src/HOL/Relation.thy
author haftmann
Sat Mar 17 08:00:18 2012 +0100 (2012-03-17)
changeset 46981 d54cea5b64e4
parent 46884 154dc6ec0041
child 46982 144d94446378
permissions -rw-r--r--
generalized INF_INT_eq, SUP_UN_eq
wenzelm@10358
     1
(*  Title:      HOL/Relation.thy
haftmann@46664
     2
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory; Stefan Berghofer, TU Muenchen
nipkow@1128
     3
*)
nipkow@1128
     4
haftmann@46664
     5
header {* Relations – as sets of pairs, and binary predicates *}
berghofe@12905
     6
nipkow@15131
     7
theory Relation
haftmann@32850
     8
imports Datatype Finite_Set
nipkow@15131
     9
begin
paulson@5978
    10
haftmann@46694
    11
text {* A preliminary: classical rules for reasoning on predicates *}
haftmann@46664
    12
noschinl@46882
    13
declare predicate1I [Pure.intro!, intro!]
noschinl@46882
    14
declare predicate1D [Pure.dest, dest]
haftmann@46664
    15
declare predicate2I [Pure.intro!, intro!]
haftmann@46664
    16
declare predicate2D [Pure.dest, dest]
haftmann@46767
    17
declare bot1E [elim!] 
haftmann@46664
    18
declare bot2E [elim!]
haftmann@46664
    19
declare top1I [intro!]
haftmann@46664
    20
declare top2I [intro!]
haftmann@46664
    21
declare inf1I [intro!]
haftmann@46664
    22
declare inf2I [intro!]
haftmann@46664
    23
declare inf1E [elim!]
haftmann@46664
    24
declare inf2E [elim!]
haftmann@46664
    25
declare sup1I1 [intro?]
haftmann@46664
    26
declare sup2I1 [intro?]
haftmann@46664
    27
declare sup1I2 [intro?]
haftmann@46664
    28
declare sup2I2 [intro?]
haftmann@46664
    29
declare sup1E [elim!]
haftmann@46664
    30
declare sup2E [elim!]
haftmann@46664
    31
declare sup1CI [intro!]
haftmann@46664
    32
declare sup2CI [intro!]
haftmann@46664
    33
declare INF1_I [intro!]
haftmann@46664
    34
declare INF2_I [intro!]
haftmann@46664
    35
declare INF1_D [elim]
haftmann@46664
    36
declare INF2_D [elim]
haftmann@46664
    37
declare INF1_E [elim]
haftmann@46664
    38
declare INF2_E [elim]
haftmann@46664
    39
declare SUP1_I [intro]
haftmann@46664
    40
declare SUP2_I [intro]
haftmann@46664
    41
declare SUP1_E [elim!]
haftmann@46664
    42
declare SUP2_E [elim!]
haftmann@46664
    43
haftmann@46694
    44
subsection {* Fundamental *}
haftmann@46664
    45
haftmann@46694
    46
subsubsection {* Relations as sets of pairs *}
haftmann@46694
    47
haftmann@46694
    48
type_synonym 'a rel = "('a * 'a) set"
haftmann@46694
    49
haftmann@46694
    50
lemma subrelI: -- {* Version of @{thm [source] subsetI} for binary relations *}
haftmann@46694
    51
  "(\<And>x y. (x, y) \<in> r \<Longrightarrow> (x, y) \<in> s) \<Longrightarrow> r \<subseteq> s"
haftmann@46694
    52
  by auto
haftmann@46694
    53
haftmann@46694
    54
lemma lfp_induct2: -- {* Version of @{thm [source] lfp_induct} for binary relations *}
haftmann@46694
    55
  "(a, b) \<in> lfp f \<Longrightarrow> mono f \<Longrightarrow>
haftmann@46694
    56
    (\<And>a b. (a, b) \<in> f (lfp f \<inter> {(x, y). P x y}) \<Longrightarrow> P a b) \<Longrightarrow> P a b"
haftmann@46694
    57
  using lfp_induct_set [of "(a, b)" f "prod_case P"] by auto
haftmann@46694
    58
haftmann@46694
    59
haftmann@46694
    60
subsubsection {* Conversions between set and predicate relations *}
haftmann@46664
    61
haftmann@46833
    62
lemma pred_equals_eq [pred_set_conv]: "(\<lambda>x. x \<in> R) = (\<lambda>x. x \<in> S) \<longleftrightarrow> R = S"
haftmann@46664
    63
  by (simp add: set_eq_iff fun_eq_iff)
haftmann@46664
    64
haftmann@46833
    65
lemma pred_equals_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) = (\<lambda>x y. (x, y) \<in> S) \<longleftrightarrow> R = S"
haftmann@46664
    66
  by (simp add: set_eq_iff fun_eq_iff)
haftmann@46664
    67
haftmann@46833
    68
lemma pred_subset_eq [pred_set_conv]: "(\<lambda>x. x \<in> R) \<le> (\<lambda>x. x \<in> S) \<longleftrightarrow> R \<subseteq> S"
haftmann@46664
    69
  by (simp add: subset_iff le_fun_def)
haftmann@46664
    70
haftmann@46833
    71
lemma pred_subset_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) \<le> (\<lambda>x y. (x, y) \<in> S) \<longleftrightarrow> R \<subseteq> S"
haftmann@46664
    72
  by (simp add: subset_iff le_fun_def)
haftmann@46664
    73
noschinl@46883
    74
lemma bot_empty_eq [pred_set_conv]: "\<bottom> = (\<lambda>x. x \<in> {})"
haftmann@46689
    75
  by (auto simp add: fun_eq_iff)
haftmann@46689
    76
noschinl@46883
    77
lemma bot_empty_eq2 [pred_set_conv]: "\<bottom> = (\<lambda>x y. (x, y) \<in> {})"
haftmann@46664
    78
  by (auto simp add: fun_eq_iff)
haftmann@46664
    79
noschinl@46883
    80
lemma top_empty_eq [pred_set_conv]: "\<top> = (\<lambda>x. x \<in> UNIV)"
noschinl@46883
    81
  by (auto simp add: fun_eq_iff)
haftmann@46689
    82
noschinl@46883
    83
lemma top_empty_eq2 [pred_set_conv]: "\<top> = (\<lambda>x y. (x, y) \<in> UNIV)"
noschinl@46883
    84
  by (auto simp add: fun_eq_iff)
haftmann@46664
    85
haftmann@46664
    86
lemma inf_Int_eq [pred_set_conv]: "(\<lambda>x. x \<in> R) \<sqinter> (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<inter> S)"
haftmann@46664
    87
  by (simp add: inf_fun_def)
haftmann@46664
    88
haftmann@46664
    89
lemma inf_Int_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) \<sqinter> (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<inter> S)"
haftmann@46664
    90
  by (simp add: inf_fun_def)
haftmann@46664
    91
haftmann@46664
    92
lemma sup_Un_eq [pred_set_conv]: "(\<lambda>x. x \<in> R) \<squnion> (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<union> S)"
haftmann@46664
    93
  by (simp add: sup_fun_def)
haftmann@46664
    94
haftmann@46664
    95
lemma sup_Un_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) \<squnion> (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<union> S)"
haftmann@46664
    96
  by (simp add: sup_fun_def)
haftmann@46664
    97
haftmann@46981
    98
lemma INF_INT_eq [pred_set_conv]: "(\<Sqinter>i\<in>S. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (\<Inter>i\<in>S. r i))"
haftmann@46981
    99
  by (simp add: fun_eq_iff)
haftmann@46981
   100
haftmann@46981
   101
lemma INF_INT_eq2 [pred_set_conv]: "(\<Sqinter>i\<in>S. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (\<Inter>i\<in>S. r i))"
haftmann@46981
   102
  by (simp add: fun_eq_iff)
haftmann@46981
   103
haftmann@46981
   104
lemma SUP_UN_eq [pred_set_conv]: "(\<Squnion>i\<in>S. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (\<Union>i\<in>S. r i))"
haftmann@46981
   105
  by (simp add: fun_eq_iff)
haftmann@46981
   106
haftmann@46981
   107
lemma SUP_UN_eq2 [pred_set_conv]: "(\<Squnion>i\<in>S. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (\<Union>i\<in>S. r i))"
haftmann@46981
   108
  by (simp add: fun_eq_iff)
haftmann@46981
   109
haftmann@46833
   110
lemma Inf_INT_eq [pred_set_conv]: "\<Sqinter>S = (\<lambda>x. x \<in> INTER S Collect)"
noschinl@46884
   111
  by (simp add: fun_eq_iff)
haftmann@46833
   112
haftmann@46833
   113
lemma INF_Int_eq [pred_set_conv]: "(\<Sqinter>i\<in>S. (\<lambda>x. x \<in> i)) = (\<lambda>x. x \<in> \<Inter>S)"
