src/HOL/Set.thy
author wenzelm
Wed Feb 28 22:05:43 2007 +0100 (2007-02-28)
changeset 22377 61610b1beedf
parent 22172 e7d6cb237b5e
child 22439 b709739c69e6
permissions -rw-r--r--
tuned ML setup;
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(*  Title:      HOL/Set.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
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*)
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header {* Set theory for higher-order logic *}
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theory Set
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imports Lattices
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begin
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text {* A set in HOL is simply a predicate. *}
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subsection {* Basic syntax *}
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global
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typedecl 'a set
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arities set :: (type) type
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consts
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  "{}"          :: "'a set"                             ("{}")
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  UNIV          :: "'a set"
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  insert        :: "'a => 'a set => 'a set"
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  Collect       :: "('a => bool) => 'a set"              -- "comprehension"
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  Int           :: "'a set => 'a set => 'a set"          (infixl 70)
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  Un            :: "'a set => 'a set => 'a set"          (infixl 65)
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  UNION         :: "'a set => ('a => 'b set) => 'b set"  -- "general union"
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  INTER         :: "'a set => ('a => 'b set) => 'b set"  -- "general intersection"
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  Union         :: "'a set set => 'a set"                -- "union of a set"
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  Inter         :: "'a set set => 'a set"                -- "intersection of a set"
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  Pow           :: "'a set => 'a set set"                -- "powerset"
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  Ball          :: "'a set => ('a => bool) => bool"      -- "bounded universal quantifiers"
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  Bex           :: "'a set => ('a => bool) => bool"      -- "bounded existential quantifiers"
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  Bex1          :: "'a set => ('a => bool) => bool"      -- "bounded unique existential quantifiers"
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  image         :: "('a => 'b) => 'a set => 'b set"      (infixr "`" 90)
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  "op :"        :: "'a => 'a set => bool"                -- "membership"
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notation
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  "op :"  ("op :") and
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  "op :"  ("(_/ : _)" [50, 51] 50)
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local
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subsection {* Additional concrete syntax *}
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abbreviation
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  range :: "('a => 'b) => 'b set" where -- "of function"
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  "range f == f ` UNIV"
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abbreviation
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  "not_mem x A == ~ (x : A)" -- "non-membership"
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notation
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  not_mem  ("op ~:") and
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  not_mem  ("(_/ ~: _)" [50, 51] 50)
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notation (xsymbols)
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  "op Int"  (infixl "\<inter>" 70) and
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  "op Un"  (infixl "\<union>" 65) and
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  "op :"  ("op \<in>") and
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  "op :"  ("(_/ \<in> _)" [50, 51] 50) and
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  not_mem  ("op \<notin>") and
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  not_mem  ("(_/ \<notin> _)" [50, 51] 50) and
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  Union  ("\<Union>_" [90] 90) and
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  Inter  ("\<Inter>_" [90] 90)
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notation (HTML output)
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  "op Int"  (infixl "\<inter>" 70) and
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  "op Un"  (infixl "\<union>" 65) and
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  "op :"  ("op \<in>") and
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  "op :"  ("(_/ \<in> _)" [50, 51] 50) and
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  not_mem  ("op \<notin>") and
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  not_mem  ("(_/ \<notin> _)" [50, 51] 50)
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syntax
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  "@Finset"     :: "args => 'a set"                       ("{(_)}")
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  "@Coll"       :: "pttrn => bool => 'a set"              ("(1{_./ _})")
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  "@SetCompr"   :: "'a => idts => bool => 'a set"         ("(1{_ |/_./ _})")
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  "@Collect"    :: "idt => 'a set => bool => 'a set"      ("(1{_ :/ _./ _})")
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  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" 10)
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  "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" 10)
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  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" 10)
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  "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" 10)
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  "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3ALL _:_./ _)" [0, 0, 10] 10)
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  "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3EX _:_./ _)" [0, 0, 10] 10)
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  "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3EX! _:_./ _)" [0, 0, 10] 10)
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  "_Bleast"       :: "id => 'a set => bool => 'a"      ("(3LEAST _:_./ _)" [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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  "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3?! _:_./ _)" [0, 0, 10] 10)
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translations
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  "{x, xs}"     == "insert x {xs}"
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  "{x}"         == "insert x {}"
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  "{x. P}"      == "Collect (%x. P)"
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  "{x:A. P}"    => "{x. x:A & P}"
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  "UN x y. B"   == "UN x. UN y. B"
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  "UN x. B"     == "UNION UNIV (%x. B)"
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  "UN x. B"     == "UN x:UNIV. B"
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  "INT x y. B"  == "INT x. INT y. B"
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  "INT x. B"    == "INTER UNIV (%x. B)"
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  "INT x. B"    == "INT x:UNIV. B"
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  "UN x:A. B"   == "UNION A (%x. B)"
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  "INT x:A. B"  == "INTER A (%x. B)"
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  "ALL x:A. P"  == "Ball A (%x. P)"
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  "EX x:A. P"   == "Bex A (%x. P)"
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  "EX! x:A. P"  == "Bex1 A (%x. P)"
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  "LEAST x:A. P" => "LEAST x. x:A & P"
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syntax (xsymbols)
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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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  "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3\<exists>!_\<in>_./ _)" [0, 0, 10] 10)
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  "_Bleast"     :: "id => 'a set => bool => 'a"           ("(3LEAST_\<in>_./ _)" [0, 0, 10] 10)
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syntax (HTML output)
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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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  "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3\<exists>!_\<in>_./ _)" [0, 0, 10] 10)
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syntax (xsymbols)
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  "@Collect"    :: "idt => 'a set => bool => 'a set"      ("(1{_ \<in>/ _./ _})")
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  "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" 10)
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  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" 10)
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  "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" 10)
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  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" 10)
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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{*
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  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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instance set :: (type) ord
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  subset_def:   "A <= B         == \<forall>x\<in>A. x \<in> B"
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  psubset_def:  "A < B          == (A::'a set) <= B & ~ A=B" ..
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abbreviation
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  subset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
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  "subset \<equiv> less"
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abbreviation
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  subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
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  "subset_eq \<equiv> less_eq"
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notation (output)
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  subset  ("op <") and
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  subset  ("(_/ < _)" [50, 51] 50) and
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  subset_eq  ("op <=") and
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  subset_eq  ("(_/ <= _)" [50, 51] 50)
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notation (xsymbols)
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  subset  ("op \<subset>") and
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  subset  ("(_/ \<subset> _)" [50, 51] 50) and
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  subset_eq  ("op \<subseteq>") and
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  subset_eq  ("(_/ \<subseteq> _)" [50, 51] 50)
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notation (HTML output)
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  subset  ("op \<subset>") and
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  subset  ("(_/ \<subset> _)" [50, 51] 50) and
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  subset_eq  ("op \<subseteq>") and
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  subset_eq  ("(_/ \<subseteq> _)" [50, 51] 50)
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abbreviation (input)
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  supset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
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  "supset \<equiv> greater"
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abbreviation (input)
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  supset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
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  "supset_eq \<equiv> greater_eq"
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notation (xsymbols)
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  supset  ("op \<supset>") and
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  supset  ("(_/ \<supset> _)" [50, 51] 50) and
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  supset_eq  ("op \<supseteq>") and
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  supset_eq  ("(_/ \<supseteq> _)" [50, 51] 50)
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subsubsection "Bounded quantifiers"
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syntax (output)
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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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  "_setleEx1"   :: "[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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  "_setleEx1"   :: "[idt, 'a, bool] => bool"   ("(3\<exists>!_\<subseteq>_./ _)" [0, 0, 10] 10)
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syntax (HOL output)
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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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  "_setleEx1"   :: "[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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  "_setleEx1"   :: "[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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 "\<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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  val Type (set_type, _) = @{typ "'a set"};
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  val All_binder = Syntax.binder_name @{const_syntax "All"};
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  val Ex_binder = Syntax.binder_name @{const_syntax "Ex"};
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  val impl = @{const_syntax "op -->"};
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  val conj = @{const_syntax "op &"};
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  val sbset = @{const_syntax "subset"};
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  val sbset_eq = @{const_syntax "subset_eq"};
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  val trans =
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   [((All_binder, impl, sbset), "_setlessAll"),
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    ((All_binder, impl, sbset_eq), "_setleAll"),
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    ((Ex_binder, conj, sbset), "_setlessEx"),
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    ((Ex_binder, conj, sbset_eq), "_setleEx")];
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  fun mk v v' c n P =
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    if v = v' andalso not (Term.exists_subterm (fn Free (x, _) => x = v | _ => false) n)
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    then Syntax.const c $ Syntax.mark_bound v' $ n $ P else raise Match;
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  fun tr' q = (q,
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    fn [Const ("_bound", _) $ Free (v, Type (T, _)), Const (c, _) $ (Const (d, _) $ (Const ("_bound", _) $ Free (v', _)) $ n) $ P] =>
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         if T = (set_type) then case AList.lookup (op =) trans (q, c, d)
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          of NONE => raise Match
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           | SOME l => mk v v' l n P
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         else raise Match
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     | _ => raise Match);
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in
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  [tr' All_binder, tr' Ex_binder]
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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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   271
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   272
    fun nvars (Const ("_idts", _) $ _ $ idts) = nvars idts + 1
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      | nvars _ = 1;
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   274
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   275
    fun setcompr_tr [e, idts, b] =
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   276
      let
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        val eq = Syntax.const "op =" $ Bound (nvars idts) $ e;
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   278
        val P = Syntax.const "op &" $ eq $ b;
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   279
        val exP = ex_tr [idts, P];
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   280
      in Syntax.const "Collect" $ Term.absdummy (dummyT, exP) end;
wenzelm@11979
   281
wenzelm@11979
   282
  in [("@SetCompr", setcompr_tr)] end;
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   283
*}
clasohm@923
   284
nipkow@13763
   285
(* To avoid eta-contraction of body: *)
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   286
print_translation {*
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   287
let
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   288
  fun btr' syn [A,Abs abs] =
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   289
    let val (x,t) = atomic_abs_tr' abs
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   290
    in Syntax.const syn $ x $ A $ t end
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   291
in
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   292
[("Ball", btr' "_Ball"),("Bex", btr' "_Bex"),
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   293
 ("UNION", btr' "@UNION"),("INTER", btr' "@INTER")]
nipkow@13763
   294
end
nipkow@13763
   295
*}
nipkow@13763
   296
nipkow@13763
   297
print_translation {*
nipkow@13763
   298
let
nipkow@13763
   299
  val ex_tr' = snd (mk_binder_tr' ("Ex", "DUMMY"));
nipkow@13763
   300
nipkow@13763
   301
  fun setcompr_tr' [Abs (abs as (_, _, P))] =
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   302
    let
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   303
      fun check (Const ("Ex", _) $ Abs (_, _, P), n) = check (P, n + 1)
nipkow@13763
   304
        | check (Const ("op &", _) $ (Const ("op =", _) $ Bound m $ e) $ P, n) =
nipkow@13763
   305
            n > 0 andalso m = n andalso not (loose_bvar1 (P, n)) andalso
nipkow@13763
   306
            ((0 upto (n - 1)) subset add_loose_bnos (e, 0, []))
nipkow@13764
   307
        | check _ = false
clasohm@923
   308
wenzelm@11979
   309
        fun tr' (_ $ abs) =
wenzelm@11979
   310
          let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' [abs]
wenzelm@11979
   311
          in Syntax.const "@SetCompr" $ e $ idts $ Q end;
nipkow@13763
   312
    in if check (P, 0) then tr' P
nipkow@15535
   313
       else let val (x as _ $ Free(xN,_), t) = atomic_abs_tr' abs
nipkow@15535
   314
                val M = Syntax.const "@Coll" $ x $ t
nipkow@15535
   315
            in case t of
nipkow@15535
   316
                 Const("op &",_)
nipkow@15535
   317
                   $ (Const("op :",_) $ (Const("_bound",_) $ Free(yN,_)) $ A)
nipkow@15535
   318
                   $ P =>
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   319
                   if xN=yN then Syntax.const "@Collect" $ x $ A $ P else M
nipkow@15535
   320
               | _ => M
nipkow@15535
   321
            end
nipkow@13763
   322
    end;
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   323
  in [("Collect", setcompr_tr')] end;
wenzelm@11979
   324
*}
wenzelm@11979
   325
wenzelm@11979
   326
wenzelm@11979
   327
subsection {* Rules and definitions *}
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   328
wenzelm@11979
   329
text {* Isomorphisms between predicates and sets. *}
clasohm@923
   330
wenzelm@11979
   331
axioms
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   332
  mem_Collect_eq: "(a : {x. P(x)}) = P(a)"
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   333
  Collect_mem_eq: "{x. x:A} = A"
wenzelm@17702
   334
finalconsts
wenzelm@17702
   335
  Collect
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   336
  "op :"
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   337
wenzelm@11979
   338
defs
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   339
  Ball_def:     "Ball A P       == ALL x. x:A --> P(x)"
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   340
  Bex_def:      "Bex A P        == EX x. x:A & P(x)"
webertj@20217
   341
  Bex1_def:     "Bex1 A P       == EX! x. x:A & P(x)"
wenzelm@11979
   342
haftmann@21333
   343
instance set :: (type) minus
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   344
  Compl_def:    "- A            == {x. ~x:A}"
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   345
  set_diff_def: "A - B          == {x. x:A & ~x:B}" ..
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   346
clasohm@923
   347
defs
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   348
  Un_def:       "A Un B         == {x. x:A | x:B}"
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   349
  Int_def:      "A Int B        == {x. x:A & x:B}"
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   350
  INTER_def:    "INTER A B      == {y. ALL x:A. y: B(x)}"
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   351
  UNION_def:    "UNION A B      == {y. EX x:A. y: B(x)}"
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   352
  Inter_def:    "Inter S        == (INT x:S. x)"
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   353
  Union_def:    "Union S        == (UN x:S. x)"
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   354
  Pow_def:      "Pow A          == {B. B <= A}"
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   355
  empty_def:    "{}             == {x. False}"
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   356
  UNIV_def:     "UNIV           == {x. True}"
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   357
  insert_def:   "insert a B     == {x. x=a} Un B"
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   358
  image_def:    "f`A            == {y. EX x:A. y = f(x)}"
wenzelm@11979
   359
wenzelm@11979
   360
wenzelm@11979
   361
subsection {* Lemmas and proof tool setup *}
wenzelm@11979
   362
wenzelm@11979
   363
subsubsection {* Relating predicates and sets *}
wenzelm@11979
   364
paulson@17085
   365
declare mem_Collect_eq [iff]  Collect_mem_eq [simp]
paulson@17085
   366
wenzelm@12257
   367
lemma CollectI: "P(a) ==> a : {x. P(x)}"
wenzelm@11979
   368
  by simp
wenzelm@11979
   369
wenzelm@11979
   370
lemma CollectD: "a : {x. P(x)} ==> P(a)"
wenzelm@11979
   371
  by simp
wenzelm@11979
   372
wenzelm@11979
   373
lemma Collect_cong: "(!!x. P x = Q x) ==> {x. P(x)} = {x. Q(x)}"
wenzelm@11979
   374
  by simp
wenzelm@11979
   375
wenzelm@12257
   376
lemmas CollectE = CollectD [elim_format]
wenzelm@11979
   377
wenzelm@11979
   378
wenzelm@11979
   379
subsubsection {* Bounded quantifiers *}
wenzelm@11979
   380
wenzelm@11979
   381
lemma ballI [intro!]: "(!!x. x:A ==> P x) ==> ALL x:A. P x"
wenzelm@11979
   382
  by (simp add: Ball_def)
wenzelm@11979
   383
wenzelm@11979
   384
lemmas strip = impI allI ballI
wenzelm@11979
   385
wenzelm@11979
   386
lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x"
wenzelm@11979
   387
  by (simp add: Ball_def)
wenzelm@11979
   388
wenzelm@11979
   389
lemma ballE [elim]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q"
wenzelm@11979
   390
  by (unfold Ball_def) blast
wenzelm@22139
   391
wenzelm@22139
   392
ML {* bind_thm ("rev_ballE", permute_prems 1 1 @{thm ballE}) *}
wenzelm@11979
   393
wenzelm@11979
   394
text {*
wenzelm@11979
   395
  \medskip This tactic takes assumptions @{prop "ALL x:A. P x"} and
wenzelm@11979
   396
  @{prop "a:A"}; creates assumption @{prop "P a"}.
wenzelm@11979
   397
*}
wenzelm@11979
   398
wenzelm@11979
   399
ML {*
wenzelm@22139
   400
  fun ball_tac i = etac @{thm ballE} i THEN contr_tac (i + 1)
wenzelm@11979
   401
*}
wenzelm@11979
   402
wenzelm@11979
   403
text {*
wenzelm@11979
   404
  Gives better instantiation for bound:
wenzelm@11979
   405
*}
wenzelm@11979
   406
wenzelm@11979
   407
ML_setup {*
wenzelm@22139
   408
  change_claset (fn cs => cs addbefore ("bspec", datac @{thm bspec} 1))
wenzelm@11979
   409
*}
wenzelm@11979
   410
wenzelm@11979
   411
lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x"
wenzelm@11979
   412
  -- {* Normally the best argument order: @{prop "P x"} constrains the
wenzelm@11979
   413
    choice of @{prop "x:A"}. *}
wenzelm@11979
   414
  by (unfold Bex_def) blast
wenzelm@11979
   415
wenzelm@13113
   416
lemma rev_bexI [intro?]: "x:A ==> P x ==> EX x:A. P x"
wenzelm@11979
   417
  -- {* The best argument order when there is only one @{prop "x:A"}. *}
wenzelm@11979
   418
  by (unfold Bex_def) blast
wenzelm@11979
   419
wenzelm@11979
   420
lemma bexCI: "(ALL x:A. ~P x ==> P a) ==> a:A ==> EX x:A. P x"
wenzelm@11979
   421
  by (unfold Bex_def) blast
wenzelm@11979
   422
wenzelm@11979
   423
lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q"
wenzelm@11979
   424
  by (unfold Bex_def) blast
wenzelm@11979
   425
wenzelm@11979
   426
lemma ball_triv [simp]: "(ALL x:A. P) = ((EX x. x:A) --> P)"
wenzelm@11979
   427
  -- {* Trival rewrite rule. *}
wenzelm@11979
   428
  by (simp add: Ball_def)
wenzelm@11979
   429
wenzelm@11979
   430
lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)"
wenzelm@11979
   431
  -- {* Dual form for existentials. *}
wenzelm@11979
   432
  by (simp add: Bex_def)
wenzelm@11979
   433
wenzelm@11979
   434
lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)"
wenzelm@11979
   435
  by blast
wenzelm@11979
   436
wenzelm@11979
   437
lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)"
wenzelm@11979
   438
  by blast
wenzelm@11979
   439
wenzelm@11979
   440
lemma bex_one_point1 [simp]: "(EX x:A. x = a & P x) = (a:A & P a)"
wenzelm@11979
   441
  by blast
wenzelm@11979
   442
wenzelm@11979
   443
lemma bex_one_point2 [simp]: "(EX x:A. a = x & P x) = (a:A & P a)"
wenzelm@11979
   444
  by blast
wenzelm@11979
   445
wenzelm@11979
   446
lemma ball_one_point1 [simp]: "(ALL x:A. x = a --> P x) = (a:A --> P a)"
wenzelm@11979
   447
  by blast
wenzelm@11979
   448
wenzelm@11979
   449
lemma ball_one_point2 [simp]: "(ALL x:A. a = x --> P x) = (a:A --> P a)"
wenzelm@11979
   450
  by blast
wenzelm@11979
   451
wenzelm@11979
   452
ML_setup {*
wenzelm@13462
   453
  local
wenzelm@22139
   454
    val unfold_bex_tac = unfold_tac @{thms "Bex_def"};
wenzelm@18328
   455
    fun prove_bex_tac ss = unfold_bex_tac ss THEN Quantifier1.prove_one_point_ex_tac;
wenzelm@11979
   456
    val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac;
wenzelm@11979
   457
wenzelm@22139
   458
    val unfold_ball_tac = unfold_tac @{thms "Ball_def"};
wenzelm@18328
   459
    fun prove_ball_tac ss = unfold_ball_tac ss THEN Quantifier1.prove_one_point_all_tac;
wenzelm@11979
   460
    val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac;
wenzelm@11979
   461
  in
wenzelm@18328
   462
    val defBEX_regroup = Simplifier.simproc (the_context ())
wenzelm@13462
   463
      "defined BEX" ["EX x:A. P x & Q x"] rearrange_bex;
wenzelm@18328
   464
    val defBALL_regroup = Simplifier.simproc (the_context ())
wenzelm@13462
   465
      "defined BALL" ["ALL x:A. P x --> Q x"] rearrange_ball;
wenzelm@11979
   466
  end;
wenzelm@13462
   467
wenzelm@13462
   468
  Addsimprocs [defBALL_regroup, defBEX_regroup];
wenzelm@11979
   469
*}
wenzelm@11979
   470
wenzelm@11979
   471
wenzelm@11979
   472
subsubsection {* Congruence rules *}
wenzelm@11979
   473
berghofe@16636
   474
lemma ball_cong:
wenzelm@11979
   475
  "A = B ==> (!!x. x:B ==> P x = Q x) ==>
wenzelm@11979
   476
    (ALL x:A. P x) = (ALL x:B. Q x)"
wenzelm@11979
   477
  by (simp add: Ball_def)
wenzelm@11979
   478
berghofe@16636
   479
lemma strong_ball_cong [cong]:
berghofe@16636
   480
  "A = B ==> (!!x. x:B =simp=> P x = Q x) ==>
berghofe@16636
   481
    (ALL x:A. P x) = (ALL x:B. Q x)"
berghofe@16636
   482
  by (simp add: simp_implies_def Ball_def)
berghofe@16636
   483
berghofe@16636
   484
lemma bex_cong:
wenzelm@11979
   485
  "A = B ==> (!!x. x:B ==> P x = Q x) ==>
wenzelm@11979
   486
    (EX x:A. P x) = (EX x:B. Q x)"
wenzelm@11979
   487
  by (simp add: Bex_def cong: conj_cong)
regensbu@1273
   488
berghofe@16636
   489
lemma strong_bex_cong [cong]:
berghofe@16636
   490
  "A = B ==> (!!x. x:B =simp=> P x = Q x) ==>
berghofe@16636
   491
    (EX x:A. P x) = (EX x:B. Q x)"
berghofe@16636
   492
  by (simp add: simp_implies_def Bex_def cong: conj_cong)
berghofe@16636
   493
wenzelm@7238
   494
wenzelm@11979
   495
subsubsection {* Subsets *}
wenzelm@11979
   496
paulson@19295
   497
lemma subsetI [atp,intro!]: "(!!x. x:A ==> x:B) ==> A \<subseteq> B"
wenzelm@11979
   498
  by (simp add: subset_def)
wenzelm@11979
   499
wenzelm@11979
   500
text {*
wenzelm@11979
   501
  \medskip Map the type @{text "'a set => anything"} to just @{typ
wenzelm@11979
   502
  'a}; for overloading constants whose first argument has type @{typ
wenzelm@11979
   503
  "'a set"}.
wenzelm@11979
   504
*}
wenzelm@11979
   505
wenzelm@12897
   506
lemma subsetD [elim]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B"
wenzelm@11979
   507
  -- {* Rule in Modus Ponens style. *}
wenzelm@11979
   508
  by (unfold subset_def) blast
wenzelm@11979
   509
wenzelm@11979
   510
declare subsetD [intro?] -- FIXME
wenzelm@11979
   511
wenzelm@12897
   512
lemma rev_subsetD: "c \<in> A ==> A \<subseteq> B ==> c \<in> B"
wenzelm@11979
   513
  -- {* The same, with reversed premises for use with @{text erule} --
wenzelm@11979
   514
      cf @{text rev_mp}. *}
wenzelm@11979
   515
  by (rule subsetD)
wenzelm@11979
   516
wenzelm@11979
   517
declare rev_subsetD [intro?] -- FIXME
wenzelm@11979
   518
wenzelm@11979
   519
text {*
wenzelm@12897
   520
  \medskip Converts @{prop "A \<subseteq> B"} to @{prop "x \<in> A ==> x \<in> B"}.
wenzelm@11979
   521
*}
wenzelm@11979
   522
wenzelm@11979
   523
ML {*
wenzelm@22139
   524
  fun impOfSubs th = th RSN (2, @{thm rev_subsetD})
wenzelm@11979
   525
*}
wenzelm@11979
   526
wenzelm@12897
   527
lemma subsetCE [elim]: "A \<subseteq>  B ==> (c \<notin> A ==> P) ==> (c \<in> B ==> P) ==> P"
wenzelm@11979
   528
  -- {* Classical elimination rule. *}
wenzelm@11979
   529
  by (unfold subset_def) blast
wenzelm@11979
   530
wenzelm@11979
   531
text {*
wenzelm@12897
   532
  \medskip Takes assumptions @{prop "A \<subseteq> B"}; @{prop "c \<in> A"} and
wenzelm@12897
   533
  creates the assumption @{prop "c \<in> B"}.
wenzelm@11979
   534
*}
wenzelm@11979
   535
wenzelm@11979
   536
ML {*
wenzelm@22139
   537
  fun set_mp_tac i = etac @{thm subsetCE} i THEN mp_tac i
wenzelm@11979
   538
*}
wenzelm@11979
   539
wenzelm@12897
   540
lemma contra_subsetD: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A"
wenzelm@11979
   541
  by blast
wenzelm@11979
   542
paulson@19175
   543
lemma subset_refl [simp,atp]: "A \<subseteq> A"
wenzelm@11979
   544
  by fast
wenzelm@11979
   545
wenzelm@12897
   546
lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C"
wenzelm@11979
   547
  by blast
clasohm@923
   548
wenzelm@2261
   549
wenzelm@11979
   550
subsubsection {* Equality *}
wenzelm@11979
   551
paulson@13865
   552
lemma set_ext: assumes prem: "(!!x. (x:A) = (x:B))" shows "A = B"
paulson@13865
   553
  apply (rule prem [THEN ext, THEN arg_cong, THEN box_equals])
paulson@13865
   554
   apply (rule Collect_mem_eq)
paulson@13865
   555
  apply (rule Collect_mem_eq)
paulson@13865
   556
  done
paulson@13865
   557
nipkow@15554
   558
(* Due to Brian Huffman *)
nipkow@15554
   559
lemma expand_set_eq: "(A = B) = (ALL x. (x:A) = (x:B))"
nipkow@15554
   560
by(auto intro:set_ext)
nipkow@15554
   561
wenzelm@12897
   562
lemma subset_antisym [intro!]: "A \<subseteq> B ==> B \<subseteq> A ==> A = B"
wenzelm@11979
   563
  -- {* Anti-symmetry of the subset relation. *}
nipkow@17589
   564
  by (iprover intro: set_ext subsetD)
wenzelm@12897
   565
wenzelm@12897
   566
lemmas equalityI [intro!] = subset_antisym
wenzelm@11979
   567
wenzelm@11979
   568
text {*
wenzelm@11979
   569
  \medskip Equality rules from ZF set theory -- are they appropriate
wenzelm@11979
   570
  here?
wenzelm@11979
   571
*}
wenzelm@11979
   572
wenzelm@12897
   573
lemma equalityD1: "A = B ==> A \<subseteq> B"
wenzelm@11979
   574
  by (simp add: subset_refl)
wenzelm@11979
   575
wenzelm@12897
   576
lemma equalityD2: "A = B ==> B \<subseteq> A"
wenzelm@11979
   577
  by (simp add: subset_refl)
wenzelm@11979
   578
wenzelm@11979
   579
text {*
wenzelm@11979
   580
  \medskip Be careful when adding this to the claset as @{text
wenzelm@11979
   581
  subset_empty} is in the simpset: @{prop "A = {}"} goes to @{prop "{}
wenzelm@12897
   582
  \<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}!
wenzelm@11979
   583
*}
wenzelm@11979
   584
wenzelm@12897
   585
lemma equalityE: "A = B ==> (A \<subseteq> B ==> B \<subseteq> A ==> P) ==> P"
wenzelm@11979
   586
  by (simp add: subset_refl)
clasohm@923
   587
wenzelm@11979
   588
lemma equalityCE [elim]:
wenzelm@12897
   589
    "A = B ==> (c \<in> A ==> c \<in> B ==> P) ==> (c \<notin> A ==> c \<notin> B ==> P) ==> P"
wenzelm@11979
   590
  by blast
wenzelm@11979
   591
wenzelm@11979
   592
lemma eqset_imp_iff: "A = B ==> (x : A) = (x : B)"
wenzelm@11979
   593
  by simp
wenzelm@11979
   594
paulson@13865
   595
lemma eqelem_imp_iff: "x = y ==> (x : A) = (y : A)"
paulson@13865
   596
  by simp
paulson@13865
   597
wenzelm@11979
   598
wenzelm@11979
   599
subsubsection {* The universal set -- UNIV *}
wenzelm@11979
   600
wenzelm@11979
   601
lemma UNIV_I [simp]: "x : UNIV"
wenzelm@11979
   602
  by (simp add: UNIV_def)
wenzelm@11979
   603
wenzelm@11979
   604
declare UNIV_I [intro]  -- {* unsafe makes it less likely to cause problems *}
wenzelm@11979
   605
wenzelm@11979
   606
lemma UNIV_witness [intro?]: "EX x. x : UNIV"
wenzelm@11979
   607
  by simp
wenzelm@11979
   608
paulson@18144
   609
lemma subset_UNIV [simp]: "A \<subseteq> UNIV"
wenzelm@11979
   610
  by (rule subsetI) (rule UNIV_I)
wenzelm@2388
   611
wenzelm@11979
   612
text {*
wenzelm@11979
   613
  \medskip Eta-contracting these two rules (to remove @{text P})
wenzelm@11979
   614
  causes them to be ignored because of their interaction with
wenzelm@11979
   615
  congruence rules.
wenzelm@11979
   616
*}
wenzelm@11979
   617
wenzelm@11979
   618
lemma ball_UNIV [simp]: "Ball UNIV P = All P"
wenzelm@11979
   619
  by (simp add: Ball_def)
wenzelm@11979
   620
wenzelm@11979
   621
lemma bex_UNIV [simp]: "Bex UNIV P = Ex P"
wenzelm@11979
   622
  by (simp add: Bex_def)
wenzelm@11979
   623
wenzelm@11979
   624
wenzelm@11979
   625
subsubsection {* The empty set *}
wenzelm@11979
   626
wenzelm@11979
   627
lemma empty_iff [simp]: "(c : {}) = False"
wenzelm@11979
   628
  by (simp add: empty_def)
wenzelm@11979
   629
wenzelm@11979
   630
lemma emptyE [elim!]: "a : {} ==> P"
wenzelm@11979
   631
  by simp
wenzelm@11979
   632
wenzelm@12897
   633
lemma empty_subsetI [iff]: "{} \<subseteq> A"
wenzelm@11979
   634
    -- {* One effect is to delete the ASSUMPTION @{prop "{} <= A"} *}
wenzelm@11979
   635
  by blast
wenzelm@11979
   636
wenzelm@12897
   637
lemma equals0I: "(!!y. y \<in> A ==> False) ==> A = {}"
wenzelm@11979
   638
  by blast
wenzelm@2388
   639
wenzelm@12897
   640
lemma equals0D: "A = {} ==> a \<notin> A"
wenzelm@11979
   641
    -- {* Use for reasoning about disjointness: @{prop "A Int B = {}"} *}
wenzelm@11979
   642
  by blast
wenzelm@11979
   643
wenzelm@11979
   644
lemma ball_empty [simp]: "Ball {} P = True"
wenzelm@11979
   645
  by (simp add: Ball_def)
wenzelm@11979
   646
wenzelm@11979
   647
lemma bex_empty [simp]: "Bex {} P = False"
wenzelm@11979
   648
  by (simp add: Bex_def)
wenzelm@11979
   649
wenzelm@11979
   650
lemma UNIV_not_empty [iff]: "UNIV ~= {}"
wenzelm@11979
   651
  by (blast elim: equalityE)
wenzelm@11979
   652
wenzelm@11979
   653
wenzelm@12023
   654
subsubsection {* The Powerset operator -- Pow *}
wenzelm@11979
   655
wenzelm@12897
   656
lemma Pow_iff [iff]: "(A \<in> Pow B) = (A \<subseteq> B)"
wenzelm@11979
   657
  by (simp add: Pow_def)
wenzelm@11979
   658
wenzelm@12897
   659
lemma PowI: "A \<subseteq> B ==> A \<in> Pow B"
wenzelm@11979
   660
  by (simp add: Pow_def)
wenzelm@11979
   661
wenzelm@12897
   662
lemma PowD: "A \<in> Pow B ==> A \<subseteq> B"
wenzelm@11979
   663
  by (simp add: Pow_def)
wenzelm@11979
   664
wenzelm@12897
   665
lemma Pow_bottom: "{} \<in> Pow B"
wenzelm@11979
   666
  by simp
wenzelm@11979
   667
wenzelm@12897
   668
lemma Pow_top: "A \<in> Pow A"
wenzelm@11979
   669
  by (simp add: subset_refl)
wenzelm@2684
   670
wenzelm@2388
   671
wenzelm@11979
   672
subsubsection {* Set complement *}
wenzelm@11979
   673
wenzelm@12897
   674
lemma Compl_iff [simp]: "(c \<in> -A) = (c \<notin> A)"
wenzelm@11979
   675
  by (unfold Compl_def) blast
wenzelm@11979
   676
wenzelm@12897
   677
lemma ComplI [intro!]: "(c \<in> A ==> False) ==> c \<in> -A"
wenzelm@11979
   678
  by (unfold Compl_def) blast
wenzelm@11979
   679
wenzelm@11979
   680
text {*
wenzelm@11979
   681
  \medskip This form, with negated conclusion, works well with the
wenzelm@11979
   682
  Classical prover.  Negated assumptions behave like formulae on the
wenzelm@11979
   683
  right side of the notional turnstile ... *}
wenzelm@11979
   684
paulson@17084
   685
lemma ComplD [dest!]: "c : -A ==> c~:A"
wenzelm@11979
   686
  by (unfold Compl_def) blast
wenzelm@11979
   687
paulson@17084
   688
lemmas ComplE = ComplD [elim_format]
wenzelm@11979
   689
wenzelm@11979
   690
wenzelm@11979
   691
subsubsection {* Binary union -- Un *}
clasohm@923
   692
wenzelm@11979
   693
lemma Un_iff [simp]: "(c : A Un B) = (c:A | c:B)"
wenzelm@11979
   694
  by (unfold Un_def) blast
wenzelm@11979
   695
wenzelm@11979
   696
lemma UnI1 [elim?]: "c:A ==> c : A Un B"
wenzelm@11979
   697
  by simp
wenzelm@11979
   698
wenzelm@11979
   699
lemma UnI2 [elim?]: "c:B ==> c : A Un B"
wenzelm@11979
   700
  by simp
clasohm@923
   701
wenzelm@11979
   702
text {*
wenzelm@11979
   703
  \medskip Classical introduction rule: no commitment to @{prop A} vs
wenzelm@11979
   704
  @{prop B}.
wenzelm@11979
   705
*}
wenzelm@11979
   706
wenzelm@11979
   707
lemma UnCI [intro!]: "(c~:B ==> c:A) ==> c : A Un B"
wenzelm@11979
   708
  by auto
wenzelm@11979
   709
wenzelm@11979
   710
lemma UnE [elim!]: "c : A Un B ==> (c:A ==> P) ==> (c:B ==> P) ==> P"
wenzelm@11979
   711
  by (unfold Un_def) blast
wenzelm@11979
   712
wenzelm@11979
   713
wenzelm@12023
   714
subsubsection {* Binary intersection -- Int *}
clasohm@923
   715
wenzelm@11979
   716
lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)"
wenzelm@11979
   717
  by (unfold Int_def) blast
wenzelm@11979
   718
wenzelm@11979
   719
lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B"
wenzelm@11979
   720
  by simp
wenzelm@11979
   721
wenzelm@11979
   722
lemma IntD1: "c : A Int B ==> c:A"
wenzelm@11979
   723
  by simp
wenzelm@11979
   724
wenzelm@11979
   725
lemma IntD2: "c : A Int B ==> c:B"
wenzelm@11979
   726
  by simp
wenzelm@11979
   727
wenzelm@11979
   728
lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P"
wenzelm@11979
   729
  by simp
wenzelm@11979
   730
wenzelm@11979
   731
wenzelm@12023
   732
subsubsection {* Set difference *}
wenzelm@11979
   733
wenzelm@11979
   734
lemma Diff_iff [simp]: "(c : A - B) = (c:A & c~:B)"
wenzelm@11979
   735
  by (unfold set_diff_def) blast
clasohm@923
   736
wenzelm@11979
   737
lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A - B"
wenzelm@11979
   738
  by simp
wenzelm@11979
   739
wenzelm@11979
   740
lemma DiffD1: "c : A - B ==> c : A"
wenzelm@11979
   741
  by simp
wenzelm@11979
   742
wenzelm@11979
   743
lemma DiffD2: "c : A - B ==> c : B ==> P"
wenzelm@11979
   744
  by simp
wenzelm@11979
   745
wenzelm@11979
   746
lemma DiffE [elim!]: "c : A - B ==> (c:A ==> c~:B ==> P) ==> P"
wenzelm@11979
   747
  by simp
wenzelm@11979
   748
wenzelm@11979
   749
wenzelm@11979
   750
subsubsection {* Augmenting a set -- insert *}
wenzelm@11979
   751
wenzelm@11979
   752
lemma insert_iff [simp]: "(a : insert b A) = (a = b | a:A)"
wenzelm@11979
   753
  by (unfold insert_def) blast
wenzelm@11979
   754
wenzelm@11979
   755
lemma insertI1: "a : insert a B"
wenzelm@11979
   756
  by simp
wenzelm@11979
   757
wenzelm@11979
   758
lemma insertI2: "a : B ==> a : insert b B"
wenzelm@11979
   759
  by simp
clasohm@923
   760
wenzelm@11979
   761
lemma insertE [elim!]: "a : insert b A ==> (a = b ==> P) ==> (a:A ==> P) ==> P"
wenzelm@11979
   762
  by (unfold insert_def) blast
wenzelm@11979
   763
wenzelm@11979
   764
lemma insertCI [intro!]: "(a~:B ==> a = b) ==> a: insert b B"
wenzelm@11979
   765
  -- {* Classical introduction rule. *}
wenzelm@11979
   766
  by auto
wenzelm@11979
   767
wenzelm@12897
   768
lemma subset_insert_iff: "(A \<subseteq> insert x B) = (if x:A then A - {x} \<subseteq> B else A \<subseteq> B)"
wenzelm@11979
   769
  by auto
wenzelm@11979
   770
wenzelm@11979
   771
wenzelm@11979
   772
subsubsection {* Singletons, using insert *}
wenzelm@11979
   773
wenzelm@11979
   774
lemma singletonI [intro!]: "a : {a}"
wenzelm@11979
   775
    -- {* Redundant? But unlike @{text insertCI}, it proves the subgoal immediately! *}
wenzelm@11979
   776
  by (rule insertI1)
wenzelm@11979
   777
paulson@17084
   778
lemma singletonD [dest!]: "b : {a} ==> b = a"
wenzelm@11979
   779
  by blast
wenzelm@11979
   780
paulson@17084
   781
lemmas singletonE = singletonD [elim_format]
wenzelm@11979
   782
wenzelm@11979
   783
lemma singleton_iff: "(b : {a}) = (b = a)"
wenzelm@11979
   784
  by blast
wenzelm@11979
   785
wenzelm@11979
   786
lemma singleton_inject [dest!]: "{a} = {b} ==> a = b"
wenzelm@11979
   787
  by blast
wenzelm@11979
   788
wenzelm@12897
   789
lemma singleton_insert_inj_eq [iff]: "({b} = insert a A) = (a = b & A \<subseteq> {b})"
wenzelm@11979
   790
  by blast
wenzelm@11979
   791
wenzelm@12897
   792
lemma singleton_insert_inj_eq' [iff]: "(insert a A = {b}) = (a = b & A \<subseteq> {b})"
wenzelm@11979
   793
  by blast
wenzelm@11979
   794
wenzelm@12897
   795
lemma subset_singletonD: "A \<subseteq> {x} ==> A = {} | A = {x}"
wenzelm@11979
   796
  by fast
wenzelm@11979
   797
wenzelm@11979
   798
lemma singleton_conv [simp]: "{x. x = a} = {a}"
wenzelm@11979
   799
  by blast
wenzelm@11979
   800
wenzelm@11979
   801
lemma singleton_conv2 [simp]: "{x. a = x} = {a}"
wenzelm@11979
   802
  by blast
clasohm@923
   803
wenzelm@12897
   804
lemma diff_single_insert: "A - {x} \<subseteq> B ==> x \<in> A ==> A \<subseteq> insert x B"
wenzelm@11979
   805
  by blast
wenzelm@11979
   806
paulson@19870
   807
lemma doubleton_eq_iff: "({a,b} = {c,d}) = (a=c & b=d | a=d & b=c)"
paulson@19870
   808
  by (blast elim: equalityE)
paulson@19870
   809
wenzelm@11979
   810
wenzelm@11979
   811
subsubsection {* Unions of families *}
wenzelm@11979
   812
wenzelm@11979
   813
text {*
wenzelm@11979
   814
  @{term [source] "UN x:A. B x"} is @{term "Union (B`A)"}.
wenzelm@11979
   815
*}
wenzelm@11979
   816
wenzelm@11979
   817
lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)"
wenzelm@11979
   818
  by (unfold UNION_def) blast
wenzelm@11979
   819
wenzelm@11979
   820
lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)"
wenzelm@11979
   821
  -- {* The order of the premises presupposes that @{term A} is rigid;
wenzelm@11979
   822
    @{term b} may be flexible. *}
wenzelm@11979
   823
  by auto
wenzelm@11979
   824
wenzelm@11979
   825
lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R"
wenzelm@11979
   826
  by (unfold UNION_def) blast
clasohm@923
   827
wenzelm@11979
   828
lemma UN_cong [cong]:
wenzelm@11979
   829
    "A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
wenzelm@11979
   830
  by (simp add: UNION_def)
wenzelm@11979
   831
wenzelm@11979
   832
wenzelm@11979
   833
subsubsection {* Intersections of families *}
wenzelm@11979
   834
wenzelm@11979
   835
text {* @{term [source] "INT x:A. B x"} is @{term "Inter (B`A)"}. *}
wenzelm@11979
   836
wenzelm@11979
   837
lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)"
wenzelm@11979
   838
  by (unfold INTER_def) blast
clasohm@923
   839
wenzelm@11979
   840
lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)"
wenzelm@11979
   841
  by (unfold INTER_def) blast
wenzelm@11979
   842
wenzelm@11979
   843
lemma INT_D [elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a"
wenzelm@11979
   844
  by auto
wenzelm@11979
   845
wenzelm@11979
   846
lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R"
wenzelm@11979
   847
  -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}
wenzelm@11979
   848
  by (unfold INTER_def) blast
wenzelm@11979
   849
wenzelm@11979
   850
lemma INT_cong [cong]:
wenzelm@11979
   851
    "A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)"
wenzelm@11979
   852
  by (simp add: INTER_def)
wenzelm@7238
   853
clasohm@923
   854
wenzelm@11979
   855
subsubsection {* Union *}
wenzelm@11979
   856
wenzelm@11979
   857
lemma Union_iff [simp]: "(A : Union C) = (EX X:C. A:X)"
wenzelm@11979
   858
  by (unfold Union_def) blast
wenzelm@11979
   859
wenzelm@11979
   860
lemma UnionI [intro]: "X:C ==> A:X ==> A : Union C"
wenzelm@11979
   861
  -- {* The order of the premises presupposes that @{term C} is rigid;
wenzelm@11979
   862
    @{term A} may be flexible. *}
wenzelm@11979
   863
  by auto
wenzelm@11979
   864
wenzelm@11979
   865
lemma UnionE [elim!]: "A : Union C ==> (!!X. A:X ==> X:C ==> R) ==> R"
wenzelm@11979
   866
  by (unfold Union_def) blast
wenzelm@11979
   867
wenzelm@11979
   868
wenzelm@11979
   869
subsubsection {* Inter *}
wenzelm@11979
   870
wenzelm@11979
   871
lemma Inter_iff [simp]: "(A : Inter C) = (ALL X:C. A:X)"
wenzelm@11979
   872
  by (unfold Inter_def) blast
wenzelm@11979
   873
wenzelm@11979
   874
lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C"
wenzelm@11979
   875
  by (simp add: Inter_def)
wenzelm@11979
   876
wenzelm@11979
   877
text {*
wenzelm@11979
   878
  \medskip A ``destruct'' rule -- every @{term X} in @{term C}
wenzelm@11979
   879
  contains @{term A} as an element, but @{prop "A:X"} can hold when
wenzelm@11979
   880
  @{prop "X:C"} does not!  This rule is analogous to @{text spec}.
wenzelm@11979
   881
*}
wenzelm@11979
   882
wenzelm@11979
   883
lemma InterD [elim]: "A : Inter C ==> X:C ==> A:X"
wenzelm@11979
   884
  by auto
wenzelm@11979
   885
wenzelm@11979
   886
lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R"
wenzelm@11979
   887
  -- {* ``Classical'' elimination rule -- does not require proving
wenzelm@11979
   888
    @{prop "X:C"}. *}
wenzelm@11979
   889
  by (unfold Inter_def) blast
wenzelm@11979
   890
wenzelm@11979
   891
text {*
wenzelm@11979
   892
  \medskip Image of a set under a function.  Frequently @{term b} does
wenzelm@11979
   893
  not have the syntactic form of @{term "f x"}.
wenzelm@11979
   894
*}
wenzelm@11979
   895
wenzelm@11979
   896
lemma image_eqI [simp, intro]: "b = f x ==> x:A ==> b : f`A"
wenzelm@11979
   897
  by (unfold image_def) blast
wenzelm@11979
   898
wenzelm@11979
   899
lemma imageI: "x : A ==> f x : f ` A"
wenzelm@11979
   900
  by (rule image_eqI) (rule refl)
wenzelm@11979
   901
wenzelm@11979
   902
lemma rev_image_eqI: "x:A ==> b = f x ==> b : f`A"
wenzelm@11979
   903
  -- {* This version's more effective when we already have the
wenzelm@11979
   904
    required @{term x}. *}
wenzelm@11979
   905
  by (unfold image_def) blast
wenzelm@11979
   906
wenzelm@11979
   907
lemma imageE [elim!]:
wenzelm@11979
   908
  "b : (%x. f x)`A ==> (!!x. b = f x ==> x:A ==> P) ==> P"
wenzelm@11979
   909
  -- {* The eta-expansion gives variable-name preservation. *}
wenzelm@11979
   910
  by (unfold image_def) blast
wenzelm@11979
   911
wenzelm@11979
   912
lemma image_Un: "f`(A Un B) = f`A Un f`B"
wenzelm@11979
   913
  by blast
wenzelm@11979
   914
wenzelm@11979
   915
lemma image_iff: "(z : f`A) = (EX x:A. z = f x)"
wenzelm@11979
   916
  by blast
wenzelm@11979
   917
wenzelm@12897
   918
lemma image_subset_iff: "(f`A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)"
wenzelm@11979
   919
  -- {* This rewrite rule would confuse users if made default. *}
wenzelm@11979
   920
  by blast
wenzelm@11979
   921
wenzelm@12897
   922
lemma subset_image_iff: "(B \<subseteq> f`A) = (EX AA. AA \<subseteq> A & B = f`AA)"
wenzelm@11979
   923
  apply safe
wenzelm@11979
   924
   prefer 2 apply fast
paulson@14208
   925
  apply (rule_tac x = "{a. a : A & f a : B}" in exI, fast)
wenzelm@11979
   926
  done
wenzelm@11979
   927
wenzelm@12897
   928
lemma image_subsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f`A \<subseteq> B"
wenzelm@11979
   929
  -- {* Replaces the three steps @{text subsetI}, @{text imageE},
wenzelm@11979
   930
    @{text hypsubst}, but breaks too many existing proofs. *}
wenzelm@11979
   931
  by blast
wenzelm@11979
   932
wenzelm@11979
   933
text {*
wenzelm@11979
   934
  \medskip Range of a function -- just a translation for image!
wenzelm@11979
   935
*}
wenzelm@11979
   936
wenzelm@12897
   937
lemma range_eqI: "b = f x ==> b \<in> range f"
wenzelm@11979
   938
  by simp
wenzelm@11979
   939
wenzelm@12897
   940
lemma rangeI: "f x \<in> range f"
wenzelm@11979
   941
  by simp
wenzelm@11979
   942
wenzelm@12897
   943
lemma rangeE [elim?]: "b \<in> range (\<lambda>x. f x) ==> (!!x. b = f x ==> P) ==> P"
wenzelm@11979
   944
  by blast
wenzelm@11979
   945
wenzelm@11979
   946
wenzelm@11979
   947
subsubsection {* Set reasoning tools *}
wenzelm@11979
   948
wenzelm@11979
   949
text {*
wenzelm@11979
   950
  Rewrite rules for boolean case-splitting: faster than @{text
wenzelm@11979
   951
  "split_if [split]"}.
wenzelm@11979
   952
*}
wenzelm@11979
   953
wenzelm@11979
   954
lemma split_if_eq1: "((if Q then x else y) = b) = ((Q --> x = b) & (~ Q --> y = b))"
wenzelm@11979
   955
  by (rule split_if)
wenzelm@11979
   956
wenzelm@11979
   957
lemma split_if_eq2: "(a = (if Q then x else y)) = ((Q --> a = x) & (~ Q --> a = y))"
wenzelm@11979
   958
  by (rule split_if)
wenzelm@11979
   959
wenzelm@11979
   960
text {*
wenzelm@11979
   961
  Split ifs on either side of the membership relation.  Not for @{text
wenzelm@11979
   962
  "[simp]"} -- can cause goals to blow up!
wenzelm@11979
   963
*}
wenzelm@11979
   964
wenzelm@11979
   965
lemma split_if_mem1: "((if Q then x else y) : b) = ((Q --> x : b) & (~ Q --> y : b))"
wenzelm@11979
   966
  by (rule split_if)
wenzelm@11979
   967
wenzelm@11979
   968
lemma split_if_mem2: "(a : (if Q then x else y)) = ((Q --> a : x) & (~ Q --> a : y))"
wenzelm@11979
   969
  by (rule split_if)
wenzelm@11979
   970
wenzelm@11979
   971
lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2
wenzelm@11979
   972
wenzelm@11979
   973
lemmas mem_simps =
wenzelm@11979
   974
  insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
wenzelm@11979
   975
  mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
wenzelm@11979
   976
  -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
wenzelm@11979
   977
wenzelm@11979
   978
(*Would like to add these, but the existing code only searches for the
wenzelm@11979
   979
  outer-level constant, which in this case is just "op :"; we instead need
wenzelm@11979
   980
  to use term-nets to associate patterns with rules.  Also, if a rule fails to
wenzelm@11979
   981
  apply, then the formula should be kept.
haftmann@19233
   982
  [("HOL.uminus", Compl_iff RS iffD1), ("HOL.minus", [Diff_iff RS iffD1]),
wenzelm@11979
   983
   ("op Int", [IntD1,IntD2]),
wenzelm@11979
   984
   ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]
wenzelm@11979
   985
 *)
wenzelm@11979
   986
wenzelm@11979
   987
ML_setup {*
wenzelm@22139
   988
  val mksimps_pairs = [("Ball", @{thms bspec})] @ mksimps_pairs;
wenzelm@17875
   989
  change_simpset (fn ss => ss setmksimps (mksimps mksimps_pairs));
wenzelm@11979
   990
*}
wenzelm@11979
   991
wenzelm@11979
   992
wenzelm@11979
   993
subsubsection {* The ``proper subset'' relation *}
wenzelm@11979
   994
wenzelm@12897
   995
lemma psubsetI [intro!]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B"
wenzelm@11979
   996
  by (unfold psubset_def) blast
wenzelm@11979
   997
paulson@13624
   998
lemma psubsetE [elim!]: 
paulson@13624
   999
    "[|A \<subset> B;  [|A \<subseteq> B; ~ (B\<subseteq>A)|] ==> R|] ==> R"
paulson@13624
  1000
  by (unfold psubset_def) blast
paulson@13624
  1001
wenzelm@11979
  1002
lemma psubset_insert_iff:
wenzelm@12897
  1003
  "(A \<subset> insert x B) = (if x \<in> B then A \<subset> B else if x \<in> A then A - {x} \<subset> B else A \<subseteq> B)"
wenzelm@12897
  1004
  by (auto simp add: psubset_def subset_insert_iff)
wenzelm@12897
  1005
wenzelm@12897
  1006
lemma psubset_eq: "(A \<subset> B) = (A \<subseteq> B & A \<noteq> B)"
wenzelm@11979
  1007
  by (simp only: psubset_def)
wenzelm@11979
  1008
wenzelm@12897
  1009
lemma psubset_imp_subset: "A \<subset> B ==> A \<subseteq> B"
wenzelm@11979
  1010
  by (simp add: psubset_eq)
wenzelm@11979
  1011
paulson@14335
  1012
lemma psubset_trans: "[| A \<subset> B; B \<subset> C |] ==> A \<subset> C"
paulson@14335
  1013
apply (unfold psubset_def)
paulson@14335
  1014
apply (auto dest: subset_antisym)
paulson@14335
  1015
done
paulson@14335
  1016
paulson@14335
  1017
lemma psubsetD: "[| A \<subset> B; c \<in> A |] ==> c \<in> B"
paulson@14335
  1018
apply (unfold psubset_def)
paulson@14335
  1019
apply (auto dest: subsetD)
paulson@14335
  1020
done
paulson@14335
  1021
wenzelm@12897
  1022
lemma psubset_subset_trans: "A \<subset> B ==> B \<subseteq> C ==> A \<subset> C"
wenzelm@11979
  1023
  by (auto simp add: psubset_eq)
wenzelm@11979
  1024
wenzelm@12897
  1025
lemma subset_psubset_trans: "A \<subseteq> B ==> B \<subset> C ==> A \<subset> C"
wenzelm@11979
  1026
  by (auto simp add: psubset_eq)
wenzelm@11979
  1027
wenzelm@12897
  1028
lemma psubset_imp_ex_mem: "A \<subset> B ==> \<exists>b. b \<in> (B - A)"
wenzelm@11979
  1029
  by (unfold psubset_def) blast
wenzelm@11979
  1030
wenzelm@11979
  1031
lemma atomize_ball:
wenzelm@12897
  1032
    "(!!x. x \<in> A ==> P x) == Trueprop (\<forall>x\<in>A. P x)"
wenzelm@11979
  1033
  by (simp only: Ball_def atomize_all atomize_imp)
wenzelm@11979
  1034
wenzelm@18832
  1035
lemmas [symmetric, rulify] = atomize_ball
wenzelm@18832
  1036
  and [symmetric, defn] = atomize_ball
wenzelm@11979
  1037
wenzelm@11979
  1038
wenzelm@11979
  1039
subsection {* Further set-theory lemmas *}
wenzelm@11979
  1040
haftmann@21384
  1041
instance set :: (type) order
haftmann@21384
  1042
  by (intro_classes,
haftmann@21384
  1043
      (assumption | rule subset_refl subset_trans subset_antisym psubset_eq)+)
haftmann@21384
  1044
wenzelm@12897
  1045
subsubsection {* Derived rules involving subsets. *}
wenzelm@12897
  1046
wenzelm@12897
  1047
text {* @{text insert}. *}
wenzelm@12897
  1048
wenzelm@12897
  1049
lemma subset_insertI: "B \<subseteq> insert a B"
wenzelm@12897
  1050
  apply (rule subsetI)
wenzelm@12897
  1051
  apply (erule insertI2)
wenzelm@12897
  1052
  done
wenzelm@12897
  1053
nipkow@14302
  1054
lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B"
nipkow@14302
  1055
by blast
nipkow@14302
  1056
wenzelm@12897
  1057
lemma subset_insert: "x \<notin> A ==> (A \<subseteq> insert x B) = (A \<subseteq> B)"
wenzelm@12897
  1058
  by blast
wenzelm@12897
  1059
wenzelm@12897
  1060
wenzelm@12897
  1061
text {* \medskip Big Union -- least upper bound of a set. *}
wenzelm@12897
  1062
wenzelm@12897
  1063
lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A"
nipkow@17589
  1064
  by (iprover intro: subsetI UnionI)
wenzelm@12897
  1065
wenzelm@12897
  1066
lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C"
nipkow@17589
  1067
  by (iprover intro: subsetI elim: UnionE dest: subsetD)
wenzelm@12897
  1068
wenzelm@12897
  1069
wenzelm@12897
  1070
text {* \medskip General union. *}
wenzelm@12897
  1071
wenzelm@12897
  1072
lemma UN_upper: "a \<in> A ==> B a \<subseteq> (\<Union>x\<in>A. B x)"
wenzelm@12897
  1073
  by blast
wenzelm@12897
  1074
wenzelm@12897
  1075
lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C"
nipkow@17589
  1076
  by (iprover intro: subsetI elim: UN_E dest: subsetD)
wenzelm@12897
  1077
wenzelm@12897
  1078
wenzelm@12897
  1079
text {* \medskip Big Intersection -- greatest lower bound of a set. *}
wenzelm@12897
  1080
wenzelm@12897
  1081
lemma Inter_lower: "B \<in> A ==> Inter A \<subseteq> B"
wenzelm@12897
  1082
  by blast
wenzelm@12897
  1083
ballarin@14551
  1084
lemma Inter_subset:
ballarin@14551
  1085
  "[| !!X. X \<in> A ==> X \<subseteq> B; A ~= {} |] ==> \<Inter>A \<subseteq> B"
ballarin@14551
  1086
  by blast
ballarin@14551
  1087
wenzelm@12897
  1088
lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A"
nipkow@17589
  1089
  by (iprover intro: InterI subsetI dest: subsetD)
wenzelm@12897
  1090
wenzelm@12897
  1091
lemma INT_lower: "a \<in> A ==> (\<Inter>x\<in>A. B x) \<subseteq> B a"
wenzelm@12897
  1092
  by blast
wenzelm@12897
  1093
wenzelm@12897
  1094
lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)"
nipkow@17589
  1095
  by (iprover intro: INT_I subsetI dest: subsetD)
wenzelm@12897
  1096
wenzelm@12897
  1097
wenzelm@12897
  1098
text {* \medskip Finite Union -- the least upper bound of two sets. *}
wenzelm@12897
  1099
wenzelm@12897
  1100
lemma Un_upper1: "A \<subseteq> A \<union> B"
wenzelm@12897
  1101
  by blast
wenzelm@12897
  1102
wenzelm@12897
  1103
lemma Un_upper2: "B \<subseteq> A \<union> B"
wenzelm@12897
  1104
  by blast
wenzelm@12897
  1105
wenzelm@12897
  1106
lemma Un_least: "A \<subseteq> C ==> B \<subseteq> C ==> A \<union> B \<subseteq> C"
wenzelm@12897
  1107
  by blast
wenzelm@12897
  1108
wenzelm@12897
  1109
wenzelm@12897
  1110
text {* \medskip Finite Intersection -- the greatest lower bound of two sets. *}
wenzelm@12897
  1111
wenzelm@12897
  1112
lemma Int_lower1: "A \<inter> B \<subseteq> A"
wenzelm@12897
  1113
  by blast
wenzelm@12897
  1114
wenzelm@12897
  1115
lemma Int_lower2: "A \<inter> B \<subseteq> B"
wenzelm@12897
  1116
  by blast
wenzelm@12897
  1117
wenzelm@12897
  1118
lemma Int_greatest: "C \<subseteq> A ==> C \<subseteq> B ==> C \<subseteq> A \<inter> B"
wenzelm@12897
  1119
  by blast
wenzelm@12897
  1120
wenzelm@12897
  1121
wenzelm@12897
  1122
text {* \medskip Set difference. *}
wenzelm@12897
  1123
wenzelm@12897
  1124
lemma Diff_subset: "A - B \<subseteq> A"
wenzelm@12897
  1125
  by blast
wenzelm@12897
  1126
nipkow@14302
  1127
lemma Diff_subset_conv: "(A - B \<subseteq> C) = (A \<subseteq> B \<union> C)"
nipkow@14302
  1128
by blast
nipkow@14302
  1129
wenzelm@12897
  1130
wenzelm@12897
  1131
text {* \medskip Monotonicity. *}
wenzelm@12897
  1132
ballarin@15206
  1133
lemma mono_Un: "mono f ==> f A \<union> f B \<subseteq> f (A \<union> B)"
paulson@16773
  1134
  by (auto simp add: mono_def)
ballarin@15206
  1135
ballarin@15206
  1136
lemma mono_Int: "mono f ==> f (A \<inter> B) \<subseteq> f A \<inter> f B"
paulson@16773
  1137
  by (auto simp add: mono_def)
wenzelm@12897
  1138
wenzelm@12897
  1139
subsubsection {* Equalities involving union, intersection, inclusion, etc. *}
wenzelm@12897
  1140
wenzelm@12897
  1141
text {* @{text "{}"}. *}
wenzelm@12897
  1142
wenzelm@12897
  1143
lemma Collect_const [simp]: "{s. P} = (if P then UNIV else {})"
wenzelm@12897
  1144
  -- {* supersedes @{text "Collect_False_empty"} *}
wenzelm@12897
  1145
  by auto
wenzelm@12897
  1146
wenzelm@12897
  1147
lemma subset_empty [simp]: "(A \<subseteq> {}) = (A = {})"
wenzelm@12897
  1148
  by blast
wenzelm@12897
  1149
wenzelm@12897
  1150
lemma not_psubset_empty [iff]: "\<not> (A < {})"
wenzelm@12897
  1151
  by (unfold psubset_def) blast
wenzelm@12897
  1152
wenzelm@12897
  1153
lemma Collect_empty_eq [simp]: "(Collect P = {}) = (\<forall>x. \<not> P x)"
nipkow@18423
  1154
by blast
nipkow@18423
  1155
nipkow@18423
  1156
lemma empty_Collect_eq [simp]: "({} = Collect P) = (\<forall>x. \<not> P x)"
nipkow@18423
  1157
by blast
wenzelm@12897
  1158
wenzelm@12897
  1159
lemma Collect_neg_eq: "{x. \<not> P x} = - {x. P x}"
wenzelm@12897
  1160
  by blast
wenzelm@12897
  1161
wenzelm@12897
  1162
lemma Collect_disj_eq: "{x. P x | Q x} = {x. P x} \<union> {x. Q x}"
wenzelm@12897
  1163
  by blast
wenzelm@12897
  1164
paulson@14812
  1165
lemma Collect_imp_eq: "{x. P x --> Q x} = -{x. P x} \<union> {x. Q x}"
paulson@14812
  1166
  by blast
paulson@14812
  1167
wenzelm@12897
  1168
lemma Collect_conj_eq: "{x. P x & Q x} = {x. P x} \<inter> {x. Q x}"
wenzelm@12897
  1169
  by blast
wenzelm@12897
  1170
wenzelm@12897
  1171
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
wenzelm@12897
  1172
  by blast
wenzelm@12897
  1173
wenzelm@12897
  1174
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
wenzelm@12897
  1175
  by blast
wenzelm@12897
  1176
wenzelm@12897
  1177
lemma Collect_ex_eq: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
wenzelm@12897
  1178
  by blast
wenzelm@12897
  1179
wenzelm@12897
  1180
lemma Collect_bex_eq: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
wenzelm@12897
  1181
  by blast
wenzelm@12897
  1182
wenzelm@12897
  1183
wenzelm@12897
  1184
text {* \medskip @{text insert}. *}
wenzelm@12897
  1185
wenzelm@12897
  1186
lemma insert_is_Un: "insert a A = {a} Un A"
wenzelm@12897
  1187
  -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a {}"} *}
wenzelm@12897
  1188
  by blast
wenzelm@12897
  1189
wenzelm@12897
  1190
lemma insert_not_empty [simp]: "insert a A \<noteq> {}"
wenzelm@12897
  1191
  by blast
wenzelm@12897
  1192
paulson@17715
  1193
lemmas empty_not_insert = insert_not_empty [symmetric, standard]
paulson@17715
  1194
declare empty_not_insert [simp]
wenzelm@12897
  1195
wenzelm@12897
  1196
lemma insert_absorb: "a \<in> A ==> insert a A = A"
wenzelm@12897
  1197
  -- {* @{text "[simp]"} causes recursive calls when there are nested inserts *}
wenzelm@12897
  1198
  -- {* with \emph{quadratic} running time *}
wenzelm@12897
  1199
  by blast
wenzelm@12897
  1200
wenzelm@12897
  1201
lemma insert_absorb2 [simp]: "insert x (insert x A) = insert x A"
wenzelm@12897
  1202
  by blast
wenzelm@12897
  1203
wenzelm@12897
  1204
lemma insert_commute: "insert x (insert y A) = insert y (insert x A)"
wenzelm@12897
  1205
  by blast
wenzelm@12897
  1206
wenzelm@12897
  1207
lemma insert_subset [simp]: "(insert x A \<subseteq> B) = (x \<in> B & A \<subseteq> B)"
wenzelm@12897
  1208
  by blast
wenzelm@12897
  1209
wenzelm@12897
  1210
lemma mk_disjoint_insert: "a \<in> A ==> \<exists>B. A = insert a B & a \<notin> B"
wenzelm@12897
  1211
  -- {* use new @{text B} rather than @{text "A - {a}"} to avoid infinite unfolding *}
paulson@14208
  1212
  apply (rule_tac x = "A - {a}" in exI, blast)
wenzelm@12897
  1213
  done
wenzelm@12897
  1214
wenzelm@12897
  1215
lemma insert_Collect: "insert a (Collect P) = {u. u \<noteq> a --> P u}"
wenzelm@12897
  1216
  by auto
wenzelm@12897
  1217
wenzelm@12897
  1218
lemma UN_insert_distrib: "u \<in> A ==> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
wenzelm@12897
  1219
  by blast
wenzelm@12897
  1220
nipkow@14302
  1221
lemma insert_inter_insert[simp]: "insert a A \<inter> insert a B = insert a (A \<inter> B)"
mehta@14742
  1222
  by blast
nipkow@14302
  1223
nipkow@13103
  1224
lemma insert_disjoint[simp]:
nipkow@13103
  1225
 "(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})"
mehta@14742
  1226
 "({} = insert a A \<inter> B) = (a \<notin> B \<and> {} = A \<inter> B)"
paulson@16773
  1227
  by auto
nipkow@13103
  1228
