* Step 1: Sum WORST_CASE(Omega(n^1),O(n^2))
    + Considered Problem:
        - Strict TRS:
            app(l,nil()) -> l
            app(cons(x,l),k) -> cons(x,app(l,k))
            app(nil(),k) -> k
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
            sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
            sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
            sum(cons(x,nil())) -> cons(x,nil())
        - Signature:
            {app/2,plus/2,sum/1} / {0/0,cons/2,nil/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app,plus,sum} and constructors {0,cons,nil,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            app(l,nil()) -> l
            app(cons(x,l),k) -> cons(x,app(l,k))
            app(nil(),k) -> k
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
            sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
            sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
            sum(cons(x,nil())) -> cons(x,nil())
        - Signature:
            {app/2,plus/2,sum/1} / {0/0,cons/2,nil/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app,plus,sum} and constructors {0,cons,nil,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          app(y,z){y -> cons(x,y)} =
            app(cons(x,y),z) ->^+ cons(x,app(y,z))
              = C[app(y,z) = app(y,z){}]

** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            app(l,nil()) -> l
            app(cons(x,l),k) -> cons(x,app(l,k))
            app(nil(),k) -> k
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
            sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
            sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
            sum(cons(x,nil())) -> cons(x,nil())
        - Signature:
            {app/2,plus/2,sum/1} / {0/0,cons/2,nil/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app,plus,sum} and constructors {0,cons,nil,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          app#(l,nil()) -> c_1()
          app#(cons(x,l),k) -> c_2(app#(l,k))
          app#(nil(),k) -> c_3()
          plus#(0(),y) -> c_4()
          plus#(s(x),y) -> c_5(plus#(x,y))
          sum#(app(l,cons(x,cons(y,k)))) -> c_6(sum#(app(l,sum(cons(x,cons(y,k)))))
                                               ,app#(l,sum(cons(x,cons(y,k))))
                                               ,sum#(cons(x,cons(y,k))))
          sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
          sum#(cons(x,nil())) -> c_8()
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: UsableRules WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            app#(l,nil()) -> c_1()
            app#(cons(x,l),k) -> c_2(app#(l,k))
            app#(nil(),k) -> c_3()
            plus#(0(),y) -> c_4()
            plus#(s(x),y) -> c_5(plus#(x,y))
            sum#(app(l,cons(x,cons(y,k)))) -> c_6(sum#(app(l,sum(cons(x,cons(y,k)))))
                                                 ,app#(l,sum(cons(x,cons(y,k))))
                                                 ,sum#(cons(x,cons(y,k))))
            sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
            sum#(cons(x,nil())) -> c_8()
        - Weak TRS:
            app(l,nil()) -> l
            app(cons(x,l),k) -> cons(x,app(l,k))
            app(nil(),k) -> k
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
            sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
            sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
            sum(cons(x,nil())) -> cons(x,nil())
        - Signature:
            {app/2,plus/2,sum/1,app#/2,plus#/2,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/3,c_7/2
            ,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,plus#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          plus(0(),y) -> y
          plus(s(x),y) -> s(plus(x,y))
          app#(l,nil()) -> c_1()
          app#(cons(x,l),k) -> c_2(app#(l,k))
          app#(nil(),k) -> c_3()
          plus#(0(),y) -> c_4()
          plus#(s(x),y) -> c_5(plus#(x,y))
          sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
          sum#(cons(x,nil())) -> c_8()
** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            app#(l,nil()) -> c_1()
            app#(cons(x,l),k) -> c_2(app#(l,k))
            app#(nil(),k) -> c_3()
            plus#(0(),y) -> c_4()
            plus#(s(x),y) -> c_5(plus#(x,y))
            sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
            sum#(cons(x,nil())) -> c_8()
        - Weak TRS:
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {app/2,plus/2,sum/1,app#/2,plus#/2,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/3,c_7/2
            ,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,plus#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,3,4,7}
        by application of
          Pre({1,3,4,7}) = {2,5,6}.
        Here rules are labelled as follows:
          1: app#(l,nil()) -> c_1()
          2: app#(cons(x,l),k) -> c_2(app#(l,k))
          3: app#(nil(),k) -> c_3()
          4: plus#(0(),y) -> c_4()
          5: plus#(s(x),y) -> c_5(plus#(x,y))
          6: sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
          7: sum#(cons(x,nil())) -> c_8()
** Step 1.b:4: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            app#(cons(x,l),k) -> c_2(app#(l,k))
            plus#(s(x),y) -> c_5(plus#(x,y))
            sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
        - Weak DPs:
            app#(l,nil()) -> c_1()
            app#(nil(),k) -> c_3()
            plus#(0(),y) -> c_4()
            sum#(cons(x,nil())) -> c_8()
        - Weak TRS:
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {app/2,plus/2,sum/1,app#/2,plus#/2,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/3,c_7/2
            ,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,plus#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:app#(cons(x,l),k) -> c_2(app#(l,k))
             -->_1 app#(nil(),k) -> c_3():5
             -->_1 app#(l,nil()) -> c_1():4
             -->_1 app#(cons(x,l),k) -> c_2(app#(l,k)):1
          
          2:S:plus#(s(x),y) -> c_5(plus#(x,y))
             -->_1 plus#(0(),y) -> c_4():6
             -->_1 plus#(s(x),y) -> c_5(plus#(x,y)):2
          
          3:S:sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
             -->_1 sum#(cons(x,nil())) -> c_8():7
             -->_2 plus#(0(),y) -> c_4():6
             -->_1 sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y)):3
             -->_2 plus#(s(x),y) -> c_5(plus#(x,y)):2
          
          4:W:app#(l,nil()) -> c_1()
             
          
          5:W:app#(nil(),k) -> c_3()
             
          
          6:W:plus#(0(),y) -> c_4()
             
          
          7:W:sum#(cons(x,nil())) -> c_8()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          7: sum#(cons(x,nil())) -> c_8()
          6: plus#(0(),y) -> c_4()
          4: app#(l,nil()) -> c_1()
          5: app#(nil(),k) -> c_3()
** Step 1.b:5: Decompose WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            app#(cons(x,l),k) -> c_2(app#(l,k))
            plus#(s(x),y) -> c_5(plus#(x,y))
            sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
        - Weak TRS:
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {app/2,plus/2,sum/1,app#/2,plus#/2,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/3,c_7/2
            ,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,plus#,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:
              app#(cons(x,l),k) -> c_2(app#(l,k))
          - Weak DPs:
              plus#(s(x),y) -> c_5(plus#(x,y))
              sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
          - Weak TRS:
              plus(0(),y) -> y
              plus(s(x),y) -> s(plus(x,y))
          - Signature:
              {app/2,plus/2,sum/1,app#/2,plus#/2,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/3
              ,c_7/2,c_8/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {app#,plus#,sum#} and constructors {0,cons,nil,s}
        
        Problem (S)
          - Strict DPs:
              plus#(s(x),y) -> c_5(plus#(x,y))
              sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
          - Weak DPs:
              app#(cons(x,l),k) -> c_2(app#(l,k))
          - Weak TRS:
              plus(0(),y) -> y
              plus(s(x),y) -> s(plus(x,y))
          - Signature:
              {app/2,plus/2,sum/1,app#/2,plus#/2,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/3
              ,c_7/2,c_8/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {app#,plus#,sum#} and constructors {0,cons,nil,s}
*** Step 1.b:5.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            app#(cons(x,l),k) -> c_2(app#(l,k))
        - Weak DPs:
            plus#(s(x),y) -> c_5(plus#(x,y))
            sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
        - Weak TRS:
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {app/2,plus/2,sum/1,app#/2,plus#/2,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/3,c_7/2
            ,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,plus#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:app#(cons(x,l),k) -> c_2(app#(l,k))
             -->_1 app#(cons(x,l),k) -> c_2(app#(l,k)):1
          
          2:W:plus#(s(x),y) -> c_5(plus#(x,y))
             -->_1 plus#(s(x),y) -> c_5(plus#(x,y)):2
          
          3:W:sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
             -->_2 plus#(s(x),y) -> c_5(plus#(x,y)):2
             -->_1 sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y)):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
          2: plus#(s(x),y) -> c_5(plus#(x,y))
*** Step 1.b:5.a:2: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            app#(cons(x,l),k) -> c_2(app#(l,k))
        - Weak TRS:
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {app/2,plus/2,sum/1,app#/2,plus#/2,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/3,c_7/2
            ,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,plus#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          app#(cons(x,l),k) -> c_2(app#(l,k))
*** Step 1.b:5.a:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            app#(cons(x,l),k) -> c_2(app#(l,k))
        - Signature:
            {app/2,plus/2,sum/1,app#/2,plus#/2,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/3,c_7/2
            ,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,plus#,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: app#(cons(x,l),k) -> c_2(app#(l,k))
          
        The strictly oriented rules are moved into the weak component.
**** Step 1.b:5.a:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            app#(cons(x,l),k) -> c_2(app#(l,k))
        - Signature:
            {app/2,plus/2,sum/1,app#/2,plus#/2,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/3,c_7/2
            ,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,plus#,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_2) = {1}
        
        Following symbols are considered usable:
          {app#,plus#,sum#}
        TcT has computed the following interpretation:
              p(0) = [1]                  
            p(app) = [0]                  
           p(cons) = [1] x2 + [2]         
            p(nil) = [8]                  
           p(plus) = [1] x1 + [2] x2 + [0]
              p(s) = [1]                  
            p(sum) = [0]                  
           p(app#) = [2] x1 + [0]         
          p(plus#) = [0]                  
           p(sum#) = [0]                  
            p(c_1) = [0]                  
            p(c_2) = [1] x1 + [0]         
            p(c_3) = [0]                  
            p(c_4) = [8]                  
            p(c_5) = [0]                  
            p(c_6) = [2] x1 + [4] x3 + [0]
            p(c_7) = [2] x1 + [4]         
            p(c_8) = [0]                  
        
        Following rules are strictly oriented:
        app#(cons(x,l),k) = [2] l + [4]   
                          > [2] l + [0]   
                          = c_2(app#(l,k))
        
        
        Following rules are (at-least) weakly oriented:
        
**** Step 1.b:5.a:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            app#(cons(x,l),k) -> c_2(app#(l,k))
        - Signature:
            {app/2,plus/2,sum/1,app#/2,plus#/2,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/3,c_7/2
            ,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,plus#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 1.b:5.a:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            app#(cons(x,l),k) -> c_2(app#(l,k))
        - Signature:
            {app/2,plus/2,sum/1,app#/2,plus#/2,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/3,c_7/2
            ,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,plus#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:app#(cons(x,l),k) -> c_2(app#(l,k))
             -->_1 app#(cons(x,l),k) -> c_2(app#(l,k)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: app#(cons(x,l),k) -> c_2(app#(l,k))
**** Step 1.b:5.a:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {app/2,plus/2,sum/1,app#/2,plus#/2,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/3,c_7/2
            ,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,plus#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 1.b:5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            plus#(s(x),y) -> c_5(plus#(x,y))
            sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
        - Weak DPs:
            app#(cons(x,l),k) -> c_2(app#(l,k))
        - Weak TRS:
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {app/2,plus/2,sum/1,app#/2,plus#/2,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/3,c_7/2
            ,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,plus#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:plus#(s(x),y) -> c_5(plus#(x,y))
             -->_1 plus#(s(x),y) -> c_5(plus#(x,y)):1
          
          2:S:sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
             -->_1 sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y)):2
             -->_2 plus#(s(x),y) -> c_5(plus#(x,y)):1
          
