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


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WORST_CASE(?,O(n^2))
* Step 1: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            b(r(x)) -> r(b(x))
            b(w(x)) -> w(b(x))
            w(r(x)) -> r(w(x))
        - Signature:
            {b/1,w/1} / {r/1}
        - Obligation:
            innermost derivational complexity wrt. signature {b,r,w}
    + Applied Processor:
        NaturalMI {miDimension = 2, miDegree = 2, 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(b) = [1 1] x1 + [0]
                 [0 1]      [0]
          p(r) = [1 0] x1 + [0]
                 [0 1]      [2]
          p(w) = [1 0] x1 + [1]
                 [0 1]      [0]
        
        Following rules are strictly oriented:
        b(r(x)) = [1 1] x + [2]
                  [0 1]     [2]
                > [1 1] x + [0]
                  [0 1]     [2]
                = r(b(x))      
        
        
        Following rules are (at-least) weakly oriented:
        b(w(x)) =  [1 1] x + [1]
                   [0 1]     [0]
                >= [1 1] x + [1]
                   [0 1]     [0]
                =  w(b(x))      
        
        w(r(x)) =  [1 0] x + [1]
                   [0 1]     [2]
                >= [1 0] x + [1]
                   [0 1]     [2]
                =  r(w(x))      
        
* Step 2: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            b(w(x)) -> w(b(x))
            w(r(x)) -> r(w(x))
        - Weak TRS:
            b(r(x)) -> r(b(x))
        - Signature:
            {b/1,w/1} / {r/1}
        - Obligation:
            innermost derivational complexity wrt. signature {b,r,w}
    + Applied Processor:
        NaturalMI {miDimension = 2, miDegree = 2, 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(b) = [1 4] x1 + [1]
                 [0 1]      [1]
          p(r) = [1 0] x1 + [1]
                 [0 1]      [0]
          p(w) = [1 5] x1 + [1]
                 [0 1]      [2]
        
        Following rules are strictly oriented:
        b(w(x)) = [1 9] x + [10]
                  [0 1]     [3] 
                > [1 9] x + [7] 
                  [0 1]     [3] 
                = w(b(x))       
        
        
        Following rules are (at-least) weakly oriented:
        b(r(x)) =  [1 4] x + [2]
                   [0 1]     [1]
                >= [1 4] x + [2]
                   [0 1]     [1]
                =  r(b(x))      
        
        w(r(x)) =  [1 5] x + [2]
                   [0 1]     [2]
                >= [1 5] x + [2]
                   [0 1]     [2]
                =  r(w(x))      
        
* Step 3: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            w(r(x)) -> r(w(x))
        - Weak TRS:
            b(r(x)) -> r(b(x))
            b(w(x)) -> w(b(x))
        - Signature:
            {b/1,w/1} / {r/1}
        - Obligation:
            innermost derivational complexity wrt. signature {b,r,w}
    + Applied Processor:
        NaturalMI {miDimension = 2, miDegree = 2, 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(b) = [1 4] x1 + [4]
                 [0 1]      [0]
          p(r) = [1 0] x1 + [2]
                 [0 1]      [2]
          p(w) = [1 4] x1 + [0]
                 [0 1]      [0]
        
        Following rules are strictly oriented:
        w(r(x)) = [1 4] x + [10]
                  [0 1]     [2] 
                > [1 4] x + [2] 
                  [0 1]     [2] 
                = r(w(x))       
        
        
        Following rules are (at-least) weakly oriented:
        b(r(x)) =  [1 4] x + [14]
                   [0 1]     [2] 
                >= [1 4] x + [6] 
                   [0 1]     [2] 
                =  r(b(x))       
        
        b(w(x)) =  [1 8] x + [4] 
                   [0 1]     [0] 
                >= [1 8] x + [4] 
                   [0 1]     [0] 
                =  w(b(x))       
        
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            b(r(x)) -> r(b(x))
            b(w(x)) -> w(b(x))
            w(r(x)) -> r(w(x))
        - Signature:
            {b/1,w/1} / {r/1}
        - Obligation:
            innermost derivational complexity wrt. signature {b,r,w}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))