/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:
            a(y,x) -> y
            a(y,c(b(a(0(),x),0()))) -> b(a(c(b(0(),y)),x),0())
            b(x,y) -> c(a(c(y),a(0(),x)))
        - Signature:
            {a/2,b/2} / {0/0,c/1}
        - Obligation:
             derivational complexity wrt. signature {0,a,b,c}
    + 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(0) = [0]                  
            p(a) = [1] x1 + [1] x2 + [0]
            p(b) = [1] x1 + [1] x2 + [1]
            p(c) = [1] x1 + [0]         
          
          Following rules are strictly oriented:
          b(x,y) = [1] x + [1] y + [1]
                 > [1] x + [1] y + [0]
                 = c(a(c(y),a(0(),x)))
          
          
          Following rules are (at-least) weakly oriented:
                           a(y,x) =  [1] x + [1] y + [0]    
                                  >= [1] y + [0]            
                                  =  y                      
          
          a(y,c(b(a(0(),x),0()))) =  [1] x + [1] y + [1]    
                                  >= [1] x + [1] y + [2]    
                                  =  b(a(c(b(0(),y)),x),0())
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            a(y,x) -> y
            a(y,c(b(a(0(),x),0()))) -> b(a(c(b(0(),y)),x),0())
        - Weak TRS:
            b(x,y) -> c(a(c(y),a(0(),x)))
        - Signature:
            {a/2,b/2} / {0/0,c/1}
        - Obligation:
             derivational complexity wrt. signature {0,a,b,c}
    + 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(0) = [0]                  
            p(a) = [1] x1 + [1] x2 + [2]
            p(b) = [1] x1 + [1] x2 + [4]
            p(c) = [1] x1 + [0]         
          
          Following rules are strictly oriented:
          a(y,x) = [1] x + [1] y + [2]
                 > [1] y + [0]        
                 = y                  
          
          
          Following rules are (at-least) weakly oriented:
          a(y,c(b(a(0(),x),0()))) =  [1] x + [1] y + [8]    
                                  >= [1] x + [1] y + [10]   
                                  =  b(a(c(b(0(),y)),x),0())
          
                           b(x,y) =  [1] x + [1] y + [4]    
                                  >= [1] x + [1] y + [4]    
                                  =  c(a(c(y),a(0(),x)))    
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: MI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            a(y,c(b(a(0(),x),0()))) -> b(a(c(b(0(),y)),x),0())
        - Weak TRS:
            a(y,x) -> y
            b(x,y) -> c(a(c(y),a(0(),x)))
        - Signature:
            {a/2,b/2} / {0/0,c/1}
        - Obligation:
             derivational complexity wrt. signature {0,a,b,c}
    + Applied Processor:
        MI {miKind = Automaton (Just 2), miDimension = 3, miUArgs = NoUArgs, miURules = NoURules, miSelector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind Automaton (Just 2):
        
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
          p(0) = [0]                            
                 [0]                            
                 [2]                            
          p(a) = [1 0 0]       [1 0 0]       [0]
                 [0 1 2] x_1 + [0 1 0] x_2 + [0]
                 [0 0 1]       [0 0 1]       [0]
          p(b) = [1 1 0]       [1 0 0]       [0]
                 [0 0 0] x_1 + [0 0 0] x_2 + [0]
                 [0 0 0]       [0 0 0]       [0]
          p(c) = [1 0 0]       [0]              
                 [0 0 0] x_1 + [0]              
                 [0 0 0]       [0]              
        
        Following rules are strictly oriented:
        a(y,c(b(a(0(),x),0()))) = [1 1 0]     [1 0 0]     [4]
                                  [0 0 0] x + [0 1 2] y + [0]
                                  [0 0 0]     [0 0 1]     [0]
                                > [1 1 0]     [1 0 0]     [0]
                                  [0 0 0] x + [0 0 0] y + [0]
                                  [0 0 0]     [0 0 0]     [0]
                                = b(a(c(b(0(),y)),x),0())    
        
        
        Following rules are (at-least) weakly oriented:
        a(y,x) =  [1 0 0]     [1 0 0]     [0]
                  [0 1 0] x + [0 1 2] y + [0]
                  [0 0 1]     [0 0 1]     [0]
               >= [1 0 0]     [0]            
                  [0 1 0] y + [0]            
                  [0 0 1]     [0]            
               =  y                          
        
        b(x,y) =  [1 1 0]     [1 0 0]     [0]
                  [0 0 0] x + [0 0 0] y + [0]
                  [0 0 0]     [0 0 0]     [0]
               >= [1 0 0]     [1 0 0]     [0]
                  [0 0 0] x + [0 0 0] y + [0]
                  [0 0 0]     [0 0 0]     [0]
               =  c(a(c(y),a(0(),x)))        
        
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a(y,x) -> y
            a(y,c(b(a(0(),x),0()))) -> b(a(c(b(0(),y)),x),0())
            b(x,y) -> c(a(c(y),a(0(),x)))
        - Signature:
            {a/2,b/2} / {0/0,c/1}
        - Obligation:
             derivational complexity wrt. signature {0,a,b,c}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))