* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
            sum1(0()) -> 0()
            sum1(s(x)) -> s(+(sum1(x),+(x,x)))
        - Signature:
            {sum/1,sum1/1} / {+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum,sum1} and constructors {+,0,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
            sum1(0()) -> 0()
            sum1(s(x)) -> s(+(sum1(x),+(x,x)))
        - Signature:
            {sum/1,sum1/1} / {+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum,sum1} and constructors {+,0,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          sum(x){x -> s(x)} =
            sum(s(x)) ->^+ +(sum(x),s(x))
              = C[sum(x) = sum(x){}]

** Step 1.b:1: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
            sum1(0()) -> 0()
            sum1(s(x)) -> s(+(sum1(x),+(x,x)))
        - Signature:
            {sum/1,sum1/1} / {+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum,sum1} 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 nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(+) = {1},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
               p(+) = [1] x1 + [0]
               p(0) = [0]         
               p(s) = [1] x1 + [7]
             p(sum) = [0]         
            p(sum1) = [4] x1 + [1]
          
          Following rules are strictly oriented:
           sum1(0()) = [1]                 
                     > [0]                 
                     = 0()                 
          
          sum1(s(x)) = [4] x + [29]        
                     > [4] x + [8]         
                     = s(+(sum1(x),+(x,x)))
          
          
          Following rules are (at-least) weakly oriented:
           sum(0()) =  [0]           
                    >= [0]           
                    =  0()           
          
          sum(s(x)) =  [0]           
                    >= [0]           
                    =  +(sum(x),s(x))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:2: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
        - Weak TRS:
            sum1(0()) -> 0()
            sum1(s(x)) -> s(+(sum1(x),+(x,x)))
        - Signature:
            {sum/1,sum1/1} / {+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum,sum1} 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 nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(+) = {1},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
               p(+) = [1] x1 + [4]
               p(0) = [7]         
               p(s) = [1] x1 + [8]
             p(sum) = [9]         
            p(sum1) = [2] x1 + [2]
          
          Following rules are strictly oriented:
          sum(0()) = [9]
                   > [7]
                   = 0()
          
          
          Following rules are (at-least) weakly oriented:
           sum(s(x)) =  [9]                 
                     >= [13]                
                     =  +(sum(x),s(x))      
          
           sum1(0()) =  [16]                
                     >= [7]                 
                     =  0()                 
          
          sum1(s(x)) =  [2] x + [18]        
                     >= [2] x + [14]        
                     =  s(+(sum1(x),+(x,x)))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:3: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            sum(s(x)) -> +(sum(x),s(x))
        - Weak TRS:
            sum(0()) -> 0()
            sum1(0()) -> 0()
            sum1(s(x)) -> s(+(sum1(x),+(x,x)))
        - Signature:
            {sum/1,sum1/1} / {+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum,sum1} 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 nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(+) = {1},
            uargs(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
               p(+) = [1] x1 + [1] 
               p(0) = [2]          
               p(s) = [1] x1 + [2] 
             p(sum) = [4] x1 + [4] 
            p(sum1) = [8] x1 + [15]
          
          Following rules are strictly oriented:
          sum(s(x)) = [4] x + [12]  
                    > [4] x + [5]   
                    = +(sum(x),s(x))
          
          
          Following rules are (at-least) weakly oriented:
            sum(0()) =  [12]                
                     >= [2]                 
                     =  0()                 
          
           sum1(0()) =  [31]                
                     >= [2]                 
                     =  0()                 
          
          sum1(s(x)) =  [8] x + [31]        
                     >= [8] x + [18]        
                     =  s(+(sum1(x),+(x,x)))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
            sum1(0()) -> 0()
            sum1(s(x)) -> s(+(sum1(x),+(x,x)))
        - Signature:
            {sum/1,sum1/1} / {+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum,sum1} and constructors {+,0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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