src/HOL/Set.thy
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explicit checks stable_finished_theory/stable_command allow parallel asynchronous command transactions; tuned;
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(*  Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel *)
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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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typedecl 'a set
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axiomatization Collect :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set" -- "comprehension"
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  and member :: "'a \<Rightarrow> 'a set \<Rightarrow> bool" -- "membership"
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where
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  mem_Collect_eq [iff, code_unfold]: "member a (Collect P) = P a"
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  and Collect_mem_eq [simp]: "Collect (\<lambda>x. member x A) = A"
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notation
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  member  ("op :") and
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  member  ("(_/ : _)" [50, 51] 50)
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abbreviation not_member where
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  "not_member x A \<equiv> ~ (x : A)" -- "non-membership"
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notation
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  not_member  ("op ~:") and
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  not_member  ("(_/ ~: _)" [50, 51] 50)
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notation (xsymbols)
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  member      ("op \<in>") and
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  member      ("(_/ \<in> _)" [50, 51] 50) and
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  not_member  ("op \<notin>") and
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  not_member  ("(_/ \<notin> _)" [50, 51] 50)
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notation (HTML output)
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  member      ("op \<in>") and
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  member      ("(_/ \<in> _)" [50, 51] 50) and
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  not_member  ("op \<notin>") and
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  not_member  ("(_/ \<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}" == "CONST Collect (%x. P)"
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syntax
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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 CollectI: "P a \<Longrightarrow> a \<in> {x. P x}"
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  by simp
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lemma CollectD: "a \<in> {x. P x} \<Longrightarrow> P a"
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  by simp
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lemma Collect_cong: "(\<And>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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simproc_setup defined_Collect ("{x. P x & Q x}") = {*
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  fn _ =>
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    Quantifier1.rearrange_Collect
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     (rtac @{thm Collect_cong} 1 THEN
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      rtac @{thm iffI} 1 THEN
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      ALLGOALS
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        (EVERY' [REPEAT_DETERM o etac @{thm conjE}, DEPTH_SOLVE_1 o ares_tac @{thms conjI}]))
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*}
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lemmas CollectE = CollectD [elim_format]
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lemma set_eqI:
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  assumes "\<And>x. x \<in> A \<longleftrightarrow> x \<in> B"
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  shows "A = B"
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proof -
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  from assms have "{x. x \<in> A} = {x. x \<in> B}" by simp
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  then show ?thesis by simp
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qed
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lemma set_eq_iff [no_atp]:
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  "A = B \<longleftrightarrow> (\<forall>x. x \<in> A \<longleftrightarrow> x \<in> B)"
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  by (auto intro:set_eqI)
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text {* Lifting of predicate class instances *}
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instantiation set :: (type) boolean_algebra
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begin
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definition less_eq_set where
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  "A \<le> B \<longleftrightarrow> (\<lambda>x. member x A) \<le> (\<lambda>x. member x B)"
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definition less_set where
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  "A < B \<longleftrightarrow> (\<lambda>x. member x A) < (\<lambda>x. member x B)"
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definition inf_set where
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  "A \<sqinter> B = Collect ((\<lambda>x. member x A) \<sqinter> (\<lambda>x. member x B))"
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definition sup_set where
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  "A \<squnion> B = Collect ((\<lambda>x. member x A) \<squnion> (\<lambda>x. member x B))"
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definition bot_set where
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  "\<bottom> = Collect \<bottom>"
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definition top_set where
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  "\<top> = Collect \<top>"
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definition uminus_set where
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  "- A = Collect (- (\<lambda>x. member x A))"
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definition minus_set where
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  "A - B = Collect ((\<lambda>x. member x A) - (\<lambda>x. member x B))"
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instance proof
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qed (simp_all add: less_eq_set_def less_set_def inf_set_def sup_set_def
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  bot_set_def top_set_def uminus_set_def minus_set_def
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  less_le_not_le inf_compl_bot sup_compl_top sup_inf_distrib1 diff_eq
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  set_eqI fun_eq_iff
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  del: inf_apply sup_apply bot_apply top_apply minus_apply uminus_apply)
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end
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text {* Set enumerations *}
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abbreviation empty :: "'a set" ("{}") where
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  "{} \<equiv> bot"
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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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definition Ball :: "'a set \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" where
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  "Ball A P \<longleftrightarrow> (\<forall>x. x \<in> A \<longrightarrow> P x)"   -- "bounded universal quantifiers"
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definition Bex :: "'a set \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" where
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  "Bex A P \<longleftrightarrow> (\<exists>x. x \<in> A \<and> P x)"   -- "bounded existential quantifiers"
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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" == "CONST Ball A (%x. P)"
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  "EX x:A. P" == "CONST Bex A (%x. P)"
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  "EX! x:A. P" => "EX! x. x:A & 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 All_binder = Mixfix.binder_name @{const_syntax All};
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  val Ex_binder = Mixfix.binder_name @{const_syntax Ex};
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  val impl = @{const_syntax HOL.implies};
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  val conj = @{const_syntax HOL.conj};
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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), @{syntax_const "_setlessAll"}),
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    ((All_binder, impl, sbset_eq), @{syntax_const "_setleAll"}),
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    ((Ex_binder, conj, sbset), @{syntax_const "_setlessEx"}),
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    ((Ex_binder, conj, sbset_eq), @{syntax_const "_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_Trans.mark_bound v' $ n $ P else raise Match;
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  fun tr' q = (q,
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        fn [Const (@{syntax_const "_bound"}, _) $ Free (v, Type (@{type_name set}, _)),
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            Const (c, _) $
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              (Const (d, _) $ (Const (@{syntax_const "_bound"}, _) $ Free (v', _)) $ n) $ P] =>
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            (case AList.lookup (op =) trans (q, c, d) of
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              NONE => raise Match
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            | SOME l => mk v v' l n P)
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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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syntax
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  "_Setcompr" :: "'a => idts => bool => 'a set"    ("(1{_ |/_./ _})")
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parse_translation {*
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  let
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    val ex_tr = snd (Syntax_Trans.mk_binder_tr ("EX ", @{const_syntax Ex}));
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    fun nvars (Const (@{syntax_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 @{const_syntax HOL.eq} $ Bound (nvars idts) $ e;
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        val P = Syntax.const @{const_syntax HOL.conj} $ eq $ b;
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        val exP = ex_tr [idts, P];
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      in Syntax.const @{const_syntax Collect} $ absdummy dummyT exP end;
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  in [(@{syntax_const "_Setcompr"}, setcompr_tr)] end;
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*}
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print_translation {*
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 [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax Ball} @{syntax_const "_Ball"},
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  Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax Bex} @{syntax_const "_Bex"}]
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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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  val ex_tr' = snd (Syntax_Trans.mk_binder_tr' (@{const_syntax 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 (@{const_syntax Ex}, _) $ Abs (_, _, P), n) = check (P, n + 1)
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        | check (Const (@{const_syntax HOL.conj}, _) $
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              (Const (@{const_syntax HOL.eq}, _) $ 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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            subset (op =) (0 upto (n - 1), 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 @{syntax_const "_Setcompr"} $ e $ idts $ Q end;
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    in
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      if check (P, 0) then tr' P
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      else
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        let
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          val (x as _ $ Free(xN, _), t) = Syntax_Trans.atomic_abs_tr' abs;
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          val M = Syntax.const @{syntax_const "_Coll"} $ x $ t;
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        in
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          case t of
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            Const (@{const_syntax HOL.conj}, _) $
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              (Const (@{const_syntax Set.member}, _) $
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                (Const (@{syntax_const "_bound"}, _) $ Free (yN, _)) $ A) $ P =>
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            if xN = yN then Syntax.const @{syntax_const "_Collect"} $ x $ A $ P else M
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          | _ => M
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        end
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    end;
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  in [(@{const_syntax Collect}, setcompr_tr')] end;
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*}
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simproc_setup defined_Bex ("EX x:A. P x & Q x") = {*
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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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  in fn _ => fn ss => Quantifier1.rearrange_bex (prove_bex_tac ss) ss end
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*}
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simproc_setup defined_All ("ALL x:A. P x --> Q x") = {*
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  let
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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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  in fn _ => fn ss => Quantifier1.rearrange_ball (prove_ball_tac ss) ss 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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lemmas strip = impI allI ballI
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lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x"
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  by (simp add: Ball_def)
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text {*
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   379
  Gives better instantiation for bound:
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*}
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26339
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declaration {* fn _ =>
46459
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   383
  Classical.map_cs (fn cs => cs addbefore ("bspec", dtac @{thm bspec} THEN' assume_tac))
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*}
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32117
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ML {*
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haftmann
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   387
structure Simpdata =
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haftmann
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   388
struct
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0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
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   390
open Simpdata;
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0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
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val mksimps_pairs = [(@{const_name Ball}, @{thms bspec})] @ mksimps_pairs;
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   393
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
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   394
end;
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
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0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
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   396
open Simpdata;
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haftmann
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diff changeset
   397
*}
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   398
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
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   399
declaration {* fn _ =>
45625
750c5a47400b modernized some old-style infix operations, which were left over from the time of ML proof scripts;
wenzelm
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diff changeset
   400
  Simplifier.map_ss (Simplifier.set_mksimps (mksimps mksimps_pairs))
32117
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
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diff changeset
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*}
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
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diff changeset
   402
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents: 32115
diff changeset
   403
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
parents: 32115
diff changeset
   404
  by (unfold Ball_def) blast
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents: 32115
diff changeset
   405
11979
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   406
lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x"
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diff changeset
   407
  -- {* Normally the best argument order: @{prop "P x"} constrains the
0a3dace545c5 converted theory "Set";
wenzelm
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diff changeset
   408
    choice of @{prop "x:A"}. *}
0a3dace545c5 converted theory "Set";
wenzelm
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diff changeset
   409
  by (unfold Bex_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   410
13113
5eb9be7b72a5 rev_bexI [intro?];
wenzelm
parents: 13103
diff changeset
   411
lemma rev_bexI [intro?]: "x:A ==> P x ==> EX x:A. P x"
11979
0a3dace545c5 converted theory "Set";
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diff changeset
   412
  -- {* The best argument order when there is only one @{prop "x:A"}. *}
0a3dace545c5 converted theory "Set";
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diff changeset
   413
  by (unfold Bex_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   414
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   415
lemma bexCI: "(ALL x:A. ~P x ==> P a) ==> a:A ==> EX x:A. P x"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   416
  by (unfold Bex_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   417
0a3dace545c5 converted theory "Set";
wenzelm
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diff changeset
   418
lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q"
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diff changeset
   419
  by (unfold Bex_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
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diff changeset
   420
0a3dace545c5 converted theory "Set";
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diff changeset
   421
lemma ball_triv [simp]: "(ALL x:A. P) = ((EX x. x:A) --> P)"
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wenzelm
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diff changeset
   422
  -- {* Trival rewrite rule. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   423
  by (simp add: Ball_def)
0a3dace545c5 converted theory "Set";
wenzelm
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diff changeset
   424
0a3dace545c5 converted theory "Set";
wenzelm
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diff changeset
   425
lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   426
  -- {* Dual form for existentials. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   427
  by (simp add: Bex_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   428
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   429
lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   430
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   431
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   432
lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   433
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   434
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   435
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
   436
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   437
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   438
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
   439
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   440
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   441
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
   442
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   443
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   444
lemma ball_one_point2 [simp]: "(ALL x:A. a = x --> P x) = (a:A --> P a)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   445
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   446
43818
fcc5d3ffb6f5 tuned lemma positions and proofs
haftmann
parents: 42459
diff changeset
   447
lemma ball_conj_distrib:
fcc5d3ffb6f5 tuned lemma positions and proofs
haftmann
parents: 42459
diff changeset
   448
  "(\<forall>x\<in>A. P x \<and> Q x) \<longleftrightarrow> ((\<forall>x\<in>A. P x) \<and> (\<forall>x\<in>A. Q x))"
fcc5d3ffb6f5 tuned lemma positions and proofs
haftmann
parents: 42459
diff changeset
   449
  by blast
fcc5d3ffb6f5 tuned lemma positions and proofs
haftmann
parents: 42459
diff changeset
   450
fcc5d3ffb6f5 tuned lemma positions and proofs
haftmann
parents: 42459
diff changeset
   451
lemma bex_disj_distrib:
fcc5d3ffb6f5 tuned lemma positions and proofs
haftmann
parents: 42459
diff changeset
   452
  "(\<exists>x\<in>A. P x \<or> Q x) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<or> (\<exists>x\<in>A. Q x))"
fcc5d3ffb6f5 tuned lemma positions and proofs
haftmann
parents: 42459
diff changeset
   453
  by blast
fcc5d3ffb6f5 tuned lemma positions and proofs
haftmann
parents: 42459
diff changeset
   454
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   455
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   456
text {* Congruence rules *}
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   457
16636
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   458
lemma ball_cong:
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   459
  "A = B ==> (!!x. x:B ==> P x = Q x) ==>
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   460
    (ALL x:A. P x) = (ALL x:B. Q x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   461
  by (simp add: Ball_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   462
16636
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   463
lemma strong_ball_cong [cong]:
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   464
  "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
   465
    (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
   466
  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
   467
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   468
lemma bex_cong:
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   469
  "A = B ==> (!!x. x:B ==> P x = Q x) ==>
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   470
    (EX x:A. P x) = (EX x:B. Q x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   471
  by (simp add: Bex_def cong: conj_cong)
1273
6960ec882bca added 8bit pragmas
regensbu
parents: 1068
diff changeset
   472
16636
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   473
lemma strong_bex_cong [cong]:
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   474
  "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
   475
    (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
   476
  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
   477
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   478
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   479
subsection {* Basic operations *}
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   480
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   481
subsubsection {* Subsets *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   482
33022
c95102496490 Removal of the unused atpset concept, the atp attribute and some related code.
paulson
parents: 32888
diff changeset
   483
lemma subsetI [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> x \<in> B) \<Longrightarrow> A \<subseteq> B"
45959
184d36538e51 `set` is now a proper type constructor; added operation for set monad
haftmann
parents: 45909
diff changeset
   484
  by (simp add: less_eq_set_def le_fun_def)
30352
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   485
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   486
text {*
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   487
  \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
   488
  '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
   489
  "'a set"}.
