src/HOL/Relation.thy
 author krauss Mon Jul 27 21:47:41 2009 +0200 (2009-07-27) changeset 32235 8f9b8d14fc9f parent 31011 506e57123cd1 child 32463 3a0a65ca2261 permissions -rw-r--r--
"more standard" argument order of relation composition (op O)
```     1 (*  Title:      HOL/Relation.thy
```
```     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     3     Copyright   1996  University of Cambridge
```
```     4 *)
```
```     5
```
```     6 header {* Relations *}
```
```     7
```
```     8 theory Relation
```
```     9 imports Finite_Set Datatype
```
```    10   (*FIXME order is important, otherwise merge problem for canonical interpretation of class monoid_mult wrt. power!*)
```
```    11 begin
```
```    12
```
```    13 subsection {* Definitions *}
```
```    14
```
```    15 definition
```
```    16   converse :: "('a * 'b) set => ('b * 'a) set"
```
```    17     ("(_^-1)" [1000] 999) where
```
```    18   "r^-1 == {(y, x). (x, y) : r}"
```
```    19
```
```    20 notation (xsymbols)
```
```    21   converse  ("(_\<inverse>)" [1000] 999)
```
```    22
```
```    23 definition
```
```    24   rel_comp  :: "[('a * 'b) set, ('b * 'c) set] => ('a * 'c) set"
```
```    25     (infixr "O" 75) where
```
```    26   "r O s == {(x,z). EX y. (x, y) : r & (y, z) : s}"
```
```    27
```
```    28 definition
```
```    29   Image :: "[('a * 'b) set, 'a set] => 'b set"
```
```    30     (infixl "``" 90) where
```
```    31   "r `` s == {y. EX x:s. (x,y):r}"
```
```    32
```
```    33 definition
```
```    34   Id :: "('a * 'a) set" where -- {* the identity relation *}
```
```    35   "Id == {p. EX x. p = (x,x)}"
```
```    36
```
```    37 definition
```
```    38   Id_on  :: "'a set => ('a * 'a) set" where -- {* diagonal: identity over a set *}
```
```    39   "Id_on A == \<Union>x\<in>A. {(x,x)}"
```
```    40
```
```    41 definition
```
```    42   Domain :: "('a * 'b) set => 'a set" where
```
```    43   "Domain r == {x. EX y. (x,y):r}"
```
```    44
```
```    45 definition
```
```    46   Range  :: "('a * 'b) set => 'b set" where
```
```    47   "Range r == Domain(r^-1)"
```
```    48
```
```    49 definition
```
```    50   Field :: "('a * 'a) set => 'a set" where
```
```    51   "Field r == Domain r \<union> Range r"
```
```    52
```
```    53 definition
```
```    54   refl_on :: "['a set, ('a * 'a) set] => bool" where -- {* reflexivity over a set *}
```
```    55   "refl_on A r == r \<subseteq> A \<times> A & (ALL x: A. (x,x) : r)"
```
```    56
```
```    57 abbreviation
```
```    58   refl :: "('a * 'a) set => bool" where -- {* reflexivity over a type *}
```
```    59   "refl == refl_on UNIV"
```
```    60
```
```    61 definition
```
```    62   sym :: "('a * 'a) set => bool" where -- {* symmetry predicate *}
```
```    63   "sym r == ALL x y. (x,y): r --> (y,x): r"
```
```    64
```
```    65 definition
```
```    66   antisym :: "('a * 'a) set => bool" where -- {* antisymmetry predicate *}
```
```    67   "antisym r == ALL x y. (x,y):r --> (y,x):r --> x=y"
```
```    68
```
```    69 definition
```
```    70   trans :: "('a * 'a) set => bool" where -- {* transitivity predicate *}
```
```    71   "trans r == (ALL x y z. (x,y):r --> (y,z):r --> (x,z):r)"
```
```    72
```
```    73 definition
