--- a/src/HOL/ex/Executable_Relation.thy Tue May 08 14:31:03 2012 +0200
+++ b/src/HOL/ex/Executable_Relation.thy Tue May 08 14:35:13 2012 +0200
@@ -2,137 +2,65 @@
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!: relcompI)
-
-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 relcomp_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 Quotient3_abs_rep[OF Quotient3_rel])
+typedef (open) 'a rel = "UNIV :: (('a * 'a) set) set"
+morphisms set_of_rel rel_of_set by simp
-lemma [simp]:
- "set_of_rel (rel_of_set R) = R"
-by (rule Quotient3_rep_abs[OF Quotient3_rel]) (rule refl)
+setup_lifting type_definition_rel
-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
+lift_definition Rel :: "'a set => ('a * 'a) set => 'a rel" is "\<lambda> X R. Id_on X Un R" .
subsubsection {* Constant definitions on relations *}
hide_const (open) converse relcomp rtrancl Image
-quotient_definition member :: "'a * 'a => 'a rel => bool" where
- "member" is "Set.member :: 'a * 'a => ('a * 'a) set => bool" done
+lift_definition member :: "'a * 'a => 'a rel => bool" is "Set.member" .
-quotient_definition converse :: "'a rel => 'a rel"
-where
- "converse" is "Relation.converse :: ('a * 'a) set => ('a * 'a) set" done
+lift_definition converse :: "'a rel => 'a rel" is "Relation.converse" .
-quotient_definition union :: "'a rel => 'a rel => 'a rel"
-where
- "union" is "Set.union :: ('a * 'a) set => ('a * 'a) set => ('a * 'a) set" done
+lift_definition union :: "'a rel => 'a rel => 'a rel" is "Set.union" .
-quotient_definition relcomp :: "'a rel => 'a rel => 'a rel"
-where
- "relcomp" is "Relation.relcomp :: ('a * 'a) set => ('a * 'a) set => ('a * 'a) set" done
+lift_definition relcomp :: "'a rel => 'a rel => 'a rel" is "Relation.relcomp" .
-quotient_definition rtrancl :: "'a rel => 'a rel"
-where
- "rtrancl" is "Transitive_Closure.rtrancl :: ('a * 'a) set => ('a * 'a) set" done
+lift_definition rtrancl :: "'a rel => 'a rel" is "Transitive_Closure.rtrancl" .
-quotient_definition Image :: "'a rel => 'a set => 'a set"
-where
- "Image" is "Relation.Image :: ('a * 'a) set => 'a set => 'a set" done
+lift_definition Image :: "'a rel => 'a set => 'a set" is "Relation.Image" .
subsubsection {* Code generation *}
-code_datatype rel
+code_datatype Rel
+
+lemma [code]:
+ "member (x, y) (Rel X R) = ((x = y \<and> x : X) \<or> (x, y) : R)"
+by transfer auto
lemma [code]:
- "member (x, y) (rel X R) = ((x = y \<and> x : X) \<or> (x, y) : R)"
-by (lifting member_raw)
+ "converse (Rel X R) = Rel X (R^-1)"
+by transfer auto
lemma [code]:
- "converse (rel X R) = rel X (R^-1)"
-by (lifting converse_raw)
+ "union (Rel X R) (Rel Y S) = Rel (X Un Y) (R Un S)"
+by transfer auto
lemma [code]:
- "union (rel X R) (rel Y S) = rel (X Un Y) (R Un S)"
-by (lifting union_raw)
-
-lemma [code]:
- "relcomp (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 relcomp_raw)
+ "relcomp (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 transfer (auto simp add: Id_on_eqI relcomp.simps)
lemma [code]:
- "rtrancl (rel X R) = rel UNIV (R^+)"
-by (lifting rtrancl_raw)
+ "rtrancl (Rel X R) = Rel UNIV (R^+)"
+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 (lifting Image_raw)
+ "Image (Rel X R) S = (X Int S) Un (R `` S)"
+by transfer auto
-quickcheck_generator rel constructors: rel
+quickcheck_generator rel constructors: Rel
lemma
"member (x, (y :: nat)) (rtrancl (union R S)) \<Longrightarrow> member (x, y) (union (rtrancl R) (rtrancl S))"