src/HOL/List.thy
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primrec definition for nth
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(*  Title:      HOL/List.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow
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    Copyright   1994 TU Muenchen
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The datatype of finite lists.
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*)
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List = Arith +
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datatype 'a list = "[]" ("[]") | "#" 'a ('a list) (infixr 65)
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consts
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  "@"         :: ['a list, 'a list] => 'a list            (infixr 65)
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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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  hd          :: 'a list => 'a
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  length      :: 'a list => nat
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  set_of_list :: 'a list => 'a set
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  list_all    :: ('a => bool) => ('a list => bool)
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  map         :: ('a=>'b) => ('a list => 'b list)
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  mem         :: ['a, 'a list] => bool                    (infixl 55)
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  nth         :: [nat, 'a list] => 'a
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  take, drop  :: [nat, 'a list] => 'a list
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  takeWhile,
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  dropWhile   :: ('a => bool) => 'a list => 'a list
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  tl,ttl      :: 'a list => 'a list
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  rev         :: 'a list => 'a list
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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"   :: [idt, 'a list, bool] => 'a list          ("(1[_:_ ./ _])")
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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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syntax (symbols)
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  "@filter"   :: [idt, 'a list, bool] => 'a list          ("(1[_\\<in>_ ./ _])")
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primrec hd list
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  "hd([]) = (@x.False)"
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  "hd(x#xs) = x"
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primrec tl list
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  "tl([]) = (@x.False)"
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  "tl(x#xs) = xs"
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primrec ttl list
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  (* a "total" version of tl: *)
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  "ttl([]) = []"
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  "ttl(x#xs) = xs"
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primrec "op mem" list
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  "x mem [] = False"
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  "x mem (y#ys) = (if y=x then True else x mem ys)"
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primrec set_of_list list
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  "set_of_list [] = {}"
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  "set_of_list (x#xs) = insert x (set_of_list xs)"
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primrec list_all list
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  list_all_Nil  "list_all P [] = True"
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  list_all_Cons "list_all P (x#xs) = (P(x) & list_all P xs)"
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primrec map list
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  "map f [] = []"
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  "map f (x#xs) = f(x)#map f xs"
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primrec "op @" list
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  "[] @ ys = ys"
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  "(x#xs)@ys = x#(xs@ys)"
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primrec rev list
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  "rev([]) = []"
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  "rev(x#xs) = rev(xs) @ [x]"
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primrec filter list
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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 foldl list
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  "foldl f a [] = a"
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  "foldl f a (x#xs) = foldl f (f a x) xs"
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primrec length list
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  "length([]) = 0"
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  "length(x#xs) = Suc(length(xs))"
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primrec concat list
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  "concat([]) = []"
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  "concat(x#xs) = x @ concat(xs)"
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primrec drop list
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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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primrec take list
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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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primrec nth nat
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  "nth 0 xs = hd xs"
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  "nth (Suc n) xs = nth n (tl xs)"
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primrec takeWhile list
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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 dropWhile list
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  "dropWhile P [] = []"
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  "dropWhile P (x#xs) = (if P x then dropWhile P xs else xs)"
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end