We consider the following Problem:

  Strict Trs:
    {  U11(tt(), N) -> activate(N)
     , U21(tt(), M, N) -> s(plus(activate(N), activate(M)))
     , and(tt(), X) -> activate(X)
     , isNat(n__0()) -> tt()
     , isNat(n__plus(V1, V2)) ->
       and(isNat(activate(V1)), n__isNat(activate(V2)))
     , isNat(n__s(V1)) -> isNat(activate(V1))
     , plus(N, 0()) -> U11(isNat(N), N)
     , plus(N, s(M)) -> U21(and(isNat(M), n__isNat(N)), M, N)
     , 0() -> n__0()
     , plus(X1, X2) -> n__plus(X1, X2)
     , isNat(X) -> n__isNat(X)
     , s(X) -> n__s(X)
     , activate(n__0()) -> 0()
     , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
     , activate(n__isNat(X)) -> isNat(X)
     , activate(n__s(X)) -> s(activate(X))
     , activate(X) -> X}
  StartTerms: basic terms
  Strategy: innermost

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

Proof:
  Arguments of following rules are not normal-forms:
  {  plus(N, s(M)) -> U21(and(isNat(M), n__isNat(N)), M, N)
   , plus(N, 0()) -> U11(isNat(N), N)}
  
  All above mentioned rules can be savely removed.
  
  We consider the following Problem:
  
    Strict Trs:
      {  U11(tt(), N) -> activate(N)
       , U21(tt(), M, N) -> s(plus(activate(N), activate(M)))
       , and(tt(), X) -> activate(X)
       , isNat(n__0()) -> tt()
       , isNat(n__plus(V1, V2)) ->
         and(isNat(activate(V1)), n__isNat(activate(V2)))
       , isNat(n__s(V1)) -> isNat(activate(V1))
       , 0() -> n__0()
       , plus(X1, X2) -> n__plus(X1, X2)
       , isNat(X) -> n__isNat(X)
       , s(X) -> n__s(X)
       , activate(n__0()) -> 0()
       , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
       , activate(n__isNat(X)) -> isNat(X)
       , activate(n__s(X)) -> s(activate(X))
       , activate(X) -> X}
    StartTerms: basic terms
    Strategy: innermost
  
  Certificate: YES(?,O(n^2))
  
  Proof:
    The weightgap principle applies, where following rules are oriented strictly:
    
    TRS Component:
      {  isNat(n__0()) -> tt()
       , 0() -> n__0()
       , plus(X1, X2) -> n__plus(X1, X2)
       , isNat(X) -> n__isNat(X)
       , s(X) -> n__s(X)
       , activate(n__0()) -> 0()}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      The following argument positions are usable:
        Uargs(U11) = {}, Uargs(activate) = {}, Uargs(U21) = {},
        Uargs(s) = {1}, Uargs(plus) = {1, 2}, Uargs(and) = {1, 2},
        Uargs(isNat) = {1}, Uargs(n__plus) = {}, Uargs(n__isNat) = {1},
        Uargs(n__s) = {}
      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
      Interpretation Functions:
       U11(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                     [1 0]      [0 0]      [1]
       tt() = [0]
              [0]
       activate(x1) = [1 0] x1 + [3]
                      [0 0]      [1]
       U21(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [1]
                         [0 0]      [0 0]      [0 0]      [1]
       s(x1) = [1 0] x1 + [1]
               [0 0]      [1]
       plus(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
                      [0 0]      [0 0]      [1]
       and(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                     [0 0]      [0 0]      [1]
       isNat(x1) = [1 0] x1 + [1]
                   [1 0]      [1]
       n__0() = [0]
                [0]
       n__plus(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                         [0 0]      [0 0]      [0]
       n__isNat(x1) = [1 0] x1 + [0]
                      [0 0]      [0]
       n__s(x1) = [1 0] x1 + [0]
                  [0 0]      [0]
       0() = [2]
             [0]
    
    The strictly oriented rules are moved into the weak component.
    
