(* Title: HOL/List.thy
ID: $Id$
Author: Tobias Nipkow
Copyright 1994 TU Muenchen
The datatype of finite lists.
*)
List = Divides +
datatype 'a list = "[]" ("[]") | "#" 'a ('a list) (infixr 65)
consts
"@" :: ['a list, 'a list] => 'a list (infixr 65)
filter :: ['a => bool, 'a list] => 'a list
concat :: 'a list list => 'a list
foldl :: [['b,'a] => 'b, 'b, 'a list] => 'b
hd :: 'a list => 'a
set :: 'a list => 'a set
list_all :: ('a => bool) => ('a list => bool)
map :: ('a=>'b) => ('a list => 'b list)
mem :: ['a, 'a list] => bool (infixl 55)
nth :: [nat, 'a list] => 'a
take, drop :: [nat, 'a list] => 'a list
takeWhile,
dropWhile :: ('a => bool) => 'a list => 'a list
tl,ttl :: 'a list => 'a list
rev :: 'a list => 'a list
syntax
(* list Enumeration *)
"@list" :: args => 'a list ("[(_)]")
(* Special syntax for filter *)
"@filter" :: [idt, 'a list, bool] => 'a list ("(1[_:_ ./ _])")
translations
"[x, xs]" == "x#[xs]"
"[x]" == "x#[]"
"[x:xs . P]" == "filter (%x.P) xs"
syntax (symbols)
"@filter" :: [idt, 'a list, bool] => 'a list ("(1[_\\<in>_ ./ _])")
consts
lists :: 'a set => 'a list set
inductive "lists A"
intrs
Nil "[]: lists A"
Cons "[| a: A; l: lists A |] ==> a#l : lists A"
(*Function "size" is overloaded for all datatypes. Users may refer to the
list version as "length".*)
syntax length :: 'a list => nat
translations "length" => "size"
primrec hd list
"hd([]) = arbitrary"
"hd(x#xs) = x"
primrec tl list
"tl([]) = arbitrary"
"tl(x#xs) = xs"
primrec ttl list
(* a "total" version of tl: *)
"ttl([]) = []"
"ttl(x#xs) = xs"
primrec "op mem" list
"x mem [] = False"
"x mem (y#ys) = (if y=x then True else x mem ys)"
primrec set list
"set [] = {}"
"set (x#xs) = insert x (set xs)"
primrec list_all list
list_all_Nil "list_all P [] = True"
list_all_Cons "list_all P (x#xs) = (P(x) & list_all P xs)"
primrec map list
"map f [] = []"
"map f (x#xs) = f(x)#map f xs"
primrec "op @" list
"[] @ ys = ys"
"(x#xs)@ys = x#(xs@ys)"
primrec rev list
"rev([]) = []"
"rev(x#xs) = rev(xs) @ [x]"
primrec filter list
"filter P [] = []"
"filter P (x#xs) = (if P x then x#filter P xs else filter P xs)"
primrec foldl list
"foldl f a [] = a"
"foldl f a (x#xs) = foldl f (f a x) xs"
primrec concat list
"concat([]) = []"
"concat(x#xs) = x @ concat(xs)"
primrec drop list
drop_Nil "drop n [] = []"
drop_Cons "drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)"
primrec take list
take_Nil "take n [] = []"
take_Cons "take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)"
primrec nth nat
"nth 0 xs = hd xs"
"nth (Suc n) xs = nth n (tl xs)"
primrec takeWhile list
"takeWhile P [] = []"
"takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])"
primrec dropWhile list
"dropWhile P [] = []"
"dropWhile P (x#xs) = (if P x then dropWhile P xs else xs)"
end