We consider the following Problem:

  Strict Trs:
    {  filter(cons(X), 0(), M) -> cons(0())
     , filter(cons(X), s(N), M) -> cons(X)
     , sieve(cons(0())) -> cons(0())
     , sieve(cons(s(N))) -> cons(s(N))
     , nats(N) -> cons(N)
     , zprimes() -> sieve(nats(s(s(0()))))}
  StartTerms: basic terms
  Strategy: innermost

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

Proof:
  We consider the following Problem:
  
    Strict Trs:
      {  filter(cons(X), 0(), M) -> cons(0())
       , filter(cons(X), s(N), M) -> cons(X)
       , sieve(cons(0())) -> cons(0())
       , sieve(cons(s(N))) -> cons(s(N))
       , nats(N) -> cons(N)
       , zprimes() -> sieve(nats(s(s(0()))))}
    StartTerms: basic terms
    Strategy: innermost
  
  Certificate: YES(?,O(n^1))
  
  Proof:
    The weightgap principle applies, where following rules are oriented strictly:
    
    TRS Component: {zprimes() -> sieve(nats(s(s(0()))))}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      The following argument positions are usable:
        Uargs(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
        Uargs(sieve) = {1}, Uargs(nats) = {}
      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
      Interpretation Functions:
       filter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [1]
                            [0 0]      [0 0]      [0 0]      [1]
       cons(x1) = [0 0] x1 + [1]
                  [0 0]      [1]
       0() = [0]
             [0]
       s(x1) = [0 0] x1 + [0]
               [0 0]      [0]
       sieve(x1) = [1 0] x1 + [0]
                   [0 0]      [1]
       nats(x1) = [0 0] x1 + [0]
                  [0 0]      [0]
       zprimes() = [2]
                   [2]
    
    The strictly oriented rules are moved into the weak component.
    
    We consider the following Problem:
    
      Strict Trs:
        {  filter(cons(X), 0(), M) -> cons(0())
         , filter(cons(X), s(N), M) -> cons(X)
         , sieve(cons(0())) -> cons(0())
         , sieve(cons(s(N))) -> cons(s(N))
         , nats(N) -> cons(N)}
      Weak Trs: {zprimes() -> sieve(nats(s(s(0()))))}
      StartTerms: basic terms
      Strategy: innermost
    
    Certificate: YES(?,O(n^1))
    
    Proof:
      The weightgap principle applies, where following rules are oriented strictly:
      
      TRS Component: {nats(N) -> cons(N)}
      
      Interpretation of nonconstant growth:
      -------------------------------------
        The following argument positions are usable:
          Uargs(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
          Uargs(sieve) = {1}, Uargs(nats) = {}
        We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
        Interpretation Functions:
         filter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [1]
                              [0 0]      [0 0]      [0 0]      [1]
         cons(x1) = [0 0] x1 + [1]
                    [0 0]      [1]
         0() = [0]
               [0]
         s(x1) = [0 0] x1 + [0]
                 [0 0]      [0]
         sieve(x1) = [1 0] x1 + [0]
                     [0 0]      [1]
         nats(x1) = [0 0] x1 + [2]
                    [0 0]      [2]
         zprimes() = [2]
                     [2]
      
      The strictly oriented rules are moved into the weak component.
      
      We consider the following Problem:
      
        Strict Trs:
          {  filter(cons(X), 0(), M) -> cons(0())
           , filter(cons(X), s(N), M) -> cons(X)
           , sieve(cons(0())) -> cons(0())
           , sieve(cons(s(N))) -> cons(s(N))}
        Weak Trs:
          {  nats(N) -> cons(N)
           , zprimes() -> sieve(nats(s(s(0()))))}
        StartTerms: basic terms
        Strategy: innermost
      
      Certificate: YES(?,O(n^1))
      
      Proof:
        The weightgap principle applies, where following rules are oriented strictly:
        
        TRS Component: {filter(cons(X), s(N), M) -> cons(X)}
        
        Interpretation of nonconstant growth:
        -------------------------------------
          The following argument positions are usable:
            Uargs(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
            Uargs(sieve) = {1}, Uargs(nats) = {}
          We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
          Interpretation Functions:
           filter(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [0 0] x3 + [1]
                                [0 0]      [0 1]      [0 0]      [1]
           cons(x1) = [0 0] x1 + [1]
                      [0 0]      [1]
           0() = [0]
                 [0]
           s(x1) = [0 0] x1 + [1]
                   [0 0]      [0]
           sieve(x1) = [1 0] x1 + [0]
                       [0 1]      [0]
           nats(x1) = [0 0] x1 + [1]
                      [0 0]      [1]
           zprimes() = [2]
                       [2]
        
        The strictly oriented rules are moved into the weak component.
        
