WORST_CASE(?,O(n^2))
* Step 1: DependencyPairs WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
        - Signature:
            {+/2,sum/1} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+,sum} and constructors {0,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          +#(x,0()) -> c_1()
          +#(x,s(y)) -> c_2(+#(x,y))
          sum#(0()) -> c_3()
          sum#(s(x)) -> c_4(+#(sum(x),s(x)),sum#(x))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            +#(x,0()) -> c_1()
            +#(x,s(y)) -> c_2(+#(x,y))
            sum#(0()) -> c_3()
            sum#(s(x)) -> c_4(+#(sum(x),s(x)),sum#(x))
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
        - Signature:
            {+/2,sum/1,+#/2,sum#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,sum#} and constructors {0,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,3}
        by application of
          Pre({1,3}) = {2,4}.
        Here rules are labelled as follows:
          1: +#(x,0()) -> c_1()
          2: +#(x,s(y)) -> c_2(+#(x,y))
          3: sum#(0()) -> c_3()
          4: sum#(s(x)) -> c_4(+#(sum(x),s(x)),sum#(x))
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            +#(x,s(y)) -> c_2(+#(x,y))
            sum#(s(x)) -> c_4(+#(sum(x),s(x)),sum#(x))
        - Weak DPs:
            +#(x,0()) -> c_1()
            sum#(0()) -> c_3()
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
        - Signature:
            {+/2,sum/1,+#/2,sum#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,sum#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:+#(x,s(y)) -> c_2(+#(x,y))
             -->_1 +#(x,0()) -> c_1():3
             -->_1 +#(x,s(y)) -> c_2(+#(x,y)):1
          
          2:S:sum#(s(x)) -> c_4(+#(sum(x),s(x)),sum#(x))
             -->_2 sum#(0()) -> c_3():4
             -->_2 sum#(s(x)) -> c_4(+#(sum(x),s(x)),sum#(x)):2
             -->_1 +#(x,s(y)) -> c_2(+#(x,y)):1
          
          3:W:+#(x,0()) -> c_1()
             
          
          4:W:sum#(0()) -> c_3()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: sum#(0()) -> c_3()
          3: +#(x,0()) -> c_1()
* Step 4: DecomposeDG WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            +#(x,s(y)) -> c_2(+#(x,y))
            sum#(s(x)) -> c_4(+#(sum(x),s(x)),sum#(x))
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
        - Signature:
            {+/2,sum/1,+#/2,sum#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,sum#} and constructors {0,s}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          sum#(s(x)) -> c_4(+#(sum(x),s(x)),sum#(x))
        and a lower component
          +#(x,s(y)) -> c_2(+#(x,y))
        Further, following extension rules are added to the lower component.
          sum#(s(x)) -> +#(sum(x),s(x))
          sum#(s(x)) -> sum#(x)
** Step 4.a:1: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(s(x)) -> c_4(+#(sum(x),s(x)),sum#(x))
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
        - Signature:
            {+/2,sum/1,+#/2,sum#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,sum#} and constructors {0,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:sum#(s(x)) -> c_4(+#(sum(x),s(x)),sum#(x))
             -->_2 sum#(s(x)) -> c_4(+#(sum(x),s(x)),sum#(x)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          sum#(s(x)) -> c_4(sum#(x))
** Step 4.a:2: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(s(x)) -> c_4(sum#(x))
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
        - Signature:
            {+/2,sum/1,+#/2,sum#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,sum#} and constructors {0,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          sum#(s(x)) -> c_4(sum#(x))
** Step 4.a:3: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(s(x)) -> c_4(sum#(x))
        - Signature:
            {+/2,sum/1,+#/2,sum#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,sum#} and constructors {0,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          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:
            all
          TcT has computed the following interpretation:
               p(+) = [0]          
               p(0) = [0]          
               p(s) = [1] x1 + [6] 
             p(sum) = [1] x1 + [1] 
              p(+#) = [2] x2 + [1] 
            p(sum#) = [3] x1 + [13]
             p(c_1) = [1]          
             p(c_2) = [1]          
             p(c_3) = [0]          
             p(c_4) = [1] x1 + [1] 
          
          Following rules are strictly oriented:
          sum#(s(x)) = [3] x + [31]
                     > [3] x + [14]
                     = c_4(sum#(x))
          
          
          Following rules are (at-least) weakly oriented:
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 4.a:4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            sum#(s(x)) -> c_4(sum#(x))
        - Signature:
            {+/2,sum/1,+#/2,sum#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,sum#} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

** Step 4.b:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            +#(x,s(y)) -> c_2(+#(x,y))
        - Weak DPs:
            sum#(s(x)) -> +#(sum(x),s(x))
            sum#(s(x)) -> sum#(x)
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
        - Signature:
            {+/2,sum/1,+#/2,sum#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,sum#} and constructors {0,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any 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:
          {+#,sum#}
        TcT has computed the following interpretation:
             p(+) = [8] x1 + [1] x2 + [4]
             p(0) = [0]                  
             p(s) = [1] x1 + [4]         
           p(sum) = [2] x1 + [1]         
            p(+#) = [1] x2 + [0]         
          p(sum#) = [1] x1 + [0]         
           p(c_1) = [0]                  
           p(c_2) = [1] x1 + [0]         
           p(c_3) = [2]                  
           p(c_4) = [1]                  
        
        Following rules are strictly oriented:
        +#(x,s(y)) = [1] y + [4] 
                   > [1] y + [0] 
                   = c_2(+#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
        sum#(s(x)) =  [1] x + [4]    
                   >= [1] x + [4]    
                   =  +#(sum(x),s(x))
        
        sum#(s(x)) =  [1] x + [4]    
                   >= [1] x + [0]    
                   =  sum#(x)        
        
** Step 4.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            +#(x,s(y)) -> c_2(+#(x,y))
            sum#(s(x)) -> +#(sum(x),s(x))
            sum#(s(x)) -> sum#(x)
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
        - Signature:
            {+/2,sum/1,+#/2,sum#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,sum#} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))