src/HOL/Relation.thy
 author nipkow Wed Aug 18 11:09:40 2004 +0200 (2004-08-18) changeset 15140 322485b816ac parent 15131 c69542757a4d child 15177 e7616269fdca permissions -rw-r--r--
import -> imports
```     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 syntax
```
```    62   reflexive :: "('a * 'a) set => bool"  -- {* reflexivity over a type *}
```
```    63 translations
```
```    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) (rules 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 trans_Id: "trans Id"
```
```    86   by (simp add: trans_def)
```
```    87
```
```    88
```
```    89 subsection {* Diagonal: identity over a set *}
```
```    90
```
```    91 lemma diag_empty [simp]: "diag {} = {}"
```
```    92   by (simp add: diag_def)
```
```    93
```
```    94 lemma diag_eqI: "a = b ==> a : A ==> (a, b) : diag A"
```
```    95   by (simp add: diag_def)
```
```    96
```
```    97 lemma diagI [intro!]: "a : A ==> (a, a) : diag A"
```
```    98   by (rule diag_eqI) (rule refl)
```
```    99
```
```   100 lemma diagE [elim!]:
```
```   101   "c : diag A ==> (!!x. x : A ==> c = (x, x) ==> P) ==> P"
```
```   102   -- {* The general elimination rule. *}
```
```   103   by (unfold diag_def) (rules elim!: UN_E singletonE)
```
```   104
```
```   105 lemma diag_iff: "((x, y) : diag A) = (x = y & x : A)"
```
```   106   by blast
```
```   107
```
```   108 lemma diag_subset_Times: "diag A \<subseteq> A \<times> A"
```
```   109   by blast
```
```   110
```
```   111
```
```   112 subsection {* Composition of two relations *}
```
```   113
```
```   114 lemma rel_compI [intro]:
```
```   115   "(a, b) : s ==> (b, c) : r ==> (a, c) : r O s"
```
```   116   by (unfold rel_comp_def) blast
```
```   117
```
```   118 lemma rel_compE [elim!]: "xz : r O s ==>
```
```   119   (!!x y z. xz = (x, z) ==> (x, y) : s ==> (y, z) : r  ==> P) ==> P"
```
```   120   by (unfold rel_comp_def) (rules elim!: CollectE splitE exE conjE)
```
```   121
```
```   122 lemma rel_compEpair:
```
```   123   "(a, c) : r O s ==> (!!y. (a, y) : s ==> (y, c) : r ==> P) ==> P"
```
```   124   by (rules elim: rel_compE Pair_inject ssubst)
```
```   125
```
```   126 lemma R_O_Id [simp]: "R O Id = R"
```
```   127   by fast
```
```   128
```
```   129 lemma Id_O_R [simp]: "Id O R = R"
```
```   130   by fast
```
```   131
```
```   132 lemma O_assoc: "(R O S) O T = R O (S O T)"
```
```   133   by blast
```
```   134
```
```   135 lemma trans_O_subset: "trans r ==> r O r \<subseteq> r"
```
```   136   by (unfold trans_def) blast
```
```   137
```
```   138 lemma rel_comp_mono: "r' \<subseteq> r ==> s' \<subseteq> s ==> (r' O s') \<subseteq> (r O s)"
```
```   139   by blast
```
```   140
```
```   141 lemma rel_comp_subset_Sigma:
```
```   142     "s \<subseteq> A \<times> B ==> r \<subseteq> B \<times> C ==> (r O s) \<subseteq> A \<times> C"
```
```   143   by blast
```
```   144
```
```   145
```
```   146 subsection {* Reflexivity *}
```
```   147
```
```   148 lemma reflI: "r \<subseteq> A \<times> A ==> (!!x. x : A ==> (x, x) : r) ==> refl A r"
```
```   149   by (unfold refl_def) (rules intro!: ballI)
```
```   150
```
```   151 lemma reflD: "refl A r ==> a : A ==> (a, a) : r"
```
```   152   by (unfold refl_def) blast
```
```   153
```
```   154
```
```   155 subsection {* Antisymmetry *}
```
```   156
```
```   157 lemma antisymI:
```
```   158   "(!!x y. (x, y) : r ==> (y, x) : r ==> x=y) ==> antisym r"
```
```   159   by (unfold antisym_def) rules
```
```   160
```
```   161 lemma antisymD: "antisym r ==> (a, b) : r ==> (b, a) : r ==> a = b"
```
```   162   by (unfold antisym_def) rules
```
```   163
```
```   164
```
```   165 subsection {* Transitivity *}
```
```   166
```
```   167 lemma transI:
```
```   168   "(!!x y z. (x, y) : r ==> (y, z) : r ==> (x, z) : r) ==> trans r"
```
```   169   by (unfold trans_def) rules
```
```   170
```
```   171 lemma transD: "trans r ==> (a, b) : r ==> (b, c) : r ==> (a, c) : r"
```
```   172   by (unfold trans_def) rules
```
```   173
```
```   174
```
```   175 subsection {* Converse *}
```
```   176
```
```   177 lemma converse_iff [iff]: "((a,b): r^-1) = ((b,a) : r)"
```
```   178   by (simp add: converse_def)
```
```   179
```
```   180 lemma converseI[sym]: "(a, b) : r ==> (b, a) : r^-1"
```
```   181   by (simp add: converse_def)
```
```   182
```
```   183 lemma converseD[sym]: "(a,b) : r^-1 ==> (b, a) : r"
```
```   184   by (simp add: converse_def)
```
```   185
```
```   186 lemma converseE [elim!]:
```
```   187   "yx : r^-1 ==> (!!x y. yx = (y, x) ==> (x, y) : r ==> P) ==> P"
```
```   188     -- {* More general than @{text converseD}, as it ``splits'' the member of the relation. *}
```
```   189   by (unfold converse_def) (rules elim!: CollectE splitE bexE)
```
```   190
```
```   191 lemma converse_converse [simp]: "(r^-1)^-1 = r"
```
```   192   by (unfold converse_def) blast
```
```   193
```
```   194 lemma converse_rel_comp: "(r O s)^-1 = s^-1 O r^-1"
```
```   195   by blast
```
```   196
```
```   197 lemma converse_Id [simp]: "Id^-1 = Id"
```
```   198   by blast
```
```   199
```
```   200 lemma converse_diag [simp]: "(diag A)^-1 = diag A"
```
```   201   by blast
```
```   202
```
```   203 lemma refl_converse: "refl A r ==> refl A (converse r)"
```
```   204   by (unfold refl_def) blast
```
```   205
```
```   206 lemma antisym_converse: "antisym (converse r) = antisym r"
```
```   207   by (unfold antisym_def) blast
```
```   208
```
```   209 lemma trans_converse: "trans (converse r) = trans r"
```
```   210   by (unfold trans_def) blast
```
```   211
```
```   212
```
```   213 subsection {* Domain *}
```
```   214
```
```   215 lemma Domain_iff: "(a : Domain r) = (EX y. (a, y) : r)"
```
```   216   by (unfold Domain_def) blast
```
```   217
```
```   218 lemma DomainI [intro]: "(a, b) : r ==> a : Domain r"
```
```   219   by (rules intro!: iffD2 [OF Domain_iff])
```
```   220
```
```   221 lemma DomainE [elim!]:
```
```   222   "a : Domain r ==> (!!y. (a, y) : r ==> P) ==> P"
```
```   223   by (rules dest!: iffD1 [OF Domain_iff])
```
```   224
```
```   225 lemma Domain_empty [simp]: "Domain {} = {}"
```
```   226   by blast
```
```   227
```
```   228 lemma Domain_insert: "Domain (insert (a, b) r) = insert a (Domain r)"
```
```   229   by blast
```
```   230
```
```   231 lemma Domain_Id [simp]: "Domain Id = UNIV"
```
```   232   by blast
```
```   233
```
```   234 lemma Domain_diag [simp]: "Domain (diag A) = A"
```
