/export/starexec/sandbox2/solver/bin/starexec_run_tct_dc /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(?,O(n^2))
* Step 1: WeightGap.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            +(x,+(y,z)) -> +(+(x,y),z)
            +(*(x,y),+(x,z)) -> *(x,+(y,z))
            +(*(x,y),+(*(x,z),u)) -> +(*(x,+(y,z)),u)
        - Signature:
            {+/2} / {*/2}
        - Obligation:
             derivational complexity wrt. signature {*,+}
    + 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(*) = [1] x1 + [1] x2 + [3]
            p(+) = [1] x1 + [1] x2 + [1]
          
          Following rules are strictly oriented:
               +(*(x,y),+(x,z)) = [2] x + [1] y + [1] z + [5]        
                                > [1] x + [1] y + [1] z + [4]        
                                = *(x,+(y,z))                        
          
          +(*(x,y),+(*(x,z),u)) = [1] u + [2] x + [1] y + [1] z + [8]
                                > [1] u + [1] x + [1] y + [1] z + [5]
                                = +(*(x,+(y,z)),u)                   
          
          
          Following rules are (at-least) weakly oriented:
          +(x,+(y,z)) =  [1] x + [1] y + [1] z + [2]
                      >= [1] x + [1] y + [1] z + [2]
                      =  +(+(x,y),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:
            +(x,+(y,z)) -> +(+(x,y),z)
        - Weak TRS:
            +(*(x,y),+(x,z)) -> *(x,+(y,z))
            +(*(x,y),+(*(x,z),u)) -> +(*(x,+(y,z)),u)
        - Signature:
            {+/2} / {*/2}
        - Obligation:
             derivational complexity wrt. signature {*,+}
    + 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(*) = [1 1] x1 + [1 1] x2 + [0]
                 [0 0]      [0 1]      [1]
          p(+) = [1 0] x1 + [1 2] x2 + [0]
                 [0 1]      [0 1]      [5]
        
        Following rules are strictly oriented:
        +(x,+(y,z)) = [1 0] x + [1 2] y + [1 4] z + [10]
                      [0 1]     [0 1]     [0 1]     [10]
                    > [1 0] x + [1 2] y + [1 2] z + [0] 
                      [0 1]     [0 1]     [0 1]     [10]
                    = +(+(x,y),z)                       
        
        
        Following rules are (at-least) weakly oriented:
             +(*(x,y),+(x,z)) =  [2 3] x + [1 1] y + [1 4] z + [10]          
                                 [0 1]     [0 1]     [0 1]     [11]          
                              >= [1 1] x + [1 1] y + [1 3] z + [5]           
                                 [0 0]     [0 1]     [0 1]     [6]           
                              =  *(x,+(y,z))                                 
        
        +(*(x,y),+(*(x,z),u)) =  [1 4] u + [2 2] x + [1 1] y + [1 3] z + [12]
                                 [0 1]     [0 0]     [0 1]     [0 1]     [12]
                              >= [1 2] u + [1 1] x + [1 1] y + [1 3] z + [5] 
                                 [0 1]     [0 0]     [0 1]     [0 1]     [11]
                              =  +(*(x,+(y,z)),u)                            
        
* Step 3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            +(x,+(y,z)) -> +(+(x,y),z)
            +(*(x,y),+(x,z)) -> *(x,+(y,z))
            +(*(x,y),+(*(x,z),u)) -> +(*(x,+(y,z)),u)
        - Signature:
            {+/2} / {*/2}
        - Obligation:
             derivational complexity wrt. signature {*,+}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))