We consider the following Problem:

  Strict Trs:
    {  and(x, false()) -> false()
     , and(x, not(false())) -> x
     , not(not(x)) -> x
     , implies(false(), y) -> not(false())
     , implies(x, false()) -> not(x)
     , implies(not(x), not(y)) -> implies(y, and(x, y))}
  StartTerms: basic terms
  Strategy: innermost

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

Proof:
  We consider the following Problem:
  
    Strict Trs:
      {  and(x, false()) -> false()
       , and(x, not(false())) -> x
       , not(not(x)) -> x
       , implies(false(), y) -> not(false())
       , implies(x, false()) -> not(x)
       , implies(not(x), not(y)) -> implies(y, and(x, y))}
    StartTerms: basic terms
    Strategy: innermost
  
  Certificate: YES(?,O(n^1))
  
  Proof:
    The weightgap principle applies, where following rules are oriented strictly:
    
    TRS Component:
      {  and(x, false()) -> false()
       , and(x, not(false())) -> x
       , implies(false(), y) -> not(false())}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      The following argument positions are usable:
        Uargs(and) = {}, Uargs(not) = {}, Uargs(implies) = {2}
      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
      Interpretation Functions:
       and(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
                     [0 1]      [0 0]      [1]
       false() = [0]
                 [0]
       not(x1) = [1 0] x1 + [0]
                 [0 1]      [0]
       implies(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                         [0 0]      [1 0]      [1]
    
    The strictly oriented rules are moved into the weak component.
    
    We consider the following Problem:
    
      Strict Trs:
        {  not(not(x)) -> x
         , implies(x, false()) -> not(x)
         , implies(not(x), not(y)) -> implies(y, and(x, y))}
      Weak Trs:
        {  and(x, false()) -> false()
         , and(x, not(false())) -> x
         , implies(false(), y) -> not(false())}
      StartTerms: basic terms
      Strategy: innermost
    
    Certificate: YES(?,O(n^1))
    
    Proof:
      The weightgap principle applies, where following rules are oriented strictly:
      
      TRS Component: {implies(x, false()) -> not(x)}
      
      Interpretation of nonconstant growth:
      -------------------------------------
        The following argument positions are usable:
          Uargs(and) = {}, Uargs(not) = {}, Uargs(implies) = {2}
        We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
        Interpretation Functions:
         and(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
                       [0 1]      [0 0]      [1]
         false() = [0]
                   [0]
         not(x1) = [1 0] x1 + [0]
                   [0 1]      [0]
         implies(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                           [0 1]      [1 0]      [1]
      
      The strictly oriented rules are moved into the weak component.
      
      We consider the following Problem:
      
        Strict Trs:
          {  not(not(x)) -> x
           , implies(not(x), not(y)) -> implies(y, and(x, y))}
        Weak Trs:
          {  implies(x, false()) -> not(x)
           , and(x, false()) -> false()
           , and(x, not(false())) -> x
           , implies(false(), y) -> not(false())}
        StartTerms: basic terms
        Strategy: innermost
      
      Certificate: YES(?,O(n^1))
      
      Proof:
        The weightgap principle applies, where following rules are oriented strictly:
        
        TRS Component: {not(not(x)) -> x}
        
        Interpretation of nonconstant growth:
        -------------------------------------
          The following argument positions are usable:
            Uargs(and) = {}, Uargs(not) = {}, Uargs(implies) = {2}
          We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
          Interpretation Functions:
           and(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
                         [0 1]      [0 0]      [1]
           false() = [0]
                     [0]
           not(x1) = [1 0] x1 + [2]
                     [0 1]      [0]
           implies(x1, x2) = [1 0] x1 + [1 0] x2 + [3]
                             [0 1]      [0 0]      [1]
        
        The strictly oriented rules are moved into the weak component.
        
        We consider the following Problem:
        
          Strict Trs: {implies(not(x), not(y)) -> implies(y, and(x, y))}
          Weak Trs:
            {  not(not(x)) -> x
             , implies(x, false()) -> not(x)
             , and(x, false()) -> false()
             , and(x, not(false())) -> x
             , implies(false(), y) -> not(false())}
          StartTerms: basic terms
          Strategy: innermost
        
        Certificate: YES(?,O(n^1))
        
        Proof:
          The weightgap principle applies, where following rules are oriented strictly:
          
          TRS Component: {implies(not(x), not(y)) -> implies(y, and(x, y))}
          
          Interpretation of nonconstant growth:
          -------------------------------------
            The following argument positions are usable:
              Uargs(and) = {}, Uargs(not) = {}, Uargs(implies) = {2}
            We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
            Interpretation Functions:
             and(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
                           [0 1]      [0 0]      [0]
             false() = [0]
                       [0]
             not(x1) = [1 0] x1 + [2]
                       [0 1]      [0]
             implies(x1, x2) = [1 0] x1 + [1 0] x2 + [3]
                               [0 1]      [0 1]      [1]
          
          The strictly oriented rules are moved into the weak component.
          
          We consider the following Problem:
          
            Weak Trs:
              {  implies(not(x), not(y)) -> implies(y, and(x, y))
               , not(not(x)) -> x
               , implies(x, false()) -> not(x)
               , and(x, false()) -> false()
               , and(x, not(false())) -> x
               , implies(false(), y) -> not(false())}
            StartTerms: basic terms
            Strategy: innermost
          
          Certificate: YES(O(1),O(1))
          
          Proof:
            We consider the following Problem:
            
              Weak Trs:
                {  implies(not(x), not(y)) -> implies(y, and(x, y))
                 , not(not(x)) -> x
                 , implies(x, false()) -> not(x)
                 , and(x, false()) -> false()
                 , and(x, not(false())) -> x
                 , implies(false(), y) -> not(false())}
              StartTerms: basic terms
              Strategy: innermost
            
            Certificate: YES(O(1),O(1))
            
            Proof:
              Empty rules are trivially bounded

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