* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            a__b() -> a()
            a__b() -> b()
            a__f(X,X) -> a__f(a(),b())
            a__f(X1,X2) -> f(X1,X2)
            mark(a()) -> a()
            mark(b()) -> a__b()
            mark(f(X1,X2)) -> a__f(mark(X1),X2)
        - Signature:
            {a__b/0,a__f/2,mark/1} / {a/0,b/0,f/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__b,a__f,mark} and constructors {a,b,f}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            a__b() -> a()
            a__b() -> b()
            a__f(X,X) -> a__f(a(),b())
            a__f(X1,X2) -> f(X1,X2)
            mark(a()) -> a()
            mark(b()) -> a__b()
            mark(f(X1,X2)) -> a__f(mark(X1),X2)
        - Signature:
            {a__b/0,a__f/2,mark/1} / {a/0,b/0,f/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__b,a__f,mark} and constructors {a,b,f}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          mark(x){x -> f(x,y)} =
            mark(f(x,y)) ->^+ a__f(mark(x),y)
              = C[mark(x) = mark(x){}]

** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a__b() -> a()
            a__b() -> b()
            a__f(X,X) -> a__f(a(),b())
            a__f(X1,X2) -> f(X1,X2)
            mark(a()) -> a()
            mark(b()) -> a__b()
            mark(f(X1,X2)) -> a__f(mark(X1),X2)
        - Signature:
            {a__b/0,a__f/2,mark/1} / {a/0,b/0,f/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__b,a__f,mark} and constructors {a,b,f}
    + 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,X) -> c_3(a__f#(a(),b()))
          a__f#(X1,X2) -> c_4()
          mark#(a()) -> c_5()
          mark#(b()) -> c_6(a__b#())
          mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a__b#() -> c_1()
            a__b#() -> c_2()
            a__f#(X,X) -> c_3(a__f#(a(),b()))
            a__f#(X1,X2) -> c_4()
            mark#(a()) -> c_5()
            mark#(b()) -> c_6(a__b#())
            mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1))
        - Weak TRS:
            a__b() -> a()
            a__b() -> b()
            a__f(X,X) -> a__f(a(),b())
            a__f(X1,X2) -> f(X1,X2)
            mark(a()) -> a()
            mark(b()) -> a__b()
            mark(f(X1,X2)) -> a__f(mark(X1),X2)
        - Signature:
            {a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,4,5}
        by application of
          Pre({1,2,4,5}) = {3,6,7}.
        Here rules are labelled as follows:
          1: a__b#() -> c_1()
          2: a__b#() -> c_2()
          3: a__f#(X,X) -> c_3(a__f#(a(),b()))
          4: a__f#(X1,X2) -> c_4()
          5: mark#(a()) -> c_5()
          6: mark#(b()) -> c_6(a__b#())
          7: mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1))
** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a__f#(X,X) -> c_3(a__f#(a(),b()))
            mark#(b()) -> c_6(a__b#())
            mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1))
        - Weak DPs:
            a__b#() -> c_1()
            a__b#() -> c_2()
            a__f#(X1,X2) -> c_4()
            mark#(a()) -> c_5()
        - Weak TRS:
            a__b() -> a()
            a__b() -> b()
            a__f(X,X) -> a__f(a(),b())
            a__f(X1,X2) -> f(X1,X2)
            mark(a()) -> a()
            mark(b()) -> a__b()
            mark(f(X1,X2)) -> a__f(mark(X1),X2)
        - Signature:
            {a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
    + 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}.
        Here rules are labelled as follows:
          1: a__f#(X,X) -> c_3(a__f#(a(),b()))
          2: mark#(b()) -> c_6(a__b#())
          3: mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1))
          4: a__b#() -> c_1()
          5: a__b#() -> c_2()
          6: a__f#(X1,X2) -> c_4()
          7: mark#(a()) -> c_5()
** Step 1.b:4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1))
        - Weak DPs:
            a__b#() -> c_1()
            a__b#() -> c_2()
            a__f#(X,X) -> c_3(a__f#(a(),b()))
            a__f#(X1,X2) -> c_4()
            mark#(a()) -> c_5()
            mark#(b()) -> c_6(a__b#())
        - Weak TRS:
            a__b() -> a()
            a__b() -> b()
            a__f(X,X) -> a__f(a(),b())
            a__f(X1,X2) -> f(X1,X2)
            mark(a()) -> a()
            mark(b()) -> a__b()
            mark(f(X1,X2)) -> a__f(mark(X1),X2)
        - Signature:
            {a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1))
             -->_2 mark#(b()) -> c_6(a__b#()):7
             -->_1 a__f#(X,X) -> c_3(a__f#(a(),b())):4
             -->_2 mark#(a()) -> c_5():6
             -->_1 a__f#(X1,X2) -> c_4():5
             -->_2 mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1)):1
          
