include "bishop_set.ma".
-coercion cic:/matita/dama/bishop_set/eq_sym.con.
+coercion eq_sym.
lemma eq_trans:∀E:bishop_set.∀x,z,y:E.x ≈ y → y ≈ z → x ≈ z ≝
λE,x,y,z.eq_trans_ E x z y.
interpretation "eq_rew" 'eqrewrite = (eq_trans _ _ _).
lemma le_rewl: ∀E:ordered_set.∀z,y,x:E. x ≈ y → x ≤ z → y ≤ z.
-intros (E z y x Exy Lxz); apply (le_transitive ???? ? Lxz);
+intros (E z y x Exy Lxz); apply (le_transitive ??? ? Lxz);
intro Xyz; apply Exy; right; assumption;
qed.
lemma le_rewr: ∀E:ordered_set.∀z,y,x:E. x ≈ y → z ≤ x → z ≤ y.
-intros (E z y x Exy Lxz); apply (le_transitive ???? Lxz);
+intros (E z y x Exy Lxz); apply (le_transitive ??? Lxz);
intro Xyz; apply Exy; left; assumption;
qed.
interpretation "ap_rewr" 'aprewriter = (ap_rewr _ _ _).
lemma exc_rewl: ∀A:ordered_set.∀x,z,y:A. x ≈ y → y ≰ z → x ≰ z.
-intros (A x z y Exy Ayz); cases (hos_cotransitive ??? x Ayz); [2:assumption]
+intros (A x z y Exy Ayz); cases (exc_cotransitive ?? x Ayz); [2:assumption]
cases Exy; right; assumption;
qed.
lemma exc_rewr: ∀A:ordered_set.∀x,z,y:A. x ≈ y → z ≰ y → z ≰ x.
-intros (A x z y Exy Azy); cases (hos_cotransitive ???x Azy); [assumption]
+intros (A x z y Exy Azy); cases (exc_cotransitive ??x Azy); [assumption]
cases (Exy); left; assumption;
qed.
notation > "'Ex'≫" non associative with precedence 50 for @{'ordered_setrewriter}.
interpretation "exc_rewr" 'ordered_setrewriter = (exc_rewr _ _ _).
-
+(*
lemma lt_rewr: ∀A:ordered_set.∀x,z,y:A. x ≈ y → z < y → z < x.
intros (A x y z E H); split; cases H;
[apply (Le≫ ? (eq_sym ??? E));|apply (Ap≫ ? E)] assumption;
interpretation "lt_rewl" 'ltrewritel = (lt_rewl _ _ _).
notation > "'Lt'≫" non associative with precedence 50 for @{'ltrewriter}.
interpretation "lt_rewr" 'ltrewriter = (lt_rewr _ _ _).
-
+*)