Theory Combine_PER

(*  Author:     Florian Haftmann, TU Muenchen *)

section ‹A combinator to build partial equivalence relations from a predicate and an equivalence relation›

theory Combine_PER
  imports Main

unbundle lattice_syntax

definition combine_per :: "('a  bool)  ('a  'a  bool)  'a  'a  bool"
  where "combine_per P R = (λx y. P x  P y)  R"

lemma combine_per_simp [simp]:
  "combine_per P R x y  P x  P y  x  y" for R (infixl "" 50)
  by (simp add: combine_per_def)

lemma combine_per_top [simp]: "combine_per  R = R"
  by (simp add: fun_eq_iff)

lemma combine_per_eq [simp]: "combine_per P HOL.eq = HOL.eq  (λx y. P x)"
  by (auto simp add: fun_eq_iff)

lemma symp_combine_per: "symp R  symp (combine_per P R)"
  by (auto simp add: symp_def sym_def combine_per_def)

lemma transp_combine_per: "transp R  transp (combine_per P R)"
  by (auto simp add: transp_def trans_def combine_per_def)

lemma combine_perI: "P x  P y  x  y  combine_per P R x y" for R (infixl "" 50)
  by (simp add: combine_per_def)

lemma symp_combine_per_symp: "symp R  symp (combine_per P R)"
  by (auto intro!: sympI elim: sympE)

lemma transp_combine_per_transp: "transp R  transp (combine_per P R)"
  by (auto intro!: transpI elim: transpE)

lemma equivp_combine_per_part_equivp [intro?]:
  fixes R (infixl "" 50)
  assumes "x. P x" and "equivp R"
  shows "part_equivp (combine_per P R)"
proof -
  from x. P x obtain x where "P x" ..
  moreover from equivp R have "x  x"
    by (rule equivp_reflp)
  ultimately have "x. P x  x  x"
    by blast
  with equivp R show ?thesis
    by (auto intro!: part_equivpI symp_combine_per_symp transp_combine_per_transp
      elim: equivpE)