removed <_,_> notation second interpretation for dependent pair, since it used

[helm.git] / helm / software / matita / contribs / dama / dama / property_exhaustivity.ma

diff --git a/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma b/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma
index b9bb51e3b212644bfef26d714417d32e7c433978..f7250f9e2e167a7a428b1d604c14638142580772 100644 (file)
--- a/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma
+++ b/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma
@@ -23,55 +23,85 @@ definition exhaustive ≝
      (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;
+  ∀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 (λn.fst (a n)).
-intros 5; change with (l ≤ fst (a n)); cases (a n); cases H; assumption;
+  ∀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. 
-    (λn.fst (a n)) ↑ x → a ↑ (sig_in ?? 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) H3); exists [apply w] assumption;
+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 → (λn:nat.fst (a n)) is_cauchy.
+    a is_cauchy → ⌊n,\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)〉));
+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 restrict_uniform_convergence:
+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).
+     ∀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 ?? (fst (a 0))); [2: apply H2;]
+    |2: apply (le_transitive ? ??? ? (H2 O));
         apply (segment_lowerbound ?l u);]
 |2: intros;
-    lapply (uparrow_upperlocated ? a (sig_in ?? x h)) as Ha1;
+    lapply (uparrow_upperlocated ? a ≪x,h≫) as Ha1;
       [2: apply segment_preserves_uparrow;split; assumption;] 
-    lapply (segment_preserves_supremum ?l u a (sig_in ??? h)) as Ha2; 
+    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); [split; assumption]
+    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