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


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WORST_CASE(?,O(n^2))
* Step 1: WeightGap.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            a(a(x1)) -> a(b(a(x1)))
            b(a(b(x1))) -> a(c(a(x1)))
        - Signature:
            {a/1,b/1} / {c/1}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,c}
    + 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) = [1] x1 + [3]
            p(b) = [1] x1 + [8]
            p(c) = [1] x1 + [5]
          
          Following rules are strictly oriented:
          b(a(b(x1))) = [1] x1 + [19]
                      > [1] x1 + [11]
                      = a(c(a(x1)))  
          
          
          Following rules are (at-least) weakly oriented:
          a(a(x1)) =  [1] x1 + [6] 
                   >= [1] x1 + [14]
                   =  a(b(a(x1)))  
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: MI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            a(a(x1)) -> a(b(a(x1)))
        - Weak TRS:
            b(a(b(x1))) -> a(c(a(x1)))
        - Signature:
            {a/1,b/1} / {c/1}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,c}
    + Applied Processor:
        MI {miKind = Automaton Nothing, miDimension = 4, miUArgs = NoUArgs, miURules = NoURules, miSelector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind Automaton Nothing:
        
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
          p(a) = [1 0 1 1]       [0]
                 [0 0 1 0] x_1 + [0]
                 [0 0 1 0]       [2]
                 [0 1 2 0]       [0]
          p(b) = [1 0 0 1]       [0]
                 [0 0 1 0] x_1 + [0]
                 [0 1 0 0]       [1]
                 [0 0 0 0]       [0]
          p(c) = [1 0 0 0]       [2]
                 [0 0 0 0] x_1 + [0]
                 [0 0 0 0]       [0]
                 [0 0 0 0]       [1]
        
        Following rules are strictly oriented:
        a(a(x1)) = [1 1 4 1]      [2]
                   [0 0 1 0] x1 + [2]
                   [0 0 1 0]      [4]
                   [0 0 3 0]      [4]
                 > [1 1 4 1]      [1]
                   [0 0 1 0] x1 + [1]
                   [0 0 1 0]      [3]
                   [0 0 3 0]      [4]
                 = a(b(a(x1)))       
        
        
        Following rules are (at-least) weakly oriented:
        b(a(b(x1))) =  [1 3 1 1]      [3]
                       [0 1 0 0] x1 + [3]
                       [0 1 0 0]      [2]
                       [0 0 0 0]      [0]
                    >= [1 0 1 1]      [3]
                       [0 0 0 0] x1 + [0]
                       [0 0 0 0]      [2]
                       [0 0 0 0]      [0]
                    =  a(c(a(x1)))       
        
* Step 3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a(a(x1)) -> a(b(a(x1)))
            b(a(b(x1))) -> a(c(a(x1)))
        - Signature:
            {a/1,b/1} / {c/1}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,c}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))