src/HOL/Set.thy
author wenzelm
Wed Dec 06 01:12:36 2006 +0100 (2006-12-06)
changeset 21669 c68717c16013
parent 21549 12eff58b56a0
child 21819 8eb82ffcdd15
permissions -rw-r--r--
removed legacy ML bindings;
     1 (*  Title:      HOL/Set.thy
     2     ID:         $Id$
     3     Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
     4 *)
     5 
     6 header {* Set theory for higher-order logic *}
     7 
     8 theory Set
     9 imports Lattices
    10 begin
    11 
    12 text {* A set in HOL is simply a predicate. *}
    13 
    14 
    15 subsection {* Basic syntax *}
    16 
    17 global
    18 
    19 typedecl 'a set
    20 arities set :: (type) type
    21 
    22 consts
    23   "{}"          :: "'a set"                             ("{}")
    24   UNIV          :: "'a set"
    25   insert        :: "'a => 'a set => 'a set"
    26   Collect       :: "('a => bool) => 'a set"              -- "comprehension"
    27   Int           :: "'a set => 'a set => 'a set"          (infixl 70)
    28   Un            :: "'a set => 'a set => 'a set"          (infixl 65)
    29   UNION         :: "'a set => ('a => 'b set) => 'b set"  -- "general union"
    30   INTER         :: "'a set => ('a => 'b set) => 'b set"  -- "general intersection"
    31   Union         :: "'a set set => 'a set"                -- "union of a set"
    32   Inter         :: "'a set set => 'a set"                -- "intersection of a set"
    33   Pow           :: "'a set => 'a set set"                -- "powerset"
    34   Ball          :: "'a set => ('a => bool) => bool"      -- "bounded universal quantifiers"
    35   Bex           :: "'a set => ('a => bool) => bool"      -- "bounded existential quantifiers"
    36   Bex1          :: "'a set => ('a => bool) => bool"      -- "bounded unique existential quantifiers"
    37   image         :: "('a => 'b) => 'a set => 'b set"      (infixr "`" 90)
    38   "op :"        :: "'a => 'a set => bool"                -- "membership"
    39 
    40 notation
    41   "op :"  ("op :") and
    42   "op :"  ("(_/ : _)" [50, 51] 50)
    43 
    44 local
    45 
    46 
    47 subsection {* Additional concrete syntax *}
    48 
    49 abbreviation
    50   range :: "('a => 'b) => 'b set" where -- "of function"
    51   "range f == f ` UNIV"
    52 
    53 abbreviation
    54   "not_mem x A == ~ (x : A)" -- "non-membership"
    55 
    56 notation
    57   not_mem  ("op ~:") and
    58   not_mem  ("(_/ ~: _)" [50, 51] 50)
    59 
    60 notation (xsymbols)
    61   "op Int"  (infixl "\<inter>" 70) and
    62   "op Un"  (infixl "\<union>" 65) and
    63   "op :"  ("op \<in>") and
    64   "op :"  ("(_/ \<in> _)" [50, 51] 50) and
    65   not_mem  ("op \<notin>") and
    66   not_mem  ("(_/ \<notin> _)" [50, 51] 50) and
    67   Union  ("\<Union>_" [90] 90) and
    68   Inter  ("\<Inter>_" [90] 90)
    69 
    70 notation (HTML output)
    71   "op Int"  (infixl "\<inter>" 70) and
    72   "op Un"  (infixl "\<union>" 65) and
    73   "op :"  ("op \<in>") and
    74   "op :"  ("(_/ \<in> _)" [50, 51] 50) and
    75   not_mem  ("op \<notin>") and
    76   not_mem  ("(_/ \<notin> _)" [50, 51] 50)
    77 
    78 syntax
    79   "@Finset"     :: "args => 'a set"                       ("{(_)}")
    80   "@Coll"       :: "pttrn => bool => 'a set"              ("(1{_./ _})")
    81   "@SetCompr"   :: "'a => idts => bool => 'a set"         ("(1{_ |/_./ _})")
    82   "@Collect"    :: "idt => 'a set => bool => 'a set"      ("(1{_ :/ _./ _})")
    83   "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" 10)
    84   "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" 10)
    85   "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" 10)
    86   "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" 10)
    87   "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3ALL _:_./ _)" [0, 0, 10] 10)
    88   "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3EX _:_./ _)" [0, 0, 10] 10)
    89   "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3EX! _:_./ _)" [0, 0, 10] 10)
    90   "_Bleast"       :: "id => 'a set => bool => 'a"      ("(3LEAST _:_./ _)" [0, 0, 10] 10)
    91 
    92 syntax (HOL)
    93   "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3! _:_./ _)" [0, 0, 10] 10)
    94   "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3? _:_./ _)" [0, 0, 10] 10)
    95   "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3?! _:_./ _)" [0, 0, 10] 10)
    96 
    97 translations
    98   "{x, xs}"     == "insert x {xs}"
    99   "{x}"         == "insert x {}"
   100   "{x. P}"      == "Collect (%x. P)"
   101   "{x:A. P}"    => "{x. x:A & P}"
   102   "UN x y. B"   == "UN x. UN y. B"
   103   "UN x. B"     == "UNION UNIV (%x. B)"
   104   "UN x. B"     == "UN x:UNIV. B"
   105   "INT x y. B"  == "INT x. INT y. B"
   106   "INT x. B"    == "INTER UNIV (%x. B)"
   107   "INT x. B"    == "INT x:UNIV. B"
   108   "UN x:A. B"   == "UNION A (%x. B)"
   109   "INT x:A. B"  == "INTER A (%x. B)"
   110   "ALL x:A. P"  == "Ball A (%x. P)"
   111   "EX x:A. P"   == "Bex A (%x. P)"
   112   "EX! x:A. P"  == "Bex1 A (%x. P)"
   113   "LEAST x:A. P" => "LEAST x. x:A & P"
   114 
   115 syntax (xsymbols)
   116   "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10)
   117   "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10)
   118   "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3\<exists>!_\<in>_./ _)" [0, 0, 10] 10)
   119   "_Bleast"     :: "id => 'a set => bool => 'a"           ("(3LEAST_\<in>_./ _)" [0, 0, 10] 10)
   120 
   121 syntax (HTML output)
   122   "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10)
   123   "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10)
   124   "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3\<exists>!_\<in>_./ _)" [0, 0, 10] 10)
   125 
   126 syntax (xsymbols)
   127   "@Collect"    :: "idt => 'a set => bool => 'a set"      ("(1{_ \<in>/ _./ _})")
   128   "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" 10)
   129   "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" 10)
   130   "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" 10)
   131   "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" 10)
   132 
   133 syntax (latex output)
   134   "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" 10)
   135   "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" 10)
   136   "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" 10)
   137   "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" 10)
   138 
   139 text{*
   140   Note the difference between ordinary xsymbol syntax of indexed
   141   unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})
   142   and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The
   143   former does not make the index expression a subscript of the
   144   union/intersection symbol because this leads to problems with nested
   145   subscripts in Proof General. *}
   146 
   147 instance set :: (type) ord
   148   subset_def:   "A <= B         == \<forall>x\<in>A. x \<in> B"
   149   psubset_def:  "A < B          == (A::'a set) <= B & ~ A=B" ..
   150 
   151 abbreviation
   152   subset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
   153   "subset == less"
   154 
   155 abbreviation
   156   subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
   157   "subset_eq == less_eq"
   158 
   159 notation (output)
   160   subset  ("op <") and
   161   subset  ("(_/ < _)" [50, 51] 50) and
   162   subset_eq  ("op <=") and
   163   subset_eq  ("(_/ <= _)" [50, 51] 50)
   164 
   165 notation (xsymbols)
   166   subset  ("op \<subset>") and
   167   subset  ("(_/ \<subset> _)" [50, 51] 50) and
   168   subset_eq  ("op \<subseteq>") and
   169   subset_eq  ("(_/ \<subseteq> _)" [50, 51] 50)
   170 
   171 notation (HTML output)
   172   subset  ("op \<subset>") and
   173   subset  ("(_/ \<subset> _)" [50, 51] 50) and
   174   subset_eq  ("op \<subseteq>") and
   175   subset_eq  ("(_/ \<subseteq> _)" [50, 51] 50)
   176 
   177 abbreviation (input)
   178   supset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"  (infixl "\<supset>" 50) where
   179   "supset == greater"
   180 
   181 abbreviation (input)
   182   supset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"  (infixl "\<supseteq>" 50) where
   183   "supset_eq == greater_eq"
   184 
   185 
   186 subsubsection "Bounded quantifiers"
   187 
   188 syntax (output)
   189   "_setlessAll" :: "[idt, 'a, bool] => bool"  ("(3ALL _<_./ _)"  [0, 0, 10] 10)
   190   "_setlessEx"  :: "[idt, 'a, bool] => bool"  ("(3EX _<_./ _)"  [0, 0, 10] 10)
   191   "_setleAll"   :: "[idt, 'a, bool] => bool"  ("(3ALL _<=_./ _)" [0, 0, 10] 10)
   192   "_setleEx"    :: "[idt, 'a, bool] => bool"  ("(3EX _<=_./ _)" [0, 0, 10] 10)
   193   "_setleEx1"   :: "[idt, 'a, bool] => bool"  ("(3EX! _<=_./ _)" [0, 0, 10] 10)
   194 
   195 syntax (xsymbols)
   196   "_setlessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<subset>_./ _)"  [0, 0, 10] 10)
   197   "_setlessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subset>_./ _)"  [0, 0, 10] 10)
   198   "_setleAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<subseteq>_./ _)" [0, 0, 10] 10)
   199   "_setleEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10)
   200   "_setleEx1"   :: "[idt, 'a, bool] => bool"   ("(3\<exists>!_\<subseteq>_./ _)" [0, 0, 10] 10)
   201 
   202 syntax (HOL output)
   203   "_setlessAll" :: "[idt, 'a, bool] => bool"   ("(3! _<_./ _)"  [0, 0, 10] 10)
   204   "_setlessEx"  :: "[idt, 'a, bool] => bool"   ("(3? _<_./ _)"  [0, 0, 10] 10)
   205   "_setleAll"   :: "[idt, 'a, bool] => bool"   ("(3! _<=_./ _)" [0, 0, 10] 10)
   206   "_setleEx"    :: "[idt, 'a, bool] => bool"   ("(3? _<=_./ _)" [0, 0, 10] 10)
   207   "_setleEx1"   :: "[idt, 'a, bool] => bool"   ("(3?! _<=_./ _)" [0, 0, 10] 10)
   208 
   209 syntax (HTML output)
   210   "_setlessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<subset>_./ _)"  [0, 0, 10] 10)
   211   "_setlessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subset>_./ _)"  [0, 0, 10] 10)
   212   "_setleAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<subseteq>_./ _)" [0, 0, 10] 10)
   213   "_setleEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10)
   214   "_setleEx1"   :: "[idt, 'a, bool] => bool"   ("(3\<exists>!_\<subseteq>_./ _)" [0, 0, 10] 10)
   215 
   216 translations
   217  "\<forall>A\<subset>B. P"   =>  "ALL A. A \<subset> B --> P"
   218  "\<exists>A\<subset>B. P"   =>  "EX A. A \<subset> B & P"
   219  "\<forall>A\<subseteq>B. P"   =>  "ALL A. A \<subseteq> B --> P"
   220  "\<exists>A\<subseteq>B. P"   =>  "EX A. A \<subseteq> B & P"
   221  "\<exists>!A\<subseteq>B. P"  =>  "EX! A. A \<subseteq> B & P"
   222 
   223 (* FIXME re-use version in Orderings.thy *)
   224 print_translation {*
   225 let
   226   fun
   227     all_tr' [Const ("_bound",_) $ Free (v,Type(T,_)), 
   228              Const("op -->",_) $ (Const ("less",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] = 
   229   (if v=v' andalso T="set"
   230    then Syntax.const "_setlessAll" $ Syntax.mark_bound v' $ n $ P
   231    else raise Match)
   232 
   233   | all_tr' [Const ("_bound",_) $ Free (v,Type(T,_)), 
   234              Const("op -->",_) $ (Const ("less_eq",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] = 
   235   (if v=v' andalso T="set"
   236    then Syntax.const "_setleAll" $ Syntax.mark_bound v' $ n $ P
   237    else raise Match);
   238 
   239   fun
   240     ex_tr' [Const ("_bound",_) $ Free (v,Type(T,_)), 
   241             Const("op &",_) $ (Const ("less",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] = 
   242   (if v=v' andalso T="set"
   243    then Syntax.const "_setlessEx" $ Syntax.mark_bound v' $ n $ P
   244    else raise Match)
   245 
   246   | ex_tr' [Const ("_bound",_) $ Free (v,Type(T,_)), 
   247             Const("op &",_) $ (Const ("less_eq",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] = 
   248   (if v=v' andalso T="set"
   249    then Syntax.const "_setleEx" $ Syntax.mark_bound v' $ n $ P
   250    else raise Match)
   251 in
   252 [("All_binder", all_tr'), ("Ex_binder", ex_tr')]
   253 end
   254 *}
   255 
   256 
   257 
   258 text {*
   259   \medskip Translate between @{text "{e | x1...xn. P}"} and @{text
   260   "{u. EX x1..xn. u = e & P}"}; @{text "{y. EX x1..xn. y = e & P}"} is
   261   only translated if @{text "[0..n] subset bvs(e)"}.
   262 *}
   263 
   264 parse_translation {*
   265   let
   266     val ex_tr = snd (mk_binder_tr ("EX ", "Ex"));
   267 
   268     fun nvars (Const ("_idts", _) $ _ $ idts) = nvars idts + 1
   269       | nvars _ = 1;
   270 
   271     fun setcompr_tr [e, idts, b] =
   272       let
   273         val eq = Syntax.const "op =" $ Bound (nvars idts) $ e;
   274         val P = Syntax.const "op &" $ eq $ b;
   275         val exP = ex_tr [idts, P];
   276       in Syntax.const "Collect" $ Term.absdummy (dummyT, exP) end;
   277 
   278   in [("@SetCompr", setcompr_tr)] end;
   279 *}
   280 
   281 (* To avoid eta-contraction of body: *)
   282 print_translation {*
   283 let
   284   fun btr' syn [A,Abs abs] =
   285     let val (x,t) = atomic_abs_tr' abs
   286     in Syntax.const syn $ x $ A $ t end
   287 in
   288 [("Ball", btr' "_Ball"),("Bex", btr' "_Bex"),
   289  ("UNION", btr' "@UNION"),("INTER", btr' "@INTER")]
   290 end
   291 *}
   292 
   293 print_translation {*
   294 let
   295   val ex_tr' = snd (mk_binder_tr' ("Ex", "DUMMY"));
   296 
   297   fun setcompr_tr' [Abs (abs as (_, _, P))] =
   298     let
   299       fun check (Const ("Ex", _) $ Abs (_, _, P), n) = check (P, n + 1)
   300         | check (Const ("op &", _) $ (Const ("op =", _) $ Bound m $ e) $ P, n) =
   301             n > 0 andalso m = n andalso not (loose_bvar1 (P, n)) andalso
   302             ((0 upto (n - 1)) subset add_loose_bnos (e, 0, []))
   303         | check _ = false
   304 
   305         fun tr' (_ $ abs) =
   306           let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' [abs]
   307           in Syntax.const "@SetCompr" $ e $ idts $ Q end;
   308     in if check (P, 0) then tr' P
   309        else let val (x as _ $ Free(xN,_), t) = atomic_abs_tr' abs
   310                 val M = Syntax.const "@Coll" $ x $ t
   311             in case t of
   312                  Const("op &",_)
   313                    $ (Const("op :",_) $ (Const("_bound",_) $ Free(yN,_)) $ A)
   314                    $ P =>
   315                    if xN=yN then Syntax.const "@Collect" $ x $ A $ P else M
   316                | _ => M
   317             end
   318     end;
   319   in [("Collect", setcompr_tr')] end;
   320 *}
   321 
   322 
   323 subsection {* Rules and definitions *}
   324 
   325 text {* Isomorphisms between predicates and sets. *}
   326 
   327 axioms
   328   mem_Collect_eq: "(a : {x. P(x)}) = P(a)"
   329   Collect_mem_eq: "{x. x:A} = A"
   330 finalconsts
   331   Collect
   332   "op :"
   333 
   334 defs
   335   Ball_def:     "Ball A P       == ALL x. x:A --> P(x)"
   336   Bex_def:      "Bex A P        == EX x. x:A & P(x)"
   337   Bex1_def:     "Bex1 A P       == EX! x. x:A & P(x)"
   338 
   339 instance set :: (type) minus
   340   Compl_def:    "- A            == {x. ~x:A}"
   341   set_diff_def: "A - B          == {x. x:A & ~x:B}" ..
   342 
   343 defs
   344   Un_def:       "A Un B         == {x. x:A | x:B}"
   345   Int_def:      "A Int B        == {x. x:A & x:B}"
   346   INTER_def:    "INTER A B      == {y. ALL x:A. y: B(x)}"
   347   UNION_def:    "UNION A B      == {y. EX x:A. y: B(x)}"
   348   Inter_def:    "Inter S        == (INT x:S. x)"
   349   Union_def:    "Union S        == (UN x:S. x)"
   350   Pow_def:      "Pow A          == {B. B <= A}"
   351   empty_def:    "{}             == {x. False}"
   352   UNIV_def:     "UNIV           == {x. True}"
   353   insert_def:   "insert a B     == {x. x=a} Un B"
   354   image_def:    "f`A            == {y. EX x:A. y = f(x)}"
   355 
   356 
   357 subsection {* Lemmas and proof tool setup *}
   358 
   359 subsubsection {* Relating predicates and sets *}
   360 
   361 declare mem_Collect_eq [iff]  Collect_mem_eq [simp]
   362 
   363 lemma CollectI: "P(a) ==> a : {x. P(x)}"
   364   by simp
   365 
   366 lemma CollectD: "a : {x. P(x)} ==> P(a)"
   367   by simp
   368 
   369 lemma Collect_cong: "(!!x. P x = Q x) ==> {x. P(x)} = {x. Q(x)}"
   370   by simp
   371 
   372 lemmas CollectE = CollectD [elim_format]
   373 
   374 
   375 subsubsection {* Bounded quantifiers *}
   376 
   377 lemma ballI [intro!]: "(!!x. x:A ==> P x) ==> ALL x:A. P x"
   378   by (simp add: Ball_def)
   379 
   380 lemmas strip = impI allI ballI
   381 
   382 lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x"
   383   by (simp add: Ball_def)
   384 
   385 lemma ballE [elim]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q"
   386   by (unfold Ball_def) blast
   387 ML {* bind_thm("rev_ballE",permute_prems 1 1 (thm "ballE")) *}
   388 
   389 text {*
   390   \medskip This tactic takes assumptions @{prop "ALL x:A. P x"} and
   391   @{prop "a:A"}; creates assumption @{prop "P a"}.
   392 *}
   393 
   394 ML {*
   395   local val ballE = thm "ballE"
   396   in fun ball_tac i = etac ballE i THEN contr_tac (i + 1) end;
   397 *}
   398 
   399 text {*
   400   Gives better instantiation for bound:
   401 *}
   402 
   403 ML_setup {*
   404   change_claset (fn cs => cs addbefore ("bspec", datac (thm "bspec") 1));
   405 *}
   406 
   407 lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x"
   408   -- {* Normally the best argument order: @{prop "P x"} constrains the
   409     choice of @{prop "x:A"}. *}
   410   by (unfold Bex_def) blast
   411 
   412 lemma rev_bexI [intro?]: "x:A ==> P x ==> EX x:A. P x"
   413   -- {* The best argument order when there is only one @{prop "x:A"}. *}
   414   by (unfold Bex_def) blast
   415 
   416 lemma bexCI: "(ALL x:A. ~P x ==> P a) ==> a:A ==> EX x:A. P x"
   417   by (unfold Bex_def) blast
   418 
   419 lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q"
   420   by (unfold Bex_def) blast
   421 
   422 lemma ball_triv [simp]: "(ALL x:A. P) = ((EX x. x:A) --> P)"
   423   -- {* Trival rewrite rule. *}
   424   by (simp add: Ball_def)
   425 
   426 lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)"
   427   -- {* Dual form for existentials. *}
   428   by (simp add: Bex_def)
   429 
   430 lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)"
   431   by blast
   432 
   433 lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)"
   434   by blast
   435 
   436 lemma bex_one_point1 [simp]: "(EX x:A. x = a & P x) = (a:A & P a)"
   437   by blast
   438 
   439 lemma bex_one_point2 [simp]: "(EX x:A. a = x & P x) = (a:A & P a)"
   440   by blast
   441 
   442 lemma ball_one_point1 [simp]: "(ALL x:A. x = a --> P x) = (a:A --> P a)"
   443   by blast
   444 
   445 lemma ball_one_point2 [simp]: "(ALL x:A. a = x --> P x) = (a:A --> P a)"
   446   by blast
   447 
   448 ML_setup {*
   449   local
   450     val unfold_bex_tac = unfold_tac [thm "Bex_def"];
   451     fun prove_bex_tac ss = unfold_bex_tac ss THEN Quantifier1.prove_one_point_ex_tac;
   452     val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac;
   453 
   454     val unfold_ball_tac = unfold_tac [thm "Ball_def"];
   455     fun prove_ball_tac ss = unfold_ball_tac ss THEN Quantifier1.prove_one_point_all_tac;
   456     val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac;
   457   in
   458     val defBEX_regroup = Simplifier.simproc (the_context ())
   459       "defined BEX" ["EX x:A. P x & Q x"] rearrange_bex;
   460     val defBALL_regroup = Simplifier.simproc (the_context ())
   461       "defined BALL" ["ALL x:A. P x --> Q x"] rearrange_ball;
   462   end;
   463 
   464   Addsimprocs [defBALL_regroup, defBEX_regroup];
   465 *}
   466 
   467 
   468 subsubsection {* Congruence rules *}
   469 
   470 lemma ball_cong:
   471   "A = B ==> (!!x. x:B ==> P x = Q x) ==>
   472     (ALL x:A. P x) = (ALL x:B. Q x)"
   473   by (simp add: Ball_def)
   474 
   475 lemma strong_ball_cong [cong]:
   476   "A = B ==> (!!x. x:B =simp=> P x = Q x) ==>
   477     (ALL x:A. P x) = (ALL x:B. Q x)"
   478   by (simp add: simp_implies_def Ball_def)
   479 
   480 lemma bex_cong:
   481   "A = B ==> (!!x. x:B ==> P x = Q x) ==>
   482     (EX x:A. P x) = (EX x:B. Q x)"
   483   by (simp add: Bex_def cong: conj_cong)
   484 
   485 lemma strong_bex_cong [cong]:
   486   "A = B ==> (!!x. x:B =simp=> P x = Q x) ==>
   487     (EX x:A. P x) = (EX x:B. Q x)"
   488   by (simp add: simp_implies_def Bex_def cong: conj_cong)
   489 
   490 
   491 subsubsection {* Subsets *}
   492 
   493 lemma subsetI [atp,intro!]: "(!!x. x:A ==> x:B) ==> A \<subseteq> B"
   494   by (simp add: subset_def)
   495 
   496 text {*
   497   \medskip Map the type @{text "'a set => anything"} to just @{typ
   498   'a}; for overloading constants whose first argument has type @{typ
   499   "'a set"}.
   500 *}
   501 
   502 lemma subsetD [elim]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B"
   503   -- {* Rule in Modus Ponens style. *}
   504   by (unfold subset_def) blast
   505 
   506 declare subsetD [intro?] -- FIXME
   507 
   508 lemma rev_subsetD: "c \<in> A ==> A \<subseteq> B ==> c \<in> B"
   509   -- {* The same, with reversed premises for use with @{text erule} --
   510       cf @{text rev_mp}. *}
   511   by (rule subsetD)
   512 
   513 declare rev_subsetD [intro?] -- FIXME
   514 
   515 text {*
   516   \medskip Converts @{prop "A \<subseteq> B"} to @{prop "x \<in> A ==> x \<in> B"}.
   517 *}
   518 
   519 ML {*
   520   local val rev_subsetD = thm "rev_subsetD"
   521   in fun impOfSubs th = th RSN (2, rev_subsetD) end;
   522 *}
   523 
   524 lemma subsetCE [elim]: "A \<subseteq>  B ==> (c \<notin> A ==> P) ==> (c \<in> B ==> P) ==> P"
   525   -- {* Classical elimination rule. *}
   526   by (unfold subset_def) blast
   527 
   528 text {*
   529   \medskip Takes assumptions @{prop "A \<subseteq> B"}; @{prop "c \<in> A"} and
   530   creates the assumption @{prop "c \<in> B"}.
   531 *}
   532 
   533 ML {*
   534   local val subsetCE = thm "subsetCE"
   535   in fun set_mp_tac i = etac subsetCE i THEN mp_tac i end;
   536 *}
   537 
   538 lemma contra_subsetD: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A"
   539   by blast
   540 
   541 lemma subset_refl [simp,atp]: "A \<subseteq> A"
   542   by fast
   543 
   544 lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C"
   545   by blast
   546 
   547 
   548 subsubsection {* Equality *}
   549 
   550 lemma set_ext: assumes prem: "(!!x. (x:A) = (x:B))" shows "A = B"
   551   apply (rule prem [THEN ext, THEN arg_cong, THEN box_equals])
   552    apply (rule Collect_mem_eq)
   553   apply (rule Collect_mem_eq)
   554   done
   555 
   556 (* Due to Brian Huffman *)
   557 lemma expand_set_eq: "(A = B) = (ALL x. (x:A) = (x:B))"
   558 by(auto intro:set_ext)
   559 
   560 lemma subset_antisym [intro!]: "A \<subseteq> B ==> B \<subseteq> A ==> A = B"
   561   -- {* Anti-symmetry of the subset relation. *}
   562   by (iprover intro: set_ext subsetD)
   563 
   564 lemmas equalityI [intro!] = subset_antisym
   565 
   566 text {*
   567   \medskip Equality rules from ZF set theory -- are they appropriate
   568   here?
   569 *}
   570 
   571 lemma equalityD1: "A = B ==> A \<subseteq> B"
   572   by (simp add: subset_refl)
   573 
   574 lemma equalityD2: "A = B ==> B \<subseteq> A"
   575   by (simp add: subset_refl)
   576 
   577 text {*
   578   \medskip Be careful when adding this to the claset as @{text
   579   subset_empty} is in the simpset: @{prop "A = {}"} goes to @{prop "{}
   580   \<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}!
   581 *}
   582 
   583 lemma equalityE: "A = B ==> (A \<subseteq> B ==> B \<subseteq> A ==> P) ==> P"
   584   by (simp add: subset_refl)
   585 
   586 lemma equalityCE [elim]:
   587     "A = B ==> (c \<in> A ==> c \<in> B ==> P) ==> (c \<notin> A ==> c \<notin> B ==> P) ==> P"
   588   by blast
   589 
   590 lemma eqset_imp_iff: "A = B ==> (x : A) = (x : B)"
   591   by simp
   592 
   593 lemma eqelem_imp_iff: "x = y ==> (x : A) = (y : A)"
   594   by simp
   595 
   596 
   597 subsubsection {* The universal set -- UNIV *}
   598 
   599 lemma UNIV_I [simp]: "x : UNIV"
   600   by (simp add: UNIV_def)
   601 
   602 declare UNIV_I [intro]  -- {* unsafe makes it less likely to cause problems *}
   603 
   604 lemma UNIV_witness [intro?]: "EX x. x : UNIV"
   605   by simp
   606 
   607 lemma subset_UNIV [simp]: "A \<subseteq> UNIV"
   608   by (rule subsetI) (rule UNIV_I)
   609 
   610 text {*
   611   \medskip Eta-contracting these two rules (to remove @{text P})
   612   causes them to be ignored because of their interaction with
   613   congruence rules.
   614 *}
   615 
   616 lemma ball_UNIV [simp]: "Ball UNIV P = All P"
   617   by (simp add: Ball_def)
   618 
   619 lemma bex_UNIV [simp]: "Bex UNIV P = Ex P"
   620   by (simp add: Bex_def)
   621 
   622 
   623 subsubsection {* The empty set *}
   624 
   625 lemma empty_iff [simp]: "(c : {}) = False"
   626   by (simp add: empty_def)
   627 
   628 lemma emptyE [elim!]: "a : {} ==> P"
   629   by simp
   630 
   631 lemma empty_subsetI [iff]: "{} \<subseteq> A"
   632     -- {* One effect is to delete the ASSUMPTION @{prop "{} <= A"} *}
   633   by blast
   634 
   635 lemma equals0I: "(!!y. y \<in> A ==> False) ==> A = {}"
   636   by blast
   637 
   638 lemma equals0D: "A = {} ==> a \<notin> A"
   639     -- {* Use for reasoning about disjointness: @{prop "A Int B = {}"} *}
   640   by blast
   641 
   642 lemma ball_empty [simp]: "Ball {} P = True"
   643   by (simp add: Ball_def)
   644 
   645 lemma bex_empty [simp]: "Bex {} P = False"
   646   by (simp add: Bex_def)
   647 
   648 lemma UNIV_not_empty [iff]: "UNIV ~= {}"
   649   by (blast elim: equalityE)
   650 
   651 
   652 subsubsection {* The Powerset operator -- Pow *}
   653 
   654 lemma Pow_iff [iff]: "(A \<in> Pow B) = (A \<subseteq> B)"
   655   by (simp add: Pow_def)
   656 
   657 lemma PowI: "A \<subseteq> B ==> A \<in> Pow B"
   658   by (simp add: Pow_def)
   659 
   660 lemma PowD: "A \<in> Pow B ==> A \<subseteq> B"
   661   by (simp add: Pow_def)
   662 
   663 lemma Pow_bottom: "{} \<in> Pow B"
   664   by simp
   665 
   666 lemma Pow_top: "A \<in> Pow A"
   667   by (simp add: subset_refl)
   668 
   669 
   670 subsubsection {* Set complement *}
   671 
   672 lemma Compl_iff [simp]: "(c \<in> -A) = (c \<notin> A)"
   673   by (unfold Compl_def) blast
   674 
   675 lemma ComplI [intro!]: "(c \<in> A ==> False) ==> c \<in> -A"
   676   by (unfold Compl_def) blast
   677 
   678 text {*
   679   \medskip This form, with negated conclusion, works well with the
   680   Classical prover.  Negated assumptions behave like formulae on the
   681   right side of the notional turnstile ... *}
   682 
   683 lemma ComplD [dest!]: "c : -A ==> c~:A"
   684   by (unfold Compl_def) blast
   685 
   686 lemmas ComplE = ComplD [elim_format]
   687 
   688 
   689 subsubsection {* Binary union -- Un *}
   690 
   691 lemma Un_iff [simp]: "(c : A Un B) = (c:A | c:B)"
   692   by (unfold Un_def) blast
   693 
   694 lemma UnI1 [elim?]: "c:A ==> c : A Un B"
   695   by simp
   696 
   697 lemma UnI2 [elim?]: "c:B ==> c : A Un B"
   698   by simp
   699 
   700 text {*
   701   \medskip Classical introduction rule: no commitment to @{prop A} vs
   702   @{prop B}.
   703 *}
   704 
   705 lemma UnCI [intro!]: "(c~:B ==> c:A) ==> c : A Un B"
   706   by auto
   707 
   708 lemma UnE [elim!]: "c : A Un B ==> (c:A ==> P) ==> (c:B ==> P) ==> P"
   709   by (unfold Un_def) blast
   710 
   711 
   712 subsubsection {* Binary intersection -- Int *}
   713 
   714 lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)"
   715   by (unfold Int_def) blast
   716 
   717 lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B"
   718   by simp
   719 
   720 lemma IntD1: "c : A Int B ==> c:A"
   721   by simp
   722 
   723 lemma IntD2: "c : A Int B ==> c:B"
   724   by simp
   725 
   726 lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P"
   727   by simp
   728 
   729 
   730 subsubsection {* Set difference *}
   731 
   732 lemma Diff_iff [simp]: "(c : A - B) = (c:A & c~:B)"
   733   by (unfold set_diff_def) blast
   734 
   735 lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A - B"
   736   by simp
   737 
   738 lemma DiffD1: "c : A - B ==> c : A"
   739   by simp
   740 
   741 lemma DiffD2: "c : A - B ==> c : B ==> P"
   742   by simp
   743 
   744 lemma DiffE [elim!]: "c : A - B ==> (c:A ==> c~:B ==> P) ==> P"
   745   by simp
   746 
   747 
   748 subsubsection {* Augmenting a set -- insert *}
   749 
   750 lemma insert_iff [simp]: "(a : insert b A) = (a = b | a:A)"
   751   by (unfold insert_def) blast
   752 
   753 lemma insertI1: "a : insert a B"
   754   by simp
   755 
   756 lemma insertI2: "a : B ==> a : insert b B"
   757   by simp
   758 
   759 lemma insertE [elim!]: "a : insert b A ==> (a = b ==> P) ==> (a:A ==> P) ==> P"
   760   by (unfold insert_def) blast
   761 
   762 lemma insertCI [intro!]: "(a~:B ==> a = b) ==> a: insert b B"
   763   -- {* Classical introduction rule. *}
   764   by auto
   765 
   766 lemma subset_insert_iff: "(A \<subseteq> insert x B) = (if x:A then A - {x} \<subseteq> B else A \<subseteq> B)"
   767   by auto
   768 
   769 
   770 subsubsection {* Singletons, using insert *}
   771 
   772 lemma singletonI [intro!]: "a : {a}"
   773     -- {* Redundant? But unlike @{text insertCI}, it proves the subgoal immediately! *}
   774   by (rule insertI1)
   775 
   776 lemma singletonD [dest!]: "b : {a} ==> b = a"
   777   by blast
   778 
   779 lemmas singletonE = singletonD [elim_format]
   780 
   781 lemma singleton_iff: "(b : {a}) = (b = a)"
   782   by blast
   783 
   784 lemma singleton_inject [dest!]: "{a} = {b} ==> a = b"
   785   by blast
   786 
   787 lemma singleton_insert_inj_eq [iff]: "({b} = insert a A) = (a = b & A \<subseteq> {b})"
   788   by blast
   789 
   790 lemma singleton_insert_inj_eq' [iff]: "(insert a A = {b}) = (a = b & A \<subseteq> {b})"
   791   by blast
   792 
   793 lemma subset_singletonD: "A \<subseteq> {x} ==> A = {} | A = {x}"
   794   by fast
   795 
   796 lemma singleton_conv [simp]: "{x. x = a} = {a}"
   797   by blast
   798 
   799 lemma singleton_conv2 [simp]: "{x. a = x} = {a}"
   800   by blast
   801 
   802 lemma diff_single_insert: "A - {x} \<subseteq> B ==> x \<in> A ==> A \<subseteq> insert x B"
   803   by blast
   804 
   805 lemma doubleton_eq_iff: "({a,b} = {c,d}) = (a=c & b=d | a=d & b=c)"
   806   by (blast elim: equalityE)
   807 
   808 
   809 subsubsection {* Unions of families *}
   810 
   811 text {*
   812   @{term [source] "UN x:A. B x"} is @{term "Union (B`A)"}.
   813 *}
   814 
   815 lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)"
   816   by (unfold UNION_def) blast
   817 
   818 lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)"
   819   -- {* The order of the premises presupposes that @{term A} is rigid;
   820     @{term b} may be flexible. *}
   821   by auto
   822 
   823 lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R"
   824   by (unfold UNION_def) blast
   825 
   826 lemma UN_cong [cong]:
   827     "A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
   828   by (simp add: UNION_def)
   829 
   830 
   831 subsubsection {* Intersections of families *}
   832 
   833 text {* @{term [source] "INT x:A. B x"} is @{term "Inter (B`A)"}. *}
   834 
   835 lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)"
   836   by (unfold INTER_def) blast
   837 
   838 lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)"
   839   by (unfold INTER_def) blast
   840 
   841 lemma INT_D [elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a"
   842   by auto
   843 
   844 lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R"
   845   -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}
   846   by (unfold INTER_def) blast
   847 
   848 lemma INT_cong [cong]:
   849     "A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)"
   850   by (simp add: INTER_def)
   851 
   852 
   853 subsubsection {* Union *}
   854 
   855 lemma Union_iff [simp]: "(A : Union C) = (EX X:C. A:X)"
   856   by (unfold Union_def) blast
   857 
   858 lemma UnionI [intro]: "X:C ==> A:X ==> A : Union C"
   859   -- {* The order of the premises presupposes that @{term C} is rigid;
   860     @{term A} may be flexible. *}
   861   by auto
   862 
   863 lemma UnionE [elim!]: "A : Union C ==> (!!X. A:X ==> X:C ==> R) ==> R"
   864   by (unfold Union_def) blast
   865 
   866 
   867 subsubsection {* Inter *}
   868 
   869 lemma Inter_iff [simp]: "(A : Inter C) = (ALL X:C. A:X)"
   870   by (unfold Inter_def) blast
   871 
   872 lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C"
   873   by (simp add: Inter_def)
   874 
   875 text {*
   876   \medskip A ``destruct'' rule -- every @{term X} in @{term C}
   877   contains @{term A} as an element, but @{prop "A:X"} can hold when
   878   @{prop "X:C"} does not!  This rule is analogous to @{text spec}.
   879 *}
   880 
   881 lemma InterD [elim]: "A : Inter C ==> X:C ==> A:X"
   882   by auto
   883 
   884 lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R"
   885   -- {* ``Classical'' elimination rule -- does not require proving
   886     @{prop "X:C"}. *}
   887   by (unfold Inter_def) blast
   888 
   889 text {*
   890   \medskip Image of a set under a function.  Frequently @{term b} does
   891   not have the syntactic form of @{term "f x"}.
   892 *}
   893 
   894 lemma image_eqI [simp, intro]: "b = f x ==> x:A ==> b : f`A"
   895   by (unfold image_def) blast
   896 
   897 lemma imageI: "x : A ==> f x : f ` A"
   898   by (rule image_eqI) (rule refl)
   899 
   900 lemma rev_image_eqI: "x:A ==> b = f x ==> b : f`A"
   901   -- {* This version's more effective when we already have the
   902     required @{term x}. *}
   903   by (unfold image_def) blast
   904 
   905 lemma imageE [elim!]:
   906   "b : (%x. f x)`A ==> (!!x. b = f x ==> x:A ==> P) ==> P"
   907   -- {* The eta-expansion gives variable-name preservation. *}
   908   by (unfold image_def) blast
   909 
   910 lemma image_Un: "f`(A Un B) = f`A Un f`B"
   911   by blast
   912 
   913 lemma image_iff: "(z : f`A) = (EX x:A. z = f x)"
   914   by blast
   915 
   916 lemma image_subset_iff: "(f`A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)"
   917   -- {* This rewrite rule would confuse users if made default. *}
   918   by blast
   919 
   920 lemma subset_image_iff: "(B \<subseteq> f`A) = (EX AA. AA \<subseteq> A & B = f`AA)"
   921   apply safe
   922    prefer 2 apply fast
   923   apply (rule_tac x = "{a. a : A & f a : B}" in exI, fast)
   924   done
   925 
   926 lemma image_subsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f`A \<subseteq> B"
   927   -- {* Replaces the three steps @{text subsetI}, @{text imageE},
   928     @{text hypsubst}, but breaks too many existing proofs. *}
   929   by blast
   930 
   931 text {*
   932   \medskip Range of a function -- just a translation for image!
   933 *}
   934 
   935 lemma range_eqI: "b = f x ==> b \<in> range f"
   936   by simp
   937 
   938 lemma rangeI: "f x \<in> range f"
   939   by simp
   940 
   941 lemma rangeE [elim?]: "b \<in> range (\<lambda>x. f x) ==> (!!x. b = f x ==> P) ==> P"
   942   by blast
   943 
   944 
   945 subsubsection {* Set reasoning tools *}
   946 
   947 text {*
   948   Rewrite rules for boolean case-splitting: faster than @{text
   949   "split_if [split]"}.
   950 *}
   951 
   952 lemma split_if_eq1: "((if Q then x else y) = b) = ((Q --> x = b) & (~ Q --> y = b))"
   953   by (rule split_if)
   954 
   955 lemma split_if_eq2: "(a = (if Q then x else y)) = ((Q --> a = x) & (~ Q --> a = y))"
   956   by (rule split_if)
   957 
   958 text {*
   959   Split ifs on either side of the membership relation.  Not for @{text
   960   "[simp]"} -- can cause goals to blow up!
   961 *}
   962 
   963 lemma split_if_mem1: "((if Q then x else y) : b) = ((Q --> x : b) & (~ Q --> y : b))"
   964   by (rule split_if)
   965 
   966 lemma split_if_mem2: "(a : (if Q then x else y)) = ((Q --> a : x) & (~ Q --> a : y))"
   967   by (rule split_if)
   968 
   969 lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2
   970 
   971 lemmas mem_simps =
   972   insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
   973   mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
   974   -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
   975 
   976 (*Would like to add these, but the existing code only searches for the
   977   outer-level constant, which in this case is just "op :"; we instead need
   978   to use term-nets to associate patterns with rules.  Also, if a rule fails to
   979   apply, then the formula should be kept.
   980   [("HOL.uminus", Compl_iff RS iffD1), ("HOL.minus", [Diff_iff RS iffD1]),
   981    ("op Int", [IntD1,IntD2]),
   982    ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]
   983  *)
   984 
   985 ML_setup {*
   986   val mksimps_pairs = [("Ball", [thm "bspec"])] @ mksimps_pairs;
   987   change_simpset (fn ss => ss setmksimps (mksimps mksimps_pairs));
   988 *}
   989 
   990 
   991 subsubsection {* The ``proper subset'' relation *}
   992 
   993 lemma psubsetI [intro!]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B"
   994   by (unfold psubset_def) blast
   995 
   996 lemma psubsetE [elim!]: 
   997     "[|A \<subset> B;  [|A \<subseteq> B; ~ (B\<subseteq>A)|] ==> R|] ==> R"
   998   by (unfold psubset_def) blast
   999 
  1000 lemma psubset_insert_iff:
  1001   "(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)"
  1002   by (auto simp add: psubset_def subset_insert_iff)
  1003 
  1004 lemma psubset_eq: "(A \<subset> B) = (A \<subseteq> B & A \<noteq> B)"
  1005   by (simp only: psubset_def)
  1006 
  1007 lemma psubset_imp_subset: "A \<subset> B ==> A \<subseteq> B"
  1008   by (simp add: psubset_eq)
  1009 
  1010 lemma psubset_trans: "[| A \<subset> B; B \<subset> C |] ==> A \<subset> C"
  1011 apply (unfold psubset_def)
  1012 apply (auto dest: subset_antisym)
  1013 done
  1014 
  1015 lemma psubsetD: "[| A \<subset> B; c \<in> A |] ==> c \<in> B"
  1016 apply (unfold psubset_def)
  1017 apply (auto dest: subsetD)
  1018 done
  1019 
  1020 lemma psubset_subset_trans: "A \<subset> B ==> B \<subseteq> C ==> A \<subset> C"
  1021   by (auto simp add: psubset_eq)
  1022 
  1023 lemma subset_psubset_trans: "A \<subseteq> B ==> B \<subset> C ==> A \<subset> C"
  1024   by (auto simp add: psubset_eq)
  1025 
  1026 lemma psubset_imp_ex_mem: "A \<subset> B ==> \<exists>b. b \<in> (B - A)"
  1027   by (unfold psubset_def) blast
  1028 
  1029 lemma atomize_ball:
  1030     "(!!x. x \<in> A ==> P x) == Trueprop (\<forall>x\<in>A. P x)"
  1031   by (simp only: Ball_def atomize_all atomize_imp)
  1032 
  1033 lemmas [symmetric, rulify] = atomize_ball
  1034   and [symmetric, defn] = atomize_ball
  1035 
  1036 
  1037 subsection {* Further set-theory lemmas *}
  1038 
  1039 instance set :: (type) order
  1040   by (intro_classes,
  1041       (assumption | rule subset_refl subset_trans subset_antisym psubset_eq)+)
  1042 
  1043 subsubsection {* Derived rules involving subsets. *}
  1044 
  1045 text {* @{text insert}. *}
  1046 
  1047 lemma subset_insertI: "B \<subseteq> insert a B"
  1048   apply (rule subsetI)
  1049   apply (erule insertI2)
  1050   done
  1051 
  1052 lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B"
  1053 by blast
  1054 
  1055 lemma subset_insert: "x \<notin> A ==> (A \<subseteq> insert x B) = (A \<subseteq> B)"
  1056   by blast
  1057 
  1058 
  1059 text {* \medskip Big Union -- least upper bound of a set. *}
  1060 
  1061 lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A"
  1062   by (iprover intro: subsetI UnionI)
  1063 
  1064 lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C"
  1065   by (iprover intro: subsetI elim: UnionE dest: subsetD)
  1066 
  1067 
  1068 text {* \medskip General union. *}
  1069 
  1070 lemma UN_upper: "a \<in> A ==> B a \<subseteq> (\<Union>x\<in>A. B x)"
  1071   by blast
  1072 
  1073 lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C"
  1074   by (iprover intro: subsetI elim: UN_E dest: subsetD)
  1075 
  1076 
  1077 text {* \medskip Big Intersection -- greatest lower bound of a set. *}
  1078 
  1079 lemma Inter_lower: "B \<in> A ==> Inter A \<subseteq> B"
  1080   by blast
  1081 
  1082 lemma Inter_subset:
  1083   "[| !!X. X \<in> A ==> X \<subseteq> B; A ~= {} |] ==> \<Inter>A \<subseteq> B"
  1084   by blast
  1085 
  1086 lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A"
  1087   by (iprover intro: InterI subsetI dest: subsetD)
  1088 
  1089 lemma INT_lower: "a \<in> A ==> (\<Inter>x\<in>A. B x) \<subseteq> B a"
  1090   by blast
  1091 
  1092 lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)"
  1093   by (iprover intro: INT_I subsetI dest: subsetD)
  1094 
  1095 
  1096 text {* \medskip Finite Union -- the least upper bound of two sets. *}
  1097 
  1098 lemma Un_upper1: "A \<subseteq> A \<union> B"
  1099   by blast
  1100 
  1101 lemma Un_upper2: "B \<subseteq> A \<union> B"
  1102   by blast
  1103 
  1104 lemma Un_least: "A \<subseteq> C ==> B \<subseteq> C ==> A \<union> B \<subseteq> C"
  1105   by blast
  1106 
  1107 
  1108 text {* \medskip Finite Intersection -- the greatest lower bound of two sets. *}
  1109 
  1110 lemma Int_lower1: "A \<inter> B \<subseteq> A"
  1111   by blast
  1112 
  1113 lemma Int_lower2: "A \<inter> B \<subseteq> B"
  1114   by blast
  1115 
  1116 lemma Int_greatest: "C \<subseteq> A ==> C \<subseteq> B ==> C \<subseteq> A \<inter> B"
  1117   by blast
  1118 
  1119 
  1120 text {* \medskip Set difference. *}
  1121 
  1122 lemma Diff_subset: "A - B \<subseteq> A"
  1123   by blast
  1124 
  1125 lemma Diff_subset_conv: "(A - B \<subseteq> C) = (A \<subseteq> B \<union> C)"
  1126 by blast
  1127 
  1128 
  1129 text {* \medskip Monotonicity. *}
  1130 
  1131 lemma mono_Un: "mono f ==> f A \<union> f B \<subseteq> f (A \<union> B)"
  1132   by (auto simp add: mono_def)
  1133 
  1134 lemma mono_Int: "mono f ==> f (A \<inter> B) \<subseteq> f A \<inter> f B"
  1135   by (auto simp add: mono_def)
  1136 
  1137 subsubsection {* Equalities involving union, intersection, inclusion, etc. *}
  1138 
  1139 text {* @{text "{}"}. *}
  1140 
  1141 lemma Collect_const [simp]: "{s. P} = (if P then UNIV else {})"
  1142   -- {* supersedes @{text "Collect_False_empty"} *}
  1143   by auto
  1144 
  1145 lemma subset_empty [simp]: "(A \<subseteq> {}) = (A = {})"
  1146   by blast
  1147 
  1148 lemma not_psubset_empty [iff]: "\<not> (A < {})"
  1149   by (unfold psubset_def) blast
  1150 
  1151 lemma Collect_empty_eq [simp]: "(Collect P = {}) = (\<forall>x. \<not> P x)"
  1152 by blast
  1153 
  1154 lemma empty_Collect_eq [simp]: "({} = Collect P) = (\<forall>x. \<not> P x)"
  1155 by blast
  1156 
  1157 lemma Collect_neg_eq: "{x. \<not> P x} = - {x. P x}"
  1158   by blast
  1159 
  1160 lemma Collect_disj_eq: "{x. P x | Q x} = {x. P x} \<union> {x. Q x}"
  1161   by blast
  1162 
  1163 lemma Collect_imp_eq: "{x. P x --> Q x} = -{x. P x} \<union> {x. Q x}"
  1164   by blast
  1165 
  1166 lemma Collect_conj_eq: "{x. P x & Q x} = {x. P x} \<inter> {x. Q x}"
  1167   by blast
  1168 
  1169 lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
  1170   by blast
  1171 
  1172 lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
  1173   by blast
  1174 
  1175 lemma Collect_ex_eq: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
  1176   by blast
  1177 
  1178 lemma Collect_bex_eq: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
  1179   by blast
  1180 
  1181 
  1182 text {* \medskip @{text insert}. *}
  1183 
  1184 lemma insert_is_Un: "insert a A = {a} Un A"
  1185   -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a {}"} *}
  1186   by blast
  1187 
  1188 lemma insert_not_empty [simp]: "insert a A \<noteq> {}"
  1189   by blast
  1190 
  1191 lemmas empty_not_insert = insert_not_empty [symmetric, standard]
  1192 declare empty_not_insert [simp]
  1193 
  1194 lemma insert_absorb: "a \<in> A ==> insert a A = A"
  1195   -- {* @{text "[simp]"} causes recursive calls when there are nested inserts *}
  1196   -- {* with \emph{quadratic} running time *}
  1197   by blast
  1198 
  1199 lemma insert_absorb2 [simp]: "insert x (insert x A) = insert x A"
  1200   by blast
  1201 
  1202 lemma insert_commute: "insert x (insert y A) = insert y (insert x A)"
  1203   by blast
  1204 
  1205 lemma insert_subset [simp]: "(insert x A \<subseteq> B) = (x \<in> B & A \<subseteq> B)"
  1206   by blast
  1207 
  1208 lemma mk_disjoint_insert: "a \<in> A ==> \<exists>B. A = insert a B & a \<notin> B"
  1209   -- {* use new @{text B} rather than @{text "A - {a}"} to avoid infinite unfolding *}
  1210   apply (rule_tac x = "A - {a}" in exI, blast)
  1211   done
  1212 
  1213 lemma insert_Collect: "insert a (Collect P) = {u. u \<noteq> a --> P u}"
  1214   by auto
  1215 
  1216 lemma UN_insert_distrib: "u \<in> A ==> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
  1217   by blast
  1218 
  1219 lemma insert_inter_insert[simp]: "insert a A \<inter> insert a B = insert a (A \<inter> B)"
  1220   by blast
  1221 
  1222 lemma insert_disjoint[simp]:
  1223  "(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})"
  1224  "({} = insert a A \<inter> B) = (a \<notin> B \<and> {} = A \<inter> B)"
  1225   by auto
  1226 
  1227 lemma disjoint_insert[simp]:
  1228  "(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})"
  1229  "({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)"
  1230   by auto
  1231 
  1232 text {* \medskip @{text image}. *}
  1233 
  1234 lemma image_empty [simp]: "f`{} = {}"
  1235   by blast
  1236 
  1237 lemma image_insert [simp]: "f ` insert a B = insert (f a) (f`B)"
  1238   by blast
  1239 
  1240 lemma image_constant: "x \<in> A ==> (\<lambda>x. c) ` A = {c}"
  1241   by auto
  1242 
  1243 lemma image_constant_conv: "(%x. c) ` A = (if A = {} then {} else {c})"
  1244 by auto
  1245 
  1246 lemma image_image: "f ` (g ` A) = (\<lambda>x. f (g x)) ` A"
  1247   by blast
  1248 
  1249 lemma insert_image [simp]: "x \<in> A ==> insert (f x) (f`A) = f`A"
  1250   by blast
  1251 
  1252 lemma image_is_empty [iff]: "(f`A = {}) = (A = {})"
  1253   by blast
  1254 
  1255 
  1256 lemma image_Collect: "f ` {x. P x} = {f x | x. P x}"
  1257   -- {* NOT suitable as a default simprule: the RHS isn't simpler than the LHS,
  1258       with its implicit quantifier and conjunction.  Also image enjoys better
  1259       equational properties than does the RHS. *}
  1260   by blast
  1261 
  1262 lemma if_image_distrib [simp]:
  1263   "(\<lambda>x. if P x then f x else g x) ` S
  1264     = (f ` (S \<inter> {x. P x})) \<union> (g ` (S \<inter> {x. \<not> P x}))"
  1265   by (auto simp add: image_def)
  1266 
  1267 lemma image_cong: "M = N ==> (!!x. x \<in> N ==> f x = g x) ==> f`M = g`N"
  1268   by (simp add: image_def)
  1269 
  1270 
  1271 text {* \medskip @{text range}. *}
  1272 
  1273 lemma full_SetCompr_eq: "{u. \<exists>x. u = f x} = range f"
  1274   by auto
  1275 
  1276 lemma range_composition [simp]: "range (\<lambda>x. f (g x)) = f`range g"
  1277 by (subst image_image, simp)
  1278 
  1279 
  1280 text {* \medskip @{text Int} *}
  1281 
  1282 lemma Int_absorb [simp]: "A \<inter> A = A"
  1283   by blast
  1284 
  1285 lemma Int_left_absorb: "A \<inter> (A \<inter> B) = A \<inter> B"
  1286   by blast
  1287 
  1288 lemma Int_commute: "A \<inter> B = B \<inter> A"
  1289   by blast
  1290 
  1291 lemma Int_left_commute: "A \<inter> (B \<inter> C) = B \<inter> (A \<inter> C)"
  1292   by blast
  1293 
  1294 lemma Int_assoc: "(A \<inter> B) \<inter> C = A \<inter> (B \<inter> C)"
  1295   by blast
  1296 
  1297 lemmas Int_ac = Int_assoc Int_left_absorb Int_commute Int_left_commute
  1298   -- {* Intersection is an AC-operator *}
  1299 
  1300 lemma Int_absorb1: "B \<subseteq> A ==> A \<inter> B = B"
  1301   by blast
  1302 
  1303 lemma Int_absorb2: "A \<subseteq> B ==> A \<inter> B = A"
  1304   by blast
  1305 
  1306 lemma Int_empty_left [simp]: "{} \<inter> B = {}"
  1307   by blast
  1308 
  1309 lemma Int_empty_right [simp]: "A \<inter> {} = {}"
  1310   by blast
  1311 
  1312 lemma disjoint_eq_subset_Compl: "(A \<inter> B = {}) = (A \<subseteq> -B)"
  1313   by blast
  1314 
  1315 lemma disjoint_iff_not_equal: "(A \<inter> B = {}) = (\<forall>x\<in>A. \<forall>y\<in>B. x \<noteq> y)"
  1316   by blast
  1317 
  1318 lemma Int_UNIV_left [simp]: "UNIV \<inter> B = B"
  1319   by blast
  1320 
  1321 lemma Int_UNIV_right [simp]: "A \<inter> UNIV = A"
  1322   by blast
  1323 
  1324 lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"
  1325   by blast
  1326 
  1327 lemma Int_Un_distrib: "A \<inter> (B \<union> C) = (A \<inter> B) \<union> (A \<inter> C)"
  1328   by blast
  1329 
  1330 lemma Int_Un_distrib2: "(B \<union> C) \<inter> A = (B \<inter> A) \<union> (C \<inter> A)"
  1331   by blast
  1332 
  1333 lemma Int_UNIV [simp]: "(A \<inter> B = UNIV) = (A = UNIV & B = UNIV)"
  1334   by blast
  1335 
  1336 lemma Int_subset_iff [simp]: "(C \<subseteq> A \<inter> B) = (C \<subseteq> A & C \<subseteq> B)"
  1337   by blast
  1338 
  1339 lemma Int_Collect: "(x \<in> A \<inter> {x. P x}) = (x \<in> A & P x)"
  1340   by blast
  1341 
  1342 
  1343 text {* \medskip @{text Un}. *}
  1344 
  1345 lemma Un_absorb [simp]: "A \<union> A = A"
  1346   by blast
  1347 
  1348 lemma Un_left_absorb: "A \<union> (A \<union> B) = A \<union> B"
  1349   by blast
  1350 
  1351 lemma Un_commute: "A \<union> B = B \<union> A"
  1352   by blast
  1353 
  1354 lemma Un_left_commute: "A \<union> (B \<union> C) = B \<union> (A \<union> C)"
  1355   by blast
  1356 
  1357 lemma Un_assoc: "(A \<union> B) \<union> C = A \<union> (B \<union> C)"
  1358   by blast
  1359 
  1360 lemmas Un_ac = Un_assoc Un_left_absorb Un_commute Un_left_commute
  1361   -- {* Union is an AC-operator *}
  1362 
  1363 lemma Un_absorb1: "A \<subseteq> B ==> A \<union> B = B"
  1364   by blast
  1365 
  1366 lemma Un_absorb2: "B \<subseteq> A ==> A \<union> B = A"
  1367   by blast
  1368 
  1369 lemma Un_empty_left [simp]: "{} \<union> B = B"
  1370   by blast
  1371 
  1372 lemma Un_empty_right [simp]: "A \<union> {} = A"
  1373   by blast
  1374 
  1375 lemma Un_UNIV_left [simp]: "UNIV \<union> B = UNIV"
  1376   by blast
  1377 
  1378 lemma Un_UNIV_right [simp]: "A \<union> UNIV = UNIV"
  1379   by blast
  1380 
  1381 lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"
  1382   by blast
  1383 
  1384 lemma Un_insert_left [simp]: "(insert a B) \<union> C = insert a (B \<union> C)"
  1385   by blast
  1386 
  1387 lemma Un_insert_right [simp]: "A \<union> (insert a B) = insert a (A \<union> B)"
  1388   by blast
  1389 
  1390 lemma Int_insert_left:
  1391     "(insert a B) Int C = (if a \<in> C then insert a (B \<inter> C) else B \<inter> C)"
  1392   by auto
  1393 
  1394 lemma Int_insert_right:
  1395     "A \<inter> (insert a B) = (if a \<in> A then insert a (A \<inter> B) else A \<inter> B)"
  1396   by auto
  1397 
  1398 lemma Un_Int_distrib: "A \<union> (B \<inter> C) = (A \<union> B) \<inter> (A \<union> C)"
  1399   by blast
  1400 
  1401 lemma Un_Int_distrib2: "(B \<inter> C) \<union> A = (B \<union> A) \<inter> (C \<union> A)"
  1402   by blast
  1403 
  1404 lemma Un_Int_crazy:
  1405     "(A \<inter> B) \<union> (B \<inter> C) \<union> (C \<inter> A) = (A \<union> B) \<inter> (B \<union> C) \<inter> (C \<union> A)"
  1406   by blast
  1407 
  1408 lemma subset_Un_eq: "(A \<subseteq> B) = (A \<union> B = B)"
  1409   by blast
  1410 
  1411 lemma Un_empty [iff]: "(A \<union> B = {}) = (A = {} & B = {})"
  1412   by blast
  1413 
  1414 lemma Un_subset_iff [simp]: "(A \<union> B \<subseteq> C) = (A \<subseteq> C & B \<subseteq> C)"
  1415   by blast
  1416 
  1417 lemma Un_Diff_Int: "(A - B) \<union> (A \<inter> B) = A"
  1418   by blast
  1419 
  1420 
  1421 text {* \medskip Set complement *}
  1422 
  1423 lemma Compl_disjoint [simp]: "A \<inter> -A = {}"
  1424   by blast
  1425 
  1426 lemma Compl_disjoint2 [simp]: "-A \<inter> A = {}"
  1427   by blast
  1428 
  1429 lemma Compl_partition: "A \<union> -A = UNIV"
  1430   by blast
  1431 
  1432 lemma Compl_partition2: "-A \<union> A = UNIV"
  1433   by blast
  1434 
  1435 lemma double_complement [simp]: "- (-A) = (A::'a set)"
  1436   by blast
  1437 
  1438 lemma Compl_Un [simp]: "-(A \<union> B) = (-A) \<inter> (-B)"
  1439   by blast
  1440 
  1441 lemma Compl_Int [simp]: "-(A \<inter> B) = (-A) \<union> (-B)"
  1442   by blast
  1443 
  1444 lemma Compl_UN [simp]: "-(\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
  1445   by blast
  1446 
  1447 lemma Compl_INT [simp]: "-(\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
  1448   by blast
  1449 
  1450 lemma subset_Compl_self_eq: "(A \<subseteq> -A) = (A = {})"
  1451   by blast
  1452 
  1453 lemma Un_Int_assoc_eq: "((A \<inter> B) \<union> C = A \<inter> (B \<union> C)) = (C \<subseteq> A)"
  1454   -- {* Halmos, Naive Set Theory, page 16. *}
  1455   by blast
  1456 
  1457 lemma Compl_UNIV_eq [simp]: "-UNIV = {}"
  1458   by blast
  1459 
  1460 lemma Compl_empty_eq [simp]: "-{} = UNIV"
  1461   by blast
  1462 
  1463 lemma Compl_subset_Compl_iff [iff]: "(-A \<subseteq> -B) = (B \<subseteq> A)"
  1464   by blast
  1465 
  1466 lemma Compl_eq_Compl_iff [iff]: "(-A = -B) = (A = (B::'a set))"
  1467   by blast
  1468 
  1469 
  1470 text {* \medskip @{text Union}. *}
  1471 
  1472 lemma Union_empty [simp]: "Union({}) = {}"
  1473   by blast
  1474 
  1475 lemma Union_UNIV [simp]: "Union UNIV = UNIV"
  1476   by blast
  1477 
  1478 lemma Union_insert [simp]: "Union (insert a B) = a \<union> \<Union>B"
  1479   by blast
  1480 
  1481 lemma Union_Un_distrib [simp]: "\<Union>(A Un B) = \<Union>A \<union> \<Union>B"
  1482   by blast
  1483 
  1484 lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
  1485   by blast
  1486 
  1487 lemma Union_empty_conv [simp]: "(\<Union>A = {}) = (\<forall>x\<in>A. x = {})"
  1488   by blast
  1489 
  1490 lemma empty_Union_conv [simp]: "({} = \<Union>A) = (\<forall>x\<in>A. x = {})"
  1491   by blast
  1492 
  1493 lemma Union_disjoint: "(\<Union>C \<inter> A = {}) = (\<forall>B\<in>C. B \<inter> A = {})"
  1494   by blast
  1495 
  1496 
  1497 text {* \medskip @{text Inter}. *}
  1498 
  1499 lemma Inter_empty [simp]: "\<Inter>{} = UNIV"
  1500   by blast
  1501 
  1502 lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}"
  1503   by blast
  1504 
  1505 lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
  1506   by blast
  1507 
  1508 lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
  1509   by blast
  1510 
  1511 lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
  1512   by blast
  1513 
  1514 lemma Inter_UNIV_conv [simp]:
  1515   "(\<Inter>A = UNIV) = (\<forall>x\<in>A. x = UNIV)"
  1516   "(UNIV = \<Inter>A) = (\<forall>x\<in>A. x = UNIV)"
  1517   by blast+
  1518 
  1519 
  1520 text {*
  1521   \medskip @{text UN} and @{text INT}.
  1522 
  1523   Basic identities: *}
  1524 
  1525 lemma UN_empty [simp]: "(\<Union>x\<in>{}. B x) = {}"
  1526   by blast
  1527 
  1528 lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"
  1529   by blast
  1530 
  1531 lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
  1532   by blast
  1533 
  1534 lemma UN_absorb: "k \<in> I ==> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
  1535   by auto
  1536 
  1537 lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"
  1538   by blast
  1539 
  1540 lemma INT_absorb: "k \<in> I ==> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
  1541   by blast
  1542 
  1543 lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
  1544   by blast
  1545 
  1546 lemma UN_Un: "(\<Union>i \<in> A \<union> B. M i) = (\<Union>i\<in>A. M i) \<union> (\<Union>i\<in>B. M i)"
  1547   by blast
  1548 
  1549 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)"
  1550   by blast
  1551 
  1552 lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
  1553   by blast
  1554 
  1555 lemma INT_subset_iff: "(B \<subseteq> (\<Inter>i\<in>I. A i)) = (\<forall>i\<in>I. B \<subseteq> A i)"
  1556   by blast
  1557 
  1558 lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
  1559   by blast
  1560 
  1561 lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"
  1562   by blast
  1563 
  1564 lemma INT_insert_distrib:
  1565     "u \<in> A ==> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
  1566   by blast
  1567 
  1568 lemma Union_image_eq [simp]: "\<Union>(B`A) = (\<Union>x\<in>A. B x)"
  1569   by blast
  1570 
  1571 lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
  1572   by blast
  1573 
  1574 lemma Inter_image_eq [simp]: "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
  1575   by blast
  1576 
  1577 lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
  1578   by auto
  1579 
  1580 lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
  1581   by auto
  1582 
  1583 lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"
  1584   by blast
  1585 
  1586 lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"
  1587   -- {* Look: it has an \emph{existential} quantifier *}
  1588   by blast
  1589 
  1590 lemma UNION_empty_conv[simp]:
  1591   "({} = (UN x:A. B x)) = (\<forall>x\<in>A. B x = {})"
  1592   "((UN x:A. B x) = {}) = (\<forall>x\<in>A. B x = {})"
  1593 by blast+
  1594 
  1595 lemma INTER_UNIV_conv[simp]:
  1596  "(UNIV = (INT x:A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
  1597  "((INT x:A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
  1598 by blast+
  1599 
  1600 
  1601 text {* \medskip Distributive laws: *}
  1602 
  1603 lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
  1604   by blast
  1605 
  1606 lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
  1607   by blast
  1608 
  1609 lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A`C) \<union> \<Union>(B`C)"
  1610   -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
  1611   -- {* Union of a family of unions *}
  1612   by blast
  1613 
  1614 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)"
  1615   -- {* Equivalent version *}
  1616   by blast
  1617 
  1618 lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
  1619   by blast
  1620 
  1621 lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A`C) \<inter> \<Inter>(B`C)"
  1622   by blast
  1623 
  1624 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)"
  1625   -- {* Equivalent version *}
  1626   by blast
  1627 
  1628 lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
  1629   -- {* Halmos, Naive Set Theory, page 35. *}
  1630   by blast
  1631 
  1632 lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
  1633   by blast
  1634 
  1635 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)"
  1636   by blast
  1637 
  1638 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)"
  1639   by blast
  1640 
  1641 
  1642 text {* \medskip Bounded quantifiers.
