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

WORST_CASE(?,O(n^2))