* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a__b() -> b()
            a__b() -> c()
            a__f(X,g(X),Y) -> a__f(Y,Y,Y)
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__g(X) -> g(X)
            a__g(b()) -> c()
            mark(b()) -> a__b()
            mark(c()) -> c()
            mark(f(X1,X2,X3)) -> a__f(X1,X2,X3)
            mark(g(X)) -> a__g(mark(X))
        - Signature:
            {a__b/0,a__f/3,a__g/1,mark/1} / {b/0,c/0,f/3,g/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__b,a__f,a__g,mark} and constructors {b,c,f,g}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          a__b#() -> c_1()
          a__b#() -> c_2()
          a__f#(X,g(X),Y) -> c_3(a__f#(Y,Y,Y))
          a__f#(X1,X2,X3) -> c_4()
          a__g#(X) -> c_5()
          a__g#(b()) -> c_6()
          mark#(b()) -> c_7(a__b#())
          mark#(c()) -> c_8()
          mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,X3))
          mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a__b#() -> c_1()
            a__b#() -> c_2()
            a__f#(X,g(X),Y) -> c_3(a__f#(Y,Y,Y))
            a__f#(X1,X2,X3) -> c_4()
            a__g#(X) -> c_5()
            a__g#(b()) -> c_6()
            mark#(b()) -> c_7(a__b#())
            mark#(c()) -> c_8()
            mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,X3))
            mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X))
        - Weak TRS:
            a__b() -> b()
            a__b() -> c()
            a__f(X,g(X),Y) -> a__f(Y,Y,Y)
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__g(X) -> g(X)
            a__g(b()) -> c()
            mark(b()) -> a__b()
            mark(c()) -> c()
            mark(f(X1,X2,X3)) -> a__f(X1,X2,X3)
            mark(g(X)) -> a__g(mark(X))
        - Signature:
            {a__b/0,a__f/3,a__g/1,mark/1,a__b#/0,a__f#/3,a__g#/1,mark#/1} / {b/0,c/0,f/3,g/1,c_1/0,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__b#,a__f#,a__g#,mark#} and constructors {b,c,f,g}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,4,5,6,8}
        by application of
          Pre({1,2,4,5,6,8}) = {3,7,9,10}.
        Here rules are labelled as follows:
          1: a__b#() -> c_1()
          2: a__b#() -> c_2()
          3: a__f#(X,g(X),Y) -> c_3(a__f#(Y,Y,Y))
          4: a__f#(X1,X2,X3) -> c_4()
          5: a__g#(X) -> c_5()
          6: a__g#(b()) -> c_6()
          7: mark#(b()) -> c_7(a__b#())
          8: mark#(c()) -> c_8()
          9: mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,X3))
          10: mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X))
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a__f#(X,g(X),Y) -> c_3(a__f#(Y,Y,Y))
            mark#(b()) -> c_7(a__b#())
            mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,X3))
            mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X))
        - Weak DPs:
            a__b#() -> c_1()
            a__b#() -> c_2()
            a__f#(X1,X2,X3) -> c_4()
            a__g#(X) -> c_5()
            a__g#(b()) -> c_6()
            mark#(c()) -> c_8()
        - Weak TRS:
            a__b() -> b()
            a__b() -> c()
            a__f(X,g(X),Y) -> a__f(Y,Y,Y)
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__g(X) -> g(X)
            a__g(b()) -> c()
            mark(b()) -> a__b()
            mark(c()) -> c()
            mark(f(X1,X2,X3)) -> a__f(X1,X2,X3)
            mark(g(X)) -> a__g(mark(X))
        - Signature:
            {a__b/0,a__f/3,a__g/1,mark/1,a__b#/0,a__f#/3,a__g#/1,mark#/1} / {b/0,c/0,f/3,g/1,c_1/0,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__b#,a__f#,a__g#,mark#} and constructors {b,c,f,g}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2}
        by application of
          Pre({1,2}) = {3,4}.
        Here rules are labelled as follows:
          1: a__f#(X,g(X),Y) -> c_3(a__f#(Y,Y,Y))
          2: mark#(b()) -> c_7(a__b#())
          3: mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,X3))
          4: mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X))
          5: a__b#() -> c_1()
          6: a__b#() -> c_2()
          7: a__f#(X1,X2,X3) -> c_4()
          8: a__g#(X) -> c_5()
          9: a__g#(b()) -> c_6()
          10: mark#(c()) -> c_8()
* Step 4: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,X3))
            mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X))
        - Weak DPs:
            a__b#() -> c_1()
            a__b#() -> c_2()
            a__f#(X,g(X),Y) -> c_3(a__f#(Y,Y,Y))
            a__f#(X1,X2,X3) -> c_4()
            a__g#(X) -> c_5()
            a__g#(b()) -> c_6()
            mark#(b()) -> c_7(a__b#())
            mark#(c()) -> c_8()
        - Weak TRS:
            a__b() -> b()
            a__b() -> c()
            a__f(X,g(X),Y) -> a__f(Y,Y,Y)
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__g(X) -> g(X)
            a__g(b()) -> c()
            mark(b()) -> a__b()
            mark(c()) -> c()
            mark(f(X1,X2,X3)) -> a__f(X1,X2,X3)
            mark(g(X)) -> a__g(mark(X))
        - Signature:
            {a__b/0,a__f/3,a__g/1,mark/1,a__b#/0,a__f#/3,a__g#/1,mark#/1} / {b/0,c/0,f/3,g/1,c_1/0,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__b#,a__f#,a__g#,mark#} and constructors {b,c,f,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: mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,X3))
          2: mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X))
          3: a__b#() -> c_1()
          4: a__b#() -> c_2()
          5: a__f#(X,g(X),Y) -> c_3(a__f#(Y,Y,Y))
          6: a__f#(X1,X2,X3) -> c_4()
          7: a__g#(X) -> c_5()
          8: a__g#(b()) -> c_6()
          9: mark#(b()) -> c_7(a__b#())
          10: mark#(c()) -> c_8()
* Step 5: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X))
        - Weak DPs:
            a__b#() -> c_1()
            a__b#() -> c_2()
            a__f#(X,g(X),Y) -> c_3(a__f#(Y,Y,Y))
            a__f#(X1,X2,X3) -> c_4()
            a__g#(X) -> c_5()
            a__g#(b()) -> c_6()
            mark#(b()) -> c_7(a__b#())
            mark#(c()) -> c_8()
            mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,X3))
        - Weak TRS:
            a__b() -> b()
            a__b() -> c()
            a__f(X,g(X),Y) -> a__f(Y,Y,Y)
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__g(X) -> g(X)
            a__g(b()) -> c()
            mark(b()) -> a__b()
            mark(c()) -> c()
            mark(f(X1,X2,X3)) -> a__f(X1,X2,X3)
            mark(g(X)) -> a__g(mark(X))
        - Signature:
            {a__b/0,a__f/3,a__g/1,mark/1,a__b#/0,a__f#/3,a__g#/1,mark#/1} / {b/0,c/0,f/3,g/1,c_1/0,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__b#,a__f#,a__g#,mark#} and constructors {b,c,f,g}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X))
             -->_2 mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,X3)):10
             -->_2 mark#(b()) -> c_7(a__b#()):8
             -->_2 mark#(c()) -> c_8():9
             -->_1 a__g#(b()) -> c_6():7
             -->_1 a__g#(X) -> c_5():6
             -->_2 mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X)):1
          
