src/HOL/List.thy
author berghofe
Tue, 30 May 2000 18:02:49 +0200
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child 9336 9ae89b9ce206
permissions -rw-r--r--
the is now defined using primrec, avoiding explicit use of arbitrary.
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(*  Title:      HOL/List.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow
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    Copyright   1994 TU Muenchen
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The datatype of finite lists.
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*)
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List = PreList +
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datatype 'a list = Nil ("[]") | Cons 'a ('a list) (infixr "#" 65)
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consts
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  "@"         :: ['a list, 'a list] => 'a list            (infixr 65)
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  filter      :: ['a => bool, 'a list] => 'a list
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  concat      :: 'a list list => 'a list
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  foldl       :: [['b,'a] => 'b, 'b, 'a list] => 'b
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  foldr       :: [['a,'b] => 'b, 'a list, 'b] => 'b
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  hd, last    :: 'a list => 'a
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  set         :: 'a list => 'a set
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  list_all    :: ('a => bool) => ('a list => bool)
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  list_all2   :: ('a => 'b => bool) => 'a list => 'b list => bool
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  map         :: ('a=>'b) => ('a list => 'b list)
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  mem         :: ['a, 'a list] => bool                    (infixl 55)
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  nth         :: ['a list, nat] => 'a			  (infixl "!" 100)
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  list_update :: 'a list => nat => 'a => 'a list
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  take, drop  :: [nat, 'a list] => 'a list
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  takeWhile,
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  dropWhile   :: ('a => bool) => 'a list => 'a list
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  tl, butlast :: 'a list => 'a list
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  rev         :: 'a list => 'a list
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  zip	      :: "'a list => 'b list => ('a * 'b) list"
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  upt         :: nat => nat => nat list                   ("(1[_../_'(])")
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  remdups     :: 'a list => 'a list
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  null, nodups :: "'a list => bool"
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  replicate   :: nat => 'a => 'a list
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nonterminals
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  lupdbinds  lupdbind
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syntax
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  (* list Enumeration *)
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  "@list"     :: args => 'a list                          ("[(_)]")
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  (* Special syntax for filter *)
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  "@filter"   :: [pttrn, 'a list, bool] => 'a list        ("(1[_:_ ./ _])")
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  (* list update *)
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  "_lupdbind"      :: ['a, 'a] => lupdbind            ("(2_ :=/ _)")
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  ""               :: lupdbind => lupdbinds           ("_")
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  "_lupdbinds"     :: [lupdbind, lupdbinds] => lupdbinds ("_,/ _")
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  "_LUpdate"       :: ['a, lupdbinds] => 'a           ("_/[(_)]" [900,0] 900)
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  upto        :: nat => nat => nat list                   ("(1[_../_])")
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translations
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  "[x, xs]"     == "x#[xs]"
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  "[x]"         == "x#[]"
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  "[x:xs . P]"  == "filter (%x. P) xs"
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  "_LUpdate xs (_lupdbinds b bs)"  == "_LUpdate (_LUpdate xs b) bs"
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  "xs[i:=x]"                       == "list_update xs i x"
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  "[i..j]" == "[i..(Suc j)(]"
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syntax (symbols)
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  "@filter"   :: [pttrn, 'a list, bool] => 'a list        ("(1[_\\<in>_ ./ _])")
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consts
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  lists        :: 'a set => 'a list set
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  inductive "lists A"
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  intrs
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    Nil  "[]: lists A"
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    Cons "[| a: A;  l: lists A |] ==> a#l : lists A"
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(*Function "size" is overloaded for all datatypes.  Users may refer to the
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  list version as "length".*)
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syntax   length :: 'a list => nat
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translations  "length"  =>  "size:: _ list => nat"
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primrec
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  "hd(x#xs) = x"
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  "tl([])   = []"
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  "tl(x#xs) = xs"
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  "null([])   = True"
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  "null(x#xs) = False"
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primrec
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  "last(x#xs) = (if xs=[] then x else last xs)"
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primrec
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  "butlast []    = []"
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  "butlast(x#xs) = (if xs=[] then [] else x#butlast xs)"
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primrec
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  "x mem []     = False"
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  "x mem (y#ys) = (if y=x then True else x mem ys)"
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  "set [] = {}"
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  "set (x#xs) = insert x (set xs)"
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primrec
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  list_all_Nil  "list_all P [] = True"
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  list_all_Cons "list_all P (x#xs) = (P(x) & list_all P xs)"
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primrec
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  "map f []     = []"
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  "map f (x#xs) = f(x)#map f xs"
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primrec
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  append_Nil  "[]    @ys = ys"
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  append_Cons "(x#xs)@ys = x#(xs@ys)"
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primrec
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  "rev([])   = []"
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  "rev(x#xs) = rev(xs) @ [x]"
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primrec
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  "filter P []     = []"
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  "filter P (x#xs) = (if P x then x#filter P xs else filter P xs)"
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primrec
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  foldl_Nil  "foldl f a [] = a"
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  foldl_Cons "foldl f a (x#xs) = foldl f (f a x) xs"
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primrec
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  "foldr f [] a     = a"
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  "foldr f (x#xs) a = f x (foldr f xs a)"
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primrec
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  "concat([])   = []"
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  "concat(x#xs) = x @ concat(xs)"
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  drop_Nil  "drop n [] = []"
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  drop_Cons "drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)"
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  (* Warning: simpset does not contain this definition but separate theorems 
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     for n=0 / n=Suc k*)
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primrec
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  take_Nil  "take n [] = []"
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  take_Cons "take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)"
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  (* Warning: simpset does not contain this definition but separate theorems 
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     for n=0 / n=Suc k*)
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primrec 
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  nth_Cons  "(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)"
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  (* Warning: simpset does not contain this definition but separate theorems 
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     for n=0 / n=Suc k*)
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primrec
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 "    [][i:=v] = []"
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 "(x#xs)[i:=v] = (case i of 0     => v # xs 
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			  | Suc j => x # xs[j:=v])"
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primrec
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  "takeWhile P []     = []"
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  "takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])"
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primrec
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  "dropWhile P []     = []"
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  "dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)"
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primrec
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  "zip xs []     = []"
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  "zip xs (y#ys) = (case xs of [] => [] | z#zs => (z,y)#zip zs ys)"
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  (* Warning: simpset does not contain this definition but separate theorems 
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     for xs=[] / xs=z#zs *)
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primrec
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  upt_0   "[i..0(] = []"
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  upt_Suc "[i..(Suc j)(] = (if i <= j then [i..j(] @ [j] else [])"
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primrec
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  "nodups []     = True"
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  "nodups (x#xs) = (x ~: set xs & nodups xs)"
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primrec
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  "remdups [] = []"
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  "remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)"
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primrec
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e2459d18ff47 changed constants mem and list_all to mere translations
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  replicate_0   "replicate  0      x = []"
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  replicate_Suc "replicate (Suc n) x = x # replicate n x"
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defs
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 list_all2_def
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 "list_all2 P xs ys == length xs = length ys & (!(x,y):set(zip xs ys). P x y)"
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(** Lexicographic orderings on lists **)
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consts
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 lexn :: "('a * 'a)set => nat => ('a list * 'a list)set"
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primrec
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"lexn r 0       = {}"
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"lexn r (Suc n) = (prod_fun (split op#) (split op#) `` (r <*lex*> lexn r n)) Int
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                  {(xs,ys). length xs = Suc n & length ys = Suc n}"
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constdefs
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 lex :: "('a * 'a)set => ('a list * 'a list)set"
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"lex r == UN n. lexn r n"
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 lexico :: "('a * 'a)set => ('a list * 'a list)set"
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"lexico r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))"
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end
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ML
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local
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(* translating size::list -> length *)
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fun size_tr' _ (Type ("fun", (Type ("list", _) :: _))) [t] =
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      Syntax.const "length" $ t
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  | size_tr' _ _ _ = raise Match;
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in
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val typed_print_translation = [("size", size_tr')];
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end;