X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fdama%2Fdama%2Fproperty_exhaustivity.ma;h=48551f85e70ab95c2867e65d4aea9b1e93fcd919;hb=f20f1ac4aea15d81599bd2283c5440fce8d4cf6a;hp=713588e7066f0feebcccdbe62f2cedc7179804bd;hpb=62571dd402d272b1632b7739607d25df3552cc04;p=helm.git

diff --git a/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma b/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma
index 713588e70..48551f85e 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:
@@ -80,27 +61,64 @@ 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:half_ordered_set.∀s:segment O.∀x:O.
+   seg_l O s (λl.l ≤≤ x) → seg_u O s (λu.x ≤≤ u) → x ∈ s.
+intros; unfold; cases (wloss_prop O); rewrite < H2;
+split; assumption;
+qed.
+
+lemma h_uparrow_to_in_segment:
+  ∀C:half_ordered_set.
+   ∀s:segment C.
+     ∀a:sequence C.
+      (∀i.a i ∈ s) →
+       ∀x:C. uparrow C a x → 
+         in_segment C s x.
+intros (C H a H1 x H2); unfold in H2; cases H2; clear H2;unfold in H3 H4; cases H4; clear H4; unfold in H2;
+cases (wloss_prop C) (W W); apply prove_in_segment; unfold; rewrite <W;simplify;
+[ apply (hle_transitive ??? x ? (H2 O)); lapply (H1 O) as K; unfold in K; rewrite <W in K;
+  cases K; unfold in H4 H6; rewrite <W in H6 H4; simplify in H4 H6; assumption;
+| intro; cases (H5 ? H4); clear H5 H4;lapply(H1 w) as K; unfold in K; rewrite<W in K;
+  cases K; unfold in H5 H4; rewrite<W in H4 H5; simplify in H4 H5; apply (H5 H6);    
+| apply (hle_transitive ??? x ?  (H2 O)); lapply (H1 0) as K; unfold in K; rewrite <W in K;
+  cases K; unfold in H4 H6; rewrite <W in H4 H6; simplify in H4 H6; assumption;
+| intro; cases (H5 ? H4); clear H5 H4;lapply(H1 w) as K; unfold in K; rewrite<W in K;
+  cases K; unfold in H5 H4; rewrite<W in H4 H5; simplify in H4 H5; apply (H4 H6);]    
+qed.
+
+notation "'uparrow_to_in_segment'" non associative with precedence 90 for @{'uparrow_to_in_segment}.
+notation "'downarrow_to_in_segment'" non associative with precedence 90 for @{'downarrow_to_in_segment}.
+
+interpretation "uparrow_to_in_segment" 'uparrow_to_in_segment = (h_uparrow_to_in_segment (os_l _)).
+interpretation "downarrow_to_in_segment" 'downarrow_to_in_segment = (h_uparrow_to_in_segment (os_r _)).
+ 
 (* 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≫.
-intros; cases H2 (Ha Hx); clear H2; cases Hx; split;
-[1: split;
-    [1: apply (supremum_is_upper_bound ? x Hx u); 
-        apply (segment_upperbound ? l);
-    |2: apply (le_transitive l ? x ? (H2 O));
-        apply (segment_lowerbound ? l u a 0);]
+    ∀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; split;
+[1: apply (uparrow_to_in_segment s ⌊n,\fst (a \sub n)⌋ ? x H2); 
+    simplify; intros; cases (a i); assumption;
 |2: intros;
     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;]
+      [2: apply (segment_preserves_uparrow s); assumption;] 
+    lapply (segment_preserves_supremum s a ≪?,h≫ H2) as Ha2; 
+    cases Ha2; clear Ha2;
+    cases (H1 a a); lapply (H5 H3 Ha1) as HaC;
+    lapply (segment_cauchy C s ? HaC) as Ha;
+    lapply (sigma_cauchy ? H  ? x ? Ha); [left; assumption]
+    apply (restric_uniform_convergence C s ≪x,h≫ a Hletin)]
 qed.
 
 lemma hint_mah1:
@@ -127,24 +145,34 @@ lemma hint_mah4:
   
 coercion hint_mah4 nocomposites.
 
+lemma hint_mah5:
+  ∀C. segment (hos_carr (os_r C)) → segment (hos_carr (os_l C)).
+  intros; assumption; qed.
+  
+coercion hint_mah5 nocomposites.
+
+lemma hint_mah6:
+  ∀C. segment (hos_carr (os_l C)) → segment (hos_carr (os_r C)).
+  intros; assumption; qed.
+
+coercion hint_mah6 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 → 
-      x∈[l,u] ∧ ∀h:x ∈ [l,u].a uniform_converges ≪x,h≫.
-intros; cases H2 (Ha Hx); clear H2; cases Hx; split;
-[1: split;
-    [2: apply (infimum_is_lower_bound ? x Hx l); 
-        apply (segment_lowerbound ? l u);
-    |1: lapply (ge_transitive ? ? x ? (H2 O)); [apply u||assumption]
-        apply (segment_upperbound ? l u a 0);]
+    ∀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; split;       
+[1: apply (downarrow_to_in_segment s ⌊n,\fst (a n)⌋ ? x); [2: apply H2]; 
+    simplify; intros; cases (a i); assumption;
 |2: intros;
     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;]
+      [2: apply (segment_preserves_downarrow s a x h H2);] 
+    lapply (segment_preserves_infimum s a ≪?,h≫ H2) as Ha2; 
+    cases Ha2; clear Ha2;
+    cases (H1 a a); lapply (H6 H3 Ha1) as HaC;
+    lapply (segment_cauchy C s ? HaC) as Ha;
+    lapply (sigma_cauchy ? H  ? x ? Ha); [right; assumption]
+    apply (restric_uniform_convergence C s ≪x,h≫ a Hletin)]
 qed.