/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: WeightGap.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            f(x) -> x
            f(c(s(x),y)) -> f(c(x,s(y)))
            f(f(x)) -> f(d(f(x)))
            g(c(x,s(y))) -> g(c(s(x),y))
        - Signature:
            {f/1,g/1} / {c/2,d/1,s/1}
        - Obligation:
             derivational complexity wrt. signature {c,d,f,g,s}
    + 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(c) = [1] x1 + [1] x2 + [0]
            p(d) = [1] x1 + [0]         
            p(f) = [1] x1 + [9]         
            p(g) = [1] x1 + [0]         
            p(s) = [1] x1 + [0]         
          
          Following rules are strictly oriented:
          f(x) = [1] x + [9]
               > [1] x + [0]
               = x          
          
          
          Following rules are (at-least) weakly oriented:
          f(c(s(x),y)) =  [1] x + [1] y + [9]
                       >= [1] x + [1] y + [9]
                       =  f(c(x,s(y)))       
          
               f(f(x)) =  [1] x + [18]       
                       >= [1] x + [18]       
                       =  f(d(f(x)))         
          
          g(c(x,s(y))) =  [1] x + [1] y + [0]
                       >= [1] x + [1] y + [0]
                       =  g(c(s(x),y))       
          
        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:
            f(c(s(x),y)) -> f(c(x,s(y)))
            f(f(x)) -> f(d(f(x)))
            g(c(x,s(y))) -> g(c(s(x),y))
        - Weak TRS:
            f(x) -> x
        - Signature:
            {f/1,g/1} / {c/2,d/1,s/1}
        - Obligation:
             derivational complexity wrt. signature {c,d,f,g,s}
    + 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(c) = [1 2] x1 + [1 3] x2 + [4]
                 [0 1]      [0 0]      [0]
          p(d) = [1 0] x1 + [5]           
                 [0 0]      [2]           
          p(f) = [1 2] x1 + [0]           
                 [0 1]      [6]           
          p(g) = [1 1] x1 + [0]           
                 [0 0]      [4]           
          p(s) = [1 0] x1 + [2]           
                 [0 1]      [1]           
        
        Following rules are strictly oriented:
        f(c(s(x),y)) = [1 4] x + [1 3] y + [10]
                       [0 1]     [0 0]     [7] 
                     > [1 4] x + [1 3] y + [9] 
                       [0 1]     [0 0]     [6] 
                     = f(c(x,s(y)))            
        
             f(f(x)) = [1 4] x + [12]          
                       [0 1]     [12]          
                     > [1 2] x + [9]           
                       [0 0]     [8]           
                     = f(d(f(x)))              
        
        
        Following rules are (at-least) weakly oriented:
                f(x) =  [1 2] x + [0]          
                        [0 1]     [6]          
                     >= [1 0] x + [0]          
                        [0 1]     [0]          
                     =  x                      
        
        g(c(x,s(y))) =  [1 3] x + [1 3] y + [9]
                        [0 0]     [0 0]     [4]
                     >= [1 3] x + [1 3] y + [9]
                        [0 0]     [0 0]     [4]
                     =  g(c(s(x),y))           
        
* Step 3: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            g(c(x,s(y))) -> g(c(s(x),y))
        - Weak TRS:
            f(x) -> x
            f(c(s(x),y)) -> f(c(x,s(y)))
            f(f(x)) -> f(d(f(x)))
        - Signature:
            {f/1,g/1} / {c/2,d/1,s/1}
        - Obligation:
             derivational complexity wrt. signature {c,d,f,g,s}
    + 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(c) = [1 0] x1 + [1 4] x2 + [0]
                 [0 1]      [0 0]      [0]
          p(d) = [1 4] x1 + [0]           
                 [0 0]      [0]           
          p(f) = [1 4] x1 + [0]           
                 [0 1]      [0]           
          p(g) = [1 0] x1 + [5]           
                 [0 0]      [1]           
          p(s) = [1 0] x1 + [0]           
                 [0 1]      [2]           
        
        Following rules are strictly oriented:
        g(c(x,s(y))) = [1 0] x + [1 4] y + [13]
                       [0 0]     [0 0]     [1] 
                     > [1 0] x + [1 4] y + [5] 
                       [0 0]     [0 0]     [1] 
                     = g(c(s(x),y))            
        
        
        Following rules are (at-least) weakly oriented:
                f(x) =  [1 4] x + [0]          
                        [0 1]     [0]          
                     >= [1 0] x + [0]          
                        [0 1]     [0]          
                     =  x                      
        
        f(c(s(x),y)) =  [1 4] x + [1 4] y + [8]
                        [0 1]     [0 0]     [2]
                     >= [1 4] x + [1 4] y + [8]
                        [0 1]     [0 0]     [0]
                     =  f(c(x,s(y)))           
        
             f(f(x)) =  [1 8] x + [0]          
                        [0 1]     [0]          
                     >= [1 8] x + [0]          
                        [0 0]     [0]          
                     =  f(d(f(x)))             
        
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(x) -> x
            f(c(s(x),y)) -> f(c(x,s(y)))
            f(f(x)) -> f(d(f(x)))
            g(c(x,s(y))) -> g(c(s(x),y))
        - Signature:
            {f/1,g/1} / {c/2,d/1,s/1}
        - Obligation:
             derivational complexity wrt. signature {c,d,f,g,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))