bir georganization, most of the structures done

[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..83dcbf7026c232d47380eeb71807fcfb34b14ce6 100644 (file)
--- a/helm/software/matita/dama/excedence.ma
+++ b/helm/software/matita/dama/excedence.ma
@@ -97,6 +97,7 @@ qed.
 lemma eq_trans:∀E:apartness.∀x,y,z:E.x ≈ y → y ≈ z → x ≈ z ≝ eq_trans_.
 
 (* 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;
@@ -160,3 +161,21 @@ 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.
+