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


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WORST_CASE(?,O(n^1))
* Step 1: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            p(f(f(x))) -> q(f(g(x)))
            p(g(g(x))) -> q(g(f(x)))
            q(f(f(x))) -> p(f(g(x)))
            q(g(g(x))) -> p(g(f(x)))
        - Signature:
            {p/1,q/1} / {f/1,g/1}
        - Obligation:
             derivational complexity wrt. signature {f,g,p,q}
    + 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 + [0] 
            p(g) = [1] x1 + [0] 
            p(p) = [1] x1 + [0] 
            p(q) = [1] x1 + [15]
          
          Following rules are strictly oriented:
          q(f(f(x))) = [1] x + [15]
                     > [1] x + [0] 
                     = p(f(g(x)))  
          
          q(g(g(x))) = [1] x + [15]
                     > [1] x + [0] 
                     = p(g(f(x)))  
          
          
          Following rules are (at-least) weakly oriented:
          p(f(f(x))) =  [1] x + [0] 
                     >= [1] x + [15]
                     =  q(f(g(x)))  
          
          p(g(g(x))) =  [1] x + [0] 
                     >= [1] x + [15]
                     =  q(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:
            p(f(f(x))) -> q(f(g(x)))
            p(g(g(x))) -> q(g(f(x)))
        - Weak TRS:
            q(f(f(x))) -> p(f(g(x)))
            q(g(g(x))) -> p(g(f(x)))
        - Signature:
            {p/1,q/1} / {f/1,g/1}
        - Obligation:
             derivational complexity wrt. signature {f,g,p,q}
    + 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 0]      [0]
                 [0 0 2] x1 + [0]
                 [0 0 0]      [1]
          p(g) = [1 0 0]      [0]
                 [0 0 0] x1 + [0]
                 [0 0 0]      [0]
          p(p) = [1 0 1]      [0]
                 [0 0 0] x1 + [0]
                 [0 0 0]      [0]
          p(q) = [1 1 0]      [0]
                 [0 0 0] x1 + [0]
                 [0 0 0]      [0]
        
        Following rules are strictly oriented:
        p(f(f(x))) = [1 0 0]     [1]
                     [0 0 0] x + [0]
                     [0 0 0]     [0]
                   > [1 0 0]     [0]
                     [0 0 0] x + [0]
                     [0 0 0]     [0]
                   = q(f(g(x)))     
        
        
        Following rules are (at-least) weakly oriented:
        p(g(g(x))) =  [1 0 0]     [0]
                      [0 0 0] x + [0]
                      [0 0 0]     [0]
                   >= [1 0 0]     [0]
                      [0 0 0] x + [0]
                      [0 0 0]     [0]
                   =  q(g(f(x)))     
        
        q(f(f(x))) =  [1 0 0]     [2]
                      [0 0 0] x + [0]
                      [0 0 0]     [0]
                   >= [1 0 0]     [1]
                      [0 0 0] x + [0]
                      [0 0 0]     [0]
                   =  p(f(g(x)))     
        
        q(g(g(x))) =  [1 0 0]     [0]
                      [0 0 0] x + [0]
                      [0 0 0]     [0]
                   >= [1 0 0]     [0]
                      [0 0 0] x + [0]
                      [0 0 0]     [0]
                   =  p(g(f(x)))     
        
* Step 3: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            p(g(g(x))) -> q(g(f(x)))
        - Weak TRS:
            p(f(f(x))) -> q(f(g(x)))
            q(f(f(x))) -> p(f(g(x)))
            q(g(g(x))) -> p(g(f(x)))
        - Signature:
            {p/1,q/1} / {f/1,g/1}
        - Obligation:
             derivational complexity wrt. signature {f,g,p,q}
    + 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 0]      [0]
                 [0 0 2] x1 + [0]
                 [0 0 0]      [1]
          p(g) = [1 0 0]      [2]
                 [0 0 0] x1 + [1]
                 [0 0 0]      [0]
          p(p) = [1 2 0]      [0]
                 [0 0 0] x1 + [0]
                 [0 0 0]      [2]
          p(q) = [1 3 0]      [0]
                 [0 0 0] x1 + [0]
                 [0 0 0]      [2]
        
        Following rules are strictly oriented:
        p(g(g(x))) = [1 0 0]     [6]
                     [0 0 0] x + [0]
                     [0 0 0]     [2]
                   > [1 0 0]     [5]
                     [0 0 0] x + [0]
                     [0 0 0]     [2]
                   = q(g(f(x)))     
        
        
        Following rules are (at-least) weakly oriented:
        p(f(f(x))) =  [1 0 0]     [4]
                      [0 0 0] x + [0]
                      [0 0 0]     [2]
                   >= [1 0 0]     [2]
                      [0 0 0] x + [0]
                      [0 0 0]     [2]
                   =  q(f(g(x)))     
        
        q(f(f(x))) =  [1 0 0]     [6]
                      [0 0 0] x + [0]
                      [0 0 0]     [2]
                   >= [1 0 0]     [2]
                      [0 0 0] x + [0]
                      [0 0 0]     [2]
                   =  p(f(g(x)))     
        
        q(g(g(x))) =  [1 0 0]     [7]
                      [0 0 0] x + [0]
                      [0 0 0]     [2]
                   >= [1 0 0]     [4]
                      [0 0 0] x + [0]
                      [0 0 0]     [2]
                   =  p(g(f(x)))     
        
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            p(f(f(x))) -> q(f(g(x)))
            p(g(g(x))) -> q(g(f(x)))
            q(f(f(x))) -> p(f(g(x)))
            q(g(g(x))) -> p(g(f(x)))
        - Signature:
            {p/1,q/1} / {f/1,g/1}
        - Obligation:
             derivational complexity wrt. signature {f,g,p,q}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))