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src/HOL/Library/Dlist.thy

author | wenzelm |

Wed Nov 30 16:27:10 2011 +0100 (2011-11-30) | |

changeset 45694 | 4a8743618257 |

parent 43764 | 366d5726de09 |

child 45927 | e0305e4f02c9 |

permissions | -rw-r--r-- |

prefer typedef without extra definition and alternative name;

tuned proofs;

tuned proofs;

1 (* Author: Florian Haftmann, TU Muenchen *)

3 header {* Lists with elements distinct as canonical example for datatype invariants *}

5 theory Dlist

6 imports Main More_List

7 begin

9 subsection {* The type of distinct lists *}

11 typedef (open) 'a dlist = "{xs::'a list. distinct xs}"

12 morphisms list_of_dlist Abs_dlist

13 proof

14 show "[] \<in> {xs. distinct xs}" by simp

15 qed

17 lemma dlist_eq_iff:

18 "dxs = dys \<longleftrightarrow> list_of_dlist dxs = list_of_dlist dys"

19 by (simp add: list_of_dlist_inject)

21 lemma dlist_eqI:

22 "list_of_dlist dxs = list_of_dlist dys \<Longrightarrow> dxs = dys"

23 by (simp add: dlist_eq_iff)

25 text {* Formal, totalized constructor for @{typ "'a dlist"}: *}

27 definition Dlist :: "'a list \<Rightarrow> 'a dlist" where

28 "Dlist xs = Abs_dlist (remdups xs)"

30 lemma distinct_list_of_dlist [simp, intro]:

31 "distinct (list_of_dlist dxs)"

32 using list_of_dlist [of dxs] by simp

34 lemma list_of_dlist_Dlist [simp]:

35 "list_of_dlist (Dlist xs) = remdups xs"

36 by (simp add: Dlist_def Abs_dlist_inverse)

38 lemma remdups_list_of_dlist [simp]:

39 "remdups (list_of_dlist dxs) = list_of_dlist dxs"

40 by simp

42 lemma Dlist_list_of_dlist [simp, code abstype]:

43 "Dlist (list_of_dlist dxs) = dxs"

44 by (simp add: Dlist_def list_of_dlist_inverse distinct_remdups_id)

47 text {* Fundamental operations: *}

49 definition empty :: "'a dlist" where

50 "empty = Dlist []"

52 definition insert :: "'a \<Rightarrow> 'a dlist \<Rightarrow> 'a dlist" where

53 "insert x dxs = Dlist (List.insert x (list_of_dlist dxs))"

55 definition remove :: "'a \<Rightarrow> 'a dlist \<Rightarrow> 'a dlist" where

56 "remove x dxs = Dlist (remove1 x (list_of_dlist dxs))"

58 definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a dlist \<Rightarrow> 'b dlist" where

59 "map f dxs = Dlist (remdups (List.map f (list_of_dlist dxs)))"

61 definition filter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a dlist \<Rightarrow> 'a dlist" where

62 "filter P dxs = Dlist (List.filter P (list_of_dlist dxs))"

65 text {* Derived operations: *}

67 definition null :: "'a dlist \<Rightarrow> bool" where

68 "null dxs = List.null (list_of_dlist dxs)"

70 definition member :: "'a dlist \<Rightarrow> 'a \<Rightarrow> bool" where

71 "member dxs = List.member (list_of_dlist dxs)"

73 definition length :: "'a dlist \<Rightarrow> nat" where

74 "length dxs = List.length (list_of_dlist dxs)"

76 definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a dlist \<Rightarrow> 'b \<Rightarrow> 'b" where

77 "fold f dxs = More_List.fold f (list_of_dlist dxs)"

79 definition foldr :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a dlist \<Rightarrow> 'b \<Rightarrow> 'b" where

80 "foldr f dxs = List.foldr f (list_of_dlist dxs)"

83 subsection {* Executable version obeying invariant *}

85 lemma list_of_dlist_empty [simp, code abstract]:

86 "list_of_dlist empty = []"

87 by (simp add: empty_def)

89 lemma list_of_dlist_insert [simp, code abstract]:

90 "list_of_dlist (insert x dxs) = List.insert x (list_of_dlist dxs)"

91 by (simp add: insert_def)

93 lemma list_of_dlist_remove [simp, code abstract]:

94 "list_of_dlist (remove x dxs) = remove1 x (list_of_dlist dxs)"

95 by (simp add: remove_def)

97 lemma list_of_dlist_map [simp, code abstract]:

98 "list_of_dlist (map f dxs) = remdups (List.map f (list_of_dlist dxs))"

99 by (simp add: map_def)

101 lemma list_of_dlist_filter [simp, code abstract]:

102 "list_of_dlist (filter P dxs) = List.filter P (list_of_dlist dxs)"

103 by (simp add: filter_def)

106 text {* Explicit executable conversion *}

108 definition dlist_of_list [simp]:

109 "dlist_of_list = Dlist"

111 lemma [code abstract]:

112 "list_of_dlist (dlist_of_list xs) = remdups xs"

113 by simp

116 text {* Equality *}

118 instantiation dlist :: (equal) equal

119 begin

121 definition "HOL.equal dxs dys \<longleftrightarrow> HOL.equal (list_of_dlist dxs) (list_of_dlist dys)"

123 instance proof

124 qed (simp add: equal_dlist_def equal list_of_dlist_inject)

126 end

128 declare equal_dlist_def [code]

130 lemma [code nbe]:

131 "HOL.equal (dxs :: 'a::equal dlist) dxs \<longleftrightarrow> True"

132 by (fact equal_refl)

135 subsection {* Induction principle and case distinction *}

137 lemma dlist_induct [case_names empty insert, induct type: dlist]:

138 assumes empty: "P empty"

139 assumes insrt: "\<And>x dxs. \<not> member dxs x \<Longrightarrow> P dxs \<Longrightarrow> P (insert x dxs)"

140 shows "P dxs"

141 proof (cases dxs)

142 case (Abs_dlist xs)

143 then have "distinct xs" and dxs: "dxs = Dlist xs" by (simp_all add: Dlist_def distinct_remdups_id)

144 from `distinct xs` have "P (Dlist xs)"

145 proof (induct xs)

146 case Nil from empty show ?case by (simp add: empty_def)

147 next

148 case (Cons x xs)

149 then have "\<not> member (Dlist xs) x" and "P (Dlist xs)"

150 by (simp_all add: member_def List.member_def)

151 with insrt have "P (insert x (Dlist xs))" .

152 with Cons show ?case by (simp add: insert_def distinct_remdups_id)

153 qed

154 with dxs show "P dxs" by simp

155 qed

157 lemma dlist_case [case_names empty insert, cases type: dlist]:

158 assumes empty: "dxs = empty \<Longrightarrow> P"

159 assumes insert: "\<And>x dys. \<not> member dys x \<Longrightarrow> dxs = insert x dys \<Longrightarrow> P"

160 shows P

161 proof (cases dxs)

162 case (Abs_dlist xs)

163 then have dxs: "dxs = Dlist xs" and distinct: "distinct xs"

164 by (simp_all add: Dlist_def distinct_remdups_id)

165 show P proof (cases xs)

166 case Nil with dxs have "dxs = empty" by (simp add: empty_def)

167 with empty show P .

168 next

169 case (Cons x xs)

170 with dxs distinct have "\<not> member (Dlist xs) x"

171 and "dxs = insert x (Dlist xs)"

172 by (simp_all add: member_def List.member_def insert_def distinct_remdups_id)

173 with insert show P .

174 qed

175 qed

178 subsection {* Functorial structure *}

180 enriched_type map: map

181 by (simp_all add: List.map.id remdups_map_remdups fun_eq_iff dlist_eq_iff)

184 hide_const (open) member fold foldr empty insert remove map filter null member length fold

186 end