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   490
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   491
30596
140b22f22071 tuned some theorem and attribute bindings
haftmann
parents: 30531
diff changeset
   492
lemma subsetD [elim, intro?]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B"
45959
184d36538e51 `set` is now a proper type constructor; added operation for set monad
haftmann
parents: 45909
diff changeset
   493
  by (simp add: less_eq_set_def le_fun_def)
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   494
  -- {* 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
   495
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35576
diff changeset
   496
lemma rev_subsetD [no_atp,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
   497
  -- {* 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
   498
      cf @{text rev_mp}. *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   499
  by (rule subsetD)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   500
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   501
text {*
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   502
  \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
   503
*}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   504
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35576
diff changeset
   505
lemma subsetCE [no_atp,elim]: "A \<subseteq> B ==> (c \<notin> A ==> P) ==> (c \<in> B ==> P) ==> P"
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   506
  -- {* Classical elimination rule. *}
45959
184d36538e51 `set` is now a proper type constructor; added operation for set monad
haftmann
parents: 45909
diff changeset
   507
  by (auto simp add: less_eq_set_def le_fun_def)
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   508
46127
af3b95160b59 cleanup of code declarations
haftmann
parents: 46036
diff changeset
   509
lemma subset_eq [code, no_atp]: "A \<le> B = (\<forall>x\<in>A. x \<in> B)" by blast
2388
d1f0505fc602 added set inclusion symbol syntax;
wenzelm
parents: 2372
diff changeset
   510
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35576
diff changeset
   511
lemma contra_subsetD [no_atp]: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A"
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   512
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   513
45121
5e495ccf6e56 Set.thy: remove redundant [simp] declarations
huffman
parents: 44744
diff changeset
   514
lemma subset_refl: "A \<subseteq> A"
5e495ccf6e56 Set.thy: remove redundant [simp] declarations
huffman
parents: 44744
diff changeset
   515
  by (fact order_refl) (* already [iff] *)
30531
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
lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C"
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   518
  by (fact order_trans)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   519
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   520
lemma set_rev_mp: "x:A ==> A \<subseteq> B ==> x:B"
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   521
  by (rule subsetD)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   522
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   523
lemma set_mp: "A \<subseteq> B ==> x:A ==> x:B"
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   524
  by (rule subsetD)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   525
46146
6baea4fca6bd incorporated various theorems from theory More_Set into corpus
haftmann
parents: 46137
diff changeset
   526
lemma subset_not_subset_eq [code]:
6baea4fca6bd incorporated various theorems from theory More_Set into corpus
haftmann
parents: 46137
diff changeset
   527
  "A \<subset> B \<longleftrightarrow> A \<subseteq> B \<and> \<not> B \<subseteq> A"
6baea4fca6bd incorporated various theorems from theory More_Set into corpus
haftmann
parents: 46137
diff changeset
   528
  by (fact less_le_not_le)
6baea4fca6bd incorporated various theorems from theory More_Set into corpus
haftmann
parents: 46137
diff changeset
   529
33044
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 33022
diff changeset
   530
lemma eq_mem_trans: "a=b ==> b \<in> A ==> a \<in> A"
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 33022
diff changeset
   531
  by simp
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 33022
diff changeset
   532
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   533
lemmas basic_trans_rules [trans] =
33044
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 33022
diff changeset
   534
  order_trans_rules set_rev_mp set_mp eq_mem_trans
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   535
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
subsubsection {* Equality *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   538
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   539
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
   540
  -- {* Anti-symmetry of the subset relation. *}
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39213
diff changeset
   541
  by (iprover intro: set_eqI subsetD)
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   542
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   543
text {*
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   544
  \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
   545
  here?
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
lemma equalityD1: "A = B ==> A \<subseteq> B"
34209
c7f621786035 killed a few warnings
krauss
parents: 33935
diff changeset
   549
  by simp
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   550
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   551
lemma equalityD2: "A = B ==> B \<subseteq> A"
34209
c7f621786035 killed a few warnings
krauss
parents: 33935
diff changeset
   552
  by simp
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   553
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   554
text {*
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   555
  \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
   556
  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
   557
  \<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}!
30352
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   558
*}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   559
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   560
lemma equalityE: "A = B ==> (A \<subseteq> B ==> B \<subseteq> A ==> P) ==> P"
34209
c7f621786035 killed a few warnings
krauss
parents: 33935
diff changeset
   561
  by simp
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   562
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   563
lemma equalityCE [elim]:
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   564
    "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
   565
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   566
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   567
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
   568
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   569
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   570
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
   571
  by simp
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
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   574
subsubsection {* The empty set *}
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   575
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   576
lemma empty_def:
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   577
  "{} = {x. False}"
45959
184d36538e51 `set` is now a proper type constructor; added operation for set monad
haftmann
parents: 45909
diff changeset
   578
  by (simp add: bot_set_def bot_fun_def)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   579
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   580
lemma empty_iff [simp]: "(c : {}) = False"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   581
  by (simp add: empty_def)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   582
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   583
lemma emptyE [elim!]: "a : {} ==> P"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   584
  by simp
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   585
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   586
lemma empty_subsetI [iff]: "{} \<subseteq> A"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   587
    -- {* One effect is to delete the ASSUMPTION @{prop "{} <= A"} *}
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   588
  by blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   589
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   590
lemma equals0I: "(!!y. y \<in> A ==> False) ==> A = {}"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   591
  by blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   592
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   593
lemma equals0D: "A = {} ==> a \<notin> A"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   594
    -- {* Use for reasoning about disjointness: @{text "A Int B = {}"} *}
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   595
  by blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   596
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   597
lemma ball_empty [simp]: "Ball {} P = True"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   598
  by (simp add: Ball_def)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   599
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   600
lemma bex_empty [simp]: "Bex {} P = False"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   601
  by (simp add: Bex_def)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   602
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   603
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   604
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
   605
32264
0be31453f698 Set.UNIV and Set.empty are mere abbreviations for top and bot
haftmann
parents: 32139
diff changeset
   606
abbreviation UNIV :: "'a set" where
0be31453f698 Set.UNIV and Set.empty are mere abbreviations for top and bot
haftmann
parents: 32139
diff changeset
   607
  "UNIV \<equiv> top"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   608
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   609
lemma UNIV_def:
32117
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents: 32115
diff changeset
   610
  "UNIV = {x. True}"
45959
184d36538e51 `set` is now a proper type constructor; added operation for set monad
haftmann
parents: 45909
diff changeset
   611
  by (simp add: top_set_def top_fun_def)
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   612
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   613
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
   614
  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
   615
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   616
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
   617
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   618
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
   619
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   620
45121
5e495ccf6e56 Set.thy: remove redundant [simp] declarations
huffman
parents: 44744
diff changeset
   621
lemma subset_UNIV: "A \<subseteq> UNIV"
5e495ccf6e56 Set.thy: remove redundant [simp] declarations
huffman
parents: 44744
diff changeset
   622
  by (fact top_greatest) (* already simp *)
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   623
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   624
text {*
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   625
  \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
   626
  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
   627
  congruence rules.
