src/HOL/Set.thy
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restructured theory Set.thy
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(*  Title:      HOL/Set.thy
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    Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
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*)
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header {* Set theory for higher-order logic *}
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theory Set
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imports Lattices
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begin
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subsection {* Basic operations *}
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subsubsection {* Comprehension and membership *}
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text {* A set in HOL is simply a predicate. *}
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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 \<Rightarrow> bool) \<Rightarrow> 'a set"
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  "op :" :: "'a => 'a set => bool"
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local
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syntax
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  "@Coll"       :: "pttrn => bool => 'a set"              ("(1{_./ _})")
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translations
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  "{x. P}"      == "Collect (%x. P)"
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notation
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  "op :"  ("op :") and
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  "op :"  ("(_/ : _)" [50, 51] 50)
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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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defs
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  Collect_def [code]: "Collect P \<equiv> P"
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  mem_def [code]: "x \<in> S \<equiv> S x"
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text {* Relating predicates and sets *}
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lemma mem_Collect_eq [iff]: "(a : {x. P(x)}) = P(a)"
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  by (simp add: Collect_def mem_def)
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lemma Collect_mem_eq [simp]: "{x. x:A} = A"
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  by (simp add: Collect_def mem_def)
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lemma CollectI: "P(a) ==> a : {x. P(x)}"
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  by simp
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lemma CollectD: "a : {x. P(x)} ==> P(a)"
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  by simp
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lemma Collect_cong: "(!!x. P x = Q x) ==> {x. P(x)} = {x. Q(x)}"
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  by simp
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lemmas CollectE = CollectD [elim_format]
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lemma set_ext: assumes prem: "(!!x. (x:A) = (x:B))" shows "A = B"
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  apply (rule prem [THEN ext, THEN arg_cong, THEN box_equals])
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   apply (rule Collect_mem_eq)
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  apply (rule Collect_mem_eq)
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  done
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(* Due to Brian Huffman *)
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lemma expand_set_eq: "(A = B) = (ALL x. (x:A) = (x:B))"
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by(auto intro:set_ext)
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lemma equalityCE [elim]:
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    "A = B ==> (c \<in> A ==> c \<in> B ==> P) ==> (c \<notin> A ==> c \<notin> B ==> P) ==> P"
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  by blast
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lemma eqset_imp_iff: "A = B ==> (x : A) = (x : B)"
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  by simp
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lemma eqelem_imp_iff: "x = y ==> (x : A) = (y : A)"
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  by simp
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subsubsection {* Subset relation, empty and universal set *}
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abbreviation
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  subset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
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  "subset \<equiv> less"
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abbreviation
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  subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
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  "subset_eq \<equiv> less_eq"
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notation (output)
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  subset  ("op <") and
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  subset  ("(_/ < _)" [50, 51] 50) and
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  subset_eq  ("op <=") and
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  subset_eq  ("(_/ <= _)" [50, 51] 50)
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notation (xsymbols)
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  subset  ("op \<subset>") and
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  subset  ("(_/ \<subset> _)" [50, 51] 50) and
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  subset_eq  ("op \<subseteq>") and
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  subset_eq  ("(_/ \<subseteq> _)" [50, 51] 50)
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notation (HTML output)
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  subset  ("op \<subset>") and
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  subset  ("(_/ \<subset> _)" [50, 51] 50) and
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  subset_eq  ("op \<subseteq>") and
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  subset_eq  ("(_/ \<subseteq> _)" [50, 51] 50)
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abbreviation (input)
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  supset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
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  "supset \<equiv> greater"
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abbreviation (input)
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  supset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
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  "supset_eq \<equiv> greater_eq"
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notation (xsymbols)
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  supset  ("op \<supset>") and
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  supset  ("(_/ \<supset> _)" [50, 51] 50) and
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  supset_eq  ("op \<supseteq>") and
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  supset_eq  ("(_/ \<supseteq> _)" [50, 51] 50)
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definition empty :: "'a set" ("{}") where
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  "empty \<equiv> {x. False}"
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definition UNIV :: "'a set" where
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  "UNIV \<equiv> {x. True}"
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lemma subsetI [atp,intro!]: "(!!x. x:A ==> x:B) ==> A \<subseteq> B"
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  by (auto simp add: mem_def intro: predicate1I)
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text {*
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  \medskip Map the type @{text "'a set => anything"} to just @{typ
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  'a}; for overloading constants whose first argument has type @{typ
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  "'a set"}.
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*}
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lemma subsetD [elim]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B"
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  -- {* Rule in Modus Ponens style. *}
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  by (unfold mem_def) blast
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declare subsetD [intro?] -- FIXME
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lemma rev_subsetD: "c \<in> A ==> A \<subseteq> B ==> c \<in> B"
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  -- {* The same, with reversed premises for use with @{text erule} --
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      cf @{text rev_mp}. *}
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  by (rule subsetD)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   167
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   168
declare rev_subsetD [intro?] -- FIXME
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   169
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   170
text {*
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   171
  \medskip Converts @{prop "A \<subseteq> B"} to @{prop "x \<in> A ==> x \<in> B"}.
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   172
*}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   173
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   174
ML {*
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   175
  fun impOfSubs th = th RSN (2, @{thm rev_subsetD})
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   176
*}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   177
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   178
lemma subsetCE [elim]: "A \<subseteq>  B ==> (c \<notin> A ==> P) ==> (c \<in> B ==> P) ==> P"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   179
  -- {* Classical elimination rule. *}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   180
  by (unfold mem_def) blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   181
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   182
text {*
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   183
  \medskip Takes assumptions @{prop "A \<subseteq> B"}; @{prop "c \<in> A"} and
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   184
  creates the assumption @{prop "c \<in> B"}.
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   185
*}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   186
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   187
ML {*
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   188
  fun set_mp_tac i = etac @{thm subsetCE} i THEN mp_tac i
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   189
*}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   190
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   191
lemma contra_subsetD: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   192
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   193
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   194
lemma subset_refl [simp,atp]: "A \<subseteq> A"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   195
  by fast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   196
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   197
lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   198
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   199
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   200
lemma subset_antisym [intro!]: "A \<subseteq> B ==> B \<subseteq> A ==> A = B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   201
  -- {* Anti-symmetry of the subset relation. *}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   202
  by (iprover intro: set_ext subsetD)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   203
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   204
text {*
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   205
  \medskip Equality rules from ZF set theory -- are they appropriate
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   206
  here?
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   207
*}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   208
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   209
lemma equalityD1: "A = B ==> A \<subseteq> B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   210
  by (simp add: subset_refl)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   211
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   212
lemma equalityD2: "A = B ==> B \<subseteq> A"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   213
  by (simp add: subset_refl)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   214
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   215
text {*
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   216
  \medskip Be careful when adding this to the claset as @{text
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   217
  subset_empty} is in the simpset: @{prop "A = {}"} goes to @{prop "{}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   218
  \<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}!
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   219
*}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   220
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   221
lemma equalityE: "A = B ==> (A \<subseteq> B ==> B \<subseteq> A ==> P) ==> P"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   222
  by (simp add: subset_refl)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   223
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   224
lemma empty_iff [simp]: "(c : {}) = False"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   225
  by (simp add: empty_def)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   226
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   227
lemma emptyE [elim!]: "a : {} ==> P"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   228
  by simp
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   229
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   230
lemma empty_subsetI [iff]: "{} \<subseteq> A"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   231
    -- {* One effect is to delete the ASSUMPTION @{prop "{} \<subseteq> A"} *}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   232
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   233
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   234
lemma bot_set_eq: "bot = {}"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   235
  by (iprover intro!: subset_antisym empty_subsetI bot_least)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   236
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   237
lemma equals0I: "(!!y. y \<in> A ==> False) ==> A = {}"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   238
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   239
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   240
lemma equals0D: "A = {} ==> a \<notin> A"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   241
    -- {* Use for reasoning about disjointness *}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   242
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   243
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   244
lemma UNIV_I [simp]: "x : UNIV"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   245
  by (simp add: UNIV_def)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   246
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   247
declare UNIV_I [intro]  -- {* unsafe makes it less likely to cause problems *}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   248
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   249
lemma UNIV_witness [intro?]: "EX x. x : UNIV"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   250
  by simp
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   251
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   252
lemma subset_UNIV [simp]: "A \<subseteq> UNIV"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   253
  by (rule subsetI) (rule UNIV_I)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   254
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   255
lemma top_set_eq: "top = UNIV"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   256
  by (iprover intro!: subset_antisym subset_UNIV top_greatest)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   257
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   258
lemma UNIV_eq_I: "(\<And>x. x \<in> A) \<Longrightarrow> UNIV = A"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   259
  by auto
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   260
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   261
lemma UNIV_not_empty [iff]: "UNIV ~= {}"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   262
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   263
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   264
lemma psubsetI [intro!,noatp]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   265
  by (unfold less_le) blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   266
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   267
lemma psubsetE [elim!,noatp]: 
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   268
    "[|A \<subset> B;  [|A \<subseteq> B; ~ (B\<subseteq>A)|] ==> R|] ==> R"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   269
  by (unfold less_le) blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   270
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   271
lemma psubset_eq: "(A \<subset> B) = (A \<subseteq> B & A \<noteq> B)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   272
  by (simp only: less_le)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   273
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   274
lemma psubset_imp_subset: "A \<subset> B ==> A \<subseteq> B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   275
  by (simp add: psubset_eq)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   276
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   277
lemma psubset_trans: "[| A \<subset> B; B \<subset> C |] ==> A \<subset> C"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   278
apply (unfold less_le)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   279
apply (auto dest: subset_antisym)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   280
done
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   281
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   282
lemma psubsetD: "[| A \<subset> B; c \<in> A |] ==> c \<in> B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   283
apply (unfold less_le)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   284
apply (auto dest: subsetD)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   285
done
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   286
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   287
lemma psubset_subset_trans: "A \<subset> B ==> B \<subseteq> C ==> A \<subset> C"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   288
  by (auto simp add: psubset_eq)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   289
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   290
lemma subset_psubset_trans: "A \<subseteq> B ==> B \<subset> C ==> A \<subset> C"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   291
  by (auto simp add: psubset_eq)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   292
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   293
lemma Collect_empty_eq [simp]: "(Collect P = {}) = (\<forall>x. \<not> P x)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   294
by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   295
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   296
lemma empty_Collect_eq [simp]: "({} = Collect P) = (\<forall>x. \<not> P x)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   297
by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   298
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   299
subsubsection {* Intersection and union *}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   300
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   301
definition Int :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Int" 70) where
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   302
  "A Int B \<equiv> {x. x \<in> A \<and> x \<in> B}"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   303
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   304
definition Un :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Un" 65) where
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   305
  "A Un B \<equiv> {x. x \<in> A \<or> x \<in> B}"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   306
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   307
notation (xsymbols)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   308
  "Int"  (infixl "\<inter>" 70) and
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   309
  "Un"  (infixl "\<union>" 65)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   310
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   311
notation (HTML output)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   312
  "Int"  (infixl "\<inter>" 70) and
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   313
  "Un"  (infixl "\<union>" 65)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   314
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   315
lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   316
  by (unfold Int_def) blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   317
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   318
lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   319
  by simp
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   320
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   321
lemma IntD1: "c : A Int B ==> c:A"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   322
  by simp
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   323
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   324
lemma IntD2: "c : A Int B ==> c:B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   325
  by simp
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   326
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   327
lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   328
  by simp
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   329
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   330
lemma Un_iff [simp]: "(c : A Un B) = (c:A | c:B)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   331
  by (unfold Un_def) blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   332
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   333
lemma UnI1 [elim?]: "c:A ==> c : A Un B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   334
  by simp
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   335
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   336
lemma UnI2 [elim?]: "c:B ==> c : A Un B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   337
  by simp
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   338
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   339
text {*
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   340
  \medskip Classical introduction rule: no commitment to @{prop A} vs
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   341
  @{prop B}.
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   342
*}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   343
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   344
lemma UnCI [intro!]: "(c~:B ==> c:A) ==> c : A Un B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   345
  by auto
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   346
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   347
lemma UnE [elim!]: "c : A Un B ==> (c:A ==> P) ==> (c:B ==> P) ==> P"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   348
  by (unfold Un_def) blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   349
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   350
lemma Int_lower1: "A \<inter> B \<subseteq> A"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   351
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   352
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   353
lemma Int_lower2: "A \<inter> B \<subseteq> B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   354
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   355
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   356
lemma Int_greatest: "C \<subseteq> A ==> C \<subseteq> B ==> C \<subseteq> A \<inter> B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   357
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   358
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   359
lemma inf_set_eq: "inf A B = A \<inter> B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   360
  apply (rule subset_antisym)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   361
  apply (rule Int_greatest)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   362
  apply (rule inf_le1)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   363
  apply (rule inf_le2)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   364
  apply (rule inf_greatest)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   365
  apply (rule Int_lower1)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   366
  apply (rule Int_lower2)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   367
  done
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   368
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   369
lemma Un_upper1: "A \<subseteq> A \<union> B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   370
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   371
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   372
lemma Un_upper2: "B \<subseteq> A \<union> B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   373
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   374
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   375
lemma Un_least: "A \<subseteq> C ==> B \<subseteq> C ==> A \<union> B \<subseteq> C"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   376
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   377
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   378
lemma sup_set_eq: "sup A B = A \<union> B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   379
  apply (rule subset_antisym)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   380
  apply (rule sup_least)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   381
  apply (rule Un_upper1)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   382
  apply (rule Un_upper2)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   383
  apply (rule Un_least)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   384
  apply (rule sup_ge1)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   385
  apply (rule sup_ge2)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   386
  done
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   387
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   388
lemma Collect_conj_eq: "{x. P x & Q x} = {x. P x} \<inter> {x. Q x}"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   389
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   390
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   391
lemma Collect_disj_eq: "{x. P x | Q x} = {x. P x} \<union> {x. Q x}"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   392
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   393
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   394
lemma subset_empty [simp]: "(A \<subseteq> {}) = (A = {})"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   395
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   396
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   397
lemma not_psubset_empty [iff]: "\<not> (A \<subset> {})"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   398
  by (unfold less_le) blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   399
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   400
lemma Collect_const [simp]: "{s. P} = (if P then UNIV else {})"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   401
  -- {* supersedes @{text "Collect_False_empty"} *}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   402
  by auto
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   403
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   404
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   405
subsubsection {* Complement and set difference *}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   406
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   407
instantiation bool :: minus
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   408
begin
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   409
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   410
definition
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   411
  bool_diff_def: "A - B \<longleftrightarrow> A \<and> \<not> B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   412
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   413
instance ..
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   414
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   415
end
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   416
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   417
instantiation "fun" :: (type, minus) minus
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   418
begin
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   419
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   420
definition
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   421
  fun_diff_def: "A - B = (\<lambda>x. A x - B x)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   422
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   423
instance ..
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   424
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   425
end
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   426
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   427
instantiation bool :: uminus
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   428
begin
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   429
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   430
definition
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   431
  bool_Compl_def: "- A \<longleftrightarrow> \<not> A"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   432
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   433
instance ..
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   434
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   435
end
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   436
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   437
instantiation "fun" :: (type, uminus) uminus
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   438
begin
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   439
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   440
definition
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   441
  fun_Compl_def: "- A = (\<lambda>x. - A x)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   442
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   443
instance ..
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   444
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   445
end
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   446
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   447
lemma Compl_iff [simp]: "(c \<in> -A) = (c \<notin> A)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   448
  by (simp add: mem_def fun_Compl_def bool_Compl_def)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   449
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   450
lemma ComplI [intro!]: "(c \<in> A ==> False) ==> c \<in> -A"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   451
  by (unfold mem_def fun_Compl_def bool_Compl_def) blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   452
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   453
text {*
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   454
  \medskip This form, with negated conclusion, works well with the
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   455
  Classical prover.  Negated assumptions behave like formulae on the
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   456
  right side of the notional turnstile ... *}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   457
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   458
lemma ComplD [dest!]: "c : -A ==> c~:A"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   459
  by (simp add: mem_def fun_Compl_def bool_Compl_def)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   460
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   461
lemmas ComplE = ComplD [elim_format]
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   462
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   463
lemma Compl_eq: "- A = {x. ~ x : A}" by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   464
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   465
lemma Diff_iff [simp]: "(c : A - B) = (c:A & c~:B)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   466
  by (simp add: mem_def fun_diff_def bool_diff_def)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   467
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   468
lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A - B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   469
  by simp
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   470
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   471
lemma DiffD1: "c : A - B ==> c : A"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   472
  by simp
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   473
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   474
lemma DiffD2: "c : A - B ==> c : B ==> P"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   475
  by simp
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   476
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   477
lemma DiffE [elim!]: "c : A - B ==> (c:A ==> c~:B ==> P) ==> P"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   478
  by simp
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   479
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   480
lemma set_diff_eq: "A - B = {x. x : A & ~ x : B}" by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   481
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   482
lemma Compl_eq_Diff_UNIV: "-A = (UNIV - A)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   483
by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   484
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   485
lemma psubset_imp_ex_mem: "A \<subset> B ==> \<exists>b. b \<in> (B - A)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   486
  by (unfold less_le) blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   487
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   488
lemma Diff_subset: "A - B \<subseteq> A"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   489
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   490
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   491
lemma Diff_subset_conv: "(A - B \<subseteq> C) = (A \<subseteq> B \<union> C)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   492
by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   493
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   494
lemma Collect_imp_eq: "{x. P x --> Q x} = -{x. P x} \<union> {x. Q x}"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   495
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   496
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   497
lemma Collect_neg_eq: "{x. \<not> P x} = - {x. P x}"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   498
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   499
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   500
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   501
subsubsection {* Set enumerations *}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   502
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   503
global
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   504
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   505
consts
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   506
  insert        :: "'a => 'a set => 'a set"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   507
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   508
local
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   509
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   510
defs
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   511
  insert_def:   "insert a B == {x. x=a} Un B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   512
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   513
syntax
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   514
  "@Finset"     :: "args => 'a set"                       ("{(_)}")
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   515
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   516
translations
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   517
  "{x, xs}"     == "insert x {xs}"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   518
  "{x}"         == "insert x {}"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   519
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   520
lemma insert_iff [simp]: "(a : insert b A) = (a = b | a:A)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   521
  by (unfold insert_def) blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   522
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   523
lemma insertI1: "a : insert a B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   524
  by simp
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   525
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   526
lemma insertI2: "a : B ==> a : insert b B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   527
  by simp
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   528
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   529
lemma insertE [elim!]: "a : insert b A ==> (a = b ==> P) ==> (a:A ==> P) ==> P"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   530
  by (unfold insert_def) blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   531
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   532
lemma insertCI [intro!]: "(a~:B ==> a = b) ==> a: insert b B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   533
  -- {* Classical introduction rule. *}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   534
  by auto
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   535
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   536
lemma subset_insert_iff: "(A \<subseteq> insert x B) = (if x:A then A - {x} \<subseteq> B else A \<subseteq> B)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   537
  by auto
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   538
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   539
lemma set_insert:
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   540
  assumes "x \<in> A"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   541
  obtains B where "A = insert x B" and "x \<notin> B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   542
proof
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   543
  from assms show "A = insert x (A - {x})" by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   544
next
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   545
  show "x \<notin> A - {x}" by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   546
qed
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   547
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   548
lemma insert_ident: "x ~: A ==> x ~: B ==> (insert x A = insert x B) = (A = B)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   549
by auto
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   550
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   551
lemma insert_is_Un: "insert a A = {a} Un A"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   552
  -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a {}"} *}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   553
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   554
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   555
lemma insert_not_empty [simp]: "insert a A \<noteq> {}"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   556
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   557
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   558
lemmas empty_not_insert = insert_not_empty [symmetric, standard]
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   559
declare empty_not_insert [simp]
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   560
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   561
lemma insert_absorb: "a \<in> A ==> insert a A = A"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   562
  -- {* @{text "[simp]"} causes recursive calls when there are nested inserts *}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   563
  -- {* with \emph{quadratic} running time *}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   564
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   565
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   566
lemma insert_absorb2 [simp]: "insert x (insert x A) = insert x A"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   567
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   568
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   569
lemma insert_commute: "insert x (insert y A) = insert y (insert x A)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   570
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   571
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   572
lemma insert_subset [simp]: "(insert x A \<subseteq> B) = (x \<in> B & A \<subseteq> B)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   573
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   574
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   575
lemma mk_disjoint_insert: "a \<in> A ==> \<exists>B. A = insert a B & a \<notin> B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   576
  -- {* use new @{text B} rather than @{text "A - {a}"} to avoid infinite unfolding *}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   577
  apply (rule_tac x = "A - {a}" in exI, blast)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   578
  done
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   579
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   580
lemma insert_Collect: "insert a (Collect P) = {u. u \<noteq> a --> P u}"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   581
  by auto
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   582
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   583
lemma insert_inter_insert[simp]: "insert a A \<inter> insert a B = insert a (A \<inter> B)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   584
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   585
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   586
lemma insert_disjoint [simp,noatp]:
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   587
 "(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   588
 "({} = insert a A \<inter> B) = (a \<notin> B \<and> {} = A \<inter> B)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   589
  by auto
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   590
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   591
lemma disjoint_insert [simp,noatp]:
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   592
 "(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   593
 "({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   594
  by auto
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   595
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   596
text {* Singletons, using insert *}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   597
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   598
lemma singletonI [intro!,noatp]: "a : {a}"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   599
    -- {* Redundant? But unlike @{text insertCI}, it proves the subgoal immediately! *}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   600
  by (rule insertI1)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   601
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   602
lemma singletonD [dest!,noatp]: "b : {a} ==> b = a"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   603
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   604
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   605
lemmas singletonE = singletonD [elim_format]
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   606
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   607
lemma singleton_iff: "(b : {a}) = (b = a)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   608
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   609
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   610
lemma singleton_inject [dest!]: "{a} = {b} ==> a = b"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   611
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   612
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   613
lemma singleton_insert_inj_eq [iff,noatp]:
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   614
     "({b} = insert a A) = (a = b & A \<subseteq> {b})"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   615
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   616
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   617
lemma singleton_insert_inj_eq' [iff,noatp]:
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   618
     "(insert a A = {b}) = (a = b & A \<subseteq> {b})"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   619
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   620
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   621
lemma subset_singletonD: "A \<subseteq> {x} ==> A = {} | A = {x}"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   622
  by fast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   623
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   624
lemma singleton_conv [simp]: "{x. x = a} = {a}"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   625
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   626
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   627
lemma singleton_conv2 [simp]: "{x. a = x} = {a}"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   628
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   629
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   630
lemma diff_single_insert: "A - {x} \<subseteq> B ==> x \<in> A ==> A \<subseteq> insert x B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   631
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   632
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   633
lemma doubleton_eq_iff: "({a,b} = {c,d}) = (a=c & b=d | a=d & b=c)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   634
  by (blast elim: equalityE)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   635
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   636
lemma psubset_insert_iff:
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   637
  "(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)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   638
  by (auto simp add: less_le subset_insert_iff)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   639
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   640
lemma subset_insertI: "B \<subseteq> insert a B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   641
  by (rule subsetI) (erule insertI2)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   642
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   643
lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   644
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   645
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   646
lemma subset_insert: "x \<notin> A ==> (A \<subseteq> insert x B) = (A \<subseteq> B)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   647
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   648
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   649
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   650
subsubsection {* Bounded quantifiers and operators *}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   651
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   652
global
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   653
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   654
consts
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   655
  Ball          :: "'a set => ('a => bool) => bool"      -- "bounded universal quantifiers"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   656
  Bex           :: "'a set => ('a => bool) => bool"      -- "bounded existential quantifiers"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   657
  Bex1          :: "'a set => ('a => bool) => bool"      -- "bounded unique existential quantifiers"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   658
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   659
local
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   660
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   661
defs
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   662
  Ball_def:     "Ball A P       == ALL x. x:A --> P(x)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   663
  Bex_def:      "Bex A P        == EX x. x:A & P(x)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   664
  Bex1_def:     "Bex1 A P       == EX! x. x:A & P(x)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   665
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   666
syntax
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   667
  "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3ALL _:_./ _)" [0, 0, 10] 10)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   668
  "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3EX _:_./ _)" [0, 0, 10] 10)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   669
  "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3EX! _:_./ _)" [0, 0, 10] 10)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   670
  "_Bleast"     :: "id => 'a set => bool => 'a"           ("(3LEAST _:_./ _)" [0, 0, 10] 10)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   671
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   672
syntax (HOL)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   673
  "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3! _:_./ _)" [0, 0, 10] 10)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   674
  "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3? _:_./ _)" [0, 0, 10] 10)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   675
  "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3?! _:_./ _)" [0, 0, 10] 10)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   676
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   677
syntax (xsymbols)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   678
  "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   679
  "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   680
  "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3\<exists>!_\<in>_./ _)" [0, 0, 10] 10)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   681
  "_Bleast"     :: "id => 'a set => bool => 'a"           ("(3LEAST_\<in>_./ _)" [0, 0, 10] 10)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   682
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   683
syntax (HTML output)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   684
  "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   685
  "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   686
  "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3\<exists>!_\<in>_./ _)" [0, 0, 10] 10)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   687
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   688
translations
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   689
  "ALL x:A. P"  == "Ball A (%x. P)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   690
  "EX x:A. P"   == "Bex A (%x. P)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   691
  "EX! x:A. P"  == "Bex1 A (%x. P)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   692
  "LEAST x:A. P" => "LEAST x. x:A & P"
14804
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   693
19656
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
   694
syntax (output)
14804
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   695
  "_setlessAll" :: "[idt, 'a, bool] => bool"  ("(3ALL _<_./ _)"  [0, 0, 10] 10)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   696
  "_setlessEx"  :: "[idt, 'a, bool] => bool"  ("(3EX _<_./ _)"  [0, 0, 10] 10)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   697
  "_setleAll"   :: "[idt, 'a, bool] => bool"  ("(3ALL _<=_./ _)" [0, 0, 10] 10)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   698
  "_setleEx"    :: "[idt, 'a, bool] => bool"  ("(3EX _<=_./ _)" [0, 0, 10] 10)
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19870
diff changeset
   699
  "_setleEx1"   :: "[idt, 'a, bool] => bool"  ("(3EX! _<=_./ _)" [0, 0, 10] 10)
14804
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   700
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   701
syntax (xsymbols)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   702
  "_setlessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<subset>_./ _)"  [0, 0, 10] 10)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   703
  "_setlessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subset>_./ _)"  [0, 0, 10] 10)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   704
  "_setleAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<subseteq>_./ _)" [0, 0, 10] 10)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   705
  "_setleEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10)
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19870
diff changeset
   706
  "_setleEx1"   :: "[idt, 'a, bool] => bool"   ("(3\<exists>!_\<subseteq>_./ _)" [0, 0, 10] 10)
14804
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   707
19656
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19637
diff changeset
   708
syntax (HOL output)
14804
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   709
  "_setlessAll" :: "[idt, 'a, bool] => bool"   ("(3! _<_./ _)"  [0, 0, 10] 10)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   710
  "_setlessEx"  :: "[idt, 'a, bool] => bool"   ("(3? _<_./ _)"  [0, 0, 10] 10)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   711
  "_setleAll"   :: "[idt, 'a, bool] => bool"   ("(3! _<=_./ _)" [0, 0, 10] 10)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   712
  "_setleEx"    :: "[idt, 'a, bool] => bool"   ("(3? _<=_./ _)" [0, 0, 10] 10)
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19870
diff changeset
   713
  "_setleEx1"   :: "[idt, 'a, bool] => bool"   ("(3?! _<=_./ _)" [0, 0, 10] 10)
14804
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   714
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   715
syntax (HTML output)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   716
  "_setlessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<subset>_./ _)"  [0, 0, 10] 10)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   717
  "_setlessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subset>_./ _)"  [0, 0, 10] 10)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   718
  "_setleAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<subseteq>_./ _)" [0, 0, 10] 10)
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   719
  "_setleEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10)
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19870
diff changeset
   720
  "_setleEx1"   :: "[idt, 'a, bool] => bool"   ("(3\<exists>!_\<subseteq>_./ _)" [0, 0, 10] 10)
14804
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   721
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   722
translations
30352
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   723
  "\<forall>A\<subset>B. P"   =>  "ALL A. A \<subset> B --> P"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   724
  "\<exists>A\<subset>B. P"   =>  "EX A. A \<subset> B & P"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   725
  "\<forall>A\<subseteq>B. P"   =>  "ALL A. A \<subseteq> B --> P"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   726
  "\<exists>A\<subseteq>B. P"   =>  "EX A. A \<subseteq> B & P"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   727
  "\<exists>!A\<subseteq>B. P"  =>  "EX! A. A \<subseteq> B & P"
14804
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   728
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   729
print_translation {*
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   730
let
22377
61610b1beedf tuned ML setup;
wenzelm
parents: 22172
diff changeset
   731
  val Type (set_type, _) = @{typ "'a set"};
61610b1beedf tuned ML setup;
wenzelm
parents: 22172
diff changeset
   732
  val All_binder = Syntax.binder_name @{const_syntax "All"};
61610b1beedf tuned ML setup;
wenzelm
parents: 22172
diff changeset
   733
  val Ex_binder = Syntax.binder_name @{const_syntax "Ex"};
61610b1beedf tuned ML setup;
wenzelm
parents: 22172
diff changeset
   734
  val impl = @{const_syntax "op -->"};
61610b1beedf tuned ML setup;
wenzelm
parents: 22172
diff changeset
   735
  val conj = @{const_syntax "op &"};
61610b1beedf tuned ML setup;
wenzelm
parents: 22172
diff changeset
   736
  val sbset = @{const_syntax "subset"};
61610b1beedf tuned ML setup;
wenzelm
parents: 22172
diff changeset
   737
  val sbset_eq = @{const_syntax "subset_eq"};
21819
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   738
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   739
  val trans =
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   740
   [((All_binder, impl, sbset), "_setlessAll"),
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   741
    ((All_binder, impl, sbset_eq), "_setleAll"),
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   742
    ((Ex_binder, conj, sbset), "_setlessEx"),
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   743
    ((Ex_binder, conj, sbset_eq), "_setleEx")];
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   744
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   745
  fun mk v v' c n P =
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   746
    if v = v' andalso not (Term.exists_subterm (fn Free (x, _) => x = v | _ => false) n)
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   747
    then Syntax.const c $ Syntax.mark_bound v' $ n $ P else raise Match;
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   748
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   749
  fun tr' q = (q,
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   750
    fn [Const ("_bound", _) $ Free (v, Type (T, _)), Const (c, _) $ (Const (d, _) $ (Const ("_bound", _) $ Free (v', _)) $ n) $ P] =>
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   751
         if T = (set_type) then case AList.lookup (op =) trans (q, c, d)
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   752
          of NONE => raise Match
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   753
           | SOME l => mk v v' l n P
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   754
         else raise Match
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   755
     | _ => raise Match);
14804
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   756
in
21819
8eb82ffcdd15 fixed syntax for bounded quantification
haftmann
parents: 21669
diff changeset
   757
  [tr' All_binder, tr' Ex_binder]
14804
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   758
end
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   759
*}
8de39d3e8eb6 Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents: 14752
diff changeset
   760
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   761
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   762
  \medskip Translate between @{text "{e | x1...xn. P}"} and @{text
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   763
  "{u. EX x1..xn. u = e & P}"}; @{text "{y. EX x1..xn. y = e & P}"} is
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   764
  only translated if @{text "[0..n] subset bvs(e)"}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   765
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   766
30352
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   767
syntax
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   768
  "@SetCompr"   :: "'a => idts => bool => 'a set"         ("(1{_ |/_./ _})")
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   769
  "@Collect"    :: "idt => 'a set => bool => 'a set"      ("(1{_ :/ _./ _})")
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   770
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   771
syntax (xsymbols)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   772
  "@Collect"    :: "idt => 'a set => bool => 'a set"      ("(1{_ \<in>/ _./ _})")
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   773
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   774
translations
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   775
  "{x:A. P}"    => "{x. x:A & P}"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   776
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   777
parse_translation {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   778
  let
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   779
    val ex_tr = snd (mk_binder_tr ("EX ", "Ex"));
3947
eb707467f8c5 adapted to qualified names;
wenzelm
parents: 3842
diff changeset
   780
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   781
    fun nvars (Const ("_idts", _) $ _ $ idts) = nvars idts + 1
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   782
      | nvars _ = 1;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   783
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   784
    fun setcompr_tr [e, idts, b] =
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   785
      let
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   786
        val eq = Syntax.const "op =" $ Bound (nvars idts) $ e;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   787
        val P = Syntax.const "op &" $ eq $ b;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   788
        val exP = ex_tr [idts, P];
17784
5cbb52f2c478 Term.absdummy;
wenzelm
parents: 17715
diff changeset
   789
      in Syntax.const "Collect" $ Term.absdummy (dummyT, exP) end;
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   790
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   791
  in [("@SetCompr", setcompr_tr)] end;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   792
*}
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   793
13763
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   794
print_translation {*
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   795
let
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   796
  val ex_tr' = snd (mk_binder_tr' ("Ex", "DUMMY"));
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   797
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   798
  fun setcompr_tr' [Abs (abs as (_, _, P))] =
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   799
    let
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   800
      fun check (Const ("Ex", _) $ Abs (_, _, P), n) = check (P, n + 1)
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   801
        | check (Const ("op &", _) $ (Const ("op =", _) $ Bound m $ e) $ P, n) =
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   802
            n > 0 andalso m = n andalso not (loose_bvar1 (P, n)) andalso
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   803
            ((0 upto (n - 1)) subset add_loose_bnos (e, 0, []))
13764
3e180bf68496 removed some problems with print translations
nipkow
parents: 13763
diff changeset
   804
        | check _ = false
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   805
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   806
        fun tr' (_ $ abs) =
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   807
          let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' [abs]
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   808
          in Syntax.const "@SetCompr" $ e $ idts $ Q end;
13763
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   809
    in if check (P, 0) then tr' P
15535
nipkow
parents: 15524
diff changeset
   810
       else let val (x as _ $ Free(xN,_), t) = atomic_abs_tr' abs
nipkow
parents: 15524
diff changeset
   811
                val M = Syntax.const "@Coll" $ x $ t
nipkow
parents: 15524
diff changeset
   812
            in case t of
nipkow
parents: 15524
diff changeset
   813
                 Const("op &",_)
nipkow
parents: 15524
diff changeset
   814
                   $ (Const("op :",_) $ (Const("_bound",_) $ Free(yN,_)) $ A)
nipkow
parents: 15524
diff changeset
   815
                   $ P =>
nipkow
parents: 15524
diff changeset
   816
                   if xN=yN then Syntax.const "@Collect" $ x $ A $ P else M
nipkow
parents: 15524
diff changeset
   817
               | _ => M
nipkow
parents: 15524
diff changeset
   818
            end
13763
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13653
diff changeset
   819
    end;
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   820
  in [("Collect", setcompr_tr')] end;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   821
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   822
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   823
lemma ballI [intro!]: "(!!x. x:A ==> P x) ==> ALL x:A. P x"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   824
  by (simp add: Ball_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   825
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   826
lemmas strip = impI allI ballI
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   827
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   828
lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   829
  by (simp add: Ball_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   830
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   831
lemma ballE [elim]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   832
  by (unfold Ball_def) blast
22139
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
   833
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
   834
ML {* bind_thm ("rev_ballE", permute_prems 1 1 @{thm ballE}) *}
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   835
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   836
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   837
  \medskip This tactic takes assumptions @{prop "ALL x:A. P x"} and
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   838
  @{prop "a:A"}; creates assumption @{prop "P a"}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   839
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   840
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   841
ML {*
22139
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
   842
  fun ball_tac i = etac @{thm ballE} i THEN contr_tac (i + 1)
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   843
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   844
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   845
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   846
  Gives better instantiation for bound:
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   847
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   848
26339
7825c83c9eff eliminated change_claset/simpset;
wenzelm
parents: 26150
diff changeset
   849
declaration {* fn _ =>
7825c83c9eff eliminated change_claset/simpset;
wenzelm
parents: 26150
diff changeset
   850
  Classical.map_cs (fn cs => cs addbefore ("bspec", datac @{thm bspec} 1))
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   851
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   852
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   853
lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   854
  -- {* Normally the best argument order: @{prop "P x"} constrains the
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   855
    choice of @{prop "x:A"}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   856
  by (unfold Bex_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   857
13113
5eb9be7b72a5 rev_bexI [intro?];
wenzelm
parents: 13103
diff changeset
   858
lemma rev_bexI [intro?]: "x:A ==> P x ==> EX x:A. P x"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   859
  -- {* The best argument order when there is only one @{prop "x:A"}. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   860
  by (unfold Bex_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   861
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   862
lemma bexCI: "(ALL x:A. ~P x ==> P a) ==> a:A ==> EX x:A. P x"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   863
  by (unfold Bex_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   864
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   865
lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   866
  by (unfold Bex_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   867
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   868
lemma ball_triv [simp]: "(ALL x:A. P) = ((EX x. x:A) --> P)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   869
  -- {* Trival rewrite rule. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   870
  by (simp add: Ball_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   871
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   872
lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   873
  -- {* Dual form for existentials. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   874
  by (simp add: Bex_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   875
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   876
lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   877
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   878
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   879
lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   880
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   881
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   882
lemma bex_one_point1 [simp]: "(EX x:A. x = a & P x) = (a:A & P a)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   883
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   884
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   885
lemma bex_one_point2 [simp]: "(EX x:A. a = x & P x) = (a:A & P a)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   886
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   887
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   888
lemma ball_one_point1 [simp]: "(ALL x:A. x = a --> P x) = (a:A --> P a)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   889
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   890
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   891
lemma ball_one_point2 [simp]: "(ALL x:A. a = x --> P x) = (a:A --> P a)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   892
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   893
26480
544cef16045b replaced 'ML_setup' by 'ML';
wenzelm
parents: 26339
diff changeset
   894
ML {*
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13421
diff changeset
   895
  local
22139
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
   896
    val unfold_bex_tac = unfold_tac @{thms "Bex_def"};
18328
841261f303a1 simprocs: static evaluation of simpset;
wenzelm
parents: 18315
diff changeset
   897
    fun prove_bex_tac ss = unfold_bex_tac ss THEN Quantifier1.prove_one_point_ex_tac;
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   898
    val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   899
22139
539a63b98f76 tuned ML setup;
wenzelm
parents: 21833
diff changeset
   900
    val unfold_ball_tac = unfold_tac @{thms "Ball_def"};
18328
841261f303a1 simprocs: static evaluation of simpset;
wenzelm
parents: 18315
diff changeset
   901
    fun prove_ball_tac ss = unfold_ball_tac ss THEN Quantifier1.prove_one_point_all_tac;
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   902
    val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   903
  in
18328
841261f303a1 simprocs: static evaluation of simpset;
wenzelm
parents: 18315
diff changeset
   904
    val defBEX_regroup = Simplifier.simproc (the_context ())
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13421
diff changeset
   905
      "defined BEX" ["EX x:A. P x & Q x"] rearrange_bex;
18328
841261f303a1 simprocs: static evaluation of simpset;
wenzelm
parents: 18315
diff changeset
   906
    val defBALL_regroup = Simplifier.simproc (the_context ())
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13421
diff changeset
   907
      "defined BALL" ["ALL x:A. P x --> Q x"] rearrange_ball;
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   908
  end;
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13421
diff changeset
   909
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13421
diff changeset
   910
  Addsimprocs [defBALL_regroup, defBEX_regroup];
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   911
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   912
30352
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   913
text {*
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   914
  \medskip Eta-contracting these four rules (to remove @{text P})
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   915
  causes them to be ignored because of their interaction with
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   916
  congruence rules.
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   917
*}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   918
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   919
lemma ball_UNIV [simp]: "Ball UNIV P = All P"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   920
  by (simp add: Ball_def)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   921
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   922
lemma bex_UNIV [simp]: "Bex UNIV P = Ex P"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   923
  by (simp add: Bex_def)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   924
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   925
lemma ball_empty [simp]: "Ball {} P = True"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   926
  by (simp add: Ball_def)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   927
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   928
lemma bex_empty [simp]: "Bex {} P = False"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   929
  by (simp add: Bex_def)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   930
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   931
text {* Congruence rules *}
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   932
16636
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   933
lemma ball_cong:
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   934
  "A = B ==> (!!x. x:B ==> P x = Q x) ==>
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   935
    (ALL x:A. P x) = (ALL x:B. Q x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   936
  by (simp add: Ball_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   937
16636
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   938
lemma strong_ball_cong [cong]:
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   939
  "A = B ==> (!!x. x:B =simp=> P x = Q x) ==>
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   940
    (ALL x:A. P x) = (ALL x:B. Q x)"
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   941
  by (simp add: simp_implies_def Ball_def)
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   942
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   943
lemma bex_cong:
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   944
  "A = B ==> (!!x. x:B ==> P x = Q x) ==>
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   945
    (EX x:A. P x) = (EX x:B. Q x)"
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   946
  by (simp add: Bex_def cong: conj_cong)
1273
6960ec882bca added 8bit pragmas
regensbu
parents: 1068
diff changeset
   947
16636
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   948
lemma strong_bex_cong [cong]:
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   949
  "A = B ==> (!!x. x:B =simp=> P x = Q x) ==>
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   950
    (EX x:A. P x) = (EX x:B. Q x)"
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   951
  by (simp add: simp_implies_def Bex_def cong: conj_cong)
1ed737a98198 Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents: 15950
diff changeset
   952
26800
dcf1dfc915a7 - Now uses Orderings as parent theory
berghofe
parents: 26732
diff changeset
   953
lemma subset_eq: "A \<le> B = (\<forall>x\<in>A. x \<in> B)" by blast
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   954
30352
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   955
lemma atomize_ball:
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   956
    "(!!x. x \<in> A ==> P x) == Trueprop (\<forall>x\<in>A. P x)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   957
  by (simp only: Ball_def atomize_all atomize_imp)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   958
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   959
lemmas [symmetric, rulify] = atomize_ball
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   960
  and [symmetric, defn] = atomize_ball
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   961
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   962
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   963
subsubsection {* Image of a set under a function. *}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   964
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   965
text {*
30352
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   966
  Frequently @{term b} does not have the syntactic form of @{term "f x"}.
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   967
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   968
30352
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   969
global
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   970
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   971
consts
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   972
  image         :: "('a => 'b) => 'a set => 'b set"      (infixr "`" 90)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   973
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   974
local
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   975
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   976
defs
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   977
  image_def [noatp]:    "f`A == {y. EX x:A. y = f(x)}"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   978
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   979
lemma image_eqI [simp, intro]: "b = f x ==> x:A ==> b : f`A"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   980
  by (unfold image_def) blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   981
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   982
lemma imageI: "x : A ==> f x : f ` A"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   983
  by (rule image_eqI) (rule refl)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   984
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   985
lemma rev_image_eqI: "x:A ==> b = f x ==> b : f`A"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   986
  -- {* This version's more effective when we already have the
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   987
    required @{term x}. *}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   988
  by (unfold image_def) blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   989
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   990
lemma imageE [elim!]:
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   991
  "b : (%x. f x)`A ==> (!!x. b = f x ==> x:A ==> P) ==> P"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   992
  -- {* The eta-expansion gives variable-name preservation. *}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   993
  by (unfold image_def) blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   994
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   995
lemma image_Un: "f`(A Un B) = f`A Un f`B"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   996
  by blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   997
30352
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   998
lemma image_iff: "(z : f`A) = (EX x:A. z = f x)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
   999
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1000
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1001
lemma image_subset_iff: "(f`A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1002
  -- {* This rewrite rule would confuse users if made default. *}
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1003
  by blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
  1004
30352
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1005
lemma subset_image_iff: "(B \<subseteq> f`A) = (EX AA. AA \<subseteq> A & B = f`AA)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1006
  apply safe
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1007
   prefer 2 apply fast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1008
  apply (rule_tac x = "{a. a : A & f a : B}" in exI, fast)
13865
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
  1009
  done
0a6bf71955b0 moved one proof, added another
paulson
parents: 13860
diff changeset
  1010
30352
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1011
lemma image_subsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f`A \<subseteq> B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1012
  -- {* Replaces the three steps @{text subsetI}, @{text imageE},
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1013
    @{text hypsubst}, but breaks too many existing proofs. *}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1014
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1015
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1016
lemma image_empty [simp]: "f`{} = {}"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1017
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1018
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1019
lemma image_insert [simp]: "f ` insert a B = insert (f a) (f`B)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1020
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1021
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1022
lemma image_constant: "x \<in> A ==> (\<lambda>x. c) ` A = {c}"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1023
  by auto
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1024
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1025
lemma image_constant_conv: "(%x. c) ` A = (if A = {} then {} else {c})"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1026
by auto
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1027
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1028
lemma image_image: "f ` (g ` A) = (\<lambda>x. f (g x)) ` A"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1029
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1030
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1031
lemma insert_image [simp]: "x \<in> A ==> insert (f x) (f`A) = f`A"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1032
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1033
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1034
lemma image_is_empty [iff]: "(f`A = {}) = (A = {})"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1035
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1036
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1037
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1038
lemma image_Collect [noatp]: "f ` {x. P x} = {f x | x. P x}"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1039
  -- {* NOT suitable as a default simprule: the RHS isn't simpler than the LHS,
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1040
      with its implicit quantifier and conjunction.  Also image enjoys better
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1041
      equational properties than does the RHS. *}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1042
  by blast
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1043
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1044
lemma if_image_distrib [simp]:
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1045
  "(\<lambda>x. if P x then f x else g x) ` S
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1046
    = (f ` (S \<inter> {x. P x})) \<union> (g ` (S \<inter> {x. \<not> P x}))"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1047
  by (auto simp add: image_def)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1048
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1049
lemma image_cong: "M = N ==> (!!x. x \<in> N ==> f x = g x) ==> f`M = g`N"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1050
  by (simp add: image_def)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1051
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1052
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1053
subsection {* Set reasoning tools *}
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1054
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1055
text {*
30352
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1056
  Rewrite rules for boolean case-splitting: faster than @{text
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1057
  "split_if [split]"}.
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1058
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1059
30352
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1060
lemma split_if_eq1: "((if Q then x else y) = b) = ((Q --> x = b) & (~ Q --> y = b))"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1061
  by (rule split_if)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1062
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1063
lemma split_if_eq2: "(a = (if Q then x else y)) = ((Q --> a = x) & (~ Q --> a = y))"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1064
  by (rule split_if)
2388
d1f0505fc602 added set inclusion symbol syntax;
wenzelm
parents: 2372
diff changeset
  1065
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1066
text {*
30352
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1067
  Split ifs on either side of the membership relation.  Not for @{text
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1068
  "[simp]"} -- can cause goals to blow up!
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1069
*}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1070
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1071
lemma split_if_mem1: "((if Q then x else y) : b) = ((Q --> x : b) & (~ Q --> y : b))"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1072
  by (rule split_if)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1073
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1074
lemma split_if_mem2: "(a : (if Q then x else y)) = ((Q --> a : x) & (~ Q --> a : y))"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1075
  by (rule split_if [where P="%S. a : S"])
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1076
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1077
lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1078
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1079
(*Would like to add these, but the existing code only searches for the
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1080
  outer-level constant, which in this case is just "op :"; we instead need
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1081
  to use term-nets to associate patterns with rules.  Also, if a rule fails to
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1082
  apply, then the formula should be kept.
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1083
  [("HOL.uminus", Compl_iff RS iffD1), ("HOL.minus", [Diff_iff RS iffD1]),
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1084
   ("Int", [IntD1,IntD2]),
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1085
   ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1086
 *)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1087
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1088
ML {*
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1089
  val mksimps_pairs = [(@{const_name Ball}, @{thms bspec})] @ mksimps_pairs;
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1090
*}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1091
declaration {* fn _ =>
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1092
  Simplifier.map_ss (fn ss => ss setmksimps (mksimps mksimps_pairs))
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1093
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1094
30352
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1095
text {* Transitivity rules for calculational reasoning *}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1096
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1097
lemma set_rev_mp: "x:A ==> A \<subseteq> B ==> x:B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1098
  by (rule subsetD)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1099
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1100
lemma set_mp: "A \<subseteq> B ==> x:A ==> x:B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1101
  by (rule subsetD)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1102
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1103
lemmas basic_trans_rules [trans] =
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1104
  order_trans_rules set_rev_mp set_mp
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1105
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1106
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1107
subsection {* Complete lattices *}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1108
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1109
notation
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1110
  less_eq  (infix "\<sqsubseteq>" 50) and
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1111
  less (infix "\<sqsubset>" 50) and
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1112
  inf  (infixl "\<sqinter>" 70) and
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1113
  sup  (infixl "\<squnion>" 65)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1114
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1115
class complete_lattice = lattice + bot + top +
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1116
  fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1117
    and Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1118
  assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1119
    and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1120
  assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1121
    and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1122
begin
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1123
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1124
lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<le> a}"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1125
  by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1126
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1127
lemma Sup_Inf:  "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<le> b}"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1128
  by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1129
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1130
lemma Inf_Univ: "\<Sqinter>UNIV = \<Squnion>{}"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1131
  unfolding Sup_Inf by auto
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1132
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1133
lemma Sup_Univ: "\<Squnion>UNIV = \<Sqinter>{}"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1134
  unfolding Inf_Sup by auto
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1135
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1136
lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1137
  by (auto intro: antisym Inf_greatest Inf_lower)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1138
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1139
lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1140
  by (auto intro: antisym Sup_least Sup_upper)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1141
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1142
lemma Inf_singleton [simp]:
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1143
  "\<Sqinter>{a} = a"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1144
  by (auto intro: antisym Inf_lower Inf_greatest)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1145
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1146
lemma Sup_singleton [simp]:
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1147
  "\<Squnion>{a} = a"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1148
  by (auto intro: antisym Sup_upper Sup_least)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1149
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1150
lemma Inf_insert_simp:
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1151
  "\<Sqinter>insert a A = (if A = {} then a else a \<sqinter> \<Sqinter>A)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1152
  by (cases "A = {}") (simp_all, simp add: Inf_insert)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1153
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1154
lemma Sup_insert_simp:
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1155
  "\<Squnion>insert a A = (if A = {} then a else a \<squnion> \<Squnion>A)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1156
  by (cases "A = {}") (simp_all, simp add: Sup_insert)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1157
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1158
lemma Inf_binary:
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1159
  "\<Sqinter>{a, b} = a \<sqinter> b"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1160
  by (simp add: Inf_insert_simp)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1161
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1162
lemma Sup_binary:
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1163
  "\<Squnion>{a, b} = a \<squnion> b"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1164
  by (simp add: Sup_insert_simp)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1165
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1166
lemma bot_def:
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1167
  "bot = \<Squnion>{}"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1168
  by (auto intro: antisym Sup_least)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1169
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1170
lemma top_def:
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1171
  "top = \<Sqinter>{}"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1172
  by (auto intro: antisym Inf_greatest)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1173
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1174
lemma sup_bot [simp]:
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1175
  "x \<squnion> bot = x"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1176
  using bot_least [of x] by (simp add: le_iff_sup sup_commute)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1177
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1178
lemma inf_top [simp]:
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1179
  "x \<sqinter> top = x"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1180
  using top_greatest [of x] by (simp add: le_iff_inf inf_commute)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1181
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1182
definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1183
  "SUPR A f == \<Squnion> (f ` A)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1184
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1185
definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1186
  "INFI A f == \<Sqinter> (f ` A)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1187
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1188
end
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1189
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1190
syntax
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1191
  "_SUP1"     :: "pttrns => 'b => 'b"           ("(3SUP _./ _)" [0, 10] 10)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1192
  "_SUP"      :: "pttrn => 'a set => 'b => 'b"  ("(3SUP _:_./ _)" [0, 10] 10)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1193
  "_INF1"     :: "pttrns => 'b => 'b"           ("(3INF _./ _)" [0, 10] 10)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1194
  "_INF"      :: "pttrn => 'a set => 'b => 'b"  ("(3INF _:_./ _)" [0, 10] 10)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1195
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1196
translations
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1197
  "SUP x y. B"   == "SUP x. SUP y. B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1198
  "SUP x. B"     == "CONST SUPR CONST UNIV (%x. B)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1199
  "SUP x. B"     == "SUP x:CONST UNIV. B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1200
  "SUP x:A. B"   == "CONST SUPR A (%x. B)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1201
  "INF x y. B"   == "INF x. INF y. B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1202
  "INF x. B"     == "CONST INFI CONST UNIV (%x. B)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1203
  "INF x. B"     == "INF x:CONST UNIV. B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1204
  "INF x:A. B"   == "CONST INFI A (%x. B)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1205
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1206
(* To avoid eta-contraction of body: *)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1207
print_translation {*
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1208
let
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1209
  fun btr' syn (A :: Abs abs :: ts) =
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1210
    let val (x,t) = atomic_abs_tr' abs
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1211
    in list_comb (Syntax.const syn $ x $ A $ t, ts) end
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1212
  val const_syntax_name = Sign.const_syntax_name @{theory} o fst o dest_Const
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1213
in
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1214
[(const_syntax_name @{term SUPR}, btr' "_SUP"),(const_syntax_name @{term "INFI"}, btr' "_INF")]
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1215
end
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1216
*}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1217
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1218
context complete_lattice
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1219
begin
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1220
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1221
lemma le_SUPI: "i : A \<Longrightarrow> M i \<le> (SUP i:A. M i)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1222
  by (auto simp add: SUPR_def intro: Sup_upper)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1223
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1224
lemma SUP_leI: "(\<And>i. i : A \<Longrightarrow> M i \<le> u) \<Longrightarrow> (SUP i:A. M i) \<le> u"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1225
  by (auto simp add: SUPR_def intro: Sup_least)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1226
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1227
lemma INF_leI: "i : A \<Longrightarrow> (INF i:A. M i) \<le> M i"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1228
  by (auto simp add: INFI_def intro: Inf_lower)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1229
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1230
lemma le_INFI: "(\<And>i. i : A \<Longrightarrow> u \<le> M i) \<Longrightarrow> u \<le> (INF i:A. M i)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1231
  by (auto simp add: INFI_def intro: Inf_greatest)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1232
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1233
lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1234
  by (auto intro: antisym SUP_leI le_SUPI)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1235
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1236
lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1237
  by (auto intro: antisym INF_leI le_INFI)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1238
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1239
end
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1240
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1241
subsubsection {* Bool as complete lattice *}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1242
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1243
instantiation bool :: complete_lattice
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1244
begin
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1245
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1246
definition
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1247
  Inf_bool_def: "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1248
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1249
definition
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1250
  Sup_bool_def: "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1251
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1252
instance
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1253
  by intro_classes (auto simp add: Inf_bool_def Sup_bool_def le_bool_def)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1254
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1255
end
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1256
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1257
lemma Inf_empty_bool [simp]:
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1258
  "\<Sqinter>{}"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1259
  unfolding Inf_bool_def by auto
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1260
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1261
lemma not_Sup_empty_bool [simp]:
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1262
  "\<not> Sup {}"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1263
  unfolding Sup_bool_def by auto
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1264
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1265
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1266
subsubsection {* Fun as complete lattice *}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1267
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1268
instantiation "fun" :: (type, complete_lattice) complete_lattice
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1269
begin
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1270
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1271
definition
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1272
  Inf_fun_def [code del]: "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1273
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1274
definition
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1275
  Sup_fun_def [code del]: "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1276
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1277
instance
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1278
  by intro_classes
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1279
    (auto simp add: Inf_fun_def Sup_fun_def le_fun_def
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1280
      intro: Inf_lower Sup_upper Inf_greatest Sup_least)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1281
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1282
end
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1283
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1284
lemma Inf_empty_fun:
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1285
  "\<Sqinter>{} = (\<lambda>_. \<Sqinter>{})"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1286
  by rule (auto simp add: Inf_fun_def)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1287
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1288
lemma Sup_empty_fun:
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1289
  "\<Squnion>{} = (\<lambda>_. \<Squnion>{})"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1290
  by rule (auto simp add: Sup_fun_def)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1291
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1292
no_notation
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1293
  less_eq  (infix "\<sqsubseteq>" 50) and
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1294
  less (infix "\<sqsubset>" 50) and
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1295
  inf  (infixl "\<sqinter>" 70) and
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1296
  sup  (infixl "\<squnion>" 65) and
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1297
  Inf  ("\<Sqinter>_" [900] 900) and
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1298
  Sup  ("\<Squnion>_" [900] 900)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1299
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1300
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1301
subsection {* Further operations *}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1302
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1303
subsubsection {* Big families as specialisation of lattice operations *}
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1304
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1305
definition INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1306
  "INTER A B \<equiv> {y. \<forall>x\<in>A. y \<in> B x}"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1307
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1308
definition UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1309
  "UNION A B \<equiv> {y. \<exists>x\<in>A. y \<in> B x}"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1310
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1311
definition Inter :: "'a set set \<Rightarrow> 'a set" where
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1312
  "Inter S \<equiv> INTER S (\<lambda>x. x)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1313
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1314
definition Union :: "'a set set \<Rightarrow> 'a set" where
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1315
  "Union S \<equiv> UNION S (\<lambda>x. x)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1316
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1317
notation (xsymbols)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1318
  Inter  ("\<Inter>_" [90] 90) and
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1319
  Union  ("\<Union>_" [90] 90)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1320
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1321
syntax
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1322
  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1323
  "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1324
  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 10] 10)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1325
  "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 10] 10)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1326
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1327
syntax (xsymbols)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1328
  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1329
  "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1330
  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 10] 10)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1331
  "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 10] 10)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1332
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1333
syntax (latex output)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1334
  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1335
  "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1336
  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1337
  "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1338
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1339
translations
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1340
  "INT x y. B"  == "INT x. INT y. B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1341
  "INT x. B"    == "CONST INTER CONST UNIV (%x. B)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1342
  "INT x. B"    == "INT x:CONST UNIV. B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1343
  "INT x:A. B"  == "CONST INTER A (%x. B)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1344
  "UN x y. B"   == "UN x. UN y. B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1345
  "UN x. B"     == "CONST UNION CONST UNIV (%x. B)"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1346
  "UN x. B"     == "UN x:CONST UNIV. B"
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1347
  "UN x:A. B"   == "CONST UNION A (%x. B)"
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
  1348
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1349
text {*
30352
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1350
  Note the difference between ordinary xsymbol syntax of indexed
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1351
  unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1352
  and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1353
  former does not make the index expression a subscript of the
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1354
  union/intersection symbol because this leads to problems with nested
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1355
  subscripts in Proof General.
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1356
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1357
30352
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1358
(* To avoid eta-contraction of body: *)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1359
(*FIXME  integrate with / factor out from similar situations*)
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1360
print_translation {*
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1361
let
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1362
  fun btr' syn [A, Abs abs] =
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1363
    let val (x, t) = atomic_abs_tr' abs
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1364
    in Syntax.const syn $ x $ A $ t end
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1365
in
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1366
[(@{const_syntax Ball}, btr' "_Ball"), (@{const_syntax Bex}, btr' "_Bex"),
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1367
 (@{const_syntax UNION}, btr' "@UNION"),(@{const_syntax INTER}, btr' "@INTER")]
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1368
end
047f183c43b0 restructured theory Set.thy
haftmann
parents: 30304
diff changeset
  1369
*}
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1370
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1371
subsubsection {* Unions of families *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1372
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1373
text {*
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1374
  @{term [source] "UN x:A. B x"} is @{term "Union (B`A)"}.
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1375
*}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1376
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 24280
diff changeset
  1377
declare UNION_def [noatp]
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 24280