src/HOL/Set.thy
 author nipkow Sat May 16 11:28:02 2009 +0200 (2009-05-16) changeset 31166 a90fe83f58ea parent 30814 10dc9bc264b7 child 31197 c1c163ec6c44 permissions -rw-r--r--
"{x. P x & x=t & Q x}" is now rewritten to "if P t & Q t then {t} else {}"
by the new simproc defColl_regroup. More precisely, the simproc pulls an
equation x=t (or t=x) out of a nest of conjunctions to the front where the
simp rule singleton_conj_conv(2) converts to "if".
     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{* Elimination of @{text"{x. \<dots> & x=t & \<dots>}"}. *}   1038   1039 lemma singleton_conj_conv[simp]: "{x. x=a & P x} = (if P a then {a} else {})"   1040 by auto   1041   1042 lemma singleton_conj_conv2[simp]: "{x. a=x & P x} = (if P a then {a} else {})"   1043 by auto   1044   1045 ML{*   1046 local   1047 val Coll_perm_tac = rtac @{thm Collect_cong} 1 THEN rtac @{thm iffI} 1 THEN   1048 ALLGOALS(EVERY'[REPEAT_DETERM o (etac @{thm conjE}),   1049 DEPTH_SOLVE_1 o (ares_tac [@{thm conjI}])])   1050 in   1051 val defColl_regroup = Simplifier.simproc (the_context ())   1052 "defined Collect" ["{x. P x & Q x}"]   1053 (Quantifier1.rearrange_Coll Coll_perm_tac)   1054 end;   1055   1056 Addsimprocs [defColl_regroup];   1057   1058 *}   1059   1060 text {*   1061 Rewrite rules for boolean case-splitting: faster than @{text   1062 "split_if [split]"}.   1063 *}   1064   1065 lemma split_if_eq1: "((if Q then x else y) = b) = ((Q --> x = b) & (~ Q --> y = b))"   1066 by (rule split_if)   1067   1068 lemma split_if_eq2: "(a = (if Q then x else y)) = ((Q --> a = x) & (~ Q --> a = y))"   1069 by (rule split_if)   1070   1071 text {*   1072 Split ifs on either side of the membership relation. Not for @{text   1073 "[simp]"} -- can cause goals to blow up!   1074 *}   1075   1076 lemma split_if_mem1: "((if Q then x else y) : b) = ((Q --> x : b) & (~ Q --> y : b))"   1077 by (rule split_if)   1078   1079 lemma split_if_mem2: "(a : (if Q then x else y)) = ((Q --> a : x) & (~ Q --> a : y))"   1080 by (rule split_if [where P="%S. a : S"])   1081   1082 lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2   1083   1084 (*Would like to add these, but the existing code only searches for the   1085 outer-level constant, which in this case is just "op :"; we instead need   1086 to use term-nets to associate patterns with rules. Also, if a rule fails to   1087 apply, then the formula should be kept.   1088 [("HOL.uminus", Compl_iff RS iffD1), ("HOL.minus", [Diff_iff RS iffD1]),   1089 ("Int", [IntD1,IntD2]),   1090 ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]   1091 *)   1092   1093 ML {*   1094 val mksimps_pairs = [(@{const_name Ball}, @{thms bspec})] @ mksimps_pairs;   1095 *}   1096 declaration {* fn _ =>   1097 Simplifier.map_ss (fn ss => ss setmksimps (mksimps mksimps_pairs))   1098 *}   1099   1100   1101 subsubsection {* The proper subset'' relation *}   1102   1103 lemma psubsetI [intro!,noatp]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B"   1104 by (unfold less_le) blast   1105   1106 lemma psubsetE [elim!,noatp]:   1107 "[|A \<subset> B; [|A \<subseteq> B; ~ (B\<subseteq>A)|] ==> R|] ==> R"   1108 by (unfold less_le) blast   1109   1110 lemma psubset_insert_iff:   1111 "(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)"   1112 by (auto simp add: less_le subset_insert_iff)   1113   1114 lemma psubset_eq: "(A \<subset> B) = (A \<subseteq> B & A \<noteq> B)"   1115 by (simp only: less_le)   1116   1117 lemma psubset_imp_subset: "A \<subset> B ==> A \<subseteq> B"   1118 by (simp add: psubset_eq)   1119   1120 lemma psubset_trans: "[| A \<subset> B; B \<subset> C |] ==> A \<subset> C"   1121 apply (unfold less_le)   1122 apply (auto dest: subset_antisym)   1123 done   1124   1125 lemma psubsetD: "[| A \<subset> B; c \<in> A |] ==> c \<in> B"   1126 apply (unfold less_le)   1127 apply (auto dest: subsetD)   1128 done   1129   1130 lemma psubset_subset_trans: "A \<subset> B ==> B \<subseteq> C ==> A \<subset> C"   1131 by (auto simp add: psubset_eq)   1132   1133 lemma subset_psubset_trans: "A \<subseteq> B ==> B \<subset> C ==> A \<subset> C"   1134 by (auto simp add: psubset_eq)   1135   1136 lemma psubset_imp_ex_mem: "A \<subset> B ==> \<exists>b. b \<in> (B - A)"   1137 by (unfold less_le) blast   1138   1139 lemma atomize_ball:   1140 "(!!x. x \<in> A ==> P x) == Trueprop (\<forall>x\<in>A. P x)"   1141 by (simp only: Ball_def atomize_all atomize_imp)   1142   1143 lemmas [symmetric, rulify] = atomize_ball   1144 and [symmetric, defn] = atomize_ball   1145   1146   1147 subsection {* Further set-theory lemmas *}   1148   1149 subsubsection {* Derived rules involving subsets. *}   1150   1151 text {* @{text insert}. *}   1152   1153 lemma subset_insertI: "B \<subseteq> insert a B"   1154 by (rule subsetI) (erule insertI2)   1155   1156 lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B"   1157 by blast   1158   1159 lemma subset_insert: "x \<notin> A ==> (A \<subseteq> insert x B) = (A \<subseteq> B)"   1160 by blast   1161   1162   1163 text {* \medskip Big Union -- least upper bound of a set. *}   1164   1165 lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A"   1166 by (iprover intro: subsetI UnionI)   1167   1168 lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C"   1169 by (iprover intro: subsetI elim: UnionE dest: subsetD)   1170   1171   1172 text {* \medskip General union. *}   1173   1174 lemma UN_upper: "a \<in> A ==> B a \<subseteq> (\<Union>x\<in>A. B x)"   1175 by blast   1176   1177 lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C"   1178 by (iprover intro: subsetI elim: UN_E dest: subsetD)   1179   1180   1181 text {* \medskip Big Intersection -- greatest lower bound of a set. *}   1182   1183 lemma Inter_lower: "B \<in> A ==> Inter A \<subseteq> B"   1184 by blast   1185   1186 lemma Inter_subset:   1187 "[| !!X. X \<in> A ==> X \<subseteq> B; A ~= {} |] ==> \<Inter>A \<subseteq> B"   1188 by blast   1189   1190 lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A"   1191 by (iprover intro: InterI subsetI dest: subsetD)   1192   1193 lemma INT_lower: "a \<in> A ==> (\<Inter>x\<in>A. B x) \<subseteq> B a"   1194 by blast   1195   1196 lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)"   1197 by (iprover intro: INT_I subsetI dest: subsetD)   1198   1199   1200 text {* \medskip Finite Union -- the least upper bound of two sets. *}   1201   1202 lemma Un_upper1: "A \<subseteq> A \<union> B"   1203 by blast   1204   1205 lemma Un_upper2: "B \<subseteq> A \<union> B"   1206 by blast   1207   1208 lemma Un_least: "A \<subseteq> C ==> B \<subseteq> C ==> A \<union> B \<subseteq> C"   1209 by blast   1210   1211   1212 text {* \medskip Finite Intersection -- the greatest lower bound of two sets. *}   1213   1214 lemma Int_lower1: "A \<inter> B \<subseteq> A"   1215 by blast   1216   1217 lemma Int_lower2: "A \<inter> B \<subseteq> B"   1218 by blast   1219   1220 lemma Int_greatest: "C \<subseteq> A ==> C \<subseteq> B ==> C \<subseteq> A \<inter> B"   1221 by blast   1222   1223   1224 text {* \medskip Set difference. *}   1225   1226 lemma Diff_subset: "A - B \<subseteq> A"   1227 by blast   1228   1229 lemma Diff_subset_conv: "(A - B \<subseteq> C) = (A \<subseteq> B \<union> C)"   1230 by blast   1231   1232   1233 subsubsection {* Equalities involving union, intersection, inclusion, etc. *}   1234   1235 text {* @{text "{}"}. *}   1236   1237 lemma Collect_const [simp]: "{s. P} = (if P then UNIV else {})"   1238 -- {* supersedes @{text "Collect_False_empty"} *}   1239 by auto   1240   1241 lemma subset_empty [simp]: "(A \<subseteq> {}) = (A = {})"   1242 by blast   1243   1244 lemma not_psubset_empty [iff]: "\<not> (A < {})"   1245 by (unfold less_le) blast   1246   1247 lemma Collect_empty_eq [simp]: "(Collect P = {}) = (\<forall>x. \<not> P x)"   1248 by blast   1249   1250 lemma empty_Collect_eq [simp]: "({} = Collect P) = (\<forall>x. \<not> P x)"   1251 by blast   1252   1253 lemma Collect_neg_eq: "{x. \<not> P x} = - {x. P x}"   1254 by blast   1255   1256 lemma Collect_disj_eq: "{x. P x | Q x} = {x. P x} \<union> {x. Q x}"   1257 by blast   1258   1259 lemma Collect_imp_eq: "{x. P x --> Q x} = -{x. P x} \<union> {x. Q x}"   1260 by blast   1261   1262 lemma Collect_conj_eq: "{x. P x & Q x} = {x. P x} \<inter> {x. Q x}"   1263 by blast   1264   1265 lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"   1266 by blast   1267   1268 lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"   1269 by blast   1270   1271 lemma Collect_ex_eq [noatp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"   1272 by blast   1273   1274 lemma Collect_bex_eq [noatp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"   1275 by blast   1276   1277   1278 text {* \medskip @{text insert}. *}   1279   1280 lemma insert_is_Un: "insert a A = {a} Un A"   1281 -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a {}"} *}   1282 by blast   1283   1284 lemma insert_not_empty [simp]: "insert a A \<noteq> {}"   1285 by blast   1286   1287 lemmas empty_not_insert = insert_not_empty [symmetric, standard]   1288 declare empty_not_insert [simp]   1289   1290 lemma insert_absorb: "a \<in> A ==> insert a A = A"   1291 -- {* @{text "[simp]"} causes recursive calls when there are nested inserts *}   1292 -- {* with \emph{quadratic} running time *}   1293 by blast   1294   1295 lemma insert_absorb2 [simp]: "insert x (insert x A) = insert x A"   1296 by blast   1297   1298 lemma insert_commute: "insert x (insert y A) = insert y (insert x A)"   1299 by blast   1300   1301 lemma insert_subset [simp]: "(insert x A \<subseteq> B) = (x \<in> B & A \<subseteq> B)"   1302 by blast   1303   1304 lemma mk_disjoint_insert: "a \<in> A ==> \<exists>B. A = insert a B & a \<notin> B"   1305 -- {* use new @{text B} rather than @{text "A - {a}"} to avoid infinite unfolding *}   1306 apply (rule_tac x = "A - {a}" in exI, blast)   1307 done   1308   1309 lemma insert_Collect: "insert a (Collect P) = {u. u \<noteq> a --> P u}"   1310 by auto   1311   1312 lemma UN_insert_distrib: "u \<in> A ==> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"   1313 by blast   1314   1315 lemma insert_inter_insert[simp]: "insert a A \<inter> insert a B = insert a (A \<inter> B)"   1316 by blast   1317   1318 lemma insert_disjoint [simp,noatp]:   1319 "(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})"   1320 "({} = insert a A \<inter> B) = (a \<notin> B \<and> {} = A \<inter> B)"   1321 by auto   1322   1323 lemma disjoint_insert [simp,noatp]:   1324 "(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})"   1325 "({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)"   1326 by auto   1327   1328 text {* \medskip @{text image}. *}   1329   1330 lemma image_empty [simp]: "f{} = {}"   1331 by blast   1332   1333 lemma image_insert [simp]: "f  insert a B = insert (f a) (fB)"   1334 by blast   1335   1336 lemma image_constant: "x \<in> A ==> (\<lambda>x. c)  A = {c}"   1337 by auto   1338   1339 lemma image_constant_conv: "(%x. c)  A = (if A = {} then {} else {c})"   1340 by auto   1341   1342 lemma image_image: "f  (g  A) = (\<lambda>x. f (g x))  A"   1343 by blast   1344   1345 lemma insert_image [simp]: "x \<in> A ==> insert (f x) (fA) = fA"   1346 by blast   1347   1348 lemma image_is_empty [iff]: "(fA = {}) = (A = {})"   1349 by blast   1350   1351   1352 lemma image_Collect [noatp]: "f  {x. P x} = {f x | x. P x}"   1353 -- {* NOT suitable as a default simprule: the RHS isn't simpler than the LHS,   1354 with its implicit quantifier and conjunction. Also image enjoys better   1355 equational properties than does the RHS. *}   1356 by blast   1357   1358 lemma if_image_distrib [simp]:   1359 "(\<lambda>x. if P x then f x else g x)  S   1360 = (f  (S \<inter> {x. P x})) \<union> (g  (S \<inter> {x. \<not> P x}))"   1361 by (auto simp add: image_def)   1362   1363 lemma image_cong: "M = N ==> (!!x. x \<in> N ==> f x = g x) ==> fM = gN"   1364 by (simp add: image_def)   1365   1366   1367 text {* \medskip @{text range}. *}   1368   1369 lemma full_SetCompr_eq [noatp]: "{u. \<exists>x. u = f x} = range f"   1370 by auto   1371   1372 lemma range_composition: "range (\<lambda>x. f (g x)) = frange g"   1373 by (subst image_image, simp)   1374   1375   1376 text {* \medskip @{text Int} *}   1377   1378 lemma Int_absorb [simp]: "A \<inter> A = A"   1379 by blast   1380   1381 lemma Int_left_absorb: "A \<inter> (A \<inter> B) = A \<inter> B"   1382 by blast   1383   1384 lemma Int_commute: "A \<inter> B = B \<inter> A"   1385 by blast   1386   1387 lemma Int_left_commute: "A \<inter> (B \<inter> C) = B \<inter> (A \<inter> C)"   1388 by blast   1389   1390 lemma Int_assoc: "(A \<inter> B) \<inter> C = A \<inter> (B \<inter> C)"   1391 by blast   1392   1393 lemmas Int_ac = Int_assoc Int_left_absorb Int_commute Int_left_commute   1394 -- {* Intersection is an AC-operator *}   1395   1396 lemma Int_absorb1: "B \<subseteq> A ==> A \<inter> B = B"   1397 by blast   1398   1399 lemma Int_absorb2: "A \<subseteq> B ==> A \<inter> B = A"   1400 by blast   1401   1402 lemma Int_empty_left [simp]: "{} \<inter> B = {}"   1403 by blast   1404   1405 lemma Int_empty_right [simp]: "A \<inter> {} = {}"   1406 by blast   1407   1408 lemma disjoint_eq_subset_Compl: "(A \<inter> B = {}) = (A \<subseteq> -B)"   1409 by blast   1410   1411 lemma disjoint_iff_not_equal: "(A \<inter> B = {}) = (\<forall>x\<in>A. \<forall>y\<in>B. x \<noteq> y)"   1412 by blast   1413   1414 lemma Int_UNIV_left [simp]: "UNIV \<inter> B = B"   1415 by blast   1416   1417 lemma Int_UNIV_right [simp]: "A \<inter> UNIV = A"   1418 by blast   1419   1420 lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"   1421 by blast   1422   1423 lemma Int_Un_distrib: "A \<inter> (B \<union> C) = (A \<inter> B) \<union> (A \<inter> C)"   1424 by blast   1425   1426 lemma Int_Un_distrib2: "(B \<union> C) \<inter> A = (B \<inter> A) \<union> (C \<inter> A)"   1427 by blast   1428   1429 lemma Int_UNIV [simp,noatp]: "(A \<inter> B = UNIV) = (A = UNIV & B = UNIV)"   1430 by blast   1431   1432 lemma Int_subset_iff [simp]: "(C \<subseteq> A \<inter> B) = (C \<subseteq> A & C \<subseteq> B)"   1433 by blast   1434   1435 lemma Int_Collect: "(x \<in> A \<inter> {x. P x}) = (x \<in> A & P x)"   1436 by blast   1437   1438   1439 text {* \medskip @{text Un}. *}   1440   1441 lemma Un_absorb [simp]: "A \<union> A = A"   1442 by blast   1443   1444 lemma Un_left_absorb: "A \<union> (A \<union> B) = A \<union> B"   1445 by blast   1446   1447 lemma Un_commute: "A \<union> B = B \<union> A"   1448 by blast   1449   1450 lemma Un_left_commute: "A \<union> (B \<union> C) = B \<union> (A \<union> C)"   1451 by blast   1452   1453 lemma Un_assoc: "(A \<union> B) \<union> C = A \<union> (B \<union> C)"   1454 by blast   1455   1456 lemmas Un_ac = Un_assoc Un_left_absorb Un_commute Un_left_commute   1457 -- {* Union is an AC-operator *}   1458   1459 lemma Un_absorb1: "A \<subseteq> B ==> A \<union> B = B"   1460 by blast   1461   1462 lemma Un_absorb2: "B \<subseteq> A ==> A \<union> B = A"   1463 by blast   1464   1465 lemma Un_empty_left [simp]: "{} \<union> B = B"   1466 by blast   1467   1468 lemma Un_empty_right [simp]: "A \<union> {} = A"   1469 by blast   1470   1471 lemma Un_UNIV_left [simp]: "UNIV \<union> B = UNIV"   1472 by blast   1473   1474 lemma Un_UNIV_right [simp]: "A \<union> UNIV = UNIV"   1475 by blast   1476   1477 lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"   1478 by blast   1479   1480 lemma Un_insert_left [simp]: "(insert a B) \<union> C = insert a (B \<union> C)"   1481 by blast   1482   1483 lemma Un_insert_right [simp]: "A \<union> (insert a B) = insert a (A \<union> B)"   1484 by blast   1485   1486 lemma Int_insert_left:   1487 "(insert a B) Int C = (if a \<in> C then insert a (B \<inter> C) else B \<inter> C)"   1488 by auto   1489   1490 lemma Int_insert_right:   1491 "A \<inter> (insert a B) = (if a \<in> A then insert a (A \<inter> B) else A \<inter> B)"   1492 by auto   1493   1494 lemma Un_Int_distrib: "A \<union> (B \<inter> C) = (A \<union> B) \<inter> (A \<union> C)"   1495 by blast   1496   1497 lemma Un_Int_distrib2: "(B \<inter> C) \<union> A = (B \<union> A) \<inter> (C \<union> A)"   1498 by blast   1499   1500 lemma Un_Int_crazy:   1501 "(A \<inter> B) \<union> (B \<inter> C) \<union> (C \<inter> A) = (A \<union> B) \<inter> (B \<union> C) \<inter> (C \<union> A)"   1502 by blast   1503   1504 lemma subset_Un_eq: "(A \<subseteq> B) = (A \<union> B = B)"   1505 by blast   1506   1507 lemma Un_empty [iff]: "(A \<union> B = {}) = (A = {} & B = {})"   1508 by blast   1509   1510 lemma Un_subset_iff [simp]: "(A \<union> B \<subseteq> C) = (A \<subseteq> C & B \<subseteq> C)"   1511 by blast   1512   1513 lemma Un_Diff_Int: "(A - B) \<union> (A \<inter> B) = A"   1514 by blast   1515   1516 lemma Diff_Int2: "A \<inter> C - B \<inter> C = A \<inter> C - B"   1517 by blast   1518   1519   1520 text {* \medskip Set complement *}   1521   1522 lemma Compl_disjoint [simp]: "A \<inter> -A = {}"   1523 by blast   1524   1525 lemma Compl_disjoint2 [simp]: "-A \<inter> A = {}"   1526 by blast   1527   1528 lemma Compl_partition: "A \<union> -A = UNIV"   1529 by blast   1530   1531 lemma Compl_partition2: "-A \<union> A = UNIV"   1532 by blast   1533   1534 lemma double_complement [simp]: "- (-A) = (A::'a set)"   1535 by blast   1536   1537 lemma Compl_Un [simp]: "-(A \<union> B) = (-A) \<inter> (-B)"   1538 by blast   1539   1540 lemma Compl_Int [simp]: "-(A \<inter> B) = (-A) \<union> (-B)"   1541 by blast   1542   1543 lemma Compl_UN [simp]: "-(\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"   1544 by blast   1545   1546 lemma Compl_INT [simp]: "-(\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"   1547 by blast   1548   1549 lemma subset_Compl_self_eq: "(A \<subseteq> -A) = (A = {})"   1550 by blast   1551   1552 lemma Un_Int_assoc_eq: "((A \<inter> B) \<union> C = A \<inter> (B \<union> C)) = (C \<subseteq> A)"   1553 -- {* Halmos, Naive Set Theory, page 16. *}   1554 by blast   1555   1556 lemma Compl_UNIV_eq [simp]: "-UNIV = {}"   1557 by blast   1558   1559 lemma Compl_empty_eq [simp]: "-{} = UNIV"   1560 by blast   1561   1562 lemma Compl_subset_Compl_iff [iff]: "(-A \<subseteq> -B) = (B \<subseteq> A)"   1563 by blast   1564   1565 lemma Compl_eq_Compl_iff [iff]: "(-A = -B) = (A = (B::'a set))"   1566 by blast   1567   1568   1569 text {* \medskip @{text Union}. *}   1570   1571 lemma Union_empty [simp]: "Union({}) = {}"   1572 by blast   1573   1574 lemma Union_UNIV [simp]: "Union UNIV = UNIV"   1575 by blast   1576   1577 lemma Union_insert [simp]: "Union (insert a B) = a \<union> \<Union>B"   1578 by blast   1579   1580 lemma Union_Un_distrib [simp]: "\<Union>(A Un B) = \<Union>A \<union> \<Union>B"   1581 by blast   1582   1583 lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"   1584 by blast   1585   1586 lemma Union_empty_conv [simp,noatp]: "(\<Union>A = {}) = (\<forall>x\<in>A. x = {})"   1587 by blast   1588   1589 lemma empty_Union_conv [simp,noatp]: "({} = \<Union>A) = (\<forall>x\<in>A. x = {})"   1590 by blast   1591   1592 lemma Union_disjoint: "(\<Union>C \<inter> A = {}) = (\<forall>B\<in>C. B \<inter> A = {})"   1593 by blast   1594   1595   1596 text {* \medskip @{text Inter}. *}   1597   1598 lemma Inter_empty [simp]: "\<Inter>{} = UNIV"   1599 by blast   1600   1601 lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}"   1602 by blast   1603   1604 lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"   1605 by blast   1606   1607 lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"   1608 by blast   1609   1610 lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"   1611 by blast   1612   1613 lemma Inter_UNIV_conv [simp,noatp]:   1614 "(\<Inter>A = UNIV) = (\<forall>x\<in>A. x = UNIV)"   1615 "(UNIV = \<Inter>A) = (\<forall>x\<in>A. x = UNIV)"   1616 by blast+   1617   1618   1619 text {*   1620 \medskip @{text UN} and @{text INT}.   