X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fdama%2Fdama%2Fproperty_exhaustivity.ma;h=713588e7066f0feebcccdbe62f2cedc7179804bd;hb=62571dd402d272b1632b7739607d25df3552cc04;hp=00f8cb25917339cc960f25c510f5b5a4a00b31c8;hpb=b284579a0c4d45bc8483f295434a465ca685f444;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 00f8cb259..713588e70 100644
--- a/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma
+++ b/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma
@@ -13,12 +13,138 @@
 (**************************************************************************)
 
 include "ordered_uniform.ma".
+include "property_sigma.ma".
 
 (* Definition 3.7 *)
-definition exhaustivity ≝
+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 3.8 *)
+
+lemma h_segment_upperbound:
+  ∀C:half_ordered_set.
+   ∀s:segment C.
+    ∀a:sequence (half_segment_ordered_set C s).
+     (seg_u C s) (upper_bound ? ⌊n,\fst (a n)⌋). 
+intros; cases (wloss_prop C); unfold; rewrite < H; simplify; intro n;
+cases (a n); simplify; unfold in H1; rewrite < H in H1; cases H1; 
+simplify in H2 H3; rewrite < H in H2 H3; assumption;
+qed.
+
+notation "'segment_upperbound'" non associative with precedence 90 for @{'segment_upperbound}.
+notation "'segment_lowerbound'" non associative with precedence 90 for @{'segment_lowerbound}.
+
+interpretation "segment_upperbound" 'segment_upperbound = (h_segment_upperbound (os_l _)).
+interpretation "segment_lowerbound" 'segment_lowerbound = (h_segment_upperbound (os_r _)).
+
+lemma h_segment_preserves_uparrow:
+  ∀C:half_ordered_set.∀s:segment C.∀a:sequence (half_segment_ordered_set C s).
+   ∀x,h. uparrow C ⌊n,\fst (a n)⌋ x → uparrow (half_segment_ordered_set C s) a ≪x,h≫.
+intros; cases H (Ha Hx); split;
+[ intro n; intro H; apply (Ha n); apply (sx2x ???? H);
+| cases Hx; split; 
+  [ intro n; intro H; apply (H1 n);apply (sx2x ???? H); 
+  | intros; cases (H2 (\fst y)); [2: apply (sx2x ???? H3);] 
+    exists [apply w] apply (x2sx ?? (a w) y H4);]]
+qed.
+
+notation "'segment_preserves_uparrow'" non associative with precedence 90 for @{'segment_preserves_uparrow}.
+notation "'segment_preserves_downarrow'" non associative with precedence 90 for @{'segment_preserves_downarrow}.
+
+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:
+  ∀C:ordered_uniform_space.∀s:‡C.∀a:sequence {[s]}.
+    a is_cauchy → ⌊n,\fst (a n)⌋ is_cauchy.
+intros 6; 
+alias symbol "pi1" (instance 3) = "pair pi1".
+alias symbol "pi2" = "pair pi2".
+apply (H (λx:{[s]} 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 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);]
+|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;]
+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 → 
+      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);]
+|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;]
+qed.