(STRATEGY
    INNERMOST)

(VAR
    X X1 X2 XS)
(DATATYPES
    A = µX.< nats, mark(X), 0, pairs, odds, ok(X) >)
(SIGNATURES
    active :: [A] -> A
    cons :: [A x A] -> A
    incr :: [A] -> A
    s :: [A] -> A
    head :: [A] -> A
    tail :: [A] -> A
    proper :: [A] -> A
    top :: [A] -> A)
(RULES
    active(nats()) -> mark(cons(0()
                               ,incr(nats())))
    active(pairs()) -> mark(cons(0()
                                ,incr(odds())))
    active(odds()) ->
      mark(incr(pairs()))
    active(incr(cons(X,XS))) ->
      mark(cons(s(X),incr(XS)))
    active(head(cons(X,XS))) ->
      mark(X)
    active(tail(cons(X,XS))) ->
      mark(XS)
    active(cons(X1,X2)) ->
      cons(active(X1),X2)
    active(incr(X)) ->
      incr(active(X))
    active(s(X)) -> s(active(X))
    active(head(X)) ->
      head(active(X))
    active(tail(X)) ->
      tail(active(X))
    cons(mark(X1),X2) ->
      mark(cons(X1,X2))
    incr(mark(X)) -> mark(incr(X))
    s(mark(X)) -> mark(s(X))
    head(mark(X)) -> mark(head(X))
    tail(mark(X)) -> mark(tail(X))
    proper(nats()) -> ok(nats())
    proper(cons(X1,X2)) ->
      cons(proper(X1),proper(X2))
    proper(0()) -> ok(0())
    proper(incr(X)) ->
      incr(proper(X))
    proper(pairs()) -> ok(pairs())
    proper(odds()) -> ok(odds())
    proper(s(X)) -> s(proper(X))
    proper(head(X)) ->
      head(proper(X))
    proper(tail(X)) ->
      tail(proper(X))
    cons(ok(X1),ok(X2)) ->
      ok(cons(X1,X2))
    incr(ok(X)) -> ok(incr(X))
    s(ok(X)) -> ok(s(X))
    head(ok(X)) -> ok(head(X))
    tail(ok(X)) -> ok(tail(X))
    top(mark(X)) -> top(proper(X))
    top(ok(X)) -> top(active(X)))