    We consider the following Problem:
    
      Strict Trs:
        {  U11(tt(), N) -> activate(N)
         , U21(tt(), M, N) -> s(plus(activate(N), activate(M)))
         , and(tt(), X) -> activate(X)
         , isNat(n__plus(V1, V2)) ->
           and(isNat(activate(V1)), n__isNat(activate(V2)))
         , isNat(n__s(V1)) -> isNat(activate(V1))
         , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
         , activate(n__isNat(X)) -> isNat(X)
         , activate(n__s(X)) -> s(activate(X))
         , activate(X) -> X}
      Weak Trs:
        {  isNat(n__0()) -> tt()
         , 0() -> n__0()
         , plus(X1, X2) -> n__plus(X1, X2)
         , isNat(X) -> n__isNat(X)
         , s(X) -> n__s(X)
         , activate(n__0()) -> 0()}
      StartTerms: basic terms
      Strategy: innermost
    
    Certificate: YES(?,O(n^2))
    
    Proof:
      The weightgap principle applies, where following rules are oriented strictly:
      
      TRS Component:
        {  U11(tt(), N) -> activate(N)
         , and(tt(), X) -> activate(X)}
      
      Interpretation of nonconstant growth:
      -------------------------------------
        The following argument positions are usable:
          Uargs(U11) = {}, Uargs(activate) = {}, Uargs(U21) = {},
          Uargs(s) = {1}, Uargs(plus) = {1, 2}, Uargs(and) = {1, 2},
          Uargs(isNat) = {1}, Uargs(n__plus) = {}, Uargs(n__isNat) = {1},
          Uargs(n__s) = {}
        We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
        Interpretation Functions:
         U11(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
                       [0 0]      [0 0]      [1]
         tt() = [0]
                [0]
         activate(x1) = [1 0] x1 + [0]
                        [0 0]      [1]
         U21(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [1]
                           [0 0]      [0 0]      [0 0]      [1]
         s(x1) = [1 0] x1 + [1]
                 [0 0]      [1]
         plus(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                        [0 0]      [0 0]      [1]
         and(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                       [0 0]      [0 0]      [1]
         isNat(x1) = [1 0] x1 + [1]
                     [0 0]      [1]
         n__0() = [0]
                  [0]
         n__plus(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                           [0 0]      [0 0]      [0]
         n__isNat(x1) = [1 0] x1 + [0]
                        [0 0]      [0]
         n__s(x1) = [1 0] x1 + [0]
                    [0 0]      [0]
         0() = [0]
               [0]
      
      The strictly oriented rules are moved into the weak component.
      
      We consider the following Problem:
      
        Strict Trs:
          {  U21(tt(), M, N) -> s(plus(activate(N), activate(M)))
           , isNat(n__plus(V1, V2)) ->
             and(isNat(activate(V1)), n__isNat(activate(V2)))
           , isNat(n__s(V1)) -> isNat(activate(V1))
           , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
           , activate(n__isNat(X)) -> isNat(X)
           , activate(n__s(X)) -> s(activate(X))
           , activate(X) -> X}
        Weak Trs:
          {  U11(tt(), N) -> activate(N)
           , and(tt(), X) -> activate(X)
           , isNat(n__0()) -> tt()
           , 0() -> n__0()
           , plus(X1, X2) -> n__plus(X1, X2)
           , isNat(X) -> n__isNat(X)
           , s(X) -> n__s(X)
           , activate(n__0()) -> 0()}
        StartTerms: basic terms
        Strategy: innermost
      
      Certificate: YES(?,O(n^2))
      
      Proof:
        The weightgap principle applies, where following rules are oriented strictly:
        
        TRS Component: {isNat(n__s(V1)) -> isNat(activate(V1))}
        
        Interpretation of nonconstant growth:
        -------------------------------------
          The following argument positions are usable:
            Uargs(U11) = {}, Uargs(activate) = {}, Uargs(U21) = {},
            Uargs(s) = {1}, Uargs(plus) = {1, 2}, Uargs(and) = {1, 2},
            Uargs(isNat) = {1}, Uargs(n__plus) = {}, Uargs(n__isNat) = {1},
            Uargs(n__s) = {}
          We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
          Interpretation Functions:
           U11(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                         [0 1]      [0 0]      [1]
           tt() = [0]
                  [0]
           activate(x1) = [1 0] x1 + [0]
                          [0 0]      [0]
           U21(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [1]
                             [0 0]      [0 0]      [0 0]      [1]
           s(x1) = [1 0] x1 + [1]
                   [0 0]      [1]
           plus(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                          [0 0]      [0 0]      [1]
           and(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                         [0 0]      [0 1]      [0]
           isNat(x1) = [1 0] x1 + [1]
                       [0 0]      [1]
           n__0() = [0]
                    [0]
           n__plus(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                             [0 0]      [0 0]      [0]
           n__isNat(x1) = [1 0] x1 + [0]
                          [0 0]      [1]
           n__s(x1) = [1 0] x1 + [1]
                      [0 0]      [0]
           0() = [0]
                 [0]
        
        The strictly oriented rules are moved into the weak component.
        
        We consider the following Problem:
        
          Strict Trs:
            {  U21(tt(), M, N) -> s(plus(activate(N), activate(M)))
             , isNat(n__plus(V1, V2)) ->
               and(isNat(activate(V1)), n__isNat(activate(V2)))
             , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
             , activate(n__isNat(X)) -> isNat(X)
             , activate(n__s(X)) -> s(activate(X))
             , activate(X) -> X}
          Weak Trs:
            {  isNat(n__s(V1)) -> isNat(activate(V1))
             , U11(tt(), N) -> activate(N)
             , and(tt(), X) -> activate(X)
             , isNat(n__0()) -> tt()
             , 0() -> n__0()
             , plus(X1, X2) -> n__plus(X1, X2)
             , isNat(X) -> n__isNat(X)
             , s(X) -> n__s(X)
             , activate(n__0()) -> 0()}
          StartTerms: basic terms
          Strategy: innermost
        
        Certificate: YES(?,O(n^2))
        
        Proof:
          The weightgap principle applies, where following rules are oriented strictly:
          
          TRS Component:
            {U21(tt(), M, N) -> s(plus(activate(N), activate(M)))}
          
          Interpretation of nonconstant growth:
          -------------------------------------
            The following argument positions are usable:
              Uargs(U11) = {}, Uargs(activate) = {}, Uargs(U21) = {},
              Uargs(s) = {1}, Uargs(plus) = {1, 2}, Uargs(and) = {1, 2},
              Uargs(isNat) = {1}, Uargs(n__plus) = {}, Uargs(n__isNat) = {1},
              Uargs(n__s) = {}
            We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
            Interpretation Functions:
             U11(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
                           [0 0]      [0 0]      [1]
             tt() = [0]
                    [1]
             activate(x1) = [1 0] x1 + [0]
                            [0 0]      [1]
             U21(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [1]
                               [0 1]      [0 0]      [0 0]      [0]
             s(x1) = [1 0] x1 + [0]
                     [0 0]      [1]
             plus(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                            [0 0]      [0 0]      [1]
             and(x1, x2) = [1 3] x1 + [1 1] x2 + [1]
                           [0 0]      [0 0]      [1]
             isNat(x1) = [1 0] x1 + [0]
                         [0 0]      [3]
             n__0() = [0]
                      [0]
             n__plus(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                               [0 0]      [0 0]      [0]
             n__isNat(x1) = [1 0] x1 + [0]
                            [0 0]      [3]
             n__s(x1) = [1 0] x1 + [0]
                        [0 0]      [0]
             0() = [0]
                   [0]
          
          The strictly oriented rules are moved into the weak component.
          
          We consider the following Problem:
          
            Strict Trs:
              {  isNat(n__plus(V1, V2)) ->
                 and(isNat(activate(V1)), n__isNat(activate(V2)))
               , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
               , activate(n__isNat(X)) -> isNat(X)
               , activate(n__s(X)) -> s(activate(X))
               , activate(X) -> X}
            Weak Trs:
              {  U21(tt(), M, N) -> s(plus(activate(N), activate(M)))
               , isNat(n__s(V1)) -> isNat(activate(V1))
               , U11(tt(), N) -> activate(N)
               , and(tt(), X) -> activate(X)
               , isNat(n__0()) -> tt()
               , 0() -> n__0()
               , plus(X1, X2) -> n__plus(X1, X2)
               , isNat(X) -> n__isNat(X)
               , s(X) -> n__s(X)
               , activate(n__0()) -> 0()}
            StartTerms: basic terms
            Strategy: innermost
          
          Certificate: YES(?,O(n^2))
          
          Proof:
            The weightgap principle applies, where following rules are oriented strictly:
            
            TRS Component:
              {  activate(n__isNat(X)) -> isNat(X)
               , activate(X) -> X}
            
            Interpretation of nonconstant growth:
            -------------------------------------
              The following argument positions are usable:
                Uargs(U11) = {}, Uargs(activate) = {}, Uargs(U21) = {},
                Uargs(s) = {1}, Uargs(plus) = {1, 2}, Uargs(and) = {1, 2},
                Uargs(isNat) = {1}, Uargs(n__plus) = {}, Uargs(n__isNat) = {1},
                Uargs(n__s) = {}
              We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
              Interpretation Functions:
               U11(x1, x2) = [0 0] x1 + [1 0] x2 + [3]
                             [0 0]      [0 1]      [1]
               tt() = [3]
                      [1]
               activate(x1) = [1 0] x1 + [2]
                              [0 1]      [1]
               U21(x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [3]
                                 [0 0]      [0 0]      [0 0]      [1]
               s(x1) = [1 0] x1 + [2]
                       [0 0]      [1]
               plus(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 0]      [0 0]      [1]
               and(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                             [0 1]      [0 1]      [0]
               isNat(x1) = [1 0] x1 + [1]
                           [0 0]      [1]
               n__0() = [2]
                        [0]
               n__plus(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
               n__isNat(x1) = [1 0] x1 + [1]
                              [0 0]      [0]
               n__s(x1) = [1 0] x1 + [2]
                          [0 0]      [0]
               0() = [2]
                     [0]
            
            The strictly oriented rules are moved into the weak component.
            
            We consider the following Problem:
            
              Strict Trs:
                {  isNat(n__plus(V1, V2)) ->
                   and(isNat(activate(V1)), n__isNat(activate(V2)))
                 , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
                 , activate(n__s(X)) -> s(activate(X))}
              Weak Trs:
                {  activate(n__isNat(X)) -> isNat(X)
                 , activate(X) -> X
                 , U21(tt(), M, N) -> s(plus(activate(N), activate(M)))
                 , isNat(n__s(V1)) -> isNat(activate(V1))
                 , U11(tt(), N) -> activate(N)
                 , and(tt(), X) -> activate(X)
                 , isNat(n__0()) -> tt()
                 , 0() -> n__0()
                 , plus(X1, X2) -> n__plus(X1, X2)
                 , isNat(X) -> n__isNat(X)
                 , s(X) -> n__s(X)
                 , activate(n__0()) -> 0()}
              StartTerms: basic terms
              Strategy: innermost
            
            Certificate: YES(?,O(n^2))
            
            Proof:
              The weightgap principle applies, where following rules are oriented strictly:
              
              TRS Component:
                {isNat(n__plus(V1, V2)) ->
                 and(isNat(activate(V1)), n__isNat(activate(V2)))}
              
              Interpretation of nonconstant growth:
              -------------------------------------
                The following argument positions are usable:
                  Uargs(U11) = {}, Uargs(activate) = {}, Uargs(U21) = {},
                  Uargs(s) = {1}, Uargs(plus) = {1, 2}, Uargs(and) = {1, 2},
                  Uargs(isNat) = {1}, Uargs(n__plus) = {}, Uargs(n__isNat) = {1},
                  Uargs(n__s) = {}
                We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
                Interpretation Functions:
                 U11(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                               [0 1]      [0 1]      [1]
                 tt() = [0]
                        [0]
                 activate(x1) = [1 0] x1 + [0]
                                [0 1]      [1]
                 U21(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [3]
                                   [0 0]      [0 0]      [0 0]      [1]
                 s(x1) = [1 0] x1 + [0]
                         [0 0]      [1]
                 plus(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
                                [0 1]      [0 0]      [0]
                 and(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                               [0 0]      [0 1]      [1]
                 isNat(x1) = [1 0] x1 + [0]
                             [0 0]      [1]
                 n__0() = [0]
                          [0]
                 n__plus(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
                                   [0 0]      [0 0]      [0]
                 n__isNat(x1) = [1 0] x1 + [0]
                                [0 0]      [0]
                 n__s(x1) = [1 0] x1 + [0]
                            [0 0]      [0]
                 0() = [0]
                       [0]
              
              The strictly oriented rules are moved into the weak component.
              
              We consider the following Problem:
              
                Strict Trs:
                  {  activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
                   , activate(n__s(X)) -> s(activate(X))}
                Weak Trs:
                  {  isNat(n__plus(V1, V2)) ->
                     and(isNat(activate(V1)), n__isNat(activate(V2)))
                   , activate(n__isNat(X)) -> isNat(X)
                   , activate(X) -> X
                   , U21(tt(), M, N) -> s(plus(activate(N), activate(M)))
                   , isNat(n__s(V1)) -> isNat(activate(V1))
                   , U11(tt(), N) -> activate(N)
                   , and(tt(), X) -> activate(X)
                   , isNat(n__0()) -> tt()
                   , 0() -> n__0()
                   , plus(X1, X2) -> n__plus(X1, X2)
                   , isNat(X) -> n__isNat(X)
                   , s(X) -> n__s(X)
                   , activate(n__0()) -> 0()}
                StartTerms: basic terms
                Strategy: innermost
              
              Certificate: YES(?,O(n^2))
              
              Proof:
                We consider the following Problem:
                
                  Strict Trs:
                    {  activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
                     , activate(n__s(X)) -> s(activate(X))}
                  Weak Trs:
                    {  isNat(n__plus(V1, V2)) ->
                       and(isNat(activate(V1)), n__isNat(activate(V2)))
                     , activate(n__isNat(X)) -> isNat(X)
                     , activate(X) -> X
                     , U21(tt(), M, N) -> s(plus(activate(N), activate(M)))
                     , isNat(n__s(V1)) -> isNat(activate(V1))
                     , U11(tt(), N) -> activate(N)
                     , and(tt(), X) -> activate(X)
                     , isNat(n__0()) -> tt()
                     , 0() -> n__0()
                     , plus(X1, X2) -> n__plus(X1, X2)
                     , isNat(X) -> n__isNat(X)
                     , s(X) -> n__s(X)
                     , activate(n__0()) -> 0()}
                  StartTerms: basic terms
                  Strategy: innermost
                
                Certificate: YES(?,O(n^2))
                
                Proof:
                  The following argument positions are usable:
                    Uargs(U11) = {}, Uargs(activate) = {}, Uargs(U21) = {},
                    Uargs(s) = {1}, Uargs(plus) = {1, 2}, Uargs(and) = {1, 2},
                    Uargs(isNat) = {1}, Uargs(n__plus) = {}, Uargs(n__isNat) = {1},
                    Uargs(n__s) = {}
                  We have the following constructor-based EDA-non-satisfying and IDA(2)-non-satisfying matrix interpretation:
                  Interpretation Functions:
                   U11(x1, x2) = [0 0 0] x1 + [2 2 0] x2 + [1]
                                 [0 0 0]      [0 2 0]      [1]
                                 [0 0 0]      [0 0 1]      [1]
                   tt() = [2]
                          [0]
                          [1]
                   activate(x1) = [1 2 0] x1 + [0]
                                  [0 1 0]      [0]
                                  [0 0 1]      [0]
                   U21(x1, x2, x3) = [0 0 0] x1 + [2 2 0] x2 + [2 2 0] x3 + [1]
                                     [0 0 2]      [0 1 0]      [0 2 0]      [2]
                                     [1 0 0]      [0 2 1]      [0 2 2]      [2]
                   s(x1) = [1 0 0] x1 + [0]
                           [0 1 0]      [2]
                           [0 1 1]      [0]
                   plus(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                                  [0 1 0]      [0 1 0]      [2]
                                  [0 1 1]      [0 1 1]      [1]
                   and(x1, x2) = [1 0 0] x1 + [1 2 0] x2 + [0]
                                 [0 0 0]      [0 1 0]      [0]
                                 [0 0 0]      [0 0 1]      [0]
                   isNat(x1) = [1 0 2] x1 + [2]
                               [0 0 0]      [1]
                               [0 0 0]      [1]
                   n__0() = [2]
                            [1]
                            [0]
                   n__plus(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                                     [0 1 0]      [0 1 0]      [2]
                                     [0 1 1]      [0 1 1]      [1]
                   n__isNat(x1) = [1 0 2] x1 + [0]
                                  [0 0 0]      [1]
                                  [0 0 0]      [1]
                   n__s(x1) = [1 0 0] x1 + [0]
                              [0 1 0]      [2]
                              [0 1 1]      [0]
                   0() = [2]
                         [1]
                         [0]

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