X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fdama%2Fdama%2Fproperty_exhaustivity.ma;h=48551f85e70ab95c2867e65d4aea9b1e93fcd919;hb=6abb01e8b00db927e16aa790354d1da57af7875b;hp=b9bb51e3b212644bfef26d714417d32e7c433978;hpb=c3b8ed2e554cbdb677729747c5b5a96112ae5169;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 b9bb51e3b..48551f85e 100644
--- a/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma
+++ b/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma
@@ -15,6 +15,52 @@
 include "ordered_uniform.ma".
 include "property_sigma.ma".
 
+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 _)).
+ 
+(* 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.       
+
 (* Definition 3.7 *)
 definition exhaustive ≝
   λC:ordered_uniform_space.
@@ -22,56 +68,111 @@ definition exhaustive ≝
      (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;
+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 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;
+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.
 
-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;
+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 →
+    ∀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 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:
+  ∀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.
     
-(* 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 hint_mah4:
+  ∀C. sequence (hos_carr (os_r C)) → sequence (hos_carr (os_l C)).
+  intros; assumption; qed.
+  
+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 3.8 *)
-lemma restrict_uniform_convergence:
+lemma restrict_uniform_convergence_downarrow:
   ∀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; 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);]
+    ∀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 (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;]
+    lapply (downarrow_lowerlocated a ≪x,h≫) as Ha1;
+      [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.
-       
\ No newline at end of file