src/HOL/Set.thy
author paulson
Thu, 29 Sep 2005 12:43:40 +0200
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a name for empty_not_insert
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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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*)
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header {* Set theory for higher-order logic *}
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theory Set
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imports LOrder
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begin
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text {* A set in HOL is simply a predicate. *}
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subsection {* Basic syntax *}
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global
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typedecl 'a set
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arities set :: (type) type
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consts
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  "{}"          :: "'a set"                             ("{}")
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  UNIV          :: "'a set"
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  insert        :: "'a => 'a set => 'a set"
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  Collect       :: "('a => bool) => 'a set"              -- "comprehension"
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  Int           :: "'a set => 'a set => 'a set"          (infixl 70)
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  Un            :: "'a set => 'a set => 'a set"          (infixl 65)
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  UNION         :: "'a set => ('a => 'b set) => 'b set"  -- "general union"
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  INTER         :: "'a set => ('a => 'b set) => 'b set"  -- "general intersection"
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  Union         :: "'a set set => 'a set"                -- "union of a set"
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  Inter         :: "'a set set => 'a set"                -- "intersection of a set"
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  Pow           :: "'a set => 'a set set"                -- "powerset"
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  Ball          :: "'a set => ('a => bool) => bool"      -- "bounded universal quantifiers"
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  Bex           :: "'a set => ('a => bool) => bool"      -- "bounded existential quantifiers"
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  image         :: "('a => 'b) => 'a set => 'b set"      (infixr "`" 90)
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syntax
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  "op :"        :: "'a => 'a set => bool"                ("op :")
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consts
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  "op :"        :: "'a => 'a set => bool"                ("(_/ : _)" [50, 51] 50)  -- "membership"
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local
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instance set :: (type) "{ord, minus}" ..
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subsection {* Additional concrete syntax *}
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syntax
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  range         :: "('a => 'b) => 'b set"             -- "of function"
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  "op ~:"       :: "'a => 'a set => bool"                 ("op ~:")  -- "non-membership"
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  "op ~:"       :: "'a => 'a set => bool"                 ("(_/ ~: _)" [50, 51] 50)
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  "@Finset"     :: "args => 'a set"                       ("{(_)}")
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  "@Coll"       :: "pttrn => bool => 'a set"              ("(1{_./ _})")
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  "@SetCompr"   :: "'a => idts => bool => 'a set"         ("(1{_ |/_./ _})")
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  "@Collect"    :: "idt => 'a set => 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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  "{x:A. P}"    => "{x. x:A & P}"
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  "UN x y. B"   == "UN x. UN y. B"
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  "UN x. B"     == "UNION UNIV (%x. B)"
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  "UN x. B"     == "UN x:UNIV. B"
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  "INT x y. B"  == "INT x. INT y. B"
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  "INT x. B"    == "INTER UNIV (%x. B)"
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  "INT x. B"    == "INT x:UNIV. B"
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  "UN x:A. B"   == "UNION A (%x. B)"
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  "INT x:A. B"  == "INTER A (%x. B)"
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  "ALL x:A. P"  == "Ball A (%x. P)"
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  "EX x:A. P"   == "Bex A (%x. P)"
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syntax (output)
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  "_setle"      :: "'a set => 'a set => bool"             ("op <=")
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  "_setle"      :: "'a set => 'a set => bool"             ("(_/ <= _)" [50, 51] 50)
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  "_setless"    :: "'a set => 'a set => bool"             ("op <")
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  "_setless"    :: "'a set => 'a set => bool"             ("(_/ < _)" [50, 51] 50)
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syntax (xsymbols)
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  "_setle"      :: "'a set => 'a set => bool"             ("op \<subseteq>")
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  "_setle"      :: "'a set => 'a set => bool"             ("(_/ \<subseteq> _)" [50, 51] 50)
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  "_setless"    :: "'a set => 'a set => bool"             ("op \<subset>")
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  "_setless"    :: "'a set => 'a set => bool"             ("(_/ \<subset> _)" [50, 51] 50)
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  "op Int"      :: "'a set => 'a set => 'a set"           (infixl "\<inter>" 70)
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  "op Un"       :: "'a set => 'a set => 'a set"           (infixl "\<union>" 65)
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  "op :"        :: "'a => 'a set => bool"                 ("op \<in>")
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  "op :"        :: "'a => 'a set => bool"                 ("(_/ \<in> _)" [50, 51] 50)
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  "op ~:"       :: "'a => 'a set => bool"                 ("op \<notin>")
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  "op ~:"       :: "'a => 'a set => bool"                 ("(_/ \<notin> _)" [50, 51] 50)
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  Union         :: "'a set set => 'a set"                 ("\<Union>_" [90] 90)
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  Inter         :: "'a set set => 'a set"                 ("\<Inter>_" [90] 90)
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  "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10)
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  "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10)
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syntax (HTML output)
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  "_setle"      :: "'a set => 'a set => bool"             ("op \<subseteq>")
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  "_setle"      :: "'a set => 'a set => bool"             ("(_/ \<subseteq> _)" [50, 51] 50)
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  "_setless"    :: "'a set => 'a set => bool"             ("op \<subset>")
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  "_setless"    :: "'a set => 'a set => bool"             ("(_/ \<subset> _)" [50, 51] 50)
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  "op Int"      :: "'a set => 'a set => 'a set"           (infixl "\<inter>" 70)
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  "op Un"       :: "'a set => 'a set => 'a set"           (infixl "\<union>" 65)
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  "op :"        :: "'a => 'a set => bool"                 ("op \<in>")
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  "op :"        :: "'a => 'a set => bool"                 ("(_/ \<in> _)" [50, 51] 50)
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  "op ~:"       :: "'a => 'a set => bool"                 ("op \<notin>")
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  "op ~:"       :: "'a => 'a set => bool"                 ("(_/ \<notin> _)" [50, 51] 50)
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  "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10)
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  "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10)
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syntax (xsymbols)
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  "@Collect"    :: "idt => 'a set => bool => 'a set"      ("(1{_ \<in>/ _./ _})")
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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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(*
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syntax (xsymbols)
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  "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" 10)
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  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" 10)
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  "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" 10)
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  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" 10)
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*)
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syntax (latex output)
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  "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" 10)
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  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" 10)
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  "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" 10)
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  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" 10)
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text{* Note the difference between ordinary xsymbol syntax of indexed
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unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})
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and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The
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former does not make the index expression a subscript of the
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union/intersection symbol because this leads to problems with nested
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subscripts in Proof General.  *}
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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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subsubsection "Bounded quantifiers"
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syntax
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  "_setlessAll" :: "[idt, 'a, bool] => bool"  ("(3ALL _<_./ _)"  [0, 0, 10] 10)
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  "_setlessEx"  :: "[idt, 'a, bool] => bool"  ("(3EX _<_./ _)"  [0, 0, 10] 10)
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  "_setleAll"   :: "[idt, 'a, bool] => bool"  ("(3ALL _<=_./ _)" [0, 0, 10] 10)
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  "_setleEx"    :: "[idt, 'a, bool] => bool"  ("(3EX _<=_./ _)" [0, 0, 10] 10)
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syntax (xsymbols)
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  "_setlessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<subset>_./ _)"  [0, 0, 10] 10)
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  "_setlessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subset>_./ _)"  [0, 0, 10] 10)
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  "_setleAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<subseteq>_./ _)" [0, 0, 10] 10)
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  "_setleEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10)
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syntax (HOL)
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  "_setlessAll" :: "[idt, 'a, bool] => bool"   ("(3! _<_./ _)"  [0, 0, 10] 10)
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  "_setlessEx"  :: "[idt, 'a, bool] => bool"   ("(3? _<_./ _)"  [0, 0, 10] 10)
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  "_setleAll"   :: "[idt, 'a, bool] => bool"   ("(3! _<=_./ _)" [0, 0, 10] 10)
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  "_setleEx"    :: "[idt, 'a, bool] => bool"   ("(3? _<=_./ _)" [0, 0, 10] 10)
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syntax (HTML output)
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  "_setlessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<subset>_./ _)"  [0, 0, 10] 10)
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  "_setlessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subset>_./ _)"  [0, 0, 10] 10)
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  "_setleAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<subseteq>_./ _)" [0, 0, 10] 10)
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  "_setleEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10)
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translations
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 "\<forall>A\<subset>B. P"   =>  "ALL A. A \<subset> B --> P"
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 "\<exists>A\<subset>B. P"    =>  "EX A. A \<subset> B & P"
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 "\<forall>A\<subseteq>B. P"  =>  "ALL A. A \<subseteq> B --> P"
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 "\<exists>A\<subseteq>B. P"   =>  "EX A. A \<subseteq> B & P"
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print_translation {*
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let
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  fun
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    all_tr' [Const ("_bound",_) $ Free (v,Type(T,_)), 
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             Const("op -->",_) $ (Const ("op <",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] = 
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  (if v=v' andalso T="set"
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   then Syntax.const "_setlessAll" $ Syntax.mark_bound v' $ n $ P
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   else raise Match)
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  | all_tr' [Const ("_bound",_) $ Free (v,Type(T,_)), 
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             Const("op -->",_) $ (Const ("op <=",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] = 
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  (if v=v' andalso T="set"
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   then Syntax.const "_setleAll" $ Syntax.mark_bound v' $ n $ P
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   else raise Match);
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  fun
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    ex_tr' [Const ("_bound",_) $ Free (v,Type(T,_)), 
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            Const("op &",_) $ (Const ("op <",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] = 
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  (if v=v' andalso T="set"
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   then Syntax.const "_setlessEx" $ Syntax.mark_bound v' $ n $ P
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   else raise Match)
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  | ex_tr' [Const ("_bound",_) $ Free (v,Type(T,_)), 
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            Const("op &",_) $ (Const ("op <=",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] = 
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  (if v=v' andalso T="set"
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   then Syntax.const "_setleEx" $ Syntax.mark_bound v' $ n $ P
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   else raise Match)
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in
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[("ALL ", all_tr'), ("EX ", ex_tr')]
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end
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*}
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text {*
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  \medskip Translate between @{text "{e | x1...xn. P}"} and @{text
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  "{u. EX x1..xn. u = e & P}"}; @{text "{y. EX x1..xn. y = e & P}"} is
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  only translated if @{text "[0..n] subset bvs(e)"}.
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*}
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parse_translation {*
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  let
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    val ex_tr = snd (mk_binder_tr ("EX ", "Ex"));
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    fun nvars (Const ("_idts", _) $ _ $ idts) = nvars idts + 1
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      | nvars _ = 1;
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    fun setcompr_tr [e, idts, b] =
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      let
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        val eq = Syntax.const "op =" $ Bound (nvars idts) $ e;
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        val P = Syntax.const "op &" $ eq $ b;
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        val exP = ex_tr [idts, P];
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      in Syntax.const "Collect" $ Abs ("", dummyT, exP) end;
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  in [("@SetCompr", setcompr_tr)] end;
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*}
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(* To avoid eta-contraction of body: *)
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print_translation {*
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let
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  fun btr' syn [A,Abs abs] =
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    let val (x,t) = atomic_abs_tr' abs
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    in Syntax.const syn $ x $ A $ t end
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in
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[("Ball", btr' "_Ball"),("Bex", btr' "_Bex"),
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 ("UNION", btr' "@UNION"),("INTER", btr' "@INTER")]
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end
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*}
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print_translation {*
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let
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  val ex_tr' = snd (mk_binder_tr' ("Ex", "DUMMY"));
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  fun setcompr_tr' [Abs (abs as (_, _, P))] =
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    let
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      fun check (Const ("Ex", _) $ Abs (_, _, P), n) = check (P, n + 1)
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        | check (Const ("op &", _) $ (Const ("op =", _) $ Bound m $ e) $ P, n) =
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            n > 0 andalso m = n andalso not (loose_bvar1 (P, n)) andalso
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            ((0 upto (n - 1)) subset add_loose_bnos (e, 0, []))
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        | check _ = false
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        fun tr' (_ $ abs) =
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          let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' [abs]
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          in Syntax.const "@SetCompr" $ e $ idts $ Q end;
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    in if check (P, 0) then tr' P
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       else let val (x as _ $ Free(xN,_), t) = atomic_abs_tr' abs
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                val M = Syntax.const "@Coll" $ x $ t
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            in case t of
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                 Const("op &",_)
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                   $ (Const("op :",_) $ (Const("_bound",_) $ Free(yN,_)) $ A)
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                   $ P =>
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                   if xN=yN then Syntax.const "@Collect" $ x $ A $ P else M
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               | _ => M
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            end
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    end;
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  in [("Collect", setcompr_tr')] end;
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*}
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subsection {* Rules and definitions *}
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text {* Isomorphisms between predicates and sets. *}
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axioms
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  mem_Collect_eq: "(a : {x. P(x)}) = P(a)"
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  Collect_mem_eq: "{x. x:A} = A"
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finalconsts
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  Collect
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  "op :"
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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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declare mem_Collect_eq [iff]  Collect_mem_eq [simp]
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lemma CollectI: "P(a) ==> a : {x. P(x)}"
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  by simp
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lemma CollectD: "a : {x. P(x)} ==> P(a)"
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  by simp
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lemma Collect_cong: "(!!x. P x = Q x) ==> {x. P(x)} = {x. Q(x)}"
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  by simp
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lemmas CollectE = CollectD [elim_format]
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subsubsection {* Bounded quantifiers *}
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lemma ballI [intro!]: "(!!x. x:A ==> P x) ==> ALL x:A. P x"
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  by (simp add: Ball_def)
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lemmas strip = impI allI ballI
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lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x"
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  by (simp add: Ball_def)
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lemma ballE [elim]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q"
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  by (unfold Ball_def) blast
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ML {* bind_thm("rev_ballE",permute_prems 1 1 (thm "ballE")) *}
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text {*
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  \medskip This tactic takes assumptions @{prop "ALL x:A. P x"} and
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  @{prop "a:A"}; creates assumption @{prop "P a"}.
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*}
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ML {*
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  local val ballE = thm "ballE"
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  in fun ball_tac i = etac ballE i THEN contr_tac (i + 1) end;
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*}
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text {*
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  Gives better instantiation for bound:
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*}
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ML_setup {*
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  claset_ref() := claset() addbefore ("bspec", datac (thm "bspec") 1);
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*}
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   385
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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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   390
13113
5eb9be7b72a5 rev_bexI [intro?];
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   391
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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   393
  by (unfold Bex_def) blast
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   394
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   395
lemma bexCI: "(ALL x:A. ~P x ==> P a) ==> a:A ==> EX x:A. P x"
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   396
  by (unfold Bex_def) blast
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   397
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   398
lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q"
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   399
  by (unfold Bex_def) blast
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   400
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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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   403
  by (simp add: Ball_def)
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diff changeset
   404
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diff changeset
   405
lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)"
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   406
  -- {* Dual form for existentials. *}
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   407
  by (simp add: Bex_def)
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diff changeset
   408
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   409
lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)"
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   410
  by blast
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   411
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   412
lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)"
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   413
  by blast
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   414
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diff changeset
   415
lemma bex_one_point1 [simp]: "(EX x:A. x = a & P x) = (a:A & P a)"
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diff changeset
   416
  by blast
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   417
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   418
lemma bex_one_point2 [simp]: "(EX x:A. a = x & P x) = (a:A & P a)"
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   419
  by blast
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   420
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   421
lemma ball_one_point1 [simp]: "(ALL x:A. x = a --> P x) = (a:A --> P a)"
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   422
  by blast
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diff changeset
   423
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lemma ball_one_point2 [simp]: "(ALL x:A. a = x --> P x) = (a:A --> P a)"
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   425
  by blast
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   426
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   427
ML_setup {*
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56610e2ba220 sane interface for simprocs;
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  local
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   429
    val Ball_def = thm "Ball_def";
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   430
    val Bex_def = thm "Bex_def";
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diff changeset
   431
17002
fb9261990ffe simprocs: Simplifier.inherit_bounds;
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   432
    val simpset = Simplifier.clear_ss HOL_basic_ss;
fb9261990ffe simprocs: Simplifier.inherit_bounds;
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   433
    fun unfold_tac ss th =
fb9261990ffe simprocs: Simplifier.inherit_bounds;
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diff changeset
   434
      ALLGOALS (full_simp_tac (Simplifier.inherit_bounds ss simpset addsimps [th]));
fb9261990ffe simprocs: Simplifier.inherit_bounds;
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diff changeset
   435
fb9261990ffe simprocs: Simplifier.inherit_bounds;
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   436
    fun prove_bex_tac ss =
fb9261990ffe simprocs: Simplifier.inherit_bounds;
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   437
      unfold_tac ss Bex_def THEN Quantifier1.prove_one_point_ex_tac;
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   438
    val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac;
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diff changeset
   439
17002
fb9261990ffe simprocs: Simplifier.inherit_bounds;
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   440
    fun prove_ball_tac ss =
fb9261990ffe simprocs: Simplifier.inherit_bounds;
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   441
      unfold_tac ss Ball_def THEN Quantifier1.prove_one_point_all_tac;
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diff changeset
   442
    val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac;
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   443
  in
13462
56610e2ba220 sane interface for simprocs;
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diff changeset
   444
    val defBEX_regroup = Simplifier.simproc (Theory.sign_of (the_context ()))
56610e2ba220 sane interface for simprocs;
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diff changeset
   445
      "defined BEX" ["EX x:A. P x & Q x"] rearrange_bex;
56610e2ba220 sane interface for simprocs;
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   446
    val defBALL_regroup = Simplifier.simproc (Theory.sign_of (the_context ()))
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13421
diff changeset
   447
      "defined BALL" ["ALL x:A. P x --> Q x"] rearrange_ball;
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   448
  end;
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13421
diff changeset
   449
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13421
diff changeset
   450
  Addsimprocs [defBALL_regroup, defBEX_regroup];
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   451
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   452
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   453
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   454
subsubsection {* Congruence rules *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   455
16636
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   456
lemma ball_cong:
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   457
  "A = B ==> (!!x. x:B ==> P x = Q x) ==>
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   458
    (ALL x:A. P x) = (ALL x:B. Q x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   459
  by (simp add: Ball_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   460
16636
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   461
lemma strong_ball_cong [cong]:
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   462
  "A = B ==> (!!x. x:B =simp=> P x = Q x) ==>
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   463
    (ALL x:A. P x) = (ALL x:B. Q x)"
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   464
  by (simp add: simp_implies_def Ball_def)
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   465
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   466
lemma bex_cong:
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   467
  "A = B ==> (!!x. x:B ==> P x = Q x) ==>
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   468
    (EX x:A. P x) = (EX x:B. Q x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   469
  by (simp add: Bex_def cong: conj_cong)
1273
6960ec882bca added 8bit pragmas
regensbu
parents: 1068
diff changeset
   470
16636
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   471
lemma strong_bex_cong [cong]:
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   472
  "A = B ==> (!!x. x:B =simp=> P x = Q x) ==>
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   473
    (EX x:A. P x) = (EX x:B. Q x)"
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   474
  by (simp add: simp_implies_def Bex_def cong: conj_cong)
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   475
7238
36e58620ffc8 replaced HOL_quantifiers flag by "HOL" print mode;
wenzelm
parents: 5931
diff changeset
   476
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   477
subsubsection {* Subsets *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   478
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   479
lemma subsetI [intro!]: "(!!x. x:A ==> x:B) ==> A \<subseteq> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   480
  by (simp add: subset_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   481
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   482
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   483
  \medskip Map the type @{text "'a set => anything"} to just @{typ
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   484
  'a}; for overloading constants whose first argument has type @{typ
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   485
  "'a set"}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   486
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   487
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   488
lemma subsetD [elim]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   489
  -- {* Rule in Modus Ponens style. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   490
  by (unfold subset_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   491
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   492
declare subsetD [intro?] -- FIXME
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   493
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   494
lemma rev_subsetD: "c \<in> A ==> A \<subseteq> B ==> c \<in> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   495
  -- {* The same, with reversed premises for use with @{text erule} --
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   496
      cf @{text rev_mp}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   497
  by (rule subsetD)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   498
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   499
declare rev_subsetD [intro?] -- FIXME
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   500
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   501
text {*
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   502
  \medskip Converts @{prop "A \<subseteq> B"} to @{prop "x \<in> A ==> x \<in> B"}.
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   503
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   504
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   505
ML {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   506
  local val rev_subsetD = thm "rev_subsetD"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   507
  in fun impOfSubs th = th RSN (2, rev_subsetD) end;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   508
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   509
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   510
lemma subsetCE [elim]: "A \<subseteq>  B ==> (c \<notin> A ==> P) ==> (c \<in> B ==> P) ==> P"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   511
  -- {* Classical elimination rule. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   512
  by (unfold subset_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   513
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   514
text {*
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   515
  \medskip Takes assumptions @{prop "A \<subseteq> B"}; @{prop "c \<in> A"} and
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   516
  creates the assumption @{prop "c \<in> B"}.
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   517
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   518
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   519
ML {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   520
  local val subsetCE = thm "subsetCE"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   521
  in fun set_mp_tac i = etac subsetCE i THEN mp_tac i end;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   522
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   523
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   524
lemma contra_subsetD: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   525
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   526
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   527
lemma subset_refl: "A \<subseteq> A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   528
  by fast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   529
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   530
lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   531
  by blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   532
2261
d926157c0a6a added "op :", "op ~:" syntax;
wenzelm
parents: 2006
diff changeset
   533
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   534
subsubsection {* Equality *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   535
13865
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   536
lemma set_ext: assumes prem: "(!!x. (x:A) = (x:B))" shows "A = B"
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   537
  apply (rule prem [THEN ext, THEN arg_cong, THEN box_equals])
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   538
   apply (rule Collect_mem_eq)
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   539
  apply (rule Collect_mem_eq)
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   540
  done
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   541
15554
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15535
diff changeset
   542
(* Due to Brian Huffman *)
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15535
diff changeset
   543
lemma expand_set_eq: "(A = B) = (ALL x. (x:A) = (x:B))"
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15535
diff changeset
   544
by(auto intro:set_ext)
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15535
diff changeset
   545
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   546
lemma subset_antisym [intro!]: "A \<subseteq> B ==> B \<subseteq> A ==> A = B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   547
  -- {* Anti-symmetry of the subset relation. *}
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17508
diff changeset
   548
  by (iprover intro: set_ext subsetD)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   549
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   550
lemmas equalityI [intro!] = subset_antisym
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   551
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   552
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   553
  \medskip Equality rules from ZF set theory -- are they appropriate
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   554
  here?
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   555
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   556
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   557
lemma equalityD1: "A = B ==> A \<subseteq> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   558
  by (simp add: subset_refl)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   559
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   560
lemma equalityD2: "A = B ==> B \<subseteq> A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   561
  by (simp add: subset_refl)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   562
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   563
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   564
  \medskip Be careful when adding this to the claset as @{text
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   565
  subset_empty} is in the simpset: @{prop "A = {}"} goes to @{prop "{}
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   566
  \<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}!
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   567
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   568
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   569
lemma equalityE: "A = B ==> (A \<subseteq> B ==> B \<subseteq> A ==> P) ==> P"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   570
  by (simp add: subset_refl)
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   571
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   572
lemma equalityCE [elim]:
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   573
    "A = B ==> (c \<in> A ==> c \<in> B ==> P) ==> (c \<notin> A ==> c \<notin> B ==> P) ==> P"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   574
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   575
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   576
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   577
  \medskip Lemma for creating induction formulae -- for "pattern
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   578
  matching" on @{text p}.  To make the induction hypotheses usable,
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   579
  apply @{text spec} or @{text bspec} to put universal quantifiers over the free
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   580
  variables in @{text p}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   581
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   582
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   583
lemma setup_induction: "p:A ==> (!!z. z:A ==> p = z --> R) ==> R"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   584
  by simp
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   585
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   586
lemma eqset_imp_iff: "A = B ==> (x : A) = (x : B)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   587
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   588
13865
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   589
lemma eqelem_imp_iff: "x = y ==> (x : A) = (y : A)"
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   590
  by simp
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   591
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   592
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   593
subsubsection {* The universal set -- UNIV *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   594
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   595
lemma UNIV_I [simp]: "x : UNIV"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   596
  by (simp add: UNIV_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   597
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   598
declare UNIV_I [intro]  -- {* unsafe makes it less likely to cause problems *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   599
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   600
lemma UNIV_witness [intro?]: "EX x. x : UNIV"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   601
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   602
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   603
lemma subset_UNIV: "A \<subseteq> UNIV"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   604
  by (rule subsetI) (rule UNIV_I)
2388
d1f0505fc602 added set inclusion symbol syntax;
wenzelm
parents: 2372
diff changeset
   605
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   606
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   607
  \medskip Eta-contracting these two rules (to remove @{text P})
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   608
  causes them to be ignored because of their interaction with
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   609
  congruence rules.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   610
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   611
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   612
lemma ball_UNIV [simp]: "Ball UNIV P = All P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   613
  by (simp add: Ball_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   614
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   615
lemma bex_UNIV [simp]: "Bex UNIV P = Ex P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   616
  by (simp add: Bex_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   617
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   618
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   619
subsubsection {* The empty set *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   620
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   621
lemma empty_iff [simp]: "(c : {}) = False"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   622
  by (simp add: empty_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   623
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   624
lemma emptyE [elim!]: "a : {} ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   625
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   626
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   627
lemma empty_subsetI [iff]: "{} \<subseteq> A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   628
    -- {* One effect is to delete the ASSUMPTION @{prop "{} <= A"} *}
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 equals0I: "(!!y. y \<in> A ==> False) ==> A = {}"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   632
  by blast
2388
d1f0505fc602 added set inclusion symbol syntax;
wenzelm
parents: 2372
diff changeset
   633
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   634
lemma equals0D: "A = {} ==> a \<notin> A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   635
    -- {* Use for reasoning about disjointness: @{prop "A Int B = {}"} *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   636
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   637
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   638
lemma ball_empty [simp]: "Ball {} P = True"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   639
  by (simp add: Ball_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   640
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   641
lemma bex_empty [simp]: "Bex {} P = False"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   642
  by (simp add: Bex_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   643
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   644
lemma UNIV_not_empty [iff]: "UNIV ~= {}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   645
  by (blast elim: equalityE)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   646
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   647
12023
wenzelm
parents: 12020
diff changeset
   648
subsubsection {* The Powerset operator -- Pow *}
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   649
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   650
lemma Pow_iff [iff]: "(A \<in> Pow B) = (A \<subseteq> B)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   651
  by (simp add: Pow_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   652
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   653
lemma PowI: "A \<subseteq> B ==> A \<in> Pow B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   654
  by (simp add: Pow_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   655
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   656
lemma PowD: "A \<in> Pow B ==> A \<subseteq> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   657
  by (simp add: Pow_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   658
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   659
lemma Pow_bottom: "{} \<in> Pow B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   660
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   661
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   662
lemma Pow_top: "A \<in> Pow A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   663
  by (simp add: subset_refl)
2684
9781d63ef063 added proper subset symbols syntax;
wenzelm
parents: 2412
diff changeset
   664
2388
d1f0505fc602 added set inclusion symbol syntax;
wenzelm
parents: 2372
diff changeset
   665
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   666
subsubsection {* Set complement *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   667
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   668
lemma Compl_iff [simp]: "(c \<in> -A) = (c \<notin> A)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   669
  by (unfold Compl_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   670
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   671
lemma ComplI [intro!]: "(c \<in> A ==> False) ==> c \<in> -A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   672
  by (unfold Compl_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   673
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   674
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   675
  \medskip This form, with negated conclusion, works well with the
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   676
  Classical prover.  Negated assumptions behave like formulae on the
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   677
  right side of the notional turnstile ... *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   678
17084
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 17002
diff changeset
   679
lemma ComplD [dest!]: "c : -A ==> c~:A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   680
  by (unfold Compl_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   681
17084
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 17002
diff changeset
   682
lemmas ComplE = ComplD [elim_format]
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   683
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   684
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   685
subsubsection {* Binary union -- Un *}
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   686
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   687
lemma Un_iff [simp]: "(c : A Un B) = (c:A | c:B)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   688
  by (unfold Un_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   689
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   690
lemma UnI1 [elim?]: "c:A ==> c : A Un B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   691
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   692
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   693
lemma UnI2 [elim?]: "c:B ==> c : A Un B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   694
  by simp
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   695
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   696
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   697
  \medskip Classical introduction rule: no commitment to @{prop A} vs
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   698
  @{prop B}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   699
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   700
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   701
lemma UnCI [intro!]: "(c~:B ==> c:A) ==> c : A Un B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   702
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   703
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   704
lemma UnE [elim!]: "c : A Un B ==> (c:A ==> P) ==> (c:B ==> P) ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   705
  by (unfold Un_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   706
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   707
12023
wenzelm
parents: 12020
diff changeset
   708
subsubsection {* Binary intersection -- Int *}
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   709
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   710
lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   711
  by (unfold Int_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   712
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   713
lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   714
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   715
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   716
lemma IntD1: "c : A Int B ==> c:A"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   717
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   718
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   719
lemma IntD2: "c : A Int B ==> c:B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   720
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   721
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   722
lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   723
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   724
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   725
12023
wenzelm
parents: 12020
diff changeset
   726
subsubsection {* Set difference *}
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   727
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   728
lemma Diff_iff [simp]: "(c : A - B) = (c:A & c~:B)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   729
  by (unfold set_diff_def) blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   730
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   731
lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A - B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   732
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   733
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   734
lemma DiffD1: "c : A - B ==> c : A"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   735
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   736
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   737
lemma DiffD2: "c : A - B ==> c : B ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   738
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   739
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   740
lemma DiffE [elim!]: "c : A - B ==> (c:A ==> c~:B ==> P) ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   741
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   742
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   743
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   744
subsubsection {* Augmenting a set -- insert *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   745
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   746
lemma insert_iff [simp]: "(a : insert b A) = (a = b | a:A)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   747
  by (unfold insert_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   748
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   749
lemma insertI1: "a : insert a B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   750
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   751
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   752
lemma insertI2: "a : B ==> a : insert b B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   753
  by simp
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   754
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   755
lemma insertE [elim!]: "a : insert b A ==> (a = b ==> P) ==> (a:A ==> P) ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   756
  by (unfold insert_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   757
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   758
lemma insertCI [intro!]: "(a~:B ==> a = b) ==> a: insert b B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   759
  -- {* Classical introduction rule. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   760
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   761
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   762
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
   763
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   764
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   765
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   766
subsubsection {* Singletons, using insert *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   767
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   768
lemma singletonI [intro!]: "a : {a}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   769
    -- {* Redundant? But unlike @{text insertCI}, it proves the subgoal immediately! *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   770
  by (rule insertI1)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   771
17084
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 17002
diff changeset
   772
lemma singletonD [dest!]: "b : {a} ==> b = a"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   773
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   774
17084
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 17002
diff changeset
   775
lemmas singletonE = singletonD [elim_format]
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   776
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   777
lemma singleton_iff: "(b : {a}) = (b = a)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   778
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   779
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   780
lemma singleton_inject [dest!]: "{a} = {b} ==> a = b"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   781
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   782
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   783
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
   784
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   785
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   786
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
   787
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   788
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   789
lemma subset_singletonD: "A \<subseteq> {x} ==> A = {} | A = {x}"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   790
  by fast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   791
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   792
lemma singleton_conv [simp]: "{x. x = a} = {a}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   793
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   794
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   795
lemma singleton_conv2 [simp]: "{x. a = x} = {a}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   796
  by blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   797
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   798
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
   799
  by blast
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
subsubsection {* Unions of families *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   803
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   804
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   805
  @{term [source] "UN x:A. B x"} is @{term "Union (B`A)"}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   806
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   807
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   808
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
   809
  by (unfold UNION_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   810
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   811
lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   812
  -- {* The order of the premises presupposes that @{term A} is rigid;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   813
    @{term b} may be flexible. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   814
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   815
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   816
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
   817
  by (unfold UNION_def) blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   818
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   819
lemma UN_cong [cong]:
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   820
    "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
   821
  by (simp add: UNION_def)
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
subsubsection {* Intersections of families *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   825
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   826
text {* @{term [source] "INT x:A. B x"} is @{term "Inter (B`A)"}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   827
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   828
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
   829
  by (unfold INTER_def) blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   830
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   831
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
   832
  by (unfold INTER_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   833
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   834
lemma INT_D [elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   835
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   836
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   837
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
   838
  -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   839
  by (unfold INTER_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   840
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   841
lemma INT_cong [cong]:
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   842
    "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
   843
  by (simp add: INTER_def)
7238
36e58620ffc8 replaced HOL_quantifiers flag by "HOL" print mode;
wenzelm
parents: 5931
diff changeset
   844
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   845
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   846
subsubsection {* Union *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   847
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   848
lemma Union_iff [simp]: "(A : Union C) = (EX X:C. A:X)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   849
  by (unfold Union_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   850
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   851
lemma UnionI [intro]: "X:C ==> A:X ==> A : Union C"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   852
  -- {* The order of the premises presupposes that @{term C} is rigid;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   853
    @{term A} may be flexible. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   854
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   855
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   856
lemma UnionE [elim!]: "A : Union C ==> (!!X. A:X ==> X:C ==> R) ==> R"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   857
  by (unfold Union_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   858
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   859
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   860
subsubsection {* Inter *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   861
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   862
lemma Inter_iff [simp]: "(A : Inter C) = (ALL X:C. A:X)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   863
  by (unfold Inter_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   864
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   865
lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   866
  by (simp add: Inter_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   867
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   868
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   869
  \medskip A ``destruct'' rule -- every @{term X} in @{term C}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   870
  contains @{term A} as an element, but @{prop "A:X"} can hold when
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   871
  @{prop "X:C"} does not!  This rule is analogous to @{text spec}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   872
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   873
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   874
lemma InterD [elim]: "A : Inter C ==> X:C ==> A:X"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   875
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   876
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   877
lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   878
  -- {* ``Classical'' elimination rule -- does not require proving
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   879
    @{prop "X:C"}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   880
  by (unfold Inter_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   881
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   882
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   883
  \medskip Image of a set under a function.  Frequently @{term b} does
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   884
  not have the syntactic form of @{term "f x"}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   885
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   886
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   887
lemma image_eqI [simp, intro]: "b = f x ==> x:A ==> b : f`A"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   888
  by (unfold image_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   889
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   890
lemma imageI: "x : A ==> f x : f ` A"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   891
  by (rule image_eqI) (rule refl)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   892
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   893
lemma rev_image_eqI: "x:A ==> b = f x ==> b : f`A"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   894
  -- {* This version's more effective when we already have the
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   895
    required @{term x}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   896
  by (unfold image_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   897
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   898
lemma imageE [elim!]:
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   899
  "b : (%x. f x)`A ==> (!!x. b = f x ==> x:A ==> P) ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   900
  -- {* The eta-expansion gives variable-name preservation. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   901
  by (unfold image_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   902
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   903
lemma image_Un: "f`(A Un B) = f`A Un f`B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   904
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   905
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   906
lemma image_iff: "(z : f`A) = (EX x:A. z = f x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   907
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   908
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   909
lemma 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
   910
  -- {* This rewrite rule would confuse users if made default. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   911
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   912
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   913
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
   914
  apply safe
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   915
   prefer 2 apply fast
14208
144f45277d5a misc tidying
paulson
parents: 14098
diff changeset
   916
  apply (rule_tac x = "{a. a : A & f a : B}" in exI, fast)
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   917
  done
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   918
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   919
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
   920
  -- {* Replaces the three steps @{text subsetI}, @{text imageE},
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   921
    @{text hypsubst}, but breaks too many existing proofs. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   922
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   923
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   924
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   925
  \medskip Range of a function -- just a translation for image!
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   926
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   927
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   928
lemma range_eqI: "b = f x ==> b \<in> range f"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   929
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   930
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   931
lemma rangeI: "f x \<in> range f"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   932
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   933
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   934
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
   935
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   936
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   937
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   938
subsubsection {* Set reasoning tools *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   939
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   940
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   941
  Rewrite rules for boolean case-splitting: faster than @{text
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   942
  "split_if [split]"}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   943
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   944
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   945
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
   946
  by (rule split_if)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   947
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   948
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
   949
  by (rule split_if)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   950
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   951
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   952
  Split ifs on either side of the membership relation.  Not for @{text
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   953
  "[simp]"} -- can cause goals to blow up!
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   954
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   955
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   956
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
   957
  by (rule split_if)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   958
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   959
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
   960
  by (rule split_if)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   961
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   962
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
   963
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   964
lemmas mem_simps =
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   965
  insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   966
  mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   967
  -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   968
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   969
(*Would like to add these, but the existing code only searches for the
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   970
  outer-level constant, which in this case is just "op :"; we instead need
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   971
  to use term-nets to associate patterns with rules.  Also, if a rule fails to
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   972
  apply, then the formula should be kept.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   973
  [("uminus", Compl_iff RS iffD1), ("op -", [Diff_iff RS iffD1]),
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   974
   ("op Int", [IntD1,IntD2]),
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   975
   ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   976
 *)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   977
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   978
ML_setup {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   979
  val mksimps_pairs = [("Ball", [thm "bspec"])] @ mksimps_pairs;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   980
  simpset_ref() := simpset() setmksimps (mksimps mksimps_pairs);
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   981
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   982
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   983
declare subset_UNIV [simp] subset_refl [simp]
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   984
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   985
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   986
subsubsection {* The ``proper subset'' relation *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   987
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   988
lemma psubsetI [intro!]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   989
  by (unfold psubset_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   990
13624
17684cf64fda added the new elim rule psubsetE
paulson
parents: 13550
diff changeset
   991
lemma psubsetE [elim!]: 
17684cf64fda added the new elim rule psubsetE
paulson
parents: 13550
diff changeset
   992
    "[|A \<subset> B;  [|A \<subseteq> B; ~ (B\<subseteq>A)|] ==> R|] ==> R"
17684cf64fda added the new elim rule psubsetE
paulson
parents: 13550
diff changeset
   993
  by (unfold psubset_def) blast
17684cf64fda added the new elim rule psubsetE
paulson
parents: 13550
diff changeset
   994
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   995
lemma psubset_insert_iff:
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   996
  "(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
   997
  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
   998
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   999
lemma psubset_eq: "(A \<subset> B) = (A \<subseteq> B & A \<noteq> B)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1000
  by (simp only: psubset_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1001
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1002
lemma psubset_imp_subset: "A \<subset> B ==> A \<subseteq> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1003
  by (simp add: psubset_eq)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1004
14335
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
  1005
lemma psubset_trans: "[| A \<subset> B; B \<subset> C |] ==> A \<subset> C"
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
  1006
apply (unfold psubset_def)
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
  1007
apply (auto dest: subset_antisym)
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
  1008
done
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
  1009
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
  1010
lemma psubsetD: "[| A \<subset> B; c \<in> A |] ==> c \<in> B"
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
  1011
apply (unfold psubset_def)
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
  1012
apply (auto dest: subsetD)
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
  1013
done
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
  1014
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1015
lemma psubset_subset_trans: "A \<subset> B ==> B \<subseteq> C ==> A \<subset> C"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1016
  by (auto simp add: psubset_eq)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1017
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1018
lemma subset_psubset_trans: "A \<subseteq> B ==> B \<subset> C ==> A \<subset> C"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1019
  by (auto simp add: psubset_eq)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1020
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1021
lemma psubset_imp_ex_mem: "A \<subset> B ==> \<exists>b. b \<in> (B - A)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1022
  by (unfold psubset_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1023
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1024
lemma atomize_ball:
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1025
    "(!!x. x \<in> A ==> P x) == Trueprop (\<forall>x\<in>A. P x)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1026
  by (simp only: Ball_def atomize_all atomize_imp)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1027
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1028
declare atomize_ball [symmetric, rulify]
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1029
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1030
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1031
subsection {* Further set-theory lemmas *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1032
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1033
subsubsection {* Derived rules involving subsets. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1034
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1035
text {* @{text insert}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1036
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1037
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
  1038
  apply (rule subsetI)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1039
  apply (erule insertI2)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1040
  done
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1041
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1042
lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1043
by blast
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1044
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1045
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
  1046
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1047
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1048
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1049
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
  1050
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1051
lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17508
diff changeset
  1052
  by (iprover intro: subsetI UnionI)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1053
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1054
lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17508
diff changeset
  1055
  by (iprover intro: subsetI elim: UnionE dest: subsetD)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1056
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1057
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1058
text {* \medskip General union. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1059
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1060
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
  1061
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1062
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1063
lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17508
diff changeset
  1064
  by (iprover intro: subsetI elim: UN_E dest: subsetD)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1065
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1066
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1067
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
  1068
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1069
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
  1070
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1071
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents: 14479
diff changeset
  1072
lemma Inter_subset:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents: 14479
diff changeset
  1073
  "[| !!X. X \<in> A ==> X \<subseteq> B; A ~= {} |] ==> \<Inter>A \<subseteq> B"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents: 14479
diff changeset
  1074
  by blast
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents: 14479
diff changeset
  1075
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1076
lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17508
diff changeset
  1077
  by (iprover intro: InterI subsetI dest: subsetD)
12897
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
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
  1080
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1081
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1082
lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17508
diff changeset
  1083
  by (iprover intro: INT_I subsetI dest: subsetD)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1084
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1085
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1086
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
  1087
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1088
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
  1089
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1090
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1091
lemma Un_upper2: "B \<subseteq> A \<union> B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1092
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1093
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1094
lemma Un_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
  1095
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1096
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
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
  1099
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1100
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
  1101
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1102
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1103
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
  1104
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1105
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1106
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
  1107
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1108
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
text {* \medskip Set difference. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1111
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1112
lemma Diff_subset: "A - B \<subseteq> A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1113
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1114
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1115
lemma Diff_subset_conv: "(A - B \<subseteq> C) = (A \<subseteq> B \<union> C)"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1116
by blast
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1117
12897
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
text {* \medskip Monotonicity. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1120
15206
09d78ec709c7 Modified locales: improved implementation of "includes".
ballarin
parents: 15140
diff changeset
  1121
lemma mono_Un: "mono f ==> f A \<union> f B \<subseteq> f (A \<union> B)"
16773
33c4d8fe6f78 tweaked
paulson
parents: 16636
diff changeset
  1122
  by (auto simp add: mono_def)
15206
09d78ec709c7 Modified locales: improved implementation of "includes".
ballarin
parents: 15140
diff changeset
  1123
09d78ec709c7 Modified locales: improved implementation of "includes".
ballarin
parents: 15140
diff changeset
  1124
lemma mono_Int: "mono f ==> f (A \<inter> B) \<subseteq> f A \<inter> f B"
16773
33c4d8fe6f78 tweaked
paulson
parents: 16636
diff changeset
  1125
  by (auto simp add: mono_def)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1126
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1127
subsubsection {* Equalities involving union, intersection, inclusion, etc. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1128
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1129
text {* @{text "{}"}. *}
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 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
  1132
  -- {* supersedes @{text "Collect_False_empty"} *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1133
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1134
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1135
lemma subset_empty [simp]: "(A \<subseteq> {}) = (A = {})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1136
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1137
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1138
lemma not_psubset_empty [iff]: "\<not> (A < {})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1139
  by (unfold psubset_def) blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1140
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1141
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
  1142
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1143
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1144
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
  1145
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1146
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1147
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
  1148
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1149
14812
4b2c006d0534 new theorem Collect_imp_eq
paulson
parents: 14804
diff changeset
  1150
lemma Collect_imp_eq: "{x. P x --> Q x} = -{x. P x} \<union> {x. Q x}"
4b2c006d0534 new theorem Collect_imp_eq
paulson
parents: 14804
diff changeset
  1151
  by blast
4b2c006d0534 new theorem Collect_imp_eq
paulson
parents: 14804
diff changeset
  1152
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1153
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
  1154
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1155
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1156
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
  1157
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1158
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1159
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
  1160
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1161
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1162
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
  1163
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1164
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1165
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
  1166
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1167
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1168
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1169
text {* \medskip @{text insert}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1170
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1171
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
  1172
  -- {* 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
  1173
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1174
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1175
lemma 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
  1176
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1177
17715
68583762ebdb a name for empty_not_insert
paulson
parents: 17702
diff changeset
  1178
lemmas empty_not_insert = insert_not_empty [symmetric, standard]
68583762ebdb a name for empty_not_insert
paulson
parents: 17702
diff changeset
  1179
declare empty_not_insert [simp]
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1180
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1181
lemma 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
  1182
  -- {* @{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
  1183
  -- {* with \emph{quadratic} running time *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1184
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1185
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1186
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
  1187
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1188
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1189
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
  1190
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1191
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1192
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
  1193
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1194
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1195
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
  1196
  -- {* use new @{text B} rather than @{text "A - {a}"} to avoid infinite unfolding *}
14208
144f45277d5a misc tidying
paulson
parents: 14098
diff changeset
  1197
  apply (rule_tac x = "A - {a}" in exI, blast)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1198
  done
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 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
  1201
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1202
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1203
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
  1204
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1205
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1206
lemma insert_inter_insert[simp]: "insert a A \<inter> insert a B = insert a (A \<inter> B)"
14742
dde816115d6a New simp rules added:
mehta
parents: 14692
diff changeset
  1207
  by blast
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1208
13103
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1209
lemma insert_disjoint[simp]:
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1210
 "(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})"
14742
dde816115d6a New simp rules added:
mehta
parents: 14692
diff changeset
  1211
 "({} = insert a A \<inter> B) = (a \<notin> B \<and> {} = A \<inter> B)"
16773
33c4d8fe6f78 tweaked
paulson
parents: 16636
diff changeset
  1212
  by auto
13103
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1213
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1214
lemma disjoint_insert[simp]:
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1215
 "(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})"
14742
dde816115d6a New simp rules added:
mehta
parents: 14692
diff changeset
  1216
 "({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)"
16773
33c4d8fe6f78 tweaked
paulson
parents: 16636
diff changeset
  1217
  by auto
14742
dde816115d6a New simp rules added:
mehta
parents: 14692
diff changeset
  1218
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1219
text {* \medskip @{text image}. *}
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 image_empty [simp]: "f`{} = {}"
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 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
  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 image_constant: "x \<in> A ==> (\<lambda>x. c) ` A = {c}"
16773
33c4d8fe6f78 tweaked
paulson
parents: 16636
diff changeset
  1228
  by auto
12897
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
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
  1231
  by blast
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 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
  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 image_is_empty [iff]: "(f`A = {}) = (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
16773
33c4d8fe6f78 tweaked
paulson
parents: 16636
diff changeset
  1239
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1240
lemma image_Collect: "f ` {x. P x} = {f x | x. P x}"
16773
33c4d8fe6f78 tweaked
paulson
parents: 16636
diff changeset
  1241
  -- {* NOT suitable as a default simprule: the RHS isn't simpler than the LHS,
33c4d8fe6f78 tweaked
paulson
parents: 16636
diff changeset
  1242
      with its implicit quantifier and conjunction.  Also image enjoys better
33c4d8fe6f78 tweaked
paulson
parents: 16636
diff changeset
  1243
      equational properties than does the RHS. *}
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1244
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1245
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1246
lemma if_image_distrib [simp]:
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1247
  "(\<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
  1248
    = (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
  1249
  by (auto simp add: image_def)
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 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
  1252
  by (simp add: image_def)
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
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1255
text {* \medskip @{text range}. *}
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 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
  1258
  by auto
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 range_composition [simp]: "range (\<lambda>x. f (g x)) = f`range g"
14208
144f45277d5a misc tidying
paulson
parents: 14098
diff changeset
  1261
by (subst image_image, simp)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1262
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
text {* \medskip @{text Int} *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1265
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1266
lemma Int_absorb [simp]: "A \<inter> A = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1267
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1268
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1269
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
  1270
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1271
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1272
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
  1273
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1274
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1275
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
  1276
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1277
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1278
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
  1279
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1280
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1281
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
  1282
  -- {* Intersection is an AC-operator *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1283
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1284
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
  1285
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1286
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1287
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
  1288
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1289
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1290
lemma Int_empty_left [simp]: "{} \<inter> B = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1291
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1292
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1293
lemma Int_empty_right [simp]: "A \<inter> {} = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1294
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1295
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1296
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
  1297
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1298
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1299
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
  1300
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1301
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1302
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
  1303
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1304
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1305
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
  1306
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1307
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1308
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
  1309
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1310
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1311
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
  1312
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1313
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1314
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
  1315
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1316
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1317
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
  1318
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1319
15102
04b0e943fcc9 new simprules Int_subset_iff and Un_subset_iff
paulson
parents: 14981
diff changeset
  1320
lemma Int_subset_iff [simp]: "(C \<subseteq> A \<inter> B) = (C \<subseteq> A & C \<subseteq> B)"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1321
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1322
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1323
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
  1324
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1325
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 {* \medskip @{text Un}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1328
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1329
lemma Un_absorb [simp]: "A \<union> A = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1330
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1331
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1332
lemma 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
  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_commute: "A \<union> B = B \<union> 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_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
  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_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
  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
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
  1345
  -- {* Union is an AC-operator *}
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 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
  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_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
  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_empty_left [simp]: "{} \<union> B = B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1354
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1355