noschinl@46884
   114
  by (simp add: fun_eq_iff)
haftmann@46833
   115
haftmann@46833
   116
lemma Inf_INT_eq2 [pred_set_conv]: "\<Sqinter>S = (\<lambda>x y. (x, y) \<in> INTER (prod_case ` S) Collect)"
noschinl@46884
   117
  by (simp add: fun_eq_iff)
haftmann@46833
   118
haftmann@46833
   119
lemma INF_Int_eq2 [pred_set_conv]: "(\<Sqinter>i\<in>S. (\<lambda>x y. (x, y) \<in> i)) = (\<lambda>x y. (x, y) \<in> \<Inter>S)"
noschinl@46884
   120
  by (simp add: fun_eq_iff)
haftmann@46833
   121
haftmann@46833
   122
lemma Sup_SUP_eq [pred_set_conv]: "\<Squnion>S = (\<lambda>x. x \<in> UNION S Collect)"
noschinl@46884
   123
  by (simp add: fun_eq_iff)
haftmann@46833
   124
haftmann@46833
   125
lemma SUP_Sup_eq [pred_set_conv]: "(\<Squnion>i\<in>S. (\<lambda>x. x \<in> i)) = (\<lambda>x. x \<in> \<Union>S)"
noschinl@46884
   126
  by (simp add: fun_eq_iff)
haftmann@46833
   127
haftmann@46833
   128
lemma Sup_SUP_eq2 [pred_set_conv]: "\<Squnion>S = (\<lambda>x y. (x, y) \<in> UNION (prod_case ` S) Collect)"
noschinl@46884
   129
  by (simp add: fun_eq_iff)
haftmann@46833
   130
haftmann@46833
   131
lemma SUP_Sup_eq2 [pred_set_conv]: "(\<Squnion>i\<in>S. (\<lambda>x y. (x, y) \<in> i)) = (\<lambda>x y. (x, y) \<in> \<Union>S)"
noschinl@46884
   132
  by (simp add: fun_eq_iff)
haftmann@46833
   133
haftmann@46664
   134
haftmann@46694
   135
subsection {* Properties of relations *}
paulson@5978
   136
haftmann@46692
   137
subsubsection {* Reflexivity *}
paulson@10786
   138
haftmann@46752
   139
definition refl_on :: "'a set \<Rightarrow> 'a rel \<Rightarrow> bool"
haftmann@46752
   140
where
haftmann@46752
   141
  "refl_on A r \<longleftrightarrow> r \<subseteq> A \<times> A \<and> (\<forall>x\<in>A. (x, x) \<in> r)"
paulson@6806
   142
haftmann@46752
   143
abbreviation refl :: "'a rel \<Rightarrow> bool"
haftmann@46752
   144
where -- {* reflexivity over a type *}
haftmann@45137
   145
  "refl \<equiv> refl_on UNIV"
nipkow@26297
   146
haftmann@46752
   147
definition reflp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
haftmann@46752
   148
where
haftmann@46694
   149
  "reflp r \<longleftrightarrow> refl {(x, y). r x y}"
haftmann@46694
   150
haftmann@46752
   151
lemma reflp_refl_eq [pred_set_conv]:
haftmann@46752
   152
  "reflp (\<lambda>x y. (x, y) \<in> r) \<longleftrightarrow> refl r" 
haftmann@46752
   153
  by (simp add: refl_on_def reflp_def)
haftmann@46752
   154
haftmann@46692
   155
lemma refl_onI: "r \<subseteq> A \<times> A ==> (!!x. x : A ==> (x, x) : r) ==> refl_on A r"
haftmann@46752
   156
  by (unfold refl_on_def) (iprover intro!: ballI)
haftmann@46692
   157
haftmann@46692
   158
lemma refl_onD: "refl_on A r ==> a : A ==> (a, a) : r"
haftmann@46752
   159
  by (unfold refl_on_def) blast
haftmann@46692
   160
haftmann@46692
   161
lemma refl_onD1: "refl_on A r ==> (x, y) : r ==> x : A"
haftmann@46752
   162
  by (unfold refl_on_def) blast
haftmann@46692
   163
haftmann@46692
   164
lemma refl_onD2: "refl_on A r ==> (x, y) : r ==> y : A"
haftmann@46752
   165
  by (unfold refl_on_def) blast
haftmann@46692
   166
haftmann@46694
   167
lemma reflpI:
haftmann@46694
   168
  "(\<And>x. r x x) \<Longrightarrow> reflp r"
haftmann@46694
   169
  by (auto intro: refl_onI simp add: reflp_def)
haftmann@46694
   170
haftmann@46694
   171
lemma reflpE:
haftmann@46694
   172
  assumes "reflp r"
haftmann@46694
   173
  obtains "r x x"
haftmann@46694
   174
  using assms by (auto dest: refl_onD simp add: reflp_def)
haftmann@46694
   175
haftmann@46692
   176
lemma refl_on_Int: "refl_on A r ==> refl_on B s ==> refl_on (A \<inter> B) (r \<inter> s)"
haftmann@46752
   177
  by (unfold refl_on_def) blast
haftmann@46752
   178
haftmann@46752
   179
lemma reflp_inf:
haftmann@46752
   180
  "reflp r \<Longrightarrow> reflp s \<Longrightarrow> reflp (r \<sqinter> s)"
haftmann@46752
   181
  by (auto intro: reflpI elim: reflpE)
haftmann@46692
   182
haftmann@46692
   183
lemma refl_on_Un: "refl_on A r ==> refl_on B s ==> refl_on (A \<union> B) (r \<union> s)"
haftmann@46752
   184
  by (unfold refl_on_def) blast
haftmann@46752
   185
haftmann@46752
   186
lemma reflp_sup:
haftmann@46752
   187
  "reflp r \<Longrightarrow> reflp s \<Longrightarrow> reflp (r \<squnion> s)"
haftmann@46752
   188
  by (auto intro: reflpI elim: reflpE)
haftmann@46692
   189
haftmann@46692
   190
lemma refl_on_INTER:
haftmann@46692
   191
  "ALL x:S. refl_on (A x) (r x) ==> refl_on (INTER S A) (INTER S r)"
haftmann@46752
   192
  by (unfold refl_on_def) fast
haftmann@46692
   193
haftmann@46692
   194
lemma refl_on_UNION:
haftmann@46692
   195
  "ALL x:S. refl_on (A x) (r x) \<Longrightarrow> refl_on (UNION S A) (UNION S r)"
haftmann@46752
   196
  by (unfold refl_on_def) blast
haftmann@46692
   197
haftmann@46752
   198
lemma refl_on_empty [simp]: "refl_on {} {}"
haftmann@46752
   199
  by (simp add:refl_on_def)
haftmann@46692
   200
haftmann@46692
   201
lemma refl_on_def' [nitpick_unfold, code]:
haftmann@46752
   202
  "refl_on A r \<longleftrightarrow> (\<forall>(x, y) \<in> r. x \<in> A \<and> y \<in> A) \<and> (\<forall>x \<in> A. (x, x) \<in> r)"
haftmann@46752
   203
  by (auto intro: refl_onI dest: refl_onD refl_onD1 refl_onD2)
haftmann@46692
   204
haftmann@46692
   205
haftmann@46694
   206
subsubsection {* Irreflexivity *}
paulson@6806
   207
haftmann@46752
   208
definition irrefl :: "'a rel \<Rightarrow> bool"
haftmann@46752
   209
where
haftmann@46752
   210
  "irrefl r \<longleftrightarrow> (\<forall>x. (x, x) \<notin> r)"
haftmann@46692
   211
haftmann@46694
   212
lemma irrefl_distinct [code]:
haftmann@46694
   213
  "irrefl r \<longleftrightarrow> (\<forall>(x, y) \<in> r. x \<noteq> y)"
haftmann@46694
   214
  by (auto simp add: irrefl_def)
haftmann@46692
   215
haftmann@46692
   216
haftmann@46692
   217
subsubsection {* Symmetry *}
haftmann@46692
   218
haftmann@46752
   219
definition sym :: "'a rel \<Rightarrow> bool"
haftmann@46752
   220
where
haftmann@46752
   221
  "sym r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r)"
haftmann@46752
   222
haftmann@46752
   223
definition symp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
haftmann@46752
   224
where
haftmann@46752
   225
  "symp r \<longleftrightarrow> (\<forall>x y. r x y \<longrightarrow> r y x)"
haftmann@46692
   226
haftmann@46752
   227
lemma symp_sym_eq [pred_set_conv]:
haftmann@46752
   228
  "symp (\<lambda>x y. (x, y) \<in> r) \<longleftrightarrow> sym r" 
haftmann@46752
   229
  by (simp add: sym_def symp_def)
haftmann@46692
   230
haftmann@46752
   231
lemma symI:
haftmann@46752
   232
  "(\<And>a b. (a, b) \<in> r \<Longrightarrow> (b, a) \<in> r) \<Longrightarrow> sym r"
haftmann@46752
   233
  by (unfold sym_def) iprover
haftmann@46694
   234
haftmann@46694
   235
lemma sympI:
haftmann@46752
   236
  "(\<And>a b. r a b \<Longrightarrow> r b a) \<Longrightarrow> symp r"
haftmann@46752
   237
  by (fact symI [to_pred])
haftmann@46752
   238
haftmann@46752
   239
lemma symE:
haftmann@46752
   240
  assumes "sym r" and "(b, a) \<in> r"
haftmann@46752
   241
  obtains "(a, b) \<in> r"
haftmann@46752
   242
  using assms by (simp add: sym_def)
haftmann@46694
   243
haftmann@46694
   244
lemma sympE:
haftmann@46752
   245
  assumes "symp r" and "r b a"
haftmann@46752
   246
  obtains "r a b"
haftmann@46752
   247
  using assms by (rule symE [to_pred])
haftmann@46752
   248
haftmann@46752
   249
lemma symD:
haftmann@46752
   250
  assumes "sym r" and "(b, a) \<in> r"
haftmann@46752
   251
  shows "(a, b) \<in> r"
haftmann@46752
   252
  using assms by (rule symE)
haftmann@46694
   253
haftmann@46752
   254
lemma sympD:
haftmann@46752
   255
  assumes "symp r" and "r b a"
haftmann@46752
   256
  shows "r a b"
haftmann@46752
   257
  using assms by (rule symD [to_pred])
haftmann@46752
   258
haftmann@46752
   259
lemma sym_Int:
haftmann@46752
   260
  "sym r \<Longrightarrow> sym s \<Longrightarrow> sym (r \<inter> s)"
haftmann@46752
   261
  by (fast intro: symI elim: symE)
haftmann@46692
   262
haftmann@46752
   263
lemma symp_inf:
haftmann@46752
   264
  "symp r \<Longrightarrow> symp s \<Longrightarrow> symp (r \<sqinter> s)"
haftmann@46752
   265
  by (fact sym_Int [to_pred])
haftmann@46752
   266
haftmann@46752
   267
lemma sym_Un:
haftmann@46752
   268
  "sym r \<Longrightarrow> sym s \<Longrightarrow> sym (r \<union> s)"
haftmann@46752
   269
  by (fast intro: symI elim: symE)
haftmann@46752
   270
haftmann@46752
   271
lemma symp_sup:
haftmann@46752
   272
  "symp r \<Longrightarrow> symp s \<Longrightarrow> symp (r \<squnion> s)"
haftmann@46752
   273
  by (fact sym_Un [to_pred])
haftmann@46692
   274
haftmann@46752
   275
lemma sym_INTER:
haftmann@46752
   276
  "\<forall>x\<in>S. sym (r x) \<Longrightarrow> sym (INTER S r)"
haftmann@46752
   277
  by (fast intro: symI elim: symE)
haftmann@46752
   278
haftmann@46752
   279
(* FIXME thm sym_INTER [to_pred] *)
haftmann@46692
   280
haftmann@46752
   281
lemma sym_UNION:
haftmann@46752
   282
  "\<forall>x\<in>S. sym (r x) \<Longrightarrow> sym (UNION S r)"
haftmann@46752
   283
  by (fast intro: symI elim: symE)
haftmann@46752
   284
haftmann@46752
   285
(* FIXME thm sym_UNION [to_pred] *)
haftmann@46692
   286
haftmann@46692
   287
haftmann@46694
   288
subsubsection {* Antisymmetry *}
haftmann@46694
   289
haftmann@46752
   290
definition antisym :: "'a rel \<Rightarrow> bool"
haftmann@46752
   291
where
haftmann@46752
   292
  "antisym r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r \<longrightarrow> x = y)"
haftmann@46752
   293
haftmann@46752
   294
abbreviation antisymP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
haftmann@46752
   295
where
haftmann@46752
   296
  "antisymP r \<equiv> antisym {(x, y). r x y}"
haftmann@46694
   297
haftmann@46694
   298
lemma antisymI:
haftmann@46694
   299
  "(!!x y. (x, y) : r ==> (y, x) : r ==> x=y) ==> antisym r"
haftmann@46752
   300
  by (unfold antisym_def) iprover
haftmann@46694
   301
haftmann@46694
   302
lemma antisymD: "antisym r ==> (a, b) : r ==> (b, a) : r ==> a = b"
haftmann@46752
   303
  by (unfold antisym_def) iprover
haftmann@46694
   304
haftmann@46694
   305
lemma antisym_subset: "r \<subseteq> s ==> antisym s ==> antisym r"
haftmann@46752
   306
  by (unfold antisym_def) blast
haftmann@46694
   307
haftmann@46694
   308
lemma antisym_empty [simp]: "antisym {}"
haftmann@46752
   309
  by (unfold antisym_def) blast
haftmann@46694
   310
haftmann@46694
   311
haftmann@46692
   312
subsubsection {* Transitivity *}
haftmann@46692
   313
haftmann@46752
   314
definition trans :: "'a rel \<Rightarrow> bool"
haftmann@46752
   315
where
haftmann@46752
   316
  "trans r \<longleftrightarrow> (\<forall>x y z. (x, y) \<in> r \<longrightarrow> (y, z) \<in> r \<longrightarrow> (x, z) \<in> r)"
haftmann@46752
   317
haftmann@46752
   318
definition transp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
haftmann@46752
   319
where
haftmann@46752
   320
  "transp r \<longleftrightarrow> (\<forall>x y z. r x y \<longrightarrow> r y z \<longrightarrow> r x z)"
haftmann@46752
   321
haftmann@46752
   322
lemma transp_trans_eq [pred_set_conv]:
haftmann@46752
   323
  "transp (\<lambda>x y. (x, y) \<in> r) \<longleftrightarrow> trans r" 
haftmann@46752
   324
  by (simp add: trans_def transp_def)
haftmann@46752
   325
haftmann@46752
   326
abbreviation transP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
haftmann@46752
   327
where -- {* FIXME drop *}
haftmann@46752
   328
  "transP r \<equiv> trans {(x, y). r x y}"
paulson@5978
   329
haftmann@46692
   330
lemma transI:
haftmann@46752
   331
  "(\<And>x y z. (x, y) \<in> r \<Longrightarrow> (y, z) \<in> r \<Longrightarrow> (x, z) \<in> r) \<Longrightarrow> trans r"
haftmann@46752
   332
  by (unfold trans_def) iprover
haftmann@46694
   333
haftmann@46694
   334
lemma transpI:
haftmann@46694
   335
  "(\<And>x y z. r x y \<Longrightarrow> r y z \<Longrightarrow> r x z) \<Longrightarrow> transp r"
haftmann@46752
   336
  by (fact transI [to_pred])
haftmann@46752
   337
haftmann@46752
   338
lemma transE:
haftmann@46752
   339
  assumes "trans r" and "(x, y) \<in> r" and "(y, z) \<in> r"
haftmann@46752
   340
  obtains "(x, z) \<in> r"
haftmann@46752
   341
  using assms by (unfold trans_def) iprover
haftmann@46752
   342
haftmann@46694
   343
lemma transpE:
haftmann@46694
   344
  assumes "transp r" and "r x y" and "r y z"
haftmann@46694
   345
  obtains "r x z"
haftmann@46752
   346
  using assms by (rule transE [to_pred])
haftmann@46752
   347
haftmann@46752
   348
lemma transD:
haftmann@46752
   349
  assumes "trans r" and "(x, y) \<in> r" and "(y, z) \<in> r"
haftmann@46752
   350
  shows "(x, z) \<in> r"
haftmann@46752
   351
  using assms by (rule transE)
haftmann@46752
   352
haftmann@46752
   353
lemma transpD:
haftmann@46752
   354
  assumes "transp r" and "r x y" and "r y z"
haftmann@46752
   355
  shows "r x z"
haftmann@46752
   356
  using assms by (rule transD [to_pred])
haftmann@46694
   357
haftmann@46752
   358
lemma trans_Int:
haftmann@46752
   359
  "trans r \<Longrightarrow> trans s \<Longrightarrow> trans (r \<inter> s)"
haftmann@46752
   360
  by (fast intro: transI elim: transE)
haftmann@46692
   361
haftmann@46752
   362
lemma transp_inf:
haftmann@46752
   363
  "transp r \<Longrightarrow> transp s \<Longrightarrow> transp (r \<sqinter> s)"
haftmann@46752
   364
  by (fact trans_Int [to_pred])
haftmann@46752
   365
haftmann@46752
   366
lemma trans_INTER:
haftmann@46752
   367
  "\<forall>x\<in>S. trans (r x) \<Longrightarrow> trans (INTER S r)"
haftmann@46752
   368
  by (fast intro: transI elim: transD)
haftmann@46752
   369
haftmann@46752
   370
(* FIXME thm trans_INTER [to_pred] *)
haftmann@46692
   371
haftmann@46694
   372
lemma trans_join [code]:
haftmann@46694
   373
  "trans r \<longleftrightarrow> (\<forall>(x, y1) \<in> r. \<forall>(y2, z) \<in> r. y1 = y2 \<longrightarrow> (x, z) \<in> r)"
haftmann@46694
   374
  by (auto simp add: trans_def)
haftmann@46692
   375
haftmann@46752
   376
lemma transp_trans:
haftmann@46752
   377
  "transp r \<longleftrightarrow> trans {(x, y). r x y}"
haftmann@46752
   378
  by (simp add: trans_def transp_def)
haftmann@46752
   379
haftmann@46692
   380
haftmann@46692
   381
subsubsection {* Totality *}
haftmann@46692
   382
haftmann@46752
   383
definition total_on :: "'a set \<Rightarrow> 'a rel \<Rightarrow> bool"
haftmann@46752
   384
where
haftmann@46752
   385
  "total_on A r \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>A. x \<noteq> y \<longrightarrow> (x, y) \<in> r \<or> (y, x) \<in> r)"
nipkow@29859
   386
nipkow@29859
   387
abbreviation "total \<equiv> total_on UNIV"
nipkow@29859
   388
haftmann@46752
   389
lemma total_on_empty [simp]: "total_on {} r"
haftmann@46752
   390
  by (simp add: total_on_def)
haftmann@46692
   391
haftmann@46692
   392
haftmann@46692
   393
subsubsection {* Single valued relations *}
haftmann@46692
   394
haftmann@46752
   395
definition single_valued :: "('a \<times> 'b) set \<Rightarrow> bool"
haftmann@46752
   396
where
haftmann@46752
   397
  "single_valued r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (\<forall>z. (x, z) \<in> r \<longrightarrow> y = z))"
haftmann@46692
   398
haftmann@46694
   399
abbreviation single_valuedP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where
haftmann@46694
   400
  "single_valuedP r \<equiv> single_valued {(x, y). r x y}"
haftmann@46694
   401
haftmann@46752
   402
lemma single_valuedI:
haftmann@46752
   403
  "ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z) ==> single_valued r"
haftmann@46752
   404
  by (unfold single_valued_def)
haftmann@46752
   405
haftmann@46752
   406
lemma single_valuedD:
haftmann@46752
   407
  "single_valued r ==> (x, y) : r ==> (x, z) : r ==> y = z"
haftmann@46752
   408
  by (simp add: single_valued_def)
haftmann@46752
   409
haftmann@46692
   410
lemma single_valued_subset:
haftmann@46692
   411
  "r \<subseteq> s ==> single_valued s ==> single_valued r"
haftmann@46752
   412
  by (unfold single_valued_def) blast
oheimb@11136
   413
berghofe@12905
   414
haftmann@46694
   415
subsection {* Relation operations *}
haftmann@46694
   416
haftmann@46664
   417
subsubsection {* The identity relation *}
berghofe@12905
   418
haftmann@46752
   419
definition Id :: "'a rel"
haftmann@46752
   420
where
haftmann@46752
   421
  "Id = {p. \<exists>x. p = (x, x)}"
haftmann@46692
   422
berghofe@12905
   423
lemma IdI [intro]: "(a, a) : Id"
haftmann@46752
   424
  by (simp add: Id_def)
berghofe@12905
   425
berghofe@12905
   426
lemma IdE [elim!]: "p : Id ==> (!!x. p = (x, x) ==> P) ==> P"
haftmann@46752
   427
  by (unfold Id_def) (iprover elim: CollectE)
berghofe@12905
   428
berghofe@12905
   429
lemma pair_in_Id_conv [iff]: "((a, b) : Id) = (a = b)"
haftmann@46752
   430
  by (unfold Id_def) blast
berghofe@12905
   431
nipkow@30198
   432
lemma refl_Id: "refl Id"
haftmann@46752
   433
  by (simp add: refl_on_def)
berghofe@12905
   434
berghofe@12905
   435
lemma antisym_Id: "antisym Id"
berghofe@12905
   436
  -- {* A strange result, since @{text Id} is also symmetric. *}
haftmann@46752
   437
  by (simp add: antisym_def)
berghofe@12905
   438
huffman@19228
   439
lemma sym_Id: "sym Id"
haftmann@46752
   440
  by (simp add: sym_def)
huffman@19228
   441
berghofe@12905
   442
lemma trans_Id: "trans Id"
haftmann@46752
   443
  by (simp add: trans_def)
berghofe@12905
   444
haftmann@46692
   445
lemma single_valued_Id [simp]: "single_valued Id"
haftmann@46692
   446
  by (unfold single_valued_def) blast
haftmann@46692
   447
haftmann@46692
   448
lemma irrefl_diff_Id [simp]: "irrefl (r - Id)"
haftmann@46692
   449
  by (simp add:irrefl_def)
haftmann@46692
   450
haftmann@46692
   451
lemma trans_diff_Id: "trans r \<Longrightarrow> antisym r \<Longrightarrow> trans (r - Id)"
haftmann@46692
   452
  unfolding antisym_def trans_def by blast
haftmann@46692
   453
haftmann@46692
   454
lemma total_on_diff_Id [simp]: "total_on A (r - Id) = total_on A r"
haftmann@46692
   455
  by (simp add: total_on_def)
haftmann@46692
   456
berghofe@12905
   457
haftmann@46664
   458
subsubsection {* Diagonal: identity over a set *}
berghofe@12905
   459
haftmann@46752
   460
definition Id_on  :: "'a set \<Rightarrow> 'a rel"
haftmann@46752
   461
where
haftmann@46752
   462
  "Id_on A = (\<Union>x\<in>A. {(x, x)})"
haftmann@46692
   463
nipkow@30198
   464
lemma Id_on_empty [simp]: "Id_on {} = {}"
haftmann@46752
   465
  by (simp add: Id_on_def) 
paulson@13812
   466
nipkow@30198
   467
lemma Id_on_eqI: "a = b ==> a : A ==> (a, b) : Id_on A"
haftmann@46752
   468
  by (simp add: Id_on_def)
berghofe@12905
   469
blanchet@35828
   470
lemma Id_onI [intro!,no_atp]: "a : A ==> (a, a) : Id_on A"
haftmann@46752
   471
  by (rule Id_on_eqI) (rule refl)
berghofe@12905
   472
nipkow@30198
   473
lemma Id_onE [elim!]:
nipkow@30198
   474
  "c : Id_on A ==> (!!x. x : A ==> c = (x, x) ==> P) ==> P"
wenzelm@12913
   475
  -- {* The general elimination rule. *}
haftmann@46752
   476
  by (unfold Id_on_def) (iprover elim!: UN_E singletonE)
berghofe@12905
   477
nipkow@30198
   478
lemma Id_on_iff: "((x, y) : Id_on A) = (x = y & x : A)"
haftmann@46752
   479
  by blast
berghofe@12905
   480
haftmann@45967
   481
lemma Id_on_def' [nitpick_unfold]:
haftmann@44278
   482
  "Id_on {x. A x} = Collect (\<lambda>(x, y). x = y \<and> A x)"
haftmann@46752
   483
  by auto
bulwahn@40923
   484
nipkow@30198
   485
lemma Id_on_subset_Times: "Id_on A \<subseteq> A \<times> A"
haftmann@46752
   486
  by blast
berghofe@12905
   487
haftmann@46692
   488
lemma refl_on_Id_on: "refl_on A (Id_on A)"
haftmann@46752
   489
  by (rule refl_onI [OF Id_on_subset_Times Id_onI])
haftmann@46692
   490
haftmann@46692
   491
lemma antisym_Id_on [simp]: "antisym (Id_on A)"
haftmann@46752
   492
  by (unfold antisym_def) blast
haftmann@46692
   493
haftmann@46692
   494
lemma sym_Id_on [simp]: "sym (Id_on A)"
haftmann@46752
   495
  by (rule symI) clarify
haftmann@46692
   496
haftmann@46692
   497
lemma trans_Id_on [simp]: "trans (Id_on A)"
haftmann@46752
   498
  by (fast intro: transI elim: transD)
haftmann@46692
   499
haftmann@46692
   500
lemma single_valued_Id_on [simp]: "single_valued (Id_on A)"
haftmann@46692
   501
  by (unfold single_valued_def) blast
haftmann@46692
   502
berghofe@12905
   503
haftmann@46694
   504
subsubsection {* Composition *}
berghofe@12905
   505
haftmann@46752
   506
inductive_set rel_comp  :: "('a \<times> 'b) set \<Rightarrow> ('b \<times> 'c) set \<Rightarrow> ('a \<times> 'c) set" (infixr "O" 75)
haftmann@46752
   507
  for r :: "('a \<times> 'b) set" and s :: "('b \<times> 'c) set"
haftmann@46694
   508
where
haftmann@46752
   509
  rel_compI [intro]: "(a, b) \<in> r \<Longrightarrow> (b, c) \<in> s \<Longrightarrow> (a, c) \<in> r O s"
haftmann@46692
   510
haftmann@46752
   511
abbreviation pred_comp (infixr "OO" 75) where
haftmann@46752
   512
  "pred_comp \<equiv> rel_compp"
berghofe@12905
   513
haftmann@46752
   514
lemmas pred_compI = rel_compp.intros
berghofe@12905
   515
haftmann@46752
   516
text {*
haftmann@46752
   517
  For historic reasons, the elimination rules are not wholly corresponding.
haftmann@46752
   518
  Feel free to consolidate this.
haftmann@46752
   519
*}
haftmann@46694
   520
haftmann@46752
   521
inductive_cases rel_compEpair: "(a, c) \<in> r O s"
haftmann@46694
   522
inductive_cases pred_compE [elim!]: "(r OO s) a c"
haftmann@46694
   523
haftmann@46752
   524
lemma rel_compE [elim!]: "xz \<in> r O s \<Longrightarrow>
haftmann@46752
   525
  (\<And>x y z. xz = (x, z) \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> (y, z) \<in> s  \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@46752
   526
  by (cases xz) (simp, erule rel_compEpair, iprover)
haftmann@46752
   527
haftmann@46752
   528
lemmas pred_comp_rel_comp_eq = rel_compp_rel_comp_eq
haftmann@46752
   529
haftmann@46752
   530
lemma R_O_Id [simp]:
haftmann@46752
   531
  "R O Id = R"
haftmann@46752
   532
  by fast
haftmann@46694
   533
haftmann@46752
   534
lemma Id_O_R [simp]:
haftmann@46752
   535
  "Id O R = R"
haftmann@46752
   536
  by fast
haftmann@46752
   537
haftmann@46752
   538
lemma rel_comp_empty1 [simp]:
haftmann@46752
   539
  "{} O R = {}"
haftmann@46752
   540
  by blast
berghofe@12905
   541
noschinl@46883
   542
lemma prod_comp_bot1 [simp]:
noschinl@46883
   543
  "\<bottom> OO R = \<bottom>"
noschinl@46883
   544
  by (fact rel_comp_empty1 [to_pred])
berghofe@12905
   545
haftmann@46752
   546
lemma rel_comp_empty2 [simp]:
haftmann@46752
   547
  "R O {} = {}"
haftmann@46752
   548
  by blast
berghofe@12905
   549
noschinl@46883
   550
lemma pred_comp_bot2 [simp]:
noschinl@46883
   551
  "R OO \<bottom> = \<bottom>"
noschinl@46883
   552
  by (fact rel_comp_empty2 [to_pred])
krauss@23185
   553
haftmann@46752
   554
lemma O_assoc:
haftmann@46752
   555
  "(R O S) O T = R O (S O T)"
haftmann@46752
   556
  by blast
haftmann@46752
   557
noschinl@46883
   558
haftmann@46752
   559
lemma pred_comp_assoc:
haftmann@46752
   560
  "(r OO s) OO t = r OO (s OO t)"
haftmann@46752
   561
  by (fact O_assoc [to_pred])
krauss@23185
   562
haftmann@46752
   563
lemma trans_O_subset:
haftmann@46752
   564
  "trans r \<Longrightarrow> r O r \<subseteq> r"
haftmann@46752
   565
  by (unfold trans_def) blast
haftmann@46752
   566
haftmann@46752
   567
lemma transp_pred_comp_less_eq:
haftmann@46752
   568
  "transp r \<Longrightarrow> r OO r \<le> r "
haftmann@46752
   569
  by (fact trans_O_subset [to_pred])
berghofe@12905
   570
haftmann@46752
   571
lemma rel_comp_mono:
haftmann@46752
   572
  "r' \<subseteq> r \<Longrightarrow> s' \<subseteq> s \<Longrightarrow> r' O s' \<subseteq> r O s"
haftmann@46752
   573
  by blast
berghofe@12905
   574
haftmann@46752
   575
lemma pred_comp_mono:
haftmann@46752
   576
  "r' \<le> r \<Longrightarrow> s' \<le> s \<Longrightarrow> r' OO s' \<le> r OO s "
haftmann@46752
   577
  by (fact rel_comp_mono [to_pred])
berghofe@12905
   578
berghofe@12905
   579
lemma rel_comp_subset_Sigma:
haftmann@46752
   580
  "r \<subseteq> A \<times> B \<Longrightarrow> s \<subseteq> B \<times> C \<Longrightarrow> r O s \<subseteq> A \<times> C"
haftmann@46752
   581
  by blast
haftmann@46752
   582
haftmann@46752
   583
lemma rel_comp_distrib [simp]:
haftmann@46752
   584
  "R O (S \<union> T) = (R O S) \<union> (R O T)" 
haftmann@46752
   585
  by auto
berghofe@12905
   586
noschinl@46882
   587
lemma pred_comp_distrib [simp]:
haftmann@46752
   588
  "R OO (S \<squnion> T) = R OO S \<squnion> R OO T"
haftmann@46752
   589
  by (fact rel_comp_distrib [to_pred])
haftmann@46752
   590
haftmann@46752
   591
lemma rel_comp_distrib2 [simp]:
haftmann@46752
   592
  "(S \<union> T) O R = (S O R) \<union> (T O R)"
haftmann@46752
   593
  by auto
krauss@28008
   594
noschinl@46882
   595
lemma pred_comp_distrib2 [simp]:
haftmann@46752
   596
  "(S \<squnion> T) OO R = S OO R \<squnion> T OO R"
haftmann@46752
   597
  by (fact rel_comp_distrib2 [to_pred])
haftmann@46752
   598
haftmann@46752
   599
lemma rel_comp_UNION_distrib:
haftmann@46752
   600
  "s O UNION I r = (\<Union>i\<in>I. s O r i) "
haftmann@46752
   601
  by auto
krauss@28008
   602
haftmann@46752
   603
(* FIXME thm rel_comp_UNION_distrib [to_pred] *)
krauss@36772
   604
haftmann@46752
   605
lemma rel_comp_UNION_distrib2:
haftmann@46752
   606
  "UNION I r O s = (\<Union>i\<in>I. r i O s) "
haftmann@46752
   607
  by auto
haftmann@46752
   608
haftmann@46752
   609
(* FIXME thm rel_comp_UNION_distrib2 [to_pred] *)
krauss@36772
   610
haftmann@46692
   611
lemma single_valued_rel_comp:
haftmann@46752
   612
  "single_valued r \<Longrightarrow> single_valued s \<Longrightarrow> single_valued (r O s)"
haftmann@46752
   613
  by (unfold single_valued_def) blast
haftmann@46752
   614
haftmann@46752
   615
lemma rel_comp_unfold:
haftmann@46752
   616
  "r O s = {(x, z). \<exists>y. (x, y) \<in> r \<and> (y, z) \<in> s}"
haftmann@46752
   617
  by (auto simp add: set_eq_iff)
berghofe@12905
   618
haftmann@46664
   619
haftmann@46664
   620
subsubsection {* Converse *}
wenzelm@12913
   621
haftmann@46752
   622
inductive_set converse :: "('a \<times> 'b) set \<Rightarrow> ('b \<times> 'a) set" ("(_^-1)" [1000] 999)
haftmann@46752
   623
  for r :: "('a \<times> 'b) set"
haftmann@46752
   624
where
haftmann@46752
   625
  "(a, b) \<in> r \<Longrightarrow> (b, a) \<in> r^-1"
haftmann@46692
   626
haftmann@46692
   627
notation (xsymbols)
haftmann@46692
   628
  converse  ("(_\<inverse>)" [1000] 999)
haftmann@46692
   629
haftmann@46752
   630
notation
haftmann@46752
   631
  conversep ("(_^--1)" [1000] 1000)
haftmann@46694
   632
haftmann@46694
   633
notation (xsymbols)
haftmann@46694
   634
  conversep  ("(_\<inverse>\<inverse>)" [1000] 1000)
haftmann@46694
   635
haftmann@46752
   636
lemma converseI [sym]:
haftmann@46752
   637
  "(a, b) \<in> r \<Longrightarrow> (b, a) \<in> r\<inverse>"
haftmann@46752
   638
  by (fact converse.intros)
haftmann@46752
   639
haftmann@46752
   640
lemma conversepI (* CANDIDATE [sym] *):
haftmann@46752
   641
  "r a b \<Longrightarrow> r\<inverse>\<inverse> b a"
haftmann@46752
   642
  by (fact conversep.intros)
haftmann@46752
   643
haftmann@46752
   644
lemma converseD [sym]:
haftmann@46752
   645
  "(a, b) \<in> r\<inverse> \<Longrightarrow> (b, a) \<in> r"
haftmann@46752
   646
  by (erule converse.cases) iprover
haftmann@46752
   647
haftmann@46752
   648
lemma conversepD (* CANDIDATE [sym] *):
haftmann@46752
   649
  "r\<inverse>\<inverse> b a \<Longrightarrow> r a b"
haftmann@46752
   650
  by (fact converseD [to_pred])
haftmann@46752
   651
haftmann@46752
   652
lemma converseE [elim!]:
haftmann@46752
   653
  -- {* More general than @{text converseD}, as it ``splits'' the member of the relation. *}
haftmann@46752
   654
  "yx \<in> r\<inverse> \<Longrightarrow> (\<And>x y. yx = (y, x) \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@46752
   655
  by (cases yx) (simp, erule converse.cases, iprover)
haftmann@46694
   656
noschinl@46882
   657
lemmas conversepE [elim!] = conversep.cases
haftmann@46752
   658
haftmann@46752
   659
lemma converse_iff [iff]:
haftmann@46752
   660
  "(a, b) \<in> r\<inverse> \<longleftrightarrow> (b, a) \<in> r"
haftmann@46752
   661
  by (auto intro: converseI)
haftmann@46752
   662
haftmann@46752
   663
lemma conversep_iff [iff]:
haftmann@46752
   664
  "r\<inverse>\<inverse> a b = r b a"
haftmann@46752
   665
  by (fact converse_iff [to_pred])
haftmann@46694
   666
haftmann@46752
   667
lemma converse_converse [simp]:
haftmann@46752
   668
  "(r\<inverse>)\<inverse> = r"
haftmann@46752
   669
  by (simp add: set_eq_iff)
haftmann@46694
   670
haftmann@46752
   671
lemma conversep_conversep [simp]:
haftmann@46752
   672
  "(r\<inverse>\<inverse>)\<inverse>\<inverse> = r"
haftmann@46752
   673
  by (fact converse_converse [to_pred])
haftmann@46752
   674
haftmann@46752
   675
lemma converse_rel_comp: "(r O s)^-1 = s^-1 O r^-1"
haftmann@46752
   676
  by blast
haftmann@46694
   677
haftmann@46694
   678
lemma converse_pred_comp: "(r OO s)^--1 = s^--1 OO r^--1"
haftmann@46694
   679
  by (iprover intro: order_antisym conversepI pred_compI
haftmann@46694
   680
    elim: pred_compE dest: conversepD)
haftmann@46694
   681
haftmann@46752
   682
lemma converse_Int: "(r \<inter> s)^-1 = r^-1 \<inter> s^-1"
haftmann@46752
   683
  by blast
haftmann@46752
   684
haftmann@46694
   685
lemma converse_meet: "(r \<sqinter> s)^--1 = r^--1 \<sqinter> s^--1"
haftmann@46694
   686
  by (simp add: inf_fun_def) (iprover intro: conversepI ext dest: conversepD)
haftmann@46694
   687
haftmann@46752
   688
lemma converse_Un: "(r \<union> s)^-1 = r^-1 \<union> s^-1"
haftmann@46752
   689
  by blast
haftmann@46752
   690
haftmann@46694
   691
lemma converse_join: "(r \<squnion> s)^--1 = r^--1 \<squnion> s^--1"
haftmann@46694
   692
  by (simp add: sup_fun_def) (iprover intro: conversepI ext dest: conversepD)
haftmann@46694
   693
huffman@19228
   694
lemma converse_INTER: "(INTER S r)^-1 = (INT x:S. (r x)^-1)"
haftmann@46752
   695
  by fast
huffman@19228
   696
huffman@19228
   697
lemma converse_UNION: "(UNION S r)^-1 = (UN x:S. (r x)^-1)"
haftmann@46752
   698
  by blast
huffman@19228
   699
berghofe@12905
   700
lemma converse_Id [simp]: "Id^-1 = Id"
haftmann@46752
   701
  by blast
berghofe@12905
   702
nipkow@30198
   703
lemma converse_Id_on [simp]: "(Id_on A)^-1 = Id_on A"
haftmann@46752
   704
  by blast
berghofe@12905
   705
nipkow@30198
   706
lemma refl_on_converse [simp]: "refl_on A (converse r) = refl_on A r"
haftmann@46752
   707
  by (unfold refl_on_def) auto
berghofe@12905
   708
huffman@19228
   709
lemma sym_converse [simp]: "sym (converse r) = sym r"
haftmann@46752
   710
  by (unfold sym_def) blast
huffman@19228
   711
huffman@19228
   712
lemma antisym_converse [simp]: "antisym (converse r) = antisym r"
haftmann@46752
   713
  by (unfold antisym_def) blast
berghofe@12905
   714
huffman@19228
   715
lemma trans_converse [simp]: "trans (converse r) = trans r"
haftmann@46752
   716
  by (unfold trans_def) blast
berghofe@12905
   717
huffman@19228
   718
lemma sym_conv_converse_eq: "sym r = (r^-1 = r)"
haftmann@46752
   719
  by (unfold sym_def) fast
huffman@19228
   720
huffman@19228
   721
lemma sym_Un_converse: "sym (r \<union> r^-1)"
haftmann@46752
   722
  by (unfold sym_def) blast
huffman@19228
   723
huffman@19228
   724
lemma sym_Int_converse: "sym (r \<inter> r^-1)"
haftmann@46752
   725
  by (unfold sym_def) blast
huffman@19228
   726
haftmann@46752
   727
lemma total_on_converse [simp]: "total_on A (r^-1) = total_on A r"
haftmann@46752
   728
  by (auto simp: total_on_def)
nipkow@29859
   729
haftmann@46692
   730
lemma finite_converse [iff]: "finite (r^-1) = finite r"
haftmann@46692
   731
  apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r")
haftmann@46692
   732
   apply simp
haftmann@46692
   733
   apply (rule iffI)
haftmann@46692
   734
    apply (erule finite_imageD [unfolded inj_on_def])
haftmann@46692
   735
    apply (simp split add: split_split)
haftmann@46692
   736
   apply (erule finite_imageI)
haftmann@46752
   737
  apply (simp add: set_eq_iff image_def, auto)
haftmann@46692
   738
  apply (rule bexI)
haftmann@46692
   739
   prefer 2 apply assumption
haftmann@46692
   740
  apply simp
haftmann@46692
   741
  done
wenzelm@12913
   742
haftmann@46752
   743
lemma conversep_noteq [simp]: "(op \<noteq>)^--1 = op \<noteq>"
haftmann@46752
   744
  by (auto simp add: fun_eq_iff)
haftmann@46752
   745
haftmann@46752
   746
lemma conversep_eq [simp]: "(op =)^--1 = op ="
haftmann@46752
   747
  by (auto simp add: fun_eq_iff)
haftmann@46752
   748
haftmann@46752
   749
lemma converse_unfold:
haftmann@46752
   750
  "r\<inverse> = {(y, x). (x, y) \<in> r}"
haftmann@46752
   751
  by (simp add: set_eq_iff)
haftmann@46752
   752
haftmann@46692
   753
haftmann@46692
   754
subsubsection {* Domain, range and field *}
haftmann@46692
   755
haftmann@46767
   756
inductive_set Domain :: "('a \<times> 'b) set \<Rightarrow> 'a set"
haftmann@46767
   757
  for r :: "('a \<times> 'b) set"
haftmann@46752
   758
where
haftmann@46767
   759
  DomainI [intro]: "(a, b) \<in> r \<Longrightarrow> a \<in> Domain r"
haftmann@46767
   760
haftmann@46767
   761
abbreviation (input) "DomainP \<equiv> Domainp"
haftmann@46767
   762
haftmann@46767
   763
lemmas DomainPI = Domainp.DomainI
haftmann@46767
   764
haftmann@46767
   765
inductive_cases DomainE [elim!]: "a \<in> Domain r"
haftmann@46767
   766
inductive_cases DomainpE [elim!]: "Domainp r a"
haftmann@46692
   767
haftmann@46767
   768
inductive_set Range :: "('a \<times> 'b) set \<Rightarrow> 'b set"
haftmann@46767
   769
  for r :: "('a \<times> 'b) set"
haftmann@46752
   770
where
haftmann@46767
   771
  RangeI [intro]: "(a, b) \<in> r \<Longrightarrow> b \<in> Range r"
haftmann@46767
   772
haftmann@46767
   773
abbreviation (input) "RangeP \<equiv> Rangep"
haftmann@46767
   774
haftmann@46767
   775
lemmas RangePI = Rangep.RangeI
haftmann@46767
   776
haftmann@46767
   777
inductive_cases RangeE [elim!]: "b \<in> Range r"
haftmann@46767
   778
inductive_cases RangepE [elim!]: "Rangep r b"
