src/HOL/Set.thy
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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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subsection {* Sets as predicates *}
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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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text {* Set comprehensions *}
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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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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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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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text {*
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Simproc for pulling @{text "x=t"} in @{text "{x. \<dots> & x=t & \<dots>}"}
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to the front (and similarly for @{text "t=x"}):
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*}
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setup {*
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let
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  val Coll_perm_tac = rtac @{thm Collect_cong} 1 THEN rtac @{thm iffI} 1 THEN
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    ALLGOALS(EVERY'[REPEAT_DETERM o (etac @{thm conjE}),
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                    DEPTH_SOLVE_1 o (ares_tac [@{thm conjI}])])
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  val defColl_regroup = Simplifier.simproc @{theory}
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    "defined Collect" ["{x. P x & Q x}"]
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    (Quantifier1.rearrange_Coll Coll_perm_tac)
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in
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  Simplifier.map_simpset (fn ss => ss addsimprocs [defColl_regroup])
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end
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*}
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lemmas CollectE = CollectD [elim_format]
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text {* Set enumerations *}
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definition empty :: "'a set" ("{}") where
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  "empty = {x. False}"
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definition insert :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set" where
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  insert_compr: "insert a B = {x. x = a \<or> x \<in> B}"
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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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subsection {* Subsets and bounded quantifiers *}
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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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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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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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5cbb52f2c478 Term.absdummy;
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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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3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
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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
15535
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   319
       else let val (x as _ $ Free(xN,_), t) = atomic_abs_tr' abs
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   320
                val M = Syntax.const "@Coll" $ x $ t
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   321
            in case t of
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   322
                 Const("op &",_)
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   323
                   $ (Const("op :",_) $ (Const("_bound",_) $ Free(yN,_)) $ A)
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   324
                   $ P =>
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   325
                   if xN=yN then Syntax.const "@Collect" $ x $ A $ P else M
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               | _ => M
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   327
            end
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    end;
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  in [("Collect", setcompr_tr')] end;
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*}
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32117
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setup {*
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let
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  val unfold_bex_tac = unfold_tac @{thms "Bex_def"};
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  fun prove_bex_tac ss = unfold_bex_tac ss THEN Quantifier1.prove_one_point_ex_tac;
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  val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac;
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  val unfold_ball_tac = unfold_tac @{thms "Ball_def"};
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  fun prove_ball_tac ss = unfold_ball_tac ss THEN Quantifier1.prove_one_point_all_tac;
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  val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac;
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  val defBEX_regroup = Simplifier.simproc @{theory}
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   341
    "defined BEX" ["EX x:A. P x & Q x"] rearrange_bex;
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  val defBALL_regroup = Simplifier.simproc @{theory}
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   343
    "defined BALL" ["ALL x:A. P x --> Q x"] rearrange_ball;
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in
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  Simplifier.map_simpset (fn ss => ss addsimprocs [defBALL_regroup, defBEX_regroup])
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   346
end
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*}
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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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   352
lemmas strip = impI allI ballI
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   353
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   354
lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x"
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   355
  by (simp add: Ball_def)
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   356
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   357
text {*
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   358
  Gives better instantiation for bound:
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   359
*}
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   360
26339
7825c83c9eff eliminated change_claset/simpset;
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   361
declaration {* fn _ =>
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   362
  Classical.map_cs (fn cs => cs addbefore ("bspec", datac @{thm bspec} 1))
11979
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   363
*}
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   364
32117
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   365
ML {*
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   366
structure Simpdata =
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   367
struct
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   368
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
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   369
open Simpdata;
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   370
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
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   371
val mksimps_pairs = [(@{const_name Ball}, @{thms bspec})] @ mksimps_pairs;
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0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
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   373
