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

** Step 1.b:1: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            average(x,s(s(s(y)))) -> s(average(s(x),y))
            average(0(),0()) -> 0()
            average(0(),s(0())) -> 0()
            average(0(),s(s(0()))) -> s(0())
            average(s(x),y) -> average(x,s(y))
        - Signature:
            {average/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {average} 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(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                  p(0) = [12]         
            p(average) = [1] x1 + [13]
                  p(s) = [1] x1 + [0] 
          
          Following rules are strictly oriented:
                average(0(),0()) = [25]  
                                 > [12]  
                                 = 0()   
          
             average(0(),s(0())) = [25]  
                                 > [12]  
                                 = 0()   
          
          average(0(),s(s(0()))) = [25]  
                                 > [12]  
                                 = s(0())
          
          
          Following rules are (at-least) weakly oriented:
          average(x,s(s(s(y)))) =  [1] x + [13]      
                                >= [1] x + [13]      
                                =  s(average(s(x),y))
          
                average(s(x),y) =  [1] x + [13]      
                                >= [1] x + [13]      
                                =  average(x,s(y))   
          
        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:
            average(x,s(s(s(y)))) -> s(average(s(x),y))
            average(s(x),y) -> average(x,s(y))
        - Weak TRS:
            average(0(),0()) -> 0()
            average(0(),s(0())) -> 0()
            average(0(),s(s(0()))) -> s(0())
        - Signature:
            {average/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {average} 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(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                  p(0) = [1]          
            p(average) = [10] x1 + [0]
                  p(s) = [1] x1 + [2] 
          
          Following rules are strictly oriented:
          average(s(x),y) = [10] x + [20]  
                          > [10] x + [0]   
                          = average(x,s(y))
          
          
          Following rules are (at-least) weakly oriented:
           average(x,s(s(s(y)))) =  [10] x + [0]      
                                 >= [10] x + [22]     
                                 =  s(average(s(x),y))
          
                average(0(),0()) =  [10]              
                                 >= [1]               
                                 =  0()               
          
             average(0(),s(0())) =  [10]              
                                 >= [1]               
                                 =  0()               
          
          average(0(),s(s(0()))) =  [10]              
                                 >= [3]               
                                 =  s(0())            
          
        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:
            average(x,s(s(s(y)))) -> s(average(s(x),y))
        - Weak TRS:
            average(0(),0()) -> 0()
            average(0(),s(0())) -> 0()
            average(0(),s(s(0()))) -> s(0())
            average(s(x),y) -> average(x,s(y))
        - Signature:
            {average/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {average} 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(s) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                  p(0) = [1]                  
            p(average) = [9] x1 + [6] x2 + [0]
                  p(s) = [1] x1 + [1]         
          
          Following rules are strictly oriented:
          average(x,s(s(s(y)))) = [9] x + [6] y + [18]
                                > [9] x + [6] y + [10]
                                = s(average(s(x),y))  
          
          
          Following rules are (at-least) weakly oriented:
                average(0(),0()) =  [15]               
                                 >= [1]                
                                 =  0()                
          
             average(0(),s(0())) =  [21]               
                                 >= [1]                
                                 =  0()                
          
          average(0(),s(s(0()))) =  [27]               
                                 >= [2]                
                                 =  s(0())             
          
                 average(s(x),y) =  [9] x + [6] y + [9]
                                 >= [9] x + [6] y + [6]
                                 =  average(x,s(y))    
          
        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:
            average(x,s(s(s(y)))) -> s(average(s(x),y))
            average(0(),0()) -> 0()
            average(0(),s(0())) -> 0()
            average(0(),s(s(0()))) -> s(0())
            average(s(x),y) -> average(x,s(y))
        - Signature:
            {average/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {average} 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))