src/HOL/Set.thy
author huffman
Sun Mar 28 12:50:38 2010 -0700 (2010-03-28)
changeset 36009 9cdbc5ffc15c
parent 35828 46cfc4b8112e
child 37387 3581483cca6c
permissions -rw-r--r--
use lattice theorems to prove set theorems
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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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global
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types 'a set = "'a => bool"
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consts
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  Collect       :: "('a => bool) => 'a set"              -- "comprehension"
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  "op :"        :: "'a => 'a set => bool"                -- "membership"
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local
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notation
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  "op :"  ("op :") and
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  "op :"  ("(_/ : _)" [50, 51] 50)
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defs
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  mem_def [code]: "x : S == S x"
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  Collect_def [code]: "Collect P == P"
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abbreviation
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  "not_mem x A == ~ (x : A)" -- "non-membership"
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notation
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  not_mem  ("op ~:") and
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  not_mem  ("(_/ ~: _)" [50, 51] 50)
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notation (xsymbols)
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  "op :"  ("op \<in>") and
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  "op :"  ("(_/ \<in> _)" [50, 51] 50) and
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  not_mem  ("op \<notin>") and
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  not_mem  ("(_/ \<notin> _)" [50, 51] 50)
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notation (HTML output)
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  "op :"  ("op \<in>") and
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  "op :"  ("(_/ \<in> _)" [50, 51] 50) and
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  not_mem  ("op \<notin>") and
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  not_mem  ("(_/ \<notin> _)" [50, 51] 50)
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text {* Set comprehensions *}
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syntax
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  "_Coll" :: "pttrn => bool => 'a set"    ("(1{_./ _})")
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translations
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  "{x. P}" == "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 mem_Collect_eq [iff]: "(a : {x. P(x)}) = P(a)"
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  by (simp add: Collect_def mem_def)
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lemma Collect_mem_eq [simp]: "{x. x:A} = A"
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  by (simp add: Collect_def mem_def)
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lemma CollectI: "P(a) ==> a : {x. P(x)}"
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  by simp
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lemma CollectD: "a : {x. P(x)} ==> P(a)"
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  by simp
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lemma Collect_cong: "(!!x. P x = Q x) ==> {x. P(x)} = {x. Q(x)}"
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  by simp
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text {*
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Simproc for pulling @{text "x=t"} in @{text "{x. \<dots> & x=t & \<dots>}"}
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to the front (and similarly for @{text "t=x"}):
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*}
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setup {*
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let
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  val Coll_perm_tac = rtac @{thm Collect_cong} 1 THEN rtac @{thm iffI} 1 THEN
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    ALLGOALS(EVERY'[REPEAT_DETERM o (etac @{thm conjE}),
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                    DEPTH_SOLVE_1 o (ares_tac [@{thm conjI}])])
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  val defColl_regroup = Simplifier.simproc @{theory}
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    "defined Collect" ["{x. P x & Q x}"]
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    (Quantifier1.rearrange_Coll Coll_perm_tac)
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in
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  Simplifier.map_simpset (fn ss => ss addsimprocs [defColl_regroup])
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end
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*}
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lemmas CollectE = CollectD [elim_format]
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text {* Set enumerations *}
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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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global
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consts
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  Ball          :: "'a set => ('a => bool) => bool"      -- "bounded universal quantifiers"
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  Bex           :: "'a set => ('a => bool) => bool"      -- "bounded existential quantifiers"
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local
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defs
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  Ball_def:     "Ball A P       == ALL x. x:A --> P(x)"
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  Bex_def:      "Bex A P        == EX x. x:A & P(x)"
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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 Type (set_type, _) = @{typ "'a set"};   (* FIXME 'a => bool (!?!) *)
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  val All_binder = Syntax.binder_name @{const_syntax All};
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  val Ex_binder = Syntax.binder_name @{const_syntax Ex};
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  val impl = @{const_syntax "op -->"};
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  val conj = @{const_syntax "op &"};
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  val sbset = @{const_syntax subset};
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  val sbset_eq = @{const_syntax subset_eq};
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  val trans =
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   [((All_binder, impl, sbset), @{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.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 (T, _)),
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            Const (c, _) $
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              (Const (d, _) $ (Const (@{syntax_const "_bound"}, _) $ Free (v', _)) $ n) $ P] =>
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            if T = set_type then
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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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            else raise Match
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         | _ => raise Match);
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in
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  [tr' All_binder, tr' Ex_binder]
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end
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*}
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text {*
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  \medskip Translate between @{text "{e | x1...xn. P}"} and @{text
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  "{u. EX x1..xn. u = e & P}"}; @{text "{y. EX x1..xn. y = e & P}"} is
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  only translated if @{text "[0..n] subset bvs(e)"}.
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*}
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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 (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 "op ="} $ Bound (nvars idts) $ e;
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        val P = Syntax.const @{const_syntax "op &"} $ eq $ b;
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        val exP = ex_tr [idts, P];
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      in Syntax.const @{const_syntax Collect} $ Term.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.preserve_binder_abs2_tr' @{const_syntax Ball} @{syntax_const "_Ball"},
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  Syntax.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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   298
  val ex_tr' = snd (mk_binder_tr' (@{const_syntax Ex}, "DUMMY"));
nipkow@13763
   299
nipkow@13763
   300
  fun setcompr_tr' [Abs (abs as (_, _, P))] =
nipkow@13763
   301
    let
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   302
      fun check (Const (@{const_syntax Ex}, _) $ Abs (_, _, P), n) = check (P, n + 1)
wenzelm@35115
   303
        | check (Const (@{const_syntax "op &"}, _) $
wenzelm@35115
   304
              (Const (@{const_syntax "op ="}, _) $ Bound m $ e) $ P, n) =
nipkow@13763
   305
            n > 0 andalso m = n andalso not (loose_bvar1 (P, n)) andalso
haftmann@33038
   306
            subset (op =) (0 upto (n - 1), add_loose_bnos (e, 0, []))
wenzelm@35115
   307
        | check _ = false;
clasohm@923
   308
wenzelm@11979
   309
        fun tr' (_ $ abs) =
wenzelm@11979
   310
          let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' [abs]
wenzelm@35115
   311
          in Syntax.const @{syntax_const "_Setcompr"} $ e $ idts $ Q end;
wenzelm@35115
   312
    in
wenzelm@35115
   313
      if check (P, 0) then tr' P
wenzelm@35115
   314
      else
wenzelm@35115
   315
        let
wenzelm@35115
   316
          val (x as _ $ Free(xN, _), t) = atomic_abs_tr' abs;
wenzelm@35115
   317
          val M = Syntax.const @{syntax_const "_Coll"} $ x $ t;
wenzelm@35115
   318
        in
wenzelm@35115
   319
          case t of
wenzelm@35115
   320
            Const (@{const_syntax "op &"}, _) $
wenzelm@35115
   321
              (Const (@{const_syntax "op :"}, _) $
wenzelm@35115
   322
                (Const (@{syntax_const "_bound"}, _) $ Free (yN, _)) $ A) $ P =>
wenzelm@35115
   323
            if xN = yN then Syntax.const @{syntax_const "_Collect"} $ x $ A $ P else M
wenzelm@35115
   324
          | _ => M
wenzelm@35115
   325
        end
nipkow@13763
   326
    end;
wenzelm@35115
   327
  in [(@{const_syntax Collect}, setcompr_tr')] end;
wenzelm@11979
   328
*}
wenzelm@11979
   329
haftmann@32117
   330
setup {*
haftmann@32117
   331
let
haftmann@32117
   332
  val unfold_bex_tac = unfold_tac @{thms "Bex_def"};
haftmann@32117
   333
  fun prove_bex_tac ss = unfold_bex_tac ss THEN Quantifier1.prove_one_point_ex_tac;
haftmann@32117
   334
  val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac;
haftmann@32117
   335
  val unfold_ball_tac = unfold_tac @{thms "Ball_def"};
haftmann@32117
   336
  fun prove_ball_tac ss = unfold_ball_tac ss THEN Quantifier1.prove_one_point_all_tac;
haftmann@32117
   337
  val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac;
haftmann@32117
   338
  val defBEX_regroup = Simplifier.simproc @{theory}
haftmann@32117
   339
    "defined BEX" ["EX x:A. P x & Q x"] rearrange_bex;
haftmann@32117
   340
  val defBALL_regroup = Simplifier.simproc @{theory}
haftmann@32117
   341
    "defined BALL" ["ALL x:A. P x --> Q x"] rearrange_ball;
haftmann@32117
   342
in
haftmann@32117
   343
  Simplifier.map_simpset (fn ss => ss addsimprocs [defBALL_regroup, defBEX_regroup])
haftmann@32117
   344
end
haftmann@32117
   345
*}
haftmann@32117
   346
wenzelm@11979
   347
lemma ballI [intro!]: "(!!x. x:A ==> P x) ==> ALL x:A. P x"
wenzelm@11979
   348
  by (simp add: Ball_def)
wenzelm@11979
   349
wenzelm@11979
   350
lemmas strip = impI allI ballI
wenzelm@11979
   351
wenzelm@11979
   352
lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x"
wenzelm@11979
   353
  by (simp add: Ball_def)
wenzelm@11979
   354
wenzelm@11979
   355
text {*
wenzelm@11979
   356
  Gives better instantiation for bound:
wenzelm@11979
   357
*}
wenzelm@11979
   358
wenzelm@26339
   359
declaration {* fn _ =>
wenzelm@26339
   360
  Classical.map_cs (fn cs => cs addbefore ("bspec", datac @{thm bspec} 1))
wenzelm@11979
   361
*}
wenzelm@11979
   362
haftmann@32117
   363
ML {*
haftmann@32117
   364
structure Simpdata =
haftmann@32117
   365
struct
haftmann@32117
   366
haftmann@32117
   367
open Simpdata;
haftmann@32117
   368
haftmann@32117
   369
val mksimps_pairs = [(@{const_name Ball}, @{thms bspec})] @ mksimps_pairs;
haftmann@32117
   370
haftmann@32117
   371
end;
haftmann@32117
   372
haftmann@32117
   373
open Simpdata;
haftmann@32117
   374
*}
haftmann@32117
   375
haftmann@32117
   376
declaration {* fn _ =>
haftmann@32117
   377
  Simplifier.map_ss (fn ss => ss setmksimps (mksimps mksimps_pairs))
haftmann@32117
   378
*}
haftmann@32117
   379
haftmann@32117
   380
lemma ballE [elim]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q"
haftmann@32117
   381
  by (unfold Ball_def) blast
haftmann@32117
   382
wenzelm@11979
   383
lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x"
wenzelm@11979
   384
  -- {* Normally the best argument order: @{prop "P x"} constrains the
wenzelm@11979
   385
    choice of @{prop "x:A"}. *}
wenzelm@11979
   386
  by (unfold Bex_def) blast
wenzelm@11979
   387
wenzelm@13113
   388
lemma rev_bexI [intro?]: "x:A ==> P x ==> EX x:A. P x"
wenzelm@11979
   389
  -- {* The best argument order when there is only one @{prop "x:A"}. *}
wenzelm@11979
   390
  by (unfold Bex_def) blast
wenzelm@11979
   391
wenzelm@11979
   392
lemma bexCI: "(ALL x:A. ~P x ==> P a) ==> a:A ==> EX x:A. P x"
wenzelm@11979
   393
  by (unfold Bex_def) blast
wenzelm@11979
   394
wenzelm@11979
   395
lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q"
wenzelm@11979
   396
  by (unfold Bex_def) blast
wenzelm@11979
   397
wenzelm@11979
   398
lemma ball_triv [simp]: "(ALL x:A. P) = ((EX x. x:A) --> P)"
wenzelm@11979
   399
  -- {* Trival rewrite rule. *}
wenzelm@11979
   400
  by (simp add: Ball_def)
wenzelm@11979
   401
wenzelm@11979
   402
lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)"
wenzelm@11979
   403
  -- {* Dual form for existentials. *}
wenzelm@11979
   404
  by (simp add: Bex_def)
wenzelm@11979
   405
wenzelm@11979
   406
lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)"
wenzelm@11979
   407
  by blast
wenzelm@11979
   408
wenzelm@11979
   409
lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)"
wenzelm@11979
   410
  by blast
wenzelm@11979
   411
wenzelm@11979
   412
lemma bex_one_point1 [simp]: "(EX x:A. x = a & P x) = (a:A & P a)"
wenzelm@11979
   413
  by blast
wenzelm@11979
   414
wenzelm@11979
   415
lemma bex_one_point2 [simp]: "(EX x:A. a = x & P x) = (a:A & P a)"
wenzelm@11979
   416
  by blast
wenzelm@11979
   417
wenzelm@11979
   418
lemma ball_one_point1 [simp]: "(ALL x:A. x = a --> P x) = (a:A --> P a)"
wenzelm@11979
   419
  by blast
wenzelm@11979
   420
wenzelm@11979
   421
lemma ball_one_point2 [simp]: "(ALL x:A. a = x --> P x) = (a:A --> P a)"
wenzelm@11979
   422
  by blast
wenzelm@11979
   423
wenzelm@11979
   424
haftmann@32081
   425
text {* Congruence rules *}
wenzelm@11979
   426
berghofe@16636
   427
lemma ball_cong:
wenzelm@11979
   428
  "A = B ==> (!!x. x:B ==> P x = Q x) ==>
wenzelm@11979
   429
    (ALL x:A. P x) = (ALL x:B. Q x)"
wenzelm@11979
   430
  by (simp add: Ball_def)
wenzelm@11979
   431
berghofe@16636
   432
lemma strong_ball_cong [cong]:
berghofe@16636
   433
  "A = B ==> (!!x. x:B =simp=> P x = Q x) ==>
berghofe@16636
   434
    (ALL x:A. P x) = (ALL x:B. Q x)"
berghofe@16636
   435
  by (simp add: simp_implies_def Ball_def)
berghofe@16636
   436
berghofe@16636
   437
lemma bex_cong:
wenzelm@11979
   438
  "A = B ==> (!!x. x:B ==> P x = Q x) ==>
wenzelm@11979
   439
    (EX x:A. P x) = (EX x:B. Q x)"
wenzelm@11979
   440
  by (simp add: Bex_def cong: conj_cong)
regensbu@1273
   441
berghofe@16636
   442
lemma strong_bex_cong [cong]:
berghofe@16636
   443
  "A = B ==> (!!x. x:B =simp=> P x = Q x) ==>
berghofe@16636
   444
    (EX x:A. P x) = (EX x:B. Q x)"
berghofe@16636
   445
  by (simp add: simp_implies_def Bex_def cong: conj_cong)
berghofe@16636
   446
haftmann@30531
   447
haftmann@32081
   448
subsection {* Basic operations *}
haftmann@32081
   449
haftmann@30531
   450
subsubsection {* Subsets *}
haftmann@30531
   451
paulson@33022
   452
lemma subsetI [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> x \<in> B) \<Longrightarrow> A \<subseteq> B"
haftmann@32888
   453
  unfolding mem_def by (rule le_funI, rule le_boolI)
haftmann@30352
   454
wenzelm@11979
   455
text {*
haftmann@30531
   456
  \medskip Map the type @{text "'a set => anything"} to just @{typ
haftmann@30531
   457
  'a}; for overloading constants whose first argument has type @{typ
haftmann@30531
   458
  "'a set"}.
wenzelm@11979
   459
*}
wenzelm@11979
   460
haftmann@30596
   461
lemma subsetD [elim, intro?]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B"
haftmann@32888
   462
  unfolding mem_def by (erule le_funE, erule le_boolE)
haftmann@30531
   463
  -- {* Rule in Modus Ponens style. *}
haftmann@30531
   464
blanchet@35828
   465
lemma rev_subsetD [no_atp,intro?]: "c \<in> A ==> A \<subseteq> B ==> c \<in> B"
haftmann@30531
   466
  -- {* The same, with reversed premises for use with @{text erule} --
haftmann@30531
   467
      cf @{text rev_mp}. *}
haftmann@30531
   468
  by (rule subsetD)
haftmann@30531
   469
wenzelm@11979
   470
text {*
haftmann@30531
   471
  \medskip Converts @{prop "A \<subseteq> B"} to @{prop "x \<in> A ==> x \<in> B"}.
haftmann@30531
   472
*}
haftmann@30531
   473
blanchet@35828
   474
lemma subsetCE [no_atp,elim]: "A \<subseteq> B ==> (c \<notin> A ==> P) ==> (c \<in> B ==> P) ==> P"
haftmann@30531
   475
  -- {* Classical elimination rule. *}
haftmann@32888
   476
  unfolding mem_def by (blast dest: le_funE le_boolE)
haftmann@30531
   477
blanchet@35828
   478
lemma subset_eq [no_atp]: "A \<le> B = (\<forall>x\<in>A. x \<in> B)" by blast
wenzelm@2388
   479
blanchet@35828
   480
lemma contra_subsetD [no_atp]: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A"
haftmann@30531
   481
  by blast
haftmann@30531
   482
paulson@33022
   483
lemma subset_refl [simp]: "A \<subseteq> A"
haftmann@32081
   484
  by (fact order_refl)
haftmann@30531
   485
haftmann@30531
   486
lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C"
haftmann@32081
   487
  by (fact order_trans)
haftmann@32081
   488
haftmann@32081
   489
lemma set_rev_mp: "x:A ==> A \<subseteq> B ==> x:B"
haftmann@32081
   490
  by (rule subsetD)
haftmann@32081
   491
haftmann@32081
   492
lemma set_mp: "A \<subseteq> B ==> x:A ==> x:B"
haftmann@32081
   493
  by (rule subsetD)
haftmann@32081
   494
paulson@33044
   495
lemma eq_mem_trans: "a=b ==> b \<in> A ==> a \<in> A"
paulson@33044
   496
  by simp
paulson@33044
   497
haftmann@32081
   498
lemmas basic_trans_rules [trans] =
paulson@33044
   499
  order_trans_rules set_rev_mp set_mp eq_mem_trans
haftmann@30531
   500
haftmann@30531
   501
haftmann@30531
   502
subsubsection {* Equality *}
haftmann@30531
   503
haftmann@30531
   504
lemma set_ext: assumes prem: "(!!x. (x:A) = (x:B))" shows "A = B"
haftmann@30531
   505
  apply (rule prem [THEN ext, THEN arg_cong, THEN box_equals])
haftmann@30531
   506
   apply (rule Collect_mem_eq)
haftmann@30531
   507
  apply (rule Collect_mem_eq)
haftmann@30531
   508
  done
haftmann@30531
   509
haftmann@30531
   510
lemma expand_set_eq: "(A = B) = (ALL x. (x:A) = (x:B))"
haftmann@30531
   511
by(auto intro:set_ext)
haftmann@30531
   512
haftmann@30531
   513
lemma subset_antisym [intro!]: "A \<subseteq> B ==> B \<subseteq> A ==> A = B"
haftmann@30531
   514
  -- {* Anti-symmetry of the subset relation. *}
haftmann@30531
   515
  by (iprover intro: set_ext subsetD)
haftmann@30531
   516
haftmann@30531
   517
text {*
haftmann@30531
   518
  \medskip Equality rules from ZF set theory -- are they appropriate
haftmann@30531
   519
  here?
haftmann@30531
   520
*}
haftmann@30531
   521
haftmann@30531
   522
lemma equalityD1: "A = B ==> A \<subseteq> B"
krauss@34209
   523
  by simp
haftmann@30531
   524
haftmann@30531
   525
lemma equalityD2: "A = B ==> B \<subseteq> A"
krauss@34209
   526
  by simp
haftmann@30531
   527
haftmann@30531
   528
text {*
haftmann@30531
   529
  \medskip Be careful when adding this to the claset as @{text
haftmann@30531
   530
  subset_empty} is in the simpset: @{prop "A = {}"} goes to @{prop "{}
haftmann@30531
   531
  \<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}!
haftmann@30352
   532
*}
haftmann@30352
   533
haftmann@30531
   534
lemma equalityE: "A = B ==> (A \<subseteq> B ==> B \<subseteq> A ==> P) ==> P"
krauss@34209
   535
  by simp
haftmann@30531
   536
haftmann@30531
   537
lemma equalityCE [elim]:
haftmann@30531
   538
    "A = B ==> (c \<in> A ==> c \<in> B ==> P) ==> (c \<notin> A ==> c \<notin> B ==> P) ==> P"
haftmann@30531
   539
  by blast
haftmann@30531
   540
haftmann@30531
   541
lemma eqset_imp_iff: "A = B ==> (x : A) = (x : B)"
haftmann@30531
   542
  by simp
haftmann@30531
   543
haftmann@30531
   544
lemma eqelem_imp_iff: "x = y ==> (x : A) = (y : A)"
haftmann@30531
   545
  by simp
haftmann@30531
   546
haftmann@30531
   547
haftmann@30531
   548
subsubsection {* The universal set -- UNIV *}
haftmann@30531
   549
haftmann@32264
   550
abbreviation UNIV :: "'a set" where
haftmann@32264
   551
  "UNIV \<equiv> top"
haftmann@32135
   552
haftmann@32135
   553
lemma UNIV_def:
haftmann@32117
   554
  "UNIV = {x. True}"
haftmann@32264
   555
  by (simp add: top_fun_eq top_bool_eq Collect_def)
haftmann@32081
   556
haftmann@30531
   557
lemma UNIV_I [simp]: "x : UNIV"
haftmann@30531
   558
  by (simp add: UNIV_def)
haftmann@30531
   559
haftmann@30531
   560
declare UNIV_I [intro]  -- {* unsafe makes it less likely to cause problems *}
haftmann@30531
   561
haftmann@30531
   562
lemma UNIV_witness [intro?]: "EX x. x : UNIV"
haftmann@30531
   563
  by simp
haftmann@30531
   564
haftmann@30531
   565
lemma subset_UNIV [simp]: "A \<subseteq> UNIV"
haftmann@30531
   566
  by (rule subsetI) (rule UNIV_I)
haftmann@30531
   567
haftmann@30531
   568
text {*
haftmann@30531
   569
  \medskip Eta-contracting these two rules (to remove @{text P})
haftmann@30531
   570
  causes them to be ignored because of their interaction with
haftmann@30531
   571
  congruence rules.
haftmann@30531
   572
*}
haftmann@30531
   573
haftmann@30531
   574
lemma ball_UNIV [simp]: "Ball UNIV P = All P"
haftmann@30531
   575
  by (simp add: Ball_def)
haftmann@30531
   576
haftmann@30531
   577
lemma bex_UNIV [simp]: "Bex UNIV P = Ex P"
haftmann@30531
   578
  by (simp add: Bex_def)
haftmann@30531
   579
haftmann@30531
   580
lemma UNIV_eq_I: "(\<And>x. x \<in> A) \<Longrightarrow> UNIV = A"
haftmann@30531
   581
  by auto
haftmann@30531
   582
haftmann@30531
   583
haftmann@30531
   584
subsubsection {* The empty set *}
haftmann@30531
   585
haftmann@32135
   586
lemma empty_def:
haftmann@32135
   587
  "{} = {x. False}"
haftmann@32264
   588
  by (simp add: bot_fun_eq bot_bool_eq Collect_def)
haftmann@32135
   589
haftmann@30531
   590
lemma empty_iff [simp]: "(c : {}) = False"
haftmann@30531
   591
  by (simp add: empty_def)
haftmann@30531
   592
haftmann@30531
   593
lemma emptyE [elim!]: "a : {} ==> P"
haftmann@30531
   594
  by simp
haftmann@30531
   595
haftmann@30531
   596
lemma empty_subsetI [iff]: "{} \<subseteq> A"
haftmann@30531
   597
    -- {* One effect is to delete the ASSUMPTION @{prop "{} <= A"} *}
haftmann@30531
   598
  by blast
haftmann@30531
   599
haftmann@30531
   600
lemma equals0I: "(!!y. y \<in> A ==> False) ==> A = {}"
haftmann@30531
   601
  by blast
haftmann@30531
   602
haftmann@30531
   603
lemma equals0D: "A = {} ==> a \<notin> A"
haftmann@32082
   604
    -- {* Use for reasoning about disjointness: @{text "A Int B = {}"} *}
haftmann@30531
   605
  by blast
haftmann@30531
   606
haftmann@30531
   607
lemma ball_empty [simp]: "Ball {} P = True"
haftmann@30531
   608
  by (simp add: Ball_def)
haftmann@30531
   609
haftmann@30531
   610
lemma bex_empty [simp]: "Bex {} P = False"
haftmann@30531
   611
  by (simp add: Bex_def)
haftmann@30531
   612
haftmann@30531
   613
lemma UNIV_not_empty [iff]: "UNIV ~= {}"
haftmann@30531
   614
  by (blast elim: equalityE)
haftmann@30531
   615
haftmann@30531
   616
haftmann@30531
   617
subsubsection {* The Powerset operator -- Pow *}
haftmann@30531
   618
haftmann@32077
   619
definition Pow :: "'a set => 'a set set" where
haftmann@32077
   620
  Pow_def: "Pow A = {B. B \<le> A}"
haftmann@32077
   621
haftmann@30531
   622
lemma Pow_iff [iff]: "(A \<in> Pow B) = (A \<subseteq> B)"
haftmann@30531
   623
  by (simp add: Pow_def)
haftmann@30531
   624
haftmann@30531
   625
lemma PowI: "A \<subseteq> B ==> A \<in> Pow B"
haftmann@30531
   626
  by (simp add: Pow_def)
haftmann@30531
   627
haftmann@30531
   628
lemma PowD: "A \<in> Pow B ==> A \<subseteq> B"
haftmann@30531
   629
  by (simp add: Pow_def)
haftmann@30531
   630
haftmann@30531
   631
lemma Pow_bottom: "{} \<in> Pow B"
haftmann@30531
   632
  by simp
haftmann@30531
   633
haftmann@30531
   634
lemma Pow_top: "A \<in> Pow A"
krauss@34209
   635
  by simp
haftmann@30531
   636
haftmann@30531
   637
haftmann@30531
   638
subsubsection {* Set complement *}
haftmann@30531
   639
haftmann@30531
   640
lemma Compl_iff [simp]: "(c \<in> -A) = (c \<notin> A)"
haftmann@30531
   641
  by (simp add: mem_def fun_Compl_def bool_Compl_def)
haftmann@30531
   642
haftmann@30531
   643
lemma ComplI [intro!]: "(c \<in> A ==> False) ==> c \<in> -A"
haftmann@30531
   644
  by (unfold mem_def fun_Compl_def bool_Compl_def) blast
clasohm@923
   645
wenzelm@11979
   646
text {*
haftmann@30531
   647
  \medskip This form, with negated conclusion, works well with the
haftmann@30531
   648
  Classical prover.  Negated assumptions behave like formulae on the
haftmann@30531
   649
  right side of the notional turnstile ... *}
haftmann@30531
   650
haftmann@30531
   651
lemma ComplD [dest!]: "c : -A ==> c~:A"
haftmann@30531
   652
  by (simp add: mem_def fun_Compl_def bool_Compl_def)
haftmann@30531
   653
haftmann@30531
   654
lemmas ComplE = ComplD [elim_format]
haftmann@30531
   655
haftmann@30531
   656
lemma Compl_eq: "- A = {x. ~ x : A}" by blast
haftmann@30531
   657
haftmann@30531
   658
haftmann@30531
   659
subsubsection {* Binary union -- Un *}
haftmann@30531
   660
haftmann@32683
   661
abbreviation union :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Un" 65) where
haftmann@32683
   662
  "op Un \<equiv> sup"
haftmann@32081
   663
haftmann@32081
   664
notation (xsymbols)
haftmann@32135
   665
  union  (infixl "\<union>" 65)
haftmann@32081
   666
haftmann@32081
   667
notation (HTML output)
haftmann@32135
   668
  union  (infixl "\<union>" 65)
haftmann@32135
   669
haftmann@32135
   670
lemma Un_def:
haftmann@32135
   671
  "A \<union> B = {x. x \<in> A \<or> x \<in> B}"
haftmann@32683
   672
  by (simp add: sup_fun_eq sup_bool_eq Collect_def mem_def)
haftmann@32081
   673
haftmann@30531
   674
lemma Un_iff [simp]: "(c : A Un B) = (c:A | c:B)"
haftmann@30531
   675
  by (unfold Un_def) blast
haftmann@30531
   676
haftmann@30531
   677
lemma UnI1 [elim?]: "c:A ==> c : A Un B"
haftmann@30531
   678
  by simp
haftmann@30531
   679
haftmann@30531
   680
lemma UnI2 [elim?]: "c:B ==> c : A Un B"
haftmann@30531
   681
  by simp
haftmann@30531
   682
haftmann@30531
   683
text {*
haftmann@30531
   684
  \medskip Classical introduction rule: no commitment to @{prop A} vs
haftmann@30531
   685
  @{prop B}.
wenzelm@11979
   686
*}
wenzelm@11979
   687
haftmann@30531
   688
lemma UnCI [intro!]: "(c~:B ==> c:A) ==> c : A Un B"
haftmann@30531
   689
  by auto
haftmann@30531
   690
haftmann@30531
   691
lemma UnE [elim!]: "c : A Un B ==> (c:A ==> P) ==> (c:B ==> P) ==> P"
haftmann@30531
   692
  by (unfold Un_def) blast
haftmann@30531
   693
haftmann@32117
   694
lemma insert_def: "insert a B = {x. x = a} \<union> B"
haftmann@32081
   695
  by (simp add: Collect_def mem_def insert_compr Un_def)
haftmann@32081
   696
haftmann@32081
   697
lemma mono_Un: "mono f \<Longrightarrow> f A \<union> f B \<subseteq> f (A \<union> B)"
haftmann@32683
   698
  by (fact mono_sup)
haftmann@32081
   699
haftmann@30531
   700
haftmann@30531
   701
subsubsection {* Binary intersection -- Int *}
haftmann@30531
   702
haftmann@32683
   703
abbreviation inter :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Int" 70) where
haftmann@32683
   704
  "op Int \<equiv> inf"
haftmann@32081
   705
haftmann@32081
   706
notation (xsymbols)
haftmann@32135
   707
  inter  (infixl "\<inter>" 70)
haftmann@32081
   708
haftmann@32081
   709
notation (HTML output)
haftmann@32135
   710
  inter  (infixl "\<inter>" 70)
haftmann@32135
   711
haftmann@32135
   712
lemma Int_def:
haftmann@32135
   713
  "A \<inter> B = {x. x \<in> A \<and> x \<in> B}"
haftmann@32683
   714
  by (simp add: inf_fun_eq inf_bool_eq Collect_def mem_def)
haftmann@32081
   715
haftmann@30531
   716
lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)"
haftmann@30531
   717
  by (unfold Int_def) blast
haftmann@30531
   718
haftmann@30531
   719
lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B"
haftmann@30531
   720
  by simp
haftmann@30531
   721
haftmann@30531
   722
lemma IntD1: "c : A Int B ==> c:A"
haftmann@30531
   723
  by simp
haftmann@30531
   724
haftmann@30531
   725
lemma IntD2: "c : A Int B ==> c:B"
haftmann@30531
   726
  by simp
haftmann@30531
   727
haftmann@30531
   728
lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P"
haftmann@30531
   729
  by simp
haftmann@30531
   730
haftmann@32081
   731
lemma mono_Int: "mono f \<Longrightarrow> f (A \<inter> B) \<subseteq> f A \<inter> f B"
haftmann@32683
   732
  by (fact mono_inf)
haftmann@32081
   733
haftmann@30531
   734
haftmann@30531
   735
subsubsection {* Set difference *}
haftmann@30531
   736
haftmann@30531
   737
lemma Diff_iff [simp]: "(c : A - B) = (c:A & c~:B)"
haftmann@30531
   738
  by (simp add: mem_def fun_diff_def bool_diff_def)
haftmann@30531
   739
haftmann@30531
   740
lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A - B"
haftmann@30531
   741
  by simp
haftmann@30531
   742
haftmann@30531
   743
lemma DiffD1: "c : A - B ==> c : A"
haftmann@30531
   744
  by simp
haftmann@30531
   745
haftmann@30531
   746
lemma DiffD2: "c : A - B ==> c : B ==> P"
haftmann@30531
   747
  by simp
haftmann@30531
   748
haftmann@30531
   749
lemma DiffE [elim!]: "c : A - B ==> (c:A ==> c~:B ==> P) ==> P"
haftmann@30531
   750
  by simp
haftmann@30531
   751
haftmann@30531
   752
lemma set_diff_eq: "A - B = {x. x : A & ~ x : B}" by blast
haftmann@30531
   753
haftmann@30531
   754
lemma Compl_eq_Diff_UNIV: "-A = (UNIV - A)"
haftmann@30531
   755
by blast
haftmann@30531
   756
haftmann@30531
   757
haftmann@31456
   758
subsubsection {* Augmenting a set -- @{const insert} *}
haftmann@30531
   759
haftmann@30531
   760
lemma insert_iff [simp]: "(a : insert b A) = (a = b | a:A)"
haftmann@30531
   761
  by (unfold insert_def) blast
haftmann@30531
   762
haftmann@30531
   763
lemma insertI1: "a : insert a B"
haftmann@30531
   764
  by simp
haftmann@30531
   765
haftmann@30531
   766
lemma insertI2: "a : B ==> a : insert b B"
haftmann@30531
   767
  by simp
haftmann@30531
   768
haftmann@30531
   769
lemma insertE [elim!]: "a : insert b A ==> (a = b ==> P) ==> (a:A ==> P) ==> P"
haftmann@30531
   770
  by (unfold insert_def) blast
haftmann@30531
   771
haftmann@30531
   772
lemma insertCI [intro!]: "(a~:B ==> a = b) ==> a: insert b B"
haftmann@30531
   773
  -- {* Classical introduction rule. *}
haftmann@30531
   774
  by auto
haftmann@30531
   775
haftmann@30531
   776
lemma subset_insert_iff: "(A \<subseteq> insert x B) = (if x:A then A - {x} \<subseteq> B else A \<subseteq> B)"
haftmann@30531
   777
  by auto
haftmann@30531
   778
haftmann@30531
   779
lemma set_insert:
haftmann@30531
   780
  assumes "x \<in> A"
haftmann@30531
   781
  obtains B where "A = insert x B" and "x \<notin> B"
haftmann@30531
   782
proof
haftmann@30531
   783
  from assms show "A = insert x (A - {x})" by blast
haftmann@30531
   784
next
haftmann@30531
   785
  show "x \<notin> A - {x}" by blast
haftmann@30531
   786
qed
haftmann@30531
   787
haftmann@30531
   788
lemma insert_ident: "x ~: A ==> x ~: B ==> (insert x A = insert x B) = (A = B)"
haftmann@30531
   789
by auto
haftmann@30531
   790
haftmann@30531
   791
subsubsection {* Singletons, using insert *}
haftmann@30531
   792
blanchet@35828
   793
lemma singletonI [intro!,no_atp]: "a : {a}"
haftmann@30531
   794
    -- {* Redundant? But unlike @{text insertCI}, it proves the subgoal immediately! *}
haftmann@30531
   795
  by (rule insertI1)
haftmann@30531
   796
blanchet@35828
   797
lemma singletonD [dest!,no_atp]: "b : {a} ==> b = a"
haftmann@30531
   798
  by blast
haftmann@30531
   799
haftmann@30531
   800
lemmas singletonE = singletonD [elim_format]
haftmann@30531
   801
haftmann@30531
   802
lemma singleton_iff: "(b : {a}) = (b = a)"
haftmann@30531
   803
  by blast
haftmann@30531
   804
haftmann@30531
   805
lemma singleton_inject [dest!]: "{a} = {b} ==> a = b"
haftmann@30531
   806
  by blast
haftmann@30531
   807
blanchet@35828
   808
lemma singleton_insert_inj_eq [iff,no_atp]:
haftmann@30531
   809
     "({b} = insert a A) = (a = b & A \<subseteq> {b})"
haftmann@30531
   810
  by blast
haftmann@30531
   811
blanchet@35828
   812
lemma singleton_insert_inj_eq' [iff,no_atp]:
haftmann@30531
   813
     "(insert a A = {b}) = (a = b & A \<subseteq> {b})"
haftmann@30531
   814
  by blast
haftmann@30531
   815
haftmann@30531
   816
lemma subset_singletonD: "A \<subseteq> {x} ==> A = {} | A = {x}"
haftmann@30531
   817
  by fast
haftmann@30531
   818
haftmann@30531
   819
lemma singleton_conv [simp]: "{x. x = a} = {a}"
haftmann@30531
   820
  by blast
haftmann@30531
   821
haftmann@30531
   822
lemma singleton_conv2 [simp]: "{x. a = x} = {a}"
haftmann@30531
   823
  by blast
haftmann@30531
   824
haftmann@30531
   825
lemma diff_single_insert: "A - {x} \<subseteq> B ==> x \<in> A ==> A \<subseteq> insert x B"
haftmann@30531
   826
  by blast
haftmann@30531
   827
haftmann@30531
   828
lemma doubleton_eq_iff: "({a,b} = {c,d}) = (a=c & b=d | a=d & b=c)"
haftmann@30531
   829
  by (blast elim: equalityE)
haftmann@30531
   830
wenzelm@11979
   831
haftmann@32077
   832
subsubsection {* Image of a set under a function *}
haftmann@32077
   833
haftmann@32077
   834
text {*
haftmann@32077
   835
  Frequently @{term b} does not have the syntactic form of @{term "f x"}.
haftmann@32077
   836
*}
haftmann@32077
   837
haftmann@32077
   838
definition image :: "('a => 'b) => 'a set => 'b set" (infixr "`" 90) where
blanchet@35828
   839
  image_def [no_atp]: "f ` A = {y. EX x:A. y = f(x)}"
haftmann@32077
   840
haftmann@32077
   841
abbreviation
haftmann@32077
   842
  range :: "('a => 'b) => 'b set" where -- "of function"
haftmann@32077
   843
  "range f == f ` UNIV"
haftmann@32077
   844
haftmann@32077
   845
lemma image_eqI [simp, intro]: "b = f x ==> x:A ==> b : f`A"
haftmann@32077
   846
  by (unfold image_def) blast
haftmann@32077
   847
haftmann@32077
   848
lemma imageI: "x : A ==> f x : f ` A"
haftmann@32077
   849
  by (rule image_eqI) (rule refl)
haftmann@32077
   850
haftmann@32077
   851
lemma rev_image_eqI: "x:A ==> b = f x ==> b : f`A"
haftmann@32077
   852
  -- {* This version's more effective when we already have the
haftmann@32077
   853
    required @{term x}. *}
haftmann@32077
   854
  by (unfold image_def) blast
haftmann@32077
   855
haftmann@32077
   856
lemma imageE [elim!]:
haftmann@32077
   857
  "b : (%x. f x)`A ==> (!!x. b = f x ==> x:A ==> P) ==> P"
haftmann@32077
   858
  -- {* The eta-expansion gives variable-name preservation. *}
haftmann@32077
   859
  by (unfold image_def) blast
haftmann@32077
   860
haftmann@32077
   861
lemma image_Un: "f`(A Un B) = f`A Un f`B"
haftmann@32077
   862
  by blast
haftmann@32077
   863
haftmann@32077
   864
lemma image_iff: "(z : f`A) = (EX x:A. z = f x)"
haftmann@32077
   865
  by blast
haftmann@32077
   866
haftmann@32077
   867
lemma image_subset_iff: "(f`A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)"
haftmann@32077
   868
  -- {* This rewrite rule would confuse users if made default. *}
haftmann@32077
   869
  by blast
haftmann@32077
   870
haftmann@32077
   871
lemma subset_image_iff: "(B \<subseteq> f`A) = (EX AA. AA \<subseteq> A & B = f`AA)"
haftmann@32077
   872
  apply safe
haftmann@32077
   873
   prefer 2 apply fast
haftmann@32077
   874
  apply (rule_tac x = "{a. a : A & f a : B}" in exI, fast)
haftmann@32077
   875
  done
haftmann@32077
   876
haftmann@32077
   877
lemma image_subsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f`A \<subseteq> B"
haftmann@32077
   878
  -- {* Replaces the three steps @{text subsetI}, @{text imageE},
haftmann@32077
   879
    @{text hypsubst}, but breaks too many existing proofs. *}
haftmann@32077
   880
  by blast
wenzelm@11979
   881
wenzelm@11979
   882
text {*
haftmann@32077
   883
  \medskip Range of a function -- just a translation for image!
haftmann@32077
   884
*}
haftmann@32077
   885
haftmann@32077
   886
lemma range_eqI: "b = f x ==> b \<in> range f"
haftmann@32077
   887
  by simp
haftmann@32077
   888
haftmann@32077
   889
lemma rangeI: "f x \<in> range f"
haftmann@32077
   890
  by simp
haftmann@32077
   891
haftmann@32077
   892
lemma rangeE [elim?]: "b \<in> range (\<lambda>x. f x) ==> (!!x. b = f x ==> P) ==> P"
haftmann@32077
   893
  by blast
haftmann@32077
   894
haftmann@32077
   895
haftmann@32117
   896
subsubsection {* Some rules with @{text "if"} *}
haftmann@32081
   897
haftmann@32081
   898
text{* Elimination of @{text"{x. \<dots> & x=t & \<dots>}"}. *}
haftmann@32081
   899
haftmann@32081
   900
lemma Collect_conv_if: "{x. x=a & P x} = (if P a then {a} else {})"
haftmann@32117
   901
  by auto
haftmann@32081
   902
haftmann@32081
   903
lemma Collect_conv_if2: "{x. a=x & P x} = (if P a then {a} else {})"
haftmann@32117
   904
  by auto
haftmann@32081
   905
haftmann@32081
   906
text {*
haftmann@32081
   907
  Rewrite rules for boolean case-splitting: faster than @{text
haftmann@32081
   908
  "split_if [split]"}.
haftmann@32081
   909
*}
haftmann@32081
   910
haftmann@32081
   911
lemma split_if_eq1: "((if Q then x else y) = b) = ((Q --> x = b) & (~ Q --> y = b))"
haftmann@32081
   912
  by (rule split_if)
haftmann@32081
   913
haftmann@32081
   914
lemma split_if_eq2: "(a = (if Q then x else y)) = ((Q --> a = x) & (~ Q --> a = y))"
haftmann@32081
   915
  by (rule split_if)
haftmann@32081
   916
haftmann@32081
   917
text {*
haftmann@32081
   918
  Split ifs on either side of the membership relation.  Not for @{text
haftmann@32081
   919
  "[simp]"} -- can cause goals to blow up!
haftmann@32081
   920
*}
haftmann@32081
   921
haftmann@32081
   922
lemma split_if_mem1: "((if Q then x else y) : b) = ((Q --> x : b) & (~ Q --> y : b))"
haftmann@32081
   923
  by (rule split_if)
haftmann@32081
   924
haftmann@32081
   925
lemma split_if_mem2: "(a : (if Q then x else y)) = ((Q --> a : x) & (~ Q --> a : y))"
haftmann@32081
   926
  by (rule split_if [where P="%S. a : S"])
haftmann@32081
   927
haftmann@32081
   928
lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2
haftmann@32081
   929
haftmann@32081
   930
(*Would like to add these, but the existing code only searches for the
haftmann@32081
   931
  outer-level constant, which in this case is just "op :"; we instead need
haftmann@32081
   932
  to use term-nets to associate patterns with rules.  Also, if a rule fails to
haftmann@32081
   933
  apply, then the formula should be kept.
haftmann@34974
   934
  [("uminus", Compl_iff RS iffD1), ("minus", [Diff_iff RS iffD1]),
haftmann@32081
   935
   ("Int", [IntD1,IntD2]),
haftmann@32081
   936
   ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]
haftmann@32081
   937
 *)
haftmann@32081
   938
haftmann@32081
   939
haftmann@32135
   940
subsection {* Further operations and lemmas *}
haftmann@32135
   941
haftmann@32135
   942
subsubsection {* The ``proper subset'' relation *}
haftmann@32135
   943
blanchet@35828
   944
lemma psubsetI [intro!,no_atp]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B"
haftmann@32135
   945
  by (unfold less_le) blast
haftmann@32135
   946
blanchet@35828
   947
lemma psubsetE [elim!,no_atp]: 
haftmann@32135
   948
    "[|A \<subset> B;  [|A \<subseteq> B; ~ (B\<subseteq>A)|] ==> R|] ==> R"
haftmann@32135
   949
  by (unfold less_le) blast
haftmann@32135
   950
haftmann@32135
   951
lemma psubset_insert_iff:
haftmann@32135
   952
  "(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)"
haftmann@32135
   953
  by (auto simp add: less_le subset_insert_iff)
haftmann@32135
   954
haftmann@32135
   955
lemma psubset_eq: "(A \<subset> B) = (A \<subseteq> B & A \<noteq> B)"
haftmann@32135
   956
  by (simp only: less_le)
haftmann@32135
   957
haftmann@32135
   958
lemma psubset_imp_subset: "A \<subset> B ==> A \<subseteq> B"
haftmann@32135
   959
  by (simp add: psubset_eq)
haftmann@32135
   960
haftmann@32135
   961
lemma psubset_trans: "[| A \<subset> B; B \<subset> C |] ==> A \<subset> C"
haftmann@32135
   962
apply (unfold less_le)
haftmann@32135
   963
apply (auto dest: subset_antisym)
haftmann@32135
   964
done
haftmann@32135
   965
haftmann@32135
   966
lemma psubsetD: "[| A \<subset> B; c \<in> A |] ==> c \<in> B"
haftmann@32135
   967
apply (unfold less_le)
haftmann@32135
   968
apply (auto dest: subsetD)
haftmann@32135
   969
done
haftmann@32135
   970
haftmann@32135
   971
lemma psubset_subset_trans: "A \<subset> B ==> B \<subseteq> C ==> A \<subset> C"
haftmann@32135
   972
  by (auto simp add: psubset_eq)
haftmann@32135
   973
haftmann@32135
   974
lemma subset_psubset_trans: "A \<subseteq> B ==> B \<subset> C ==> A \<subset> C"
haftmann@32135
   975
  by (auto simp add: psubset_eq)
haftmann@32135
   976
haftmann@32135
   977
lemma psubset_imp_ex_mem: "A \<subset> B ==> \<exists>b. b \<in> (B - A)"
haftmann@32135
   978
  by (unfold less_le) blast
haftmann@32135
   979
haftmann@32135
   980
lemma atomize_ball:
haftmann@32135
   981
    "(!!x. x \<in> A ==> P x) == Trueprop (\<forall>x\<in>A. P x)"
haftmann@32135
   982
  by (simp only: Ball_def atomize_all atomize_imp)
haftmann@32135
   983
haftmann@32135
   984
lemmas [symmetric, rulify] = atomize_ball
haftmann@32135
   985
  and [symmetric, defn] = atomize_ball
haftmann@32135
   986
haftmann@32135
   987
subsubsection {* Derived rules involving subsets. *}
haftmann@32135
   988
haftmann@32135
   989
text {* @{text insert}. *}
haftmann@32135
   990
haftmann@32135
   991
lemma subset_insertI: "B \<subseteq> insert a B"
haftmann@32135
   992
  by (rule subsetI) (erule insertI2)
haftmann@32135
   993
haftmann@32135
   994
lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B"
haftmann@32135
   995
  by blast
haftmann@32135
   996
haftmann@32135
   997
lemma subset_insert: "x \<notin> A ==> (A \<subseteq> insert x B) = (A \<subseteq> B)"
haftmann@32135
   998
  by blast
haftmann@32135
   999
haftmann@32135
  1000
haftmann@32135
  1001
text {* \medskip Finite Union -- the least upper bound of two sets. *}
haftmann@32135
  1002
haftmann@32135
  1003
lemma Un_upper1: "A \<subseteq> A \<union> B"
huffman@36009
  1004
  by (fact sup_ge1)
haftmann@32135
  1005
haftmann@32135
  1006
lemma Un_upper2: "B \<subseteq> A \<union> B"
huffman@36009
  1007
  by (fact sup_ge2)
haftmann@32135
  1008
haftmann@32135
  1009
lemma Un_least: "A \<subseteq> C ==> B \<subseteq> C ==> A \<union> B \<subseteq> C"
huffman@36009
  1010
  by (fact sup_least)
haftmann@32135
  1011
haftmann@32135
  1012
haftmann@32135
  1013
text {* \medskip Finite Intersection -- the greatest lower bound of two sets. *}
haftmann@32135
  1014
haftmann@32135
  1015
lemma Int_lower1: "A \<inter> B \<subseteq> A"
huffman@36009
  1016
  by (fact inf_le1)
haftmann@32135
  1017
haftmann@32135
  1018
lemma Int_lower2: "A \<inter> B \<subseteq> B"
huffman@36009
  1019
  by (fact inf_le2)
haftmann@32135
  1020
haftmann@32135
  1021
lemma Int_greatest: "C \<subseteq> A ==> C \<subseteq> B ==> C \<subseteq> A \<inter> B"
huffman@36009
  1022
  by (fact inf_greatest)
haftmann@32135
  1023
haftmann@32135
  1024
haftmann@32135
  1025
text {* \medskip Set difference. *}
haftmann@32135
  1026
haftmann@32135
  1027
lemma Diff_subset: "A - B \<subseteq> A"
haftmann@32135
  1028
  by blast
haftmann@32135
  1029
haftmann@32135
  1030
lemma Diff_subset_conv: "(A - B \<subseteq> C) = (A \<subseteq> B \<union> C)"
haftmann@32135
  1031
by blast
haftmann@32135
  1032
haftmann@32135
  1033
haftmann@32135
  1034
subsubsection {* Equalities involving union, intersection, inclusion, etc. *}
haftmann@32135
  1035
haftmann@32135
  1036
text {* @{text "{}"}. *}
haftmann@32135
  1037
haftmann@32135
  1038
lemma Collect_const [simp]: "{s. P} = (if P then UNIV else {})"
haftmann@32135
  1039
  -- {* supersedes @{text "Collect_False_empty"} *}
haftmann@32135
  1040
  by auto
haftmann@32135
  1041
haftmann@32135
  1042
lemma subset_empty [simp]: "(A \<subseteq> {}) = (A = {})"
haftmann@32135
  1043
  by blast
haftmann@32135
  1044
haftmann@32135
  1045
lemma not_psubset_empty [iff]: "\<not> (A < {})"
haftmann@32135
  1046
  by (unfold less_le) blast
haftmann@32135
  1047
haftmann@32135
  1048
lemma Collect_empty_eq [simp]: "(Collect P = {}) = (\<forall>x. \<not> P x)"
haftmann@32135
  1049
by blast
haftmann@32135
  1050
haftmann@32135
  1051
lemma empty_Collect_eq [simp]: "({} = Collect P) = (\<forall>x. \<not> P x)"
haftmann@32135
  1052
by blast
haftmann@32135
  1053
haftmann@32135
  1054
lemma Collect_neg_eq: "{x. \<not> P x} = - {x. P x}"
haftmann@32135
  1055
  by blast
haftmann@32135
  1056
haftmann@32135
  1057
lemma Collect_disj_eq: "{x. P x | Q x} = {x. P x} \<union> {x. Q x}"
haftmann@32135
  1058
  by blast
haftmann@32135
  1059
haftmann@32135
  1060
lemma Collect_imp_eq: "{x. P x --> Q x} = -{x. P x} \<union> {x. Q x}"
haftmann@32135
  1061
  by blast
haftmann@32135
  1062
haftmann@32135
  1063
lemma Collect_conj_eq: "{x. P x & Q x} = {x. P x} \<inter> {x. Q x}"
haftmann@32135
  1064
  by blast
haftmann@32135
  1065
haftmann@32135
  1066
haftmann@32135
  1067
text {* \medskip @{text insert}. *}
haftmann@32135
  1068
haftmann@32135
  1069
lemma insert_is_Un: "insert a A = {a} Un A"
haftmann@32135
  1070
  -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a {}"} *}
haftmann@32135
  1071
  by blast
haftmann@32135
  1072
haftmann@32135
  1073
lemma insert_not_empty [simp]: "insert a A \<noteq> {}"
haftmann@32135
  1074
  by blast
haftmann@32135
  1075
haftmann@32135
  1076
lemmas empty_not_insert = insert_not_empty [symmetric, standard]
haftmann@32135
  1077
declare empty_not_insert [simp]
haftmann@32135
  1078
haftmann@32135
  1079
lemma insert_absorb: "a \<in> A ==> insert a A = A"
haftmann@32135
  1080
  -- {* @{text "[simp]"} causes recursive calls when there are nested inserts *}
haftmann@32135
  1081
  -- {* with \emph{quadratic} running time *}
haftmann@32135
  1082
  by blast
haftmann@32135
  1083
haftmann@32135
  1084
lemma insert_absorb2 [simp]: "insert x (insert x A) = insert x A"
haftmann@32135
  1085
  by blast
haftmann@32135
  1086
haftmann@32135
  1087
lemma insert_commute: "insert x (insert y A) = insert y (insert x A)"
haftmann@32135
  1088
  by blast
haftmann@32135
  1089
haftmann@32135
  1090
lemma insert_subset [simp]: "(insert x A \<subseteq> B) = (x \<in> B & A \<subseteq> B)"
haftmann@32135
  1091
  by blast
haftmann@32135
  1092
haftmann@32135
  1093
lemma mk_disjoint_insert: "a \<in> A ==> \<exists>B. A = insert a B & a \<notin> B"
haftmann@32135
  1094
  -- {* use new @{text B} rather than @{text "A - {a}"} to avoid infinite unfolding *}
haftmann@32135
  1095
  apply (rule_tac x = "A - {a}" in exI, blast)
haftmann@32135
  1096
  done
haftmann@32135
  1097
haftmann@32135
  1098
lemma insert_Collect: "insert a (Collect P) = {u. u \<noteq> a --> P u}"
haftmann@32135
  1099
  by auto
haftmann@32135
  1100
haftmann@32135
  1101
lemma insert_inter_insert[simp]: "insert a A \<inter> insert a B = insert a (A \<inter> B)"
haftmann@32135
  1102
  by blast
haftmann@32135
  1103
blanchet@35828
  1104
lemma insert_disjoint [simp,no_atp]:
haftmann@32135
  1105
 "(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})"
haftmann@32135
  1106
 "({} = insert a A \<inter> B) = (a \<notin> B \<and> {} = A \<inter> B)"
haftmann@32135
  1107
  by auto
haftmann@32135
  1108
blanchet@35828
  1109
lemma disjoint_insert [simp,no_atp]:
haftmann@32135
  1110
 "(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})"
haftmann@32135
  1111
 "({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)"
haftmann@32135
  1112
  by auto
haftmann@32135
  1113
haftmann@32135
  1114
text {* \medskip @{text image}. *}
haftmann@32135
  1115
haftmann@32135
  1116
lemma image_empty [simp]: "f`{} = {}"
haftmann@32135
  1117
  by blast
haftmann@32135
  1118
haftmann@32135
  1119
lemma image_insert [simp]: "f ` insert a B = insert (f a) (f`B)"
haftmann@32135
  1120
  by blast
haftmann@32135
  1121
haftmann@32135
  1122
lemma image_constant: "x \<in> A ==> (\<lambda>x. c) ` A = {c}"
haftmann@32135
  1123
  by auto
haftmann@32135
  1124
haftmann@32135
  1125
lemma image_constant_conv: "(%x. c) ` A = (if A = {} then {} else {c})"
haftmann@32135
  1126
by auto
haftmann@32135
  1127
haftmann@32135
  1128
lemma image_image: "f ` (g ` A) = (\<lambda>x. f (g x)) ` A"
haftmann@32135
  1129
by blast
haftmann@32135
  1130
haftmann@32135
  1131
lemma insert_image [simp]: "x \<in> A ==> insert (f x) (f`A) = f`A"
haftmann@32135
  1132
by blast
haftmann@32135
  1133
haftmann@32135
  1134
lemma image_is_empty [iff]: "(f`A = {}) = (A = {})"
haftmann@32135
  1135
by blast
haftmann@32135
  1136
haftmann@32135
  1137
lemma empty_is_image[iff]: "({} = f ` A) = (A = {})"
haftmann@32135
  1138
by blast
haftmann@32135
  1139
haftmann@32135
  1140
blanchet@35828
  1141
lemma image_Collect [no_atp]: "f ` {x. P x} = {f x | x. P x}"
haftmann@32135
  1142
  -- {* NOT suitable as a default simprule: the RHS isn't simpler than the LHS,
haftmann@32135
  1143
      with its implicit quantifier and conjunction.  Also image enjoys better
haftmann@32135
  1144
      equational properties than does the RHS. *}
haftmann@32135
  1145
  by blast
haftmann@32135
  1146
haftmann@32135
  1147
lemma if_image_distrib [simp]:
haftmann@32135
  1148
  "(\<lambda>x. if P x then f x else g x) ` S
haftmann@32135
  1149
    = (f ` (S \<inter> {x. P x})) \<union> (g ` (S \<inter> {x. \<not> P x}))"
haftmann@32135
  1150
  by (auto simp add: image_def)
haftmann@32135
  1151
haftmann@32135
  1152
lemma image_cong: "M = N ==> (!!x. x \<in> N ==> f x = g x) ==> f`M = g`N"
haftmann@32135
  1153
  by (simp add: image_def)
haftmann@32135
  1154
haftmann@32135
  1155
haftmann@32135
  1156
text {* \medskip @{text range}. *}
haftmann@32135
  1157
blanchet@35828
  1158
lemma full_SetCompr_eq [no_atp]: "{u. \<exists>x. u = f x} = range f"
haftmann@32135
  1159
  by auto
haftmann@32135
  1160
haftmann@32135
  1161
lemma range_composition: "range (\<lambda>x. f (g x)) = f`range g"
haftmann@32135
  1162
by (subst image_image, simp)
haftmann@32135
  1163
haftmann@32135
  1164
haftmann@32135
  1165
text {* \medskip @{text Int} *}
haftmann@32135
  1166
haftmann@32135
  1167
lemma Int_absorb [simp]: "A \<inter> A = A"
huffman@36009
  1168
  by (fact inf_idem)
haftmann@32135
  1169
haftmann@32135
  1170
lemma Int_left_absorb: "A \<inter> (A \<inter> B) = A \<inter> B"
huffman@36009
  1171
  by (fact inf_left_idem)
haftmann@32135
  1172
haftmann@32135
  1173
lemma Int_commute: "A \<inter> B = B \<inter> A"
huffman@36009
  1174
  by (fact inf_commute)
haftmann@32135
  1175
haftmann@32135
  1176
lemma Int_left_commute: "A \<inter> (B \<inter> C) = B \<inter> (A \<inter> C)"
huffman@36009
  1177
  by (fact inf_left_commute)
haftmann@32135
  1178
haftmann@32135
  1179
lemma Int_assoc: "(A \<inter> B) \<inter> C = A \<inter> (B \<inter> C)"
huffman@36009
  1180
  by (fact inf_assoc)
haftmann@32135
  1181
haftmann@32135
  1182
lemmas Int_ac = Int_assoc Int_left_absorb Int_commute Int_left_commute
haftmann@32135
  1183
  -- {* Intersection is an AC-operator *}
haftmann@32135
  1184
haftmann@32135
  1185
lemma Int_absorb1: "B \<subseteq> A ==> A \<inter> B = B"
huffman@36009
  1186
  by (fact inf_absorb2)
haftmann@32135
  1187
haftmann@32135
  1188
lemma Int_absorb2: "A \<subseteq> B ==> A \<inter> B = A"
huffman@36009
  1189
  by (fact inf_absorb1)
haftmann@32135
  1190
haftmann@32135
  1191
lemma Int_empty_left [simp]: "{} \<inter> B = {}"
huffman@36009
  1192
  by (fact inf_bot_left)
haftmann@32135
  1193
haftmann@32135
  1194
lemma Int_empty_right [simp]: "A \<inter> {} = {}"
huffman@36009
  1195
  by (fact inf_bot_right)
haftmann@32135
  1196
haftmann@32135
  1197
lemma disjoint_eq_subset_Compl: "(A \<inter> B = {}) = (A \<subseteq> -B)"
haftmann@32135
  1198
  by blast
haftmann@32135
  1199
haftmann@32135
  1200
lemma disjoint_iff_not_equal: "(A \<inter> B = {}) = (\<forall>x\<in>A. \<forall>y\<in>B. x \<noteq> y)"
haftmann@32135
  1201
  by blast
haftmann@32135
  1202
haftmann@32135
  1203
lemma Int_UNIV_left [simp]: "UNIV \<inter> B = B"
huffman@36009
  1204
  by (fact inf_top_left)
haftmann@32135
  1205
haftmann@32135
  1206
lemma Int_UNIV_right [simp]: "A \<inter> UNIV = A"
huffman@36009
  1207
  by (fact inf_top_right)
haftmann@32135
  1208
haftmann@32135
  1209
lemma Int_Un_distrib: "A \<inter> (B \<union> C) = (A \<inter> B) \<union> (A \<inter> C)"
huffman@36009
  1210
  by (fact inf_sup_distrib1)
haftmann@32135
  1211
haftmann@32135
  1212
lemma Int_Un_distrib2: "(B \<union> C) \<inter> A = (B \<inter> A) \<union> (C \<inter> A)"
huffman@36009
  1213
  by (fact inf_sup_distrib2)
haftmann@32135
  1214
blanchet@35828
  1215
lemma Int_UNIV [simp,no_atp]: "(A \<inter> B = UNIV) = (A = UNIV & B = UNIV)"
huffman@36009
  1216
  by (fact inf_eq_top_iff)
haftmann@32135
  1217
haftmann@32135
  1218
lemma Int_subset_iff [simp]: "(C \<subseteq> A \<inter> B) = (C \<subseteq> A & C \<subseteq> B)"
huffman@36009
  1219
  by (fact le_inf_iff)
haftmann@32135
  1220
haftmann@32135
  1221
lemma Int_Collect: "(x \<in> A \<inter> {x. P x}) = (x \<in> A & P x)"
haftmann@32135
  1222
  by blast
haftmann@32135
  1223
haftmann@32135
  1224
haftmann@32135
  1225
text {* \medskip @{text Un}. *}
haftmann@32135
  1226
haftmann@32135
  1227
lemma Un_absorb [simp]: "A \<union> A = A"
huffman@36009
  1228
  by (fact sup_idem)
haftmann@32135
  1229
haftmann@32135
  1230
lemma Un_left_absorb: "A \<union> (A \<union> B) = A \<union> B"
huffman@36009
  1231
  by (fact sup_left_idem)
haftmann@32135
  1232
haftmann@32135
  1233
lemma Un_commute: "A \<union> B = B \<union> A"
huffman@36009
  1234
  by (fact sup_commute)
haftmann@32135
  1235
haftmann@32135
  1236
lemma Un_left_commute: "A \<union> (B \<union> C) = B \<union> (A \<union> C)"
huffman@36009
  1237
  by (fact sup_left_commute)
haftmann@32135
  1238
haftmann@32135
  1239
lemma Un_assoc: "(A \<union> B) \<union> C = A \<union> (B \<union> C)"
huffman@36009
  1240
  by (fact sup_assoc)
haftmann@32135
  1241
haftmann@32135
  1242
lemmas Un_ac = Un_assoc Un_left_absorb Un_commute Un_left_commute
haftmann@32135
  1243
  -- {* Union is an AC-operator *}
haftmann@32135
  1244
haftmann@32135
  1245
lemma Un_absorb1: "A \<subseteq> B ==> A \<union> B = B"
huffman@36009
  1246
  by (fact sup_absorb2)
haftmann@32135
  1247
haftmann@32135
  1248
lemma Un_absorb2: "B \<subseteq> A ==> A \<union> B = A"
huffman@36009
  1249
  by (fact sup_absorb1)
haftmann@32135
  1250
haftmann@32135
  1251
lemma Un_empty_left [simp]: "{} \<union> B = B"
huffman@36009
  1252
  by (fact sup_bot_left)
haftmann@32135
  1253
haftmann@32135
  1254
lemma Un_empty_right [simp]: "A \<union> {} = A"
huffman@36009
  1255
  by (fact sup_bot_right)
haftmann@32135
  1256
haftmann@32135
  1257
lemma Un_UNIV_left [simp]: "UNIV \<union> B = UNIV"
huffman@36009
  1258
  by (fact sup_top_left)
haftmann@32135
  1259
haftmann@32135
  1260
lemma Un_UNIV_right [simp]: "A \<union> UNIV = UNIV"
huffman@36009
  1261
  by (fact sup_top_right)
haftmann@32135
  1262
haftmann@32135
  1263
lemma Un_insert_left [simp]: "(insert a B) \<union> C = insert a (B \<union> C)"
haftmann@32135
  1264
  by blast
haftmann@32135
  1265
haftmann@32135
  1266
lemma Un_insert_right [simp]: "A \<union> (insert a B) = insert a (A \<union> B)"
haftmann@32135
  1267
  by blast
haftmann@32135
  1268
haftmann@32135
  1269
lemma Int_insert_left:
haftmann@32135
  1270
    "(insert a B) Int C = (if a \<in> C then insert a (B \<inter> C) else B \<inter> C)"
haftmann@32135
  1271
  by auto
haftmann@32135
  1272
nipkow@32456
  1273
lemma Int_insert_left_if0[simp]:
nipkow@32456
  1274
    "a \<notin> C \<Longrightarrow> (insert a B) Int C = B \<inter> C"
nipkow@32456
  1275
  by auto
nipkow@32456
  1276
nipkow@32456
  1277
lemma Int_insert_left_if1[simp]:
nipkow@32456
  1278
    "a \<in> C \<Longrightarrow> (insert a B) Int C = insert a (B Int C)"
nipkow@32456
  1279
  by auto
nipkow@32456
  1280
haftmann@32135
  1281
lemma Int_insert_right:
haftmann@32135
  1282
    "A \<inter> (insert a B) = (if a \<in> A then insert a (A \<inter> B) else A \<inter> B)"
haftmann@32135
  1283
  by auto
haftmann@32135
  1284
nipkow@32456
  1285
lemma Int_insert_right_if0[simp]:
nipkow@32456
  1286
    "a \<notin> A \<Longrightarrow> A Int (insert a B) = A Int B"
nipkow@32456
  1287
  by auto
nipkow@32456
  1288
nipkow@32456
  1289
lemma Int_insert_right_if1[simp]:
nipkow@32456
  1290
    "a \<in> A \<Longrightarrow> A Int (insert a B) = insert a (A Int B)"
nipkow@32456
  1291
  by auto
nipkow@32456
  1292
haftmann@32135
  1293
lemma Un_Int_distrib: "A \<union> (B \<inter> C) = (A \<union> B) \<inter> (A \<union> C)"
huffman@36009
  1294
  by (fact sup_inf_distrib1)
haftmann@32135
  1295
haftmann@32135
  1296
lemma Un_Int_distrib2: "(B \<inter> C) \<union> A = (B \<union> A) \<inter> (C \<union> A)"
huffman@36009
  1297
  by (fact sup_inf_distrib2)
haftmann@32135
  1298
haftmann@32135
  1299
lemma Un_Int_crazy:
haftmann@32135
  1300
    "(A \<inter> B) \<union> (B \<inter> C) \<union> (C \<inter> A) = (A \<union> B) \<inter> (B \<union> C) \<inter> (C \<union> A)"
haftmann@32135
  1301
  by blast
haftmann@32135
  1302
haftmann@32135
  1303
lemma subset_Un_eq: "(A \<subseteq> B) = (A \<union> B = B)"
huffman@36009
  1304
  by (fact le_iff_sup)
haftmann@32135
  1305
haftmann@32135
  1306
lemma Un_empty [iff]: "(A \<union> B = {}) = (A = {} & B = {})"
huffman@36009
  1307
  by (fact sup_eq_bot_iff)
haftmann@32135
  1308
haftmann@32135
  1309
lemma Un_subset_iff [simp]: "(A \<union> B \<subseteq> C) = (A \<subseteq> C & B \<subseteq> C)"
huffman@36009
  1310
  by (fact le_sup_iff)
haftmann@32135
  1311
haftmann@32135
  1312
lemma Un_Diff_Int: "(A - B) \<union> (A \<inter> B) = A"
haftmann@32135
  1313
  by blast
haftmann@32135
  1314
haftmann@32135
  1315
lemma Diff_Int2: "A \<inter> C - B \<inter> C = A \<inter> C - B"
haftmann@32135
  1316
  by blast
haftmann@32135
  1317
haftmann@32135
  1318
haftmann@32135
  1319
text {* \medskip Set complement *}
haftmann@32135
  1320
haftmann@32135
  1321
lemma Compl_disjoint [simp]: "A \<inter> -A = {}"
huffman@36009
  1322
  by (fact inf_compl_bot)
haftmann@32135
  1323
haftmann@32135
  1324
lemma Compl_disjoint2 [simp]: "-A \<inter> A = {}"
huffman@36009
  1325
  by (fact compl_inf_bot)
haftmann@32135
  1326
haftmann@32135
  1327
lemma Compl_partition: "A \<union> -A = UNIV"
huffman@36009
  1328
  by (fact sup_compl_top)
haftmann@32135
  1329
haftmann@32135
  1330
lemma Compl_partition2: "-A \<union> A = UNIV"
huffman@36009
  1331
  by (fact compl_sup_top)
haftmann@32135
  1332
haftmann@32135
  1333
lemma double_complement [simp]: "- (-A) = (A::'a set)"
huffman@36009
  1334
  by (fact double_compl)
haftmann@32135
  1335
haftmann@32135
  1336
lemma Compl_Un [simp]: "-(A \<union> B) = (-A) \<inter> (-B)"
huffman@36009
  1337
  by (fact compl_sup)
haftmann@32135
  1338
haftmann@32135
  1339
lemma Compl_Int [simp]: "-(A \<inter> B) = (-A) \<union> (-B)"
huffman@36009
  1340
  by (fact compl_inf)
haftmann@32135
  1341
haftmann@32135
  1342
lemma subset_Compl_self_eq: "(A \<subseteq> -A) = (A = {})"
haftmann@32135
  1343
  by blast
haftmann@32135
  1344
haftmann@32135
  1345
lemma Un_Int_assoc_eq: "((A \<inter> B) \<union> C = A \<inter> (B \<union> C)) = (C \<subseteq> A)"
haftmann@32135
  1346
  -- {* Halmos, Naive Set Theory, page 16. *}
haftmann@32135
  1347
  by blast
haftmann@32135
  1348
haftmann@32135
  1349
lemma Compl_UNIV_eq [simp]: "-UNIV = {}"
huffman@36009
  1350
  by (fact compl_top_eq)
haftmann@32135
  1351
haftmann@32135
  1352
lemma Compl_empty_eq [simp]: "-{} = UNIV"
huffman@36009
  1353
  by (fact compl_bot_eq)
haftmann@32135
  1354
haftmann@32135
  1355
lemma Compl_subset_Compl_iff [iff]: "(-A \<subseteq> -B) = (B \<subseteq> A)"
huffman@36009
  1356
  by (fact compl_le_compl_iff)
haftmann@32135
  1357
haftmann@32135
  1358
lemma Compl_eq_Compl_iff [iff]: "(-A = -B) = (A = (B::'a set))"
huffman@36009
  1359
  by (fact compl_eq_compl_iff)
haftmann@32135
  1360
haftmann@32135
  1361
text {* \medskip Bounded quantifiers.
haftmann@32135
  1362
haftmann@32135
  1363
  The following are not added to the default simpset because
haftmann@32135
  1364
  (a) they duplicate the body and (b) there are no similar rules for @{text Int}. *}
haftmann@32135
  1365
haftmann@32135
  1366
lemma ball_Un: "(\<forall>x \<in> A \<union> B. P x) = ((\<forall>x\<in>A. P x) & (\<forall>x\<in>B. P x))"
haftmann@32135
  1367
  by blast
haftmann@32135
  1368
haftmann@32135
  1369
lemma bex_Un: "(\<exists>x \<in> A \<union> B. P x) = ((\<exists>x\<in>A. P x) | (\<exists>x\<in>B. P x))"
haftmann@32135
  1370
  by blast
haftmann@32135
  1371
haftmann@32135
  1372
haftmann@32135
  1373
text {* \medskip Set difference. *}
haftmann@32135
  1374
haftmann@32135
  1375
lemma Diff_eq: "A - B = A \<inter> (-B)"
haftmann@32135
  1376
  by blast
haftmann@32135
  1377
blanchet@35828
  1378
lemma Diff_eq_empty_iff [simp,no_atp]: "(A - B = {}) = (A \<subseteq> B)"
haftmann@32135
  1379
  by blast
haftmann@32135
  1380
haftmann@32135
  1381
lemma Diff_cancel [simp]: "A - A = {}"
haftmann@32135
  1382
  by blast
haftmann@32135
  1383
haftmann@32135
  1384
lemma Diff_idemp [simp]: "(A - B) - B = A - (B::'a set)"
haftmann@32135
  1385
by blast
haftmann@32135
  1386
haftmann@32135
  1387
lemma Diff_triv: "A \<inter> B = {} ==> A - B = A"
haftmann@32135
  1388
  by (blast elim: equalityE)
haftmann@32135
  1389
haftmann@32135
  1390
lemma empty_Diff [simp]: "{} - A = {}"
haftmann@32135
  1391
  by blast
haftmann@32135
  1392
haftmann@32135
  1393
lemma Diff_empty [simp]: "A - {} = A"
haftmann@32135
  1394
  by blast
haftmann@32135
  1395
haftmann@32135
  1396
lemma Diff_UNIV [simp]: "A - UNIV = {}"
haftmann@32135
  1397
  by blast
haftmann@32135
  1398
blanchet@35828
  1399
lemma Diff_insert0 [simp,no_atp]: "x \<notin> A ==> A - insert x B = A - B"
haftmann@32135
  1400
  by blast
haftmann@32135
  1401
haftmann@32135
  1402
lemma Diff_insert: "A - insert a B = A - B - {a}"
haftmann@32135
  1403
  -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a 0"} *}
haftmann@32135
  1404
  by blast
haftmann@32135
  1405
haftmann@32135
  1406
lemma Diff_insert2: "A - insert a B = A - {a} - B"
haftmann@32135
  1407
  -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a 0"} *}
haftmann@32135
  1408
  by blast
haftmann@32135
  1409
haftmann@32135
  1410
lemma insert_Diff_if: "insert x A - B = (if x \<in> B then A - B else insert x (A - B))"
haftmann@32135
  1411
  by auto
haftmann@32135
  1412
haftmann@32135
  1413
lemma insert_Diff1 [simp]: "x \<in> B ==> insert x A - B = A - B"
haftmann@32135
  1414
  by blast
haftmann@32135
  1415
haftmann@32135
  1416
lemma insert_Diff_single[simp]: "insert a (A - {a}) = insert a A"
haftmann@32135
  1417
by blast
haftmann@32135
  1418
haftmann@32135
  1419
lemma insert_Diff: "a \<in> A ==> insert a (A - {a}) = A"
haftmann@32135
  1420
  by blast
haftmann@32135
  1421
haftmann@32135
  1422
lemma Diff_insert_absorb: "x \<notin> A ==> (insert x A) - {x} = A"
haftmann@32135
  1423
  by auto
haftmann@32135
  1424
haftmann@32135
  1425
lemma Diff_disjoint [simp]: "A \<inter> (B - A) = {}"
haftmann@32135
  1426
  by blast
haftmann@32135
  1427
haftmann@32135
  1428
lemma Diff_partition: "A \<subseteq> B ==> A \<union> (B - A) = B"
haftmann@32135
  1429
  by blast
haftmann@32135
  1430
haftmann@32135
  1431
lemma double_diff: "A \<subseteq> B ==> B \<subseteq> C ==> B - (C - A) = A"
haftmann@32135
  1432
  by blast
haftmann@32135
  1433
haftmann@32135
  1434
lemma Un_Diff_cancel [simp]: "A \<union> (B - A) = A \<union> B"
haftmann@32135
  1435
  by blast
haftmann@32135
  1436
haftmann@32135
  1437
lemma Un_Diff_cancel2 [simp]: "(B - A) \<union> A = B \<union> A"
haftmann@32135
  1438
  by blast
haftmann@32135
  1439
haftmann@32135
  1440
lemma Diff_Un: "A - (B \<union> C) = (A - B) \<inter> (A - C)"
haftmann@32135
  1441
  by blast
haftmann@32135
  1442
haftmann@32135
  1443
lemma Diff_Int: "A - (B \<inter> C) = (A - B) \<union> (A - C)"
haftmann@32135
  1444
  by blast
haftmann@32135
  1445
haftmann@32135
  1446
lemma Un_Diff: "(A \<union> B) - C = (A - C) \<union> (B - C)"
haftmann@32135
  1447
  by blast
haftmann@32135
  1448
haftmann@32135
  1449
lemma Int_Diff: "(A \<inter> B) - C = A \<inter> (B - C)"
haftmann@32135
  1450
  by blast
haftmann@32135
  1451
haftmann@32135
  1452
lemma Diff_Int_distrib: "C \<inter> (A - B) = (C \<inter> A) - (C \<inter> B)"
haftmann@32135
  1453
  by blast
haftmann@32135
  1454
haftmann@32135
  1455
lemma Diff_Int_distrib2: "(A - B) \<inter> C = (A \<inter> C) - (B \<inter> C)"
haftmann@32135
  1456
  by blast
haftmann@32135
  1457
haftmann@32135
  1458
lemma Diff_Compl [simp]: "A - (- B) = A \<inter> B"
haftmann@32135
  1459
  by auto
haftmann@32135
  1460
haftmann@32135
  1461
lemma Compl_Diff_eq [simp]: "- (A - B) = -A \<union> B"
haftmann@32135
  1462
  by blast
haftmann@32135
  1463
haftmann@32135
  1464
haftmann@32135
  1465
text {* \medskip Quantification over type @{typ bool}. *}
haftmann@32135
  1466
haftmann@32135
  1467
lemma bool_induct: "P True \<Longrightarrow> P False \<Longrightarrow> P x"
haftmann@32135
  1468
  by (cases x) auto
haftmann@32135
  1469
haftmann@32135
  1470
lemma all_bool_eq: "(\<forall>b. P b) \<longleftrightarrow> P True \<and> P False"
haftmann@32135
  1471
  by (auto intro: bool_induct)
haftmann@32135
  1472
haftmann@32135
  1473
lemma bool_contrapos: "P x \<Longrightarrow> \<not> P False \<Longrightarrow> P True"
haftmann@32135
  1474
  by (cases x) auto
haftmann@32135
  1475
haftmann@32135
  1476
lemma ex_bool_eq: "(\<exists>b. P b) \<longleftrightarrow> P True \<or> P False"
haftmann@32135
  1477
  by (auto intro: bool_contrapos)
haftmann@32135
  1478
haftmann@32135
  1479
text {* \medskip @{text Pow} *}
haftmann@32135
  1480
haftmann@32135
  1481
lemma Pow_empty [simp]: "Pow {} = {{}}"
haftmann@32135
  1482
  by (auto simp add: Pow_def)
haftmann@32135
  1483
haftmann@32135
  1484
lemma Pow_insert: "Pow (insert a A) = Pow A \<union> (insert a ` Pow A)"
haftmann@32135
  1485
  by (blast intro: image_eqI [where ?x = "u - {a}", standard])
haftmann@32135
  1486
haftmann@32135
  1487
lemma Pow_Compl: "Pow (- A) = {-B | B. A \<in> Pow B}"
haftmann@32135
  1488
  by (blast intro: exI [where ?x = "- u", standard])
haftmann@32135
  1489
haftmann@32135
  1490
lemma Pow_UNIV [simp]: "Pow UNIV = UNIV"
haftmann@32135
  1491
  by blast
haftmann@32135
  1492
haftmann@32135
  1493
lemma Un_Pow_subset: "Pow A \<union> Pow B \<subseteq> Pow (A \<union> B)"
haftmann@32135
  1494
  by blast
haftmann@32135
  1495
haftmann@32135
  1496
lemma Pow_Int_eq [simp]: "Pow (A \<inter> B) = Pow A \<inter> Pow B"
haftmann@32135
  1497
  by blast
haftmann@32135
  1498
haftmann@32135
  1499
haftmann@32135
  1500
text {* \medskip Miscellany. *}
haftmann@32135
  1501
haftmann@32135
  1502
lemma set_eq_subset: "(A = B) = (A \<subseteq> B & B \<subseteq> A)"
haftmann@32135
  1503
  by blast
haftmann@32135
  1504
haftmann@32135
  1505
lemma subset_iff: "(A \<subseteq> B) = (\<forall>t. t \<in> A --> t \<in> B)"
haftmann@32135
  1506
  by blast
haftmann@32135
  1507
haftmann@32135
  1508
lemma subset_iff_psubset_eq: "(A \<subseteq> B) = ((A \<subset> B) | (A = B))"
haftmann@32135
  1509
  by (unfold less_le) blast
haftmann@32135
  1510
haftmann@32135
  1511
lemma all_not_in_conv [simp]: "(\<forall>x. x \<notin> A) = (A = {})"
haftmann@32135
  1512
  by blast
haftmann@32135
  1513
haftmann@32135
  1514
lemma ex_in_conv: "(\<exists>x. x \<in> A) = (A \<noteq> {})"
haftmann@32135
  1515
  by blast
haftmann@32135
  1516
haftmann@32135
  1517
lemma distinct_lemma: "f x \<noteq> f y ==> x \<noteq> y"
haftmann@32135
  1518
  by iprover
haftmann@32135
  1519
haftmann@32135
  1520
haftmann@32135
  1521
subsubsection {* Monotonicity of various operations *}
haftmann@32135
  1522
haftmann@32135
  1523
lemma image_mono: "A \<subseteq> B ==> f`A \<subseteq> f`B"
haftmann@32135
  1524
  by blast
haftmann@32135
  1525
haftmann@32135
  1526
lemma Pow_mono: "A \<subseteq> B ==> Pow A \<subseteq> Pow B"
haftmann@32135
  1527
  by blast
haftmann@32135
  1528
haftmann@32135
  1529
lemma insert_mono: "C \<subseteq> D ==> insert a C \<subseteq> insert a D"
haftmann@32135
  1530
  by blast
haftmann@32135
  1531
haftmann@32135
  1532
lemma Un_mono: "A \<subseteq> C ==> B \<subseteq> D ==> A \<union> B \<subseteq> C \<union> D"
huffman@36009
  1533
  by (fact sup_mono)
haftmann@32135
  1534
haftmann@32135
  1535
lemma Int_mono: "A \<subseteq> C ==> B \<subseteq> D ==> A \<inter> B \<subseteq> C \<inter> D"
huffman@36009
  1536
  by (fact inf_mono)
haftmann@32135
  1537
haftmann@32135
  1538
lemma Diff_mono: "A \<subseteq> C ==> D \<subseteq> B ==> A - B \<subseteq> C - D"
haftmann@32135
  1539
  by blast
haftmann@32135
  1540
haftmann@32135
  1541
lemma Compl_anti_mono: "A \<subseteq> B ==> -B \<subseteq> -A"
huffman@36009
  1542
  by (fact compl_mono)
haftmann@32135
  1543
haftmann@32135
  1544
text {* \medskip Monotonicity of implications. *}
haftmann@32135
  1545
haftmann@32135
  1546
lemma in_mono: "A \<subseteq> B ==> x \<in> A --> x \<in> B"
haftmann@32135
  1547
  apply (rule impI)
haftmann@32135
  1548
  apply (erule subsetD, assumption)
haftmann@32135
  1549
  done
haftmann@32135
  1550
haftmann@32135
  1551
lemma conj_mono: "P1 --> Q1 ==> P2 --> Q2 ==> (P1 & P2) --> (Q1 & Q2)"
haftmann@32135
  1552
  by iprover
haftmann@32135
  1553
haftmann@32135
  1554
lemma disj_mono: "P1 --> Q1 ==> P2 --> Q2 ==> (P1 | P2) --> (Q1 | Q2)"
haftmann@32135
  1555
  by iprover
haftmann@32135
  1556
haftmann@32135
  1557
lemma imp_mono: "Q1 --> P1 ==> P2 --> Q2 ==> (P1 --> P2) --> (Q1 --> Q2)"
haftmann@32135
  1558
  by iprover
haftmann@32135
  1559
haftmann@32135
  1560
lemma imp_refl: "P --> P" ..
haftmann@32135
  1561
berghofe@33935
  1562
lemma not_mono: "Q --> P ==> ~ P --> ~ Q"
berghofe@33935
  1563
  by iprover
berghofe@33935
  1564
haftmann@32135
  1565
lemma ex_mono: "(!!x. P x --> Q x) ==> (EX x. P x) --> (EX x. Q x)"
haftmann@32135
  1566
  by iprover
haftmann@32135
  1567
haftmann@32135
  1568
lemma all_mono: "(!!x. P x --> Q x) ==> (ALL x. P x) --> (ALL x. Q x)"
haftmann@32135
  1569
  by iprover
haftmann@32135
  1570
haftmann@32135
  1571
lemma Collect_mono: "(!!x. P x --> Q x) ==> Collect P \<subseteq> Collect Q"
haftmann@32135
  1572
  by blast
haftmann@32135
  1573
haftmann@32135
  1574
lemma Int_Collect_mono:
haftmann@32135
  1575
    "A \<subseteq> B ==> (!!x. x \<in> A ==> P x --> Q x) ==> A \<inter> Collect P \<subseteq> B \<inter> Collect Q"
haftmann@32135
  1576
  by blast
haftmann@32135
  1577
haftmann@32135
  1578
lemmas basic_monos =
haftmann@32135
  1579
  subset_refl imp_refl disj_mono conj_mono
haftmann@32135
  1580
  ex_mono Collect_mono in_mono
haftmann@32135
  1581
haftmann@32135
  1582
lemma eq_to_mono: "a = b ==> c = d ==> b --> d ==> a --> c"
haftmann@32135
  1583
  by iprover
haftmann@32135
  1584
haftmann@32135
  1585
haftmann@32135
  1586
subsubsection {* Inverse image of a function *}
haftmann@32135
  1587
haftmann@35416
  1588
definition vimage :: "('a => 'b) => 'b set => 'a set" (infixr "-`" 90) where
haftmann@32135
  1589
  [code del]: "f -` B == {x. f x : B}"
haftmann@32135
  1590
haftmann@32135
  1591
lemma vimage_eq [simp]: "(a : f -` B) = (f a : B)"
haftmann@32135
  1592
  by (unfold vimage_def) blast
haftmann@32135
  1593
haftmann@32135
  1594
lemma vimage_singleton_eq: "(a : f -` {b}) = (f a = b)"
haftmann@32135
  1595
  by simp
haftmann@32135
  1596
haftmann@32135
  1597
lemma vimageI [intro]: "f a = b ==> b:B ==> a : f -` B"
haftmann@32135
  1598
  by (unfold vimage_def) blast
haftmann@32135
  1599
haftmann@32135
  1600
lemma vimageI2: "f a : A ==> a : f -` A"
haftmann@32135
  1601
  by (unfold vimage_def) fast
haftmann@32135
  1602
haftmann@32135
  1603
lemma vimageE [elim!]: "a: f -` B ==> (!!x. f a = x ==> x:B ==> P) ==> P"
haftmann@32135
  1604
  by (unfold vimage_def) blast
haftmann@32135
  1605
haftmann@32135
  1606
lemma vimageD: "a : f -` A ==> f a : A"
haftmann@32135
  1607
  by (unfold vimage_def) fast
haftmann@32135
  1608
haftmann@32135
  1609
lemma vimage_empty [simp]: "f -` {} = {}"
haftmann@32135
  1610
  by blast
haftmann@32135
  1611
haftmann@32135
  1612
lemma vimage_Compl: "f -` (-A) = -(f -` A)"
haftmann@32135
  1613
  by blast
haftmann@32135
  1614
haftmann@32135
  1615
lemma vimage_Un [simp]: "f -` (A Un B) = (f -` A) Un (f -` B)"
haftmann@32135
  1616
  by blast
haftmann@32135
  1617
haftmann@32135
  1618
lemma vimage_Int [simp]: "f -` (A Int B) = (f -` A) Int (f -` B)"
haftmann@32135
  1619
  by fast
haftmann@32135
  1620
haftmann@32135
  1621
lemma vimage_Collect_eq [simp]: "f -` Collect P = {y. P (f y)}"
haftmann@32135
  1622
  by blast
haftmann@32135
  1623
haftmann@32135
  1624
lemma vimage_Collect: "(!!x. P (f x) = Q x) ==> f -` (Collect P) = Collect Q"
haftmann@32135
  1625
  by blast
haftmann@32135
  1626
haftmann@32135
  1627
lemma vimage_insert: "f-`(insert a B) = (f-`{a}) Un (f-`B)"
haftmann@32135
  1628
  -- {* NOT suitable for rewriting because of the recurrence of @{term "{a}"}. *}
haftmann@32135
  1629
  by blast
haftmann@32135
  1630
haftmann@32135
  1631
lemma vimage_Diff: "f -` (A - B) = (f -` A) - (f -` B)"
haftmann@32135
  1632
  by blast
haftmann@32135
  1633
haftmann@32135
  1634
lemma vimage_UNIV [simp]: "f -` UNIV = UNIV"
haftmann@32135
  1635
  by blast
haftmann@32135
  1636
haftmann@32135
  1637
lemma vimage_mono: "A \<subseteq> B ==> f -` A \<subseteq> f -` B"
haftmann@32135
  1638
  -- {* monotonicity *}
haftmann@32135
  1639
  by blast
haftmann@32135
  1640
blanchet@35828
  1641
lemma vimage_image_eq [no_atp]: "f -` (f ` A) = {y. EX x:A. f x = f y}"
haftmann@32135
  1642
by (blast intro: sym)
haftmann@32135
  1643
haftmann@32135
  1644
lemma image_vimage_subset: "f ` (f -` A) <= A"
haftmann@32135
  1645
by blast
haftmann@32135
  1646
haftmann@32135
  1647
lemma image_vimage_eq [simp]: "f ` (f -` A) = A Int range f"
haftmann@32135
  1648
by blast
haftmann@32135
  1649
paulson@33533
  1650
lemma vimage_const [simp]: "((\<lambda>x. c) -` A) = (if c \<in> A then UNIV else {})"
paulson@33533
  1651
  by auto
paulson@33533
  1652
paulson@33533
  1653
lemma vimage_if [simp]: "((\<lambda>x. if x \<in> B then c else d) -` A) = 
paulson@33533
  1654
   (if c \<in> A then (if d \<in> A then UNIV else B)
paulson@33533
  1655
    else if d \<in> A then -B else {})"  
paulson@33533
  1656
  by (auto simp add: vimage_def) 
paulson@33533
  1657
hoelzl@35576
  1658
lemma vimage_inter_cong:
hoelzl@35576
  1659
  "(\<And> w. w \<in> S \<Longrightarrow> f w = g w) \<Longrightarrow> f -` y \<inter> S = g -` y \<inter> S"
hoelzl@35576
  1660
  by auto
hoelzl@35576
  1661
haftmann@32135
  1662
lemma image_Int_subset: "f`(A Int B) <= f`A Int f`B"
haftmann@32135
  1663
by blast
haftmann@32135
  1664
haftmann@32135
  1665
lemma image_diff_subset: "f`A - f`B <= f`(A - B)"
haftmann@32135
  1666
by blast
haftmann@32135
  1667
haftmann@32135
  1668
haftmann@32135
  1669
subsubsection {* Getting the Contents of a Singleton Set *}
haftmann@32135
  1670
haftmann@32135
  1671
definition contents :: "'a set \<Rightarrow> 'a" where
haftmann@32135
  1672
  [code del]: "contents X = (THE x. X = {x})"
haftmann@32135
  1673
haftmann@32135
  1674
lemma contents_eq [simp]: "contents {x} = x"
haftmann@32135
  1675
  by (simp add: contents_def)
haftmann@32135
  1676
haftmann@32135
  1677
haftmann@32135
  1678
subsubsection {* Least value operator *}
haftmann@32135
  1679
haftmann@32135
  1680
lemma Least_mono:
haftmann@32135
  1681
  "mono (f::'a::order => 'b::order) ==> EX x:S. ALL y:S. x <= y
haftmann@32135
  1682
    ==> (LEAST y. y : f ` S) = f (LEAST x. x : S)"
haftmann@32135
  1683
    -- {* Courtesy of Stephan Merz *}
haftmann@32135
  1684
  apply clarify
haftmann@32135
  1685
  apply (erule_tac P = "%x. x : S" in LeastI2_order, fast)
haftmann@32135
  1686
  apply (rule LeastI2_order)
haftmann@32135
  1687
  apply (auto elim: monoD intro!: order_antisym)
haftmann@32135
  1688
  done
haftmann@32135
  1689
haftmann@32135
  1690
subsection {* Misc *}
haftmann@32135
  1691
haftmann@32135
  1692
text {* Rudimentary code generation *}
haftmann@32135
  1693
haftmann@32135
  1694
lemma insert_code [code]: "insert y A x \<longleftrightarrow> y = x \<or> A x"
haftmann@32135
  1695
  by (auto simp add: insert_compr Collect_def mem_def)
haftmann@32135
  1696
haftmann@32135
  1697
lemma vimage_code [code]: "(f -` A) x = A (f x)"
haftmann@32135
  1698
  by (simp add: vimage_def Collect_def mem_def)
haftmann@32135
  1699
haftmann@32135
  1700
haftmann@32135
  1701
text {* Misc theorem and ML bindings *}
haftmann@32135
  1702
haftmann@32135
  1703
lemmas equalityI = subset_antisym
haftmann@32135
  1704
haftmann@32135
  1705
ML {*
haftmann@32135
  1706
val Ball_def = @{thm Ball_def}
haftmann@32135
  1707
val Bex_def = @{thm Bex_def}
haftmann@32135
  1708
val CollectD = @{thm CollectD}
haftmann@32135
  1709
val CollectE = @{thm CollectE}
haftmann@32135
  1710
val CollectI = @{thm CollectI}
haftmann@32135
  1711
val Collect_conj_eq = @{thm Collect_conj_eq}
haftmann@32135
  1712
val Collect_mem_eq = @{thm Collect_mem_eq}
haftmann@32135
  1713
val IntD1 = @{thm IntD1}
haftmann@32135
  1714
val IntD2 = @{thm IntD2}
haftmann@32135
  1715
val IntE = @{thm IntE}
haftmann@32135
  1716
val IntI = @{thm IntI}
haftmann@32135
  1717
val Int_Collect = @{thm Int_Collect}
haftmann@32135
  1718
val UNIV_I = @{thm UNIV_I}
haftmann@32135
  1719
val UNIV_witness = @{thm UNIV_witness}
haftmann@32135
  1720
val UnE = @{thm UnE}
haftmann@32135
  1721
val UnI1 = @{thm UnI1}
haftmann@32135
  1722
val UnI2 = @{thm UnI2}
haftmann@32135
  1723
val ballE = @{thm ballE}
haftmann@32135
  1724
val ballI = @{thm ballI}
haftmann@32135
  1725
val bexCI = @{thm bexCI}
haftmann@32135
  1726
val bexE = @{thm bexE}
haftmann@32135
  1727
val bexI = @{thm bexI}
haftmann@32135
  1728
val bex_triv = @{thm bex_triv}
haftmann@32135
  1729
val bspec = @{thm bspec}
haftmann@32135
  1730
val contra_subsetD = @{thm contra_subsetD}
haftmann@32135
  1731
val distinct_lemma = @{thm distinct_lemma}
haftmann@32135
  1732
val eq_to_mono = @{thm eq_to_mono}
haftmann@32135
  1733
val equalityCE = @{thm equalityCE}
haftmann@32135
  1734
val equalityD1 = @{thm equalityD1}
haftmann@32135
  1735
val equalityD2 = @{thm equalityD2}
haftmann@32135
  1736
val equalityE = @{thm equalityE}
haftmann@32135
  1737
val equalityI = @{thm equalityI}
haftmann@32135
  1738
val imageE = @{thm imageE}
haftmann@32135
  1739
val imageI = @{thm imageI}
haftmann@32135
  1740
val image_Un = @{thm image_Un}
haftmann@32135
  1741
val image_insert = @{thm image_insert}
haftmann@32135
  1742
val insert_commute = @{thm insert_commute}
haftmann@32135
  1743
val insert_iff = @{thm insert_iff}
haftmann@32135
  1744
val mem_Collect_eq = @{thm mem_Collect_eq}
haftmann@32135
  1745
val rangeE = @{thm rangeE}
haftmann@32135
  1746
val rangeI = @{thm rangeI}
haftmann@32135
  1747
val range_eqI = @{thm range_eqI}
haftmann@32135
  1748
val subsetCE = @{thm subsetCE}
haftmann@32135
  1749
val subsetD = @{thm subsetD}
haftmann@32135
  1750
val subsetI = @{thm subsetI}
haftmann@32135
  1751
val subset_refl = @{thm subset_refl}
haftmann@32135
  1752
val subset_trans = @{thm subset_trans}
haftmann@32135
  1753
val vimageD = @{thm vimageD}
haftmann@32135
  1754
val vimageE = @{thm vimageE}
haftmann@32135
  1755
val vimageI = @{thm vimageI}
haftmann@32135
  1756
val vimageI2 = @{thm vimageI2}
haftmann@32135
  1757
val vimage_Collect = @{thm vimage_Collect}
haftmann@32135
  1758
val vimage_Int = @{thm vimage_Int}
haftmann@32135
  1759
val vimage_Un = @{thm vimage_Un}
haftmann@32135
  1760
*}
haftmann@32135
  1761
haftmann@32077
  1762
end