src/HOL/Set.thy
author wenzelm
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(*  Title:      HOL/Set.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
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    License:    GPL (GNU GENERAL PUBLIC LICENSE)
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*)
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header {* Set theory for higher-order logic *}
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theory Set = HOL:
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text {* A set in HOL is simply a predicate. *}
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subsection {* Basic syntax *}
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global
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typedecl 'a set
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arities set :: (type) type
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consts
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  "{}"          :: "'a set"                             ("{}")
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  UNIV          :: "'a set"
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  insert        :: "'a => 'a set => 'a set"
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  Collect       :: "('a => bool) => 'a set"              -- "comprehension"
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  Int           :: "'a set => 'a set => 'a set"          (infixl 70)
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  Un            :: "'a set => 'a set => 'a set"          (infixl 65)
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  UNION         :: "'a set => ('a => 'b set) => 'b set"  -- "general union"
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  INTER         :: "'a set => ('a => 'b set) => 'b set"  -- "general intersection"
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  Union         :: "'a set set => 'a set"                -- "union of a set"
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  Inter         :: "'a set set => 'a set"                -- "intersection of a set"
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  Pow           :: "'a set => 'a set set"                -- "powerset"
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  Ball          :: "'a set => ('a => bool) => bool"      -- "bounded universal quantifiers"
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  Bex           :: "'a set => ('a => bool) => bool"      -- "bounded existential quantifiers"
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  image         :: "('a => 'b) => 'a set => 'b set"      (infixr "`" 90)
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syntax
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  "op :"        :: "'a => 'a set => bool"                ("op :")
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consts
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  "op :"        :: "'a => 'a set => bool"                ("(_/ : _)" [50, 51] 50)  -- "membership"
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local
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instance set :: (type) "{ord, minus}" ..
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subsection {* Additional concrete syntax *}
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syntax
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  range         :: "('a => 'b) => 'b set"             -- "of function"
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  "op ~:"       :: "'a => 'a set => bool"                 ("op ~:")  -- "non-membership"
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  "op ~:"       :: "'a => 'a set => bool"                 ("(_/ ~: _)" [50, 51] 50)
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  "@Finset"     :: "args => 'a set"                       ("{(_)}")
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  "@Coll"       :: "pttrn => bool => 'a set"              ("(1{_./ _})")
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  "@SetCompr"   :: "'a => idts => bool => 'a set"         ("(1{_ |/_./ _})")
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  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" 10)
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  "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" 10)
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  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" 10)
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  "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" 10)
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  "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3ALL _:_./ _)" [0, 0, 10] 10)
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  "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3EX _:_./ _)" [0, 0, 10] 10)
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syntax (HOL)
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  "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3! _:_./ _)" [0, 0, 10] 10)
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  "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3? _:_./ _)" [0, 0, 10] 10)
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translations
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  "range f"     == "f`UNIV"
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  "x ~: y"      == "~ (x : y)"
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  "{x, xs}"     == "insert x {xs}"
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  "{x}"         == "insert x {}"
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  "{x. P}"      == "Collect (%x. P)"
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  "UN x y. B"   == "UN x. UN y. B"
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  "UN x. B"     == "UNION UNIV (%x. B)"
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  "UN x. B"     == "UN x:UNIV. B"
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  "INT x y. B"  == "INT x. INT y. B"
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  "INT x. B"    == "INTER UNIV (%x. B)"
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  "INT x. B"    == "INT x:UNIV. B"
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  "UN x:A. B"   == "UNION A (%x. B)"
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  "INT x:A. B"  == "INTER A (%x. B)"
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  "ALL x:A. P"  == "Ball A (%x. P)"
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  "EX x:A. P"   == "Bex A (%x. P)"
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syntax (output)
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  "_setle"      :: "'a set => 'a set => bool"             ("op <=")
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  "_setle"      :: "'a set => 'a set => bool"             ("(_/ <= _)" [50, 51] 50)
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  "_setless"    :: "'a set => 'a set => bool"             ("op <")
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  "_setless"    :: "'a set => 'a set => bool"             ("(_/ < _)" [50, 51] 50)
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syntax (xsymbols)
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  "_setle"      :: "'a set => 'a set => bool"             ("op \<subseteq>")
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  "_setle"      :: "'a set => 'a set => bool"             ("(_/ \<subseteq> _)" [50, 51] 50)
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  "_setless"    :: "'a set => 'a set => bool"             ("op \<subset>")
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  "_setless"    :: "'a set => 'a set => bool"             ("(_/ \<subset> _)" [50, 51] 50)
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  "op Int"      :: "'a set => 'a set => 'a set"           (infixl "\<inter>" 70)
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  "op Un"       :: "'a set => 'a set => 'a set"           (infixl "\<union>" 65)
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  "op :"        :: "'a => 'a set => bool"                 ("op \<in>")
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  "op :"        :: "'a => 'a set => bool"                 ("(_/ \<in> _)" [50, 51] 50)
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  "op ~:"       :: "'a => 'a set => bool"                 ("op \<notin>")
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  "op ~:"       :: "'a => 'a set => bool"                 ("(_/ \<notin> _)" [50, 51] 50)
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  Union         :: "'a set set => 'a set"                 ("\<Union>_" [90] 90)
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  Inter         :: "'a set set => 'a set"                 ("\<Inter>_" [90] 90)
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  "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10)
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  "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10)
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syntax (HTML output)
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  "_setle"      :: "'a set => 'a set => bool"             ("op \<subseteq>")
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  "_setle"      :: "'a set => 'a set => bool"             ("(_/ \<subseteq> _)" [50, 51] 50)
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  "_setless"    :: "'a set => 'a set => bool"             ("op \<subset>")
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  "_setless"    :: "'a set => 'a set => bool"             ("(_/ \<subset> _)" [50, 51] 50)
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  "op Int"      :: "'a set => 'a set => 'a set"           (infixl "\<inter>" 70)
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  "op Un"       :: "'a set => 'a set => 'a set"           (infixl "\<union>" 65)
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  "op :"        :: "'a => 'a set => bool"                 ("op \<in>")
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  "op :"        :: "'a => 'a set => bool"                 ("(_/ \<in> _)" [50, 51] 50)
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  "op ~:"       :: "'a => 'a set => bool"                 ("op \<notin>")
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  "op ~:"       :: "'a => 'a set => bool"                 ("(_/ \<notin> _)" [50, 51] 50)
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  "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10)
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  "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10)
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syntax (input)
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  "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" 10)
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  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" 10)
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  "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" 10)
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  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" 10)
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syntax (xsymbols)
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  "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>()\<^bsub>_\<^esub>/ _)" 10)
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  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>()\<^bsub>_\<^esub>/ _)" 10)
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  "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>()\<^bsub>_\<in>_\<^esub>/ _)" 10)
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  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>()\<^bsub>_\<in>_\<^esub>/ _)" 10)
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translations
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  "op \<subseteq>" => "op <= :: _ set => _ set => bool"
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  "op \<subset>" => "op <  :: _ set => _ set => bool"
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typed_print_translation {*
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  let
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    fun le_tr' _ (Type ("fun", (Type ("set", _) :: _))) ts =
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          list_comb (Syntax.const "_setle", ts)
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      | le_tr' _ _ _ = raise Match;
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    fun less_tr' _ (Type ("fun", (Type ("set", _) :: _))) ts =
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          list_comb (Syntax.const "_setless", ts)
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      | less_tr' _ _ _ = raise Match;
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  in [("op <=", le_tr'), ("op <", less_tr')] end
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*}
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text {*
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  \medskip Translate between @{text "{e | x1...xn. P}"} and @{text
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  "{u. EX x1..xn. u = e & P}"}; @{text "{y. EX x1..xn. y = e & P}"} is
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  only translated if @{text "[0..n] subset bvs(e)"}.
