/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:
            duplicate(Cons(x,xs)) -> Cons(x,Cons(x,duplicate(xs)))
            duplicate(Nil()) -> Nil()
            goal(x) -> duplicate(x)
        - Signature:
            {duplicate/1,goal/1} / {Cons/2,Nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {duplicate,goal} and constructors {Cons,Nil}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            duplicate(Cons(x,xs)) -> Cons(x,Cons(x,duplicate(xs)))
            duplicate(Nil()) -> Nil()
            goal(x) -> duplicate(x)
        - Signature:
            {duplicate/1,goal/1} / {Cons/2,Nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {duplicate,goal} and constructors {Cons,Nil}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
*** Step 1.a:1.a:1: Ara.  MAYBE
    + Considered Problem:
        - Strict TRS:
            duplicate(Cons(x,xs)) -> Cons(x,Cons(x,duplicate(xs)))
            duplicate(Nil()) -> Nil()
            goal(x) -> duplicate(x)
        - Signature:
            {duplicate/1,goal/1} / {Cons/2,Nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {duplicate,goal} and constructors {Cons,Nil}
    + Applied Processor:
        Ara {minDegree = 1, maxDegree = 3, araTimeout = 15, araRuleShifting = Just 1, isBestCase = True, mkCompletelyDefined = False, verboseOutput = False}
    + Details:
        Signatures used:
        ----------------
          F (TrsFun "Cons") :: ["A"(0) x "A"(1)] -(1)-> "A"(1)
          F (TrsFun "Cons") :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
          F (TrsFun "Nil") :: [] -(0)-> "A"(1)
          F (TrsFun "Nil") :: [] -(0)-> "A"(0)
          F (TrsFun "duplicate") :: ["A"(1)] -(1)-> "A"(0)
          F (TrsFun "goal") :: ["A"(1)] -(1)-> "A"(0)
          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:
          duplicate(Cons(x,xs)) -> Cons(x,Cons(x,duplicate(xs)))
          duplicate(Nil()) -> Nil()
          goal(x) -> duplicate(x)
          main(x1) -> goal(x1)
        2. Weak:
          
        

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

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            duplicate(Cons(x,xs)) -> Cons(x,Cons(x,duplicate(xs)))
            duplicate(Nil()) -> Nil()
            goal(x) -> duplicate(x)
        - Signature:
            {duplicate/1,goal/1} / {Cons/2,Nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {duplicate,goal} and constructors {Cons,Nil}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs))
          duplicate#(Nil()) -> c_2()
          goal#(x) -> c_3(duplicate#(x))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs))
            duplicate#(Nil()) -> c_2()
            goal#(x) -> c_3(duplicate#(x))
        - Weak TRS:
            duplicate(Cons(x,xs)) -> Cons(x,Cons(x,duplicate(xs)))
            duplicate(Nil()) -> Nil()
            goal(x) -> duplicate(x)
        - Signature:
            {duplicate/1,goal/1,duplicate#/1,goal#/1} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {duplicate#,goal#} and constructors {Cons,Nil}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2}
        by application of
          Pre({2}) = {1,3}.
        Here rules are labelled as follows:
          1: duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs))
          2: duplicate#(Nil()) -> c_2()
          3: goal#(x) -> c_3(duplicate#(x))
** Step 1.b:3: RemoveWeakSuffixes.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs))
            goal#(x) -> c_3(duplicate#(x))
        - Weak DPs:
            duplicate#(Nil()) -> c_2()
        - Weak TRS:
            duplicate(Cons(x,xs)) -> Cons(x,Cons(x,duplicate(xs)))
            duplicate(Nil()) -> Nil()
            goal(x) -> duplicate(x)
        - Signature:
            {duplicate/1,goal/1,duplicate#/1,goal#/1} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {duplicate#,goal#} and constructors {Cons,Nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs))
             -->_1 duplicate#(Nil()) -> c_2():3
             -->_1 duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs)):1
          
          2:S:goal#(x) -> c_3(duplicate#(x))
             -->_1 duplicate#(Nil()) -> c_2():3
             -->_1 duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs)):1
          
          3:W:duplicate#(Nil()) -> c_2()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: duplicate#(Nil()) -> c_2()
** Step 1.b:4: RemoveHeads.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs))
            goal#(x) -> c_3(duplicate#(x))
        - Weak TRS:
            duplicate(Cons(x,xs)) -> Cons(x,Cons(x,duplicate(xs)))
            duplicate(Nil()) -> Nil()
            goal(x) -> duplicate(x)
        - Signature:
            {duplicate/1,goal/1,duplicate#/1,goal#/1} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {duplicate#,goal#} and constructors {Cons,Nil}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs))
           -->_1 duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs)):1
        
        2:S:goal#(x) -> c_3(duplicate#(x))
           -->_1 duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs)):1
        
        
        Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
        
        [(2,goal#(x) -> c_3(duplicate#(x)))]
** Step 1.b:5: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs))
        - Weak TRS:
            duplicate(Cons(x,xs)) -> Cons(x,Cons(x,duplicate(xs)))
            duplicate(Nil()) -> Nil()
            goal(x) -> duplicate(x)
        - Signature:
            {duplicate/1,goal/1,duplicate#/1,goal#/1} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {duplicate#,goal#} and constructors {Cons,Nil}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs))
** Step 1.b:6: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs))
        - Signature:
            {duplicate/1,goal/1,duplicate#/1,goal#/1} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {duplicate#,goal#} and constructors {Cons,Nil}
    + 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_1) = {1}
        
        Following symbols are considered usable:
          {duplicate#,goal#}
        TcT has computed the following interpretation:
                p(Cons) = [1] x2 + [8]
                 p(Nil) = [8]         
           p(duplicate) = [1]         
                p(goal) = [1] x1 + [1]
          p(duplicate#) = [1] x1 + [0]
               p(goal#) = [1] x1 + [8]
                 p(c_1) = [1] x1 + [0]
                 p(c_2) = [0]         
                 p(c_3) = [1] x1 + [0]
        
        Following rules are strictly oriented:
        duplicate#(Cons(x,xs)) = [1] xs + [8]       
                               > [1] xs + [0]       
                               = c_1(duplicate#(xs))
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 1.b:7: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            duplicate#(Cons(x,xs)) -> c_1(duplicate#(xs))
        - Signature:
            {duplicate/1,goal/1,duplicate#/1,goal#/1} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {duplicate#,goal#} and constructors {Cons,Nil}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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