src/HOL/Set.thy
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Mon, 20 Jul 2009 08:31:12 +0200
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(*  Title:      HOL/Set.thy
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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 definitions and syntax *}
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global
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types 'a set = "'a => bool"
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consts
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  Collect       :: "('a => bool) => 'a set"              -- "comprehension"
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  "op :"        :: "'a => 'a set => bool"                -- "membership"
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local
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notation
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  "op :"  ("op :") and
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  "op :"  ("(_/ : _)" [50, 51] 50)
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defs
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  mem_def [code]: "x : S == S x"
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  Collect_def [code]: "Collect P == P"
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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 :"  ("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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notation (HTML output)
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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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  "@Coll"       :: "pttrn => bool => 'a set"              ("(1{_./ _})")
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translations
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  "{x. P}"      == "Collect (%x. P)"
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definition Int :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Int" 70) where
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  "A Int B \<equiv> {x. x \<in> A \<and> x \<in> B}"
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definition Un :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Un" 65) where
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  "A Un B \<equiv> {x. x \<in> A \<or> x \<in> B}"
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notation (xsymbols)
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  "Int"  (infixl "\<inter>" 70) and
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  "Un"  (infixl "\<union>" 65)
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notation (HTML output)
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  "Int"  (infixl "\<inter>" 70) and
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  "Un"  (infixl "\<union>" 65)
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definition empty :: "'a set" ("{}") where
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  "empty \<equiv> {x. False}"
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definition insert :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set" where
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  "insert a B \<equiv> {x. x = a} \<union> B"
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definition UNIV :: "'a set" where
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  "UNIV \<equiv> {x. True}"
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syntax
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  "@Finset"     :: "args => 'a set"                       ("{(_)}")
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translations
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  "{x, xs}"     == "CONST insert x {xs}"
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  "{x}"         == "CONST insert x {}"
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global
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consts
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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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local
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defs
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  Ball_def:     "Ball A P       == ALL x. x:A --> P(x)"
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  Bex_def:      "Bex A P        == EX x. x:A & P(x)"
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  Bex1_def:     "Bex1 A P       == EX! x. x:A & P(x)"
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syntax
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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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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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translations
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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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subsection {* Additional concrete syntax *}
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syntax
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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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syntax (xsymbols)
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  "@Collect"    :: "idt => 'a set => bool => 'a set"      ("(1{_ \<in>/ _./ _})")
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translations
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  "{x:A. P}"    => "{x. x:A & P}"
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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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    fun nvars (Const ("_idts", _) $ _ $ idts) = nvars idts + 1
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      | nvars _ = 1;
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    fun setcompr_tr [e, idts, b] =
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      let
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        val eq = Syntax.const "op =" $ Bound (nvars idts) $ e;
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        val P = Syntax.const "op &" $ eq $ b;
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        val exP = ex_tr [idts, P];
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      in Syntax.const "Collect" $ Term.absdummy (dummyT, exP) end;
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  in [("@SetCompr", setcompr_tr)] end;
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*}
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(* To avoid eta-contraction of body: *)
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print_translation {*
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let
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  fun btr' syn [A, Abs abs] =
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    let val (x, t) = atomic_abs_tr' abs
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    in Syntax.const syn $ x $ A $ t end
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in [(@{const_syntax Ball}, btr' "_Ball"), (@{const_syntax Bex}, btr' "_Bex")] end
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*}
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print_translation {*
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let
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  val ex_tr' = snd (mk_binder_tr' ("Ex", "DUMMY"));
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  fun setcompr_tr' [Abs (abs as (_, _, P))] =
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    let
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      fun check (Const ("Ex", _) $ Abs (_, _, P), n) = check (P, n + 1)
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        | check (Const ("op &", _) $ (Const ("op =", _) $ Bound m $ e) $ P, n) =
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            n > 0 andalso m = n andalso not (loose_bvar1 (P, n)) andalso
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            ((0 upto (n - 1)) subset add_loose_bnos (e, 0, []))
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        | check _ = false
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        fun tr' (_ $ abs) =
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          let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' [abs]
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          in Syntax.const "@SetCompr" $ e $ idts $ Q end;
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    in if check (P, 0) then tr' P
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       else let val (x as _ $ Free(xN,_), t) = atomic_abs_tr' abs
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                val M = Syntax.const "@Coll" $ x $ t
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            in case t of
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                 Const("op &",_)
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                   $ (Const("op :",_) $ (Const("_bound",_) $ Free(yN,_)) $ A)
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                   $ P =>
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                   if xN=yN then Syntax.const "@Collect" $ x $ A $ P else M
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               | _ => M
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            end
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    end;
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  in [("Collect", setcompr_tr')] end;
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*}
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subsection {* Lemmas and proof tool setup *}
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subsubsection {* Relating predicates and sets *}
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lemma mem_Collect_eq [iff]: "(a : {x. P(x)}) = P(a)"
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  by (simp add: Collect_def mem_def)
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lemma Collect_mem_eq [simp]: "{x. x:A} = A"
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  by (simp add: Collect_def mem_def)
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lemma CollectI: "P(a) ==> a : {x. P(x)}"
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  by simp
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lemma CollectD: "a : {x. P(x)} ==> P(a)"
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  by simp
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lemma Collect_cong: "(!!x. P x = Q x) ==> {x. P(x)} = {x. Q(x)}"
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  by simp
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lemmas CollectE = CollectD [elim_format]
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subsubsection {* Bounded quantifiers *}
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lemma ballI [intro!]: "(!!x. x:A ==> P x) ==> ALL x:A. P x"
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  by (simp add: Ball_def)
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lemmas strip = impI allI ballI
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lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x"
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  by (simp add: Ball_def)
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lemma ballE [elim]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q"
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  by (unfold Ball_def) blast
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ML {* bind_thm ("rev_ballE", Thm.permute_prems 1 1 @{thm ballE}) *}
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text {*
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  \medskip This tactic takes assumptions @{prop "ALL x:A. P x"} and
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  @{prop "a:A"}; creates assumption @{prop "P a"}.
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*}
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ML {*
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  fun ball_tac i = etac @{thm ballE} i THEN contr_tac (i + 1)
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*}
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text {*
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  Gives better instantiation for bound:
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*}
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declaration {* fn _ =>
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  Classical.map_cs (fn cs => cs addbefore ("bspec", datac @{thm bspec} 1))
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*}
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lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x"
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  -- {* Normally the best argument order: @{prop "P x"} constrains the
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    choice of @{prop "x:A"}. *}
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  by (unfold Bex_def) blast
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lemma rev_bexI [intro?]: "x:A ==> P x ==> EX x:A. P x"
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  -- {* The best argument order when there is only one @{prop "x:A"}. *}
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  by (unfold Bex_def) blast
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   380
lemma bexCI: "(ALL x:A. ~P x ==> P a) ==> a:A ==> EX x:A. P x"
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  by (unfold Bex_def) blast
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   382
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   383
lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q"
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   384
  by (unfold Bex_def) blast
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   385
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   386
lemma ball_triv [simp]: "(ALL x:A. P) = ((EX x. x:A) --> P)"
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  -- {* Trival rewrite rule. *}
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   388
  by (simp add: Ball_def)
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   389
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   390
lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)"
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   391
  -- {* Dual form for existentials. *}
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   392
  by (simp add: Bex_def)
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   393
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   394
lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)"
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   395
  by blast
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   396
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   397
lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)"
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   398
  by blast
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   399
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   400
lemma bex_one_point1 [simp]: "(EX x:A. x = a & P x) = (a:A & P a)"
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   401
  by blast
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   402
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diff changeset
   403
lemma bex_one_point2 [simp]: "(EX x:A. a = x & P x) = (a:A & P a)"
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diff changeset
   404
  by blast
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diff changeset
   405
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   406
lemma ball_one_point1 [simp]: "(ALL x:A. x = a --> P x) = (a:A --> P a)"
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   407
  by blast
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diff changeset
   408
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   409
lemma ball_one_point2 [simp]: "(ALL x:A. a = x --> P x) = (a:A --> P a)"
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diff changeset
   410
  by blast
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   411
26480
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   412
ML {*
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   413
  local
22139
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diff changeset
   414
    val unfold_bex_tac = unfold_tac @{thms "Bex_def"};
18328
841261f303a1 simprocs: static evaluation of simpset;
wenzelm
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diff changeset
   415
    fun prove_bex_tac ss = unfold_bex_tac ss THEN Quantifier1.prove_one_point_ex_tac;
11979
0a3dace545c5 converted theory "Set";
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diff changeset
   416
    val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac;
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diff changeset
   417
22139
539a63b98f76 tuned ML setup;
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diff changeset
   418
    val unfold_ball_tac = unfold_tac @{thms "Ball_def"};
18328
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   419
    fun prove_ball_tac ss = unfold_ball_tac ss THEN Quantifier1.prove_one_point_all_tac;
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diff changeset
   420
    val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac;
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   421
  in
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   422
    val defBEX_regroup = Simplifier.simproc (the_context ())
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   423
      "defined BEX" ["EX x:A. P x & Q x"] rearrange_bex;
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   424
    val defBALL_regroup = Simplifier.simproc (the_context ())
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   425
      "defined BALL" ["ALL x:A. P x --> Q x"] rearrange_ball;
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   426
  end;
13462
56610e2ba220 sane interface for simprocs;
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diff changeset
   427
56610e2ba220 sane interface for simprocs;
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   428
  Addsimprocs [defBALL_regroup, defBEX_regroup];
11979
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   429
*}
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diff changeset
   430
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
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diff changeset
   431
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
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   432
subsubsection {* Congruence rules *}
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diff changeset
   433
16636
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
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   434
lemma ball_cong:
11979
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diff changeset
   435
  "A = B ==> (!!x. x:B ==> P x = Q x) ==>
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diff changeset
   436
    (ALL x:A. P x) = (ALL x:B. Q x)"
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diff changeset
   437
  by (simp add: Ball_def)
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diff changeset
   438
16636
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   439
lemma strong_ball_cong [cong]:
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   440
  "A = B ==> (!!x. x:B =simp=> P x = Q x) ==>
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   441
    (ALL x:A. P x) = (ALL x:B. Q x)"
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   442
  by (simp add: simp_implies_def Ball_def)
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   443
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   444
lemma bex_cong:
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   445
  "A = B ==> (!!x. x:B ==> P x = Q x) ==>
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   446
    (EX x:A. P x) = (EX x:B. Q x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   447
  by (simp add: Bex_def cong: conj_cong)
1273
6960ec882bca added 8bit pragmas
regensbu
parents: 1068
diff changeset
   448
16636
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   449
lemma strong_bex_cong [cong]:
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   450
  "A = B ==> (!!x. x:B =simp=> P x = Q x) ==>
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   451
    (EX x:A. P x) = (EX x:B. Q x)"
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   452
  by (simp add: simp_implies_def Bex_def cong: conj_cong)
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   453
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   454
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   455
subsubsection {* Subsets *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   456
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   457
lemma subsetI [atp,intro!]: "(!!x. x:A ==> x:B) ==> A \<subseteq> B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   458
  by (auto simp add: mem_def intro: predicate1I)
30352
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   459
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   460
text {*
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   461
  \medskip Map the type @{text "'a set => anything"} to just @{typ
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   462
  'a}; for overloading constants whose first argument has type @{typ
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   463
  "'a set"}.
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   464
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   465
30596
140b22f22071 tuned some theorem and attribute bindings
haftmann
parents: 30531
diff changeset
   466
lemma subsetD [elim, intro?]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B"
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   467
  -- {* Rule in Modus Ponens style. *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   468
  by (unfold mem_def) blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   469
30596
140b22f22071 tuned some theorem and attribute bindings
haftmann
parents: 30531
diff changeset
   470
lemma rev_subsetD [intro?]: "c \<in> A ==> A \<subseteq> B ==> c \<in> B"
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   471
  -- {* The same, with reversed premises for use with @{text erule} --
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   472
      cf @{text rev_mp}. *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   473
  by (rule subsetD)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   474
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   475
text {*
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   476
  \medskip Converts @{prop "A \<subseteq> B"} to @{prop "x \<in> A ==> x \<in> B"}.
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   477
*}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   478
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   479
ML {*
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   480
  fun impOfSubs th = th RSN (2, @{thm rev_subsetD})
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   481
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   482
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   483
lemma subsetCE [elim]: "A \<subseteq>  B ==> (c \<notin> A ==> P) ==> (c \<in> B ==> P) ==> P"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   484
  -- {* Classical elimination rule. *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   485
  by (unfold mem_def) blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   486
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   487
lemma subset_eq: "A \<le> B = (\<forall>x\<in>A. x \<in> B)" by blast
2388
d1f0505fc602 added set inclusion symbol syntax;
wenzelm
parents: 2372
diff changeset
   488
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   489
text {*
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   490
  \medskip Takes assumptions @{prop "A \<subseteq> B"}; @{prop "c \<in> A"} and
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   491
  creates the assumption @{prop "c \<in> B"}.
30352
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   492
*}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   493
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   494
ML {*
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   495
  fun set_mp_tac i = etac @{thm subsetCE} i THEN mp_tac i
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   496
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   497
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   498
lemma contra_subsetD: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   499
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   500
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   501
lemma subset_refl [simp,atp]: "A \<subseteq> A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   502
  by fast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   503
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   504
lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   505
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   506
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   507
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   508
subsubsection {* Equality *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   509
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   510
lemma set_ext: assumes prem: "(!!x. (x:A) = (x:B))" shows "A = B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   511
  apply (rule prem [THEN ext, THEN arg_cong, THEN box_equals])
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   512
   apply (rule Collect_mem_eq)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   513
  apply (rule Collect_mem_eq)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   514
  done
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   515
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   516
(* Due to Brian Huffman *)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   517
lemma expand_set_eq: "(A = B) = (ALL x. (x:A) = (x:B))"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   518
by(auto intro:set_ext)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   519
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   520
lemma subset_antisym [intro!]: "A \<subseteq> B ==> B \<subseteq> A ==> A = B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   521
  -- {* Anti-symmetry of the subset relation. *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   522
  by (iprover intro: set_ext subsetD)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   523
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   524
text {*
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   525
  \medskip Equality rules from ZF set theory -- are they appropriate
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   526
  here?
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   527
*}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   528
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   529
lemma equalityD1: "A = B ==> A \<subseteq> B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   530
  by (simp add: subset_refl)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   532
lemma equalityD2: "A = B ==> B \<subseteq> A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   533
  by (simp add: subset_refl)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   534
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   535
text {*
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   536
  \medskip Be careful when adding this to the claset as @{text
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   537
  subset_empty} is in the simpset: @{prop "A = {}"} goes to @{prop "{}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   538
  \<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}!
30352
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   539
*}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   540
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   541
lemma equalityE: "A = B ==> (A \<subseteq> B ==> B \<subseteq> A ==> P) ==> P"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   542
  by (simp add: subset_refl)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   543
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   544
lemma equalityCE [elim]:
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   545
    "A = B ==> (c \<in> A ==> c \<in> B ==> P) ==> (c \<notin> A ==> c \<notin> B ==> P) ==> P"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   546
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   547
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   548
lemma eqset_imp_iff: "A = B ==> (x : A) = (x : B)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   549
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   550
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   551
lemma eqelem_imp_iff: "x = y ==> (x : A) = (y : A)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   552
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   553
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   554
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   555
subsubsection {* The universal set -- UNIV *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   556
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   557
lemma UNIV_I [simp]: "x : UNIV"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   558
  by (simp add: UNIV_def)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   559
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   560
declare UNIV_I [intro]  -- {* unsafe makes it less likely to cause problems *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   561
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   562
lemma UNIV_witness [intro?]: "EX x. x : UNIV"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   563
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   564
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   565
lemma subset_UNIV [simp]: "A \<subseteq> UNIV"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   566
  by (rule subsetI) (rule UNIV_I)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   567
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   568
text {*
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   569
  \medskip Eta-contracting these two rules (to remove @{text P})
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   570
  causes them to be ignored because of their interaction with
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   571
  congruence rules.
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   572
*}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   573
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   574
lemma ball_UNIV [simp]: "Ball UNIV P = All P"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   575
  by (simp add: Ball_def)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   576
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   577
lemma bex_UNIV [simp]: "Bex UNIV P = Ex P"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   578
  by (simp add: Bex_def)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   579
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   580
lemma UNIV_eq_I: "(\<And>x. x \<in> A) \<Longrightarrow> UNIV = A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   581
  by auto
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   582
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   583
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   584
subsubsection {* The empty set *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   585
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   586
lemma empty_iff [simp]: "(c : {}) = False"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   587
  by (simp add: empty_def)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   588
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   589
lemma emptyE [elim!]: "a : {} ==> P"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   590
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   591
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   592
lemma empty_subsetI [iff]: "{} \<subseteq> A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   593
    -- {* One effect is to delete the ASSUMPTION @{prop "{} <= A"} *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   594
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   595
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   596
lemma equals0I: "(!!y. y \<in> A ==> False) ==> A = {}"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   597
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   598
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   599
lemma equals0D: "A = {} ==> a \<notin> A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   600
    -- {* Use for reasoning about disjointness: @{prop "A Int B = {}"} *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   601
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   602
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   603
lemma ball_empty [simp]: "Ball {} P = True"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   604
  by (simp add: Ball_def)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   605
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   606
lemma bex_empty [simp]: "Bex {} P = False"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   607
  by (simp add: Bex_def)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   608
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   609
lemma UNIV_not_empty [iff]: "UNIV ~= {}"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   610
  by (blast elim: equalityE)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   611
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   612
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   613
subsubsection {* The Powerset operator -- Pow *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   614
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   615
definition Pow :: "'a set => 'a set set" where
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   616
  Pow_def: "Pow A = {B. B \<le> A}"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   617
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   618
lemma Pow_iff [iff]: "(A \<in> Pow B) = (A \<subseteq> B)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   619
  by (simp add: Pow_def)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   620
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   621
lemma PowI: "A \<subseteq> B ==> A \<in> Pow B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   622
  by (simp add: Pow_def)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   623
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   624
lemma PowD: "A \<in> Pow B ==> A \<subseteq> B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   625
  by (simp add: Pow_def)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   626
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   627
lemma Pow_bottom: "{} \<in> Pow B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   628
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   629
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   630
lemma Pow_top: "A \<in> Pow A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   631
  by (simp add: subset_refl)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   632
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   633
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   634
subsubsection {* Set complement *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   635
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   636
lemma Compl_iff [simp]: "(c \<in> -A) = (c \<notin> A)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   637
  by (simp add: mem_def fun_Compl_def bool_Compl_def)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   638
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   639
lemma ComplI [intro!]: "(c \<in> A ==> False) ==> c \<in> -A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   640
  by (unfold mem_def fun_Compl_def bool_Compl_def) blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   641
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   642
text {*
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   643
  \medskip This form, with negated conclusion, works well with the
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   644
  Classical prover.  Negated assumptions behave like formulae on the
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   645
  right side of the notional turnstile ... *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   646
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   647
lemma ComplD [dest!]: "c : -A ==> c~:A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   648
  by (simp add: mem_def fun_Compl_def bool_Compl_def)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   649
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   650
lemmas ComplE = ComplD [elim_format]
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   651
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   652
lemma Compl_eq: "- A = {x. ~ x : A}" by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   653
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   654
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   655
subsubsection {* Binary union -- Un *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   656
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   657
lemma Un_iff [simp]: "(c : A Un B) = (c:A | c:B)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   658
  by (unfold Un_def) blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   659
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   660
lemma UnI1 [elim?]: "c:A ==> c : A Un B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   661
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   662
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   663
lemma UnI2 [elim?]: "c:B ==> c : A Un B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   664
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   665
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   666
text {*
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   667
  \medskip Classical introduction rule: no commitment to @{prop A} vs
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   668
  @{prop B}.
