some work on uniformity

[helm.git] / helm / software / matita / contribs / dama / dama / bishop_set.ma

diff --git a/helm/software/matita/contribs/dama/dama/bishop_set.ma b/helm/software/matita/contribs/dama/dama/bishop_set.ma
index b877e094cd63d26dfc68045622e576a6c31b0d52..1e7436af9a3e97d347a47dc259c726537e2a06be 100644 (file)
--- a/helm/software/matita/contribs/dama/dama/bishop_set.ma
+++ b/helm/software/matita/contribs/dama/dama/bishop_set.ma
@@ -72,3 +72,29 @@ intros 5 (E x y Lxy Lyx); intro H;
 cases H; [apply Lxy;|apply Lyx] assumption;
 qed.
 
+lemma le_le_eq: ∀E:ordered_set.∀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.
+
+definition lt ≝ λE:ordered_set.λa,b:E. a ≤ b ∧ a # b.
+
+interpretation "ordered sets less than" 'lt a b = 
+  (cic:/matita/dama/bishop_set/lt.con _ a b).
+
+lemma lt_coreflexive: ∀E.coreflexive ? (lt E).
+intros 2 (E x); intro H; cases H (_ ABS);
+apply (bs_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;
+[1: cases (os_cotransitive ??? y H1) (X X); [cases (Lxy X)|cases (os_coreflexive ?? X)]
+|2: cases (os_cotransitive ??? x H2) (X X); [right;assumption|cases (Lxy X)]]
+qed.
+
+theorem lt_to_excess: ∀E:ordered_set.∀a,b:E. (a < b) → (b ≰ a).
+intros (E a b Lab); cases Lab (LEab Aab); cases Aab (H H);[cases (LEab H)]
+assumption;
+qed.