/export/starexec/sandbox2/solver/bin/starexec_run_tct_rci /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(?,O(n^1))
* Step 1: Sum WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            +(0(),y) -> y
            +(s(x),0()) -> s(x)
            +(s(x),s(y)) -> s(+(s(x),+(y,0())))
        - Signature:
            {+/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+} and constructors {0,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            +(0(),y) -> y
            +(s(x),0()) -> s(x)
            +(s(x),s(y)) -> s(+(s(x),+(y,0())))
        - Signature:
            {+/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+} and constructors {0,s}
    + Applied Processor:
        NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(linear):
        The following argument positions are considered usable:
          uargs(+) = {2},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {+}
        TcT has computed the following interpretation:
          p(+) = x1 + 8*x2
          p(0) = 0        
          p(s) = 3 + x1   
        
        Following rules are strictly oriented:
        +(s(x),s(y)) = 27 + x + 8*y       
                     > 6 + x + 8*y        
                     = s(+(s(x),+(y,0())))
        
        
        Following rules are (at-least) weakly oriented:
           +(0(),y) =  8*y  
                    >= y    
                    =  y    
        
        +(s(x),0()) =  3 + x
                    >= 3 + x
                    =  s(x) 
        
* Step 3: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            +(0(),y) -> y
            +(s(x),0()) -> s(x)
        - Weak TRS:
            +(s(x),s(y)) -> s(+(s(x),+(y,0())))
        - Signature:
            {+/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+} and constructors {0,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(+) = {2},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          {+}
        TcT has computed the following interpretation:
          p(+) = [1] x1 + [8] x2 + [2]
          p(0) = [0]                  
          p(s) = [1] x1 + [3]         
        
        Following rules are strictly oriented:
           +(0(),y) = [8] y + [2]
                    > [1] y + [0]
                    = y          
        
        +(s(x),0()) = [1] x + [5]
                    > [1] x + [3]
                    = s(x)       
        
        
        Following rules are (at-least) weakly oriented:
        +(s(x),s(y)) =  [1] x + [8] y + [29]
                     >= [1] x + [8] y + [24]
                     =  s(+(s(x),+(y,0()))) 
        
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),0()) -> s(x)
            +(s(x),s(y)) -> s(+(s(x),+(y,0())))
        - Signature:
            {+/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))