src/HOL/Set.thy
author haftmann
Mon Jul 20 09:52:09 2009 +0200 (2009-07-20)
changeset 32078 1c14f77201d4
parent 32075 e8e0fb5da77a
parent 32077 3698947146b2
child 32081 1b7a901e2edc
permissions -rw-r--r--
merged
     1 (*  Title:      HOL/Set.thy
     2     Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
     3 *)
     4 
     5 header {* Set theory for higher-order logic *}
     6 
     7 theory Set
     8 imports Lattices
     9 begin
    10 
    11 text {* A set in HOL is simply a predicate. *}
    12 
    13 subsection {* Basic definitions and syntax *}
    14 
    15 global
    16 
    17 types 'a set = "'a => bool"
    18 
    19 consts
    20   Collect       :: "('a => bool) => 'a set"              -- "comprehension"
    21   "op :"        :: "'a => 'a set => bool"                -- "membership"
    22 
    23 local
    24 
    25 notation
    26   "op :"  ("op :") and
    27   "op :"  ("(_/ : _)" [50, 51] 50)
    28 
    29 defs
    30   mem_def [code]: "x : S == S x"
    31   Collect_def [code]: "Collect P == P"
    32 
    33 abbreviation
    34   "not_mem x A == ~ (x : A)" -- "non-membership"
    35 
    36 notation
    37   not_mem  ("op ~:") and
    38   not_mem  ("(_/ ~: _)" [50, 51] 50)
    39 
    40 notation (xsymbols)
    41   "op :"  ("op \<in>") and
    42   "op :"  ("(_/ \<in> _)" [50, 51] 50) and
    43   not_mem  ("op \<notin>") and
    44   not_mem  ("(_/ \<notin> _)" [50, 51] 50)
    45 
    46 notation (HTML output)
    47   "op :"  ("op \<in>") and
    48   "op :"  ("(_/ \<in> _)" [50, 51] 50) and
    49   not_mem  ("op \<notin>") and
    50   not_mem  ("(_/ \<notin> _)" [50, 51] 50)
    51 
    52 syntax
    53   "@Coll"       :: "pttrn => bool => 'a set"              ("(1{_./ _})")
    54 
    55 translations
    56   "{x. P}"      == "Collect (%x. P)"
    57 
    58 definition Int :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Int" 70) where
    59   "A Int B \<equiv> {x. x \<in> A \<and> x \<in> B}"
    60 
    61 definition Un :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Un" 65) where
    62   "A Un B \<equiv> {x. x \<in> A \<or> x \<in> B}"
    63 
    64 notation (xsymbols)
    65   "Int"  (infixl "\<inter>" 70) and
    66   "Un"  (infixl "\<union>" 65)
    67 
    68 notation (HTML output)
    69   "Int"  (infixl "\<inter>" 70) and
    70   "Un"  (infixl "\<union>" 65)
    71 
    72 definition empty :: "'a set" ("{}") where
    73   "empty \<equiv> {x. False}"
    74 
    75 definition insert :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set" where
    76   "insert a B \<equiv> {x. x = a} \<union> B"
    77 
    78 definition UNIV :: "'a set" where
    79   "UNIV \<equiv> {x. True}"
    80 
    81 syntax
    82   "@Finset"     :: "args => 'a set"                       ("{(_)}")
    83 
    84 translations
    85   "{x, xs}"     == "CONST insert x {xs}"
    86   "{x}"         == "CONST insert x {}"
    87 
    88 global
    89 
    90 consts
    91   Ball          :: "'a set => ('a => bool) => bool"      -- "bounded universal quantifiers"
    92   Bex           :: "'a set => ('a => bool) => bool"      -- "bounded existential quantifiers"
    93   Bex1          :: "'a set => ('a => bool) => bool"      -- "bounded unique existential quantifiers"
    94 
    95 local
    96 
    97 defs
    98   Ball_def:     "Ball A P       == ALL x. x:A --> P(x)"
    99   Bex_def:      "Bex A P        == EX x. x:A & P(x)"
   100   Bex1_def:     "Bex1 A P       == EX! x. x:A & P(x)"
   101 
   102 syntax
   103   "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3ALL _:_./ _)" [0, 0, 10] 10)
   104   "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3EX _:_./ _)" [0, 0, 10] 10)
   105   "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3EX! _:_./ _)" [0, 0, 10] 10)
   106   "_Bleast"     :: "id => 'a set => bool => 'a"           ("(3LEAST _:_./ _)" [0, 0, 10] 10)
   107 
   108 syntax (HOL)
   109   "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3! _:_./ _)" [0, 0, 10] 10)
   110   "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3? _:_./ _)" [0, 0, 10] 10)
   111   "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3?! _:_./ _)" [0, 0, 10] 10)
   112 
   113 syntax (xsymbols)
   114   "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10)
   115   "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10)
   116   "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3\<exists>!_\<in>_./ _)" [0, 0, 10] 10)
   117   "_Bleast"     :: "id => 'a set => bool => 'a"           ("(3LEAST_\<in>_./ _)" [0, 0, 10] 10)
   118 
   119 syntax (HTML output)
   120   "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10)
   121   "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10)
   122   "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3\<exists>!_\<in>_./ _)" [0, 0, 10] 10)
   123 
   124 translations
   125   "ALL x:A. P"  == "Ball A (%x. P)"
   126   "EX x:A. P"   == "Bex A (%x. P)"
   127   "EX! x:A. P"  == "Bex1 A (%x. P)"
   128   "LEAST x:A. P" => "LEAST x. x:A & P"
   129 
   130 
   131 subsection {* Additional concrete syntax *}
   132 
   133 syntax
   134   "@SetCompr"   :: "'a => idts => bool => 'a set"         ("(1{_ |/_./ _})")
   135   "@Collect"    :: "idt => 'a set => bool => 'a set"      ("(1{_ :/ _./ _})")
   136 
   137 syntax (xsymbols)
   138   "@Collect"    :: "idt => 'a set => bool => 'a set"      ("(1{_ \<in>/ _./ _})")
   139 
   140 translations
   141   "{x:A. P}"    => "{x. x:A & P}"
   142 
   143 abbreviation
   144   subset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
   145   "subset \<equiv> less"
   146 
   147 abbreviation
   148   subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
   149   "subset_eq \<equiv> less_eq"
   150 
   151 notation (output)
   152   subset  ("op <") and
   153   subset  ("(_/ < _)" [50, 51] 50) and
   154   subset_eq  ("op <=") and
   155   subset_eq  ("(_/ <= _)" [50, 51] 50)
   156 
   157 notation (xsymbols)
   158   subset  ("op \<subset>") and
   159   subset  ("(_/ \<subset> _)" [50, 51] 50) and
   160   subset_eq  ("op \<subseteq>") and
   161   subset_eq  ("(_/ \<subseteq> _)" [50, 51] 50)
   162 
   163 notation (HTML output)
   164   subset  ("op \<subset>") and
   165   subset  ("(_/ \<subset> _)" [50, 51] 50) and
   166   subset_eq  ("op \<subseteq>") and
   167   subset_eq  ("(_/ \<subseteq> _)" [50, 51] 50)
   168 
   169 abbreviation (input)
   170   supset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
   171   "supset \<equiv> greater"
   172 
   173 abbreviation (input)
   174   supset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
   175   "supset_eq \<equiv> greater_eq"
   176 
   177 notation (xsymbols)
   178   supset  ("op \<supset>") and
   179   supset  ("(_/ \<supset> _)" [50, 51] 50) and
   180   supset_eq  ("op \<supseteq>") and
   181   supset_eq  ("(_/ \<supseteq> _)" [50, 51] 50)
   182 
   183 
   184 
   185 subsubsection "Bounded quantifiers"
   186 
   187 syntax (output)
   188   "_setlessAll" :: "[idt, 'a, bool] => bool"  ("(3ALL _<_./ _)"  [0, 0, 10] 10)
   189   "_setlessEx"  :: "[idt, 'a, bool] => bool"  ("(3EX _<_./ _)"  [0, 0, 10] 10)
   190   "_setleAll"   :: "[idt, 'a, bool] => bool"  ("(3ALL _<=_./ _)" [0, 0, 10] 10)
   191   "_setleEx"    :: "[idt, 'a, bool] => bool"  ("(3EX _<=_./ _)" [0, 0, 10] 10)
   192   "_setleEx1"   :: "[idt, 'a, bool] => bool"  ("(3EX! _<=_./ _)" [0, 0, 10] 10)
   193 
   194 syntax (xsymbols)
   195   "_setlessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<subset>_./ _)"  [0, 0, 10] 10)
   196   "_setlessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subset>_./ _)"  [0, 0, 10] 10)
   197   "_setleAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<subseteq>_./ _)" [0, 0, 10] 10)
   198   "_setleEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10)
   199   "_setleEx1"   :: "[idt, 'a, bool] => bool"   ("(3\<exists>!_\<subseteq>_./ _)" [0, 0, 10] 10)
   200 
   201 syntax (HOL output)
   202   "_setlessAll" :: "[idt, 'a, bool] => bool"   ("(3! _<_./ _)"  [0, 0, 10] 10)
   203   "_setlessEx"  :: "[idt, 'a, bool] => bool"   ("(3? _<_./ _)"  [0, 0, 10] 10)
   204   "_setleAll"   :: "[idt, 'a, bool] => bool"   ("(3! _<=_./ _)" [0, 0, 10] 10)
   205   "_setleEx"    :: "[idt, 'a, bool] => bool"   ("(3? _<=_./ _)" [0, 0, 10] 10)
   206   "_setleEx1"   :: "[idt, 'a, bool] => bool"   ("(3?! _<=_./ _)" [0, 0, 10] 10)
   207 
   208 syntax (HTML output)
   209   "_setlessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<subset>_./ _)"  [0, 0, 10] 10)
   210   "_setlessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subset>_./ _)"  [0, 0, 10] 10)
   211   "_setleAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<subseteq>_./ _)" [0, 0, 10] 10)
   212   "_setleEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10)
   213   "_setleEx1"   :: "[idt, 'a, bool] => bool"   ("(3\<exists>!_\<subseteq>_./ _)" [0, 0, 10] 10)
   214 
   215 translations
   216  "\<forall>A\<subset>B. P"   =>  "ALL A. A \<subset> B --> P"
   217  "\<exists>A\<subset>B. P"   =>  "EX A. A \<subset> B & P"
   218  "\<forall>A\<subseteq>B. P"   =>  "ALL A. A \<subseteq> B --> P"
   219  "\<exists>A\<subseteq>B. P"   =>  "EX A. A \<subseteq> B & P"
   220  "\<exists>!A\<subseteq>B. P"  =>  "EX! A. A \<subseteq> B & P"
   221 
   222 print_translation {*
   223 let
   224   val Type (set_type, _) = @{typ "'a set"};
   225   val All_binder = Syntax.binder_name @{const_syntax "All"};
   226   val Ex_binder = Syntax.binder_name @{const_syntax "Ex"};
   227   val impl = @{const_syntax "op -->"};
   228   val conj = @{const_syntax "op &"};
   229   val sbset = @{const_syntax "subset"};
   230   val sbset_eq = @{const_syntax "subset_eq"};
   231 
   232   val trans =
   233    [((All_binder, impl, sbset), "_setlessAll"),
   234     ((All_binder, impl, sbset_eq), "_setleAll"),
   235     ((Ex_binder, conj, sbset), "_setlessEx"),
   236     ((Ex_binder, conj, sbset_eq), "_setleEx")];
   237 
   238   fun mk v v' c n P =
   239     if v = v' andalso not (Term.exists_subterm (fn Free (x, _) => x = v | _ => false) n)
   240     then Syntax.const c $ Syntax.mark_bound v' $ n $ P else raise Match;
   241 
   242   fun tr' q = (q,
   243     fn [Const ("_bound", _) $ Free (v, Type (T, _)), Const (c, _) $ (Const (d, _) $ (Const ("_bound", _) $ Free (v', _)) $ n) $ P] =>
   244          if T = (set_type) then case AList.lookup (op =) trans (q, c, d)
   245           of NONE => raise Match
   246            | SOME l => mk v v' l n P
   247          else raise Match
   248      | _ => raise Match);
   249 in
   250   [tr' All_binder, tr' Ex_binder]
   251 end
   252 *}
   253 
   254 
   255 text {*
   256   \medskip Translate between @{text "{e | x1...xn. P}"} and @{text
   257   "{u. EX x1..xn. u = e & P}"}; @{text "{y. EX x1..xn. y = e & P}"} is
   258   only translated if @{text "[0..n] subset bvs(e)"}.
   259 *}
   260 
   261 parse_translation {*
   262   let
   263     val ex_tr = snd (mk_binder_tr ("EX ", "Ex"));
   264 
   265     fun nvars (Const ("_idts", _) $ _ $ idts) = nvars idts + 1
   266       | nvars _ = 1;
   267 
   268     fun setcompr_tr [e, idts, b] =
   269       let
   270         val eq = Syntax.const "op =" $ Bound (nvars idts) $ e;
   271         val P = Syntax.const "op &" $ eq $ b;
   272         val exP = ex_tr [idts, P];
   273       in Syntax.const "Collect" $ Term.absdummy (dummyT, exP) end;
   274 
   275   in [("@SetCompr", setcompr_tr)] end;
   276 *}
   277 
   278 (* To avoid eta-contraction of body: *)
   279 print_translation {*
   280 let
   281   fun btr' syn [A, Abs abs] =
   282     let val (x, t) = atomic_abs_tr' abs
   283     in Syntax.const syn $ x $ A $ t end
   284 in [(@{const_syntax Ball}, btr' "_Ball"), (@{const_syntax Bex}, btr' "_Bex")] end
   285 *}
   286 
   287 print_translation {*
   288 let
   289   val ex_tr' = snd (mk_binder_tr' ("Ex", "DUMMY"));
   290 
   291   fun setcompr_tr' [Abs (abs as (_, _, P))] =
   292     let
   293       fun check (Const ("Ex", _) $ Abs (_, _, P), n) = check (P, n + 1)
   294         | check (Const ("op &", _) $ (Const ("op =", _) $ Bound m $ e) $ P, n) =
   295             n > 0 andalso m = n andalso not (loose_bvar1 (P, n)) andalso
   296             ((0 upto (n - 1)) subset add_loose_bnos (e, 0, []))
   297         | check _ = false
   298 
   299         fun tr' (_ $ abs) =
   300           let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' [abs]
   301           in Syntax.const "@SetCompr" $ e $ idts $ Q end;
   302     in if check (P, 0) then tr' P
   303        else let val (x as _ $ Free(xN,_), t) = atomic_abs_tr' abs
   304                 val M = Syntax.const "@Coll" $ x $ t
   305             in case t of
   306                  Const("op &",_)
   307                    $ (Const("op :",_) $ (Const("_bound",_) $ Free(yN,_)) $ A)
   308                    $ P =>
   309                    if xN=yN then Syntax.const "@Collect" $ x $ A $ P else M
   310                | _ => M
   311             end
   312     end;
   313   in [("Collect", setcompr_tr')] end;
   314 *}
   315 
   316 
   317 subsection {* Lemmas and proof tool setup *}
   318 
   319 subsubsection {* Relating predicates and sets *}
   320 
   321 lemma mem_Collect_eq [iff]: "(a : {x. P(x)}) = P(a)"
   322   by (simp add: Collect_def mem_def)
   323 
   324 lemma Collect_mem_eq [simp]: "{x. x:A} = A"
   325   by (simp add: Collect_def mem_def)
   326 
   327 lemma CollectI: "P(a) ==> a : {x. P(x)}"
   328   by simp
   329 
   330 lemma CollectD: "a : {x. P(x)} ==> P(a)"
   331   by simp
   332 
   333 lemma Collect_cong: "(!!x. P x = Q x) ==> {x. P(x)} = {x. Q(x)}"
   334   by simp
   335 
   336 lemmas CollectE = CollectD [elim_format]
   337 
   338 
   339 subsubsection {* Bounded quantifiers *}
   340 
   341 lemma ballI [intro!]: "(!!x. x:A ==> P x) ==> ALL x:A. P x"
   342   by (simp add: Ball_def)
   343 
   344 lemmas strip = impI allI ballI
   345 
   346 lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x"
   347   by (simp add: Ball_def)
   348 
   349 lemma ballE [elim]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q"
   350   by (unfold Ball_def) blast
   351 
   352 ML {* bind_thm ("rev_ballE", Thm.permute_prems 1 1 @{thm ballE}) *}
   353 
   354 text {*
   355   \medskip This tactic takes assumptions @{prop "ALL x:A. P x"} and
   356   @{prop "a:A"}; creates assumption @{prop "P a"}.
   357 *}
   358 
   359 ML {*
   360   fun ball_tac i = etac @{thm ballE} i THEN contr_tac (i + 1)
   361 *}
   362 
   363 text {*
   364   Gives better instantiation for bound:
   365 *}
   366 
   367 declaration {* fn _ =>
   368   Classical.map_cs (fn cs => cs addbefore ("bspec", datac @{thm bspec} 1))
   369 *}
   370 
   371 lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x"
   372   -- {* Normally the best argument order: @{prop "P x"} constrains the
   373     choice of @{prop "x:A"}. *}
   374   by (unfold Bex_def) blast
   375 
   376 lemma rev_bexI [intro?]: "x:A ==> P x ==> EX x:A. P x"
   377   -- {* The best argument order when there is only one @{prop "x:A"}. *}
   378   by (unfold Bex_def) blast
   379 
   380 lemma bexCI: "(ALL x:A. ~P x ==> P a) ==> a:A ==> EX x:A. P x"
   381   by (unfold Bex_def) blast
   382 
   383 lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q"
   384   by (unfold Bex_def) blast
   385 
   386 lemma ball_triv [simp]: "(ALL x:A. P) = ((EX x. x:A) --> P)"
   387   -- {* Trival rewrite rule. *}
   388   by (simp add: Ball_def)
   389 
   390 lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)"
   391   -- {* Dual form for existentials. *}
   392   by (simp add: Bex_def)
   393 
   394 lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)"
   395   by blast
   396 
   397 lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)"
   398   by blast
   399 
   400 lemma bex_one_point1 [simp]: "(EX x:A. x = a & P x) = (a:A & P a)"
   401   by blast
   402 
   403 lemma bex_one_point2 [simp]: "(EX x:A. a = x & P x) = (a:A & P a)"
   404   by blast
   405 
   406 lemma ball_one_point1 [simp]: "(ALL x:A. x = a --> P x) = (a:A --> P a)"
   407   by blast
   408 
   409 lemma ball_one_point2 [simp]: "(ALL x:A. a = x --> P x) = (a:A --> P a)"
   410   by blast
   411 
   412 ML {*
   413   local
   414     val unfold_bex_tac = unfold_tac @{thms "Bex_def"};
   415     fun prove_bex_tac ss = unfold_bex_tac ss THEN Quantifier1.prove_one_point_ex_tac;
   416     val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac;
   417 
   418     val unfold_ball_tac = unfold_tac @{thms "Ball_def"};
   419     fun prove_ball_tac ss = unfold_ball_tac ss THEN Quantifier1.prove_one_point_all_tac;
   420     val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac;
   421   in
   422     val defBEX_regroup = Simplifier.simproc @{theory}
   423       "defined BEX" ["EX x:A. P x & Q x"] rearrange_bex;
   424     val defBALL_regroup = Simplifier.simproc @{theory}
   425       "defined BALL" ["ALL x:A. P x --> Q x"] rearrange_ball;
   426   end;
   427 
   428   Addsimprocs [defBALL_regroup, defBEX_regroup];
   429 *}
   430 
   431 
   432 subsubsection {* Congruence rules *}
   433 
   434 lemma ball_cong:
   435   "A = B ==> (!!x. x:B ==> P x = Q x) ==>
   436     (ALL x:A. P x) = (ALL x:B. Q x)"
   437   by (simp add: Ball_def)
   438 
   439 lemma strong_ball_cong [cong]:
   440   "A = B ==> (!!x. x:B =simp=> P x = Q x) ==>
   441     (ALL x:A. P x) = (ALL x:B. Q x)"
   442   by (simp add: simp_implies_def Ball_def)
   443 
   444 lemma bex_cong:
   445   "A = B ==> (!!x. x:B ==> P x = Q x) ==>
   446     (EX x:A. P x) = (EX x:B. Q x)"
   447   by (simp add: Bex_def cong: conj_cong)
   448 
   449 lemma strong_bex_cong [cong]:
   450   "A = B ==> (!!x. x:B =simp=> P x = Q x) ==>
   451     (EX x:A. P x) = (EX x:B. Q x)"
   452   by (simp add: simp_implies_def Bex_def cong: conj_cong)
   453 
   454 
   455 subsubsection {* Subsets *}
   456 
   457 lemma subsetI [atp,intro!]: "(!!x. x:A ==> x:B) ==> A \<subseteq> B"
   458   by (auto simp add: mem_def intro: predicate1I)
   459 
   460 text {*
   461   \medskip Map the type @{text "'a set => anything"} to just @{typ
   462   'a}; for overloading constants whose first argument has type @{typ
   463   "'a set"}.
   464 *}
   465 
   466 lemma subsetD [elim, intro?]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B"
   467   -- {* Rule in Modus Ponens style. *}
   468   by (unfold mem_def) blast
   469 
   470 lemma rev_subsetD [intro?]: "c \<in> A ==> A \<subseteq> B ==> c \<in> B"
   471   -- {* The same, with reversed premises for use with @{text erule} --
   472       cf @{text rev_mp}. *}
   473   by (rule subsetD)
   474 
   475 text {*
   476   \medskip Converts @{prop "A \<subseteq> B"} to @{prop "x \<in> A ==> x \<in> B"}.
   477 *}
   478 
   479 ML {*
   480   fun impOfSubs th = th RSN (2, @{thm rev_subsetD})
   481 *}
   482 
   483 lemma subsetCE [elim]: "A \<subseteq>  B ==> (c \<notin> A ==> P) ==> (c \<in> B ==> P) ==> P"
   484   -- {* Classical elimination rule. *}
   485   by (unfold mem_def) blast
   486 
   487 lemma subset_eq: "A \<le> B = (\<forall>x\<in>A. x \<in> B)" by blast
   488 
   489 text {*
   490   \medskip Takes assumptions @{prop "A \<subseteq> B"}; @{prop "c \<in> A"} and
   491   creates the assumption @{prop "c \<in> B"}.
   492 *}
   493 
   494 ML {*
   495   fun set_mp_tac i = etac @{thm subsetCE} i THEN mp_tac i
   496 *}
   497 
   498 lemma contra_subsetD: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A"
   499   by blast
   500 
   501 lemma subset_refl [simp,atp]: "A \<subseteq> A"
   502   by fast
   503 
   504 lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C"
   505   by blast
   506 
   507 
   508 subsubsection {* Equality *}
   509 
   510 lemma set_ext: assumes prem: "(!!x. (x:A) = (x:B))" shows "A = B"
   511   apply (rule prem [THEN ext, THEN arg_cong, THEN box_equals])
   512    apply (rule Collect_mem_eq)
   513   apply (rule Collect_mem_eq)
   514   done
   515 
   516 (* Due to Brian Huffman *)
   517 lemma expand_set_eq: "(A = B) = (ALL x. (x:A) = (x:B))"
   518 by(auto intro:set_ext)
   519 
   520 lemma subset_antisym [intro!]: "A \<subseteq> B ==> B \<subseteq> A ==> A = B"
   521   -- {* Anti-symmetry of the subset relation. *}
   522   by (iprover intro: set_ext subsetD)
   523 
   524 text {*
   525   \medskip Equality rules from ZF set theory -- are they appropriate
   526   here?
   527 *}
   528 
   529 lemma equalityD1: "A = B ==> A \<subseteq> B"
   530   by (simp add: subset_refl)
   531 
   532 lemma equalityD2: "A = B ==> B \<subseteq> A"
   533   by (simp add: subset_refl)
   534 
   535 text {*
   536   \medskip Be careful when adding this to the claset as @{text
   537   subset_empty} is in the simpset: @{prop "A = {}"} goes to @{prop "{}
   538   \<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}!
   539 *}
   540 
   541 lemma equalityE: "A = B ==> (A \<subseteq> B ==> B \<subseteq> A ==> P) ==> P"
   542   by (simp add: subset_refl)
   543 
   544 lemma equalityCE [elim]:
   545     "A = B ==> (c \<in> A ==> c \<in> B ==> P) ==> (c \<notin> A ==> c \<notin> B ==> P) ==> P"
   546   by blast
   547 
   548 lemma eqset_imp_iff: "A = B ==> (x : A) = (x : B)"
   549   by simp
   550 
   551 lemma eqelem_imp_iff: "x = y ==> (x : A) = (y : A)"
   552   by simp
   553 
   554 
   555 subsubsection {* The universal set -- UNIV *}
   556 
   557 lemma UNIV_I [simp]: "x : UNIV"
   558   by (simp add: UNIV_def)
   559 
   560 declare UNIV_I [intro]  -- {* unsafe makes it less likely to cause problems *}
   561 
   562 lemma UNIV_witness [intro?]: "EX x. x : UNIV"
   563   by simp
   564 
   565 lemma subset_UNIV [simp]: "A \<subseteq> UNIV"
   566   by (rule subsetI) (rule UNIV_I)
   567 
   568 text {*
   569   \medskip Eta-contracting these two rules (to remove @{text P})
   570   causes them to be ignored because of their interaction with
   571   congruence rules.
   572 *}
   573 
   574 lemma ball_UNIV [simp]: "Ball UNIV P = All P"
   575   by (simp add: Ball_def)
   576 
   577 lemma bex_UNIV [simp]: "Bex UNIV P = Ex P"
   578   by (simp add: Bex_def)
   579 
   580 lemma UNIV_eq_I: "(\<And>x. x \<in> A) \<Longrightarrow> UNIV = A"
   581   by auto
   582 
   583 
   584 subsubsection {* The empty set *}
   585 
   586 lemma empty_iff [simp]: "(c : {}) = False"
   587   by (simp add: empty_def)
   588 
   589 lemma emptyE [elim!]: "a : {} ==> P"
   590   by simp
   591 
   592 lemma empty_subsetI [iff]: "{} \<subseteq> A"
   593     -- {* One effect is to delete the ASSUMPTION @{prop "{} <= A"} *}
   594   by blast
   595 
   596 lemma equals0I: "(!!y. y \<in> A ==> False) ==> A = {}"
   597   by blast
   598 
   599 lemma equals0D: "A = {} ==> a \<notin> A"
   600     -- {* Use for reasoning about disjointness: @{prop "A Int B = {}"} *}
   601   by blast
   602 
   603 lemma ball_empty [simp]: "Ball {} P = True"
   604   by (simp add: Ball_def)
   605 
   606 lemma bex_empty [simp]: "Bex {} P = False"
   607   by (simp add: Bex_def)
   608 
   609 lemma UNIV_not_empty [iff]: "UNIV ~= {}"
   610   by (blast elim: equalityE)
   611 
   612 
   613 subsubsection {* The Powerset operator -- Pow *}
   614 
   615 definition Pow :: "'a set => 'a set set" where
   616   Pow_def: "Pow A = {B. B \<le> A}"
   617 
   618 lemma Pow_iff [iff]: "(A \<in> Pow B) = (A \<subseteq> B)"
   619   by (simp add: Pow_def)
   620 
   621 lemma PowI: "A \<subseteq> B ==> A \<in> Pow B"
   622   by (simp add: Pow_def)
   623 
   624 lemma PowD: "A \<in> Pow B ==> A \<subseteq> B"
   625   by (simp add: Pow_def)
   626 
   627 lemma Pow_bottom: "{} \<in> Pow B"
   628   by simp
   629 
   630 lemma Pow_top: "A \<in> Pow A"
   631   by (simp add: subset_refl)
   632 
   633 
   634 subsubsection {* Set complement *}
   635 
   636 lemma Compl_iff [simp]: "(c \<in> -A) = (c \<notin> A)"
   637   by (simp add: mem_def fun_Compl_def bool_Compl_def)
   638 
   639 lemma ComplI [intro!]: "(c \<in> A ==> False) ==> c \<in> -A"
   640   by (unfold mem_def fun_Compl_def bool_Compl_def) blast
   641 
   642 text {*
   643   \medskip This form, with negated conclusion, works well with the
   644   Classical prover.  Negated assumptions behave like formulae on the
   645   right side of the notional turnstile ... *}
   646 
   647 lemma ComplD [dest!]: "c : -A ==> c~:A"
   648   by (simp add: mem_def fun_Compl_def bool_Compl_def)
   649 
   650 lemmas ComplE = ComplD [elim_format]
   651 
   652 lemma Compl_eq: "- A = {x. ~ x : A}" by blast
   653 
   654 
   655 subsubsection {* Binary union -- Un *}
   656 
   657 lemma Un_iff [simp]: "(c : A Un B) = (c:A | c:B)"
   658   by (unfold Un_def) blast
   659 
   660 lemma UnI1 [elim?]: "c:A ==> c : A Un B"
   661   by simp
   662 
   663 lemma UnI2 [elim?]: "c:B ==> c : A Un B"
   664   by simp
   665 
   666 text {*
   667   \medskip Classical introduction rule: no commitment to @{prop A} vs
   668   @{prop B}.
   669 *}
   670 
   671 lemma UnCI [intro!]: "(c~:B ==> c:A) ==> c : A Un B"
   672   by auto
   673 
   674 lemma UnE [elim!]: "c : A Un B ==> (c:A ==> P) ==> (c:B ==> P) ==> P"
   675   by (unfold Un_def) blast
   676 
   677 
   678 subsubsection {* Binary intersection -- Int *}
   679 
   680 lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)"
   681   by (unfold Int_def) blast
   682 
   683 lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B"
   684   by simp
   685 
   686 lemma IntD1: "c : A Int B ==> c:A"
   687   by simp
   688 
   689 lemma IntD2: "c : A Int B ==> c:B"
   690   by simp
   691 
   692 lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P"
   693   by simp
   694 
   695 
   696 subsubsection {* Set difference *}
   697 
   698 lemma Diff_iff [simp]: "(c : A - B) = (c:A & c~:B)"
   699   by (simp add: mem_def fun_diff_def bool_diff_def)
   700 
   701 lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A - B"
   702   by simp
   703 
   704 lemma DiffD1: "c : A - B ==> c : A"
   705   by simp
   706 
   707 lemma DiffD2: "c : A - B ==> c : B ==> P"
   708   by simp
   709 
   710 lemma DiffE [elim!]: "c : A - B ==> (c:A ==> c~:B ==> P) ==> P"
   711   by simp
   712 
   713 lemma set_diff_eq: "A - B = {x. x : A & ~ x : B}" by blast
   714 
   715 lemma Compl_eq_Diff_UNIV: "-A = (UNIV - A)"
   716 by blast
   717 
   718 
   719 subsubsection {* Augmenting a set -- @{const insert} *}
   720 
   721 lemma insert_iff [simp]: "(a : insert b A) = (a = b | a:A)"
   722   by (unfold insert_def) blast
   723 
   724 lemma insertI1: "a : insert a B"
   725   by simp
   726 
   727 lemma insertI2: "a : B ==> a : insert b B"
   728   by simp
   729 
   730 lemma insertE [elim!]: "a : insert b A ==> (a = b ==> P) ==> (a:A ==> P) ==> P"
   731   by (unfold insert_def) blast
   732 
   733 lemma insertCI [intro!]: "(a~:B ==> a = b) ==> a: insert b B"
   734   -- {* Classical introduction rule. *}
   735   by auto
   736 
   737 lemma subset_insert_iff: "(A \<subseteq> insert x B) = (if x:A then A - {x} \<subseteq> B else A \<subseteq> B)"
   738   by auto
   739 
   740 lemma set_insert:
   741   assumes "x \<in> A"
   742   obtains B where "A = insert x B" and "x \<notin> B"
   743 proof
   744   from assms show "A = insert x (A - {x})" by blast
   745 next
   746   show "x \<notin> A - {x}" by blast
   747 qed
   748 
   749 lemma insert_ident: "x ~: A ==> x ~: B ==> (insert x A = insert x B) = (A = B)"
   750 by auto
   751 
   752 subsubsection {* Singletons, using insert *}
   753 
   754 lemma singletonI [intro!,noatp]: "a : {a}"
   755     -- {* Redundant? But unlike @{text insertCI}, it proves the subgoal immediately! *}
   756   by (rule insertI1)
   757 
   758 lemma singletonD [dest!,noatp]: "b : {a} ==> b = a"
   759   by blast
   760 
   761 lemmas singletonE = singletonD [elim_format]
   762 
   763 lemma singleton_iff: "(b : {a}) = (b = a)"
   764   by blast
   765 
   766 lemma singleton_inject [dest!]: "{a} = {b} ==> a = b"
   767   by blast
   768 
   769 lemma singleton_insert_inj_eq [iff,noatp]:
   770      "({b} = insert a A) = (a = b & A \<subseteq> {b})"
   771   by blast
   772 
   773 lemma singleton_insert_inj_eq' [iff,noatp]:
   774      "(insert a A = {b}) = (a = b & A \<subseteq> {b})"
   775   by blast
   776 
   777 lemma subset_singletonD: "A \<subseteq> {x} ==> A = {} | A = {x}"
   778   by fast
   779 
   780 lemma singleton_conv [simp]: "{x. x = a} = {a}"
   781   by blast
   782 
   783 lemma singleton_conv2 [simp]: "{x. a = x} = {a}"
   784   by blast
   785 
   786 lemma diff_single_insert: "A - {x} \<subseteq> B ==> x \<in> A ==> A \<subseteq> insert x B"
   787   by blast
   788 
   789 lemma doubleton_eq_iff: "({a,b} = {c,d}) = (a=c & b=d | a=d & b=c)"
   790   by (blast elim: equalityE)
   791 
   792 
   793 subsubsection {* Image of a set under a function *}
   794 
   795 text {*
   796   Frequently @{term b} does not have the syntactic form of @{term "f x"}.
   797 *}
   798 
   799 definition image :: "('a => 'b) => 'a set => 'b set" (infixr "`" 90) where
   800   image_def [noatp]: "f ` A = {y. EX x:A. y = f(x)}"
   801 
   802 abbreviation
   803   range :: "('a => 'b) => 'b set" where -- "of function"
   804   "range f == f ` UNIV"
   805 
   806 lemma image_eqI [simp, intro]: "b = f x ==> x:A ==> b : f`A"
   807   by (unfold image_def) blast
   808 
   809 lemma imageI: "x : A ==> f x : f ` A"
   810   by (rule image_eqI) (rule refl)
   811 
   812 lemma rev_image_eqI: "x:A ==> b = f x ==> b : f`A"
   813   -- {* This version's more effective when we already have the
   814     required @{term x}. *}
   815   by (unfold image_def) blast
   816 
   817 lemma imageE [elim!]:
   818   "b : (%x. f x)`A ==> (!!x. b = f x ==> x:A ==> P) ==> P"
   819   -- {* The eta-expansion gives variable-name preservation. *}
   820   by (unfold image_def) blast
   821 
   822 lemma image_Un: "f`(A Un B) = f`A Un f`B"
   823   by blast
   824 
   825 lemma image_iff: "(z : f`A) = (EX x:A. z = f x)"
   826   by blast
   827 
   828 lemma image_subset_iff: "(f`A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)"
   829   -- {* This rewrite rule would confuse users if made default. *}
   830   by blast
   831 
   832 lemma subset_image_iff: "(B \<subseteq> f`A) = (EX AA. AA \<subseteq> A & B = f`AA)"
   833   apply safe
   834    prefer 2 apply fast
   835   apply (rule_tac x = "{a. a : A & f a : B}" in exI, fast)
   836   done
   837 
   838 lemma image_subsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f`A \<subseteq> B"
   839   -- {* Replaces the three steps @{text subsetI}, @{text imageE},
   840     @{text hypsubst}, but breaks too many existing proofs. *}
   841   by blast
   842 
   843 text {*
   844   \medskip Range of a function -- just a translation for image!
   845 *}
   846 
   847 lemma range_eqI: "b = f x ==> b \<in> range f"
   848   by simp
   849 
   850 lemma rangeI: "f x \<in> range f"
   851   by simp
   852 
   853 lemma rangeE [elim?]: "b \<in> range (\<lambda>x. f x) ==> (!!x. b = f x ==> P) ==> P"
   854   by blast
   855 
   856 
   857 subsection {* Complete lattices *}
   858 
   859 notation
   860   less_eq  (infix "\<sqsubseteq>" 50) and
   861   less (infix "\<sqsubset>" 50) and
   862   inf  (infixl "\<sqinter>" 70) and
   863   sup  (infixl "\<squnion>" 65)
   864 
   865 class complete_lattice = lattice + bot + top +
   866   fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
   867     and Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
   868   assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"
   869      and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"
   870   assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A"
   871      and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z"
   872 begin
   873 
   874 lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<le> a}"
   875   by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
   876 
   877 lemma Sup_Inf:  "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<le> b}"
   878   by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
   879 
   880 lemma Inf_Univ: "\<Sqinter>UNIV = \<Squnion>{}"
   881   unfolding Sup_Inf by (auto simp add: UNIV_def)
   882 
   883 lemma Sup_Univ: "\<Squnion>UNIV = \<Sqinter>{}"
   884   unfolding Inf_Sup by (auto simp add: UNIV_def)
   885 
   886 lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
   887   by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower)
   888 
   889 lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
   890   by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper)
   891 
   892 lemma Inf_singleton [simp]:
   893   "\<Sqinter>{a} = a"
   894   by (auto intro: antisym Inf_lower Inf_greatest)
   895 
   896 lemma Sup_singleton [simp]:
   897   "\<Squnion>{a} = a"
   898   by (auto intro: antisym Sup_upper Sup_least)
   899 
   900 lemma Inf_insert_simp:
   901   "\<Sqinter>insert a A = (if A = {} then a else a \<sqinter> \<Sqinter>A)"
   902   by (cases "A = {}") (simp_all, simp add: Inf_insert)
   903 
   904 lemma Sup_insert_simp:
   905   "\<Squnion>insert a A = (if A = {} then a else a \<squnion> \<Squnion>A)"
   906   by (cases "A = {}") (simp_all, simp add: Sup_insert)
   907 
   908 lemma Inf_binary:
   909   "\<Sqinter>{a, b} = a \<sqinter> b"
   910   by (auto simp add: Inf_insert_simp)
   911 
   912 lemma Sup_binary:
   913   "\<Squnion>{a, b} = a \<squnion> b"
   914   by (auto simp add: Sup_insert_simp)
   915 
   916 lemma bot_def:
   917   "bot = \<Squnion>{}"
   918   by (auto intro: antisym Sup_least)
   919 
   920 lemma top_def:
   921   "top = \<Sqinter>{}"
   922   by (auto intro: antisym Inf_greatest)
   923 
   924 lemma sup_bot [simp]:
   925   "x \<squnion> bot = x"
   926   using bot_least [of x] by (simp add: le_iff_sup sup_commute)
   927 
   928 lemma inf_top [simp]:
   929   "x \<sqinter> top = x"
   930   using top_greatest [of x] by (simp add: le_iff_inf inf_commute)
   931 
   932 definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
   933   "SUPR A f == \<Squnion> (f ` A)"
   934 
   935 definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
   936   "INFI A f == \<Sqinter> (f ` A)"
   937 
   938 end
   939 
   940 syntax
   941   "_SUP1"     :: "pttrns => 'b => 'b"           ("(3SUP _./ _)" [0, 10] 10)
   942   "_SUP"      :: "pttrn => 'a set => 'b => 'b"  ("(3SUP _:_./ _)" [0, 10] 10)
   943   "_INF1"     :: "pttrns => 'b => 'b"           ("(3INF _./ _)" [0, 10] 10)
   944   "_INF"      :: "pttrn => 'a set => 'b => 'b"  ("(3INF _:_./ _)" [0, 10] 10)
   945 
   946 translations
   947   "SUP x y. B"   == "SUP x. SUP y. B"
   948   "SUP x. B"     == "CONST SUPR CONST UNIV (%x. B)"
   949   "SUP x. B"     == "SUP x:CONST UNIV. B"
   950   "SUP x:A. B"   == "CONST SUPR A (%x. B)"
   951   "INF x y. B"   == "INF x. INF y. B"
   952   "INF x. B"     == "CONST INFI CONST UNIV (%x. B)"
   953   "INF x. B"     == "INF x:CONST UNIV. B"
   954   "INF x:A. B"   == "CONST INFI A (%x. B)"
   955 
   956 (* To avoid eta-contraction of body: *)
   957 print_translation {*
   958 let
   959   fun btr' syn (A :: Abs abs :: ts) =
   960     let val (x,t) = atomic_abs_tr' abs
   961     in list_comb (Syntax.const syn $ x $ A $ t, ts) end
   962   val const_syntax_name = Sign.const_syntax_name @{theory} o fst o dest_Const
   963 in
   964 [(const_syntax_name @{term SUPR}, btr' "_SUP"),(const_syntax_name @{term "INFI"}, btr' "_INF")]
   965 end
   966 *}
   967 
   968 context complete_lattice
   969 begin
   970 
   971 lemma le_SUPI: "i : A \<Longrightarrow> M i \<le> (SUP i:A. M i)"
   972   by (auto simp add: SUPR_def intro: Sup_upper)
   973 
   974 lemma SUP_leI: "(\<And>i. i : A \<Longrightarrow> M i \<le> u) \<Longrightarrow> (SUP i:A. M i) \<le> u"
   975   by (auto simp add: SUPR_def intro: Sup_least)
   976 
   977 lemma INF_leI: "i : A \<Longrightarrow> (INF i:A. M i) \<le> M i"
   978   by (auto simp add: INFI_def intro: Inf_lower)
   979 
   980 lemma le_INFI: "(\<And>i. i : A \<Longrightarrow> u \<le> M i) \<Longrightarrow> u \<le> (INF i:A. M i)"
   981   by (auto simp add: INFI_def intro: Inf_greatest)
   982 
   983 lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M"
   984   by (auto intro: antisym SUP_leI le_SUPI)
   985 
   986 lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M"
   987   by (auto intro: antisym INF_leI le_INFI)
   988 
   989 end
   990 
   991 
   992 subsection {* Bool as complete lattice *}
   993 
   994 instantiation bool :: complete_lattice
   995 begin
   996 
   997 definition
   998   Inf_bool_def: "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)"
   999 
  1000 definition
  1001   Sup_bool_def: "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)"
  1002 
  1003 instance proof
  1004 qed (auto simp add: Inf_bool_def Sup_bool_def le_bool_def)
  1005 
  1006 end
  1007 
  1008 lemma Inf_empty_bool [simp]:
  1009   "\<Sqinter>{}"
  1010   unfolding Inf_bool_def by auto
  1011 
  1012 lemma not_Sup_empty_bool [simp]:
  1013   "\<not> \<Squnion>{}"
  1014   unfolding Sup_bool_def by auto
  1015 
  1016 
  1017 subsection {* Fun as complete lattice *}
  1018 
  1019 instantiation "fun" :: (type, complete_lattice) complete_lattice
  1020 begin
  1021 
  1022 definition
  1023   Inf_fun_def [code del]: "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})"
  1024 
  1025 definition
  1026   Sup_fun_def [code del]: "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})"
  1027 
  1028 instance proof
  1029 qed (auto simp add: Inf_fun_def Sup_fun_def le_fun_def
  1030   intro: Inf_lower Sup_upper Inf_greatest Sup_least)
  1031 
  1032 end
  1033 
  1034 lemma Inf_empty_fun:
  1035   "\<Sqinter>{} = (\<lambda>_. \<Sqinter>{})"
  1036   by rule (simp add: Inf_fun_def, simp add: empty_def)
  1037 
  1038 lemma Sup_empty_fun:
  1039   "\<Squnion>{} = (\<lambda>_. \<Squnion>{})"
  1040   by rule (simp add: Sup_fun_def, simp add: empty_def)
  1041 
  1042 
  1043 subsection {* Set as lattice *}
  1044 
  1045 definition INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
  1046   "INTER A B \<equiv> {y. \<forall>x\<in>A. y \<in> B x}"
  1047 
  1048 definition UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
  1049   "UNION A B \<equiv> {y. \<exists>x\<in>A. y \<in> B x}"
  1050 
  1051 definition Inter :: "'a set set \<Rightarrow> 'a set" where
  1052   "Inter S \<equiv> INTER S (\<lambda>x. x)"
  1053 
  1054 definition Union :: "'a set set \<Rightarrow> 'a set" where
  1055   "Union S \<equiv> UNION S (\<lambda>x. x)"
  1056 
  1057 notation (xsymbols)
  1058   Inter  ("\<Inter>_" [90] 90) and
  1059   Union  ("\<Union>_" [90] 90)
  1060 
  1061 syntax
  1062   "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
  1063   "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
  1064   "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 10] 10)
  1065   "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 10] 10)
  1066 
  1067 syntax (xsymbols)
  1068   "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
  1069   "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
  1070   "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 10] 10)
  1071   "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 10] 10)
  1072 
  1073 syntax (latex output)
  1074   "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
  1075   "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
  1076   "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10)
  1077   "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10)
  1078 
  1079 translations
  1080   "INT x y. B"  == "INT x. INT y. B"
  1081   "INT x. B"    == "CONST INTER CONST UNIV (%x. B)"
  1082   "INT x. B"    == "INT x:CONST UNIV. B"
  1083   "INT x:A. B"  == "CONST INTER A (%x. B)"
  1084   "UN x y. B"   == "UN x. UN y. B"
  1085   "UN x. B"     == "CONST UNION CONST UNIV (%x. B)"
  1086   "UN x. B"     == "UN x:CONST UNIV. B"
  1087   "UN x:A. B"   == "CONST UNION A (%x. B)"
  1088 
  1089 text {*
  1090   Note the difference between ordinary xsymbol syntax of indexed
  1091   unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})
  1092   and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The
  1093   former does not make the index expression a subscript of the
  1094   union/intersection symbol because this leads to problems with nested
  1095   subscripts in Proof General.
  1096 *}
  1097 
  1098 (* To avoid eta-contraction of body: *)
  1099 print_translation {*
  1100 let
  1101   fun btr' syn [A, Abs abs] =
  1102     let val (x, t) = atomic_abs_tr' abs
  1103     in Syntax.const syn $ x $ A $ t end
  1104 in [(@{const_syntax UNION}, btr' "@UNION"),(@{const_syntax INTER}, btr' "@INTER")] end
  1105 *}
  1106 
  1107 lemma Inter_image_eq [simp]:
  1108   "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
  1109   by (auto simp add: Inter_def INTER_def image_def)
  1110 
  1111 lemma Union_image_eq [simp]:
  1112   "\<Union>(B`A) = (\<Union>x\<in>A. B x)"
  1113   by (auto simp add: Union_def UNION_def image_def)
  1114 
  1115 lemma inf_set_eq: "A \<sqinter> B = A \<inter> B"
  1116   by (simp add: inf_fun_eq inf_bool_eq Int_def Collect_def mem_def)
  1117 
  1118 lemma sup_set_eq: "A \<squnion> B = A \<union> B"
  1119   by (simp add: sup_fun_eq sup_bool_eq Un_def Collect_def mem_def)
  1120 
  1121 lemma mono_Int: "mono f \<Longrightarrow> f (A \<inter> B) \<subseteq> f A \<inter> f B"
  1122   apply (fold inf_set_eq sup_set_eq)
  1123   apply (erule mono_inf)
  1124   done
  1125 
  1126 lemma mono_Un: "mono f \<Longrightarrow> f A \<union> f B \<subseteq> f (A \<union> B)"
  1127   apply (fold inf_set_eq sup_set_eq)
  1128   apply (erule mono_sup)
  1129   done
  1130 
  1131 lemma top_set_eq: "top = UNIV"
  1132   by (iprover intro!: subset_antisym subset_UNIV top_greatest)
  1133 
  1134 lemma bot_set_eq: "bot = {}"
  1135   by (iprover intro!: subset_antisym empty_subsetI bot_least)
  1136 
  1137 lemma Inter_eq:
  1138   "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
  1139   by (simp add: Inter_def INTER_def)
  1140 
  1141 lemma Union_eq:
  1142   "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
  1143   by (simp add: Union_def UNION_def)
  1144 
  1145 lemma Inf_set_eq:
  1146   "\<Sqinter>S = \<Inter>S"
  1147 proof (rule set_ext)
  1148   fix x
  1149   have "(\<forall>Q\<in>{P. \<exists>A\<in>S. P \<longleftrightarrow> x \<in> A}. Q) \<longleftrightarrow> (\<forall>A\<in>S. x \<in> A)"
  1150     by auto
  1151   then show "x \<in> \<Sqinter>S \<longleftrightarrow> x \<in> \<Inter>S"
  1152     by (simp add: Inter_eq Inf_fun_def Inf_bool_def) (simp add: mem_def)
  1153 qed
  1154 
  1155 lemma Sup_set_eq:
  1156   "\<Squnion>S = \<Union>S"
  1157 proof (rule set_ext)
  1158   fix x
  1159   have "(\<exists>Q\<in>{P. \<exists>A\<in>S. P \<longleftrightarrow> x \<in> A}. Q) \<longleftrightarrow> (\<exists>A\<in>S. x \<in> A)"
  1160     by auto
  1161   then show "x \<in> \<Squnion>S \<longleftrightarrow> x \<in> \<Union>S"
  1162     by (simp add: Union_eq Sup_fun_def Sup_bool_def) (simp add: mem_def)
  1163 qed
  1164 
  1165 lemma INFI_set_eq:
  1166   "(INF x:S. f x) = (\<Inter>x\<in>S. f x)"
  1167   by (simp add: INFI_def Inf_set_eq)
  1168 
  1169 lemma SUPR_set_eq:
  1170   "(SUP x:S. f x) = (\<Union>x\<in>S. f x)"
  1171   by (simp add: SUPR_def Sup_set_eq)
  1172   
  1173 no_notation
  1174   less_eq  (infix "\<sqsubseteq>" 50) and
  1175   less (infix "\<sqsubset>" 50) and
  1176   inf  (infixl "\<sqinter>" 70) and
  1177   sup  (infixl "\<squnion>" 65) and
  1178   Inf  ("\<Sqinter>_" [900] 900) and
  1179   Sup  ("\<Squnion>_" [900] 900)
  1180 
  1181 
  1182 subsubsection {* Unions of families *}
  1183 
  1184 declare UNION_def [noatp]
  1185 
  1186 lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)"
  1187   by (unfold UNION_def) blast
  1188 
  1189 lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)"
  1190   -- {* The order of the premises presupposes that @{term A} is rigid;
  1191     @{term b} may be flexible. *}
  1192   by auto
  1193 
  1194 lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R"
  1195   by (unfold UNION_def) blast
  1196 
  1197 lemma UN_cong [cong]:
  1198     "A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
  1199   by (simp add: UNION_def)
  1200 
  1201 lemma strong_UN_cong:
  1202     "A = B ==> (!!x. x:B =simp=> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
  1203   by (simp add: UNION_def simp_implies_def)
  1204 
  1205 lemma image_eq_UN: "f`A = (UN x:A. {f x})"
  1206   by blast
  1207 
  1208 
  1209 subsubsection {* Intersections of families *}
  1210 
  1211 lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)"
  1212   by (unfold INTER_def) blast
  1213 
  1214 lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)"
  1215   by (unfold INTER_def) blast
  1216 
  1217 lemma INT_D [elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a"
  1218   by auto
  1219 
  1220 lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R"
  1221   -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}
  1222   by (unfold INTER_def) blast
  1223 
  1224 lemma INT_cong [cong]:
  1225     "A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)"
  1226   by (simp add: INTER_def)
  1227 
  1228 
  1229 subsubsection {* Union *}
  1230 
  1231 lemma Union_iff [simp,noatp]: "(A : Union C) = (EX X:C. A:X)"
  1232   by (unfold Union_def) blast
  1233 
  1234 lemma UnionI [intro]: "X:C ==> A:X ==> A : Union C"
  1235   -- {* The order of the premises presupposes that @{term C} is rigid;
  1236     @{term A} may be flexible. *}
  1237   by auto
  1238 
  1239 lemma UnionE [elim!]: "A : Union C ==> (!!X. A:X ==> X:C ==> R) ==> R"
  1240   by (unfold Union_def) blast
  1241 
  1242 
  1243 subsubsection {* Inter *}
  1244 
  1245 lemma Inter_iff [simp,noatp]: "(A : Inter C) = (ALL X:C. A:X)"
  1246   by (unfold Inter_def) blast
  1247 
  1248 lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C"
  1249   by (simp add: Inter_def)
  1250 
  1251 text {*
  1252   \medskip A ``destruct'' rule -- every @{term X} in @{term C}
  1253   contains @{term A} as an element, but @{prop "A:X"} can hold when
  1254   @{prop "X:C"} does not!  This rule is analogous to @{text spec}.
  1255 *}
  1256 
  1257 lemma InterD [elim]: "A : Inter C ==> X:C ==> A:X"
  1258   by auto
  1259 
  1260 lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R"
  1261   -- {* ``Classical'' elimination rule -- does not require proving
  1262     @{prop "X:C"}. *}
  1263   by (unfold Inter_def) blast
  1264 
  1265 
  1266 subsubsection {* Set reasoning tools *}
  1267 
  1268 text{* Elimination of @{text"{x. \<dots> & x=t & \<dots>}"}. *}
  1269 
  1270 lemma Collect_conv_if: "{x. x=a & P x} = (if P a then {a} else {})"
  1271 by auto
  1272 
  1273 lemma Collect_conv_if2: "{x. a=x & P x} = (if P a then {a} else {})"
  1274 by auto
  1275 
  1276 text {*
  1277 Simproc for pulling @{text "x=t"} in @{text "{x. \<dots> & x=t & \<dots>}"}
  1278 to the front (and similarly for @{text "t=x"}):
  1279 *}
  1280 
  1281 ML{*
  1282   local
  1283     val Coll_perm_tac = rtac @{thm Collect_cong} 1 THEN rtac @{thm iffI} 1 THEN
  1284     ALLGOALS(EVERY'[REPEAT_DETERM o (etac @{thm conjE}),
  1285                     DEPTH_SOLVE_1 o (ares_tac [@{thm conjI}])])
  1286   in
  1287     val defColl_regroup = Simplifier.simproc @{theory}
  1288       "defined Collect" ["{x. P x & Q x}"]
  1289       (Quantifier1.rearrange_Coll Coll_perm_tac)
  1290   end;
  1291 
  1292   Addsimprocs [defColl_regroup];
  1293 *}
  1294 
  1295 text {*
  1296   Rewrite rules for boolean case-splitting: faster than @{text
  1297   "split_if [split]"}.
  1298 *}
  1299 
  1300 lemma split_if_eq1: "((if Q then x else y) = b) = ((Q --> x = b) & (~ Q --> y = b))"
  1301   by (rule split_if)
  1302 
  1303 lemma split_if_eq2: "(a = (if Q then x else y)) = ((Q --> a = x) & (~ Q --> a = y))"
  1304   by (rule split_if)
  1305 
  1306 text {*
  1307   Split ifs on either side of the membership relation.  Not for @{text
  1308   "[simp]"} -- can cause goals to blow up!
  1309 *}
  1310 
  1311 lemma split_if_mem1: "((if Q then x else y) : b) = ((Q --> x : b) & (~ Q --> y : b))"
  1312   by (rule split_if)
  1313 
  1314 lemma split_if_mem2: "(a : (if Q then x else y)) = ((Q --> a : x) & (~ Q --> a : y))"
  1315   by (rule split_if [where P="%S. a : S"])
  1316 
  1317 lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2
  1318 
  1319 (*Would like to add these, but the existing code only searches for the
  1320   outer-level constant, which in this case is just "op :"; we instead need
  1321   to use term-nets to associate patterns with rules.  Also, if a rule fails to
  1322   apply, then the formula should be kept.
