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))
  , p(s(x)) -> x
  , f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), 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))
  , p^#(s(x)) -> c_7()
  , f^#(s(x)) ->
    c_8(f^#(-(p(*(s(x), s(x))), *(s(x), s(x)))),
        -^#(p(*(s(x), s(x))), *(s(x), s(x))),
        p^#(*(s(x), s(x))),
        *^#(s(x), s(x)),
        *^#(s(x), 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))
  , p^#(s(x)) -> c_7()
  , f^#(s(x)) ->
    c_8(f^#(-(p(*(s(x), s(x))), *(s(x), s(x)))),
        -^#(p(*(s(x), s(x))), *(s(x), s(x))),
        p^#(*(s(x), s(x))),
        *^#(s(x), s(x)),
        *^#(s(x), 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))
  , p(s(x)) -> x
  , f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x)))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {1,3,5,7} by applications
of Pre({1,3,5,7}) = {2,4,6,8}. 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: p^#(s(x)) -> c_7()
    , 8: f^#(s(x)) ->
         c_8(f^#(-(p(*(s(x), s(x))), *(s(x), s(x)))),
             -^#(p(*(s(x), s(x))), *(s(x), s(x))),
             p^#(*(s(x), s(x))),
             *^#(s(x), s(x)),
             *^#(s(x), 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_8(f^#(-(p(*(s(x), s(x))), *(s(x), s(x)))),
        -^#(p(*(s(x), s(x))), *(s(x), s(x))),
        p^#(*(s(x), s(x))),
        *^#(s(x), s(x)),
        *^#(s(x), s(x))) }
Weak DPs:
  { -^#(x, 0()) -> c_1()
  , +^#(0(), y) -> c_3()
  , *^#(x, 0()) -> c_5()
  , p^#(s(x)) -> c_7() }
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))
  , p(s(x)) -> x
  , f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), 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()
, p^#(s(x)) -> c_7() }

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_8(f^#(-(p(*(s(x), s(x))), *(s(x), s(x)))),
        -^#(p(*(s(x), s(x))), *(s(x), s(x))),
        p^#(*(s(x), s(x))),
        *^#(s(x), s(x)),
        *^#(s(x), 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))
  , p(s(x)) -> x
  , f(s(x)) -> f(-(p(*(s(x), s(x))), *(s(x), s(x)))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

Due to missing edges in the dependency-graph, the right-hand sides
of following rules could be simplified:

  { f^#(s(x)) ->
    c_8(f^#(-(p(*(s(x), s(x))), *(s(x), s(x)))),
        -^#(p(*(s(x), s(x))), *(s(x), s(x))),
        p^#(*(s(x), s(x))),
        *^#(s(x), s(x)),
        *^#(s(x), s(x))) }

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

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

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

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

The input cannot be shown compatible

Arrrr..