src/HOL/Library/Dlist.thy
author huffman
Fri Mar 30 12:32:35 2012 +0200 (2012-03-30)
changeset 47220 52426c62b5d0
parent 46565 ad21900e0ee9
child 48282 39bfb2844b9e
permissions -rw-r--r--
replace lemmas eval_nat_numeral with a simpler reformulation
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(* Author: Florian Haftmann, TU Muenchen *)
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header {* Lists with elements distinct as canonical example for datatype invariants *}
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theory Dlist
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imports Main
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begin
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subsection {* The type of distinct lists *}
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typedef (open) 'a dlist = "{xs::'a list. distinct xs}"
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  morphisms list_of_dlist Abs_dlist
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proof
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  show "[] \<in> {xs. distinct xs}" by simp
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qed
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lemma dlist_eq_iff:
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  "dxs = dys \<longleftrightarrow> list_of_dlist dxs = list_of_dlist dys"
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  by (simp add: list_of_dlist_inject)
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lemma dlist_eqI:
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  "list_of_dlist dxs = list_of_dlist dys \<Longrightarrow> dxs = dys"
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  by (simp add: dlist_eq_iff)
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text {* Formal, totalized constructor for @{typ "'a dlist"}: *}
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definition Dlist :: "'a list \<Rightarrow> 'a dlist" where
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  "Dlist xs = Abs_dlist (remdups xs)"
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lemma distinct_list_of_dlist [simp, intro]:
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  "distinct (list_of_dlist dxs)"
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  using list_of_dlist [of dxs] by simp
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lemma list_of_dlist_Dlist [simp]:
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  "list_of_dlist (Dlist xs) = remdups xs"
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  by (simp add: Dlist_def Abs_dlist_inverse)
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lemma remdups_list_of_dlist [simp]:
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  "remdups (list_of_dlist dxs) = list_of_dlist dxs"
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  by simp
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lemma Dlist_list_of_dlist [simp, code abstype]:
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  "Dlist (list_of_dlist dxs) = dxs"
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  by (simp add: Dlist_def list_of_dlist_inverse distinct_remdups_id)
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text {* Fundamental operations: *}
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definition empty :: "'a dlist" where
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  "empty = Dlist []"
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definition insert :: "'a \<Rightarrow> 'a dlist \<Rightarrow> 'a dlist" where
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  "insert x dxs = Dlist (List.insert x (list_of_dlist dxs))"
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definition remove :: "'a \<Rightarrow> 'a dlist \<Rightarrow> 'a dlist" where
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  "remove x dxs = Dlist (remove1 x (list_of_dlist dxs))"
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definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a dlist \<Rightarrow> 'b dlist" where
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  "map f dxs = Dlist (remdups (List.map f (list_of_dlist dxs)))"
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definition filter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a dlist \<Rightarrow> 'a dlist" where
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  "filter P dxs = Dlist (List.filter P (list_of_dlist dxs))"
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text {* Derived operations: *}
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definition null :: "'a dlist \<Rightarrow> bool" where
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  "null dxs = List.null (list_of_dlist dxs)"
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definition member :: "'a dlist \<Rightarrow> 'a \<Rightarrow> bool" where
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  "member dxs = List.member (list_of_dlist dxs)"
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definition length :: "'a dlist \<Rightarrow> nat" where
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  "length dxs = List.length (list_of_dlist dxs)"
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definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a dlist \<Rightarrow> 'b \<Rightarrow> 'b" where
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  "fold f dxs = List.fold f (list_of_dlist dxs)"
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definition foldr :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a dlist \<Rightarrow> 'b \<Rightarrow> 'b" where
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  "foldr f dxs = List.foldr f (list_of_dlist dxs)"
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subsection {* Executable version obeying invariant *}
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lemma list_of_dlist_empty [simp, code abstract]:
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  "list_of_dlist empty = []"
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  by (simp add: empty_def)
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lemma list_of_dlist_insert [simp, code abstract]:
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  "list_of_dlist (insert x dxs) = List.insert x (list_of_dlist dxs)"
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  by (simp add: insert_def)
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lemma list_of_dlist_remove [simp, code abstract]:
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  "list_of_dlist (remove x dxs) = remove1 x (list_of_dlist dxs)"
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  by (simp add: remove_def)
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lemma list_of_dlist_map [simp, code abstract]:
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  "list_of_dlist (map f dxs) = remdups (List.map f (list_of_dlist dxs))"
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  by (simp add: map_def)
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lemma list_of_dlist_filter [simp, code abstract]:
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  "list_of_dlist (filter P dxs) = List.filter P (list_of_dlist dxs)"
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  by (simp add: filter_def)
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text {* Explicit executable conversion *}
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definition dlist_of_list [simp]:
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  "dlist_of_list = Dlist"
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lemma [code abstract]:
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  "list_of_dlist (dlist_of_list xs) = remdups xs"
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  by simp
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text {* Equality *}
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instantiation dlist :: (equal) equal
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begin
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definition "HOL.equal dxs dys \<longleftrightarrow> HOL.equal (list_of_dlist dxs) (list_of_dlist dys)"
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instance proof
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qed (simp add: equal_dlist_def equal list_of_dlist_inject)
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end
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declare equal_dlist_def [code]
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lemma [code nbe]:
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  "HOL.equal (dxs :: 'a::equal dlist) dxs \<longleftrightarrow> True"
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  by (fact equal_refl)
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subsection {* Induction principle and case distinction *}
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lemma dlist_induct [case_names empty insert, induct type: dlist]:
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  assumes empty: "P empty"
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  assumes insrt: "\<And>x dxs. \<not> member dxs x \<Longrightarrow> P dxs \<Longrightarrow> P (insert x dxs)"
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  shows "P dxs"
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proof (cases dxs)
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  case (Abs_dlist xs)
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  then have "distinct xs" and dxs: "dxs = Dlist xs" by (simp_all add: Dlist_def distinct_remdups_id)
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  from `distinct xs` have "P (Dlist xs)"
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  proof (induct xs)
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    case Nil from empty show ?case by (simp add: empty_def)
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  next
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    case (Cons x xs)
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    then have "\<not> member (Dlist xs) x" and "P (Dlist xs)"
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      by (simp_all add: member_def List.member_def)
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    with insrt have "P (insert x (Dlist xs))" .
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    with Cons show ?case by (simp add: insert_def distinct_remdups_id)
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  qed
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  with dxs show "P dxs" by simp
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qed
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lemma dlist_case [case_names empty insert, cases type: dlist]:
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  assumes empty: "dxs = empty \<Longrightarrow> P"
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  assumes insert: "\<And>x dys. \<not> member dys x \<Longrightarrow> dxs = insert x dys \<Longrightarrow> P"
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  shows P
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proof (cases dxs)
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  case (Abs_dlist xs)
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  then have dxs: "dxs = Dlist xs" and distinct: "distinct xs"
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    by (simp_all add: Dlist_def distinct_remdups_id)
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  show P proof (cases xs)
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    case Nil with dxs have "dxs = empty" by (simp add: empty_def) 
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    with empty show P .
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  next
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    case (Cons x xs)
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    with dxs distinct have "\<not> member (Dlist xs) x"
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      and "dxs = insert x (Dlist xs)"
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      by (simp_all add: member_def List.member_def insert_def distinct_remdups_id)
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    with insert show P .
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  qed
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qed
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subsection {* Functorial structure *}
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enriched_type map: map
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  by (simp_all add: List.map.id remdups_map_remdups fun_eq_iff dlist_eq_iff)
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subsection {* Quickcheck generators *}
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quickcheck_generator dlist predicate: distinct constructors: empty, insert
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hide_const (open) member fold foldr empty insert remove map filter null member length fold
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end