lebesgue completely dualized

[helm.git] / helm / software / matita / contribs / dama / dama / property_exhaustivity.ma

diff --git a/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma b/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma
index d04ec1acca886c457ae4370f3a5fcdc722120a9d..41674ec004dd3ccdff0700922112746089fa72e2 100644 (file)
--- a/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma
+++ b/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma
@@ -34,7 +34,7 @@ qed.
 
 lemma segment_preserves_uparrow:
   ∀C:ordered_set.∀l,u:C.∀a:sequence {[l,u]}.∀x,h. 
-    â\8c\8an,\fst (a n)â\8c\8b â\86\91 x â\86\92 a â\86\91 â\8c©x,hâ\8cª.
+    â\8c\8an,\fst (a n)â\8c\8b â\86\91 x â\86\92 a â\86\91 â\89ªx,hâ\89«.
 intros; cases H (Ha Hx); split [apply Ha] cases Hx; 
 split; [apply H1] intros;
 cases (H2 (\fst y)); [2: apply H3;] exists [apply w] assumption;
@@ -42,7 +42,7 @@ qed.
 
 lemma segment_preserves_downarrow:
   ∀C:ordered_set.∀l,u:C.∀a:sequence {[l,u]}.∀x,h. 
-    â\8c\8an,\fst (a n)â\8c\8b â\86\93 x â\86\92 a â\86\93 â\8c©x,hâ\8cª.
+    â\8c\8an,\fst (a n)â\8c\8b â\86\93 x â\86\92 a â\86\93 â\89ªx,hâ\89«.
 intros; cases H (Ha Hx); split [apply Ha] cases Hx; 
 split; [apply H1] intros;
 cases (H2 (\fst y));[2:apply H3]; exists [apply w] assumption;
@@ -55,53 +55,77 @@ lemma segment_cauchy:
 intros 7; 
 alias symbol "pi1" (instance 3) = "pair pi1".
 alias symbol "pi2" = "pair pi2".
-apply (H (λx:{[l,u]} square.U 〈\fst (\fst x),\fst (\snd x)〉));
+apply (H (λx:{[l,u]} squareB.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 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 → 
-      xâ\88\88[l,u] â\88§ â\88\80h:x â\88\88 [l,u].a uniform_converges â\8c©x,hâ\8cª.
+      xâ\88\88[l,u] â\88§ â\88\80h:x â\88\88 [l,u].a uniform_converges â\89ªx,hâ\89«.
 intros; cases H2 (Ha Hx); clear H2; cases Hx; split;
 [1: split;
-    [1: apply (supremum_is_upper_bound C ?? Hx u); 
+    [1: apply (supremum_is_upper_bound ? x Hx u); 
         apply (segment_upperbound ? l);
-    |2: apply (le_transitive ? ??? ? (H2 O));
-        apply (segment_lowerbound ?l u);]
+    |2: apply (le_transitive l ? x ? (H2 O));
+        apply (segment_lowerbound ? l u a 0);]
 |2: intros;
-    lapply (uparrow_upperlocated ? a 〈x,h〉) as Ha1;
-      [2: apply segment_preserves_uparrow;split; assumption;] 
-    lapply (segment_preserves_supremum ? l u a 〈?,h〉) as Ha2; 
+    lapply (uparrow_upperlocated a ≪x,h≫) as Ha1;
+      [2: apply (segment_preserves_uparrow C l u);split; assumption;] 
+    lapply (segment_preserves_supremum C l u a ≪?,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); [left; split; assumption]
     apply restric_uniform_convergence; assumption;]
 qed.
-      
+
+lemma hint_mah1:
+  ∀C. Type_OF_ordered_uniform_space1 C → hos_carr (os_r C).
+  intros; assumption; qed.
+  
+coercion hint_mah1 nocomposites.
+
+lemma hint_mah2:
+  ∀C. sequence (hos_carr (os_l C)) → sequence (hos_carr (os_r C)).
+  intros; assumption; qed.
+  
+coercion hint_mah2 nocomposites.
+
+lemma hint_mah3:
+  ∀C. Type_OF_ordered_uniform_space C → hos_carr (os_r C).
+  intros; assumption; qed.
+  
+coercion hint_mah3 nocomposites.
+    
+lemma hint_mah4:
+  ∀C. sequence (hos_carr (os_r C)) → sequence (hos_carr (os_l C)).
+  intros; assumption; qed.
+  
+coercion hint_mah4 nocomposites.
+
 lemma restrict_uniform_convergence_downarrow:
   ∀C:ordered_uniform_space.property_sigma C →
     ∀l,u:C.exhaustive {[l,u]} →
-     ∀a:sequence {[l,u]}.∀x:C. ⌊n,\fst (a n)⌋ ↓ x → 
-      xâ\88\88[l,u] â\88§ â\88\80h:x â\88\88 [l,u].a uniform_converges â\8c©x,hâ\8cª.
+     ∀a:sequence {[l,u]}.∀x: C. ⌊n,\fst (a n)⌋ ↓ x → 
+      xâ\88\88[l,u] â\88§ â\88\80h:x â\88\88 [l,u].a uniform_converges â\89ªx,hâ\89«.
 intros; cases H2 (Ha Hx); clear H2; cases Hx; split;
 [1: split;
-    [2: apply (infimum_is_lower_bound C ?? Hx l); 
+    [2: apply (infimum_is_lower_bound ? x Hx l); 
         apply (segment_lowerbound ? l u);
-    |1: apply (le_transitive ???? (H2 O));
-        apply (segment_upperbound ? l u);]
+    |1: lapply (ge_transitive ? ? x ? (H2 O)); [apply u||assumption]
+        apply (segment_upperbound ? l u a 0);]
 |2: intros;
-    lapply (downarrow_lowerlocated ? a 〈x,h〉) as Ha1;
-      [2: apply segment_preserves_downarrow;split; assumption;] 
-    lapply (segment_preserves_infimum ?l u a 〈?,h〉) as Ha2; 
+    lapply (downarrow_lowerlocated a ≪x,h≫) as Ha1;
+      [2: apply (segment_preserves_downarrow ? l u);split; assumption;]
+          lapply (segment_preserves_infimum C l u a ≪x,h≫) as Ha2; 
       [2:split; assumption]; cases Ha2; clear Ha2;
     cases (H1 a a); lapply (H7 H4 Ha1) as HaC;
     lapply (segment_cauchy ? l u ? HaC) as Ha;
     lapply (sigma_cauchy ? H  ? x ? Ha); [right; split; assumption]
     apply restric_uniform_convergence; assumption;]
-qed.
\ No newline at end of file
+qed.