/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:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            +(0(),s(y)) -> s(y)
            s(+(0(),y)) -> s(y)
        - Signature:
            {+/2,s/1} / {0/0}
        - Obligation:
            innermost derivational complexity wrt. signature {+,0,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(+) = [1] x1 + [1] x2 + [2]
          p(0) = [2]                  
          p(s) = [1] x1 + [0]         
        
        Following rules are strictly oriented:
           +(x,0()) = [1] x + [4]
                    > [1] x + [0]
                    = x          
        
        +(0(),s(y)) = [1] y + [4]
                    > [1] y + [0]
                    = s(y)       
        
        s(+(0(),y)) = [1] y + [4]
                    > [1] y + [0]
                    = s(y)       
        
        
        Following rules are (at-least) weakly oriented:
        +(x,s(y)) =  [1] x + [1] y + [2]
                  >= [1] x + [1] y + [2]
                  =  s(+(x,y))          
        
* Step 2: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            +(x,s(y)) -> s(+(x,y))
        - Weak TRS:
            +(x,0()) -> x
            +(0(),s(y)) -> s(y)
            s(+(0(),y)) -> s(y)
        - Signature:
            {+/2,s/1} / {0/0}
        - Obligation:
            innermost derivational complexity wrt. signature {+,0,s}
    + 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(+) = [1 0] x1 + [1 1] x2 + [0]
                 [0 1]      [0 1]      [0]
          p(0) = [0]                      
                 [0]                      
          p(s) = [1 0] x1 + [0]           
                 [0 1]      [1]           
        
        Following rules are strictly oriented:
        +(x,s(y)) = [1 0] x + [1 1] y + [1]
                    [0 1]     [0 1]     [1]
                  > [1 0] x + [1 1] y + [0]
                    [0 1]     [0 1]     [1]
                  = s(+(x,y))              
        
        
        Following rules are (at-least) weakly oriented:
           +(x,0()) =  [1 0] x + [0]
                       [0 1]     [0]
                    >= [1 0] x + [0]
                       [0 1]     [0]
                    =  x            
        
        +(0(),s(y)) =  [1 1] y + [1]
                       [0 1]     [1]
                    >= [1 0] y + [0]
                       [0 1]     [1]
                    =  s(y)         
        
        s(+(0(),y)) =  [1 1] y + [0]
                       [0 1]     [1]
                    >= [1 0] y + [0]
                       [0 1]     [1]
                    =  s(y)         
        
* Step 3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            +(0(),s(y)) -> s(y)
            s(+(0(),y)) -> s(y)
        - Signature:
            {+/2,s/1} / {0/0}
        - Obligation:
            innermost derivational complexity wrt. signature {+,0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))