
wenzelm@13462: (*  Title:      HOL/List.thy
wenzelm@13462:     ID:         $Id$
wenzelm@13462:     Author:     Tobias Nipkow
clasohm@923: *)
clasohm@923: 
wenzelm@13114: header {* The datatype of finite lists *}
wenzelm@13122: 
nipkow@15131: theory List
nipkow@15140: imports PreList
nipkow@15131: begin
clasohm@923: 
wenzelm@13142: datatype 'a list =
wenzelm@13366:     Nil    ("[]")
wenzelm@13366:   | Cons 'a  "'a list"    (infixr "#" 65)
clasohm@923: 
clasohm@923: consts
wenzelm@13366:   "@" :: "'a list => 'a list => 'a list"    (infixr 65)
wenzelm@13366:   filter:: "('a => bool) => 'a list => 'a list"
wenzelm@13366:   concat:: "'a list list => 'a list"
wenzelm@13366:   foldl :: "('b => 'a => 'b) => 'b => 'a list => 'b"
wenzelm@13366:   foldr :: "('a => 'b => 'b) => 'a list => 'b => 'b"
wenzelm@13366:   hd:: "'a list => 'a"
wenzelm@13366:   tl:: "'a list => 'a list"
wenzelm@13366:   last:: "'a list => 'a"
wenzelm@13366:   butlast :: "'a list => 'a list"
wenzelm@13366:   set :: "'a list => 'a set"
wenzelm@13366:   list_all:: "('a => bool) => ('a list => bool)"
wenzelm@13366:   list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool"
wenzelm@13366:   map :: "('a=>'b) => ('a list => 'b list)"
wenzelm@13366:   mem :: "'a => 'a list => bool"    (infixl 55)
wenzelm@13366:   nth :: "'a list => nat => 'a"    (infixl "!" 100)
wenzelm@13366:   list_update :: "'a list => nat => 'a => 'a list"
wenzelm@13366:   take:: "nat => 'a list => 'a list"
wenzelm@13366:   drop:: "nat => 'a list => 'a list"
wenzelm@13366:   takeWhile :: "('a => bool) => 'a list => 'a list"
wenzelm@13366:   dropWhile :: "('a => bool) => 'a list => 'a list"
wenzelm@13366:   rev :: "'a list => 'a list"
wenzelm@13366:   zip :: "'a list => 'b list => ('a * 'b) list"
wenzelm@13366:   upt :: "nat => nat => nat list" ("(1[_../_'(])")
wenzelm@13366:   remdups :: "'a list => 'a list"
nipkow@15110:   remove1 :: "'a => 'a list => 'a list"
wenzelm@13366:   null:: "'a list => bool"
wenzelm@13366:   "distinct":: "'a list => bool"
wenzelm@13366:   replicate :: "nat => 'a => 'a list"
clasohm@923: 
nipkow@13146: nonterminals lupdbinds lupdbind
nipkow@5077: 
clasohm@923: syntax
wenzelm@13366:   -- {* list Enumeration *}
wenzelm@13366:   "@list" :: "args => 'a list"    ("[(_)]")
clasohm@923: 
wenzelm@13366:   -- {* Special syntax for filter *}
wenzelm@13366:   "@filter" :: "[pttrn, 'a list, bool] => 'a list"    ("(1[_:_./ _])")
clasohm@923: 
wenzelm@13366:   -- {* list update *}
wenzelm@13366:   "_lupdbind":: "['a, 'a] => lupdbind"    ("(2_ :=/ _)")
wenzelm@13366:   "" :: "lupdbind => lupdbinds"    ("_")
wenzelm@13366:   "_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds"    ("_,/ _")
wenzelm@13366:   "_LUpdate" :: "['a, lupdbinds] => 'a"    ("_/[(_)]" [900,0] 900)
nipkow@5077: 
wenzelm@13366:   upto:: "nat => nat => nat list"    ("(1[_../_])")
nipkow@5427: 
clasohm@923: translations
wenzelm@13366:   "[x, xs]" == "x#[xs]"
wenzelm@13366:   "[x]" == "x#[]"
wenzelm@13366:   "[x:xs . P]"== "filter (%x. P) xs"
clasohm@923: 
wenzelm@13366:   "_LUpdate xs (_lupdbinds b bs)"== "_LUpdate (_LUpdate xs b) bs"
wenzelm@13366:   "xs[i:=x]" == "list_update xs i x"
nipkow@5077: 
wenzelm@13366:   "[i..j]" == "[i..(Suc j)(]"
nipkow@5427: 
nipkow@5427: 
wenzelm@12114: syntax (xsymbols)
wenzelm@13366:   "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])")
kleing@14565: syntax (HTML output)
kleing@14565:   "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])")
paulson@3342: 
paulson@3342: 
wenzelm@13142: text {*
wenzelm@14589:   Function @{text size} is overloaded for all datatypes. Users may
wenzelm@13366:   refer to the list version as @{text length}. *}
wenzelm@13142: 
wenzelm@13142: syntax length :: "'a list => nat"
wenzelm@13142: translations "length" => "size :: _ list => nat"
wenzelm@13114: 
wenzelm@13142: typed_print_translation {*
wenzelm@13366:   let
wenzelm@13366:     fun size_tr' _ (Type ("fun", (Type ("list", _) :: _))) [t] =
wenzelm@13366:           Syntax.const "length" $ t
wenzelm@13366:       | size_tr' _ _ _ = raise Match;
wenzelm@13366:   in [("size", size_tr')] end
wenzelm@13114: *}
paulson@3437: 
berghofe@5183: primrec
nipkow@13145: "hd(x#xs) = x"
berghofe@5183: primrec
nipkow@13145: "tl([]) = []"
nipkow@13145: "tl(x#xs) = xs"
berghofe@5183: primrec
nipkow@13145: "null([]) = True"
nipkow@13145: "null(x#xs) = False"
paulson@8972: primrec
nipkow@13145: "last(x#xs) = (if xs=[] then x else last xs)"
berghofe@5183: primrec
nipkow@13145: "butlast []= []"
nipkow@13145: "butlast(x#xs) = (if xs=[] then [] else x#butlast xs)"
berghofe@5183: primrec
nipkow@13145: "x mem [] = False"
nipkow@13145: "x mem (y#ys) = (if y=x then True else x mem ys)"
oheimb@5518: primrec
nipkow@13145: "set [] = {}"
nipkow@13145: "set (x#xs) = insert x (set xs)"
berghofe@5183: primrec
nipkow@13145: list_all_Nil:"list_all P [] = True"
nipkow@13145: list_all_Cons: "list_all P (x#xs) = (P(x) \<and> list_all P xs)"
oheimb@5518: primrec
nipkow@13145: "map f [] = []"
nipkow@13145: "map f (x#xs) = f(x)#map f xs"
berghofe@5183: primrec
nipkow@13145: append_Nil:"[]@ys = ys"
nipkow@13145: append_Cons: "(x#xs)@ys = x#(xs@ys)"
berghofe@5183: primrec
nipkow@13145: "rev([]) = []"
nipkow@13145: "rev(x#xs) = rev(xs) @ [x]"
berghofe@5183: primrec
nipkow@13145: "filter P [] = []"
nipkow@13145: "filter P (x#xs) = (if P x then x#filter P xs else filter P xs)"
berghofe@5183: primrec
nipkow@13145: foldl_Nil:"foldl f a [] = a"
nipkow@13145: foldl_Cons: "foldl f a (x#xs) = foldl f (f a x) xs"
berghofe@5183: primrec
nipkow@13145: "foldr f [] a = a"
nipkow@13145: "foldr f (x#xs) a = f x (foldr f xs a)"
paulson@8000: primrec
nipkow@13145: "concat([]) = []"
nipkow@13145: "concat(x#xs) = x @ concat(xs)"
berghofe@5183: primrec
nipkow@13145: drop_Nil:"drop n [] = []"
nipkow@13145: drop_Cons: "drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)"
nipkow@13145: -- {* Warning: simpset does not contain this definition *}
nipkow@13145: -- {* but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
berghofe@5183: primrec
nipkow@13145: take_Nil:"take n [] = []"
nipkow@13145: take_Cons: "take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)"
nipkow@13145: -- {* Warning: simpset does not contain this definition *}
nipkow@13145: -- {* but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
berghofe@5183: primrec
nipkow@13145: nth_Cons:"(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)"
nipkow@13145: -- {* Warning: simpset does not contain this definition *}
nipkow@13145: -- {* but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
wenzelm@13142: primrec
nipkow@13145: "[][i:=v] = []"
nipkow@13145: "(x#xs)[i:=v] =
nipkow@13145: (case i of 0 => v # xs
nipkow@13145: | Suc j => x # xs[j:=v])"
berghofe@5183: primrec
nipkow@13145: "takeWhile P [] = []"
nipkow@13145: "takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])"
berghofe@5183: primrec
nipkow@13145: "dropWhile P [] = []"
nipkow@13145: "dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)"
berghofe@5183: primrec
nipkow@13145: "zip xs [] = []"
nipkow@13145: zip_Cons: "zip xs (y#ys) = (case xs of [] => [] | z#zs => (z,y)#zip zs ys)"
nipkow@13145: -- {* Warning: simpset does not contain this definition *}
nipkow@13145: -- {* but separate theorems for @{text "xs = []"} and @{text "xs = z # zs"} *}
nipkow@5427: primrec
nipkow@13145: upt_0: "[i..0(] = []"
nipkow@13145: upt_Suc: "[i..(Suc j)(] = (if i <= j then [i..j(] @ [j] else [])"
berghofe@5183: primrec
nipkow@13145: "distinct [] = True"
nipkow@13145: "distinct (x#xs) = (x ~: set xs \<and> distinct xs)"
berghofe@5183: primrec
nipkow@13145: "remdups [] = []"
nipkow@13145: "remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)"
berghofe@5183: primrec
nipkow@15110: "remove1 x [] = []"
nipkow@15110: "remove1 x (y#xs) = (if x=y then xs else y # remove1 x xs)"
nipkow@15110: primrec
nipkow@13147: replicate_0: "replicate 0 x = []"
nipkow@13145: replicate_Suc: "replicate (Suc n) x = x # replicate n x"
nipkow@8115: defs
wenzelm@13114:  list_all2_def:
wenzelm@13142:  "list_all2 P xs ys == length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y)"
nipkow@8115: 
paulson@3196: 
wenzelm@13142: subsection {* Lexicographic orderings on lists *}
nipkow@5281: 
nipkow@5281: consts
nipkow@13145: lexn :: "('a * 'a)set => nat => ('a list * 'a list)set"
nipkow@5281: primrec
nipkow@13145: "lexn r 0 = {}"
nipkow@13145: "lexn r (Suc n) =
nipkow@13145: (prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs) ` (r <*lex*> lexn r n)) Int
nipkow@13145: {(xs,ys). length xs = Suc n \<and> length ys = Suc n}"
nipkow@5281: 
nipkow@5281: constdefs
nipkow@13145: lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
nipkow@13145: "lex r == \<Union>n. lexn r n"
nipkow@5281: 
nipkow@13145: lexico :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
nipkow@13145: "lexico r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))"
paulson@9336: 
nipkow@13145: sublist :: "'a list => nat set => 'a list"
nipkow@13145: "sublist xs A == map fst (filter (%p. snd p : A) (zip xs [0..size xs(]))"
nipkow@5281: 
wenzelm@13114: 
wenzelm@13142: lemma not_Cons_self [simp]: "xs \<noteq> 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 \<noteq> []) = (\<exists>y ys. xs = y # ys)"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma length_induct:
nipkow@13145: "(!!xs. \<forall>ys. length ys < length xs --> P ys ==> P xs) ==> P xs"
nipkow@13145: by (rule measure_induct [of length]) rules
wenzelm@13114: 
wenzelm@13114: 
wenzelm@13142: subsection {* @{text lists}: the list-forming operator over sets *}
wenzelm@13114: 
wenzelm@13142: consts lists :: "'a set => 'a list set"
wenzelm@13142: inductive "lists A"
paulson@15168:  intros
paulson@15168:   Nil [intro!]: "[]: lists A"
paulson@15168:   Cons [intro!]: "[| a: A;l: lists A|] ==> a#l : lists A"
wenzelm@13114: 
wenzelm@13142: inductive_cases listsE [elim!]: "x#l : lists A"
wenzelm@13114: 
wenzelm@13366: lemma lists_mono [mono]: "A \<subseteq> B ==> lists A \<subseteq> lists B"
nipkow@13145: by (unfold lists.defs) (blast intro!: lfp_mono)
wenzelm@13114: 
berghofe@13883: lemma lists_IntI:
berghofe@13883:   assumes l: "l: lists A" shows "l: lists B ==> l: lists (A Int B)" using l
berghofe@13883:   by induct blast+
wenzelm@13142: 
wenzelm@13142: lemma lists_Int_eq [simp]: "lists (A \<inter> B) = lists A \<inter> lists B"
paulson@15113: proof (rule mono_Int [THEN equalityI])
paulson@15113:   show "mono lists" by (simp add: mono_def lists_mono)
paulson@15113:   show "lists A \<inter> lists B \<subseteq> lists (A \<inter> B)" by (blast intro: lists_IntI)
paulson@15113: qed
wenzelm@13114: 
wenzelm@13142: lemma append_in_lists_conv [iff]:
paulson@15113:      "(xs @ ys : lists A) = (xs : lists A \<and> ys : lists A)"
nipkow@13145: by (induct xs) auto
wenzelm@13142: 
wenzelm@13142: 
wenzelm@13142: subsection {* @{text length} *}
wenzelm@13114: 
wenzelm@13142: text {*
nipkow@13145: Needs to come before @{text "@"} because of theorem @{text
nipkow@13145: append_eq_append_conv}.
wenzelm@13142: *}
wenzelm@13114: 
wenzelm@13142: lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma length_map [simp]: "length (map f xs) = length xs"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma length_rev [simp]: "length (rev xs) = length xs"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma length_tl [simp]: "length (tl xs) = length xs - 1"
nipkow@13145: by (cases xs) auto
wenzelm@13114: 
wenzelm@13142: lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13114: lemma length_Suc_conv:
nipkow@13145: "(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
nipkow@13145: by (induct xs) auto
wenzelm@13142: 
nipkow@14025: lemma Suc_length_conv:
nipkow@14025: "(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
paulson@14208: apply (induct xs, simp, simp)
nipkow@14025: apply blast
nipkow@14025: done
nipkow@14025: 
oheimb@14099: lemma impossible_Cons [rule_format]: 
oheimb@14099:   "length xs <= length ys --> xs = x # ys = False"
paulson@14208: apply (induct xs, auto)
oheimb@14099: done
oheimb@14099: 
nipkow@14247: lemma list_induct2[consumes 1]: "\<And>ys.
nipkow@14247:  \<lbrakk> length xs = length ys;
nipkow@14247:    P [] [];
nipkow@14247:    \<And>x xs y ys. \<lbrakk> length xs = length ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
nipkow@14247:  \<Longrightarrow> 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: 
wenzelm@13142: subsection {* @{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 = [] \<and> ys = [])"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> 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 \<or> length us = length vs
berghofe@13883:  ==> (xs@us = ys@vs) = (xs=ys \<and> 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 \<and> 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 \<noteq> [] ==> 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 \<noteq> [] ==> 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 \<noteq> [] ==> 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@14300: 
wenzelm@13142: text {* Trivial rules for solving @{text "@"}-equations automatically. *}
wenzelm@13114: 
wenzelm@13114: lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
nipkow@13145: by simp
wenzelm@13114: 
wenzelm@13142: lemma Cons_eq_appendI:
nipkow@13145: "[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
nipkow@13145: by (drule sym) simp
wenzelm@13114: 
wenzelm@13142: lemma append_eq_appendI:
nipkow@13145: "[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
nipkow@13145: by (drule sym) simp
wenzelm@13114: 
wenzelm@13114: 
wenzelm@13142: text {*
nipkow@13145: Simplification procedure for all list equalities.
nipkow@13145: Currently only tries to rearrange @{text "@"} to see if
nipkow@13145: - both lists end in a singleton list,
nipkow@13145: - or both lists end in the same list.
wenzelm@13142: *}
wenzelm@13142: 
wenzelm@13142: ML_setup {*
nipkow@3507: local
nipkow@3507: 
wenzelm@13122: val append_assoc = thm "append_assoc";
wenzelm@13122: val append_Nil = thm "append_Nil";
wenzelm@13122: val append_Cons = thm "append_Cons";
wenzelm@13122: val append1_eq_conv = thm "append1_eq_conv";
wenzelm@13122: val append_same_eq = thm "append_same_eq";
wenzelm@13122: 
wenzelm@13114: fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
wenzelm@13462:   (case xs of Const("List.list.Nil",_) => cons | _ => last xs)
wenzelm@13462:   | last (Const("List.op @",_) $ _ $ ys) = last ys
wenzelm@13462:   | last t = t;
wenzelm@13114: 
wenzelm@13114: fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
wenzelm@13462:   | list1 _ = false;
wenzelm@13114: 
wenzelm@13114: fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
wenzelm@13462:   (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs)
wenzelm@13462:   | butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys
wenzelm@13462:   | butlast xs = Const("List.list.Nil",fastype_of xs);
wenzelm@13114: 
wenzelm@13114: val rearr_tac =
wenzelm@13462:   simp_tac (HOL_basic_ss addsimps [append_assoc, append_Nil, append_Cons]);
wenzelm@13114: 
wenzelm@13114: fun list_eq sg _ (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
wenzelm@13462:   let
wenzelm@13462:     val lastl = last lhs and lastr = last rhs;
wenzelm@13462:     fun rearr conv =
wenzelm@13462:       let
wenzelm@13462:         val lhs1 = butlast lhs and rhs1 = butlast rhs;
wenzelm@13462:         val Type(_,listT::_) = eqT
wenzelm@13462:         val appT = [listT,listT] ---> listT
wenzelm@13462:         val app = Const("List.op @",appT)
wenzelm@13462:         val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
wenzelm@13480:         val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
wenzelm@13480:         val thm = Tactic.prove sg [] [] eq (K (rearr_tac 1));
wenzelm@13462:       in Some ((conv RS (thm RS trans)) RS eq_reflection) end;
wenzelm@13114: 
wenzelm@13462:   in
wenzelm@13462:     if list1 lastl andalso list1 lastr then rearr append1_eq_conv
wenzelm@13462:     else if lastl aconv lastr then rearr append_same_eq
wenzelm@13462:     else None
wenzelm@13462:   end;
wenzelm@13462: 
wenzelm@13114: in
wenzelm@13462: 
wenzelm@13462: val list_eq_simproc =
wenzelm@13462:   Simplifier.simproc (Theory.sign_of (the_context ())) "list_eq" ["(xs::'a list) = ys"] list_eq;
wenzelm@13462: 
wenzelm@13114: end;
wenzelm@13114: 
wenzelm@13114: Addsimprocs [list_eq_simproc];
wenzelm@13114: *}
wenzelm@13114: 
wenzelm@13114: 
wenzelm@13142: subsection {* @{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 (\<lambda>x. x) = (\<lambda>xs. xs)"
nipkow@13145: by (rule ext, induct_tac xs) auto
wenzelm@13114: 
wenzelm@13142: lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma map_compose: "map (f o g) xs = map f (map g xs)"
nipkow@13145: by (induct xs) (auto simp add: o_def)
wenzelm@13114: 
wenzelm@13142: lemma rev_map: "rev (map f xs) = map f (rev xs)"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
nipkow@13737: lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
nipkow@13737: by (induct xs) auto
nipkow@13737: 
wenzelm@13366: lemma map_cong [recdef_cong]:
nipkow@13145: "xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"
nipkow@13145: -- {* a congruence rule for @{text map} *}
nipkow@13737: by simp
wenzelm@13114: 
wenzelm@13142: lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
nipkow@13145: by (cases xs) auto
wenzelm@13114: 
wenzelm@13142: lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
nipkow@13145: by (cases xs) auto
wenzelm@13114: 
nipkow@14025: lemma map_eq_Cons_conv[iff]:
nipkow@14025:  "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
nipkow@13145: by (cases xs) auto
wenzelm@13114: 
nipkow@14025: lemma Cons_eq_map_conv[iff]:
nipkow@14025:  "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
nipkow@14025: by (cases ys) auto
nipkow@14025: 
nipkow@14111: lemma ex_map_conv:
nipkow@14111:   "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
nipkow@14111: by(induct ys, auto)
nipkow@14111: 
nipkow@15110: lemma map_eq_imp_length_eq:
nipkow@15110:   "!!xs. map f xs = map f ys ==> length xs = length ys"
nipkow@15110: apply (induct ys)
nipkow@15110:  apply simp
nipkow@15110: apply(simp (no_asm_use))
nipkow@15110: apply clarify
nipkow@15110: apply(simp (no_asm_use))
nipkow@15110: apply fast
nipkow@15110: done
nipkow@15110: 
nipkow@15110: lemma map_inj_on:
nipkow@15110:  "[| map f xs = map f ys; inj_on f (set xs Un set ys) |]
nipkow@15110:   ==> xs = ys"
nipkow@15110: apply(frule map_eq_imp_length_eq)
nipkow@15110: apply(rotate_tac -1)
nipkow@15110: apply(induct rule:list_induct2)
nipkow@15110:  apply simp
nipkow@15110: apply(simp)
nipkow@15110: apply (blast intro:sym)
nipkow@15110: done
nipkow@15110: 
nipkow@15110: lemma inj_on_map_eq_map:
nipkow@15110:  "inj_on f (set xs Un set ys) \<Longrightarrow> (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 \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@14339: by(blast dest:map_injective)
nipkow@14339: 
wenzelm@13114: lemma inj_mapI: "inj f ==> inj (map f)"
paulson@13585: by (rules dest: map_injective injD intro: inj_onI)
wenzelm@13114: 
wenzelm@13114: lemma inj_mapD: "inj (map f) ==> inj f"
paulson@14208: apply (unfold inj_on_def, clarify)
nipkow@13145: apply (erule_tac x = "[x]" in ballE)
paulson@14208:  apply (erule_tac x = "[y]" in ballE, simp, blast)
nipkow@13145: apply blast
nipkow@13145: done
wenzelm@13114: 
nipkow@14339: lemma inj_map[iff]: "inj (map f) = inj f"
nipkow@13145: by (blast dest: inj_mapD intro: inj_mapI)
wenzelm@13114: 
kleing@14343: lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
kleing@14343: by (induct xs, auto)
wenzelm@13114: 
nipkow@14402: lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> 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 \<Longrightarrow> 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 \<Longrightarrow> map snd (zip xs ys) = ys"
nipkow@15110: by (induct rule:list_induct2, simp_all)
nipkow@15110: 
nipkow@15110: 
wenzelm@13142: subsection {* @{text rev} *}
wenzelm@13114: 
wenzelm@13142: lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma rev_is_rev_conv [iff]: "!!ys. (rev xs = rev ys) = (xs = ys)"
paulson@14208: apply (induct xs, force)
paulson@14208: apply (case_tac ys, simp, force)
nipkow@13145: done
wenzelm@13114: 
wenzelm@13366: lemma rev_induct [case_names Nil snoc]:
wenzelm@13366:   "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
nipkow@13145: apply(subst rev_rev_ident[symmetric])
nipkow@13145: apply(rule_tac list = "rev xs" in list.induct, simp_all)
nipkow@13145: done
wenzelm@13114: 
nipkow@13145: ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility"
wenzelm@13114: 
wenzelm@13366: lemma rev_exhaust [case_names Nil snoc]:
wenzelm@13366:   "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
nipkow@13145: by (induct xs rule: rev_induct) auto
wenzelm@13114: 
wenzelm@13366: lemmas rev_cases = rev_exhaust
wenzelm@13366: 
wenzelm@13114: 
wenzelm@13142: subsection {* @{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 \<union> set ys)"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
oheimb@14099: lemma hd_in_set: "l = x#xs \<Longrightarrow> x\<in>set l"
paulson@14208: by (case_tac l, auto)
oheimb@14099: 
wenzelm@13142: lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
nipkow@13145: by auto
wenzelm@13114: 
oheimb@14099: lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
oheimb@14099: by auto
oheimb@14099: 
wenzelm@13142: lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
nipkow@15245: lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
nipkow@15245: by(induct xs) auto
nipkow@15245: 
wenzelm@13142: lemma set_rev [simp]: "set (rev xs) = set xs"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma set_map [simp]: "set (map f xs) = f`(set xs)"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma set_upt [simp]: "set[i..j(] = {k. i \<le> k \<and> 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) = (\<exists>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 \<in> set (a # xs)"
paulson@15113:     with prems show "\<exists>ys zs. a # xs = ys @ x # zs"
paulson@15113:       by (simp, blast intro: Cons_eq_appendI)
paulson@15113:   next
paulson@15113:     assume "\<exists>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 \<in> set (a # xs)" 
paulson@15113:       by (cases ys, auto simp add: eq)
paulson@15113:   qed
paulson@15113: qed
wenzelm@13142: 
wenzelm@13142: lemma in_lists_conv_set: "(xs : lists A) = (\<forall>x \<in> set xs. x : A)"
nipkow@13145: -- {* eliminate @{text lists} in favour of @{text set} *}
nipkow@13145: by (induct xs) auto
wenzelm@13142: 
wenzelm@13142: lemma in_listsD [dest!]: "xs \<in> lists A ==> \<forall>x\<in>set xs. x \<in> A"
nipkow@13145: by (rule in_lists_conv_set [THEN iffD1])
wenzelm@13142: 
wenzelm@13142: lemma in_listsI [intro!]: "\<forall>x\<in>set xs. x \<in> A ==> xs \<in> lists A"
nipkow@13145: by (rule in_lists_conv_set [THEN iffD2])
wenzelm@13114: 
paulson@15168: lemma lists_UNIV [simp]: "lists UNIV = UNIV"
paulson@15168: by auto
paulson@15168: 
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) \<le> length xs"
kleing@14388: by (induct xs) (auto simp add: card_insert_if)
wenzelm@13114: 
paulson@15168: 
paulson@15168: subsection{*Sets of Lists*}
paulson@15168: 
paulson@15168: text{*Resembles a Cartesian product: it denotes the set of lists with head
paulson@15168:   drawn from @{term A} and tail drawn from @{term Xs}.*}
paulson@15168: constdefs
paulson@15168:   set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set"
paulson@15168:    "set_Cons A XS == {z. \<exists>x xs. z = x#xs & x \<in> A & xs \<in> XS}"
paulson@15168: 
paulson@15168: lemma [simp]: "set_Cons A {[]} = (%x. [x])`A"
paulson@15168: by (auto simp add: set_Cons_def)
paulson@15168: 
paulson@15168: text{*Yields the set of lists, all of the same length as the argument and
paulson@15168: with elements drawn from the corresponding element of the argument.*}
paulson@15168: consts  listset :: "'a set list \<Rightarrow> 'a list set"
paulson@15168: primrec
paulson@15168:    "listset []    = {[]}"
paulson@15168:    "listset(A#As) = set_Cons A (listset As)"
paulson@15168: 
paulson@15168: 
paulson@15176: subsection{*Lifting a Relation on List Elements to the Lists*}
paulson@15176: consts  listrel :: "('a * 'a)set => ('a list * 'a list)set"
paulson@15176: 
paulson@15176: inductive "listrel(r)"
paulson@15176:  intros
paulson@15176:    Nil:  "([],[]) \<in> listrel r"
paulson@15176:    Cons: "[| (x,y) \<in> r; (xs,ys) \<in> listrel r |] ==> (x#xs, y#ys) \<in> listrel r"
paulson@15176: 
paulson@15176: inductive_cases listrel_Nil1 [elim!]: "([],xs) \<in> listrel r"
paulson@15176: inductive_cases listrel_Nil2 [elim!]: "(xs,[]) \<in> listrel r"
paulson@15176: inductive_cases listrel_Cons1 [elim!]: "(y#ys,xs) \<in> listrel r"
paulson@15176: inductive_cases listrel_Cons2 [elim!]: "(xs,y#ys) \<in> listrel r"
paulson@15176: 
paulson@15176: 
paulson@15176: lemma listrel_mono: "r \<subseteq> s \<Longrightarrow> listrel r \<subseteq> listrel s"
paulson@15176: apply clarify  
paulson@15176: apply (erule listrel.induct)
paulson@15176: apply (blast intro: listrel.intros)+
paulson@15176: done
paulson@15176: 
paulson@15176: lemma listrel_subset: "r \<subseteq> A \<times> A \<Longrightarrow> listrel r \<subseteq> lists A \<times> lists A"
paulson@15176: apply clarify 
paulson@15176: apply (erule listrel.induct, auto) 
paulson@15176: done
paulson@15176: 
paulson@15176: lemma listrel_refl: "refl A r \<Longrightarrow> refl (lists A) (listrel r)" 
paulson@15176: apply (simp add: refl_def listrel_subset Ball_def)
paulson@15176: apply (rule allI) 
paulson@15176: apply (induct_tac x) 
paulson@15176: apply (auto intro: listrel.intros)
paulson@15176: done
paulson@15176: 
paulson@15176: lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)" 
paulson@15176: apply (auto simp add: sym_def)
paulson@15176: apply (erule listrel.induct) 
paulson@15176: apply (blast intro: listrel.intros)+
paulson@15176: done
paulson@15176: 
paulson@15176: lemma listrel_trans: "trans r \<Longrightarrow> trans (listrel r)" 
paulson@15176: apply (simp add: trans_def)
paulson@15176: apply (intro allI) 
paulson@15176: apply (rule impI) 
paulson@15176: apply (erule listrel.induct) 
paulson@15176: apply (blast intro: listrel.intros)+
paulson@15176: done
paulson@15176: 
paulson@15176: theorem equiv_listrel: "equiv A r \<Longrightarrow> equiv (lists A) (listrel r)"
paulson@15176: by (simp add: equiv_def listrel_refl listrel_sym listrel_trans) 
paulson@15176: 
paulson@15176: lemma listrel_Nil [simp]: "listrel r `` {[]} = {[]}"
paulson@15176: by (blast intro: listrel.intros)
paulson@15176: 
paulson@15176: lemma listrel_Cons:
paulson@15176:      "listrel r `` {x#xs} = set_Cons (r``{x}) (listrel r `` {xs})";
paulson@15176: by (auto simp add: set_Cons_def intro: listrel.intros) 
paulson@15176: 
paulson@15176: 
wenzelm@13142: subsection {* @{text mem} *}
wenzelm@13114: 
wenzelm@13114: lemma set_mem_eq: "(x mem xs) = (x : set xs)"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13114: 
wenzelm@13142: subsection {* @{text list_all} *}
wenzelm@13114: 
wenzelm@13142: lemma list_all_conv: "list_all P xs = (\<forall>x \<in> set xs. P x)"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma list_all_append [simp]:
nipkow@13145: "list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13114: 
wenzelm@13142: subsection {* @{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: 
wenzelm@13142: lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
nipkow@15246: lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
nipkow@13145: by (induct xs) (auto simp add: le_SucI)
wenzelm@13114: 
wenzelm@13142: lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
nipkow@13145: by auto
wenzelm@13114: 
nipkow@15246: lemma length_filter_less:
nipkow@15246:   "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> 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: 
wenzelm@13142: subsection {* @{text concat} *}
wenzelm@13114: 
wenzelm@13142: lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma concat_eq_Nil_conv [iff]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145: by (induct xss) auto
wenzelm@13114: 
wenzelm@13142: lemma Nil_eq_concat_conv [iff]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145: by (induct xss) auto
wenzelm@13114: 
wenzelm@13142: lemma set_concat [simp]: "set (concat xs) = \<Union>(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: 
wenzelm@13142: subsection {* @{text nth} *}
wenzelm@13114: 
wenzelm@13142: lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
nipkow@13145: by auto
wenzelm@13114: 
wenzelm@13142: lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
nipkow@13145: by auto
wenzelm@13114: 
wenzelm@13142: declare nth.simps [simp del]
wenzelm@13114: 
wenzelm@13114: lemma nth_append:
nipkow@13145: "!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
paulson@14208: apply (induct "xs", simp)
paulson@14208: apply (case_tac n, auto)
nipkow@13145: done
wenzelm@13114: 
nipkow@14402: lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
nipkow@14402: by (induct "xs") auto
nipkow@14402: 
nipkow@14402: lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
nipkow@14402: by (induct "xs") auto
nipkow@14402: 
wenzelm@13142: lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)"
paulson@14208: apply (induct xs, simp)
paulson@14208: apply (case_tac n, auto)
nipkow@13145: done
wenzelm@13114: 
wenzelm@13142: lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
paulson@14208: apply (induct_tac xs, simp, simp)
nipkow@13145: apply safe
paulson@14208: apply (rule_tac x = 0 in exI, simp)
paulson@14208:  apply (rule_tac x = "Suc i" in exI, simp)
paulson@14208: apply (case_tac i, simp)
nipkow@13145: apply (rename_tac j)
paulson@14208: apply (rule_tac x = j in exI, simp)
nipkow@13145: done
wenzelm@13114: 
nipkow@13145: lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
nipkow@13145: by (auto simp add: set_conv_nth)
wenzelm@13114: 
wenzelm@13142: lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
nipkow@13145: by (auto simp add: set_conv_nth)
wenzelm@13114: 
wenzelm@13114: lemma all_nth_imp_all_set:
nipkow@13145: "[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
nipkow@13145: by (auto simp add: set_conv_nth)
wenzelm@13114: 
wenzelm@13114: lemma all_set_conv_all_nth:
nipkow@13145: "(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
nipkow@13145: by (auto simp add: set_conv_nth)
wenzelm@13114: 
wenzelm@13114: 
wenzelm@13142: subsection {* @{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 \<noteq> j ==> xs[i:=x]!j = xs!j"
nipkow@13145: by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114: 
wenzelm@13142: lemma list_update_overwrite [simp]:
nipkow@13145: "!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]"
nipkow@13145: by (induct xs) (auto split: nat.split)
wenzelm@13114: 
nipkow@14402: lemma list_update_id[simp]: "!!i. i < length xs ==> xs[i := xs!i] = xs"
paulson@14208: apply (induct xs, simp)
