/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^1))
* Step 1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(h(x)) -> f(i(x))
            g(i(x)) -> g(h(x))
            h(a()) -> b()
            i(a()) -> b()
        - Signature:
            {f/1,g/1,h/1,i/1} / {a/0,b/0}
        - Obligation:
             derivational complexity wrt. signature {a,b,f,g,h,i}
    + 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(a) = [9]         
          p(b) = [1]         
          p(f) = [1] x1 + [1]
          p(g) = [1] x1 + [5]
          p(h) = [1] x1 + [0]
          p(i) = [1] x1 + [0]
        
        Following rules are strictly oriented:
        h(a()) = [9]
               > [1]
               = b()
        
        i(a()) = [9]
               > [1]
               = b()
        
        
        Following rules are (at-least) weakly oriented:
        f(h(x)) =  [1] x + [1]
                >= [1] x + [1]
                =  f(i(x))    
        
        g(i(x)) =  [1] x + [5]
                >= [1] x + [5]
                =  g(h(x))    
        
* Step 2: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(h(x)) -> f(i(x))
            g(i(x)) -> g(h(x))
        - Weak TRS:
            h(a()) -> b()
            i(a()) -> b()
        - Signature:
            {f/1,g/1,h/1,i/1} / {a/0,b/0}
        - Obligation:
             derivational complexity wrt. signature {a,b,f,g,h,i}
    + 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) = [6]          
            p(b) = [12]         
            p(f) = [1] x1 + [0] 
            p(g) = [1] x1 + [0] 
            p(h) = [1] x1 + [6] 
            p(i) = [1] x1 + [15]
          
          Following rules are strictly oriented:
          g(i(x)) = [1] x + [15]
                  > [1] x + [6] 
                  = g(h(x))     
          
          
          Following rules are (at-least) weakly oriented:
          f(h(x)) =  [1] x + [6] 
                  >= [1] x + [15]
                  =  f(i(x))     
          
           h(a()) =  [12]        
                  >= [12]        
                  =  b()         
          
           i(a()) =  [21]        
                  >= [12]        
                  =  b()         
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(h(x)) -> f(i(x))
        - Weak TRS:
            g(i(x)) -> g(h(x))
            h(a()) -> b()
            i(a()) -> b()
        - Signature:
            {f/1,g/1,h/1,i/1} / {a/0,b/0}
        - Obligation:
             derivational complexity wrt. signature {a,b,f,g,h,i}
    + 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) = [4]            
                 [5]            
          p(b) = [0]            
                 [0]            
          p(f) = [1 12] x1 + [0]
                 [0  0]      [3]
          p(g) = [1 4] x1 + [10]
                 [0 0]      [2] 
          p(h) = [1 1] x1 + [0] 
                 [0 0]      [2] 
          p(i) = [1 1] x1 + [8] 
                 [0 0]      [0] 
        
        Following rules are strictly oriented:
        f(h(x)) = [1 1] x + [24]
                  [0 0]     [3] 
                > [1 1] x + [8] 
                  [0 0]     [3] 
                = f(i(x))       
        
        
        Following rules are (at-least) weakly oriented:
        g(i(x)) =  [1 1] x + [18]
                   [0 0]     [2] 
                >= [1 1] x + [18]
                   [0 0]     [2] 
                =  g(h(x))       
        
         h(a()) =  [9]           
                   [2]           
                >= [0]           
                   [0]           
                =  b()           
        
         i(a()) =  [17]          
                   [0]           
                >= [0]           
                   [0]           
                =  b()           
        
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(h(x)) -> f(i(x))
            g(i(x)) -> g(h(x))
            h(a()) -> b()
            i(a()) -> b()
        - Signature:
            {f/1,g/1,h/1,i/1} / {a/0,b/0}
        - Obligation:
             derivational complexity wrt. signature {a,b,f,g,h,i}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))