Beginning of Section A307168
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
L25Variable F2 : set → set
L26Hypothesis HF2 : ∀x0 ∈ int, F2 x0 ∈ int
L28Hypothesis HG2 : G2 ∈ int
L30Hypothesis HH2 : H2 ∈ int
L31Variable U2 : set → set → set
L32Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, U2 x0 x1 ∈ int
L34Hypothesis HV2 : V2 ∈ int
L36Hypothesis HI0 : I0 ∈ int
L38Hypothesis HJ0 : J0 ∈ int
L39Variable U0 : set → set → set → set
L40Hypothesis HU0 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U0 x0 x1 x2 ∈ int
L41Variable V0 : set → set → set → set
L42Hypothesis HV0 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V0 x0 x1 x2 ∈ int
L43Variable W0 : set → set
L44Hypothesis HW0 : ∀x0 ∈ int, W0 x0 ∈ int
L45Variable SMALL : set → set
L46Hypothesis HSMALL : ∀x0 ∈ int, SMALL x0 ∈ int
L47Variable F3 : set → set → set
L48Hypothesis HF3 : ∀x0 ∈ int, ∀x1 ∈ int, F3 x0 x1 ∈ int
L49Variable G3 : set → set
L50Hypothesis HG3 : ∀x0 ∈ int, G3 x0 ∈ int
L51Variable H3 : set → set
L52Hypothesis HH3 : ∀x0 ∈ int, H3 x0 ∈ int
L54Hypothesis HI3 : I3 ∈ int
L56Hypothesis HJ3 : J3 ∈ int
L57Variable U3 : set → set → set → set
L58Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U3 x0 x1 x2 ∈ int
L59Variable V3 : set → set → set → set
L60Hypothesis HV3 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V3 x0 x1 x2 ∈ int
L61Variable W3 : set → set
L62Hypothesis HW3 : ∀x0 ∈ int, W3 x0 ∈ int
L63Variable FAST : set → set
L64Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L65Hypothesis H1 : (∀X ∈ int, ((F1 X) = (X * X)))
L66Hypothesis H2 : (G1 = 2)
L67Hypothesis H3 : (H1 = 2)
L68Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))
L69Hypothesis H5 : (V1 = (U1 G1 H1))
L70Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = ((V1 * X) + - Y))))
L71Hypothesis H7 : (∀X ∈ int, ((G0 X) = X))
L72Hypothesis H8 : (∀X ∈ int, ((H0 X) = X))
L73Hypothesis H9 : (∀X ∈ int, ((F2 X) = (1 + (X + X))))
L74Hypothesis H10 : (G2 = 2)
L75Hypothesis H11 : (H2 = 2)
L76Hypothesis H12 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y))))))
L77Hypothesis H13 : (V2 = (U2 G2 H2))
L78Hypothesis H14 : (I0 = V2)
L79Hypothesis H15 : (J0 = (2 + 2))
L80Hypothesis H16 : (∀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)))))))
L81Hypothesis H17 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V0 X Y Z) = (if (X <= 0) then Z else (G0 (U0 (X + - 1) Y Z)))))))
L82Hypothesis H18 : (∀X ∈ int, ((W0 X) = (U0 (H0 X) I0 J0)))
L83Hypothesis H19 : (∀X ∈ int, ((SMALL X) = (W0 X)))
L84Hypothesis H20 : (∀X ∈ int, (∀Y ∈ int, ((F3 X Y) = (((2 * (2 * (2 + 2))) * X) + - Y))))
L85Hypothesis H21 : (∀X ∈ int, ((G3 X) = X))
L86Hypothesis H22 : (∀X ∈ int, ((H3 X) = X))
L87Hypothesis H23 : (I3 = (1 + (2 + (2 * (2 + 2)))))
L88Hypothesis H24 : (J3 = (2 + 2))
L89Hypothesis H25 : (∀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)))))))
L90Hypothesis H26 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V3 X Y Z) = (if (X <= 0) then Z else (G3 (U3 (X + - 1) Y Z)))))))
L91Hypothesis H27 : (∀X ∈ int, ((W3 X) = (U3 (H3 X) I3 J3)))
L92Hypothesis H28 : (∀X ∈ int, ((FAST X) = (W3 X)))
L93Theorem. (
A307168)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.