/export/starexec/sandbox/solver/bin/starexec_run_tct_dc /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(?,O(n^2))
* Step 1: WeightGap WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            div(X,e()) -> i(X)
            div(div(X,Y),Z) -> div(Y,div(i(X),Z))
            i(div(X,Y)) -> div(Y,X)
        - Signature:
            {div/2,i/1} / {e/0}
        - Obligation:
             derivational complexity wrt. signature {div,e,i}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind triangular matrix interpretation:
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
            p(div) = [1] x1 + [1] x2 + [6]
              p(e) = [0]                  
              p(i) = [1] x1 + [1]         
          
          Following rules are strictly oriented:
           div(X,e()) = [1] X + [6]        
                      > [1] X + [1]        
                      = i(X)               
          
          i(div(X,Y)) = [1] X + [1] Y + [7]
                      > [1] X + [1] Y + [6]
                      = div(Y,X)           
          
          
          Following rules are (at-least) weakly oriented:
          div(div(X,Y),Z) =  [1] X + [1] Y + [1] Z + [12]
                          >= [1] X + [1] Y + [1] Z + [13]
                          =  div(Y,div(i(X),Z))          
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: NaturalMI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            div(div(X,Y),Z) -> div(Y,div(i(X),Z))
        - Weak TRS:
            div(X,e()) -> i(X)
            i(div(X,Y)) -> div(Y,X)
        - Signature:
            {div/2,i/1} / {e/0}
        - Obligation:
             derivational complexity wrt. signature {div,e,i}
    + Applied Processor:
        NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind triangular matrix interpretation:
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
          p(div) = [1 1] x1 + [1 0] x2 + [0]
                   [0 1]      [0 1]      [1]
            p(e) = [6]                      
                   [0]                      
            p(i) = [1 1] x1 + [0]           
                   [0 1]      [0]           
        
        Following rules are strictly oriented:
        div(div(X,Y),Z) = [1 2] X + [1 1] Y + [1 0] Z + [1]
                          [0 1]     [0 1]     [0 1]     [2]
                        > [1 2] X + [1 1] Y + [1 0] Z + [0]
                          [0 1]     [0 1]     [0 1]     [2]
                        = div(Y,div(i(X),Z))               
        
        
        Following rules are (at-least) weakly oriented:
         div(X,e()) =  [1 1] X + [6]          
                       [0 1]     [1]          
                    >= [1 1] X + [0]          
                       [0 1]     [0]          
                    =  i(X)                   
        
        i(div(X,Y)) =  [1 2] X + [1 1] Y + [1]
                       [0 1]     [0 1]     [1]
                    >= [1 0] X + [1 1] Y + [0]
                       [0 1]     [0 1]     [1]
                    =  div(Y,X)               
        
* Step 3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            div(X,e()) -> i(X)
            div(div(X,Y),Z) -> div(Y,div(i(X),Z))
            i(div(X,Y)) -> div(Y,X)
        - Signature:
            {div/2,i/1} / {e/0}
        - Obligation:
             derivational complexity wrt. signature {div,e,i}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))