Suppose you are given the following axioms:

1. 0 ≤ 3.
2. 7 ≤ 9.
3. ∀x x≤ x.
4. ∀x x≤ x + 0.
5. ∀x x+ 0 ≤ x.
6. ∀ x, y x + y ≤ y + x.
7. ∀ w, x, y, z w ≤ y ∧ x ≤ z ⇒ w + x ≤ y + z.
8. ∀ x, y, z x ≤ y ∧ y ≤ z ⇒ x ≤ z

a. Give a backward-chaining proof of the sentence 7 ≤ 3 + 9. (Be sure, of course, to use only the axioms given here, not anything else you may know about arithmetic.) Show only the steps that leads to success, not the irrelevant steps.

b. Give a forward-chaining proof of the sentence 7 ≤ 3 + 9. Again, show only the steps that lead to success.