/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: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(x,y) -> h(x,y)
            f(x,y) -> h(y,x)
            h(x,x) -> x
        - Signature:
            {f/2,h/2} / {}
        - Obligation:
            innermost derivational complexity wrt. signature {f,h}
    + 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(f) = [1] x1 + [1] x2 + [1]
          p(h) = [1] x1 + [1] x2 + [1]
        
        Following rules are strictly oriented:
        h(x,x) = [2] x + [1]
               > [1] x + [0]
               = x          
        
        
        Following rules are (at-least) weakly oriented:
        f(x,y) =  [1] x + [1] y + [1]
               >= [1] x + [1] y + [1]
               =  h(x,y)             
        
        f(x,y) =  [1] x + [1] y + [1]
               >= [1] x + [1] y + [1]
               =  h(y,x)             
        
* Step 2: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(x,y) -> h(x,y)
            f(x,y) -> h(y,x)
        - Weak TRS:
            h(x,x) -> x
        - Signature:
            {f/2,h/2} / {}
        - Obligation:
            innermost derivational complexity wrt. signature {f,h}
    + 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 + [1] x2 + [1]
            p(h) = [1] x1 + [1] x2 + [0]
          
          Following rules are strictly oriented:
          f(x,y) = [1] x + [1] y + [1]
                 > [1] x + [1] y + [0]
                 = h(x,y)             
          
          f(x,y) = [1] x + [1] y + [1]
                 > [1] x + [1] y + [0]
                 = h(y,x)             
          
          
          Following rules are (at-least) weakly oriented:
          h(x,x) =  [2] x + [0]
                 >= [1] x + [0]
                 =  x          
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(x,y) -> h(x,y)
            f(x,y) -> h(y,x)
            h(x,x) -> x
        - Signature:
            {f/2,h/2} / {}
        - Obligation:
            innermost derivational complexity wrt. signature {f,h}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))