src/HOL/Relation.thy
 changeset 12905 bbbae3f359e6 parent 12487 bbd564190c9b child 12913 5ac498bffb6b
```     1.1 --- a/src/HOL/Relation.thy	Wed Feb 20 00:55:42 2002 +0100
1.2 +++ b/src/HOL/Relation.thy	Wed Feb 20 15:47:42 2002 +0100
1.3 @@ -4,13 +4,15 @@
1.4      Copyright   1996  University of Cambridge
1.5  *)
1.6
1.7 -Relation = Product_Type +
1.9 +
1.10 +theory Relation = Product_Type:
1.11
1.12  constdefs
1.13    converse :: "('a * 'b) set => ('b * 'a) set"    ("(_^-1)"  999)
1.14    "r^-1 == {(y, x). (x, y) : r}"
1.15  syntax (xsymbols)
1.16 -  converse :: "('a * 'b) set => ('b * 'a) set"    ("(_\\<inverse>)"  999)
1.17 +  converse :: "('a * 'b) set => ('b * 'a) set"    ("(_\<inverse>)"  999)
1.18
1.19  constdefs
1.20    rel_comp  :: "[('b * 'c) set, ('a * 'b) set] => ('a * 'c) set"  (infixr "O" 60)
1.21 @@ -19,10 +21,10 @@
1.22    Image :: "[('a * 'b) set, 'a set] => 'b set"                (infixl "``" 90)
1.23      "r `` s == {y. ? x:s. (x,y):r}"
1.24
1.25 -  Id    :: "('a * 'a) set"                            (*the identity relation*)
1.26 +  Id    :: "('a * 'a) set"  -- {* the identity relation *}
1.27      "Id == {p. ? x. p = (x,x)}"
1.28
1.29 -  diag  :: "'a set => ('a * 'a) set"          (*diagonal: identity over a set*)
1.30 +  diag  :: "'a set => ('a * 'a) set"  -- {* diagonal: identity over a set *}
1.31      "diag(A) == UN x:A. {(x,x)}"
1.32
1.33    Domain :: "('a * 'b) set => 'a set"
1.34 @@ -34,16 +36,16 @@
1.35    Field :: "('a * 'a) set => 'a set"
1.36      "Field r == Domain r Un Range r"
1.37
1.38 -  refl   :: "['a set, ('a * 'a) set] => bool" (*reflexivity over a set*)
1.39 +  refl   :: "['a set, ('a * 'a) set] => bool"  -- {* reflexivity over a set *}
1.40      "refl A r == r <= A <*> A & (ALL x: A. (x,x) : r)"
1.41
1.42 -  sym    :: "('a * 'a) set => bool"             (*symmetry predicate*)
1.43 +  sym    :: "('a * 'a) set => bool"  -- {* symmetry predicate *}
1.44      "sym(r) == ALL x y. (x,y): r --> (y,x): r"
1.45
1.46 -  antisym:: "('a * 'a) set => bool"          (*antisymmetry predicate*)
1.47 +  antisym:: "('a * 'a) set => bool"  -- {* antisymmetry predicate *}
1.48      "antisym(r) == ALL x y. (x,y):r --> (y,x):r --> x=y"
1.49
1.50 -  trans  :: "('a * 'a) set => bool"          (*transitivity predicate*)
1.51 +  trans  :: "('a * 'a) set => bool"  -- {* transitivity predicate *}
1.52      "trans(r) == (!x y z. (x,y):r --> (y,z):r --> (x,z):r)"
1.53
1.54    single_valued :: "('a * 'b) set => bool"
1.55 @@ -56,8 +58,329 @@
1.56      "inv_image r f == {(x,y). (f(x), f(y)) : r}"
1.57
1.58  syntax
1.59 -  reflexive :: "('a * 'a) set => bool"       (*reflexivity over a type*)
