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


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WORST_CASE(?,O(n^1))
* Step 1: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(a(),f(b(),f(a(),x))) -> f(a(),f(b(),f(b(),f(a(),x))))
            f(b(),f(b(),f(b(),x))) -> f(b(),f(b(),x))
        - Signature:
            {f/2} / {a/0,b/0}
        - Obligation:
             derivational complexity wrt. signature {a,b,f}
    + 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) = [0]                  
            p(b) = [1]                  
            p(f) = [1] x1 + [1] x2 + [0]
          
          Following rules are strictly oriented:
          f(b(),f(b(),f(b(),x))) = [1] x + [3]    
                                 > [1] x + [2]    
                                 = f(b(),f(b(),x))
          
          
          Following rules are (at-least) weakly oriented:
          f(a(),f(b(),f(a(),x))) =  [1] x + [1]                  
                                 >= [1] x + [2]                  
                                 =  f(a(),f(b(),f(b(),f(a(),x))))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: Bounds.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(a(),f(b(),f(a(),x))) -> f(a(),f(b(),f(b(),f(a(),x))))
        - Weak TRS:
            f(b(),f(b(),f(b(),x))) -> f(b(),f(b(),x))
        - Signature:
            {f/2} / {a/0,b/0}
        - Obligation:
             derivational complexity wrt. signature {a,b,f}
    + Applied Processor:
        Bounds {initialAutomaton = minimal, enrichment = match}
    + Details:
        The problem is match-bounded by 1.
        The enriched problem is compatible with follwoing automaton.
          a_0() -> 1
          a_1() -> 2
          a_1() -> 8
          b_0() -> 1
          b_1() -> 4
          b_1() -> 6
          f_0(1,1) -> 1
          f_1(2,3) -> 1
          f_1(4,5) -> 3
          f_1(6,7) -> 5
          f_1(8,1) -> 7
          f_1(8,3) -> 7
* Step 3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(a(),f(b(),f(a(),x))) -> f(a(),f(b(),f(b(),f(a(),x))))
            f(b(),f(b(),f(b(),x))) -> f(b(),f(b(),x))
        - Signature:
            {f/2} / {a/0,b/0}
        - Obligation:
             derivational complexity wrt. signature {a,b,f}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))