          2:W:a__b#() -> c_1()
             
          
          3:W:a__b#() -> c_2()
             
          
          4:W:a__f#(X,g(X),Y) -> c_3(a__f#(Y,Y,Y))
             -->_1 a__f#(X1,X2,X3) -> c_4():5
          
          5:W:a__f#(X1,X2,X3) -> c_4()
             
          
          6:W:a__g#(X) -> c_5()
             
          
          7:W:a__g#(b()) -> c_6()
             
          
          8:W:mark#(b()) -> c_7(a__b#())
             -->_1 a__b#() -> c_2():3
             -->_1 a__b#() -> c_1():2
          
          9:W:mark#(c()) -> c_8()
             
          
          10:W:mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,X3))
             -->_1 a__f#(X1,X2,X3) -> c_4():5
             -->_1 a__f#(X,g(X),Y) -> c_3(a__f#(Y,Y,Y)):4
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: a__g#(X) -> c_5()
          7: a__g#(b()) -> c_6()
          9: mark#(c()) -> c_8()
          8: mark#(b()) -> c_7(a__b#())
          2: a__b#() -> c_1()
          3: a__b#() -> c_2()
          10: mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,X3))
          4: a__f#(X,g(X),Y) -> c_3(a__f#(Y,Y,Y))
          5: a__f#(X1,X2,X3) -> c_4()
* Step 6: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X))
        - Weak TRS:
            a__b() -> b()
            a__b() -> c()
            a__f(X,g(X),Y) -> a__f(Y,Y,Y)
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__g(X) -> g(X)
            a__g(b()) -> c()
            mark(b()) -> a__b()
            mark(c()) -> c()
            mark(f(X1,X2,X3)) -> a__f(X1,X2,X3)
            mark(g(X)) -> a__g(mark(X))
        - Signature:
            {a__b/0,a__f/3,a__g/1,mark/1,a__b#/0,a__f#/3,a__g#/1,mark#/1} / {b/0,c/0,f/3,g/1,c_1/0,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__b#,a__f#,a__g#,mark#} and constructors {b,c,f,g}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X))
             -->_2 mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          mark#(g(X)) -> c_10(mark#(X))
* Step 7: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mark#(g(X)) -> c_10(mark#(X))
        - Weak TRS:
            a__b() -> b()
            a__b() -> c()
            a__f(X,g(X),Y) -> a__f(Y,Y,Y)
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__g(X) -> g(X)
            a__g(b()) -> c()
            mark(b()) -> a__b()
            mark(c()) -> c()
            mark(f(X1,X2,X3)) -> a__f(X1,X2,X3)
            mark(g(X)) -> a__g(mark(X))
        - Signature:
            {a__b/0,a__f/3,a__g/1,mark/1,a__b#/0,a__f#/3,a__g#/1,mark#/1} / {b/0,c/0,f/3,g/1,c_1/0,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__b#,a__f#,a__g#,mark#} and constructors {b,c,f,g}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          mark#(g(X)) -> c_10(mark#(X))
* Step 8: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mark#(g(X)) -> c_10(mark#(X))
        - Signature:
            {a__b/0,a__f/3,a__g/1,mark/1,a__b#/0,a__f#/3,a__g#/1,mark#/1} / {b/0,c/0,f/3,g/1,c_1/0,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__b#,a__f#,a__g#,mark#} and constructors {b,c,f,g}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: mark#(g(X)) -> c_10(mark#(X))
          
        The strictly oriented rules are moved into the weak component.
** Step 8.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mark#(g(X)) -> c_10(mark#(X))
        - Signature:
            {a__b/0,a__f/3,a__g/1,mark/1,a__b#/0,a__f#/3,a__g#/1,mark#/1} / {b/0,c/0,f/3,g/1,c_1/0,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__b#,a__f#,a__g#,mark#} and constructors {b,c,f,g}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_10) = {1}
        
        Following symbols are considered usable:
          {a__b#,a__f#,a__g#,mark#}
        TcT has computed the following interpretation:
           p(a__b) = [0]                  
           p(a__f) = [1] x2 + [1] x3 + [0]
           p(a__g) = [1] x1 + [1]         
              p(b) = [0]                  
              p(c) = [1]                  
              p(f) = [1]                  
              p(g) = [1] x1 + [6]         
           p(mark) = [0]                  
          p(a__b#) = [1]                  
          p(a__f#) = [1] x3 + [1]         
          p(a__g#) = [1] x1 + [1]         
          p(mark#) = [4] x1 + [0]         
            p(c_1) = [8]                  
            p(c_2) = [0]                  
            p(c_3) = [2] x1 + [1]         
            p(c_4) = [1]                  
            p(c_5) = [8]                  
            p(c_6) = [0]                  
            p(c_7) = [0]                  
            p(c_8) = [1]                  
            p(c_9) = [1]                  
           p(c_10) = [1] x1 + [15]        
        
        Following rules are strictly oriented:
        mark#(g(X)) = [4] X + [24]  
                    > [4] X + [15]  
                    = c_10(mark#(X))
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 8.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            mark#(g(X)) -> c_10(mark#(X))
        - Signature:
            {a__b/0,a__f/3,a__g/1,mark/1,a__b#/0,a__f#/3,a__g#/1,mark#/1} / {b/0,c/0,f/3,g/1,c_1/0,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__b#,a__f#,a__g#,mark#} and constructors {b,c,f,g}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

** Step 8.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            mark#(g(X)) -> c_10(mark#(X))
        - Signature:
            {a__b/0,a__f/3,a__g/1,mark/1,a__b#/0,a__f#/3,a__g#/1,mark#/1} / {b/0,c/0,f/3,g/1,c_1/0,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__b#,a__f#,a__g#,mark#} and constructors {b,c,f,g}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:mark#(g(X)) -> c_10(mark#(X))
             -->_1 mark#(g(X)) -> c_10(mark#(X)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: mark#(g(X)) -> c_10(mark#(X))
** Step 8.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {a__b/0,a__f/3,a__g/1,mark/1,a__b#/0,a__f#/3,a__g#/1,mark#/1} / {b/0,c/0,f/3,g/1,c_1/0,c_2/0,c_3/1,c_4/0
            ,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__b#,a__f#,a__g#,mark#} and constructors {b,c,f,g}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))