--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Induct/SList.thy Wed May 07 12:50:26 1997 +0200
@@ -0,0 +1,119 @@
+(* 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
+ set_of_list :: ('a list => 'a set)
+ mem :: ['a, 'a list] => bool (infixl 55)
+ 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 filter *)
+ "@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"
+
+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)
+ (%g. List_case c (%x y. d x y (g y))) M"
+
+ 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)"
+
+ set_of_list_def "set_of_list xs == list_rec xs {} (%x l r. insert x r)"
+
+ mem_def "x mem xs ==
+ list_rec xs False (%y ys r. if y=x then True else 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