/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(Omega(n^1),O(n^1))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(g(X)) -> f(X)
        - Signature:
            {f/1} / {g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {g}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            f(g(X)) -> f(X)
        - Signature:
            {f/1} / {g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {g}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
*** Step 1.a:1.a:1: Ara.  MAYBE
    + Considered Problem:
        - Strict TRS:
            f(g(X)) -> f(X)
        - Signature:
            {f/1} / {g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {g}
    + 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"(1)] -(0)-> "A"(0)
          F (TrsFun "g") :: ["A"(1)] -(1)-> "A"(1)
          F (TrsFun "main") :: ["A"(1)] -(1)-> "A"(0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          f(g(X)) -> f(X)
          main(x1) -> f(x1)
        2. Weak:
          
        

*** Step 1.a:1.b:1: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            f(g(X)) -> f(X)
        - Signature:
            {f/1} / {g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {g}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          f(x){x -> g(x)} =
            f(g(x)) ->^+ f(x)
              = C[f(x) = f(x){}]

** Step 1.b:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(g(X)) -> f(X)
        - Signature:
            {f/1} / {g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {g}
    + 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:
          none
        
        Following symbols are considered usable:
          {f}
        TcT has computed the following interpretation:
          p(f) = [1] x1 + [0]
          p(g) = [1] x1 + [4]
        
        Following rules are strictly oriented:
        f(g(X)) = [1] X + [4]
                > [1] X + [0]
                = f(X)       
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 1.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(g(X)) -> f(X)
        - Signature:
            {f/1} / {g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {g}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^1))