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
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   630
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
   631
  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
   632
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   633
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
   634
  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
   635
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   636
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
   637
  by auto
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   638
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   639
lemma 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
   640
  by (blast elim: equalityE)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   641
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   642
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   643
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
   644
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
   645
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
   646
  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
   647
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   648
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
   649
  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
   650
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   651
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
   652
  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
   653
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   654
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
   655
  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
   656
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   657
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
   658
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   659
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   660
lemma Pow_top: "A \<in> Pow A"
34209
c7f621786035 killed a few warnings
krauss
parents: 33935
diff changeset
   661
  by simp
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   662
40703
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39910
diff changeset
   663
lemma Pow_not_empty: "Pow A \<noteq> {}"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39910
diff changeset
   664
  using Pow_top by blast
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   665
41076
a7fba340058c primitive definitions of bot/top/inf/sup for bool and fun are named with canonical suffix `_def` rather than `_eq`;
haftmann
parents: 40872
diff changeset
   666
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   667
subsubsection {* Set complement *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   668
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   669
lemma Compl_iff [simp]: "(c \<in> -A) = (c \<notin> A)"
45959
184d36538e51 `set` is now a proper type constructor; added operation for set monad
haftmann
parents: 45909
diff changeset
   670
  by (simp add: fun_Compl_def uminus_set_def)
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   671
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   672
lemma ComplI [intro!]: "(c \<in> A ==> False) ==> c \<in> -A"
45959
184d36538e51 `set` is now a proper type constructor; added operation for set monad
haftmann
parents: 45909
diff changeset
   673
  by (simp add: fun_Compl_def uminus_set_def) blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   674
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   675
text {*
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   676
  \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
   677
  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
   678
  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
   679
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   680
lemma ComplD [dest!]: "c : -A ==> c~:A"
45959
184d36538e51 `set` is now a proper type constructor; added operation for set monad
haftmann
parents: 45909
diff changeset
   681
  by simp
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   682
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   683
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
   684
45959
184d36538e51 `set` is now a proper type constructor; added operation for set monad
haftmann
parents: 45909
diff changeset
   685
lemma Compl_eq: "- A = {x. ~ x : A}"
184d36538e51 `set` is now a proper type constructor; added operation for set monad
haftmann
parents: 45909
diff changeset
   686
  by blast
30531
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
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   689
subsubsection {* Binary intersection *}
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   690
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   691
abbreviation inter :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Int" 70) where
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   692
  "op Int \<equiv> inf"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   693
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   694
notation (xsymbols)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   695
  inter  (infixl "\<inter>" 70)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   696
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   697
notation (HTML output)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   698
  inter  (infixl "\<inter>" 70)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   699
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   700
lemma Int_def:
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   701
  "A \<inter> B = {x. x \<in> A \<and> x \<in> B}"
45959
184d36538e51 `set` is now a proper type constructor; added operation for set monad
haftmann
parents: 45909
diff changeset
   702
  by (simp add: inf_set_def inf_fun_def)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   703
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   704
lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   705
  by (unfold Int_def) blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   706
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   707
lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   708
  by simp
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   709
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   710
lemma IntD1: "c : A Int B ==> c:A"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   711
  by simp
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   712
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   713
lemma IntD2: "c : A Int B ==> c:B"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   714
  by simp
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   715
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   716
lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   717
  by simp
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   718
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   719
lemma mono_Int: "mono f \<Longrightarrow> f (A \<inter> B) \<subseteq> f A \<inter> f B"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   720
  by (fact mono_inf)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   721
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   722
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41076
diff changeset
   723
subsubsection {* Binary union *}
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   724
32683
7c1fe854ca6a inter and union are mere abbreviations for inf and sup
haftmann
parents: 32456
diff changeset
   725
abbreviation union :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Un" 65) where
41076
a7fba340058c primitive definitions of bot/top/inf/sup for bool and fun are named with canonical suffix `_def` rather than `_eq`;
haftmann
parents: 40872
diff changeset
   726
  "union \<equiv> sup"
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   727
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   728
notation (xsymbols)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   729
  union  (infixl "\<union>" 65)
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   730
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   731
notation (HTML output)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   732
  union  (infixl "\<union>" 65)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   733
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   734
lemma Un_def:
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   735
  "A \<union> B = {x. x \<in> A \<or> x \<in> B}"
45959
184d36538e51 `set` is now a proper type constructor; added operation for set monad
haftmann
parents: 45909
diff changeset
   736
  by (simp add: sup_set_def sup_fun_def)
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   737
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   738
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
   739
  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
   740
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   741
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
   742
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   743
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   744
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
   745
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   746
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   747
text {*
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   748
  \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
   749
  @{prop B}.
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   750
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   751
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   752
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
   753
  by auto
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   754
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   755
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
   756
  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
   757
32117
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents: 32115
diff changeset
   758
lemma insert_def: "insert a B = {x. x = a} \<union> B"
45959
184d36538e51 `set` is now a proper type constructor; added operation for set monad
haftmann
parents: 45909
diff changeset
   759
  by (simp add: insert_compr Un_def)
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   760
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   761
lemma mono_Un: "mono f \<Longrightarrow> f A \<union> f B \<subseteq> f (A \<union> B)"
32683
7c1fe854ca6a inter and union are mere abbreviations for inf and sup
haftmann
parents: 32456
diff changeset
   762
  by (fact mono_sup)
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   763
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   764
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   765
subsubsection {* Set difference *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   766
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   767
lemma Diff_iff [simp]: "(c : A - B) = (c:A & c~:B)"
45959
184d36538e51 `set` is now a proper type constructor; added operation for set monad
haftmann
parents: 45909
diff changeset
   768
  by (simp add: minus_set_def fun_diff_def)
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   769
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   770
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
   771
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   772
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   773
lemma 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
   774
  by simp
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 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
   777
  by simp
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 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
   780
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   781
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   782
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
   783
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   784
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
   785
by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   786
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   787
31456
55edadbd43d5 insert now qualified and with authentic syntax
haftmann
parents: 31197
diff changeset
   788
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
   789
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   790
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
   791
  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
   792
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   793
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
   794
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   795
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   796
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
   797
  by simp
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   798
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   799
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
   800
  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
   801
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   802
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
   803
  -- {* Classical introduction rule. *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   804
  by auto
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   805
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   806
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
   807
  by auto
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   808
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   809
lemma set_insert:
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   810
  assumes "x \<in> A"
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   811
  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
   812
proof
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   813
  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
   814
next
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   815
  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
   816
qed
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   817
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   818
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
   819
by auto
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   820
44744
bdf8eb8f126b added new lemmas
nipkow
parents: 44490
diff changeset
   821
lemma insert_eq_iff: assumes "a \<notin> A" "b \<notin> B"
bdf8eb8f126b added new lemmas
nipkow
parents: 44490
diff changeset
   822
shows "insert a A = insert b B \<longleftrightarrow>
bdf8eb8f126b added new lemmas
nipkow
parents: 44490
diff changeset
   823
  (if a=b then A=B else \<exists>C. A = insert b C \<and> b \<notin> C \<and> B = insert a C \<and> a \<notin> C)"
bdf8eb8f126b added new lemmas
nipkow
parents: 44490
diff changeset
   824
  (is "?L \<longleftrightarrow> ?R")
bdf8eb8f126b added new lemmas
nipkow
parents: 44490
diff changeset
   825
proof
bdf8eb8f126b added new lemmas
nipkow
parents: 44490
diff changeset
   826
  assume ?L
bdf8eb8f126b added new lemmas
nipkow
parents: 44490
diff changeset
   827
  show ?R
bdf8eb8f126b added new lemmas
nipkow
parents: 44490
diff changeset
   828
  proof cases
bdf8eb8f126b added new lemmas
nipkow
parents: 44490
diff changeset
   829
    assume "a=b" with assms `?L` show ?R by (simp add: insert_ident)
bdf8eb8f126b added new lemmas
nipkow
parents: 44490
diff changeset
   830
  next
bdf8eb8f126b added new lemmas
nipkow
parents: 44490
diff changeset
   831
    assume "a\<noteq>b"
bdf8eb8f126b added new lemmas
nipkow
parents: 44490
diff changeset
   832
    let ?C = "A - {b}"
bdf8eb8f126b added new lemmas
nipkow
parents: 44490
diff changeset
   833
    have "A = insert b ?C \<and> b \<notin> ?C \<and> B = insert a ?C \<and> a \<notin> ?C"
bdf8eb8f126b added new lemmas
nipkow
parents: 44490
diff changeset
   834
      using assms `?L` `a\<noteq>b` by auto
bdf8eb8f126b added new lemmas
nipkow
parents: 44490
diff changeset
   835
    thus ?R using `a\<noteq>b` by auto
bdf8eb8f126b added new lemmas
nipkow
parents: 44490
diff changeset
   836
  qed
bdf8eb8f126b added new lemmas
nipkow
parents: 44490
diff changeset
   837
next
46128
53e7cc599f58 interaction of set operations for execution and membership predicate
haftmann
parents: 46127
diff changeset
   838
  assume ?R thus ?L by (auto split: if_splits)
44744
bdf8eb8f126b added new lemmas
nipkow
parents: 44490
diff changeset
   839
qed
bdf8eb8f126b added new lemmas
nipkow
parents: 44490
diff changeset
   840
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   841
subsubsection {* Singletons, using insert *}
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   842
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35576
diff changeset
   843
lemma singletonI [intro!,no_atp]: "a : {a}"
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   844
    -- {* 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
   845
  by (rule insertI1)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   846
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35576
diff changeset
   847
lemma singletonD [dest!,no_atp]: "b : {a} ==> b = a"
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   848
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   849
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   850
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
   851
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   852
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
   853
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   854
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   855
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
   856
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   857
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35576
diff changeset
   858
lemma singleton_insert_inj_eq [iff,no_atp]:
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   859
     "({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
   860
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   861
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35576
diff changeset
   862
lemma singleton_insert_inj_eq' [iff,no_atp]:
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   863
     "(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
   864
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   865
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   866
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
   867
  by fast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   868
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   869
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
   870
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   871
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   872
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
   873
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   874
46504
cd4832aa2229 removing unnecessary premise from diff_single_insert
bulwahn
parents: 46459
diff changeset
   875
lemma diff_single_insert: "A - {x} \<subseteq> B ==> A \<subseteq> insert x B"
30531
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   876
  by blast
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   877
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   878
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
   879
  by (blast elim: equalityE)
ab3d61baf66a reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents: 30352
diff changeset
   880
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   881
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
   882
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
   883
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
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
   885
  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
   886
*}
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
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
definition image :: "('a => 'b) => 'a set => 'b set" (infixr "`" 90) where
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35576
diff changeset
   889
  image_def [no_atp]: "f ` A = {y. EX x:A. y = f(x)}"
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
   890
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
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
   892
  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
   893
  "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
   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
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
   896
  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
   897
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   898
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
   899
  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
   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
   901
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
   902
  -- {* 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
   903
    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
   904
  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
   905
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   906
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
   907
  "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
   908
  -- {* 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
   909
  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
   910
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   911
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
   912
  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
   913
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   914
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
   915
  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
   916
38648
52ea97d95e4b "no_atp" a few facts that often lead to unsound proofs
blanchet
parents: 37767
diff changeset
   917
lemma image_subset_iff [no_atp]: "(f`A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)"
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
   918
  -- {* 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
   919
  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
   920
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   921
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
   922
  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
   923
   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
   924
  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
   925
  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
   926
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   927
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
   928
  -- {* 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
   929
    @{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
   930
  by blast
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   931
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   932
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
   933
  \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
   934
*}
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   935
43898
935359fd8210 moved lemmas to appropriate theory
haftmann
parents: 43866
diff changeset
   936
lemma image_ident [simp]: "(%x. x) ` Y = Y"
935359fd8210 moved lemmas to appropriate theory
haftmann
parents: 43866
diff changeset
   937
  by blast
935359fd8210 moved lemmas to appropriate theory
haftmann
parents: 43866
diff changeset
   938
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
   939
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
   940
  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
   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
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
   943
  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
   944
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
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
   946
  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
   947
32117
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents: 32115
diff changeset
   948
subsubsection {* Some rules with @{text "if"} *}
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   949
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   950
text{* Elimination of @{text"{x. \<dots> & x=t & \<dots>}"}. *}
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   951
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   952
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
   953
  by auto
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   954
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   955
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
   956
  by auto
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   957
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   958
text {*
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   959
  Rewrite rules for boolean case-splitting: faster than @{text
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   960
  "split_if [split]"}.
