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: DecomposeDG 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:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          fac#(s(x)) -> c_1(times#(fac(p(s(x))),s(x)),fac#(p(s(x))),p#(s(x)))
        and a lower component
          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))
        Further, following extension rules are added to the lower component.
          fac#(s(x)) -> fac#(p(s(x)))
          fac#(s(x)) -> p#(s(x))
          fac#(s(x)) -> times#(fac(p(s(x))),s(x))
** Step 4.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            fac#(s(x)) -> c_1(times#(fac(p(s(x))),s(x)),fac#(p(s(x))),p#(s(x)))
        - 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:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: fac#(s(x)) -> c_1(times#(fac(p(s(x))),s(x)),fac#(p(s(x))),p#(s(x)))
          
        The strictly oriented rules are moved into the weak component.
*** Step 4.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            fac#(s(x)) -> c_1(times#(fac(p(s(x))),s(x)),fac#(p(s(x))),p#(s(x)))
        - 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:
        NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        The following argument positions are considered usable:
          uargs(c_1) = {1,2}
        
        Following symbols are considered usable:
          {p,fac#,p#,plus#,times#}
        TcT has computed the following interpretation:
               p(0) = [1]                                       
                      [1]                                       
                      [1]                                       
             p(fac) = [0 0 0]      [0]                          
                      [0 0 1] x1 + [1]                          
                      [0 0 0]      [0]                          
               p(p) = [1 1 0]      [0]                          
                      [1 0 0] x1 + [0]                          
                      [0 0 0]      [1]                          
            p(plus) = [0 0 1]      [0 0 1]      [1]             
                      [0 1 0] x1 + [1 0 1] x2 + [1]             
                      [0 0 0]      [0 0 0]      [0]             
               p(s) = [0 1 0]      [0]                          
                      [0 1 0] x1 + [1]                          
                      [0 0 0]      [0]                          
           p(times) = [1 0 0]      [0 0 0]      [1]             
                      [0 1 0] x1 + [0 0 1] x2 + [1]             
                      [0 1 0]      [0 0 0]      [1]             
            p(fac#) = [0 1 0]      [0]                          
                      [1 0 0] x1 + [1]                          
                      [1 1 0]      [1]                          
              p(p#) = [0 0 0]      [1]                          
                      [0 1 0] x1 + [0]                          
                      [1 1 0]      [0]                          
           p(plus#) = [0]                                       
                      [0]                                       
                      [0]                                       
          p(times#) = [0 0 0]      [0]                          
                      [0 0 0] x2 + [0]                          
                      [1 0 0]      [1]                          
             p(c_1) = [1 0 0]      [1 0 0]      [0 0 0]      [0]
                      [0 0 0] x1 + [0 0 0] x2 + [0 1 0] x3 + [0]
                      [0 0 0]      [0 0 0]      [0 0 1]      [1]
             p(c_2) = [0]                                       
                      [0]                                       
                      [0]                                       
             p(c_3) = [0]                                       
                      [0]                                       
                      [0]                                       
             p(c_4) = [0]                                       
                      [0]                                       
                      [0]                                       
             p(c_5) = [0]                                       
                      [0]                                       
                      [0]                                       
             p(c_6) = [0]                                       
                      [0]                                       
                      [0]                                       
             p(c_7) = [0]                                       
                      [0]                                       
                      [0]                                       
             p(c_8) = [0]                                       
                      [0]                                       
                      [0]                                       
        
        Following rules are strictly oriented:
        fac#(s(x)) = [0 1 0]     [1]                                      
                     [0 1 0] x + [1]                                      
                     [0 2 0]     [2]                                      
                   > [0 1 0]     [0]                                      
                     [0 1 0] x + [1]                                      
                     [0 2 0]     [2]                                      
                   = c_1(times#(fac(p(s(x))),s(x)),fac#(p(s(x))),p#(s(x)))
        
        
        Following rules are (at-least) weakly oriented:
         p(s(0())) =  [3]            
                      [1]            
                      [1]            
                   >= [1]            
                      [1]            
                      [1]            
                   =  0()            
        
        p(s(s(x))) =  [0 2 0]     [3]
                      [0 1 0] x + [1]
                      [0 0 0]     [1]
                   >= [0 1 0]     [0]
                      [0 1 0] x + [1]
                      [0 0 0]     [0]
                   =  s(p(s(x)))     
        
*** Step 4.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            fac#(s(x)) -> c_1(times#(fac(p(s(x))),s(x)),fac#(p(s(x))),p#(s(x)))
        - 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:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 4.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            fac#(s(x)) -> c_1(times#(fac(p(s(x))),s(x)),fac#(p(s(x))),p#(s(x)))
        - 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:W:fac#(s(x)) -> c_1(times#(fac(p(s(x))),s(x)),fac#(p(s(x))),p#(s(x)))
             -->_2 fac#(s(x)) -> c_1(times#(fac(p(s(x))),s(x)),fac#(p(s(x))),p#(s(x))):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: fac#(s(x)) -> c_1(times#(fac(p(s(x))),s(x)),fac#(p(s(x))),p#(s(x)))
*** Step 4.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - 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 already closed. The intended complexity is O(1).

** Step 4.b:1: PredecessorEstimationCP MAYBE
    + Considered Problem:
        - Strict DPs:
            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:
            fac#(s(x)) -> fac#(p(s(x)))
            fac#(s(x)) -> p#(s(x))
            fac#(s(x)) -> times#(fac(p(s(x))),s(x))
        - 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:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: p#(s(s(x))) -> c_3(p#(s(x)))
          
        The strictly oriented rules are moved into the weak component.
*** Step 4.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            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:
            fac#(s(x)) -> fac#(p(s(x)))
            fac#(s(x)) -> p#(s(x))
            fac#(s(x)) -> times#(fac(p(s(x))),s(x))
        - 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:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_3) = {1},
          uargs(c_5) = {1},
          uargs(c_8) = {1,2}
        
        Following symbols are considered usable:
          {p,fac#,p#,plus#,times#}
        TcT has computed the following interpretation:
               p(0) = [8]                  
             p(fac) = [2] x1 + [0]         
               p(p) = [1] x1 + [0]         
            p(plus) = [0]                  
               p(s) = [1] x1 + [8]         
           p(times) = [0]                  
            p(fac#) = [2] x1 + [3]         
              p(p#) = [1] x1 + [0]         
           p(plus#) = [0]                  
          p(times#) = [0]                  
             p(c_1) = [2] x3 + [8]         
             p(c_2) = [1]                  
             p(c_3) = [1] x1 + [4]         
             p(c_4) = [8]                  
             p(c_5) = [2] x1 + [0]         
             p(c_6) = [1]                  
             p(c_7) = [1]                  
             p(c_8) = [2] x1 + [1] x2 + [0]
        
        Following rules are strictly oriented:
        p#(s(s(x))) = [1] x + [16] 
                    > [1] x + [12] 
                    = c_3(p#(s(x)))
        
        
        Following rules are (at-least) weakly oriented:
            fac#(s(x)) =  [2] x + [19]                        
                       >= [2] x + [19]                        
                       =  fac#(p(s(x)))                       
        
            fac#(s(x)) =  [2] x + [19]                        
                       >= [1] x + [8]                         
                       =  p#(s(x))                            
        
            fac#(s(x)) =  [2] x + [19]                        
                       >= [0]                                 
                       =  times#(fac(p(s(x))),s(x))           
        
         plus#(x,s(y)) =  [0]                                 
                       >= [0]                                 
                       =  c_5(plus#(x,y))                     
        
        times#(s(x),y) =  [0]                                 
                       >= [0]                                 
                       =  c_8(plus#(times(x,y),y),times#(x,y))
        
             p(s(0())) =  [16]                                
                       >= [8]                                 
                       =  0()                                 
        
            p(s(s(x))) =  [1] x + [16]                        
                       >= [1] x + [16]                        
                       =  s(p(s(x)))                          
        
*** Step 4.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            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:
            fac#(s(x)) -> fac#(p(s(x)))
            fac#(s(x)) -> p#(s(x))
            fac#(s(x)) -> times#(fac(p(s(x))),s(x))
            p#(s(s(x))) -> c_3(p#(s(x)))
        - 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:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 4.b:1.b:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            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:
            fac#(s(x)) -> fac#(p(s(x)))
            fac#(s(x)) -> p#(s(x))
            fac#(s(x)) -> times#(fac(p(s(x))),s(x))
            p#(s(s(x))) -> c_3(p#(s(x)))
        - 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:plus#(x,s(y)) -> c_5(plus#(x,y))
             -->_1 plus#(x,s(y)) -> c_5(plus#(x,y)):1
          
          2:S:times#(s(x),y) -> c_8(plus#(times(x,y),y),times#(x,y))
             -->_2 times#(s(x),y) -> c_8(plus#(times(x,y),y),times#(x,y)):2
             -->_1 plus#(x,s(y)) -> c_5(plus#(x,y)):1
          
          3:W:fac#(s(x)) -> fac#(p(s(x)))
             -->_1 fac#(s(x)) -> times#(fac(p(s(x))),s(x)):5
             -->_1 fac#(s(x)) -> p#(s(x)):4
             -->_1 fac#(s(x)) -> fac#(p(s(x))):3
          
          4:W:fac#(s(x)) -> p#(s(x))
             -->_1 p#(s(s(x))) -> c_3(p#(s(x))):6
          
          5:W:fac#(s(x)) -> times#(fac(p(s(x))),s(x))
             -->_1 times#(s(x),y) -> c_8(plus#(times(x,y),y),times#(x,y)):2
          
          6:W:p#(s(s(x))) -> c_3(p#(s(x)))
             -->_1 p#(s(s(x))) -> c_3(p#(s(x))):6
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: fac#(s(x)) -> p#(s(x))
          6: p#(s(s(x))) -> c_3(p#(s(x)))
*** Step 4.b:1.b:2: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          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:
          fac#(s(x)) -> fac#(p(s(x)))
          fac#(s(x)) -> times#(fac(p(s(x))),s(x))
      - 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