added do_heavy_checks default

[helm.git] / helm / software / matita / dama / excedence.ma

diff --git a/helm/software/matita/dama/excedence.ma b/helm/software/matita/dama/excedence.ma
index c2a5ffd4fde20049391060de83b85f2cd22cf310..fa6848e120314b987eaa0f61bf81cec57d42dea6 100644 (file)
--- a/helm/software/matita/dama/excedence.ma
+++ b/helm/software/matita/dama/excedence.ma
@@ -52,9 +52,11 @@ record apartness : Type ≝ {
 }.
 
 notation "a break # b" non associative with precedence 50 for @{ 'apart $a $b}.
-interpretation "axiomatic apartness" 'apart x y = 
+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.
@@ -84,7 +86,7 @@ intros (E); unfold; intros (x y Exy); unfold; unfold; intros (Ayx); apply Exy;
 apply ap_symmetric; assumption; 
 qed.
 
-lemma eq_sym:∀E:apartness.∀x,y:E.x ≈ y → y ≈ x := eq_sym_.
+lemma eq_sym:∀E:apartness.∀x,y:E.x ≈ y → y ≈ x ≝ eq_sym_.
 
 coercion cic:/matita/excedence/eq_sym.con.
 
@@ -94,9 +96,17 @@ 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,y,z:E.x ≈ y → y ≈ z → x ≈ z ≝ eq_trans_.
+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;
@@ -136,11 +146,23 @@ 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));
@@ -160,3 +182,44 @@ 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.
+