haftmann@46692
   779
haftmann@46752
   780
definition Field :: "'a rel \<Rightarrow> 'a set"
haftmann@46752
   781
where
haftmann@46692
   782
  "Field r = Domain r \<union> Range r"
berghofe@12905
   783
haftmann@46694
   784
lemma Domain_fst [code]:
haftmann@46694
   785
  "Domain r = fst ` r"
haftmann@46767
   786
  by force
haftmann@46767
   787
haftmann@46767
   788
lemma Range_snd [code]:
haftmann@46767
   789
  "Range r = snd ` r"
haftmann@46767
   790
  by force
haftmann@46767
   791
haftmann@46767
   792
lemma fst_eq_Domain: "fst ` R = Domain R"
haftmann@46767
   793
  by force
haftmann@46767
   794
haftmann@46767
   795
lemma snd_eq_Range: "snd ` R = Range R"
haftmann@46767
   796
  by force
haftmann@46694
   797
haftmann@46694
   798
lemma Domain_empty [simp]: "Domain {} = {}"
haftmann@46767
   799
  by auto
haftmann@46767
   800
haftmann@46767
   801
lemma Range_empty [simp]: "Range {} = {}"
haftmann@46767
   802
  by auto
haftmann@46767
   803
haftmann@46767
   804
lemma Field_empty [simp]: "Field {} = {}"
haftmann@46767
   805
  by (simp add: Field_def)
haftmann@46694
   806
haftmann@46694
   807
lemma Domain_empty_iff: "Domain r = {} \<longleftrightarrow> r = {}"
haftmann@46694
   808
  by auto
haftmann@46694
   809
haftmann@46767
   810
lemma Range_empty_iff: "Range r = {} \<longleftrightarrow> r = {}"
haftmann@46767
   811
  by auto
haftmann@46767
   812
noschinl@46882
   813
lemma Domain_insert [simp]: "Domain (insert (a, b) r) = insert a (Domain r)"
haftmann@46767
   814
  by blast
haftmann@46767
   815
noschinl@46882
   816
lemma Range_insert [simp]: "Range (insert (a, b) r) = insert b (Range r)"
haftmann@46767
   817
  by blast
haftmann@46767
   818
haftmann@46767
   819
lemma Field_insert [simp]: "Field (insert (a, b) r) = {a, b} \<union> Field r"
noschinl@46884
   820
  by (auto simp add: Field_def)
haftmann@46767
   821
haftmann@46767
   822
lemma Domain_iff: "a \<in> Domain r \<longleftrightarrow> (\<exists>y. (a, y) \<in> r)"
haftmann@46767
   823
  by blast
haftmann@46767
   824
haftmann@46767
   825
lemma Range_iff: "a \<in> Range r \<longleftrightarrow> (\<exists>y. (y, a) \<in> r)"
haftmann@46694
   826
  by blast
haftmann@46694
   827
haftmann@46694
   828
lemma Domain_Id [simp]: "Domain Id = UNIV"
haftmann@46694
   829
  by blast
haftmann@46694
   830
haftmann@46767
   831
lemma Range_Id [simp]: "Range Id = UNIV"
haftmann@46767
   832
  by blast
haftmann@46767
   833
haftmann@46694
   834
lemma Domain_Id_on [simp]: "Domain (Id_on A) = A"
haftmann@46694
   835
  by blast
haftmann@46694
   836
haftmann@46767
   837
lemma Range_Id_on [simp]: "Range (Id_on A) = A"
haftmann@46767
   838
  by blast
haftmann@46767
   839
haftmann@46767
   840
lemma Domain_Un_eq: "Domain (A \<union> B) = Domain A \<union> Domain B"
haftmann@46694
   841
  by blast
haftmann@46694
   842
haftmann@46767
   843
lemma Range_Un_eq: "Range (A \<union> B) = Range A \<union> Range B"
haftmann@46767
   844
  by blast
haftmann@46767
   845
haftmann@46767
   846
lemma Field_Un [simp]: "Field (r \<union> s) = Field r \<union> Field s"
haftmann@46767
   847
  by (auto simp: Field_def)
haftmann@46767
   848
haftmann@46767
   849
lemma Domain_Int_subset: "Domain (A \<inter> B) \<subseteq> Domain A \<inter> Domain B"
haftmann@46694
   850
  by blast
haftmann@46694
   851
haftmann@46767
   852
lemma Range_Int_subset: "Range (A \<inter> B) \<subseteq> Range A \<inter> Range B"
haftmann@46767
   853
  by blast
haftmann@46767
   854
haftmann@46767
   855
lemma Domain_Diff_subset: "Domain A - Domain B \<subseteq> Domain (A - B)"
haftmann@46767
   856
  by blast
haftmann@46767
   857
haftmann@46767
   858
lemma Range_Diff_subset: "Range A - Range B \<subseteq> Range (A - B)"
haftmann@46694
   859
  by blast
haftmann@46694
   860
haftmann@46767
   861
lemma Domain_Union: "Domain (\<Union>S) = (\<Union>A\<in>S. Domain A)"
haftmann@46694
   862
  by blast
haftmann@46694
   863
haftmann@46767
   864
lemma Range_Union: "Range (\<Union>S) = (\<Union>A\<in>S. Range A)"
haftmann@46767
   865
  by blast
haftmann@46767
   866
haftmann@46767
   867
lemma Field_Union [simp]: "Field (\<Union>R) = \<Union>(Field ` R)"
haftmann@46767
   868
  by (auto simp: Field_def)
haftmann@46767
   869
haftmann@46752
   870
lemma Domain_converse [simp]: "Domain (r\<inverse>) = Range r"
haftmann@46752
   871
  by auto
haftmann@46694
   872
haftmann@46767
   873
lemma Range_converse [simp]: "Range (r\<inverse>) = Domain r"
haftmann@46694
   874
  by blast
haftmann@46694
   875
haftmann@46767
   876
lemma Field_converse [simp]: "Field (r\<inverse>) = Field r"
haftmann@46767
   877
  by (auto simp: Field_def)
haftmann@46767
   878
haftmann@46767
   879
lemma Domain_Collect_split [simp]: "Domain {(x, y). P x y} = {x. EX y. P x y}"
haftmann@46767
   880
  by auto
haftmann@46767
   881
haftmann@46767
   882
lemma Range_Collect_split [simp]: "Range {(x, y). P x y} = {y. EX x. P x y}"
haftmann@46767
   883
  by auto
haftmann@46767
   884
haftmann@46767
   885
lemma finite_Domain: "finite r \<Longrightarrow> finite (Domain r)"
noschinl@46884
   886
  by (induct set: finite) auto
haftmann@46767
   887
haftmann@46767
   888
lemma finite_Range: "finite r \<Longrightarrow> finite (Range r)"
noschinl@46884
   889
  by (induct set: finite) auto
haftmann@46767
   890
haftmann@46767
   891
lemma finite_Field: "finite r \<Longrightarrow> finite (Field r)"
haftmann@46767
   892
  by (simp add: Field_def finite_Domain finite_Range)
haftmann@46767
   893
haftmann@46767
   894
lemma Domain_mono: "r \<subseteq> s \<Longrightarrow> Domain r \<subseteq> Domain s"
haftmann@46767
   895
  by blast
haftmann@46767
   896
haftmann@46767
   897
lemma Range_mono: "r \<subseteq> s \<Longrightarrow> Range r \<subseteq> Range s"
haftmann@46767
   898
  by blast
haftmann@46767
   899
haftmann@46767
   900
lemma mono_Field: "r \<subseteq> s \<Longrightarrow> Field r \<subseteq> Field s"
haftmann@46767
   901
  by (auto simp: Field_def Domain_def Range_def)
haftmann@46767
   902
haftmann@46767
   903
lemma Domain_unfold:
haftmann@46767
   904
  "Domain r = {x. \<exists>y. (x, y) \<in> r}"
haftmann@46767
   905
  by blast
haftmann@46694
   906
haftmann@46694
   907
lemma Domain_dprod [simp]: "Domain (dprod r s) = uprod (Domain r) (Domain s)"
haftmann@46694
   908
  by auto
haftmann@46694
   909
haftmann@46694
   910
lemma Domain_dsum [simp]: "Domain (dsum r s) = usum (Domain r) (Domain s)"
haftmann@46694
   911
  by auto
haftmann@46694
   912
berghofe@12905
   913
haftmann@46664
