From 3f80c07b790d3abdb3aafcd1da33f14fa90ada25 Mon Sep 17 00:00:00 2001
From: Enrico Tassi <enrico.tassi@inria.fr>
Date: Mon, 27 Oct 2008 20:16:34 +0000
Subject: [PATCH] WIP

---
 .../contribs/dama/dama/ordered_uniform.ma     |  4 +-
 .../dama/dama/property_exhaustivity.ma        | 84 ++++++++++++++-----
 .../matita/contribs/dama/dama/supremum.ma     |  4 +-
 3 files changed, 65 insertions(+), 27 deletions(-)

diff --git a/helm/software/matita/contribs/dama/dama/ordered_uniform.ma b/helm/software/matita/contribs/dama/dama/ordered_uniform.ma
index 1dba15a7c..36ba2d287 100644
--- a/helm/software/matita/contribs/dama/dama/ordered_uniform.ma
+++ b/helm/software/matita/contribs/dama/dama/ordered_uniform.ma
@@ -202,7 +202,7 @@ intros (O s); apply mk_ordered_uniform_space;
 ]  
 qed.
 
-interpretation "Ordered uniform space segment" 'segment_set a = 
+interpretation "Ordered uniform space segment" 'segset a = 
  (segment_ordered_uniform_space _ a).
 
 (* Lemma 3.2 *)
@@ -210,7 +210,7 @@ alias symbol "pi1" = "exT \fst".
 lemma restric_uniform_convergence:
  ∀O:ordered_uniform_space.∀s:‡O.
   ∀x:{[s]}.
-   ∀a:sequence (segment_ordered_uniform_space O s).
+   ∀a:sequence {[s]}.
     (⌊n, \fst (a n)⌋ : sequence O) uniform_converges (\fst x) → 
       a uniform_converges x.
 intros 7; cases H1; cases H2; clear H2 H1;
diff --git a/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma b/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma
index 713588e70..fc7121349 100644
--- a/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma
+++ b/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma
@@ -15,13 +15,6 @@
 include "ordered_uniform.ma".
 include "property_sigma.ma".
 
-(* Definition 3.7 *)
-definition exhaustive ≝
-  λC:ordered_uniform_space.
-   ∀a,b:sequence C.
-     (a is_increasing → a is_upper_located → a is_cauchy) ∧
-     (b is_decreasing → b is_lower_located → b is_cauchy).
-
 lemma h_segment_upperbound:
   ∀C:half_ordered_set.
    ∀s:segment C.
@@ -54,18 +47,6 @@ notation "'segment_preserves_downarrow'" non associative with precedence 90 for
 
 interpretation "segment_preserves_uparrow" 'segment_preserves_uparrow = (h_segment_preserves_uparrow (os_l _)).
 interpretation "segment_preserves_downarrow" 'segment_preserves_downarrow = (h_segment_preserves_uparrow (os_r _)).
-
-lemma hint_pippo:
- ∀C,s.
-  sequence
-   (Type_of_ordered_set
-    (segment_ordered_set
-     (ordered_set_OF_ordered_uniform_space C) s))
- →  
- sequence (Type_OF_uniform_space (segment_ordered_uniform_space C s)). intros; assumption;
-qed. 
-
-coercion hint_pippo nocomposites.
  
 (* Fact 2.18 *)
 lemma segment_cauchy:
@@ -80,14 +61,71 @@ exists [apply U] split; [assumption;]
 intro; cases b; intros; simplify; split; intros; assumption;
 qed.       
 
+(* Definition 3.7 *)
+definition exhaustive ≝
+  λC:ordered_uniform_space.
+   ∀a,b:sequence C.
+     (a is_increasing → a is_upper_located → a is_cauchy) ∧
+     (b is_decreasing → b is_lower_located → b is_cauchy).
+
+lemma prove_in_segment: 
+ ∀O:ordered_set.∀s:segment (os_l O).∀x:O.
+   𝕝_s (λl.l ≤ x) → 𝕦_s (λu.x ≤ u) → x ∈ s.
+intros; unfold; cases (wloss_prop (os_l O)); rewrite < H2;
+split; assumption;
+qed.
+
+lemma under_wloss_upperbound: 
+ ∀C:half_ordered_set.∀s:segment C.∀a:sequence C.
+  seg_u C s (upper_bound C a) → 
+    ∀i.seg_u C s (λu.a i ≤≤ u).
+intros; unfold in H; unfold;
+cases (wloss_prop C); rewrite <H1 in H ⊢ %;
+apply (H i);
+qed. 
+
+
 (* Lemma 3.8 NON DUALIZZATO *)
 lemma restrict_uniform_convergence_uparrow:
   ∀C:ordered_uniform_space.property_sigma C →
-    ∀l,u:C.exhaustive {[l,u]} →
-     ∀a:sequence {[l,u]}.∀x:C. ⌊n,\fst (a n)⌋ ↑ x → 
-      x∈[l,u] ∧ ∀h:x ∈ [l,u].a uniform_converges ≪x,h≫.
+    ∀s:segment (os_l C).exhaustive (segment_ordered_uniform_space C s) →
+     ∀a:sequence (segment_ordered_uniform_space C s).
+      ∀x:C. ⌊n,\fst (a n)⌋ ↑ x → 
+       in_segment (os_l C) s x ∧ ∀h:x ∈ s.a uniform_converges ≪x,h≫.
 intros; cases H2 (Ha Hx); clear H2; cases Hx; split;
-[1: split;
+[1: apply prove_in_segment; 
+    lapply depth=0 (under_wloss_upperbound (os_l C) ?? (segment_upperbound s a) O) as W1;
+    lapply depth=0 (under_wloss_upperbound (os_r C) ?? (h_segment_upperbound (os_r C) s a) O) as W2;
+    lapply (H2 O); simplify in Hletin; simplify in W2 W1;
+    cases a in Hletin W2 W1; simplify; cases (f O); simplify; intros;
+    whd in H6:(? % ? ? ? ?);
+    simplify in H6:(%);
+    cases (wloss_prop (os_l C)); rewrite <H8 in H5 H6 ⊢ %;
+    [ change in H6 with (le (os_l C) (seg_l_ (os_l C) s) w);
+      apply (le_transitive ??? H6 H7);
+    | apply (le_transitive (seg_u_ (os_l C) s) w x H6 H7);
+    |  
+      lapply depth=0 (supremum_is_upper_bound ? x Hx (seg_u_ (os_l C) s)) as K;    
+      lapply depth=0 (under_wloss_upperbound (os_l C) ?? (segment_upperbound s a));
+      apply K; intro; lapply (Hletin n); unfold; unfold in Hletin1;
+      rewrite < H8 in Hletin1; intro; apply Hletin1; clear Hletin1; apply H9;
+    | lapply depth=0 (h_supremum_is_upper_bound (os_r C) ⌊n,\fst (a n)⌋ x Hx (seg_l_ (os_r C) s)) as K;    
+      lapply depth=0 (under_wloss_upperbound (os_r C) ?? (h_segment_upperbound (os_r C) s a));
+      apply K; intro; lapply (Hletin n); unfold; unfold in Hletin1;
+whd in Hletin1:(? % ? ? ? ?);
+simplify in Hletin1:(%);
+      rewrite < H8 in Hletin1; intro; apply Hletin1; clear Hletin1; apply H9;
+  
+      
+        apply (segment_upperbound ? l);
+    generalize in match (H2 O); generalize in match Hx; unfold supremum;
+    unfold upper_bound; whd in ⊢ (?→%→?); rewrite < H4;
+    split; unfold; rewrite < H4; simplify;
+      [1: lapply (infimum_is_lower_bound ? ? Hx u); 
+
+
+
+split;
     [1: apply (supremum_is_upper_bound ? x Hx u); 
         apply (segment_upperbound ? l);
     |2: apply (le_transitive l ? x ? (H2 O));
diff --git a/helm/software/matita/contribs/dama/dama/supremum.ma b/helm/software/matita/contribs/dama/dama/supremum.ma
index 8ee17fc7d..bad35177b 100644
--- a/helm/software/matita/contribs/dama/dama/supremum.ma
+++ b/helm/software/matita/contribs/dama/dama/supremum.ma
@@ -209,10 +209,10 @@ notation "𝕝 \sub term 90 s p" non associative with precedence 45 for @{'low $
  
 definition seg_u ≝
  λO:half_ordered_set.λs:segment O.λP: O → CProp.
-   wloss O ? (λl,u.P u) (seg_u_ ? s) (seg_l_ ? s).
+   wloss O ? (λl,u.P l) (seg_u_ ? s) (seg_l_ ? s).
 definition seg_l ≝
  λO:half_ordered_set.λs:segment O.λP: O → CProp.
-   wloss O ? (λl,u.P u) (seg_l_ ? s) (seg_u_ ? s). 
+   wloss O ? (λl,u.P l) (seg_l_ ? s) (seg_u_ ? s). 
  
 interpretation "uppper" 'upp s P = (seg_u (os_l _) s P).
 interpretation "lower" 'low s P = (seg_l (os_l _) s P).
-- 
2.39.2