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src/HOL/List.thy

author | haftmann |

Tue, 31 Oct 2006 14:58:14 +0100 | |

changeset 21126 | 4dbc3ccbaab0 |

parent 21113 | 5b76e541cc0a |

child 21131 | a447addc14af |

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

adapted seralizer syntax

(* Title: HOL/List.thy ID: $Id$ Author: Tobias Nipkow *) header {* The datatype of finite lists *} theory List imports PreList FunDef begin datatype 'a list = Nil ("[]") | Cons 'a "'a list" (infixr "#" 65) subsection{*Basic list processing functions*} consts "@" :: "'a list => 'a list => 'a list" (infixr 65) filter:: "('a => bool) => 'a list => 'a list" concat:: "'a list list => 'a list" foldl :: "('b => 'a => 'b) => 'b => 'a list => 'b" foldr :: "('a => 'b => 'b) => 'a list => 'b => 'b" hd:: "'a list => 'a" tl:: "'a list => 'a list" last:: "'a list => 'a" butlast :: "'a list => 'a list" set :: "'a list => 'a set" map :: "('a=>'b) => ('a list => 'b list)" nth :: "'a list => nat => 'a" (infixl "!" 100) list_update :: "'a list => nat => 'a => 'a list" take:: "nat => 'a list => 'a list" drop:: "nat => 'a list => 'a list" takeWhile :: "('a => bool) => 'a list => 'a list" dropWhile :: "('a => bool) => 'a list => 'a list" rev :: "'a list => 'a list" zip :: "'a list => 'b list => ('a * 'b) list" upt :: "nat => nat => nat list" ("(1[_..</_'])") remdups :: "'a list => 'a list" remove1 :: "'a => 'a list => 'a list" "distinct":: "'a list => bool" replicate :: "nat => 'a => 'a list" splice :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" abbreviation upto:: "nat => nat => nat list" ("(1[_../_])") "[i..j] == [i..<(Suc j)]" nonterminals lupdbinds lupdbind syntax -- {* list Enumeration *} "@list" :: "args => 'a list" ("[(_)]") -- {* Special syntax for filter *} "@filter" :: "[pttrn, 'a list, bool] => 'a list" ("(1[_:_./ _])") -- {* list update *} "_lupdbind":: "['a, 'a] => lupdbind" ("(2_ :=/ _)") "" :: "lupdbind => lupdbinds" ("_") "_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds" ("_,/ _") "_LUpdate" :: "['a, lupdbinds] => 'a" ("_/[(_)]" [900,0] 900) translations "[x, xs]" == "x#[xs]" "[x]" == "x#[]" "[x:xs . P]"== "filter (%x. P) xs" "_LUpdate xs (_lupdbinds b bs)"== "_LUpdate (_LUpdate xs b) bs" "xs[i:=x]" == "list_update xs i x" syntax (xsymbols) "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])") syntax (HTML output) "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])") text {* Function @{text size} is overloaded for all datatypes. Users may refer to the list version as @{text length}. *} abbreviation length :: "'a list => nat" "length == size" primrec "hd(x#xs) = x" primrec "tl([]) = []" "tl(x#xs) = xs" primrec "last(x#xs) = (if xs=[] then x else last xs)" primrec "butlast []= []" "butlast(x#xs) = (if xs=[] then [] else x#butlast xs)" primrec "set [] = {}" "set (x#xs) = insert x (set xs)" primrec "map f [] = []" "map f (x#xs) = f(x)#map f xs" primrec append_Nil:"[]@ys = ys" append_Cons: "(x#xs)@ys = x#(xs@ys)" primrec "rev([]) = []" "rev(x#xs) = rev(xs) @ [x]" primrec "filter P [] = []" "filter P (x#xs) = (if P x then x#filter P xs else filter P xs)" primrec foldl_Nil:"foldl f a [] = a" foldl_Cons: "foldl f a (x#xs) = foldl f (f a x) xs" primrec "foldr f [] a = a" "foldr f (x#xs) a = f x (foldr f xs a)" primrec "concat([]) = []" "concat(x#xs) = x @ concat(xs)" primrec drop_Nil:"drop n [] = []" drop_Cons: "drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)" -- {*Warning: simpset does not contain this definition, but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *} primrec take_Nil:"take n [] = []" take_Cons: "take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)" -- {*Warning: simpset does not contain this definition, but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *} primrec nth_Cons:"(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)" -- {*Warning: simpset does not contain this definition, but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *} primrec "[][i:=v] = []" "(x#xs)[i:=v] = (case i of 0 => v # xs | Suc j => x # xs[j:=v])" primrec "takeWhile P [] = []" "takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])" primrec "dropWhile P [] = []" "dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)" primrec "zip xs [] = []" zip_Cons: "zip xs (y#ys) = (case xs of [] => [] | z#zs => (z,y)#zip zs ys)" -- {*Warning: simpset does not contain this definition, but separate theorems for @{text "xs = []"} and @{text "xs = z # zs"} *} primrec upt_0: "[i..<0] = []" upt_Suc: "[i..<(Suc j)] = (if i <= j then [i..<j] @ [j] else [])" primrec "distinct [] = True" "distinct (x#xs) = (x ~: set xs \<and> distinct xs)" primrec "remdups [] = []" "remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)" primrec "remove1 x [] = []" "remove1 x (y#xs) = (if x=y then xs else y # remove1 x xs)" primrec replicate_0: "replicate 0 x = []" replicate_Suc: "replicate (Suc n) x = x # replicate n x" definition rotate1 :: "'a list \<Rightarrow> 'a list" rotate1_def: "rotate1 xs = (case xs of [] \<Rightarrow> [] | x#xs \<Rightarrow> xs @ [x])" rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" rotate_def: "rotate n = rotate1 ^ n" list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool" list_all2_def: "list_all2 P xs ys = (length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y))" sublist :: "'a list => nat set => 'a list" sublist_def: "sublist xs A = map fst (filter (\<lambda>p. snd p \<in> A) (zip xs [0..<size xs]))" primrec "splice [] ys = ys" "splice (x#xs) ys = (if ys=[] then x#xs else x # hd ys # splice xs (tl ys))" -- {*Warning: simpset does not contain the second eqn but a derived one. *} subsubsection {* @{const Nil} and @{const Cons} *} lemma not_Cons_self [simp]: "xs \<noteq> x # xs" by (induct xs) auto lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric] lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)" by (induct xs) auto lemma length_induct: "(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs" by (rule measure_induct [of length]) iprover subsubsection {* @{const length} *} text {* Needs to come before @{text "@"} because of theorem @{text append_eq_append_conv}. *} lemma length_append [simp]: "length (xs @ ys) = length xs + length ys" by (induct xs) auto lemma length_map [simp]: "length (map f xs) = length xs" by (induct xs) auto lemma length_rev [simp]: "length (rev xs) = length xs" by (induct xs) auto lemma length_tl [simp]: "length (tl xs) = length xs - 1" by (cases xs) auto lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])" by (induct xs) auto lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])" by (induct xs) auto lemma length_Suc_conv: "(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)" by (induct xs) auto lemma Suc_length_conv: "(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)" apply (induct xs, simp, simp) apply blast done lemma impossible_Cons [rule_format]: "length xs <= length ys --> xs = x # ys = False" apply (induct xs) apply auto done lemma list_induct2[consumes 1]: "\<And>ys. \<lbrakk> length xs = length ys; P [] []; \<And>x xs y ys. \<lbrakk> length xs = length ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk> \<Longrightarrow> P xs ys" apply(induct xs) apply simp apply(case_tac ys) apply simp apply(simp) done subsubsection {* @{text "@"} -- append *} lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)" by (induct xs) auto lemma append_Nil2 [simp]: "xs @ [] = xs" by (induct xs) auto lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])" by (induct xs) auto lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])" by (induct xs) auto lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])" by (induct xs) auto lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])" by (induct xs) auto lemma append_eq_append_conv [simp]: "!!ys. length xs = length ys \<or> length us = length vs ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)" apply (induct xs) apply (case_tac ys, simp, force) apply (case_tac ys, force, simp) done lemma append_eq_append_conv2: "!!ys zs ts. (xs @ ys = zs @ ts) = (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)" apply (induct xs) apply fastsimp apply(case_tac zs) apply simp apply fastsimp done lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)" by simp lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)" by simp lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)" by simp lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])" using append_same_eq [of _ _ "[]"] by auto lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])" using append_same_eq [of "[]"] by auto lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs" by (induct xs) auto lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)" by (induct xs) auto lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs" by (simp add: hd_append split: list.split) lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)" by (simp split: list.split) lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys" by (simp add: tl_append split: list.split) lemma Cons_eq_append_conv: "x#xs = ys@zs = (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))" by(cases ys) auto lemma append_eq_Cons_conv: "(ys@zs = x#xs) = (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))" by(cases ys) auto text {* Trivial rules for solving @{text "@"}-equations automatically. *} lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys" by simp lemma Cons_eq_appendI: "[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs" by (drule sym) simp lemma append_eq_appendI: "[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us" by (drule sym) simp text {* Simplification procedure for all list equalities. Currently only tries to rearrange @{text "@"} to see if - both lists end in a singleton list, - or both lists end in the same list. *} ML_setup {* local val append_assoc = thm "append_assoc"; val append_Nil = thm "append_Nil"; val append_Cons = thm "append_Cons"; val append1_eq_conv = thm "append1_eq_conv"; val append_same_eq = thm "append_same_eq"; fun last (cons as Const("List.list.Cons",_) $ _ $ xs) = (case xs of Const("List.list.Nil",_) => cons | _ => last xs) | last (Const("List.op @",_) $ _ $ ys) = last ys | last t = t; fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true | list1 _ = false; fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) = (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs) | butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys | butlast xs = Const("List.list.Nil",fastype_of xs); val rearr_ss = HOL_basic_ss addsimps [append_assoc, append_Nil, append_Cons]; fun list_eq ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) = let val lastl = last lhs and lastr = last rhs; fun rearr conv = let val lhs1 = butlast lhs and rhs1 = butlast rhs; val Type(_,listT::_) = eqT val appT = [listT,listT] ---> listT val app = Const("List.op @",appT) val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr) val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2)); val thm = Goal.prove (Simplifier.the_context ss) [] [] eq (K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1)); in SOME ((conv RS (thm RS trans)) RS eq_reflection) end; in if list1 lastl andalso list1 lastr then rearr append1_eq_conv else if lastl aconv lastr then rearr append_same_eq else NONE end; in val list_eq_simproc = Simplifier.simproc (the_context ()) "list_eq" ["(xs::'a list) = ys"] (K list_eq); end; Addsimprocs [list_eq_simproc]; *} subsubsection {* @{text map} *} lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs" by (induct xs) simp_all lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)" by (rule ext, induct_tac xs) auto lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys" by (induct xs) auto lemma map_compose: "map (f o g) xs = map f (map g xs)" by (induct xs) (auto simp add: o_def) lemma rev_map: "rev (map f xs) = map f (rev xs)" by (induct xs) auto lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)" by (induct xs) auto lemma map_cong [fundef_cong, recdef_cong]: "xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys" -- {* a congruence rule for @{text map} *} by simp lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])" by (cases xs) auto lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])" by (cases xs) auto lemma map_eq_Cons_conv: "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)" by (cases xs) auto lemma Cons_eq_map_conv: "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)" by (cases ys) auto lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1] lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1] declare map_eq_Cons_D [dest!] Cons_eq_map_D [dest!] lemma ex_map_conv: "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)" by(induct ys, auto simp add: Cons_eq_map_conv) lemma map_eq_imp_length_eq: "!!xs. map f xs = map f ys ==> length xs = length ys" apply (induct ys) apply simp apply(simp (no_asm_use)) apply clarify apply(simp (no_asm_use)) apply fast done lemma map_inj_on: "[| map f xs = map f ys; inj_on f (set xs Un set ys) |] ==> xs = ys" apply(frule map_eq_imp_length_eq) apply(rotate_tac -1) apply(induct rule:list_induct2) apply simp apply(simp) apply (blast intro:sym) done lemma inj_on_map_eq_map: "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)" by(blast dest:map_inj_on) lemma map_injective: "!!xs. map f xs = map f ys ==> inj f ==> xs = ys" by (induct ys) (auto dest!:injD) lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)" by(blast dest:map_injective) lemma inj_mapI: "inj f ==> inj (map f)" by (iprover dest: map_injective injD intro: inj_onI) lemma inj_mapD: "inj (map f) ==> inj f" apply (unfold inj_on_def, clarify) apply (erule_tac x = "[x]" in ballE) apply (erule_tac x = "[y]" in ballE, simp, blast) apply blast done lemma inj_map[iff]: "inj (map f) = inj f" by (blast dest: inj_mapD intro: inj_mapI) lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A" apply(rule inj_onI) apply(erule map_inj_on) apply(blast intro:inj_onI dest:inj_onD) done lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs" by (induct xs, auto) lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs" by (induct xs) auto lemma map_fst_zip[simp]: "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs" by (induct rule:list_induct2, simp_all) lemma map_snd_zip[simp]: "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys" by (induct rule:list_induct2, simp_all) subsubsection {* @{text rev} *} lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs" by (induct xs) auto lemma rev_rev_ident [simp]: "rev (rev xs) = xs" by (induct xs) auto lemma rev_swap: "(rev xs = ys) = (xs = rev ys)" by auto lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])" by (induct xs) auto lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])" by (induct xs) auto lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])" by (cases xs) auto lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])" by (cases xs) auto lemma rev_is_rev_conv [iff]: "(rev xs = rev ys) = (xs = ys)" apply (induct xs arbitrary: ys, force) apply (case_tac ys, simp, force) done lemma inj_on_rev[iff]: "inj_on rev A" by(simp add:inj_on_def) lemma rev_induct [case_names Nil snoc]: "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs" apply(simplesubst rev_rev_ident[symmetric]) apply(rule_tac list = "rev xs" in list.induct, simp_all) done ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility" lemma rev_exhaust [case_names Nil snoc]: "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P" by (induct xs rule: rev_induct) auto lemmas rev_cases = rev_exhaust lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])" by(rule rev_cases[of xs]) auto subsubsection {* @{text set} *} lemma finite_set [iff]: "finite (set xs)" by (induct xs) auto lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)" by (induct xs) auto lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs" by(cases xs) auto lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)" by auto lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" by auto lemma set_empty [iff]: "(set xs = {}) = (xs = [])" by (induct xs) auto lemma set_empty2[iff]: "({} = set xs) = (xs = [])" by(induct xs) auto lemma set_rev [simp]: "set (rev xs) = set xs" by (induct xs) auto lemma set_map [simp]: "set (map f xs) = f`(set xs)" by (induct xs) auto lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}" by (induct xs) auto lemma set_upt [simp]: "set[i..<j] = {k. i \<le> k \<and> k < j}" apply (induct j, simp_all) apply (erule ssubst, auto) done lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)" proof (induct xs) case Nil show ?case by simp case (Cons a xs) show ?case proof assume "x \<in> set (a # xs)" with prems show "\<exists>ys zs. a # xs = ys @ x # zs" by (simp, blast intro: Cons_eq_appendI) next assume "\<exists>ys zs. a # xs = ys @ x # zs" then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast show "x \<in> set (a # xs)" by (cases ys, auto simp add: eq) qed qed lemma in_set_conv_decomp_first: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)" proof (induct xs) case Nil show ?case by simp next case (Cons a xs) show ?case proof cases assume "x = a" thus ?case using Cons by force next assume "x \<noteq> a" show ?case proof assume "x \<in> set (a # xs)" from prems show "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys" by(fastsimp intro!: Cons_eq_appendI) next assume "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys" then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast show "x \<in> set (a # xs)" by (cases ys, auto simp add: eq) qed qed qed lemmas split_list = in_set_conv_decomp[THEN iffD1, standard] lemmas split_list_first = in_set_conv_decomp_first[THEN iffD1, standard] lemma finite_list: "finite A ==> EX l. set l = A" apply (erule finite_induct, auto) apply (rule_tac x="x#l" in exI, auto) done lemma card_length: "card (set xs) \<le> length xs" by (induct xs) (auto simp add: card_insert_if) subsubsection {* @{text filter} *} lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys" by (induct xs) auto lemma rev_filter: "rev (filter P xs) = filter P (rev xs)" by (induct xs) simp_all lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs" by (induct xs) auto lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs" by (induct xs) (auto simp add: le_SucI) lemma sum_length_filter_compl: "length(filter P xs) + length(filter (%x. ~P x) xs) = length xs" by(induct xs) simp_all lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs" by (induct xs) auto lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []" by (induct xs) auto lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" by (induct xs) simp_all lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)" apply (induct xs) apply auto apply(cut_tac P=P and xs=xs in length_filter_le) apply simp done lemma filter_map: "filter P (map f xs) = map f (filter (P o f) xs)" by (induct xs) simp_all lemma length_filter_map[simp]: "length (filter P (map f xs)) = length(filter (P o f) xs)" by (simp add:filter_map) lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs" by auto lemma length_filter_less: "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs" proof (induct xs) case Nil thus ?case by simp next case (Cons x xs) thus ?case apply (auto split:split_if_asm) using length_filter_le[of P xs] apply arith done qed lemma length_filter_conv_card: "length(filter p xs) = card{i. i < length xs & p(xs!i)}" proof (induct xs) case Nil thus ?case by simp next case (Cons x xs) let ?S = "{i. i < length xs & p(xs!i)}" have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite) show ?case (is "?l = card ?S'") proof (cases) assume "p x" hence eq: "?S' = insert 0 (Suc ` ?S)" by(auto simp add: nth_Cons image_def split:nat.split elim:lessE) have "length (filter p (x # xs)) = Suc(card ?S)" using Cons by simp also have "\<dots> = Suc(card(Suc ` ?S))" using fin by (simp add: card_image inj_Suc) also have "\<dots> = card ?S'" using eq fin by (simp add:card_insert_if) (simp add:image_def) finally show ?thesis . next assume "\<not> p x" hence eq: "?S' = Suc ` ?S" by(auto simp add: nth_Cons image_def split:nat.split elim:lessE) have "length (filter p (x # xs)) = card ?S" using Cons by simp also have "\<dots> = card(Suc ` ?S)" using fin by (simp add: card_image inj_Suc) also have "\<dots> = card ?S'" using eq fin by (simp add:card_insert_if) finally show ?thesis . qed qed lemma Cons_eq_filterD: "x#xs = filter P ys \<Longrightarrow> \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs" (is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs") proof(induct ys) case Nil thus ?case by simp next case (Cons y ys) show ?case (is "\<exists>x. ?Q x") proof cases assume Py: "P y" show ?thesis proof cases assume xy: "x = y" show ?thesis proof from Py xy Cons(2) show "?Q []" by simp qed next assume "x \<noteq> y" with Py Cons(2) show ?thesis by simp qed next assume Py: "\<not> P y" with Cons obtain us vs where 1 : "?P (y#ys) (y#us) vs" by fastsimp show ?thesis (is "? us. ?Q us") proof show "?Q (y#us)" using 1 by simp qed qed qed lemma filter_eq_ConsD: "filter P ys = x#xs \<Longrightarrow> \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs" by(rule Cons_eq_filterD) simp lemma filter_eq_Cons_iff: "(filter P ys = x#xs) = (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)" by(auto dest:filter_eq_ConsD) lemma Cons_eq_filter_iff: "(x#xs = filter P ys) = (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)" by(auto dest:Cons_eq_filterD) lemma filter_cong[fundef_cong, recdef_cong]: "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys" apply simp apply(erule thin_rl) by (induct ys) simp_all subsubsection {* @{text concat} *} lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys" by (induct xs) auto lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])" by (induct xss) auto lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])" by (induct xss) auto lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)" by (induct xs) auto lemma map_concat: "map f (concat xs) = concat (map (map f) xs)" by (induct xs) auto lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)" by (induct xs) auto lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))" by (induct xs) auto subsubsection {* @{text nth} *} lemma nth_Cons_0 [simp]: "(x # xs)!0 = x" by auto lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n" by auto declare nth.simps [simp del] lemma nth_append: "!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))" apply (induct "xs", simp) apply (case_tac n, auto) done lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x" by (induct "xs") auto lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n" by (induct "xs") auto lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)" apply (induct xs, simp) apply (case_tac n, auto) done lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0" by(cases xs) simp_all lemma list_eq_iff_nth_eq: "!!ys. (xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))" apply(induct xs) apply simp apply blast apply(case_tac ys) apply simp apply(simp add:nth_Cons split:nat.split)apply blast done lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}" apply (induct xs, simp, simp) apply safe apply (rule_tac x = 0 in exI, simp) apply (rule_tac x = "Suc i" in exI, simp) apply (case_tac i, simp) apply (rename_tac j) apply (rule_tac x = j in exI, simp) done lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)" by(auto simp:set_conv_nth) lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)" by (auto simp add: set_conv_nth) lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs" by (auto simp add: set_conv_nth) lemma all_nth_imp_all_set: "[| !i < length xs. P(xs!i); x : set xs|] ==> P x" by (auto simp add: set_conv_nth) lemma all_set_conv_all_nth: "(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))" by (auto simp add: set_conv_nth) subsubsection {* @{text list_update} *} lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs" by (induct xs) (auto split: nat.split) lemma nth_list_update: "!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)" by (induct xs) (auto simp add: nth_Cons split: nat.split) lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x" by (simp add: nth_list_update) lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j" by (induct xs) (auto simp add: nth_Cons split: nat.split) lemma list_update_overwrite [simp]: "!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]" by (induct xs) (auto split: nat.split) lemma list_update_id[simp]: "!!i. i < length xs ==> xs[i := xs!i] = xs" apply (induct xs, simp) apply(simp split:nat.splits) done lemma list_update_beyond[simp]: "\<And>i. length xs \<le> i \<Longrightarrow> xs[i:=x] = xs" apply (induct xs) apply simp apply (case_tac i) apply simp_all done lemma list_update_same_conv: "!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)" by (induct xs) (auto split: nat.split) lemma list_update_append1: "!!i. i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys" apply (induct xs, simp) apply(simp split:nat.split) done lemma list_update_append: "!!n. (xs @ ys) [n:= x] = (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))" by (induct xs) (auto split:nat.splits) lemma list_update_length [simp]: "(xs @ x # ys)[length xs := y] = (xs @ y # ys)" by (induct xs, auto) lemma update_zip: "!!i xy xs. length xs = length ys ==> (zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])" by (induct ys) (auto, case_tac xs, auto split: nat.split) lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)" by (induct xs) (auto split: nat.split) lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A" by (blast dest!: set_update_subset_insert [THEN subsetD]) lemma set_update_memI: "!!n. n < length xs \<Longrightarrow> x \<in> set (xs[n := x])" by (induct xs) (auto split:nat.splits) subsubsection {* @{text last} and @{text butlast} *} lemma last_snoc [simp]: "last (xs @ [x]) = x" by (induct xs) auto lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs" by (induct xs) auto lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x" by(simp add:last.simps) lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs" by(simp add:last.simps) lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)" by (induct xs) (auto) lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs" by(simp add:last_append) lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys" by(simp add:last_append) lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs" by(rule rev_exhaust[of xs]) simp_all lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs" by(cases xs) simp_all lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as" by (induct as) auto lemma length_butlast [simp]: "length (butlast xs) = length xs - 1" by (induct xs rule: rev_induct) auto lemma butlast_append: "!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)" by (induct xs) auto lemma append_butlast_last_id [simp]: "xs \<noteq> [] ==> butlast xs @ [last xs] = xs" by (induct xs) auto lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs" by (induct xs) (auto split: split_if_asm) lemma in_set_butlast_appendI: "x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))" by (auto dest: in_set_butlastD simp add: butlast_append) lemma last_drop[simp]: "!!n. n < length xs \<Longrightarrow> last (drop n xs) = last xs" apply (induct xs) apply simp apply (auto split:nat.split) done lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)" by(induct xs)(auto simp:neq_Nil_conv) subsubsection {* @{text take} and @{text drop} *} lemma take_0 [simp]: "take 0 xs = []" by (induct xs) auto lemma drop_0 [simp]: "drop 0 xs = xs" by (induct xs) auto lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs" by simp lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs" by simp declare take_Cons [simp del] and drop_Cons [simp del] lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)" by(clarsimp simp add:neq_Nil_conv) lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)" by(cases xs, simp_all) lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)" by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split) lemma nth_via_drop: "!!n. drop n xs = y#ys \<Longrightarrow> xs!n = y" apply (induct xs, simp) apply(simp add:drop_Cons nth_Cons split:nat.splits) done lemma take_Suc_conv_app_nth: "!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]" apply (induct xs, simp) apply (case_tac i, auto) done lemma drop_Suc_conv_tl: "!!i. i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs" apply (induct xs, simp) apply (case_tac i, auto) done lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n" by (induct n) (auto, case_tac xs, auto) lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)" by (induct n) (auto, case_tac xs, auto) lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs" by (induct n) (auto, case_tac xs, auto) lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []" by (induct n) (auto, case_tac xs, auto) lemma take_append [simp]: "!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)" by (induct n) (auto, case_tac xs, auto) lemma drop_append [simp]: "!