/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^1))
* Step 1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a(x1) -> b(b(x1))
            b(b(b(x1))) -> a(x1)
        - Signature:
            {a/1,b/1} / {}
        - Obligation:
             derivational complexity wrt. signature {a,b}
    + 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) = [1] x1 + [24]
          p(b) = [1] x1 + [8] 
        
        Following rules are strictly oriented:
        a(x1) = [1] x1 + [24]
              > [1] x1 + [16]
              = b(b(x1))     
        
        
        Following rules are (at-least) weakly oriented:
        b(b(b(x1))) =  [1] x1 + [24]
                    >= [1] x1 + [24]
                    =  a(x1)        
        
* Step 2: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            b(b(b(x1))) -> a(x1)
        - Weak TRS:
            a(x1) -> b(b(x1))
        - Signature:
            {a/1,b/1} / {}
        - Obligation:
             derivational complexity wrt. signature {a,b}
    + 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) = [1] x1 + [4]
          p(b) = [1] x1 + [2]
        
        Following rules are strictly oriented:
        b(b(b(x1))) = [1] x1 + [6]
                    > [1] x1 + [4]
                    = a(x1)       
        
        
        Following rules are (at-least) weakly oriented:
        a(x1) =  [1] x1 + [4]
              >= [1] x1 + [4]
              =  b(b(x1))    
        
* Step 3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a(x1) -> b(b(x1))
            b(b(b(x1))) -> a(x1)
        - Signature:
            {a/1,b/1} / {}
        - Obligation:
             derivational complexity wrt. signature {a,b}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))