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*}
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parse_translation {*
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  let
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    val ex_tr = snd (mk_binder_tr ("EX ", "Ex"));
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    fun nvars (Const ("_idts", _) $ _ $ idts) = nvars idts + 1
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      | nvars _ = 1;
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    fun setcompr_tr [e, idts, b] =
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      let
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        val eq = Syntax.const "op =" $ Bound (nvars idts) $ e;
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        val P = Syntax.const "op &" $ eq $ b;
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        val exP = ex_tr [idts, P];
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      in Syntax.const "Collect" $ Abs ("", dummyT, exP) end;
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  in [("@SetCompr", setcompr_tr)] end;
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*}
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(* To avoid eta-contraction of body: *)
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print_translation {*
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let
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  fun btr' syn [A,Abs abs] =
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    let val (x,t) = atomic_abs_tr' abs
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    in Syntax.const syn $ x $ A $ t end
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in
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[("Ball", btr' "_Ball"),("Bex", btr' "_Bex"),
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 ("UNION", btr' "@UNION"),("INTER", btr' "@INTER")]
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end
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*}
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print_translation {*
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let
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  val ex_tr' = snd (mk_binder_tr' ("Ex", "DUMMY"));
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  fun setcompr_tr' [Abs (abs as (_, _, P))] =
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    let
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      fun check (Const ("Ex", _) $ Abs (_, _, P), n) = check (P, n + 1)
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        | check (Const ("op &", _) $ (Const ("op =", _) $ Bound m $ e) $ P, n) =
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            n > 0 andalso m = n andalso not (loose_bvar1 (P, n)) andalso
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            ((0 upto (n - 1)) subset add_loose_bnos (e, 0, []))
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        | check _ = false
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        fun tr' (_ $ abs) =
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          let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' [abs]
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          in Syntax.const "@SetCompr" $ e $ idts $ Q end;
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    in if check (P, 0) then tr' P
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       else let val (x,t) = atomic_abs_tr' abs
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            in Syntax.const "@Coll" $ x $ t end
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    end;
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  in [("Collect", setcompr_tr')] end;
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*}
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subsection {* Rules and definitions *}
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text {* Isomorphisms between predicates and sets. *}
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axioms
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  mem_Collect_eq [iff]: "(a : {x. P(x)}) = P(a)"
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  Collect_mem_eq [simp]: "{x. x:A} = A"
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defs
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  Ball_def:     "Ball A P       == ALL x. x:A --> P(x)"
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  Bex_def:      "Bex A P        == EX x. x:A & P(x)"
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defs (overloaded)
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  subset_def:   "A <= B         == ALL x:A. x:B"
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  psubset_def:  "A < B          == (A::'a set) <= B & ~ A=B"
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  Compl_def:    "- A            == {x. ~x:A}"
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  set_diff_def: "A - B          == {x. x:A & ~x:B}"
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defs
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  Un_def:       "A Un B         == {x. x:A | x:B}"
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  Int_def:      "A Int B        == {x. x:A & x:B}"
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  INTER_def:    "INTER A B      == {y. ALL x:A. y: B(x)}"
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  UNION_def:    "UNION A B      == {y. EX x:A. y: B(x)}"
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  Inter_def:    "Inter S        == (INT x:S. x)"
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  Union_def:    "Union S        == (UN x:S. x)"
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  Pow_def:      "Pow A          == {B. B <= A}"
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  empty_def:    "{}             == {x. False}"
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  UNIV_def:     "UNIV           == {x. True}"
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  insert_def:   "insert a B     == {x. x=a} Un B"
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  image_def:    "f`A            == {y. EX x:A. y = f(x)}"
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subsection {* Lemmas and proof tool setup *}
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subsubsection {* Relating predicates and sets *}
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lemma CollectI: "P(a) ==> a : {x. P(x)}"
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  by simp
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lemma CollectD: "a : {x. P(x)} ==> P(a)"
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  by simp
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lemma Collect_cong: "(!!x. P x = Q x) ==> {x. P(x)} = {x. Q(x)}"
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  by simp
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lemmas CollectE = CollectD [elim_format]
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subsubsection {* Bounded quantifiers *}
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lemma ballI [intro!]: "(!!x. x:A ==> P x) ==> ALL x:A. P x"
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  by (simp add: Ball_def)
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lemmas strip = impI allI ballI
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lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x"
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  by (simp add: Ball_def)
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lemma ballE [elim]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q"
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  by (unfold Ball_def) blast
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ML {* bind_thm("rev_ballE",permute_prems 1 1 (thm "ballE")) *}
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text {*
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  \medskip This tactic takes assumptions @{prop "ALL x:A. P x"} and
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  @{prop "a:A"}; creates assumption @{prop "P a"}.
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*}
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ML {*
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  local val ballE = thm "ballE"
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  in fun ball_tac i = etac ballE i THEN contr_tac (i + 1) end;
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*}
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text {*
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  Gives better instantiation for bound:
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*}
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ML_setup {*
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  claset_ref() := claset() addbefore ("bspec", datac (thm "bspec") 1);
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*}
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lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x"
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  -- {* Normally the best argument order: @{prop "P x"} constrains the
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    choice of @{prop "x:A"}. *}
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  by (unfold Bex_def) blast
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lemma rev_bexI [intro?]: "x:A ==> P x ==> EX x:A. P x"
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  -- {* The best argument order when there is only one @{prop "x:A"}. *}
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  by (unfold Bex_def) blast
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lemma bexCI: "(ALL x:A. ~P x ==> P a) ==> a:A ==> EX x:A. P x"
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  by (unfold Bex_def) blast
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lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q"
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  by (unfold Bex_def) blast
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lemma ball_triv [simp]: "(ALL x:A. P) = ((EX x. x:A) --> P)"
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  -- {* Trival rewrite rule. *}
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  by (simp add: Ball_def)
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lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)"
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  -- {* Dual form for existentials. *}
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  by (simp add: Bex_def)
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lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)"
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  by blast
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lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)"
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  by blast
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lemma bex_one_point1 [simp]: "(EX x:A. x = a & P x) = (a:A & P a)"
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  by blast
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lemma bex_one_point2 [simp]: "(EX x:A. a = x & P x) = (a:A & P a)"
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  by blast
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lemma ball_one_point1 [simp]: "(ALL x:A. x = a --> P x) = (a:A --> P a)"
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  by blast
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lemma ball_one_point2 [simp]: "(ALL x:A. a = x --> P x) = (a:A --> P a)"
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  by blast
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ML_setup {*
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  local
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    val Ball_def = thm "Ball_def";
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    val Bex_def = thm "Bex_def";
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    val prove_bex_tac =
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      rewrite_goals_tac [Bex_def] THEN Quantifier1.prove_one_point_ex_tac;
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    val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac;
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    val prove_ball_tac =
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      rewrite_goals_tac [Ball_def] THEN Quantifier1.prove_one_point_all_tac;
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    val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac;
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  in
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    val defBEX_regroup = Simplifier.simproc (Theory.sign_of (the_context ()))
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      "defined BEX" ["EX x:A. P x & Q x"] rearrange_bex;
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    val defBALL_regroup = Simplifier.simproc (Theory.sign_of (the_context ()))
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      "defined BALL" ["ALL x:A. P x --> Q x"] rearrange_ball;
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  end;
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  Addsimprocs [defBALL_regroup, defBEX_regroup];
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*}
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subsubsection {* Congruence rules *}
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lemma ball_cong [cong]:
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  "A = B ==> (!!x. x:B ==> P x = Q x) ==>
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    (ALL x:A. P x) = (ALL x:B. Q x)"
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  by (simp add: Ball_def)
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lemma bex_cong [cong]:
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   364
  "A = B ==> (!!x. x:B ==> P x = Q x) ==>
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    (EX x:A. P x) = (EX x:B. Q x)"
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  by (simp add: Bex_def cong: conj_cong)
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7238
36e58620ffc8 replaced HOL_quantifiers flag by "HOL" print mode;
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subsubsection {* Subsets *}
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   370
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lemma subsetI [intro!]: "(!!x. x:A ==> x:B) ==> A \<subseteq> B"
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  by (simp add: subset_def)
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text {*
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  \medskip Map the type @{text "'a set => anything"} to just @{typ
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  'a}; for overloading constants whose first argument has type @{typ
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  "'a set"}.
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*}
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lemma subsetD [elim]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B"
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  -- {* Rule in Modus Ponens style. *}
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  by (unfold subset_def) blast
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   383
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   384
declare subsetD [intro?] -- FIXME
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   385
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lemma rev_subsetD: "c \<in> A ==> A \<subseteq> B ==> c \<in> B"
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  -- {* The same, with reversed premises for use with @{text erule} --
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   388
      cf @{text rev_mp}. *}
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   389
  by (rule subsetD)
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   390
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   391
declare rev_subsetD [intro?] -- FIXME
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   392
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text {*
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  \medskip Converts @{prop "A \<subseteq> B"} to @{prop "x \<in> A ==> x \<in> B"}.
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*}
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   396
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   397
ML {*
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   398
  local val rev_subsetD = thm "rev_subsetD"
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   399
  in fun impOfSubs th = th RSN (2, rev_subsetD) end;
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   400
*}
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   401
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   402
lemma subsetCE [elim]: "A \<subseteq>  B ==> (c \<notin> A ==> P) ==> (c \<in> B ==> P) ==> P"
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   403
  -- {* Classical elimination rule. *}
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   404
  by (unfold subset_def) blast
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   405
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   406
text {*
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   407
  \medskip Takes assumptions @{prop "A \<subseteq> B"}; @{prop "c \<in> A"} and
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   408
  creates the assumption @{prop "c \<in> B"}.
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   409
*}
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   410
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   411
ML {*
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   412
  local val subsetCE = thm "subsetCE"
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   413
  in fun set_mp_tac i = etac subsetCE i THEN mp_tac i end;
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   414
*}
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   415
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   416
lemma contra_subsetD: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A"
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   417
  by blast
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   418
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   419
lemma subset_refl: "A \<subseteq> A"
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   420
  by fast
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   421
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f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
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   422
lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C"
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   423
  by blast
923
ff1574a81019 new version of HOL with curried function application
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parents:
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   424
2261
d926157c0a6a added "op :", "op ~:" syntax;
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diff changeset
   425
11979
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   426
subsubsection {* Equality *}
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   427
13865
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   428
lemma set_ext: assumes prem: "(!!x. (x:A) = (x:B))" shows "A = B"
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paulson
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   429
  apply (rule prem [THEN ext, THEN arg_cong, THEN box_equals])
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paulson
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diff changeset
   430
   apply (rule Collect_mem_eq)
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paulson
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diff changeset
   431
  apply (rule Collect_mem_eq)
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paulson
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diff changeset
   432
  done
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diff changeset
   433
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   434
lemma subset_antisym [intro!]: "A \<subseteq> B ==> B \<subseteq> A ==> A = B"
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diff changeset
   435
  -- {* Anti-symmetry of the subset relation. *}
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diff changeset
   436
  by (rules intro: set_ext subsetD)
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diff changeset
   437
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
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diff changeset
   438
lemmas equalityI [intro!] = subset_antisym
11979
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   439
0a3dace545c5 converted theory "Set";
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   440
text {*
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diff changeset
   441
  \medskip Equality rules from ZF set theory -- are they appropriate
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diff changeset
   442
  here?
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   443
*}
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diff changeset
   444
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diff changeset
   445
lemma equalityD1: "A = B ==> A \<subseteq> B"
11979
0a3dace545c5 converted theory "Set";
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diff changeset
   446
  by (simp add: subset_refl)
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diff changeset
   447
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
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diff changeset
   448
lemma equalityD2: "A = B ==> B \<subseteq> A"
11979
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diff changeset
   449
  by (simp add: subset_refl)
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diff changeset
   450
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   451
text {*
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diff changeset
   452
  \medskip Be careful when adding this to the claset as @{text
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diff changeset
   453
  subset_empty} is in the simpset: @{prop "A = {}"} goes to @{prop "{}
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f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
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diff changeset
   454
  \<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}!
