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

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a() -> n__a()
            activate(X) -> X
            activate(n__a()) -> a()
            activate(n__f(X)) -> f(X)
            activate(n__g(X)) -> g(activate(X))
            f(X) -> n__f(X)
            f(n__f(n__a())) -> f(n__g(n__f(n__a())))
            g(X) -> n__g(X)
        - Signature:
            {a/0,activate/1,f/1,g/1} / {n__a/0,n__f/1,n__g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a,activate,f,g} and constructors {n__a,n__f,n__g}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          a#() -> c_1()
          activate#(X) -> c_2()
          activate#(n__a()) -> c_3(a#())
          activate#(n__f(X)) -> c_4(f#(X))
          activate#(n__g(X)) -> c_5(g#(activate(X)),activate#(X))
          f#(X) -> c_6()
          f#(n__f(n__a())) -> c_7(f#(n__g(n__f(n__a()))))
          g#(X) -> c_8()
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a#() -> c_1()
            activate#(X) -> c_2()
            activate#(n__a()) -> c_3(a#())
            activate#(n__f(X)) -> c_4(f#(X))
            activate#(n__g(X)) -> c_5(g#(activate(X)),activate#(X))
            f#(X) -> c_6()
            f#(n__f(n__a())) -> c_7(f#(n__g(n__f(n__a()))))
            g#(X) -> c_8()
        - Weak TRS:
            a() -> n__a()
            activate(X) -> X
            activate(n__a()) -> a()
            activate(n__f(X)) -> f(X)
            activate(n__g(X)) -> g(activate(X))
            f(X) -> n__f(X)
            f(n__f(n__a())) -> f(n__g(n__f(n__a())))
            g(X) -> n__g(X)
        - Signature:
            {a/0,activate/1,f/1,g/1,a#/0,activate#/1,f#/1,g#/1} / {n__a/0,n__f/1,n__g/1,c_1/0,c_2/0,c_3/1,c_4/1,c_5/2
            ,c_6/0,c_7/1,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,activate#,f#,g#} and constructors {n__a,n__f,n__g}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,6,8}
        by application of
          Pre({1,2,6,8}) = {3,4,5,7}.
        Here rules are labelled as follows:
          1: a#() -> c_1()
          2: activate#(X) -> c_2()
          3: activate#(n__a()) -> c_3(a#())
          4: activate#(n__f(X)) -> c_4(f#(X))
          5: activate#(n__g(X)) -> c_5(g#(activate(X)),activate#(X))
          6: f#(X) -> c_6()
          7: f#(n__f(n__a())) -> c_7(f#(n__g(n__f(n__a()))))
          8: g#(X) -> c_8()
** Step 1.b:3: PredecessorEstimation.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__a()) -> c_3(a#())
            activate#(n__f(X)) -> c_4(f#(X))
            activate#(n__g(X)) -> c_5(g#(activate(X)),activate#(X))
            f#(n__f(n__a())) -> c_7(f#(n__g(n__f(n__a()))))
        - Weak DPs:
            a#() -> c_1()
            activate#(X) -> c_2()
            f#(X) -> c_6()
            g#(X) -> c_8()
        - Weak TRS:
            a() -> n__a()
            activate(X) -> X
            activate(n__a()) -> a()
            activate(n__f(X)) -> f(X)
            activate(n__g(X)) -> g(activate(X))
            f(X) -> n__f(X)
            f(n__f(n__a())) -> f(n__g(n__f(n__a())))
            g(X) -> n__g(X)
        - Signature:
            {a/0,activate/1,f/1,g/1,a#/0,activate#/1,f#/1,g#/1} / {n__a/0,n__f/1,n__g/1,c_1/0,c_2/0,c_3/1,c_4/1,c_5/2
            ,c_6/0,c_7/1,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,activate#,f#,g#} and constructors {n__a,n__f,n__g}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,4}
        by application of
          Pre({1,4}) = {2,3}.
        Here rules are labelled as follows:
          1: activate#(n__a()) -> c_3(a#())
          2: activate#(n__f(X)) -> c_4(f#(X))
          3: activate#(n__g(X)) -> c_5(g#(activate(X)),activate#(X))
          4: f#(n__f(n__a())) -> c_7(f#(n__g(n__f(n__a()))))
          5: a#() -> c_1()
          6: activate#(X) -> c_2()
          7: f#(X) -> c_6()
          8: g#(X) -> c_8()
** Step 1.b:4: PredecessorEstimation.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__f(X)) -> c_4(f#(X))
            activate#(n__g(X)) -> c_5(g#(activate(X)),activate#(X))
        - Weak DPs:
            a#() -> c_1()
            activate#(X) -> c_2()
            activate#(n__a()) -> c_3(a#())
            f#(X) -> c_6()
            f#(n__f(n__a())) -> c_7(f#(n__g(n__f(n__a()))))
            g#(X) -> c_8()
        - Weak TRS:
            a() -> n__a()
            activate(X) -> X
            activate(n__a()) -> a()
            activate(n__f(X)) -> f(X)
            activate(n__g(X)) -> g(activate(X))
            f(X) -> n__f(X)
            f(n__f(n__a())) -> f(n__g(n__f(n__a())))
            g(X) -> n__g(X)
        - Signature:
            {a/0,activate/1,f/1,g/1,a#/0,activate#/1,f#/1,g#/1} / {n__a/0,n__f/1,n__g/1,c_1/0,c_2/0,c_3/1,c_4/1,c_5/2
            ,c_6/0,c_7/1,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,activate#,f#,g#} and constructors {n__a,n__f,n__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: activate#(n__f(X)) -> c_4(f#(X))
          2: activate#(n__g(X)) -> c_5(g#(activate(X)),activate#(X))
          3: a#() -> c_1()
          4: activate#(X) -> c_2()
          5: activate#(n__a()) -> c_3(a#())
          6: f#(X) -> c_6()
          7: f#(n__f(n__a())) -> c_7(f#(n__g(n__f(n__a()))))
          8: g#(X) -> c_8()
** Step 1.b:5: RemoveWeakSuffixes.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__g(X)) -> c_5(g#(activate(X)),activate#(X))
        - Weak DPs:
            a#() -> c_1()
            activate#(X) -> c_2()
            activate#(n__a()) -> c_3(a#())
            activate#(n__f(X)) -> c_4(f#(X))
            f#(X) -> c_6()
            f#(n__f(n__a())) -> c_7(f#(n__g(n__f(n__a()))))
            g#(X) -> c_8()
        - Weak TRS:
            a() -> n__a()
            activate(X) -> X
            activate(n__a()) -> a()
            activate(n__f(X)) -> f(X)
            activate(n__g(X)) -> g(activate(X))
            f(X) -> n__f(X)
            f(n__f(n__a())) -> f(n__g(n__f(n__a())))
            g(X) -> n__g(X)
        - Signature:
            {a/0,activate/1,f/1,g/1,a#/0,activate#/1,f#/1,g#/1} / {n__a/0,n__f/1,n__g/1,c_1/0,c_2/0,c_3/1,c_4/1,c_5/2
            ,c_6/0,c_7/1,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,activate#,f#,g#} and constructors {n__a,n__f,n__g}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:activate#(n__g(X)) -> c_5(g#(activate(X)),activate#(X))
             -->_2 activate#(n__f(X)) -> c_4(f#(X)):5
             -->_2 activate#(n__a()) -> c_3(a#()):4
             -->_1 g#(X) -> c_8():8
             -->_2 activate#(X) -> c_2():3
             -->_2 activate#(n__g(X)) -> c_5(g#(activate(X)),activate#(X)):1
          
          2:W:a#() -> c_1()
             
          
          3:W:activate#(X) -> c_2()
             
          
          4:W:activate#(n__a()) -> c_3(a#())
             -->_1 a#() -> c_1():2
          
          5:W:activate#(n__f(X)) -> c_4(f#(X))
             -->_1 f#(n__f(n__a())) -> c_7(f#(n__g(n__f(n__a())))):7
             -->_1 f#(X) -> c_6():6
          
          6:W:f#(X) -> c_6()
             
          
          7:W:f#(n__f(n__a())) -> c_7(f#(n__g(n__f(n__a()))))
             -->_1 f#(X) -> c_6():6
          
          8:W:g#(X) -> c_8()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: activate#(X) -> c_2()
          8: g#(X) -> c_8()
          4: activate#(n__a()) -> c_3(a#())
          2: a#() -> c_1()
          5: activate#(n__f(X)) -> c_4(f#(X))
          7: f#(n__f(n__a())) -> c_7(f#(n__g(n__f(n__a()))))
          6: f#(X) -> c_6()
** Step 1.b:6: SimplifyRHS.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__g(X)) -> c_5(g#(activate(X)),activate#(X))
        - Weak TRS:
            a() -> n__a()
            activate(X) -> X
            activate(n__a()) -> a()
            activate(n__f(X)) -> f(X)
            activate(n__g(X)) -> g(activate(X))
            f(X) -> n__f(X)
            f(n__f(n__a())) -> f(n__g(n__f(n__a())))
            g(X) -> n__g(X)
        - Signature:
            {a/0,activate/1,f/1,g/1,a#/0,activate#/1,f#/1,g#/1} / {n__a/0,n__f/1,n__g/1,c_1/0,c_2/0,c_3/1,c_4/1,c_5/2
            ,c_6/0,c_7/1,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,activate#,f#,g#} and constructors {n__a,n__f,n__g}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:activate#(n__g(X)) -> c_5(g#(activate(X)),activate#(X))
             -->_2 activate#(n__g(X)) -> c_5(g#(activate(X)),activate#(X)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          activate#(n__g(X)) -> c_5(activate#(X))
** Step 1.b:7: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__g(X)) -> c_5(activate#(X))
        - Weak TRS:
            a() -> n__a()
            activate(X) -> X
            activate(n__a()) -> a()
            activate(n__f(X)) -> f(X)
            activate(n__g(X)) -> g(activate(X))
            f(X) -> n__f(X)
            f(n__f(n__a())) -> f(n__g(n__f(n__a())))
            g(X) -> n__g(X)
        - Signature:
            {a/0,activate/1,f/1,g/1,a#/0,activate#/1,f#/1,g#/1} / {n__a/0,n__f/1,n__g/1,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1
            ,c_6/0,c_7/1,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,activate#,f#,g#} and constructors {n__a,n__f,n__g}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          activate#(n__g(X)) -> c_5(activate#(X))
** Step 1.b:8: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__g(X)) -> c_5(activate#(X))
        - Signature:
            {a/0,activate/1,f/1,g/1,a#/0,activate#/1,f#/1,g#/1} / {n__a/0,n__f/1,n__g/1,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1
            ,c_6/0,c_7/1,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,activate#,f#,g#} and constructors {n__a,n__f,n__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_5) = {1}
        
        Following symbols are considered usable:
          {a#,activate#,f#,g#}
        TcT has computed the following interpretation:
                  p(a) = [0]          
           p(activate) = [0]          
                  p(f) = [1] x1 + [2] 
                  p(g) = [1] x1 + [0] 
               p(n__a) = [0]          
               p(n__f) = [0]          
               p(n__g) = [1] x1 + [2] 
                 p(a#) = [2]          
          p(activate#) = [8] x1 + [0] 
                 p(f#) = [1] x1 + [1] 
                 p(g#) = [2]          
                p(c_1) = [1]          
                p(c_2) = [0]          
                p(c_3) = [1]          
                p(c_4) = [0]          
                p(c_5) = [1] x1 + [12]
                p(c_6) = [1]          
                p(c_7) = [8] x1 + [2] 
                p(c_8) = [1]          
        
        Following rules are strictly oriented:
        activate#(n__g(X)) = [8] X + [16]     
                           > [8] X + [12]     
                           = c_5(activate#(X))
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 1.b:9: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            activate#(n__g(X)) -> c_5(activate#(X))
        - Signature:
            {a/0,activate/1,f/1,g/1,a#/0,activate#/1,f#/1,g#/1} / {n__a/0,n__f/1,n__g/1,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1
            ,c_6/0,c_7/1,c_8/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,activate#,f#,g#} and constructors {n__a,n__f,n__g}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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