1621   1622 Basic identities: *}   1623   1624 lemma UN_empty [simp,noatp]: "(\<Union>x\<in>{}. B x) = {}"   1625 by blast   1626   1627 lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"   1628 by blast   1629   1630 lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"   1631 by blast   1632   1633 lemma UN_absorb: "k \<in> I ==> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"   1634 by auto   1635   1636 lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"   1637 by blast   1638   1639 lemma INT_absorb: "k \<in> I ==> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"   1640 by blast   1641   1642 lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"   1643 by blast   1644   1645 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)"   1646 by blast   1647   1648 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)"   1649 by blast   1650   1651 lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"   1652 by blast   1653   1654 lemma INT_subset_iff: "(B \<subseteq> (\<Inter>i\<in>I. A i)) = (\<forall>i\<in>I. B \<subseteq> A i)"   1655 by blast   1656   1657 lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"   1658 by blast   1659   1660 lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"   1661 by blast   1662   1663 lemma INT_insert_distrib:   1664 "u \<in> A ==> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"   1665 by blast   1666   1667 lemma Union_image_eq [simp]: "\<Union>(BA) = (\<Union>x\<in>A. B x)"   1668 by blast   1669   1670 lemma image_Union: "f  \<Union>S = (\<Union>x\<in>S. f  x)"   1671 by blast   1672   1673 lemma Inter_image_eq [simp]: "\<Inter>(BA) = (\<Inter>x\<in>A. B x)"   1674 by blast   1675   1676 lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"   1677 by auto   1678   1679 lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"   1680 by auto   1681   1682 lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"   1683 by blast   1684   1685 lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"   1686 -- {* Look: it has an \emph{existential} quantifier *}   1687 by blast   1688   1689 lemma UNION_empty_conv[simp]:   1690 "({} = (UN x:A. B x)) = (\<forall>x\<in>A. B x = {})"   1691 "((UN x:A. B x) = {}) = (\<forall>x\<in>A. B x = {})"   1692 by blast+   1693   1694 lemma INTER_UNIV_conv[simp]:   1695 "(UNIV = (INT x:A. B x)) = (\<forall>x\<in>A. B x = UNIV)"   1696 "((INT x:A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"   1697 by blast+   1698   1699   1700 text {* \medskip Distributive laws: *}   1701   1702 lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"   1703 by blast   1704   1705 lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"   1706 by blast   1707   1708 lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(AC) \<union> \<Union>(BC)"   1709 -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}   1710 -- {* Union of a family of unions *}   1711 by blast   1712   1713 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)"   1714 -- {* Equivalent version *}   1715 by blast   1716   1717 lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"   1718 by blast   1719   1720 lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(AC) \<inter> \<Inter>(BC)"   1721 by blast   1722   1723 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)"   1724 -- {* Equivalent version *}   1725 by blast   1726   1727 lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"   1728 -- {* Halmos, Naive Set Theory, page 35. *}   1729 by blast   1730   1731 lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"   1732 by blast   1733   1734 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)"   1735 by blast   1736   1737 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)"   1738 by blast   1739   1740   1741 text {* \medskip Bounded quantifiers.   1742   1743 The following are not added to the default simpset because   1744 (a) they duplicate the body and (b) there are no similar rules for @{text Int}. *}   1745   1746 lemma ball_Un: "(\<forall>x \<in> A \<union> B. P x) = ((\<forall>x\<in>A. P x) & (\<forall>x\<in>B. P x))"   1747 by blast   1748   1749 lemma bex_Un: "(\<exists>x \<in> A \<union> B. P x) = ((\<exists>x\<in>A. P x) | (\<exists>x\<in>B. P x))"   1750 by blast   1751   1752 lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) = (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"   1753 by blast   1754   1755 lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) = (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"   1756 by blast   1757   1758   1759 text {* \medskip Set difference. *}   1760   1761 lemma Diff_eq: "A - B = A \<inter> (-B)"   1762 by blast   1763   1764 lemma Diff_eq_empty_iff [simp,noatp]: "(A - B = {}) = (A \<subseteq> B)"   1765 by blast   1766   1767 lemma Diff_cancel [simp]: "A - A = {}"   1768 by blast   1769   1770 lemma Diff_idemp [simp]: "(A - B) - B = A - (B::'a set)"   1771 by blast   1772   1773 lemma Diff_triv: "A \<inter> B = {} ==> A - B = A"   1774 by (blast elim: equalityE)   1775   1776 lemma empty_Diff [simp]: "{} - A = {}"   1777 by blast   1778   1779 lemma Diff_empty [simp]: "A - {} = A"   1780 by blast   1781   1782 lemma Diff_UNIV [simp]: "A - UNIV = {}"   1783 by blast   1784   1785 lemma Diff_insert0 [simp,noatp]: "x \<notin> A ==> A - insert x B = A - B"   1786 by blast   1787   1788 lemma Diff_insert: "A - insert a B = A - B - {a}"   1789 -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a 0"} *}   1790 by blast   1791   1792 lemma Diff_insert2: "A - insert a B = A - {a} - B"   1793 -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a 0"} *}   1794 by blast   1795   1796 lemma insert_Diff_if: "insert x A - B = (if x \<in> B then A - B else insert x (A - B))"   1797 by auto   1798   1799 lemma insert_Diff1 [simp]: "x \<in> B ==> insert x A - B = A - B"   1800 by blast   1801   1802 lemma insert_Diff_single[simp]: "insert a (A - {a}) = insert a A"   1803 by blast   1804   1805 lemma insert_Diff: "a \<in> A ==> insert a (A - {a}) = A"   1806 by blast   1807   1808 lemma Diff_insert_absorb: "x \<notin> A ==> (insert x A) - {x} = A"   1809 by auto   1810   1811 lemma Diff_disjoint [simp]: "A \<inter> (B - A) = {}"   1812 by blast   1813   1814 lemma Diff_partition: "A \<subseteq> B ==> A \<union> (B - A) = B"   1815 by blast   1816   1817 lemma double_diff: "A \<subseteq> B ==> B \<subseteq> C ==> B - (C - A) = A"   1818 by blast   1819   1820 lemma Un_Diff_cancel [simp]: "A \<union> (B - A) = A \<union> B"   1821 by blast   1822   1823 lemma Un_Diff_cancel2 [simp]: "(B - A) \<union> A = B \<union> A"   1824 by blast   1825   1826 lemma Diff_Un: "A - (B \<union> C) = (A - B) \<inter> (A - C)"   1827 by blast   1828   1829 lemma Diff_Int: "A - (B \<inter> C) = (A - B) \<union> (A - C)"   1830 by blast   1831   1832 lemma Un_Diff: "(A \<union> B) - C = (A - C) \<union> (B - C)"   1833 by blast   1834   1835 lemma Int_Diff: "(A \<inter> B) - C = A \<inter> (B - C)"   1836 by blast   1837   1838 lemma Diff_Int_distrib: "C \<inter> (A - B) = (C \<inter> A) - (C \<inter> B)"   1839 by blast   1840   1841 lemma Diff_Int_distrib2: "(A - B) \<inter> C = (A \<inter> C) - (B \<inter> C)"   1842 by blast   1843   1844 lemma Diff_Compl [simp]: "A - (- B) = A \<inter> B"   1845 by auto   1846   1847 lemma Compl_Diff_eq [simp]: "- (A - B) = -A \<union> B"   1848 by blast   1849   1850   1851 text {* \medskip Quantification over type @{typ bool}. *}   1852   1853 lemma bool_induct: "P True \<Longrightarrow> P False \<Longrightarrow> P x"   1854 by (cases x) auto   1855   1856 lemma all_bool_eq: "(\<forall>b. P b) \<longleftrightarrow> P True \<and> P False"   1857 by (auto intro: bool_induct)   1858   1859 lemma bool_contrapos: "P x \<Longrightarrow> \<not> P False \<Longrightarrow> P True"   1860 by (cases x) auto   1861   1862 lemma ex_bool_eq: "(\<exists>b. P b) \<longleftrightarrow> P True \<or> P False"   1863 by (auto intro: bool_contrapos)   1864   1865 lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"   1866 by (auto simp add: split_if_mem2)   1867   1868 lemma UN_bool_eq: "(\<Union>b::bool. A b) = (A True \<union> A False)"   1869 by (auto intro: bool_contrapos)   1870   1871 lemma INT_bool_eq: "(\<Inter>b::bool. A b) = (A True \<inter> A False)"   1872 by (auto intro: bool_induct)   1873   1874 text {* \medskip @{text Pow} *}   1875   1876 lemma Pow_empty [simp]: "Pow {} = {{}}"   1877 by (auto simp add: Pow_def)   1878   1879 lemma Pow_insert: "Pow (insert a A) = Pow A \<union> (insert a  Pow A)"   1880 by (blast intro: image_eqI [where ?x = "u - {a}", standard])   1881   1882 lemma Pow_Compl: "Pow (- A) = {-B | B. A \<in> Pow B}"   1883 by (blast intro: exI [where ?x = "- u", standard])   1884   1885 lemma Pow_UNIV [simp]: "Pow UNIV = UNIV"   1886 by blast   1887   1888 lemma Un_Pow_subset: "Pow A \<union> Pow B \<subseteq> Pow (A \<union> B)"   1889 by blast   1890   1891 lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"   1892 by blast   1893   1894 lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"   1895 by blast   1896   1897 lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"   1898 by blast   1899   1900 lemma Pow_Int_eq [simp]: "Pow (A \<inter> B) = Pow A \<inter> Pow B"   1901 by blast   1902   1903 lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"   1904 by blast   1905   1906   1907 text {* \medskip Miscellany. *}   1908   1909 lemma set_eq_subset: "(A = B) = (A \<subseteq> B & B \<subseteq> A)"   1910 by blast   1911   1912 lemma subset_iff: "(A \<subseteq> B) = (\<forall>t. t \<in> A --> t \<in> B)"   1913 by blast   1914   1915 lemma subset_iff_psubset_eq: "(A \<subseteq> B) = ((A \<subset> B) | (A = B))"   1916 by (unfold less_le) blast   1917   1918 lemma all_not_in_conv [simp]: "(\<forall>x. x \<notin> A) = (A = {})"   1919 by blast   1920   1921 lemma ex_in_conv: "(\<exists>x. x \<in> A) = (A \<noteq> {})"   1922 by blast   1923   1924 lemma distinct_lemma: "f x \<noteq> f y ==> x \<noteq> y"   1925 by iprover   1926   1927   1928 text {* \medskip Miniscoping: pushing in quantifiers and big Unions   1929 and Intersections. *}   1930   1931 lemma UN_simps [simp]:   1932 "!!a B C. (UN x:C. insert a (B x)) = (if C={} then {} else insert a (UN x:C. B x))"   1933 "!!A B C. (UN x:C. A x Un B) = ((if C={} then {} else (UN x:C. A x) Un B))"   1934 "!!A B C. (UN x:C. A Un B x) = ((if C={} then {} else A Un (UN x:C. B x)))"   1935 "!!A B C. (UN x:C. A x Int B) = ((UN x:C. A x) Int B)"   1936 "!!A B C. (UN x:C. A Int B x) = (A Int (UN x:C. B x))"   1937 "!!A B C. (UN x:C. A x - B) = ((UN x:C. A x) - B)"   1938 "!!A B C. (UN x:C. A - B x) = (A - (INT x:C. B x))"   1939 "!!A B. (UN x: Union A. B x) = (UN y:A. UN x:y. B x)"   1940 "!!A B C. (UN z: UNION A B. C z) = (UN x:A. UN z: B(x). C z)"   1941 "!!A B f. (UN x:fA. B x) = (UN a:A. B (f a))"   1942 by auto   1943   1944 lemma INT_simps [simp]:   1945 "!!A B C. (INT x:C. A x Int B) = (if C={} then UNIV else (INT x:C. A x) Int B)"   1946 "!!A B C. (INT x:C. A Int B x) = (if C={} then UNIV else A Int (INT x:C. B x))"   1947 "!!A B C. (INT x:C. A x - B) = (if C={} then UNIV else (INT x:C. A x) - B)"   1948 "!!A B C. (INT x:C. A - B x) = (if C={} then UNIV else A - (UN x:C. B x))"   1949 "!!a B C. (INT x:C. insert a (B x)) = insert a (INT x:C. B x)"   1950 "!!A B C. (INT x:C. A x Un B) = ((INT x:C. A x) Un B)"   1951 "!!A B C. (INT x:C. A Un B x) = (A Un (INT x:C. B x))"   1952 "!!A B. (INT x: Union A. B x) = (INT y:A. INT x:y. B x)"   1953 "!!A B C. (INT z: UNION A B. C z) = (INT x:A. INT z: B(x). C z)"   1954 "!!A B f. (INT x:fA. B x) = (INT a:A. B (f a))"   1955 by auto   1956   1957 lemma ball_simps [simp,noatp]:   1958 "!!A P Q. (ALL x:A. P x | Q) = ((ALL x:A. P x) | Q)"   1959 "!!A P Q. (ALL x:A. P | Q x) = (P | (ALL x:A. Q x))"   1960 "!!A P Q. (ALL x:A. P --> Q x) = (P --> (ALL x:A. Q x))"   1961 "!!A P Q. (ALL x:A. P x --> Q) = ((EX x:A. P x) --> Q)"   1962 "!!P. (ALL x:{}. P x) = True"   1963 "!!P. (ALL x:UNIV. P x) = (ALL x. P x)"   1964 "!!a B P. (ALL x:insert a B. P x) = (P a & (ALL x:B. P x))"   1965 "!!A P. (ALL x:Union A. P x) = (ALL y:A. ALL x:y. P x)"   1966 "!!A B P. (ALL x: UNION A B. P x) = (ALL a:A. ALL x: B a. P x)"   1967 "!!P Q. (ALL x:Collect Q. P x) = (ALL x. Q x --> P x)"   1968 "!!A P f. (ALL x:fA. P x) = (ALL x:A. P (f x))"   1969 "!!A P. (~(ALL x:A. P x)) = (EX x:A. ~P x)"   1970 by auto   1971   1972 lemma bex_simps [simp,noatp]:   1973 "!!A P Q. (EX x:A. P x & Q) = ((EX x:A. P x) & Q)"   1974 "!!A P Q. (EX x:A. P & Q x) = (P & (EX x:A. Q x))"   1975 "!!P. (EX x:{}. P x) = False"   1976 "!!P. (EX x:UNIV. P x) = (EX x. P x)"   1977 "!!a B P. (EX x:insert a B. P x) = (P(a) | (EX x:B. P x))"   1978 "!!A P. (EX x:Union A. P x) = (EX y:A. EX x:y. P x)"   1979 "!!A B P. (EX x: UNION A B. P x) = (EX a:A. EX x:B a. P x)"   1980 "!!P Q. (EX x:Collect Q. P x) = (EX x. Q x & P x)"   1981 "!!A P f. (EX x:fA. P x) = (EX x:A. P (f x))"   1982 "!!A P. (~(EX x:A. P x)) = (ALL x:A. ~P x)"   1983 by auto   1984   1985 lemma ball_conj_distrib:   1986 "(ALL x:A. P x & Q x) = ((ALL x:A. P x) & (ALL x:A. Q x))"   1987 by blast   1988   1989 lemma bex_disj_distrib:   1990 "(EX x:A. P x | Q x) = ((EX x:A. P x) | (EX x:A. Q x))"   1991 by blast   1992   1993   1994 text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}   1995   1996 lemma UN_extend_simps:   1997 "!!a B C. insert a (UN x:C. B x) = (if C={} then {a} else (UN x:C. insert a (B x)))"   1998 "!!A B C. (UN x:C. A x) Un B = (if C={} then B else (UN x:C. A x Un B))"   1999 "!!A B C. A Un (UN x:C. B x) = (if C={} then A else (UN x:C. A Un B x))"   2000 "!!A B C. ((UN x:C. A x) Int B) = (UN x:C. A x Int B)"   2001 "!!A B C. (A Int (UN x:C. B x)) = (UN x:C. A Int B x)"   2002 "!!A B C. ((UN x:C. A x) - B) = (UN x:C. A x - B)"   2003 "!!A B C. (A - (INT x:C. B x)) = (UN x:C. A - B x)"   2004 "!!A B. (UN y:A. UN x:y. B x) = (UN x: Union A. B x)"   2005 "!!A B C. (UN x:A. UN z: B(x). C z) = (UN z: UNION A B. C z)"   2006 "!!A B f. (UN a:A. B (f a)) = (UN x:fA. B x)"   2007 by auto   2008   2009 lemma INT_extend_simps:   2010 "!!A B C. (INT x:C. A x) Int B = (if C={} then B else (INT x:C. A x Int B))"   2011 "!!A B C. A Int (INT x:C. B x) = (if C={} then A else (INT x:C. A Int B x))"   2012 "!!A B C. (INT x:C. A x) - B = (if C={} then UNIV-B else (INT x:C. A x - B))"   2013 "!!A B C. A - (UN x:C. B x) = (if C={} then A else (INT x:C. A - B x))"   2014 "!!a B C. insert a (INT x:C. B x) = (INT x:C. insert a (B x))"   2015 "!!A B C. ((INT x:C. A x) Un B) = (INT x:C. A x Un B)"   2016 "!!A B C. A Un (INT x:C. B x) = (INT x:C. A Un B x)"   2017 "!!A B. (INT y:A. INT x:y. B x) = (INT x: Union A. B x)"   2018 "!!A B C. (INT x:A. INT z: B(x). C z) = (INT z: UNION A B. C z)"   2019 "!!A B f. (INT a:A. B (f a)) = (INT x:fA. B x)"   2020 by auto   2021   2022   2023 subsubsection {* Monotonicity of various operations *}   2024   2025 lemma image_mono: "A \<subseteq> B ==> fA \<subseteq> fB"   2026 by blast   2027   2028 lemma Pow_mono: "A \<subseteq> B ==> Pow A \<subseteq> Pow B"   2029 by blast   2030   2031 lemma Union_mono: "A \<subseteq> B ==> \<Union>A \<subseteq> \<Union>B"   2032 by blast   2033   2034 lemma Inter_anti_mono: "B \<subseteq> A ==> \<Inter>A \<subseteq> \<Inter>B"   2035 by blast   2036   2037 lemma UN_mono:   2038 "A \<subseteq> B ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>   2039 (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"   2040 by (blast dest: subsetD)   2041   2042 lemma INT_anti_mono:   2043 "B \<subseteq> A ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>   2044 (\<Inter>x\<in>A. f x) \<subseteq> (\<Inter>x\<in>A. g x)"   2045 -- {* The last inclusion is POSITIVE! *}   2046 by (blast dest: subsetD)   2047   2048 lemma insert_mono: "C \<subseteq> D ==> insert a C \<subseteq> insert a D"   2049 by blast   2050   2051 lemma Un_mono: "A \<subseteq> C ==> B \<subseteq> D ==> A \<union> B \<subseteq> C \<union> D"   2052 by blast   2053   2054 lemma Int_mono: "A \<subseteq> C ==> B \<subseteq> D ==> A \<inter> B \<subseteq> C \<inter> D"   2055 by blast   2056   2057 lemma Diff_mono: "A \<subseteq> C ==> D \<subseteq> B ==> A - B \<subseteq> C - D"   2058 by blast   2059   2060 lemma Compl_anti_mono: "A \<subseteq> B ==> -B \<subseteq> -A"   2061 by blast   2062   2063 text {* \medskip Monotonicity of implications. *}   2064   2065 lemma in_mono: "A \<subseteq> B ==> x \<in> A --> x \<in> B"   2066 apply (rule impI)   2067 apply (erule subsetD, assumption)   2068 done   2069   2070 lemma conj_mono: "P1 --> Q1 ==> P2 --> Q2 ==> (P1 & P2) --> (Q1 & Q2)"   2071 by iprover   2072   2073 lemma disj_mono: "P1 --> Q1 ==> P2 --> Q2 ==> (P1 | P2) --> (Q1 | Q2)"   2074 by iprover   2075   2076 lemma imp_mono: "Q1 --> P1 ==> P2 --> Q2 ==> (P1 --> P2) --> (Q1 --> Q2)"   2077 by iprover   2078   2079 lemma imp_refl: "P --> P" ..   2080   2081 lemma ex_mono: "(!!x. P x --> Q x) ==> (EX x. P x) --> (EX x. Q x)"   2082 by iprover   2083   2084 lemma all_mono: "(!!x. P x --> Q x) ==> (ALL x. P x) --> (ALL x. Q x)"   2085 by iprover   2086   2087 lemma Collect_mono: "(!!x. P x --> Q x) ==> Collect P \<subseteq> Collect Q"   2088 by blast   2089   2090 lemma Int_Collect_mono:   2091 "A \<subseteq> B ==> (!!x. x \<in> A ==> P x --> Q x) ==> A \<inter> Collect P \<subseteq> B \<inter> Collect Q"   2092 by blast   2093   2094 lemmas basic_monos =   2095 subset_refl imp_refl disj_mono conj_mono   2096 ex_mono Collect_mono in_mono   2097   2098 lemma eq_to_mono: "a = b ==> c = d ==> b --> d ==> a --> c"   2099 by iprover   2100   2101 lemma eq_to_mono2: "a = b ==> c = d ==> ~ b --> ~ d ==> ~ a --> ~ c"   2102 by iprover   2103   2104   2105 subsection {* Inverse image of a function *}   2106   2107 constdefs   2108 vimage :: "('a => 'b) => 'b set => 'a set" (infixr "-" 90)   2109 [code del]: "f - B == {x. f x : B}"   2110   2111   2112 subsubsection {* Basic rules *}   2113   2114 lemma vimage_eq [simp]: "(a : f - B) = (f a : B)"   2115 by (unfold vimage_def) blast   2116   2117 lemma vimage_singleton_eq: "(a : f - {b}) = (f a = b)"   2118 by simp   2119   2120 lemma vimageI [intro]: "f a = b ==> b:B ==> a : f - B"   2121 by (unfold vimage_def) blast   2122   2123 lemma vimageI2: "f a : A ==> a : f - A"   2124 by (unfold vimage_def) fast   2125   2126 lemma vimageE [elim!]: "a: f - B ==> (!!x. f a = x ==> x:B ==> P) ==> P"   2127 by (unfold vimage_def) blast   2128   2129 lemma vimageD: "a : f - A ==> f a : A"   2130 by (unfold vimage_def) fast   2131   2132   2133 subsubsection {* Equations *}   2134   2135 lemma vimage_empty [simp]: "f - {} = {}"   2136 by blast   2137   2138 lemma vimage_Compl: "f - (-A) = -(f - A)"   2139 by blast   2140   2141 lemma vimage_Un [simp]: "f - (A Un B) = (f - A) Un (f - B)"   2142 by blast   2143   2144 lemma vimage_Int [simp]: "f - (A Int B) = (f - A) Int (f - B)"   2145 by fast   2146   2147 lemma vimage_Union: "f - (Union A) = (UN X:A. f - X)"   2148 by blast   2149   2150 lemma vimage_UN: "f-(UN x:A. B x) = (UN x:A. f - B x)"   2151 by blast   2152   2153 lemma vimage_INT: "f-(INT x:A. B x) = (INT x:A. f - B x)"   2154 by blast   2155   2156 lemma vimage_Collect_eq [simp]: "f - Collect P = {y. P (f y)}"   2157 by blast   2158   2159 lemma vimage_Collect: "(!!x. P (f x) = Q x) ==> f - (Collect P) = Collect Q"   2160 by blast   2161   2162 lemma vimage_insert: "f-(insert a B) = (f-{a}) Un (f-B)"   2163 -- {* NOT suitable for rewriting because of the recurrence of @{term "{a}"}. *}   2164 by blast   2165   2166 lemma vimage_Diff: "f - (A - B) = (f - A) - (f - B)"   2167 by blast   2168   2169 lemma vimage_UNIV [simp]: "f - UNIV = UNIV"   2170 by blast   2171   2172 lemma vimage_eq_UN: "f-B = (UN y: B. f-{y})"   2173 -- {* NOT suitable for rewriting *}   2174 by blast   2175   2176 lemma vimage_mono: "A \<subseteq> B ==> f - A \<subseteq> f - B"   2177 -- {* monotonicity *}   2178 by blast   2179   2180 lemma vimage_image_eq [noatp]: "f - (f  A) = {y. EX x:A. f x = f y}"   2181 by (blast intro: sym)   2182   2183 lemma image_vimage_subset: "f  (f - A) <= A"   2184 by blast   2185   2186 lemma image_vimage_eq [simp]: "f  (f - A) = A Int range f"   2187 by blast   2188   2189 lemma image_Int_subset: "f(A Int B) <= fA Int fB"   2190 by blast   2191   2192 lemma image_diff_subset: "fA - fB <= f(A - B)"   2193 by blast   2194   2195 lemma image_UN: "(f  (UNION A B)) = (UN x:A.(f  (B x)))"   2196 by blast   2197   2198   2199 subsection {* Getting the Contents of a Singleton Set *}   2200   2201 definition contents :: "'a set \<Rightarrow> 'a" where   2202 [code del]: "contents X = (THE x. X = {x})"   2203   2204 lemma contents_eq [simp]: "contents {x} = x"   2205 by (simp add: contents_def)   2206   2207   2208 subsection {* Transitivity rules for calculational reasoning *}   2209   2210 lemma set_rev_mp: "x:A ==> A \<subseteq> B ==> x:B"   2211 by (rule subsetD)   2212   2213 lemma set_mp: "A \<subseteq> B ==> x:A ==> x:B"   2214 by (rule subsetD)   2215   2216 lemmas basic_trans_rules [trans] =   2217 order_trans_rules set_rev_mp set_mp   2218   2219   2220 subsection {* Least value operator *}   2221   2222 lemma Least_mono:   2223 "mono (f::'a::order => 'b::order) ==> EX x:S. ALL y:S. x <= y   2224 ==> (LEAST y. y : f  S) = f (LEAST x. x : S)"   2225 -- {* Courtesy of Stephan Merz *}   2226 apply clarify   2227 apply (erule_tac P = "%x. x : S" in LeastI2_order, fast)   2228 apply (rule LeastI2_order)   2229 apply (auto elim: monoD intro!: order_antisym)   2230 done   2231   2232   2233 subsection {* Rudimentary code generation *}   2234   2235 lemma empty_code [code]: "{} x \<longleftrightarrow> False"   2236 unfolding empty_def Collect_def ..   2237   2238 lemma UNIV_code [code]: "UNIV x \<longleftrightarrow> True"   2239 unfolding UNIV_def Collect_def ..   2240   2241 lemma insert_code [code]: "insert y A x \<longleftrightarrow> y = x \<or> A x"   2242 unfolding insert_def Collect_def mem_def Un_def by auto   2243   2244 lemma inter_code [code]: "(A \<inter> B) x \<longleftrightarrow> A x \<and> B x"   2245 unfolding Int_def Collect_def mem_def ..   2246   2247 lemma union_code [code]: "(A \<union> B) x \<longleftrightarrow> A x \<or> B x"   2248 unfolding Un_def Collect_def mem_def ..   2249   2250 lemma vimage_code [code]: "(f - A) x = A (f x)"   2251 unfolding vimage_def Collect_def mem_def ..   2252   2253   2254 subsection {* Complete lattices *}   2255   2256 notation   2257 less_eq (infix "\<sqsubseteq>" 50) and   2258 less (infix "\<sqsubset>" 50) and   2259 inf (infixl "\<sqinter>" 70) and   2260 sup (infixl "\<squnion>" 65)   2261   2262 class complete_lattice = lattice + bot + top +   2263 fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)   2264 and Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)   2265 assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"   2266 and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"   2267 assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A"   2268 and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z"   2269 begin   2270   2271 lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<le> a}"   2272 by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)   2273   2274 lemma Sup_Inf: "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<le> b}"   2275 by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)   2276   2277 lemma Inf_Univ: "\<Sqinter>UNIV = \<Squnion>{}"   2278 unfolding Sup_Inf by auto   2279   2280 lemma Sup_Univ: "\<Squnion>UNIV = \<Sqinter>{}"   2281 unfolding Inf_Sup by auto   2282   2283 lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"   2284 by (auto intro: antisym Inf_greatest Inf_lower)   2285   2286 lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"   2287 by (auto intro: antisym Sup_least Sup_upper)   2288   2289 lemma Inf_singleton [simp]:   2290 "\<Sqinter>{a} = a"   2291 by (auto intro: antisym Inf_lower Inf_greatest)   2292   2293 lemma Sup_singleton [simp]:   2294 "\<Squnion>{a} = a"   2295 by (auto intro: antisym Sup_upper Sup_least)   2296   2297 lemma Inf_insert_simp:   2298 "\<Sqinter>insert a A = (if A = {} then a else a \<sqinter> \<Sqinter>A)"   2299 by (cases "A = {}") (simp_all, simp add: Inf_insert)   2300   2301 lemma Sup_insert_simp:   2302 "\<Squnion>insert a A = (if A = {} then a else a \<squnion> \<Squnion>A)"   2303 by (cases "A = {}") (simp_all, simp add: Sup_insert)   2304   2305 lemma Inf_binary:   2306 "\<Sqinter>{a, b} = a \<sqinter> b"   2307 by (simp add: Inf_insert_simp)   2308   2309 lemma Sup_binary:   2310 "\<Squnion>{a, b} = a \<squnion> b"   2311 by (simp add: Sup_insert_simp)   2312   2313 lemma bot_def:   2314 "bot = \<Squnion>{}"   2315 by (auto intro: antisym Sup_least)   2316   2317 lemma top_def:   2318 "top = \<Sqinter>{}"   2319 by (auto intro: antisym Inf_greatest)   2320   2321 lemma sup_bot [simp]:   2322 "x \<squnion> bot = x"   2323 using bot_least [of x] by (simp add: le_iff_sup sup_commute)   2324   2325 lemma inf_top [simp]:   2326 "x \<sqinter> top = x"   2327 using top_greatest [of x] by (simp add: le_iff_inf inf_commute)   2328   2329 definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where   2330 "SUPR A f == \<Squnion> (f  A)"   2331   2332 definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where   2333 "INFI A f == \<Sqinter> (f  A)"   2334   2335 end   2336   2337 syntax   2338 "_SUP1" :: "pttrns => 'b => 'b" ("(3SUP _./ _)" [0, 10] 10)   2339 "_SUP" :: "pttrn => 'a set => 'b => 'b" ("(3SUP _:_./ _)" [0, 10] 10)   2340 "_INF1" :: "pttrns => 'b => 'b" ("(3INF _./ _)" [0, 10] 10)   2341 "_INF" :: "pttrn => 'a set => 'b => 'b" ("(3INF _:_./ _)" [0, 10] 10)   2342   2343 translations   2344 "SUP x y. B" == "SUP x. SUP y. B"   2345 "SUP x. B" == "CONST SUPR CONST UNIV (%x. B)"   2346 "SUP x. B" == "SUP x:CONST UNIV. B"   2347 "SUP x:A. B" == "CONST SUPR A (%x. B)"   2348 "INF x y. B" == "INF x. INF y. B"   2349 "INF x. B" == "CONST INFI CONST UNIV (%x. B)"   2350 "INF x. B" == "INF x:CONST UNIV. B"   2351 "INF x:A. B" == "CONST INFI A (%x. B)"   2352   2353 (* To avoid eta-contraction of body: *)   2354 print_translation {*   2355 let   2356 fun btr' syn (A :: Abs abs :: ts) =   2357 let val (x,t) = atomic_abs_tr' abs   2358 in list_comb (Syntax.const syn$ x $A$ t, ts) end

  2359   val const_syntax_name = Sign.const_syntax_name @{theory} o fst o dest_Const

  2360 in

  2361 [(const_syntax_name @{term SUPR}, btr' "_SUP"),(const_syntax_name @{term "INFI"}, btr' "_INF")]

  2362 end

  2363 *}

  2364

  2365 context complete_lattice

  2366 begin

  2367

  2368 lemma le_SUPI: "i : A \<Longrightarrow> M i \<le> (SUP i:A. M i)"

  2369   by (auto simp add: SUPR_def intro: Sup_upper)

  2370

  2371 lemma SUP_leI: "(\<And>i. i : A \<Longrightarrow> M i \<le> u) \<Longrightarrow> (SUP i:A. M i) \<le> u"

  2372   by (auto simp add: SUPR_def intro: Sup_least)

  2373

  2374 lemma INF_leI: "i : A \<Longrightarrow> (INF i:A. M i) \<le> M i"

  2375   by (auto simp add: INFI_def intro: Inf_lower)

  2376

  2377 lemma le_INFI: "(\<And>i. i : A \<Longrightarrow> u \<le> M i) \<Longrightarrow> u \<le> (INF i:A. M i)"

  2378   by (auto simp add: INFI_def intro: Inf_greatest)

  2379

  2380 lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M"

  2381   by (auto intro: antisym SUP_leI le_SUPI)

  2382

  2383 lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M"

  2384   by (auto intro: antisym INF_leI le_INFI)

  2385

  2386 end

  2387

  2388

  2389 subsection {* Bool as complete lattice *}

  2390

  2391 instantiation bool :: complete_lattice

  2392 begin

  2393

  2394 definition

  2395   Inf_bool_def: "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)"

  2396

  2397 definition

  2398   Sup_bool_def: "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)"

  2399

  2400 instance

  2401   by intro_classes (auto simp add: Inf_bool_def Sup_bool_def le_bool_def)

  2402

  2403 end

  2404

  2405 lemma Inf_empty_bool [simp]:

  2406   "\<Sqinter>{}"

  2407   unfolding Inf_bool_def by auto

  2408

  2409 lemma not_Sup_empty_bool [simp]:

  2410   "\<not> \<Squnion>{}"

  2411   unfolding Sup_bool_def by auto

  2412

  2413

  2414 subsection {* Fun as complete lattice *}

  2415

  2416 instantiation "fun" :: (type, complete_lattice) complete_lattice

  2417 begin

  2418

  2419 definition

  2420   Inf_fun_def [code del]: "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})"

  2421

  2422 definition

  2423   Sup_fun_def [code del]: "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})"

  2424

  2425 instance

  2426   by intro_classes

  2427     (auto simp add: Inf_fun_def Sup_fun_def le_fun_def

  2428       intro: Inf_lower Sup_upper Inf_greatest Sup_least)

  2429

  2430 end

  2431

  2432 lemma Inf_empty_fun:

  2433   "\<Sqinter>{} = (\<lambda>_. \<Sqinter>{})"

  2434   by rule (auto simp add: Inf_fun_def)

  2435

  2436 lemma Sup_empty_fun:

  2437   "\<Squnion>{} = (\<lambda>_. \<Squnion>{})"

  2438   by rule (auto simp add: Sup_fun_def)

  2439

  2440

  2441 subsection {* Set as lattice *}

  2442

  2443 lemma inf_set_eq: "A \<sqinter> B = A \<inter> B"

  2444   apply (rule subset_antisym)

  2445   apply (rule Int_greatest)

  2446   apply (rule inf_le1)

  2447   apply (rule inf_le2)

  2448   apply (rule inf_greatest)

  2449   apply (rule Int_lower1)

  2450   apply (rule Int_lower2)

  2451   done

  2452

  2453 lemma sup_set_eq: "A \<squnion> B = A \<union> B"

  2454   apply (rule subset_antisym)

  2455   apply (rule sup_least)

  2456   apply (rule Un_upper1)

  2457   apply (rule Un_upper2)

  2458   apply (rule Un_least)

  2459   apply (rule sup_ge1)

  2460   apply (rule sup_ge2)

  2461   done

  2462

  2463 lemma mono_Int: "mono f \<Longrightarrow> f (A \<inter> B) \<subseteq> f A \<inter> f B"

  2464   apply (fold inf_set_eq sup_set_eq)

  2465   apply (erule mono_inf)

  2466   done

  2467

  2468 lemma mono_Un: "mono f \<Longrightarrow> f A \<union> f B \<subseteq> f (A \<union> B)"

  2469   apply (fold inf_set_eq sup_set_eq)

  2470   apply (erule mono_sup)

  2471   done

  2472

  2473 lemma Inf_set_eq: "\<Sqinter>S = \<Inter>S"

  2474   apply (rule subset_antisym)

  2475   apply (rule Inter_greatest)

  2476   apply (erule Inf_lower)

  2477   apply (rule Inf_greatest)

  2478   apply (erule Inter_lower)

  2479   done

  2480

  2481 lemma Sup_set_eq: "\<Squnion>S = \<Union>S"

  2482   apply (rule subset_antisym)

  2483   apply (rule Sup_least)

  2484   apply (erule Union_upper)

  2485   apply (rule Union_least)

  2486   apply (erule Sup_upper)

  2487   done

  2488

  2489 lemma top_set_eq: "top = UNIV"

  2490   by (iprover intro!: subset_antisym subset_UNIV top_greatest)

  2491

  2492 lemma bot_set_eq: "bot = {}"

  2493   by (iprover intro!: subset_antisym empty_subsetI bot_least)

  2494

  2495 no_notation

  2496   less_eq  (infix "\<sqsubseteq>" 50) and

  2497   less (infix "\<sqsubset>" 50) and

  2498   inf  (infixl "\<sqinter>" 70) and

  2499   sup  (infixl "\<squnion>" 65) and

  2500   Inf  ("\<Sqinter>_" [900] 900) and

  2501   Sup  ("\<Squnion>_" [900] 900)

  2502

  2503

  2504 subsection {* Misc theorem and ML bindings *}

  2505

  2506 lemmas equalityI = subset_antisym

  2507 lemmas mem_simps =

  2508   insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff

  2509   mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff

  2510   -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}

  2511

  2512 ML {*

  2513 val Ball_def = @{thm Ball_def}

  2514 val Bex_def = @{thm Bex_def}

  2515 val CollectD = @{thm CollectD}

  2516 val CollectE = @{thm CollectE}

  2517 val CollectI = @{thm CollectI}

  2518 val Collect_conj_eq = @{thm Collect_conj_eq}

  2519 val Collect_mem_eq = @{thm Collect_mem_eq}

  2520 val IntD1 = @{thm IntD1}

  2521 val IntD2 = @{thm IntD2}

  2522 val IntE = @{thm IntE}

  2523 val IntI = @{thm IntI}

  2524 val Int_Collect = @{thm Int_Collect}

  2525 val UNIV_I = @{thm UNIV_I}

  2526 val UNIV_witness = @{thm UNIV_witness}

  2527 val UnE = @{thm UnE}

  2528 val UnI1 = @{thm UnI1}

  2529 val UnI2 = @{thm UnI2}

  2530 val ballE = @{thm ballE}

  2531 val ballI = @{thm ballI}

  2532 val bexCI = @{thm bexCI}

  2533 val bexE = @{thm bexE}

  2534 val bexI = @{thm bexI}

  2535 val bex_triv = @{thm bex_triv}

  2536 val bspec = @{thm bspec}

  2537 val contra_subsetD = @{thm contra_subsetD}

  2538 val distinct_lemma = @{thm distinct_lemma}

  2539 val eq_to_mono = @{thm eq_to_mono}

  2540 val eq_to_mono2 = @{thm eq_to_mono2}

  2541 val equalityCE = @{thm equalityCE}

  2542 val equalityD1 = @{thm equalityD1}

  2543 val equalityD2 = @{thm equalityD2}

  2544 val equalityE = @{thm equalityE}

  2545 val equalityI = @{thm equalityI}

  2546 val imageE = @{thm imageE}

  2547 val imageI = @{thm imageI}

  2548 val image_Un = @{thm image_Un}

  2549 val image_insert = @{thm image_insert}

  2550 val insert_commute = @{thm insert_commute}

  2551 val insert_iff = @{thm insert_iff}

  2552 val mem_Collect_eq = @{thm mem_Collect_eq}

  2553 val rangeE = @{thm rangeE}

  2554 val rangeI = @{thm rangeI}

  2555 val range_eqI = @{thm range_eqI}

  2556 val subsetCE = @{thm subsetCE}

  2557 val subsetD = @{thm subsetD}

  2558 val subsetI = @{thm subsetI}

  2559 val subset_refl = @{thm subset_refl}

  2560 val subset_trans = @{thm subset_trans}

  2561 val vimageD = @{thm vimageD}

  2562 val vimageE = @{thm vimageE}

  2563 val vimageI = @{thm vimageI}

  2564 val vimageI2 = @{thm vimageI2}

  2565 val vimage_Collect = @{thm vimage_Collect}

  2566 val vimage_Int = @{thm vimage_Int}

  2567 val vimage_Un = @{thm vimage_Un}

  2568 *}

  2569

  2570 end
`