src/HOL/Set.thy
author paulson
Thu Jan 01 21:47:07 2004 +0100 (2004-01-01)
changeset 14335 9c0b5e081037
parent 14302 6c24235e8d5d
child 14381 1189a8212a12
permissions -rw-r--r--
conversion of Real/PReal to Isar script;
type "complex" is now in class "field"
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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 ..
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instance set :: (type) 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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  "@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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  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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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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   300
  by blast
wenzelm@11979
   301
wenzelm@11979
   302
lemma bex_one_point2 [simp]: "(EX x:A. a = x & P x) = (a:A & P a)"
wenzelm@11979
   303
  by blast
wenzelm@11979
   304
wenzelm@11979
   305
lemma ball_one_point1 [simp]: "(ALL x:A. x = a --> P x) = (a:A --> P a)"
wenzelm@11979
   306
  by blast
wenzelm@11979
   307
wenzelm@11979
   308
lemma ball_one_point2 [simp]: "(ALL x:A. a = x --> P x) = (a:A --> P a)"
wenzelm@11979
   309
  by blast
wenzelm@11979
   310
wenzelm@11979
   311
ML_setup {*
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   312
  local
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   313
    val Ball_def = thm "Ball_def";
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   314
    val Bex_def = thm "Bex_def";
wenzelm@11979
   315
wenzelm@11979
   316
    val prove_bex_tac =
wenzelm@11979
   317
      rewrite_goals_tac [Bex_def] THEN Quantifier1.prove_one_point_ex_tac;
wenzelm@11979
   318
    val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac;
wenzelm@11979
   319
wenzelm@11979
   320
    val prove_ball_tac =
wenzelm@11979
   321
      rewrite_goals_tac [Ball_def] THEN Quantifier1.prove_one_point_all_tac;
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   322
    val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac;
wenzelm@11979
   323
  in
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   324
    val defBEX_regroup = Simplifier.simproc (Theory.sign_of (the_context ()))
wenzelm@13462
   325
      "defined BEX" ["EX x:A. P x & Q x"] rearrange_bex;
wenzelm@13462
   326
    val defBALL_regroup = Simplifier.simproc (Theory.sign_of (the_context ()))
wenzelm@13462
   327
      "defined BALL" ["ALL x:A. P x --> Q x"] rearrange_ball;
wenzelm@11979
   328
  end;
wenzelm@13462
   329
wenzelm@13462
   330
  Addsimprocs [defBALL_regroup, defBEX_regroup];
wenzelm@11979
   331
*}
wenzelm@11979
   332
wenzelm@11979
   333
wenzelm@11979
   334
subsubsection {* Congruence rules *}
wenzelm@11979
   335
wenzelm@11979
   336
lemma ball_cong [cong]:
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   337
  "A = B ==> (!!x. x:B ==> P x = Q x) ==>
wenzelm@11979
   338
    (ALL x:A. P x) = (ALL x:B. Q x)"
wenzelm@11979
   339
  by (simp add: Ball_def)
wenzelm@11979
   340
wenzelm@11979
   341
lemma bex_cong [cong]:
wenzelm@11979
   342
  "A = B ==> (!!x. x:B ==> P x = Q x) ==>
wenzelm@11979
   343
    (EX x:A. P x) = (EX x:B. Q x)"
wenzelm@11979
   344
  by (simp add: Bex_def cong: conj_cong)
regensbu@1273
   345
wenzelm@7238
   346
wenzelm@11979
   347
subsubsection {* Subsets *}
wenzelm@11979
   348
wenzelm@12897
   349
lemma subsetI [intro!]: "(!!x. x:A ==> x:B) ==> A \<subseteq> B"
wenzelm@11979
   350
  by (simp add: subset_def)
wenzelm@11979
   351
wenzelm@11979
   352
text {*
wenzelm@11979
   353
  \medskip Map the type @{text "'a set => anything"} to just @{typ
wenzelm@11979
   354
  'a}; for overloading constants whose first argument has type @{typ
wenzelm@11979
   355
  "'a set"}.
wenzelm@11979
   356
*}
wenzelm@11979
   357
wenzelm@12897
   358
lemma subsetD [elim]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B"
wenzelm@11979
   359
  -- {* Rule in Modus Ponens style. *}
wenzelm@11979
   360
  by (unfold subset_def) blast
wenzelm@11979
   361
wenzelm@11979
   362
declare subsetD [intro?] -- FIXME
wenzelm@11979
   363
wenzelm@12897
   364
lemma rev_subsetD: "c \<in> A ==> A \<subseteq> B ==> c \<in> B"
wenzelm@11979
   365
  -- {* The same, with reversed premises for use with @{text erule} --
wenzelm@11979
   366
      cf @{text rev_mp}. *}
wenzelm@11979
   367
  by (rule subsetD)
wenzelm@11979
   368
wenzelm@11979
   369
declare rev_subsetD [intro?] -- FIXME
wenzelm@11979
   370
wenzelm@11979
   371
text {*
wenzelm@12897
   372
  \medskip Converts @{prop "A \<subseteq> B"} to @{prop "x \<in> A ==> x \<in> B"}.
wenzelm@11979
   373
*}
wenzelm@11979
   374
wenzelm@11979
   375
ML {*
wenzelm@11979
   376
  local val rev_subsetD = thm "rev_subsetD"
wenzelm@11979
   377
  in fun impOfSubs th = th RSN (2, rev_subsetD) end;
wenzelm@11979
   378
*}
wenzelm@11979
   379
wenzelm@12897
   380
lemma subsetCE [elim]: "A \<subseteq>  B ==> (c \<notin> A ==> P) ==> (c \<in> B ==> P) ==> P"
wenzelm@11979
   381
  -- {* Classical elimination rule. *}
wenzelm@11979
   382
  by (unfold subset_def) blast
wenzelm@11979
   383
wenzelm@11979
   384
text {*
wenzelm@12897
   385
  \medskip Takes assumptions @{prop "A \<subseteq> B"}; @{prop "c \<in> A"} and
wenzelm@12897
   386
  creates the assumption @{prop "c \<in> B"}.
wenzelm@11979
   387
*}
wenzelm@11979
   388
wenzelm@11979
   389
ML {*
wenzelm@11979
   390
  local val subsetCE = thm "subsetCE"
wenzelm@11979
   391
  in fun set_mp_tac i = etac subsetCE i THEN mp_tac i end;
wenzelm@11979
   392
*}
wenzelm@11979
   393
wenzelm@12897
   394
lemma contra_subsetD: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A"
wenzelm@11979
   395
  by blast
wenzelm@11979
   396
wenzelm@12897
   397
lemma subset_refl: "A \<subseteq> A"
wenzelm@11979
   398
  by fast
wenzelm@11979
   399
wenzelm@12897
   400
lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C"
wenzelm@11979
   401
  by blast
clasohm@923
   402
wenzelm@2261
   403
wenzelm@11979
   404
subsubsection {* Equality *}
wenzelm@11979
   405
paulson@13865
   406
lemma set_ext: assumes prem: "(!!x. (x:A) = (x:B))" shows "A = B"
paulson@13865
   407
  apply (rule prem [THEN ext, THEN arg_cong, THEN box_equals])
paulson@13865
   408
   apply (rule Collect_mem_eq)
paulson@13865
   409
  apply (rule Collect_mem_eq)
paulson@13865
   410
  done
paulson@13865
   411
wenzelm@12897
   412
lemma subset_antisym [intro!]: "A \<subseteq> B ==> B \<subseteq> A ==> A = B"
wenzelm@11979
   413
  -- {* Anti-symmetry of the subset relation. *}
wenzelm@12897
   414
  by (rules intro: set_ext subsetD)
wenzelm@12897
   415
wenzelm@12897
   416
lemmas equalityI [intro!] = subset_antisym
wenzelm@11979
   417
wenzelm@11979
   418
text {*
wenzelm@11979
   419
  \medskip Equality rules from ZF set theory -- are they appropriate
wenzelm@11979
   420
  here?
wenzelm@11979
   421
*}
wenzelm@11979
   422
wenzelm@12897
   423
lemma equalityD1: "A = B ==> A \<subseteq> B"
wenzelm@11979
   424
  by (simp add: subset_refl)
wenzelm@11979
   425
wenzelm@12897
   426
lemma equalityD2: "A = B ==> B \<subseteq> A"
wenzelm@11979
   427
  by (simp add: subset_refl)
wenzelm@11979
   428
wenzelm@11979
   429
text {*
wenzelm@11979
   430
  \medskip Be careful when adding this to the claset as @{text
wenzelm@11979
   431
  subset_empty} is in the simpset: @{prop "A = {}"} goes to @{prop "{}
wenzelm@12897
   432
  \<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}!
wenzelm@11979
   433
*}
wenzelm@11979
   434
wenzelm@12897
   435
lemma equalityE: "A = B ==> (A \<subseteq> B ==> B \<subseteq> A ==> P) ==> P"
wenzelm@11979
   436
  by (simp add: subset_refl)
clasohm@923
   437
wenzelm@11979
   438
lemma equalityCE [elim]:
wenzelm@12897
   439
    "A = B ==> (c \<in> A ==> c \<in> B ==> P) ==> (c \<notin> A ==> c \<notin> B ==> P) ==> P"
wenzelm@11979
   440
  by blast
wenzelm@11979
   441
wenzelm@11979
   442
text {*
wenzelm@11979
   443
  \medskip Lemma for creating induction formulae -- for "pattern
wenzelm@11979
   444
  matching" on @{text p}.  To make the induction hypotheses usable,
wenzelm@11979
   445
  apply @{text spec} or @{text bspec} to put universal quantifiers over the free
wenzelm@11979
   446
  variables in @{text p}.
wenzelm@11979
   447
*}
wenzelm@11979
   448
wenzelm@11979
   449
lemma setup_induction: "p:A ==> (!!z. z:A ==> p = z --> R) ==> R"
wenzelm@11979
   450
  by simp
clasohm@923
   451
wenzelm@11979
   452
lemma eqset_imp_iff: "A = B ==> (x : A) = (x : B)"
wenzelm@11979
   453
  by simp
wenzelm@11979
   454
paulson@13865
   455
lemma eqelem_imp_iff: "x = y ==> (x : A) = (y : A)"
paulson@13865
   456
  by simp
paulson@13865
   457
wenzelm@11979
   458
wenzelm@11979
   459
subsubsection {* The universal set -- UNIV *}
wenzelm@11979
   460
wenzelm@11979
   461
lemma UNIV_I [simp]: "x : UNIV"
wenzelm@11979
   462
  by (simp add: UNIV_def)
wenzelm@11979
   463
wenzelm@11979
   464
declare UNIV_I [intro]  -- {* unsafe makes it less likely to cause problems *}
wenzelm@11979
   465
wenzelm@11979
   466
lemma UNIV_witness [intro?]: "EX x. x : UNIV"
wenzelm@11979
   467
  by simp
wenzelm@11979
   468
wenzelm@12897
   469
lemma subset_UNIV: "A \<subseteq> UNIV"
wenzelm@11979
   470
  by (rule subsetI) (rule UNIV_I)
wenzelm@2388
   471
wenzelm@11979
   472
text {*
wenzelm@11979
   473
  \medskip Eta-contracting these two rules (to remove @{text P})
wenzelm@11979
   474
  causes them to be ignored because of their interaction with
wenzelm@11979
   475
  congruence rules.
wenzelm@11979
   476
*}
wenzelm@11979
   477
wenzelm@11979
   478
lemma ball_UNIV [simp]: "Ball UNIV P = All P"
wenzelm@11979
   479
  by (simp add: Ball_def)
wenzelm@11979
   480
wenzelm@11979
   481
lemma bex_UNIV [simp]: "Bex UNIV P = Ex P"
wenzelm@11979
   482
  by (simp add: Bex_def)
wenzelm@11979
   483
wenzelm@11979
   484
wenzelm@11979
   485
subsubsection {* The empty set *}
wenzelm@11979
   486
wenzelm@11979
   487
lemma empty_iff [simp]: "(c : {}) = False"
wenzelm@11979
   488
  by (simp add: empty_def)
wenzelm@11979
   489
wenzelm@11979
   490
lemma emptyE [elim!]: "a : {} ==> P"
wenzelm@11979
   491
  by simp
wenzelm@11979
   492
wenzelm@12897
   493
lemma empty_subsetI [iff]: "{} \<subseteq> A"
wenzelm@11979
   494
    -- {* One effect is to delete the ASSUMPTION @{prop "{} <= A"} *}
wenzelm@11979
   495
  by blast
wenzelm@11979
   496
wenzelm@12897
   497
lemma equals0I: "(!!y. y \<in> A ==> False) ==> A = {}"
wenzelm@11979
   498
  by blast
wenzelm@2388
   499
wenzelm@12897
   500
lemma equals0D: "A = {} ==> a \<notin> A"
wenzelm@11979
   501
    -- {* Use for reasoning about disjointness: @{prop "A Int B = {}"} *}
wenzelm@11979
   502
  by blast
wenzelm@11979
   503
wenzelm@11979
   504
lemma ball_empty [simp]: "Ball {} P = True"
wenzelm@11979
   505
  by (simp add: Ball_def)
wenzelm@11979
   506
wenzelm@11979
   507
lemma bex_empty [simp]: "Bex {} P = False"
wenzelm@11979
   508
  by (simp add: Bex_def)
wenzelm@11979
   509
wenzelm@11979
   510
lemma UNIV_not_empty [iff]: "UNIV ~= {}"
wenzelm@11979
   511
  by (blast elim: equalityE)
wenzelm@11979
   512
wenzelm@11979
   513
wenzelm@12023
   514
subsubsection {* The Powerset operator -- Pow *}
wenzelm@11979
   515
wenzelm@12897
   516
lemma Pow_iff [iff]: "(A \<in> Pow B) = (A \<subseteq> B)"
wenzelm@11979
   517
  by (simp add: Pow_def)
wenzelm@11979
   518
wenzelm@12897
   519
lemma PowI: "A \<subseteq> B ==> A \<in> Pow B"
wenzelm@11979
   520
  by (simp add: Pow_def)
wenzelm@11979
   521
wenzelm@12897
   522
lemma PowD: "A \<in> Pow B ==> A \<subseteq> B"
wenzelm@11979
   523
  by (simp add: Pow_def)
wenzelm@11979
   524
wenzelm@12897
   525
lemma Pow_bottom: "{} \<in> Pow B"
wenzelm@11979
   526
  by simp
wenzelm@11979
   527
wenzelm@12897
   528
lemma Pow_top: "A \<in> Pow A"
wenzelm@11979
   529
  by (simp add: subset_refl)
wenzelm@2684
   530
wenzelm@2388
   531
wenzelm@11979
   532
subsubsection {* Set complement *}
wenzelm@11979
   533
wenzelm@12897
   534
lemma Compl_iff [simp]: "(c \<in> -A) = (c \<notin> A)"
wenzelm@11979
   535
  by (unfold Compl_def) blast
wenzelm@11979
   536
wenzelm@12897
   537
lemma ComplI [intro!]: "(c \<in> A ==> False) ==> c \<in> -A"
wenzelm@11979
   538
  by (unfold Compl_def) blast
wenzelm@11979
   539
wenzelm@11979
   540
text {*
wenzelm@11979
   541
  \medskip This form, with negated conclusion, works well with the
wenzelm@11979
   542
  Classical prover.  Negated assumptions behave like formulae on the
wenzelm@11979
   543
  right side of the notional turnstile ... *}
wenzelm@11979
   544
wenzelm@11979
   545
lemma ComplD: "c : -A ==> c~:A"
wenzelm@11979
   546
  by (unfold Compl_def) blast
wenzelm@11979
   547
wenzelm@11979
   548
lemmas ComplE [elim!] = ComplD [elim_format]
wenzelm@11979
   549
wenzelm@11979
   550
wenzelm@11979
   551
subsubsection {* Binary union -- Un *}
clasohm@923
   552
wenzelm@11979
   553
lemma Un_iff [simp]: "(c : A Un B) = (c:A | c:B)"
wenzelm@11979
   554
  by (unfold Un_def) blast
wenzelm@11979
   555
wenzelm@11979
   556
lemma UnI1 [elim?]: "c:A ==> c : A Un B"
wenzelm@11979
   557
  by simp
wenzelm@11979
   558
wenzelm@11979
   559
lemma UnI2 [elim?]: "c:B ==> c : A Un B"
wenzelm@11979
   560
  by simp
clasohm@923
   561
wenzelm@11979
   562
text {*
wenzelm@11979
   563
  \medskip Classical introduction rule: no commitment to @{prop A} vs
wenzelm@11979
   564
  @{prop B}.
wenzelm@11979
   565
*}
wenzelm@11979
   566
wenzelm@11979
   567
lemma UnCI [intro!]: "(c~:B ==> c:A) ==> c : A Un B"
wenzelm@11979
   568
  by auto
wenzelm@11979
   569
wenzelm@11979
   570
lemma UnE [elim!]: "c : A Un B ==> (c:A ==> P) ==> (c:B ==> P) ==> P"
wenzelm@11979
   571
  by (unfold Un_def) blast
wenzelm@11979
   572
wenzelm@11979
   573
wenzelm@12023
   574
subsubsection {* Binary intersection -- Int *}
clasohm@923
   575
wenzelm@11979
   576
lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)"
wenzelm@11979
   577
  by (unfold Int_def) blast
wenzelm@11979
   578
wenzelm@11979
   579
lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B"
wenzelm@11979
   580
  by simp
wenzelm@11979
   581
wenzelm@11979
   582
lemma IntD1: "c : A Int B ==> c:A"
wenzelm@11979
   583
  by simp
wenzelm@11979
   584
wenzelm@11979
   585
lemma IntD2: "c : A Int B ==> c:B"
wenzelm@11979
   586
  by simp
wenzelm@11979
   587
wenzelm@11979
   588
lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P"
wenzelm@11979
   589
  by simp
wenzelm@11979
   590
wenzelm@11979
   591
wenzelm@12023
   592
subsubsection {* Set difference *}
wenzelm@11979
   593
wenzelm@11979
   594
lemma Diff_iff [simp]: "(c : A - B) = (c:A & c~:B)"
wenzelm@11979
   595
  by (unfold set_diff_def) blast
clasohm@923
   596
wenzelm@11979
   597
lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A - B"
wenzelm@11979
   598
  by simp
wenzelm@11979
   599
wenzelm@11979
   600
lemma DiffD1: "c : A - B ==> c : A"
wenzelm@11979
   601
  by simp
wenzelm@11979
   602
wenzelm@11979
   603
lemma DiffD2: "c : A - B ==> c : B ==> P"
wenzelm@11979
   604
  by simp
wenzelm@11979
   605
wenzelm@11979
   606
lemma DiffE [elim!]: "c : A - B ==> (c:A ==> c~:B ==> P) ==> P"
wenzelm@11979
   607
  by simp
wenzelm@11979
   608
wenzelm@11979
   609
wenzelm@11979
   610
subsubsection {* Augmenting a set -- insert *}
wenzelm@11979
   611
wenzelm@11979
   612
lemma insert_iff [simp]: "(a : insert b A) = (a = b | a:A)"
wenzelm@11979
   613
  by (unfold insert_def) blast
wenzelm@11979
   614
wenzelm@11979
   615
lemma insertI1: "a : insert a B"
wenzelm@11979
   616
  by simp
wenzelm@11979
   617
wenzelm@11979
   618
lemma insertI2: "a : B ==> a : insert b B"
wenzelm@11979
   619
  by simp
clasohm@923
   620
wenzelm@11979
   621
lemma insertE [elim!]: "a : insert b A ==> (a = b ==> P) ==> (a:A ==> P) ==> P"
wenzelm@11979
   622
  by (unfold insert_def) blast
wenzelm@11979
   623
wenzelm@11979
   624
lemma insertCI [intro!]: "(a~:B ==> a = b) ==> a: insert b B"
wenzelm@11979
   625
  -- {* Classical introduction rule. *}
wenzelm@11979
   626
  by auto
wenzelm@11979
   627
wenzelm@12897
   628
lemma subset_insert_iff: "(A \<subseteq> insert x B) = (if x:A then A - {x} \<subseteq> B else A \<subseteq> B)"
wenzelm@11979
   629
  by auto
wenzelm@11979
   630
wenzelm@11979
   631
wenzelm@11979
   632
subsubsection {* Singletons, using insert *}
wenzelm@11979
   633
wenzelm@11979
   634
lemma singletonI [intro!]: "a : {a}"
wenzelm@11979
   635
    -- {* Redundant? But unlike @{text insertCI}, it proves the subgoal immediately! *}
wenzelm@11979
   636
  by (rule insertI1)
wenzelm@11979
   637
wenzelm@11979
   638
lemma singletonD: "b : {a} ==> b = a"
wenzelm@11979
   639
  by blast
wenzelm@11979
   640
wenzelm@11979
   641
lemmas singletonE [elim!] = singletonD [elim_format]
wenzelm@11979
   642
wenzelm@11979
   643
lemma singleton_iff: "(b : {a}) = (b = a)"
wenzelm@11979
   644
  by blast
wenzelm@11979
   645
wenzelm@11979
   646
lemma singleton_inject [dest!]: "{a} = {b} ==> a = b"
wenzelm@11979
   647
  by blast
wenzelm@11979
   648
wenzelm@12897
   649
lemma singleton_insert_inj_eq [iff]: "({b} = insert a A) = (a = b & A \<subseteq> {b})"
wenzelm@11979
   650
  by blast
wenzelm@11979
   651
wenzelm@12897
   652
lemma singleton_insert_inj_eq' [iff]: "(insert a A = {b}) = (a = b & A \<subseteq> {b})"
wenzelm@11979
   653
  by blast
wenzelm@11979
   654
wenzelm@12897
   655
lemma subset_singletonD: "A \<subseteq> {x} ==> A = {} | A = {x}"
wenzelm@11979
   656
  by fast
wenzelm@11979
   657
wenzelm@11979
   658
lemma singleton_conv [simp]: "{x. x = a} = {a}"
wenzelm@11979
   659
  by blast
wenzelm@11979
   660
wenzelm@11979
   661
lemma singleton_conv2 [simp]: "{x. a = x} = {a}"
wenzelm@11979
   662
  by blast
clasohm@923
   663
wenzelm@12897
   664
lemma diff_single_insert: "A - {x} \<subseteq> B ==> x \<in> A ==> A \<subseteq> insert x B"
wenzelm@11979
   665
  by blast
wenzelm@11979
   666
wenzelm@11979
   667
wenzelm@11979
   668
subsubsection {* Unions of families *}
wenzelm@11979
   669
wenzelm@11979
   670
text {*
wenzelm@11979
   671
  @{term [source] "UN x:A. B x"} is @{term "Union (B`A)"}.
wenzelm@11979
   672
*}
wenzelm@11979
   673
wenzelm@11979
   674
lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)"
wenzelm@11979
   675
  by (unfold UNION_def) blast
wenzelm@11979
   676
wenzelm@11979
   677
lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)"
wenzelm@11979
   678
  -- {* The order of the premises presupposes that @{term A} is rigid;
wenzelm@11979
   679
    @{term b} may be flexible. *}
wenzelm@11979
   680
  by auto
wenzelm@11979
   681
wenzelm@11979
   682
lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R"
wenzelm@11979
   683
  by (unfold UNION_def) blast
clasohm@923
   684
wenzelm@11979
   685
lemma UN_cong [cong]:
wenzelm@11979
   686
    "A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
wenzelm@11979
   687
  by (simp add: UNION_def)
wenzelm@11979
   688
wenzelm@11979
   689
wenzelm@11979
   690
subsubsection {* Intersections of families *}
wenzelm@11979
   691
wenzelm@11979
   692
text {* @{term [source] "INT x:A. B x"} is @{term "Inter (B`A)"}. *}
wenzelm@11979
   693
wenzelm@11979
   694
lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)"
wenzelm@11979
   695
  by (unfold INTER_def) blast
clasohm@923
   696
wenzelm@11979
   697
lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)"
wenzelm@11979
   698
  by (unfold INTER_def) blast
wenzelm@11979
   699
wenzelm@11979
   700
lemma INT_D [elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a"
wenzelm@11979
   701
  by auto
wenzelm@11979
   702
wenzelm@11979
   703
lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R"
wenzelm@11979
   704
  -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}
wenzelm@11979
   705
  by (unfold INTER_def) blast
wenzelm@11979
   706
wenzelm@11979
   707
lemma INT_cong [cong]:
wenzelm@11979
   708
    "A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)"
wenzelm@11979
   709
  by (simp add: INTER_def)
wenzelm@7238
   710
clasohm@923
   711
wenzelm@11979
   712
subsubsection {* Union *}
wenzelm@11979
   713
wenzelm@11979
   714
lemma Union_iff [simp]: "(A : Union C) = (EX X:C. A:X)"
wenzelm@11979
   715
  by (unfold Union_def) blast
wenzelm@11979
   716
wenzelm@11979
   717
lemma UnionI [intro]: "X:C ==> A:X ==> A : Union C"
wenzelm@11979
   718
  -- {* The order of the premises presupposes that @{term C} is rigid;
wenzelm@11979
   719
    @{term A} may be flexible. *}
wenzelm@11979
   720
  by auto
wenzelm@11979
   721
wenzelm@11979
   722
lemma UnionE [elim!]: "A : Union C ==> (!!X. A:X ==> X:C ==> R) ==> R"
wenzelm@11979
   723
  by (unfold Union_def) blast
wenzelm@11979
   724
wenzelm@11979
   725
wenzelm@11979
   726
subsubsection {* Inter *}
wenzelm@11979
   727
wenzelm@11979
   728
lemma Inter_iff [simp]: "(A : Inter C) = (ALL X:C. A:X)"
wenzelm@11979
   729
  by (unfold Inter_def) blast
wenzelm@11979
   730
wenzelm@11979
   731
lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C"
wenzelm@11979
   732
  by (simp add: Inter_def)
wenzelm@11979
   733
wenzelm@11979
   734
text {*
wenzelm@11979
   735
  \medskip A ``destruct'' rule -- every @{term X} in @{term C}
wenzelm@11979
   736
  contains @{term A} as an element, but @{prop "A:X"} can hold when
wenzelm@11979
   737
  @{prop "X:C"} does not!  This rule is analogous to @{text spec}.
wenzelm@11979
   738
*}
wenzelm@11979
   739
wenzelm@11979
   740
lemma InterD [elim]: "A : Inter C ==> X:C ==> A:X"
wenzelm@11979
   741
  by auto
wenzelm@11979
   742
wenzelm@11979
   743
lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R"
wenzelm@11979
   744
  -- {* ``Classical'' elimination rule -- does not require proving
wenzelm@11979
   745
    @{prop "X:C"}. *}
wenzelm@11979
   746
  by (unfold Inter_def) blast
wenzelm@11979
   747
wenzelm@11979
   748
text {*
wenzelm@11979
   749
  \medskip Image of a set under a function.  Frequently @{term b} does
wenzelm@11979
   750
  not have the syntactic form of @{term "f x"}.
wenzelm@11979
   751
*}
wenzelm@11979
   752
wenzelm@11979
   753
lemma image_eqI [simp, intro]: "b = f x ==> x:A ==> b : f`A"
wenzelm@11979
   754
  by (unfold image_def) blast
wenzelm@11979
   755
wenzelm@11979
   756
lemma imageI: "x : A ==> f x : f ` A"
wenzelm@11979
   757
  by (rule image_eqI) (rule refl)
wenzelm@11979
   758
wenzelm@11979
   759
lemma rev_image_eqI: "x:A ==> b = f x ==> b : f`A"
wenzelm@11979
   760
  -- {* This version's more effective when we already have the
wenzelm@11979
   761
    required @{term x}. *}
wenzelm@11979
   762
  by (unfold image_def) blast
wenzelm@11979
   763
wenzelm@11979
   764
lemma imageE [elim!]:
wenzelm@11979
   765
  "b : (%x. f x)`A ==> (!!x. b = f x ==> x:A ==> P) ==> P"
wenzelm@11979
   766
  -- {* The eta-expansion gives variable-name preservation. *}
wenzelm@11979
   767
  by (unfold image_def) blast
wenzelm@11979
   768
wenzelm@11979
   769
lemma image_Un: "f`(A Un B) = f`A Un f`B"
wenzelm@11979
   770
  by blast
wenzelm@11979
   771
wenzelm@11979
   772
lemma image_iff: "(z : f`A) = (EX x:A. z = f x)"
wenzelm@11979
   773
  by blast
wenzelm@11979
   774
wenzelm@12897
   775
lemma image_subset_iff: "(f`A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)"
wenzelm@11979
   776
  -- {* This rewrite rule would confuse users if made default. *}
wenzelm@11979
   777
  by blast
wenzelm@11979
   778
wenzelm@12897
   779
lemma subset_image_iff: "(B \<subseteq> f`A) = (EX AA. AA \<subseteq> A & B = f`AA)"
wenzelm@11979
   780
  apply safe
wenzelm@11979
   781
   prefer 2 apply fast
paulson@14208
   782
  apply (rule_tac x = "{a. a : A & f a : B}" in exI, fast)
wenzelm@11979
   783
  done
wenzelm@11979
   784
wenzelm@12897
   785
lemma image_subsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f`A \<subseteq> B"
wenzelm@11979
   786
  -- {* Replaces the three steps @{text subsetI}, @{text imageE},
wenzelm@11979
   787
    @{text hypsubst}, but breaks too many existing proofs. *}
wenzelm@11979
   788
  by blast
wenzelm@11979
   789
wenzelm@11979
   790
text {*
wenzelm@11979
   791
  \medskip Range of a function -- just a translation for image!
wenzelm@11979
   792
*}
wenzelm@11979
   793
wenzelm@12897
   794
lemma range_eqI: "b = f x ==> b \<in> range f"
wenzelm@11979
   795
  by simp
wenzelm@11979
   796
wenzelm@12897
   797
lemma rangeI: "f x \<in> range f"
wenzelm@11979
   798
  by simp
wenzelm@11979
   799
wenzelm@12897
   800
lemma rangeE [elim?]: "b \<in> range (\<lambda>x. f x) ==> (!!x. b = f x ==> P) ==> P"
wenzelm@11979
   801
  by blast
wenzelm@11979
   802
wenzelm@11979
   803
wenzelm@11979
   804
subsubsection {* Set reasoning tools *}
wenzelm@11979
   805
wenzelm@11979
   806
text {*
wenzelm@11979
   807
  Rewrite rules for boolean case-splitting: faster than @{text
wenzelm@11979
   808
  "split_if [split]"}.
wenzelm@11979
   809
*}
wenzelm@11979
   810
wenzelm@11979
   811
lemma split_if_eq1: "((if Q then x else y) = b) = ((Q --> x = b) & (~ Q --> y = b))"
wenzelm@11979
   812
  by (rule split_if)
wenzelm@11979
   813
wenzelm@11979
   814
lemma split_if_eq2: "(a = (if Q then x else y)) = ((Q --> a = x) & (~ Q --> a = y))"
wenzelm@11979
   815
  by (rule split_if)
wenzelm@11979
   816
wenzelm@11979
   817
text {*
wenzelm@11979
   818
  Split ifs on either side of the membership relation.  Not for @{text
wenzelm@11979
   819
  "[simp]"} -- can cause goals to blow up!
wenzelm@11979
   820
*}
wenzelm@11979
   821
wenzelm@11979
   822
lemma split_if_mem1: "((if Q then x else y) : b) = ((Q --> x : b) & (~ Q --> y : b))"
wenzelm@11979
   823
  by (rule split_if)
wenzelm@11979
   824
wenzelm@11979
   825
lemma split_if_mem2: "(a : (if Q then x else y)) = ((Q --> a : x) & (~ Q --> a : y))"
wenzelm@11979
   826
  by (rule split_if)
wenzelm@11979
   827
wenzelm@11979
   828
lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2
wenzelm@11979
   829
wenzelm@11979
   830
lemmas mem_simps =
wenzelm@11979
   831
  insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
wenzelm@11979
   832
  mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
wenzelm@11979
   833
  -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
wenzelm@11979
   834
wenzelm@11979
   835
(*Would like to add these, but the existing code only searches for the
wenzelm@11979
   836
  outer-level constant, which in this case is just "op :"; we instead need
wenzelm@11979
   837
  to use term-nets to associate patterns with rules.  Also, if a rule fails to
wenzelm@11979
   838
  apply, then the formula should be kept.
wenzelm@11979
   839
  [("uminus", Compl_iff RS iffD1), ("op -", [Diff_iff RS iffD1]),
wenzelm@11979
   840
   ("op Int", [IntD1,IntD2]),
wenzelm@11979
   841
   ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]
wenzelm@11979
   842
 *)
wenzelm@11979
   843
wenzelm@11979
   844
ML_setup {*
wenzelm@11979
   845
  val mksimps_pairs = [("Ball", [thm "bspec"])] @ mksimps_pairs;
wenzelm@11979
   846
  simpset_ref() := simpset() setmksimps (mksimps mksimps_pairs);
wenzelm@11979
   847
*}
wenzelm@11979
   848
wenzelm@11979
   849
declare subset_UNIV [simp] subset_refl [simp]
wenzelm@11979
   850
wenzelm@11979
   851
wenzelm@11979
   852
subsubsection {* The ``proper subset'' relation *}
wenzelm@11979
   853
wenzelm@12897
   854
lemma psubsetI [intro!]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B"
wenzelm@11979
   855
  by (unfold psubset_def) blast
wenzelm@11979
   856
paulson@13624
   857
lemma psubsetE [elim!]: 
paulson@13624
   858
    "[|A \<subset> B;  [|A \<subseteq> B; ~ (B\<subseteq>A)|] ==> R|] ==> R"
paulson@13624
   859
  by (unfold psubset_def) blast
paulson@13624
   860
wenzelm@11979
   861
lemma psubset_insert_iff:
wenzelm@12897
   862
  "(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)"
wenzelm@12897
   863
  by (auto simp add: psubset_def subset_insert_iff)
wenzelm@12897
   864
wenzelm@12897
   865
lemma psubset_eq: "(A \<subset> B) = (A \<subseteq> B & A \<noteq> B)"
wenzelm@11979
   866
  by (simp only: psubset_def)
wenzelm@11979
   867
wenzelm@12897
   868
lemma psubset_imp_subset: "A \<subset> B ==> A \<subseteq> B"
wenzelm@11979
   869
  by (simp add: psubset_eq)
wenzelm@11979
   870
paulson@14335
   871
lemma psubset_trans: "[| A \<subset> B; B \<subset> C |] ==> A \<subset> C"
paulson@14335
   872
apply (unfold psubset_def)
paulson@14335
   873
apply (auto dest: subset_antisym)
paulson@14335
   874
done
paulson@14335
   875
paulson@14335
   876
lemma psubsetD: "[| A \<subset> B; c \<in> A |] ==> c \<in> B"
paulson@14335
   877
apply (unfold psubset_def)
paulson@14335
   878
apply (auto dest: subsetD)
paulson@14335
   879
done
paulson@14335
   880
wenzelm@12897
   881
lemma psubset_subset_trans: "A \<subset> B ==> B \<subseteq> C ==> A \<subset> C"
wenzelm@11979
   882
  by (auto simp add: psubset_eq)
wenzelm@11979
   883
wenzelm@12897
   884
lemma subset_psubset_trans: "A \<subseteq> B ==> B \<subset> C ==> A \<subset> C"
wenzelm@11979
   885
  by (auto simp add: psubset_eq)
wenzelm@11979
   886
wenzelm@12897
   887
lemma psubset_imp_ex_mem: "A \<subset> B ==> \<exists>b. b \<in> (B - A)"
wenzelm@11979
   888
  by (unfold psubset_def) blast
wenzelm@11979
   889
wenzelm@11979
   890
lemma atomize_ball:
wenzelm@12897
   891
    "(!!x. x \<in> A ==> P x) == Trueprop (\<forall>x\<in>A. P x)"
wenzelm@11979
   892
  by (simp only: Ball_def atomize_all atomize_imp)
wenzelm@11979
   893
wenzelm@11979
   894
declare atomize_ball [symmetric, rulify]
wenzelm@11979
   895
wenzelm@11979
   896
wenzelm@11979
   897
subsection {* Further set-theory lemmas *}
wenzelm@11979
   898
wenzelm@12897
   899
subsubsection {* Derived rules involving subsets. *}
wenzelm@12897
   900
wenzelm@12897
   901
text {* @{text insert}. *}
wenzelm@12897
   902
wenzelm@12897
   903
lemma subset_insertI: "B \<subseteq> insert a B"
wenzelm@12897
   904
  apply (rule subsetI)
wenzelm@12897
   905
  apply (erule insertI2)
wenzelm@12897
   906
  done
wenzelm@12897
   907
nipkow@14302
   908
lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B"
nipkow@14302
   909
by blast
nipkow@14302
   910
wenzelm@12897
   911
lemma subset_insert: "x \<notin> A ==> (A \<subseteq> insert x B) = (A \<subseteq> B)"
wenzelm@12897
   912
  by blast
wenzelm@12897
   913
wenzelm@12897
   914
wenzelm@12897
   915
text {* \medskip Big Union -- least upper bound of a set. *}
wenzelm@12897
   916
wenzelm@12897
   917
lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A"
wenzelm@12897
   918
  by (rules intro: subsetI UnionI)
wenzelm@12897
   919
wenzelm@12897
   920
lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C"
wenzelm@12897
   921
  by (rules intro: subsetI elim: UnionE dest: subsetD)
wenzelm@12897
   922
wenzelm@12897
   923
wenzelm@12897
   924
text {* \medskip General union. *}
wenzelm@12897
   925
wenzelm@12897
   926
lemma UN_upper: "a \<in> A ==> B a \<subseteq> (\<Union>x\<in>A. B x)"
wenzelm@12897
   927
  by blast
wenzelm@12897
   928
wenzelm@12897
   929
lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C"
wenzelm@12897
   930
  by (rules intro: subsetI elim: UN_E dest: subsetD)
wenzelm@12897
   931
wenzelm@12897
   932
wenzelm@12897
   933
text {* \medskip Big Intersection -- greatest lower bound of a set. *}
wenzelm@12897
   934
wenzelm@12897
   935
lemma Inter_lower: "B \<in> A ==> Inter A \<subseteq> B"
wenzelm@12897
   936
  by blast
wenzelm@12897
   937
wenzelm@12897
   938
lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A"
wenzelm@12897
   939
  by (rules intro: InterI subsetI dest: subsetD)
wenzelm@12897
   940
wenzelm@12897
   941
lemma INT_lower: "a \<in> A ==> (\<Inter>x\<in>A. B x) \<subseteq> B a"
wenzelm@12897
   942
  by blast
wenzelm@12897
   943
wenzelm@12897
   944
lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)"
wenzelm@12897
   945
  by (rules intro: INT_I subsetI dest: subsetD)
wenzelm@12897
   946
wenzelm@12897
   947
wenzelm@12897
   948
text {* \medskip Finite Union -- the least upper bound of two sets. *}
wenzelm@12897
   949
wenzelm@12897
   950
lemma Un_upper1: "A \<subseteq> A \<union> B"
wenzelm@12897
   951
  by blast
wenzelm@12897
   952
wenzelm@12897
   953
lemma Un_upper2: "B \<subseteq> A \<union> B"
wenzelm@12897
   954
  by blast
wenzelm@12897
   955
wenzelm@12897
   956
lemma Un_least: "A \<subseteq> C ==> B \<subseteq> C ==> A \<union> B \<subseteq> C"
wenzelm@12897
   957
  by blast
wenzelm@12897
   958
wenzelm@12897
   959
wenzelm@12897
   960
text {* \medskip Finite Intersection -- the greatest lower bound of two sets. *}
wenzelm@12897
   961
wenzelm@12897
   962
lemma Int_lower1: "A \<inter> B \<subseteq> A"
wenzelm@12897
   963
  by blast
wenzelm@12897
   964
wenzelm@12897
   965
lemma Int_lower2: "A \<inter> B \<subseteq> B"
wenzelm@12897
   966
  by blast
wenzelm@12897
   967
wenzelm@12897
   968
lemma Int_greatest: "C \<subseteq> A ==> C \<subseteq> B ==> C \<subseteq> A \<inter> B"
wenzelm@12897
   969
  by blast
wenzelm@12897
   970
wenzelm@12897
   971
wenzelm@12897
   972
text {* \medskip Set difference. *}
wenzelm@12897
   973
wenzelm@12897
   974
lemma Diff_subset: "A - B \<subseteq> A"
wenzelm@12897
   975
  by blast
wenzelm@12897
   976
nipkow@14302
   977
lemma Diff_subset_conv: "(A - B \<subseteq> C) = (A \<subseteq> B \<union> C)"
nipkow@14302
   978
by blast
nipkow@14302
   979
wenzelm@12897
   980
wenzelm@12897
   981
text {* \medskip Monotonicity. *}
wenzelm@12897
   982
wenzelm@13421
   983
lemma mono_Un: includes mono shows "f A \<union> f B \<subseteq> f (A \<union> B)"
wenzelm@12897
   984
  apply (rule Un_least)
wenzelm@13421
   985
   apply (rule Un_upper1 [THEN mono])
wenzelm@13421
   986
  apply (rule Un_upper2 [THEN mono])
wenzelm@12897
   987
  done
wenzelm@12897
   988
wenzelm@13421
   989
lemma mono_Int: includes mono shows "f (A \<inter> B) \<subseteq> f A \<inter> f B"
wenzelm@12897
   990
  apply (rule Int_greatest)
wenzelm@13421
   991
   apply (rule Int_lower1 [THEN mono])
wenzelm@13421
   992
  apply (rule Int_lower2 [THEN mono])
wenzelm@12897
   993
  done
wenzelm@12897
   994
wenzelm@12897
   995
wenzelm@12897
   996
subsubsection {* Equalities involving union, intersection, inclusion, etc. *}
wenzelm@12897
   997
wenzelm@12897
   998
text {* @{text "{}"}. *}
wenzelm@12897
   999
wenzelm@12897
  1000
lemma Collect_const [simp]: "{s. P} = (if P then UNIV else {})"
wenzelm@12897
  1001
  -- {* supersedes @{text "Collect_False_empty"} *}
wenzelm@12897
  1002
  by auto
wenzelm@12897
  1003
wenzelm@12897
  1004
lemma subset_empty [simp]: "(A \<subseteq> {}) = (A = {})"
wenzelm@12897
  1005
  by blast
wenzelm@12897
  1006
wenzelm@12897
  1007
lemma not_psubset_empty [iff]: "\<not> (A < {})"
wenzelm@12897
  1008
  by (unfold psubset_def) blast
wenzelm@12897
  1009
wenzelm@12897
  1010
lemma Collect_empty_eq [simp]: "(Collect P = {}) = (\<forall>x. \<not> P x)"
wenzelm@12897
  1011
  by auto
wenzelm@12897
  1012
wenzelm@12897
  1013
lemma Collect_neg_eq: "{x. \<not> P x} = - {x. P x}"
wenzelm@12897
  1014
  by blast
wenzelm@12897
  1015
wenzelm@12897
  1016
lemma Collect_disj_eq: "{x. P x | Q x} = {x. P x} \<union> {x. Q x}"
wenzelm@12897
  1017
  by blast
wenzelm@12897
  1018
wenzelm@12897
  1019
lemma Collect_conj_eq: "{x. P x & Q x} = {x. P x} \<inter> {x. Q x}"
wenzelm@12897
  1020
  by blast
wenzelm@12897
