Added "nth" and some lemmas.
(* 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
null :: "'a list => bool"
hd :: "'a list => 'a"
tl,ttl :: "'a list => 'a list"
mem :: "['a, 'a list] => bool" (infixl 55)
list_all :: "('a => bool) => ('a list => bool)"
map :: "('a=>'b) => ('a list => 'b list)"
"@" :: "['a list, 'a list] => 'a list" (infixr 65)
filter :: "['a => bool, 'a list] => 'a list"
foldl :: "[['b,'a] => 'b, 'b, 'a list] => 'b"
length :: "'a list => nat"
nth :: "[nat, 'a list] => 'a"
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 null list
null_Nil "null([]) = True"
null_Cons "null(x#xs) = False"
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, True, 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 filter list
filter_Nil "filter(P,[]) = []"
filter_Cons "filter(P,x#xs) = if(P(x), x#filter(P,xs), 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))"
defs
nth_def "nth(n) == nat_rec(n, hd, %m r xs. r(tl(xs)))"
end