1.60 +  reflexive :: "('a * 'a) set => bool"  -- {* reflexivity over a type *}
1.61  translations
1.62    "reflexive" == "refl UNIV"
1.63
1.64 +
1.65 +subsection {* Identity relation *}
1.66 +
1.67 +lemma IdI [intro]: "(a, a) : Id"
1.68 +  by (simp add: Id_def)
1.69 +
1.70 +lemma IdE [elim!]: "p : Id ==> (!!x. p = (x, x) ==> P) ==> P"
1.71 +  by (unfold Id_def) (rules elim: CollectE)
1.72 +
1.73 +lemma pair_in_Id_conv [iff]: "((a, b) : Id) = (a = b)"
1.74 +  by (unfold Id_def) blast
1.75 +
1.76 +lemma reflexive_Id: "reflexive Id"
1.77 +  by (simp add: refl_def)
1.78 +
1.79 +lemma antisym_Id: "antisym Id"
1.80 +  -- {* A strange result, since @{text Id} is also symmetric. *}
1.81 +  by (simp add: antisym_def)
1.82 +
1.83 +lemma trans_Id: "trans Id"
1.84 +  by (simp add: trans_def)
1.85 +
1.86 +
1.87 +subsection {* Diagonal relation: identity restricted to some set *}
1.88 +
1.89 +lemma diag_eqI: "a = b ==> a : A ==> (a, b) : diag A"
1.90 +  by (simp add: diag_def)
1.91 +
1.92 +lemma diagI [intro!]: "a : A ==> (a, a) : diag A"
1.93 +  by (rule diag_eqI) (rule refl)
1.94 +
1.95 +lemma diagE [elim!]:
1.96 +  "c : diag A ==> (!!x. x : A ==> c = (x, x) ==> P) ==> P"
1.97 +  -- {* The general elimination rule *}
1.98 +  by (unfold diag_def) (rules elim!: UN_E singletonE)
1.99 +
1.100 +lemma diag_iff: "((x, y) : diag A) = (x = y & x : A)"
1.101 +  by blast
1.102 +
1.103 +lemma diag_subset_Times: "diag A <= A <*> A"
1.104 +  by blast
1.105 +
1.106 +
1.107 +subsection {* Composition of two relations *}
1.108 +
1.109 +lemma rel_compI [intro]:
1.110 +  "(a, b) : s ==> (b, c) : r ==> (a, c) : r O s"
1.111 +  by (unfold rel_comp_def) blast
1.112 +
1.113 +lemma rel_compE [elim!]: "xz : r O s ==>
1.114 +  (!!x y z. xz = (x, z) ==> (x, y) : s ==> (y, z) : r  ==> P) ==> P"
1.115 +  by (unfold rel_comp_def) (rules elim!: CollectE splitE exE conjE)
1.116 +
1.117 +lemma rel_compEpair:
1.118 +  "(a, c) : r O s ==> (!!y. (a, y) : s ==> (y, c) : r ==> P) ==> P"
1.119 +  by (rules elim: rel_compE Pair_inject ssubst)
1.120 +
1.121 +lemma R_O_Id [simp]: "R O Id = R"
1.122 +  by fast
1.123 +
1.124 +lemma Id_O_R [simp]: "Id O R = R"
1.125 +  by fast
1.126 +
1.127 +lemma O_assoc: "(R O S) O T = R O (S O T)"
1.128 +  by blast
1.129 +
1.130 +lemma trans_O_subset: "trans r ==> r O r <= r"
1.131 +  by (unfold trans_def) blast
1.132 +
1.133 +lemma rel_comp_mono: "[| r'<=r; s'<=s |] ==> (r' O s') <= (r O s)"
1.134 +  by blast
1.135 +
1.136 +lemma rel_comp_subset_Sigma:
1.137 +  "[| s <= A <*> B;  r <= B <*> C |] ==> (r O s) <= A <*> C"
1.138 +  by blast
1.139 +
1.140 +subsection {* Natural deduction for refl(r) *}
1.141 +
1.142 +lemma reflI: "r <= A <*> A ==> (!!x. x : A ==> (x, x) : r) ==> refl A r"
