(* Title: HOL/ex/SList.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
Definition of type 'a list (strict lists) 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
*)
SList = Sexp +
types
'a list
arities
list :: (term) term
consts
list :: "'a item set => 'a item set"
Rep_list :: "'a list => 'a item"
Abs_list :: "'a item => 'a list"
NIL :: "'a item"
CONS :: "['a item, 'a item] => 'a item"
Nil :: "'a list"
"#" :: "['a, 'a list] => 'a list" (infixr 65)
List_case :: "['b, ['a item, 'a item]=>'b, 'a item] => 'b"
List_rec :: "['a item, 'b, ['a item, 'a item, 'b]=>'b] => 'b"
list_case :: "['b, ['a, 'a list]=>'b, 'a list] => 'b"
list_rec :: "['a list, 'b, ['a, 'a list, 'b]=>'b] => 'b"
Rep_map :: "('b => 'a item) => ('b list => 'a item)"
Abs_map :: "('a item => 'b) => 'a item => '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" (infixr 65)
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 a (%y ys.b) xs"
"[x:xs . P]" == "filter (%x.P) xs"
"Alls x:xs.P" == "list_all (%x.P) xs"
defs
(* Defining the Concrete Constructors *)
NIL_def "NIL == In0(Numb(0))"
CONS_def "CONS M N == In1(M $ N)"
inductive "list(A)"
intrs
NIL_I "NIL: list(A)"
CONS_I "[| a: A; M: list(A) |] ==> CONS a M : list(A)"
rules
(* 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"
defs
(* 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 c d == Case (%x.c) (Split d)"
(* list Recursion -- the trancl is Essential; see list.ML *)
List_rec_def
"List_rec M c d == wfrec (trancl pred_sexp) M \
\ (List_case (%g.c) (%x y g. 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 then True else 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) then x#r else r)"
list_case_def "list_case a f xs == list_rec xs a (%x xs r.f x xs)"
end