From bde2f037924b8854d5ed4e6b133c306156a1fcf5 Mon Sep 17 00:00:00 2001
From: Enrico Tassi <enrico.tassi@inria.fr>
Date: Sun, 26 Oct 2008 15:54:40 +0000
Subject: [PATCH] ...

---
 .../contribs/dama/dama/ordered_uniform.ma     | 82 ++++++++++++++-----
 1 file changed, 60 insertions(+), 22 deletions(-)

diff --git a/helm/software/matita/contribs/dama/dama/ordered_uniform.ma b/helm/software/matita/contribs/dama/dama/ordered_uniform.ma
index 63c966056..7b93454ef 100644
--- a/helm/software/matita/contribs/dama/dama/ordered_uniform.ma
+++ b/helm/software/matita/contribs/dama/dama/ordered_uniform.ma
@@ -109,13 +109,24 @@ 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.
@@ -127,43 +138,70 @@ 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; 
+    
+lemma ls2l: 
+  ∀O:ordered_set.∀s:‡O.∀x,y:os_l (square_ordered_set (segment_ordered_set O s)).
+    le (os_l (square_ordered_set (segment_ordered_set O s))) x y → 
+    \fst x ≤ \fst y.
+intros 4; cases x (a1 a2); cases y (b1 b2); clear x y; 
+intros 2 (H K); apply H; clear H;
+simplify in K:(? ? ? %);
+simplify in K:(? ? % ?);
+whd in ⊢ (? (? (% ?)) ? ?);
+whd in ⊢ (? (? ((λ_:?.? ? ? (? ? (? (% ?)))) ?)) ? ?);
+simplify;
+whd in K:(% ? ? ?);
+simplify in K:(%);
+whd in ⊢ (? (% ?) ? ? ? ?);
+simplify in ⊢ (? (% ?) ? ? ? ?);
+simplify in ⊢ (? % ? ? ? ?);
+simplify in ⊢ (%);
+cases (wloss_prop (os_l (segment_ordered_set O s)));
+rewrite <H in K ⊢ %;
+whd in ⊢ (? % ? ?);
+simplify in ⊢ (%); 
+unfold hos_excess; do 2 rewrite <H; 
+
+SQUARE SEEMS NOT DUAL!
+
+cases a1 (a _); cases a2 (b _);
+cases b1 (c _); cases b2 (d _); clear a1 a2 b1 b2; simplify;
+intros 2 (H K); apply H; clear H; apply K;           
+    
     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);]]
+    [1: apply (unrestrict O s ??? HuU); apply H1;
+    |2: apply (restrict O s ??? HuU Cu);]]
 qed.
 
 interpretation "Ordered uniform space segment" 'segment_set a b = 
-- 
2.39.2