src/HOL/Set.thy
 author haftmann Thu Dec 29 10:47:54 2011 +0100 (2011-12-29) changeset 46026 83caa4f4bd56 parent 45986 c9e50153e5ae child 46036 6a86cc88b02f permissions -rw-r--r--
semiring_numeral_0_eq_0, semiring_numeral_1_eq_1 now [simp], superseeding corresponding simp rules on type nat
     1 (*  Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel *)

     2

     3 header {* Set theory for higher-order logic *}

     4

     5 theory Set

     6 imports Lattices

     7 begin

     8

     9 subsection {* Sets as predicates *}

    10

    11 typedecl 'a set

    12

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

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

    15 where

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

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

    18

    19 notation

    20   member  ("op :") and

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

    22

    23 abbreviation not_member where

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

    25

    26 notation

    27   not_member  ("op ~:") and

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

    29

    30 notation (xsymbols)

    31   member      ("op \<in>") and

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

    33   not_member  ("op \<notin>") and

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

    35

    36 notation (HTML output)

    37   member      ("op \<in>") and

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

    39   not_member  ("op \<notin>") and

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

    41

    42

    43 text {* Set comprehensions *}

    44

    45 syntax

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

    47 translations

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

    49

    50 syntax

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

    52 syntax (xsymbols)

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

    54 translations

    55   "{x:A. P}" => "{x. x:A & P}"

    56

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

    58   by simp

    59

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

    61   by simp

    62

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

    64   by simp

    65

    66 text {*

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

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

    69 *}

    70

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

    72   fn _ =>

    73     Quantifier1.rearrange_Collect

    74      (rtac @{thm Collect_cong} 1 THEN

    75       rtac @{thm iffI} 1 THEN

    76       ALLGOALS

    77         (EVERY' [REPEAT_DETERM o etac @{thm conjE}, DEPTH_SOLVE_1 o ares_tac @{thms conjI}]))

    78 *}

    79

    80 lemmas CollectE = CollectD [elim_format]

    81

    82 lemma set_eqI:

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

    84   shows "A = B"

    85 proof -

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

    87   then show ?thesis by simp

    88 qed

    89

    90 lemma set_eq_iff [no_atp]:

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

    92   by (auto intro:set_eqI)

    93

    94 text {* Lifting of predicate class instances *}

    95

    96 instantiation set :: (type) boolean_algebra

    97 begin

    98

    99 definition less_eq_set where

   100   "less_eq_set A B = less_eq (\<lambda>x. member x A) (\<lambda>x. member x B)"

   101

   102 definition less_set where

   103   "less_set A B = less (\<lambda>x. member x A) (\<lambda>x. member x B)"

   104

   105 definition inf_set where

   106   "inf_set A B = Collect (inf (\<lambda>x. member x A) (\<lambda>x. member x B))"

   107

   108 definition sup_set where

   109   "sup_set A B = Collect (sup (\<lambda>x. member x A) (\<lambda>x. member x B))"

   110

   111 definition bot_set where

   112   "bot = Collect bot"

   113

   114 definition top_set where

   115   "top = Collect top"

   116

   117 definition uminus_set where

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

   119

   120 definition minus_set where

   121   "minus_set A B = Collect (minus (\<lambda>x. member x A) (\<lambda>x. member x B))"

   122

   123 instance proof

   124 qed (simp_all add: less_eq_set_def less_set_def inf_set_def sup_set_def

   125   bot_set_def top_set_def uminus_set_def minus_set_def

   126   less_le_not_le inf_compl_bot sup_compl_top sup_inf_distrib1 diff_eq

   127   set_eqI fun_eq_iff)

   128

   129 end

   130

   131 text {* Set enumerations *}

   132

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

   134   "{} \<equiv> bot"

   135

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

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

   138

   139 syntax

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

   141 translations

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

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

   144

   145

   146 subsection {* Subsets and bounded quantifiers *}

   147

   148 abbreviation

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

   150   "subset \<equiv> less"

   151

   152 abbreviation

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

   154   "subset_eq \<equiv> less_eq"

   155

   156 notation (output)

   157   subset  ("op <") and

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

   159   subset_eq  ("op <=") and

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

   161

   162 notation (xsymbols)

   163   subset  ("op \<subset>") and

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

   165   subset_eq  ("op \<subseteq>") and

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

   167

   168 notation (HTML output)

   169   subset  ("op \<subset>") and

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

   171   subset_eq  ("op \<subseteq>") and

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

   173

   174 abbreviation (input)

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

   176   "supset \<equiv> greater"

   177

   178 abbreviation (input)

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

   180   "supset_eq \<equiv> greater_eq"

   181

   182 notation (xsymbols)

   183   supset  ("op \<supset>") and

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

   185   supset_eq  ("op \<supseteq>") and

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

   187

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

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

   190

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

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

   193

   194 syntax

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

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

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

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

   199

   200 syntax (HOL)

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

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

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

   204

   205 syntax (xsymbols)

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

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

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

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

   210

   211 syntax (HTML output)

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

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

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

   215

   216 translations

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

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

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

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

   221

   222 syntax (output)

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

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

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

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

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

   228

   229 syntax (xsymbols)

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

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

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

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

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

   235

   236 syntax (HOL output)

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

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

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

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

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

   242

   243 syntax (HTML output)

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

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

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

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

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

   249

   250 translations

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

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

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

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

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

   256

   257 print_translation {*

   258 let

   259   val Type (set_type, _) = @{typ "'a set"};   (* FIXME 'a => bool (!?!) *)

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

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

   262   val impl = @{const_syntax HOL.implies};

   263   val conj = @{const_syntax HOL.conj};

   264   val sbset = @{const_syntax subset};

   265   val sbset_eq = @{const_syntax subset_eq};

   266

   267   val trans =

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

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

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

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

   272

   273   fun mk v v' c n P =

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

   275     then Syntax.const c $Syntax_Trans.mark_bound v'$ n $P else raise Match;   276   277 fun tr' q = (q,   278 fn [Const (@{syntax_const "_bound"}, _)$ Free (v, Type (T, _)),

   279             Const (c, _) $  280 (Const (d, _)$ (Const (@{syntax_const "_bound"}, _) $Free (v', _))$ n) $P] =>   281 if T = set_type then   282 (case AList.lookup (op =) trans (q, c, d) of   283 NONE => raise Match   284 | SOME l => mk v v' l n P)   285 else raise Match   286 | _ => raise Match);   287 in   288 [tr' All_binder, tr' Ex_binder]   289 end   290 *}   291   292   293 text {*   294 \medskip Translate between @{text "{e | x1...xn. P}"} and @{text   295 "{u. EX x1..xn. u = e & P}"}; @{text "{y. EX x1..xn. y = e & P}"} is   296 only translated if @{text "[0..n] subset bvs(e)"}.   297 *}   298   299 syntax   300 "_Setcompr" :: "'a => idts => bool => 'a set" ("(1{_ |/_./ _})")   301   302 parse_translation {*   303 let   304 val ex_tr = snd (Syntax_Trans.mk_binder_tr ("EX ", @{const_syntax Ex}));   305   306 fun nvars (Const (@{syntax_const "_idts"}, _)$ _ $idts) = nvars idts + 1   307 | nvars _ = 1;   308   309 fun setcompr_tr [e, idts, b] =   310 let   311 val eq = Syntax.const @{const_syntax HOL.eq}$ Bound (nvars idts) $e;   312 val P = Syntax.const @{const_syntax HOL.conj}$ eq $b;   313 val exP = ex_tr [idts, P];   314 in Syntax.const @{const_syntax Collect}$ absdummy dummyT exP end;

   315

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

   317 *}

   318

   319 print_translation {*

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

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

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

   323

   324 print_translation {*

   325 let

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

   327

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

   329     let

   330       fun check (Const (@{const_syntax Ex}, _) $Abs (_, _, P), n) = check (P, n + 1)   331 | check (Const (@{const_syntax HOL.conj}, _)$

