src/HOL/List.thy
author nipkow
Fri May 15 10:01:57 2009 +0200 (2009-05-15)
changeset 31159 bac0d673b6d6
parent 31154 f919b8e67413
child 31201 3dde56615750
permissions -rw-r--r--
new lemma
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(*  Title:      HOL/List.thy
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    Author:     Tobias Nipkow
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*)
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header {* The datatype of finite lists *}
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theory List
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imports Plain Presburger Recdef ATP_Linkup
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uses ("Tools/list_code.ML")
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begin
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datatype 'a list =
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    Nil    ("[]")
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  | Cons 'a  "'a list"    (infixr "#" 65)
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subsection{*Basic list processing functions*}
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consts
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  filter:: "('a => bool) => 'a list => 'a list"
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  concat:: "'a list list => 'a list"
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  foldl :: "('b => 'a => 'b) => 'b => 'a list => 'b"
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  foldr :: "('a => 'b => 'b) => 'a list => 'b => 'b"
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  hd:: "'a list => 'a"
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  tl:: "'a list => 'a list"
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  last:: "'a list => 'a"
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  butlast :: "'a list => 'a list"
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  set :: "'a list => 'a set"
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  map :: "('a=>'b) => ('a list => 'b list)"
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  listsum ::  "'a list => 'a::monoid_add"
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  list_update :: "'a list => nat => 'a => 'a list"
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  take:: "nat => 'a list => 'a list"
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  drop:: "nat => 'a list => 'a list"
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  takeWhile :: "('a => bool) => 'a list => 'a list"
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  dropWhile :: "('a => bool) => 'a list => 'a list"
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  rev :: "'a list => 'a list"
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  zip :: "'a list => 'b list => ('a * 'b) list"
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  upt :: "nat => nat => nat list" ("(1[_..</_'])")
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  remdups :: "'a list => 'a list"
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  remove1 :: "'a => 'a list => 'a list"
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  removeAll :: "'a => 'a list => 'a list"
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  "distinct":: "'a list => bool"
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  replicate :: "nat => 'a => 'a list"
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  splice :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
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nonterminals lupdbinds lupdbind
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syntax
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  -- {* list Enumeration *}
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  "@list" :: "args => 'a list"    ("[(_)]")
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  -- {* Special syntax for filter *}
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"    ("(1[_<-_./ _])")
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  -- {* list update *}
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  "_lupdbind":: "['a, 'a] => lupdbind"    ("(2_ :=/ _)")
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  "" :: "lupdbind => lupdbinds"    ("_")
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  "_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds"    ("_,/ _")
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  "_LUpdate" :: "['a, lupdbinds] => 'a"    ("_/[(_)]" [900,0] 900)
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translations
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  "[x, xs]" == "x#[xs]"
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  "[x]" == "x#[]"
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  "[x<-xs . P]"== "filter (%x. P) xs"
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  "_LUpdate xs (_lupdbinds b bs)"== "_LUpdate (_LUpdate xs b) bs"
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  "xs[i:=x]" == "list_update xs i x"
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syntax (xsymbols)
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<leftarrow>_ ./ _])")
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syntax (HTML output)
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<leftarrow>_ ./ _])")
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text {*
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  Function @{text size} is overloaded for all datatypes. Users may
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  refer to the list version as @{text length}. *}
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abbreviation
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  length :: "'a list => nat" where
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  "length == size"
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primrec
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  "hd(x#xs) = x"
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primrec
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  "tl([]) = []"
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  "tl(x#xs) = xs"
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primrec
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  "last(x#xs) = (if xs=[] then x else last xs)"
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primrec
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  "butlast []= []"
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  "butlast(x#xs) = (if xs=[] then [] else x#butlast xs)"
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primrec
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  "set [] = {}"
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  "set (x#xs) = insert x (set xs)"
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primrec
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  "map f [] = []"
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  "map f (x#xs) = f(x)#map f xs"
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primrec
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  append :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixr "@" 65)
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where
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  append_Nil:"[] @ ys = ys"
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  | append_Cons: "(x#xs) @ ys = x # xs @ ys"
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primrec
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  "rev([]) = []"
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  "rev(x#xs) = rev(xs) @ [x]"
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primrec
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  "filter P [] = []"
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  "filter P (x#xs) = (if P x then x#filter P xs else filter P xs)"
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primrec
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  foldl_Nil:"foldl f a [] = a"
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  foldl_Cons: "foldl f a (x#xs) = foldl f (f a x) xs"
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primrec
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  "foldr f [] a = a"
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  "foldr f (x#xs) a = f x (foldr f xs a)"
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primrec
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  "concat([]) = []"
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  "concat(x#xs) = x @ concat(xs)"
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primrec
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"listsum [] = 0"
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"listsum (x # xs) = x + listsum xs"
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primrec
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  drop_Nil:"drop n [] = []"
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  drop_Cons: "drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec
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  take_Nil:"take n [] = []"
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  take_Cons: "take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec nth :: "'a list => nat => 'a" (infixl "!" 100) where
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  nth_Cons: "(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec
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  "[][i:=v] = []"
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  "(x#xs)[i:=v] = (case i of 0 => v # xs | Suc j => x # xs[j:=v])"
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primrec
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  "takeWhile P [] = []"
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  "takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])"
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primrec
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  "dropWhile P [] = []"
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  "dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)"
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primrec
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  "zip xs [] = []"
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  zip_Cons: "zip xs (y#ys) = (case xs of [] => [] | z#zs => (z,y)#zip zs ys)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "xs = []"} and @{text "xs = z # zs"} *}
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primrec
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  upt_0: "[i..<0] = []"
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  upt_Suc: "[i..<(Suc j)] = (if i <= j then [i..<j] @ [j] else [])"
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primrec
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  "distinct [] = True"
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  "distinct (x#xs) = (x ~: set xs \<and> distinct xs)"
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primrec
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  "remdups [] = []"
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  "remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)"
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primrec
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  "remove1 x [] = []"
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  "remove1 x (y#xs) = (if x=y then xs else y # remove1 x xs)"
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primrec
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  "removeAll x [] = []"
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  "removeAll x (y#xs) = (if x=y then removeAll x xs else y # removeAll x xs)"
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primrec
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  replicate_0: "replicate 0 x = []"
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  replicate_Suc: "replicate (Suc n) x = x # replicate n x"
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definition
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  rotate1 :: "'a list \<Rightarrow> 'a list" where
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  "rotate1 xs = (case xs of [] \<Rightarrow> [] | x#xs \<Rightarrow> xs @ [x])"
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definition
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  rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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  "rotate n = rotate1 ^^ n"
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definition
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  list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool" where
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  [code del]: "list_all2 P xs ys =
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    (length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y))"
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definition
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  sublist :: "'a list => nat set => 'a list" where
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  "sublist xs A = map fst (filter (\<lambda>p. snd p \<in> A) (zip xs [0..<size xs]))"
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primrec
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  "splice [] ys = ys"
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  "splice (x#xs) ys = (if ys=[] then x#xs else x # hd ys # splice xs (tl ys))"
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    -- {*Warning: simpset does not contain the second eqn but a derived one. *}
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text{*
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\begin{figure}[htbp]
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\fbox{
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\begin{tabular}{l}
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@{lemma "[a,b]@[c,d] = [a,b,c,d]" by simp}\\
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@{lemma "length [a,b,c] = 3" by simp}\\
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@{lemma "set [a,b,c] = {a,b,c}" by simp}\\
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@{lemma "map f [a,b,c] = [f a, f b, f c]" by simp}\\
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@{lemma "rev [a,b,c] = [c,b,a]" by simp}\\
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@{lemma "hd [a,b,c,d] = a" by simp}\\
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@{lemma "tl [a,b,c,d] = [b,c,d]" by simp}\\
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@{lemma "last [a,b,c,d] = d" by simp}\\
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@{lemma "butlast [a,b,c,d] = [a,b,c]" by simp}\\
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@{lemma[source] "filter (\<lambda>n::nat. n<2) [0,2,1] = [0,1]" by simp}\\
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@{lemma "concat [[a,b],[c,d,e],[],[f]] = [a,b,c,d,e,f]" by simp}\\
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@{lemma "foldl f x [a,b,c] = f (f (f x a) b) c" by simp}\\
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@{lemma "foldr f [a,b,c] x = f a (f b (f c x))" by simp}\\
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@{lemma "zip [a,b,c] [x,y,z] = [(a,x),(b,y),(c,z)]" by simp}\\
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@{lemma "zip [a,b] [x,y,z] = [(a,x),(b,y)]" by simp}\\
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@{lemma "splice [a,b,c] [x,y,z] = [a,x,b,y,c,z]" by simp}\\
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@{lemma "splice [a,b,c,d] [x,y] = [a,x,b,y,c,d]" by simp}\\
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@{lemma "take 2 [a,b,c,d] = [a,b]" by simp}\\
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@{lemma "take 6 [a,b,c,d] = [a,b,c,d]" by simp}\\
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@{lemma "drop 2 [a,b,c,d] = [c,d]" by simp}\\
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@{lemma "drop 6 [a,b,c,d] = []" by simp}\\
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@{lemma "takeWhile (%n::nat. n<3) [1,2,3,0] = [1,2]" by simp}\\
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@{lemma "dropWhile (%n::nat. n<3) [1,2,3,0] = [3,0]" by simp}\\
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@{lemma "distinct [2,0,1::nat]" by simp}\\
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@{lemma "remdups [2,0,2,1::nat,2] = [0,1,2]" by simp}\\
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@{lemma "remove1 2 [2,0,2,1::nat,2] = [0,2,1,2]" by simp}\\
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@{lemma "removeAll 2 [2,0,2,1::nat,2] = [0,1]" by simp}\\
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@{lemma "nth [a,b,c,d] 2 = c" by simp}\\
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@{lemma "[a,b,c,d][2 := x] = [a,b,x,d]" by simp}\\
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@{lemma "sublist [a,b,c,d,e] {0,2,3} = [a,c,d]" by (simp add:sublist_def)}\\
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@{lemma "rotate1 [a,b,c,d] = [b,c,d,a]" by (simp add:rotate1_def)}\\
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@{lemma "rotate 3 [a,b,c,d] = [d,a,b,c]" by (simp add:rotate1_def rotate_def nat_number)}\\
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@{lemma "replicate 4 a = [a,a,a,a]" by (simp add:nat_number)}\\
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@{lemma "[2..<5] = [2,3,4]" by (simp add:nat_number)}\\
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@{lemma "listsum [1,2,3::nat] = 6" by simp}
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\end{tabular}}
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\caption{Characteristic examples}
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\label{fig:Characteristic}
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\end{figure}
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Figure~\ref{fig:Characteristic} shows characteristic examples
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that should give an intuitive understanding of the above functions.
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*}
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text{* The following simple sort functions are intended for proofs,
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not for efficient implementations. *}
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context linorder
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begin
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fun sorted :: "'a list \<Rightarrow> bool" where
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"sorted [] \<longleftrightarrow> True" |
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"sorted [x] \<longleftrightarrow> True" |
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"sorted (x#y#zs) \<longleftrightarrow> x <= y \<and> sorted (y#zs)"
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primrec insort :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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"insort x [] = [x]" |
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"insort x (y#ys) = (if x <= y then (x#y#ys) else y#(insort x ys))"
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primrec sort :: "'a list \<Rightarrow> 'a list" where
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"sort [] = []" |
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"sort (x#xs) = insort x (sort xs)"
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end
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subsubsection {* List comprehension *}
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text{* Input syntax for Haskell-like list comprehension notation.
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Typical example: @{text"[(x,y). x \<leftarrow> xs, y \<leftarrow> ys, x \<noteq> y]"},
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the list of all pairs of distinct elements from @{text xs} and @{text ys}.
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The syntax is as in Haskell, except that @{text"|"} becomes a dot
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(like in Isabelle's set comprehension): @{text"[e. x \<leftarrow> xs, \<dots>]"} rather than
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\verb![e| x <- xs, ...]!.
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The qualifiers after the dot are
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\begin{description}
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\item[generators] @{text"p \<leftarrow> xs"},
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 where @{text p} is a pattern and @{text xs} an expression of list type, or
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\item[guards] @{text"b"}, where @{text b} is a boolean expression.
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%\item[local bindings] @ {text"let x = e"}.
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\end{description}
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Just like in Haskell, list comprehension is just a shorthand. To avoid
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misunderstandings, the translation into desugared form is not reversed
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upon output. Note that the translation of @{text"[e. x \<leftarrow> xs]"} is
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optmized to @{term"map (%x. e) xs"}.
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It is easy to write short list comprehensions which stand for complex
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expressions. During proofs, they may become unreadable (and
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mangled). In such cases it can be advisable to introduce separate
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definitions for the list comprehensions in question.  *}
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   312
nipkow@23209
   313
(*
nipkow@23240
   314
Proper theorem proving support would be nice. For example, if
nipkow@23192
   315
@{text"set[f x y. x \<leftarrow> xs, y \<leftarrow> ys, P x y]"}
nipkow@23192
   316
produced something like
nipkow@23209
   317
@{term"{z. EX x: set xs. EX y:set ys. P x y \<and> z = f x y}"}.
nipkow@23209
   318
*)
nipkow@23209
   319
nipkow@23240
   320
nonterminals lc_qual lc_quals
nipkow@23192
   321
nipkow@23192
   322
syntax
nipkow@23240
   323
"_listcompr" :: "'a \<Rightarrow> lc_qual \<Rightarrow> lc_quals \<Rightarrow> 'a list"  ("[_ . __")
nipkow@24349
   324
"_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ <- _")
nipkow@23240
   325
"_lc_test" :: "bool \<Rightarrow> lc_qual" ("_")
nipkow@24476
   326
(*"_lc_let" :: "letbinds => lc_qual"  ("let _")*)
nipkow@23240
   327
"_lc_end" :: "lc_quals" ("]")
nipkow@23240
   328
"_lc_quals" :: "lc_qual \<Rightarrow> lc_quals \<Rightarrow> lc_quals" (", __")
nipkow@24349
   329
"_lc_abs" :: "'a => 'b list => 'b list"
nipkow@23192
   330
nipkow@24476
   331
(* These are easier than ML code but cannot express the optimized
nipkow@24476
   332
   translation of [e. p<-xs]
nipkow@23192
   333
translations
nipkow@24349
   334
"[e. p<-xs]" => "concat(map (_lc_abs p [e]) xs)"
nipkow@23240
   335
"_listcompr e (_lc_gen p xs) (_lc_quals Q Qs)"
nipkow@24349
   336
 => "concat (map (_lc_abs p (_listcompr e Q Qs)) xs)"
nipkow@23240
   337
"[e. P]" => "if P then [e] else []"
nipkow@23240
   338
"_listcompr e (_lc_test P) (_lc_quals Q Qs)"
nipkow@23240
   339
 => "if P then (_listcompr e Q Qs) else []"
nipkow@24349
   340
"_listcompr e (_lc_let b) (_lc_quals Q Qs)"
nipkow@24349
   341
 => "_Let b (_listcompr e Q Qs)"
nipkow@24476
   342
*)
nipkow@23240
   343
nipkow@23279
   344
syntax (xsymbols)
nipkow@24349
   345
"_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _")
nipkow@23279
   346
syntax (HTML output)
nipkow@24349
   347
"_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _")
nipkow@24349
   348
nipkow@24349
   349
parse_translation (advanced) {*
nipkow@24349
   350
let
nipkow@24476
   351
  val NilC = Syntax.const @{const_name Nil};
nipkow@24476
   352
  val ConsC = Syntax.const @{const_name Cons};
nipkow@24476
   353
  val mapC = Syntax.const @{const_name map};
nipkow@24476
   354
  val concatC = Syntax.const @{const_name concat};
nipkow@24476
   355
  val IfC = Syntax.const @{const_name If};
nipkow@24476
   356
  fun singl x = ConsC $ x $ NilC;
nipkow@24476
   357
nipkow@24476
   358
   fun pat_tr ctxt p e opti = (* %x. case x of p => e | _ => [] *)
nipkow@24349
   359
    let
wenzelm@29281
   360
      val x = Free (Name.variant (fold Term.add_free_names [p, e] []) "x", dummyT);
nipkow@24476
   361
      val e = if opti then singl e else e;
nipkow@24476
   362
      val case1 = Syntax.const "_case1" $ p $ e;
nipkow@24349
   363
      val case2 = Syntax.const "_case1" $ Syntax.const Term.dummy_patternN
nipkow@24476
   364
                                        $ NilC;
nipkow@24349
   365
      val cs = Syntax.const "_case2" $ case1 $ case2
nipkow@24349
   366
      val ft = DatatypeCase.case_tr false DatatypePackage.datatype_of_constr
nipkow@24349
   367
                 ctxt [x, cs]
nipkow@24349
   368
    in lambda x ft end;
nipkow@24349
   369
nipkow@24476
   370
  fun abs_tr ctxt (p as Free(s,T)) e opti =
nipkow@24349
   371
        let val thy = ProofContext.theory_of ctxt;
nipkow@24349
   372
            val s' = Sign.intern_const thy s
nipkow@24476
   373
        in if Sign.declared_const thy s'
nipkow@24476
   374
           then (pat_tr ctxt p e opti, false)
nipkow@24476
   375
           else (lambda p e, true)
nipkow@24349
   376
        end
nipkow@24476
   377
    | abs_tr ctxt p e opti = (pat_tr ctxt p e opti, false);
nipkow@24476
   378
nipkow@24476
   379
  fun lc_tr ctxt [e, Const("_lc_test",_)$b, qs] =
nipkow@24476
   380
        let val res = case qs of Const("_lc_end",_) => singl e
nipkow@24476
   381
                      | Const("_lc_quals",_)$q$qs => lc_tr ctxt [e,q,qs];
nipkow@24476
   382
        in IfC $ b $ res $ NilC end
nipkow@24476
   383
    | lc_tr ctxt [e, Const("_lc_gen",_) $ p $ es, Const("_lc_end",_)] =
nipkow@24476
   384
        (case abs_tr ctxt p e true of
nipkow@24476
   385
           (f,true) => mapC $ f $ es
nipkow@24476
   386
         | (f, false) => concatC $ (mapC $ f $ es))
nipkow@24476
   387
    | lc_tr ctxt [e, Const("_lc_gen",_) $ p $ es, Const("_lc_quals",_)$q$qs] =
nipkow@24476
   388
        let val e' = lc_tr ctxt [e,q,qs];
nipkow@24476
   389
        in concatC $ (mapC $ (fst(abs_tr ctxt p e' false)) $ es) end
nipkow@24476
   390
nipkow@24476
   391
in [("_listcompr", lc_tr)] end
nipkow@24349
   392
*}
nipkow@23279
   393
nipkow@23240
   394
(*
nipkow@23240
   395
term "[(x,y,z). b]"
nipkow@24476
   396
term "[(x,y,z). x\<leftarrow>xs]"
nipkow@24476
   397
term "[e x y. x\<leftarrow>xs, y\<leftarrow>ys]"
nipkow@24476
   398
term "[(x,y,z). x<a, x>b]"
nipkow@24476
   399
term "[(x,y,z). x\<leftarrow>xs, x>b]"
nipkow@24476
   400
term "[(x,y,z). x<a, x\<leftarrow>xs]"
nipkow@24349
   401
term "[(x,y). Cons True x \<leftarrow> xs]"
nipkow@24349
   402
term "[(x,y,z). Cons x [] \<leftarrow> xs]"
nipkow@23240
   403
term "[(x,y,z). x<a, x>b, x=d]"
nipkow@23240
   404
term "[(x,y,z). x<a, x>b, y\<leftarrow>ys]"
nipkow@23240
   405
term "[(x,y,z). x<a, x\<leftarrow>xs,y>b]"
nipkow@23240
   406
term "[(x,y,z). x<a, x\<leftarrow>xs, y\<leftarrow>ys]"
nipkow@23240
   407
term "[(x,y,z). x\<leftarrow>xs, x>b, y<a]"
nipkow@23240
   408
term "[(x,y,z). x\<leftarrow>xs, x>b, y\<leftarrow>ys]"
nipkow@23240
   409
term "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,y>x]"
nipkow@23240
   410
term "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,z\<leftarrow>zs]"
nipkow@24349
   411
term "[(x,y). x\<leftarrow>xs, let xx = x+x, y\<leftarrow>ys, y \<noteq> xx]"
nipkow@23192
   412
*)
nipkow@23192
   413
haftmann@21061
   414
subsubsection {* @{const Nil} and @{const Cons} *}
haftmann@21061
   415
haftmann@21061
   416
lemma not_Cons_self [simp]:
haftmann@21061
   417
  "xs \<noteq> x # xs"
nipkow@13145
   418
by (induct xs) auto
wenzelm@13114
   419
wenzelm@13142
   420
lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric]
wenzelm@13114
   421
wenzelm@13142
   422
lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
nipkow@13145
   423
by (induct xs) auto
wenzelm@13114
   424
wenzelm@13142
   425
lemma length_induct:
haftmann@21061
   426
  "(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs"
nipkow@17589
   427
by (rule measure_induct [of length]) iprover
wenzelm@13114
   428
wenzelm@13114
   429
haftmann@21061
   430
subsubsection {* @{const length} *}
wenzelm@13114
   431
wenzelm@13142
   432
text {*
haftmann@21061
   433
  Needs to come before @{text "@"} because of theorem @{text
haftmann@21061
   434
  append_eq_append_conv}.
wenzelm@13142
   435
*}
wenzelm@13114
   436
wenzelm@13142
   437
lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
nipkow@13145
   438
by (induct xs) auto
wenzelm@13114
   439
wenzelm@13142
   440
lemma length_map [simp]: "length (map f xs) = length xs"
nipkow@13145
   441
by (induct xs) auto
wenzelm@13114
   442
wenzelm@13142
   443
lemma length_rev [simp]: "length (rev xs) = length xs"
nipkow@13145
   444
by (induct xs) auto
wenzelm@13114
   445
wenzelm@13142
   446
lemma length_tl [simp]: "length (tl xs) = length xs - 1"
nipkow@13145
   447
by (cases xs) auto
wenzelm@13114
   448
wenzelm@13142
   449
lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
nipkow@13145
   450
by (induct xs) auto
wenzelm@13114
   451
wenzelm@13142
   452
lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
nipkow@13145
   453
by (induct xs) auto
wenzelm@13114
   454
nipkow@23479
   455
lemma length_pos_if_in_set: "x : set xs \<Longrightarrow> length xs > 0"
nipkow@23479
   456
by auto
nipkow@23479
   457
wenzelm@13114
   458
lemma length_Suc_conv:
nipkow@13145
   459
"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
nipkow@13145
   460
by (induct xs) auto
wenzelm@13142
   461
nipkow@14025
   462
lemma Suc_length_conv:
nipkow@14025
   463
"(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
paulson@14208
   464
apply (induct xs, simp, simp)
nipkow@14025
   465
apply blast
nipkow@14025
   466
done
nipkow@14025
   467
wenzelm@25221
   468
lemma impossible_Cons: "length xs <= length ys ==> xs = x # ys = False"
wenzelm@25221
   469
  by (induct xs) auto
wenzelm@25221
   470
haftmann@26442
   471
lemma list_induct2 [consumes 1, case_names Nil Cons]:
haftmann@26442
   472
  "length xs = length ys \<Longrightarrow> P [] [] \<Longrightarrow>
haftmann@26442
   473
   (\<And>x xs y ys. length xs = length ys \<Longrightarrow> P xs ys \<Longrightarrow> P (x#xs) (y#ys))
haftmann@26442
   474
   \<Longrightarrow> P xs ys"
haftmann@26442
   475
proof (induct xs arbitrary: ys)
haftmann@26442
   476
  case Nil then show ?case by simp
haftmann@26442
   477
next
haftmann@26442
   478
  case (Cons x xs ys) then show ?case by (cases ys) simp_all
haftmann@26442
   479
qed
haftmann@26442
   480
haftmann@26442
   481
lemma list_induct3 [consumes 2, case_names Nil Cons]:
haftmann@26442
   482
  "length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> P [] [] [] \<Longrightarrow>
haftmann@26442
   483
   (\<And>x xs y ys z zs. length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> P xs ys zs \<Longrightarrow> P (x#xs) (y#ys) (z#zs))
haftmann@26442
   484
   \<Longrightarrow> P xs ys zs"
haftmann@26442
   485
proof (induct xs arbitrary: ys zs)
haftmann@26442
   486
  case Nil then show ?case by simp
haftmann@26442
   487
next
haftmann@26442
   488
  case (Cons x xs ys zs) then show ?case by (cases ys, simp_all)
haftmann@26442
   489
    (cases zs, simp_all)
haftmann@26442
   490
qed
wenzelm@13114
   491
krauss@22493
   492
lemma list_induct2': 
krauss@22493
   493
  "\<lbrakk> P [] [];
krauss@22493
   494
  \<And>x xs. P (x#xs) [];
krauss@22493
   495
  \<And>y ys. P [] (y#ys);
krauss@22493
   496
   \<And>x xs y ys. P xs ys  \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
krauss@22493
   497
 \<Longrightarrow> P xs ys"
krauss@22493
   498
by (induct xs arbitrary: ys) (case_tac x, auto)+
krauss@22493
   499
nipkow@22143
   500
lemma neq_if_length_neq: "length xs \<noteq> length ys \<Longrightarrow> (xs = ys) == False"
nipkow@24349
   501
by (rule Eq_FalseI) auto
wenzelm@24037
   502
wenzelm@24037
   503
simproc_setup list_neq ("(xs::'a list) = ys") = {*
nipkow@22143
   504
(*
nipkow@22143
   505
Reduces xs=ys to False if xs and ys cannot be of the same length.
nipkow@22143
   506
This is the case if the atomic sublists of one are a submultiset
nipkow@22143
   507
of those of the other list and there are fewer Cons's in one than the other.
nipkow@22143
   508
*)
wenzelm@24037
   509
wenzelm@24037
   510
let
nipkow@22143
   511
huffman@29856
   512
fun len (Const(@{const_name Nil},_)) acc = acc
huffman@29856
   513
  | len (Const(@{const_name Cons},_) $ _ $ xs) (ts,n) = len xs (ts,n+1)
huffman@29856
   514
  | len (Const(@{const_name append},_) $ xs $ ys) acc = len xs (len ys acc)
huffman@29856
   515
  | len (Const(@{const_name rev},_) $ xs) acc = len xs acc
huffman@29856
   516
  | len (Const(@{const_name map},_) $ _ $ xs) acc = len xs acc
nipkow@22143
   517
  | len t (ts,n) = (t::ts,n);
nipkow@22143
   518
wenzelm@24037
   519
fun list_neq _ ss ct =
nipkow@22143
   520
  let
wenzelm@24037
   521
    val (Const(_,eqT) $ lhs $ rhs) = Thm.term_of ct;
nipkow@22143
   522
    val (ls,m) = len lhs ([],0) and (rs,n) = len rhs ([],0);
nipkow@22143
   523
    fun prove_neq() =
nipkow@22143
   524
      let
nipkow@22143
   525
        val Type(_,listT::_) = eqT;
haftmann@22994
   526
        val size = HOLogic.size_const listT;
nipkow@22143
   527
        val eq_len = HOLogic.mk_eq (size $ lhs, size $ rhs);
nipkow@22143
   528
        val neq_len = HOLogic.mk_Trueprop (HOLogic.Not $ eq_len);
nipkow@22143
   529
        val thm = Goal.prove (Simplifier.the_context ss) [] [] neq_len
haftmann@22633
   530
          (K (simp_tac (Simplifier.inherit_context ss @{simpset}) 1));
haftmann@22633
   531
      in SOME (thm RS @{thm neq_if_length_neq}) end
nipkow@22143
   532
  in
wenzelm@23214
   533
    if m < n andalso submultiset (op aconv) (ls,rs) orelse
wenzelm@23214
   534
       n < m andalso submultiset (op aconv) (rs,ls)
nipkow@22143
   535
    then prove_neq() else NONE
nipkow@22143
   536
  end;
wenzelm@24037
   537
in list_neq end;
nipkow@22143
   538
*}
nipkow@22143
   539
nipkow@22143
   540
nipkow@15392
   541
subsubsection {* @{text "@"} -- append *}
wenzelm@13114
   542
wenzelm@13142
   543
lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
nipkow@13145
   544
by (induct xs) auto
wenzelm@13114
   545
wenzelm@13142
   546
lemma append_Nil2 [simp]: "xs @ [] = xs"
nipkow@13145
   547
by (induct xs) auto
nipkow@3507
   548
wenzelm@13142
   549
lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
nipkow@13145
   550
by (induct xs) auto
wenzelm@13114
   551
wenzelm@13142
   552
lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
nipkow@13145
   553
by (induct xs) auto
wenzelm@13114
   554
wenzelm@13142
   555
lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
nipkow@13145
   556
by (induct xs) auto
wenzelm@13114
   557
wenzelm@13142
   558
lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
nipkow@13145
   559
by (induct xs) auto
wenzelm@13114
   560
wenzelm@25221
   561
lemma append_eq_append_conv [simp, noatp]:
nipkow@24526
   562
 "length xs = length ys \<or> length us = length vs
berghofe@13883
   563
 ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
nipkow@24526
   564
apply (induct xs arbitrary: ys)
paulson@14208
   565
 apply (case_tac ys, simp, force)
paulson@14208
   566
apply (case_tac ys, force, simp)
nipkow@13145
   567
done
wenzelm@13142
   568
nipkow@24526
   569
lemma append_eq_append_conv2: "(xs @ ys = zs @ ts) =
nipkow@24526
   570
  (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
nipkow@24526
   571
apply (induct xs arbitrary: ys zs ts)
nipkow@14495
   572
 apply fastsimp
nipkow@14495
   573
apply(case_tac zs)
nipkow@14495
   574
 apply simp
nipkow@14495
   575
apply fastsimp
nipkow@14495
   576
done
nipkow@14495
   577
wenzelm@13142
   578
lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"
nipkow@13145
   579
by simp
wenzelm@13142
   580
wenzelm@13142
   581
lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
nipkow@13145
   582
by simp
wenzelm@13114
   583
wenzelm@13142
   584
lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"
nipkow@13145
   585
by simp
wenzelm@13114
   586
wenzelm@13142
   587
lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
nipkow@13145
   588
using append_same_eq [of _ _ "[]"] by auto
nipkow@3507
   589
wenzelm@13142
   590
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
nipkow@13145
   591
using append_same_eq [of "[]"] by auto
wenzelm@13114
   592
paulson@24286
   593
lemma hd_Cons_tl [simp,noatp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
nipkow@13145
   594
by (induct xs) auto
wenzelm@13114
   595
wenzelm@13142
   596
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
nipkow@13145
   597
by (induct xs) auto
wenzelm@13114
   598
wenzelm@13142
   599
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
nipkow@13145
   600
by (simp add: hd_append split: list.split)
wenzelm@13114
   601
wenzelm@13142
   602
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
nipkow@13145
   603
by (simp split: list.split)
wenzelm@13114
   604
wenzelm@13142
   605
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
nipkow@13145
   606
by (simp add: tl_append split: list.split)
wenzelm@13114
   607
wenzelm@13114
   608
nipkow@14300
   609
lemma Cons_eq_append_conv: "x#xs = ys@zs =
nipkow@14300
   610
 (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
nipkow@14300
   611
by(cases ys) auto
nipkow@14300
   612
nipkow@15281
   613
lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
nipkow@15281
   614
 (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
nipkow@15281
   615
by(cases ys) auto
nipkow@15281
   616
nipkow@14300
   617
wenzelm@13142
   618
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
wenzelm@13114
   619
wenzelm@13114
   620
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
nipkow@13145
   621
by simp
wenzelm@13114
   622
wenzelm@13142
   623
lemma Cons_eq_appendI:
nipkow@13145
   624
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
nipkow@13145
   625
by (drule sym) simp
wenzelm@13114
   626
wenzelm@13142
   627
lemma append_eq_appendI:
nipkow@13145
   628
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
nipkow@13145
   629
by (drule sym) simp
wenzelm@13114
   630
wenzelm@13114
   631
wenzelm@13142
   632
text {*
nipkow@13145
   633
Simplification procedure for all list equalities.
nipkow@13145
   634
Currently only tries to rearrange @{text "@"} to see if
nipkow@13145
   635
- both lists end in a singleton list,
nipkow@13145
   636
- or both lists end in the same list.
wenzelm@13142
   637
*}
wenzelm@13142
   638
wenzelm@26480
   639
ML {*
nipkow@3507
   640
local
nipkow@3507
   641
huffman@29856
   642
fun last (cons as Const(@{const_name Cons},_) $ _ $ xs) =
huffman@29856
   643
  (case xs of Const(@{const_name Nil},_) => cons | _ => last xs)
huffman@29856
   644
  | last (Const(@{const_name append},_) $ _ $ ys) = last ys
wenzelm@13462
   645
  | last t = t;
wenzelm@13114
   646
huffman@29856
   647
fun list1 (Const(@{const_name Cons},_) $ _ $ Const(@{const_name Nil},_)) = true
wenzelm@13462
   648
  | list1 _ = false;
wenzelm@13114
   649
huffman@29856
   650
fun butlast ((cons as Const(@{const_name Cons},_) $ x) $ xs) =
huffman@29856
   651
  (case xs of Const(@{const_name Nil},_) => xs | _ => cons $ butlast xs)
huffman@29856
   652
  | butlast ((app as Const(@{const_name append},_) $ xs) $ ys) = app $ butlast ys
huffman@29856
   653
  | butlast xs = Const(@{const_name Nil},fastype_of xs);
wenzelm@13114
   654
haftmann@22633
   655
val rearr_ss = HOL_basic_ss addsimps [@{thm append_assoc},
haftmann@22633
   656
  @{thm append_Nil}, @{thm append_Cons}];
wenzelm@16973
   657
wenzelm@20044
   658
fun list_eq ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
wenzelm@13462
   659
  let
wenzelm@13462
   660
    val lastl = last lhs and lastr = last rhs;
wenzelm@13462
   661
    fun rearr conv =
wenzelm@13462
   662
      let
wenzelm@13462
   663
        val lhs1 = butlast lhs and rhs1 = butlast rhs;
wenzelm@13462
   664
        val Type(_,listT::_) = eqT
wenzelm@13462
   665
        val appT = [listT,listT] ---> listT
huffman@29856
   666
        val app = Const(@{const_name append},appT)
wenzelm@13462
   667
        val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
wenzelm@13480
   668
        val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
wenzelm@20044
   669
        val thm = Goal.prove (Simplifier.the_context ss) [] [] eq
wenzelm@17877
   670
          (K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1));
skalberg@15531
   671
      in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;
wenzelm@13114
   672
wenzelm@13462
   673
  in
haftmann@22633
   674
    if list1 lastl andalso list1 lastr then rearr @{thm append1_eq_conv}
haftmann@22633
   675
    else if lastl aconv lastr then rearr @{thm append_same_eq}
skalberg@15531
   676
    else NONE
wenzelm@13462
   677
  end;
wenzelm@13462
   678
wenzelm@13114
   679
in
wenzelm@13462
   680
wenzelm@13462
   681
val list_eq_simproc =
wenzelm@28262
   682
  Simplifier.simproc (the_context ()) "list_eq" ["(xs::'a list) = ys"] (K list_eq);
wenzelm@13462
   683
wenzelm@13114
   684
end;
wenzelm@13114
   685
wenzelm@13114
   686
Addsimprocs [list_eq_simproc];
wenzelm@13114
   687
*}
wenzelm@13114
   688
wenzelm@13114
   689
nipkow@15392
   690
subsubsection {* @{text map} *}
wenzelm@13114
   691
wenzelm@13142
   692
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
nipkow@13145
   693
by (induct xs) simp_all
wenzelm@13114
   694
wenzelm@13142
   695
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
nipkow@13145
   696
by (rule ext, induct_tac xs) auto
wenzelm@13114
   697
wenzelm@13142
   698
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
nipkow@13145
   699
by (induct xs) auto
wenzelm@13114
   700
wenzelm@13142
   701
lemma map_compose: "map (f o g) xs = map f (map g xs)"
nipkow@13145
   702
by (induct xs) (auto simp add: o_def)
wenzelm@13114
   703
wenzelm@13142
   704
lemma rev_map: "rev (map f xs) = map f (rev xs)"
nipkow@13145
   705
by (induct xs) auto
wenzelm@13114
   706
nipkow@13737
   707
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
nipkow@13737
   708
by (induct xs) auto
nipkow@13737
   709
krauss@19770
   710
lemma map_cong [fundef_cong, recdef_cong]:
nipkow@13145
   711
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"
nipkow@13145
   712
-- {* a congruence rule for @{text map} *}
nipkow@13737
   713
by simp
wenzelm@13114
   714
wenzelm@13142
   715
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
nipkow@13145
   716
by (cases xs) auto
wenzelm@13114
   717
wenzelm@13142
   718
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
nipkow@13145
   719
by (cases xs) auto
wenzelm@13114
   720
paulson@18447
   721
lemma map_eq_Cons_conv:
nipkow@14025
   722
 "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
nipkow@13145
   723
by (cases xs) auto
wenzelm@13114
   724
paulson@18447
   725
lemma Cons_eq_map_conv:
nipkow@14025
   726
 "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
nipkow@14025
   727
by (cases ys) auto
nipkow@14025
   728
paulson@18447
   729
lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1]
paulson@18447
   730
lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1]
paulson@18447
   731
declare map_eq_Cons_D [dest!]  Cons_eq_map_D [dest!]
paulson@18447
   732
nipkow@14111
   733
lemma ex_map_conv:
nipkow@14111
   734
  "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
paulson@18447
   735
by(induct ys, auto simp add: Cons_eq_map_conv)
nipkow@14111
   736
nipkow@15110
   737
lemma map_eq_imp_length_eq:
haftmann@26734
   738
  assumes "map f xs = map f ys"
haftmann@26734
   739
  shows "length xs = length ys"
haftmann@26734
   740
using assms proof (induct ys arbitrary: xs)
haftmann@26734
   741
  case Nil then show ?case by simp
haftmann@26734
   742
next
haftmann@26734
   743
  case (Cons y ys) then obtain z zs where xs: "xs = z # zs" by auto
haftmann@26734
   744
  from Cons xs have "map f zs = map f ys" by simp
haftmann@26734
   745
  moreover with Cons have "length zs = length ys" by blast
haftmann@26734
   746
  with xs show ?case by simp
haftmann@26734
   747
qed
haftmann@26734
   748
  
nipkow@15110
   749
lemma map_inj_on:
nipkow@15110
   750
 "[| map f xs = map f ys; inj_on f (set xs Un set ys) |]
nipkow@15110
   751
  ==> xs = ys"
nipkow@15110
   752
apply(frule map_eq_imp_length_eq)
nipkow@15110
   753
apply(rotate_tac -1)
nipkow@15110
   754
apply(induct rule:list_induct2)
nipkow@15110
   755
 apply simp
nipkow@15110
   756
apply(simp)
nipkow@15110
   757
apply (blast intro:sym)
nipkow@15110
   758
done
nipkow@15110
   759
nipkow@15110
   760
lemma inj_on_map_eq_map:
nipkow@15110
   761
 "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@15110
   762
by(blast dest:map_inj_on)
nipkow@15110
   763
wenzelm@13114
   764
lemma map_injective:
nipkow@24526
   765
 "map f xs = map f ys ==> inj f ==> xs = ys"
nipkow@24526
   766
by (induct ys arbitrary: xs) (auto dest!:injD)
wenzelm@13114
   767
nipkow@14339
   768
lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@14339
   769
by(blast dest:map_injective)
nipkow@14339
   770
wenzelm@13114
   771
lemma inj_mapI: "inj f ==> inj (map f)"
nipkow@17589
   772
by (iprover dest: map_injective injD intro: inj_onI)
wenzelm@13114
   773
wenzelm@13114
   774
lemma inj_mapD: "inj (map f) ==> inj f"
paulson@14208
   775
apply (unfold inj_on_def, clarify)
nipkow@13145
   776
apply (erule_tac x = "[x]" in ballE)
paulson@14208
   777
 apply (erule_tac x = "[y]" in ballE, simp, blast)
nipkow@13145
   778
apply blast
nipkow@13145
   779
done
wenzelm@13114
   780
nipkow@14339
   781
lemma inj_map[iff]: "inj (map f) = inj f"
nipkow@13145
   782
by (blast dest: inj_mapD intro: inj_mapI)
wenzelm@13114
   783
nipkow@15303
   784
lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
nipkow@15303
   785
apply(rule inj_onI)
nipkow@15303
   786
apply(erule map_inj_on)
nipkow@15303
   787
apply(blast intro:inj_onI dest:inj_onD)
nipkow@15303
   788
done
nipkow@15303
   789
kleing@14343
   790
lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
kleing@14343
   791
by (induct xs, auto)
wenzelm@13114
   792
nipkow@14402
   793
lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
nipkow@14402
   794
by (induct xs) auto
nipkow@14402
   795
nipkow@15110
   796
lemma map_fst_zip[simp]:
nipkow@15110
   797
  "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"
nipkow@15110
   798
by (induct rule:list_induct2, simp_all)
nipkow@15110
   799
nipkow@15110
   800
lemma map_snd_zip[simp]:
nipkow@15110
   801
  "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"
nipkow@15110
   802
by (induct rule:list_induct2, simp_all)
nipkow@15110
   803
nipkow@15110
   804
nipkow@15392
   805
subsubsection {* @{text rev} *}
wenzelm@13114
   806
wenzelm@13142
   807
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
nipkow@13145
   808
by (induct xs) auto
wenzelm@13114
   809
wenzelm@13142
   810
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
nipkow@13145
   811
by (induct xs) auto
wenzelm@13114
   812
kleing@15870
   813
lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"
kleing@15870
   814
by auto
kleing@15870
   815
wenzelm@13142
   816
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
nipkow@13145
   817
by (induct xs) auto
wenzelm@13114
   818
wenzelm@13142
   819
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
nipkow@13145
   820
by (induct xs) auto
wenzelm@13114
   821
kleing@15870
   822
lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"
kleing@15870
   823
by (cases xs) auto
kleing@15870
   824
kleing@15870
   825
lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"
kleing@15870
   826
by (cases xs) auto
kleing@15870
   827
haftmann@21061
   828
lemma rev_is_rev_conv [iff]: "(rev xs = rev ys) = (xs = ys)"
haftmann@21061
   829
apply (induct xs arbitrary: ys, force)
paulson@14208
   830
apply (case_tac ys, simp, force)
nipkow@13145
   831
done
wenzelm@13114
   832
nipkow@15439
   833
lemma inj_on_rev[iff]: "inj_on rev A"
nipkow@15439
   834
by(simp add:inj_on_def)
nipkow@15439
   835
wenzelm@13366
   836
lemma rev_induct [case_names Nil snoc]:
wenzelm@13366
   837
  "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
berghofe@15489
   838
apply(simplesubst rev_rev_ident[symmetric])
nipkow@13145
   839
apply(rule_tac list = "rev xs" in list.induct, simp_all)
nipkow@13145
   840
done
wenzelm@13114
   841
wenzelm@13366
   842
lemma rev_exhaust [case_names Nil snoc]:
wenzelm@13366
   843
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
nipkow@13145
   844
by (induct xs rule: rev_induct) auto
wenzelm@13114
   845
wenzelm@13366
   846
lemmas rev_cases = rev_exhaust
wenzelm@13366
   847
nipkow@18423
   848
lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])"
nipkow@18423
   849
by(rule rev_cases[of xs]) auto
nipkow@18423
   850
wenzelm@13114
   851
nipkow@15392
   852
subsubsection {* @{text set} *}
wenzelm@13114
   853
wenzelm@13142
   854
lemma finite_set [iff]: "finite (set xs)"
nipkow@13145
   855
by (induct xs) auto
wenzelm@13114
   856
wenzelm@13142
   857
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
nipkow@13145
   858
by (induct xs) auto
wenzelm@13114
   859
nipkow@17830
   860
lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"
nipkow@17830
   861
by(cases xs) auto
oheimb@14099
   862
wenzelm@13142
   863
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
nipkow@13145
   864
by auto
wenzelm@13114
   865
oheimb@14099
   866
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
oheimb@14099
   867
by auto
oheimb@14099
   868
wenzelm@13142
   869
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
nipkow@13145
   870
by (induct xs) auto
wenzelm@13114
   871
nipkow@15245
   872
lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
nipkow@15245
   873
by(induct xs) auto
nipkow@15245
   874
wenzelm@13142
   875
lemma set_rev [simp]: "set (rev xs) = set xs"
nipkow@13145
   876
by (induct xs) auto
wenzelm@13114
   877
wenzelm@13142
   878
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
nipkow@13145
   879
by (induct xs) auto
wenzelm@13114
   880
wenzelm@13142
   881
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
nipkow@13145
   882
by (induct xs) auto
wenzelm@13114
   883
nipkow@15425
   884
lemma set_upt [simp]: "set[i..<j] = {k. i \<le> k \<and> k < j}"
paulson@14208
   885
apply (induct j, simp_all)
paulson@14208
   886
apply (erule ssubst, auto)
nipkow@13145
   887
done
wenzelm@13114
   888
wenzelm@13142
   889
wenzelm@25221
   890
lemma split_list: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs"
nipkow@18049
   891
proof (induct xs)
nipkow@26073
   892
  case Nil thus ?case by simp
nipkow@26073
   893
next
nipkow@26073
   894
  case Cons thus ?case by (auto intro: Cons_eq_appendI)
nipkow@26073
   895
qed
nipkow@26073
   896
haftmann@26734
   897
lemma in_set_conv_decomp: "x \<in> set xs \<longleftrightarrow> (\<exists>ys zs. xs = ys @ x # zs)"
haftmann@26734
   898
  by (auto elim: split_list)
nipkow@26073
   899
nipkow@26073
   900
lemma split_list_first: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys"
nipkow@26073
   901
proof (induct xs)
nipkow@26073
   902
  case Nil thus ?case by simp
nipkow@18049
   903
next
nipkow@18049
   904
  case (Cons a xs)
nipkow@18049
   905
  show ?case
nipkow@18049
   906
  proof cases
wenzelm@25221
   907
    assume "x = a" thus ?case using Cons by fastsimp
nipkow@18049
   908
  next
nipkow@26073
   909
    assume "x \<noteq> a" thus ?case using Cons by(fastsimp intro!: Cons_eq_appendI)
nipkow@26073
   910
  qed
nipkow@26073
   911
qed
nipkow@26073
   912
nipkow@26073
   913
lemma in_set_conv_decomp_first:
nipkow@26073
   914
  "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
haftmann@26734
   915
  by (auto dest!: split_list_first)
nipkow@26073
   916
nipkow@26073
   917
lemma split_list_last: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs"
nipkow@26073
   918
proof (induct xs rule:rev_induct)
nipkow@26073
   919
  case Nil thus ?case by simp
nipkow@26073
   920
next
nipkow@26073
   921
  case (snoc a xs)
nipkow@26073
   922
  show ?case
nipkow@26073
   923
  proof cases
nipkow@26073
   924
    assume "x = a" thus ?case using snoc by simp (metis ex_in_conv set_empty2)
nipkow@26073
   925
  next
nipkow@26073
   926
    assume "x \<noteq> a" thus ?case using snoc by fastsimp
nipkow@18049
   927
  qed
nipkow@18049
   928
qed
nipkow@18049
   929
nipkow@26073
   930
lemma in_set_conv_decomp_last:
nipkow@26073
   931
  "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs)"
haftmann@26734
   932
  by (auto dest!: split_list_last)
nipkow@26073
   933
nipkow@26073
   934
lemma split_list_prop: "\<exists>x \<in> set xs. P x \<Longrightarrow> \<exists>ys x zs. xs = ys @ x # zs & P x"
nipkow@26073
   935
proof (induct xs)
nipkow@26073
   936
  case Nil thus ?case by simp
nipkow@26073
   937
next
nipkow@26073
   938
  case Cons thus ?case
nipkow@26073
   939
    by(simp add:Bex_def)(metis append_Cons append.simps(1))
nipkow@26073
   940
qed
nipkow@26073
   941
nipkow@26073
   942
lemma split_list_propE:
haftmann@26734
   943
  assumes "\<exists>x \<in> set xs. P x"
haftmann@26734
   944
  obtains ys x zs where "xs = ys @ x # zs" and "P x"
haftmann@26734
   945
using split_list_prop [OF assms] by blast
nipkow@26073
   946
nipkow@26073
   947
lemma split_list_first_prop:
nipkow@26073
   948
  "\<exists>x \<in> set xs. P x \<Longrightarrow>
nipkow@26073
   949
   \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y)"
haftmann@26734
   950
proof (induct xs)
nipkow@26073
   951
  case Nil thus ?case by simp
nipkow@26073
   952
next
nipkow@26073
   953
  case (Cons x xs)
nipkow@26073
   954
  show ?case
nipkow@26073
   955
  proof cases
nipkow@26073
   956
    assume "P x"
haftmann@26734
   957
    thus ?thesis by simp
haftmann@26734
   958
      (metis Un_upper1 contra_subsetD in_set_conv_decomp_first self_append_conv2 set_append)
nipkow@26073
   959
  next
nipkow@26073
   960
    assume "\<not> P x"
nipkow@26073
   961
    hence "\<exists>x\<in>set xs. P x" using Cons(2) by simp
nipkow@26073
   962
    thus ?thesis using `\<not> P x` Cons(1) by (metis append_Cons set_ConsD)
nipkow@26073
   963
  qed
nipkow@26073
   964
qed
nipkow@26073
   965
nipkow@26073
   966
lemma split_list_first_propE:
haftmann@26734
   967
  assumes "\<exists>x \<in> set xs. P x"
haftmann@26734
   968
  obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>y \<in> set ys. \<not> P y"
haftmann@26734
   969
using split_list_first_prop [OF assms] by blast
nipkow@26073
   970
nipkow@26073
   971
lemma split_list_first_prop_iff:
nipkow@26073
   972
  "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
nipkow@26073
   973
   (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y))"
haftmann@26734
   974
by (rule, erule split_list_first_prop) auto
nipkow@26073
   975
nipkow@26073
   976
lemma split_list_last_prop:
nipkow@26073
   977
  "\<exists>x \<in> set xs. P x \<Longrightarrow>
nipkow@26073
   978
   \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z)"
nipkow@26073
   979
proof(induct xs rule:rev_induct)
nipkow@26073
   980
  case Nil thus ?case by simp
nipkow@26073
   981
next
nipkow@26073
   982
  case (snoc x xs)
nipkow@26073
   983
  show ?case
nipkow@26073
   984
  proof cases
nipkow@26073
   985
    assume "P x" thus ?thesis by (metis emptyE set_empty)
nipkow@26073
   986
  next
nipkow@26073
   987
    assume "\<not> P x"
nipkow@26073
   988
    hence "\<exists>x\<in>set xs. P x" using snoc(2) by simp
nipkow@26073
   989
    thus ?thesis using `\<not> P x` snoc(1) by fastsimp
nipkow@26073
   990
  qed
nipkow@26073
   991
qed
nipkow@26073
   992
nipkow@26073
   993
lemma split_list_last_propE:
haftmann@26734
   994
  assumes "\<exists>x \<in> set xs. P x"
haftmann@26734
   995
  obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>z \<in> set zs. \<not> P z"
haftmann@26734
   996
using split_list_last_prop [OF assms] by blast
nipkow@26073
   997
nipkow@26073
   998
lemma split_list_last_prop_iff:
nipkow@26073
   999
  "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
nipkow@26073
  1000
   (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z))"
haftmann@26734
  1001
by (metis split_list_last_prop [where P=P] in_set_conv_decomp)
nipkow@26073
  1002
nipkow@26073
  1003
lemma finite_list: "finite A ==> EX xs. set xs = A"
haftmann@26734
  1004
  by (erule finite_induct)
haftmann@26734
  1005
    (auto simp add: set.simps(2) [symmetric] simp del: set.simps(2))
paulson@13508
  1006
kleing@14388
  1007
lemma card_length: "card (set xs) \<le> length xs"
kleing@14388
  1008
by (induct xs) (auto simp add: card_insert_if)
wenzelm@13114
  1009
haftmann@26442
  1010
lemma set_minus_filter_out:
haftmann@26442
  1011
  "set xs - {y} = set (filter (\<lambda>x. \<not> (x = y)) xs)"
haftmann@26442
  1012
  by (induct xs) auto
paulson@15168
  1013
nipkow@15392
  1014
subsubsection {* @{text filter} *}
wenzelm@13114
  1015
wenzelm@13142
  1016
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
nipkow@13145
  1017
by (induct xs) auto
wenzelm@13114
  1018
nipkow@15305
  1019
lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
nipkow@15305
  1020
by (induct xs) simp_all
nipkow@15305
  1021
wenzelm@13142
  1022
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
nipkow@13145
  1023
by (induct xs) auto
wenzelm@13114
  1024
nipkow@16998
  1025
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
nipkow@16998
  1026
by (induct xs) (auto simp add: le_SucI)
nipkow@16998
  1027
nipkow@18423
  1028
lemma sum_length_filter_compl:
nipkow@18423
  1029
  "length(filter P xs) + length(filter (%x. ~P x) xs) = length xs"
nipkow@18423
  1030
by(induct xs) simp_all
nipkow@18423
  1031
wenzelm@13142
  1032
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
nipkow@13145
  1033
by (induct xs) auto
wenzelm@13114
  1034
wenzelm@13142
  1035
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
nipkow@13145
  1036
by (induct xs) auto
wenzelm@13114
  1037
nipkow@16998
  1038
lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" 
nipkow@24349
  1039
by (induct xs) simp_all
nipkow@16998
  1040
nipkow@16998
  1041
lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
nipkow@16998
  1042
apply (induct xs)
nipkow@16998
  1043
 apply auto
nipkow@16998
  1044
apply(cut_tac P=P and xs=xs in length_filter_le)
nipkow@16998
  1045
apply simp
nipkow@16998
  1046
done
wenzelm@13114
  1047
nipkow@16965
  1048
lemma filter_map:
nipkow@16965
  1049
  "filter P (map f xs) = map f (filter (P o f) xs)"
nipkow@16965
  1050
by (induct xs) simp_all
nipkow@16965
  1051
nipkow@16965
  1052
lemma length_filter_map[simp]:
nipkow@16965
  1053
  "length (filter P (map f xs)) = length(filter (P o f) xs)"
nipkow@16965
  1054
by (simp add:filter_map)
nipkow@16965
  1055
wenzelm@13142
  1056
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
nipkow@13145
  1057
by auto
wenzelm@13114
  1058
nipkow@15246
  1059
lemma length_filter_less:
nipkow@15246
  1060
  "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
nipkow@15246
  1061
proof (induct xs)
nipkow@15246
  1062
  case Nil thus ?case by simp
nipkow@15246
  1063
next
nipkow@15246
  1064
  case (Cons x xs) thus ?case
nipkow@15246
  1065
    apply (auto split:split_if_asm)
nipkow@15246
  1066
    using length_filter_le[of P xs] apply arith
nipkow@15246
  1067
  done
nipkow@15246
  1068
qed
wenzelm@13114
  1069
nipkow@15281
  1070
lemma length_filter_conv_card:
nipkow@15281
  1071
 "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
nipkow@15281
  1072
proof (induct xs)
nipkow@15281
  1073
  case Nil thus ?case by simp
nipkow@15281
  1074
next
nipkow@15281
  1075
  case (Cons x xs)
nipkow@15281
  1076
  let ?S = "{i. i < length xs & p(xs!i)}"
nipkow@15281
  1077
  have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
nipkow@15281
  1078
  show ?case (is "?l = card ?S'")
nipkow@15281
  1079
  proof (cases)
nipkow@15281
  1080
    assume "p x"
nipkow@15281
  1081
    hence eq: "?S' = insert 0 (Suc ` ?S)"
nipkow@25162
  1082
      by(auto simp: image_def split:nat.split dest:gr0_implies_Suc)
nipkow@15281
  1083
    have "length (filter p (x # xs)) = Suc(card ?S)"
wenzelm@23388
  1084
      using Cons `p x` by simp
nipkow@15281
  1085
    also have "\<dots> = Suc(card(Suc ` ?S))" using fin
nipkow@15281
  1086
      by (simp add: card_image inj_Suc)
nipkow@15281
  1087
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
  1088
      by (simp add:card_insert_if) (simp add:image_def)
nipkow@15281
  1089
    finally show ?thesis .
nipkow@15281
  1090
  next
nipkow@15281
  1091
    assume "\<not> p x"
nipkow@15281
  1092
    hence eq: "?S' = Suc ` ?S"
nipkow@25162
  1093
      by(auto simp add: image_def split:nat.split elim:lessE)
nipkow@15281
  1094
    have "length (filter p (x # xs)) = card ?S"
wenzelm@23388
  1095
      using Cons `\<not> p x` by simp
nipkow@15281
  1096
    also have "\<dots> = card(Suc ` ?S)" using fin
nipkow@15281
  1097
      by (simp add: card_image inj_Suc)
nipkow@15281
  1098
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
  1099
      by (simp add:card_insert_if)
nipkow@15281
  1100
    finally show ?thesis .
nipkow@15281
  1101
  qed
nipkow@15281
  1102
qed
nipkow@15281
  1103
nipkow@17629
  1104
lemma Cons_eq_filterD:
nipkow@17629
  1105
 "x#xs = filter P ys \<Longrightarrow>
nipkow@17629
  1106
  \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
wenzelm@19585
  1107
  (is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs")
nipkow@17629
  1108
proof(induct ys)
nipkow@17629
  1109
  case Nil thus ?case by simp
nipkow@17629
  1110
next
nipkow@17629
  1111
  case (Cons y ys)
nipkow@17629
  1112
  show ?case (is "\<exists>x. ?Q x")
nipkow@17629
  1113
  proof cases
nipkow@17629
  1114
    assume Py: "P y"
nipkow@17629
  1115
    show ?thesis
nipkow@17629
  1116
    proof cases
wenzelm@25221
  1117
      assume "x = y"
wenzelm@25221
  1118
      with Py Cons.prems have "?Q []" by simp
wenzelm@25221
  1119
      then show ?thesis ..
nipkow@17629
  1120
    next
wenzelm@25221
  1121
      assume "x \<noteq> y"
wenzelm@25221
  1122
      with Py Cons.prems show ?thesis by simp
nipkow@17629
  1123
    qed
nipkow@17629
  1124
  next
wenzelm@25221
  1125
    assume "\<not> P y"
wenzelm@25221
  1126
    with Cons obtain us vs where "?P (y#ys) (y#us) vs" by fastsimp
wenzelm@25221
  1127
    then have "?Q (y#us)" by simp
wenzelm@25221
  1128
    then show ?thesis ..
nipkow@17629
  1129
  qed
nipkow@17629
  1130
qed
nipkow@17629
  1131
nipkow@17629
  1132
lemma filter_eq_ConsD:
nipkow@17629
  1133
 "filter P ys = x#xs \<Longrightarrow>
nipkow@17629
  1134
  \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
nipkow@17629
  1135
by(rule Cons_eq_filterD) simp
nipkow@17629
  1136
nipkow@17629
  1137
lemma filter_eq_Cons_iff:
nipkow@17629
  1138
 "(filter P ys = x#xs) =
nipkow@17629
  1139
  (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
nipkow@17629
  1140
by(auto dest:filter_eq_ConsD)
nipkow@17629
  1141
nipkow@17629
  1142
lemma Cons_eq_filter_iff:
nipkow@17629
  1143
 "(x#xs = filter P ys) =
nipkow@17629
  1144
  (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
nipkow@17629
  1145
by(auto dest:Cons_eq_filterD)
nipkow@17629
  1146
krauss@19770
  1147
lemma filter_cong[fundef_cong, recdef_cong]:
nipkow@17501
  1148
 "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys"
nipkow@17501
  1149
apply simp
nipkow@17501
  1150
apply(erule thin_rl)
nipkow@17501
  1151
by (induct ys) simp_all
nipkow@17501
  1152
nipkow@15281
  1153
haftmann@26442
  1154
subsubsection {* List partitioning *}
haftmann@26442
  1155
haftmann@26442
  1156
primrec partition :: "('a \<Rightarrow> bool) \<Rightarrow>'a list \<Rightarrow> 'a list \<times> 'a list" where
haftmann@26442
  1157
  "partition P [] = ([], [])"
haftmann@26442
  1158
  | "partition P (x # xs) = 
haftmann@26442
  1159
      (let (yes, no) = partition P xs
haftmann@26442
  1160
      in if P x then (x # yes, no) else (yes, x # no))"
haftmann@26442
  1161
haftmann@26442
  1162
lemma partition_filter1:
haftmann@26442
  1163
    "fst (partition P xs) = filter P xs"
haftmann@26442
  1164
by (induct xs) (auto simp add: Let_def split_def)
haftmann@26442
  1165
haftmann@26442
  1166
lemma partition_filter2:
haftmann@26442
  1167
    "snd (partition P xs) = filter (Not o P) xs"
haftmann@26442
  1168
by (induct xs) (auto simp add: Let_def split_def)
haftmann@26442
  1169
haftmann@26442
  1170
lemma partition_P:
haftmann@26442
  1171
  assumes "partition P xs = (yes, no)"
haftmann@26442
  1172
  shows "(\<forall>p \<in> set yes.  P p) \<and> (\<forall>p  \<in> set no. \<not> P p)"
haftmann@26442
  1173
proof -
haftmann@26442
  1174
  from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
haftmann@26442
  1175
    by simp_all
haftmann@26442
  1176
  then show ?thesis by (simp_all add: partition_filter1 partition_filter2)
haftmann@26442
  1177
qed
haftmann@26442
  1178
haftmann@26442
  1179
lemma partition_set:
haftmann@26442
  1180
  assumes "partition P xs = (yes, no)"
haftmann@26442
  1181
  shows "set yes \<union> set no = set xs"
haftmann@26442
  1182
proof -
haftmann@26442
  1183
  from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
haftmann@26442
  1184
    by simp_all
haftmann@26442
  1185
  then show ?thesis by (auto simp add: partition_filter1 partition_filter2) 
haftmann@26442
  1186
qed
haftmann@26442
  1187
haftmann@26442
  1188
nipkow@15392
  1189
subsubsection {* @{text concat} *}
wenzelm@13114
  1190
wenzelm@13142
  1191
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
nipkow@13145
  1192
by (induct xs) auto
wenzelm@13114
  1193
paulson@18447
  1194
lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
  1195
by (induct xss) auto
wenzelm@13114
  1196
paulson@18447
  1197
lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
  1198
by (induct xss) auto
wenzelm@13114
  1199
nipkow@24308
  1200
lemma set_concat [simp]: "set (concat xs) = (UN x:set xs. set x)"
nipkow@13145
  1201
by (induct xs) auto
wenzelm@13114
  1202
nipkow@24476
  1203
lemma concat_map_singleton[simp]: "concat(map (%x. [f x]) xs) = map f xs"
nipkow@24349
  1204
by (induct xs) auto
nipkow@24349
  1205
wenzelm@13142
  1206
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
nipkow@13145
  1207
by (induct xs) auto
wenzelm@13114
  1208
wenzelm@13142
  1209
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
nipkow@13145
  1210
by (induct xs) auto
wenzelm@13114
  1211
wenzelm@13142
  1212
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
nipkow@13145
  1213
by (induct xs) auto
wenzelm@13114
  1214
wenzelm@13114
  1215
nipkow@15392
  1216
subsubsection {* @{text nth} *}
wenzelm@13114
  1217
haftmann@29827
  1218
lemma nth_Cons_0 [simp, code]: "(x # xs)!0 = x"
nipkow@13145
  1219
by auto
wenzelm@13114
  1220
haftmann@29827
  1221
lemma nth_Cons_Suc [simp, code]: "(x # xs)!(Suc n) = xs!n"
nipkow@13145
  1222
by auto
wenzelm@13114
  1223
wenzelm@13142
  1224
declare nth.simps [simp del]
wenzelm@13114
  1225
wenzelm@13114
  1226
lemma nth_append:
nipkow@24526
  1227
  "(xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
nipkow@24526
  1228
apply (induct xs arbitrary: n, simp)
paulson@14208
  1229
apply (case_tac n, auto)
nipkow@13145
  1230
done
wenzelm@13114
  1231
nipkow@14402
  1232
lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
wenzelm@25221
  1233
by (induct xs) auto
nipkow@14402
  1234
nipkow@14402
  1235
lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
wenzelm@25221
  1236
by (induct xs) auto
nipkow@14402
  1237
nipkow@24526
  1238
lemma nth_map [simp]: "n < length xs ==> (map f xs)!n = f(xs!n)"
nipkow@24526
  1239
apply (induct xs arbitrary: n, simp)
paulson@14208
  1240
apply (case_tac n, auto)
nipkow@13145
  1241
done
wenzelm@13114
  1242
nipkow@18423
  1243
lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0"
nipkow@18423
  1244
by(cases xs) simp_all
nipkow@18423
  1245
nipkow@18049
  1246
nipkow@18049
  1247
lemma list_eq_iff_nth_eq:
nipkow@24526
  1248
 "(xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))"
nipkow@24526
  1249
apply(induct xs arbitrary: ys)
paulson@24632
  1250
 apply force
nipkow@18049
  1251
apply(case_tac ys)
nipkow@18049
  1252
 apply simp
nipkow@18049
  1253
apply(simp add:nth_Cons split:nat.split)apply blast
nipkow@18049
  1254
done
nipkow@18049
  1255
wenzelm@13142
  1256
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
paulson@15251
  1257
apply (induct xs, simp, simp)
nipkow@13145
  1258
apply safe
paulson@24632
  1259
apply (metis nat_case_0 nth.simps zero_less_Suc)
paulson@24632
  1260
apply (metis less_Suc_eq_0_disj nth_Cons_Suc)
paulson@14208
  1261
apply (case_tac i, simp)
paulson@24632
  1262
apply (metis diff_Suc_Suc nat_case_Suc nth.simps zero_less_diff)
nipkow@13145
  1263
done
wenzelm@13114
  1264
nipkow@17501
  1265
lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)"
nipkow@17501
  1266
by(auto simp:set_conv_nth)
nipkow@17501
  1267
nipkow@13145
  1268
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
nipkow@13145
  1269
by (auto simp add: set_conv_nth)
wenzelm@13114
  1270
wenzelm@13142
  1271
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
nipkow@13145
  1272
by (auto simp add: set_conv_nth)
wenzelm@13114
  1273
wenzelm@13114
  1274
lemma all_nth_imp_all_set:
nipkow@13145
  1275
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
nipkow@13145
  1276
by (auto simp add: set_conv_nth)
wenzelm@13114
  1277
wenzelm@13114
  1278
lemma all_set_conv_all_nth:
nipkow@13145
  1279
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
nipkow@13145
  1280
by (auto simp add: set_conv_nth)
wenzelm@13114
  1281
kleing@25296
  1282
lemma rev_nth:
kleing@25296
  1283
  "n < size xs \<Longrightarrow> rev xs ! n = xs ! (length xs - Suc n)"
kleing@25296
  1284
proof (induct xs arbitrary: n)
kleing@25296
  1285
  case Nil thus ?case by simp
kleing@25296
  1286
next
kleing@25296
  1287
  case (Cons x xs)
kleing@25296
  1288
  hence n: "n < Suc (length xs)" by simp
kleing@25296
  1289
  moreover
kleing@25296
  1290
  { assume "n < length xs"
kleing@25296
  1291
    with n obtain n' where "length xs - n = Suc n'"
kleing@25296
  1292
      by (cases "length xs - n", auto)
kleing@25296
  1293
    moreover
kleing@25296
  1294
    then have "length xs - Suc n = n'" by simp
kleing@25296
  1295
    ultimately
kleing@25296
  1296
    have "xs ! (length xs - Suc n) = (x # xs) ! (length xs - n)" by simp
kleing@25296
  1297
  }
kleing@25296
  1298
  ultimately
kleing@25296
  1299
  show ?case by (clarsimp simp add: Cons nth_append)
kleing@25296
  1300
qed
wenzelm@13114
  1301
nipkow@31159
  1302
lemma Skolem_list_nth:
nipkow@31159
  1303
  "(ALL i<k. EX x. P i x) = (EX xs. size xs = k & (ALL i<k. P i (xs!i)))"
nipkow@31159
  1304
  (is "_ = (EX xs. ?P k xs)")
nipkow@31159
  1305
proof(induct k)
nipkow@31159
  1306
  case 0 show ?case by simp
nipkow@31159
  1307
next
nipkow@31159
  1308
  case (Suc k)
nipkow@31159
  1309
  show ?case (is "?L = ?R" is "_ = (EX xs. ?P' xs)")
nipkow@31159
  1310
  proof
nipkow@31159
  1311
    assume "?R" thus "?L" using Suc by auto
nipkow@31159
  1312
  next
nipkow@31159
  1313
    assume "?L"
nipkow@31159
  1314
    with Suc obtain x xs where "?P k xs & P k x" by (metis less_Suc_eq)
nipkow@31159
  1315
    hence "?P'(xs@[x])" by(simp add:nth_append less_Suc_eq)
nipkow@31159
  1316
    thus "?R" ..
nipkow@31159
  1317
  qed
nipkow@31159
  1318
qed
nipkow@31159
  1319
nipkow@31159
  1320
nipkow@15392
  1321
subsubsection {* @{text list_update} *}
wenzelm@13114
  1322
nipkow@24526
  1323
lemma length_list_update [simp]: "length(xs[i:=x]) = length xs"
nipkow@24526
  1324
by (induct xs arbitrary: i) (auto split: nat.split)
wenzelm@13114
  1325
wenzelm@13114
  1326
lemma nth_list_update:
nipkow@24526
  1327
"i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
nipkow@24526
  1328
by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
  1329
wenzelm@13142
  1330
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
nipkow@13145
  1331
by (simp add: nth_list_update)
wenzelm@13114
  1332
nipkow@24526
  1333
lemma nth_list_update_neq [simp]: "i \<noteq> j ==> xs[i:=x]!j = xs!j"
nipkow@24526
  1334
by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
  1335
nipkow@24526
  1336
lemma list_update_id[simp]: "xs[i := xs!i] = xs"
nipkow@24526
  1337
by (induct xs arbitrary: i) (simp_all split:nat.splits)
nipkow@24526
  1338
nipkow@24526
  1339
lemma list_update_beyond[simp]: "length xs \<le> i \<Longrightarrow> xs[i:=x] = xs"
nipkow@24526
  1340
apply (induct xs arbitrary: i)
nipkow@17501
  1341
 apply simp
nipkow@17501
  1342
apply (case_tac i)
nipkow@17501
  1343
apply simp_all
nipkow@17501
  1344
done
nipkow@17501
  1345
nipkow@31077
  1346
lemma list_update_nonempty[simp]: "xs[k:=x] = [] \<longleftrightarrow> xs=[]"
nipkow@31077
  1347
by(metis length_0_conv length_list_update)
nipkow@31077
  1348
wenzelm@13114
  1349
lemma list_update_same_conv:
nipkow@24526
  1350
"i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
nipkow@24526
  1351
by (induct xs arbitrary: i) (auto split: nat.split)
wenzelm@13114
  1352
nipkow@14187
  1353
lemma list_update_append1:
nipkow@24526
  1354
 "i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
nipkow@24526
  1355
apply (induct xs arbitrary: i, simp)
nipkow@14187
  1356
apply(simp split:nat.split)
nipkow@14187
  1357
done
nipkow@14187
  1358
kleing@15868
  1359
lemma list_update_append:
nipkow@24526
  1360
  "(xs @ ys) [n:= x] = 
kleing@15868
  1361
  (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))"
nipkow@24526
  1362
by (induct xs arbitrary: n) (auto split:nat.splits)
kleing@15868
  1363
nipkow@14402
  1364
lemma list_update_length [simp]:
nipkow@14402
  1365
 "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
nipkow@14402
  1366
by (induct xs, auto)
nipkow@14402
  1367
wenzelm@13114
  1368
lemma update_zip:
nipkow@31080
  1369
  "(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
nipkow@24526
  1370
by (induct ys arbitrary: i xy xs) (auto, case_tac xs, auto split: nat.split)
nipkow@24526
  1371
nipkow@24526
  1372
lemma set_update_subset_insert: "set(xs[i:=x]) <= insert x (set xs)"
nipkow@24526
  1373
by (induct xs arbitrary: i) (auto split: nat.split)
wenzelm@13114
  1374
wenzelm@13114
  1375
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
nipkow@13145
  1376
by (blast dest!: set_update_subset_insert [THEN subsetD])
wenzelm@13114
  1377
nipkow@24526
  1378
lemma set_update_memI: "n < length xs \<Longrightarrow> x \<in> set (xs[n := x])"
nipkow@24526
  1379
by (induct xs arbitrary: n) (auto split:nat.splits)
kleing@15868
  1380
nipkow@31077
  1381
lemma list_update_overwrite[simp]:
haftmann@24796
  1382
  "xs [i := x, i := y] = xs [i := y]"
nipkow@31077
  1383
apply (induct xs arbitrary: i) apply simp
nipkow@31077
  1384
apply (case_tac i, simp_all)
haftmann@24796
  1385
done
haftmann@24796
  1386
haftmann@24796
  1387
lemma list_update_swap:
haftmann@24796
  1388
  "i \<noteq> i' \<Longrightarrow> xs [i := x, i' := x'] = xs [i' := x', i := x]"
haftmann@24796
  1389
apply (induct xs arbitrary: i i')
haftmann@24796
  1390
apply simp
haftmann@24796
  1391
apply (case_tac i, case_tac i')
haftmann@24796
  1392
apply auto
haftmann@24796
  1393
apply (case_tac i')
haftmann@24796
  1394
apply auto
haftmann@24796
  1395
done
haftmann@24796
  1396
haftmann@29827
  1397
lemma list_update_code [code]:
haftmann@29827
  1398
  "[][i := y] = []"
haftmann@29827
  1399
  "(x # xs)[0 := y] = y # xs"
haftmann@29827
  1400
  "(x # xs)[Suc i := y] = x # xs[i := y]"
haftmann@29827
  1401
  by simp_all
haftmann@29827
  1402
wenzelm@13114
  1403
nipkow@15392
  1404
subsubsection {* @{text last} and @{text butlast} *}
wenzelm@13114
  1405
wenzelm@13142
  1406
lemma last_snoc [simp]: "last (xs @ [x]) = x"
nipkow@13145
  1407
by (induct xs) auto
wenzelm@13114
  1408
wenzelm@13142
  1409
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
nipkow@13145
  1410
by (induct xs) auto
wenzelm@13114
  1411
nipkow@14302
  1412
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
nipkow@14302
  1413
by(simp add:last.simps)
nipkow@14302
  1414
nipkow@14302
  1415
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
nipkow@14302
  1416
by(simp add:last.simps)
nipkow@14302
  1417
nipkow@14302
  1418
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
nipkow@14302
  1419
by (induct xs) (auto)
nipkow@14302
  1420
nipkow@14302
  1421
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
nipkow@14302
  1422
by(simp add:last_append)
nipkow@14302
  1423
nipkow@14302
  1424
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
nipkow@14302
  1425
by(simp add:last_append)
nipkow@14302
  1426
nipkow@17762
  1427
lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs"
nipkow@17762
  1428
by(rule rev_exhaust[of xs]) simp_all
nipkow@17762
  1429
nipkow@17762
  1430
lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs"
nipkow@17762
  1431
by(cases xs) simp_all
nipkow@17762
  1432
nipkow@17765
  1433
lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as"
nipkow@17765
  1434
by (induct as) auto
nipkow@17762
  1435
wenzelm@13142
  1436
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
nipkow@13145
  1437
by (induct xs rule: rev_induct) auto
wenzelm@13114
  1438
wenzelm@13114
  1439
lemma butlast_append:
nipkow@24526
  1440
  "butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
nipkow@24526
  1441
by (induct xs arbitrary: ys) auto
wenzelm@13114
  1442
wenzelm@13142
  1443
lemma append_butlast_last_id [simp]:
nipkow@13145
  1444
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
nipkow@13145
  1445
by (induct xs) auto
wenzelm@13114
  1446
wenzelm@13142
  1447
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
nipkow@13145
  1448
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1449
wenzelm@13114
  1450
lemma in_set_butlast_appendI:
nipkow@13145
  1451
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
nipkow@13145
  1452
by (auto dest: in_set_butlastD simp add: butlast_append)
wenzelm@13114
  1453
nipkow@24526
  1454
lemma last_drop[simp]: "n < length xs \<Longrightarrow> last (drop n xs) = last xs"
nipkow@24526
  1455
apply (induct xs arbitrary: n)
nipkow@17501
  1456
 apply simp
nipkow@17501
  1457
apply (auto split:nat.split)
nipkow@17501
  1458
done
nipkow@17501
  1459
huffman@30128
  1460
lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)"
nipkow@17589
  1461
by(induct xs)(auto simp:neq_Nil_conv)
nipkow@17589
  1462
huffman@30128
  1463
lemma butlast_conv_take: "butlast xs = take (length xs - 1) xs"
huffman@26584
  1464
by (induct xs, simp, case_tac xs, simp_all)
huffman@26584
  1465
nipkow@31077
  1466
lemma last_list_update:
nipkow@31077
  1467
  "xs \<noteq> [] \<Longrightarrow> last(xs[k:=x]) = (if k = size xs - 1 then x else last xs)"
nipkow@31077
  1468
by (auto simp: last_conv_nth)
nipkow@31077
  1469
nipkow@31077
  1470
lemma butlast_list_update:
nipkow@31077
  1471
  "butlast(xs[k:=x]) =
nipkow@31077
  1472
 (if k = size xs - 1 then butlast xs else (butlast xs)[k:=x])"
nipkow@31077
  1473
apply(cases xs rule:rev_cases)
nipkow@31077
  1474
apply simp
nipkow@31077
  1475
apply(simp add:list_update_append split:nat.splits)
nipkow@31077
  1476
done
nipkow@31077
  1477
haftmann@24796
  1478
nipkow@15392
  1479
subsubsection {* @{text take} and @{text drop} *}
wenzelm@13114
  1480
wenzelm@13142
  1481
lemma take_0 [simp]: "take 0 xs = []"
nipkow@13145
  1482
by (induct xs) auto
wenzelm@13114
  1483
wenzelm@13142
  1484
lemma drop_0 [simp]: "drop 0 xs = xs"
nipkow@13145
  1485
by (induct xs) auto
wenzelm@13114
  1486
wenzelm@13142
  1487
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
nipkow@13145
  1488
by simp
wenzelm@13114
  1489
wenzelm@13142
  1490
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
nipkow@13145
  1491
by simp
wenzelm@13114
  1492
wenzelm@13142
  1493
declare take_Cons [simp del] and drop_Cons [simp del]
wenzelm@13114
  1494
huffman@30128
  1495
lemma take_1_Cons [simp]: "take 1 (x # xs) = [x]"
huffman@30128
  1496
  unfolding One_nat_def by simp
huffman@30128
  1497
huffman@30128
  1498
lemma drop_1_Cons [simp]: "drop 1 (x # xs) = xs"
huffman@30128
  1499
  unfolding One_nat_def by simp
huffman@30128
  1500
nipkow@15110
  1501
lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
nipkow@15110
  1502
by(clarsimp simp add:neq_Nil_conv)
nipkow@15110
  1503
nipkow@14187
  1504
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
nipkow@14187
  1505
by(cases xs, simp_all)
nipkow@14187
  1506
huffman@26584
  1507
lemma take_tl: "take n (tl xs) = tl (take (Suc n) xs)"
huffman@26584
  1508
by (induct xs arbitrary: n) simp_all
huffman@26584
  1509
nipkow@24526
  1510
lemma drop_tl: "drop n (tl xs) = tl(drop n xs)"
nipkow@24526
  1511
by(induct xs arbitrary: n, simp_all add:drop_Cons drop_Suc split:nat.split)
nipkow@24526
  1512
huffman@26584
  1513
lemma tl_take: "tl (take n xs) = take (n - 1) (tl xs)"
huffman@26584
  1514
by (cases n, simp, cases xs, auto)
huffman@26584
  1515
huffman@26584
  1516
lemma tl_drop: "tl (drop n xs) = drop n (tl xs)"
huffman@26584
  1517
by (simp only: drop_tl)
huffman@26584
  1518
nipkow@24526
  1519
lemma nth_via_drop: "drop n xs = y#ys \<Longrightarrow> xs!n = y"
nipkow@24526
  1520
apply (induct xs arbitrary: n, simp)
nipkow@14187
  1521
apply(simp add:drop_Cons nth_Cons split:nat.splits)
nipkow@14187
  1522
done
nipkow@14187
  1523
nipkow@13913
  1524
lemma take_Suc_conv_app_nth:
nipkow@24526
  1525
  "i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
nipkow@24526
  1526
apply (induct xs arbitrary: i, simp)
paulson@14208
  1527
apply (case_tac i, auto)
nipkow@13913
  1528
done
nipkow@13913
  1529
mehta@14591
  1530
lemma drop_Suc_conv_tl:
nipkow@24526
  1531
  "i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
nipkow@24526
  1532
apply (induct xs arbitrary: i, simp)
mehta@14591
  1533
apply (case_tac i, auto)
mehta@14591
  1534
done
mehta@14591
  1535
nipkow@24526
  1536
lemma length_take [simp]: "length (take n xs) = min (length xs) n"
nipkow@24526
  1537
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1538
nipkow@24526
  1539
lemma length_drop [simp]: "length (drop n xs) = (length xs - n)"
nipkow@24526
  1540
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1541
nipkow@24526
  1542
lemma take_all [simp]: "length xs <= n ==> take n xs = xs"
nipkow@24526
  1543
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1544
nipkow@24526
  1545
lemma drop_all [simp]: "length xs <= n ==> drop n xs = []"
nipkow@24526
  1546
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
wenzelm@13114
  1547
wenzelm@13142
  1548
lemma take_append [simp]:
nipkow@24526
  1549
  "take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
nipkow@24526
  1550
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
wenzelm@13114
  1551
wenzelm@13142
  1552
lemma drop_append [simp]:
nipkow@24526
  1553
  "drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
nipkow@24526
  1554
by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526
  1555
nipkow@24526
  1556
lemma take_take [simp]: "take n (take m xs) = take (min n m) xs"
nipkow@24526
  1557
apply (induct m arbitrary: xs n, auto)
paulson@14208
  1558
apply (case_tac xs, auto)
nipkow@15236
  1559
apply (case_tac n, auto)
nipkow@13145
  1560
done
wenzelm@13114
  1561
nipkow@24526
  1562
lemma drop_drop [simp]: "drop n (drop m xs) = drop (n + m) xs"
nipkow@24526
  1563
apply (induct m arbitrary: xs, auto)
paulson@14208
  1564
apply (case_tac xs, auto)
nipkow@13145
  1565
done
wenzelm@13114
  1566
nipkow@24526
  1567
lemma take_drop: "take n (drop m xs) = drop m (take (n + m) xs)"
nipkow@24526
  1568
apply (induct m arbitrary: xs n, auto)
paulson@14208
  1569
apply (case_tac xs, auto)
nipkow@13145
  1570
done
wenzelm@13114
  1571
nipkow@24526
  1572
lemma drop_take: "drop n (take m xs) = take (m-n) (drop n xs)"
nipkow@24526
  1573
apply(induct xs arbitrary: m n)
nipkow@14802
  1574
 apply simp
nipkow@14802
  1575
apply(simp add: take_Cons drop_Cons split:nat.split)
nipkow@14802
  1576
done
nipkow@14802
  1577
nipkow@24526
  1578
lemma append_take_drop_id [simp]: "take n xs @ drop n xs = xs"
nipkow@24526
  1579
apply (induct n arbitrary: xs, auto)
paulson@14208
  1580
apply (case_tac xs, auto)
nipkow@13145
  1581
done
wenzelm@13114
  1582
nipkow@24526
  1583
lemma take_eq_Nil[simp]: "(take n xs = []) = (n = 0 \<or> xs = [])"
nipkow@24526
  1584
apply(induct xs arbitrary: n)
nipkow@15110
  1585
 apply simp
nipkow@15110
  1586
apply(simp add:take_Cons split:nat.split)
nipkow@15110
  1587
done
nipkow@15110
  1588
nipkow@24526
  1589
lemma drop_eq_Nil[simp]: "(drop n xs = []) = (length xs <= n)"
nipkow@24526
  1590
apply(induct xs arbitrary: n)
nipkow@15110
  1591
apply simp
nipkow@15110
  1592
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1593
done
nipkow@15110
  1594
nipkow@24526
  1595
lemma take_map: "take n (map f xs) = map f (take n xs)"
nipkow@24526
  1596
apply (induct n arbitrary: xs, auto)
paulson@14208
  1597
apply (case_tac xs, auto)
nipkow@13145
  1598
done
wenzelm@13114
  1599
nipkow@24526
  1600
lemma drop_map: "drop n (map f xs) = map f (drop n xs)"
nipkow@24526
  1601
apply (induct n arbitrary: xs, auto)
paulson@14208
  1602
apply (case_tac xs, auto)
nipkow@13145
  1603
done
wenzelm@13114
  1604
nipkow@24526
  1605
lemma rev_take: "rev (take i xs) = drop (length xs - i) (rev xs)"
nipkow@24526
  1606
apply (induct xs arbitrary: i, auto)
paulson@14208
  1607
apply (case_tac i, auto)
nipkow@13145
  1608
done
wenzelm@13114
  1609
nipkow@24526
  1610
lemma rev_drop: "rev (drop i xs) = take (length xs - i) (rev xs)"
nipkow@24526
  1611
apply (induct xs arbitrary: i, auto)
paulson@14208
  1612
apply (case_tac i, auto)
nipkow@13145
  1613
done
wenzelm@13114
  1614
nipkow@24526
  1615
lemma nth_take [simp]: "i < n ==> (take n xs)!i = xs!i"
nipkow@24526
  1616
apply (induct xs arbitrary: i n, auto)
paulson@14208
  1617
apply (case_tac n, blast)
paulson@14208
  1618
apply (case_tac i, auto)
nipkow@13145
  1619
done
wenzelm@13114
  1620
wenzelm@13142
  1621
lemma nth_drop [simp]:
nipkow@24526
  1622
  "n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
nipkow@24526
  1623
apply (induct n arbitrary: xs i, auto)
paulson@14208
  1624
apply (case_tac xs, auto)
nipkow@13145
  1625
done
nipkow@3507
  1626
huffman@26584
  1627
lemma butlast_take:
huffman@30128
  1628
  "n <= length xs ==> butlast (take n xs) = take (n - 1) xs"
huffman@26584
  1629
by (simp add: butlast_conv_take min_max.inf_absorb1 min_max.inf_absorb2)
huffman@26584
  1630
huffman@26584
  1631
lemma butlast_drop: "butlast (drop n xs) = drop n (butlast xs)"
huffman@30128
  1632
by (simp add: butlast_conv_take drop_take add_ac)
huffman@26584
  1633
huffman@26584
  1634
lemma take_butlast: "n < length xs ==> take n (butlast xs) = take n xs"
huffman@26584
  1635
by (simp add: butlast_conv_take min_max.inf_absorb1)
huffman@26584
  1636
huffman@26584
  1637
lemma drop_butlast: "drop n (butlast xs) = butlast (drop n xs)"
huffman@30128
  1638
by (simp add: butlast_conv_take drop_take add_ac)
huffman@26584
  1639
nipkow@18423
  1640
lemma hd_drop_conv_nth: "\<lbrakk> xs \<noteq> []; n < length xs \<rbrakk> \<Longrightarrow> hd(drop n xs) = xs!n"
nipkow@18423
  1641
by(simp add: hd_conv_nth)
nipkow@18423
  1642
nipkow@24526
  1643
lemma set_take_subset: "set(take n xs) \<subseteq> set xs"
nipkow@24526
  1644
by(induct xs arbitrary: n)(auto simp:take_Cons split:nat.split)
nipkow@24526
  1645
nipkow@24526
  1646
lemma set_drop_subset: "set(drop n xs) \<subseteq> set xs"
nipkow@24526
  1647
by(induct xs arbitrary: n)(auto simp:drop_Cons split:nat.split)
nipkow@14025
  1648
nipkow@14187
  1649
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1650
using set_take_subset by fast
nipkow@14187
  1651
nipkow@14187
  1652
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1653
using set_drop_subset by fast
nipkow@14187
  1654
wenzelm@13114
  1655
lemma append_eq_conv_conj:
nipkow@24526
  1656
  "(xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
nipkow@24526
  1657
apply (induct xs arbitrary: zs, simp, clarsimp)
paulson@14208
  1658
apply (case_tac zs, auto)
nipkow@13145
  1659
done
wenzelm@13142
  1660
nipkow@24526
  1661
lemma take_add: 
nipkow@24526
  1662
  "i+j \<le> length(xs) \<Longrightarrow> take (i+j) xs = take i xs @ take j (drop i xs)"
nipkow@24526
  1663
apply (induct xs arbitrary: i, auto) 
nipkow@24526
  1664
apply (case_tac i, simp_all)
paulson@14050
  1665
done
paulson@14050
  1666
nipkow@14300
  1667
lemma append_eq_append_conv_if:
nipkow@24526
  1668
 "(xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
nipkow@14300
  1669
  (if size xs\<^isub>1 \<le> size ys\<^isub>1
nipkow@14300
  1670
   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
  1671
   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@24526
  1672
apply(induct xs\<^isub>1 arbitrary: ys\<^isub>1)
nipkow@14300
  1673
 apply simp
nipkow@14300
  1674
apply(case_tac ys\<^isub>1)
nipkow@14300
  1675
apply simp_all
nipkow@14300
  1676
done
nipkow@14300
  1677
nipkow@15110
  1678
lemma take_hd_drop:
huffman@30079
  1679
  "n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (Suc n) xs"
nipkow@24526
  1680
apply(induct xs arbitrary: n)
nipkow@15110
  1681
apply simp
nipkow@15110
  1682
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1683
done
nipkow@15110
  1684
nipkow@17501
  1685
lemma id_take_nth_drop:
nipkow@17501
  1686
 "i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs" 
nipkow@17501
  1687
proof -
nipkow@17501
  1688
  assume si: "i < length xs"
nipkow@17501
  1689
  hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto
nipkow@17501
  1690
  moreover
nipkow@17501
  1691
  from si have "take (Suc i) xs = take i xs @ [xs!i]"
nipkow@17501
  1692
    apply (rule_tac take_Suc_conv_app_nth) by arith
nipkow@17501
  1693
  ultimately show ?thesis by auto
nipkow@17501
  1694
qed
nipkow@17501
  1695
  
nipkow@17501
  1696
lemma upd_conv_take_nth_drop:
nipkow@17501
  1697
 "i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  1698
proof -
nipkow@17501
  1699
  assume i: "i < length xs"
nipkow@17501
  1700
  have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]"
nipkow@17501
  1701
    by(rule arg_cong[OF id_take_nth_drop[OF i]])
nipkow@17501
  1702
  also have "\<dots> = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  1703
    using i by (simp add: list_update_append)
nipkow@17501
  1704
  finally show ?thesis .
nipkow@17501
  1705
qed
nipkow@17501
  1706
haftmann@24796
  1707
lemma nth_drop':
haftmann@24796
  1708
  "i < length xs \<Longrightarrow> xs ! i # drop (Suc i) xs = drop i xs"
haftmann@24796
  1709
apply (induct i arbitrary: xs)
haftmann@24796
  1710
apply (simp add: neq_Nil_conv)
haftmann@24796
  1711
apply (erule exE)+
haftmann@24796
  1712
apply simp
haftmann@24796
  1713
apply (case_tac xs)
haftmann@24796
  1714
apply simp_all
haftmann@24796
  1715
done
haftmann@24796
  1716
wenzelm@13114
  1717
nipkow@15392
  1718
subsubsection {* @{text takeWhile} and @{text dropWhile} *}
wenzelm@13114
  1719
wenzelm@13142
  1720
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
nipkow@13145
  1721
by (induct xs) auto
wenzelm@13114
  1722
wenzelm@13142
  1723
lemma takeWhile_append1 [simp]:
nipkow@13145
  1724
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
nipkow@13145
  1725
by (induct xs) auto
wenzelm@13114
  1726
wenzelm@13142
  1727
lemma takeWhile_append2 [simp]:
nipkow@13145
  1728
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
nipkow@13145
  1729
by (induct xs) auto
wenzelm@13114
  1730
wenzelm@13142
  1731
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
nipkow@13145
  1732
by (induct xs) auto
wenzelm@13114
  1733
wenzelm@13142
  1734
lemma dropWhile_append1 [simp]:
nipkow@13145
  1735
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
nipkow@13145
  1736
by (induct xs) auto
wenzelm@13114
  1737
wenzelm@13142
  1738
lemma dropWhile_append2 [simp]:
nipkow@13145
  1739
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
nipkow@13145
  1740
by (induct xs) auto
wenzelm@13114
  1741
krauss@23971
  1742
lemma set_takeWhileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
nipkow@13145
  1743
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1744
nipkow@13913
  1745
lemma takeWhile_eq_all_conv[simp]:
nipkow@13913
  1746
 "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1747
by(induct xs, auto)
nipkow@13913
  1748
nipkow@13913
  1749
lemma dropWhile_eq_Nil_conv[simp]:
nipkow@13913
  1750
 "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1751
by(induct xs, auto)
nipkow@13913
  1752
nipkow@13913
  1753
lemma dropWhile_eq_Cons_conv:
nipkow@13913
  1754
 "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
nipkow@13913
  1755
by(induct xs, auto)
nipkow@13913
  1756
nipkow@31077
  1757
lemma distinct_takeWhile[simp]: "distinct xs ==> distinct (takeWhile P xs)"
nipkow@31077
  1758
by (induct xs) (auto dest: set_takeWhileD)
nipkow@31077
  1759
nipkow@31077
  1760
lemma distinct_dropWhile[simp]: "distinct xs ==> distinct (dropWhile P xs)"
nipkow@31077
  1761
by (induct xs) auto
nipkow@31077
  1762
nipkow@31077
  1763
nipkow@17501
  1764
text{* The following two lemmmas could be generalized to an arbitrary
nipkow@17501
  1765
property. *}
nipkow@17501
  1766
nipkow@17501
  1767
lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  1768
 takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"
nipkow@17501
  1769
by(induct xs) (auto simp: takeWhile_tail[where l="[]"])
nipkow@17501
  1770
nipkow@17501
  1771
lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  1772
  dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)"
nipkow@17501
  1773
apply(induct xs)
nipkow@17501
  1774
 apply simp
nipkow@17501
  1775
apply auto
nipkow@17501
  1776
apply(subst dropWhile_append2)
nipkow@17501
  1777
apply auto
nipkow@17501
  1778
done
nipkow@17501
  1779
nipkow@18423
  1780
lemma takeWhile_not_last:
nipkow@18423
  1781
 "\<lbrakk> xs \<noteq> []; distinct xs\<rbrakk> \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs"
nipkow@18423
  1782
apply(induct xs)
nipkow@18423
  1783
 apply simp
nipkow@18423
  1784
apply(case_tac xs)
nipkow@18423
  1785
apply(auto)
nipkow@18423
  1786
done
nipkow@18423
  1787
krauss@19770
  1788
lemma takeWhile_cong [fundef_cong, recdef_cong]:
krauss@18336
  1789
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  1790
  ==> takeWhile P l = takeWhile Q k"
nipkow@24349
  1791
by (induct k arbitrary: l) (simp_all)
krauss@18336
  1792
krauss@19770
  1793
lemma dropWhile_cong [fundef_cong, recdef_cong]:
krauss@18336
  1794
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  1795
  ==> dropWhile P l = dropWhile Q k"
nipkow@24349
  1796
by (induct k arbitrary: l, simp_all)
krauss@18336
  1797
wenzelm@13114
  1798
nipkow@15392
  1799
subsubsection {* @{text zip} *}
wenzelm@13114
  1800
wenzelm@13142
  1801
lemma zip_Nil [simp]: "zip [] ys = []"
nipkow@13145
  1802
by (induct ys) auto
wenzelm@13114
  1803
wenzelm@13142
  1804
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
nipkow@13145
  1805
by simp
wenzelm@13114
  1806
wenzelm@13142
  1807
declare zip_Cons [simp del]
wenzelm@13114
  1808
nipkow@15281
  1809
lemma zip_Cons1:
nipkow@15281
  1810
 "zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)"
nipkow@15281
  1811
by(auto split:list.split)
nipkow@15281
  1812
wenzelm@13142
  1813
lemma length_zip [simp]:
krauss@22493
  1814
"length (zip xs ys) = min (length xs) (length ys)"
krauss@22493
  1815
by (induct xs ys rule:list_induct2') auto
wenzelm@13114
  1816
wenzelm@13114
  1817
lemma zip_append1:
krauss@22493
  1818
"zip (xs @ ys) zs =
nipkow@13145
  1819
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
krauss@22493
  1820
by (induct xs zs rule:list_induct2') auto
wenzelm@13114
  1821
wenzelm@13114
  1822
lemma zip_append2:
krauss@22493
  1823
"zip xs (ys @ zs) =
nipkow@13145
  1824
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
krauss@22493
  1825
by (induct xs ys rule:list_induct2') auto
wenzelm@13114
  1826
wenzelm@13142
  1827
lemma zip_append [simp]:
wenzelm@13142
  1828
 "[| length xs = length us; length ys = length vs |] ==>
nipkow@13145
  1829
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
nipkow@13145
  1830
by (simp add: zip_append1)
wenzelm@13114
  1831
wenzelm@13114
  1832
lemma zip_rev:
nipkow@14247
  1833
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
nipkow@14247
  1834
by (induct rule:list_induct2, simp_all)
wenzelm@13114
  1835
nipkow@23096
  1836
lemma map_zip_map:
nipkow@23096
  1837
 "map f (zip (map g xs) ys) = map (%(x,y). f(g x, y)) (zip xs ys)"
nipkow@23096
  1838
apply(induct xs arbitrary:ys) apply simp
nipkow@23096
  1839
apply(case_tac ys)
nipkow@23096
  1840
apply simp_all
nipkow@23096
  1841
done
nipkow@23096
  1842
nipkow@23096
  1843
lemma map_zip_map2:
nipkow@23096
  1844
 "map f (zip xs (map g ys)) = map (%(x,y). f(x, g y)) (zip xs ys)"
nipkow@23096
  1845
apply(induct xs arbitrary:ys) apply simp
nipkow@23096
  1846
apply(case_tac ys)
nipkow@23096
  1847
apply simp_all
nipkow@23096
  1848
done
nipkow@23096
  1849
nipkow@31080
  1850
text{* Courtesy of Andreas Lochbihler: *}
nipkow@31080
  1851
lemma zip_same_conv_map: "zip xs xs = map (\<lambda>x. (x, x)) xs"
nipkow@31080
  1852
by(induct xs) auto
nipkow@31080
  1853
wenzelm@13142
  1854
lemma nth_zip [simp]:
nipkow@24526
  1855
"[| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
nipkow@24526
  1856
apply (induct ys arbitrary: i xs, simp)
nipkow@13145
  1857
apply (case_tac xs)
nipkow@13145
  1858
 apply (simp_all add: nth.simps split: nat.split)
nipkow@13145
  1859
done
wenzelm@13114
  1860
wenzelm@13114
  1861
lemma set_zip:
nipkow@13145
  1862
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
nipkow@31080
  1863
by(simp add: set_conv_nth cong: rev_conj_cong)
wenzelm@13114
  1864
wenzelm@13114
  1865
lemma zip_update:
nipkow@31080
  1866
  "zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
nipkow@31080
  1867
by(rule sym, simp add: update_zip)
wenzelm@13114
  1868
wenzelm@13142
  1869
lemma zip_replicate [simp]:
nipkow@24526
  1870
  "zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
nipkow@24526
  1871
apply (induct i arbitrary: j, auto)
paulson@14208
  1872
apply (case_tac j, auto)
nipkow@13145
  1873
done
wenzelm@13114
  1874
nipkow@19487
  1875
lemma take_zip:
nipkow@24526
  1876
  "take n (zip xs ys) = zip (take n xs) (take n ys)"
nipkow@24526
  1877
apply (induct n arbitrary: xs ys)
nipkow@19487
  1878
 apply simp
nipkow@19487
  1879
apply (case_tac xs, simp)
nipkow@19487
  1880
apply (case_tac ys, simp_all)
nipkow@19487
  1881
done
nipkow@19487
  1882
nipkow@19487
  1883
lemma drop_zip:
nipkow@24526
  1884
  "drop n (zip xs ys) = zip (drop n xs) (drop n ys)"
nipkow@24526
  1885
apply (induct n arbitrary: xs ys)
nipkow@19487
  1886
 apply simp
nipkow@19487
  1887
apply (case_tac xs, simp)
nipkow@19487
  1888
apply (case_tac ys, simp_all)
nipkow@19487
  1889
done
nipkow@19487
  1890
krauss@22493
  1891
lemma set_zip_leftD:
krauss@22493
  1892
  "(x,y)\<in> set (zip xs ys) \<Longrightarrow> x \<in> set xs"
krauss@22493
  1893
by (induct xs ys rule:list_induct2') auto
krauss@22493
  1894
krauss@22493
  1895
lemma set_zip_rightD:
krauss@22493
  1896
  "(x,y)\<in> set (zip xs ys) \<Longrightarrow> y \<in> set ys"
krauss@22493
  1897
by (induct xs ys rule:list_induct2') auto
wenzelm@13142
  1898
nipkow@23983
  1899
lemma in_set_zipE:
nipkow@23983
  1900
  "(x,y) : set(zip xs ys) \<Longrightarrow> (\<lbrakk> x : set xs; y : set ys \<rbrakk> \<Longrightarrow> R) \<Longrightarrow> R"
nipkow@23983
  1901
by(blast dest: set_zip_leftD set_zip_rightD)
nipkow@23983
  1902
haftmann@29829
  1903
lemma zip_map_fst_snd:
haftmann@29829
  1904
  "zip (map fst zs) (map snd zs) = zs"
haftmann@29829
  1905
  by (induct zs) simp_all
haftmann@29829
  1906
haftmann@29829
  1907
lemma zip_eq_conv:
haftmann@29829
  1908
  "length xs = length ys \<Longrightarrow> zip xs ys = zs \<longleftrightarrow> map fst zs = xs \<and> map snd zs = ys"
haftmann@29829
  1909
  by (auto simp add: zip_map_fst_snd)
haftmann@29829
  1910
haftmann@29829
  1911
nipkow@15392
  1912
subsubsection {* @{text list_all2} *}
wenzelm@13114
  1913
kleing@14316
  1914
lemma list_all2_lengthD [intro?]: 
kleing@14316
  1915
  "list_all2 P xs ys ==> length xs = length ys"
nipkow@24349
  1916
by (simp add: list_all2_def)
haftmann@19607
  1917
haftmann@19787
  1918
lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])"
nipkow@24349
  1919
by (simp add: list_all2_def)
haftmann@19607
  1920
haftmann@19787
  1921
lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])"
nipkow@24349
  1922
by (simp add: list_all2_def)
haftmann@19607
  1923
haftmann@19607
  1924
lemma list_all2_Cons [iff, code]:
haftmann@19607
  1925
  "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
nipkow@24349
  1926
by (auto simp add: list_all2_def)
wenzelm@13114
  1927
wenzelm@13114
  1928
lemma list_all2_Cons1:
nipkow@13145
  1929
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
nipkow@13145
  1930
by (cases ys) auto
wenzelm@13114
  1931
wenzelm@13114
  1932
lemma list_all2_Cons2:
nipkow@13145
  1933
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
nipkow@13145
  1934
by (cases xs) auto
wenzelm@13114
  1935
wenzelm@13142
  1936
lemma list_all2_rev [iff]:
nipkow@13145
  1937
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
nipkow@13145
  1938
by (simp add: list_all2_def zip_rev cong: conj_cong)
wenzelm@13114
  1939
kleing@13863
  1940
lemma list_all2_rev1:
kleing@13863
  1941
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
kleing@13863
  1942
by (subst list_all2_rev [symmetric]) simp
kleing@13863
  1943
wenzelm@13114
  1944
lemma list_all2_append1:
nipkow@13145
  1945
"list_all2 P (xs @ ys) zs =
nipkow@13145
  1946
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
nipkow@13145
  1947
list_all2 P xs us \<and> list_all2 P ys vs)"
nipkow@13145
  1948
apply (simp add: list_all2_def zip_append1)
nipkow@13145
  1949
apply (rule iffI)
nipkow@13145
  1950
 apply (rule_tac x = "take (length xs) zs" in exI)
nipkow@13145
  1951
 apply (rule_tac x = "drop (length xs) zs" in exI)
paulson@14208
  1952
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1953
apply (simp add: ball_Un)
nipkow@13145
  1954
done
wenzelm@13114
  1955
wenzelm@13114
  1956
lemma list_all2_append2:
nipkow@13145
  1957
"list_all2 P xs (ys @ zs) =
nipkow@13145
  1958
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
nipkow@13145
  1959
list_all2 P us ys \<and> list_all2 P vs zs)"
nipkow@13145
  1960
apply (simp add: list_all2_def zip_append2)
nipkow@13145
  1961
apply (rule iffI)
nipkow@13145
  1962
 apply (rule_tac x = "take (length ys) xs" in exI)
nipkow@13145
  1963
 apply (rule_tac x = "drop (length ys) xs" in exI)
paulson@14208
  1964
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1965
apply (simp add: ball_Un)
nipkow@13145
  1966
done
wenzelm@13114
  1967
kleing@13863
  1968
lemma list_all2_append:
nipkow@14247
  1969
  "length xs = length ys \<Longrightarrow>
nipkow@14247
  1970
  list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
nipkow@14247
  1971
by (induct rule:list_induct2, simp_all)
kleing@13863
  1972
kleing@13863
  1973
lemma list_all2_appendI [intro?, trans]:
kleing@13863
  1974
  "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
nipkow@24349
  1975
by (simp add: list_all2_append list_all2_lengthD)
kleing@13863
  1976
wenzelm@13114
  1977
lemma list_all2_conv_all_nth:
nipkow@13145
  1978
"list_all2 P xs ys =
nipkow@13145
  1979
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
nipkow@13145
  1980
by (force simp add: list_all2_def set_zip)
wenzelm@13114
  1981
berghofe@13883
  1982
lemma list_all2_trans:
berghofe@13883
  1983
  assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
berghofe@13883
  1984
  shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
berghofe@13883
  1985
        (is "!!bs cs. PROP ?Q as bs cs")
berghofe@13883
  1986
proof (induct as)
berghofe@13883
  1987
  fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
berghofe@13883
  1988
  show "!!cs. PROP ?Q (x # xs) bs cs"
berghofe@13883
  1989
  proof (induct bs)
berghofe@13883
  1990
    fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
berghofe@13883
  1991
    show "PROP ?Q (x # xs) (y # ys) cs"
berghofe@13883
  1992
      by (induct cs) (auto intro: tr I1 I2)
berghofe@13883
  1993
  qed simp
berghofe@13883
  1994
qed simp
berghofe@13883
  1995
kleing@13863
  1996
lemma list_all2_all_nthI [intro?]:
kleing@13863
  1997
  "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
nipkow@24349
  1998
by (simp add: list_all2_conv_all_nth)
kleing@13863
  1999
paulson@14395
  2000
lemma list_all2I:
paulson@14395
  2001
  "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
nipkow@24349
  2002
by (simp add: list_all2_def)
paulson@14395
  2003
kleing@14328
  2004
lemma list_all2_nthD:
kleing@13863
  2005
  "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
nipkow@24349
  2006
by (simp add: list_all2_conv_all_nth)
kleing@13863
  2007
nipkow@14302
  2008
lemma list_all2_nthD2:
nipkow@14302
  2009
  "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
nipkow@24349
  2010
by (frule list_all2_lengthD) (auto intro: list_all2_nthD)
nipkow@14302
  2011
kleing@13863
  2012
lemma list_all2_map1: 
kleing@13863
  2013
  "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
nipkow@24349
  2014
by (simp add: list_all2_conv_all_nth)
kleing@13863
  2015
kleing@13863
  2016
lemma list_all2_map2: 
kleing@13863
  2017
  "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
nipkow@24349
  2018
by (auto simp add: list_all2_conv_all_nth)
kleing@13863
  2019
kleing@14316
  2020
lemma list_all2_refl [intro?]:
kleing@13863
  2021
  "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
nipkow@24349
  2022
by (simp add: list_all2_conv_all_nth)
kleing@13863
  2023
kleing@13863
  2024
lemma list_all2_update_cong:
kleing@13863
  2025
  "\<lbrakk> i<size xs; list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
nipkow@24349
  2026
by (simp add: list_all2_conv_all_nth nth_list_update)
kleing@13863
  2027
kleing@13863
  2028
lemma list_all2_update_cong2:
kleing@13863
  2029
  "\<lbrakk>list_all2 P xs ys; P x y; i < length ys\<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
nipkow@24349
  2030
by (simp add: list_all2_lengthD list_all2_update_cong)
kleing@13863
  2031
nipkow@14302
  2032
lemma list_all2_takeI [simp,intro?]:
nipkow@24526
  2033
  "list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
nipkow@24526
  2034
apply (induct xs arbitrary: n ys)
nipkow@24526
  2035
 apply simp
nipkow@24526
  2036
apply (clarsimp simp add: list_all2_Cons1)
nipkow@24526
  2037
apply (case_tac n)
nipkow@24526
  2038
apply auto
nipkow@24526
  2039
done
nipkow@14302
  2040
nipkow@14302
  2041
lemma list_all2_dropI [simp,intro?]:
nipkow@24526
  2042
  "list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
nipkow@24526
  2043
apply (induct as arbitrary: n bs, simp)
nipkow@24526
  2044
apply (clarsimp simp add: list_all2_Cons1)
nipkow@24526
  2045
apply (case_tac n, simp, simp)
nipkow@24526
  2046
done
kleing@13863
  2047
kleing@14327
  2048
lemma list_all2_mono [intro?]:
nipkow@24526
  2049
  "list_all2 P xs ys \<Longrightarrow> (\<And>xs ys. P xs ys \<Longrightarrow> Q xs ys) \<Longrightarrow> list_all2 Q xs ys"
nipkow@24526
  2050
apply (induct xs arbitrary: ys, simp)
nipkow@24526
  2051
apply (case_tac ys, auto)
nipkow@24526
  2052
done
kleing@13863
  2053
haftmann@22551
  2054
lemma list_all2_eq:
haftmann@22551
  2055
  "xs = ys \<longleftrightarrow> list_all2 (op =) xs ys"
nipkow@24349
  2056
by (induct xs ys rule: list_induct2') auto
haftmann@22551
  2057
wenzelm@13142
  2058
nipkow@15392
  2059
subsubsection {* @{text foldl} and @{text foldr} *}
wenzelm@13142
  2060
wenzelm@13142
  2061
lemma foldl_append [simp]:
nipkow@24526
  2062
  "foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
nipkow@24526
  2063
by (induct xs arbitrary: a) auto
wenzelm@13142
  2064
nipkow@14402
  2065
lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
nipkow@14402
  2066
by (induct xs) auto
nipkow@14402
  2067
nipkow@23096
  2068
lemma foldr_map: "foldr g (map f xs) a = foldr (g o f) xs a"
nipkow@23096
  2069
by(induct xs) simp_all
nipkow@23096
  2070
nipkow@24449
  2071
text{* For efficient code generation: avoid intermediate list. *}
nipkow@24449
  2072
lemma foldl_map[code unfold]:
nipkow@24449
  2073
  "foldl g a (map f xs) = foldl (%a x. g a (f x)) a xs"
nipkow@23096
  2074
by(induct xs arbitrary:a) simp_all
nipkow@23096
  2075
krauss@19770
  2076
lemma foldl_cong [fundef_cong, recdef_cong]:
krauss@18336
  2077
  "[| a = b; l = k; !!a x. x : set l ==> f a x = g a x |] 
krauss@18336
  2078
  ==> foldl f a l = foldl g b k"
nipkow@24349
  2079
by (induct k arbitrary: a b l) simp_all
krauss@18336
  2080
krauss@19770
  2081
lemma foldr_cong [fundef_cong, recdef_cong]:
krauss@18336
  2082
  "[| a = b; l = k; !!a x. x : set l ==> f x a = g x a |] 
krauss@18336
  2083
  ==> foldr f l a = foldr g k b"
nipkow@24349
  2084
by (induct k arbitrary: a b l) simp_all
krauss@18336
  2085
nipkow@24449
  2086
lemma (in semigroup_add) foldl_assoc:
haftmann@25062
  2087
shows "foldl op+ (x+y) zs = x + (foldl op+ y zs)"
nipkow@24449
  2088
by (induct zs arbitrary: y) (simp_all add:add_assoc)
nipkow@24449
  2089
nipkow@24449
  2090
lemma (in monoid_add) foldl_absorb0:
haftmann@25062
  2091
shows "x + (foldl op+ 0 zs) = foldl op+ x zs"
nipkow@24449
  2092
by (induct zs) (simp_all add:foldl_assoc)
nipkow@24449
  2093
nipkow@24449
  2094
nipkow@23096
  2095
text{* The ``First Duality Theorem'' in Bird \& Wadler: *}
nipkow@23096
  2096
nipkow@23096
  2097
lemma foldl_foldr1_lemma:
nipkow@23096
  2098
 "foldl op + a xs = a + foldr op + xs (0\<Colon>'a::monoid_add)"
nipkow@23096
  2099
by (induct xs arbitrary: a) (auto simp:add_assoc)
nipkow@23096
  2100
nipkow@23096
  2101
corollary foldl_foldr1:
nipkow@23096
  2102
 "foldl op + 0 xs = foldr op + xs (0\<Colon>'a::monoid_add)"
nipkow@23096
  2103
by (simp add:foldl_foldr1_lemma)
nipkow@23096
  2104
nipkow@23096
  2105
nipkow@23096
  2106
text{* The ``Third Duality Theorem'' in Bird \& Wadler: *}
nipkow@23096
  2107
nipkow@14402
  2108
lemma foldr_foldl: "foldr f xs a = foldl (%x y. f y x) a (rev xs)"
nipkow@14402
  2109
by (induct xs) auto
nipkow@14402
  2110
nipkow@14402
  2111
lemma foldl_foldr: "foldl f a xs = foldr (%x y. f y x) (rev xs) a"
nipkow@14402
  2112
by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"])
nipkow@14402
  2113
haftmann@25062
  2114
lemma (in ab_semigroup_add) foldr_conv_foldl: "foldr op + xs a = foldl op + a xs"
chaieb@24471
  2115
  by (induct xs, auto simp add: foldl_assoc add_commute)
chaieb@24471
  2116
wenzelm@13142
  2117
text {*
nipkow@13145
  2118
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
nipkow@13145
  2119
difficult to use because it requires an additional transitivity step.
wenzelm@13142
  2120
*}
wenzelm@13142
  2121
nipkow@24526
  2122
lemma start_le_sum: "(m::nat) <= n ==> m <= foldl (op +) n ns"
nipkow@24526
  2123
by (induct ns arbitrary: n) auto
nipkow@24526
  2124
nipkow@24526
  2125
lemma elem_le_sum: "(n::nat) : set ns ==> n <= foldl (op +) 0 ns"
nipkow@13145
  2126
by (force intro: start_le_sum simp add: in_set_conv_decomp)
wenzelm@13142
  2127
wenzelm@13142
  2128
lemma sum_eq_0_conv [iff]:
nipkow@24526
  2129
  "(foldl (op +) (m::nat) ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"
nipkow@24526
  2130
by (induct ns arbitrary: m) auto
wenzelm@13114
  2131
chaieb@24471
  2132
lemma foldr_invariant: 
chaieb@24471
  2133
  "\<lbrakk>Q x ; \<forall> x\<in> set xs. P x; \<forall> x y. P x \<and> Q y \<longrightarrow> Q (f x y) \<rbrakk> \<Longrightarrow> Q (foldr f xs x)"
chaieb@24471
  2134
  by (induct xs, simp_all)
chaieb@24471
  2135
chaieb@24471
  2136
lemma foldl_invariant: 
chaieb@24471
  2137
  "\<lbrakk>Q x ; \<forall> x\<in> set xs. P x; \<forall> x y. P x \<and> Q y \<longrightarrow> Q (f y x) \<rbrakk> \<Longrightarrow> Q (foldl f x xs)"
chaieb@24471
  2138
  by (induct xs arbitrary: x, simp_all)
chaieb@24471
  2139
nipkow@24449
  2140
text{* @{const foldl} and @{text concat} *}
nipkow@24449
  2141
nipkow@24449
  2142
lemma foldl_conv_concat:
haftmann@29782
  2143
  "foldl (op @) xs xss = xs @ concat xss"
haftmann@29782
  2144
proof (induct xss arbitrary: xs)
haftmann@29782
  2145
  case Nil show ?case by simp
haftmann@29782
  2146
next
haftmann@29782
  2147
  interpret monoid_add "[]" "op @" proof qed simp_all
haftmann@29782
  2148
  case Cons then show ?case by (simp add: foldl_absorb0)
haftmann@29782
  2149
qed
haftmann@29782
  2150
haftmann@29782
  2151
lemma concat_conv_foldl: "concat xss = foldl (op @) [] xss"
haftmann@29782
  2152
  by (simp add: foldl_conv_concat)
haftmann@29782
  2153
nipkow@24449
  2154
nipkow@23096
  2155
subsubsection {* List summation: @{const listsum} and @{text"\<Sum>"}*}
nipkow@23096
  2156
haftmann@26442
  2157
lemma listsum_append [simp]: "listsum (xs @ ys) = listsum xs + listsum ys"
nipkow@24449
  2158
by (induct xs) (simp_all add:add_assoc)
nipkow@24449
  2159
haftmann@26442
  2160
lemma listsum_rev [simp]:
haftmann@26442
  2161
  fixes xs :: "'a\<Colon>comm_monoid_add list"
haftmann@26442
  2162
  shows "listsum (rev xs) = listsum xs"
nipkow@24449
  2163
by (induct xs) (simp_all add:add_ac)
nipkow@24449
  2164
nipkow@31022
  2165
lemma listsum_map_remove1:
nipkow@31022
  2166
fixes f :: "'a \<Rightarrow> ('b::comm_monoid_add)"
nipkow@31022
  2167
shows "x : set xs \<Longrightarrow> listsum(map f xs) = f x + listsum(map f (remove1 x xs))"
nipkow@31022
  2168
by (induct xs)(auto simp add:add_ac)
nipkow@31022
  2169
nipkow@31022
  2170
lemma list_size_conv_listsum:
nipkow@31022
  2171
  "list_size f xs = listsum (map f xs) + size xs"
nipkow@31022
  2172
by(induct xs) auto
nipkow@31022
  2173
haftmann@26442
  2174
lemma listsum_foldr: "listsum xs = foldr (op +) xs 0"
haftmann@26442
  2175
by (induct xs) auto
haftmann@26442
  2176
haftmann@26442
  2177
lemma length_concat: "length (concat xss) = listsum (map length xss)"
haftmann@26442
  2178
by (induct xss) simp_all
nipkow@23096
  2179
nipkow@24449
  2180
text{* For efficient code generation ---
nipkow@24449
  2181
       @{const listsum} is not tail recursive but @{const foldl} is. *}
nipkow@24449
  2182
lemma listsum[code unfold]: "listsum xs = foldl (op +) 0 xs"
nipkow@23096
  2183
by(simp add:listsum_foldr foldl_foldr1)
nipkow@23096
  2184
nipkow@31077
  2185
lemma distinct_listsum_conv_Setsum:
nipkow@31077
  2186
  "distinct xs \<Longrightarrow> listsum xs = Setsum(set xs)"
nipkow@31077
  2187
by (induct xs) simp_all
nipkow@31077
  2188
nipkow@24449
  2189
nipkow@23096
  2190
text{* Some syntactic sugar for summing a function over a list: *}
nipkow@23096
  2191
nipkow@23096
  2192
syntax
nipkow@23096
  2193
  "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3SUM _<-_. _)" [0, 51, 10] 10)
nipkow@23096
  2194
syntax (xsymbols)
nipkow@23096
  2195
  "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
nipkow@23096
  2196
syntax (HTML output)
nipkow@23096
  2197
  "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
nipkow@23096
  2198
nipkow@23096
  2199
translations -- {* Beware of argument permutation! *}
nipkow@23096
  2200
  "SUM x<-xs. b" == "CONST listsum (map (%x. b) xs)"
nipkow@23096
  2201
  "\<Sum>x\<leftarrow>xs. b" == "CONST listsum (map (%x. b) xs)"
nipkow@23096
  2202
haftmann@26442
  2203
lemma listsum_triv: "(\<Sum>x\<leftarrow>xs. r) = of_nat (length xs) * r"
haftmann@26442
  2204
  by (induct xs) (simp_all add: left_distrib)
haftmann@26442
  2205
nipkow@23096
  2206
lemma listsum_0 [simp]: "(\<Sum>x\<leftarrow>xs. 0) = 0"
haftmann@26442
  2207
  by (induct xs) (simp_all add: left_distrib)
nipkow@23096
  2208
nipkow@23096
  2209
text{* For non-Abelian groups @{text xs} needs to be reversed on one side: *}
nipkow@23096
  2210
lemma uminus_listsum_map:
haftmann@26442
  2211
  fixes f :: "'a \<Rightarrow> 'b\<Colon>ab_group_add"
haftmann@26442
  2212
  shows "- listsum (map f xs) = (listsum (map (uminus o f) xs))"
haftmann@26442
  2213
by (induct xs) simp_all
nipkow@23096
  2214
wenzelm@13114
  2215
nipkow@24645
  2216
subsubsection {* @{text upt} *}
wenzelm@13114
  2217
nipkow@17090
  2218
lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])"
nipkow@17090
  2219
-- {* simp does not terminate! *}
nipkow@13145
  2220
by (induct j) auto
wenzelm@13142
  2221
nipkow@15425
  2222
lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []"
nipkow@13145
  2223
by (subst upt_rec) simp
wenzelm@13114
  2224
nipkow@15425
  2225
lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)"
nipkow@15281
  2226
by(induct j)simp_all
nipkow@15281
  2227
nipkow@15281
  2228
lemma upt_eq_Cons_conv:
nipkow@24526
  2229
 "([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)"
nipkow@24526
  2230
apply(induct j arbitrary: x xs)
nipkow@15281
  2231
 apply simp
nipkow@15281
  2232
apply(clarsimp simp add: append_eq_Cons_conv)
nipkow@15281
  2233
apply arith
nipkow@15281
  2234
done
nipkow@15281
  2235
nipkow@15425
  2236
lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]"
nipkow@13145
  2237
-- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
nipkow@13145
  2238
by simp
wenzelm@13114
  2239
nipkow@15425
  2240
lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]"
haftmann@26734
  2241
  by (simp add: upt_rec)
wenzelm@13114
  2242
nipkow@15425
  2243
lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]"
nipkow@13145
  2244
-- {* LOOPS as a simprule, since @{text "j <= j"}. *}
nipkow@13145
  2245
by (induct k) auto
wenzelm@13114
  2246
nipkow@15425
  2247
lemma length_upt [simp]: "length [i..<j] = j - i"
nipkow@13145
  2248
by (induct j) (auto simp add: Suc_diff_le)
wenzelm@13114
  2249
nipkow@15425
  2250
lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k"
nipkow@13145
  2251
apply (induct j)
nipkow@13145
  2252
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
nipkow@13145
  2253
done
wenzelm@13114
  2254
nipkow@17906
  2255
nipkow@17906
  2256
lemma hd_upt[simp]: "i < j \<Longrightarrow> hd[i..<j] = i"
nipkow@17906
  2257
by(simp add:upt_conv_Cons)
nipkow@17906
  2258
nipkow@17906
  2259
lemma last_upt[simp]: "i < j \<Longrightarrow> last[i..<j] = j - 1"
nipkow@17906
  2260
apply(cases j)
nipkow@17906
  2261
 apply simp
nipkow@17906
  2262
by(simp add:upt_Suc_append)
nipkow@17906
  2263
nipkow@24526
  2264
lemma take_upt [simp]: "i+m <= n ==> take m [i..<n] = [i..<i+m]"
nipkow@24526
  2265
apply (induct m arbitrary: i, simp)
nipkow@13145
  2266
apply (subst upt_rec)
nipkow@13145
  2267
apply (rule sym)
nipkow@13145
  2268
apply (subst upt_rec)
nipkow@13145
  2269
apply (simp del: upt.simps)
nipkow@13145
  2270
done
nipkow@3507
  2271
nipkow@17501
  2272
lemma drop_upt[simp]: "drop m [i..<j] = [i+m..<j]"
nipkow@17501
  2273
apply(induct j)
nipkow@17501
  2274
apply auto
nipkow@17501
  2275
done
nipkow@17501
  2276
nipkow@24645
  2277
lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..<Suc n]"
nipkow@13145
  2278
by (induct n) auto
wenzelm@13114
  2279
nipkow@24526
  2280
lemma nth_map_upt: "i < n-m ==> (map f [m..<n]) ! i = f(m+i)"
nipkow@24526
  2281
apply (induct n m  arbitrary: i rule: diff_induct)
nipkow@13145
  2282
prefer 3 apply (subst map_Suc_upt[symmetric])
nipkow@13145
  2283
apply (auto simp add: less_diff_conv nth_upt)
nipkow@13145
  2284
done
wenzelm@13114
  2285
berghofe@13883
  2286
lemma nth_take_lemma:
nipkow@24526
  2287
  "k <= length xs ==> k <= length ys ==>
berghofe@13883
  2288
     (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
nipkow@24526
  2289
apply (atomize, induct k arbitrary: xs ys)
paulson@14208
  2290
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
nipkow@13145
  2291
txt {* Both lists must be non-empty *}
paulson@14208
  2292
apply (case_tac xs, simp)
paulson@14208
  2293
apply (case_tac ys, clarify)
nipkow@13145
  2294
 apply (simp (no_asm_use))
nipkow@13145
  2295
apply clarify
nipkow@13145
  2296
txt {* prenexing's needed, not miniscoping *}
nipkow@13145
  2297
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
nipkow@13145
  2298
apply blast
nipkow@13145
  2299
done
wenzelm@13114
  2300
wenzelm@13114
  2301
lemma nth_equalityI:
wenzelm@13114
  2302
 "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
nipkow@13145
  2303
apply (frule nth_take_lemma [OF le_refl eq_imp_le])
nipkow@13145
  2304
apply (simp_all add: take_all)
nipkow@13145
  2305
done
wenzelm@13142
  2306
haftmann@24796
  2307
lemma map_nth:
haftmann@24796
  2308
  "map (\<lambda>i. xs ! i) [0..<length xs] = xs"
haftmann@24796
  2309
  by (rule nth_equalityI, auto)
haftmann@24796
  2310
kleing@13863
  2311
(* needs nth_equalityI *)
kleing@13863
  2312
lemma list_all2_antisym:
kleing@13863
  2313
  "\<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
  2314
  \<Longrightarrow> xs = ys"
kleing@13863
  2315
  apply (simp add: list_all2_conv_all_nth) 
paulson@14208
  2316
  apply (rule nth_equalityI, blast, simp)
kleing@13863
  2317
  done
kleing@13863
  2318
wenzelm@13142
  2319
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
nipkow@13145
  2320
-- {* The famous take-lemma. *}
nipkow@13145
  2321
apply (drule_tac x = "max (length xs) (length ys)" in spec)
nipkow@13145
  2322
apply (simp add: le_max_iff_disj take_all)
nipkow@13145
  2323
done
wenzelm@13142
  2324
wenzelm@13142
  2325
nipkow@15302
  2326
lemma take_Cons':
nipkow@15302
  2327
     "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
nipkow@15302
  2328
by (cases n) simp_all
nipkow@15302
  2329
nipkow@15302
  2330
lemma drop_Cons':
nipkow@15302
  2331
     "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
nipkow@15302
  2332
by (cases n) simp_all
nipkow@15302
  2333
nipkow@15302
  2334
lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
nipkow@15302
  2335
by (cases n) simp_all
nipkow@15302
  2336
paulson@18622
  2337
lemmas take_Cons_number_of = take_Cons'[of "number_of v",standard]
paulson@18622
  2338
lemmas drop_Cons_number_of = drop_Cons'[of "number_of v",standard]
paulson@18622
  2339
lemmas nth_Cons_number_of = nth_Cons'[of _ _ "number_of v",standard]
paulson@18622
  2340
paulson@18622
  2341
declare take_Cons_number_of [simp] 
paulson@18622
  2342
        drop_Cons_number_of [simp] 
paulson@18622
  2343
        nth_Cons_number_of [simp] 
nipkow@15302
  2344
nipkow@15302
  2345
nipkow@15392
  2346
subsubsection {* @{text "distinct"} and @{text remdups} *}
wenzelm@13142
  2347
wenzelm@13142
  2348
lemma distinct_append [simp]:
nipkow@13145
  2349
"distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
nipkow@13145
  2350
by (induct xs) auto
wenzelm@13142
  2351
nipkow@15305
  2352
lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs"
nipkow@15305
  2353
by(induct xs) auto
nipkow@15305
  2354
wenzelm@13142
  2355
lemma set_remdups [simp]: "set (remdups xs) = set xs"
nipkow@13145
  2356
by (induct xs) (auto simp add: insert_absorb)
wenzelm@13142
  2357
wenzelm@13142
  2358
lemma distinct_remdups [iff]: "distinct (remdups xs)"
nipkow@13145
  2359
by (induct xs) auto
wenzelm@13142
  2360
nipkow@25287
  2361
lemma distinct_remdups_id: "distinct xs ==> remdups xs = xs"
nipkow@25287
  2362
by (induct xs, auto)
nipkow@25287
  2363
haftmann@26734
  2364
lemma remdups_id_iff_distinct [simp]: "remdups xs = xs \<longleftrightarrow> distinct xs"
haftmann@26734
  2365
by (metis distinct_remdups distinct_remdups_id)
nipkow@25287
  2366
nipkow@24566
  2367
lemma finite_distinct_list: "finite A \<Longrightarrow> EX xs. set xs = A & distinct xs"
paulson@24632
  2368
by (metis distinct_remdups finite_list set_remdups)
nipkow@24566
  2369
paulson@15072
  2370
lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"
nipkow@24349
  2371
by (induct x, auto) 
paulson@15072
  2372
paulson@15072
  2373
lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"
nipkow@24349
  2374
by (induct x, auto)
paulson@15072
  2375
nipkow@15245
  2376
lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs"
nipkow@15245
  2377
by (induct xs) auto
nipkow@15245
  2378
nipkow@15245
  2379
lemma length_remdups_eq[iff]:
nipkow@15245
  2380
  "(length (remdups xs) = length xs) = (remdups xs = xs)"
nipkow@15245
  2381
apply(induct xs)
nipkow@15245
  2382
 apply auto
nipkow@15245
  2383
apply(subgoal_tac "length (remdups xs) <= length xs")
nipkow@15245
  2384
 apply arith
nipkow@15245
  2385
apply(rule length_remdups_leq)
nipkow@15245
  2386
done
nipkow@15245
  2387
nipkow@18490
  2388
nipkow@18490
  2389
lemma distinct_map:
nipkow@18490
  2390
  "distinct(map f xs) = (distinct xs & inj_on f (set xs))"
nipkow@18490
  2391
by (induct xs) auto
nipkow@18490
  2392
nipkow@18490
  2393
wenzelm@13142
  2394
lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
nipkow@13145
  2395
by (induct xs) auto
wenzelm@13114
  2396
nipkow@17501
  2397
lemma distinct_upt[simp]: "distinct[i..<j]"
nipkow@17501
  2398
by (induct j) auto
nipkow@17501
  2399
nipkow@24526
  2400
lemma distinct_take[simp]: "distinct xs \<Longrightarrow> distinct (take i xs)"
nipkow@24526
  2401
apply(induct xs arbitrary: i)
nipkow@17501
  2402
 apply simp
nipkow@17501
  2403
apply (case_tac i)
nipkow@17501
  2404
 apply simp_all
nipkow@17501
  2405
apply(blast dest:in_set_takeD)
nipkow@17501
  2406
done
nipkow@17501
  2407
nipkow@24526
  2408
lemma distinct_drop[simp]: "distinct xs \<Longrightarrow> distinct (drop i xs)"
nipkow@24526
  2409
apply(induct xs arbitrary: i)
nipkow@17501
  2410
 apply simp
nipkow@17501
  2411
apply (case_tac i)
nipkow@17501
  2412
 apply simp_all
nipkow@17501
  2413
done
nipkow@17501
  2414
nipkow@17501
  2415
lemma distinct_list_update:
nipkow@17501
  2416
assumes d: "distinct xs" and a: "a \<notin> set xs - {xs!i}"
nipkow@17501
  2417
shows "distinct (xs[i:=a])"
nipkow@17501
  2418
proof (cases "i < length xs")
nipkow@17501
  2419
  case True
nipkow@17501
  2420
  with a have "a \<notin> set (take i xs @ xs ! i # drop (Suc i) xs) - {xs!i}"
nipkow@17501
  2421
    apply (drule_tac id_take_nth_drop) by simp
nipkow@17501
  2422
  with d True show ?thesis
nipkow@17501
  2423
    apply (simp add: upd_conv_take_nth_drop)
nipkow@17501
  2424
    apply (drule subst [OF id_take_nth_drop]) apply assumption
nipkow@17501
  2425
    apply simp apply (cases "a = xs!i") apply simp by blast
nipkow@17501
  2426
next
nipkow@17501
  2427
  case False with d show ?thesis by auto
nipkow@17501
  2428
qed
nipkow@17501
  2429
nipkow@17501
  2430
nipkow@17501
  2431
text {* It is best to avoid this indexed version of distinct, but
nipkow@17501
  2432
sometimes it is useful. *}
nipkow@17501
  2433
wenzelm@13142
  2434
lemma distinct_conv_nth:
nipkow@17501
  2435
"distinct xs = (\<forall>i < size xs. \<forall>j < size xs. i \<noteq> j --> xs!i \<noteq> xs!j)"
paulson@15251
  2436
apply (induct xs, simp, simp)
paulson@14208
  2437
apply (rule iffI, clarsimp)
nipkow@13145
  2438
 apply (case_tac i)
paulson@14208
  2439
apply (case_tac j, simp)
nipkow@13145
  2440
apply (simp add: set_conv_nth)
nipkow@13145
  2441
 apply (case_tac j)
paulson@24648
  2442
apply (clarsimp simp add: set_conv_nth, simp) 
nipkow@13145
  2443
apply (rule conjI)
paulson@24648
  2444
(*TOO SLOW
paulson@24632
  2445
apply (metis Zero_neq_Suc gr0_conv_Suc in_set_conv_nth lessI less_trans_Suc nth_Cons' nth_Cons_Suc)
paulson@24648
  2446
*)
paulson@24648
  2447
 apply (clarsimp simp add: set_conv_nth)
paulson@24648
  2448
 apply (erule_tac x = 0 in allE, simp)
paulson@24648
  2449
 apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
wenzelm@25130
  2450
(*TOO SLOW
paulson@24632
  2451
apply (metis Suc_Suc_eq lessI less_trans_Suc nth_Cons_Suc)
wenzelm@25130
  2452
*)
wenzelm@25130
  2453
apply (erule_tac x = "Suc i" in allE, simp)
wenzelm@25130
  2454
apply (erule_tac x = "Suc j" in allE, simp)
nipkow@13145
  2455
done
wenzelm@13114
  2456
nipkow@18490
  2457
lemma nth_eq_iff_index_eq:
nipkow@18490
  2458
 "\<lbrakk> distinct xs; i < length xs; j < length xs \<rbrakk> \<Longrightarrow> (xs!i = xs!j) = (i = j)"
nipkow@18490
  2459
by(auto simp: distinct_conv_nth)
nipkow@18490
  2460
nipkow@15110
  2461
lemma distinct_card: "distinct xs ==> card (set xs) = size xs"
nipkow@24349
  2462
by (induct xs) auto
kleing@14388
  2463
nipkow@15110
  2464
lemma card_distinct: "card (set xs) = size xs ==> distinct xs"
kleing@14388
  2465
proof (induct xs)
kleing@14388
  2466
  case Nil thus ?case by simp
kleing@14388
  2467
next
kleing@14388
  2468
  case (Cons x xs)
kleing@14388
  2469
  show ?case
kleing@14388
  2470
  proof (cases "x \<in> set xs")
kleing@14388
  2471
    case False with Cons show ?thesis by simp
kleing@14388
  2472
  next
kleing@14388
  2473
    case True with Cons.prems
kleing@14388
  2474
    have "card (set xs) = Suc (length xs)" 
kleing@14388
  2475
      by (simp add: card_insert_if split: split_if_asm)
kleing@14388
  2476
    moreover have "card (set xs) \<le> length xs" by (rule card_length)
kleing@14388
  2477
    ultimately have False by simp
kleing@14388
  2478
    thus ?thesis ..
kleing@14388
  2479
  qed
kleing@14388
  2480
qed
kleing@14388
  2481
nipkow@25287
  2482
lemma not_distinct_decomp: "~ distinct ws ==> EX xs ys zs y. ws = xs@[y]@ys@[y]@zs"
nipkow@25287
  2483
apply (induct n == "length ws" arbitrary:ws) apply simp
nipkow@25287
  2484
apply(case_tac ws) apply simp
nipkow@25287
  2485
apply (simp split:split_if_asm)
nipkow@25287
  2486
apply (metis Cons_eq_appendI eq_Nil_appendI split_list)
nipkow@25287
  2487
done
nipkow@18490
  2488
nipkow@18490
  2489
lemma length_remdups_concat:
nipkow@18490
  2490
 "length(remdups(concat xss)) = card(\<Union>xs \<in> set xss. set xs)"
nipkow@24308
  2491
by(simp add: set_concat distinct_card[symmetric])
nipkow@17906
  2492
nipkow@17906
  2493
nipkow@15392
  2494
subsubsection {* @{text remove1} *}
nipkow@15110
  2495
nipkow@18049
  2496
lemma remove1_append:
nipkow@18049
  2497
  "remove1 x (xs @ ys) =
nipkow@18049
  2498
  (if x \<in> set xs then remove1 x xs @ ys else xs @ remove1 x ys)"
nipkow@18049
  2499
by (induct xs) auto
nipkow@18049
  2500
nipkow@23479
  2501
lemma in_set_remove1[simp]:
nipkow@23479
  2502
  "a \<noteq> b \<Longrightarrow> a : set(remove1 b xs) = (a : set xs)"
nipkow@23479
  2503
apply (induct xs)
nipkow@23479
  2504
apply auto
nipkow@23479
  2505
done
nipkow@23479
  2506
nipkow@15110
  2507
lemma set_remove1_subset: "set(remove1 x xs) <= set xs"
nipkow@15110
  2508
apply(induct xs)
nipkow@15110
  2509
 apply simp
nipkow@15110
  2510
apply simp
nipkow@15110
  2511
apply blast
nipkow@15110
  2512
done
nipkow@15110
  2513
paulson@17724
  2514
lemma set_remove1_eq [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"
nipkow@15110
  2515
apply(induct xs)
nipkow@15110
  2516
 apply simp
nipkow@15110
  2517
apply simp
nipkow@15110
  2518
apply blast
nipkow@15110
  2519
done
nipkow@15110
  2520
nipkow@23479
  2521
lemma length_remove1:
huffman@30128
  2522
  "length(remove1 x xs) = (if x : set xs then length xs - 1 else length xs)"
nipkow@23479
  2523
apply (induct xs)
nipkow@23479
  2524
 apply (auto dest!:length_pos_if_in_set)
nipkow@23479
  2525
done
nipkow@23479
  2526
nipkow@18049
  2527
lemma remove1_filter_not[simp]:
nipkow@18049
  2528
  "\<not> P x \<Longrightarrow> remove1 x (filter P xs) = filter P xs"
nipkow@18049
  2529
by(induct xs) auto
nipkow@18049
  2530
nipkow@15110
  2531
lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"
nipkow@15110
  2532
apply(insert set_remove1_subset)
nipkow@15110
  2533
apply fast
nipkow@15110
  2534
done
nipkow@15110
  2535
nipkow@15110
  2536
lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)"
nipkow@15110
  2537
by (induct xs) simp_all
nipkow@15110
  2538
wenzelm@13114
  2539
nipkow@27693
  2540
subsubsection {* @{text removeAll} *}
nipkow@27693
  2541
nipkow@27693
  2542
lemma removeAll_append[simp]:
nipkow@27693
  2543
  "removeAll x (xs @ ys) = removeAll x xs @ removeAll x ys"
nipkow@27693
  2544
by (induct xs) auto
nipkow@27693
  2545
nipkow@27693
  2546
lemma set_removeAll[simp]: "set(removeAll x xs) = set xs - {x}"
nipkow@27693
  2547
by (induct xs) auto
nipkow@27693
  2548
nipkow@27693
  2549
lemma removeAll_id[simp]: "x \<notin> set xs \<Longrightarrow> removeAll x xs = xs"
nipkow@27693
  2550
by (induct xs) auto
nipkow@27693
  2551
nipkow@27693
  2552
(* Needs count:: 'a \<Rightarrow> a' list \<Rightarrow> nat
nipkow@27693
  2553
lemma length_removeAll:
nipkow@27693
  2554
  "length(removeAll x xs) = length xs - count x xs"
nipkow@27693
  2555
*)
nipkow@27693
  2556
nipkow@27693
  2557
lemma removeAll_filter_not[simp]:
nipkow@27693
  2558
  "\<not> P x \<Longrightarrow> removeAll x (filter P xs) = filter P xs"
nipkow@27693
  2559
by(induct xs) auto
nipkow@27693
  2560
nipkow@27693
  2561
nipkow@27693
  2562
lemma distinct_remove1_removeAll:
nipkow@27693
  2563
  "distinct xs ==> remove1 x xs = removeAll x xs"
nipkow@27693
  2564
by (induct xs) simp_all
nipkow@27693
  2565
nipkow@27693
  2566
lemma map_removeAll_inj_on: "inj_on f (insert x (set xs)) \<Longrightarrow>
nipkow@27693
  2567
  map f (removeAll x xs) = removeAll (f x) (map f xs)"
nipkow@27693
  2568
by (induct xs) (simp_all add:inj_on_def)
nipkow@27693
  2569
nipkow@27693
  2570
lemma map_removeAll_inj: "inj f \<Longrightarrow>
nipkow@27693
  2571
  map f (removeAll x xs) = removeAll (f x) (map f xs)"
nipkow@27693
  2572
by(metis map_removeAll_inj_on subset_inj_on subset_UNIV)
nipkow@27693
  2573
nipkow@27693
  2574
nipkow@15392
  2575
subsubsection {* @{text replicate} *}
wenzelm@13114
  2576
wenzelm@13142
  2577
lemma length_replicate [simp]: "length (replicate n x) = n"
nipkow@13145
  2578
by (induct n) auto
nipkow@13124
  2579
wenzelm@13142
  2580
lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
nipkow@13145
  2581
by (induct n) auto
wenzelm@13114
  2582
wenzelm@13114
  2583
lemma replicate_app_Cons_same:
nipkow@13145
  2584
"(replicate n x) @ (x # xs) = x # replicate n x @ xs"
nipkow@13145
  2585
by (induct n) auto
wenzelm@13114
  2586
wenzelm@13142
  2587
lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
paulson@14208
  2588
apply (induct n, simp)
nipkow@13145
  2589
apply (simp add: replicate_app_Cons_same)
nipkow@13145
  2590
done
wenzelm@13114
  2591
wenzelm@13142
  2592
lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
nipkow@13145
  2593
by (induct n) auto
wenzelm@13114
  2594
nipkow@16397
  2595
text{* Courtesy of Matthias Daum: *}
nipkow@16397
  2596
lemma append_replicate_commute:
nipkow@16397
  2597
  "replicate n x @ replicate k x = replicate k x @ replicate n x"
nipkow@16397
  2598
apply (simp add: replicate_add [THEN sym])
nipkow@16397
  2599
apply (simp add: add_commute)
nipkow@16397
  2600
done
nipkow@16397
  2601
nipkow@31080
  2602
text{* Courtesy of Andreas Lochbihler: *}
nipkow@31080
  2603
lemma filter_replicate:
nipkow@31080
  2604
  "filter P (replicate n x) = (if P x then replicate n x else [])"
nipkow@31080
  2605
by(induct n) auto
nipkow@31080
  2606
wenzelm@13142
  2607
lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
nipkow@13145
  2608
by (induct n) auto
wenzelm@13114
  2609
wenzelm@13142
  2610
lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"
nipkow@13145
  2611
by (induct n) auto
wenzelm@13114
  2612
wenzelm@13142
  2613
lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
nipkow@13145
  2614
by (atomize (full), induct n) auto
wenzelm@13114
  2615
nipkow@24526
  2616
lemma nth_replicate[simp]: "i < n ==> (replicate n x)!i = x"
nipkow@24526
  2617
apply (induct n arbitrary: i, simp)
nipkow@13145
  2618
apply (simp add: nth_Cons split: nat.split)
nipkow@13145
  2619
done
wenzelm@13114
  2620
nipkow@16397
  2621
text{* Courtesy of Matthias Daum (2 lemmas): *}
nipkow@16397
  2622
lemma take_replicate[simp]: "take i (replicate k x) = replicate (min i k) x"
nipkow@16397
  2623
apply (case_tac "k \<le> i")
nipkow@16397
  2624
 apply  (simp add: min_def)
nipkow@16397
  2625
apply (drule not_leE)
nipkow@16397
  2626
apply (simp add: min_def)
nipkow@16397
  2627
apply (subgoal_tac "replicate k x = replicate i x @ replicate (k - i) x")
nipkow@16397
  2628
 apply  simp
nipkow@16397
  2629
apply (simp add: replicate_add [symmetric])
nipkow@16397
  2630
done
nipkow@16397
  2631
nipkow@24526
  2632
lemma drop_replicate[simp]: "drop i (replicate k x) = replicate (k-i) x"
nipkow@24526
  2633
apply (induct k arbitrary: i)
nipkow@16397
  2634
 apply simp
nipkow@16397
  2635
apply clarsimp
nipkow@16397
  2636
apply (case_tac i)
nipkow@16397
  2637
 apply simp
nipkow@16397
  2638
apply clarsimp
nipkow@16397
  2639
done
nipkow@16397
  2640
nipkow@16397
  2641
wenzelm@13142
  2642
lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
nipkow@13145
  2643
by (induct n) auto
wenzelm@13114
  2644
wenzelm@13142
  2645
lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
nipkow@13145
  2646
by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
wenzelm@13114
  2647
wenzelm@13142
  2648
lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
nipkow@13145
  2649
by auto
wenzelm@13114
  2650
wenzelm@13142
  2651
lemma in_set_replicateD: "x : set (replicate n y) ==> x = y"
nipkow@13145
  2652
by (simp add: set_replicate_conv_if split: split_if_asm)
wenzelm@13114
  2653
haftmann@24796
  2654
lemma replicate_append_same:
haftmann@24796
  2655
  "replicate i x @ [x] = x # replicate i x"
haftmann@24796
  2656
  by (induct i) simp_all
haftmann@24796
  2657
haftmann@24796
  2658
lemma map_replicate_trivial:
haftmann@24796
  2659
  "map (\<lambda>i. x) [0..<i] = replicate i x"
haftmann@24796
  2660
  by (induct i) (simp_all add: replicate_append_same)
haftmann@24796
  2661
wenzelm@13114
  2662
nipkow@28642
  2663
lemma replicate_empty[simp]: "(replicate n x = []) \<longleftrightarrow> n=0"
nipkow@28642
  2664
by (induct n) auto
nipkow@28642
  2665
nipkow@28642
  2666
lemma empty_replicate[simp]: "([] = replicate n x) \<longleftrightarrow> n=0"
nipkow@28642
  2667
by (induct n) auto
nipkow@28642
  2668
nipkow@28642
  2669
lemma replicate_eq_replicate[simp]:
nipkow@28642
  2670
  "(replicate m x = replicate n y) \<longleftrightarrow> (m=n & (m\<noteq>0 \<longrightarrow> x=y))"
nipkow@28642
  2671
apply(induct m arbitrary: n)
nipkow@28642
  2672
 apply simp
nipkow@28642
  2673
apply(induct_tac n)
nipkow@28642
  2674
apply auto
nipkow@28642
  2675
done
nipkow@28642
  2676
nipkow@28642
  2677
nipkow@15392
  2678
subsubsection{*@{text rotate1} and @{text rotate}*}
nipkow@15302
  2679
nipkow@15302
  2680
lemma rotate_simps[simp]: "rotate1 [] = [] \<and> rotate1 (x#xs) = xs @ [x]"
nipkow@15302
  2681
by(simp add:rotate1_def)
nipkow@15302
  2682
nipkow@15302
  2683
lemma rotate0[simp]: "rotate 0 = id"
nipkow@15302
  2684
by(simp add:rotate_def)
nipkow@15302
  2685
nipkow@15302
  2686
lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)"
nipkow@15302
  2687
by(simp add:rotate_def)
nipkow@15302
  2688
nipkow@15302
  2689
lemma rotate_add:
nipkow@15302
  2690
  "rotate (m+n) = rotate m o rotate n"
nipkow@15302
  2691
by(simp add:rotate_def funpow_add)
nipkow@15302
  2692
nipkow@15302
  2693
lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs"
nipkow@15302
  2694
by(simp add:rotate_add)
nipkow@15302
  2695
nipkow@18049
  2696
lemma rotate1_rotate_swap: "rotate1 (rotate n xs) = rotate n (rotate1 xs)"
nipkow@18049
  2697
by(simp add:rotate_def funpow_swap1)
nipkow@18049
  2698
nipkow@15302
  2699
lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs"
nipkow@15302
  2700
by(cases xs) simp_all
nipkow@15302
  2701
nipkow@15302
  2702
lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs"
nipkow@15302
  2703
apply(induct n)
nipkow@15302
  2704
 apply simp
nipkow@15302
  2705
apply (simp add:rotate_def)
nipkow@13145
  2706
done
wenzelm@13114
  2707
nipkow@15302
  2708
lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]"
nipkow@15302
  2709
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2710
nipkow@15302
  2711
lemma rotate_drop_take:
nipkow@15302
  2712
  "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"
nipkow@15302
  2713
apply(induct n)
nipkow@15302
  2714
 apply simp
nipkow@15302
  2715
apply(simp add:rotate_def)
nipkow@15302
  2716
apply(cases "xs = []")
nipkow@15302
  2717
 apply (simp)
nipkow@15302
  2718
apply(case_tac "n mod length xs = 0")
nipkow@15302
  2719
 apply(simp add:mod_Suc)
nipkow@15302
  2720
 apply(simp add: rotate1_hd_tl drop_Suc take_Suc)
nipkow@15302
  2721
apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric]
nipkow@15302
  2722
                take_hd_drop linorder_not_le)
nipkow@13145
  2723
done
wenzelm@13114
  2724
nipkow@15302
  2725
lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs"
nipkow@15302
  2726
by(simp add:rotate_drop_take)
nipkow@15302
  2727
nipkow@15302
  2728
lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs"
nipkow@15302
  2729
by(simp add:rotate_drop_take)
nipkow@15302
  2730
nipkow@15302
  2731
lemma length_rotate1[simp]: "length(rotate1 xs) = length xs"
nipkow@15302
  2732
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2733
nipkow@24526
  2734
lemma length_rotate[simp]: "length(rotate n xs) = length xs"
nipkow@24526
  2735
by (induct n arbitrary: xs) (simp_all add:rotate_def)
nipkow@15302
  2736
nipkow@15302
  2737
lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs"
nipkow@15302
  2738
by(simp add:rotate1_def split:list.split) blast
nipkow@15302
  2739
nipkow@15302
  2740
lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs"
nipkow@15302
  2741
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  2742
nipkow@15302
  2743
lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)"
nipkow@15302
  2744
by(simp add:rotate_drop_take take_map drop_map)
nipkow@15302
  2745
nipkow@15302
  2746
lemma set_rotate1[simp]: "set(rotate1 xs) = set xs"
nipkow@15302
  2747
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2748
nipkow@15302
  2749
lemma set_rotate[simp]: "set(rotate n xs) = set xs"
nipkow@15302
  2750
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  2751
nipkow@15302
  2752
lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])"
nipkow@15302
  2753
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2754
nipkow@15302
  2755
lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])"