/export/starexec/sandbox2/solver/bin/starexec_run_tct_dci /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(?,O(n^1))
* Step 1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(a()) -> g(h(a()))
            h(g(x)) -> g(h(f(x)))
            k(x,h(x),a()) -> h(x)
            k(f(x),y,x) -> f(x)
        - Signature:
            {f/1,h/1,k/3} / {a/0,g/1}
        - Obligation:
            innermost derivational complexity wrt. signature {a,f,g,h,k}
    + 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) = [4]                           
          p(f) = [1] x1 + [0]                  
          p(g) = [1] x1 + [0]                  
          p(h) = [1] x1 + [0]                  
          p(k) = [1] x1 + [1] x2 + [1] x3 + [8]
        
        Following rules are strictly oriented:
        k(x,h(x),a()) = [2] x + [12]       
                      > [1] x + [0]        
                      = h(x)               
        
          k(f(x),y,x) = [2] x + [1] y + [8]
                      > [1] x + [0]        
                      = f(x)               
        
        
        Following rules are (at-least) weakly oriented:
         f(a()) =  [4]        
                >= [4]        
                =  g(h(a()))  
        
        h(g(x)) =  [1] x + [0]
                >= [1] x + [0]
                =  g(h(f(x))) 
        
* Step 2: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(a()) -> g(h(a()))
            h(g(x)) -> g(h(f(x)))
        - Weak TRS:
            k(x,h(x),a()) -> h(x)
            k(f(x),y,x) -> f(x)
        - Signature:
            {f/1,h/1,k/3} / {a/0,g/1}
        - Obligation:
            innermost derivational complexity wrt. signature {a,f,g,h,k}
    + 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) = [10]                          
            p(f) = [1] x1 + [2]                  
            p(g) = [1] x1 + [0]                  
            p(h) = [1] x1 + [1]                  
            p(k) = [1] x1 + [1] x2 + [1] x3 + [0]
          
          Following rules are strictly oriented:
          f(a()) = [12]     
                 > [11]     
                 = g(h(a()))
          
          
          Following rules are (at-least) weakly oriented:
                h(g(x)) =  [1] x + [1]        
                        >= [1] x + [3]        
                        =  g(h(f(x)))         
          
          k(x,h(x),a()) =  [2] x + [11]       
                        >= [1] x + [1]        
                        =  h(x)               
          
            k(f(x),y,x) =  [2] x + [1] y + [2]
                        >= [1] x + [2]        
                        =  f(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:
            h(g(x)) -> g(h(f(x)))
        - Weak TRS:
            f(a()) -> g(h(a()))
            k(x,h(x),a()) -> h(x)
            k(f(x),y,x) -> f(x)
        - Signature:
            {f/1,h/1,k/3} / {a/0,g/1}
        - Obligation:
            innermost derivational complexity wrt. signature {a,f,g,h,k}
    + 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(a) = [5]                                       
                 [0]                                       
                 [1]                                       
          p(f) = [1 0 6]      [0]                          
                 [0 0 1] x1 + [0]                          
                 [0 0 0]      [0]                          
          p(g) = [1 0 7]      [1]                          
                 [0 0 0] x1 + [1]                          
                 [0 0 0]      [0]                          
          p(h) = [1 1 0]      [5]                          
                 [0 0 0] x1 + [1]                          
                 [0 0 0]      [0]                          
          p(k) = [1 2 4]      [1 1 0]      [1 0 0]      [2]
                 [0 0 4] x1 + [0 0 1] x2 + [0 0 1] x3 + [5]
                 [0 0 0]      [0 0 0]      [0 0 0]      [0]
        
        Following rules are strictly oriented:
        h(g(x)) = [1 0 7]     [7]
                  [0 0 0] x + [1]
                  [0 0 0]     [0]
                > [1 0 7]     [6]
                  [0 0 0] x + [1]
                  [0 0 0]     [0]
                = g(h(f(x)))     
        
        
        Following rules are (at-least) weakly oriented:
               f(a()) =  [11]                       
                         [1]                        
                         [0]                        
                      >= [11]                       
                         [1]                        
                         [0]                        
                      =  g(h(a()))                  
        
        k(x,h(x),a()) =  [2 3 4]     [13]           
                         [0 0 4] x + [6]            
                         [0 0 0]     [0]            
                      >= [1 1 0]     [5]            
                         [0 0 0] x + [1]            
                         [0 0 0]     [0]            
                      =  h(x)                       
        
          k(f(x),y,x) =  [2 0 8]     [1 1 0]     [2]
                         [0 0 1] x + [0 0 1] y + [5]
                         [0 0 0]     [0 0 0]     [0]
                      >= [1 0 6]     [0]            
                         [0 0 1] x + [0]            
                         [0 0 0]     [0]            
                      =  f(x)                       
        
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(a()) -> g(h(a()))
            h(g(x)) -> g(h(f(x)))
            k(x,h(x),a()) -> h(x)
            k(f(x),y,x) -> f(x)
        - Signature:
            {f/1,h/1,k/3} / {a/0,g/1}
        - Obligation:
            innermost derivational complexity wrt. signature {a,f,g,h,k}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))