   332               (Const (@{const_syntax HOL.eq}, _) $Bound m$ e) $P, n) =   333 n > 0 andalso m = n andalso not (loose_bvar1 (P, n)) andalso   334 subset (op =) (0 upto (n - 1), add_loose_bnos (e, 0, []))   335 | check _ = false;   336   337 fun tr' (_$ abs) =

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

   339           in Syntax.const @{syntax_const "_Setcompr"} $e$ idts $Q end;   340 in   341 if check (P, 0) then tr' P   342 else   343 let   344 val (x as _$ Free(xN, _), t) = Syntax_Trans.atomic_abs_tr' abs;

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

   346         in

   347           case t of

   348             Const (@{const_syntax HOL.conj}, _) $  349 (Const (@{const_syntax Set.member}, _)$

   350                 (Const (@{syntax_const "_bound"}, _) $Free (yN, _))$ A) $P =>   351 if xN = yN then Syntax.const @{syntax_const "_Collect"}$ x $A$ P else M

   352           | _ => M

   353         end

   354     end;

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

   356 *}

   357

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

   359   let

   360     val unfold_bex_tac = unfold_tac @{thms Bex_def};

   361     fun prove_bex_tac ss = unfold_bex_tac ss THEN Quantifier1.prove_one_point_ex_tac;

   362   in fn _ => fn ss => Quantifier1.rearrange_bex (prove_bex_tac ss) ss end

   363 *}

   364

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

   366   let

   367     val unfold_ball_tac = unfold_tac @{thms Ball_def};

   368     fun prove_ball_tac ss = unfold_ball_tac ss THEN Quantifier1.prove_one_point_all_tac;

   369   in fn _ => fn ss => Quantifier1.rearrange_ball (prove_ball_tac ss) ss end

   370 *}

   371

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

   373   by (simp add: Ball_def)

   374

   375 lemmas strip = impI allI ballI

   376

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

   378   by (simp add: Ball_def)

   379

   380 text {*

   381   Gives better instantiation for bound:

   382 *}

   383

   384 declaration {* fn _ =>

   385   Classical.map_cs (fn cs => cs addbefore ("bspec", datac @{thm bspec} 1))

   386 *}

   387

   388 ML {*

   389 structure Simpdata =

   390 struct

   391

   392 open Simpdata;

   393

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

   395

   396 end;

   397

   398 open Simpdata;

   399 *}

   400

   401 declaration {* fn _ =>

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

   403 *}

   404

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

   406   by (unfold Ball_def) blast

   407

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

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

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

   411   by (unfold Bex_def) blast

   412

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

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

   415   by (unfold Bex_def) blast

   416

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

   418   by (unfold Bex_def) blast

   419

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

   421   by (unfold Bex_def) blast

   422

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

   424   -- {* Trival rewrite rule. *}

   425   by (simp add: Ball_def)

   426

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

   428   -- {* Dual form for existentials. *}

   429   by (simp add: Bex_def)

   430

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

   432   by blast

   433

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

   435   by blast

   436

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

   438   by blast

   439

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

   441   by blast

   442

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

   444   by blast

   445

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

   447   by blast

   448

   449 lemma ball_conj_distrib:

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

   451   by blast

   452

   453 lemma bex_disj_distrib:

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

   455   by blast

   456

   457

   458 text {* Congruence rules *}

   459

   460 lemma ball_cong:

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

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

   463   by (simp add: Ball_def)

   464

   465 lemma strong_ball_cong [cong]:

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

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

   468   by (simp add: simp_implies_def Ball_def)

   469

   470 lemma bex_cong:

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

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

   473   by (simp add: Bex_def cong: conj_cong)

   474

   475 lemma strong_bex_cong [cong]:

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

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

   478   by (simp add: simp_implies_def Bex_def cong: conj_cong)

   479

   480

   481 subsection {* Basic operations *}

   482

   483 subsubsection {* Subsets *}

   484

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

   486   by (simp add: less_eq_set_def le_fun_def)

   487

   488 text {*

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

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

   491   "'a set"}.

   492 *}

   493

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

   495   by (simp add: less_eq_set_def le_fun_def)

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

   497

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

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

   500       cf @{text rev_mp}. *}

   501   by (rule subsetD)

   502

   503 text {*

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

   505 *}

   506

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

   508   -- {* Classical elimination rule. *}

   509   by (auto simp add: less_eq_set_def le_fun_def)

   510

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

   512

   513 lemma contra_subsetD [no_atp]: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A"

   514   by blast

   515

   516 lemma subset_refl: "A \<subseteq> A"

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

   518

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

   520   by (fact order_trans)

   521

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

   523   by (rule subsetD)

   524

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

   526   by (rule subsetD)

   527

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

   529   by simp

   530

   531 lemmas basic_trans_rules [trans] =

   532   order_trans_rules set_rev_mp set_mp eq_mem_trans

   533

   534

   535 subsubsection {* Equality *}

   536

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

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

   539   by (iprover intro: set_eqI subsetD)

   540

   541 text {*

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

   543   here?

   544 *}

   545

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

   547   by simp

   548

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

   550   by simp

   551

   552 text {*

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

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

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

   556 *}

   557

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

   559   by simp

   560

   561 lemma equalityCE [elim]:

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

   563   by blast

   564

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

   566   by simp

   567

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

   569   by simp

   570

   571

   572 subsubsection {* The empty set *}

   573

   574 lemma empty_def:

   575   "{} = {x. False}"

   576   by (simp add: bot_set_def bot_fun_def)

   577

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

   579   by (simp add: empty_def)

   580

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

   582   by simp

   583

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

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

   586   by blast

   587

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

   589   by blast

   590

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

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

   593   by blast

   594

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

   596   by (simp add: Ball_def)

   597

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

   599   by (simp add: Bex_def)

   600

   601

   602 subsubsection {* The universal set -- UNIV *}

   603

   604 abbreviation UNIV :: "'a set" where

   605   "UNIV \<equiv> top"

   606

   607 lemma UNIV_def:

   608   "UNIV = {x. True}"

   609   by (simp add: top_set_def top_fun_def)

   610

   611 lemma UNIV_I [simp]: "x : UNIV"

   612   by (simp add: UNIV_def)

   613

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

   615

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

   617   by simp

   618

   619 lemma subset_UNIV: "A \<subseteq> UNIV"

   620   by (fact top_greatest) (* already simp *)

   621

   622 text {*

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

   624   causes them to be ignored because of their interaction with

   625   congruence rules.

   626 *}

   627

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

   629   by (simp add: Ball_def)

   630

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

   632   by (simp add: Bex_def)

   633

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

   635   by auto

   636

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

   638   by (blast elim: equalityE)

   639

   640

   641 subsubsection {* The Powerset operator -- Pow *}

   642

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

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

   645

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

   647   by (simp add: Pow_def)

   648

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

   650   by (simp add: Pow_def)

   651

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

   653   by (simp add: Pow_def)

   654

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

   656   by simp

   657

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

   659   by simp

   660

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

   662   using Pow_top by blast

   663

   664

   665 subsubsection {* Set complement *}

   666

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

   668   by (simp add: fun_Compl_def uminus_set_def)

   669

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

   671   by (simp add: fun_Compl_def uminus_set_def) blast

   672

   673 text {*

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

   675   Classical prover.  Negated assumptions behave like formulae on the

   676   right side of the notional turnstile ... *}

   677

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

   679   by simp

   680

   681 lemmas ComplE = ComplD [elim_format]

   682

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

   684   by blast

   685

   686

   687 subsubsection {* Binary intersection *}

   688

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

   690   "op Int \<equiv> inf"

   691

   692 notation (xsymbols)

   693   inter  (infixl "\<inter>" 70)

   694

   695 notation (HTML output)

   696   inter  (infixl "\<inter>" 70)

   697

   698 lemma Int_def:

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

   700   by (simp add: inf_set_def inf_fun_def)

   701

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

   703   by (unfold Int_def) blast

   704

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

   706   by simp

   707

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

   709   by simp

   710

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

   712   by simp

   713

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

   715   by simp

   716

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

   718   by (fact mono_inf)

   719

   720

   721 subsubsection {* Binary union *}

   722

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

   724   "union \<equiv> sup"

   725

   726 notation (xsymbols)

   727   union  (infixl "\<union>" 65)

   728

   729 notation (HTML output)

   730   union  (infixl "\<union>" 65)

   731

   732 lemma Un_def:

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

   734   by (simp add: sup_set_def sup_fun_def)

   735

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

   737   by (unfold Un_def) blast

   738

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

   740   by simp

   741

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

   743   by simp

   744

   745 text {*

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

   747   @{prop B}.

