From: Enrico Tassi <enrico.tassi@inria.fr>
Date: Mon, 14 Jan 2008 14:58:26 +0000 (+0000)
Subject: fixed a pulback and proved 3.17
X-Git-Tag: make_still_working~5670
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=954ed2eaf305a60d7e046206472bc0397a421ad2;p=helm.git

fixed a pulback and proved 3.17
---

diff --git a/helm/software/matita/dama/metric_lattice.ma b/helm/software/matita/dama/metric_lattice.ma
index 36742163b..40f3d41eb 100644
--- a/helm/software/matita/dama/metric_lattice.ma
+++ b/helm/software/matita/dama/metric_lattice.ma
@@ -19,12 +19,26 @@ include "lattice.ma".
 
 record mlattice_ (R : todgroup) : Type ≝ {
   ml_mspace_: metric_space R;
-  ml_lattice:> lattice;
-  ml_with_: ms_carr ? ml_mspace_ = ap_carr (l_carr ml_lattice)
+  ml_lattice_: lattice;
+  ml_with_: ms_carr ? ml_mspace_ = l_carr ml_lattice_;
+  ml_with2_: l_carr ml_lattice_ = apart_of_metric_space ? ml_mspace_ 
 }.
 
+lemma ml_lattice: ∀R.mlattice_ R → lattice.
+intros (R ml); apply (mk_lattice (apart_of_metric_space ? (ml_mspace_ ? ml))); try unfold eq;
+cases (ml_with2_ ? ml);
+[apply (join (ml_lattice_ ? ml));|apply (meet (ml_lattice_ ? ml));
+|apply (join_refl (ml_lattice_ R ml));| apply (meet_refl (ml_lattice_ ? ml));
+|apply (join_comm (ml_lattice_ ? ml));| apply (meet_comm (ml_lattice_ ? ml));
+|apply (join_assoc (ml_lattice_ ? ml));|apply (meet_assoc (ml_lattice_ ? ml));
+|apply (absorbjm (ml_lattice_ ? ml)); |apply (absorbmj (ml_lattice_ ? ml));
+|apply (strong_extj (ml_lattice_ ? ml));|apply (strong_extm (ml_lattice_ ? ml));]
+qed.
+
+coercion cic:/matita/metric_lattice/ml_lattice.con.
+
 lemma ml_mspace: ∀R.mlattice_ R → metric_space R.
-intros (R ml); apply (mk_metric_space R ml); unfold Type_OF_mlattice_;
+intros (R ml); apply (mk_metric_space R ml);
 cases (ml_with_ ? ml); simplify;
 [apply (metric ? (ml_mspace_ ? ml));|apply (mpositive ? (ml_mspace_ ? ml));
 |apply (mreflexive ? (ml_mspace_ ? ml));|apply (msymmetric ? (ml_mspace_ ? ml));
@@ -107,4 +121,4 @@ qed.
 (* 3.22 sup debole (più piccolo dei maggioranti) *)
 (* 3.23 conclusion: δ x sup(...) ≈ 0 *)
 (* 3.25 vero nel reticolo e basta (niente δ) *)
-(* 3.36 conclusion: δ x y ≈ 0 *)
\ No newline at end of file
+(* 3.36 conclusion: δ x y ≈ 0 *)
diff --git a/helm/software/matita/dama/metric_space.ma b/helm/software/matita/dama/metric_space.ma
index d36e90a68..595407de8 100644
--- a/helm/software/matita/dama/metric_space.ma
+++ b/helm/software/matita/dama/metric_space.ma
@@ -23,7 +23,6 @@ record metric_space (R : todgroup) : Type ≝ {
   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
-  (* manca qualcosa per essere una metrica e non una semimetrica *)
 }.
 
 notation < "\nbsp \delta a \nbsp b" non associative with precedence 80 for @{ 'delta2 $a $b}.
@@ -49,3 +48,6 @@ intros (R ms); apply (mk_apartness ? (apart_metric ? ms)); unfold apart_metric;
 qed.
     
 coercion cic:/matita/metric_space/apart_of_metric_space.con.
+
+lemma ap2delta: ∀R.∀m:metric_space R.∀x,y:m.x#y → 0 < δ x y.
+intros 2 (R m); cases m 0; simplify; intros; assumption; qed. 
diff --git a/helm/software/matita/dama/sandwich_corollary.ma b/helm/software/matita/dama/sandwich_corollary.ma
new file mode 100644
index 000000000..f41b65028
--- /dev/null
+++ b/helm/software/matita/dama/sandwich_corollary.ma
@@ -0,0 +1,45 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 "sandwich.ma".
+
+include "metric_lattice.ma".
+
+(* 3.17 *)
+lemma tends_uniq:
+  ∀R.∀ml:mlattice R.∀xn:sequence ml.
+  ∀x,y:apart_of_metric_space ? ml. 
+  (* BUG: it inserts a compesoed coercion instead of an hand made one, 
+     what to do? prefer the human made one or allow to kill a coercion? 
+   *)
+     xn ⇝ x → xn ⇝ y → x ≈ y.
+intros (R ml xn x y H1 H2); unfold tends0 in H1 H2; unfold d2s in H1 H2;
+intro Axy; lapply (ap2delta R ml x y Axy) as ge0;
+cases (H1 (δ x y/1) (divide_preserves_lt ??? ge0)) (n1 Hn1); clear H1; 
+cases (H2 (δ x y/1) (divide_preserves_lt ??? ge0)) (n2 Hn2); clear H2;
+letin N ≝ (S (n2 + n1));
+cases (Hn1 N ?) (H1 H2); [apply (ltwr ? n2); rewrite < sym_plus; apply le_n;]
+cases (Hn2 N ?) (H3 H4); [apply (ltwl ? n1); rewrite < sym_plus; apply le_n;]
+clear H1 H3 Hn2 Hn1 N ge0 Axy; lapply (mtineq ?? x y (xn (S (n2+n1)))) as H5;
+cut ( δx (xn (S (n2+n1)))+ δ(xn (S (n2+n1))) y <   δx y/1 + δ(xn (S (n2+n1))) y) as H6;[2:
+  apply flt_plusr; apply (Lt≪ ? (msymmetric ????)); assumption]
+lapply (le_lt_transitive ???? H5 H6) as H7; clear H6;
+cut (δx y/1+ δ(xn (S (n2+n1))) y < δx y/1+  δx y/1) as H6; [2:apply flt_plusl; assumption]
+lapply (lt_transitive ???? H7 H6) as ABS; clear H6 H7 H4 H5 H2 n1 n2 xn;
+lapply (divpow ? (δ x y) 1) as D; lapply (Lt≪ ? (eq_sym ??? D) ABS) as H;
+change in H with ( δx y/1+ δx y/1< δx y/1+ δx y/1);
+apply (lt_coreflexive ?? H);
+qed.
+
+