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diff changeset
   455
*}
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diff changeset
   456
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diff changeset
   457
lemma equalityE: "A = B ==> (A \<subseteq> B ==> B \<subseteq> A ==> P) ==> P"
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diff changeset
   458
  by (simp add: subset_refl)
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   459
11979
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diff changeset
   460
lemma equalityCE [elim]:
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   461
    "A = B ==> (c \<in> A ==> c \<in> B ==> P) ==> (c \<notin> A ==> c \<notin> B ==> P) ==> P"
11979
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   462
  by blast
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diff changeset
   463
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   464
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   465
  \medskip Lemma for creating induction formulae -- for "pattern
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   466
  matching" on @{text p}.  To make the induction hypotheses usable,
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   467
  apply @{text spec} or @{text bspec} to put universal quantifiers over the free
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   468
  variables in @{text p}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   469
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   470
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   471
lemma setup_induction: "p:A ==> (!!z. z:A ==> p = z --> R) ==> R"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   472
  by simp
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   473
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   474
lemma eqset_imp_iff: "A = B ==> (x : A) = (x : B)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   475
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   476
13865
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   477
lemma eqelem_imp_iff: "x = y ==> (x : A) = (y : A)"
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   478
  by simp
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   479
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   480
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   481
subsubsection {* The universal set -- UNIV *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   482
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   483
lemma UNIV_I [simp]: "x : UNIV"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   484
  by (simp add: UNIV_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   485
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   486
declare UNIV_I [intro]  -- {* unsafe makes it less likely to cause problems *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   487
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   488
lemma UNIV_witness [intro?]: "EX x. x : UNIV"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   489
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   490
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   491
lemma subset_UNIV: "A \<subseteq> UNIV"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   492
  by (rule subsetI) (rule UNIV_I)
2388
d1f0505fc602 added set inclusion symbol syntax;
wenzelm
parents: 2372
diff changeset
   493
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   494
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   495
  \medskip Eta-contracting these two rules (to remove @{text P})
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   496
  causes them to be ignored because of their interaction with
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   497
  congruence rules.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   498
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   499
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   500
lemma ball_UNIV [simp]: "Ball UNIV P = All P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   501
  by (simp add: Ball_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   502
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   503
lemma bex_UNIV [simp]: "Bex UNIV P = Ex P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   504
  by (simp add: Bex_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   505
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   506
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   507
subsubsection {* The empty set *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   508
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   509
lemma empty_iff [simp]: "(c : {}) = False"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   510
  by (simp add: empty_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   511
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   512
lemma emptyE [elim!]: "a : {} ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   513
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   514
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   515
lemma empty_subsetI [iff]: "{} \<subseteq> A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   516
    -- {* One effect is to delete the ASSUMPTION @{prop "{} <= A"} *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   517
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   518
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   519
lemma equals0I: "(!!y. y \<in> A ==> False) ==> A = {}"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   520
  by blast
2388
d1f0505fc602 added set inclusion symbol syntax;
wenzelm
parents: 2372
diff changeset
   521
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   522
lemma equals0D: "A = {} ==> a \<notin> A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   523
    -- {* Use for reasoning about disjointness: @{prop "A Int B = {}"} *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   524
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   525
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   526
lemma ball_empty [simp]: "Ball {} P = True"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   527
  by (simp add: Ball_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   528
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   529
lemma bex_empty [simp]: "Bex {} P = False"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   530
  by (simp add: Bex_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   531
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   532
lemma UNIV_not_empty [iff]: "UNIV ~= {}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   533
  by (blast elim: equalityE)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   534
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   535
12023
wenzelm
parents: 12020
diff changeset
   536
subsubsection {* The Powerset operator -- Pow *}
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   537
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   538
lemma Pow_iff [iff]: "(A \<in> Pow B) = (A \<subseteq> B)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   539
  by (simp add: Pow_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   540
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   541
lemma PowI: "A \<subseteq> B ==> A \<in> Pow B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   542
  by (simp add: Pow_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   543
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   544
lemma PowD: "A \<in> Pow B ==> A \<subseteq> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   545
  by (simp add: Pow_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   546
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   547
lemma Pow_bottom: "{} \<in> Pow B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   548
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   549
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   550
lemma Pow_top: "A \<in> Pow A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   551
  by (simp add: subset_refl)
2684
9781d63ef063 added proper subset symbols syntax;
wenzelm
parents: 2412
diff changeset
   552
2388
d1f0505fc602 added set inclusion symbol syntax;
wenzelm
parents: 2372
diff changeset
   553
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   554
subsubsection {* Set complement *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   555
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   556
lemma Compl_iff [simp]: "(c \<in> -A) = (c \<notin> A)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   557
  by (unfold Compl_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   558
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   559
lemma ComplI [intro!]: "(c \<in> A ==> False) ==> c \<in> -A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   560
  by (unfold Compl_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   561
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   562
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   563
  \medskip This form, with negated conclusion, works well with the
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   564
  Classical prover.  Negated assumptions behave like formulae on the
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   565
  right side of the notional turnstile ... *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   566
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   567
lemma ComplD: "c : -A ==> c~:A"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   568
  by (unfold Compl_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   569
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   570
lemmas ComplE [elim!] = ComplD [elim_format]
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   571
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   572
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   573
subsubsection {* Binary union -- Un *}
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   574
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   575
lemma Un_iff [simp]: "(c : A Un B) = (c:A | c:B)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   576
  by (unfold Un_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   577
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   578
lemma UnI1 [elim?]: "c:A ==> c : A Un B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   579
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   580
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   581
lemma UnI2 [elim?]: "c:B ==> c : A Un B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   582
  by simp
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   583
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   584
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   585
  \medskip Classical introduction rule: no commitment to @{prop A} vs
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   586
  @{prop B}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   587
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   588
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   589
lemma UnCI [intro!]: "(c~:B ==> c:A) ==> c : A Un B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   590
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   591
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   592
lemma UnE [elim!]: "c : A Un B ==> (c:A ==> P) ==> (c:B ==> P) ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   593
  by (unfold Un_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   594
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   595
12023
wenzelm
parents: 12020
diff changeset
   596
subsubsection {* Binary intersection -- Int *}
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   597
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   598
lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   599
  by (unfold Int_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   600
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   601
lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   602
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   603
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   604
lemma IntD1: "c : A Int B ==> c:A"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   605
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   606
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   607
lemma IntD2: "c : A Int B ==> c:B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   608
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   609
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   610
lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   611
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   612
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   613
12023
wenzelm
parents: 12020
diff changeset
   614
subsubsection {* Set difference *}
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   615
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   616
lemma Diff_iff [simp]: "(c : A - B) = (c:A & c~:B)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   617
  by (unfold set_diff_def) blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   618
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   619
lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A - B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   620
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   621
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   622
lemma DiffD1: "c : A - B ==> c : A"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   623
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   624
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   625
lemma DiffD2: "c : A - B ==> c : B ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   626
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   627
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   628
lemma DiffE [elim!]: "c : A - B ==> (c:A ==> c~:B ==> P) ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   629
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   630
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   631
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   632
subsubsection {* Augmenting a set -- insert *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   633
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   634
lemma insert_iff [simp]: "(a : insert b A) = (a = b | a:A)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   635
  by (unfold insert_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   636
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   637
lemma insertI1: "a : insert a B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   638
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   639
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   640
lemma insertI2: "a : B ==> a : insert b B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   641
  by simp
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   642
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   643
lemma insertE [elim!]: "a : insert b A ==> (a = b ==> P) ==> (a:A ==> P) ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   644
  by (unfold insert_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   645
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   646
lemma insertCI [intro!]: "(a~:B ==> a = b) ==> a: insert b B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   647
  -- {* Classical introduction rule. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   648
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   649
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   650
lemma subset_insert_iff: "(A \<subseteq> insert x B) = (if x:A then A - {x} \<subseteq> B else A \<subseteq> B)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   651
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   652
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   653
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   654
subsubsection {* Singletons, using insert *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   655
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   656
lemma singletonI [intro!]: "a : {a}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   657
    -- {* Redundant? But unlike @{text insertCI}, it proves the subgoal immediately! *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   658
  by (rule insertI1)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   659
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   660
lemma singletonD: "b : {a} ==> b = a"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   661
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   662
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   663
lemmas singletonE [elim!] = singletonD [elim_format]
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   664
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   665
lemma singleton_iff: "(b : {a}) = (b = a)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   666
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   667
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   668
lemma singleton_inject [dest!]: "{a} = {b} ==> a = b"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   669
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   670
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   671
lemma singleton_insert_inj_eq [iff]: "({b} = insert a A) = (a = b & A \<subseteq> {b})"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   672
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   673
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   674
lemma singleton_insert_inj_eq' [iff]: "(insert a A = {b}) = (a = b & A \<subseteq> {b})"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   675
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   676
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   677
lemma subset_singletonD: "A \<subseteq> {x} ==> A = {} | A = {x}"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   678
  by fast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   679
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   680
lemma singleton_conv [simp]: "{x. x = a} = {a}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   681
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   682
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   683
lemma singleton_conv2 [simp]: "{x. a = x} = {a}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   684
  by blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   685
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   686
lemma diff_single_insert: "A - {x} \<subseteq> B ==> x \<in> A ==> A \<subseteq> insert x B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   687
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   688
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   689
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   690
subsubsection {* Unions of families *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   691
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   692
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   693
  @{term [source] "UN x:A. B x"} is @{term "Union (B`A)"}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   694
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   695
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   696
lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   697
  by (unfold UNION_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   698
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   699
lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   700
  -- {* The order of the premises presupposes that @{term A} is rigid;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   701
    @{term b} may be flexible. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   702
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   703
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   704
lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   705
  by (unfold UNION_def) blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   706
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   707
lemma UN_cong [cong]:
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   708
    "A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   709
  by (simp add: UNION_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   710
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   711
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   712
subsubsection {* Intersections of families *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   713
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   714
text {* @{term [source] "INT x:A. B x"} is @{term "Inter (B`A)"}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   715
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   716
lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   717
  by (unfold INTER_def) blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   718
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   719
lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   720
  by (unfold INTER_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   721
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   722
lemma INT_D [elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   723
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   724
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   725
lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   726
  -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   727
  by (unfold INTER_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   728
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   729
lemma INT_cong [cong]:
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   730
    "A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   731
  by (simp add: INTER_def)
7238
36e58620ffc8 replaced HOL_quantifiers flag by "HOL" print mode;
wenzelm
parents: 5931
diff changeset
   732
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   733
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   734
subsubsection {* Union *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   735
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   736
lemma Union_iff [simp]: "(A : Union C) = (EX X:C. A:X)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   737
  by (unfold Union_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   738
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   739
lemma UnionI [intro]: "X:C ==> A:X ==> A : Union C"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   740
  -- {* The order of the premises presupposes that @{term C} is rigid;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   741
    @{term A} may be flexible. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   742
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   743
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   744
lemma UnionE [elim!]: "A : Union C ==> (!!X. A:X ==> X:C ==> R) ==> R"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   745
  by (unfold Union_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   746
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   747
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   748
subsubsection {* Inter *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   749
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   750
lemma Inter_iff [simp]: "(A : Inter C) = (ALL X:C. A:X)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   751
  by (unfold Inter_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   752
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   753
lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   754
  by (simp add: Inter_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   755
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   756
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   757
  \medskip A ``destruct'' rule -- every @{term X} in @{term C}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   758
  contains @{term A} as an element, but @{prop "A:X"} can hold when
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   759
  @{prop "X:C"} does not!  This rule is analogous to @{text spec}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   760
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   761
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   762
lemma InterD [elim]: "A : Inter C ==> X:C ==> A:X"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   763
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   764
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   765
lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   766
  -- {* ``Classical'' elimination rule -- does not require proving
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   767
    @{prop "X:C"}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   768
  by (unfold Inter_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   769
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   770
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   771
  \medskip Image of a set under a function.  Frequently @{term b} does
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   772
  not have the syntactic form of @{term "f x"}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   773
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   774
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   775
lemma image_eqI [simp, intro]: "b = f x ==> x:A ==> b : f`A"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   776
  by (unfold image_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   777
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   778
lemma imageI: "x : A ==> f x : f ` A"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   779
  by (rule image_eqI) (rule refl)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   780
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   781
lemma rev_image_eqI: "x:A ==> b = f x ==> b : f`A"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   782
  -- {* This version's more effective when we already have the
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   783
    required @{term x}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   784
  by (unfold image_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   785
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   786
lemma imageE [elim!]:
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   787
  "b : (%x. f x)`A ==> (!!x. b = f x ==> x:A ==> P) ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   788
  -- {* The eta-expansion gives variable-name preservation. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   789
  by (unfold image_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   790
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   791
lemma image_Un: "f`(A Un B) = f`A Un f`B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   792
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   793
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   794
lemma image_iff: "(z : f`A) = (EX x:A. z = f x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   795
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   796
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   797
lemma image_subset_iff: "(f`A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   798
  -- {* This rewrite rule would confuse users if made default. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   799
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   800
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   801
lemma subset_image_iff: "(B \<subseteq> f`A) = (EX AA. AA \<subseteq> A & B = f`AA)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   802
  apply safe
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   803
   prefer 2 apply fast
14208
144f45277d5a misc tidying
paulson
parents: 14098
diff changeset
   804
  apply (rule_tac x = "{a. a : A & f a : B}" in exI, fast)
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   805
  done
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   806
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   807
lemma image_subsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f`A \<subseteq> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   808
  -- {* Replaces the three steps @{text subsetI}, @{text imageE},
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   809
    @{text hypsubst}, but breaks too many existing proofs. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   810
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   811
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   812
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   813
  \medskip Range of a function -- just a translation for image!
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   814
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   815
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   816
lemma range_eqI: "b = f x ==> b \<in> range f"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   817
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   818
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   819
lemma rangeI: "f x \<in> range f"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   820
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   821
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   822
lemma rangeE [elim?]: "b \<in> range (\<lambda>x. f x) ==> (!!x. b = f x ==> P) ==> P"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   823
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   824
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   825
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   826
subsubsection {* Set reasoning tools *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   827
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   828
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   829
  Rewrite rules for boolean case-splitting: faster than @{text
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   830
  "split_if [split]"}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   831
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   832
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   833
lemma split_if_eq1: "((if Q then x else y) = b) = ((Q --> x = b) & (~ Q --> y = b))"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   834
  by (rule split_if)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   835
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   836
lemma split_if_eq2: "(a = (if Q then x else y)) = ((Q --> a = x) & (~ Q --> a = y))"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   837
  by (rule split_if)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   838
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   839
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   840
  Split ifs on either side of the membership relation.  Not for @{text
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   841
  "[simp]"} -- can cause goals to blow up!
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   842
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   843
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   844
lemma split_if_mem1: "((if Q then x else y) : b) = ((Q --> x : b) & (~ Q --> y : b))"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   845
  by (rule split_if)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   846
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   847
lemma split_if_mem2: "(a : (if Q then x else y)) = ((Q --> a : x) & (~ Q --> a : y))"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   848
  by (rule split_if)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   849
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   850
lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   851
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   852
lemmas mem_simps =
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   853
  insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   854
  mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   855
  -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   856
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   857
(*Would like to add these, but the existing code only searches for the
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   858
  outer-level constant, which in this case is just "op :"; we instead need
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   859
  to use term-nets to associate patterns with rules.  Also, if a rule fails to
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   860
  apply, then the formula should be kept.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   861
  [("uminus", Compl_iff RS iffD1), ("op -", [Diff_iff RS iffD1]),
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   862
   ("op Int", [IntD1,IntD2]),
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   863
   ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   864
 *)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   865
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   866
ML_setup {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   867
  val mksimps_pairs = [("Ball", [thm "bspec"])] @ mksimps_pairs;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   868
  simpset_ref() := simpset() setmksimps (mksimps mksimps_pairs);
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   869
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   870
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   871
declare subset_UNIV [simp] subset_refl [simp]
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   872
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   873
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   874
subsubsection {* The ``proper subset'' relation *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   875
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   876
lemma psubsetI [intro!]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   877
  by (unfold psubset_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   878
13624
17684cf64fda added the new elim rule psubsetE
paulson
parents: 13550
diff changeset
   879
lemma psubsetE [elim!]: 
17684cf64fda added the new elim rule psubsetE
paulson
parents: 13550
diff changeset
   880
    "[|A \<subset> B;  [|A \<subseteq> B; ~ (B\<subseteq>A)|] ==> R|] ==> R"
17684cf64fda added the new elim rule psubsetE
paulson
parents: 13550
diff changeset
   881
  by (unfold psubset_def) blast
17684cf64fda added the new elim rule psubsetE
paulson
parents: 13550
diff changeset
   882
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   883
lemma psubset_insert_iff:
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   884
  "(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)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   885
  by (auto simp add: psubset_def subset_insert_iff)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   886
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   887
lemma psubset_eq: "(A \<subset> B) = (A \<subseteq> B & A \<noteq> B)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   888
  by (simp only: psubset_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   889
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   890
lemma psubset_imp_subset: "A \<subset> B ==> A \<subseteq> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   891
  by (simp add: psubset_eq)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   892
14335
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
   893
lemma psubset_trans: "[| A \<subset> B; B \<subset> C |] ==> A \<subset> C"
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
   894
apply (unfold psubset_def)
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
   895
apply (auto dest: subset_antisym)
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
   896
done
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
   897
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
   898
lemma psubsetD: "[| A \<subset> B; c \<in> A |] ==> c \<in> B"
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
   899
apply (unfold psubset_def)
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
   900
apply (auto dest: subsetD)
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
   901
done
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
   902
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   903
lemma psubset_subset_trans: "A \<subset> B ==> B \<subseteq> C ==> A \<subset> C"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   904
  by (auto simp add: psubset_eq)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   905
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   906
lemma subset_psubset_trans: "A \<subseteq> B ==> B \<subset> C ==> A \<subset> C"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   907
  by (auto simp add: psubset_eq)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   908
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   909
lemma psubset_imp_ex_mem: "A \<subset> B ==> \<exists>b. b \<in> (B - A)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   910
  by (unfold psubset_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   911
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   912
lemma atomize_ball:
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   913
    "(!!x. x \<in> A ==> P x) == Trueprop (\<forall>x\<in>A. P x)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   914
  by (simp only: Ball_def atomize_all atomize_imp)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   915
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   916
declare atomize_ball [symmetric, rulify]
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   917
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   918
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   919
subsection {* Further set-theory lemmas *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   920
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   921
subsubsection {* Derived rules involving subsets. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   922
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   923
text {* @{text insert}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   924
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   925
lemma subset_insertI: "B \<subseteq> insert a B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   926
  apply (rule subsetI)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   927
  apply (erule insertI2)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   928
  done
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   929
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
   930
lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
   931
by blast
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
   932
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   933
lemma subset_insert: "x \<notin> A ==> (A \<subseteq> insert x B) = (A \<subseteq> B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   934
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   935
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   936
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   937
text {* \medskip Big Union -- least upper bound of a set. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   938
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   939
lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   940
  by (rules intro: subsetI UnionI)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   941
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   942
lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   943
  by (rules intro: subsetI elim: UnionE dest: subsetD)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   944
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   945
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   946
text {* \medskip General union. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   947
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   948
lemma UN_upper: "a \<in> A ==> B a \<subseteq> (\<Union>x\<in>A. B x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   949
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   950
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   951
lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   952
  by (rules intro: subsetI elim: UN_E dest: subsetD)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   953
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   954
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   955
text {* \medskip Big Intersection -- greatest lower bound of a set. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   956
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   957
lemma Inter_lower: "B \<in> A ==> Inter A \<subseteq> B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   958
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   959
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents: 14479
diff changeset
   960
lemma Inter_subset:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents: 14479
diff changeset
   961
  "[| !!X. X \<in> A ==> X \<subseteq> B; A ~= {} |] ==> \<Inter>A \<subseteq> B"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents: 14479
diff changeset
   962
  by blast
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents: 14479
diff changeset
   963
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   964
lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   965
  by (rules intro: InterI subsetI dest: subsetD)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   966
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   967
lemma INT_lower: "a \<in> A ==> (\<Inter>x\<in>A. B x) \<subseteq> B a"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   968
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   969
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   970
lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   971
  by (rules intro: INT_I subsetI dest: subsetD)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   972
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   973
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   974
text {* \medskip Finite Union -- the least upper bound of two sets. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   975
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   976
lemma Un_upper1: "A \<subseteq> A \<union> B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   977
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   978
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   979
lemma Un_upper2: "B \<subseteq> A \<union> B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   980
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   981
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   982
lemma Un_least: "A \<subseteq> C ==> B \<subseteq> C ==> A \<union> B \<subseteq> C"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   983
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   984
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   985
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   986
text {* \medskip Finite Intersection -- the greatest lower bound of two sets. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   987
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   988
lemma Int_lower1: "A \<inter> B \<subseteq> A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   989
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   990
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   991
lemma Int_lower2: "A \<inter> B \<subseteq> B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   992
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   993
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   994
lemma Int_greatest: "C \<subseteq> A ==> C \<subseteq> B ==> C \<subseteq> A \<inter> B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   995
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   996
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   997
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   998
text {* \medskip Set difference. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   999
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1000
lemma Diff_subset: "A - B \<subseteq> A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1001
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1002
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1003
lemma Diff_subset_conv: "(A - B \<subseteq> C) = (A \<subseteq> B \<union> C)"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1004
by blast
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1005
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1006
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1007
text {* \medskip Monotonicity. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1008
13421
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13113
diff changeset
  1009
lemma mono_Un: includes mono shows "f A \<union> f B \<subseteq> f (A \<union> B)"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1010
  apply (rule Un_least)
13421
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13113
diff changeset
  1011
   apply (rule Un_upper1 [THEN mono])
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13113
diff changeset
  1012
  apply (rule Un_upper2 [THEN mono])
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1013
  done
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1014
13421
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13113
diff changeset
  1015
lemma mono_Int: includes mono shows "f (A \<inter> B) \<subseteq> f A \<inter> f B"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1016
  apply (rule Int_greatest)
13421
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13113
diff changeset
  1017
   apply (rule Int_lower1 [THEN mono])
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13113
diff changeset
  1018
  apply (rule Int_lower2 [THEN mono])
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1019
  done
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1020
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1021
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1022
subsubsection {* Equalities involving union, intersection, inclusion, etc. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1023
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1024
text {* @{text "{}"}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1025
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1026
lemma Collect_const [simp]: "{s. P} = (if P then UNIV else {})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1027
  -- {* supersedes @{text "Collect_False_empty"} *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1028
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1029
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1030
lemma subset_empty [simp]: "(A \<subseteq> {}) = (A = {})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1031
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1032
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1033
lemma not_psubset_empty [iff]: "\<not> (A < {})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1034
  by (unfold psubset_def) blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1035
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1036
lemma Collect_empty_eq [simp]: "(Collect P = {}) = (\<forall>x. \<not> P x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1037
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1038
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1039
lemma Collect_neg_eq: "{x. \<not> P x} = - {x. P x}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1040
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1041
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1042
lemma Collect_disj_eq: "{x. P x | Q x} = {x. P x} \<union> {x. Q x}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1043
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1044
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1045
lemma Collect_conj_eq: "{x. P x & Q x} = {x. P x} \<inter> {x. Q x}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1046
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1047
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1048
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1049
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1050
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1051
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1052
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1053
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1054
lemma Collect_ex_eq: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1055
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1056
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1057
lemma Collect_bex_eq: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1058
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1059
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1060
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1061
text {* \medskip @{text insert}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1062
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1063
lemma insert_is_Un: "insert a A = {a} Un A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1064
  -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a {}"} *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1065
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1066
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1067
lemma insert_not_empty [simp]: "insert a A \<noteq> {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1068
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1069
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1070
lemmas empty_not_insert [simp] = insert_not_empty [symmetric, standard]
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1071
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1072
lemma insert_absorb: "a \<in> A ==> insert a A = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1073
  -- {* @{text "[simp]"} causes recursive calls when there are nested inserts *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1074
  -- {* with \emph{quadratic} running time *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1075
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1076
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1077
lemma insert_absorb2 [simp]: "insert x (insert x A) = insert x A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1078
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1079
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1080
lemma insert_commute: "insert x (insert y A) = insert y (insert x A)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1081
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1082
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1083
lemma insert_subset [simp]: "(insert x A \<subseteq> B) = (x \<in> B & A \<subseteq> B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1084
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1085
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1086
lemma mk_disjoint_insert: "a \<in> A ==> \<exists>B. A = insert a B & a \<notin> B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1087
  -- {* use new @{text B} rather than @{text "A - {a}"} to avoid infinite unfolding *}
14208
144f45277d5a misc tidying
paulson
parents: 14098
diff changeset
  1088
  apply (rule_tac x = "A - {a}" in exI, blast)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1089
  done
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1090
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1091
lemma insert_Collect: "insert a (Collect P) = {u. u \<noteq> a --> P u}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1092
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1093
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1094
lemma UN_insert_distrib: "u \<in> A ==> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1095
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1096
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1097
lemma insert_inter_insert[simp]: "insert a A \<inter> insert a B = insert a (A \<inter> B)"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1098
by blast
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1099
13103
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1100
lemma insert_disjoint[simp]:
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1101
 "(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})"
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1102
by blast
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1103
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1104
lemma disjoint_insert[simp]:
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1105
 "(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})"
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1106
by blast
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1107
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1108
text {* \medskip @{text image}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1109
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1110
lemma image_empty [simp]: "f`{} = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1111
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1112
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1113
lemma image_insert [simp]: "f ` insert a B = insert (f a) (f`B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1114
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1115
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1116
lemma image_constant: "x \<in> A ==> (\<lambda>x. c) ` A = {c}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1117
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1118
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1119
lemma image_image: "f ` (g ` A) = (\<lambda>x. f (g x)) ` A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1120
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1121
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1122
lemma insert_image [simp]: "x \<in> A ==> insert (f x) (f`A) = f`A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1123
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1124
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1125
lemma image_is_empty [iff]: "(f`A = {}) = (A = {})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1126
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1127
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1128
lemma image_Collect: "f ` {x. P x} = {f x | x. P x}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1129
  -- {* NOT suitable as a default simprule: the RHS isn't simpler than the LHS, *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1130
  -- {* with its implicit quantifier and conjunction.  Also image enjoys better *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1131
  -- {* equational properties than does the RHS. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1132
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1133
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1134
lemma if_image_distrib [simp]:
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1135
  "(\<lambda>x. if P x then f x else g x) ` S
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1136
    = (f ` (S \<inter> {x. P x})) \<union> (g ` (S \<inter> {x. \<not> P x}))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1137
  by (auto simp add: image_def)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1138
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1139
lemma image_cong: "M = N ==> (!!x. x \<in> N ==> f x = g x) ==> f`M = g`N"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1140
  by (simp add: image_def)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1141
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1142
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1143
text {* \medskip @{text range}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1144
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1145
lemma full_SetCompr_eq: "{u. \<exists>x. u = f x} = range f"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1146
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1147
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1148
lemma range_composition [simp]: "range (\<lambda>x. f (g x)) = f`range g"
14208
144f45277d5a misc tidying
paulson
parents: 14098
diff changeset
  1149
by (subst image_image, simp)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1150
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1151
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1152
text {* \medskip @{text Int} *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1153
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1154
lemma Int_absorb [simp]: "A \<inter> A = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1155
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1156
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1157
lemma Int_left_absorb: "A \<inter> (A \<inter> B) = A \<inter> B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1158
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1159
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1160
lemma Int_commute: "A \<inter> B = B \<inter> A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1161
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1162
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1163
lemma Int_left_commute: "A \<inter> (B \<inter> C) = B \<inter> (A \<inter> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1164
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1165
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1166
lemma Int_assoc: "(A \<inter> B) \<inter> C = A \<inter> (B \<inter> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1167
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1168
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1169
lemmas Int_ac = Int_assoc Int_left_absorb Int_commute Int_left_commute
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1170
  -- {* Intersection is an AC-operator *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1171
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1172
lemma Int_absorb1: "B \<subseteq> A ==> A \<inter> B = B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1173
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1174
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1175
lemma Int_absorb2: "A \<subseteq> B ==> A \<inter> B = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1176
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1177
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1178
lemma Int_empty_left [simp]: "{} \<inter> B = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1179
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1180
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1181
