X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fdama%2Fordered_sets.ma;h=a78d6118428014f95e79bde34fa0f2bf907d9abc;hb=d5fa086d0324953f6f6ec7955f7eaa60796eb69d;hp=b8c6952c7912bcdf0a1439b23fc58ad08e174404;hpb=314a6b26e04acc304b542b3b2f6c8fb0bbddd5b8;p=helm.git

diff --git a/helm/software/matita/dama/ordered_sets.ma b/helm/software/matita/dama/ordered_sets.ma
index b8c6952c7..a78d61184 100644
--- a/helm/software/matita/dama/ordered_sets.ma
+++ b/helm/software/matita/dama/ordered_sets.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/ordered_sets/".
+set "baseuri" "cic:/matita/excedence/".
 
 include "higher_order_defs/relations.ma".
 include "nat/plus.ma".
@@ -27,12 +27,12 @@ record excedence : Type ≝ {
 }.
 
 interpretation "excedence" 'nleq a b =
- (cic:/matita/ordered_sets/exc_relation.con _ a b). 
+ (cic:/matita/excedence/exc_relation.con _ a b). 
 
 definition le ≝ λE:excedence.λa,b:E. ¬ (a ≰ b).
 
 interpretation "ordered sets less or equal than" 'leq a b = 
- (cic:/matita/ordered_sets/le.con _ a b).
+ (cic:/matita/excedence/le.con _ a b).
 
 lemma le_reflexive: ∀E.reflexive ? (le E).
 intros (E); unfold; cases E; simplify; intros (x); apply (H x);
@@ -46,7 +46,7 @@ qed.
 definition apart ≝ λE:excedence.λa,b:E. a ≰ b ∨ b ≰ a.
 
 notation "a # b" non associative with precedence 50 for @{ 'apart $a $b}.
-interpretation "apartness" 'apart a b = (cic:/matita/ordered_sets/apart.con _ a b). 
+interpretation "apartness" 'apart a b = (cic:/matita/excedence/apart.con _ a b). 
 
 lemma apart_coreflexive: ∀E.coreflexive ? (apart E).
 intros (E); unfold; cases E; simplify; clear E; intros (x); unfold;
@@ -67,7 +67,7 @@ definition eq ≝ λE:excedence.λa,b:E. ¬ (a # b).
 
 notation "a ≈ b" non associative with precedence 50 for @{ 'napart $a $b}.    
 interpretation "alikeness" 'napart a b =
-  (cic:/matita/ordered_sets/eq.con _ a b). 
+  (cic:/matita/excedence/eq.con _ a b). 
 
 lemma eq_reflexive:∀E. reflexive ? (eq E).
 intros (E); unfold; cases E (T f cRf _); simplify; unfold Not; intros (x H);
@@ -94,7 +94,7 @@ qed.
 definition lt ≝ λE:excedence.λa,b:E. a ≤ b ∧ a # b.
 
 interpretation "ordered sets less than" 'lt a b =
- (cic:/matita/ordered_sets/lt.con _ a b).
+ (cic:/matita/excedence/lt.con _ a b).
 
 lemma lt_coreflexive: ∀E.coreflexive ? (lt E).
 intros (E); unfold; unfold Not; intros (x H); cases H (_ ABS); 
@@ -111,286 +111,7 @@ lapply (exc_coreflexive E) as r; unfold coreflexive in r;
 |2: lapply (c ?? x H2) as H3; cases H3 (H4 H4); [right; assumption|cases (Lxy H4)]]
 qed.
 
-theorem mah: ∀E:excedence.∀a,b:E. (a < b) → (b ≰ a).
+theorem lt_to_excede: ∀E:excedence.∀a,b:E. (a < b) → (b ≰ a).
 intros (E a b Lab); cases Lab (LEab Aab);
 cases Aab (H H); [cases (LEab H)] fold normalize (b ≰ a); assumption; (* BUG *)  
 qed.
-
--- altro file
-opposto TH è assioma per ordine totale.
-
--- 
-
-
-
-
-
-
-
-record is_order_relation (C:Type) (le:C→C→Prop) (eq:C→C→Prop) : Type ≝ { 
-  or_reflexive: reflexive ? le;
-  or_transitive: transitive ? le;
-  or_antisimmetric: antisymmetric ? le eq
-}.
-
-record ordered_set: Type ≝ { 
-  os_carr:> excedence;
-  os_order_relation_properties:> is_order_relation ? (le os_carr) (apart os_carr)
-}.
-
-ordered_set.
-
-E
-
-E
-
-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
-  ].
-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_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 (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 (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 (O:ordered_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 ≝
- { 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 ≝
- { ib_bounded_below:> is_bounded_below ? a;
-   ib_bounded_above:> is_bounded_above ? a
- }.
-
-record bounded_below_sequence (O:ordered_set) : Type ≝
- { bbs_seq:1> nat→O;
-   bbs_is_bounded_below:> is_bounded_below ? bbs_seq
- }.
-
-record bounded_above_sequence (O:ordered_set) : Type ≝
- { bas_seq:1> nat→O;
-   bas_is_bounded_above:> is_bounded_above ? bas_seq
- }.
-
-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
- }.
-
-definition bounded_below_sequence_of_bounded_sequence ≝
- 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 ≝
- 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 ≝
- 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)
- }.