MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            *(x,0()) -> 0()
            *(0(),x) -> 0()
            *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
            +(x,0()) -> x
            +(0(),x) -> x
            +(s(x),s(y)) -> s(s(+(x,y)))
            prod(cons(x,l)) -> *(x,prod(l))
            prod(nil()) -> s(0())
            sum(cons(x,l)) -> +(x,sum(l))
            sum(nil()) -> 0()
        - Signature:
            {*/2,+/2,prod/1,sum/1} / {0/0,cons/2,nil/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*,+,prod,sum} and constructors {0,cons,nil,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          *#(x,0()) -> c_1()
          *#(0(),x) -> c_2()
          *#(s(x),s(y)) -> c_3(+#(*(x,y),+(x,y)),*#(x,y),+#(x,y))
          +#(x,0()) -> c_4()
          +#(0(),x) -> c_5()
          +#(s(x),s(y)) -> c_6(+#(x,y))
          prod#(cons(x,l)) -> c_7(*#(x,prod(l)),prod#(l))
          prod#(nil()) -> c_8()
          sum#(cons(x,l)) -> c_9(+#(x,sum(l)),sum#(l))
          sum#(nil()) -> c_10()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            *#(x,0()) -> c_1()
            *#(0(),x) -> c_2()
            *#(s(x),s(y)) -> c_3(+#(*(x,y),+(x,y)),*#(x,y),+#(x,y))
            +#(x,0()) -> c_4()
            +#(0(),x) -> c_5()
            +#(s(x),s(y)) -> c_6(+#(x,y))
            prod#(cons(x,l)) -> c_7(*#(x,prod(l)),prod#(l))
            prod#(nil()) -> c_8()
            sum#(cons(x,l)) -> c_9(+#(x,sum(l)),sum#(l))
            sum#(nil()) -> c_10()
        - Weak TRS:
            *(x,0()) -> 0()
            *(0(),x) -> 0()
            *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
            +(x,0()) -> x
            +(0(),x) -> x
            +(s(x),s(y)) -> s(s(+(x,y)))
            prod(cons(x,l)) -> *(x,prod(l))
            prod(nil()) -> s(0())
            sum(cons(x,l)) -> +(x,sum(l))
            sum(nil()) -> 0()
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/2,c_8/0,c_9/2,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,4,5,8,10}
        by application of
          Pre({1,2,4,5,8,10}) = {3,6,7,9}.
        Here rules are labelled as follows:
          1: *#(x,0()) -> c_1()
          2: *#(0(),x) -> c_2()
          3: *#(s(x),s(y)) -> c_3(+#(*(x,y),+(x,y)),*#(x,y),+#(x,y))
          4: +#(x,0()) -> c_4()
          5: +#(0(),x) -> c_5()
          6: +#(s(x),s(y)) -> c_6(+#(x,y))
          7: prod#(cons(x,l)) -> c_7(*#(x,prod(l)),prod#(l))
          8: prod#(nil()) -> c_8()
          9: sum#(cons(x,l)) -> c_9(+#(x,sum(l)),sum#(l))
          10: sum#(nil()) -> c_10()
* Step 3: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            *#(s(x),s(y)) -> c_3(+#(*(x,y),+(x,y)),*#(x,y),+#(x,y))
            +#(s(x),s(y)) -> c_6(+#(x,y))
            prod#(cons(x,l)) -> c_7(*#(x,prod(l)),prod#(l))
            sum#(cons(x,l)) -> c_9(+#(x,sum(l)),sum#(l))
        - Weak DPs:
            *#(x,0()) -> c_1()
            *#(0(),x) -> c_2()
            +#(x,0()) -> c_4()
            +#(0(),x) -> c_5()
            prod#(nil()) -> c_8()
            sum#(nil()) -> c_10()
        - Weak TRS:
            *(x,0()) -> 0()
            *(0(),x) -> 0()
            *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
            +(x,0()) -> x
            +(0(),x) -> x
            +(s(x),s(y)) -> s(s(+(x,y)))
            prod(cons(x,l)) -> *(x,prod(l))
            prod(nil()) -> s(0())
            sum(cons(x,l)) -> +(x,sum(l))
            sum(nil()) -> 0()
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/2,c_8/0,c_9/2,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:*#(s(x),s(y)) -> c_3(+#(*(x,y),+(x,y)),*#(x,y),+#(x,y))
             -->_3 +#(s(x),s(y)) -> c_6(+#(x,y)):2
             -->_1 +#(s(x),s(y)) -> c_6(+#(x,y)):2
             -->_3 +#(0(),x) -> c_5():8
             -->_1 +#(0(),x) -> c_5():8
             -->_3 +#(x,0()) -> c_4():7
             -->_1 +#(x,0()) -> c_4():7
             -->_2 *#(0(),x) -> c_2():6
             -->_2 *#(x,0()) -> c_1():5
             -->_2 *#(s(x),s(y)) -> c_3(+#(*(x,y),+(x,y)),*#(x,y),+#(x,y)):1
          
          2:S:+#(s(x),s(y)) -> c_6(+#(x,y))
             -->_1 +#(0(),x) -> c_5():8
             -->_1 +#(x,0()) -> c_4():7
             -->_1 +#(s(x),s(y)) -> c_6(+#(x,y)):2
          
          3:S:prod#(cons(x,l)) -> c_7(*#(x,prod(l)),prod#(l))
             -->_2 prod#(nil()) -> c_8():9
             -->_1 *#(0(),x) -> c_2():6
             -->_1 *#(x,0()) -> c_1():5
             -->_2 prod#(cons(x,l)) -> c_7(*#(x,prod(l)),prod#(l)):3
             -->_1 *#(s(x),s(y)) -> c_3(+#(*(x,y),+(x,y)),*#(x,y),+#(x,y)):1
          
          4:S:sum#(cons(x,l)) -> c_9(+#(x,sum(l)),sum#(l))
             -->_2 sum#(nil()) -> c_10():10
             -->_1 +#(0(),x) -> c_5():8
             -->_1 +#(x,0()) -> c_4():7
             -->_2 sum#(cons(x,l)) -> c_9(+#(x,sum(l)),sum#(l)):4
             -->_1 +#(s(x),s(y)) -> c_6(+#(x,y)):2
          
          5:W:*#(x,0()) -> c_1()
             
          
          6:W:*#(0(),x) -> c_2()
             
          
          7:W:+#(x,0()) -> c_4()
             
          
          8:W:+#(0(),x) -> c_5()
             
          
          9:W:prod#(nil()) -> c_8()
             
          
          10:W:sum#(nil()) -> c_10()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          10: sum#(nil()) -> c_10()
          9: prod#(nil()) -> c_8()
          5: *#(x,0()) -> c_1()
          6: *#(0(),x) -> c_2()
          7: +#(x,0()) -> c_4()
          8: +#(0(),x) -> c_5()
* Step 4: Decompose MAYBE
    + Considered Problem:
        - Strict DPs:
            *#(s(x),s(y)) -> c_3(+#(*(x,y),+(x,y)),*#(x,y),+#(x,y))
            +#(s(x),s(y)) -> c_6(+#(x,y))
            prod#(cons(x,l)) -> c_7(*#(x,prod(l)),prod#(l))
            sum#(cons(x,l)) -> c_9(+#(x,sum(l)),sum#(l))
        - Weak TRS:
            *(x,0()) -> 0()
            *(0(),x) -> 0()
            *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
            +(x,0()) -> x
            +(0(),x) -> x
            +(s(x),s(y)) -> s(s(+(x,y)))
            prod(cons(x,l)) -> *(x,prod(l))
            prod(nil()) -> s(0())
            sum(cons(x,l)) -> +(x,sum(l))
            sum(nil()) -> 0()
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/2,c_8/0,c_9/2,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    + Details:
        We analyse the complexity of following sub-problems (R) and (S).
        Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
        
        Problem (R)
          - Strict DPs:
              *#(s(x),s(y)) -> c_3(+#(*(x,y),+(x,y)),*#(x,y),+#(x,y))
              +#(s(x),s(y)) -> c_6(+#(x,y))
          - Weak DPs:
              prod#(cons(x,l)) -> c_7(*#(x,prod(l)),prod#(l))
              sum#(cons(x,l)) -> c_9(+#(x,sum(l)),sum#(l))
          - Weak TRS:
              *(x,0()) -> 0()
              *(0(),x) -> 0()
              *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
              +(x,0()) -> x
              +(0(),x) -> x
              +(s(x),s(y)) -> s(s(+(x,y)))
              prod(cons(x,l)) -> *(x,prod(l))
              prod(nil()) -> s(0())
              sum(cons(x,l)) -> +(x,sum(l))
              sum(nil()) -> 0()
          - Signature:
              {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0
              ,c_6/1,c_7/2,c_8/0,c_9/2,c_10/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
        
        Problem (S)
          - Strict DPs:
              prod#(cons(x,l)) -> c_7(*#(x,prod(l)),prod#(l))
              sum#(cons(x,l)) -> c_9(+#(x,sum(l)),sum#(l))
          - Weak DPs:
              *#(s(x),s(y)) -> c_3(+#(*(x,y),+(x,y)),*#(x,y),+#(x,y))
              +#(s(x),s(y)) -> c_6(+#(x,y))
          - Weak TRS:
              *(x,0()) -> 0()
              *(0(),x) -> 0()
              *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
              +(x,0()) -> x
              +(0(),x) -> x
              +(s(x),s(y)) -> s(s(+(x,y)))
              prod(cons(x,l)) -> *(x,prod(l))
              prod(nil()) -> s(0())
              sum(cons(x,l)) -> +(x,sum(l))
              sum(nil()) -> 0()
          - Signature:
              {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0
              ,c_6/1,c_7/2,c_8/0,c_9/2,c_10/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
** Step 4.a:1: DecomposeDG MAYBE
    + Considered Problem:
        - Strict DPs:
            *#(s(x),s(y)) -> c_3(+#(*(x,y),+(x,y)),*#(x,y),+#(x,y))
            +#(s(x),s(y)) -> c_6(+#(x,y))
        - Weak DPs:
            prod#(cons(x,l)) -> c_7(*#(x,prod(l)),prod#(l))
            sum#(cons(x,l)) -> c_9(+#(x,sum(l)),sum#(l))
        - Weak TRS:
            *(x,0()) -> 0()
            *(0(),x) -> 0()
            *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
            +(x,0()) -> x
            +(0(),x) -> x
            +(s(x),s(y)) -> s(s(+(x,y)))
            prod(cons(x,l)) -> *(x,prod(l))
            prod(nil()) -> s(0())
            sum(cons(x,l)) -> +(x,sum(l))
            sum(nil()) -> 0()
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/2,c_8/0,c_9/2,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,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
          prod#(cons(x,l)) -> c_7(*#(x,prod(l)),prod#(l))
          sum#(cons(x,l)) -> c_9(+#(x,sum(l)),sum#(l))
        and a lower component
          *#(s(x),s(y)) -> c_3(+#(*(x,y),+(x,y)),*#(x,y),+#(x,y))
          +#(s(x),s(y)) -> c_6(+#(x,y))
        Further, following extension rules are added to the lower component.
          prod#(cons(x,l)) -> *#(x,prod(l))
          prod#(cons(x,l)) -> prod#(l)
          sum#(cons(x,l)) -> +#(x,sum(l))
          sum#(cons(x,l)) -> sum#(l)
*** Step 4.a:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            prod#(cons(x,l)) -> c_7(*#(x,prod(l)),prod#(l))
            sum#(cons(x,l)) -> c_9(+#(x,sum(l)),sum#(l))
        - Weak TRS:
            *(x,0()) -> 0()
            *(0(),x) -> 0()
            *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
            +(x,0()) -> x
            +(0(),x) -> x
            +(s(x),s(y)) -> s(s(+(x,y)))
            prod(cons(x,l)) -> *(x,prod(l))
            prod(nil()) -> s(0())
            sum(cons(x,l)) -> +(x,sum(l))
            sum(nil()) -> 0()
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/2,c_8/0,c_9/2,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,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:
          2: sum#(cons(x,l)) -> c_9(+#(x,sum(l)),sum#(l))
          
