X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fcontribs%2Fdama%2Fdama%2Fmetric_space.ma;fp=matita%2Fcontribs%2Fdama%2Fdama%2Fmetric_space.ma;h=2266fe9e9618a08a56e5a8a91e3033e9123818a8;hp=0000000000000000000000000000000000000000;hb=f61af501fb4608cc4fb062a0864c774e677f0d76;hpb=58ae1809c352e71e7b5530dc41e2bfc834e1aef1

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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ordered_divisible_group.ma".
+
+record metric_space (R: todgroup) : Type ≝ {
+  ms_carr :> Type;
+  metric: ms_carr → ms_carr → R;
+  mpositive: ∀a,b:ms_carr. 0 ≤ metric a b; 
+  mreflexive: ∀a. metric a a ≈ 0;
+  msymmetric: ∀a,b. metric a b ≈ metric b a;
+  mtineq: ∀a,b,c:ms_carr. metric a b ≤ metric a c + metric c b
+}.
+
+notation < "\nbsp \delta a \nbsp b" non associative with precedence 80 for @{ 'delta2 $a $b}.
+interpretation "metric" 'delta2 a b = (cic:/matita/metric_space/metric.con _ _ a b).
+notation "\delta" non associative with precedence 80 for @{ 'delta }.
+interpretation "metric" 'delta = (cic:/matita/metric_space/metric.con _ _).
+
+lemma apart_of_metric_space: ∀R.metric_space R → apartness.
+intros (R ms); apply (mk_apartness ? (λa,b:ms.0 < δ a b)); unfold;
+[1: intros 2 (x H); cases H (H1 H2); clear H; 
+    lapply (Ap≫ ? (eq_sym ??? (mreflexive ??x)) H2);
+    apply (ap_coreflexive R 0); assumption;
+|2: intros (x y H); cases H; split;
+    [1: apply (Le≫ ? (msymmetric ????)); assumption
+    |2: apply (Ap≫ ? (msymmetric ????)); assumption]
+|3: simplify; intros (x y z H); elim H (LExy Axy);
+    lapply (mtineq ?? x y z) as H1; elim (ap2exc ??? Axy) (H2 H2); [cases (LExy H2)]
+    clear LExy; lapply (lt_le_transitive ???? H H1) as LT0;
+    apply (lt0plus_orlt ????? LT0); apply mpositive;]
+qed.
+    
+lemma ap2delta: ∀R.∀m:metric_space R.∀x,y:m.ap_apart (apart_of_metric_space ? m) x y → 0 < δ x y.
+intros 2 (R m); cases m 0; simplify; intros; assumption; qed.