  1021
wenzelm@12897
  1022
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
wenzelm@12897
  1023
  by blast
wenzelm@12897
  1024
wenzelm@12897
  1025
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
wenzelm@12897
  1026
  by blast
wenzelm@12897
  1027
wenzelm@12897
  1028
lemma Collect_ex_eq: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
wenzelm@12897
  1029
  by blast
wenzelm@12897
  1030
wenzelm@12897
  1031
lemma Collect_bex_eq: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
wenzelm@12897
  1032
  by blast
wenzelm@12897
  1033
wenzelm@12897
  1034
wenzelm@12897
  1035
text {* \medskip @{text insert}. *}
wenzelm@12897
  1036
wenzelm@12897
  1037
lemma insert_is_Un: "insert a A = {a} Un A"
wenzelm@12897
  1038
  -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a {}"} *}
wenzelm@12897
  1039
  by blast
wenzelm@12897
  1040
wenzelm@12897
  1041
lemma insert_not_empty [simp]: "insert a A \<noteq> {}"
wenzelm@12897
  1042
  by blast
wenzelm@12897
  1043
wenzelm@12897
  1044
lemmas empty_not_insert [simp] = insert_not_empty [symmetric, standard]
wenzelm@12897
  1045
wenzelm@12897
  1046
lemma insert_absorb: "a \<in> A ==> insert a A = A"
wenzelm@12897
  1047
  -- {* @{text "[simp]"} causes recursive calls when there are nested inserts *}
wenzelm@12897
  1048
  -- {* with \emph{quadratic} running time *}
wenzelm@12897
  1049
  by blast
wenzelm@12897
  1050
wenzelm@12897
  1051
lemma insert_absorb2 [simp]: "insert x (insert x A) = insert x A"
wenzelm@12897
  1052
  by blast
wenzelm@12897
  1053
wenzelm@12897
  1054
lemma insert_commute: "insert x (insert y A) = insert y (insert x A)"
wenzelm@12897
  1055
  by blast
wenzelm@12897
  1056
wenzelm@12897
  1057
lemma insert_subset [simp]: "(insert x A \<subseteq> B) = (x \<in> B & A \<subseteq> B)"
wenzelm@12897
  1058
  by blast
wenzelm@12897
  1059
wenzelm@12897
  1060
lemma mk_disjoint_insert: "a \<in> A ==> \<exists>B. A = insert a B & a \<notin> B"
wenzelm@12897
  1061
  -- {* use new @{text B} rather than @{text "A - {a}"} to avoid infinite unfolding *}
paulson@14208
  1062
  apply (rule_tac x = "A - {a}" in exI, blast)
wenzelm@12897
  1063
  done
wenzelm@12897
  1064
wenzelm@12897
  1065
lemma insert_Collect: "insert a (Collect P) = {u. u \<noteq> a --> P u}"
wenzelm@12897
  1066
  by auto
wenzelm@12897
  1067
wenzelm@12897
  1068
lemma UN_insert_distrib: "u \<in> A ==> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
wenzelm@12897
  1069
  by blast
wenzelm@12897
  1070
nipkow@14302
  1071
lemma insert_inter_insert[simp]: "insert a A \<inter> insert a B = insert a (A \<inter> B)"
nipkow@14302
  1072
by blast
nipkow@14302
  1073
nipkow@13103
  1074
lemma insert_disjoint[simp]:
nipkow@13103
  1075
 "(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})"
nipkow@13103
  1076
by blast
nipkow@13103
  1077
nipkow@13103
  1078
lemma disjoint_insert[simp]:
nipkow@13103
  1079
 "(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})"
nipkow@13103
  1080
by blast
wenzelm@12897
  1081
wenzelm@12897
  1082
text {* \medskip @{text image}. *}
wenzelm@12897
  1083
wenzelm@12897
  1084
lemma image_empty [simp]: "f`{} = {}"
wenzelm@12897
  1085
  by blast
wenzelm@12897
  1086
wenzelm@12897
  1087
lemma image_insert [simp]: "f ` insert a B = insert (f a) (f`B)"
wenzelm@12897
  1088
  by blast
wenzelm@12897
  1089
wenzelm@12897
  1090
lemma image_constant: "x \<in> A ==> (\<lambda>x. c) ` A = {c}"
wenzelm@12897
  1091
  by blast
wenzelm@12897
  1092
wenzelm@12897
  1093
lemma image_image: "f ` (g ` A) = (\<lambda>x. f (g x)) ` A"
wenzelm@12897
  1094
  by blast
wenzelm@12897
  1095
wenzelm@12897
  1096
lemma insert_image [simp]: "x \<in> A ==> insert (f x) (f`A) = f`A"
wenzelm@12897
  1097
  by blast
wenzelm@12897
  1098
wenzelm@12897
  1099
lemma image_is_empty [iff]: "(f`A = {}) = (A = {})"
wenzelm@12897
  1100
  by blast
wenzelm@12897
  1101
wenzelm@12897
  1102
lemma image_Collect: "f ` {x. P x} = {f x | x. P x}"
wenzelm@12897
  1103
  -- {* NOT suitable as a default simprule: the RHS isn't simpler than the LHS, *}
wenzelm@12897
  1104
  -- {* with its implicit quantifier and conjunction.  Also image enjoys better *}
wenzelm@12897
  1105
  -- {* equational properties than does the RHS. *}
wenzelm@12897
  1106
  by blast
wenzelm@12897
  1107
wenzelm@12897
  1108
lemma if_image_distrib [simp]:
wenzelm@12897
  1109
  "(\<lambda>x. if P x then f x else g x) ` S
wenzelm@12897
  1110
    = (f ` (S \<inter> {x. P x})) \<union> (g ` (S \<inter> {x. \<not> P x}))"
wenzelm@12897
  1111
  by (auto simp add: image_def)
wenzelm@12897
  1112
wenzelm@12897
  1113
lemma image_cong: "M = N ==> (!!x. x \<in> N ==> f x = g x) ==> f`M = g`N"
wenzelm@12897
  1114
  by (simp add: image_def)
wenzelm@12897
  1115
wenzelm@12897
  1116
wenzelm@12897
  1117
text {* \medskip @{text range}. *}
wenzelm@12897
  1118
wenzelm@12897
  1119
lemma full_SetCompr_eq: "{u. \<exists>x. u = f x} = range f"
wenzelm@12897
  1120
  by auto
wenzelm@12897
  1121
wenzelm@12897
  1122
lemma range_composition [simp]: "range (\<lambda>x. f (g x)) = f`range g"
paulson@14208
  1123
by (subst image_image, simp)
wenzelm@12897
  1124
wenzelm@12897
  1125
wenzelm@12897
  1126
text {* \medskip @{text Int} *}
wenzelm@12897
  1127
wenzelm@12897
  1128
lemma Int_absorb [simp]: "A \<inter> A = A"
wenzelm@12897
  1129
  by blast
wenzelm@12897
  1130
wenzelm@12897
  1131
lemma Int_left_absorb: "A \<inter> (A \<inter> B) = A \<inter> B"
wenzelm@12897
  1132
  by blast
wenzelm@12897
  1133
wenzelm@12897
  1134
lemma Int_commute: "A \<inter> B = B \<inter> A"
wenzelm@12897
  1135
  by blast
wenzelm@12897
  1136
wenzelm@12897
  1137
lemma Int_left_commute: "A \<inter> (B \<inter> C) = B \<inter> (A \<inter> C)"
wenzelm@12897
  1138
  by blast
wenzelm@12897
  1139
wenzelm@12897
  1140
lemma Int_assoc: "(A \<inter> B) \<inter> C = A \<inter> (B \<inter> C)"
wenzelm@12897
  1141
  by blast
wenzelm@12897
  1142
wenzelm@12897
  1143
lemmas Int_ac = Int_assoc Int_left_absorb Int_commute Int_left_commute
wenzelm@12897
  1144
  -- {* Intersection is an AC-operator *}
wenzelm@12897
  1145
wenzelm@12897
  1146
lemma Int_absorb1: "B \<subseteq> A ==> A \<inter> B = B"
wenzelm@12897
  1147
  by blast
wenzelm@12897
  1148
wenzelm@12897
  1149
lemma Int_absorb2: "A \<subseteq> B ==> A \<inter> B = A"
wenzelm@12897
  1150
  by blast
wenzelm@12897
  1151
wenzelm@12897
  1152
lemma Int_empty_left [simp]: "{} \<inter> B = {}"
wenzelm@12897
  1153
  by blast
wenzelm@12897
  1154
wenzelm@12897
  1155
lemma Int_empty_right [simp]: "A \<inter> {} = {}"
wenzelm@12897
  1156
  by blast
wenzelm@12897
  1157
wenzelm@12897
  1158
lemma disjoint_eq_subset_Compl: "(A \<inter> B = {}) = (A \<subseteq> -B)"
wenzelm@12897
  1159
  by blast
wenzelm@12897
  1160
wenzelm@12897
  1161
lemma disjoint_iff_not_equal: "(A \<inter> B = {}) = (\<forall>x\<in>A. \<forall>y\<in>B. x \<noteq> y)"
wenzelm@12897
  1162
  by blast
wenzelm@12897
  1163
wenzelm@12897
  1164
lemma Int_UNIV_left [simp]: "UNIV \<inter> B = B"
wenzelm@12897
  1165
  by blast
wenzelm@12897
  1166
wenzelm@12897
  1167
lemma Int_UNIV_right [simp]: "A \<inter> UNIV = A"
wenzelm@12897
  1168
  by blast
wenzelm@12897
  1169
wenzelm@12897
  1170
lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"
wenzelm@12897
  1171
  by blast
wenzelm@12897
  1172
wenzelm@12897
  1173
lemma Int_Un_distrib: "A \<inter> (B \<union> C) = (A \<inter> B) \<union> (A \<inter> C)"
wenzelm@12897
  1174
  by blast
wenzelm@12897
  1175
wenzelm@12897
  1176
lemma Int_Un_distrib2: "(B \<union> C) \<inter> A = (B \<inter> A) \<union> (C \<inter> A)"
wenzelm@12897
  1177
  by blast
wenzelm@12897
  1178
wenzelm@12897
  1179
lemma Int_UNIV [simp]: "(A \<inter> B = UNIV) = (A = UNIV & B = UNIV)"
wenzelm@12897
  1180
  by blast
wenzelm@12897
  1181
wenzelm@12897
  1182
lemma Int_subset_iff: "(C \<subseteq> A \<inter> B) = (C \<subseteq> A & C \<subseteq> B)"
wenzelm@12897
  1183
  by blast
wenzelm@12897
  1184
wenzelm@12897
  1185
lemma Int_Collect: "(x \<in> A \<inter> {x. P x}) = (x \<in> A & P x)"
wenzelm@12897
  1186
  by blast
wenzelm@12897
  1187
wenzelm@12897
  1188
wenzelm@12897
  1189
text {* \medskip @{text Un}. *}
wenzelm@12897
  1190
wenzelm@12897
  1191
lemma Un_absorb [simp]: "A \<union> A = A"
wenzelm@12897
  1192
  by blast
wenzelm@12897
  1193
wenzelm@12897
  1194
lemma Un_left_absorb: "A \<union> (A \<union> B) = A \<union> B"
wenzelm@12897
  1195
  by blast
wenzelm@12897
  1196
wenzelm@12897
  1197
lemma Un_commute: "A \<union> B = B \<union> A"
wenzelm@12897
  1198
  by blast
wenzelm@12897
  1199
wenzelm@12897
  1200
lemma Un_left_commute: "A \<union> (B \<union> C) = B \<union> (A \<union> C)"
wenzelm@12897
  1201
  by blast
wenzelm@12897
  1202
wenzelm@12897
  1203
lemma Un_assoc: "(A \<union> B) \<union> C = A \<union> (B \<union> C)"
wenzelm@12897
  1204
  by blast
wenzelm@12897
  1205
wenzelm@12897
  1206
lemmas Un_ac = Un_assoc Un_left_absorb Un_commute Un_left_commute
wenzelm@12897
  1207
  -- {* Union is an AC-operator *}
wenzelm@12897
  1208
wenzelm@12897
  1209
lemma Un_absorb1: "A \<subseteq> B ==> A \<union> B = B"
wenzelm@12897
  1210
  by blast
wenzelm@12897
  1211
wenzelm@12897
  1212
lemma Un_absorb2: "B \<subseteq> A ==> A \<union> B = A"
wenzelm@12897
  1213
  by blast
wenzelm@12897
  1214
wenzelm@12897
  1215
lemma Un_empty_left [simp]: "{} \<union> B = B"
wenzelm@12897
  1216
  by blast
wenzelm@12897
  1217
wenzelm@12897
  1218
lemma Un_empty_right [simp]: "A \<union> {} = A"
wenzelm@12897
  1219
  by blast
wenzelm@12897
  1220
wenzelm@12897
  1221
lemma Un_UNIV_left [simp]: "UNIV \<union> B = UNIV"
wenzelm@12897
  1222
  by blast
wenzelm@12897
  1223
wenzelm@12897
  1224
lemma Un_UNIV_right [simp]: "A \<union> UNIV = UNIV"
wenzelm@12897
  1225
  by blast
wenzelm@12897
  1226
wenzelm@12897
  1227
lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"
wenzelm@12897
  1228
  by blast
wenzelm@12897
  1229
wenzelm@12897
  1230
lemma Un_insert_left [simp]: "(insert a B) \<union> C = insert a (B \<union> C)"
wenzelm@12897
  1231
  by blast
wenzelm@12897
  1232
wenzelm@12897
  1233
lemma Un_insert_right [simp]: "A \<union> (insert a B) = insert a (A \<union> B)"
wenzelm@12897
  1234
  by blast
wenzelm@12897
  1235
wenzelm@12897
  1236
lemma Int_insert_left:
wenzelm@12897
  1237
    "(insert a B) Int C = (if a \<in> C then insert a (B \<inter> C) else B \<inter> C)"
wenzelm@12897
  1238
  by auto
wenzelm@12897
  1239
wenzelm@12897
  1240
lemma Int_insert_right:
wenzelm@12897
  1241
    "A \<inter> (insert a B) = (if a \<in> A then insert a (A \<inter> B) else A \<inter> B)"
wenzelm@12897
  1242
  by auto
wenzelm@12897
  1243
wenzelm@12897
  1244
lemma Un_Int_distrib: "A \<union> (B \<inter> C) = (A \<union> B) \<inter> (A \<union> C)"
wenzelm@12897
  1245
  by blast
wenzelm@12897
  1246
wenzelm@12897
  1247
lemma Un_Int_distrib2: "(B \<inter> C) \<union> A = (B \<union> A) \<inter> (C \<union> A)"
wenzelm@12897
  1248
  by blast
wenzelm@12897
  1249
wenzelm@12897
  1250
lemma Un_Int_crazy:
wenzelm@12897
  1251
    "(A \<inter> B) \<union> (B \<inter> C) \<union> (C \<inter> A) = (A \<union> B) \<inter> (B \<union> C) \<inter> (C \<union> A)"
