ int, F2 x0 ∈ int
L44Hypothesis HG2 : G2 ∈ int
L45Variable F4 : set → set → set
L46Hypothesis HF4 : ∀x0 ∈ int, ∀x1 ∈ int, F4 x0 x1 ∈ int
L47Variable G4 : set → set → set
L48Hypothesis HG4 : ∀x0 ∈ int, ∀x1 ∈ int, G4 x0 x1 ∈ int
L49Variable H4 : set → set
L50Hypothesis HH4 : ∀x0 ∈ int, H4 x0 ∈ int
L52Hypothesis HI4 : I4 ∈ int
L54Hypothesis HJ4 : J4 ∈ int
L55Variable U4 : set → set → set → set
L56Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U4 x0 x1 x2 ∈ int
L57Variable V4 : set → set → set → set
L58Hypothesis HV4 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V4 x0 x1 x2 ∈ int
L59Variable W4 : set → set
L60Hypothesis HW4 : ∀x0 ∈ int, W4 x0 ∈ int
L61Variable H2 : set → set
L62Hypothesis HH2 : ∀x0 ∈ int, H2 x0 ∈ int
L63Variable U2 : set → set → set
L64Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, U2 x0 x1 ∈ int
L65Variable V2 : set → set
L66Hypothesis HV2 : ∀x0 ∈ int, V2 x0 ∈ int
L67Variable FAST : set → set
L68Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L69Hypothesis H1 : (∀X ∈ int, ((F0 X) = ((X + X) + X)))
L70Hypothesis H2 : (∀X ∈ int, ((G0 X) = ((X + X) + X)))
L71Hypothesis H3 : (∀X ∈ int, ((F1 X) = (1 + (2 * (X + X)))))
L72Hypothesis H4 : (G1 = 2)
L73Hypothesis H5 : (H1 = 2)
L74Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))
L75Hypothesis H7 : (V1 = (U1 G1 H1))
L76Hypothesis H8 : (H0 = V1)
L77Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y))))))
L78Hypothesis H10 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))
L79Hypothesis H11 : (∀X ∈ int, ((SMALL X) = (V0 X)))
L80Hypothesis H12 : (∀X ∈ int, ((F3 X) = (X * X)))
L81Hypothesis H13 : (G3 = 1)
L82Hypothesis H14 : (H3 = (2 + (2 + 2)))
L83Hypothesis H15 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y))))))
L84Hypothesis H16 : (V3 = (U3 G3 H3))
L85Hypothesis H17 : (∀X ∈ int, ((F2 X) = (((X * X) * (1 + V3)) * X)))
L86Hypothesis H18 : (G2 = 1)
L87Hypothesis H19 : (∀X ∈ int, (∀Y ∈ int, ((F4 X Y) = (X * Y))))
L88Hypothesis H20 : (∀X ∈ int, (∀Y ∈ int, ((G4 X Y) = Y)))
L89Hypothesis H21 : (∀X ∈ int, ((H4 X) = X))
L90Hypothesis H22 : (I4 = 1)
L91Hypothesis H23 : (J4 = (1 + 2))
L92Hypothesis H24 : (∀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)))))))
L93Hypothesis H25 : (∀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)))))))
L94Hypothesis H26 : (∀X ∈ int, ((W4 X) = (U4 (H4 X) I4 J4)))
L95Hypothesis H27 : (∀X ∈ int, ((H2 X) = (W4 X)))
L96Hypothesis H28 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y))))))
L97Hypothesis H29 : (∀X ∈ int, ((V2 X) = (U2 G2 (H2 X))))
L98Hypothesis H30 : (∀X ∈ int, ((FAST X) = (V2 X)))
L99Theorem. (
A306472)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.