src/HOL/Set.thy
author wenzelm
Fri Dec 17 17:43:54 2010 +0100 (2010-12-17)
changeset 41229 d797baa3d57c
parent 41107 8795cd75965e
child 42163 392fd6c4669c
permissions -rw-r--r--
replaced command 'nonterminals' by slightly modernized version 'nonterminal';
     1 (*  Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel *)
     2 
     3 header {* Set theory for higher-order logic *}
     4 
     5 theory Set
     6 imports Lattices
     7 begin
     8 
     9 subsection {* Sets as predicates *}
    10 
    11 types 'a set = "'a \<Rightarrow> bool"
    12 
    13 definition Collect :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set" where -- "comprehension"
    14   "Collect P = P"
    15 
    16 definition member :: "'a \<Rightarrow> 'a set \<Rightarrow> bool" where -- "membership"
    17   mem_def: "member x A = A x"
    18 
    19 notation
    20   member  ("op :") and
    21   member  ("(_/ : _)" [50, 51] 50)
    22 
    23 abbreviation not_member where
    24   "not_member x A \<equiv> ~ (x : A)" -- "non-membership"
    25 
    26 notation
    27   not_member  ("op ~:") and
    28   not_member  ("(_/ ~: _)" [50, 51] 50)
    29 
    30 notation (xsymbols)
    31   member      ("op \<in>") and
    32   member      ("(_/ \<in> _)" [50, 51] 50) and
    33   not_member  ("op \<notin>") and
    34   not_member  ("(_/ \<notin> _)" [50, 51] 50)
    35 
    36 notation (HTML output)
    37   member      ("op \<in>") and
    38   member      ("(_/ \<in> _)" [50, 51] 50) and
    39   not_member  ("op \<notin>") and
    40   not_member  ("(_/ \<notin> _)" [50, 51] 50)
    41 
    42 
    43 
    44 text {* Set comprehensions *}
    45 
    46 syntax
    47   "_Coll" :: "pttrn => bool => 'a set"    ("(1{_./ _})")
    48 translations
    49   "{x. P}" == "CONST Collect (%x. P)"
    50 
    51 syntax
    52   "_Collect" :: "idt => 'a set => bool => 'a set"    ("(1{_ :/ _./ _})")
    53 syntax (xsymbols)
    54   "_Collect" :: "idt => 'a set => bool => 'a set"    ("(1{_ \<in>/ _./ _})")
    55 translations
    56   "{x:A. P}" => "{x. x:A & P}"
    57 
    58 lemma mem_Collect_eq [iff]: "a \<in> {x. P x} = P a"
    59   by (simp add: Collect_def mem_def)
    60 
    61 lemma Collect_mem_eq [simp]: "{x. x \<in> A} = A"
    62   by (simp add: Collect_def mem_def)
    63 
    64 lemma CollectI: "P a \<Longrightarrow> a \<in> {x. P x}"
    65   by simp
    66 
    67 lemma CollectD: "a \<in> {x. P x} \<Longrightarrow> P a"
    68   by simp
    69 
    70 lemma Collect_cong: "(\<And>x. P x = Q x) ==> {x. P x} = {x. Q x}"
    71   by simp
    72 
    73 text {*
    74 Simproc for pulling @{text "x=t"} in @{text "{x. \<dots> & x=t & \<dots>}"}
    75 to the front (and similarly for @{text "t=x"}):
    76 *}
    77 
    78 setup {*
    79 let
    80   val Coll_perm_tac = rtac @{thm Collect_cong} 1 THEN rtac @{thm iffI} 1 THEN
    81     ALLGOALS(EVERY'[REPEAT_DETERM o (etac @{thm conjE}),
    82                     DEPTH_SOLVE_1 o (ares_tac [@{thm conjI}])])
    83   val defColl_regroup = Simplifier.simproc_global @{theory}
    84     "defined Collect" ["{x. P x & Q x}"]
    85     (Quantifier1.rearrange_Coll Coll_perm_tac)
    86 in
    87   Simplifier.map_simpset (fn ss => ss addsimprocs [defColl_regroup])
    88 end
    89 *}
    90 
    91 lemmas CollectE = CollectD [elim_format]
    92 
    93 lemma set_eqI:
    94   assumes "\<And>x. x \<in> A \<longleftrightarrow> x \<in> B"
    95   shows "A = B"
    96 proof -
    97   from assms have "{x. x \<in> A} = {x. x \<in> B}" by simp
    98   then show ?thesis by simp
    99 qed
   100 
   101 lemma set_eq_iff [no_atp]:
   102   "A = B \<longleftrightarrow> (\<forall>x. x \<in> A \<longleftrightarrow> x \<in> B)"
   103   by (auto intro:set_eqI)
   104 
   105 text {* Set enumerations *}
   106 
   107 abbreviation empty :: "'a set" ("{}") where
   108   "{} \<equiv> bot"
   109 
   110 definition insert :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set" where
   111   insert_compr: "insert a B = {x. x = a \<or> x \<in> B}"
   112 
   113 syntax
   114   "_Finset" :: "args => 'a set"    ("{(_)}")
   115 translations
   116   "{x, xs}" == "CONST insert x {xs}"
   117   "{x}" == "CONST insert x {}"
   118 
   119 
   120 subsection {* Subsets and bounded quantifiers *}
   121 
   122 abbreviation
   123   subset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
   124   "subset \<equiv> less"
   125 
   126 abbreviation
   127   subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
   128   "subset_eq \<equiv> less_eq"
   129 
   130 notation (output)
   131   subset  ("op <") and
   132   subset  ("(_/ < _)" [50, 51] 50) and
   133   subset_eq  ("op <=") and
   134   subset_eq  ("(_/ <= _)" [50, 51] 50)
   135 
   136 notation (xsymbols)
   137   subset  ("op \<subset>") and
   138   subset  ("(_/ \<subset> _)" [50, 51] 50) and
   139   subset_eq  ("op \<subseteq>") and
   140   subset_eq  ("(_/ \<subseteq> _)" [50, 51] 50)
   141 
   142 notation (HTML output)
   143   subset  ("op \<subset>") and
   144   subset  ("(_/ \<subset> _)" [50, 51] 50) and
   145   subset_eq  ("op \<subseteq>") and
   146   subset_eq  ("(_/ \<subseteq> _)" [50, 51] 50)
   147 
   148 abbreviation (input)
   149   supset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
   150   "supset \<equiv> greater"
   151 
   152 abbreviation (input)
   153   supset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
   154   "supset_eq \<equiv> greater_eq"
   155 
   156 notation (xsymbols)
   157   supset  ("op \<supset>") and
   158   supset  ("(_/ \<supset> _)" [50, 51] 50) and
   159   supset_eq  ("op \<supseteq>") and
   160   supset_eq  ("(_/ \<supseteq> _)" [50, 51] 50)
   161 
   162 definition Ball :: "'a set \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" where
   163   "Ball A P \<longleftrightarrow> (\<forall>x. x \<in> A \<longrightarrow> P x)"   -- "bounded universal quantifiers"
   164 
   165 definition Bex :: "'a set \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" where
   166   "Bex A P \<longleftrightarrow> (\<exists>x. x \<in> A \<and> P x)"   -- "bounded existential quantifiers"
   167 
   168 syntax
   169   "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3ALL _:_./ _)" [0, 0, 10] 10)
   170   "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3EX _:_./ _)" [0, 0, 10] 10)
   171   "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3EX! _:_./ _)" [0, 0, 10] 10)
   172   "_Bleast"     :: "id => 'a set => bool => 'a"           ("(3LEAST _:_./ _)" [0, 0, 10] 10)
   173 
   174 syntax (HOL)
   175   "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3! _:_./ _)" [0, 0, 10] 10)
   176   "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3? _:_./ _)" [0, 0, 10] 10)
   177   "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3?! _:_./ _)" [0, 0, 10] 10)
   178 
   179 syntax (xsymbols)
   180   "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10)
   181   "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10)
   182   "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3\<exists>!_\<in>_./ _)" [0, 0, 10] 10)
   183   "_Bleast"     :: "id => 'a set => bool => 'a"           ("(3LEAST_\<in>_./ _)" [0, 0, 10] 10)
   184 
   185 syntax (HTML output)
   186   "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10)
   187   "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10)
   188   "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3\<exists>!_\<in>_./ _)" [0, 0, 10] 10)
   189 
   190 translations
   191   "ALL x:A. P" == "CONST Ball A (%x. P)"
   192   "EX x:A. P" == "CONST Bex A (%x. P)"
   193   "EX! x:A. P" => "EX! x. x:A & P"
   194   "LEAST x:A. P" => "LEAST x. x:A & P"
   195 
   196 syntax (output)
   197   "_setlessAll" :: "[idt, 'a, bool] => bool"  ("(3ALL _<_./ _)"  [0, 0, 10] 10)
   198   "_setlessEx"  :: "[idt, 'a, bool] => bool"  ("(3EX _<_./ _)"  [0, 0, 10] 10)
   199   "_setleAll"   :: "[idt, 'a, bool] => bool"  ("(3ALL _<=_./ _)" [0, 0, 10] 10)
   200   "_setleEx"    :: "[idt, 'a, bool] => bool"  ("(3EX _<=_./ _)" [0, 0, 10] 10)
   201   "_setleEx1"   :: "[idt, 'a, bool] => bool"  ("(3EX! _<=_./ _)" [0, 0, 10] 10)
   202 
   203 syntax (xsymbols)
   204   "_setlessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<subset>_./ _)"  [0, 0, 10] 10)
   205   "_setlessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subset>_./ _)"  [0, 0, 10] 10)
   206   "_setleAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<subseteq>_./ _)" [0, 0, 10] 10)
   207   "_setleEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10)
   208   "_setleEx1"   :: "[idt, 'a, bool] => bool"   ("(3\<exists>!_\<subseteq>_./ _)" [0, 0, 10] 10)
   209 
   210 syntax (HOL output)
   211   "_setlessAll" :: "[idt, 'a, bool] => bool"   ("(3! _<_./ _)"  [0, 0, 10] 10)
   212   "_setlessEx"  :: "[idt, 'a, bool] => bool"   ("(3? _<_./ _)"  [0, 0, 10] 10)
   213   "_setleAll"   :: "[idt, 'a, bool] => bool"   ("(3! _<=_./ _)" [0, 0, 10] 10)
   214   "_setleEx"    :: "[idt, 'a, bool] => bool"   ("(3? _<=_./ _)" [0, 0, 10] 10)
   215   "_setleEx1"   :: "[idt, 'a, bool] => bool"   ("(3?! _<=_./ _)" [0, 0, 10] 10)
   216 
   217 syntax (HTML output)
   218   "_setlessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<subset>_./ _)"  [0, 0, 10] 10)
   219   "_setlessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subset>_./ _)"  [0, 0, 10] 10)
   220   "_setleAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<subseteq>_./ _)" [0, 0, 10] 10)
   221   "_setleEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10)
   222   "_setleEx1"   :: "[idt, 'a, bool] => bool"   ("(3\<exists>!_\<subseteq>_./ _)" [0, 0, 10] 10)
   223 
   224 translations
   225  "\<forall>A\<subset>B. P"   =>  "ALL A. A \<subset> B --> P"
   226  "\<exists>A\<subset>B. P"   =>  "EX A. A \<subset> B & P"
   227  "\<forall>A\<subseteq>B. P"   =>  "ALL A. A \<subseteq> B --> P"
   228  "\<exists>A\<subseteq>B. P"   =>  "EX A. A \<subseteq> B & P"
   229  "\<exists>!A\<subseteq>B. P"  =>  "EX! A. A \<subseteq> B & P"
   230 
   231 print_translation {*
   232 let
   233   val Type (set_type, _) = @{typ "'a set"};   (* FIXME 'a => bool (!?!) *)
   234   val All_binder = Syntax.binder_name @{const_syntax All};
   235   val Ex_binder = Syntax.binder_name @{const_syntax Ex};
   236   val impl = @{const_syntax HOL.implies};
   237   val conj = @{const_syntax HOL.conj};
   238   val sbset = @{const_syntax subset};
   239   val sbset_eq = @{const_syntax subset_eq};
   240 
   241   val trans =
   242    [((All_binder, impl, sbset), @{syntax_const "_setlessAll"}),
   243     ((All_binder, impl, sbset_eq), @{syntax_const "_setleAll"}),
   244     ((Ex_binder, conj, sbset), @{syntax_const "_setlessEx"}),
   245     ((Ex_binder, conj, sbset_eq), @{syntax_const "_setleEx"})];
   246 
   247   fun mk v v' c n P =
   248     if v = v' andalso not (Term.exists_subterm (fn Free (x, _) => x = v | _ => false) n)
   249     then Syntax.const c $ Syntax.mark_bound v' $ n $ P else raise Match;
   250 
   251   fun tr' q = (q,
   252         fn [Const (@{syntax_const "_bound"}, _) $ Free (v, Type (T, _)),
   253             Const (c, _) $
   254               (Const (d, _) $ (Const (@{syntax_const "_bound"}, _) $ Free (v', _)) $ n) $ P] =>
   255             if T = set_type then
   256               (case AList.lookup (op =) trans (q, c, d) of
   257                 NONE => raise Match
   258               | SOME l => mk v v' l n P)
   259             else raise Match
   260          | _ => raise Match);
   261 in
   262   [tr' All_binder, tr' Ex_binder]
   263 end
   264 *}
   265 
   266 
   267 text {*
   268   \medskip Translate between @{text "{e | x1...xn. P}"} and @{text
   269   "{u. EX x1..xn. u = e & P}"}; @{text "{y. EX x1..xn. y = e & P}"} is
   270   only translated if @{text "[0..n] subset bvs(e)"}.
   271 *}
   272 
   273 syntax
   274   "_Setcompr" :: "'a => idts => bool => 'a set"    ("(1{_ |/_./ _})")
   275 
   276 parse_translation {*
   277   let
   278     val ex_tr = snd (mk_binder_tr ("EX ", @{const_syntax Ex}));
   279 
   280     fun nvars (Const (@{syntax_const "_idts"}, _) $ _ $ idts) = nvars idts + 1
   281       | nvars _ = 1;
   282 
   283     fun setcompr_tr [e, idts, b] =
   284       let
   285         val eq = Syntax.const @{const_syntax HOL.eq} $ Bound (nvars idts) $ e;
   286         val P = Syntax.const @{const_syntax HOL.conj} $ eq $ b;
   287         val exP = ex_tr [idts, P];
   288       in Syntax.const @{const_syntax Collect} $ Term.absdummy (dummyT, exP) end;
   289 
   290   in [(@{syntax_const "_Setcompr"}, setcompr_tr)] end;
   291 *}
   292 
   293 print_translation {*
   294  [Syntax.preserve_binder_abs2_tr' @{const_syntax Ball} @{syntax_const "_Ball"},
   295   Syntax.preserve_binder_abs2_tr' @{const_syntax Bex} @{syntax_const "_Bex"}]
   296 *} -- {* to avoid eta-contraction of body *}
   297 
   298 print_translation {*
   299 let
   300   val ex_tr' = snd (mk_binder_tr' (@{const_syntax Ex}, "DUMMY"));
   301 
   302   fun setcompr_tr' [Abs (abs as (_, _, P))] =
   303     let
   304       fun check (Const (@{const_syntax Ex}, _) $ Abs (_, _, P), n) = check (P, n + 1)
   305         | check (Const (@{const_syntax HOL.conj}, _) $
   306               (Const (@{const_syntax HOL.eq}, _) $ Bound m $ e) $ P, n) =
   307             n > 0 andalso m = n andalso not (loose_bvar1 (P, n)) andalso
   308             subset (op =) (0 upto (n - 1), add_loose_bnos (e, 0, []))
   309         | check _ = false;
   310 
   311         fun tr' (_ $ abs) =
   312           let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' [abs]
   313           in Syntax.const @{syntax_const "_Setcompr"} $ e $ idts $ Q end;
   314     in
   315       if check (P, 0) then tr' P
   316       else
   317         let
   318           val (x as _ $ Free(xN, _), t) = atomic_abs_tr' abs;
   319           val M = Syntax.const @{syntax_const "_Coll"} $ x $ t;
   320         in
   321           case t of
   322             Const (@{const_syntax HOL.conj}, _) $
   323               (Const (@{const_syntax Set.member}, _) $
   324                 (Const (@{syntax_const "_bound"}, _) $ Free (yN, _)) $ A) $ P =>
   325             if xN = yN then Syntax.const @{syntax_const "_Collect"} $ x $ A $ P else M
   326           | _ => M
   327         end
   328     end;
   329   in [(@{const_syntax Collect}, setcompr_tr')] end;
   330 *}
   331 
   332 setup {*
   333 let
   334   val unfold_bex_tac = unfold_tac @{thms "Bex_def"};
   335   fun prove_bex_tac ss = unfold_bex_tac ss THEN Quantifier1.prove_one_point_ex_tac;
   336   val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac;
   337   val unfold_ball_tac = unfold_tac @{thms "Ball_def"};
   338   fun prove_ball_tac ss = unfold_ball_tac ss THEN Quantifier1.prove_one_point_all_tac;
   339   val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac;
   340   val defBEX_regroup = Simplifier.simproc_global @{theory}
   341     "defined BEX" ["EX x:A. P x & Q x"] rearrange_bex;
   342   val defBALL_regroup = Simplifier.simproc_global @{theory}
   343     "defined BALL" ["ALL x:A. P x --> Q x"] rearrange_ball;
   344 in
   345   Simplifier.map_simpset (fn ss => ss addsimprocs [defBALL_regroup, defBEX_regroup])
   346 end
   347 *}
   348 
   349 lemma ballI [intro!]: "(!!x. x:A ==> P x) ==> ALL x:A. P x"
   350   by (simp add: Ball_def)
   351 
   352 lemmas strip = impI allI ballI
   353 
   354 lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x"
   355   by (simp add: Ball_def)
   356 
   357 text {*
   358   Gives better instantiation for bound:
   359 *}
   360 
   361 declaration {* fn _ =>
   362   Classical.map_cs (fn cs => cs addbefore ("bspec", datac @{thm bspec} 1))
   363 *}
   364 
   365 ML {*
   366 structure Simpdata =
   367 struct
   368 
   369 open Simpdata;
   370 
   371 val mksimps_pairs = [(@{const_name Ball}, @{thms bspec})] @ mksimps_pairs;
   372 
   373 end;
   374 
   375 open Simpdata;
   376 *}
   377 
   378 declaration {* fn _ =>
   379   Simplifier.map_ss (fn ss => ss setmksimps (mksimps mksimps_pairs))
   380 *}
   381 
   382 lemma ballE [elim]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q"
   383   by (unfold Ball_def) blast
   384 
   385 lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x"
   386   -- {* Normally the best argument order: @{prop "P x"} constrains the
   387     choice of @{prop "x:A"}. *}
   388   by (unfold Bex_def) blast
   389 
   390 lemma rev_bexI [intro?]: "x:A ==> P x ==> EX x:A. P x"
   391   -- {* The best argument order when there is only one @{prop "x:A"}. *}
   392   by (unfold Bex_def) blast
   393 
   394 lemma bexCI: "(ALL x:A. ~P x ==> P a) ==> a:A ==> EX x:A. P x"
   395   by (unfold Bex_def) blast
   396 
   397 lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q"
   398   by (unfold Bex_def) blast
   399 
   400 lemma ball_triv [simp]: "(ALL x:A. P) = ((EX x. x:A) --> P)"
   401   -- {* Trival rewrite rule. *}
   402   by (simp add: Ball_def)
   403 
   404 lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)"
   405   -- {* Dual form for existentials. *}
   406   by (simp add: Bex_def)
   407 
   408 lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)"
   409   by blast
   410 
   411 lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)"
   412   by blast
   413 
   414 lemma bex_one_point1 [simp]: "(EX x:A. x = a & P x) = (a:A & P a)"
   415   by blast
   416 
   417 lemma bex_one_point2 [simp]: "(EX x:A. a = x & P x) = (a:A & P a)"
   418   by blast
   419 
   420 lemma ball_one_point1 [simp]: "(ALL x:A. x = a --> P x) = (a:A --> P a)"
   421   by blast
   422 
   423 lemma ball_one_point2 [simp]: "(ALL x:A. a = x --> P x) = (a:A --> P a)"
   424   by blast
   425 
   426 
   427 text {* Congruence rules *}
   428 
   429 lemma ball_cong:
   430   "A = B ==> (!!x. x:B ==> P x = Q x) ==>
   431     (ALL x:A. P x) = (ALL x:B. Q x)"
   432   by (simp add: Ball_def)
   433 
   434 lemma strong_ball_cong [cong]:
   435   "A = B ==> (!!x. x:B =simp=> P x = Q x) ==>
   436     (ALL x:A. P x) = (ALL x:B. Q x)"
   437   by (simp add: simp_implies_def Ball_def)
   438 
   439 lemma bex_cong:
   440   "A = B ==> (!!x. x:B ==> P x = Q x) ==>
   441     (EX x:A. P x) = (EX x:B. Q x)"
   442   by (simp add: Bex_def cong: conj_cong)
   443 
   444 lemma strong_bex_cong [cong]:
   445   "A = B ==> (!!x. x:B =simp=> P x = Q x) ==>
   446     (EX x:A. P x) = (EX x:B. Q x)"
   447   by (simp add: simp_implies_def Bex_def cong: conj_cong)
   448 
   449 
   450 subsection {* Basic operations *}
   451 
   452 subsubsection {* Subsets *}
   453 
   454 lemma subsetI [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> x \<in> B) \<Longrightarrow> A \<subseteq> B"
   455   unfolding mem_def by (rule le_funI, rule le_boolI)
   456 
   457 text {*
   458   \medskip Map the type @{text "'a set => anything"} to just @{typ
   459   'a}; for overloading constants whose first argument has type @{typ
   460   "'a set"}.