wenzelm@12897
  1252
  by blast
wenzelm@12897
  1253
wenzelm@12897
  1254
lemma subset_Un_eq: "(A \<subseteq> B) = (A \<union> B = B)"
wenzelm@12897
  1255
  by blast
wenzelm@12897
  1256
wenzelm@12897
  1257
lemma Un_empty [iff]: "(A \<union> B = {}) = (A = {} & B = {})"
wenzelm@12897
  1258
  by blast
wenzelm@12897
  1259
wenzelm@12897
  1260
lemma Un_subset_iff: "(A \<union> B \<subseteq> C) = (A \<subseteq> C & B \<subseteq> C)"
wenzelm@12897
  1261
  by blast
wenzelm@12897
  1262
wenzelm@12897
  1263
lemma Un_Diff_Int: "(A - B) \<union> (A \<inter> B) = A"
wenzelm@12897
  1264
  by blast
wenzelm@12897
  1265
wenzelm@12897
  1266
wenzelm@12897
  1267
text {* \medskip Set complement *}
wenzelm@12897
  1268
wenzelm@12897
  1269
lemma Compl_disjoint [simp]: "A \<inter> -A = {}"
wenzelm@12897
  1270
  by blast
wenzelm@12897
  1271
wenzelm@12897
  1272
lemma Compl_disjoint2 [simp]: "-A \<inter> A = {}"
wenzelm@12897
  1273
  by blast
wenzelm@12897
  1274
paulson@13818
  1275
lemma Compl_partition: "A \<union> -A = UNIV"
paulson@13818
  1276
  by blast
paulson@13818
  1277
paulson@13818
  1278
lemma Compl_partition2: "-A \<union> A = UNIV"
wenzelm@12897
  1279
  by blast
wenzelm@12897
  1280
wenzelm@12897
  1281
lemma double_complement [simp]: "- (-A) = (A::'a set)"
wenzelm@12897
  1282
  by blast
wenzelm@12897
  1283
wenzelm@12897
  1284
lemma Compl_Un [simp]: "-(A \<union> B) = (-A) \<inter> (-B)"
wenzelm@12897
  1285
  by blast
wenzelm@12897
  1286
wenzelm@12897
  1287
lemma Compl_Int [simp]: "-(A \<inter> B) = (-A) \<union> (-B)"
wenzelm@12897
  1288
  by blast
wenzelm@12897
  1289
wenzelm@12897
  1290
lemma Compl_UN [simp]: "-(\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
wenzelm@12897
  1291
  by blast
wenzelm@12897
  1292
wenzelm@12897
  1293
lemma Compl_INT [simp]: "-(\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
wenzelm@12897
  1294
  by blast
wenzelm@12897
  1295
wenzelm@12897
  1296
lemma subset_Compl_self_eq: "(A \<subseteq> -A) = (A = {})"
wenzelm@12897
  1297
  by blast
wenzelm@12897
  1298
wenzelm@12897
  1299
lemma Un_Int_assoc_eq: "((A \<inter> B) \<union> C = A \<inter> (B \<union> C)) = (C \<subseteq> A)"
wenzelm@12897
  1300
  -- {* Halmos, Naive Set Theory, page 16. *}
wenzelm@12897
  1301
  by blast
wenzelm@12897
  1302
wenzelm@12897
  1303
lemma Compl_UNIV_eq [simp]: "-UNIV = {}"
wenzelm@12897
  1304
  by blast
wenzelm@12897
  1305
wenzelm@12897
  1306
lemma Compl_empty_eq [simp]: "-{} = UNIV"
wenzelm@12897
  1307
  by blast
wenzelm@12897
  1308
wenzelm@12897
  1309
lemma Compl_subset_Compl_iff [iff]: "(-A \<subseteq> -B) = (B \<subseteq> A)"
wenzelm@12897
  1310
  by blast
wenzelm@12897
  1311
wenzelm@12897
  1312
lemma Compl_eq_Compl_iff [iff]: "(-A = -B) = (A = (B::'a set))"
wenzelm@12897
  1313
  by blast
wenzelm@12897
  1314
wenzelm@12897
  1315
wenzelm@12897
  1316
text {* \medskip @{text Union}. *}
wenzelm@12897
  1317
wenzelm@12897
  1318
lemma Union_empty [simp]: "Union({}) = {}"
wenzelm@12897
  1319
  by blast
wenzelm@12897
  1320
wenzelm@12897
  1321
lemma Union_UNIV [simp]: "Union UNIV = UNIV"
wenzelm@12897
  1322
  by blast
wenzelm@12897
  1323
wenzelm@12897
  1324
lemma Union_insert [simp]: "Union (insert a B) = a \<union> \<Union>B"
wenzelm@12897
  1325
  by blast
wenzelm@12897
  1326
wenzelm@12897
  1327
lemma Union_Un_distrib [simp]: "\<Union>(A Un B) = \<Union>A \<union> \<Union>B"
wenzelm@12897
  1328
  by blast
wenzelm@12897
  1329
wenzelm@12897
  1330
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
wenzelm@12897
  1331
  by blast
wenzelm@12897
  1332
wenzelm@12897
  1333
lemma Union_empty_conv [iff]: "(\<Union>A = {}) = (\<forall>x\<in>A. x = {})"
nipkow@13653
  1334
  by blast
nipkow@13653
  1335
nipkow@13653
  1336
lemma empty_Union_conv [iff]: "({} = \<Union>A) = (\<forall>x\<in>A. x = {})"
nipkow@13653
  1337
  by blast
wenzelm@12897
  1338
wenzelm@12897
  1339
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) = (\<forall>B\<in>C. B \<inter> A = {})"
wenzelm@12897
  1340
  by blast
wenzelm@12897
  1341
wenzelm@12897
  1342
wenzelm@12897
  1343
text {* \medskip @{text Inter}. *}
wenzelm@12897
  1344
wenzelm@12897
  1345
lemma Inter_empty [simp]: "\<Inter>{} = UNIV"
wenzelm@12897
  1346
  by blast
wenzelm@12897
  1347
wenzelm@12897
  1348
lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}"
wenzelm@12897
  1349
  by blast
wenzelm@12897
  1350
wenzelm@12897
  1351
lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
wenzelm@12897
  1352
  by blast
wenzelm@12897
  1353
wenzelm@12897
  1354
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
wenzelm@12897
  1355
  by blast
wenzelm@12897
  1356
wenzelm@12897
  1357
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
wenzelm@12897
  1358
  by blast
wenzelm@12897
  1359
nipkow@13653
  1360
lemma Inter_UNIV_conv [iff]:
nipkow@13653
  1361
  "(\<Inter>A = UNIV) = (\<forall>x\<in>A. x = UNIV)"
nipkow@13653
  1362
  "(UNIV = \<Inter>A) = (\<forall>x\<in>A. x = UNIV)"
paulson@14208
  1363
  by blast+
nipkow@13653
  1364
wenzelm@12897
  1365
wenzelm@12897
  1366
text {*
wenzelm@12897
  1367
  \medskip @{text UN} and @{text INT}.
wenzelm@12897
  1368
wenzelm@12897
  1369
  Basic identities: *}
wenzelm@12897
  1370
wenzelm@12897
  1371
lemma UN_empty [simp]: "(\<Union>x\<in>{}. B x) = {}"
wenzelm@12897
  1372
  by blast
wenzelm@12897
  1373
wenzelm@12897
  1374
lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"
wenzelm@12897
  1375
  by blast
wenzelm@12897
  1376
wenzelm@12897
  1377
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
wenzelm@12897
  1378
  by blast
wenzelm@12897
  1379
wenzelm@12897
  1380
lemma UN_absorb: "k \<in> I ==> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
wenzelm@12897
  1381
  by blast
wenzelm@12897
  1382
wenzelm@12897
  1383
lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"
wenzelm@12897
  1384
  by blast
wenzelm@12897
  1385
wenzelm@12897
  1386
lemma INT_absorb: "k \<in> I ==> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
wenzelm@12897
  1387
  by blast
wenzelm@12897
  1388
wenzelm@12897
  1389
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
wenzelm@12897
  1390
  by blast
wenzelm@12897
  1391
wenzelm@12897
  1392
lemma UN_Un: "(\<Union>i \<in> A \<union> B. M i) = (\<Union>i\<in>A. M i) \<union> (\<Union>i\<in>B. M i)"
wenzelm@12897
  1393
  by blast
wenzelm@12897
  1394
wenzelm@12897
  1395
lemma UN_UN_flatten: "(\<Union>x \<in> (\<Union>y\<in>A. B y). C x) = (\<Union>y\<in>A. \<Union>x\<in>B y. C x)"
wenzelm@12897
  1396
  by blast
wenzelm@12897
  1397
wenzelm@12897
  1398
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
wenzelm@12897
  1399
  by blast
wenzelm@12897
  1400
wenzelm@12897
  1401
lemma INT_subset_iff: "(B \<subseteq> (\<Inter>i\<in>I. A i)) = (\<forall>i\<in>I. B \<subseteq> A i)"
wenzelm@12897
  1402
  by blast
wenzelm@12897
  1403
wenzelm@12897
  1404
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
wenzelm@12897
  1405
  by blast
wenzelm@12897
  1406
wenzelm@12897
  1407
lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"
wenzelm@12897
  1408
  by blast
wenzelm@12897
  1409
wenzelm@12897
  1410
lemma INT_insert_distrib:
wenzelm@12897
  1411
    "u \<in> A ==> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
wenzelm@12897
  1412
  by blast
wenzelm@12897
  1413
wenzelm@12897
  1414
lemma Union_image_eq [simp]: "\<Union>(B`A) = (\<Union>x\<in>A. B x)"
wenzelm@12897
  1415
  by blast
wenzelm@12897
  1416
wenzelm@12897
  1417
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
wenzelm@12897
  1418
  by blast
wenzelm@12897
  1419
wenzelm@12897
  1420
lemma Inter_image_eq [simp]: "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
wenzelm@12897
  1421
  by blast
wenzelm@12897
  1422
wenzelm@12897
  1423
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
wenzelm@12897
  1424
  by auto
wenzelm@12897
  1425
wenzelm@12897
  1426
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
wenzelm@12897
  1427
  by auto
wenzelm@12897
  1428
wenzelm@12897
  1429
lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"
wenzelm@12897
  1430
  by blast
wenzelm@12897
  1431
wenzelm@12897
  1432
lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"
wenzelm@12897
  1433
  -- {* Look: it has an \emph{existential} quantifier *}
wenzelm@12897
  1434
  by blast
wenzelm@12897
  1435
nipkow@13653
  1436
lemma UNION_empty_conv[iff]:
nipkow@13653
  1437
  "({} = (UN x:A. B x)) = (\<forall>x\<in>A. B x = {})"
nipkow@13653
  1438
  "((UN x:A. B x) = {}) = (\<forall>x\<in>A. B x = {})"
nipkow@13653
  1439
by blast+
nipkow@13653
  1440
nipkow@13653
  1441
lemma INTER_UNIV_conv[iff]:
nipkow@13653
  1442
 "(UNIV = (INT x:A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
nipkow@13653
  1443
 "((INT x:A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
nipkow@13653
  1444
by blast+
wenzelm@12897
  1445
wenzelm@12897
  1446
wenzelm@12897
  1447
text {* \medskip Distributive laws: *}
wenzelm@12897
  1448
wenzelm@12897
  1449
lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
wenzelm@12897
  1450
  by blast
wenzelm@12897
  1451
wenzelm@12897
  1452
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
wenzelm@12897
  1453
  by blast
wenzelm@12897
  1454
wenzelm@12897
  1455
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A`C) \<union> \<Union>(B`C)"
wenzelm@12897
  1456
  -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
wenzelm@12897
  1457
  -- {* Union of a family of unions *}
wenzelm@12897
  1458
  by blast
wenzelm@12897
  1459
wenzelm@12897
  1460
lemma UN_Un_distrib: "(\<Union>i\<in>I. A i \<union> B i) = (\<Union>i\<in>I. A i) \<union> (\<Union>i\<in>I. B i)"
wenzelm@12897
  1461
  -- {* Equivalent version *}
wenzelm@12897
  1462
  by blast
wenzelm@12897
  1463
wenzelm@12897
  1464
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
wenzelm@12897
  1465
  by blast
wenzelm@12897
  1466
wenzelm@12897
  1467
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A`C) \<inter> \<Inter>(B`C)"
wenzelm@12897
  1468
  by blast
wenzelm@12897
  1469
wenzelm@12897
  1470
lemma INT_Int_distrib: "(\<Inter>i\<in>I. A i \<inter> B i) = (\<Inter>i\<in>I. A i) \<inter> (\<Inter>i\<in>I. B i)"
wenzelm@12897
  1471
  -- {* Equivalent version *}
wenzelm@12897
  1472
  by blast
wenzelm@12897
  1473
wenzelm@12897
  1474
lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
wenzelm@12897
  1475
  -- {* Halmos, Naive Set Theory, page 35. *}
wenzelm@12897
  1476
  by blast
wenzelm@12897
  1477
wenzelm@12897
  1478
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
wenzelm@12897
  1479
  by blast
wenzelm@12897
  1480
wenzelm@12897
  1481
lemma Int_UN_distrib2: "(\<Union>i\<in>I. A i) \<inter> (\<Union>j\<in>J. B j) = (\<Union>i\<in>I. \<Union>j\<in>J. A i \<inter> B j)"
wenzelm@12897
  1482
  by blast
wenzelm@12897
  1483
wenzelm@12897
  1484
lemma Un_INT_distrib2: "(\<Inter>i\<in>I. A i) \<union> (\<Inter>j\<in>J. B j) = (\<Inter>i\<in>I. \<Inter>j\<in>J. A i \<union> B j)"
wenzelm@12897
  1485
  by blast
wenzelm@12897
  1486
wenzelm@12897
  1487
wenzelm@12897
  1488
text {* \medskip Bounded quantifiers.
wenzelm@12897
  1489
wenzelm@12897
  1490
  The following are not added to the default simpset because
wenzelm@12897
  1491
  (a) they duplicate the body and (b) there are no similar rules for @{text Int}. *}
wenzelm@12897
  1492
wenzelm@12897
  1493
lemma ball_Un: "(\<forall>x \<in> A \<union> B. P x) = ((\<forall>x\<in>A. P x) & (\<forall>x\<in>B. P x))"
wenzelm@12897
  1494
  by blast
wenzelm@12897
  1495
wenzelm@12897
  1496
lemma bex_Un: "(\<exists>x \<in> A \<union> B. P x) = ((\<exists>x\<in>A. P x) | (\<exists>x\<in>B. P x))"
wenzelm@12897
  1497
  by blast
wenzelm@12897
  1498
wenzelm@12897
  1499
lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) = (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
wenzelm@12897
  1500
  by blast
wenzelm@12897
  1501
wenzelm@12897
  1502
lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) = (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
wenzelm@12897
  1503
  by blast
wenzelm@12897
  1504
wenzelm@12897
  1505
wenzelm@12897
  1506
text {* \medskip Set difference. *}
wenzelm@12897
  1507
wenzelm@12897
  1508
lemma Diff_eq: "A - B = A \<inter> (-B)"
wenzelm@12897
  1509
  by blast
wenzelm@12897
  1510
wenzelm@12897
  1511
lemma Diff_eq_empty_iff [simp]: "(A - B = {}) = (A \<subseteq> B)"
wenzelm@12897
  1512
  by blast
wenzelm@12897
  1513
wenzelm@12897
  1514
lemma Diff_cancel [simp]: "A - A = {}"
wenzelm@12897
  1515
  by blast
wenzelm@12897
  1516
nipkow@14302
  1517
lemma Diff_idemp [simp]: "(A - B) - B = A - (B::'a set)"
nipkow@14302
  1518
by blast
nipkow@14302
  1519
wenzelm@12897
  1520
lemma Diff_triv: "A \<inter> B = {} ==> A - B = A"
wenzelm@12897
  1521
  by (blast elim: equalityE)
wenzelm@12897
  1522
wenzelm@12897
  1523
lemma empty_Diff [simp]: "{} - A = {}"
wenzelm@12897
  1524
  by blast
wenzelm@12897
  1525
wenzelm@12897
  1526
lemma Diff_empty [simp]: "A - {} = A"
wenzelm@12897
  1527
  by blast
wenzelm@12897
  1528
wenzelm@12897
  1529
lemma Diff_UNIV [simp]: "A - UNIV = {}"
wenzelm@12897
  1530
  by blast
wenzelm@12897
  1531
wenzelm@12897
  1532
lemma Diff_insert0 [simp]: "x \<notin> A ==> A - insert x B = A - B"
wenzelm@12897
  1533
  by blast
wenzelm@12897
  1534
wenzelm@12897
  1535
lemma Diff_insert: "A - insert a B = A - B - {a}"
wenzelm@12897
  1536
  -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a 0"} *}
wenzelm@12897
  1537
  by blast
wenzelm@12897
  1538
wenzelm@12897
  1539
lemma Diff_insert2: "A - insert a B = A - {a} - B"
wenzelm@12897
  1540
  -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a 0"} *}
wenzelm@12897
  1541
  by blast
wenzelm@12897
  1542
wenzelm@12897
  1543
lemma insert_Diff_if: "insert x A - B = (if x \<in> B then A - B else insert x (A - B))"
wenzelm@12897
  1544
  by auto
wenzelm@12897
  1545
wenzelm@12897
  1546
lemma insert_Diff1 [simp]: "x \<in> B ==> insert x A - B = A - B"
wenzelm@12897
  1547
  by blast
wenzelm@12897
  1548
nipkow@14302
  1549
lemma insert_Diff_single[simp]: "insert a (A - {a}) = insert a A"
nipkow@14302
  1550
by blast
nipkow@14302
  1551
wenzelm@12897
  1552
lemma insert_Diff: "a \<in> A ==> insert a (A - {a}) = A"
wenzelm@12897
  1553
  by blast
wenzelm@12897
  1554
wenzelm@12897
  1555
lemma Diff_insert_absorb: "x \<notin> A ==> (insert x A) - {x} = A"
wenzelm@12897
  1556
  by auto
wenzelm@12897
  1557
wenzelm@12897
  1558
lemma Diff_disjoint [simp]: "A \<inter> (B - A) = {}"
wenzelm@12897
  1559
  by blast
wenzelm@12897
  1560
wenzelm@12897
  1561
lemma Diff_partition: "A \<subseteq> B ==> A \<union> (B - A) = B"
wenzelm@12897
  1562
  by blast
wenzelm@12897
  1563
wenzelm@12897
  1564
lemma double_diff: "A \<subseteq> B ==> B \<subseteq> C ==> B - (C - A) = A"
wenzelm@12897
  1565
  by blast
wenzelm@12897
  1566
wenzelm@12897
  1567
lemma Un_Diff_cancel [simp]: "A \<union> (B - A) = A \<union> B"
wenzelm@12897
  1568
  by blast
wenzelm@12897
  1569
wenzelm@12897
  1570
lemma Un_Diff_cancel2 [simp]: "(B - A) \<union> A = B \<union> A"
wenzelm@12897
  1571
  by blast
wenzelm@12897
  1572
wenzelm@12897
  1573
lemma Diff_Un: "A - (B \<union> C) = (A - B) \<inter> (A - C)"
wenzelm@12897
  1574
  by blast
wenzelm@12897
  1575
wenzelm@12897
  1576
lemma Diff_Int: "A - (B \<inter> C) = (A - B) \<union> (A - C)"
wenzelm@12897
  1577
  by blast
wenzelm@12897
  1578
wenzelm@12897
  1579
lemma Un_Diff: "(A \<union> B) - C = (A - C) \<union> (B - C)"
wenzelm@12897
  1580
  by blast
wenzelm@12897
  1581
wenzelm@12897
  1582
lemma Int_Diff: "(A \<inter> B) - C = A \<inter> (B - C)"
wenzelm@12897
  1583
  by blast
wenzelm@12897
  1584
wenzelm@12897
  1585
lemma Diff_Int_distrib: "C \<inter> (A - B) = (C \<inter> A) - (C \<inter> B)"
wenzelm@12897
  1586
  by blast
wenzelm@12897
  1587
wenzelm@12897
  1588
lemma Diff_Int_distrib2: "(A - B) \<inter> C = (A \<inter> C) - (B \<inter> C)"
wenzelm@12897
  1589
  by blast
wenzelm@12897
  1590
wenzelm@12897
  1591
lemma Diff_Compl [simp]: "A - (- B) = A \<inter> B"
wenzelm@12897
  1592
  by auto
wenzelm@12897
  1593
wenzelm@12897
  1594
lemma Compl_Diff_eq [simp]: "- (A - B) = -A \<union> B"
wenzelm@12897
  1595
  by blast
wenzelm@12897
  1596
wenzelm@12897
  1597
wenzelm@12897
  1598
text {* \medskip Quantification over type @{typ bool}. *}
wenzelm@12897
  1599
wenzelm@12897
  1600
lemma all_bool_eq: "(\<forall>b::bool. P b) = (P True & P False)"
wenzelm@12897
  1601
  apply auto
paulson@14208
  1602
  apply (tactic {* case_tac "b" 1 *}, auto)
wenzelm@12897
  1603
  done
wenzelm@12897
  1604
wenzelm@12897
  1605
lemma bool_induct: "P True \<Longrightarrow> P False \<Longrightarrow> P x"
wenzelm@12897
  1606
  by (rule conjI [THEN all_bool_eq [THEN iffD2], THEN spec])
wenzelm@12897
  1607
wenzelm@12897
  1608
lemma ex_bool_eq: "(\<exists>b::bool. P b) = (P True | P False)"
wenzelm@12897
  1609
  apply auto
paulson@14208
  1610
  apply (tactic {* case_tac "b" 1 *}, auto)
wenzelm@12897
  1611
  done
wenzelm@12897
  1612
wenzelm@12897
  1613
lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
wenzelm@12897
  1614
  by (auto simp add: split_if_mem2)
wenzelm@12897
  1615
wenzelm@12897
  1616
lemma UN_bool_eq: "(\<Union>b::bool. A b) = (A True \<union> A False)"
wenzelm@12897
  1617
  apply auto
paulson@14208
  1618
  apply (tactic {* case_tac "b" 1 *}, auto)
wenzelm@12897
  1619
  done
wenzelm@12897
  1620
wenzelm@12897
  1621
lemma INT_bool_eq: "(\<Inter>b::bool. A b) = (A True \<inter> A False)"
wenzelm@12897
  1622
  apply auto
paulson@14208
  1623
  apply (tactic {* case_tac "b" 1 *}, auto)
wenzelm@12897
  1624
  done
wenzelm@12897
  1625
wenzelm@12897
  1626
wenzelm@12897
  1627
text {* \medskip @{text Pow} *}
wenzelm@12897
  1628
wenzelm@12897
  1629
lemma Pow_empty [simp]: "Pow {} = {{}}"
wenzelm@12897
  1630
  by (auto simp add: Pow_def)
wenzelm@12897
  1631
wenzelm@12897
  1632
lemma Pow_insert: "Pow (insert a A) = Pow A \<union> (insert a ` Pow A)"
wenzelm@12897
  1633
  by (blast intro: image_eqI [where ?x = "u - {a}", standard])
wenzelm@12897
  1634
wenzelm@12897
  1635
lemma Pow_Compl: "Pow (- A) = {-B | B. A \<in> Pow B}"
wenzelm@12897
  1636
  by (blast intro: exI [where ?x = "- u", standard])
wenzelm@12897
  1637
wenzelm@12897
  1638
lemma Pow_UNIV [simp]: "Pow UNIV = UNIV"
wenzelm@12897
  1639
  by blast
wenzelm@12897
  1640
wenzelm@12897
  1641
lemma Un_Pow_subset: "Pow A \<union> Pow B \<subseteq> Pow (A \<union> B)"
wenzelm@12897
  1642
  by blast
wenzelm@12897
  1643
wenzelm@12897
  1644
lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
wenzelm@12897
  1645
  by blast
wenzelm@12897
  1646
wenzelm@12897
  1647
lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
wenzelm@12897
  1648
  by blast
wenzelm@12897
  1649
wenzelm@12897
  1650
lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
wenzelm@12897
  1651
  by blast
wenzelm@12897
  1652
wenzelm@12897
  1653
lemma Pow_Int_eq [simp]: "Pow (A \<inter> B) = Pow A \<inter> Pow B"
wenzelm@12897
  1654
  by blast
wenzelm@12897
  1655
wenzelm@12897
  1656
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
wenzelm@12897
  1657
  by blast
wenzelm@12897
  1658
wenzelm@12897
  1659
wenzelm@12897
  1660
text {* \medskip Miscellany. *}
wenzelm@12897
  1661
wenzelm@12897
  1662
lemma set_eq_subset: "(A = B) = (A \<subseteq> B & B \<subseteq> A)"
wenzelm@12897
  1663
  by blast
wenzelm@12897
  1664
wenzelm@12897
  1665
lemma subset_iff: "(A \<subseteq> B) = (\<forall>t. t \<in> A --> t \<in> B)"
wenzelm@12897
  1666
  by blast
wenzelm@12897
  1667
wenzelm@12897
  1668
lemma subset_iff_psubset_eq: "(A \<subseteq> B) = ((A \<subset> B) | (A = B))"
wenzelm@12897
  1669
  by (unfold psubset_def) blast
wenzelm@12897
  1670
wenzelm@12897
  1671
lemma all_not_in_conv [iff]: "(\<forall>x. x \<notin> A) = (A = {})"
wenzelm@12897
  1672
  by blast
wenzelm@12897
  1673
paulson@13831
  1674
lemma ex_in_conv: "(\<exists>x. x \<in> A) = (A \<noteq> {})"
paulson@13831
  1675
  by blast
paulson@13831
  1676
wenzelm@12897
  1677
lemma distinct_lemma: "f x \<noteq> f y ==> x \<noteq> y"
wenzelm@12897
  1678
  by rules
wenzelm@12897
  1679
wenzelm@12897
  1680
paulson@13860
  1681
text {* \medskip Miniscoping: pushing in quantifiers and big Unions
paulson@13860
  1682
           and Intersections. *}
wenzelm@12897
  1683
wenzelm@12897
  1684
lemma UN_simps [simp]:
wenzelm@12897
  1685
  "!!a B C. (UN x:C. insert a (B x)) = (if C={} then {} else insert a (UN x:C. B x))"
wenzelm@12897
  1686
  "!!A B C. (UN x:C. A x Un B)   = ((if C={} then {} else (UN x:C. A x) Un B))"
wenzelm@12897
  1687
  "!!A B C. (UN x:C. A Un B x)   = ((if C={} then {} else A Un (UN x:C. B x)))"
wenzelm@12897
  1688
  "!!A B C. (UN x:C. A x Int B)  = ((UN x:C. A x) Int B)"
wenzelm@12897
  1689
  "!!A B C. (UN x:C. A Int B x)  = (A Int (UN x:C. B x))"
wenzelm@12897
  1690
  "!!A B C. (UN x:C. A x - B)    = ((UN x:C. A x) - B)"
wenzelm@12897
  1691
  "!!A B C. (UN x:C. A - B x)    = (A - (INT x:C. B x))"
wenzelm@12897
  1692
  "!!A B. (UN x: Union A. B x) = (UN y:A. UN x:y. B x)"
wenzelm@12897
  1693
  "!!A B C. (UN z: UNION A B. C z) = (UN  x:A. UN z: B(x). C z)"
wenzelm@12897
  1694
  "!!A B f. (UN x:f`A. B x)     = (UN a:A. B (f a))"
wenzelm@12897
  1695
  by auto
wenzelm@12897
  1696
wenzelm@12897
  1697
lemma INT_simps [simp]:
wenzelm@12897
  1698
  "!!A B C. (INT x:C. A x Int B) = (if C={} then UNIV else (INT x:C. A x) Int B)"
wenzelm@12897
  1699
  "!!A B C. (INT x:C. A Int B x) = (if C={} then UNIV else A Int (INT x:C. B x))"
wenzelm@12897
  1700
  "!!A B C. (INT x:C. A x - B)   = (if C={} then UNIV else (INT x:C. A x) - B)"
wenzelm@12897
  1701
  "!!A B C. (INT x:C. A - B x)   = (if C={} then UNIV else A - (UN x:C. B x))"
wenzelm@12897
  1702
  "!!a B C. (INT x:C. insert a (B x)) = insert a (INT x:C. B x)"
