src/HOL/Relation.thy
 author krauss Tue Sep 26 17:33:04 2006 +0200 (2006-09-26) changeset 20716 a6686a8e1b68 parent 19656 09be06943252 child 21210 c17fd2df4e9e permissions -rw-r--r--
Changed precedence of "op O" (relation composition) from 60 to 75.
```     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 definition
```
```    16   converse :: "('a * 'b) set => ('b * 'a) set"    ("(_^-1)" [1000] 999)
```
```    17   "r^-1 == {(y, x). (x, y) : r}"
```
```    18
```
```    19 const_syntax (xsymbols)
```
```    20   converse  ("(_\<inverse>)" [1000] 999)
```
```    21
```
```    22 definition
```
```    23   rel_comp  :: "[('b * 'c) set, ('a * 'b) set] => ('a * 'c) set"  (infixr "O" 75)
```
```    24   "r O s == {(x,z). EX y. (x, y) : s & (y, z) : r}"
```
```    25
```
```    26   Image :: "[('a * 'b) set, 'a set] => 'b set"                (infixl "``" 90)
```
```    27   "r `` s == {y. EX x:s. (x,y):r}"
```
```    28
```
```    29   Id    :: "('a * 'a) set"  -- {* the identity relation *}
```
```    30   "Id == {p. EX x. p = (x,x)}"
```
```    31
```
```    32   diag  :: "'a set => ('a * 'a) set"  -- {* diagonal: identity over a set *}
```
```    33   "diag A == \<Union>x\<in>A. {(x,x)}"
```
```    34
```
```    35   Domain :: "('a * 'b) set => 'a set"
```
```    36   "Domain r == {x. EX y. (x,y):r}"
```
```    37
```
```    38   Range  :: "('a * 'b) set => 'b set"
```
```    39   "Range r == Domain(r^-1)"
```
```    40
```
```    41   Field :: "('a * 'a) set => 'a set"
```
```    42   "Field r == Domain r \<union> Range r"
```
```    43
```
```    44   refl   :: "['a set, ('a * 'a) set] => bool"  -- {* reflexivity over a set *}
```
```    45   "refl A r == r \<subseteq> A \<times> A & (ALL x: A. (x,x) : r)"
```
```    46
```
```    47   sym    :: "('a * 'a) set => bool"  -- {* symmetry predicate *}
```
```    48   "sym r == ALL x y. (x,y): r --> (y,x): r"
```
```    49
```
```    50   antisym:: "('a * 'a) set => bool"  -- {* antisymmetry predicate *}
```
```    51   "antisym r == ALL x y. (x,y):r --> (y,x):r --> x=y"
```
```    52
```
```    53   trans  :: "('a * 'a) set => bool"  -- {* transitivity predicate *}
```
```    54   "trans r == (ALL x y z. (x,y):r --> (y,z):r --> (x,z):r)"
```
```    55
```
```    56   single_valued :: "('a * 'b) set => bool"
```
```    57   "single_valued r == ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z)"
```
```    58
```
```    59   inv_image :: "('b * 'b) set => ('a => 'b) => ('a * 'a) set"
```
```    60   "inv_image r f == {(x, y). (f x, f y) : r}"
```
```    61
```
```    62 abbreviation
```
```    63   reflexive :: "('a * 'a) set => bool"  -- {* reflexivity over a type *}
```
```    64   "reflexive == refl UNIV"
```
```    65
```
```    66
```
```    67 subsection {* The identity relation *}
```
```    68
```
```    69 lemma IdI [intro]: "(a, a) : Id"
```
```    70   by (simp add: Id_def)
```
```    71
```
```    72 lemma IdE [elim!]: "p : Id ==> (!!x. p = (x, x) ==> P) ==> P"
```
```    73   by (unfold Id_def) (iprover elim: CollectE)
```
```    74
```
```    75 lemma pair_in_Id_conv [iff]: "((a, b) : Id) = (a = b)"
```
```    76   by (unfold Id_def) blast
```
```    77
```
```    78 lemma reflexive_Id: "reflexive Id"
```
```    79   by (simp add: refl_def)
```
```    80
```
```    81 lemma antisym_Id: "antisym Id"
```
```    82   -- {* A strange result, since @{text Id} is also symmetric. *}
```
```    83   by (simp add: antisym_def)
