adding an example for a datatype refinement which would allow rtrancl to be executable on an infinite type
theory Executable_Relation
imports Main
begin
text {*
Current problem: rtrancl is not executable on an infinite type.
*}
lemma
"(x, (y :: nat)) : rtrancl (R Un S) \<Longrightarrow> (x, y) : (rtrancl R) Un (rtrancl S)"
(* quickcheck[exhaustive] fails ! *)
oops
code_thms rtrancl
hide_const (open) rtrancl trancl
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)
no_notation
Set.member ("(_/ : _)" [50, 51] 50)
quotient_definition member :: "'a * 'a => 'a rel => bool" where
"member" is "Set.member :: 'a * 'a => ('a * 'a) set => bool"
notation
member ("(_/ : _)" [50, 51] 50)
quotient_definition union :: "'a rel => 'a rel => 'a rel" where
"union" is "Set.union :: ('a * 'a) set => ('a * 'a) set => ('a * 'a) set"
quotient_definition rtrancl :: "'a rel => 'a rel" where
"rtrancl" is "Transitive_Closure.rtrancl :: ('a * 'a) set => ('a * 'a) set"
definition reflcl_raw
where "reflcl_raw R = R \<union> Id"
quotient_definition reflcl :: "('a * 'a) set => 'a rel" where
"reflcl" is "reflcl_raw :: ('a * 'a) set => ('a * 'a) set"
code_datatype reflcl rel_of_set
lemma member_code[code]:
"(x, y) : rel_of_set R = Set.member (x, y) R"
"(x, y) : reflcl R = ((x = y) \<or> Set.member (x, y) R)"
unfolding member_def reflcl_def reflcl_raw_def map_fun_def_raw o_def id_def
by auto
lemma union_code[code]:
"union (rel_of_set R) (rel_of_set S) = rel_of_set (Set.union R S)"
"union (reflcl R) (rel_of_set S) = reflcl (Set.union R S)"
"union (reflcl R) (reflcl S) = reflcl (Set.union R S)"
"union (rel_of_set R) (reflcl S) = reflcl (Set.union R S)"
unfolding union_def reflcl_def reflcl_raw_def map_fun_def_raw o_def id_def
by (auto intro: arg_cong[where f=rel_of_set])
lemma rtrancl_code[code]:
"rtrancl (rel_of_set R) = reflcl (Transitive_Closure.trancl R)"
"rtrancl (reflcl R) = reflcl (Transitive_Closure.trancl R)"
unfolding rtrancl_def reflcl_def reflcl_raw_def map_fun_def_raw o_def id_def
by (auto intro: arg_cong[where f=rel_of_set])
quickcheck_generator rel constructors: rel_of_set
lemma
"(x, (y :: nat)) : rtrancl (union R S) \<Longrightarrow> (x, y) : (union (rtrancl R) (rtrancl S))"
quickcheck[exhaustive]
oops
end