   748 *}

   749

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

   751   by auto

   752

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

   754   by (unfold Un_def) blast

   755

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

   757   by (simp add: insert_compr Un_def)

   758

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

   760   by (fact mono_sup)

   761

   762

   763 subsubsection {* Set difference *}

   764

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

   766   by (simp add: minus_set_def fun_diff_def)

   767

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

   769   by simp

   770

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

   772   by simp

   773

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

   775   by simp

   776

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

   778   by simp

   779

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

   781

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

   783 by blast

   784

   785

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

   787

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

   789   by (unfold insert_def) blast

   790

   791 lemma insertI1: "a : insert a B"

   792   by simp

   793

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

   795   by simp

   796

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

   798   by (unfold insert_def) blast

   799

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

   801   -- {* Classical introduction rule. *}

   802   by auto

   803

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

   805   by auto

   806

   807 lemma set_insert:

   808   assumes "x \<in> A"

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

   810 proof

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

   812 next

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

   814 qed

   815

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

   817 by auto

   818

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

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

   821   (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)"

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

   823 proof

   824   assume ?L

   825   show ?R

   826   proof cases

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

   828   next

   829     assume "a\<noteq>b"

   830     let ?C = "A - {b}"

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

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

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

   834   qed

   835 next

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

   837 qed

   838

   839 subsubsection {* Singletons, using insert *}

   840

   841 lemma singletonI [intro!,no_atp]: "a : {a}"

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

   843   by (rule insertI1)

   844

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

   846   by blast

   847

   848 lemmas singletonE = singletonD [elim_format]

   849

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

   851   by blast

   852

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

   854   by blast

   855

   856 lemma singleton_insert_inj_eq [iff,no_atp]:

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

   858   by blast

   859

   860 lemma singleton_insert_inj_eq' [iff,no_atp]:

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

   862   by blast

   863

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

   865   by fast

   866

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

   868   by blast

   869

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

   871   by blast

   872

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

   874   by blast

   875

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

   877   by (blast elim: equalityE)

   878

   879

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

   881

   882 text {*

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

   884 *}

   885

   886 definition image :: "('a => 'b) => 'a set => 'b set" (infixr "" 90) where

   887   image_def [no_atp]: "f  A = {y. EX x:A. y = f(x)}"

   888

   889 abbreviation

   890   range :: "('a => 'b) => 'b set" where -- "of function"

   891   "range f == f  UNIV"

   892

   893 lemma image_eqI [simp, intro]: "b = f x ==> x:A ==> b : fA"

   894   by (unfold image_def) blast

   895

   896 lemma imageI: "x : A ==> f x : f  A"

   897   by (rule image_eqI) (rule refl)

   898

   899 lemma rev_image_eqI: "x:A ==> b = f x ==> b : fA"

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

   901     required @{term x}. *}

   902   by (unfold image_def) blast

   903

   904 lemma imageE [elim!]:

   905   "b : (%x. f x)A ==> (!!x. b = f x ==> x:A ==> P) ==> P"

   906   -- {* The eta-expansion gives variable-name preservation. *}

   907   by (unfold image_def) blast

   908

   909 lemma image_Un: "f(A Un B) = fA Un fB"

   910   by blast

   911

   912 lemma image_iff: "(z : fA) = (EX x:A. z = f x)"

   913   by blast

   914

   915 lemma image_subset_iff [no_atp]: "(fA \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)"

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

   917   by blast

   918

   919 lemma subset_image_iff: "(B \<subseteq> fA) = (EX AA. AA \<subseteq> A & B = fAA)"

   920   apply safe

   921    prefer 2 apply fast

   922   apply (rule_tac x = "{a. a : A & f a : B}" in exI, fast)

   923   done

   924

   925 lemma image_subsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> fA \<subseteq> B"

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

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

   928   by blast

   929

   930 text {*

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

   932 *}

   933

   934 lemma image_ident [simp]: "(%x. x)  Y = Y"

   935   by blast

   936

   937 lemma range_eqI: "b = f x ==> b \<in> range f"

   938   by simp

   939

   940 lemma rangeI: "f x \<in> range f"

   941   by simp

   942

   943 lemma rangeE [elim?]: "b \<in> range (\<lambda>x. f x) ==> (!!x. b = f x ==> P) ==> P"

   944   by blast

   945

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

   947

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

   949

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

   951   by auto

   952

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

   954   by auto

   955

   956 text {*

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

   958   "split_if [split]"}.

   959 *}

   960

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

   962   by (rule split_if)

   963

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

   965   by (rule split_if)

   966

   967 text {*

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

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

   970 *}

   971

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

   973   by (rule split_if)

   974

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

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

   977

   978 lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2

   979

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

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

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

   983   apply, then the formula should be kept.

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

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

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

   987  *)

   988

   989

   990 subsection {* Further operations and lemmas *}

   991

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

   993

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

   995   by (unfold less_le) blast

   996

   997 lemma psubsetE [elim!,no_atp]:

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

   999   by (unfold less_le) blast

  1000

  1001 lemma psubset_insert_iff:

  1002   "(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)"

  1003   by (auto simp add: less_le subset_insert_iff)

  1004

  1005 lemma psubset_eq: "(A \<subset> B) = (A \<subseteq> B & A \<noteq> B)"

  1006   by (simp only: less_le)

  1007

  1008 lemma psubset_imp_subset: "A \<subset> B ==> A \<subseteq> B"

  1009   by (simp add: psubset_eq)

  1010

  1011 lemma psubset_trans: "[| A \<subset> B; B \<subset> C |] ==> A \<subset> C"

  1012 apply (unfold less_le)

  1013 apply (auto dest: subset_antisym)

  1014 done

  1015

  1016 lemma psubsetD: "[| A \<subset> B; c \<in> A |] ==> c \<in> B"

  1017 apply (unfold less_le)

  1018 apply (auto dest: subsetD)

  1019 done

  1020

  1021 lemma psubset_subset_trans: "A \<subset> B ==> B \<subseteq> C ==> A \<subset> C"

  1022   by (auto simp add: psubset_eq)

  1023

  1024 lemma subset_psubset_trans: "A \<subseteq> B ==> B \<subset> C ==> A \<subset> C"

  1025   by (auto simp add: psubset_eq)

  1026

  1027 lemma psubset_imp_ex_mem: "A \<subset> B ==> \<exists>b. b \<in> (B - A)"

  1028   by (unfold less_le) blast

  1029

  1030 lemma atomize_ball:

  1031     "(!!x. x \<in> A ==> P x) == Trueprop (\<forall>x\<in>A. P x)"

  1032   by (simp only: Ball_def atomize_all atomize_imp)

  1033

  1034 lemmas [symmetric, rulify] = atomize_ball

  1035   and [symmetric, defn] = atomize_ball

  1036

  1037 lemma image_Pow_mono:

  1038   assumes "f  A \<le> B"

  1039   shows "(image f)  (Pow A) \<le> Pow B"

  1040 using assms by blast

  1041

  1042 lemma image_Pow_surj:

  1043   assumes "f  A = B"

  1044   shows "(image f)  (Pow A) = Pow B"

  1045 using assms unfolding Pow_def proof(auto)

  1046   fix Y assume *: "Y \<le> f  A"

  1047   obtain X where X_def: "X = {x \<in> A. f x \<in> Y}" by blast

  1048   have "f  X = Y \<and> X \<le> A" unfolding X_def using * by auto

  1049   thus "Y \<in> (image f)  {X. X \<le> A}" by blast

  1050 qed

  1051

  1052 subsubsection {* Derived rules involving subsets. *}

  1053

  1054 text {* @{text insert}. *}

  1055

  1056 lemma subset_insertI: "B \<subseteq> insert a B"

  1057   by (rule subsetI) (erule insertI2)

  1058

  1059 lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B"

  1060   by blast

  1061

  1062 lemma subset_insert: "x \<notin> A ==> (A \<subseteq> insert x B) = (A \<subseteq> B)"

  1063   by blast

  1064

  1065

  1066 text {* \medskip Finite Union -- the least upper bound of two sets. *}

  1067

  1068 lemma Un_upper1: "A \<subseteq> A \<union> B"

  1069   by (fact sup_ge1)

  1070

  1071 lemma Un_upper2: "B \<subseteq> A \<union> B"

  1072   by (fact sup_ge2)

  1073

  1074 lemma Un_least: "A \<subseteq> C ==> B \<subseteq> C ==> A \<union> B \<subseteq> C"

  1075   by (fact sup_least)

  1076

  1077

  1078 text {* \medskip Finite Intersection -- the greatest lower bound of two sets. *}

  1079

  1080 lemma Int_lower1: "A \<inter> B \<subseteq> A"

  1081   by (fact inf_le1)

  1082

  1083 lemma Int_lower2: "A \<inter> B \<subseteq> B"

  1084   by (fact inf_le2)

  1085

  1086 lemma Int_greatest: "C \<subseteq> A ==> C \<subseteq> B ==> C \<subseteq> A \<inter> B"

  1087   by (fact inf_greatest)

  1088

  1089

  1090 text {* \medskip Set difference. *}

  1091

  1092 lemma Diff_subset: "A - B \<subseteq> A"

  1093   by blast

  1094

  1095 lemma Diff_subset_conv: "(A - B \<subseteq> C) = (A \<subseteq> B \<union> C)"

  1096 by blast

  1097

  1098

  1099 subsubsection {* Equalities involving union, intersection, inclusion, etc. *}

  1100

  1101 text {* @{text "{}"}. *}

  1102

  1103 lemma Collect_const [simp]: "{s. P} = (if P then UNIV else {})"

  1104   -- {* supersedes @{text "Collect_False_empty"} *}

  1105   by auto

  1106

  1107 lemma subset_empty [simp]: "(A \<subseteq> {}) = (A = {})"

  1108   by (fact bot_unique)

  1109

  1110 lemma not_psubset_empty [iff]: "\<not> (A < {})"

  1111   by (fact not_less_bot) (* FIXME: already simp *)

  1112

  1113 lemma Collect_empty_eq [simp]: "(Collect P = {}) = (\<forall>x. \<not> P x)"

  1114 by blast

  1115

  1116 lemma empty_Collect_eq [simp]: "({} = Collect P) = (\<forall>x. \<not> P x)"

  1117 by blast

  1118

  1119 lemma Collect_neg_eq: "{x. \<not> P x} = - {x. P x}"

  1120   by blast

  1121

  1122 lemma Collect_disj_eq: "{x. P x | Q x} = {x. P x} \<union> {x. Q x}"

  1123   by blast

  1124

  1125 lemma Collect_imp_eq: "{x. P x --> Q x} = -{x. P x} \<union> {x. Q x}"

  1126   by blast

  1127

  1128 lemma Collect_conj_eq: "{x. P x & Q x} = {x. P x} \<inter> {x. Q x}"

  1129   by blast

  1130

  1131

  1132 text {* \medskip @{text insert}. *}

  1133

  1134 lemma insert_is_Un: "insert a A = {a} Un A"

  1135   -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a {}"} *}

  1136   by blast

  1137

  1138 lemma insert_not_empty [simp]: "insert a A \<noteq> {}"

  1139   by blast

  1140

  1141 lemmas empty_not_insert = insert_not_empty [symmetric]

  1142 declare empty_not_insert [simp]

  1143

  1144 lemma insert_absorb: "a \<in> A ==> insert a A = A"

  1145   -- {* @{text "[simp]"} causes recursive calls when there are nested inserts *}

  1146   -- {* with \emph{quadratic} running time *}

  1147   by blast

  1148

  1149 lemma insert_absorb2 [simp]: "insert x (insert x A) = insert x A"

  1150   by blast

  1151

  1152 lemma insert_commute: "insert x (insert y A) = insert y (insert x A)"

  1153   by blast

  1154

  1155 lemma insert_subset [simp]: "(insert x A \<subseteq> B) = (x \<in> B & A \<subseteq> B)"

  1156   by blast

  1157

  1158 lemma mk_disjoint_insert: "a \<in> A ==> \<exists>B. A = insert a B & a \<notin> B"

  1159   -- {* use new @{text B} rather than @{text "A - {a}"} to avoid infinite unfolding *}

  1160   apply (rule_tac x = "A - {a}" in exI, blast)

  1161   done

  1162

  1163 lemma insert_Collect: "insert a (Collect P) = {u. u \<noteq> a --> P u}"

  1164   by auto

  1165

  1166 lemma insert_inter_insert[simp]: "insert a A \<inter> insert a B = insert a (A \<inter> B)"

  1167   by blast

  1168

  1169 lemma insert_disjoint [simp,no_atp]:

  1170  "(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})"

  1171  "({} = insert a A \<inter> B) = (a \<notin> B \<and> {} = A \<inter> B)"

  1172   by auto

  1173

  1174 lemma disjoint_insert [simp,no_atp]:

  1175  "(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})"

  1176  "({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)"

  1177   by auto

  1178

  1179 text {* \medskip @{text image}. *}

  1180

  1181 lemma image_empty [simp]: "f{} = {}"

  1182   by blast

  1183

  1184 lemma image_insert [simp]: "f  insert a B = insert (f a) (fB)"

  1185   by blast

  1186

  1187 lemma image_constant: "x \<in> A ==> (\<lambda>x. c)  A = {c}"

  1188   by auto

  1189

  1190 lemma image_constant_conv: "(%x. c)  A = (if A = {} then {} else {c})"

  1191 by auto

  1192

  1193 lemma image_image: "f  (g  A) = (\<lambda>x. f (g x))  A"

  1194 by blast

  1195

  1196 lemma insert_image [simp]: "x \<in> A ==> insert (f x) (fA) = fA"

  1197 by blast

  1198

  1199 lemma image_is_empty [iff]: "(fA = {}) = (A = {})"

  1200 by blast

  1201

  1202 lemma empty_is_image[iff]: "({} = f  A) = (A = {})"

  1203 by blast

  1204

  1205

  1206 lemma image_Collect [no_atp]: "f  {x. P x} = {f x | x. P x}"

  1207   -- {* NOT suitable as a default simprule: the RHS isn't simpler than the LHS,

  1208       with its implicit quantifier and conjunction.  Also image enjoys better

  1209       equational properties than does the RHS. *}

  1210   by blast

  1211

  1212 lemma if_image_distrib [simp]:

  1213   "(\<lambda>x. if P x then f x else g x)  S

  1214     = (f  (S \<inter> {x. P x})) \<union> (g  (S \<inter> {x. \<not> P x}))"

  1215   by (auto simp add: image_def)

  1216

  1217 lemma image_cong: "M = N ==> (!!x. x \<in> N ==> f x = g x) ==> fM = gN"

  1218   by (simp add: image_def)

  1219

  1220 lemma image_Int_subset: "f(A Int B) <= fA Int fB"

  1221 by blast

  1222

  1223 lemma image_diff_subset: "fA - fB <= f(A - B)"

  1224 by blast

  1225

  1226

  1227 text {* \medskip @{text range}. *}

  1228

  1229 lemma full_SetCompr_eq [no_atp]: "{u. \<exists>x. u = f x} = range f"

  1230   by auto

  1231

  1232 lemma range_composition: "range (\<lambda>x. f (g x)) = frange g"

  1233 by (subst image_image, simp)

  1234

  1235

  1236 text {* \medskip @{text Int} *}

  1237

  1238 lemma Int_absorb: "A \<inter> A = A"

  1239   by (fact inf_idem) (* already simp *)

  1240

  1241 lemma Int_left_absorb: "A \<inter> (A \<inter> B) = A \<inter> B"

  1242   by (fact inf_left_idem)

  1243

  1244 lemma Int_commute: "A \<inter> B = B \<inter> A"

  1245   by (fact inf_commute)

  1246

  1247 lemma Int_left_commute: "A \<inter> (B \<inter> C) = B \<inter> (A \<inter> C)"

  1248   by (fact inf_left_commute)

  1249

  1250 lemma Int_assoc: "(A \<inter> B) \<inter> C = A \<inter> (B \<inter> C)"

  1251   by (fact inf_assoc)

  1252

  1253 lemmas Int_ac = Int_assoc Int_left_absorb Int_commute Int_left_commute

  1254   -- {* Intersection is an AC-operator *}

  1255

  1256 lemma Int_absorb1: "B \<subseteq> A ==> A \<inter> B = B"

  1257   by (fact inf_absorb2)

  1258

  1259 lemma Int_absorb2: "A \<subseteq> B ==> A \<inter> B = A"

  1260   by (fact inf_absorb1)

  1261

  1262 lemma Int_empty_left: "{} \<inter> B = {}"

  1263   by (fact inf_bot_left) (* already simp *)

  1264

  1265 lemma Int_empty_right: "A \<inter> {} = {}"

  1266   by (fact inf_bot_right) (* already simp *)

  1267

  1268 lemma disjoint_eq_subset_Compl: "(A \<inter> B = {}) = (A \<subseteq> -B)"

  1269   by blast

  1270

  1271 lemma disjoint_iff_not_equal: "(A \<inter> B = {}) = (\<forall>x\<in>A. \<forall>y\<in>B. x \<noteq> y)"

  1272   by blast

  1273

  1274 lemma Int_UNIV_left: "UNIV \<inter> B = B"

  1275   by (fact inf_top_left) (* already simp *)

  1276

  1277 lemma Int_UNIV_right: "A \<inter> UNIV = A"

  1278   by (fact inf_top_right) (* already simp *)

  1279

  1280 lemma Int_Un_distrib: "A \<inter> (B \<union> C) = (A \<inter> B) \<union> (A \<inter> C)"

  1281   by (fact inf_sup_distrib1)

  1282

  1283 lemma Int_Un_distrib2: "(B \<union> C) \<inter> A = (B \<inter> A) \<union> (C \<inter> A)"

  1284   by (fact inf_sup_distrib2)

  1285

  1286 lemma Int_UNIV [simp,no_atp]: "(A \<inter> B = UNIV) = (A = UNIV & B = UNIV)"

  1287   by (fact inf_eq_top_iff) (* already simp *)

  1288

  1289 lemma Int_subset_iff [no_atp, simp]: "(C \<subseteq> A \<inter> B) = (C \<subseteq> A & C \<subseteq> B)"

  1290   by (fact le_inf_iff)

  1291

  1292 lemma Int_Collect: "(x \<in> A \<inter> {x. P x}) = (x \<in> A & P x)"

  1293   by blast

  1294

  1295

  1296 text {* \medskip @{text Un}. *}

  1297

  1298 lemma Un_absorb: "A \<union> A = A"

  1299   by (fact sup_idem) (* already simp *)

  1300

  1301 lemma Un_left_absorb: "A \<union> (A \<union> B) = A \<union> B"

  1302   by (fact sup_left_idem)

  1303

  1304 lemma Un_commute: "A \<union> B = B \<union> A"

  1305   by (fact sup_commute)

  1306

  1307 lemma Un_left_commute: "A \<union> (B \<union> C) = B \<union> (A \<union> C)"

  1308   by (fact sup_left_commute)

  1309

  1310 lemma Un_assoc: "(A \<union> B) \<union> C = A \<union> (B \<union> C)"

  1311   by (fact sup_assoc)

  1312

  1313 lemmas Un_ac = Un_assoc Un_left_absorb Un_commute Un_left_commute

  1314   -- {* Union is an AC-operator *}

  1315

  1316 lemma Un_absorb1: "A \<subseteq> B ==> A \<union> B = B"

  1317   by (fact sup_absorb2)

  1318

  1319 lemma Un_absorb2: "B \<subseteq> A ==> A \<union> B = A"

  1320   by (fact sup_absorb1)

  1321

  1322 lemma Un_empty_left: "{} \<union> B = B"

  1323   by (fact sup_bot_left) (* already simp *)

  1324

  1325 lemma Un_empty_right: "A \<union> {} = A"

  1326   by (fact sup_bot_right) (* already simp *)

  1327

  1328 lemma Un_UNIV_left: "UNIV \<union> B = UNIV"

  1329   by (fact sup_top_left) (* already simp *)

  1330

  1331 lemma Un_UNIV_right: "A \<union> UNIV = UNIV"

  1332   by (fact sup_top_right) (* already simp *)

  1333

  1334 lemma Un_insert_left [simp]: "(insert a B) \<union> C = insert a (B \<union> C)"

  1335   by blast

  1336

  1337 lemma Un_insert_right [simp]: "A \<union> (insert a B) = insert a (A \<union> B)"

  1338   by blast

  1339

  1340 lemma Int_insert_left:

  1341     "(insert a B) Int C = (if a \<in> C then insert a (B \<inter> C) else B \<inter> C)"

  1342   by auto

  1343

  1344 lemma Int_insert_left_if0[simp]:

  1345     "a \<notin> C \<Longrightarrow> (insert a B) Int C = B \<inter> C"

  1346   by auto

  1347

  1348 lemma Int_insert_left_if1[simp]:

  1349     "a \<in> C \<Longrightarrow> (insert a B) Int C = insert a (B Int C)"

  1350   by auto

  1351

  1352 lemma Int_insert_right:

  1353     "A \<inter> (insert a B) = (if a \<in> A then insert a (A \<inter> B) else A \<inter> B)"

  1354   by auto

  1355

  1356 lemma Int_insert_right_if0[simp]:

  1357     "a \<notin> A \<Longrightarrow> A Int (insert a B) = A Int B"

  1358   by auto

  1359

  1360 lemma Int_insert_right_if1[simp]:

  1361     "a \<in> A \<Longrightarrow> A Int (insert a B) = insert a (A Int B)"

  1362   by auto

  1363

  1364 lemma Un_Int_distrib: "A \<union> (B \<inter> C) = (A \<union> B) \<inter> (A \<union> C)"

  1365   by (fact sup_inf_distrib1)

  1366

  1367 lemma Un_Int_distrib2: "(B \<inter> C) \<union> A = (B \<union> A) \<inter> (C \<union> A)"

  1368   by (fact sup_inf_distrib2)

  1369

  1370 lemma Un_Int_crazy:

  1371     "(A \<inter> B) \<union> (B \<inter> C) \<union> (C \<inter> A) = (A \<union> B) \<inter> (B \<union> C) \<inter> (C \<union> A)"

  1372   by blast

  1373

  1374 lemma subset_Un_eq: "(A \<subseteq> B) = (A \<union> B = B)"

  1375   by (fact le_iff_sup)

  1376

  1377 lemma Un_empty [iff]: "(A \<union> B = {}) = (A = {} & B = {})"

  1378   by (fact sup_eq_bot_iff) (* FIXME: already simp *)

  1379

  1380 lemma Un_subset_iff [no_atp, simp]: "(A \<union> B \<subseteq> C) = (A \<subseteq> C & B \<subseteq> C)"

  1381   by (fact le_sup_iff)

  1382

  1383 lemma Un_Diff_Int: "(A - B) \<union> (A \<inter> B) = A"

  1384   by blast

  1385

  1386 lemma Diff_Int2: "A \<inter> C - B \<inter> C = A \<inter> C - B"

  1387   by blast

  1388

  1389

  1390 text {* \medskip Set complement *}

  1391

  1392 lemma Compl_disjoint [simp]: "A \<inter> -A = {}"

  1393   by (fact inf_compl_bot)

  1394

  1395 lemma Compl_disjoint2 [simp]: "-A \<inter> A = {}"

  1396   by (fact compl_inf_bot)

  1397

  1398 lemma Compl_partition: "A \<union> -A = UNIV"

  1399   by (fact sup_compl_top)

  1400

  1401 lemma Compl_partition2: "-A \<union> A = UNIV"

  1402   by (fact compl_sup_top)

  1403

  1404 lemma double_complement: "- (-A) = (A::'a set)"

  1405   by (fact double_compl) (* already simp *)

  1406

  1407 lemma Compl_Un: "-(A \<union> B) = (-A) \<inter> (-B)"

  1408   by (fact compl_sup) (* already simp *)

  1409

  1410 lemma Compl_Int: "-(A \<inter> B) = (-A) \<union> (-B)"

  1411   by (fact compl_inf) (* already simp *)

  1412

  1413 lemma subset_Compl_self_eq: "(A \<subseteq> -A) = (A = {})"

  1414   by blast

  1415

  1416 lemma Un_Int_assoc_eq: "((A \<inter> B) \<union> C = A \<inter> (B \<union> C)) = (C \<subseteq> A)"

  1417   -- {* Halmos, Naive Set Theory, page 16. *}

  1418   by blast

  1419

  1420 lemma Compl_UNIV_eq: "-UNIV = {}"

  1421   by (fact compl_top_eq) (* already simp *)

  1422

  1423 lemma Compl_empty_eq: "-{} = UNIV"

  1424   by (fact compl_bot_eq) (* already simp *)

  1425

  1426 lemma Compl_subset_Compl_iff [iff]: "(-A \<subseteq> -B) = (B \<subseteq> A)"

  1427   by (fact compl_le_compl_iff) (* FIXME: already simp *)

  1428

  1429 lemma Compl_eq_Compl_iff [iff]: "(-A = -B) = (A = (B::'a set))"

  1430   by (fact compl_eq_compl_iff) (* FIXME: already simp *)

  1431

  1432 lemma Compl_insert: "- insert x A = (-A) - {x}"

  1433   by blast

  1434

  1435 text {* \medskip Bounded quantifiers.

  1436

  1437   The following are not added to the default simpset because

  1438   (a) they duplicate the body and (b) there are no similar rules for @{text Int}. *}

  1439

  1440 lemma ball_Un: "(\<forall>x \<in> A \<union> B. P x) = ((\<forall>x\<in>A. P x) & (\<forall>x\<in>B. P x))"

  1441   by blast

  1442

  1443 lemma bex_Un: "(\<exists>x \<in> A \<union> B. P x) = ((\<exists>x\<in>A. P x) | (\<exists>x\<in>B. P x))"

  1444   by blast

  1445

  1446

  1447 text {* \medskip Set difference. *}

  1448

  1449 lemma Diff_eq: "A - B = A \<inter> (-B)"

  1450   by blast

  1451

  1452 lemma Diff_eq_empty_iff [simp,no_atp]: "(A - B = {}) = (A \<subseteq> B)"

  1453   by blast

  1454

  1455 lemma Diff_cancel [simp]: "A - A = {}"

  1456   by blast

  1457

  1458 lemma Diff_idemp [simp]: "(A - B) - B = A - (B::'a set)"

  1459 by blast

  1460

  1461 lemma Diff_triv: "A \<inter> B = {} ==> A - B = A"

  1462   by (blast elim: equalityE)

  1463

  1464 lemma empty_Diff [simp]: "{} - A = {}"

  1465   by blast

  1466

  1467 lemma Diff_empty [simp]: "A - {} = A"

  1468   by blast

  1469

  1470 lemma Diff_UNIV [simp]: "A - UNIV = {}"

  1471   by blast

  1472

  1473 lemma Diff_insert0 [simp,no_atp]: "x \<notin> A ==> A - insert x B = A - B"

  1474   by blast

  1475

  1476 lemma Diff_insert: "A - insert a B = A - B - {a}"

  1477   -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a 0"} *}

  1478   by blast

  1479

  1480 lemma Diff_insert2: "A - insert a B = A - {a} - B"

  1481   -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a 0"} *}

  1482   by blast

  1483

  1484 lemma insert_Diff_if: "insert x A - B = (if x \<in> B then A - B else insert x (A - B))"

  1485   by auto

  1486

  1487 lemma insert_Diff1 [simp]: "x \<in> B ==> insert x A - B = A - B"

  1488   by blast

  1489

  1490 lemma insert_Diff_single[simp]: "insert a (A - {a}) = insert a A"

  1491 by blast

  1492

  1493 lemma insert_Diff: "a \<in> A ==> insert a (A - {a}) = A"

  1494   by blast

  1495

  1496 lemma Diff_insert_absorb: "x \<notin> A ==> (insert x A) - {x} = A"

  1497   by auto

  1498

  1499 lemma Diff_disjoint [simp]: "A \<inter> (B - A) = {}"

  1500   by blast

  1501

  1502 lemma Diff_partition: "A \<subseteq> B ==> A \<union> (B - A) = B"

  1503   by blast

  1504

  1505 lemma double_diff: "A \<subseteq> B ==> B \<subseteq> C ==> B - (C - A) = A"

  1506   by blast

  1507

  1508 lemma Un_Diff_cancel [simp]: "A \<union> (B - A) = A \<union> B"

  1509   by blast

  1510

  1511 lemma Un_Diff_cancel2 [simp]: "(B - A) \<union> A = B \<union> A"

  1512   by blast

  1513

  1514 lemma Diff_Un: "A - (B \<union> C) = (A - B) \<inter> (A - C)"

  1515   by blast

  1516

  1517 lemma Diff_Int: "A - (B \<inter> C) = (A - B) \<union> (A - C)"

  1518   by blast

  1519

  1520 lemma Un_Diff: "(A \<union> B) - C = (A - C) \<union> (B - C)"

  1521   by blast

  1522

  1523 lemma Int_Diff: "(A \<inter> B) - C = A \<inter> (B - C)"

  1524   by blast

  1525

  1526 lemma Diff_Int_distrib: "C \<inter> (A - B) = (C \<inter> A) - (C \<inter> B)"

  1527   by blast

  1528

  1529 lemma Diff_Int_distrib2: "(A - B) \<inter> C = (A \<inter> C) - (B \<inter> C)"

  1530   by blast

  1531

  1532 lemma Diff_Compl [simp]: "A - (- B) = A \<inter> B"

  1533   by auto

  1534

  1535 lemma Compl_Diff_eq [simp]: "- (A - B) = -A \<union> B"

  1536   by blast

  1537

  1538

  1539 text {* \medskip Quantification over type @{typ bool}. *}

  1540

  1541 lemma bool_induct: "P True \<Longrightarrow> P False \<Longrightarrow> P x"

  1542   by (cases x) auto

  1543

  1544 lemma all_bool_eq: "(\<forall>b. P b) \<longleftrightarrow> P True \<and> P False"

  1545   by (auto intro: bool_induct)

  1546

  1547 lemma bool_contrapos: "P x \<Longrightarrow> \<not> P False \<Longrightarrow> P True"

  1548   by (cases x) auto

  1549

  1550 lemma ex_bool_eq: "(\<exists>b. P b) \<longleftrightarrow> P True \<or> P False"

  1551   by (auto intro: bool_contrapos)

  1552

  1553 lemma UNIV_bool [no_atp]: "UNIV = {False, True}"

  1554   by (auto intro: bool_induct)

  1555

  1556 text {* \medskip @{text Pow} *}

  1557

  1558 lemma Pow_empty [simp]: "Pow {} = {{}}"

  1559   by (auto simp add: Pow_def)

  1560

  1561 lemma Pow_insert: "Pow (insert a A) = Pow A \<union> (insert a  Pow A)"

  1562   by (blast intro: image_eqI [where ?x = "u - {a}", standard])

  1563

  1564 lemma Pow_Compl: "Pow (- A) = {-B | B. A \<in> Pow B}"

  1565   by (blast intro: exI [where ?x = "- u", standard])

  1566

  1567 lemma Pow_UNIV [simp]: "Pow UNIV = UNIV"

  1568   by blast

  1569

  1570 lemma Un_Pow_subset: "Pow A \<union> Pow B \<subseteq> Pow (A \<union> B)"

  1571   by blast

  1572

  1573 lemma Pow_Int_eq [simp]: "Pow (A \<inter> B) = Pow A \<inter> Pow B"

  1574   by blast

  1575

  1576

  1577 text {* \medskip Miscellany. *}

  1578

  1579 lemma set_eq_subset: "(A = B) = (A \<subseteq> B & B \<subseteq> A)"

  1580   by blast

  1581

  1582 lemma subset_iff [no_atp]: "(A \<subseteq> B) = (\<forall>t. t \<in> A --> t \<in> B)"

  1583   by blast

  1584

  1585 lemma subset_iff_psubset_eq: "(A \<subseteq> B) = ((A \<subset> B) | (A = B))"

  1586   by (unfold less_le) blast

  1587

  1588 lemma all_not_in_conv [simp]: "(\<forall>x. x \<notin> A) = (A = {})"

  1589   by blast

  1590

  1591 lemma ex_in_conv: "(\<exists>x. x \<in> A) = (A \<noteq> {})"

  1592   by blast

  1593

  1594 lemma ball_simps [simp, no_atp]:

  1595   "\<And>A P Q. (\<forall>x\<in>A. P x \<or> Q) \<longleftrightarrow> ((\<forall>x\<in>A. P x) \<or> Q)"

  1596   "\<And>A P Q. (\<forall>x\<in>A. P \<or> Q x) \<longleftrightarrow> (P \<or> (\<forall>x\<in>A. Q x))"

  1597   "\<And>A P Q. (\<forall>x\<in>A. P \<longrightarrow> Q x) \<longleftrightarrow> (P \<longrightarrow> (\<forall>x\<in>A. Q x))"

  1598   "\<And>A P Q. (\<forall>x\<in>A. P x \<longrightarrow> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<longrightarrow> Q)"

  1599   "\<And>P. (\<forall>x\<in>{}. P x) \<longleftrightarrow> True"

  1600   "\<And>P. (\<forall>x\<in>UNIV. P x) \<longleftrightarrow> (\<forall>x. P x)"

  1601   "\<And>a B P. (\<forall>x\<in>insert a B. P x) \<longleftrightarrow> (P a \<and> (\<forall>x\<in>B. P x))"

  1602   "\<And>P Q. (\<forall>x\<in>Collect Q. P x) \<longleftrightarrow> (\<forall>x. Q x \<longrightarrow> P x)"

  1603   "\<And>A P f. (\<forall>x\<in>fA. P x) \<longleftrightarrow> (\<forall>x\<in>A. P (f x))"

  1604   "\<And>A P. (\<not> (\<forall>x\<in>A. P x)) \<longleftrightarrow> (\<exists>x\<in>A. \<not> P x)"

  1605   by auto

  1606

  1607 lemma bex_simps [simp, no_atp]:

  1608   "\<And>A P Q. (\<exists>x\<in>A. P x \<and> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<and> Q)"

  1609   "\<And>A P Q. (\<exists>x\<in>A. P \<and> Q x) \<longleftrightarrow> (P \<and> (\<exists>x\<in>A. Q x))"

  1610   "\<And>P. (\<exists>x\<in>{}. P x) \<longleftrightarrow> False"

  1611   "\<And>P. (\<exists>x\<in>UNIV. P x) \<longleftrightarrow> (\<exists>x. P x)"

  1612   "\<And>a B P. (\<exists>x\<in>insert a B. P x) \<longleftrightarrow> (P a | (\<exists>x\<in>B. P x))"

  1613   "\<And>P Q. (\<exists>x\<in>Collect Q. P x) \<longleftrightarrow> (\<exists>x. Q x \<and> P x)"

  1614   "\<And>A P f. (\<exists>x\<in>fA. P x) \<longleftrightarrow> (\<exists>x\<in>A. P (f x))"

  1615   "\<And>A P. (\<not>(\<exists>x\<in>A. P x)) \<longleftrightarrow> (\<forall>x\<in>A. \<not> P x)"

  1616   by auto

  1617

  1618

  1619 subsubsection {* Monotonicity of various operations *}

  1620

  1621 lemma image_mono: "A \<subseteq> B ==> fA \<subseteq> fB"

  1622   by blast

  1623

  1624 lemma Pow_mono: "A \<subseteq> B ==> Pow A \<subseteq> Pow B"

  1625   by blast

  1626

  1627 lemma insert_mono: "C \<subseteq> D ==> insert a C \<subseteq> insert a D"

  1628   by blast

  1629

  1630 lemma Un_mono: "A \<subseteq> C ==> B \<subseteq> D ==> A \<union> B \<subseteq> C \<union> D"

  1631   by (fact sup_mono)

  1632

  1633 lemma Int_mono: "A \<subseteq> C ==> B \<subseteq> D ==> A \<inter> B \<subseteq> C \<inter> D"

  1634   by (fact inf_mono)

  1635

  1636 lemma Diff_mono: "A \<subseteq> C ==> D \<subseteq> B ==> A - B \<subseteq> C - D"

  1637   by blast

  1638

  1639 lemma Compl_anti_mono: "A \<subseteq> B ==> -B \<subseteq> -A"

  1640   by (fact compl_mono)

  1641

  1642 text {* \medskip Monotonicity of implications. *}

  1643

  1644 lemma in_mono: "A \<subseteq> B ==> x \<in> A --> x \<in> B"

  1645   apply (rule impI)

  1646   apply (erule subsetD, assumption)

  1647   done

  1648

  1649 lemma conj_mono: "P1 --> Q1 ==> P2 --> Q2 ==> (P1 & P2) --> (Q1 & Q2)"

  1650   by iprover

  1651

  1652 lemma disj_mono: "P1 --> Q1 ==> P2 --> Q2 ==> (P1 | P2) --> (Q1 | Q2)"

  1653   by iprover

  1654

  1655 lemma imp_mono: "Q1 --> P1 ==> P2 --> Q2 ==> (P1 --> P2) --> (Q1 --> Q2)"

  1656   by iprover

  1657

  1658 lemma imp_refl: "P --> P" ..

  1659

  1660 lemma not_mono: "Q --> P ==> ~ P --> ~ Q"

  1661   by iprover

  1662

  1663 lemma ex_mono: "(!!x. P x --> Q x) ==> (EX x. P x) --> (EX x. Q x)"

  1664   by iprover

  1665

  1666 lemma all_mono: "(!!x. P x --> Q x) ==> (ALL x. P x) --> (ALL x. Q x)"

  1667   by iprover

  1668

  1669 lemma Collect_mono: "(!!x. P x --> Q x) ==> Collect P \<subseteq> Collect Q"

  1670   by blast

  1671

  1672 lemma Int_Collect_mono:

  1673     "A \<subseteq> B ==> (!!x. x \<in> A ==> P x --> Q x) ==> A \<inter> Collect P \<subseteq> B \<inter> Collect Q"

  1674   by blast

  1675

  1676 lemmas basic_monos =

  1677   subset_refl imp_refl disj_mono conj_mono

  1678   ex_mono Collect_mono in_mono

  1679

  1680 lemma eq_to_mono: "a = b ==> c = d ==> b --> d ==> a --> c"

  1681   by iprover

  1682

  1683

  1684 subsubsection {* Inverse image of a function *}

  1685

  1686 definition vimage :: "('a => 'b) => 'b set => 'a set" (infixr "-" 90) where

  1687   "f - B == {x. f x : B}"

  1688

  1689 lemma vimage_eq [simp]: "(a : f - B) = (f a : B)"

  1690   by (unfold vimage_def) blast

  1691

  1692 lemma vimage_singleton_eq: "(a : f - {b}) = (f a = b)"

  1693   by simp

  1694

  1695 lemma vimageI [intro]: "f a = b ==> b:B ==> a : f - B"

  1696   by (unfold vimage_def) blast

  1697

  1698 lemma vimageI2: "f a : A ==> a : f - A"

  1699   by (unfold vimage_def) fast

  1700

  1701 lemma vimageE [elim!]: "a: f - B ==> (!!x. f a = x ==> x:B ==> P) ==> P"

  1702   by (unfold vimage_def) blast

  1703

  1704 lemma vimageD: "a : f - A ==> f a : A"

  1705   by (unfold vimage_def) fast

  1706

  1707 lemma vimage_empty [simp]: "f - {} = {}"

  1708   by blast

  1709

  1710 lemma vimage_Compl: "f - (-A) = -(f - A)"

  1711   by blast

  1712

  1713 lemma vimage_Un [simp]: "f - (A Un B) = (f - A) Un (f - B)"

  1714   by blast

  1715

  1716 lemma vimage_Int [simp]: "f - (A Int B) = (f - A) Int (f - B)"

  1717   by fast

  1718

  1719 lemma vimage_Collect_eq [simp]: "f - Collect P = {y. P (f y)}"

  1720   by blast

  1721

  1722 lemma vimage_Collect: "(!!x. P (f x) = Q x) ==> f - (Collect P) = Collect Q"

  1723   by blast

  1724

  1725 lemma vimage_insert: "f-(insert a B) = (f-{a}) Un (f-B)"

  1726   -- {* NOT suitable for rewriting because of the recurrence of @{term "{a}"}. *}

  1727   by blast

  1728

  1729 lemma vimage_Diff: "f - (A - B) = (f - A) - (f - B)"

  1730   by blast

  1731

  1732 lemma vimage_UNIV [simp]: "f - UNIV = UNIV"

  1733   by blast

  1734

  1735 lemma vimage_mono: "A \<subseteq> B ==> f - A \<subseteq> f - B"

  1736   -- {* monotonicity *}

  1737   by blast

  1738

  1739 lemma vimage_image_eq [no_atp]: "f - (f  A) = {y. EX x:A. f x = f y}"

  1740 by (blast intro: sym)

  1741

  1742 lemma image_vimage_subset: "f  (f - A) <= A"

  1743 by blast

  1744

  1745 lemma image_vimage_eq [simp]: "f  (f - A) = A Int range f"

  1746 by blast

  1747

  1748 lemma vimage_const [simp]: "((\<lambda>x. c) - A) = (if c \<in> A then UNIV else {})"

  1749   by auto

  1750

  1751 lemma vimage_if [simp]: "((\<lambda>x. if x \<in> B then c else d) - A) =

  1752    (if c \<in> A then (if d \<in> A then UNIV else B)

  1753     else if d \<in> A then -B else {})"

  1754   by (auto simp add: vimage_def)

  1755

  1756 lemma vimage_inter_cong:

  1757   "(\<And> w. w \<in> S \<Longrightarrow> f w = g w) \<Longrightarrow> f - y \<inter> S = g - y \<inter> S"

  1758   by auto

  1759

  1760 lemma vimage_ident [simp]: "(%x. x) - Y = Y"

  1761   by blast

  1762

  1763

  1764 subsubsection {* Getting the Contents of a Singleton Set *}

  1765

  1766 definition the_elem :: "'a set \<Rightarrow> 'a" where

  1767   "the_elem X = (THE x. X = {x})"

  1768

  1769 lemma the_elem_eq [simp]: "the_elem {x} = x"

  1770   by (simp add: the_elem_def)

  1771

  1772

  1773 subsubsection {* Least value operator *}

  1774

  1775 lemma Least_mono:

  1776   "mono (f::'a::order => 'b::order) ==> EX x:S. ALL y:S. x <= y

  1777     ==> (LEAST y. y : f  S) = f (LEAST x. x : S)"

  1778     -- {* Courtesy of Stephan Merz *}

  1779   apply clarify

  1780   apply (erule_tac P = "%x. x : S" in LeastI2_order, fast)

  1781   apply (rule LeastI2_order)

  1782   apply (auto elim: monoD intro!: order_antisym)

  1783   done

  1784

  1785

  1786 subsubsection {* Monad operation *}

  1787

  1788 definition bind :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where

  1789   "bind A f = {x. \<exists>B \<in> fA. x \<in> B}"

  1790

  1791 hide_const (open) bind

  1792

  1793

  1794 subsubsection {* Operations for execution *}

  1795

  1796 definition is_empty :: "'a set \<Rightarrow> bool" where

  1797   "is_empty A \<longleftrightarrow> A = {}"

  1798

  1799 hide_const (open) is_empty

  1800

  1801 definition remove :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set" where

  1802   "remove x A = A - {x}"

  1803

  1804 hide_const (open) remove

  1805

  1806 definition project :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set" where

  1807   [code_abbrev]: "project P A = {a \<in> A. P a}"

  1808

  1809 hide_const (open) project

  1810

  1811 instantiation set :: (equal) equal

  1812 begin

  1813

  1814 definition

  1815   "HOL.equal A B \<longleftrightarrow> A \<subseteq> B \<and> B \<subseteq> A"

  1816

  1817 instance proof

  1818 qed (auto simp add: equal_set_def)

  1819

  1820 end

  1821

  1822 text {* Misc *}

  1823

  1824 hide_const (open) member not_member

  1825

  1826 lemmas equalityI = subset_antisym

  1827

  1828 ML {*

  1829 val Ball_def = @{thm Ball_def}

  1830 val Bex_def = @{thm Bex_def}

  1831 val CollectD = @{thm CollectD}

  1832 val CollectE = @{thm CollectE}

  1833 val CollectI = @{thm CollectI}

  1834 val Collect_conj_eq = @{thm Collect_conj_eq}

  1835 val Collect_mem_eq = @{thm Collect_mem_eq}

  1836 val IntD1 = @{thm IntD1}

  1837 val IntD2 = @{thm IntD2}

  1838 val IntE = @{thm IntE}

  1839 val IntI = @{thm IntI}

  1840 val Int_Collect = @{thm Int_Collect}

  1841 val UNIV_I = @{thm UNIV_I}

  1842 val UNIV_witness = @{thm UNIV_witness}

  1843 val UnE = @{thm UnE}

  1844 val UnI1 = @{thm UnI1}

  1845 val UnI2 = @{thm UnI2}

  1846 val ballE = @{thm ballE}

  1847 val ballI = @{thm ballI}

  1848 val bexCI = @{thm bexCI}

  1849 val bexE = @{thm bexE}

  1850 val bexI = @{thm bexI}

  1851 val bex_triv = @{thm bex_triv}

  1852 val bspec = @{thm bspec}

  1853 val contra_subsetD = @{thm contra_subsetD}

  1854 val equalityCE = @{thm equalityCE}

  1855 val equalityD1 = @{thm equalityD1}

  1856 val equalityD2 = @{thm equalityD2}

  1857 val equalityE = @{thm equalityE}

  1858 val equalityI = @{thm equalityI}

  1859 val imageE = @{thm imageE}

  1860 val imageI = @{thm imageI}

  1861 val image_Un = @{thm image_Un}

  1862 val image_insert = @{thm image_insert}

  1863 val insert_commute = @{thm insert_commute}

  1864 val insert_iff = @{thm insert_iff}

  1865 val mem_Collect_eq = @{thm mem_Collect_eq}

  1866 val rangeE = @{thm rangeE}

  1867 val rangeI = @{thm rangeI}

  1868 val range_eqI = @{thm range_eqI}

  1869 val subsetCE = @{thm subsetCE}

  1870 val subsetD = @{thm subsetD}

  1871 val subsetI = @{thm subsetI}

  1872 val subset_refl = @{thm subset_refl}

  1873 val subset_trans = @{thm subset_trans}

  1874 val vimageD = @{thm vimageD}

  1875 val vimageE = @{thm vimageE}

  1876 val vimageI = @{thm vimageI}

  1877 val vimageI2 = @{thm vimageI2}

  1878 val vimage_Collect = @{thm vimage_Collect}

  1879 val vimage_Int = @{thm vimage_Int}

  1880 val vimage_Un = @{thm vimage_Un}

  1881 *}

  1882

  1883 end