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)
  , p(s(x)) -> x
  , f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))
  , f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x)))) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following weak dependency pairs:

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

We replace rewrite rules by usable rules:

  Strict Usable Rules:
    { -(x, 0()) -> x
    , -(s(x), s(y)) -> -(x, y)
    , p(s(x)) -> x }

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))
  , p^#(s(x)) -> c_3()
  , f^#(x, s(y)) -> c_4(f^#(p(-(x, s(y))), p(-(s(y), x))))
  , f^#(s(x), y) -> c_5(f^#(p(-(s(x), y)), p(-(y, s(x))))) }
Strict Trs:
  { -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, y)
  , p(s(x)) -> x }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The weightgap principle applies (using the following constant
growth matrix-interpretation)

The following argument positions are usable:
  Uargs(p) = {1}, Uargs(c_2) = {1}, Uargs(f^#) = {1, 2},
  Uargs(c_4) = {1}, Uargs(c_5) = {1}

TcT has computed following constructor-restricted matrix
interpretation.

    [-](x1, x2) = [1] x1 + [1]         
                                       
            [0] = [2]                  
                                       
        [s](x1) = [1] x1 + [1]         
                                       
        [p](x1) = [1] x1 + [0]         
                                       
  [-^#](x1, x2) = [2] x1 + [2] x2 + [2]
                                       
          [c_1] = [1]                  
                                       
      [c_2](x1) = [1] x1 + [1]         
                                       
      [p^#](x1) = [2] x1 + [1]         
                                       
          [c_3] = [2]                  
                                       
  [f^#](x1, x2) = [2] x1 + [1] x2 + [1]
                                       
      [c_4](x1) = [1] x1 + [1]         
                                       
      [c_5](x1) = [1] x1 + [1]         

This order satisfies following ordering constraints:

      [-(x, 0())] = [1] x + [1]
                  > [1] x + [0]
                  = [x]        
                               
  [-(s(x), s(y))] = [1] x + [2]
                  > [1] x + [1]
                  = [-(x, y)]  
                               
        [p(s(x))] = [1] x + [1]
                  > [1] x + [0]
                  = [x]        
                               

Further, it can be verified that all rules not oriented are covered by the weightgap condition.

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

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

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

Strict DPs:
  { f^#(x, s(y)) -> c_4(f^#(p(-(x, s(y))), p(-(s(y), x))))
  , f^#(s(x), y) -> c_5(f^#(p(-(s(x), y)), p(-(y, s(x))))) }
Weak Trs:
  { -(x, 0()) -> x
  , -(s(x), s(y)) -> -(x, y)
  , p(s(x)) -> x }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..