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 haftmann@22844: imports PreList wenzelm@21754: uses "Tools/string_syntax.ML" 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: 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: map :: "('a=>'b) => ('a list => 'b list)" nipkow@23096: listsum :: "'a list => 'a::monoid_add" 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: "distinct":: "'a list => bool" wenzelm@13366: replicate :: "nat => 'a => 'a list" nipkow@19390: splice :: "'a list \ 'a list \ 'a list" nipkow@23279: partition :: "('a \ bool) \'a list \ (('a list) \ ('a list))" nipkow@22830: allpairs :: "('a \ 'b \ 'c) \ 'a list \ 'b list \ 'c list" nipkow@15302: wenzelm@19363: abbreviation wenzelm@21404: upto:: "nat => nat => nat list" ("(1[_../_])") where wenzelm@19363: "[i..j] == [i..<(Suc j)]" wenzelm@19302: 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 *} nipkow@23279: "@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: clasohm@923: translations wenzelm@13366: "[x, xs]" == "x#[xs]" wenzelm@13366: "[x]" == "x#[]" nipkow@23279: "[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@5427: wenzelm@12114: syntax (xsymbols) nipkow@23279: "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\_ ./ _])") kleing@14565: syntax (HTML output) nipkow@23279: "@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@19363: abbreviation wenzelm@21404: length :: "'a list => nat" where wenzelm@19363: "length == size" 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: "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: "set [] = {}" paulson@15307: "set (x#xs) = insert x (set xs)" paulson@15307: berghofe@5183: primrec paulson@15307: "map f [] = []" paulson@15307: "map f (x#xs) = f(x)#map f xs" paulson@15307: haftmann@23235: function (*authentic syntax for append -- revert to primrec haftmann@23235: as soon as "authentic" primrec is available*) haftmann@23235: append :: "'a list \ 'a list \ 'a list" (infixr "@" 65) haftmann@23235: where haftmann@23235: append_Nil: "[] @ ys = ys" haftmann@23235: | append_Cons: "(x # xs) @ ys = x # (xs @ ys)" haftmann@23235: by (auto, case_tac a, auto) haftmann@23235: termination by (relation "measure (size o fst)") auto 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 nipkow@23096: "listsum [] = 0" nipkow@23096: "listsum (x # xs) = x + listsum xs" nipkow@23096: nipkow@23096: 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: haftmann@21061: definition wenzelm@21404: rotate1 :: "'a list \ 'a list" where wenzelm@21404: "rotate1 xs = (case xs of [] \ [] | x#xs \ xs @ [x])" wenzelm@21404: wenzelm@21404: definition wenzelm@21404: rotate :: "nat \ 'a list \ 'a list" where wenzelm@21404: "rotate n = rotate1 ^ n" wenzelm@21404: wenzelm@21404: definition wenzelm@21404: list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool" where wenzelm@21404: "list_all2 P xs ys = haftmann@21061: (length xs = length ys \ (\(x, y) \ set (zip xs ys). P x y))" wenzelm@21404: wenzelm@21404: definition wenzelm@21404: sublist :: "'a list => nat set => 'a list" where wenzelm@21404: "sublist xs A = map fst (filter (\p. snd p \ A) (zip xs [0.. xs, y \ ys, x \ y]"}, the list of all pairs of distinct elements from @{text xs} and @{text ys}. nipkow@23209: nipkow@23240: There are two differences to Haskell. The general synatx is nipkow@23240: @{text"[e. p \ xs, \]"} rather than \verb![x| x <- xs, ...]!. Patterns in nipkow@23240: generators can only be tuples (at the moment). nipkow@23240: nipkow@23240: To avoid misunderstandings, the translation is not reversed upon nipkow@23240: output. You can add the inverse translations in your own theory if you nipkow@23240: desire. nipkow@23240: nipkow@23240: Hint: formulae containing complex list comprehensions may become quite nipkow@23240: unreadable after the simplifier has finished with them. It can be nipkow@23240: helpful to introduce definitions for such list comprehensions and nipkow@23240: treat them separately in suitable lemmas. nipkow@23209: *} nipkow@23209: (* nipkow@23240: Proper theorem proving support would be nice. For example, if nipkow@23192: @{text"set[f x y. x \ xs, y \ ys, P x y]"} nipkow@23192: produced something like nipkow@23209: @{term"{z. EX x: set xs. EX y:set ys. P x y \ z = f x y}"}. nipkow@23209: *) nipkow@23209: nipkow@23240: nonterminals lc_qual lc_quals nipkow@23192: nipkow@23192: syntax nipkow@23240: "_listcompr" :: "'a \ lc_qual \ lc_quals \ 'a list" ("[_ . __") nipkow@23240: "_lc_gen" :: "pttrn \ 'a list \ lc_qual" ("_ <- _") nipkow@23240: "_lc_test" :: "bool \ lc_qual" ("_") nipkow@23240: "_lc_end" :: "lc_quals" ("]") nipkow@23240: "_lc_quals" :: "lc_qual \ lc_quals \ lc_quals" (", __") nipkow@23192: nipkow@23192: translations nipkow@23279: "[e. p<-xs]" => "map (%p. e) xs" nipkow@23240: "_listcompr e (_lc_gen p xs) (_lc_quals Q Qs)" nipkow@23240: => "concat (map (%p. _listcompr e Q Qs) xs)" nipkow@23240: "[e. P]" => "if P then [e] else []" nipkow@23240: "_listcompr e (_lc_test P) (_lc_quals Q Qs)" nipkow@23240: => "if P then (_listcompr e Q Qs) else []" nipkow@23240: nipkow@23279: syntax (xsymbols) nipkow@23279: "_lc_gen" :: "pttrn \ 'a list \ lc_qual" ("_ \ _") nipkow@23279: syntax (HTML output) nipkow@23279: "_lc_gen" :: "pttrn \ 'a list \ lc_qual" ("_ \ _") nipkow@23279: nipkow@23279: nipkow@23240: (* nipkow@23240: term "[(x,y,z). b]" nipkow@23240: term "[(x,y,z). x \ xs]" nipkow@23240: term "[(x,y,z). xb]" nipkow@23240: term "[(x,y,z). xxs]" nipkow@23240: term "[(x,y,z). x\xs, x>b]" nipkow@23240: term "[(x,y,z). x\xs, y\ys]" nipkow@23240: term "[(x,y,z). xb, x=d]" nipkow@23240: term "[(x,y,z). xb, y\ys]" nipkow@23240: term "[(x,y,z). xxs,y>b]" nipkow@23240: term "[(x,y,z). xxs, y\ys]" nipkow@23240: term "[(x,y,z). x\xs, x>b, yxs, x>b, y\ys]" nipkow@23240: term "[(x,y,z). x\xs, y\ys,y>x]" nipkow@23240: term "[(x,y,z). x\xs, y\ys,z\zs]" nipkow@23192: *) nipkow@23192: nipkow@23240: haftmann@21061: subsubsection {* @{const Nil} and @{const Cons} *} haftmann@21061: haftmann@21061: lemma not_Cons_self [simp]: haftmann@21061: "xs \ 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: haftmann@21061: "(\xs. \ys. length ys < length xs \ P ys \ P xs) \ P xs" nipkow@17589: by (rule measure_induct [of length]) iprover wenzelm@13114: wenzelm@13114: haftmann@21061: subsubsection {* @{const length} *} wenzelm@13114: wenzelm@13142: text {* haftmann@21061: Needs to come before @{text "@"} because of theorem @{text haftmann@21061: 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: nipkow@22828: lemma length_allpairs[simp]: nipkow@22830: "length(allpairs f xs ys) = length xs * length ys" nipkow@22828: by(induct xs) auto nipkow@22828: 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" wenzelm@20503: apply (induct xs) wenzelm@20503: apply 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: krauss@22493: lemma list_induct2': krauss@22493: "\ P [] []; krauss@22493: \x xs. P (x#xs) []; krauss@22493: \y ys. P [] (y#ys); krauss@22493: \x xs y ys. P xs ys \ P (x#xs) (y#ys) \ krauss@22493: \ P xs ys" krauss@22493: by (induct xs arbitrary: ys) (case_tac x, auto)+ krauss@22493: nipkow@22143: lemma neq_if_length_neq: "length xs \ length ys \ (xs = ys) == False" nipkow@22143: apply(rule Eq_FalseI) nipkow@22143: by auto nipkow@22143: nipkow@22143: (* nipkow@22143: Reduces xs=ys to False if xs and ys cannot be of the same length. nipkow@22143: This is the case if the atomic sublists of one are a submultiset nipkow@22143: of those of the other list and there are fewer Cons's in one than the other. nipkow@22143: *) nipkow@22143: ML_setup {* nipkow@22143: local nipkow@22143: nipkow@22143: fun len (Const("List.list.Nil",_)) acc = acc nipkow@22143: | len (Const("List.list.Cons",_) $ _ $ xs) (ts,n) = len xs (ts,n+1) haftmann@23029: | len (Const("List.append",_) $ xs $ ys) acc = len xs (len ys acc) nipkow@22143: | len (Const("List.rev",_) $ xs) acc = len xs acc nipkow@22143: | len (Const("List.map",_) $ _ $ xs) acc = len xs acc nipkow@22143: | len t (ts,n) = (t::ts,n); nipkow@22143: nipkow@22143: fun list_eq ss (Const(_,eqT) $ lhs $ rhs) = nipkow@22143: let nipkow@22143: val (ls,m) = len lhs ([],0) and (rs,n) = len rhs ([],0); nipkow@22143: fun prove_neq() = nipkow@22143: let nipkow@22143: val Type(_,listT::_) = eqT; haftmann@22994: val size = HOLogic.size_const listT; nipkow@22143: val eq_len = HOLogic.mk_eq (size $ lhs, size $ rhs); nipkow@22143: val neq_len = HOLogic.mk_Trueprop (HOLogic.Not $ eq_len); nipkow@22143: val thm = Goal.prove (Simplifier.the_context ss) [] [] neq_len haftmann@22633: (K (simp_tac (Simplifier.inherit_context ss @{simpset}) 1)); haftmann@22633: in SOME (thm RS @{thm neq_if_length_neq}) end nipkow@22143: in wenzelm@23214: if m < n andalso submultiset (op aconv) (ls,rs) orelse wenzelm@23214: n < m andalso submultiset (op aconv) (rs,ls) nipkow@22143: then prove_neq() else NONE nipkow@22143: end; nipkow@22143: nipkow@22143: in nipkow@22143: nipkow@22143: val list_neq_simproc = haftmann@22633: Simplifier.simproc @{theory} "list_neq" ["(xs::'a list) = ys"] (K list_eq); nipkow@22143: nipkow@22143: end; nipkow@22143: nipkow@22143: Addsimprocs [list_neq_simproc]; nipkow@22143: *} nipkow@22143: nipkow@22143: 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@13114: fun last (cons as Const("List.list.Cons",_) $ _ $ xs) = wenzelm@13462: (case xs of Const("List.list.Nil",_) => cons | _ => last xs) haftmann@23029: | last (Const("List.append",_) $ _ $ 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) haftmann@23029: | butlast ((app as Const("List.append",_) $ xs) $ ys) = app $ butlast ys wenzelm@13462: | butlast xs = Const("List.list.Nil",fastype_of xs); wenzelm@13114: haftmann@22633: val rearr_ss = HOL_basic_ss addsimps [@{thm append_assoc}, haftmann@22633: @{thm append_Nil}, @{thm append_Cons}]; wenzelm@16973: wenzelm@20044: fun list_eq ss (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 haftmann@23029: val app = Const("List.append",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@20044: val thm = Goal.prove (Simplifier.the_context ss) [] [] eq wenzelm@17877: (K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1)); skalberg@15531: in SOME ((conv RS (thm RS trans)) RS eq_reflection) end; wenzelm@13114: wenzelm@13462: in haftmann@22633: if list1 lastl andalso list1 lastr then rearr @{thm append1_eq_conv} haftmann@22633: else if lastl aconv lastr then rearr @{thm append_same_eq} skalberg@15531: else NONE wenzelm@13462: end; wenzelm@13462: wenzelm@13114: in wenzelm@13462: wenzelm@13462: val list_eq_simproc = haftmann@22633: Simplifier.simproc @{theory} "list_eq" ["(xs::'a list) = ys"] (K 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: krauss@19770: lemma map_cong [fundef_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: paulson@18447: lemma map_eq_Cons_conv: 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: paulson@18447: lemma Cons_eq_map_conv: 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: paulson@18447: lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1] paulson@18447: lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1] paulson@18447: declare map_eq_Cons_D [dest!] Cons_eq_map_D [dest!] paulson@18447: nipkow@14111: lemma ex_map_conv: nipkow@14111: "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)" paulson@18447: by(induct ys, auto simp add: Cons_eq_map_conv) 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)" nipkow@17589: by (iprover 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: kleing@15870: lemma rev_swap: "(rev xs = ys) = (xs = rev ys)" kleing@15870: by auto kleing@15870: 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: kleing@15870: lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])" kleing@15870: by (cases xs) auto kleing@15870: kleing@15870: lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])" kleing@15870: by (cases xs) auto kleing@15870: haftmann@21061: lemma rev_is_rev_conv [iff]: "(rev xs = rev ys) = (xs = ys)" haftmann@21061: apply (induct xs arbitrary: ys, 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: nipkow@18423: lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])" nipkow@18423: by(rule rev_cases[of xs]) auto nipkow@18423: 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: nipkow@17830: lemma hd_in_set[simp]: "xs \ [] \ hd xs : set xs" nipkow@17830: by(cases xs) 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: nipkow@22828: lemma set_allpairs[simp]: nipkow@22830: "set(allpairs f xs ys) = {z. EX x : set xs. EX y : set ys. z = f x y}" nipkow@22828: by(induct xs) auto nipkow@22828: 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: nipkow@18049: lemma in_set_conv_decomp_first: nipkow@18049: "(x : set xs) = (\ys zs. xs = ys @ x # zs \ x \ set ys)" nipkow@18049: proof (induct xs) nipkow@18049: case Nil show ?case by simp nipkow@18049: next nipkow@18049: case (Cons a xs) nipkow@18049: show ?case nipkow@18049: proof cases nipkow@18049: assume "x = a" thus ?case using Cons by force nipkow@18049: next nipkow@18049: assume "x \ a" nipkow@18049: show ?case nipkow@18049: proof nipkow@18049: assume "x \ set (a # xs)" nipkow@18049: from prems show "\ys zs. a # xs = ys @ x # zs \ x \ set ys" nipkow@18049: by(fastsimp intro!: Cons_eq_appendI) nipkow@18049: next nipkow@18049: assume "\ys zs. a # xs = ys @ x # zs \ x \ set ys" nipkow@18049: then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast nipkow@18049: show "x \ set (a # xs)" by (cases ys, auto simp add: eq) nipkow@18049: qed nipkow@18049: qed nipkow@18049: qed nipkow@18049: nipkow@18049: lemmas split_list = in_set_conv_decomp[THEN iffD1, standard] nipkow@18049: lemmas split_list_first = in_set_conv_decomp_first[THEN iffD1, standard] nipkow@18049: nipkow@18049: 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@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: nipkow@16998: lemma length_filter_le [simp]: "length (filter P xs) \ length xs" nipkow@16998: by (induct xs) (auto simp add: le_SucI) nipkow@16998: nipkow@18423: lemma sum_length_filter_compl: nipkow@18423: "length(filter P xs) + length(filter (%x. ~P x) xs) = length xs" nipkow@18423: by(induct xs) simp_all nipkow@18423: 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@16998: lemma filter_empty_conv: "(filter P xs = []) = (\x\set xs. \ P x)" nipkow@16998: by (induct xs) simp_all nipkow@16998: nipkow@16998: lemma filter_id_conv: "(filter P xs = xs) = (\x\set xs. P x)" nipkow@16998: apply (induct xs) nipkow@16998: apply auto nipkow@16998: apply(cut_tac P=P and xs=xs in length_filter_le) nipkow@16998: apply simp nipkow@16998: done wenzelm@13114: nipkow@16965: lemma filter_map: nipkow@16965: "filter P (map f xs) = map f (filter (P o f) xs)" nipkow@16965: by (induct xs) simp_all nipkow@16965: nipkow@16965: lemma length_filter_map[simp]: nipkow@16965: "length (filter P (map f xs)) = length(filter (P o f) xs)" nipkow@16965: by (simp add:filter_map) nipkow@16965: 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@17629: lemma Cons_eq_filterD: nipkow@17629: "x#xs = filter P ys \ nipkow@17629: \us vs. ys = us @ x # vs \ (\u\set us. \ P u) \ P x \ xs = filter P vs" wenzelm@19585: (is "_ \ \us vs. ?P ys us vs") nipkow@17629: proof(induct ys) nipkow@17629: case Nil thus ?case by simp nipkow@17629: next nipkow@17629: case (Cons y ys) nipkow@17629: show ?case (is "\x. ?Q x") nipkow@17629: proof cases nipkow@17629: assume Py: "P y" nipkow@17629: show ?thesis nipkow@17629: proof cases nipkow@17629: assume xy: "x = y" nipkow@17629: show ?thesis nipkow@17629: proof from Py xy Cons(2) show "?Q []" by simp qed nipkow@17629: next nipkow@17629: assume "x \ y" with Py Cons(2) show ?thesis by simp nipkow@17629: qed nipkow@17629: next nipkow@17629: assume Py: "\ P y" nipkow@17629: with Cons obtain us vs where 1 : "?P (y#ys) (y#us) vs" by fastsimp nipkow@17629: show ?thesis (is "? us. ?Q us") nipkow@17629: proof show "?Q (y#us)" using 1 by simp qed nipkow@17629: qed nipkow@17629: qed nipkow@17629: nipkow@17629: lemma filter_eq_ConsD: nipkow@17629: "filter P ys = x#xs \ nipkow@17629: \us vs. ys = us @ x # vs \ (\u\set us. \ P u) \ P x \ xs = filter P vs" nipkow@17629: by(rule Cons_eq_filterD) simp nipkow@17629: nipkow@17629: lemma filter_eq_Cons_iff: nipkow@17629: "(filter P ys = x#xs) = nipkow@17629: (\us vs. ys = us @ x # vs \ (\u\set us. \ P u) \ P x \ xs = filter P vs)" nipkow@17629: by(auto dest:filter_eq_ConsD) nipkow@17629: nipkow@17629: lemma Cons_eq_filter_iff: nipkow@17629: "(x#xs = filter P ys) = nipkow@17629: (\us vs. ys = us @ x # vs \ (\u\set us. \ P u) \ P x \ xs = filter P vs)" nipkow@17629: by(auto dest:Cons_eq_filterD) nipkow@17629: krauss@19770: lemma filter_cong[fundef_cong, recdef_cong]: nipkow@17501: "xs = ys \ (\x. x \ set ys \ P x = Q x) \ filter P xs = filter Q ys" nipkow@17501: apply simp nipkow@17501: apply(erule thin_rl) nipkow@17501: by (induct ys) simp_all nipkow@17501: 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: paulson@18447: lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\xs \ set xss. xs = [])" nipkow@13145: by (induct xss) auto wenzelm@13114: paulson@18447: lemma Nil_eq_concat_conv [simp]: "([] = 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: nipkow@18423: lemma hd_conv_nth: "xs \ [] \ hd xs = xs!0" nipkow@18423: by(cases xs) simp_all nipkow@18423: nipkow@18049: nipkow@18049: lemma list_eq_iff_nth_eq: nipkow@18049: "!!ys. (xs = ys) = (length xs = length ys \ (ALL i set xs) = (\i < length xs. xs!i = x)" nipkow@17501: by(auto simp:set_conv_nth) nipkow@17501: 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: nipkow@17501: lemma list_update_beyond[simp]: "\i. length xs \ i \ xs[i:=x] = xs" nipkow@17501: apply (induct xs) nipkow@17501: apply simp nipkow@17501: apply (case_tac i) nipkow@17501: apply simp_all nipkow@17501: done nipkow@17501: 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: kleing@15868: lemma list_update_append: kleing@15868: "!!n. (xs @ ys) [n:= x] = kleing@15868: (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))" kleing@15868: by (induct xs) (auto split:nat.splits) kleing@15868: 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: kleing@15868: lemma set_update_memI: "!!n. n < length xs \ x \ set (xs[n := x])" kleing@15868: by (induct xs) (auto split:nat.splits) kleing@15868: 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@17762: lemma hd_rev: "xs \ [] \ hd(rev xs) = last xs" nipkow@17762: by(rule rev_exhaust[of xs]) simp_all nipkow@17762: nipkow@17762: lemma last_rev: "xs \ [] \ last(rev xs) = hd xs" nipkow@17762: by(cases xs) simp_all nipkow@17762: nipkow@17765: lemma last_in_set[simp]: "as \ [] \ last as \ set as" nipkow@17765: by (induct as) auto nipkow@17762: 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: nipkow@17501: lemma last_drop[simp]: "!!n. n < length xs \ last (drop n xs) = last xs" nipkow@17501: apply (induct xs) nipkow@17501: apply simp nipkow@17501: apply (auto split:nat.split) nipkow@17501: done nipkow@17501: nipkow@17589: lemma last_conv_nth: "xs\[] \ last xs = xs!(length xs - 1)" nipkow@17589: by(induct xs)(auto simp:neq_Nil_conv) nipkow@17589: 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@18423: lemma hd_drop_conv_nth: "\ xs \ []; n < length xs \ \ hd(drop n xs) = xs!n" nipkow@18423: by(simp add: hd_conv_nth) nipkow@18423: 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: nipkow@17501: lemma id_take_nth_drop: nipkow@17501: "i < length xs \ xs = take i xs @ xs!i # drop (Suc i) xs" nipkow@17501: proof - nipkow@17501: assume si: "i < length xs" nipkow@17501: hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto nipkow@17501: moreover nipkow@17501: from si have "take (Suc i) xs = take i xs @ [xs!i]" nipkow@17501: apply (rule_tac take_Suc_conv_app_nth) by arith nipkow@17501: ultimately show ?thesis by auto nipkow@17501: qed nipkow@17501: nipkow@17501: lemma upd_conv_take_nth_drop: nipkow@17501: "i < length xs \ xs[i:=a] = take i xs @ a # drop (Suc i) xs" nipkow@17501: proof - nipkow@17501: assume i: "i < length xs" nipkow@17501: have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]" nipkow@17501: by(rule arg_cong[OF id_take_nth_drop[OF i]]) nipkow@17501: also have "\ = take i xs @ a # drop (Suc i) xs" nipkow@17501: using i by (simp add: list_update_append) nipkow@17501: finally show ?thesis . nipkow@17501: qed nipkow@17501: 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: nipkow@17501: text{* The following two lemmmas could be generalized to an arbitrary nipkow@17501: property. *} nipkow@17501: nipkow@17501: lemma takeWhile_neq_rev: "\distinct xs; x \ set xs\ \ nipkow@17501: takeWhile (\y. y \ x) (rev xs) = rev (tl (dropWhile (\y. y \ x) xs))" nipkow@17501: by(induct xs) (auto simp: takeWhile_tail[where l="[]"]) nipkow@17501: nipkow@17501: lemma dropWhile_neq_rev: "\distinct xs; x \ set xs\ \ nipkow@17501: dropWhile (\y. y \ x) (rev xs) = x # rev (takeWhile (\y. y \ x) xs)" nipkow@17501: apply(induct xs) nipkow@17501: apply simp nipkow@17501: apply auto nipkow@17501: apply(subst dropWhile_append2) nipkow@17501: apply auto nipkow@17501: done nipkow@17501: nipkow@18423: lemma takeWhile_not_last: nipkow@18423: "\ xs \ []; distinct xs\ \ takeWhile (\y. y \ last xs) xs = butlast xs" nipkow@18423: apply(induct xs) nipkow@18423: apply simp nipkow@18423: apply(case_tac xs) nipkow@18423: apply(auto) nipkow@18423: done nipkow@18423: krauss@19770: lemma takeWhile_cong [fundef_cong, recdef_cong]: krauss@18336: "[| l = k; !!x. x : set l ==> P x = Q x |] krauss@18336: ==> takeWhile P l = takeWhile Q k" wenzelm@20503: by (induct k arbitrary: l) (simp_all) krauss@18336: krauss@19770: lemma dropWhile_cong [fundef_cong, recdef_cong]: krauss@18336: "[| l = k; !!x. x : set l ==> P x = Q x |] krauss@18336: ==> dropWhile P l = dropWhile Q k" wenzelm@20503: by (induct k arbitrary: l, simp_all) krauss@18336: 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]: krauss@22493: "length (zip xs ys) = min (length xs) (length ys)" krauss@22493: by (induct xs ys rule:list_induct2') auto wenzelm@13114: wenzelm@13114: lemma zip_append1: krauss@22493: "zip (xs @ ys) zs = nipkow@13145: zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)" krauss@22493: by (induct xs zs rule:list_induct2') auto wenzelm@13114: wenzelm@13114: lemma zip_append2: krauss@22493: "zip xs (ys @ zs) = nipkow@13145: zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs" krauss@22493: by (induct xs ys rule:list_induct2') auto 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: nipkow@23096: lemma map_zip_map: nipkow@23096: "map f (zip (map g xs) ys) = map (%(x,y). f(g x, y)) (zip xs ys)" nipkow@23096: apply(induct xs arbitrary:ys) apply simp nipkow@23096: apply(case_tac ys) nipkow@23096: apply simp_all nipkow@23096: done nipkow@23096: nipkow@23096: lemma map_zip_map2: nipkow@23096: "map f (zip xs (map g ys)) = map (%(x,y). f(x, g y)) (zip xs ys)" nipkow@23096: apply(induct xs arbitrary:ys) apply simp nipkow@23096: apply(case_tac ys) nipkow@23096: apply simp_all nipkow@23096: done nipkow@23096: 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: nipkow@19487: lemma take_zip: nipkow@19487: "!!xs ys. take n (zip xs ys) = zip (take n xs) (take n ys)" nipkow@19487: apply (induct n) nipkow@19487: apply simp nipkow@19487: apply (case_tac xs, simp) nipkow@19487: apply (case_tac ys, simp_all) nipkow@19487: done nipkow@19487: nipkow@19487: lemma drop_zip: nipkow@19487: "!!xs ys. drop n (zip xs ys) = zip (drop n xs) (drop n ys)" nipkow@19487: apply (induct n) nipkow@19487: apply simp nipkow@19487: apply (case_tac xs, simp) nipkow@19487: apply (case_tac ys, simp_all) nipkow@19487: done nipkow@19487: krauss@22493: lemma set_zip_leftD: krauss@22493: "(x,y)\ set (zip xs ys) \ x \ set xs" krauss@22493: by (induct xs ys rule:list_induct2') auto krauss@22493: krauss@22493: lemma set_zip_rightD: krauss@22493: "(x,y)\ set (zip xs ys) \ y \ set ys" krauss@22493: by (induct xs ys rule:list_induct2') auto 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" haftmann@19607: by (simp add: list_all2_def) haftmann@19607: haftmann@19787: lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])" haftmann@19607: by (simp add: list_all2_def) haftmann@19607: haftmann@19787: lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])" haftmann@19787: by (simp add: list_all2_def) haftmann@19607: haftmann@19607: lemma list_all2_Cons [iff, code]: haftmann@19607: "list_all2 P (x # xs) (y # ys) = (P x y \ list_all2 P xs ys)" haftmann@19607: 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: haftmann@22551: lemma list_all2_eq: haftmann@22551: "xs = ys \ list_all2 (op =) xs ys" haftmann@22551: by (induct xs ys rule: list_induct2') auto haftmann@22551: 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@23096: lemma foldr_map: "foldr g (map f xs) a = foldr (g o f) xs a" nipkow@23096: by(induct xs) simp_all nipkow@23096: nipkow@23096: lemma foldl_map: "foldl g a (map f xs) = foldl (%a x. g a (f x)) a xs" nipkow@23096: by(induct xs arbitrary:a) simp_all nipkow@23096: krauss@19770: lemma foldl_cong [fundef_cong, recdef_cong]: krauss@18336: "[| a = b; l = k; !!a x. x : set l ==> f a x = g a x |] krauss@18336: ==> foldl f a l = foldl g b k" wenzelm@20503: by (induct k arbitrary: a b l) simp_all krauss@18336: krauss@19770: lemma foldr_cong [fundef_cong, recdef_cong]: krauss@18336: "[| a = b; l = k; !!a x. x : set l ==> f x a = g x a |] krauss@18336: ==> foldr f l a = foldr g k b" wenzelm@20503: by (induct k arbitrary: a b l) simp_all krauss@18336: nipkow@23096: text{* The ``First Duality Theorem'' in Bird \& Wadler: *} nipkow@23096: nipkow@23096: lemma foldl_foldr1_lemma: nipkow@23096: "foldl op + a xs = a + foldr op + xs (0\'a::monoid_add)" nipkow@23096: by (induct xs arbitrary: a) (auto simp:add_assoc) nipkow@23096: nipkow@23096: corollary foldl_foldr1: nipkow@23096: "foldl op + 0 xs = foldr op + xs (0\'a::monoid_add)" nipkow@23096: by (simp add:foldl_foldr1_lemma) nipkow@23096: nipkow@23096: nipkow@23096: text{* The ``Third Duality Theorem'' in Bird \& Wadler: *} nipkow@23096: 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: nipkow@23096: subsubsection {* List summation: @{const listsum} and @{text"\"}*} nipkow@23096: nipkow@23096: lemma listsum_foldr: nipkow@23096: "listsum xs = foldr (op +) xs 0" nipkow@23096: by(induct xs) auto nipkow@23096: nipkow@23096: (* for efficient code generation *) nipkow@23096: lemma listsum[code]: "listsum xs = foldl (op +) 0 xs" nipkow@23096: by(simp add:listsum_foldr foldl_foldr1) nipkow@23096: nipkow@23096: text{* Some syntactic sugar for summing a function over a list: *} nipkow@23096: nipkow@23096: syntax nipkow@23096: "_listsum" :: "pttrn => 'a list => 'b => 'b" ("(3SUM _<-_. _)" [0, 51, 10] 10) nipkow@23096: syntax (xsymbols) nipkow@23096: "_listsum" :: "pttrn => 'a list => 'b => 'b" ("(3\_\_. _)" [0, 51, 10] 10) nipkow@23096: syntax (HTML output) nipkow@23096: "_listsum" :: "pttrn => 'a list => 'b => 'b" ("(3\_\_. _)" [0, 51, 10] 10) nipkow@23096: nipkow@23096: translations -- {* Beware of argument permutation! *} nipkow@23096: "SUM x<-xs. b" == "CONST listsum (map (%x. b) xs)" nipkow@23096: "\x\xs. b" == "CONST listsum (map (%x. b) xs)" nipkow@23096: nipkow@23096: lemma listsum_0 [simp]: "(\x\xs. 0) = 0" nipkow@23096: by (induct xs) simp_all nipkow@23096: nipkow@23096: text{* For non-Abelian groups @{text xs} needs to be reversed on one side: *} nipkow@23096: lemma uminus_listsum_map: nipkow@23096: "- listsum (map f xs) = (listsum (map (uminus o f) xs) :: 'a::ab_group_add)" nipkow@23096: by(induct xs) simp_all nipkow@23096: wenzelm@13114: nipkow@15392: subsubsection {* @{text upto} *} wenzelm@13114: nipkow@17090: lemma upt_rec[code]: "[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.. hd[i.. last[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: paulson@18622: lemmas take_Cons_number_of = take_Cons'[of "number_of v",standard] paulson@18622: lemmas drop_Cons_number_of = drop_Cons'[of "number_of v",standard] paulson@18622: lemmas nth_Cons_number_of = nth_Cons'[of _ _ "number_of v",standard] paulson@18622: paulson@18622: declare take_Cons_number_of [simp] paulson@18622: drop_Cons_number_of [simp] paulson@18622: nth_Cons_number_of [simp] 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: nipkow@18490: nipkow@18490: lemma distinct_map: nipkow@18490: "distinct(map f xs) = (distinct xs & inj_on f (set xs))" nipkow@18490: by (induct xs) auto nipkow@18490: nipkow@18490: wenzelm@13142: lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)" nipkow@13145: by (induct xs) auto wenzelm@13114: nipkow@17501: lemma distinct_upt[simp]: "distinct[i..i. distinct xs \ distinct (take i xs)" nipkow@17501: apply(induct xs) nipkow@17501: apply simp nipkow@17501: apply (case_tac i) nipkow@17501: apply simp_all nipkow@17501: apply(blast dest:in_set_takeD) nipkow@17501: done nipkow@17501: nipkow@17501: lemma distinct_drop[simp]: "\i. distinct xs \ distinct (drop i xs)" nipkow@17501: apply(induct xs) nipkow@17501: apply simp nipkow@17501: apply (case_tac i) nipkow@17501: apply simp_all nipkow@17501: done nipkow@17501: nipkow@17501: lemma distinct_list_update: nipkow@17501: assumes d: "distinct xs" and a: "a \ set xs - {xs!i}" nipkow@17501: shows "distinct (xs[i:=a])" nipkow@17501: proof (cases "i < length xs") nipkow@17501: case True nipkow@17501: with a have "a \ set (take i xs @ xs ! i # drop (Suc i) xs) - {xs!i}" nipkow@17501: apply (drule_tac id_take_nth_drop) by simp nipkow@17501: with d True show ?thesis nipkow@17501: apply (simp add: upd_conv_take_nth_drop) nipkow@17501: apply (drule subst [OF id_take_nth_drop]) apply assumption nipkow@17501: apply simp apply (cases "a = xs!i") apply simp by blast nipkow@17501: next nipkow@17501: case False with d show ?thesis by auto nipkow@17501: qed nipkow@17501: nipkow@17501: nipkow@17501: text {* It is best to avoid this indexed version of distinct, but nipkow@17501: sometimes it is useful. *} nipkow@17501: wenzelm@13142: lemma distinct_conv_nth: nipkow@17501: "distinct xs = (\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@17501: apply (erule_tac x = 0 in allE, simp) paulson@14208: apply (erule_tac x = "Suc i" in allE, simp, clarsimp) nipkow@17501: apply (erule_tac x = "Suc i" in allE, simp) paulson@14208: apply (erule_tac x = "Suc j" in allE, simp) nipkow@13145: done wenzelm@13114: nipkow@18490: lemma nth_eq_iff_index_eq: nipkow@18490: "\ distinct xs; i < length xs; j < length xs \ \ (xs!i = xs!j) = (i = j)" nipkow@18490: by(auto simp: distinct_conv_nth) nipkow@18490: 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@18490: nipkow@18490: lemma length_remdups_concat: nipkow@18490: "length(remdups(concat xss)) = card(\xs \ set xss. set xs)" nipkow@18490: by(simp add: distinct_card[symmetric]) nipkow@17906: nipkow@17906: nipkow@15392: subsubsection {* @{text remove1} *} nipkow@15110: nipkow@18049: lemma remove1_append: nipkow@18049: "remove1 x (xs @ ys) = nipkow@18049: (if x \ set xs then remove1 x xs @ ys else xs @ remove1 x ys)" nipkow@18049: by (induct xs) auto nipkow@18049: 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: paulson@17724: lemma set_remove1_eq [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@18049: lemma remove1_filter_not[simp]: nipkow@18049: "\ P x \ remove1 x (filter P xs) = filter P xs" nipkow@18049: by(induct xs) auto nipkow@18049: 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: nipkow@16397: text{* Courtesy of Matthias Daum: *} nipkow@16397: lemma append_replicate_commute: nipkow@16397: "replicate n x @ replicate k x = replicate k x @ replicate n x" nipkow@16397: apply (simp add: replicate_add [THEN sym]) nipkow@16397: apply (simp add: add_commute) nipkow@16397: done nipkow@16397: 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: nipkow@16397: text{* Courtesy of Matthias Daum (2 lemmas): *} nipkow@16397: lemma take_replicate[simp]: "take i (replicate k x) = replicate (min i k) x" nipkow@16397: apply (case_tac "k \ i") nipkow@16397: apply (simp add: min_def) nipkow@16397: apply (drule not_leE) nipkow@16397: apply (simp add: min_def) nipkow@16397: apply (subgoal_tac "replicate k x = replicate i x @ replicate (k - i) x") nipkow@16397: apply simp nipkow@16397: apply (simp add: replicate_add [symmetric]) nipkow@16397: done nipkow@16397: nipkow@16397: lemma drop_replicate[simp]: "!!i. drop i (replicate k x) = replicate (k-i) x" nipkow@16397: apply (induct k) nipkow@16397: apply simp nipkow@16397: apply clarsimp nipkow@16397: apply (case_tac i) nipkow@16397: apply simp nipkow@16397: apply clarsimp nipkow@16397: done nipkow@16397: nipkow@16397: 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@18049: lemma rotate1_rotate_swap: "rotate1 (rotate n xs) = rotate n (rotate1 xs)" nipkow@18049: by(simp add:rotate_def funpow_swap1) nipkow@18049: 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: nipkow@18423: lemma hd_rotate_conv_nth: "xs \ [] \ hd(rotate n xs) = xs!(n mod length xs)" nipkow@18423: apply(simp add:rotate_drop_take hd_append hd_drop_conv_nth hd_conv_nth) nipkow@18423: apply(subgoal_tac "length xs \ 0") nipkow@18423: prefer 2 apply simp nipkow@18423: using mod_less_divisor[of "length xs" n] by arith nipkow@18423: 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@23279: "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 {..s. distinct xs \ nipkow@17501: filter (%x. x \ set(sublist xs s)) xs = sublist xs s" nipkow@17501: proof (induct xs) nipkow@17501: case Nil thus ?case by simp nipkow@17501: next nipkow@17501: case (Cons a xs) nipkow@17501: moreover hence "!x. x: set xs \ x \ a" by auto nipkow@17501: ultimately show ?case by(simp add: sublist_Cons cong:filter_cong) nipkow@17501: qed nipkow@17501: wenzelm@13114: nipkow@19390: subsubsection {* @{const splice} *} nipkow@19390: haftmann@19607: lemma splice_Nil2 [simp, code]: nipkow@19390: "splice xs [] = xs" nipkow@19390: by (cases xs) simp_all nipkow@19390: haftmann@19607: lemma splice_Cons_Cons [simp, code]: nipkow@19390: "splice (x#xs) (y#ys) = x # y # splice xs ys" nipkow@19390: by simp nipkow@19390: haftmann@19607: declare splice.simps(2) [simp del, code del] nipkow@19390: nipkow@22793: lemma length_splice[simp]: "!!ys. length(splice xs ys) = length xs + length ys" nipkow@22793: apply(induct xs) apply simp nipkow@22793: apply(case_tac ys) nipkow@22793: apply auto nipkow@22793: done nipkow@22793: nipkow@22828: subsubsection {* @{const allpairs} *} nipkow@22828: nipkow@22940: lemma allpairs_conv_concat: nipkow@22940: "allpairs f xs ys = concat(map (%x. map (f x) ys) xs)" nipkow@22940: by(induct xs) auto nipkow@22940: nipkow@22828: lemma allpairs_append: nipkow@22830: "allpairs f (xs @ ys) zs = allpairs f xs zs @ allpairs f ys zs" nipkow@22828: by(induct xs) auto nipkow@22828: nipkow@15392: subsubsection {* @{text lists}: the list-forming operator over sets *} nipkow@15302: berghofe@22262: inductive2 berghofe@22262: listsp :: "('a \ bool) \ 'a list \ bool" berghofe@22262: for A :: "'a \ bool" berghofe@22262: where berghofe@22262: Nil [intro!]: "listsp A []" berghofe@22262: | Cons [intro!]: "[| A a; listsp A l |] ==> listsp A (a # l)" berghofe@22262: berghofe@22262: constdefs berghofe@22262: lists :: "'a set => 'a list set" berghofe@22262: "lists A == Collect (listsp (member A))" berghofe@22262: berghofe@22262: lemma listsp_lists_eq [pred_set_conv]: "listsp (member A) = member (lists A)" berghofe@22262: by (simp add: lists_def) berghofe@22262: berghofe@22262: lemmas lists_intros [intro!] = listsp.intros [to_set] berghofe@22262: berghofe@22262: lemmas lists_induct [consumes 1, case_names Nil Cons, induct set: lists] = berghofe@22262: listsp.induct [to_set] berghofe@22262: berghofe@22262: inductive_cases2 listspE [elim!]: "listsp A (x # l)" berghofe@22262: berghofe@22262: lemmas listsE [elim!] = listspE [to_set] berghofe@22262: berghofe@22262: lemma listsp_mono [mono2]: "A \ B ==> listsp A \ listsp B" berghofe@22262: by (clarify, erule listsp.induct, blast+) berghofe@22262: berghofe@22262: lemmas lists_mono [mono] = listsp_mono [to_set] berghofe@22262: haftmann@22422: lemma listsp_infI: haftmann@22422: assumes l: "listsp A l" shows "listsp B l ==> listsp (inf A B) l" using l nipkow@15302: by induct blast+ nipkow@15302: haftmann@22422: lemmas lists_IntI = listsp_infI [to_set] haftmann@22422: haftmann@22422: lemma listsp_inf_eq [simp]: "listsp (inf A B) = inf (listsp A) (listsp B)" haftmann@22422: proof (rule mono_inf [where f=listsp, THEN order_antisym]) berghofe@22262: show "mono listsp" by (simp add: mono_def listsp_mono) haftmann@22422: show "inf (listsp A) (listsp B) \ listsp (inf A B)" by (blast intro: listsp_infI) kleing@14388: qed kleing@14388: haftmann@22422: lemmas listsp_conj_eq [simp] = listsp_inf_eq [simplified inf_fun_eq inf_bool_eq] haftmann@22422: haftmann@22422: lemmas lists_Int_eq [simp] = listsp_inf_eq [to_set] berghofe@22262: berghofe@22262: lemma append_in_listsp_conv [iff]: berghofe@22262: "(listsp A (xs @ ys)) = (listsp A xs \ listsp A ys)" nipkow@15302: by (induct xs) auto nipkow@15302: berghofe@22262: lemmas append_in_lists_conv [iff] = append_in_listsp_conv [to_set] berghofe@22262: berghofe@22262: lemma in_listsp_conv_set: "(listsp A xs) = (\x \ set xs. A x)" berghofe@22262: -- {* eliminate @{text listsp} in favour of @{text set} *} nipkow@15302: by (induct xs) auto nipkow@15302: berghofe@22262: lemmas in_lists_conv_set = in_listsp_conv_set [to_set] berghofe@22262: berghofe@22262: lemma in_listspD [dest!]: "listsp A xs ==> \x\set xs. A x" berghofe@22262: by (rule in_listsp_conv_set [THEN iffD1]) berghofe@22262: berghofe@22262: lemmas in_listsD [dest!] = in_listspD [to_set] berghofe@22262: berghofe@22262: lemma in_listspI [intro!]: "\x\set xs. A x ==> listsp A xs" berghofe@22262: by (rule in_listsp_conv_set [THEN iffD2]) berghofe@22262: berghofe@22262: lemmas in_listsI [intro!] = in_listspI [to_set] nipkow@15302: nipkow@15302: lemma lists_UNIV [simp]: "lists UNIV = UNIV" nipkow@15302: by auto nipkow@15302: nipkow@17086: nipkow@17086: nipkow@17086: subsubsection{* Inductive definition for membership *} nipkow@17086: berghofe@22262: inductive2 ListMem :: "'a \ 'a list \ bool" berghofe@22262: where berghofe@22262: elem: "ListMem x (x # xs)" berghofe@22262: | insert: "ListMem x xs \ ListMem x (y # xs)" berghofe@22262: berghofe@22262: lemma ListMem_iff: "(ListMem x xs) = (x \ set xs)" nipkow@17086: apply (rule iffI) nipkow@17086: apply (induct set: ListMem) nipkow@17086: apply auto nipkow@17086: apply (induct xs) nipkow@17086: apply (auto intro: ListMem.intros) nipkow@17086: done nipkow@17086: nipkow@17086: nipkow@17086: 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: paulson@17724: lemma set_Cons_sing_Nil [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: nipkow@15693: lenlex :: "('a \ 'a) set => ('a list \ 'a list) set" nipkow@15693: "lenlex 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@18423: apply (induct n, simp) 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@15693: lemma wf_lenlex [intro!]: "wf r ==> wf (lenlex r)" nipkow@15693: by (unfold lenlex_def) blast nipkow@15693: nipkow@15693: lemma lenlex_conv: nipkow@15693: "lenlex r = {(xs,ys). length xs < length ys | paulson@15656: length xs = length ys \ (xs, ys) : lex r}" nipkow@19623: by (simp add: lenlex_def diag_def lex_prod_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: paulson@18447: lemma Cons_in_lex [simp]: 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. nipkow@17090: 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: krauss@21103: subsection {* Lexicographic combination of measure functions *} krauss@21103: krauss@21103: text {* These are useful for termination proofs *} krauss@21103: krauss@21103: definition krauss@21103: "measures fs = inv_image (lex less_than) (%a. map (%f. f a) fs)" krauss@21103: krauss@21106: lemma wf_measures[recdef_wf, simp]: "wf (measures fs)" krauss@21103: unfolding measures_def krauss@21103: by blast krauss@21103: krauss@21103: lemma in_measures[simp]: krauss@21103: "(x, y) \ measures [] = False" krauss@21103: "(x, y) \ measures (f # fs) krauss@21103: = (f x < f y \ (f x = f y \ (x, y) \ measures fs))" krauss@21103: unfolding measures_def krauss@21103: by auto krauss@21103: krauss@21103: lemma measures_less: "f x < f y ==> (x, y) \ measures (f#fs)" krauss@21103: by simp krauss@21103: krauss@21103: lemma measures_lesseq: "f x <= f y ==> (x, y) \ measures fs ==> (x, y) \ measures (f#fs)" krauss@21103: by auto krauss@21103: krauss@21211: (* install the lexicographic_order method and the "fun" command *) bulwahn@21131: use "Tools/function_package/lexicographic_order.ML" krauss@21211: use "Tools/function_package/fundef_datatype.ML" krauss@21211: setup LexicographicOrder.setup krauss@21211: setup FundefDatatype.setup krauss@21211: krauss@21103: nipkow@15392: subsubsection{*Lifting a Relation on List Elements to the Lists*} nipkow@15302: berghofe@22262: inductive2 berghofe@22262: list_all2' :: "('a \ 'b \ bool) \ 'a list \ 'b list \ bool" berghofe@22262: for r :: "'a \ 'b \ bool" berghofe@22262: where berghofe@22262: Nil: "list_all2' r [] []" berghofe@22262: | Cons: "[| r x y; list_all2' r xs ys |] ==> list_all2' r (x#xs) (y#ys)" berghofe@22262: berghofe@22262: constdefs berghofe@22262: listrel :: "('a * 'b) set => ('a list * 'b list) set" berghofe@22262: "listrel r == Collect2 (list_all2' (member2 r))" berghofe@22262: berghofe@22262: lemma list_all2_listrel_eq [pred_set_conv]: berghofe@22262: "list_all2' (member2 r) = member2 (listrel r)" berghofe@22262: by (simp add: listrel_def) berghofe@22262: berghofe@22262: lemmas listrel_induct [consumes 1, case_names Nil Cons, induct set: listrel] = berghofe@22262: list_all2'.induct [to_set] berghofe@22262: berghofe@22262: lemmas listrel_intros = list_all2'.intros [to_set] berghofe@22262: berghofe@22262: inductive_cases2 listrel_Nil1 [to_set, elim!]: "list_all2' r [] xs" berghofe@22262: inductive_cases2 listrel_Nil2 [to_set, elim!]: "list_all2' r xs []" berghofe@22262: inductive_cases2 listrel_Cons1 [to_set, elim!]: "list_all2' r (y#ys) xs" berghofe@22262: inductive_cases2 listrel_Cons2 [to_set, elim!]: "list_all2' r xs (y#ys)" nipkow@15302: nipkow@15302: nipkow@15302: lemma listrel_mono: "r \ s \ listrel r \ listrel s" nipkow@15302: apply clarify berghofe@22262: apply (erule listrel_induct) berghofe@22262: 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 berghofe@22262: 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) berghofe@22262: 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) berghofe@22262: apply (erule listrel_induct) berghofe@22262: 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) berghofe@22262: apply (erule listrel_induct) berghofe@22262: 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 `` {[]} = {[]}" berghofe@22262: 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})"; berghofe@22262: 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@21754: setup StringSyntax.setup wenzelm@13366: haftmann@20453: haftmann@21061: subsection {* Code generator *} haftmann@21061: haftmann@21061: subsubsection {* Setup *} berghofe@15064: berghofe@16770: types_code berghofe@16770: "list" ("_ list") berghofe@16770: attach (term_of) {* wenzelm@21760: fun term_of_list f T = HOLogic.mk_list T o map f; berghofe@16770: *} berghofe@16770: attach (test) {* 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@16770: *} berghofe@16770: "char" ("string") berghofe@16770: attach (term_of) {* haftmann@21455: val term_of_char = HOLogic.mk_char; berghofe@16770: *} berghofe@16770: attach (test) {* berghofe@15064: fun gen_char i = chr (random_range (ord "a") (Int.min (ord "a" + i, ord "z"))); berghofe@15064: *} berghofe@15064: berghofe@15064: consts_code "Cons" ("(_ ::/ _)") berghofe@15064: haftmann@20453: code_type list haftmann@20453: (SML "_ list") haftmann@21911: (OCaml "_ list") haftmann@21113: (Haskell "![_]") haftmann@20453: haftmann@22799: code_reserved SML haftmann@22799: list haftmann@22799: haftmann@22799: code_reserved OCaml haftmann@22799: list haftmann@22799: haftmann@20453: code_const Nil haftmann@21113: (SML "[]") haftmann@21911: (OCaml "[]") haftmann@21113: (Haskell "[]") haftmann@20453: haftmann@21911: setup {* haftmann@22799: fold (fn target => CodegenSerializer.add_pretty_list target haftmann@22799: @{const_name Nil} @{const_name Cons} haftmann@22799: ) ["SML", "OCaml", "Haskell"] haftmann@21911: *} haftmann@21911: haftmann@22799: code_instance list :: eq haftmann@22799: (Haskell -) haftmann@20588: haftmann@21455: code_const "op = \ 'a\eq list \ 'a list \ bool" haftmann@20588: (Haskell infixl 4 "==") haftmann@20588: haftmann@20453: setup {* haftmann@20453: let haftmann@20453: haftmann@20453: fun list_codegen thy defs gr dep thyname b t = haftmann@20453: let val (gr', ps) = foldl_map (Codegen.invoke_codegen thy defs dep thyname false) haftmann@20453: (gr, HOLogic.dest_list t) haftmann@20453: in SOME (gr', Pretty.list "[" "]" ps) end handle TERM _ => NONE; haftmann@20453: haftmann@20453: fun char_codegen thy defs gr dep thyname b t = berghofe@22539: case Option.map chr (try HOLogic.dest_char t) of berghofe@22539: SOME c => SOME (gr, Pretty.quote (Pretty.str (ML_Syntax.print_char c))) haftmann@20453: | NONE => NONE; haftmann@20453: haftmann@20453: in haftmann@20453: Codegen.add_codegen "list_codegen" list_codegen haftmann@20453: #> Codegen.add_codegen "char_codegen" char_codegen haftmann@20453: end; haftmann@20453: *} berghofe@15064: haftmann@21061: haftmann@21061: subsubsection {* Generation of efficient code *} haftmann@21061: haftmann@21061: consts haftmann@21079: memberl :: "'a \ 'a list \ bool" (infixl "mem" 55) haftmann@21061: null:: "'a list \ bool" haftmann@21061: list_inter :: "'a list \ 'a list \ 'a list" haftmann@21061: list_ex :: "('a \ bool) \ 'a list \ bool" haftmann@21061: list_all :: "('a \ bool) \ ('a list \ bool)" haftmann@21061: filtermap :: "('a \ 'b option) \ 'a list \ 'b list" haftmann@21061: map_filter :: "('a \ 'b) \ ('a \ bool) \ 'a list \ 'b list" haftmann@21061: haftmann@21061: primrec haftmann@21061: "x mem [] = False" haftmann@21079: "x mem (y#ys) = (x = y \ x mem ys)" haftmann@21061: haftmann@21061: primrec haftmann@21061: "null [] = True" haftmann@21061: "null (x#xs) = False" haftmann@21061: haftmann@21061: primrec haftmann@21061: "list_inter [] bs = []" haftmann@21061: "list_inter (a#as) bs = haftmann@21061: (if a \ set bs then a # list_inter as bs else list_inter as bs)" haftmann@21061: haftmann@21061: primrec haftmann@21061: "list_all P [] = True" haftmann@21061: "list_all P (x#xs) = (P x \ list_all P xs)" haftmann@21061: haftmann@21061: primrec haftmann@21061: "list_ex P [] = False" haftmann@21061: "list_ex P (x#xs) = (P x \ list_ex P xs)" haftmann@21061: haftmann@21061: primrec haftmann@21061: "filtermap f [] = []" haftmann@21061: "filtermap f (x#xs) = haftmann@21061: (case f x of None \ filtermap f xs haftmann@21061: | Some y \ y # filtermap f xs)" haftmann@21061: haftmann@21061: primrec haftmann@21061: "map_filter f P [] = []" haftmann@21061: "map_filter f P (x#xs) = haftmann@21061: (if P x then f x # map_filter f P xs else map_filter f P xs)" haftmann@21061: nipkow@23096: haftmann@21061: text {* wenzelm@21754: Only use @{text mem} for generating executable code. Otherwise use wenzelm@21754: @{prop "x : set xs"} instead --- it is much easier to reason about. haftmann@21061: The same is true for @{const list_all} and @{const list_ex}: write haftmann@21061: @{text "\x\set xs"} and @{text "\x\set xs"} instead because the HOL wenzelm@21754: quantifiers are aleady known to the automatic provers. In fact, the wenzelm@21754: declarations in the code subsection make sure that @{text "\"}, wenzelm@21754: @{text "\x\set xs"} and @{text "\x\set xs"} are implemented wenzelm@21754: efficiently. haftmann@21061: haftmann@21061: Efficient emptyness check is implemented by @{const null}. haftmann@21061: haftmann@23060: The functions @{const filtermap} and @{const map_filter} are just haftmann@23060: there to generate efficient code. Do not use wenzelm@21754: them for modelling and proving. haftmann@21061: *} haftmann@21061: haftmann@23060: lemma rev_foldl_cons [code]: haftmann@23060: "rev xs = foldl (\xs x. x # xs) [] xs" haftmann@23060: proof (induct xs) haftmann@23060: case Nil then show ?case by simp haftmann@23060: next haftmann@23060: case Cons haftmann@23060: { haftmann@23060: fix x xs ys haftmann@23060: have "foldl (\xs x. x # xs) ys xs @ [x] haftmann@23060: = foldl (\xs x. x # xs) (ys @ [x]) xs" haftmann@23060: by (induct xs arbitrary: ys) auto haftmann@23060: } haftmann@23060: note aux = this haftmann@23060: show ?case by (induct xs) (auto simp add: Cons aux) haftmann@23060: qed haftmann@23060: haftmann@21061: lemma mem_iff [normal post]: haftmann@22422: "x mem xs \ x \ set xs" haftmann@21061: by (induct xs) auto haftmann@21061: haftmann@22799: lemmas in_set_code [code unfold] = mem_iff [symmetric] haftmann@21061: haftmann@21061: lemma empty_null [code inline]: haftmann@22422: "xs = [] \ null xs" haftmann@21061: by (cases xs) simp_all haftmann@21061: haftmann@21061: lemmas null_empty [normal post] = haftmann@21061: empty_null [symmetric] haftmann@21061: haftmann@21061: lemma list_inter_conv: haftmann@21061: "set (list_inter xs ys) = set xs \ set ys" haftmann@21061: by (induct xs) auto haftmann@21061: haftmann@21061: lemma list_all_iff [normal post]: haftmann@22422: "list_all P xs \ (\x \ set xs. P x)" haftmann@21061: by (induct xs) auto haftmann@21061: haftmann@22799: lemmas list_ball_code [code unfold] = list_all_iff [symmetric] haftmann@21061: haftmann@21061: lemma list_all_append [simp]: haftmann@22422: "list_all P (xs @ ys) \ (list_all P xs \ list_all P ys)" haftmann@21061: by (induct xs) auto haftmann@21061: haftmann@21061: lemma list_all_rev [simp]: haftmann@22422: "list_all P (rev xs) \ list_all P xs" haftmann@21061: by (simp add: list_all_iff) haftmann@21061: haftmann@22506: lemma list_all_length: haftmann@22506: "list_all P xs \ (\n < length xs. P (xs ! n))" haftmann@22506: unfolding list_all_iff by (auto intro: all_nth_imp_all_set) haftmann@22506: haftmann@21061: lemma list_ex_iff [normal post]: haftmann@22422: "list_ex P xs \ (\x \ set xs. P x)" haftmann@21061: by (induct xs) simp_all haftmann@21061: haftmann@21061: lemmas list_bex_code [code unfold] = haftmann@22799: list_ex_iff [symmetric] haftmann@21061: haftmann@22506: lemma list_ex_length: haftmann@22506: "list_ex P xs \ (\n < length xs. P (xs ! n))" haftmann@22506: unfolding list_ex_iff set_conv_nth by auto haftmann@22506: haftmann@21061: lemma filtermap_conv: haftmann@21061: "filtermap f xs = map (\x. the (f x)) (filter (\x. f x \ None) xs)" haftmann@21061: by (induct xs) (simp_all split: option.split) haftmann@21061: haftmann@21061: lemma map_filter_conv [simp]: haftmann@21061: "map_filter f P xs = map f (filter P xs)" haftmann@21061: by (induct xs) auto haftmann@21061: nipkow@23096: lemma [code inline]: "listsum (map f xs) = foldl (%n x. n + f x) 0 xs" nipkow@23096: by(simp add:listsum_foldr foldl_map[symmetric] foldl_foldr1) nipkow@23096: nipkow@23096: nipkow@23096: text {* Code for bounded quantification over nats. *} haftmann@21891: haftmann@22799: lemma atMost_upto [code unfold]: haftmann@21891: "{..n} = set [0..n]" haftmann@21891: by auto haftmann@22799: haftmann@22799: lemma atLeast_upt [code unfold]: haftmann@21891: "{..mnat. P m) \ (\m \ {0..mnat. P m) \ (\m \ {0..m\n\nat. P m) \ (\m \ {0..n}. P m)" haftmann@21891: by auto haftmann@22799: haftmann@22799: lemma ex_nat_less [code unfold]: haftmann@21891: "(\m\n\nat. P m) \ (\m \ {0..n}. P m)" haftmann@21891: by auto haftmann@22799: chaieb@23245: lemma foldl_append_append: "\ ss ss'. foldl (op @) (ss@ss') xs = ss@(foldl (op @) ss' xs)" chaieb@23245: by (induct xs, simp_all) chaieb@23245: chaieb@23245: lemma foldl_append_map_set: "\ ss. set (foldl (op @) ss (map f xs)) = set ss \ UNION (set xs) (\x. set (f x))" chaieb@23245: proof(induct xs) chaieb@23245: case Nil thus ?case by simp chaieb@23245: next chaieb@23245: case (Cons x xs ss) chaieb@23245: have "set (foldl op @ ss (map f (x # xs))) = set (foldl op @ (ss @ f x) (map f xs))" by simp chaieb@23245: also have "\ = set (ss@ f x) \ UNION (set xs) (\x. set (f x))" using prems by simp chaieb@23245: also have "\= set ss \ set (f x) \ UNION (set xs) (\x. set (f x))" by simp chaieb@23245: also have "\ = set ss \ UNION (set (x#xs)) (\x. set (f x))" chaieb@23245: by (simp add: Un_assoc) chaieb@23245: finally show ?case by simp chaieb@23245: qed chaieb@23245: chaieb@23245: lemma foldl_append_map_Nil_set: chaieb@23245: "set (foldl (op @) [] (map f xs)) = UNION (set xs) (\x. set (f x))" chaieb@23245: using foldl_append_map_set[where ss="[]" and xs="xs" and f="f"] by simp chaieb@23245: chaieb@23246: primrec chaieb@23246: "partition P [] = ([],[])" chaieb@23246: "partition P (x#xs) = chaieb@23246: (let (yes,no) = partition P xs chaieb@23246: in (if (P x) then ((x#yes),no) else (yes,(x#no))))" chaieb@23246: chaieb@23246: lemma partition_P: chaieb@23246: "partition P xs = (yes,no) \ (\p\ set yes. P p) \ (\p\ set no. \ P p)" chaieb@23246: proof(induct xs arbitrary: yes no rule: partition.induct) chaieb@23246: case (Cons a as yes no) chaieb@23246: have "\ y n. partition P as = (y,n)" by auto chaieb@23246: then obtain "y" "n" where yn_def: "partition P as = (y,n)" by blast chaieb@23246: have "P a \ \ P a" by simp chaieb@23246: moreover chaieb@23246: { assume "P a" chaieb@23246: hence "partition P (a#as) = (a#y,n)" chaieb@23246: by (auto simp add: Let_def yn_def) chaieb@23246: hence "yes = a#y" using prems by auto chaieb@23246: with prems have ?case by simp chaieb@23246: } chaieb@23246: moreover chaieb@23246: { assume "\ P a" chaieb@23246: hence "partition P (a#as) = (y,a#n)" chaieb@23246: by (auto simp add: Let_def yn_def) chaieb@23246: hence "no = a#n" using prems by auto chaieb@23246: with prems have ?case by simp chaieb@23246: } chaieb@23246: ultimately show ?case by blast chaieb@23246: qed simp_all chaieb@23246: chaieb@23246: lemma partition_filter1: chaieb@23246: " fst (partition P xs) = filter P xs " chaieb@23246: by (induct xs rule: partition.induct) (auto simp add: Let_def split_def) chaieb@23246: chaieb@23246: lemma partition_filter2: chaieb@23246: "snd (partition P xs) = filter (Not o P ) xs " chaieb@23246: by (induct xs rule: partition.induct) (auto simp add: Let_def split_def) chaieb@23246: chaieb@23246: lemma partition_set: "partition P xs = (yes,no) \ set yes \ set no = set xs" chaieb@23246: proof- chaieb@23246: fix yes no chaieb@23246: assume A: "partition P xs = (yes,no)" chaieb@23246: have "set xs = {x. x \ set xs \ P x} \ {x. x \ set xs \ \ P x}" by auto chaieb@23246: also have "\ = set (List.filter P xs) Un (set (List.filter (Not o P) xs))" by auto chaieb@23246: also have "\ = set (fst (partition P xs)) Un set (snd (partition P xs))" chaieb@23246: using partition_filter1[where xs="xs" and P="P"] chaieb@23246: partition_filter2[where xs="xs" and P="P"] by auto chaieb@23246: finally show "set yes Un set no = set xs" using A by simp chaieb@23246: qed chaieb@23246: haftmann@22799: end