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   669
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   670
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   671
lemma UnCI [intro!]: "(c~:B ==> c:A) ==> c : A Un B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   672
  by auto
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   673
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   674
lemma UnE [elim!]: "c : A Un B ==> (c:A ==> P) ==> (c:B ==> P) ==> P"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   675
  by (unfold Un_def) blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   676
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   677
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   678
subsubsection {* Binary intersection -- Int *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   679
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   680
lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   681
  by (unfold Int_def) blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   682
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   683
lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   684
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   685
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   686
lemma IntD1: "c : A Int B ==> c:A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   687
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   688
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   689
lemma IntD2: "c : A Int B ==> c:B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   690
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   691
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   692
lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   693
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   694
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   695
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   696
subsubsection {* Set difference *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   697
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   698
lemma Diff_iff [simp]: "(c : A - B) = (c:A & c~:B)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   699
  by (simp add: mem_def fun_diff_def bool_diff_def)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   700
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   701
lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A - B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   702
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   703
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   704
lemma DiffD1: "c : A - B ==> c : A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   705
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   706
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   707
lemma DiffD2: "c : A - B ==> c : B ==> P"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   708
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   709
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   710
lemma DiffE [elim!]: "c : A - B ==> (c:A ==> c~:B ==> P) ==> P"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   711
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   712
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   713
lemma set_diff_eq: "A - B = {x. x : A & ~ x : B}" by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   714
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   715
lemma Compl_eq_Diff_UNIV: "-A = (UNIV - A)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   716
by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   717
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   718
31456
55edadbd43d5 insert now qualified and with authentic syntax
haftmann
parents: 31197
diff changeset
   719
subsubsection {* Augmenting a set -- @{const insert} *}
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   720
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   721
lemma insert_iff [simp]: "(a : insert b A) = (a = b | a:A)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   722
  by (unfold insert_def) blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   723
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   724
lemma insertI1: "a : insert a B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   725
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   726
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   727
lemma insertI2: "a : B ==> a : insert b B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   728
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   729
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   730
lemma insertE [elim!]: "a : insert b A ==> (a = b ==> P) ==> (a:A ==> P) ==> P"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   731
  by (unfold insert_def) blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   732
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   733
lemma insertCI [intro!]: "(a~:B ==> a = b) ==> a: insert b B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   734
  -- {* Classical introduction rule. *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   735
  by auto
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   736
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   737
lemma subset_insert_iff: "(A \<subseteq> insert x B) = (if x:A then A - {x} \<subseteq> B else A \<subseteq> B)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   738
  by auto
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   739
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   740
lemma set_insert:
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   741
  assumes "x \<in> A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   742
  obtains B where "A = insert x B" and "x \<notin> B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   743
proof
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   744
  from assms show "A = insert x (A - {x})" by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   745
next
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   746
  show "x \<notin> A - {x}" by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   747
qed
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   748
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   749
lemma insert_ident: "x ~: A ==> x ~: B ==> (insert x A = insert x B) = (A = B)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   750
by auto
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   751
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   752
subsubsection {* Singletons, using insert *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   753
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   754
lemma singletonI [intro!,noatp]: "a : {a}"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   755
    -- {* Redundant? But unlike @{text insertCI}, it proves the subgoal immediately! *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   756
  by (rule insertI1)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   757
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   758
lemma singletonD [dest!,noatp]: "b : {a} ==> b = a"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   759
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   760
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   761
lemmas singletonE = singletonD [elim_format]
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   762
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   763
lemma singleton_iff: "(b : {a}) = (b = a)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   764
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   765
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   766
lemma singleton_inject [dest!]: "{a} = {b} ==> a = b"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   767
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   768
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   769
lemma singleton_insert_inj_eq [iff,noatp]:
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   770
     "({b} = insert a A) = (a = b & A \<subseteq> {b})"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   771
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   772
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   773
lemma singleton_insert_inj_eq' [iff,noatp]:
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   774
     "(insert a A = {b}) = (a = b & A \<subseteq> {b})"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   775
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   776
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   777
lemma subset_singletonD: "A \<subseteq> {x} ==> A = {} | A = {x}"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   778
  by fast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   779
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   780
lemma singleton_conv [simp]: "{x. x = a} = {a}"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   781
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   782
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   783
lemma singleton_conv2 [simp]: "{x. a = x} = {a}"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   784
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   785
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   786
lemma diff_single_insert: "A - {x} \<subseteq> B ==> x \<in> A ==> A \<subseteq> insert x B"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   787
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   788
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   789
lemma doubleton_eq_iff: "({a,b} = {c,d}) = (a=c & b=d | a=d & b=c)"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   790
  by (blast elim: equalityE)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   791
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   792
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   793
subsubsection {* Image of a set under a function *}
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   794
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   795
text {*
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   796
  Frequently @{term b} does not have the syntactic form of @{term "f x"}.
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   797
*}
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   798
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   799
definition image :: "('a => 'b) => 'a set => 'b set" (infixr "`" 90) where
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   800
  image_def [noatp]: "f ` A = {y. EX x:A. y = f(x)}"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   801
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   802
abbreviation
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   803
  range :: "('a => 'b) => 'b set" where -- "of function"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   804
  "range f == f ` UNIV"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   805
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   806
lemma image_eqI [simp, intro]: "b = f x ==> x:A ==> b : f`A"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   807
  by (unfold image_def) blast
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   808
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   809
lemma imageI: "x : A ==> f x : f ` A"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   810
  by (rule image_eqI) (rule refl)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   811
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   812
lemma rev_image_eqI: "x:A ==> b = f x ==> b : f`A"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   813
  -- {* This version's more effective when we already have the
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   814
    required @{term x}. *}
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   815
  by (unfold image_def) blast
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   816
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   817
lemma imageE [elim!]:
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   818
  "b : (%x. f x)`A ==> (!!x. b = f x ==> x:A ==> P) ==> P"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   819
  -- {* The eta-expansion gives variable-name preservation. *}
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   820
  by (unfold image_def) blast
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   821
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   822
lemma image_Un: "f`(A Un B) = f`A Un f`B"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   823
  by blast
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   824
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   825
lemma image_iff: "(z : f`A) = (EX x:A. z = f x)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   826
  by blast
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   827
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   828
lemma image_subset_iff: "(f`A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   829
  -- {* This rewrite rule would confuse users if made default. *}
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   830
  by blast
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   831
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   832
lemma subset_image_iff: "(B \<subseteq> f`A) = (EX AA. AA \<subseteq> A & B = f`AA)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   833
  apply safe
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   834
   prefer 2 apply fast
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   835
  apply (rule_tac x = "{a. a : A & f a : B}" in exI, fast)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   836
  done
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   837
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   838
lemma image_subsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f`A \<subseteq> B"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   839
  -- {* Replaces the three steps @{text subsetI}, @{text imageE},
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   840
    @{text hypsubst}, but breaks too many existing proofs. *}
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   841
  by blast
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   842
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   843
text {*
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   844
  \medskip Range of a function -- just a translation for image!
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   845
*}
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   846
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   847
lemma range_eqI: "b = f x ==> b \<in> range f"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   848
  by simp
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   849
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   850
lemma rangeI: "f x \<in> range f"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   851
  by simp
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   852
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   853
lemma rangeE [elim?]: "b \<in> range (\<lambda>x. f x) ==> (!!x. b = f x ==> P) ==> P"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   854
  by blast
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   855
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   856
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   857
subsection {* Complete lattices *}
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   858
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   859
notation
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   860
  less_eq  (infix "\<sqsubseteq>" 50) and
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   861
  less (infix "\<sqsubset>" 50) and
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   862
  inf  (infixl "\<sqinter>" 70) and
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   863
  sup  (infixl "\<squnion>" 65)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   864
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   865
class complete_lattice = lattice + bot + top +
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   866
  fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   867
    and Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   868
  assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   869
     and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   870
  assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   871
     and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   872
begin
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   873
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   874
lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<le> a}"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   875
  by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   876
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   877
lemma Sup_Inf:  "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<le> b}"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   878
  by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   879
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   880
lemma Inf_Univ: "\<Sqinter>UNIV = \<Squnion>{}"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   881
  unfolding Sup_Inf by (auto simp add: UNIV_def)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   882
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   883
lemma Sup_Univ: "\<Squnion>UNIV = \<Sqinter>{}"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   884
  unfolding Inf_Sup by (auto simp add: UNIV_def)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   885
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   886
lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   887
  by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   888
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   889
lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   890
  by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   891
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   892
lemma Inf_singleton [simp]:
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   893
  "\<Sqinter>{a} = a"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   894
  by (auto intro: antisym Inf_lower Inf_greatest)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   895
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   896
lemma Sup_singleton [simp]:
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   897
  "\<Squnion>{a} = a"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   898
  by (auto intro: antisym Sup_upper Sup_least)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   899
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   900
lemma Inf_insert_simp:
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   901
  "\<Sqinter>insert a A = (if A = {} then a else a \<sqinter> \<Sqinter>A)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   902
  by (cases "A = {}") (simp_all, simp add: Inf_insert)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   903
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   904
lemma Sup_insert_simp:
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   905
  "\<Squnion>insert a A = (if A = {} then a else a \<squnion> \<Squnion>A)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   906
  by (cases "A = {}") (simp_all, simp add: Sup_insert)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   907
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   908
lemma Inf_binary:
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   909
  "\<Sqinter>{a, b} = a \<sqinter> b"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   910
  by (auto simp add: Inf_insert_simp)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   911
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   912
lemma Sup_binary:
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   913
  "\<Squnion>{a, b} = a \<squnion> b"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   914
  by (auto simp add: Sup_insert_simp)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   915
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   916
lemma bot_def:
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   917
  "bot = \<Squnion>{}"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   918
  by (auto intro: antisym Sup_least)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   919
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   920
lemma top_def:
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   921
  "top = \<Sqinter>{}"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   922
  by (auto intro: antisym Inf_greatest)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   923
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   924
lemma sup_bot [simp]:
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   925
  "x \<squnion> bot = x"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   926
  using bot_least [of x] by (simp add: le_iff_sup sup_commute)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   927
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   928
lemma inf_top [simp]:
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   929
  "x \<sqinter> top = x"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   930
  using top_greatest [of x] by (simp add: le_iff_inf inf_commute)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   931
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   932
definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   933
  "SUPR A f == \<Squnion> (f ` A)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   934
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   935
definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   936
  "INFI A f == \<Sqinter> (f ` A)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   937
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   938
end
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   939
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   940
syntax
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   941
  "_SUP1"     :: "pttrns => 'b => 'b"           ("(3SUP _./ _)" [0, 10] 10)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   942
  "_SUP"      :: "pttrn => 'a set => 'b => 'b"  ("(3SUP _:_./ _)" [0, 10] 10)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   943
  "_INF1"     :: "pttrns => 'b => 'b"           ("(3INF _./ _)" [0, 10] 10)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   944
  "_INF"      :: "pttrn => 'a set => 'b => 'b"  ("(3INF _:_./ _)" [0, 10] 10)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   945
