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


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


WORST_CASE(?,O(n^2))
* Step 1: NaturalMI 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:
        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(c) = [1] x1 + [1] x2 + [8]
          p(d) = [1] x1 + [0]         
          p(f) = [1] x1 + [8]         
          p(g) = [1] x1 + [0]         
          p(s) = [1] x1 + [10]        
        
        Following rules are strictly oriented:
        f(x) = [1] x + [8]
             > [1] x + [0]
             = x          
        
        
        Following rules are (at-least) weakly oriented:
        f(c(s(x),y)) =  [1] x + [1] y + [26]
                     >= [1] x + [1] y + [26]
                     =  f(c(x,s(y)))        
        
             f(f(x)) =  [1] x + [16]        
                     >= [1] x + [16]        
                     =  f(d(f(x)))          
        
        g(c(x,s(y))) =  [1] x + [1] y + [18]
                     >= [1] x + [1] y + [18]
                     =  g(c(s(x),y))        
        
* 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))