src/HOL/List.thy

Simplified primrec definitions.

(*  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 setOfList 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
  setOfList :: ('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 setOfList list
  "setOfList [] = {}"
  "setOfList (x#xs) = insert x (setOfList 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