YES

TRS:

  and(false(),false()) -> false()
  and(true(),false()) -> false()
  and(false(),true()) -> false()
  and(true(),true()) -> true()
  eq(nil(),nil()) -> true()
  eq(cons(T,L),nil()) -> false()
  eq(nil(),cons(T,L)) -> false()
  eq(cons(T,L),cons(Tp,Lp)) -> and(eq(T,Tp),eq(L,Lp))
  eq(var(L),var(Lp)) -> eq(L,Lp)
  eq(var(L),apply(T,S)) -> false()
  eq(var(L),lambda(X,T)) -> false()
  eq(apply(T,S),var(L)) -> false()
  eq(apply(T,S),apply(Tp,Sp)) -> and(eq(T,Tp),eq(S,Sp))
  eq(apply(T,S),lambda(X,Tp)) -> false()
  eq(lambda(X,T),var(L)) -> false()
  eq(lambda(X,T),apply(Tp,Sp)) -> false()
  eq(lambda(X,T),lambda(Xp,Tp)) -> and(eq(T,Tp),eq(X,Xp))
  if(true(),var(K),var(L)) -> var(K)
  if(false(),var(K),var(L)) -> var(L)
  ren(var(L),var(K),var(Lp)) -> if(eq(L,Lp),var(K),var(Lp))
  ren(X,Y,apply(T,S)) -> apply(ren(X,Y,T),ren(X,Y,S))
  ren(X,Y,lambda(Z,T)) -> lambda(var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),ren(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T)))

linear polynomial interpretations on N:

  and_A(x1,x2) = 1
  and#_A(x1,x2) = 3
  false_A = 1
  false#_A = 1
  true_A = 1
  true#_A = 2
  eq_A(x1,x2) = 1
  eq#_A(x1,x2) = 4
  nil_A = 1
  nil#_A = 3
  cons_A(x1,x2) = 1
  cons#_A(x1,x2) = 2
  var_A(x1) = 1
  var#_A(x1) = 0
  apply_A(x1,x2) = x1 + x2 + 1
  apply#_A(x1,x2) = 2
  lambda_A(x1,x2) = x2 + 1
  lambda#_A(x1,x2) = 4
  if_A(x1,x2,x3) = x2
  if#_A(x1,x2,x3) = x3
  ren_A(x1,x2,x3) = x3
  ren#_A(x1,x2,x3) = x3 + 4

precedence:

  ren > eq = cons = lambda = if > and = nil = var > true > false > apply