src/HOL/Relation.thy
 changeset 46694 0988b22e2626 parent 46692 1f8b766224f6 child 46696 28a01ea3523a
```     1.1 --- a/src/HOL/Relation.thy	Sun Feb 26 21:25:54 2012 +0100
1.2 +++ b/src/HOL/Relation.thy	Sun Feb 26 21:26:28 2012 +0100
1.3 @@ -8,7 +8,7 @@
1.4  imports Datatype Finite_Set
1.5  begin
1.6
1.7 -subsection {* Classical rules for reasoning on predicates *}
1.8 +text {* A preliminary: classical rules for reasoning on predicates *}
1.9
1.10  (* CANDIDATE declare predicate1I [Pure.intro!, intro!] *)
1.11  declare predicate1D [Pure.dest?, dest?]
1.12 @@ -42,8 +42,23 @@
1.13  declare SUP1_E [elim!]
1.14  declare SUP2_E [elim!]
1.15
1.16 +subsection {* Fundamental *}
1.17
1.18 -subsection {* Conversions between set and predicate relations *}
1.19 +subsubsection {* Relations as sets of pairs *}
1.20 +
1.21 +type_synonym 'a rel = "('a * 'a) set"
1.22 +
1.23 +lemma subrelI: -- {* Version of @{thm [source] subsetI} for binary relations *}
1.24 +  "(\<And>x y. (x, y) \<in> r \<Longrightarrow> (x, y) \<in> s) \<Longrightarrow> r \<subseteq> s"
1.25 +  by auto
1.26 +
1.27 +lemma lfp_induct2: -- {* Version of @{thm [source] lfp_induct} for binary relations *}
1.28 +  "(a, b) \<in> lfp f \<Longrightarrow> mono f \<Longrightarrow>
1.29 +    (\<And>a b. (a, b) \<in> f (lfp f \<inter> {(x, y). P x y}) \<Longrightarrow> P a b) \<Longrightarrow> P a b"
1.30 +  using lfp_induct_set [of "(a, b)" f "prod_case P"] by auto
1.31 +
1.32 +
1.33 +subsubsection {* Conversions between set and predicate relations *}
1.34
1.35  lemma pred_equals_eq [pred_set_conv]: "((\<lambda>x. x \<in> R) = (\<lambda>x. x \<in> S)) \<longleftrightarrow> (R = S)"
1.36    by (simp add: set_eq_iff fun_eq_iff)
1.37 @@ -94,30 +109,21 @@
1.38    by (simp add: SUP_apply fun_eq_iff)
1.39
1.40
1.41 -subsection {* Relations as sets of pairs *}
1.42 -
1.43 -type_synonym 'a rel = "('a * 'a) set"
1.44 -
1.45 -lemma subrelI: -- {* Version of @{thm [source] subsetI} for binary relations *}
1.46 -  "(\<And>x y. (x, y) \<in> r \<Longrightarrow> (x, y) \<in> s) \<Longrightarrow> r \<subseteq> s"
1.47 -  by auto
1.48 -
1.49 -lemma lfp_induct2: -- {* Version of @{thm [source] lfp_induct} for binary relations *}
1.50 -  "(a, b) \<in> lfp f \<Longrightarrow> mono f \<Longrightarrow>
1.51 -    (\<And>a b. (a, b) \<in> f (lfp f \<inter> {(x, y). P x y}) \<Longrightarrow> P a b) \<Longrightarrow> P a b"
1.52 -  using lfp_induct_set [of "(a, b)" f "prod_case P"] by auto
1.53 -
1.54 +subsection {* Properties of relations *}
1.55
1.56  subsubsection {* Reflexivity *}
1.57
1.58  definition
1.59 -  refl_on :: "['a set, ('a * 'a) set] => bool" where -- {* reflexivity over a set *}
1.60 +  refl_on :: "'a set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> bool" where
1.61    "refl_on A r \<longleftrightarrow> r \<subseteq> A \<times> A & (ALL x: A. (x,x) : r)"
1.62
1.63  abbreviation
1.64    refl :: "('a * 'a) set => bool" where -- {* reflexivity over a type *}
1.65    "refl \<equiv> refl_on UNIV"
1.66
1.67 +definition reflp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
1.68 +  "reflp r \<longleftrightarrow> refl {(x, y). r x y}"
1.69 +
1.70  lemma refl_onI: "r \<subseteq> A \<times> A ==> (!!x. x : A ==> (x, x) : r) ==> refl_on A r"
1.71  by (unfold refl_on_def) (iprover intro!: ballI)
1.72
1.73 @@ -130,6 +136,15 @@
1.74  lemma refl_onD2: "refl_on A r ==> (x, y) : r ==> y : A"
1.75  by (unfold refl_on_def) blast
1.76
1.77 +lemma reflpI:
1.78 +  "(\<And>x. r x x) \<Longrightarrow> reflp r"
1.79 +  by (auto intro: refl_onI simp add: reflp_def)
1.80 +
1.81 +lemma reflpE:
1.82 +  assumes "reflp r"
1.83 +  obtains "r x x"
1.84 +  using assms by (auto dest: refl_onD simp add: reflp_def)
1.85 +
1.86  lemma refl_on_Int: "refl_on A r ==> refl_on B s ==> refl_on (A \<inter> B) (r \<inter> s)"
1.87  by (unfold refl_on_def) blast
1.88
1.89 @@ -152,30 +167,21 @@
1.90  by (auto intro: refl_onI dest: refl_onD refl_onD1 refl_onD2)
1.91
1.92
1.93 -subsubsection {* Antisymmetry *}
1.94 +subsubsection {* Irreflexivity *}
1.95
1.96  definition
1.97 -  antisym :: "('a * 'a) set => bool" where -- {* antisymmetry predicate *}
1.98 -  "antisym r \<longleftrightarrow> (ALL x y. (x,y):r --> (y,x):r --> x=y)"
1.99 -
1.100 -lemma antisymI:
1.101 -  "(!!x y. (x, y) : r ==> (y, x) : r ==> x=y) ==> antisym r"
1.102 -by (unfold antisym_def) iprover
1.103 +  irrefl :: "('a * 'a) set => bool" where
1.104 +  "irrefl r \<longleftrightarrow> (\<forall>x. (x,x) \<notin> r)"
1.105
1.106 -lemma antisymD: "antisym r ==> (a, b) : r ==> (b, a) : r ==> a = b"
1.107 -by (unfold antisym_def) iprover
1.108 -
1.109 -lemma antisym_subset: "r \<subseteq> s ==> antisym s ==> antisym r"
1.110 -by (unfold antisym_def) blast
1.111 -
1.112 -lemma antisym_empty [simp]: "antisym {}"
1.113 -by (unfold antisym_def) blast
1.114 +lemma irrefl_distinct [code]:
1.115 +  "irrefl r \<longleftrightarrow> (\<forall>(x, y) \<in> r. x \<noteq> y)"
1.116 +  by (auto simp add: irrefl_def)
1.117
1.118
1.119  subsubsection {* Symmetry *}
1.120
1.121  definition
1.122 -  sym :: "('a * 'a) set => bool" where -- {* symmetry predicate *}
1.123 +  sym :: "('a * 'a) set => bool" where
1.124    "sym r \<longleftrightarrow> (ALL x y. (x,y): r --> (y,x): r)"
1.125
1.126  lemma symI: "(!!a b. (a, b) : r ==> (b, a) : r) ==> sym r"
1.127 @@ -184,6 +190,18 @@
1.128  lemma symD: "sym r ==> (a, b) : r ==> (b, a) : r"
1.129  by (unfold sym_def, blast)
1.130
1.131 +definition symp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
1.132 +  "symp r \<longleftrightarrow> sym {(x, y). r x y}"
1.133 +
1.134 +lemma sympI:
1.135 +  "(\<And>x y. r x y \<Longrightarrow> r y x) \<Longrightarrow> symp r"
1.136 +  by (auto intro: symI simp add: symp_def)
1.137 +
1.138 +lemma sympE:
1.139 +  assumes "symp r" and "r x y"
1.140 +  obtains "r y x"
1.141 +  using assms by (auto dest: symD simp add: symp_def)
1.142 +
1.143  lemma sym_Int: "sym r ==> sym s ==> sym (r \<inter> s)"
1.144  by (fast intro: symI dest: symD)
1.145
1.146 @@ -197,16 +215,35 @@
1.147  by (fast intro: symI dest: symD)
1.148
1.149
1.150 +subsubsection {* Antisymmetry *}
1.151 +
1.152 +definition
1.153 +  antisym :: "('a * 'a) set => bool" where
1.154 +  "antisym r \<longleftrightarrow> (ALL x y. (x,y):r --> (y,x):r --> x=y)"
1.155 +
1.156 +lemma antisymI:
1.157 +  "(!!x y. (x, y) : r ==> (y, x) : r ==> x=y) ==> antisym r"
1.158 +by (unfold antisym_def) iprover
1.159 +
1.160 +lemma antisymD: "antisym r ==> (a, b) : r ==> (b, a) : r ==> a = b"
1.161 +by (unfold antisym_def) iprover
1.162 +
1.163 +abbreviation antisymP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
1.164 +  "antisymP r \<equiv> antisym {(x, y). r x y}"
1.165 +
1.166 +lemma antisym_subset: "r \<subseteq> s ==> antisym s ==> antisym r"
1.167 +by (unfold antisym_def) blast
1.168 +
1.169 +lemma antisym_empty [simp]: "antisym {}"
1.170 +by (unfold antisym_def) blast
1.171 +
1.172 +
1.173  subsubsection {* Transitivity *}
1.174
1.175  definition
1.176 -  trans :: "('a * 'a) set => bool" where -- {* transitivity predicate *}
1.177 +  trans :: "('a * 'a) set => bool" where
1.178    "trans r \<longleftrightarrow> (ALL x y z. (x,y):r --> (y,z):r --> (x,z):r)"
1.179
1.180 -lemma trans_join [code]:
1.181 -  "trans r \<longleftrightarrow> (\<forall>(x, y1) \<in> r. \<forall>(y2, z) \<in> r. y1 = y2 \<longrightarrow> (x, z) \<in> r)"
