(* Author: Florian Haftmann, TU Muenchen *)

 header {* Preorders with explicit equivalence relation *}

 theory Preorder

 imports 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 ~~") and

   equiv ("(_/ ~~ _)" [51, 51] 50)

 notation (xsymbols)

   equiv ("op \<approx>") and

   equiv ("(_/ \<approx> _)"  [51, 51] 50)

 notation (HTML output)

   equiv ("op \<approx>") and

   equiv ("(_/ \<approx> _)"  [51, 51] 50)

 lemma refl [iff]:

   "x \<approx> x"

   unfolding equiv_def by simp

 lemma trans:

   "x \<approx> y \<Longrightarrow> y \<approx> z \<Longrightarrow> x \<approx> z"

   unfolding equiv_def by (auto intro: order_trans)

 lemma antisym:

   "x \<le> y \<Longrightarrow> y \<le> x \<Longrightarrow> x \<approx> y"

   unfolding 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