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys" by (induct n) (auto, case_tac xs, auto) lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs" apply (induct m, auto) apply (case_tac xs, auto) apply (case_tac n, auto) done lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs" apply (induct m, auto) apply (case_tac xs, auto) done lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)" apply (induct m, auto) apply (case_tac xs, auto) done lemma drop_take: "!!m n. drop n (take m xs) = take (m-n) (drop n xs)" apply(induct xs) apply simp apply(simp add: take_Cons drop_Cons split:nat.split) done lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs" apply (induct n, auto) apply (case_tac xs, auto) done lemma take_eq_Nil[simp]: "!!n. (take n xs = []) = (n = 0 \<or> xs = [])" apply(induct xs) apply simp apply(simp add:take_Cons split:nat.split) done lemma drop_eq_Nil[simp]: "!!n. (drop n xs = []) = (length xs <= n)" apply(induct xs) apply simp apply(simp add:drop_Cons split:nat.split) done lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)" apply (induct n, auto) apply (case_tac xs, auto) done lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)" apply (induct n, auto) apply (case_tac xs, auto) done lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)" apply (induct xs, auto) apply (case_tac i, auto) done lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)" apply (induct xs, auto) apply (case_tac i, auto) done lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i" apply (induct xs, auto) apply (case_tac n, blast) apply (case_tac i, auto) done lemma nth_drop [simp]: "!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)" apply (induct n, auto) apply (case_tac xs, auto) done lemma hd_drop_conv_nth: "\<lbrakk> xs \<noteq> []; n < length xs \<rbrakk> \<Longrightarrow> hd(drop n xs) = xs!n" by(simp add: hd_conv_nth) lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs" by(induct xs)(auto simp:take_Cons split:nat.split) lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> set xs" by(induct xs)(auto simp:drop_Cons split:nat.split) lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs" using set_take_subset by fast lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs" using set_drop_subset by fast lemma append_eq_conv_conj: "!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)" apply (induct xs, simp, clarsimp) apply (case_tac zs, auto) done lemma take_add [rule_format]: "\<forall>i. i+j \<le> length(xs) --> take (i+j) xs = take i xs @ take j (drop i xs)" apply (induct xs, auto) apply (case_tac i, simp_all) done lemma append_eq_append_conv_if: "!! ys\<^isub>1. (xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) = (if size xs\<^isub>1 \<le> size ys\<^isub>1 then xs\<^isub>1 = take (size xs\<^isub>1) ys\<^isub>1 \<and> xs\<^isub>2 = drop (size xs\<^isub>1) ys\<^isub>1 @ ys\<^isub>2 else take (size ys\<^isub>1) xs\<^isub>1 = ys\<^isub>1 \<and> drop (size ys\<^isub>1) xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>2)" apply(induct xs\<^isub>1) apply simp apply(case_tac ys\<^isub>1) apply simp_all done lemma take_hd_drop: "!!n. n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (n+1) xs" apply(induct xs) apply simp apply(simp add:drop_Cons split:nat.split) done lemma id_take_nth_drop: "i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs" proof - assume si: "i < length xs" hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto moreover from si have "take (Suc i) xs = take i xs @ [xs!i]" apply (rule_tac take_Suc_conv_app_nth) by arith ultimately show ?thesis by auto qed lemma upd_conv_take_nth_drop: "i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs" proof - assume i: "i < length xs" have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]" by(rule arg_cong[OF id_take_nth_drop[OF i]]) also have "\<dots> = take i xs @ a # drop (Suc i) xs" using i by (simp add: list_update_append) finally show ?thesis . qed subsubsection {* @{text takeWhile} and @{text dropWhile} *} lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs" by (induct xs) auto lemma takeWhile_append1 [simp]: "[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs" by (induct xs) auto lemma takeWhile_append2 [simp]: "(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys" by (induct xs) auto lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs" by (induct xs) auto lemma dropWhile_append1 [simp]: "[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys" by (induct xs) auto lemma dropWhile_append2 [simp]: "(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys" by (induct xs) auto lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x" by (induct xs) (auto split: split_if_asm) lemma takeWhile_eq_all_conv[simp]: "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)" by(induct xs, auto) lemma dropWhile_eq_Nil_conv[simp]: "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)" by(induct xs, auto) lemma dropWhile_eq_Cons_conv: "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)" by(induct xs, auto) text{* The following two lemmmas could be generalized to an arbitrary property. *} lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))" by(induct xs) (auto simp: takeWhile_tail[where l="[]"]) lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow> dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)" apply(induct xs) apply simp apply auto apply(subst dropWhile_append2) apply auto done lemma takeWhile_not_last: "\<lbrakk> xs \<noteq> []; distinct xs\<rbrakk> \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs" apply(induct xs) apply simp apply(case_tac xs) apply(auto) done lemma takeWhile_cong [fundef_cong, recdef_cong]: "[| l = k; !!x. x : set l ==> P x = Q x |] ==> takeWhile P l = takeWhile Q k" by (induct k arbitrary: l) (simp_all) lemma dropWhile_cong [fundef_cong, recdef_cong]: "[| l = k; !!x. x : set l ==> P x = Q x |] ==> dropWhile P l = dropWhile Q k" by (induct k arbitrary: l, simp_all) subsubsection {* @{text zip} *} lemma zip_Nil [simp]: "zip [] ys = []" by (induct ys) auto lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys" by simp declare zip_Cons [simp del] lemma zip_Cons1: "zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)" by(auto split:list.split) lemma length_zip [simp]: "!!xs. length (zip xs ys) = min (length xs) (length ys)" apply (induct ys, simp) apply (case_tac xs, auto) done lemma zip_append1: "!!xs. zip (xs @ ys) zs = zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)" apply (induct zs, simp) apply (case_tac xs, simp_all) done lemma zip_append2: "!!ys. zip xs (ys @ zs) = zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs" apply (induct xs, simp) apply (case_tac ys, simp_all) done lemma zip_append [simp]: "[| length xs = length us; length ys = length vs |] ==> zip (xs@ys) (us@vs) = zip xs us @ zip ys vs" by (simp add: zip_append1) lemma zip_rev: "length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)" by (induct rule:list_induct2, simp_all) lemma nth_zip [simp]: "!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)" apply (induct ys, simp) apply (case_tac xs) apply (simp_all add: nth.simps split: nat.split) done lemma set_zip: "set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}" by (simp add: set_conv_nth cong: rev_conj_cong) lemma zip_update: "length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]" by (rule sym, simp add: update_zip) lemma zip_replicate [simp]: "!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)" apply (induct i, auto) apply (case_tac j, auto) done lemma take_zip: "!!xs ys. take n (zip xs ys) = zip (take n xs) (take n ys)" apply (induct n) apply simp apply (case_tac xs, simp) apply (case_tac ys, simp_all) done lemma drop_zip: "!!xs ys. drop n (zip xs ys) = zip (drop n xs) (drop n ys)" apply (induct n) apply simp apply (case_tac xs, simp) apply (case_tac ys, simp_all) done subsubsection {* @{text list_all2} *} lemma list_all2_lengthD [intro?]: "list_all2 P xs ys ==> length xs = length ys" by (simp add: list_all2_def) lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])" by (simp add: list_all2_def) lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])" by (simp add: list_all2_def) lemma list_all2_Cons [iff, code]: "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)" by (auto simp add: list_all2_def) lemma list_all2_Cons1: "list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)" by (cases ys) auto lemma list_all2_Cons2: "list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)" by (cases xs) auto lemma list_all2_rev [iff]: "list_all2 P (rev xs) (rev ys) = list_all2 P xs ys" by (simp add: list_all2_def zip_rev cong: conj_cong) lemma list_all2_rev1: "list_all2 P (rev xs) ys = list_all2 P xs (rev ys)" by (subst list_all2_rev [symmetric]) simp lemma list_all2_append1: "list_all2 P (xs @ ys) zs = (EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and> list_all2 P xs us \<and> list_all2 P ys vs)" apply (simp add: list_all2_def zip_append1) apply (rule iffI) apply (rule_tac x = "take (length xs) zs" in exI) apply (rule_tac x = "drop (length xs) zs" in exI) apply (force split: nat_diff_split simp add: min_def, clarify) apply (simp add: ball_Un) done lemma list_all2_append2: "list_all2 P xs (ys @ zs) = (EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and> list_all2 P us ys \<and> list_all2 P vs zs)" apply (simp add: list_all2_def zip_append2) apply (rule iffI) apply (rule_tac x = "take (length ys) xs" in exI) apply (rule_tac x = "drop (length ys) xs" in exI) apply (force split: nat_diff_split simp add: min_def, clarify) apply (simp add: ball_Un) done lemma list_all2_append: "length xs = length ys \<Longrightarrow> list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)" by (induct rule:list_induct2, simp_all) lemma list_all2_appendI [intro?, trans]: "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)" by (simp add: list_all2_append list_all2_lengthD) lemma list_all2_conv_all_nth: "list_all2 P xs ys = (length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))" by (force simp add: list_all2_def set_zip) lemma list_all2_trans: assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c" shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs" (is "!!bs cs. PROP ?Q as bs cs") proof (induct as) fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs" show "!!cs. PROP ?Q (x # xs) bs cs" proof (induct bs) fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs" show "PROP ?Q (x # xs) (y # ys) cs" by (induct cs) (auto intro: tr I1 I2) qed simp qed simp lemma list_all2_all_nthI [intro?]: "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b" by (simp add: list_all2_conv_all_nth) lemma list_all2I: "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b" by (simp add: list_all2_def) lemma list_all2_nthD: "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)" by (simp add: list_all2_conv_all_nth) lemma list_all2_nthD2: "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)" by (frule list_all2_lengthD) (auto intro: list_all2_nthD) lemma list_all2_map1: "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs" by (simp add: list_all2_conv_all_nth) lemma list_all2_map2: "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs" by (auto simp add: list_all2_conv_all_nth) lemma list_all2_refl [intro?]: "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs" by (simp add: list_all2_conv_all_nth) lemma list_all2_update_cong: "\<lbrakk> i<size xs; list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])" by (simp add: list_all2_conv_all_nth nth_list_update) lemma list_all2_update_cong2: "\<lbrakk>list_all2 P xs ys; P x y; i < length ys\<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])" by (simp add: list_all2_lengthD list_all2_update_cong) lemma list_all2_takeI [simp,intro?]: "\<And>n ys. list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)" apply (induct xs) apply simp apply (clarsimp simp add: list_all2_Cons1) apply (case_tac n) apply auto done lemma list_all2_dropI [simp,intro?]: "\<And>n bs. list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)" apply (induct as, simp) apply (clarsimp simp add: list_all2_Cons1) apply (case_tac n, simp, simp) done lemma list_all2_mono [intro?]: "\<And>y. list_all2 P x y \<Longrightarrow> (\<And>x y. P x y \<Longrightarrow> Q x y) \<Longrightarrow> list_all2 Q x y" apply (induct x, simp) apply (case_tac y, auto) done subsubsection {* @{text foldl} and @{text foldr} *} lemma foldl_append [simp]: "!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys" by (induct xs) auto lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)" by (induct xs) auto lemma foldl_cong [fundef_cong, recdef_cong]: "[| a = b; l = k; !!a x. x : set l ==> f a x = g a x |] ==> foldl f a l = foldl g b k" by (induct k arbitrary: a b l) simp_all lemma foldr_cong [fundef_cong, recdef_cong]: "[| a = b; l = k; !!a x. x : set l ==> f x a = g x a |] ==> foldr f l a = foldr g k b" by (induct k arbitrary: a b l) simp_all lemma foldr_foldl: "foldr f xs a = foldl (%x y. f y x) a (rev xs)" by (induct xs) auto lemma foldl_foldr: "foldl f a xs = foldr (%x y. f y x) (rev xs) a" by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"]) text {* Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more difficult to use because it requires an additional transitivity step. *} lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns" by (induct ns) auto lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns" by (force intro: start_le_sum simp add: in_set_conv_decomp) lemma sum_eq_0_conv [iff]: "!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))" by (induct ns) auto subsubsection {* @{text upto} *} lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])" -- {* simp does not terminate! *} by (induct j) auto lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []" by (subst upt_rec) simp lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)" by(induct j)simp_all lemma upt_eq_Cons_conv: "!!x xs. ([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)" apply(induct j) apply simp apply(clarsimp simp add: append_eq_Cons_conv) apply arith done lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]" -- {* Only needed if @{text upt_Suc} is deleted from the simpset. *} by simp lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]" apply(rule trans) apply(subst upt_rec) prefer 2 apply (rule refl, simp) done lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]" -- {* LOOPS as a simprule, since @{text "j <= j"}. *} by (induct k) auto lemma length_upt [simp]: "length [i..<j] = j - i" by (induct j) (auto simp add: Suc_diff_le) lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k" apply (induct j) apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split) done lemma hd_upt[simp]: "i < j \<Longrightarrow> hd[i..<j] = i" by(simp add:upt_conv_Cons) lemma last_upt[simp]: "i < j \<Longrightarrow> last[i..<j] = j - 1" apply(cases j) apply simp by(simp add:upt_Suc_append) lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..<n] = [i..<i+m]" apply (induct m, simp) apply (subst upt_rec) apply (rule sym) apply (subst upt_rec) apply (simp del: upt.simps) done lemma drop_upt[simp]: "drop m [i..<j] = [i+m..<j]" apply(induct j) apply auto done lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..n]" by (induct n) auto lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..<n]) ! i = f(m+i)" apply (induct n m rule: diff_induct) prefer 3 apply (subst map_Suc_upt[symmetric]) apply (auto simp add: less_diff_conv nth_upt) done lemma nth_take_lemma: "!!xs ys. k <= length xs ==> k <= length ys ==> (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys" apply (atomize, induct k) apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify) txt {* Both lists must be non-empty *} apply (case_tac xs, simp) apply (case_tac ys, clarify) apply (simp (no_asm_use)) apply clarify txt {* prenexing's needed, not miniscoping *} apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps) apply blast done lemma nth_equalityI: "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys" apply (frule nth_take_lemma [OF le_refl eq_imp_le]) apply (simp_all add: take_all) done (* needs nth_equalityI *) lemma list_all2_antisym: "\<lbrakk> (\<And>x y. \<lbrakk>P x y; Q y x\<rbrakk> \<Longrightarrow> x = y); list_all2 P xs ys; list_all2 Q ys xs \<rbrakk> \<Longrightarrow> xs = ys" apply (simp add: list_all2_conv_all_nth) apply (rule nth_equalityI, blast, simp) done lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys" -- {* The famous take-lemma. *} apply (drule_tac x = "max (length xs) (length ys)" in spec) apply (simp add: le_max_iff_disj take_all) done lemma take_Cons': "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)" by (cases n) simp_all lemma drop_Cons': "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)" by (cases n) simp_all lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))" by (cases n) simp_all lemmas take_Cons_number_of = take_Cons'[of "number_of v",standard] lemmas drop_Cons_number_of = drop_Cons'[of "number_of v",standard] lemmas nth_Cons_number_of = nth_Cons'[of _ _ "number_of v",standard] declare take_Cons_number_of [simp] drop_Cons_number_of [simp] nth_Cons_number_of [simp] subsubsection {* @{text "distinct"} and @{text remdups} *} lemma distinct_append [simp]: "distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})" by (induct xs) auto lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs" by(induct xs) auto lemma set_remdups [simp]: "set (remdups xs) = set xs" by (induct xs) (auto simp add: insert_absorb) lemma distinct_remdups [iff]: "distinct (remdups xs)" by (induct xs) auto lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])" by (induct x, auto) lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])" by (induct x, auto) lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs" by (induct xs) auto lemma length_remdups_eq[iff]: "(length (remdups xs) = length xs) = (remdups xs = xs)" apply(induct xs) apply auto apply(subgoal_tac "length (remdups xs) <= length xs") apply arith apply(rule length_remdups_leq) done lemma distinct_map: "distinct(map f xs) = (distinct xs & inj_on f (set xs))" by (induct xs) auto lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)" by (induct xs) auto lemma distinct_upt[simp]: "distinct[i..<j]" by (induct j) auto lemma distinct_take[simp]: "\<And>i. distinct xs \<Longrightarrow> distinct (take i xs)" apply(induct xs) apply simp apply (case_tac i) apply simp_all apply(blast dest:in_set_takeD) done lemma distinct_drop[simp]: "\<And>i. distinct xs \<Longrightarrow> distinct (drop i xs)" apply(induct xs) apply simp apply (case_tac i) apply simp_all done lemma distinct_list_update: assumes d: "distinct xs" and a: "a \<notin> set xs - {xs!i}" shows "distinct (xs[i:=a])" proof (cases "i < length xs") case True with a have "a \<notin> set (take i xs @ xs ! i # drop (Suc i) xs) - {xs!i}" apply (drule_tac id_take_nth_drop) by simp with d True show ?thesis apply (simp add: upd_conv_take_nth_drop) apply (drule subst [OF id_take_nth_drop]) apply assumption apply simp apply (cases "a = xs!i") apply simp by blast next case False with d show ?thesis by auto qed text {* It is best to avoid this indexed version of distinct, but sometimes it is useful. *} lemma distinct_conv_nth: "distinct xs = (\<forall>i < size xs. \<forall>j < size xs. i \<noteq> j --> xs!i \<noteq> xs!j)" apply (induct xs, simp, simp) apply (rule iffI, clarsimp) apply (case_tac i) apply (case_tac j, simp) apply (simp add: set_conv_nth) apply (case_tac j) apply (clarsimp simp add: set_conv_nth, simp) apply (rule conjI) apply (clarsimp simp add: set_conv_nth) apply (erule_tac x = 0 in allE, simp) apply (erule_tac x = "Suc i" in allE, simp, clarsimp) apply (erule_tac x = "Suc i" in allE, simp) apply (erule_tac x = "Suc j" in allE, simp) done lemma nth_eq_iff_index_eq: "\<lbrakk> distinct xs; i < length xs; j < length xs \<rbrakk> \<Longrightarrow> (xs!i = xs!j) = (i = j)" by(auto simp: distinct_conv_nth) lemma distinct_card: "distinct xs ==> card (set xs) = size xs" by (induct xs) auto lemma card_distinct: "card (set xs) = size xs ==> distinct xs" proof (induct xs) case Nil thus ?case by simp next case (Cons x xs) show ?case proof (cases "x \<in> set xs") case False with Cons show ?thesis by simp next case True with Cons.prems have "card (set xs) = Suc (length xs)" by (simp add: card_insert_if split: split_if_asm) moreover have "card (set xs) \<le> length xs" by (rule card_length) ultimately have False by simp thus ?thesis .. qed qed lemma length_remdups_concat: "length(remdups(concat xss)) = card(\<Union>xs \<in> set xss. set xs)" by(simp add: distinct_card[symmetric]) subsubsection {* @{text remove1} *} lemma remove1_append: "remove1 x (xs @ ys) = (if x \<in> set xs then remove1 x xs @ ys else xs @ remove1 x ys)" by (induct xs) auto lemma set_remove1_subset: "set(remove1 x xs) <= set xs" apply(induct xs) apply simp apply simp apply blast done lemma set_remove1_eq [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}" apply(induct xs) apply simp apply simp apply blast done lemma remove1_filter_not[simp]: "\<not> P x \<Longrightarrow> remove1 x (filter P xs) = filter P xs" by(induct xs) auto lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)" apply(insert set_remove1_subset) apply fast done lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)" by (induct xs) simp_all subsubsection {* @{text replicate} *} lemma length_replicate [simp]: "length (replicate n x) = n" by (induct n) auto lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)" by (induct n) auto lemma replicate_app_Cons_same: "(replicate n x) @ (x # xs) = x # replicate n x @ xs" by (induct n) auto lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x" apply (induct n, simp) apply (simp add: replicate_app_Cons_same) done lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x" by (induct n) auto text{* Courtesy of Matthias Daum: *} lemma append_replicate_commute: "replicate n x @ replicate k x = replicate k x @ replicate n x" apply (simp add: replicate_add [THEN sym]) apply (simp add: add_commute) done lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x" by (induct n) auto lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x" by (induct n) auto lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x" by (atomize (full), induct n) auto lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x" apply (induct n, simp) apply (simp add: nth_Cons split: nat.split) done text{* Courtesy of Matthias Daum (2 lemmas): *} lemma take_replicate[simp]: "take i (replicate k x) = replicate (min i k) x" apply (case_tac "k \<le> i") apply (simp add: min_def) apply (drule not_leE) apply (simp add: min_def) apply (subgoal_tac "replicate k x = replicate i x @ replicate (k - i) x") apply simp apply (simp add: replicate_add [symmetric]) done lemma drop_replicate[simp]: "!!i. drop i (replicate k x) = replicate (k-i) x" apply (induct k) apply simp apply clarsimp apply (case_tac i) apply simp apply clarsimp done lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}" by (induct n) auto lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}" by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc) lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})" by auto lemma in_set_replicateD: "x : set (replicate n y) ==> x = y" by (simp add: set_replicate_conv_if split: split_if_asm) subsubsection{*@{text rotate1} and @{text rotate}*} lemma rotate_simps[simp]: "rotate1 [] = [] \<and> rotate1 (x#xs) = xs @ [x]" by(simp add:rotate1_def) lemma rotate0[simp]: "rotate 0 = id" by(simp add:rotate_def) lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)" by(simp add:rotate_def) lemma rotate_add: "rotate (m+n) = rotate m o rotate n" by(simp add:rotate_def funpow_add) lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs" by(simp add:rotate_add) lemma rotate1_rotate_swap: "rotate1 (rotate n xs) = rotate n (rotate1 xs)" by(simp add:rotate_def funpow_swap1) lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs" by(cases xs) simp_all lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs" apply(induct n) apply simp apply (simp add:rotate_def) done lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]" by(simp add:rotate1_def split:list.split) lemma rotate_drop_take: "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs" apply(induct n) apply simp apply(simp add:rotate_def) apply(cases "xs = []") apply (simp) apply(case_tac "n mod length xs = 0") apply(simp add:mod_Suc) apply(simp add: rotate1_hd_tl drop_Suc take_Suc) apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric] take_hd_drop linorder_not_le) done lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs" by(simp add:rotate_drop_take) lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs" by(simp add:rotate_drop_take) lemma length_rotate1[simp]: "length(rotate1 xs) = length xs" by(simp add:rotate1_def split:list.split) lemma length_rotate[simp]: "!!xs. length(rotate n xs) = length xs" by (induct n) (simp_all add:rotate_def) lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs" by(simp add:rotate1_def split:list.split) blast lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs" by (induct n) (simp_all add:rotate_def) lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)" by(simp add:rotate_drop_take take_map drop_map) lemma set_rotate1[simp]: "set(rotate1 xs) = set xs" by(simp add:rotate1_def split:list.split) lemma set_rotate[simp]: "set(rotate n xs) = set xs" by (induct n) (simp_all add:rotate_def) lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])" by(simp add:rotate1_def split:list.split) lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])" by (induct n) (simp_all add:rotate_def) lemma rotate_rev: "rotate n (rev xs) = rev(rotate (length xs - (n mod length xs)) xs)" apply(simp add:rotate_drop_take rev_drop rev_take) apply(cases "length xs = 0") apply simp apply(cases "n mod length xs = 0") apply simp apply(simp add:rotate_drop_take rev_drop rev_take) done lemma hd_rotate_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd(rotate n xs) = xs!(n mod length xs)" apply(simp add:rotate_drop_take hd_append hd_drop_conv_nth hd_conv_nth) apply(subgoal_tac "length xs \<noteq> 0") prefer 2 apply simp using mod_less_divisor[of "length xs" n] by arith subsubsection {* @{text sublist} --- a generalization of @{text nth} to sets *} lemma sublist_empty [simp]: "sublist xs {} = []" by (auto simp add: sublist_def) lemma sublist_nil [simp]: "sublist [] A = []" by (auto simp add: sublist_def) lemma length_sublist: "length(sublist xs I) = card{i. i < length xs \<and> i : I}" by(simp add: sublist_def length_filter_conv_card cong:conj_cong) lemma sublist_shift_lemma_Suc: "!!is. map fst (filter (%p. P(Suc(snd p))) (zip xs is)) = map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))" apply(induct xs) apply simp apply (case_tac "is") apply simp apply simp done lemma sublist_shift_lemma: "map fst [p:zip xs [i..<i + length xs] . snd p : A] = map fst [p:zip xs [0..<length xs] . snd p + i : A]" by (induct xs rule: rev_induct) (simp_all add: add_commute) lemma sublist_append: "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}" apply (unfold sublist_def) apply (induct l' rule: rev_induct, simp) apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma) apply (simp add: add_commute) done lemma sublist_Cons: "sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}" apply (induct l rule: rev_induct) apply (simp add: sublist_def) apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append) done lemma set_sublist: "!!I. set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}" apply(induct xs) apply simp apply(auto simp add:sublist_Cons nth_Cons split:nat.split elim: lessE) apply(erule lessE) apply auto apply(erule lessE) apply auto done lemma set_sublist_subset: "set(sublist xs I) \<subseteq> set xs" by(auto simp add:set_sublist) lemma notin_set_sublistI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(sublist xs I)" by(auto simp add:set_sublist) lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs" by(auto simp add:set_sublist) lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])" by (simp add: sublist_Cons) lemma distinct_sublistI[simp]: "!!I. distinct xs \<Longrightarrow> distinct(sublist xs I)" apply(induct xs) apply simp apply(auto simp add:sublist_Cons) done lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l" apply (induct l rule: rev_induct, simp) apply (simp split: nat_diff_split add: sublist_append) done lemma filter_in_sublist: "\<And>s. distinct xs \<Longrightarrow> filter (%x. x \<in> set(sublist xs s)) xs = sublist xs s" proof (induct xs) case Nil thus ?case by simp next case (Cons a xs) moreover hence "!x. x: set xs \<longrightarrow> x \<noteq> a" by auto ultimately show ?case by(simp add: sublist_Cons cong:filter_cong) qed subsubsection {* @{const splice} *} lemma splice_Nil2 [simp, code]: "splice xs [] = xs" by (cases xs) simp_all lemma splice_Cons_Cons [simp, code]: "splice (x#xs) (y#ys) = x # y # splice xs ys" by simp declare splice.simps(2) [simp del, code del] subsubsection{*Sets of Lists*} subsubsection {* @{text lists}: the list-forming operator over sets *} consts lists :: "'a set => 'a list set" inductive "lists A" intros Nil [intro!]: "[]: lists A" Cons [intro!]: "[| a: A;l: lists A|] ==> a#l : lists A" inductive_cases listsE [elim!]: "x#l : lists A" lemma lists_mono [mono]: "A \<subseteq> B ==> lists A \<subseteq> lists B" by (unfold lists.defs) (blast intro!: lfp_mono) lemma lists_IntI: assumes l: "l: lists A" shows "l: lists B ==> l: lists (A Int B)" using l by induct blast+ lemma lists_Int_eq [simp]: "lists (A \<inter> B) = lists A \<inter> lists B" proof (rule mono_Int [THEN equalityI]) show "mono lists" by (simp add: mono_def lists_mono) show "lists A \<inter> lists B \<subseteq> lists (A \<inter> B)" by (blast intro: lists_IntI) qed lemma append_in_lists_conv [iff]: "(xs @ ys : lists A) = (xs : lists A \<and> ys : lists A)" by (induct xs) auto lemma in_lists_conv_set: "(xs : lists A) = (\<forall>x \<in> set xs. x : A)" -- {* eliminate @{text lists} in favour of @{text set} *} by (induct xs) auto lemma in_listsD [dest!]: "xs \<in> lists A ==> \<forall>x\<in>set xs. x \<in> A" by (rule in_lists_conv_set [THEN iffD1]) lemma in_listsI [intro!]: "\<forall>x\<in>set xs. x \<in> A ==> xs \<in> lists A" by (rule in_lists_conv_set [THEN iffD2]) lemma lists_UNIV [simp]: "lists UNIV = UNIV" by auto subsubsection{* Inductive definition for membership *} consts ListMem :: "('a \<times> 'a list)set" inductive ListMem intros elem: "(x,x#xs) \<in> ListMem" insert: "(x,xs) \<in> ListMem \<Longrightarrow> (x,y#xs) \<in> ListMem" lemma ListMem_iff: "((x,xs) \<in> ListMem) = (x \<in> set xs)" apply (rule iffI) apply (induct set: ListMem) apply auto apply (induct xs) apply (auto intro: ListMem.intros) done subsubsection{*Lists as Cartesian products*} text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from @{term A} and tail drawn from @{term Xs}.*} constdefs set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set" "set_Cons A XS == {z. \<exists>x xs. z = x#xs & x \<in> A & xs \<in> XS}" lemma set_Cons_sing_Nil [simp]: "set_Cons A {[]} = (%x. [x])`A" by (auto simp add: set_Cons_def) text{*Yields the set of lists, all of the same length as the argument and with elements drawn from the corresponding element of the argument.*} consts listset :: "'a set list \<Rightarrow> 'a list set" primrec "listset [] = {[]}" "listset(A#As) = set_Cons A (listset As)" subsection{*Relations on Lists*} subsubsection {* Length Lexicographic Ordering *} text{*These orderings preserve well-foundedness: shorter lists precede longer lists. These ordering are not used in dictionaries.*} consts lexn :: "('a * 'a)set => nat => ('a list * 'a list)set" --{*The lexicographic ordering for lists of the specified length*} primrec "lexn r 0 = {}" "lexn r (Suc n) = (prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs) ` (r <*lex*> lexn r n)) Int {(xs,ys). length xs = Suc n \<and> length ys = Suc n}" constdefs lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set" "lex r == \<Union>n. lexn r n" --{*Holds only between lists of the same length*} lenlex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set" "lenlex r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))" --{*Compares lists by their length and then lexicographically*} lemma wf_lexn: "wf r ==> wf (lexn r n)" apply (induct n, simp, simp) apply(rule wf_subset) prefer 2 apply (rule Int_lower1) apply(rule wf_prod_fun_image) prefer 2 apply (rule inj_onI, auto) done lemma lexn_length: "!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n" by (induct n) auto lemma wf_lex [intro!]: "wf r ==> wf (lex r)" apply (unfold lex_def) apply (rule wf_UN) apply (blast intro: wf_lexn, clarify) apply (rename_tac m n) apply (subgoal_tac "m \<noteq> n") prefer 2 apply blast apply (blast dest: lexn_length not_sym) done lemma lexn_conv: "lexn r n = {(xs,ys). length xs = n \<and> length ys = n \<and> (\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}" apply (induct n, simp) apply (simp add: image_Collect lex_prod_def, safe, blast) apply (rule_tac x = "ab # xys" in exI, simp) apply (case_tac xys, simp_all, blast) done lemma lex_conv: "lex r = {(xs,ys). length xs = length ys \<and> (\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}" by (force simp add: lex_def lexn_conv) lemma wf_lenlex [intro!]: "wf r ==> wf (lenlex r)" by (unfold lenlex_def) blast lemma lenlex_conv: "lenlex r = {(xs,ys). length xs < length ys | length xs = length ys \<and> (xs, ys) : lex r}" by (simp add: lenlex_def diag_def lex_prod_def inv_image_def) lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r" by (simp add: lex_conv) lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r" by (simp add:lex_conv) lemma Cons_in_lex [simp]: "((x # xs, y # ys) : lex r) = ((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)" apply (simp add: lex_conv) apply (rule iffI) prefer 2 apply (blast intro: Cons_eq_appendI, clarify) apply (case_tac xys, simp, simp) apply blast done subsubsection {* Lexicographic Ordering *} text {* Classical lexicographic ordering on lists, ie. "a" < "ab" < "b". This ordering does \emph{not} preserve well-foundedness. Author: N. Voelker, March 2005. *} constdefs lexord :: "('a * 'a)set \<Rightarrow> ('a list * 'a list) set" "lexord r == {(x,y). \<exists> a v. y = x @ a # v \<or> (\<exists> u a b v w. (a,b) \<in> r \<and> x = u @ (a # v) \<and> y = u @ (b # w))}" lemma lexord_Nil_left[simp]: "([],y) \<in> lexord r = (\<exists> a x. y = a # x)" by (unfold lexord_def, induct_tac y, auto) lemma lexord_Nil_right[simp]: "(x,[]) \<notin> lexord r" by (unfold lexord_def, induct_tac x, auto) lemma lexord_cons_cons[simp]: "((a # x, b # y) \<in> lexord r) = ((a,b)\<in> r | (a = b & (x,y)\<in> lexord r))" apply (unfold lexord_def, safe, simp_all) apply (case_tac u, simp, simp) apply (case_tac u, simp, clarsimp, blast, blast, clarsimp) apply (erule_tac x="b # u" in allE) by force lemmas lexord_simps = lexord_Nil_left lexord_Nil_right lexord_cons_cons lemma lexord_append_rightI: "\<exists> b z. y = b # z \<Longrightarrow> (x, x @ y) \<in> lexord r" by (induct_tac x, auto) lemma lexord_append_left_rightI: "(a,b) \<in> r \<Longrightarrow> (u @ a # x, u @ b # y) \<in> lexord r" by (induct_tac u, auto) lemma lexord_append_leftI: " (u,v) \<in> lexord r \<Longrightarrow> (x @ u, x @ v) \<in> lexord r" by (induct x, auto) lemma lexord_append_leftD: "\<lbrakk> (x @ u, x @ v) \<in> lexord r; (! a. (a,a) \<notin> r) \<rbrakk> \<Longrightarrow> (u,v) \<in> lexord r" by (erule rev_mp, induct_tac x, auto) lemma lexord_take_index_conv: "((x,y) : lexord r) = ((length x < length y \<and> take (length x) y = x) \<or> (\<exists>i. i < min(length x)(length y) & take i x = take i y & (x!i,y!i) \<in> r))" apply (unfold lexord_def Let_def, clarsimp) apply (rule_tac f = "(% a b. a \<or> b)" in arg_cong2) apply auto apply (rule_tac x="hd (drop (length x) y)" in exI) apply (rule_tac x="tl (drop (length x) y)" in exI) apply (erule subst, simp add: min_def) apply (rule_tac x ="length u" in exI, simp) apply (rule_tac x ="take i x" in exI) apply (rule_tac x ="x ! i" in exI) apply (rule_tac x ="y ! i" in exI, safe) apply (rule_tac x="drop (Suc i) x" in exI) apply (drule sym, simp add: drop_Suc_conv_tl) apply (rule_tac x="drop (Suc i) y" in exI) by (simp add: drop_Suc_conv_tl) -- {* lexord is extension of partial ordering List.lex *} lemma lexord_lex: " (x,y) \<in> lex r = ((x,y) \<in> lexord r \<and> length x = length y)" apply (rule_tac x = y in spec) apply (induct_tac x, clarsimp) by (clarify, case_tac x, simp, force) lemma lexord_irreflexive: "(! x. (x,x) \<notin> r) \<Longrightarrow> (y,y) \<notin> lexord r" by (induct y, auto) lemma lexord_trans: "\<lbrakk> (x, y) \<in> lexord r; (y, z) \<in> lexord r; trans r \<rbrakk> \<Longrightarrow> (x, z) \<in> lexord r" apply (erule rev_mp)+ apply (rule_tac x = x in spec) apply (rule_tac x = z in spec) apply ( induct_tac y, simp, clarify) apply (case_tac xa, erule ssubst) apply (erule allE, erule allE) -- {* avoid simp recursion *} apply (case_tac x, simp, simp) apply (case_tac x, erule allE, erule allE, simp) apply (erule_tac x = listb in allE) apply (erule_tac x = lista in allE, simp) apply (unfold trans_def) by blast lemma lexord_transI: "trans r \<Longrightarrow> trans (lexord r)" by (rule transI, drule lexord_trans, blast) lemma lexord_linear: "(! a b. (a,b)\<in> r | a = b | (b,a) \<in> r) \<Longrightarrow> (x,y) : lexord r | x = y | (y,x) : lexord r" apply (rule_tac x = y in spec) apply (induct_tac x, rule allI) apply (case_tac x, simp, simp) apply (rule allI, case_tac x, simp, simp) by blast subsection {* Lexicographic combination of measure functions *} text {* These are useful for termination proofs *} definition "measures fs = inv_image (lex less_than) (%a. map (%f. f a) fs)" lemma wf_measures[recdef_wf, simp]: "wf (measures fs)" unfolding measures_def by blast lemma in_measures[simp]: "(x, y) \<in> measures [] = False" "(x, y) \<in> measures (f # fs) = (f x < f y \<or> (f x = f y \<and> (x, y) \<in> measures fs))" unfolding measures_def by auto lemma measures_less: "f x < f y ==> (x, y) \<in> measures (f#fs)" by simp lemma measures_lesseq: "f x <= f y ==> (x, y) \<in> measures fs ==> (x, y) \<in> measures (f#fs)" by auto subsubsection{*Lifting a Relation on List Elements to the Lists*} consts listrel :: "('a * 'a)set => ('a list * 'a list)set" inductive "listrel(r)" intros Nil: "([],[]) \<in> listrel r" Cons: "[| (x,y) \<in> r; (xs,ys) \<in> listrel r |] ==> (x#xs, y#ys) \<in> listrel r" inductive_cases listrel_Nil1 [elim!]: "([],xs) \<in> listrel r" inductive_cases listrel_Nil2 [elim!]: "(xs,[]) \<in> listrel r" inductive_cases listrel_Cons1 [elim!]: "(y#ys,xs) \<in> listrel r" inductive_cases listrel_Cons2 [elim!]: "(xs,y#ys) \<in> listrel r" lemma listrel_mono: "r \<subseteq> s \<Longrightarrow> listrel r \<subseteq> listrel s" apply clarify apply (erule listrel.induct) apply (blast intro: listrel.intros)+ done lemma listrel_subset: "r \<subseteq> A \<times> A \<Longrightarrow> listrel r \<subseteq> lists A \<times> lists A" apply clarify apply (erule listrel.induct, auto) done lemma listrel_refl: "refl A r \<Longrightarrow> refl (lists A) (listrel r)" apply (simp add: refl_def listrel_subset Ball_def) apply (rule allI) apply (induct_tac x) apply (auto intro: listrel.intros) done lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)" apply (auto simp add: sym_def) apply (erule listrel.induct) apply (blast intro: listrel.intros)+ done lemma listrel_trans: "trans r \<Longrightarrow> trans (listrel r)" apply (simp add: trans_def) apply (intro allI) apply (rule impI) apply (erule listrel.induct) apply (blast intro: listrel.intros)+ done theorem equiv_listrel: "equiv A r \<Longrightarrow> equiv (lists A) (listrel r)" by (simp add: equiv_def listrel_refl listrel_sym listrel_trans) lemma listrel_Nil [simp]: "listrel r `` {[]} = {[]}" by (blast intro: listrel.intros) lemma listrel_Cons: "listrel r `` {x#xs} = set_Cons (r``{x}) (listrel r `` {xs})"; by (auto simp add: set_Cons_def intro: listrel.intros) subsection{*Miscellany*} subsubsection {* Characters and strings *} datatype nibble = Nibble0 | Nibble1 | Nibble2 | Nibble3 | Nibble4 | Nibble5 | Nibble6 | Nibble7 | Nibble8 | Nibble9 | NibbleA | NibbleB | NibbleC | NibbleD | NibbleE | NibbleF datatype char = Char nibble nibble -- "Note: canonical order of character encoding coincides with standard term ordering" types string = "char list" syntax "_Char" :: "xstr => char" ("CHR _") "_String" :: "xstr => string" ("_") parse_ast_translation {* let val constants = Syntax.Appl o map Syntax.Constant; fun mk_nib n = "Nibble" ^ chr (n + (if n <= 9 then ord "0" else ord "A" - 10)); fun mk_char c = if Symbol.is_ascii c andalso Symbol.is_printable c then constants ["Char", mk_nib (ord c div 16), mk_nib (ord c mod 16)] else error ("Printable ASCII character expected: " ^ quote c); fun mk_string [] = Syntax.Constant "Nil" | mk_string (c :: cs) = Syntax.Appl [Syntax.Constant "Cons", mk_char c, mk_string cs]; fun char_ast_tr [Syntax.Variable xstr] = (case Syntax.explode_xstr xstr of [c] => mk_char c | _ => error ("Single character expected: " ^ xstr)) | char_ast_tr asts = raise AST ("char_ast_tr", asts); fun string_ast_tr [Syntax.Variable xstr] = (case Syntax.explode_xstr xstr of [] => constants [Syntax.constrainC, "Nil", "string"] | cs => mk_string cs) | string_ast_tr asts = raise AST ("string_tr", asts); in [("_Char", char_ast_tr), ("_String", string_ast_tr)] end; *} ML {* structure HOList = struct local val thy = the_context (); in val typ_string = Type (Sign.intern_type thy "string", []); fun typ_list ty = Type (Sign.intern_type thy "list", [ty]); fun term_list ty f [] = Const (Sign.intern_const thy "Nil", typ_list ty) | term_list ty f (x::xs) = Const (Sign.intern_const thy "Cons", ty --> typ_list ty --> typ_list ty) $ f x $ term_list ty f xs; end; fun int_of_nibble h = if "0" <= h andalso h <= "9" then ord h - ord "0" else if "A" <= h andalso h <= "F" then ord h - ord "A" + 10 else raise Match; fun nibble_of_int i = if i <= 9 then chr (ord "0" + i) else chr (ord "A" + i - 10); fun dest_char (Const ("List.char.Char", _) $ c1 $ c2) = let fun dest_nibble (Const (s, _)) = (int_of_nibble o unprefix "List.nibble.Nibble") s | dest_nibble _ = raise Match; in (SOME (dest_nibble c1 * 16 + dest_nibble c2) handle Fail _ => NONE | Match => NONE) end | dest_char _ = NONE; val print_list = Pretty.enum "," "[" "]"; fun print_char c = let val i = ord c in if i < 32 then prefix "\\" (string_of_int i) else c end; val print_string = quote; fun term_string s = let val ty_n = Type ("List.nibble", []); val ty_c = Type ("List.char", []); val ty_l = Type ("List.list", [ty_c]); fun mk_nib n = Const ("List.nibble.Nibble" ^ chr (n + (if n <= 9 then ord "0" else ord "A" - 10)), ty_n); fun mk_char c = if Symbol.is_ascii c andalso Symbol.is_printable c then Const ("List.char.Char", ty_n --> ty_n --> ty_c) $ mk_nib (ord c div 16) $ mk_nib (ord c mod 16) else error ("Printable ASCII character expected: " ^ quote c); fun mk_string c t = Const ("List.list.Cons", ty_c --> ty_l --> ty_l) $ mk_char c $ t; in fold_rev mk_string (explode s) (Const ("List.list.Nil", ty_l)) end; end; *} print_ast_translation {* let fun dest_nib (Syntax.Constant c) = (case explode c of ["N", "i", "b", "b", "l", "e", h] => HOList.int_of_nibble h | _ => raise Match) | dest_nib _ = raise Match; fun dest_chr c1 c2 = let val c = chr (dest_nib c1 * 16 + dest_nib c2) in if Symbol.is_printable c then c else raise Match end; fun dest_char (Syntax.Appl [Syntax.Constant "Char", c1, c2]) = dest_chr c1 c2 | dest_char _ = raise Match; fun xstr cs = Syntax.Appl [Syntax.Constant "_xstr", Syntax.Variable (Syntax.implode_xstr cs)]; fun char_ast_tr' [c1, c2] = Syntax.Appl [Syntax.Constant "_Char", xstr [dest_chr c1 c2]] | char_ast_tr' _ = raise Match; fun list_ast_tr' [args] = Syntax.Appl [Syntax.Constant "_String", xstr (map dest_char (Syntax.unfold_ast "_args" args))] | list_ast_tr' ts = raise Match; in [("Char", char_ast_tr'), ("@list", list_ast_tr')] end; *} subsection {* Code generator *} subsubsection {* Setup *} types_code "list" ("_ list") attach (term_of) {* val term_of_list = HOLogic.mk_list; *} attach (test) {* fun gen_list' aG i j = frequency [(i, fn () => aG j :: gen_list' aG (i-1) j), (1, fn () => [])] () and gen_list aG i = gen_list' aG i i; *} "char" ("string") attach (term_of) {* val nibbleT = Type ("List.nibble", []); fun term_of_char c = Const ("List.char.Char", nibbleT --> nibbleT --> Type ("List.char", [])) $ Const ("List.nibble.Nibble" ^ HOList.nibble_of_int (ord c div 16), nibbleT) $ Const ("List.nibble.Nibble" ^ HOList.nibble_of_int (ord c mod 16), nibbleT); *} attach (test) {* fun gen_char i = chr (random_range (ord "a") (Int.min (ord "a" + i, ord "z"))); *} consts_code "Cons" ("(_ ::/ _)") code_type list (SML "_ list") (Haskell "![_]") code_const Nil (SML "[]") (Haskell "[]") code_type char (SML "char") (Haskell "Char") code_const Char (SML "!((_),/ (_))") (Haskell "!((_),/ (_))") code_instance list :: eq and char :: eq (Haskell - and -) code_const "Code_Generator.eq \<Colon> 'a\<Colon>eq list \<Rightarrow> 'a list \<Rightarrow> bool" (Haskell infixl 4 "==") code_const "Code_Generator.eq \<Colon> char \<Rightarrow> char \<Rightarrow> bool" (Haskell infixl 4 "==") code_reserved SML list char code_reserved Haskell Char setup {* let fun list_codegen thy defs gr dep thyname b t = let val (gr', ps) = foldl_map (Codegen.invoke_codegen thy defs dep thyname false) (gr, HOLogic.dest_list t) in SOME (gr', Pretty.list "[" "]" ps) end handle TERM _ => NONE; fun char_codegen thy defs gr dep thyname b t = case (Option.map chr o HOList.dest_char) t of SOME c => if Symbol.is_printable c then SOME (gr, (Pretty.quote o Pretty.str) c) else NONE | NONE => NONE; in Codegen.add_codegen "list_codegen" list_codegen #> Codegen.add_codegen "char_codegen" char_codegen #> CodegenSerializer.add_pretty_list "SML" "List.list.Nil" "List.list.Cons" HOList.print_list NONE (7, "::") #> CodegenSerializer.add_pretty_list "Haskell" "List.list.Nil" "List.list.Cons" HOList.print_list (SOME (HOList.print_char, HOList.print_string)) (5, ":") #> CodegenPackage.add_appconst ("List.char.Char", CodegenPackage.appgen_char HOList.dest_char) end; *} subsubsection {* Generation of efficient code *} consts memberl :: "'a \<Rightarrow> 'a list \<Rightarrow> bool" (infixl "mem" 55) null:: "'a list \<Rightarrow> bool" list_inter :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" list_ex :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" list_all :: "('a \<Rightarrow> bool) \<Rightarrow> ('a list \<Rightarrow> bool)" itrev :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" filtermap :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a list \<Rightarrow> 'b list" map_filter :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'b list" primrec "x mem [] = False" "x mem (y#ys) = (x = y \<or> x mem ys)" primrec "null [] = True" "null (x#xs) = False" primrec "list_inter [] bs = []" "list_inter (a#as) bs = (if a \<in> set bs then a # list_inter as bs else list_inter as bs)" primrec "list_all P [] = True" "list_all P (x#xs) = (P x \<and> list_all P xs)" primrec "list_ex P [] = False" "list_ex P (x#xs) = (P x \<or> list_ex P xs)" primrec "filtermap f [] = []" "filtermap f (x#xs) = (case f x of None \<Rightarrow> filtermap f xs | Some y \<Rightarrow> y # filtermap f xs)" primrec "map_filter f P [] = []" "map_filter f P (x#xs) = (if P x then f x # map_filter f P xs else map_filter f P xs)" primrec "itrev [] ys = ys" "itrev (x#xs) ys = itrev xs (x#ys)" text {* Only use @{text mem} for generating executable code. Otherwise use @{prop "x : set xs"} instead --- it is much easier to reason about. The same is true for @{const list_all} and @{const list_ex}: write @{text "\<forall>x\<in>set xs"} and @{text "\<exists>x\<in>set xs"} instead because the HOL quantifiers are aleady known to the automatic provers. In fact, the declarations in the code subsection make sure that @{text "\<in>"}, @{text "\<forall>x\<in>set xs"} and @{text "\<exists>x\<in>set xs"} are implemented efficiently. Efficient emptyness check is implemented by @{const null}. The functions @{const itrev}, @{const filtermap} and @{const map_filter} are just there to generate efficient code. Do not use them for modelling and proving. *} lemma mem_iff [normal post]: "(x mem xs) = (x \<in> set xs)" by (induct xs) auto lemmas in_set_code [code unfold] = mem_iff [symmetric, THEN eq_reflection] lemma empty_null [code inline]: "(xs = []) = null xs" by (cases xs) simp_all lemmas null_empty [normal post] = empty_null [symmetric] lemma list_inter_conv: "set (list_inter xs ys) = set xs \<inter> set ys" by (induct xs) auto lemma list_all_iff [normal post]: "list_all P xs = (\<forall>x \<in> set xs. P x)" by (induct xs) auto lemmas list_ball_code [code unfold] = list_all_iff [symmetric, THEN eq_reflection] lemma list_all_append [simp]: "list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)" by (induct xs) auto lemma list_all_rev [simp]: "list_all P (rev xs) = list_all P xs" by (simp add: list_all_iff) lemma list_ex_iff [normal post]: "list_ex P xs = (\<exists>x \<in> set xs. P x)" by (induct xs) simp_all lemmas list_bex_code [code unfold] = list_ex_iff [symmetric, THEN eq_reflection] lemma itrev [simp, normal post]: "itrev xs ys = rev xs @ ys" by (induct xs arbitrary: ys) simp_all lemma filtermap_conv: "filtermap f xs = map (\<lambda>x. the (f x)) (filter (\<lambda>x. f x \<noteq> None) xs)" by (induct xs) (simp_all split: option.split) lemma map_filter_conv [simp]: "map_filter f P xs = map f (filter P xs)" by (induct xs) auto lemma rev_code [code func, code unfold]: "rev xs == itrev xs []" by simp end