From c3b8ed2e554cbdb677729747c5b5a96112ae5169 Mon Sep 17 00:00:00 2001
From: Enrico Tassi <enrico.tassi@inria.fr>
Date: Mon, 9 Jun 2008 10:24:09 +0000
Subject: [PATCH] exhaustivity completed

---
 .../contribs/dama/dama/ordered_uniform.ma     |  1 -
 .../dama/dama/property_exhaustivity.ma        | 53 +++++++++++++++++--
 .../matita/contribs/dama/dama/uniform.ma      |  2 -
 3 files changed, 48 insertions(+), 8 deletions(-)

diff --git a/helm/software/matita/contribs/dama/dama/ordered_uniform.ma b/helm/software/matita/contribs/dama/dama/ordered_uniform.ma
index 63f263d27..3bdfbe978 100644
--- a/helm/software/matita/contribs/dama/dama/ordered_uniform.ma
+++ b/helm/software/matita/contribs/dama/dama/ordered_uniform.ma
@@ -153,4 +153,3 @@ qed.
 definition order_continuity ≝
   λC:ordered_uniform_space.∀a:sequence C.∀x:C.
     (a ↑ x → a uniform_converges x) ∧ (a ↓ x → a uniform_converges x).
-
diff --git a/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma b/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma
index 173739f27..b9bb51e3b 100644
--- a/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma
+++ b/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma
@@ -21,14 +21,57 @@ definition exhaustive ≝
    ∀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 segment_upperbound:
+  ∀C:ordered_set.∀l,u:C.∀a:sequence {[l,u]}.u is_upper_bound (λn.fst (a n)).
+intros 5; change with (fst (a n) ≤ u); cases (a n); cases H; assumption;
+qed.
+
+lemma segment_lowerbound:
+  ∀C:ordered_set.∀l,u:C.∀a:sequence {[l,u]}.l is_lower_bound (λn.fst (a n)).
+intros 5; change with (l ≤ fst (a n)); cases (a n); cases H; assumption;
+qed.
+
+lemma segment_preserves_uparrow:
+  ∀C:ordered_set.∀l,u:C.∀a:sequence {[l,u]}.∀x,h. 
+    (λn.fst (a n)) ↑ x → a ↑ (sig_in ?? x h).
+intros; cases H (Ha Hx); split [apply Ha] cases Hx; 
+split; [apply H1] intros;
+cases (H2 (fst y) H3); exists [apply w] assumption;
+qed.
+    
+(* Fact 2.18 *)
+lemma segment_cauchy:
+  ∀C:ordered_uniform_space.∀l,u:C.∀a:sequence {[l,u]}.
+    a is_cauchy → (λn:nat.fst (a n)) is_cauchy.
+intros 7; 
+alias symbol "pi1" (instance 3) = "pair pi1".
+apply (H (λx:{[l,u]} square.U 〈fst (fst x),fst (snd x)〉));
+(unfold segment_ordered_uniform_space; simplify);
+exists [apply U] split; [assumption;]
+intro; cases b; intros; simplify; split; intros; assumption;
+qed.       
+
 (* Lemma 3.8 *)
-lemma xxx:
+lemma restrict_uniform_convergence:
   ∀C:ordered_uniform_space.property_sigma C →
     ∀l,u:C.exhaustive {[l,u]} →
      ∀a:sequence {[l,u]}.∀x:C. (λn.fst (a n)) ↑ x → 
       x∈[l,u] ∧ ∀h:x ∈ [l,u].a uniform_converges (sig_in ?? x h).
-intros; split;
-[1: (* manca fact 2.5 *)
-      
+intros; cases H2 (Ha Hx); clear H2; cases Hx; split;
+[1: split;
+    [1: apply (supremum_is_upper_bound C ?? Hx u); 
+        apply (segment_upperbound ? l);
+    |2: apply (le_transitive ?? (fst (a 0))); [2: apply H2;]
+        apply (segment_lowerbound ?l u);]
+|2: intros;
+    lapply (uparrow_upperlocated ? a (sig_in ?? x h)) as Ha1;
+      [2: apply segment_preserves_uparrow;split; assumption;] 
+    lapply (segment_preserves_supremum ?l u a (sig_in ??? h)) as Ha2; 
+      [2:split; assumption]; cases Ha2; clear Ha2;
+    cases (H1 a a); lapply (H6 H4 Ha1) as HaC;
+    lapply (segment_cauchy ? l u ? HaC) as Ha;
+    lapply (sigma_cauchy ? H  ? x ? Ha); [split; assumption]
+    apply restric_uniform_convergence; assumption;]
+qed.
        
\ No newline at end of file
diff --git a/helm/software/matita/contribs/dama/dama/uniform.ma b/helm/software/matita/contribs/dama/dama/uniform.ma
index d5d714726..41b593ae4 100644
--- a/helm/software/matita/contribs/dama/dama/uniform.ma
+++ b/helm/software/matita/contribs/dama/dama/uniform.ma
@@ -139,5 +139,3 @@ interpretation "explicit big set" 'explicitset t =
 definition restrict_uniformity ≝
   λC:uniform_space.λX:C→Prop.
    {U:C square → Prop| (U ⊆ {x:C square|X (fst x) ∧ X(snd x)}) ∧ us_unifbase C U}.
-
-
-- 
2.39.2