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