YES

TRS:

  and(true(),y) -> y
  and(false(),y) -> false()
  eq(nil(),nil()) -> true()
  eq(cons(t,l),nil()) -> false()
  eq(nil(),cons(t,l)) -> false()
  eq(cons(t,l),cons(t',l')) -> and(eq(t,t'),eq(l,l'))
  eq(var(l),var(l')) -> eq(l,l')
  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(t',s')) -> and(eq(t,t'),eq(s,s'))
  eq(apply(t,s),lambda(x,t)) -> false()
  eq(lambda(x,t),var(l)) -> false()
  eq(lambda(x,t),apply(t,s)) -> false()
  eq(lambda(x,t),lambda(x',t')) -> and(eq(x,x'),eq(t,t'))
  if(true(),var(k),var(l')) -> var(k)
  if(false(),var(k),var(l')) -> var(l')
  ren(var(l),var(k),var(l')) -> if(eq(l,l'),var(k),var(l'))
  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) = x2
  and#_A(x1,x2) = 2
  true_A = 1
  true#_A = 0
  false_A = 0
  false#_A = 1
  eq_A(x1,x2) = 1
  eq#_A(x1,x2) = 3
  nil_A = 1
  nil#_A = 2
  cons_A(x1,x2) = x1
  cons#_A(x1,x2) = x2 + 2
  var_A(x1) = 1
  var#_A(x1) = 6
  apply_A(x1,x2) = x1 + x2 + 1
  apply#_A(x1,x2) = 0
  lambda_A(x1,x2) = x2 + 4
  lambda#_A(x1,x2) = 3
  if_A(x1,x2,x3) = x3
  if#_A(x1,x2,x3) = 0
  ren_A(x1,x2,x3) = x3
  ren#_A(x1,x2,x3) = x2 + x3 + 3

precedence:

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