```
```    84
```
```    85 lemma sym_Id: "sym Id"
```
```    86   by (simp add: sym_def)
```
```    87
```
```    88 lemma trans_Id: "trans Id"
```
```    89   by (simp add: trans_def)
```
```    90
```
```    91
```
```    92 subsection {* Diagonal: identity over a set *}
```
```    93
```
```    94 lemma diag_empty [simp]: "diag {} = {}"
```
```    95   by (simp add: diag_def)
```
```    96
```
```    97 lemma diag_eqI: "a = b ==> a : A ==> (a, b) : diag A"
```
```    98   by (simp add: diag_def)
```
```    99
```
```   100 lemma diagI [intro!]: "a : A ==> (a, a) : diag A"
```
```   101   by (rule diag_eqI) (rule refl)
```
```   102
```
```   103 lemma diagE [elim!]:
```
```   104   "c : diag A ==> (!!x. x : A ==> c = (x, x) ==> P) ==> P"
```
```   105   -- {* The general elimination rule. *}
```
```   106   by (unfold diag_def) (iprover elim!: UN_E singletonE)
```
```   107
```
```   108 lemma diag_iff: "((x, y) : diag A) = (x = y & x : A)"
```
```   109   by blast
```
```   110
```
```   111 lemma diag_subset_Times: "diag A \<subseteq> A \<times> A"
```
```   112   by blast
```
```   113
```
```   114
```
```   115 subsection {* Composition of two relations *}
```
```   116
```
```   117 lemma rel_compI [intro]:
```
```   118   "(a, b) : s ==> (b, c) : r ==> (a, c) : r O s"
```
```   119   by (unfold rel_comp_def) blast
```
```   120
```
```   121 lemma rel_compE [elim!]: "xz : r O s ==>
```
```   122   (!!x y z. xz = (x, z) ==> (x, y) : s ==> (y, z) : r  ==> P) ==> P"
```
```   123   by (unfold rel_comp_def) (iprover elim!: CollectE splitE exE conjE)
```
```   124
```
```   125 lemma rel_compEpair:
```
```   126   "(a, c) : r O s ==> (!!y. (a, y) : s ==> (y, c) : r ==> P) ==> P"
```
```   127   by (iprover elim: rel_compE Pair_inject ssubst)
```
```   128
```
```   129 lemma R_O_Id [simp]: "R O Id = R"
```
```   130   by fast
```
```   131
```
```   132 lemma Id_O_R [simp]: "Id O R = R"
```
```   133   by fast
```
```   134
```
```   135 lemma O_assoc: "(R O S) O T = R O (S O T)"
```
```   136   by blast
```
```   137
```
```   138 lemma trans_O_subset: "trans r ==> r O r \<subseteq> r"
```
```   139   by (unfold trans_def) blast
```
```   140
```
```   141 lemma rel_comp_mono: "r' \<subseteq> r ==> s' \<subseteq> s ==> (r' O s') \<subseteq> (r O s)"
```
```   142   by blast
```
```   143
```
```   144 lemma rel_comp_subset_Sigma:
```
```   145     "s \<subseteq> A \<times> B ==> r \<subseteq> B \<times> C ==> (r O s) \<subseteq> A \<times> C"
```
```   146   by blast
```
```   147
```
```   148
```
```   149 subsection {* Reflexivity *}
```
```   150
```
```   151 lemma reflI: "r \<subseteq> A \<times> A ==> (!!x. x : A ==> (x, x) : r) ==> refl A r"
```
```   152   by (unfold refl_def) (iprover intro!: ballI)
```
```   153
```
```   154 lemma reflD: "refl A r ==> a : A ==> (a, a) : r"
```
```   155   by (unfold refl_def) blast
```
```   156
```
```   157 lemma reflD1: "refl A r ==> (x, y) : r ==> x : A"
```
```   158   by (unfold refl_def) blast
```
```   159
```
```   160 lemma reflD2: "refl A r ==> (x, y) : r ==> y : A"
```
```   161   by (unfold refl_def) blast
```
```   162
```
```   163 lemma refl_Int: "refl A r ==> refl B s ==> refl (A \<inter> B) (r \<inter> s)"
```
```   164   by (unfold refl_def) blast
```
```   165
```
```   166 lemma refl_Un: "refl A r ==> refl B s ==> refl (A \<union> B) (r \<union> s)"
```
```   167   by (unfold refl_def) blast
```
```   168
```
```   169 lemma refl_INTER:
```
```   170   "ALL x:S. refl (A x) (r x) ==> refl (INTER S A) (INTER S r)"
```
```   171   by (unfold refl_def) fast
```
```   172
```
```   173 lemma refl_UNION:
```
```   174   "ALL x:S. refl (A x) (r x) \<Longrightarrow> refl (UNION S A) (UNION S r)"
```
```   175   by (unfold refl_def) blast
```
```   176
```
```   177 lemma refl_diag: "refl A (diag A)"
```
```   178   by (rule reflI [OF diag_subset_Times diagI])
```
```   179
```
```   180
```
```   181 subsection {* Antisymmetry *}
```
```   182
```
```   183 lemma antisymI:
```
```   184   "(!!x y. (x, y) : r ==> (y, x) : r ==> x=y) ==> antisym r"
```
```   185   by (unfold antisym_def) iprover
```
```   186
```
```   187 lemma antisymD: "antisym r ==> (a, b) : r ==> (b, a) : r ==> a = b"
```
```   188   by (unfold antisym_def) iprover
```
```   189
```
```   190 lemma antisym_subset: "r \<subseteq> s ==> antisym s ==> antisym r"
```
```   191   by (unfold antisym_def) blast
```
```   192
```
```   193 lemma antisym_empty [simp]: "antisym {}"
```
```   194   by (unfold antisym_def) blast
```
```   195
```
```   196 lemma antisym_diag [simp]: "antisym (diag A)"
```
```   197   by (unfold antisym_def) blast
```
```   198
```
```   199
```
```   200 subsection {* Symmetry *}
```
```   201
```
```   202 lemma symI: "(!!a b. (a, b) : r ==> (b, a) : r) ==> sym r"
```
```   203   by (unfold sym_def) iprover
```
```   204
```
```   205 lemma symD: "sym r ==> (a, b) : r ==> (b, a) : r"
```
```   206   by (unfold sym_def, blast)
```
```   207
```
```   208 lemma sym_Int: "sym r ==> sym s ==> sym (r \<inter> s)"
```
```   209   by (fast intro: symI dest: symD)
```
```   210
```
```   211 lemma sym_Un: "sym r ==> sym s ==> sym (r \<union> s)"
```
```   212   by (fast intro: symI dest: symD)
```
```   213
```
```   214 lemma sym_INTER: "ALL x:S. sym (r x) ==> sym (INTER S r)"
```
```   215   by (fast intro: symI dest: symD)
```
```   216
```
```   217 lemma sym_UNION: "ALL x:S. sym (r x) ==> sym (UNION S r)"
```
```   218   by (fast intro: symI dest: symD)
```
```   219
```
```   220 lemma sym_diag [simp]: "sym (diag A)"
```
```   221   by (rule symI) clarify
```
```   222
```
```   223
```
```   224 subsection {* Transitivity *}
```
```   225
```
```   226 lemma transI:
```
```   227   "(!!x y z. (x, y) : r ==> (y, z) : r ==> (x, z) : r) ==> trans r"
```
```   228   by (unfold trans_def) iprover
```
```   229
```
```   230 lemma transD: "trans r ==> (a, b) : r ==> (b, c) : r ==> (a, c) : r"
```
```   231   by (unfold trans_def) iprover
```
```   232
```
```   233 lemma trans_Int: "trans r ==> trans s ==> trans (r \<inter> s)"
```
```   234   by (fast intro: transI elim: transD)
```
```   235
```
```   236 lemma trans_INTER: "ALL x:S. trans (r x) ==> trans (INTER S r)"
```
```   237   by (fast intro: transI elim: transD)
```
```   238
```
```   239 lemma trans_diag [simp]: "trans (diag A)"
```
```   240   by (fast intro: transI elim: transD)
```
```   241
```
```   242
```
```   243 subsection {* Converse *}
```
```   244
```
```   245 lemma converse_iff [iff]: "((a,b): r^-1) = ((b,a) : r)"
```
```   246   by (simp add: converse_def)
```
```   247
```
```   248 lemma converseI[sym]: "(a, b) : r ==> (b, a) : r^-1"
```
```   249   by (simp add: converse_def)
```
```   250
```
```   251 lemma converseD[sym]: "(a,b) : r^-1 ==> (b, a) : r"
```
```   252   by (simp add: converse_def)
```
```   253
```
```   254 lemma converseE [elim!]:
```
```   255   "yx : r^-1 ==> (!!x y. yx = (y, x) ==> (x, y) : r ==> P) ==> P"
```
```   256     -- {* More general than @{text converseD}, as it ``splits'' the member of the relation. *}
```
```   257   by (unfold converse_def) (iprover elim!: CollectE splitE bexE)
```
```   258
```
```   259 lemma converse_converse [simp]: "(r^-1)^-1 = r"
```
```   260   by (unfold converse_def) blast
```
```   261
```
```   262 lemma converse_rel_comp: "(r O s)^-1 = s^-1 O r^-1"
```
```   263   by blast
```
```   264
```
```   265 lemma converse_Int: "(r \<inter> s)^-1 = r^-1 \<inter> s^-1"
```
```   266   by blast
```
```   267
```
```   268 lemma converse_Un: "(r \<union> s)^-1 = r^-1 \<union> s^-1"
```
```   269   by blast
```
```   270
```
```   271 lemma converse_INTER: "(INTER S r)^-1 = (INT x:S. (r x)^-1)"
```
```   272   by fast
```
```   273
```
```   274 lemma converse_UNION: "(UNION S r)^-1 = (UN x:S. (r x)^-1)"
```
```   275   by blast
```
```   276
```
```   277 lemma converse_Id [simp]: "Id^-1 = Id"
```
```   278   by blast
```
```   279
```
```   280 lemma converse_diag [simp]: "(diag A)^-1 = diag A"
```
```   281   by blast
```
```   282
```
```   283 lemma refl_converse [simp]: "refl A (converse r) = refl A r"
```
```   284   by (unfold refl_def) auto
```
```   285
```
```   286 lemma sym_converse [simp]: "sym (converse r) = sym r"
```
```   287   by (unfold sym_def) blast
```
```   288
```
```   289 lemma antisym_converse [simp]: "antisym (converse r) = antisym r"
```
```   290   by (unfold antisym_def) blast
```
```   291
```
```   292 lemma trans_converse [simp]: "trans (converse r) = trans r"
```
```   293   by (unfold trans_def) blast
```
```   294
```
```   295 lemma sym_conv_converse_eq: "sym r = (r^-1 = r)"
```
```   296   by (unfold sym_def) fast
```
```   297
```
```   298 lemma sym_Un_converse: "sym (r \<union> r^-1)"
```
```   299   by (unfold sym_def) blast
```
```   300
```
```   301 lemma sym_Int_converse: "sym (r \<inter> r^-1)"
```
```   302   by (unfold sym_def) blast
```
```   303
```
```   304
```
```   305 subsection {* Domain *}
```
```   306
```
```   307 lemma Domain_iff: "(a : Domain r) = (EX y. (a, y) : r)"
```
```   308   by (unfold Domain_def) blast
```
```   309
```
```   310 lemma DomainI [intro]: "(a, b) : r ==> a : Domain r"
```
```   311   by (iprover intro!: iffD2 [OF Domain_iff])
```
```   312
```
```   313 lemma DomainE [elim!]:
```
```   314   "a : Domain r ==> (!!y. (a, y) : r ==> P) ==> P"
```
```   315   by (iprover dest!: iffD1 [OF Domain_iff])
```
```   316
```
```   317 lemma Domain_empty [simp]: "Domain {} = {}"
```
```   318   by blast
```
```   319
```
```   320 lemma Domain_insert: "Domain (insert (a, b) r) = insert a (Domain r)"
```
```   321   by blast
```
```   322
```
```   323 lemma Domain_Id [simp]: "Domain Id = UNIV"
```
```   324   by blast
```
```   325
```
```   326 lemma Domain_diag [simp]: "Domain (diag A) = A"
```
```   327   by blast
```
```   328
```
```   329 lemma Domain_Un_eq: "Domain(A \<union> B) = Domain(A) \<union> Domain(B)"
```
```   330   by blast
```
```   331
```
```   332 lemma Domain_Int_subset: "Domain(A \<inter> B) \<subseteq> Domain(A) \<inter> Domain(B)"
```
```   333   by blast
```
```   334
```
```   335 lemma Domain_Diff_subset: "Domain(A) - Domain(B) \<subseteq> Domain(A - B)"
```
```   336   by blast
```
