MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            @(Cons(x,xs),ys) -> Cons(x,@(xs,ys))
            @(Nil(),ys) -> ys
            binom(Cons(x,xs),Cons(x',xs')) -> @(binom(xs,xs'),binom(xs,Cons(x',xs')))
            binom(Cons(x,xs),Nil()) -> Cons(Nil(),Nil())
            binom(Nil(),k) -> Cons(Nil(),Nil())
            goal(x,y) -> binom(x,y)
        - Signature:
            {@/2,binom/2,goal/2} / {Cons/2,Nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@,binom,goal} and constructors {Cons,Nil}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          @#(Cons(x,xs),ys) -> c_1(@#(xs,ys))
          @#(Nil(),ys) -> c_2()
          binom#(Cons(x,xs),Cons(x',xs')) -> c_3(@#(binom(xs,xs'),binom(xs,Cons(x',xs')))
                                                ,binom#(xs,xs')
                                                ,binom#(xs,Cons(x',xs')))
          binom#(Cons(x,xs),Nil()) -> c_4()
          binom#(Nil(),k) -> c_5()
          goal#(x,y) -> c_6(binom#(x,y))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            @#(Cons(x,xs),ys) -> c_1(@#(xs,ys))
            @#(Nil(),ys) -> c_2()
            binom#(Cons(x,xs),Cons(x',xs')) -> c_3(@#(binom(xs,xs'),binom(xs,Cons(x',xs')))
                                                  ,binom#(xs,xs')
                                                  ,binom#(xs,Cons(x',xs')))
            binom#(Cons(x,xs),Nil()) -> c_4()
            binom#(Nil(),k) -> c_5()
            goal#(x,y) -> c_6(binom#(x,y))
        - Weak TRS:
            @(Cons(x,xs),ys) -> Cons(x,@(xs,ys))
            @(Nil(),ys) -> ys
            binom(Cons(x,xs),Cons(x',xs')) -> @(binom(xs,xs'),binom(xs,Cons(x',xs')))
            binom(Cons(x,xs),Nil()) -> Cons(Nil(),Nil())
            binom(Nil(),k) -> Cons(Nil(),Nil())
            goal(x,y) -> binom(x,y)
        - Signature:
            {@/2,binom/2,goal/2,@#/2,binom#/2,goal#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@#,binom#,goal#} and constructors {Cons,Nil}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          @(Cons(x,xs),ys) -> Cons(x,@(xs,ys))
          @(Nil(),ys) -> ys
          binom(Cons(x,xs),Cons(x',xs')) -> @(binom(xs,xs'),binom(xs,Cons(x',xs')))
          binom(Cons(x,xs),Nil()) -> Cons(Nil(),Nil())
          binom(Nil(),k) -> Cons(Nil(),Nil())
          @#(Cons(x,xs),ys) -> c_1(@#(xs,ys))
          @#(Nil(),ys) -> c_2()
          binom#(Cons(x,xs),Cons(x',xs')) -> c_3(@#(binom(xs,xs'),binom(xs,Cons(x',xs')))
                                                ,binom#(xs,xs')
                                                ,binom#(xs,Cons(x',xs')))
          binom#(Cons(x,xs),Nil()) -> c_4()
          binom#(Nil(),k) -> c_5()
          goal#(x,y) -> c_6(binom#(x,y))
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            @#(Cons(x,xs),ys) -> c_1(@#(xs,ys))
            @#(Nil(),ys) -> c_2()
            binom#(Cons(x,xs),Cons(x',xs')) -> c_3(@#(binom(xs,xs'),binom(xs,Cons(x',xs')))
                                                  ,binom#(xs,xs')
                                                  ,binom#(xs,Cons(x',xs')))
            binom#(Cons(x,xs),Nil()) -> c_4()
            binom#(Nil(),k) -> c_5()
            goal#(x,y) -> c_6(binom#(x,y))
        - Weak TRS:
            @(Cons(x,xs),ys) -> Cons(x,@(xs,ys))
            @(Nil(),ys) -> ys
            binom(Cons(x,xs),Cons(x',xs')) -> @(binom(xs,xs'),binom(xs,Cons(x',xs')))
            binom(Cons(x,xs),Nil()) -> Cons(Nil(),Nil())
            binom(Nil(),k) -> Cons(Nil(),Nil())
        - Signature:
            {@/2,binom/2,goal/2,@#/2,binom#/2,goal#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@#,binom#,goal#} and constructors {Cons,Nil}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,4,5}
        by application of
          Pre({2,4,5}) = {1,3,6}.
        Here rules are labelled as follows:
          1: @#(Cons(x,xs),ys) -> c_1(@#(xs,ys))
          2: @#(Nil(),ys) -> c_2()
          3: binom#(Cons(x,xs),Cons(x',xs')) -> c_3(@#(binom(xs,xs'),binom(xs,Cons(x',xs')))
                                                   ,binom#(xs,xs')
                                                   ,binom#(xs,Cons(x',xs')))
          4: binom#(Cons(x,xs),Nil()) -> c_4()
          5: binom#(Nil(),k) -> c_5()
          6: goal#(x,y) -> c_6(binom#(x,y))
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            @#(Cons(x,xs),ys) -> c_1(@#(xs,ys))
            binom#(Cons(x,xs),Cons(x',xs')) -> c_3(@#(binom(xs,xs'),binom(xs,Cons(x',xs')))
                                                  ,binom#(xs,xs')
                                                  ,binom#(xs,Cons(x',xs')))
            goal#(x,y) -> c_6(binom#(x,y))
        - Weak DPs:
            @#(Nil(),ys) -> c_2()
            binom#(Cons(x,xs),Nil()) -> c_4()
            binom#(Nil(),k) -> c_5()
        - Weak TRS:
            @(Cons(x,xs),ys) -> Cons(x,@(xs,ys))
            @(Nil(),ys) -> ys
            binom(Cons(x,xs),Cons(x',xs')) -> @(binom(xs,xs'),binom(xs,Cons(x',xs')))
            binom(Cons(x,xs),Nil()) -> Cons(Nil(),Nil())
            binom(Nil(),k) -> Cons(Nil(),Nil())
        - Signature:
            {@/2,binom/2,goal/2,@#/2,binom#/2,goal#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@#,binom#,goal#} and constructors {Cons,Nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:@#(Cons(x,xs),ys) -> c_1(@#(xs,ys))
             -->_1 @#(Nil(),ys) -> c_2():4
             -->_1 @#(Cons(x,xs),ys) -> c_1(@#(xs,ys)):1
          
