We consider the following Problem:

  Strict Trs:
    {  first(0(), X) -> nil()
     , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
     , from(X) -> cons(X, n__from(n__s(X)))
     , first(X1, X2) -> n__first(X1, X2)
     , from(X) -> n__from(X)
     , s(X) -> n__s(X)
     , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
     , activate(n__from(X)) -> from(activate(X))
     , activate(n__s(X)) -> s(activate(X))
     , activate(X) -> X}
  StartTerms: basic terms
  Strategy: innermost

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

Proof:
  Arguments of following rules are not normal-forms:
  {first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))}
  
  All above mentioned rules can be savely removed.
  
  We consider the following Problem:
  
    Strict Trs:
      {  first(0(), X) -> nil()
       , from(X) -> cons(X, n__from(n__s(X)))
       , first(X1, X2) -> n__first(X1, X2)
       , from(X) -> n__from(X)
       , s(X) -> n__s(X)
       , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
       , activate(n__from(X)) -> from(activate(X))
       , activate(n__s(X)) -> s(activate(X))
       , activate(X) -> X}
    StartTerms: basic terms
    Strategy: innermost
  
  Certificate: YES(?,O(n^1))
  
  Proof:
    The weightgap principle applies, where following rules are oriented strictly:
    
    TRS Component: {first(0(), X) -> nil()}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      The following argument positions are usable:
        Uargs(first) = {1, 2}, Uargs(s) = {1}, Uargs(cons) = {},
        Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(from) = {1},
        Uargs(n__from) = {}, Uargs(n__s) = {}
      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
      Interpretation Functions:
       first(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                       [0 0]      [0 0]      [1]
       0() = [0]
             [0]
       nil() = [0]
               [0]
       s(x1) = [1 0] x1 + [0]
               [0 0]      [1]
       cons(x1, x2) = [1 1] x1 + [0 0] x2 + [1]
                      [0 0]      [0 0]      [1]
       n__first(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                          [0 1]      [0 1]      [0]
       activate(x1) = [1 1] x1 + [1]
                      [0 0]      [1]
       from(x1) = [1 1] x1 + [1]
                  [0 0]      [1]
       n__from(x1) = [0 0] x1 + [0]
                     [1 1]      [0]
       n__s(x1) = [0 0] x1 + [0]
                  [1 1]      [0]
    
    The strictly oriented rules are moved into the weak component.
    
    We consider the following Problem:
    
      Strict Trs:
        {  from(X) -> cons(X, n__from(n__s(X)))
         , first(X1, X2) -> n__first(X1, X2)
         , from(X) -> n__from(X)
         , s(X) -> n__s(X)
         , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
         , activate(n__from(X)) -> from(activate(X))
         , activate(n__s(X)) -> s(activate(X))
         , activate(X) -> X}
      Weak Trs: {first(0(), X) -> nil()}
      StartTerms: basic terms
      Strategy: innermost
    
    Certificate: YES(?,O(n^1))
    
    Proof:
      The weightgap principle applies, where following rules are oriented strictly:
      
      TRS Component: {activate(n__s(X)) -> s(activate(X))}
      
      Interpretation of nonconstant growth:
      -------------------------------------
        The following argument positions are usable:
          Uargs(first) = {1, 2}, Uargs(s) = {1}, Uargs(cons) = {},
          Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(from) = {1},
          Uargs(n__from) = {}, Uargs(n__s) = {}
        We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
        Interpretation Functions:
         first(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                         [0 0]      [0 0]      [1]
         0() = [0]
               [0]
         nil() = [0]
                 [0]
         s(x1) = [1 0] x1 + [0]
                 [0 0]      [1]
         cons(x1, x2) = [1 1] x1 + [0 0] x2 + [1]
                        [0 0]      [0 0]      [1]
         n__first(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                            [0 1]      [0 1]      [0]
         activate(x1) = [1 1] x1 + [1]
                        [0 0]      [1]
         from(x1) = [1 1] x1 + [1]
                    [0 0]      [1]
         n__from(x1) = [0 0] x1 + [0]
                       [1 1]      [0]
         n__s(x1) = [0 0] x1 + [0]
                    [1 1]      [2]
      
      The strictly oriented rules are moved into the weak component.
      
