X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fdama%2Fordered_sets.ma;h=1ca191339b697514c39b171ed9111ef8c6a1003c;hb=e560ae80f1f1cece95e0eda87daae3814d9413c5;hp=612d92aa276f3607b0dcdf43a3c00819b36a6056;hpb=dd3157d36216486d914a97cfff7a9cd34f009ffe;p=helm.git

diff --git a/matita/dama/ordered_sets.ma b/matita/dama/ordered_sets.ma
index 612d92aa2..1ca191339 100644
--- a/matita/dama/ordered_sets.ma
+++ b/matita/dama/ordered_sets.ma
@@ -18,35 +18,27 @@ include "higher_order_defs/relations.ma".
 include "nat/plus.ma".
 include "constructive_connectives.ma".
 
-record pre_ordered_set (C:Type) : Type ≝
- { le_:C→C→Prop }.
-
-definition carrier_of_pre_ordered_set ≝ λC:Type.λO:pre_ordered_set C.C.
-
-coercion cic:/matita/ordered_sets/carrier_of_pre_ordered_set.con.
-
-definition os_le: ∀C.∀O:pre_ordered_set C.O → O → Prop ≝ le_.
-
-interpretation "Ordered Sets le" 'leq a b =
- (cic:/matita/ordered_sets/os_le.con _ _ a b).
-
 definition cotransitive ≝
  λC:Type.λle:C→C→Prop.∀x,y,z:C. le x y → le x z ∨ le z y. 
 
 definition antisimmetric ≝
  λC:Type.λle:C→C→Prop.∀x,y:C. le x y → le y x → x=y.
 
-record is_order_relation (C) (O:pre_ordered_set C) : Type ≝
- { or_reflexive: reflexive ? (os_le ? O);
-   or_transitive: transitive ? (os_le ? O);
-   or_antisimmetric: antisimmetric ? (os_le ? O)
+record is_order_relation (C:Type) (le:C→C→Prop) : Type ≝
+ { or_reflexive: reflexive ? le;
+   or_transitive: transitive ? le;
+   or_antisimmetric: antisimmetric ? le
  }.
 
-record ordered_set (C:Type): Type ≝
- { os_pre_ordered_set:> pre_ordered_set C;
-   os_order_relation_properties:> is_order_relation ? os_pre_ordered_set
+record ordered_set: Type ≝
+ { os_carrier:> Type;
+   os_le: os_carrier → os_carrier → Prop;
+   os_order_relation_properties:> is_order_relation ? os_le
  }.
 
+interpretation "Ordered Sets le" 'leq a b =
+ (cic:/matita/ordered_sets/os_le.con _ a b).
+
 theorem antisimmetric_to_cotransitive_to_transitive:
  ∀C.∀le:relation C. antisimmetric ? le → cotransitive ? le →
   transitive ? le.
@@ -60,139 +52,139 @@ theorem antisimmetric_to_cotransitive_to_transitive:
   ].
 qed.
 
-definition is_increasing ≝ λC.λO:ordered_set C.λa:nat→O.∀n:nat.a n ≤ a (S n).
-definition is_decreasing ≝ λC.λO:ordered_set C.λa:nat→O.∀n:nat.a (S n) ≤ a n.
+definition is_increasing ≝ λO:ordered_set.λa:nat→O.∀n:nat.a n ≤ a (S n).
+definition is_decreasing ≝ λO:ordered_set.λa:nat→O.∀n:nat.a (S n) ≤ a n.
 
-definition is_upper_bound ≝ λC.λO:ordered_set C.λa:nat→O.λu:O.∀n:nat.a n ≤ u.
-definition is_lower_bound ≝ λC.λO:ordered_set C.λa:nat→O.λu:O.∀n:nat.u ≤ a n.
+definition is_upper_bound ≝ λO:ordered_set.λa:nat→O.λu:O.∀n:nat.a n ≤ u.
+definition is_lower_bound ≝ λO:ordered_set.λa:nat→O.λu:O.∀n:nat.u ≤ a n.
 
-record is_sup (C:Type) (O:ordered_set C) (a:nat→O) (u:O) : Prop ≝
- { sup_upper_bound: is_upper_bound ? O a u; 
-   sup_least_upper_bound: ∀v:O. is_upper_bound ? O a v → u≤v
+record is_sup (O:ordered_set) (a:nat→O) (u:O) : Prop ≝
+ { sup_upper_bound: is_upper_bound O a u; 
+   sup_least_upper_bound: ∀v:O. is_upper_bound O a v → u≤v
  }.
 
-record is_inf (C:Type) (O:ordered_set C) (a:nat→O) (u:O) : Prop ≝
- { inf_lower_bound: is_lower_bound ? O a u; 
-   inf_greatest_lower_bound: ∀v:O. is_lower_bound ? O a v → v≤u
+record is_inf (O:ordered_set) (a:nat→O) (u:O) : Prop ≝
+ { inf_lower_bound: is_lower_bound O a u; 
+   inf_greatest_lower_bound: ∀v:O. is_lower_bound O a v → v≤u
  }.
 
-record is_bounded_below (C:Type) (O:ordered_set C) (a:nat→O) : Type ≝
+record is_bounded_below (O:ordered_set) (a:nat→O) : Type ≝
  { ib_lower_bound: O;
-   ib_lower_bound_is_lower_bound: is_lower_bound ? ? a ib_lower_bound
+   ib_lower_bound_is_lower_bound: is_lower_bound ? a ib_lower_bound
  }.
 
-record is_bounded_above (C:Type) (O:ordered_set C) (a:nat→O) : Type ≝
+record is_bounded_above (O:ordered_set) (a:nat→O) : Type ≝
  { ib_upper_bound: O;
-   ib_upper_bound_is_upper_bound: is_upper_bound ? ? a ib_upper_bound
+   ib_upper_bound_is_upper_bound: is_upper_bound ? a ib_upper_bound
  }.
 
-record is_bounded (C:Type) (O:ordered_set C) (a:nat→O) : Type ≝
- { ib_bounded_below:> is_bounded_below ? ? a;
-   ib_bounded_above:> is_bounded_above ? ? a
+record is_bounded (O:ordered_set) (a:nat→O) : Type ≝
+ { ib_bounded_below:> is_bounded_below ? a;
+   ib_bounded_above:> is_bounded_above ? a
  }.
 
-record bounded_below_sequence (C:Type) (O:ordered_set C) : Type ≝
+record bounded_below_sequence (O:ordered_set) : Type ≝
  { bbs_seq:1> nat→O;
-   bbs_is_bounded_below:> is_bounded_below ? ? bbs_seq
+   bbs_is_bounded_below:> is_bounded_below ? bbs_seq
  }.
 
-record bounded_above_sequence (C:Type) (O:ordered_set C) : Type ≝
+record bounded_above_sequence (O:ordered_set) : Type ≝
  { bas_seq:1> nat→O;
-   bas_is_bounded_above:> is_bounded_above ? ? bas_seq
+   bas_is_bounded_above:> is_bounded_above ? bas_seq
  }.
 
-record bounded_sequence (C:Type) (O:ordered_set C) : Type ≝
+record bounded_sequence (O:ordered_set) : Type ≝
  { bs_seq:1> nat → O;
-   bs_is_bounded_below: is_bounded_below ? ? bs_seq;
-   bs_is_bounded_above: is_bounded_above ? ? bs_seq
+   bs_is_bounded_below: is_bounded_below ? bs_seq;
+   bs_is_bounded_above: is_bounded_above ? bs_seq
  }.
 
 definition bounded_below_sequence_of_bounded_sequence ≝
- λC.λO:ordered_set C.λb:bounded_sequence ? O.
-  mk_bounded_below_sequence ? ? b (bs_is_bounded_below ? ? b).
+ λO:ordered_set.λb:bounded_sequence O.
+  mk_bounded_below_sequence ? b (bs_is_bounded_below ? b).
 
 coercion cic:/matita/ordered_sets/bounded_below_sequence_of_bounded_sequence.con.
 
 definition bounded_above_sequence_of_bounded_sequence ≝
- λC.λO:ordered_set C.λb:bounded_sequence ? O.
-  mk_bounded_above_sequence ? ? b (bs_is_bounded_above ? ? b).
+ λO:ordered_set.λb:bounded_sequence O.
+  mk_bounded_above_sequence ? b (bs_is_bounded_above ? b).
 
 coercion cic:/matita/ordered_sets/bounded_above_sequence_of_bounded_sequence.con.
 
 definition lower_bound ≝
- λC.λO:ordered_set C.λb:bounded_below_sequence ? O.
-  ib_lower_bound ? ? b (bbs_is_bounded_below ? ? b).
+ λO:ordered_set.λb:bounded_below_sequence O.
+  ib_lower_bound ? b (bbs_is_bounded_below ? b).
 
 lemma lower_bound_is_lower_bound:
- ∀C.∀O:ordered_set C.∀b:bounded_below_sequence ? O.
-  is_lower_bound ? ? b (lower_bound ? ? b).
+ ∀O:ordered_set.∀b:bounded_below_sequence O.
+  is_lower_bound ? b (lower_bound ? b).
  intros;
  unfold lower_bound;
  apply ib_lower_bound_is_lower_bound.
 qed.
 
 definition upper_bound ≝
- λC.λO:ordered_set C.λb:bounded_above_sequence ? O.
-  ib_upper_bound ? ? b (bas_is_bounded_above ? ? b).
+ λO:ordered_set.λb:bounded_above_sequence O.
+  ib_upper_bound ? b (bas_is_bounded_above ? b).
 
 lemma upper_bound_is_upper_bound:
- ∀C.∀O:ordered_set C.∀b:bounded_above_sequence ? O.
-  is_upper_bound ? ? b (upper_bound ? ? b).
+ ∀O:ordered_set.∀b:bounded_above_sequence O.
+  is_upper_bound ? b (upper_bound ? b).
  intros;
  unfold upper_bound;
  apply ib_upper_bound_is_upper_bound.
 qed.
 
-record is_dedekind_sigma_complete (C:Type) (O:ordered_set C) : Type ≝
- { dsc_inf: ∀a:nat→O.∀m:O. is_lower_bound ? ? a m → ex ? (λs:O.is_inf ? O a s);
+record is_dedekind_sigma_complete (O:ordered_set) : Type ≝
+ { dsc_inf: ∀a:nat→O.∀m:O. is_lower_bound ? a m → ex ? (λs:O.is_inf O a s);
    dsc_inf_proof_irrelevant:
-    ∀a:nat→O.∀m,m':O.∀p:is_lower_bound ? ? a m.∀p':is_lower_bound ? ? a m'.
+    ∀a:nat→O.∀m,m':O.∀p:is_lower_bound ? a m.∀p':is_lower_bound ? a m'.
      (match dsc_inf a m p with [ ex_intro s _ ⇒ s ]) =
      (match dsc_inf a m' p' with [ ex_intro s' _ ⇒ s' ]); 
-   dsc_sup: ∀a:nat→O.∀m:O. is_upper_bound ? ? a m → ex ? (λs:O.is_sup ? O a s);
+   dsc_sup: ∀a:nat→O.∀m:O. is_upper_bound ? a m → ex ? (λs:O.is_sup O a s);
    dsc_sup_proof_irrelevant:
-    ∀a:nat→O.∀m,m':O.∀p:is_upper_bound ? ? a m.∀p':is_upper_bound ? ? a m'.
+    ∀a:nat→O.∀m,m':O.∀p:is_upper_bound ? a m.∀p':is_upper_bound ? a m'.
      (match dsc_sup a m p with [ ex_intro s _ ⇒ s ]) =
      (match dsc_sup a m' p' with [ ex_intro s' _ ⇒ s' ])    
  }.
 
-record dedekind_sigma_complete_ordered_set (C:Type) : Type ≝
- { dscos_ordered_set:> ordered_set C;
+record dedekind_sigma_complete_ordered_set : Type ≝
+ { dscos_ordered_set:> ordered_set;
    dscos_dedekind_sigma_complete_properties:>
-    is_dedekind_sigma_complete ? dscos_ordered_set
+    is_dedekind_sigma_complete dscos_ordered_set
  }.
 
 definition inf:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
-  bounded_below_sequence ? O → O.
+ ∀O:dedekind_sigma_complete_ordered_set.
+  bounded_below_sequence O → O.
  intros;
  elim
-  (dsc_inf ? O (dscos_dedekind_sigma_complete_properties ? O) b);
+  (dsc_inf O (dscos_dedekind_sigma_complete_properties O) b);
   [ apply a
-  | apply (lower_bound ? ? b)
+  | apply (lower_bound ? b)
   | apply lower_bound_is_lower_bound
   ]
 qed.
 
 lemma inf_is_inf:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
-  ∀a:bounded_below_sequence ? O.
-   is_inf ? ? a (inf ? ? a).
+ ∀O:dedekind_sigma_complete_ordered_set.
+  ∀a:bounded_below_sequence O.
+   is_inf ? a (inf ? a).
  intros;
  unfold inf;
  simplify;
- elim (dsc_inf C O (dscos_dedekind_sigma_complete_properties C O) a
-(lower_bound C O a) (lower_bound_is_lower_bound C O a));
+ elim (dsc_inf O (dscos_dedekind_sigma_complete_properties O) a
+(lower_bound O a) (lower_bound_is_lower_bound O a));
  simplify;
  assumption.
 qed.
 
 lemma inf_proof_irrelevant:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
-  ∀a,a':bounded_below_sequence ? O.
-   bbs_seq ? ? a = bbs_seq ? ? a' →
-    inf ? ? a = inf ? ? a'.
- intros 4;
+ ∀O:dedekind_sigma_complete_ordered_set.
+  ∀a,a':bounded_below_sequence O.
+   bbs_seq ? a = bbs_seq ? a' →
+    inf ? a = inf ? a'.
+ intros 3;
  elim a 0;
  elim a';
  simplify in H;
@@ -201,43 +193,43 @@ lemma inf_proof_irrelevant:
  rewrite > H;
  intro;
  simplify;
- rewrite < (dsc_inf_proof_irrelevant C O O f (ib_lower_bound ? ? f i2)
-  (ib_lower_bound ? ? f i) (ib_lower_bound_is_lower_bound ? ? f i2)
-  (ib_lower_bound_is_lower_bound ? ? f i));
+ rewrite < (dsc_inf_proof_irrelevant O O f (ib_lower_bound ? f i)
+  (ib_lower_bound ? f i2) (ib_lower_bound_is_lower_bound ? f i)
+  (ib_lower_bound_is_lower_bound ? f i2));
  reflexivity.
 qed.
 
 definition sup:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
-  bounded_above_sequence ? O → O.
+ ∀O:dedekind_sigma_complete_ordered_set.
+  bounded_above_sequence O → O.
  intros;
  elim
-  (dsc_sup ? O (dscos_dedekind_sigma_complete_properties ? O) b);
+  (dsc_sup O (dscos_dedekind_sigma_complete_properties O) b);
   [ apply a
-  | apply (upper_bound ? ? b)
+  | apply (upper_bound ? b)
   | apply upper_bound_is_upper_bound
   ].
 qed.
 
 lemma sup_is_sup:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
-  ∀a:bounded_above_sequence ? O.
-   is_sup ? ? a (sup ? ? a).
+ ∀O:dedekind_sigma_complete_ordered_set.
+  ∀a:bounded_above_sequence O.
+   is_sup ? a (sup ? a).
  intros;
  unfold sup;
  simplify;
- elim (dsc_sup C O (dscos_dedekind_sigma_complete_properties C O) a
-(upper_bound C O a) (upper_bound_is_upper_bound C O a));
+ elim (dsc_sup O (dscos_dedekind_sigma_complete_properties O) a
+(upper_bound O a) (upper_bound_is_upper_bound O a));
  simplify;
  assumption.
 qed.
 
