MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            a__f(X) -> f(X)
            a__f(0()) -> cons(0(),f(s(0())))
            a__f(s(0())) -> a__f(a__p(s(0())))
            a__p(X) -> p(X)
            a__p(s(X)) -> mark(X)
            mark(0()) -> 0()
            mark(cons(X1,X2)) -> cons(mark(X1),X2)
            mark(f(X)) -> a__f(mark(X))
            mark(p(X)) -> a__p(mark(X))
            mark(s(X)) -> s(mark(X))
        - Signature:
            {a__f/1,a__p/1,mark/1} / {0/0,cons/2,f/1,p/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__f,a__p,mark} and constructors {0,cons,f,p,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          a__f#(X) -> c_1()
          a__f#(0()) -> c_2()
          a__f#(s(0())) -> c_3(a__f#(a__p(s(0()))),a__p#(s(0())))
          a__p#(X) -> c_4()
          a__p#(s(X)) -> c_5(mark#(X))
          mark#(0()) -> c_6()
          mark#(cons(X1,X2)) -> c_7(mark#(X1))
          mark#(f(X)) -> c_8(a__f#(mark(X)),mark#(X))
          mark#(p(X)) -> c_9(a__p#(mark(X)),mark#(X))
          mark#(s(X)) -> c_10(mark#(X))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            a__f#(X) -> c_1()
            a__f#(0()) -> c_2()
            a__f#(s(0())) -> c_3(a__f#(a__p(s(0()))),a__p#(s(0())))
            a__p#(X) -> c_4()
            a__p#(s(X)) -> c_5(mark#(X))
            mark#(0()) -> c_6()
            mark#(cons(X1,X2)) -> c_7(mark#(X1))
            mark#(f(X)) -> c_8(a__f#(mark(X)),mark#(X))
            mark#(p(X)) -> c_9(a__p#(mark(X)),mark#(X))
            mark#(s(X)) -> c_10(mark#(X))
        - Weak TRS:
            a__f(X) -> f(X)
            a__f(0()) -> cons(0(),f(s(0())))
            a__f(s(0())) -> a__f(a__p(s(0())))
            a__p(X) -> p(X)
            a__p(s(X)) -> mark(X)
            mark(0()) -> 0()
            mark(cons(X1,X2)) -> cons(mark(X1),X2)
            mark(f(X)) -> a__f(mark(X))
            mark(p(X)) -> a__p(mark(X))
            mark(s(X)) -> s(mark(X))
        - Signature:
            {a__f/1,a__p/1,mark/1,a__f#/1,a__p#/1,mark#/1} / {0/0,cons/2,f/1,p/1,s/1,c_1/0,c_2/0,c_3/2,c_4/0,c_5/1,c_6/0
            ,c_7/1,c_8/2,c_9/2,c_10/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__f#,a__p#,mark#} and constructors {0,cons,f,p,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,4,6}
        by application of
          Pre({1,2,4,6}) = {3,5,7,8,9,10}.
        Here rules are labelled as follows:
          1: a__f#(X) -> c_1()
          2: a__f#(0()) -> c_2()
          3: a__f#(s(0())) -> c_3(a__f#(a__p(s(0()))),a__p#(s(0())))
          4: a__p#(X) -> c_4()
          5: a__p#(s(X)) -> c_5(mark#(X))
          6: mark#(0()) -> c_6()
          7: mark#(cons(X1,X2)) -> c_7(mark#(X1))
          8: mark#(f(X)) -> c_8(a__f#(mark(X)),mark#(X))
          9: mark#(p(X)) -> c_9(a__p#(mark(X)),mark#(X))
          10: mark#(s(X)) -> c_10(mark#(X))
* Step 3: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            a__f#(s(0())) -> c_3(a__f#(a__p(s(0()))),a__p#(s(0())))
            a__p#(s(X)) -> c_5(mark#(X))
            mark#(cons(X1,X2)) -> c_7(mark#(X1))
            mark#(f(X)) -> c_8(a__f#(mark(X)),mark#(X))
            mark#(p(X)) -> c_9(a__p#(mark(X)),mark#(X))
            mark#(s(X)) -> c_10(mark#(X))
        - Weak DPs:
            a__f#(X) -> c_1()
            a__f#(0()) -> c_2()
            a__p#(X) -> c_4()
            mark#(0()) -> c_6()
        - Weak TRS:
            a__f(X) -> f(X)
            a__f(0()) -> cons(0(),f(s(0())))
            a__f(s(0())) -> a__f(a__p(s(0())))
            a__p(X) -> p(X)
            a__p(s(X)) -> mark(X)
            mark(0()) -> 0()
            mark(cons(X1,X2)) -> cons(mark(X1),X2)
            mark(f(X)) -> a__f(mark(X))
            mark(p(X)) -> a__p(mark(X))
            mark(s(X)) -> s(mark(X))
        - Signature:
            {a__f/1,a__p/1,mark/1,a__f#/1,a__p#/1,mark#/1} / {0/0,cons/2,f/1,p/1,s/1,c_1/0,c_2/0,c_3/2,c_4/0,c_5/1,c_6/0
            ,c_7/1,c_8/2,c_9/2,c_10/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__f#,a__p#,mark#} and constructors {0,cons,f,p,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:a__f#(s(0())) -> c_3(a__f#(a__p(s(0()))),a__p#(s(0())))
             -->_2 a__p#(s(X)) -> c_5(mark#(X)):2
             -->_2 a__p#(X) -> c_4():9
             -->_1 a__f#(0()) -> c_2():8
             -->_1 a__f#(X) -> c_1():7
             -->_1 a__f#(s(0())) -> c_3(a__f#(a__p(s(0()))),a__p#(s(0()))):1
          
          2:S:a__p#(s(X)) -> c_5(mark#(X))
             -->_1 mark#(s(X)) -> c_10(mark#(X)):6
             -->_1 mark#(p(X)) -> c_9(a__p#(mark(X)),mark#(X)):5
             -->_1 mark#(f(X)) -> c_8(a__f#(mark(X)),mark#(X)):4
             -->_1 mark#(cons(X1,X2)) -> c_7(mark#(X1)):3
             -->_1 mark#(0()) -> c_6():10
          
          3:S:mark#(cons(X1,X2)) -> c_7(mark#(X1))
             -->_1 mark#(s(X)) -> c_10(mark#(X)):6
             -->_1 mark#(p(X)) -> c_9(a__p#(mark(X)),mark#(X)):5
             -->_1 mark#(f(X)) -> c_8(a__f#(mark(X)),mark#(X)):4
             -->_1 mark#(0()) -> c_6():10
             -->_1 mark#(cons(X1,X2)) -> c_7(mark#(X1)):3
          
          4:S:mark#(f(X)) -> c_8(a__f#(mark(X)),mark#(X))
             -->_2 mark#(s(X)) -> c_10(mark#(X)):6
             -->_2 mark#(p(X)) -> c_9(a__p#(mark(X)),mark#(X)):5
             -->_2 mark#(0()) -> c_6():10
             -->_1 a__f#(0()) -> c_2():8
             -->_1 a__f#(X) -> c_1():7
             -->_2 mark#(f(X)) -> c_8(a__f#(mark(X)),mark#(X)):4
             -->_2 mark#(cons(X1,X2)) -> c_7(mark#(X1)):3
             -->_1 a__f#(s(0())) -> c_3(a__f#(a__p(s(0()))),a__p#(s(0()))):1
          
          5:S:mark#(p(X)) -> c_9(a__p#(mark(X)),mark#(X))
             -->_2 mark#(s(X)) -> c_10(mark#(X)):6
             -->_2 mark#(0()) -> c_6():10
             -->_1 a__p#(X) -> c_4():9
             -->_2 mark#(p(X)) -> c_9(a__p#(mark(X)),mark#(X)):5
             -->_2 mark#(f(X)) -> c_8(a__f#(mark(X)),mark#(X)):4
             -->_2 mark#(cons(X1,X2)) -> c_7(mark#(X1)):3
             -->_1 a__p#(s(X)) -> c_5(mark#(X)):2
          
          6:S:mark#(s(X)) -> c_10(mark#(X))
             -->_1 mark#(0()) -> c_6():10
             -->_1 mark#(s(X)) -> c_10(mark#(X)):6
             -->_1 mark#(p(X)) -> c_9(a__p#(mark(X)),mark#(X)):5
             -->_1 mark#(f(X)) -> c_8(a__f#(mark(X)),mark#(X)):4
             -->_1 mark#(cons(X1,X2)) -> c_7(mark#(X1)):3
          
          7:W:a__f#(X) -> c_1()
             
          
          8:W:a__f#(0()) -> c_2()
             
          
          9:W:a__p#(X) -> c_4()
             
          
          10:W:mark#(0()) -> c_6()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          7: a__f#(X) -> c_1()
          8: a__f#(0()) -> c_2()
          9: a__p#(X) -> c_4()
          10: mark#(0()) -> c_6()
* Step 4: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          a__f#(s(0())) -> c_3(a__f#(a__p(s(0()))),a__p#(s(0())))
          a__p#(s(X)) -> c_5(mark#(X))
          mark#(cons(X1,X2)) -> c_7(mark#(X1))
          mark#(f(X)) -> c_8(a__f#(mark(X)),mark#(X))
          mark#(p(X)) -> c_9(a__p#(mark(X)),mark#(X))
          mark#(s(X)) -> c_10(mark#(X))
      - Weak TRS:
          a__f(X) -> f(X)
          a__f(0()) -> cons(0(),f(s(0())))
          a__f(s(0())) -> a__f(a__p(s(0())))
          a__p(X) -> p(X)
          a__p(s(X)) -> mark(X)
          mark(0()) -> 0()
          mark(cons(X1,X2)) -> cons(mark(X1),X2)
          mark(f(X)) -> a__f(mark(X))
          mark(p(X)) -> a__p(mark(X))
          mark(s(X)) -> s(mark(X))
      - Signature:
          {a__f/1,a__p/1,mark/1,a__f#/1,a__p#/1,mark#/1} / {0/0,cons/2,f/1,p/1,s/1,c_1/0,c_2/0,c_3/2,c_4/0,c_5/1,c_6/0
          ,c_7/1,c_8/2,c_9/2,c_10/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {a__f#,a__p#,mark#} and constructors {0,cons,f,p,s}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE