X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fdama%2Fdama%2Fordered_uniform.ma;h=e724dc2e6ba4b7613b011973f483f89109096b63;hb=1ee5193677b8e2a80d4f068ee79ecac335de1196;hp=c9b5e7da6ecf6fded8e8250e1bba68ae2126a2e2;hpb=3c1ca5620048ad842144fba291f8bc5f0dca7061;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 c9b5e7da6..e724dc2e6 100644
--- a/helm/software/matita/contribs/dama/dama/ordered_uniform.ma
+++ b/helm/software/matita/contribs/dama/dama/ordered_uniform.ma
@@ -30,120 +30,165 @@ 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_convex: ∀U.us_unifbase ous_stuff U → convex ous_stuff U
 }.   
+(*
+definition Type_of_ordered_uniform_space : ordered_uniform_space → Type.
+intro; compose ordered_set_OF_ordered_uniform_space with os_l.
+apply (hos_carr (f o));
+qed.
+
+definition Type_of_ordered_uniform_space_dual : ordered_uniform_space → Type.
+intro; compose ordered_set_OF_ordered_uniform_space with os_r.
+apply (hos_carr (f o));
+qed.
 
-lemma segment_square_of_O_square: 
-  ∀O:ordered_set.∀u,v:O.∀x:O square.
-   fst x ∈ [u,v] → snd x ∈ [u,v] → {[u,v]} square.
-intros; split; exists; [1: apply (fst x) |3: apply (snd x)] assumption;
+coercion Type_of_ordered_uniform_space_dual.
+coercion Type_of_ordered_uniform_space.
+*)
+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.
 
-coercion cic:/matita/dama/ordered_uniform/segment_square_of_O_square.con 0 2.
+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 segment_square_of_ordered_set_square: 
+  ∀O:ordered_set.∀u,v:O.∀x:O squareO.
+   \fst x ∈ [u,v] → \snd x ∈ [u,v] → {[u,v]} squareO.
+intros; split; exists; [1: apply (\fst x) |3: apply (\snd x)] assumption;
+qed.
 
-alias symbol "pi1" (instance 4) = "sigma pi1".
-alias symbol "pi1" (instance 2) = "sigma pi1".
-lemma O_square_of_segment_square : 
- ∀O:ordered_set.∀u,v:O.{[u,v]} square → O square ≝ 
- λO:ordered_set.λu,v:O.λb:{[u,v]} square.〈fst(fst b),fst(snd b)〉.
+coercion segment_square_of_ordered_set_square with 0 2 nocomposites.
 
-coercion cic:/matita/dama/ordered_uniform/O_square_of_segment_square.con.
+alias symbol "pi1" (instance 4) = "exT \fst".
+alias symbol "pi1" (instance 2) = "exT \fst".
+lemma ordered_set_square_of_segment_square : 
+ ∀O:ordered_set.∀u,v:O.{[u,v]} squareO → O squareO ≝ 
+  λO:ordered_set.λu,v:O.λb:{[u,v]} squareO.〈\fst(\fst b),\fst(\snd b)〉.
+
+coercion ordered_set_square_of_segment_square nocomposites.
 
 lemma restriction_agreement : 
-  ∀O:ordered_uniform_space.∀l,r:O.∀P:{[l,r]} square → Prop.∀OP:O square → Prop.Prop.
-apply(λO:ordered_uniform_space.λl,r:O.λP:{[l,r]} square → Prop.λOP:O square → Prop.
-    ∀b:O square.∀H1,H2.
-      (P b → OP b) ∧ (OP b → P b));
+  ∀O:ordered_uniform_space.∀l,r:O.∀P:{[l,r]} squareO → Prop.∀OP:O squareO → Prop.Prop.
+apply(λO:ordered_uniform_space.λl,r:O.
+       λP:{[l,r]} 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.∀l,r:O.∀U,u.∀x:{[l,r]} square.
+lemma unrestrict: ∀O:ordered_uniform_space.∀l,r:O.∀U,u.∀x:{[l,r]} squareO.
   restriction_agreement ? l r U u → U x → u x.
-intros 7; cases x (b b1); cases b; cases b1; 
-cases (H 〈x1,x2〉 H1 H2) (L _); intros; apply L; assumption;
+intros 7; 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.∀l,r:O.∀U,u.∀x:{[l,r]} square.
+lemma restrict: ∀O:ordered_uniform_space.∀l,r:O.∀U,u.∀x:{[l,r]} squareO.
   restriction_agreement ? l r U u → u x → U x.
-intros 6; cases x (b b1); cases b; cases b1; intros (X); 
-cases (X 〈x1,x2〉 H H1) (_ R); apply R; assumption;
+intros 6; 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.∀l,r:O.∀U:{[l,r]} square → Prop.∀u.
+  ∀O:ordered_uniform_space.∀l,r:O.
+   ∀U:{[l,r]} squareO → Prop.∀u:O squareO → Prop.
     restriction_agreement ? l r U u →
-    restriction_agreement ? l r (inv U) (inv u).
+    restriction_agreement ? l r (\inv U) (\inv u).
 intros 9; split; intro;
-[1: apply (unrestrict ????? (segment_square_of_O_square ??? 〈snd b,fst b〉 H2 H1) H H3);
-|2: apply (restrict ????? (segment_square_of_O_square ??? 〈snd b,fst b〉 H2 H1) H H3);]
+[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 bs_of_ss: 
- ∀O:ordered_set.∀u,v:O.{[u,v]} square → (bishop_set_of_ordered_set O) square ≝ 
- λO:ordered_set.λu,v:O.λb:{[u,v]} square.〈fst(fst b),fst(snd b)〉.
 
-notation < "x \sub \neq" non associative with precedence 91 for @{'bsss $x}.
+lemma bs2_of_bss2: 
+ ∀O:ordered_set.∀u,v:O.(bishop_set_of_ordered_set {[u,v]}) squareB → (bishop_set_of_ordered_set O) squareB ≝ 
+  λO:ordered_set.λu,v:O.λb:{[u,v]} squareO.〈\fst(\fst b),\fst(\snd b)〉.
+
+coercion bs2_of_bss2 nocomposites.
+
+(*
+notation < "x \sub \neq" with precedence 91 for @{'bsss $x}.
 interpretation "bs_of_ss" 'bsss x = (bs_of_ss _ _ _ x).
+*)
+
+(*
+lemma ss_of_bs: 
+ ∀O:ordered_set.∀u,v:O.
+  ∀b:O squareO.\fst b ∈ [u,v] → \snd b ∈ [u,v] → {[u,v]} squareO ≝ 
+ λO:ordered_set.λu,v:O.
+  λb:O squareB.λH1,H2.〈≪\fst b,H1≫,≪\snd b,H2≫〉.
+*)
+
+(*
+notation < "x \sub \nleq" with precedence 91 for @{'ssbs $x}.
+interpretation "ss_of_bs" 'ssbs x = (ss_of_bs _ _ _ x _ _).
+*)
 