```   337
```
```   338 lemma Domain_Union: "Domain (Union S) = (\<Union>A\<in>S. Domain A)"
```
```   339   by blast
```
```   340
```
```   341 lemma Domain_mono: "r \<subseteq> s ==> Domain r \<subseteq> Domain s"
```
```   342   by blast
```
```   343
```
```   344
```
```   345 subsection {* Range *}
```
```   346
```
```   347 lemma Range_iff: "(a : Range r) = (EX y. (y, a) : r)"
```
```   348   by (simp add: Domain_def Range_def)
```
```   349
```
```   350 lemma RangeI [intro]: "(a, b) : r ==> b : Range r"
```
```   351   by (unfold Range_def) (iprover intro!: converseI DomainI)
```
```   352
```
```   353 lemma RangeE [elim!]: "b : Range r ==> (!!x. (x, b) : r ==> P) ==> P"
```
```   354   by (unfold Range_def) (iprover elim!: DomainE dest!: converseD)
```
```   355
```
```   356 lemma Range_empty [simp]: "Range {} = {}"
```
```   357   by blast
```
```   358
```
```   359 lemma Range_insert: "Range (insert (a, b) r) = insert b (Range r)"
```
```   360   by blast
```
```   361
```
```   362 lemma Range_Id [simp]: "Range Id = UNIV"
```
```   363   by blast
```
```   364
```
```   365 lemma Range_diag [simp]: "Range (diag A) = A"
```
```   366   by auto
```
```   367
```
```   368 lemma Range_Un_eq: "Range(A \<union> B) = Range(A) \<union> Range(B)"
```
```   369   by blast
```
```   370
```
```   371 lemma Range_Int_subset: "Range(A \<inter> B) \<subseteq> Range(A) \<inter> Range(B)"
```
```   372   by blast
```
```   373
```
```   374 lemma Range_Diff_subset: "Range(A) - Range(B) \<subseteq> Range(A - B)"
```
```   375   by blast
```
```   376
```
```   377 lemma Range_Union: "Range (Union S) = (\<Union>A\<in>S. Range A)"
```
```   378   by blast
```
```   379
```
```   380
```
```   381 subsection {* Image of a set under a relation *}
```
```   382
```
```   383 lemma Image_iff: "(b : r``A) = (EX x:A. (x, b) : r)"
```
```   384   by (simp add: Image_def)
```
```   385
```
```   386 lemma Image_singleton: "r``{a} = {b. (a, b) : r}"
```
```   387   by (simp add: Image_def)
```
```   388
```
```   389 lemma Image_singleton_iff [iff]: "(b : r``{a}) = ((a, b) : r)"
```
```   390   by (rule Image_iff [THEN trans]) simp
```
```   391
```
```   392 lemma ImageI [intro]: "(a, b) : r ==> a : A ==> b : r``A"
```
```   393   by (unfold Image_def) blast
```
```   394
```
```   395 lemma ImageE [elim!]:
```
```   396     "b : r `` A ==> (!!x. (x, b) : r ==> x : A ==> P) ==> P"
```
```   397   by (unfold Image_def) (iprover elim!: CollectE bexE)
```
```   398
```
```   399 lemma rev_ImageI: "a : A ==> (a, b) : r ==> b : r `` A"
```
```   400   -- {* This version's more effective when we already have the required @{text a} *}
```
```   401   by blast
```
```   402
```
```   403 lemma Image_empty [simp]: "R``{} = {}"
```
```   404   by blast
```
```   405
```
```   406 lemma Image_Id [simp]: "Id `` A = A"
```
```   407   by blast
```
```   408
```
```   409 lemma Image_diag [simp]: "diag A `` B = A \<inter> B"
```
```   410   by blast
```
```   411
```
```   412 lemma Image_Int_subset: "R `` (A \<inter> B) \<subseteq> R `` A \<inter> R `` B"
```
```   413   by blast
```
```   414
```
```   415 lemma Image_Int_eq:
```
```   416      "single_valued (converse R) ==> R `` (A \<inter> B) = R `` A \<inter> R `` B"
```
```   417   by (simp add: single_valued_def, blast)
```
```   418
```
```   419 lemma Image_Un: "R `` (A \<union> B) = R `` A \<union> R `` B"
```
```   420   by blast
```
```   421
```
```   422 lemma Un_Image: "(R \<union> S) `` A = R `` A \<union> S `` A"
```
```   423   by blast
```
```   424
```
```   425 lemma Image_subset: "r \<subseteq> A \<times> B ==> r``C \<subseteq> B"
```
```   426   by (iprover intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2)
```
```   427
```
```   428 lemma Image_eq_UN: "r``B = (\<Union>y\<in> B. r``{y})"
```
```   429   -- {* NOT suitable for rewriting *}
```
```   430   by blast
```
```   431
```
```   432 lemma Image_mono: "r' \<subseteq> r ==> A' \<subseteq> A ==> (r' `` A') \<subseteq> (r `` A)"
```
```   433   by blast
```
```   434
```
```   435 lemma Image_UN: "(r `` (UNION A B)) = (\<Union>x\<in>A. r `` (B x))"
```
```   436   by blast
```
```   437
```
```   438 lemma Image_INT_subset: "(r `` INTER A B) \<subseteq> (\<Inter>x\<in>A. r `` (B x))"
```
```   439   by blast
```
```   440
```
```   441 text{*Converse inclusion requires some assumptions*}
```
```   442 lemma Image_INT_eq:
```
```   443      "[|single_valued (r\<inverse>); A\<noteq>{}|] ==> r `` INTER A B = (\<Inter>x\<in>A. r `` B x)"
```
```   444 apply (rule equalityI)
```
```   445  apply (rule Image_INT_subset)
```
```   446 apply  (simp add: single_valued_def, blast)
```
```   447 done
```
```   448
```
```   449 lemma Image_subset_eq: "(r``A \<subseteq> B) = (A \<subseteq> - ((r^-1) `` (-B)))"
```
```   450   by blast
```
```   451
```
```   452
```
```   453 subsection {* Single valued relations *}
```
```   454
```
```   455 lemma single_valuedI:
```
```   456   "ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z) ==> single_valued r"
```
```   457   by (unfold single_valued_def)
```
```   458
```
```   459 lemma single_valuedD:
```
```   460   "single_valued r ==> (x, y) : r ==> (x, z) : r ==> y = z"
```
```   461   by (simp add: single_valued_def)
```
```   462
```
```   463 lemma single_valued_rel_comp:
```
```   464   "single_valued r ==> single_valued s ==> single_valued (r O s)"
```
```   465   by (unfold single_valued_def) blast
```
```   466
```
```   467 lemma single_valued_subset:
```
```   468   "r \<subseteq> s ==> single_valued s ==> single_valued r"
```
```   469   by (unfold single_valued_def) blast
```
```   470
```
```   471 lemma single_valued_Id [simp]: "single_valued Id"
```
```   472   by (unfold single_valued_def) blast
```
```   473
```
```   474 lemma single_valued_diag [simp]: "single_valued (diag A)"
```
```   475   by (unfold single_valued_def) blast
```
```   476
```
```   477
```
```   478 subsection {* Graphs given by @{text Collect} *}
```
```   479
```
```   480 lemma Domain_Collect_split [simp]: "Domain{(x,y). P x y} = {x. EX y. P x y}"
```
```   481   by auto
```
```   482
```
```   483 lemma Range_Collect_split [simp]: "Range{(x,y). P x y} = {y. EX x. P x y}"
```
```   484   by auto
```
```   485
```
```   486 lemma Image_Collect_split [simp]: "{(x,y). P x y} `` A = {y. EX x:A. P x y}"
```
```   487   by auto
```
```   488
```
```   489
```
```   490 subsection {* Inverse image *}
```
```   491
```
```   492 lemma sym_inv_image: "sym r ==> sym (inv_image r f)"
```
```   493   by (unfold sym_def inv_image_def) blast
```
```   494
```
```   495 lemma trans_inv_image: "trans r ==> trans (inv_image r f)"
```
```   496   apply (unfold trans_def inv_image_def)
```
```   497   apply (simp (no_asm))
```
```   498   apply blast
```
```   499   done
```
```   500
```
```   501 end
```