--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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.