 lemma segment_ordered_uniform_space: 
   ∀O:ordered_uniform_space.∀u,v:O.ordered_uniform_space.
 intros (O l r); apply mk_ordered_uniform_space;
 [1: apply (mk_ordered_uniform_space_ {[l,r]});
     [1: alias symbol "and" = "constructive and".
-        letin f ≝ (λP:{[l,r]} square → Prop. ∃OP:O square → Prop.
+        letin f ≝ (λP:{[l,r]} squareO → Prop. ∃OP:O squareO → Prop.
                     (us_unifbase O OP) ∧ restriction_agreement ??? P OP);
         apply (mk_uniform_space (bishop_set_of_ordered_set {[l,r]}) f);
-        [1: intros (U H); intro x; unfold mk_set; simplify; 
+        [1: intros (U H); intro x; simplify; 
             cases H (w Hw); cases Hw (Gw Hwp); clear H Hw; intro Hm;
-            lapply (us_phi1 ?? Gw x Hm) as IH;
-            apply (restrict ?????? Hwp IH);
+            lapply (us_phi1 O w Gw x Hm) as IH;
+            apply (restrict ? l r ??? 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 ??? Gu Gv) (w HW); cases HW (Gw Hw); clear HW;
-            exists; [apply (λb:{[l,r]} square.w b)] split;
+            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;
             [1: unfold f; simplify; clearbody f;
                 exists; [apply w]; split; [assumption] intro b; simplify;
-                unfold segment_square_of_O_square; (* ??? *)
+                unfold segment_square_of_ordered_set_square;
                 cases b; intros; split; intros; assumption;
-            |2: intros 2 (x Hx); unfold mk_set; cases (Hw ? Hx); split;
-                [apply (restrict ?????? HuU H)|apply (restrict ?????? HvV H1);]]
+            |2: intros 2 (x Hx); cases (Hw ? Hx); split;
+                [apply (restrict O l r ??? HuU H)|apply (restrict O l r ??? HvV H1);]]
         |3: intros (U Hu); cases Hu (u HU); cases HU (Gu HuU); clear Hu HU;
-            cases (us_phi3 ?? Gu) (w HW); cases HW (Gw Hwu); clear HW;
-            exists; [apply (λx:{[l,r]} square.w x)] split;
+            cases (us_phi3 O u Gu) (w HW); cases HW (Gw Hwu); clear HW;
+            exists; [apply (λx:{[l,r]} squareB.w x)] split;
             [1: exists;[apply w];split;[assumption] intros; simplify; intro;
-                unfold segment_square_of_O_square; (* ??? *)
+                unfold segment_square_of_ordered_set_square;
                 cases b; intros; split; intro; assumption;
-            |2: intros 2 (x Hx); apply (restrict ?????? HuU); apply Hwu; 
-                cases Hx (m Hm); exists[apply (fst m)] apply Hm;]
+            |2: intros 2 (x Hx); apply (restrict O l r ??? 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 ?? Gu x) (Hul Hur);
+            cases (us_phi4 O u Gu x) (Hul Hur);
             split; intros; 
-            [1: apply (restrict ?????? (invert_restriction_agreement ????? HuU));
-                apply Hul; apply (unrestrict ?????? HuU H);
-            |2: apply (restrict ?????? HuU); apply Hur; 
-                apply (unrestrict ?????? (invert_restriction_agreement ????? HuU) H);]]
+            [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);]]
     |2: simplify; reflexivity;]
 |2: simplify; unfold convex; intros;
     cases H (u HU); cases HU (Gu HuU); clear HU H; 
-    lapply (ous_convex ?? Gu (bs_of_ss ? l r p) ? H2 (bs_of_ss ? l r y) H3) as Cu;
-    [1: apply (unrestrict ?????? HuU); apply H1;
-    |2: apply (restrict ?????? HuU Cu);]]
+    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);]]
 qed.
 
 interpretation "Ordered uniform space segment" 'segment_set a b = 
  (segment_ordered_uniform_space _ a b).
 
 (* Lemma 3.2 *)
-alias symbol "pi1" = "sigma pi1".
+alias symbol "pi1" = "exT \fst".
 lemma restric_uniform_convergence:
  ∀O:ordered_uniform_space.∀l,u:O.
-  ∀x:{[l,u]}.
-   ∀a:sequence {[l,u]}.
-     (λn.fst (a n)) uniform_converges (fst x) → 
+  ∀x:(segment_ordered_uniform_space O l u).
+   ∀a:sequence (segment_ordered_uniform_space O l u).
+     uniform_converge (segment_ordered_uniform_space O l u) 
+     (mk_seq O (λn:nat.\fst (a n))) (\fst x) → True. 
       a uniform_converges x.
 intros 8; cases H1; cases H2; clear H2 H1;
 cases (H ? H3) (m Hm); exists [apply m]; intros;