(*  Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel *)

 section {* Set theory for higher-order logic *}

 theory Set

 imports Lattices

 begin

 subsection {* Sets as predicates *}

 typedecl 'a set

 axiomatization Collect :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set" -- "comprehension"

   and member :: "'a \<Rightarrow> 'a set \<Rightarrow> bool" -- "membership"

 where

   mem_Collect_eq [iff, code_unfold]: "member a (Collect P) = P a"

   and Collect_mem_eq [simp]: "Collect (\<lambda>x. member x A) = A"

 notation

   member  ("op :") and

   member  ("(_/ : _)" [51, 51] 50)

 abbreviation not_member where

   "not_member x A \<equiv> ~ (x : A)" -- "non-membership"

 notation

   not_member  ("op ~:") and

   not_member  ("(_/ ~: _)" [51, 51] 50)

 notation (xsymbols)

   member      ("op \<in>") and

   member      ("(_/ \<in> _)" [51, 51] 50) and

   not_member  ("op \<notin>") and

   not_member  ("(_/ \<notin> _)" [51, 51] 50)

 notation (HTML output)

   member      ("op \<in>") and

   member      ("(_/ \<in> _)" [51, 51] 50) and

   not_member  ("op \<notin>") and

   not_member  ("(_/ \<notin> _)" [51, 51] 50)

 text {* Set comprehensions *}

 syntax

   "_Coll" :: "pttrn => bool => 'a set"    ("(1{_./ _})")

 translations

   "{x. P}" == "CONST Collect (%x. P)"

 syntax

   "_Collect" :: "pttrn => 'a set => bool => 'a set"    ("(1{_ :/ _./ _})")

 syntax (xsymbols)

   "_Collect" :: "pttrn => 'a set => bool => 'a set"    ("(1{_ \<in>/ _./ _})")

 translations

   "{p:A. P}" => "CONST Collect (%p. p:A & P)"

 lemma CollectI: "P a \<Longrightarrow> a \<in> {x. P x}"

   by simp

 lemma CollectD: "a \<in> {x. P x} \<Longrightarrow> P a"

   by simp

 lemma Collect_cong: "(\<And>x. P x = Q x) ==> {x. P x} = {x. Q x}"

   by simp

 text {*

 Simproc for pulling @{text "x=t"} in @{text "{x. \<dots> & x=t & \<dots>}"}

 to the front (and similarly for @{text "t=x"}):

 *}

 simproc_setup defined_Collect ("{x. P x & Q x}") = {*

   fn _ => Quantifier1.rearrange_Collect

     (fn ctxt =>

       resolve_tac ctxt @{thms Collect_cong} 1 THEN

       resolve_tac ctxt @{thms iffI} 1 THEN

       ALLGOALS

         (EVERY' [REPEAT_DETERM o eresolve_tac ctxt @{thms conjE},

           DEPTH_SOLVE_1 o (assume_tac ctxt ORELSE' resolve_tac ctxt @{thms conjI})]))

 *}

 lemmas CollectE = CollectD [elim_format]

 lemma set_eqI:

   assumes "\<And>x. x \<in> A \<longleftrightarrow> x \<in> B"

   shows "A = B"

 proof -

   from assms have "{x. x \<in> A} = {x. x \<in> B}" by simp

   then show ?thesis by simp

 qed

 lemma set_eq_iff:

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

   by (auto intro:set_eqI)

 text {* Lifting of predicate class instances *}

 instantiation set :: (type) boolean_algebra

 begin

 definition less_eq_set where

   "A \<le> B \<longleftrightarrow> (\<lambda>x. member x A) \<le> (\<lambda>x. member x B)"

 definition less_set where

   "A < B \<longleftrightarrow> (\<lambda>x. member x A) < (\<lambda>x. member x B)"

 definition inf_set where

   "A \<sqinter> B = Collect ((\<lambda>x. member x A) \<sqinter> (\<lambda>x. member x B))"

 definition sup_set where

   "A \<squnion> B = Collect ((\<lambda>x. member x A) \<squnion> (\<lambda>x. member x B))"

 definition bot_set where

   "\<bottom> = Collect \<bottom>"

 definition top_set where

   "\<top> = Collect \<top>"

 definition uminus_set where

   "- A = Collect (- (\<lambda>x. member x A))"

 definition minus_set where

   "A - B = Collect ((\<lambda>x. member x A) - (\<lambda>x. member x B))"

 instance proof

 qed (simp_all add: less_eq_set_def less_set_def inf_set_def sup_set_def

   bot_set_def top_set_def uminus_set_def minus_set_def

   less_le_not_le inf_compl_bot sup_compl_top sup_inf_distrib1 diff_eq

   set_eqI fun_eq_iff

   del: inf_apply sup_apply bot_apply top_apply minus_apply uminus_apply)

 end

 text {* Set enumerations *}

 abbreviation empty :: "'a set" ("{}") where

   "{} \<equiv> bot"

 definition insert :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set" where

   insert_compr: "insert a B = {x. x = a \<or> x \<in> B}"

 syntax

   "_Finset" :: "args => 'a set"    ("{(_)}")

 translations

   "{x, xs}" == "CONST insert x {xs}"

   "{x}" == "CONST insert x {}"

 subsection {* Subsets and bounded quantifiers *}

 abbreviation

   subset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where

   "subset \<equiv> less"

 abbreviation

   subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where

   "subset_eq \<equiv> less_eq"

 notation (output)

   subset  ("op <") and

   subset  ("(_/ < _)" [51, 51] 50) and

   subset_eq  ("op <=") and

   subset_eq  ("(_/ <= _)" [51, 51] 50)

 notation (xsymbols)

   subset  ("op \<subset>") and

   subset  ("(_/ \<subset> _)" [51, 51] 50) and

   subset_eq  ("op \<subseteq>") and

   subset_eq  ("(_/ \<subseteq> _)" [51, 51] 50)

 notation (HTML output)

   subset  ("op \<subset>") and

   subset  ("(_/ \<subset> _)" [51, 51] 50) and

   subset_eq  ("op \<subseteq>") and

   subset_eq  ("(_/ \<subseteq> _)" [51, 51] 50)

 abbreviation (input)

   supset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where

   "supset \<equiv> greater"

 abbreviation (input)

   supset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where

   "supset_eq \<equiv> greater_eq"

 notation (xsymbols)

   supset  ("op \<supset>") and

   supset  ("(_/ \<supset> _)" [51, 51] 50) and

   supset_eq  ("op \<supseteq>") and

   supset_eq  ("(_/ \<supseteq> _)" [51, 51] 50)

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

   "Ball A P \<longleftrightarrow> (\<forall>x. x \<in> A \<longrightarrow> P x)"   -- "bounded universal quantifiers"

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

   "Bex A P \<longleftrightarrow> (\<exists>x. x \<in> A \<and> P x)"   -- "bounded existential quantifiers"

 syntax

   "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3ALL _:_./ _)" [0, 0, 10] 10)

   "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3EX _:_./ _)" [0, 0, 10] 10)

   "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3EX! _:_./ _)" [0, 0, 10] 10)

   "_Bleast"     :: "id => 'a set => bool => 'a"           ("(3LEAST _:_./ _)" [0, 0, 10] 10)

 syntax (HOL)

   "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3! _:_./ _)" [0, 0, 10] 10)

   "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3? _:_./ _)" [0, 0, 10] 10)

   "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3?! _:_./ _)" [0, 0, 10] 10)

 syntax (xsymbols)

   "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10)

   "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10)

   "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3\<exists>!_\<in>_./ _)" [0, 0, 10] 10)

   "_Bleast"     :: "id => 'a set => bool => 'a"           ("(3LEAST_\<in>_./ _)" [0, 0, 10] 10)

 syntax (HTML output)

   "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10)

   "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10)

   "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3\<exists>!_\<in>_./ _)" [0, 0, 10] 10)

 translations

   "ALL x:A. P" == "CONST Ball A (%x. P)"

   "EX x:A. P" == "CONST Bex A (%x. P)"

   "EX! x:A. P" => "EX! x. x:A & P"

   "LEAST x:A. P" => "LEAST x. x:A & P"

 syntax (output)

   "_setlessAll" :: "[idt, 'a, bool] => bool"  ("(3ALL _<_./ _)"  [0, 0, 10] 10)

   "_setlessEx"  :: "[idt, 'a, bool] => bool"  ("(3EX _<_./ _)"  [0, 0, 10] 10)

   "_setleAll"   :: "[idt, 'a, bool] => bool"  ("(3ALL _<=_./ _)" [0, 0, 10] 10)

   "_setleEx"    :: "[idt, 'a, bool] => bool"  ("(3EX _<=_./ _)" [0, 0, 10] 10)

   "_setleEx1"   :: "[idt, 'a, bool] => bool"  ("(3EX! _<=_./ _)" [0, 0, 10] 10)

 syntax (xsymbols)

   "_setlessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<subset>_./ _)"  [0, 0, 10] 10)

   "_setlessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subset>_./ _)"  [0, 0, 10] 10)

   "_setleAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<subseteq>_./ _)" [0, 0, 10] 10)

   "_setleEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10)

   "_setleEx1"   :: "[idt, 'a, bool] => bool"   ("(3\<exists>!_\<subseteq>_./ _)" [0, 0, 10] 10)

 syntax (HOL output)

   "_setlessAll" :: "[idt, 'a, bool] => bool"   ("(3! _<_./ _)"  [0, 0, 10] 10)

   "_setlessEx"  :: "[idt, 'a, bool] => bool"   ("(3? _<_./ _)"  [0, 0, 10] 10)

