(* 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.
*)
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"
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_Nil "hd([]) = (@x.False)"
hd_Cons "hd(x#xs) = x"
primrec tl list
tl_Nil "tl([]) = (@x.False)"
tl_Cons "tl(x#xs) = xs"
primrec ttl list
(* a "total" version of tl: *)
ttl_Nil "ttl([]) = []"
ttl_Cons "ttl(x#xs) = xs"
primrec "op mem" list
mem_Nil "x mem [] = False"
mem_Cons "x mem (y#ys) = (if y=x then True else x mem ys)"
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_Nil "map f [] = []"
map_Cons "map f (x#xs) = f(x)#map f xs"
primrec "op @" list
append_Nil "[] @ ys = ys"
append_Cons "(x#xs)@ys = x#(xs@ys)"
primrec rev list
rev_Nil "rev([]) = []"
rev_Cons "rev(x#xs) = rev(xs) @ [x]"
primrec filter list
filter_Nil "filter P [] = []"
filter_Cons "filter P (x#xs) = (if P x then x#filter P xs else filter P xs)"
primrec foldl list
foldl_Nil "foldl f a [] = a"
foldl_Cons "foldl f a (x#xs) = foldl f (f a x) xs"
primrec length list
length_Nil "length([]) = 0"
length_Cons "length(x#xs) = Suc(length(xs))"
primrec flat list
flat_Nil "flat([]) = []"
flat_Cons "flat(x#xs) = x @ flat(xs)"
defs
drop_def "drop n == nat_rec n (%xs.xs) \
\ (%m r xs. case xs of [] => [] | y#ys => r ys)"
take_def "take n == nat_rec n (%xs.[]) \
\ (%m r xs. case xs of [] => [] | y#ys => y # r ys)"
nth_def "nth(n) == nat_rec n hd (%m r xs. r(tl(xs)))"
end