/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:
            g(x,h(y,z)) -> h(g(x,y),z)
            g(f(x,y),z) -> f(x,g(y,z))
            g(h(x,y),z) -> g(x,f(y,z))
        - Signature:
            {g/2} / {f/2,h/2}
        - Obligation:
            innermost derivational complexity wrt. signature {f,g,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 + [0] 
            p(g) = [1] x1 + [1] x2 + [14]
            p(h) = [1] x1 + [1] x2 + [1] 
          
          Following rules are strictly oriented:
          g(h(x,y),z) = [1] x + [1] y + [1] z + [15]
                      > [1] x + [1] y + [1] z + [14]
                      = g(x,f(y,z))                 
          
          
          Following rules are (at-least) weakly oriented:
          g(x,h(y,z)) =  [1] x + [1] y + [1] z + [15]
                      >= [1] x + [1] y + [1] z + [15]
                      =  h(g(x,y),z)                 
          
          g(f(x,y),z) =  [1] x + [1] y + [1] z + [14]
                      >= [1] x + [1] y + [1] z + [14]
                      =  f(x,g(y,z))                 
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            g(x,h(y,z)) -> h(g(x,y),z)
            g(f(x,y),z) -> f(x,g(y,z))
        - Weak TRS:
            g(h(x,y),z) -> g(x,f(y,z))
        - Signature:
            {g/2} / {f/2,h/2}
        - Obligation:
            innermost derivational complexity wrt. signature {f,g,h}
    + 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(f) = [1 4] x1 + [1 0] x2 + [1]
                 [0 0]      [0 1]      [2]
          p(g) = [1 4] x1 + [1 0] x2 + [0]
                 [0 1]      [0 0]      [0]
          p(h) = [1 0] x1 + [1 4] x2 + [1]
                 [0 1]      [0 0]      [0]
        
        Following rules are strictly oriented:
        g(f(x,y),z) = [1 4] x + [1 4] y + [1 0] z + [9]
                      [0 0]     [0 1]     [0 0]     [2]
                    > [1 4] x + [1 4] y + [1 0] z + [1]
                      [0 0]     [0 1]     [0 0]     [2]
                    = f(x,g(y,z))                      
        
        
        Following rules are (at-least) weakly oriented:
        g(x,h(y,z)) =  [1 4] x + [1 0] y + [1 4] z + [1]
                       [0 1]     [0 0]     [0 0]     [0]
                    >= [1 4] x + [1 0] y + [1 4] z + [1]
                       [0 1]     [0 0]     [0 0]     [0]
                    =  h(g(x,y),z)                      
        
        g(h(x,y),z) =  [1 4] x + [1 4] y + [1 0] z + [1]
                       [0 1]     [0 0]     [0 0]     [0]
                    >= [1 4] x + [1 4] y + [1 0] z + [1]
                       [0 1]     [0 0]     [0 0]     [0]
                    =  g(x,f(y,z))                      
        
* Step 3: MI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            g(x,h(y,z)) -> h(g(x,y),z)
        - Weak TRS:
            g(f(x,y),z) -> f(x,g(y,z))
            g(h(x,y),z) -> g(x,f(y,z))
        - Signature:
            {g/2} / {f/2,h/2}
        - Obligation:
            innermost derivational complexity wrt. signature {f,g,h}
    + Applied Processor:
        MI {miKind = Automaton Nothing, miDimension = 2, 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(f) = [1 0] x_1 + [1 0] x_2 + [0]
                 [0 0]       [0 0]       [0]
          p(g) = [1 0] x_1 + [1 4] x_2 + [0]
                 [0 0]       [0 1]       [6]
          p(h) = [1 0] x_1 + [1 0] x_2 + [0]
                 [0 1]       [0 0]       [2]
        
        Following rules are strictly oriented:
        g(x,h(y,z)) = [1 0] x + [1 4] y + [1 0] z + [8]
                      [0 0]     [0 1]     [0 0]     [8]
                    > [1 0] x + [1 4] y + [1 0] z + [0]
                      [0 0]     [0 1]     [0 0]     [8]
                    = h(g(x,y),z)                      
        
        
        Following rules are (at-least) weakly oriented:
        g(f(x,y),z) =  [1 0] x + [1 0] y + [1 4] z + [0]
                       [0 0]     [0 0]     [0 1]     [6]
                    >= [1 0] x + [1 0] y + [1 4] z + [0]
                       [0 0]     [0 0]     [0 0]     [0]
                    =  f(x,g(y,z))                      
        
        g(h(x,y),z) =  [1 0] x + [1 0] y + [1 4] z + [0]
                       [0 0]     [0 0]     [0 1]     [6]
                    >= [1 0] x + [1 0] y + [1 0] z + [0]
                       [0 0]     [0 0]     [0 0]     [6]
                    =  g(x,f(y,z))                      
        
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            g(x,h(y,z)) -> h(g(x,y),z)
            g(f(x,y),z) -> f(x,g(y,z))
            g(h(x,y),z) -> g(x,f(y,z))
        - Signature:
            {g/2} / {f/2,h/2}
        - Obligation:
            innermost derivational complexity wrt. signature {f,g,h}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))