src/HOL/Relation.thy
 author obua Mon Apr 10 16:00:34 2006 +0200 (2006-04-10) changeset 19404 9bf2cdc9e8e8 parent 19363 667b5ea637dd child 19656 09be06943252 permissions -rw-r--r--
Moved stuff from Ring_and_Field to Matrix
```     1 (*  Title:      HOL/Relation.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     4     Copyright   1996  University of Cambridge
```
```     5 *)
```
```     6
```
```     7 header {* Relations *}
```
```     8
```
```     9 theory Relation
```
```    10 imports Product_Type
```
```    11 begin
```
```    12
```
```    13 subsection {* Definitions *}
```
```    14
```
```    15 constdefs
```
```    16   converse :: "('a * 'b) set => ('b * 'a) set"    ("(_^-1)" [1000] 999)
```
```    17   "r^-1 == {(y, x). (x, y) : r}"
```
```    18 syntax (xsymbols)
```
```    19   converse :: "('a * 'b) set => ('b * 'a) set"    ("(_\<inverse>)" [1000] 999)
```
```    20
```
```    21 constdefs
```
```    22   rel_comp  :: "[('b * 'c) set, ('a * 'b) set] => ('a * 'c) set"  (infixr "O" 60)
```
```    23   "r O s == {(x,z). EX y. (x, y) : s & (y, z) : r}"
```
```    24
```
```    25   Image :: "[('a * 'b) set, 'a set] => 'b set"                (infixl "``" 90)
```
```    26   "r `` s == {y. EX x:s. (x,y):r}"
```
```    27
```
```    28   Id    :: "('a * 'a) set"  -- {* the identity relation *}
```
```    29   "Id == {p. EX x. p = (x,x)}"
```
```    30
```
```    31   diag  :: "'a set => ('a * 'a) set"  -- {* diagonal: identity over a set *}
```
```    32   "diag A == \<Union>x\<in>A. {(x,x)}"
```
```    33
```
```    34   Domain :: "('a * 'b) set => 'a set"
```
```    35   "Domain r == {x. EX y. (x,y):r}"
```
```    36
```
```    37   Range  :: "('a * 'b) set => 'b set"
```
```    38   "Range r == Domain(r^-1)"
```
```    39
```
```    40   Field :: "('a * 'a) set => 'a set"
```
```    41   "Field r == Domain r \<union> Range r"
```
```    42
```
```    43   refl   :: "['a set, ('a * 'a) set] => bool"  -- {* reflexivity over a set *}
```
```    44   "refl A r == r \<subseteq> A \<times> A & (ALL x: A. (x,x) : r)"
```
```    45
```
```    46   sym    :: "('a * 'a) set => bool"  -- {* symmetry predicate *}
```
```    47   "sym r == ALL x y. (x,y): r --> (y,x): r"
```
```    48
```
```    49   antisym:: "('a * 'a) set => bool"  -- {* antisymmetry predicate *}
```
```    50   "antisym r == ALL x y. (x,y):r --> (y,x):r --> x=y"
```
```    51
```
```    52   trans  :: "('a * 'a) set => bool"  -- {* transitivity predicate *}
```
```    53   "trans r == (ALL x y z. (x,y):r --> (y,z):r --> (x,z):r)"
```
```    54
```
```    55   single_valued :: "('a * 'b) set => bool"
```
```    56   "single_valued r == ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z)"
```
```    57
```
```    58   inv_image :: "('b * 'b) set => ('a => 'b) => ('a * 'a) set"
```
```    59   "inv_image r f == {(x, y). (f x, f y) : r}"
```
```    60
```
```    61 abbreviation
```
```    62   reflexive :: "('a * 'a) set => bool"  -- {* reflexivity over a type *}
```
```    63   "reflexive == refl UNIV"
```
```    64
```
```    65
```
```    66 subsection {* The identity relation *}
```
```    67
```
```    68 lemma IdI [intro]: "(a, a) : Id"
```
```    69   by (simp add: Id_def)
```
```    70
```
```    71 lemma IdE [elim!]: "p : Id ==> (!!x. p = (x, x) ==> P) ==> P"
```
```    72   by (unfold Id_def) (iprover elim: CollectE)
```
```    73
```
```    74 lemma pair_in_Id_conv [iff]: "((a, b) : Id) = (a = b)"
```
```    75   by (unfold Id_def) blast