      We consider the following Problem:
      
        Strict Trs:
          {  from(X) -> cons(X, n__from(n__s(X)))
           , first(X1, X2) -> n__first(X1, X2)
           , from(X) -> n__from(X)
           , s(X) -> n__s(X)
           , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
           , activate(n__from(X)) -> from(activate(X))
           , activate(X) -> X}
        Weak Trs:
          {  activate(n__s(X)) -> s(activate(X))
           , first(0(), X) -> nil()}
        StartTerms: basic terms
        Strategy: innermost
      
      Certificate: YES(?,O(n^1))
      
      Proof:
        The weightgap principle applies, where following rules are oriented strictly:
        
        TRS Component: {from(X) -> cons(X, n__from(n__s(X)))}
        
        Interpretation of nonconstant growth:
        -------------------------------------
          The following argument positions are usable:
            Uargs(first) = {1, 2}, Uargs(s) = {1}, Uargs(cons) = {},
            Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(from) = {1},
            Uargs(n__from) = {}, Uargs(n__s) = {}
          We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
          Interpretation Functions:
           first(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                           [0 0]      [0 0]      [1]
           0() = [0]
                 [0]
           nil() = [0]
                   [0]
           s(x1) = [1 0] x1 + [0]
                   [0 0]      [1]
           cons(x1, x2) = [1 1] x1 + [0 0] x2 + [1]
                          [0 0]      [0 0]      [1]
           n__first(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 1]      [0 1]      [0]
           activate(x1) = [1 1] x1 + [1]
                          [0 0]      [1]
           from(x1) = [1 1] x1 + [3]
                      [0 0]      [1]
           n__from(x1) = [0 0] x1 + [0]
                         [1 1]      [0]
           n__s(x1) = [0 0] x1 + [0]
                      [1 1]      [0]
        
        The strictly oriented rules are moved into the weak component.
        
        We consider the following Problem:
        
          Strict Trs:
            {  first(X1, X2) -> n__first(X1, X2)
             , from(X) -> n__from(X)
             , s(X) -> n__s(X)
             , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
             , activate(n__from(X)) -> from(activate(X))
             , activate(X) -> X}
          Weak Trs:
            {  from(X) -> cons(X, n__from(n__s(X)))
             , activate(n__s(X)) -> s(activate(X))
             , first(0(), X) -> nil()}
          StartTerms: basic terms
          Strategy: innermost
        
        Certificate: YES(?,O(n^1))
        
        Proof:
          The weightgap principle applies, where following rules are oriented strictly:
          
          TRS Component: {first(X1, X2) -> n__first(X1, X2)}
          
          Interpretation of nonconstant growth:
          -------------------------------------
            The following argument positions are usable:
              Uargs(first) = {1, 2}, Uargs(s) = {1}, Uargs(cons) = {},
              Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(from) = {1},
              Uargs(n__from) = {}, Uargs(n__s) = {}
            We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
            Interpretation Functions:
             first(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                             [0 0]      [0 0]      [1]
             0() = [0]
                   [0]
             nil() = [0]
                     [0]
             s(x1) = [1 0] x1 + [0]
                     [0 0]      [1]
             cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [1]
             n__first(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                                [0 0]      [0 0]      [0]
             activate(x1) = [1 0] x1 + [1]
                            [0 0]      [1]
             from(x1) = [1 0] x1 + [0]
                        [0 0]      [1]
             n__from(x1) = [1 0] x1 + [0]
                           [0 0]      [0]
             n__s(x1) = [1 0] x1 + [0]
                        [0 0]      [0]
          
          The strictly oriented rules are moved into the weak component.
          
          We consider the following Problem:
          
            Strict Trs:
              {  from(X) -> n__from(X)
               , s(X) -> n__s(X)
               , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
               , activate(n__from(X)) -> from(activate(X))
               , activate(X) -> X}
            Weak Trs:
              {  first(X1, X2) -> n__first(X1, X2)
               , from(X) -> cons(X, n__from(n__s(X)))
               , activate(n__s(X)) -> s(activate(X))
               , first(0(), X) -> nil()}
            StartTerms: basic terms
            Strategy: innermost
          
          Certificate: YES(?,O(n^1))
          
          Proof:
            The weightgap principle applies, where following rules are oriented strictly:
            
            TRS Component: {from(X) -> n__from(X)}
            
            Interpretation of nonconstant growth:
            -------------------------------------
              The following argument positions are usable:
                Uargs(first) = {1, 2}, Uargs(s) = {1}, Uargs(cons) = {},
                Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(from) = {1},
                Uargs(n__from) = {}, Uargs(n__s) = {}
              We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
              Interpretation Functions:
               first(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                               [0 0]      [0 0]      [1]
               0() = [0]
                     [0]
               nil() = [0]
                       [0]
               s(x1) = [1 0] x1 + [0]
                       [0 1]      [0]
               cons(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
                              [1 1]      [0 0]      [1]
               n__first(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
               activate(x1) = [1 0] x1 + [1]
                              [1 0]      [3]
               from(x1) = [1 0] x1 + [2]
                          [1 1]      [1]
               n__from(x1) = [1 0] x1 + [1]
                             [0 0]      [0]
               n__s(x1) = [1 0] x1 + [0]
                          [0 0]      [0]
            
            The strictly oriented rules are moved into the weak component.
            
            We consider the following Problem:
            
              Strict Trs:
                {  s(X) -> n__s(X)
                 , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
                 , activate(n__from(X)) -> from(activate(X))
                 , activate(X) -> X}
              Weak Trs:
                {  from(X) -> n__from(X)
                 , first(X1, X2) -> n__first(X1, X2)
                 , from(X) -> cons(X, n__from(n__s(X)))
                 , activate(n__s(X)) -> s(activate(X))
                 , first(0(), X) -> nil()}
              StartTerms: basic terms
              Strategy: innermost
            
            Certificate: YES(?,O(n^1))
            
            Proof:
              The weightgap principle applies, where following rules are oriented strictly:
              
              TRS Component: {activate(X) -> X}
              
              Interpretation of nonconstant growth:
              -------------------------------------
                The following argument positions are usable:
                  Uargs(first) = {1, 2}, Uargs(s) = {1}, Uargs(cons) = {},
                  Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(from) = {1},
                  Uargs(n__from) = {}, Uargs(n__s) = {}
                We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
                Interpretation Functions:
                 first(x1, x2) = [1 0] x1 + [1 1] x2 + [0]
                                 [0 0]      [0 0]      [1]
                 0() = [0]
                       [0]
                 nil() = [0]
                         [0]
                 s(x1) = [1 0] x1 + [0]
                         [0 1]      [0]
                 cons(x1, x2) = [0 1] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [1]
                 n__first(x1, x2) = [1 0] x1 + [1 1] x2 + [0]
                                    [0 0]      [0 0]      [1]
                 activate(x1) = [1 0] x1 + [1]
                                [0 1]      [3]
                 from(x1) = [1 3] x1 + [0]
                            [0 0]      [1]
                 n__from(x1) = [1 3] x1 + [0]
                               [0 0]      [1]
                 n__s(x1) = [1 0] x1 + [0]
                            [0 1]      [0]
              
              The strictly oriented rules are moved into the weak component.
              
              We consider the following Problem:
              
                Strict Trs:
                  {  s(X) -> n__s(X)
                   , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
                   , activate(n__from(X)) -> from(activate(X))}
                Weak Trs:
                  {  activate(X) -> X
                   , from(X) -> n__from(X)
                   , first(X1, X2) -> n__first(X1, X2)
                   , from(X) -> cons(X, n__from(n__s(X)))
                   , activate(n__s(X)) -> s(activate(X))
                   , first(0(), X) -> nil()}
                StartTerms: basic terms
                Strategy: innermost
              
              Certificate: YES(?,O(n^1))
              
              Proof:
                We consider the following Problem:
                
                  Strict Trs:
                    {  s(X) -> n__s(X)
                     , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
                     , activate(n__from(X)) -> from(activate(X))}
                  Weak Trs:
                    {  activate(X) -> X
                     , from(X) -> n__from(X)
                     , first(X1, X2) -> n__first(X1, X2)
                     , from(X) -> cons(X, n__from(n__s(X)))
                     , activate(n__s(X)) -> s(activate(X))
                     , first(0(), X) -> nil()}
                  StartTerms: basic terms
                  Strategy: innermost
                
                Certificate: YES(?,O(n^1))
                
                Proof:
                  The following argument positions are usable:
                    Uargs(first) = {1, 2}, Uargs(s) = {1}, Uargs(cons) = {},
                    Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(from) = {1},
                    Uargs(n__from) = {}, Uargs(n__s) = {}
                  We have the following constructor-based EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
                  Interpretation Functions:
                   first(x1, x2) = [1 0] x1 + [1 0] x2 + [3]
                                   [0 1]      [0 1]      [2]
                   0() = [0]
                         [0]
                   nil() = [0]
                           [0]
                   s(x1) = [1 0] x1 + [1]
                           [0 1]      [1]
                   cons(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
                                  [0 0]      [0 0]      [1]
                   n__first(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
                                      [0 1]      [0 1]      [2]
                   activate(x1) = [1 1] x1 + [0]
                                  [0 1]      [0]
                   from(x1) = [1 0] x1 + [3]
                              [0 1]      [2]
                   n__from(x1) = [1 0] x1 + [2]
                                 [0 1]      [2]
                   n__s(x1) = [1 0] x1 + [0]
                              [0 1]      [1]

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