   461 *}
   462 
   463 lemma subsetD [elim, intro?]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B"
   464   unfolding mem_def by (erule le_funE, erule le_boolE)
   465   -- {* Rule in Modus Ponens style. *}
   466 
   467 lemma rev_subsetD [no_atp,intro?]: "c \<in> A ==> A \<subseteq> B ==> c \<in> B"
   468   -- {* The same, with reversed premises for use with @{text erule} --
   469       cf @{text rev_mp}. *}
   470   by (rule subsetD)
   471 
   472 text {*
   473   \medskip Converts @{prop "A \<subseteq> B"} to @{prop "x \<in> A ==> x \<in> B"}.
   474 *}
   475 
   476 lemma subsetCE [no_atp,elim]: "A \<subseteq> B ==> (c \<notin> A ==> P) ==> (c \<in> B ==> P) ==> P"
   477   -- {* Classical elimination rule. *}
   478   unfolding mem_def by (blast dest: le_funE le_boolE)
   479 
   480 lemma subset_eq [no_atp]: "A \<le> B = (\<forall>x\<in>A. x \<in> B)" by blast
   481 
   482 lemma contra_subsetD [no_atp]: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A"
   483   by blast
   484 
   485 lemma subset_refl [simp]: "A \<subseteq> A"
   486   by (fact order_refl)
   487 
   488 lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C"
   489   by (fact order_trans)
   490 
   491 lemma set_rev_mp: "x:A ==> A \<subseteq> B ==> x:B"
   492   by (rule subsetD)
   493 
   494 lemma set_mp: "A \<subseteq> B ==> x:A ==> x:B"
   495   by (rule subsetD)
   496 
   497 lemma eq_mem_trans: "a=b ==> b \<in> A ==> a \<in> A"
   498   by simp
   499 
   500 lemmas basic_trans_rules [trans] =
   501   order_trans_rules set_rev_mp set_mp eq_mem_trans
   502 
   503 
   504 subsubsection {* Equality *}
   505 
   506 lemma subset_antisym [intro!]: "A \<subseteq> B ==> B \<subseteq> A ==> A = B"
   507   -- {* Anti-symmetry of the subset relation. *}
   508   by (iprover intro: set_eqI subsetD)
   509 
   510 text {*
   511   \medskip Equality rules from ZF set theory -- are they appropriate
   512   here?
   513 *}
   514 
   515 lemma equalityD1: "A = B ==> A \<subseteq> B"
   516   by simp
   517 
   518 lemma equalityD2: "A = B ==> B \<subseteq> A"
   519   by simp
   520 
   521 text {*
   522   \medskip Be careful when adding this to the claset as @{text
   523   subset_empty} is in the simpset: @{prop "A = {}"} goes to @{prop "{}
   524   \<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}!
   525 *}
   526 
   527 lemma equalityE: "A = B ==> (A \<subseteq> B ==> B \<subseteq> A ==> P) ==> P"
   528   by simp
   529 
   530 lemma equalityCE [elim]:
   531     "A = B ==> (c \<in> A ==> c \<in> B ==> P) ==> (c \<notin> A ==> c \<notin> B ==> P) ==> P"
   532   by blast
   533 
   534 lemma eqset_imp_iff: "A = B ==> (x : A) = (x : B)"
   535   by simp
   536 
   537 lemma eqelem_imp_iff: "x = y ==> (x : A) = (y : A)"
   538   by simp
   539 
   540 
   541 subsubsection {* The empty set *}
   542 
   543 lemma empty_def:
   544   "{} = {x. False}"
   545   by (simp add: bot_fun_def bot_bool_def Collect_def)
   546 
   547 lemma empty_iff [simp]: "(c : {}) = False"
   548   by (simp add: empty_def)
   549 
   550 lemma emptyE [elim!]: "a : {} ==> P"
   551   by simp
   552 
   553 lemma empty_subsetI [iff]: "{} \<subseteq> A"
   554     -- {* One effect is to delete the ASSUMPTION @{prop "{} <= A"} *}
   555   by blast
   556 
   557 lemma equals0I: "(!!y. y \<in> A ==> False) ==> A = {}"
   558   by blast
   559 
   560 lemma equals0D: "A = {} ==> a \<notin> A"
   561     -- {* Use for reasoning about disjointness: @{text "A Int B = {}"} *}
   562   by blast
   563 
   564 lemma ball_empty [simp]: "Ball {} P = True"
   565   by (simp add: Ball_def)
   566 
   567 lemma bex_empty [simp]: "Bex {} P = False"
   568   by (simp add: Bex_def)
   569 
   570 
   571 subsubsection {* The universal set -- UNIV *}
   572 
   573 abbreviation UNIV :: "'a set" where
   574   "UNIV \<equiv> top"
   575 
   576 lemma UNIV_def:
   577   "UNIV = {x. True}"
   578   by (simp add: top_fun_def top_bool_def Collect_def)
   579 
   580 lemma UNIV_I [simp]: "x : UNIV"
   581   by (simp add: UNIV_def)
   582 
   583 declare UNIV_I [intro]  -- {* unsafe makes it less likely to cause problems *}
   584 
   585 lemma UNIV_witness [intro?]: "EX x. x : UNIV"
   586   by simp
   587 
   588 lemma subset_UNIV [simp]: "A \<subseteq> UNIV"
   589   by (rule subsetI) (rule UNIV_I)
   590 
   591 text {*
   592   \medskip Eta-contracting these two rules (to remove @{text P})
   593   causes them to be ignored because of their interaction with
   594   congruence rules.
   595 *}
   596 
   597 lemma ball_UNIV [simp]: "Ball UNIV P = All P"
   598   by (simp add: Ball_def)
   599 
   600 lemma bex_UNIV [simp]: "Bex UNIV P = Ex P"
   601   by (simp add: Bex_def)
   602 
   603 lemma UNIV_eq_I: "(\<And>x. x \<in> A) \<Longrightarrow> UNIV = A"
   604   by auto
   605 
   606 lemma UNIV_not_empty [iff]: "UNIV ~= {}"
   607   by (blast elim: equalityE)
   608 
   609 
   610 subsubsection {* The Powerset operator -- Pow *}
   611 
   612 definition Pow :: "'a set => 'a set set" where
   613   Pow_def: "Pow A = {B. B \<le> A}"
   614 
   615 lemma Pow_iff [iff]: "(A \<in> Pow B) = (A \<subseteq> B)"
   616   by (simp add: Pow_def)
   617 
   618 lemma PowI: "A \<subseteq> B ==> A \<in> Pow B"
   619   by (simp add: Pow_def)
   620 
   621 lemma PowD: "A \<in> Pow B ==> A \<subseteq> B"
   622   by (simp add: Pow_def)
   623 
   624 lemma Pow_bottom: "{} \<in> Pow B"
   625   by simp
   626 
   627 lemma Pow_top: "A \<in> Pow A"
   628   by simp
   629 
   630 lemma Pow_not_empty: "Pow A \<noteq> {}"
   631   using Pow_top by blast
   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 intersection *}
   656 
   657 abbreviation inter :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Int" 70) where
   658   "op Int \<equiv> inf"
   659 
   660 notation (xsymbols)
   661   inter  (infixl "\<inter>" 70)
   662 
   663 notation (HTML output)
   664   inter  (infixl "\<inter>" 70)
   665 
   666 lemma Int_def:
   667   "A \<inter> B = {x. x \<in> A \<and> x \<in> B}"
   668   by (simp add: inf_fun_def inf_bool_def Collect_def mem_def)
   669 
   670 lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)"
   671   by (unfold Int_def) blast
   672 
   673 lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B"
   674   by simp
   675 
   676 lemma IntD1: "c : A Int B ==> c:A"
   677   by simp
   678 
   679 lemma IntD2: "c : A Int B ==> c:B"
   680   by simp
   681 
   682 lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P"
   683   by simp
   684 
   685 lemma mono_Int: "mono f \<Longrightarrow> f (A \<inter> B) \<subseteq> f A \<inter> f B"
   686   by (fact mono_inf)
   687 
   688 
   689 subsubsection {* Binary union *}
   690 
   691 abbreviation union :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Un" 65) where
   692   "union \<equiv> sup"
   693 
   694 notation (xsymbols)
   695   union  (infixl "\<union>" 65)
   696 
   697 notation (HTML output)
   698   union  (infixl "\<union>" 65)
   699 
   700 lemma Un_def:
   701   "A \<union> B = {x. x \<in> A \<or> x \<in> B}"
   702   by (simp add: sup_fun_def sup_bool_def Collect_def mem_def)
   703 
   704 lemma Un_iff [simp]: "(c : A Un B) = (c:A | c:B)"
   705   by (unfold Un_def) blast
   706 
   707 lemma UnI1 [elim?]: "c:A ==> c : A Un B"
   708   by simp
   709 
   710 lemma UnI2 [elim?]: "c:B ==> c : A Un B"
   711   by simp
   712 
   713 text {*
   714   \medskip Classical introduction rule: no commitment to @{prop A} vs
   715   @{prop B}.
   716 *}
   717 
   718 lemma UnCI [intro!]: "(c~:B ==> c:A) ==> c : A Un B"
   719   by auto
   720 
   721 lemma UnE [elim!]: "c : A Un B ==> (c:A ==> P) ==> (c:B ==> P) ==> P"
   722   by (unfold Un_def) blast
   723 
   724 lemma insert_def: "insert a B = {x. x = a} \<union> B"
   725   by (simp add: Collect_def mem_def insert_compr Un_def)
   726 
   727 lemma mono_Un: "mono f \<Longrightarrow> f A \<union> f B \<subseteq> f (A \<union> B)"
   728   by (fact mono_sup)
   729 
   730 
   731 subsubsection {* Set difference *}
   732 
   733 lemma Diff_iff [simp]: "(c : A - B) = (c:A & c~:B)"
   734   by (simp add: mem_def fun_diff_def bool_diff_def)
   735 
   736 lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A - B"
   737   by simp
   738 
   739 lemma DiffD1: "c : A - B ==> c : A"
   740   by simp
   741 
   742 lemma DiffD2: "c : A - B ==> c : B ==> P"
   743   by simp
   744 
   745 lemma DiffE [elim!]: "c : A - B ==> (c:A ==> c~:B ==> P) ==> P"
   746   by simp
   747 
   748 lemma set_diff_eq: "A - B = {x. x : A & ~ x : B}" by blast
   749 
   750 lemma Compl_eq_Diff_UNIV: "-A = (UNIV - A)"
   751 by blast
   752 
   753 
   754 subsubsection {* Augmenting a set -- @{const insert} *}
   755 
   756 lemma insert_iff [simp]: "(a : insert b A) = (a = b | a:A)"
   757   by (unfold insert_def) blast
   758 
   759 lemma insertI1: "a : insert a B"
   760   by simp
   761 
   762 lemma insertI2: "a : B ==> a : insert b B"
   763   by simp
   764 
   765 lemma insertE [elim!]: "a : insert b A ==> (a = b ==> P) ==> (a:A ==> P) ==> P"
   766   by (unfold insert_def) blast
   767 
   768 lemma insertCI [intro!]: "(a~:B ==> a = b) ==> a: insert b B"
   769   -- {* Classical introduction rule. *}
   770   by auto
   771 
   772 lemma subset_insert_iff: "(A \<subseteq> insert x B) = (if x:A then A - {x} \<subseteq> B else A \<subseteq> B)"
   773   by auto
   774 
   775 lemma set_insert:
   776   assumes "x \<in> A"
   777   obtains B where "A = insert x B" and "x \<notin> B"
   778 proof
   779   from assms show "A = insert x (A - {x})" by blast
   780 next
   781   show "x \<notin> A - {x}" by blast
   782 qed
   783 
   784 lemma insert_ident: "x ~: A ==> x ~: B ==> (insert x A = insert x B) = (A = B)"
   785 by auto
   786 
   787 subsubsection {* Singletons, using insert *}
   788 
   789 lemma singletonI [intro!,no_atp]: "a : {a}"
   790     -- {* Redundant? But unlike @{text insertCI}, it proves the subgoal immediately! *}
   791   by (rule insertI1)
   792 
   793 lemma singletonD [dest!,no_atp]: "b : {a} ==> b = a"
   794   by blast
   795 
   796 lemmas singletonE = singletonD [elim_format]
   797 
   798 lemma singleton_iff: "(b : {a}) = (b = a)"
   799   by blast
   800 
   801 lemma singleton_inject [dest!]: "{a} = {b} ==> a = b"
   802   by blast
   803 
   804 lemma singleton_insert_inj_eq [iff,no_atp]:
   805      "({b} = insert a A) = (a = b & A \<subseteq> {b})"
   806   by blast
   807 
   808 lemma singleton_insert_inj_eq' [iff,no_atp]:
   809      "(insert a A = {b}) = (a = b & A \<subseteq> {b})"
   810   by blast
   811 
   812 lemma subset_singletonD: "A \<subseteq> {x} ==> A = {} | A = {x}"
   813   by fast
   814 
   815 lemma singleton_conv [simp]: "{x. x = a} = {a}"
   816   by blast
   817 
   818 lemma singleton_conv2 [simp]: "{x. a = x} = {a}"
   819   by blast
   820 
   821 lemma diff_single_insert: "A - {x} \<subseteq> B ==> x \<in> A ==> A \<subseteq> insert x B"
   822   by blast
   823 
   824 lemma doubleton_eq_iff: "({a,b} = {c,d}) = (a=c & b=d | a=d & b=c)"
   825   by (blast elim: equalityE)
   826 
   827 
   828 subsubsection {* Image of a set under a function *}
   829 
   830 text {*
   831   Frequently @{term b} does not have the syntactic form of @{term "f x"}.
   832 *}
   833 
   834 definition image :: "('a => 'b) => 'a set => 'b set" (infixr "`" 90) where
   835   image_def [no_atp]: "f ` A = {y. EX x:A. y = f(x)}"
   836 
   837 abbreviation
   838   range :: "('a => 'b) => 'b set" where -- "of function"
   839   "range f == f ` UNIV"
   840 
   841 lemma image_eqI [simp, intro]: "b = f x ==> x:A ==> b : f`A"
   842   by (unfold image_def) blast
   843 
   844 lemma imageI: "x : A ==> f x : f ` A"
   845   by (rule image_eqI) (rule refl)
   846 
   847 lemma rev_image_eqI: "x:A ==> b = f x ==> b : f`A"
   848   -- {* This version's more effective when we already have the
   849     required @{term x}. *}
   850   by (unfold image_def) blast
   851 
   852 lemma imageE [elim!]:
   853   "b : (%x. f x)`A ==> (!!x. b = f x ==> x:A ==> P) ==> P"
   854   -- {* The eta-expansion gives variable-name preservation. *}
   855   by (unfold image_def) blast
   856 
   857 lemma image_Un: "f`(A Un B) = f`A Un f`B"
   858   by blast
   859 
   860 lemma image_iff: "(z : f`A) = (EX x:A. z = f x)"
   861   by blast
   862 
   863 lemma image_subset_iff [no_atp]: "(f`A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)"
   864   -- {* This rewrite rule would confuse users if made default. *}
   865   by blast
   866 
   867 lemma subset_image_iff: "(B \<subseteq> f`A) = (EX AA. AA \<subseteq> A & B = f`AA)"
   868   apply safe
   869    prefer 2 apply fast
   870   apply (rule_tac x = "{a. a : A & f a : B}" in exI, fast)
   871   done
   872 
   873 lemma image_subsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f`A \<subseteq> B"
   874   -- {* Replaces the three steps @{text subsetI}, @{text imageE},
   875     @{text hypsubst}, but breaks too many existing proofs. *}
   876   by blast
   877 
   878 text {*
   879   \medskip Range of a function -- just a translation for image!
   880 *}
   881 
   882 lemma range_eqI: "b = f x ==> b \<in> range f"
   883   by simp
   884 
   885 lemma rangeI: "f x \<in> range f"
   886   by simp
   887 
   888 lemma rangeE [elim?]: "b \<in> range (\<lambda>x. f x) ==> (!!x. b = f x ==> P) ==> P"
   889   by blast
   890 
   891 subsubsection {* Some rules with @{text "if"} *}
   892 
   893 text{* Elimination of @{text"{x. \<dots> & x=t & \<dots>}"}. *}
   894 
   895 lemma Collect_conv_if: "{x. x=a & P x} = (if P a then {a} else {})"
   896   by auto
   897 
   898 lemma Collect_conv_if2: "{x. a=x & P x} = (if P a then {a} else {})"
   899   by auto
   900 
   901 text {*
   902   Rewrite rules for boolean case-splitting: faster than @{text
   903   "split_if [split]"}.
   904 *}
   905 
   906 lemma split_if_eq1: "((if Q then x else y) = b) = ((Q --> x = b) & (~ Q --> y = b))"
   907   by (rule split_if)
   908 
   909 lemma split_if_eq2: "(a = (if Q then x else y)) = ((Q --> a = x) & (~ Q --> a = y))"
   910   by (rule split_if)
   911 
   912 text {*
   913   Split ifs on either side of the membership relation.  Not for @{text
   914   "[simp]"} -- can cause goals to blow up!
   915 *}
   916 
   917 lemma split_if_mem1: "((if Q then x else y) : b) = ((Q --> x : b) & (~ Q --> y : b))"
   918   by (rule split_if)
   919 
   920 lemma split_if_mem2: "(a : (if Q then x else y)) = ((Q --> a : x) & (~ Q --> a : y))"
   921   by (rule split_if [where P="%S. a : S"])
   922 
   923 lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2
   924 
   925 (*Would like to add these, but the existing code only searches for the
   926   outer-level constant, which in this case is just Set.member; we instead need
   927   to use term-nets to associate patterns with rules.  Also, if a rule fails to
   928   apply, then the formula should be kept.
