src/HOL/ex/Executable_Relation.thy

clarified context;

theory Executable_Relation
imports Main
begin

subsection \<open>A dedicated type for relations\<close>

subsubsection \<open>Definition of the dedicated type for relations\<close>

typedef 'a rel = "UNIV :: (('a * 'a) set) set"
morphisms set_of_rel rel_of_set by simp

setup_lifting type_definition_rel

lift_definition Rel :: "'a set => ('a * 'a) set => 'a rel" is "\<lambda> X R. Id_on X Un R" .

subsubsection \<open>Constant definitions on relations\<close>

hide_const (open) converse relcomp rtrancl Image

lift_definition member :: "'a * 'a => 'a rel => bool" is "Set.member" .

lift_definition converse :: "'a rel => 'a rel" is "Relation.converse" .

lift_definition union :: "'a rel => 'a rel => 'a rel" is "Set.union" .

lift_definition relcomp :: "'a rel => 'a rel => 'a rel" is "Relation.relcomp" .

lift_definition rtrancl :: "'a rel => 'a rel" is "Transitive_Closure.rtrancl" .

lift_definition Image :: "'a rel => 'a set => 'a set" is "Relation.Image" .

subsubsection \<open>Code generation\<close>

code_datatype Rel

lemma [code]:
  "member (x, y) (Rel X R) = ((x = y \<and> x \<in> X) \<or> (x, y) \<in> R)"
by transfer auto

lemma [code]:
  "converse (Rel X R) = Rel X (R\<inverse>)"
by transfer auto

lemma [code]:
  "union (Rel X R) (Rel Y S) = Rel (X Un Y) (R Un S)"
by transfer auto

lemma [code]:
   "relcomp (Rel X R) (Rel Y S) = Rel (X \<inter> Y) (Set.filter (\<lambda>(x, y). y \<in> Y) R \<union> (Set.filter (\<lambda>(x, y). x \<in> X) S \<union> R O S))"
by transfer (auto simp add: Id_on_eqI relcomp.simps)

lemma [code]:
  "rtrancl (Rel X R) = Rel UNIV (R\<^sup>+)"
apply transfer
apply auto
apply (metis Id_on_iff Un_commute UNIV_I rtrancl_Un_separatorE rtrancl_eq_or_trancl)
by (metis in_rtrancl_UnI trancl_into_rtrancl)

lemma [code]:
  "Image (Rel X R) S = (X Int S) Un (R `` S)"
by transfer auto

quickcheck_generator rel constructors: Rel

lemma
  "member (x, (y :: nat)) (rtrancl (union R S)) \<Longrightarrow> member (x, y) (union (rtrancl R) (rtrancl S))"
quickcheck[exhaustive, expect = counterexample]
oops

end