definition lt ≝ λE:excess.λa,b:E. a ≤ b ∧ a # b.
-interpretation "ordered sets less than" 'lt a b = (cic:/matita/dama/excess/lt.con _ a b).
+interpretation "ordered sets less than" 'lt a b = (lt _ a b).
lemma lt_coreflexive: ∀E.coreflexive ? (lt E).
intros 2 (E x); intro H; cases H (_ ABS);
qed.
notation > "'Lt'≪" non associative with precedence 50 for @{'ltrewritel}.
-interpretation "lt_rewl" 'ltrewritel = (cic:/matita/dama/excess/lt_rewl.con _ _ _).
+interpretation "lt_rewl" 'ltrewritel = (lt_rewl _ _ _).
notation > "'Lt'≫" non associative with precedence 50 for @{'ltrewriter}.
-interpretation "lt_rewr" 'ltrewriter = (cic:/matita/dama/excess/lt_rewr.con _ _ _).
+interpretation "lt_rewr" 'ltrewriter = (lt_rewr _ _ _).
lemma lt_le_transitive: ∀A:excess.∀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)]