src/HOL/Set.thy
author nipkow
Fri, 18 Oct 2002 09:53:02 +0200
changeset 13653 ef123b9e8089
parent 13624 17684cf64fda
child 13763 f94b569cd610
permissions -rw-r--r--
Added a few thms about UN/INT/{}/UNIV
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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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  "INT x y. B"  == "INT x. INT y. B"
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  "INT x. B"    == "INTER UNIV (%x. 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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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 (_, _, 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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              if 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, [])) then ()
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              else raise Match;
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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 check (P, 0); tr' P 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 set_ext: assumes prem: "(!!x. (x:A) = (x:B))" shows "A = B"
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  apply (rule prem [THEN ext, THEN arg_cong, THEN box_equals])
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   apply (rule Collect_mem_eq)
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  apply (rule Collect_mem_eq)
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  done
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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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text {*
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  \medskip This tactic takes assumptions @{prop "ALL x:A. P x"} and
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  @{prop "a:A"}; creates assumption @{prop "P a"}.
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*}
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ML {*
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  local val ballE = thm "ballE"
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  in fun ball_tac i = etac ballE i THEN contr_tac (i + 1) end;
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*}
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text {*
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  Gives better instantiation for bound:
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*}
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ML_setup {*
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  claset_ref() := claset() addbefore ("bspec", datac (thm "bspec") 1);
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*}
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lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x"
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  -- {* Normally the best argument order: @{prop "P x"} constrains the
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    choice of @{prop "x:A"}. *}
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  by (unfold Bex_def) blast
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lemma rev_bexI [intro?]: "x:A ==> P x ==> EX x:A. P x"
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  -- {* The best argument order when there is only one @{prop "x:A"}. *}
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  by (unfold Bex_def) blast
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lemma bexCI: "(ALL x:A. ~P x ==> P a) ==> a:A ==> EX x:A. P x"
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  by (unfold Bex_def) blast
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lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q"
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  by (unfold Bex_def) blast
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lemma ball_triv [simp]: "(ALL x:A. P) = ((EX x. x:A) --> P)"
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  -- {* Trival rewrite rule. *}
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  by (simp add: Ball_def)
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lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)"
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  -- {* Dual form for existentials. *}
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  by (simp add: Bex_def)
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lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)"
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  by blast
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lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)"
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  by blast
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lemma bex_one_point1 [simp]: "(EX x:A. x = a & P x) = (a:A & P a)"
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  by blast
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lemma bex_one_point2 [simp]: "(EX x:A. a = x & P x) = (a:A & P a)"
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  by blast
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lemma ball_one_point1 [simp]: "(ALL x:A. x = a --> P x) = (a:A --> P a)"
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  by blast
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lemma ball_one_point2 [simp]: "(ALL x:A. a = x --> P x) = (a:A --> P a)"
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  by blast
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ML_setup {*
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  local
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    val Ball_def = thm "Ball_def";
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    val Bex_def = thm "Bex_def";
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    val prove_bex_tac =
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      rewrite_goals_tac [Bex_def] THEN Quantifier1.prove_one_point_ex_tac;
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    val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac;
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    val prove_ball_tac =
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      rewrite_goals_tac [Ball_def] THEN Quantifier1.prove_one_point_all_tac;
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    val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac;
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  in
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    val defBEX_regroup = Simplifier.simproc (Theory.sign_of (the_context ()))
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      "defined BEX" ["EX x:A. P x & Q x"] rearrange_bex;
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    val defBALL_regroup = Simplifier.simproc (Theory.sign_of (the_context ()))
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      "defined BALL" ["ALL x:A. P x --> Q x"] rearrange_ball;
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  end;
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  Addsimprocs [defBALL_regroup, defBEX_regroup];
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*}
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subsubsection {* Congruence rules *}
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lemma ball_cong [cong]:
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  "A = B ==> (!!x. x:B ==> P x = Q x) ==>
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    (ALL x:A. P x) = (ALL x:B. Q x)"
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  by (simp add: Ball_def)
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lemma bex_cong [cong]:
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  "A = B ==> (!!x. x:B ==> P x = Q x) ==>
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    (EX x:A. P x) = (EX x:B. Q x)"
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  by (simp add: Bex_def cong: conj_cong)
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subsubsection {* Subsets *}
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lemma subsetI [intro!]: "(!!x. x:A ==> x:B) ==> A \<subseteq> B"
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  by (simp add: subset_def)
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text {*
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  \medskip Map the type @{text "'a set => anything"} to just @{typ
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  'a}; for overloading constants whose first argument has type @{typ
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  "'a set"}.
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*}
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lemma subsetD [elim]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B"
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  -- {* Rule in Modus Ponens style. *}
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  by (unfold subset_def) blast
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declare subsetD [intro?] -- FIXME
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lemma rev_subsetD: "c \<in> A ==> A \<subseteq> B ==> c \<in> B"
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  -- {* The same, with reversed premises for use with @{text erule} --
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      cf @{text rev_mp}. *}
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  by (rule subsetD)
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declare rev_subsetD [intro?] -- FIXME
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text {*
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  \medskip Converts @{prop "A \<subseteq> B"} to @{prop "x \<in> A ==> x \<in> B"}.
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*}
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ML {*
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   364
  local val rev_subsetD = thm "rev_subsetD"
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  in fun impOfSubs th = th RSN (2, rev_subsetD) end;
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*}
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lemma subsetCE [elim]: "A \<subseteq>  B ==> (c \<notin> A ==> P) ==> (c \<in> B ==> P) ==> P"
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  -- {* Classical elimination rule. *}
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  by (unfold subset_def) blast
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text {*
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  \medskip Takes assumptions @{prop "A \<subseteq> B"}; @{prop "c \<in> A"} and
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  creates the assumption @{prop "c \<in> B"}.
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*}
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ML {*
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   378
  local val subsetCE = thm "subsetCE"
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  in fun set_mp_tac i = etac subsetCE i THEN mp_tac i end;
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*}
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lemma contra_subsetD: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A"
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  by blast
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lemma subset_refl: "A \<subseteq> A"
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  by fast
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lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C"
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  by blast
923
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2261
d926157c0a6a added "op :", "op ~:" syntax;
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subsubsection {* Equality *}
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   393
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lemma subset_antisym [intro!]: "A \<subseteq> B ==> B \<subseteq> A ==> A = B"
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  -- {* Anti-symmetry of the subset relation. *}
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  by (rules intro: set_ext subsetD)
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lemmas equalityI [intro!] = subset_antisym
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text {*
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   401
  \medskip Equality rules from ZF set theory -- are they appropriate
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   402
  here?
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   403
*}
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   404
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   405
lemma equalityD1: "A = B ==> A \<subseteq> B"
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   406
  by (simp add: subset_refl)
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   407
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   408
lemma equalityD2: "A = B ==> B \<subseteq> A"
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   409
  by (simp add: subset_refl)
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   410
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   411
text {*
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   412
  \medskip Be careful when adding this to the claset as @{text
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   413
  subset_empty} is in the simpset: @{prop "A = {}"} goes to @{prop "{}
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diff changeset
   414
  \<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}!
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   415
*}
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   416
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   417
lemma equalityE: "A = B ==> (A \<subseteq> B ==> B \<subseteq> A ==> P) ==> P"
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   418
  by (simp add: subset_refl)
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   420
lemma equalityCE [elim]:
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diff changeset
   421
    "A = B ==> (c \<in> A ==> c \<in> B ==> P) ==> (c \<notin> A ==> c \<notin> B ==> P) ==> P"
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diff changeset
   422
  by blast
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   423
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   424
text {*
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diff changeset
   425
  \medskip Lemma for creating induction formulae -- for "pattern
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diff changeset
   426
  matching" on @{text p}.  To make the induction hypotheses usable,
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diff changeset
   427
  apply @{text spec} or @{text bspec} to put universal quantifiers over the free
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diff changeset
   428
  variables in @{text p}.
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diff changeset
   429
*}
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   430
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diff changeset
   431
lemma setup_induction: "p:A ==> (!!z. z:A ==> p = z --> R) ==> R"
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diff changeset
   432
  by simp
923
ff1574a81019 new version of HOL with curried function application
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   434
lemma eqset_imp_iff: "A = B ==> (x : A) = (x : B)"
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diff changeset
   435
  by simp
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   436
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   437
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   438
subsubsection {* The universal set -- UNIV *}
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   439
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   440
lemma UNIV_I [simp]: "x : UNIV"
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   441
  by (simp add: UNIV_def)
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   442
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   443
declare UNIV_I [intro]  -- {* unsafe makes it less likely to cause problems *}
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   444
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   445
lemma UNIV_witness [intro?]: "EX x. x : UNIV"
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diff changeset
   446
  by simp
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   447
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lemma subset_UNIV: "A \<subseteq> UNIV"
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diff changeset
   449
  by (rule subsetI) (rule UNIV_I)
2388
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diff changeset
   450
11979
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   451
text {*
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diff changeset
   452
  \medskip Eta-contracting these two rules (to remove @{text P})
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diff changeset
   453
  causes them to be ignored because of their interaction with
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diff changeset
   454
  congruence rules.
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diff changeset
   455
*}
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diff changeset
   456
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diff changeset
   457
lemma ball_UNIV [simp]: "Ball UNIV P = All P"
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diff changeset
   458
  by (simp add: Ball_def)
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diff changeset
   459
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   460
lemma bex_UNIV [simp]: "Bex UNIV P = Ex P"
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diff changeset
   461
  by (simp add: Bex_def)
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   462
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   463
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   464
subsubsection {* The empty set *}
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   465
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   466
lemma empty_iff [simp]: "(c : {}) = False"
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diff changeset
   467
  by (simp add: empty_def)
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diff changeset
   468
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diff changeset
   469
lemma emptyE [elim!]: "a : {} ==> P"
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diff changeset
   470
  by simp
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   471
12897
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   472
lemma empty_subsetI [iff]: "{} \<subseteq> A"
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   473
    -- {* One effect is to delete the ASSUMPTION @{prop "{} <= A"} *}
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   474
  by blast
0a3dace545c5 converted theory "Set";
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diff changeset
   475
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   476
lemma equals0I: "(!!y. y \<in> A ==> False) ==> A = {}"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   477
  by blast
2388
d1f0505fc602 added set inclusion symbol syntax;
wenzelm
parents: 2372
diff changeset
   478
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   479
lemma equals0D: "A = {} ==> a \<notin> A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   480
    -- {* Use for reasoning about disjointness: @{prop "A Int B = {}"} *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   481
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   482
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   483
lemma ball_empty [simp]: "Ball {} P = True"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   484
  by (simp add: Ball_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   485
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   486
lemma bex_empty [simp]: "Bex {} P = False"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   487
  by (simp add: Bex_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   488
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   489
lemma UNIV_not_empty [iff]: "UNIV ~= {}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   490
  by (blast elim: equalityE)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   491
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   492
12023
wenzelm
parents: 12020
diff changeset
   493
subsubsection {* The Powerset operator -- Pow *}
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   494
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   495
lemma Pow_iff [iff]: "(A \<in> Pow B) = (A \<subseteq> B)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   496
  by (simp add: Pow_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   497
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   498
lemma PowI: "A \<subseteq> B ==> A \<in> Pow B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   499
  by (simp add: Pow_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   500
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   501
lemma PowD: "A \<in> Pow B ==> A \<subseteq> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   502
  by (simp add: Pow_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   503
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   504
lemma Pow_bottom: "{} \<in> Pow B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   505
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   506
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   507
lemma Pow_top: "A \<in> Pow A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   508
  by (simp add: subset_refl)
2684
9781d63ef063 added proper subset symbols syntax;
wenzelm
parents: 2412
diff changeset
   509
2388
d1f0505fc602 added set inclusion symbol syntax;
wenzelm
parents: 2372
diff changeset
   510
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   511
subsubsection {* Set complement *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   512
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   513
lemma Compl_iff [simp]: "(c \<in> -A) = (c \<notin> A)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   514
  by (unfold Compl_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   515
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   516
lemma ComplI [intro!]: "(c \<in> A ==> False) ==> c \<in> -A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   517
  by (unfold Compl_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   518
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   519
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   520
  \medskip This form, with negated conclusion, works well with the
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   521
  Classical prover.  Negated assumptions behave like formulae on the
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   522
  right side of the notional turnstile ... *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   523
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   524
lemma ComplD: "c : -A ==> c~:A"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   525
  by (unfold Compl_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   526
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   527
lemmas ComplE [elim!] = ComplD [elim_format]
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   528
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   529
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   530
subsubsection {* Binary union -- Un *}
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   531
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   532
lemma Un_iff [simp]: "(c : A Un B) = (c:A | c:B)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   533
  by (unfold Un_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   534
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   535
lemma UnI1 [elim?]: "c:A ==> c : A Un B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   536
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   537
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   538
lemma UnI2 [elim?]: "c:B ==> c : A Un B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   539
  by simp
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   540
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   541
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   542
  \medskip Classical introduction rule: no commitment to @{prop A} vs
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   543
  @{prop B}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   544
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   545
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   546
lemma UnCI [intro!]: "(c~:B ==> c:A) ==> c : A Un B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   547
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   548
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   549
lemma UnE [elim!]: "c : A Un B ==> (c:A ==> P) ==> (c:B ==> P) ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   550
  by (unfold Un_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   551
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   552
12023
wenzelm
parents: 12020
diff changeset
   553
subsubsection {* Binary intersection -- Int *}
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   554
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   555
lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   556
  by (unfold Int_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   557
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   558
lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   559
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   560
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   561
lemma IntD1: "c : A Int B ==> c:A"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   562
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   563
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   564
lemma IntD2: "c : A Int B ==> c:B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   565
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   566
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   567
lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   568
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   569
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   570
12023
wenzelm
parents: 12020
diff changeset
   571
subsubsection {* Set difference *}
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   572
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   573
lemma Diff_iff [simp]: "(c : A - B) = (c:A & c~:B)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   574
  by (unfold set_diff_def) blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   575
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   576
lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A - B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   577
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   578
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   579
lemma DiffD1: "c : A - B ==> c : A"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   580
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   581
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   582
lemma DiffD2: "c : A - B ==> c : B ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   583
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   584
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   585
lemma DiffE [elim!]: "c : A - B ==> (c:A ==> c~:B ==> P) ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   586
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   587
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   588
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   589
subsubsection {* Augmenting a set -- insert *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   590
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   591
lemma insert_iff [simp]: "(a : insert b A) = (a = b | a:A)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   592
  by (unfold insert_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   593
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   594
lemma insertI1: "a : insert a B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   595
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   596
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   597
lemma insertI2: "a : B ==> a : insert b B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   598
  by simp
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   599
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   600
lemma insertE [elim!]: "a : insert b A ==> (a = b ==> P) ==> (a:A ==> P) ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   601
  by (unfold insert_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   602
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   603
lemma insertCI [intro!]: "(a~:B ==> a = b) ==> a: insert b B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   604
  -- {* Classical introduction rule. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   605
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   606
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   607
lemma subset_insert_iff: "(A \<subseteq> insert x B) = (if x:A then A - {x} \<subseteq> B else A \<subseteq> B)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   608
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   609
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   610
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   611
subsubsection {* Singletons, using insert *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   612
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   613
lemma singletonI [intro!]: "a : {a}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   614
    -- {* Redundant? But unlike @{text insertCI}, it proves the subgoal immediately! *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   615
  by (rule insertI1)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   616
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   617
lemma singletonD: "b : {a} ==> b = a"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   618
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   619
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   620
lemmas singletonE [elim!] = singletonD [elim_format]
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   621
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   622
lemma singleton_iff: "(b : {a}) = (b = a)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   623
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   624
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   625
lemma singleton_inject [dest!]: "{a} = {b} ==> a = b"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   626
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   627
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   628
lemma singleton_insert_inj_eq [iff]: "({b} = insert a A) = (a = b & A \<subseteq> {b})"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   629
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   630
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   631
lemma singleton_insert_inj_eq' [iff]: "(insert a A = {b}) = (a = b & A \<subseteq> {b})"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   632
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   633
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   634
lemma subset_singletonD: "A \<subseteq> {x} ==> A = {} | A = {x}"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   635
  by fast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   636
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   637
lemma singleton_conv [simp]: "{x. x = a} = {a}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   638
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   639
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   640
lemma singleton_conv2 [simp]: "{x. a = x} = {a}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   641
  by blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   642
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   643
lemma diff_single_insert: "A - {x} \<subseteq> B ==> x \<in> A ==> A \<subseteq> insert x B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   644
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   645
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   646
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   647
subsubsection {* Unions of families *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   648
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   649
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   650
  @{term [source] "UN x:A. B x"} is @{term "Union (B`A)"}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   651
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   652
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   653
lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   654
  by (unfold UNION_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   655
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   656
lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   657
  -- {* The order of the premises presupposes that @{term A} is rigid;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   658
    @{term b} may be flexible. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   659
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   660
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   661
lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   662
  by (unfold UNION_def) blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   663
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   664
lemma UN_cong [cong]:
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   665
    "A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   666
  by (simp add: UNION_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   667
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   668
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   669
subsubsection {* Intersections of families *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   670
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   671
text {* @{term [source] "INT x:A. B x"} is @{term "Inter (B`A)"}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   672
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   673
lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   674
  by (unfold INTER_def) blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   675
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   676
lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   677
  by (unfold INTER_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   678
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   679
lemma INT_D [elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   680
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   681
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   682
lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   683
  -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   684
  by (unfold INTER_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   685
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   686
lemma INT_cong [cong]:
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   687
    "A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   688
  by (simp add: INTER_def)
7238
36e58620ffc8 replaced HOL_quantifiers flag by "HOL" print mode;
wenzelm
parents: 5931
diff changeset
   689
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   690
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   691
subsubsection {* Union *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   692
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   693
lemma Union_iff [simp]: "(A : Union C) = (EX X:C. A:X)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   694
  by (unfold Union_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   695
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   696
lemma UnionI [intro]: "X:C ==> A:X ==> A : Union C"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   697
  -- {* The order of the premises presupposes that @{term C} is rigid;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   698
    @{term A} may be flexible. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   699
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   700
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   701
lemma UnionE [elim!]: "A : Union C ==> (!!X. A:X ==> X:C ==> R) ==> R"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   702
  by (unfold Union_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   703
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   704
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   705
subsubsection {* Inter *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   706
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   707
lemma Inter_iff [simp]: "(A : Inter C) = (ALL X:C. A:X)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   708
  by (unfold Inter_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   709
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   710
lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   711
  by (simp add: Inter_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   712
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   713
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   714
  \medskip A ``destruct'' rule -- every @{term X} in @{term C}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   715
  contains @{term A} as an element, but @{prop "A:X"} can hold when
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   716
  @{prop "X:C"} does not!  This rule is analogous to @{text spec}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   717
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   718
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   719
lemma InterD [elim]: "A : Inter C ==> X:C ==> A:X"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   720
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   721
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   722
lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   723
  -- {* ``Classical'' elimination rule -- does not require proving
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   724
    @{prop "X:C"}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   725
  by (unfold Inter_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   726
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   727
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   728
  \medskip Image of a set under a function.  Frequently @{term b} does
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   729
  not have the syntactic form of @{term "f x"}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   730
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   731
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   732
lemma image_eqI [simp, intro]: "b = f x ==> x:A ==> b : f`A"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   733
  by (unfold image_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   734
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   735
lemma imageI: "x : A ==> f x : f ` A"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   736
  by (rule image_eqI) (rule refl)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   737
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   738
lemma rev_image_eqI: "x:A ==> b = f x ==> b : f`A"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   739
  -- {* This version's more effective when we already have the
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   740
    required @{term x}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   741
  by (unfold image_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   742
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   743
lemma imageE [elim!]:
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   744
  "b : (%x. f x)`A ==> (!!x. b = f x ==> x:A ==> P) ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   745
  -- {* The eta-expansion gives variable-name preservation. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   746
  by (unfold image_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   747
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   748
lemma image_Un: "f`(A Un B) = f`A Un f`B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   749
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   750
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   751
lemma image_iff: "(z : f`A) = (EX x:A. z = f x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   752
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   753
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   754
lemma image_subset_iff: "(f`A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   755
  -- {* This rewrite rule would confuse users if made default. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   756
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   757
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   758
lemma subset_image_iff: "(B \<subseteq> f`A) = (EX AA. AA \<subseteq> A & B = f`AA)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   759
  apply safe
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   760
   prefer 2 apply fast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   761
  apply (rule_tac x = "{a. a : A & f a : B}" in exI)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   762
  apply fast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   763
  done
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   764
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   765
lemma image_subsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f`A \<subseteq> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   766
  -- {* Replaces the three steps @{text subsetI}, @{text imageE},
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   767
    @{text hypsubst}, but breaks too many existing proofs. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   768
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   769
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   770
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   771
  \medskip Range of a function -- just a translation for image!
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   772
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   773
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   774
lemma range_eqI: "b = f x ==> b \<in> range f"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   775
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   776
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   777
lemma rangeI: "f x \<in> range f"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   778
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   779
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   780
lemma rangeE [elim?]: "b \<in> range (\<lambda>x. f x) ==> (!!x. b = f x ==> P) ==> P"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   781
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   782
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   783
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   784
subsubsection {* Set reasoning tools *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   785
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   786
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   787
  Rewrite rules for boolean case-splitting: faster than @{text
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   788
  "split_if [split]"}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   789
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   790
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   791
lemma split_if_eq1: "((if Q then x else y) = b) = ((Q --> x = b) & (~ Q --> y = b))"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   792
  by (rule split_if)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   793
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   794
lemma split_if_eq2: "(a = (if Q then x else y)) = ((Q --> a = x) & (~ Q --> a = y))"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   795
  by (rule split_if)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   796
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   797
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   798
  Split ifs on either side of the membership relation.  Not for @{text
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   799
  "[simp]"} -- can cause goals to blow up!
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   800
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   801
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   802
lemma split_if_mem1: "((if Q then x else y) : b) = ((Q --> x : b) & (~ Q --> y : b))"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   803
  by (rule split_if)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   804
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   805
lemma split_if_mem2: "(a : (if Q then x else y)) = ((Q --> a : x) & (~ Q --> a : y))"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   806
  by (rule split_if)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   807
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   808
lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   809
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   810
lemmas mem_simps =
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   811
  insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   812
  mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   813
  -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   814
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   815
(*Would like to add these, but the existing code only searches for the
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   816
  outer-level constant, which in this case is just "op :"; we instead need
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   817
  to use term-nets to associate patterns with rules.  Also, if a rule fails to
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   818
  apply, then the formula should be kept.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   819
  [("uminus", Compl_iff RS iffD1), ("op -", [Diff_iff RS iffD1]),
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   820
   ("op Int", [IntD1,IntD2]),
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   821
   ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   822
 *)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   823
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   824
ML_setup {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   825
  val mksimps_pairs = [("Ball", [thm "bspec"])] @ mksimps_pairs;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   826
  simpset_ref() := simpset() setmksimps (mksimps mksimps_pairs);
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   827
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   828
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   829
declare subset_UNIV [simp] subset_refl [simp]
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   830
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   831
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   832
subsubsection {* The ``proper subset'' relation *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   833
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   834
lemma psubsetI [intro!]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   835
  by (unfold psubset_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   836
13624
17684cf64fda added the new elim rule psubsetE
paulson
parents: 13550
diff changeset
   837
lemma psubsetE [elim!]: 
17684cf64fda added the new elim rule psubsetE
paulson
parents: 13550
diff changeset
   838
    "[|A \<subset> B;  [|A \<subseteq> B; ~ (B\<subseteq>A)|] ==> R|] ==> R"
17684cf64fda added the new elim rule psubsetE
paulson
parents: 13550
diff changeset
   839
  by (unfold psubset_def) blast
17684cf64fda added the new elim rule psubsetE
paulson
parents: 13550
diff changeset
   840
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   841
lemma psubset_insert_iff:
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   842
  "(A \<subset> insert x B) = (if x \<in> B then A \<subset> B else if x \<in> A then A - {x} \<subset> B else A \<subseteq> B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   843
  by (auto simp add: psubset_def subset_insert_iff)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   844
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   845
lemma psubset_eq: "(A \<subset> B) = (A \<subseteq> B & A \<noteq> B)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   846
  by (simp only: psubset_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   847
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   848
lemma psubset_imp_subset: "A \<subset> B ==> A \<subseteq> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   849
  by (simp add: psubset_eq)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   850
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   851
lemma psubset_subset_trans: "A \<subset> B ==> B \<subseteq> C ==> A \<subset> C"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   852
  by (auto simp add: psubset_eq)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   853
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   854
lemma subset_psubset_trans: "A \<subseteq> B ==> B \<subset> C ==> A \<subset> C"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   855
  by (auto simp add: psubset_eq)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   856
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   857
lemma psubset_imp_ex_mem: "A \<subset> B ==> \<exists>b. b \<in> (B - A)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   858
  by (unfold psubset_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   859
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   860
lemma atomize_ball:
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   861
    "(!!x. x \<in> A ==> P x) == Trueprop (\<forall>x\<in>A. P x)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   862
  by (simp only: Ball_def atomize_all atomize_imp)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   863
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   864
declare atomize_ball [symmetric, rulify]
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   865
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   866
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   867
subsection {* Further set-theory lemmas *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   868
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   869
subsubsection {* Derived rules involving subsets. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   870
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   871
text {* @{text insert}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   872
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   873
lemma subset_insertI: "B \<subseteq> insert a B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   874
  apply (rule subsetI)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   875
  apply (erule insertI2)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   876
  done
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   877
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   878
lemma subset_insert: "x \<notin> A ==> (A \<subseteq> insert x B) = (A \<subseteq> B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   879
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   880
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   881
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   882
text {* \medskip Big Union -- least upper bound of a set. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   883
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   884
lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   885
  by (rules intro: subsetI UnionI)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   886
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   887
lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   888
  by (rules intro: subsetI elim: UnionE dest: subsetD)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   889
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   890
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   891
text {* \medskip General union. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   892
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   893
lemma UN_upper: "a \<in> A ==> B a \<subseteq> (\<Union>x\<in>A. B x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   894
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   895
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   896
lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   897
  by (rules intro: subsetI elim: UN_E dest: subsetD)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   898
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   899
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   900
text {* \medskip Big Intersection -- greatest lower bound of a set. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   901
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   902
lemma Inter_lower: "B \<in> A ==> Inter A \<subseteq> B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   903
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   904
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   905
lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   906
  by (rules intro: InterI subsetI dest: subsetD)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   907
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   908
lemma INT_lower: "a \<in> A ==> (\<Inter>x\<in>A. B x) \<subseteq> B a"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   909
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   910
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   911
lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   912
  by (rules intro: INT_I subsetI dest: subsetD)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   913
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   914
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   915
text {* \medskip Finite Union -- the least upper bound of two sets. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   916
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   917
lemma Un_upper1: "A \<subseteq> A \<union> B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   918
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   919
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   920
lemma Un_upper2: "B \<subseteq> A \<union> B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   921
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   922
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   923
lemma Un_least: "A \<subseteq> C ==> B \<subseteq> C ==> A \<union> B \<subseteq> C"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   924
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   925
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   926
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   927
text {* \medskip Finite Intersection -- the greatest lower bound of two sets. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   928
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   929
lemma Int_lower1: "A \<inter> B \<subseteq> A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   930
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   931
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   932
lemma Int_lower2: "A \<inter> B \<subseteq> B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   933
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   934
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   935
lemma Int_greatest: "C \<subseteq> A ==> C \<subseteq> B ==> C \<subseteq> A \<inter> B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   936
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   937
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   938
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   939
text {* \medskip Set difference. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   940
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   941
lemma Diff_subset: "A - B \<subseteq> A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   942
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   943
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   944
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   945
text {* \medskip Monotonicity. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   946
13421
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13113
diff changeset
   947
lemma mono_Un: includes mono shows "f A \<union> f B \<subseteq> f (A \<union> B)"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   948
  apply (rule Un_least)
13421
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13113
diff changeset
   949
   apply (rule Un_upper1 [THEN mono])
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13113
diff changeset
   950
  apply (rule Un_upper2 [THEN mono])
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   951
  done
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   952
13421
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13113
diff changeset
   953
lemma mono_Int: includes mono shows "f (A \<inter> B) \<subseteq> f A \<inter> f B"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   954
  apply (rule Int_greatest)
13421
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13113
diff changeset
   955
   apply (rule Int_lower1 [THEN mono])
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13113
diff changeset
   956
  apply (rule Int_lower2 [THEN mono])
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   957
  done
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   958
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   959
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   960
subsubsection {* Equalities involving union, intersection, inclusion, etc. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   961
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   962
text {* @{text "{}"}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   963
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   964
lemma Collect_const [simp]: "{s. P} = (if P then UNIV else {})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   965
  -- {* supersedes @{text "Collect_False_empty"} *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   966
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   967
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   968
lemma subset_empty [simp]: "(A \<subseteq> {}) = (A = {})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   969
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   970
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   971
lemma not_psubset_empty [iff]: "\<not> (A < {})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   972
  by (unfold psubset_def) blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   973
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   974
lemma Collect_empty_eq [simp]: "(Collect P = {}) = (\<forall>x. \<not> P x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   975
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   976
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   977
lemma Collect_neg_eq: "{x. \<not> P x} = - {x. P x}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   978
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   979
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   980
lemma Collect_disj_eq: "{x. P x | Q x} = {x. P x} \<union> {x. Q x}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   981
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   982
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   983
lemma Collect_conj_eq: "{x. P x & Q x} = {x. P x} \<inter> {x. Q x}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   984
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   985
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   986
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   987
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   988
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   989
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   990
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   991
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   992
lemma Collect_ex_eq: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   993
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   994
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   995
lemma Collect_bex_eq: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   996
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   997
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   998
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   999
text {* \medskip @{text insert}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1000
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1001
lemma insert_is_Un: "insert a A = {a} Un A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1002
  -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a {}"} *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1003
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1004
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1005
lemma insert_not_empty [simp]: "insert a A \<noteq> {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1006
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1007
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1008
lemmas empty_not_insert [simp] = insert_not_empty [symmetric, standard]
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1009
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1010
lemma insert_absorb: "a \<in> A ==> insert a A = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1011
  -- {* @{text "[simp]"} causes recursive calls when there are nested inserts *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1012
  -- {* with \emph{quadratic} running time *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1013
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1014
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1015
lemma insert_absorb2 [simp]: "insert x (insert x A) = insert x A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1016
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1017
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1018
lemma insert_commute: "insert x (insert y A) = insert y (insert x A)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1019
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1020
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1021
lemma insert_subset [simp]: "(insert x A \<subseteq> B) = (x \<in> B & A \<subseteq> B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1022
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1023
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1024
lemma mk_disjoint_insert: "a \<in> A ==> \<exists>B. A = insert a B & a \<notin> B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1025
  -- {* use new @{text B} rather than @{text "A - {a}"} to avoid infinite unfolding *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1026
  apply (rule_tac x = "A - {a}" in exI)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1027
  apply blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1028
  done
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1029
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1030
lemma insert_Collect: "insert a (Collect P) = {u. u \<noteq> a --> P u}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1031
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1032
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1033
lemma UN_insert_distrib: "u \<in> A ==> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1034
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1035
13103
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1036
lemma insert_disjoint[simp]:
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1037
 "(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})"
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1038
by blast
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1039
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1040
lemma disjoint_insert[simp]:
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1041
 "(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})"
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1042
by blast
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1043
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1044
text {* \medskip @{text image}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1045
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1046
lemma image_empty [simp]: "f`{} = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1047
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1048
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1049
lemma image_insert [simp]: "f ` insert a B = insert (f a) (f`B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1050
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1051
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1052
lemma image_constant: "x \<in> A ==> (\<lambda>x. c) ` A = {c}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1053
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1054
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1055
lemma image_image: "f ` (g ` A) = (\<lambda>x. f (g x)) ` A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1056
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1057
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1058
lemma insert_image [simp]: "x \<in> A ==> insert (f x) (f`A) = f`A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1059
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1060
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1061
lemma image_is_empty [iff]: "(f`A = {}) = (A = {})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1062
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1063
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1064
lemma image_Collect: "f ` {x. P x} = {f x | x. P x}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1065
  -- {* NOT suitable as a default simprule: the RHS isn't simpler than the LHS, *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1066
  -- {* with its implicit quantifier and conjunction.  Also image enjoys better *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1067
  -- {* equational properties than does the RHS. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1068
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1069
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1070
lemma if_image_distrib [simp]:
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1071
  "(\<lambda>x. if P x then f x else g x) ` S
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1072
    = (f ` (S \<inter> {x. P x})) \<union> (g ` (S \<inter> {x. \<not> P x}))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1073
  by (auto simp add: image_def)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1074
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1075
lemma image_cong: "M = N ==> (!!x. x \<in> N ==> f x = g x) ==> f`M = g`N"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1076
  by (simp add: image_def)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1077
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1078
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1079
text {* \medskip @{text range}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1080
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1081
lemma full_SetCompr_eq: "{u. \<exists>x. u = f x} = range f"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1082
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1083
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1084
lemma range_composition [simp]: "range (\<lambda>x. f (g x)) = f`range g"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1085
  apply (subst image_image)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1086
  apply simp
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1087
  done
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1088
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1089
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1090
text {* \medskip @{text Int} *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1091
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1092
lemma Int_absorb [simp]: "A \<inter> A = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1093
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1094
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1095
lemma Int_left_absorb: "A \<inter> (A \<inter> B) = A \<inter> B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1096
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1097
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1098
lemma Int_commute: "A \<inter> B = B \<inter> A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1099
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1100
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1101
lemma Int_left_commute: "A \<inter> (B \<inter> C) = B \<inter> (A \<inter> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1102
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1103
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1104
lemma Int_assoc: "(A \<inter> B) \<inter> C = A \<inter> (B \<inter> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1105
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1106
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1107
lemmas Int_ac = Int_assoc Int_left_absorb Int_commute Int_left_commute
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1108
  -- {* Intersection is an AC-operator *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1109
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1110
lemma Int_absorb1: "B \<subseteq> A ==> A \<inter> B = B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1111
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1112
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1113
lemma Int_absorb2: "A \<subseteq> B ==> A \<inter> B = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1114
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1115
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1116
lemma Int_empty_left [simp]: "{} \<inter> B = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1117
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1118
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1119
lemma Int_empty_right [simp]: "A \<inter> {} = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1120
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1121
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1122
lemma disjoint_eq_subset_Compl: "(A \<inter> B = {}) = (A \<subseteq> -B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1123
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1124
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1125
lemma disjoint_iff_not_equal: "(A \<inter> B = {}) = (\<forall>x\<in>A. \<forall>y\<in>B. x \<noteq> y)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1126
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1127
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1128
lemma Int_UNIV_left [simp]: "UNIV \<inter> B = B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1129
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1130
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1131
lemma Int_UNIV_right [simp]: "A \<inter> UNIV = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1132
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1133
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1134
lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1135
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1136
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1137
lemma Int_Un_distrib: "A \<inter> (B \<union> C) = (A \<inter> B) \<union> (A \<inter> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1138
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1139
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1140
lemma Int_Un_distrib2: "(B \<union> C) \<inter> A = (B \<inter> A) \<union> (C \<inter> A)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1141
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1142
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1143
lemma Int_UNIV [simp]: "(A \<inter> B = UNIV) = (A = UNIV & B = UNIV)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1144
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1145
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1146
lemma Int_subset_iff: "(C \<subseteq> A \<inter> B) = (C \<subseteq> A & C \<subseteq> B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1147
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1148
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1149
lemma Int_Collect: "(x \<in> A \<inter> {x. P x}) = (x \<in> A & P x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1150
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1151
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1152
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1153
text {* \medskip @{text Un}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1154
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1155
lemma Un_absorb [simp]: "A \<union> A = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1156
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1157
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1158
lemma Un_left_absorb: "A \<union> (A \<union> B) = A \<union> B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1159
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1160
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1161
lemma Un_commute: "A \<union> B = B \<union> A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1162
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1163
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1164
lemma Un_left_commute: "A \<union> (B \<union> C) = B \<union> (A \<union> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1165
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1166
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1167
lemma Un_assoc: "(A \<union> B) \<union> C = A \<union> (B \<union> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1168
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1169
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1170
lemmas Un_ac = Un_assoc Un_left_absorb Un_commute Un_left_commute
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1171
  -- {* Union is an AC-operator *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1172
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1173
lemma Un_absorb1: "A \<subseteq> B ==> A \<union> B = B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1174
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1175
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1176
lemma Un_absorb2: "B \<subseteq> A ==> A \<union> B = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1177
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1178
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1179
lemma Un_empty_left [simp]: "{} \<union> B = B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1180
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1181
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1182
lemma Un_empty_right [simp]: "A \<union> {} = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1183
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1184
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1185
lemma Un_UNIV_left [simp]: "UNIV \<union> B = UNIV"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1186
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1187
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1188
lemma Un_UNIV_right [simp]: "A \<union> UNIV = UNIV"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1189
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1190
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1191
lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1192
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1193
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1194
lemma Un_insert_left [simp]: "(insert a B) \<union> C = insert a (B \<union> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1195
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1196
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1197
lemma Un_insert_right [simp]: "A \<union> (insert a B) = insert a (A \<union> B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1198
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1199
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1200
lemma Int_insert_left:
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1201
    "(insert a B) Int C = (if a \<in> C then insert a (B \<inter> C) else B \<inter> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1202
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1203
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1204
lemma Int_insert_right:
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1205
    "A \<inter> (insert a B) = (if a \<in> A then insert a (A \<inter> B) else A \<inter> B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1206
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1207
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1208
lemma Un_Int_distrib: "A \<union> (B \<inter> C) = (A \<union> B) \<inter> (A \<union> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1209
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1210
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1211
lemma Un_Int_distrib2: "(B \<inter> C) \<union> A = (B \<union> A) \<inter> (C \<union> A)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1212
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1213
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1214
lemma Un_Int_crazy:
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1215