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   946
translations
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   947
  "SUP x y. B"   == "SUP x. SUP y. B"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   948
  "SUP x. B"     == "CONST SUPR CONST UNIV (%x. B)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   949
  "SUP x. B"     == "SUP x:CONST UNIV. B"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   950
  "SUP x:A. B"   == "CONST SUPR A (%x. B)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   951
  "INF x y. B"   == "INF x. INF y. B"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   952
  "INF x. B"     == "CONST INFI CONST UNIV (%x. B)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   953
  "INF x. B"     == "INF x:CONST UNIV. B"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   954
  "INF x:A. B"   == "CONST INFI A (%x. B)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   955
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   956
(* To avoid eta-contraction of body: *)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   957
print_translation {*
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   958
let
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   959
  fun btr' syn (A :: Abs abs :: ts) =
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   960
    let val (x,t) = atomic_abs_tr' abs
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   961
    in list_comb (Syntax.const syn $ x $ A $ t, ts) end
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   962
  val const_syntax_name = Sign.const_syntax_name @{theory} o fst o dest_Const
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   963
in
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   964
[(const_syntax_name @{term SUPR}, btr' "_SUP"),(const_syntax_name @{term "INFI"}, btr' "_INF")]
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   965
end
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   966
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   967
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   968
context complete_lattice
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   969
begin
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   970
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   971
lemma le_SUPI: "i : A \<Longrightarrow> M i \<le> (SUP i:A. M i)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   972
  by (auto simp add: SUPR_def intro: Sup_upper)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   973
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   974
lemma SUP_leI: "(\<And>i. i : A \<Longrightarrow> M i \<le> u) \<Longrightarrow> (SUP i:A. M i) \<le> u"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   975
  by (auto simp add: SUPR_def intro: Sup_least)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   976
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   977
lemma INF_leI: "i : A \<Longrightarrow> (INF i:A. M i) \<le> M i"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   978
  by (auto simp add: INFI_def intro: Inf_lower)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   979
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   980
lemma le_INFI: "(\<And>i. i : A \<Longrightarrow> u \<le> M i) \<Longrightarrow> u \<le> (INF i:A. M i)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   981
  by (auto simp add: INFI_def intro: Inf_greatest)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   982
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   983
lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   984
  by (auto intro: antisym SUP_leI le_SUPI)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   985
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   986
lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   987
  by (auto intro: antisym INF_leI le_INFI)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   988
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   989
end
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   990
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   991
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   992
subsection {* Bool as complete lattice *}
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   993
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   994
instantiation bool :: complete_lattice
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   995
begin
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   996
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   997
definition
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   998
  Inf_bool_def: "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   999
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1000
definition
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1001
  Sup_bool_def: "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1002
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1003
instance proof
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1004
qed (auto simp add: Inf_bool_def Sup_bool_def le_bool_def)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1005
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1006
end
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1007
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1008
lemma Inf_empty_bool [simp]:
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1009
  "\<Sqinter>{}"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1010
  unfolding Inf_bool_def by auto
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1011
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1012
lemma not_Sup_empty_bool [simp]:
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1013
  "\<not> \<Squnion>{}"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1014
  unfolding Sup_bool_def by auto
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1015
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1016
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1017
subsection {* Fun as complete lattice *}
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1018
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1019
instantiation "fun" :: (type, complete_lattice) complete_lattice
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1020
begin
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1021
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1022
definition
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1023
  Inf_fun_def [code del]: "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1024
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1025
definition
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1026
  Sup_fun_def [code del]: "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1027
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1028
instance proof
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1029
qed (auto simp add: Inf_fun_def Sup_fun_def le_fun_def
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1030
  intro: Inf_lower Sup_upper Inf_greatest Sup_least)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1031
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1032
end
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1033
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1034
lemma Inf_empty_fun:
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1035
  "\<Sqinter>{} = (\<lambda>_. \<Sqinter>{})"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1036
  by rule (simp add: Inf_fun_def, simp add: empty_def)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1037
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1038
lemma Sup_empty_fun:
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1039
  "\<Squnion>{} = (\<lambda>_. \<Squnion>{})"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1040
  by rule (simp add: Sup_fun_def, simp add: empty_def)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1041
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1042
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1043
subsection {* Set as lattice *}
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1044
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1045
definition INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1046
  "INTER A B \<equiv> {y. \<forall>x\<in>A. y \<in> B x}"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1047
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1048
definition UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1049
  "UNION A B \<equiv> {y. \<exists>x\<in>A. y \<in> B x}"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1050
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1051
definition Inter :: "'a set set \<Rightarrow> 'a set" where
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1052
  "Inter S \<equiv> INTER S (\<lambda>x. x)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1053
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1054
definition Union :: "'a set set \<Rightarrow> 'a set" where
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1055
  "Union S \<equiv> UNION S (\<lambda>x. x)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1056
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1057
notation (xsymbols)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1058
  Inter  ("\<Inter>_" [90] 90) and
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1059
  Union  ("\<Union>_" [90] 90)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1060
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1061
syntax
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1062
  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1063
  "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1064
  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 10] 10)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1065
  "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 10] 10)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1066
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1067
syntax (xsymbols)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1068
  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1069
  "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1070
  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 10] 10)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1071
  "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 10] 10)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1072
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1073
syntax (latex output)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1074
  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1075
  "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1076
  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1077
  "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1078
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1079
translations
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1080
  "INT x y. B"  == "INT x. INT y. B"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1081
  "INT x. B"    == "CONST INTER CONST UNIV (%x. B)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1082
  "INT x. B"    == "INT x:CONST UNIV. B"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1083
  "INT x:A. B"  == "CONST INTER A (%x. B)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1084
  "UN x y. B"   == "UN x. UN y. B"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1085
  "UN x. B"     == "CONST UNION CONST UNIV (%x. B)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1086
  "UN x. B"     == "UN x:CONST UNIV. B"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1087
  "UN x:A. B"   == "CONST UNION A (%x. B)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1088
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1089
text {*
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1090
  Note the difference between ordinary xsymbol syntax of indexed
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1091
  unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1092
  and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1093
  former does not make the index expression a subscript of the
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1094
  union/intersection symbol because this leads to problems with nested
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1095
  subscripts in Proof General.
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1096
*}
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1097
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1098
(* To avoid eta-contraction of body: *)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1099
print_translation {*
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1100
let
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1101
  fun btr' syn [A, Abs abs] =
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1102
    let val (x, t) = atomic_abs_tr' abs
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1103
    in Syntax.const syn $ x $ A $ t end
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1104
in [(@{const_syntax UNION}, btr' "@UNION"),(@{const_syntax INTER}, btr' "@INTER")] end
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1105
*}
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1106
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1107
lemma Inter_image_eq [simp]:
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1108
  "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1109
  by (auto simp add: Inter_def INTER_def image_def)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1110
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1111
lemma Union_image_eq [simp]:
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1112
  "\<Union>(B`A) = (\<Union>x\<in>A. B x)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1113
  by (auto simp add: Union_def UNION_def image_def)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1114
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1115
lemma inf_set_eq: "A \<sqinter> B = A \<inter> B"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1116
  by (simp add: inf_fun_eq inf_bool_eq Int_def Collect_def mem_def)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1117
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1118
lemma sup_set_eq: "A \<squnion> B = A \<union> B"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1119
  by (simp add: sup_fun_eq sup_bool_eq Un_def Collect_def mem_def)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1120
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1121
lemma mono_Int: "mono f \<Longrightarrow> f (A \<inter> B) \<subseteq> f A \<inter> f B"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1122
  apply (fold inf_set_eq sup_set_eq)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1123
  apply (erule mono_inf)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1124
  done
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1125
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1126
lemma mono_Un: "mono f \<Longrightarrow> f A \<union> f B \<subseteq> f (A \<union> B)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1127
  apply (fold inf_set_eq sup_set_eq)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1128
  apply (erule mono_sup)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1129
  done
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1130
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1131
lemma top_set_eq: "top = UNIV"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1132
  by (iprover intro!: subset_antisym subset_UNIV top_greatest)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1133
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1134
lemma bot_set_eq: "bot = {}"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1135
  by (iprover intro!: subset_antisym empty_subsetI bot_least)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1136
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1137
lemma Inter_eq:
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1138
  "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1139
  by (simp add: Inter_def INTER_def)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1140
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1141
lemma Union_eq:
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1142
  "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1143
  by (simp add: Union_def UNION_def)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1144
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1145
lemma Inf_set_eq:
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1146
  "\<Sqinter>S = \<Inter>S"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1147
proof (rule set_ext)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1148
  fix x
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1149
  have "(\<forall>Q\<in>{P. \<exists>A\<in>S. P \<longleftrightarrow> x \<in> A}. Q) \<longleftrightarrow> (\<forall>A\<in>S. x \<in> A)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1150
    by auto
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1151
  then show "x \<in> \<Sqinter>S \<longleftrightarrow> x \<in> \<Inter>S"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1152
    by (simp add: Inter_eq Inf_fun_def Inf_bool_def) (simp add: mem_def)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1153
qed
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1154
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1155
lemma Sup_set_eq:
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1156
  "\<Squnion>S = \<Union>S"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1157
proof (rule set_ext)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1158
  fix x
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1159
  have "(\<exists>Q\<in>{P. \<exists>A\<in>S. P \<longleftrightarrow> x \<in> A}. Q) \<longleftrightarrow> (\<exists>A\<in>S. x \<in> A)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1160
    by auto
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1161
  then show "x \<in> \<Squnion>S \<longleftrightarrow> x \<in> \<Union>S"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1162
    by (simp add: Union_eq Sup_fun_def Sup_bool_def) (simp add: mem_def)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1163
qed
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1164
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1165
lemma INFI_set_eq:
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1166
  "(INF x:S. f x) = (\<Inter>x\<in>S. f x)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1167
  by (simp add: INFI_def Inf_set_eq)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1168
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1169
lemma SUPR_set_eq:
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1170
  "(SUP x:S. f x) = (\<Union>x\<in>S. f x)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1171
  by (simp add: SUPR_def Sup_set_eq)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1172
  
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1173
no_notation
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1174
  less_eq  (infix "\<sqsubseteq>" 50) and
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1175
  less (infix "\<sqsubset>" 50) and
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1176
  inf  (infixl "\<sqinter>" 70) and
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1177
  sup  (infixl "\<squnion>" 65) and
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1178
  Inf  ("\<Sqinter>_" [900] 900) and
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1179
  Sup  ("\<Squnion>_" [900] 900)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1180
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1181
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1182
subsubsection {* Unions of families *}
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1183
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 24280
diff changeset
  1184
declare UNION_def [noatp]
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 24280
diff changeset
  1185
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1186
lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1187
  by (unfold UNION_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1188
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1189
lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1190
  -- {* The order of the premises presupposes that @{term A} is rigid;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1191
    @{term b} may be flexible. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1192
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1193
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1194
lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1195
  by (unfold UNION_def) blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
  1196
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1197
lemma UN_cong [cong]:
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1198
    "A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1199
  by (simp add: UNION_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1200
29691
9f03b5f847cd Added strong congruence rule for UN.
berghofe
parents: 28562
diff changeset
  1201
lemma strong_UN_cong:
9f03b5f847cd Added strong congruence rule for UN.
berghofe
parents: 28562
diff changeset
  1202
    "A = B ==> (!!x. x:B =simp=> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
9f03b5f847cd Added strong congruence rule for UN.
berghofe
parents: 28562
diff changeset
  1203
  by (simp add: UNION_def simp_implies_def)
9f03b5f847cd Added strong congruence rule for UN.
berghofe
parents: 28562
diff changeset
  1204
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1205
lemma image_eq_UN: "f`A = (UN x:A. {f x})"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1206
  by blast
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1207
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1208
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1209
subsubsection {* Intersections of families *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1210
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1211
lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1212
  by (unfold INTER_def) blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
  1213
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1214
lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1215
  by (unfold INTER_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1216
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1217
lemma INT_D [elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1218
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1219
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1220
lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1221
  -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1222
  by (unfold INTER_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1223
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1224
lemma INT_cong [cong]:
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1225
    "A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1226
  by (simp add: INTER_def)
7238
36e58620ffc8 replaced HOL_quantifiers flag by "HOL" print mode;
wenzelm
parents: 5931
diff changeset
  1227
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
  1228
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1229
subsubsection {* Union *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1230
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 24280
diff changeset
  1231
lemma Union_iff [simp,noatp]: "(A : Union C) = (EX X:C. A:X)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1232
  by (unfold Union_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1233
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1234
lemma UnionI [intro]: "X:C ==> A:X ==> A : Union C"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1235
  -- {* The order of the premises presupposes that @{term C} is rigid;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1236
    @{term A} may be flexible. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1237
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1238
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1239
lemma UnionE [elim!]: "A : Union C ==> (!!X. A:X ==> X:C ==> R) ==> R"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1240
  by (unfold Union_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1241
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1242
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1243
subsubsection {* Inter *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1244
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 24280
diff changeset
  1245
lemma Inter_iff [simp,noatp]: "(A : Inter C) = (ALL X:C. A:X)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1246
  by (unfold Inter_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1247
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1248
lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1249
  by (simp add: Inter_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1250
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1251
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1252
  \medskip A ``destruct'' rule -- every @{term X} in @{term C}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1253
  contains @{term A} as an element, but @{prop "A:X"} can hold when
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1254
  @{prop "X:C"} does not!  This rule is analogous to @{text spec}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1255
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1256
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1257
lemma InterD [elim]: "A : Inter C ==> X:C ==> A:X"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1258
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1259
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1260
lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1261
  -- {* ``Classical'' elimination rule -- does not require proving
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1262
    @{prop "X:C"}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1263
  by (unfold Inter_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1264
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1265
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1266
subsubsection {* Set reasoning tools *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1267
31166
a90fe83f58ea "{x. P x & x=t & Q x}" is now rewritten to "if P t & Q t then {t} else {}"
nipkow
parents: 30814
diff changeset
  1268
text{* Elimination of @{text"{x. \<dots> & x=t & \<dots>}"}. *}
a90fe83f58ea "{x. P x & x=t & Q x}" is now rewritten to "if P t & Q t then {t} else {}"
nipkow
parents: 30814
diff changeset
  1269
31197
c1c163ec6c44 fine-tuned elimination of comprehensions involving x=t.
