MAYBE
MAYBE
TRS:
 {
                       0(#()) -> #(),
                    +(x, #()) -> x,
                    +(#(), x) -> x,
                +(0(x), 0(y)) -> 0(+(x, y)),
                +(0(x), 1(y)) -> 1(+(x, y)),
                +(+(x, y), z) -> +(x, +(y, z)),
                +(1(x), 0(y)) -> 1(+(x, y)),
                +(1(x), 1(y)) -> 0(+(+(x, y), 1(#()))),
                    -(x, #()) -> x,
                    -(#(), x) -> #(),
                -(0(x), 0(y)) -> 0(-(x, y)),
                -(0(x), 1(y)) -> 1(-(-(x, y), 1(#()))),
                -(1(x), 0(y)) -> 1(-(x, y)),
                -(1(x), 1(y)) -> 0(-(x, y)),
                 not(false()) -> true(),
                  not(true()) -> false(),
            if(false(), x, y) -> y,
             if(true(), x, y) -> x,
                 eq(#(), #()) -> true(),
                eq(#(), 0(y)) -> eq(#(), y),
                eq(#(), 1(y)) -> false(),
                eq(0(x), #()) -> eq(x, #()),
               eq(0(x), 0(y)) -> eq(x, y),
               eq(0(x), 1(y)) -> false(),
                eq(1(x), #()) -> false(),
               eq(1(x), 0(y)) -> false(),
               eq(1(x), 1(y)) -> eq(x, y),
                   ge(x, #()) -> true(),
                ge(#(), 0(x)) -> ge(#(), x),
                ge(#(), 1(x)) -> false(),
               ge(0(x), 0(y)) -> ge(x, y),
               ge(0(x), 1(y)) -> not(ge(y, x)),
               ge(1(x), 0(y)) -> ge(x, y),
               ge(1(x), 1(y)) -> ge(x, y),
                    log'(#()) -> #(),
                   log'(0(x)) -> if(ge(x, 1(#())), +(log'(x), 1(#())), #()),
                   log'(1(x)) -> +(log'(x), 1(#())),
                       log(x) -> -(log'(x), 1(#())),
                *(x, +(y, z)) -> +(*(x, y), *(x, z)),
                    *(#(), x) -> #(),
                   *(0(x), y) -> 0(*(x, y)),
                   *(1(x), y) -> +(0(*(x, y)), y),
                *(*(x, y), z) -> *(x, *(y, z)),
                app(nil(), l) -> l,
         app(cons(x, l1), l2) -> cons(x, app(l1, l2)),
             sum(app(l1, l2)) -> +(sum(l1), sum(l2)),
                   sum(nil()) -> 0(#()),
              sum(cons(x, l)) -> +(x, sum(l)),
            prod(app(l1, l2)) -> *(prod(l1), prod(l2)),
                  prod(nil()) -> 1(#()),
             prod(cons(x, l)) -> *(x, prod(l)),
                mem(x, nil()) -> false(),
           mem(x, cons(y, l)) -> if(eq(x, y), true(), mem(x, l)),
              inter(x, nil()) -> nil(),
       inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3)),
       inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1),
       inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3)),
              inter(nil(), x) -> nil(),
       inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2),
  ifinter(false(), x, l1, l2) -> inter(l1, l2),
   ifinter(true(), x, l1, l2) -> cons(x, inter(l1, l2))
 }
 DUP: We consider a duplicating system.
  Trs:
   {
                         0(#()) -> #(),
                      +(x, #()) -> x,
                      +(#(), x) -> x,
                  +(0(x), 0(y)) -> 0(+(x, y)),
                  +(0(x), 1(y)) -> 1(+(x, y)),
                  +(+(x, y), z) -> +(x, +(y, z)),
                  +(1(x), 0(y)) -> 1(+(x, y)),
                  +(1(x), 1(y)) -> 0(+(+(x, y), 1(#()))),
                      -(x, #()) -> x,
                      -(#(), x) -> #(),
                  -(0(x), 0(y)) -> 0(-(x, y)),
                  -(0(x), 1(y)) -> 1(-(-(x, y), 1(#()))),
                  -(1(x), 0(y)) -> 1(-(x, y)),
                  -(1(x), 1(y)) -> 0(-(x, y)),
                   not(false()) -> true(),
                    not(true()) -> false(),
              if(false(), x, y) -> y,
               if(true(), x, y) -> x,
                   eq(#(), #()) -> true(),
                  eq(#(), 0(y)) -> eq(#(), y),
                  eq(#(), 1(y)) -> false(),
                  eq(0(x), #()) -> eq(x, #()),
                 eq(0(x), 0(y)) -> eq(x, y),
                 eq(0(x), 1(y)) -> false(),
                  eq(1(x), #()) -> false(),
                 eq(1(x), 0(y)) -> false(),
                 eq(1(x), 1(y)) -> eq(x, y),
                     ge(x, #()) -> true(),
                  ge(#(), 0(x)) -> ge(#(), x),
                  ge(#(), 1(x)) -> false(),
                 ge(0(x), 0(y)) -> ge(x, y),
                 ge(0(x), 1(y)) -> not(ge(y, x)),
                 ge(1(x), 0(y)) -> ge(x, y),
                 ge(1(x), 1(y)) -> ge(x, y),
                      log'(#()) -> #(),
                     log'(0(x)) -> if(ge(x, 1(#())), +(log'(x), 1(#())), #()),
                     log'(1(x)) -> +(log'(x), 1(#())),
                         log(x) -> -(log'(x), 1(#())),
                  *(x, +(y, z)) -> +(*(x, y), *(x, z)),
                      *(#(), x) -> #(),
                     *(0(x), y) -> 0(*(x, y)),
                     *(1(x), y) -> +(0(*(x, y)), y),
                  *(*(x, y), z) -> *(x, *(y, z)),
                  app(nil(), l) -> l,
           app(cons(x, l1), l2) -> cons(x, app(l1, l2)),
               sum(app(l1, l2)) -> +(sum(l1), sum(l2)),
                     sum(nil()) -> 0(#()),
                sum(cons(x, l)) -> +(x, sum(l)),
              prod(app(l1, l2)) -> *(prod(l1), prod(l2)),
                    prod(nil()) -> 1(#()),
               prod(cons(x, l)) -> *(x, prod(l)),
                  mem(x, nil()) -> false(),
             mem(x, cons(y, l)) -> if(eq(x, y), true(), mem(x, l)),
                inter(x, nil()) -> nil(),
         inter(l1, app(l2, l3)) -> app(inter(l1, l2), inter(l1, l3)),
         inter(l1, cons(x, l2)) -> ifinter(mem(x, l1), x, l2, l1),
         inter(app(l1, l2), l3) -> app(inter(l1, l3), inter(l2, l3)),
                inter(nil(), x) -> nil(),
         inter(cons(x, l1), l2) -> ifinter(mem(x, l2), x, l1, l2),
    ifinter(false(), x, l1, l2) -> inter(l1, l2),
     ifinter(true(), x, l1, l2) -> cons(x, inter(l1, l2))
   }
  Fail