author | nipkow |
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(* Author: Florian Haftmann, TU Muenchen |
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Author: Andreas Lochbihler, ETH Zurich *) |
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section \<open>Lists with elements distinct as canonical example for datatype invariants\<close> |
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theory Dlist |
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imports Confluent_Quotient |
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begin |
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subsection \<open>The type of distinct lists\<close> |
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typedef '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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context begin |
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qualified definition dlist_eq where "dlist_eq = BNF_Def.vimage2p remdups remdups (=)" |
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qualified lemma equivp_dlist_eq: "equivp dlist_eq" |
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unfolding dlist_eq_def by(rule equivp_vimage2p)(rule identity_equivp) |
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qualified definition abs_dlist :: "'a list \<Rightarrow> 'a dlist" where "abs_dlist = Abs_dlist o remdups" |
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definition qcr_dlist :: "'a list \<Rightarrow> 'a dlist \<Rightarrow> bool" where "qcr_dlist x y \<longleftrightarrow> y = abs_dlist x" |
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qualified lemma Quotient_dlist_remdups: "Quotient dlist_eq abs_dlist list_of_dlist qcr_dlist" |
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unfolding Quotient_def dlist_eq_def qcr_dlist_def vimage2p_def abs_dlist_def |
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by (auto simp add: fun_eq_iff Abs_dlist_inject |
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list_of_dlist[simplified] list_of_dlist_inverse distinct_remdups_id) |
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end |
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locale Quotient_dlist begin |
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setup_lifting Dlist.Quotient_dlist_remdups Dlist.equivp_dlist_eq[THEN equivp_reflp2] |
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end |
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setup_lifting type_definition_dlist |
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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 \<open>Formal, totalized constructor for \<^typ>\<open>'a dlist\<close>:\<close> |
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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 \<open>Fundamental operations:\<close> |
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context |
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begin |
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qualified definition empty :: "'a dlist" where |
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"empty = Dlist []" |
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qualified 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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qualified 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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qualified 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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qualified 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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qualified definition rotate :: "nat \<Rightarrow> 'a dlist \<Rightarrow> 'a dlist" where |
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"rotate n dxs = Dlist (List.rotate n (list_of_dlist dxs))" |
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end |
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text \<open>Derived operations:\<close> |
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context |
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begin |
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qualified definition null :: "'a dlist \<Rightarrow> bool" where |
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"null dxs = List.null (list_of_dlist dxs)" |
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qualified 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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qualified definition length :: "'a dlist \<Rightarrow> nat" where |
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"length dxs = List.length (list_of_dlist dxs)" |
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qualified 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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qualified 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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end |
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subsection \<open>Executable version obeying invariant\<close> |
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lemma list_of_dlist_empty [simp, code abstract]: |
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"list_of_dlist Dlist.empty = []" |
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by (simp add: Dlist.empty_def) |
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lemma list_of_dlist_insert [simp, code abstract]: |
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"list_of_dlist (Dlist.insert x dxs) = List.insert x (list_of_dlist dxs)" |
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by (simp add: Dlist.insert_def) |
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lemma list_of_dlist_remove [simp, code abstract]: |
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"list_of_dlist (Dlist.remove x dxs) = remove1 x (list_of_dlist dxs)" |
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by (simp add: Dlist.remove_def) |
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lemma list_of_dlist_map [simp, code abstract]: |
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"list_of_dlist (Dlist.map f dxs) = remdups (List.map f (list_of_dlist dxs))" |
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by (simp add: Dlist.map_def) |
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lemma list_of_dlist_filter [simp, code abstract]: |
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"list_of_dlist (Dlist.filter P dxs) = List.filter P (list_of_dlist dxs)" |
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by (simp add: Dlist.filter_def) |
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lemma list_of_dlist_rotate [simp, code abstract]: |
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"list_of_dlist (Dlist.rotate n dxs) = List.rotate n (list_of_dlist dxs)" |
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by (simp add: Dlist.rotate_def) |
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text \<open>Explicit executable conversion\<close> |
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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 \<open>Equality\<close> |
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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 |
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by standard (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]: "HOL.equal (dxs :: 'a::equal dlist) dxs \<longleftrightarrow> True" |
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by (fact equal_refl) |
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subsection \<open>Induction principle and case distinction\<close> |
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lemma dlist_induct [case_names empty insert, induct type: dlist]: |
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assumes empty: "P Dlist.empty" |
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assumes insrt: "\<And>x dxs. \<not> Dlist.member dxs x \<Longrightarrow> P dxs \<Longrightarrow> P (Dlist.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" |
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by (simp_all add: Dlist_def distinct_remdups_id) |
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from \<open>distinct xs\<close> have "P (Dlist xs)" |
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proof (induct xs) |
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case Nil from empty show ?case by (simp add: Dlist.empty_def) |
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next |
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case (Cons x xs) |
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then have "\<not> Dlist.member (Dlist xs) x" and "P (Dlist xs)" |
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by (simp_all add: Dlist.member_def List.member_def) |
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with insrt have "P (Dlist.insert x (Dlist xs))" . |
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with Cons show ?case by (simp add: Dlist.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 [cases type: dlist]: |
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obtains (empty) "dxs = Dlist.empty" |
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| (insert) x dys where "\<not> Dlist.member dys x" and "dxs = Dlist.insert x dys" |
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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 thesis |
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proof (cases xs) |
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case Nil with dxs |
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have "dxs = Dlist.empty" by (simp add: Dlist.empty_def) |
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with empty show ?thesis . |
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next |
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case (Cons x xs) |
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with dxs distinct have "\<not> Dlist.member (Dlist xs) x" |
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and "dxs = Dlist.insert x (Dlist xs)" |
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by (simp_all add: Dlist.member_def List.member_def Dlist.insert_def distinct_remdups_id) |
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with insert show ?thesis . |
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qed |
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qed |
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subsection \<open>Functorial structure\<close> |
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functor map: map |
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by (simp_all add: remdups_map_remdups fun_eq_iff dlist_eq_iff) |
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subsection \<open>Quickcheck generators\<close> |
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quickcheck_generator dlist predicate: distinct constructors: Dlist.empty, Dlist.insert |
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subsection \<open>BNF instance\<close> |
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context begin |
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qualified inductive double :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" where |
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"double (xs @ ys) (xs @ x # ys)" if "x \<in> set ys" |
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qualified lemma strong_confluentp_double: "strong_confluentp double" |
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proof |
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fix xs ys zs :: "'a list" |
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assume ys: "double xs ys" and zs: "double xs zs" |
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consider (left) as y bs z cs where "xs = as @ bs @ cs" "ys = as @ y # bs @ cs" "zs = as @ bs @ z # cs" "y \<in> set (bs @ cs)" "z \<in> set cs" |
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| (right) as y bs z cs where "xs = as @ bs @ cs" "ys = as @ bs @ y # cs" "zs = as @ z # bs @ cs" "y \<in> set cs" "z \<in> set (bs @ cs)" |
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proof - |
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show thesis using ys zs |
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by(clarsimp simp add: double.simps append_eq_append_conv2)(auto intro: that) |
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qed |
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then show "\<exists>us. double\<^sup>*\<^sup>* ys us \<and> double\<^sup>=\<^sup>= zs us" |
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proof cases |
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case left |
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let ?us = "as @ y # bs @ z # cs" |
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have "double ys ?us" "double zs ?us" using left |
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by(auto 4 4 simp add: double.simps)(metis append_Cons append_assoc)+ |
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then show ?thesis by blast |
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next |
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case right |
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let ?us = "as @ z # bs @ y # cs" |
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have "double ys ?us" "double zs ?us" using right |
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by(auto 4 4 simp add: double.simps)(metis append_Cons append_assoc)+ |
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then show ?thesis by blast |
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qed |
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qed |
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qualified lemma double_Cons1 [simp]: "double xs (x # xs)" if "x \<in> set xs" |
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using double.intros[of x xs "[]"] that by simp |
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qualified lemma double_Cons_same [simp]: "double xs ys \<Longrightarrow> double (x # xs) (x # ys)" |
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by(auto simp add: double.simps Cons_eq_append_conv) |
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qualified lemma doubles_Cons_same: "double\<^sup>*\<^sup>* xs ys \<Longrightarrow> double\<^sup>*\<^sup>* (x # xs) (x # ys)" |
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by(induction rule: rtranclp_induct)(auto intro: rtranclp.rtrancl_into_rtrancl) |
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qualified lemma remdups_into_doubles: "double\<^sup>*\<^sup>* (remdups xs) xs" |
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by(induction xs)(auto intro: doubles_Cons_same rtranclp.rtrancl_into_rtrancl) |
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qualified lemma dlist_eq_into_doubles: "Dlist.dlist_eq \<le> equivclp double" |
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by(auto 4 4 simp add: Dlist.dlist_eq_def vimage2p_def |
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intro: equivclp_trans converse_rtranclp_into_equivclp rtranclp_into_equivclp remdups_into_doubles) |
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qualified lemma factor_double_map: "double (map f xs) ys \<Longrightarrow> \<exists>zs. Dlist.dlist_eq xs zs \<and> ys = map f zs \<and> set zs \<subseteq> set xs" |
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by(auto simp add: double.simps Dlist.dlist_eq_def vimage2p_def map_eq_append_conv) |
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(metis (no_types, opaque_lifting) list.simps(9) map_append remdups.simps(2) remdups_append2 set_append set_eq_subset set_remdups) |
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qualified lemma dlist_eq_set_eq: "Dlist.dlist_eq xs ys \<Longrightarrow> set xs = set ys" |
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by(simp add: Dlist.dlist_eq_def vimage2p_def)(metis set_remdups) |
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qualified lemma dlist_eq_map_respect: "Dlist.dlist_eq xs ys \<Longrightarrow> Dlist.dlist_eq (map f xs) (map f ys)" |
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by(clarsimp simp add: Dlist.dlist_eq_def vimage2p_def)(metis remdups_map_remdups) |
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qualified lemma confluent_quotient_dlist: |
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"confluent_quotient double Dlist.dlist_eq Dlist.dlist_eq Dlist.dlist_eq Dlist.dlist_eq Dlist.dlist_eq |
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(map fst) (map snd) (map fst) (map snd) list_all2 list_all2 list_all2 set set" |
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by(unfold_locales)(auto intro: strong_confluentp_imp_confluentp strong_confluentp_double |
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dest: factor_double_map dlist_eq_into_doubles[THEN predicate2D] dlist_eq_set_eq |
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simp add: list.in_rel list.rel_compp dlist_eq_map_respect Dlist.equivp_dlist_eq equivp_imp_transp) |
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lifting_update dlist.lifting |
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lifting_forget dlist.lifting |
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end |
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context begin |
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interpretation Quotient_dlist: Quotient_dlist . |
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lift_bnf (plugins del: code) 'a dlist |
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subgoal for A B by(rule confluent_quotient.subdistributivity[OF Dlist.confluent_quotient_dlist]) |
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subgoal by(force dest: Dlist.dlist_eq_set_eq intro: equivp_reflp[OF Dlist.equivp_dlist_eq]) |
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done |
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qualified lemma list_of_dlist_transfer[transfer_rule]: |
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"bi_unique R \<Longrightarrow> (rel_fun (Quotient_dlist.pcr_dlist R) (list_all2 R)) remdups list_of_dlist" |
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unfolding rel_fun_def Quotient_dlist.pcr_dlist_def qcr_dlist_def Dlist.abs_dlist_def |
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by (auto simp: Abs_dlist_inverse intro!: remdups_transfer[THEN rel_funD]) |
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lemma list_of_dlist_map_dlist[simp]: |
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"list_of_dlist (map_dlist f xs) = remdups (map f (list_of_dlist xs))" |
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by transfer (auto simp: remdups_map_remdups) |
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end |
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end |