wenzelm@12897
  1703
  "!!A B C. (INT x:C. A x Un B)  = ((INT x:C. A x) Un B)"
wenzelm@12897
  1704
  "!!A B C. (INT x:C. A Un B x)  = (A Un (INT x:C. B x))"
wenzelm@12897
  1705
  "!!A B. (INT x: Union A. B x) = (INT y:A. INT x:y. B x)"
wenzelm@12897
  1706
  "!!A B C. (INT z: UNION A B. C z) = (INT x:A. INT z: B(x). C z)"
wenzelm@12897
  1707
  "!!A B f. (INT x:f`A. B x)    = (INT a:A. B (f a))"
wenzelm@12897
  1708
  by auto
wenzelm@12897
  1709
wenzelm@12897
  1710
lemma ball_simps [simp]:
wenzelm@12897
  1711
  "!!A P Q. (ALL x:A. P x | Q) = ((ALL x:A. P x) | Q)"
wenzelm@12897
  1712
  "!!A P Q. (ALL x:A. P | Q x) = (P | (ALL x:A. Q x))"
wenzelm@12897
  1713
  "!!A P Q. (ALL x:A. P --> Q x) = (P --> (ALL x:A. Q x))"
wenzelm@12897
  1714
  "!!A P Q. (ALL x:A. P x --> Q) = ((EX x:A. P x) --> Q)"
wenzelm@12897
  1715
  "!!P. (ALL x:{}. P x) = True"
wenzelm@12897
  1716
  "!!P. (ALL x:UNIV. P x) = (ALL x. P x)"
wenzelm@12897
  1717
  "!!a B P. (ALL x:insert a B. P x) = (P a & (ALL x:B. P x))"
wenzelm@12897
  1718
  "!!A P. (ALL x:Union A. P x) = (ALL y:A. ALL x:y. P x)"
wenzelm@12897
  1719
  "!!A B P. (ALL x: UNION A B. P x) = (ALL a:A. ALL x: B a. P x)"
wenzelm@12897
  1720
  "!!P Q. (ALL x:Collect Q. P x) = (ALL x. Q x --> P x)"
wenzelm@12897
  1721
  "!!A P f. (ALL x:f`A. P x) = (ALL x:A. P (f x))"
wenzelm@12897
  1722
  "!!A P. (~(ALL x:A. P x)) = (EX x:A. ~P x)"
wenzelm@12897
  1723
  by auto
wenzelm@12897
  1724
wenzelm@12897
  1725
lemma bex_simps [simp]:
wenzelm@12897
  1726
  "!!A P Q. (EX x:A. P x & Q) = ((EX x:A. P x) & Q)"
wenzelm@12897
  1727
  "!!A P Q. (EX x:A. P & Q x) = (P & (EX x:A. Q x))"
wenzelm@12897
  1728
  "!!P. (EX x:{}. P x) = False"
wenzelm@12897
  1729
  "!!P. (EX x:UNIV. P x) = (EX x. P x)"
wenzelm@12897
  1730
  "!!a B P. (EX x:insert a B. P x) = (P(a) | (EX x:B. P x))"
wenzelm@12897
  1731
  "!!A P. (EX x:Union A. P x) = (EX y:A. EX x:y. P x)"
wenzelm@12897
  1732
  "!!A B P. (EX x: UNION A B. P x) = (EX a:A. EX x:B a. P x)"
wenzelm@12897
  1733
  "!!P Q. (EX x:Collect Q. P x) = (EX x. Q x & P x)"
wenzelm@12897
  1734
  "!!A P f. (EX x:f`A. P x) = (EX x:A. P (f x))"
wenzelm@12897
  1735
  "!!A P. (~(EX x:A. P x)) = (ALL x:A. ~P x)"
wenzelm@12897
  1736
  by auto
wenzelm@12897
  1737
wenzelm@12897
  1738
lemma ball_conj_distrib:
wenzelm@12897
  1739
  "(ALL x:A. P x & Q x) = ((ALL x:A. P x) & (ALL x:A. Q x))"
wenzelm@12897
  1740
  by blast
wenzelm@12897
  1741
wenzelm@12897
  1742
lemma bex_disj_distrib:
wenzelm@12897
  1743
  "(EX x:A. P x | Q x) = ((EX x:A. P x) | (EX x:A. Q x))"
wenzelm@12897
  1744
  by blast
wenzelm@12897
  1745
wenzelm@12897
  1746
paulson@13860
  1747
text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
paulson@13860
  1748
paulson@13860
  1749
lemma UN_extend_simps:
paulson@13860
  1750
  "!!a B C. insert a (UN x:C. B x) = (if C={} then {a} else (UN x:C. insert a (B x)))"
paulson@13860
  1751
  "!!A B C. (UN x:C. A x) Un B    = (if C={} then B else (UN x:C. A x Un B))"
paulson@13860
  1752
  "!!A B C. A Un (UN x:C. B x)   = (if C={} then A else (UN x:C. A Un B x))"
paulson@13860
  1753
  "!!A B C. ((UN x:C. A x) Int B) = (UN x:C. A x Int B)"
paulson@13860
  1754
  "!!A B C. (A Int (UN x:C. B x)) = (UN x:C. A Int B x)"
paulson@13860
  1755
  "!!A B C. ((UN x:C. A x) - B) = (UN x:C. A x - B)"
paulson@13860
  1756
  "!!A B C. (A - (INT x:C. B x)) = (UN x:C. A - B x)"
paulson@13860
  1757
  "!!A B. (UN y:A. UN x:y. B x) = (UN x: Union A. B x)"
paulson@13860
  1758
  "!!A B C. (UN  x:A. UN z: B(x). C z) = (UN z: UNION A B. C z)"
paulson@13860
  1759
  "!!A B f. (UN a:A. B (f a)) = (UN x:f`A. B x)"
paulson@13860
  1760
  by auto
paulson@13860
  1761
paulson@13860
  1762
lemma INT_extend_simps:
paulson@13860
  1763
  "!!A B C. (INT x:C. A x) Int B = (if C={} then B else (INT x:C. A x Int B))"
paulson@13860
  1764
  "!!A B C. A Int (INT x:C. B x) = (if C={} then A else (INT x:C. A Int B x))"
paulson@13860
  1765
  "!!A B C. (INT x:C. A x) - B   = (if C={} then UNIV-B else (INT x:C. A x - B))"
paulson@13860
  1766
  "!!A B C. A - (UN x:C. B x)   = (if C={} then A else (INT x:C. A - B x))"
paulson@13860
  1767
  "!!a B C. insert a (INT x:C. B x) = (INT x:C. insert a (B x))"
paulson@13860
  1768
  "!!A B C. ((INT x:C. A x) Un B)  = (INT x:C. A x Un B)"
paulson@13860
  1769
  "!!A B C. A Un (INT x:C. B x)  = (INT x:C. A Un B x)"
paulson@13860
  1770
  "!!A B. (INT y:A. INT x:y. B x) = (INT x: Union A. B x)"
paulson@13860
  1771
  "!!A B C. (INT x:A. INT z: B(x). C z) = (INT z: UNION A B. C z)"
paulson@13860
  1772
  "!!A B f. (INT a:A. B (f a))    = (INT x:f`A. B x)"
paulson@13860
  1773
  by auto
paulson@13860
  1774
paulson@13860
  1775
wenzelm@12897
  1776
subsubsection {* Monotonicity of various operations *}
wenzelm@12897
  1777
wenzelm@12897
  1778
lemma image_mono: "A \<subseteq> B ==> f`A \<subseteq> f`B"
wenzelm@12897
  1779
  by blast
wenzelm@12897
  1780
wenzelm@12897
  1781
lemma Pow_mono: "A \<subseteq> B ==> Pow A \<subseteq> Pow B"
wenzelm@12897
  1782
  by blast
wenzelm@12897
  1783
wenzelm@12897
  1784
lemma Union_mono: "A \<subseteq> B ==> \<Union>A \<subseteq> \<Union>B"
wenzelm@12897
  1785
  by blast
wenzelm@12897
  1786
wenzelm@12897
  1787
lemma Inter_anti_mono: "B \<subseteq> A ==> \<Inter>A \<subseteq> \<Inter>B"
wenzelm@12897
  1788
  by blast
wenzelm@12897
  1789
wenzelm@12897
  1790
lemma UN_mono:
wenzelm@12897
  1791
  "A \<subseteq> B ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
wenzelm@12897
  1792
    (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
wenzelm@12897
  1793
  by (blast dest: subsetD)
wenzelm@12897
  1794
wenzelm@12897
  1795
lemma INT_anti_mono:
wenzelm@12897
  1796
  "B \<subseteq> A ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
wenzelm@12897
  1797
    (\<Inter>x\<in>A. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
wenzelm@12897
  1798
  -- {* The last inclusion is POSITIVE! *}
wenzelm@12897
  1799
  by (blast dest: subsetD)
wenzelm@12897
  1800
wenzelm@12897
  1801
lemma insert_mono: "C \<subseteq> D ==> insert a C \<subseteq> insert a D"
wenzelm@12897
  1802
  by blast
wenzelm@12897
  1803
wenzelm@12897
  1804
lemma Un_mono: "A \<subseteq> C ==> B \<subseteq> D ==> A \<union> B \<subseteq> C \<union> D"
wenzelm@12897
  1805
  by blast
wenzelm@12897
  1806
wenzelm@12897
  1807
lemma Int_mono: "A \<subseteq> C ==> B \<subseteq> D ==> A \<inter> B \<subseteq> C \<inter> D"
wenzelm@12897
  1808
  by blast
wenzelm@12897
  1809
wenzelm@12897
  1810
lemma Diff_mono: "A \<subseteq> C ==> D \<subseteq> B ==> A - B \<subseteq> C - D"
wenzelm@12897
  1811
  by blast
wenzelm@12897
  1812
wenzelm@12897
  1813
lemma Compl_anti_mono: "A \<subseteq> B ==> -B \<subseteq> -A"
wenzelm@12897
  1814
  by blast
wenzelm@12897
  1815
wenzelm@12897
  1816
text {* \medskip Monotonicity of implications. *}
wenzelm@12897
  1817
wenzelm@12897
  1818
lemma in_mono: "A \<subseteq> B ==> x \<in> A --> x \<in> B"
wenzelm@12897
  1819
  apply (rule impI)
paulson@14208
  1820
  apply (erule subsetD, assumption)
wenzelm@12897
  1821
  done
wenzelm@12897
  1822
wenzelm@12897
  1823
lemma conj_mono: "P1 --> Q1 ==> P2 --> Q2 ==> (P1 & P2) --> (Q1 & Q2)"
wenzelm@12897
  1824
  by rules
wenzelm@12897
  1825
wenzelm@12897
  1826
lemma disj_mono: "P1 --> Q1 ==> P2 --> Q2 ==> (P1 | P2) --> (Q1 | Q2)"
wenzelm@12897
  1827
  by rules
wenzelm@12897
  1828
wenzelm@12897
  1829
lemma imp_mono: "Q1 --> P1 ==> P2 --> Q2 ==> (P1 --> P2) --> (Q1 --> Q2)"
wenzelm@12897
  1830
  by rules
wenzelm@12897
  1831
wenzelm@12897
  1832
lemma imp_refl: "P --> P" ..
wenzelm@12897
  1833
wenzelm@12897
  1834
lemma ex_mono: "(!!x. P x --> Q x) ==> (EX x. P x) --> (EX x. Q x)"
wenzelm@12897
  1835
  by rules
wenzelm@12897
  1836
wenzelm@12897
  1837
lemma all_mono: "(!!x. P x --> Q x) ==> (ALL x. P x) --> (ALL x. Q x)"
wenzelm@12897
  1838
  by rules
wenzelm@12897
  1839
wenzelm@12897
  1840
lemma Collect_mono: "(!!x. P x --> Q x) ==> Collect P \<subseteq> Collect Q"
wenzelm@12897
  1841
  by blast
wenzelm@12897
  1842
wenzelm@12897
  1843
lemma Int_Collect_mono:
wenzelm@12897
  1844
    "A \<subseteq> B ==> (!!x. x \<in> A ==> P x --> Q x) ==> A \<inter> Collect P \<subseteq> B \<inter> Collect Q"
wenzelm@12897
  1845
  by blast
wenzelm@12897
  1846
wenzelm@12897
  1847
lemmas basic_monos =
wenzelm@12897
  1848
  subset_refl imp_refl disj_mono conj_mono
wenzelm@12897
  1849
  ex_mono Collect_mono in_mono
wenzelm@12897
  1850
wenzelm@12897
  1851
lemma eq_to_mono: "a = b ==> c = d ==> b --> d ==> a --> c"
wenzelm@12897
  1852
  by rules
wenzelm@12897
  1853
wenzelm@12897
  1854
lemma eq_to_mono2: "a = b ==> c = d ==> ~ b --> ~ d ==> ~ a --> ~ c"
wenzelm@12897
  1855
  by rules
wenzelm@11979
  1856
wenzelm@11982
  1857
lemma Least_mono:
wenzelm@11982
  1858
  "mono (f::'a::order => 'b::order) ==> EX x:S. ALL y:S. x <= y
wenzelm@11982
  1859
    ==> (LEAST y. y : f ` S) = f (LEAST x. x : S)"
wenzelm@11982
  1860
    -- {* Courtesy of Stephan Merz *}
wenzelm@11982
  1861
  apply clarify
paulson@14208
  1862
  apply (erule_tac P = "%x. x : S" in LeastI2, fast)
wenzelm@11982
  1863
  apply (rule LeastI2)
wenzelm@11982
  1864
  apply (auto elim: monoD intro!: order_antisym)
wenzelm@11982
  1865
  done
wenzelm@11982
  1866
wenzelm@12020
  1867
wenzelm@12257
  1868
subsection {* Inverse image of a function *}
wenzelm@12257
  1869
wenzelm@12257
  1870
constdefs
wenzelm@12257
  1871
  vimage :: "('a => 'b) => 'b set => 'a set"    (infixr "-`" 90)
wenzelm@12257
  1872
  "f -` B == {x. f x : B}"
wenzelm@12257
  1873
wenzelm@12257
  1874
wenzelm@12257
  1875
subsubsection {* Basic rules *}
wenzelm@12257
  1876
wenzelm@12257
  1877
lemma vimage_eq [simp]: "(a : f -` B) = (f a : B)"
wenzelm@12257
  1878
  by (unfold vimage_def) blast
wenzelm@12257
  1879
wenzelm@12257
  1880
lemma vimage_singleton_eq: "(a : f -` {b}) = (f a = b)"
wenzelm@12257
  1881
  by simp
wenzelm@12257
  1882
wenzelm@12257
  1883
lemma vimageI [intro]: "f a = b ==> b:B ==> a : f -` B"
wenzelm@12257
  1884
  by (unfold vimage_def) blast
wenzelm@12257
  1885
wenzelm@12257
  1886
lemma vimageI2: "f a : A ==> a : f -` A"
wenzelm@12257
  1887
  by (unfold vimage_def) fast
wenzelm@12257
  1888
wenzelm@12257
  1889
lemma vimageE [elim!]: "a: f -` B ==> (!!x. f a = x ==> x:B ==> P) ==> P"
wenzelm@12257
  1890
  by (unfold vimage_def) blast
wenzelm@12257
  1891
wenzelm@12257
  1892
lemma vimageD: "a : f -` A ==> f a : A"
wenzelm@12257
  1893
  by (unfold vimage_def) fast
wenzelm@12257
  1894
wenzelm@12257
  1895
wenzelm@12257
  1896
subsubsection {* Equations *}
wenzelm@12257
  1897
wenzelm@12257
  1898
lemma vimage_empty [simp]: "f -` {} = {}"
wenzelm@12257
  1899
  by blast
wenzelm@12257
  1900
wenzelm@12257
  1901
lemma vimage_Compl: "f -` (-A) = -(f -` A)"
wenzelm@12257
  1902
  by blast
wenzelm@12257
  1903
wenzelm@12257
  1904
lemma vimage_Un [simp]: "f -` (A Un B) = (f -` A) Un (f -` B)"
wenzelm@12257
  1905
  by blast
wenzelm@12257
  1906
wenzelm@12257
  1907
lemma vimage_Int [simp]: "f -` (A Int B) = (f -` A) Int (f -` B)"
wenzelm@12257
  1908
  by fast
wenzelm@12257
  1909
wenzelm@12257
  1910
lemma vimage_Union: "f -` (Union A) = (UN X:A. f -` X)"
wenzelm@12257
  1911
  by blast
wenzelm@12257
  1912
wenzelm@12257
  1913
lemma vimage_UN: "f-`(UN x:A. B x) = (UN x:A. f -` B x)"
wenzelm@12257
  1914
  by blast
wenzelm@12257
  1915
wenzelm@12257
  1916
lemma vimage_INT: "f-`(INT x:A. B x) = (INT x:A. f -` B x)"
wenzelm@12257
  1917
  by blast
wenzelm@12257
  1918
wenzelm@12257
  1919
lemma vimage_Collect_eq [simp]: "f -` Collect P = {y. P (f y)}"
wenzelm@12257
  1920
  by blast
wenzelm@12257
  1921
wenzelm@12257
  1922
lemma vimage_Collect: "(!!x. P (f x) = Q x) ==> f -` (Collect P) = Collect Q"
wenzelm@12257
  1923
  by blast
wenzelm@12257
  1924
wenzelm@12257
  1925
lemma vimage_insert: "f-`(insert a B) = (f-`{a}) Un (f-`B)"
wenzelm@12257
  1926
  -- {* NOT suitable for rewriting because of the recurrence of @{term "{a}"}. *}
wenzelm@12257
  1927
  by blast
wenzelm@12257
  1928
wenzelm@12257
  1929
lemma vimage_Diff: "f -` (A - B) = (f -` A) - (f -` B)"
wenzelm@12257
  1930
  by blast
wenzelm@12257
  1931
wenzelm@12257
  1932
lemma vimage_UNIV [simp]: "f -` UNIV = UNIV"
wenzelm@12257
  1933
  by blast
wenzelm@12257
  1934
wenzelm@12257
  1935
lemma vimage_eq_UN: "f-`B = (UN y: B. f-`{y})"
wenzelm@12257
  1936
  -- {* NOT suitable for rewriting *}
wenzelm@12257
  1937
  by blast
wenzelm@12257
  1938
wenzelm@12897
  1939
lemma vimage_mono: "A \<subseteq> B ==> f -` A \<subseteq> f -` B"
wenzelm@12257
  1940
  -- {* monotonicity *}
wenzelm@12257
  1941
  by blast
wenzelm@12257
  1942
wenzelm@12257
  1943
wenzelm@12023
  1944
subsection {* Transitivity rules for calculational reasoning *}
wenzelm@12020
  1945
wenzelm@12020
  1946
lemma forw_subst: "a = b ==> P b ==> P a"
wenzelm@12020
  1947
  by (rule ssubst)
wenzelm@12020
  1948
wenzelm@12020
  1949
lemma back_subst: "P a ==> a = b ==> P b"
wenzelm@12020
  1950
  by (rule subst)
wenzelm@12020
  1951
wenzelm@12897
  1952
lemma set_rev_mp: "x:A ==> A \<subseteq> B ==> x:B"
wenzelm@12020
  1953
  by (rule subsetD)
wenzelm@12020
  1954
wenzelm@12897
  1955
lemma set_mp: "A \<subseteq> B ==> x:A ==> x:B"
wenzelm@12020
  1956
  by (rule subsetD)
wenzelm@12020
  1957
wenzelm@12020
  1958
lemma order_neq_le_trans: "a ~= b ==> (a::'a::order) <= b ==> a < b"
wenzelm@12020
  1959
  by (simp add: order_less_le)
wenzelm@12020
  1960
wenzelm@12020
  1961
lemma order_le_neq_trans: "(a::'a::order) <= b ==> a ~= b ==> a < b"
wenzelm@12020
  1962
  by (simp add: order_less_le)
wenzelm@12020
  1963
wenzelm@12020
  1964
lemma order_less_asym': "(a::'a::order) < b ==> b < a ==> P"
wenzelm@12020
  1965
  by (rule order_less_asym)
wenzelm@12020
  1966
wenzelm@12020
  1967
lemma ord_le_eq_trans: "a <= b ==> b = c ==> a <= c"
wenzelm@12020
  1968
  by (rule subst)
wenzelm@12020
  1969
wenzelm@12020
  1970
lemma ord_eq_le_trans: "a = b ==> b <= c ==> a <= c"
wenzelm@12020
  1971
  by (rule ssubst)
wenzelm@12020
  1972
wenzelm@12020
  1973
lemma ord_less_eq_trans: "a < b ==> b = c ==> a < c"
wenzelm@12020
  1974
  by (rule subst)
wenzelm@12020
  1975
wenzelm@12020
  1976
lemma ord_eq_less_trans: "a = b ==> b < c ==> a < c"
wenzelm@12020
  1977
  by (rule ssubst)
wenzelm@12020
  1978
wenzelm@12020
  1979
lemma order_less_subst2: "(a::'a::order) < b ==> f b < (c::'c::order) ==>
wenzelm@12020
  1980
  (!!x y. x < y ==> f x < f y) ==> f a < c"
wenzelm@12020
  1981
proof -
wenzelm@12020
  1982
  assume r: "!!x y. x < y ==> f x < f y"
wenzelm@12020
  1983
  assume "a < b" hence "f a < f b" by (rule r)
wenzelm@12020
  1984
  also assume "f b < c"
wenzelm@12020
  1985
  finally (order_less_trans) show ?thesis .
wenzelm@12020
  1986
qed
wenzelm@12020
  1987
wenzelm@12020
  1988
lemma order_less_subst1: "(a::'a::order) < f b ==> (b::'b::order) < c ==>
wenzelm@12020
  1989
  (!!x y. x < y ==> f x < f y) ==> a < f c"
wenzelm@12020
  1990
proof -
wenzelm@12020
  1991
  assume r: "!!x y. x < y ==> f x < f y"
wenzelm@12020
  1992
  assume "a < f b"
wenzelm@12020
  1993
  also assume "b < c" hence "f b < f c" by (rule r)
wenzelm@12020
  1994
  finally (order_less_trans) show ?thesis .
wenzelm@12020
  1995
qed
wenzelm@12020
  1996
wenzelm@12020
  1997
lemma order_le_less_subst2: "(a::'a::order) <= b ==> f b < (c::'c::order) ==>
wenzelm@12020
  1998
  (!!x y. x <= y ==> f x <= f y) ==> f a < c"
wenzelm@12020
  1999
proof -
wenzelm@12020
  2000
  assume r: "!!x y. x <= y ==> f x <= f y"
wenzelm@12020
  2001
  assume "a <= b" hence "f a <= f b" by (rule r)
wenzelm@12020
  2002
  also assume "f b < c"
wenzelm@12020
  2003
  finally (order_le_less_trans) show ?thesis .
wenzelm@12020
  2004
qed
wenzelm@12020
  2005
wenzelm@12020
  2006
lemma order_le_less_subst1: "(a::'a::order) <= f b ==> (b::'b::order) < c ==>
wenzelm@12020
  2007
  (!!x y. x < y ==> f x < f y) ==> a < f c"
wenzelm@12020
  2008
proof -
wenzelm@12020
  2009
  assume r: "!!x y. x < y ==> f x < f y"
wenzelm@12020
  2010
  assume "a <= f b"
wenzelm@12020
  2011
  also assume "b < c" hence "f b < f c" by (rule r)
wenzelm@12020
  2012
  finally (order_le_less_trans) show ?thesis .
wenzelm@12020
  2013
qed
wenzelm@12020
  2014
wenzelm@12020
  2015
lemma order_less_le_subst2: "(a::'a::order) < b ==> f b <= (c::'c::order) ==>
wenzelm@12020
  2016
  (!!x y. x < y ==> f x < f y) ==> f a < c"
wenzelm@12020
  2017
proof -
wenzelm@12020
  2018
  assume r: "!!x y. x < y ==> f x < f y"
wenzelm@12020
  2019
  assume "a < b" hence "f a < f b" by (rule r)
wenzelm@12020
  2020
  also assume "f b <= c"
wenzelm@12020
  2021
  finally (order_less_le_trans) show ?thesis .
wenzelm@12020
  2022
qed
wenzelm@12020
  2023
wenzelm@12020
  2024
lemma order_less_le_subst1: "(a::'a::order) < f b ==> (b::'b::order) <= c ==>
wenzelm@12020
  2025
  (!!x y. x <= y ==> f x <= f y) ==> a < f c"
wenzelm@12020
  2026
proof -
wenzelm@12020
  2027
  assume r: "!!x y. x <= y ==> f x <= f y"
wenzelm@12020
  2028
  assume "a < f b"
wenzelm@12020
  2029
  also assume "b <= c" hence "f b <= f c" by (rule r)
wenzelm@12020
  2030
  finally (order_less_le_trans) show ?thesis .
wenzelm@12020
  2031
qed
wenzelm@12020
  2032
wenzelm@12020
  2033
lemma order_subst1: "(a::'a::order) <= f b ==> (b::'b::order) <= c ==>
wenzelm@12020
  2034
  (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
wenzelm@12020
  2035
proof -
wenzelm@12020
  2036
  assume r: "!!x y. x <= y ==> f x <= f y"
wenzelm@12020
  2037
  assume "a <= f b"
wenzelm@12020
  2038
  also assume "b <= c" hence "f b <= f c" by (rule r)
wenzelm@12020
  2039
  finally (order_trans) show ?thesis .
wenzelm@12020
  2040
qed
wenzelm@12020
  2041
wenzelm@12020
  2042
lemma order_subst2: "(a::'a::order) <= b ==> f b <= (c::'c::order) ==>
wenzelm@12020
  2043
  (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
wenzelm@12020
  2044
proof -
wenzelm@12020
  2045
  assume r: "!!x y. x <= y ==> f x <= f y"
wenzelm@12020
  2046
  assume "a <= b" hence "f a <= f b" by (rule r)
wenzelm@12020
  2047
  also assume "f b <= c"
wenzelm@12020
  2048
  finally (order_trans) show ?thesis .
wenzelm@12020
  2049
qed
wenzelm@12020
  2050
wenzelm@12020
  2051
lemma ord_le_eq_subst: "a <= b ==> f b = c ==>
wenzelm@12020
  2052
  (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
wenzelm@12020
  2053
proof -
wenzelm@12020
  2054
  assume r: "!!x y. x <= y ==> f x <= f y"
wenzelm@12020
  2055
  assume "a <= b" hence "f a <= f b" by (rule r)
wenzelm@12020
  2056
  also assume "f b = c"
wenzelm@12020
  2057
  finally (ord_le_eq_trans) show ?thesis .
wenzelm@12020
  2058
qed
wenzelm@12020
  2059
wenzelm@12020
  2060
lemma ord_eq_le_subst: "a = f b ==> b <= c ==>
wenzelm@12020
  2061
  (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
wenzelm@12020
  2062
proof -
wenzelm@12020
  2063
  assume r: "!!x y. x <= y ==> f x <= f y"
wenzelm@12020
  2064
  assume "a = f b"
wenzelm@12020
  2065
  also assume "b <= c" hence "f b <= f c" by (rule r)
wenzelm@12020
  2066
  finally (ord_eq_le_trans) show ?thesis .
wenzelm@12020
  2067
qed
wenzelm@12020
  2068
wenzelm@12020
  2069
lemma ord_less_eq_subst: "a < b ==> f b = c ==>
wenzelm@12020
  2070
  (!!x y. x < y ==> f x < f y) ==> f a < c"
wenzelm@12020
  2071
proof -
wenzelm@12020
  2072
  assume r: "!!x y. x < y ==> f x < f y"
wenzelm@12020
  2073
  assume "a < b" hence "f a < f b" by (rule r)
wenzelm@12020
  2074
  also assume "f b = c"
wenzelm@12020
  2075
  finally (ord_less_eq_trans) show ?thesis .
wenzelm@12020
  2076
qed
wenzelm@12020
  2077
wenzelm@12020
  2078
lemma ord_eq_less_subst: "a = f b ==> b < c ==>
wenzelm@12020
  2079
  (!!x y. x < y ==> f x < f y) ==> a < f c"
wenzelm@12020
  2080
proof -
wenzelm@12020
  2081
  assume r: "!!x y. x < y ==> f x < f y"
wenzelm@12020
  2082
  assume "a = f b"
wenzelm@12020
  2083
  also assume "b < c" hence "f b < f c" by (rule r)
wenzelm@12020
  2084
  finally (ord_eq_less_trans) show ?thesis .
wenzelm@12020
  2085
qed
wenzelm@12020
  2086
wenzelm@12020
  2087
text {*
wenzelm@12020
  2088
  Note that this list of rules is in reverse order of priorities.
wenzelm@12020
  2089
*}
wenzelm@12020
  2090
wenzelm@12020
  2091
lemmas basic_trans_rules [trans] =
wenzelm@12020
  2092
  order_less_subst2
wenzelm@12020
  2093
  order_less_subst1
wenzelm@12020
  2094
  order_le_less_subst2
wenzelm@12020
  2095
  order_le_less_subst1
wenzelm@12020
  2096
  order_less_le_subst2
wenzelm@12020
  2097
  order_less_le_subst1
wenzelm@12020
  2098
  order_subst2
wenzelm@12020
  2099
  order_subst1
wenzelm@12020
  2100
  ord_le_eq_subst
wenzelm@12020
  2101
  ord_eq_le_subst
wenzelm@12020
  2102
  ord_less_eq_subst
wenzelm@12020
  2103
  ord_eq_less_subst
wenzelm@12020
  2104
  forw_subst
wenzelm@12020
  2105
  back_subst
wenzelm@12020
  2106
  rev_mp
wenzelm@12020
  2107
  mp
wenzelm@12020
  2108
  set_rev_mp
wenzelm@12020
  2109
  set_mp
wenzelm@12020
  2110
  order_neq_le_trans
wenzelm@12020
  2111
  order_le_neq_trans
wenzelm@12020
  2112
  order_less_trans
wenzelm@12020
  2113
  order_less_asym'
wenzelm@12020
  2114
  order_le_less_trans
wenzelm@12020
  2115
  order_less_le_trans
wenzelm@12020
  2116
  order_trans
wenzelm@12020
  2117
  order_antisym
wenzelm@12020
  2118
  ord_le_eq_trans
wenzelm@12020
  2119
  ord_eq_le_trans
wenzelm@12020
  2120
  ord_less_eq_trans
wenzelm@12020
  2121
  ord_eq_less_trans
wenzelm@12020
  2122
  trans
wenzelm@12020
  2123
wenzelm@11979
  2124
end