From 858e703335b47065529d481891863d380a2156d7 Mon Sep 17 00:00:00 2001
From: Enrico Tassi <enrico.tassi@inria.fr>
Date: Mon, 27 Oct 2008 08:46:46 +0000
Subject: [PATCH] big lemma done

---
 .../contribs/dama/dama/ordered_uniform.ma     | 59 +++++++------------
 .../matita/contribs/dama/dama/supremum.ma     | 15 +++--
 2 files changed, 30 insertions(+), 44 deletions(-)

diff --git a/helm/software/matita/contribs/dama/dama/ordered_uniform.ma b/helm/software/matita/contribs/dama/dama/ordered_uniform.ma
index 7b93454ef..1f4f6f686 100644
--- a/helm/software/matita/contribs/dama/dama/ordered_uniform.ma
+++ b/helm/software/matita/contribs/dama/dama/ordered_uniform.ma
@@ -34,7 +34,7 @@ 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: ∀U.us_unifbase ous_stuff U → convex (os_l ous_stuff) U
 }.   
 
 definition half_ordered_set_OF_ordered_uniform_space : ordered_uniform_space → half_ordered_set.
@@ -167,41 +167,22 @@ intros (O s); apply mk_ordered_uniform_space;
             |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 s ??? HuU); apply H1;
-    |2: apply (restrict O s ??? 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 ?? 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);]]  
 qed.
 
 interpretation "Ordered uniform space segment" 'segment_set a b = 
@@ -210,9 +191,9 @@ interpretation "Ordered uniform space segment" 'segment_set a b =
 (* 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;
diff --git a/helm/software/matita/contribs/dama/dama/supremum.ma b/helm/software/matita/contribs/dama/dama/supremum.ma
index 54e9d98f0..7bb684fd8 100644
--- a/helm/software/matita/contribs/dama/dama/supremum.ma
+++ b/helm/software/matita/contribs/dama/dama/supremum.ma
@@ -363,13 +363,18 @@ qed.
 alias symbol "pi2" = "pair pi2".
 alias symbol "pi1" = "pair pi1".
 definition square_segment ≝ 
-  λO:ordered_set.λa,b:O.λx: O squareO.
-    And4 (\fst x ≤ b) (a ≤ \fst x) (\snd x ≤ b) (a ≤ \snd x).
+  λO:half_ordered_set.λs:segment O.λx: square_half_ordered_set O.
+    in_segment ? s (\fst x) ∧ in_segment ? s (\snd x).
  
 definition convex ≝
-  λO:ordered_set.λU:O squareO → Prop.
-    ∀p.U p → \fst p ≤ \snd p → ∀y. 
-      square_segment O (\fst p) (\snd p) y → U y.
+  λO:half_ordered_set.λU:square_half_ordered_set O → Prop.
+    ∀s.U s → le O (\fst s) (\snd s) → 
+     ∀y. 
+       le O (\fst y) (\snd s) → 
+       le O (\fst s) (\fst y) →
+       le O (\snd y) (\snd s) →
+       le O (\fst y) (\snd y) →
+       U y.
   
 (* Definition 2.11 *)  
 definition upper_located ≝
-- 
2.39.2