          2:W:a__b#() -> c_1()
             
          
          3:W:a__b#() -> c_2()
             
          
          4:W:a__f#(X,X) -> c_3(a__f#(a(),b()))
             -->_1 a__f#(X1,X2) -> c_4():5
          
          5:W:a__f#(X1,X2) -> c_4()
             
          
          6:W:mark#(a()) -> c_5()
             
          
          7:W:mark#(b()) -> c_6(a__b#())
             -->_1 a__b#() -> c_2():3
             -->_1 a__b#() -> c_1():2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: mark#(a()) -> c_5()
          4: a__f#(X,X) -> c_3(a__f#(a(),b()))
          5: a__f#(X1,X2) -> c_4()
          7: mark#(b()) -> c_6(a__b#())
          2: a__b#() -> c_1()
          3: a__b#() -> c_2()
** Step 1.b:5: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1))
        - Weak TRS:
            a__b() -> a()
            a__b() -> b()
            a__f(X,X) -> a__f(a(),b())
            a__f(X1,X2) -> f(X1,X2)
            mark(a()) -> a()
            mark(b()) -> a__b()
            mark(f(X1,X2)) -> a__f(mark(X1),X2)
        - Signature:
            {a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1))
             -->_2 mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          mark#(f(X1,X2)) -> c_7(mark#(X1))
** Step 1.b:6: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mark#(f(X1,X2)) -> c_7(mark#(X1))
        - Weak TRS:
            a__b() -> a()
            a__b() -> b()
            a__f(X,X) -> a__f(a(),b())
            a__f(X1,X2) -> f(X1,X2)
            mark(a()) -> a()
            mark(b()) -> a__b()
            mark(f(X1,X2)) -> a__f(mark(X1),X2)
        - Signature:
            {a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          mark#(f(X1,X2)) -> c_7(mark#(X1))
** Step 1.b:7: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mark#(f(X1,X2)) -> c_7(mark#(X1))
        - Signature:
            {a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
    + 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#(f(X1,X2)) -> c_7(mark#(X1))
          
        The strictly oriented rules are moved into the weak component.
*** Step 1.b:7.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mark#(f(X1,X2)) -> c_7(mark#(X1))
        - Signature:
            {a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
    + 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_7) = {1}
        
        Following symbols are considered usable:
          {a__b#,a__f#,mark#}
        TcT has computed the following interpretation:
              p(a) = [2]          
           p(a__b) = [1]          
           p(a__f) = [1]          
              p(b) = [0]          
              p(f) = [1] x1 + [1] 
           p(mark) = [2]          
          p(a__b#) = [1]          
          p(a__f#) = [8] x2 + [1] 
          p(mark#) = [1] x1 + [10]
            p(c_1) = [1]          
            p(c_2) = [0]          
            p(c_3) = [1] x1 + [1] 
            p(c_4) = [0]          
            p(c_5) = [1]          
            p(c_6) = [2] x1 + [4] 
            p(c_7) = [1] x1 + [0] 
        
        Following rules are strictly oriented:
        mark#(f(X1,X2)) = [1] X1 + [11] 
                        > [1] X1 + [10] 
                        = c_7(mark#(X1))
        
        
        Following rules are (at-least) weakly oriented:
        
*** Step 1.b:7.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            mark#(f(X1,X2)) -> c_7(mark#(X1))
        - Signature:
            {a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 1.b:7.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            mark#(f(X1,X2)) -> c_7(mark#(X1))
        - Signature:
            {a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:mark#(f(X1,X2)) -> c_7(mark#(X1))
             -->_1 mark#(f(X1,X2)) -> c_7(mark#(X1)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: mark#(f(X1,X2)) -> c_7(mark#(X1))
*** Step 1.b:7.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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