added ordered sets

diff --git a/matita/dama/ordered_sets.ma b/matita/dama/ordered_sets.ma
new file mode 100644 (file)
index 0000000..5ae4f56
--- /dev/null
+++ b/matita/dama/ordered_sets.ma
@@ -0,0 +1,283 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/ordered_sets/".
+
+include "ordered_sets.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: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: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: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: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: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: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: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: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:pordered_set) : Type ≝
+ { bbs_seq:1> nat→O;
+   bbs_is_bounded_below:> is_bounded_below ? bbs_seq
+ }.
+
+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: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: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: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.
+
+definition lower_bound ≝
+ λO:ordered_set.λb:bounded_below_sequence O.
+  ib_lower_bound ? b (bbs_is_bounded_below ? b).
+
+lemma lower_bound_is_lower_bound:
+ ∀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 ≝
+ λO:ordered_set.λb:bounded_above_sequence O.
+  ib_upper_bound ? b (bas_is_bounded_above ? b).
+
+lemma upper_bound_is_upper_bound:
+ ∀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.
+
+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).
+
+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.
+
+record cotransitively_ordered_set: Type :=
+ { cos_ordered_set :> ordered_set;
+   cos_cotransitive: cotransitive ? (os_le cos_ordered_set)
+ }.