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   961
*}
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   962
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   963
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
   964
  by (rule split_if)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   965
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   966
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
   967
  by (rule split_if)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   968
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   969
text {*
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   970
  Split ifs on either side of the membership relation.  Not for @{text
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   971
  "[simp]"} -- can cause goals to blow up!
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   972
*}
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   973
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   974
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
   975
  by (rule split_if)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   976
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   977
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
   978
  by (rule split_if [where P="%S. a : S"])
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   979
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   980
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
   981
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   982
(*Would like to add these, but the existing code only searches for the
37677
c5a8b612e571 qualified constants Set.member and Set.Collect
haftmann
parents: 37387
diff changeset
   983
  outer-level constant, which in this case is just Set.member; we instead need
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   984
  to use term-nets to associate patterns with rules.  Also, if a rule fails to
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   985
  apply, then the formula should be kept.
34974
18b41bba42b5 new theory Algebras.thy for generic algebraic structures
haftmann
parents: 34209
diff changeset
   986
  [("uminus", Compl_iff RS iffD1), ("minus", [Diff_iff RS iffD1]),
32081
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   987
   ("Int", [IntD1,IntD2]),
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   988
   ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   989
 *)
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   990
1b7a901e2edc refined outline structure
haftmann
parents: 32078
diff changeset
   991
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   992
subsection {* Further operations and lemmas *}
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   993
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   994
subsubsection {* The ``proper subset'' relation *}
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   995
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35576
diff changeset
   996
lemma psubsetI [intro!,no_atp]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   997
  by (unfold less_le) blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   998
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35576
diff changeset
   999
lemma psubsetE [elim!,no_atp]: 
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1000
    "[|A \<subset> B;  [|A \<subseteq> B; ~ (B\<subseteq>A)|] ==> R|] ==> R"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1001
  by (unfold less_le) blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1002
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1003
lemma psubset_insert_iff:
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1004
  "(A \<subset> insert x B) = (if x \<in> B then A \<subset> B else if x \<in> A then A - {x} \<subset> B else A \<subseteq> B)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1005
  by (auto simp add: less_le subset_insert_iff)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1006
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1007
lemma psubset_eq: "(A \<subset> B) = (A \<subseteq> B & A \<noteq> B)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1008
  by (simp only: less_le)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1009
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1010
lemma psubset_imp_subset: "A \<subset> B ==> A \<subseteq> B"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1011
  by (simp add: psubset_eq)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1012
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1013
lemma psubset_trans: "[| A \<subset> B; B \<subset> C |] ==> A \<subset> C"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1014
apply (unfold less_le)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1015
apply (auto dest: subset_antisym)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1016
done
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1017
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1018
lemma psubsetD: "[| A \<subset> B; c \<in> A |] ==> c \<in> B"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1019
apply (unfold less_le)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1020
apply (auto dest: subsetD)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1021
done
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1022
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1023
lemma psubset_subset_trans: "A \<subset> B ==> B \<subseteq> C ==> A \<subset> C"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1024
  by (auto simp add: psubset_eq)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1025
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1026
lemma subset_psubset_trans: "A \<subseteq> B ==> B \<subset> C ==> A \<subset> C"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1027
  by (auto simp add: psubset_eq)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1028
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1029
lemma psubset_imp_ex_mem: "A \<subset> B ==> \<exists>b. b \<in> (B - A)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1030
  by (unfold less_le) blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1031
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1032
lemma atomize_ball:
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1033
    "(!!x. x \<in> A ==> P x) == Trueprop (\<forall>x\<in>A. P x)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1034
  by (simp only: Ball_def atomize_all atomize_imp)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1035
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1036
lemmas [symmetric, rulify] = atomize_ball
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1037
  and [symmetric, defn] = atomize_ball
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1038
40703
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39910
diff changeset
  1039
lemma image_Pow_mono:
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39910
diff changeset
  1040
  assumes "f ` A \<le> B"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39910
diff changeset
  1041
  shows "(image f) ` (Pow A) \<le> Pow B"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39910
diff changeset
  1042
using assms by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39910
diff changeset
  1043
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39910
diff changeset
  1044
lemma image_Pow_surj:
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39910
diff changeset
  1045
  assumes "f ` A = B"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39910
diff changeset
  1046
  shows "(image f) ` (Pow A) = Pow B"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39910
diff changeset
  1047
using assms unfolding Pow_def proof(auto)
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39910
diff changeset
  1048
  fix Y assume *: "Y \<le> f ` A"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39910
diff changeset
  1049
  obtain X where X_def: "X = {x \<in> A. f x \<in> Y}" by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39910
diff changeset
  1050
  have "f ` X = Y \<and> X \<le> A" unfolding X_def using * by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39910
diff changeset
  1051
  thus "Y \<in> (image f) ` {X. X \<le> A}" by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39910
diff changeset
  1052
qed
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39910
diff changeset
  1053
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1054
subsubsection {* Derived rules involving subsets. *}
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1055
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1056
text {* @{text insert}. *}
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1057
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1058
lemma subset_insertI: "B \<subseteq> insert a B"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1059
  by (rule subsetI) (erule insertI2)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1060
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1061
lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1062
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1063
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1064
lemma subset_insert: "x \<notin> A ==> (A \<subseteq> insert x B) = (A \<subseteq> B)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1065
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1066
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1067
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1068
text {* \medskip Finite Union -- the least upper bound of two sets. *}
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1069
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1070
lemma Un_upper1: "A \<subseteq> A \<union> B"
36009
9cdbc5ffc15c use lattice theorems to prove set theorems
huffman
parents: 35828
diff changeset
  1071
  by (fact sup_ge1)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1072
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1073
lemma Un_upper2: "B \<subseteq> A \<union> B"