   "_setleAll"   :: "[idt, 'a, bool] => bool"   ("(3! _<=_./ _)" [0, 0, 10] 10)

   "_setleEx"    :: "[idt, 'a, bool] => bool"   ("(3? _<=_./ _)" [0, 0, 10] 10)

   "_setleEx1"   :: "[idt, 'a, bool] => bool"   ("(3?! _<=_./ _)" [0, 0, 10] 10)

 syntax (HTML output)

   "_setlessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<subset>_./ _)"  [0, 0, 10] 10)

   "_setlessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subset>_./ _)"  [0, 0, 10] 10)

   "_setleAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<subseteq>_./ _)" [0, 0, 10] 10)

   "_setleEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10)

   "_setleEx1"   :: "[idt, 'a, bool] => bool"   ("(3\<exists>!_\<subseteq>_./ _)" [0, 0, 10] 10)

 translations

  "\<forall>A\<subset>B. P"   =>  "ALL A. A \<subset> B --> P"

  "\<exists>A\<subset>B. P"   =>  "EX A. A \<subset> B & P"

  "\<forall>A\<subseteq>B. P"   =>  "ALL A. A \<subseteq> B --> P"

  "\<exists>A\<subseteq>B. P"   =>  "EX A. A \<subseteq> B & P"

  "\<exists>!A\<subseteq>B. P"  =>  "EX! A. A \<subseteq> B & P"

 print_translation {*

   let

     val All_binder = Mixfix.binder_name @{const_syntax All};

     val Ex_binder = Mixfix.binder_name @{const_syntax Ex};

     val impl = @{const_syntax HOL.implies};

     val conj = @{const_syntax HOL.conj};

     val sbset = @{const_syntax subset};

     val sbset_eq = @{const_syntax subset_eq};

     val trans =

      [((All_binder, impl, sbset), @{syntax_const "_setlessAll"}),

       ((All_binder, impl, sbset_eq), @{syntax_const "_setleAll"}),

       ((Ex_binder, conj, sbset), @{syntax_const "_setlessEx"}),

       ((Ex_binder, conj, sbset_eq), @{syntax_const "_setleEx"})];

     fun mk v (v', T) c n P =

       if v = v' andalso not (Term.exists_subterm (fn Free (x, _) => x = v | _ => false) n)

       then Syntax.const c $ Syntax_Trans.mark_bound_body (v', T) $ n $ P

       else raise Match;

     fun tr' q = (q, fn _ =>

       (fn [Const (@{syntax_const "_bound"}, _) $ Free (v, Type (@{type_name set}, _)),

           Const (c, _) $

             (Const (d, _) $ (Const (@{syntax_const "_bound"}, _) $ Free (v', T)) $ n) $ P] =>

           (case AList.lookup (op =) trans (q, c, d) of

             NONE => raise Match

           | SOME l => mk v (v', T) l n P)

         | _ => raise Match));

   in

     [tr' All_binder, tr' Ex_binder]

   end

 *}

 text {*

   \medskip Translate between @{text "{e | x1...xn. P}"} and @{text

   "{u. EX x1..xn. u = e & P}"}; @{text "{y. EX x1..xn. y = e & P}"} is

   only translated if @{text "[0..n] subset bvs(e)"}.

 *}

 syntax

   "_Setcompr" :: "'a => idts => bool => 'a set"    ("(1{_ |/_./ _})")

 parse_translation {*

   let

     val ex_tr = snd (Syntax_Trans.mk_binder_tr ("EX ", @{const_syntax Ex}));

     fun nvars (Const (@{syntax_const "_idts"}, _) $ _ $ idts) = nvars idts + 1

       | nvars _ = 1;

     fun setcompr_tr ctxt [e, idts, b] =

       let

         val eq = Syntax.const @{const_syntax HOL.eq} $ Bound (nvars idts) $ e;

         val P = Syntax.const @{const_syntax HOL.conj} $ eq $ b;

         val exP = ex_tr ctxt [idts, P];

       in Syntax.const @{const_syntax Collect} $ absdummy dummyT exP end;

   in [(@{syntax_const "_Setcompr"}, setcompr_tr)] end;

 *}

 print_translation {*

  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax Ball} @{syntax_const "_Ball"},

   Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax Bex} @{syntax_const "_Bex"}]

 *} -- {* to avoid eta-contraction of body *}

 print_translation {*

 let

   val ex_tr' = snd (Syntax_Trans.mk_binder_tr' (@{const_syntax Ex}, "DUMMY"));

   fun setcompr_tr' ctxt [Abs (abs as (_, _, P))] =

     let

       fun check (Const (@{const_syntax Ex}, _) $ Abs (_, _, P), n) = check (P, n + 1)

         | check (Const (@{const_syntax HOL.conj}, _) $

               (Const (@{const_syntax HOL.eq}, _) $ Bound m $ e) $ P, n) =

             n > 0 andalso m = n andalso not (loose_bvar1 (P, n)) andalso

             subset (op =) (0 upto (n - 1), add_loose_bnos (e, 0, []))

         | check _ = false;

         fun tr' (_ $ abs) =

           let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' ctxt [abs]

           in Syntax.const @{syntax_const "_Setcompr"} $ e $ idts $ Q end;

     in

       if check (P, 0) then tr' P

       else

         let

           val (x as _ $ Free(xN, _), t) = Syntax_Trans.atomic_abs_tr' abs;

           val M = Syntax.const @{syntax_const "_Coll"} $ x $ t;

         in

           case t of

             Const (@{const_syntax HOL.conj}, _) $

               (Const (@{const_syntax Set.member}, _) $

                 (Const (@{syntax_const "_bound"}, _) $ Free (yN, _)) $ A) $ P =>

             if xN = yN then Syntax.const @{syntax_const "_Collect"} $ x $ A $ P else M

           | _ => M

         end

     end;

   in [(@{const_syntax Collect}, setcompr_tr')] end;

 *}

 simproc_setup defined_Bex ("EX x:A. P x & Q x") = {*

   fn _ => Quantifier1.rearrange_bex

     (fn ctxt =>

       unfold_tac ctxt @{thms Bex_def} THEN

       Quantifier1.prove_one_point_ex_tac ctxt)

 *}

 simproc_setup defined_All ("ALL x:A. P x --> Q x") = {*

   fn _ => Quantifier1.rearrange_ball

     (fn ctxt =>

       unfold_tac ctxt @{thms Ball_def} THEN

       Quantifier1.prove_one_point_all_tac ctxt)

 *}

 lemma ballI [intro!]: "(!!x. x:A ==> P x) ==> ALL x:A. P x"

   by (simp add: Ball_def)

 lemmas strip = impI allI ballI

 lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x"

   by (simp add: Ball_def)

 text {*

   Gives better instantiation for bound:

 *}

 setup {*

   map_theory_claset (fn ctxt =>

     ctxt addbefore ("bspec", fn ctxt' => dresolve_tac ctxt' @{thms bspec} THEN' assume_tac ctxt'))

 *}

 ML {*

 structure Simpdata =

 struct

 open Simpdata;

 val mksimps_pairs = [(@{const_name Ball}, @{thms bspec})] @ mksimps_pairs;

 end;

 open Simpdata;

 *}

 declaration {* fn _ =>

   Simplifier.map_ss (Simplifier.set_mksimps (mksimps mksimps_pairs))

 *}

 lemma ballE [elim]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q"

   by (unfold Ball_def) blast

 lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x"

   -- {* Normally the best argument order: @{prop "P x"} constrains the

     choice of @{prop "x:A"}. *}

   by (unfold Bex_def) blast

 lemma rev_bexI [intro?]: "x:A ==> P x ==> EX x:A. P x"

   -- {* The best argument order when there is only one @{prop "x:A"}. *}

   by (unfold Bex_def) blast

 lemma bexCI: "(ALL x:A. ~P x ==> P a) ==> a:A ==> EX x:A. P x"

   by (unfold Bex_def) blast

 lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q"

   by (unfold Bex_def) blast

 lemma ball_triv [simp]: "(ALL x:A. P) = ((EX x. x:A) --> P)"

   -- {* Trival rewrite rule. *}

   by (simp add: Ball_def)

 lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)"

   -- {* Dual form for existentials. *}

   by (simp add: Bex_def)

 lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)"

   by blast

 lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)"

   by blast

 lemma bex_one_point1 [simp]: "(EX x:A. x = a & P x) = (a:A & P a)"

   by blast

 lemma bex_one_point2 [simp]: "(EX x:A. a = x & P x) = (a:A & P a)"

   by blast

 lemma ball_one_point1 [simp]: "(ALL x:A. x = a --> P x) = (a:A --> P a)"

   by blast

 lemma ball_one_point2 [simp]: "(ALL x:A. a = x --> P x) = (a:A --> P a)"

   by blast

 lemma ball_conj_distrib:

   "(\<forall>x\<in>A. P x \<and> Q x) \<longleftrightarrow> ((\<forall>x\<in>A. P x) \<and> (\<forall>x\<in>A. Q x))"

   by blast

 lemma bex_disj_distrib:

   "(\<exists>x\<in>A. P x \<or> Q x) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<or> (\<exists>x\<in>A. Q x))"

   by blast

 text {* Congruence rules *}

 lemma ball_cong:

   "A = B ==> (!!x. x:B ==> P x = Q x) ==>

     (ALL x:A. P x) = (ALL x:B. Q x)"

   by (simp add: Ball_def)

 lemma strong_ball_cong [cong]:

   "A = B ==> (!!x. x:B =simp=> P x = Q x) ==>

     (ALL x:A. P x) = (ALL x:B. Q x)"

   by (simp add: simp_implies_def Ball_def)

 lemma bex_cong:

   "A = B ==> (!!x. x:B ==> P x = Q x) ==>

     (EX x:A. P x) = (EX x:B. Q x)"

   by (simp add: Bex_def cong: conj_cong)

 lemma strong_bex_cong [cong]:

   "A = B ==> (!!x. x:B =simp=> P x = Q x) ==>

     (EX x:A. P x) = (EX x:B. Q x)"

   by (simp add: simp_implies_def Bex_def cong: conj_cong)

 lemma bex1_def: "(\<exists>!x\<in>X. P x) \<longleftrightarrow> (\<exists>x\<in>X. P x) \<and> (\<forall>x\<in>X. \<forall>y\<in>X. P x \<longrightarrow> P y \<longrightarrow> x = y)"

   by auto

 subsection {* Basic operations *}

 subsubsection {* Subsets *}

 lemma subsetI [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> x \<in> B) \<Longrightarrow> A \<subseteq> B"

   by (simp add: less_eq_set_def le_fun_def)

 text {*

   \medskip Map the type @{text "'a set => anything"} to just @{typ

   'a}; for overloading constants whose first argument has type @{typ

   "'a set"}.