```
```    76
```
```    77 lemma reflexive_Id: "reflexive Id"
```
```    78   by (simp add: refl_def)
```
```    79
```
```    80 lemma antisym_Id: "antisym Id"
```
```    81   -- {* A strange result, since @{text Id} is also symmetric. *}
```
```    82   by (simp add: antisym_def)
```
```    83
```
```    84 lemma sym_Id: "sym Id"
```
```    85   by (simp add: sym_def)
```
```    86
```
```    87 lemma trans_Id: "trans Id"
```
```    88   by (simp add: trans_def)
```
```    89
```
```    90
```
```    91 subsection {* Diagonal: identity over a set *}
```
```    92
```
```    93 lemma diag_empty [simp]: "diag {} = {}"
```
```    94   by (simp add: diag_def)
```
```    95
```
```    96 lemma diag_eqI: "a = b ==> a : A ==> (a, b) : diag A"
```
```    97   by (simp add: diag_def)
```
```    98
```
```    99 lemma diagI [intro!]: "a : A ==> (a, a) : diag A"
```
```   100   by (rule diag_eqI) (rule refl)
```
```   101
```
```   102 lemma diagE [elim!]:
```
```   103   "c : diag A ==> (!!x. x : A ==> c = (x, x) ==> P) ==> P"
```
```   104   -- {* The general elimination rule. *}
```
```   105   by (unfold diag_def) (iprover elim!: UN_E singletonE)
```
```   106
```
```   107 lemma diag_iff: "((x, y) : diag A) = (x = y & x : A)"
```
```   108   by blast
```
```   109
```
```   110 lemma diag_subset_Times: "diag A \<subseteq> A \<times> A"
```
```   111   by blast
```
```   112
```
```   113
```
```   114 subsection {* Composition of two relations *}
```
```   115
```
```   116 lemma rel_compI [intro]:
```
```   117   "(a, b) : s ==> (b, c) : r ==> (a, c) : r O s"
```
```   118   by (unfold rel_comp_def) blast
```
```   119
```
```   120 lemma rel_compE [elim!]: "xz : r O s ==>
```
```   121   (!!x y z. xz = (x, z) ==> (x, y) : s ==> (y, z) : r  ==> P) ==> P"
```
```   122   by (unfold rel_comp_def) (iprover elim!: CollectE splitE exE conjE)
```
```   123
```
```   124 lemma rel_compEpair:
```
```   125   "(a, c) : r O s ==> (!!y. (a, y) : s ==> (y, c) : r ==> P) ==> P"
```
```   126   by (iprover elim: rel_compE Pair_inject ssubst)
```
```   127
```
```   128 lemma R_O_Id [simp]: "R O Id = R"
```
```   129   by fast
```
```   130
```
```   131 lemma Id_O_R [simp]: "Id O R = R"
```
```   132   by fast
```
```   133
```
```   134 lemma O_assoc: "(R O S) O T = R O (S O T)"
```
```   135   by blast
```
```   136
```
```   137 lemma trans_O_subset: "trans r ==> r O r \<subseteq> r"
```
```   138   by (unfold trans_def) blast
```
```   139
```
```   140 lemma rel_comp_mono: "r' \<subseteq> r ==> s' \<subseteq> s ==> (r' O s') \<subseteq> (r O s)"
```
```   141   by blast
```
```   142
```
```   143 lemma rel_comp_subset_Sigma:
```
```   144     "s \<subseteq> A \<times> B ==> r \<subseteq> B \<times> C ==> (r O s) \<subseteq> A \<times> C"
```
```   145   by blast
```
```   146
```
```   147
```
```   148 subsection {* Reflexivity *}
```
```   149
```
```   150 lemma reflI: "r \<subseteq> A \<times> A ==> (!!x. x : A ==> (x, x) : r) ==> refl A r"
```
```   151   by (unfold refl_def) (iprover intro!: ballI)
```
```   152
```
```   153 lemma reflD: "refl A r ==> a : A ==> (a, a) : r"
```
```   154   by (unfold refl_def) blast
```
```   155
```
```   156 lemma reflD1: "refl A r ==> (x, y) : r ==> x : A"
```
```   157   by (unfold refl_def) blast
```
```   158
```
```   159 lemma reflD2: "refl A r ==> (x, y) : r ==> y : A"
```
```   160   by (unfold refl_def) blast
```
```   161
```
```   162 lemma refl_Int: "refl A r ==> refl B s ==> refl (A \<inter> B) (r \<inter> s)"
```
```   163   by (unfold refl_def) blast
```
```   164
```
```   165 lemma refl_Un: "refl A r ==> refl B s ==> refl (A \<union> B) (r \<union> s)"
```
```   166   by (unfold refl_def) blast
```
```   167
```
```   168 lemma refl_INTER:
```
```   169   "ALL x:S. refl (A x) (r x) ==> refl (INTER S A) (INTER S r)"
```
```   170   by (unfold refl_def) fast
```
```   171
```
```   172 lemma refl_UNION:
```
```   173   "ALL x:S. refl (A x) (r x) \<Longrightarrow> refl (UNION S A) (UNION S r)"
```
```   174   by (unfold refl_def) blast
```
```   175
```
```   176 lemma refl_diag: "refl A (diag A)"
```
```   177   by (rule reflI [OF diag_subset_Times diagI])
```
```   178
```
```   179
```
```   180 subsection {* Antisymmetry *}
```
```   181
```
```   182 lemma antisymI:
```
```   183   "(!!x y. (x, y) : r ==> (y, x) : r ==> x=y) ==> antisym r"
```
```   184   by (unfold antisym_def) iprover
```
```   185
```
```   186 lemma antisymD: "antisym r ==> (a, b) : r ==> (b, a) : r ==> a = b"
```
```   187   by (unfold antisym_def) iprover
```
```   188
```
```   189 lemma antisym_subset: "r \<subseteq> s ==> antisym s ==> antisym r"
```
```   190   by (unfold antisym_def) blast
```
```   191
```
```   192 lemma antisym_empty [simp]: "antisym {}"
```
```   193   by (unfold antisym_def) blast
```
```   194
```
```   195 lemma antisym_diag [simp]: "antisym (diag A)"
```
```   196   by (unfold antisym_def) blast
```
```   197
```
```   198
```
```   199 subsection {* Symmetry *}
```
```   200
```
```   201 lemma symI: "(!!a b. (a, b) : r ==> (b, a) : r) ==> sym r"
```
```   202   by (unfold sym_def) iprover
```
```   203
```
```   204 lemma symD: "sym r ==> (a, b) : r ==> (b, a) : r"
```
```   205   by (unfold sym_def, blast)
```
```   206
```
```   207 lemma sym_Int: "sym r ==> sym s ==> sym (r \<inter> s)"
```
```   208   by (fast intro: symI dest: symD)
```
```   209
```
```   210 lemma sym_Un: "sym r ==> sym s ==> sym (r \<union> s)"
```
```   211   by (fast intro: symI dest: symD)
```
```   212
```
```   213 lemma sym_INTER: "ALL x:S. sym (r x) ==> sym (INTER S r)"
```
```   214   by (fast intro: symI dest: symD)
```
```   215
```
```   216 lemma sym_UNION: "ALL x:S. sym (r x) ==> sym (UNION S r)"
```
```   217   by (fast intro: symI dest: symD)
```
```   218
```
```   219 lemma sym_diag [simp]: "sym (diag A)"
```
```   220   by (rule symI) clarify
```
```   221
```
```   222
```
```   223 subsection {* Transitivity *}
```
```   224
```
```   225 lemma transI:
```
```   226   "(!!x y z. (x, y) : r ==> (y, z) : r ==> (x, z) : r) ==> trans r"
```
```   227   by (unfold trans_def) iprover
```
```   228
```
```   229 lemma transD: "trans r ==> (a, b) : r ==> (b, c) : r ==> (a, c) : r"
```
```   230   by (unfold trans_def) iprover
```
```   231
```
```   232 lemma trans_Int: "trans r ==> trans s ==> trans (r \<inter> s)"
```
```   233   by (fast intro: transI elim: transD)
```
```   234
```
```   235 lemma trans_INTER: "ALL x:S. trans (r x) ==> trans (INTER S r)"
```
```   236   by (fast intro: transI elim: transD)
```
```   237
```
```   238 lemma trans_diag [simp]: "trans (diag A)"
```
```   239   by (fast intro: transI elim: transD)
```
```   240
```
```   241
```
```   242 subsection {* Converse *}
```
```   243
```
```   244 lemma converse_iff [iff]: "((a,b): r^-1) = ((b,a) : r)"
```
```   245   by (simp add: converse_def)
```
```   246
```
```   247 lemma converseI[sym]: "(a, b) : r ==> (b, a) : r^-1"
```
```   248   by (simp add: converse_def)
```
```   249
```
```   250 lemma converseD[sym]: "(a,b) : r^-1 ==> (b, a) : r"
```
```   251   by (simp add: converse_def)
```
```   252
```
```   253 lemma converseE [elim!]:
```
```   254   "yx : r^-1 ==> (!!x y. yx = (y, x) ==> (x, y) : r ==> P) ==> P"
```
```   255     -- {* More general than @{text converseD}, as it ``splits'' the member of the relation. *}
```
```   256   by (unfold converse_def) (iprover elim!: CollectE splitE bexE)
```
```   257
```
```   258 lemma converse_converse [simp]: "(r^-1)^-1 = r"
```
```   259   by (unfold converse_def) blast
```
```   260
```
```   261 lemma converse_rel_comp: "(r O s)^-1 = s^-1 O r^-1"
```
```   262   by blast
```
```   263
```
```   264 lemma converse_Int: "(r \<inter> s)^-1 = r^-1 \<inter> s^-1"
```
```   265   by blast
```
```   266
```
```   267 lemma converse_Un: "(r \<union> s)^-1 = r^-1 \<union> s^-1"
```
```   268   by blast
```
```   269
```
```   270 lemma converse_INTER: "(INTER S r)^-1 = (INT x:S. (r x)^-1)"
```
```   271   by fast
```
```   272
```
```   273 lemma converse_UNION: "(UNION S r)^-1 = (UN x:S. (r x)^-1)"
```
```   274   by blast
```
```   275
```
```   276 lemma converse_Id [simp]: "Id^-1 = Id"
```
```   277   by blast
```
```   278
```
```   279 lemma converse_diag [simp]: "(diag A)^-1 = diag A"
```
```   280   by blast
```
```   281
```
```   282 lemma refl_converse [simp]: "refl A (converse r) = refl A r"
```
```   283   by (unfold refl_def) auto
```
```   284
```
```   285 lemma sym_converse [simp]: "sym (converse r) = sym r"
```
```   286   by (unfold sym_def) blast
```
```   287
```
```   288 lemma antisym_converse [simp]: "antisym (converse r) = antisym r"
```
```   289   by (unfold antisym_def) blast
```
```   290
```
```   291 lemma trans_converse [simp]: "trans (converse r) = trans r"
```
```   292   by (unfold trans_def) blast
```
```   293
```
```   294 lemma sym_conv_converse_eq: "sym r = (r^-1 = r)"
```
```   295   by (unfold sym_def) fast
```
```   296
```
```   297 lemma sym_Un_converse: "sym (r \<union> r^-1)"
```
```   298   by (unfold sym_def) blast
```
```   299
```
```   300 lemma sym_Int_converse: "sym (r \<inter> r^-1)"
```
```   301   by (unfold sym_def) blast
```
```   302
```
```   303
```
```   304 subsection {* Domain *}
```
```   305
```
```   306 lemma Domain_iff: "(a : Domain r) = (EX y. (a, y) : r)"
```
```   307   by (unfold Domain_def) blast
```
```   308
```
```   309 lemma DomainI [intro]: "(a, b) : r ==> a : Domain r"
```
```   310   by (iprover intro!: iffD2 [OF Domain_iff])
```
```   311
```
```   312 lemma DomainE [elim!]:
```
```   313   "a : Domain r ==> (!!y. (a, y) : r ==> P) ==> P"
```
```   314   by (iprover dest!: iffD1 [OF Domain_iff])
```
```   315
```
```   316 lemma Domain_empty [simp]: "Domain {} = {}"
```
```   317   by blast
```
```   318
```
```   319 lemma Domain_insert: "Domain (insert (a, b) r) = insert a (Domain r)"
```
```   320   by blast
```
```   321
```
```   322 lemma Domain_Id [simp]: "Domain Id = UNIV"
```
```   323   by blast
```
```   324
```
```   325 lemma Domain_diag [simp]: "Domain (diag A) = A"
```
```   326   by blast
```
```   327
```
```   328 lemma Domain_Un_eq: "Domain(A \<union> B) = Domain(A) \<union> Domain(B)"
```
```   329   by blast
```
```   330
```
```   331 lemma Domain_Int_subset: "Domain(A \<inter> B) \<subseteq> Domain(A) \<inter> Domain(B)"
```
```   332   by blast
```
```   333
```
```   334 lemma Domain_Diff_subset: "Domain(A) - Domain(B) \<subseteq> Domain(A - B)"
```
```   335   by blast
```
```   336
```
```   337 lemma Domain_Union: "Domain (Union S) = (\<Union>A\<in>S. Domain A)"
```
```   338   by blast
```
```   339
```
```   340 lemma Domain_mono: "r \<subseteq> s ==> Domain r \<subseteq> Domain s"
```
```   341   by blast
```
```   342
```
```   343
```
```   344 subsection {* Range *}
```
```   345
```
```   346 lemma Range_iff: "(a : Range r) = (EX y. (y, a) : r)"
```
```   347   by (simp add: Domain_def Range_def)
```
```   348
```
```   349 lemma RangeI [intro]: "(a, b) : r ==> b : Range r"
```
```   350   by (unfold Range_def) (iprover intro!: converseI DomainI)
```
```   351
```
```   352 lemma RangeE [elim!]: "b : Range r ==> (!!x. (x, b) : r ==> P) ==> P"
```
```   353   by (unfold Range_def) (iprover elim!: DomainE dest!: converseD)
```
```   354
```
```   355 lemma Range_empty [simp]: "Range {} = {}"
```
```   356   by blast
```
```   357
```
```   358 lemma Range_insert: "Range (insert (a, b) r) = insert b (Range r)"
```
```   359   by blast
```
```   360
```
```   361 lemma Range_Id [simp]: "Range Id = UNIV"
```
```   362   by blast
```
```   363
```
```   364 lemma Range_diag [simp]: "Range (diag A) = A"
```
```   365   by auto
```
```   366
```
```   367 lemma Range_Un_eq: "Range(A \<union> B) = Range(A) \<union> Range(B)"
```
```   368   by blast
```
```   369
```
```   370 lemma Range_Int_subset: "Range(A \<inter> B) \<subseteq> Range(A) \<inter> Range(B)"
```
```   371   by blast
```
```   372
```
```   373 lemma Range_Diff_subset: "Range(A) - Range(B) \<subseteq> Range(A - B)"
```
```   374   by blast
```
```   375
```
```   376 lemma Range_Union: "Range (Union S) = (\<Union>A\<in>S. Range A)"
```
```   377   by blast
```
```   378
```
```   379
```
```   380 subsection {* Image of a set under a relation *}
```
```   381
```
```   382 lemma Image_iff: "(b : r``A) = (EX x:A. (x, b) : r)"
```
```   383   by (simp add: Image_def)
```
```   384
```
```   385 lemma Image_singleton: "r``{a} = {b. (a, b) : r}"
```
```   386   by (simp add: Image_def)
```
```   387
```
```   388 lemma Image_singleton_iff [iff]: "(b : r``{a}) = ((a, b) : r)"
```
```   389   by (rule Image_iff [THEN trans]) simp
```
```   390
```
```   391 lemma ImageI [intro]: "(a, b) : r ==> a : A ==> b : r``A"
```
```   392   by (unfold Image_def) blast
```
```   393
```
```   394 lemma ImageE [elim!]:
```
```   395     "b : r `` A ==> (!!x. (x, b) : r ==> x : A ==> P) ==> P"
```
```   396   by (unfold Image_def) (iprover elim!: CollectE bexE)
```
```   397
```
```   398 lemma rev_ImageI: "a : A ==> (a, b) : r ==> b : r `` A"
```
```   399   -- {* This version's more effective when we already have the required @{text a} *}
```
```   400   by blast
```
```   401
```
```   402 lemma Image_empty [simp]: "R``{} = {}"
```
```   403   by blast
```
```   404
```
```   405 lemma Image_Id [simp]: "Id `` A = A"
```
```   406   by blast
```
```   407
```
```   408 lemma Image_diag [simp]: "diag A `` B = A \<inter> B"
```
```   409   by blast
```
```   410
```
```   411 lemma Image_Int_subset: "R `` (A \<inter> B) \<subseteq> R `` A \<inter> R `` B"
```
```   412   by blast
```
```   413
```
```   414 lemma Image_Int_eq:
```
```   415      "single_valued (converse R) ==> R `` (A \<inter> B) = R `` A \<inter> R `` B"
```
```   416   by (simp add: single_valued_def, blast)
```
```   417
```
```   418 lemma Image_Un: "R `` (A \<union> B) = R `` A \<union> R `` B"
```
```   419   by blast
```
```   420
```
```   421 lemma Un_Image: "(R \<union> S) `` A = R `` A \<union> S `` A"
```
```   422   by blast
```
```   423
```
```   424 lemma Image_subset: "r \<subseteq> A \<times> B ==> r``C \<subseteq> B"
```
```   425   by (iprover intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2)
```
```   426
```
```   427 lemma Image_eq_UN: "r``B = (\<Union>y\<in> B. r``{y})"
```
```   428   -- {* NOT suitable for rewriting *}
```
```   429   by blast
```
```   430
```
```   431 lemma Image_mono: "r' \<subseteq> r ==> A' \<subseteq> A ==> (r' `` A') \<subseteq> (r `` A)"
```
```   432   by blast
```
```   433
```
```   434 lemma Image_UN: "(r `` (UNION A B)) = (\<Union>x\<in>A. r `` (B x))"
```
```   435   by blast
```
```   436
```
```   437 lemma Image_INT_subset: "(r `` INTER A B) \<subseteq> (\<Inter>x\<in>A. r `` (B x))"
```
```   438   by blast
```
```   439
```
```   440 text{*Converse inclusion requires some assumptions*}
```
```   441 lemma Image_INT_eq:
```
```   442      "[|single_valued (r\<inverse>); A\<noteq>{}|] ==> r `` INTER A B = (\<Inter>x\<in>A. r `` B x)"
```
```   443 apply (rule equalityI)
```
```   444  apply (rule Image_INT_subset)
```
```   445 apply  (simp add: single_valued_def, blast)
```
```   446 done
```
```   447
```
```   448 lemma Image_subset_eq: "(r``A \<subseteq> B) = (A \<subseteq> - ((r^-1) `` (-B)))"
```
```   449   by blast
```
```   450
```
```   451
```
```   452 subsection {* Single valued relations *}
```
```   453
```
```   454 lemma single_valuedI:
```
```   455   "ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z) ==> single_valued r"
```
```   456   by (unfold single_valued_def)
```
```   457
```
```   458 lemma single_valuedD:
```
```   459   "single_valued r ==> (x, y) : r ==> (x, z) : r ==> y = z"
```
```   460   by (simp add: single_valued_def)
```
```   461
```
```   462 lemma single_valued_rel_comp:
```
```   463   "single_valued r ==> single_valued s ==> single_valued (r O s)"
```
```   464   by (unfold single_valued_def) blast
```
```   465
```
```   466 lemma single_valued_subset:
```
```   467   "r \<subseteq> s ==> single_valued s ==> single_valued r"
```
```   468   by (unfold single_valued_def) blast
```
```   469
```
```   470 lemma single_valued_Id [simp]: "single_valued Id"
```
```   471   by (unfold single_valued_def) blast
```
```   472
```
```   473 lemma single_valued_diag [simp]: "single_valued (diag A)"
```
```   474   by (unfold single_valued_def) blast
```
```   475
```
```   476
```
```   477 subsection {* Graphs given by @{text Collect} *}
```
```   478
```
```   479 lemma Domain_Collect_split [simp]: "Domain{(x,y). P x y} = {x. EX y. P x y}"
```
```   480   by auto
```
```   481
```
```   482 lemma Range_Collect_split [simp]: "Range{(x,y). P x y} = {y. EX x. P x y}"
```
```   483   by auto
```
```   484
```
```   485 lemma Image_Collect_split [simp]: "{(x,y). P x y} `` A = {y. EX x:A. P x y}"
```
```   486   by auto
```
```   487
```
```   488
```
```   489 subsection {* Inverse image *}
```
```   490
```
```   491 lemma sym_inv_image: "sym r ==> sym (inv_image r f)"
```
```   492   by (unfold sym_def inv_image_def) blast
```
```   493
```
```   494 lemma trans_inv_image: "trans r ==> trans (inv_image r f)"
```
```   495   apply (unfold trans_def inv_image_def)
```
```   496   apply (simp (no_asm))
```
```   497   apply blast
```
```   498   done
```
```   499
```
```   500 end
```