/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: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            *(x,0()) -> 0()
            *(*(x,y),z) -> *(x,*(y,z))
            *(1(),y) -> y
            *(i(x),x) -> 1()
        - Signature:
            {*/2} / {0/0,1/0,i/1}
        - Obligation:
             derivational complexity wrt. signature {*,0,1,i}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, 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] x1 + [1] x2 + [10]
          p(0) = [0]                   
          p(1) = [13]                  
          p(i) = [1] x1 + [3]          
        
        Following rules are strictly oriented:
        *(x,0()) = [1] x + [10]
                 > [0]         
                 = 0()         
        
        *(1(),y) = [1] y + [23]
                 > [1] y + [0] 
                 = y           
        
        
        Following rules are (at-least) weakly oriented:
        *(*(x,y),z) =  [1] x + [1] y + [1] z + [20]
                    >= [1] x + [1] y + [1] z + [20]
                    =  *(x,*(y,z))                 
        
          *(i(x),x) =  [2] x + [13]                
                    >= [13]                        
                    =  1()                         
        
* Step 2: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            *(*(x,y),z) -> *(x,*(y,z))
            *(i(x),x) -> 1()
        - Weak TRS:
            *(x,0()) -> 0()
            *(1(),y) -> y
        - Signature:
            {*/2} / {0/0,1/0,i/1}
        - Obligation:
             derivational complexity wrt. signature {*,0,1,i}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, 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] x1 + [1] x2 + [0]
          p(0) = [0]                  
          p(1) = [7]                  
          p(i) = [1] x1 + [12]        
        
        Following rules are strictly oriented:
        *(i(x),x) = [2] x + [12]
                  > [7]         
                  = 1()         
        
        
        Following rules are (at-least) weakly oriented:
           *(x,0()) =  [1] x + [0]                
                    >= [0]                        
                    =  0()                        
        
        *(*(x,y),z) =  [1] x + [1] y + [1] z + [0]
                    >= [1] x + [1] y + [1] z + [0]
                    =  *(x,*(y,z))                
        
           *(1(),y) =  [1] y + [7]                
                    >= [1] y + [0]                
                    =  y                          
        
* Step 3: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            *(*(x,y),z) -> *(x,*(y,z))
        - Weak TRS:
            *(x,0()) -> 0()
            *(1(),y) -> y
            *(i(x),x) -> 1()
        - Signature:
            {*/2} / {0/0,1/0,i/1}
        - Obligation:
             derivational complexity wrt. signature {*,0,1,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(*) = [1 6] x1 + [1 0] x2 + [0]
                 [0 1]      [0 1]      [1]
          p(0) = [7]                      
                 [3]                      
          p(1) = [0]                      
                 [0]                      
          p(i) = [1 0] x1 + [0]           
                 [0 1]      [1]           
        
        Following rules are strictly oriented:
        *(*(x,y),z) = [1 12] x + [1 6] y + [1 0] z + [6]
                      [0  1]     [0 1]     [0 1]     [2]
                    > [1 6] x + [1 6] y + [1 0] z + [0] 
                      [0 1]     [0 1]     [0 1]     [2] 
                    = *(x,*(y,z))                       
        
        
        Following rules are (at-least) weakly oriented:
         *(x,0()) =  [1 6] x + [7]
                     [0 1]     [4]
                  >= [7]          
                     [3]          
                  =  0()          
        
         *(1(),y) =  [1 0] y + [0]
                     [0 1]     [1]
                  >= [1 0] y + [0]
                     [0 1]     [0]
                  =  y            
        
        *(i(x),x) =  [2 6] x + [6]
                     [0 2]     [2]
                  >= [0]          
                     [0]          
                  =  1()          
        
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            *(x,0()) -> 0()
            *(*(x,y),z) -> *(x,*(y,z))
            *(1(),y) -> y
            *(i(x),x) -> 1()
        - Signature:
            {*/2} / {0/0,1/0,i/1}
        - Obligation:
             derivational complexity wrt. signature {*,0,1,i}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))