end;
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   374
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
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   375
open Simpdata;
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haftmann
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   376
*}
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   377
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
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   378
declaration {* fn _ =>
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haftmann
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diff changeset
   379
  Simplifier.map_ss (fn ss => ss setmksimps (mksimps mksimps_pairs))
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haftmann
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   380
*}
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   381
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
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   382
lemma ballE [elim]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q"
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
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diff changeset
   383
  by (unfold Ball_def) blast
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
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diff changeset
   384
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   385
lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x"
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diff changeset
   386
  -- {* Normally the best argument order: @{prop "P x"} constrains the
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wenzelm
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diff changeset
   387
    choice of @{prop "x:A"}. *}
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diff changeset
   388
  by (unfold Bex_def) blast
0a3dace545c5 converted theory "Set";
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diff changeset
   389
13113
5eb9be7b72a5 rev_bexI [intro?];
wenzelm
parents: 13103
diff changeset
   390
lemma rev_bexI [intro?]: "x:A ==> P x ==> EX x:A. P x"
11979
0a3dace545c5 converted theory "Set";
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   391
  -- {* The best argument order when there is only one @{prop "x:A"}. *}
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diff changeset
   392
  by (unfold Bex_def) blast
0a3dace545c5 converted theory "Set";
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diff changeset
   393
0a3dace545c5 converted theory "Set";
wenzelm
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diff changeset
   394
lemma bexCI: "(ALL x:A. ~P x ==> P a) ==> a:A ==> EX x:A. P x"
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wenzelm
parents: 11752
diff changeset
   395
  by (unfold Bex_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
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diff changeset
   396
0a3dace545c5 converted theory "Set";
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diff changeset
   397
lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q"
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wenzelm
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diff changeset
   398
  by (unfold Bex_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   399
0a3dace545c5 converted theory "Set";
wenzelm
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diff changeset
   400
lemma ball_triv [simp]: "(ALL x:A. P) = ((EX x. x:A) --> P)"
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wenzelm
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diff changeset
   401
  -- {* Trival rewrite rule. *}
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wenzelm
parents: 11752
diff changeset
   402
  by (simp add: Ball_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   403
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   404
lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   405
  -- {* Dual form for existentials. *}
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wenzelm
parents: 11752
diff changeset
   406
  by (simp add: Bex_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   407
0a3dace545c5 converted theory "Set";
wenzelm
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diff changeset
   408
lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   409
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   410
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   411
lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   412
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   413
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   414
lemma bex_one_point1 [simp]: "(EX x:A. x = a & P x) = (a:A & P a)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   415
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   416
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   417
lemma bex_one_point2 [simp]: "(EX x:A. a = x & P x) = (a:A & P a)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   418
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   419
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   420
lemma ball_one_point1 [simp]: "(ALL x:A. x = a --> P x) = (a:A --> P a)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   421
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   422
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   423
lemma ball_one_point2 [simp]: "(ALL x:A. a = x --> P x) = (a:A --> P a)"
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wenzelm
parents: 11752
diff changeset
   424
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   425
0a3dace545c5 converted theory "Set";
wenzelm
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diff changeset
   426
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   427
text {* Congruence rules *}
11979
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wenzelm
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diff changeset
   428
16636
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   429
lemma ball_cong:
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   430
  "A = B ==> (!!x. x:B ==> P x = Q x) ==>
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   431
    (ALL x:A. P x) = (ALL x:B. Q x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   432
  by (simp add: Ball_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   433
16636
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   434
lemma strong_ball_cong [cong]:
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   435
  "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
   436
    (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
   437
  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
   438
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   439
lemma bex_cong:
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   440
  "A = B ==> (!!x. x:B ==> P x = Q x) ==>
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   441
    (EX x:A. P x) = (EX x:B. Q x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   442
  by (simp add: Bex_def cong: conj_cong)
1273
6960ec882bca added 8bit pragmas
regensbu
parents: 1068
diff changeset
   443
16636
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   444
lemma strong_bex_cong [cong]:
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   445
  "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
   446
    (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
   447
  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
   448
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   449
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   450
subsection {* Basic operations *}
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   451
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   452
subsubsection {* Subsets *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   453
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   454
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
   455
  by (auto simp add: mem_def intro: predicate1I)
30352
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   456
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   457
text {*
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   458
  \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
   459
  '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
   460
  "'a set"}.
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   461
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   462
30596
140b22f22071 tuned some theorem and attribute bindings
haftmann
parents: 30531
diff changeset
   463
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
   464
  -- {* 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
   465
  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
   466
30596
140b22f22071 tuned some theorem and attribute bindings
haftmann
parents: 30531
diff changeset
   467
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
   468
  -- {* 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
   469
      cf @{text rev_mp}. *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   470
  by (rule subsetD)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   471
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   472
text {*
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   473
  \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
   474
*}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   475
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   476
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
   477
  -- {* Classical elimination rule. *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   478
  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
   479
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   480
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
   481
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   482
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
   483
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   484
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   485
lemma subset_refl [simp,atp]: "A \<subseteq> A"
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   486
  by (fact order_refl)
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   487
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   488
lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C"
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   489
  by (fact order_trans)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   490
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   491
lemma set_rev_mp: "x:A ==> A \<subseteq> B ==> x:B"
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   492
  by (rule subsetD)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   493
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   494
lemma set_mp: "A \<subseteq> B ==> x:A ==> x:B"
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   495
  by (rule subsetD)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   496
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   497
lemmas basic_trans_rules [trans] =
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   498
  order_trans_rules set_rev_mp set_mp
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   499
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
subsubsection {* Equality *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   502
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   503
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
   504
  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
   505
   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
   506
  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
   507
  done
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   508
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   509
(* Due to Brian Huffman *)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   510
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
   511
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
   512
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   513
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
   514
  -- {* 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
   515
  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
   516
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   517
text {*
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   518
  \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
   519
  here?
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   520
*}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   521
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   522
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
   523
  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
   524
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   525
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
   526
  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
   527
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   528
text {*
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   529
  \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
   530
  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
   531
  \<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}!
30352
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   532
*}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   533
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   534
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
   535
  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
   536
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   537
lemma equalityCE [elim]:
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   538
    "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
   539
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   540
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   541
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
   542
  by simp
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 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
   545
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   546
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
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
   549
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   550
definition UNIV :: "'a set" where
32117
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents: 32115
diff changeset
   551
  "UNIV = {x. True}"
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   552
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   553
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
   554
  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
   555
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   556
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
   557
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   558
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
   559
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   560
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   561
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
   562
  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
   563
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   564
lemma top_set_eq: "top = UNIV"
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   565
  by (iprover intro!: subset_antisym subset_UNIV top_greatest)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   566
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   567
text {*
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   568
  \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
   569
  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
   570
  congruence rules.
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   571
*}
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
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
   574
  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
   575
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   576
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
   577
  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
   578
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   579
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
   580
  by auto
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   581
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
subsubsection {* The empty set *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   584
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   585
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
   586
  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
   587
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   588
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
   589
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   590
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   591
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
   592
    -- {* 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
   593
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   594
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   595
lemma bot_set_eq: "bot = {}"
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   596
  by (iprover intro!: subset_antisym empty_subsetI bot_least)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   597
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   598
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
   599
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   600
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   601
lemma equals0D: "A = {} ==> a \<notin> A"
32082
90d03908b3d7 less digestible
haftmann
parents: 32081
diff changeset
   602
    -- {* Use for reasoning about disjointness: @{text "A Int B = {}"} *}
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   603
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   604
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   605
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
   606
  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
   607
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   608
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
   609
  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
   610
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   611
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
   612
  by (blast elim: equalityE)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   613
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   614
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   615
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
   616
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
   617
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
   618
  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
   619
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   620
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
   621
  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
   622
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   623
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
   624
  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
   625
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   626
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
   627
  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
   628
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   629
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
   630
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   631
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   632
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
   633
  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
   634
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
subsubsection {* Set complement *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   637
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   638
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
   639
  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
   640
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   641
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
   642
  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
   643
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   644
text {*
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   645
  \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
   646
  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
   647
  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
   648
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   649
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
   650
  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
   651
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   652
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
   653
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   654
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
   655
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
subsubsection {* Binary union -- Un *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   658
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   659
definition Un :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Un" 65) where
32117
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents: 32115
diff changeset
   660
  "A Un B = {x. x \<in> A \<or> x \<in> B}"
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   661
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   662
notation (xsymbols)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   663
  "Un"  (infixl "\<union>" 65)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   664
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   665
notation (HTML output)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   666
  "Un"  (infixl "\<union>" 65)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   667
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   668
lemma sup_set_eq: "sup A B = A \<union> B"
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   669
  by (simp add: sup_fun_eq sup_bool_eq Un_def Collect_def mem_def)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
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 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
   672
  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
   673
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   674
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
   675
  by simp
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
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
   678
  by simp
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
text {*
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   681
  \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
   682
  @{prop B}.
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   683
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   684
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   685
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
   686
  by auto
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   687
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   688
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
   689
  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
   690
32117
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents: 32115
diff changeset
   691
lemma insert_def: "insert a B = {x. x = a} \<union> B"
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   692
  by (simp add: Collect_def mem_def insert_compr Un_def)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   693
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   694
lemma mono_Un: "mono f \<Longrightarrow> f A \<union> f B \<subseteq> f (A \<union> B)"
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   695
  apply (fold sup_set_eq)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   696
  apply (erule mono_sup)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   697
  done
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   698
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   699
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   700
subsubsection {* Binary intersection -- Int *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   701
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   702
definition Int :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Int" 70) where
32117
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents: 32115
diff changeset
   703
  "A Int B = {x. x \<in> A \<and> x \<in> B}"
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   704
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   705
notation (xsymbols)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   706
  "Int"  (infixl "\<inter>" 70)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   707
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   708
notation (HTML output)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   709
  "Int"  (infixl "\<inter>" 70)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   710
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   711
lemma inf_set_eq: "inf A B = A \<inter> B"
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   712
  by (simp add: inf_fun_eq inf_bool_eq Int_def Collect_def mem_def)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   713
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   714
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
   715
  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
   716
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   717
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
   718
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   719
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   720
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
   721
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   722
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   723
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
   724
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   725
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   726
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
   727
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   728
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   729
lemma mono_Int: "mono f \<Longrightarrow> f (A \<inter> B) \<subseteq> f A \<inter> f B"
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   730
  apply (fold inf_set_eq)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   731
  apply (erule mono_inf)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   732
  done
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   733
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   734
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   735
subsubsection {* Set difference *}
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 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
   738
  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
   739
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   740
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
   741
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   742
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   743
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
   744
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   745
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   746
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
   747
  by simp
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 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
   750
  by simp
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
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
   753
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   754
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
   755
by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   756
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   757
31456
55edadbd43d5 insert now qualified and with authentic syntax
haftmann
parents: 31197
diff changeset
   758
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
   759
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   760
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
   761
  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
   762
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   763
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
   764
  by simp
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 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
   767
  by simp
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 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
   770
  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
   771
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   772
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
   773
  -- {* Classical introduction rule. *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   774
  by auto
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   775
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   776
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
   777
  by auto
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   778
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   779
lemma set_insert:
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   780
  assumes "x \<in> A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   781
  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
   782
proof
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   783
  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
   784
next
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   785
  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
   786
qed
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   787
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   788
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
   789
by auto
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   790
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   791
subsubsection {* Singletons, using insert *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   792
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   793
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
   794
    -- {* 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
   795
  by (rule insertI1)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   796
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   797
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
   798
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   799
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   800
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
   801
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   802
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
   803
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   804
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   805
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
   806
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   807
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   808
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
   809
     "({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
   810
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   811
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   812
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
   813
     "(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
   814
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   815
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   816
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
   817
  by fast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   818
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   819
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
   820
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   821
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   822
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
   823
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   824
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   825
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
   826
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   827
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   828
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
   829
  by (blast elim: equalityE)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   830
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   831
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
   832
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
   833
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
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
   835
  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
   836
*}
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
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
   839
  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
   840
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
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
   842
  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
   843
  "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
   844
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
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
   846
  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
   847
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
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
   849
  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
   850
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
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
   852
  -- {* 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
   853
    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
   854
  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
   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
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
   857
  "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
   858
  -- {* 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
   859
  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
   860
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
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
   862
  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
   863
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
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
   865
  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
   866
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
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
   868
  -- {* 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
   869
  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
   870
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
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
   872
  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
   873
   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
   874
  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
   875
  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
   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 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
   878
  -- {* 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
   879
    @{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
   880
  by blast
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   881
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   882
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
   883
  \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
   884
*}
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 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
   887
  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
   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 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
   890
  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
   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 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
   893
  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
   894
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
32117
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents: 32115
diff changeset
   896
subsubsection {* Some rules with @{text "if"} *}
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   897
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   898
text{* Elimination of @{text"{x. \<dots> & x=t & \<dots>}"}. *}
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   899
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   900
lemma Collect_conv_if: "{x. x=a & P x} = (if P a then {a} else {})"
32117
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents: 32115
diff changeset
   901
  by auto
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   902
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   903
lemma Collect_conv_if2: "{x. a=x & P x} = (if P a then {a} else {})"
32117
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents: 32115
diff changeset
   904
  by auto
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   905
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   906
text {*
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   907
  Rewrite rules for boolean case-splitting: faster than @{text
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   908
  "split_if [split]"}.
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   909
*}
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   910
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   911
lemma split_if_eq1: "((if Q then x else y) = b) = ((Q --> x = b) & (~ Q --> y = b))"
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   912
  by (rule split_if)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   913
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   914
lemma split_if_eq2: "(a = (if Q then x else y)) = ((Q --> a = x) & (~ Q --> a = y))"
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   915
  by (rule split_if)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   916
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   917
text {*
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   918
  Split ifs on either side of the membership relation.  Not for @{text
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   919
  "[simp]"} -- can cause goals to blow up!
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   920
*}
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   921
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   922
lemma split_if_mem1: "((if Q then x else y) : b) = ((Q --> x : b) & (~ Q --> y : b))"
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   923
  by (rule split_if)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   924
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   925
lemma split_if_mem2: "(a : (if Q then x else y)) = ((Q --> a : x) & (~ Q --> a : y))"
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   926
  by (rule split_if [where P="%S. a : S"])
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   927
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   928
lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   929
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   930
(*Would like to add these, but the existing code only searches for the
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   931
  outer-level constant, which in this case is just "op :"; we instead need
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   932
  to use term-nets to associate patterns with rules.  Also, if a rule fails to
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   933
  apply, then the formula should be kept.
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   934
  [("HOL.uminus", Compl_iff RS iffD1), ("HOL.minus", [Diff_iff RS iffD1]),
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   935
   ("Int", [IntD1,IntD2]),
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   936
   ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   937
 *)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   938
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   939
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
   940
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
   941
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
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
   943
  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
   944
  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
   945
  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
   946
  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
   947
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
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
   949
  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
   950
    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
   951
  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
   952
     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
   953
  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
   954
     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
   955
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
   956
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
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
   958
  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
   959
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
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
   961
  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
   962
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
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
   964
  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
   965
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   966
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
   967
  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
   968
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
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
   970
  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
   971
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
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
   973
  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
   974
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
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
   976
  "\<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
   977
  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
   978
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
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
   980
  "\<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
   981
  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
   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 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
   984
  "\<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
   985
  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
   986
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
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
   988
  "\<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
   989
  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
   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
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
   992
  "\<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
   993
  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
   994
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
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
   996
  "\<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
   997
  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
   998
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
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
  1000
  "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
  1001
  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
  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
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
  1004
  "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
  1005
  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
  1006
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
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
  1008
  "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
  1009
  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
  1010
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
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
  1012
  "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
  1013
  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
  1014
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
definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
32117
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents: 32115
diff changeset
  1016
  "SUPR A f = \<Squnion> (f ` A)"
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
  1017
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
definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
32117
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents: 32115
diff changeset
  1019
  "INFI A f = \<Sqinter> (f ` A)"
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
  1020
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
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
  1022
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
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
  1024
  "_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
  1025
  "_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
  1026
  "_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
  1027
  "_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
  1028
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
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
  1030
  "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
  1031
  "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
  1032
  "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
  1033
  "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
  1034
  "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
  1035
  "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
  1036
  "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
  1037
  "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
  1038
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
(* 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
  1040
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
  1041
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
  1042
  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
  1043
    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
  1044
    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
  1045
  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
  1046
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
  1047
[(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
  1048
end
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1049
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1050
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
  1051
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
  1052
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
  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
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
  1055
  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
  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
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
  1058
  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
  1059
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
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
  1061
  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
  1062
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
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
  1064
  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
  1065
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
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
  1067
  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
  1068
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
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
  1070
  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
  1071
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
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
  1073
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
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
  1075
subsubsection {* @{typ bool} and @{typ "_ \<Rightarrow> _"} as complete lattice *}
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
  1076
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
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
  1078
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
  1079
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
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
  1081
  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
  1082
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
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
  1084
  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
  1085
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
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
  1087
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
  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
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
  1090
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
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
  1092
  "\<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
  1093
  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
  1094
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
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
  1096
  "\<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
  1097
  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
  1098
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
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
  1100
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
  1101
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
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
  1103
  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
  1104
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
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
  1106
  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
  1107
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
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
  1109
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
  1110
  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
  1111
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
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
  1113
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
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
  1115
  "\<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
  1116
  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
  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_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
  1119
  "\<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
  1120
  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
  1121
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
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1123
subsubsection {* Union *}
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1124
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1125
definition Union :: "'a set set \<Rightarrow> 'a set" where
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1126
  Union_eq [code del]: "Union A = {x. \<exists>B \<in> A. x \<in> B}"
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1127
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1128
notation (xsymbols)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1129
  Union  ("\<Union>_" [90] 90)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1130
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1131
lemma Sup_set_eq:
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1132
  "\<Squnion>S = \<Union>S"
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1133
proof (rule set_ext)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1134
  fix x
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1135
  have "(\<exists>Q\<in>{P. \<exists>A\<in>S. P \<longleftrightarrow> x \<in> A}. Q) \<longleftrightarrow> (\<exists>A\<in>S. x \<in> A)"
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1136
    by auto
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1137
  then show "x \<in> \<Squnion>S \<longleftrightarrow> x \<in> \<Union>S"
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1138
    by (simp add: Sup_fun_def Sup_bool_def Union_eq) (simp add: mem_def)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1139
qed
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1140
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1141
lemma Union_iff [simp, noatp]:
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1142
  "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1143
  by (unfold Union_eq) blast
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1144
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1145
lemma UnionI [intro]:
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1146
  "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1147
  -- {* The order of the premises presupposes that @{term C} is rigid;
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1148
    @{term A} may be flexible. *}
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1149
  by auto
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1150
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1151
lemma UnionE [elim!]:
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1152
  "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A\<in>X \<Longrightarrow> X\<in>C \<Longrightarrow> R) \<Longrightarrow> R"
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1153
  by auto
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1154
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1155
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
  1156
subsubsection {* Unions of families *}
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
  1157
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
definition UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1159
  UNION_eq_Union_image: "UNION A B = \<Union>(B`A)"
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
  1160
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
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
  1162
  "@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
  1163
  "@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
  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
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
  1166
  "@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
  1167
  "@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
  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
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
  1170
  "@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
  1171
  "@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
  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
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
  1174
  "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
  1175
  "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
  1176
  "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
  1177
  "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
  1178
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
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
  1180
  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
  1181
  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
  1182
  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
  1183
  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
  1184
  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
  1185
  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
  1186
*}
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1187
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1188
(* 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
  1189
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
  1190
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
  1191
  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
  1192
    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
  1193
    in Syntax.const syn $ x $ A $ t end
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
  1194
in [(@{const_syntax UNION}, btr' "@UNION")] end
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
  1195
*}
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
  1196
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1197
lemma SUPR_set_eq:
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1198
  "(SUP x:S. f x) = (\<Union>x\<in>S. f x)"
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1199
  by (simp add: SUPR_def UNION_eq_Union_image Sup_set_eq)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1200
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1201
lemma Union_def:
32117
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents: 32115
diff changeset
  1202
  "\<Union>S = (\<Union>x\<in>S. x)"
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1203
  by (simp add: UNION_eq_Union_image image_def)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1204
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1205
lemma UNION_def [noatp]:
32117
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents: 32115
diff changeset
  1206
  "UNION A B = {y. \<exists>x\<in>A. y \<in> B x}"
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents: 32115
diff changeset
  1207
  by (auto simp add: UNION_eq_Union_image Union_eq)
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1208
  
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1209
lemma Union_image_eq [simp]:
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1210
  "\<Union>(B`A) = (\<Union>x\<in>A. B x)"
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1211
  by (rule sym) (fact UNION_eq_Union_image)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1212
  
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1213
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
  1214
  by (unfold UNION_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1215
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1216
lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1217
  -- {* The order of the premises presupposes that @{term A} is rigid;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1218
    @{term b} may be flexible. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1219
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1220
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1221
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
  1222
  by (unfold UNION_def) blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
  1223
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1224
lemma UN_cong [cong]:
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1225
    "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
  1226
  by (simp add: UNION_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1227
29691
9f03b5f847cd Added strong congruence rule for UN.