```
```    74 irrefl :: "('a * 'a) set => bool" where
```
```    75 "irrefl r \<equiv> \<forall>x. (x,x) \<notin> r"
```
```    76
```
```    77 definition
```
```    78 total_on :: "'a set => ('a * 'a) set => bool" where
```
```    79 "total_on A r \<equiv> \<forall>x\<in>A.\<forall>y\<in>A. x\<noteq>y \<longrightarrow> (x,y)\<in>r \<or> (y,x)\<in>r"
```
```    80
```
```    81 abbreviation "total \<equiv> total_on UNIV"
```
```    82
```
```    83 definition
```
```    84   single_valued :: "('a * 'b) set => bool" where
```
```    85   "single_valued r == ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z)"
```
```    86
```
```    87 definition
```
```    88   inv_image :: "('b * 'b) set => ('a => 'b) => ('a * 'a) set" where
```
```    89   "inv_image r f == {(x, y). (f x, f y) : r}"
```
```    90
```
```    91
```
```    92 subsection {* The identity relation *}
```
```    93
```
```    94 lemma IdI [intro]: "(a, a) : Id"
```
```    95 by (simp add: Id_def)
```
```    96
```
```    97 lemma IdE [elim!]: "p : Id ==> (!!x. p = (x, x) ==> P) ==> P"
```
```    98 by (unfold Id_def) (iprover elim: CollectE)
```
```    99
```
```   100 lemma pair_in_Id_conv [iff]: "((a, b) : Id) = (a = b)"
```
```   101 by (unfold Id_def) blast
```
```   102
```
```   103 lemma refl_Id: "refl Id"
```
```   104 by (simp add: refl_on_def)
```
```   105
```
```   106 lemma antisym_Id: "antisym Id"
```
```   107   -- {* A strange result, since @{text Id} is also symmetric. *}
```
```   108 by (simp add: antisym_def)
```
```   109
```
```   110 lemma sym_Id: "sym Id"
```
```   111 by (simp add: sym_def)
```
```   112
```
```   113 lemma trans_Id: "trans Id"
```
```   114 by (simp add: trans_def)
```
```   115
```
```   116
```
```   117 subsection {* Diagonal: identity over a set *}
```
```   118
```
```   119 lemma Id_on_empty [simp]: "Id_on {} = {}"
```
```   120 by (simp add: Id_on_def)
```
```   121
```
```   122 lemma Id_on_eqI: "a = b ==> a : A ==> (a, b) : Id_on A"
```
```   123 by (simp add: Id_on_def)
```
```   124
```
```   125 lemma Id_onI [intro!,noatp]: "a : A ==> (a, a) : Id_on A"
```
```   126 by (rule Id_on_eqI) (rule refl)
```
```   127
```
```   128 lemma Id_onE [elim!]:
```
```   129   "c : Id_on A ==> (!!x. x : A ==> c = (x, x) ==> P) ==> P"
```
```   130   -- {* The general elimination rule. *}
```
```   131 by (unfold Id_on_def) (iprover elim!: UN_E singletonE)
```
```   132
```
```   133 lemma Id_on_iff: "((x, y) : Id_on A) = (x = y & x : A)"
```
```   134 by blast
```
```   135
```
```   136 lemma Id_on_subset_Times: "Id_on A \<subseteq> A \<times> A"
```
```   137 by blast
```
```   138
```
```   139
```
```   140 subsection {* Composition of two relations *}
```
```   141
```
```   142 lemma rel_compI [intro]:
```
```   143   "(a, b) : r ==> (b, c) : s ==> (a, c) : r O s"
```
```   144 by (unfold rel_comp_def) blast
```
```   145
```
```   146 lemma rel_compE [elim!]: "xz : r O s ==>
```
```   147   (!!x y z. xz = (x, z) ==> (x, y) : r ==> (y, z) : s  ==> P) ==> P"
```
```   148 by (unfold rel_comp_def) (iprover elim!: CollectE splitE exE conjE)
```
```   149
```
```   150 lemma rel_compEpair:
```
```   151   "(a, c) : r O s ==> (!!y. (a, y) : r ==> (y, c) : s ==> P) ==> P"
```
