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


--------------------------------------------------------------------------------


WORST_CASE(?,O(n^2))
* Step 1: WeightGap.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            minus(f(x,y)) -> f(minus(y),minus(x))
            minus(h(x)) -> h(minus(x))
            minus(minus(x)) -> x
        - Signature:
            {minus/1} / {f/2,h/1}
        - Obligation:
             derivational complexity wrt. signature {f,h,minus}
    + 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(f) = [1] x1 + [1] x2 + [0]
                p(h) = [1] x1 + [0]         
            p(minus) = [1] x1 + [2]         
          
          Following rules are strictly oriented:
          minus(minus(x)) = [1] x + [4]
                          > [1] x + [0]
                          = x          
          
          
          Following rules are (at-least) weakly oriented:
          minus(f(x,y)) =  [1] x + [1] y + [2] 
                        >= [1] x + [1] y + [4] 
                        =  f(minus(y),minus(x))
          
            minus(h(x)) =  [1] x + [2]         
                        >= [1] x + [2]         
                        =  h(minus(x))         
          
        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:
            minus(f(x,y)) -> f(minus(y),minus(x))
            minus(h(x)) -> h(minus(x))
        - Weak TRS:
            minus(minus(x)) -> x
        - Signature:
            {minus/1} / {f/2,h/1}
        - Obligation:
             derivational complexity wrt. signature {f,h,minus}
    + 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(f) = [1 0] x1 + [1 0] x2 + [7]
                     [0 1]      [0 1]      [2]
              p(h) = [1 0] x1 + [0]           
                     [0 1]      [0]           
          p(minus) = [1 4] x1 + [0]           
                     [0 1]      [0]           
        
        Following rules are strictly oriented:
        minus(f(x,y)) = [1 4] x + [1 4] y + [15]
                        [0 1]     [0 1]     [2] 
                      > [1 4] x + [1 4] y + [7] 
                        [0 1]     [0 1]     [2] 
                      = f(minus(y),minus(x))    
        
        
        Following rules are (at-least) weakly oriented:
            minus(h(x)) =  [1 4] x + [0]
                           [0 1]     [0]
                        >= [1 4] x + [0]
                           [0 1]     [0]
                        =  h(minus(x))  
        
        minus(minus(x)) =  [1 8] x + [0]
                           [0 1]     [0]
                        >= [1 0] x + [0]
                           [0 1]     [0]
                        =  x            
        
* Step 3: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            minus(h(x)) -> h(minus(x))
        - Weak TRS:
            minus(f(x,y)) -> f(minus(y),minus(x))
            minus(minus(x)) -> x
        - Signature:
            {minus/1} / {f/2,h/1}
        - Obligation:
             derivational complexity wrt. signature {f,h,minus}
    + 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(f) = [1 0] x1 + [1 0] x2 + [4]
                     [0 1]      [0 1]      [6]
              p(h) = [1 0] x1 + [0]           
                     [0 1]      [1]           
          p(minus) = [1 1] x1 + [0]           
                     [0 1]      [0]           
        
        Following rules are strictly oriented:
        minus(h(x)) = [1 1] x + [1]
                      [0 1]     [1]
                    > [1 1] x + [0]
                      [0 1]     [1]
                    = h(minus(x))  
        
        
        Following rules are (at-least) weakly oriented:
          minus(f(x,y)) =  [1 1] x + [1 1] y + [10]
                           [0 1]     [0 1]     [6] 
                        >= [1 1] x + [1 1] y + [4] 
                           [0 1]     [0 1]     [6] 
                        =  f(minus(y),minus(x))    
        
        minus(minus(x)) =  [1 2] x + [0]           
                           [0 1]     [0]           
                        >= [1 0] x + [0]           
                           [0 1]     [0]           
                        =  x                       
        
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            minus(f(x,y)) -> f(minus(y),minus(x))
            minus(h(x)) -> h(minus(x))
            minus(minus(x)) -> x
        - Signature:
            {minus/1} / {f/2,h/1}
        - Obligation:
             derivational complexity wrt. signature {f,h,minus}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))