 *}

 lemma subsetD [elim, intro?]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B"

   by (simp add: less_eq_set_def le_fun_def)

   -- {* Rule in Modus Ponens style. *}

 lemma rev_subsetD [intro?]: "c \<in> A ==> A \<subseteq> B ==> c \<in> B"

   -- {* The same, with reversed premises for use with @{text erule} --

       cf @{text rev_mp}. *}

   by (rule subsetD)

 text {*

   \medskip Converts @{prop "A \<subseteq> B"} to @{prop "x \<in> A ==> x \<in> B"}.

 *}

 lemma subsetCE [elim]: "A \<subseteq> B ==> (c \<notin> A ==> P) ==> (c \<in> B ==> P) ==> P"

   -- {* Classical elimination rule. *}

   by (auto simp add: less_eq_set_def le_fun_def)

 lemma subset_eq: "A \<le> B = (\<forall>x\<in>A. x \<in> B)" by blast

 lemma contra_subsetD: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A"

   by blast

 lemma subset_refl: "A \<subseteq> A"

   by (fact order_refl) (* already [iff] *)

 lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C"

   by (fact order_trans)

 lemma set_rev_mp: "x:A ==> A \<subseteq> B ==> x:B"

   by (rule subsetD)

 lemma set_mp: "A \<subseteq> B ==> x:A ==> x:B"

   by (rule subsetD)

 lemma subset_not_subset_eq [code]:

   "A \<subset> B \<longleftrightarrow> A \<subseteq> B \<and> \<not> B \<subseteq> A"

   by (fact less_le_not_le)

 lemma eq_mem_trans: "a=b ==> b \<in> A ==> a \<in> A"

   by simp

 lemmas basic_trans_rules [trans] =

   order_trans_rules set_rev_mp set_mp eq_mem_trans

 subsubsection {* Equality *}

 lemma subset_antisym [intro!]: "A \<subseteq> B ==> B \<subseteq> A ==> A = B"

   -- {* Anti-symmetry of the subset relation. *}

   by (iprover intro: set_eqI subsetD)

 text {*

   \medskip Equality rules from ZF set theory -- are they appropriate

   here?

 *}

 lemma equalityD1: "A = B ==> A \<subseteq> B"

   by simp

 lemma equalityD2: "A = B ==> B \<subseteq> A"

   by simp

 text {*

   \medskip Be careful when adding this to the claset as @{text

   subset_empty} is in the simpset: @{prop "A = {}"} goes to @{prop "{}

   \<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}!

 *}

 lemma equalityE: "A = B ==> (A \<subseteq> B ==> B \<subseteq> A ==> P) ==> P"

   by simp

 lemma equalityCE [elim]:

     "A = B ==> (c \<in> A ==> c \<in> B ==> P) ==> (c \<notin> A ==> c \<notin> B ==> P) ==> P"

   by blast

 lemma eqset_imp_iff: "A = B ==> (x : A) = (x : B)"

   by simp

 lemma eqelem_imp_iff: "x = y ==> (x : A) = (y : A)"

   by simp

 subsubsection {* The empty set *}

 lemma empty_def:

   "{} = {x. False}"

   by (simp add: bot_set_def bot_fun_def)

 lemma empty_iff [simp]: "(c : {}) = False"

   by (simp add: empty_def)

 lemma emptyE [elim!]: "a : {} ==> P"

   by simp

 lemma empty_subsetI [iff]: "{} \<subseteq> A"

     -- {* One effect is to delete the ASSUMPTION @{prop "{} <= A"} *}

   by blast

 lemma equals0I: "(!!y. y \<in> A ==> False) ==> A = {}"

   by blast

 lemma equals0D: "A = {} ==> a \<notin> A"

     -- {* Use for reasoning about disjointness: @{text "A Int B = {}"} *}

   by blast

 lemma ball_empty [simp]: "Ball {} P = True"

   by (simp add: Ball_def)

 lemma bex_empty [simp]: "Bex {} P = False"

   by (simp add: Bex_def)

 subsubsection {* The universal set -- UNIV *}

 abbreviation UNIV :: "'a set" where

   "UNIV \<equiv> top"

 lemma UNIV_def:

   "UNIV = {x. True}"

   by (simp add: top_set_def top_fun_def)

 lemma UNIV_I [simp]: "x : UNIV"

   by (simp add: UNIV_def)

 declare UNIV_I [intro]  -- {* unsafe makes it less likely to cause problems *}

 lemma UNIV_witness [intro?]: "EX x. x : UNIV"

   by simp

 lemma subset_UNIV: "A \<subseteq> UNIV"

   by (fact top_greatest) (* already simp *)

 text {*

   \medskip Eta-contracting these two rules (to remove @{text P})

   causes them to be ignored because of their interaction with

   congruence rules.

 *}

 lemma ball_UNIV [simp]: "Ball UNIV P = All P"

   by (simp add: Ball_def)

 lemma bex_UNIV [simp]: "Bex UNIV P = Ex P"

   by (simp add: Bex_def)

 lemma UNIV_eq_I: "(\<And>x. x \<in> A) \<Longrightarrow> UNIV = A"

   by auto

 lemma UNIV_not_empty [iff]: "UNIV ~= {}"

   by (blast elim: equalityE)

 lemma empty_not_UNIV[simp]: "{} \<noteq> UNIV"

 by blast

 subsubsection {* The Powerset operator -- Pow *}

 definition Pow :: "'a set => 'a set set" where

   Pow_def: "Pow A = {B. B \<le> A}"

 lemma Pow_iff [iff]: "(A \<in> Pow B) = (A \<subseteq> B)"

   by (simp add: Pow_def)

 lemma PowI: "A \<subseteq> B ==> A \<in> Pow B"

   by (simp add: Pow_def)

 lemma PowD: "A \<in> Pow B ==> A \<subseteq> B"

   by (simp add: Pow_def)

 lemma Pow_bottom: "{} \<in> Pow B"

   by simp

 lemma Pow_top: "A \<in> Pow A"

   by simp

 lemma Pow_not_empty: "Pow A \<noteq> {}"

   using Pow_top by blast

 subsubsection {* Set complement *}

 lemma Compl_iff [simp]: "(c \<in> -A) = (c \<notin> A)"

   by (simp add: fun_Compl_def uminus_set_def)

 lemma ComplI [intro!]: "(c \<in> A ==> False) ==> c \<in> -A"

   by (simp add: fun_Compl_def uminus_set_def) blast

 text {*

   \medskip This form, with negated conclusion, works well with the

   Classical prover.  Negated assumptions behave like formulae on the

   right side of the notional turnstile ... *}

 lemma ComplD [dest!]: "c : -A ==> c~:A"

   by simp

 lemmas ComplE = ComplD [elim_format]

 lemma Compl_eq: "- A = {x. ~ x : A}"

   by blast

 subsubsection {* Binary intersection *}

 abbreviation inter :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Int" 70) where

   "op Int \<equiv> inf"

 notation (xsymbols)

   inter  (infixl "\<inter>" 70)

 notation (HTML output)

   inter  (infixl "\<inter>" 70)

 lemma Int_def:

   "A \<inter> B = {x. x \<in> A \<and> x \<in> B}"

   by (simp add: inf_set_def inf_fun_def)

 lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)"

   by (unfold Int_def) blast

 lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B"

   by simp

 lemma IntD1: "c : A Int B ==> c:A"

   by simp

 lemma IntD2: "c : A Int B ==> c:B"

   by simp

 lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P"

   by simp

 lemma mono_Int: "mono f \<Longrightarrow> f (A \<inter> B) \<subseteq> f A \<inter> f B"

   by (fact mono_inf)

 subsubsection {* Binary union *}

 abbreviation union :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Un" 65) where

   "union \<equiv> sup"

 notation (xsymbols)

   union  (infixl "\<union>" 65)

 notation (HTML output)

   union  (infixl "\<union>" 65)

 lemma Un_def:

   "A \<union> B = {x. x \<in> A \<or> x \<in> B}"

   by (simp add: sup_set_def sup_fun_def)

 lemma Un_iff [simp]: "(c : A Un B) = (c:A | c:B)"

   by (unfold Un_def) blast

 lemma UnI1 [elim?]: "c:A ==> c : A Un B"

   by simp

 lemma UnI2 [elim?]: "c:B ==> c : A Un B"

   by simp

 text {*

   \medskip Classical introduction rule: no commitment to @{prop A} vs

   @{prop B}.

 *}

 lemma UnCI [intro!]: "(c~:B ==> c:A) ==> c : A Un B"

   by auto

 lemma UnE [elim!]: "c : A Un B ==> (c:A ==> P) ==> (c:B ==> P) ==> P"

   by (unfold Un_def) blast

 lemma insert_def: "insert a B = {x. x = a} \<union> B"

   by (simp add: insert_compr Un_def)

 lemma mono_Un: "mono f \<Longrightarrow> f A \<union> f B \<subseteq> f (A \<union> B)"

   by (fact mono_sup)

 subsubsection {* Set difference *}

 lemma Diff_iff [simp]: "(c : A - B) = (c:A & c~:B)"

   by (simp add: minus_set_def fun_diff_def)

 lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A - B"

   by simp

 lemma DiffD1: "c : A - B ==> c : A"

   by simp

 lemma DiffD2: "c : A - B ==> c : B ==> P"

   by simp

 lemma DiffE [elim!]: "c : A - B ==> (c:A ==> c~:B ==> P) ==> P"

   by simp

 lemma set_diff_eq: "A - B = {x. x : A & ~ x : B}" by blast

 lemma Compl_eq_Diff_UNIV: "-A = (UNIV - A)"

 by blast

 subsubsection {* Augmenting a set -- @{const insert} *}

 lemma insert_iff [simp]: "(a : insert b A) = (a = b | a:A)"

   by (unfold insert_def) blast

 lemma insertI1: "a : insert a B"

   by simp

 lemma insertI2: "a : B ==> a : insert b B"

   by simp

 lemma insertE [elim!]: "a : insert b A ==> (a = b ==> P) ==> (a:A ==> P) ==> P"

   by (unfold insert_def) blast

 lemma insertCI [intro!]: "(a~:B ==> a = b) ==> a: insert b B"

   -- {* Classical introduction rule. *}

   by auto

 lemma subset_insert_iff: "(A \<subseteq> insert x B) = (if x:A then A - {x} \<subseteq> B else A \<subseteq> B)"

   by auto

 lemma set_insert:

   assumes "x \<in> A"

   obtains B where "A = insert x B" and "x \<notin> B"

 proof

   from assms show "A = insert x (A - {x})" by blast

 next

   show "x \<notin> A - {x}" by blast

 qed

 lemma insert_ident: "x ~: A ==> x ~: B ==> (insert x A = insert x B) = (A = B)"

 by auto

 lemma insert_eq_iff: assumes "a \<notin> A" "b \<notin> B"

 shows "insert a A = insert b B \<longleftrightarrow>

   (if a=b then A=B else \<exists>C. A = insert b C \<and> b \<notin> C \<and> B = insert a C \<and> a \<notin> C)"

   (is "?L \<longleftrightarrow> ?R")

 proof

   assume ?L

   show ?R

   proof cases

     assume "a=b" with assms `?L` show ?R by (simp add: insert_ident)

   next

     assume "a\<noteq>b"

     let ?C = "A - {b}"

     have "A = insert b ?C \<and> b \<notin> ?C \<and> B = insert a ?C \<and> a \<notin> ?C"

       using assms `?L` `a\<noteq>b` by auto

     thus ?R using `a\<noteq>b` by auto

   qed

 next

   assume ?R thus ?L by (auto split: if_splits)

 qed

 lemma insert_UNIV: "insert x UNIV = UNIV"

 by auto

 subsubsection {* Singletons, using insert *}

 lemma singletonI [intro!]: "a : {a}"

     -- {* Redundant? But unlike @{text insertCI}, it proves the subgoal immediately! *}

   by (rule insertI1)

 lemma singletonD [dest!]: "b : {a} ==> b = a"

   by blast

 lemmas singletonE = singletonD [elim_format]

 lemma singleton_iff: "(b : {a}) = (b = a)"

   by blast

 lemma singleton_inject [dest!]: "{a} = {b} ==> a = b"

   by blast

 lemma singleton_insert_inj_eq [iff]:

      "({b} = insert a A) = (a = b & A \<subseteq> {b})"

   by blast

 lemma singleton_insert_inj_eq' [iff]:

      "(insert a A = {b}) = (a = b & A \<subseteq> {b})"

   by blast

 lemma subset_singletonD: "A \<subseteq> {x} ==> A = {} | A = {x}"

   by fast

 lemma singleton_conv [simp]: "{x. x = a} = {a}"

   by blast

 lemma singleton_conv2 [simp]: "{x. a = x} = {a}"

   by blast

 lemma diff_single_insert: "A - {x} \<subseteq> B ==> A \<subseteq> insert x B"

   by blast

 lemma doubleton_eq_iff: "({a,b} = {c,d}) = (a=c & b=d | a=d & b=c)"

   by (blast elim: equalityE)

 lemma Un_singleton_iff:

   "(A \<union> B = {x}) = (A = {} \<and> B = {x} \<or> A = {x} \<and> B = {} \<or> A = {x} \<and> B = {x})"

 by auto

 lemma singleton_Un_iff:

   "({x} = A \<union> B) = (A = {} \<and> B = {x} \<or> A = {x} \<and> B = {} \<or> A = {x} \<and> B = {x})"

 by auto

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

 text {*

   Frequently @{term b} does not have the syntactic form of @{term "f x"}.

 *}

 definition image :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set" (infixr "`" 90)

 where

   "f ` A = {y. \<exists>x\<in>A. y = f x}"

 lemma image_eqI [simp, intro]:

   "b = f x \<Longrightarrow> x \<in> A \<Longrightarrow> b \<in> f ` A"

   by (unfold image_def) blast

 lemma imageI:

   "x \<in> A \<Longrightarrow> f x \<in> f ` A"

   by (rule image_eqI) (rule refl)

 lemma rev_image_eqI:

   "x \<in> A \<Longrightarrow> b = f x \<Longrightarrow> b \<in> f ` A"

   -- {* This version's more effective when we already have the

     required @{term x}. *}

   by (rule image_eqI)

 lemma imageE [elim!]:

   assumes "b \<in> (\<lambda>x. f x) ` A" -- {* The eta-expansion gives variable-name preservation. *}

   obtains x where "b = f x" and "x \<in> A"

   using assms by (unfold image_def) blast

 lemma Compr_image_eq:

   "{x \<in> f ` A. P x} = f ` {x \<in> A. P (f x)}"

   by auto

 lemma image_Un:

   "f ` (A \<union> B) = f ` A \<union> f ` B"

   by blast

 lemma image_iff:

   "z \<in> f ` A \<longleftrightarrow> (\<exists>x\<in>A. z = f x)"

   by blast

 lemma image_subsetI:

   "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B) \<Longrightarrow> f ` A \<subseteq> B"

   -- {* Replaces the three steps @{text subsetI}, @{text imageE},

     @{text hypsubst}, but breaks too many existing proofs. *}

   by blast

 lemma image_subset_iff:

   "f ` A \<subseteq> B \<longleftrightarrow> (\<forall>x\<in>A. f x \<in> B)"

   -- {* This rewrite rule would confuse users if made default. *}

   by blast

 lemma subset_imageE:

   assumes "B \<subseteq> f ` A"

   obtains C where "C \<subseteq> A" and "B = f ` C"

 proof -

   from assms have "B = f ` {a \<in> A. f a \<in> B}" by fast

   moreover have "{a \<in> A. f a \<in> B} \<subseteq> A" by blast

   ultimately show thesis by (blast intro: that)

 qed

 lemma subset_image_iff:

   "B \<subseteq> f ` A \<longleftrightarrow> (\<exists>AA\<subseteq>A. B = f ` AA)"

   by (blast elim: subset_imageE)

 lemma image_ident [simp]:

   "(\<lambda>x. x) ` Y = Y"

   by blast

 lemma image_empty [simp]:

   "f ` {} = {}"

   by blast

 lemma image_insert [simp]:

   "f ` insert a B = insert (f a) (f ` B)"

   by blast

 lemma image_constant:

   "x \<in> A \<Longrightarrow> (\<lambda>x. c) ` A = {c}"

   by auto

 lemma image_constant_conv:

   "(\<lambda>x. c) ` A = (if A = {} then {} else {c})"

   by auto

 lemma image_image:

   "f ` (g ` A) = (\<lambda>x. f (g x)) ` A"

   by blast

 lemma insert_image [simp]:

   "x \<in> A ==> insert (f x) (f ` A) = f ` A"

   by blast

 lemma image_is_empty [iff]:

   "f ` A = {} \<longleftrightarrow> A = {}"

   by blast

 lemma empty_is_image [iff]:

   "{} = f ` A \<longleftrightarrow> A = {}"

   by blast

 lemma image_Collect:

   "f ` {x. P x} = {f x | x. P x}"

   -- {* NOT suitable as a default simprule: the RHS isn't simpler than the LHS,

       with its implicit quantifier and conjunction.  Also image enjoys better

       equational properties than does the RHS. *}

   by blast

 lemma if_image_distrib [simp]:

   "(\<lambda>x. if P x then f x else g x) ` S

     = (f ` (S \<inter> {x. P x})) \<union> (g ` (S \<inter> {x. \<not> P x}))"

   by auto

 lemma image_cong:

   "M = N \<Longrightarrow> (\<And>x. x \<in> N \<Longrightarrow> f x = g x) \<Longrightarrow> f ` M = g ` N"

   by (simp add: image_def)

 lemma image_Int_subset:

   "f ` (A \<inter> B) \<subseteq> f ` A \<inter> f ` B"

   by blast

 lemma image_diff_subset:

   "f ` A - f ` B \<subseteq> f ` (A - B)"

   by blast

 lemma Setcompr_eq_image: "{f x | x. x \<in> A} = f ` A"

   by blast

 lemma ball_imageD:

   assumes "\<forall>x\<in>f ` A. P x"

   shows "\<forall>x\<in>A. P (f x)"

   using assms by simp

 lemma bex_imageD:

   assumes "\<exists>x\<in>f ` A. P x"

   shows "\<exists>x\<in>A. P (f x)"

   using assms by auto

 text {*

   \medskip Range of a function -- just a translation for image!

 *}

 abbreviation range :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b set"

 where -- "of function"

   "range f \<equiv> f ` UNIV"

 lemma range_eqI:

   "b = f x \<Longrightarrow> b \<in> range f"

   by simp

 lemma rangeI:

   "f x \<in> range f"

   by simp

 lemma rangeE [elim?]:

   "b \<in> range (\<lambda>x. f x) \<Longrightarrow> (\<And>x. b = f x \<Longrightarrow> P) \<Longrightarrow> P"

   by (rule imageE)

 lemma full_SetCompr_eq:

   "{u. \<exists>x. u = f x} = range f"

   by auto

 lemma range_composition:

   "range (\<lambda>x. f (g x)) = f ` range g"

   by auto

 subsubsection {* Some rules with @{text "if"} *}

 text{* Elimination of @{text"{x. \<dots> & x=t & \<dots>}"}. *}

 lemma Collect_conv_if: "{x. x=a & P x} = (if P a then {a} else {})"

   by auto

 lemma Collect_conv_if2: "{x. a=x & P x} = (if P a then {a} else {})"

   by auto

 text {*

   Rewrite rules for boolean case-splitting: faster than @{text

   "split_if [split]"}.

 *}

 lemma split_if_eq1: "((if Q then x else y) = b) = ((Q --> x = b) & (~ Q --> y = b))"

   by (rule split_if)

 lemma split_if_eq2: "(a = (if Q then x else y)) = ((Q --> a = x) & (~ Q --> a = y))"

   by (rule split_if)

 text {*

   Split ifs on either side of the membership relation.  Not for @{text

   "[simp]"} -- can cause goals to blow up!

 *}

 lemma split_if_mem1: "((if Q then x else y) : b) = ((Q --> x : b) & (~ Q --> y : b))"

   by (rule split_if)

 lemma split_if_mem2: "(a : (if Q then x else y)) = ((Q --> a : x) & (~ Q --> a : y))"

   by (rule split_if [where P="%S. a : S"])

 lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2

 (*Would like to add these, but the existing code only searches for the

   outer-level constant, which in this case is just Set.member; we instead need

   to use term-nets to associate patterns with rules.  Also, if a rule fails to

   apply, then the formula should be kept.

   [("uminus", Compl_iff RS iffD1), ("minus", [Diff_iff RS iffD1]),

    ("Int", [IntD1,IntD2]),

    ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]

  *)

 subsection {* Further operations and lemmas *}

 subsubsection {* The ``proper subset'' relation *}

 lemma psubsetI [intro!]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B"

   by (unfold less_le) blast

 lemma psubsetE [elim!]:

     "[|A \<subset> B;  [|A \<subseteq> B; ~ (B\<subseteq>A)|] ==> R|] ==> R"

   by (unfold less_le) blast

 lemma psubset_insert_iff:

   "(A \<subset> insert x B) = (if x \<in> B then A \<subset> B else if x \<in> A then A - {x} \<subset> B else A \<subseteq> B)"

   by (auto simp add: less_le subset_insert_iff)

 lemma psubset_eq: "(A \<subset> B) = (A \<subseteq> B & A \<noteq> B)"

   by (simp only: less_le)

 lemma psubset_imp_subset: "A \<subset> B ==> A \<subseteq> B"

   by (simp add: psubset_eq)

 lemma psubset_trans: "[| A \<subset> B; B \<subset> C |] ==> A \<subset> C"

 apply (unfold less_le)

 apply (auto dest: subset_antisym)

 done

 lemma psubsetD: "[| A \<subset> B; c \<in> A |] ==> c \<in> B"

 apply (unfold less_le)

 apply (auto dest: subsetD)

 done

 lemma psubset_subset_trans: "A \<subset> B ==> B \<subseteq> C ==> A \<subset> C"

   by (auto simp add: psubset_eq)

 lemma subset_psubset_trans: "A \<subseteq> B ==> B \<subset> C ==> A \<subset> C"

   by (auto simp add: psubset_eq)

 lemma psubset_imp_ex_mem: "A \<subset> B ==> \<exists>b. b \<in> (B - A)"

   by (unfold less_le) blast

 lemma atomize_ball:

     "(!!x. x \<in> A ==> P x) == Trueprop (\<forall>x\<in>A. P x)"

   by (simp only: Ball_def atomize_all atomize_imp)

 lemmas [symmetric, rulify] = atomize_ball

   and [symmetric, defn] = atomize_ball

 lemma image_Pow_mono:

   assumes "f ` A \<subseteq> B"

   shows "image f ` Pow A \<subseteq> Pow B"

   using assms by blast

 lemma image_Pow_surj:

   assumes "f ` A = B"

   shows "image f ` Pow A = Pow B"

   using assms by (blast elim: subset_imageE)

 subsubsection {* Derived rules involving subsets. *}

 text {* @{text insert}. *}

 lemma subset_insertI: "B \<subseteq> insert a B"

   by (rule subsetI) (erule insertI2)

 lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B"

   by blast

 lemma subset_insert: "x \<notin> A ==> (A \<subseteq> insert x B) = (A \<subseteq> B)"

   by blast

 text {* \medskip Finite Union -- the least upper bound of two sets. *}

 lemma Un_upper1: "A \<subseteq> A \<union> B"

   by (fact sup_ge1)

 lemma Un_upper2: "B \<subseteq> A \<union> B"

   by (fact sup_ge2)

 lemma Un_least: "A \<subseteq> C ==> B \<subseteq> C ==> A \<union> B \<subseteq> C"

   by (fact sup_least)

 text {* \medskip Finite Intersection -- the greatest lower bound of two sets. *}

 lemma Int_lower1: "A \<inter> B \<subseteq> A"

   by (fact inf_le1)

 lemma Int_lower2: "A \<inter> B \<subseteq> B"

   by (fact inf_le2)

 lemma Int_greatest: "C \<subseteq> A ==> C \<subseteq> B ==> C \<subseteq> A \<inter> B"

   by (fact inf_greatest)

 text {* \medskip Set difference. *}

 lemma Diff_subset: "A - B \<subseteq> A"

   by blast

 lemma Diff_subset_conv: "(A - B \<subseteq> C) = (A \<subseteq> B \<union> C)"

 by blast

 subsubsection {* Equalities involving union, intersection, inclusion, etc. *}

 text {* @{text "{}"}. *}

 lemma Collect_const [simp]: "{s. P} = (if P then UNIV else {})"

   -- {* supersedes @{text "Collect_False_empty"} *}

   by auto

 lemma subset_empty [simp]: "(A \<subseteq> {}) = (A = {})"

   by (fact bot_unique)

 lemma not_psubset_empty [iff]: "\<not> (A < {})"

   by (fact not_less_bot) (* FIXME: already simp *)

 lemma Collect_empty_eq [simp]: "(Collect P = {}) = (\<forall>x. \<not> P x)"

 by blast

 lemma empty_Collect_eq [simp]: "({} = Collect P) = (\<forall>x. \<not> P x)"

 by blast

 lemma Collect_neg_eq: "{x. \<not> P x} = - {x. P x}"

   by blast

 lemma Collect_disj_eq: "{x. P x | Q x} = {x. P x} \<union> {x. Q x}"

   by blast

 lemma Collect_imp_eq: "{x. P x --> Q x} = -{x. P x} \<union> {x. Q x}"

   by blast

 lemma Collect_conj_eq: "{x. P x & Q x} = {x. P x} \<inter> {x. Q x}"

   by blast

 lemma Collect_mono_iff: "Collect P \<subseteq> Collect Q \<longleftrightarrow> (\<forall>x. P x \<longrightarrow> Q x)"

   by blast

 text {* \medskip @{text insert}. *}

 lemma insert_is_Un: "insert a A = {a} Un A"

   -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a {}"} *}

   by blast

 lemma insert_not_empty [simp]: "insert a A \<noteq> {}"

   by blast

 lemmas empty_not_insert = insert_not_empty [symmetric]

 declare empty_not_insert [simp]

 lemma insert_absorb: "a \<in> A ==> insert a A = A"

   -- {* @{text "[simp]"} causes recursive calls when there are nested inserts *}

   -- {* with \emph{quadratic} running time *}

   by blast

 lemma insert_absorb2 [simp]: "insert x (insert x A) = insert x A"

   by blast

 lemma insert_commute: "insert x (insert y A) = insert y (insert x A)"

   by blast

 lemma insert_subset [simp]: "(insert x A \<subseteq> B) = (x \<in> B & A \<subseteq> B)"

   by blast

 lemma mk_disjoint_insert: "a \<in> A ==> \<exists>B. A = insert a B & a \<notin> B"

   -- {* use new @{text B} rather than @{text "A - {a}"} to avoid infinite unfolding *}

   apply (rule_tac x = "A - {a}" in exI, blast)

   done

 lemma insert_Collect: "insert a (Collect P) = {u. u \<noteq> a --> P u}"

   by auto

 lemma insert_inter_insert[simp]: "insert a A \<inter> insert a B = insert a (A \<inter> B)"

   by blast

 lemma insert_disjoint [simp]:

  "(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})"

  "({} = insert a A \<inter> B) = (a \<notin> B \<and> {} = A \<inter> B)"

   by auto

 lemma disjoint_insert [simp]:

  "(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})"

  "({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)"

   by auto

 text {* \medskip @{text Int} *}

 lemma Int_absorb: "A \<inter> A = A"

   by (fact inf_idem) (* already simp *)

 lemma Int_left_absorb: "A \<inter> (A \<inter> B) = A \<inter> B"

   by (fact inf_left_idem)

 lemma Int_commute: "A \<inter> B = B \<inter> A"

   by (fact inf_commute)

 lemma Int_left_commute: "A \<inter> (B \<inter> C) = B \<inter> (A \<inter> C)"

   by (fact inf_left_commute)

 lemma Int_assoc: "(A \<inter> B) \<inter> C = A \<inter> (B \<inter> C)"

   by (fact inf_assoc)

 lemmas Int_ac = Int_assoc Int_left_absorb Int_commute Int_left_commute

   -- {* Intersection is an AC-operator *}

 lemma Int_absorb1: "B \<subseteq> A ==> A \<inter> B = B"

   by (fact inf_absorb2)

 lemma Int_absorb2: "A \<subseteq> B ==> A \<inter> B = A"

   by (fact inf_absorb1)

 lemma Int_empty_left: "{} \<inter> B = {}"

   by (fact inf_bot_left) (* already simp *)

 lemma Int_empty_right: "A \<inter> {} = {}"

   by (fact inf_bot_right) (* already simp *)

 lemma disjoint_eq_subset_Compl: "(A \<inter> B = {}) = (A \<subseteq> -B)"

   by blast

 lemma disjoint_iff_not_equal: "(A \<inter> B = {}) = (\<forall>x\<in>A. \<forall>y\<in>B. x \<noteq> y)"

   by blast

 lemma Int_UNIV_left: "UNIV \<inter> B = B"

   by (fact inf_top_left) (* already simp *)

 lemma Int_UNIV_right: "A \<inter> UNIV = A"

   by (fact inf_top_right) (* already simp *)

 lemma Int_Un_distrib: "A \<inter> (B \<union> C) = (A \<inter> B) \<union> (A \<inter> C)"

   by (fact inf_sup_distrib1)

 lemma Int_Un_distrib2: "(B \<union> C) \<inter> A = (B \<inter> A) \<union> (C \<inter> A)"

   by (fact inf_sup_distrib2)

 lemma Int_UNIV [simp]: "(A \<inter> B = UNIV) = (A = UNIV & B = UNIV)"

   by (fact inf_eq_top_iff) (* already simp *)

 lemma Int_subset_iff [simp]: "(C \<subseteq> A \<inter> B) = (C \<subseteq> A & C \<subseteq> B)"

   by (fact le_inf_iff)

 lemma Int_Collect: "(x \<in> A \<inter> {x. P x}) = (x \<in> A & P x)"

   by blast

 text {* \medskip @{text Un}. *}

 lemma Un_absorb: "A \<union> A = A"

   by (fact sup_idem) (* already simp *)

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

   by (fact sup_left_idem)

 lemma Un_commute: "A \<union> B = B \<union> A"

   by (fact sup_commute)

 lemma Un_left_commute: "A \<union> (B \<union> C) = B \<union> (A \<union> C)"

   by (fact sup_left_commute)

 lemma Un_assoc: "(A \<union> B) \<union> C = A \<union> (B \<union> C)"

   by (fact sup_assoc)

 lemmas Un_ac = Un_assoc Un_left_absorb Un_commute Un_left_commute

   -- {* Union is an AC-operator *}

 lemma Un_absorb1: "A \<subseteq> B ==> A \<union> B = B"

   by (fact sup_absorb2)

 lemma Un_absorb2: "B \<subseteq> A ==> A \<union> B = A"

   by (fact sup_absorb1)

 lemma Un_empty_left: "{} \<union> B = B"

   by (fact sup_bot_left) (* already simp *)

 lemma Un_empty_right: "A \<union> {} = A"

   by (fact sup_bot_right) (* already simp *)

 lemma Un_UNIV_left: "UNIV \<union> B = UNIV"

   by (fact sup_top_left) (* already simp *)

 lemma Un_UNIV_right: "A \<union> UNIV = UNIV"

   by (fact sup_top_right) (* already simp *)

 lemma Un_insert_left [simp]: "(insert a B) \<union> C = insert a (B \<union> C)"

   by blast

 lemma Un_insert_right [simp]: "A \<union> (insert a B) = insert a (A \<union> B)"

   by blast

 lemma Int_insert_left:

     "(insert a B) Int C = (if a \<in> C then insert a (B \<inter> C) else B \<inter> C)"

   by auto

 lemma Int_insert_left_if0[simp]:

     "a \<notin> C \<Longrightarrow> (insert a B) Int C = B \<inter> C"

   by auto

 lemma Int_insert_left_if1[simp]:

     "a \<in> C \<Longrightarrow> (insert a B) Int C = insert a (B Int C)"

   by auto

 lemma Int_insert_right:

     "A \<inter> (insert a B) = (if a \<in> A then insert a (A \<inter> B) else A \<inter> B)"

   by auto

 lemma Int_insert_right_if0[simp]:

     "a \<notin> A \<Longrightarrow> A Int (insert a B) = A Int B"

   by auto

 lemma Int_insert_right_if1[simp]:

     "a \<in> A \<Longrightarrow> A Int (insert a B) = insert a (A Int B)"

   by auto

 lemma Un_Int_distrib: "A \<union> (B \<inter> C) = (A \<union> B) \<inter> (A \<union> C)"

   by (fact sup_inf_distrib1)

 lemma Un_Int_distrib2: "(B \<inter> C) \<union> A = (B \<union> A) \<inter> (C \<union> A)"

   by (fact sup_inf_distrib2)

 lemma Un_Int_crazy:

     "(A \<inter> B) \<union> (B \<inter> C) \<union> (C \<inter> A) = (A \<union> B) \<inter> (B \<union> C) \<inter> (C \<union> A)"

   by blast

 lemma subset_Un_eq: "(A \<subseteq> B) = (A \<union> B = B)"

   by (fact le_iff_sup)

 lemma Un_empty [iff]: "(A \<union> B = {}) = (A = {} & B = {})"

   by (fact sup_eq_bot_iff) (* FIXME: already simp *)

 lemma Un_subset_iff [simp]: "(A \<union> B \<subseteq> C) = (A \<subseteq> C & B \<subseteq> C)"

   by (fact le_sup_iff)

 lemma Un_Diff_Int: "(A - B) \<union> (A \<inter> B) = A"

   by blast

 lemma Diff_Int2: "A \<inter> C - B \<inter> C = A \<inter> C - B"

   by blast

 text {* \medskip Set complement *}

 lemma Compl_disjoint [simp]: "A \<inter> -A = {}"

   by (fact inf_compl_bot)

 lemma Compl_disjoint2 [simp]: "-A \<inter> A = {}"

   by (fact compl_inf_bot)

 lemma Compl_partition: "A \<union> -A = UNIV"

   by (fact sup_compl_top)

 lemma Compl_partition2: "-A \<union> A = UNIV"

   by (fact compl_sup_top)

 lemma double_complement: "- (-A) = (A::'a set)"

   by (fact double_compl) (* already simp *)

 lemma Compl_Un: "-(A \<union> B) = (-A) \<inter> (-B)"

   by (fact compl_sup) (* already simp *)

 lemma Compl_Int: "-(A \<inter> B) = (-A) \<union> (-B)"

   by (fact compl_inf) (* already simp *)

 lemma subset_Compl_self_eq: "(A \<subseteq> -A) = (A = {})"

   by blast

 lemma Un_Int_assoc_eq: "((A \<inter> B) \<union> C = A \<inter> (B \<union> C)) = (C \<subseteq> A)"

   -- {* Halmos, Naive Set Theory, page 16. *}

   by blast

 lemma Compl_UNIV_eq: "-UNIV = {}"

   by (fact compl_top_eq) (* already simp *)

 lemma Compl_empty_eq: "-{} = UNIV"

   by (fact compl_bot_eq) (* already simp *)

 lemma Compl_subset_Compl_iff [iff]: "(-A \<subseteq> -B) = (B \<subseteq> A)"

   by (fact compl_le_compl_iff) (* FIXME: already simp *)

 lemma Compl_eq_Compl_iff [iff]: "(-A = -B) = (A = (B::'a set))"

   by (fact compl_eq_compl_iff) (* FIXME: already simp *)

 lemma Compl_insert: "- insert x A = (-A) - {x}"

   by blast

 text {* \medskip Bounded quantifiers.

   The following are not added to the default simpset because

   (a) they duplicate the body and (b) there are no similar rules for @{text Int}. *}

 lemma ball_Un: "(\<forall>x \<in> A \<union> B. P x) = ((\<forall>x\<in>A. P x) & (\<forall>x\<in>B. P x))"

   by blast

 lemma bex_Un: "(\<exists>x \<in> A \<union> B. P x) = ((\<exists>x\<in>A. P x) | (\<exists>x\<in>B. P x))"

   by blast

 text {* \medskip Set difference. *}

 lemma Diff_eq: "A - B = A \<inter> (-B)"

   by blast

 lemma Diff_eq_empty_iff [simp]: "(A - B = {}) = (A \<subseteq> B)"

   by blast

 lemma Diff_cancel [simp]: "A - A = {}"

   by blast

 lemma Diff_idemp [simp]: "(A - B) - B = A - (B::'a set)"

 by blast

 lemma Diff_triv: "A \<inter> B = {} ==> A - B = A"

   by (blast elim: equalityE)

 lemma empty_Diff [simp]: "{} - A = {}"

   by blast

 lemma Diff_empty [simp]: "A - {} = A"

   by blast

 lemma Diff_UNIV [simp]: "A - UNIV = {}"

   by blast

 lemma Diff_insert0 [simp]: "x \<notin> A ==> A - insert x B = A - B"

   by blast

 lemma Diff_insert: "A - insert a B = A - B - {a}"

   -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a 0"} *}

   by blast

 lemma Diff_insert2: "A - insert a B = A - {a} - B"

   -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a 0"} *}

   by blast

 lemma insert_Diff_if: "insert x A - B = (if x \<in> B then A - B else insert x (A - B))"

   by auto

 lemma insert_Diff1 [simp]: "x \<in> B ==> insert x A - B = A - B"

   by blast

 lemma insert_Diff_single[simp]: "insert a (A - {a}) = insert a A"

 by blast

 lemma insert_Diff: "a \<in> A ==> insert a (A - {a}) = A"

   by blast

 lemma Diff_insert_absorb: "x \<notin> A ==> (insert x A) - {x} = A"

   by auto

 lemma Diff_disjoint [simp]: "A \<inter> (B - A) = {}"

   by blast

 lemma Diff_partition: "A \<subseteq> B ==> A \<union> (B - A) = B"

   by blast

 lemma double_diff: "A \<subseteq> B ==> B \<subseteq> C ==> B - (C - A) = A"

   by blast

 lemma Un_Diff_cancel [simp]: "A \<union> (B - A) = A \<union> B"

   by blast

 lemma Un_Diff_cancel2 [simp]: "(B - A) \<union> A = B \<union> A"

   by blast

 lemma Diff_Un: "A - (B \<union> C) = (A - B) \<inter> (A - C)"

   by blast

 lemma Diff_Int: "A - (B \<inter> C) = (A - B) \<union> (A - C)"

   by blast

 lemma Un_Diff: "(A \<union> B) - C = (A - C) \<union> (B - C)"

   by blast

 lemma Int_Diff: "(A \<inter> B) - C = A \<inter> (B - C)"

   by blast

 lemma Diff_Int_distrib: "C \<inter> (A - B) = (C \<inter> A) - (C \<inter> B)"

   by blast

 lemma Diff_Int_distrib2: "(A - B) \<inter> C = (A \<inter> C) - (B \<inter> C)"

   by blast

 lemma Diff_Compl [simp]: "A - (- B) = A \<inter> B"

   by auto

 lemma Compl_Diff_eq [simp]: "- (A - B) = -A \<union> B"

   by blast

 text {* \medskip Quantification over type @{typ bool}. *}

 lemma bool_induct: "P True \<Longrightarrow> P False \<Longrightarrow> P x"

   by (cases x) auto

 lemma all_bool_eq: "(\<forall>b. P b) \<longleftrightarrow> P True \<and> P False"

   by (auto intro: bool_induct)

 lemma bool_contrapos: "P x \<Longrightarrow> \<not> P False \<Longrightarrow> P True"

   by (cases x) auto

 lemma ex_bool_eq: "(\<exists>b. P b) \<longleftrightarrow> P True \<or> P False"

   by (auto intro: bool_contrapos)

 lemma UNIV_bool: "UNIV = {False, True}"

   by (auto intro: bool_induct)

 text {* \medskip @{text Pow} *}

 lemma Pow_empty [simp]: "Pow {} = {{}}"

   by (auto simp add: Pow_def)

 lemma Pow_singleton_iff [simp]: "Pow X = {Y} \<longleftrightarrow> X = {} \<and> Y = {}"

 by blast

 lemma Pow_insert: "Pow (insert a A) = Pow A \<union> (insert a ` Pow A)"

   by (blast intro: image_eqI [where ?x = "u - {a}" for u])

 lemma Pow_Compl: "Pow (- A) = {-B | B. A \<in> Pow B}"

   by (blast intro: exI [where ?x = "- u" for u])

 lemma Pow_UNIV [simp]: "Pow UNIV = UNIV"

   by blast

 lemma Un_Pow_subset: "Pow A \<union> Pow B \<subseteq> Pow (A \<union> B)"

   by blast

 lemma Pow_Int_eq [simp]: "Pow (A \<inter> B) = Pow A \<inter> Pow B"

   by blast

 text {* \medskip Miscellany. *}

 lemma set_eq_subset: "(A = B) = (A \<subseteq> B & B \<subseteq> A)"

   by blast

 lemma subset_iff: "(A \<subseteq> B) = (\<forall>t. t \<in> A --> t \<in> B)"

   by blast

 lemma subset_iff_psubset_eq: "(A \<subseteq> B) = ((A \<subset> B) | (A = B))"

   by (unfold less_le) blast

 lemma all_not_in_conv [simp]: "(\<forall>x. x \<notin> A) = (A = {})"

   by blast

 lemma ex_in_conv: "(\<exists>x. x \<in> A) = (A \<noteq> {})"

   by blast

 lemma ball_simps [simp, no_atp]:

   "\<And>A P Q. (\<forall>x\<in>A. P x \<or> Q) \<longleftrightarrow> ((\<forall>x\<in>A. P x) \<or> Q)"

   "\<And>A P Q. (\<forall>x\<in>A. P \<or> Q x) \<longleftrightarrow> (P \<or> (\<forall>x\<in>A. Q x))"

   "\<And>A P Q. (\<forall>x\<in>A. P \<longrightarrow> Q x) \<longleftrightarrow> (P \<longrightarrow> (\<forall>x\<in>A. Q x))"

   "\<And>A P Q. (\<forall>x\<in>A. P x \<longrightarrow> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<longrightarrow> Q)"

   "\<And>P. (\<forall>x\<in>{}. P x) \<longleftrightarrow> True"

   "\<And>P. (\<forall>x\<in>UNIV. P x) \<longleftrightarrow> (\<forall>x. P x)"

   "\<And>a B P. (\<forall>x\<in>insert a B. P x) \<longleftrightarrow> (P a \<and> (\<forall>x\<in>B. P x))"

   "\<And>P Q. (\<forall>x\<in>Collect Q. P x) \<longleftrightarrow> (\<forall>x. Q x \<longrightarrow> P x)"

   "\<And>A P f. (\<forall>x\<in>f`A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P (f x))"

   "\<And>A P. (\<not> (\<forall>x\<in>A. P x)) \<longleftrightarrow> (\<exists>x\<in>A. \<not> P x)"

   by auto

 lemma bex_simps [simp, no_atp]:

   "\<And>A P Q. (\<exists>x\<in>A. P x \<and> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<and> Q)"

   "\<And>A P Q. (\<exists>x\<in>A. P \<and> Q x) \<longleftrightarrow> (P \<and> (\<exists>x\<in>A. Q x))"

   "\<And>P. (\<exists>x\<in>{}. P x) \<longleftrightarrow> False"

   "\<And>P. (\<exists>x\<in>UNIV. P x) \<longleftrightarrow> (\<exists>x. P x)"

   "\<And>a B P. (\<exists>x\<in>insert a B. P x) \<longleftrightarrow> (P a | (\<exists>x\<in>B. P x))"

   "\<And>P Q. (\<exists>x\<in>Collect Q. P x) \<longleftrightarrow> (\<exists>x. Q x \<and> P x)"

   "\<And>A P f. (\<exists>x\<in>f`A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P (f x))"

   "\<And>A P. (\<not>(\<exists>x\<in>A. P x)) \<longleftrightarrow> (\<forall>x\<in>A. \<not> P x)"

   by auto

 subsubsection {* Monotonicity of various operations *}

 lemma image_mono: "A \<subseteq> B ==> f`A \<subseteq> f`B"

   by blast

 lemma Pow_mono: "A \<subseteq> B ==> Pow A \<subseteq> Pow B"

   by blast

 lemma insert_mono: "C \<subseteq> D ==> insert a C \<subseteq> insert a D"

   by blast

 lemma Un_mono: "A \<subseteq> C ==> B \<subseteq> D ==> A \<union> B \<subseteq> C \<union> D"

   by (fact sup_mono)

 lemma Int_mono: "A \<subseteq> C ==> B \<subseteq> D ==> A \<inter> B \<subseteq> C \<inter> D"

   by (fact inf_mono)

 lemma Diff_mono: "A \<subseteq> C ==> D \<subseteq> B ==> A - B \<subseteq> C - D"

   by blast

 lemma Compl_anti_mono: "A \<subseteq> B ==> -B \<subseteq> -A"

   by (fact compl_mono)

 text {* \medskip Monotonicity of implications. *}

 lemma in_mono: "A \<subseteq> B ==> x \<in> A --> x \<in> B"

   apply (rule impI)

   apply (erule subsetD, assumption)

   done

 lemma conj_mono: "P1 --> Q1 ==> P2 --> Q2 ==> (P1 & P2) --> (Q1 & Q2)"

   by iprover

 lemma disj_mono: "P1 --> Q1 ==> P2 --> Q2 ==> (P1 | P2) --> (Q1 | Q2)"

   by iprover

 lemma imp_mono: "Q1 --> P1 ==> P2 --> Q2 ==> (P1 --> P2) --> (Q1 --> Q2)"

   by iprover

 lemma imp_refl: "P --> P" ..

 lemma not_mono: "Q --> P ==> ~ P --> ~ Q"

   by iprover

 lemma ex_mono: "(!!x. P x --> Q x) ==> (EX x. P x) --> (EX x. Q x)"

   by iprover

 lemma all_mono: "(!!x. P x --> Q x) ==> (ALL x. P x) --> (ALL x. Q x)"

   by iprover

 lemma Collect_mono: "(!!x. P x --> Q x) ==> Collect P \<subseteq> Collect Q"

   by blast

 lemma Int_Collect_mono:

     "A \<subseteq> B ==> (!!x. x \<in> A ==> P x --> Q x) ==> A \<inter> Collect P \<subseteq> B \<inter> Collect Q"

   by blast

 lemmas basic_monos =

   subset_refl imp_refl disj_mono conj_mono

   ex_mono Collect_mono in_mono

 lemma eq_to_mono: "a = b ==> c = d ==> b --> d ==> a --> c"

   by iprover

 subsubsection {* Inverse image of a function *}

 definition vimage :: "('a => 'b) => 'b set => 'a set" (infixr "-`" 90) where

   "f -` B == {x. f x : B}"

 lemma vimage_eq [simp]: "(a : f -` B) = (f a : B)"

   by (unfold vimage_def) blast

 lemma vimage_singleton_eq: "(a : f -` {b}) = (f a = b)"

   by simp

 lemma vimageI [intro]: "f a = b ==> b:B ==> a : f -` B"

   by (unfold vimage_def) blast

 lemma vimageI2: "f a : A ==> a : f -` A"

   by (unfold vimage_def) fast

 lemma vimageE [elim!]: "a: f -` B ==> (!!x. f a = x ==> x:B ==> P) ==> P"

   by (unfold vimage_def) blast

 lemma vimageD: "a : f -` A ==> f a : A"

   by (unfold vimage_def) fast

 lemma vimage_empty [simp]: "f -` {} = {}"

   by blast

 lemma vimage_Compl: "f -` (-A) = -(f -` A)"

   by blast

 lemma vimage_Un [simp]: "f -` (A Un B) = (f -` A) Un (f -` B)"

   by blast

 lemma vimage_Int [simp]: "f -` (A Int B) = (f -` A) Int (f -` B)"

   by fast

 lemma vimage_Collect_eq [simp]: "f -` Collect P = {y. P (f y)}"

   by blast

 lemma vimage_Collect: "(!!x. P (f x) = Q x) ==> f -` (Collect P) = Collect Q"

   by blast

 lemma vimage_insert: "f-`(insert a B) = (f-`{a}) Un (f-`B)"

   -- {* NOT suitable for rewriting because of the recurrence of @{term "{a}"}. *}

   by blast

 lemma vimage_Diff: "f -` (A - B) = (f -` A) - (f -` B)"

   by blast

 lemma vimage_UNIV [simp]: "f -` UNIV = UNIV"

   by blast

 lemma vimage_mono: "A \<subseteq> B ==> f -` A \<subseteq> f -` B"

   -- {* monotonicity *}

   by blast

 lemma vimage_image_eq: "f -` (f ` A) = {y. EX x:A. f x = f y}"

 by (blast intro: sym)

 lemma image_vimage_subset: "f ` (f -` A) <= A"

 by blast

 lemma image_vimage_eq [simp]: "f ` (f -` A) = A Int range f"

 by blast

 lemma image_subset_iff_subset_vimage: "f ` A \<subseteq> B \<longleftrightarrow> A \<subseteq> f -` B"

   by blast

 lemma vimage_const [simp]: "((\<lambda>x. c) -` A) = (if c \<in> A then UNIV else {})"

   by auto

 lemma vimage_if [simp]: "((\<lambda>x. if x \<in> B then c else d) -` A) =

    (if c \<in> A then (if d \<in> A then UNIV else B)

     else if d \<in> A then -B else {})"

   by (auto simp add: vimage_def)

 lemma vimage_inter_cong:

   "(\<And> w. w \<in> S \<Longrightarrow> f w = g w) \<Longrightarrow> f -` y \<inter> S = g -` y \<inter> S"

   by auto

 lemma vimage_ident [simp]: "(%x. x) -` Y = Y"

   by blast

 subsubsection {* Getting the Contents of a Singleton Set *}

 definition the_elem :: "'a set \<Rightarrow> 'a" where

   "the_elem X = (THE x. X = {x})"

 lemma the_elem_eq [simp]: "the_elem {x} = x"

   by (simp add: the_elem_def)

 lemma the_elem_image_unique:

   assumes "A \<noteq> {}"

   assumes *: "\<And>y. y \<in> A \<Longrightarrow> f y = f x"

   shows "the_elem (f ` A) = f x"

 unfolding the_elem_def proof (rule the1_equality)

   from `A \<noteq> {}` obtain y where "y \<in> A" by auto

   with * have "f x = f y" by simp

   with `y \<in> A` have "f x \<in> f ` A" by blast

   with * show "f ` A = {f x}" by auto

   then show "\<exists>!x. f ` A = {x}" by auto

 qed

 subsubsection {* Least value operator *}

 lemma Least_mono:

   "mono (f::'a::order => 'b::order) ==> EX x:S. ALL y:S. x <= y

     ==> (LEAST y. y : f ` S) = f (LEAST x. x : S)"

     -- {* Courtesy of Stephan Merz *}

   apply clarify

   apply (erule_tac P = "%x. x : S" in LeastI2_order, fast)

   apply (rule LeastI2_order)

   apply (auto elim: monoD intro!: order_antisym)

   done

 subsubsection {* Monad operation *}

 definition bind :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where

   "bind A f = {x. \<exists>B \<in> f`A. x \<in> B}"

 hide_const (open) bind

 lemma bind_bind:

   fixes A :: "'a set"

   shows "Set.bind (Set.bind A B) C = Set.bind A (\<lambda>x. Set.bind (B x) C)"

   by (auto simp add: bind_def)

 lemma empty_bind [simp]:

   "Set.bind {} f = {}"

   by (simp add: bind_def)

 lemma nonempty_bind_const:

   "A \<noteq> {} \<Longrightarrow> Set.bind A (\<lambda>_. B) = B"

   by (auto simp add: bind_def)

 lemma bind_const: "Set.bind A (\<lambda>_. B) = (if A = {} then {} else B)"

   by (auto simp add: bind_def)

 lemma bind_singleton_conv_image: "Set.bind A (\<lambda>x. {f x}) = f ` A"

   by(auto simp add: bind_def)

 subsubsection {* Operations for execution *}

 definition is_empty :: "'a set \<Rightarrow> bool" where

   [code_abbrev]: "is_empty A \<longleftrightarrow> A = {}"

 hide_const (open) is_empty

 definition remove :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set" where

   [code_abbrev]: "remove x A = A - {x}"

 hide_const (open) remove

 lemma member_remove [simp]:

   "x \<in> Set.remove y A \<longleftrightarrow> x \<in> A \<and> x \<noteq> y"

   by (simp add: remove_def)

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

   [code_abbrev]: "filter P A = {a \<in> A. P a}"

 hide_const (open) filter

 lemma member_filter [simp]:

   "x \<in> Set.filter P A \<longleftrightarrow> x \<in> A \<and> P x"

   by (simp add: filter_def)

 instantiation set :: (equal) equal

 begin

 definition

   "HOL.equal A B \<longleftrightarrow> A \<subseteq> B \<and> B \<subseteq> A"

 instance proof

 qed (auto simp add: equal_set_def)

 end

 text {* Misc *}

 hide_const (open) member not_member

 lemmas equalityI = subset_antisym

 ML {*

 val Ball_def = @{thm Ball_def}

 val Bex_def = @{thm Bex_def}

 val CollectD = @{thm CollectD}

 val CollectE = @{thm CollectE}

 val CollectI = @{thm CollectI}

 val Collect_conj_eq = @{thm Collect_conj_eq}

 val Collect_mem_eq = @{thm Collect_mem_eq}

 val IntD1 = @{thm IntD1}

 val IntD2 = @{thm IntD2}

 val IntE = @{thm IntE}

 val IntI = @{thm IntI}

 val Int_Collect = @{thm Int_Collect}

 val UNIV_I = @{thm UNIV_I}

 val UNIV_witness = @{thm UNIV_witness}

 val UnE = @{thm UnE}

 val UnI1 = @{thm UnI1}

 val UnI2 = @{thm UnI2}

 val ballE = @{thm ballE}

 val ballI = @{thm ballI}

 val bexCI = @{thm bexCI}

 val bexE = @{thm bexE}

 val bexI = @{thm bexI}

 val bex_triv = @{thm bex_triv}

 val bspec = @{thm bspec}

 val contra_subsetD = @{thm contra_subsetD}

 val equalityCE = @{thm equalityCE}

 val equalityD1 = @{thm equalityD1}

 val equalityD2 = @{thm equalityD2}

 val equalityE = @{thm equalityE}

 val equalityI = @{thm equalityI}

 val imageE = @{thm imageE}

 val imageI = @{thm imageI}

 val image_Un = @{thm image_Un}

 val image_insert = @{thm image_insert}

 val insert_commute = @{thm insert_commute}

 val insert_iff = @{thm insert_iff}

 val mem_Collect_eq = @{thm mem_Collect_eq}

 val rangeE = @{thm rangeE}

 val rangeI = @{thm rangeI}

 val range_eqI = @{thm range_eqI}

 val subsetCE = @{thm subsetCE}

 val subsetD = @{thm subsetD}

 val subsetI = @{thm subsetI}

 val subset_refl = @{thm subset_refl}

 val subset_trans = @{thm subset_trans}

 val vimageD = @{thm vimageD}

 val vimageE = @{thm vimageE}

 val vimageI = @{thm vimageI}

 val vimageI2 = @{thm vimageI2}

 val vimage_Collect = @{thm vimage_Collect}

 val vimage_Int = @{thm vimage_Int}

 val vimage_Un = @{thm vimage_Un}

 *}

 end