/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^1))
* Step 1: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            b(b(x1)) -> c(d(x1))
            c(x1) -> g(x1)
            c(c(x1)) -> d(d(d(x1)))
            d(d(x1)) -> c(f(x1))
            d(d(d(x1))) -> g(c(x1))
            f(x1) -> a(g(x1))
            g(x1) -> d(a(b(x1)))
            g(g(x1)) -> b(c(x1))
        - Signature:
            {b/1,c/1,d/1,f/1,g/1} / {a/1}
        - Obligation:
             derivational complexity wrt. signature {a,b,c,d,f,g}
    + 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(a) = [1] x1 + [13]
            p(b) = [1] x1 + [0] 
            p(c) = [1] x1 + [0] 
            p(d) = [1] x1 + [0] 
            p(f) = [1] x1 + [15]
            p(g) = [1] x1 + [0] 
          
          Following rules are strictly oriented:
          f(x1) = [1] x1 + [15]
                > [1] x1 + [13]
                = a(g(x1))     
          
          
          Following rules are (at-least) weakly oriented:
             b(b(x1)) =  [1] x1 + [0] 
                      >= [1] x1 + [0] 
                      =  c(d(x1))     
          
                c(x1) =  [1] x1 + [0] 
                      >= [1] x1 + [0] 
                      =  g(x1)        
          
             c(c(x1)) =  [1] x1 + [0] 
                      >= [1] x1 + [0] 
                      =  d(d(d(x1)))  
          
             d(d(x1)) =  [1] x1 + [0] 
                      >= [1] x1 + [15]
                      =  c(f(x1))     
          
          d(d(d(x1))) =  [1] x1 + [0] 
                      >= [1] x1 + [0] 
                      =  g(c(x1))     
          
                g(x1) =  [1] x1 + [0] 
                      >= [1] x1 + [13]
                      =  d(a(b(x1)))  
          
             g(g(x1)) =  [1] x1 + [0] 
                      >= [1] x1 + [0] 
                      =  b(c(x1))     
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            b(b(x1)) -> c(d(x1))
            c(x1) -> g(x1)
            c(c(x1)) -> d(d(d(x1)))
            d(d(x1)) -> c(f(x1))
            d(d(d(x1))) -> g(c(x1))
            g(x1) -> d(a(b(x1)))
            g(g(x1)) -> b(c(x1))
        - Weak TRS:
            f(x1) -> a(g(x1))
        - Signature:
            {b/1,c/1,d/1,f/1,g/1} / {a/1}
        - Obligation:
             derivational complexity wrt. signature {a,b,c,d,f,g}
    + 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(a) = [1] x1 + [9] 
            p(b) = [1] x1 + [10]
            p(c) = [1] x1 + [8] 
            p(d) = [1] x1 + [0] 
            p(f) = [1] x1 + [10]
            p(g) = [1] x1 + [0] 
          
          Following rules are strictly oriented:
          b(b(x1)) = [1] x1 + [20]
                   > [1] x1 + [8] 
                   = c(d(x1))     
          
             c(x1) = [1] x1 + [8] 
                   > [1] x1 + [0] 
                   = g(x1)        
          
          c(c(x1)) = [1] x1 + [16]
                   > [1] x1 + [0] 
                   = d(d(d(x1)))  
          
          
          Following rules are (at-least) weakly oriented:
             d(d(x1)) =  [1] x1 + [0] 
                      >= [1] x1 + [18]
                      =  c(f(x1))     
          
          d(d(d(x1))) =  [1] x1 + [0] 
                      >= [1] x1 + [8] 
                      =  g(c(x1))     
          
                f(x1) =  [1] x1 + [10]
                      >= [1] x1 + [9] 
                      =  a(g(x1))     
          
                g(x1) =  [1] x1 + [0] 
                      >= [1] x1 + [19]
                      =  d(a(b(x1)))  
          
             g(g(x1)) =  [1] x1 + [0] 
                      >= [1] x1 + [18]
                      =  b(c(x1))     
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            d(d(x1)) -> c(f(x1))
            d(d(d(x1))) -> g(c(x1))
            g(x1) -> d(a(b(x1)))
            g(g(x1)) -> b(c(x1))
        - Weak TRS:
            b(b(x1)) -> c(d(x1))
            c(x1) -> g(x1)
            c(c(x1)) -> d(d(d(x1)))
            f(x1) -> a(g(x1))
        - Signature:
            {b/1,c/1,d/1,f/1,g/1} / {a/1}
        - Obligation:
             derivational complexity wrt. signature {a,b,c,d,f,g}
    + 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(a) = [1] x1 + [7] 
            p(b) = [1] x1 + [8] 
            p(c) = [1] x1 + [10]
            p(d) = [1] x1 + [5] 
            p(f) = [1] x1 + [7] 
            p(g) = [1] x1 + [0] 
          
          Following rules are strictly oriented:
          d(d(d(x1))) = [1] x1 + [15]
                      > [1] x1 + [10]
                      = g(c(x1))     
          
          
          Following rules are (at-least) weakly oriented:
          b(b(x1)) =  [1] x1 + [16]
                   >= [1] x1 + [15]
                   =  c(d(x1))     
          
             c(x1) =  [1] x1 + [10]
                   >= [1] x1 + [0] 
                   =  g(x1)        
          
          c(c(x1)) =  [1] x1 + [20]
                   >= [1] x1 + [15]
                   =  d(d(d(x1)))  
          
          d(d(x1)) =  [1] x1 + [10]
                   >= [1] x1 + [17]
                   =  c(f(x1))     
          
             f(x1) =  [1] x1 + [7] 
                   >= [1] x1 + [7] 
                   =  a(g(x1))     
          
             g(x1) =  [1] x1 + [0] 
                   >= [1] x1 + [20]
                   =  d(a(b(x1)))  
          
          g(g(x1)) =  [1] x1 + [0] 
                   >= [1] x1 + [18]
                   =  b(c(x1))     
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            d(d(x1)) -> c(f(x1))
            g(x1) -> d(a(b(x1)))
            g(g(x1)) -> b(c(x1))
        - Weak TRS:
            b(b(x1)) -> c(d(x1))
            c(x1) -> g(x1)
            c(c(x1)) -> d(d(d(x1)))
            d(d(d(x1))) -> g(c(x1))
            f(x1) -> a(g(x1))
        - Signature:
            {b/1,c/1,d/1,f/1,g/1} / {a/1}
        - Obligation:
             derivational complexity wrt. signature {a,b,c,d,f,g}
    + 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(a) = [1] x1 + [0]
            p(b) = [1] x1 + [8]
            p(c) = [1] x1 + [3]
            p(d) = [1] x1 + [2]
            p(f) = [1] x1 + [0]
            p(g) = [1] x1 + [0]
          
          Following rules are strictly oriented:
          d(d(x1)) = [1] x1 + [4]
                   > [1] x1 + [3]
                   = c(f(x1))    
          
          
          Following rules are (at-least) weakly oriented:
             b(b(x1)) =  [1] x1 + [16]
                      >= [1] x1 + [5] 
                      =  c(d(x1))     
          
                c(x1) =  [1] x1 + [3] 
                      >= [1] x1 + [0] 
                      =  g(x1)        
          
             c(c(x1)) =  [1] x1 + [6] 
                      >= [1] x1 + [6] 
                      =  d(d(d(x1)))  
          
          d(d(d(x1))) =  [1] x1 + [6] 
                      >= [1] x1 + [3] 
                      =  g(c(x1))     
          
                f(x1) =  [1] x1 + [0] 
                      >= [1] x1 + [0] 
                      =  a(g(x1))     
          
                g(x1) =  [1] x1 + [0] 
                      >= [1] x1 + [10]
                      =  d(a(b(x1)))  
          
             g(g(x1)) =  [1] x1 + [0] 
                      >= [1] x1 + [11]
                      =  b(c(x1))     
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            g(x1) -> d(a(b(x1)))
            g(g(x1)) -> b(c(x1))
        - Weak TRS:
            b(b(x1)) -> c(d(x1))
            c(x1) -> g(x1)
            c(c(x1)) -> d(d(d(x1)))
            d(d(x1)) -> c(f(x1))
            d(d(d(x1))) -> g(c(x1))
            f(x1) -> a(g(x1))
        - Signature:
            {b/1,c/1,d/1,f/1,g/1} / {a/1}
        - Obligation:
             derivational complexity wrt. signature {a,b,c,d,f,g}
    + Applied Processor:
        NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind triangular matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
          p(a) = [1 0] x1 + [0]
                 [0 0]      [0]
          p(b) = [1 1] x1 + [1]
                 [0 0]      [4]
          p(c) = [1 1] x1 + [2]
                 [0 0]      [4]
          p(d) = [1 1] x1 + [0]
                 [0 0]      [4]
          p(f) = [1 1] x1 + [2]
                 [0 0]      [0]
          p(g) = [1 1] x1 + [2]
                 [0 0]      [4]
        
        Following rules are strictly oriented:
           g(x1) = [1 1] x1 + [2]
                   [0 0]      [4]
                 > [1 1] x1 + [1]
                   [0 0]      [4]
                 = d(a(b(x1)))   
        
        g(g(x1)) = [1 1] x1 + [8]
                   [0 0]      [4]
                 > [1 1] x1 + [7]
                   [0 0]      [4]
                 = b(c(x1))      
        
        
        Following rules are (at-least) weakly oriented:
           b(b(x1)) =  [1 1] x1 + [6]
                       [0 0]      [4]
                    >= [1 1] x1 + [6]
                       [0 0]      [4]
                    =  c(d(x1))      
        
              c(x1) =  [1 1] x1 + [2]
                       [0 0]      [4]
                    >= [1 1] x1 + [2]
                       [0 0]      [4]
                    =  g(x1)         
        
           c(c(x1)) =  [1 1] x1 + [8]
                       [0 0]      [4]
                    >= [1 1] x1 + [8]
                       [0 0]      [4]
                    =  d(d(d(x1)))   
        
           d(d(x1)) =  [1 1] x1 + [4]
                       [0 0]      [4]
                    >= [1 1] x1 + [4]
                       [0 0]      [4]
                    =  c(f(x1))      
        
        d(d(d(x1))) =  [1 1] x1 + [8]
                       [0 0]      [4]
                    >= [1 1] x1 + [8]
                       [0 0]      [4]
                    =  g(c(x1))      
        
              f(x1) =  [1 1] x1 + [2]
                       [0 0]      [0]
                    >= [1 1] x1 + [2]
                       [0 0]      [0]
                    =  a(g(x1))      
        
* Step 6: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            b(b(x1)) -> c(d(x1))
            c(x1) -> g(x1)
            c(c(x1)) -> d(d(d(x1)))
            d(d(x1)) -> c(f(x1))
            d(d(d(x1))) -> g(c(x1))
            f(x1) -> a(g(x1))
            g(x1) -> d(a(b(x1)))
            g(g(x1)) -> b(c(x1))
        - Signature:
            {b/1,c/1,d/1,f/1,g/1} / {a/1}
        - Obligation:
             derivational complexity wrt. signature {a,b,c,d,f,g}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))