  1323   [("HOL.uminus", Compl_iff RS iffD1), ("HOL.minus", [Diff_iff RS iffD1]),
  1324    ("Int", [IntD1,IntD2]),
  1325    ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]
  1326  *)
  1327 
  1328 ML {*
  1329   val mksimps_pairs = [(@{const_name Ball}, @{thms bspec})] @ mksimps_pairs;
  1330 *}
  1331 declaration {* fn _ =>
  1332   Simplifier.map_ss (fn ss => ss setmksimps (mksimps mksimps_pairs))
  1333 *}
  1334 
  1335 
  1336 subsubsection {* The ``proper subset'' relation *}
  1337 
  1338 lemma psubsetI [intro!,noatp]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B"
  1339   by (unfold less_le) blast
  1340 
  1341 lemma psubsetE [elim!,noatp]: 
  1342     "[|A \<subset> B;  [|A \<subseteq> B; ~ (B\<subseteq>A)|] ==> R|] ==> R"
  1343   by (unfold less_le) blast
  1344 
  1345 lemma psubset_insert_iff:
  1346   "(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)"
  1347   by (auto simp add: less_le subset_insert_iff)
  1348 
  1349 lemma psubset_eq: "(A \<subset> B) = (A \<subseteq> B & A \<noteq> B)"
  1350   by (simp only: less_le)
  1351 
  1352 lemma psubset_imp_subset: "A \<subset> B ==> A \<subseteq> B"
  1353   by (simp add: psubset_eq)
  1354 
  1355 lemma psubset_trans: "[| A \<subset> B; B \<subset> C |] ==> A \<subset> C"
  1356 apply (unfold less_le)
  1357 apply (auto dest: subset_antisym)
  1358 done
  1359 
  1360 lemma psubsetD: "[| A \<subset> B; c \<in> A |] ==> c \<in> B"
  1361 apply (unfold less_le)
  1362 apply (auto dest: subsetD)
  1363 done
  1364 
  1365 lemma psubset_subset_trans: "A \<subset> B ==> B \<subseteq> C ==> A \<subset> C"
  1366   by (auto simp add: psubset_eq)
  1367 
  1368 lemma subset_psubset_trans: "A \<subseteq> B ==> B \<subset> C ==> A \<subset> C"
  1369   by (auto simp add: psubset_eq)
  1370 
  1371 lemma psubset_imp_ex_mem: "A \<subset> B ==> \<exists>b. b \<in> (B - A)"
  1372   by (unfold less_le) blast
  1373 
  1374 lemma atomize_ball:
  1375     "(!!x. x \<in> A ==> P x) == Trueprop (\<forall>x\<in>A. P x)"
  1376   by (simp only: Ball_def atomize_all atomize_imp)
  1377 
  1378 lemmas [symmetric, rulify] = atomize_ball
  1379   and [symmetric, defn] = atomize_ball
  1380 
  1381 
  1382 subsection {* Further set-theory lemmas *}
  1383 
  1384 subsubsection {* Derived rules involving subsets. *}
  1385 
  1386 text {* @{text insert}. *}
  1387 
  1388 lemma subset_insertI: "B \<subseteq> insert a B"
  1389   by (rule subsetI) (erule insertI2)
  1390 
  1391 lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B"
  1392   by blast
  1393 
  1394 lemma subset_insert: "x \<notin> A ==> (A \<subseteq> insert x B) = (A \<subseteq> B)"
  1395   by blast
  1396 
  1397 
  1398 text {* \medskip Big Union -- least upper bound of a set. *}
  1399 
  1400 lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A"
  1401   by (iprover intro: subsetI UnionI)
  1402 
  1403 lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C"
  1404   by (iprover intro: subsetI elim: UnionE dest: subsetD)
  1405 
  1406 
  1407 text {* \medskip General union. *}
  1408 
  1409 lemma UN_upper: "a \<in> A ==> B a \<subseteq> (\<Union>x\<in>A. B x)"
  1410   by blast
  1411 
  1412 lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C"
  1413   by (iprover intro: subsetI elim: UN_E dest: subsetD)
  1414 
  1415 
  1416 text {* \medskip Big Intersection -- greatest lower bound of a set. *}
  1417 
  1418 lemma Inter_lower: "B \<in> A ==> Inter A \<subseteq> B"
  1419   by blast
  1420 
  1421 lemma Inter_subset:
  1422   "[| !!X. X \<in> A ==> X \<subseteq> B; A ~= {} |] ==> \<Inter>A \<subseteq> B"
  1423   by blast
  1424 
  1425 lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A"
  1426   by (iprover intro: InterI subsetI dest: subsetD)
  1427 
  1428 lemma INT_lower: "a \<in> A ==> (\<Inter>x\<in>A. B x) \<subseteq> B a"
  1429   by blast
  1430 
  1431 lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)"
  1432   by (iprover intro: INT_I subsetI dest: subsetD)
  1433 
  1434 
  1435 text {* \medskip Finite Union -- the least upper bound of two sets. *}
  1436 
  1437 lemma Un_upper1: "A \<subseteq> A \<union> B"
  1438   by blast
  1439 
  1440 lemma Un_upper2: "B \<subseteq> A \<union> B"
  1441   by blast
  1442 
  1443 lemma Un_least: "A \<subseteq> C ==> B \<subseteq> C ==> A \<union> B \<subseteq> C"
  1444   by blast
  1445 
  1446 
  1447 text {* \medskip Finite Intersection -- the greatest lower bound of two sets. *}
  1448 
  1449 lemma Int_lower1: "A \<inter> B \<subseteq> A"
  1450   by blast
  1451 
  1452 lemma Int_lower2: "A \<inter> B \<subseteq> B"
  1453   by blast
  1454 
  1455 lemma Int_greatest: "C \<subseteq> A ==> C \<subseteq> B ==> C \<subseteq> A \<inter> B"
  1456   by blast
  1457 
  1458 
  1459 text {* \medskip Set difference. *}
  1460 
  1461 lemma Diff_subset: "A - B \<subseteq> A"
  1462   by blast
  1463 
  1464 lemma Diff_subset_conv: "(A - B \<subseteq> C) = (A \<subseteq> B \<union> C)"
  1465 by blast
  1466 
  1467 
  1468 subsubsection {* Equalities involving union, intersection, inclusion, etc. *}
  1469 
  1470 text {* @{text "{}"}. *}
  1471 
  1472 lemma Collect_const [simp]: "{s. P} = (if P then UNIV else {})"
  1473   -- {* supersedes @{text "Collect_False_empty"} *}
  1474   by auto
  1475 
  1476 lemma subset_empty [simp]: "(A \<subseteq> {}) = (A = {})"
  1477   by blast
  1478 
  1479 lemma not_psubset_empty [iff]: "\<not> (A < {})"
  1480   by (unfold less_le) blast
  1481 
  1482 lemma Collect_empty_eq [simp]: "(Collect P = {}) = (\<forall>x. \<not> P x)"
  1483 by blast
  1484 
  1485 lemma empty_Collect_eq [simp]: "({} = Collect P) = (\<forall>x. \<not> P x)"
  1486 by blast
  1487 
  1488 lemma Collect_neg_eq: "{x. \<not> P x} = - {x. P x}"
  1489   by blast
  1490 
  1491 lemma Collect_disj_eq: "{x. P x | Q x} = {x. P x} \<union> {x. Q x}"
  1492   by blast
  1493 
  1494 lemma Collect_imp_eq: "{x. P x --> Q x} = -{x. P x} \<union> {x. Q x}"
  1495   by blast
  1496 
  1497 lemma Collect_conj_eq: "{x. P x & Q x} = {x. P x} \<inter> {x. Q x}"
  1498   by blast
  1499 
  1500 lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
  1501   by blast
  1502 
  1503 lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
  1504   by blast
  1505 
  1506 lemma Collect_ex_eq [noatp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
  1507   by blast
  1508 
  1509 lemma Collect_bex_eq [noatp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
  1510   by blast
  1511 
  1512 
  1513 text {* \medskip @{text insert}. *}
  1514 
  1515 lemma insert_is_Un: "insert a A = {a} Un A"
  1516   -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a {}"} *}
  1517   by blast
  1518 
  1519 lemma insert_not_empty [simp]: "insert a A \<noteq> {}"
  1520   by blast
  1521 
  1522 lemmas empty_not_insert = insert_not_empty [symmetric, standard]
  1523 declare empty_not_insert [simp]
  1524 
  1525 lemma insert_absorb: "a \<in> A ==> insert a A = A"
  1526   -- {* @{text "[simp]"} causes recursive calls when there are nested inserts *}
  1527   -- {* with \emph{quadratic} running time *}
  1528   by blast
  1529 
  1530 lemma insert_absorb2 [simp]: "insert x (insert x A) = insert x A"
  1531   by blast
  1532 
  1533 lemma insert_commute: "insert x (insert y A) = insert y (insert x A)"
  1534   by blast
  1535 
  1536 lemma insert_subset [simp]: "(insert x A \<subseteq> B) = (x \<in> B & A \<subseteq> B)"
  1537   by blast
  1538 
  1539 lemma mk_disjoint_insert: "a \<in> A ==> \<exists>B. A = insert a B & a \<notin> B"
  1540   -- {* use new @{text B} rather than @{text "A - {a}"} to avoid infinite unfolding *}
  1541   apply (rule_tac x = "A - {a}" in exI, blast)
  1542   done
  1543 
  1544 lemma insert_Collect: "insert a (Collect P) = {u. u \<noteq> a --> P u}"
  1545   by auto
  1546 
  1547 lemma UN_insert_distrib: "u \<in> A ==> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
  1548   by blast
  1549 
  1550 lemma insert_inter_insert[simp]: "insert a A \<inter> insert a B = insert a (A \<inter> B)"
  1551   by blast
  1552 
  1553 lemma insert_disjoint [simp,noatp]:
  1554  "(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})"
  1555  "({} = insert a A \<inter> B) = (a \<notin> B \<and> {} = A \<inter> B)"
  1556   by auto
  1557 
  1558 lemma disjoint_insert [simp,noatp]:
  1559  "(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})"
  1560  "({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)"
  1561   by auto
  1562 
  1563 text {* \medskip @{text image}. *}
  1564 
  1565 lemma image_empty [simp]: "f`{} = {}"
  1566   by blast
  1567 
  1568 lemma image_insert [simp]: "f ` insert a B = insert (f a) (f`B)"
  1569   by blast
  1570 
  1571 lemma image_constant: "x \<in> A ==> (\<lambda>x. c) ` A = {c}"
  1572   by auto
  1573 
  1574 lemma image_constant_conv: "(%x. c) ` A = (if A = {} then {} else {c})"
  1575 by auto
  1576 
  1577 lemma image_image: "f ` (g ` A) = (\<lambda>x. f (g x)) ` A"
  1578 by blast
  1579 
  1580 lemma insert_image [simp]: "x \<in> A ==> insert (f x) (f`A) = f`A"
  1581 by blast
  1582 
  1583 lemma image_is_empty [iff]: "(f`A = {}) = (A = {})"
  1584 by blast
  1585 
  1586 lemma empty_is_image[iff]: "({} = f ` A) = (A = {})"
  1587 by blast
  1588 
  1589 
  1590 lemma image_Collect [noatp]: "f ` {x. P x} = {f x | x. P x}"
  1591   -- {* NOT suitable as a default simprule: the RHS isn't simpler than the LHS,
  1592       with its implicit quantifier and conjunction.  Also image enjoys better
  1593       equational properties than does the RHS. *}
  1594   by blast
  1595 
  1596 lemma if_image_distrib [simp]:
  1597   "(\<lambda>x. if P x then f x else g x) ` S
  1598     = (f ` (S \<inter> {x. P x})) \<union> (g ` (S \<inter> {x. \<not> P x}))"
  1599   by (auto simp add: image_def)
  1600 
  1601 lemma image_cong: "M = N ==> (!!x. x \<in> N ==> f x = g x) ==> f`M = g`N"
  1602   by (simp add: image_def)
  1603 
  1604 
  1605 text {* \medskip @{text range}. *}
  1606 
  1607 lemma full_SetCompr_eq [noatp]: "{u. \<exists>x. u = f x} = range f"
  1608   by auto
  1609 
  1610 lemma range_composition: "range (\<lambda>x. f (g x)) = f`range g"
  1611 by (subst image_image, simp)
  1612 
  1613 
  1614 text {* \medskip @{text Int} *}
  1615 
  1616 lemma Int_absorb [simp]: "A \<inter> A = A"
  1617   by blast
  1618 
  1619 lemma Int_left_absorb: "A \<inter> (A \<inter> B) = A \<inter> B"
  1620   by blast
  1621 
  1622 lemma Int_commute: "A \<inter> B = B \<inter> A"
  1623   by blast
  1624 
  1625 lemma Int_left_commute: "A \<inter> (B \<inter> C) = B \<inter> (A \<inter> C)"
  1626   by blast
  1627 
  1628 lemma Int_assoc: "(A \<inter> B) \<inter> C = A \<inter> (B \<inter> C)"
  1629   by blast
  1630 
  1631 lemmas Int_ac = Int_assoc Int_left_absorb Int_commute Int_left_commute
  1632   -- {* Intersection is an AC-operator *}
  1633 
  1634 lemma Int_absorb1: "B \<subseteq> A ==> A \<inter> B = B"
  1635   by blast
  1636 
  1637 lemma Int_absorb2: "A \<subseteq> B ==> A \<inter> B = A"
  1638   by blast
  1639 
  1640 lemma Int_empty_left [simp]: "{} \<inter> B = {}"
  1641   by blast
  1642 
  1643 lemma Int_empty_right [simp]: "A \<inter> {} = {}"
  1644   by blast
  1645 
  1646 lemma disjoint_eq_subset_Compl: "(A \<inter> B = {}) = (A \<subseteq> -B)"
  1647   by blast
  1648 
  1649 lemma disjoint_iff_not_equal: "(A \<inter> B = {}) = (\<forall>x\<in>A. \<forall>y\<in>B. x \<noteq> y)"
  1650   by blast
  1651 
  1652 lemma Int_UNIV_left [simp]: "UNIV \<inter> B = B"
  1653   by blast
  1654 
  1655 lemma Int_UNIV_right [simp]: "A \<inter> UNIV = A"
  1656   by blast
  1657 
  1658 lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"
  1659   by blast
  1660 
  1661 lemma Int_Un_distrib: "A \<inter> (B \<union> C) = (A \<inter> B) \<union> (A \<inter> C)"
  1662   by blast
  1663 
  1664 lemma Int_Un_distrib2: "(B \<union> C) \<inter> A = (B \<inter> A) \<union> (C \<inter> A)"
  1665   by blast
  1666 
  1667 lemma Int_UNIV [simp,noatp]: "(A \<inter> B = UNIV) = (A = UNIV & B = UNIV)"
  1668   by blast
  1669 
  1670 lemma Int_subset_iff [simp]: "(C \<subseteq> A \<inter> B) = (C \<subseteq> A & C \<subseteq> B)"
  1671   by blast
  1672 
  1673 lemma Int_Collect: "(x \<in> A \<inter> {x. P x}) = (x \<in> A & P x)"
  1674   by blast
  1675 
  1676 
  1677 text {* \medskip @{text Un}. *}
  1678 
  1679 lemma Un_absorb [simp]: "A \<union> A = A"
  1680   by blast
  1681 
  1682 lemma Un_left_absorb: "A \<union> (A \<union> B) = A \<union> B"
  1683   by blast
  1684 
  1685 lemma Un_commute: "A \<union> B = B \<union> A"
  1686   by blast
  1687 
  1688 lemma Un_left_commute: "A \<union> (B \<union> C) = B \<union> (A \<union> C)"
  1689   by blast
  1690 
  1691 lemma Un_assoc: "(A \<union> B) \<union> C = A \<union> (B \<union> C)"
  1692   by blast
  1693 
  1694 lemmas Un_ac = Un_assoc Un_left_absorb Un_commute Un_left_commute
  1695   -- {* Union is an AC-operator *}
  1696 
  1697 lemma Un_absorb1: "A \<subseteq> B ==> A \<union> B = B"
  1698   by blast
  1699 
  1700 lemma Un_absorb2: "B \<subseteq> A ==> A \<union> B = A"
  1701   by blast
  1702 
  1703 lemma Un_empty_left [simp]: "{} \<union> B = B"
  1704   by blast
  1705 
  1706 lemma Un_empty_right [simp]: "A \<union> {} = A"
  1707   by blast
  1708 
  1709 lemma Un_UNIV_left [simp]: "UNIV \<union> B = UNIV"
  1710   by blast
  1711 
  1712 lemma Un_UNIV_right [simp]: "A \<union> UNIV = UNIV"
  1713   by blast
  1714 
  1715 lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"
  1716   by blast
  1717 
  1718 lemma Un_insert_left [simp]: "(insert a B) \<union> C = insert a (B \<union> C)"
  1719   by blast
  1720 
  1721 lemma Un_insert_right [simp]: "A \<union> (insert a B) = insert a (A \<union> B)"
  1722   by blast
  1723 
  1724 lemma Int_insert_left:
  1725     "(insert a B) Int C = (if a \<in> C then insert a (B \<inter> C) else B \<inter> C)"
  1726   by auto
  1727 
  1728 lemma Int_insert_right:
  1729     "A \<inter> (insert a B) = (if a \<in> A then insert a (A \<inter> B) else A \<inter> B)"
  1730   by auto
  1731 
  1732 lemma Un_Int_distrib: "A \<union> (B \<inter> C) = (A \<union> B) \<inter> (A \<union> C)"
  1733   by blast
  1734 
  1735 lemma Un_Int_distrib2: "(B \<inter> C) \<union> A = (B \<union> A) \<inter> (C \<union> A)"
  1736   by blast
  1737 
  1738 lemma Un_Int_crazy:
  1739     "(A \<inter> B) \<union> (B \<inter> C) \<union> (C \<inter> A) = (A \<union> B) \<inter> (B \<union> C) \<inter> (C \<union> A)"
  1740   by blast
  1741 
  1742 lemma subset_Un_eq: "(A \<subseteq> B) = (A \<union> B = B)"
  1743   by blast
  1744 
  1745 lemma Un_empty [iff]: "(A \<union> B = {}) = (A = {} & B = {})"
  1746   by blast
  1747 
  1748 lemma Un_subset_iff [simp]: "(A \<union> B \<subseteq> C) = (A \<subseteq> C & B \<subseteq> C)"
  1749   by blast
  1750 
  1751 lemma Un_Diff_Int: "(A - B) \<union> (A \<inter> B) = A"
  1752   by blast
  1753 
  1754 lemma Diff_Int2: "A \<inter> C - B \<inter> C = A \<inter> C - B"
  1755   by blast
  1756 
  1757 
  1758 text {* \medskip Set complement *}
  1759 
  1760 lemma Compl_disjoint [simp]: "A \<inter> -A = {}"
  1761   by blast
  1762 
  1763 lemma Compl_disjoint2 [simp]: "-A \<inter> A = {}"
  1764   by blast
  1765 
  1766 lemma Compl_partition: "A \<union> -A = UNIV"
  1767   by blast
  1768 
  1769 lemma Compl_partition2: "-A \<union> A = UNIV"
  1770   by blast
  1771 
  1772 lemma double_complement [simp]: "- (-A) = (A::'a set)"
  1773   by blast
  1774 
  1775 lemma Compl_Un [simp]: "-(A \<union> B) = (-A) \<inter> (-B)"
  1776   by blast
  1777 
  1778 lemma Compl_Int [simp]: "-(A \<inter> B) = (-A) \<union> (-B)"
  1779   by blast
  1780 
  1781 lemma Compl_UN [simp]: "-(\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
  1782   by blast
  1783 
  1784 lemma Compl_INT [simp]: "-(\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
  1785   by blast
  1786 
  1787 lemma subset_Compl_self_eq: "(A \<subseteq> -A) = (A = {})"
  1788   by blast
  1789 
  1790 lemma Un_Int_assoc_eq: "((A \<inter> B) \<union> C = A \<inter> (B \<union> C)) = (C \<subseteq> A)"
  1791   -- {* Halmos, Naive Set Theory, page 16. *}
  1792   by blast
  1793 
  1794 lemma Compl_UNIV_eq [simp]: "-UNIV = {}"
  1795   by blast
  1796 
  1797 lemma Compl_empty_eq [simp]: "-{} = UNIV"
  1798   by blast
  1799 
  1800 lemma Compl_subset_Compl_iff [iff]: "(-A \<subseteq> -B) = (B \<subseteq> A)"
  1801   by blast
  1802 
  1803 lemma Compl_eq_Compl_iff [iff]: "(-A = -B) = (A = (B::'a set))"
  1804   by blast
  1805 
  1806 
  1807 text {* \medskip @{text Union}. *}
  1808 
  1809 lemma Union_empty [simp]: "Union({}) = {}"
  1810   by blast
  1811 
  1812 lemma Union_UNIV [simp]: "Union UNIV = UNIV"
  1813   by blast
  1814 
  1815 lemma Union_insert [simp]: "Union (insert a B) = a \<union> \<Union>B"
  1816   by blast
  1817 
  1818 lemma Union_Un_distrib [simp]: "\<Union>(A Un B) = \<Union>A \<union> \<Union>B"
  1819   by blast
  1820 
  1821 lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
  1822   by blast
  1823 
  1824 lemma Union_empty_conv [simp,noatp]: "(\<Union>A = {}) = (\<forall>x\<in>A. x = {})"
  1825   by blast
  1826 
  1827 lemma empty_Union_conv [simp,noatp]: "({} = \<Union>A) = (\<forall>x\<in>A. x = {})"
  1828   by blast
  1829 
  1830 lemma Union_disjoint: "(\<Union>C \<inter> A = {}) = (\<forall>B\<in>C. B \<inter> A = {})"
  1831   by blast
  1832 
  1833 
  1834 text {* \medskip @{text Inter}. *}
  1835 
  1836 lemma Inter_empty [simp]: "\<Inter>{} = UNIV"
  1837   by blast
  1838 
  1839 lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}"
  1840   by blast
  1841 
  1842 lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
  1843   by blast
  1844 
  1845 lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
  1846   by blast
  1847 
  1848 lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
  1849   by blast
  1850 
  1851 lemma Inter_UNIV_conv [simp,noatp]:
  1852   "(\<Inter>A = UNIV) = (\<forall>x\<in>A. x = UNIV)"
  1853   "(UNIV = \<Inter>A) = (\<forall>x\<in>A. x = UNIV)"
  1854   by blast+
  1855 
  1856 
  1857 text {*
  1858   \medskip @{text UN} and @{text INT}.
  1859 
  1860   Basic identities: *}
  1861 
  1862 lemma UN_empty [simp,noatp]: "(\<Union>x\<in>{}. B x) = {}"
  1863   by blast
  1864 
  1865 lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"
  1866   by blast
  1867 
  1868 lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
  1869   by blast
  1870 
  1871 lemma UN_absorb: "k \<in> I ==> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
  1872   by auto
  1873 
  1874 lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"
  1875   by blast
  1876 
  1877 lemma INT_absorb: "k \<in> I ==> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
  1878   by blast
  1879 
  1880 lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
  1881   by blast
  1882 
  1883 lemma UN_Un[simp]: "(\<Union>i \<in> A \<union> B. M i) = (\<Union>i\<in>A. M i) \<union> (\<Union>i\<in>B. M i)"
  1884   by blast
  1885 
  1886 lemma UN_UN_flatten: "(\<Union>x \<in> (\<Union>y\<in>A. B y). C x) = (\<Union>y\<in>A. \<Union>x\<in>B y. C x)"
  1887   by blast
  1888 
  1889 lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
  1890   by blast
  1891 
  1892 lemma INT_subset_iff: "(B \<subseteq> (\<Inter>i\<in>I. A i)) = (\<forall>i\<in>I. B \<subseteq> A i)"
  1893   by blast
  1894 
  1895 lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
  1896   by blast
  1897 
  1898 lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"
  1899   by blast
  1900 
  1901 lemma INT_insert_distrib:
  1902     "u \<in> A ==> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
  1903   by blast
  1904 
  1905 lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
  1906   by blast
  1907 
  1908 lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
  1909   by auto
  1910 
  1911 lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
  1912   by auto
  1913 
  1914 lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"
  1915   by blast
  1916 
  1917 lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"
  1918   -- {* Look: it has an \emph{existential} quantifier *}
  1919   by blast
  1920 
  1921 lemma UNION_empty_conv[simp]:
  1922   "({} = (UN x:A. B x)) = (\<forall>x\<in>A. B x = {})"
  1923   "((UN x:A. B x) = {}) = (\<forall>x\<in>A. B x = {})"
  1924 by blast+
  1925 
  1926 lemma INTER_UNIV_conv[simp]:
  1927  "(UNIV = (INT x:A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
  1928  "((INT x:A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
  1929 by blast+
  1930 
  1931 
  1932 text {* \medskip Distributive laws: *}
  1933 
  1934 lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
  1935   by blast
  1936 
  1937 lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
  1938   by blast
  1939 
  1940 lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A`C) \<union> \<Union>(B`C)"
  1941   -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
  1942   -- {* Union of a family of unions *}
  1943   by blast
  1944 
  1945 lemma UN_Un_distrib: "(\<Union>i\<in>I. A i \<union> B i) = (\<Union>i\<in>I. A i) \<union> (\<Union>i\<in>I. B i)"
  1946   -- {* Equivalent version *}
  1947   by blast
  1948 
  1949 lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
  1950   by blast
  1951 
  1952 lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A`C) \<inter> \<Inter>(B`C)"
  1953   by blast
  1954 
  1955 lemma INT_Int_distrib: "(\<Inter>i\<in>I. A i \<inter> B i) = (\<Inter>i\<in>I. A i) \<inter> (\<Inter>i\<in>I. B i)"
  1956   -- {* Equivalent version *}
  1957   by blast
  1958 
  1959 lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
  1960   -- {* Halmos, Naive Set Theory, page 35. *}
  1961   by blast
  1962 
  1963 lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
  1964   by blast
  1965 
  1966 lemma Int_UN_distrib2: "(\<Union>i\<in>I. A i) \<inter> (\<Union>j\<in>J. B j) = (\<Union>i\<in>I. \<Union>j\<in>J. A i \<inter> B j)"
  1967   by blast
  1968 
  1969 lemma Un_INT_distrib2: "(\<Inter>i\<in>I. A i) \<union> (\<Inter>j\<in>J. B j) = (\<Inter>i\<in>I. \<Inter>j\<in>J. A i \<union> B j)"
  1970   by blast
  1971 
  1972 
  1973 text {* \medskip Bounded quantifiers.
  1974 
  1975   The following are not added to the default simpset because
  1976   (a) they duplicate the body and (b) there are no similar rules for @{text Int}. *}
  1977 
  1978 lemma ball_Un: "(\<forall>x \<in> A \<union> B. P x) = ((\<forall>x\<in>A. P x) & (\<forall>x\<in>B. P x))"
  1979   by blast
  1980 
  1981 lemma bex_Un: "(\<exists>x \<in> A \<union> B. P x) = ((\<exists>x\<in>A. P x) | (\<exists>x\<in>B. P x))"
  1982   by blast
  1983 
  1984 lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) = (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
  1985   by blast
  1986 
  1987 lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) = (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
  1988   by blast
  1989 
  1990 
  1991 text {* \medskip Set difference. *}
  1992 
  1993 lemma Diff_eq: "A - B = A \<inter> (-B)"
  1994   by blast
  1995 
  1996 lemma Diff_eq_empty_iff [simp,noatp]: "(A - B = {}) = (A \<subseteq> B)"
  1997   by blast
  1998 
  1999 lemma Diff_cancel [simp]: "A - A = {}"
  2000   by blast
  2001 
  2002 lemma Diff_idemp [simp]: "(A - B) - B = A - (B::'a set)"
  2003 by blast
  2004 
  2005 lemma Diff_triv: "A \<inter> B = {} ==> A - B = A"
  2006   by (blast elim: equalityE)
  2007 
  2008 lemma empty_Diff [simp]: "{} - A = {}"
  2009   by blast
  2010 
  2011 lemma Diff_empty [simp]: "A - {} = A"
  2012   by blast
  2013 
  2014 lemma Diff_UNIV [simp]: "A - UNIV = {}"
  2015   by blast
  2016 
  2017 lemma Diff_insert0 [simp,noatp]: "x \<notin> A ==> A - insert x B = A - B"
  2018   by blast
  2019 
  2020 lemma Diff_insert: "A - insert a B = A - B - {a}"
  2021   -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a 0"} *}
  2022   by blast
  2023 
  2024 lemma Diff_insert2: "A - insert a B = A - {a} - B"
  2025   -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a 0"} *}
  2026   by blast
  2027 
  2028 lemma insert_Diff_if: "insert x A - B = (if x \<in> B then A - B else insert x (A - B))"
  2029   by auto
  2030 
  2031 lemma insert_Diff1 [simp]: "x \<in> B ==> insert x A - B = A - B"
  2032   by blast
  2033 
  2034 lemma insert_Diff_single[simp]: "insert a (A - {a}) = insert a A"
  2035 by blast
  2036 
  2037 lemma insert_Diff: "a \<in> A ==> insert a (A - {a}) = A"
  2038   by blast
  2039 
  2040 lemma Diff_insert_absorb: "x \<notin> A ==> (insert x A) - {x} = A"
  2041   by auto
  2042 
  2043 lemma Diff_disjoint [simp]: "A \<inter> (B - A) = {}"
  2044   by blast
  2045 
  2046 lemma Diff_partition: "A \<subseteq> B ==> A \<union> (B - A) = B"
  2047   by blast
  2048 
  2049 lemma double_diff: "A \<subseteq> B ==> B \<subseteq> C ==> B - (C - A) = A"
  2050   by blast
  2051 
  2052 lemma Un_Diff_cancel [simp]: "A \<union> (B - A) = A \<union> B"
  2053   by blast
  2054 
  2055 lemma Un_Diff_cancel2 [simp]: "(B - A) \<union> A = B \<union> A"
  2056   by blast
  2057 
  2058 lemma Diff_Un: "A - (B \<union> C) = (A - B) \<inter> (A - C)"
  2059   by blast
  2060 
  2061 lemma Diff_Int: "A - (B \<inter> C) = (A - B) \<union> (A - C)"
  2062   by blast
  2063 
  2064 lemma Un_Diff: "(A \<union> B) - C = (A - C) \<union> (B - C)"
  2065   by blast
  2066 
  2067 lemma Int_Diff: "(A \<inter> B) - C = A \<inter> (B - C)"
  2068   by blast
  2069 
  2070 lemma Diff_Int_distrib: "C \<inter> (A - B) = (C \<inter> A) - (C \<inter> B)"
  2071   by blast
  2072 
  2073 lemma Diff_Int_distrib2: "(A - B) \<inter> C = (A \<inter> C) - (B \<inter> C)"
  2074   by blast
  2075 
  2076 lemma Diff_Compl [simp]: "A - (- B) = A \<inter> B"
  2077   by auto
  2078 
  2079 lemma Compl_Diff_eq [simp]: "- (A - B) = -A \<union> B"
  2080   by blast
  2081 
  2082 
  2083 text {* \medskip Quantification over type @{typ bool}. *}
  2084 
  2085 lemma bool_induct: "P True \<Longrightarrow> P False \<Longrightarrow> P x"
  2086   by (cases x) auto
  2087 
  2088 lemma all_bool_eq: "(\<forall>b. P b) \<longleftrightarrow> P True \<and> P False"
  2089   by (auto intro: bool_induct)
  2090 
  2091 lemma bool_contrapos: "P x \<Longrightarrow> \<not> P False \<Longrightarrow> P True"
  2092   by (cases x) auto
  2093 
  2094 lemma ex_bool_eq: "(\<exists>b. P b) \<longleftrightarrow> P True \<or> P False"
  2095   by (auto intro: bool_contrapos)
  2096 
  2097 lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
  2098   by (auto simp add: split_if_mem2)
  2099 
  2100 lemma UN_bool_eq: "(\<Union>b::bool. A b) = (A True \<union> A False)"
  2101   by (auto intro: bool_contrapos)
  2102 
  2103 lemma INT_bool_eq: "(\<Inter>b::bool. A b) = (A True \<inter> A False)"
  2104   by (auto intro: bool_induct)
  2105 
  2106 text {* \medskip @{text Pow} *}
  2107 
  2108 lemma Pow_empty [simp]: "Pow {} = {{}}"
  2109   by (auto simp add: Pow_def)
  2110 
  2111 lemma Pow_insert: "Pow (insert a A) = Pow A \<union> (insert a ` Pow A)"
  2112   by (blast intro: image_eqI [where ?x = "u - {a}", standard])
  2113 
  2114 lemma Pow_Compl: "Pow (- A) = {-B | B. A \<in> Pow B}"
  2115   by (blast intro: exI [where ?x = "- u", standard])
  2116 
  2117 lemma Pow_UNIV [simp]: "Pow UNIV = UNIV"
  2118   by blast
  2119 
  2120 lemma Un_Pow_subset: "Pow A \<union> Pow B \<subseteq> Pow (A \<union> B)"
  2121   by blast
  2122 
  2123 lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
  2124   by blast
  2125 
  2126 lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
  2127   by blast
  2128 
  2129 lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
  2130   by blast
  2131 
  2132 lemma Pow_Int_eq [simp]: "Pow (A \<inter> B) = Pow A \<inter> Pow B"
  2133   by blast
  2134 
  2135 lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
  2136   by blast
  2137 
  2138 
  2139 text {* \medskip Miscellany. *}
  2140 
  2141 lemma set_eq_subset: "(A = B) = (A \<subseteq> B & B \<subseteq> A)"
  2142   by blast
  2143 
  2144 lemma subset_iff: "(A \<subseteq> B) = (\<forall>t. t \<in> A --> t \<in> B)"
  2145   by blast
  2146 
  2147 lemma subset_iff_psubset_eq: "(A \<subseteq> B) = ((A \<subset> B) | (A = B))"
  2148   by (unfold less_le) blast
  2149 
  2150 lemma all_not_in_conv [simp]: "(\<forall>x. x \<notin> A) = (A = {})"
  2151   by blast
  2152 
  2153 lemma ex_in_conv: "(\<exists>x. x \<in> A) = (A \<noteq> {})"
  2154   by blast
  2155 
  2156 lemma distinct_lemma: "f x \<noteq> f y ==> x \<noteq> y"
  2157   by iprover
  2158 
  2159 
  2160 text {* \medskip Miniscoping: pushing in quantifiers and big Unions
  2161            and Intersections. *}
  2162 
  2163 lemma UN_simps [simp]:
  2164   "!!a B C. (UN x:C. insert a (B x)) = (if C={} then {} else insert a (UN x:C. B x))"
  2165   "!!A B C. (UN x:C. A x Un B)   = ((if C={} then {} else (UN x:C. A x) Un B))"
  2166   "!!A B C. (UN x:C. A Un B x)   = ((if C={} then {} else A Un (UN x:C. B x)))"
  2167   "!!A B C. (UN x:C. A x Int B)  = ((UN x:C. A x) Int B)"
  2168   "!!A B C. (UN x:C. A Int B x)  = (A Int (UN x:C. B x))"
  2169   "!!A B C. (UN x:C. A x - B)    = ((UN x:C. A x) - B)"
  2170   "!!A B C. (UN x:C. A - B x)    = (A - (INT x:C. B x))"
  2171   "!!A B. (UN x: Union A. B x) = (UN y:A. UN x:y. B x)"
  2172   "!!A B C. (UN z: UNION A B. C z) = (UN  x:A. UN z: B(x). C z)"
  2173   "!!A B f. (UN x:f`A. B x)     = (UN a:A. B (f a))"
  2174   by auto
  2175 
  2176 lemma INT_simps [simp]:
  2177   "!!A B C. (INT x:C. A x Int B) = (if C={} then UNIV else (INT x:C. A x) Int B)"
  2178   "!!A B C. (INT x:C. A Int B x) = (if C={} then UNIV else A Int (INT x:C. B x))"
  2179   "!!A B C. (INT x:C. A x - B)   = (if C={} then UNIV else (INT x:C. A x) - B)"
  2180   "!!A B C. (INT x:C. A - B x)   = (if C={} then UNIV else A - (UN x:C. B x))"
  2181   "!!a B C. (INT x:C. insert a (B x)) = insert a (INT x:C. B x)"
  2182   "!!A B C. (INT x:C. A x Un B)  = ((INT x:C. A x) Un B)"
  2183   "!!A B C. (INT x:C. A Un B x)  = (A Un (INT x:C. B x))"
  2184   "!!A B. (INT x: Union A. B x) = (INT y:A. INT x:y. B x)"
  2185   "!!A B C. (INT z: UNION A B. C z) = (INT x:A. INT z: B(x). C z)"
  2186   "!!A B f. (INT x:f`A. B x)    = (INT a:A. B (f a))"
  2187   by auto
  2188 
  2189 lemma ball_simps [simp,noatp]:
  2190   "!!A P Q. (ALL x:A. P x | Q) = ((ALL x:A. P x) | Q)"
  2191   "!!A P Q. (ALL x:A. P | Q x) = (P | (ALL x:A. Q x))"
  2192   "!!A P Q. (ALL x:A. P --> Q x) = (P --> (ALL x:A. Q x))"
  2193   "!!A P Q. (ALL x:A. P x --> Q) = ((EX x:A. P x) --> Q)"
  2194   "!!P. (ALL x:{}. P x) = True"
  2195   "!!P. (ALL x:UNIV. P x) = (ALL x. P x)"
  2196   "!!a B P. (ALL x:insert a B. P x) = (P a & (ALL x:B. P x))"
  2197   "!!A P. (ALL x:Union A. P x) = (ALL y:A. ALL x:y. P x)"
  2198   "!!A B P. (ALL x: UNION A B. P x) = (ALL a:A. ALL x: B a. P x)"
  2199   "!!P Q. (ALL x:Collect Q. P x) = (ALL x. Q x --> P x)"
  2200   "!!A P f. (ALL x:f`A. P x) = (ALL x:A. P (f x))"
  2201   "!!A P. (~(ALL x:A. P x)) = (EX x:A. ~P x)"
  2202   by auto
  2203 
  2204 lemma bex_simps [simp,noatp]:
  2205   "!!A P Q. (EX x:A. P x & Q) = ((EX x:A. P x) & Q)"
  2206   "!!A P Q. (EX x:A. P & Q x) = (P & (EX x:A. Q x))"
  2207   "!!P. (EX x:{}. P x) = False"
  2208   "!!P. (EX x:UNIV. P x) = (EX x. P x)"
  2209   "!!a B P. (EX x:insert a B. P x) = (P(a) | (EX x:B. P x))"
  2210   "!!A P. (EX x:Union A. P x) = (EX y:A. EX x:y. P x)"
  2211   "!!A B P. (EX x: UNION A B. P x) = (EX a:A. EX x:B a. P x)"
  2212   "!!P Q. (EX x:Collect Q. P x) = (EX x. Q x & P x)"
  2213   "!!A P f. (EX x:f`A. P x) = (EX x:A. P (f x))"
  2214   "!!A P. (~(EX x:A. P x)) = (ALL x:A. ~P x)"
  2215   by auto
  2216 
  2217 lemma ball_conj_distrib:
  2218   "(ALL x:A. P x & Q x) = ((ALL x:A. P x) & (ALL x:A. Q x))"
  2219   by blast
  2220 
  2221 lemma bex_disj_distrib:
  2222   "(EX x:A. P x | Q x) = ((EX x:A. P x) | (EX x:A. Q x))"
  2223   by blast
  2224 
  2225 
  2226 text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
  2227 
  2228 lemma UN_extend_simps:
  2229   "!!a B C. insert a (UN x:C. B x) = (if C={} then {a} else (UN x:C. insert a (B x)))"
  2230   "!!A B C. (UN x:C. A x) Un B    = (if C={} then B else (UN x:C. A x Un B))"
  2231   "!!A B C. A Un (UN x:C. B x)   = (if C={} then A else (UN x:C. A Un B x))"
  2232   "!!A B C. ((UN x:C. A x) Int B) = (UN x:C. A x Int B)"
  2233   "!!A B C. (A Int (UN x:C. B x)) = (UN x:C. A Int B x)"
  2234   "!!A B C. ((UN x:C. A x) - B) = (UN x:C. A x - B)"
  2235   "!!A B C. (A - (INT x:C. B x)) = (UN x:C. A - B x)"
  2236   "!!A B. (UN y:A. UN x:y. B x) = (UN x: Union A. B x)"
  2237   "!!A B C. (UN  x:A. UN z: B(x). C z) = (UN z: UNION A B. C z)"
  2238   "!!A B f. (UN a:A. B (f a)) = (UN x:f`A. B x)"
  2239   by auto
  2240 
  2241 lemma INT_extend_simps:
  2242   "!!A B C. (INT x:C. A x) Int B = (if C={} then B else (INT x:C. A x Int B))"
  2243   "!!A B C. A Int (INT x:C. B x) = (if C={} then A else (INT x:C. A Int B x))"
  2244   "!!A B C. (INT x:C. A x) - B   = (if C={} then UNIV-B else (INT x:C. A x - B))"
  2245   "!!A B C. A - (UN x:C. B x)   = (if C={} then A else (INT x:C. A - B x))"
  2246   "!!a B C. insert a (INT x:C. B x) = (INT x:C. insert a (B x))"
  2247   "!!A B C. ((INT x:C. A x) Un B)  = (INT x:C. A x Un B)"
  2248   "!!A B C. A Un (INT x:C. B x)  = (INT x:C. A Un B x)"
  2249   "!!A B. (INT y:A. INT x:y. B x) = (INT x: Union A. B x)"
  2250   "!!A B C. (INT x:A. INT z: B(x). C z) = (INT z: UNION A B. C z)"
  2251   "!!A B f. (INT a:A. B (f a))    = (INT x:f`A. B x)"
  2252   by auto
  2253 
  2254 
  2255 subsubsection {* Monotonicity of various operations *}
  2256 
  2257 lemma image_mono: "A \<subseteq> B ==> f`A \<subseteq> f`B"
  2258   by blast
  2259 
  2260 lemma Pow_mono: "A \<subseteq> B ==> Pow A \<subseteq> Pow B"
  2261   by blast
  2262 
  2263 lemma Union_mono: "A \<subseteq> B ==> \<Union>A \<subseteq> \<Union>B"
  2264   by blast
  2265 
  2266 lemma Inter_anti_mono: "B \<subseteq> A ==> \<Inter>A \<subseteq> \<Inter>B"
  2267   by blast
  2268 
  2269 lemma UN_mono:
  2270   "A \<subseteq> B ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
  2271     (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
  2272   by (blast dest: subsetD)
  2273 
  2274 lemma INT_anti_mono:
  2275   "B \<subseteq> A ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
  2276     (\<Inter>x\<in>A. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
  2277   -- {* The last inclusion is POSITIVE! *}
  2278   by (blast dest: subsetD)
  2279 
  2280 lemma insert_mono: "C \<subseteq> D ==> insert a C \<subseteq> insert a D"
  2281   by blast
  2282 
  2283 lemma Un_mono: "A \<subseteq> C ==> B \<subseteq> D ==> A \<union> B \<subseteq> C \<union> D"
  2284   by blast
  2285 
  2286 lemma Int_mono: "A \<subseteq> C ==> B \<subseteq> D ==> A \<inter> B \<subseteq> C \<inter> D"
  2287   by blast
  2288 
  2289 lemma Diff_mono: "A \<subseteq> C ==> D \<subseteq> B ==> A - B \<subseteq> C - D"
  2290   by blast
  2291 
  2292 lemma Compl_anti_mono: "A \<subseteq> B ==> -B \<subseteq> -A"
  2293   by blast
  2294 
  2295 text {* \medskip Monotonicity of implications. *}
  2296 
  2297 lemma in_mono: "A \<subseteq> B ==> x \<in> A --> x \<in> B"
  2298   apply (rule impI)
  2299   apply (erule subsetD, assumption)
  2300   done
  2301 
  2302 lemma conj_mono: "P1 --> Q1 ==> P2 --> Q2 ==> (P1 & P2) --> (Q1 & Q2)"
  2303   by iprover
  2304 
  2305 lemma disj_mono: "P1 --> Q1 ==> P2 --> Q2 ==> (P1 | P2) --> (Q1 | Q2)"
  2306   by iprover
  2307 
  2308 lemma imp_mono: "Q1 --> P1 ==> P2 --> Q2 ==> (P1 --> P2) --> (Q1 --> Q2)"
  2309   by iprover
  2310 
  2311 lemma imp_refl: "P --> P" ..
  2312 
  2313 lemma ex_mono: "(!!x. P x --> Q x) ==> (EX x. P x) --> (EX x. Q x)"
  2314   by iprover
  2315 
  2316 lemma all_mono: "(!!x. P x --> Q x) ==> (ALL x. P x) --> (ALL x. Q x)"
  2317   by iprover
  2318 
  2319 lemma Collect_mono: "(!!x. P x --> Q x) ==> Collect P \<subseteq> Collect Q"
  2320   by blast
  2321 
  2322 lemma Int_Collect_mono:
  2323     "A \<subseteq> B ==> (!!x. x \<in> A ==> P x --> Q x) ==> A \<inter> Collect P \<subseteq> B \<inter> Collect Q"
  2324   by blast
  2325 
  2326 lemmas basic_monos =
  2327   subset_refl imp_refl disj_mono conj_mono
  2328   ex_mono Collect_mono in_mono
  2329 
  2330 lemma eq_to_mono: "a = b ==> c = d ==> b --> d ==> a --> c"
  2331   by iprover
  2332 
  2333 lemma eq_to_mono2: "a = b ==> c = d ==> ~ b --> ~ d ==> ~ a --> ~ c"
  2334   by iprover
  2335 
  2336 
  2337 subsection {* Inverse image of a function *}
  2338 
  2339 constdefs
  2340   vimage :: "('a => 'b) => 'b set => 'a set"    (infixr "-`" 90)
  2341   [code del]: "f -` B == {x. f x : B}"
  2342 
  2343 
  2344 subsubsection {* Basic rules *}
  2345 
  2346 lemma vimage_eq [simp]: "(a : f -` B) = (f a : B)"
  2347   by (unfold vimage_def) blast
  2348 
  2349 lemma vimage_singleton_eq: "(a : f -` {b}) = (f a = b)"
  2350   by simp
  2351 
  2352 lemma vimageI [intro]: "f a = b ==> b:B ==> a : f -` B"
  2353   by (unfold vimage_def) blast
  2354 
  2355 lemma vimageI2: "f a : A ==> a : f -` A"
  2356   by (unfold vimage_def) fast
  2357 
  2358 lemma vimageE [elim!]: "a: f -` B ==> (!!x. f a = x ==> x:B ==> P) ==> P"
  2359   by (unfold vimage_def) blast
  2360 
  2361 lemma vimageD: "a : f -` A ==> f a : A"
  2362   by (unfold vimage_def) fast
  2363 
  2364 
  2365 subsubsection {* Equations *}
  2366 
  2367 lemma vimage_empty [simp]: "f -` {} = {}"
  2368   by blast
  2369 
  2370 lemma vimage_Compl: "f -` (-A) = -(f -` A)"
  2371   by blast
  2372 
  2373 lemma vimage_Un [simp]: "f -` (A Un B) = (f -` A) Un (f -` B)"
  2374   by blast
  2375 
  2376 lemma vimage_Int [simp]: "f -` (A Int B) = (f -` A) Int (f -` B)"
  2377   by fast
  2378 
  2379 lemma vimage_Union: "f -` (Union A) = (UN X:A. f -` X)"
  2380   by blast
  2381 
  2382 lemma vimage_UN: "f-`(UN x:A. B x) = (UN x:A. f -` B x)"
  2383   by blast
  2384 
  2385 lemma vimage_INT: "f-`(INT x:A. B x) = (INT x:A. f -` B x)"
  2386   by blast
  2387 
  2388 lemma vimage_Collect_eq [simp]: "f -` Collect P = {y. P (f y)}"
  2389   by blast
  2390 
  2391 lemma vimage_Collect: "(!!x. P (f x) = Q x) ==> f -` (Collect P) = Collect Q"
  2392   by blast
  2393 
  2394 lemma vimage_insert: "f-`(insert a B) = (f-`{a}) Un (f-`B)"
  2395   -- {* NOT suitable for rewriting because of the recurrence of @{term "{a}"}. *}
  2396   by blast
  2397 
  2398 lemma vimage_Diff: "f -` (A - B) = (f -` A) - (f -` B)"
  2399   by blast
  2400 
  2401 lemma vimage_UNIV [simp]: "f -` UNIV = UNIV"
  2402   by blast
  2403 
  2404 lemma vimage_eq_UN: "f-`B = (UN y: B. f-`{y})"
  2405   -- {* NOT suitable for rewriting *}
  2406   by blast
  2407 
  2408 lemma vimage_mono: "A \<subseteq> B ==> f -` A \<subseteq> f -` B"
  2409   -- {* monotonicity *}
  2410   by blast
  2411 
  2412 lemma vimage_image_eq [noatp]: "f -` (f ` A) = {y. EX x:A. f x = f y}"
  2413 by (blast intro: sym)
  2414 
  2415 lemma image_vimage_subset: "f ` (f -` A) <= A"
  2416 by blast
  2417 
  2418 lemma image_vimage_eq [simp]: "f ` (f -` A) = A Int range f"
  2419 by blast
  2420 
  2421 lemma image_Int_subset: "f`(A Int B) <= f`A Int f`B"
  2422 by blast
  2423 
  2424 lemma image_diff_subset: "f`A - f`B <= f`(A - B)"
  2425 by blast
  2426 
  2427 lemma image_UN: "(f ` (UNION A B)) = (UN x:A.(f ` (B x)))"
  2428 by blast
  2429 
  2430 
  2431 subsection {* Getting the Contents of a Singleton Set *}
  2432 
  2433 definition contents :: "'a set \<Rightarrow> 'a" where
  2434   [code del]: "contents X = (THE x. X = {x})"
  2435 
  2436 lemma contents_eq [simp]: "contents {x} = x"
  2437   by (simp add: contents_def)
  2438 
  2439 
  2440 subsection {* Transitivity rules for calculational reasoning *}
  2441 
  2442 lemma set_rev_mp: "x:A ==> A \<subseteq> B ==> x:B"
  2443   by (rule subsetD)
  2444 
  2445 lemma set_mp: "A \<subseteq> B ==> x:A ==> x:B"
  2446   by (rule subsetD)
  2447 
  2448 lemmas basic_trans_rules [trans] =
  2449   order_trans_rules set_rev_mp set_mp
  2450 
  2451 
  2452 subsection {* Least value operator *}
  2453 
  2454 lemma Least_mono:
  2455   "mono (f::'a::order => 'b::order) ==> EX x:S. ALL y:S. x <= y
  2456     ==> (LEAST y. y : f ` S) = f (LEAST x. x : S)"
  2457     -- {* Courtesy of Stephan Merz *}
  2458   apply clarify
  2459   apply (erule_tac P = "%x. x : S" in LeastI2_order, fast)
  2460   apply (rule LeastI2_order)
  2461   apply (auto elim: monoD intro!: order_antisym)
  2462   done
  2463 
  2464 
  2465 subsection {* Rudimentary code generation *}
  2466 
  2467 lemma empty_code [code]: "{} x \<longleftrightarrow> False"
  2468   unfolding empty_def Collect_def ..
  2469 
  2470 lemma UNIV_code [code]: "UNIV x \<longleftrightarrow> True"
  2471   unfolding UNIV_def Collect_def ..
  2472 
  2473 lemma insert_code [code]: "insert y A x \<longleftrightarrow> y = x \<or> A x"
  2474   unfolding insert_def Collect_def mem_def Un_def by auto
  2475 
  2476 lemma inter_code [code]: "(A \<inter> B) x \<longleftrightarrow> A x \<and> B x"
  2477   unfolding Int_def Collect_def mem_def ..
  2478 
  2479 lemma union_code [code]: "(A \<union> B) x \<longleftrightarrow> A x \<or> B x"
  2480   unfolding Un_def Collect_def mem_def ..
  2481 
  2482 lemma vimage_code [code]: "(f -` A) x = A (f x)"
  2483   unfolding vimage_def Collect_def mem_def ..
  2484 
  2485 
  2486 subsection {* Misc theorem and ML bindings *}
  2487 
  2488 lemmas equalityI = subset_antisym
  2489 lemmas mem_simps =
  2490   insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
  2491   mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
  2492   -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
  2493 
  2494 ML {*
  2495 val Ball_def = @{thm Ball_def}
  2496 val Bex_def = @{thm Bex_def}
  2497 val CollectD = @{thm CollectD}
  2498 val CollectE = @{thm CollectE}
  2499 val CollectI = @{thm CollectI}
  2500 val Collect_conj_eq = @{thm Collect_conj_eq}
  2501 val Collect_mem_eq = @{thm Collect_mem_eq}
  2502 val IntD1 = @{thm IntD1}
  2503 val IntD2 = @{thm IntD2}
  2504 val IntE = @{thm IntE}
  2505 val IntI = @{thm IntI}
  2506 val Int_Collect = @{thm Int_Collect}
  2507 val UNIV_I = @{thm UNIV_I}
  2508 val UNIV_witness = @{thm UNIV_witness}
  2509 val UnE = @{thm UnE}
  2510 val UnI1 = @{thm UnI1}
  2511 val UnI2 = @{thm UnI2}
  2512 val ballE = @{thm ballE}
  2513 val ballI = @{thm ballI}
  2514 val bexCI = @{thm bexCI}
  2515 val bexE = @{thm bexE}
  2516 val bexI = @{thm bexI}
  2517 val bex_triv = @{thm bex_triv}
  2518 val bspec = @{thm bspec}
  2519 val contra_subsetD = @{thm contra_subsetD}
  2520 val distinct_lemma = @{thm distinct_lemma}
  2521 val eq_to_mono = @{thm eq_to_mono}
  2522 val eq_to_mono2 = @{thm eq_to_mono2}
  2523 val equalityCE = @{thm equalityCE}
  2524 val equalityD1 = @{thm equalityD1}
  2525 val equalityD2 = @{thm equalityD2}
  2526 val equalityE = @{thm equalityE}
  2527 val equalityI = @{thm equalityI}
  2528 val imageE = @{thm imageE}
  2529 val imageI = @{thm imageI}
  2530 val image_Un = @{thm image_Un}
  2531 val image_insert = @{thm image_insert}
  2532 val insert_commute = @{thm insert_commute}
  2533 val insert_iff = @{thm insert_iff}
  2534 val mem_Collect_eq = @{thm mem_Collect_eq}
  2535 val rangeE = @{thm rangeE}
  2536 val rangeI = @{thm rangeI}
  2537 val range_eqI = @{thm range_eqI}
  2538 val subsetCE = @{thm subsetCE}
  2539 val subsetD = @{thm subsetD}
  2540 val subsetI = @{thm subsetI}
  2541 val subset_refl = @{thm subset_refl}
  2542 val subset_trans = @{thm subset_trans}
  2543 val vimageD = @{thm vimageD}
  2544 val vimageE = @{thm vimageE}
  2545 val vimageI = @{thm vimageI}
  2546 val vimageI2 = @{thm vimageI2}
  2547 val vimage_Collect = @{thm vimage_Collect}
  2548 val vimage_Int = @{thm vimage_Int}
  2549 val vimage_Un = @{thm vimage_Un}
  2550 *}
  2551 
  2552 end