1.182 -  by (auto simp add: trans_def)
1.183 -
1.184  lemma transI:
1.185    "(!!x y z. (x, y) : r ==> (y, z) : r ==> (x, z) : r) ==> trans r"
1.186  by (unfold trans_def) iprover
1.187 @@ -214,22 +251,30 @@
1.188  lemma transD: "trans r ==> (a, b) : r ==> (b, c) : r ==> (a, c) : r"
1.189  by (unfold trans_def) iprover
1.190
1.191 +abbreviation transP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
1.192 +  "transP r \<equiv> trans {(x, y). r x y}"
1.193 +
1.194 +definition transp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
1.195 +  "transp r \<longleftrightarrow> trans {(x, y). r x y}"
1.196 +
1.197 +lemma transpI:
1.198 +  "(\<And>x y z. r x y \<Longrightarrow> r y z \<Longrightarrow> r x z) \<Longrightarrow> transp r"
1.199 +  by (auto intro: transI simp add: transp_def)
1.200 +
1.201 +lemma transpE:
1.202 +  assumes "transp r" and "r x y" and "r y z"
1.203 +  obtains "r x z"
1.204 +  using assms by (auto dest: transD simp add: transp_def)
1.205 +
1.206  lemma trans_Int: "trans r ==> trans s ==> trans (r \<inter> s)"
1.207  by (fast intro: transI elim: transD)
1.208
1.209  lemma trans_INTER: "ALL x:S. trans (r x) ==> trans (INTER S r)"
1.210  by (fast intro: transI elim: transD)
1.211
1.212 -
1.213 -subsubsection {* Irreflexivity *}
1.214 -
1.215 -definition
1.216 -  irrefl :: "('a * 'a) set => bool" where
1.217 -  "irrefl r \<longleftrightarrow> (\<forall>x. (x,x) \<notin> r)"
1.218 -
1.219 -lemma irrefl_distinct [code]:
1.220 -  "irrefl r \<longleftrightarrow> (\<forall>(x, y) \<in> r. x \<noteq> y)"
1.221 -  by (auto simp add: irrefl_def)
1.222 +lemma trans_join [code]:
1.223 +  "trans r \<longleftrightarrow> (\<forall>(x, y1) \<in> r. \<forall>(y2, z) \<in> r. y1 = y2 \<longrightarrow> (x, z) \<in> r)"
1.224 +  by (auto simp add: trans_def)
1.225
1.226
1.227  subsubsection {* Totality *}
1.228 @@ -258,15 +303,20 @@
1.229    "single_valued r ==> (x, y) : r ==> (x, z) : r ==> y = z"
1.231
1.232 +abbreviation single_valuedP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where
1.233 +  "single_valuedP r \<equiv> single_valued {(x, y). r x y}"
1.234 +
1.235  lemma single_valued_subset:
1.236    "r \<subseteq> s ==> single_valued s ==> single_valued r"
1.237  by (unfold single_valued_def) blast
1.238
1.239
1.240 +subsection {* Relation operations *}
1.241 +
1.242  subsubsection {* The identity relation *}
1.243
1.244  definition
1.245 -  Id :: "('a * 'a) set" where -- {* the identity relation *}
1.246 +  Id :: "('a * 'a) set" where
1.247    "Id = {p. EX x. p = (x,x)}"
1.248
1.249  lemma IdI [intro]: "(a, a) : Id"
1.250 @@ -307,7 +357,7 @@
1.251  subsubsection {* Diagonal: identity over a set *}
1.252
1.253  definition
1.254 -  Id_on  :: "'a set => ('a * 'a) set" where -- {* diagonal: identity over a set *}
1.255 +  Id_on  :: "'a set => ('a * 'a) set" where
1.256    "Id_on A = (\<Union>x\<in>A. {(x,x)})"
1.257
1.258  lemma Id_on_empty [simp]: "Id_on {} = {}"
1.259 @@ -350,12 +400,11 @@
1.260    by (unfold single_valued_def) blast
1.261
1.262
1.263 -subsubsection {* Composition of two relations *}
1.264 +subsubsection {* Composition *}
1.265
1.266 -definition
1.267 -  rel_comp  :: "[('a * 'b) set, ('b * 'c) set] => ('a * 'c) set"
1.268 -    (infixr "O" 75) where
1.269 -  "r O s = {(x,z). EX y. (x, y) : r & (y, z) : s}"
1.270 +definition rel_comp  :: "('a \<times> 'b) set \<Rightarrow> ('b \<times> 'c) set \<Rightarrow> ('a * 'c) set" (infixr "O" 75)
1.271 +where
1.272 +  "r O s = {(x, z). \<exists>y. (x, y) \<in> r \<and> (y, z) \<in> s}"
1.273
1.274  lemma rel_compI [intro]:
1.275    "(a, b) : r ==> (b, c) : s ==> (a, c) : r O s"
1.276 @@ -365,6 +414,17 @@
1.277    (!!x y z. xz = (x, z) ==> (x, y) : r ==> (y, z) : s  ==> P) ==> P"
1.278  by (unfold rel_comp_def) (iprover elim!: CollectE splitE exE conjE)
1.279
1.280 +inductive pred_comp :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool" (infixr "OO" 75)
1.281 +for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" and s :: "'b \<Rightarrow> 'c \<Rightarrow> bool"
1.282 +where
1.283 +  pred_compI [intro]: "r a b \<Longrightarrow> s b c \<Longrightarrow> (r OO s) a c"
1.284 +
1.285 +inductive_cases pred_compE [elim!]: "(r OO s) a c"
1.286 +
1.287 +lemma pred_comp_rel_comp_eq [pred_set_conv]:
1.288 +  "((\<lambda>x y. (x, y) \<in> r) OO (\<lambda>x y. (x, y) \<in> s)) = (\<lambda>x y. (x, y) \<in> r O s)"
1.289 +  by (auto simp add: fun_eq_iff)
1.290 +
1.291  lemma rel_compEpair:
1.292    "(a, c) : r O s ==> (!!y. (a, y) : r ==> (y, c) : s ==> P) ==> P"
1.293  by (iprover elim: rel_compE Pair_inject ssubst)
1.294 @@ -421,19 +481,58 @@
1.295  notation (xsymbols)
1.296    converse  ("(_\<inverse>)" [1000] 999)
1.297
1.298 -lemma converse_iff [iff]: "((a,b): r^-1) = ((b,a) : r)"
1.300 +lemma converseI [sym]: "(a, b) : r ==> (b, a) : r^-1"
1.301 +  by (simp add: converse_def)
1.302
1.303 -lemma converseI[sym]: "(a, b) : r ==> (b, a) : r^-1"
1.305 -
1.306 -lemma converseD[sym]: "(a,b) : r^-1 ==> (b, a) : r"
1.308 +lemma converseD [sym]: "(a,b) : r^-1 ==> (b, a) : r"
1.309 +  by (simp add: converse_def)
1.310
1.311  lemma converseE [elim!]:
1.312    "yx : r^-1 ==> (!!x y. yx = (y, x) ==> (x, y) : r ==> P) ==> P"
1.313      -- {* More general than @{text converseD}, as it ``splits'' the member of the relation. *}
1.314 -by (unfold converse_def) (iprover elim!: CollectE splitE bexE)
1.315 +  by (unfold converse_def) (iprover elim!: CollectE splitE bexE)
1.316 +
1.317 +lemma converse_iff [iff]: "((a,b): r^-1) = ((b,a) : r)"
1.318 +  by (simp add: converse_def)
1.319 +
1.320 +inductive conversep :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(_^--1)" [1000] 1000)
1.321 +  for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" where
1.322 +  conversepI: "r a b \<Longrightarrow> r^--1 b a"
1.323 +
1.324 +notation (xsymbols)
1.325 +  conversep  ("(_\<inverse>\<inverse>)" [1000] 1000)
1.326 +
1.327 +lemma conversepD:
1.328 +  assumes ab: "r^--1 a b"
1.329 +  shows "r b a" using ab
1.330 +  by cases simp
1.331 +
1.332 +lemma conversep_iff [iff]: "r^--1 a b = r b a"
1.333 +  by (iprover intro: conversepI dest: conversepD)
1.334 +
1.335 +lemma conversep_converse_eq [pred_set_conv]:
1.336 +  "(\<lambda>x y. (x, y) \<in> r)^--1 = (\<lambda>x y. (x, y) \<in> r^-1)"
1.337 +  apply (auto simp add: fun_eq_iff)
1.338 +  oops
1.339 +
1.340 +lemma conversep_conversep [simp]: "(r^--1)^--1 = r"
1.341 +  by (iprover intro: order_antisym conversepI dest: conversepD)
1.342 +
1.343 +lemma converse_pred_comp: "(r OO s)^--1 = s^--1 OO r^--1"
1.344 +  by (iprover intro: order_antisym conversepI pred_compI
1.345 +    elim: pred_compE dest: conversepD)
1.346 +
1.347 +lemma converse_meet: "(r \<sqinter> s)^--1 = r^--1 \<sqinter> s^--1"
1.348 +  by (simp add: inf_fun_def) (iprover intro: conversepI ext dest: conversepD)
1.349 +
1.350 +lemma converse_join: "(r \<squnion> s)^--1 = r^--1 \<squnion> s^--1"
1.351 +  by (simp add: sup_fun_def) (iprover intro: conversepI ext dest: conversepD)
1.352 +
1.353 +lemma conversep_noteq [simp]: "(op \<noteq>)^--1 = op \<noteq>"
1.354 +  by (auto simp add: fun_eq_iff)
1.355 +
1.356 +lemma conversep_eq [simp]: "(op =)^--1 = op ="
1.357 +  by (auto simp add: fun_eq_iff)
1.358
1.359  lemma converse_converse [simp]: "(r^-1)^-1 = r"
1.360  by (unfold converse_def) blast
1.361 @@ -523,58 +622,6 @@
1.362    "a : Domain r ==> (!!y. (a, y) : r ==> P) ==> P"
1.363  by (iprover dest!: iffD1 [OF Domain_iff])
1.364
1.365 -lemma Domain_fst [code]:
1.366 -  "Domain r = fst ` r"
1.367 -  by (auto simp add: image_def Bex_def)
1.368 -
1.369 -lemma Domain_empty [simp]: "Domain {} = {}"
1.370 -by blast
1.371 -
1.372 -lemma Domain_empty_iff: "Domain r = {} \<longleftrightarrow> r = {}"
1.373 -  by auto
1.374 -
1.375 -lemma Domain_insert: "Domain (insert (a, b) r) = insert a (Domain r)"
1.376 -by blast
1.377 -
1.378 -lemma Domain_Id [simp]: "Domain Id = UNIV"
1.379 -by blast
1.380 -
1.381 -lemma Domain_Id_on [simp]: "Domain (Id_on A) = A"
1.382 -by blast
1.383 -
1.384 -lemma Domain_Un_eq: "Domain(A \<union> B) = Domain(A) \<union> Domain(B)"
1.385 -by blast
1.386 -
1.387 -lemma Domain_Int_subset: "Domain(A \<inter> B) \<subseteq> Domain(A) \<inter> Domain(B)"
1.388 -by blast
1.389 -
1.390 -lemma Domain_Diff_subset: "Domain(A) - Domain(B) \<subseteq> Domain(A - B)"
1.391 -by blast
1.392 -
1.393 -lemma Domain_Union: "Domain (Union S) = (\<Union>A\<in>S. Domain A)"
1.394 -by blast
1.395 -
1.396 -lemma Domain_converse[simp]: "Domain(r^-1) = Range r"
1.397 -by(auto simp:Range_def)
1.398 -
1.399 -lemma Domain_mono: "r \<subseteq> s ==> Domain r \<subseteq> Domain s"
1.400 -by blast
1.401 -
1.402 -lemma fst_eq_Domain: "fst ` R = Domain R"
1.403 -  by force
1.404 -
1.405 -lemma Domain_dprod [simp]: "Domain (dprod r s) = uprod (Domain r) (Domain s)"
1.406 -by auto
1.407 -
1.408 -lemma Domain_dsum [simp]: "Domain (dsum r s) = usum (Domain r) (Domain s)"
1.409 -by auto
1.410 -
1.411 -lemma Domain_Collect_split [simp]: "Domain {(x, y). P x y} = {x. EX y. P x y}"
1.412 -by auto
1.413 -
1.414 -lemma finite_Domain: "finite r ==> finite (Domain r)"
1.415 -  by (induct set: finite) (auto simp add: Domain_insert)
1.416 -
1.417  lemma Range_iff: "(a : Range r) = (EX y. (y, a) : r)"
1.418  by (simp add: Domain_def Range_def)
1.419
1.420 @@ -584,66 +631,136 @@
1.421  lemma RangeE [elim!]: "b : Range r ==> (!!x. (x, b) : r ==> P) ==> P"
1.422  by (unfold Range_def) (iprover elim!: DomainE dest!: converseD)
1.423
1.424 +inductive DomainP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"
1.425 +  for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" where
1.426 +  DomainPI [intro]: "r a b \<Longrightarrow> DomainP r a"
1.427 +
1.428 +inductive_cases DomainPE [elim!]: "DomainP r a"
1.429 +
1.430 +lemma DomainP_Domain_eq [pred_set_conv]: "DomainP (\<lambda>x y. (x, y) \<in> r) = (\<lambda>x. x \<in> Domain r)"
1.431 +  by (blast intro!: Orderings.order_antisym predicate1I)
1.432 +
1.433 +inductive RangeP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"
1.434 +  for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" where
1.435 +  RangePI [intro]: "r a b \<Longrightarrow> RangeP r b"
1.436 +
1.437 +inductive_cases RangePE [elim!]: "RangeP r b"
1.438 +
1.439 +lemma RangeP_Range_eq [pred_set_conv]: "RangeP (\<lambda>x y. (x, y) \<in> r) = (\<lambda>x. x \<in> Range r)"
1.440 +  by (auto intro!: Orderings.order_antisym predicate1I)
1.441 +
1.442 +lemma Domain_fst [code]:
1.443 +  "Domain r = fst ` r"
1.444 +  by (auto simp add: image_def Bex_def)
1.445 +
1.446 +lemma Domain_empty [simp]: "Domain {} = {}"
1.447 +  by blast
1.448 +
1.449 +lemma Domain_empty_iff: "Domain r = {} \<longleftrightarrow> r = {}"
1.450 +  by auto
1.451 +
1.452 +lemma Domain_insert: "Domain (insert (a, b) r) = insert a (Domain r)"
1.453 +  by blast
1.454 +
1.455 +lemma Domain_Id [simp]: "Domain Id = UNIV"
1.456 +  by blast
1.457 +
1.458 +lemma Domain_Id_on [simp]: "Domain (Id_on A) = A"
1.459 +  by blast
1.460 +
