We consider the following Problem:

  Strict Trs:
    {  terms(N) -> cons(recip(sqr(N)))
     , sqr(0()) -> 0()
     , sqr(s()) -> s()
     , dbl(0()) -> 0()
     , dbl(s()) -> s()
     , add(0(), X) -> X
     , add(s(), Y) -> s()
     , first(0(), X) -> nil()
     , first(s(), cons(Y)) -> cons(Y)}
  StartTerms: basic terms
  Strategy: innermost

Certificate: YES(?,O(n^1))

Proof:
  We consider the following Problem:
  
    Strict Trs:
      {  terms(N) -> cons(recip(sqr(N)))
       , sqr(0()) -> 0()
       , sqr(s()) -> s()
       , dbl(0()) -> 0()
       , dbl(s()) -> s()
       , add(0(), X) -> X
       , add(s(), Y) -> s()
       , first(0(), X) -> nil()
       , first(s(), cons(Y)) -> cons(Y)}
    StartTerms: basic terms
    Strategy: innermost
  
  Certificate: YES(?,O(n^1))
  
  Proof:
    The weightgap principle applies, where following rules are oriented strictly:
    
    TRS Component:
      {  terms(N) -> cons(recip(sqr(N)))
       , dbl(0()) -> 0()
       , dbl(s()) -> s()
       , add(s(), Y) -> s()
       , first(0(), X) -> nil()
       , first(s(), cons(Y)) -> cons(Y)}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      The following argument positions are usable:
        Uargs(terms) = {}, Uargs(cons) = {1}, Uargs(recip) = {1},
        Uargs(sqr) = {}, Uargs(dbl) = {}, Uargs(add) = {},
        Uargs(first) = {}
      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
      Interpretation Functions:
       terms(x1) = [1 0] x1 + [2]
                   [1 0]      [2]
       cons(x1) = [1 0] x1 + [1]
                  [0 0]      [1]
       recip(x1) = [1 0] x1 + [0]
                   [0 0]      [0]
       sqr(x1) = [0 0] x1 + [0]
                 [0 0]      [1]
       0() = [0]
             [0]
       s() = [0]
             [0]
       dbl(x1) = [0 0] x1 + [1]
                 [0 0]      [1]
       add(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
                     [0 0]      [0 0]      [1]
       first(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
                       [0 0]      [1 0]      [1]
       nil() = [0]
               [0]
    
    The strictly oriented rules are moved into the weak component.
    
    We consider the following Problem:
    
      Strict Trs:
        {  sqr(0()) -> 0()
         , sqr(s()) -> s()
         , add(0(), X) -> X}
      Weak Trs:
        {  terms(N) -> cons(recip(sqr(N)))
         , dbl(0()) -> 0()
         , dbl(s()) -> s()
         , add(s(), Y) -> s()
         , first(0(), X) -> nil()
         , first(s(), cons(Y)) -> cons(Y)}
      StartTerms: basic terms
      Strategy: innermost
    
    Certificate: YES(?,O(n^1))
    
    Proof:
      The weightgap principle applies, where following rules are oriented strictly:
      
      TRS Component:
        {  sqr(0()) -> 0()
         , sqr(s()) -> s()}
      
      Interpretation of nonconstant growth:
      -------------------------------------
        The following argument positions are usable:
          Uargs(terms) = {}, Uargs(cons) = {1}, Uargs(recip) = {1},
          Uargs(sqr) = {}, Uargs(dbl) = {}, Uargs(add) = {},
          Uargs(first) = {}
        We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
        Interpretation Functions:
         terms(x1) = [0 0] x1 + [2]
                     [0 0]      [2]
         cons(x1) = [1 0] x1 + [0]
                    [0 0]      [1]
         recip(x1) = [1 0] x1 + [0]
                     [0 0]      [0]
         sqr(x1) = [0 0] x1 + [1]
                   [0 0]      [1]
         0() = [0]
               [0]
         s() = [0]
               [0]
         dbl(x1) = [0 0] x1 + [1]
                   [0 0]      [1]
         add(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
                       [0 0]      [0 0]      [1]
         first(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
                         [0 0]      [0 0]      [1]
         nil() = [0]
                 [0]
      
      The strictly oriented rules are moved into the weak component.
      
      We consider the following Problem:
      
        Strict Trs: {add(0(), X) -> X}
        Weak Trs:
          {  sqr(0()) -> 0()
           , sqr(s()) -> s()
           , terms(N) -> cons(recip(sqr(N)))
           , dbl(0()) -> 0()
           , dbl(s()) -> s()
           , add(s(), Y) -> s()
           , first(0(), X) -> nil()
           , first(s(), cons(Y)) -> cons(Y)}
        StartTerms: basic terms
        Strategy: innermost
      
      Certificate: YES(?,O(n^1))
      
      Proof:
        The weightgap principle applies, where following rules are oriented strictly:
        
        TRS Component: {add(0(), X) -> X}
        
        Interpretation of nonconstant growth:
        -------------------------------------
          The following argument positions are usable:
            Uargs(terms) = {}, Uargs(cons) = {1}, Uargs(recip) = {1},
            Uargs(sqr) = {}, Uargs(dbl) = {}, Uargs(add) = {},
            Uargs(first) = {}
          We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
          Interpretation Functions:
           terms(x1) = [1 0] x1 + [1]
                       [0 1]      [2]
           cons(x1) = [1 0] x1 + [0]
                      [0 0]      [1]
           recip(x1) = [1 0] x1 + [0]
                       [0 0]      [0]
           sqr(x1) = [0 0] x1 + [1]
                     [0 0]      [1]
           0() = [0]
                 [0]
           s() = [0]
                 [0]
           dbl(x1) = [0 0] x1 + [1]
                     [0 0]      [1]
           add(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
                         [0 0]      [0 1]      [1]
           first(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
                           [0 0]      [0 0]      [1]
           nil() = [0]
                   [0]
        
        The strictly oriented rules are moved into the weak component.
        
        We consider the following Problem:
        
          Weak Trs:
            {  add(0(), X) -> X
             , sqr(0()) -> 0()
             , sqr(s()) -> s()
             , terms(N) -> cons(recip(sqr(N)))
             , dbl(0()) -> 0()
             , dbl(s()) -> s()
             , add(s(), Y) -> s()
             , first(0(), X) -> nil()
             , first(s(), cons(Y)) -> cons(Y)}
          StartTerms: basic terms
          Strategy: innermost
        
        Certificate: YES(O(1),O(1))
        
        Proof:
          We consider the following Problem:
          
            Weak Trs:
              {  add(0(), X) -> X
               , sqr(0()) -> 0()
               , sqr(s()) -> s()
               , terms(N) -> cons(recip(sqr(N)))
               , dbl(0()) -> 0()
               , dbl(s()) -> s()
               , add(s(), Y) -> s()
               , first(0(), X) -> nil()
               , first(s(), cons(Y)) -> cons(Y)}
            StartTerms: basic terms
            Strategy: innermost
          
          Certificate: YES(O(1),O(1))
          
          Proof:
            Empty rules are trivially bounded

Hurray, we answered YES(?,O(n^1))