We consider the following Problem:

  Strict Trs:
    {  first(0(), X) -> nil()
     , first(s(X), cons(Y)) -> cons(Y)
     , from(X) -> cons(X)}
  StartTerms: basic terms
  Strategy: innermost

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

Proof:
  We consider the following Problem:
  
    Strict Trs:
      {  first(0(), X) -> nil()
       , first(s(X), cons(Y)) -> cons(Y)
       , from(X) -> cons(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()
       , first(s(X), cons(Y)) -> cons(Y)}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      The following argument positions are usable:
        Uargs(first) = {}, Uargs(s) = {}, Uargs(cons) = {},
        Uargs(from) = {}
      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
      Interpretation Functions:
       first(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
                       [0 0]      [0 0]      [1]
       0() = [0]
             [0]
       nil() = [0]
               [0]
       s(x1) = [0 0] x1 + [0]
               [0 0]      [0]
       cons(x1) = [0 0] x1 + [0]
                  [0 0]      [0]
       from(x1) = [0 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) -> cons(X)}
      Weak Trs:
        {  first(0(), X) -> nil()
         , first(s(X), 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: {from(X) -> cons(X)}
      
      Interpretation of nonconstant growth:
      -------------------------------------
        The following argument positions are usable:
          Uargs(first) = {}, Uargs(s) = {}, Uargs(cons) = {},
          Uargs(from) = {}
        We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
        Interpretation Functions:
         first(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
                         [0 0]      [0 0]      [1]
         0() = [0]
               [0]
         nil() = [0]
                 [0]
         s(x1) = [0 0] x1 + [0]
                 [0 0]      [0]
         cons(x1) = [0 0] x1 + [0]
                    [0 0]      [0]
         from(x1) = [0 0] x1 + [2]
                    [0 0]      [0]
      
      The strictly oriented rules are moved into the weak component.
      
      We consider the following Problem:
      
        Weak Trs:
          {  from(X) -> cons(X)
           , first(0(), X) -> nil()
           , first(s(X), cons(Y)) -> cons(Y)}
        StartTerms: basic terms
        Strategy: innermost
      
      Certificate: YES(O(1),O(1))
      
      Proof:
        We consider the following Problem:
        
          Weak Trs:
            {  from(X) -> cons(X)
             , first(0(), X) -> nil()
             , first(s(X), 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))