MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            first(0(),X) -> nil()
            first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
            terms(N) -> cons(recip(sqr(N)),terms(s(N)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1} / {0/0,cons/2,nil/0,recip/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add,dbl,first,sqr,terms} and constructors {0,cons,nil
            ,recip,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          add#(0(),X) -> c_1()
          add#(s(X),Y) -> c_2(add#(X,Y))
          dbl#(0()) -> c_3()
          dbl#(s(X)) -> c_4(dbl#(X))
          first#(0(),X) -> c_5()
          first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
          sqr#(0()) -> c_7()
          sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
          terms#(N) -> c_9(sqr#(N),terms#(s(N)))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            add#(0(),X) -> c_1()
            add#(s(X),Y) -> c_2(add#(X,Y))
            dbl#(0()) -> c_3()
            dbl#(s(X)) -> c_4(dbl#(X))
            first#(0(),X) -> c_5()
            first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
            sqr#(0()) -> c_7()
            sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
            terms#(N) -> c_9(sqr#(N),terms#(s(N)))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            first(0(),X) -> nil()
            first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
            terms(N) -> cons(recip(sqr(N)),terms(s(N)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/3,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          add(0(),X) -> X
          add(s(X),Y) -> s(add(X,Y))
          dbl(0()) -> 0()
          dbl(s(X)) -> s(s(dbl(X)))
          sqr(0()) -> 0()
          sqr(s(X)) -> s(add(sqr(X),dbl(X)))
          add#(0(),X) -> c_1()
          add#(s(X),Y) -> c_2(add#(X,Y))
          dbl#(0()) -> c_3()
          dbl#(s(X)) -> c_4(dbl#(X))
          first#(0(),X) -> c_5()
          first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
          sqr#(0()) -> c_7()
          sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
          terms#(N) -> c_9(sqr#(N),terms#(s(N)))
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            add#(0(),X) -> c_1()
            add#(s(X),Y) -> c_2(add#(X,Y))
            dbl#(0()) -> c_3()
            dbl#(s(X)) -> c_4(dbl#(X))
            first#(0(),X) -> c_5()
            first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
            sqr#(0()) -> c_7()
            sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
            terms#(N) -> c_9(sqr#(N),terms#(s(N)))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/3,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,3,5,7}
        by application of
          Pre({1,3,5,7}) = {2,4,6,8,9}.
        Here rules are labelled as follows:
          1: add#(0(),X) -> c_1()
          2: add#(s(X),Y) -> c_2(add#(X,Y))
          3: dbl#(0()) -> c_3()
          4: dbl#(s(X)) -> c_4(dbl#(X))
          5: first#(0(),X) -> c_5()
          6: first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
          7: sqr#(0()) -> c_7()
          8: sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
          9: terms#(N) -> c_9(sqr#(N),terms#(s(N)))
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            add#(s(X),Y) -> c_2(add#(X,Y))
            dbl#(s(X)) -> c_4(dbl#(X))
            first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
            sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
            terms#(N) -> c_9(sqr#(N),terms#(s(N)))
        - Weak DPs:
            add#(0(),X) -> c_1()
            dbl#(0()) -> c_3()
            first#(0(),X) -> c_5()
            sqr#(0()) -> c_7()
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/3,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:add#(s(X),Y) -> c_2(add#(X,Y))
             -->_1 add#(0(),X) -> c_1():6
             -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
          
          2:S:dbl#(s(X)) -> c_4(dbl#(X))
             -->_1 dbl#(0()) -> c_3():7
             -->_1 dbl#(s(X)) -> c_4(dbl#(X)):2
          
          3:S:first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
             -->_1 first#(0(),X) -> c_5():8
             -->_1 first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)):3
          
          4:S:sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
             -->_2 sqr#(0()) -> c_7():9
             -->_3 dbl#(0()) -> c_3():7
             -->_1 add#(0(),X) -> c_1():6
             -->_2 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):4
             -->_3 dbl#(s(X)) -> c_4(dbl#(X)):2
             -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
          
          5:S:terms#(N) -> c_9(sqr#(N),terms#(s(N)))
             -->_1 sqr#(0()) -> c_7():9
             -->_2 terms#(N) -> c_9(sqr#(N),terms#(s(N))):5
             -->_1 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):4
          
          6:W:add#(0(),X) -> c_1()
             
          
          7:W:dbl#(0()) -> c_3()
             
          
          8:W:first#(0(),X) -> c_5()
             
          
          9:W:sqr#(0()) -> c_7()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          9: sqr#(0()) -> c_7()
          8: first#(0(),X) -> c_5()
          7: dbl#(0()) -> c_3()
          6: add#(0(),X) -> c_1()
* Step 5: Decompose MAYBE
    + Considered Problem:
        - Strict DPs:
            add#(s(X),Y) -> c_2(add#(X,Y))
            dbl#(s(X)) -> c_4(dbl#(X))
            first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
            sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
            terms#(N) -> c_9(sqr#(N),terms#(s(N)))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/3,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + 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:
              add#(s(X),Y) -> c_2(add#(X,Y))
          - Weak DPs:
              dbl#(s(X)) -> c_4(dbl#(X))
              first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
              sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
              terms#(N) -> c_9(sqr#(N),terms#(s(N)))
          - Weak TRS:
              add(0(),X) -> X
              add(s(X),Y) -> s(add(X,Y))
              dbl(0()) -> 0()
              dbl(s(X)) -> s(s(dbl(X)))
              sqr(0()) -> 0()
              sqr(s(X)) -> s(add(sqr(X),dbl(X)))
          - Signature:
              {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
              ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/3,c_9/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
              ,nil,recip,s}
        
        Problem (S)
          - Strict DPs:
              dbl#(s(X)) -> c_4(dbl#(X))
              first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
              sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
              terms#(N) -> c_9(sqr#(N),terms#(s(N)))
          - Weak DPs:
              add#(s(X),Y) -> c_2(add#(X,Y))
          - Weak TRS:
              add(0(),X) -> X
              add(s(X),Y) -> s(add(X,Y))
              dbl(0()) -> 0()
              dbl(s(X)) -> s(s(dbl(X)))
              sqr(0()) -> 0()
              sqr(s(X)) -> s(add(sqr(X),dbl(X)))
          - Signature:
              {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
              ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/3,c_9/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
              ,nil,recip,s}
** Step 5.a:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            add#(s(X),Y) -> c_2(add#(X,Y))
        - Weak DPs:
            dbl#(s(X)) -> c_4(dbl#(X))
            first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
            sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
            terms#(N) -> c_9(sqr#(N),terms#(s(N)))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/3,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:add#(s(X),Y) -> c_2(add#(X,Y))
             -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
          
          2:W:dbl#(s(X)) -> c_4(dbl#(X))
             -->_1 dbl#(s(X)) -> c_4(dbl#(X)):2
          
          3:W:first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
             -->_1 first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)):3
          
          4:W:sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
             -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
             -->_3 dbl#(s(X)) -> c_4(dbl#(X)):2
             -->_2 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):4
          
          5:W:terms#(N) -> c_9(sqr#(N),terms#(s(N)))
             -->_1 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):4
             -->_2 terms#(N) -> c_9(sqr#(N),terms#(s(N))):5
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
          2: dbl#(s(X)) -> c_4(dbl#(X))
** Step 5.a:2: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            add#(s(X),Y) -> c_2(add#(X,Y))
        - Weak DPs:
            sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
            terms#(N) -> c_9(sqr#(N),terms#(s(N)))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/3,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:add#(s(X),Y) -> c_2(add#(X,Y))
             -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
          
          4:W:sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
             -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
             -->_2 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):4
          
          5:W:terms#(N) -> c_9(sqr#(N),terms#(s(N)))
             -->_1 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):4
             -->_2 terms#(N) -> c_9(sqr#(N),terms#(s(N))):5
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X))
** Step 5.a:3: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          add#(s(X),Y) -> c_2(add#(X,Y))
      - Weak DPs:
          sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X))
          terms#(N) -> c_9(sqr#(N),terms#(s(N)))
      - Weak TRS:
          add(0(),X) -> X
          add(s(X),Y) -> s(add(X,Y))
          dbl(0()) -> 0()
          dbl(s(X)) -> s(s(dbl(X)))
          sqr(0()) -> 0()
          sqr(s(X)) -> s(add(sqr(X),dbl(X)))
      - Signature:
          {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
          ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/2,c_9/2}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
          ,nil,recip,s}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
** Step 5.b:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            dbl#(s(X)) -> c_4(dbl#(X))
            first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
            sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
            terms#(N) -> c_9(sqr#(N),terms#(s(N)))
        - Weak DPs:
            add#(s(X),Y) -> c_2(add#(X,Y))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/3,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:dbl#(s(X)) -> c_4(dbl#(X))
             -->_1 dbl#(s(X)) -> c_4(dbl#(X)):1
          
          2:S:first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
             -->_1 first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)):2
          
          3:S:sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
             -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):5
             -->_2 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):3
             -->_3 dbl#(s(X)) -> c_4(dbl#(X)):1
          
          4:S:terms#(N) -> c_9(sqr#(N),terms#(s(N)))
             -->_2 terms#(N) -> c_9(sqr#(N),terms#(s(N))):4
             -->_1 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):3
          
          5:W:add#(s(X),Y) -> c_2(add#(X,Y))
             -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):5
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: add#(s(X),Y) -> c_2(add#(X,Y))
** Step 5.b:2: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            dbl#(s(X)) -> c_4(dbl#(X))
            first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
            sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
            terms#(N) -> c_9(sqr#(N),terms#(s(N)))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/3,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:dbl#(s(X)) -> c_4(dbl#(X))
             -->_1 dbl#(s(X)) -> c_4(dbl#(X)):1
          
          2:S:first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
             -->_1 first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)):2
          
          3:S:sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
             -->_2 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):3
             -->_3 dbl#(s(X)) -> c_4(dbl#(X)):1
          
          4:S:terms#(N) -> c_9(sqr#(N),terms#(s(N)))
             -->_2 terms#(N) -> c_9(sqr#(N),terms#(s(N))):4
             -->_1 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):3
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
** Step 5.b:3: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            dbl#(s(X)) -> c_4(dbl#(X))
            first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
            sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
            terms#(N) -> c_9(sqr#(N),terms#(s(N)))
        - Weak TRS:
            add(0(),X) -> X
            add(s(X),Y) -> s(add(X,Y))
            dbl(0()) -> 0()
            dbl(s(X)) -> s(s(dbl(X)))
            sqr(0()) -> 0()
            sqr(s(X)) -> s(add(sqr(X),dbl(X)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/2,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          dbl#(s(X)) -> c_4(dbl#(X))
          first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
          sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
          terms#(N) -> c_9(sqr#(N),terms#(s(N)))
** Step 5.b:4: Decompose MAYBE
    + Considered Problem:
        - Strict DPs:
            dbl#(s(X)) -> c_4(dbl#(X))
            first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
            sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
            terms#(N) -> c_9(sqr#(N),terms#(s(N)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/2,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + 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:
              dbl#(s(X)) -> c_4(dbl#(X))
          - Weak DPs:
              first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
              sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
              terms#(N) -> c_9(sqr#(N),terms#(s(N)))
          - Signature:
              {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
              ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/2,c_9/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
              ,nil,recip,s}
        
        Problem (S)
          - Strict DPs:
              first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
              sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
              terms#(N) -> c_9(sqr#(N),terms#(s(N)))
          - Weak DPs:
              dbl#(s(X)) -> c_4(dbl#(X))
          - Signature:
              {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
              ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/2,c_9/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
              ,nil,recip,s}
*** Step 5.b:4.a:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            dbl#(s(X)) -> c_4(dbl#(X))
        - Weak DPs:
            first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
            sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
            terms#(N) -> c_9(sqr#(N),terms#(s(N)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/2,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:dbl#(s(X)) -> c_4(dbl#(X))
             -->_1 dbl#(s(X)) -> c_4(dbl#(X)):1
          
          2:W:first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
             -->_1 first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)):2
          
          3:W:sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
             -->_2 dbl#(s(X)) -> c_4(dbl#(X)):1
             -->_1 sqr#(s(X)) -> c_8(sqr#(X),dbl#(X)):3
          
