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 "sandwich.ma".
17 include "metric_lattice.ma".
21 ∀R.∀ml:mlattice R.∀xn:sequence ml.
23 (* BUG: it inserts a compesoed coercion instead of an hand made one,
24 what to do? prefer the human made one or allow to kill a coercion?
26 xn ⇝ x → xn ⇝ y → δ x y ≈ 0.
27 intros (R ml xn x y H1 H2); unfold tends0 in H1 H2; unfold d2s in H1 H2;
28 intro Axy; lapply (ap_le_to_lt ??? (ap_symmetric ??? Axy) (mpositive ? ml ??)) as ge0;
29 cases (H1 (δ x y/1) (divide_preserves_lt ??? ge0)) (n1 Hn1); clear H1;
30 cases (H2 (δ x y/1) (divide_preserves_lt ??? ge0)) (n2 Hn2); clear H2;
31 letin N ≝ (S (n2 + n1));
32 cases (Hn1 N ?) (H1 H2); [apply (ltwr ? n2); rewrite < sym_plus; apply le_n;]
33 cases (Hn2 N ?) (H3 H4); [apply (ltwl ? n1); rewrite < sym_plus; apply le_n;]
34 clear H1 H3 Hn2 Hn1 N ge0 Axy; lapply (mtineq ?? x y (xn (S (n2+n1)))) as H5;
35 cut ( δx (xn (S (n2+n1)))+ δ(xn (S (n2+n1))) y < δx y/1 + δ(xn (S (n2+n1))) y) as H6;[2:
36 apply flt_plusr; apply (Lt≪ ? (msymmetric ????)); assumption]
37 lapply (le_lt_transitive ???? H5 H6) as H7; clear H6;
38 cut (δx y/1+ δ(xn (S (n2+n1))) y < δx y/1+ δx y/1) as H6; [2:apply flt_plusl; assumption]
39 lapply (lt_transitive ???? H7 H6) as ABS; clear H6 H7 H4 H5 H2 n1 n2 xn;
40 lapply (divpow ? (δ x y) 1) as D; lapply (Lt≪ ? (eq_sym ??? D) ABS) as H;
41 change in H with ( δx y/1+ δx y/1< δx y/1+ δx y/1);
42 apply (lt_coreflexive ?? H);