 lemma sup_proof_irrelevant:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
-  ∀a,a':bounded_above_sequence ? O.
-   bas_seq ? ? a = bas_seq ? ? a' →
-    sup ? ? a = sup ? ? a'.
- intros 4;
+ ∀O:dedekind_sigma_complete_ordered_set.
+  ∀a,a':bounded_above_sequence O.
+   bas_seq ? a = bas_seq ? a' →
+    sup ? a = sup ? a'.
+ intros 3;
  elim a 0;
  elim a';
  simplify in H;
@@ -246,18 +238,18 @@ lemma sup_proof_irrelevant:
  rewrite > H;
  intro;
  simplify;
- rewrite < (dsc_sup_proof_irrelevant C O O f (ib_upper_bound ? ? f i2)
-  (ib_upper_bound ? ? f i) (ib_upper_bound_is_upper_bound ? ? f i2)
-  (ib_upper_bound_is_upper_bound ? ? f i));
+ rewrite < (dsc_sup_proof_irrelevant O O f (ib_upper_bound ? f i2)
+  (ib_upper_bound ? f i) (ib_upper_bound_is_upper_bound ? f i2)
+  (ib_upper_bound_is_upper_bound ? f i));
  reflexivity.
 qed.
 
 axiom daemon: False.
 
 theorem inf_le_sup:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
-  ∀a:bounded_sequence ? O. inf ? ? a ≤ sup ? ? a.
- intros (C O');
+ ∀O:dedekind_sigma_complete_ordered_set.
+  ∀a:bounded_sequence O. inf ? a ≤ sup ? a.
+ intros (O');
  apply (or_transitive ? ? O' ? (a O));
   [ elim daemon (*apply (inf_lower_bound ? ? ? ? (inf_is_inf ? ? a))*)
   | elim daemon (*apply (sup_upper_bound ? ? ? ? (sup_is_sup ? ? a))*)
@@ -265,75 +257,75 @@ theorem inf_le_sup:
 qed.
 
 lemma inf_respects_le:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
-  ∀a:bounded_below_sequence ? O.∀m:O.
-   is_upper_bound ? ? a m → inf ? ? a ≤ m.
- intros (C O');
- apply (or_transitive ? ? O' ? (sup ? ? (mk_bounded_sequence ? ? a ? ?)));
-  [ apply (bbs_is_bounded_below ? ? a)
-  | apply (mk_is_bounded_above ? ? ? m H)
+ ∀O:dedekind_sigma_complete_ordered_set.
+  ∀a:bounded_below_sequence O.∀m:O.
+   is_upper_bound ? a m → inf ? a ≤ m.
+ intros (O');
+ apply (or_transitive ? ? O' ? (sup ? (mk_bounded_sequence ? a ? ?)));
+  [ apply (bbs_is_bounded_below ? a)
+  | apply (mk_is_bounded_above ? ? m H)
   | apply inf_le_sup 
   | apply
-     (sup_least_upper_bound ? ? ? ?
-      (sup_is_sup ? ? (mk_bounded_sequence C O' a a
-        (mk_is_bounded_above C O' a m H))));
+     (sup_least_upper_bound ? ? ?
+      (sup_is_sup ? (mk_bounded_sequence O' a a
+        (mk_is_bounded_above O' a m H))));
     assumption
   ].
 qed. 
 
 definition is_sequentially_monotone ≝
- λC.λO:ordered_set C.λf:O→O.
-  ∀a:nat→O.∀p:is_increasing ? ? a.
-   is_increasing ? ? (λi.f (a i)).
+ λO:ordered_set.λf:O→O.
+  ∀a:nat→O.∀p:is_increasing ? a.
+   is_increasing ? (λi.f (a i)).
 
-record is_order_continuous (C)
- (O:dedekind_sigma_complete_ordered_set C) (f:O→O) : Prop
+record is_order_continuous
+ (O:dedekind_sigma_complete_ordered_set) (f:O→O) : Prop
 ≝
- { ioc_is_sequentially_monotone: is_sequentially_monotone ? ? f;
+ { ioc_is_sequentially_monotone: is_sequentially_monotone ? f;
    ioc_is_upper_bound_f_sup:
-    ∀a:bounded_above_sequence ? O.
-     is_upper_bound ? ? (λi.f (a i)) (f (sup ? ? a));
+    ∀a:bounded_above_sequence O.
+     is_upper_bound ? (λi.f (a i)) (f (sup ? a));
    ioc_respects_sup:
-    ∀a:bounded_above_sequence ? O.
-     is_increasing ? ? a →
-      f (sup ? ? a) =
-       sup ? ? (mk_bounded_above_sequence ? ? (λi.f (a i))
-        (mk_is_bounded_above ? ? ? (f (sup ? ? a))
+    ∀a:bounded_above_sequence O.
+     is_increasing ? a →
+      f (sup ? a) =
+       sup ? (mk_bounded_above_sequence ? (λi.f (a i))
+        (mk_is_bounded_above ? ? (f (sup ? a))
          (ioc_is_upper_bound_f_sup a)));
    ioc_is_lower_bound_f_inf:
-    ∀a:bounded_below_sequence ? O.
-     is_lower_bound ? ? (λi.f (a i)) (f (inf ? ? a));
+    ∀a:bounded_below_sequence O.
+     is_lower_bound ? (λi.f (a i)) (f (inf ? a));
    ioc_respects_inf:
-    ∀a:bounded_below_sequence ? O.
-     is_decreasing ? ? a →
-      f (inf ? ? a) =
-       inf ? ? (mk_bounded_below_sequence ? ? (λi.f (a i))
-        (mk_is_bounded_below ? ? ? (f (inf ? ? a))
+    ∀a:bounded_below_sequence O.
+     is_decreasing ? a →
+      f (inf ? a) =
+       inf ? (mk_bounded_below_sequence ? (λi.f (a i))
+        (mk_is_bounded_below ? ? (f (inf ? a))
          (ioc_is_lower_bound_f_inf a)))   
  }.
 
 theorem tail_inf_increasing:
- ∀C.∀O:dedekind_sigma_complete_ordered_set C.
-  ∀a:bounded_below_sequence ? O.
-   let y ≝ λi.mk_bounded_below_sequence ? ? (λj.a (i+j)) ? in
-   let x ≝ λi.inf ? ? (y i) in
-    is_increasing ? ? x.
- [ apply (mk_is_bounded_below ? ? ? (ib_lower_bound ? ? a a));
+ ∀O:dedekind_sigma_complete_ordered_set.
+  ∀a:bounded_below_sequence O.
+   let y ≝ λi.mk_bounded_below_sequence ? (λj.a (i+j)) ? in
+   let x ≝ λi.inf ? (y i) in
+    is_increasing ? x.
+ [ apply (mk_is_bounded_below ? ? (ib_lower_bound ? a a));
    simplify;
    intro;
-   apply (ib_lower_bound_is_lower_bound ? ? a a)
+   apply (ib_lower_bound_is_lower_bound ? a a)
  | intros;
    unfold is_increasing;
    intro;
-   unfold x in ⊢ (? ? ? ? %);
-   apply (inf_greatest_lower_bound ? ? ? ? (inf_is_inf ? ? (y (S n))));
-   change with (is_lower_bound ? ? (y (S n)) (inf ? ? (y n)));
+   unfold x in ⊢ (? ? ? %);
+   apply (inf_greatest_lower_bound ? ? ? (inf_is_inf ? (y (S n))));
+   change with (is_lower_bound ? (y (S n)) (inf ? (y n)));
    unfold is_lower_bound;
    intro;
-   generalize in match (inf_lower_bound ? ? ? ? (inf_is_inf ? ? (y n)) (S n1));
+   generalize in match (inf_lower_bound ? ? ? (inf_is_inf ? (y n)) (S n1));
    (*CSC: coercion per FunClass inserita a mano*)
-   suppose (inf ? ? (y n) ≤ bbs_seq ? ? (y n) (S n1)) (H);
-   cut (bbs_seq ? ? (y n) (S n1) = bbs_seq ? ? (y (S n)) n1);
+   suppose (inf ? (y n) ≤ bbs_seq ? (y n) (S n1)) (H);
+   cut (bbs_seq ? (y n) (S n1) = bbs_seq ? (y (S n)) n1);
     [ rewrite < Hcut;
       assumption
     | unfold y;
@@ -344,47 +336,249 @@ theorem tail_inf_increasing:
 qed.
 
 definition is_liminf:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
-  bounded_below_sequence ? O → O → Prop.
+ ∀O:dedekind_sigma_complete_ordered_set.
+  bounded_below_sequence O → O → Prop.
  intros;
  apply
-  (is_sup ? ? (λi.inf ? ? (mk_bounded_below_sequence ? ? (λj.b (i+j)) ?)) t);
- apply (mk_is_bounded_below ? ? ? (ib_lower_bound ? ? b b));
+  (is_sup ? (λi.inf ? (mk_bounded_below_sequence ? (λj.b (i+j)) ?)) t);
+ apply (mk_is_bounded_below ? ? (ib_lower_bound ? b b));
  simplify;
  intros;
- apply (ib_lower_bound_is_lower_bound ? ? b b).
+ apply (ib_lower_bound_is_lower_bound ? b b).
 qed.  
 
 definition liminf:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
-  bounded_sequence ? O → O.
+ ∀O:dedekind_sigma_complete_ordered_set.
+  bounded_sequence O → O.
  intros;
  apply
-  (sup ? ?
-   (mk_bounded_above_sequence ? ?
-     (λi.inf ? ?
-       (mk_bounded_below_sequence ? ?
+  (sup ?
+   (mk_bounded_above_sequence ?
+     (λi.inf ?
+       (mk_bounded_below_sequence ?
          (λj.b (i+j)) ?)) ?));
-  [ apply (mk_is_bounded_below ? ? ? (ib_lower_bound ? ? b b));
+  [ apply (mk_is_bounded_below ? ? (ib_lower_bound ? b b));
     simplify;
     intros;
-    apply (ib_lower_bound_is_lower_bound ? ? b b)
-  | apply (mk_is_bounded_above ? ? ? (ib_upper_bound ? ? b b));
+    apply (ib_lower_bound_is_lower_bound ? b b)
+  | apply (mk_is_bounded_above ? ? (ib_upper_bound ? b b));
     unfold is_upper_bound;
     intro;
     change with
-     (inf C O
-  (mk_bounded_below_sequence C O (\lambda j:nat.b (n+j))
-   (mk_is_bounded_below C O (\lambda j:nat.b (n+j)) (ib_lower_bound C O b b)
-    (\lambda j:nat.ib_lower_bound_is_lower_bound C O b b (n+j))))
-\leq ib_upper_bound C O b b);
-    apply (inf_respects_le ? O);
+     (inf O
+  (mk_bounded_below_sequence O (\lambda j:nat.b (n+j))
+   (mk_is_bounded_below O (\lambda j:nat.b (n+j)) (ib_lower_bound O b b)
+    (\lambda j:nat.ib_lower_bound_is_lower_bound O b b (n+j))))
+\leq ib_upper_bound O b b);
+    apply (inf_respects_le O);
     simplify;
     intro;
-    apply (ib_upper_bound_is_upper_bound ? ? b b)
+    apply (ib_upper_bound_is_upper_bound ? b b)
   ].
 qed.
 