36009
9cdbc5ffc15c use lattice theorems to prove set theorems
huffman
parents: 35828
diff changeset
  1074
  by (fact sup_ge2)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1075
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1076
lemma Un_least: "A \<subseteq> C ==> B \<subseteq> C ==> A \<union> B \<subseteq> C"
36009
9cdbc5ffc15c use lattice theorems to prove set theorems
huffman
parents: 35828
diff changeset
  1077
  by (fact sup_least)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1078
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1079
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1080
text {* \medskip Finite Intersection -- the greatest lower bound of two sets. *}
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1081
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1082
lemma Int_lower1: "A \<inter> B \<subseteq> A"
36009
9cdbc5ffc15c use lattice theorems to prove set theorems
huffman
parents: 35828
diff changeset
  1083
  by (fact inf_le1)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1084
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1085
lemma Int_lower2: "A \<inter> B \<subseteq> B"
36009
9cdbc5ffc15c use lattice theorems to prove set theorems
huffman
parents: 35828
diff changeset
  1086
  by (fact inf_le2)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1087
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1088
lemma Int_greatest: "C \<subseteq> A ==> C \<subseteq> B ==> C \<subseteq> A \<inter> B"
36009
9cdbc5ffc15c use lattice theorems to prove set theorems
huffman
parents: 35828
diff changeset
  1089
  by (fact inf_greatest)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1090
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1091
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1092
text {* \medskip Set difference. *}
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1093
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1094
lemma Diff_subset: "A - B \<subseteq> A"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1095
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1096
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1097
lemma Diff_subset_conv: "(A - B \<subseteq> C) = (A \<subseteq> B \<union> C)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1098
by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1099
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1100
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1101
subsubsection {* Equalities involving union, intersection, inclusion, etc. *}
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1102
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1103
text {* @{text "{}"}. *}
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1104
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1105
lemma Collect_const [simp]: "{s. P} = (if P then UNIV else {})"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1106
  -- {* supersedes @{text "Collect_False_empty"} *}
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1107
  by auto
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1108
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1109
lemma subset_empty [simp]: "(A \<subseteq> {}) = (A = {})"
45121
5e495ccf6e56 Set.thy: remove redundant [simp] declarations
huffman
parents: 44744
diff changeset
  1110
  by (fact bot_unique)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1111
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1112
lemma not_psubset_empty [iff]: "\<not> (A < {})"
45121
5e495ccf6e56 Set.thy: remove redundant [simp] declarations
huffman
parents: 44744
diff changeset
  1113
  by (fact not_less_bot) (* FIXME: already simp *)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1114
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1115
lemma Collect_empty_eq [simp]: "(Collect P = {}) = (\<forall>x. \<not> P x)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1116
by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1117
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1118
lemma empty_Collect_eq [simp]: "({} = Collect P) = (\<forall>x. \<not> P x)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1119
by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1120
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1121
lemma Collect_neg_eq: "{x. \<not> P x} = - {x. P x}"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1122
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1123
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1124
lemma Collect_disj_eq: "{x. P x | Q x} = {x. P x} \<union> {x. Q x}"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1125
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1126
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1127
lemma Collect_imp_eq: "{x. P x --> Q x} = -{x. P x} \<union> {x. Q x}"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1128
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1129
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1130
lemma Collect_conj_eq: "{x. P x & Q x} = {x. P x} \<inter> {x. Q x}"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1131
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1132
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1133
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1134
text {* \medskip @{text insert}. *}
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1136
lemma insert_is_Un: "insert a A = {a} Un A"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1137
  -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a {}"} *}
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1138
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1139
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1140
lemma insert_not_empty [simp]: "insert a A \<noteq> {}"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1141
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1142
45607
16b4f5774621 eliminated obsolete "standard";
wenzelm
parents: 45152
diff changeset
  1143
lemmas empty_not_insert = insert_not_empty [symmetric]
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1144
declare empty_not_insert [simp]
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1145
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1146
lemma insert_absorb: "a \<in> A ==> insert a A = A"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1147
  -- {* @{text "[simp]"} causes recursive calls when there are nested inserts *}
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1148
  -- {* with \emph{quadratic} running time *}
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1149
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1150
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1151
lemma insert_absorb2 [simp]: "insert x (insert x A) = insert x A"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1152
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1153
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1154
lemma insert_commute: "insert x (insert y A) = insert y (insert x A)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1155
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1156
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1157
lemma insert_subset [simp]: "(insert x A \<subseteq> B) = (x \<in> B & A \<subseteq> B)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1158
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1159
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1160
lemma mk_disjoint_insert: "a \<in> A ==> \<exists>B. A = insert a B & a \<notin> B"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1161
  -- {* use new @{text B} rather than @{text "A - {a}"} to avoid infinite unfolding *}
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1162
  apply (rule_tac x = "A - {a}" in exI, blast)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1163
  done
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1164
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1165
lemma insert_Collect: "insert a (Collect P) = {u. u \<noteq> a --> P u}"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1166
  by auto
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1167
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1168
lemma insert_inter_insert[simp]: "insert a A \<inter> insert a B = insert a (A \<inter> B)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1169
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1170
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35576
diff changeset
  1171
lemma insert_disjoint [simp,no_atp]:
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1172
 "(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1173
 "({} = insert a A \<inter> B) = (a \<notin> B \<and> {} = A \<inter> B)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1174
  by auto
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1175
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35576
diff changeset
  1176
lemma disjoint_insert [simp,no_atp]:
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1177
 "(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1178
 "({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1179
  by auto
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1180
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1181
text {* \medskip @{text image}. *}
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1182
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1183
lemma image_empty [simp]: "f`{} = {}"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1184
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1185
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1186
lemma image_insert [simp]: "f ` insert a B = insert (f a) (f`B)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1187
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1188
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1189
lemma image_constant: "x \<in> A ==> (\<lambda>x. c) ` A = {c}"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1190
  by auto
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1191
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1192
lemma image_constant_conv: "(%x. c) ` A = (if A = {} then {} else {c})"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1193
by auto
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1194
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1195
lemma image_image: "f ` (g ` A) = (\<lambda>x. f (g x)) ` A"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1196
by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1197
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1198
lemma insert_image [simp]: "x \<in> A ==> insert (f x) (f`A) = f`A"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1199
by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1200
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1201
lemma image_is_empty [iff]: "(f`A = {}) = (A = {})"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1202
by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1203
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1204
lemma empty_is_image[iff]: "({} = f ` A) = (A = {})"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1205
by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1206
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1207
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35576
diff changeset
  1208
lemma image_Collect [no_atp]: "f ` {x. P x} = {f x | x. P x}"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1209
  -- {* NOT suitable as a default simprule: the RHS isn't simpler than the LHS,
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1210
      with its implicit quantifier and conjunction.  Also image enjoys better
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1211
      equational properties than does the RHS. *}
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1212
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1213
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1214
lemma if_image_distrib [simp]:
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1215
  "(\<lambda>x. if P x then f x else g x) ` S
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1216
    = (f ` (S \<inter> {x. P x})) \<union> (g ` (S \<inter> {x. \<not> P x}))"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1217
  by (auto simp add: image_def)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1218
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1219
lemma image_cong: "M = N ==> (!!x. x \<in> N ==> f x = g x) ==> f`M = g`N"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1220
  by (simp add: image_def)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1221
43898
935359fd8210 moved lemmas to appropriate theory
haftmann
parents: 43866
diff changeset
  1222
lemma image_Int_subset: "f`(A Int B) <= f`A Int f`B"
935359fd8210 moved lemmas to appropriate theory
haftmann
parents: 43866
diff changeset
  1223
by blast
935359fd8210 moved lemmas to appropriate theory
haftmann
parents: 43866
diff changeset
  1224
935359fd8210 moved lemmas to appropriate theory
haftmann
parents: 43866
diff changeset
  1225
lemma image_diff_subset: "f`A - f`B <= f`(A - B)"
935359fd8210 moved lemmas to appropriate theory
haftmann
parents: 43866
diff changeset
  1226
by blast
935359fd8210 moved lemmas to appropriate theory
haftmann
parents: 43866
diff changeset
  1227
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1228
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1229
text {* \medskip @{text range}. *}
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1230
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35576
diff changeset
  1231
lemma full_SetCompr_eq [no_atp]: "{u. \<exists>x. u = f x} = range f"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1232
  by auto
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1233
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1234
lemma range_composition: "range (\<lambda>x. f (g x)) = f`range g"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1235
by (subst image_image, simp)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1236
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1237
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1238
text {* \medskip @{text Int} *}
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1239
45121
5e495ccf6e56 Set.thy: remove redundant [simp] declarations
huffman
parents: 44744
diff changeset
  1240
lemma Int_absorb: "A \<inter> A = A"
5e495ccf6e56 Set.thy: remove redundant [simp] declarations
huffman
parents: 44744
diff changeset
  1241
  by (fact inf_idem) (* already simp *)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1242
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1243
lemma Int_left_absorb: "A \<inter> (A \<inter> B) = A \<inter> B"
36009
9cdbc5ffc15c use lattice theorems to prove set theorems
huffman
parents: 35828
diff changeset
  1244
  by (fact inf_left_idem)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1245
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1246
lemma Int_commute: "A \<inter> B = B \<inter> A"
36009
9cdbc5ffc15c use lattice theorems to prove set theorems
huffman
parents: 35828
diff changeset
  1247
  by (fact inf_commute)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1248
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1249
lemma Int_left_commute: "A \<inter> (B \<inter> C) = B \<inter> (A \<inter> C)"
36009
9cdbc5ffc15c use lattice theorems to prove set theorems
huffman
parents: 35828
diff changeset
  1250
  by (fact inf_left_commute)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1251
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1252
lemma Int_assoc: "(A \<inter> B) \<inter> C = A \<inter> (B \<inter> C)"
36009
9cdbc5ffc15c use lattice theorems to prove set theorems
huffman
parents: 35828
diff changeset
  1253
  by (fact inf_assoc)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1254
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1255
lemmas Int_ac = Int_assoc Int_left_absorb Int_commute Int_left_commute
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1256
  -- {* Intersection is an AC-operator *}
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1257
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1258
lemma Int_absorb1: "B \<subseteq> A ==> A \<inter> B = B"
36009
9cdbc5ffc15c use lattice theorems to prove set theorems
huffman
parents: 35828
diff changeset
  1259
  by (fact inf_absorb2)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1260
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1261
lemma Int_absorb2: "A \<subseteq> B ==> A \<inter> B = A"
36009
9cdbc5ffc15c use lattice theorems to prove set theorems
huffman
parents: 35828
diff changeset
  1262
  by (fact inf_absorb1)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1263
45121
5e495ccf6e56 Set.thy: remove redundant [simp] declarations
huffman
parents: 44744
diff changeset
  1264
lemma Int_empty_left: "{} \<inter> B = {}"
5e495ccf6e56 Set.thy: remove redundant [simp] declarations
huffman
parents: 44744
diff changeset
  1265
  by (fact inf_bot_left) (* already simp *)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1266
45121
5e495ccf6e56 Set.thy: remove redundant [simp] declarations
huffman
parents: 44744
diff changeset
  1267
lemma Int_empty_right: "A \<inter> {} = {}"
5e495ccf6e56 Set.thy: remove redundant [simp] declarations
huffman
parents: 44744
diff changeset
  1268
  by (fact inf_bot_right) (* already simp *)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1269
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1270
lemma disjoint_eq_subset_Compl: "(A \<inter> B = {}) = (A \<subseteq> -B)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1271
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1272
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1273
lemma disjoint_iff_not_equal: "(A \<inter> B = {}) = (\<forall>x\<in>A. \<forall>y\<in>B. x \<noteq> y)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1274
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1275
45121
5e495ccf6e56 Set.thy: remove redundant [simp] declarations
huffman
parents: 44744
diff changeset
  1276
lemma Int_UNIV_left: "UNIV \<inter> B = B"
5e495ccf6e56 Set.thy: remove redundant [simp] declarations
huffman
parents: 44744
diff changeset
  1277
  by (fact inf_top_left) (* already simp *)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1278
45121
5e495ccf6e56 Set.thy: remove redundant [simp] declarations
huffman
parents: 44744
diff changeset
  1279
lemma Int_UNIV_right: "A \<inter> UNIV = A"
5e495ccf6e56 Set.thy: remove redundant [simp] declarations
huffman
parents: 44744
diff changeset
  1280
  by (fact inf_top_right) (* already simp *)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1281
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1282
lemma Int_Un_distrib: "A \<inter> (B \<union> C) = (A \<inter> B) \<union> (A \<inter> C)"
36009
9cdbc5ffc15c use lattice theorems to prove set theorems
huffman
parents: 35828
diff changeset
  1283
  by (fact inf_sup_distrib1)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1284
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1285
lemma Int_Un_distrib2: "(B \<union> C) \<inter> A = (B \<inter> A) \<union> (C \<inter> A)"
36009
9cdbc5ffc15c use lattice theorems to prove set theorems
huffman
parents: 35828
diff changeset
  1286
  by (fact inf_sup_distrib2)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1287
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35576
diff changeset
  1288
lemma Int_UNIV [simp,no_atp]: "(A \<inter> B = UNIV) = (A = UNIV & B = UNIV)"
45121
5e495ccf6e56 Set.thy: remove redundant [simp] declarations
huffman
parents: 44744
diff changeset
  1289
  by (fact inf_eq_top_iff) (* already simp *)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1290
38648
52ea97d95e4b "no_atp" a few facts that often lead to unsound proofs
blanchet
parents: 37767
diff changeset
  1291
lemma Int_subset_iff [no_atp, simp]: "(C \<subseteq> A \<inter> B) = (C \<subseteq> A & C \<subseteq> B)"
36009
9cdbc5ffc15c use lattice theorems to prove set theorems
huffman
parents: 35828