MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            f(0()) -> 0()
            f(s(x)) -> s(f(f(p(s(x)))))
            p(s(x)) -> x
        - Signature:
            {f/1,p/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,p} and constructors {0,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          f#(0()) -> c_1()
          f#(s(x)) -> c_2(f#(f(p(s(x)))),f#(p(s(x))),p#(s(x)))
          p#(s(x)) -> c_3()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            f#(0()) -> c_1()
            f#(s(x)) -> c_2(f#(f(p(s(x)))),f#(p(s(x))),p#(s(x)))
            p#(s(x)) -> c_3()
        - Weak TRS:
            f(0()) -> 0()
            f(s(x)) -> s(f(f(p(s(x)))))
            p(s(x)) -> x
        - Signature:
            {f/1,p/1,f#/1,p#/1} / {0/0,s/1,c_1/0,c_2/3,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,p#} and constructors {0,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,3}
        by application of
          Pre({1,3}) = {2}.
        Here rules are labelled as follows:
          1: f#(0()) -> c_1()
          2: f#(s(x)) -> c_2(f#(f(p(s(x)))),f#(p(s(x))),p#(s(x)))
          3: p#(s(x)) -> c_3()
* Step 3: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            f#(s(x)) -> c_2(f#(f(p(s(x)))),f#(p(s(x))),p#(s(x)))
        - Weak DPs:
            f#(0()) -> c_1()
            p#(s(x)) -> c_3()
        - Weak TRS:
            f(0()) -> 0()
            f(s(x)) -> s(f(f(p(s(x)))))
            p(s(x)) -> x
        - Signature:
            {f/1,p/1,f#/1,p#/1} / {0/0,s/1,c_1/0,c_2/3,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,p#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f#(s(x)) -> c_2(f#(f(p(s(x)))),f#(p(s(x))),p#(s(x)))
             -->_3 p#(s(x)) -> c_3():3
             -->_2 f#(0()) -> c_1():2
             -->_1 f#(0()) -> c_1():2
             -->_2 f#(s(x)) -> c_2(f#(f(p(s(x)))),f#(p(s(x))),p#(s(x))):1
             -->_1 f#(s(x)) -> c_2(f#(f(p(s(x)))),f#(p(s(x))),p#(s(x))):1
          
          2:W:f#(0()) -> c_1()
             
          
          3:W:p#(s(x)) -> c_3()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: f#(0()) -> c_1()
          3: p#(s(x)) -> c_3()
* Step 4: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            f#(s(x)) -> c_2(f#(f(p(s(x)))),f#(p(s(x))),p#(s(x)))
        - Weak TRS:
            f(0()) -> 0()
            f(s(x)) -> s(f(f(p(s(x)))))
            p(s(x)) -> x
        - Signature:
            {f/1,p/1,f#/1,p#/1} / {0/0,s/1,c_1/0,c_2/3,c_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,p#} and constructors {0,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:f#(s(x)) -> c_2(f#(f(p(s(x)))),f#(p(s(x))),p#(s(x)))
             -->_2 f#(s(x)) -> c_2(f#(f(p(s(x)))),f#(p(s(x))),p#(s(x))):1
             -->_1 f#(s(x)) -> c_2(f#(f(p(s(x)))),f#(p(s(x))),p#(s(x))):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          f#(s(x)) -> c_2(f#(f(p(s(x)))),f#(p(s(x))))
* Step 5: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          f#(s(x)) -> c_2(f#(f(p(s(x)))),f#(p(s(x))))
      - Weak TRS:
          f(0()) -> 0()
          f(s(x)) -> s(f(f(p(s(x)))))
          p(s(x)) -> x
      - Signature:
          {f/1,p/1,f#/1,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {f#,p#} and constructors {0,s}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE