src/HOL/Set.thy
 author haftmann Thu Mar 19 14:08:41 2009 +0100 (2009-03-19) changeset 30596 140b22f22071 parent 30531 ab3d61baf66a child 30814 10dc9bc264b7 permissions -rw-r--r--
tuned some theorem and attribute bindings
     1 (*  Title:      HOL/Set.thy

     2     Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel

     3 *)

     4

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

     6

     7 theory Set

     8 imports Lattices

     9 begin

    10

    11 text {* A set in HOL is simply a predicate. *}

    12

    13

    14 subsection {* Basic syntax *}

    15

    16 global

    17

    18 types 'a set = "'a => bool"

    19

    20 consts

    21   Collect       :: "('a => bool) => 'a set"              -- "comprehension"

    22   "op :"        :: "'a => 'a set => bool"                -- "membership"

    23   insert        :: "'a => 'a set => 'a set"

    24   Ball          :: "'a set => ('a => bool) => bool"      -- "bounded universal quantifiers"

    25   Bex           :: "'a set => ('a => bool) => bool"      -- "bounded existential quantifiers"

    26   Bex1          :: "'a set => ('a => bool) => bool"      -- "bounded unique existential quantifiers"

    27   Pow           :: "'a set => 'a set set"                -- "powerset"

    28   image         :: "('a => 'b) => 'a set => 'b set"      (infixr "" 90)

    29

    30 local

    31

    32 notation

    33   "op :"  ("op :") and

    34   "op :"  ("(_/ : _)" [50, 51] 50)

    35

    36 abbreviation

    37   "not_mem x A == ~ (x : A)" -- "non-membership"

    38

    39 notation

    40   not_mem  ("op ~:") and

    41   not_mem  ("(_/ ~: _)" [50, 51] 50)

    42

    43 notation (xsymbols)

    44   "op :"  ("op \<in>") and

    45   "op :"  ("(_/ \<in> _)" [50, 51] 50) and

    46   not_mem  ("op \<notin>") and

    47   not_mem  ("(_/ \<notin> _)" [50, 51] 50)

    48

    49 notation (HTML output)

    50   "op :"  ("op \<in>") and

    51   "op :"  ("(_/ \<in> _)" [50, 51] 50) and

    52   not_mem  ("op \<notin>") and

    53   not_mem  ("(_/ \<notin> _)" [50, 51] 50)

    54

    55 syntax

    56   "@Coll"       :: "pttrn => bool => 'a set"              ("(1{_./ _})")

    57

    58 translations

    59   "{x. P}"      == "Collect (%x. P)"

    60

    61 definition empty :: "'a set" ("{}") where

    62   "empty \<equiv> {x. False}"

    63

    64 definition UNIV :: "'a set" where

    65   "UNIV \<equiv> {x. True}"

    66

    67 syntax

    68   "@Finset"     :: "args => 'a set"                       ("{(_)}")

    69

    70 translations

    71   "{x, xs}"     == "insert x {xs}"

    72   "{x}"         == "insert x {}"

    73

    74 definition Int :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Int" 70) where

    75   "A Int B \<equiv> {x. x \<in> A \<and> x \<in> B}"

    76

    77 definition Un :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Un" 65) where

    78   "A Un B \<equiv> {x. x \<in> A \<or> x \<in> B}"

    79

    80 notation (xsymbols)

    81   "Int"  (infixl "\<inter>" 70) and

    82   "Un"  (infixl "\<union>" 65)

    83

    84 notation (HTML output)

    85   "Int"  (infixl "\<inter>" 70) and

    86   "Un"  (infixl "\<union>" 65)

    87

    88 syntax

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

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

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

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

    93

    94 syntax (HOL)

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

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

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

    98

    99 syntax (xsymbols)

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

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

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

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

   104

   105 syntax (HTML output)

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

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

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

   109

   110 translations

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

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

   113   "EX! x:A. P"  == "Bex1 A (%x. P)"

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

   115

   116 definition INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where

   117   "INTER A B \<equiv> {y. \<forall>x\<in>A. y \<in> B x}"

   118

   119 definition UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where

   120   "UNION A B \<equiv> {y. \<exists>x\<in>A. y \<in> B x}"

   121

   122 definition Inter :: "'a set set \<Rightarrow> 'a set" where

   123   "Inter S \<equiv> INTER S (\<lambda>x. x)"

   124

   125 definition Union :: "'a set set \<Rightarrow> 'a set" where

   126   "Union S \<equiv> UNION S (\<lambda>x. x)"

   127

   128 notation (xsymbols)

   129   Inter  ("\<Inter>_" [90] 90) and

   130   Union  ("\<Union>_" [90] 90)

   131

   132

   133 subsection {* Additional concrete syntax *}

   134

   135 syntax

   136   "@SetCompr"   :: "'a => idts => bool => 'a set"         ("(1{_ |/_./ _})")

   137   "@Collect"    :: "idt => 'a set => bool => 'a set"      ("(1{_ :/ _./ _})")

   138   "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)

   139   "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)

   140   "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 10] 10)

   141   "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 10] 10)

   142

   143 syntax (xsymbols)

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

   145   "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)

   146   "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)

   147   "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 10] 10)

   148   "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 10] 10)

   149

   150 syntax (latex output)

   151   "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)

   152   "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)

   153   "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10)

   154   "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10)

   155

   156 translations

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

   158   "INT x y. B"  == "INT x. INT y. B"

   159   "INT x. B"    == "CONST INTER CONST UNIV (%x. B)"

   160   "INT x. B"    == "INT x:CONST UNIV. B"

   161   "INT x:A. B"  == "CONST INTER A (%x. B)"

   162   "UN x y. B"   == "UN x. UN y. B"

   163   "UN x. B"     == "CONST UNION CONST UNIV (%x. B)"

   164   "UN x. B"     == "UN x:CONST UNIV. B"

   165   "UN x:A. B"   == "CONST UNION A (%x. B)"

   166

   167 text {*

   168   Note the difference between ordinary xsymbol syntax of indexed

   169   unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})

   170   and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The

   171   former does not make the index expression a subscript of the

   172   union/intersection symbol because this leads to problems with nested

   173   subscripts in Proof General.

   174 *}

   175

   176 abbreviation

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

   178   "subset \<equiv> less"

   179

   180 abbreviation

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

   182   "subset_eq \<equiv> less_eq"

   183

   184 notation (output)

   185   subset  ("op <") and

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

   187   subset_eq  ("op <=") and

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

   189

   190 notation (xsymbols)

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

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

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

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

   195

   196 notation (HTML output)

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

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

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

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

   201

   202 abbreviation (input)

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

   204   "supset \<equiv> greater"

   205

   206 abbreviation (input)

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

   208   "supset_eq \<equiv> greater_eq"

   209

   210 notation (xsymbols)

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

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

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

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

   215

   216 abbreviation

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

   218   "range f == f  UNIV"

   219

   220

   221 subsubsection "Bounded quantifiers"

   222

   223 syntax (output)

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

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

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

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

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

   229

   230 syntax (xsymbols)

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

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

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

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

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

   236

   237 syntax (HOL output)

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

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

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

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

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

   243

   244 syntax (HTML output)

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

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

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

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

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

   250

   251 translations

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

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

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

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

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

   257

   258 print_translation {*

   259 let

   260   val Type (set_type, _) = @{typ "'a set"};

   261   val All_binder = Syntax.binder_name @{const_syntax "All"};

   262   val Ex_binder = Syntax.binder_name @{const_syntax "Ex"};

   263   val impl = @{const_syntax "op -->"};

   264   val conj = @{const_syntax "op &"};

   265   val sbset = @{const_syntax "subset"};

   266   val sbset_eq = @{const_syntax "subset_eq"};

   267

   268   val trans =

   269    [((All_binder, impl, sbset), "_setlessAll"),

   270     ((All_binder, impl, sbset_eq), "_setleAll"),

   271     ((Ex_binder, conj, sbset), "_setlessEx"),

   272     ((Ex_binder, conj, sbset_eq), "_setleEx")];

   273

   274   fun mk v v' c n P =

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

   276     then Syntax.const c $Syntax.mark_bound v'$ n $P else raise Match;   277   278 fun tr' q = (q,   279 fn [Const ("_bound", _)$ Free (v, Type (T, _)), Const (c, _) $(Const (d, _)$ (Const ("_bound", _) $Free (v', _))$ n) $P] =>   280 if T = (set_type) then case AList.lookup (op =) trans (q, c, d)   281 of NONE => raise Match   282 | SOME l => mk v v' l n P   283 else raise Match   284 | _ => raise Match);   285 in   286 [tr' All_binder, tr' Ex_binder]   287 end   288 *}   289   290   291 text {*   292 \medskip Translate between @{text "{e | x1...xn. P}"} and @{text   293 "{u. EX x1..xn. u = e & P}"}; @{text "{y. EX x1..xn. y = e & P}"} is   294 only translated if @{text "[0..n] subset bvs(e)"}.   295 *}   296   297 parse_translation {*   298 let   299 val ex_tr = snd (mk_binder_tr ("EX ", "Ex"));   300   301 fun nvars (Const ("_idts", _)$ _ $idts) = nvars idts + 1   302 | nvars _ = 1;   303   304 fun setcompr_tr [e, idts, b] =   305 let   306 val eq = Syntax.const "op ="$ Bound (nvars idts) $e;   307 val P = Syntax.const "op &"$ eq $b;   308 val exP = ex_tr [idts, P];   309 in Syntax.const "Collect"$ Term.absdummy (dummyT, exP) end;

   310

   311   in [("@SetCompr", setcompr_tr)] end;

   312 *}

   313

   314 (* To avoid eta-contraction of body: *)

   315 print_translation {*

   316 let

   317   fun btr' syn [A, Abs abs] =

   318     let val (x, t) = atomic_abs_tr' abs

   319     in Syntax.const syn $x$ A $t end   320 in   321 [(@{const_syntax Ball}, btr' "_Ball"), (@{const_syntax Bex}, btr' "_Bex"),   322 (@{const_syntax UNION}, btr' "@UNION"),(@{const_syntax INTER}, btr' "@INTER")]   323 end   324 *}   325   326 print_translation {*   327 let   328 val ex_tr' = snd (mk_binder_tr' ("Ex", "DUMMY"));   329   330 fun setcompr_tr' [Abs (abs as (_, _, P))] =   331 let   332 fun check (Const ("Ex", _)$ Abs (_, _, P), n) = check (P, n + 1)

   333         | check (Const ("op &", _) $(Const ("op =", _)$ Bound m $e)$ P, n) =

   334             n > 0 andalso m = n andalso not (loose_bvar1 (P, n)) andalso

   335             ((0 upto (n - 1)) subset add_loose_bnos (e, 0, []))

   336         | check _ = false

   337

   338         fun tr' (_ $abs) =   339 let val _$ idts $(_$ (_ $_$ e) $Q) = ex_tr' [abs]   340 in Syntax.const "@SetCompr"$ e $idts$ Q end;

   341     in if check (P, 0) then tr' P

   342        else let val (x as _ $Free(xN,_), t) = atomic_abs_tr' abs   343 val M = Syntax.const "@Coll"$ x $t   344 in case t of   345 Const("op &",_)   346$ (Const("op :",_) $(Const("_bound",_)$ Free(yN,_)) $A)   347$ P =>

   348                    if xN=yN then Syntax.const "@Collect" $x$ A $P else M   349 | _ => M   350 end   351 end;   352 in [("Collect", setcompr_tr')] end;   353 *}   354   355   356 subsection {* Rules and definitions *}   357   358 text {* Isomorphisms between predicates and sets. *}   359   360 defs   361 mem_def [code]: "x : S == S x"   362 Collect_def [code]: "Collect P == P"   363   364 defs   365 Ball_def: "Ball A P == ALL x. x:A --> P(x)"   366 Bex_def: "Bex A P == EX x. x:A & P(x)"   367 Bex1_def: "Bex1 A P == EX! x. x:A & P(x)"   368   369 instantiation "fun" :: (type, minus) minus   370 begin   371   372 definition   373 fun_diff_def: "A - B = (%x. A x - B x)"   374   375 instance ..   376   377 end   378   379 instantiation bool :: minus   380 begin   381   382 definition   383 bool_diff_def: "A - B = (A & ~ B)"   384   385 instance ..   386   387 end   388   389 instantiation "fun" :: (type, uminus) uminus   390 begin   391   392 definition   393 fun_Compl_def: "- A = (%x. - A x)"   394   395 instance ..   396   397 end   398   399 instantiation bool :: uminus   400 begin   401   402 definition   403 bool_Compl_def: "- A = (~ A)"   404   405 instance ..   406   407 end   408   409 defs   410 Pow_def: "Pow A == {B. B <= A}"   411 insert_def: "insert a B == {x. x=a} Un B"   412 image_def: "fA == {y. EX x:A. y = f(x)}"   413   414   415 subsection {* Lemmas and proof tool setup *}   416   417 subsubsection {* Relating predicates and sets *}   418   419 lemma mem_Collect_eq [iff]: "(a : {x. P(x)}) = P(a)"   420 by (simp add: Collect_def mem_def)   421   422 lemma Collect_mem_eq [simp]: "{x. x:A} = A"   423 by (simp add: Collect_def mem_def)   424   425 lemma CollectI: "P(a) ==> a : {x. P(x)}"   426 by simp   427   428 lemma CollectD: "a : {x. P(x)} ==> P(a)"   429 by simp   430   431 lemma Collect_cong: "(!!x. P x = Q x) ==> {x. P(x)} = {x. Q(x)}"   432 by simp   433   434 lemmas CollectE = CollectD [elim_format]   435   436   437 subsubsection {* Bounded quantifiers *}   438   439 lemma ballI [intro!]: "(!!x. x:A ==> P x) ==> ALL x:A. P x"   440 by (simp add: Ball_def)   441   442 lemmas strip = impI allI ballI   443   444 lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x"   445 by (simp add: Ball_def)   446   447 lemma ballE [elim]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q"   448 by (unfold Ball_def) blast   449   450 ML {* bind_thm ("rev_ballE", permute_prems 1 1 @{thm ballE}) *}   451   452 text {*   453 \medskip This tactic takes assumptions @{prop "ALL x:A. P x"} and   454 @{prop "a:A"}; creates assumption @{prop "P a"}.   455 *}   456   457 ML {*   458 fun ball_tac i = etac @{thm ballE} i THEN contr_tac (i + 1)   459 *}   460   461 text {*   462 Gives better instantiation for bound:   463 *}   464   465 declaration {* fn _ =>   466 Classical.map_cs (fn cs => cs addbefore ("bspec", datac @{thm bspec} 1))   467 *}   468   469 lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x"   470 -- {* Normally the best argument order: @{prop "P x"} constrains the   471 choice of @{prop "x:A"}. *}   472 by (unfold Bex_def) blast   473   474 lemma rev_bexI [intro?]: "x:A ==> P x ==> EX x:A. P x"   475 -- {* The best argument order when there is only one @{prop "x:A"}. *}   476 by (unfold Bex_def) blast   477   478 lemma bexCI: "(ALL x:A. ~P x ==> P a) ==> a:A ==> EX x:A. P x"   479 by (unfold Bex_def) blast   480   481 lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q"   482 by (unfold Bex_def) blast   483   484 lemma ball_triv [simp]: "(ALL x:A. P) = ((EX x. x:A) --> P)"   485 -- {* Trival rewrite rule. *}   486 by (simp add: Ball_def)   487   488 lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)"   489 -- {* Dual form for existentials. *}   490 by (simp add: Bex_def)   491   492 lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)"   493 by blast   494   495 lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)"   496 by blast   497   498 lemma bex_one_point1 [simp]: "(EX x:A. x = a & P x) = (a:A & P a)"   499 by blast   500   501 lemma bex_one_point2 [simp]: "(EX x:A. a = x & P x) = (a:A & P a)"   502 by blast   503   504 lemma ball_one_point1 [simp]: "(ALL x:A. x = a --> P x) = (a:A --> P a)"   505 by blast   506   507 lemma ball_one_point2 [simp]: "(ALL x:A. a = x --> P x) = (a:A --> P a)"   508 by blast   509   510 ML {*   511 local   512 val unfold_bex_tac = unfold_tac @{thms "Bex_def"};   513 fun prove_bex_tac ss = unfold_bex_tac ss THEN Quantifier1.prove_one_point_ex_tac;   514 val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac;   515   516 val unfold_ball_tac = unfold_tac @{thms "Ball_def"};   517 fun prove_ball_tac ss = unfold_ball_tac ss THEN Quantifier1.prove_one_point_all_tac;   518 val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac;   519 in   520 val defBEX_regroup = Simplifier.simproc (the_context ())   521 "defined BEX" ["EX x:A. P x & Q x"] rearrange_bex;   522 val defBALL_regroup = Simplifier.simproc (the_context ())   523 "defined BALL" ["ALL x:A. P x --> Q x"] rearrange_ball;   524 end;   525   526 Addsimprocs [defBALL_regroup, defBEX_regroup];   527 *}   528   529   530 subsubsection {* Congruence rules *}   531   532 lemma ball_cong:   533 "A = B ==> (!!x. x:B ==> P x = Q x) ==>   534 (ALL x:A. P x) = (ALL x:B. Q x)"   535 by (simp add: Ball_def)   536   537 lemma strong_ball_cong [cong]:   538 "A = B ==> (!!x. x:B =simp=> P x = Q x) ==>   539 (ALL x:A. P x) = (ALL x:B. Q x)"   540 by (simp add: simp_implies_def Ball_def)   541   542 lemma bex_cong:   543 "A = B ==> (!!x. x:B ==> P x = Q x) ==>   544 (EX x:A. P x) = (EX x:B. Q x)"   545 by (simp add: Bex_def cong: conj_cong)   546   547 lemma strong_bex_cong [cong]:   548 "A = B ==> (!!x. x:B =simp=> P x = Q x) ==>   549 (EX x:A. P x) = (EX x:B. Q x)"   550 by (simp add: simp_implies_def Bex_def cong: conj_cong)   551   552   553 subsubsection {* Subsets *}   554   555 lemma subsetI [atp,intro!]: "(!!x. x:A ==> x:B) ==> A \<subseteq> B"   556 by (auto simp add: mem_def intro: predicate1I)   557   558 text {*   559 \medskip Map the type @{text "'a set => anything"} to just @{typ   560 'a}; for overloading constants whose first argument has type @{typ   561 "'a set"}.   562 *}   563   564 lemma subsetD [elim, intro?]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B"   565 -- {* Rule in Modus Ponens style. *}   566 by (unfold mem_def) blast   567   568 lemma rev_subsetD [intro?]: "c \<in> A ==> A \<subseteq> B ==> c \<in> B"   569 -- {* The same, with reversed premises for use with @{text erule} --   570 cf @{text rev_mp}. *}   571 by (rule subsetD)   572   573 text {*   574 \medskip Converts @{prop "A \<subseteq> B"} to @{prop "x \<in> A ==> x \<in> B"}.   575 *}   576   577 ML {*   578 fun impOfSubs th = th RSN (2, @{thm rev_subsetD})   579 *}   580   581 lemma subsetCE [elim]: "A \<subseteq> B ==> (c \<notin> A ==> P) ==> (c \<in> B ==> P) ==> P"   582 -- {* Classical elimination rule. *}   583 by (unfold mem_def) blast   584   585 lemma subset_eq: "A \<le> B = (\<forall>x\<in>A. x \<in> B)" by blast   586   587 text {*   588 \medskip Takes assumptions @{prop "A \<subseteq> B"}; @{prop "c \<in> A"} and   589 creates the assumption @{prop "c \<in> B"}.   590 *}   591   592 ML {*   593 fun set_mp_tac i = etac @{thm subsetCE} i THEN mp_tac i   594 *}   595   596 lemma contra_subsetD: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A"   597 by blast   598   599 lemma subset_refl [simp,atp]: "A \<subseteq> A"   600 by fast   601   602 lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C"   603 by blast   604   605   606 subsubsection {* Equality *}   607   608 lemma set_ext: assumes prem: "(!!x. (x:A) = (x:B))" shows "A = B"   609 apply (rule prem [THEN ext, THEN arg_cong, THEN box_equals])   610 apply (rule Collect_mem_eq)   611 apply (rule Collect_mem_eq)   612 done   613   614 (* Due to Brian Huffman *)   615 lemma expand_set_eq: "(A = B) = (ALL x. (x:A) = (x:B))"   616 by(auto intro:set_ext)   617   618 lemma subset_antisym [intro!]: "A \<subseteq> B ==> B \<subseteq> A ==> A = B"   619 -- {* Anti-symmetry of the subset relation. *}   620 by (iprover intro: set_ext subsetD)   621   622 text {*   623 \medskip Equality rules from ZF set theory -- are they appropriate   624 here?   625 *}   626   627 lemma equalityD1: "A = B ==> A \<subseteq> B"   628 by (simp add: subset_refl)   629   630 lemma equalityD2: "A = B ==> B \<subseteq> A"   631 by (simp add: subset_refl)   632   633 text {*   634 \medskip Be careful when adding this to the claset as @{text   635 subset_empty} is in the simpset: @{prop "A = {}"} goes to @{prop "{}   636 \<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}!   637 *}   638   639 lemma equalityE: "A = B ==> (A \<subseteq> B ==> B \<subseteq> A ==> P) ==> P"   640 by (simp add: subset_refl)   641   642 lemma equalityCE [elim]:   643 "A = B ==> (c \<in> A ==> c \<in> B ==> P) ==> (c \<notin> A ==> c \<notin> B ==> P) ==> P"   644 by blast   645   646 lemma eqset_imp_iff: "A = B ==> (x : A) = (x : B)"   647 by simp   648   649 lemma eqelem_imp_iff: "x = y ==> (x : A) = (y : A)"   650 by simp   651   652   653 subsubsection {* The universal set -- UNIV *}   654   655 lemma UNIV_I [simp]: "x : UNIV"   656 by (simp add: UNIV_def)   657   658 declare UNIV_I [intro] -- {* unsafe makes it less likely to cause problems *}   659   660 lemma UNIV_witness [intro?]: "EX x. x : UNIV"   661 by simp   662   663 lemma subset_UNIV [simp]: "A \<subseteq> UNIV"   664 by (rule subsetI) (rule UNIV_I)   665   666 text {*   667 \medskip Eta-contracting these two rules (to remove @{text P})   668 causes them to be ignored because of their interaction with   669 congruence rules.   670 *}   671   672 lemma ball_UNIV [simp]: "Ball UNIV P = All P"   673 by (simp add: Ball_def)   674   675 lemma bex_UNIV [simp]: "Bex UNIV P = Ex P"   676 by (simp add: Bex_def)   677   678 lemma UNIV_eq_I: "(\<And>x. x \<in> A) \<Longrightarrow> UNIV = A"   679 by auto   680   681   682 subsubsection {* The empty set *}   683   684 lemma empty_iff [simp]: "(c : {}) = False"   685 by (simp add: empty_def)   686   687 lemma emptyE [elim!]: "a : {} ==> P"   688 by simp   689   690 lemma empty_subsetI [iff]: "{} \<subseteq> A"   691 -- {* One effect is to delete the ASSUMPTION @{prop "{} <= A"} *}   692 by blast   693   694 lemma equals0I: "(!!y. y \<in> A ==> False) ==> A = {}"   695 by blast   696   697 lemma equals0D: "A = {} ==> a \<notin> A"   698 -- {* Use for reasoning about disjointness: @{prop "A Int B = {}"} *}   699 by blast   700   701 lemma ball_empty [simp]: "Ball {} P = True"   702 by (simp add: Ball_def)   703   704 lemma bex_empty [simp]: "Bex {} P = False"   705 by (simp add: Bex_def)   706   707 lemma UNIV_not_empty [iff]: "UNIV ~= {}"   708 by (blast elim: equalityE)   709   710   711 subsubsection {* The Powerset operator -- Pow *}   712   713 lemma Pow_iff [iff]: "(A \<in> Pow B) = (A \<subseteq> B)"   714 by (simp add: Pow_def)   715   716 lemma PowI: "A \<subseteq> B ==> A \<in> Pow B"   717 by (simp add: Pow_def)   718   719 lemma PowD: "A \<in> Pow B ==> A \<subseteq> B"   720 by (simp add: Pow_def)   721   722 lemma Pow_bottom: "{} \<in> Pow B"   723 by simp   724   725 lemma Pow_top: "A \<in> Pow A"   726 by (simp add: subset_refl)   727   728   729 subsubsection {* Set complement *}   730   731 lemma Compl_iff [simp]: "(c \<in> -A) = (c \<notin> A)"   732 by (simp add: mem_def fun_Compl_def bool_Compl_def)   733   734 lemma ComplI [intro!]: "(c \<in> A ==> False) ==> c \<in> -A"   735 by (unfold mem_def fun_Compl_def bool_Compl_def) blast   736   737 text {*   738 \medskip This form, with negated conclusion, works well with the   739 Classical prover. Negated assumptions behave like formulae on the   740 right side of the notional turnstile ... *}   741   742 lemma ComplD [dest!]: "c : -A ==> c~:A"   743 by (simp add: mem_def fun_Compl_def bool_Compl_def)   744   745 lemmas ComplE = ComplD [elim_format]   746   747 lemma Compl_eq: "- A = {x. ~ x : A}" by blast   748   749   750 subsubsection {* Binary union -- Un *}   751   752 lemma Un_iff [simp]: "(c : A Un B) = (c:A | c:B)"   753 by (unfold Un_def) blast   754   755 lemma UnI1 [elim?]: "c:A ==> c : A Un B"   756 by simp   757   758 lemma UnI2 [elim?]: "c:B ==> c : A Un B"   759 by simp   760   761 text {*   762 \medskip Classical introduction rule: no commitment to @{prop A} vs   763 @{prop B}.   764 *}   765   766 lemma UnCI [intro!]: "(c~:B ==> c:A) ==> c : A Un B"   767 by auto   768   769 lemma UnE [elim!]: "c : A Un B ==> (c:A ==> P) ==> (c:B ==> P) ==> P"   770 by (unfold Un_def) blast   771   772   773 subsubsection {* Binary intersection -- Int *}   774   775 lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)"   776 by (unfold Int_def) blast   777   778 lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B"   779 by simp   780   781 lemma IntD1: "c : A Int B ==> c:A"   782 by simp   783   784 lemma IntD2: "c : A Int B ==> c:B"   785 by simp   786   787 lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P"   788 by simp   789   790   791 subsubsection {* Set difference *}   792   793 lemma Diff_iff [simp]: "(c : A - B) = (c:A & c~:B)"   794 by (simp add: mem_def fun_diff_def bool_diff_def)   795   796 lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A - B"   797 by simp   798   799 lemma DiffD1: "c : A - B ==> c : A"   800 by simp   801   802 lemma DiffD2: "c : A - B ==> c : B ==> P"   803 by simp   804   805 lemma DiffE [elim!]: "c : A - B ==> (c:A ==> c~:B ==> P) ==> P"   806 by simp   807   808 lemma set_diff_eq: "A - B = {x. x : A & ~ x : B}" by blast   809   810 lemma Compl_eq_Diff_UNIV: "-A = (UNIV - A)"   811 by blast   812   813   814 subsubsection {* Augmenting a set -- insert *}   815   816 lemma insert_iff [simp]: "(a : insert b A) = (a = b | a:A)"   817 by (unfold insert_def) blast   818   819 lemma insertI1: "a : insert a B"   820 by simp   821   822 lemma insertI2: "a : B ==> a : insert b B"   823 by simp   824   825 lemma insertE [elim!]: "a : insert b A ==> (a = b ==> P) ==> (a:A ==> P) ==> P"   826 by (unfold insert_def) blast   827   828 lemma insertCI [intro!]: "(a~:B ==> a = b) ==> a: insert b B"   829 -- {* Classical introduction rule. *}   830 by auto   831   832 lemma subset_insert_iff: "(A \<subseteq> insert x B) = (if x:A then A - {x} \<subseteq> B else A \<subseteq> B)"   833 by auto   834   835 lemma set_insert:   836 assumes "x \<in> A"   837 obtains B where "A = insert x B" and "x \<notin> B"   838 proof   839 from assms show "A = insert x (A - {x})" by blast   840 next   841 show "x \<notin> A - {x}" by blast   842 qed   843   844 lemma insert_ident: "x ~: A ==> x ~: B ==> (insert x A = insert x B) = (A = B)"   845 by auto   846   847 subsubsection {* Singletons, using insert *}   848   849 lemma singletonI [intro!,noatp]: "a : {a}"   850 -- {* Redundant? But unlike @{text insertCI}, it proves the subgoal immediately! *}   851 by (rule insertI1)   852   853 lemma singletonD [dest!,noatp]: "b : {a} ==> b = a"   854 by blast   855   856 lemmas singletonE = singletonD [elim_format]   857   858 lemma singleton_iff: "(b : {a}) = (b = a)"   859 by blast   860   861 lemma singleton_inject [dest!]: "{a} = {b} ==> a = b"   862 by blast   863   864 lemma singleton_insert_inj_eq [iff,noatp]:   865 "({b} = insert a A) = (a = b & A \<subseteq> {b})"   866 by blast   867   868 lemma singleton_insert_inj_eq' [iff,noatp]:   869 "(insert a A = {b}) = (a = b & A \<subseteq> {b})"   870 by blast   871   872 lemma subset_singletonD: "A \<subseteq> {x} ==> A = {} | A = {x}"   873 by fast   874   875 lemma singleton_conv [simp]: "{x. x = a} = {a}"   876 by blast   877   878 lemma singleton_conv2 [simp]: "{x. a = x} = {a}"   879 by blast   880   881 lemma diff_single_insert: "A - {x} \<subseteq> B ==> x \<in> A ==> A \<subseteq> insert x B"   882 by blast   883   884 lemma doubleton_eq_iff: "({a,b} = {c,d}) = (a=c & b=d | a=d & b=c)"   885 by (blast elim: equalityE)   886   887   888 subsubsection {* Unions of families *}   889   890 text {*   891 @{term [source] "UN x:A. B x"} is @{term "Union (BA)"}.   892 *}   893   894 declare UNION_def [noatp]   895   896 lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)"   897 by (unfold UNION_def) blast   898   899 lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)"   900 -- {* The order of the premises presupposes that @{term A} is rigid;   901 @{term b} may be flexible. *}   902 by auto   903   904 lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R"   905 by (unfold UNION_def) blast   906   907 lemma UN_cong [cong]:   908 "A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"   909 by (simp add: UNION_def)   910   911 lemma strong_UN_cong:   912 "A = B ==> (!!x. x:B =simp=> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"   913 by (simp add: UNION_def simp_implies_def)   914   915   916 subsubsection {* Intersections of families *}   917   918 text {* @{term [source] "INT x:A. B x"} is @{term "Inter (BA)"}. *}   919   920 lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)"   921 by (unfold INTER_def) blast   922   923 lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)"   924 by (unfold INTER_def) blast   925   926 lemma INT_D [elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a"   927 by auto   928   929 lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R"   930 -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}   931 by (unfold INTER_def) blast   932   933 lemma INT_cong [cong]:   934 "A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)"   935 by (simp add: INTER_def)   936   937   938 subsubsection {* Union *}   939   940 lemma Union_iff [simp,noatp]: "(A : Union C) = (EX X:C. A:X)"   941 by (unfold Union_def) blast   942   943 lemma UnionI [intro]: "X:C ==> A:X ==> A : Union C"   944 -- {* The order of the premises presupposes that @{term C} is rigid;   945 @{term A} may be flexible. *}   946 by auto   947   948 lemma UnionE [elim!]: "A : Union C ==> (!!X. A:X ==> X:C ==> R) ==> R"   949 by (unfold Union_def) blast   950   951   952 subsubsection {* Inter *}   953   954 lemma Inter_iff [simp,noatp]: "(A : Inter C) = (ALL X:C. A:X)"   955 by (unfold Inter_def) blast   956   957 lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C"   958 by (simp add: Inter_def)   959   960 text {*   961 \medskip A destruct'' rule -- every @{term X} in @{term C}   962 contains @{term A} as an element, but @{prop "A:X"} can hold when   963 @{prop "X:C"} does not! This rule is analogous to @{text spec}.   964 *}   965   966 lemma InterD [elim]: "A : Inter C ==> X:C ==> A:X"   967 by auto   968   969 lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R"   970 -- {* Classical'' elimination rule -- does not require proving   971 @{prop "X:C"}. *}   972 by (unfold Inter_def) blast   973   974 text {*   975 \medskip Image of a set under a function. Frequently @{term b} does   976 not have the syntactic form of @{term "f x"}.   977 *}   978   979 declare image_def [noatp]   980   981 lemma image_eqI [simp, intro]: "b = f x ==> x:A ==> b : fA"   982 by (unfold image_def) blast   983   984 lemma imageI: "x : A ==> f x : f  A"   985 by (rule image_eqI) (rule refl)   986   987 lemma rev_image_eqI: "x:A ==> b = f x ==> b : fA"   988 -- {* This version's more effective when we already have the   989 required @{term x}. *}   990 by (unfold image_def) blast   991   992 lemma imageE [elim!]:   993 "b : (%x. f x)A ==> (!!x. b = f x ==> x:A ==> P) ==> P"   994 -- {* The eta-expansion gives variable-name preservation. *}   995 by (unfold image_def) blast   996   997 lemma image_Un: "f(A Un B) = fA Un fB"   998 by blast   999   1000 lemma image_eq_UN: "fA = (UN x:A. {f x})"   1001 by blast   1002   1003 lemma image_iff: "(z : fA) = (EX x:A. z = f x)"   1004 by blast   1005   1006 lemma image_subset_iff: "(fA \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)"   1007 -- {* This rewrite rule would confuse users if made default. *}   1008 by blast   1009   1010 lemma subset_image_iff: "(B \<subseteq> fA) = (EX AA. AA \<subseteq> A & B = fAA)"   1011 apply safe   1012 prefer 2 apply fast   1013 apply (rule_tac x = "{a. a : A & f a : B}" in exI, fast)   1014 done   1015   1016 lemma image_subsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> fA \<subseteq> B"   1017 -- {* Replaces the three steps @{text subsetI}, @{text imageE},   1018 @{text hypsubst}, but breaks too many existing proofs. *}   1019 by blast   1020   1021 text {*   1022 \medskip Range of a function -- just a translation for image!   1023 *}   1024   1025 lemma range_eqI: "b = f x ==> b \<in> range f"   1026 by simp   1027   1028 lemma rangeI: "f x \<in> range f"   1029 by simp   1030   1031 lemma rangeE [elim?]: "b \<in> range (\<lambda>x. f x) ==> (!!x. b = f x ==> P) ==> P"   1032 by blast   1033   1034   1035 subsubsection {* Set reasoning tools *}   1036   1037 text {*   1038 Rewrite rules for boolean case-splitting: faster than @{text   1039 "split_if [split]"}.   1040 *}   1041   1042 lemma split_if_eq1: "((if Q then x else y) = b) = ((Q --> x = b) & (~ Q --> y = b))"   1043 by (rule split_if)   1044   1045 lemma split_if_eq2: "(a = (if Q then x else y)) = ((Q --> a = x) & (~ Q --> a = y))"   1046 by (rule split_if)   1047   1048 text {*   1049 Split ifs on either side of the membership relation. Not for @{text   1050 "[simp]"} -- can cause goals to blow up!   1051 *}   1052   1053 lemma split_if_mem1: "((if Q then x else y) : b) = ((Q --> x : b) & (~ Q --> y : b))"   1054 by (rule split_if)   1055   1056 lemma split_if_mem2: "(a : (if Q then x else y)) = ((Q --> a : x) & (~ Q --> a : y))"   1057 by (rule split_if [where P="%S. a : S"])   1058   1059 lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2   1060   1061 (*Would like to add these, but the existing code only searches for the   1062 outer-level constant, which in this case is just "op :"; we instead need   1063 to use term-nets to associate patterns with rules. Also, if a rule fails to   1064 apply, then the formula should be kept.   1065 [("HOL.uminus", Compl_iff RS iffD1), ("HOL.minus", [Diff_iff RS iffD1]),   1066 ("Int", [IntD1,IntD2]),   1067 ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]   1068 *)   1069   1070 ML {*   1071 val mksimps_pairs = [(@{const_name Ball}, @{thms bspec})] @ mksimps_pairs;   1072 *}   1073 declaration {* fn _ =>   1074 Simplifier.map_ss (fn ss => ss setmksimps (mksimps mksimps_pairs))   1075 *}   1076   1077   1078 subsubsection {* The proper subset'' relation *}   1079   1080 lemma psubsetI [intro!,noatp]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B"   1081 by (unfold less_le) blast   1082   1083 lemma psubsetE [elim!,noatp]:   1084 "[|A \<subset> B; [|A \<subseteq> B; ~ (B\<subseteq>A)|] ==> R|] ==> R"   1085 by (unfold less_le) blast   1086   1087 lemma psubset_insert_iff:   1088 "(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)"   1089 by (auto simp add: less_le subset_insert_iff)   1090   1091 lemma psubset_eq: "(A \<subset> B) = (A \<subseteq> B & A \<noteq> B)"   1092 by (simp only: less_le)   1093   1094 lemma psubset_imp_subset: "A \<subset> B ==> A \<subseteq> B"   1095 by (simp add: psubset_eq)   1096   1097 lemma psubset_trans: "[| A \<subset> B; B \<subset> C |] ==> A \<subset> C"   1098 apply (unfold less_le)   1099 apply (auto dest: subset_antisym)   1100 done   1101   1102 lemma psubsetD: "[| A \<subset> B; c \<in> A |] ==> c \<in> B"   1103 apply (unfold less_le)   1104 apply (auto dest: subsetD)   1105 done   1106   1107 lemma psubset_subset_trans: "A \<subset> B ==> B \<subseteq> C ==> A \<subset> C"   1108 by (auto simp add: psubset_eq)   1109   1110 lemma subset_psubset_trans: "A \<subseteq> B ==> B \<subset> C ==> A \<subset> C"   1111 by (auto simp add: psubset_eq)   1112   1113 lemma psubset_imp_ex_mem: "A \<subset> B ==> \<exists>b. b \<in> (B - A)"   1114 by (unfold less_le) blast   1115   1116 lemma atomize_ball:   1117 "(!!x. x \<in> A ==> P x) == Trueprop (\<forall>x\<in>A. P x)"   1118 by (simp only: Ball_def atomize_all atomize_imp)   1119   1120 lemmas [symmetric, rulify] = atomize_ball   1121 and [symmetric, defn] = atomize_ball   1122   1123   1124 subsection {* Further set-theory lemmas *}   1125   1126 subsubsection {* Derived rules involving subsets. *}   1127   1128 text {* @{text insert}. *}   1129   1130 lemma subset_insertI: "B \<subseteq> insert a B"   1131 by (rule subsetI) (erule insertI2)   1132   1133 lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B"   1134 by blast   1135   1136 lemma subset_insert: "x \<notin> A ==> (A \<subseteq> insert x B) = (A \<subseteq> B)"   1137 by blast   1138   1139   1140 text {* \medskip Big Union -- least upper bound of a set. *}   1141   1142 lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A"   1143 by (iprover intro: subsetI UnionI)   1144   1145 lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C"   1146 by (iprover intro: subsetI elim: UnionE dest: subsetD)   1147   1148   1149 text {* \medskip General union. *}   1150   1151 lemma UN_upper: "a \<in> A ==> B a \<subseteq> (\<Union>x\<in>A. B x)"   1152 by blast   1153   1154 lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C"   1155 by (iprover intro: subsetI elim: UN_E dest: subsetD)   1156   1157   1158 text {* \medskip Big Intersection -- greatest lower bound of a set. *}   1159   1160 lemma Inter_lower: "B \<in> A ==> Inter A \<subseteq> B"   1161 by blast   1162   1163 lemma Inter_subset:   1164 "[| !!X. X \<in> A ==> X \<subseteq> B; A ~= {} |] ==> \<Inter>A \<subseteq> B"   1165 by blast   1166   1167 lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A"   1168 by (iprover intro: InterI subsetI dest: subsetD)   1169   1170 lemma INT_lower: "a \<in> A ==> (\<Inter>x\<in>A. B x) \<subseteq> B a"   1171 by blast   1172   1173 lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)"   1174 by (iprover intro: INT_I subsetI dest: subsetD)   1175   1176   1177 text {* \medskip Finite Union -- the least upper bound of two sets. *}   1178   1179 lemma Un_upper1: "A \<subseteq> A \<union> B"   1180 by blast   1181   1182 lemma Un_upper2: "B \<subseteq> A \<union> B"   1183 by blast   1184   1185 lemma Un_least: "A \<subseteq> C ==> B \<subseteq> C ==> A \<union> B \<subseteq> C"   1186 by blast   1187   1188   1189 text {* \medskip Finite Intersection -- the greatest lower bound of two sets. *}   1190   1191 lemma Int_lower1: "A \<inter> B \<subseteq> A"   1192 by blast   1193   1194 lemma Int_lower2: "A \<inter> B \<subseteq> B"   1195 by blast   1196   1197 lemma Int_greatest: "C \<subseteq> A ==> C \<subseteq> B ==> C \<subseteq> A \<inter> B"   1198 by blast   1199   1200   1201 text {* \medskip Set difference. *}   1202   1203 lemma Diff_subset: "A - B \<subseteq> A"   1204 by blast   1205   1206 lemma Diff_subset_conv: "(A - B \<subseteq> C) = (A \<subseteq> B \<union> C)"   1207 by blast   1208   1209   1210 subsubsection {* Equalities involving union, intersection, inclusion, etc. *}   1211   1212 text {* @{text "{}"}. *}   1213   1214 lemma Collect_const [simp]: "{s. P} = (if P then UNIV else {})"   1215 -- {* supersedes @{text "Collect_False_empty"} *}   1216 by auto   1217   1218 lemma subset_empty [simp]: "(A \<subseteq> {}) = (A = {})"   1219 by blast   1220   1221 lemma not_psubset_empty [iff]: "\<not> (A < {})"   1222 by (unfold less_le) blast   1223   1224 lemma Collect_empty_eq [simp]: "(Collect P = {}) = (\<forall>x. \<not> P x)"   1225 by blast   1226   1227 lemma empty_Collect_eq [simp]: "({} = Collect P) = (\<forall>x. \<not> P x)"   1228 by blast   1229   1230 lemma Collect_neg_eq: "{x. \<not> P x} = - {x. P x}"   1231 by blast   1232   1233 lemma Collect_disj_eq: "{x. P x | Q x} = {x. P x} \<union> {x. Q x}"   1234 by blast   1235   1236 lemma Collect_imp_eq: "{x. P x --> Q x} = -{x. P x} \<union> {x. Q x}"   1237 by blast   1238   1239 lemma Collect_conj_eq: "{x. P x & Q x} = {x. P x} \<inter> {x. Q x}"   1240 by blast   1241   1242 lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"   1243 by blast   1244   1245 lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"   1246 by blast   1247   1248 lemma Collect_ex_eq [noatp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"   1249 by blast   1250   1251 lemma Collect_bex_eq [noatp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"   1252 by blast   1253   1254   1255 text {* \medskip @{text insert}. *}   1256   1257 lemma insert_is_Un: "insert a A = {a} Un A"   1258 -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a {}"} *}   1259 by blast   1260   1261 lemma insert_not_empty [simp]: "insert a A \<noteq> {}"   1262 by blast   1263   1264 lemmas empty_not_insert = insert_not_empty [symmetric, standard]   1265 declare empty_not_insert [simp]   1266   1267 lemma insert_absorb: "a \<in> A ==> insert a A = A"   1268 -- {* @{text "[simp]"} causes recursive calls when there are nested inserts *}   1269 -- {* with \emph{quadratic} running time *}   1270 by blast   1271   1272 lemma insert_absorb2 [simp]: "insert x (insert x A) = insert x A"   1273 by blast   1274   1275 lemma insert_commute: "insert x (insert y A) = insert y (insert x A)"   1276 by blast   1277   1278 lemma insert_subset [simp]: "(insert x A \<subseteq> B) = (x \<in> B & A \<subseteq> B)"   1279 by blast   1280   1281 lemma mk_disjoint_insert: "a \<in> A ==> \<exists>B. A = insert a B & a \<notin> B"   1282 -- {* use new @{text B} rather than @{text "A - {a}"} to avoid infinite unfolding *}   1283 apply (rule_tac x = "A - {a}" in exI, blast)   1284 done   1285   1286 lemma insert_Collect: "insert a (Collect P) = {u. u \<noteq> a --> P u}"   1287 by auto   1288   1289 lemma UN_insert_distrib: "u \<in> A ==> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"   1290 by blast   1291   1292 lemma insert_inter_insert[simp]: "insert a A \<inter> insert a B = insert a (A \<inter> B)"   1293 by blast   1294   1295 lemma insert_disjoint [simp,noatp]:   1296 "(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})"   1297 "({} = insert a A \<inter> B) = (a \<notin> B \<and> {} = A \<inter> B)"   1298 by auto   1299   1300 lemma disjoint_insert [simp,noatp]:   1301 "(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})"   1302 "({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)"   1303 by auto   1304   1305 text {* \medskip @{text image}. *}   1306   1307 lemma image_empty [simp]: "f{} = {}"   1308 by blast   1309   1310 lemma image_insert [simp]: "f  insert a B = insert (f a) (fB)"   1311 by blast   1312   1313 lemma image_constant: "x \<in> A ==> (\<lambda>x. c)  A = {c}"   1314 by auto   1315   1316 lemma image_constant_conv: "(%x. c)  A = (if A = {} then {} else {c})"   1317 by auto   1318   1319 lemma image_image: "f  (g  A) = (\<lambda>x. f (g x))  A"   1320 by blast   1321   1322 lemma insert_image [simp]: "x \<in> A ==> insert (f x) (fA) = fA"   1323 by blast   1324   1325 lemma image_is_empty [iff]: "(fA = {}) = (A = {})"   1326 by blast   1327   1328   1329 lemma image_Collect [noatp]: "f  {x. P x} = {f x | x. P x}"   1330 -- {* NOT suitable as a default simprule: the RHS isn't simpler than the LHS,   1331 with its implicit quantifier and conjunction. Also image enjoys better   1332 equational properties than does the RHS. *}   1333 by blast   1334   1335 lemma if_image_distrib [simp]:   1336 "(\<lambda>x. if P x then f x else g x)  S   1337 = (f  (S \<inter> {x. P x})) \<union> (g  (S \<inter> {x. \<not> P x}))"   1338 by (auto simp add: image_def)   1339   1340 lemma image_cong: "M = N ==> (!!x. x \<in> N ==> f x = g x) ==> fM = gN"   1341 by (simp add: image_def)   1342   1343   1344 text {* \medskip @{text range}. *}   1345   1346 lemma full_SetCompr_eq [noatp]: "{u. \<exists>x. u = f x} = range f"   1347 by auto   1348   1349 lemma range_composition: "range (\<lambda>x. f (g x)) = frange g"   1350 by (subst image_image, simp)   1351   1352   1353 text {* \medskip @{text Int} *}   1354   1355 lemma Int_absorb [simp]: "A \<inter> A = A"   1356 by blast   1357   1358 lemma Int_left_absorb: "A \<inter> (A \<inter> B) = A \<inter> B"   1359 by blast   1360   1361 lemma Int_commute: "A \<inter> B = B \<inter> A"   1362 by blast   1363   1364 lemma Int_left_commute: "A \<inter> (B \<inter> C) = B \<inter> (A \<inter> C)"   1365 by blast   1366   1367 lemma Int_assoc: "(A \<inter> B) \<inter> C = A \<inter> (B \<inter> C)"   1368 by blast   1369   1370 lemmas Int_ac = Int_assoc Int_left_absorb Int_commute Int_left_commute   1371 -- {* Intersection is an AC-operator *}   1372   1373 lemma Int_absorb1: "B \<subseteq> A ==> A \<inter> B = B"   1374 by blast   1375   1376 lemma Int_absorb2: "A \<subseteq> B ==> A \<inter> B = A"   1377 by blast   1378   1379 lemma Int_empty_left [simp]: "{} \<inter> B = {}"   1380 by blast   1381   1382 lemma Int_empty_right [simp]: "A \<inter> {} = {}"   1383 by blast   1384   1385 lemma disjoint_eq_subset_Compl: "(A \<inter> B = {}) = (A \<subseteq> -B)"   1386 by blast   1387   1388 lemma disjoint_iff_not_equal: "(A \<inter> B = {}) = (\<forall>x\<in>A. \<forall>y\<in>B. x \<noteq> y)"   1389 by blast   1390   1391 lemma Int_UNIV_left [simp]: "UNIV \<inter> B = B"   1392 by blast   1393   1394 lemma Int_UNIV_right [simp]: "A \<inter> UNIV = A"   1395 by blast   1396   1397 lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"   1398 by blast   1399   1400 lemma Int_Un_distrib: "A \<inter> (B \<union> C) = (A \<inter> B) \<union> (A \<inter> C)"   1401 by blast   1402   1403 lemma Int_Un_distrib2: "(B \<union> C) \<inter> A = (B \<inter> A) \<union> (C \<inter> A)"   1404 by blast   1405   1406 lemma Int_UNIV [simp,noatp]: "(A \<inter> B = UNIV) = (A = UNIV & B = UNIV)"   1407 by blast   1408   1409 lemma Int_subset_iff [simp]: "(C \<subseteq> A \<inter> B) = (C \<subseteq> A & C \<subseteq> B)"   1410 by blast   1411   1412 lemma Int_Collect: "(x \<in> A \<inter> {x. P x}) = (x \<in> A & P x)"   1413 by blast   1414   1415   1416 text {* \medskip @{text Un}. *}   1417   1418 lemma Un_absorb [simp]: "A \<union> A = A"   1419 by blast   1420   1421 lemma Un_left_absorb: "A \<union> (A \<union> B) = A \<union> B"   1422 by blast   1423   1424 lemma Un_commute: "A \<union> B = B \<union> A"   1425 by blast   1426   1427 lemma Un_left_commute: "A \<union> (B \<union> C) = B \<union> (A \<union> C)"   1428 by blast   1429   1430 lemma Un_assoc: "(A \<union> B) \<union> C = A \<union> (B \<union> C)"   1431 by blast   1432   1433 lemmas Un_ac = Un_assoc Un_left_absorb Un_commute Un_left_commute   1434 -- {* Union is an AC-operator *}   1435   1436 lemma Un_absorb1: "A \<subseteq> B ==> A \<union> B = B"   1437 by blast   1438   1439 lemma Un_absorb2: "B \<subseteq> A ==> A \<union> B = A"   1440 by blast   1441   1442 lemma Un_empty_left [simp]: "{} \<union> B = B"   1443 by blast   1444   1445 lemma Un_empty_right [simp]: "A \<union> {} = A"   1446 by blast   1447   1448 lemma Un_UNIV_left [simp]: "UNIV \<union> B = UNIV"   1449 by blast   1450   1451 lemma Un_UNIV_right [simp]: "A \<union> UNIV = UNIV"   1452 by blast   1453   1454 lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"   1455 by blast   1456   1457 lemma Un_insert_left [simp]: "(insert a B) \<union> C = insert a (B \<union> C)"   1458 by blast   1459   1460 lemma Un_insert_right [simp]: "A \<union> (insert a B) = insert a (A \<union> B)"   1461 by blast   1462   1463 lemma Int_insert_left:   1464 "(insert a B) Int C = (if a \<in> C then insert a (B \<inter> C) else B \<inter> C)"   1465 by auto   1466   1467 lemma Int_insert_right:   1468 "A \<inter> (insert a B) = (if a \<in> A then insert a (A \<inter> B) else A \<inter> B)"   1469 by auto   1470   1471 lemma Un_Int_distrib: "A \<union> (B \<inter> C) = (A \<union> B) \<inter> (A \<union> C)"   1472 by blast   1473   1474 lemma Un_Int_distrib2: "(B \<inter> C) \<union> A = (B \<union> A) \<inter> (C \<union> A)"   1475 by blast   1476   1477 lemma Un_Int_crazy:   1478 "(A \<inter> B) \<union> (B \<inter> C) \<union> (C \<inter> A) = (A \<union> B) \<inter> (B \<union> C) \<inter> (C \<union> A)"   1479 by blast   1480   1481 lemma subset_Un_eq: "(A \<subseteq> B) = (A \<union> B = B)"   1482 by blast   1483   1484 lemma Un_empty [iff]: "(A \<union> B = {}) = (A = {} & B = {})"   1485 by blast   1486   1487 lemma Un_subset_iff [simp]: "(A \<union> B \<subseteq> C) = (A \<subseteq> C & B \<subseteq> C)"   1488 by blast   1489   1490 lemma Un_Diff_Int: "(A - B) \<union> (A \<inter> B) = A"   1491 by blast   1492   1493 lemma Diff_Int2: "A \<inter> C - B \<inter> C = A \<inter> C - B"   1494 by blast   1495   1496   1497 text {* \medskip Set complement *}   1498   1499 lemma Compl_disjoint [simp]: "A \<inter> -A = {}"   1500 by blast   1501   1502 lemma Compl_disjoint2 [simp]: "-A \<inter> A = {}"   1503 by blast   1504   1505 lemma Compl_partition: "A \<union> -A = UNIV"   1506 by blast   1507   1508 lemma Compl_partition2: "-A \<union> A = UNIV"   1509 by blast   1510   1511 lemma double_complement [simp]: "- (-A) = (A::'a set)"   1512 by blast   1513   1514 lemma Compl_Un [simp]: "-(A \<union> B) = (-A) \<inter> (-B)"   1515 by blast   1516   1517 lemma Compl_Int [simp]: "-(A \<inter> B) = (-A) \<union> (-B)"   1518 by blast   1519   1520 lemma Compl_UN [simp]: "-(\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"   1521 by blast   1522   1523 lemma Compl_INT [simp]: "-(\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"   1524 by blast   1525   1526 lemma subset_Compl_self_eq: "(A \<subseteq> -A) = (A = {})"   1527 by blast   1528   1529 lemma Un_Int_assoc_eq: "((A \<inter> B) \<union> C = A \<inter> (B \<union> C)) = (C \<subseteq> A)"   1530 -- {* Halmos, Naive Set Theory, page 16. *}   1531 by blast   1532   1533 lemma Compl_UNIV_eq [simp]: "-UNIV = {}"   1534 by blast   1535   1536 lemma Compl_empty_eq [simp]: "-{} = UNIV"   1537 by blast   1538   1539 lemma Compl_subset_Compl_iff [iff]: "(-A \<subseteq> -B) = (B \<subseteq> A)"   1540 by blast   1541   1542 lemma Compl_eq_Compl_iff [iff]: "(-A = -B) = (A = (B::'a set))"   1543 by blast   1544   1545   1546 text {* \medskip @{text Union}. *}   1547   1548 lemma Union_empty [simp]: "Union({}) = {}"   1549 by blast   1550   1551 lemma Union_UNIV [simp]: "Union UNIV = UNIV"   1552 by blast   1553   1554 lemma Union_insert [simp]: "Union (insert a B) = a \<union> \<Union>B"   1555 by blast   1556   1557 lemma Union_Un_distrib [simp]: "\<Union>(A Un B) = \<Union>A \<union> \<Union>B"   1558 by blast   1559   1560 lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"   1561 by blast   1562   1563 lemma Union_empty_conv [simp,noatp]: "(\<Union>A = {}) = (\<forall>x\<in>A. x = {})"   1564 by blast   1565   1566 lemma empty_Union_conv [simp,noatp]: "({} = \<Union>A) = (\<forall>x\<in>A. x = {})"   1567 by blast   1568   1569 lemma Union_disjoint: "(\<Union>C \<inter> A = {}) = (\<forall>B\<in>C. B \<inter> A = {})"   1570 by blast   1571   1572   1573 text {* \medskip @{text Inter}. *}   1574   1575 lemma Inter_empty [simp]: "\<Inter>{} = UNIV"   1576 by blast   1577   1578 lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}"   1579 by blast   1580   1581 lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"   1582 by blast   1583   1584 lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"   1585 by blast   1586   1587 lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"   1588 by blast   1589   1590 lemma Inter_UNIV_conv [simp,noatp]:   1591 "(\<Inter>A = UNIV) = (\<forall>x\<in>A. x = UNIV)"   1592 "(UNIV = \<Inter>A) = (\<forall>x\<in>A. x = UNIV)"   1593 by blast+   1594   1595   1596 text {*   1597 \medskip @{text UN} and @{text INT}.   1598   1599 Basic identities: *}   1600   1601 lemma UN_empty [simp,noatp]: "(\<Union>x\<in>{}. B x) = {}"   1602 by blast   1603   1604 lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"   1605 by blast   1606   1607 lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"   1608 by blast   1609   1610 lemma UN_absorb: "k \<in> I ==> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"   1611 by auto   1612   1613 lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"   1614 by blast   1615   1616 lemma INT_absorb: "k \<in> I ==> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"   1617 by blast   1618   1619 lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"   1620 by blast   1621   1622 lemma UN_Un[simp]: "(\<Union>i \<in> A \<union> B. M i) = (\<Union>i\<in>A. M i) \<union> (\<Union>i\<in>B. M i)"   1623 by blast   1624   1625 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)"   1626 by blast   1627   1628 lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"   1629 by blast   1630   1631 lemma INT_subset_iff: "(B \<subseteq> (\<Inter>i\<in>I. A i)) = (\<forall>i\<in>I. B \<subseteq> A i)"   1632 by blast   1633   1634 lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"   1635 by blast   1636   1637 lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"   1638 by blast   1639   1640 lemma INT_insert_distrib:   1641 "u \<in> A ==> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"   1642 by blast   1643   1644 lemma Union_image_eq [simp]: "\<Union>(BA) = (\<Union>x\<in>A. B x)"   1645 by blast   1646   1647 lemma image_Union: "f  \<Union>S = (\<Union>x\<in>S. f  x)"   1648 by blast   1649   1650 lemma Inter_image_eq [simp]: "\<Inter>(BA) = (\<Inter>x\<in>A. B x)"   1651 by blast   1652   1653 lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"   1654 by auto   1655   1656 lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"   1657 by auto   1658   1659 lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"   1660 by blast   1661   1662 lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"   1663 -- {* Look: it has an \emph{existential} quantifier *}   1664 by blast   1665   1666 lemma UNION_empty_conv[simp]:   1667 "({} = (UN x:A. B x)) = (\<forall>x\<in>A. B x = {})"   1668 "((UN x:A. B x) = {}) = (\<forall>x\<in>A. B x = {})"   1669 by blast+   1670   1671 lemma INTER_UNIV_conv[simp]:   1672 "(UNIV = (INT x:A. B x)) = (\<forall>x\<in>A. B x = UNIV)"   1673 "((INT x:A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"   1674 by blast+   1675   1676   1677 text {* \medskip Distributive laws: *}   1678   1679 lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"   1680 by blast   1681   1682 lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"   1683 by blast   1684   1685 lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(AC) \<union> \<Union>(BC)"   1686 -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}   1687 -- {* Union of a family of unions *}   1688 by blast   1689   1690 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)"   1691 -- {* Equivalent version *}   1692 by blast   1693   1694 lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"   1695 by blast   1696   1697 lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(AC) \<inter> \<Inter>(BC)"   1698 by blast   1699   1700 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)"   1701 -- {* Equivalent version *}   1702 by blast   1703   1704 lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"   1705 -- {* Halmos, Naive Set Theory, page 35. *}   1706 by blast   1707   1708 lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"   1709 by blast   1710   1711 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)"   1712 by blast   1713   1714 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)"   1715 by blast   1716   1717   1718 text {* \medskip Bounded quantifiers.   1719   1720 The following are not added to the default simpset because   1721 (a) they duplicate the body and (b) there are no similar rules for @{text Int}. *}   1722   1723 lemma ball_Un: "(\<forall>x \<in> A \<union> B. P x) = ((\<forall>x\<in>A. P x) & (\<forall>x\<in>B. P x))"   1724 by blast   1725   1726 lemma bex_Un: "(\<exists>x \<in> A \<union> B. P x) = ((\<exists>x\<in>A. P x) | (\<exists>x\<in>B. P x))"   1727 by blast   1728   1729 lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) = (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"   1730 by blast   1731   1732 lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) = (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"   1733 by blast   1734   1735   1736 text {* \medskip Set difference. *}   1737   1738 lemma Diff_eq: "A - B = A \<inter> (-B)"   1739 by blast   1740   1741 lemma Diff_eq_empty_iff [simp,noatp]: "(A - B = {}) = (A \<subseteq> B)"   1742 by blast   1743   1744 lemma Diff_cancel [simp]: "A - A = {}"   1745 by blast   1746   1747 lemma Diff_idemp [simp]: "(A - B) - B = A - (B::'a set)"   1748 by blast   1749   1750 lemma Diff_triv: "A \<inter> B = {} ==> A - B = A"   1751 by (blast elim: equalityE)   1752   1753 lemma empty_Diff [simp]: "{} - A = {}"   1754 by blast   1755   1756 lemma Diff_empty [simp]: "A - {} = A"   1757 by blast   1758   1759 lemma Diff_UNIV [simp]: "A - UNIV = {}"   1760 by blast   1761   1762 lemma Diff_insert0 [simp,noatp]: "x \<notin> A ==> A - insert x B = A - B"   1763 by blast   1764   1765 lemma Diff_insert: "A - insert a B = A - B - {a}"   1766 -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a 0"} *}   1767 by blast   1768   1769 lemma Diff_insert2: "A - insert a B = A - {a} - B"   1770 -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a 0"} *}   1771 by blast   1772   1773 lemma insert_Diff_if: "insert x A - B = (if x \<in> B then A - B else insert x (A - B))"   1774 by auto   1775   1776 lemma insert_Diff1 [simp]: "x \<in> B ==> insert x A - B = A - B"   1777 by blast   1778   1779 lemma insert_Diff_single[simp]: "insert a (A - {a}) = insert a A"   1780 by blast   1781   1782 lemma insert_Diff: "a \<in> A ==> insert a (A - {a}) = A"   1783 by blast   1784   1785 lemma Diff_insert_absorb: "x \<notin> A ==> (insert x A) - {x} = A"   1786 by auto   1787   1788 lemma Diff_disjoint [simp]: "A \<inter> (B - A) = {}"   1789 by blast   1790   1791 lemma Diff_partition: "A \<subseteq> B ==> A \<union> (B - A) = B"   1792 by blast   1793   1794 lemma double_diff: "A \<subseteq> B ==> B \<subseteq> C ==> B - (C - A) = A"   1795 by blast   1796   1797 lemma Un_Diff_cancel [simp]: "A \<union> (B - A) = A \<union> B"   1798 by blast   1799   1800 lemma Un_Diff_cancel2 [simp]: "(B - A) \<union> A = B \<union> A"   1801 by blast   1802   1803 lemma Diff_Un: "A - (B \<union> C) = (A - B) \<inter> (A - C)"   1804 by blast   1805   1806 lemma Diff_Int: "A - (B \<inter> C) = (A - B) \<union> (A - C)"   1807 by blast   1808   1809 lemma Un_Diff: "(A \<union> B) - C = (A - C) \<union> (B - C)"   1810 by blast   1811   1812 lemma Int_Diff: "(A \<inter> B) - C = A \<inter> (B - C)"   1813 by blast   1814   1815 lemma Diff_Int_distrib: "C \<inter> (A - B) = (C \<inter> A) - (C \<inter> B)"   1816 by blast   1817   1818 lemma Diff_Int_distrib2: "(A - B) \<inter> C = (A \<inter> C) - (B \<inter> C)"   1819 by blast   1820   1821 lemma Diff_Compl [simp]: "A - (- B) = A \<inter> B"   1822 by auto   1823   1824 lemma Compl_Diff_eq [simp]: "- (A - B) = -A \<union> B"   1825 by blast   1826   1827   1828 text {* \medskip Quantification over type @{typ bool}. *}   1829   1830 lemma bool_induct: "P True \<Longrightarrow> P False \<Longrightarrow> P x"   1831 by (cases x) auto   1832   1833 lemma all_bool_eq: "(\<forall>b. P b) \<longleftrightarrow> P True \<and> P False"   1834 by (auto intro: bool_induct)   1835   1836 lemma bool_contrapos: "P x \<Longrightarrow> \<not> P False \<Longrightarrow> P True"   1837 by (cases x) auto   1838   1839 lemma ex_bool_eq: "(\<exists>b. P b) \<longleftrightarrow> P True \<or> P False"   1840 by (auto intro: bool_contrapos)   1841   1842 lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"   1843 by (auto simp add: split_if_mem2)   1844   1845 lemma UN_bool_eq: "(\<Union>b::bool. A b) = (A True \<union> A False)"   1846 by (auto intro: bool_contrapos)   1847   1848 lemma INT_bool_eq: "(\<Inter>b::bool. A b) = (A True \<inter> A False)"   1849 by (auto intro: bool_induct)   1850   1851 text {* \medskip @{text Pow} *}   1852   1853 lemma Pow_empty [simp]: "Pow {} = {{}}"   1854 by (auto simp add: Pow_def)   1855   1856 lemma Pow_insert: "Pow (insert a A) = Pow A \<union> (insert a  Pow A)"   1857 by (blast intro: image_eqI [where ?x = "u - {a}", standard])   1858   1859 lemma Pow_Compl: "Pow (- A) = {-B | B. A \<in> Pow B}"   1860 by (blast intro: exI [where ?x = "- u", standard])   1861   1862 lemma Pow_UNIV [simp]: "Pow UNIV = UNIV"   1863 by blast   1864   1865 lemma Un_Pow_subset: "Pow A \<union> Pow B \<subseteq> Pow (A \<union> B)"   1866 by blast   1867   1868 lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"   1869 by blast   1870   1871 lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"   1872 by blast   1873   1874 lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"   1875 by blast   1876   1877 lemma Pow_Int_eq [simp]: "Pow (A \<inter> B) = Pow A \<inter> Pow B"   1878 by blast   1879   1880 lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"   1881 by blast   1882   1883   1884 text {* \medskip Miscellany. *}   1885   1886 lemma set_eq_subset: "(A = B) = (A \<subseteq> B & B \<subseteq> A)"   1887 by blast   1888   1889 lemma subset_iff: "(A \<subseteq> B) = (\<forall>t. t \<in> A --> t \<in> B)"   1890 by blast   1891   1892 lemma subset_iff_psubset_eq: "(A \<subseteq> B) = ((A \<subset> B) | (A = B))"   1893 by (unfold less_le) blast   1894   1895 lemma all_not_in_conv [simp]: "(\<forall>x. x \<notin> A) = (A = {})"   1896 by blast   1897   1898 lemma ex_in_conv: "(\<exists>x. x \<in> A) = (A \<noteq> {})"   1899 by blast   1900   1901 lemma distinct_lemma: "f x \<noteq> f y ==> x \<noteq> y"   1902 by iprover   1903   1904   1905 text {* \medskip Miniscoping: pushing in quantifiers and big Unions   1906 and Intersections. *}   1907   1908 lemma UN_simps [simp]:   1909 "!!a B C. (UN x:C. insert a (B x)) = (if C={} then {} else insert a (UN x:C. B x))"   1910 "!!A B C. (UN x:C. A x Un B) = ((if C={} then {} else (UN x:C. A x) Un B))"   1911 "!!A B C. (UN x:C. A Un B x) = ((if C={} then {} else A Un (UN x:C. B x)))"   1912 "!!A B C. (UN x:C. A x Int B) = ((UN x:C. A x) Int B)"   1913 "!!A B C. (UN x:C. A Int B x) = (A Int (UN x:C. B x))"   1914 "!!A B C. (UN x:C. A x - B) = ((UN x:C. A x) - B)"   1915 "!!A B C. (UN x:C. A - B x) = (A - (INT x:C. B x))"   1916 "!!A B. (UN x: Union A. B x) = (UN y:A. UN x:y. B x)"   1917 "!!A B C. (UN z: UNION A B. C z) = (UN x:A. UN z: B(x). C z)"   1918 "!!A B f. (UN x:fA. B x) = (UN a:A. B (f a))"   1919 by auto   1920   1921 lemma INT_simps [simp]:   1922 "!!A B C. (INT x:C. A x Int B) = (if C={} then UNIV else (INT x:C. A x) Int B)"   1923 "!!A B C. (INT x:C. A Int B x) = (if C={} then UNIV else A Int (INT x:C. B x))"   1924 "!!A B C. (INT x:C. A x - B) = (if C={} then UNIV else (INT x:C. A x) - B)"   1925 "!!A B C. (INT x:C. A - B x) = (if C={} then UNIV else A - (UN x:C. B x))"   1926 "!!a B C. (INT x:C. insert a (B x)) = insert a (INT x:C. B x)"   1927 "!!A B C. (INT x:C. A x Un B) = ((INT x:C. A x) Un B)"   1928 "!!A B C. (INT x:C. A Un B x) = (A Un (INT x:C. B x))"   1929 "!!A B. (INT x: Union A. B x) = (INT y:A. INT x:y. B x)"   1930 "!!A B C. (INT z: UNION A B. C z) = (INT x:A. INT z: B(x). C z)"   1931 "!!A B f. (INT x:fA. B x) = (INT a:A. B (f a))"   1932 by auto   1933   1934 lemma ball_simps [simp,noatp]:   1935 "!!A P Q. (ALL x:A. P x | Q) = ((ALL x:A. P x) | Q)"   1936 "!!A P Q. (ALL x:A. P | Q x) = (P | (ALL x:A. Q x))"   1937 "!!A P Q. (ALL x:A. P --> Q x) = (P --> (ALL x:A. Q x))"   1938 "!!A P Q. (ALL x:A. P x --> Q) = ((EX x:A. P x) --> Q)"   1939 "!!P. (ALL x:{}. P x) = True"   1940 "!!P. (ALL x:UNIV. P x) = (ALL x. P x)"   1941 "!!a B P. (ALL x:insert a B. P x) = (P a & (ALL x:B. P x))"   1942 "!!A P. (ALL x:Union A. P x) = (ALL y:A. ALL x:y. P x)"   1943 "!!A B P. (ALL x: UNION A B. P x) = (ALL a:A. ALL x: B a. P x)"   1944 "!!P Q. (ALL x:Collect Q. P x) = (ALL x. Q x --> P x)"   1945 "!!A P f. (ALL x:fA. P x) = (ALL x:A. P (f x))"   1946 "!!A P. (~(ALL x:A. P x)) = (EX x:A. ~P x)"   1947 by auto   1948   1949 lemma bex_simps [simp,noatp]:   1950 "!!A P Q. (EX x:A. P x & Q) = ((EX x:A. P x) & Q)"   1951 "!!A P Q. (EX x:A. P & Q x) = (P & (EX x:A. Q x))"   1952 "!!P. (EX x:{}. P x) = False"   1953 "!!P. (EX x:UNIV. P x) = (EX x. P x)"   1954 "!!a B P. (EX x:insert a B. P x) = (P(a) | (EX x:B. P x))"   1955 "!!A P. (EX x:Union A. P x) = (EX y:A. EX x:y. P x)"   1956 "!!A B P. (EX x: UNION A B. P x) = (EX a:A. EX x:B a. P x)"   1957 "!!P Q. (EX x:Collect Q. P x) = (EX x. Q x & P x)"   1958 "!!A P f. (EX x:fA. P x) = (EX x:A. P (f x))"   1959 "!!A P. (~(EX x:A. P x)) = (ALL x:A. ~P x)"   1960 by auto   1961   1962 lemma ball_conj_distrib:   1963 "(ALL x:A. P x & Q x) = ((ALL x:A. P x) & (ALL x:A. Q x))"   1964 by blast   1965   1966 lemma bex_disj_distrib:   1967 "(EX x:A. P x | Q x) = ((EX x:A. P x) | (EX x:A. Q x))"   1968 by blast   1969   1970   1971 text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}   1972   1973 lemma UN_extend_simps:   1974 "!!a B C. insert a (UN x:C. B x) = (if C={} then {a} else (UN x:C. insert a (B x)))"   1975 "!!A B C. (UN x:C. A x) Un B = (if C={} then B else (UN x:C. A x Un B))"   1976 "!!A B C. A Un (UN x:C. B x) = (if C={} then A else (UN x:C. A Un B x))"   1977 "!!A B C. ((UN x:C. A x) Int B) = (UN x:C. A x Int B)"   1978 "!!A B C. (A Int (UN x:C. B x)) = (UN x:C. A Int B x)"   1979 "!!A B C. ((UN x:C. A x) - B) = (UN x:C. A x - B)"   1980 "!!A B C. (A - (INT x:C. B x)) = (UN x:C. A - B x)"   1981 "!!A B. (UN y:A. UN x:y. B x) = (UN x: Union A. B x)"   1982 "!!A B C. (UN x:A. UN z: B(x). C z) = (UN z: UNION A B. C z)"   1983 "!!A B f. (UN a:A. B (f a)) = (UN x:fA. B x)"   1984 by auto   1985   1986 lemma INT_extend_simps:   1987 "!!A B C. (INT x:C. A x) Int B = (if C={} then B else (INT x:C. A x Int B))"   1988 "!!A B C. A Int (INT x:C. B x) = (if C={} then A else (INT x:C. A Int B x))"   1989 "!!A B C. (INT x:C. A x) - B = (if C={} then UNIV-B else (INT x:C. A x - B))"   1990 "!!A B C. A - (UN x:C. B x) = (if C={} then A else (INT x:C. A - B x))"   1991 "!!a B C. insert a (INT x:C. B x) = (INT x:C. insert a (B x))"   1992 "!!A B C. ((INT x:C. A x) Un B) = (INT x:C. A x Un B)"   1993 "!!A B C. A Un (INT x:C. B x) = (INT x:C. A Un B x)"   1994 "!!A B. (INT y:A. INT x:y. B x) = (INT x: Union A. B x)"   1995 "!!A B C. (INT x:A. INT z: B(x). C z) = (INT z: UNION A B. C z)"   1996 "!!A B f. (INT a:A. B (f a)) = (INT x:fA. B x)"   1997 by auto   1998   1999   2000 subsubsection {* Monotonicity of various operations *}   2001   2002 lemma image_mono: "A \<subseteq> B ==> fA \<subseteq> fB"   2003 by blast   2004   2005 lemma Pow_mono: "A \<subseteq> B ==> Pow A \<subseteq> Pow B"   2006 by blast   2007   2008 lemma Union_mono: "A \<subseteq> B ==> \<Union>A \<subseteq> \<Union>B"   2009 by blast   2010   2011 lemma Inter_anti_mono: "B \<subseteq> A ==> \<Inter>A \<subseteq> \<Inter>B"   2012 by blast   2013   2014 lemma UN_mono:   2015 "A \<subseteq> B ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>   2016 (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"   2017 by (blast dest: subsetD)   2018   2019 lemma INT_anti_mono:   2020 "B \<subseteq> A ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>   2021 (\<Inter>x\<in>A. f x) \<subseteq> (\<Inter>x\<in>A. g x)"   2022 -- {* The last inclusion is POSITIVE! *}   2023 by (blast dest: subsetD)   2024   2025 lemma insert_mono: "C \<subseteq> D ==> insert a C \<subseteq> insert a D"   2026 by blast   2027   2028 lemma Un_mono: "A \<subseteq> C ==> B \<subseteq> D ==> A \<union> B \<subseteq> C \<union> D"   2029 by blast   2030   2031 lemma Int_mono: "A \<subseteq> C ==> B \<subseteq> D ==> A \<inter> B \<subseteq> C \<inter> D"   2032 by blast   2033   2034 lemma Diff_mono: "A \<subseteq> C ==> D \<subseteq> B ==> A - B \<subseteq> C - D"   2035 by blast   2036   2037 lemma Compl_anti_mono: "A \<subseteq> B ==> -B \<subseteq> -A"   2038 by blast   2039   2040 text {* \medskip Monotonicity of implications. *}   2041   2042 lemma in_mono: "A \<subseteq> B ==> x \<in> A --> x \<in> B"   2043 apply (rule impI)   2044 apply (erule subsetD, assumption)   2045 done   2046   2047 lemma conj_mono: "P1 --> Q1 ==> P2 --> Q2 ==> (P1 & P2) --> (Q1 & Q2)"   2048 by iprover   2049   2050 lemma disj_mono: "P1 --> Q1 ==> P2 --> Q2 ==> (P1 | P2) --> (Q1 | Q2)"   2051 by iprover   2052   2053 lemma imp_mono: "Q1 --> P1 ==> P2 --> Q2 ==> (P1 --> P2) --> (Q1 --> Q2)"   2054 by iprover   2055   2056 lemma imp_refl: "P --> P" ..   2057   2058 lemma ex_mono: "(!!x. P x --> Q x) ==> (EX x. P x) --> (EX x. Q x)"   2059 by iprover   2060   2061 lemma all_mono: "(!!x. P x --> Q x) ==> (ALL x. P x) --> (ALL x. Q x)"   2062 by iprover   2063   2064 lemma Collect_mono: "(!!x. P x --> Q x) ==> Collect P \<subseteq> Collect Q"   2065 by blast   2066   2067 lemma Int_Collect_mono:   2068 "A \<subseteq> B ==> (!!x. x \<in> A ==> P x --> Q x) ==> A \<inter> Collect P \<subseteq> B \<inter> Collect Q"   2069 by blast   2070   2071 lemmas basic_monos =   2072 subset_refl imp_refl disj_mono conj_mono   2073 ex_mono Collect_mono in_mono   2074   2075 lemma eq_to_mono: "a = b ==> c = d ==> b --> d ==> a --> c"   2076 by iprover   2077   2078 lemma eq_to_mono2: "a = b ==> c = d ==> ~ b --> ~ d ==> ~ a --> ~ c"   2079 by iprover   2080   2081   2082 subsection {* Inverse image of a function *}   2083   2084 constdefs   2085 vimage :: "('a => 'b) => 'b set => 'a set" (infixr "-" 90)   2086 [code del]: "f - B == {x. f x : B}"   2087   2088   2089 subsubsection {* Basic rules *}   2090   2091 lemma vimage_eq [simp]: "(a : f - B) = (f a : B)"   2092 by (unfold vimage_def) blast   2093   2094 lemma vimage_singleton_eq: "(a : f - {b}) = (f a = b)"   2095 by simp   2096   2097 lemma vimageI [intro]: "f a = b ==> b:B ==> a : f - B"   2098 by (unfold vimage_def) blast   2099   2100 lemma vimageI2: "f a : A ==> a : f - A"   2101 by (unfold vimage_def) fast   2102   2103 lemma vimageE [elim!]: "a: f - B ==> (!!x. f a = x ==> x:B ==> P) ==> P"   2104 by (unfold vimage_def) blast   2105   2106 lemma vimageD: "a : f - A ==> f a : A"   2107 by (unfold vimage_def) fast   2108   2109   2110 subsubsection {* Equations *}   2111   2112 lemma vimage_empty [simp]: "f - {} = {}"   2113 by blast   2114   2115 lemma vimage_Compl: "f - (-A) = -(f - A)"   2116 by blast   2117   2118 lemma vimage_Un [simp]: "f - (A Un B) = (f - A) Un (f - B)"   2119 by blast   2120   2121 lemma vimage_Int [simp]: "f - (A Int B) = (f - A) Int (f - B)"   2122 by fast   2123   2124 lemma vimage_Union: "f - (Union A) = (UN X:A. f - X)"   2125 by blast   2126   2127 lemma vimage_UN: "f-(UN x:A. B x) = (UN x:A. f - B x)"   2128 by blast   2129   2130 lemma vimage_INT: "f-(INT x:A. B x) = (INT x:A. f - B x)"   2131 by blast   2132   2133 lemma vimage_Collect_eq [simp]: "f - Collect P = {y. P (f y)}"   2134 by blast   2135   2136 lemma vimage_Collect: "(!!x. P (f x) = Q x) ==> f - (Collect P) = Collect Q"   2137 by blast   2138   2139 lemma vimage_insert: "f-(insert a B) = (f-{a}) Un (f-B)"   2140 -- {* NOT suitable for rewriting because of the recurrence of @{term "{a}"}. *}   2141 by blast   2142   2143 lemma vimage_Diff: "f - (A - B) = (f - A) - (f - B)"   2144 by blast   2145   2146 lemma vimage_UNIV [simp]: "f - UNIV = UNIV"   2147 by blast   2148   2149 lemma vimage_eq_UN: "f-B = (UN y: B. f-{y})"   2150 -- {* NOT suitable for rewriting *}   2151 by blast   2152   2153 lemma vimage_mono: "A \<subseteq> B ==> f - A \<subseteq> f - B"   2154 -- {* monotonicity *}   2155 by blast   2156   2157 lemma vimage_image_eq [noatp]: "f - (f  A) = {y. EX x:A. f x = f y}"   2158 by (blast intro: sym)   2159   2160 lemma image_vimage_subset: "f  (f - A) <= A"   2161 by blast   2162   2163 lemma image_vimage_eq [simp]: "f  (f - A) = A Int range f"   2164 by blast   2165   2166 lemma image_Int_subset: "f(A Int B) <= fA Int fB"   2167 by blast   2168   2169 lemma image_diff_subset: "fA - fB <= f(A - B)"   2170 by blast   2171   2172 lemma image_UN: "(f  (UNION A B)) = (UN x:A.(f  (B x)))"   2173 by blast   2174   2175   2176 subsection {* Getting the Contents of a Singleton Set *}   2177   2178 definition contents :: "'a set \<Rightarrow> 'a" where   2179 [code del]: "contents X = (THE x. X = {x})"   2180   2181 lemma contents_eq [simp]: "contents {x} = x"   2182 by (simp add: contents_def)   2183   2184   2185 subsection {* Transitivity rules for calculational reasoning *}   2186   2187 lemma set_rev_mp: "x:A ==> A \<subseteq> B ==> x:B"   2188 by (rule subsetD)   2189   2190 lemma set_mp: "A \<subseteq> B ==> x:A ==> x:B"   2191 by (rule subsetD)   2192   2193 lemmas basic_trans_rules [trans] =   2194 order_trans_rules set_rev_mp set_mp   2195   2196   2197 subsection {* Least value operator *}   2198   2199 lemma Least_mono:   2200 "mono (f::'a::order => 'b::order) ==> EX x:S. ALL y:S. x <= y   2201 ==> (LEAST y. y : f  S) = f (LEAST x. x : S)"   2202 -- {* Courtesy of Stephan Merz *}   2203 apply clarify   2204 apply (erule_tac P = "%x. x : S" in LeastI2_order, fast)   2205 apply (rule LeastI2_order)   2206 apply (auto elim: monoD intro!: order_antisym)   2207 done   2208   2209   2210 subsection {* Rudimentary code generation *}   2211   2212 lemma empty_code [code]: "{} x \<longleftrightarrow> False"   2213 unfolding empty_def Collect_def ..   2214   2215 lemma UNIV_code [code]: "UNIV x \<longleftrightarrow> True"   2216 unfolding UNIV_def Collect_def ..   2217   2218 lemma insert_code [code]: "insert y A x \<longleftrightarrow> y = x \<or> A x"   2219 unfolding insert_def Collect_def mem_def Un_def by auto   2220   2221 lemma inter_code [code]: "(A \<inter> B) x \<longleftrightarrow> A x \<and> B x"   2222 unfolding Int_def Collect_def mem_def ..   2223   2224 lemma union_code [code]: "(A \<union> B) x \<longleftrightarrow> A x \<or> B x"   2225 unfolding Un_def Collect_def mem_def ..   2226   2227 lemma vimage_code [code]: "(f - A) x = A (f x)"   2228 unfolding vimage_def Collect_def mem_def ..   2229   2230   2231 subsection {* Complete lattices *}   2232   2233 notation   2234 less_eq (infix "\<sqsubseteq>" 50) and   2235 less (infix "\<sqsubset>" 50) and   2236 inf (infixl "\<sqinter>" 70) and   2237 sup (infixl "\<squnion>" 65)   2238   2239 class complete_lattice = lattice + bot + top +   2240 fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)   2241 and Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)   2242 assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"   2243 and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"   2244 assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A"   2245 and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z"   2246 begin   2247   2248 lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<le> a}"   2249 by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)   2250   2251 lemma Sup_Inf: "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<le> b}"   2252 by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)   2253   2254 lemma Inf_Univ: "\<Sqinter>UNIV = \<Squnion>{}"   2255 unfolding Sup_Inf by auto   2256   2257 lemma Sup_Univ: "\<Squnion>UNIV = \<Sqinter>{}"   2258 unfolding Inf_Sup by auto   2259   2260 lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"   2261 by (auto intro: antisym Inf_greatest Inf_lower)   2262   2263 lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"   2264 by (auto intro: antisym Sup_least Sup_upper)   2265   2266 lemma Inf_singleton [simp]:   2267 "\<Sqinter>{a} = a"   2268 by (auto intro: antisym Inf_lower Inf_greatest)   2269   2270 lemma Sup_singleton [simp]:   2271 "\<Squnion>{a} = a"   2272 by (auto intro: antisym Sup_upper Sup_least)   2273   2274 lemma Inf_insert_simp:   2275 "\<Sqinter>insert a A = (if A = {} then a else a \<sqinter> \<Sqinter>A)"   2276 by (cases "A = {}") (simp_all, simp add: Inf_insert)   2277   2278 lemma Sup_insert_simp:   2279 "\<Squnion>insert a A = (if A = {} then a else a \<squnion> \<Squnion>A)"   2280 by (cases "A = {}") (simp_all, simp add: Sup_insert)   2281   2282 lemma Inf_binary:   2283 "\<Sqinter>{a, b} = a \<sqinter> b"   2284 by (simp add: Inf_insert_simp)   2285   2286 lemma Sup_binary:   2287 "\<Squnion>{a, b} = a \<squnion> b"   2288 by (simp add: Sup_insert_simp)   2289   2290 lemma bot_def:   2291 "bot = \<Squnion>{}"   2292 by (auto intro: antisym Sup_least)   2293   2294 lemma top_def:   2295 "top = \<Sqinter>{}"   2296 by (auto intro: antisym Inf_greatest)   2297   2298 lemma sup_bot [simp]:   2299 "x \<squnion> bot = x"   2300 using bot_least [of x] by (simp add: le_iff_sup sup_commute)   2301   2302 lemma inf_top [simp]:   2303 "x \<sqinter> top = x"   2304 using top_greatest [of x] by (simp add: le_iff_inf inf_commute)   2305   2306 definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where   2307 "SUPR A f == \<Squnion> (f  A)"   2308   2309 definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where   2310 "INFI A f == \<Sqinter> (f  A)"   2311   2312 end   2313   2314 syntax   2315 "_SUP1" :: "pttrns => 'b => 'b" ("(3SUP _./ _)" [0, 10] 10)   2316 "_SUP" :: "pttrn => 'a set => 'b => 'b" ("(3SUP _:_./ _)" [0, 10] 10)   2317 "_INF1" :: "pttrns => 'b => 'b" ("(3INF _./ _)" [0, 10] 10)   2318 "_INF" :: "pttrn => 'a set => 'b => 'b" ("(3INF _:_./ _)" [0, 10] 10)   2319   2320 translations   2321 "SUP x y. B" == "SUP x. SUP y. B"   2322 "SUP x. B" == "CONST SUPR CONST UNIV (%x. B)"   2323 "SUP x. B" == "SUP x:CONST UNIV. B"   2324 "SUP x:A. B" == "CONST SUPR A (%x. B)"   2325 "INF x y. B" == "INF x. INF y. B"   2326 "INF x. B" == "CONST INFI CONST UNIV (%x. B)"   2327 "INF x. B" == "INF x:CONST UNIV. B"   2328 "INF x:A. B" == "CONST INFI A (%x. B)"   2329   2330 (* To avoid eta-contraction of body: *)   2331 print_translation {*   2332 let   2333 fun btr' syn (A :: Abs abs :: ts) =   2334 let val (x,t) = atomic_abs_tr' abs   2335 in list_comb (Syntax.const syn$ x $A$ t, ts) end

  2336   val const_syntax_name = Sign.const_syntax_name @{theory} o fst o dest_Const

  2337 in

  2338 [(const_syntax_name @{term SUPR}, btr' "_SUP"),(const_syntax_name @{term "INFI"}, btr' "_INF")]

  2339 end

  2340 *}

  2341

  2342 context complete_lattice

  2343 begin

  2344

  2345 lemma le_SUPI: "i : A \<Longrightarrow> M i \<le> (SUP i:A. M i)"

  2346   by (auto simp add: SUPR_def intro: Sup_upper)

  2347

  2348 lemma SUP_leI: "(\<And>i. i : A \<Longrightarrow> M i \<le> u) \<Longrightarrow> (SUP i:A. M i) \<le> u"

  2349   by (auto simp add: SUPR_def intro: Sup_least)

  2350

  2351 lemma INF_leI: "i : A \<Longrightarrow> (INF i:A. M i) \<le> M i"

  2352   by (auto simp add: INFI_def intro: Inf_lower)

  2353

  2354 lemma le_INFI: "(\<And>i. i : A \<Longrightarrow> u \<le> M i) \<Longrightarrow> u \<le> (INF i:A. M i)"

  2355   by (auto simp add: INFI_def intro: Inf_greatest)

  2356

  2357 lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M"

  2358   by (auto intro: antisym SUP_leI le_SUPI)

  2359

  2360 lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M"

  2361   by (auto intro: antisym INF_leI le_INFI)

  2362

  2363 end

  2364

  2365

  2366 subsection {* Bool as complete lattice *}

  2367

  2368 instantiation bool :: complete_lattice

  2369 begin

  2370

  2371 definition

  2372   Inf_bool_def: "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)"

  2373

  2374 definition

  2375   Sup_bool_def: "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)"

  2376

  2377 instance

  2378   by intro_classes (auto simp add: Inf_bool_def Sup_bool_def le_bool_def)

  2379

  2380 end

  2381

  2382 lemma Inf_empty_bool [simp]:

  2383   "\<Sqinter>{}"

  2384   unfolding Inf_bool_def by auto

  2385

  2386 lemma not_Sup_empty_bool [simp]:

  2387   "\<not> Sup {}"

  2388   unfolding Sup_bool_def by auto

  2389

  2390

  2391 subsection {* Fun as complete lattice *}

  2392

  2393 instantiation "fun" :: (type, complete_lattice) complete_lattice

  2394 begin

  2395

  2396 definition

  2397   Inf_fun_def [code del]: "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})"

  2398

  2399 definition

  2400   Sup_fun_def [code del]: "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})"

  2401

  2402 instance

  2403   by intro_classes

  2404     (auto simp add: Inf_fun_def Sup_fun_def le_fun_def

  2405       intro: Inf_lower Sup_upper Inf_greatest Sup_least)

  2406

  2407 end

  2408

  2409 lemma Inf_empty_fun:

  2410   "\<Sqinter>{} = (\<lambda>_. \<Sqinter>{})"

  2411   by rule (auto simp add: Inf_fun_def)

  2412

  2413 lemma Sup_empty_fun:

  2414   "\<Squnion>{} = (\<lambda>_. \<Squnion>{})"

  2415   by rule (auto simp add: Sup_fun_def)

  2416

  2417

  2418 subsection {* Set as lattice *}

  2419

  2420 lemma inf_set_eq: "A \<sqinter> B = A \<inter> B"

  2421   apply (rule subset_antisym)

  2422   apply (rule Int_greatest)

  2423   apply (rule inf_le1)

  2424   apply (rule inf_le2)

  2425   apply (rule inf_greatest)

  2426   apply (rule Int_lower1)

  2427   apply (rule Int_lower2)

  2428   done

  2429

  2430 lemma sup_set_eq: "A \<squnion> B = A \<union> B"

  2431   apply (rule subset_antisym)

  2432   apply (rule sup_least)

  2433   apply (rule Un_upper1)

  2434   apply (rule Un_upper2)

  2435   apply (rule Un_least)

  2436   apply (rule sup_ge1)

  2437   apply (rule sup_ge2)

  2438   done

  2439

  2440 lemma mono_Int: "mono f \<Longrightarrow> f (A \<inter> B) \<subseteq> f A \<inter> f B"

  2441   apply (fold inf_set_eq sup_set_eq)

  2442   apply (erule mono_inf)

  2443   done

  2444

  2445 lemma mono_Un: "mono f \<Longrightarrow> f A \<union> f B \<subseteq> f (A \<union> B)"

  2446   apply (fold inf_set_eq sup_set_eq)

  2447   apply (erule mono_sup)

  2448   done

  2449

  2450 lemma Inf_set_eq: "\<Sqinter>S = \<Inter>S"

  2451   apply (rule subset_antisym)

  2452   apply (rule Inter_greatest)

  2453   apply (erule Inf_lower)

  2454   apply (rule Inf_greatest)

  2455   apply (erule Inter_lower)

  2456   done

  2457

  2458 lemma Sup_set_eq: "\<Squnion>S = \<Union>S"

  2459   apply (rule subset_antisym)

  2460   apply (rule Sup_least)

  2461   apply (erule Union_upper)

  2462   apply (rule Union_least)

  2463   apply (erule Sup_upper)

  2464   done

  2465

  2466 lemma top_set_eq: "top = UNIV"

  2467   by (iprover intro!: subset_antisym subset_UNIV top_greatest)

  2468

  2469 lemma bot_set_eq: "bot = {}"

  2470   by (iprover intro!: subset_antisym empty_subsetI bot_least)

  2471

  2472 no_notation

  2473   less_eq  (infix "\<sqsubseteq>" 50) and

  2474   less (infix "\<sqsubset>" 50) and

  2475   inf  (infixl "\<sqinter>" 70) and

  2476   sup  (infixl "\<squnion>" 65) and

  2477   Inf  ("\<Sqinter>_" [900] 900) and

  2478   Sup  ("\<Squnion>_" [900] 900)

  2479

  2480

  2481 subsection {* Misc theorem and ML bindings *}

  2482

  2483 lemmas equalityI = subset_antisym

  2484 lemmas mem_simps =

  2485   insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff

  2486   mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff

  2487   -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}

  2488

  2489 ML {*

  2490 val Ball_def = @{thm Ball_def}

  2491 val Bex_def = @{thm Bex_def}

  2492 val CollectD = @{thm CollectD}

  2493 val CollectE = @{thm CollectE}

  2494 val CollectI = @{thm CollectI}

  2495 val Collect_conj_eq = @{thm Collect_conj_eq}

  2496 val Collect_mem_eq = @{thm Collect_mem_eq}

  2497 val IntD1 = @{thm IntD1}

  2498 val IntD2 = @{thm IntD2}

  2499 val IntE = @{thm IntE}

  2500 val IntI = @{thm IntI}

  2501 val Int_Collect = @{thm Int_Collect}

  2502 val UNIV_I = @{thm UNIV_I}

  2503 val UNIV_witness = @{thm UNIV_witness}

  2504 val UnE = @{thm UnE}

  2505 val UnI1 = @{thm UnI1}

  2506 val UnI2 = @{thm UnI2}

  2507 val ballE = @{thm ballE}

  2508 val ballI = @{thm ballI}

  2509 val bexCI = @{thm bexCI}

  2510 val bexE = @{thm bexE}

  2511 val bexI = @{thm bexI}

  2512 val bex_triv = @{thm bex_triv}

  2513 val bspec = @{thm bspec}

  2514 val contra_subsetD = @{thm contra_subsetD}

  2515 val distinct_lemma = @{thm distinct_lemma}

  2516 val eq_to_mono = @{thm eq_to_mono}

  2517 val eq_to_mono2 = @{thm eq_to_mono2}

  2518 val equalityCE = @{thm equalityCE}

  2519 val equalityD1 = @{thm equalityD1}

  2520 val equalityD2 = @{thm equalityD2}

  2521 val equalityE = @{thm equalityE}

  2522 val equalityI = @{thm equalityI}

  2523 val imageE = @{thm imageE}

  2524 val imageI = @{thm imageI}

  2525 val image_Un = @{thm image_Un}

  2526 val image_insert = @{thm image_insert}

  2527 val insert_commute = @{thm insert_commute}

  2528 val insert_iff = @{thm insert_iff}

  2529 val mem_Collect_eq = @{thm mem_Collect_eq}

  2530 val rangeE = @{thm rangeE}

  2531 val rangeI = @{thm rangeI}

  2532 val range_eqI = @{thm range_eqI}

  2533 val subsetCE = @{thm subsetCE}

  2534 val subsetD = @{thm subsetD}

  2535 val subsetI = @{thm subsetI}

  2536 val subset_refl = @{thm subset_refl}

  2537 val subset_trans = @{thm subset_trans}

  2538 val vimageD = @{thm vimageD}

  2539 val vimageE = @{thm vimageE}

  2540 val vimageI = @{thm vimageI}

  2541 val vimageI2 = @{thm vimageI2}

  2542 val vimage_Collect = @{thm vimage_Collect}

  2543 val vimage_Int = @{thm vimage_Int}

  2544 val vimage_Un = @{thm vimage_Un}

  2545 *}

  2546

  2547 end
`