1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 include "arithmetics/nat.ma".
18 naxiom nat_to_Q: nat → Q.
19 ncoercion nat_to_Q : ∀x:nat.Q ≝ nat_to_Q on _x:nat to Q.
20 naxiom Qplus: Q → Q → Q.
21 naxiom Qtimes: Q → Q → Q.
22 naxiom Qdivides: Q → Q → Q.
23 interpretation "Q plus" 'plus x y = (Qplus x y).
24 interpretation "Q times" 'times x y = (Qtimes x y).
25 interpretation "Q divides" 'divide x y = (Qdivides x y).
26 naxiom Qtimes_distr: ∀n,m:nat.∀q:Q. (n * q + m * q) = (plus n m) * q.
27 naxiom Qmult_one: ∀q:Q. 1 * q = q.
28 naxiom Qdivides_mult: ∀q1,q2. (q1 * q2) / q1 = q2.
30 (*ndefinition aaa ≝ Qtimes_distr.
31 ndefinition bbb ≝ Qmult_one.
32 ndefinition ccc ≝ Qdivides_mult.*)
34 naxiom disjoint: Q → Q → Q → Q → bool.
36 ncoinductive locate : Q → Q → Prop ≝
37 L: ∀l,u. locate l ((2 * l + u) / 3) → locate l u
38 | H: ∀l,u. locate ((l + 2 * u) / 3) u → locate l u.
40 ndefinition R ≝ ∃l,u:Q. locate l u.
42 nlet corec Q_to_locate q : locate q q ≝ L q q ?.
43 (*NOT WORKING: nrewrite < (Qmult_one q) in ⊢ (? ? %); *)
44 ncut (locate q q = locate q ((2*q+q)/3))
45 [##2: #H; ncases H; (*NOT WORKING: nrewrite > H;*) napply Q_to_locate
47 ncut ((2 * q + q) = ((2 * q + 1 * q))); //; #H;
48 ncut (q = ((2 * q + q) / 3)); //;
49 ncut ((2 * q + q) = ((2 * q + 1 * q))); //; #K;
50 ncut (2 * q + 1 * q = 3 * q); /4/ ]
53 ndefinition Q_to_R : Q → R.
57 nlet corec locate_add (l1,u1:?) (r1: locate l1 u1) (l2,u2:?) (r2: locate l2 u2) :
58 locate (l1 + l2) (u1 + u2) ≝ ?.
59 ncases r1; ncases r2; #l2; #u2; #r2; #l1; #u1; #r1
60 [ @1; napply locate_add;
62 nlet corec apart (l1,u1) (r1: locate l1 u1) (l2,u2) (r2: locate l2 u2) : CProp[0] ≝
63 match disjoint l1 u1 l2 u2 with