--- a/ex/LList.thy Tue Oct 24 14:59:17 1995 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,145 +0,0 @@
-(* Title: HOL/LList.thy
- ID: $Id$
- Author: Lawrence C Paulson, Cambridge University Computer Laboratory
- Copyright 1994 University of Cambridge
-
-Definition of type 'a llist by a greatest fixed point
-
-Shares NIL, CONS, List_case with List.thy
-
-Still needs filter and flatten functions -- hard because they need
-bounds on the amount of lookahead required.
-
-Could try (but would it work for the gfp analogue of term?)
- LListD_Fun_def "LListD_Fun(A) == (%Z.diag({Numb(0)}) <++> diag(A) <**> Z)"
-
-A nice but complex example would be [ML for the Working Programmer, page 176]
- from(1) = enumerate (Lmap (Lmap(pack), makeqq(from(1),from(1))))
-
-Previous definition of llistD_Fun was explicit:
- llistD_Fun_def
- "llistD_Fun(r) ==
- {<LNil,LNil>} Un
- (UN x. (split(%l1 l2.<LCons(x,l1),LCons(x,l2)>))``r)"
-*)
-
-LList = Gfp + SList +
-
-types
- 'a llist
-
-arities
- llist :: (term)term
-
-consts
- list_Fun :: "['a item set, 'a item set] => 'a item set"
- LListD_Fun ::
- "[('a item * 'a item)set, ('a item * 'a item)set] =>
- ('a item * 'a item)set"
-
- llist :: "'a item set => 'a item set"
- LListD :: "('a item * 'a item)set => ('a item * 'a item)set"
- llistD_Fun :: "('a llist * 'a llist)set => ('a llist * 'a llist)set"
-
- Rep_llist :: "'a llist => 'a item"
- Abs_llist :: "'a item => 'a llist"
- LNil :: "'a llist"
- LCons :: "['a, 'a llist] => 'a llist"
-
- llist_case :: "['b, ['a, 'a llist]=>'b, 'a llist] => 'b"
-
- LList_corec_fun :: "[nat, 'a=>unit+('b item * 'a), 'a] => 'b item"
- LList_corec :: "['a, 'a => unit + ('b item * 'a)] => 'b item"
- llist_corec :: "['a, 'a => unit + ('b * 'a)] => 'b llist"
-
- Lmap :: "('a item => 'b item) => ('a item => 'b item)"
- lmap :: "('a=>'b) => ('a llist => 'b llist)"
-
- iterates :: "['a => 'a, 'a] => 'a llist"
-
- Lconst :: "'a item => 'a item"
- Lappend :: "['a item, 'a item] => 'a item"
- lappend :: "['a llist, 'a llist] => 'a llist"
-
-
-coinductive "llist(A)"
- intrs
- NIL_I "NIL: llist(A)"
- CONS_I "[| a: A; M: llist(A) |] ==> CONS(a,M) : llist(A)"
-
-coinductive "LListD(r)"
- intrs
- NIL_I "<NIL, NIL> : LListD(r)"
- CONS_I "[| <a,b>: r; <M,N> : LListD(r)
- |] ==> <CONS(a,M), CONS(b,N)> : LListD(r)"
-
-defs
- (*Now used exclusively for abbreviating the coinduction rule*)
- list_Fun_def "list_Fun(A,X) ==
- {z. z = NIL | (? M a. z = CONS(a, M) & a : A & M : X)}"
-
- LListD_Fun_def "LListD_Fun(r,X) ==
- {z. z = <NIL, NIL> |
- (? M N a b. z = <CONS(a, M), CONS(b, N)> &
- <a, b> : r & <M, N> : X)}"
-
- (*defining the abstract constructors*)
- LNil_def "LNil == Abs_llist(NIL)"
- LCons_def "LCons(x,xs) == Abs_llist(CONS(Leaf(x), Rep_llist(xs)))"
-
- llist_case_def
- "llist_case(c,d,l) ==
- List_case(c, %x y. d(Inv(Leaf,x), Abs_llist(y)), Rep_llist(l))"
-
- LList_corec_fun_def
- "LList_corec_fun(k,f) ==
- nat_rec(k, %x. {},
- %j r x. sum_case(%u.NIL, split(%z w. CONS(z, r(w))), f(x)))"
-
- LList_corec_def
- "LList_corec(a,f) == UN k. LList_corec_fun(k,f,a)"
-
- llist_corec_def
- "llist_corec(a,f) ==
- Abs_llist(LList_corec(a, %z.sum_case(%x.Inl(x),
- split(%v w. Inr(<Leaf(v), w>)), f(z))))"
-
- llistD_Fun_def
- "llistD_Fun(r) ==
- prod_fun(Abs_llist,Abs_llist) ``
- LListD_Fun(diag(range(Leaf)),
- prod_fun(Rep_llist,Rep_llist) `` r)"
-
- Lconst_def "Lconst(M) == lfp(%N. CONS(M, N))"
-
- Lmap_def
- "Lmap(f,M) == LList_corec(M, List_case(Inl(Unity), %x M'. Inr(<f(x), M'>)))"
-
- lmap_def
- "lmap(f,l) == llist_corec(l, llist_case(Inl(Unity), %y z. Inr(<f(y), z>)))"
-
- iterates_def "iterates(f,a) == llist_corec(a, %x. Inr(<x, f(x)>))"
-
-(*Append generates its result by applying f, where
- f(<NIL,NIL>) = Inl(Unity)
- f(<NIL, CONS(N1,N2)>) = Inr(<N1, <NIL,N2>)
- f(<CONS(M1,M2), N>) = Inr(<M1, <M2,N>)
-*)
-
- Lappend_def
- "Lappend(M,N) == LList_corec(<M,N>,
- split(List_case(List_case(Inl(Unity), %N1 N2. Inr(<N1, <NIL,N2>>)),
- %M1 M2 N. Inr(<M1, <M2,N>>))))"
-
- lappend_def
- "lappend(l,n) == llist_corec(<l,n>,
- split(llist_case(llist_case(Inl(Unity), %n1 n2. Inr(<n1, <LNil,n2>>)),
- %l1 l2 n. Inr(<l1, <l2,n>>))))"
-
-rules
- (*faking a type definition...*)
- Rep_llist "Rep_llist(xs): llist(range(Leaf))"
- Rep_llist_inverse "Abs_llist(Rep_llist(xs)) = xs"
- Abs_llist_inverse "M: llist(range(Leaf)) ==> Rep_llist(Abs_llist(M)) = M"
-
-end