1.143 +  by (unfold refl_def) (rules intro!: ballI)
1.144 +
1.145 +lemma reflD: "refl A r ==> a : A ==> (a, a) : r"
1.146 +  by (unfold refl_def) blast
1.147 +
1.148 +subsection {* Natural deduction for antisym(r) *}
1.149 +
1.150 +lemma antisymI:
1.151 +  "(!!x y. (x, y) : r ==> (y, x) : r ==> x=y) ==> antisym r"
1.152 +  by (unfold antisym_def) rules
1.153 +
1.154 +lemma antisymD: "antisym r ==> (a, b) : r ==> (b, a) : r ==> a = b"
1.155 +  by (unfold antisym_def) rules
1.156 +
1.157 +subsection {* Natural deduction for trans(r) *}
1.158 +
1.159 +lemma transI:
1.160 +  "(!!x y z. (x, y) : r ==> (y, z) : r ==> (x, z) : r) ==> trans r"
1.161 +  by (unfold trans_def) rules
1.162 +
1.163 +lemma transD: "trans r ==> (a, b) : r ==> (b, c) : r ==> (a, c) : r"
1.164 +  by (unfold trans_def) rules
1.165 +
1.166 +subsection {* Natural deduction for r^-1 *}
1.167 +
1.168 +lemma converse_iff [iff]: "((a,b): r^-1) = ((b,a):r)"
1.169 +  by (simp add: converse_def)
1.170 +
1.171 +lemma converseI: "(a,b):r ==> (b,a): r^-1"
1.172 +  by (simp add: converse_def)
1.173 +
1.174 +lemma converseD: "(a,b) : r^-1 ==> (b,a) : r"
1.175 +  by (simp add: converse_def)
1.176 +
1.177 +(*More general than converseD, as it "splits" the member of the relation*)
1.178 +
1.179 +lemma converseE [elim!]:
1.180 +  "yx : r^-1 ==> (!!x y. yx = (y, x) ==> (x, y) : r ==> P) ==> P"
1.181 +  by (unfold converse_def) (rules elim!: CollectE splitE bexE)
1.182 +
1.183 +lemma converse_converse [simp]: "(r^-1)^-1 = r"
1.184 +  by (unfold converse_def) blast
1.185 +
1.186 +lemma converse_rel_comp: "(r O s)^-1 = s^-1 O r^-1"
1.187 +  by blast
1.188 +
1.189 +lemma converse_Id [simp]: "Id^-1 = Id"
1.190 +  by blast
1.191 +
1.192 +lemma converse_diag [simp]: "(diag A) ^-1 = diag A"
1.193 +  by blast
1.194 +
1.195 +lemma refl_converse: "refl A r ==> refl A (converse r)"
1.196 +  by (unfold refl_def) blast
1.197 +
1.198 +lemma antisym_converse: "antisym (converse r) = antisym r"
1.199 +  by (unfold antisym_def) blast
1.200 +
1.201 +lemma trans_converse: "trans (converse r) = trans r"
1.202 +  by (unfold trans_def) blast
1.203 +
1.204 +subsection {* Domain *}
1.205 +
1.206 +lemma Domain_iff: "(a : Domain r) = (EX y. (a, y) : r)"
1.207 +  by (unfold Domain_def) blast
1.208 +
1.209 +lemma DomainI [intro]: "(a, b) : r ==> a : Domain r"
1.210 +  by (rules intro!: iffD2 [OF Domain_iff])
1.211 +
1.212 +lemma DomainE [elim!]:
1.213 +  "a : Domain r ==> (!!y. (a, y) : r ==> P) ==> P"
1.214 +  by (rules dest!: iffD1 [OF Domain_iff])
1.215 +
1.216 +lemma Domain_empty [simp]: "Domain {} = {}"
1.217 +  by blast
1.218 +
1.219 +lemma Domain_insert: "Domain (insert (a, b) r) = insert a (Domain r)"
1.220 +  by blast
1.221 +
1.222 +lemma Domain_Id [simp]: "Domain Id = UNIV"
1.223 +  by blast
1.224 +
1.225 +lemma Domain_diag [simp]: "Domain (diag A) = A"
1.226 +  by blast
1.227 +
1.228 +lemma Domain_Un_eq: "Domain(A Un B) = Domain(A) Un Domain(B)"
1.229 +  by blast
1.230 +
1.231 +lemma Domain_Int_subset: "Domain(A Int B) <= Domain(A) Int Domain(B)"
1.232 +  by blast
1.233 +
1.234 +lemma Domain_Diff_subset: "Domain(A) - Domain(B) <= Domain(A - B)"
1.235 +  by blast
1.236 +
1.237 +lemma Domain_Union: "Domain (Union S) = (UN A:S. Domain A)"
1.238 +  by blast
1.239 +
1.240 +lemma Domain_mono: "r <= s ==> Domain r <= Domain s"
1.241 +  by blast
1.242 +
1.243 +
1.244 +subsection {* Range *}
1.245 +
1.246 +lemma Range_iff: "(a : Range r) = (EX y. (y, a) : r)"
1.247 +  by (simp add: Domain_def Range_def)
1.248 +
1.249 +lemma RangeI [intro]: "(a, b) : r ==> b : Range r"
1.250 +  by (unfold Range_def) (rules intro!: converseI DomainI)
1.251 +
1.252 +lemma RangeE [elim!]: "b : Range r ==> (!!x. (x, b) : r ==> P) ==> P"
1.253 +  by (unfold Range_def) (rules elim!: DomainE dest!: converseD)
1.254 +
1.255 +lemma Range_empty [simp]: "Range {} = {}"
1.256 +  by blast
1.257 +
1.258 +lemma Range_insert: "Range (insert (a, b) r) = insert b (Range r)"
1.259 +  by blast
1.260 +
1.261 +lemma Range_Id [simp]: "Range Id = UNIV"
1.262 +  by blast
1.263 +
1.264 +lemma Range_diag [simp]: "Range (diag A) = A"
1.265 +  by auto
1.266 +
1.267 +lemma Range_Un_eq: "Range(A Un B) = Range(A) Un Range(B)"
1.268 +  by blast
1.269 +
1.270 +lemma Range_Int_subset: "Range(A Int B) <= Range(A) Int Range(B)"
1.271 +  by blast
1.272 +
1.273 +lemma Range_Diff_subset: "Range(A) - Range(B) <= Range(A - B)"
1.274 +  by blast
1.275 +
1.276 +lemma Range_Union: "Range (Union S) = (UN A:S. Range A)"
1.277 +  by blast
1.278 +
1.279 +
1.280 +subsection {* Image of a set under a relation *}
1.281 +
1.282 +ML {* overload_1st_set "Relation.Image" *}
1.283 +
1.284 +lemma Image_iff: "(b : r``A) = (EX x:A. (x,b):r)"
1.285 +  by (simp add: Image_def)
1.286 +
1.287 +lemma Image_singleton: "r``{a} = {b. (a,b):r}"
1.288 +  by (simp add: Image_def)
1.289 +
1.290 +lemma Image_singleton_iff [iff]: "(b : r``{a}) = ((a,b):r)"
1.291 +  by (rule Image_iff [THEN trans]) simp
1.292 +
1.293 +lemma ImageI [intro]: "[| (a,b): r;  a:A |] ==> b : r``A"
1.294 +  by (unfold Image_def) blast
1.295 +
1.296 +lemma ImageE [elim!]:
1.297 +  "b : r `` A ==> (!!x. (x, b) : r ==> x : A ==> P) ==> P"
1.298 +  by (unfold Image_def) (rules elim!: CollectE bexE)
1.299 +
1.300 +lemma rev_ImageI: "a : A ==> (a, b) : r ==> b : r `` A"
1.301 +  -- {* This version's more effective when we already have the required @{text a} *}
1.302 +  by blast
1.303 +
1.304 +lemma Image_empty [simp]: "R``{} = {}"
1.305 +  by blast
1.306 +
1.307 +lemma Image_Id [simp]: "Id `` A = A"
1.308 +  by blast
1.309 +
1.310 +lemma Image_diag [simp]: "diag A `` B = A Int B"
1.311 +  by blast
1.312 +
1.313 +lemma Image_Int_subset: "R `` (A Int B) <= R `` A Int R `` B"
1.314 +  by blast
1.315 +
1.316 +lemma Image_Un: "R `` (A Un B) = R `` A Un R `` B"
1.317 +  by blast
1.318 +
1.319 +lemma Image_subset: "r <= A <*> B ==> r``C <= B"
1.320 +  by (rules intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2)
1.321 +
1.322 +lemma Image_eq_UN: "r``B = (UN y: B. r``{y})"
1.323 +  -- {* NOT suitable for rewriting *}
1.324 +  by blast
1.325 +
1.326 +lemma Image_mono: "[| r'<=r; A'<=A |] ==> (r' `` A') <= (r `` A)"
1.327 +  by blast
1.328 +
1.329 +lemma Image_UN: "(r `` (UNION A B)) = (UN x:A.(r `` (B x)))"
1.330 +  by blast
1.331 +
1.332 +lemma Image_INT_subset: "(r `` (INTER A B)) <= (INT x:A.(r `` (B x)))"
1.333 +  -- {* Converse inclusion fails *}
1.334 +  by blast
1.335 +
1.336 +lemma Image_subset_eq: "(r``A <= B) = (A <= - ((r^-1) `` (-B)))"
1.337 +  by blast
1.338 +
1.339 +subsection "single_valued"
1.340 +
1.341 +lemma single_valuedI:
1.342 +  "ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z) ==> single_valued r"
1.343 +  by (unfold single_valued_def)
1.344 +
1.345 +lemma single_valuedD:
1.346 +  "single_valued r ==> (x, y) : r ==> (x, z) : r ==> y = z"
1.347 +  by (simp add: single_valued_def)
1.348 +
1.349 +
1.350 +subsection {* Graphs given by @{text Collect} *}
1.351 +
1.352 +lemma Domain_Collect_split [simp]: "Domain{(x,y). P x y} = {x. EX y. P x y}"
1.353 +  by auto
1.354 +
1.355 +lemma Range_Collect_split [simp]: "Range{(x,y). P x y} = {y. EX x. P x y}"
1.356 +  by auto
1.357 +
1.358 +lemma Image_Collect_split [simp]: "{(x,y). P x y} `` A = {y. EX x:A. P x y}"
1.359 +  by auto
1.360 +
1.361 +
1.362 +subsection {* Composition of function and relation *}
1.363 +
1.364 +lemma fun_rel_comp_mono: "A <= B ==> fun_rel_comp f A <= fun_rel_comp f B"
1.365 +  by (unfold fun_rel_comp_def) fast
1.366 +
1.367 +lemma fun_rel_comp_unique:
1.368 +  "ALL x. EX! y. (f x, y) : R ==> EX! g. g : fun_rel_comp f R"
1.369 +  apply (unfold fun_rel_comp_def)
1.370 +  apply (rule_tac a = "%x. THE y. (f x, y) : R" in ex1I)
1.371 +  apply (fast dest!: theI')
1.372 +  apply (fast intro: ext the1_equality [symmetric])
1.373 +  done
1.374 +
1.375 +
1.376 +subsection "inverse image"
1.377 +
1.378 +lemma trans_inv_image:
1.379 +  "trans r ==> trans (inv_image r f)"
1.380 +  apply (unfold trans_def inv_image_def)
1.381 +  apply (simp (no_asm))
1.382 +  apply blast
1.383 +  done
1.384 +
1.385  end
```