ordered_sets are built with excedence

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

diff --git a/matita/dama/ordered_sets.ma b/matita/dama/ordered_sets.ma
index dcae29e18f3d8b7c26de5255525ce349c600a60c..3424edbb57ee42a840a565488c821d667892ae5f 100644 (file)
--- a/matita/dama/ordered_sets.ma
+++ b/matita/dama/ordered_sets.ma
@@ -14,99 +14,91 @@
 
 set "baseuri" "cic:/matita/ordered_sets/".
 
-include "higher_order_defs/relations.ma".
-include "nat/plus.ma".
-include "constructive_connectives.ma".
-
-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:Type) (le:C→C→Prop) : Type ≝
- { or_reflexive: reflexive ? le;
-   or_transitive: transitive ? le;
-   or_antisimmetric: antisimmetric ? le
- }.
-
-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).
-
+include "excedence.ma".
+
+record is_porder_relation (C:Type) (le:C→C→Prop) (eq:C→C→Prop) : Type ≝ { 
+  por_reflexive: reflexive ? le;
+  por_transitive: transitive ? le;
+  por_antisimmetric: antisymmetric ? le eq
+}.
+
+record pordered_set: Type ≝ { 
+  pos_carr:> excedence;
+  pos_order_relation_properties:> is_porder_relation ? (le pos_carr) (eq pos_carr)
+}.
+
+lemma pordered_set_of_excedence: excedence → pordered_set.
+intros (E); apply (mk_pordered_set E); apply (mk_is_porder_relation);
+[apply le_reflexive|apply le_transitive|apply le_antisymmetric]
+qed. 
+
+definition total_order : ∀E:excedence. Type ≝
+  λE:excedence. ∀a,b:E. a ≰ b → a < b.
+
+alias id "transitive" = "cic:/matita/higher_order_defs/relations/transitive.con".
+alias id "cotransitive" = "cic:/matita/higher_order_defs/relations/cotransitive.con".
+alias id "antisymmetric" = "cic:/matita/higher_order_defs/relations/antisymmetric.con".
 theorem antisimmetric_to_cotransitive_to_transitive:
- ∀C.∀le:relation C. antisimmetric ? le → cotransitive ? le →
-  transitive ? le.
- intros;
- unfold transitive;
- intros;
- elim (c ? ? z H1);
-  [ assumption
-  | rewrite < (H ? ? H2 t);
-    assumption
-  ].
+ ∀C:Type.∀le:C→C→Prop. antisymmetric ? le → cotransitive ? le → transitive ? le.  
+intros (T f Af cT); unfold transitive; intros (x y z fxy fyz);
+lapply (cT ? ? fxy z) as H; cases H; [assumption] cases (Af ? ? fyz H1);
 qed.
 
-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_increasing ≝ λO:pordered_set.λa:nat→O.∀n:nat.a n ≤ a (S n).
+definition is_decreasing ≝ λO:pordered_set.λa:nat→O.∀n:nat.a (S n) ≤ 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.
+definition is_upper_bound ≝ λO:pordered_set.λa:nat→O.λu:O.∀n:nat.a n ≤ u.
+definition is_lower_bound ≝ λO:pordered_set.λa:nat→O.λu:O.∀n:nat.u ≤ a n.
 
-record is_sup (O:ordered_set) (a:nat→O) (u:O) : Prop ≝
+record is_sup (O:pordered_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 (O:ordered_set) (a:nat→O) (u:O) : Prop ≝
+record is_inf (O:pordered_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 (O:ordered_set) (a:nat→O) : Type ≝
+record is_bounded_below (O:pordered_set) (a:nat→O) : Type ≝
  { ib_lower_bound: O;
    ib_lower_bound_is_lower_bound: is_lower_bound ? a ib_lower_bound
  }.
 
-record is_bounded_above (O:ordered_set) (a:nat→O) : Type ≝
+record is_bounded_above (O:pordered_set) (a:nat→O) : Type ≝
  { ib_upper_bound: O;
    ib_upper_bound_is_upper_bound: is_upper_bound ? a ib_upper_bound
  }.
 
-record is_bounded (O:ordered_set) (a:nat→O) : Type ≝
+record is_bounded (O:pordered_set) (a:nat→O) : Type ≝
  { ib_bounded_below:> is_bounded_below ? a;
    ib_bounded_above:> is_bounded_above ? a
  }.
 
-record bounded_below_sequence (O:ordered_set) : Type ≝
+record bounded_below_sequence (O:pordered_set) : Type ≝
  { bbs_seq:1> nat→O;
    bbs_is_bounded_below:> is_bounded_below ? bbs_seq
  }.
 
-record bounded_above_sequence (O:ordered_set) : Type ≝
+record bounded_above_sequence (O:pordered_set) : Type ≝
  { bas_seq:1> nat→O;
    bas_is_bounded_above:> is_bounded_above ? bas_seq
  }.
 
-record bounded_sequence (O:ordered_set) : Type ≝
+record bounded_sequence (O:pordered_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
  }.
 
 definition bounded_below_sequence_of_bounded_sequence ≝
- λO:ordered_set.λb:bounded_sequence O.
+ λO:pordered_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 ≝
- λO:ordered_set.λb:bounded_sequence O.
+ λO:pordered_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.