Exercise 7.11 For each of the following entailments, give a derivation in Elf (if it holds in intuitionistic logic) or indicate that it does not hold.

1. ∀x. A(x) ∧ B(x) ⊢ (∀x. A(x)) ∧ (∀x. B(x)).

2. (∀x. A(x)) ∧ (∀x. B(x)) ⊢ ∀x. A(x) ∧ B(x).

3. ∀x. ⊥ ⊢ ⊥.

4. ⊥ ⊢ ∀x. ⊥.

5. ∀x. ∀y. A(x, y) ⊢ ∀y. ∀x. A(x, y).

6. ∀x. ¬A(x) ⊢ ¬∃x. A(x).

7. ¬∃x. A(x) ⊢ ∀x. ¬A(x).

8. ∃x. A(x) ∨ B(x) ⊢ (∃x. A(x)) ∨ (∃x. B(x)).

9. (∃x. A(x)) ⊃ B ⊢ ∀x. (A(x) ⊃ B).

10. ∀x. (A(x) ⊃ B) ⊢ (∃x. A(x)) ⊃ B.

11. (∀x. A(x)) ⊃ B ⊢ ∃x. (A(x) ⊃ B).

12. ∃x. (A(x) ⊃ B) ⊢ (∀x. A(x)) ⊃ B. Exercise 7.12 Show that the system of natural deduction extended with the rule of excluded middle XM is equivalent to the system of natural deduction extended with proof by contradiction ⊥_{C} (see page 242). Implement this proof in Elf.