From f71d9d74c33e4ab3d86dc8244a7bdd3576bb0ac0 Mon Sep 17 00:00:00 2001
From: Enrico Tassi <enrico.tassi@inria.fr>
Date: Sat, 25 Oct 2008 17:11:56 +0000
Subject: [PATCH] ...

---
 .../contribs/dama/dama/ordered_uniform.ma     | 79 ++++++++++++-------
 .../matita/contribs/dama/dama/supremum.ma     | 10 +--
 2 files changed, 54 insertions(+), 35 deletions(-)

diff --git a/helm/software/matita/contribs/dama/dama/ordered_uniform.ma b/helm/software/matita/contribs/dama/dama/ordered_uniform.ma
index 701651cb1..63c966056 100644
--- a/helm/software/matita/contribs/dama/dama/ordered_uniform.ma
+++ b/helm/software/matita/contribs/dama/dama/ordered_uniform.ma
@@ -49,9 +49,17 @@ 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.∀u,v:O.∀x:O squareO.
-   \fst x ∈ [u,v] → \snd x ∈ [u,v] → {[u,v]} squareO.
+  ∀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.
 
@@ -60,59 +68,70 @@ 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.∀u,v:O.{[u,v]} squareO → O squareO ≝ 
-  λO:ordered_set.λu,v:O.λb:{[u,v]} squareO.〈\fst(\fst b),\fst(\snd b)〉.
+ ∀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.∀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.
+  ∀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.∀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 (w1 H1); cases b1 (w2 H2); clear b b1 x; 
+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.∀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 (w1 H1); cases b1 (w2 H2); clear b1 b x;
+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.∀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).
-intros 9; 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);]
+  ∀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.∀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)〉.
+ ∀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 xxx :
+ ∀O,s,x.bs2_of_bss2 (ordered_set_OF_ordered_uniform_space O) s x
+              =
+              x.
+intros; reflexivity;
+*)
+
 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]});
+  ∀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:{[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);
+        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 Hm) as IH;
-            apply (restrict ? l r ??? Hwp IH); 
+            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);]
+
+            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;
diff --git a/helm/software/matita/contribs/dama/dama/supremum.ma b/helm/software/matita/contribs/dama/dama/supremum.ma
index 7a52e5f06..54e9d98f0 100644
--- a/helm/software/matita/contribs/dama/dama/supremum.ma
+++ b/helm/software/matita/contribs/dama/dama/supremum.ma
@@ -350,15 +350,15 @@ notation "'segment_preserves_infimum'" non associative with precedence 90 for @{
 interpretation "segment_preserves_supremum" 'segment_preserves_supremum = (h_segment_preserves_supremum (os_l _)).
 interpretation "segment_preserves_infimum" 'segment_preserves_infimum = (h_segment_preserves_supremum (os_r _)).
 
-(* TEST, ma quanto godo! *)
-lemma segment_preserves_infimum2:
+(*
+test segment_preserves_infimum2:
   ∀O:ordered_set.∀s:‡O.∀a:sequence {[s]}.∀x:{[s]}. 
     ⌊n,\fst (a n)⌋ is_decreasing ∧ 
     (\fst x) is_infimum ⌊n,\fst (a n)⌋ → a ↓ x.
 intros; apply (segment_preserves_infimum s a x H);
 qed.
 *)
-           
+       
 (* Definition 2.10 *)
 alias symbol "pi2" = "pair pi2".
 alias symbol "pi1" = "pair pi1".
@@ -394,8 +394,8 @@ interpretation "Ordered set lower locatedness" 'lower_located s =
 lemma h_uparrow_upperlocated:
   ∀C:half_ordered_set.∀a:sequence C.∀u:C.uparrow ? a u → upper_located ? a.
 intros (C a u H); cases H (H2 H3); clear H; intros 3 (x y Hxy);
-cases H3 (H4 H5); clear H3; cases (hos_cotransitive ??? u Hxy) (W W);
-[2: cases (H5 ? W) (w Hw); left; exists [apply w] assumption;
+cases H3 (H4 H5); clear H3; cases (hos_cotransitive C y x u Hxy) (W W);
+[2: cases (H5 x W) (w Hw); left; exists [apply w] assumption;
 |1: right; exists [apply u]; split; [apply W|apply H4]]
 qed.
 
-- 
2.39.2