lemma Int_empty_right [simp]: "A \<inter> {} = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1182
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1183
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1184
lemma disjoint_eq_subset_Compl: "(A \<inter> B = {}) = (A \<subseteq> -B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1185
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1186
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1187
lemma disjoint_iff_not_equal: "(A \<inter> B = {}) = (\<forall>x\<in>A. \<forall>y\<in>B. x \<noteq> y)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1188
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1189
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1190
lemma Int_UNIV_left [simp]: "UNIV \<inter> B = B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1191
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1192
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1193
lemma Int_UNIV_right [simp]: "A \<inter> UNIV = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1194
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1195
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1196
lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1197
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1198
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1199
lemma Int_Un_distrib: "A \<inter> (B \<union> C) = (A \<inter> B) \<union> (A \<inter> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1200
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1201
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1202
lemma Int_Un_distrib2: "(B \<union> C) \<inter> A = (B \<inter> A) \<union> (C \<inter> A)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1203
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1204
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1205
lemma Int_UNIV [simp]: "(A \<inter> B = UNIV) = (A = UNIV & B = UNIV)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1206
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1207
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1208
lemma Int_subset_iff: "(C \<subseteq> A \<inter> B) = (C \<subseteq> A & C \<subseteq> B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1209
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1210
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1211
lemma Int_Collect: "(x \<in> A \<inter> {x. P x}) = (x \<in> A & P x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1212
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1213
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1214
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1215
text {* \medskip @{text Un}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1216
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1217
lemma Un_absorb [simp]: "A \<union> A = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1218
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1219
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1220
lemma Un_left_absorb: "A \<union> (A \<union> B) = A \<union> B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1221
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1222
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1223
lemma Un_commute: "A \<union> B = B \<union> A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1224
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1225
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1226
lemma Un_left_commute: "A \<union> (B \<union> C) = B \<union> (A \<union> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1227
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1228
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1229
lemma Un_assoc: "(A \<union> B) \<union> C = A \<union> (B \<union> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1230
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1231
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1232
lemmas Un_ac = Un_assoc Un_left_absorb Un_commute Un_left_commute
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1233
  -- {* Union is an AC-operator *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1234
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1235
lemma Un_absorb1: "A \<subseteq> B ==> A \<union> B = B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1236
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1237
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1238
lemma Un_absorb2: "B \<subseteq> A ==> A \<union> B = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1239
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1240
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1241
lemma Un_empty_left [simp]: "{} \<union> B = B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1242
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1243
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1244
lemma Un_empty_right [simp]: "A \<union> {} = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1245
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1246
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1247
lemma Un_UNIV_left [simp]: "UNIV \<union> B = UNIV"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1248
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1249
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1250
lemma Un_UNIV_right [simp]: "A \<union> UNIV = UNIV"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1251
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1252
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1253
lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1254
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1255
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1256
lemma Un_insert_left [simp]: "(insert a B) \<union> C = insert a (B \<union> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1257
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1258
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1259
lemma Un_insert_right [simp]: "A \<union> (insert a B) = insert a (A \<union> B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1260
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1261
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1262
lemma Int_insert_left:
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1263
    "(insert a B) Int C = (if a \<in> C then insert a (B \<inter> C) else B \<inter> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1264
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1265
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1266
lemma Int_insert_right:
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1267
    "A \<inter> (insert a B) = (if a \<in> A then insert a (A \<inter> B) else A \<inter> B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1268
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1269
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1270
lemma Un_Int_distrib: "A \<union> (B \<inter> C) = (A \<union> B) \<inter> (A \<union> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1271
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1272
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1273
lemma Un_Int_distrib2: "(B \<inter> C) \<union> A = (B \<union> A) \<inter> (C \<union> A)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1274
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1275
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1276
lemma Un_Int_crazy:
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1277
    "(A \<inter> B) \<union> (B \<inter> C) \<union> (C \<inter> A) = (A \<union> B) \<inter> (B \<union> C) \<inter> (C \<union> A)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1278
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1279
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1280
lemma subset_Un_eq: "(A \<subseteq> B) = (A \<union> B = B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1281
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1282
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1283
lemma Un_empty [iff]: "(A \<union> B = {}) = (A = {} & B = {})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1284
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1285
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1286
lemma Un_subset_iff: "(A \<union> B \<subseteq> C) = (A \<subseteq> C & B \<subseteq> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1287
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1288
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1289
lemma Un_Diff_Int: "(A - B) \<union> (A \<inter> B) = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1290
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1291
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1292
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1293
text {* \medskip Set complement *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1294
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1295
lemma Compl_disjoint [simp]: "A \<inter> -A = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1296
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1297
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1298
lemma Compl_disjoint2 [simp]: "-A \<inter> A = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1299
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1300
13818
274fda8cca4b new theorem Compl_partition2
paulson
parents: 13764
diff changeset
  1301
lemma Compl_partition: "A \<union> -A = UNIV"
274fda8cca4b new theorem Compl_partition2
paulson
parents: 13764
diff changeset
  1302
  by blast
274fda8cca4b new theorem Compl_partition2
paulson
parents: 13764
diff changeset
  1303
274fda8cca4b new theorem Compl_partition2
paulson
parents: 13764
diff changeset
  1304
lemma Compl_partition2: "-A \<union> A = UNIV"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1305
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1306
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1307
lemma double_complement [simp]: "- (-A) = (A::'a set)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1308
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1309
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1310
lemma Compl_Un [simp]: "-(A \<union> B) = (-A) \<inter> (-B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1311
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1312
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1313
lemma Compl_Int [simp]: "-(A \<inter> B) = (-A) \<union> (-B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1314
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1315
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1316
lemma Compl_UN [simp]: "-(\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1317
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1318
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1319
lemma Compl_INT [simp]: "-(\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1320
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1321
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1322
lemma subset_Compl_self_eq: "(A \<subseteq> -A) = (A = {})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1323
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1324
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1325
lemma Un_Int_assoc_eq: "((A \<inter> B) \<union> C = A \<inter> (B \<union> C)) = (C \<subseteq> A)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1326
  -- {* Halmos, Naive Set Theory, page 16. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1327
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1328
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1329
lemma Compl_UNIV_eq [simp]: "-UNIV = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1330
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1331
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1332
lemma Compl_empty_eq [simp]: "-{} = UNIV"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1333
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1334
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1335
lemma Compl_subset_Compl_iff [iff]: "(-A \<subseteq> -B) = (B \<subseteq> A)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1336
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1337
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1338
lemma Compl_eq_Compl_iff [iff]: "(-A = -B) = (A = (B::'a set))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1339
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1340
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1341
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1342
text {* \medskip @{text Union}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1343
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1344
lemma Union_empty [simp]: "Union({}) = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1345
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1346
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1347
lemma Union_UNIV [simp]: "Union UNIV = UNIV"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1348
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1349
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1350
lemma Union_insert [simp]: "Union (insert a B) = a \<union> \<Union>B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1351
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1352
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1353
lemma Union_Un_distrib [simp]: "\<Union>(A Un B) = \<Union>A \<union> \<Union>B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1354
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1355
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1356
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1357
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1358
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1359
lemma Union_empty_conv [iff]: "(\<Union>A = {}) = (\<forall>x\<in>A. x = {})"
13653
ef123b9e8089 Added a few thms about UN/INT/{}/UNIV
nipkow
parents: 13624
diff changeset
  1360
  by blast