/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:
            f(1()) -> f(g(1()))
            f(f(x)) -> f(x)
            g(0()) -> g(f(0()))
            g(g(x)) -> g(x)
        - Signature:
            {f/1,g/1} / {0/0,1/0}
        - Obligation:
             derivational complexity wrt. signature {0,1,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(0) = [9]         
            p(1) = [8]         
            p(f) = [1] x1 + [0]
            p(g) = [1] x1 + [4]
          
          Following rules are strictly oriented:
          g(g(x)) = [1] x + [8]
                  > [1] x + [4]
                  = g(x)       
          
          
          Following rules are (at-least) weakly oriented:
           f(1()) =  [8]        
                  >= [12]       
                  =  f(g(1()))  
          
          f(f(x)) =  [1] x + [0]
                  >= [1] x + [0]
                  =  f(x)       
          
           g(0()) =  [13]       
                  >= [13]       
                  =  g(f(0()))  
          
        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:
            f(1()) -> f(g(1()))
            f(f(x)) -> f(x)
            g(0()) -> g(f(0()))
        - Weak TRS:
            g(g(x)) -> g(x)
        - Signature:
            {f/1,g/1} / {0/0,1/0}
        - Obligation:
             derivational complexity wrt. signature {0,1,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(0) = [1]         
            p(1) = [5]         
            p(f) = [1] x1 + [2]
            p(g) = [1] x1 + [0]
          
          Following rules are strictly oriented:
          f(f(x)) = [1] x + [4]
                  > [1] x + [2]
                  = f(x)       
          
          
          Following rules are (at-least) weakly oriented:
           f(1()) =  [7]        
                  >= [7]        
                  =  f(g(1()))  
          
           g(0()) =  [1]        
                  >= [3]        
                  =  g(f(0()))  
          
          g(g(x)) =  [1] x + [0]
                  >= [1] x + [0]
                  =  g(x)       
          
        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(1()) -> f(g(1()))
            g(0()) -> g(f(0()))
        - Weak TRS:
            f(f(x)) -> f(x)
            g(g(x)) -> g(x)
        - Signature:
            {f/1,g/1} / {0/0,1/0}
        - Obligation:
             derivational complexity wrt. signature {0,1,f,g}
    + Applied Processor:
        NaturalMI {miDimension = 3, 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(0) = [3]             
                 [0]             
                 [5]             
          p(1) = [2]             
                 [2]             
                 [0]             
          p(f) = [1 4 0]      [0]
                 [0 0 0] x1 + [2]
                 [0 0 0]      [2]
          p(g) = [1 3 2]      [0]
                 [0 0 0] x1 + [0]
                 [0 0 0]      [5]
        
        Following rules are strictly oriented:
        f(1()) = [10]     
                 [2]      
                 [2]      
               > [8]      
                 [2]      
                 [2]      
               = f(g(1()))
        
        
        Following rules are (at-least) weakly oriented:
        f(f(x)) =  [1 4 0]     [8] 
                   [0 0 0] x + [2] 
                   [0 0 0]     [2] 
                >= [1 4 0]     [0] 
                   [0 0 0] x + [2] 
                   [0 0 0]     [2] 
                =  f(x)            
        
         g(0()) =  [13]            
                   [0]             
                   [5]             
                >= [13]            
                   [0]             
                   [5]             
                =  g(f(0()))       
        
        g(g(x)) =  [1 3 2]     [10]
                   [0 0 0] x + [0] 
                   [0 0 0]     [5] 
                >= [1 3 2]     [0] 
                   [0 0 0] x + [0] 
                   [0 0 0]     [5] 
                =  g(x)            
        
* Step 4: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            g(0()) -> g(f(0()))
        - Weak TRS:
            f(1()) -> f(g(1()))
            f(f(x)) -> f(x)
            g(g(x)) -> g(x)
        - Signature:
            {f/1,g/1} / {0/0,1/0}
        - Obligation:
             derivational complexity wrt. signature {0,1,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(0) = [0]           
                 [12]          
          p(1) = [4]           
                 [0]           
          p(f) = [1 1] x1 + [0]
                 [0 0]      [0]
          p(g) = [1 2] x1 + [0]
                 [0 0]      [0]
        
        Following rules are strictly oriented:
        g(0()) = [24]     
                 [0]      
               > [12]     
                 [0]      
               = g(f(0()))
        
        
        Following rules are (at-least) weakly oriented:
         f(1()) =  [4]          
                   [0]          
                >= [4]          
                   [0]          
                =  f(g(1()))    
        
        f(f(x)) =  [1 1] x + [0]
                   [0 0]     [0]
                >= [1 1] x + [0]
                   [0 0]     [0]
                =  f(x)         
        
        g(g(x)) =  [1 2] x + [0]
                   [0 0]     [0]
                >= [1 2] x + [0]
                   [0 0]     [0]
                =  g(x)         
        
* Step 5: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(1()) -> f(g(1()))
            f(f(x)) -> f(x)
            g(0()) -> g(f(0()))
            g(g(x)) -> g(x)
        - Signature:
            {f/1,g/1} / {0/0,1/0}
        - Obligation:
             derivational complexity wrt. signature {0,1,f,g}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))