          2:S:binom#(Cons(x,xs),Cons(x',xs')) -> c_3(@#(binom(xs,xs'),binom(xs,Cons(x',xs')))
                                                    ,binom#(xs,xs')
                                                    ,binom#(xs,Cons(x',xs')))
             -->_3 binom#(Nil(),k) -> c_5():6
             -->_2 binom#(Nil(),k) -> c_5():6
             -->_2 binom#(Cons(x,xs),Nil()) -> c_4():5
             -->_1 @#(Nil(),ys) -> c_2():4
             -->_3 binom#(Cons(x,xs),Cons(x',xs')) -> c_3(@#(binom(xs,xs'),binom(xs,Cons(x',xs')))
                                                         ,binom#(xs,xs')
                                                         ,binom#(xs,Cons(x',xs'))):2
             -->_2 binom#(Cons(x,xs),Cons(x',xs')) -> c_3(@#(binom(xs,xs'),binom(xs,Cons(x',xs')))
                                                         ,binom#(xs,xs')
                                                         ,binom#(xs,Cons(x',xs'))):2
             -->_1 @#(Cons(x,xs),ys) -> c_1(@#(xs,ys)):1
          
          3:S:goal#(x,y) -> c_6(binom#(x,y))
             -->_1 binom#(Nil(),k) -> c_5():6
             -->_1 binom#(Cons(x,xs),Nil()) -> c_4():5
             -->_1 binom#(Cons(x,xs),Cons(x',xs')) -> c_3(@#(binom(xs,xs'),binom(xs,Cons(x',xs')))
                                                         ,binom#(xs,xs')
                                                         ,binom#(xs,Cons(x',xs'))):2
          
          4:W:@#(Nil(),ys) -> c_2()
             
          
          5:W:binom#(Cons(x,xs),Nil()) -> c_4()
             
          
          6:W:binom#(Nil(),k) -> c_5()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: binom#(Cons(x,xs),Nil()) -> c_4()
          6: binom#(Nil(),k) -> c_5()
          4: @#(Nil(),ys) -> c_2()
* Step 5: RemoveHeads MAYBE
    + Considered Problem:
        - Strict DPs:
            @#(Cons(x,xs),ys) -> c_1(@#(xs,ys))
            binom#(Cons(x,xs),Cons(x',xs')) -> c_3(@#(binom(xs,xs'),binom(xs,Cons(x',xs')))
                                                  ,binom#(xs,xs')
                                                  ,binom#(xs,Cons(x',xs')))
            goal#(x,y) -> c_6(binom#(x,y))
        - Weak TRS:
            @(Cons(x,xs),ys) -> Cons(x,@(xs,ys))
            @(Nil(),ys) -> ys
            binom(Cons(x,xs),Cons(x',xs')) -> @(binom(xs,xs'),binom(xs,Cons(x',xs')))
            binom(Cons(x,xs),Nil()) -> Cons(Nil(),Nil())
            binom(Nil(),k) -> Cons(Nil(),Nil())
        - Signature:
            {@/2,binom/2,goal/2,@#/2,binom#/2,goal#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@#,binom#,goal#} and constructors {Cons,Nil}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:@#(Cons(x,xs),ys) -> c_1(@#(xs,ys))
           -->_1 @#(Cons(x,xs),ys) -> c_1(@#(xs,ys)):1
        
        2:S:binom#(Cons(x,xs),Cons(x',xs')) -> c_3(@#(binom(xs,xs'),binom(xs,Cons(x',xs')))
                                                  ,binom#(xs,xs')
                                                  ,binom#(xs,Cons(x',xs')))
           -->_3 binom#(Cons(x,xs),Cons(x',xs')) -> c_3(@#(binom(xs,xs'),binom(xs,Cons(x',xs')))
                                                       ,binom#(xs,xs')
                                                       ,binom#(xs,Cons(x',xs'))):2
           -->_2 binom#(Cons(x,xs),Cons(x',xs')) -> c_3(@#(binom(xs,xs'),binom(xs,Cons(x',xs')))
                                                       ,binom#(xs,xs')
                                                       ,binom#(xs,Cons(x',xs'))):2
           -->_1 @#(Cons(x,xs),ys) -> c_1(@#(xs,ys)):1
        
        3:S:goal#(x,y) -> c_6(binom#(x,y))
           -->_1 binom#(Cons(x,xs),Cons(x',xs')) -> c_3(@#(binom(xs,xs'),binom(xs,Cons(x',xs')))
                                                       ,binom#(xs,xs')
                                                       ,binom#(xs,Cons(x',xs'))):2
        
        
        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).
        
        [(3,goal#(x,y) -> c_6(binom#(x,y)))]
* Step 6: Decompose MAYBE
    + Considered Problem:
        - Strict DPs:
            @#(Cons(x,xs),ys) -> c_1(@#(xs,ys))
            binom#(Cons(x,xs),Cons(x',xs')) -> c_3(@#(binom(xs,xs'),binom(xs,Cons(x',xs')))
                                                  ,binom#(xs,xs')
                                                  ,binom#(xs,Cons(x',xs')))
        - Weak TRS:
            @(Cons(x,xs),ys) -> Cons(x,@(xs,ys))
            @(Nil(),ys) -> ys
            binom(Cons(x,xs),Cons(x',xs')) -> @(binom(xs,xs'),binom(xs,Cons(x',xs')))
            binom(Cons(x,xs),Nil()) -> Cons(Nil(),Nil())
            binom(Nil(),k) -> Cons(Nil(),Nil())
        - Signature:
            {@/2,binom/2,goal/2,@#/2,binom#/2,goal#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@#,binom#,goal#} and constructors {Cons,Nil}
    + 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:
              @#(Cons(x,xs),ys) -> c_1(@#(xs,ys))
          - Weak DPs:
              binom#(Cons(x,xs),Cons(x',xs')) -> c_3(@#(binom(xs,xs'),binom(xs,Cons(x',xs')))
                                                    ,binom#(xs,xs')
                                                    ,binom#(xs,Cons(x',xs')))
          - Weak TRS:
              @(Cons(x,xs),ys) -> Cons(x,@(xs,ys))
              @(Nil(),ys) -> ys
              binom(Cons(x,xs),Cons(x',xs')) -> @(binom(xs,xs'),binom(xs,Cons(x',xs')))
              binom(Cons(x,xs),Nil()) -> Cons(Nil(),Nil())
              binom(Nil(),k) -> Cons(Nil(),Nil())
          - Signature:
              {@/2,binom/2,goal/2,@#/2,binom#/2,goal#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {@#,binom#,goal#} and constructors {Cons,Nil}
        
        Problem (S)
          - Strict DPs:
              binom#(Cons(x,xs),Cons(x',xs')) -> c_3(@#(binom(xs,xs'),binom(xs,Cons(x',xs')))
                                                    ,binom#(xs,xs')
                                                    ,binom#(xs,Cons(x',xs')))
          - Weak DPs:
              @#(Cons(x,xs),ys) -> c_1(@#(xs,ys))
          - Weak TRS:
              @(Cons(x,xs),ys) -> Cons(x,@(xs,ys))
              @(Nil(),ys) -> ys
              binom(Cons(x,xs),Cons(x',xs')) -> @(binom(xs,xs'),binom(xs,Cons(x',xs')))
              binom(Cons(x,xs),Nil()) -> Cons(Nil(),Nil())
              binom(Nil(),k) -> Cons(Nil(),Nil())
          - Signature:
              {@/2,binom/2,goal/2,@#/2,binom#/2,goal#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {@#,binom#,goal#} and constructors {Cons,Nil}
** Step 6.a:1: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          @#(Cons(x,xs),ys) -> c_1(@#(xs,ys))
      - Weak DPs:
          binom#(Cons(x,xs),Cons(x',xs')) -> c_3(@#(binom(xs,xs'),binom(xs,Cons(x',xs')))
                                                ,binom#(xs,xs')
                                                ,binom#(xs,Cons(x',xs')))
      - Weak TRS:
          @(Cons(x,xs),ys) -> Cons(x,@(xs,ys))
          @(Nil(),ys) -> ys
          binom(Cons(x,xs),Cons(x',xs')) -> @(binom(xs,xs'),binom(xs,Cons(x',xs')))
          binom(Cons(x,xs),Nil()) -> Cons(Nil(),Nil())
          binom(Nil(),k) -> Cons(Nil(),Nil())
      - Signature:
          {@/2,binom/2,goal/2,@#/2,binom#/2,goal#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {@#,binom#,goal#} and constructors {Cons,Nil}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
** Step 6.b:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            binom#(Cons(x,xs),Cons(x',xs')) -> c_3(@#(binom(xs,xs'),binom(xs,Cons(x',xs')))
                                                  ,binom#(xs,xs')
                                                  ,binom#(xs,Cons(x',xs')))
        - Weak DPs:
            @#(Cons(x,xs),ys) -> c_1(@#(xs,ys))
        - Weak TRS:
            @(Cons(x,xs),ys) -> Cons(x,@(xs,ys))
            @(Nil(),ys) -> ys
            binom(Cons(x,xs),Cons(x',xs')) -> @(binom(xs,xs'),binom(xs,Cons(x',xs')))
            binom(Cons(x,xs),Nil()) -> Cons(Nil(),Nil())
            binom(Nil(),k) -> Cons(Nil(),Nil())
        - Signature:
            {@/2,binom/2,goal/2,@#/2,binom#/2,goal#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@#,binom#,goal#} and constructors {Cons,Nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:binom#(Cons(x,xs),Cons(x',xs')) -> c_3(@#(binom(xs,xs'),binom(xs,Cons(x',xs')))
                                                    ,binom#(xs,xs')
                                                    ,binom#(xs,Cons(x',xs')))
             -->_1 @#(Cons(x,xs),ys) -> c_1(@#(xs,ys)):2
             -->_3 binom#(Cons(x,xs),Cons(x',xs')) -> c_3(@#(binom(xs,xs'),binom(xs,Cons(x',xs')))
                                                         ,binom#(xs,xs')
                                                         ,binom#(xs,Cons(x',xs'))):1
             -->_2 binom#(Cons(x,xs),Cons(x',xs')) -> c_3(@#(binom(xs,xs'),binom(xs,Cons(x',xs')))
                                                         ,binom#(xs,xs')
                                                         ,binom#(xs,Cons(x',xs'))):1
          
          2:W:@#(Cons(x,xs),ys) -> c_1(@#(xs,ys))
             -->_1 @#(Cons(x,xs),ys) -> c_1(@#(xs,ys)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: @#(Cons(x,xs),ys) -> c_1(@#(xs,ys))
** Step 6.b:2: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            binom#(Cons(x,xs),Cons(x',xs')) -> c_3(@#(binom(xs,xs'),binom(xs,Cons(x',xs')))
                                                  ,binom#(xs,xs')
                                                  ,binom#(xs,Cons(x',xs')))
        - Weak TRS:
            @(Cons(x,xs),ys) -> Cons(x,@(xs,ys))
            @(Nil(),ys) -> ys
            binom(Cons(x,xs),Cons(x',xs')) -> @(binom(xs,xs'),binom(xs,Cons(x',xs')))
            binom(Cons(x,xs),Nil()) -> Cons(Nil(),Nil())
            binom(Nil(),k) -> Cons(Nil(),Nil())
        - Signature:
            {@/2,binom/2,goal/2,@#/2,binom#/2,goal#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/3,c_4/0,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@#,binom#,goal#} and constructors {Cons,Nil}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:binom#(Cons(x,xs),Cons(x',xs')) -> c_3(@#(binom(xs,xs'),binom(xs,Cons(x',xs')))
                                                    ,binom#(xs,xs')
                                                    ,binom#(xs,Cons(x',xs')))
             -->_3 binom#(Cons(x,xs),Cons(x',xs')) -> c_3(@#(binom(xs,xs'),binom(xs,Cons(x',xs')))
                                                         ,binom#(xs,xs')
                                                         ,binom#(xs,Cons(x',xs'))):1
             -->_2 binom#(Cons(x,xs),Cons(x',xs')) -> c_3(@#(binom(xs,xs'),binom(xs,Cons(x',xs')))
                                                         ,binom#(xs,xs')
                                                         ,binom#(xs,Cons(x',xs'))):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          binom#(Cons(x,xs),Cons(x',xs')) -> c_3(binom#(xs,xs'),binom#(xs,Cons(x',xs')))
** Step 6.b:3: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            binom#(Cons(x,xs),Cons(x',xs')) -> c_3(binom#(xs,xs'),binom#(xs,Cons(x',xs')))
        - Weak TRS:
            @(Cons(x,xs),ys) -> Cons(x,@(xs,ys))
            @(Nil(),ys) -> ys
            binom(Cons(x,xs),Cons(x',xs')) -> @(binom(xs,xs'),binom(xs,Cons(x',xs')))
            binom(Cons(x,xs),Nil()) -> Cons(Nil(),Nil())
            binom(Nil(),k) -> Cons(Nil(),Nil())
        - Signature:
            {@/2,binom/2,goal/2,@#/2,binom#/2,goal#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {@#,binom#,goal#} and constructors {Cons,Nil}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          binom#(Cons(x,xs),Cons(x',xs')) -> c_3(binom#(xs,xs'),binom#(xs,Cons(x',xs')))
** Step 6.b:4: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          binom#(Cons(x,xs),Cons(x',xs')) -> c_3(binom#(xs,xs'),binom#(xs,Cons(x',xs')))
      - Signature:
          {@/2,binom/2,goal/2,@#/2,binom#/2,goal#/2} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0,c_6/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {@#,binom#,goal#} and constructors {Cons,Nil}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE