(* Title: HOL/list
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
Definition of type 'a list by a least fixed point
We use List(A) == lfp(%Z. {NUMB(0)} <+> A <*> Z)
and not List == lfp(%Z. {NUMB(0)} <+> range(Leaf) <*> Z)
so that List can serve as a "functor" for defining other recursive types
*)
List = Sexp +
types
list 1
arities
list :: (term) term
consts
List_Fun :: "['a node set set, 'a node set set] => 'a node set set"
List :: "'a node set set => 'a node set set"
Rep_List :: "'a list => 'a node set"
Abs_List :: "'a node set => 'a list"
NIL :: "'a node set"
CONS :: "['a node set, 'a node set] => 'a node set"
Nil :: "'a list"
"#" :: "['a, 'a list] => 'a list" (infixr 65)
List_case :: "['a node set, 'b, ['a node set, 'a node set]=>'b] => 'b"
List_rec :: "['a node set, 'b, ['a node set, 'a node set, 'b]=>'b] => 'b"
list_rec :: "['a list, 'b, ['a, 'a list, 'b]=>'b] => 'b"
Rep_map :: "('b => 'a node set) => ('b list => 'a node set)"
Abs_map :: "('a node set => 'b) => 'a node set => 'b list"
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" (infixl 65)
list_case :: "['a list, 'b, ['a, 'a list]=>'b] => 'b"
filter :: "['a => bool, 'a list] => 'a list"
(* List Enumeration *)
"[]" :: "'a list" ("[]")
"@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#[]"
"[]" == "Nil"
"case xs of Nil => a | y#ys => b" == "list_case(xs,a,%y ys.b)"
"[x:xs . P]" == "filter(%x.P,xs)"
"Alls x:xs.P" == "list_all(%x.P,xs)"
rules
List_Fun_def "List_Fun(A) == (%Z. {Numb(0)} <+> A <*> Z)"
List_def "List(A) == lfp(List_Fun(A))"
(* Faking a Type Definition ... *)
Rep_List "Rep_List(xs): List(range(Leaf))"
Rep_List_inverse "Abs_List(Rep_List(xs)) = xs"
Abs_List_inverse "M: List(range(Leaf)) ==> Rep_List(Abs_List(M)) = M"
(* Defining the Concrete Constructors *)
NIL_def "NIL == In0(Numb(0))"
CONS_def "CONS(M, N) == In1(M $ N)"
(* Defining the Abstract Constructors *)
Nil_def "Nil == Abs_List(NIL)"
Cons_def "x#xs == Abs_List(CONS(Leaf(x), Rep_List(xs)))"
List_case_def "List_case(M, c, d) == Case(M, %x.c, %u. Split(u, %x y.d(x, y)))"
(* List Recursion -- the trancl is Essential; see list.ML *)
List_rec_def
"List_rec(M, c, d) == wfrec(trancl(pred_Sexp), M, \
\ %z g. List_case(z, c, %x y. d(x, y, g(y))))"
list_rec_def
"list_rec(l, c, d) == \
\ List_rec(Rep_List(l), c, %x y r. d(Inv(Leaf, x), Abs_List(y), r))"
(* Generalized Map Functionals *)
Rep_map_def
"Rep_map(f, xs) == list_rec(xs, NIL, %x l r. CONS(f(x), r))"
Abs_map_def
"Abs_map(g, M) == List_rec(M, Nil, %N L r. g(N)#r)"
null_def "null(xs) == list_rec(xs, True, %x xs r.False)"
hd_def "hd(xs) == list_rec(xs, @x.True, %x xs r.x)"
tl_def "tl(xs) == list_rec(xs, @xs.True, %x xs r.xs)"
(* a total version of tl: *)
ttl_def "ttl(xs) == list_rec(xs, [], %x xs r.xs)"
mem_def "x mem xs == \
\ list_rec(xs, False, %y ys r. if(y=x, True, r))"
list_all_def "list_all(P, xs) == list_rec(xs, True, %x l r. P(x) & r)"
map_def "map(f, xs) == list_rec(xs, [], %x l r. f(x)#r)"
append_def "xs@ys == list_rec(xs, ys, %x l r. x#r)"
filter_def "filter(P,xs) == \
\ list_rec(xs, [], %x xs r. if(P(x), x#r, r))"
list_case_def "list_case(xs, a, f) == list_rec(xs, a, %x xs r.f(x, xs))"
end