(* Author: Florian Haftmann, TU Muenchen *)
section \<open>Preorders with explicit equivalence relation\<close>
theory Preorder
imports HOL.Orderings
begin
class preorder_equiv = preorder
begin
definition equiv :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
where "equiv x y \<longleftrightarrow> x \<le> y \<and> y \<le> x"
notation
equiv ("op \<approx>") and
equiv ("(_/ \<approx> _)" [51, 51] 50)
lemma refl [iff]: "x \<approx> x"
by (simp add: equiv_def)
lemma trans: "x \<approx> y \<Longrightarrow> y \<approx> z \<Longrightarrow> x \<approx> z"
by (auto simp: equiv_def intro: order_trans)
lemma antisym: "x \<le> y \<Longrightarrow> y \<le> x \<Longrightarrow> x \<approx> y"
by (simp only: equiv_def)
lemma less_le: "x < y \<longleftrightarrow> x \<le> y \<and> \<not> x \<approx> y"
by (auto simp add: equiv_def less_le_not_le)
lemma le_less: "x \<le> y \<longleftrightarrow> x < y \<or> x \<approx> y"
by (auto simp add: equiv_def less_le)
lemma le_imp_less_or_eq: "x \<le> y \<Longrightarrow> x < y \<or> x \<approx> y"
by (simp add: less_le)
lemma less_imp_not_eq: "x < y \<Longrightarrow> x \<approx> y \<longleftrightarrow> False"
by (simp add: less_le)
lemma less_imp_not_eq2: "x < y \<Longrightarrow> y \<approx> x \<longleftrightarrow> False"
by (simp add: equiv_def less_le)
lemma neq_le_trans: "\<not> a \<approx> b \<Longrightarrow> a \<le> b \<Longrightarrow> a < b"
by (simp add: less_le)
lemma le_neq_trans: "a \<le> b \<Longrightarrow> \<not> a \<approx> b \<Longrightarrow> a < b"
by (simp add: less_le)
lemma antisym_conv: "y \<le> x \<Longrightarrow> x \<le> y \<longleftrightarrow> x \<approx> y"
by (simp add: equiv_def)
end
end