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