nipkow@13103
  1229
lemma disjoint_insert[simp]:
nipkow@13103
  1230
 "(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})"
mehta@14742
  1231
 "({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)"
paulson@16773
  1232
  by auto
mehta@14742
  1233
wenzelm@12897
  1234
text {* \medskip @{text image}. *}
wenzelm@12897
  1235
wenzelm@12897
  1236
lemma image_empty [simp]: "f`{} = {}"
wenzelm@12897
  1237
  by blast
wenzelm@12897
  1238
wenzelm@12897
  1239
lemma image_insert [simp]: "f ` insert a B = insert (f a) (f`B)"
wenzelm@12897
  1240
  by blast
wenzelm@12897
  1241
wenzelm@12897
  1242
lemma image_constant: "x \<in> A ==> (\<lambda>x. c) ` A = {c}"
paulson@16773
  1243
  by auto
wenzelm@12897
  1244
nipkow@21316
  1245
lemma image_constant_conv: "(%x. c) ` A = (if A = {} then {} else {c})"
nipkow@21312
  1246
by auto
nipkow@21312
  1247
wenzelm@12897
  1248
lemma image_image: "f ` (g ` A) = (\<lambda>x. f (g x)) ` A"
wenzelm@12897
  1249
  by blast
wenzelm@12897
  1250
wenzelm@12897
  1251
lemma insert_image [simp]: "x \<in> A ==> insert (f x) (f`A) = f`A"
wenzelm@12897
  1252
  by blast
wenzelm@12897
  1253
wenzelm@12897
  1254
lemma image_is_empty [iff]: "(f`A = {}) = (A = {})"
wenzelm@12897
  1255
  by blast
wenzelm@12897
  1256
paulson@16773
  1257
wenzelm@12897
  1258
lemma image_Collect: "f ` {x. P x} = {f x | x. P x}"
paulson@16773
  1259
  -- {* NOT suitable as a default simprule: the RHS isn't simpler than the LHS,
paulson@16773
  1260
      with its implicit quantifier and conjunction.  Also image enjoys better
paulson@16773
  1261
      equational properties than does the RHS. *}
wenzelm@12897
  1262
  by blast
wenzelm@12897
  1263
wenzelm@12897
  1264
lemma if_image_distrib [simp]:
wenzelm@12897
  1265
  "(\<lambda>x. if P x then f x else g x) ` S
wenzelm@12897
  1266
    = (f ` (S \<inter> {x. P x})) \<union> (g ` (S \<inter> {x. \<not> P x}))"
wenzelm@12897
  1267
  by (auto simp add: image_def)
wenzelm@12897
  1268
wenzelm@12897
  1269
lemma image_cong: "M = N ==> (!!x. x \<in> N ==> f x = g x) ==> f`M = g`N"
wenzelm@12897
  1270
  by (simp add: image_def)
wenzelm@12897
  1271
wenzelm@12897
  1272
wenzelm@12897
  1273
text {* \medskip @{text range}. *}
wenzelm@12897
  1274
wenzelm@12897
  1275
lemma full_SetCompr_eq: "{u. \<exists>x. u = f x} = range f"
wenzelm@12897
  1276
  by auto
wenzelm@12897
  1277
wenzelm@12897
  1278
lemma range_composition [simp]: "range (\<lambda>x. f (g x)) = f`range g"
paulson@14208
  1279
by (subst image_image, simp)
wenzelm@12897
  1280
wenzelm@12897
  1281
wenzelm@12897
  1282
text {* \medskip @{text Int} *}
wenzelm@12897
  1283
wenzelm@12897
  1284
lemma Int_absorb [simp]: "A \<inter> A = A"
wenzelm@12897
  1285
  by blast
wenzelm@12897
  1286
wenzelm@12897
  1287
lemma Int_left_absorb: "A \<inter> (A \<inter> B) = A \<inter> B"
wenzelm@12897
  1288
  by blast
wenzelm@12897
  1289
wenzelm@12897
  1290
lemma Int_commute: "A \<inter> B = B \<inter> A"
wenzelm@12897
  1291
  by blast
wenzelm@12897
  1292
wenzelm@12897
  1293
lemma Int_left_commute: "A \<inter> (B \<inter> C) = B \<inter> (A \<inter> C)"
wenzelm@12897
  1294
  by blast
wenzelm@12897
  1295
wenzelm@12897
  1296
lemma Int_assoc: "(A \<inter> B) \<inter> C = A \<inter> (B \<inter> C)"
wenzelm@12897
  1297
  by blast
wenzelm@12897
  1298
wenzelm@12897
  1299
lemmas Int_ac = Int_assoc Int_left_absorb Int_commute Int_left_commute
wenzelm@12897
  1300
  -- {* Intersection is an AC-operator *}
wenzelm@12897
  1301
wenzelm@12897
  1302
lemma Int_absorb1: "B \<subseteq> A ==> A \<inter> B = B"
wenzelm@12897
  1303
  by blast
wenzelm@12897
  1304
wenzelm@12897
  1305
lemma Int_absorb2: "A \<subseteq> B ==> A \<inter> B = A"
wenzelm@12897
  1306
  by blast
wenzelm@12897
  1307
wenzelm@12897
  1308
lemma Int_empty_left [simp]: "{} \<inter> B = {}"
wenzelm@12897
  1309
  by blast
wenzelm@12897
  1310
wenzelm@12897
  1311
lemma Int_empty_right [simp]: "A \<inter> {} = {}"
wenzelm@12897
  1312
  by blast
wenzelm@12897
  1313
wenzelm@12897
  1314
lemma disjoint_eq_subset_Compl: "(A \<inter> B = {}) = (A \<subseteq> -B)"
wenzelm@12897
  1315
  by blast
wenzelm@12897
  1316
wenzelm@12897
  1317
lemma disjoint_iff_not_equal: "(A \<inter> B = {}) = (\<forall>x\<in>A. \<forall>y\<in>B. x \<noteq> y)"
wenzelm@12897
  1318
  by blast
wenzelm@12897
  1319
wenzelm@12897
  1320
lemma Int_UNIV_left [simp]: "UNIV \<inter> B = B"
wenzelm@12897
  1321
  by blast
wenzelm@12897
  1322
wenzelm@12897
  1323
lemma Int_UNIV_right [simp]: "A \<inter> UNIV = A"
wenzelm@12897
  1324
  by blast
wenzelm@12897
  1325
wenzelm@12897
  1326
lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"
wenzelm@12897
  1327
  by blast
wenzelm@12897
  1328
wenzelm@12897
  1329
lemma Int_Un_distrib: "A \<inter> (B \<union> C) = (A \<inter> B) \<union> (A \<inter> C)"
wenzelm@12897
  1330
  by blast
wenzelm@12897
  1331
wenzelm@12897
  1332
lemma Int_Un_distrib2: "(B \<union> C) \<inter> A = (B \<inter> A) \<union> (C \<inter> A)"
wenzelm@12897
  1333
  by blast
wenzelm@12897
  1334
wenzelm@12897
  1335
lemma Int_UNIV [simp]: "(A \<inter> B = UNIV) = (A = UNIV & B = UNIV)"
wenzelm@12897
  1336
  by blast
wenzelm@12897
  1337
paulson@15102
  1338
lemma Int_subset_iff [simp]: "(C \<subseteq> A \<inter> B) = (C \<subseteq> A & C \<subseteq> B)"
wenzelm@12897
  1339
  by blast
wenzelm@12897
  1340
wenzelm@12897
  1341
lemma Int_Collect: "(x \<in> A \<inter> {x. P x}) = (x \<in> A & P x)"
wenzelm@12897
  1342
  by blast
wenzelm@12897
  1343
wenzelm@12897
  1344
wenzelm@12897
  1345
text {* \medskip @{text Un}. *}
wenzelm@12897
  1346
wenzelm@12897
  1347
lemma Un_absorb [simp]: "A \<union> A = A"
wenzelm@12897
  1348
  by blast
wenzelm@12897
  1349
wenzelm@12897
  1350
lemma Un_left_absorb: "A \<union> (A \<union> B) = A \<union> B"
wenzelm@12897
  1351
  by blast
wenzelm@12897
  1352
wenzelm@12897
  1353
lemma Un_commute: "A \<union> B = B \<union> A"
wenzelm@12897
  1354
  by blast
wenzelm@12897
  1355
wenzelm@12897
  1356
lemma Un_left_commute: "A \<union> (B \<union> C) = B \<union> (A \<union> C)"
wenzelm@12897
  1357
  by blast
wenzelm@12897
  1358
wenzelm@12897
  1359
lemma Un_assoc: "(A \<union> B) \<union> C = A \<union> (B \<union> C)"
wenzelm@12897
  1360
  by blast
wenzelm@12897
  1361
wenzelm@12897
  1362
lemmas Un_ac = Un_assoc Un_left_absorb Un_commute Un_left_commute
wenzelm@12897
  1363
  -- {* Union is an AC-operator *}
wenzelm@12897
  1364
wenzelm@12897
  1365
lemma Un_absorb1: "A \<subseteq> B ==> A \<union> B = B"
wenzelm@12897
  1366
  by blast
wenzelm@12897
  1367
wenzelm@12897
  1368
lemma Un_absorb2: "B \<subseteq> A ==> A \<union> B = A"
wenzelm@12897
  1369
  by blast
wenzelm@12897
  1370
wenzelm@12897
  1371
lemma Un_empty_left [simp]: "{} \<union> B = B"
wenzelm@12897
  1372
  by blast
wenzelm@12897
  1373
wenzelm@12897
  1374
lemma Un_empty_right [simp]: "A \<union> {} = A"
wenzelm@12897
  1375
  by blast
wenzelm@12897
  1376
wenzelm@12897
  1377
lemma Un_UNIV_left [simp]: "UNIV \<union> B = UNIV"
wenzelm@12897
  1378
  by blast
wenzelm@12897
  1379
wenzelm@12897
  1380
lemma Un_UNIV_right [simp]: "A \<union> UNIV = UNIV"
wenzelm@12897
  1381
  by blast
wenzelm@12897
  1382
wenzelm@12897
  1383
lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"
wenzelm@12897
  1384
  by blast
wenzelm@12897
  1385
wenzelm@12897
  1386
lemma Un_insert_left [simp]: "(insert a B) \<union> C = insert a (B \<union> C)"
wenzelm@12897
  1387
  by blast
wenzelm@12897
  1388
wenzelm@12897
  1389
lemma Un_insert_right [simp]: "A \<union> (insert a B) = insert a (A \<union> B)"
wenzelm@12897
  1390
  by blast
wenzelm@12897
  1391
wenzelm@12897
  1392
lemma Int_insert_left:
wenzelm@12897
  1393
    "(insert a B) Int C = (if a \<in> C then insert a (B \<inter> C) else B \<inter> C)"
wenzelm@12897
  1394
  by auto
wenzelm@12897
  1395
wenzelm@12897
  1396
lemma Int_insert_right:
wenzelm@12897
  1397
    "A \<inter> (insert a B) = (if a \<in> A then insert a (A \<inter> B) else A \<inter> B)"
wenzelm@12897
  1398
  by auto
wenzelm@12897
  1399
wenzelm@12897
  1400
lemma Un_Int_distrib: "A \<union> (B \<inter> C) = (A \<union> B) \<inter> (A \<union> C)"
wenzelm@12897
  1401
  by blast
wenzelm@12897
  1402
wenzelm@12897
  1403
lemma Un_Int_distrib2: "(B \<inter> C) \<union> A = (B \<union> A) \<inter> (C \<union> A)"
wenzelm@12897
  1404
  by blast
wenzelm@12897
  1405
wenzelm@12897
  1406
lemma Un_Int_crazy:
wenzelm@12897
  1407
    "(A \<inter> B) \<union> (B \<inter> C) \<union> (C \<inter> A) = (A \<union> B) \<inter> (B \<union> C) \<inter> (C \<union> A)"
wenzelm@12897
  1408
  by blast
wenzelm@12897
  1409
wenzelm@12897
  1410
lemma subset_Un_eq: "(A \<subseteq> B) = (A \<union> B = B)"
wenzelm@12897
  1411
  by blast
wenzelm@12897
  1412
wenzelm@12897
  1413
lemma Un_empty [iff]: "(A \<union> B = {}) = (A = {} & B = {})"
wenzelm@12897
  1414
  by blast
paulson@15102
  1415
paulson@15102
  1416
lemma Un_subset_iff [simp]: "(A \<union> B \<subseteq> C) = (A \<subseteq> C & B \<subseteq> C)"
wenzelm@12897
  1417
  by blast
wenzelm@12897
  1418
wenzelm@12897
  1419
lemma Un_Diff_Int: "(A - B) \<union> (A \<inter> B) = A"
wenzelm@12897
  1420
  by blast
wenzelm@12897
  1421
paulson@22172
  1422
lemma Diff_Int2: "A \<inter> C - B \<inter> C = A \<inter> C - B"
paulson@22172
  1423
  by blast
paulson@22172
  1424
wenzelm@12897
  1425
wenzelm@12897
  1426
text {* \medskip Set complement *}
wenzelm@12897
  1427
wenzelm@12897
  1428
lemma Compl_disjoint [simp]: "A \<inter> -A = {}"
wenzelm@12897
  1429
  by blast
wenzelm@12897
  1430
wenzelm@12897
  1431
lemma Compl_disjoint2 [simp]: "-A \<inter> A = {}"
wenzelm@12897
  1432
  by blast
wenzelm@12897
  1433
paulson@13818
  1434
lemma Compl_partition: "A \<union> -A = UNIV"
paulson@13818
  1435
  by blast
paulson@13818
  1436
paulson@13818
  1437
lemma Compl_partition2: "-A \<union> A = UNIV"
wenzelm@12897
  1438
  by blast
wenzelm@12897
  1439
wenzelm@12897
  1440
lemma double_complement [simp]: "- (-A) = (A::'a set)"
wenzelm@12897
  1441
  by blast
wenzelm@12897
  1442
wenzelm@12897
  1443
lemma Compl_Un [simp]: "-(A \<union> B) = (-A) \<inter> (-B)"
wenzelm@12897
  1444
  by blast
wenzelm@12897
  1445
wenzelm@12897
  1446
lemma Compl_Int [simp]: "-(A \<inter> B) = (-A) \<union> (-B)"
wenzelm@12897
  1447
  by blast
wenzelm@12897
  1448
wenzelm@12897
  1449
lemma Compl_UN [simp]: "-(\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
wenzelm@12897
  1450
  by blast
wenzelm@12897
  1451
wenzelm@12897
  1452
lemma Compl_INT [simp]: "-(\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
wenzelm@12897
  1453
  by blast
wenzelm@12897
  1454
wenzelm@12897
  1455
lemma subset_Compl_self_eq: "(A \<subseteq> -A) = (A = {})"
wenzelm@12897
  1456
  by blast
wenzelm@12897
  1457
wenzelm@12897
  1458
lemma Un_Int_assoc_eq: "((A \<inter> B) \<union> C = A \<inter> (B \<union> C)) = (C \<subseteq> A)"
wenzelm@12897
  1459
  -- {* Halmos, Naive Set Theory, page 16. *}
wenzelm@12897
  1460
  by blast
wenzelm@12897
  1461
wenzelm@12897
  1462
lemma Compl_UNIV_eq [simp]: "-UNIV = {}"
wenzelm@12897
  1463
  by blast
wenzelm@12897
  1464
wenzelm@12897
  1465
lemma Compl_empty_eq [simp]: "-{} = UNIV"
wenzelm@12897
  1466
  by blast
wenzelm@12897
  1467
wenzelm@12897
  1468
lemma Compl_subset_Compl_iff [iff]: "(-A \<subseteq> -B) = (B \<subseteq> A)"
wenzelm@12897
  1469
  by blast
wenzelm@12897
  1470
wenzelm@12897
  1471
lemma Compl_eq_Compl_iff [iff]: "(-A = -B) = (A = (B::'a set))"
wenzelm@12897
  1472
  by blast
wenzelm@12897
  1473
wenzelm@12897
  1474
wenzelm@12897
  1475
text {* \medskip @{text Union}. *}
wenzelm@12897
  1476
wenzelm@12897
  1477
lemma Union_empty [simp]: "Union({}) = {}"
wenzelm@12897
  1478
  by blast
wenzelm@12897
  1479
wenzelm@12897
  1480
lemma Union_UNIV [simp]: "Union UNIV = UNIV"
wenzelm@12897
  1481
  by blast
wenzelm@12897
  1482
wenzelm@12897
  1483
lemma Union_insert [simp]: "Union (insert a B) = a \<union> \<Union>B"
wenzelm@12897
  1484
  by blast
wenzelm@12897
  1485
wenzelm@12897
  1486
lemma Union_Un_distrib [simp]: "\<Union>(A Un B) = \<Union>A \<union> \<Union>B"
wenzelm@12897
  1487
  by blast
wenzelm@12897
  1488
wenzelm@12897
  1489
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
wenzelm@12897
  1490
  by blast
wenzelm@12897
  1491
paulson@18447
  1492
lemma Union_empty_conv [simp]: "(\<Union>A = {}) = (\<forall>x\<in>A. x = {})"
nipkow@13653
  1493
  by blast
nipkow@13653
  1494
paulson@18447
  1495
lemma empty_Union_conv [simp]: "({} = \<Union>A) = (\<forall>x\<in>A. x = {})"
nipkow@13653
  1496
  by blast
wenzelm@12897
  1497
wenzelm@12897
  1498
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) = (\<forall>B\<in>C. B \<inter> A = {})"
wenzelm@12897
  1499
  by blast
wenzelm@12897
  1500
wenzelm@12897
  1501
wenzelm@12897
  1502
text {* \medskip @{text Inter}. *}
wenzelm@12897
  1503
wenzelm@12897
  1504
lemma Inter_empty [simp]: "\<Inter>{} = UNIV"
wenzelm@12897
  1505
  by blast
wenzelm@12897
  1506
wenzelm@12897
  1507
lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}"
wenzelm@12897
  1508
  by blast
wenzelm@12897
  1509
wenzelm@12897
  1510
lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
wenzelm@12897
  1511
  by blast
wenzelm@12897
  1512
wenzelm@12897
  1513
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
wenzelm@12897
  1514
  by blast
wenzelm@12897
  1515
wenzelm@12897
  1516
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
wenzelm@12897
  1517
  by blast
wenzelm@12897
  1518
paulson@18447
  1519
lemma Inter_UNIV_conv [simp]:
nipkow@13653
  1520
  "(\<Inter>A = UNIV) = (\<forall>x\<in>A. x = UNIV)"
nipkow@13653
  1521
  "(UNIV = \<Inter>A) = (\<forall>x\<in>A. x = UNIV)"
paulson@14208
  1522
  by blast+
nipkow@13653
  1523
wenzelm@12897
  1524
wenzelm@12897
  1525
text {*
wenzelm@12897
  1526
  \medskip @{text UN} and @{text INT}.
wenzelm@12897
  1527
wenzelm@12897
  1528
  Basic identities: *}
wenzelm@12897
  1529
wenzelm@12897
  1530
lemma UN_empty [simp]: "(\<Union>x\<in>{}. B x) = {}"
wenzelm@12897
  1531
  by blast
wenzelm@12897
  1532
wenzelm@12897
  1533
lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"
wenzelm@12897
  1534
  by blast
wenzelm@12897
  1535
wenzelm@12897
  1536
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
wenzelm@12897
  1537
  by blast
wenzelm@12897
  1538
wenzelm@12897
  1539
lemma UN_absorb: "k \<in> I ==> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
paulson@15102
  1540
  by auto
wenzelm@12897
  1541
wenzelm@12897
  1542
lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"
wenzelm@12897
  1543
  by blast
wenzelm@12897
  1544
wenzelm@12897
  1545
lemma INT_absorb: "k \<in> I ==> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
wenzelm@12897
  1546
  by blast
wenzelm@12897
  1547
wenzelm@12897
  1548
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
wenzelm@12897
  1549
  by blast
wenzelm@12897
  1550
wenzelm@12897
  1551
lemma UN_Un: "(\<Union>i \<in> A \<union> B. M i) = (\<Union>i\<in>A. M i) \<union> (\<Union>i\<in>B. M i)"
wenzelm@12897
  1552
  by blast
wenzelm@12897
  1553
wenzelm@12897
  1554
lemma UN_UN_flatten: "(\<Union>x \<in> (\<Union>y\<in>A. B y). C x) = (\<Union>y\<in>A. \<Union>x\<in>B y. C x)"
wenzelm@12897
  1555
  by blast
wenzelm@12897
  1556
wenzelm@12897
  1557
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
wenzelm@12897
  1558
  by blast
wenzelm@12897
  1559
wenzelm@12897
  1560
lemma INT_subset_iff: "(B \<subseteq> (\<Inter>i\<in>I. A i)) = (\<forall>i\<in>I. B \<subseteq> A i)"
wenzelm@12897
  1561
  by blast
wenzelm@12897
  1562
wenzelm@12897
  1563
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
wenzelm@12897
  1564
  by blast
wenzelm@12897
  1565
wenzelm@12897
  1566
lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"
wenzelm@12897
  1567
  by blast
wenzelm@12897
  1568
wenzelm@12897
  1569
lemma INT_insert_distrib:
wenzelm@12897
  1570
    "u \<in> A ==> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
wenzelm@12897
  1571
  by blast
wenzelm@12897
  1572
wenzelm@12897
  1573
lemma Union_image_eq [simp]: "\<Union>(B`A) = (\<Union>x\<in>A. B x)"
wenzelm@12897
  1574
  by blast
wenzelm@12897
  1575
wenzelm@12897
  1576
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
wenzelm@12897
  1577
  by blast
wenzelm@12897
  1578
wenzelm@12897
  1579
lemma Inter_image_eq [simp]: "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
wenzelm@12897
  1580
  by blast
wenzelm@12897
  1581
wenzelm@12897
  1582
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
wenzelm@12897
  1583
  by auto
wenzelm@12897
  1584
wenzelm@12897
  1585
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
wenzelm@12897
  1586
  by auto
wenzelm@12897
  1587
wenzelm@12897
  1588
lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"
wenzelm@12897
  1589
  by blast
wenzelm@12897
  1590
wenzelm@12897
  1591
lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"
wenzelm@12897
  1592
  -- {* Look: it has an \emph{existential} quantifier *}
wenzelm@12897
  1593
  by blast
wenzelm@12897
  1594
paulson@18447
  1595
lemma UNION_empty_conv[simp]:
nipkow@13653
  1596
  "({} = (UN x:A. B x)) = (\<forall>x\<in>A. B x = {})"
nipkow@13653
  1597
  "((UN x:A. B x) = {}) = (\<forall>x\<in>A. B x = {})"
nipkow@13653
  1598
by blast+
nipkow@13653
  1599
paulson@18447
  1600
lemma INTER_UNIV_conv[simp]:
nipkow@13653
  1601
 "(UNIV = (INT x:A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
nipkow@13653
  1602
 "((INT x:A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
nipkow@13653
  1603
by blast+
wenzelm@12897
  1604
wenzelm@12897
  1605
wenzelm@12897
  1606
text {* \medskip Distributive laws: *}
wenzelm@12897
  1607
wenzelm@12897
  1608
lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
wenzelm@12897
  1609
  by blast
wenzelm@12897
  1610
wenzelm@12897
  1611
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
wenzelm@12897
  1612
  by blast
wenzelm@12897
  1613
wenzelm@12897
  1614
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A`C) \<union> \<Union>(B`C)"
wenzelm@12897
  1615
  -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
wenzelm@12897
  1616
  -- {* Union of a family of unions *}
wenzelm@12897
  1617
  by blast
wenzelm@12897
  1618
wenzelm@12897
  1619
lemma UN_Un_distrib: "(\<Union>i\<in>I. A i \<union> B i) = (\<Union>i\<in>I. A i) \<union> (\<Union>i\<in>I. B i)"
wenzelm@12897
  1620
  -- {* Equivalent version *}
wenzelm@12897
  1621
  by blast
wenzelm@12897
  1622
wenzelm@12897
  1623
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
wenzelm@12897
  1624
  by blast
wenzelm@12897
  1625
wenzelm@12897
  1626
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A`C) \<inter> \<Inter>(B`C)"
wenzelm@12897
  1627
  by blast
wenzelm@12897
  1628
wenzelm@12897
  1629
lemma INT_Int_distrib: "(\<Inter>i\<in>I. A i \<inter> B i) = (\<Inter>i\<in>I. A i) \<inter> (\<Inter>i\<in>I. B i)"
wenzelm@12897
  1630
  -- {* Equivalent version *}
wenzelm@12897
  1631
  by blast
wenzelm@12897
  1632
wenzelm@12897
  1633
lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
wenzelm@12897
  1634
  -- {* Halmos, Naive Set Theory, page 35. *}
wenzelm@12897
  1635
  by blast
wenzelm@12897
  1636
wenzelm@12897
  1637
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
wenzelm@12897
  1638
  by blast
wenzelm@12897
  1639
wenzelm@12897
  1640
lemma Int_UN_distrib2: "(\<Union>i\<in>I. A i) \<inter> (\<Union>j\<in>J. B j) = (\<Union>i\<in>I. \<Union>j\<in>J. A i \<inter> B j)"
wenzelm@12897
  1641
  by blast
wenzelm@12897
  1642
wenzelm@12897
  1643
lemma Un_INT_distrib2: "(\<Inter>i\<in>I. A i) \<union> (\<Inter>j\<in>J. B j) = (\<Inter>i\<in>I. \<Inter>j\<in>J. A i \<union> B j)"
wenzelm@12897
  1644
  by blast
wenzelm@12897
  1645
wenzelm@12897
  1646
wenzelm@12897
  1647
text {* \medskip Bounded quantifiers.
wenzelm@12897
  1648
wenzelm@12897
  1649
  The following are not added to the default simpset because
wenzelm@12897
  1650
  (a) they duplicate the body and (b) there are no similar rules for @{text Int}. *}
wenzelm@12897
  1651
wenzelm@12897
  1652
lemma ball_Un: "(\<forall>x \<in> A \<union> B. P x) = ((\<forall>x\<in>A. P x) & (\<forall>x\<in>B. P x))"
wenzelm@12897
  1653
  by blast
wenzelm@12897
  1654
wenzelm@12897
  1655
lemma bex_Un: "(\<exists>x \<in> A \<union> B. P x) = ((\<exists>x\<in>A. P x) | (\<exists>x\<in>B. P x))"
wenzelm@12897
  1656
  by blast
wenzelm@12897
  1657
wenzelm@12897
  1658
lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) = (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
wenzelm@12897
  1659
  by blast
wenzelm@12897
  1660
wenzelm@12897
  1661
lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) = (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
wenzelm@12897
  1662
  by blast
wenzelm@12897
  1663
wenzelm@12897
  1664
wenzelm@12897
  1665
text {* \medskip Set difference. *}
wenzelm@12897
  1666
wenzelm@12897
  1667
lemma Diff_eq: "A - B = A \<inter> (-B)"
wenzelm@12897
  1668
  by blast
wenzelm@12897
  1669
wenzelm@12897
  1670
lemma Diff_eq_empty_iff [simp]: "(A - B = {}) = (A \<subseteq> B)"
wenzelm@12897
  1671
  by blast
wenzelm@12897
  1672
wenzelm@12897
  1673
lemma Diff_cancel [simp]: "A - A = {}"
wenzelm@12897
  1674
  by blast
wenzelm@12897
  1675
nipkow@14302
  1676
lemma Diff_idemp [simp]: "(A - B) - B = A - (B::'a set)"
nipkow@14302
  1677
by blast
nipkow@14302
  1678
wenzelm@12897
  1679
lemma Diff_triv: "A \<inter> B = {} ==> A - B = A"
wenzelm@12897
  1680
  by (blast elim: equalityE)
wenzelm@12897
  1681
wenzelm@12897
  1682
lemma empty_Diff [simp]: "{} - A = {}"
wenzelm@12897
  1683
  by blast
wenzelm@12897
  1684
wenzelm@12897
  1685
lemma Diff_empty [simp]: "A - {} = A"
wenzelm@12897
  1686
  by blast
wenzelm@12897
  1687
wenzelm@12897
  1688
lemma Diff_UNIV [simp]: "A - UNIV = {}"
wenzelm@12897
  1689
  by blast
wenzelm@12897
  1690
wenzelm@12897
  1691
lemma Diff_insert0 [simp]: "x \<notin> A ==> A - insert x B = A - B"
wenzelm@12897
  1692
  by blast
wenzelm@12897
  1693
wenzelm@12897
  1694
lemma Diff_insert: "A - insert a B = A - B - {a}"
wenzelm@12897
  1695
  -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a 0"} *}
wenzelm@12897
  1696
  by blast
wenzelm@12897
  1697
wenzelm@12897
  1698
lemma Diff_insert2: "A - insert a B = A - {a} - B"
wenzelm@12897
  1699
  -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a 0"} *}
wenzelm@12897
  1700
  by blast
wenzelm@12897
  1701
wenzelm@12897
  1702
lemma insert_Diff_if: "insert x A - B = (if x \<in> B then A - B else insert x (A - B))"
wenzelm@12897
  1703
  by auto
wenzelm@12897
  1704
wenzelm@12897
  1705
lemma insert_Diff1 [simp]: "x \<in> B ==> insert x A - B = A - B"
wenzelm@12897
  1706
  by blast
wenzelm@12897
  1707
nipkow@14302
  1708
lemma insert_Diff_single[simp]: "insert a (A - {a}) = insert a A"
nipkow@14302
  1709
by blast
nipkow@14302
  1710
wenzelm@12897
  1711
lemma insert_Diff: "a \<in> A ==> insert a (A - {a}) = A"
wenzelm@12897
  1712
  by blast
wenzelm@12897
  1713
wenzelm@12897
  1714
lemma Diff_insert_absorb: "x \<notin> A ==> (insert x A) - {x} = A"
wenzelm@12897
  1715
  by auto
wenzelm@12897
  1716
wenzelm@12897
  1717
lemma Diff_disjoint [simp]: "A \<inter> (B - A) = {}"
wenzelm@12897
  1718
  by blast
wenzelm@12897
  1719
wenzelm@12897
  1720
lemma Diff_partition: "A \<subseteq> B ==> A \<union> (B - A) = B"
wenzelm@12897
  1721
  by blast
wenzelm@12897
  1722
wenzelm@12897
  1723
lemma double_diff: "A \<subseteq> B ==> B \<subseteq> C ==> B - (C - A) = A"
wenzelm@12897
  1724
  by blast
wenzelm@12897
  1725
wenzelm@12897
  1726
lemma Un_Diff_cancel [simp]: "A \<union> (B - A) = A \<union> B"
wenzelm@12897
  1727
  by blast
wenzelm@12897
  1728
wenzelm@12897
  1729
lemma Un_Diff_cancel2 [simp]: "(B - A) \<union> A = B \<union> A"
wenzelm@12897
  1730
  by blast
wenzelm@12897
  1731
wenzelm@12897
  1732
lemma Diff_Un: "A - (B \<union> C) = (A - B) \<inter> (A - C)"
wenzelm@12897
  1733
  by blast
wenzelm@12897
  1734
wenzelm@12897
  1735
lemma Diff_Int: "A - (B \<inter> C) = (A - B) \<union> (A - C)"
wenzelm@12897
  1736
  by blast
wenzelm@12897
  1737
wenzelm@12897
  1738
lemma Un_Diff: "(A \<union> B) - C = (A - C) \<union> (B - C)"
wenzelm@12897
  1739
  by blast
wenzelm@12897
  1740
wenzelm@12897
  1741
lemma Int_Diff: "(A \<inter> B) - C = A \<inter> (B - C)"
wenzelm@12897
  1742
  by blast
wenzelm@12897
  1743
wenzelm@12897
  1744
lemma Diff_Int_distrib: "C \<inter> (A - B) = (C \<inter> A) - (C \<inter> B)"
wenzelm@12897
  1745
  by blast
wenzelm@12897
  1746
wenzelm@12897
  1747
lemma Diff_Int_distrib2: "(A - B) \<inter> C = (A \<inter> C) - (B \<inter> C)"
wenzelm@12897
  1748
  by blast
wenzelm@12897
  1749
wenzelm@12897
  1750
lemma Diff_Compl [simp]: "A - (- B) = A \<inter> B"
wenzelm@12897
  1751
  by auto
wenzelm@12897
  1752
wenzelm@12897
  1753
lemma Compl_Diff_eq [simp]: "- (A - B) = -A \<union> B"
wenzelm@12897
  1754
  by blast
wenzelm@12897
  1755
wenzelm@12897
  1756
wenzelm@12897
  1757
text {* \medskip Quantification over type @{typ bool}. *}
wenzelm@12897
  1758
wenzelm@12897
  1759
lemma bool_induct: "P True \<Longrightarrow> P False \<Longrightarrow> P x"
haftmann@21549
  1760
  by (cases x) auto
haftmann@21549
  1761
haftmann@21549
  1762
lemma all_bool_eq: "(\<forall>b. P b) \<longleftrightarrow> P True \<and> P False"
haftmann@21549
  1763
  by (auto intro: bool_induct)
haftmann@21549
  1764
haftmann@21549
  1765
lemma bool_contrapos: "P x \<Longrightarrow> \<not> P False \<Longrightarrow> P True"
haftmann@21549
  1766
  by (cases x) auto
haftmann@21549
  1767
haftmann@21549
  1768
lemma ex_bool_eq: "(\<exists>b. P b) \<longleftrightarrow> P True \<or> P False"
haftmann@21549
  1769
  by (auto intro: bool_contrapos)
wenzelm@12897
  1770
wenzelm@12897
  1771
lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
wenzelm@12897
  1772
  by (auto simp add: split_if_mem2)
wenzelm@12897
  1773
wenzelm@12897
  1774
lemma UN_bool_eq: "(\<Union>b::bool. A b) = (A True \<union> A False)"
haftmann@21549
  1775
  by (auto intro: bool_contrapos)
wenzelm@12897
  1776
wenzelm@12897
  1777
lemma INT_bool_eq: "(\<Inter>b::bool. A b) = (A True \<inter> A False)"
haftmann@21549
  1778
  by (auto intro: bool_induct)
wenzelm@12897
  1779
wenzelm@12897
  1780
text {* \medskip @{text Pow} *}
wenzelm@12897
  1781
wenzelm@12897
  1782
lemma Pow_empty [simp]: "Pow {} = {{}}"
wenzelm@12897
  1783
  by (auto simp add: Pow_def)
wenzelm@12897
  1784
wenzelm@12897
  1785
lemma Pow_insert: "Pow (insert a A) = Pow A \<union> (insert a ` Pow A)"
wenzelm@12897
  1786
  by (blast intro: image_eqI [where ?x = "u - {a}", standard])
wenzelm@12897
  1787
wenzelm@12897
  1788
lemma Pow_Compl: "Pow (- A) = {-B | B. A \<in> Pow B}"
wenzelm@12897
  1789
  by (blast intro: exI [where ?x = "- u", standard])
wenzelm@12897
  1790
wenzelm@12897
  1791
lemma Pow_UNIV [simp]: "Pow UNIV = UNIV"
wenzelm@12897
  1792
  by blast
wenzelm@12897
  1793
wenzelm@12897
  1794
lemma Un_Pow_subset: "Pow A \<union> Pow B \<subseteq> Pow (A \<union> B)"
wenzelm@12897
  1795
  by blast
wenzelm@12897
  1796
wenzelm@12897
  1797
lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
wenzelm@12897
  1798
  by blast
wenzelm@12897
  1799
wenzelm@12897
  1800
lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
wenzelm@12897
  1801
  by blast
wenzelm@12897
  1802
wenzelm@12897
  1803
lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
wenzelm@12897
  1804
  by blast
wenzelm@12897
  1805
wenzelm@12897
  1806
lemma Pow_Int_eq [simp]: "Pow (A \<inter> B) = Pow A \<inter> Pow B"
wenzelm@12897
  1807
  by blast
wenzelm@12897
  1808
wenzelm@12897
  1809
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
wenzelm@12897
  1810
  by blast
wenzelm@12897
  1811
wenzelm@12897
  1812
wenzelm@12897
  1813
text {* \medskip Miscellany. *}
wenzelm@12897
  1814
wenzelm@12897
  1815
lemma set_eq_subset: "(A = B) = (A \<subseteq> B & B \<subseteq> A)"
wenzelm@12897
  1816
  by blast
wenzelm@12897
  1817
wenzelm@12897
  1818
lemma subset_iff: "(A \<subseteq> B) = (\<forall>t. t \<in> A --> t \<in> B)"
wenzelm@12897
  1819
  by blast
wenzelm@12897
  1820
wenzelm@12897
  1821
lemma subset_iff_psubset_eq: "(A \<subseteq> B) = ((A \<subset> B) | (A = B))"
wenzelm@12897
  1822
  by (unfold psubset_def) blast
wenzelm@12897
  1823
paulson@18447
  1824
lemma all_not_in_conv [simp]: "(\<forall>x. x \<notin> A) = (A = {})"
wenzelm@12897
  1825
  by blast
wenzelm@12897
  1826
paulson@13831
  1827
lemma ex_in_conv: "(\<exists>x. x \<in> A) = (A \<noteq> {})"
paulson@13831
  1828
  by blast
paulson@13831
  1829
wenzelm@12897
  1830
lemma distinct_lemma: "f x \<noteq> f y ==> x \<noteq> y"
nipkow@17589
  1831
  by iprover
wenzelm@12897
  1832
wenzelm@12897
  1833
paulson@13860
  1834
text {* \medskip Miniscoping: pushing in quantifiers and big Unions
paulson@13860
  1835
           and Intersections. *}
wenzelm@12897
  1836
wenzelm@12897
  1837
lemma UN_simps [simp]:
wenzelm@12897
  1838
  "!!a B C. (UN x:C. insert a (B x)) = (if C={} then {} else insert a (UN x:C. B x))"
wenzelm@12897
  1839
  "!!A B C. (UN x:C. A x Un B)   = ((if C={} then {} else (UN x:C. A x) Un B))"
wenzelm@12897
  1840
  "!!A B C. (UN x:C. A Un B x)   = ((if C={} then {} else A Un (UN x:C. B x)))"
wenzelm@12897
  1841
  "!!A B C. (UN x:C. A x Int B)  = ((UN x:C. A x) Int B)"
wenzelm@12897
  1842
  "!!A B C. (UN x:C. A Int B x)  = (A Int (UN x:C. B x))"
wenzelm@12897
  1843
  "!!A B C. (UN x:C. A x - B)    = ((UN x:C. A x) - B)"
wenzelm@12897
  1844
  "!!A B C. (UN x:C. A - B x)    = (A - (INT x:C. B x))"
wenzelm@12897
  1845
  "!!A B. (UN x: Union A. B x) = (UN y:A. UN x:y. B x)"
wenzelm@12897
  1846
  "!!A B C. (UN z: UNION A B. C z) = (UN  x:A. UN z: B(x). C z)"
wenzelm@12897
  1847
  "!!A B f. (UN x:f`A. B x)     = (UN a:A. B (f a))"
wenzelm@12897
  1848
  by auto
wenzelm@12897
  1849
wenzelm@12897
  1850
lemma INT_simps [simp]:
wenzelm@12897
  1851
  "!!A B C. (INT x:C. A x Int B) = (if C={} then UNIV else (INT x:C. A x) Int B)"
wenzelm@12897
  1852
  "!!A B C. (INT x:C. A Int B x) = (if C={} then UNIV else A Int (INT x:C. B x))"
wenzelm@12897
  1853
  "!!A B C. (INT x:C. A x - B)   = (if C={} then UNIV else (INT x:C. A x) - B)"
wenzelm@12897
  1854
  "!!A B C. (INT x:C. A - B x)   = (if C={} then UNIV else A - (UN x:C. B x))"
wenzelm@12897
  1855
  "!!a B C. (INT x:C. insert a (B x)) = insert a (INT x:C. B x)"
wenzelm@12897
  1856
  "!!A B C. (INT x:C. A x Un B)  = ((INT x:C. A x) Un B)"
wenzelm@12897
  1857
  "!!A B C. (INT x:C. A Un B x)  = (A Un (INT x:C. B x))"
wenzelm@12897
  1858
  "!!A B. (INT x: Union A. B x) = (INT y:A. INT x:y. B x)"
wenzelm@12897
  1859
  "!!A B C. (INT z: UNION A B. C z) = (INT x:A. INT z: B(x). C z)"
wenzelm@12897
  1860
  "!!A B f. (INT x:f`A. B x)    = (INT a:A. B (f a))"
wenzelm@12897
  1861
  by auto
wenzelm@12897
  1862
wenzelm@12897
  1863
lemma ball_simps [simp]:
wenzelm@12897
  1864
  "!!A P Q. (ALL x:A. P x | Q) = ((ALL x:A. P x) | Q)"
wenzelm@12897
  1865
  "!!A P Q. (ALL x:A. P | Q x) = (P | (ALL x:A. Q x))"
wenzelm@12897
  1866
  "!!A P Q. (ALL x:A. P --> Q x) = (P --> (ALL x:A. Q x))"
wenzelm@12897
  1867
  "!!A P Q. (ALL x:A. P x --> Q) = ((EX x:A. P x) --> Q)"
wenzelm@12897
  1868
  "!!P. (ALL x:{}. P x) = True"
wenzelm@12897
  1869
  "!!P. (ALL x:UNIV. P x) = (ALL x. P x)"
wenzelm@12897
  1870
  "!!a B P. (ALL x:insert a B. P x) = (P a & (ALL x:B. P x))"
wenzelm@12897
  1871
  "!!A P. (ALL x:Union A. P x) = (ALL y:A. ALL x:y. P x)"
wenzelm@12897
  1872
  "!!A B P. (ALL x: UNION A B. P x) = (ALL a:A. ALL x: B a. P x)"
wenzelm@12897
  1873
  "!!P Q. (ALL x:Collect Q. P x) = (ALL x. Q x --> P x)"
wenzelm@12897
  1874
  "!!A P f. (ALL x:f`A. P x) = (ALL x:A. P (f x))"
wenzelm@12897
  1875
  "!!A P. (~(ALL x:A. P x)) = (EX x:A. ~P x)"
wenzelm@12897
  1876
  by auto
wenzelm@12897
  1877
wenzelm@12897
  1878
lemma bex_simps [simp]:
wenzelm@12897
  1879
  "!!A P Q. (EX x:A. P x & Q) = ((EX x:A. P x) & Q)"
wenzelm@12897
  1880
  "!!A P Q. (EX x:A. P & Q x) = (P & (EX x:A. Q x))"
wenzelm@12897
  1881
  "!!P. (EX x:{}. P x) = False"
wenzelm@12897
  1882
  "!!P. (EX x:UNIV. P x) = (EX x. P x)"
wenzelm@12897
  1883
  "!!a B P. (EX x:insert a B. P x) = (P(a) | (EX x:B. P x))"
wenzelm@12897
  1884
  "!!A P. (EX x:Union A. P x) = (EX y:A. EX x:y. P x)"
wenzelm@12897
  1885
  "!!A B P. (EX x: UNION A B. P x) = (EX a:A. EX x:B a. P x)"
wenzelm@12897
  1886
  "!!P Q. (EX x:Collect Q. P x) = (EX x. Q x & P x)"
wenzelm@12897
  1887
  "!!A P f. (EX x:f`A. P x) = (EX x:A. P (f x))"
wenzelm@12897
  1888
  "!!A P. (~(EX x:A. P x)) = (ALL x:A. ~P x)"
wenzelm@12897
  1889
  by auto
wenzelm@12897
  1890
wenzelm@12897
  1891
lemma ball_conj_distrib:
wenzelm@12897
  1892
  "(ALL x:A. P x & Q x) = ((ALL x:A. P x) & (ALL x:A. Q x))"
wenzelm@12897
  1893
  by blast
wenzelm@12897
  1894
wenzelm@12897
  1895
lemma bex_disj_distrib:
wenzelm@12897
  1896
  "(EX x:A. P x | Q x) = ((EX x:A. P x) | (EX x:A. Q x))"
wenzelm@12897
  1897
  by blast
wenzelm@12897
  1898
wenzelm@12897
  1899
paulson@13860
  1900
text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
paulson@13860
  1901
paulson@13860
  1902
lemma UN_extend_simps:
paulson@13860
  1903
  "!!a B C. insert a (UN x:C. B x) = (if C={} then {a} else (UN x:C. insert a (B x)))"
paulson@13860
  1904
  "!!A B C. (UN x:C. A x) Un B    = (if C={} then B else (UN x:C. A x Un B))"
paulson@13860
  1905
  "!!A B C. A Un (UN x:C. B x)   = (if C={} then A else (UN x:C. A Un B x))"
paulson@13860
  1906
  "!!A B C. ((UN x:C. A x) Int B) = (UN x:C. A x Int B)"
paulson@13860
  1907
  "!!A B C. (A Int (UN x:C. B x)) = (UN x:C. A Int B x)"
paulson@13860
  1908
  "!!A B C. ((UN x:C. A x) - B) = (UN x:C. A x - B)"
paulson@13860
  1909
  "!!A B C. (A - (INT x:C. B x)) = (UN x:C. A - B x)"
paulson@13860
  1910
  "!!A B. (UN y:A. UN x:y. B x) = (UN x: Union A. B x)"
paulson@13860
  1911
  "!!A B C. (UN  x:A. UN z: B(x). C z) = (UN z: UNION A B. C z)"
paulson@13860
  1912
  "!!A B f. (UN a:A. B (f a)) = (UN x:f`A. B x)"
paulson@13860
  1913
  by auto
paulson@13860
  1914
paulson@13860
  1915
lemma INT_extend_simps:
paulson@13860
  1916
  "!!A B C. (INT x:C. A x) Int B = (if C={} then B else (INT x:C. A x Int B))"
paulson@13860
  1917
  "!!A B C. A Int (INT x:C. B x) = (if C={} then A else (INT x:C. A Int B x))"
paulson@13860
  1918
  "!!A B C. (INT x:C. A x) - B   = (if C={} then UNIV-B else (INT x:C. A x - B))"
paulson@13860
  1919
  "!!A B C. A - (UN x:C. B x)   = (if C={} then A else (INT x:C. A - B x))"
paulson@13860
  1920
  "!!a B C. insert a (INT x:C. B x) = (INT x:C. insert a (B x))"
paulson@13860
  1921
  "!!A B C. ((INT x:C. A x) Un B)  = (INT x:C. A x Un B)"
paulson@13860
  1922
  "!!A B C. A Un (INT x:C. B x)  = (INT x:C. A Un B x)"
paulson@13860
  1923
  "!!A B. (INT y:A. INT x:y. B x) = (INT x: Union A. B x)"
paulson@13860
  1924
  "!!A B C. (INT x:A. INT z: B(x). C z) = (INT z: UNION A B. C z)"
paulson@13860
  1925
  "!!A B f. (INT a:A. B (f a))    = (INT x:f`A. B x)"
paulson@13860
  1926
  by auto
paulson@13860
  1927
paulson@13860
  1928
wenzelm@12897
  1929
subsubsection {* Monotonicity of various operations *}
wenzelm@12897
  1930
wenzelm@12897
  1931
lemma image_mono: "A \<subseteq> B ==> f`A \<subseteq> f`B"
wenzelm@12897
  1932
  by blast
wenzelm@12897
  1933
wenzelm@12897
  1934
lemma Pow_mono: "A \<subseteq> B ==> Pow A \<subseteq> Pow B"
wenzelm@12897
  1935
  by blast
wenzelm@12897
  1936
wenzelm@12897
  1937
lemma Union_mono: "A \<subseteq> B ==> \<Union>A \<subseteq> \<Union>B"
wenzelm@12897
  1938
  by blast
wenzelm@12897
  1939
wenzelm@12897
  1940
lemma Inter_anti_mono: "B \<subseteq> A ==> \<Inter>A \<subseteq> \<Inter>B"
wenzelm@12897
  1941
  by blast
wenzelm@12897
  1942
wenzelm@12897
  1943
lemma UN_mono:
wenzelm@12897
  1944
  "A \<subseteq> B ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
wenzelm@12897
  1945
    (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
wenzelm@12897
  1946
  by (blast dest: subsetD)
wenzelm@12897
  1947
wenzelm@12897
  1948
lemma INT_anti_mono:
wenzelm@12897
  1949
  "B \<subseteq> A ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
wenzelm@12897
  1950
    (\<Inter>x\<in>A. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
wenzelm@12897
  1951
  -- {* The last inclusion is POSITIVE! *}
wenzelm@12897
  1952
  by (blast dest: subsetD)
wenzelm@12897
  1953
wenzelm@12897
  1954
lemma insert_mono: "C \<subseteq> D ==> insert a C \<subseteq> insert a D"
wenzelm@12897
  1955
  by blast
wenzelm@12897
  1956
wenzelm@12897
  1957
lemma Un_mono: "A \<subseteq> C ==> B \<subseteq> D ==> A \<union> B \<subseteq> C \<union> D"
wenzelm@12897
  1958
  by blast
wenzelm@12897
  1959
wenzelm@12897
  1960
lemma Int_mono: "A \<subseteq> C ==> B \<subseteq> D ==> A \<inter> B \<subseteq> C \<inter> D"
wenzelm@12897
  1961
  by blast
wenzelm@12897
  1962
wenzelm@12897
  1963
lemma Diff_mono: "A \<subseteq> C ==> D \<subseteq> B ==> A - B \<subseteq> C - D"
wenzelm@12897
  1964
  by blast
wenzelm@12897
  1965
wenzelm@12897
  1966
lemma Compl_anti_mono: "A \<subseteq> B ==> -B \<subseteq> -A"
wenzelm@12897
  1967
  by blast
wenzelm@12897
  1968
wenzelm@12897
  1969
text {* \medskip Monotonicity of implications. *}
wenzelm@12897
  1970
wenzelm@12897
  1971
lemma in_mono: "A \<subseteq> B ==> x \<in> A --> x \<in> B"
wenzelm@12897
  1972
  apply (rule impI)
paulson@14208
  1973
  apply (erule subsetD, assumption)
wenzelm@12897
  1974
  done
wenzelm@12897
  1975
wenzelm@12897
  1976
lemma conj_mono: "P1 --> Q1 ==> P2 --> Q2 ==> (P1 & P2) --> (Q1 & Q2)"
nipkow@17589
  1977
  by iprover
wenzelm@12897
  1978
wenzelm@12897
  1979
lemma disj_mono: "P1 --> Q1 ==> P2 --> Q2 ==> (P1 | P2) --> (Q1 | Q2)"
nipkow@17589
  1980
  by iprover
wenzelm@12897
  1981
wenzelm@12897
  1982
lemma imp_mono: "Q1 --> P1 ==> P2 --> Q2 ==> (P1 --> P2) --> (Q1 --> Q2)"
nipkow@17589
  1983
  by iprover
wenzelm@12897
  1984
wenzelm@12897
  1985
lemma imp_refl: "P --> P" ..
wenzelm@12897
  1986
wenzelm@12897
  1987
lemma ex_mono: "(!!x. P x --> Q x) ==> (EX x. P x) --> (EX x. Q x)"
nipkow@17589
  1988
  by iprover
wenzelm@12897
  1989
wenzelm@12897
  1990
lemma all_mono: "(!!x. P x --> Q x) ==> (ALL x. P x) --> (ALL x. Q x)"
nipkow@17589
  1991
  by iprover
wenzelm@12897
  1992
wenzelm@12897
  1993
lemma Collect_mono: "(!!x. P x --> Q x) ==> Collect P \<subseteq> Collect Q"
wenzelm@12897
  1994
  by blast
wenzelm@12897
  1995
wenzelm@12897
  1996
lemma Int_Collect_mono:
wenzelm@12897
  1997
    "A \<subseteq> B ==> (!!x. x \<in> A ==> P x --> Q x) ==> A \<inter> Collect P \<subseteq> B \<inter> Collect Q"
wenzelm@12897
  1998
  by blast
wenzelm@12897
  1999
wenzelm@12897
  2000
lemmas basic_monos =
wenzelm@12897
  2001
  subset_refl imp_refl disj_mono conj_mono
wenzelm@12897
  2002
  ex_mono Collect_mono in_mono
wenzelm@12897
  2003
wenzelm@12897
  2004
lemma eq_to_mono: "a = b ==> c = d ==> b --> d ==> a --> c"
nipkow@17589
  2005
  by iprover
wenzelm@12897
  2006
wenzelm@12897
  2007
lemma eq_to_mono2: "a = b ==> c = d ==> ~ b --> ~ d ==> ~ a --> ~ c"
nipkow@17589
  2008
  by iprover
wenzelm@11979
  2009
wenzelm@11982
  2010
lemma Least_mono:
wenzelm@11982
  2011
  "mono (f::'a::order => 'b::order) ==> EX x:S. ALL y:S. x <= y
wenzelm@11982
  2012
    ==> (LEAST y. y : f ` S) = f (LEAST x. x : S)"
wenzelm@11982
  2013
    -- {* Courtesy of Stephan Merz *}
wenzelm@11982
  2014
  apply clarify
nipkow@15950
  2015
  apply (erule_tac P = "%x. x : S" in LeastI2_order, fast)
nipkow@15950
  2016
  apply (rule LeastI2_order)
wenzelm@11982
  2017
  apply (auto elim: monoD intro!: order_antisym)
wenzelm@11982
  2018
  done
wenzelm@11982
  2019
wenzelm@12020
  2020
wenzelm@12257
  2021
subsection {* Inverse image of a function *}
wenzelm@12257
  2022
wenzelm@12257
  2023
constdefs
wenzelm@12257
  2024
  vimage :: "('a => 'b) => 'b set => 'a set"    (infixr "-`" 90)
wenzelm@12257
  2025
  "f -` B == {x. f x : B}"
wenzelm@12257
  2026
wenzelm@12257
  2027
wenzelm@12257
  2028
subsubsection {* Basic rules *}
wenzelm@12257
  2029
wenzelm@12257
  2030
lemma vimage_eq [simp]: "(a : f -` B) = (f a : B)"
wenzelm@12257
  2031
  by (unfold vimage_def) blast
wenzelm@12257
  2032
wenzelm@12257
  2033
lemma vimage_singleton_eq: "(a : f -` {b}) = (f a = b)"
wenzelm@12257
  2034
  by simp
wenzelm@12257
  2035
wenzelm@12257
  2036
lemma vimageI [intro]: "f a = b ==> b:B ==> a : f -` B"
wenzelm@12257
  2037
  by (unfold vimage_def) blast
wenzelm@12257
  2038
wenzelm@12257
  2039
lemma vimageI2: "f a : A ==> a : f -` A"
wenzelm@12257
  2040
  by (unfold vimage_def) fast
wenzelm@12257
  2041
wenzelm@12257
  2042
lemma vimageE [elim!]: "a: f -` B ==> (!!x. f a = x ==> x:B ==> P) ==> P"
wenzelm@12257
  2043
  by (unfold vimage_def) blast
wenzelm@12257
  2044
wenzelm@12257
  2045
lemma vimageD: "a : f -` A ==> f a : A"
wenzelm@12257
  2046
  by (unfold vimage_def) fast
wenzelm@12257
  2047
wenzelm@12257
  2048
wenzelm@12257
  2049
subsubsection {* Equations *}
wenzelm@12257
  2050
wenzelm@12257
  2051
lemma vimage_empty [simp]: "f -` {} = {}"
wenzelm@12257
  2052
  by blast
wenzelm@12257
  2053
wenzelm@12257
  2054
lemma vimage_Compl: "f -` (-A) = -(f -` A)"
wenzelm@12257
  2055
  by blast
wenzelm@12257
  2056
wenzelm@12257
  2057
lemma vimage_Un [simp]: "f -` (A Un B) = (f -` A) Un (f -` B)"
wenzelm@12257
  2058
  by blast
wenzelm@12257
  2059
wenzelm@12257
  2060
lemma vimage_Int [simp]: "f -` (A Int B) = (f -` A) Int (f -` B)"
wenzelm@12257
  2061
  by fast
wenzelm@12257
  2062
wenzelm@12257
  2063
lemma vimage_Union: "f -` (Union A) = (UN X:A. f -` X)"
wenzelm@12257
  2064
  by blast
wenzelm@12257
  2065
wenzelm@12257
  2066
lemma vimage_UN: "f-`(UN x:A. B x) = (UN x:A. f -` B x)"
wenzelm@12257
  2067
  by blast
wenzelm@12257
  2068
wenzelm@12257
  2069
lemma vimage_INT: "f-`(INT x:A. B x) = (INT x:A. f -` B x)"
wenzelm@12257
  2070
  by blast
wenzelm@12257
  2071
wenzelm@12257
  2072
lemma vimage_Collect_eq [simp]: "f -` Collect P = {y. P (f y)}"
wenzelm@12257
  2073
  by blast
wenzelm@12257
  2074
wenzelm@12257
  2075
lemma vimage_Collect: "(!!x. P (f x) = Q x) ==> f -` (Collect P) = Collect Q"
wenzelm@12257
  2076
  by blast
wenzelm@12257
  2077
wenzelm@12257
  2078
lemma vimage_insert: "f-`(insert a B) = (f-`{a}) Un (f-`B)"
wenzelm@12257
  2079
  -- {* NOT suitable for rewriting because of the recurrence of @{term "{a}"}. *}
wenzelm@12257
  2080
  by blast
wenzelm@12257
  2081
wenzelm@12257
  2082
lemma vimage_Diff: "f -` (A - B) = (f -` A) - (f -` B)"
wenzelm@12257
  2083
  by blast
wenzelm@12257
  2084
wenzelm@12257
  2085
lemma vimage_UNIV [simp]: "f -` UNIV = UNIV"
wenzelm@12257
  2086
  by blast
wenzelm@12257
  2087
wenzelm@12257
  2088
lemma vimage_eq_UN: "f-`B = (UN y: B. f-`{y})"
wenzelm@12257
  2089
  -- {* NOT suitable for rewriting *}
wenzelm@12257
  2090
  by blast
wenzelm@12257
  2091
wenzelm@12897
  2092
lemma vimage_mono: "A \<subseteq> B ==> f -` A \<subseteq> f -` B"
wenzelm@12257
  2093
  -- {* monotonicity *}
wenzelm@12257
  2094
  by blast
wenzelm@12257
  2095
wenzelm@12257
  2096
paulson@14479
  2097
subsection {* Getting the Contents of a Singleton Set *}
paulson@14479
  2098
paulson@14479
  2099
constdefs
paulson@14479
  2100
  contents :: "'a set => 'a"
paulson@14479
  2101
   "contents X == THE x. X = {x}"
paulson@14479
  2102
paulson@14479
  2103
lemma contents_eq [simp]: "contents {x} = x"
paulson@14479
  2104
by (simp add: contents_def)
paulson@14479
  2105
paulson@14479
  2106
wenzelm@12023
  2107
subsection {* Transitivity rules for calculational reasoning *}
wenzelm@12020
  2108
wenzelm@12897
  2109
lemma set_rev_mp: "x:A ==> A \<subseteq> B ==> x:B"
wenzelm@12020
  2110
  by (rule subsetD)
wenzelm@12020
  2111
wenzelm@12897
  2112
lemma set_mp: "A \<subseteq> B ==> x:A ==> x:B"
wenzelm@12020
  2113
  by (rule subsetD)
wenzelm@12020
  2114
wenzelm@12020
  2115
lemmas basic_trans_rules [trans] =
haftmann@21384
  2116
  order_trans_rules set_rev_mp set_mp
wenzelm@12020
  2117
wenzelm@21669
  2118
wenzelm@21669
  2119
subsection {* Basic ML bindings *}
wenzelm@21669
  2120
wenzelm@21669
  2121
ML {*
wenzelm@22139
  2122
val Ball_def = @{thm Ball_def}
wenzelm@22139
  2123
val Bex_def = @{thm Bex_def}
wenzelm@22139
  2124
val CollectD = @{thm CollectD}
wenzelm@22139
  2125
val CollectE = @{thm CollectE}
wenzelm@22139
  2126
val CollectI = @{thm CollectI}
wenzelm@22139
  2127
val Collect_conj_eq = @{thm Collect_conj_eq}
wenzelm@22139
  2128
val Collect_mem_eq = @{thm Collect_mem_eq}
wenzelm@22139
  2129
val IntD1 = @{thm IntD1}
wenzelm@22139
  2130
val IntD2 = @{thm IntD2}
wenzelm@22139
  2131
val IntE = @{thm IntE}
wenzelm@22139
  2132
val IntI = @{thm IntI}
wenzelm@22139
  2133
val Int_Collect = @{thm Int_Collect}
wenzelm@22139
  2134
val UNIV_I = @{thm UNIV_I}
wenzelm@22139
  2135
val UNIV_witness = @{thm UNIV_witness}
wenzelm@22139
  2136
val UnE = @{thm UnE}
wenzelm@22139
  2137
val UnI1 = @{thm UnI1}
wenzelm@22139
  2138
val UnI2 = @{thm UnI2}
wenzelm@22139
  2139
val ballE = @{thm ballE}
wenzelm@22139
  2140
val ballI = @{thm ballI}
wenzelm@22139
  2141
val bexCI = @{thm bexCI}
wenzelm@22139
  2142
val bexE = @{thm bexE}
wenzelm@22139
  2143
val bexI = @{thm bexI}
wenzelm@22139
  2144
val bex_triv = @{thm bex_triv}
wenzelm@22139
  2145
val bspec = @{thm bspec}
wenzelm@22139
  2146
val contra_subsetD = @{thm contra_subsetD}
wenzelm@22139
  2147
val distinct_lemma = @{thm distinct_lemma}
wenzelm@22139
  2148
val eq_to_mono = @{thm eq_to_mono}
wenzelm@22139
  2149
val eq_to_mono2 = @{thm eq_to_mono2}
wenzelm@22139
  2150
val equalityCE = @{thm equalityCE}
wenzelm@22139
  2151
val equalityD1 = @{thm equalityD1}
wenzelm@22139
  2152
val equalityD2 = @{thm equalityD2}
wenzelm@22139
  2153
val equalityE = @{thm equalityE}
wenzelm@22139
  2154
val equalityI = @{thm equalityI}
wenzelm@22139
  2155
val imageE = @{thm imageE}
wenzelm@22139
  2156
val imageI = @{thm imageI}
wenzelm@22139
  2157
val image_Un = @{thm image_Un}
wenzelm@22139
  2158
val image_insert = @{thm image_insert}
wenzelm@22139
  2159
val insert_commute = @{thm insert_commute}
wenzelm@22139
  2160
val insert_iff = @{thm insert_iff}
wenzelm@22139
  2161
val mem_Collect_eq = @{thm mem_Collect_eq}
wenzelm@22139
  2162
val rangeE = @{thm rangeE}
wenzelm@22139
  2163
val rangeI = @{thm rangeI}
wenzelm@22139
  2164
val range_eqI = @{thm range_eqI}
wenzelm@22139
  2165
val subsetCE = @{thm subsetCE}
wenzelm@22139
  2166
val subsetD = @{thm subsetD}
wenzelm@22139
  2167
val subsetI = @{thm subsetI}
wenzelm@22139
  2168
val subset_refl = @{thm subset_refl}
wenzelm@22139
  2169
val subset_trans = @{thm subset_trans}
wenzelm@22139
  2170
val vimageD = @{thm vimageD}
wenzelm@22139
  2171
val vimageE = @{thm vimageE}
wenzelm@22139
  2172
val vimageI = @{thm vimageI}
wenzelm@22139
  2173
val vimageI2 = @{thm vimageI2}
wenzelm@22139
  2174
val vimage_Collect = @{thm vimage_Collect}
wenzelm@22139
  2175
val vimage_Int = @{thm vimage_Int}
wenzelm@22139
  2176
val vimage_Un = @{thm vimage_Un}
wenzelm@21669
  2177
*}
wenzelm@21669
  2178
wenzelm@11979
  2179
end