src/HOL/Set.thy
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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 = HOL:
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text {* A set in HOL is simply a predicate. *}
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subsection {* Basic syntax *}
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global
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typedecl 'a set
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arities set :: (type) type
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consts
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  "{}"          :: "'a set"                             ("{}")
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  UNIV          :: "'a set"
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  insert        :: "'a => 'a set => 'a set"
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  Collect       :: "('a => bool) => 'a set"              -- "comprehension"
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  Int           :: "'a set => 'a set => 'a set"          (infixl 70)
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  Un            :: "'a set => 'a set => 'a set"          (infixl 65)
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  UNION         :: "'a set => ('a => 'b set) => 'b set"  -- "general union"
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  INTER         :: "'a set => ('a => 'b set) => 'b set"  -- "general intersection"
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  Union         :: "'a set set => 'a set"                -- "union of a set"
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  Inter         :: "'a set set => 'a set"                -- "intersection of a set"
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  Pow           :: "'a set => 'a set set"                -- "powerset"
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  Ball          :: "'a set => ('a => bool) => bool"      -- "bounded universal quantifiers"
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  Bex           :: "'a set => ('a => bool) => bool"      -- "bounded existential quantifiers"
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  image         :: "('a => 'b) => 'a set => 'b set"      (infixr "`" 90)
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syntax
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  "op :"        :: "'a => 'a set => bool"                ("op :")
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consts
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  "op :"        :: "'a => 'a set => bool"                ("(_/ : _)" [50, 51] 50)  -- "membership"
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local
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instance set :: (type) "{ord, minus}" ..
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subsection {* Additional concrete syntax *}
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syntax
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  range         :: "('a => 'b) => 'b set"             -- "of function"
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  "op ~:"       :: "'a => 'a set => bool"                 ("op ~:")  -- "non-membership"
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  "op ~:"       :: "'a => 'a set => bool"                 ("(_/ ~: _)" [50, 51] 50)
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  "@Finset"     :: "args => 'a set"                       ("{(_)}")
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  "@Coll"       :: "pttrn => bool => 'a set"              ("(1{_./ _})")
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  "@SetCompr"   :: "'a => idts => bool => 'a set"         ("(1{_ |/_./ _})")
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  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" 10)
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  "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" 10)
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  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" 10)
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  "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" 10)
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  "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3ALL _:_./ _)" [0, 0, 10] 10)
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  "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3EX _:_./ _)" [0, 0, 10] 10)
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syntax (HOL)
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  "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3! _:_./ _)" [0, 0, 10] 10)
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  "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3? _:_./ _)" [0, 0, 10] 10)
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translations
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  "range f"     == "f`UNIV"
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  "x ~: y"      == "~ (x : y)"
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  "{x, xs}"     == "insert x {xs}"
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  "{x}"         == "insert x {}"
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  "{x. P}"      == "Collect (%x. P)"
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  "UN x y. B"   == "UN x. UN y. B"
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  "UN x. B"     == "UNION UNIV (%x. B)"
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  "UN x. B"     == "UN x:UNIV. B"
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  "INT x y. B"  == "INT x. INT y. B"
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  "INT x. B"    == "INTER UNIV (%x. B)"
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  "INT x. B"    == "INT x:UNIV. B"
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  "UN x:A. B"   == "UNION A (%x. B)"
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  "INT x:A. B"  == "INTER A (%x. B)"
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  "ALL x:A. P"  == "Ball A (%x. P)"
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  "EX x:A. P"   == "Bex A (%x. P)"
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syntax (output)
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  "_setle"      :: "'a set => 'a set => bool"             ("op <=")
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  "_setle"      :: "'a set => 'a set => bool"             ("(_/ <= _)" [50, 51] 50)
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  "_setless"    :: "'a set => 'a set => bool"             ("op <")
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  "_setless"    :: "'a set => 'a set => bool"             ("(_/ < _)" [50, 51] 50)
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syntax (xsymbols)
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  "_setle"      :: "'a set => 'a set => bool"             ("op \<subseteq>")
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  "_setle"      :: "'a set => 'a set => bool"             ("(_/ \<subseteq> _)" [50, 51] 50)
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  "_setless"    :: "'a set => 'a set => bool"             ("op \<subset>")
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  "_setless"    :: "'a set => 'a set => bool"             ("(_/ \<subset> _)" [50, 51] 50)
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  "op Int"      :: "'a set => 'a set => 'a set"           (infixl "\<inter>" 70)
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  "op Un"       :: "'a set => 'a set => 'a set"           (infixl "\<union>" 65)
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  "op :"        :: "'a => 'a set => bool"                 ("op \<in>")
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  "op :"        :: "'a => 'a set => bool"                 ("(_/ \<in> _)" [50, 51] 50)
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  "op ~:"       :: "'a => 'a set => bool"                 ("op \<notin>")
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  "op ~:"       :: "'a => 'a set => bool"                 ("(_/ \<notin> _)" [50, 51] 50)
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  Union         :: "'a set set => 'a set"                 ("\<Union>_" [90] 90)
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  Inter         :: "'a set set => 'a set"                 ("\<Inter>_" [90] 90)
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  "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10)
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  "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10)
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syntax (HTML output)
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  "_setle"      :: "'a set => 'a set => bool"             ("op \<subseteq>")
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  "_setle"      :: "'a set => 'a set => bool"             ("(_/ \<subseteq> _)" [50, 51] 50)
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  "_setless"    :: "'a set => 'a set => bool"             ("op \<subset>")
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  "_setless"    :: "'a set => 'a set => bool"             ("(_/ \<subset> _)" [50, 51] 50)
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  "op Int"      :: "'a set => 'a set => 'a set"           (infixl "\<inter>" 70)
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  "op Un"       :: "'a set => 'a set => 'a set"           (infixl "\<union>" 65)
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  "op :"        :: "'a => 'a set => bool"                 ("op \<in>")
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  "op :"        :: "'a => 'a set => bool"                 ("(_/ \<in> _)" [50, 51] 50)
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  "op ~:"       :: "'a => 'a set => bool"                 ("op \<notin>")
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  "op ~:"       :: "'a => 'a set => bool"                 ("(_/ \<notin> _)" [50, 51] 50)
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  "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10)
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  "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10)
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syntax (xsymbols)
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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,t) = atomic_abs_tr' abs
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            in Syntax.const "@Coll" $ x $ t end
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    end;
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  in [("Collect", setcompr_tr')] end;
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*}
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subsection {* Rules and definitions *}
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text {* Isomorphisms between predicates and sets. *}
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axioms
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  mem_Collect_eq [iff]: "(a : {x. P(x)}) = P(a)"
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  Collect_mem_eq [simp]: "{x. x:A} = A"
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defs
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  Ball_def:     "Ball A P       == ALL x. x:A --> P(x)"
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  Bex_def:      "Bex A P        == EX x. x:A & P(x)"
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defs (overloaded)
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  subset_def:   "A <= B         == ALL x:A. x:B"
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  psubset_def:  "A < B          == (A::'a set) <= B & ~ A=B"
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  Compl_def:    "- A            == {x. ~x:A}"
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  set_diff_def: "A - B          == {x. x:A & ~x:B}"
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defs
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  Un_def:       "A Un B         == {x. x:A | x:B}"
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  Int_def:      "A Int B        == {x. x:A & x:B}"
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  INTER_def:    "INTER A B      == {y. ALL x:A. y: B(x)}"
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  UNION_def:    "UNION A B      == {y. EX x:A. y: B(x)}"
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  Inter_def:    "Inter S        == (INT x:S. x)"
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  Union_def:    "Union S        == (UN x:S. x)"
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  Pow_def:      "Pow A          == {B. B <= A}"
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  empty_def:    "{}             == {x. False}"
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  UNIV_def:     "UNIV           == {x. True}"
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  insert_def:   "insert a B     == {x. x=a} Un B"
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  image_def:    "f`A            == {y. EX x:A. y = f(x)}"
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subsection {* Lemmas and proof tool setup *}
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subsubsection {* Relating predicates and sets *}
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lemma CollectI: "P(a) ==> a : {x. P(x)}"
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  by simp
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lemma CollectD: "a : {x. P(x)} ==> P(a)"
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  by simp
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lemma Collect_cong: "(!!x. P x = Q x) ==> {x. P(x)} = {x. Q(x)}"
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  by simp
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lemmas CollectE = CollectD [elim_format]
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subsubsection {* Bounded quantifiers *}
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lemma ballI [intro!]: "(!!x. x:A ==> P x) ==> ALL x:A. P x"
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  by (simp add: Ball_def)
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lemmas strip = impI allI ballI
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lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x"
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  by (simp add: Ball_def)
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lemma ballE [elim]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q"
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  by (unfold Ball_def) blast
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ML {* bind_thm("rev_ballE",permute_prems 1 1 (thm "ballE")) *}
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text {*
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  \medskip This tactic takes assumptions @{prop "ALL x:A. P x"} and
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  @{prop "a:A"}; creates assumption @{prop "P a"}.
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*}
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ML {*
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  local val ballE = thm "ballE"
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  in fun ball_tac i = etac ballE i THEN contr_tac (i + 1) end;
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*}
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text {*
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  Gives better instantiation for bound:
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*}
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ML_setup {*
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  claset_ref() := claset() addbefore ("bspec", datac (thm "bspec") 1);
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*}
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lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x"
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  -- {* Normally the best argument order: @{prop "P x"} constrains the
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    choice of @{prop "x:A"}. *}
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  by (unfold Bex_def) blast
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lemma rev_bexI [intro?]: "x:A ==> P x ==> EX x:A. P x"
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  -- {* The best argument order when there is only one @{prop "x:A"}. *}
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  by (unfold Bex_def) blast
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lemma bexCI: "(ALL x:A. ~P x ==> P a) ==> a:A ==> EX x:A. P x"
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  by (unfold Bex_def) blast
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lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q"
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  by (unfold Bex_def) blast
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   384
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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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   387
  by (simp add: Ball_def)
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   388
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lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)"
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   390
  -- {* Dual form for existentials. *}
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   391
  by (simp add: Bex_def)
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   392
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lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)"
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  by blast
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   395
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lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)"
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  by blast
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   398
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   399
lemma bex_one_point1 [simp]: "(EX x:A. x = a & P x) = (a:A & P a)"
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  by blast
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   401
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   402
lemma bex_one_point2 [simp]: "(EX x:A. a = x & P x) = (a:A & P a)"
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   403
  by blast
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   404
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   405
lemma ball_one_point1 [simp]: "(ALL x:A. x = a --> P x) = (a:A --> P a)"
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  by blast
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   407
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   408
lemma ball_one_point2 [simp]: "(ALL x:A. a = x --> P x) = (a:A --> P a)"
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   409
  by blast
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   410
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   411
ML_setup {*
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  local
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   413
    val Ball_def = thm "Ball_def";
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   414
    val Bex_def = thm "Bex_def";
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   415
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   416
    val prove_bex_tac =
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   417
      rewrite_goals_tac [Bex_def] THEN Quantifier1.prove_one_point_ex_tac;
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    val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac;
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   419
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   420
    val prove_ball_tac =
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   421
      rewrite_goals_tac [Ball_def] THEN Quantifier1.prove_one_point_all_tac;
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   422
    val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac;
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   423
  in
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   424
    val defBEX_regroup = Simplifier.simproc (Theory.sign_of (the_context ()))
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   425
      "defined BEX" ["EX x:A. P x & Q x"] rearrange_bex;
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   426
    val defBALL_regroup = Simplifier.simproc (Theory.sign_of (the_context ()))
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   427
      "defined BALL" ["ALL x:A. P x --> Q x"] rearrange_ball;
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   428
  end;
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   429
56610e2ba220 sane interface for simprocs;
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   430
  Addsimprocs [defBALL_regroup, defBEX_regroup];
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   431
*}
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   432
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   433
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   434
subsubsection {* Congruence rules *}
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   435
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   436
lemma ball_cong [cong]:
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   437
  "A = B ==> (!!x. x:B ==> P x = Q x) ==>
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   438
    (ALL x:A. P x) = (ALL x:B. Q x)"
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   439
  by (simp add: Ball_def)
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   440
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   441
lemma bex_cong [cong]:
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   442
  "A = B ==> (!!x. x:B ==> P x = Q x) ==>
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diff changeset
   443
    (EX x:A. P x) = (EX x:B. Q x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   444
  by (simp add: Bex_def cong: conj_cong)
1273
6960ec882bca added 8bit pragmas
regensbu
parents: 1068
diff changeset
   445
7238
36e58620ffc8 replaced HOL_quantifiers flag by "HOL" print mode;
wenzelm
parents: 5931
diff changeset
   446
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   447
subsubsection {* Subsets *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   448
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   449
lemma subsetI [intro!]: "(!!x. x:A ==> x:B) ==> A \<subseteq> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   450
  by (simp add: subset_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   451
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   452
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   453
  \medskip Map the type @{text "'a set => anything"} to just @{typ
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   454
  'a}; for overloading constants whose first argument has type @{typ
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   455
  "'a set"}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   456
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   457
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   458
lemma subsetD [elim]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   459
  -- {* Rule in Modus Ponens style. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   460
  by (unfold subset_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   461
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   462
declare subsetD [intro?] -- FIXME
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   463
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   464
lemma rev_subsetD: "c \<in> A ==> A \<subseteq> B ==> c \<in> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   465
  -- {* The same, with reversed premises for use with @{text erule} --
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   466
      cf @{text rev_mp}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   467
  by (rule subsetD)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   468
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   469
declare rev_subsetD [intro?] -- FIXME
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   470
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   471
text {*
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   472
  \medskip Converts @{prop "A \<subseteq> B"} to @{prop "x \<in> A ==> x \<in> B"}.
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   473
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   474
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   475
ML {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   476
  local val rev_subsetD = thm "rev_subsetD"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   477
  in fun impOfSubs th = th RSN (2, rev_subsetD) end;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   478
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   479
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   480
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
   481
  -- {* Classical elimination rule. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   482
  by (unfold subset_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   483
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   484
text {*
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   485
  \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
   486
  creates the assumption @{prop "c \<in> B"}.
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   487
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   488
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   489
ML {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   490
  local val subsetCE = thm "subsetCE"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   491
  in fun set_mp_tac i = etac subsetCE i THEN mp_tac i end;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   492
*}
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 contra_subsetD: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   495
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   496
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   497
lemma subset_refl: "A \<subseteq> A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   498
  by fast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   499
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   500
lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   501
  by blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   502
2261
d926157c0a6a added "op :", "op ~:" syntax;
wenzelm
parents: 2006
diff changeset
   503
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   504
subsubsection {* Equality *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   505
13865
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   506
lemma set_ext: assumes prem: "(!!x. (x:A) = (x:B))" shows "A = B"
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   507
  apply (rule prem [THEN ext, THEN arg_cong, THEN box_equals])
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   508
   apply (rule Collect_mem_eq)
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   509
  apply (rule Collect_mem_eq)
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   510
  done
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   511
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   512
lemma subset_antisym [intro!]: "A \<subseteq> B ==> B \<subseteq> A ==> A = B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   513
  -- {* Anti-symmetry of the subset relation. *}
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   514
  by (rules intro: set_ext subsetD)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   515
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   516
lemmas equalityI [intro!] = subset_antisym
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   517
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   518
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   519
  \medskip Equality rules from ZF set theory -- are they appropriate
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   520
  here?
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   521
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   522
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   523
lemma equalityD1: "A = B ==> A \<subseteq> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   524
  by (simp add: subset_refl)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   525
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   526
lemma equalityD2: "A = B ==> B \<subseteq> A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   527
  by (simp add: subset_refl)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   528
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   529
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   530
  \medskip Be careful when adding this to the claset as @{text
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   531
  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
   532
  \<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}!
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   533
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   534
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   535
lemma equalityE: "A = B ==> (A \<subseteq> B ==> B \<subseteq> A ==> P) ==> P"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   536
  by (simp add: subset_refl)
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   537
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   538
lemma equalityCE [elim]:
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   539
    "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
   540
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   541
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   542
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   543
  \medskip Lemma for creating induction formulae -- for "pattern
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   544
  matching" on @{text p}.  To make the induction hypotheses usable,
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   545
  apply @{text spec} or @{text bspec} to put universal quantifiers over the free
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   546
  variables in @{text p}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   547
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   548
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   549
lemma setup_induction: "p:A ==> (!!z. z:A ==> p = z --> R) ==> R"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   550
  by simp
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   551
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   552
lemma eqset_imp_iff: "A = B ==> (x : A) = (x : B)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   553
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   554
13865
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   555
lemma eqelem_imp_iff: "x = y ==> (x : A) = (y : A)"
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   556
  by simp
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
   557
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   558
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   559
subsubsection {* The universal set -- UNIV *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   560
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   561
lemma UNIV_I [simp]: "x : UNIV"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   562
  by (simp add: UNIV_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   563
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   564
declare UNIV_I [intro]  -- {* unsafe makes it less likely to cause problems *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   565
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   566
lemma UNIV_witness [intro?]: "EX x. x : UNIV"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   567
  by simp
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 subset_UNIV: "A \<subseteq> UNIV"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   570
  by (rule subsetI) (rule UNIV_I)
2388
d1f0505fc602 added set inclusion symbol syntax;
wenzelm
parents: 2372
diff changeset
   571
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   572
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   573
  \medskip Eta-contracting these two rules (to remove @{text P})
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   574
  causes them to be ignored because of their interaction with
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   575
  congruence rules.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   576
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   577
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   578
lemma ball_UNIV [simp]: "Ball UNIV P = All P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   579
  by (simp add: Ball_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   580
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   581
lemma bex_UNIV [simp]: "Bex UNIV P = Ex P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   582
  by (simp add: Bex_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   583
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   584
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   585
subsubsection {* The empty set *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   586
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   587
lemma empty_iff [simp]: "(c : {}) = False"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   588
  by (simp add: empty_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   589
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   590
lemma emptyE [elim!]: "a : {} ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   591
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   592
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   593
lemma empty_subsetI [iff]: "{} \<subseteq> A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   594
    -- {* One effect is to delete the ASSUMPTION @{prop "{} <= A"} *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   595
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   596
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   597
lemma equals0I: "(!!y. y \<in> A ==> False) ==> A = {}"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   598
  by blast
2388
d1f0505fc602 added set inclusion symbol syntax;
wenzelm
parents: 2372
diff changeset
   599
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   600
lemma equals0D: "A = {} ==> a \<notin> A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   601
    -- {* Use for reasoning about disjointness: @{prop "A Int B = {}"} *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   602
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   603
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   604
lemma ball_empty [simp]: "Ball {} P = True"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   605
  by (simp add: Ball_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   606
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   607
lemma bex_empty [simp]: "Bex {} P = False"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   608
  by (simp add: Bex_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   609
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   610
lemma UNIV_not_empty [iff]: "UNIV ~= {}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   611
  by (blast elim: equalityE)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   612
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   613
12023
wenzelm
parents: 12020
diff changeset
   614
subsubsection {* The Powerset operator -- Pow *}
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   615
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   616
lemma Pow_iff [iff]: "(A \<in> Pow B) = (A \<subseteq> B)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   617
  by (simp add: Pow_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   618
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   619
lemma PowI: "A \<subseteq> B ==> A \<in> Pow B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   620
  by (simp add: Pow_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   621
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   622
lemma PowD: "A \<in> Pow B ==> A \<subseteq> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   623
  by (simp add: Pow_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   624
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   625
lemma Pow_bottom: "{} \<in> Pow B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   626
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   627
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   628
lemma Pow_top: "A \<in> Pow A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   629
  by (simp add: subset_refl)
2684
9781d63ef063 added proper subset symbols syntax;
wenzelm
parents: 2412
diff changeset
   630
2388
d1f0505fc602 added set inclusion symbol syntax;
wenzelm
parents: 2372
diff changeset
   631
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   632
subsubsection {* Set complement *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   633
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   634
lemma Compl_iff [simp]: "(c \<in> -A) = (c \<notin> A)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   635
  by (unfold Compl_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   636
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   637
lemma ComplI [intro!]: "(c \<in> A ==> False) ==> c \<in> -A"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   638
  by (unfold Compl_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   639
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   640
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   641
  \medskip This form, with negated conclusion, works well with the
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   642
  Classical prover.  Negated assumptions behave like formulae on the
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   643
  right side of the notional turnstile ... *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   644
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   645
lemma ComplD: "c : -A ==> c~:A"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   646
  by (unfold Compl_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   647
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   648
lemmas ComplE [elim!] = ComplD [elim_format]
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   649
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   650
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   651
subsubsection {* Binary union -- Un *}
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   652
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   653
lemma Un_iff [simp]: "(c : A Un B) = (c:A | c:B)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   654
  by (unfold Un_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   655
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   656
lemma UnI1 [elim?]: "c:A ==> c : A Un B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   657
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   658
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   659
lemma UnI2 [elim?]: "c:B ==> c : A Un B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   660
  by simp
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   661
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   662
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   663
  \medskip Classical introduction rule: no commitment to @{prop A} vs
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   664
  @{prop B}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   665
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   666
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   667
lemma UnCI [intro!]: "(c~:B ==> c:A) ==> c : A Un B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   668
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   669
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   670
lemma UnE [elim!]: "c : A Un B ==> (c:A ==> P) ==> (c:B ==> P) ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   671
  by (unfold Un_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   672
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   673
12023
wenzelm
parents: 12020
diff changeset
   674
subsubsection {* Binary intersection -- Int *}
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   675
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   676
lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   677
  by (unfold Int_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   678
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   679
lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   680
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   681
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   682
lemma IntD1: "c : A Int B ==> c:A"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   683
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   684
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   685
lemma IntD2: "c : A Int B ==> c:B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   686
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   687
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   688
lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   689
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   690
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   691
12023
wenzelm
parents: 12020
diff changeset
   692
subsubsection {* Set difference *}
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   693
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   694
lemma Diff_iff [simp]: "(c : A - B) = (c:A & c~:B)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   695
  by (unfold set_diff_def) blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   696
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   697
lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A - B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   698
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   699
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   700
lemma DiffD1: "c : A - B ==> c : A"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   701
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   702
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   703
lemma DiffD2: "c : A - B ==> c : B ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   704
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   705
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   706
lemma DiffE [elim!]: "c : A - B ==> (c:A ==> c~:B ==> P) ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   707
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   708
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   709
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   710
subsubsection {* Augmenting a set -- insert *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   711
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   712
lemma insert_iff [simp]: "(a : insert b A) = (a = b | a:A)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   713
  by (unfold insert_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   714
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   715
lemma insertI1: "a : insert a B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   716
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   717
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   718
lemma insertI2: "a : B ==> a : insert b B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   719
  by simp
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   720
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   721
lemma insertE [elim!]: "a : insert b A ==> (a = b ==> P) ==> (a:A ==> P) ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   722
  by (unfold insert_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   723
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   724
lemma insertCI [intro!]: "(a~:B ==> a = b) ==> a: insert b B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   725
  -- {* Classical introduction rule. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   726
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   727
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   728
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
   729
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   730
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   731
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   732
subsubsection {* Singletons, using insert *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   733
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   734
lemma singletonI [intro!]: "a : {a}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   735
    -- {* Redundant? But unlike @{text insertCI}, it proves the subgoal immediately! *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   736
  by (rule insertI1)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   737
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   738
lemma singletonD: "b : {a} ==> b = a"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   739
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   740
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   741
lemmas singletonE [elim!] = singletonD [elim_format]
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   742
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   743
lemma singleton_iff: "(b : {a}) = (b = a)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   744
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   745
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   746
lemma singleton_inject [dest!]: "{a} = {b} ==> a = b"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   747
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   748
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   749
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
   750
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   751
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   752
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
   753
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   754
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   755
lemma subset_singletonD: "A \<subseteq> {x} ==> A = {} | A = {x}"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   756
  by fast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   757
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   758
lemma singleton_conv [simp]: "{x. x = a} = {a}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   759
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   760
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   761
lemma singleton_conv2 [simp]: "{x. a = x} = {a}"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   762
  by blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   763
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   764
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
   765
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   766
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   767
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   768
subsubsection {* Unions of families *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   769
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   770
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   771
  @{term [source] "UN x:A. B x"} is @{term "Union (B`A)"}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   772
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   773
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   774
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
   775
  by (unfold UNION_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   776
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   777
lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   778
  -- {* The order of the premises presupposes that @{term A} is rigid;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   779
    @{term b} may be flexible. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   780
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   781
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   782
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
   783
  by (unfold UNION_def) blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   784
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   785
lemma UN_cong [cong]:
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   786
    "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
   787
  by (simp add: UNION_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   788
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   789
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   790
subsubsection {* Intersections of families *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   791
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   792
text {* @{term [source] "INT x:A. B x"} is @{term "Inter (B`A)"}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   793
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   794
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
   795
  by (unfold INTER_def) blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   796
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   797
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
   798
  by (unfold INTER_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   799
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   800
lemma INT_D [elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   801
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   802
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   803
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
   804
  -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   805
  by (unfold INTER_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   806
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   807
lemma INT_cong [cong]:
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   808
    "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
   809
  by (simp add: INTER_def)
7238
36e58620ffc8 replaced HOL_quantifiers flag by "HOL" print mode;
wenzelm
parents: 5931
diff changeset
   810
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   811
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   812
subsubsection {* Union *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   813
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   814
lemma Union_iff [simp]: "(A : Union C) = (EX X:C. A:X)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   815
  by (unfold Union_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   816
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   817
lemma UnionI [intro]: "X:C ==> A:X ==> A : Union C"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   818
  -- {* The order of the premises presupposes that @{term C} is rigid;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   819
    @{term A} may be flexible. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   820
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   821
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   822
lemma UnionE [elim!]: "A : Union C ==> (!!X. A:X ==> X:C ==> R) ==> R"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   823
  by (unfold Union_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   824
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   825
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   826
subsubsection {* Inter *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   827
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   828
lemma Inter_iff [simp]: "(A : Inter C) = (ALL X:C. A:X)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   829
  by (unfold Inter_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   830
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   831
lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   832
  by (simp add: Inter_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   833
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   834
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   835
  \medskip A ``destruct'' rule -- every @{term X} in @{term C}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   836
  contains @{term A} as an element, but @{prop "A:X"} can hold when
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   837
  @{prop "X:C"} does not!  This rule is analogous to @{text spec}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   838
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   839
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   840
lemma InterD [elim]: "A : Inter C ==> X:C ==> A:X"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   841
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   842
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   843
lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   844
  -- {* ``Classical'' elimination rule -- does not require proving
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   845
    @{prop "X:C"}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   846
  by (unfold Inter_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   847
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   848
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   849
  \medskip Image of a set under a function.  Frequently @{term b} does
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   850
  not have the syntactic form of @{term "f x"}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   851
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   852
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   853
lemma image_eqI [simp, intro]: "b = f x ==> x:A ==> b : f`A"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   854
  by (unfold image_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   855
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   856
lemma imageI: "x : A ==> f x : f ` A"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   857
  by (rule image_eqI) (rule refl)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   858
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   859
lemma rev_image_eqI: "x:A ==> b = f x ==> b : f`A"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   860
  -- {* This version's more effective when we already have the
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   861
    required @{term x}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   862
  by (unfold image_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   863
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   864
lemma imageE [elim!]:
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   865
  "b : (%x. f x)`A ==> (!!x. b = f x ==> x:A ==> P) ==> P"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   866
  -- {* The eta-expansion gives variable-name preservation. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   867
  by (unfold image_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   868
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   869
lemma image_Un: "f`(A Un B) = f`A Un f`B"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   870
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   871
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   872
lemma image_iff: "(z : f`A) = (EX x:A. z = f x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   873
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   874
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   875
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
   876
  -- {* This rewrite rule would confuse users if made default. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   877
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   878
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   879
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
   880
  apply safe
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   881
   prefer 2 apply fast
14208
144f45277d5a misc tidying
paulson
parents: 14098
diff changeset
   882
  apply (rule_tac x = "{a. a : A & f a : B}" in exI, fast)
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   883
  done
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   884
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   885
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
   886
  -- {* Replaces the three steps @{text subsetI}, @{text imageE},
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   887
    @{text hypsubst}, but breaks too many existing proofs. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   888
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   889
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   890
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   891
  \medskip Range of a function -- just a translation for image!
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   892
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   893
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   894
lemma range_eqI: "b = f x ==> b \<in> range f"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   895
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   896
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   897
lemma rangeI: "f x \<in> range f"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   898
  by simp
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   899
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   900
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
   901
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   902
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   903
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   904
subsubsection {* Set reasoning tools *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   905
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   906
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   907
  Rewrite rules for boolean case-splitting: faster than @{text
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   908
  "split_if [split]"}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   909
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   910
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   911
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
   912
  by (rule split_if)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   913
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   914
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
   915
  by (rule split_if)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   916
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   917
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   918
  Split ifs on either side of the membership relation.  Not for @{text
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   919
  "[simp]"} -- can cause goals to blow up!
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   920
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   921
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   922
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
   923
  by (rule split_if)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   924
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   925
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
   926
  by (rule split_if)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   927
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   928
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
   929
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   930
lemmas mem_simps =
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   931
  insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   932
  mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   933
  -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   934
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   935
(*Would like to add these, but the existing code only searches for the
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   936
  outer-level constant, which in this case is just "op :"; we instead need
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   937
  to use term-nets to associate patterns with rules.  Also, if a rule fails to
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   938
  apply, then the formula should be kept.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   939
  [("uminus", Compl_iff RS iffD1), ("op -", [Diff_iff RS iffD1]),
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   940
   ("op Int", [IntD1,IntD2]),
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   941
   ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   942
 *)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   943
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   944
ML_setup {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   945
  val mksimps_pairs = [("Ball", [thm "bspec"])] @ mksimps_pairs;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   946
  simpset_ref() := simpset() setmksimps (mksimps mksimps_pairs);
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   947
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   948
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   949
declare subset_UNIV [simp] subset_refl [simp]
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   950
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   951
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   952
subsubsection {* The ``proper subset'' relation *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   953
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   954
lemma psubsetI [intro!]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   955
  by (unfold psubset_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   956
13624
17684cf64fda added the new elim rule psubsetE
paulson
parents: 13550
diff changeset
   957
lemma psubsetE [elim!]: 
17684cf64fda added the new elim rule psubsetE
paulson
parents: 13550
diff changeset
   958
    "[|A \<subset> B;  [|A \<subseteq> B; ~ (B\<subseteq>A)|] ==> R|] ==> R"
17684cf64fda added the new elim rule psubsetE
paulson
parents: 13550
diff changeset
   959
  by (unfold psubset_def) blast
17684cf64fda added the new elim rule psubsetE
paulson
parents: 13550
diff changeset
   960
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   961
lemma psubset_insert_iff:
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   962
  "(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
   963
  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
   964
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   965
lemma psubset_eq: "(A \<subset> B) = (A \<subseteq> B & A \<noteq> B)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   966
  by (simp only: psubset_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   967
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   968
lemma psubset_imp_subset: "A \<subset> B ==> A \<subseteq> B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   969
  by (simp add: psubset_eq)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   970
14335
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
   971
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
   972
apply (unfold psubset_def)
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
   973
apply (auto dest: subset_antisym)
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
   974
done
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
   975
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
   976
lemma psubsetD: "[| A \<subset> B; c \<in> A |] ==> c \<in> B"
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
   977
apply (unfold psubset_def)
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
   978
apply (auto dest: subsetD)
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
   979
done
9c0b5e081037 conversion of Real/PReal to Isar script;
paulson
parents: 14302
diff changeset
   980
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   981
lemma psubset_subset_trans: "A \<subset> B ==> B \<subseteq> C ==> A \<subset> C"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   982
  by (auto simp add: psubset_eq)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   983
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   984
lemma subset_psubset_trans: "A \<subseteq> B ==> B \<subset> C ==> A \<subset> C"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   985
  by (auto simp add: psubset_eq)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   986
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   987
lemma psubset_imp_ex_mem: "A \<subset> B ==> \<exists>b. b \<in> (B - A)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   988
  by (unfold psubset_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   989
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   990
lemma atomize_ball:
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   991
    "(!!x. x \<in> A ==> P x) == Trueprop (\<forall>x\<in>A. P x)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   992
  by (simp only: Ball_def atomize_all atomize_imp)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   993
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   994
declare atomize_ball [symmetric, rulify]
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   995
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   996
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   997
subsection {* Further set-theory lemmas *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   998
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   999
subsubsection {* Derived rules involving subsets. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1000
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1001
text {* @{text insert}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1002
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1003
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
  1004
  apply (rule subsetI)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1005
  apply (erule insertI2)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1006
  done
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1007
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1008
lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1009
by blast
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1010
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1011
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
  1012
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1013
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1014
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1015
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
  1016
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1017
lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1018
  by (rules intro: subsetI UnionI)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1019
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1020
lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1021
  by (rules intro: subsetI elim: UnionE dest: subsetD)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1022
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1023
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1024
text {* \medskip General union. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1025
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1026
lemma 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
  1027
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1028
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1029
lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1030
  by (rules intro: subsetI elim: UN_E dest: subsetD)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1031
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1032
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1033
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
  1034
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1035
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
  1036
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1037
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents: 14479
diff changeset
  1038
lemma Inter_subset:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents: 14479
diff changeset
  1039
  "[| !!X. X \<in> A ==> X \<subseteq> B; A ~= {} |] ==> \<Inter>A \<subseteq> B"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents: 14479
diff changeset
  1040
  by blast
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents: 14479
diff changeset
  1041
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1042
lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1043
  by (rules intro: InterI subsetI dest: subsetD)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1044
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1045
lemma 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
  1046
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1047
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1048
lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1049
  by (rules intro: INT_I subsetI dest: subsetD)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1050
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1051
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1052
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
  1053
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1054
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
  1055
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1056
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1057
lemma Un_upper2: "B \<subseteq> A \<union> B"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1058
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1059
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1060
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
  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
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1064
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
  1065
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1066
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
  1067
  by blast
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 Int_lower2: "A \<inter> B \<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
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1072
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
  1073
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1074
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1075
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1076
text {* \medskip Set difference. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1077
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1078
lemma Diff_subset: "A - B \<subseteq> A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1079
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1080
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1081
lemma Diff_subset_conv: "(A - B \<subseteq> C) = (A \<subseteq> B \<union> C)"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1082
by blast
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1083
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
text {* \medskip Monotonicity. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1086
13421
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13113
diff changeset
  1087
lemma mono_Un: includes mono shows "f A \<union> f B \<subseteq> f (A \<union> B)"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1088
  apply (rule Un_least)
13421
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13113
diff changeset
  1089
   apply (rule Un_upper1 [THEN mono])
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13113
diff changeset
  1090
  apply (rule Un_upper2 [THEN mono])
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1091
  done
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1092
13421
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13113
diff changeset
  1093
lemma mono_Int: includes mono shows "f (A \<inter> B) \<subseteq> f A \<inter> f B"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1094
  apply (rule Int_greatest)
13421
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13113
diff changeset
  1095
   apply (rule Int_lower1 [THEN mono])
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13113
diff changeset
  1096
  apply (rule Int_lower2 [THEN mono])
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1097
  done
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1098
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
subsubsection {* Equalities involving union, intersection, inclusion, etc. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1101
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1102
text {* @{text "{}"}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1103
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1104
lemma 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
  1105
  -- {* supersedes @{text "Collect_False_empty"} *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1106
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1107
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1108
lemma subset_empty [simp]: "(A \<subseteq> {}) = (A = {})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1109
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1110
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1111
lemma not_psubset_empty [iff]: "\<not> (A < {})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1112
  by (unfold psubset_def) blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1113
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1114
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
  1115
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1116
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1117
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
  1118
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1119
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1120
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
  1121
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1122
14812
4b2c006d0534 new theorem Collect_imp_eq
paulson
parents: 14804
diff changeset
  1123
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
  1124
  by blast
4b2c006d0534 new theorem Collect_imp_eq
paulson
parents: 14804
diff changeset
  1125
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1126
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
  1127
  by blast
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
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
  1130
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1131
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1132
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
  1133
  by blast
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 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
  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 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
  1139
  by 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
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1142
text {* \medskip @{text insert}. *}
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 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
  1145
  -- {* 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
  1146
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1147
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1148
lemma 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
  1149
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1150
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1151
lemmas empty_not_insert [simp] = insert_not_empty [symmetric, standard]
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1152
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1153
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
  1154
  -- {* @{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
  1155
  -- {* with \emph{quadratic} running time *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1156
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1157
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1158
lemma 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
  1159
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1160
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1161
lemma 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
  1162
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1163
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1164
lemma 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
  1165
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1166
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1167
lemma 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
  1168
  -- {* use new @{text B} rather than @{text "A - {a}"} to avoid infinite unfolding *}
14208
144f45277d5a misc tidying
paulson
parents: 14098
diff changeset
  1169
  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
  1170
  done
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1171
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1172
lemma 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
  1173
  by auto
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 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
  1176
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1177
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1178
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
  1179
  by blast
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14208
diff changeset
  1180
13103
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1181
lemma insert_disjoint[simp]:
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1182
 "(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})"
14742
dde816115d6a New simp rules added:
mehta
parents: 14692
diff changeset
  1183
 "({} = insert a A \<inter> B) = (a \<notin> B \<and> {} = A \<inter> B)"
dde816115d6a New simp rules added:
mehta
parents: 14692
diff changeset
  1184
by auto
13103
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1185
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1186
lemma disjoint_insert[simp]:
66659a4b16f6 Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents: 12937
diff changeset
  1187
 "(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})"
14742
dde816115d6a New simp rules added:
mehta
parents: 14692
diff changeset
  1188
 "({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)"
dde816115d6a New simp rules added:
mehta
parents: 14692
diff changeset
  1189
by auto
dde816115d6a New simp rules added:
mehta
parents: 14692
diff changeset
  1190
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1191
text {* \medskip @{text image}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1192
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1193
lemma image_empty [simp]: "f`{} = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1194
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1195
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1196
lemma 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
  1197
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1198
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1199
lemma image_constant: "x \<in> A ==> (\<lambda>x. c) ` A = {c}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1200
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1201
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1202
lemma 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
  1203
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1204
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1205
lemma 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
  1206
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1207
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1208
lemma image_is_empty [iff]: "(f`A = {}) = (A = {})"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1209
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1210
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1211
lemma image_Collect: "f ` {x. P x} = {f x | x. P x}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1212
  -- {* NOT suitable as a default simprule: the RHS isn't simpler than the LHS, *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1213
  -- {* with its implicit quantifier and conjunction.  Also image enjoys better *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1214
  -- {* equational properties than does the RHS. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1215
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1216
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1217
lemma if_image_distrib [simp]:
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1218
  "(\<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
  1219
    = (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
  1220
  by (auto simp add: image_def)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1221
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1222
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
  1223
  by (simp add: image_def)
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1224
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1225
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1226
text {* \medskip @{text range}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1227
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1228
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
  1229
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1230
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1231
lemma range_composition [simp]: "range (\<lambda>x. f (g x)) = f`range g"
14208
144f45277d5a misc tidying
paulson
parents: 14098
diff changeset
  1232
by (subst image_image, simp)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1233
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1234
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1235
text {* \medskip @{text Int} *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1236
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1237
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
  1238
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1239
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1240
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
  1241
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1242
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1243
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
  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 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
  1247
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1248
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1249
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
  1250
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1251
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1252
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
  1253
  -- {* Intersection is an AC-operator *}
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
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
  1256
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1257
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1258
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
  1259
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1260
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1261
lemma Int_empty_left [simp]: "{} \<inter> B = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1262
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1263
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1264
lemma Int_empty_right [simp]: "A \<inter> {} = {}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1265
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1266
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1267
lemma 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
  1268
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1269
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1270
lemma 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
  1271
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1272
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1273
lemma 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
  1274
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1275
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1276
lemma 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
  1277
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1278
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1279
lemma 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
  1280
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1281
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1282
lemma 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
  1283
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1284
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1285
lemma 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
  1286
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1287
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1288
lemma 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
  1289
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1290
15102
04b0e943fcc9 new simprules Int_subset_iff and Un_subset_iff
paulson
parents: 14981
diff changeset
  1291
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
  1292
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1293
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1294
lemma 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
  1295
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1296
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1297
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1298
text {* \medskip @{text Un}. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1299
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1300
lemma Un_absorb [simp]: "A \<union> A = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1301
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1302
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1303
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
  1304
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1305
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1306
lemma 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
  1307
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1308
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1309
lemma 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
  1310
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1311
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1312
lemma 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
  1313
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1314
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1315
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
  1316
  -- {* Union is an AC-operator *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1317
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1318
lemma 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
  1319
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1320
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1321
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
  1322
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1323
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1324
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
  1325
  by blast
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
lemma Un_empty_right [simp]: "A \<union> {} = A"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1328
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1329
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1330
lemma Un_UNIV_left [simp]: "UNIV \<union> B = UNIV"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1331
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1332
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1333
lemma Un_UNIV_right [simp]: "A \<union> UNIV = UNIV"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1334
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1335
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1336
lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1337
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1338
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1339
lemma Un_insert_left [simp]: "(insert a B) \<union> C = insert a (B \<union> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1340
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1341
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1342
lemma Un_insert_right [simp]: "A \<union> (insert a B) = insert a (A \<union> B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1343
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1344
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1345
lemma Int_insert_left:
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1346
    "(insert a B) Int C = (if a \<in> C then insert a (B \<inter> C) else B \<inter> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1347
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1348
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1349
lemma Int_insert_right:
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1350
    "A \<inter> (insert a B) = (if a \<in> A then insert a (A \<inter> B) else A \<inter> B)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1351
  by auto
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_Int_distrib: "A \<union> (B \<inter> C) = (A \<union> B) \<inter> (A \<union> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1354
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1355
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1356
lemma Un_Int_distrib2: "(B \<inter> C) \<union> A = (B \<union> A) \<inter> (C \<union> A)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1357
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1358