We consider the following Problem:

  Strict Trs:
    {  p(s(x)) -> x
     , fac(0()) -> s(0())
     , fac(s(x)) -> times(s(x), fac(p(s(x))))}
  StartTerms: basic terms
  Strategy: innermost

Certificate: YES(?,O(n^2))

Proof:
  We consider the following Problem:
  
    Strict Trs:
      {  p(s(x)) -> x
       , fac(0()) -> s(0())
       , fac(s(x)) -> times(s(x), fac(p(s(x))))}
    StartTerms: basic terms
    Strategy: innermost
  
  Certificate: YES(?,O(n^2))
  
  Proof:
    The weightgap principle applies, where following rules are oriented strictly:
    
    TRS Component: {fac(0()) -> s(0())}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      The following argument positions are usable:
        Uargs(p) = {}, Uargs(s) = {}, Uargs(fac) = {1}, Uargs(times) = {2}
      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
      Interpretation Functions:
       p(x1) = [1 0] x1 + [1]
               [1 0]      [1]
       s(x1) = [1 0] x1 + [0]
               [0 0]      [1]
       fac(x1) = [1 0] x1 + [1]
                 [0 0]      [1]
       0() = [0]
             [0]
       times(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
                       [0 0]      [0 0]      [1]
    
    The strictly oriented rules are moved into the weak component.
    
    We consider the following Problem:
    
      Strict Trs:
        {  p(s(x)) -> x
         , fac(s(x)) -> times(s(x), fac(p(s(x))))}
      Weak Trs: {fac(0()) -> s(0())}
      StartTerms: basic terms
      Strategy: innermost
    
    Certificate: YES(?,O(n^2))
    
    Proof:
      The weightgap principle applies, where following rules are oriented strictly:
      
      TRS Component: {p(s(x)) -> x}
      
      Interpretation of nonconstant growth:
      -------------------------------------
        The following argument positions are usable:
          Uargs(p) = {}, Uargs(s) = {}, Uargs(fac) = {1}, Uargs(times) = {2}
        We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
        Interpretation Functions:
         p(x1) = [1 0] x1 + [1]
                 [0 1]      [2]
         s(x1) = [1 0] x1 + [0]
                 [0 1]      [2]
         fac(x1) = [1 0] x1 + [0]
                   [0 0]      [2]
         0() = [0]
               [0]
         times(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                         [0 0]      [0 0]      [1]
      
      The strictly oriented rules are moved into the weak component.
      
      We consider the following Problem:
      
        Strict Trs: {fac(s(x)) -> times(s(x), fac(p(s(x))))}
        Weak Trs:
          {  p(s(x)) -> x
           , fac(0()) -> s(0())}
        StartTerms: basic terms
        Strategy: innermost
      
      Certificate: YES(?,O(n^2))
      
      Proof:
        We consider the following Problem:
        
          Strict Trs: {fac(s(x)) -> times(s(x), fac(p(s(x))))}
          Weak Trs:
            {  p(s(x)) -> x
             , fac(0()) -> s(0())}
          StartTerms: basic terms
          Strategy: innermost
        
        Certificate: YES(?,O(n^2))
        
        Proof:
          The following argument positions are usable:
            Uargs(p) = {}, Uargs(s) = {}, Uargs(fac) = {1}, Uargs(times) = {2}
          We have the following constructor-based EDA-non-satisfying and IDA(2)-non-satisfying matrix interpretation:
          Interpretation Functions:
           p(x1) = [1 0 0] x1 + [0]
                   [0 0 1]      [0]
                   [2 0 0]      [1]
           s(x1) = [1 0 2] x1 + [0]
                   [0 1 0]      [2]
                   [0 1 0]      [0]
           fac(x1) = [2 2 0] x1 + [0]
                     [0 1 0]      [2]
                     [0 0 0]      [2]
           0() = [0]
                 [2]
                 [0]
           times(x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [1]
                           [0 0 0]      [0 0 0]      [1]
                           [0 0 0]      [0 0 0]      [0]

Hurray, we answered YES(?,O(n^2))