WORST_CASE(?,O(n^1))
* Step 1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            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 runtime complexity wrt. defined symbols {sqr,sum} and constructors {*,+,0,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        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 + [0]
            p(0) = [2]                  
            p(s) = [1] x1 + [0]         
          p(sqr) = [0]                  
          p(sum) = [8] x1 + [0]         
        
        Following rules are strictly oriented:
        sum(0()) = [16]
                 > [2] 
                 = 0() 
        
        
        Following rules are (at-least) weakly oriented:
           sqr(x) =  [0]                   
                  >= [0]                   
                  =  *(x,x)                
        
        sum(s(x)) =  [8] x + [0]           
                  >= [8] x + [0]           
                  =  +(*(s(x),s(x)),sum(x))
        
        sum(s(x)) =  [8] x + [0]           
                  >= [8] x + [0]           
                  =  +(sqr(s(x)),sum(x))   
        
* Step 2: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            sqr(x) -> *(x,x)
            sum(s(x)) -> +(*(s(x),s(x)),sum(x))
            sum(s(x)) -> +(sqr(s(x)),sum(x))
        - Weak TRS:
            sum(0()) -> 0()
        - Signature:
            {sqr/1,sum/1} / {*/2,+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sqr,sum} 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(+) = {1,2}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
              p(*) = [3]                  
              p(+) = [1] x1 + [1] x2 + [7]
              p(0) = [0]                  
              p(s) = [0]                  
            p(sqr) = [1] x1 + [4]         
            p(sum) = [1]                  
          
          Following rules are strictly oriented:
          sqr(x) = [1] x + [4]
                 > [3]        
                 = *(x,x)     
          
          
          Following rules are (at-least) weakly oriented:
           sum(0()) =  [1]                   
                    >= [0]                   
                    =  0()                   
          
          sum(s(x)) =  [1]                   
                    >= [11]                  
                    =  +(*(s(x),s(x)),sum(x))
          
          sum(s(x)) =  [1]                   
                    >= [12]                  
                    =  +(sqr(s(x)),sum(x))   
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            sum(s(x)) -> +(*(s(x),s(x)),sum(x))
            sum(s(x)) -> +(sqr(s(x)),sum(x))
        - Weak TRS:
            sqr(x) -> *(x,x)
            sum(0()) -> 0()
        - Signature:
            {sqr/1,sum/1} / {*/2,+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sqr,sum} 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(+) = {1,2}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
              p(*) = [1]                  
              p(+) = [1] x1 + [1] x2 + [0]
              p(0) = [10]                 
              p(s) = [1] x1 + [2]         
            p(sqr) = [1]                  
            p(sum) = [1] x1 + [0]         
          
          Following rules are strictly oriented:
          sum(s(x)) = [1] x + [2]           
                    > [1] x + [1]           
                    = +(*(s(x),s(x)),sum(x))
          
          sum(s(x)) = [1] x + [2]           
                    > [1] x + [1]           
                    = +(sqr(s(x)),sum(x))   
          
          
          Following rules are (at-least) weakly oriented:
            sqr(x) =  [1]   
                   >= [1]   
                   =  *(x,x)
          
          sum(0()) =  [10]  
                   >= [10]  
                   =  0()   
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            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 runtime complexity wrt. defined symbols {sqr,sum} and constructors {*,+,0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))