1.461 +lemma Domain_Un_eq: "Domain(A \<union> B) = Domain(A) \<union> Domain(B)"
1.462 +  by blast
1.463 +
1.464 +lemma Domain_Int_subset: "Domain(A \<inter> B) \<subseteq> Domain(A) \<inter> Domain(B)"
1.465 +  by blast
1.466 +
1.467 +lemma Domain_Diff_subset: "Domain(A) - Domain(B) \<subseteq> Domain(A - B)"
1.468 +  by blast
1.469 +
1.470 +lemma Domain_Union: "Domain (Union S) = (\<Union>A\<in>S. Domain A)"
1.471 +  by blast
1.472 +
1.473 +lemma Domain_converse[simp]: "Domain(r^-1) = Range r"
1.474 +  by(auto simp: Range_def)
1.475 +
1.476 +lemma Domain_mono: "r \<subseteq> s ==> Domain r \<subseteq> Domain s"
1.477 +  by blast
1.478 +
1.479 +lemma fst_eq_Domain: "fst ` R = Domain R"
1.480 +  by force
1.481 +
1.482 +lemma Domain_dprod [simp]: "Domain (dprod r s) = uprod (Domain r) (Domain s)"
1.483 +  by auto
1.484 +
1.485 +lemma Domain_dsum [simp]: "Domain (dsum r s) = usum (Domain r) (Domain s)"
1.486 +  by auto
1.487 +
1.488 +lemma Domain_Collect_split [simp]: "Domain {(x, y). P x y} = {x. EX y. P x y}"
1.489 +  by auto
1.490 +
1.491 +lemma finite_Domain: "finite r ==> finite (Domain r)"
1.492 +  by (induct set: finite) (auto simp add: Domain_insert)
1.493 +
1.494  lemma Range_snd [code]:
1.495    "Range r = snd ` r"
1.496    by (auto simp add: image_def Bex_def)
1.497
1.498  lemma Range_empty [simp]: "Range {} = {}"
1.499 -by blast
1.500 +  by blast
1.501
1.502  lemma Range_empty_iff: "Range r = {} \<longleftrightarrow> r = {}"
1.503    by auto
1.504
1.505  lemma Range_insert: "Range (insert (a, b) r) = insert b (Range r)"
1.506 -by blast
1.507 +  by blast
1.508
1.509  lemma Range_Id [simp]: "Range Id = UNIV"
1.510 -by blast
1.511 +  by blast
1.512
1.513  lemma Range_Id_on [simp]: "Range (Id_on A) = A"
1.514 -by auto
1.515 +  by auto
1.516
1.517  lemma Range_Un_eq: "Range(A \<union> B) = Range(A) \<union> Range(B)"
1.518 -by blast
1.519 +  by blast
1.520
1.521  lemma Range_Int_subset: "Range(A \<inter> B) \<subseteq> Range(A) \<inter> Range(B)"
1.522 -by blast
1.523 +  by blast
1.524
1.525  lemma Range_Diff_subset: "Range(A) - Range(B) \<subseteq> Range(A - B)"
1.526 -by blast
1.527 +  by blast
1.528
1.529  lemma Range_Union: "Range (Union S) = (\<Union>A\<in>S. Range A)"
1.530 -by blast
1.531 +  by blast
1.532
1.533 -lemma Range_converse[simp]: "Range(r^-1) = Domain r"
1.534 -by blast
1.535 +lemma Range_converse [simp]: "Range(r^-1) = Domain r"
1.536 +  by blast
1.537
1.538  lemma snd_eq_Range: "snd ` R = Range R"
1.539    by force
1.540
1.541  lemma Range_Collect_split [simp]: "Range {(x, y). P x y} = {y. EX x. P x y}"
1.542 -by auto
1.543 +  by auto
1.544
1.545  lemma finite_Range: "finite r ==> finite (Range r)"
1.546    by (induct set: finite) (auto simp add: Range_insert)
1.547
1.548  lemma mono_Field: "r \<subseteq> s \<Longrightarrow> Field r \<subseteq> Field s"
1.549 -by(auto simp:Field_def Domain_def Range_def)
1.550 +  by (auto simp: Field_def Domain_def Range_def)
1.551
1.552  lemma Field_empty[simp]: "Field {} = {}"
1.553 -by(auto simp:Field_def)
1.554 +  by (auto simp: Field_def)
1.555
1.556 -lemma Field_insert[simp]: "Field (insert (a,b) r) = {a,b} \<union> Field r"
1.557 -by(auto simp:Field_def)
1.558 +lemma Field_insert [simp]: "Field (insert (a, b) r) = {a, b} \<union> Field r"
1.559 +  by (auto simp: Field_def)
1.560
1.561 -lemma Field_Un[simp]: "Field (r \<union> s) = Field r \<union> Field s"
1.562 -by(auto simp:Field_def)
1.563 +lemma Field_Un [simp]: "Field (r \<union> s) = Field r \<union> Field s"
1.564 +  by (auto simp: Field_def)
1.565
1.566 -lemma Field_Union[simp]: "Field (\<Union>R) = \<Union>(Field ` R)"
1.567 -by(auto simp:Field_def)
1.568 +lemma Field_Union [simp]: "Field (\<Union>R) = \<Union>(Field ` R)"
1.569 +  by (auto simp: Field_def)
1.570
1.571 -lemma Field_converse[simp]: "Field(r^-1) = Field r"
1.572 -by(auto simp:Field_def)
1.573 +lemma Field_converse [simp]: "Field(r^-1) = Field r"
1.574 +  by (auto simp: Field_def)
1.575
1.576  lemma finite_Field: "finite r ==> finite (Field r)"
1.577    -- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *}
1.578 @@ -740,6 +857,12 @@
1.579    inv_image :: "('b * 'b) set => ('a => 'b) => ('a * 'a) set" where
1.580    "inv_image r f = {(x, y). (f x, f y) : r}"
1.581
1.582 +definition inv_imagep :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" where
1.583 +  "inv_imagep r f = (\<lambda>x y. r (f x) (f y))"
1.584 +
1.585 +lemma [pred_set_conv]: "inv_imagep (\<lambda>x y. (x, y) \<in> r) f = (\<lambda>x y. (x, y) \<in> inv_image r f)"
1.586 +  by (simp add: inv_image_def inv_imagep_def)
1.587 +
1.588  lemma sym_inv_image: "sym r ==> sym (inv_image r f)"
1.589  by (unfold sym_def inv_image_def) blast
1.590
1.591 @@ -755,95 +878,6 @@
1.592  lemma converse_inv_image[simp]: "(inv_image R f)^-1 = inv_image (R^-1) f"
1.593  unfolding inv_image_def converse_def by auto
1.594
1.595 -
1.596 -subsection {* Relations as binary predicates *}
1.597 -
1.598 -subsubsection {* Composition *}
1.599 -
1.600 -inductive pred_comp  :: "['a \<Rightarrow> 'b \<Rightarrow> bool, 'b \<Rightarrow> 'c \<Rightarrow> bool] \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool" (infixr "OO" 75)
1.601 -  for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" and s :: "'b \<Rightarrow> 'c \<Rightarrow> bool" where
1.602 -  pred_compI [intro]: "r a b \<Longrightarrow> s b c \<Longrightarrow> (r OO s) a c"
1.603 -
1.604 -inductive_cases pred_compE [elim!]: "(r OO s) a c"
1.605 -
1.606 -lemma pred_comp_rel_comp_eq [pred_set_conv]:
1.607 -  "((\<lambda>x y. (x, y) \<in> r) OO (\<lambda>x y. (x, y) \<in> s)) = (\<lambda>x y. (x, y) \<in> r O s)"
1.608 -  by (auto simp add: fun_eq_iff)
1.609 -
1.610 -
1.611 -subsubsection {* Converse *}
1.612 -
1.613 -inductive conversep :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(_^--1)" [1000] 1000)
1.614 -  for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" where
1.615 -  conversepI: "r a b \<Longrightarrow> r^--1 b a"
1.616 -
1.617 -notation (xsymbols)
1.618 -  conversep  ("(_\<inverse>\<inverse>)" [1000] 1000)
1.619 -
1.620 -lemma conversepD:
1.621 -  assumes ab: "r^--1 a b"
1.622 -  shows "r b a" using ab
1.623 -  by cases simp
1.624 -
1.625 -lemma conversep_iff [iff]: "r^--1 a b = r b a"
1.626 -  by (iprover intro: conversepI dest: conversepD)
1.627 -
1.628 -lemma conversep_converse_eq [pred_set_conv]:
1.629 -  "(\<lambda>x y. (x, y) \<in> r)^--1 = (\<lambda>x y. (x, y) \<in> r^-1)"
1.630 -  by (auto simp add: fun_eq_iff)
1.631 -
1.632 -lemma conversep_conversep [simp]: "(r^--1)^--1 = r"
1.633 -  by (iprover intro: order_antisym conversepI dest: conversepD)
1.634 -
1.635 -lemma converse_pred_comp: "(r OO s)^--1 = s^--1 OO r^--1"
1.636 -  by (iprover intro: order_antisym conversepI pred_compI
1.637 -    elim: pred_compE dest: conversepD)
1.638 -
1.639 -lemma converse_meet: "(r \<sqinter> s)^--1 = r^--1 \<sqinter> s^--1"
1.640 -  by (simp add: inf_fun_def) (iprover intro: conversepI ext dest: conversepD)
1.641 -
1.642 -lemma converse_join: "(r \<squnion> s)^--1 = r^--1 \<squnion> s^--1"
1.643 -  by (simp add: sup_fun_def) (iprover intro: conversepI ext dest: conversepD)
1.644 -
1.645 -lemma conversep_noteq [simp]: "(op \<noteq>)^--1 = op \<noteq>"
1.646 -  by (auto simp add: fun_eq_iff)
1.647 -
1.648 -lemma conversep_eq [simp]: "(op =)^--1 = op ="
1.649 -  by (auto simp add: fun_eq_iff)
1.650 -
1.651 -
1.652 -subsubsection {* Domain *}
1.653 -
1.654 -inductive DomainP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"
1.655 -  for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" where
1.656 -  DomainPI [intro]: "r a b \<Longrightarrow> DomainP r a"
1.657 -
1.658 -inductive_cases DomainPE [elim!]: "DomainP r a"
1.659 -
1.660 -lemma DomainP_Domain_eq [pred_set_conv]: "DomainP (\<lambda>x y. (x, y) \<in> r) = (\<lambda>x. x \<in> Domain r)"
1.661 -  by (blast intro!: Orderings.order_antisym predicate1I)
1.662 -
1.663 -
1.664 -subsubsection {* Range *}
1.665 -
1.666 -inductive RangeP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"
1.667 -  for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" where
1.668 -  RangePI [intro]: "r a b \<Longrightarrow> RangeP r b"
1.669 -
1.670 -inductive_cases RangePE [elim!]: "RangeP r b"
1.671 -
1.672 -lemma RangeP_Range_eq [pred_set_conv]: "RangeP (\<lambda>x y. (x, y) \<in> r) = (\<lambda>x. x \<in> Range r)"
1.673 -  by (blast intro!: Orderings.order_antisym predicate1I)
1.674 -
1.675 -
1.676 -subsubsection {* Inverse image *}
1.677 -
1.678 -definition inv_imagep :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" where
1.679 -  "inv_imagep r f = (\<lambda>x y. r (f x) (f y))"
1.680 -
1.681 -lemma [pred_set_conv]: "inv_imagep (\<lambda>x y. (x, y) \<in> r) f = (\<lambda>x y. (x, y) \<in> inv_image r f)"
1.682 -  by (simp add: inv_image_def inv_imagep_def)
1.683 -
1.684  lemma in_inv_imagep [simp]: "inv_imagep r f x y = r (f x) (f y)"
1.686
1.687 @@ -858,55 +892,5 @@
1.688
1.689  lemmas Powp_mono [mono] = Pow_mono [to_pred]
1.690
1.691 -
1.692 -subsubsection {* Properties of predicate relations *}
1.693 -
1.694 -abbreviation antisymP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
1.695 -  "antisymP r \<equiv> antisym {(x, y). r x y}"
1.696 -
1.697 -abbreviation transP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
1.698 -  "transP r \<equiv> trans {(x, y). r x y}"
1.699 -
1.700 -abbreviation single_valuedP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where
1.701 -  "single_valuedP r \<equiv> single_valued {(x, y). r x y}"
1.702 -
1.703 -(*FIXME inconsistencies: abbreviations vs. definitions, suffix `P` vs. suffix `p`*)
1.704 -
1.705 -definition reflp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
1.706 -  "reflp r \<longleftrightarrow> refl {(x, y). r x y}"
1.707 -
1.708 -definition symp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
1.709 -  "symp r \<longleftrightarrow> sym {(x, y). r x y}"
1.710 -
1.711 -definition transp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
1.712 -  "transp r \<longleftrightarrow> trans {(x, y). r x y}"
1.713 -
1.714 -lemma reflpI:
1.715 -  "(\<And>x. r x x) \<Longrightarrow> reflp r"
1.716 -  by (auto intro: refl_onI simp add: reflp_def)
1.717 -
1.718 -lemma reflpE:
1.719 -  assumes "reflp r"
1.720 -  obtains "r x x"
1.721 -  using assms by (auto dest: refl_onD simp add: reflp_def)
1.722 -
1.723 -lemma sympI:
1.724 -  "(\<And>x y. r x y \<Longrightarrow> r y x) \<Longrightarrow> symp r"
1.725 -  by (auto intro: symI simp add: symp_def)
1.726 -
1.727 -lemma sympE:
1.728 -  assumes "symp r" and "r x y"
1.729 -  obtains "r y x"
1.730 -  using assms by (auto dest: symD simp add: symp_def)
1.731 -
1.732 -lemma transpI:
1.733 -  "(\<And>x y z. r x y \<Longrightarrow> r y z \<Longrightarrow> r x z) \<Longrightarrow> transp r"
1.734 -  by (auto intro: transI simp add: transp_def)
1.735 -
1.736 -lemma transpE:
1.737 -  assumes "transp r" and "r x y" and "r y z"
1.738 -  obtains "r x z"
1.739 -  using assms by (auto dest: transD simp add: transp_def)
1.740 -
1.741  end
1.742
```