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))
            half(0()) -> 0()
            half(dbl(X)) -> X
            half(s(0())) -> 0()
            half(s(s(X))) -> s(half(X))
            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,half/1,sqr/1,terms/1} / {0/0,cons/2,nil/0,recip/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add,dbl,first,half,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))
          half#(0()) -> c_7()
          half#(dbl(X)) -> c_8()
          half#(s(0())) -> c_9()
          half#(s(s(X))) -> c_10(half#(X))
          sqr#(0()) -> c_11()
          sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
          terms#(N) -> c_13(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))
            half#(0()) -> c_7()
            half#(dbl(X)) -> c_8()
            half#(s(0())) -> c_9()
            half#(s(s(X))) -> c_10(half#(X))
            sqr#(0()) -> c_11()
            sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
            terms#(N) -> c_13(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))
            half(0()) -> 0()
            half(dbl(X)) -> X
            half(s(0())) -> 0()
            half(s(s(X))) -> s(half(X))
            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,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,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_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,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))
          half#(0()) -> c_7()
          half#(dbl(X)) -> c_8()
          half#(s(0())) -> c_9()
          half#(s(s(X))) -> c_10(half#(X))
          sqr#(0()) -> c_11()
          sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
          terms#(N) -> c_13(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))
            half#(0()) -> c_7()
            half#(dbl(X)) -> c_8()
            half#(s(0())) -> c_9()
            half#(s(s(X))) -> c_10(half#(X))
            sqr#(0()) -> c_11()
            sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
            terms#(N) -> c_13(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,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,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_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,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
          {1,3,5,7,8,9,11}
        by application of
          Pre({1,3,5,7,8,9,11}) = {2,4,6,10,12,13}.
        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: half#(0()) -> c_7()
          8: half#(dbl(X)) -> c_8()
          9: half#(s(0())) -> c_9()
          10: half#(s(s(X))) -> c_10(half#(X))
          11: sqr#(0()) -> c_11()
          12: sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
          13: terms#(N) -> c_13(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))
            half#(s(s(X))) -> c_10(half#(X))
            sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
            terms#(N) -> c_13(sqr#(N),terms#(s(N)))
        - Weak DPs:
            add#(0(),X) -> c_1()
            dbl#(0()) -> c_3()
            first#(0(),X) -> c_5()
            half#(0()) -> c_7()
            half#(dbl(X)) -> c_8()
            half#(s(0())) -> c_9()
            sqr#(0()) -> c_11()
        - 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,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,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_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,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():7
             -->_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():8
             -->_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():9
             -->_1 first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)):3
          
          4:S:half#(s(s(X))) -> c_10(half#(X))
             -->_1 half#(s(0())) -> c_9():12
             -->_1 half#(dbl(X)) -> c_8():11
             -->_1 half#(0()) -> c_7():10
             -->_1 half#(s(s(X))) -> c_10(half#(X)):4
          
          5:S:sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
             -->_2 sqr#(0()) -> c_11():13
             -->_3 dbl#(0()) -> c_3():8
             -->_1 add#(0(),X) -> c_1():7
             -->_2 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):5
             -->_3 dbl#(s(X)) -> c_4(dbl#(X)):2
             -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
          
          6:S:terms#(N) -> c_13(sqr#(N),terms#(s(N)))
             -->_1 sqr#(0()) -> c_11():13
             -->_2 terms#(N) -> c_13(sqr#(N),terms#(s(N))):6
             -->_1 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):5
          
          7:W:add#(0(),X) -> c_1()
             
          
          8:W:dbl#(0()) -> c_3()
             
          
          9:W:first#(0(),X) -> c_5()
             
          
          10:W:half#(0()) -> c_7()
             
          
          11:W:half#(dbl(X)) -> c_8()
             
          
          12:W:half#(s(0())) -> c_9()
             
          
          13:W:sqr#(0()) -> c_11()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          13: sqr#(0()) -> c_11()
          10: half#(0()) -> c_7()
          11: half#(dbl(X)) -> c_8()
          12: half#(s(0())) -> c_9()
          9: first#(0(),X) -> c_5()
          8: dbl#(0()) -> c_3()
          7: add#(0(),X) -> c_1()
* Step 5: Failure 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))
          half#(s(s(X))) -> c_10(half#(X))
          sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
          terms#(N) -> c_13(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,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,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_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/2}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0
          ,cons,nil,recip,s}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE