*** 1 Progress [(O(1),O(n^2))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        +(x,0()) -> x
        +(x,s(y)) -> s(+(x,y))
        sum(0()) -> 0()
        sum(s(x)) -> +(sum(x),s(x))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {+/2,sum/1} / {0/0,s/1}
      Obligation:
        Full
        basic terms: {+,sum}/{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(+) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
            p(+) = [1] x1 + [5]
            p(0) = [0]         
            p(s) = [1] x1 + [8]
          p(sum) = [2] x1 + [0]
        
        Following rules are strictly oriented:
         +(x,0()) = [1] x + [5]   
                  > [1] x + [0]   
                  = x             
        
        sum(s(x)) = [2] x + [16]  
                  > [2] x + [5]   
                  = +(sum(x),s(x))
        
        
        Following rules are (at-least) weakly oriented:
        +(x,s(y)) =  [1] x + [5] 
                  >= [1] x + [13]
                  =  s(+(x,y))   
        
         sum(0()) =  [0]         
                  >= [0]         
                  =  0()         
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1 Progress [(O(1),O(n^2))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        +(x,s(y)) -> s(+(x,y))
        sum(0()) -> 0()
      Weak DP Rules:
        
      Weak TRS Rules:
        +(x,0()) -> x
        sum(s(x)) -> +(sum(x),s(x))
      Signature:
        {+/2,sum/1} / {0/0,s/1}
      Obligation:
        Full
        basic terms: {+,sum}/{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(+) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
            p(+) = [1] x1 + [0]
            p(0) = [0]         
            p(s) = [1] x1 + [0]
          p(sum) = [7]         
        
        Following rules are strictly oriented:
        sum(0()) = [7]
                 > [0]
                 = 0()
        
        
        Following rules are (at-least) weakly oriented:
         +(x,0()) =  [1] x + [0]   
                  >= [1] x + [0]   
                  =  x             
        
        +(x,s(y)) =  [1] x + [0]   
                  >= [1] x + [0]   
                  =  s(+(x,y))     
        
        sum(s(x)) =  [7]           
                  >= [7]           
                  =  +(sum(x),s(x))
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1 Progress [(O(1),O(n^2))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        +(x,s(y)) -> s(+(x,y))
      Weak DP Rules:
        
      Weak TRS Rules:
        +(x,0()) -> x
        sum(0()) -> 0()
        sum(s(x)) -> +(sum(x),s(x))
      Signature:
        {+/2,sum/1} / {0/0,s/1}
      Obligation:
        Full
        basic terms: {+,sum}/{0,s}
    Applied Processor:
      NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules, greedy = NoGreedy}
    Proof:
      We apply a polynomial interpretation of kind constructor-based(mixed(2)):
      The following argument positions are considered usable:
        uargs(+) = {1},
        uargs(s) = {1}
      
      Following symbols are considered usable:
        {}
      TcT has computed the following interpretation:
          p(+) = x1 + 3*x2  
          p(0) = 0          
          p(s) = 1 + x1     
        p(sum) = x1 + 2*x1^2
      
      Following rules are strictly oriented:
      +(x,s(y)) = 3 + x + 3*y
                > 1 + x + 3*y
                = s(+(x,y))  
      
      
      Following rules are (at-least) weakly oriented:
       +(x,0()) =  x              
                >= x              
                =  x              
      
       sum(0()) =  0              
                >= 0              
                =  0()            
      
      sum(s(x)) =  3 + 5*x + 2*x^2
                >= 3 + 4*x + 2*x^2
                =  +(sum(x),s(x)) 
      
*** 1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        +(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:
        Full
        basic terms: {+,sum}/{0,s}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).