   929   [("uminus", Compl_iff RS iffD1), ("minus", [Diff_iff RS iffD1]),
   930    ("Int", [IntD1,IntD2]),
   931    ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]
   932  *)
   933 
   934 
   935 subsection {* Further operations and lemmas *}
   936 
   937 subsubsection {* The ``proper subset'' relation *}
   938 
   939 lemma psubsetI [intro!,no_atp]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B"
   940   by (unfold less_le) blast
   941 
   942 lemma psubsetE [elim!,no_atp]: 
   943     "[|A \<subset> B;  [|A \<subseteq> B; ~ (B\<subseteq>A)|] ==> R|] ==> R"
   944   by (unfold less_le) blast
   945 
   946 lemma psubset_insert_iff:
   947   "(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)"
   948   by (auto simp add: less_le subset_insert_iff)
   949 
   950 lemma psubset_eq: "(A \<subset> B) = (A \<subseteq> B & A \<noteq> B)"
   951   by (simp only: less_le)
   952 
   953 lemma psubset_imp_subset: "A \<subset> B ==> A \<subseteq> B"
   954   by (simp add: psubset_eq)
   955 
   956 lemma psubset_trans: "[| A \<subset> B; B \<subset> C |] ==> A \<subset> C"
   957 apply (unfold less_le)
   958 apply (auto dest: subset_antisym)
   959 done
   960 
   961 lemma psubsetD: "[| A \<subset> B; c \<in> A |] ==> c \<in> B"
   962 apply (unfold less_le)
   963 apply (auto dest: subsetD)
   964 done
   965 
   966 lemma psubset_subset_trans: "A \<subset> B ==> B \<subseteq> C ==> A \<subset> C"
   967   by (auto simp add: psubset_eq)
   968 
   969 lemma subset_psubset_trans: "A \<subseteq> B ==> B \<subset> C ==> A \<subset> C"
   970   by (auto simp add: psubset_eq)
   971 
   972 lemma psubset_imp_ex_mem: "A \<subset> B ==> \<exists>b. b \<in> (B - A)"
   973   by (unfold less_le) blast
   974 
   975 lemma atomize_ball:
   976     "(!!x. x \<in> A ==> P x) == Trueprop (\<forall>x\<in>A. P x)"
   977   by (simp only: Ball_def atomize_all atomize_imp)
   978 
   979 lemmas [symmetric, rulify] = atomize_ball
   980   and [symmetric, defn] = atomize_ball
   981 
   982 lemma image_Pow_mono:
   983   assumes "f ` A \<le> B"
   984   shows "(image f) ` (Pow A) \<le> Pow B"
   985 using assms by blast
   986 
   987 lemma image_Pow_surj:
   988   assumes "f ` A = B"
   989   shows "(image f) ` (Pow A) = Pow B"
   990 using assms unfolding Pow_def proof(auto)
   991   fix Y assume *: "Y \<le> f ` A"
   992   obtain X where X_def: "X = {x \<in> A. f x \<in> Y}" by blast
   993   have "f ` X = Y \<and> X \<le> A" unfolding X_def using * by auto
   994   thus "Y \<in> (image f) ` {X. X \<le> A}" by blast
   995 qed
   996 
   997 subsubsection {* Derived rules involving subsets. *}
   998 
   999 text {* @{text insert}. *}
  1000 
  1001 lemma subset_insertI: "B \<subseteq> insert a B"
  1002   by (rule subsetI) (erule insertI2)
  1003 
  1004 lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B"
  1005   by blast
  1006 
  1007 lemma subset_insert: "x \<notin> A ==> (A \<subseteq> insert x B) = (A \<subseteq> B)"
  1008   by blast
  1009 
  1010 
  1011 text {* \medskip Finite Union -- the least upper bound of two sets. *}
  1012 
  1013 lemma Un_upper1: "A \<subseteq> A \<union> B"
  1014   by (fact sup_ge1)
  1015 
  1016 lemma Un_upper2: "B \<subseteq> A \<union> B"
  1017   by (fact sup_ge2)
  1018 
  1019 lemma Un_least: "A \<subseteq> C ==> B \<subseteq> C ==> A \<union> B \<subseteq> C"
  1020   by (fact sup_least)
  1021 
  1022 
  1023 text {* \medskip Finite Intersection -- the greatest lower bound of two sets. *}
  1024 
  1025 lemma Int_lower1: "A \<inter> B \<subseteq> A"
  1026   by (fact inf_le1)
  1027 
  1028 lemma Int_lower2: "A \<inter> B \<subseteq> B"
  1029   by (fact inf_le2)
  1030 
  1031 lemma Int_greatest: "C \<subseteq> A ==> C \<subseteq> B ==> C \<subseteq> A \<inter> B"
  1032   by (fact inf_greatest)
  1033 
  1034 
  1035 text {* \medskip Set difference. *}
  1036 
  1037 lemma Diff_subset: "A - B \<subseteq> A"
  1038   by blast
  1039 
  1040 lemma Diff_subset_conv: "(A - B \<subseteq> C) = (A \<subseteq> B \<union> C)"
  1041 by blast
  1042 
  1043 
  1044 subsubsection {* Equalities involving union, intersection, inclusion, etc. *}
  1045 
  1046 text {* @{text "{}"}. *}
  1047 
  1048 lemma Collect_const [simp]: "{s. P} = (if P then UNIV else {})"
  1049   -- {* supersedes @{text "Collect_False_empty"} *}
  1050   by auto
  1051 
  1052 lemma subset_empty [simp]: "(A \<subseteq> {}) = (A = {})"
  1053   by blast
  1054 
  1055 lemma not_psubset_empty [iff]: "\<not> (A < {})"
  1056   by (unfold less_le) blast
  1057 
  1058 lemma Collect_empty_eq [simp]: "(Collect P = {}) = (\<forall>x. \<not> P x)"
  1059 by blast
  1060 
  1061 lemma empty_Collect_eq [simp]: "({} = Collect P) = (\<forall>x. \<not> P x)"
  1062 by blast
  1063 
  1064 lemma Collect_neg_eq: "{x. \<not> P x} = - {x. P x}"
  1065   by blast
  1066 
  1067 lemma Collect_disj_eq: "{x. P x | Q x} = {x. P x} \<union> {x. Q x}"
  1068   by blast
  1069 
  1070 lemma Collect_imp_eq: "{x. P x --> Q x} = -{x. P x} \<union> {x. Q x}"
  1071   by blast
  1072 
  1073 lemma Collect_conj_eq: "{x. P x & Q x} = {x. P x} \<inter> {x. Q x}"
  1074   by blast
  1075 
  1076 
  1077 text {* \medskip @{text insert}. *}
  1078 
  1079 lemma insert_is_Un: "insert a A = {a} Un A"
  1080   -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a {}"} *}
  1081   by blast
  1082 
  1083 lemma insert_not_empty [simp]: "insert a A \<noteq> {}"
  1084   by blast
  1085 
  1086 lemmas empty_not_insert = insert_not_empty [symmetric, standard]
  1087 declare empty_not_insert [simp]
  1088 
  1089 lemma insert_absorb: "a \<in> A ==> insert a A = A"
  1090   -- {* @{text "[simp]"} causes recursive calls when there are nested inserts *}
  1091   -- {* with \emph{quadratic} running time *}
  1092   by blast
  1093 
  1094 lemma insert_absorb2 [simp]: "insert x (insert x A) = insert x A"
  1095   by blast
  1096 
  1097 lemma insert_commute: "insert x (insert y A) = insert y (insert x A)"
  1098   by blast
  1099 
  1100 lemma insert_subset [simp]: "(insert x A \<subseteq> B) = (x \<in> B & A \<subseteq> B)"
  1101   by blast
  1102 
  1103 lemma mk_disjoint_insert: "a \<in> A ==> \<exists>B. A = insert a B & a \<notin> B"
  1104   -- {* use new @{text B} rather than @{text "A - {a}"} to avoid infinite unfolding *}
  1105   apply (rule_tac x = "A - {a}" in exI, blast)
  1106   done
  1107 
  1108 lemma insert_Collect: "insert a (Collect P) = {u. u \<noteq> a --> P u}"
  1109   by auto
  1110 
  1111 lemma insert_inter_insert[simp]: "insert a A \<inter> insert a B = insert a (A \<inter> B)"
  1112   by blast
  1113 
  1114 lemma insert_disjoint [simp,no_atp]:
  1115  "(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})"
  1116  "({} = insert a A \<inter> B) = (a \<notin> B \<and> {} = A \<inter> B)"
  1117   by auto
  1118 
  1119 lemma disjoint_insert [simp,no_atp]:
  1120  "(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})"
  1121  "({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)"
  1122   by auto
  1123 
  1124 text {* \medskip @{text image}. *}
  1125 
  1126 lemma image_empty [simp]: "f`{} = {}"
  1127   by blast
  1128 
  1129 lemma image_insert [simp]: "f ` insert a B = insert (f a) (f`B)"
  1130   by blast
  1131 
  1132 lemma image_constant: "x \<in> A ==> (\<lambda>x. c) ` A = {c}"
  1133   by auto
  1134 
  1135 lemma image_constant_conv: "(%x. c) ` A = (if A = {} then {} else {c})"
  1136 by auto
  1137 
  1138 lemma image_image: "f ` (g ` A) = (\<lambda>x. f (g x)) ` A"
  1139 by blast
  1140 
  1141 lemma insert_image [simp]: "x \<in> A ==> insert (f x) (f`A) = f`A"
  1142 by blast
  1143 
  1144 lemma image_is_empty [iff]: "(f`A = {}) = (A = {})"
  1145 by blast
  1146 
  1147 lemma empty_is_image[iff]: "({} = f ` A) = (A = {})"
  1148 by blast
  1149 
  1150 
  1151 lemma image_Collect [no_atp]: "f ` {x. P x} = {f x | x. P x}"
  1152   -- {* NOT suitable as a default simprule: the RHS isn't simpler than the LHS,
  1153       with its implicit quantifier and conjunction.  Also image enjoys better
  1154       equational properties than does the RHS. *}
  1155   by blast
  1156 
  1157 lemma if_image_distrib [simp]:
  1158   "(\<lambda>x. if P x then f x else g x) ` S
  1159     = (f ` (S \<inter> {x. P x})) \<union> (g ` (S \<inter> {x. \<not> P x}))"
  1160   by (auto simp add: image_def)
  1161 
  1162 lemma image_cong: "M = N ==> (!!x. x \<in> N ==> f x = g x) ==> f`M = g`N"
  1163   by (simp add: image_def)
  1164 
  1165 
  1166 text {* \medskip @{text range}. *}
  1167 
  1168 lemma full_SetCompr_eq [no_atp]: "{u. \<exists>x. u = f x} = range f"
  1169   by auto
  1170 
  1171 lemma range_composition: "range (\<lambda>x. f (g x)) = f`range g"
  1172 by (subst image_image, simp)
  1173 
  1174 
  1175 text {* \medskip @{text Int} *}
  1176 
  1177 lemma Int_absorb [simp]: "A \<inter> A = A"
  1178   by (fact inf_idem)
  1179 
  1180 lemma Int_left_absorb: "A \<inter> (A \<inter> B) = A \<inter> B"
  1181   by (fact inf_left_idem)
  1182 
  1183 lemma Int_commute: "A \<inter> B = B \<inter> A"
  1184   by (fact inf_commute)
  1185 
  1186 lemma Int_left_commute: "A \<inter> (B \<inter> C) = B \<inter> (A \<inter> C)"
  1187   by (fact inf_left_commute)
  1188 
  1189 lemma Int_assoc: "(A \<inter> B) \<inter> C = A \<inter> (B \<inter> C)"
  1190   by (fact inf_assoc)
  1191 
  1192 lemmas Int_ac = Int_assoc Int_left_absorb Int_commute Int_left_commute
  1193   -- {* Intersection is an AC-operator *}
  1194 
  1195 lemma Int_absorb1: "B \<subseteq> A ==> A \<inter> B = B"
  1196   by (fact inf_absorb2)
  1197 
  1198 lemma Int_absorb2: "A \<subseteq> B ==> A \<inter> B = A"
  1199   by (fact inf_absorb1)
  1200 
  1201 lemma Int_empty_left [simp]: "{} \<inter> B = {}"
  1202   by (fact inf_bot_left)
  1203 
  1204 lemma Int_empty_right [simp]: "A \<inter> {} = {}"
  1205   by (fact inf_bot_right)
  1206 
  1207 lemma disjoint_eq_subset_Compl: "(A \<inter> B = {}) = (A \<subseteq> -B)"
  1208   by blast
  1209 
  1210 lemma disjoint_iff_not_equal: "(A \<inter> B = {}) = (\<forall>x\<in>A. \<forall>y\<in>B. x \<noteq> y)"
  1211   by blast
  1212 
  1213 lemma Int_UNIV_left [simp]: "UNIV \<inter> B = B"
  1214   by (fact inf_top_left)
  1215 
  1216 lemma Int_UNIV_right [simp]: "A \<inter> UNIV = A"
  1217   by (fact inf_top_right)
  1218 
  1219 lemma Int_Un_distrib: "A \<inter> (B \<union> C) = (A \<inter> B) \<union> (A \<inter> C)"
  1220   by (fact inf_sup_distrib1)
  1221 
  1222 lemma Int_Un_distrib2: "(B \<union> C) \<inter> A = (B \<inter> A) \<union> (C \<inter> A)"
  1223   by (fact inf_sup_distrib2)
  1224 
  1225 lemma Int_UNIV [simp,no_atp]: "(A \<inter> B = UNIV) = (A = UNIV & B = UNIV)"
  1226   by (fact inf_eq_top_iff)
  1227 
  1228 lemma Int_subset_iff [no_atp, simp]: "(C \<subseteq> A \<inter> B) = (C \<subseteq> A & C \<subseteq> B)"
  1229   by (fact le_inf_iff)
  1230 
  1231 lemma Int_Collect: "(x \<in> A \<inter> {x. P x}) = (x \<in> A & P x)"
  1232   by blast
  1233 
  1234 
  1235 text {* \medskip @{text Un}. *}
  1236 
  1237 lemma Un_absorb [simp]: "A \<union> A = A"
  1238   by (fact sup_idem)
  1239 
  1240 lemma Un_left_absorb: "A \<union> (A \<union> B) = A \<union> B"
  1241   by (fact sup_left_idem)
  1242 
  1243 lemma Un_commute: "A \<union> B = B \<union> A"
  1244   by (fact sup_commute)
  1245 
  1246 lemma Un_left_commute: "A \<union> (B \<union> C) = B \<union> (A \<union> C)"
  1247   by (fact sup_left_commute)
  1248 
  1249 lemma Un_assoc: "(A \<union> B) \<union> C = A \<union> (B \<union> C)"
  1250   by (fact sup_assoc)
  1251 
  1252 lemmas Un_ac = Un_assoc Un_left_absorb Un_commute Un_left_commute
  1253   -- {* Union is an AC-operator *}
  1254 
  1255 lemma Un_absorb1: "A \<subseteq> B ==> A \<union> B = B"
  1256   by (fact sup_absorb2)
  1257 
  1258 lemma Un_absorb2: "B \<subseteq> A ==> A \<union> B = A"
  1259   by (fact sup_absorb1)
  1260 
  1261 lemma Un_empty_left [simp]: "{} \<union> B = B"
  1262   by (fact sup_bot_left)
  1263 
  1264 lemma Un_empty_right [simp]: "A \<union> {} = A"
  1265   by (fact sup_bot_right)
  1266 
  1267 lemma Un_UNIV_left [simp]: "UNIV \<union> B = UNIV"
  1268   by (fact sup_top_left)
  1269 
  1270 lemma Un_UNIV_right [simp]: "A \<union> UNIV = UNIV"
  1271   by (fact sup_top_right)
  1272 
  1273 lemma Un_insert_left [simp]: "(insert a B) \<union> C = insert a (B \<union> C)"
  1274   by blast
  1275 
  1276 lemma Un_insert_right [simp]: "A \<union> (insert a B) = insert a (A \<union> B)"
  1277   by blast
  1278 
  1279 lemma Int_insert_left:
  1280     "(insert a B) Int C = (if a \<in> C then insert a (B \<inter> C) else B \<inter> C)"
  1281   by auto
  1282 
  1283 lemma Int_insert_left_if0[simp]:
  1284     "a \<notin> C \<Longrightarrow> (insert a B) Int C = B \<inter> C"
  1285   by auto
  1286 
  1287 lemma Int_insert_left_if1[simp]:
  1288     "a \<in> C \<Longrightarrow> (insert a B) Int C = insert a (B Int C)"
  1289   by auto
  1290 
  1291 lemma Int_insert_right:
  1292     "A \<inter> (insert a B) = (if a \<in> A then insert a (A \<inter> B) else A \<inter> B)"
  1293   by auto
  1294 
  1295 lemma Int_insert_right_if0[simp]:
  1296     "a \<notin> A \<Longrightarrow> A Int (insert a B) = A Int B"
  1297   by auto
  1298 
  1299 lemma Int_insert_right_if1[simp]:
  1300     "a \<in> A \<Longrightarrow> A Int (insert a B) = insert a (A Int B)"
  1301   by auto
  1302 
  1303 lemma Un_Int_distrib: "A \<union> (B \<inter> C) = (A \<union> B) \<inter> (A \<union> C)"
  1304   by (fact sup_inf_distrib1)
  1305 
  1306 lemma Un_Int_distrib2: "(B \<inter> C) \<union> A = (B \<union> A) \<inter> (C \<union> A)"
  1307   by (fact sup_inf_distrib2)
  1308 
  1309 lemma Un_Int_crazy:
  1310     "(A \<inter> B) \<union> (B \<inter> C) \<union> (C \<inter> A) = (A \<union> B) \<inter> (B \<union> C) \<inter> (C \<union> A)"
  1311   by blast
  1312 
  1313 lemma subset_Un_eq: "(A \<subseteq> B) = (A \<union> B = B)"
  1314   by (fact le_iff_sup)
  1315 
  1316 lemma Un_empty [iff]: "(A \<union> B = {}) = (A = {} & B = {})"
  1317   by (fact sup_eq_bot_iff)
  1318 
  1319 lemma Un_subset_iff [no_atp, simp]: "(A \<union> B \<subseteq> C) = (A \<subseteq> C & B \<subseteq> C)"
  1320   by (fact le_sup_iff)
  1321 
  1322 lemma Un_Diff_Int: "(A - B) \<union> (A \<inter> B) = A"
  1323   by blast
  1324 
  1325 lemma Diff_Int2: "A \<inter> C - B \<inter> C = A \<inter> C - B"
  1326   by blast
  1327 
  1328 
  1329 text {* \medskip Set complement *}
  1330 
  1331 lemma Compl_disjoint [simp]: "A \<inter> -A = {}"
  1332   by (fact inf_compl_bot)
  1333 
  1334 lemma Compl_disjoint2 [simp]: "-A \<inter> A = {}"
  1335   by (fact compl_inf_bot)
  1336 
  1337 lemma Compl_partition: "A \<union> -A = UNIV"
  1338   by (fact sup_compl_top)
  1339 
  1340 lemma Compl_partition2: "-A \<union> A = UNIV"
  1341   by (fact compl_sup_top)
  1342 
  1343 lemma double_complement [simp]: "- (-A) = (A::'a set)"
  1344   by (fact double_compl)
  1345 
  1346 lemma Compl_Un [simp]: "-(A \<union> B) = (-A) \<inter> (-B)"
  1347   by (fact compl_sup)
  1348 
  1349 lemma Compl_Int [simp]: "-(A \<inter> B) = (-A) \<union> (-B)"
  1350   by (fact compl_inf)
  1351 
  1352 lemma subset_Compl_self_eq: "(A \<subseteq> -A) = (A = {})"
  1353   by blast
  1354 
  1355 lemma Un_Int_assoc_eq: "((A \<inter> B) \<union> C = A \<inter> (B \<union> C)) = (C \<subseteq> A)"
  1356   -- {* Halmos, Naive Set Theory, page 16. *}
  1357   by blast
  1358 
  1359 lemma Compl_UNIV_eq [simp]: "-UNIV = {}"
  1360   by (fact compl_top_eq)
  1361 
  1362 lemma Compl_empty_eq [simp]: "-{} = UNIV"
  1363   by (fact compl_bot_eq)
  1364 
  1365 lemma Compl_subset_Compl_iff [iff]: "(-A \<subseteq> -B) = (B \<subseteq> A)"
  1366   by (fact compl_le_compl_iff)
  1367 
  1368 lemma Compl_eq_Compl_iff [iff]: "(-A = -B) = (A = (B::'a set))"
  1369   by (fact compl_eq_compl_iff)
  1370 
  1371 text {* \medskip Bounded quantifiers.
  1372 
  1373   The following are not added to the default simpset because
  1374   (a) they duplicate the body and (b) there are no similar rules for @{text Int}. *}
  1375 
  1376 lemma ball_Un: "(\<forall>x \<in> A \<union> B. P x) = ((\<forall>x\<in>A. P x) & (\<forall>x\<in>B. P x))"
  1377   by blast
  1378 
  1379 lemma bex_Un: "(\<exists>x \<in> A \<union> B. P x) = ((\<exists>x\<in>A. P x) | (\<exists>x\<in>B. P x))"
  1380   by blast
  1381 
  1382 
  1383 text {* \medskip Set difference. *}
  1384 
  1385 lemma Diff_eq: "A - B = A \<inter> (-B)"
  1386   by blast
  1387 
  1388 lemma Diff_eq_empty_iff [simp,no_atp]: "(A - B = {}) = (A \<subseteq> B)"
  1389   by blast
  1390 
  1391 lemma Diff_cancel [simp]: "A - A = {}"
  1392   by blast
  1393 
  1394 lemma Diff_idemp [simp]: "(A - B) - B = A - (B::'a set)"
  1395 by blast
  1396 
  1397 lemma Diff_triv: "A \<inter> B = {} ==> A - B = A"
  1398   by (blast elim: equalityE)
  1399 
  1400 lemma empty_Diff [simp]: "{} - A = {}"
  1401   by blast
  1402 
  1403 lemma Diff_empty [simp]: "A - {} = A"
  1404   by blast
  1405 
  1406 lemma Diff_UNIV [simp]: "A - UNIV = {}"
  1407   by blast
  1408 
  1409 lemma Diff_insert0 [simp,no_atp]: "x \<notin> A ==> A - insert x B = A - B"
  1410   by blast
  1411 
  1412 lemma Diff_insert: "A - insert a B = A - B - {a}"
  1413   -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a 0"} *}
  1414   by blast
  1415 
  1416 lemma Diff_insert2: "A - insert a B = A - {a} - B"
  1417   -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a 0"} *}
  1418   by blast
  1419 
  1420 lemma insert_Diff_if: "insert x A - B = (if x \<in> B then A - B else insert x (A - B))"
  1421   by auto
  1422 
  1423 lemma insert_Diff1 [simp]: "x \<in> B ==> insert x A - B = A - B"
  1424   by blast
  1425 
  1426 lemma insert_Diff_single[simp]: "insert a (A - {a}) = insert a A"
  1427 by blast
  1428 
  1429 lemma insert_Diff: "a \<in> A ==> insert a (A - {a}) = A"
  1430   by blast
  1431 
  1432 lemma Diff_insert_absorb: "x \<notin> A ==> (insert x A) - {x} = A"
  1433   by auto
  1434 
  1435 lemma Diff_disjoint [simp]: "A \<inter> (B - A) = {}"
  1436   by blast
  1437 
  1438 lemma Diff_partition: "A \<subseteq> B ==> A \<union> (B - A) = B"
  1439   by blast
  1440 
  1441 lemma double_diff: "A \<subseteq> B ==> B \<subseteq> C ==> B - (C - A) = A"
  1442   by blast
  1443 
  1444 lemma Un_Diff_cancel [simp]: "A \<union> (B - A) = A \<union> B"
  1445   by blast
  1446 
  1447 lemma Un_Diff_cancel2 [simp]: "(B - A) \<union> A = B \<union> A"
  1448   by blast
  1449 
  1450 lemma Diff_Un: "A - (B \<union> C) = (A - B) \<inter> (A - C)"
  1451   by blast
  1452 
  1453 lemma Diff_Int: "A - (B \<inter> C) = (A - B) \<union> (A - C)"
  1454   by blast
  1455 
  1456 lemma Un_Diff: "(A \<union> B) - C = (A - C) \<union> (B - C)"
  1457   by blast
  1458 
  1459 lemma Int_Diff: "(A \<inter> B) - C = A \<inter> (B - C)"
  1460   by blast
  1461 
  1462 lemma Diff_Int_distrib: "C \<inter> (A - B) = (C \<inter> A) - (C \<inter> B)"
  1463   by blast
  1464 
  1465 lemma Diff_Int_distrib2: "(A - B) \<inter> C = (A \<inter> C) - (B \<inter> C)"
  1466   by blast
  1467 
  1468 lemma Diff_Compl [simp]: "A - (- B) = A \<inter> B"
  1469   by auto
  1470 
  1471 lemma Compl_Diff_eq [simp]: "- (A - B) = -A \<union> B"
  1472   by blast
  1473 
  1474 
  1475 text {* \medskip Quantification over type @{typ bool}. *}
  1476 
  1477 lemma bool_induct: "P True \<Longrightarrow> P False \<Longrightarrow> P x"
  1478   by (cases x) auto
  1479 
  1480 lemma all_bool_eq: "(\<forall>b. P b) \<longleftrightarrow> P True \<and> P False"
  1481   by (auto intro: bool_induct)
  1482 
  1483 lemma bool_contrapos: "P x \<Longrightarrow> \<not> P False \<Longrightarrow> P True"
  1484   by (cases x) auto
  1485 
  1486 lemma ex_bool_eq: "(\<exists>b. P b) \<longleftrightarrow> P True \<or> P False"
  1487   by (auto intro: bool_contrapos)
  1488 
  1489 text {* \medskip @{text Pow} *}
  1490 
  1491 lemma Pow_empty [simp]: "Pow {} = {{}}"
  1492   by (auto simp add: Pow_def)
  1493 
  1494 lemma Pow_insert: "Pow (insert a A) = Pow A \<union> (insert a ` Pow A)"
  1495   by (blast intro: image_eqI [where ?x = "u - {a}", standard])
  1496 
  1497 lemma Pow_Compl: "Pow (- A) = {-B | B. A \<in> Pow B}"
  1498   by (blast intro: exI [where ?x = "- u", standard])
  1499 
  1500 lemma Pow_UNIV [simp]: "Pow UNIV = UNIV"
  1501   by blast
  1502 
  1503 lemma Un_Pow_subset: "Pow A \<union> Pow B \<subseteq> Pow (A \<union> B)"
  1504   by blast
  1505 
  1506 lemma Pow_Int_eq [simp]: "Pow (A \<inter> B) = Pow A \<inter> Pow B"
  1507   by blast
  1508 
  1509 
  1510 text {* \medskip Miscellany. *}
  1511 
  1512 lemma set_eq_subset: "(A = B) = (A \<subseteq> B & B \<subseteq> A)"
  1513   by blast
  1514 
  1515 lemma subset_iff [no_atp]: "(A \<subseteq> B) = (\<forall>t. t \<in> A --> t \<in> B)"
  1516   by blast
  1517 
  1518 lemma subset_iff_psubset_eq: "(A \<subseteq> B) = ((A \<subset> B) | (A = B))"
  1519   by (unfold less_le) blast
  1520 
  1521 lemma all_not_in_conv [simp]: "(\<forall>x. x \<notin> A) = (A = {})"
  1522   by blast
  1523 
  1524 lemma ex_in_conv: "(\<exists>x. x \<in> A) = (A \<noteq> {})"
  1525   by blast
  1526 
  1527 lemma distinct_lemma: "f x \<noteq> f y ==> x \<noteq> y"
  1528   by iprover
  1529 
  1530 
  1531 subsubsection {* Monotonicity of various operations *}
  1532 
  1533 lemma image_mono: "A \<subseteq> B ==> f`A \<subseteq> f`B"
  1534   by blast
  1535 
  1536 lemma Pow_mono: "A \<subseteq> B ==> Pow A \<subseteq> Pow B"
  1537   by blast
  1538 
  1539 lemma insert_mono: "C \<subseteq> D ==> insert a C \<subseteq> insert a D"
  1540   by blast
  1541 
  1542 lemma Un_mono: "A \<subseteq> C ==> B \<subseteq> D ==> A \<union> B \<subseteq> C \<union> D"
  1543   by (fact sup_mono)
  1544 
  1545 lemma Int_mono: "A \<subseteq> C ==> B \<subseteq> D ==> A \<inter> B \<subseteq> C \<inter> D"
  1546   by (fact inf_mono)
  1547 
  1548 lemma Diff_mono: "A \<subseteq> C ==> D \<subseteq> B ==> A - B \<subseteq> C - D"
  1549   by blast
  1550 
  1551 lemma Compl_anti_mono: "A \<subseteq> B ==> -B \<subseteq> -A"
  1552   by (fact compl_mono)
  1553 
  1554 text {* \medskip Monotonicity of implications. *}
  1555 
  1556 lemma in_mono: "A \<subseteq> B ==> x \<in> A --> x \<in> B"
  1557   apply (rule impI)
  1558   apply (erule subsetD, assumption)
  1559   done
  1560 
  1561 lemma conj_mono: "P1 --> Q1 ==> P2 --> Q2 ==> (P1 & P2) --> (Q1 & Q2)"
  1562   by iprover
  1563 
  1564 lemma disj_mono: "P1 --> Q1 ==> P2 --> Q2 ==> (P1 | P2) --> (Q1 | Q2)"
  1565   by iprover
  1566 
  1567 lemma imp_mono: "Q1 --> P1 ==> P2 --> Q2 ==> (P1 --> P2) --> (Q1 --> Q2)"
  1568   by iprover
  1569 
  1570 lemma imp_refl: "P --> P" ..
  1571 
  1572 lemma not_mono: "Q --> P ==> ~ P --> ~ Q"
  1573   by iprover
  1574 
  1575 lemma ex_mono: "(!!x. P x --> Q x) ==> (EX x. P x) --> (EX x. Q x)"
  1576   by iprover
  1577 
  1578 lemma all_mono: "(!!x. P x --> Q x) ==> (ALL x. P x) --> (ALL x. Q x)"
  1579   by iprover
  1580 
  1581 lemma Collect_mono: "(!!x. P x --> Q x) ==> Collect P \<subseteq> Collect Q"
  1582   by blast
  1583 
  1584 lemma Int_Collect_mono:
  1585     "A \<subseteq> B ==> (!!x. x \<in> A ==> P x --> Q x) ==> A \<inter> Collect P \<subseteq> B \<inter> Collect Q"
  1586   by blast
  1587 
  1588 lemmas basic_monos =
  1589   subset_refl imp_refl disj_mono conj_mono
  1590   ex_mono Collect_mono in_mono
  1591 
  1592 lemma eq_to_mono: "a = b ==> c = d ==> b --> d ==> a --> c"
  1593   by iprover
  1594 
  1595 
  1596 subsubsection {* Inverse image of a function *}
  1597 
  1598 definition vimage :: "('a => 'b) => 'b set => 'a set" (infixr "-`" 90) where
  1599   "f -` B == {x. f x : B}"
  1600 
  1601 lemma vimage_eq [simp]: "(a : f -` B) = (f a : B)"
  1602   by (unfold vimage_def) blast
  1603 
  1604 lemma vimage_singleton_eq: "(a : f -` {b}) = (f a = b)"
  1605   by simp
  1606 
  1607 lemma vimageI [intro]: "f a = b ==> b:B ==> a : f -` B"
  1608   by (unfold vimage_def) blast
  1609 
  1610 lemma vimageI2: "f a : A ==> a : f -` A"
  1611   by (unfold vimage_def) fast
  1612 
  1613 lemma vimageE [elim!]: "a: f -` B ==> (!!x. f a = x ==> x:B ==> P) ==> P"
  1614   by (unfold vimage_def) blast
  1615 
  1616 lemma vimageD: "a : f -` A ==> f a : A"
  1617   by (unfold vimage_def) fast
  1618 
  1619 lemma vimage_empty [simp]: "f -` {} = {}"
  1620   by blast
  1621 
  1622 lemma vimage_Compl: "f -` (-A) = -(f -` A)"
  1623   by blast
  1624 
  1625 lemma vimage_Un [simp]: "f -` (A Un B) = (f -` A) Un (f -` B)"
  1626   by blast
  1627 
  1628 lemma vimage_Int [simp]: "f -` (A Int B) = (f -` A) Int (f -` B)"
  1629   by fast
  1630 
  1631 lemma vimage_Collect_eq [simp]: "f -` Collect P = {y. P (f y)}"
  1632   by blast
  1633 
  1634 lemma vimage_Collect: "(!!x. P (f x) = Q x) ==> f -` (Collect P) = Collect Q"
  1635   by blast
  1636 
  1637 lemma vimage_insert: "f-`(insert a B) = (f-`{a}) Un (f-`B)"
  1638   -- {* NOT suitable for rewriting because of the recurrence of @{term "{a}"}. *}
  1639   by blast
  1640 
  1641 lemma vimage_Diff: "f -` (A - B) = (f -` A) - (f -` B)"
  1642   by blast
  1643 
  1644 lemma vimage_UNIV [simp]: "f -` UNIV = UNIV"
  1645   by blast
  1646 
  1647 lemma vimage_mono: "A \<subseteq> B ==> f -` A \<subseteq> f -` B"
  1648   -- {* monotonicity *}
  1649   by blast
  1650 
  1651 lemma vimage_image_eq [no_atp]: "f -` (f ` A) = {y. EX x:A. f x = f y}"
  1652 by (blast intro: sym)
  1653 
  1654 lemma image_vimage_subset: "f ` (f -` A) <= A"
  1655 by blast
  1656 
  1657 lemma image_vimage_eq [simp]: "f ` (f -` A) = A Int range f"
  1658 by blast
  1659 
  1660 lemma vimage_const [simp]: "((\<lambda>x. c) -` A) = (if c \<in> A then UNIV else {})"
  1661   by auto
  1662 
  1663 lemma vimage_if [simp]: "((\<lambda>x. if x \<in> B then c else d) -` A) = 
  1664    (if c \<in> A then (if d \<in> A then UNIV else B)
  1665     else if d \<in> A then -B else {})"  
  1666   by (auto simp add: vimage_def) 
  1667 
  1668 lemma vimage_inter_cong:
  1669   "(\<And> w. w \<in> S \<Longrightarrow> f w = g w) \<Longrightarrow> f -` y \<inter> S = g -` y \<inter> S"
  1670   by auto
  1671 
  1672 lemma image_Int_subset: "f`(A Int B) <= f`A Int f`B"
  1673 by blast
  1674 
  1675 lemma image_diff_subset: "f`A - f`B <= f`(A - B)"
  1676 by blast
  1677 
  1678 
  1679 subsubsection {* Getting the Contents of a Singleton Set *}
  1680 
  1681 definition the_elem :: "'a set \<Rightarrow> 'a" where
  1682   "the_elem X = (THE x. X = {x})"
  1683 
  1684 lemma the_elem_eq [simp]: "the_elem {x} = x"
  1685   by (simp add: the_elem_def)
  1686 
  1687 
  1688 subsubsection {* Least value operator *}
  1689 
  1690 lemma Least_mono:
  1691   "mono (f::'a::order => 'b::order) ==> EX x:S. ALL y:S. x <= y
  1692     ==> (LEAST y. y : f ` S) = f (LEAST x. x : S)"
  1693     -- {* Courtesy of Stephan Merz *}
  1694   apply clarify
  1695   apply (erule_tac P = "%x. x : S" in LeastI2_order, fast)
  1696   apply (rule LeastI2_order)
  1697   apply (auto elim: monoD intro!: order_antisym)
  1698   done
  1699 
  1700 subsection {* Misc *}
  1701 
  1702 text {* Rudimentary code generation *}
  1703 
  1704 lemma insert_code [code]: "insert y A x \<longleftrightarrow> y = x \<or> A x"
  1705   by (auto simp add: insert_compr Collect_def mem_def)
  1706 
  1707 lemma vimage_code [code]: "(f -` A) x = A (f x)"
  1708   by (simp add: vimage_def Collect_def mem_def)
  1709 
  1710 hide_const (open) member
  1711 
  1712 text {* Misc theorem and ML bindings *}
  1713 
  1714 lemmas equalityI = subset_antisym
  1715 
  1716 ML {*
  1717 val Ball_def = @{thm Ball_def}
  1718 val Bex_def = @{thm Bex_def}
  1719 val CollectD = @{thm CollectD}
  1720 val CollectE = @{thm CollectE}
  1721 val CollectI = @{thm CollectI}
  1722 val Collect_conj_eq = @{thm Collect_conj_eq}
  1723 val Collect_mem_eq = @{thm Collect_mem_eq}
  1724 val IntD1 = @{thm IntD1}
  1725 val IntD2 = @{thm IntD2}
  1726 val IntE = @{thm IntE}
  1727 val IntI = @{thm IntI}
  1728 val Int_Collect = @{thm Int_Collect}
  1729 val UNIV_I = @{thm UNIV_I}
  1730 val UNIV_witness = @{thm UNIV_witness}
  1731 val UnE = @{thm UnE}
  1732 val UnI1 = @{thm UnI1}
  1733 val UnI2 = @{thm UnI2}
  1734 val ballE = @{thm ballE}
  1735 val ballI = @{thm ballI}
  1736 val bexCI = @{thm bexCI}
  1737 val bexE = @{thm bexE}
  1738 val bexI = @{thm bexI}
  1739 val bex_triv = @{thm bex_triv}
  1740 val bspec = @{thm bspec}
  1741 val contra_subsetD = @{thm contra_subsetD}
  1742 val distinct_lemma = @{thm distinct_lemma}
  1743 val eq_to_mono = @{thm eq_to_mono}
  1744 val equalityCE = @{thm equalityCE}
  1745 val equalityD1 = @{thm equalityD1}
  1746 val equalityD2 = @{thm equalityD2}
  1747 val equalityE = @{thm equalityE}
  1748 val equalityI = @{thm equalityI}
  1749 val imageE = @{thm imageE}
  1750 val imageI = @{thm imageI}
  1751 val image_Un = @{thm image_Un}
  1752 val image_insert = @{thm image_insert}
  1753 val insert_commute = @{thm insert_commute}
  1754 val insert_iff = @{thm insert_iff}
  1755 val mem_Collect_eq = @{thm mem_Collect_eq}
  1756 val rangeE = @{thm rangeE}
  1757 val rangeI = @{thm rangeI}
  1758 val range_eqI = @{thm range_eqI}
  1759 val subsetCE = @{thm subsetCE}
  1760 val subsetD = @{thm subsetD}
  1761 val subsetI = @{thm subsetI}
  1762 val subset_refl = @{thm subset_refl}
  1763 val subset_trans = @{thm subset_trans}
  1764 val vimageD = @{thm vimageD}
  1765 val vimageE = @{thm vimageE}
  1766 val vimageI = @{thm vimageI}
  1767 val vimageI2 = @{thm vimageI2}
  1768 val vimage_Collect = @{thm vimage_Collect}
  1769 val vimage_Int = @{thm vimage_Int}
  1770 val vimage_Un = @{thm vimage_Un}
  1771 *}
  1772 
  1773 end