        The strictly oriented rules are moved into the weak component.
**** Step 4.a:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            prod#(cons(x,l)) -> c_7(*#(x,prod(l)),prod#(l))
            sum#(cons(x,l)) -> c_9(+#(x,sum(l)),sum#(l))
        - Weak TRS:
            *(x,0()) -> 0()
            *(0(),x) -> 0()
            *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
            +(x,0()) -> x
            +(0(),x) -> x
            +(s(x),s(y)) -> s(s(+(x,y)))
            prod(cons(x,l)) -> *(x,prod(l))
            prod(nil()) -> s(0())
            sum(cons(x,l)) -> +(x,sum(l))
            sum(nil()) -> 0()
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/2,c_8/0,c_9/2,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,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_7) = {1,2},
          uargs(c_9) = {1,2}
        
        Following symbols are considered usable:
          {*#,+#,prod#,sum#}
        TcT has computed the following interpretation:
              p(*) = [1] x1 + [1] x2 + [2]         
              p(+) = [6] x1 + [4] x2 + [0]         
              p(0) = [0]                           
           p(cons) = [1] x1 + [1] x2 + [4]         
            p(nil) = [1]                           
           p(prod) = [0]                           
              p(s) = [1] x1 + [1]                  
            p(sum) = [2]                           
             p(*#) = [0]                           
             p(+#) = [4] x1 + [1]                  
          p(prod#) = [0]                           
           p(sum#) = [4] x1 + [12]                 
            p(c_1) = [4]                           
            p(c_2) = [2]                           
            p(c_3) = [2] x1 + [1] x2 + [1] x3 + [0]
            p(c_4) = [1]                           
            p(c_5) = [0]                           
            p(c_6) = [2]                           
            p(c_7) = [8] x1 + [4] x2 + [0]         
            p(c_8) = [2]                           
            p(c_9) = [1] x1 + [1] x2 + [8]         
           p(c_10) = [1]                           
        
        Following rules are strictly oriented:
        sum#(cons(x,l)) = [4] l + [4] x + [28]     
                        > [4] l + [4] x + [21]     
                        = c_9(+#(x,sum(l)),sum#(l))
        
        
        Following rules are (at-least) weakly oriented:
        prod#(cons(x,l)) =  [0]                        
                         >= [0]                        
                         =  c_7(*#(x,prod(l)),prod#(l))
        
**** Step 4.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            prod#(cons(x,l)) -> c_7(*#(x,prod(l)),prod#(l))
        - Weak DPs:
            sum#(cons(x,l)) -> c_9(+#(x,sum(l)),sum#(l))
        - Weak TRS:
            *(x,0()) -> 0()
            *(0(),x) -> 0()
            *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
            +(x,0()) -> x
            +(0(),x) -> x
            +(s(x),s(y)) -> s(s(+(x,y)))
            prod(cons(x,l)) -> *(x,prod(l))
            prod(nil()) -> s(0())
            sum(cons(x,l)) -> +(x,sum(l))
            sum(nil()) -> 0()
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/2,c_8/0,c_9/2,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 4.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            prod#(cons(x,l)) -> c_7(*#(x,prod(l)),prod#(l))
        - Weak DPs:
            sum#(cons(x,l)) -> c_9(+#(x,sum(l)),sum#(l))
        - Weak TRS:
            *(x,0()) -> 0()
            *(0(),x) -> 0()
            *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
            +(x,0()) -> x
            +(0(),x) -> x
            +(s(x),s(y)) -> s(s(+(x,y)))
            prod(cons(x,l)) -> *(x,prod(l))
            prod(nil()) -> s(0())
            sum(cons(x,l)) -> +(x,sum(l))
            sum(nil()) -> 0()
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/2,c_8/0,c_9/2,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:prod#(cons(x,l)) -> c_7(*#(x,prod(l)),prod#(l))
             -->_2 prod#(cons(x,l)) -> c_7(*#(x,prod(l)),prod#(l)):1
          
          2:W:sum#(cons(x,l)) -> c_9(+#(x,sum(l)),sum#(l))
             -->_2 sum#(cons(x,l)) -> c_9(+#(x,sum(l)),sum#(l)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: sum#(cons(x,l)) -> c_9(+#(x,sum(l)),sum#(l))
**** Step 4.a:1.a:1.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            prod#(cons(x,l)) -> c_7(*#(x,prod(l)),prod#(l))
        - Weak TRS:
            *(x,0()) -> 0()
            *(0(),x) -> 0()
            *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
            +(x,0()) -> x
            +(0(),x) -> x
            +(s(x),s(y)) -> s(s(+(x,y)))
            prod(cons(x,l)) -> *(x,prod(l))
            prod(nil()) -> s(0())
            sum(cons(x,l)) -> +(x,sum(l))
            sum(nil()) -> 0()
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/2,c_8/0,c_9/2,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:prod#(cons(x,l)) -> c_7(*#(x,prod(l)),prod#(l))
             -->_2 prod#(cons(x,l)) -> c_7(*#(x,prod(l)),prod#(l)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          prod#(cons(x,l)) -> c_7(prod#(l))
**** Step 4.a:1.a:1.b:3: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            prod#(cons(x,l)) -> c_7(prod#(l))
        - Weak TRS:
            *(x,0()) -> 0()
            *(0(),x) -> 0()
            *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
            +(x,0()) -> x
            +(0(),x) -> x
            +(s(x),s(y)) -> s(s(+(x,y)))
            prod(cons(x,l)) -> *(x,prod(l))
            prod(nil()) -> s(0())
            sum(cons(x,l)) -> +(x,sum(l))
            sum(nil()) -> 0()
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/1,c_8/0,c_9/2,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          prod#(cons(x,l)) -> c_7(prod#(l))
**** Step 4.a:1.a:1.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            prod#(cons(x,l)) -> c_7(prod#(l))
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/1,c_8/0,c_9/2,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,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: prod#(cons(x,l)) -> c_7(prod#(l))
          
        The strictly oriented rules are moved into the weak component.
***** Step 4.a:1.a:1.b:4.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            prod#(cons(x,l)) -> c_7(prod#(l))
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/1,c_8/0,c_9/2,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,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_7) = {1}
        
        Following symbols are considered usable:
          {*#,+#,prod#,sum#}
        TcT has computed the following interpretation:
              p(*) = [1]                  
              p(+) = [2] x1 + [1]         
              p(0) = [1]                  
           p(cons) = [1] x2 + [8]         
            p(nil) = [1]                  
           p(prod) = [1] x1 + [4]         
              p(s) = [1] x1 + [0]         
            p(sum) = [0]                  
             p(*#) = [1] x1 + [1] x2 + [0]
             p(+#) = [8] x1 + [1]         
          p(prod#) = [1] x1 + [8]         
           p(sum#) = [1] x1 + [1]         
            p(c_1) = [0]                  
            p(c_2) = [2]                  
            p(c_3) = [1] x2 + [1] x3 + [0]
            p(c_4) = [2]                  
            p(c_5) = [1]                  
            p(c_6) = [0]                  
            p(c_7) = [1] x1 + [0]         
            p(c_8) = [2]                  
            p(c_9) = [8] x2 + [1]         
           p(c_10) = [1]                  
        
        Following rules are strictly oriented:
        prod#(cons(x,l)) = [1] l + [16] 
                         > [1] l + [8]  
                         = c_7(prod#(l))
        
        
        Following rules are (at-least) weakly oriented:
        
***** Step 4.a:1.a:1.b:4.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            prod#(cons(x,l)) -> c_7(prod#(l))
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/1,c_8/0,c_9/2,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

***** Step 4.a:1.a:1.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            prod#(cons(x,l)) -> c_7(prod#(l))
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/1,c_8/0,c_9/2,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:prod#(cons(x,l)) -> c_7(prod#(l))
             -->_1 prod#(cons(x,l)) -> c_7(prod#(l)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: prod#(cons(x,l)) -> c_7(prod#(l))
***** Step 4.a:1.a:1.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/1,c_8/0,c_9/2,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 4.a:1.b:1: DecomposeDG MAYBE
    + Considered Problem:
        - Strict DPs:
            *#(s(x),s(y)) -> c_3(+#(*(x,y),+(x,y)),*#(x,y),+#(x,y))
            +#(s(x),s(y)) -> c_6(+#(x,y))
        - Weak DPs:
            prod#(cons(x,l)) -> *#(x,prod(l))
            prod#(cons(x,l)) -> prod#(l)
            sum#(cons(x,l)) -> +#(x,sum(l))
            sum#(cons(x,l)) -> sum#(l)
        - Weak TRS:
            *(x,0()) -> 0()
            *(0(),x) -> 0()
            *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
            +(x,0()) -> x
            +(0(),x) -> x
            +(s(x),s(y)) -> s(s(+(x,y)))
            prod(cons(x,l)) -> *(x,prod(l))
            prod(nil()) -> s(0())
            sum(cons(x,l)) -> +(x,sum(l))
            sum(nil()) -> 0()
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/2,c_8/0,c_9/2,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,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
          *#(s(x),s(y)) -> c_3(+#(*(x,y),+(x,y)),*#(x,y),+#(x,y))
          prod#(cons(x,l)) -> *#(x,prod(l))
          prod#(cons(x,l)) -> prod#(l)
          sum#(cons(x,l)) -> +#(x,sum(l))
          sum#(cons(x,l)) -> sum#(l)
        and a lower component
          +#(s(x),s(y)) -> c_6(+#(x,y))
        Further, following extension rules are added to the lower component.
          *#(s(x),s(y)) -> *#(x,y)
          *#(s(x),s(y)) -> +#(x,y)
          *#(s(x),s(y)) -> +#(*(x,y),+(x,y))
          prod#(cons(x,l)) -> *#(x,prod(l))
          prod#(cons(x,l)) -> prod#(l)
          sum#(cons(x,l)) -> +#(x,sum(l))
          sum#(cons(x,l)) -> sum#(l)
**** Step 4.a:1.b:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            *#(s(x),s(y)) -> c_3(+#(*(x,y),+(x,y)),*#(x,y),+#(x,y))
            sum#(cons(x,l)) -> +#(x,sum(l))
        - Weak DPs:
            prod#(cons(x,l)) -> *#(x,prod(l))
            prod#(cons(x,l)) -> prod#(l)
            sum#(cons(x,l)) -> sum#(l)
        - Weak TRS:
            *(x,0()) -> 0()
            *(0(),x) -> 0()
            *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
            +(x,0()) -> x
            +(0(),x) -> x
            +(s(x),s(y)) -> s(s(+(x,y)))
            prod(cons(x,l)) -> *(x,prod(l))
            prod(nil()) -> s(0())
            sum(cons(x,l)) -> +(x,sum(l))
            sum(nil()) -> 0()
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/2,c_8/0,c_9/2,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,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:
          2: sum#(cons(x,l)) -> +#(x,sum(l))
          
        The strictly oriented rules are moved into the weak component.
***** Step 4.a:1.b:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            *#(s(x),s(y)) -> c_3(+#(*(x,y),+(x,y)),*#(x,y),+#(x,y))
            sum#(cons(x,l)) -> +#(x,sum(l))
        - Weak DPs:
            prod#(cons(x,l)) -> *#(x,prod(l))
            prod#(cons(x,l)) -> prod#(l)
            sum#(cons(x,l)) -> sum#(l)
        - Weak TRS:
            *(x,0()) -> 0()
            *(0(),x) -> 0()
            *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
            +(x,0()) -> x
            +(0(),x) -> x
            +(s(x),s(y)) -> s(s(+(x,y)))
            prod(cons(x,l)) -> *(x,prod(l))
            prod(nil()) -> s(0())
            sum(cons(x,l)) -> +(x,sum(l))
            sum(nil()) -> 0()
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/2,c_8/0,c_9/2,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,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) = {1,2}
        
        Following symbols are considered usable:
          {*#,+#,prod#,sum#}
        TcT has computed the following interpretation:
              p(*) = [4] x1 + [1]         
              p(+) = [9]                  
              p(0) = [5]                  
           p(cons) = [4]                  
            p(nil) = [1]                  
           p(prod) = [9]                  
              p(s) = [1] x1 + [0]         
            p(sum) = [4] x1 + [8]         
             p(*#) = [0]                  
             p(+#) = [0]                  
          p(prod#) = [0]                  
           p(sum#) = [8]                  
            p(c_1) = [1]                  
            p(c_2) = [0]                  
            p(c_3) = [8] x1 + [8] x2 + [0]
            p(c_4) = [0]                  
            p(c_5) = [8]                  
            p(c_6) = [1]                  
            p(c_7) = [1] x1 + [1]         
            p(c_8) = [0]                  
            p(c_9) = [1] x1 + [2] x2 + [1]
           p(c_10) = [0]                  
        
        Following rules are strictly oriented:
        sum#(cons(x,l)) = [8]         
                        > [0]         
                        = +#(x,sum(l))
        
        
        Following rules are (at-least) weakly oriented:
           *#(s(x),s(y)) =  [0]                                   
                         >= [0]                                   
                         =  c_3(+#(*(x,y),+(x,y)),*#(x,y),+#(x,y))
        
        prod#(cons(x,l)) =  [0]                                   
                         >= [0]                                   
                         =  *#(x,prod(l))                         
        
        prod#(cons(x,l)) =  [0]                                   
                         >= [0]                                   
                         =  prod#(l)                              
        
         sum#(cons(x,l)) =  [8]                                   
                         >= [8]                                   
                         =  sum#(l)                               
        
***** Step 4.a:1.b:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            *#(s(x),s(y)) -> c_3(+#(*(x,y),+(x,y)),*#(x,y),+#(x,y))
        - Weak DPs:
            prod#(cons(x,l)) -> *#(x,prod(l))
            prod#(cons(x,l)) -> prod#(l)
            sum#(cons(x,l)) -> +#(x,sum(l))
            sum#(cons(x,l)) -> sum#(l)
        - Weak TRS:
            *(x,0()) -> 0()
            *(0(),x) -> 0()
            *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
            +(x,0()) -> x
            +(0(),x) -> x
            +(s(x),s(y)) -> s(s(+(x,y)))
            prod(cons(x,l)) -> *(x,prod(l))
            prod(nil()) -> s(0())
            sum(cons(x,l)) -> +(x,sum(l))
            sum(nil()) -> 0()
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/2,c_8/0,c_9/2,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

***** Step 4.a:1.b:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            *#(s(x),s(y)) -> c_3(+#(*(x,y),+(x,y)),*#(x,y),+#(x,y))
        - Weak DPs:
            prod#(cons(x,l)) -> *#(x,prod(l))
            prod#(cons(x,l)) -> prod#(l)
            sum#(cons(x,l)) -> +#(x,sum(l))
            sum#(cons(x,l)) -> sum#(l)
        - Weak TRS:
            *(x,0()) -> 0()
            *(0(),x) -> 0()
            *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
            +(x,0()) -> x
            +(0(),x) -> x
            +(s(x),s(y)) -> s(s(+(x,y)))
            prod(cons(x,l)) -> *(x,prod(l))
            prod(nil()) -> s(0())
            sum(cons(x,l)) -> +(x,sum(l))
            sum(nil()) -> 0()
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/2,c_8/0,c_9/2,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:*#(s(x),s(y)) -> c_3(+#(*(x,y),+(x,y)),*#(x,y),+#(x,y))
             -->_2 *#(s(x),s(y)) -> c_3(+#(*(x,y),+(x,y)),*#(x,y),+#(x,y)):1
          
          2:W:prod#(cons(x,l)) -> *#(x,prod(l))
             -->_1 *#(s(x),s(y)) -> c_3(+#(*(x,y),+(x,y)),*#(x,y),+#(x,y)):1
          
          3:W:prod#(cons(x,l)) -> prod#(l)
             -->_1 prod#(cons(x,l)) -> prod#(l):3
             -->_1 prod#(cons(x,l)) -> *#(x,prod(l)):2
          
          4:W:sum#(cons(x,l)) -> +#(x,sum(l))
             
          
          5:W:sum#(cons(x,l)) -> sum#(l)
             -->_1 sum#(cons(x,l)) -> sum#(l):5
             -->_1 sum#(cons(x,l)) -> +#(x,sum(l)):4
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: sum#(cons(x,l)) -> sum#(l)
          4: sum#(cons(x,l)) -> +#(x,sum(l))
***** Step 4.a:1.b:1.a:1.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            *#(s(x),s(y)) -> c_3(+#(*(x,y),+(x,y)),*#(x,y),+#(x,y))
        - Weak DPs:
            prod#(cons(x,l)) -> *#(x,prod(l))
            prod#(cons(x,l)) -> prod#(l)
        - Weak TRS:
            *(x,0()) -> 0()
            *(0(),x) -> 0()
            *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
            +(x,0()) -> x
            +(0(),x) -> x
            +(s(x),s(y)) -> s(s(+(x,y)))
            prod(cons(x,l)) -> *(x,prod(l))
            prod(nil()) -> s(0())
            sum(cons(x,l)) -> +(x,sum(l))
            sum(nil()) -> 0()
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/2,c_8/0,c_9/2,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:*#(s(x),s(y)) -> c_3(+#(*(x,y),+(x,y)),*#(x,y),+#(x,y))
             -->_2 *#(s(x),s(y)) -> c_3(+#(*(x,y),+(x,y)),*#(x,y),+#(x,y)):1
          
          2:W:prod#(cons(x,l)) -> *#(x,prod(l))
             -->_1 *#(s(x),s(y)) -> c_3(+#(*(x,y),+(x,y)),*#(x,y),+#(x,y)):1
          
          3:W:prod#(cons(x,l)) -> prod#(l)
             -->_1 prod#(cons(x,l)) -> prod#(l):3
             -->_1 prod#(cons(x,l)) -> *#(x,prod(l)):2
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          *#(s(x),s(y)) -> c_3(*#(x,y))
***** Step 4.a:1.b:1.a:1.b:3: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            *#(s(x),s(y)) -> c_3(*#(x,y))
        - Weak DPs:
            prod#(cons(x,l)) -> *#(x,prod(l))
            prod#(cons(x,l)) -> prod#(l)
        - Weak TRS:
            *(x,0()) -> 0()
            *(0(),x) -> 0()
            *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
            +(x,0()) -> x
            +(0(),x) -> x
            +(s(x),s(y)) -> s(s(+(x,y)))
            prod(cons(x,l)) -> *(x,prod(l))
            prod(nil()) -> s(0())
            sum(cons(x,l)) -> +(x,sum(l))
            sum(nil()) -> 0()
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1
            ,c_7/2,c_8/0,c_9/2,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          *(x,0()) -> 0()
          *(0(),x) -> 0()
          *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
          +(x,0()) -> x
          +(0(),x) -> x
          +(s(x),s(y)) -> s(s(+(x,y)))
          prod(cons(x,l)) -> *(x,prod(l))
          prod(nil()) -> s(0())
          *#(s(x),s(y)) -> c_3(*#(x,y))
          prod#(cons(x,l)) -> *#(x,prod(l))
          prod#(cons(x,l)) -> prod#(l)
***** Step 4.a:1.b:1.a:1.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            *#(s(x),s(y)) -> c_3(*#(x,y))
        - Weak DPs:
            prod#(cons(x,l)) -> *#(x,prod(l))
            prod#(cons(x,l)) -> prod#(l)
        - Weak TRS:
            *(x,0()) -> 0()
            *(0(),x) -> 0()
            *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
            +(x,0()) -> x
            +(0(),x) -> x
            +(s(x),s(y)) -> s(s(+(x,y)))
            prod(cons(x,l)) -> *(x,prod(l))
            prod(nil()) -> s(0())
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1
            ,c_7/2,c_8/0,c_9/2,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,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: *#(s(x),s(y)) -> c_3(*#(x,y))
          
        The strictly oriented rules are moved into the weak component.
****** Step 4.a:1.b:1.a:1.b:4.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            *#(s(x),s(y)) -> c_3(*#(x,y))
        - Weak DPs:
            prod#(cons(x,l)) -> *#(x,prod(l))
            prod#(cons(x,l)) -> prod#(l)
        - Weak TRS:
            *(x,0()) -> 0()
            *(0(),x) -> 0()
            *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
            +(x,0()) -> x
            +(0(),x) -> x
            +(s(x),s(y)) -> s(s(+(x,y)))
            prod(cons(x,l)) -> *(x,prod(l))
            prod(nil()) -> s(0())
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1
            ,c_7/2,c_8/0,c_9/2,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,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}
        
        Following symbols are considered usable:
          {*#,+#,prod#,sum#}
        TcT has computed the following interpretation:
              p(*) = [4] x1 + [0]         
              p(+) = [0]                  
              p(0) = [1]                  
           p(cons) = [1] x1 + [1] x2 + [0]
            p(nil) = [0]                  
           p(prod) = [8] x1 + [2]         
              p(s) = [1] x1 + [4]         
            p(sum) = [0]                  
             p(*#) = [2] x1 + [0]         
             p(+#) = [0]                  
          p(prod#) = [8] x1 + [0]         
           p(sum#) = [0]                  
            p(c_1) = [0]                  
            p(c_2) = [2]                  
            p(c_3) = [1] x1 + [6]         
            p(c_4) = [0]                  
            p(c_5) = [0]                  
            p(c_6) = [1] x1 + [1]         
            p(c_7) = [2] x1 + [1] x2 + [1]
            p(c_8) = [0]                  
            p(c_9) = [1]                  
           p(c_10) = [1]                  
        
        Following rules are strictly oriented:
        *#(s(x),s(y)) = [2] x + [8] 
                      > [2] x + [6] 
                      = c_3(*#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
        prod#(cons(x,l)) =  [8] l + [8] x + [0]
                         >= [2] x + [0]        
                         =  *#(x,prod(l))      
        
        prod#(cons(x,l)) =  [8] l + [8] x + [0]
                         >= [8] l + [0]        
                         =  prod#(l)           
        
****** Step 4.a:1.b:1.a:1.b:4.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            *#(s(x),s(y)) -> c_3(*#(x,y))
            prod#(cons(x,l)) -> *#(x,prod(l))
            prod#(cons(x,l)) -> prod#(l)
        - Weak TRS:
            *(x,0()) -> 0()
            *(0(),x) -> 0()
            *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
            +(x,0()) -> x
            +(0(),x) -> x
            +(s(x),s(y)) -> s(s(+(x,y)))
            prod(cons(x,l)) -> *(x,prod(l))
            prod(nil()) -> s(0())
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1
            ,c_7/2,c_8/0,c_9/2,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

****** Step 4.a:1.b:1.a:1.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            *#(s(x),s(y)) -> c_3(*#(x,y))
            prod#(cons(x,l)) -> *#(x,prod(l))
            prod#(cons(x,l)) -> prod#(l)
        - Weak TRS:
            *(x,0()) -> 0()
            *(0(),x) -> 0()
            *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
            +(x,0()) -> x
            +(0(),x) -> x
            +(s(x),s(y)) -> s(s(+(x,y)))
            prod(cons(x,l)) -> *(x,prod(l))
            prod(nil()) -> s(0())
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1
            ,c_7/2,c_8/0,c_9/2,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:*#(s(x),s(y)) -> c_3(*#(x,y))
             -->_1 *#(s(x),s(y)) -> c_3(*#(x,y)):1
          
          2:W:prod#(cons(x,l)) -> *#(x,prod(l))
             -->_1 *#(s(x),s(y)) -> c_3(*#(x,y)):1
          
          3:W:prod#(cons(x,l)) -> prod#(l)
             -->_1 prod#(cons(x,l)) -> prod#(l):3
             -->_1 prod#(cons(x,l)) -> *#(x,prod(l)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: prod#(cons(x,l)) -> prod#(l)
          2: prod#(cons(x,l)) -> *#(x,prod(l))
          1: *#(s(x),s(y)) -> c_3(*#(x,y))
****** Step 4.a:1.b:1.a:1.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            *(x,0()) -> 0()
            *(0(),x) -> 0()
            *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
            +(x,0()) -> x
            +(0(),x) -> x
            +(s(x),s(y)) -> s(s(+(x,y)))
            prod(cons(x,l)) -> *(x,prod(l))
            prod(nil()) -> s(0())
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1
            ,c_7/2,c_8/0,c_9/2,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

**** Step 4.a:1.b:1.b:1: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          +#(s(x),s(y)) -> c_6(+#(x,y))
      - Weak DPs:
          *#(s(x),s(y)) -> *#(x,y)
          *#(s(x),s(y)) -> +#(x,y)
          *#(s(x),s(y)) -> +#(*(x,y),+(x,y))
          prod#(cons(x,l)) -> *#(x,prod(l))
          prod#(cons(x,l)) -> prod#(l)
          sum#(cons(x,l)) -> +#(x,sum(l))
          sum#(cons(x,l)) -> sum#(l)
      - Weak TRS:
          *(x,0()) -> 0()
          *(0(),x) -> 0()
          *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
          +(x,0()) -> x
          +(0(),x) -> x
          +(s(x),s(y)) -> s(s(+(x,y)))
          prod(cons(x,l)) -> *(x,prod(l))
          prod(nil()) -> s(0())
          sum(cons(x,l)) -> +(x,sum(l))
          sum(nil()) -> 0()
      - Signature:
          {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
          ,c_7/2,c_8/0,c_9/2,c_10/0}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
** Step 4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            prod#(cons(x,l)) -> c_7(*#(x,prod(l)),prod#(l))
            sum#(cons(x,l)) -> c_9(+#(x,sum(l)),sum#(l))
        - Weak DPs:
            *#(s(x),s(y)) -> c_3(+#(*(x,y),+(x,y)),*#(x,y),+#(x,y))
            +#(s(x),s(y)) -> c_6(+#(x,y))
        - Weak TRS:
            *(x,0()) -> 0()
            *(0(),x) -> 0()
            *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
            +(x,0()) -> x
            +(0(),x) -> x
            +(s(x),s(y)) -> s(s(+(x,y)))
            prod(cons(x,l)) -> *(x,prod(l))
            prod(nil()) -> s(0())
            sum(cons(x,l)) -> +(x,sum(l))
            sum(nil()) -> 0()
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/2,c_8/0,c_9/2,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:prod#(cons(x,l)) -> c_7(*#(x,prod(l)),prod#(l))
             -->_1 *#(s(x),s(y)) -> c_3(+#(*(x,y),+(x,y)),*#(x,y),+#(x,y)):3
             -->_2 prod#(cons(x,l)) -> c_7(*#(x,prod(l)),prod#(l)):1
          
          2:S:sum#(cons(x,l)) -> c_9(+#(x,sum(l)),sum#(l))
             -->_1 +#(s(x),s(y)) -> c_6(+#(x,y)):4
             -->_2 sum#(cons(x,l)) -> c_9(+#(x,sum(l)),sum#(l)):2
          
          3:W:*#(s(x),s(y)) -> c_3(+#(*(x,y),+(x,y)),*#(x,y),+#(x,y))
             -->_3 +#(s(x),s(y)) -> c_6(+#(x,y)):4
             -->_1 +#(s(x),s(y)) -> c_6(+#(x,y)):4
             -->_2 *#(s(x),s(y)) -> c_3(+#(*(x,y),+(x,y)),*#(x,y),+#(x,y)):3
          
          4:W:+#(s(x),s(y)) -> c_6(+#(x,y))
             -->_1 +#(s(x),s(y)) -> c_6(+#(x,y)):4
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: *#(s(x),s(y)) -> c_3(+#(*(x,y),+(x,y)),*#(x,y),+#(x,y))
          4: +#(s(x),s(y)) -> c_6(+#(x,y))
** Step 4.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            prod#(cons(x,l)) -> c_7(*#(x,prod(l)),prod#(l))
            sum#(cons(x,l)) -> c_9(+#(x,sum(l)),sum#(l))
        - Weak TRS:
            *(x,0()) -> 0()
            *(0(),x) -> 0()
            *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
            +(x,0()) -> x
            +(0(),x) -> x
            +(s(x),s(y)) -> s(s(+(x,y)))
            prod(cons(x,l)) -> *(x,prod(l))
            prod(nil()) -> s(0())
            sum(cons(x,l)) -> +(x,sum(l))
            sum(nil()) -> 0()
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/2,c_8/0,c_9/2,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:prod#(cons(x,l)) -> c_7(*#(x,prod(l)),prod#(l))
             -->_2 prod#(cons(x,l)) -> c_7(*#(x,prod(l)),prod#(l)):1
          
          2:S:sum#(cons(x,l)) -> c_9(+#(x,sum(l)),sum#(l))
             -->_2 sum#(cons(x,l)) -> c_9(+#(x,sum(l)),sum#(l)):2
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          prod#(cons(x,l)) -> c_7(prod#(l))
          sum#(cons(x,l)) -> c_9(sum#(l))
** Step 4.b:3: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            prod#(cons(x,l)) -> c_7(prod#(l))
            sum#(cons(x,l)) -> c_9(sum#(l))
        - Weak TRS:
            *(x,0()) -> 0()
            *(0(),x) -> 0()
            *(s(x),s(y)) -> s(+(*(x,y),+(x,y)))
            +(x,0()) -> x
            +(0(),x) -> x
            +(s(x),s(y)) -> s(s(+(x,y)))
            prod(cons(x,l)) -> *(x,prod(l))
            prod(nil()) -> s(0())
            sum(cons(x,l)) -> +(x,sum(l))
            sum(nil()) -> 0()
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/1,c_8/0,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          prod#(cons(x,l)) -> c_7(prod#(l))
          sum#(cons(x,l)) -> c_9(sum#(l))
** Step 4.b:4: Decompose WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            prod#(cons(x,l)) -> c_7(prod#(l))
            sum#(cons(x,l)) -> c_9(sum#(l))
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/1,c_8/0,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    + Details:
        We analyse the complexity of following sub-problems (R) and (S).
        Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
        
