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src/HOL/List.thy

author | wenzelm |

Fri, 16 Apr 2004 13:51:04 +0200 | |

changeset 14589 | feae7b5fd425 |

parent 14565 | c6dc17aab88a |

child 14591 | 7be4d5dadf15 |

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

tuned document;

(* Title: HOL/List.thy ID: $Id$ Author: Tobias Nipkow License: GPL (GNU GENERAL PUBLIC LICENSE) *) header {* The datatype of finite lists *} theory List = PreList: datatype 'a list = Nil ("[]") | Cons 'a "'a list" (infixr "#" 65) 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" list_all:: "('a => bool) => ('a list => bool)" list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool" map :: "('a=>'b) => ('a list => 'b list)" mem :: "'a => 'a list => bool" (infixl 55) 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" null:: "'a list => bool" "distinct":: "'a list => bool" replicate :: "nat => 'a => 'a list" 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) upto:: "nat => nat => nat list" ("(1[_../_])") 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" "[i..j]" == "[i..(Suc j)(]" 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}. *} syntax length :: "'a list => nat" translations "length" => "size :: _ list => nat" typed_print_translation {* let fun size_tr' _ (Type ("fun", (Type ("list", _) :: _))) [t] = Syntax.const "length" $ t | size_tr' _ _ _ = raise Match; in [("size", size_tr')] end *} primrec "hd(x#xs) = x" primrec "tl([]) = []" "tl(x#xs) = xs" primrec "null([]) = True" "null(x#xs) = False" primrec "last(x#xs) = (if xs=[] then x else last xs)" primrec "butlast []= []" "butlast(x#xs) = (if xs=[] then [] else x#butlast xs)" primrec "x mem [] = False" "x mem (y#ys) = (if y=x then True else x mem ys)" primrec "set [] = {}" "set (x#xs) = insert x (set xs)" primrec list_all_Nil:"list_all P [] = True" list_all_Cons: "list_all P (x#xs) = (P(x) \<and> list_all P 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 replicate_0: "replicate 0 x = []" replicate_Suc: "replicate (Suc n) x = x # replicate n x" defs list_all2_def: "list_all2 P xs ys == length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y)" subsection {* Lexicographic orderings on lists *} consts lexn :: "('a * 'a)set => nat => ('a list * 'a list)set" 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" lexico :: "('a \<times> 'a) set => ('a list \<times> 'a list) set" "lexico r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))" sublist :: "'a list => nat set => 'a list" "sublist xs A == map fst (filter (%p. snd p : A) (zip xs [0..size xs(]))" 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: "(!!xs. \<forall>ys. length ys < length xs --> P ys ==> P xs) ==> P xs" by (rule measure_induct [of length]) rules subsection {* @{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" apply (rule mono_Int [THEN equalityI]) apply (simp add: mono_def lists_mono) apply (blast intro!: lists_IntI) done lemma append_in_lists_conv [iff]: "(xs @ ys : lists A) = (xs : lists A \<and> ys : lists A)" by (induct xs) auto subsection {* @{text 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, 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 subsection {* @{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 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_tac = simp_tac (HOL_basic_ss addsimps [append_assoc, append_Nil, append_Cons]); fun list_eq sg _ (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 = Tactic.prove sg [] [] eq (K (rearr_tac 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 (Theory.sign_of (the_context ())) "list_eq" ["(xs::'a list) = ys"] list_eq; end; Addsimprocs [list_eq_simproc]; *} subsection {* @{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 [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[iff]: "(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[iff]: "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)" by (cases ys) auto lemma ex_map_conv: "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)" by(induct ys, auto) 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 (rules 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 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 subsection {* @{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_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_is_rev_conv [iff]: "!!ys. (rev xs = rev ys) = (xs = ys)" apply (induct xs, force) apply (case_tac ys, simp, force) done lemma rev_induct [case_names Nil snoc]: "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs" apply(subst 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 subsection {* @{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: "l = x#xs \<Longrightarrow> x\<in>set l" by (case_tac l, 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_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)" apply (induct xs, simp, simp) apply (rule iffI) apply (blast intro: eq_Nil_appendI Cons_eq_appendI) apply (erule exE)+ apply (case_tac ys, auto) done 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 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) subsection {* @{text mem} *} lemma set_mem_eq: "(x mem xs) = (x : set xs)" by (induct xs) auto subsection {* @{text list_all} *} lemma list_all_conv: "list_all P xs = (\<forall>x \<in> set xs. P x)" by (induct xs) auto lemma list_all_append [simp]: "list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)" by (induct xs) auto subsection {* @{text filter} *} lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys" by (induct xs) auto lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs" by (induct xs) auto 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 length_filter [simp]: "length (filter P xs) \<le> length xs" by (induct xs) (auto simp add: le_SucI) lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs" by auto subsection {* @{text concat} *} lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys" by (induct xs) auto lemma concat_eq_Nil_conv [iff]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])" by (induct xss) auto lemma Nil_eq_concat_conv [iff]: "([] = 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 subsection {* @{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 set_conv_nth: "set xs = {xs!i | i. i < length xs}" apply (induct_tac 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 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) subsection {* @{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_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_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]) subsection {* @{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 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) subsection {* @{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 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 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 na, 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 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_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 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 subsection {* @{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) subsection {* @{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 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 subsection {* @{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]: "list_all2 P [] ys = (ys = [])" by (simp add: list_all2_def) lemma list_all2_Nil2[iff]: "list_all2 P xs [] = (xs = [])" by (simp add: list_all2_def) lemma list_all2_Cons [iff]: "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 subsection {* @{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 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 subsection {* @{text upto} *} lemma upt_rec: "[i..j(] = (if i<j then i#[Suc i..j(] else [])" -- {* Does not terminate! *} by (induct j) auto lemma upt_conv_Nil [simp]: "j <= i ==> [i..j(] = []" by (subst upt_rec) simp 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 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 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 subsection {* @{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 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 distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)" by (induct xs) auto text {* It is best to avoid this indexed version of distinct, but sometimes it is useful. *} lemma distinct_conv_nth: "distinct xs = (\<forall>i j. i < size xs \<and> j < size xs \<and> i \<noteq> j --> xs!i \<noteq> xs!j)" apply (induct_tac 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) apply (erule_tac x = "Suc i" in allE, simp, clarsimp) apply (erule_tac x = "Suc i" in allE) apply (erule_tac x = "Suc j" in allE, simp) done lemma distinct_card: "distinct xs \<Longrightarrow> card (set xs) = size xs" by (induct xs) auto lemma card_distinct: "card (set xs) = size xs \<Longrightarrow> 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 subsection {* @{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 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 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) subsection {* Lexicographic orderings on lists *} lemma wf_lexn: "wf r ==> wf (lexn r n)" apply (induct_tac 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_tac n, simp, blast) 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_lexico [intro!]: "wf r ==> wf (lexico r)" by (unfold lexico_def) blast lemma lexico_conv: "lexico r = {(xs,ys). length xs < length ys | length xs = length ys \<and> (xs, ys) : lex r}" by (simp add: lexico_def diag_def lex_prod_def measure_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 [iff]: "((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 subsection {* @{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 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 sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])" by (simp add: sublist_Cons) 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 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 [simp] = take_Cons'[of "number_of v",standard] drop_Cons'[of "number_of v",standard] nth_Cons'[of _ _ "number_of v",standard] lemma distinct_card: "distinct xs \<Longrightarrow> card (set xs) = size xs" by (induct xs) auto lemma card_length: "card (set xs) \<le> length xs" by (induct xs) (auto simp add: card_insert_if) lemma "card (set xs) = size xs \<Longrightarrow> 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 subsection {* 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; *} print_ast_translation {* let fun dest_nib (Syntax.Constant c) = (case explode c of ["N", "i", "b", "b", "l", "e", 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 | _ => 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; *} end