    "(A \<inter> B) \<union> (B \<inter> C) \<union> (C \<inter> A) = (A \<union> B) \<inter> (B \<union> C) \<inter> (C \<union> A)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1216
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1217
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1218
lemma subset_Un_eq: "(A \<subseteq> B) = (A \<union> B = B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1219
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1220
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1221
lemma Un_empty [iff]: "(A \<union> B = {}) = (A = {} & B = {})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1222
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1223
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1224
lemma Un_subset_iff: "(A \<union> B \<subseteq> C) = (A \<subseteq> C & B \<subseteq> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1225
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1226
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1227
lemma Un_Diff_Int: "(A - B) \<union> (A \<inter> B) = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1228
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1229
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1230
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1231
text {* \medskip Set complement *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1232
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1233
lemma Compl_disjoint [simp]: "A \<inter> -A = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1234
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1235
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1236
lemma Compl_disjoint2 [simp]: "-A \<inter> A = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1237
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1238
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1239
lemma Compl_partition: "A \<union> (-A) = UNIV"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1240
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1241
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1242
lemma double_complement [simp]: "- (-A) = (A::'a set)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1243
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1244
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1245
lemma Compl_Un [simp]: "-(A \<union> B) = (-A) \<inter> (-B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1246
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1247
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1248
lemma Compl_Int [simp]: "-(A \<inter> B) = (-A) \<union> (-B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1249
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1250
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1251
lemma Compl_UN [simp]: "-(\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1252
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1253
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1254
lemma Compl_INT [simp]: "-(\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1255
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1256
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1257
lemma subset_Compl_self_eq: "(A \<subseteq> -A) = (A = {})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1258
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1259
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1260
lemma Un_Int_assoc_eq: "((A \<inter> B) \<union> C = A \<inter> (B \<union> C)) = (C \<subseteq> A)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1261
  -- {* Halmos, Naive Set Theory, page 16. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1262
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1263
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1264
lemma Compl_UNIV_eq [simp]: "-UNIV = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1265
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1266
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1267
lemma Compl_empty_eq [simp]: "-{} = UNIV"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1268
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1269
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1270
lemma Compl_subset_Compl_iff [iff]: "(-A \<subseteq> -B) = (B \<subseteq> A)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1271
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1272
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1273
lemma Compl_eq_Compl_iff [iff]: "(-A = -B) = (A = (B::'a set))"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1274
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1275
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1276
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1277
text {* \medskip @{text Union}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1278
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1279
lemma Union_empty [simp]: "Union({}) = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1280
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1281
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1282
lemma Union_UNIV [simp]: "Union UNIV = UNIV"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1283
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1284
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1285
lemma Union_insert [simp]: "Union (insert a B) = a \<union> \<Union>B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1286
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1287
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1288
lemma Union_Un_distrib [simp]: "\<Union>(A Un B) = \<Union>A \<union> \<Union>B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1289
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1290
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1291
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1292
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1293
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1294
lemma Union_empty_conv [iff]: "(\<Union>A = {}) = (\<forall>x\<in>A. x = {})"
13653
ef123b9e8089 Added a few thms about UN/INT/{}/UNIV
nipkow
parents: 13624
diff changeset
  1295
  by blast
ef123b9e8089 Added a few thms about UN/INT/{}/UNIV
nipkow
parents: 13624
diff changeset
  1296
ef123b9e8089 Added a few thms about UN/INT/{}/UNIV
nipkow
parents: 13624
diff changeset
  1297
lemma empty_Union_conv [iff]: "({} = \<Union>A) = (\<forall>x\<in>A. x = {})"
ef123b9e8089 Added a few thms about UN/INT/{}/UNIV
nipkow
parents: 13624
diff changeset
  1298
  by blast
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1299
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1300
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) = (\<forall>B\<in>C. B \<inter> A = {})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1301
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1302
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1303
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1304
text {* \medskip @{text Inter}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1305
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1306
lemma Inter_empty [simp]: "\<Inter>{} = UNIV"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1307
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1308
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1309
lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1310
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1311
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1312
lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1313
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1314
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1315
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1316
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1317
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1318
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1319
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1320
13653
ef123b9e8089 Added a few thms about UN/INT/{}/UNIV
nipkow
parents: 13624
diff changeset
  1321
lemma Inter_UNIV_conv [iff]:
ef123b9e8089 Added a few thms about UN/INT/{}/UNIV
nipkow
parents: 13624
diff changeset
  1322
  "(\<Inter>A = UNIV) = (\<forall>x\<in>A. x = UNIV)"
ef123b9e8089 Added a few thms about UN/INT/{}/UNIV
nipkow
parents: 13624
diff changeset
  1323
  "(UNIV = \<Inter>A) = (\<forall>x\<in>A. x = UNIV)"
ef123b9e8089 Added a few thms about UN/INT/{}/UNIV
nipkow
parents: 13624
diff changeset
  1324
  by(blast)+
ef123b9e8089 Added a few thms about UN/INT/{}/UNIV
nipkow
parents: 13624
diff changeset
  1325
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1326
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1327
text {*
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1328
  \medskip @{text UN} and @{text INT}.
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1329
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1330
  Basic identities: *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1331
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1332
lemma UN_empty [simp]: "(\<Union>x\<in>{}. B x) = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1333
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1334
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1335
lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1336
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1337
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1338
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1339
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1340
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1341
lemma UN_absorb: "k \<in> I ==> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1342
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1343
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1344
lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1345
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1346
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1347
lemma INT_absorb: "k \<in> I ==> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1348
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1349
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1350
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1351
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1352
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1353
lemma UN_Un: "(\<Union>i \<in> A \<union> B. M i) = (\<Union>i\<in>A. M i) \<union> (\<Union>i\<in>B. M i)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1354
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1355
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1356
lemma 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)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1357
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1358
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1359
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1360
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1361
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1362
lemma INT_subset_iff: "(B \<subseteq> (\<Inter>i\<in>I. A i)) = (\<forall>i\<in>I. B \<subseteq> A i)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633