WORST_CASE(?,O(n^1))
* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a__c() -> b()
            a__c() -> c()
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__f(b(),X,c()) -> a__f(X,a__c(),X)
            mark(b()) -> b()
            mark(c()) -> a__c()
            mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
        - Signature:
            {a__c/0,a__f/3,mark/1} / {b/0,c/0,f/3}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__c,a__f,mark} and constructors {b,c,f}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          a__c#() -> c_1()
          a__c#() -> c_2()
          a__f#(X1,X2,X3) -> c_3()
          a__f#(b(),X,c()) -> c_4(a__f#(X,a__c(),X),a__c#())
          mark#(b()) -> c_5()
          mark#(c()) -> c_6(a__c#())
          mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3),mark#(X2))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a__c#() -> c_1()
            a__c#() -> c_2()
            a__f#(X1,X2,X3) -> c_3()
            a__f#(b(),X,c()) -> c_4(a__f#(X,a__c(),X),a__c#())
            mark#(b()) -> c_5()
            mark#(c()) -> c_6(a__c#())
            mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3),mark#(X2))
        - Weak TRS:
            a__c() -> b()
            a__c() -> c()
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__f(b(),X,c()) -> a__f(X,a__c(),X)
            mark(b()) -> b()
            mark(c()) -> a__c()
            mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
        - Signature:
            {a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0,c_6/1,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__c#,a__f#,mark#} and constructors {b,c,f}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,3,5}
        by application of
          Pre({1,2,3,5}) = {4,6,7}.
        Here rules are labelled as follows:
          1: a__c#() -> c_1()
          2: a__c#() -> c_2()
          3: a__f#(X1,X2,X3) -> c_3()
          4: a__f#(b(),X,c()) -> c_4(a__f#(X,a__c(),X),a__c#())
          5: mark#(b()) -> c_5()
          6: mark#(c()) -> c_6(a__c#())
          7: mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3),mark#(X2))
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a__f#(b(),X,c()) -> c_4(a__f#(X,a__c(),X),a__c#())
            mark#(c()) -> c_6(a__c#())
            mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3),mark#(X2))
        - Weak DPs:
            a__c#() -> c_1()
            a__c#() -> c_2()
            a__f#(X1,X2,X3) -> c_3()
            mark#(b()) -> c_5()
        - Weak TRS:
            a__c() -> b()
            a__c() -> c()
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__f(b(),X,c()) -> a__f(X,a__c(),X)
            mark(b()) -> b()
            mark(c()) -> a__c()
            mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
        - Signature:
            {a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0,c_6/1,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__c#,a__f#,mark#} and constructors {b,c,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#(b(),X,c()) -> c_4(a__f#(X,a__c(),X),a__c#())
          2: mark#(c()) -> c_6(a__c#())
          3: mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3),mark#(X2))
          4: a__c#() -> c_1()
          5: a__c#() -> c_2()
          6: a__f#(X1,X2,X3) -> c_3()
          7: mark#(b()) -> c_5()
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3),mark#(X2))
        - Weak DPs:
            a__c#() -> c_1()
            a__c#() -> c_2()
            a__f#(X1,X2,X3) -> c_3()
            a__f#(b(),X,c()) -> c_4(a__f#(X,a__c(),X),a__c#())
            mark#(b()) -> c_5()
            mark#(c()) -> c_6(a__c#())
        - Weak TRS:
            a__c() -> b()
            a__c() -> c()
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__f(b(),X,c()) -> a__f(X,a__c(),X)
            mark(b()) -> b()
            mark(c()) -> a__c()
            mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
        - Signature:
            {a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0,c_6/1,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__c#,a__f#,mark#} and constructors {b,c,f}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3),mark#(X2))
             -->_2 mark#(c()) -> c_6(a__c#()):7
             -->_1 a__f#(b(),X,c()) -> c_4(a__f#(X,a__c(),X),a__c#()):5
             -->_2 mark#(b()) -> c_5():6
             -->_1 a__f#(X1,X2,X3) -> c_3():4
             -->_2 mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3),mark#(X2)):1
          
          2:W:a__c#() -> c_1()
             
          
          3:W:a__c#() -> c_2()
             
          
          4:W:a__f#(X1,X2,X3) -> c_3()
             
          
          5:W:a__f#(b(),X,c()) -> c_4(a__f#(X,a__c(),X),a__c#())
             -->_1 a__f#(X1,X2,X3) -> c_3():4
             -->_2 a__c#() -> c_2():3
             -->_2 a__c#() -> c_1():2
          
          6:W:mark#(b()) -> c_5()
             
          
          7:W:mark#(c()) -> c_6(a__c#())
             -->_1 a__c#() -> c_2():3
             -->_1 a__c#() -> 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#(b()) -> c_5()
          5: a__f#(b(),X,c()) -> c_4(a__f#(X,a__c(),X),a__c#())
          4: a__f#(X1,X2,X3) -> c_3()
          7: mark#(c()) -> c_6(a__c#())
          2: a__c#() -> c_1()
          3: a__c#() -> c_2()
* Step 5: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3),mark#(X2))
        - Weak TRS:
            a__c() -> b()
            a__c() -> c()
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__f(b(),X,c()) -> a__f(X,a__c(),X)
            mark(b()) -> b()
            mark(c()) -> a__c()
            mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
        - Signature:
            {a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0,c_6/1,c_7/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__c#,a__f#,mark#} and constructors {b,c,f}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3),mark#(X2))
             -->_2 mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3),mark#(X2)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          mark#(f(X1,X2,X3)) -> c_7(mark#(X2))
* Step 6: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mark#(f(X1,X2,X3)) -> c_7(mark#(X2))
        - Weak TRS:
            a__c() -> b()
            a__c() -> c()
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__f(b(),X,c()) -> a__f(X,a__c(),X)
            mark(b()) -> b()
            mark(c()) -> a__c()
            mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
        - Signature:
            {a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0,c_6/1,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__c#,a__f#,mark#} and constructors {b,c,f}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          mark#(f(X1,X2,X3)) -> c_7(mark#(X2))
* Step 7: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mark#(f(X1,X2,X3)) -> c_7(mark#(X2))
        - Signature:
            {a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0,c_6/1,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__c#,a__f#,mark#} and constructors {b,c,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,X3)) -> c_7(mark#(X2))
          
        The strictly oriented rules are moved into the weak component.
** Step 7.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mark#(f(X1,X2,X3)) -> c_7(mark#(X2))
        - Signature:
            {a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0,c_6/1,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__c#,a__f#,mark#} and constructors {b,c,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__c#,a__f#,mark#}
        TcT has computed the following interpretation:
           p(a__c) = [1]                  
           p(a__f) = [2]                  
              p(b) = [0]                  
              p(c) = [8]                  
              p(f) = [1] x2 + [1] x3 + [6]
           p(mark) = [1] x1 + [0]         
          p(a__c#) = [1]                  
          p(a__f#) = [1] x2 + [1]         
          p(mark#) = [4] x1 + [0]         
            p(c_1) = [2]                  
            p(c_2) = [4]                  
            p(c_3) = [1]                  
            p(c_4) = [2] x1 + [8]         
            p(c_5) = [2]                  
            p(c_6) = [2] x1 + [1]         
            p(c_7) = [1] x1 + [15]        
        
        Following rules are strictly oriented:
        mark#(f(X1,X2,X3)) = [4] X2 + [4] X3 + [24]
                           > [4] X2 + [15]         
                           = c_7(mark#(X2))        
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 7.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            mark#(f(X1,X2,X3)) -> c_7(mark#(X2))
        - Signature:
            {a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0,c_6/1,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__c#,a__f#,mark#} and constructors {b,c,f}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

** Step 7.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            mark#(f(X1,X2,X3)) -> c_7(mark#(X2))
        - Signature:
            {a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0,c_6/1,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__c#,a__f#,mark#} and constructors {b,c,f}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:mark#(f(X1,X2,X3)) -> c_7(mark#(X2))
             -->_1 mark#(f(X1,X2,X3)) -> c_7(mark#(X2)):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,X3)) -> c_7(mark#(X2))
** Step 7.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0,c_6/1,c_7/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__c#,a__f#,mark#} and constructors {b,c,f}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))