/export/starexec/sandbox/solver/bin/starexec_run_tct_rc /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__b() -> a()
            a__b() -> b()
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__f(a(),X,X) -> a__f(X,a__b(),b())
            mark(a()) -> a()
            mark(b()) -> a__b()
            mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
        - Signature:
            {a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3}
        - Obligation:
             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: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            a__b() -> a()
            a__b() -> b()
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__f(a(),X,X) -> a__f(X,a__b(),b())
            mark(a()) -> a()
            mark(b()) -> a__b()
            mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
        - Signature:
            {a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3}
        - Obligation:
             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:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            a__b() -> a()
            a__b() -> b()
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__f(a(),X,X) -> a__f(X,a__b(),b())
            mark(a()) -> a()
            mark(b()) -> a__b()
            mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
        - Signature:
            {a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3}
        - Obligation:
             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(y){y -> f(x,y,z)} =
            mark(f(x,y,z)) ->^+ a__f(x,mark(y),z)
              = C[mark(y) = mark(y){}]

** Step 1.b:1: ToInnermost.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a__b() -> a()
            a__b() -> b()
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__f(a(),X,X) -> a__f(X,a__b(),b())
            mark(a()) -> a()
            mark(b()) -> a__b()
            mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
        - Signature:
            {a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3}
        - Obligation:
             runtime complexity wrt. defined symbols {a__b,a__f,mark} and constructors {a,b,f}
    + Applied Processor:
        ToInnermost
    + Details:
        switch to innermost, as the system is overlay and right linear and does not contain weak rules
** Step 1.b:2: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a__b() -> a()
            a__b() -> b()
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__f(a(),X,X) -> a__f(X,a__b(),b())
            mark(a()) -> a()
            mark(b()) -> a__b()
            mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
        - Signature:
            {a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a__b,a__f,mark} and constructors {a,b,f}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        Strict DPs
          a__b#() -> c_1()
          a__b#() -> c_2()
          a__f#(X1,X2,X3) -> c_3()
          a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
          mark#(a()) -> c_5()
          mark#(b()) -> c_6(a__b#())
          mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:3: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a__b#() -> c_1()
            a__b#() -> c_2()
            a__f#(X1,X2,X3) -> c_3()
            a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
            mark#(a()) -> c_5()
            mark#(b()) -> c_6(a__b#())
            mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
        - Strict TRS:
            a__b() -> a()
            a__b() -> b()
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__f(a(),X,X) -> a__f(X,a__b(),b())
            mark(a()) -> a()
            mark(b()) -> a__b()
            mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
        - Signature:
            {a__b/0,a__f/3,mark/1,a__b#/0,a__f#/3,mark#/1} / {a/0,b/0,f/3,c_1/0,c_2/0,c_3/0,c_4/1,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:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(a__f) = {2},
            uargs(a__f#) = {2},
            uargs(c_4) = {1},
            uargs(c_6) = {1},
            uargs(c_7) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                p(a) = [3]                           
             p(a__b) = [8]                           
             p(a__f) = [3] x1 + [1] x2 + [4] x3 + [2]
                p(b) = [0]                           
                p(f) = [1] x1 + [1] x2 + [1] x3 + [1]
             p(mark) = [4] x1 + [10]                 
            p(a__b#) = [1]                           
            p(a__f#) = [1] x2 + [4] x3 + [2]         
            p(mark#) = [4] x1 + [4]                  
              p(c_1) = [1]                           
              p(c_2) = [1]                           
              p(c_3) = [0]                           
              p(c_4) = [1] x1 + [8]                  
              p(c_5) = [1]                           
              p(c_6) = [1] x1 + [1]                  
              p(c_7) = [1] x1 + [4]                  
          
          Following rules are strictly oriented:
            a__f#(X1,X2,X3) = [1] X2 + [4] X3 + [2]          
                            > [0]                            
                            = c_3()                          
          
                 mark#(a()) = [16]                           
                            > [1]                            
                            = c_5()                          
          
                 mark#(b()) = [4]                            
                            > [2]                            
                            = c_6(a__b#())                   
          
                     a__b() = [8]                            
                            > [3]                            
                            = a()                            
          
                     a__b() = [8]                            
                            > [0]                            
                            = b()                            
          
             a__f(X1,X2,X3) = [3] X1 + [1] X2 + [4] X3 + [2] 
                            > [1] X1 + [1] X2 + [1] X3 + [1] 
                            = f(X1,X2,X3)                    
          
              a__f(a(),X,X) = [5] X + [11]                   
                            > [3] X + [10]                   
                            = a__f(X,a__b(),b())             
          
                  mark(a()) = [22]                           
                            > [3]                            
                            = a()                            
          
                  mark(b()) = [10]                           
                            > [8]                            
                            = a__b()                         
          
          mark(f(X1,X2,X3)) = [4] X1 + [4] X2 + [4] X3 + [14]
                            > [3] X1 + [4] X2 + [4] X3 + [12]
                            = a__f(X1,mark(X2),X3)           
          
          
          Following rules are (at-least) weakly oriented:
                     a__b#() =  [1]                           
                             >= [1]                           
                             =  c_1()                         
          
                     a__b#() =  [1]                           
                             >= [1]                           
                             =  c_2()                         
          
              a__f#(a(),X,X) =  [5] X + [2]                   
                             >= [18]                          
                             =  c_4(a__f#(X,a__b(),b()))      
          
          mark#(f(X1,X2,X3)) =  [4] X1 + [4] X2 + [4] X3 + [8]
                             >= [4] X2 + [4] X3 + [16]        
                             =  c_7(a__f#(X1,mark(X2),X3))    
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:4: RemoveWeakSuffixes.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a__b#() -> c_1()
            a__b#() -> c_2()
            a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
            mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
        - Weak DPs:
            a__f#(X1,X2,X3) -> c_3()
            mark#(a()) -> c_5()
            mark#(b()) -> c_6(a__b#())
        - Weak TRS:
            a__b() -> a()
            a__b() -> b()
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__f(a(),X,X) -> a__f(X,a__b(),b())
            mark(a()) -> a()
            mark(b()) -> a__b()
            mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
        - Signature:
            {a__b/0,a__f/3,mark/1,a__b#/0,a__f#/3,mark#/1} / {a/0,b/0,f/3,c_1/0,c_2/0,c_3/0,c_4/1,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:S:a__b#() -> c_1()
             
          
          2:S:a__b#() -> c_2()
             
          
          3:S:a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
             -->_1 a__f#(X1,X2,X3) -> c_3():5
             -->_1 a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b())):3
          
          4:S:mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
             -->_1 a__f#(X1,X2,X3) -> c_3():5
             -->_1 a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b())):3
          
          5:W:a__f#(X1,X2,X3) -> c_3()
             
          
          6:W:mark#(a()) -> c_5()
             
          
          7:W:mark#(b()) -> c_6(a__b#())
             -->_1 a__b#() -> c_2():2
             -->_1 a__b#() -> c_1():1
          
        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()
          5: a__f#(X1,X2,X3) -> c_3()
** Step 1.b:5: RemoveHeads.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a__b#() -> c_1()
            a__b#() -> c_2()
            a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
            mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
        - Weak DPs:
            mark#(b()) -> c_6(a__b#())
        - Weak TRS:
            a__b() -> a()
            a__b() -> b()
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__f(a(),X,X) -> a__f(X,a__b(),b())
            mark(a()) -> a()
            mark(b()) -> a__b()
            mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
        - Signature:
            {a__b/0,a__f/3,mark/1,a__b#/0,a__f#/3,mark#/1} / {a/0,b/0,f/3,c_1/0,c_2/0,c_3/0,c_4/1,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:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:a__b#() -> c_1()
           
        
        2:S:a__b#() -> c_2()
           
        
        3:S:a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
           -->_1 a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b())):3
        
        4:S:mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
           -->_1 a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b())):3
        
        7:W:mark#(b()) -> c_6(a__b#())
           -->_1 a__b#() -> c_2():2
           -->_1 a__b#() -> c_1():1
        
        
        Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
        
        [(7,mark#(b()) -> c_6(a__b#()))]
** Step 1.b:6: RemoveHeads.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a__b#() -> c_1()
            a__b#() -> c_2()
            a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
            mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
        - Weak TRS:
            a__b() -> a()
            a__b() -> b()
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__f(a(),X,X) -> a__f(X,a__b(),b())
            mark(a()) -> a()
            mark(b()) -> a__b()
            mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
        - Signature:
            {a__b/0,a__f/3,mark/1,a__b#/0,a__f#/3,mark#/1} / {a/0,b/0,f/3,c_1/0,c_2/0,c_3/0,c_4/1,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:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:a__b#() -> c_1()
           
        
        2:S:a__b#() -> c_2()
           
        
        3:S:a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
           -->_1 a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b())):3
        
        4:S:mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
           -->_1 a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b())):3
        
        
        Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
        
        [(1,a__b#() -> c_1()),(2,a__b#() -> c_2())]
** Step 1.b:7: Decompose.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
            mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
        - Weak TRS:
            a__b() -> a()
            a__b() -> b()
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__f(a(),X,X) -> a__f(X,a__b(),b())
            mark(a()) -> a()
            mark(b()) -> a__b()
            mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
        - Signature:
            {a__b/0,a__f/3,mark/1,a__b#/0,a__f#/3,mark#/1} / {a/0,b/0,f/3,c_1/0,c_2/0,c_3/0,c_4/1,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:
        Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    + Details:
        We analyse the complexity of following sub-problems (R) and (S).
        Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
        
        Problem (R)
          - Strict DPs:
              a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
          - Weak DPs:
              mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
          - Weak TRS:
              a__b() -> a()
              a__b() -> b()
              a__f(X1,X2,X3) -> f(X1,X2,X3)
              a__f(a(),X,X) -> a__f(X,a__b(),b())
              mark(a()) -> a()
              mark(b()) -> a__b()
              mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
          - Signature:
              {a__b/0,a__f/3,mark/1,a__b#/0,a__f#/3,mark#/1} / {a/0,b/0,f/3,c_1/0,c_2/0,c_3/0,c_4/1,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}
        
        Problem (S)
          - Strict DPs:
              mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
          - Weak DPs:
              a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
          - Weak TRS:
              a__b() -> a()
              a__b() -> b()
              a__f(X1,X2,X3) -> f(X1,X2,X3)
              a__f(a(),X,X) -> a__f(X,a__b(),b())
              mark(a()) -> a()
              mark(b()) -> a__b()
              mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
          - Signature:
              {a__b/0,a__f/3,mark/1,a__b#/0,a__f#/3,mark#/1} / {a/0,b/0,f/3,c_1/0,c_2/0,c_3/0,c_4/1,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}
*** Step 1.b:7.a:1: PredecessorEstimationCP.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
        - Weak DPs:
            mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
        - Weak TRS:
            a__b() -> a()
            a__b() -> b()
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__f(a(),X,X) -> a__f(X,a__b(),b())
            mark(a()) -> a()
            mark(b()) -> a__b()
            mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
        - Signature:
            {a__b/0,a__f/3,mark/1,a__b#/0,a__f#/3,mark#/1} / {a/0,b/0,f/3,c_1/0,c_2/0,c_3/0,c_4/1,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:
          3: a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
          
        Consider the set of all dependency pairs
          3: a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
          4: mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
        Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1))
        BEST_CASE TIME (?,?)
        SPACE(?,?)on application of the dependency pairs
          {3}
        These cover all (indirect) predecessors of dependency pairs
          {3,4}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
**** Step 1.b:7.a:1.a:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
        - Weak DPs:
            mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
        - Weak TRS:
            a__b() -> a()
            a__b() -> b()
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__f(a(),X,X) -> a__f(X,a__b(),b())
            mark(a()) -> a()
            mark(b()) -> a__b()
            mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
        - Signature:
            {a__b/0,a__f/3,mark/1,a__b#/0,a__f#/3,mark#/1} / {a/0,b/0,f/3,c_1/0,c_2/0,c_3/0,c_4/1,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_4) = {1},
          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) = [0]                  
           p(a__f) = [5] x1 + [2] x2 + [6]
              p(b) = [0]                  
              p(f) = [1] x1 + [1] x3 + [0]
           p(mark) = [5] x1 + [9]         
          p(a__b#) = [1]                  
          p(a__f#) = [2] x1 + [8] x3 + [0]
          p(mark#) = [8] x1 + [1]         
            p(c_1) = [0]                  
            p(c_2) = [2]                  
            p(c_3) = [4]                  
            p(c_4) = [2] x1 + [0]         
            p(c_5) = [1]                  
            p(c_6) = [1] x1 + [1]         
            p(c_7) = [1] x1 + [0]         
        
        Following rules are strictly oriented:
        a__f#(a(),X,X) = [8] X + [4]             
                       > [4] X + [0]             
                       = c_4(a__f#(X,a__b(),b()))
        
        
        Following rules are (at-least) weakly oriented:
        mark#(f(X1,X2,X3)) =  [8] X1 + [8] X3 + [1]     
                           >= [2] X1 + [8] X3 + [0]     
                           =  c_7(a__f#(X1,mark(X2),X3))
        
**** Step 1.b:7.a:1.a:2: Assumption.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
            mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
        - Weak TRS:
            a__b() -> a()
            a__b() -> b()
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__f(a(),X,X) -> a__f(X,a__b(),b())
            mark(a()) -> a()
            mark(b()) -> a__b()
            mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
        - Signature:
            {a__b/0,a__f/3,mark/1,a__b#/0,a__f#/3,mark#/1} / {a/0,b/0,f/3,c_1/0,c_2/0,c_3/0,c_4/1,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, timeBCUB = Unknown, timeBCLB = Unknown}}
    + Details:
        ()

**** Step 1.b:7.a:1.b:1: RemoveWeakSuffixes.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
            mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
        - Weak TRS:
            a__b() -> a()
            a__b() -> b()
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__f(a(),X,X) -> a__f(X,a__b(),b())
            mark(a()) -> a()
            mark(b()) -> a__b()
            mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
        - Signature:
            {a__b/0,a__f/3,mark/1,a__b#/0,a__f#/3,mark#/1} / {a/0,b/0,f/3,c_1/0,c_2/0,c_3/0,c_4/1,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:a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
             -->_1 a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b())):1
          
          2:W:mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
             -->_1 a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b())):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
          1: a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
**** Step 1.b:7.a:1.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a__b() -> a()
            a__b() -> b()
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__f(a(),X,X) -> a__f(X,a__b(),b())
            mark(a()) -> a()
            mark(b()) -> a__b()
            mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
        - Signature:
            {a__b/0,a__f/3,mark/1,a__b#/0,a__f#/3,mark#/1} / {a/0,b/0,f/3,c_1/0,c_2/0,c_3/0,c_4/1,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).

*** Step 1.b:7.b:1: PredecessorEstimation.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
        - Weak DPs:
            a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
        - Weak TRS:
            a__b() -> a()
            a__b() -> b()
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__f(a(),X,X) -> a__f(X,a__b(),b())
            mark(a()) -> a()
            mark(b()) -> a__b()
            mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
        - Signature:
            {a__b/0,a__f/3,mark/1,a__b#/0,a__f#/3,mark#/1} / {a/0,b/0,f/3,c_1/0,c_2/0,c_3/0,c_4/1,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:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1}
        by application of
          Pre({1}) = {}.
        Here rules are labelled as follows:
          1: mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
          2: a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
*** Step 1.b:7.b:2: RemoveWeakSuffixes.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
            mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
        - Weak TRS:
            a__b() -> a()
            a__b() -> b()
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__f(a(),X,X) -> a__f(X,a__b(),b())
            mark(a()) -> a()
            mark(b()) -> a__b()
            mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
        - Signature:
            {a__b/0,a__f/3,mark/1,a__b#/0,a__f#/3,mark#/1} / {a/0,b/0,f/3,c_1/0,c_2/0,c_3/0,c_4/1,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:a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
             -->_1 a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b())):1
          
          2:W:mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
             -->_1 a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b())):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3))
          1: a__f#(a(),X,X) -> c_4(a__f#(X,a__b(),b()))
*** Step 1.b:7.b:3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a__b() -> a()
            a__b() -> b()
            a__f(X1,X2,X3) -> f(X1,X2,X3)
            a__f(a(),X,X) -> a__f(X,a__b(),b())
            mark(a()) -> a()
            mark(b()) -> a__b()
            mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
        - Signature:
            {a__b/0,a__f/3,mark/1,a__b#/0,a__f#/3,mark#/1} / {a/0,b/0,f/3,c_1/0,c_2/0,c_3/0,c_4/1,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))