theory Executable_Relation

imports Main

begin

subsection {* Preliminaries on the raw type of relations *}

definition rel_raw :: "'a set => ('a * 'a) set => ('a * 'a) set"

where

  "rel_raw X R = Id_on X Un R"

lemma member_raw:

  "(x, y) : (rel_raw X R) = ((x = y \<and> x : X) \<or> (x, y) : R)"

unfolding rel_raw_def by auto

lemma Id_raw:

  "Id = rel_raw UNIV {}"

unfolding rel_raw_def by auto

lemma converse_raw:

  "converse (rel_raw X R) = rel_raw X (converse R)"

unfolding rel_raw_def by auto

lemma union_raw:

  "(rel_raw X R) Un (rel_raw Y S) = rel_raw (X Un Y) (R Un S)"

unfolding rel_raw_def by auto

lemma comp_Id_on:

  "Id_on X O R = Set.project (%(x, y). x : X) R"

by (auto intro!: rel_compI)

lemma comp_Id_on':

  "R O Id_on X = Set.project (%(x, y). y : X) R"

by auto

lemma project_Id_on:

  "Set.project (%(x, y). x : X) (Id_on Y) = Id_on (X Int Y)"

by auto

lemma rel_comp_raw:

  "(rel_raw X R) O (rel_raw Y S) = rel_raw (X Int Y) (Set.project (%(x, y). y : Y) R Un (Set.project (%(x, y). x : X) S Un R O S))"

unfolding rel_raw_def

apply simp

apply (simp add: comp_Id_on)

apply (simp add: project_Id_on)

apply (simp add: comp_Id_on')

apply auto

done

lemma rtrancl_raw:

  "(rel_raw X R)^* = rel_raw UNIV (R^+)"

unfolding rel_raw_def

apply auto

apply (metis Id_on_iff Un_commute iso_tuple_UNIV_I rtrancl_Un_separatorE rtrancl_eq_or_trancl)

by (metis in_rtrancl_UnI trancl_into_rtrancl)

lemma Image_raw:

  "(rel_raw X R) `` S = (X Int S) Un (R `` S)"

unfolding rel_raw_def by auto

subsection {* A dedicated type for relations *}

subsubsection {* Definition of the dedicated type for relations *}

quotient_type 'a rel = "('a * 'a) set" / "(op =)"

morphisms set_of_rel rel_of_set by (metis identity_equivp)

lemma [simp]:

  "rel_of_set (set_of_rel S) = S"

by (rule Quotient_abs_rep[OF Quotient_rel])

lemma [simp]:

  "set_of_rel (rel_of_set R) = R"

by (rule Quotient_rep_abs[OF Quotient_rel]) (rule refl)

lemmas rel_raw_of_set_eqI[intro!] = arg_cong[where f="rel_of_set"]

quotient_definition rel where "rel :: 'a set => ('a * 'a) set => 'a rel" is rel_raw done

subsubsection {* Constant definitions on relations *}

hide_const (open) converse rel_comp rtrancl Image

quotient_definition member :: "'a * 'a => 'a rel => bool" where

  "member" is "Set.member :: 'a * 'a => ('a * 'a) set => bool" done

quotient_definition converse :: "'a rel => 'a rel"

where

  "converse" is "Relation.converse :: ('a * 'a) set => ('a * 'a) set" done

quotient_definition union :: "'a rel => 'a rel => 'a rel"

where

  "union" is "Set.union :: ('a * 'a) set => ('a * 'a) set => ('a * 'a) set" done

quotient_definition rel_comp :: "'a rel => 'a rel => 'a rel"

where

  "rel_comp" is "Relation.rel_comp :: ('a * 'a) set => ('a * 'a) set => ('a * 'a) set" done

quotient_definition rtrancl :: "'a rel => 'a rel"

where

  "rtrancl" is "Transitive_Closure.rtrancl :: ('a * 'a) set => ('a * 'a) set" done

quotient_definition Image :: "'a rel => 'a set => 'a set"

where

  "Image" is "Relation.Image :: ('a * 'a) set => 'a set => 'a set" done

subsubsection {* Code generation *}

code_datatype rel

lemma [code]:

  "member (x, y) (rel X R) = ((x = y \<and> x : X) \<or> (x, y) : R)"

by (lifting member_raw)

lemma [code]:

  "converse (rel X R) = rel X (R^-1)"

by (lifting converse_raw)

lemma [code]:

  "union (rel X R) (rel Y S) = rel (X Un Y) (R Un S)"

by (lifting union_raw)

lemma [code]:

   "rel_comp (rel X R) (rel Y S) = rel (X Int Y) (Set.project (%(x, y). y : Y) R Un (Set.project (%(x, y). x : X) S Un R O S))"

by (lifting rel_comp_raw)

lemma [code]:

  "rtrancl (rel X R) = rel UNIV (R^+)"

by (lifting rtrancl_raw)

lemma [code]:

  "Image (rel X R) S = (X Int S) Un (R `` S)"

by (lifting Image_raw)

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