          4:W:terms#(N) -> c_9(sqr#(N),terms#(s(N)))
             -->_1 sqr#(s(X)) -> c_8(sqr#(X),dbl#(X)):3
             -->_2 terms#(N) -> c_9(sqr#(N),terms#(s(N))):4
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
*** Step 5.b:4.a:2: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          dbl#(s(X)) -> c_4(dbl#(X))
      - Weak DPs:
          sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
          terms#(N) -> c_9(sqr#(N),terms#(s(N)))
      - Signature:
          {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
          ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/2,c_9/2}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
          ,nil,recip,s}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
*** Step 5.b:4.b:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
            sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
            terms#(N) -> c_9(sqr#(N),terms#(s(N)))
        - Weak DPs:
            dbl#(s(X)) -> c_4(dbl#(X))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/2,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
             -->_1 first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)):1
          
          2:S:sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
             -->_2 dbl#(s(X)) -> c_4(dbl#(X)):4
             -->_1 sqr#(s(X)) -> c_8(sqr#(X),dbl#(X)):2
          
          3:S:terms#(N) -> c_9(sqr#(N),terms#(s(N)))
             -->_2 terms#(N) -> c_9(sqr#(N),terms#(s(N))):3
             -->_1 sqr#(s(X)) -> c_8(sqr#(X),dbl#(X)):2
          
          4:W:dbl#(s(X)) -> c_4(dbl#(X))
             -->_1 dbl#(s(X)) -> c_4(dbl#(X)):4
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: dbl#(s(X)) -> c_4(dbl#(X))
*** Step 5.b:4.b:2: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
            sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
            terms#(N) -> c_9(sqr#(N),terms#(s(N)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/2,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
             -->_1 first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)):1
          
          2:S:sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
             -->_1 sqr#(s(X)) -> c_8(sqr#(X),dbl#(X)):2
          
          3:S:terms#(N) -> c_9(sqr#(N),terms#(s(N)))
             -->_2 terms#(N) -> c_9(sqr#(N),terms#(s(N))):3
             -->_1 sqr#(s(X)) -> c_8(sqr#(X),dbl#(X)):2
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          sqr#(s(X)) -> c_8(sqr#(X))
*** Step 5.b:4.b:3: Decompose MAYBE
    + Considered Problem:
        - Strict DPs:
            first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
            sqr#(s(X)) -> c_8(sqr#(X))
            terms#(N) -> c_9(sqr#(N),terms#(s(N)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + 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:
              first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
          - Weak DPs:
              sqr#(s(X)) -> c_8(sqr#(X))
              terms#(N) -> c_9(sqr#(N),terms#(s(N)))
          - Signature:
              {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
              ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
              ,nil,recip,s}
        
        Problem (S)
          - Strict DPs:
              sqr#(s(X)) -> c_8(sqr#(X))
              terms#(N) -> c_9(sqr#(N),terms#(s(N)))
          - Weak DPs:
              first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
          - Signature:
              {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
              ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
              ,nil,recip,s}
**** Step 5.b:4.b:3.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
        - Weak DPs:
            sqr#(s(X)) -> c_8(sqr#(X))
            terms#(N) -> c_9(sqr#(N),terms#(s(N)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
             -->_1 first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)):1
          
          2:W:sqr#(s(X)) -> c_8(sqr#(X))
             -->_1 sqr#(s(X)) -> c_8(sqr#(X)):2
          
          3:W:terms#(N) -> c_9(sqr#(N),terms#(s(N)))
             -->_1 sqr#(s(X)) -> c_8(sqr#(X)):2
             -->_2 terms#(N) -> c_9(sqr#(N),terms#(s(N))):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: terms#(N) -> c_9(sqr#(N),terms#(s(N)))
          2: sqr#(s(X)) -> c_8(sqr#(X))
**** Step 5.b:4.b:3.a:2: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + 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: first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
          
        The strictly oriented rules are moved into the weak component.
***** Step 5.b:4.b:3.a:2.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + 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_6) = {1}
        
        Following symbols are considered usable:
          {add#,dbl#,first#,sqr#,terms#}
        TcT has computed the following interpretation:
               p(0) = [0]                  
             p(add) = [0]                  
            p(cons) = [1] x2 + [2]         
             p(dbl) = [4] x1 + [8]         
           p(first) = [2] x1 + [1] x2 + [1]
             p(nil) = [1]                  
           p(recip) = [0]                  
               p(s) = [1] x1 + [0]         
             p(sqr) = [1] x1 + [0]         
           p(terms) = [8] x1 + [0]         
            p(add#) = [2] x1 + [1]         
            p(dbl#) = [1]                  
          p(first#) = [8] x1 + [4] x2 + [0]
            p(sqr#) = [0]                  
          p(terms#) = [1] x1 + [1]         
             p(c_1) = [0]                  
             p(c_2) = [1] x1 + [0]         
             p(c_3) = [0]                  
             p(c_4) = [4]                  
             p(c_5) = [0]                  
             p(c_6) = [1] x1 + [0]         
             p(c_7) = [8]                  
             p(c_8) = [2] x1 + [1]         
             p(c_9) = [1] x2 + [1]         
        
        Following rules are strictly oriented:
        first#(s(X),cons(Y,Z)) = [8] X + [4] Z + [8]
                               > [8] X + [4] Z + [0]
                               = c_6(first#(X,Z))   
        
        
        Following rules are (at-least) weakly oriented:
        
***** Step 5.b:4.b:3.a:2.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

***** Step 5.b:4.b:3.a:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
             -->_1 first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
***** Step 5.b:4.b:3.a:2.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

**** Step 5.b:4.b:3.b:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            sqr#(s(X)) -> c_8(sqr#(X))
            terms#(N) -> c_9(sqr#(N),terms#(s(N)))
        - Weak DPs:
            first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:sqr#(s(X)) -> c_8(sqr#(X))
             -->_1 sqr#(s(X)) -> c_8(sqr#(X)):1
          
          2:S:terms#(N) -> c_9(sqr#(N),terms#(s(N)))
             -->_2 terms#(N) -> c_9(sqr#(N),terms#(s(N))):2
             -->_1 sqr#(s(X)) -> c_8(sqr#(X)):1
          
          3:W:first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
             -->_1 first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z))
**** Step 5.b:4.b:3.b:2: Decompose MAYBE
    + Considered Problem:
        - Strict DPs:
            sqr#(s(X)) -> c_8(sqr#(X))
            terms#(N) -> c_9(sqr#(N),terms#(s(N)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + 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:
              sqr#(s(X)) -> c_8(sqr#(X))
          - Weak DPs:
              terms#(N) -> c_9(sqr#(N),terms#(s(N)))
          - Signature:
              {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
              ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
              ,nil,recip,s}
        
        Problem (S)
          - Strict DPs:
              terms#(N) -> c_9(sqr#(N),terms#(s(N)))
          - Weak DPs:
              sqr#(s(X)) -> c_8(sqr#(X))
          - Signature:
              {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
              ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
              ,nil,recip,s}
***** Step 5.b:4.b:3.b:2.a:1: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          sqr#(s(X)) -> c_8(sqr#(X))
      - Weak DPs:
          terms#(N) -> c_9(sqr#(N),terms#(s(N)))
      - Signature:
          {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
          ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/2}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
          ,nil,recip,s}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
***** Step 5.b:4.b:3.b:2.b:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            terms#(N) -> c_9(sqr#(N),terms#(s(N)))
        - Weak DPs:
            sqr#(s(X)) -> c_8(sqr#(X))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:terms#(N) -> c_9(sqr#(N),terms#(s(N)))
             -->_1 sqr#(s(X)) -> c_8(sqr#(X)):2
             -->_2 terms#(N) -> c_9(sqr#(N),terms#(s(N))):1
          
          2:W:sqr#(s(X)) -> c_8(sqr#(X))
             -->_1 sqr#(s(X)) -> c_8(sqr#(X)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: sqr#(s(X)) -> c_8(sqr#(X))
***** Step 5.b:4.b:3.b:2.b:2: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            terms#(N) -> c_9(sqr#(N),terms#(s(N)))
        - Signature:
            {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
            ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
            ,nil,recip,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:terms#(N) -> c_9(sqr#(N),terms#(s(N)))
             -->_2 terms#(N) -> c_9(sqr#(N),terms#(s(N))):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          terms#(N) -> c_9(terms#(s(N)))
***** Step 5.b:4.b:3.b:2.b:3: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          terms#(N) -> c_9(terms#(s(N)))
      - Signature:
          {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/2,nil/0,recip/1,s/1
          ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
          ,nil,recip,s}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE