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


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


WORST_CASE(?,O(n^1))
* Step 1: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(f(X)) -> f(g(f(g(f(X)))))
            f(g(f(X))) -> f(g(X))
        - Signature:
            {f/1} / {g/1}
        - Obligation:
            innermost derivational complexity wrt. signature {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(f) = [1] x1 + [4]
            p(g) = [1] x1 + [8]
          
          Following rules are strictly oriented:
          f(g(f(X))) = [1] X + [16]
                     > [1] X + [12]
                     = f(g(X))     
          
          
          Following rules are (at-least) weakly oriented:
          f(f(X)) =  [1] X + [8]     
                  >= [1] X + [28]    
                  =  f(g(f(g(f(X)))))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(f(X)) -> f(g(f(g(f(X)))))
        - Weak TRS:
            f(g(f(X))) -> f(g(X))
        - Signature:
            {f/1} / {g/1}
        - Obligation:
            innermost derivational complexity wrt. signature {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(f) = [1 0 1]      [0]
                 [0 0 0] x1 + [0]
                 [0 0 0]      [1]
          p(g) = [1 0 0]      [0]
                 [0 0 0] x1 + [1]
                 [0 0 0]      [0]
        
        Following rules are strictly oriented:
        f(f(X)) = [1 0 1]     [1] 
                  [0 0 0] X + [0] 
                  [0 0 0]     [1] 
                > [1 0 1]     [0] 
                  [0 0 0] X + [0] 
                  [0 0 0]     [1] 
                = f(g(f(g(f(X)))))
        
        
        Following rules are (at-least) weakly oriented:
        f(g(f(X))) =  [1 0 1]     [0]
                      [0 0 0] X + [0]
                      [0 0 0]     [1]
                   >= [1 0 0]     [0]
                      [0 0 0] X + [0]
                      [0 0 0]     [1]
                   =  f(g(X))        
        
* Step 3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(f(X)) -> f(g(f(g(f(X)))))
            f(g(f(X))) -> f(g(X))
        - Signature:
            {f/1} / {g/1}
        - Obligation:
            innermost derivational complexity wrt. signature {f,g}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))