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