MAYBE

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

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

We add following weak dependency pairs:

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

and mark the set of starting terms.

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

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

We replace rewrite rules by usable rules:

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

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

Strict DPs:
  { f^#(s(x)) -> c_1(f^#(p(s(x))))
  , f^#(0()) -> c_2()
  , p^#(s(x)) -> c_3() }
Strict Trs: { 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(f^#) = {1}, Uargs(c_1) = {1}

TcT has computed following constructor-restricted matrix
interpretation.

    [s](x1) = [1] x1 + [2]
                          
    [p](x1) = [1] x1 + [0]
                          
        [0] = [1]         
                          
  [f^#](x1) = [2] x1 + [2]
                          
  [c_1](x1) = [1] x1 + [1]
                          
      [c_2] = [1]         
                          
  [p^#](x1) = [1] x1 + [1]
                          
      [c_3] = [2]         

This order satisfies following ordering constraints:

  [p(s(x))] = [1] x + [2]
            > [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^#(s(x)) -> c_1(f^#(p(s(x)))) }
Weak DPs:
  { f^#(0()) -> c_2()
  , p^#(s(x)) -> c_3() }
Weak Trs: { 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.

{ f^#(0()) -> c_2()
, p^#(s(x)) -> c_3() }

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

Strict DPs: { f^#(s(x)) -> c_1(f^#(p(s(x)))) }
Weak Trs: { p(s(x)) -> x }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..