--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/ex/SList.thy Fri Dec 02 16:09:49 1994 +0100
@@ -0,0 +1,120 @@
+(* 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, 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(a, f, xs) == list_rec(xs, a, %x xs r.f(x, xs))"
+
+end