X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fdama%2Fexcedence.ma;h=fa6848e120314b987eaa0f61bf81cec57d42dea6;hb=c0f86a886451a0df3b42a1435e21b5def9f34792;hp=59b8baa4ed1668cae080c58dcaec9cb1fcc99f39;hpb=624a7c13a2ed22ed2535690074c7a08e18de7f13;p=helm.git

diff --git a/helm/software/matita/dama/excedence.ma b/helm/software/matita/dama/excedence.ma
index 59b8baa4e..fa6848e12 100644
--- a/helm/software/matita/dama/excedence.ma
+++ b/helm/software/matita/dama/excedence.ma
@@ -21,7 +21,7 @@ include "constructive_higher_order_relations.ma".
 
 record excedence : Type ≝ {
   exc_carr:> Type;
-  exc_relation: exc_carr → exc_carr → Prop;
+  exc_relation: exc_carr → exc_carr → Type;
   exc_coreflexive: coreflexive ? exc_relation;
   exc_cotransitive: cotransitive ? exc_relation 
 }.
@@ -51,10 +51,12 @@ record apartness : Type ≝ {
   ap_cotransitive: cotransitive ? ap_apart
 }.
 
-notation "a # b" non associative with precedence 50 for @{ 'apart $a $b}.
-interpretation "axiomatic apartness" 'apart x y = 
+notation "a break # b" non associative with precedence 50 for @{ 'apart $a $b}.
+interpretation "apartness" 'apart x y = 
   (cic:/matita/excedence/ap_apart.con _ x y).
 
+definition strong_ext ≝ λA:apartness.λop:A→A.∀x,y. op x # op y → x # y.
+
 definition apart ≝ λE:excedence.λa,b:E. a ≰ b ∨ b ≰ a.
 
 definition apart_of_excedence: excedence → apartness.
@@ -63,7 +65,7 @@ intros (E); apply (mk_apartness E (apart E));
     intros (H1); apply (H x); cases H1; assumption;
 |2: unfold; intros(x y H); cases H; clear H; [right|left] assumption;
 |3: 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;
+    cases Axy (H H); lapply (cTf ? ? z H) as H1; cases H1; clear Axy H1;
     [left; left|right; left|right; right|left; right] assumption]
 qed.
 
@@ -71,7 +73,7 @@ coercion cic:/matita/excedence/apart_of_excedence.con.
 
 definition eq ≝ λA:apartness.λa,b:A. ¬ (a # b).
 
-notation "a ≈ b" non associative with precedence 50 for @{ 'napart $a $b}.    
+notation "a break ≈ b" non associative with precedence 50 for @{ 'napart $a $b}.    
 interpretation "alikeness" 'napart a b =
   (cic:/matita/excedence/eq.con _ a b). 
 
@@ -79,17 +81,32 @@ lemma eq_reflexive:∀E. reflexive ? (eq E).
 intros (E); unfold; intros (x); apply ap_coreflexive; 
 qed.
 
-lemma eq_symmetric:∀E.symmetric ? (eq E).
+lemma eq_sym_:∀E.symmetric ? (eq E).
 intros (E); unfold; intros (x y Exy); unfold; unfold; intros (Ayx); apply Exy;
 apply ap_symmetric; assumption; 
 qed.
 
-lemma eq_transitive: ∀E.transitive ? (eq E).
+lemma eq_sym:∀E:apartness.∀x,y:E.x ≈ y → y ≈ x ≝ eq_sym_.
+
+coercion cic:/matita/excedence/eq_sym.con.
+
+lemma eq_trans_: ∀E.transitive ? (eq E).
 (* bug. intros k deve fare whd quanto basta *)
 intros 6 (E x y z Exy Eyz); intro Axy; cases (ap_cotransitive ???y Axy); 
 [apply Exy|apply Eyz] assumption.
 qed.
+
+lemma eq_trans:∀E:apartness.∀x,z,y:E.x ≈ y → y ≈ z → x ≈ z ≝ 
+  λE,x,y,z.eq_trans_ E x z y.
+
+notation > "'Eq'≈" non associative with precedence 50 for
+ @{'eqrewrite}.
+ 
+interpretation "eq_rew" 'eqrewrite = 
+ (cic:/matita/excedence/eq_trans.con _ _ _).
+
 (* BUG: vedere se ricapita *)
+alias id "antisymmetric" = "cic:/matita/constructive_higher_order_relations/antisymmetric.con".
 lemma le_antisymmetric: ∀E.antisymmetric ? (le E) (eq ?).
 intros 5 (E x y Lxy Lyx); intro H;
 cases H; [apply Lxy;|apply Lyx] assumption;
@@ -120,12 +137,8 @@ intros (E a b Lab); cases Lab (LEab Aab);
 cases Aab (H H); [cases (LEab H)] fold normalize (b ≰ a); assumption; (* BUG *)  
 qed.
 
-theorem le_le_to_eq: ∀E:excedence.∀x,y:E. x ≤ y → y ≤ x → x ≈ y.
-intros 6 (E x y L1 L2 H); cases H; [apply (L1 H1)|apply (L2 H1)]
-qed.
-
 lemma unfold_apart: ∀E:excedence. ∀x,y:E. x ≰ y ∨ y ≰ x → x # y.
-unfold apart_of_excedence; unfold apart; simplify; intros; assumption;
+intros; assumption;
 qed.
 
 lemma le_rewl: ∀E:excedence.∀z,y,x:E. x ≈ y → x ≤ z → y ≤ z.
@@ -133,7 +146,80 @@ intros (E z y x Exy Lxz); apply (le_transitive ???? ? Lxz);
 intro Xyz; apply Exy; apply unfold_apart; right; assumption;
 qed.
 
+notation > "'Ex'≪" non associative with precedence 50 for
+ @{'excedencerewritel}.
+ 
+interpretation "exc_rewl" 'excedencerewritel = 
+ (cic:/matita/excedence/exc_rewl.con _ _ _).
+
 lemma le_rewr: ∀E:excedence.∀z,y,x:E. x ≈ y → z ≤ x → z ≤ y.
 intros (E z y x Exy Lxz); apply (le_transitive ???? Lxz);
 intro Xyz; apply Exy; apply unfold_apart; left; assumption;
 qed.
+
+notation > "'Ex'≫" non associative with precedence 50 for
+ @{'excedencerewriter}.
+ 
+interpretation "exc_rewr" 'excedencerewriter = 
+ (cic:/matita/excedence/exc_rewr.con _ _ _).
+
+lemma ap_rewl: ∀A:apartness.∀x,z,y:A. x ≈ y → y # z → x # z.
+intros (A x z y Exy Ayz); cases (ap_cotransitive ???x Ayz); [2:assumption]
+cases (Exy (ap_symmetric ??? a));
+qed.
+  
+lemma ap_rewr: ∀A:apartness.∀x,z,y:A. x ≈ y → z # y → z # x.
+intros (A x z y Exy Azy); apply ap_symmetric; apply (ap_rewl ???? Exy);
+apply ap_symmetric; assumption;
+qed.
+
+lemma exc_rewl: ∀A:excedence.∀x,z,y:A. x ≈ y → y ≰ z → x ≰ z.
+intros (A x z y Exy Ayz); elim (exc_cotransitive ???x Ayz); [2:assumption]
+cases Exy; right; assumption;
+qed.
+  
+lemma exc_rewr: ∀A:excedence.∀x,z,y:A. x ≈ y → z ≰ y → z ≰ x.
+intros (A x z y Exy Azy); elim (exc_cotransitive ???x Azy); [assumption]
+elim (Exy); left; assumption;
+qed.
+
+lemma lt_rewr: ∀A:excedence.∀x,z,y:A. x ≈ y → z < y → z < x.
+intros (A x y z E H); split; elim H; 
+[apply (le_rewr ???? (eq_sym ??? E));|apply (ap_rewr ???? E)] assumption;
+qed.
+
+lemma lt_rewl: ∀A:excedence.∀x,z,y:A. x ≈ y → y < z → x < z.
+intros (A x y z E H); split; elim H; 
+[apply (le_rewl ???? (eq_sym ??? E));| apply (ap_rewl ???? E);] assumption;
+qed.
+
+lemma lt_le_transitive: ∀A:excedence.∀x,y,z:A.x < y → y ≤ z → x < z.
+intros (A x y z LT LE); cases LT (LEx APx); split; [apply (le_transitive ???? LEx LE)]
+whd in LE LEx APx; cases APx (EXx EXx); [cases (LEx EXx)]
+cases (exc_cotransitive ??? z EXx) (EXz EXz); [cases (LE EXz)]
+right; assumption;
+qed.
+
+lemma le_lt_transitive: ∀A:excedence.∀x,y,z:A.x ≤ y → y < z → x < z.
+intros (A x y z LE LT); cases LT (LEx APx); split; [apply (le_transitive ???? LE LEx)]
+whd in LE LEx APx; cases APx (EXx EXx); [cases (LEx EXx)]
+cases (exc_cotransitive ??? x EXx) (EXz EXz); [right; assumption]
+cases LE; assumption;
+qed.
+    
+lemma le_le_eq: ∀E:excedence.∀a,b:E. a ≤ b → b ≤ a → a ≈ b.
+intros (E x y L1 L2); intro H; cases H; [apply L1|apply L2] assumption;
+qed.
+
+lemma eq_le_le: ∀E:excedence.∀a,b:E. a ≈ b → a ≤ b ∧ b ≤ a.
+intros (E x y H); unfold apart_of_excedence in H; unfold apart in H;
+simplify in H; split; intro; apply H; [left|right] assumption.
+qed.
+
+lemma ap_le_to_lt: ∀E:excedence.∀a,c:E.c # a → c ≤ a → c < a.
+intros; split; assumption;
+qed.
+
+definition total_order_property : ∀E:excedence. Type ≝
+  λE:excedence. ∀a,b:E. a ≰ b → b < a.
+