/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:
            f(s(X)) -> f(X)
            g(cons(0(),Y)) -> g(Y)
            g(cons(s(X),Y)) -> s(X)
            h(cons(X,Y)) -> h(g(cons(X,Y)))
        - Signature:
            {f/1,g/1,h/1} / {0/0,cons/2,s/1}
        - Obligation:
             derivational complexity wrt. signature {0,cons,f,g,h,s}
    + 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(0) = [2]                  
            p(cons) = [1] x1 + [1] x2 + [1]
               p(f) = [1] x1 + [0]         
               p(g) = [1] x1 + [0]         
               p(h) = [1] x1 + [0]         
               p(s) = [1] x1 + [5]         
          
          Following rules are strictly oriented:
                  f(s(X)) = [1] X + [5]        
                          > [1] X + [0]        
                          = f(X)               
          
           g(cons(0(),Y)) = [1] Y + [3]        
                          > [1] Y + [0]        
                          = g(Y)               
          
          g(cons(s(X),Y)) = [1] X + [1] Y + [6]
                          > [1] X + [5]        
                          = s(X)               
          
          
          Following rules are (at-least) weakly oriented:
          h(cons(X,Y)) =  [1] X + [1] Y + [1]
                       >= [1] X + [1] Y + [1]
                       =  h(g(cons(X,Y)))    
          
        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:
            h(cons(X,Y)) -> h(g(cons(X,Y)))
        - Weak TRS:
            f(s(X)) -> f(X)
            g(cons(0(),Y)) -> g(Y)
            g(cons(s(X),Y)) -> s(X)
        - Signature:
            {f/1,g/1,h/1} / {0/0,cons/2,s/1}
        - Obligation:
             derivational complexity wrt. signature {0,cons,f,g,h,s}
    + Applied Processor:
        NaturalMI {miDimension = 2, 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(0) = [4]                      
                    [1]                      
          p(cons) = [1 0] x1 + [1 0] x2 + [0]
                    [0 0]      [0 0]      [1]
             p(f) = [1 4] x1 + [6]           
                    [0 0]      [0]           
             p(g) = [1 0] x1 + [1]           
                    [0 0]      [0]           
             p(h) = [1 3] x1 + [0]           
                    [0 0]      [0]           
             p(s) = [1 4] x1 + [7]           
                    [0 0]      [0]           
        
        Following rules are strictly oriented:
        h(cons(X,Y)) = [1 0] X + [1 0] Y + [3]
                       [0 0]     [0 0]     [0]
                     > [1 0] X + [1 0] Y + [1]
                       [0 0]     [0 0]     [0]
                     = h(g(cons(X,Y)))        
        
        
        Following rules are (at-least) weakly oriented:
                f(s(X)) =  [1 4] X + [13]         
                           [0 0]     [0]          
                        >= [1 4] X + [6]          
                           [0 0]     [0]          
                        =  f(X)                   
        
         g(cons(0(),Y)) =  [1 0] Y + [5]          
                           [0 0]     [0]          
                        >= [1 0] Y + [1]          
                           [0 0]     [0]          
                        =  g(Y)                   
        
        g(cons(s(X),Y)) =  [1 4] X + [1 0] Y + [8]
                           [0 0]     [0 0]     [0]
                        >= [1 4] X + [7]          
                           [0 0]     [0]          
                        =  s(X)                   
        
* Step 3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(s(X)) -> f(X)
            g(cons(0(),Y)) -> g(Y)
            g(cons(s(X),Y)) -> s(X)
            h(cons(X,Y)) -> h(g(cons(X,Y)))
        - Signature:
            {f/1,g/1,h/1} / {0/0,cons/2,s/1}
        - Obligation:
             derivational complexity wrt. signature {0,cons,f,g,h,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))