Code extraction unbranched again.

[helm.git] / matita / dama / ordered_sets2.ma

diff --git a/matita/dama/ordered_sets2.ma b/matita/dama/ordered_sets2.ma
index b597d372c078ea561cff569ccfaaec660517f238..a8050ec210282e424f0aef5d9e15b1d3c5753b76 100644 (file)
--- a/matita/dama/ordered_sets2.ma
+++ b/matita/dama/ordered_sets2.ma
@@ -17,11 +17,11 @@ set "baseuri" "cic:/matita/ordered_sets2".
 include "ordered_sets.ma".
 
 theorem le_f_inf_inf_f:
- ∀C.∀O':dedekind_sigma_complete_ordered_set C.
-  ∀f:O'→O'. ∀H:is_order_continuous ? ? f.
-   ∀a:bounded_below_sequence ? O'.
+ ∀O':dedekind_sigma_complete_ordered_set.
+  ∀f:O'→O'. ∀H:is_order_continuous ? f.
+   ∀a:bounded_below_sequence O'.
     let p := ? in
-     f (inf ? ? a) ≤ inf ? ? (mk_bounded_below_sequence ? ? (λi. f (a i)) p).
+     f (inf ? a) ≤ inf ? (mk_bounded_below_sequence ? (λi. f (a i)) p).
  [ apply mk_is_bounded_below;
     [2: apply ioc_is_lower_bound_f_inf;
         assumption
@@ -29,64 +29,104 @@ theorem le_f_inf_inf_f:
     ] 
  | intros;
    clearbody p;
-   apply (inf_greatest_lower_bound ? ? ? ? (inf_is_inf ? ? ?));
+   apply (inf_greatest_lower_bound ? ? ? (inf_is_inf ? ?));
    simplify;
    intro;
-   letin b := (λi.match i with [ O ⇒ inf ? ? a | S _ ⇒ a n]);
+   letin b := (λi.match i with [ O ⇒ inf ? a | S _ ⇒ a n]);
    change with (f (b O) ≤ f (b (S O)));
-   apply (ioc_is_sequentially_monotone ? ? ? H);
+   apply (ioc_is_sequentially_monotone ? ? H);
    simplify;
    clear b;
    intro;
    elim n1; simplify;
-    [ apply (inf_lower_bound ? ? ? ? (inf_is_inf ? ? ?));
-    | apply (or_reflexive ? O' (dscos_ordered_set ? O'))
+    [ apply (inf_lower_bound ? ? ? (inf_is_inf ? ?));
+    | apply (or_reflexive O' ? (dscos_ordered_set O'))
     ]
  ].
 qed.
 
 theorem le_to_le_sup_sup:
- ∀C.∀O':dedekind_sigma_complete_ordered_set C.
-  ∀a,b:bounded_above_sequence ? O'.
-   (∀i.a i ≤ b i) → sup ? ? a ≤ sup ? ? b.
+ ∀O':dedekind_sigma_complete_ordered_set.
+  ∀a,b:bounded_above_sequence O'.
+   (∀i.a i ≤ b i) → sup ? a ≤ sup ? b.
  intros;
- apply (sup_least_upper_bound ? ? ? ? (sup_is_sup ? ? a));
+ apply (sup_least_upper_bound ? ? ? (sup_is_sup ? a));
  unfold;
  intro;
  apply (or_transitive ? ? O');
   [2: apply H
   | skip
-  | apply (sup_upper_bound ? ? ? ? (sup_is_sup ? ? b))
+  | apply (sup_upper_bound ? ? ? (sup_is_sup ? b))
   ].
 qed. 
 
 interpretation "mk_bounded_sequence" 'hide_everything_but a
 = (cic:/matita/ordered_sets/bounded_sequence.ind#xpointer(1/1/1) _ _ a _ _).
 
+lemma reduce_bas_seq:
+ ∀O:ordered_set.∀a:nat→O.∀p.∀i.
+  bas_seq ? (mk_bounded_above_sequence ? a p) i = a i.
+ intros;
+ reflexivity.
+qed.
+
+(*lemma reduce_bbs_seq:
+ ∀C.∀O:ordered_set C.∀a:nat→O.∀p.∀i.
+  bbs_seq ? ? (mk_bounded_below_sequence ? ? a p) i = a i.
+ intros;
+ reflexivity.
+qed.*)
+
+axiom inf_extensional:
+ ∀O:dedekind_sigma_complete_ordered_set.
+  ∀a,b:bounded_below_sequence O.
+   (∀i.a i = b i) → inf ? a = inf O b.
+   
+lemma eq_to_le: ∀O:ordered_set.∀x,y:O.x=y → x ≤ y.
+ intros;
+ rewrite > H;
+ apply (or_reflexive ? ? O).
+qed.
+
 theorem fatou:
- ∀C.∀O':dedekind_sigma_complete_ordered_set C.
-  ∀f:O'→O'. ∀H:is_order_continuous ? ? f.
-   ∀a:bounded_sequence ? O'.
+ ∀O':dedekind_sigma_complete_ordered_set.
+  ∀f:O'→O'. ∀H:is_order_continuous ? f.
+   ∀a:bounded_sequence O'.
     let pb := ? in
     let pa := ? in
-     f (liminf ? ? a) ≤ liminf ? ? (mk_bounded_sequence ? ? (λi. f (a i)) pb pa).
- [ letin bas ≝ (bounded_above_sequence_of_bounded_sequence ? ? a);
+     f (liminf ? a) ≤ liminf ? (mk_bounded_sequence ? (λi. f (a i)) pb pa).
+ [ letin bas ≝ (bounded_above_sequence_of_bounded_sequence ? a);
    apply mk_is_bounded_above;
-    [2: apply (ioc_is_upper_bound_f_sup ? ? ? H bas)
+    [2: apply (ioc_is_upper_bound_f_sup ? ? H bas)
     | skip
     ]
- | letin bbs ≝ (bounded_below_sequence_of_bounded_sequence ? ? a);
+ | letin bbs ≝ (bounded_below_sequence_of_bounded_sequence ? a);
    apply mk_is_bounded_below;
-    [2: apply (ioc_is_lower_bound_f_inf ? ? ? H bbs)
+    [2: apply (ioc_is_lower_bound_f_inf ? ? H bbs)
     | skip
     ] 
  | intros;
-   rewrite > eq_f_liminf_sup_f_inf;
+   rewrite > eq_f_liminf_sup_f_inf in ⊢ (? ? % ?);
     [ unfold liminf;
       apply le_to_le_sup_sup;
-      elim daemon (*(* f inf < inf f *)
-      apply le_f_inf_inf_f*)
+      intro;
+      rewrite > reduce_bas_seq;
+      rewrite > reduce_bas_seq;
+      apply (or_transitive ? ? O');
+       [2: apply le_f_inf_inf_f;
+           assumption
+       | skip
+       | apply eq_to_le;
+         apply inf_extensional;
+         intro;
+         reflexivity
+       ]
     | assumption
     ]
  ].
 qed.
+
+record cotransitively_ordered_set: Type :=
+ { cos_ordered_set :> ordered_set;
+   cos_cotransitive: cotransitive ? (os_le cos_ordered_set)
+ }.