          3:W:app#(cons(x,l),k) -> c_2(app#(l,k))
             -->_1 app#(cons(x,l),k) -> c_2(app#(l,k)):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: app#(cons(x,l),k) -> c_2(app#(l,k))
*** Step 1.b:5.b:2: Decompose WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            plus#(s(x),y) -> c_5(plus#(x,y))
            sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
        - Weak TRS:
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {app/2,plus/2,sum/1,app#/2,plus#/2,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/3,c_7/2
            ,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,plus#,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:
              plus#(s(x),y) -> c_5(plus#(x,y))
          - Weak DPs:
              sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
          - Weak TRS:
              plus(0(),y) -> y
              plus(s(x),y) -> s(plus(x,y))
          - Signature:
              {app/2,plus/2,sum/1,app#/2,plus#/2,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/3
              ,c_7/2,c_8/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {app#,plus#,sum#} and constructors {0,cons,nil,s}
        
        Problem (S)
          - Strict DPs:
              sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
          - Weak DPs:
              plus#(s(x),y) -> c_5(plus#(x,y))
          - Weak TRS:
              plus(0(),y) -> y
              plus(s(x),y) -> s(plus(x,y))
          - Signature:
              {app/2,plus/2,sum/1,app#/2,plus#/2,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/3
              ,c_7/2,c_8/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {app#,plus#,sum#} and constructors {0,cons,nil,s}
**** Step 1.b:5.b:2.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            plus#(s(x),y) -> c_5(plus#(x,y))
        - Weak DPs:
            sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
        - Weak TRS:
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {app/2,plus/2,sum/1,app#/2,plus#/2,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/3,c_7/2
            ,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,plus#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: plus#(s(x),y) -> c_5(plus#(x,y))
          
        The strictly oriented rules are moved into the weak component.
***** Step 1.b:5.b:2.a:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            plus#(s(x),y) -> c_5(plus#(x,y))
        - Weak DPs:
            sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
        - Weak TRS:
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {app/2,plus/2,sum/1,app#/2,plus#/2,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/3,c_7/2
            ,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,plus#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, 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 polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(c_5) = {1},
          uargs(c_7) = {1,2}
        
        Following symbols are considered usable:
          {plus,app#,plus#,sum#}
        TcT has computed the following interpretation:
              p(0) = 0                 
            p(app) = 2 + 4*x1*x2 + x1^2
           p(cons) = 1 + x1 + x2       
            p(nil) = 0                 
           p(plus) = x1 + x2           
              p(s) = 1 + x1            
            p(sum) = 4 + x1^2          
           p(app#) = 2                 
          p(plus#) = 1 + 4*x1          
           p(sum#) = 5 + 2*x1^2        
            p(c_1) = 0                 
            p(c_2) = 1                 
            p(c_3) = 0                 
            p(c_4) = 1                 
            p(c_5) = x1                
            p(c_6) = 1 + x1 + x2       
            p(c_7) = x1 + x2           
            p(c_8) = 0                 
        
        Following rules are strictly oriented:
        plus#(s(x),y) = 5 + 4*x        
                      > 1 + 4*x        
                      = c_5(plus#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
        sum#(cons(x,cons(y,l))) =  13 + 8*l + 4*l*x + 4*l*y + 2*l^2 + 8*x + 4*x*y + 2*x^2 + 8*y + 2*y^2
                                >= 8 + 4*l + 4*l*x + 4*l*y + 2*l^2 + 8*x + 4*x*y + 2*x^2 + 4*y + 2*y^2 
                                =  c_7(sum#(cons(plus(x,y),l)),plus#(x,y))                             
        
                    plus(0(),y) =  y                                                                   
                                >= y                                                                   
                                =  y                                                                   
        
                   plus(s(x),y) =  1 + x + y                                                           
                                >= 1 + x + y                                                           
                                =  s(plus(x,y))                                                        
        
***** Step 1.b:5.b:2.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            plus#(s(x),y) -> c_5(plus#(x,y))
            sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
        - Weak TRS:
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {app/2,plus/2,sum/1,app#/2,plus#/2,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/3,c_7/2
            ,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,plus#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

***** Step 1.b:5.b:2.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            plus#(s(x),y) -> c_5(plus#(x,y))
            sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
        - Weak TRS:
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {app/2,plus/2,sum/1,app#/2,plus#/2,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/3,c_7/2
            ,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,plus#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:plus#(s(x),y) -> c_5(plus#(x,y))
             -->_1 plus#(s(x),y) -> c_5(plus#(x,y)):1
          
          2:W:sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
             -->_1 sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y)):2
             -->_2 plus#(s(x),y) -> c_5(plus#(x,y)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
          1: plus#(s(x),y) -> c_5(plus#(x,y))
***** Step 1.b:5.b:2.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {app/2,plus/2,sum/1,app#/2,plus#/2,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/3,c_7/2
            ,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,plus#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

**** Step 1.b:5.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
        - Weak DPs:
            plus#(s(x),y) -> c_5(plus#(x,y))
        - Weak TRS:
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {app/2,plus/2,sum/1,app#/2,plus#/2,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/3,c_7/2
            ,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,plus#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
             -->_2 plus#(s(x),y) -> c_5(plus#(x,y)):2
             -->_1 sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y)):1
          
          2:W:plus#(s(x),y) -> c_5(plus#(x,y))
             -->_1 plus#(s(x),y) -> c_5(plus#(x,y)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: plus#(s(x),y) -> c_5(plus#(x,y))
**** Step 1.b:5.b:2.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
        - Weak TRS:
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {app/2,plus/2,sum/1,app#/2,plus#/2,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/3,c_7/2
            ,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,plus#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
             -->_1 sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)))
**** Step 1.b:5.b:2.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)))
        - Weak TRS:
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {app/2,plus/2,sum/1,app#/2,plus#/2,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/3,c_7/1
            ,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,plus#,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,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)))
          
        The strictly oriented rules are moved into the weak component.
***** Step 1.b:5.b:2.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)))
        - Weak TRS:
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {app/2,plus/2,sum/1,app#/2,plus#/2,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/3,c_7/1
            ,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,plus#,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:
          {plus,app#,plus#,sum#}
        TcT has computed the following interpretation:
              p(0) = [0]                  
            p(app) = [1] x1 + [1] x2 + [0]
           p(cons) = [1] x1 + [1] x2 + [3]
            p(nil) = [1]                  
           p(plus) = [1] x1 + [1] x2 + [1]
              p(s) = [1]                  
            p(sum) = [1] x1 + [2]         
           p(app#) = [2] x1 + [1]         
          p(plus#) = [1] x1 + [1]         
           p(sum#) = [4] x1 + [0]         
            p(c_1) = [1]                  
            p(c_2) = [1] x1 + [1]         
            p(c_3) = [1]                  
            p(c_4) = [1]                  
            p(c_5) = [2] x1 + [2]         
            p(c_6) = [8] x1 + [4]         
            p(c_7) = [1] x1 + [4]         
            p(c_8) = [0]                  
        
        Following rules are strictly oriented:
        sum#(cons(x,cons(y,l))) = [4] l + [4] x + [4] y + [24]
                                > [4] l + [4] x + [4] y + [20]
                                = c_7(sum#(cons(plus(x,y),l)))
        
        
        Following rules are (at-least) weakly oriented:
         plus(0(),y) =  [1] y + [1] 
                     >= [1] y + [0] 
                     =  y           
        
        plus(s(x),y) =  [1] y + [2] 
                     >= [1]         
                     =  s(plus(x,y))
        
***** Step 1.b:5.b:2.b:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)))
        - Weak TRS:
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {app/2,plus/2,sum/1,app#/2,plus#/2,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/3,c_7/1
            ,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,plus#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

***** Step 1.b:5.b:2.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)))
        - Weak TRS:
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {app/2,plus/2,sum/1,app#/2,plus#/2,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/3,c_7/1
            ,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,plus#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)))
             -->_1 sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),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,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)))
***** Step 1.b:5.b:2.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            plus(0(),y) -> y
            plus(s(x),y) -> s(plus(x,y))
        - Signature:
            {app/2,plus/2,sum/1,app#/2,plus#/2,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/3,c_7/1
            ,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {app#,plus#,sum#} and constructors {0,cons,nil,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^2))