X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fdama%2Fdama%2Fproperty_exhaustivity.ma;h=7145b42886c802cc4d75574a77e3c64906e357ea;hb=bb7af347df386afcd3ea2adea8e7e982e3a5a253;hp=41674ec004dd3ccdff0700922112746089fa72e2;hpb=82d281529c1a9450ac213a058e7f8c0e228026fa;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 41674ec00..7145b4288 100644
--- a/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma
+++ b/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma
@@ -15,73 +15,103 @@
 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 segment_upperbound:
-  ∀C:ordered_set.∀l,u:C.∀a:sequence {[l,u]}.u is_upper_bound ⌊n,\fst (a n)⌋.
-intros 5; change with (\fst (a n) ≤ u); cases (a n); cases H; assumption;
+lemma h_segment_upperbound:
+  ∀C:half_ordered_set.
+   ∀s:segment C.
+    ∀a:sequence (half_segment_ordered_set C s).
+      upper_bound ? ⌊n,\fst (a n)⌋ (seg_u C s). 
+intros 4; simplify; cases (a n); simplify; unfold in H;
+cases (wloss_prop C); rewrite < H1 in H; simplify; cases H; 
+assumption;
 qed.
 
-lemma segment_lowerbound:
-  ∀C:ordered_set.∀l,u:C.∀a:sequence {[l,u]}.l is_lower_bound ⌊n,\fst (a n)⌋.
-intros 5; change with (l ≤ \fst (a n)); cases (a n); cases H; assumption;
-qed.
+notation "'segment_upperbound'" non associative with precedence 90 for @{'segment_upperbound}.
+notation "'segment_lowerbound'" non associative with precedence 90 for @{'segment_lowerbound}.
 
-lemma segment_preserves_uparrow:
-  ∀C:ordered_set.∀l,u:C.∀a:sequence {[l,u]}.∀x,h. 
-    ⌊n,\fst (a n)⌋ ↑ x → a ↑ ≪x,h≫.
-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;
-qed.
+interpretation "segment_upperbound" 'segment_upperbound = (h_segment_upperbound (os_l _)).
+interpretation "segment_lowerbound" 'segment_lowerbound = (h_segment_upperbound (os_r _)).
 
-lemma segment_preserves_downarrow:
-  ∀C:ordered_set.∀l,u:C.∀a:sequence {[l,u]}.∀x,h. 
-    ⌊n,\fst (a n)⌋ ↓ x → a ↓ ≪x,h≫.
-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;
+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 _)).
+ 
 (* Fact 2.18 *)
 lemma segment_cauchy:
-  ∀C:ordered_uniform_space.∀l,u:C.∀a:sequence {[l,u]}.
+  ∀C:ordered_uniform_space.∀s:‡C.∀a:sequence {[s]}.
     a is_cauchy → ⌊n,\fst (a n)⌋ is_cauchy.
-intros 7; 
+intros 6; 
 alias symbol "pi1" (instance 3) = "pair pi1".
 alias symbol "pi2" = "pair pi2".
-apply (H (λx:{[l,u]} squareB.U 〈\fst (\fst x),\fst (\snd x)〉));
+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.       
 
+(* 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_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; 
+[ apply (hle_transitive ??? x ? (H2 O)); lapply (H1 O) as K; unfold in K; rewrite <W in K;
+  cases K; unfold in H4 H6; apply H4;
+| 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; apply H5; apply 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; apply H6;
+| 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; 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:
@@ -108,24 +138,34 @@ 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.