(* Title: HOL/List.thy
ID: $Id$
Author: Tobias Nipkow
Copyright 1994 TU Muenchen
Definition of type 'a list as a datatype. This allows primrec to work.
TODO: delete list_all from this theory and from ex/Sorting, etc.
use set_of_list instead
*)
List = Arith +
datatype 'a list = "[]" ("[]") | "#" 'a ('a list) (infixr 65)
consts
"@" :: ['a list, 'a list] => 'a list (infixr 65)
drop :: [nat, 'a list] => 'a list
filter :: ['a => bool, 'a list] => 'a list
flat :: 'a list list => 'a list
foldl :: [['b,'a] => 'b, 'b, 'a list] => 'b
hd :: 'a list => 'a
length :: 'a list => nat
set_of_list :: ('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 :: [nat, '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 list_all and filter *)
"@Alls" :: [idt, 'a list, bool] => bool ("(2Alls _:_./ _)" 10)
"@filter" :: [idt, 'a list, bool] => 'a list ("(1[_:_ ./ _])")
translations
"[x, xs]" == "x#[xs]"
"[x]" == "x#[]"
"[x:xs . P]" == "filter (%x.P) xs"
"Alls x:xs.P" == "list_all (%x.P) xs"
primrec hd list
"hd([]) = (@x.False)"
"hd(x#xs) = x"
primrec tl list
"tl([]) = (@x.False)"
"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_of_list list
"set_of_list [] = {}"
"set_of_list (x#xs) = insert x (set_of_list 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 length list
"length([]) = 0"
"length(x#xs) = Suc(length(xs))"
primrec flat list
"flat([]) = []"
"flat(x#xs) = x @ flat(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)"
defs
nth_def "nth(n) == nat_rec hd (%m r xs. r(tl(xs))) n"
end