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=c33fae30b4ce40198b8e1889ea1c1b58697cd567;hp=974405214746fa6acf4ccf4866d65fd2056f391d;hpb=1509e99ac3aba0e725ac7ced7db20d5d23ea276a;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 974405214..36ba2d287 100644
--- a/helm/software/matita/contribs/dama/dama/ordered_uniform.ma
+++ b/helm/software/matita/contribs/dama/dama/ordered_uniform.ma
@@ -30,33 +30,207 @@ unfold bishop_set_OF_ordered_uniform_space_;
 |5: cases (with_ X); simplify; apply (us_phi4 (ous_us_ X))]
 qed.
 
-coercion cic:/matita/dama/ordered_uniform/ous_unifspace.con.
+coercion ous_unifspace.
 
 record ordered_uniform_space : Type ≝ {
   ous_stuff :> ordered_uniform_space_;
-  ous_prop1: ∀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.
+intro; compose ordered_set_OF_ordered_uniform_space with os_l. apply (f o);
+qed.
+
+definition invert_os_relation ≝
+  λC:ordered_set.λU:C squareO → Prop.
+    λx:C squareO. U 〈\snd x,\fst x〉.
+
+interpretation "relation invertion" 'invert a = (invert_os_relation _ a).
+interpretation "relation invertion" 'invert_symbol = (invert_os_relation _).
+interpretation "relation invertion" 'invert_appl a x = (invert_os_relation _ a x).
+
+lemma hint_segment: ∀O.
+ segment (Type_of_ordered_set O) →
+ segment (hos_carr (os_l O)).
+intros; assumption;
+qed.
+
+coercion hint_segment nocomposites.
+
+lemma segment_square_of_ordered_set_square: 
+  ∀O:ordered_set.∀s:‡O.∀x:O squareO.
+   \fst x ∈ s → \snd x ∈ s → {[s]} squareO.
+intros; split; exists; [1: apply (\fst x) |3: apply (\snd x)] assumption;
+qed.
+
+coercion segment_square_of_ordered_set_square with 0 2 nocomposites.
+
+alias symbol "pi1" (instance 4) = "exT \fst".
+alias symbol "pi1" (instance 2) = "exT \fst".
+lemma ordered_set_square_of_segment_square : 
+ ∀O:ordered_set.∀s:‡O.{[s]} squareO → O squareO ≝ 
+  λO:ordered_set.λs:‡O.λb:{[s]} squareO.〈\fst(\fst b),\fst(\snd b)〉.
+
+coercion ordered_set_square_of_segment_square nocomposites.
+
+lemma restriction_agreement : 
+  ∀O:ordered_uniform_space.∀s:‡O.∀P:{[s]} squareO → Prop.∀OP:O squareO → Prop.Prop.
+apply(λO:ordered_uniform_space.λs:‡O.
+       λP:{[s]} squareO → Prop. λOP:O squareO → Prop.
+          ∀b:O squareO.∀H1,H2.(P b → OP b) ∧ (OP b → P b));
+[5,7: apply H1|6,8:apply H2]skip;
+qed.
+
+lemma unrestrict: ∀O:ordered_uniform_space.∀s:‡O.∀U,u.∀x:{[s]} squareO.
+  restriction_agreement ? s U u → U x → u x.
+intros 6; cases x (b b1); cases b (w1 H1); cases b1 (w2 H2); clear b b1 x; 
+cases (H 〈w1,w2〉 H1 H2) (L _); intro Uw; apply L; apply Uw;
+qed.
+
+lemma restrict: ∀O:ordered_uniform_space.∀s:‡O.∀U,u.∀x:{[s]} squareO.
+  restriction_agreement ? s U u → u x → U x.
+intros 5; cases x (b b1); cases b (w1 H1); cases b1 (w2 H2); clear b1 b x;
+intros (Ra uw); cases (Ra 〈w1,w2〉 H1 H2) (_ R); apply R; apply uw;
+qed.
+
+lemma invert_restriction_agreement: 
+  ∀O:ordered_uniform_space.∀s:‡O.
+   ∀U:{[s]} squareO → Prop.∀u:O squareO → Prop.
+    restriction_agreement ? s U u →
+    restriction_agreement ? s (\inv U) (\inv u).
+intros 8; split; intro;
+[1: apply (unrestrict ???? (segment_square_of_ordered_set_square ?? 〈\snd b,\fst b〉 H2 H1) H H3);
+|2: apply (restrict ???? (segment_square_of_ordered_set_square ?? 〈\snd b,\fst b〉 H2 H1) H H3);]
+qed. 
+
+lemma bs2_of_bss2: 
+ ∀O:ordered_set.∀s:‡O.(bishop_set_of_ordered_set {[s]}) squareB → (bishop_set_of_ordered_set O) squareB ≝ 
+  λO:ordered_set.λs:‡O.λb:{[s]} squareO.〈\fst(\fst b),\fst(\snd b)〉.
+
+coercion bs2_of_bss2 nocomposites.
+
+
+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.∀u,v:O.ordered_uniform_space.
-intros (O u v); apply mk_ordered_uniform_space;
-[1: apply mk_ordered_uniform_space_;
-    [1: apply (mk_ordered_set (sigma ? (λx.x ∈ [u,v])));
-        [1: intros (x y); apply (fst x ≰ fst y);
-        |2: intros 1; cases x; simplify; apply os_coreflexive;
-        |3: intros 3; cases x; cases y; cases z; simplify; apply os_cotransitive]
-    |2: apply (mk_uniform_space (bishop_set_of_ordered_set (mk_ordered_set (sigma ? (λx.x ∈ [u,v])) ???)));
-    |3: apply refl_eq;
+  ∀O:ordered_uniform_space.∀s:‡O.ordered_uniform_space.
+intros (O s); apply mk_ordered_uniform_space;
+[1: apply (mk_ordered_uniform_space_ {[s]});
+    [1: alias symbol "and" = "constructive and".
+        letin f ≝ (λP:{[s]} squareO → Prop. ∃OP:O squareO → Prop.
+                    (us_unifbase O OP) ∧ restriction_agreement ?? P OP);
+        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 (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:{[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 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:{[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 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 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 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" 'segset a = 
+ (segment_ordered_uniform_space _ a).
 
 (* Lemma 3.2 *)
-lemma foo:
- ∀O:ordered_uniform_space.∀l,u:O.
-  ∀x:(segment_ordered_uniform_space O l u).
-   ∀a:sequence (segment_ordered_uniform_space O l u).
-     (* (λn.fst (a n)) uniform_converges (fst x) → *) 
+alias symbol "pi1" = "exT \fst".
+lemma restric_uniform_convergence:
+ ∀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.
-     
-      
\ No newline at end of file
+intros 7; cases H1; cases H2; clear H2 H1;
+cases (H ? H3) (m Hm); exists [apply m]; intros; 
+apply (restrict ? s ??? H4); apply (Hm ? H1);
+qed.
+
+definition order_continuity ≝
+  λC:ordered_uniform_space.∀a:sequence C.∀x:C.
+    (a ↑ x → a uniform_converges x) ∧ (a ↓ x → a uniform_converges x).
+
+lemma hint_boh1: ∀O. Type_OF_ordered_uniform_space O → hos_carr (os_l O).
+intros; assumption;
+qed.
+
+coercion hint_boh1 nocomposites. 
+
+lemma hint_boh2: ∀O:ordered_uniform_space. hos_carr (os_l O) → Type_OF_ordered_uniform_space O.
+intros; assumption;
+qed.
+
+coercion hint_boh2 nocomposites. 
+