wenzelm@13462: (* Title: HOL/List.thy wenzelm@13462: ID: $Id$ wenzelm@13462: Author: Tobias Nipkow clasohm@923: *) clasohm@923: wenzelm@13114: header {* The datatype of finite lists *} wenzelm@13122: nipkow@15131: theory List nipkow@15140: imports PreList nipkow@15131: begin clasohm@923: wenzelm@13142: datatype 'a list = wenzelm@13366: Nil ("[]") wenzelm@13366: | Cons 'a "'a list" (infixr "#" 65) clasohm@923: nipkow@15392: subsection{*Basic list processing functions*} nipkow@15302: clasohm@923: consts wenzelm@13366: "@" :: "'a list => 'a list => 'a list" (infixr 65) wenzelm@13366: filter:: "('a => bool) => 'a list => 'a list" wenzelm@13366: concat:: "'a list list => 'a list" wenzelm@13366: foldl :: "('b => 'a => 'b) => 'b => 'a list => 'b" wenzelm@13366: foldr :: "('a => 'b => 'b) => 'a list => 'b => 'b" wenzelm@13366: hd:: "'a list => 'a" wenzelm@13366: tl:: "'a list => 'a list" wenzelm@13366: last:: "'a list => 'a" wenzelm@13366: butlast :: "'a list => 'a list" wenzelm@13366: set :: "'a list => 'a set" wenzelm@13366: list_all:: "('a => bool) => ('a list => bool)" wenzelm@13366: list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool" nipkow@15439: list_ex :: "('a \ bool) \ 'a list \ bool" wenzelm@13366: map :: "('a=>'b) => ('a list => 'b list)" wenzelm@13366: mem :: "'a => 'a list => bool" (infixl 55) wenzelm@13366: nth :: "'a list => nat => 'a" (infixl "!" 100) wenzelm@13366: list_update :: "'a list => nat => 'a => 'a list" wenzelm@13366: take:: "nat => 'a list => 'a list" wenzelm@13366: drop:: "nat => 'a list => 'a list" wenzelm@13366: takeWhile :: "('a => bool) => 'a list => 'a list" wenzelm@13366: dropWhile :: "('a => bool) => 'a list => 'a list" wenzelm@13366: rev :: "'a list => 'a list" wenzelm@13366: zip :: "'a list => 'b list => ('a * 'b) list" nipkow@15425: upt :: "nat => nat => nat list" ("(1[_.. 'a list" nipkow@15110: remove1 :: "'a => 'a list => 'a list" wenzelm@13366: null:: "'a list => bool" wenzelm@13366: "distinct":: "'a list => bool" wenzelm@13366: replicate :: "nat => 'a => 'a list" nipkow@15302: rotate1 :: "'a list \ 'a list" nipkow@15302: rotate :: "nat \ 'a list \ 'a list" nipkow@15302: sublist :: "'a list => nat set => 'a list" nipkow@15302: clasohm@923: nipkow@13146: nonterminals lupdbinds lupdbind nipkow@5077: clasohm@923: syntax wenzelm@13366: -- {* list Enumeration *} wenzelm@13366: "@list" :: "args => 'a list" ("[(_)]") clasohm@923: wenzelm@13366: -- {* Special syntax for filter *} wenzelm@13366: "@filter" :: "[pttrn, 'a list, bool] => 'a list" ("(1[_:_./ _])") clasohm@923: wenzelm@13366: -- {* list update *} wenzelm@13366: "_lupdbind":: "['a, 'a] => lupdbind" ("(2_ :=/ _)") wenzelm@13366: "" :: "lupdbind => lupdbinds" ("_") wenzelm@13366: "_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds" ("_,/ _") wenzelm@13366: "_LUpdate" :: "['a, lupdbinds] => 'a" ("_/[(_)]" [900,0] 900) nipkow@5077: wenzelm@13366: upto:: "nat => nat => nat list" ("(1[_../_])") nipkow@5427: clasohm@923: translations wenzelm@13366: "[x, xs]" == "x#[xs]" wenzelm@13366: "[x]" == "x#[]" wenzelm@13366: "[x:xs . P]"== "filter (%x. P) xs" clasohm@923: wenzelm@13366: "_LUpdate xs (_lupdbinds b bs)"== "_LUpdate (_LUpdate xs b) bs" wenzelm@13366: "xs[i:=x]" == "list_update xs i x" nipkow@5077: nipkow@15425: "[i..j]" == "[i..<(Suc j)]" nipkow@5427: nipkow@5427: wenzelm@12114: syntax (xsymbols) wenzelm@13366: "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\_ ./ _])") kleing@14565: syntax (HTML output) kleing@14565: "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\_ ./ _])") paulson@3342: paulson@3342: wenzelm@13142: text {* wenzelm@14589: Function @{text size} is overloaded for all datatypes. Users may wenzelm@13366: refer to the list version as @{text length}. *} wenzelm@13142: wenzelm@13142: syntax length :: "'a list => nat" wenzelm@13142: translations "length" => "size :: _ list => nat" wenzelm@13114: wenzelm@13142: typed_print_translation {* wenzelm@13366: let wenzelm@13366: fun size_tr' _ (Type ("fun", (Type ("list", _) :: _))) [t] = wenzelm@13366: Syntax.const "length" $ t wenzelm@13366: | size_tr' _ _ _ = raise Match; wenzelm@13366: in [("size", size_tr')] end wenzelm@13114: *} paulson@3437: nipkow@15302: berghofe@5183: primrec paulson@15307: "hd(x#xs) = x" paulson@15307: berghofe@5183: primrec paulson@15307: "tl([]) = []" paulson@15307: "tl(x#xs) = xs" paulson@15307: berghofe@5183: primrec paulson@15307: "null([]) = True" paulson@15307: "null(x#xs) = False" paulson@15307: paulson@8972: primrec paulson@15307: "last(x#xs) = (if xs=[] then x else last xs)" paulson@15307: berghofe@5183: primrec paulson@15307: "butlast []= []" paulson@15307: "butlast(x#xs) = (if xs=[] then [] else x#butlast xs)" paulson@15307: berghofe@5183: primrec paulson@15307: "x mem [] = False" paulson@15307: "x mem (y#ys) = (if y=x then True else x mem ys)" paulson@15307: oheimb@5518: primrec paulson@15307: "set [] = {}" paulson@15307: "set (x#xs) = insert x (set xs)" paulson@15307: berghofe@5183: primrec paulson@15307: list_all_Nil:"list_all P [] = True" paulson@15307: list_all_Cons: "list_all P (x#xs) = (P(x) \ list_all P xs)" paulson@15307: berghofe@5183: primrec nipkow@15439: "list_ex P [] = False" nipkow@15439: "list_ex P (x#xs) = (P x \ list_ex P xs)" nipkow@15439: nipkow@15439: primrec paulson@15307: "map f [] = []" paulson@15307: "map f (x#xs) = f(x)#map f xs" paulson@15307: berghofe@5183: primrec paulson@15307: append_Nil:"[]@ys = ys" paulson@15307: append_Cons: "(x#xs)@ys = x#(xs@ys)" paulson@15307: berghofe@5183: primrec paulson@15307: "rev([]) = []" paulson@15307: "rev(x#xs) = rev(xs) @ [x]" paulson@15307: berghofe@5183: primrec paulson@15307: "filter P [] = []" paulson@15307: "filter P (x#xs) = (if P x then x#filter P xs else filter P xs)" paulson@15307: berghofe@5183: primrec paulson@15307: foldl_Nil:"foldl f a [] = a" paulson@15307: foldl_Cons: "foldl f a (x#xs) = foldl f (f a x) xs" paulson@15307: paulson@8000: primrec paulson@15307: "foldr f [] a = a" paulson@15307: "foldr f (x#xs) a = f x (foldr f xs a)" paulson@15307: berghofe@5183: primrec paulson@15307: "concat([]) = []" paulson@15307: "concat(x#xs) = x @ concat(xs)" paulson@15307: berghofe@5183: primrec paulson@15307: drop_Nil:"drop n [] = []" paulson@15307: drop_Cons: "drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)" paulson@15307: -- {*Warning: simpset does not contain this definition, but separate paulson@15307: theorems for @{text "n = 0"} and @{text "n = Suc k"} *} paulson@15307: berghofe@5183: primrec paulson@15307: take_Nil:"take n [] = []" paulson@15307: take_Cons: "take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)" paulson@15307: -- {*Warning: simpset does not contain this definition, but separate paulson@15307: theorems for @{text "n = 0"} and @{text "n = Suc k"} *} paulson@15307: wenzelm@13142: primrec paulson@15307: nth_Cons:"(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)" paulson@15307: -- {*Warning: simpset does not contain this definition, but separate paulson@15307: theorems for @{text "n = 0"} and @{text "n = Suc k"} *} paulson@15307: berghofe@5183: primrec paulson@15307: "[][i:=v] = []" paulson@15307: "(x#xs)[i:=v] = (case i of 0 => v # xs | Suc j => x # xs[j:=v])" paulson@15307: paulson@15307: primrec paulson@15307: "takeWhile P [] = []" paulson@15307: "takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])" paulson@15307: berghofe@5183: primrec paulson@15307: "dropWhile P [] = []" paulson@15307: "dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)" paulson@15307: berghofe@5183: primrec paulson@15307: "zip xs [] = []" paulson@15307: zip_Cons: "zip xs (y#ys) = (case xs of [] => [] | z#zs => (z,y)#zip zs ys)" paulson@15307: -- {*Warning: simpset does not contain this definition, but separate paulson@15307: theorems for @{text "xs = []"} and @{text "xs = z # zs"} *} paulson@15307: nipkow@5427: primrec nipkow@15425: upt_0: "[i..<0] = []" nipkow@15425: upt_Suc: "[i..<(Suc j)] = (if i <= j then [i.. distinct xs)" paulson@15307: berghofe@5183: primrec paulson@15307: "remdups [] = []" paulson@15307: "remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)" paulson@15307: berghofe@5183: primrec paulson@15307: "remove1 x [] = []" paulson@15307: "remove1 x (y#xs) = (if x=y then xs else y # remove1 x xs)" paulson@15307: nipkow@15110: primrec paulson@15307: replicate_0: "replicate 0 x = []" paulson@15307: replicate_Suc: "replicate (Suc n) x = x # replicate n x" paulson@15307: nipkow@8115: defs nipkow@15302: rotate1_def: "rotate1 xs == (case xs of [] \ [] | x#xs \ xs @ [x])" nipkow@15302: rotate_def: "rotate n == rotate1 ^ n" nipkow@15302: nipkow@15302: list_all2_def: nipkow@15302: "list_all2 P xs ys == nipkow@15302: length xs = length ys \ (\(x, y) \ set (zip xs ys). P x y)" nipkow@15302: nipkow@15302: sublist_def: nipkow@15425: "sublist xs A == map fst (filter (%p. snd p : A) (zip xs [0.. x # xs" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric] wenzelm@13114: wenzelm@13142: lemma neq_Nil_conv: "(xs \ []) = (\y ys. xs = y # ys)" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma length_induct: nipkow@13145: "(!!xs. \ys. length ys < length xs --> P ys ==> P xs) ==> P xs" nipkow@13145: by (rule measure_induct [of length]) rules wenzelm@13114: wenzelm@13114: nipkow@15392: subsubsection {* @{text length} *} wenzelm@13114: wenzelm@13142: text {* nipkow@13145: Needs to come before @{text "@"} because of theorem @{text nipkow@13145: append_eq_append_conv}. wenzelm@13142: *} wenzelm@13114: wenzelm@13142: lemma length_append [simp]: "length (xs @ ys) = length xs + length ys" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma length_map [simp]: "length (map f xs) = length xs" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma length_rev [simp]: "length (rev xs) = length xs" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma length_tl [simp]: "length (tl xs) = length xs - 1" nipkow@13145: by (cases xs) auto wenzelm@13114: wenzelm@13142: lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \ [])" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13114: lemma length_Suc_conv: nipkow@13145: "(length xs = Suc n) = (\y ys. xs = y # ys \ length ys = n)" nipkow@13145: by (induct xs) auto wenzelm@13142: nipkow@14025: lemma Suc_length_conv: nipkow@14025: "(Suc n = length xs) = (\y ys. xs = y # ys \ length ys = n)" paulson@14208: apply (induct xs, simp, simp) nipkow@14025: apply blast nipkow@14025: done nipkow@14025: oheimb@14099: lemma impossible_Cons [rule_format]: oheimb@14099: "length xs <= length ys --> xs = x # ys = False" paulson@14208: apply (induct xs, auto) oheimb@14099: done oheimb@14099: nipkow@14247: lemma list_induct2[consumes 1]: "\ys. nipkow@14247: \ length xs = length ys; nipkow@14247: P [] []; nipkow@14247: \x xs y ys. \ length xs = length ys; P xs ys \ \ P (x#xs) (y#ys) \ nipkow@14247: \ P xs ys" nipkow@14247: apply(induct xs) nipkow@14247: apply simp nipkow@14247: apply(case_tac ys) nipkow@14247: apply simp nipkow@14247: apply(simp) nipkow@14247: done wenzelm@13114: nipkow@15392: subsubsection {* @{text "@"} -- append *} wenzelm@13114: wenzelm@13142: lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma append_Nil2 [simp]: "xs @ [] = xs" nipkow@13145: by (induct xs) auto nipkow@3507: wenzelm@13142: lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \ ys = [])" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \ ys = [])" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])" nipkow@13145: by (induct xs) auto wenzelm@13114: berghofe@13883: lemma append_eq_append_conv [simp]: berghofe@13883: "!!ys. length xs = length ys \ length us = length vs berghofe@13883: ==> (xs@us = ys@vs) = (xs=ys \ us=vs)" berghofe@13883: apply (induct xs) paulson@14208: apply (case_tac ys, simp, force) paulson@14208: apply (case_tac ys, force, simp) nipkow@13145: done wenzelm@13142: nipkow@14495: lemma append_eq_append_conv2: "!!ys zs ts. nipkow@14495: (xs @ ys = zs @ ts) = nipkow@14495: (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)" nipkow@14495: apply (induct xs) nipkow@14495: apply fastsimp nipkow@14495: apply(case_tac zs) nipkow@14495: apply simp nipkow@14495: apply fastsimp nipkow@14495: done nipkow@14495: wenzelm@13142: lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)" nipkow@13145: by simp wenzelm@13142: wenzelm@13142: lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \ x = y)" nipkow@13145: by simp wenzelm@13114: wenzelm@13142: lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)" nipkow@13145: by simp wenzelm@13114: wenzelm@13142: lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])" nipkow@13145: using append_same_eq [of _ _ "[]"] by auto nipkow@3507: wenzelm@13142: lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])" nipkow@13145: using append_same_eq [of "[]"] by auto wenzelm@13114: wenzelm@13142: lemma hd_Cons_tl [simp]: "xs \ [] ==> hd xs # tl xs = xs" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma hd_append2 [simp]: "xs \ [] ==> hd (xs @ ys) = hd xs" nipkow@13145: by (simp add: hd_append split: list.split) wenzelm@13114: wenzelm@13142: lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)" nipkow@13145: by (simp split: list.split) wenzelm@13114: wenzelm@13142: lemma tl_append2 [simp]: "xs \ [] ==> tl (xs @ ys) = tl xs @ ys" nipkow@13145: by (simp add: tl_append split: list.split) wenzelm@13114: wenzelm@13114: nipkow@14300: lemma Cons_eq_append_conv: "x#xs = ys@zs = nipkow@14300: (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))" nipkow@14300: by(cases ys) auto nipkow@14300: nipkow@15281: lemma append_eq_Cons_conv: "(ys@zs = x#xs) = nipkow@15281: (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))" nipkow@15281: by(cases ys) auto nipkow@15281: nipkow@14300: wenzelm@13142: text {* Trivial rules for solving @{text "@"}-equations automatically. *} wenzelm@13114: wenzelm@13114: lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys" nipkow@13145: by simp wenzelm@13114: wenzelm@13142: lemma Cons_eq_appendI: nipkow@13145: "[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs" nipkow@13145: by (drule sym) simp wenzelm@13114: wenzelm@13142: lemma append_eq_appendI: nipkow@13145: "[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us" nipkow@13145: by (drule sym) simp wenzelm@13114: wenzelm@13114: wenzelm@13142: text {* nipkow@13145: Simplification procedure for all list equalities. nipkow@13145: Currently only tries to rearrange @{text "@"} to see if nipkow@13145: - both lists end in a singleton list, nipkow@13145: - or both lists end in the same list. wenzelm@13142: *} wenzelm@13142: wenzelm@13142: ML_setup {* nipkow@3507: local nipkow@3507: wenzelm@13122: val append_assoc = thm "append_assoc"; wenzelm@13122: val append_Nil = thm "append_Nil"; wenzelm@13122: val append_Cons = thm "append_Cons"; wenzelm@13122: val append1_eq_conv = thm "append1_eq_conv"; wenzelm@13122: val append_same_eq = thm "append_same_eq"; wenzelm@13122: wenzelm@13114: fun last (cons as Const("List.list.Cons",_) $ _ $ xs) = wenzelm@13462: (case xs of Const("List.list.Nil",_) => cons | _ => last xs) wenzelm@13462: | last (Const("List.op @",_) $ _ $ ys) = last ys wenzelm@13462: | last t = t; wenzelm@13114: wenzelm@13114: fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true wenzelm@13462: | list1 _ = false; wenzelm@13114: wenzelm@13114: fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) = wenzelm@13462: (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs) wenzelm@13462: | butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys wenzelm@13462: | butlast xs = Const("List.list.Nil",fastype_of xs); wenzelm@13114: wenzelm@13114: val rearr_tac = wenzelm@13462: simp_tac (HOL_basic_ss addsimps [append_assoc, append_Nil, append_Cons]); wenzelm@13114: wenzelm@13114: fun list_eq sg _ (F as (eq as Const(_,eqT)) $ lhs $ rhs) = wenzelm@13462: let wenzelm@13462: val lastl = last lhs and lastr = last rhs; wenzelm@13462: fun rearr conv = wenzelm@13462: let wenzelm@13462: val lhs1 = butlast lhs and rhs1 = butlast rhs; wenzelm@13462: val Type(_,listT::_) = eqT wenzelm@13462: val appT = [listT,listT] ---> listT wenzelm@13462: val app = Const("List.op @",appT) wenzelm@13462: val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr) wenzelm@13480: val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2)); wenzelm@13480: val thm = Tactic.prove sg [] [] eq (K (rearr_tac 1)); skalberg@15531: in SOME ((conv RS (thm RS trans)) RS eq_reflection) end; wenzelm@13114: wenzelm@13462: in wenzelm@13462: if list1 lastl andalso list1 lastr then rearr append1_eq_conv wenzelm@13462: else if lastl aconv lastr then rearr append_same_eq skalberg@15531: else NONE wenzelm@13462: end; wenzelm@13462: wenzelm@13114: in wenzelm@13462: wenzelm@13462: val list_eq_simproc = wenzelm@13462: Simplifier.simproc (Theory.sign_of (the_context ())) "list_eq" ["(xs::'a list) = ys"] list_eq; wenzelm@13462: wenzelm@13114: end; wenzelm@13114: wenzelm@13114: Addsimprocs [list_eq_simproc]; wenzelm@13114: *} wenzelm@13114: wenzelm@13114: nipkow@15392: subsubsection {* @{text map} *} wenzelm@13114: wenzelm@13142: lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs" nipkow@13145: by (induct xs) simp_all wenzelm@13114: wenzelm@13142: lemma map_ident [simp]: "map (\x. x) = (\xs. xs)" nipkow@13145: by (rule ext, induct_tac xs) auto wenzelm@13114: wenzelm@13142: lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma map_compose: "map (f o g) xs = map f (map g xs)" nipkow@13145: by (induct xs) (auto simp add: o_def) wenzelm@13114: wenzelm@13142: lemma rev_map: "rev (map f xs) = map f (rev xs)" nipkow@13145: by (induct xs) auto wenzelm@13114: nipkow@13737: lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)" nipkow@13737: by (induct xs) auto nipkow@13737: wenzelm@13366: lemma map_cong [recdef_cong]: nipkow@13145: "xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys" nipkow@13145: -- {* a congruence rule for @{text map} *} nipkow@13737: by simp wenzelm@13114: wenzelm@13142: lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])" nipkow@13145: by (cases xs) auto wenzelm@13114: wenzelm@13142: lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])" nipkow@13145: by (cases xs) auto wenzelm@13114: nipkow@14025: lemma map_eq_Cons_conv[iff]: nipkow@14025: "(map f xs = y#ys) = (\z zs. xs = z#zs \ f z = y \ map f zs = ys)" nipkow@13145: by (cases xs) auto wenzelm@13114: nipkow@14025: lemma Cons_eq_map_conv[iff]: nipkow@14025: "(x#xs = map f ys) = (\z zs. ys = z#zs \ x = f z \ xs = map f zs)" nipkow@14025: by (cases ys) auto nipkow@14025: nipkow@14111: lemma ex_map_conv: nipkow@14111: "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)" nipkow@14111: by(induct ys, auto) nipkow@14111: nipkow@15110: lemma map_eq_imp_length_eq: nipkow@15110: "!!xs. map f xs = map f ys ==> length xs = length ys" nipkow@15110: apply (induct ys) nipkow@15110: apply simp nipkow@15110: apply(simp (no_asm_use)) nipkow@15110: apply clarify nipkow@15110: apply(simp (no_asm_use)) nipkow@15110: apply fast nipkow@15110: done nipkow@15110: nipkow@15110: lemma map_inj_on: nipkow@15110: "[| map f xs = map f ys; inj_on f (set xs Un set ys) |] nipkow@15110: ==> xs = ys" nipkow@15110: apply(frule map_eq_imp_length_eq) nipkow@15110: apply(rotate_tac -1) nipkow@15110: apply(induct rule:list_induct2) nipkow@15110: apply simp nipkow@15110: apply(simp) nipkow@15110: apply (blast intro:sym) nipkow@15110: done nipkow@15110: nipkow@15110: lemma inj_on_map_eq_map: nipkow@15110: "inj_on f (set xs Un set ys) \ (map f xs = map f ys) = (xs = ys)" nipkow@15110: by(blast dest:map_inj_on) nipkow@15110: wenzelm@13114: lemma map_injective: nipkow@14338: "!!xs. map f xs = map f ys ==> inj f ==> xs = ys" nipkow@14338: by (induct ys) (auto dest!:injD) wenzelm@13114: nipkow@14339: lemma inj_map_eq_map[simp]: "inj f \ (map f xs = map f ys) = (xs = ys)" nipkow@14339: by(blast dest:map_injective) nipkow@14339: wenzelm@13114: lemma inj_mapI: "inj f ==> inj (map f)" paulson@13585: by (rules dest: map_injective injD intro: inj_onI) wenzelm@13114: wenzelm@13114: lemma inj_mapD: "inj (map f) ==> inj f" paulson@14208: apply (unfold inj_on_def, clarify) nipkow@13145: apply (erule_tac x = "[x]" in ballE) paulson@14208: apply (erule_tac x = "[y]" in ballE, simp, blast) nipkow@13145: apply blast nipkow@13145: done wenzelm@13114: nipkow@14339: lemma inj_map[iff]: "inj (map f) = inj f" nipkow@13145: by (blast dest: inj_mapD intro: inj_mapI) wenzelm@13114: nipkow@15303: lemma inj_on_mapI: "inj_on f (\(set ` A)) \ inj_on (map f) A" nipkow@15303: apply(rule inj_onI) nipkow@15303: apply(erule map_inj_on) nipkow@15303: apply(blast intro:inj_onI dest:inj_onD) nipkow@15303: done nipkow@15303: kleing@14343: lemma map_idI: "(\x. x \ set xs \ f x = x) \ map f xs = xs" kleing@14343: by (induct xs, auto) wenzelm@13114: nipkow@14402: lemma map_fun_upd [simp]: "y \ set xs \ map (f(y:=v)) xs = map f xs" nipkow@14402: by (induct xs) auto nipkow@14402: nipkow@15110: lemma map_fst_zip[simp]: nipkow@15110: "length xs = length ys \ map fst (zip xs ys) = xs" nipkow@15110: by (induct rule:list_induct2, simp_all) nipkow@15110: nipkow@15110: lemma map_snd_zip[simp]: nipkow@15110: "length xs = length ys \ map snd (zip xs ys) = ys" nipkow@15110: by (induct rule:list_induct2, simp_all) nipkow@15110: nipkow@15110: nipkow@15392: subsubsection {* @{text rev} *} wenzelm@13114: wenzelm@13142: lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma rev_rev_ident [simp]: "rev (rev xs) = xs" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma rev_is_rev_conv [iff]: "!!ys. (rev xs = rev ys) = (xs = ys)" paulson@14208: apply (induct xs, force) paulson@14208: apply (case_tac ys, simp, force) nipkow@13145: done wenzelm@13114: nipkow@15439: lemma inj_on_rev[iff]: "inj_on rev A" nipkow@15439: by(simp add:inj_on_def) nipkow@15439: wenzelm@13366: lemma rev_induct [case_names Nil snoc]: wenzelm@13366: "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs" berghofe@15489: apply(simplesubst rev_rev_ident[symmetric]) nipkow@13145: apply(rule_tac list = "rev xs" in list.induct, simp_all) nipkow@13145: done wenzelm@13114: nipkow@13145: ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility" wenzelm@13114: wenzelm@13366: lemma rev_exhaust [case_names Nil snoc]: wenzelm@13366: "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P" nipkow@13145: by (induct xs rule: rev_induct) auto wenzelm@13114: wenzelm@13366: lemmas rev_cases = rev_exhaust wenzelm@13366: wenzelm@13114: nipkow@15392: subsubsection {* @{text set} *} wenzelm@13114: wenzelm@13142: lemma finite_set [iff]: "finite (set xs)" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma set_append [simp]: "set (xs @ ys) = (set xs \ set ys)" nipkow@13145: by (induct xs) auto wenzelm@13114: oheimb@14099: lemma hd_in_set: "l = x#xs \ x\set l" paulson@14208: by (case_tac l, auto) oheimb@14099: wenzelm@13142: lemma set_subset_Cons: "set xs \ set (x # xs)" nipkow@13145: by auto wenzelm@13114: oheimb@14099: lemma set_ConsD: "y \ set (x # xs) \ y=x \ y \ set xs" oheimb@14099: by auto oheimb@14099: wenzelm@13142: lemma set_empty [iff]: "(set xs = {}) = (xs = [])" nipkow@13145: by (induct xs) auto wenzelm@13114: nipkow@15245: lemma set_empty2[iff]: "({} = set xs) = (xs = [])" nipkow@15245: by(induct xs) auto nipkow@15245: wenzelm@13142: lemma set_rev [simp]: "set (rev xs) = set xs" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma set_map [simp]: "set (map f xs) = f`(set xs)" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \ P x}" nipkow@13145: by (induct xs) auto wenzelm@13114: nipkow@15425: lemma set_upt [simp]: "set[i.. k \ k < j}" paulson@14208: apply (induct j, simp_all) paulson@14208: apply (erule ssubst, auto) nipkow@13145: done wenzelm@13114: wenzelm@13142: lemma in_set_conv_decomp: "(x : set xs) = (\ys zs. xs = ys @ x # zs)" paulson@15113: proof (induct xs) paulson@15113: case Nil show ?case by simp paulson@15113: case (Cons a xs) paulson@15113: show ?case paulson@15113: proof paulson@15113: assume "x \ set (a # xs)" paulson@15113: with prems show "\ys zs. a # xs = ys @ x # zs" paulson@15113: by (simp, blast intro: Cons_eq_appendI) paulson@15113: next paulson@15113: assume "\ys zs. a # xs = ys @ x # zs" paulson@15113: then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast paulson@15113: show "x \ set (a # xs)" paulson@15113: by (cases ys, auto simp add: eq) paulson@15113: qed paulson@15113: qed wenzelm@13142: paulson@13508: lemma finite_list: "finite A ==> EX l. set l = A" paulson@13508: apply (erule finite_induct, auto) paulson@13508: apply (rule_tac x="x#l" in exI, auto) paulson@13508: done paulson@13508: kleing@14388: lemma card_length: "card (set xs) \ length xs" kleing@14388: by (induct xs) (auto simp add: card_insert_if) wenzelm@13114: paulson@15168: nipkow@15439: subsubsection {* @{text mem}, @{text list_all} and @{text list_ex} *} wenzelm@13114: nipkow@15302: text{* Only use @{text mem} for generating executable code. Otherwise nipkow@15439: use @{prop"x : set xs"} instead --- it is much easier to reason about. nipkow@15439: The same is true for @{text list_all} and @{text list_ex}: write nipkow@15439: @{text"\x\set xs"} and @{text"\x\set xs"} instead because the HOL nipkow@15439: quantifiers are aleady known to the automatic provers. For the purpose nipkow@15439: of generating executable code use the theorems @{text set_mem_eq}, nipkow@15439: @{text list_all_conv} and @{text list_ex_iff} to get rid off or nipkow@15439: introduce the combinators. *} nipkow@15302: wenzelm@13114: lemma set_mem_eq: "(x mem xs) = (x : set xs)" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma list_all_conv: "list_all P xs = (\x \ set xs. P x)" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma list_all_append [simp]: nipkow@13145: "list_all P (xs @ ys) = (list_all P xs \ list_all P ys)" nipkow@13145: by (induct xs) auto wenzelm@13114: kleing@15426: lemma list_all_rev [simp]: "list_all P (rev xs) = list_all P xs" kleing@15426: by (simp add: list_all_conv) kleing@15426: nipkow@15439: lemma list_ex_iff: "list_ex P xs = (\x \ set xs. P x)" nipkow@15439: by (induct xs) simp_all kleing@15426: wenzelm@13114: nipkow@15392: subsubsection {* @{text filter} *} wenzelm@13114: wenzelm@13142: lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys" nipkow@13145: by (induct xs) auto wenzelm@13114: nipkow@15305: lemma rev_filter: "rev (filter P xs) = filter P (rev xs)" nipkow@15305: by (induct xs) simp_all nipkow@15305: wenzelm@13142: lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\x. Q x \ P x) xs" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma filter_True [simp]: "\x \ set xs. P x ==> filter P xs = xs" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma filter_False [simp]: "\x \ set xs. \ P x ==> filter P xs = []" nipkow@13145: by (induct xs) auto wenzelm@13114: nipkow@15246: lemma length_filter_le [simp]: "length (filter P xs) \ length xs" nipkow@13145: by (induct xs) (auto simp add: le_SucI) wenzelm@13114: wenzelm@13142: lemma filter_is_subset [simp]: "set (filter P xs) \ set xs" nipkow@13145: by auto wenzelm@13114: nipkow@15246: lemma length_filter_less: nipkow@15246: "\ x : set xs; ~ P x \ \ length(filter P xs) < length xs" nipkow@15246: proof (induct xs) nipkow@15246: case Nil thus ?case by simp nipkow@15246: next nipkow@15246: case (Cons x xs) thus ?case nipkow@15246: apply (auto split:split_if_asm) nipkow@15246: using length_filter_le[of P xs] apply arith nipkow@15246: done nipkow@15246: qed wenzelm@13114: nipkow@15281: lemma length_filter_conv_card: nipkow@15281: "length(filter p xs) = card{i. i < length xs & p(xs!i)}" nipkow@15281: proof (induct xs) nipkow@15281: case Nil thus ?case by simp nipkow@15281: next nipkow@15281: case (Cons x xs) nipkow@15281: let ?S = "{i. i < length xs & p(xs!i)}" nipkow@15281: have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite) nipkow@15281: show ?case (is "?l = card ?S'") nipkow@15281: proof (cases) nipkow@15281: assume "p x" nipkow@15281: hence eq: "?S' = insert 0 (Suc ` ?S)" nipkow@15281: by(auto simp add: nth_Cons image_def split:nat.split elim:lessE) nipkow@15281: have "length (filter p (x # xs)) = Suc(card ?S)" nipkow@15281: using Cons by simp nipkow@15281: also have "\ = Suc(card(Suc ` ?S))" using fin nipkow@15281: by (simp add: card_image inj_Suc) nipkow@15281: also have "\ = card ?S'" using eq fin nipkow@15281: by (simp add:card_insert_if) (simp add:image_def) nipkow@15281: finally show ?thesis . nipkow@15281: next nipkow@15281: assume "\ p x" nipkow@15281: hence eq: "?S' = Suc ` ?S" nipkow@15281: by(auto simp add: nth_Cons image_def split:nat.split elim:lessE) nipkow@15281: have "length (filter p (x # xs)) = card ?S" nipkow@15281: using Cons by simp nipkow@15281: also have "\ = card(Suc ` ?S)" using fin nipkow@15281: by (simp add: card_image inj_Suc) nipkow@15281: also have "\ = card ?S'" using eq fin nipkow@15281: by (simp add:card_insert_if) nipkow@15281: finally show ?thesis . nipkow@15281: qed nipkow@15281: qed nipkow@15281: nipkow@15281: nipkow@15392: subsubsection {* @{text concat} *} wenzelm@13114: wenzelm@13142: lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma concat_eq_Nil_conv [iff]: "(concat xss = []) = (\xs \ set xss. xs = [])" nipkow@13145: by (induct xss) auto wenzelm@13114: wenzelm@13142: lemma Nil_eq_concat_conv [iff]: "([] = concat xss) = (\xs \ set xss. xs = [])" nipkow@13145: by (induct xss) auto wenzelm@13114: wenzelm@13142: lemma set_concat [simp]: "set (concat xs) = \(set ` set xs)" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma map_concat: "map f (concat xs) = concat (map (map f) xs)" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13114: nipkow@15392: subsubsection {* @{text nth} *} wenzelm@13114: wenzelm@13142: lemma nth_Cons_0 [simp]: "(x # xs)!0 = x" nipkow@13145: by auto wenzelm@13114: wenzelm@13142: lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n" nipkow@13145: by auto wenzelm@13114: wenzelm@13142: declare nth.simps [simp del] wenzelm@13114: wenzelm@13114: lemma nth_append: nipkow@13145: "!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))" paulson@14208: apply (induct "xs", simp) paulson@14208: apply (case_tac n, auto) nipkow@13145: done wenzelm@13114: nipkow@14402: lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x" nipkow@14402: by (induct "xs") auto nipkow@14402: nipkow@14402: lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n" nipkow@14402: by (induct "xs") auto nipkow@14402: wenzelm@13142: lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)" paulson@14208: apply (induct xs, simp) paulson@14208: apply (case_tac n, auto) nipkow@13145: done wenzelm@13114: wenzelm@13142: lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}" paulson@15251: apply (induct xs, simp, simp) nipkow@13145: apply safe paulson@14208: apply (rule_tac x = 0 in exI, simp) paulson@14208: apply (rule_tac x = "Suc i" in exI, simp) paulson@14208: apply (case_tac i, simp) nipkow@13145: apply (rename_tac j) paulson@14208: apply (rule_tac x = j in exI, simp) nipkow@13145: done wenzelm@13114: nipkow@13145: lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)" nipkow@13145: by (auto simp add: set_conv_nth) wenzelm@13114: wenzelm@13142: lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs" nipkow@13145: by (auto simp add: set_conv_nth) wenzelm@13114: wenzelm@13114: lemma all_nth_imp_all_set: nipkow@13145: "[| !i < length xs. P(xs!i); x : set xs|] ==> P x" nipkow@13145: by (auto simp add: set_conv_nth) wenzelm@13114: wenzelm@13114: lemma all_set_conv_all_nth: nipkow@13145: "(\x \ set xs. P x) = (\i. i < length xs --> P (xs ! i))" nipkow@13145: by (auto simp add: set_conv_nth) wenzelm@13114: wenzelm@13114: nipkow@15392: subsubsection {* @{text list_update} *} wenzelm@13114: wenzelm@13142: lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs" nipkow@13145: by (induct xs) (auto split: nat.split) wenzelm@13114: wenzelm@13114: lemma nth_list_update: nipkow@13145: "!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)" nipkow@13145: by (induct xs) (auto simp add: nth_Cons split: nat.split) wenzelm@13114: wenzelm@13142: lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x" nipkow@13145: by (simp add: nth_list_update) wenzelm@13114: wenzelm@13142: lemma nth_list_update_neq [simp]: "!!i j. i \ j ==> xs[i:=x]!j = xs!j" nipkow@13145: by (induct xs) (auto simp add: nth_Cons split: nat.split) wenzelm@13114: wenzelm@13142: lemma list_update_overwrite [simp]: nipkow@13145: "!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]" nipkow@13145: by (induct xs) (auto split: nat.split) wenzelm@13114: nipkow@14402: lemma list_update_id[simp]: "!!i. i < length xs ==> xs[i := xs!i] = xs" paulson@14208: apply (induct xs, simp) nipkow@14187: apply(simp split:nat.splits) nipkow@14187: done nipkow@14187: wenzelm@13114: lemma list_update_same_conv: nipkow@13145: "!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)" nipkow@13145: by (induct xs) (auto split: nat.split) wenzelm@13114: nipkow@14187: lemma list_update_append1: nipkow@14187: "!!i. i < size xs \ (xs @ ys)[i:=x] = xs[i:=x] @ ys" paulson@14208: apply (induct xs, simp) nipkow@14187: apply(simp split:nat.split) nipkow@14187: done nipkow@14187: nipkow@14402: lemma list_update_length [simp]: nipkow@14402: "(xs @ x # ys)[length xs := y] = (xs @ y # ys)" nipkow@14402: by (induct xs, auto) nipkow@14402: wenzelm@13114: lemma update_zip: nipkow@13145: "!!i xy xs. length xs = length ys ==> nipkow@13145: (zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])" nipkow@13145: by (induct ys) (auto, case_tac xs, auto split: nat.split) wenzelm@13114: wenzelm@13114: lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)" nipkow@13145: by (induct xs) (auto split: nat.split) wenzelm@13114: wenzelm@13114: lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A" nipkow@13145: by (blast dest!: set_update_subset_insert [THEN subsetD]) wenzelm@13114: wenzelm@13114: nipkow@15392: subsubsection {* @{text last} and @{text butlast} *} wenzelm@13114: wenzelm@13142: lemma last_snoc [simp]: "last (xs @ [x]) = x" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs" nipkow@13145: by (induct xs) auto wenzelm@13114: nipkow@14302: lemma last_ConsL: "xs = [] \ last(x#xs) = x" nipkow@14302: by(simp add:last.simps) nipkow@14302: nipkow@14302: lemma last_ConsR: "xs \ [] \ last(x#xs) = last xs" nipkow@14302: by(simp add:last.simps) nipkow@14302: nipkow@14302: lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)" nipkow@14302: by (induct xs) (auto) nipkow@14302: nipkow@14302: lemma last_appendL[simp]: "ys = [] \ last(xs @ ys) = last xs" nipkow@14302: by(simp add:last_append) nipkow@14302: nipkow@14302: lemma last_appendR[simp]: "ys \ [] \ last(xs @ ys) = last ys" nipkow@14302: by(simp add:last_append) nipkow@14302: nipkow@14302: wenzelm@13142: lemma length_butlast [simp]: "length (butlast xs) = length xs - 1" nipkow@13145: by (induct xs rule: rev_induct) auto wenzelm@13114: wenzelm@13114: lemma butlast_append: nipkow@13145: "!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma append_butlast_last_id [simp]: nipkow@13145: "xs \ [] ==> butlast xs @ [last xs] = xs" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs" nipkow@13145: by (induct xs) (auto split: split_if_asm) wenzelm@13114: wenzelm@13114: lemma in_set_butlast_appendI: nipkow@13145: "x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))" nipkow@13145: by (auto dest: in_set_butlastD simp add: butlast_append) wenzelm@13114: wenzelm@13142: nipkow@15392: subsubsection {* @{text take} and @{text drop} *} wenzelm@13114: wenzelm@13142: lemma take_0 [simp]: "take 0 xs = []" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma drop_0 [simp]: "drop 0 xs = xs" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs" nipkow@13145: by simp wenzelm@13114: wenzelm@13142: lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs" nipkow@13145: by simp wenzelm@13114: wenzelm@13142: declare take_Cons [simp del] and drop_Cons [simp del] wenzelm@13114: nipkow@15110: lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)" nipkow@15110: by(clarsimp simp add:neq_Nil_conv) nipkow@15110: nipkow@14187: lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)" nipkow@14187: by(cases xs, simp_all) nipkow@14187: nipkow@14187: lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)" nipkow@14187: by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split) nipkow@14187: nipkow@14187: lemma nth_via_drop: "!!n. drop n xs = y#ys \ xs!n = y" paulson@14208: apply (induct xs, simp) nipkow@14187: apply(simp add:drop_Cons nth_Cons split:nat.splits) nipkow@14187: done nipkow@14187: nipkow@13913: lemma take_Suc_conv_app_nth: nipkow@13913: "!!i. i < length xs \ take (Suc i) xs = take i xs @ [xs!i]" paulson@14208: apply (induct xs, simp) paulson@14208: apply (case_tac i, auto) nipkow@13913: done nipkow@13913: mehta@14591: lemma drop_Suc_conv_tl: mehta@14591: "!!i. i < length xs \ (xs!i) # (drop (Suc i) xs) = drop i xs" mehta@14591: apply (induct xs, simp) mehta@14591: apply (case_tac i, auto) mehta@14591: done mehta@14591: wenzelm@13142: lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n" nipkow@13145: by (induct n) (auto, case_tac xs, auto) wenzelm@13114: wenzelm@13142: lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)" nipkow@13145: by (induct n) (auto, case_tac xs, auto) wenzelm@13114: wenzelm@13142: lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs" nipkow@13145: by (induct n) (auto, case_tac xs, auto) wenzelm@13114: wenzelm@13142: lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []" nipkow@13145: by (induct n) (auto, case_tac xs, auto) wenzelm@13114: wenzelm@13142: lemma take_append [simp]: nipkow@13145: "!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)" nipkow@13145: by (induct n) (auto, case_tac xs, auto) wenzelm@13114: wenzelm@13142: lemma drop_append [simp]: nipkow@13145: "!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys" nipkow@13145: by (induct n) (auto, case_tac xs, auto) wenzelm@13114: wenzelm@13142: lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs" paulson@14208: apply (induct m, auto) paulson@14208: apply (case_tac xs, auto) nipkow@15236: apply (case_tac n, auto) nipkow@13145: done wenzelm@13114: wenzelm@13142: lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs" paulson@14208: apply (induct m, auto) paulson@14208: apply (case_tac xs, auto) nipkow@13145: done wenzelm@13114: wenzelm@13114: lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)" paulson@14208: apply (induct m, auto) paulson@14208: apply (case_tac xs, auto) nipkow@13145: done wenzelm@13114: nipkow@14802: lemma drop_take: "!!m n. drop n (take m xs) = take (m-n) (drop n xs)" nipkow@14802: apply(induct xs) nipkow@14802: apply simp nipkow@14802: apply(simp add: take_Cons drop_Cons split:nat.split) nipkow@14802: done nipkow@14802: wenzelm@13142: lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs" paulson@14208: apply (induct n, auto) paulson@14208: apply (case_tac xs, auto) nipkow@13145: done wenzelm@13114: nipkow@15110: lemma take_eq_Nil[simp]: "!!n. (take n xs = []) = (n = 0 \ xs = [])" nipkow@15110: apply(induct xs) nipkow@15110: apply simp nipkow@15110: apply(simp add:take_Cons split:nat.split) nipkow@15110: done nipkow@15110: nipkow@15110: lemma drop_eq_Nil[simp]: "!!n. (drop n xs = []) = (length xs <= n)" nipkow@15110: apply(induct xs) nipkow@15110: apply simp nipkow@15110: apply(simp add:drop_Cons split:nat.split) nipkow@15110: done nipkow@15110: wenzelm@13114: lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)" paulson@14208: apply (induct n, auto) paulson@14208: apply (case_tac xs, auto) nipkow@13145: done wenzelm@13114: wenzelm@13142: lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)" paulson@14208: apply (induct n, auto) paulson@14208: apply (case_tac xs, auto) nipkow@13145: done wenzelm@13114: wenzelm@13114: lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)" paulson@14208: apply (induct xs, auto) paulson@14208: apply (case_tac i, auto) nipkow@13145: done wenzelm@13114: wenzelm@13114: lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)" paulson@14208: apply (induct xs, auto) paulson@14208: apply (case_tac i, auto) nipkow@13145: done wenzelm@13114: wenzelm@13142: lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i" paulson@14208: apply (induct xs, auto) paulson@14208: apply (case_tac n, blast) paulson@14208: apply (case_tac i, auto) nipkow@13145: done wenzelm@13114: wenzelm@13142: lemma nth_drop [simp]: nipkow@13145: "!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)" paulson@14208: apply (induct n, auto) paulson@14208: apply (case_tac xs, auto) nipkow@13145: done nipkow@3507: nipkow@14025: lemma set_take_subset: "\n. set(take n xs) \ set xs" nipkow@14025: by(induct xs)(auto simp:take_Cons split:nat.split) nipkow@14025: nipkow@14025: lemma set_drop_subset: "\n. set(drop n xs) \ set xs" nipkow@14025: by(induct xs)(auto simp:drop_Cons split:nat.split) nipkow@14025: nipkow@14187: lemma in_set_takeD: "x : set(take n xs) \ x : set xs" nipkow@14187: using set_take_subset by fast nipkow@14187: nipkow@14187: lemma in_set_dropD: "x : set(drop n xs) \ x : set xs" nipkow@14187: using set_drop_subset by fast nipkow@14187: wenzelm@13114: lemma append_eq_conv_conj: nipkow@13145: "!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \ ys = drop (length xs) zs)" paulson@14208: apply (induct xs, simp, clarsimp) paulson@14208: apply (case_tac zs, auto) nipkow@13145: done wenzelm@13142: paulson@14050: lemma take_add [rule_format]: paulson@14050: "\i. i+j \ length(xs) --> take (i+j) xs = take i xs @ take j (drop i xs)" paulson@14050: apply (induct xs, auto) paulson@14050: apply (case_tac i, simp_all) paulson@14050: done paulson@14050: nipkow@14300: lemma append_eq_append_conv_if: nipkow@14300: "!! ys\<^isub>1. (xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) = nipkow@14300: (if size xs\<^isub>1 \ size ys\<^isub>1 nipkow@14300: then xs\<^isub>1 = take (size xs\<^isub>1) ys\<^isub>1 \ xs\<^isub>2 = drop (size xs\<^isub>1) ys\<^isub>1 @ ys\<^isub>2 nipkow@14300: else take (size ys\<^isub>1) xs\<^isub>1 = ys\<^isub>1 \ drop (size ys\<^isub>1) xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>2)" nipkow@14300: apply(induct xs\<^isub>1) nipkow@14300: apply simp nipkow@14300: apply(case_tac ys\<^isub>1) nipkow@14300: apply simp_all nipkow@14300: done nipkow@14300: nipkow@15110: lemma take_hd_drop: nipkow@15110: "!!n. n < length xs \ take n xs @ [hd (drop n xs)] = take (n+1) xs" nipkow@15110: apply(induct xs) nipkow@15110: apply simp nipkow@15110: apply(simp add:drop_Cons split:nat.split) nipkow@15110: done nipkow@15110: wenzelm@13114: nipkow@15392: subsubsection {* @{text takeWhile} and @{text dropWhile} *} wenzelm@13114: wenzelm@13142: lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma takeWhile_append1 [simp]: nipkow@13145: "[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma takeWhile_append2 [simp]: nipkow@13145: "(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma takeWhile_tail: "\ P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma dropWhile_append1 [simp]: nipkow@13145: "[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma dropWhile_append2 [simp]: nipkow@13145: "(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys" nipkow@13145: by (induct xs) auto wenzelm@13114: wenzelm@13142: lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \ P x" nipkow@13145: by (induct xs) (auto split: split_if_asm) wenzelm@13114: nipkow@13913: lemma takeWhile_eq_all_conv[simp]: nipkow@13913: "(takeWhile P xs = xs) = (\x \ set xs. P x)" nipkow@13913: by(induct xs, auto) nipkow@13913: nipkow@13913: lemma dropWhile_eq_Nil_conv[simp]: nipkow@13913: "(dropWhile P xs = []) = (\x \ set xs. P x)" nipkow@13913: by(induct xs, auto) nipkow@13913: nipkow@13913: lemma dropWhile_eq_Cons_conv: nipkow@13913: "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \ P y)" nipkow@13913: by(induct xs, auto) nipkow@13913: wenzelm@13114: nipkow@15392: subsubsection {* @{text zip} *} wenzelm@13114: wenzelm@13142: lemma zip_Nil [simp]: "zip [] ys = []" nipkow@13145: by (induct ys) auto wenzelm@13114: wenzelm@13142: lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys" nipkow@13145: by simp wenzelm@13114: wenzelm@13142: declare zip_Cons [simp del] wenzelm@13114: nipkow@15281: lemma zip_Cons1: nipkow@15281: "zip (x#xs) ys = (case ys of [] \ [] | y#ys \ (x,y)#zip xs ys)" nipkow@15281: by(auto split:list.split) nipkow@15281: wenzelm@13142: lemma length_zip [simp]: nipkow@13145: "!!xs. length (zip xs ys) = min (length xs) (length ys)" paulson@14208: apply (induct ys, simp) paulson@14208: apply (case_tac xs, auto) nipkow@13145: done wenzelm@13114: wenzelm@13114: lemma zip_append1: nipkow@13145: "!!xs. zip (xs @ ys) zs = nipkow@13145: zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)" paulson@14208: apply (induct zs, simp) paulson@14208: apply (case_tac xs, simp_all) nipkow@13145: done wenzelm@13114: wenzelm@13114: lemma zip_append2: nipkow@13145: "!!ys. zip xs (ys @ zs) = nipkow@13145: zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs" paulson@14208: apply (induct xs, simp) paulson@14208: apply (case_tac ys, simp_all) nipkow@13145: done wenzelm@13114: wenzelm@13142: lemma zip_append [simp]: wenzelm@13142: "[| length xs = length us; length ys = length vs |] ==> nipkow@13145: zip (xs@ys) (us@vs) = zip xs us @ zip ys vs" nipkow@13145: by (simp add: zip_append1) wenzelm@13114: wenzelm@13114: lemma zip_rev: nipkow@14247: "length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)" nipkow@14247: by (induct rule:list_induct2, simp_all) wenzelm@13114: wenzelm@13142: lemma nth_zip [simp]: nipkow@13145: "!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)" paulson@14208: apply (induct ys, simp) nipkow@13145: apply (case_tac xs) nipkow@13145: apply (simp_all add: nth.simps split: nat.split) nipkow@13145: done wenzelm@13114: wenzelm@13114: lemma set_zip: nipkow@13145: "set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}" nipkow@13145: by (simp add: set_conv_nth cong: rev_conj_cong) wenzelm@13114: wenzelm@13114: lemma zip_update: nipkow@13145: "length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]" nipkow@13145: by (rule sym, simp add: update_zip) wenzelm@13114: wenzelm@13142: lemma zip_replicate [simp]: nipkow@13145: "!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)" paulson@14208: apply (induct i, auto) paulson@14208: apply (case_tac j, auto) nipkow@13145: done wenzelm@13114: wenzelm@13142: nipkow@15392: subsubsection {* @{text list_all2} *} wenzelm@13114: kleing@14316: lemma list_all2_lengthD [intro?]: kleing@14316: "list_all2 P xs ys ==> length xs = length ys" nipkow@13145: by (simp add: list_all2_def) wenzelm@13114: wenzelm@13142: lemma list_all2_Nil [iff]: "list_all2 P [] ys = (ys = [])" nipkow@13145: by (simp add: list_all2_def) wenzelm@13114: wenzelm@13142: lemma list_all2_Nil2[iff]: "list_all2 P xs [] = (xs = [])" nipkow@13145: by (simp add: list_all2_def) wenzelm@13114: wenzelm@13142: lemma list_all2_Cons [iff]: nipkow@13145: "list_all2 P (x # xs) (y # ys) = (P x y \ list_all2 P xs ys)" nipkow@13145: by (auto simp add: list_all2_def) wenzelm@13114: wenzelm@13114: lemma list_all2_Cons1: nipkow@13145: "list_all2 P (x # xs) ys = (\z zs. ys = z # zs \ P x z \ list_all2 P xs zs)" nipkow@13145: by (cases ys) auto wenzelm@13114: wenzelm@13114: lemma list_all2_Cons2: nipkow@13145: "list_all2 P xs (y # ys) = (\z zs. xs = z # zs \ P z y \ list_all2 P zs ys)" nipkow@13145: by (cases xs) auto wenzelm@13114: wenzelm@13142: lemma list_all2_rev [iff]: nipkow@13145: "list_all2 P (rev xs) (rev ys) = list_all2 P xs ys" nipkow@13145: by (simp add: list_all2_def zip_rev cong: conj_cong) wenzelm@13114: kleing@13863: lemma list_all2_rev1: kleing@13863: "list_all2 P (rev xs) ys = list_all2 P xs (rev ys)" kleing@13863: by (subst list_all2_rev [symmetric]) simp kleing@13863: wenzelm@13114: lemma list_all2_append1: nipkow@13145: "list_all2 P (xs @ ys) zs = nipkow@13145: (EX us vs. zs = us @ vs \ length us = length xs \ length vs = length ys \ nipkow@13145: list_all2 P xs us \ list_all2 P ys vs)" nipkow@13145: apply (simp add: list_all2_def zip_append1) nipkow@13145: apply (rule iffI) nipkow@13145: apply (rule_tac x = "take (length xs) zs" in exI) nipkow@13145: apply (rule_tac x = "drop (length xs) zs" in exI) paulson@14208: apply (force split: nat_diff_split simp add: min_def, clarify) nipkow@13145: apply (simp add: ball_Un) nipkow@13145: done wenzelm@13114: wenzelm@13114: lemma list_all2_append2: nipkow@13145: "list_all2 P xs (ys @ zs) = nipkow@13145: (EX us vs. xs = us @ vs \ length us = length ys \ length vs = length zs \ nipkow@13145: list_all2 P us ys \ list_all2 P vs zs)" nipkow@13145: apply (simp add: list_all2_def zip_append2) nipkow@13145: apply (rule iffI) nipkow@13145: apply (rule_tac x = "take (length ys) xs" in exI) nipkow@13145: apply (rule_tac x = "drop (length ys) xs" in exI) paulson@14208: apply (force split: nat_diff_split simp add: min_def, clarify) nipkow@13145: apply (simp add: ball_Un) nipkow@13145: done wenzelm@13114: kleing@13863: lemma list_all2_append: nipkow@14247: "length xs = length ys \ nipkow@14247: list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \ list_all2 P us vs)" nipkow@14247: by (induct rule:list_induct2, simp_all) kleing@13863: kleing@13863: lemma list_all2_appendI [intro?, trans]: kleing@13863: "\ list_all2 P a b; list_all2 P c d \ \ list_all2 P (a@c) (b@d)" kleing@13863: by (simp add: list_all2_append list_all2_lengthD) kleing@13863: wenzelm@13114: lemma list_all2_conv_all_nth: nipkow@13145: "list_all2 P xs ys = nipkow@13145: (length xs = length ys \ (\i < length xs. P (xs!i) (ys!i)))" nipkow@13145: by (force simp add: list_all2_def set_zip) wenzelm@13114: berghofe@13883: lemma list_all2_trans: berghofe@13883: assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c" berghofe@13883: shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs" berghofe@13883: (is "!!bs cs. PROP ?Q as bs cs") berghofe@13883: proof (induct as) berghofe@13883: fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs" berghofe@13883: show "!!cs. PROP ?Q (x # xs) bs cs" berghofe@13883: proof (induct bs) berghofe@13883: fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs" berghofe@13883: show "PROP ?Q (x # xs) (y # ys) cs" berghofe@13883: by (induct cs) (auto intro: tr I1 I2) berghofe@13883: qed simp berghofe@13883: qed simp berghofe@13883: kleing@13863: lemma list_all2_all_nthI [intro?]: kleing@13863: "length a = length b \ (\n. n < length a \ P (a!n) (b!n)) \ list_all2 P a b" kleing@13863: by (simp add: list_all2_conv_all_nth) kleing@13863: paulson@14395: lemma list_all2I: paulson@14395: "\x \ set (zip a b). split P x \ length a = length b \ list_all2 P a b" paulson@14395: by (simp add: list_all2_def) paulson@14395: kleing@14328: lemma list_all2_nthD: kleing@13863: "\ list_all2 P xs ys; p < size xs \ \ P (xs!p) (ys!p)" kleing@13863: by (simp add: list_all2_conv_all_nth) kleing@13863: nipkow@14302: lemma list_all2_nthD2: nipkow@14302: "\list_all2 P xs ys; p < size ys\ \ P (xs!p) (ys!p)" nipkow@14302: by (frule list_all2_lengthD) (auto intro: list_all2_nthD) nipkow@14302: kleing@13863: lemma list_all2_map1: kleing@13863: "list_all2 P (map f as) bs = list_all2 (\x y. P (f x) y) as bs" kleing@13863: by (simp add: list_all2_conv_all_nth) kleing@13863: kleing@13863: lemma list_all2_map2: kleing@13863: "list_all2 P as (map f bs) = list_all2 (\x y. P x (f y)) as bs" kleing@13863: by (auto simp add: list_all2_conv_all_nth) kleing@13863: kleing@14316: lemma list_all2_refl [intro?]: kleing@13863: "(\x. P x x) \ list_all2 P xs xs" kleing@13863: by (simp add: list_all2_conv_all_nth) kleing@13863: kleing@13863: lemma list_all2_update_cong: kleing@13863: "\ i \ list_all2 P (xs[i:=x]) (ys[i:=y])" kleing@13863: by (simp add: list_all2_conv_all_nth nth_list_update) kleing@13863: kleing@13863: lemma list_all2_update_cong2: kleing@13863: "\list_all2 P xs ys; P x y; i < length ys\ \ list_all2 P (xs[i:=x]) (ys[i:=y])" kleing@13863: by (simp add: list_all2_lengthD list_all2_update_cong) kleing@13863: nipkow@14302: lemma list_all2_takeI [simp,intro?]: nipkow@14302: "\n ys. list_all2 P xs ys \ list_all2 P (take n xs) (take n ys)" nipkow@14302: apply (induct xs) nipkow@14302: apply simp nipkow@14302: apply (clarsimp simp add: list_all2_Cons1) nipkow@14302: apply (case_tac n) nipkow@14302: apply auto nipkow@14302: done nipkow@14302: nipkow@14302: lemma list_all2_dropI [simp,intro?]: kleing@13863: "\n bs. list_all2 P as bs \ list_all2 P (drop n as) (drop n bs)" paulson@14208: apply (induct as, simp) kleing@13863: apply (clarsimp simp add: list_all2_Cons1) paulson@14208: apply (case_tac n, simp, simp) kleing@13863: done kleing@13863: kleing@14327: lemma list_all2_mono [intro?]: kleing@13863: "\y. list_all2 P x y \ (\x y. P x y \ Q x y) \ list_all2 Q x y" paulson@14208: apply (induct x, simp) paulson@14208: apply (case_tac y, auto) kleing@13863: done kleing@13863: wenzelm@13142: nipkow@15392: subsubsection {* @{text foldl} and @{text foldr} *} wenzelm@13142: wenzelm@13142: lemma foldl_append [simp]: nipkow@13145: "!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys" nipkow@13145: by (induct xs) auto wenzelm@13142: nipkow@14402: lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)" nipkow@14402: by (induct xs) auto nipkow@14402: nipkow@14402: lemma foldr_foldl: "foldr f xs a = foldl (%x y. f y x) a (rev xs)" nipkow@14402: by (induct xs) auto nipkow@14402: nipkow@14402: lemma foldl_foldr: "foldl f a xs = foldr (%x y. f y x) (rev xs) a" nipkow@14402: by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"]) nipkow@14402: wenzelm@13142: text {* nipkow@13145: Note: @{text "n \ foldl (op +) n ns"} looks simpler, but is more nipkow@13145: difficult to use because it requires an additional transitivity step. wenzelm@13142: *} wenzelm@13142: wenzelm@13142: lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns" nipkow@13145: by (induct ns) auto wenzelm@13142: wenzelm@13142: lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns" nipkow@13145: by (force intro: start_le_sum simp add: in_set_conv_decomp) wenzelm@13142: wenzelm@13142: lemma sum_eq_0_conv [iff]: nipkow@13145: "!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \ (\n \ set ns. n = 0))" nipkow@13145: by (induct ns) auto wenzelm@13114: wenzelm@13114: nipkow@15392: subsubsection {* @{text upto} *} wenzelm@13114: nipkow@15425: lemma upt_rec: "[i.. [i.. j <= i)" nipkow@15281: by(induct j)simp_all nipkow@15281: nipkow@15281: lemma upt_eq_Cons_conv: nipkow@15425: "!!x xs. ([i.. [i..<(Suc j)] = [i.. [i.. [i.. [i.. take m [i.. (map f [m.. k <= length ys ==> berghofe@13883: (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys" berghofe@13883: apply (atomize, induct k) paulson@14208: apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify) nipkow@13145: txt {* Both lists must be non-empty *} paulson@14208: apply (case_tac xs, simp) paulson@14208: apply (case_tac ys, clarify) nipkow@13145: apply (simp (no_asm_use)) nipkow@13145: apply clarify nipkow@13145: txt {* prenexing's needed, not miniscoping *} nipkow@13145: apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps) nipkow@13145: apply blast nipkow@13145: done wenzelm@13114: wenzelm@13114: lemma nth_equalityI: wenzelm@13114: "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys" nipkow@13145: apply (frule nth_take_lemma [OF le_refl eq_imp_le]) nipkow@13145: apply (simp_all add: take_all) nipkow@13145: done wenzelm@13142: kleing@13863: (* needs nth_equalityI *) kleing@13863: lemma list_all2_antisym: kleing@13863: "\ (\x y. \P x y; Q y x\ \ x = y); list_all2 P xs ys; list_all2 Q ys xs \ kleing@13863: \ xs = ys" kleing@13863: apply (simp add: list_all2_conv_all_nth) paulson@14208: apply (rule nth_equalityI, blast, simp) kleing@13863: done kleing@13863: wenzelm@13142: lemma take_equalityI: "(\i. take i xs = take i ys) ==> xs = ys" nipkow@13145: -- {* The famous take-lemma. *} nipkow@13145: apply (drule_tac x = "max (length xs) (length ys)" in spec) nipkow@13145: apply (simp add: le_max_iff_disj take_all) nipkow@13145: done wenzelm@13142: wenzelm@13142: nipkow@15302: lemma take_Cons': nipkow@15302: "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)" nipkow@15302: by (cases n) simp_all nipkow@15302: nipkow@15302: lemma drop_Cons': nipkow@15302: "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)" nipkow@15302: by (cases n) simp_all nipkow@15302: nipkow@15302: lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))" nipkow@15302: by (cases n) simp_all nipkow@15302: nipkow@15302: lemmas [simp] = take_Cons'[of "number_of v",standard] nipkow@15302: drop_Cons'[of "number_of v",standard] nipkow@15302: nth_Cons'[of _ _ "number_of v",standard] nipkow@15302: nipkow@15302: nipkow@15392: subsubsection {* @{text "distinct"} and @{text remdups} *} wenzelm@13142: wenzelm@13142: lemma distinct_append [simp]: nipkow@13145: "distinct (xs @ ys) = (distinct xs \ distinct ys \ set xs \ set ys = {})" nipkow@13145: by (induct xs) auto wenzelm@13142: nipkow@15305: lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs" nipkow@15305: by(induct xs) auto nipkow@15305: wenzelm@13142: lemma set_remdups [simp]: "set (remdups xs) = set xs" nipkow@13145: by (induct xs) (auto simp add: insert_absorb) wenzelm@13142: wenzelm@13142: lemma distinct_remdups [iff]: "distinct (remdups xs)" nipkow@13145: by (induct xs) auto wenzelm@13142: paulson@15072: lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])" paulson@15251: by (induct x, auto) paulson@15072: paulson@15072: lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])" paulson@15251: by (induct x, auto) paulson@15072: nipkow@15245: lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs" nipkow@15245: by (induct xs) auto nipkow@15245: nipkow@15245: lemma length_remdups_eq[iff]: nipkow@15245: "(length (remdups xs) = length xs) = (remdups xs = xs)" nipkow@15245: apply(induct xs) nipkow@15245: apply auto nipkow@15245: apply(subgoal_tac "length (remdups xs) <= length xs") nipkow@15245: apply arith nipkow@15245: apply(rule length_remdups_leq) nipkow@15245: done nipkow@15245: wenzelm@13142: lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)" nipkow@13145: by (induct xs) auto wenzelm@13114: nipkow@15304: lemma distinct_map_filterI: nipkow@15304: "distinct(map f xs) \ distinct(map f (filter P xs))" nipkow@15304: apply(induct xs) nipkow@15304: apply simp nipkow@15304: apply force nipkow@15304: done nipkow@15304: wenzelm@13142: text {* nipkow@13145: It is best to avoid this indexed version of distinct, but sometimes nipkow@13145: it is useful. *} wenzelm@13142: lemma distinct_conv_nth: nipkow@13145: "distinct xs = (\i j. i < size xs \ j < size xs \ i \ j --> xs!i \ xs!j)" paulson@15251: apply (induct xs, simp, simp) paulson@14208: apply (rule iffI, clarsimp) nipkow@13145: apply (case_tac i) paulson@14208: apply (case_tac j, simp) nipkow@13145: apply (simp add: set_conv_nth) nipkow@13145: apply (case_tac j) paulson@14208: apply (clarsimp simp add: set_conv_nth, simp) nipkow@13145: apply (rule conjI) nipkow@13145: apply (clarsimp simp add: set_conv_nth) nipkow@13145: apply (erule_tac x = 0 in allE) paulson@14208: apply (erule_tac x = "Suc i" in allE, simp, clarsimp) nipkow@13145: apply (erule_tac x = "Suc i" in allE) paulson@14208: apply (erule_tac x = "Suc j" in allE, simp) nipkow@13145: done wenzelm@13114: nipkow@15110: lemma distinct_card: "distinct xs ==> card (set xs) = size xs" kleing@14388: by (induct xs) auto kleing@14388: nipkow@15110: lemma card_distinct: "card (set xs) = size xs ==> distinct xs" kleing@14388: proof (induct xs) kleing@14388: case Nil thus ?case by simp kleing@14388: next kleing@14388: case (Cons x xs) kleing@14388: show ?case kleing@14388: proof (cases "x \ set xs") kleing@14388: case False with Cons show ?thesis by simp kleing@14388: next kleing@14388: case True with Cons.prems kleing@14388: have "card (set xs) = Suc (length xs)" kleing@14388: by (simp add: card_insert_if split: split_if_asm) kleing@14388: moreover have "card (set xs) \ length xs" by (rule card_length) kleing@14388: ultimately have False by simp kleing@14388: thus ?thesis .. kleing@14388: qed kleing@14388: qed kleing@14388: nipkow@15110: lemma inj_on_setI: "distinct(map f xs) ==> inj_on f (set xs)" nipkow@15110: apply(induct xs) nipkow@15110: apply simp nipkow@15110: apply fastsimp nipkow@15110: done nipkow@15110: nipkow@15110: lemma inj_on_set_conv: nipkow@15110: "distinct xs \ inj_on f (set xs) = distinct(map f xs)" nipkow@15110: apply(induct xs) nipkow@15110: apply simp nipkow@15110: apply fastsimp nipkow@15110: done nipkow@15110: nipkow@15110: nipkow@15392: subsubsection {* @{text remove1} *} nipkow@15110: nipkow@15110: lemma set_remove1_subset: "set(remove1 x xs) <= set xs" nipkow@15110: apply(induct xs) nipkow@15110: apply simp nipkow@15110: apply simp nipkow@15110: apply blast nipkow@15110: done nipkow@15110: nipkow@15110: lemma [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}" nipkow@15110: apply(induct xs) nipkow@15110: apply simp nipkow@15110: apply simp nipkow@15110: apply blast nipkow@15110: done nipkow@15110: nipkow@15110: lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)" nipkow@15110: apply(insert set_remove1_subset) nipkow@15110: apply fast nipkow@15110: done nipkow@15110: nipkow@15110: lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)" nipkow@15110: by (induct xs) simp_all nipkow@15110: wenzelm@13114: nipkow@15392: subsubsection {* @{text replicate} *} wenzelm@13114: wenzelm@13142: lemma length_replicate [simp]: "length (replicate n x) = n" nipkow@13145: by (induct n) auto nipkow@13124: wenzelm@13142: lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)" nipkow@13145: by (induct n) auto wenzelm@13114: wenzelm@13114: lemma replicate_app_Cons_same: nipkow@13145: "(replicate n x) @ (x # xs) = x # replicate n x @ xs" nipkow@13145: by (induct n) auto wenzelm@13114: wenzelm@13142: lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x" paulson@14208: apply (induct n, simp) nipkow@13145: apply (simp add: replicate_app_Cons_same) nipkow@13145: done wenzelm@13114: wenzelm@13142: lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x" nipkow@13145: by (induct n) auto wenzelm@13114: wenzelm@13142: lemma hd_replicate [simp]: "n \ 0 ==> hd (replicate n x) = x" nipkow@13145: by (induct n) auto wenzelm@13114: wenzelm@13142: lemma tl_replicate [simp]: "n \ 0 ==> tl (replicate n x) = replicate (n - 1) x" nipkow@13145: by (induct n) auto wenzelm@13114: wenzelm@13142: lemma last_replicate [simp]: "n \ 0 ==> last (replicate n x) = x" nipkow@13145: by (atomize (full), induct n) auto wenzelm@13114: wenzelm@13142: lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x" paulson@14208: apply (induct n, simp) nipkow@13145: apply (simp add: nth_Cons split: nat.split) nipkow@13145: done wenzelm@13114: wenzelm@13142: lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}" nipkow@13145: by (induct n) auto wenzelm@13114: wenzelm@13142: lemma set_replicate [simp]: "n \ 0 ==> set (replicate n x) = {x}" nipkow@13145: by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc) wenzelm@13114: wenzelm@13142: lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})" nipkow@13145: by auto wenzelm@13114: wenzelm@13142: lemma in_set_replicateD: "x : set (replicate n y) ==> x = y" nipkow@13145: by (simp add: set_replicate_conv_if split: split_if_asm) wenzelm@13114: wenzelm@13114: nipkow@15392: subsubsection{*@{text rotate1} and @{text rotate}*} nipkow@15302: nipkow@15302: lemma rotate_simps[simp]: "rotate1 [] = [] \ rotate1 (x#xs) = xs @ [x]" nipkow@15302: by(simp add:rotate1_def) nipkow@15302: nipkow@15302: lemma rotate0[simp]: "rotate 0 = id" nipkow@15302: by(simp add:rotate_def) nipkow@15302: nipkow@15302: lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)" nipkow@15302: by(simp add:rotate_def) nipkow@15302: nipkow@15302: lemma rotate_add: nipkow@15302: "rotate (m+n) = rotate m o rotate n" nipkow@15302: by(simp add:rotate_def funpow_add) nipkow@15302: nipkow@15302: lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs" nipkow@15302: by(simp add:rotate_add) nipkow@15302: nipkow@15302: lemma rotate1_length01[simp]: "length xs <= 1 \ rotate1 xs = xs" nipkow@15302: by(cases xs) simp_all nipkow@15302: nipkow@15302: lemma rotate_length01[simp]: "length xs <= 1 \ rotate n xs = xs" nipkow@15302: apply(induct n) nipkow@15302: apply simp nipkow@15302: apply (simp add:rotate_def) nipkow@13145: done wenzelm@13114: nipkow@15302: lemma rotate1_hd_tl: "xs \ [] \ rotate1 xs = tl xs @ [hd xs]" nipkow@15302: by(simp add:rotate1_def split:list.split) nipkow@15302: nipkow@15302: lemma rotate_drop_take: nipkow@15302: "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs" nipkow@15302: apply(induct n) nipkow@15302: apply simp nipkow@15302: apply(simp add:rotate_def) nipkow@15302: apply(cases "xs = []") nipkow@15302: apply (simp) nipkow@15302: apply(case_tac "n mod length xs = 0") nipkow@15302: apply(simp add:mod_Suc) nipkow@15302: apply(simp add: rotate1_hd_tl drop_Suc take_Suc) nipkow@15302: apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric] nipkow@15302: take_hd_drop linorder_not_le) nipkow@13145: done wenzelm@13114: nipkow@15302: lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs" nipkow@15302: by(simp add:rotate_drop_take) nipkow@15302: nipkow@15302: lemma rotate_id[simp]: "n mod length xs = 0 \ rotate n xs = xs" nipkow@15302: by(simp add:rotate_drop_take) nipkow@15302: nipkow@15302: lemma length_rotate1[simp]: "length(rotate1 xs) = length xs" nipkow@15302: by(simp add:rotate1_def split:list.split) nipkow@15302: nipkow@15302: lemma length_rotate[simp]: "!!xs. length(rotate n xs) = length xs" nipkow@15302: by (induct n) (simp_all add:rotate_def) nipkow@15302: nipkow@15302: lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs" nipkow@15302: by(simp add:rotate1_def split:list.split) blast nipkow@15302: nipkow@15302: lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs" nipkow@15302: by (induct n) (simp_all add:rotate_def) nipkow@15302: nipkow@15302: lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)" nipkow@15302: by(simp add:rotate_drop_take take_map drop_map) nipkow@15302: nipkow@15302: lemma set_rotate1[simp]: "set(rotate1 xs) = set xs" nipkow@15302: by(simp add:rotate1_def split:list.split) nipkow@15302: nipkow@15302: lemma set_rotate[simp]: "set(rotate n xs) = set xs" nipkow@15302: by (induct n) (simp_all add:rotate_def) nipkow@15302: nipkow@15302: lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])" nipkow@15302: by(simp add:rotate1_def split:list.split) nipkow@15302: nipkow@15302: lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])" nipkow@15302: by (induct n) (simp_all add:rotate_def) wenzelm@13114: nipkow@15439: lemma rotate_rev: nipkow@15439: "rotate n (rev xs) = rev(rotate (length xs - (n mod length xs)) xs)" nipkow@15439: apply(simp add:rotate_drop_take rev_drop rev_take) nipkow@15439: apply(cases "length xs = 0") nipkow@15439: apply simp nipkow@15439: apply(cases "n mod length xs = 0") nipkow@15439: apply simp nipkow@15439: apply(simp add:rotate_drop_take rev_drop rev_take) nipkow@15439: done nipkow@15439: wenzelm@13114: nipkow@15392: subsubsection {* @{text sublist} --- a generalization of @{text nth} to sets *} wenzelm@13114: wenzelm@13142: lemma sublist_empty [simp]: "sublist xs {} = []" nipkow@13145: by (auto simp add: sublist_def) wenzelm@13114: wenzelm@13142: lemma sublist_nil [simp]: "sublist [] A = []" nipkow@13145: by (auto simp add: sublist_def) wenzelm@13114: nipkow@15281: lemma length_sublist: nipkow@15281: "length(sublist xs I) = card{i. i < length xs \ i : I}" nipkow@15281: by(simp add: sublist_def length_filter_conv_card cong:conj_cong) nipkow@15281: nipkow@15281: lemma sublist_shift_lemma_Suc: nipkow@15281: "!!is. map fst (filter (%p. P(Suc(snd p))) (zip xs is)) = nipkow@15281: map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))" nipkow@15281: apply(induct xs) nipkow@15281: apply simp nipkow@15281: apply (case_tac "is") nipkow@15281: apply simp nipkow@15281: apply simp nipkow@15281: done nipkow@15281: wenzelm@13114: lemma sublist_shift_lemma: nipkow@15425: "map fst [p:zip xs [i.. i \ I}" nipkow@15281: apply(induct xs) nipkow@15281: apply simp nipkow@15281: apply(auto simp add:sublist_Cons nth_Cons split:nat.split elim: lessE) nipkow@15281: apply(erule lessE) nipkow@15281: apply auto nipkow@15281: apply(erule lessE) nipkow@15281: apply auto nipkow@15281: done nipkow@15281: nipkow@15281: lemma set_sublist_subset: "set(sublist xs I) \ set xs" nipkow@15281: by(auto simp add:set_sublist) nipkow@15281: nipkow@15281: lemma notin_set_sublistI[simp]: "x \ set xs \ x \ set(sublist xs I)" nipkow@15281: by(auto simp add:set_sublist) nipkow@15281: nipkow@15281: lemma in_set_sublistD: "x \ set(sublist xs I) \ x \ set xs" nipkow@15281: by(auto simp add:set_sublist) nipkow@15281: wenzelm@13142: lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])" nipkow@13145: by (simp add: sublist_Cons) wenzelm@13114: nipkow@15281: nipkow@15281: lemma distinct_sublistI[simp]: "!!I. distinct xs \ distinct(sublist xs I)" nipkow@15281: apply(induct xs) nipkow@15281: apply simp nipkow@15281: apply(auto simp add:sublist_Cons) nipkow@15281: done nipkow@15281: nipkow@15281: nipkow@15045: lemma sublist_upt_eq_take [simp]: "sublist l {.. 'a list set" nipkow@15302: inductive "lists A" nipkow@15302: intros nipkow@15302: Nil [intro!]: "[]: lists A" nipkow@15302: Cons [intro!]: "[| a: A;l: lists A|] ==> a#l : lists A" nipkow@15302: nipkow@15302: inductive_cases listsE [elim!]: "x#l : lists A" nipkow@15302: nipkow@15302: lemma lists_mono [mono]: "A \ B ==> lists A \ lists B" nipkow@15302: by (unfold lists.defs) (blast intro!: lfp_mono) nipkow@15302: nipkow@15302: lemma lists_IntI: nipkow@15302: assumes l: "l: lists A" shows "l: lists B ==> l: lists (A Int B)" using l nipkow@15302: by induct blast+ nipkow@15302: nipkow@15302: lemma lists_Int_eq [simp]: "lists (A \ B) = lists A \ lists B" nipkow@15302: proof (rule mono_Int [THEN equalityI]) nipkow@15302: show "mono lists" by (simp add: mono_def lists_mono) nipkow@15302: show "lists A \ lists B \ lists (A \ B)" by (blast intro: lists_IntI) kleing@14388: qed kleing@14388: nipkow@15302: lemma append_in_lists_conv [iff]: nipkow@15302: "(xs @ ys : lists A) = (xs : lists A \ ys : lists A)" nipkow@15302: by (induct xs) auto nipkow@15302: nipkow@15302: lemma in_lists_conv_set: "(xs : lists A) = (\x \ set xs. x : A)" nipkow@15302: -- {* eliminate @{text lists} in favour of @{text set} *} nipkow@15302: by (induct xs) auto nipkow@15302: nipkow@15302: lemma in_listsD [dest!]: "xs \ lists A ==> \x\set xs. x \ A" nipkow@15302: by (rule in_lists_conv_set [THEN iffD1]) nipkow@15302: nipkow@15302: lemma in_listsI [intro!]: "\x\set xs. x \ A ==> xs \ lists A" nipkow@15302: by (rule in_lists_conv_set [THEN iffD2]) nipkow@15302: nipkow@15302: lemma lists_UNIV [simp]: "lists UNIV = UNIV" nipkow@15302: by auto nipkow@15302: nipkow@15392: subsubsection{*Lists as Cartesian products*} nipkow@15302: nipkow@15302: text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from nipkow@15302: @{term A} and tail drawn from @{term Xs}.*} nipkow@15302: nipkow@15302: constdefs nipkow@15302: set_Cons :: "'a set \ 'a list set \ 'a list set" nipkow@15302: "set_Cons A XS == {z. \x xs. z = x#xs & x \ A & xs \ XS}" nipkow@15302: nipkow@15302: lemma [simp]: "set_Cons A {[]} = (%x. [x])`A" nipkow@15302: by (auto simp add: set_Cons_def) nipkow@15302: nipkow@15302: text{*Yields the set of lists, all of the same length as the argument and nipkow@15302: with elements drawn from the corresponding element of the argument.*} nipkow@15302: nipkow@15302: consts listset :: "'a set list \ 'a list set" nipkow@15302: primrec nipkow@15302: "listset [] = {[]}" nipkow@15302: "listset(A#As) = set_Cons A (listset As)" nipkow@15302: nipkow@15302: paulson@15656: subsection{*Relations on Lists*} paulson@15656: paulson@15656: subsubsection {* Length Lexicographic Ordering *} paulson@15656: paulson@15656: text{*These orderings preserve well-foundedness: shorter lists paulson@15656: precede longer lists. These ordering are not used in dictionaries.*} paulson@15656: paulson@15656: consts lexn :: "('a * 'a)set => nat => ('a list * 'a list)set" paulson@15656: --{*The lexicographic ordering for lists of the specified length*} nipkow@15302: primrec paulson@15656: "lexn r 0 = {}" paulson@15656: "lexn r (Suc n) = paulson@15656: (prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs) ` (r <*lex*> lexn r n)) Int paulson@15656: {(xs,ys). length xs = Suc n \ length ys = Suc n}" nipkow@15302: nipkow@15302: constdefs paulson@15656: lex :: "('a \ 'a) set => ('a list \ 'a list) set" paulson@15656: "lex r == \n. lexn r n" paulson@15656: --{*Holds only between lists of the same length*} paulson@15656: paulson@15656: lexico :: "('a \ 'a) set => ('a list \ 'a list) set" paulson@15656: "lexico r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))" paulson@15656: --{*Compares lists by their length and then lexicographically*} nipkow@15302: nipkow@15302: nipkow@15302: lemma wf_lexn: "wf r ==> wf (lexn r n)" nipkow@15302: apply (induct n, simp, simp) nipkow@15302: apply(rule wf_subset) nipkow@15302: prefer 2 apply (rule Int_lower1) nipkow@15302: apply(rule wf_prod_fun_image) nipkow@15302: prefer 2 apply (rule inj_onI, auto) nipkow@15302: done nipkow@15302: nipkow@15302: lemma lexn_length: nipkow@15302: "!!xs ys. (xs, ys) : lexn r n ==> length xs = n \ length ys = n" nipkow@15302: by (induct n) auto nipkow@15302: nipkow@15302: lemma wf_lex [intro!]: "wf r ==> wf (lex r)" nipkow@15302: apply (unfold lex_def) nipkow@15302: apply (rule wf_UN) nipkow@15302: apply (blast intro: wf_lexn, clarify) nipkow@15302: apply (rename_tac m n) nipkow@15302: apply (subgoal_tac "m \ n") nipkow@15302: prefer 2 apply blast nipkow@15302: apply (blast dest: lexn_length not_sym) nipkow@15302: done nipkow@15302: nipkow@15302: lemma lexn_conv: paulson@15656: "lexn r n = paulson@15656: {(xs,ys). length xs = n \ length ys = n \ paulson@15656: (\xys x y xs' ys'. xs= xys @ x#xs' \ ys= xys @ y # ys' \ (x, y):r)}" nipkow@15302: apply (induct n, simp, blast) nipkow@15302: apply (simp add: image_Collect lex_prod_def, safe, blast) nipkow@15302: apply (rule_tac x = "ab # xys" in exI, simp) nipkow@15302: apply (case_tac xys, simp_all, blast) nipkow@15302: done nipkow@15302: nipkow@15302: lemma lex_conv: paulson@15656: "lex r = paulson@15656: {(xs,ys). length xs = length ys \ paulson@15656: (\xys x y xs' ys'. xs = xys @ x # xs' \ ys = xys @ y # ys' \ (x, y):r)}" nipkow@15302: by (force simp add: lex_def lexn_conv) nipkow@15302: nipkow@15302: lemma wf_lexico [intro!]: "wf r ==> wf (lexico r)" nipkow@15302: by (unfold lexico_def) blast nipkow@15302: nipkow@15302: lemma lexico_conv: paulson@15656: "lexico r = {(xs,ys). length xs < length ys | paulson@15656: length xs = length ys \ (xs, ys) : lex r}" nipkow@15302: by (simp add: lexico_def diag_def lex_prod_def measure_def inv_image_def) nipkow@15302: nipkow@15302: lemma Nil_notin_lex [iff]: "([], ys) \ lex r" nipkow@15302: by (simp add: lex_conv) nipkow@15302: nipkow@15302: lemma Nil2_notin_lex [iff]: "(xs, []) \ lex r" nipkow@15302: by (simp add:lex_conv) nipkow@15302: nipkow@15302: lemma Cons_in_lex [iff]: paulson@15656: "((x # xs, y # ys) : lex r) = paulson@15656: ((x, y) : r \ length xs = length ys | x = y \ (xs, ys) : lex r)" nipkow@15302: apply (simp add: lex_conv) nipkow@15302: apply (rule iffI) nipkow@15302: prefer 2 apply (blast intro: Cons_eq_appendI, clarify) nipkow@15302: apply (case_tac xys, simp, simp) nipkow@15302: apply blast nipkow@15302: done nipkow@15302: nipkow@15302: paulson@15656: subsubsection {* Lexicographic Ordering *} paulson@15656: paulson@15656: text {* Classical lexicographic ordering on lists, ie. "a" < "ab" < "b". paulson@15656: This ordering does \emph{not} preserve well-foundedness. paulson@15656: Author: N. Voelker, March 2005 *} paulson@15656: paulson@15656: constdefs paulson@15656: lexord :: "('a * 'a)set \ ('a list * 'a list) set" paulson@15656: "lexord r == {(x,y). \ a v. y = x @ a # v \ paulson@15656: (\ u a b v w. (a,b) \ r \ x = u @ (a # v) \ y = u @ (b # w))}" paulson@15656: paulson@15656: lemma lexord_Nil_left[simp]: "([],y) \ lexord r = (\ a x. y = a # x)" paulson@15656: by (unfold lexord_def, induct_tac y, auto) paulson@15656: paulson@15656: lemma lexord_Nil_right[simp]: "(x,[]) \ lexord r" paulson@15656: by (unfold lexord_def, induct_tac x, auto) paulson@15656: paulson@15656: lemma lexord_cons_cons[simp]: paulson@15656: "((a # x, b # y) \ lexord r) = ((a,b)\ r | (a = b & (x,y)\ lexord r))" paulson@15656: apply (unfold lexord_def, safe, simp_all) paulson@15656: apply (case_tac u, simp, simp) paulson@15656: apply (case_tac u, simp, clarsimp, blast, blast, clarsimp) paulson@15656: apply (erule_tac x="b # u" in allE) paulson@15656: by force paulson@15656: paulson@15656: lemmas lexord_simps = lexord_Nil_left lexord_Nil_right lexord_cons_cons paulson@15656: paulson@15656: lemma lexord_append_rightI: "\ b z. y = b # z \ (x, x @ y) \ lexord r" paulson@15656: by (induct_tac x, auto) paulson@15656: paulson@15656: lemma lexord_append_left_rightI: paulson@15656: "(a,b) \ r \ (u @ a # x, u @ b # y) \ lexord r" paulson@15656: by (induct_tac u, auto) paulson@15656: paulson@15656: lemma lexord_append_leftI: " (u,v) \ lexord r \ (x @ u, x @ v) \ lexord r" paulson@15656: by (induct x, auto) paulson@15656: paulson@15656: lemma lexord_append_leftD: paulson@15656: "\ (x @ u, x @ v) \ lexord r; (! a. (a,a) \ r) \ \ (u,v) \ lexord r" paulson@15656: by (erule rev_mp, induct_tac x, auto) paulson@15656: paulson@15656: lemma lexord_take_index_conv: paulson@15656: "((x,y) : lexord r) = paulson@15656: ((length x < length y \ take (length x) y = x) \ paulson@15656: (\i. i < min(length x)(length y) & take i x = take i y & (x!i,y!i) \ r))" paulson@15656: apply (unfold lexord_def Let_def, clarsimp) paulson@15656: apply (rule_tac f = "(% a b. a \ b)" in arg_cong2) paulson@15656: apply auto paulson@15656: apply (rule_tac x="hd (drop (length x) y)" in exI) paulson@15656: apply (rule_tac x="tl (drop (length x) y)" in exI) paulson@15656: apply (erule subst, simp add: min_def) paulson@15656: apply (rule_tac x ="length u" in exI, simp) paulson@15656: apply (rule_tac x ="take i x" in exI) paulson@15656: apply (rule_tac x ="x ! i" in exI) paulson@15656: apply (rule_tac x ="y ! i" in exI, safe) paulson@15656: apply (rule_tac x="drop (Suc i) x" in exI) paulson@15656: apply (drule sym, simp add: drop_Suc_conv_tl) paulson@15656: apply (rule_tac x="drop (Suc i) y" in exI) paulson@15656: by (simp add: drop_Suc_conv_tl) paulson@15656: paulson@15656: -- {* lexord is extension of partial ordering List.lex *} paulson@15656: lemma lexord_lex: " (x,y) \ lex r = ((x,y) \ lexord r \ length x = length y)" paulson@15656: apply (rule_tac x = y in spec) paulson@15656: apply (induct_tac x, clarsimp) paulson@15656: by (clarify, case_tac x, simp, force) paulson@15656: paulson@15656: lemma lexord_irreflexive: "(! x. (x,x) \ r) \ (y,y) \ lexord r" paulson@15656: by (induct y, auto) paulson@15656: paulson@15656: lemma lexord_trans: paulson@15656: "\ (x, y) \ lexord r; (y, z) \ lexord r; trans r \ \ (x, z) \ lexord r" paulson@15656: apply (erule rev_mp)+ paulson@15656: apply (rule_tac x = x in spec) paulson@15656: apply (rule_tac x = z in spec) paulson@15656: apply ( induct_tac y, simp, clarify) paulson@15656: apply (case_tac xa, erule ssubst) paulson@15656: apply (erule allE, erule allE) -- {* avoid simp recursion *} paulson@15656: apply (case_tac x, simp, simp) paulson@15656: apply (case_tac x, erule allE, erule allE, simp) paulson@15656: apply (erule_tac x = listb in allE) paulson@15656: apply (erule_tac x = lista in allE, simp) paulson@15656: apply (unfold trans_def) paulson@15656: by blast paulson@15656: paulson@15656: lemma lexord_transI: "trans r \ trans (lexord r)" paulson@15656: by (rule transI, drule lexord_trans, blast) paulson@15656: paulson@15656: lemma lexord_linear: "(! a b. (a,b)\ r | a = b | (b,a) \ r) \ (x,y) : lexord r | x = y | (y,x) : lexord r" paulson@15656: apply (rule_tac x = y in spec) paulson@15656: apply (induct_tac x, rule allI) paulson@15656: apply (case_tac x, simp, simp) paulson@15656: apply (rule allI, case_tac x, simp, simp) paulson@15656: by blast paulson@15656: paulson@15656: nipkow@15392: subsubsection{*Lifting a Relation on List Elements to the Lists*} nipkow@15302: nipkow@15302: consts listrel :: "('a * 'a)set => ('a list * 'a list)set" nipkow@15302: nipkow@15302: inductive "listrel(r)" nipkow@15302: intros nipkow@15302: Nil: "([],[]) \ listrel r" nipkow@15302: Cons: "[| (x,y) \ r; (xs,ys) \ listrel r |] ==> (x#xs, y#ys) \ listrel r" nipkow@15302: nipkow@15302: inductive_cases listrel_Nil1 [elim!]: "([],xs) \ listrel r" nipkow@15302: inductive_cases listrel_Nil2 [elim!]: "(xs,[]) \ listrel r" nipkow@15302: inductive_cases listrel_Cons1 [elim!]: "(y#ys,xs) \ listrel r" nipkow@15302: inductive_cases listrel_Cons2 [elim!]: "(xs,y#ys) \ listrel r" nipkow@15302: nipkow@15302: nipkow@15302: lemma listrel_mono: "r \ s \ listrel r \ listrel s" nipkow@15302: apply clarify nipkow@15302: apply (erule listrel.induct) nipkow@15302: apply (blast intro: listrel.intros)+ nipkow@15302: done nipkow@15302: nipkow@15302: lemma listrel_subset: "r \ A \ A \ listrel r \ lists A \ lists A" nipkow@15302: apply clarify nipkow@15302: apply (erule listrel.induct, auto) nipkow@15302: done nipkow@15302: nipkow@15302: lemma listrel_refl: "refl A r \ refl (lists A) (listrel r)" nipkow@15302: apply (simp add: refl_def listrel_subset Ball_def) nipkow@15302: apply (rule allI) nipkow@15302: apply (induct_tac x) nipkow@15302: apply (auto intro: listrel.intros) nipkow@15302: done nipkow@15302: nipkow@15302: lemma listrel_sym: "sym r \ sym (listrel r)" nipkow@15302: apply (auto simp add: sym_def) nipkow@15302: apply (erule listrel.induct) nipkow@15302: apply (blast intro: listrel.intros)+ nipkow@15302: done nipkow@15302: nipkow@15302: lemma listrel_trans: "trans r \ trans (listrel r)" nipkow@15302: apply (simp add: trans_def) nipkow@15302: apply (intro allI) nipkow@15302: apply (rule impI) nipkow@15302: apply (erule listrel.induct) nipkow@15302: apply (blast intro: listrel.intros)+ nipkow@15302: done nipkow@15302: nipkow@15302: theorem equiv_listrel: "equiv A r \ equiv (lists A) (listrel r)" nipkow@15302: by (simp add: equiv_def listrel_refl listrel_sym listrel_trans) nipkow@15302: nipkow@15302: lemma listrel_Nil [simp]: "listrel r `` {[]} = {[]}" nipkow@15302: by (blast intro: listrel.intros) nipkow@15302: nipkow@15302: lemma listrel_Cons: nipkow@15302: "listrel r `` {x#xs} = set_Cons (r``{x}) (listrel r `` {xs})"; nipkow@15302: by (auto simp add: set_Cons_def intro: listrel.intros) nipkow@15302: nipkow@15302: nipkow@15392: subsection{*Miscellany*} nipkow@15392: nipkow@15392: subsubsection {* Characters and strings *} wenzelm@13366: wenzelm@13366: datatype nibble = wenzelm@13366: Nibble0 | Nibble1 | Nibble2 | Nibble3 | Nibble4 | Nibble5 | Nibble6 | Nibble7 wenzelm@13366: | Nibble8 | Nibble9 | NibbleA | NibbleB | NibbleC | NibbleD | NibbleE | NibbleF wenzelm@13366: wenzelm@13366: datatype char = Char nibble nibble wenzelm@13366: -- "Note: canonical order of character encoding coincides with standard term ordering" wenzelm@13366: wenzelm@13366: types string = "char list" wenzelm@13366: wenzelm@13366: syntax wenzelm@13366: "_Char" :: "xstr => char" ("CHR _") wenzelm@13366: "_String" :: "xstr => string" ("_") wenzelm@13366: wenzelm@13366: parse_ast_translation {* wenzelm@13366: let wenzelm@13366: val constants = Syntax.Appl o map Syntax.Constant; wenzelm@13366: wenzelm@13366: fun mk_nib n = "Nibble" ^ chr (n + (if n <= 9 then ord "0" else ord "A" - 10)); wenzelm@13366: fun mk_char c = wenzelm@13366: if Symbol.is_ascii c andalso Symbol.is_printable c then wenzelm@13366: constants ["Char", mk_nib (ord c div 16), mk_nib (ord c mod 16)] wenzelm@13366: else error ("Printable ASCII character expected: " ^ quote c); wenzelm@13366: wenzelm@13366: fun mk_string [] = Syntax.Constant "Nil" wenzelm@13366: | mk_string (c :: cs) = Syntax.Appl [Syntax.Constant "Cons", mk_char c, mk_string cs]; wenzelm@13366: wenzelm@13366: fun char_ast_tr [Syntax.Variable xstr] = wenzelm@13366: (case Syntax.explode_xstr xstr of wenzelm@13366: [c] => mk_char c wenzelm@13366: | _ => error ("Single character expected: " ^ xstr)) wenzelm@13366: | char_ast_tr asts = raise AST ("char_ast_tr", asts); wenzelm@13366: wenzelm@13366: fun string_ast_tr [Syntax.Variable xstr] = wenzelm@13366: (case Syntax.explode_xstr xstr of wenzelm@13366: [] => constants [Syntax.constrainC, "Nil", "string"] wenzelm@13366: | cs => mk_string cs) wenzelm@13366: | string_ast_tr asts = raise AST ("string_tr", asts); wenzelm@13366: in [("_Char", char_ast_tr), ("_String", string_ast_tr)] end; wenzelm@13366: *} wenzelm@13366: berghofe@15064: ML {* berghofe@15064: fun int_of_nibble h = berghofe@15064: if "0" <= h andalso h <= "9" then ord h - ord "0" berghofe@15064: else if "A" <= h andalso h <= "F" then ord h - ord "A" + 10 berghofe@15064: else raise Match; berghofe@15064: berghofe@15064: fun nibble_of_int i = berghofe@15064: if i <= 9 then chr (ord "0" + i) else chr (ord "A" + i - 10); berghofe@15064: *} berghofe@15064: wenzelm@13366: print_ast_translation {* wenzelm@13366: let wenzelm@13366: fun dest_nib (Syntax.Constant c) = wenzelm@13366: (case explode c of berghofe@15064: ["N", "i", "b", "b", "l", "e", h] => int_of_nibble h wenzelm@13366: | _ => raise Match) wenzelm@13366: | dest_nib _ = raise Match; wenzelm@13366: wenzelm@13366: fun dest_chr c1 c2 = wenzelm@13366: let val c = chr (dest_nib c1 * 16 + dest_nib c2) wenzelm@13366: in if Symbol.is_printable c then c else raise Match end; wenzelm@13366: wenzelm@13366: fun dest_char (Syntax.Appl [Syntax.Constant "Char", c1, c2]) = dest_chr c1 c2 wenzelm@13366: | dest_char _ = raise Match; wenzelm@13366: wenzelm@13366: fun xstr cs = Syntax.Appl [Syntax.Constant "_xstr", Syntax.Variable (Syntax.implode_xstr cs)]; wenzelm@13366: wenzelm@13366: fun char_ast_tr' [c1, c2] = Syntax.Appl [Syntax.Constant "_Char", xstr [dest_chr c1 c2]] wenzelm@13366: | char_ast_tr' _ = raise Match; wenzelm@13366: wenzelm@13366: fun list_ast_tr' [args] = Syntax.Appl [Syntax.Constant "_String", wenzelm@13366: xstr (map dest_char (Syntax.unfold_ast "_args" args))] wenzelm@13366: | list_ast_tr' ts = raise Match; wenzelm@13366: in [("Char", char_ast_tr'), ("@list", list_ast_tr')] end; wenzelm@13366: *} wenzelm@13366: nipkow@15392: subsubsection {* Code generator setup *} berghofe@15064: berghofe@15064: ML {* berghofe@15064: local berghofe@15064: berghofe@15064: fun list_codegen thy gr dep b t = berghofe@15064: let val (gr', ps) = foldl_map (Codegen.invoke_codegen thy dep false) berghofe@15064: (gr, HOLogic.dest_list t) skalberg@15531: in SOME (gr', Pretty.list "[" "]" ps) end handle TERM _ => NONE; berghofe@15064: berghofe@15064: fun dest_nibble (Const (s, _)) = int_of_nibble (unprefix "List.nibble.Nibble" s) berghofe@15064: | dest_nibble _ = raise Match; berghofe@15064: berghofe@15064: fun char_codegen thy gr dep b (Const ("List.char.Char", _) $ c1 $ c2) = berghofe@15064: (let val c = chr (dest_nibble c1 * 16 + dest_nibble c2) skalberg@15531: in if Symbol.is_printable c then SOME (gr, Pretty.quote (Pretty.str c)) skalberg@15531: else NONE skalberg@15570: end handle Fail _ => NONE | Match => NONE) skalberg@15531: | char_codegen thy gr dep b _ = NONE; berghofe@15064: berghofe@15064: in berghofe@15064: berghofe@15064: val list_codegen_setup = berghofe@15064: [Codegen.add_codegen "list_codegen" list_codegen, berghofe@15064: Codegen.add_codegen "char_codegen" char_codegen]; berghofe@15064: berghofe@15064: end; berghofe@15064: berghofe@15064: val term_of_list = HOLogic.mk_list; berghofe@15064: berghofe@15064: fun gen_list' aG i j = frequency berghofe@15064: [(i, fn () => aG j :: gen_list' aG (i-1) j), (1, fn () => [])] () berghofe@15064: and gen_list aG i = gen_list' aG i i; berghofe@15064: berghofe@15064: val nibbleT = Type ("List.nibble", []); berghofe@15064: berghofe@15064: fun term_of_char c = berghofe@15064: Const ("List.char.Char", nibbleT --> nibbleT --> Type ("List.char", [])) $ berghofe@15064: Const ("List.nibble.Nibble" ^ nibble_of_int (ord c div 16), nibbleT) $ berghofe@15064: Const ("List.nibble.Nibble" ^ nibble_of_int (ord c mod 16), nibbleT); berghofe@15064: berghofe@15064: fun gen_char i = chr (random_range (ord "a") (Int.min (ord "a" + i, ord "z"))); berghofe@15064: *} berghofe@15064: berghofe@15064: types_code berghofe@15064: "list" ("_ list") berghofe@15064: "char" ("string") berghofe@15064: berghofe@15064: consts_code "Cons" ("(_ ::/ _)") berghofe@15064: berghofe@15064: setup list_codegen_setup berghofe@15064: wenzelm@13122: end