theory Executable_Relation
imports Main
begin
subsection ‹A dedicated type
for relations
›
subsubsection
‹Definition of the dedicated type
for relations
›
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 "\ X R. Id_on X Un R" .
subsubsection
‹Constant definitions on relations
›
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
‹Code generation
›
code_datatype Rel
lemma [code]:
"member (x, y) (Rel X R) = ((x = y \ x \ X) \ (x, y) \ R)"
by transfer auto
lemma [code]:
"converse (Rel X R) = Rel X (R\)"
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 \ Y) (Set.filter (\(x, y). y \ Y) R \ (Set.filter (\(x, y). x \ X) S \ 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)) \ member (x, y) (union (rtrancl R) (rtrancl S))"
quickcheck[exhaustive, expect = counterexample]
oops
end
