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:    "fA            == {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 (BA)"}.

   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 (BA)"}. *}

   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 : fA"

   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 : fA"

   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) = fA Un fB"

   911   by blast

   912

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

   914   by blast

   915

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

   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) ==> fA \<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) (fB)"

  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) (fA) = fA"

  1250   by blast

  1251

  1252 lemma image_is_empty [iff]: "(fA = {}) = (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) ==> fM = gN"

  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)) = frange 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>(BA) = (\<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>(BA) = (\<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>(AC) \<union> \<Union>(BC)"

  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>(AC) \<inter> \<Inter>(BC)"

  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:fA. 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:fA. 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:fA. 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:fA. 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:fA. 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:fA. B x)"

  1921   by auto

  1922

  1923

  1924 subsubsection {* Monotonicity of various operations *}

  1925

  1926 lemma image_mono: "A \<subseteq> B ==> fA \<subseteq> fB"

  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
`