/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: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(X,X) -> c(X)
            f(X,c(X)) -> f(s(X),X)
            f(s(X),X) -> f(X,a(X))
        - Signature:
            {f/2} / {a/1,c/1,s/1}
        - Obligation:
             derivational complexity wrt. signature {a,c,f,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, 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(a) = [1] x1 + [0]         
          p(c) = [1] x1 + [0]         
          p(f) = [1] x1 + [1] x2 + [8]
          p(s) = [1] x1 + [0]         
        
        Following rules are strictly oriented:
        f(X,X) = [2] X + [8]
               > [1] X + [0]
               = c(X)       
        
        
        Following rules are (at-least) weakly oriented:
        f(X,c(X)) =  [2] X + [8]
                  >= [2] X + [8]
                  =  f(s(X),X)  
        
        f(s(X),X) =  [2] X + [8]
                  >= [2] X + [8]
                  =  f(X,a(X))  
        
* Step 2: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(X,c(X)) -> f(s(X),X)
            f(s(X),X) -> f(X,a(X))
        - Weak TRS:
            f(X,X) -> c(X)
        - Signature:
            {f/2} / {a/1,c/1,s/1}
        - Obligation:
             derivational complexity wrt. signature {a,c,f,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(a) = [1] x1 + [8]         
            p(c) = [1] x1 + [0]         
            p(f) = [1] x1 + [1] x2 + [0]
            p(s) = [1] x1 + [9]         
          
          Following rules are strictly oriented:
          f(s(X),X) = [2] X + [9]
                    > [2] X + [8]
                    = f(X,a(X))  
          
          
          Following rules are (at-least) weakly oriented:
             f(X,X) =  [2] X + [0]
                    >= [1] X + [0]
                    =  c(X)       
          
          f(X,c(X)) =  [2] X + [0]
                    >= [2] X + [9]
                    =  f(s(X),X)  
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(X,c(X)) -> f(s(X),X)
        - Weak TRS:
            f(X,X) -> c(X)
            f(s(X),X) -> f(X,a(X))
        - Signature:
            {f/2} / {a/1,c/1,s/1}
        - Obligation:
             derivational complexity wrt. signature {a,c,f,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, 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(a) = [1] x1 + [0]         
          p(c) = [1] x1 + [5]         
          p(f) = [1] x1 + [1] x2 + [9]
          p(s) = [1] x1 + [0]         
        
        Following rules are strictly oriented:
        f(X,c(X)) = [2] X + [14]
                  > [2] X + [9] 
                  = f(s(X),X)   
        
        
        Following rules are (at-least) weakly oriented:
           f(X,X) =  [2] X + [9]
                  >= [1] X + [5]
                  =  c(X)       
        
        f(s(X),X) =  [2] X + [9]
                  >= [2] X + [9]
                  =  f(X,a(X))  
        
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(X,X) -> c(X)
            f(X,c(X)) -> f(s(X),X)
            f(s(X),X) -> f(X,a(X))
        - Signature:
            {f/2} / {a/1,c/1,s/1}
        - Obligation:
             derivational complexity wrt. signature {a,c,f,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))