We consider the following Problem:

  Strict Trs:
    {  a(lambda(x), y) -> lambda(a(x, p(1(), a(y, t()))))
     , a(p(x, y), z) -> p(a(x, z), a(y, z))
     , a(a(x, y), z) -> a(x, a(y, z))
     , lambda(x) -> x
     , a(x, y) -> x
     , a(x, y) -> y
     , p(x, y) -> x
     , p(x, y) -> y}
  StartTerms: basic terms
  Strategy: innermost

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

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

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