YES

Problem:
 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))
 a(id(),x) -> x
 a(1(),id()) -> 1()
 a(t(),id()) -> t()
 a(1(),p(x,y)) -> x
 a(t(),p(x,y)) -> y

Proof:
 DP Processor:
  DPs:
   a#(lambda(x),y) -> a#(y,t())
   a#(lambda(x),y) -> a#(x,p(1(),a(y,t())))
   a#(p(x,y),z) -> a#(y,z)
   a#(p(x,y),z) -> a#(x,z)
   a#(a(x,y),z) -> a#(y,z)
   a#(a(x,y),z) -> a#(x,a(y,z))
  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))
   a(id(),x) -> x
   a(1(),id()) -> 1()
   a(t(),id()) -> t()
   a(1(),p(x,y)) -> x
   a(t(),p(x,y)) -> y
  Maximal Polynomial Processor:
   usable rules:
    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))
    a(id(),x) -> x
    a(1(),id()) -> 1()
    a(t(),id()) -> t()
    a(1(),p(x,y)) -> x
    a(t(),p(x,y)) -> y
   interpretations:
    a#(x0, x1) = max(0, 0 + 1 + x0 + -1 + x1)
    id = 0
    p(x0, x1) = max(0, x0, x1)
    t = 0
    1 = 0
    a(x0, x1) = max(0, 0 + x0 + x1)
    lambda(x0) = max(0, 1 + x0)
   problem:
    DPs:
     a#(p(x,y),z) -> a#(y,z)
     a#(p(x,y),z) -> a#(x,z)
     a#(a(x,y),z) -> a#(y,z)
     a#(a(x,y),z) -> a#(x,a(y,z))
    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))
     a(id(),x) -> x
     a(1(),id()) -> 1()
     a(t(),id()) -> t()
     a(1(),p(x,y)) -> x
     a(t(),p(x,y)) -> y
   Restore Modifier:
    DPs:
     a#(p(x,y),z) -> a#(y,z)
     a#(p(x,y),z) -> a#(x,z)
     a#(a(x,y),z) -> a#(y,z)
     a#(a(x,y),z) -> a#(x,a(y,z))
    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))
     a(id(),x) -> x
     a(1(),id()) -> 1()
     a(t(),id()) -> t()
     a(1(),p(x,y)) -> x
     a(t(),p(x,y)) -> y
    Subterm Criterion Processor:
     simple projection:
      pi(a#) = 0
     problem:
      DPs:
       
      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))
       a(id(),x) -> x
       a(1(),id()) -> 1()
       a(t(),id()) -> t()
       a(1(),p(x,y)) -> x
       a(t(),p(x,y)) -> y
     Qed