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


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


WORST_CASE(?,O(n^2))
* Step 1: NaturalMI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            f(a,cons(x,k)) -> f(cons(x,a),k)
            f(a,empty()) -> g(a,empty())
            g(cons(x,k),d) -> g(k,cons(x,d))
            g(empty(),d) -> d
        - Signature:
            {f/2,g/2} / {cons/2,empty/0}
        - Obligation:
             derivational complexity wrt. signature {cons,empty,f,g}
    + 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(cons) = [1] x1 + [1] x2 + [2]
          p(empty) = [8]                  
              p(f) = [1] x1 + [1] x2 + [0]
              p(g) = [1] x1 + [1] x2 + [0]
        
        Following rules are strictly oriented:
        g(empty(),d) = [1] d + [8]
                     > [1] d + [0]
                     = d          
        
        
        Following rules are (at-least) weakly oriented:
        f(a,cons(x,k)) =  [1] a + [1] k + [1] x + [2]
                       >= [1] a + [1] k + [1] x + [2]
                       =  f(cons(x,a),k)             
        
          f(a,empty()) =  [1] a + [8]                
                       >= [1] a + [8]                
                       =  g(a,empty())               
        
        g(cons(x,k),d) =  [1] d + [1] k + [1] x + [2]
                       >= [1] d + [1] k + [1] x + [2]
                       =  g(k,cons(x,d))             
        
* Step 2: NaturalMI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            f(a,cons(x,k)) -> f(cons(x,a),k)
            f(a,empty()) -> g(a,empty())
            g(cons(x,k),d) -> g(k,cons(x,d))
        - Weak TRS:
            g(empty(),d) -> d
        - Signature:
            {f/2,g/2} / {cons/2,empty/0}
        - Obligation:
             derivational complexity wrt. signature {cons,empty,f,g}
    + 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(cons) = [1] x1 + [1] x2 + [2]
          p(empty) = [0]                  
              p(f) = [1] x1 + [1] x2 + [1]
              p(g) = [1] x1 + [1] x2 + [0]
        
        Following rules are strictly oriented:
        f(a,empty()) = [1] a + [1] 
                     > [1] a + [0] 
                     = g(a,empty())
        
        
        Following rules are (at-least) weakly oriented:
        f(a,cons(x,k)) =  [1] a + [1] k + [1] x + [3]
                       >= [1] a + [1] k + [1] x + [3]
                       =  f(cons(x,a),k)             
        
        g(cons(x,k),d) =  [1] d + [1] k + [1] x + [2]
                       >= [1] d + [1] k + [1] x + [2]
                       =  g(k,cons(x,d))             
        
          g(empty(),d) =  [1] d + [0]                
                       >= [1] d + [0]                
                       =  d                          
        
* Step 3: NaturalMI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            f(a,cons(x,k)) -> f(cons(x,a),k)
            g(cons(x,k),d) -> g(k,cons(x,d))
        - Weak TRS:
            f(a,empty()) -> g(a,empty())
            g(empty(),d) -> d
        - Signature:
            {f/2,g/2} / {cons/2,empty/0}
        - Obligation:
             derivational complexity wrt. signature {cons,empty,f,g}
    + 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(cons) = [1 0] x1 + [1 0] x2 + [0]
                     [0 0]      [0 1]      [2]
          p(empty) = [0]                      
                     [0]                      
              p(f) = [1 4] x1 + [1 7] x2 + [0]
                     [0 1]      [0 1]      [3]
              p(g) = [1 1] x1 + [1 0] x2 + [0]
                     [0 1]      [0 1]      [1]
        
        Following rules are strictly oriented:
        f(a,cons(x,k)) = [1 4] a + [1 7] k + [1 0] x + [14]
                         [0 1]     [0 1]     [0 0]     [5] 
                       > [1 4] a + [1 7] k + [1 0] x + [8] 
                         [0 1]     [0 1]     [0 0]     [5] 
                       = f(cons(x,a),k)                    
        
        g(cons(x,k),d) = [1 0] d + [1 1] k + [1 0] x + [2] 
                         [0 1]     [0 1]     [0 0]     [3] 
                       > [1 0] d + [1 1] k + [1 0] x + [0] 
                         [0 1]     [0 1]     [0 0]     [3] 
                       = g(k,cons(x,d))                    
        
        
        Following rules are (at-least) weakly oriented:
        f(a,empty()) =  [1 4] a + [0]
                        [0 1]     [3]
                     >= [1 1] a + [0]
                        [0 1]     [1]
                     =  g(a,empty()) 
        
        g(empty(),d) =  [1 0] d + [0]
                        [0 1]     [1]
                     >= [1 0] d + [0]
                        [0 1]     [0]
                     =  d            
        
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(a,cons(x,k)) -> f(cons(x,a),k)
            f(a,empty()) -> g(a,empty())
            g(cons(x,k),d) -> g(k,cons(x,d))
            g(empty(),d) -> d
        - Signature:
            {f/2,g/2} / {cons/2,empty/0}
        - Obligation:
             derivational complexity wrt. signature {cons,empty,f,g}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))