berghofe
parents: 28562
diff changeset
  1228
lemma strong_UN_cong:
9f03b5f847cd Added strong congruence rule for UN.
berghofe
parents: 28562
diff changeset
  1229
    "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
  1230
  by (simp add: UNION_def simp_implies_def)
9f03b5f847cd Added strong congruence rule for UN.
berghofe
parents: 28562
diff changeset
  1231
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
  1232
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
  1233
  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
  1234
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1235
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1236
subsubsection {* Inter *}
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1237
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1238
definition Inter :: "'a set set \<Rightarrow> 'a set" where
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1239
  Inter_eq [code del]: "Inter A = {x. \<forall>B \<in> A. x \<in> B}"
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1240
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1241
notation (xsymbols)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1242
  Inter  ("\<Inter>_" [90] 90)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1243
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1244
lemma Inf_set_eq:
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1245
  "\<Sqinter>S = \<Inter>S"
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1246
proof (rule set_ext)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1247
  fix x
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1248
  have "(\<forall>Q\<in>{P. \<exists>A\<in>S. P \<longleftrightarrow> x \<in> A}. Q) \<longleftrightarrow> (\<forall>A\<in>S. x \<in> A)"
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1249
    by auto
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1250
  then show "x \<in> \<Sqinter>S \<longleftrightarrow> x \<in> \<Inter>S"
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1251
    by (simp add: Inter_eq Inf_fun_def Inf_bool_def) (simp add: mem_def)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1252
qed
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1253
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1254
lemma Inter_iff [simp,noatp]: "(A : Inter C) = (ALL X:C. A:X)"
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1255
  by (unfold Inter_eq) blast
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1256
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1257
lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C"
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1258
  by (simp add: Inter_eq)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1259
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1260
text {*
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1261
  \medskip A ``destruct'' rule -- every @{term X} in @{term C}
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1262
  contains @{term A} as an element, but @{prop "A:X"} can hold when
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1263
  @{prop "X:C"} does not!  This rule is analogous to @{text spec}.
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1264
*}
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1265
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1266
lemma InterD [elim]: "A : Inter C ==> X:C ==> A:X"
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1267
  by auto
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1268
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1269
lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R"
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1270
  -- {* ``Classical'' elimination rule -- does not require proving
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1271
    @{prop "X:C"}. *}
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1272
  by (unfold Inter_eq) blast
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1273
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1274
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1275
subsubsection {* Intersections of families *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1276
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
  1277
definition INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1278
  INTER_eq_Inter_image: "INTER A B = \<Inter>(B`A)"
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
  1279
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
  1280
syntax
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
  1281
  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
  1282
  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 10] 10)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
  1283
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
  1284
syntax (xsymbols)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
  1285
  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
  1286
  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 10] 10)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
  1287
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
  1288
syntax (latex output)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
  1289
  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
  1290
  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
  1291
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
  1292
translations
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
  1293
  "INT x y. B"  == "INT x. INT y. B"
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
  1294
  "INT x. B"    == "CONST INTER CONST UNIV (%x. B)"
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
  1295
  "INT x. B"    == "INT x:CONST UNIV. B"
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
  1296
  "INT x:A. B"  == "CONST INTER A (%x. B)"
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
  1297
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
  1298
(* To avoid eta-contraction of body: *)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
  1299
print_translation {*
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
  1300
let
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
  1301
  fun btr' syn [A, Abs abs] =
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
  1302
    let val (x, t) = atomic_abs_tr' abs
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
  1303
    in Syntax.const syn $ x $ A $ t end
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
  1304
in [(@{const_syntax INTER}, btr' "@INTER")] end
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
  1305
*}
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
  1306
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1307
lemma INFI_set_eq:
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1308
  "(INF x:S. f x) = (\<Inter>x\<in>S. f x)"
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1309
  by (simp add: INFI_def INTER_eq_Inter_image Inf_set_eq)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1310
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
  1311
lemma Inter_def:
32117
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents: 32115
diff changeset
  1312
  "Inter S = INTER S (\<lambda>x. x)"