  1643 
  1644   The following are not added to the default simpset because
  1645   (a) they duplicate the body and (b) there are no similar rules for @{text Int}. *}
  1646 
  1647 lemma ball_Un: "(\<forall>x \<in> A \<union> B. P x) = ((\<forall>x\<in>A. P x) & (\<forall>x\<in>B. P x))"
  1648   by blast
  1649 
  1650 lemma bex_Un: "(\<exists>x \<in> A \<union> B. P x) = ((\<exists>x\<in>A. P x) | (\<exists>x\<in>B. P x))"
  1651   by blast
  1652 
  1653 lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) = (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
  1654   by blast
  1655 
  1656 lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) = (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
  1657   by blast
  1658 
  1659 
  1660 text {* \medskip Set difference. *}
  1661 
  1662 lemma Diff_eq: "A - B = A \<inter> (-B)"
  1663   by blast
  1664 
  1665 lemma Diff_eq_empty_iff [simp]: "(A - B = {}) = (A \<subseteq> B)"
  1666   by blast
  1667 
  1668 lemma Diff_cancel [simp]: "A - A = {}"
  1669   by blast
  1670 
  1671 lemma Diff_idemp [simp]: "(A - B) - B = A - (B::'a set)"
  1672 by blast
  1673 
  1674 lemma Diff_triv: "A \<inter> B = {} ==> A - B = A"
  1675   by (blast elim: equalityE)
  1676 
  1677 lemma empty_Diff [simp]: "{} - A = {}"
  1678   by blast
  1679 
  1680 lemma Diff_empty [simp]: "A - {} = A"
  1681   by blast
  1682 
  1683 lemma Diff_UNIV [simp]: "A - UNIV = {}"
  1684   by blast
  1685 
  1686 lemma Diff_insert0 [simp]: "x \<notin> A ==> A - insert x B = A - B"
  1687   by blast
  1688 
  1689 lemma Diff_insert: "A - insert a B = A - B - {a}"
  1690   -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a 0"} *}
  1691   by blast
  1692 
  1693 lemma Diff_insert2: "A - insert a B = A - {a} - B"
  1694   -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a 0"} *}
  1695   by blast
  1696 
  1697 lemma insert_Diff_if: "insert x A - B = (if x \<in> B then A - B else insert x (A - B))"
  1698   by auto
  1699 
  1700 lemma insert_Diff1 [simp]: "x \<in> B ==> insert x A - B = A - B"
  1701   by blast
  1702 
  1703 lemma insert_Diff_single[simp]: "insert a (A - {a}) = insert a A"
  1704 by blast
  1705 
  1706 lemma insert_Diff: "a \<in> A ==> insert a (A - {a}) = A"
  1707   by blast
  1708 
  1709 lemma Diff_insert_absorb: "x \<notin> A ==> (insert x A) - {x} = A"
  1710   by auto
  1711 
  1712 lemma Diff_disjoint [simp]: "A \<inter> (B - A) = {}"
  1713   by blast
  1714 
  1715 lemma Diff_partition: "A \<subseteq> B ==> A \<union> (B - A) = B"
  1716   by blast
  1717 
  1718 lemma double_diff: "A \<subseteq> B ==> B \<subseteq> C ==> B - (C - A) = A"
  1719   by blast
  1720 
  1721 lemma Un_Diff_cancel [simp]: "A \<union> (B - A) = A \<union> B"
  1722   by blast
  1723 
  1724 lemma Un_Diff_cancel2 [simp]: "(B - A) \<union> A = B \<union> A"
  1725   by blast
  1726 
  1727 lemma Diff_Un: "A - (B \<union> C) = (A - B) \<inter> (A - C)"
  1728   by blast
  1729 
  1730 lemma Diff_Int: "A - (B \<inter> C) = (A - B) \<union> (A - C)"
  1731   by blast
  1732 
  1733 lemma Un_Diff: "(A \<union> B) - C = (A - C) \<union> (B - C)"
  1734   by blast
  1735 
  1736 lemma Int_Diff: "(A \<inter> B) - C = A \<inter> (B - C)"
  1737   by blast
  1738 
  1739 lemma Diff_Int_distrib: "C \<inter> (A - B) = (C \<inter> A) - (C \<inter> B)"
  1740   by blast
  1741 
  1742 lemma Diff_Int_distrib2: "(A - B) \<inter> C = (A \<inter> C) - (B \<inter> C)"
  1743   by blast
  1744 
  1745 lemma Diff_Compl [simp]: "A - (- B) = A \<inter> B"
  1746   by auto
  1747 
  1748 lemma Compl_Diff_eq [simp]: "- (A - B) = -A \<union> B"
  1749   by blast
  1750 
  1751 
  1752 text {* \medskip Quantification over type @{typ bool}. *}
  1753 
  1754 lemma bool_induct: "P True \<Longrightarrow> P False \<Longrightarrow> P x"
  1755   by (cases x) auto
  1756 
  1757 lemma all_bool_eq: "(\<forall>b. P b) \<longleftrightarrow> P True \<and> P False"
  1758   by (auto intro: bool_induct)
  1759 
  1760 lemma bool_contrapos: "P x \<Longrightarrow> \<not> P False \<Longrightarrow> P True"
  1761   by (cases x) auto
  1762 
  1763 lemma ex_bool_eq: "(\<exists>b. P b) \<longleftrightarrow> P True \<or> P False"
  1764   by (auto intro: bool_contrapos)
  1765 
  1766 lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
  1767   by (auto simp add: split_if_mem2)
  1768 
  1769 lemma UN_bool_eq: "(\<Union>b::bool. A b) = (A True \<union> A False)"
  1770   by (auto intro: bool_contrapos)
  1771 
  1772 lemma INT_bool_eq: "(\<Inter>b::bool. A b) = (A True \<inter> A False)"
  1773   by (auto intro: bool_induct)
  1774 
  1775 text {* \medskip @{text Pow} *}
  1776 
  1777 lemma Pow_empty [simp]: "Pow {} = {{}}"
  1778   by (auto simp add: Pow_def)
  1779 
  1780 lemma Pow_insert: "Pow (insert a A) = Pow A \<union> (insert a ` Pow A)"
  1781   by (blast intro: image_eqI [where ?x = "u - {a}", standard])
  1782 
  1783 lemma Pow_Compl: "Pow (- A) = {-B | B. A \<in> Pow B}"
  1784   by (blast intro: exI [where ?x = "- u", standard])
  1785 
  1786 lemma Pow_UNIV [simp]: "Pow UNIV = UNIV"
  1787   by blast
  1788 
  1789 lemma Un_Pow_subset: "Pow A \<union> Pow B \<subseteq> Pow (A \<union> B)"
  1790   by blast
  1791 
  1792 lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
  1793   by blast
  1794 
  1795 lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
  1796   by blast
  1797 
  1798 lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
  1799   by blast
  1800 
  1801 lemma Pow_Int_eq [simp]: "Pow (A \<inter> B) = Pow A \<inter> Pow B"
  1802   by blast
  1803 
  1804 lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
  1805   by blast
  1806 
  1807 
  1808 text {* \medskip Miscellany. *}
  1809 
  1810 lemma set_eq_subset: "(A = B) = (A \<subseteq> B & B \<subseteq> A)"
  1811   by blast
  1812 
  1813 lemma subset_iff: "(A \<subseteq> B) = (\<forall>t. t \<in> A --> t \<in> B)"
  1814   by blast
  1815 
  1816 lemma subset_iff_psubset_eq: "(A \<subseteq> B) = ((A \<subset> B) | (A = B))"
  1817   by (unfold psubset_def) blast
  1818 
  1819 lemma all_not_in_conv [simp]: "(\<forall>x. x \<notin> A) = (A = {})"
  1820   by blast
  1821 
  1822 lemma ex_in_conv: "(\<exists>x. x \<in> A) = (A \<noteq> {})"
  1823   by blast
  1824 
  1825 lemma distinct_lemma: "f x \<noteq> f y ==> x \<noteq> y"
  1826   by iprover
  1827 
  1828 
  1829 text {* \medskip Miniscoping: pushing in quantifiers and big Unions
  1830            and Intersections. *}
  1831 
  1832 lemma UN_simps [simp]:
  1833   "!!a B C. (UN x:C. insert a (B x)) = (if C={} then {} else insert a (UN x:C. B x))"
  1834   "!!A B C. (UN x:C. A x Un B)   = ((if C={} then {} else (UN x:C. A x) Un B))"
  1835   "!!A B C. (UN x:C. A Un B x)   = ((if C={} then {} else A Un (UN x:C. B x)))"
  1836   "!!A B C. (UN x:C. A x Int B)  = ((UN x:C. A x) Int B)"
  1837   "!!A B C. (UN x:C. A Int B x)  = (A Int (UN x:C. B x))"
  1838   "!!A B C. (UN x:C. A x - B)    = ((UN x:C. A x) - B)"
  1839   "!!A B C. (UN x:C. A - B x)    = (A - (INT x:C. B x))"
  1840   "!!A B. (UN x: Union A. B x) = (UN y:A. UN x:y. B x)"
  1841   "!!A B C. (UN z: UNION A B. C z) = (UN  x:A. UN z: B(x). C z)"
  1842   "!!A B f. (UN x:f`A. B x)     = (UN a:A. B (f a))"
  1843   by auto
  1844 
  1845 lemma INT_simps [simp]:
  1846   "!!A B C. (INT x:C. A x Int B) = (if C={} then UNIV else (INT x:C. A x) Int B)"
  1847   "!!A B C. (INT x:C. A Int B x) = (if C={} then UNIV else A Int (INT x:C. B x))"
  1848   "!!A B C. (INT x:C. A x - B)   = (if C={} then UNIV else (INT x:C. A x) - B)"
  1849   "!!A B C. (INT x:C. A - B x)   = (if C={} then UNIV else A - (UN x:C. B x))"
  1850   "!!a B C. (INT x:C. insert a (B x)) = insert a (INT x:C. B x)"
  1851   "!!A B C. (INT x:C. A x Un B)  = ((INT x:C. A x) Un B)"
  1852   "!!A B C. (INT x:C. A Un B x)  = (A Un (INT x:C. B x))"
  1853   "!!A B. (INT x: Union A. B x) = (INT y:A. INT x:y. B x)"
  1854   "!!A B C. (INT z: UNION A B. C z) = (INT x:A. INT z: B(x). C z)"
  1855   "!!A B f. (INT x:f`A. B x)    = (INT a:A. B (f a))"
  1856   by auto
  1857 
  1858 lemma ball_simps [simp]:
  1859   "!!A P Q. (ALL x:A. P x | Q) = ((ALL x:A. P x) | Q)"
  1860   "!!A P Q. (ALL x:A. P | Q x) = (P | (ALL x:A. Q x))"
  1861   "!!A P Q. (ALL x:A. P --> Q x) = (P --> (ALL x:A. Q x))"
  1862   "!!A P Q. (ALL x:A. P x --> Q) = ((EX x:A. P x) --> Q)"
  1863   "!!P. (ALL x:{}. P x) = True"
  1864   "!!P. (ALL x:UNIV. P x) = (ALL x. P x)"
  1865   "!!a B P. (ALL x:insert a B. P x) = (P a & (ALL x:B. P x))"
  1866   "!!A P. (ALL x:Union A. P x) = (ALL y:A. ALL x:y. P x)"
  1867   "!!A B P. (ALL x: UNION A B. P x) = (ALL a:A. ALL x: B a. P x)"
  1868   "!!P Q. (ALL x:Collect Q. P x) = (ALL x. Q x --> P x)"
  1869   "!!A P f. (ALL x:f`A. P x) = (ALL x:A. P (f x))"
  1870   "!!A P. (~(ALL x:A. P x)) = (EX x:A. ~P x)"
  1871   by auto
  1872 
  1873 lemma bex_simps [simp]:
  1874   "!!A P Q. (EX x:A. P x & Q) = ((EX x:A. P x) & Q)"
  1875   "!!A P Q. (EX x:A. P & Q x) = (P & (EX x:A. Q x))"
  1876   "!!P. (EX x:{}. P x) = False"
  1877   "!!P. (EX x:UNIV. P x) = (EX x. P x)"
  1878   "!!a B P. (EX x:insert a B. P x) = (P(a) | (EX x:B. P x))"
  1879   "!!A P. (EX x:Union A. P x) = (EX y:A. EX x:y. P x)"
  1880   "!!A B P. (EX x: UNION A B. P x) = (EX a:A. EX x:B a. P x)"
  1881   "!!P Q. (EX x:Collect Q. P x) = (EX x. Q x & P x)"
  1882   "!!A P f. (EX x:f`A. P x) = (EX x:A. P (f x))"
  1883   "!!A P. (~(EX x:A. P x)) = (ALL x:A. ~P x)"
  1884   by auto
  1885 
  1886 lemma ball_conj_distrib:
  1887   "(ALL x:A. P x & Q x) = ((ALL x:A. P x) & (ALL x:A. Q x))"
  1888   by blast
  1889 
  1890 lemma bex_disj_distrib:
  1891   "(EX x:A. P x | Q x) = ((EX x:A. P x) | (EX x:A. Q x))"
  1892   by blast
  1893 
  1894 
  1895 text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
  1896 
  1897 lemma UN_extend_simps:
  1898   "!!a B C. insert a (UN x:C. B x) = (if C={} then {a} else (UN x:C. insert a (B x)))"
  1899   "!!A B C. (UN x:C. A x) Un B    = (if C={} then B else (UN x:C. A x Un B))"
  1900   "!!A B C. A Un (UN x:C. B x)   = (if C={} then A else (UN x:C. A Un B x))"
  1901   "!!A B C. ((UN x:C. A x) Int B) = (UN x:C. A x Int B)"
  1902   "!!A B C. (A Int (UN x:C. B x)) = (UN x:C. A Int B x)"
  1903   "!!A B C. ((UN x:C. A x) - B) = (UN x:C. A x - B)"
  1904   "!!A B C. (A - (INT x:C. B x)) = (UN x:C. A - B x)"
  1905   "!!A B. (UN y:A. UN x:y. B x) = (UN x: Union A. B x)"
  1906   "!!A B C. (UN  x:A. UN z: B(x). C z) = (UN z: UNION A B. C z)"
  1907   "!!A B f. (UN a:A. B (f a)) = (UN x:f`A. B x)"
  1908   by auto
  1909 
  1910 lemma INT_extend_simps:
  1911   "!!A B C. (INT x:C. A x) Int B = (if C={} then B else (INT x:C. A x Int B))"
  1912   "!!A B C. A Int (INT x:C. B x) = (if C={} then A else (INT x:C. A Int B x))"
  1913   "!!A B C. (INT x:C. A x) - B   = (if C={} then UNIV-B else (INT x:C. A x - B))"
  1914   "!!A B C. A - (UN x:C. B x)   = (if C={} then A else (INT x:C. A - B x))"
  1915   "!!a B C. insert a (INT x:C. B x) = (INT x:C. insert a (B x))"
  1916   "!!A B C. ((INT x:C. A x) Un B)  = (INT x:C. A x Un B)"
  1917   "!!A B C. A Un (INT x:C. B x)  = (INT x:C. A Un B x)"
  1918   "!!A B. (INT y:A. INT x:y. B x) = (INT x: Union A. B x)"
  1919   "!!A B C. (INT x:A. INT z: B(x). C z) = (INT z: UNION A B. C z)"
  1920   "!!A B f. (INT a:A. B (f a))    = (INT x:f`A. B x)"
  1921   by auto
  1922 
  1923 
  1924 subsubsection {* Monotonicity of various operations *}
  1925 
  1926 lemma image_mono: "A \<subseteq> B ==> f`A \<subseteq> f`B"
  1927   by blast
  1928 
  1929 lemma Pow_mono: "A \<subseteq> B ==> Pow A \<subseteq> Pow B"
  1930   by blast
  1931 
  1932 lemma Union_mono: "A \<subseteq> B ==> \<Union>A \<subseteq> \<Union>B"
  1933   by blast
  1934 
  1935 lemma Inter_anti_mono: "B \<subseteq> A ==> \<Inter>A \<subseteq> \<Inter>B"
  1936   by blast
  1937 
  1938 lemma UN_mono:
  1939   "A \<subseteq> B ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
  1940     (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
  1941   by (blast dest: subsetD)
  1942 
  1943 lemma INT_anti_mono:
  1944   "B \<subseteq> A ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
  1945     (\<Inter>x\<in>A. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
  1946   -- {* The last inclusion is POSITIVE! *}
  1947   by (blast dest: subsetD)
  1948 
  1949 lemma insert_mono: "C \<subseteq> D ==> insert a C \<subseteq> insert a D"
  1950   by blast
  1951 
  1952 lemma Un_mono: "A \<subseteq> C ==> B \<subseteq> D ==> A \<union> B \<subseteq> C \<union> D"
  1953   by blast
  1954 
  1955 lemma Int_mono: "A \<subseteq> C ==> B \<subseteq> D ==> A \<inter> B \<subseteq> C \<inter> D"
  1956   by blast
  1957 
  1958 lemma Diff_mono: "A \<subseteq> C ==> D \<subseteq> B ==> A - B \<subseteq> C - D"
  1959   by blast
  1960 
  1961 lemma Compl_anti_mono: "A \<subseteq> B ==> -B \<subseteq> -A"
  1962   by blast
  1963 
  1964 text {* \medskip Monotonicity of implications. *}
  1965 
  1966 lemma in_mono: "A \<subseteq> B ==> x \<in> A --> x \<in> B"
  1967   apply (rule impI)
  1968   apply (erule subsetD, assumption)
  1969   done
  1970 
  1971 lemma conj_mono: "P1 --> Q1 ==> P2 --> Q2 ==> (P1 & P2) --> (Q1 & Q2)"
  1972   by iprover
  1973 
  1974 lemma disj_mono: "P1 --> Q1 ==> P2 --> Q2 ==> (P1 | P2) --> (Q1 | Q2)"
  1975   by iprover
  1976 
  1977 lemma imp_mono: "Q1 --> P1 ==> P2 --> Q2 ==> (P1 --> P2) --> (Q1 --> Q2)"
  1978   by iprover
  1979 
  1980 lemma imp_refl: "P --> P" ..
  1981 
  1982 lemma ex_mono: "(!!x. P x --> Q x) ==> (EX x. P x) --> (EX x. Q x)"
  1983   by iprover
  1984 
  1985 lemma all_mono: "(!!x. P x --> Q x) ==> (ALL x. P x) --> (ALL x. Q x)"
  1986   by iprover
  1987 
  1988 lemma Collect_mono: "(!!x. P x --> Q x) ==> Collect P \<subseteq> Collect Q"
  1989   by blast
  1990 
  1991 lemma Int_Collect_mono:
  1992     "A \<subseteq> B ==> (!!x. x \<in> A ==> P x --> Q x) ==> A \<inter> Collect P \<subseteq> B \<inter> Collect Q"
  1993   by blast
  1994 
  1995 lemmas basic_monos =
  1996   subset_refl imp_refl disj_mono conj_mono
  1997   ex_mono Collect_mono in_mono
  1998 
  1999 lemma eq_to_mono: "a = b ==> c = d ==> b --> d ==> a --> c"
  2000   by iprover
  2001 
  2002 lemma eq_to_mono2: "a = b ==> c = d ==> ~ b --> ~ d ==> ~ a --> ~ c"
  2003   by iprover
  2004 
  2005 lemma Least_mono:
  2006   "mono (f::'a::order => 'b::order) ==> EX x:S. ALL y:S. x <= y
  2007     ==> (LEAST y. y : f ` S) = f (LEAST x. x : S)"
  2008     -- {* Courtesy of Stephan Merz *}
  2009   apply clarify
  2010   apply (erule_tac P = "%x. x : S" in LeastI2_order, fast)
  2011   apply (rule LeastI2_order)
  2012   apply (auto elim: monoD intro!: order_antisym)
  2013   done
  2014 
  2015 
  2016 subsection {* Inverse image of a function *}
  2017 
  2018 constdefs
  2019   vimage :: "('a => 'b) => 'b set => 'a set"    (infixr "-`" 90)
  2020   "f -` B == {x. f x : B}"
  2021 
  2022 
  2023 subsubsection {* Basic rules *}
  2024 
  2025 lemma vimage_eq [simp]: "(a : f -` B) = (f a : B)"
  2026   by (unfold vimage_def) blast
  2027 
  2028 lemma vimage_singleton_eq: "(a : f -` {b}) = (f a = b)"
  2029   by simp
  2030 
  2031 lemma vimageI [intro]: "f a = b ==> b:B ==> a : f -` B"
  2032   by (unfold vimage_def) blast
  2033 
  2034 lemma vimageI2: "f a : A ==> a : f -` A"
  2035   by (unfold vimage_def) fast
  2036 
  2037 lemma vimageE [elim!]: "a: f -` B ==> (!!x. f a = x ==> x:B ==> P) ==> P"
  2038   by (unfold vimage_def) blast
  2039 
  2040 lemma vimageD: "a : f -` A ==> f a : A"
  2041   by (unfold vimage_def) fast
  2042 
  2043 
  2044 subsubsection {* Equations *}
  2045 
  2046 lemma vimage_empty [simp]: "f -` {} = {}"
  2047   by blast
  2048 
  2049 lemma vimage_Compl: "f -` (-A) = -(f -` A)"
  2050   by blast
  2051 
  2052 lemma vimage_Un [simp]: "f -` (A Un B) = (f -` A) Un (f -` B)"
  2053   by blast
  2054 
  2055 lemma vimage_Int [simp]: "f -` (A Int B) = (f -` A) Int (f -` B)"
  2056   by fast
  2057 
  2058 lemma vimage_Union: "f -` (Union A) = (UN X:A. f -` X)"
  2059   by blast
  2060 
  2061 lemma vimage_UN: "f-`(UN x:A. B x) = (UN x:A. f -` B x)"
  2062   by blast
  2063 
  2064 lemma vimage_INT: "f-`(INT x:A. B x) = (INT x:A. f -` B x)"
  2065   by blast
  2066 
  2067 lemma vimage_Collect_eq [simp]: "f -` Collect P = {y. P (f y)}"
  2068   by blast
  2069 
  2070 lemma vimage_Collect: "(!!x. P (f x) = Q x) ==> f -` (Collect P) = Collect Q"
  2071   by blast
  2072 
  2073 lemma vimage_insert: "f-`(insert a B) = (f-`{a}) Un (f-`B)"
  2074   -- {* NOT suitable for rewriting because of the recurrence of @{term "{a}"}. *}
  2075   by blast
  2076 
  2077 lemma vimage_Diff: "f -` (A - B) = (f -` A) - (f -` B)"
  2078   by blast
  2079 
  2080 lemma vimage_UNIV [simp]: "f -` UNIV = UNIV"
  2081   by blast
  2082 
  2083 lemma vimage_eq_UN: "f-`B = (UN y: B. f-`{y})"
  2084   -- {* NOT suitable for rewriting *}
  2085   by blast
  2086 
  2087 lemma vimage_mono: "A \<subseteq> B ==> f -` A \<subseteq> f -` B"
  2088   -- {* monotonicity *}
  2089   by blast
  2090 
  2091 
  2092 subsection {* Getting the Contents of a Singleton Set *}
  2093 
  2094 constdefs
  2095   contents :: "'a set => 'a"
  2096    "contents X == THE x. X = {x}"
  2097 
  2098 lemma contents_eq [simp]: "contents {x} = x"
  2099 by (simp add: contents_def)
  2100 
  2101 
  2102 subsection {* Transitivity rules for calculational reasoning *}
  2103 
  2104 lemma set_rev_mp: "x:A ==> A \<subseteq> B ==> x:B"
  2105   by (rule subsetD)
  2106 
  2107 lemma set_mp: "A \<subseteq> B ==> x:A ==> x:B"
  2108   by (rule subsetD)
  2109 
  2110 lemmas basic_trans_rules [trans] =
  2111   order_trans_rules set_rev_mp set_mp
  2112 
  2113 
  2114 subsection {* Basic ML bindings *}
  2115 
  2116 ML {*
  2117 val Ball_def = thm "Ball_def";
  2118 val Bex_def = thm "Bex_def";
  2119 val CollectD = thm "CollectD";
  2120 val CollectE = thm "CollectE";
  2121 val CollectI = thm "CollectI";
  2122 val Collect_conj_eq = thm "Collect_conj_eq";
  2123 val Collect_mem_eq = thm "Collect_mem_eq";
  2124 val IntD1 = thm "IntD1";
  2125 val IntD2 = thm "IntD2";
  2126 val IntE = thm "IntE";
  2127 val IntI = thm "IntI";
  2128 val Int_Collect = thm "Int_Collect";
  2129 val UNIV_I = thm "UNIV_I";
  2130 val UNIV_witness = thm "UNIV_witness";
  2131 val UnE = thm "UnE";
  2132 val UnI1 = thm "UnI1";
  2133 val UnI2 = thm "UnI2";
  2134 val ballE = thm "ballE";
  2135 val ballI = thm "ballI";
  2136 val bexCI = thm "bexCI";
  2137 val bexE = thm "bexE";
  2138 val bexI = thm "bexI";
  2139 val bex_triv = thm "bex_triv";
  2140 val bspec = thm "bspec";
  2141 val contra_subsetD = thm "contra_subsetD";
  2142 val distinct_lemma = thm "distinct_lemma";
  2143 val eq_to_mono = thm "eq_to_mono";
  2144 val eq_to_mono2 = thm "eq_to_mono2";
  2145 val equalityCE = thm "equalityCE";
  2146 val equalityD1 = thm "equalityD1";
  2147 val equalityD2 = thm "equalityD2";
  2148 val equalityE = thm "equalityE";
  2149 val equalityI = thm "equalityI";
  2150 val imageE = thm "imageE";
  2151 val imageI = thm "imageI";
  2152 val image_Un = thm "image_Un";
  2153 val image_insert = thm "image_insert";
  2154 val insert_commute = thm "insert_commute";
  2155 val insert_iff = thm "insert_iff";
  2156 val mem_Collect_eq = thm "mem_Collect_eq";
  2157 val rangeE = thm "rangeE";
  2158 val rangeI = thm "rangeI";
  2159 val range_eqI = thm "range_eqI";
  2160 val subsetCE = thm "subsetCE";
  2161 val subsetD = thm "subsetD";
  2162 val subsetI = thm "subsetI";
  2163 val subset_refl = thm "subset_refl";
  2164 val subset_trans = thm "subset_trans";
  2165 val vimageD = thm "vimageD";
  2166 val vimageE = thm "vimageE";
  2167 val vimageI = thm "vimageI";
  2168 val vimageI2 = thm "vimageI2";
  2169 val vimage_Collect = thm "vimage_Collect";
  2170 val vimage_Int = thm "vimage_Int";
  2171 val vimage_Un = thm "vimage_Un";
  2172 *}
  2173 
  2174 end