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


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WORST_CASE(?,O(n^1))
* Step 1: Sum.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            g(f(x),y) -> f(h(x,y))
            h(x,y) -> g(x,f(y))
        - Signature:
            {g/2,h/2} / {f/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {g,h} and constructors {f}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  MAYBE
    + Considered Problem:
        - Strict TRS:
            g(f(x),y) -> f(h(x,y))
            h(x,y) -> g(x,f(y))
        - Signature:
            {g/2,h/2} / {f/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {g,h} and constructors {f}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: Ara.  MAYBE
    + Considered Problem:
        - Strict TRS:
            g(f(x),y) -> f(h(x,y))
            h(x,y) -> g(x,f(y))
        - Signature:
            {g/2,h/2} / {f/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {g,h} and constructors {f}
    + Applied Processor:
        Ara {minDegree = 1, maxDegree = 3, araTimeout = 15, araRuleShifting = Just 1, isBestCase = True, mkCompletelyDefined = False, verboseOutput = False}
    + Details:
        Signatures used:
        ----------------
          F (TrsFun "f") :: ["A"(0, 0, 0)] -(0)-> "A"(0, 0, 0)
          F (TrsFun "f") :: ["A"(1, 2, 1)] -(0)-> "A"(0, 1, 1)
          F (TrsFun "g") :: ["A"(0, 0, 0) x "A"(0, 1, 1)] -(1)-> "A"(0, 0, 0)
          F (TrsFun "h") :: ["A"(0, 0, 0) x "A"(1, 1, 1)] -(1)-> "A"(0, 0, 0)
          F (TrsFun "main") :: ["A"(0, 0, 0) x "A"(1, 1, 1)] -(1)-> "A"(0, 0, 0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          g(f(x),y) -> f(h(x,y))
          h(x,y) -> g(x,f(y))
          main(x1,x2) -> h(x1,x2)
        2. Weak:
          
        

** Step 1.b:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            g(f(x),y) -> f(h(x,y))
            h(x,y) -> g(x,f(y))
        - Signature:
            {g/2,h/2} / {f/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {g,h} and constructors {f}
    + 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(f) = {1}
        
        Following symbols are considered usable:
          {g,h}
        TcT has computed the following interpretation:
          p(f) = [1] x1 + [2]
          p(g) = [8] x1 + [8]
          p(h) = [8] x1 + [9]
        
        Following rules are strictly oriented:
        g(f(x),y) = [8] x + [24]
                  > [8] x + [11]
                  = f(h(x,y))   
        
           h(x,y) = [8] x + [9] 
                  > [8] x + [8] 
                  = g(x,f(y))   
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 1.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            g(f(x),y) -> f(h(x,y))
            h(x,y) -> g(x,f(y))
        - Signature:
            {g/2,h/2} / {f/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {g,h} and constructors {f}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))