```   235   by blast
```
```   236
```
```   237 lemma Domain_Un_eq: "Domain(A \<union> B) = Domain(A) \<union> Domain(B)"
```
```   238   by blast
```
```   239
```
```   240 lemma Domain_Int_subset: "Domain(A \<inter> B) \<subseteq> Domain(A) \<inter> Domain(B)"
```
```   241   by blast
```
```   242
```
```   243 lemma Domain_Diff_subset: "Domain(A) - Domain(B) \<subseteq> Domain(A - B)"
```
```   244   by blast
```
```   245
```
```   246 lemma Domain_Union: "Domain (Union S) = (\<Union>A\<in>S. Domain A)"
```
```   247   by blast
```
```   248
```
```   249 lemma Domain_mono: "r \<subseteq> s ==> Domain r \<subseteq> Domain s"
```
```   250   by blast
```
```   251
```
```   252
```
```   253 subsection {* Range *}
```
```   254
```
```   255 lemma Range_iff: "(a : Range r) = (EX y. (y, a) : r)"
```
```   256   by (simp add: Domain_def Range_def)
```
```   257
```
```   258 lemma RangeI [intro]: "(a, b) : r ==> b : Range r"
```
```   259   by (unfold Range_def) (rules intro!: converseI DomainI)
```
```   260
```
```   261 lemma RangeE [elim!]: "b : Range r ==> (!!x. (x, b) : r ==> P) ==> P"
```
```   262   by (unfold Range_def) (rules elim!: DomainE dest!: converseD)
```
```   263
```
```   264 lemma Range_empty [simp]: "Range {} = {}"
```
```   265   by blast
```
```   266
```
```   267 lemma Range_insert: "Range (insert (a, b) r) = insert b (Range r)"
```
```   268   by blast
```
```   269
```
```   270 lemma Range_Id [simp]: "Range Id = UNIV"
```
```   271   by blast
```
```   272
```
```   273 lemma Range_diag [simp]: "Range (diag A) = A"
```
```   274   by auto
```
```   275
```
```   276 lemma Range_Un_eq: "Range(A \<union> B) = Range(A) \<union> Range(B)"
```
```   277   by blast
```
```   278
```
```   279 lemma Range_Int_subset: "Range(A \<inter> B) \<subseteq> Range(A) \<inter> Range(B)"
```
```   280   by blast
```
```   281
```
```   282 lemma Range_Diff_subset: "Range(A) - Range(B) \<subseteq> Range(A - B)"
```
```   283   by blast
```
```   284
```
```   285 lemma Range_Union: "Range (Union S) = (\<Union>A\<in>S. Range A)"
```
```   286   by blast
```
```   287
```
```   288
```
```   289 subsection {* Image of a set under a relation *}
```
```   290
```
```   291 lemma Image_iff: "(b : r``A) = (EX x:A. (x, b) : r)"
```
```   292   by (simp add: Image_def)
```
```   293
```
```   294 lemma Image_singleton: "r``{a} = {b. (a, b) : r}"
```
```   295   by (simp add: Image_def)
```
```   296
```
```   297 lemma Image_singleton_iff [iff]: "(b : r``{a}) = ((a, b) : r)"
```
```   298   by (rule Image_iff [THEN trans]) simp
```
```   299
```
```   300 lemma ImageI [intro]: "(a, b) : r ==> a : A ==> b : r``A"
```
```   301   by (unfold Image_def) blast
```
```   302
```
```   303 lemma ImageE [elim!]:
```
```   304     "b : r `` A ==> (!!x. (x, b) : r ==> x : A ==> P) ==> P"
```
```   305   by (unfold Image_def) (rules elim!: CollectE bexE)
```
```   306
```
```   307 lemma rev_ImageI: "a : A ==> (a, b) : r ==> b : r `` A"
```
```   308   -- {* This version's more effective when we already have the required @{text a} *}
```
```   309   by blast
```
```   310
```
```   311 lemma Image_empty [simp]: "R``{} = {}"
```
```   312   by blast
```
```   313
```
```   314 lemma Image_Id [simp]: "Id `` A = A"
```
```   315   by blast
```
```   316
```
```   317 lemma Image_diag [simp]: "diag A `` B = A \<inter> B"
```
```   318   by blast
```
```   319
```
```   320 lemma Image_Int_subset: "R `` (A \<inter> B) \<subseteq> R `` A \<inter> R `` B"
```
```   321   by blast
```
```   322
```
```   323 lemma Image_Int_eq:
```
```   324      "single_valued (converse R) ==> R `` (A \<inter> B) = R `` A \<inter> R `` B"
```
```   325   by (simp add: single_valued_def, blast)
```
```   326
```
```   327 lemma Image_Un: "R `` (A \<union> B) = R `` A \<union> R `` B"
```
```   328   by blast
```
```   329
```
```   330 lemma Un_Image: "(R \<union> S) `` A = R `` A \<union> S `` A"
```
```   331   by blast
```
```   332
```
```   333 lemma Image_subset: "r \<subseteq> A \<times> B ==> r``C \<subseteq> B"
```
```   334   by (rules intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2)
```
```   335
```
```   336 lemma Image_eq_UN: "r``B = (\<Union>y\<in> B. r``{y})"
```
```   337   -- {* NOT suitable for rewriting *}
```
```   338   by blast
```
```   339
```
```   340 lemma Image_mono: "r' \<subseteq> r ==> A' \<subseteq> A ==> (r' `` A') \<subseteq> (r `` A)"
```
```   341   by blast
```
```   342
```
```   343 lemma Image_UN: "(r `` (UNION A B)) = (\<Union>x\<in>A. r `` (B x))"
```
```   344   by blast
```
```   345
```
```   346 lemma Image_INT_subset: "(r `` INTER A B) \<subseteq> (\<Inter>x\<in>A. r `` (B x))"
```
```   347   by blast
```
```   348
```
```   349 text{*Converse inclusion requires some assumptions*}
```
```   350 lemma Image_INT_eq:
```
```   351      "[|single_valued (r\<inverse>); A\<noteq>{}|] ==> r `` INTER A B = (\<Inter>x\<in>A. r `` B x)"
```
```   352 apply (rule equalityI)
```
```   353  apply (rule Image_INT_subset)
```
```   354 apply  (simp add: single_valued_def, blast)
```
```   355 done
```
```   356
```
```   357 lemma Image_subset_eq: "(r``A \<subseteq> B) = (A \<subseteq> - ((r^-1) `` (-B)))"
```
```   358   by blast
```
```   359
```
```   360
```
```   361 subsection {* Single valued relations *}
```
```   362
```
```   363 lemma single_valuedI:
```
```   364   "ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z) ==> single_valued r"
```
```   365   by (unfold single_valued_def)
```
```   366
```
```   367 lemma single_valuedD:
```
```   368   "single_valued r ==> (x, y) : r ==> (x, z) : r ==> y = z"
```
```   369   by (simp add: single_valued_def)
```
```   370
```
```   371
```
```   372 subsection {* Graphs given by @{text Collect} *}
```
```   373
```
```   374 lemma Domain_Collect_split [simp]: "Domain{(x,y). P x y} = {x. EX y. P x y}"
```
```   375   by auto
```
```   376
```
```   377 lemma Range_Collect_split [simp]: "Range{(x,y). P x y} = {y. EX x. P x y}"
```
```   378   by auto
```
```   379
```
```   380 lemma Image_Collect_split [simp]: "{(x,y). P x y} `` A = {y. EX x:A. P x y}"
```
```   381   by auto
```
```   382
```
```   383
```
```   384 subsection {* Inverse image *}
```
```   385
```
```   386 lemma trans_inv_image: "trans r ==> trans (inv_image r f)"
```
```   387   apply (unfold trans_def inv_image_def)
```
```   388   apply (simp (no_asm))
```
```   389   apply blast
```
```   390   done
```
```   391
```
```   392 end
```