* Step 1: DependencyPairs WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict TRS:
            sum(cons(0(),x),y) -> sum(x,y)
            sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y))
            sum(nil(),y) -> y
            weight(cons(n,cons(m,x))) -> weight(sum(cons(n,cons(m,x)),cons(0(),x)))
            weight(cons(n,nil())) -> n
        - Signature:
            {sum/2,weight/1} / {0/0,cons/2,nil/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum,weight} and constructors {0,cons,nil,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          sum#(cons(0(),x),y) -> c_1(sum#(x,y))
          sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y)))
          sum#(nil(),y) -> c_3()
          weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
                                           ,sum#(cons(n,cons(m,x)),cons(0(),x)))
          weight#(cons(n,nil())) -> c_5()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            sum#(cons(0(),x),y) -> c_1(sum#(x,y))
            sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y)))
            sum#(nil(),y) -> c_3()
            weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
                                             ,sum#(cons(n,cons(m,x)),cons(0(),x)))
            weight#(cons(n,nil())) -> c_5()
        - Weak TRS:
            sum(cons(0(),x),y) -> sum(x,y)
            sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y))
            sum(nil(),y) -> y
            weight(cons(n,cons(m,x))) -> weight(sum(cons(n,cons(m,x)),cons(0(),x)))
            weight(cons(n,nil())) -> n
        - Signature:
            {sum/2,weight/1,sum#/2,weight#/1} / {0/0,cons/2,nil/0,s/1,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,weight#} and constructors {0,cons,nil,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          sum(cons(0(),x),y) -> sum(x,y)
          sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y))
          sum(nil(),y) -> y
          sum#(cons(0(),x),y) -> c_1(sum#(x,y))
          sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y)))
          sum#(nil(),y) -> c_3()
          weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
                                           ,sum#(cons(n,cons(m,x)),cons(0(),x)))
          weight#(cons(n,nil())) -> c_5()
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            sum#(cons(0(),x),y) -> c_1(sum#(x,y))
            sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y)))
            sum#(nil(),y) -> c_3()
            weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
                                             ,sum#(cons(n,cons(m,x)),cons(0(),x)))
            weight#(cons(n,nil())) -> c_5()
        - Weak TRS:
            sum(cons(0(),x),y) -> sum(x,y)
            sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y))
            sum(nil(),y) -> y
        - Signature:
            {sum/2,weight/1,sum#/2,weight#/1} / {0/0,cons/2,nil/0,s/1,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,weight#} and constructors {0,cons,nil,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {3,5}
        by application of
          Pre({3,5}) = {1,4}.
        Here rules are labelled as follows:
          1: sum#(cons(0(),x),y) -> c_1(sum#(x,y))
          2: sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y)))
          3: sum#(nil(),y) -> c_3()
          4: weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
                                              ,sum#(cons(n,cons(m,x)),cons(0(),x)))
          5: weight#(cons(n,nil())) -> c_5()
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            sum#(cons(0(),x),y) -> c_1(sum#(x,y))
            sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y)))
            weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
                                             ,sum#(cons(n,cons(m,x)),cons(0(),x)))
        - Weak DPs:
            sum#(nil(),y) -> c_3()
            weight#(cons(n,nil())) -> c_5()
        - Weak TRS:
            sum(cons(0(),x),y) -> sum(x,y)
            sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y))
            sum(nil(),y) -> y
        - Signature:
            {sum/2,weight/1,sum#/2,weight#/1} / {0/0,cons/2,nil/0,s/1,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,weight#} and constructors {0,cons,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:sum#(cons(0(),x),y) -> c_1(sum#(x,y))
             -->_1 sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y))):2
             -->_1 sum#(nil(),y) -> c_3():4
             -->_1 sum#(cons(0(),x),y) -> c_1(sum#(x,y)):1
          
          2:S:sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y)))
             -->_1 sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y))):2
             -->_1 sum#(cons(0(),x),y) -> c_1(sum#(x,y)):1
          
          3:S:weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
                                               ,sum#(cons(n,cons(m,x)),cons(0(),x)))
             -->_1 weight#(cons(n,nil())) -> c_5():5
             -->_1 weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
                                                    ,sum#(cons(n,cons(m,x)),cons(0(),x))):3
             -->_2 sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y))):2
             -->_2 sum#(cons(0(),x),y) -> c_1(sum#(x,y)):1
          
          4:W:sum#(nil(),y) -> c_3()
             
          
          5:W:weight#(cons(n,nil())) -> c_5()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: weight#(cons(n,nil())) -> c_5()
          4: sum#(nil(),y) -> c_3()
* Step 5: Decompose WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            sum#(cons(0(),x),y) -> c_1(sum#(x,y))
            sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y)))
            weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
                                             ,sum#(cons(n,cons(m,x)),cons(0(),x)))
        - Weak TRS:
            sum(cons(0(),x),y) -> sum(x,y)
            sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y))
            sum(nil(),y) -> y
        - Signature:
            {sum/2,weight/1,sum#/2,weight#/1} / {0/0,cons/2,nil/0,s/1,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,weight#} 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:
              sum#(cons(0(),x),y) -> c_1(sum#(x,y))
              sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y)))
          - Weak DPs:
              weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
                                               ,sum#(cons(n,cons(m,x)),cons(0(),x)))
          - Weak TRS:
              sum(cons(0(),x),y) -> sum(x,y)
              sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y))
              sum(nil(),y) -> y
          - Signature:
              {sum/2,weight/1,sum#/2,weight#/1} / {0/0,cons/2,nil/0,s/1,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {sum#,weight#} and constructors {0,cons,nil,s}
        
        Problem (S)
          - Strict DPs:
              weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
                                               ,sum#(cons(n,cons(m,x)),cons(0(),x)))
          - Weak DPs:
              sum#(cons(0(),x),y) -> c_1(sum#(x,y))
              sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y)))
          - Weak TRS:
              sum(cons(0(),x),y) -> sum(x,y)
              sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y))
              sum(nil(),y) -> y
          - Signature:
              {sum/2,weight/1,sum#/2,weight#/1} / {0/0,cons/2,nil/0,s/1,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {sum#,weight#} and constructors {0,cons,nil,s}
** Step 5.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            sum#(cons(0(),x),y) -> c_1(sum#(x,y))
            sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y)))
        - Weak DPs:
            weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
                                             ,sum#(cons(n,cons(m,x)),cons(0(),x)))
        - Weak TRS:
            sum(cons(0(),x),y) -> sum(x,y)
            sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y))
            sum(nil(),y) -> y
        - Signature:
            {sum/2,weight/1,sum#/2,weight#/1} / {0/0,cons/2,nil/0,s/1,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,weight#} and constructors {0,cons,nil,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: sum#(cons(0(),x),y) -> c_1(sum#(x,y))
          
        The strictly oriented rules are moved into the weak component.
*** Step 5.a:1.a:1: NaturalMI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            sum#(cons(0(),x),y) -> c_1(sum#(x,y))
            sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y)))
        - Weak DPs:
            weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
                                             ,sum#(cons(n,cons(m,x)),cons(0(),x)))
        - Weak TRS:
            sum(cons(0(),x),y) -> sum(x,y)
            sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y))
            sum(nil(),y) -> y
        - Signature:
            {sum/2,weight/1,sum#/2,weight#/1} / {0/0,cons/2,nil/0,s/1,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,weight#} and constructors {0,cons,nil,s}
    + Applied Processor:
        NaturalMI {miDimension = 2, miDegree = 2, 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_1) = {1},
          uargs(c_2) = {1},
          uargs(c_4) = {1,2}
        
        Following symbols are considered usable:
          {sum,sum#,weight#}
        TcT has computed the following interpretation:
                p(0) = [2]                      
                       [0]                      
             p(cons) = [1 1] x2 + [0]           
                       [0 1]      [1]           
              p(nil) = [0]                      
                       [1]                      
                p(s) = [1]                      
                       [0]                      
              p(sum) = [1 0] x2 + [0]           
                       [0 1]      [0]           
           p(weight) = [2 0] x1 + [0]           
                       [0 1]      [0]           
             p(sum#) = [0 1] x1 + [0 0] x2 + [2]
                       [0 0]      [3 2]      [0]
          p(weight#) = [2 2] x1 + [0]           
                       [0 0]      [1]           
              p(c_1) = [1 0] x1 + [0]           
                       [0 0]      [0]           
              p(c_2) = [1 0] x1 + [0]           
                       [0 1]      [0]           
              p(c_3) = [0]                      
                       [1]                      
              p(c_4) = [1 0] x1 + [1 0] x2 + [0]
                       [0 0]      [0 0]      [0]
              p(c_5) = [2]                      
                       [0]                      
        
        Following rules are strictly oriented:
        sum#(cons(0(),x),y) = [0 1] x + [0 0] y + [3]
                              [0 0]     [3 2]     [0]
                            > [0 1] x + [2]          
                              [0 0]     [0]          
                            = c_1(sum#(x,y))         
        
        
        Following rules are (at-least) weakly oriented:
        sum#(cons(s(n),x),cons(m,y)) =  [0 1] x + [0 0] y + [3]                                                             
                                        [0 0]     [3 5]     [2]                                                             
                                     >= [0 1] x + [0 0] y + [3]                                                             
                                        [0 0]     [3 5]     [2]                                                             
                                     =  c_2(sum#(cons(n,x),cons(s(m),y)))                                                   
        
          weight#(cons(n,cons(m,x))) =  [2 6] x + [6]                                                                       
                                        [0 0]     [1]                                                                       
                                     >= [2 5] x + [6]                                                                       
                                        [0 0]     [0]                                                                       
                                     =  c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x))),sum#(cons(n,cons(m,x)),cons(0(),x)))
        
                  sum(cons(0(),x),y) =  [1 0] y + [0]                                                                       
                                        [0 1]     [0]                                                                       
                                     >= [1 0] y + [0]                                                                       
                                        [0 1]     [0]                                                                       
                                     =  sum(x,y)                                                                            
        
         sum(cons(s(n),x),cons(m,y)) =  [1 1] y + [0]                                                                       
                                        [0 1]     [1]                                                                       
                                     >= [1 1] y + [0]                                                                       
                                        [0 1]     [1]                                                                       
                                     =  sum(cons(n,x),cons(s(m),y))                                                         
        
                        sum(nil(),y) =  [1 0] y + [0]                                                                       
                                        [0 1]     [0]                                                                       
                                     >= [1 0] y + [0]                                                                       
                                        [0 1]     [0]                                                                       
                                     =  y                                                                                   
        
*** Step 5.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y)))
        - Weak DPs:
            sum#(cons(0(),x),y) -> c_1(sum#(x,y))
            weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
                                             ,sum#(cons(n,cons(m,x)),cons(0(),x)))
        - Weak TRS:
            sum(cons(0(),x),y) -> sum(x,y)
            sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y))
            sum(nil(),y) -> y
        - Signature:
            {sum/2,weight/1,sum#/2,weight#/1} / {0/0,cons/2,nil/0,s/1,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,weight#} and constructors {0,cons,nil,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 5.a:1.b:1: DecomposeDG WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y)))
        - Weak DPs:
            sum#(cons(0(),x),y) -> c_1(sum#(x,y))
            weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
                                             ,sum#(cons(n,cons(m,x)),cons(0(),x)))
        - Weak TRS:
            sum(cons(0(),x),y) -> sum(x,y)
            sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y))
            sum(nil(),y) -> y
        - Signature:
            {sum/2,weight/1,sum#/2,weight#/1} / {0/0,cons/2,nil/0,s/1,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,weight#} 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
          weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
                                           ,sum#(cons(n,cons(m,x)),cons(0(),x)))
        and a lower component
          sum#(cons(0(),x),y) -> c_1(sum#(x,y))
          sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y)))
        Further, following extension rules are added to the lower component.
          weight#(cons(n,cons(m,x))) -> sum#(cons(n,cons(m,x)),cons(0(),x))
          weight#(cons(n,cons(m,x))) -> weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
**** Step 5.a:1.b:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
                                             ,sum#(cons(n,cons(m,x)),cons(0(),x)))
        - Weak TRS:
            sum(cons(0(),x),y) -> sum(x,y)
            sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y))
            sum(nil(),y) -> y
        - Signature:
            {sum/2,weight/1,sum#/2,weight#/1} / {0/0,cons/2,nil/0,s/1,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,weight#} 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: weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
                                              ,sum#(cons(n,cons(m,x)),cons(0(),x)))
          
        The strictly oriented rules are moved into the weak component.
***** Step 5.a:1.b:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
                                             ,sum#(cons(n,cons(m,x)),cons(0(),x)))
        - Weak TRS:
            sum(cons(0(),x),y) -> sum(x,y)
            sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y))
            sum(nil(),y) -> y
        - Signature:
            {sum/2,weight/1,sum#/2,weight#/1} / {0/0,cons/2,nil/0,s/1,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,weight#} 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_4) = {1}
        
        Following symbols are considered usable:
          {sum,sum#,weight#}
        TcT has computed the following interpretation:
                p(0) = [1]                  
             p(cons) = [1] x1 + [1] x2 + [6]
              p(nil) = [1]                  
                p(s) = [0]                  
              p(sum) = [1] x2 + [1]         
           p(weight) = [2]                  
             p(sum#) = [1]                  
          p(weight#) = [1] x1 + [0]         
              p(c_1) = [2]                  
              p(c_2) = [4] x1 + [1]         
              p(c_3) = [0]                  
              p(c_4) = [1] x1 + [3] x2 + [0]
              p(c_5) = [1]                  
        
        Following rules are strictly oriented:
        weight#(cons(n,cons(m,x))) = [1] m + [1] n + [1] x + [12]                                                        
                                   > [1] x + [11]                                                                        
                                   = c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x))),sum#(cons(n,cons(m,x)),cons(0(),x)))
        
        
        Following rules are (at-least) weakly oriented:
                 sum(cons(0(),x),y) =  [1] y + [1]                
                                    >= [1] y + [1]                
                                    =  sum(x,y)                   
        
        sum(cons(s(n),x),cons(m,y)) =  [1] m + [1] y + [7]        
                                    >= [1] y + [7]                
                                    =  sum(cons(n,x),cons(s(m),y))
        
                       sum(nil(),y) =  [1] y + [1]                
                                    >= [1] y + [0]                
                                    =  y                          
        
***** Step 5.a:1.b:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
                                             ,sum#(cons(n,cons(m,x)),cons(0(),x)))
        - Weak TRS:
            sum(cons(0(),x),y) -> sum(x,y)
            sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y))
            sum(nil(),y) -> y
        - Signature:
            {sum/2,weight/1,sum#/2,weight#/1} / {0/0,cons/2,nil/0,s/1,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,weight#} and constructors {0,cons,nil,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

***** Step 5.a:1.b:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
                                             ,sum#(cons(n,cons(m,x)),cons(0(),x)))
        - Weak TRS:
            sum(cons(0(),x),y) -> sum(x,y)
            sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y))
            sum(nil(),y) -> y
        - Signature:
            {sum/2,weight/1,sum#/2,weight#/1} / {0/0,cons/2,nil/0,s/1,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,weight#} and constructors {0,cons,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
                                               ,sum#(cons(n,cons(m,x)),cons(0(),x)))
             -->_1 weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
                                                    ,sum#(cons(n,cons(m,x)),cons(0(),x))):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
                                              ,sum#(cons(n,cons(m,x)),cons(0(),x)))
***** Step 5.a:1.b:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            sum(cons(0(),x),y) -> sum(x,y)
            sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y))
            sum(nil(),y) -> y
        - Signature:
            {sum/2,weight/1,sum#/2,weight#/1} / {0/0,cons/2,nil/0,s/1,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,weight#} and constructors {0,cons,nil,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

**** Step 5.a:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y)))
        - Weak DPs:
            sum#(cons(0(),x),y) -> c_1(sum#(x,y))
            weight#(cons(n,cons(m,x))) -> sum#(cons(n,cons(m,x)),cons(0(),x))
            weight#(cons(n,cons(m,x))) -> weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
        - Weak TRS:
            sum(cons(0(),x),y) -> sum(x,y)
            sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y))
            sum(nil(),y) -> y
        - Signature:
            {sum/2,weight/1,sum#/2,weight#/1} / {0/0,cons/2,nil/0,s/1,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,weight#} and constructors {0,cons,nil,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y)))
          
        The strictly oriented rules are moved into the weak component.
***** Step 5.a:1.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y)))
        - Weak DPs:
            sum#(cons(0(),x),y) -> c_1(sum#(x,y))
            weight#(cons(n,cons(m,x))) -> sum#(cons(n,cons(m,x)),cons(0(),x))
            weight#(cons(n,cons(m,x))) -> weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
        - Weak TRS:
            sum(cons(0(),x),y) -> sum(x,y)
            sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y))
            sum(nil(),y) -> y
        - Signature:
            {sum/2,weight/1,sum#/2,weight#/1} / {0/0,cons/2,nil/0,s/1,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,weight#} and constructors {0,cons,nil,s}
    + Applied Processor:
        NaturalMI {miDimension = 2, miDegree = 2, 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_1) = {1},
          uargs(c_2) = {1}
        
        Following symbols are considered usable:
          {sum,sum#,weight#}
        TcT has computed the following interpretation:
                p(0) = [0]                      
                       [0]                      
             p(cons) = [0 1] x1 + [1 2] x2 + [2]
                       [0 1]      [0 1]      [0]
              p(nil) = [1]                      
                       [0]                      
                p(s) = [0 1] x1 + [0]           
                       [0 1]      [2]           
              p(sum) = [0 1] x1 + [1 0] x2 + [2]
                       [0 2]      [0 2]      [2]
           p(weight) = [0 0] x1 + [2]           
                       [2 1]      [2]           
             p(sum#) = [1 0] x1 + [0]           
                       [0 0]      [0]           
          p(weight#) = [1 0] x1 + [0]           
                       [0 0]      [0]           
              p(c_1) = [1 0] x1 + [0]           
                       [0 0]      [0]           
              p(c_2) = [1 0] x1 + [0]           
                       [0 0]      [0]           
              p(c_3) = [0]                      
                       [0]                      
              p(c_4) = [1]                      
                       [2]                      
              p(c_5) = [0]                      
                       [1]                      
        
        Following rules are strictly oriented:
        sum#(cons(s(n),x),cons(m,y)) = [0 1] n + [1 2] x + [4]          
                                       [0 0]     [0 0]     [0]          
                                     > [0 1] n + [1 2] x + [2]          
                                       [0 0]     [0 0]     [0]          
                                     = c_2(sum#(cons(n,x),cons(s(m),y)))
        
        
        Following rules are (at-least) weakly oriented:
                sum#(cons(0(),x),y) =  [1 2] x + [2]                              
                                       [0 0]     [0]                              
                                    >= [1 0] x + [0]                              
                                       [0 0]     [0]                              
                                    =  c_1(sum#(x,y))                             
        
         weight#(cons(n,cons(m,x))) =  [0 3] m + [0 1] n + [1 4] x + [4]          
                                       [0 0]     [0 0]     [0 0]     [0]          
                                    >= [0 3] m + [0 1] n + [1 4] x + [4]          
                                       [0 0]     [0 0]     [0 0]     [0]          
                                    =  sum#(cons(n,cons(m,x)),cons(0(),x))        
        
         weight#(cons(n,cons(m,x))) =  [0 3] m + [0 1] n + [1 4] x + [4]          
                                       [0 0]     [0 0]     [0 0]     [0]          
                                    >= [0 1] m + [0 1] n + [1 3] x + [4]          
                                       [0 0]     [0 0]     [0 0]     [0]          
                                    =  weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
        
                 sum(cons(0(),x),y) =  [0 1] x + [1 0] y + [2]                    
                                       [0 2]     [0 2]     [2]                    
                                    >= [0 1] x + [1 0] y + [2]                    
                                       [0 2]     [0 2]     [2]                    
                                    =  sum(x,y)                                   
        
        sum(cons(s(n),x),cons(m,y)) =  [0 1] m + [0 1] n + [0 1] x + [1 2] y + [6]
                                       [0 2]     [0 2]     [0 2]     [0 2]     [6]
                                    >= [0 1] m + [0 1] n + [0 1] x + [1 2] y + [6]
                                       [0 2]     [0 2]     [0 2]     [0 2]     [6]
                                    =  sum(cons(n,x),cons(s(m),y))                
        
                       sum(nil(),y) =  [1 0] y + [2]                              
                                       [0 2]     [2]                              
                                    >= [1 0] y + [0]                              
                                       [0 1]     [0]                              
                                    =  y                                          
        
***** Step 5.a:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            sum#(cons(0(),x),y) -> c_1(sum#(x,y))
            sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y)))
            weight#(cons(n,cons(m,x))) -> sum#(cons(n,cons(m,x)),cons(0(),x))
            weight#(cons(n,cons(m,x))) -> weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
        - Weak TRS:
            sum(cons(0(),x),y) -> sum(x,y)
            sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y))
            sum(nil(),y) -> y
        - Signature:
            {sum/2,weight/1,sum#/2,weight#/1} / {0/0,cons/2,nil/0,s/1,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,weight#} and constructors {0,cons,nil,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

***** Step 5.a:1.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            sum#(cons(0(),x),y) -> c_1(sum#(x,y))
            sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y)))
            weight#(cons(n,cons(m,x))) -> sum#(cons(n,cons(m,x)),cons(0(),x))
            weight#(cons(n,cons(m,x))) -> weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
        - Weak TRS:
            sum(cons(0(),x),y) -> sum(x,y)
            sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y))
            sum(nil(),y) -> y
        - Signature:
            {sum/2,weight/1,sum#/2,weight#/1} / {0/0,cons/2,nil/0,s/1,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,weight#} and constructors {0,cons,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:sum#(cons(0(),x),y) -> c_1(sum#(x,y))
             -->_1 sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y))):2
             -->_1 sum#(cons(0(),x),y) -> c_1(sum#(x,y)):1
          
          2:W:sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y)))
             -->_1 sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y))):2
             -->_1 sum#(cons(0(),x),y) -> c_1(sum#(x,y)):1
          
          3:W:weight#(cons(n,cons(m,x))) -> sum#(cons(n,cons(m,x)),cons(0(),x))
             -->_1 sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y))):2
             -->_1 sum#(cons(0(),x),y) -> c_1(sum#(x,y)):1
          
          4:W:weight#(cons(n,cons(m,x))) -> weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
             -->_1 weight#(cons(n,cons(m,x))) -> weight#(sum(cons(n,cons(m,x)),cons(0(),x))):4
             -->_1 weight#(cons(n,cons(m,x))) -> sum#(cons(n,cons(m,x)),cons(0(),x)):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: weight#(cons(n,cons(m,x))) -> weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
          3: weight#(cons(n,cons(m,x))) -> sum#(cons(n,cons(m,x)),cons(0(),x))
          1: sum#(cons(0(),x),y) -> c_1(sum#(x,y))
          2: sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y)))
***** Step 5.a:1.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            sum(cons(0(),x),y) -> sum(x,y)
            sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y))
            sum(nil(),y) -> y
        - Signature:
            {sum/2,weight/1,sum#/2,weight#/1} / {0/0,cons/2,nil/0,s/1,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,weight#} and constructors {0,cons,nil,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
                                             ,sum#(cons(n,cons(m,x)),cons(0(),x)))
        - Weak DPs:
            sum#(cons(0(),x),y) -> c_1(sum#(x,y))
            sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y)))
        - Weak TRS:
            sum(cons(0(),x),y) -> sum(x,y)
            sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y))
            sum(nil(),y) -> y
        - Signature:
            {sum/2,weight/1,sum#/2,weight#/1} / {0/0,cons/2,nil/0,s/1,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,weight#} and constructors {0,cons,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
                                               ,sum#(cons(n,cons(m,x)),cons(0(),x)))
             -->_2 sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y))):3
             -->_2 sum#(cons(0(),x),y) -> c_1(sum#(x,y)):2
             -->_1 weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
                                                    ,sum#(cons(n,cons(m,x)),cons(0(),x))):1
          
          2:W:sum#(cons(0(),x),y) -> c_1(sum#(x,y))
             -->_1 sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y))):3
             -->_1 sum#(cons(0(),x),y) -> c_1(sum#(x,y)):2
          
          3:W:sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y)))
             -->_1 sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y))):3
             -->_1 sum#(cons(0(),x),y) -> c_1(sum#(x,y)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: sum#(cons(s(n),x),cons(m,y)) -> c_2(sum#(cons(n,x),cons(s(m),y)))
          2: sum#(cons(0(),x),y) -> c_1(sum#(x,y))
** Step 5.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
                                             ,sum#(cons(n,cons(m,x)),cons(0(),x)))
        - Weak TRS:
            sum(cons(0(),x),y) -> sum(x,y)
            sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y))
            sum(nil(),y) -> y
        - Signature:
            {sum/2,weight/1,sum#/2,weight#/1} / {0/0,cons/2,nil/0,s/1,c_1/1,c_2/1,c_3/0,c_4/2,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,weight#} and constructors {0,cons,nil,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
                                               ,sum#(cons(n,cons(m,x)),cons(0(),x)))
             -->_1 weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x)))
                                                    ,sum#(cons(n,cons(m,x)),cons(0(),x))):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x))))
** Step 5.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x))))
        - Weak TRS:
            sum(cons(0(),x),y) -> sum(x,y)
            sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y))
            sum(nil(),y) -> y
        - Signature:
            {sum/2,weight/1,sum#/2,weight#/1} / {0/0,cons/2,nil/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,weight#} 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: weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x))))
          
        The strictly oriented rules are moved into the weak component.
*** Step 5.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x))))
        - Weak TRS:
            sum(cons(0(),x),y) -> sum(x,y)
            sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y))
            sum(nil(),y) -> y
        - Signature:
            {sum/2,weight/1,sum#/2,weight#/1} / {0/0,cons/2,nil/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,weight#} 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_4) = {1}
        
        Following symbols are considered usable:
          {sum,sum#,weight#}
        TcT has computed the following interpretation:
                p(0) = [8]                  
             p(cons) = [1] x2 + [1]         
              p(nil) = [0]                  
                p(s) = [2]                  
              p(sum) = [1] x2 + [0]         
           p(weight) = [1]                  
             p(sum#) = [1] x1 + [1] x2 + [1]
          p(weight#) = [8] x1 + [0]         
              p(c_1) = [2]                  
              p(c_2) = [8] x1 + [0]         
              p(c_3) = [1]                  
              p(c_4) = [1] x1 + [0]         
              p(c_5) = [0]                  
        
        Following rules are strictly oriented:
        weight#(cons(n,cons(m,x))) = [8] x + [16]                                    
                                   > [8] x + [8]                                     
                                   = c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x))))
        
        
        Following rules are (at-least) weakly oriented:
                 sum(cons(0(),x),y) =  [1] y + [0]                
                                    >= [1] y + [0]                
                                    =  sum(x,y)                   
        
        sum(cons(s(n),x),cons(m,y)) =  [1] y + [1]                
                                    >= [1] y + [1]                
                                    =  sum(cons(n,x),cons(s(m),y))
        
                       sum(nil(),y) =  [1] y + [0]                
                                    >= [1] y + [0]                
                                    =  y                          
        
*** Step 5.b:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x))))
        - Weak TRS:
            sum(cons(0(),x),y) -> sum(x,y)
            sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y))
            sum(nil(),y) -> y
        - Signature:
            {sum/2,weight/1,sum#/2,weight#/1} / {0/0,cons/2,nil/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,weight#} and constructors {0,cons,nil,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 5.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x))))
        - Weak TRS:
            sum(cons(0(),x),y) -> sum(x,y)
            sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y))
            sum(nil(),y) -> y
        - Signature:
            {sum/2,weight/1,sum#/2,weight#/1} / {0/0,cons/2,nil/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,weight#} and constructors {0,cons,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x))))
             -->_1 weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x)))):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: weight#(cons(n,cons(m,x))) -> c_4(weight#(sum(cons(n,cons(m,x)),cons(0(),x))))
*** Step 5.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            sum(cons(0(),x),y) -> sum(x,y)
            sum(cons(s(n),x),cons(m,y)) -> sum(cons(n,x),cons(s(m),y))
            sum(nil(),y) -> y
        - Signature:
            {sum/2,weight/1,sum#/2,weight#/1} / {0/0,cons/2,nil/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,weight#} and constructors {0,cons,nil,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^3))