*** 1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        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))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {average/2} / {0/0,s/1}
      Obligation:
        Full
        basic terms: {average}/{0,s}
    Applied Processor:
      ToInnermost
    Proof:
      switch to innermost, as the system is overlay and right linear and does not contain weak rules
*** 1.1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        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))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {average/2} / {0/0,s/1}
      Obligation:
        Innermost
        basic terms: {average}/{0,s}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    Proof:
      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:
          {}
        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.
*** 1.1.1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        average(x,s(s(s(y)))) -> s(average(s(x),y))
        average(s(x),y) -> average(x,s(y))
      Weak DP Rules:
        
      Weak TRS Rules:
        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
        basic terms: {average}/{0,s}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    Proof:
      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:
          {}
        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.
*** 1.1.1.1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        average(x,s(s(s(y)))) -> s(average(s(x),y))
      Weak DP Rules:
        
      Weak TRS Rules:
        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
        basic terms: {average}/{0,s}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    Proof:
      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:
          {}
        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.
*** 1.1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        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
        basic terms: {average}/{0,s}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).