src/HOL/Set.thy
 author wenzelm Tue May 07 19:54:04 2002 +0200 (2002-05-07) changeset 13113 5eb9be7b72a5 parent 13103 66659a4b16f6 child 13421 8fcdf4a26468 permissions -rw-r--r--
rev_bexI [intro?];
     1 (*  Title:      HOL/Set.thy

     2     ID:         $Id$

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

     4     License:    GPL (GNU GENERAL PUBLIC LICENSE)

     5 *)

     6

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

     8

     9 theory Set = HOL:

    10

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

    12

    13

    14 subsection {* Basic syntax *}

    15

    16 global

    17

    18 typedecl 'a set

    19 arities set :: (type) type

    20

    21 consts

    22   "{}"          :: "'a set"                             ("{}")

    23   UNIV          :: "'a set"

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

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

    26   Int           :: "'a set => 'a set => 'a set"          (infixl 70)

    27   Un            :: "'a set => 'a set => 'a set"          (infixl 65)

    28   UNION         :: "'a set => ('a => 'b set) => 'b set"  -- "general union"

    29   INTER         :: "'a set => ('a => 'b set) => 'b set"  -- "general intersection"

    30   Union         :: "'a set set => 'a set"                -- "union of a set"

    31   Inter         :: "'a set set => 'a set"                -- "intersection of a set"

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

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

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

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

    36

    37 syntax

    38   "op :"        :: "'a => 'a set => bool"                ("op :")

    39 consts

    40   "op :"        :: "'a => 'a set => bool"                ("(_/ : _)" [50, 51] 50)  -- "membership"

    41

    42 local

    43

    44 instance set :: (type) ord ..

    45 instance set :: (type) minus ..

    46

    47

    48 subsection {* Additional concrete syntax *}

    49

    50 syntax

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

    52

    53   "op ~:"       :: "'a => 'a set => bool"                 ("op ~:")  -- "non-membership"

    54   "op ~:"       :: "'a => 'a set => bool"                 ("(_/ ~: _)" [50, 51] 50)

    55

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

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

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

    59

    60   "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" 10)

    61   "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" 10)

    62   "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" 10)

    63   "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" 10)

    64

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

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

    67

    68 syntax (HOL)

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

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

    71

    72 translations

    73   "range f"     == "fUNIV"

    74   "x ~: y"      == "~ (x : y)"

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

    76   "{x}"         == "insert x {}"

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

    78   "UN x y. B"   == "UN x. UN y. B"

    79   "UN x. B"     == "UNION UNIV (%x. B)"

    80   "INT x y. B"  == "INT x. INT y. B"

    81   "INT x. B"    == "INTER UNIV (%x. B)"

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

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

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

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

    86

    87 syntax (output)

    88   "_setle"      :: "'a set => 'a set => bool"             ("op <=")

    89   "_setle"      :: "'a set => 'a set => bool"             ("(_/ <= _)" [50, 51] 50)

    90   "_setless"    :: "'a set => 'a set => bool"             ("op <")

    91   "_setless"    :: "'a set => 'a set => bool"             ("(_/ < _)" [50, 51] 50)

    92

    93 syntax (xsymbols)

    94   "_setle"      :: "'a set => 'a set => bool"             ("op \<subseteq>")

    95   "_setle"      :: "'a set => 'a set => bool"             ("(_/ \<subseteq> _)" [50, 51] 50)

    96   "_setless"    :: "'a set => 'a set => bool"             ("op \<subset>")

    97   "_setless"    :: "'a set => 'a set => bool"             ("(_/ \<subset> _)" [50, 51] 50)

    98   "op Int"      :: "'a set => 'a set => 'a set"           (infixl "\<inter>" 70)

    99   "op Un"       :: "'a set => 'a set => 'a set"           (infixl "\<union>" 65)

   100   "op :"        :: "'a => 'a set => bool"                 ("op \<in>")

   101   "op :"        :: "'a => 'a set => bool"                 ("(_/ \<in> _)" [50, 51] 50)

   102   "op ~:"       :: "'a => 'a set => bool"                 ("op \<notin>")

   103   "op ~:"       :: "'a => 'a set => bool"                 ("(_/ \<notin> _)" [50, 51] 50)

   104   "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" 10)

   105   "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" 10)

   106   "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" 10)

   107   "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" 10)

   108   Union         :: "'a set set => 'a set"                 ("\<Union>_" [90] 90)

   109   Inter         :: "'a set set => 'a set"                 ("\<Inter>_" [90] 90)

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

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

   112

   113 translations

   114   "op \<subseteq>" => "op <= :: _ set => _ set => bool"

   115   "op \<subset>" => "op <  :: _ set => _ set => bool"

   116

   117

   118 typed_print_translation {*

   119   let

   120     fun le_tr' _ (Type ("fun", (Type ("set", _) :: _))) ts =

   121           list_comb (Syntax.const "_setle", ts)

   122       | le_tr' _ _ _ = raise Match;

   123

   124     fun less_tr' _ (Type ("fun", (Type ("set", _) :: _))) ts =

   125           list_comb (Syntax.const "_setless", ts)

   126       | less_tr' _ _ _ = raise Match;

   127   in [("op <=", le_tr'), ("op <", less_tr')] end

   128 *}

   129

   130 text {*

   131   \medskip Translate between @{text "{e | x1...xn. P}"} and @{text

   132   "{u. EX x1..xn. u = e & P}"}; @{text "{y. EX x1..xn. y = e & P}"} is

   133   only translated if @{text "[0..n] subset bvs(e)"}.

   134 *}

   135

   136 parse_translation {*

   137   let

   138     val ex_tr = snd (mk_binder_tr ("EX ", "Ex"));

   139

   140     fun nvars (Const ("_idts", _) $_$ idts) = nvars idts + 1

   141       | nvars _ = 1;

   142

   143     fun setcompr_tr [e, idts, b] =

   144       let

   145         val eq = Syntax.const "op =" $Bound (nvars idts)$ e;

   146         val P = Syntax.const "op &" $eq$ b;

   147         val exP = ex_tr [idts, P];

   148       in Syntax.const "Collect" $Abs ("", dummyT, exP) end;   149   150 in [("@SetCompr", setcompr_tr)] end;   151 *}   152   153 print_translation {*   154 let   155 val ex_tr' = snd (mk_binder_tr' ("Ex", "DUMMY"));   156   157 fun setcompr_tr' [Abs (_, _, P)] =   158 let   159 fun check (Const ("Ex", _)$ Abs (_, _, P), n) = check (P, n + 1)

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

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

   162                 ((0 upto (n - 1)) subset add_loose_bnos (e, 0, [])) then ()

   163               else raise Match;

   164

   165         fun tr' (_ $abs) =   166 let val _$ idts $(_$ (_ $_$ e) $Q) = ex_tr' [abs]   167 in Syntax.const "@SetCompr"$ e $idts$ Q end;

   168       in check (P, 0); tr' P end;

   169   in [("Collect", setcompr_tr')] end;

   170 *}

   171

   172

   173 subsection {* Rules and definitions *}

   174

   175 text {* Isomorphisms between predicates and sets. *}

   176

   177 axioms

   178   mem_Collect_eq [iff]: "(a : {x. P(x)}) = P(a)"

   179   Collect_mem_eq [simp]: "{x. x:A} = A"

   180

   181 defs

   182   Ball_def:     "Ball A P       == ALL x. x:A --> P(x)"

   183   Bex_def:      "Bex A P        == EX x. x:A & P(x)"

   184

   185 defs (overloaded)

   186   subset_def:   "A <= B         == ALL x:A. x:B"

   187   psubset_def:  "A < B          == (A::'a set) <= B & ~ A=B"

   188   Compl_def:    "- A            == {x. ~x:A}"

   189   set_diff_def: "A - B          == {x. x:A & ~x:B}"

   190

   191 defs

   192   Un_def:       "A Un B         == {x. x:A | x:B}"

   193   Int_def:      "A Int B        == {x. x:A & x:B}"

   194   INTER_def:    "INTER A B      == {y. ALL x:A. y: B(x)}"

   195   UNION_def:    "UNION A B      == {y. EX x:A. y: B(x)}"

   196   Inter_def:    "Inter S        == (INT x:S. x)"

   197   Union_def:    "Union S        == (UN x:S. x)"

   198   Pow_def:      "Pow A          == {B. B <= A}"

   199   empty_def:    "{}             == {x. False}"

   200   UNIV_def:     "UNIV           == {x. True}"

   201   insert_def:   "insert a B     == {x. x=a} Un B"

   202   image_def:    "fA            == {y. EX x:A. y = f(x)}"

   203

   204

   205 subsection {* Lemmas and proof tool setup *}

   206

   207 subsubsection {* Relating predicates and sets *}

   208

   209 lemma CollectI: "P(a) ==> a : {x. P(x)}"

   210   by simp

   211

   212 lemma CollectD: "a : {x. P(x)} ==> P(a)"

   213   by simp

   214

   215 lemma set_ext: assumes prem: "(!!x. (x:A) = (x:B))" shows "A = B"

   216   apply (rule prem [THEN ext, THEN arg_cong, THEN box_equals])

   217    apply (rule Collect_mem_eq)

   218   apply (rule Collect_mem_eq)

   219   done

   220

   221 lemma Collect_cong: "(!!x. P x = Q x) ==> {x. P(x)} = {x. Q(x)}"

   222   by simp

   223

   224 lemmas CollectE = CollectD [elim_format]

   225

   226

   227 subsubsection {* Bounded quantifiers *}

   228

   229 lemma ballI [intro!]: "(!!x. x:A ==> P x) ==> ALL x:A. P x"

   230   by (simp add: Ball_def)

   231

   232 lemmas strip = impI allI ballI

   233

   234 lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x"

   235   by (simp add: Ball_def)

   236

   237 lemma ballE [elim]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q"

   238   by (unfold Ball_def) blast

   239

   240 text {*

   241   \medskip This tactic takes assumptions @{prop "ALL x:A. P x"} and

   242   @{prop "a:A"}; creates assumption @{prop "P a"}.

