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

*** Step 1.a:1.b:1: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            f(x,a()) -> x
            f(x,g(y)) -> f(g(x),y)
        - Signature:
            {f/2} / {a/0,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {a,g}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          f(x,y){y -> g(y)} =
            f(x,g(y)) ->^+ f(g(x),y)
              = C[f(g(x),y) = f(x,y){x -> g(x)}]

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(x,a()) -> x
            f(x,g(y)) -> f(g(x),y)
        - Signature:
            {f/2} / {a/0,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {a,g}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          f#(x,a()) -> c_1()
          f#(x,g(y)) -> c_2(f#(g(x),y))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(x,a()) -> c_1()
            f#(x,g(y)) -> c_2(f#(g(x),y))
        - Weak TRS:
            f(x,a()) -> x
            f(x,g(y)) -> f(g(x),y)
        - Signature:
            {f/2,f#/2} / {a/0,g/1,c_1/0,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {a,g}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1}
        by application of
          Pre({1}) = {2}.
        Here rules are labelled as follows:
          1: f#(x,a()) -> c_1()
          2: f#(x,g(y)) -> c_2(f#(g(x),y))
** Step 1.b:3: RemoveWeakSuffixes.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(x,g(y)) -> c_2(f#(g(x),y))
        - Weak DPs:
            f#(x,a()) -> c_1()
        - Weak TRS:
            f(x,a()) -> x
            f(x,g(y)) -> f(g(x),y)
        - Signature:
            {f/2,f#/2} / {a/0,g/1,c_1/0,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {a,g}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f#(x,g(y)) -> c_2(f#(g(x),y))
             -->_1 f#(x,a()) -> c_1():2
             -->_1 f#(x,g(y)) -> c_2(f#(g(x),y)):1
          
          2:W:f#(x,a()) -> c_1()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: f#(x,a()) -> c_1()
** Step 1.b:4: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(x,g(y)) -> c_2(f#(g(x),y))
        - Weak TRS:
            f(x,a()) -> x
            f(x,g(y)) -> f(g(x),y)
        - Signature:
            {f/2,f#/2} / {a/0,g/1,c_1/0,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {a,g}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          f#(x,g(y)) -> c_2(f#(g(x),y))
** Step 1.b:5: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(x,g(y)) -> c_2(f#(g(x),y))
        - Signature:
            {f/2,f#/2} / {a/0,g/1,c_1/0,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {a,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:
          uargs(c_2) = {1}
        
        Following symbols are considered usable:
          {f#}
        TcT has computed the following interpretation:
            p(a) = [1]                   
            p(f) = [2] x1 + [1] x2 + [0] 
            p(g) = [1] x1 + [1]          
           p(f#) = [3] x1 + [8] x2 + [11]
          p(c_1) = [0]                   
          p(c_2) = [1] x1 + [3]          
        
        Following rules are strictly oriented:
        f#(x,g(y)) = [3] x + [8] y + [19]
                   > [3] x + [8] y + [17]
                   = c_2(f#(g(x),y))     
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 1.b:6: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f#(x,g(y)) -> c_2(f#(g(x),y))
        - Signature:
            {f/2,f#/2} / {a/0,g/1,c_1/0,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {a,g}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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