MAYBE

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict Trs:
  { minus(x, x) -> 0()
  , minus(x, 0()) -> x
  , minus(0(), x) -> 0()
  , minus(s(x), s(y)) -> minus(x, y)
  , div(0(), s(y)) -> 0()
  , div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
  , f(x, 0(), b) -> x
  , f(x, s(y), b) -> div(f(x, minus(s(y), s(0())), b), b) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { minus^#(x, x) -> c_1()
  , minus^#(x, 0()) -> c_2()
  , minus^#(0(), x) -> c_3()
  , minus^#(s(x), s(y)) -> c_4(minus^#(x, y))
  , div^#(0(), s(y)) -> c_5()
  , div^#(s(x), s(y)) -> c_6(div^#(minus(x, y), s(y)), minus^#(x, y))
  , f^#(x, 0(), b) -> c_7()
  , f^#(x, s(y), b) ->
    c_8(div^#(f(x, minus(s(y), s(0())), b), b),
        f^#(x, minus(s(y), s(0())), b),
        minus^#(s(y), s(0()))) }

and mark the set of starting terms.

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { minus^#(x, x) -> c_1()
  , minus^#(x, 0()) -> c_2()
  , minus^#(0(), x) -> c_3()
  , minus^#(s(x), s(y)) -> c_4(minus^#(x, y))
  , div^#(0(), s(y)) -> c_5()
  , div^#(s(x), s(y)) -> c_6(div^#(minus(x, y), s(y)), minus^#(x, y))
  , f^#(x, 0(), b) -> c_7()
  , f^#(x, s(y), b) ->
    c_8(div^#(f(x, minus(s(y), s(0())), b), b),
        f^#(x, minus(s(y), s(0())), b),
        minus^#(s(y), s(0()))) }
Weak Trs:
  { minus(x, x) -> 0()
  , minus(x, 0()) -> x
  , minus(0(), x) -> 0()
  , minus(s(x), s(y)) -> minus(x, y)
  , div(0(), s(y)) -> 0()
  , div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
  , f(x, 0(), b) -> x
  , f(x, s(y), b) -> div(f(x, minus(s(y), s(0())), b), b) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {1,2,3,5,7} by
applications of Pre({1,2,3,5,7}) = {4,6,8}. Here rules are labeled
as follows:

  DPs:
    { 1: minus^#(x, x) -> c_1()
    , 2: minus^#(x, 0()) -> c_2()
    , 3: minus^#(0(), x) -> c_3()
    , 4: minus^#(s(x), s(y)) -> c_4(minus^#(x, y))
    , 5: div^#(0(), s(y)) -> c_5()
    , 6: div^#(s(x), s(y)) ->
         c_6(div^#(minus(x, y), s(y)), minus^#(x, y))
    , 7: f^#(x, 0(), b) -> c_7()
    , 8: f^#(x, s(y), b) ->
         c_8(div^#(f(x, minus(s(y), s(0())), b), b),
             f^#(x, minus(s(y), s(0())), b),
             minus^#(s(y), s(0()))) }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { minus^#(s(x), s(y)) -> c_4(minus^#(x, y))
  , div^#(s(x), s(y)) -> c_6(div^#(minus(x, y), s(y)), minus^#(x, y))
  , f^#(x, s(y), b) ->
    c_8(div^#(f(x, minus(s(y), s(0())), b), b),
        f^#(x, minus(s(y), s(0())), b),
        minus^#(s(y), s(0()))) }
Weak DPs:
  { minus^#(x, x) -> c_1()
  , minus^#(x, 0()) -> c_2()
  , minus^#(0(), x) -> c_3()
  , div^#(0(), s(y)) -> c_5()
  , f^#(x, 0(), b) -> c_7() }
Weak Trs:
  { minus(x, x) -> 0()
  , minus(x, 0()) -> x
  , minus(0(), x) -> 0()
  , minus(s(x), s(y)) -> minus(x, y)
  , div(0(), s(y)) -> 0()
  , div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
  , f(x, 0(), b) -> x
  , f(x, s(y), b) -> div(f(x, minus(s(y), s(0())), b), b) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ minus^#(x, x) -> c_1()
, minus^#(x, 0()) -> c_2()
, minus^#(0(), x) -> c_3()
, div^#(0(), s(y)) -> c_5()
, f^#(x, 0(), b) -> c_7() }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { minus^#(s(x), s(y)) -> c_4(minus^#(x, y))
  , div^#(s(x), s(y)) -> c_6(div^#(minus(x, y), s(y)), minus^#(x, y))
  , f^#(x, s(y), b) ->
    c_8(div^#(f(x, minus(s(y), s(0())), b), b),
        f^#(x, minus(s(y), s(0())), b),
        minus^#(s(y), s(0()))) }
Weak Trs:
  { minus(x, x) -> 0()
  , minus(x, 0()) -> x
  , minus(0(), x) -> 0()
  , minus(s(x), s(y)) -> minus(x, y)
  , div(0(), s(y)) -> 0()
  , div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
  , f(x, 0(), b) -> x
  , f(x, s(y), b) -> div(f(x, minus(s(y), s(0())), b), b) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..