nipkow@14187: apply(simp split:nat.splits)
nipkow@14187: done
nipkow@14187: 
wenzelm@13114: lemma list_update_same_conv:
nipkow@13145: "!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
nipkow@13145: by (induct xs) (auto split: nat.split)
wenzelm@13114: 
nipkow@14187: lemma list_update_append1:
nipkow@14187:  "!!i. i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
paulson@14208: apply (induct xs, simp)
nipkow@14187: apply(simp split:nat.split)
nipkow@14187: done
nipkow@14187: 
nipkow@14402: lemma list_update_length [simp]:
nipkow@14402:  "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
nipkow@14402: by (induct xs, auto)
nipkow@14402: 
wenzelm@13114: lemma update_zip:
nipkow@13145: "!!i xy xs. length xs = length ys ==>
nipkow@13145: (zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
nipkow@13145: by (induct ys) (auto, case_tac xs, auto split: nat.split)
wenzelm@13114: 
wenzelm@13114: lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)"
nipkow@13145: by (induct xs) (auto split: nat.split)
wenzelm@13114: 
wenzelm@13114: lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
nipkow@13145: by (blast dest!: set_update_subset_insert [THEN subsetD])
wenzelm@13114: 
wenzelm@13114: 
wenzelm@13142: subsection {* @{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 = [] \<Longrightarrow> last(x#xs) = x"
nipkow@14302: by(simp add:last.simps)
nipkow@14302: 
nipkow@14302: lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> 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 = [] \<Longrightarrow> last(xs @ ys) = last xs"
nipkow@14302: by(simp add:last_append)
nipkow@14302: 
nipkow@14302: lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
nipkow@14302: by(simp add:last_append)
nipkow@14302: 
nipkow@14302: 
wenzelm@13142: lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
nipkow@13145: by (induct xs rule: rev_induct) auto
wenzelm@13114: 
wenzelm@13114: lemma butlast_append:
nipkow@13145: "!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma append_butlast_last_id [simp]:
nipkow@13145: "xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
nipkow@13145: by (induct xs) (auto split: split_if_asm)
wenzelm@13114: 
wenzelm@13114: lemma in_set_butlast_appendI:
nipkow@13145: "x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
nipkow@13145: by (auto dest: in_set_butlastD simp add: butlast_append)
wenzelm@13114: 
wenzelm@13142: 
wenzelm@13142: subsection {* @{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 \<Longrightarrow> 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 \<Longrightarrow> 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 \<Longrightarrow> (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 \<or> xs = [])"
nipkow@15110: apply(induct xs)
nipkow@15110:  apply simp
nipkow@15110: apply(simp add:take_Cons split:nat.split)
nipkow@15110: done
nipkow@15110: 
nipkow@15110: lemma drop_eq_Nil[simp]: "!!n. (drop n xs = []) = (length xs <= n)"
nipkow@15110: apply(induct xs)
nipkow@15110: apply simp
nipkow@15110: apply(simp add:drop_Cons split:nat.split)
nipkow@15110: done
nipkow@15110: 
wenzelm@13114: lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)"
paulson@14208: apply (induct n, auto)
paulson@14208: apply (case_tac xs, auto)
nipkow@13145: done
wenzelm@13114: 
wenzelm@13142: lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)"
paulson@14208: apply (induct n, auto)
paulson@14208: apply (case_tac xs, auto)
nipkow@13145: done
wenzelm@13114: 
wenzelm@13114: lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)"
paulson@14208: apply (induct xs, auto)
paulson@14208: apply (case_tac i, auto)
nipkow@13145: done
wenzelm@13114: 
wenzelm@13114: lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)"
paulson@14208: apply (induct xs, auto)
paulson@14208: apply (case_tac i, auto)
nipkow@13145: done
wenzelm@13114: 
wenzelm@13142: lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i"
paulson@14208: apply (induct xs, auto)
paulson@14208: apply (case_tac n, blast)
paulson@14208: apply (case_tac i, auto)
nipkow@13145: done
wenzelm@13114: 
wenzelm@13142: lemma nth_drop [simp]:
nipkow@13145: "!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
paulson@14208: apply (induct n, auto)
paulson@14208: apply (case_tac xs, auto)
nipkow@13145: done
nipkow@3507: 
nipkow@14025: lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs"
nipkow@14025: by(induct xs)(auto simp:take_Cons split:nat.split)
nipkow@14025: 
nipkow@14025: lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> 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) \<Longrightarrow> x : set xs"
nipkow@14187: using set_take_subset by fast
nipkow@14187: 
nipkow@14187: lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> 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 \<and> 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:     "\<forall>i. i+j \<le> 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 \<le> size ys\<^isub>1
nipkow@14300:    then xs\<^isub>1 = take (size xs\<^isub>1) ys\<^isub>1 \<and> xs\<^isub>2 = drop (size xs\<^isub>1) ys\<^isub>1 @ ys\<^isub>2
nipkow@14300:    else take (size ys\<^isub>1) xs\<^isub>1 = ys\<^isub>1 \<and> drop (size ys\<^isub>1) xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>2)"
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 \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (n+1) xs"
nipkow@15110: apply(induct xs)
nipkow@15110: apply simp
nipkow@15110: apply(simp add:drop_Cons split:nat.split)
nipkow@15110: done
nipkow@15110: 
wenzelm@13114: 
wenzelm@13142: subsection {* @{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: "\<not> 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 \<and> 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) = (\<forall>x \<in> 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 = []) = (\<forall>x \<in> 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 & \<not> P y)"
nipkow@13913: by(induct xs, auto)
nipkow@13913: 
wenzelm@13114: 
wenzelm@13142: subsection {* @{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: 
wenzelm@13142: lemma length_zip [simp]:
nipkow@13145: "!!xs. length (zip xs ys) = min (length xs) (length ys)"
paulson@14208: apply (induct ys, simp)
paulson@14208: apply (case_tac xs, auto)
nipkow@13145: done
wenzelm@13114: 
wenzelm@13114: lemma zip_append1:
nipkow@13145: "!!xs. zip (xs @ ys) zs =
nipkow@13145: zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
paulson@14208: apply (induct zs, simp)
paulson@14208: apply (case_tac xs, simp_all)
nipkow@13145: done
wenzelm@13114: 
wenzelm@13114: lemma zip_append2:
nipkow@13145: "!!ys. zip xs (ys @ zs) =
nipkow@13145: zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
paulson@14208: apply (induct xs, simp)
paulson@14208: apply (case_tac ys, simp_all)
nipkow@13145: done
wenzelm@13114: 
wenzelm@13142: lemma zip_append [simp]:
wenzelm@13142:  "[| length xs = length us; length ys = length vs |] ==>
nipkow@13145: zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
nipkow@13145: by (simp add: zip_append1)
wenzelm@13114: 
wenzelm@13114: lemma zip_rev:
nipkow@14247: "length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
nipkow@14247: by (induct rule:list_induct2, simp_all)
wenzelm@13114: 
wenzelm@13142: lemma nth_zip [simp]:
nipkow@13145: "!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
paulson@14208: apply (induct ys, simp)
nipkow@13145: apply (case_tac xs)
nipkow@13145:  apply (simp_all add: nth.simps split: nat.split)
nipkow@13145: done
wenzelm@13114: 
wenzelm@13114: lemma set_zip:
nipkow@13145: "set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
nipkow@13145: by (simp add: set_conv_nth cong: rev_conj_cong)
wenzelm@13114: 
wenzelm@13114: lemma zip_update:
nipkow@13145: "length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
nipkow@13145: by (rule sym, simp add: update_zip)
wenzelm@13114: 
wenzelm@13142: lemma zip_replicate [simp]:
nipkow@13145: "!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
paulson@14208: apply (induct i, auto)
paulson@14208: apply (case_tac j, auto)
nipkow@13145: done
wenzelm@13114: 
wenzelm@13142: 
wenzelm@13142: subsection {* @{text list_all2} *}
wenzelm@13114: 
kleing@14316: lemma list_all2_lengthD [intro?]: 
kleing@14316:   "list_all2 P xs ys ==> length xs = length ys"
nipkow@13145: by (simp add: list_all2_def)
wenzelm@13114: 
wenzelm@13142: lemma list_all2_Nil [iff]: "list_all2 P [] ys = (ys = [])"
nipkow@13145: by (simp add: list_all2_def)
wenzelm@13114: 
wenzelm@13142: lemma list_all2_Nil2[iff]: "list_all2 P xs [] = (xs = [])"
nipkow@13145: by (simp add: list_all2_def)
wenzelm@13114: 
wenzelm@13142: lemma list_all2_Cons [iff]:
nipkow@13145: "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
nipkow@13145: by (auto simp add: list_all2_def)
wenzelm@13114: 
wenzelm@13114: lemma list_all2_Cons1:
nipkow@13145: "list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> 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) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> 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 \<and> length us = length xs \<and> length vs = length ys \<and>
nipkow@13145: list_all2 P xs us \<and> 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 \<and> length us = length ys \<and> length vs = length zs \<and>
nipkow@13145: list_all2 P us ys \<and> 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 \<Longrightarrow>
nipkow@14247:   list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> 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:   "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> 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 \<and> (\<forall>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 \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
kleing@13863:   by (simp add: list_all2_conv_all_nth)
kleing@13863: 
paulson@14395: lemma list_all2I:
paulson@14395:   "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
paulson@14395:   by (simp add: list_all2_def)
paulson@14395: 
kleing@14328: lemma list_all2_nthD:
kleing@13863:   "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> 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:   "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> 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 (\<lambda>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 (\<lambda>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:   "(\<And>x. P x x) \<Longrightarrow> 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:   "\<lbrakk> i<size xs; list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> 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:   "\<lbrakk>list_all2 P xs ys; P x y; i < length ys\<rbrakk> \<Longrightarrow> 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:   "\<And>n ys. list_all2 P xs ys \<Longrightarrow> 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:   "\<And>n bs. list_all2 P as bs \<Longrightarrow> 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:   "\<And>y. list_all2 P x y \<Longrightarrow> (\<And>x y. P x y \<Longrightarrow> Q x y) \<Longrightarrow> list_all2 Q x y"
paulson@14208:   apply (induct x, simp)
paulson@14208:   apply (case_tac y, auto)
kleing@13863:   done
kleing@13863: 
wenzelm@13142: 
nipkow@14402: subsection {* @{text foldl} and @{text foldr} *}
wenzelm@13142: 
wenzelm@13142: lemma foldl_append [simp]:
nipkow@13145: "!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
nipkow@13145: by (induct xs) auto
wenzelm@13142: 
nipkow@14402: lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
nipkow@14402: by (induct xs) auto
nipkow@14402: 
nipkow@14402: lemma foldr_foldl: "foldr f xs a = foldl (%x y. f y x) a (rev xs)"
nipkow@14402: by (induct xs) auto
nipkow@14402: 
nipkow@14402: lemma foldl_foldr: "foldl f a xs = foldr (%x y. f y x) (rev xs) a"
nipkow@14402: by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"])
nipkow@14402: 
wenzelm@13142: text {*
nipkow@13145: Note: @{text "n \<le> 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 \<and> (\<forall>n \<in> set ns. n = 0))"
nipkow@13145: by (induct ns) auto
wenzelm@13114: 
wenzelm@13114: 
wenzelm@13142: subsection {* @{text upto} *}
wenzelm@13114: 
wenzelm@13142: lemma upt_rec: "[i..j(] = (if i<j then i#[Suc i..j(] else [])"
nipkow@13145: -- {* Does not terminate! *}
nipkow@13145: by (induct j) auto
wenzelm@13142: 
wenzelm@13142: lemma upt_conv_Nil [simp]: "j <= i ==> [i..j(] = []"
nipkow@13145: by (subst upt_rec) simp
wenzelm@13114: 
wenzelm@13142: lemma upt_Suc_append: "i <= j ==> [i..(Suc j)(] = [i..j(]@[j]"
nipkow@13145: -- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
nipkow@13145: by simp
wenzelm@13114: 
wenzelm@13142: lemma upt_conv_Cons: "i < j ==> [i..j(] = i # [Suc i..j(]"
nipkow@13145: apply(rule trans)
nipkow@13145: apply(subst upt_rec)
paulson@14208:  prefer 2 apply (rule refl, simp)
nipkow@13145: done
wenzelm@13114: 
wenzelm@13142: lemma upt_add_eq_append: "i<=j ==> [i..j+k(] = [i..j(]@[j..j+k(]"
nipkow@13145: -- {* LOOPS as a simprule, since @{text "j <= j"}. *}
nipkow@13145: by (induct k) auto
wenzelm@13114: 
wenzelm@13142: lemma length_upt [simp]: "length [i..j(] = j - i"
nipkow@13145: by (induct j) (auto simp add: Suc_diff_le)
wenzelm@13114: 
wenzelm@13142: lemma nth_upt [simp]: "i + k < j ==> [i..j(] ! k = i + k"
nipkow@13145: apply (induct j)
nipkow@13145: apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
nipkow@13145: done
wenzelm@13114: 
wenzelm@13142: lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..n(] = [i..i+m(]"
paulson@14208: apply (induct m, simp)
nipkow@13145: apply (subst upt_rec)
nipkow@13145: apply (rule sym)
nipkow@13145: apply (subst upt_rec)
nipkow@13145: apply (simp del: upt.simps)
nipkow@13145: done
nipkow@3507: 
wenzelm@13114: lemma map_Suc_upt: "map Suc [m..n(] = [Suc m..n]"
nipkow@13145: by (induct n) auto
wenzelm@13114: 
wenzelm@13114: lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..n(]) ! i = f(m+i)"
nipkow@13145: apply (induct n m rule: diff_induct)
nipkow@13145: prefer 3 apply (subst map_Suc_upt[symmetric])
nipkow@13145: apply (auto simp add: less_diff_conv nth_upt)
nipkow@13145: done
wenzelm@13114: 
berghofe@13883: lemma nth_take_lemma:
berghofe@13883:   "!!xs ys. k <= length xs ==> 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:   "\<lbrakk> (\<And>x y. \<lbrakk>P x y; Q y x\<rbrakk> \<Longrightarrow> x = y); list_all2 P xs ys; list_all2 Q ys xs \<rbrakk> 
kleing@13863:   \<Longrightarrow> 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: "(\<forall>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: 
wenzelm@13142: subsection {* @{text "distinct"} and @{text remdups} *}
wenzelm@13142: 
wenzelm@13142: lemma distinct_append [simp]:
nipkow@13145: "distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
nipkow@13145: by (induct xs) auto
wenzelm@13142: 
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@15072:   by (induct_tac x, auto) 
paulson@15072: 
paulson@15072: lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"
paulson@15072:   by (induct_tac x, auto)
paulson@15072: 
nipkow@15245: lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs"
nipkow@15245: by (induct xs) auto
nipkow@15245: 
nipkow@15245: lemma length_remdups_eq[iff]:
nipkow@15245:   "(length (remdups xs) = length xs) = (remdups xs = xs)"
nipkow@15245: apply(induct xs)
nipkow@15245:  apply auto
nipkow@15245: apply(subgoal_tac "length (remdups xs) <= length xs")
nipkow@15245:  apply arith
nipkow@15245: apply(rule length_remdups_leq)
nipkow@15245: done
nipkow@15245: 
wenzelm@13142: lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: text {*
nipkow@13145: It is best to avoid this indexed version of distinct, but sometimes
nipkow@13145: it is useful. *}
wenzelm@13142: lemma distinct_conv_nth:
nipkow@13145: "distinct xs = (\<forall>i j. i < size xs \<and> j < size xs \<and> i \<noteq> j --> xs!i \<noteq> xs!j)"
paulson@14208: apply (induct_tac xs, simp, simp)
paulson@14208: apply (rule iffI, clarsimp)
nipkow@13145:  apply (case_tac i)
paulson@14208: apply (case_tac j, simp)
nipkow@13145: apply (simp add: set_conv_nth)
nipkow@13145:  apply (case_tac j)
paulson@14208: apply (clarsimp simp add: set_conv_nth, simp)
nipkow@13145: apply (rule conjI)
nipkow@13145:  apply (clarsimp simp add: set_conv_nth)
nipkow@13145:  apply (erule_tac x = 0 in allE)
paulson@14208:  apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
nipkow@13145: apply (erule_tac x = "Suc i" in allE)
paulson@14208: apply (erule_tac x = "Suc j" in allE, simp)
nipkow@13145: done
wenzelm@13114: 
nipkow@15110: lemma distinct_card: "distinct xs ==> card (set xs) = size xs"
kleing@14388:   by (induct xs) auto
kleing@14388: 
nipkow@15110: lemma card_distinct: "card (set xs) = size xs ==> distinct xs"
kleing@14388: proof (induct xs)
kleing@14388:   case Nil thus ?case by simp
kleing@14388: next
kleing@14388:   case (Cons x xs)
kleing@14388:   show ?case
kleing@14388:   proof (cases "x \<in> 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) \<le> length xs" by (rule card_length)
kleing@14388:     ultimately have False by simp
kleing@14388:     thus ?thesis ..
kleing@14388:   qed
kleing@14388: qed
kleing@14388: 
nipkow@15110: lemma inj_on_setI: "distinct(map f xs) ==> inj_on f (set xs)"
nipkow@15110: apply(induct xs)
nipkow@15110:  apply simp
nipkow@15110: apply fastsimp
nipkow@15110: done
nipkow@15110: 
nipkow@15110: lemma inj_on_set_conv:
nipkow@15110:  "distinct xs \<Longrightarrow> inj_on f (set xs) = distinct(map f xs)"
nipkow@15110: apply(induct xs)
nipkow@15110:  apply simp
nipkow@15110: apply fastsimp
nipkow@15110: done
nipkow@15110: 
nipkow@15110: 
nipkow@15110: subsection {* @{text remove1} *}
nipkow@15110: 
nipkow@15110: lemma set_remove1_subset: "set(remove1 x xs) <= set xs"
nipkow@15110: apply(induct xs)
nipkow@15110:  apply simp
nipkow@15110: apply simp
nipkow@15110: apply blast
nipkow@15110: done
nipkow@15110: 
nipkow@15110: lemma [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"
nipkow@15110: apply(induct xs)
nipkow@15110:  apply simp
nipkow@15110: apply simp
nipkow@15110: apply blast
nipkow@15110: done
nipkow@15110: 
nipkow@15110: lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"
nipkow@15110: apply(insert set_remove1_subset)
nipkow@15110: apply fast
nipkow@15110: done
nipkow@15110: 
nipkow@15110: lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)"
nipkow@15110: by (induct xs) simp_all
nipkow@15110: 
wenzelm@13114: 
wenzelm@13142: subsection {* @{text replicate} *}
wenzelm@13114: 
wenzelm@13142: lemma length_replicate [simp]: "length (replicate n x) = n"
nipkow@13145: by (induct n) auto
nipkow@13124: 
wenzelm@13142: lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
nipkow@13145: by (induct n) auto
wenzelm@13114: 
wenzelm@13114: lemma replicate_app_Cons_same:
nipkow@13145: "(replicate n x) @ (x # xs) = x # replicate n x @ xs"
nipkow@13145: by (induct n) auto
wenzelm@13114: 
wenzelm@13142: lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
paulson@14208: apply (induct n, simp)
nipkow@13145: apply (simp add: replicate_app_Cons_same)
nipkow@13145: done
wenzelm@13114: 
wenzelm@13142: lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
nipkow@13145: by (induct n) auto
wenzelm@13114: 
wenzelm@13142: lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
nipkow@13145: by (induct n) auto
wenzelm@13114: 
wenzelm@13142: lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"
nipkow@13145: by (induct n) auto
wenzelm@13114: 
wenzelm@13142: lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
nipkow@13145: by (atomize (full), induct n) auto
wenzelm@13114: 
wenzelm@13142: lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x"
paulson@14208: apply (induct n, simp)
nipkow@13145: apply (simp add: nth_Cons split: nat.split)
nipkow@13145: done
wenzelm@13114: 
wenzelm@13142: lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
nipkow@13145: by (induct n) auto
wenzelm@13114: 
wenzelm@13142: lemma set_replicate [simp]: "n \<noteq> 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: 
oheimb@14099: subsection {* Lexicographic orderings on lists *}
nipkow@3507: 
wenzelm@13142: lemma wf_lexn: "wf r ==> wf (lexn r n)"
paulson@14208: apply (induct_tac n, simp, simp)
nipkow@13145: apply(rule wf_subset)
nipkow@13145:  prefer 2 apply (rule Int_lower1)
nipkow@13145: apply(rule wf_prod_fun_image)
paulson@14208:  prefer 2 apply (rule inj_onI, auto)
nipkow@13145: done
wenzelm@13114: 
wenzelm@13114: lemma lexn_length:
paulson@15168:      "!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
nipkow@13145: by (induct n) auto
wenzelm@13114: 
wenzelm@13142: lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
nipkow@13145: apply (unfold lex_def)
nipkow@13145: apply (rule wf_UN)
paulson@14208: apply (blast intro: wf_lexn, clarify)
nipkow@13145: apply (rename_tac m n)
nipkow@13145: apply (subgoal_tac "m \<noteq> n")
nipkow@13145:  prefer 2 apply blast
nipkow@13145: apply (blast dest: lexn_length not_sym)
nipkow@13145: done
wenzelm@13114: 
wenzelm@13114: lemma lexn_conv:
nipkow@13145: "lexn r n =
nipkow@13145: {(xs,ys). length xs = n \<and> length ys = n \<and>
nipkow@13145: (\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
paulson@14208: apply (induct_tac n, simp, blast)
paulson@14208: apply (simp add: image_Collect lex_prod_def, safe, blast)
paulson@14208:  apply (rule_tac x = "ab # xys" in exI, simp)
paulson@14208: apply (case_tac xys, simp_all, blast)
nipkow@13145: done
wenzelm@13114: 
wenzelm@13114: lemma lex_conv:
nipkow@13145: "lex r =
nipkow@13145: {(xs,ys). length xs = length ys \<and>
nipkow@13145: (\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
nipkow@13145: by (force simp add: lex_def lexn_conv)
wenzelm@13114: 
wenzelm@13142: lemma wf_lexico [intro!]: "wf r ==> wf (lexico r)"
nipkow@13145: by (unfold lexico_def) blast
wenzelm@13114: 
wenzelm@13114: lemma lexico_conv:
nipkow@13145: "lexico r = {(xs,ys). length xs < length ys |
nipkow@13145: length xs = length ys \<and> (xs, ys) : lex r}"
nipkow@13145: by (simp add: lexico_def diag_def lex_prod_def measure_def inv_image_def)
wenzelm@13114: 
wenzelm@13142: lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
nipkow@13145: by (simp add: lex_conv)
wenzelm@13114: 
wenzelm@13142: lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
nipkow@13145: by (simp add:lex_conv)
wenzelm@13114: 
wenzelm@13142: lemma Cons_in_lex [iff]:
nipkow@13145: "((x # xs, y # ys) : lex r) =
nipkow@13145: ((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
nipkow@13145: apply (simp add: lex_conv)
nipkow@13145: apply (rule iffI)
paulson@14208:  prefer 2 apply (blast intro: Cons_eq_appendI, clarify)
paulson@14208: apply (case_tac xys, simp, simp)
nipkow@13145: apply blast
nipkow@13145: done
wenzelm@13114: 
wenzelm@13114: 
wenzelm@13142: subsection {* @{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: 
wenzelm@13114: lemma sublist_shift_lemma:
paulson@15168:      "map fst [p:zip xs [i..i + length xs(] . snd p : A] =
paulson@15168:       map fst [p:zip xs [0..length xs(] . snd p + i : A]"
nipkow@13145: by (induct xs rule: rev_induct) (simp_all add: add_commute)
wenzelm@13114: 
wenzelm@13114: lemma sublist_append:
paulson@15168:      "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
nipkow@13145: apply (unfold sublist_def)
paulson@14208: apply (induct l' rule: rev_induct, simp)
nipkow@13145: apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma)
nipkow@13145: apply (simp add: add_commute)
nipkow@13145: done
wenzelm@13114: 
wenzelm@13114: lemma sublist_Cons:
nipkow@13145: "sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
nipkow@13145: apply (induct l rule: rev_induct)
nipkow@13145:  apply (simp add: sublist_def)
nipkow@13145: apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
nipkow@13145: done
wenzelm@13114: 
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@15045: lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l"
paulson@14208: apply (induct l rule: rev_induct, simp)
nipkow@13145: apply (simp split: nat_diff_split add: sublist_append)
nipkow@13145: done
wenzelm@13114: 
wenzelm@13114: 
wenzelm@13142: lemma take_Cons':
paulson@15168:      "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
nipkow@13145: by (cases n) simp_all
wenzelm@13114: 
wenzelm@13142: lemma drop_Cons':
paulson@15168:      "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
nipkow@13145: by (cases n) simp_all
wenzelm@13114: 
wenzelm@13142: lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
nipkow@13145: by (cases n) simp_all
wenzelm@13142: 
nipkow@13145: lemmas [simp] = take_Cons'[of "number_of v",standard]
nipkow@13145:                 drop_Cons'[of "number_of v",standard]
nipkow@13145:                 nth_Cons'[of _ _ "number_of v",standard]
nipkow@3507: 
wenzelm@13462: 
kleing@14388: lemma "card (set xs) = size xs \<Longrightarrow> 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 \<in> 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) \<le> 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: 
wenzelm@13366: subsection {* Characters and strings *}
wenzelm@13366: 
wenzelm@13366: datatype nibble =
wenzelm@13366:     Nibble0 | Nibble1 | Nibble2 | Nibble3 | Nibble4 | Nibble5 | Nibble6 | Nibble7
wenzelm@13366:   | Nibble8 | Nibble9 | NibbleA | NibbleB | NibbleC | NibbleD | NibbleE | NibbleF
wenzelm@13366: 
wenzelm@13366: datatype char = Char nibble nibble
wenzelm@13366:   -- "Note: canonical order of character encoding coincides with standard term ordering"
wenzelm@13366: 
wenzelm@13366: types string = "char list"
wenzelm@13366: 
wenzelm@13366: syntax
wenzelm@13366:   "_Char" :: "xstr => char"    ("CHR _")
wenzelm@13366:   "_String" :: "xstr => string"    ("_")
wenzelm@13366: 
wenzelm@13366: parse_ast_translation {*
wenzelm@13366:   let
wenzelm@13366:     val constants = Syntax.Appl o map Syntax.Constant;
wenzelm@13366: 
wenzelm@13366:     fun mk_nib n = "Nibble" ^ chr (n + (if n <= 9 then ord "0" else ord "A" - 10));
wenzelm@13366:     fun mk_char c =
wenzelm@13366:       if Symbol.is_ascii c andalso Symbol.is_printable c then
wenzelm@13366:         constants ["Char", mk_nib (ord c div 16), mk_nib (ord c mod 16)]
wenzelm@13366:       else error ("Printable ASCII character expected: " ^ quote c);
wenzelm@13366: 
wenzelm@13366:     fun mk_string [] = Syntax.Constant "Nil"
wenzelm@13366:       | mk_string (c :: cs) = Syntax.Appl [Syntax.Constant "Cons", mk_char c, mk_string cs];
wenzelm@13366: 
wenzelm@13366:     fun char_ast_tr [Syntax.Variable xstr] =
wenzelm@13366:         (case Syntax.explode_xstr xstr of
wenzelm@13366:           [c] => mk_char c
wenzelm@13366:         | _ => error ("Single character expected: " ^ xstr))
wenzelm@13366:       | char_ast_tr asts = raise AST ("char_ast_tr", asts);
wenzelm@13366: 
wenzelm@13366:     fun string_ast_tr [Syntax.Variable xstr] =
wenzelm@13366:         (case Syntax.explode_xstr xstr of
wenzelm@13366:           [] => constants [Syntax.constrainC, "Nil", "string"]
wenzelm@13366:         | cs => mk_string cs)
wenzelm@13366:       | string_ast_tr asts = raise AST ("string_tr", asts);
wenzelm@13366:   in [("_Char", char_ast_tr), ("_String", string_ast_tr)] end;
wenzelm@13366: *}
wenzelm@13366: 
berghofe@15064: ML {*
berghofe@15064: fun int_of_nibble h =
berghofe@15064:   if "0" <= h andalso h <= "9" then ord h - ord "0"
berghofe@15064:   else if "A" <= h andalso h <= "F" then ord h - ord "A" + 10
berghofe@15064:   else raise Match;
berghofe@15064: 
berghofe@15064: fun nibble_of_int i =
berghofe@15064:   if i <= 9 then chr (ord "0" + i) else chr (ord "A" + i - 10);
berghofe@15064: *}
berghofe@15064: 
wenzelm@13366: print_ast_translation {*
wenzelm@13366:   let
wenzelm@13366:     fun dest_nib (Syntax.Constant c) =
wenzelm@13366:         (case explode c of
berghofe@15064:           ["N", "i", "b", "b", "l", "e", h] => int_of_nibble h
wenzelm@13366:         | _ => raise Match)
wenzelm@13366:       | dest_nib _ = raise Match;
wenzelm@13366: 
wenzelm@13366:     fun dest_chr c1 c2 =
wenzelm@13366:       let val c = chr (dest_nib c1 * 16 + dest_nib c2)
wenzelm@13366:       in if Symbol.is_printable c then c else raise Match end;
wenzelm@13366: 
wenzelm@13366:     fun dest_char (Syntax.Appl [Syntax.Constant "Char", c1, c2]) = dest_chr c1 c2
wenzelm@13366:       | dest_char _ = raise Match;
wenzelm@13366: 
wenzelm@13366:     fun xstr cs = Syntax.Appl [Syntax.Constant "_xstr", Syntax.Variable (Syntax.implode_xstr cs)];
wenzelm@13366: 
wenzelm@13366:     fun char_ast_tr' [c1, c2] = Syntax.Appl [Syntax.Constant "_Char", xstr [dest_chr c1 c2]]
wenzelm@13366:       | char_ast_tr' _ = raise Match;
wenzelm@13366: 
wenzelm@13366:     fun list_ast_tr' [args] = Syntax.Appl [Syntax.Constant "_String",
wenzelm@13366:             xstr (map dest_char (Syntax.unfold_ast "_args" args))]
wenzelm@13366:       | list_ast_tr' ts = raise Match;
wenzelm@13366:   in [("Char", char_ast_tr'), ("@list", list_ast_tr')] end;
wenzelm@13366: *}
wenzelm@13366: 
berghofe@15064: subsection {* Code generator setup *}
berghofe@15064: 
berghofe@15064: ML {*
berghofe@15064: local
berghofe@15064: 
berghofe@15064: fun list_codegen thy gr dep b t =
berghofe@15064:   let val (gr', ps) = foldl_map (Codegen.invoke_codegen thy dep false)
berghofe@15064:     (gr, HOLogic.dest_list t)
berghofe@15064:   in Some (gr', Pretty.list "[" "]" ps) end handle TERM _ => None;
berghofe@15064: 
berghofe@15064: fun dest_nibble (Const (s, _)) = int_of_nibble (unprefix "List.nibble.Nibble" s)
berghofe@15064:   | dest_nibble _ = raise Match;
berghofe@15064: 
berghofe@15064: fun char_codegen thy gr dep b (Const ("List.char.Char", _) $ c1 $ c2) =
berghofe@15064:     (let val c = chr (dest_nibble c1 * 16 + dest_nibble c2)
berghofe@15064:      in if Symbol.is_printable c then Some (gr, Pretty.quote (Pretty.str c))
berghofe@15064:        else None
berghofe@15064:      end handle LIST _ => None | Match => None)
berghofe@15064:   | char_codegen thy gr dep b _ = None;
berghofe@15064: 
berghofe@15064: in
berghofe@15064: 
berghofe@15064: val list_codegen_setup =
berghofe@15064:   [Codegen.add_codegen "list_codegen" list_codegen,
berghofe@15064:    Codegen.add_codegen "char_codegen" char_codegen];
berghofe@15064: 
berghofe@15064: end;
berghofe@15064: 
berghofe@15064: val term_of_list = HOLogic.mk_list;
berghofe@15064: 
berghofe@15064: fun gen_list' aG i j = frequency
berghofe@15064:   [(i, fn () => aG j :: gen_list' aG (i-1) j), (1, fn () => [])] ()
berghofe@15064: and gen_list aG i = gen_list' aG i i;
berghofe@15064: 
berghofe@15064: val nibbleT = Type ("List.nibble", []);
berghofe@15064: 
berghofe@15064: fun term_of_char c =
berghofe@15064:   Const ("List.char.Char", nibbleT --> nibbleT --> Type ("List.char", [])) $
berghofe@15064:     Const ("List.nibble.Nibble" ^ nibble_of_int (ord c div 16), nibbleT) $
berghofe@15064:     Const ("List.nibble.Nibble" ^ nibble_of_int (ord c mod 16), nibbleT);
berghofe@15064: 
berghofe@15064: fun gen_char i = chr (random_range (ord "a") (Int.min (ord "a" + i, ord "z")));
berghofe@15064: *}
berghofe@15064: 
berghofe@15064: types_code
berghofe@15064:   "list" ("_ list")
berghofe@15064:   "char" ("string")
berghofe@15064: 
berghofe@15064: consts_code "Cons" ("(_ ::/ _)")
berghofe@15064: 
berghofe@15064: setup list_codegen_setup
berghofe@15064: 
wenzelm@13122: end




