src/HOL/ex/Executable_Relation.thy
 author griff Tue Apr 03 17:26:30 2012 +0900 (2012-04-03) changeset 47433 07f4bf913230 parent 47097 987cb55cac44 child 47435 e1b761c216ac permissions -rw-r--r--
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
1 theory Executable_Relation
2 imports Main
3 begin
5 subsection {* Preliminaries on the raw type of relations *}
7 definition rel_raw :: "'a set => ('a * 'a) set => ('a * 'a) set"
8 where
9   "rel_raw X R = Id_on X Un R"
11 lemma member_raw:
12   "(x, y) : (rel_raw X R) = ((x = y \<and> x : X) \<or> (x, y) : R)"
13 unfolding rel_raw_def by auto
16 lemma Id_raw:
17   "Id = rel_raw UNIV {}"
18 unfolding rel_raw_def by auto
20 lemma converse_raw:
21   "converse (rel_raw X R) = rel_raw X (converse R)"
22 unfolding rel_raw_def by auto
24 lemma union_raw:
25   "(rel_raw X R) Un (rel_raw Y S) = rel_raw (X Un Y) (R Un S)"
26 unfolding rel_raw_def by auto
28 lemma comp_Id_on:
29   "Id_on X O R = Set.project (%(x, y). x : X) R"
30 by (auto intro!: relcompI)
32 lemma comp_Id_on':
33   "R O Id_on X = Set.project (%(x, y). y : X) R"
34 by auto
36 lemma project_Id_on:
37   "Set.project (%(x, y). x : X) (Id_on Y) = Id_on (X Int Y)"
38 by auto
40 lemma relcomp_raw:
41   "(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))"
42 unfolding rel_raw_def
43 apply simp
44 apply (simp add: comp_Id_on)
45 apply (simp add: project_Id_on)
46 apply (simp add: comp_Id_on')
47 apply auto
48 done
50 lemma rtrancl_raw:
51   "(rel_raw X R)^* = rel_raw UNIV (R^+)"
52 unfolding rel_raw_def
53 apply auto
54 apply (metis Id_on_iff Un_commute iso_tuple_UNIV_I rtrancl_Un_separatorE rtrancl_eq_or_trancl)
55 by (metis in_rtrancl_UnI trancl_into_rtrancl)
57 lemma Image_raw:
58   "(rel_raw X R) `` S = (X Int S) Un (R `` S)"
59 unfolding rel_raw_def by auto
61 subsection {* A dedicated type for relations *}
63 subsubsection {* Definition of the dedicated type for relations *}
65 quotient_type 'a rel = "('a * 'a) set" / "(op =)"
66 morphisms set_of_rel rel_of_set by (metis identity_equivp)
68 lemma [simp]:
69   "rel_of_set (set_of_rel S) = S"
70 by (rule Quotient_abs_rep[OF Quotient_rel])
72 lemma [simp]:
73   "set_of_rel (rel_of_set R) = R"
74 by (rule Quotient_rep_abs[OF Quotient_rel]) (rule refl)
76 lemmas rel_raw_of_set_eqI[intro!] = arg_cong[where f="rel_of_set"]
78 quotient_definition rel where "rel :: 'a set => ('a * 'a) set => 'a rel" is rel_raw done
80 subsubsection {* Constant definitions on relations *}
82 hide_const (open) converse relcomp rtrancl Image
84 quotient_definition member :: "'a * 'a => 'a rel => bool" where
85   "member" is "Set.member :: 'a * 'a => ('a * 'a) set => bool" done
87 quotient_definition converse :: "'a rel => 'a rel"
88 where
89   "converse" is "Relation.converse :: ('a * 'a) set => ('a * 'a) set" done
91 quotient_definition union :: "'a rel => 'a rel => 'a rel"
92 where
93   "union" is "Set.union :: ('a * 'a) set => ('a * 'a) set => ('a * 'a) set" done
95 quotient_definition relcomp :: "'a rel => 'a rel => 'a rel"
96 where
97   "relcomp" is "Relation.relcomp :: ('a * 'a) set => ('a * 'a) set => ('a * 'a) set" done
99 quotient_definition rtrancl :: "'a rel => 'a rel"
100 where
101   "rtrancl" is "Transitive_Closure.rtrancl :: ('a * 'a) set => ('a * 'a) set" done
103 quotient_definition Image :: "'a rel => 'a set => 'a set"
104 where
105   "Image" is "Relation.Image :: ('a * 'a) set => 'a set => 'a set" done
107 subsubsection {* Code generation *}
109 code_datatype rel
111 lemma [code]:
112   "member (x, y) (rel X R) = ((x = y \<and> x : X) \<or> (x, y) : R)"
113 by (lifting member_raw)
115 lemma [code]:
116   "converse (rel X R) = rel X (R^-1)"
117 by (lifting converse_raw)
119 lemma [code]:
120   "union (rel X R) (rel Y S) = rel (X Un Y) (R Un S)"
121 by (lifting union_raw)
123 lemma [code]:
124    "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))"
125 by (lifting relcomp_raw)
127 lemma [code]:
128   "rtrancl (rel X R) = rel UNIV (R^+)"
129 by (lifting rtrancl_raw)
131 lemma [code]:
132   "Image (rel X R) S = (X Int S) Un (R `` S)"
133 by (lifting Image_raw)
135 quickcheck_generator rel constructors: rel
137 lemma
138   "member (x, (y :: nat)) (rtrancl (union R S)) \<Longrightarrow> member (x, y) (union (rtrancl R) (rtrancl S))"
139 quickcheck[exhaustive, expect = counterexample]
140 oops
142 end