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

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

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