MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            fac(s(x)) -> times(fac(p(s(x))),s(x))
            p(s(0())) -> 0()
            p(s(s(x))) -> s(p(s(x)))
            plus(x,0()) -> x
            plus(x,s(y)) -> s(plus(x,y))
            times(x,0()) -> 0()
            times(0(),y) -> 0()
            times(s(x),y) -> plus(times(x,y),y)
        - Signature:
            {fac/1,p/1,plus/2,times/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fac,p,plus,times} and constructors {0,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          fac#(s(x)) -> c_1(times#(fac(p(s(x))),s(x)),fac#(p(s(x))),p#(s(x)))
          p#(s(0())) -> c_2()
          p#(s(s(x))) -> c_3(p#(s(x)))
          plus#(x,0()) -> c_4()
          plus#(x,s(y)) -> c_5(plus#(x,y))
          times#(x,0()) -> c_6()
          times#(0(),y) -> c_7()
          times#(s(x),y) -> c_8(plus#(times(x,y),y),times#(x,y))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            fac#(s(x)) -> c_1(times#(fac(p(s(x))),s(x)),fac#(p(s(x))),p#(s(x)))
            p#(s(0())) -> c_2()
            p#(s(s(x))) -> c_3(p#(s(x)))
            plus#(x,0()) -> c_4()
            plus#(x,s(y)) -> c_5(plus#(x,y))
            times#(x,0()) -> c_6()
            times#(0(),y) -> c_7()
            times#(s(x),y) -> c_8(plus#(times(x,y),y),times#(x,y))
        - Weak TRS:
            fac(s(x)) -> times(fac(p(s(x))),s(x))
            p(s(0())) -> 0()
            p(s(s(x))) -> s(p(s(x)))
            plus(x,0()) -> x
            plus(x,s(y)) -> s(plus(x,y))
            times(x,0()) -> 0()
            times(0(),y) -> 0()
            times(s(x),y) -> plus(times(x,y),y)
        - Signature:
            {fac/1,p/1,plus/2,times/2,fac#/1,p#/1,plus#/2,times#/2} / {0/0,s/1,c_1/3,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/0
            ,c_8/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fac#,p#,plus#,times#} and constructors {0,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,4,6,7}
        by application of
          Pre({2,4,6,7}) = {1,3,5,8}.
        Here rules are labelled as follows:
          1: fac#(s(x)) -> c_1(times#(fac(p(s(x))),s(x)),fac#(p(s(x))),p#(s(x)))
          2: p#(s(0())) -> c_2()
          3: p#(s(s(x))) -> c_3(p#(s(x)))
          4: plus#(x,0()) -> c_4()
          5: plus#(x,s(y)) -> c_5(plus#(x,y))
          6: times#(x,0()) -> c_6()
          7: times#(0(),y) -> c_7()
          8: times#(s(x),y) -> c_8(plus#(times(x,y),y),times#(x,y))
* Step 3: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            fac#(s(x)) -> c_1(times#(fac(p(s(x))),s(x)),fac#(p(s(x))),p#(s(x)))
            p#(s(s(x))) -> c_3(p#(s(x)))
            plus#(x,s(y)) -> c_5(plus#(x,y))
            times#(s(x),y) -> c_8(plus#(times(x,y),y),times#(x,y))
        - Weak DPs:
            p#(s(0())) -> c_2()
            plus#(x,0()) -> c_4()
            times#(x,0()) -> c_6()
            times#(0(),y) -> c_7()
        - Weak TRS:
            fac(s(x)) -> times(fac(p(s(x))),s(x))
            p(s(0())) -> 0()
            p(s(s(x))) -> s(p(s(x)))
            plus(x,0()) -> x
            plus(x,s(y)) -> s(plus(x,y))
            times(x,0()) -> 0()
            times(0(),y) -> 0()
            times(s(x),y) -> plus(times(x,y),y)
        - Signature:
            {fac/1,p/1,plus/2,times/2,fac#/1,p#/1,plus#/2,times#/2} / {0/0,s/1,c_1/3,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/0
            ,c_8/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {fac#,p#,plus#,times#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:fac#(s(x)) -> c_1(times#(fac(p(s(x))),s(x)),fac#(p(s(x))),p#(s(x)))
             -->_1 times#(s(x),y) -> c_8(plus#(times(x,y),y),times#(x,y)):4
             -->_3 p#(s(s(x))) -> c_3(p#(s(x))):2
             -->_1 times#(0(),y) -> c_7():8
             -->_3 p#(s(0())) -> c_2():5
             -->_2 fac#(s(x)) -> c_1(times#(fac(p(s(x))),s(x)),fac#(p(s(x))),p#(s(x))):1
          
          2:S:p#(s(s(x))) -> c_3(p#(s(x)))
             -->_1 p#(s(0())) -> c_2():5
             -->_1 p#(s(s(x))) -> c_3(p#(s(x))):2
          
          3:S:plus#(x,s(y)) -> c_5(plus#(x,y))
             -->_1 plus#(x,0()) -> c_4():6
             -->_1 plus#(x,s(y)) -> c_5(plus#(x,y)):3
          
          4:S:times#(s(x),y) -> c_8(plus#(times(x,y),y),times#(x,y))
             -->_2 times#(0(),y) -> c_7():8
             -->_2 times#(x,0()) -> c_6():7
             -->_1 plus#(x,0()) -> c_4():6
             -->_2 times#(s(x),y) -> c_8(plus#(times(x,y),y),times#(x,y)):4
             -->_1 plus#(x,s(y)) -> c_5(plus#(x,y)):3
          
          5:W:p#(s(0())) -> c_2()
             
          
          6:W:plus#(x,0()) -> c_4()
             
          
          7:W:times#(x,0()) -> c_6()
             
          
          8:W:times#(0(),y) -> c_7()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: p#(s(0())) -> c_2()
          6: plus#(x,0()) -> c_4()
          7: times#(x,0()) -> c_6()
          8: times#(0(),y) -> c_7()
* Step 4: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          fac#(s(x)) -> c_1(times#(fac(p(s(x))),s(x)),fac#(p(s(x))),p#(s(x)))
          p#(s(s(x))) -> c_3(p#(s(x)))
          plus#(x,s(y)) -> c_5(plus#(x,y))
          times#(s(x),y) -> c_8(plus#(times(x,y),y),times#(x,y))
      - Weak TRS:
          fac(s(x)) -> times(fac(p(s(x))),s(x))
          p(s(0())) -> 0()
          p(s(s(x))) -> s(p(s(x)))
          plus(x,0()) -> x
          plus(x,s(y)) -> s(plus(x,y))
          times(x,0()) -> 0()
          times(0(),y) -> 0()
          times(s(x),y) -> plus(times(x,y),y)
      - Signature:
          {fac/1,p/1,plus/2,times/2,fac#/1,p#/1,plus#/2,times#/2} / {0/0,s/1,c_1/3,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/0
          ,c_8/2}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {fac#,p#,plus#,times#} and constructors {0,s}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE