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
```
```     4
```
```     5 subsection {* Preliminaries on the raw type of relations *}
```
```     6
```
```     7 definition rel_raw :: "'a set => ('a * 'a) set => ('a * 'a) set"
```
```     8 where
```
```     9   "rel_raw X R = Id_on X Un R"
```
```    10
```
```    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
```
```    14
```
```    15
```
```    16 lemma Id_raw:
```
```    17   "Id = rel_raw UNIV {}"
```
```    18 unfolding rel_raw_def by auto
```
```    19
```
```    20 lemma converse_raw:
```
```    21   "converse (rel_raw X R) = rel_raw X (converse R)"
```
```    22 unfolding rel_raw_def by auto
```
```    23
```
```    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
```
```    27
```
```    28 lemma comp_Id_on:
```
```    29   "Id_on X O R = Set.project (%(x, y). x : X) R"
```
```    30 by (auto intro!: relcompI)
```
```    31
```
```    32 lemma comp_Id_on':
```
```    33   "R O Id_on X = Set.project (%(x, y). y : X) R"
```
```    34 by auto
```
```    35
```
```    36 lemma project_Id_on:
```
```    37   "Set.project (%(x, y). x : X) (Id_on Y) = Id_on (X Int Y)"
```
```    38 by auto
```
```    39
```
```    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
```
```    49
```
```    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)
```
```    56
```
```    57 lemma Image_raw:
```
```    58   "(rel_raw X R) `` S = (X Int S) Un (R `` S)"
```
```    59 unfolding rel_raw_def by auto
```
```    60
```
```    61 subsection {* A dedicated type for relations *}
```
```    62
```
```    63 subsubsection {* Definition of the dedicated type for relations *}
```
```    64
```
```    65 quotient_type 'a rel = "('a * 'a) set" / "(op =)"
```
```    66 morphisms set_of_rel rel_of_set by (metis identity_equivp)
```
```    67
```
```    68 lemma [simp]:
```
```    69   "rel_of_set (set_of_rel S) = S"
```
```    70 by (rule Quotient_abs_rep[OF Quotient_rel])
```
```    71
```
```    72 lemma [simp]:
```
```    73   "set_of_rel (rel_of_set R) = R"
```
```    74 by (rule Quotient_rep_abs[OF Quotient_rel]) (rule refl)
```
```    75
```
```    76 lemmas rel_raw_of_set_eqI[intro!] = arg_cong[where f="rel_of_set"]
```
```    77
```
```    78 quotient_definition rel where "rel :: 'a set => ('a * 'a) set => 'a rel" is rel_raw done
```
```    79
```
```    80 subsubsection {* Constant definitions on relations *}
```
```    81
```
```    82 hide_const (open) converse relcomp rtrancl Image
```
```    83
```
```    84 quotient_definition member :: "'a * 'a => 'a rel => bool" where
```
```    85   "member" is "Set.member :: 'a * 'a => ('a * 'a) set => bool" done
```
```    86
```
```    87 quotient_definition converse :: "'a rel => 'a rel"
```
```    88 where
```
```    89   "converse" is "Relation.converse :: ('a * 'a) set => ('a * 'a) set" done
```
```    90
```
```    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
```
```    94
```
```    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
```
```    98
```
```    99 quotient_definition rtrancl :: "'a rel => 'a rel"
```
```   100 where
```
```   101   "rtrancl" is "Transitive_Closure.rtrancl :: ('a * 'a) set => ('a * 'a) set" done
```
```   102
```
```   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
```
```   106
```
```   107 subsubsection {* Code generation *}
```
```   108
```
```   109 code_datatype rel
```
```   110
```
```   111 lemma [code]:
```
```   112   "member (x, y) (rel X R) = ((x = y \<and> x : X) \<or> (x, y) : R)"
```
```   113 by (lifting member_raw)
```
```   114
```
```   115 lemma [code]:
```
```   116   "converse (rel X R) = rel X (R^-1)"
```
```   117 by (lifting converse_raw)
```
```   118
```
```   119 lemma [code]:
```
```   120   "union (rel X R) (rel Y S) = rel (X Un Y) (R Un S)"
```
```   121 by (lifting union_raw)
```
```   122
```
```   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)
```
```   126
```
```   127 lemma [code]:
```
```   128   "rtrancl (rel X R) = rel UNIV (R^+)"
```
```   129 by (lifting rtrancl_raw)
```
```   130
```
```   131 lemma [code]:
```
```   132   "Image (rel X R) S = (X Int S) Un (R `` S)"
```
```   133 by (lifting Image_raw)
```
```   134
```
```   135 quickcheck_generator rel constructors: rel
```
```   136
```
```   137 lemma
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```   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
```
```   141
```
```   142 end
```