   914
subsubsection {* Image of a set under a relation *}
berghofe@12905
   915
haftmann@46752
   916
definition Image :: "('a \<times> 'b) set \<Rightarrow> 'a set \<Rightarrow> 'b set" (infixl "``" 90)
haftmann@46752
   917
where
haftmann@46752
   918
  "r `` s = {y. \<exists>x\<in>s. (x, y) \<in> r}"
haftmann@46692
   919
blanchet@35828
   920
declare Image_def [no_atp]
paulson@24286
   921
wenzelm@12913
   922
lemma Image_iff: "(b : r``A) = (EX x:A. (x, b) : r)"
haftmann@46752
   923
  by (simp add: Image_def)
berghofe@12905
   924
wenzelm@12913
   925
lemma Image_singleton: "r``{a} = {b. (a, b) : r}"
haftmann@46752
   926
  by (simp add: Image_def)
berghofe@12905
   927
wenzelm@12913
   928
lemma Image_singleton_iff [iff]: "(b : r``{a}) = ((a, b) : r)"
haftmann@46752
   929
  by (rule Image_iff [THEN trans]) simp
berghofe@12905
   930
blanchet@35828
   931
lemma ImageI [intro,no_atp]: "(a, b) : r ==> a : A ==> b : r``A"
haftmann@46752
   932
  by (unfold Image_def) blast
berghofe@12905
   933
berghofe@12905
   934
lemma ImageE [elim!]:
haftmann@46752
   935
  "b : r `` A ==> (!!x. (x, b) : r ==> x : A ==> P) ==> P"
haftmann@46752
   936
  by (unfold Image_def) (iprover elim!: CollectE bexE)
berghofe@12905
   937
berghofe@12905
   938
lemma rev_ImageI: "a : A ==> (a, b) : r ==> b : r `` A"
berghofe@12905
   939
  -- {* This version's more effective when we already have the required @{text a} *}
haftmann@46752
   940
  by blast
berghofe@12905
   941
berghofe@12905
   942
lemma Image_empty [simp]: "R``{} = {}"
haftmann@46752
   943
  by blast
berghofe@12905
   944
berghofe@12905
   945
lemma Image_Id [simp]: "Id `` A = A"
haftmann@46752
   946
  by blast
berghofe@12905
   947
nipkow@30198
   948
lemma Image_Id_on [simp]: "Id_on A `` B = A \<inter> B"
haftmann@46752
   949
  by blast
paulson@13830
   950
paulson@13830
   951
lemma Image_Int_subset: "R `` (A \<inter> B) \<subseteq> R `` A \<inter> R `` B"
haftmann@46752
   952
  by blast
berghofe@12905
   953
paulson@13830
   954
lemma Image_Int_eq:
haftmann@46767
   955
  "single_valued (converse R) ==> R `` (A \<inter> B) = R `` A \<inter> R `` B"
haftmann@46767
   956
  by (simp add: single_valued_def, blast) 
berghofe@12905
   957
paulson@13830
   958
lemma Image_Un: "R `` (A \<union> B) = R `` A \<union> R `` B"
haftmann@46752
   959
  by blast
berghofe@12905
   960
paulson@13812
   961
lemma Un_Image: "(R \<union> S) `` A = R `` A \<union> S `` A"
haftmann@46752
   962
  by blast
paulson@13812
   963
wenzelm@12913
   964
lemma Image_subset: "r \<subseteq> A \<times> B ==> r``C \<subseteq> B"
haftmann@46752
   965
  by (iprover intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2)
berghofe@12905
   966
paulson@13830
   967
lemma Image_eq_UN: "r``B = (\<Union>y\<in> B. r``{y})"
berghofe@12905
   968
  -- {* NOT suitable for rewriting *}
haftmann@46752
   969
  by blast
berghofe@12905
   970
wenzelm@12913
   971
lemma Image_mono: "r' \<subseteq> r ==> A' \<subseteq> A ==> (r' `` A') \<subseteq> (r `` A)"
haftmann@46752
   972
  by blast
berghofe@12905
   973
paulson@13830
   974
lemma Image_UN: "(r `` (UNION A B)) = (\<Union>x\<in>A. r `` (B x))"
haftmann@46752
   975
  by blast
paulson@13830
   976
paulson@13830
   977
lemma Image_INT_subset: "(r `` INTER A B) \<subseteq> (\<Inter>x\<in>A. r `` (B x))"
haftmann@46752
   978
  by blast
berghofe@12905
   979
paulson@13830
   980
text{*Converse inclusion requires some assumptions*}
paulson@13830
   981
lemma Image_INT_eq:
paulson@13830
   982
     "[|single_valued (r\<inverse>); A\<noteq>{}|] ==> r `` INTER A B = (\<Inter>x\<in>A. r `` B x)"
paulson@13830
   983
apply (rule equalityI)
paulson@13830
   984
 apply (rule Image_INT_subset) 
paulson@13830
   985
apply  (simp add: single_valued_def, blast)
paulson@13830
   986
done
berghofe@12905
   987
wenzelm@12913
   988
lemma Image_subset_eq: "(r``A \<subseteq> B) = (A \<subseteq> - ((r^-1) `` (-B)))"
haftmann@46752
   989
  by blast
berghofe@12905
   990
haftmann@46692
   991
lemma Image_Collect_split [simp]: "{(x, y). P x y} `` A = {y. EX x:A. P x y}"
haftmann@46752
   992
  by auto
berghofe@12905
   993
berghofe@12905
   994
haftmann@46664
   995
subsubsection {* Inverse image *}
berghofe@12905
   996
haftmann@46752
   997
definition inv_image :: "'b rel \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a rel"
haftmann@46752
   998
where
haftmann@46752
   999
  "inv_image r f = {(x, y). (f x, f y) \<in> r}"
haftmann@46692
  1000
haftmann@46752
  1001
definition inv_imagep :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
haftmann@46752
  1002
where
haftmann@46694
  1003
  "inv_imagep r f = (\<lambda>x y. r (f x) (f y))"
haftmann@46694
  1004
haftmann@46694
  1005
lemma [pred_set_conv]: "inv_imagep (\<lambda>x y. (x, y) \<in> r) f = (\<lambda>x y. (x, y) \<in> inv_image r f)"
haftmann@46694
  1006
  by (simp add: inv_image_def inv_imagep_def)
haftmann@46694
  1007
huffman@19228
  1008
lemma sym_inv_image: "sym r ==> sym (inv_image r f)"
haftmann@46752
  1009
  by (unfold sym_def inv_image_def) blast
huffman@19228
  1010
wenzelm@12913
  1011
lemma trans_inv_image: "trans r ==> trans (inv_image r f)"
berghofe@12905
  1012
  apply (unfold trans_def inv_image_def)
berghofe@12905
  1013
  apply (simp (no_asm))
berghofe@12905
  1014
  apply blast
berghofe@12905
  1015
  done
berghofe@12905
  1016
krauss@32463
  1017
lemma in_inv_image[simp]: "((x,y) : inv_image r f) = ((f x, f y) : r)"
krauss@32463
  1018
  by (auto simp:inv_image_def)
krauss@32463
  1019
krauss@33218
  1020
lemma converse_inv_image[simp]: "(inv_image R f)^-1 = inv_image (R^-1) f"
haftmann@46752
  1021
  unfolding inv_image_def converse_unfold by auto
krauss@33218
  1022
haftmann@46664
  1023
lemma in_inv_imagep [simp]: "inv_imagep r f x y = r (f x) (f y)"
haftmann@46664
  1024
  by (simp add: inv_imagep_def)
haftmann@46664
  1025
haftmann@46664
  1026
haftmann@46664
  1027
subsubsection {* Powerset *}
haftmann@46664
  1028
haftmann@46752
  1029
definition Powp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"
haftmann@46752
  1030
where
haftmann@46664
  1031
  "Powp A = (\<lambda>B. \<forall>x \<in> B. A x)"
haftmann@46664
  1032
haftmann@46664
  1033
lemma Powp_Pow_eq [pred_set_conv]: "Powp (\<lambda>x. x \<in> A) = (\<lambda>x. x \<in> Pow A)"
haftmann@46664
  1034
  by (auto simp add: Powp_def fun_eq_iff)
haftmann@46664
  1035
haftmann@46664
  1036
lemmas Powp_mono [mono] = Pow_mono [to_pred]
haftmann@46664
  1037
nipkow@1128
  1038
end
haftmann@46689
  1039