We consider the following Problem:

  Strict Trs:
    {  minus(x, 0()) -> x
     , minus(s(x), s(y)) -> minus(x, y)
     , quot(0(), s(y)) -> 0()
     , quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))}
  StartTerms: basic terms
  Strategy: innermost

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

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

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