src/HOL/ex/Executable_Relation.thy

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/ex/Executable_Relation.thy	Thu Feb 02 10:12:11 2012 +0100
@@ -0,0 +1,79 @@
+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