(* Title: HOL/llist.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
Definition of type 'a llist by a greatest fixed point
Shares NIL, CONS, List_Fun, 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. (%z.split(z, %l1 l2.<LCons(x,l1),LCons(x,l2)>))``r)"
*)
LList = Gfp + List +
types llist 1
arities llist :: (term)term
consts
LList :: "'a node set set => 'a node set set"
LListD_Fun ::
"[('a node set * 'a node set)set, ('a node set * 'a node set)set] => \
\ ('a node set * 'a node set)set"
LListD ::
"('a node set * 'a node set)set => ('a node set * 'a node set)set"
llistD_Fun :: "('a llist * 'a llist)set => ('a llist * 'a llist)set"
Rep_LList :: "'a llist => 'a node set"
Abs_LList :: "'a node set => 'a llist"
LNil :: "'a llist"
LCons :: "['a, 'a llist] => 'a llist"
llist_case :: "['a llist, 'b, ['a, 'a llist]=>'b] => 'b"
LList_corec_fun :: "[nat, 'a=>unit+('b node set * 'a), 'a] => 'b node set"
LList_corec :: "['a, 'a => unit + ('b node set * 'a)] => 'b node set"
llist_corec :: "['a, 'a => unit + ('b * 'a)] => 'b llist"
Lmap :: "('a node set => 'b node set) => ('a node set => 'b node set)"
lmap :: "('a=>'b) => ('a llist => 'b llist)"
iterates :: "['a => 'a, 'a] => 'a llist"
Lconst :: "'a node set => 'a node set"
Lappend :: "['a node set, 'a node set] => 'a node set"
lappend :: "['a llist, 'a llist] => 'a llist"
rules
LListD_Fun_def "LListD_Fun(r) == (%Z.diag({Numb(0)}) <++> r <**> Z)"
LList_def "LList(A) == gfp(List_Fun(A))"
LListD_def "LListD(r) == gfp(LListD_Fun(r))"
(*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"
(*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(l,c,d) == \
\ List_case(Rep_LList(l), c, %x y. d(Inv(Leaf,x), Abs_LList(y)))"
LList_corec_fun_def
"LList_corec_fun(k,f) == \
\ nat_rec(k, %x. {}, \
\ %j r x. sum_case(f(x), %u.NIL, \
\ %v. split(v, %z w. CONS(z, r(w)))))"
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(f(z), %x.Inl(x), \
\ %y.split(y, %v w. Inr(<Leaf(v), w>)))))"
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, %M. List_case(M, Inl(Unity), %x M'. Inr(<f(x), M'>)))"
lmap_def
"lmap(f,l) == \
\ llist_corec(l, %l. llist_case(l, 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>, %p. split(p, \
\ %M N. List_case(M, List_case(N, Inl(Unity), \
\ %N1 N2. Inr(<N1, <NIL,N2>>)), \
\ %M1 M2. Inr(<M1, <M2,N>>))))"
lappend_def
"lappend(l,n) == llist_corec(<l,n>, %p. split(p, \
\ %l n. llist_case(l, llist_case(n, Inl(Unity), \
\ %n1 n2. Inr(<n1, <LNil,n2>>)), \
\ %l1 l2. Inr(<l1, <l2,n>>))))"
end