```   152 by (iprover elim: rel_compE Pair_inject ssubst)
```
```   153
```
```   154 lemma R_O_Id [simp]: "R O Id = R"
```
```   155 by fast
```
```   156
```
```   157 lemma Id_O_R [simp]: "Id O R = R"
```
```   158 by fast
```
```   159
```
```   160 lemma rel_comp_empty1[simp]: "{} O R = {}"
```
```   161 by blast
```
```   162
```
```   163 lemma rel_comp_empty2[simp]: "R O {} = {}"
```
```   164 by blast
```
```   165
```
```   166 lemma O_assoc: "(R O S) O T = R O (S O T)"
```
```   167 by blast
```
```   168
```
```   169 lemma trans_O_subset: "trans r ==> r O r \<subseteq> r"
```
```   170 by (unfold trans_def) blast
```
```   171
```
```   172 lemma rel_comp_mono: "r' \<subseteq> r ==> s' \<subseteq> s ==> (r' O s') \<subseteq> (r O s)"
```
```   173 by blast
```
```   174
```
```   175 lemma rel_comp_subset_Sigma:
```
```   176     "r \<subseteq> A \<times> B ==> s \<subseteq> B \<times> C ==> (r O s) \<subseteq> A \<times> C"
```
```   177 by blast
```
```   178
```
```   179 lemma rel_comp_distrib[simp]: "R O (S \<union> T) = (R O S) \<union> (R O T)"
```
```   180 by auto
```
```   181
```
```   182 lemma rel_comp_distrib2[simp]: "(S \<union> T) O R = (S O R) \<union> (T O R)"
```
```   183 by auto
```
```   184
```
```   185
```
```   186 subsection {* Reflexivity *}
```
```   187
```
```   188 lemma refl_onI: "r \<subseteq> A \<times> A ==> (!!x. x : A ==> (x, x) : r) ==> refl_on A r"
```
```   189 by (unfold refl_on_def) (iprover intro!: ballI)
```
```   190
```
```   191 lemma refl_onD: "refl_on A r ==> a : A ==> (a, a) : r"
```
```   192 by (unfold refl_on_def) blast
```
```   193
```
```   194 lemma refl_onD1: "refl_on A r ==> (x, y) : r ==> x : A"
```
```   195 by (unfold refl_on_def) blast
```
```   196
```
```   197 lemma refl_onD2: "refl_on A r ==> (x, y) : r ==> y : A"
```
```   198 by (unfold refl_on_def) blast
```
```   199
```
```   200 lemma refl_on_Int: "refl_on A r ==> refl_on B s ==> refl_on (A \<inter> B) (r \<inter> s)"
```
```   201 by (unfold refl_on_def) blast
```
```   202
```
```   203 lemma refl_on_Un: "refl_on A r ==> refl_on B s ==> refl_on (A \<union> B) (r \<union> s)"
```
```   204 by (unfold refl_on_def) blast
```
```   205
```
```   206 lemma refl_on_INTER:
```
```   207   "ALL x:S. refl_on (A x) (r x) ==> refl_on (INTER S A) (INTER S r)"
```
```   208 by (unfold refl_on_def) fast
```
```   209
```
```   210 lemma refl_on_UNION:
```
```   211   "ALL x:S. refl_on (A x) (r x) \<Longrightarrow> refl_on (UNION S A) (UNION S r)"
```
```   212 by (unfold refl_on_def) blast
```
```   213
```
```   214 lemma refl_on_empty[simp]: "refl_on {} {}"
```
```   215 by(simp add:refl_on_def)
```
```   216
```
```   217 lemma refl_on_Id_on: "refl_on A (Id_on A)"
```
```   218 by (rule refl_onI [OF Id_on_subset_Times Id_onI])
```
```   219
```
```   220
```
```   221 subsection {* Antisymmetry *}
```
```   222
```
```   223 lemma antisymI:
```
```   224   "(!!x y. (x, y) : r ==> (y, x) : r ==> x=y) ==> antisym r"
```
```   225 by (unfold antisym_def) iprover
```
```   226
```
```   227 lemma antisymD: "antisym r ==> (a, b) : r ==> (b, a) : r ==> a = b"
```
```   228 by (unfold antisym_def) iprover
```
```   229
```
```   230 lemma antisym_subset: "r \<subseteq> s ==> antisym s ==> antisym r"
```
```   231 by (unfold antisym_def) blast
```
```   232
```
```   233 lemma antisym_empty [simp]: "antisym {}"
```
```   234 by (unfold antisym_def) blast
```
```   235
```
```   236 lemma antisym_Id_on [simp]: "antisym (Id_on A)"
```
```   237 by (unfold antisym_def) blast
```
```   238
```
```   239
```
```   240 subsection {* Symmetry *}
```
```   241
```
```   242 lemma symI: "(!!a b. (a, b) : r ==> (b, a) : r) ==> sym r"
```
```   243 by (unfold sym_def) iprover
```
```   244
```
```   245 lemma symD: "sym r ==> (a, b) : r ==> (b, a) : r"
```
```   246 by (unfold sym_def, blast)
```
```   247
```
```   248 lemma sym_Int: "sym r ==> sym s ==> sym (r \<inter> s)"
```
```   249 by (fast intro: symI dest: symD)
```
```   250
```
```   251 lemma sym_Un: "sym r ==> sym s ==> sym (r \<union> s)"
```
```   252 by (fast intro: symI dest: symD)
```
```   253
```
```   254 lemma sym_INTER: "ALL x:S. sym (r x) ==> sym (INTER S r)"
```
```   255 by (fast intro: symI dest: symD)
```
```   256
```
```   257 lemma sym_UNION: "ALL x:S. sym (r x) ==> sym (UNION S r)"
```
```   258 by (fast intro: symI dest: symD)
```
```   259
```
```   260 lemma sym_Id_on [simp]: "sym (Id_on A)"
```
```   261 by (rule symI) clarify
```
```   262
```
```   263
```
```   264 subsection {* Transitivity *}
```
```   265
```
```   266 lemma transI:
```
```   267   "(!!x y z. (x, y) : r ==> (y, z) : r ==> (x, z) : r) ==> trans r"
```
```   268 by (unfold trans_def) iprover
```
```   269
```
```   270 lemma transD: "trans r ==> (a, b) : r ==> (b, c) : r ==> (a, c) : r"
```
```   271 by (unfold trans_def) iprover
```
```   272
```
```   273 lemma trans_Int: "trans r ==> trans s ==> trans (r \<inter> s)"
```
```   274 by (fast intro: transI elim: transD)
```
```   275
```
```   276 lemma trans_INTER: "ALL x:S. trans (r x) ==> trans (INTER S r)"
```
```   277 by (fast intro: transI elim: transD)
```
```   278
```
```   279 lemma trans_Id_on [simp]: "trans (Id_on A)"
```
```   280 by (fast intro: transI elim: transD)
```
```   281
```
```   282 lemma trans_diff_Id: " trans r \<Longrightarrow> antisym r \<Longrightarrow> trans (r-Id)"
```
```   283 unfolding antisym_def trans_def by blast
```
```   284
```
```   285 subsection {* Irreflexivity *}
```
```   286
```
```   287 lemma irrefl_diff_Id[simp]: "irrefl(r-Id)"
```
```   288 by(simp add:irrefl_def)
```
```   289
```
```   290 subsection {* Totality *}
```
```   291
```
```   292 lemma total_on_empty[simp]: "total_on {} r"
```
```   293 by(simp add:total_on_def)
```
```   294
```
```   295 lemma total_on_diff_Id[simp]: "total_on A (r-Id) = total_on A r"
```
```   296 by(simp add: total_on_def)
```
```   297
```
```   298 subsection {* Converse *}
```
```   299
```
```   300 lemma converse_iff [iff]: "((a,b): r^-1) = ((b,a) : r)"
```
```   301 by (simp add: converse_def)
```
```   302
```
```   303 lemma converseI[sym]: "(a, b) : r ==> (b, a) : r^-1"
```
```   304 by (simp add: converse_def)
```
```   305
```
```   306 lemma converseD[sym]: "(a,b) : r^-1 ==> (b, a) : r"
```
```   307 by (simp add: converse_def)
```
```   308
```
```   309 lemma converseE [elim!]:
```
```   310   "yx : r^-1 ==> (!!x y. yx = (y, x) ==> (x, y) : r ==> P) ==> P"
```
```   311     -- {* More general than @{text converseD}, as it ``splits'' the member of the relation. *}
```
```   312 by (unfold converse_def) (iprover elim!: CollectE splitE bexE)
```
```   313
```
```   314 lemma converse_converse [simp]: "(r^-1)^-1 = r"
```
```   315 by (unfold converse_def) blast
```
```   316
```
```   317 lemma converse_rel_comp: "(r O s)^-1 = s^-1 O r^-1"
```
```   318 by blast
```
```   319
```
```   320 lemma converse_Int: "(r \<inter> s)^-1 = r^-1 \<inter> s^-1"
```
```   321 by blast
```
```   322
```
```   323 lemma converse_Un: "(r \<union> s)^-1 = r^-1 \<union> s^-1"
```
```   324 by blast
```
```   325
```
```   326 lemma converse_INTER: "(INTER S r)^-1 = (INT x:S. (r x)^-1)"
```
```   327 by fast
```
```   328
```
```   329 lemma converse_UNION: "(UNION S r)^-1 = (UN x:S. (r x)^-1)"
```
```   330 by blast
```
```   331
```
```   332 lemma converse_Id [simp]: "Id^-1 = Id"
```
```   333 by blast
```
```   334
```
```   335 lemma converse_Id_on [simp]: "(Id_on A)^-1 = Id_on A"
```
```   336 by blast
```
```   337
```
```   338 lemma refl_on_converse [simp]: "refl_on A (converse r) = refl_on A r"
```
```   339 by (unfold refl_on_def) auto
```
```   340
```
```   341 lemma sym_converse [simp]: "sym (converse r) = sym r"
```
```   342 by (unfold sym_def) blast
```
```   343
```
```   344 lemma antisym_converse [simp]: "antisym (converse r) = antisym r"
```
```   345 by (unfold antisym_def) blast
```
```   346
```
```   347 lemma trans_converse [simp]: "trans (converse r) = trans r"
```
```   348 by (unfold trans_def) blast
```
```   349
```
```   350 lemma sym_conv_converse_eq: "sym r = (r^-1 = r)"
```
```   351 by (unfold sym_def) fast
```
```   352
```
```   353 lemma sym_Un_converse: "sym (r \<union> r^-1)"
```
```   354 by (unfold sym_def) blast
```
```   355
```
```   356 lemma sym_Int_converse: "sym (r \<inter> r^-1)"
```
```   357 by (unfold sym_def) blast
```
```   358
```
```   359 lemma total_on_converse[simp]: "total_on A (r^-1) = total_on A r"
```
```   360 by (auto simp: total_on_def)
```
```   361
```
```   362
```
```   363 subsection {* Domain *}
```
```   364
```
```   365 declare Domain_def [noatp]
```
```   366
```
```   367 lemma Domain_iff: "(a : Domain r) = (EX y. (a, y) : r)"
```
```   368 by (unfold Domain_def) blast
```
```   369
```
```   370 lemma DomainI [intro]: "(a, b) : r ==> a : Domain r"
```
```   371 by (iprover intro!: iffD2 [OF Domain_iff])
```
```   372
```
```   373 lemma DomainE [elim!]:
```
```   374   "a : Domain r ==> (!!y. (a, y) : r ==> P) ==> P"
```
```   375 by (iprover dest!: iffD1 [OF Domain_iff])
```
```   376
```
```   377 lemma Domain_empty [simp]: "Domain {} = {}"
```
```   378 by blast
```
```   379
```
```   380 lemma Domain_insert: "Domain (insert (a, b) r) = insert a (Domain r)"
```
```   381 by blast
```
```   382
```
```   383 lemma Domain_Id [simp]: "Domain Id = UNIV"
```
```   384 by blast
```
```   385
```
```   386 lemma Domain_Id_on [simp]: "Domain (Id_on A) = A"
```
```   387 by blast
```
```   388
```
```   389 lemma Domain_Un_eq: "Domain(A \<union> B) = Domain(A) \<union> Domain(B)"
```
```   390 by blast
```
```   391
```
```   392 lemma Domain_Int_subset: "Domain(A \<inter> B) \<subseteq> Domain(A) \<inter> Domain(B)"
```
```   393 by blast
```
```   394
```
```   395 lemma Domain_Diff_subset: "Domain(A) - Domain(B) \<subseteq> Domain(A - B)"
```
```   396 by blast
```
```   397
```
```   398 lemma Domain_Union: "Domain (Union S) = (\<Union>A\<in>S. Domain A)"
```
```   399 by blast
```
```   400
```
```   401 lemma Domain_converse[simp]: "Domain(r^-1) = Range r"
```
```   402 by(auto simp:Range_def)
```
```   403
```
```   404 lemma Domain_mono: "r \<subseteq> s ==> Domain r \<subseteq> Domain s"
```
```   405 by blast
```
```   406
```
```   407 lemma fst_eq_Domain: "fst ` R = Domain R";
```
```   408 by (auto intro!:image_eqI)
```
```   409
```
```   410 lemma Domain_dprod [simp]: "Domain (dprod r s) = uprod (Domain r) (Domain s)"
```
```   411 by auto
```
```   412
```
```   413 lemma Domain_dsum [simp]: "Domain (dsum r s) = usum (Domain r) (Domain s)"
```
```   414 by auto
```
```   415
```
```   416
```
```   417 subsection {* Range *}
```
```   418
```
```   419 lemma Range_iff: "(a : Range r) = (EX y. (y, a) : r)"
```
```   420 by (simp add: Domain_def Range_def)
```
```   421
```
```   422 lemma RangeI [intro]: "(a, b) : r ==> b : Range r"
```
```   423 by (unfold Range_def) (iprover intro!: converseI DomainI)
```
```   424
```
```   425 lemma RangeE [elim!]: "b : Range r ==> (!!x. (x, b) : r ==> P) ==> P"
```
```   426 by (unfold Range_def) (iprover elim!: DomainE dest!: converseD)
```
```   427
```
```   428 lemma Range_empty [simp]: "Range {} = {}"
```
```   429 by blast
```
```   430
```
```   431 lemma Range_insert: "Range (insert (a, b) r) = insert b (Range r)"
```
```   432 by blast
```
```   433
```
```   434 lemma Range_Id [simp]: "Range Id = UNIV"
```
```   435 by blast
```
```   436
```
```   437 lemma Range_Id_on [simp]: "Range (Id_on A) = A"
```
```   438 by auto
```
```   439
```
```   440 lemma Range_Un_eq: "Range(A \<union> B) = Range(A) \<union> Range(B)"
```
```   441 by blast
```
```   442
```
```   443 lemma Range_Int_subset: "Range(A \<inter> B) \<subseteq> Range(A) \<inter> Range(B)"
```
```   444 by blast
```
```   445
```
```   446 lemma Range_Diff_subset: "Range(A) - Range(B) \<subseteq> Range(A - B)"
```
```   447 by blast
```
```   448
```
```   449 lemma Range_Union: "Range (Union S) = (\<Union>A\<in>S. Range A)"
```
```   450 by blast
```
```   451
```
```   452 lemma Range_converse[simp]: "Range(r^-1) = Domain r"
```
```   453 by blast
```
```   454
```
```   455 lemma snd_eq_Range: "snd ` R = Range R";
```
```   456 by (auto intro!:image_eqI)
```
```   457
```
```   458
```
```   459 subsection {* Field *}
```
```   460
```
```   461 lemma mono_Field: "r \<subseteq> s \<Longrightarrow> Field r \<subseteq> Field s"
```
```   462 by(auto simp:Field_def Domain_def Range_def)
```
```   463
```
```   464 lemma Field_empty[simp]: "Field {} = {}"
```
```   465 by(auto simp:Field_def)
```
```   466
```
```   467 lemma Field_insert[simp]: "Field (insert (a,b) r) = {a,b} \<union> Field r"
```
```   468 by(auto simp:Field_def)
```
```   469
```
```   470 lemma Field_Un[simp]: "Field (r \<union> s) = Field r \<union> Field s"
```
```   471 by(auto simp:Field_def)
```
```   472
```
```   473 lemma Field_Union[simp]: "Field (\<Union>R) = \<Union>(Field ` R)"
```
```   474 by(auto simp:Field_def)
```
```   475
```
```   476 lemma Field_converse[simp]: "Field(r^-1) = Field r"
```
```   477 by(auto simp:Field_def)
```
```   478
```
```   479
```
```   480 subsection {* Image of a set under a relation *}
```
```   481
```
```   482 declare Image_def [noatp]
```
```   483
```
```   484 lemma Image_iff: "(b : r``A) = (EX x:A. (x, b) : r)"
```
```   485 by (simp add: Image_def)
```
```   486
```
```   487 lemma Image_singleton: "r``{a} = {b. (a, b) : r}"
```
```   488 by (simp add: Image_def)
```
```   489
```
```   490 lemma Image_singleton_iff [iff]: "(b : r``{a}) = ((a, b) : r)"
```
```   491 by (rule Image_iff [THEN trans]) simp
```
```   492
```
```   493 lemma ImageI [intro,noatp]: "(a, b) : r ==> a : A ==> b : r``A"
```
```   494 by (unfold Image_def) blast
```
```   495
```
```   496 lemma ImageE [elim!]:
```
```   497     "b : r `` A ==> (!!x. (x, b) : r ==> x : A ==> P) ==> P"
```
```   498 by (unfold Image_def) (iprover elim!: CollectE bexE)
```
```   499
```
```   500 lemma rev_ImageI: "a : A ==> (a, b) : r ==> b : r `` A"
```
```   501   -- {* This version's more effective when we already have the required @{text a} *}
```
```   502 by blast
```
```   503
```
```   504 lemma Image_empty [simp]: "R``{} = {}"
```
```   505 by blast
```
```   506
```
```   507 lemma Image_Id [simp]: "Id `` A = A"
```
```   508 by blast
```
```   509
```
```   510 lemma Image_Id_on [simp]: "Id_on A `` B = A \<inter> B"
```
```   511 by blast
```
```   512
```
```   513 lemma Image_Int_subset: "R `` (A \<inter> B) \<subseteq> R `` A \<inter> R `` B"
```
```   514 by blast
```
```   515
```
```   516 lemma Image_Int_eq:
```
```   517      "single_valued (converse R) ==> R `` (A \<inter> B) = R `` A \<inter> R `` B"
```
```   518 by (simp add: single_valued_def, blast)
```
```   519
```
```   520 lemma Image_Un: "R `` (A \<union> B) = R `` A \<union> R `` B"
```
```   521 by blast
```
```   522
```
```   523 lemma Un_Image: "(R \<union> S) `` A = R `` A \<union> S `` A"
```
```   524 by blast
```
```   525
```
```   526 lemma Image_subset: "r \<subseteq> A \<times> B ==> r``C \<subseteq> B"
```
```   527 by (iprover intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2)
```
```   528
```
```   529 lemma Image_eq_UN: "r``B = (\<Union>y\<in> B. r``{y})"
```
```   530   -- {* NOT suitable for rewriting *}
```
```   531 by blast
```
```   532
```
```   533 lemma Image_mono: "r' \<subseteq> r ==> A' \<subseteq> A ==> (r' `` A') \<subseteq> (r `` A)"
```
```   534 by blast
```
```   535
```
```   536 lemma Image_UN: "(r `` (UNION A B)) = (\<Union>x\<in>A. r `` (B x))"
```
```   537 by blast
```
```   538
```
```   539 lemma Image_INT_subset: "(r `` INTER A B) \<subseteq> (\<Inter>x\<in>A. r `` (B x))"
```
```   540 by blast
```
```   541
```
```   542 text{*Converse inclusion requires some assumptions*}
```
```   543 lemma Image_INT_eq:
```
```   544      "[|single_valued (r\<inverse>); A\<noteq>{}|] ==> r `` INTER A B = (\<Inter>x\<in>A. r `` B x)"
```
```   545 apply (rule equalityI)
```
```   546  apply (rule Image_INT_subset)
```
```   547 apply  (simp add: single_valued_def, blast)
```
```   548 done
```
```   549
```
```   550 lemma Image_subset_eq: "(r``A \<subseteq> B) = (A \<subseteq> - ((r^-1) `` (-B)))"
```
```   551 by blast
```
```   552
```
```   553
```
```   554 subsection {* Single valued relations *}
```
```   555
```
```   556 lemma single_valuedI:
```
```   557   "ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z) ==> single_valued r"
```
```   558 by (unfold single_valued_def)
```
```   559
```
```   560 lemma single_valuedD:
```
```   561   "single_valued r ==> (x, y) : r ==> (x, z) : r ==> y = z"
```
```   562 by (simp add: single_valued_def)
```
```   563
```
```   564 lemma single_valued_rel_comp:
```
```   565   "single_valued r ==> single_valued s ==> single_valued (r O s)"
```
```   566 by (unfold single_valued_def) blast
```
```   567
```
```   568 lemma single_valued_subset:
```
```   569   "r \<subseteq> s ==> single_valued s ==> single_valued r"
```
```   570 by (unfold single_valued_def) blast
```
```   571
```
```   572 lemma single_valued_Id [simp]: "single_valued Id"
```
```   573 by (unfold single_valued_def) blast
```
```   574
```
```   575 lemma single_valued_Id_on [simp]: "single_valued (Id_on A)"
```
```   576 by (unfold single_valued_def) blast
```
```   577
```
```   578
```
```   579 subsection {* Graphs given by @{text Collect} *}
```
```   580
```
```   581 lemma Domain_Collect_split [simp]: "Domain{(x,y). P x y} = {x. EX y. P x y}"
```
```   582 by auto
```
```   583
```
```   584 lemma Range_Collect_split [simp]: "Range{(x,y). P x y} = {y. EX x. P x y}"
```
```   585 by auto
```
```   586
```
```   587 lemma Image_Collect_split [simp]: "{(x,y). P x y} `` A = {y. EX x:A. P x y}"
```
```   588 by auto
```
```   589
```
```   590
```
```   591 subsection {* Inverse image *}
```
```   592
```
```   593 lemma sym_inv_image: "sym r ==> sym (inv_image r f)"
```
```   594 by (unfold sym_def inv_image_def) blast
```
```   595
```
```   596 lemma trans_inv_image: "trans r ==> trans (inv_image r f)"
```
```   597   apply (unfold trans_def inv_image_def)
```
```   598   apply (simp (no_asm))
```
```   599   apply blast
```
```   600   done
```
```   601
```
```   602
```
```   603 subsection {* Finiteness *}
```
```   604
```
```   605 lemma finite_converse [iff]: "finite (r^-1) = finite r"
```
```   606   apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r")
```
```   607    apply simp
```
```   608    apply (rule iffI)
```
```   609     apply (erule finite_imageD [unfolded inj_on_def])
```
```   610     apply (simp split add: split_split)
```
```   611    apply (erule finite_imageI)
```
```   612   apply (simp add: converse_def image_def, auto)
```
```   613   apply (rule bexI)
```
```   614    prefer 2 apply assumption
```
```   615   apply simp
```
```   616   done
```
```   617
```
```   618 text {* \paragraph{Finiteness of transitive closure} (Thanks to Sidi
```
```   619 Ehmety) *}
```
```   620
```
```   621 lemma finite_Field: "finite r ==> finite (Field r)"
```
```   622   -- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *}
```
```   623   apply (induct set: finite)
```
```   624    apply (auto simp add: Field_def Domain_insert Range_insert)
```
```   625   done
```
```   626
```
```   627
```
```   628 subsection {* Version of @{text lfp_induct} for binary relations *}
```
```   629
```
```   630 lemmas lfp_induct2 =
```
```   631   lfp_induct_set [of "(a, b)", split_format (complete)]
```
```   632
```
```   633 end
```