nipkow
parents: 31166
diff changeset
  1270
lemma Collect_conv_if: "{x. x=a & P x} = (if P a then {a} else {})"
c1c163ec6c44 fine-tuned elimination of comprehensions involving x=t.
nipkow
parents: 31166
diff changeset
  1271
by auto
c1c163ec6c44 fine-tuned elimination of comprehensions involving x=t.
nipkow
parents: 31166
diff changeset
  1272
c1c163ec6c44 fine-tuned elimination of comprehensions involving x=t.
nipkow
parents: 31166
diff changeset
  1273
lemma Collect_conv_if2: "{x. a=x & P x} = (if P a then {a} else {})"
31166
a90fe83f58ea "{x. P x & x=t & Q x}" is now rewritten to "if P t & Q t then {t} else {}"
nipkow
parents: 30814
diff changeset
  1274
by auto
a90fe83f58ea "{x. P x & x=t & Q x}" is now rewritten to "if P t & Q t then {t} else {}"
nipkow
parents: 30814
diff changeset
  1275
31197
c1c163ec6c44 fine-tuned elimination of comprehensions involving x=t.
nipkow
parents: 31166
diff changeset
  1276
text {*
c1c163ec6c44 fine-tuned elimination of comprehensions involving x=t.
nipkow
parents: 31166
diff changeset
  1277
Simproc for pulling @{text "x=t"} in @{text "{x. \<dots> & x=t & \<dots>}"}
c1c163ec6c44 fine-tuned elimination of comprehensions involving x=t.
nipkow
parents: 31166
diff changeset
  1278
to the front (and similarly for @{text "t=x"}):
c1c163ec6c44 fine-tuned elimination of comprehensions involving x=t.
nipkow
parents: 31166
diff changeset
  1279
*}
31166
a90fe83f58ea "{x. P x & x=t & Q x}" is now rewritten to "if P t & Q t then {t} else {}"
nipkow
parents: 30814
diff changeset
  1280
a90fe83f58ea "{x. P x & x=t & Q x}" is now rewritten to "if P t & Q t then {t} else {}"
nipkow
parents: 30814
diff changeset
  1281
ML{*
a90fe83f58ea "{x. P x & x=t & Q x}" is now rewritten to "if P t & Q t then {t} else {}"
nipkow
parents: 30814
diff changeset
  1282
  local
a90fe83f58ea "{x. P x & x=t & Q x}" is now rewritten to "if P t & Q t then {t} else {}"
nipkow
parents: 30814
diff changeset
  1283
    val Coll_perm_tac = rtac @{thm Collect_cong} 1 THEN rtac @{thm iffI} 1 THEN
a90fe83f58ea "{x. P x & x=t & Q x}" is now rewritten to "if P t & Q t then {t} else {}"
nipkow
parents: 30814
diff changeset
  1284
    ALLGOALS(EVERY'[REPEAT_DETERM o (etac @{thm conjE}),
a90fe83f58ea "{x. P x & x=t & Q x}" is now rewritten to "if P t & Q t then {t} else {}"
nipkow
parents: 30814
diff changeset
  1285
                    DEPTH_SOLVE_1 o (ares_tac [@{thm conjI}])])
a90fe83f58ea "{x. P x & x=t & Q x}" is now rewritten to "if P t & Q t then {t} else {}"
nipkow
parents: 30814
diff changeset
  1286
  in
a90fe83f58ea "{x. P x & x=t & Q x}" is now rewritten to "if P t & Q t then {t} else {}"
nipkow
parents: 30814
diff changeset
  1287
    val defColl_regroup = Simplifier.simproc (the_context ())
a90fe83f58ea "{x. P x & x=t & Q x}" is now rewritten to "if P t & Q t then {t} else {}"
nipkow
parents: 30814
diff changeset
  1288
      "defined Collect" ["{x. P x & Q x}"]
a90fe83f58ea "{x. P x & x=t & Q x}" is now rewritten to "if P t & Q t then {t} else {}"
nipkow
parents: 30814
diff changeset
  1289
      (Quantifier1.rearrange_Coll Coll_perm_tac)
a90fe83f58ea "{x. P x & x=t & Q x}" is now rewritten to "if P t & Q t then {t} else {}"
nipkow
parents: 30814
diff changeset
  1290
  end;
a90fe83f58ea "{x. P x & x=t & Q x}" is now rewritten to "if P t & Q t then {t} else {}"
nipkow
parents: 30814
diff changeset
  1291
a90fe83f58ea "{x. P x & x=t & Q x}" is now rewritten to "if P t & Q t then {t} else {}"
nipkow
parents: 30814
diff changeset
  1292
  Addsimprocs [defColl_regroup];
a90fe83f58ea "{x. P x & x=t & Q x}" is now rewritten to "if P t & Q t then {t} else {}"
nipkow
parents: 30814
diff changeset
  1293
*}
a90fe83f58ea "{x. P x & x=t & Q x}" is now rewritten to "if P t & Q t then {t} else {}"
nipkow
parents: 30814
diff changeset
  1294
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1295
text {*
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1296
  Rewrite rules for boolean case-splitting: faster than @{text
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1297
  "split_if [split]"}.
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
  1298
*}