/export/starexec/sandbox2/solver/bin/starexec_run_tct_dci /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:
            merge(x,nil()) -> x
            merge(++(x,y),++(u(),v())) -> ++(x,merge(y,++(u(),v())))
            merge(++(x,y),++(u(),v())) -> ++(u(),merge(++(x,y),v()))
            merge(nil(),y) -> y
        - Signature:
            {merge/2} / {++/2,nil/0,u/0,v/0}
        - Obligation:
            innermost derivational complexity wrt. signature {++,merge,nil,u,v}
    + 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 + [10]
            p(merge) = [1] x1 + [1] x2 + [0] 
              p(nil) = [1]                   
                p(u) = [0]                   
                p(v) = [3]                   
          
          Following rules are strictly oriented:
          merge(x,nil()) = [1] x + [1]
                         > [1] x + [0]
                         = x          
          
          merge(nil(),y) = [1] y + [1]
                         > [1] y + [0]
                         = y          
          
          
          Following rules are (at-least) weakly oriented:
          merge(++(x,y),++(u(),v())) =  [1] x + [1] y + [23]      
                                     >= [1] x + [1] y + [23]      
                                     =  ++(x,merge(y,++(u(),v())))
          
          merge(++(x,y),++(u(),v())) =  [1] x + [1] y + [23]      
                                     >= [1] x + [1] y + [23]      
                                     =  ++(u(),merge(++(x,y),v()))
          
        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:
            merge(++(x,y),++(u(),v())) -> ++(x,merge(y,++(u(),v())))
            merge(++(x,y),++(u(),v())) -> ++(u(),merge(++(x,y),v()))
        - Weak TRS:
            merge(x,nil()) -> x
            merge(nil(),y) -> y
        - Signature:
            {merge/2} / {++/2,nil/0,u/0,v/0}
        - Obligation:
            innermost derivational complexity wrt. signature {++,merge,nil,u,v}
    + 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 0] x2 + [2]
                     [0 1]      [0 0]      [0]
          p(merge) = [1 0] x1 + [1 1] x2 + [2]
                     [0 1]      [0 1]      [0]
            p(nil) = [1]                      
                     [4]                      
              p(u) = [5]                      
                     [1]                      
              p(v) = [0]                      
                     [0]                      
        
        Following rules are strictly oriented:
        merge(++(x,y),++(u(),v())) = [1 1] x + [1 0] y + [13]  
                                     [0 1]     [0 0]     [1]   
                                   > [1 1] x + [1 0] y + [12]  
                                     [0 0]     [0 0]     [1]   
                                   = ++(u(),merge(++(x,y),v()))
        
        
        Following rules are (at-least) weakly oriented:
                    merge(x,nil()) =  [1 0] x + [7]             
                                      [0 1]     [4]             
                                   >= [1 0] x + [0]             
                                      [0 1]     [0]             
                                   =  x                         
        
        merge(++(x,y),++(u(),v())) =  [1 1] x + [1 0] y + [13]  
                                      [0 1]     [0 0]     [1]   
                                   >= [1 1] x + [1 0] y + [13]  
                                      [0 1]     [0 0]     [0]   
                                   =  ++(x,merge(y,++(u(),v())))
        
                    merge(nil(),y) =  [1 1] y + [3]             
                                      [0 1]     [4]             
                                   >= [1 0] y + [0]             
                                      [0 1]     [0]             
                                   =  y                         
        
* Step 3: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            merge(++(x,y),++(u(),v())) -> ++(x,merge(y,++(u(),v())))
        - Weak TRS:
            merge(x,nil()) -> x
            merge(++(x,y),++(u(),v())) -> ++(u(),merge(++(x,y),v()))
            merge(nil(),y) -> y
        - Signature:
            {merge/2} / {++/2,nil/0,u/0,v/0}
        - Obligation:
            innermost derivational complexity wrt. signature {++,merge,nil,u,v}
    + 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 0] x1 + [1 0] x2 + [0]
                     [0 1]      [0 1]      [1]
          p(merge) = [1 4] x1 + [1 0] x2 + [4]
                     [0 1]      [0 1]      [5]
            p(nil) = [1]                      
                     [0]                      
              p(u) = [0]                      
                     [1]                      
              p(v) = [1]                      
                     [0]                      
        
        Following rules are strictly oriented:
        merge(++(x,y),++(u(),v())) = [1 4] x + [1 4] y + [9]   
                                     [0 1]     [0 1]     [8]   
                                   > [1 0] x + [1 4] y + [5]   
                                     [0 1]     [0 1]     [8]   
                                   = ++(x,merge(y,++(u(),v())))
        
        
        Following rules are (at-least) weakly oriented:
                    merge(x,nil()) =  [1 4] x + [5]             
                                      [0 1]     [5]             
                                   >= [1 0] x + [0]             
                                      [0 1]     [0]             
                                   =  x                         
        
        merge(++(x,y),++(u(),v())) =  [1 4] x + [1 4] y + [9]   
                                      [0 1]     [0 1]     [8]   
                                   >= [1 4] x + [1 4] y + [9]   
                                      [0 1]     [0 1]     [8]   
                                   =  ++(u(),merge(++(x,y),v()))
        
                    merge(nil(),y) =  [1 0] y + [5]             
                                      [0 1]     [5]             
                                   >= [1 0] y + [0]             
                                      [0 1]     [0]             
                                   =  y                         
        
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            merge(x,nil()) -> x
            merge(++(x,y),++(u(),v())) -> ++(x,merge(y,++(u(),v())))
            merge(++(x,y),++(u(),v())) -> ++(u(),merge(++(x,y),v()))
            merge(nil(),y) -> y
        - Signature:
            {merge/2} / {++/2,nil/0,u/0,v/0}
        - Obligation:
            innermost derivational complexity wrt. signature {++,merge,nil,u,v}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))