(*  Title:      HOL/Relation.thy

     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory; Stefan Berghofer, TU Muenchen

 *)

 header {* Relations – as sets of pairs, and binary predicates *}

 theory Relation

 imports Datatype Finite_Set

 begin

 text {* A preliminary: classical rules for reasoning on predicates *}

 (* CANDIDATE declare predicate1I [Pure.intro!, intro!] *)

 declare predicate1D [Pure.dest?, dest?]

 (* CANDIDATE declare predicate1D [Pure.dest, dest] *)

 declare predicate2I [Pure.intro!, intro!]

 declare predicate2D [Pure.dest, dest]

 declare bot1E [elim!]

 declare bot2E [elim!]

 declare top1I [intro!]

 declare top2I [intro!]

 declare inf1I [intro!]

 declare inf2I [intro!]

 declare inf1E [elim!]

 declare inf2E [elim!]

 declare sup1I1 [intro?]

 declare sup2I1 [intro?]

 declare sup1I2 [intro?]

 declare sup2I2 [intro?]

 declare sup1E [elim!]

 declare sup2E [elim!]

 declare sup1CI [intro!]

 declare sup2CI [intro!]

 declare INF1_I [intro!]

 declare INF2_I [intro!]

 declare INF1_D [elim]

 declare INF2_D [elim]

 declare INF1_E [elim]

 declare INF2_E [elim]

 declare SUP1_I [intro]

 declare SUP2_I [intro]

 declare SUP1_E [elim!]

 declare SUP2_E [elim!]

 subsection {* Fundamental *}

 subsubsection {* Relations as sets of pairs *}

 type_synonym 'a rel = "('a * 'a) set"

 lemma subrelI: -- {* Version of @{thm [source] subsetI} for binary relations *}

   "(\<And>x y. (x, y) \<in> r \<Longrightarrow> (x, y) \<in> s) \<Longrightarrow> r \<subseteq> s"

   by auto

 lemma lfp_induct2: -- {* Version of @{thm [source] lfp_induct} for binary relations *}

   "(a, b) \<in> lfp f \<Longrightarrow> mono f \<Longrightarrow>

     (\<And>a b. (a, b) \<in> f (lfp f \<inter> {(x, y). P x y}) \<Longrightarrow> P a b) \<Longrightarrow> P a b"

   using lfp_induct_set [of "(a, b)" f "prod_case P"] by auto

 subsubsection {* Conversions between set and predicate relations *}

 lemma pred_equals_eq [pred_set_conv]: "((\<lambda>x. x \<in> R) = (\<lambda>x. x \<in> S)) \<longleftrightarrow> (R = S)"

   by (simp add: set_eq_iff fun_eq_iff)

 lemma pred_equals_eq2 [pred_set_conv]: "((\<lambda>x y. (x, y) \<in> R) = (\<lambda>x y. (x, y) \<in> S)) \<longleftrightarrow> (R = S)"

   by (simp add: set_eq_iff fun_eq_iff)

 lemma pred_subset_eq [pred_set_conv]: "((\<lambda>x. x \<in> R) \<le> (\<lambda>x. x \<in> S)) \<longleftrightarrow> (R \<subseteq> S)"

   by (simp add: subset_iff le_fun_def)

 lemma pred_subset_eq2 [pred_set_conv]: "((\<lambda>x y. (x, y) \<in> R) \<le> (\<lambda>x y. (x, y) \<in> S)) \<longleftrightarrow> (R \<subseteq> S)"

   by (simp add: subset_iff le_fun_def)

 lemma bot_empty_eq (* CANDIDATE [pred_set_conv] *): "\<bottom> = (\<lambda>x. x \<in> {})"

   by (auto simp add: fun_eq_iff)

 lemma bot_empty_eq2 (* CANDIDATE [pred_set_conv] *): "\<bottom> = (\<lambda>x y. (x, y) \<in> {})"

   by (auto simp add: fun_eq_iff)

 (* CANDIDATE lemma top_empty_eq [pred_set_conv]: "\<top> = (\<lambda>x. x \<in> UNIV)"

   by (auto simp add: fun_eq_iff) *)

 (* CANDIDATE lemma top_empty_eq2 [pred_set_conv]: "\<top> = (\<lambda>x y. (x, y) \<in> UNIV)"

   by (auto simp add: fun_eq_iff) *)

 lemma inf_Int_eq [pred_set_conv]: "(\<lambda>x. x \<in> R) \<sqinter> (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<inter> S)"

   by (simp add: inf_fun_def)

 lemma inf_Int_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) \<sqinter> (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<inter> S)"

   by (simp add: inf_fun_def)

 lemma sup_Un_eq [pred_set_conv]: "(\<lambda>x. x \<in> R) \<squnion> (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<union> S)"

   by (simp add: sup_fun_def)

 lemma sup_Un_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) \<squnion> (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<union> S)"

   by (simp add: sup_fun_def)

 lemma INF_INT_eq (* CANDIDATE [pred_set_conv] *): "(\<Sqinter>i. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (\<Inter>i. r i))"

   by (simp add: INF_apply fun_eq_iff)

 lemma INF_INT_eq2 (* CANDIDATE [pred_set_conv] *): "(\<Sqinter>i. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (\<Inter>i. r i))"

   by (simp add: INF_apply fun_eq_iff)

 lemma SUP_UN_eq [pred_set_conv]: "(\<Squnion>i. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (\<Union>i. r i))"

   by (simp add: SUP_apply fun_eq_iff)

 lemma SUP_UN_eq2 [pred_set_conv]: "(\<Squnion>i. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (\<Union>i. r i))"

   by (simp add: SUP_apply fun_eq_iff)

 subsection {* Properties of relations *}

 subsubsection {* Reflexivity *}

 definition refl_on :: "'a set \<Rightarrow> 'a rel \<Rightarrow> bool"

 where

   "refl_on A r \<longleftrightarrow> r \<subseteq> A \<times> A \<and> (\<forall>x\<in>A. (x, x) \<in> r)"

 abbreviation refl :: "'a rel \<Rightarrow> bool"

 where -- {* reflexivity over a type *}

   "refl \<equiv> refl_on UNIV"

 definition reflp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"

 where

   "reflp r \<longleftrightarrow> refl {(x, y). r x y}"

 lemma reflp_refl_eq [pred_set_conv]:

   "reflp (\<lambda>x y. (x, y) \<in> r) \<longleftrightarrow> refl r" 

   by (simp add: refl_on_def reflp_def)

 lemma refl_onI: "r \<subseteq> A \<times> A ==> (!!x. x : A ==> (x, x) : r) ==> refl_on A r"

   by (unfold refl_on_def) (iprover intro!: ballI)

 lemma refl_onD: "refl_on A r ==> a : A ==> (a, a) : r"

   by (unfold refl_on_def) blast

 lemma refl_onD1: "refl_on A r ==> (x, y) : r ==> x : A"

   by (unfold refl_on_def) blast

 lemma refl_onD2: "refl_on A r ==> (x, y) : r ==> y : A"

   by (unfold refl_on_def) blast

 lemma reflpI:

   "(\<And>x. r x x) \<Longrightarrow> reflp r"

   by (auto intro: refl_onI simp add: reflp_def)

 lemma reflpE:

   assumes "reflp r"

   obtains "r x x"

   using assms by (auto dest: refl_onD simp add: reflp_def)

 lemma refl_on_Int: "refl_on A r ==> refl_on B s ==> refl_on (A \<inter> B) (r \<inter> s)"

   by (unfold refl_on_def) blast

 lemma reflp_inf:

   "reflp r \<Longrightarrow> reflp s \<Longrightarrow> reflp (r \<sqinter> s)"

   by (auto intro: reflpI elim: reflpE)

 lemma refl_on_Un: "refl_on A r ==> refl_on B s ==> refl_on (A \<union> B) (r \<union> s)"

   by (unfold refl_on_def) blast

 lemma reflp_sup:

   "reflp r \<Longrightarrow> reflp s \<Longrightarrow> reflp (r \<squnion> s)"

   by (auto intro: reflpI elim: reflpE)

 lemma refl_on_INTER:

   "ALL x:S. refl_on (A x) (r x) ==> refl_on (INTER S A) (INTER S r)"

   by (unfold refl_on_def) fast

 lemma refl_on_UNION:

   "ALL x:S. refl_on (A x) (r x) \<Longrightarrow> refl_on (UNION S A) (UNION S r)"

   by (unfold refl_on_def) blast

 lemma refl_on_empty [simp]: "refl_on {} {}"

   by (simp add:refl_on_def)

 lemma refl_on_def' [nitpick_unfold, code]:

   "refl_on A r \<longleftrightarrow> (\<forall>(x, y) \<in> r. x \<in> A \<and> y \<in> A) \<and> (\<forall>x \<in> A. (x, x) \<in> r)"

   by (auto intro: refl_onI dest: refl_onD refl_onD1 refl_onD2)

 subsubsection {* Irreflexivity *}

 definition irrefl :: "'a rel \<Rightarrow> bool"

 where

   "irrefl r \<longleftrightarrow> (\<forall>x. (x, x) \<notin> r)"

 lemma irrefl_distinct [code]:

   "irrefl r \<longleftrightarrow> (\<forall>(x, y) \<in> r. x \<noteq> y)"

   by (auto simp add: irrefl_def)

 subsubsection {* Symmetry *}

 definition sym :: "'a rel \<Rightarrow> bool"

 where

   "sym r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r)"

 definition symp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"

 where

   "symp r \<longleftrightarrow> (\<forall>x y. r x y \<longrightarrow> r y x)"

 lemma symp_sym_eq [pred_set_conv]:

   "symp (\<lambda>x y. (x, y) \<in> r) \<longleftrightarrow> sym r" 

   by (simp add: sym_def symp_def)

 lemma symI:

   "(\<And>a b. (a, b) \<in> r \<Longrightarrow> (b, a) \<in> r) \<Longrightarrow> sym r"

   by (unfold sym_def) iprover

 lemma sympI:

   "(\<And>a b. r a b \<Longrightarrow> r b a) \<Longrightarrow> symp r"

   by (fact symI [to_pred])

 lemma symE:

   assumes "sym r" and "(b, a) \<in> r"

   obtains "(a, b) \<in> r"

   using assms by (simp add: sym_def)

 lemma sympE:

   assumes "symp r" and "r b a"

   obtains "r a b"

   using assms by (rule symE [to_pred])

 lemma symD:

   assumes "sym r" and "(b, a) \<in> r"

   shows "(a, b) \<in> r"

   using assms by (rule symE)

 lemma sympD:

   assumes "symp r" and "r b a"

   shows "r a b"

   using assms by (rule symD [to_pred])

 lemma sym_Int:

   "sym r \<Longrightarrow> sym s \<Longrightarrow> sym (r \<inter> s)"

   by (fast intro: symI elim: symE)

 lemma symp_inf:

   "symp r \<Longrightarrow> symp s \<Longrightarrow> symp (r \<sqinter> s)"

   by (fact sym_Int [to_pred])

 lemma sym_Un:

   "sym r \<Longrightarrow> sym s \<Longrightarrow> sym (r \<union> s)"

   by (fast intro: symI elim: symE)

 lemma symp_sup:

   "symp r \<Longrightarrow> symp s \<Longrightarrow> symp (r \<squnion> s)"

   by (fact sym_Un [to_pred])

 lemma sym_INTER:

   "\<forall>x\<in>S. sym (r x) \<Longrightarrow> sym (INTER S r)"

   by (fast intro: symI elim: symE)

 (* FIXME thm sym_INTER [to_pred] *)

 lemma sym_UNION:

   "\<forall>x\<in>S. sym (r x) \<Longrightarrow> sym (UNION S r)"

   by (fast intro: symI elim: symE)

 (* FIXME thm sym_UNION [to_pred] *)

 subsubsection {* Antisymmetry *}

 definition antisym :: "'a rel \<Rightarrow> bool"

 where

   "antisym r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r \<longrightarrow> x = y)"

 abbreviation antisymP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"

 where

   "antisymP r \<equiv> antisym {(x, y). r x y}"

 lemma antisymI:

   "(!!x y. (x, y) : r ==> (y, x) : r ==> x=y) ==> antisym r"

   by (unfold antisym_def) iprover

 lemma antisymD: "antisym r ==> (a, b) : r ==> (b, a) : r ==> a = b"

   by (unfold antisym_def) iprover

 lemma antisym_subset: "r \<subseteq> s ==> antisym s ==> antisym r"

   by (unfold antisym_def) blast

 lemma antisym_empty [simp]: "antisym {}"

   by (unfold antisym_def) blast

 subsubsection {* Transitivity *}

 definition trans :: "'a rel \<Rightarrow> bool"

 where

   "trans r \<longleftrightarrow> (\<forall>x y z. (x, y) \<in> r \<longrightarrow> (y, z) \<in> r \<longrightarrow> (x, z) \<in> r)"

 definition transp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"

 where

   "transp r \<longleftrightarrow> (\<forall>x y z. r x y \<longrightarrow> r y z \<longrightarrow> r x z)"

 lemma transp_trans_eq [pred_set_conv]:

   "transp (\<lambda>x y. (x, y) \<in> r) \<longleftrightarrow> trans r" 

   by (simp add: trans_def transp_def)

 abbreviation transP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"

 where -- {* FIXME drop *}

   "transP r \<equiv> trans {(x, y). r x y}"

 lemma transI:

   "(\<And>x y z. (x, y) \<in> r \<Longrightarrow> (y, z) \<in> r \<Longrightarrow> (x, z) \<in> r) \<Longrightarrow> trans r"

   by (unfold trans_def) iprover

 lemma transpI:

   "(\<And>x y z. r x y \<Longrightarrow> r y z \<Longrightarrow> r x z) \<Longrightarrow> transp r"

   by (fact transI [to_pred])

 lemma transE:

   assumes "trans r" and "(x, y) \<in> r" and "(y, z) \<in> r"

   obtains "(x, z) \<in> r"

   using assms by (unfold trans_def) iprover

 lemma transpE:

   assumes "transp r" and "r x y" and "r y z"

   obtains "r x z"

   using assms by (rule transE [to_pred])

 lemma transD:

   assumes "trans r" and "(x, y) \<in> r" and "(y, z) \<in> r"

   shows "(x, z) \<in> r"

   using assms by (rule transE)

 lemma transpD:

   assumes "transp r" and "r x y" and "r y z"

   shows "r x z"

   using assms by (rule transD [to_pred])

 lemma trans_Int:

   "trans r \<Longrightarrow> trans s \<Longrightarrow> trans (r \<inter> s)"

   by (fast intro: transI elim: transE)

 lemma transp_inf:

   "transp r \<Longrightarrow> transp s \<Longrightarrow> transp (r \<sqinter> s)"

   by (fact trans_Int [to_pred])

 lemma trans_INTER:

   "\<forall>x\<in>S. trans (r x) \<Longrightarrow> trans (INTER S r)"

   by (fast intro: transI elim: transD)

 (* FIXME thm trans_INTER [to_pred] *)

 lemma trans_join [code]:

   "trans r \<longleftrightarrow> (\<forall>(x, y1) \<in> r. \<forall>(y2, z) \<in> r. y1 = y2 \<longrightarrow> (x, z) \<in> r)"

   by (auto simp add: trans_def)

 lemma transp_trans:

   "transp r \<longleftrightarrow> trans {(x, y). r x y}"

   by (simp add: trans_def transp_def)

 subsubsection {* Totality *}

 definition total_on :: "'a set \<Rightarrow> 'a rel \<Rightarrow> bool"

 where

   "total_on A r \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>A. x \<noteq> y \<longrightarrow> (x, y) \<in> r \<or> (y, x) \<in> r)"

 abbreviation "total \<equiv> total_on UNIV"

 lemma total_on_empty [simp]: "total_on {} r"

   by (simp add: total_on_def)

 subsubsection {* Single valued relations *}

 definition single_valued :: "('a \<times> 'b) set \<Rightarrow> bool"

 where

   "single_valued r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (\<forall>z. (x, z) \<in> r \<longrightarrow> y = z))"

 abbreviation single_valuedP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where

   "single_valuedP r \<equiv> single_valued {(x, y). r x y}"

 lemma single_valuedI:

   "ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z) ==> single_valued r"

   by (unfold single_valued_def)

 lemma single_valuedD:

   "single_valued r ==> (x, y) : r ==> (x, z) : r ==> y = z"

   by (simp add: single_valued_def)

 lemma single_valued_subset:

   "r \<subseteq> s ==> single_valued s ==> single_valued r"

   by (unfold single_valued_def) blast

 subsection {* Relation operations *}

 subsubsection {* The identity relation *}

 definition Id :: "'a rel"

 where

   "Id = {p. \<exists>x. p = (x, x)}"

 lemma IdI [intro]: "(a, a) : Id"

   by (simp add: Id_def)

 lemma IdE [elim!]: "p : Id ==> (!!x. p = (x, x) ==> P) ==> P"

   by (unfold Id_def) (iprover elim: CollectE)

 lemma pair_in_Id_conv [iff]: "((a, b) : Id) = (a = b)"

   by (unfold Id_def) blast

 lemma refl_Id: "refl Id"

   by (simp add: refl_on_def)

 lemma antisym_Id: "antisym Id"

   -- {* A strange result, since @{text Id} is also symmetric. *}

   by (simp add: antisym_def)

 lemma sym_Id: "sym Id"

   by (simp add: sym_def)

 lemma trans_Id: "trans Id"

   by (simp add: trans_def)

 lemma single_valued_Id [simp]: "single_valued Id"

   by (unfold single_valued_def) blast

 lemma irrefl_diff_Id [simp]: "irrefl (r - Id)"

   by (simp add:irrefl_def)

 lemma trans_diff_Id: "trans r \<Longrightarrow> antisym r \<Longrightarrow> trans (r - Id)"

   unfolding antisym_def trans_def by blast

 lemma total_on_diff_Id [simp]: "total_on A (r - Id) = total_on A r"

   by (simp add: total_on_def)

 subsubsection {* Diagonal: identity over a set *}

 definition Id_on  :: "'a set \<Rightarrow> 'a rel"

 where

   "Id_on A = (\<Union>x\<in>A. {(x, x)})"

 lemma Id_on_empty [simp]: "Id_on {} = {}"

   by (simp add: Id_on_def) 

 lemma Id_on_eqI: "a = b ==> a : A ==> (a, b) : Id_on A"

   by (simp add: Id_on_def)

 lemma Id_onI [intro!,no_atp]: "a : A ==> (a, a) : Id_on A"

   by (rule Id_on_eqI) (rule refl)

 lemma Id_onE [elim!]:

   "c : Id_on A ==> (!!x. x : A ==> c = (x, x) ==> P) ==> P"

   -- {* The general elimination rule. *}

   by (unfold Id_on_def) (iprover elim!: UN_E singletonE)

 lemma Id_on_iff: "((x, y) : Id_on A) = (x = y & x : A)"

   by blast

 lemma Id_on_def' [nitpick_unfold]:

   "Id_on {x. A x} = Collect (\<lambda>(x, y). x = y \<and> A x)"

   by auto

 lemma Id_on_subset_Times: "Id_on A \<subseteq> A \<times> A"

   by blast

 lemma refl_on_Id_on: "refl_on A (Id_on A)"

   by (rule refl_onI [OF Id_on_subset_Times Id_onI])

 lemma antisym_Id_on [simp]: "antisym (Id_on A)"

   by (unfold antisym_def) blast

 lemma sym_Id_on [simp]: "sym (Id_on A)"

   by (rule symI) clarify

 lemma trans_Id_on [simp]: "trans (Id_on A)"

   by (fast intro: transI elim: transD)

 lemma single_valued_Id_on [simp]: "single_valued (Id_on A)"

   by (unfold single_valued_def) blast

 subsubsection {* Composition *}

 inductive_set rel_comp  :: "('a \<times> 'b) set \<Rightarrow> ('b \<times> 'c) set \<Rightarrow> ('a \<times> 'c) set" (infixr "O" 75)

   for r :: "('a \<times> 'b) set" and s :: "('b \<times> 'c) set"

 where

   rel_compI [intro]: "(a, b) \<in> r \<Longrightarrow> (b, c) \<in> s \<Longrightarrow> (a, c) \<in> r O s"

 abbreviation pred_comp (infixr "OO" 75) where

   "pred_comp \<equiv> rel_compp"

 lemmas pred_compI = rel_compp.intros

 text {*

   For historic reasons, the elimination rules are not wholly corresponding.

   Feel free to consolidate this.

 *}

 inductive_cases rel_compEpair: "(a, c) \<in> r O s"

 inductive_cases pred_compE [elim!]: "(r OO s) a c"

 lemma rel_compE [elim!]: "xz \<in> r O s \<Longrightarrow>

   (\<And>x y z. xz = (x, z) \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> (y, z) \<in> s  \<Longrightarrow> P) \<Longrightarrow> P"

   by (cases xz) (simp, erule rel_compEpair, iprover)

 lemmas pred_comp_rel_comp_eq = rel_compp_rel_comp_eq

 lemma R_O_Id [simp]:

   "R O Id = R"

   by fast

 lemma Id_O_R [simp]:

   "Id O R = R"

   by fast

 lemma rel_comp_empty1 [simp]:

   "{} O R = {}"

   by blast

 (* CANDIDATE lemma pred_comp_bot1 [simp]:

   ""

   by (fact rel_comp_empty1 [to_pred]) *)

 lemma rel_comp_empty2 [simp]:

   "R O {} = {}"

   by blast

 (* CANDIDATE lemma pred_comp_bot2 [simp]:

   ""

   by (fact rel_comp_empty2 [to_pred]) *)

 lemma O_assoc:

   "(R O S) O T = R O (S O T)"

   by blast

 lemma pred_comp_assoc:

   "(r OO s) OO t = r OO (s OO t)"

   by (fact O_assoc [to_pred])

 lemma trans_O_subset:

   "trans r \<Longrightarrow> r O r \<subseteq> r"

   by (unfold trans_def) blast

 lemma transp_pred_comp_less_eq:

   "transp r \<Longrightarrow> r OO r \<le> r "

   by (fact trans_O_subset [to_pred])

 lemma rel_comp_mono:

   "r' \<subseteq> r \<Longrightarrow> s' \<subseteq> s \<Longrightarrow> r' O s' \<subseteq> r O s"

   by blast

 lemma pred_comp_mono:

   "r' \<le> r \<Longrightarrow> s' \<le> s \<Longrightarrow> r' OO s' \<le> r OO s "

   by (fact rel_comp_mono [to_pred])

 lemma rel_comp_subset_Sigma:

   "r \<subseteq> A \<times> B \<Longrightarrow> s \<subseteq> B \<times> C \<Longrightarrow> r O s \<subseteq> A \<times> C"

   by blast

 lemma rel_comp_distrib [simp]:

   "R O (S \<union> T) = (R O S) \<union> (R O T)" 

   by auto

 lemma pred_comp_distrib (* CANDIDATE [simp] *):

   "R OO (S \<squnion> T) = R OO S \<squnion> R OO T"

   by (fact rel_comp_distrib [to_pred])

 lemma rel_comp_distrib2 [simp]:

   "(S \<union> T) O R = (S O R) \<union> (T O R)"

   by auto

 lemma pred_comp_distrib2 (* CANDIDATE [simp] *):

   "(S \<squnion> T) OO R = S OO R \<squnion> T OO R"

   by (fact rel_comp_distrib2 [to_pred])

 lemma rel_comp_UNION_distrib:

   "s O UNION I r = (\<Union>i\<in>I. s O r i) "

   by auto

 (* FIXME thm rel_comp_UNION_distrib [to_pred] *)

 lemma rel_comp_UNION_distrib2:

   "UNION I r O s = (\<Union>i\<in>I. r i O s) "

   by auto

 (* FIXME thm rel_comp_UNION_distrib2 [to_pred] *)

 lemma single_valued_rel_comp:

   "single_valued r \<Longrightarrow> single_valued s \<Longrightarrow> single_valued (r O s)"

   by (unfold single_valued_def) blast

 lemma rel_comp_unfold:

   "r O s = {(x, z). \<exists>y. (x, y) \<in> r \<and> (y, z) \<in> s}"

   by (auto simp add: set_eq_iff)

 subsubsection {* Converse *}

 inductive_set converse :: "('a \<times> 'b) set \<Rightarrow> ('b \<times> 'a) set" ("(_^-1)" [1000] 999)

   for r :: "('a \<times> 'b) set"

 where

   "(a, b) \<in> r \<Longrightarrow> (b, a) \<in> r^-1"

 notation (xsymbols)

   converse  ("(_\<inverse>)" [1000] 999)

 notation

   conversep ("(_^--1)" [1000] 1000)

 notation (xsymbols)

   conversep  ("(_\<inverse>\<inverse>)" [1000] 1000)

 lemma converseI [sym]:

   "(a, b) \<in> r \<Longrightarrow> (b, a) \<in> r\<inverse>"

   by (fact converse.intros)

 lemma conversepI (* CANDIDATE [sym] *):

   "r a b \<Longrightarrow> r\<inverse>\<inverse> b a"

   by (fact conversep.intros)

 lemma converseD [sym]:

   "(a, b) \<in> r\<inverse> \<Longrightarrow> (b, a) \<in> r"

   by (erule converse.cases) iprover

 lemma conversepD (* CANDIDATE [sym] *):

   "r\<inverse>\<inverse> b a \<Longrightarrow> r a b"

   by (fact converseD [to_pred])

 lemma converseE [elim!]:

   -- {* More general than @{text converseD}, as it ``splits'' the member of the relation. *}

   "yx \<in> r\<inverse> \<Longrightarrow> (\<And>x y. yx = (y, x) \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> P) \<Longrightarrow> P"

   by (cases yx) (simp, erule converse.cases, iprover)

 lemmas conversepE (* CANDIDATE [elim!] *) = conversep.cases

 lemma converse_iff [iff]:

   "(a, b) \<in> r\<inverse> \<longleftrightarrow> (b, a) \<in> r"

   by (auto intro: converseI)

 lemma conversep_iff [iff]:

   "r\<inverse>\<inverse> a b = r b a"

   by (fact converse_iff [to_pred])

 lemma converse_converse [simp]:

   "(r\<inverse>)\<inverse> = r"

   by (simp add: set_eq_iff)

 lemma conversep_conversep [simp]:

   "(r\<inverse>\<inverse>)\<inverse>\<inverse> = r"

   by (fact converse_converse [to_pred])

 lemma converse_rel_comp: "(r O s)^-1 = s^-1 O r^-1"

   by blast

 lemma converse_pred_comp: "(r OO s)^--1 = s^--1 OO r^--1"

   by (iprover intro: order_antisym conversepI pred_compI

     elim: pred_compE dest: conversepD)

 lemma converse_Int: "(r \<inter> s)^-1 = r^-1 \<inter> s^-1"

   by blast

 lemma converse_meet: "(r \<sqinter> s)^--1 = r^--1 \<sqinter> s^--1"

   by (simp add: inf_fun_def) (iprover intro: conversepI ext dest: conversepD)

 lemma converse_Un: "(r \<union> s)^-1 = r^-1 \<union> s^-1"

   by blast

 lemma converse_join: "(r \<squnion> s)^--1 = r^--1 \<squnion> s^--1"

   by (simp add: sup_fun_def) (iprover intro: conversepI ext dest: conversepD)

 lemma converse_INTER: "(INTER S r)^-1 = (INT x:S. (r x)^-1)"

   by fast

 lemma converse_UNION: "(UNION S r)^-1 = (UN x:S. (r x)^-1)"

   by blast

 lemma converse_Id [simp]: "Id^-1 = Id"

   by blast

 lemma converse_Id_on [simp]: "(Id_on A)^-1 = Id_on A"

   by blast

 lemma refl_on_converse [simp]: "refl_on A (converse r) = refl_on A r"

   by (unfold refl_on_def) auto

 lemma sym_converse [simp]: "sym (converse r) = sym r"

   by (unfold sym_def) blast

 lemma antisym_converse [simp]: "antisym (converse r) = antisym r"

   by (unfold antisym_def) blast

 lemma trans_converse [simp]: "trans (converse r) = trans r"

   by (unfold trans_def) blast

 lemma sym_conv_converse_eq: "sym r = (r^-1 = r)"

   by (unfold sym_def) fast

 lemma sym_Un_converse: "sym (r \<union> r^-1)"

   by (unfold sym_def) blast

 lemma sym_Int_converse: "sym (r \<inter> r^-1)"

   by (unfold sym_def) blast

 lemma total_on_converse [simp]: "total_on A (r^-1) = total_on A r"

   by (auto simp: total_on_def)

 lemma finite_converse [iff]: "finite (r^-1) = finite r"

   apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r")

    apply simp

    apply (rule iffI)

     apply (erule finite_imageD [unfolded inj_on_def])

     apply (simp split add: split_split)

    apply (erule finite_imageI)

   apply (simp add: set_eq_iff image_def, auto)

   apply (rule bexI)

    prefer 2 apply assumption

   apply simp

   done

 lemma conversep_noteq [simp]: "(op \<noteq>)^--1 = op \<noteq>"

   by (auto simp add: fun_eq_iff)

 lemma conversep_eq [simp]: "(op =)^--1 = op ="

   by (auto simp add: fun_eq_iff)

 lemma converse_unfold:

   "r\<inverse> = {(y, x). (x, y) \<in> r}"

   by (simp add: set_eq_iff)

 subsubsection {* Domain, range and field *}

 definition Domain :: "('a \<times> 'b) set \<Rightarrow> 'a set"

 where

   "Domain r = {x. \<exists>y. (x, y) \<in> r}"

 definition Range  :: "('a \<times> 'b) set \<Rightarrow> 'b set"

 where

   "Range r = Domain (r\<inverse>)"

 definition Field :: "'a rel \<Rightarrow> 'a set"

 where

   "Field r = Domain r \<union> Range r"

 declare Domain_def [no_atp]

 lemma Domain_iff: "(a : Domain r) = (EX y. (a, y) : r)"

   by (unfold Domain_def) blast

 lemma DomainI [intro]: "(a, b) : r ==> a : Domain r"

   by (iprover intro!: iffD2 [OF Domain_iff])

 lemma DomainE [elim!]:

   "a : Domain r ==> (!!y. (a, y) : r ==> P) ==> P"

   by (iprover dest!: iffD1 [OF Domain_iff])

 lemma Range_iff: "(a : Range r) = (EX y. (y, a) : r)"

   by (simp add: Domain_def Range_def)

 lemma RangeI [intro]: "(a, b) : r ==> b : Range r"

   by (unfold Range_def) (iprover intro!: converseI DomainI)

 lemma RangeE [elim!]: "b : Range r ==> (!!x. (x, b) : r ==> P) ==> P"

   by (unfold Range_def) (iprover elim!: DomainE dest!: converseD)

 inductive DomainP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"

   for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"

 where

   DomainPI [intro]: "r a b \<Longrightarrow> DomainP r a"

 inductive_cases DomainPE [elim!]: "DomainP r a"

 lemma DomainP_Domain_eq [pred_set_conv]: "DomainP (\<lambda>x y. (x, y) \<in> r) = (\<lambda>x. x \<in> Domain r)"

   by (blast intro!: Orderings.order_antisym predicate1I)

 inductive RangeP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"

   for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"

 where

   RangePI [intro]: "r a b \<Longrightarrow> RangeP r b"

 inductive_cases RangePE [elim!]: "RangeP r b"

 lemma RangeP_Range_eq [pred_set_conv]: "RangeP (\<lambda>x y. (x, y) \<in> r) = (\<lambda>x. x \<in> Range r)"

   by (auto intro!: Orderings.order_antisym predicate1I)

 lemma Domain_fst [code]:

   "Domain r = fst ` r"

   by (auto simp add: image_def Bex_def)

 lemma Domain_empty [simp]: "Domain {} = {}"

   by blast

 lemma Domain_empty_iff: "Domain r = {} \<longleftrightarrow> r = {}"

   by auto

 lemma Domain_insert: "Domain (insert (a, b) r) = insert a (Domain r)"

   by blast

 lemma Domain_Id [simp]: "Domain Id = UNIV"

   by blast

 lemma Domain_Id_on [simp]: "Domain (Id_on A) = A"

   by blast

 lemma Domain_Un_eq: "Domain(A \<union> B) = Domain(A) \<union> Domain(B)"

   by blast

 lemma Domain_Int_subset: "Domain(A \<inter> B) \<subseteq> Domain(A) \<inter> Domain(B)"

   by blast

 lemma Domain_Diff_subset: "Domain(A) - Domain(B) \<subseteq> Domain(A - B)"

   by blast

 lemma Domain_Union: "Domain (Union S) = (\<Union>A\<in>S. Domain A)"

   by blast

 lemma Domain_converse [simp]: "Domain (r\<inverse>) = Range r"

   by auto

 lemma Domain_mono: "r \<subseteq> s ==> Domain r \<subseteq> Domain s"

   by blast

 lemma fst_eq_Domain: "fst ` R = Domain R"

   by force

 lemma Domain_dprod [simp]: "Domain (dprod r s) = uprod (Domain r) (Domain s)"

   by auto

 lemma Domain_dsum [simp]: "Domain (dsum r s) = usum (Domain r) (Domain s)"

   by auto

 lemma Domain_Collect_split [simp]: "Domain {(x, y). P x y} = {x. EX y. P x y}"

   by auto

 lemma finite_Domain: "finite r ==> finite (Domain r)"

   by (induct set: finite) (auto simp add: Domain_insert)

 lemma Range_snd [code]:

   "Range r = snd ` r"

   by (auto simp add: image_def Bex_def)

 lemma Range_empty [simp]: "Range {} = {}"

   by blast

 lemma Range_empty_iff: "Range r = {} \<longleftrightarrow> r = {}"

   by auto

 lemma Range_insert: "Range (insert (a, b) r) = insert b (Range r)"

   by blast

 lemma Range_Id [simp]: "Range Id = UNIV"

   by blast

 lemma Range_Id_on [simp]: "Range (Id_on A) = A"

   by auto

 lemma Range_Un_eq: "Range(A \<union> B) = Range(A) \<union> Range(B)"

   by blast

 lemma Range_Int_subset: "Range(A \<inter> B) \<subseteq> Range(A) \<inter> Range(B)"

   by blast

 lemma Range_Diff_subset: "Range(A) - Range(B) \<subseteq> Range(A - B)"

   by blast

 lemma Range_Union: "Range (Union S) = (\<Union>A\<in>S. Range A)"

   by blast

 lemma Range_converse [simp]: "Range (r\<inverse>) = Domain r"

   by blast

 lemma snd_eq_Range: "snd ` R = Range R"

   by force

 lemma Range_Collect_split [simp]: "Range {(x, y). P x y} = {y. EX x. P x y}"

   by auto

 lemma finite_Range: "finite r ==> finite (Range r)"

   by (induct set: finite) (auto simp add: Range_insert)

 lemma mono_Field: "r \<subseteq> s \<Longrightarrow> Field r \<subseteq> Field s"

   by (auto simp: Field_def Domain_def Range_def)

 lemma Field_empty[simp]: "Field {} = {}"

   by (auto simp: Field_def)

 lemma Field_insert [simp]: "Field (insert (a, b) r) = {a, b} \<union> Field r"

   by (auto simp: Field_def)

 lemma Field_Un [simp]: "Field (r \<union> s) = Field r \<union> Field s"

   by (auto simp: Field_def)

 lemma Field_Union [simp]: "Field (\<Union>R) = \<Union>(Field ` R)"

   by (auto simp: Field_def)

 lemma Field_converse [simp]: "Field(r^-1) = Field r"

   by (auto simp: Field_def)

 lemma finite_Field: "finite r ==> finite (Field r)"

   -- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *}

   apply (induct set: finite)

    apply (auto simp add: Field_def Domain_insert Range_insert)

   done

 lemma Domain_unfold:

   "Domain r = {x. \<exists>y. (x, y) \<in> r}"

   by (fact Domain_def)

 subsubsection {* Image of a set under a relation *}

 definition Image :: "('a \<times> 'b) set \<Rightarrow> 'a set \<Rightarrow> 'b set" (infixl "``" 90)

 where

   "r `` s = {y. \<exists>x\<in>s. (x, y) \<in> r}"

 declare Image_def [no_atp]

 lemma Image_iff: "(b : r``A) = (EX x:A. (x, b) : r)"

   by (simp add: Image_def)

 lemma Image_singleton: "r``{a} = {b. (a, b) : r}"

   by (simp add: Image_def)

 lemma Image_singleton_iff [iff]: "(b : r``{a}) = ((a, b) : r)"

   by (rule Image_iff [THEN trans]) simp

 lemma ImageI [intro,no_atp]: "(a, b) : r ==> a : A ==> b : r``A"

   by (unfold Image_def) blast

 lemma ImageE [elim!]:

   "b : r `` A ==> (!!x. (x, b) : r ==> x : A ==> P) ==> P"

   by (unfold Image_def) (iprover elim!: CollectE bexE)

 lemma rev_ImageI: "a : A ==> (a, b) : r ==> b : r `` A"

   -- {* This version's more effective when we already have the required @{text a} *}

   by blast

 lemma Image_empty [simp]: "R``{} = {}"

   by blast

 lemma Image_Id [simp]: "Id `` A = A"

   by blast

 lemma Image_Id_on [simp]: "Id_on A `` B = A \<inter> B"

   by blast

 lemma Image_Int_subset: "R `` (A \<inter> B) \<subseteq> R `` A \<inter> R `` B"

   by blast

 lemma Image_Int_eq:

      "single_valued (converse R) ==> R `` (A \<inter> B) = R `` A \<inter> R `` B"

      by (simp add: single_valued_def, blast) 

 lemma Image_Un: "R `` (A \<union> B) = R `` A \<union> R `` B"

   by blast

 lemma Un_Image: "(R \<union> S) `` A = R `` A \<union> S `` A"

   by blast

 lemma Image_subset: "r \<subseteq> A \<times> B ==> r``C \<subseteq> B"

   by (iprover intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2)

 lemma Image_eq_UN: "r``B = (\<Union>y\<in> B. r``{y})"

   -- {* NOT suitable for rewriting *}

   by blast

 lemma Image_mono: "r' \<subseteq> r ==> A' \<subseteq> A ==> (r' `` A') \<subseteq> (r `` A)"

   by blast

 lemma Image_UN: "(r `` (UNION A B)) = (\<Union>x\<in>A. r `` (B x))"

   by blast

 lemma Image_INT_subset: "(r `` INTER A B) \<subseteq> (\<Inter>x\<in>A. r `` (B x))"

   by blast

 text{*Converse inclusion requires some assumptions*}

 lemma Image_INT_eq:

      "[|single_valued (r\<inverse>); A\<noteq>{}|] ==> r `` INTER A B = (\<Inter>x\<in>A. r `` B x)"

 apply (rule equalityI)

  apply (rule Image_INT_subset) 

 apply  (simp add: single_valued_def, blast)

 done

 lemma Image_subset_eq: "(r``A \<subseteq> B) = (A \<subseteq> - ((r^-1) `` (-B)))"

   by blast

 lemma Image_Collect_split [simp]: "{(x, y). P x y} `` A = {y. EX x:A. P x y}"

   by auto

 subsubsection {* Inverse image *}

 definition inv_image :: "'b rel \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a rel"

 where

   "inv_image r f = {(x, y). (f x, f y) \<in> r}"

 definition inv_imagep :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"

 where

   "inv_imagep r f = (\<lambda>x y. r (f x) (f y))"

 lemma [pred_set_conv]: "inv_imagep (\<lambda>x y. (x, y) \<in> r) f = (\<lambda>x y. (x, y) \<in> inv_image r f)"

   by (simp add: inv_image_def inv_imagep_def)

 lemma sym_inv_image: "sym r ==> sym (inv_image r f)"

   by (unfold sym_def inv_image_def) blast

 lemma trans_inv_image: "trans r ==> trans (inv_image r f)"

   apply (unfold trans_def inv_image_def)

   apply (simp (no_asm))

   apply blast

   done

 lemma in_inv_image[simp]: "((x,y) : inv_image r f) = ((f x, f y) : r)"

   by (auto simp:inv_image_def)

 lemma converse_inv_image[simp]: "(inv_image R f)^-1 = inv_image (R^-1) f"

   unfolding inv_image_def converse_unfold by auto

 lemma in_inv_imagep [simp]: "inv_imagep r f x y = r (f x) (f y)"

   by (simp add: inv_imagep_def)

 subsubsection {* Powerset *}

 definition Powp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"

 where

   "Powp A = (\<lambda>B. \<forall>x \<in> B. A x)"

 lemma Powp_Pow_eq [pred_set_conv]: "Powp (\<lambda>x. x \<in> A) = (\<lambda>x. x \<in> Pow A)"

   by (auto simp add: Powp_def fun_eq_iff)

 lemmas Powp_mono [mono] = Pow_mono [to_pred]

 end