Beginning of Section A16292
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 F2 : set → set
L14Hypothesis HF2 : ∀x0 ∈ int, F2 x0 ∈ int
L15Variable G2 : set → set → set
L16Hypothesis HG2 : ∀x0 ∈ int, ∀x1 ∈ int, G2 x0 x1 ∈ int
L18Hypothesis HH2 : H2 ∈ int
L19Variable U2 : set → set → set
L20Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, U2 x0 x1 ∈ int
L21Variable V2 : set → set → set
L22Hypothesis HV2 : ∀x0 ∈ int, ∀x1 ∈ int, V2 x0 x1 ∈ int
L23Variable F1 : set → set → set
L24Hypothesis HF1 : ∀x0 ∈ int, ∀x1 ∈ int, F1 x0 x1 ∈ int
L25Variable G1 : set → set
L26Hypothesis HG1 : ∀x0 ∈ int, G1 x0 ∈ int
L28Hypothesis HH1 : H1 ∈ int
L29Variable U1 : set → set → set
L30Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, U1 x0 x1 ∈ int
L31Variable V1 : set → set
L32Hypothesis HV1 : ∀x0 ∈ int, V1 x0 ∈ int
L33Variable H0 : set → set
L34Hypothesis HH0 : ∀x0 ∈ int, H0 x0 ∈ int
L35Variable U0 : set → set → set
L36Hypothesis HU0 : ∀x0 ∈ int, ∀x1 ∈ int, U0 x0 x1 ∈ int
L37Variable V0 : set → set
L38Hypothesis HV0 : ∀x0 ∈ int, V0 x0 ∈ int
L39Variable SMALL : set → set
L40Hypothesis HSMALL : ∀x0 ∈ int, SMALL x0 ∈ int
L41Variable F3 : set → set → set
L42Hypothesis HF3 : ∀x0 ∈ int, ∀x1 ∈ int, F3 x0 x1 ∈ int
L43Variable G3 : set → set → set
L44Hypothesis HG3 : ∀x0 ∈ int, ∀x1 ∈ int, G3 x0 x1 ∈ int
L45Variable H3 : set → set
L46Hypothesis HH3 : ∀x0 ∈ int, H3 x0 ∈ int
L48Hypothesis HI3 : I3 ∈ int
L50Hypothesis HJ3 : J3 ∈ int
L51Variable U3 : set → set → set → set
L52Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U3 x0 x1 x2 ∈ int
L53Variable V3 : set → set → set → set
L54Hypothesis HV3 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V3 x0 x1 x2 ∈ int
L55Variable W3 : set → set
L56Hypothesis HW3 : ∀x0 ∈ int, W3 x0 ∈ int
L57Variable F4 : set → set
L58Hypothesis HF4 : ∀x0 ∈ int, F4 x0 ∈ int
L59Variable G4 : set → set
L60Hypothesis HG4 : ∀x0 ∈ int, G4 x0 ∈ int
L62Hypothesis HH4 : H4 ∈ int
L63Variable U4 : set → set → set
L64Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, U4 x0 x1 ∈ int
L65Variable V4 : set → set
L66Hypothesis HV4 : ∀x0 ∈ int, V4 x0 ∈ int
L67Variable FAST : set → set
L68Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L69Hypothesis H1 : (∀X ∈ int, ((F0 X) = (X + X)))
L70Hypothesis H2 : (∀X ∈ int, ((G0 X) = X))
L71Hypothesis H3 : (∀X ∈ int, ((F2 X) = (1 + ((2 * (X + X)) + X))))
L72Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((G2 X Y) = Y)))
L73Hypothesis H5 : (H2 = 1)
L74Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y))))))
L75Hypothesis H7 : (∀X ∈ int, (∀Y ∈ int, ((V2 X Y) = (U2 (G2 X Y) H2))))
L76Hypothesis H8 : (∀X ∈ int, (∀Y ∈ int, ((F1 X Y) = (((V2 X Y) + X) + X))))
L77Hypothesis H9 : (∀X ∈ int, ((G1 X) = X))
L78Hypothesis H10 : (H1 = 1)
L79Hypothesis H11 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y) X)))))
L80Hypothesis H12 : (∀X ∈ int, ((V1 X) = (U1 (G1 X) H1)))
L81Hypothesis H13 : (∀X ∈ int, ((H0 X) = (V1 X)))
L82Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y))))))
L83Hypothesis H15 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) (H0 X))))
L84Hypothesis H16 : (∀X ∈ int, ((SMALL X) = (V0 X)))
L85Hypothesis H17 : (∀X ∈ int, (∀Y ∈ int, ((F3 X Y) = (((2 * ((X + X) + Y)) + - 1) + X))))
L86Hypothesis H18 : (∀X ∈ int, (∀Y ∈ int, ((G3 X Y) = (Y + Y))))
L87Hypothesis H19 : (∀X ∈ int, ((H3 X) = X))
L88Hypothesis H20 : (I3 = 1)
L89Hypothesis H21 : (J3 = 2)
L90Hypothesis H22 : (∀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)))))))
L91Hypothesis H23 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V3 X Y Z) = (if (X <= 0) then Z else (G3 (U3 (X + - 1) Y Z) (V3 (X + - 1) Y Z)))))))
L92Hypothesis H24 : (∀X ∈ int, ((W3 X) = (U3 (H3 X) I3 J3)))
L93Hypothesis H25 : (∀X ∈ int, ((F4 X) = (X + X)))
L94Hypothesis H26 : (∀X ∈ int, ((G4 X) = X))
L95Hypothesis H27 : (H4 = 1)
L96Hypothesis H28 : (∀X ∈ int, (∀Y ∈ int, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y))))))
L97Hypothesis H29 : (∀X ∈ int, ((V4 X) = (U4 (G4 X) H4)))
L98Hypothesis H30 : (∀X ∈ int, ((FAST X) = ((W3 X) * (V4 X))))
L99Theorem. (
A16292)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.