X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fdama%2Fdama%2Fproperty_exhaustivity.ma;h=f7250f9e2e167a7a428b1d604c14638142580772;hb=6b61a9e6698a7c1936adf217b599e34e65a5e4c9;hp=173739f27853b69db8dcbe8dd89fa5e5b9b48346;hpb=7e9bcf66cf96265ea78b3de6dc95ff3c9b75f83d;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 173739f27..f7250f9e2 100644
--- a/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma
+++ b/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma
@@ -21,14 +21,87 @@ definition exhaustive ≝
    ∀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;
+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.
+
+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.
+
+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;
+qed.
+    
+(* Fact 2.18 *)
+lemma segment_cauchy:
+  ∀C:ordered_uniform_space.∀l,u:C.∀a:sequence {[l,u]}.
+    a is_cauchy → ⌊n,\fst (a n)⌋ is_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)〉));
+(unfold segment_ordered_uniform_space; simplify);
+exists [apply U] split; [assumption;]
+intro; cases b; intros; simplify; split; intros; assumption;
+qed.       
+
 (* Lemma 3.8 *)
-lemma xxx:
+lemma restrict_uniform_convergence_uparrow:
   ∀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; split;
-[1: (* manca fact 2.5 *)
+     ∀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 C ?? Hx u); 
+        apply (segment_upperbound ? l);
+    |2: apply (le_transitive ? ??? ? (H2 O));
+        apply (segment_lowerbound ?l u);]
+|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; 
+      [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.
       
-       
\ No newline at end of file
+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 C ?? Hx l); 
+        apply (segment_lowerbound ? l u);
+    |1: apply (le_transitive ???? (H2 O));
+        apply (segment_upperbound ? l u);]
+|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; 
+      [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