+definition reverse_ordered_set: ordered_set → ordered_set.
+ intros;
+ apply mk_ordered_set;
+  [2:apply (λx,y:o.y ≤ x)
+  | skip
+  | apply mk_is_order_relation;
+     [ simplify;
+       intros;
+       apply (or_reflexive ? ? o)
+     | simplify;
+       intros;
+       apply (or_transitive ? ? o);
+        [2: apply H1
+        | skip
+        | assumption
+        ] 
+     | simplify;
+       intros;
+       apply (or_antisimmetric ? ? o);
+       assumption
+     ]
+  ].
+qed.
+ 
+interpretation "Ordered set ge" 'geq a b =
+ (cic:/matita/ordered_sets/os_le.con _
+  (cic:/matita/ordered_sets/os_pre_ordered_set.con _
+   (cic:/matita/ordered_sets/reverse_ordered_set.con _ _)) a b).
+
+lemma is_lower_bound_reverse_is_upper_bound:
+ ∀O:ordered_set.∀a:nat→O.∀l:O.
+  is_lower_bound O a l → is_upper_bound (reverse_ordered_set O) a l.
+ intros;
+ unfold;
+ intro;
+ unfold;
+ unfold reverse_ordered_set;
+ simplify;
+ apply H.
+qed.
+
+lemma is_upper_bound_reverse_is_lower_bound:
+ ∀O:ordered_set.∀a:nat→O.∀l:O.
+  is_upper_bound O a l → is_lower_bound (reverse_ordered_set O) a l.
+ intros;
+ unfold;
+ intro;
+ unfold;
+ unfold reverse_ordered_set;
+ simplify;
+ apply H.
+qed.
+
+lemma reverse_is_lower_bound_is_upper_bound:
+ ∀O:ordered_set.∀a:nat→O.∀l:O.
+  is_lower_bound (reverse_ordered_set O) a l → is_upper_bound O a l.
+ intros;
+ unfold in H;
+ unfold reverse_ordered_set in H;
+ apply H.
+qed.
+
+lemma reverse_is_upper_bound_is_lower_bound:
+ ∀O:ordered_set.∀a:nat→O.∀l:O.
+  is_upper_bound (reverse_ordered_set O) a l → is_lower_bound O a l.
+ intros;
+ unfold in H;
+ unfold reverse_ordered_set in H;
+ apply H.
+qed.
+
+
+lemma is_inf_to_reverse_is_sup:
+ ∀O:ordered_set.∀a:bounded_below_sequence O.∀l:O.
+  is_inf O a l → is_sup (reverse_ordered_set O) a l.
+ intros;
+ apply (mk_is_sup (reverse_ordered_set O));
+  [ apply is_lower_bound_reverse_is_upper_bound;
+    apply inf_lower_bound;
+    assumption
+  | intros;
+    change in v with (os_carrier O);
+    change with (v ≤ l);
+    apply (inf_greatest_lower_bound ? ? ? H);
+    apply reverse_is_upper_bound_is_lower_bound;
+    assumption
+  ].
+qed.
+ 
+lemma is_sup_to_reverse_is_inf:
+ ∀O:ordered_set.∀a:bounded_above_sequence O.∀l:O.
+  is_sup O a l → is_inf (reverse_ordered_set O) a l.
+ intros;
+ apply (mk_is_inf (reverse_ordered_set O));
+  [ apply is_upper_bound_reverse_is_lower_bound;
+    apply sup_upper_bound;
+    assumption
+  | intros;
+    change in v with (os_carrier O);
+    change with (l ≤ v);
+    apply (sup_least_upper_bound ? ? ? H);
+    apply reverse_is_lower_bound_is_upper_bound;
+    assumption
+  ].
+qed.
+
+lemma reverse_is_sup_to_is_inf:
+ ∀O:ordered_set.∀a:bounded_above_sequence O.∀l:O.
+  is_sup (reverse_ordered_set O) a l → is_inf O a l.
+ intros;
+ apply mk_is_inf;
+  [ apply reverse_is_upper_bound_is_lower_bound;
+    change in l with (os_carrier (reverse_ordered_set O));
+    apply sup_upper_bound;
+    assumption
+  | intros;
+    change in l with (os_carrier (reverse_ordered_set O));
+    change in v with (os_carrier (reverse_ordered_set O));
+    change with (os_le (reverse_ordered_set O) l v);
+    apply (sup_least_upper_bound ? ? ? H);
+    change in v with (os_carrier O);
+    apply is_lower_bound_reverse_is_upper_bound;
+    assumption
+  ].
+qed.
+
+lemma reverse_is_inf_to_is_sup:
+ ∀O:ordered_set.∀a:bounded_above_sequence O.∀l:O.
+  is_inf (reverse_ordered_set O) a l → is_sup O a l.
+ intros;
+ apply mk_is_sup;
+  [ apply reverse_is_lower_bound_is_upper_bound;
+    change in l with (os_carrier (reverse_ordered_set O));
+    apply (inf_lower_bound ? ? ? H)
+  | intros;
+    change in l with (os_carrier (reverse_ordered_set O));
+    change in v with (os_carrier (reverse_ordered_set O));
+    change with (os_le (reverse_ordered_set O) v l);
+    apply (inf_greatest_lower_bound ? ? ? H);
+    change in v with (os_carrier O);
+    apply is_upper_bound_reverse_is_lower_bound;
+    assumption
+  ].
+qed.
+
+
+definition reverse_dedekind_sigma_complete_ordered_set:
+ dedekind_sigma_complete_ordered_set → dedekind_sigma_complete_ordered_set.
+ intros;
+ apply mk_dedekind_sigma_complete_ordered_set;
+  [ apply (reverse_ordered_set d)
+  | elim daemon
+    (*apply mk_is_dedekind_sigma_complete;
+     [ intros;
+       elim (dsc_sup ? ? d a m) 0;
+        [ generalize in match H; clear H;
+          generalize in match m; clear m;
+          elim d;
+          simplify in a1;
+          simplify;
+          change in a1 with (Type_OF_ordered_set ? (reverse_ordered_set ? o)); 
+          apply (ex_intro ? ? a1);
+          simplify in H1;
+          change in m with (Type_OF_ordered_set ? o);
+          apply (is_sup_to_reverse_is_inf ? ? ? ? H1)
+        | generalize in match H; clear H;
+          generalize in match m; clear m;
+          elim d;
+          simplify;
+          change in t with (Type_OF_ordered_set ? o);
+          simplify in t;
+          apply reverse_is_lower_bound_is_upper_bound;
+          assumption
+        ]
+     | apply is_sup_reverse_is_inf;
+     | apply m
+     | 
+     ]*)
+  ].
+qed.
+
+definition reverse_bounded_sequence:
+ ∀O:dedekind_sigma_complete_ordered_set.
+  bounded_sequence O →
+   bounded_sequence (reverse_dedekind_sigma_complete_ordered_set O).
+ intros;
+ apply mk_bounded_sequence;
+  [ apply bs_seq;
+    unfold reverse_dedekind_sigma_complete_ordered_set;
+    simplify;
+    elim daemon
+  | elim daemon
+  | elim daemon
+  ].
+qed.
+
+definition limsup ≝
+ λO:dedekind_sigma_complete_ordered_set.
+  λa:bounded_sequence O.
+   liminf (reverse_dedekind_sigma_complete_ordered_set O)
+    (reverse_bounded_sequence O a).
+
 notation "hvbox(〈a〉)"
  non associative with precedence 45
 for @{ 'hide_everything_but $a }.
@@ -396,12 +590,12 @@ interpretation "mk_bounded_below_sequence" 'hide_everything_but a
 = (cic:/matita/ordered_sets/bounded_below_sequence.ind#xpointer(1/1/1) _ _ a _).
 
 theorem eq_f_sup_sup_f:
- ∀C.∀O':dedekind_sigma_complete_ordered_set C.
-  ∀f:O'→O'. ∀H:is_order_continuous ? ? f.
-   ∀a:bounded_above_sequence ? O'.
-    ∀p:is_increasing ? ? a.
-     f (sup ? ? a) = sup ? ? (mk_bounded_above_sequence ? ? (λi.f (a i)) ?).
- [ apply (mk_is_bounded_above ? ? ? (f (sup ? ? a))); 
+ ∀O':dedekind_sigma_complete_ordered_set.
+  ∀f:O'→O'. ∀H:is_order_continuous ? f.
+   ∀a:bounded_above_sequence O'.
+    ∀p:is_increasing ? a.
+     f (sup ? a) = sup ? (mk_bounded_above_sequence ? (λi.f (a i)) ?).
+ [ apply (mk_is_bounded_above ? ? (f (sup ? a))); 
    apply ioc_is_upper_bound_f_sup;
    assumption
  | intros;
@@ -411,83 +605,83 @@ theorem eq_f_sup_sup_f:
 qed.
 
 theorem eq_f_sup_sup_f':
- ∀C.∀O':dedekind_sigma_complete_ordered_set C.
-  ∀f:O'→O'. ∀H:is_order_continuous ? ? f.
-   ∀a:bounded_above_sequence ? O'.
-    ∀p:is_increasing ? ? a.
-     ∀p':is_bounded_above ? ? (λi.f (a i)).
-      f (sup ? ? a) = sup ? ? (mk_bounded_above_sequence ? ? (λi.f (a i)) p').
+ ∀O':dedekind_sigma_complete_ordered_set.
+  ∀f:O'→O'. ∀H:is_order_continuous ? f.
+   ∀a:bounded_above_sequence O'.
+    ∀p:is_increasing ? a.
+     ∀p':is_bounded_above ? (λi.f (a i)).
+      f (sup ? a) = sup ? (mk_bounded_above_sequence ? (λi.f (a i)) p').
  intros;
- rewrite > (eq_f_sup_sup_f ? ? f H a H1);
+ rewrite > (eq_f_sup_sup_f ? f H a H1);
  apply sup_proof_irrelevant;
  reflexivity.
 qed.
 
 theorem eq_f_liminf_sup_f_inf:
- ∀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 p1 := ? in
-    f (liminf ? ? a) =
-     sup ? ?
-      (mk_bounded_above_sequence ? ?
-        (λi.f (inf ? ?
-          (mk_bounded_below_sequence ? ?
+    f (liminf ? a) =
+     sup ?
+      (mk_bounded_above_sequence ?
+        (λi.f (inf ?
+          (mk_bounded_below_sequence ?
             (λj.a (i+j))
             ?)))
         p1).
- [ apply (mk_is_bounded_below ? ? ? (ib_lower_bound ? ? a a));
+ [ apply (mk_is_bounded_below ? ? (ib_lower_bound ? a a));
    simplify;
    intro;
-   apply (ib_lower_bound_is_lower_bound ? ? a a)
- | apply (mk_is_bounded_above ? ? ? (f (sup ? ? a)));
+   apply (ib_lower_bound_is_lower_bound ? a a)
+ | apply (mk_is_bounded_above ? ? (f (sup ? a)));
    unfold is_upper_bound;
    intro;
    apply (or_transitive ? ? O' ? (f (a n)));
-    [ generalize in match (ioc_is_lower_bound_f_inf ? ? ? H);
+    [ generalize in match (ioc_is_lower_bound_f_inf ? ? H);
       intro H1;
       simplify in H1;
-      rewrite > (plus_n_O n) in ⊢ (? ? ? ? (? (? ? ? ? %)));
-      apply (H1 (mk_bounded_below_sequence C O' (\lambda j:nat.a (n+j))
-(mk_is_bounded_below C O' (\lambda j:nat.a (n+j)) (ib_lower_bound C O' a a)
- (\lambda j:nat.ib_lower_bound_is_lower_bound C O' a a (n+j)))) O);
+      rewrite > (plus_n_O n) in ⊢ (? ? ? (? (? ? ? %)));
+      apply (H1 (mk_bounded_below_sequence O' (\lambda j:nat.a (n+j))
+(mk_is_bounded_below O' (\lambda j:nat.a (n+j)) (ib_lower_bound O' a a)
+ (\lambda j:nat.ib_lower_bound_is_lower_bound O' a a (n+j)))) O);
     | elim daemon (*apply (ioc_is_upper_bound_f_sup ? ? ? H)*)
     ]
  | intros;
    unfold liminf;
    clearbody p1;
    generalize in match (\lambda n:nat
-.inf_respects_le C O'
- (mk_bounded_below_sequence C O' (\lambda j:nat.a (plus n j))
-  (mk_is_bounded_below C O' (\lambda j:nat.a (plus n j))
-   (ib_lower_bound C O' a a)
-   (\lambda j:nat.ib_lower_bound_is_lower_bound C O' a a (plus n j))))
- (ib_upper_bound C O' a a)
- (\lambda n1:nat.ib_upper_bound_is_upper_bound C O' a a (plus n n1)));
+.inf_respects_le O'
+ (mk_bounded_below_sequence O' (\lambda j:nat.a (plus n j))
+  (mk_is_bounded_below O' (\lambda j:nat.a (plus n j))
+   (ib_lower_bound O' a a)
+   (\lambda j:nat.ib_lower_bound_is_lower_bound O' a a (plus n j))))
+ (ib_upper_bound O' a a)
+ (\lambda n1:nat.ib_upper_bound_is_upper_bound O' a a (plus n n1)));
    intro p2;
-   apply (eq_f_sup_sup_f' ? ? f H (mk_bounded_above_sequence C O'
+   apply (eq_f_sup_sup_f' ? f H (mk_bounded_above_sequence O'
 (\lambda i:nat
- .inf C O'
-  (mk_bounded_below_sequence C O' (\lambda j:nat.a (plus i j))
-   (mk_is_bounded_below C O' (\lambda j:nat.a (plus i j))
-    (ib_lower_bound C O' a a)
-    (\lambda n:nat.ib_lower_bound_is_lower_bound C O' a a (plus i n)))))
-(mk_is_bounded_above C O'
+ .inf O'
+  (mk_bounded_below_sequence O' (\lambda j:nat.a (plus i j))
+   (mk_is_bounded_below O' (\lambda j:nat.a (plus i j))
+    (ib_lower_bound O' a a)
+    (\lambda n:nat.ib_lower_bound_is_lower_bound O' a a (plus i n)))))
+(mk_is_bounded_above O'
  (\lambda i:nat
-  .inf C O'
-   (mk_bounded_below_sequence C O' (\lambda j:nat.a (plus i j))
-    (mk_is_bounded_below C O' (\lambda j:nat.a (plus i j))
-     (ib_lower_bound C O' a a)
-     (\lambda n:nat.ib_lower_bound_is_lower_bound C O' a a (plus i n)))))
- (ib_upper_bound C O' a a) p2)));
+  .inf O'
+   (mk_bounded_below_sequence O' (\lambda j:nat.a (plus i j))
+    (mk_is_bounded_below O' (\lambda j:nat.a (plus i j))
+     (ib_lower_bound O' a a)
+     (\lambda n:nat.ib_lower_bound_is_lower_bound O' a a (plus i n)))))
+ (ib_upper_bound O' a a) p2)));
    unfold bas_seq;
    change with
-    (is_increasing ? ? (\lambda i:nat
-.inf C O'
- (mk_bounded_below_sequence C O' (\lambda j:nat.a (plus i j))
-  (mk_is_bounded_below C O' (\lambda j:nat.a (plus i j))
-   (ib_lower_bound C O' a a)
-   (\lambda n:nat.ib_lower_bound_is_lower_bound C O' a a (plus i n))))));
+    (is_increasing ? (\lambda i:nat
+.inf O'
+ (mk_bounded_below_sequence O' (\lambda j:nat.a (plus i j))
+  (mk_is_bounded_below O' (\lambda j:nat.a (plus i j))
+   (ib_lower_bound O' a a)
+   (\lambda n:nat.ib_lower_bound_is_lower_bound O' a a (plus i n))))));
    apply tail_inf_increasing
  ].
 qed.
@@ -495,7 +689,7 @@ qed.
 
 
 
-definition lt ≝ λC.λO:ordered_set C.λa,b:O.a ≤ b ∧ a ≠ b.
+definition lt ≝ λO:ordered_set.λa,b:O.a ≤ b ∧ a ≠ b.
 
 interpretation "Ordered set lt" 'lt a b =
- (cic:/matita/ordered_sets/lt.con _ _ a b).
\ No newline at end of file
+ (cic:/matita/ordered_sets/lt.con _ a b).