(* Author: Florian Haftmann, TU Muenchen Author: Andreas Lochbihler, ETH Zurich *) section ‹Lists with elements distinct as canonical example for datatype invariants› theory Dlist imports Main begin subsection ‹The type of distinct lists› typedef 'a dlist = "{xs::'a list. distinct xs}" morphisms list_of_dlist Abs_dlist proof show "[] ∈ {xs. distinct xs}" by simp qed setup_lifting type_definition_dlist lemma dlist_eq_iff: "dxs = dys ⟷ list_of_dlist dxs = list_of_dlist dys" by (simp add: list_of_dlist_inject) lemma dlist_eqI: "list_of_dlist dxs = list_of_dlist dys ⟹ dxs = dys" by (simp add: dlist_eq_iff) text ‹Formal, totalized constructor for @{typ "'a dlist"}:› definition Dlist :: "'a list ⇒ 'a dlist" where "Dlist xs = Abs_dlist (remdups xs)" lemma distinct_list_of_dlist [simp, intro]: "distinct (list_of_dlist dxs)" using list_of_dlist [of dxs] by simp lemma list_of_dlist_Dlist [simp]: "list_of_dlist (Dlist xs) = remdups xs" by (simp add: Dlist_def Abs_dlist_inverse) lemma remdups_list_of_dlist [simp]: "remdups (list_of_dlist dxs) = list_of_dlist dxs" by simp lemma Dlist_list_of_dlist [simp, code abstype]: "Dlist (list_of_dlist dxs) = dxs" by (simp add: Dlist_def list_of_dlist_inverse distinct_remdups_id) text ‹Fundamental operations:› context begin qualified definition empty :: "'a dlist" where "empty = Dlist []" qualified definition insert :: "'a ⇒ 'a dlist ⇒ 'a dlist" where "insert x dxs = Dlist (List.insert x (list_of_dlist dxs))" qualified definition remove :: "'a ⇒ 'a dlist ⇒ 'a dlist" where "remove x dxs = Dlist (remove1 x (list_of_dlist dxs))" qualified definition map :: "('a ⇒ 'b) ⇒ 'a dlist ⇒ 'b dlist" where "map f dxs = Dlist (remdups (List.map f (list_of_dlist dxs)))" qualified definition filter :: "('a ⇒ bool) ⇒ 'a dlist ⇒ 'a dlist" where "filter P dxs = Dlist (List.filter P (list_of_dlist dxs))" end text ‹Derived operations:› context begin qualified definition null :: "'a dlist ⇒ bool" where "null dxs = List.null (list_of_dlist dxs)" qualified definition member :: "'a dlist ⇒ 'a ⇒ bool" where "member dxs = List.member (list_of_dlist dxs)" qualified definition length :: "'a dlist ⇒ nat" where "length dxs = List.length (list_of_dlist dxs)" qualified definition fold :: "('a ⇒ 'b ⇒ 'b) ⇒ 'a dlist ⇒ 'b ⇒ 'b" where "fold f dxs = List.fold f (list_of_dlist dxs)" qualified definition foldr :: "('a ⇒ 'b ⇒ 'b) ⇒ 'a dlist ⇒ 'b ⇒ 'b" where "foldr f dxs = List.foldr f (list_of_dlist dxs)" end subsection ‹Executable version obeying invariant› lemma list_of_dlist_empty [simp, code abstract]: "list_of_dlist Dlist.empty = []" by (simp add: Dlist.empty_def) lemma list_of_dlist_insert [simp, code abstract]: "list_of_dlist (Dlist.insert x dxs) = List.insert x (list_of_dlist dxs)" by (simp add: Dlist.insert_def) lemma list_of_dlist_remove [simp, code abstract]: "list_of_dlist (Dlist.remove x dxs) = remove1 x (list_of_dlist dxs)" by (simp add: Dlist.remove_def) lemma list_of_dlist_map [simp, code abstract]: "list_of_dlist (Dlist.map f dxs) = remdups (List.map f (list_of_dlist dxs))" by (simp add: Dlist.map_def) lemma list_of_dlist_filter [simp, code abstract]: "list_of_dlist (Dlist.filter P dxs) = List.filter P (list_of_dlist dxs)" by (simp add: Dlist.filter_def) text ‹Explicit executable conversion› definition dlist_of_list [simp]: "dlist_of_list = Dlist" lemma [code abstract]: "list_of_dlist (dlist_of_list xs) = remdups xs" by simp text ‹Equality› instantiation dlist :: (equal) equal begin definition "HOL.equal dxs dys ⟷ HOL.equal (list_of_dlist dxs) (list_of_dlist dys)" instance by standard (simp add: equal_dlist_def equal list_of_dlist_inject) end declare equal_dlist_def [code] lemma [code nbe]: "HOL.equal (dxs :: 'a::equal dlist) dxs ⟷ True" by (fact equal_refl) subsection ‹Induction principle and case distinction› lemma dlist_induct [case_names empty insert, induct type: dlist]: assumes empty: "P Dlist.empty" assumes insrt: "⋀x dxs. ¬ Dlist.member dxs x ⟹ P dxs ⟹ P (Dlist.insert x dxs)" shows "P dxs" proof (cases dxs) case (Abs_dlist xs) then have "distinct xs" and dxs: "dxs = Dlist xs" by (simp_all add: Dlist_def distinct_remdups_id) from ‹distinct xs› have "P (Dlist xs)" proof (induct xs) case Nil from empty show ?case by (simp add: Dlist.empty_def) next case (Cons x xs) then have "¬ Dlist.member (Dlist xs) x" and "P (Dlist xs)" by (simp_all add: Dlist.member_def List.member_def) with insrt have "P (Dlist.insert x (Dlist xs))" . with Cons show ?case by (simp add: Dlist.insert_def distinct_remdups_id) qed with dxs show "P dxs" by simp qed lemma dlist_case [cases type: dlist]: obtains (empty) "dxs = Dlist.empty" | (insert) x dys where "¬ Dlist.member dys x" and "dxs = Dlist.insert x dys" proof (cases dxs) case (Abs_dlist xs) then have dxs: "dxs = Dlist xs" and distinct: "distinct xs" by (simp_all add: Dlist_def distinct_remdups_id) show thesis proof (cases xs) case Nil with dxs have "dxs = Dlist.empty" by (simp add: Dlist.empty_def) with empty show ?thesis . next case (Cons x xs) with dxs distinct have "¬ Dlist.member (Dlist xs) x" and "dxs = Dlist.insert x (Dlist xs)" by (simp_all add: Dlist.member_def List.member_def Dlist.insert_def distinct_remdups_id) with insert show ?thesis . qed qed subsection ‹Functorial structure› functor map: map by (simp_all add: remdups_map_remdups fun_eq_iff dlist_eq_iff) subsection ‹Quickcheck generators› quickcheck_generator dlist predicate: distinct constructors: Dlist.empty, Dlist.insert subsection ‹BNF instance› context begin qualified fun wpull :: "('a × 'b) list ⇒ ('b × 'c) list ⇒ ('a × 'c) list" where "wpull [] ys = []" | "wpull xs [] = []" | "wpull ((a, b) # xs) ((b', c) # ys) = (if b ∈ snd ` set xs then (a, the (map_of (rev ((b', c) # ys)) b)) # wpull xs ((b', c) # ys) else if b' ∈ fst ` set ys then (the (map_of (map prod.swap (rev ((a, b) # xs))) b'), c) # wpull ((a, b) # xs) ys else (a, c) # wpull xs ys)" qualified lemma wpull_eq_Nil_iff [simp]: "wpull xs ys = [] ⟷ xs = [] ∨ ys = []" by(cases "(xs, ys)" rule: wpull.cases) simp_all qualified lemma wpull_induct [consumes 1, case_names Nil left[xs eq in_set IH] right[xs ys eq in_set IH] step[xs ys eq IH] ]: assumes eq: "remdups (map snd xs) = remdups (map fst ys)" and Nil: "P [] []" and left: "⋀a b xs b' c ys. ⟦ b ∈ snd ` set xs; remdups (map snd xs) = remdups (map fst ((b', c) # ys)); (b, the (map_of (rev ((b', c) # ys)) b)) ∈ set ((b', c) # ys); P xs ((b', c) # ys) ⟧ ⟹ P ((a, b) # xs) ((b', c) # ys)" and right: "⋀a b xs b' c ys. ⟦ b ∉ snd ` set xs; b' ∈ fst ` set ys; remdups (map snd ((a, b) # xs)) = remdups (map fst ys); (the (map_of (map prod.swap (rev ((a, b) #xs))) b'), b') ∈ set ((a, b) # xs); P ((a, b) # xs) ys ⟧ ⟹ P ((a, b) # xs) ((b', c) # ys)" and step: "⋀a b xs c ys. ⟦ b ∉ snd ` set xs; b ∉ fst ` set ys; remdups (map snd xs) = remdups (map fst ys); P xs ys ⟧ ⟹ P ((a, b) # xs) ((b, c) # ys)" shows "P xs ys" using eq proof(induction xs ys rule: wpull.induct) case 1 thus ?case by(simp add: Nil) next case 2 thus ?case by(simp split: split_if_asm) next case Cons: (3 a b xs b' c ys) let ?xs = "(a, b) # xs" and ?ys = "(b', c) # ys" consider (xs) "b ∈ snd ` set xs" | (ys) "b ∉ snd ` set xs" "b' ∈ fst ` set ys" | (step) "b ∉ snd ` set xs" "b' ∉ fst ` set ys" by auto thus ?case proof cases case xs with Cons.prems have eq: "remdups (map snd xs) = remdups (map fst ?ys)" by auto from xs eq have "b ∈ fst ` set ?ys" by (metis list.set_map set_remdups) hence "map_of (rev ?ys) b ≠ None" unfolding map_of_eq_None_iff by auto then obtain c' where "map_of (rev ?ys) b = Some c'" by blast then have "(b, the (map_of (rev ?ys) b)) ∈ set ?ys" by(auto dest: map_of_SomeD split: split_if_asm) from xs eq this Cons.IH(1)[OF xs eq] show ?thesis by(rule left) next case ys from ys Cons.prems have eq: "remdups (map snd ?xs) = remdups (map fst ys)" by auto from ys eq have "b' ∈ snd ` set ?xs" by (metis list.set_map set_remdups) hence "map_of (map prod.swap (rev ?xs)) b' ≠ None" unfolding map_of_eq_None_iff by(auto simp add: image_image) then obtain a' where "map_of (map prod.swap (rev ?xs)) b' = Some a'" by blast then have "(the (map_of (map prod.swap (rev ?xs)) b'), b') ∈ set ?xs" by(auto dest: map_of_SomeD split: split_if_asm) from ys eq this Cons.IH(2)[OF ys eq] show ?thesis by(rule right) next case *: step hence "remdups (map snd xs) = remdups (map fst ys)" "b = b'" using Cons.prems by auto from * this(1) Cons.IH(3)[OF * this(1)] show ?thesis unfolding ‹b = b'› by(rule step) qed qed qualified lemma set_wpull_subset: assumes "remdups (map snd xs) = remdups (map fst ys)" shows "set (wpull xs ys) ⊆ set xs O set ys" using assms by(induction xs ys rule: wpull_induct) auto qualified lemma set_fst_wpull: assumes "remdups (map snd xs) = remdups (map fst ys)" shows "fst ` set (wpull xs ys) = fst ` set xs" using assms by(induction xs ys rule: wpull_induct)(auto intro: rev_image_eqI) qualified lemma set_snd_wpull: assumes "remdups (map snd xs) = remdups (map fst ys)" shows "snd ` set (wpull xs ys) = snd ` set ys" using assms by(induction xs ys rule: wpull_induct)(auto intro: rev_image_eqI) qualified lemma wpull: assumes "distinct xs" and "distinct ys" and "set xs ⊆ {(x, y). R x y}" and "set ys ⊆ {(x, y). S x y}" and eq: "remdups (map snd xs) = remdups (map fst ys)" shows "∃zs. distinct zs ∧ set zs ⊆ {(x, y). (R OO S) x y} ∧ remdups (map fst zs) = remdups (map fst xs) ∧ remdups (map snd zs) = remdups (map snd ys)" proof(intro exI conjI) let ?zs = "remdups (wpull xs ys)" show "distinct ?zs" by simp show "set ?zs ⊆ {(x, y). (R OO S) x y}" using assms(3-4) set_wpull_subset[OF eq] by fastforce show "remdups (map fst ?zs) = remdups (map fst xs)" unfolding remdups_map_remdups using eq by(induction xs ys rule: wpull_induct)(auto simp add: set_fst_wpull intro: rev_image_eqI) show "remdups (map snd ?zs) = remdups (map snd ys)" unfolding remdups_map_remdups using eq by(induction xs ys rule: wpull_induct)(auto simp add: set_snd_wpull intro: rev_image_eqI) qed qualified lift_definition set :: "'a dlist ⇒ 'a set" is List.set . qualified lemma map_transfer [transfer_rule]: "(rel_fun op = (rel_fun (pcr_dlist op =) (pcr_dlist op =))) (λf x. remdups (List.map f x)) Dlist.map" by(simp add: rel_fun_def dlist.pcr_cr_eq cr_dlist_def Dlist.map_def remdups_remdups) bnf "'a dlist" map: Dlist.map sets: set bd: natLeq wits: Dlist.empty unfolding OO_Grp_alt mem_Collect_eq subgoal by(rule ext)(simp add: dlist_eq_iff) subgoal by(rule ext)(simp add: dlist_eq_iff remdups_map_remdups) subgoal by(simp add: dlist_eq_iff set_def cong: list.map_cong) subgoal by(simp add: set_def fun_eq_iff) subgoal by(simp add: natLeq_card_order) subgoal by(simp add: natLeq_cinfinite) subgoal by(rule ordLess_imp_ordLeq)(simp add: finite_iff_ordLess_natLeq[symmetric] set_def) subgoal by(rule predicate2I)(transfer; auto simp add: wpull) subgoal by(rule refl) subgoal by(simp add: set_def) done lifting_update dlist.lifting lifting_forget dlist.lifting end end