MAYBE
TRS:
 {                   minus(x, 0()) -> x,
             minus(minus(x, y), z) -> minus(x, plus(y, z)),
                 minus(s(x), s(y)) -> minus(x, y),
                   quot(0(), s(y)) -> 0(),
                  quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))),
                      plus(0(), y) -> y,
                     plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)),
                     plus(s(x), y) -> s(plus(x, y)),
                  plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))),
                 plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))),
                   plus(zero(), y) -> y,
                     app(l, nil()) -> l,
                     app(nil(), k) -> k,
                app(cons(x, l), k) -> cons(x, app(l, k)),
  sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k))))),
               sum(cons(x, nil())) -> cons(x, nil()),
          sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l)),
                  if(true(), x, y) -> x,
                 if(false(), x, y) -> y,
                    gt(s(x), s(y)) -> gt(x, y),
                  gt(s(x), zero()) -> true(),
                     gt(zero(), y) -> false(),
                            not(x) -> if(x, false(), true()),
                             id(x) -> x}
 DP:
  Strict:
   {           minus#(minus(x, y), z) -> minus#(x, plus(y, z)),
               minus#(minus(x, y), z) -> plus#(y, z),
                   minus#(s(x), s(y)) -> minus#(x, y),
                    quot#(s(x), s(y)) -> minus#(x, y),
                    quot#(s(x), s(y)) -> quot#(minus(x, y), s(y)),
                       plus#(s(x), x) -> plus#(if(gt(x, x), id(x), id(x)), s(x)),
                       plus#(s(x), x) -> if#(gt(x, x), id(x), id(x)),
                       plus#(s(x), x) -> gt#(x, x),
                       plus#(s(x), x) -> id#(x),
                       plus#(s(x), y) -> plus#(x, y),
                    plus#(s(x), s(y)) -> plus#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))),
                    plus#(s(x), s(y)) -> if#(gt(x, y), x, y),
                    plus#(s(x), s(y)) -> if#(not(gt(x, y)), id(x), id(y)),
                    plus#(s(x), s(y)) -> gt#(x, y),
                    plus#(s(x), s(y)) -> not#(gt(x, y)),
                    plus#(s(x), s(y)) -> id#(x),
                    plus#(s(x), s(y)) -> id#(y),
                   plus#(id(x), s(y)) -> plus#(x, if(gt(s(y), y), y, s(y))),
                   plus#(id(x), s(y)) -> if#(gt(s(y), y), y, s(y)),
                   plus#(id(x), s(y)) -> gt#(s(y), y),
                  app#(cons(x, l), k) -> app#(l, k),
    sum#(app(l, cons(x, cons(y, k)))) -> app#(l, sum(cons(x, cons(y, k)))),
    sum#(app(l, cons(x, cons(y, k)))) -> sum#(app(l, sum(cons(x, cons(y, k))))),
    sum#(app(l, cons(x, cons(y, k)))) -> sum#(cons(x, cons(y, k))),
            sum#(cons(x, cons(y, l))) -> plus#(x, y),
            sum#(cons(x, cons(y, l))) -> sum#(cons(plus(x, y), l)),
                      gt#(s(x), s(y)) -> gt#(x, y),
                              not#(x) -> if#(x, false(), true())}
  Weak:
  {                   minus(x, 0()) -> x,
              minus(minus(x, y), z) -> minus(x, plus(y, z)),
                  minus(s(x), s(y)) -> minus(x, y),
                    quot(0(), s(y)) -> 0(),
                   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))),
                       plus(0(), y) -> y,
                      plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)),
                      plus(s(x), y) -> s(plus(x, y)),
                   plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))),
                  plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))),
                    plus(zero(), y) -> y,
                      app(l, nil()) -> l,
                      app(nil(), k) -> k,
                 app(cons(x, l), k) -> cons(x, app(l, k)),
   sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k))))),
                sum(cons(x, nil())) -> cons(x, nil()),
           sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l)),
                   if(true(), x, y) -> x,
                  if(false(), x, y) -> y,
                     gt(s(x), s(y)) -> gt(x, y),
                   gt(s(x), zero()) -> true(),
                      gt(zero(), y) -> false(),
                             not(x) -> if(x, false(), true()),
                              id(x) -> x}
  EDG:
   {
    (sum#(app(l, cons(x, cons(y, k)))) -> app#(l, sum(cons(x, cons(y, k)))), app#(cons(x, l), k) -> app#(l, k))
    (sum#(app(l, cons(x, cons(y, k)))) -> sum#(cons(x, cons(y, k))), sum#(cons(x, cons(y, l))) -> sum#(cons(plus(x, y), l)))
    (sum#(app(l, cons(x, cons(y, k)))) -> sum#(cons(x, cons(y, k))), sum#(cons(x, cons(y, l))) -> plus#(x, y))
    (plus#(id(x), s(y)) -> gt#(s(y), y), gt#(s(x), s(y)) -> gt#(x, y))
    (sum#(cons(x, cons(y, l))) -> sum#(cons(plus(x, y), l)), sum#(cons(x, cons(y, l))) -> sum#(cons(plus(x, y), l)))
    (sum#(cons(x, cons(y, l))) -> sum#(cons(plus(x, y), l)), sum#(cons(x, cons(y, l))) -> plus#(x, y))
    (plus#(s(x), s(y)) -> plus#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))), plus#(id(x), s(y)) -> gt#(s(y), y))
    (plus#(s(x), s(y)) -> plus#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))), plus#(id(x), s(y)) -> if#(gt(s(y), y), y, s(y)))
    (plus#(s(x), s(y)) -> plus#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))), plus#(id(x), s(y)) -> plus#(x, if(gt(s(y), y), y, s(y))))
    (plus#(s(x), s(y)) -> plus#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))), plus#(s(x), s(y)) -> id#(y))
    (plus#(s(x), s(y)) -> plus#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))), plus#(s(x), s(y)) -> id#(x))
    (plus#(s(x), s(y)) -> plus#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))), plus#(s(x), s(y)) -> not#(gt(x, y)))
    (plus#(s(x), s(y)) -> plus#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))), plus#(s(x), s(y)) -> gt#(x, y))
    (plus#(s(x), s(y)) -> plus#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))), plus#(s(x), s(y)) -> if#(not(gt(x, y)), id(x), id(y)))
    (plus#(s(x), s(y)) -> plus#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))), plus#(s(x), s(y)) -> if#(gt(x, y), x, y))
    (plus#(s(x), s(y)) -> plus#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))), plus#(s(x), s(y)) -> plus#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))
    (plus#(s(x), s(y)) -> plus#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))), plus#(s(x), y) -> plus#(x, y))
    (plus#(s(x), s(y)) -> plus#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))), plus#(s(x), x) -> id#(x))
    (plus#(s(x), s(y)) -> plus#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))), plus#(s(x), x) -> gt#(x, x))
    (plus#(s(x), s(y)) -> plus#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))), plus#(s(x), x) -> if#(gt(x, x), id(x), id(x)))
    (plus#(s(x), s(y)) -> plus#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))), plus#(s(x), x) -> plus#(if(gt(x, x), id(x), id(x)), s(x)))
    (minus#(minus(x, y), z) -> minus#(x, plus(y, z)), minus#(s(x), s(y)) -> minus#(x, y))
    (minus#(minus(x, y), z) -> minus#(x, plus(y, z)), minus#(minus(x, y), z) -> plus#(y, z))
    (minus#(minus(x, y), z) -> minus#(x, plus(y, z)), minus#(minus(x, y), z) -> minus#(x, plus(y, z)))
    (quot#(s(x), s(y)) -> minus#(x, y), minus#(s(x), s(y)) -> minus#(x, y))
    (quot#(s(x), s(y)) -> minus#(x, y), minus#(minus(x, y), z) -> plus#(y, z))
    (quot#(s(x), s(y)) -> minus#(x, y), minus#(minus(x, y), z) -> minus#(x, plus(y, z)))
    (plus#(s(x), s(y)) -> gt#(x, y), gt#(s(x), s(y)) -> gt#(x, y))
    (gt#(s(x), s(y)) -> gt#(x, y), gt#(s(x), s(y)) -> gt#(x, y))
    (app#(cons(x, l), k) -> app#(l, k), app#(cons(x, l), k) -> app#(l, k))
    (plus#(s(x), x) -> plus#(if(gt(x, x), id(x), id(x)), s(x)), plus#(id(x), s(y)) -> gt#(s(y), y))
    (plus#(s(x), x) -> plus#(if(gt(x, x), id(x), id(x)), s(x)), plus#(id(x), s(y)) -> if#(gt(s(y), y), y, s(y)))
    (plus#(s(x), x) -> plus#(if(gt(x, x), id(x), id(x)), s(x)), plus#(id(x), s(y)) -> plus#(x, if(gt(s(y), y), y, s(y))))
    (plus#(s(x), x) -> plus#(if(gt(x, x), id(x), id(x)), s(x)), plus#(s(x), s(y)) -> id#(y))
    (plus#(s(x), x) -> plus#(if(gt(x, x), id(x), id(x)), s(x)), plus#(s(x), s(y)) -> id#(x))
    (plus#(s(x), x) -> plus#(if(gt(x, x), id(x), id(x)), s(x)), plus#(s(x), s(y)) -> not#(gt(x, y)))
    (plus#(s(x), x) -> plus#(if(gt(x, x), id(x), id(x)), s(x)), plus#(s(x), s(y)) -> gt#(x, y))
    (plus#(s(x), x) -> plus#(if(gt(x, x), id(x), id(x)), s(x)), plus#(s(x), s(y)) -> if#(not(gt(x, y)), id(x), id(y)))
    (plus#(s(x), x) -> plus#(if(gt(x, x), id(x), id(x)), s(x)), plus#(s(x), s(y)) -> if#(gt(x, y), x, y))
    (plus#(s(x), x) -> plus#(if(gt(x, x), id(x), id(x)), s(x)), plus#(s(x), s(y)) -> plus#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))
    (plus#(s(x), x) -> plus#(if(gt(x, x), id(x), id(x)), s(x)), plus#(s(x), y) -> plus#(x, y))
    (plus#(s(x), x) -> plus#(if(gt(x, x), id(x), id(x)), s(x)), plus#(s(x), x) -> id#(x))
    (plus#(s(x), x) -> plus#(if(gt(x, x), id(x), id(x)), s(x)), plus#(s(x), x) -> gt#(x, x))
    (plus#(s(x), x) -> plus#(if(gt(x, x), id(x), id(x)), s(x)), plus#(s(x), x) -> if#(gt(x, x), id(x), id(x)))
    (plus#(s(x), x) -> plus#(if(gt(x, x), id(x), id(x)), s(x)), plus#(s(x), x) -> plus#(if(gt(x, x), id(x), id(x)), s(x)))
    (quot#(s(x), s(y)) -> quot#(minus(x, y), s(y)), quot#(s(x), s(y)) -> minus#(x, y))
    (quot#(s(x), s(y)) -> quot#(minus(x, y), s(y)), quot#(s(x), s(y)) -> quot#(minus(x, y), s(y)))
    (sum#(cons(x, cons(y, l))) -> plus#(x, y), plus#(s(x), x) -> plus#(if(gt(x, x), id(x), id(x)), s(x)))
    (sum#(cons(x, cons(y, l))) -> plus#(x, y), plus#(s(x), x) -> if#(gt(x, x), id(x), id(x)))
    (sum#(cons(x, cons(y, l))) -> plus#(x, y), plus#(s(x), x) -> gt#(x, x))
    (sum#(cons(x, cons(y, l))) -> plus#(x, y), plus#(s(x), x) -> id#(x))
    (sum#(cons(x, cons(y, l))) -> plus#(x, y), plus#(s(x), y) -> plus#(x, y))
    (sum#(cons(x, cons(y, l))) -> plus#(x, y), plus#(s(x), s(y)) -> plus#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))
    (sum#(cons(x, cons(y, l))) -> plus#(x, y), plus#(s(x), s(y)) -> if#(gt(x, y), x, y))
    (sum#(cons(x, cons(y, l))) -> plus#(x, y), plus#(s(x), s(y)) -> if#(not(gt(x, y)), id(x), id(y)))
    (sum#(cons(x, cons(y, l))) -> plus#(x, y), plus#(s(x), s(y)) -> gt#(x, y))
    (sum#(cons(x, cons(y, l))) -> plus#(x, y), plus#(s(x), s(y)) -> not#(gt(x, y)))
    (sum#(cons(x, cons(y, l))) -> plus#(x, y), plus#(s(x), s(y)) -> id#(x))
    (sum#(cons(x, cons(y, l))) -> plus#(x, y), plus#(s(x), s(y)) -> id#(y))
    (sum#(cons(x, cons(y, l))) -> plus#(x, y), plus#(id(x), s(y)) -> plus#(x, if(gt(s(y), y), y, s(y))))
    (sum#(cons(x, cons(y, l))) -> plus#(x, y), plus#(id(x), s(y)) -> if#(gt(s(y), y), y, s(y)))
    (sum#(cons(x, cons(y, l))) -> plus#(x, y), plus#(id(x), s(y)) -> gt#(s(y), y))
    (plus#(s(x), y) -> plus#(x, y), plus#(s(x), x) -> plus#(if(gt(x, x), id(x), id(x)), s(x)))
    (plus#(s(x), y) -> plus#(x, y), plus#(s(x), x) -> if#(gt(x, x), id(x), id(x)))
    (plus#(s(x), y) -> plus#(x, y), plus#(s(x), x) -> gt#(x, x))
    (plus#(s(x), y) -> plus#(x, y), plus#(s(x), x) -> id#(x))
    (plus#(s(x), y) -> plus#(x, y), plus#(s(x), y) -> plus#(x, y))
    (plus#(s(x), y) -> plus#(x, y), plus#(s(x), s(y)) -> plus#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))
    (plus#(s(x), y) -> plus#(x, y), plus#(s(x), s(y)) -> if#(gt(x, y), x, y))
    (plus#(s(x), y) -> plus#(x, y), plus#(s(x), s(y)) -> if#(not(gt(x, y)), id(x), id(y)))
    (plus#(s(x), y) -> plus#(x, y), plus#(s(x), s(y)) -> gt#(x, y))
    (plus#(s(x), y) -> plus#(x, y), plus#(s(x), s(y)) -> not#(gt(x, y)))
    (plus#(s(x), y) -> plus#(x, y), plus#(s(x), s(y)) -> id#(x))
    (plus#(s(x), y) -> plus#(x, y), plus#(s(x), s(y)) -> id#(y))
    (plus#(s(x), y) -> plus#(x, y), plus#(id(x), s(y)) -> plus#(x, if(gt(s(y), y), y, s(y))))
    (plus#(s(x), y) -> plus#(x, y), plus#(id(x), s(y)) -> if#(gt(s(y), y), y, s(y)))
    (plus#(s(x), y) -> plus#(x, y), plus#(id(x), s(y)) -> gt#(s(y), y))
    (minus#(s(x), s(y)) -> minus#(x, y), minus#(minus(x, y), z) -> minus#(x, plus(y, z)))
    (minus#(s(x), s(y)) -> minus#(x, y), minus#(minus(x, y), z) -> plus#(y, z))
    (minus#(s(x), s(y)) -> minus#(x, y), minus#(s(x), s(y)) -> minus#(x, y))
    (plus#(id(x), s(y)) -> plus#(x, if(gt(s(y), y), y, s(y))), plus#(s(x), x) -> plus#(if(gt(x, x), id(x), id(x)), s(x)))
    (plus#(id(x), s(y)) -> plus#(x, if(gt(s(y), y), y, s(y))), plus#(s(x), x) -> if#(gt(x, x), id(x), id(x)))
    (plus#(id(x), s(y)) -> plus#(x, if(gt(s(y), y), y, s(y))), plus#(s(x), x) -> gt#(x, x))
    (plus#(id(x), s(y)) -> plus#(x, if(gt(s(y), y), y, s(y))), plus#(s(x), x) -> id#(x))
    (plus#(id(x), s(y)) -> plus#(x, if(gt(s(y), y), y, s(y))), plus#(s(x), y) -> plus#(x, y))
    (plus#(id(x), s(y)) -> plus#(x, if(gt(s(y), y), y, s(y))), plus#(s(x), s(y)) -> plus#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))
    (plus#(id(x), s(y)) -> plus#(x, if(gt(s(y), y), y, s(y))), plus#(s(x), s(y)) -> if#(gt(x, y), x, y))
    (plus#(id(x), s(y)) -> plus#(x, if(gt(s(y), y), y, s(y))), plus#(s(x), s(y)) -> if#(not(gt(x, y)), id(x), id(y)))
    (plus#(id(x), s(y)) -> plus#(x, if(gt(s(y), y), y, s(y))), plus#(s(x), s(y)) -> gt#(x, y))
    (plus#(id(x), s(y)) -> plus#(x, if(gt(s(y), y), y, s(y))), plus#(s(x), s(y)) -> not#(gt(x, y)))
    (plus#(id(x), s(y)) -> plus#(x, if(gt(s(y), y), y, s(y))), plus#(s(x), s(y)) -> id#(x))
    (plus#(id(x), s(y)) -> plus#(x, if(gt(s(y), y), y, s(y))), plus#(s(x), s(y)) -> id#(y))
    (plus#(id(x), s(y)) -> plus#(x, if(gt(s(y), y), y, s(y))), plus#(id(x), s(y)) -> plus#(x, if(gt(s(y), y), y, s(y))))
    (plus#(id(x), s(y)) -> plus#(x, if(gt(s(y), y), y, s(y))), plus#(id(x), s(y)) -> if#(gt(s(y), y), y, s(y)))
    (plus#(id(x), s(y)) -> plus#(x, if(gt(s(y), y), y, s(y))), plus#(id(x), s(y)) -> gt#(s(y), y))
    (sum#(app(l, cons(x, cons(y, k)))) -> sum#(app(l, sum(cons(x, cons(y, k))))), sum#(app(l, cons(x, cons(y, k)))) -> app#(l, sum(cons(x, cons(y, k)))))
    (sum#(app(l, cons(x, cons(y, k)))) -> sum#(app(l, sum(cons(x, cons(y, k))))), sum#(app(l, cons(x, cons(y, k)))) -> sum#(app(l, sum(cons(x, cons(y, k))))))
    (sum#(app(l, cons(x, cons(y, k)))) -> sum#(app(l, sum(cons(x, cons(y, k))))), sum#(app(l, cons(x, cons(y, k)))) -> sum#(cons(x, cons(y, k))))
    (sum#(app(l, cons(x, cons(y, k)))) -> sum#(app(l, sum(cons(x, cons(y, k))))), sum#(cons(x, cons(y, l))) -> plus#(x, y))
    (sum#(app(l, cons(x, cons(y, k)))) -> sum#(app(l, sum(cons(x, cons(y, k))))), sum#(cons(x, cons(y, l))) -> sum#(cons(plus(x, y), l)))
    (plus#(s(x), s(y)) -> not#(gt(x, y)), not#(x) -> if#(x, false(), true()))
    (minus#(minus(x, y), z) -> plus#(y, z), plus#(s(x), x) -> plus#(if(gt(x, x), id(x), id(x)), s(x)))
    (minus#(minus(x, y), z) -> plus#(y, z), plus#(s(x), x) -> if#(gt(x, x), id(x), id(x)))
    (minus#(minus(x, y), z) -> plus#(y, z), plus#(s(x), x) -> gt#(x, x))
    (minus#(minus(x, y), z) -> plus#(y, z), plus#(s(x), x) -> id#(x))
    (minus#(minus(x, y), z) -> plus#(y, z), plus#(s(x), y) -> plus#(x, y))
    (minus#(minus(x, y), z) -> plus#(y, z), plus#(s(x), s(y)) -> plus#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))
    (minus#(minus(x, y), z) -> plus#(y, z), plus#(s(x), s(y)) -> if#(gt(x, y), x, y))
    (minus#(minus(x, y), z) -> plus#(y, z), plus#(s(x), s(y)) -> if#(not(gt(x, y)), id(x), id(y)))
    (minus#(minus(x, y), z) -> plus#(y, z), plus#(s(x), s(y)) -> gt#(x, y))
    (minus#(minus(x, y), z) -> plus#(y, z), plus#(s(x), s(y)) -> not#(gt(x, y)))
    (minus#(minus(x, y), z) -> plus#(y, z), plus#(s(x), s(y)) -> id#(x))
    (minus#(minus(x, y), z) -> plus#(y, z), plus#(s(x), s(y)) -> id#(y))
    (minus#(minus(x, y), z) -> plus#(y, z), plus#(id(x), s(y)) -> plus#(x, if(gt(s(y), y), y, s(y))))
    (minus#(minus(x, y), z) -> plus#(y, z), plus#(id(x), s(y)) -> if#(gt(s(y), y), y, s(y)))
    (minus#(minus(x, y), z) -> plus#(y, z), plus#(id(x), s(y)) -> gt#(s(y), y))
    (plus#(s(x), x) -> gt#(x, x), gt#(s(x), s(y)) -> gt#(x, y))
   }
   SCCS:
    Scc:
     {gt#(s(x), s(y)) -> gt#(x, y)}
    Scc:
     {sum#(cons(x, cons(y, l))) -> sum#(cons(plus(x, y), l))}
    Scc:
     {sum#(app(l, cons(x, cons(y, k)))) -> sum#(app(l, sum(cons(x, cons(y, k)))))}
    Scc:
     {app#(cons(x, l), k) -> app#(l, k)}
    Scc:
     {    plus#(s(x), x) -> plus#(if(gt(x, x), id(x), id(x)), s(x)),
          plus#(s(x), y) -> plus#(x, y),
       plus#(s(x), s(y)) -> plus#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))),
      plus#(id(x), s(y)) -> plus#(x, if(gt(s(y), y), y, s(y)))}
    Scc:
     {quot#(s(x), s(y)) -> quot#(minus(x, y), s(y))}
    Scc:
     {minus#(minus(x, y), z) -> minus#(x, plus(y, z)),
          minus#(s(x), s(y)) -> minus#(x, y)}
    SCC:
     Strict:
      {gt#(s(x), s(y)) -> gt#(x, y)}
     Weak:
     {                   minus(x, 0()) -> x,
                 minus(minus(x, y), z) -> minus(x, plus(y, z)),
                     minus(s(x), s(y)) -> minus(x, y),
                       quot(0(), s(y)) -> 0(),
                      quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))),
                          plus(0(), y) -> y,
                         plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)),
                         plus(s(x), y) -> s(plus(x, y)),
                      plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))),
                     plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))),
                       plus(zero(), y) -> y,
                         app(l, nil()) -> l,
                         app(nil(), k) -> k,
                    app(cons(x, l), k) -> cons(x, app(l, k)),
      sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k))))),
                   sum(cons(x, nil())) -> cons(x, nil()),
              sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l)),
                      if(true(), x, y) -> x,
                     if(false(), x, y) -> y,
                        gt(s(x), s(y)) -> gt(x, y),
                      gt(s(x), zero()) -> true(),
                         gt(zero(), y) -> false(),
                                not(x) -> if(x, false(), true()),
                                 id(x) -> x}
     SPSC:
      Simple Projection:
       pi(gt#) = 0
      Strict:
       {}
      Qed
    SCC:
     Strict:
      {sum#(cons(x, cons(y, l))) -> sum#(cons(plus(x, y), l))}
     Weak:
     {                   minus(x, 0()) -> x,
                 minus(minus(x, y), z) -> minus(x, plus(y, z)),
                     minus(s(x), s(y)) -> minus(x, y),
                       quot(0(), s(y)) -> 0(),
                      quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))),
                          plus(0(), y) -> y,
                         plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)),
                         plus(s(x), y) -> s(plus(x, y)),
                      plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))),
                     plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))),
                       plus(zero(), y) -> y,
                         app(l, nil()) -> l,
                         app(nil(), k) -> k,
                    app(cons(x, l), k) -> cons(x, app(l, k)),
      sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k))))),
                   sum(cons(x, nil())) -> cons(x, nil()),
              sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l)),
                      if(true(), x, y) -> x,
                     if(false(), x, y) -> y,
                        gt(s(x), s(y)) -> gt(x, y),
                      gt(s(x), zero()) -> true(),
                         gt(zero(), y) -> false(),
                                not(x) -> if(x, false(), true()),
                                 id(x) -> x}
     POLY:
      Argument Filtering:
       pi(false) = [], pi(true) = [], pi(zero) = [], pi(id) = [], pi(not) = [], pi(gt) = [], pi(if) = [], pi(sum#) = 0, pi(sum) = [], pi(cons) = [1], pi(nil) = [], pi(app) = [], pi(plus) = [], pi(quot) = [], pi(s) = [], pi(0) = [], pi(minus) = []
      Usable Rules:
       {}
      Interpretation:
       [cons](x0) = x0 + 1
      Strict:
       {}
      Weak:
       {                   minus(x, 0()) -> x,
                   minus(minus(x, y), z) -> minus(x, plus(y, z)),
                       minus(s(x), s(y)) -> minus(x, y),
                         quot(0(), s(y)) -> 0(),
                        quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))),
                            plus(0(), y) -> y,
                           plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)),
                           plus(s(x), y) -> s(plus(x, y)),
                        plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))),
                       plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))),
                         plus(zero(), y) -> y,
                           app(l, nil()) -> l,
                           app(nil(), k) -> k,
                      app(cons(x, l), k) -> cons(x, app(l, k)),
        sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k))))),
                     sum(cons(x, nil())) -> cons(x, nil()),
                sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l)),
                        if(true(), x, y) -> x,
                       if(false(), x, y) -> y,
                          gt(s(x), s(y)) -> gt(x, y),
                        gt(s(x), zero()) -> true(),
                           gt(zero(), y) -> false(),
                                  not(x) -> if(x, false(), true()),
                                   id(x) -> x}
      Qed
    SCC:
     Strict:
      {sum#(app(l, cons(x, cons(y, k)))) -> sum#(app(l, sum(cons(x, cons(y, k)))))}
     Weak:
     {                   minus(x, 0()) -> x,
                 minus(minus(x, y), z) -> minus(x, plus(y, z)),
                     minus(s(x), s(y)) -> minus(x, y),
                       quot(0(), s(y)) -> 0(),
                      quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))),
                          plus(0(), y) -> y,
                         plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)),
                         plus(s(x), y) -> s(plus(x, y)),
                      plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))),
                     plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))),
                       plus(zero(), y) -> y,
                         app(l, nil()) -> l,
                         app(nil(), k) -> k,
                    app(cons(x, l), k) -> cons(x, app(l, k)),
      sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k))))),
                   sum(cons(x, nil())) -> cons(x, nil()),
              sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l)),
                      if(true(), x, y) -> x,
                     if(false(), x, y) -> y,
                        gt(s(x), s(y)) -> gt(x, y),
                      gt(s(x), zero()) -> true(),
                         gt(zero(), y) -> false(),
                                not(x) -> if(x, false(), true()),
                                 id(x) -> x}
     POLY:
      Argument Filtering:
       pi(false) = [], pi(true) = [], pi(zero) = [], pi(id) = [], pi(not) = [], pi(gt) = [], pi(if) = [], pi(sum#) = 0, pi(sum) = [], pi(cons) = [1], pi(nil) = [], pi(app) = [0,1], pi(plus) = [], pi(quot) = [], pi(s) = [], pi(0) = [], pi(minus) = []
      Usable Rules:
       {}
      Interpretation:
       [cons](x0) = x0 + 1,
       [app](x0, x1) = x0 + x1,
       [sum] = 1
      Strict:
       {}
      Weak:
       {                   minus(x, 0()) -> x,
                   minus(minus(x, y), z) -> minus(x, plus(y, z)),
                       minus(s(x), s(y)) -> minus(x, y),
                         quot(0(), s(y)) -> 0(),
                        quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))),
                            plus(0(), y) -> y,
                           plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)),
                           plus(s(x), y) -> s(plus(x, y)),
                        plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))),
                       plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))),
                         plus(zero(), y) -> y,
                           app(l, nil()) -> l,
                           app(nil(), k) -> k,
                      app(cons(x, l), k) -> cons(x, app(l, k)),
        sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k))))),
                     sum(cons(x, nil())) -> cons(x, nil()),
                sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l)),
                        if(true(), x, y) -> x,
                       if(false(), x, y) -> y,
                          gt(s(x), s(y)) -> gt(x, y),
                        gt(s(x), zero()) -> true(),
                           gt(zero(), y) -> false(),
                                  not(x) -> if(x, false(), true()),
                                   id(x) -> x}
      Qed
    SCC:
     Strict:
      {app#(cons(x, l), k) -> app#(l, k)}
     Weak:
     {                   minus(x, 0()) -> x,
                 minus(minus(x, y), z) -> minus(x, plus(y, z)),
                     minus(s(x), s(y)) -> minus(x, y),
                       quot(0(), s(y)) -> 0(),
                      quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))),
                          plus(0(), y) -> y,
                         plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)),
                         plus(s(x), y) -> s(plus(x, y)),
                      plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))),
                     plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))),
                       plus(zero(), y) -> y,
                         app(l, nil()) -> l,
                         app(nil(), k) -> k,
                    app(cons(x, l), k) -> cons(x, app(l, k)),
      sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k))))),
                   sum(cons(x, nil())) -> cons(x, nil()),
              sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l)),
                      if(true(), x, y) -> x,
                     if(false(), x, y) -> y,
                        gt(s(x), s(y)) -> gt(x, y),
                      gt(s(x), zero()) -> true(),
                         gt(zero(), y) -> false(),
                                not(x) -> if(x, false(), true()),
                                 id(x) -> x}
     SPSC:
      Simple Projection:
       pi(app#) = 0
      Strict:
       {}
      Qed
    SCC:
     Strict:
      {    plus#(s(x), x) -> plus#(if(gt(x, x), id(x), id(x)), s(x)),
           plus#(s(x), y) -> plus#(x, y),
        plus#(s(x), s(y)) -> plus#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))),
       plus#(id(x), s(y)) -> plus#(x, if(gt(s(y), y), y, s(y)))}
     Weak:
     {                   minus(x, 0()) -> x,
                 minus(minus(x, y), z) -> minus(x, plus(y, z)),
                     minus(s(x), s(y)) -> minus(x, y),
                       quot(0(), s(y)) -> 0(),
                      quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))),
                          plus(0(), y) -> y,
                         plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)),
                         plus(s(x), y) -> s(plus(x, y)),
                      plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))),
                     plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))),
                       plus(zero(), y) -> y,
                         app(l, nil()) -> l,
                         app(nil(), k) -> k,
                    app(cons(x, l), k) -> cons(x, app(l, k)),
      sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k))))),
                   sum(cons(x, nil())) -> cons(x, nil()),
              sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l)),
                      if(true(), x, y) -> x,
                     if(false(), x, y) -> y,
                        gt(s(x), s(y)) -> gt(x, y),
                      gt(s(x), zero()) -> true(),
                         gt(zero(), y) -> false(),
                                not(x) -> if(x, false(), true()),
                                 id(x) -> x}
     Fail
    SCC:
     Strict:
      {quot#(s(x), s(y)) -> quot#(minus(x, y), s(y))}
     Weak:
     {                   minus(x, 0()) -> x,
                 minus(minus(x, y), z) -> minus(x, plus(y, z)),
                     minus(s(x), s(y)) -> minus(x, y),
                       quot(0(), s(y)) -> 0(),
                      quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))),
                          plus(0(), y) -> y,
                         plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)),
                         plus(s(x), y) -> s(plus(x, y)),
                      plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))),
                     plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))),
                       plus(zero(), y) -> y,
                         app(l, nil()) -> l,
                         app(nil(), k) -> k,
                    app(cons(x, l), k) -> cons(x, app(l, k)),
      sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k))))),
                   sum(cons(x, nil())) -> cons(x, nil()),
              sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l)),
                      if(true(), x, y) -> x,
                     if(false(), x, y) -> y,
                        gt(s(x), s(y)) -> gt(x, y),
                      gt(s(x), zero()) -> true(),
                         gt(zero(), y) -> false(),
                                not(x) -> if(x, false(), true()),
                                 id(x) -> x}
     POLY:
      Argument Filtering:
       pi(false) = [], pi(true) = [], pi(zero) = [], pi(id) = [], pi(not) = [], pi(gt) = [], pi(if) = [], pi(sum) = [], pi(cons) = [], pi(nil) = [], pi(app) = [], pi(plus) = [], pi(quot#) = [0], pi(quot) = [], pi(s) = [0], pi(0) = [], pi(minus) = 0
      Usable Rules:
       {}
      Interpretation:
       [quot#](x0) = x0 + 1,
       [s](x0) = x0 + 1
      Strict:
       {}
      Weak:
       {                   minus(x, 0()) -> x,
                   minus(minus(x, y), z) -> minus(x, plus(y, z)),
                       minus(s(x), s(y)) -> minus(x, y),
                         quot(0(), s(y)) -> 0(),
                        quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))),
                            plus(0(), y) -> y,
                           plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)),
                           plus(s(x), y) -> s(plus(x, y)),
                        plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))),
                       plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))),
                         plus(zero(), y) -> y,
                           app(l, nil()) -> l,
                           app(nil(), k) -> k,
                      app(cons(x, l), k) -> cons(x, app(l, k)),
        sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k))))),
                     sum(cons(x, nil())) -> cons(x, nil()),
                sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l)),
                        if(true(), x, y) -> x,
                       if(false(), x, y) -> y,
                          gt(s(x), s(y)) -> gt(x, y),
                        gt(s(x), zero()) -> true(),
                           gt(zero(), y) -> false(),
                                  not(x) -> if(x, false(), true()),
                                   id(x) -> x}
      Qed
    SCC:
     Strict:
      {minus#(minus(x, y), z) -> minus#(x, plus(y, z)),
           minus#(s(x), s(y)) -> minus#(x, y)}
     Weak:
     {                   minus(x, 0()) -> x,
                 minus(minus(x, y), z) -> minus(x, plus(y, z)),
                     minus(s(x), s(y)) -> minus(x, y),
                       quot(0(), s(y)) -> 0(),
                      quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))),
                          plus(0(), y) -> y,
                         plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)),
                         plus(s(x), y) -> s(plus(x, y)),
                      plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))),
                     plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))),
                       plus(zero(), y) -> y,
                         app(l, nil()) -> l,
                         app(nil(), k) -> k,
                    app(cons(x, l), k) -> cons(x, app(l, k)),
      sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k))))),
                   sum(cons(x, nil())) -> cons(x, nil()),
              sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l)),
                      if(true(), x, y) -> x,
                     if(false(), x, y) -> y,
                        gt(s(x), s(y)) -> gt(x, y),
                      gt(s(x), zero()) -> true(),
                         gt(zero(), y) -> false(),
                                not(x) -> if(x, false(), true()),
                                 id(x) -> x}
     SPSC:
      Simple Projection:
       pi(minus#) = 0
      Strict:
       {minus#(s(x), s(y)) -> minus#(x, y)}
      EDG:
       {(minus#(s(x), s(y)) -> minus#(x, y), minus#(s(x), s(y)) -> minus#(x, y))}
       SCCS:
        Scc:
         {minus#(s(x), s(y)) -> minus#(x, y)}
        SCC:
         Strict:
          {minus#(s(x), s(y)) -> minus#(x, y)}
         Weak:
         {                   minus(x, 0()) -> x,
                     minus(minus(x, y), z) -> minus(x, plus(y, z)),
                         minus(s(x), s(y)) -> minus(x, y),
                           quot(0(), s(y)) -> 0(),
                          quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))),
                              plus(0(), y) -> y,
                             plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x)),
                             plus(s(x), y) -> s(plus(x, y)),
                          plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))),
                         plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y)))),
                           plus(zero(), y) -> y,
                             app(l, nil()) -> l,
                             app(nil(), k) -> k,
                        app(cons(x, l), k) -> cons(x, app(l, k)),
          sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k))))),
                       sum(cons(x, nil())) -> cons(x, nil()),
                  sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l)),
                          if(true(), x, y) -> x,
                         if(false(), x, y) -> y,
                            gt(s(x), s(y)) -> gt(x, y),
                          gt(s(x), zero()) -> true(),
                             gt(zero(), y) -> false(),
                                    not(x) -> if(x, false(), true()),
                                     id(x) -> x}
         SPSC:
          Simple Projection:
           pi(minus#) = 1
          Strict:
           {}
          Qed