X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fdama%2Fdama%2Fordered_uniform.ma;h=36ba2d287503736707f4911fe6407d6fcee08d27;hb=66155917bd62757397bc324029d0baade2cc281f;hp=63c966056db54a2f07ad19786ba1329b2722e1c6;hpb=f71d9d74c33e4ab3d86dc8244a7bdd3576bb0ac0;p=helm.git

diff --git a/helm/software/matita/contribs/dama/dama/ordered_uniform.ma b/helm/software/matita/contribs/dama/dama/ordered_uniform.ma
index 63c966056..36ba2d287 100644
--- a/helm/software/matita/contribs/dama/dama/ordered_uniform.ma
+++ b/helm/software/matita/contribs/dama/dama/ordered_uniform.ma
@@ -34,7 +34,8 @@ coercion ous_unifspace.
 
 record ordered_uniform_space : Type ≝ {
   ous_stuff :> ordered_uniform_space_;
-  ous_convex: ∀U.us_unifbase ous_stuff U → convex ous_stuff U
+  ous_convex_l: ∀U.us_unifbase ous_stuff U → convex (os_l ous_stuff) U;
+  ous_convex_r: ∀U.us_unifbase ous_stuff U → convex (os_r ous_stuff) U
 }.   
 
 definition half_ordered_set_OF_ordered_uniform_space : ordered_uniform_space → half_ordered_set.
@@ -109,13 +110,24 @@ lemma bs2_of_bss2:
 
 coercion bs2_of_bss2 nocomposites.
 
-(*
-lemma xxx :
- ∀O,s,x.bs2_of_bss2 (ordered_set_OF_ordered_uniform_space O) s x
-              =
-              x.
-intros; reflexivity;
-*)
+
+lemma a2sa :
+ ∀O:ordered_uniform_space.∀s:‡(ordered_set_OF_ordered_uniform_space O).
+ ∀x:
+  bs_carr
+  (square_bishop_set
+   (bishop_set_of_ordered_set
+     (segment_ordered_set 
+       (ordered_set_OF_ordered_uniform_space O) s))).
+ (\fst x) ≈ (\snd x) →
+ (\fst (bs2_of_bss2 (ordered_set_OF_ordered_uniform_space O) s x))
+ ≈
+ (\snd (bs2_of_bss2 (ordered_set_OF_ordered_uniform_space O) s x)).
+intros 3; cases x (a b); clear x; cases a (x Hx); cases b (y Hy); clear a b;
+simplify; intros 2 (K H); apply K; clear K; whd; whd in H; cases H;
+[left|right] apply x2sx; assumption;
+qed.
+
 
 lemma segment_ordered_uniform_space: 
   ∀O:ordered_uniform_space.∀s:‡O.ordered_uniform_space.
@@ -127,59 +139,83 @@ intros (O s); apply mk_ordered_uniform_space;
         apply (mk_uniform_space (bishop_set_of_ordered_set {[s]}) f);
         [1: intros (U H); intro x; simplify; 
             cases H (w Hw); cases Hw (Gw Hwp); clear H Hw; intro Hm;
-            lapply (us_phi1 O w Gw x) as IH;[2:intro;apply Hm;cases H; clear H;
-              [left;apply (x2sx ? s (\fst x) (\snd x) H1);
-              |right;apply (x2sx ? s ?? H1);]
-
+            lapply (us_phi1 O w Gw x (a2sa ??? Hm)) as IH;
             apply (restrict ? s ??? Hwp IH); 
         |2: intros (U V HU HV); cases HU (u Hu); cases HV (v Hv); clear HU HV;
             cases Hu (Gu HuU); cases Hv (Gv HvV); clear Hu Hv;
             cases (us_phi2 O u v Gu Gv) (w HW); cases HW (Gw Hw); clear HW;
-            exists; [apply (λb:{[l,r]} squareB.w b)] split;
+            exists; [apply (λb:{[s]} squareB.w b)] split;
             [1: unfold f; simplify; clearbody f;
                 exists; [apply w]; split; [assumption] intro b; simplify;
                 unfold segment_square_of_ordered_set_square;
                 cases b; intros; split; intros; assumption;
             |2: intros 2 (x Hx); cases (Hw ? Hx); split;
-                [apply (restrict O l r ??? HuU H)|apply (restrict O l r ??? HvV H1);]]
+                [apply (restrict O s ??? HuU H)|apply (restrict O s ??? HvV H1);]]
         |3: intros (U Hu); cases Hu (u HU); cases HU (Gu HuU); clear Hu HU;
             cases (us_phi3 O u Gu) (w HW); cases HW (Gw Hwu); clear HW;
-            exists; [apply (λx:{[l,r]} squareB.w x)] split;
+            exists; [apply (λx:{[s]} squareB.w x)] split;
             [1: exists;[apply w];split;[assumption] intros; simplify; intro;
                 unfold segment_square_of_ordered_set_square;
                 cases b; intros; split; intro; assumption;
-            |2: intros 2 (x Hx); apply (restrict O l r ??? HuU); apply Hwu; 
+            |2: intros 2 (x Hx); apply (restrict O s ??? HuU); apply Hwu; 
                 cases Hx (m Hm); exists[apply (\fst m)] apply Hm;]
         |4: intros (U HU x); cases HU (u Hu); cases Hu (Gu HuU); clear HU Hu;
             cases (us_phi4 O u Gu x) (Hul Hur);
             split; intros; 
-            [1: lapply (invert_restriction_agreement O l r ?? HuU) as Ra;
-                apply (restrict O l r ?? x Ra);
-                apply Hul; apply (unrestrict O l r ??? HuU H);
-            |2: apply (restrict O l r ??? HuU); apply Hur; 
-                apply (unrestrict O l r ??? (invert_restriction_agreement O l r ?? HuU) H);]]
+            [1: lapply (invert_restriction_agreement O s ?? HuU) as Ra;
+                apply (restrict O s ?? x Ra);
+                apply Hul; apply (unrestrict O s ??? HuU H);
+            |2: apply (restrict O s ??? HuU); apply Hur; 
+                apply (unrestrict O s ??? (invert_restriction_agreement O s ?? HuU) H);]]
     |2: simplify; reflexivity;]
-|2: simplify; unfold convex; intros;
-    cases H (u HU); cases HU (Gu HuU); clear HU H; 
-    lapply (ous_convex ?? Gu p ? H2 y H3) as Cu;
-    [1: apply (unrestrict O l r ??? HuU); apply H1;
-    |2: apply (restrict O l r ??? HuU Cu);]]
+|2: simplify; unfold convex; intros 3; cases s1; intros;
+    (* TODO: x2sx is for ≰, we need one for ≤ *) 
+    cases H (u HU); cases HU (Gu HuU); clear HU H;
+    lapply depth=0 (ous_convex_l ?? Gu 〈\fst h,\fst h1〉 ???????) as K3;
+    [2: intro; apply H2; apply (x2sx (os_l O) s h h1 H);
+    |3: apply 〈\fst (\fst y),\fst (\snd y)〉;
+    |4: intro; change in H with (\fst (\fst y) ≰ \fst h1); apply H3;
+        apply (x2sx (os_l O) s (\fst y) h1 H);
+    |5: change with (\fst h ≤ \fst (\fst y)); intro; apply H4;
+        apply (x2sx (os_l O) s h (\fst y) H);
+    |6: change with (\fst (\snd y) ≤ \fst h1); intro; apply H5;
+        apply (x2sx (os_l O) s (\snd y) h1 H);
+    |7: change with (\fst (\fst y) ≤ \fst (\snd y)); intro; apply H6;
+        apply (x2sx (os_l O) s (\fst y) (\snd y) H);
+    |8: apply (restrict O s U u y HuU K3);
+    |1: apply (unrestrict O s ?? 〈h,h1〉 HuU H1);]
+|3: simplify; unfold convex; intros 3; cases s1; intros; (* TODO *)
+    cases H (u HU); cases HU (Gu HuU); clear HU H;
+    lapply depth=0 (ous_convex_r ?? Gu 〈\fst h,\fst h1〉 ???????) as K3;
+    [2: intro; apply H2; apply (x2sx (os_r O) s h h1 H);
+    |3: apply 〈\fst (\fst y),\fst (\snd y)〉;
+    |4: intro; (*change in H with (\fst (\fst y) ≱ \fst h1);*) apply H3;
+        apply (x2sx (os_r O) s (\fst y) h1 H);
+    |5: (*change with (\fst h ≥ \fst (\fst y));*) intro; apply H4;
+        apply (x2sx (os_r O) s h (\fst y) H);
+    |6: (*change with (\fst (\snd y) ≤ \fst h1);*) intro; apply H5;
+        apply (x2sx (os_r O) s (\snd y) h1 H);
+    |7: (*change with (\fst (\fst y) ≤ \fst (\snd y));*) intro; apply H6;
+        apply (x2sx (os_r O) s (\fst y) (\snd y) H);
+    |8: apply (restrict O s U u y HuU K3);
+    |1: apply (unrestrict O s ?? 〈h,h1〉 HuU H1);]
+]  
 qed.
 
-interpretation "Ordered uniform space segment" 'segment_set a b = 
- (segment_ordered_uniform_space _ a b).
+interpretation "Ordered uniform space segment" 'segset a = 
+ (segment_ordered_uniform_space _ a).
 
 (* Lemma 3.2 *)
 alias symbol "pi1" = "exT \fst".
 lemma restric_uniform_convergence:
- ∀O:ordered_uniform_space.∀l,u:O.
-  ∀x:{[l,u]}.
-   ∀a:sequence {[l,u]}.
+ ∀O:ordered_uniform_space.∀s:‡O.
+  ∀x:{[s]}.
+   ∀a:sequence {[s]}.
     (⌊n, \fst (a n)⌋ : sequence O) uniform_converges (\fst x) → 
       a uniform_converges x.
-intros 8; cases H1; cases H2; clear H2 H1;
+intros 7; cases H1; cases H2; clear H2 H1;
 cases (H ? H3) (m Hm); exists [apply m]; intros; 
-apply (restrict ? l u ??? H4); apply (Hm ? H1);
+apply (restrict ? s ??? H4); apply (Hm ? H1);
 qed.
 
 definition order_continuity ≝