(* Title: HOL/Map.thy Author: Tobias Nipkow, based on a theory by David von Oheimb Copyright 1997-2003 TU Muenchen The datatype of "maps"; strongly resembles maps in VDM. *) section ‹Maps› theory Map imports List abbrevs "(=" = "⊆⇩_{m}" begin type_synonym ('a, 'b) "map" = "'a ⇒ 'b option" (infixr "⇀" 0) abbreviation empty :: "'a ⇀ 'b" where "empty ≡ λx. None" definition map_comp :: "('b ⇀ 'c) ⇒ ('a ⇀ 'b) ⇒ ('a ⇀ 'c)" (infixl "∘⇩_{m}" 55) where "f ∘⇩_{m}g = (λk. case g k of None ⇒ None | Some v ⇒ f v)" definition map_add :: "('a ⇀ 'b) ⇒ ('a ⇀ 'b) ⇒ ('a ⇀ 'b)" (infixl "++" 100) where "m1 ++ m2 = (λx. case m2 x of None ⇒ m1 x | Some y ⇒ Some y)" definition restrict_map :: "('a ⇀ 'b) ⇒ 'a set ⇒ ('a ⇀ 'b)" (infixl "|`" 110) where "m|`A = (λx. if x ∈ A then m x else None)" notation (latex output) restrict_map ("_↾⇘_⇙" [111,110] 110) definition dom :: "('a ⇀ 'b) ⇒ 'a set" where "dom m = {a. m a ≠ None}" definition ran :: "('a ⇀ 'b) ⇒ 'b set" where "ran m = {b. ∃a. m a = Some b}" definition graph :: "('a ⇀ 'b) ⇒ ('a × 'b) set" where "graph m = {(a, b) | a b. m a = Some b}" definition map_le :: "('a ⇀ 'b) ⇒ ('a ⇀ 'b) ⇒ bool" (infix "⊆⇩_{m}" 50) where "(m⇩_{1}⊆⇩_{m}m⇩_{2}) ⟷ (∀a ∈ dom m⇩_{1}. m⇩_{1}a = m⇩_{2}a)" nonterminal maplets and maplet syntax "_maplet" :: "['a, 'a] ⇒ maplet" ("_ /↦/ _") "_maplets" :: "['a, 'a] ⇒ maplet" ("_ /[↦]/ _") "" :: "maplet ⇒ maplets" ("_") "_Maplets" :: "[maplet, maplets] ⇒ maplets" ("_,/ _") "_MapUpd" :: "['a ⇀ 'b, maplets] ⇒ 'a ⇀ 'b" ("_/'(_')" [900, 0] 900) "_Map" :: "maplets ⇒ 'a ⇀ 'b" ("(1[_])") syntax (ASCII) "_maplet" :: "['a, 'a] ⇒ maplet" ("_ /|->/ _") "_maplets" :: "['a, 'a] ⇒ maplet" ("_ /[|->]/ _") translations "_MapUpd m (_Maplets xy ms)" ⇌ "_MapUpd (_MapUpd m xy) ms" "_MapUpd m (_maplet x y)" ⇌ "m(x := CONST Some y)" "_Map ms" ⇌ "_MapUpd (CONST empty) ms" "_Map (_Maplets ms1 ms2)" ↽ "_MapUpd (_Map ms1) ms2" "_Maplets ms1 (_Maplets ms2 ms3)" ↽ "_Maplets (_Maplets ms1 ms2) ms3" primrec map_of :: "('a × 'b) list ⇒ 'a ⇀ 'b" where "map_of [] = empty" | "map_of (p # ps) = (map_of ps)(fst p ↦ snd p)" definition map_upds :: "('a ⇀ 'b) ⇒ 'a list ⇒ 'b list ⇒ 'a ⇀ 'b" where "map_upds m xs ys = m ++ map_of (rev (zip xs ys))" translations "_MapUpd m (_maplets x y)" ⇌ "CONST map_upds m x y" lemma map_of_Cons_code [code]: "map_of [] k = None" "map_of ((l, v) # ps) k = (if l = k then Some v else map_of ps k)" by simp_all subsection ‹@{term [source] empty}› lemma empty_upd_none [simp]: "empty(x := None) = empty" by (rule ext) simp subsection ‹@{term [source] map_upd}› lemma map_upd_triv: "t k = Some x ⟹ t(k↦x) = t" by (rule ext) simp lemma map_upd_nonempty [simp]: "t(k↦x) ≠ empty" proof assume "t(k ↦ x) = empty" then have "(t(k ↦ x)) k = None" by simp then show False by simp qed lemma map_upd_eqD1: assumes "m(a↦x) = n(a↦y)" shows "x = y" proof - from assms have "(m(a↦x)) a = (n(a↦y)) a" by simp then show ?thesis by simp qed lemma map_upd_Some_unfold: "((m(a↦b)) x = Some y) = (x = a ∧ b = y ∨ x ≠ a ∧ m x = Some y)" by auto lemma image_map_upd [simp]: "x ∉ A ⟹ m(x ↦ y) ` A = m ` A" by auto lemma finite_range_updI: assumes "finite (range f)" shows "finite (range (f(a↦b)))" proof - have "range (f(a↦b)) ⊆ insert (Some b) (range f)" by auto then show ?thesis by (rule finite_subset) (use assms in auto) qed subsection ‹@{term [source] map_of}› lemma map_of_eq_empty_iff [simp]: "map_of xys = empty ⟷ xys = []" proof show "map_of xys = empty ⟹ xys = []" by (induction xys) simp_all qed simp lemma empty_eq_map_of_iff [simp]: "empty = map_of xys ⟷ xys = []" by(subst eq_commute) simp lemma map_of_eq_None_iff: "(map_of xys x = None) = (x ∉ fst ` (set xys))" by (induct xys) simp_all lemma map_of_eq_Some_iff [simp]: "distinct(map fst xys) ⟹ (map_of xys x = Some y) = ((x,y) ∈ set xys)" proof (induct xys) case (Cons xy xys) then show ?case by (cases xy) (auto simp flip: map_of_eq_None_iff) qed auto lemma Some_eq_map_of_iff [simp]: "distinct(map fst xys) ⟹ (Some y = map_of xys x) = ((x,y) ∈ set xys)" by (auto simp del: map_of_eq_Some_iff simp: map_of_eq_Some_iff [symmetric]) lemma map_of_is_SomeI [simp]: "⟦distinct(map fst xys); (x,y) ∈ set xys⟧ ⟹ map_of xys x = Some y" by simp lemma map_of_zip_is_None [simp]: "length xs = length ys ⟹ (map_of (zip xs ys) x = None) = (x ∉ set xs)" by (induct rule: list_induct2) simp_all lemma map_of_zip_is_Some: assumes "length xs = length ys" shows "x ∈ set xs ⟷ (∃y. map_of (zip xs ys) x = Some y)" using assms by (induct rule: list_induct2) simp_all lemma map_of_zip_upd: fixes x :: 'a and xs :: "'a list" and ys zs :: "'b list" assumes "length ys = length xs" and "length zs = length xs" and "x ∉ set xs" and "map_of (zip xs ys)(x ↦ y) = map_of (zip xs zs)(x ↦ z)" shows "map_of (zip xs ys) = map_of (zip xs zs)" proof fix x' :: 'a show "map_of (zip xs ys) x' = map_of (zip xs zs) x'" proof (cases "x = x'") case True from assms True map_of_zip_is_None [of xs ys x'] have "map_of (zip xs ys) x' = None" by simp moreover from assms True map_of_zip_is_None [of xs zs x'] have "map_of (zip xs zs) x' = None" by simp ultimately show ?thesis by simp next case False from assms have "(map_of (zip xs ys)(x ↦ y)) x' = (map_of (zip xs zs)(x ↦ z)) x'" by auto with False show ?thesis by simp qed qed lemma map_of_zip_inject: assumes "length ys = length xs" and "length zs = length xs" and dist: "distinct xs" and map_of: "map_of (zip xs ys) = map_of (zip xs zs)" shows "ys = zs" using assms(1) assms(2)[symmetric] using dist map_of proof (induct ys xs zs rule: list_induct3) case Nil show ?case by simp next case (Cons y ys x xs z zs) from ‹map_of (zip (x#xs) (y#ys)) = map_of (zip (x#xs) (z#zs))› have map_of: "map_of (zip xs ys)(x ↦ y) = map_of (zip xs zs)(x ↦ z)" by simp from Cons have "length ys = length xs" and "length zs = length xs" and "x ∉ set xs" by simp_all then have "map_of (zip xs ys) = map_of (zip xs zs)" using map_of by (rule map_of_zip_upd) with Cons.hyps ‹distinct (x # xs)› have "ys = zs" by simp moreover from map_of have "y = z" by (rule map_upd_eqD1) ultimately show ?case by simp qed lemma map_of_zip_nth: assumes "length xs = length ys" assumes "distinct xs" assumes "i < length ys" shows "map_of (zip xs ys) (xs ! i) = Some (ys ! i)" using assms proof (induct arbitrary: i rule: list_induct2) case Nil then show ?case by simp next case (Cons x xs y ys) then show ?case using less_Suc_eq_0_disj by auto qed lemma map_of_zip_map: "map_of (zip xs (map f xs)) = (λx. if x ∈ set xs then Some (f x) else None)" by (induct xs) (simp_all add: fun_eq_iff) lemma finite_range_map_of: "finite (range (map_of xys))" proof (induct xys) case (Cons a xys) then show ?case using finite_range_updI by fastforce qed auto lemma map_of_SomeD: "map_of xs k = Some y ⟹ (k, y) ∈ set xs" by (induct xs) (auto split: if_splits) lemma map_of_mapk_SomeI: "inj f ⟹ map_of t k = Some x ⟹ map_of (map (case_prod (λk. Pair (f k))) t) (f k) = Some x" by (induct t) (auto simp: inj_eq) lemma weak_map_of_SomeI: "(k, x) ∈ set l ⟹ ∃x. map_of l k = Some x" by (induct l) auto lemma map_of_filter_in: "map_of xs k = Some z ⟹ P k z ⟹ map_of (filter (case_prod P) xs) k = Some z" by (induct xs) auto lemma map_of_map: "map_of (map (λ(k, v). (k, f v)) xs) = map_option f ∘ map_of xs" by (induct xs) (auto simp: fun_eq_iff) lemma dom_map_option: "dom (λk. map_option (f k) (m k)) = dom m" by (simp add: dom_def) lemma dom_map_option_comp [simp]: "dom (map_option g ∘ m) = dom m" using dom_map_option [of "λ_. g" m] by (simp add: comp_def) subsection ‹\<^const>‹map_option› related› lemma map_option_o_empty [simp]: "map_option f ∘ empty = empty" by (rule ext) simp lemma map_option_o_map_upd [simp]: "map_option f ∘ m(a↦b) = (map_option f ∘ m)(a↦f b)" by (rule ext) simp subsection ‹@{term [source] map_comp} related› lemma map_comp_empty [simp]: "m ∘⇩_{m}empty = empty" "empty ∘⇩_{m}m = empty" by (auto simp: map_comp_def split: option.splits) lemma map_comp_simps [simp]: "m2 k = None ⟹ (m1 ∘⇩_{m}m2) k = None" "m2 k = Some k' ⟹ (m1 ∘⇩_{m}m2) k = m1 k'" by (auto simp: map_comp_def) lemma map_comp_Some_iff: "((m1 ∘⇩_{m}m2) k = Some v) = (∃k'. m2 k = Some k' ∧ m1 k' = Some v)" by (auto simp: map_comp_def split: option.splits) lemma map_comp_None_iff: "((m1 ∘⇩_{m}m2) k = None) = (m2 k = None ∨ (∃k'. m2 k = Some k' ∧ m1 k' = None)) " by (auto simp: map_comp_def split: option.splits) subsection ‹‹++›› lemma map_add_empty[simp]: "m ++ empty = m" by(simp add: map_add_def) lemma empty_map_add[simp]: "empty ++ m = m" by (rule ext) (simp add: map_add_def split: option.split) lemma map_add_assoc[simp]: "m1 ++ (m2 ++ m3) = (m1 ++ m2) ++ m3" by (rule ext) (simp add: map_add_def split: option.split) lemma map_add_Some_iff: "((m ++ n) k = Some x) = (n k = Some x ∨ n k = None ∧ m k = Some x)" by (simp add: map_add_def split: option.split) lemma map_add_SomeD [dest!]: "(m ++ n) k = Some x ⟹ n k = Some x ∨ n k = None ∧ m k = Some x" by (rule map_add_Some_iff [THEN iffD1]) lemma map_add_find_right [simp]: "n k = Some xx ⟹ (m ++ n) k = Some xx" by (subst map_add_Some_iff) fast lemma map_add_None [iff]: "((m ++ n) k = None) = (n k = None ∧ m k = None)" by (simp add: map_add_def split: option.split) lemma map_add_upd[simp]: "f ++ g(x↦y) = (f ++ g)(x↦y)" by (rule ext) (simp add: map_add_def) lemma map_add_upds[simp]: "m1 ++ (m2(xs[↦]ys)) = (m1++m2)(xs[↦]ys)" by (simp add: map_upds_def) lemma map_add_upd_left: "m∉dom e2 ⟹ e1(m ↦ u1) ++ e2 = (e1 ++ e2)(m ↦ u1)" by (rule ext) (auto simp: map_add_def dom_def split: option.split) lemma map_of_append[simp]: "map_of (xs @ ys) = map_of ys ++ map_of xs" unfolding map_add_def proof (induct xs) case (Cons a xs) then show ?case by (force split: option.split) qed auto lemma finite_range_map_of_map_add: "finite (range f) ⟹ finite (range (f ++ map_of l))" proof (induct l) case (Cons a l) then show ?case by (metis finite_range_updI map_add_upd map_of.simps(2)) qed auto lemma inj_on_map_add_dom [iff]: "inj_on (m ++ m') (dom m') = inj_on m' (dom m')" by (fastforce simp: map_add_def dom_def inj_on_def split: option.splits) lemma map_upds_fold_map_upd: "m(ks[↦]vs) = foldl (λm (k, v). m(k ↦ v)) m (zip ks vs)" unfolding map_upds_def proof (rule sym, rule zip_obtain_same_length) fix ks :: "'a list" and vs :: "'b list" assume "length ks = length vs" then show "foldl (λm (k, v). m(k↦v)) m (zip ks vs) = m ++ map_of (rev (zip ks vs))" by(induct arbitrary: m rule: list_induct2) simp_all qed lemma map_add_map_of_foldr: "m ++ map_of ps = foldr (λ(k, v) m. m(k ↦ v)) ps m" by (induct ps) (auto simp: fun_eq_iff map_add_def) subsection ‹@{term [source] restrict_map}› lemma restrict_map_to_empty [simp]: "m|`{} = empty" by (simp add: restrict_map_def) lemma restrict_map_insert: "f |` (insert a A) = (f |` A)(a := f a)" by (auto simp: restrict_map_def) lemma restrict_map_empty [simp]: "empty|`D = empty" by (simp add: restrict_map_def) lemma restrict_in [simp]: "x ∈ A ⟹ (m|`A) x = m x" by (simp add: restrict_map_def) lemma restrict_out [simp]: "x ∉ A ⟹ (m|`A) x = None" by (simp add: restrict_map_def) lemma ran_restrictD: "y ∈ ran (m|`A) ⟹ ∃x∈A. m x = Some y" by (auto simp: restrict_map_def ran_def split: if_split_asm) lemma dom_restrict [simp]: "dom (m|`A) = dom m ∩ A" by (auto simp: restrict_map_def dom_def split: if_split_asm) lemma restrict_upd_same [simp]: "m(x↦y)|`(-{x}) = m|`(-{x})" by (rule ext) (auto simp: restrict_map_def) lemma restrict_restrict [simp]: "m|`A|`B = m|`(A∩B)" by (rule ext) (auto simp: restrict_map_def) lemma restrict_fun_upd [simp]: "m(x := y)|`D = (if x ∈ D then (m|`(D-{x}))(x := y) else m|`D)" by (simp add: restrict_map_def fun_eq_iff) lemma fun_upd_None_restrict [simp]: "(m|`D)(x := None) = (if x ∈ D then m|`(D - {x}) else m|`D)" by (simp add: restrict_map_def fun_eq_iff) lemma fun_upd_restrict: "(m|`D)(x := y) = (m|`(D-{x}))(x := y)" by (simp add: restrict_map_def fun_eq_iff) lemma fun_upd_restrict_conv [simp]: "x ∈ D ⟹ (m|`D)(x := y) = (m|`(D-{x}))(x := y)" by (rule fun_upd_restrict) lemma map_of_map_restrict: "map_of (map (λk. (k, f k)) ks) = (Some ∘ f) |` set ks" by (induct ks) (simp_all add: fun_eq_iff restrict_map_insert) lemma restrict_complement_singleton_eq: "f |` (- {x}) = f(x := None)" by auto subsection ‹@{term [source] map_upds}› lemma map_upds_Nil1 [simp]: "m([] [↦] bs) = m" by (simp add: map_upds_def) lemma map_upds_Nil2 [simp]: "m(as [↦] []) = m" by (simp add:map_upds_def) lemma map_upds_Cons [simp]: "m(a#as [↦] b#bs) = (m(a↦b))(as[↦]bs)" by (simp add:map_upds_def) lemma map_upds_append1 [simp]: "size xs < size ys ⟹ m(xs@[x] [↦] ys) = m(xs [↦] ys)(x ↦ ys!size xs)" proof (induct xs arbitrary: ys m) case Nil then show ?case by (auto simp: neq_Nil_conv) next case (Cons a xs) then show ?case by (cases ys) auto qed lemma map_upds_list_update2_drop [simp]: "size xs ≤ i ⟹ m(xs[↦]ys[i:=y]) = m(xs[↦]ys)" proof (induct xs arbitrary: m ys i) case Nil then show ?case by auto next case (Cons a xs) then show ?case by (cases ys) (use Cons in ‹auto split: nat.split›) qed text ‹Something weirdly sensitive about this proof, which needs only four lines in apply style› lemma map_upd_upds_conv_if: "(f(x↦y))(xs [↦] ys) = (if x ∈ set(take (length ys) xs) then f(xs [↦] ys) else (f(xs [↦] ys))(x↦y))" proof (induct xs arbitrary: x y ys f) case (Cons a xs) show ?case proof (cases ys) case (Cons z zs) then show ?thesis using Cons.hyps apply (auto split: if_split simp: fun_upd_twist) using Cons.hyps apply fastforce+ done qed auto qed auto lemma map_upds_twist [simp]: "a ∉ set as ⟹ m(a↦b)(as[↦]bs) = m(as[↦]bs)(a↦b)" using set_take_subset by (fastforce simp add: map_upd_upds_conv_if) lemma map_upds_apply_nontin [simp]: "x ∉ set xs ⟹ (f(xs[↦]ys)) x = f x" proof (induct xs arbitrary: ys) case (Cons a xs) then show ?case by (cases ys) (auto simp: map_upd_upds_conv_if) qed auto lemma fun_upds_append_drop [simp]: "size xs = size ys ⟹ m(xs@zs[↦]ys) = m(xs[↦]ys)" proof (induct xs arbitrary: ys) case (Cons a xs) then show ?case by (cases ys) (auto simp: map_upd_upds_conv_if) qed auto lemma fun_upds_append2_drop [simp]: "size xs = size ys ⟹ m(xs[↦]ys@zs) = m(xs[↦]ys)" proof (induct xs arbitrary: ys) case (Cons a xs) then show ?case by (cases ys) (auto simp: map_upd_upds_conv_if) qed auto lemma restrict_map_upds[simp]: "⟦ length xs = length ys; set xs ⊆ D ⟧ ⟹ m(xs [↦] ys)|`D = (m|`(D - set xs))(xs [↦] ys)" proof (induct xs arbitrary: m ys) case (Cons a xs) then show ?case proof (cases ys) case (Cons z zs) with Cons.hyps Cons.prems show ?thesis apply (simp add: insert_absorb flip: Diff_insert) apply (auto simp add: map_upd_upds_conv_if) done qed auto qed auto subsection ‹@{term [source] dom}› lemma dom_eq_empty_conv [simp]: "dom f = {} ⟷ f = empty" by (auto simp: dom_def) lemma domI: "m a = Some b ⟹ a ∈ dom m" by (simp add: dom_def) (* declare domI [intro]? *) lemma domD: "a ∈ dom m ⟹ ∃b. m a = Some b" by (cases "m a") (auto simp add: dom_def) lemma domIff [iff, simp del, code_unfold]: "a ∈ dom m ⟷ m a ≠ None" by (simp add: dom_def) lemma dom_empty [simp]: "dom empty = {}" by (simp add: dom_def) lemma dom_fun_upd [simp]: "dom(f(x := y)) = (if y = None then dom f - {x} else insert x (dom f))" by (auto simp: dom_def) lemma dom_if: "dom (λx. if P x then f x else g x) = dom f ∩ {x. P x} ∪ dom g ∩ {x. ¬ P x}" by (auto split: if_splits) lemma dom_map_of_conv_image_fst: "dom (map_of xys) = fst ` set xys" by (induct xys) (auto simp add: dom_if) lemma dom_map_of_zip [simp]: "length xs = length ys ⟹ dom (map_of (zip xs ys)) = set xs" by (induct rule: list_induct2) (auto simp: dom_if) lemma finite_dom_map_of: "finite (dom (map_of l))" by (induct l) (auto simp: dom_def insert_Collect [symmetric]) lemma dom_map_upds [simp]: "dom(m(xs[↦]ys)) = set(take (length ys) xs) ∪ dom m" proof (induct xs arbitrary: ys) case (Cons a xs) then show ?case by (cases ys) (auto simp: map_upd_upds_conv_if) qed auto lemma dom_map_add [simp]: "dom (m ++ n) = dom n ∪ dom m" by (auto simp: dom_def) lemma dom_override_on [simp]: "dom (override_on f g A) = (dom f - {a. a ∈ A - dom g}) ∪ {a. a ∈ A ∩ dom g}" by (auto simp: dom_def override_on_def) lemma map_add_comm: "dom m1 ∩ dom m2 = {} ⟹ m1 ++ m2 = m2 ++ m1" by (rule ext) (force simp: map_add_def dom_def split: option.split) lemma map_add_dom_app_simps: "m ∈ dom l2 ⟹ (l1 ++ l2) m = l2 m" "m ∉ dom l1 ⟹ (l1 ++ l2) m = l2 m" "m ∉ dom l2 ⟹ (l1 ++ l2) m = l1 m" by (auto simp add: map_add_def split: option.split_asm) lemma dom_const [simp]: "dom (λx. Some (f x)) = UNIV" by auto (* Due to John Matthews - could be rephrased with dom *) lemma finite_map_freshness: "finite (dom (f :: 'a ⇀ 'b)) ⟹ ¬ finite (UNIV :: 'a set) ⟹ ∃x. f x = None" by (bestsimp dest: ex_new_if_finite) lemma dom_minus: "f x = None ⟹ dom f - insert x A = dom f - A" unfolding dom_def by simp lemma insert_dom: "f x = Some y ⟹ insert x (dom f) = dom f" unfolding dom_def by auto lemma map_of_map_keys: "set xs = dom m ⟹ map_of (map (λk. (k, the (m k))) xs) = m" by (rule ext) (auto simp add: map_of_map_restrict restrict_map_def) lemma map_of_eqI: assumes set_eq: "set (map fst xs) = set (map fst ys)" assumes map_eq: "∀k∈set (map fst xs). map_of xs k = map_of ys k" shows "map_of xs = map_of ys" proof (rule ext) fix k show "map_of xs k = map_of ys k" proof (cases "map_of xs k") case None then have "k ∉ set (map fst xs)" by (simp add: map_of_eq_None_iff) with set_eq have "k ∉ set (map fst ys)" by simp then have "map_of ys k = None" by (simp add: map_of_eq_None_iff) with None show ?thesis by simp next case (Some v) then have "k ∈ set (map fst xs)" by (auto simp add: dom_map_of_conv_image_fst [symmetric]) with map_eq show ?thesis by auto qed qed lemma map_of_eq_dom: assumes "map_of xs = map_of ys" shows "fst ` set xs = fst ` set ys" proof - from assms have "dom (map_of xs) = dom (map_of ys)" by simp then show ?thesis by (simp add: dom_map_of_conv_image_fst) qed lemma finite_set_of_finite_maps: assumes "finite A" "finite B" shows "finite {m. dom m = A ∧ ran m ⊆ B}" (is "finite ?S") proof - let ?S' = "{m. ∀x. (x ∈ A ⟶ m x ∈ Some ` B) ∧ (x ∉ A ⟶ m x = None)}" have "?S = ?S'" proof show "?S ⊆ ?S'" by (auto simp: dom_def ran_def image_def) show "?S' ⊆ ?S" proof fix m assume "m ∈ ?S'" hence 1: "dom m = A" by force hence 2: "ran m ⊆ B" using ‹m ∈ ?S'› by (auto simp: dom_def ran_def) from 1 2 show "m ∈ ?S" by blast qed qed with assms show ?thesis by(simp add: finite_set_of_finite_funs) qed subsection ‹@{term [source] ran}› lemma ranI: "m a = Some b ⟹ b ∈ ran m" by (auto simp: ran_def) (* declare ranI [intro]? *) lemma ran_empty [simp]: "ran empty = {}" by (auto simp: ran_def) lemma ran_map_upd [simp]: "m a = None ⟹ ran(m(a↦b)) = insert b (ran m)" unfolding ran_def by force lemma fun_upd_None_if_notin_dom[simp]: "k ∉ dom m ⟹ m(k := None) = m" by auto lemma ran_map_upd_Some: "⟦ m x = Some y; inj_on m (dom m); z ∉ ran m ⟧ ⟹ ran(m(x := Some z)) = ran m - {y} ∪ {z}" by(force simp add: ran_def domI inj_onD) lemma ran_map_add: assumes "dom m1 ∩ dom m2 = {}" shows "ran (m1 ++ m2) = ran m1 ∪ ran m2" proof show "ran (m1 ++ m2) ⊆ ran m1 ∪ ran m2" unfolding ran_def by auto next show "ran m1 ∪ ran m2 ⊆ ran (m1 ++ m2)" proof - have "(m1 ++ m2) x = Some y" if "m1 x = Some y" for x y using assms map_add_comm that by fastforce moreover have "(m1 ++ m2) x = Some y" if "m2 x = Some y" for x y using assms that by auto ultimately show ?thesis unfolding ran_def by blast qed qed lemma finite_ran: assumes "finite (dom p)" shows "finite (ran p)" proof - have "ran p = (λx. the (p x)) ` dom p" unfolding ran_def by force from this ‹finite (dom p)› show ?thesis by auto qed lemma ran_distinct: assumes dist: "distinct (map fst al)" shows "ran (map_of al) = snd ` set al" using assms proof (induct al) case Nil then show ?case by simp next case (Cons kv al) then have "ran (map_of al) = snd ` set al" by simp moreover from Cons.prems have "map_of al (fst kv) = None" by (simp add: map_of_eq_None_iff) ultimately show ?case by (simp only: map_of.simps ran_map_upd) simp qed lemma ran_map_of_zip: assumes "length xs = length ys" "distinct xs" shows "ran (map_of (zip xs ys)) = set ys" using assms by (simp add: ran_distinct set_map[symmetric]) lemma ran_map_option: "ran (λx. map_option f (m x)) = f ` ran m" by (auto simp add: ran_def) subsection ‹@{term [source] graph}› lemma graph_empty[simp]: "graph empty = {}" unfolding graph_def by simp lemma in_graphI: "m k = Some v ⟹ (k, v) ∈ graph m" unfolding graph_def by blast lemma in_graphD: "(k, v) ∈ graph m ⟹ m k = Some v" unfolding graph_def by blast lemma graph_map_upd[simp]: "graph (m(k ↦ v)) = insert (k, v) (graph (m(k := None)))" unfolding graph_def by (auto split: if_splits) lemma graph_fun_upd_None: "graph (m(k := None)) = {e ∈ graph m. fst e ≠ k}" unfolding graph_def by (auto split: if_splits) lemma graph_restrictD: assumes "(k, v) ∈ graph (m |` A)" shows "k ∈ A" and "m k = Some v" using assms unfolding graph_def by (auto simp: restrict_map_def split: if_splits) lemma graph_map_comp[simp]: "graph (m1 ∘⇩_{m}m2) = graph m2 O graph m1" unfolding graph_def by (auto simp: map_comp_Some_iff relcomp_unfold) lemma graph_map_add: "dom m1 ∩ dom m2 = {} ⟹ graph (m1 ++ m2) = graph m1 ∪ graph m2" unfolding graph_def using map_add_comm by force lemma graph_eq_to_snd_dom: "graph m = (λx. (x, the (m x))) ` dom m" unfolding graph_def dom_def by force lemma fst_graph_eq_dom: "fst ` graph m = dom m" unfolding graph_eq_to_snd_dom by force lemma graph_domD: "x ∈ graph m ⟹ fst x ∈ dom m" using fst_graph_eq_dom by (metis imageI) lemma snd_graph_ran: "snd ` graph m = ran m" unfolding graph_def ran_def by force lemma graph_ranD: "x ∈ graph m ⟹ snd x ∈ ran m" using snd_graph_ran by (metis imageI) lemma finite_graph_map_of: "finite (graph (map_of al))" unfolding graph_eq_to_snd_dom finite_dom_map_of using finite_dom_map_of by blast lemma graph_map_of_if_distinct_dom: "distinct (map fst al) ⟹ graph (map_of al) = set al" unfolding graph_def by auto lemma finite_graph_iff_finite_dom[simp]: "finite (graph m) = finite (dom m)" by (metis graph_eq_to_snd_dom finite_imageI fst_graph_eq_dom) lemma inj_on_fst_graph: "inj_on fst (graph m)" unfolding graph_def inj_on_def by force subsection ‹‹map_le›› lemma map_le_empty [simp]: "empty ⊆⇩_{m}g" by (simp add: map_le_def) lemma upd_None_map_le [simp]: "f(x := None) ⊆⇩_{m}f" by (force simp add: map_le_def) lemma map_le_upd[simp]: "f ⊆⇩_{m}g ==> f(a := b) ⊆⇩_{m}g(a := b)" by (fastforce simp add: map_le_def) lemma map_le_imp_upd_le [simp]: "m1 ⊆⇩_{m}m2 ⟹ m1(x := None) ⊆⇩_{m}m2(x ↦ y)" by (force simp add: map_le_def) lemma map_le_upds [simp]: "f ⊆⇩_{m}g ⟹ f(as [↦] bs) ⊆⇩_{m}g(as [↦] bs)" proof (induct as arbitrary: f g bs) case (Cons a as) then show ?case by (cases bs) (use Cons in auto) qed auto lemma map_le_implies_dom_le: "(f ⊆⇩_{m}g) ⟹ (dom f ⊆ dom g)" by (fastforce simp add: map_le_def dom_def) lemma map_le_refl [simp]: "f ⊆⇩_{m}f" by (simp add: map_le_def) lemma map_le_trans[trans]: "⟦ m1 ⊆⇩_{m}m2; m2 ⊆⇩_{m}m3⟧ ⟹ m1 ⊆⇩_{m}m3" by (auto simp add: map_le_def dom_def) lemma map_le_antisym: "⟦ f ⊆⇩_{m}g; g ⊆⇩_{m}f ⟧ ⟹ f = g" unfolding map_le_def by (metis ext domIff) lemma map_le_map_add [simp]: "f ⊆⇩_{m}g ++ f" by (fastforce simp: map_le_def) lemma map_le_iff_map_add_commute: "f ⊆⇩_{m}f ++ g ⟷ f ++ g = g ++ f" by (fastforce simp: map_add_def map_le_def fun_eq_iff split: option.splits) lemma map_add_le_mapE: "f ++ g ⊆⇩_{m}h ⟹ g ⊆⇩_{m}h" by (fastforce simp: map_le_def map_add_def dom_def) lemma map_add_le_mapI: "⟦ f ⊆⇩_{m}h; g ⊆⇩_{m}h ⟧ ⟹ f ++ g ⊆⇩_{m}h" by (auto simp: map_le_def map_add_def dom_def split: option.splits) lemma map_add_subsumed1: "f ⊆⇩_{m}g ⟹ f++g = g" by (simp add: map_add_le_mapI map_le_antisym) lemma map_add_subsumed2: "f ⊆⇩_{m}g ⟹ g++f = g" by (metis map_add_subsumed1 map_le_iff_map_add_commute) lemma dom_eq_singleton_conv: "dom f = {x} ⟷ (∃v. f = [x ↦ v])" (is "?lhs ⟷ ?rhs") proof assume ?rhs then show ?lhs by (auto split: if_split_asm) next assume ?lhs then obtain v where v: "f x = Some v" by auto show ?rhs proof show "f = [x ↦ v]" proof (rule map_le_antisym) show "[x ↦ v] ⊆⇩_{m}f" using v by (auto simp add: map_le_def) show "f ⊆⇩_{m}[x ↦ v]" using ‹dom f = {x}› ‹f x = Some v› by (auto simp add: map_le_def) qed qed qed lemma map_add_eq_empty_iff[simp]: "(f++g = empty) ⟷ f = empty ∧ g = empty" by (metis map_add_None) lemma empty_eq_map_add_iff[simp]: "(empty = f++g) ⟷ f = empty ∧ g = empty" by(subst map_add_eq_empty_iff[symmetric])(rule eq_commute) subsection ‹Various› lemma set_map_of_compr: assumes distinct: "distinct (map fst xs)" shows "set xs = {(k, v). map_of xs k = Some v}" using assms proof (induct xs) case Nil then show ?case by simp next case (Cons x xs) obtain k v where "x = (k, v)" by (cases x) blast with Cons.prems have "k ∉ dom (map_of xs)" by (simp add: dom_map_of_conv_image_fst) then have *: "insert (k, v) {(k, v). map_of xs k = Some v} = {(k', v'). (map_of xs(k ↦ v)) k' = Some v'}" by (auto split: if_splits) from Cons have "set xs = {(k, v). map_of xs k = Some v}" by simp with * ‹x = (k, v)› show ?case by simp qed lemma eq_key_imp_eq_value: "v1 = v2" if "distinct (map fst xs)" "(k, v1) ∈ set xs" "(k, v2) ∈ set xs" proof - from that have "inj_on fst (set xs)" by (simp add: distinct_map) moreover have "fst (k, v1) = fst (k, v2)" by simp ultimately have "(k, v1) = (k, v2)" by (rule inj_onD) (fact that)+ then show ?thesis by simp qed lemma map_of_inject_set: assumes distinct: "distinct (map fst xs)" "distinct (map fst ys)" shows "map_of xs = map_of ys ⟷ set xs = set ys" (is "?lhs ⟷ ?rhs") proof assume ?lhs moreover from ‹distinct (map fst xs)› have "set xs = {(k, v). map_of xs k = Some v}" by (rule set_map_of_compr) moreover from ‹distinct (map fst ys)› have "set ys = {(k, v). map_of ys k = Some v}" by (rule set_map_of_compr) ultimately show ?rhs by simp next assume ?rhs show ?lhs proof fix k show "map_of xs k = map_of ys k" proof (cases "map_of xs k") case None with ‹?rhs› have "map_of ys k = None" by (simp add: map_of_eq_None_iff) with None show ?thesis by simp next case (Some v) with distinct ‹?rhs› have "map_of ys k = Some v" by simp with Some show ?thesis by simp qed qed qed lemma finite_Map_induct[consumes 1, case_names empty update]: assumes "finite (dom m)" assumes "P Map.empty" assumes "⋀k v m. finite (dom m) ⟹ k ∉ dom m ⟹ P m ⟹ P (m(k ↦ v))" shows "P m" using assms(1) proof(induction "dom m" arbitrary: m rule: finite_induct) case empty then show ?case using assms(2) unfolding dom_def by simp next case (insert x F) then have "finite (dom (m(x:=None)))" "x ∉ dom (m(x:=None))" "P (m(x:=None))" by (metis Diff_insert_absorb dom_fun_upd)+ with assms(3)[OF this] show ?case by (metis fun_upd_triv fun_upd_upd option.exhaust) qed hide_const (open) Map.empty Map.graph end