renamed ordered sets into excedence.ma

diff --git a/matita/dama/excedence.ma b/matita/dama/excedence.ma
new file mode 100644 (file)
index 0000000..a78d611
--- /dev/null
+++ b/matita/dama/excedence.ma
@@ -0,0 +1,117 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/excedence/".
+
+include "higher_order_defs/relations.ma".
+include "nat/plus.ma".
+include "constructive_connectives.ma".
+include "constructive_higher_order_relations.ma".
+
+record excedence : Type ≝ {
+  exc_carr:> Type;
+  exc_relation: exc_carr → exc_carr → Prop;
+  exc_coreflexive: coreflexive ? exc_relation;
+  exc_cotransitive: cotransitive ? exc_relation 
+}.
+
+interpretation "excedence" 'nleq 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/excedence/le.con _ a b).
+
+lemma le_reflexive: ∀E.reflexive ? (le E).
+intros (E); unfold; cases E; simplify; intros (x); apply (H x);
+qed.
+
+lemma le_transitive: ∀E.transitive ? (le E).
+intros (E); unfold; cases E; simplify; unfold Not; intros (x y z Rxy Ryz H2); 
+cases (c x z y H2) (H4 H5); clear H2; [exact (Rxy H4)|exact (Ryz H5)] 
+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/excedence/apart.con _ a b). 
+
+lemma apart_coreflexive: ∀E.coreflexive ? (apart E).
+intros (E); unfold; cases E; simplify; clear E; intros (x); unfold;
+intros (H1); apply (H x); cases H1; assumption;
+qed.
+
+lemma apart_symmetric: ∀E.symmetric ? (apart E).
+intros (E); unfold; intros(x y H); cases H; clear H; [right|left] assumption; 
+qed.
+
+lemma apart_cotrans: ∀E. cotransitive ? (apart E).
+intros (E); unfold; cases E (T f _ cTf); simplify; intros (x y z Axy);
+cases Axy (H); lapply (cTf ? ? z H) as H1; cases H1; clear Axy H1;
+[left; left|right; left|right; right|left; right] assumption.
+qed.
+
+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/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);
+apply (cRf x); cases H; assumption;
+qed.
+
+lemma eq_symmetric:∀E.symmetric ? (eq E).
+intros (E); unfold; unfold eq; unfold Not;
+intros (x y H1 H2); apply H1; cases H2; [right|left] assumption; 
+qed.
+
+lemma eq_transitive: ∀E.transitive ? (eq E).
+intros (E); unfold; cases E (T f _ cTf); simplify; unfold Not; 
+intros (x y z H1 H2 H3); cases H3 (H4 H4); clear E H3; lapply (cTf ? ? y H4) as H5;
+cases H5; clear H5 H4 cTf; [1,4: apply H1|*:apply H2] clear H1 H2;
+[1,3:left|*:right] assumption;
+qed.
+
+lemma le_antisymmetric: ∀E.antisymmetric ? (le E) (eq E).
+intros (E); unfold; intros (x y Lxy Lyx); unfold; unfold; intros (H);
+cases H; [apply Lxy;|apply Lyx] assumption;
+qed.
+
+definition lt ≝ λE:excedence.λa,b:E. a ≤ b ∧ a # b.
+
+interpretation "ordered sets less than" 'lt 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); 
+apply (apart_coreflexive ? x ABS);
+qed.
+
+lemma lt_transitive: ∀E.transitive ? (lt E).
+intros (E); unfold; intros (x y z H1 H2); cases H1 (Lxy Axy); cases H2 (Lyz Ayz); 
+split; [apply (le_transitive ???? Lxy Lyz)] clear H1 H2;
+cases Axy (H1 H1); cases Ayz (H2 H2); [1:cases (Lxy H1)|3:cases (Lyz H2)]
+clear Axy Ayz;lapply (exc_cotransitive E) as c; unfold cotransitive in c;
+lapply (exc_coreflexive E) as r; unfold coreflexive in r;
+[1: lapply (c ?? y H1) as H3; cases H3 (H4 H4); [cases (Lxy H4)|cases (r ? H4)]
+|2: lapply (c ?? x H2) as H3; cases H3 (H4 H4); [right; assumption|cases (Lxy H4)]]
+qed.
+
+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.

diff --git a/matita/dama/ordered_sets.ma b/matita/dama/ordered_sets.ma
deleted file mode 100644 (file)
index a78d611..0000000
--- a/matita/dama/ordered_sets.ma
+++ /dev/null
@@ -1,117 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||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/excedence/".
-
-include "higher_order_defs/relations.ma".
-include "nat/plus.ma".
-include "constructive_connectives.ma".
-include "constructive_higher_order_relations.ma".
-
-record excedence : Type ≝ {
-  exc_carr:> Type;
-  exc_relation: exc_carr → exc_carr → Prop;
-  exc_coreflexive: coreflexive ? exc_relation;
-  exc_cotransitive: cotransitive ? exc_relation 
-}.
-
-interpretation "excedence" 'nleq 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/excedence/le.con _ a b).
-
-lemma le_reflexive: ∀E.reflexive ? (le E).
-intros (E); unfold; cases E; simplify; intros (x); apply (H x);
-qed.
-
-lemma le_transitive: ∀E.transitive ? (le E).
-intros (E); unfold; cases E; simplify; unfold Not; intros (x y z Rxy Ryz H2); 
-cases (c x z y H2) (H4 H5); clear H2; [exact (Rxy H4)|exact (Ryz H5)] 
-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/excedence/apart.con _ a b). 
-
-lemma apart_coreflexive: ∀E.coreflexive ? (apart E).
-intros (E); unfold; cases E; simplify; clear E; intros (x); unfold;
-intros (H1); apply (H x); cases H1; assumption;
-qed.
-
-lemma apart_symmetric: ∀E.symmetric ? (apart E).
-intros (E); unfold; intros(x y H); cases H; clear H; [right|left] assumption; 
-qed.
-
-lemma apart_cotrans: ∀E. cotransitive ? (apart E).
-intros (E); unfold; cases E (T f _ cTf); simplify; intros (x y z Axy);
-cases Axy (H); lapply (cTf ? ? z H) as H1; cases H1; clear Axy H1;
-[left; left|right; left|right; right|left; right] assumption.
-qed.
-
-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/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);
-apply (cRf x); cases H; assumption;
-qed.
-
-lemma eq_symmetric:∀E.symmetric ? (eq E).
-intros (E); unfold; unfold eq; unfold Not;
-intros (x y H1 H2); apply H1; cases H2; [right|left] assumption; 
-qed.
-
-lemma eq_transitive: ∀E.transitive ? (eq E).
-intros (E); unfold; cases E (T f _ cTf); simplify; unfold Not; 
-intros (x y z H1 H2 H3); cases H3 (H4 H4); clear E H3; lapply (cTf ? ? y H4) as H5;
-cases H5; clear H5 H4 cTf; [1,4: apply H1|*:apply H2] clear H1 H2;
-[1,3:left|*:right] assumption;
-qed.
-
-lemma le_antisymmetric: ∀E.antisymmetric ? (le E) (eq E).
-intros (E); unfold; intros (x y Lxy Lyx); unfold; unfold; intros (H);
-cases H; [apply Lxy;|apply Lyx] assumption;
-qed.
-
-definition lt ≝ λE:excedence.λa,b:E. a ≤ b ∧ a # b.
-
-interpretation "ordered sets less than" 'lt 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); 
-apply (apart_coreflexive ? x ABS);
-qed.
-
-lemma lt_transitive: ∀E.transitive ? (lt E).
-intros (E); unfold; intros (x y z H1 H2); cases H1 (Lxy Axy); cases H2 (Lyz Ayz); 
-split; [apply (le_transitive ???? Lxy Lyz)] clear H1 H2;
-cases Axy (H1 H1); cases Ayz (H2 H2); [1:cases (Lxy H1)|3:cases (Lyz H2)]
-clear Axy Ayz;lapply (exc_cotransitive E) as c; unfold cotransitive in c;
-lapply (exc_coreflexive E) as r; unfold coreflexive in r;
-[1: lapply (c ?? y H1) as H3; cases H3 (H4 H4); [cases (Lxy H4)|cases (r ? H4)]
-|2: lapply (c ?? x H2) as H3; cases H3 (H4 H4); [right; assumption|cases (Lxy H4)]]
-qed.
-
-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.