   243 *}

   244

   245 ML {*

   246   local val ballE = thm "ballE"

   247   in fun ball_tac i = etac ballE i THEN contr_tac (i + 1) end;

   248 *}

   249

   250 text {*

   251   Gives better instantiation for bound:

   252 *}

   253

   254 ML_setup {*

   255   claset_ref() := claset() addbefore ("bspec", datac (thm "bspec") 1);

   256 *}

   257

   258 lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x"

   259   -- {* Normally the best argument order: @{prop "P x"} constrains the

   260     choice of @{prop "x:A"}. *}

   261   by (unfold Bex_def) blast

   262

   263 lemma rev_bexI [intro?]: "x:A ==> P x ==> EX x:A. P x"

   264   -- {* The best argument order when there is only one @{prop "x:A"}. *}

   265   by (unfold Bex_def) blast

   266

   267 lemma bexCI: "(ALL x:A. ~P x ==> P a) ==> a:A ==> EX x:A. P x"

   268   by (unfold Bex_def) blast

   269

   270 lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q"

   271   by (unfold Bex_def) blast

   272

   273 lemma ball_triv [simp]: "(ALL x:A. P) = ((EX x. x:A) --> P)"

   274   -- {* Trival rewrite rule. *}

   275   by (simp add: Ball_def)

   276

   277 lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)"

   278   -- {* Dual form for existentials. *}

   279   by (simp add: Bex_def)

   280

   281 lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)"

   282   by blast

   283

   284 lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)"

   285   by blast

   286

   287 lemma bex_one_point1 [simp]: "(EX x:A. x = a & P x) = (a:A & P a)"

   288   by blast

   289

   290 lemma bex_one_point2 [simp]: "(EX x:A. a = x & P x) = (a:A & P a)"

   291   by blast

   292

   293 lemma ball_one_point1 [simp]: "(ALL x:A. x = a --> P x) = (a:A --> P a)"

   294   by blast

   295

   296 lemma ball_one_point2 [simp]: "(ALL x:A. a = x --> P x) = (a:A --> P a)"

   297   by blast

   298

   299 ML_setup {*

   300   let

   301     val Ball_def = thm "Ball_def";

   302     val Bex_def = thm "Bex_def";

   303

   304     val ex_pattern = Thm.read_cterm (Theory.sign_of (the_context ()))

   305       ("EX x:A. P x & Q x", HOLogic.boolT);

   306

   307     val prove_bex_tac =

   308       rewrite_goals_tac [Bex_def] THEN Quantifier1.prove_one_point_ex_tac;

   309     val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac;

   310

   311     val all_pattern = Thm.read_cterm (Theory.sign_of (the_context ()))

   312       ("ALL x:A. P x --> Q x", HOLogic.boolT);

   313

   314     val prove_ball_tac =

   315       rewrite_goals_tac [Ball_def] THEN Quantifier1.prove_one_point_all_tac;

   316     val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac;

   317

   318     val defBEX_regroup = mk_simproc "defined BEX" [ex_pattern] rearrange_bex;

   319     val defBALL_regroup = mk_simproc "defined BALL" [all_pattern] rearrange_ball;

   320   in

   321     Addsimprocs [defBALL_regroup, defBEX_regroup]

   322   end;

   323 *}

   324

   325

   326 subsubsection {* Congruence rules *}

   327

   328 lemma ball_cong [cong]:

   329   "A = B ==> (!!x. x:B ==> P x = Q x) ==>

   330     (ALL x:A. P x) = (ALL x:B. Q x)"

   331   by (simp add: Ball_def)

   332

   333 lemma bex_cong [cong]:

   334   "A = B ==> (!!x. x:B ==> P x = Q x) ==>

   335     (EX x:A. P x) = (EX x:B. Q x)"

   336   by (simp add: Bex_def cong: conj_cong)

   337

   338

   339 subsubsection {* Subsets *}

   340

   341 lemma subsetI [intro!]: "(!!x. x:A ==> x:B) ==> A \<subseteq> B"

   342   by (simp add: subset_def)

   343

   344 text {*

   345   \medskip Map the type @{text "'a set => anything"} to just @{typ

   346   'a}; for overloading constants whose first argument has type @{typ

   347   "'a set"}.

   348 *}

   349

   350 ML {*

   351   fun overload_1st_set s = Blast.overloaded (s, HOLogic.dest_setT o domain_type);

   352 *}

   353

   354 ML "

   355   (* While (:) is not, its type must be kept

   356     for overloading of = to work. *)

   357   Blast.overloaded (\"op :\", domain_type);

   358

   359   overload_1st_set \"Ball\";            (*need UNION, INTER also?*)

   360   overload_1st_set \"Bex\";

   361

   362   (*Image: retain the type of the set being expressed*)

   363   Blast.overloaded (\"image\", domain_type);

   364 "

   365

   366 lemma subsetD [elim]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B"

   367   -- {* Rule in Modus Ponens style. *}

   368   by (unfold subset_def) blast

   369

   370 declare subsetD [intro?] -- FIXME

   371

   372 lemma rev_subsetD: "c \<in> A ==> A \<subseteq> B ==> c \<in> B"

   373   -- {* The same, with reversed premises for use with @{text erule} --

   374       cf @{text rev_mp}. *}

   375   by (rule subsetD)

   376

   377 declare rev_subsetD [intro?] -- FIXME

   378

   379 text {*

   380   \medskip Converts @{prop "A \<subseteq> B"} to @{prop "x \<in> A ==> x \<in> B"}.

   381 *}

   382

   383 ML {*

   384   local val rev_subsetD = thm "rev_subsetD"

   385   in fun impOfSubs th = th RSN (2, rev_subsetD) end;

   386 *}

   387

   388 lemma subsetCE [elim]: "A \<subseteq>  B ==> (c \<notin> A ==> P) ==> (c \<in> B ==> P) ==> P"

   389   -- {* Classical elimination rule. *}

   390   by (unfold subset_def) blast

   391

   392 text {*

   393   \medskip Takes assumptions @{prop "A \<subseteq> B"}; @{prop "c \<in> A"} and

   394   creates the assumption @{prop "c \<in> B"}.

   395 *}

   396

   397 ML {*

   398   local val subsetCE = thm "subsetCE"

   399   in fun set_mp_tac i = etac subsetCE i THEN mp_tac i end;

   400 *}

   401

   402 lemma contra_subsetD: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A"

   403   by blast

   404

   405 lemma subset_refl: "A \<subseteq> A"

   406   by fast

   407

   408 lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C"

   409   by blast

   410

   411

   412 subsubsection {* Equality *}

   413

   414 lemma subset_antisym [intro!]: "A \<subseteq> B ==> B \<subseteq> A ==> A = B"

   415   -- {* Anti-symmetry of the subset relation. *}

   416   by (rules intro: set_ext subsetD)

   417

   418 lemmas equalityI [intro!] = subset_antisym

   419

   420 text {*

   421   \medskip Equality rules from ZF set theory -- are they appropriate

   422   here?

   423 *}

   424

   425 lemma equalityD1: "A = B ==> A \<subseteq> B"

   426   by (simp add: subset_refl)

   427

   428 lemma equalityD2: "A = B ==> B \<subseteq> A"

   429   by (simp add: subset_refl)

   430

   431 text {*

   432   \medskip Be careful when adding this to the claset as @{text

   433   subset_empty} is in the simpset: @{prop "A = {}"} goes to @{prop "{}

   434   \<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}!

   435 *}

   436

   437 lemma equalityE: "A = B ==> (A \<subseteq> B ==> B \<subseteq> A ==> P) ==> P"

   438   by (simp add: subset_refl)

   439

   440 lemma equalityCE [elim]:

   441     "A = B ==> (c \<in> A ==> c \<in> B ==> P) ==> (c \<notin> A ==> c \<notin> B ==> P) ==> P"

   442   by blast

   443

   444 text {*

   445   \medskip Lemma for creating induction formulae -- for "pattern

   446   matching" on @{text p}.  To make the induction hypotheses usable,

   447   apply @{text spec} or @{text bspec} to put universal quantifiers over the free

   448   variables in @{text p}.

   449 *}

   450

   451 lemma setup_induction: "p:A ==> (!!z. z:A ==> p = z --> R) ==> R"

   452   by simp

   453

   454 lemma eqset_imp_iff: "A = B ==> (x : A) = (x : B)"

   455   by simp

   456

   457

   458 subsubsection {* The universal set -- UNIV *}

   459

   460 lemma UNIV_I [simp]: "x : UNIV"

   461   by (simp add: UNIV_def)

   462

   463 declare UNIV_I [intro]  -- {* unsafe makes it less likely to cause problems *}

   464

   465 lemma UNIV_witness [intro?]: "EX x. x : UNIV"

   466   by simp

   467

   468 lemma subset_UNIV: "A \<subseteq> UNIV"

   469   by (rule subsetI) (rule UNIV_I)

   470

   471 text {*

   472   \medskip Eta-contracting these two rules (to remove @{text P})

   473   causes them to be ignored because of their interaction with

   474   congruence rules.

   475 *}

   476

   477 lemma ball_UNIV [simp]: "Ball UNIV P = All P"

   478   by (simp add: Ball_def)

   479

   480 lemma bex_UNIV [simp]: "Bex UNIV P = Ex P"

   481   by (simp add: Bex_def)

   482

   483

   484 subsubsection {* The empty set *}

   485

   486 lemma empty_iff [simp]: "(c : {}) = False"

   487   by (simp add: empty_def)

   488

   489 lemma emptyE [elim!]: "a : {} ==> P"

   490   by simp

   491

   492 lemma empty_subsetI [iff]: "{} \<subseteq> A"

   493     -- {* One effect is to delete the ASSUMPTION @{prop "{} <= A"} *}

   494   by blast

   495

   496 lemma equals0I: "(!!y. y \<in> A ==> False) ==> A = {}"

   497   by blast

   498

   499 lemma equals0D: "A = {} ==> a \<notin> A"

   500     -- {* Use for reasoning about disjointness: @{prop "A Int B = {}"} *}

   501   by blast

   502

   503 lemma ball_empty [simp]: "Ball {} P = True"

   504   by (simp add: Ball_def)

   505

   506 lemma bex_empty [simp]: "Bex {} P = False"

   507   by (simp add: Bex_def)

   508

   509 lemma UNIV_not_empty [iff]: "UNIV ~= {}"

   510   by (blast elim: equalityE)

   511

   512

   513 subsubsection {* The Powerset operator -- Pow *}

   514

   515 lemma Pow_iff [iff]: "(A \<in> Pow B) = (A \<subseteq> B)"

   516   by (simp add: Pow_def)

   517

   518 lemma PowI: "A \<subseteq> B ==> A \<in> Pow B"

   519   by (simp add: Pow_def)

   520

   521 lemma PowD: "A \<in> Pow B ==> A \<subseteq> B"

   522   by (simp add: Pow_def)

   523

   524 lemma Pow_bottom: "{} \<in> Pow B"

   525   by simp

   526

   527 lemma Pow_top: "A \<in> Pow A"

   528   by (simp add: subset_refl)

   529

   530

   531 subsubsection {* Set complement *}

   532

   533 lemma Compl_iff [simp]: "(c \<in> -A) = (c \<notin> A)"

   534   by (unfold Compl_def) blast

   535

   536 lemma ComplI [intro!]: "(c \<in> A ==> False) ==> c \<in> -A"

   537   by (unfold Compl_def) blast

   538

   539 text {*

   540   \medskip This form, with negated conclusion, works well with the

   541   Classical prover.  Negated assumptions behave like formulae on the

   542   right side of the notional turnstile ... *}

   543

   544 lemma ComplD: "c : -A ==> c~:A"

   545   by (unfold Compl_def) blast

   546

   547 lemmas ComplE [elim!] = ComplD [elim_format]

   548

   549

   550 subsubsection {* Binary union -- Un *}

   551

   552 lemma Un_iff [simp]: "(c : A Un B) = (c:A | c:B)"

   553   by (unfold Un_def) blast

   554

   555 lemma UnI1 [elim?]: "c:A ==> c : A Un B"

   556   by simp

   557

   558 lemma UnI2 [elim?]: "c:B ==> c : A Un B"

   559   by simp

   560

   561 text {*

   562   \medskip Classical introduction rule: no commitment to @{prop A} vs

   563   @{prop B}.

   564 *}

   565

   566 lemma UnCI [intro!]: "(c~:B ==> c:A) ==> c : A Un B"

   567   by auto

   568

   569 lemma UnE [elim!]: "c : A Un B ==> (c:A ==> P) ==> (c:B ==> P) ==> P"

   570   by (unfold Un_def) blast

   571

   572

   573 subsubsection {* Binary intersection -- Int *}

   574

   575 lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)"

   576   by (unfold Int_def) blast

   577

   578 lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B"

   579   by simp

   580

   581 lemma IntD1: "c : A Int B ==> c:A"

   582   by simp

   583

   584 lemma IntD2: "c : A Int B ==> c:B"

   585   by simp

   586

   587 lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P"

   588   by simp

   589

   590

   591 subsubsection {* Set difference *}

   592

   593 lemma Diff_iff [simp]: "(c : A - B) = (c:A & c~:B)"

   594   by (unfold set_diff_def) blast

   595

   596 lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A - B"

   597   by simp

   598

   599 lemma DiffD1: "c : A - B ==> c : A"

   600   by simp

   601

   602 lemma DiffD2: "c : A - B ==> c : B ==> P"

   603   by simp

   604

   605 lemma DiffE [elim!]: "c : A - B ==> (c:A ==> c~:B ==> P) ==> P"

   606   by simp

   607

   608

   609 subsubsection {* Augmenting a set -- insert *}

   610

   611 lemma insert_iff [simp]: "(a : insert b A) = (a = b | a:A)"

   612   by (unfold insert_def) blast

   613

   614 lemma insertI1: "a : insert a B"

   615   by simp

   616

   617 lemma insertI2: "a : B ==> a : insert b B"

   618   by simp

   619

   620 lemma insertE [elim!]: "a : insert b A ==> (a = b ==> P) ==> (a:A ==> P) ==> P"

   621   by (unfold insert_def) blast

   622

   623 lemma insertCI [intro!]: "(a~:B ==> a = b) ==> a: insert b B"

   624   -- {* Classical introduction rule. *}

   625   by auto

   626

   627 lemma subset_insert_iff: "(A \<subseteq> insert x B) = (if x:A then A - {x} \<subseteq> B else A \<subseteq> B)"

   628   by auto

   629

   630

   631 subsubsection {* Singletons, using insert *}

   632

   633 lemma singletonI [intro!]: "a : {a}"

   634     -- {* Redundant? But unlike @{text insertCI}, it proves the subgoal immediately! *}

   635   by (rule insertI1)

   636

   637 lemma singletonD: "b : {a} ==> b = a"

   638   by blast

   639

   640 lemmas singletonE [elim!] = singletonD [elim_format]

   641

   642 lemma singleton_iff: "(b : {a}) = (b = a)"

   643   by blast

   644

   645 lemma singleton_inject [dest!]: "{a} = {b} ==> a = b"

   646   by blast

   647

   648 lemma singleton_insert_inj_eq [iff]: "({b} = insert a A) = (a = b & A \<subseteq> {b})"

   649   by blast

   650

   651 lemma singleton_insert_inj_eq' [iff]: "(insert a A = {b}) = (a = b & A \<subseteq> {b})"

   652   by blast

   653

   654 lemma subset_singletonD: "A \<subseteq> {x} ==> A = {} | A = {x}"

   655   by fast

   656

   657 lemma singleton_conv [simp]: "{x. x = a} = {a}"

   658   by blast

   659

   660 lemma singleton_conv2 [simp]: "{x. a = x} = {a}"

   661   by blast

   662

   663 lemma diff_single_insert: "A - {x} \<subseteq> B ==> x \<in> A ==> A \<subseteq> insert x B"

   664   by blast

   665

   666

   667 subsubsection {* Unions of families *}

   668

   669 text {*

   670   @{term [source] "UN x:A. B x"} is @{term "Union (BA)"}.

   671 *}

   672

   673 lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)"

   674   by (unfold UNION_def) blast

   675

   676 lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)"

   677   -- {* The order of the premises presupposes that @{term A} is rigid;

   678     @{term b} may be flexible. *}

   679   by auto

   680

   681 lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R"

   682   by (unfold UNION_def) blast

   683

   684 lemma UN_cong [cong]:

   685     "A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"

   686   by (simp add: UNION_def)

   687

   688

   689 subsubsection {* Intersections of families *}

   690

   691 text {* @{term [source] "INT x:A. B x"} is @{term "Inter (BA)"}. *}

   692

   693 lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)"

   694   by (unfold INTER_def) blast

   695

   696 lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)"

   697   by (unfold INTER_def) blast

   698

   699 lemma INT_D [elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a"

   700   by auto

   701

   702 lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R"

   703   -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}

   704   by (unfold INTER_def) blast

   705

   706 lemma INT_cong [cong]:

   707     "A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)"

   708   by (simp add: INTER_def)

   709

   710

   711 subsubsection {* Union *}

   712

   713 lemma Union_iff [simp]: "(A : Union C) = (EX X:C. A:X)"

   714   by (unfold Union_def) blast

   715

   716 lemma UnionI [intro]: "X:C ==> A:X ==> A : Union C"

   717   -- {* The order of the premises presupposes that @{term C} is rigid;

   718     @{term A} may be flexible. *}

   719   by auto

   720

   721 lemma UnionE [elim!]: "A : Union C ==> (!!X. A:X ==> X:C ==> R) ==> R"

   722   by (unfold Union_def) blast

   723

   724

   725 subsubsection {* Inter *}

   726

   727 lemma Inter_iff [simp]: "(A : Inter C) = (ALL X:C. A:X)"

   728   by (unfold Inter_def) blast

   729

   730 lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C"

   731   by (simp add: Inter_def)

   732

   733 text {*

   734   \medskip A destruct'' rule -- every @{term X} in @{term C}

   735   contains @{term A} as an element, but @{prop "A:X"} can hold when

   736   @{prop "X:C"} does not!  This rule is analogous to @{text spec}.

   737 *}

   738

   739 lemma InterD [elim]: "A : Inter C ==> X:C ==> A:X"

   740   by auto

   741

   742 lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R"

   743   -- {* Classical'' elimination rule -- does not require proving

   744     @{prop "X:C"}. *}

   745   by (unfold Inter_def) blast

   746

   747 text {*

   748   \medskip Image of a set under a function.  Frequently @{term b} does

   749   not have the syntactic form of @{term "f x"}.

   750 *}

   751

   752 lemma image_eqI [simp, intro]: "b = f x ==> x:A ==> b : fA"

   753   by (unfold image_def) blast

   754

   755 lemma imageI: "x : A ==> f x : f  A"

   756   by (rule image_eqI) (rule refl)

   757

   758 lemma rev_image_eqI: "x:A ==> b = f x ==> b : fA"

   759   -- {* This version's more effective when we already have the

   760     required @{term x}. *}

   761   by (unfold image_def) blast

   762

   763 lemma imageE [elim!]:

   764   "b : (%x. f x)A ==> (!!x. b = f x ==> x:A ==> P) ==> P"

   765   -- {* The eta-expansion gives variable-name preservation. *}

   766   by (unfold image_def) blast

   767

   768 lemma image_Un: "f(A Un B) = fA Un fB"

   769   by blast

   770

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

   772   by blast

   773

   774 lemma image_subset_iff: "(fA \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)"

   775   -- {* This rewrite rule would confuse users if made default. *}

   776   by blast

   777

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

   779   apply safe

   780    prefer 2 apply fast

   781   apply (rule_tac x = "{a. a : A & f a : B}" in exI)

   782   apply fast

   783   done

   784

   785 lemma image_subsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> fA \<subseteq> B"

   786   -- {* Replaces the three steps @{text subsetI}, @{text imageE},

   787     @{text hypsubst}, but breaks too many existing proofs. *}

   788   by blast

   789

   790 text {*

   791   \medskip Range of a function -- just a translation for image!

   792 *}

   793

   794 lemma range_eqI: "b = f x ==> b \<in> range f"

   795   by simp

   796

   797 lemma rangeI: "f x \<in> range f"

   798   by simp

   799

   800 lemma rangeE [elim?]: "b \<in> range (\<lambda>x. f x) ==> (!!x. b = f x ==> P) ==> P"

   801   by blast

   802

   803

   804 subsubsection {* Set reasoning tools *}

   805

   806 text {*

   807   Rewrite rules for boolean case-splitting: faster than @{text

   808   "split_if [split]"}.

   809 *}

   810

   811 lemma split_if_eq1: "((if Q then x else y) = b) = ((Q --> x = b) & (~ Q --> y = b))"

   812   by (rule split_if)

   813

   814 lemma split_if_eq2: "(a = (if Q then x else y)) = ((Q --> a = x) & (~ Q --> a = y))"

   815   by (rule split_if)

   816

   817 text {*

   818   Split ifs on either side of the membership relation.  Not for @{text

   819   "[simp]"} -- can cause goals to blow up!

   820 *}

   821

   822 lemma split_if_mem1: "((if Q then x else y) : b) = ((Q --> x : b) & (~ Q --> y : b))"

   823   by (rule split_if)

   824

   825 lemma split_if_mem2: "(a : (if Q then x else y)) = ((Q --> a : x) & (~ Q --> a : y))"

   826   by (rule split_if)

   827

   828 lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2

   829

   830 lemmas mem_simps =

   831   insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff

   832   mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff

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

   834

   835 (*Would like to add these, but the existing code only searches for the

   836   outer-level constant, which in this case is just "op :"; we instead need

   837   to use term-nets to associate patterns with rules.  Also, if a rule fails to

   838   apply, then the formula should be kept.

   839   [("uminus", Compl_iff RS iffD1), ("op -", [Diff_iff RS iffD1]),

   840    ("op Int", [IntD1,IntD2]),

   841    ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]

   842  *)

   843

   844 ML_setup {*

   845   val mksimps_pairs = [("Ball", [thm "bspec"])] @ mksimps_pairs;

   846   simpset_ref() := simpset() setmksimps (mksimps mksimps_pairs);

   847 *}

   848

   849 declare subset_UNIV [simp] subset_refl [simp]

   850

   851

   852 subsubsection {* The proper subset'' relation *}

   853

   854 lemma psubsetI [intro!]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B"

   855   by (unfold psubset_def) blast

   856

   857 lemma psubset_insert_iff:

   858   "(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)"

   859   by (auto simp add: psubset_def subset_insert_iff)

   860

   861 lemma psubset_eq: "(A \<subset> B) = (A \<subseteq> B & A \<noteq> B)"

   862   by (simp only: psubset_def)

   863

   864 lemma psubset_imp_subset: "A \<subset> B ==> A \<subseteq> B"

   865   by (simp add: psubset_eq)

   866

   867 lemma psubset_subset_trans: "A \<subset> B ==> B \<subseteq> C ==> A \<subset> C"

   868   by (auto simp add: psubset_eq)

   869

   870 lemma subset_psubset_trans: "A \<subseteq> B ==> B \<subset> C ==> A \<subset> C"

   871   by (auto simp add: psubset_eq)

   872

   873 lemma psubset_imp_ex_mem: "A \<subset> B ==> \<exists>b. b \<in> (B - A)"

   874   by (unfold psubset_def) blast

   875

   876 lemma atomize_ball:

   877     "(!!x. x \<in> A ==> P x) == Trueprop (\<forall>x\<in>A. P x)"

   878   by (simp only: Ball_def atomize_all atomize_imp)

   879

   880 declare atomize_ball [symmetric, rulify]

   881

   882

   883 subsection {* Further set-theory lemmas *}

   884

   885 subsubsection {* Derived rules involving subsets. *}

   886

   887 text {* @{text insert}. *}

   888

   889 lemma subset_insertI: "B \<subseteq> insert a B"

   890   apply (rule subsetI)

   891   apply (erule insertI2)

   892   done

   893

   894 lemma subset_insert: "x \<notin> A ==> (A \<subseteq> insert x B) = (A \<subseteq> B)"

   895   by blast

   896

   897

   898 text {* \medskip Big Union -- least upper bound of a set. *}

   899

   900 lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A"

   901   by (rules intro: subsetI UnionI)

   902

   903 lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C"

   904   by (rules intro: subsetI elim: UnionE dest: subsetD)

   905

   906

   907 text {* \medskip General union. *}

   908

   909 lemma UN_upper: "a \<in> A ==> B a \<subseteq> (\<Union>x\<in>A. B x)"

   910   by blast

   911

   912 lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C"

   913   by (rules intro: subsetI elim: UN_E dest: subsetD)

   914

   915

   916 text {* \medskip Big Intersection -- greatest lower bound of a set. *}

   917

   918 lemma Inter_lower: "B \<in> A ==> Inter A \<subseteq> B"

   919   by blast

   920

   921 lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A"

   922   by (rules intro: InterI subsetI dest: subsetD)

   923

   924 lemma INT_lower: "a \<in> A ==> (\<Inter>x\<in>A. B x) \<subseteq> B a"

   925   by blast

   926

   927 lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)"

   928   by (rules intro: INT_I subsetI dest: subsetD)

   929

   930

   931 text {* \medskip Finite Union -- the least upper bound of two sets. *}

   932

   933 lemma Un_upper1: "A \<subseteq> A \<union> B"

   934   by blast

   935

   936 lemma Un_upper2: "B \<subseteq> A \<union> B"

   937   by blast

   938

   939 lemma Un_least: "A \<subseteq> C ==> B \<subseteq> C ==> A \<union> B \<subseteq> C"

   940   by blast

   941

   942

   943 text {* \medskip Finite Intersection -- the greatest lower bound of two sets. *}

   944

   945 lemma Int_lower1: "A \<inter> B \<subseteq> A"

   946   by blast

   947

   948 lemma Int_lower2: "A \<inter> B \<subseteq> B"

   949   by blast

   950

   951 lemma Int_greatest: "C \<subseteq> A ==> C \<subseteq> B ==> C \<subseteq> A \<inter> B"

   952   by blast

   953

   954

   955 text {* \medskip Set difference. *}

   956

   957 lemma Diff_subset: "A - B \<subseteq> A"

   958   by blast

   959

   960

   961 text {* \medskip Monotonicity. *}

   962

   963 lemma mono_Un: "mono f ==> f A \<union> f B \<subseteq> f (A \<union> B)"

   964   apply (rule Un_least)

   965    apply (erule Un_upper1 [THEN [2] monoD])

   966   apply (erule Un_upper2 [THEN [2] monoD])

   967   done

   968

   969 lemma mono_Int: "mono f ==> f (A \<inter> B) \<subseteq> f A \<inter> f B"

   970   apply (rule Int_greatest)

   971    apply (erule Int_lower1 [THEN [2] monoD])

   972   apply (erule Int_lower2 [THEN [2] monoD])

   973   done

   974

   975

   976 subsubsection {* Equalities involving union, intersection, inclusion, etc. *}

   977

   978 text {* @{text "{}"}. *}

   979

   980 lemma Collect_const [simp]: "{s. P} = (if P then UNIV else {})"

   981   -- {* supersedes @{text "Collect_False_empty"} *}

   982   by auto

   983

   984 lemma subset_empty [simp]: "(A \<subseteq> {}) = (A = {})"

   985   by blast

   986

   987 lemma not_psubset_empty [iff]: "\<not> (A < {})"

   988   by (unfold psubset_def) blast

   989

   990 lemma Collect_empty_eq [simp]: "(Collect P = {}) = (\<forall>x. \<not> P x)"

   991   by auto

   992

   993 lemma Collect_neg_eq: "{x. \<not> P x} = - {x. P x}"

   994   by blast

   995

   996 lemma Collect_disj_eq: "{x. P x | Q x} = {x. P x} \<union> {x. Q x}"

   997   by blast

   998

   999 lemma Collect_conj_eq: "{x. P x & Q x} = {x. P x} \<inter> {x. Q x}"

  1000   by blast

  1001

  1002 lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"

  1003   by blast

  1004

  1005 lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"

  1006   by blast

  1007

  1008 lemma Collect_ex_eq: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"

  1009   by blast

  1010

  1011 lemma Collect_bex_eq: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"

  1012   by blast

  1013

  1014

  1015 text {* \medskip @{text insert}. *}

  1016

  1017 lemma insert_is_Un: "insert a A = {a} Un A"

  1018   -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a {}"} *}

  1019   by blast

  1020

  1021 lemma insert_not_empty [simp]: "insert a A \<noteq> {}"

  1022   by blast

  1023

  1024 lemmas empty_not_insert [simp] = insert_not_empty [symmetric, standard]

  1025

  1026 lemma insert_absorb: "a \<in> A ==> insert a A = A"

  1027   -- {* @{text "[simp]"} causes recursive calls when there are nested inserts *}

  1028   -- {* with \emph{quadratic} running time *}

  1029   by blast

  1030

  1031 lemma insert_absorb2 [simp]: "insert x (insert x A) = insert x A"

  1032   by blast

  1033

  1034 lemma insert_commute: "insert x (insert y A) = insert y (insert x A)"

  1035   by blast

  1036

  1037 lemma insert_subset [simp]: "(insert x A \<subseteq> B) = (x \<in> B & A \<subseteq> B)"

  1038   by blast

  1039

  1040 lemma mk_disjoint_insert: "a \<in> A ==> \<exists>B. A = insert a B & a \<notin> B"

  1041   -- {* use new @{text B} rather than @{text "A - {a}"} to avoid infinite unfolding *}

  1042   apply (rule_tac x = "A - {a}" in exI)

  1043   apply blast

  1044   done

  1045

  1046 lemma insert_Collect: "insert a (Collect P) = {u. u \<noteq> a --> P u}"

  1047   by auto

  1048

  1049 lemma UN_insert_distrib: "u \<in> A ==> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"

  1050   by blast

  1051

  1052 lemma insert_disjoint[simp]:

  1053  "(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})"

  1054 by blast

  1055

  1056 lemma disjoint_insert[simp]:

  1057  "(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})"

  1058 by blast

  1059

  1060 text {* \medskip @{text image}. *}

  1061

  1062 lemma image_empty [simp]: "f{} = {}"

  1063   by blast

  1064

  1065 lemma image_insert [simp]: "f  insert a B = insert (f a) (fB)"

  1066   by blast

  1067

  1068 lemma image_constant: "x \<in> A ==> (\<lambda>x. c)  A = {c}"

  1069   by blast

  1070

  1071 lemma image_image: "f  (g  A) = (\<lambda>x. f (g x))  A"

  1072   by blast

  1073

  1074 lemma insert_image [simp]: "x \<in> A ==> insert (f x) (fA) = fA"

  1075   by blast

  1076

  1077 lemma image_is_empty [iff]: "(fA = {}) = (A = {})"

  1078   by blast

  1079

  1080 lemma image_Collect: "f  {x. P x} = {f x | x. P x}"

  1081   -- {* NOT suitable as a default simprule: the RHS isn't simpler than the LHS, *}

  1082   -- {* with its implicit quantifier and conjunction.  Also image enjoys better *}

  1083   -- {* equational properties than does the RHS. *}

  1084   by blast

  1085

  1086 lemma if_image_distrib [simp]:

  1087   "(\<lambda>x. if P x then f x else g x)  S

  1088     = (f  (S \<inter> {x. P x})) \<union> (g  (S \<inter> {x. \<not> P x}))"

  1089   by (auto simp add: image_def)

  1090

  1091 lemma image_cong: "M = N ==> (!!x. x \<in> N ==> f x = g x) ==> fM = gN"

  1092   by (simp add: image_def)

  1093

  1094

  1095 text {* \medskip @{text range}. *}

  1096

  1097 lemma full_SetCompr_eq: "{u. \<exists>x. u = f x} = range f"

  1098   by auto

  1099

  1100 lemma range_composition [simp]: "range (\<lambda>x. f (g x)) = frange g"

  1101   apply (subst image_image)

  1102   apply simp

  1103   done

  1104

  1105

  1106 text {* \medskip @{text Int} *}

  1107

  1108 lemma Int_absorb [simp]: "A \<inter> A = A"

  1109   by blast

  1110

  1111 lemma Int_left_absorb: "A \<inter> (A \<inter> B) = A \<inter> B"

  1112   by blast

  1113

  1114 lemma Int_commute: "A \<inter> B = B \<inter> A"

  1115   by blast

  1116

  1117 lemma Int_left_commute: "A \<inter> (B \<inter> C) = B \<inter> (A \<inter> C)"

  1118   by blast

  1119

  1120 lemma Int_assoc: "(A \<inter> B) \<inter> C = A \<inter> (B \<inter> C)"

  1121   by blast

  1122

  1123 lemmas Int_ac = Int_assoc Int_left_absorb Int_commute Int_left_commute

  1124   -- {* Intersection is an AC-operator *}

  1125

  1126 lemma Int_absorb1: "B \<subseteq> A ==> A \<inter> B = B"

  1127   by blast

  1128

  1129 lemma Int_absorb2: "A \<subseteq> B ==> A \<inter> B = A"

  1130   by blast

  1131

  1132 lemma Int_empty_left [simp]: "{} \<inter> B = {}"

  1133   by blast

  1134

  1135 lemma Int_empty_right [simp]: "A \<inter> {} = {}"

  1136   by blast

  1137

  1138 lemma disjoint_eq_subset_Compl: "(A \<inter> B = {}) = (A \<subseteq> -B)"

  1139   by blast

  1140

  1141 lemma disjoint_iff_not_equal: "(A \<inter> B = {}) = (\<forall>x\<in>A. \<forall>y\<in>B. x \<noteq> y)"

  1142   by blast

  1143

  1144 lemma Int_UNIV_left [simp]: "UNIV \<inter> B = B"

  1145   by blast

  1146

  1147 lemma Int_UNIV_right [simp]: "A \<inter> UNIV = A"

  1148   by blast

  1149

  1150 lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"

  1151   by blast

  1152

  1153 lemma Int_Un_distrib: "A \<inter> (B \<union> C) = (A \<inter> B) \<union> (A \<inter> C)"

  1154   by blast

  1155

  1156 lemma Int_Un_distrib2: "(B \<union> C) \<inter> A = (B \<inter> A) \<union> (C \<inter> A)"

  1157   by blast

  1158

  1159 lemma Int_UNIV [simp]: "(A \<inter> B = UNIV) = (A = UNIV & B = UNIV)"

  1160   by blast

  1161

  1162 lemma Int_subset_iff: "(C \<subseteq> A \<inter> B) = (C \<subseteq> A & C \<subseteq> B)"

  1163   by blast

  1164

  1165 lemma Int_Collect: "(x \<in> A \<inter> {x. P x}) = (x \<in> A & P x)"

  1166   by blast

  1167

  1168

  1169 text {* \medskip @{text Un}. *}

  1170

  1171 lemma Un_absorb [simp]: "A \<union> A = A"

  1172   by blast

  1173

  1174 lemma Un_left_absorb: "A \<union> (A \<union> B) = A \<union> B"

  1175   by blast

  1176

  1177 lemma Un_commute: "A \<union> B = B \<union> A"

  1178   by blast

  1179

  1180 lemma Un_left_commute: "A \<union> (B \<union> C) = B \<union> (A \<union> C)"

  1181   by blast

  1182

  1183 lemma Un_assoc: "(A \<union> B) \<union> C = A \<union> (B \<union> C)"

  1184   by blast

  1185

  1186 lemmas Un_ac = Un_assoc Un_left_absorb Un_commute Un_left_commute

  1187   -- {* Union is an AC-operator *}

  1188

  1189 lemma Un_absorb1: "A \<subseteq> B ==> A \<union> B = B"

  1190   by blast

  1191

  1192 lemma Un_absorb2: "B \<subseteq> A ==> A \<union> B = A"

  1193   by blast

  1194

  1195 lemma Un_empty_left [simp]: "{} \<union> B = B"

  1196   by blast

  1197

  1198 lemma Un_empty_right [simp]: "A \<union> {} = A"

  1199   by blast

  1200

  1201 lemma Un_UNIV_left [simp]: "UNIV \<union> B = UNIV"

  1202   by blast

  1203

  1204 lemma Un_UNIV_right [simp]: "A \<union> UNIV = UNIV"

  1205   by blast

  1206

  1207 lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"

  1208   by blast

  1209

  1210 lemma Un_insert_left [simp]: "(insert a B) \<union> C = insert a (B \<union> C)"

  1211   by blast

  1212

  1213 lemma Un_insert_right [simp]: "A \<union> (insert a B) = insert a (A \<union> B)"

  1214   by blast

  1215

  1216 lemma Int_insert_left:

  1217     "(insert a B) Int C = (if a \<in> C then insert a (B \<inter> C) else B \<inter> C)"

  1218   by auto

  1219

  1220 lemma Int_insert_right:

  1221     "A \<inter> (insert a B) = (if a \<in> A then insert a (A \<inter> B) else A \<inter> B)"

  1222   by auto

  1223

  1224 lemma Un_Int_distrib: "A \<union> (B \<inter> C) = (A \<union> B) \<inter> (A \<union> C)"

  1225   by blast

  1226

  1227 lemma Un_Int_distrib2: "(B \<inter> C) \<union> A = (B \<union> A) \<inter> (C \<union> A)"

  1228   by blast

  1229

  1230 lemma Un_Int_crazy:

  1231     "(A \<inter> B) \<union> (B \<inter> C) \<union> (C \<inter> A) = (A \<union> B) \<inter> (B \<union> C) \<inter> (C \<union> A)"

  1232   by blast

  1233

  1234 lemma subset_Un_eq: "(A \<subseteq> B) = (A \<union> B = B)"

  1235   by blast

  1236

  1237 lemma Un_empty [iff]: "(A \<union> B = {}) = (A = {} & B = {})"

  1238   by blast

  1239

  1240 lemma Un_subset_iff: "(A \<union> B \<subseteq> C) = (A \<subseteq> C & B \<subseteq> C)"

  1241   by blast

  1242

  1243 lemma Un_Diff_Int: "(A - B) \<union> (A \<inter> B) = A"

  1244   by blast

  1245

  1246

  1247 text {* \medskip Set complement *}

  1248

  1249 lemma Compl_disjoint [simp]: "A \<inter> -A = {}"

  1250   by blast

  1251

  1252 lemma Compl_disjoint2 [simp]: "-A \<inter> A = {}"

  1253   by blast

  1254

  1255 lemma Compl_partition: "A \<union> (-A) = UNIV"

  1256   by blast

  1257

  1258 lemma double_complement [simp]: "- (-A) = (A::'a set)"

  1259   by blast

  1260

  1261 lemma Compl_Un [simp]: "-(A \<union> B) = (-A) \<inter> (-B)"

  1262   by blast

  1263

  1264 lemma Compl_Int [simp]: "-(A \<inter> B) = (-A) \<union> (-B)"

  1265   by blast

  1266

  1267 lemma Compl_UN [simp]: "-(\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"

  1268   by blast

  1269

  1270 lemma Compl_INT [simp]: "-(\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"

  1271   by blast

  1272

  1273 lemma subset_Compl_self_eq: "(A \<subseteq> -A) = (A = {})"

  1274   by blast

  1275

  1276 lemma Un_Int_assoc_eq: "((A \<inter> B) \<union> C = A \<inter> (B \<union> C)) = (C \<subseteq> A)"

  1277   -- {* Halmos, Naive Set Theory, page 16. *}

  1278   by blast

  1279

  1280 lemma Compl_UNIV_eq [simp]: "-UNIV = {}"

  1281   by blast

  1282

  1283 lemma Compl_empty_eq [simp]: "-{} = UNIV"

  1284   by blast

  1285

  1286 lemma Compl_subset_Compl_iff [iff]: "(-A \<subseteq> -B) = (B \<subseteq> A)"

  1287   by blast

  1288

  1289 lemma Compl_eq_Compl_iff [iff]: "(-A = -B) = (A = (B::'a set))"

  1290   by blast

  1291

  1292

  1293 text {* \medskip @{text Union}. *}

  1294

  1295 lemma Union_empty [simp]: "Union({}) = {}"

  1296   by blast

  1297

  1298 lemma Union_UNIV [simp]: "Union UNIV = UNIV"

  1299   by blast

  1300

  1301 lemma Union_insert [simp]: "Union (insert a B) = a \<union> \<Union>B"

  1302   by blast

  1303

  1304 lemma Union_Un_distrib [simp]: "\<Union>(A Un B) = \<Union>A \<union> \<Union>B"

  1305   by blast

  1306

  1307 lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"

  1308   by blast

  1309

  1310 lemma Union_empty_conv [iff]: "(\<Union>A = {}) = (\<forall>x\<in>A. x = {})"

  1311   by auto

  1312

  1313 lemma Union_disjoint: "(\<Union>C \<inter> A = {}) = (\<forall>B\<in>C. B \<inter> A = {})"

  1314   by blast

  1315

  1316

  1317 text {* \medskip @{text Inter}. *}

  1318

  1319 lemma Inter_empty [simp]: "\<Inter>{} = UNIV"

  1320   by blast

  1321

  1322 lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}"

  1323   by blast

  1324

  1325 lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"

  1326   by blast

  1327

  1328 lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"

  1329   by blast

  1330

  1331 lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"

  1332   by blast

  1333

  1334

  1335 text {*

  1336   \medskip @{text UN} and @{text INT}.

  1337

  1338   Basic identities: *}

  1339

  1340 lemma UN_empty [simp]: "(\<Union>x\<in>{}. B x) = {}"

  1341   by blast

  1342

  1343 lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"

  1344   by blast

  1345

  1346 lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"

  1347   by blast

  1348

  1349 lemma UN_absorb: "k \<in> I ==> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"

  1350   by blast

  1351

  1352 lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"

  1353   by blast

  1354

  1355 lemma INT_absorb: "k \<in> I ==> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"

  1356   by blast

  1357

  1358 lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"

  1359   by blast

  1360

  1361 lemma UN_Un: "(\<Union>i \<in> A \<union> B. M i) = (\<Union>i\<in>A. M i) \<union> (\<Union>i\<in>B. M i)"

  1362   by blast

  1363

  1364 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)"

  1365   by blast

  1366

  1367 lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"

  1368   by blast

  1369

  1370 lemma INT_subset_iff: "(B \<subseteq> (\<Inter>i\<in>I. A i)) = (\<forall>i\<in>I. B \<subseteq> A i)"

  1371   by blast

  1372

  1373 lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"

  1374   by blast

  1375

  1376 lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"

  1377   by blast

  1378

  1379 lemma INT_insert_distrib:

  1380     "u \<in> A ==> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"

  1381   by blast

  1382

  1383 lemma Union_image_eq [simp]: "\<Union>(BA) = (\<Union>x\<in>A. B x)"

  1384   by blast

  1385

  1386 lemma image_Union: "f  \<Union>S = (\<Union>x\<in>S. f  x)"

  1387   by blast

  1388

  1389 lemma Inter_image_eq [simp]: "\<Inter>(BA) = (\<Inter>x\<in>A. B x)"

  1390   by blast

  1391

  1392 lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"

  1393   by auto

  1394

  1395 lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"

  1396   by auto

  1397

  1398 lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"

  1399   by blast

  1400

  1401 lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"

  1402   -- {* Look: it has an \emph{existential} quantifier *}

  1403   by blast

  1404

  1405 lemma UN_empty3 [iff]: "(UNION A B = {}) = (\<forall>x\<in>A. B x = {})"

  1406   by auto

  1407

  1408

  1409 text {* \medskip Distributive laws: *}

  1410

  1411 lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"

  1412   by blast

  1413

  1414 lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"

  1415   by blast

  1416

  1417 lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(AC) \<union> \<Union>(BC)"

  1418   -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}

  1419   -- {* Union of a family of unions *}

  1420   by blast

  1421

  1422 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)"

  1423   -- {* Equivalent version *}

  1424   by blast

  1425

  1426 lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"

  1427   by blast

  1428

  1429 lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(AC) \<inter> \<Inter>(BC)"

  1430   by blast

  1431

  1432 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)"

  1433   -- {* Equivalent version *}

  1434   by blast

  1435

  1436 lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"

  1437   -- {* Halmos, Naive Set Theory, page 35. *}

  1438   by blast

  1439

  1440 lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"

  1441   by blast

  1442

  1443 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)"

  1444   by blast

  1445

  1446 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)"

  1447   by blast

  1448

  1449

  1450 text {* \medskip Bounded quantifiers.

  1451

  1452   The following are not added to the default simpset because

  1453   (a) they duplicate the body and (b) there are no similar rules for @{text Int}. *}

  1454

  1455 lemma ball_Un: "(\<forall>x \<in> A \<union> B. P x) = ((\<forall>x\<in>A. P x) & (\<forall>x\<in>B. P x))"

  1456   by blast

  1457

  1458 lemma bex_Un: "(\<exists>x \<in> A \<union> B. P x) = ((\<exists>x\<in>A. P x) | (\<exists>x\<in>B. P x))"

  1459   by blast

  1460

  1461 lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) = (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"

  1462   by blast

  1463

  1464 lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) = (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"

  1465   by blast

  1466

  1467

  1468 text {* \medskip Set difference. *}

  1469

  1470 lemma Diff_eq: "A - B = A \<inter> (-B)"

  1471   by blast

  1472

  1473 lemma Diff_eq_empty_iff [simp]: "(A - B = {}) = (A \<subseteq> B)"

  1474   by blast

  1475

  1476 lemma Diff_cancel [simp]: "A - A = {}"

  1477   by blast

  1478

  1479 lemma Diff_triv: "A \<inter> B = {} ==> A - B = A"

  1480   by (blast elim: equalityE)

  1481

  1482 lemma empty_Diff [simp]: "{} - A = {}"

  1483   by blast

  1484

  1485 lemma Diff_empty [simp]: "A - {} = A"

  1486   by blast

  1487

  1488 lemma Diff_UNIV [simp]: "A - UNIV = {}"

  1489   by blast

  1490

  1491 lemma Diff_insert0 [simp]: "x \<notin> A ==> A - insert x B = A - B"

  1492   by blast

  1493

  1494 lemma Diff_insert: "A - insert a B = A - B - {a}"

  1495   -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a 0"} *}

  1496   by blast

  1497

  1498 lemma Diff_insert2: "A - insert a B = A - {a} - B"

  1499   -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a 0"} *}

  1500   by blast

  1501

  1502 lemma insert_Diff_if: "insert x A - B = (if x \<in> B then A - B else insert x (A - B))"

  1503   by auto

  1504

  1505 lemma insert_Diff1 [simp]: "x \<in> B ==> insert x A - B = A - B"

  1506   by blast

  1507

  1508 lemma insert_Diff: "a \<in> A ==> insert a (A - {a}) = A"

  1509   by blast

  1510

  1511 lemma Diff_insert_absorb: "x \<notin> A ==> (insert x A) - {x} = A"

  1512   by auto

  1513

  1514 lemma Diff_disjoint [simp]: "A \<inter> (B - A) = {}"

  1515   by blast

  1516

  1517 lemma Diff_partition: "A \<subseteq> B ==> A \<union> (B - A) = B"

  1518   by blast

  1519

  1520 lemma double_diff: "A \<subseteq> B ==> B \<subseteq> C ==> B - (C - A) = A"

  1521   by blast

  1522

  1523 lemma Un_Diff_cancel [simp]: "A \<union> (B - A) = A \<union> B"

  1524   by blast

  1525

  1526 lemma Un_Diff_cancel2 [simp]: "(B - A) \<union> A = B \<union> A"

  1527   by blast

  1528

  1529 lemma Diff_Un: "A - (B \<union> C) = (A - B) \<inter> (A - C)"

  1530   by blast

  1531

  1532 lemma Diff_Int: "A - (B \<inter> C) = (A - B) \<union> (A - C)"

  1533   by blast

  1534

  1535 lemma Un_Diff: "(A \<union> B) - C = (A - C) \<union> (B - C)"

  1536   by blast

  1537

  1538 lemma Int_Diff: "(A \<inter> B) - C = A \<inter> (B - C)"

  1539   by blast

  1540

  1541 lemma Diff_Int_distrib: "C \<inter> (A - B) = (C \<inter> A) - (C \<inter> B)"

  1542   by blast

  1543

  1544 lemma Diff_Int_distrib2: "(A - B) \<inter> C = (A \<inter> C) - (B \<inter> C)"

  1545   by blast

  1546

  1547 lemma Diff_Compl [simp]: "A - (- B) = A \<inter> B"

  1548   by auto

  1549

  1550 lemma Compl_Diff_eq [simp]: "- (A - B) = -A \<union> B"

  1551   by blast

  1552

  1553

  1554 text {* \medskip Quantification over type @{typ bool}. *}

  1555

  1556 lemma all_bool_eq: "(\<forall>b::bool. P b) = (P True & P False)"

  1557   apply auto

  1558   apply (tactic {* case_tac "b" 1 *})

  1559    apply auto

  1560   done

  1561

  1562 lemma bool_induct: "P True \<Longrightarrow> P False \<Longrightarrow> P x"

  1563   by (rule conjI [THEN all_bool_eq [THEN iffD2], THEN spec])

  1564

  1565 lemma ex_bool_eq: "(\<exists>b::bool. P b) = (P True | P False)"

  1566   apply auto

  1567   apply (tactic {* case_tac "b" 1 *})

  1568    apply auto

  1569   done

  1570

  1571 lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"

  1572   by (auto simp add: split_if_mem2)

  1573

  1574 lemma UN_bool_eq: "(\<Union>b::bool. A b) = (A True \<union> A False)"

  1575   apply auto

  1576   apply (tactic {* case_tac "b" 1 *})

  1577    apply auto

  1578   done

  1579

  1580 lemma INT_bool_eq: "(\<Inter>b::bool. A b) = (A True \<inter> A False)"

  1581   apply auto

  1582   apply (tactic {* case_tac "b" 1 *})

  1583   apply auto

  1584   done

  1585

  1586

  1587 text {* \medskip @{text Pow} *}

  1588

  1589 lemma Pow_empty [simp]: "Pow {} = {{}}"

  1590   by (auto simp add: Pow_def)

  1591

  1592 lemma Pow_insert: "Pow (insert a A) = Pow A \<union> (insert a  Pow A)"

  1593   by (blast intro: image_eqI [where ?x = "u - {a}", standard])

  1594

  1595 lemma Pow_Compl: "Pow (- A) = {-B | B. A \<in> Pow B}"

  1596   by (blast intro: exI [where ?x = "- u", standard])

  1597

  1598 lemma Pow_UNIV [simp]: "Pow UNIV = UNIV"

  1599   by blast

  1600

  1601 lemma Un_Pow_subset: "Pow A \<union> Pow B \<subseteq> Pow (A \<union> B)"

  1602   by blast

  1603

  1604 lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"

  1605   by blast

  1606

  1607 lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"

  1608   by blast

  1609

  1610 lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"

  1611   by blast

  1612

  1613 lemma Pow_Int_eq [simp]: "Pow (A \<inter> B) = Pow A \<inter> Pow B"

  1614   by blast

  1615

  1616 lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"

  1617   by blast

  1618

  1619

  1620 text {* \medskip Miscellany. *}

  1621

  1622 lemma set_eq_subset: "(A = B) = (A \<subseteq> B & B \<subseteq> A)"

  1623   by blast

  1624

  1625 lemma subset_iff: "(A \<subseteq> B) = (\<forall>t. t \<in> A --> t \<in> B)"

  1626   by blast

  1627

  1628 lemma subset_iff_psubset_eq: "(A \<subseteq> B) = ((A \<subset> B) | (A = B))"

  1629   by (unfold psubset_def) blast

  1630

  1631 lemma all_not_in_conv [iff]: "(\<forall>x. x \<notin> A) = (A = {})"

  1632   by blast

  1633

  1634 lemma distinct_lemma: "f x \<noteq> f y ==> x \<noteq> y"

  1635   by rules

  1636

  1637

  1638 text {* \medskip Miniscoping: pushing in big Unions and Intersections. *}

  1639

  1640 lemma UN_simps [simp]:

  1641   "!!a B C. (UN x:C. insert a (B x)) = (if C={} then {} else insert a (UN x:C. B x))"

  1642   "!!A B C. (UN x:C. A x Un B)   = ((if C={} then {} else (UN x:C. A x) Un B))"

  1643   "!!A B C. (UN x:C. A Un B x)   = ((if C={} then {} else A Un (UN x:C. B x)))"

  1644   "!!A B C. (UN x:C. A x Int B)  = ((UN x:C. A x) Int B)"

  1645   "!!A B C. (UN x:C. A Int B x)  = (A Int (UN x:C. B x))"

  1646   "!!A B C. (UN x:C. A x - B)    = ((UN x:C. A x) - B)"

  1647   "!!A B C. (UN x:C. A - B x)    = (A - (INT x:C. B x))"

  1648   "!!A B. (UN x: Union A. B x) = (UN y:A. UN x:y. B x)"

  1649   "!!A B C. (UN z: UNION A B. C z) = (UN  x:A. UN z: B(x). C z)"

  1650   "!!A B f. (UN x:fA. B x)     = (UN a:A. B (f a))"

  1651   by auto

  1652

  1653 lemma INT_simps [simp]:

  1654   "!!A B C. (INT x:C. A x Int B) = (if C={} then UNIV else (INT x:C. A x) Int B)"

  1655   "!!A B C. (INT x:C. A Int B x) = (if C={} then UNIV else A Int (INT x:C. B x))"

  1656   "!!A B C. (INT x:C. A x - B)   = (if C={} then UNIV else (INT x:C. A x) - B)"

  1657   "!!A B C. (INT x:C. A - B x)   = (if C={} then UNIV else A - (UN x:C. B x))"

  1658   "!!a B C. (INT x:C. insert a (B x)) = insert a (INT x:C. B x)"

  1659   "!!A B C. (INT x:C. A x Un B)  = ((INT x:C. A x) Un B)"

  1660   "!!A B C. (INT x:C. A Un B x)  = (A Un (INT x:C. B x))"

  1661   "!!A B. (INT x: Union A. B x) = (INT y:A. INT x:y. B x)"

  1662   "!!A B C. (INT z: UNION A B. C z) = (INT x:A. INT z: B(x). C z)"

  1663   "!!A B f. (INT x:fA. B x)    = (INT a:A. B (f a))"

  1664   by auto

  1665

  1666 lemma ball_simps [simp]:

  1667   "!!A P Q. (ALL x:A. P x | Q) = ((ALL x:A. P x) | Q)"

  1668   "!!A P Q. (ALL x:A. P | Q x) = (P | (ALL x:A. Q x))"

  1669   "!!A P Q. (ALL x:A. P --> Q x) = (P --> (ALL x:A. Q x))"

  1670   "!!A P Q. (ALL x:A. P x --> Q) = ((EX x:A. P x) --> Q)"

  1671   "!!P. (ALL x:{}. P x) = True"

  1672   "!!P. (ALL x:UNIV. P x) = (ALL x. P x)"

  1673   "!!a B P. (ALL x:insert a B. P x) = (P a & (ALL x:B. P x))"

  1674   "!!A P. (ALL x:Union A. P x) = (ALL y:A. ALL x:y. P x)"

  1675   "!!A B P. (ALL x: UNION A B. P x) = (ALL a:A. ALL x: B a. P x)"

  1676   "!!P Q. (ALL x:Collect Q. P x) = (ALL x. Q x --> P x)"

  1677   "!!A P f. (ALL x:fA. P x) = (ALL x:A. P (f x))"

  1678   "!!A P. (~(ALL x:A. P x)) = (EX x:A. ~P x)"

  1679   by auto

  1680

  1681 lemma bex_simps [simp]:

  1682   "!!A P Q. (EX x:A. P x & Q) = ((EX x:A. P x) & Q)"

  1683   "!!A P Q. (EX x:A. P & Q x) = (P & (EX x:A. Q x))"

  1684   "!!P. (EX x:{}. P x) = False"

  1685   "!!P. (EX x:UNIV. P x) = (EX x. P x)"

  1686   "!!a B P. (EX x:insert a B. P x) = (P(a) | (EX x:B. P x))"

  1687   "!!A P. (EX x:Union A. P x) = (EX y:A. EX x:y. P x)"

  1688   "!!A B P. (EX x: UNION A B. P x) = (EX a:A. EX x:B a. P x)"

  1689   "!!P Q. (EX x:Collect Q. P x) = (EX x. Q x & P x)"

  1690   "!!A P f. (EX x:fA. P x) = (EX x:A. P (f x))"

  1691   "!!A P. (~(EX x:A. P x)) = (ALL x:A. ~P x)"

  1692   by auto

  1693

  1694 lemma ball_conj_distrib:

  1695   "(ALL x:A. P x & Q x) = ((ALL x:A. P x) & (ALL x:A. Q x))"

  1696   by blast

  1697

  1698 lemma bex_disj_distrib:

  1699   "(EX x:A. P x | Q x) = ((EX x:A. P x) | (EX x:A. Q x))"

  1700   by blast

  1701

  1702

  1703 subsubsection {* Monotonicity of various operations *}

  1704

  1705 lemma image_mono: "A \<subseteq> B ==> fA \<subseteq> fB"

  1706   by blast

  1707

  1708 lemma Pow_mono: "A \<subseteq> B ==> Pow A \<subseteq> Pow B"

  1709   by blast

  1710

  1711 lemma Union_mono: "A \<subseteq> B ==> \<Union>A \<subseteq> \<Union>B"

  1712   by blast

  1713

  1714 lemma Inter_anti_mono: "B \<subseteq> A ==> \<Inter>A \<subseteq> \<Inter>B"

  1715   by blast

  1716

  1717 lemma UN_mono:

  1718   "A \<subseteq> B ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>

  1719     (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"

  1720   by (blast dest: subsetD)

  1721

  1722 lemma INT_anti_mono:

  1723   "B \<subseteq> A ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>

  1724     (\<Inter>x\<in>A. f x) \<subseteq> (\<Inter>x\<in>A. g x)"

  1725   -- {* The last inclusion is POSITIVE! *}

  1726   by (blast dest: subsetD)

  1727

  1728 lemma insert_mono: "C \<subseteq> D ==> insert a C \<subseteq> insert a D"

  1729   by blast

  1730

  1731 lemma Un_mono: "A \<subseteq> C ==> B \<subseteq> D ==> A \<union> B \<subseteq> C \<union> D"

  1732   by blast

  1733

  1734 lemma Int_mono: "A \<subseteq> C ==> B \<subseteq> D ==> A \<inter> B \<subseteq> C \<inter> D"

  1735   by blast

  1736

  1737 lemma Diff_mono: "A \<subseteq> C ==> D \<subseteq> B ==> A - B \<subseteq> C - D"

  1738   by blast

  1739

  1740 lemma Compl_anti_mono: "A \<subseteq> B ==> -B \<subseteq> -A"

  1741   by blast

  1742

  1743 text {* \medskip Monotonicity of implications. *}

  1744

  1745 lemma in_mono: "A \<subseteq> B ==> x \<in> A --> x \<in> B"

  1746   apply (rule impI)

  1747   apply (erule subsetD)

  1748   apply assumption

  1749   done

  1750

  1751 lemma conj_mono: "P1 --> Q1 ==> P2 --> Q2 ==> (P1 & P2) --> (Q1 & Q2)"

  1752   by rules

  1753

  1754 lemma disj_mono: "P1 --> Q1 ==> P2 --> Q2 ==> (P1 | P2) --> (Q1 | Q2)"

  1755   by rules

  1756

  1757 lemma imp_mono: "Q1 --> P1 ==> P2 --> Q2 ==> (P1 --> P2) --> (Q1 --> Q2)"

  1758   by rules

  1759

  1760 lemma imp_refl: "P --> P" ..

  1761

  1762 lemma ex_mono: "(!!x. P x --> Q x) ==> (EX x. P x) --> (EX x. Q x)"

  1763   by rules

  1764

  1765 lemma all_mono: "(!!x. P x --> Q x) ==> (ALL x. P x) --> (ALL x. Q x)"

  1766   by rules

  1767

  1768 lemma Collect_mono: "(!!x. P x --> Q x) ==> Collect P \<subseteq> Collect Q"

  1769   by blast

  1770

  1771 lemma Int_Collect_mono:

  1772     "A \<subseteq> B ==> (!!x. x \<in> A ==> P x --> Q x) ==> A \<inter> Collect P \<subseteq> B \<inter> Collect Q"

  1773   by blast

  1774

  1775 lemmas basic_monos =

  1776   subset_refl imp_refl disj_mono conj_mono

  1777   ex_mono Collect_mono in_mono

  1778

  1779 lemma eq_to_mono: "a = b ==> c = d ==> b --> d ==> a --> c"

  1780   by rules

  1781

  1782 lemma eq_to_mono2: "a = b ==> c = d ==> ~ b --> ~ d ==> ~ a --> ~ c"

  1783   by rules

  1784

  1785 lemma Least_mono:

  1786   "mono (f::'a::order => 'b::order) ==> EX x:S. ALL y:S. x <= y

  1787     ==> (LEAST y. y : f  S) = f (LEAST x. x : S)"

  1788     -- {* Courtesy of Stephan Merz *}

  1789   apply clarify

  1790   apply (erule_tac P = "%x. x : S" in LeastI2)

  1791    apply fast

  1792   apply (rule LeastI2)

  1793   apply (auto elim: monoD intro!: order_antisym)

  1794   done

  1795

  1796

  1797 subsection {* Inverse image of a function *}

  1798

  1799 constdefs

  1800   vimage :: "('a => 'b) => 'b set => 'a set"    (infixr "-" 90)

  1801   "f - B == {x. f x : B}"

  1802

  1803

  1804 subsubsection {* Basic rules *}

  1805

  1806 lemma vimage_eq [simp]: "(a : f - B) = (f a : B)"

  1807   by (unfold vimage_def) blast

  1808

  1809 lemma vimage_singleton_eq: "(a : f - {b}) = (f a = b)"

  1810   by simp

  1811

  1812 lemma vimageI [intro]: "f a = b ==> b:B ==> a : f - B"

  1813   by (unfold vimage_def) blast

  1814

  1815 lemma vimageI2: "f a : A ==> a : f - A"

  1816   by (unfold vimage_def) fast

  1817

  1818 lemma vimageE [elim!]: "a: f - B ==> (!!x. f a = x ==> x:B ==> P) ==> P"

  1819   by (unfold vimage_def) blast

  1820

  1821 lemma vimageD: "a : f - A ==> f a : A"

  1822   by (unfold vimage_def) fast

  1823

  1824

  1825 subsubsection {* Equations *}

  1826

  1827 lemma vimage_empty [simp]: "f - {} = {}"

  1828   by blast

  1829

  1830 lemma vimage_Compl: "f - (-A) = -(f - A)"

  1831   by blast

  1832

  1833 lemma vimage_Un [simp]: "f - (A Un B) = (f - A) Un (f - B)"

  1834   by blast

  1835

  1836 lemma vimage_Int [simp]: "f - (A Int B) = (f - A) Int (f - B)"

  1837   by fast

  1838

  1839 lemma vimage_Union: "f - (Union A) = (UN X:A. f - X)"

  1840   by blast

  1841

  1842 lemma vimage_UN: "f-(UN x:A. B x) = (UN x:A. f - B x)"

  1843   by blast

  1844

  1845 lemma vimage_INT: "f-(INT x:A. B x) = (INT x:A. f - B x)"

  1846   by blast

  1847

  1848 lemma vimage_Collect_eq [simp]: "f - Collect P = {y. P (f y)}"

  1849   by blast

  1850

  1851 lemma vimage_Collect: "(!!x. P (f x) = Q x) ==> f - (Collect P) = Collect Q"

  1852   by blast

  1853

  1854 lemma vimage_insert: "f-(insert a B) = (f-{a}) Un (f-B)"

  1855   -- {* NOT suitable for rewriting because of the recurrence of @{term "{a}"}. *}

  1856   by blast

  1857

  1858 lemma vimage_Diff: "f - (A - B) = (f - A) - (f - B)"

  1859   by blast

  1860

  1861 lemma vimage_UNIV [simp]: "f - UNIV = UNIV"

  1862   by blast

  1863

  1864 lemma vimage_eq_UN: "f-B = (UN y: B. f-{y})"

  1865   -- {* NOT suitable for rewriting *}

  1866   by blast

  1867

  1868 lemma vimage_mono: "A \<subseteq> B ==> f - A \<subseteq> f - B"

  1869   -- {* monotonicity *}

  1870   by blast

  1871

  1872

  1873 subsection {* Transitivity rules for calculational reasoning *}

  1874

  1875 lemma forw_subst: "a = b ==> P b ==> P a"

  1876   by (rule ssubst)

  1877

  1878 lemma back_subst: "P a ==> a = b ==> P b"

  1879   by (rule subst)

  1880

  1881 lemma set_rev_mp: "x:A ==> A \<subseteq> B ==> x:B"

  1882   by (rule subsetD)

  1883

  1884 lemma set_mp: "A \<subseteq> B ==> x:A ==> x:B"

  1885   by (rule subsetD)

  1886

  1887 lemma order_neq_le_trans: "a ~= b ==> (a::'a::order) <= b ==> a < b"

  1888   by (simp add: order_less_le)

  1889

  1890 lemma order_le_neq_trans: "(a::'a::order) <= b ==> a ~= b ==> a < b"

  1891   by (simp add: order_less_le)

  1892

  1893 lemma order_less_asym': "(a::'a::order) < b ==> b < a ==> P"

  1894   by (rule order_less_asym)

  1895

  1896 lemma ord_le_eq_trans: "a <= b ==> b = c ==> a <= c"

  1897   by (rule subst)

  1898

  1899 lemma ord_eq_le_trans: "a = b ==> b <= c ==> a <= c"

  1900   by (rule ssubst)

  1901

  1902 lemma ord_less_eq_trans: "a < b ==> b = c ==> a < c"

  1903   by (rule subst)

  1904

  1905 lemma ord_eq_less_trans: "a = b ==> b < c ==> a < c"

  1906   by (rule ssubst)

  1907

  1908 lemma order_less_subst2: "(a::'a::order) < b ==> f b < (c::'c::order) ==>

  1909   (!!x y. x < y ==> f x < f y) ==> f a < c"

  1910 proof -

  1911   assume r: "!!x y. x < y ==> f x < f y"

  1912   assume "a < b" hence "f a < f b" by (rule r)

  1913   also assume "f b < c"

  1914   finally (order_less_trans) show ?thesis .

  1915 qed

  1916

  1917 lemma order_less_subst1: "(a::'a::order) < f b ==> (b::'b::order) < c ==>

  1918   (!!x y. x < y ==> f x < f y) ==> a < f c"

  1919 proof -

  1920   assume r: "!!x y. x < y ==> f x < f y"

  1921   assume "a < f b"

  1922   also assume "b < c" hence "f b < f c" by (rule r)

  1923   finally (order_less_trans) show ?thesis .

  1924 qed

  1925

  1926 lemma order_le_less_subst2: "(a::'a::order) <= b ==> f b < (c::'c::order) ==>

  1927   (!!x y. x <= y ==> f x <= f y) ==> f a < c"

  1928 proof -

  1929   assume r: "!!x y. x <= y ==> f x <= f y"

  1930   assume "a <= b" hence "f a <= f b" by (rule r)

  1931   also assume "f b < c"

  1932   finally (order_le_less_trans) show ?thesis .

  1933 qed

  1934

  1935 lemma order_le_less_subst1: "(a::'a::order) <= f b ==> (b::'b::order) < c ==>

  1936   (!!x y. x < y ==> f x < f y) ==> a < f c"

  1937 proof -

  1938   assume r: "!!x y. x < y ==> f x < f y"

  1939   assume "a <= f b"

  1940   also assume "b < c" hence "f b < f c" by (rule r)

  1941   finally (order_le_less_trans) show ?thesis .

  1942 qed

  1943

  1944 lemma order_less_le_subst2: "(a::'a::order) < b ==> f b <= (c::'c::order) ==>

  1945   (!!x y. x < y ==> f x < f y) ==> f a < c"

  1946 proof -

  1947   assume r: "!!x y. x < y ==> f x < f y"

  1948   assume "a < b" hence "f a < f b" by (rule r)

  1949   also assume "f b <= c"

  1950   finally (order_less_le_trans) show ?thesis .

  1951 qed

  1952

  1953 lemma order_less_le_subst1: "(a::'a::order) < f b ==> (b::'b::order) <= c ==>

  1954   (!!x y. x <= y ==> f x <= f y) ==> a < f c"

  1955 proof -

  1956   assume r: "!!x y. x <= y ==> f x <= f y"

  1957   assume "a < f b"

  1958   also assume "b <= c" hence "f b <= f c" by (rule r)

  1959   finally (order_less_le_trans) show ?thesis .

  1960 qed

  1961

  1962 lemma order_subst1: "(a::'a::order) <= f b ==> (b::'b::order) <= c ==>

  1963   (!!x y. x <= y ==> f x <= f y) ==> a <= f c"

  1964 proof -

  1965   assume r: "!!x y. x <= y ==> f x <= f y"

  1966   assume "a <= f b"

  1967   also assume "b <= c" hence "f b <= f c" by (rule r)

  1968   finally (order_trans) show ?thesis .

  1969 qed

  1970

  1971 lemma order_subst2: "(a::'a::order) <= b ==> f b <= (c::'c::order) ==>

  1972   (!!x y. x <= y ==> f x <= f y) ==> f a <= c"

  1973 proof -

  1974   assume r: "!!x y. x <= y ==> f x <= f y"

  1975   assume "a <= b" hence "f a <= f b" by (rule r)

  1976   also assume "f b <= c"

  1977   finally (order_trans) show ?thesis .

  1978 qed

  1979

  1980 lemma ord_le_eq_subst: "a <= b ==> f b = c ==>

  1981   (!!x y. x <= y ==> f x <= f y) ==> f a <= c"

  1982 proof -

  1983   assume r: "!!x y. x <= y ==> f x <= f y"

  1984   assume "a <= b" hence "f a <= f b" by (rule r)

  1985   also assume "f b = c"

  1986   finally (ord_le_eq_trans) show ?thesis .

  1987 qed

  1988

  1989 lemma ord_eq_le_subst: "a = f b ==> b <= c ==>

  1990   (!!x y. x <= y ==> f x <= f y) ==> a <= f c"

  1991 proof -

  1992   assume r: "!!x y. x <= y ==> f x <= f y"

  1993   assume "a = f b"

  1994   also assume "b <= c" hence "f b <= f c" by (rule r)

  1995   finally (ord_eq_le_trans) show ?thesis .

  1996 qed

  1997

  1998 lemma ord_less_eq_subst: "a < b ==> f b = c ==>

  1999   (!!x y. x < y ==> f x < f y) ==> f a < c"

  2000 proof -

  2001   assume r: "!!x y. x < y ==> f x < f y"

  2002   assume "a < b" hence "f a < f b" by (rule r)

  2003   also assume "f b = c"

  2004   finally (ord_less_eq_trans) show ?thesis .

  2005 qed

  2006

  2007 lemma ord_eq_less_subst: "a = f b ==> b < c ==>

  2008   (!!x y. x < y ==> f x < f y) ==> a < f c"

  2009 proof -

  2010   assume r: "!!x y. x < y ==> f x < f y"

  2011   assume "a = f b"

  2012   also assume "b < c" hence "f b < f c" by (rule r)

  2013   finally (ord_eq_less_trans) show ?thesis .

  2014 qed

  2015

  2016 text {*

  2017   Note that this list of rules is in reverse order of priorities.

  2018 *}

  2019

  2020 lemmas basic_trans_rules [trans] =

  2021   order_less_subst2

  2022   order_less_subst1

  2023   order_le_less_subst2

  2024   order_le_less_subst1

  2025   order_less_le_subst2

  2026   order_less_le_subst1

  2027   order_subst2

  2028   order_subst1

  2029   ord_le_eq_subst

  2030   ord_eq_le_subst

  2031   ord_less_eq_subst

  2032   ord_eq_less_subst

  2033   forw_subst

  2034   back_subst

  2035   rev_mp

  2036   mp

  2037   set_rev_mp

  2038   set_mp

  2039   order_neq_le_trans

  2040   order_le_neq_trans

  2041   order_less_trans

  2042   order_less_asym'

  2043   order_le_less_trans

  2044   order_less_le_trans

  2045   order_trans

  2046   order_antisym

  2047   ord_le_eq_trans

  2048   ord_eq_le_trans

  2049   ord_less_eq_trans

  2050   ord_eq_less_trans

  2051   trans

  2052

  2053 end