        We consider the following Problem:
        
          Strict Trs:
            {  filter(cons(X), 0(), M) -> cons(0())
             , sieve(cons(0())) -> cons(0())
             , sieve(cons(s(N))) -> cons(s(N))}
          Weak Trs:
            {  filter(cons(X), s(N), M) -> cons(X)
             , nats(N) -> cons(N)
             , zprimes() -> sieve(nats(s(s(0()))))}
          StartTerms: basic terms
          Strategy: innermost
        
        Certificate: YES(?,O(n^1))
        
        Proof:
          The weightgap principle applies, where following rules are oriented strictly:
          
          TRS Component:
            {  sieve(cons(0())) -> cons(0())
             , sieve(cons(s(N))) -> cons(s(N))}
          
          Interpretation of nonconstant growth:
          -------------------------------------
            The following argument positions are usable:
              Uargs(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
              Uargs(sieve) = {1}, Uargs(nats) = {}
            We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
            Interpretation Functions:
             filter(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [1]
                                  [0 0]      [1 0]      [1 0]      [1]
             cons(x1) = [0 0] x1 + [1]
                        [0 0]      [1]
             0() = [0]
                   [0]
             s(x1) = [0 0] x1 + [0]
                     [0 0]      [0]
             sieve(x1) = [1 0] x1 + [2]
                         [1 0]      [0]
             nats(x1) = [0 0] x1 + [1]
                        [0 0]      [2]
             zprimes() = [3]
                         [2]
          
          The strictly oriented rules are moved into the weak component.
          
          We consider the following Problem:
          
            Strict Trs: {filter(cons(X), 0(), M) -> cons(0())}
            Weak Trs:
              {  sieve(cons(0())) -> cons(0())
               , sieve(cons(s(N))) -> cons(s(N))
               , filter(cons(X), s(N), M) -> cons(X)
               , nats(N) -> cons(N)
               , zprimes() -> sieve(nats(s(s(0()))))}
            StartTerms: basic terms
            Strategy: innermost
          
          Certificate: YES(?,O(n^1))
          
          Proof:
            The weightgap principle applies, where following rules are oriented strictly:
            
            TRS Component: {filter(cons(X), 0(), M) -> cons(0())}
            
            Interpretation of nonconstant growth:
            -------------------------------------
              The following argument positions are usable:
                Uargs(filter) = {}, Uargs(cons) = {}, Uargs(s) = {},
                Uargs(sieve) = {1}, Uargs(nats) = {}
              We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
              Interpretation Functions:
               filter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [1]
                                    [0 0]      [0 0]      [0 0]      [1]
               cons(x1) = [0 0] x1 + [0]
                          [0 0]      [1]
               0() = [0]
                     [0]
               s(x1) = [0 0] x1 + [0]
                       [0 0]      [0]
               sieve(x1) = [1 0] x1 + [1]
                           [0 0]      [1]
               nats(x1) = [0 0] x1 + [0]
                          [0 0]      [2]
               zprimes() = [2]
                           [2]
            
            The strictly oriented rules are moved into the weak component.
            
            We consider the following Problem:
            
              Weak Trs:
                {  filter(cons(X), 0(), M) -> cons(0())
                 , sieve(cons(0())) -> cons(0())
                 , sieve(cons(s(N))) -> cons(s(N))
                 , filter(cons(X), s(N), M) -> cons(X)
                 , nats(N) -> cons(N)
                 , zprimes() -> sieve(nats(s(s(0()))))}
              StartTerms: basic terms
              Strategy: innermost
            
            Certificate: YES(O(1),O(1))
            
            Proof:
              We consider the following Problem:
              
                Weak Trs:
                  {  filter(cons(X), 0(), M) -> cons(0())
                   , sieve(cons(0())) -> cons(0())
                   , sieve(cons(s(N))) -> cons(s(N))
                   , filter(cons(X), s(N), M) -> cons(X)
                   , nats(N) -> cons(N)
                   , zprimes() -> sieve(nats(s(s(0()))))}
                StartTerms: basic terms
                Strategy: innermost
              
              Certificate: YES(O(1),O(1))
              
              Proof:
                Empty rules are trivially bounded

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