MAYBE

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

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

We add following dependency tuples:

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

and mark the set of starting terms.

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

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

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

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

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

Strict DPs:
  { -^#(s(x), s(y)) -> c_2(-^#(x, y))
  , +^#(s(x), y) -> c_4(+^#(x, y))
  , *^#(x, s(y)) -> c_6(+^#(x, *(x, y)), *^#(x, y))
  , f^#(s(x)) ->
    c_7(f^#(-(+(*(s(x), s(x)), *(s(x), s(s(s(0()))))),
              *(s(s(x)), s(s(x))))),
        -^#(+(*(s(x), s(x)), *(s(x), s(s(s(0()))))), *(s(s(x)), s(s(x)))),
        +^#(*(s(x), s(x)), *(s(x), s(s(s(0()))))),
        *^#(s(x), s(x)),
        *^#(s(x), s(s(s(0())))),
        *^#(s(s(x)), s(s(x)))) }
Weak DPs:
  { -^#(x, 0()) -> c_1()
  , +^#(0(), y) -> c_3()
  , *^#(x, 0()) -> c_5() }
Weak Trs:
  { -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, y)
  , +(0(), y) -> y
  , +(s(x), y) -> s(+(x, y))
  , *(x, 0()) -> 0()
  , *(x, s(y)) -> +(x, *(x, y))
  , f(s(x)) ->
    f(-(+(*(s(x), s(x)), *(s(x), s(s(s(0()))))),
        *(s(s(x)), s(s(x))))) }
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.

{ -^#(x, 0()) -> c_1()
, +^#(0(), y) -> c_3()
, *^#(x, 0()) -> c_5() }

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

Strict DPs:
  { -^#(s(x), s(y)) -> c_2(-^#(x, y))
  , +^#(s(x), y) -> c_4(+^#(x, y))
  , *^#(x, s(y)) -> c_6(+^#(x, *(x, y)), *^#(x, y))
  , f^#(s(x)) ->
    c_7(f^#(-(+(*(s(x), s(x)), *(s(x), s(s(s(0()))))),
              *(s(s(x)), s(s(x))))),
        -^#(+(*(s(x), s(x)), *(s(x), s(s(s(0()))))), *(s(s(x)), s(s(x)))),
        +^#(*(s(x), s(x)), *(s(x), s(s(s(0()))))),
        *^#(s(x), s(x)),
        *^#(s(x), s(s(s(0())))),
        *^#(s(s(x)), s(s(x)))) }
Weak Trs:
  { -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, y)
  , +(0(), y) -> y
  , +(s(x), y) -> s(+(x, y))
  , *(x, 0()) -> 0()
  , *(x, s(y)) -> +(x, *(x, y))
  , f(s(x)) ->
    f(-(+(*(s(x), s(x)), *(s(x), s(s(s(0()))))),
        *(s(s(x)), s(s(x))))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { -(x, 0()) -> x
    , -(s(x), s(y)) -> -(x, y)
    , +(0(), y) -> y
    , +(s(x), y) -> s(+(x, y))
    , *(x, 0()) -> 0()
    , *(x, s(y)) -> +(x, *(x, y)) }

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

Strict DPs:
  { -^#(s(x), s(y)) -> c_2(-^#(x, y))
  , +^#(s(x), y) -> c_4(+^#(x, y))
  , *^#(x, s(y)) -> c_6(+^#(x, *(x, y)), *^#(x, y))
  , f^#(s(x)) ->
    c_7(f^#(-(+(*(s(x), s(x)), *(s(x), s(s(s(0()))))),
              *(s(s(x)), s(s(x))))),
        -^#(+(*(s(x), s(x)), *(s(x), s(s(s(0()))))), *(s(s(x)), s(s(x)))),
        +^#(*(s(x), s(x)), *(s(x), s(s(s(0()))))),
        *^#(s(x), s(x)),
        *^#(s(x), s(s(s(0())))),
        *^#(s(s(x)), s(s(x)))) }
Weak Trs:
  { -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, y)
  , +(0(), y) -> y
  , +(s(x), y) -> s(+(x, y))
  , *(x, 0()) -> 0()
  , *(x, s(y)) -> +(x, *(x, y)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..