        Problem (R)
          - Strict DPs:
              prod#(cons(x,l)) -> c_7(prod#(l))
          - Weak DPs:
              sum#(cons(x,l)) -> c_9(sum#(l))
          - Signature:
              {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0
              ,c_6/1,c_7/1,c_8/0,c_9/1,c_10/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
        
        Problem (S)
          - Strict DPs:
              sum#(cons(x,l)) -> c_9(sum#(l))
          - Weak DPs:
              prod#(cons(x,l)) -> c_7(prod#(l))
          - Signature:
              {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0
              ,c_6/1,c_7/1,c_8/0,c_9/1,c_10/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
*** Step 4.b:4.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            prod#(cons(x,l)) -> c_7(prod#(l))
        - Weak DPs:
            sum#(cons(x,l)) -> c_9(sum#(l))
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/1,c_8/0,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:prod#(cons(x,l)) -> c_7(prod#(l))
             -->_1 prod#(cons(x,l)) -> c_7(prod#(l)):1
          
          2:W:sum#(cons(x,l)) -> c_9(sum#(l))
             -->_1 sum#(cons(x,l)) -> c_9(sum#(l)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: sum#(cons(x,l)) -> c_9(sum#(l))
*** Step 4.b:4.a:2: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            prod#(cons(x,l)) -> c_7(prod#(l))
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/1,c_8/0,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,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: prod#(cons(x,l)) -> c_7(prod#(l))
          
        The strictly oriented rules are moved into the weak component.
**** Step 4.b:4.a:2.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            prod#(cons(x,l)) -> c_7(prod#(l))
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/1,c_8/0,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,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_7) = {1}
        
        Following symbols are considered usable:
          {*#,+#,prod#,sum#}
        TcT has computed the following interpretation:
              p(*) = [4] x2 + [8]         
              p(+) = [1]                  
              p(0) = [0]                  
           p(cons) = [1] x1 + [1] x2 + [4]
            p(nil) = [0]                  
           p(prod) = [4]                  
              p(s) = [0]                  
            p(sum) = [2] x1 + [1]         
             p(*#) = [1] x1 + [2]         
             p(+#) = [2] x1 + [8]         
          p(prod#) = [4] x1 + [0]         
           p(sum#) = [1] x1 + [2]         
            p(c_1) = [0]                  
            p(c_2) = [1]                  
            p(c_3) = [2] x1 + [4]         
            p(c_4) = [2]                  
            p(c_5) = [1]                  
            p(c_6) = [2] x1 + [0]         
            p(c_7) = [1] x1 + [14]        
            p(c_8) = [1]                  
            p(c_9) = [4] x1 + [2]         
           p(c_10) = [1]                  
        
        Following rules are strictly oriented:
        prod#(cons(x,l)) = [4] l + [4] x + [16]
                         > [4] l + [14]        
                         = c_7(prod#(l))       
        
        
        Following rules are (at-least) weakly oriented:
        
**** Step 4.b:4.a:2.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            prod#(cons(x,l)) -> c_7(prod#(l))
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/1,c_8/0,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 4.b:4.a:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            prod#(cons(x,l)) -> c_7(prod#(l))
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/1,c_8/0,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:prod#(cons(x,l)) -> c_7(prod#(l))
             -->_1 prod#(cons(x,l)) -> c_7(prod#(l)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: prod#(cons(x,l)) -> c_7(prod#(l))
**** Step 4.b:4.a:2.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/1,c_8/0,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 4.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(cons(x,l)) -> c_9(sum#(l))
        - Weak DPs:
            prod#(cons(x,l)) -> c_7(prod#(l))
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/1,c_8/0,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:sum#(cons(x,l)) -> c_9(sum#(l))
             -->_1 sum#(cons(x,l)) -> c_9(sum#(l)):1
          
          2:W:prod#(cons(x,l)) -> c_7(prod#(l))
             -->_1 prod#(cons(x,l)) -> c_7(prod#(l)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: prod#(cons(x,l)) -> c_7(prod#(l))
*** Step 4.b:4.b:2: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(cons(x,l)) -> c_9(sum#(l))
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/1,c_8/0,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,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: sum#(cons(x,l)) -> c_9(sum#(l))
          
        The strictly oriented rules are moved into the weak component.
**** Step 4.b:4.b:2.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(cons(x,l)) -> c_9(sum#(l))
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/1,c_8/0,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,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_9) = {1}
        
        Following symbols are considered usable:
          {*#,+#,prod#,sum#}
        TcT has computed the following interpretation:
              p(*) = [2]                  
              p(+) = [2] x1 + [1] x2 + [1]
              p(0) = [1]                  
           p(cons) = [1] x1 + [1] x2 + [2]
            p(nil) = [1]                  
           p(prod) = [1] x1 + [0]         
              p(s) = [0]                  
            p(sum) = [1] x1 + [2]         
             p(*#) = [2]                  
             p(+#) = [1] x1 + [0]         
          p(prod#) = [1]                  
           p(sum#) = [2] x1 + [0]         
            p(c_1) = [0]                  
            p(c_2) = [0]                  
            p(c_3) = [1] x1 + [0]         
            p(c_4) = [8]                  
            p(c_5) = [4]                  
            p(c_6) = [4] x1 + [0]         
            p(c_7) = [2] x1 + [0]         
            p(c_8) = [0]                  
            p(c_9) = [1] x1 + [0]         
           p(c_10) = [1]                  
        
        Following rules are strictly oriented:
        sum#(cons(x,l)) = [2] l + [2] x + [4]
                        > [2] l + [0]        
                        = c_9(sum#(l))       
        
        
        Following rules are (at-least) weakly oriented:
        
**** Step 4.b:4.b:2.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            sum#(cons(x,l)) -> c_9(sum#(l))
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/1,c_8/0,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 4.b:4.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            sum#(cons(x,l)) -> c_9(sum#(l))
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/1,c_8/0,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:sum#(cons(x,l)) -> c_9(sum#(l))
             -->_1 sum#(cons(x,l)) -> c_9(sum#(l)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: sum#(cons(x,l)) -> c_9(sum#(l))
**** Step 4.b:4.b:2.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {*/2,+/2,prod/1,sum/1,*#/2,+#/2,prod#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1
            ,c_7/1,c_8/0,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,prod#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

MAYBE