In what follows, all exercises cited by number come from the end of Chapter 9 of Gries & Schneider. In proving a numbered theorem, use only axioms, theorems, and metatheorems that have a smaller number. Keep in mind that the General Laws of Quantification (as expressed in the theorems of Chapter 8) apply to universal and existential quantification.
1. Without making use of Theorem (9.3), prove
Hints: Theorem (9.4a) is proved in the book; (3.60)
2. Do Exercise 9.1, which can be restated as follows:
Suppose that, in place of Axiom (9.5), Gries & Schneider had offered this seemingly weaker version:
Axiom (9.5') Provided that there are no free occurrences of x in P,
Use (9.5') to prove (9.5):
(9.5) Provided that there are no free occurrences of x in P,
Hint: Use Trading (9.3).
3. Do Exercise 9.7, which asks for a proof of Theorem (9.10).
Hint: Don't forget about the theorems in Chapter 8.
4. Do Exercise 9.12, which asks for a proof of Theorem (9.18a) (Generalized DeMorgan).
Hint: Use Axiom (9.17), which defines existential quantification in terms of universal quantification.
5. Do Exercise 9.17, which asks for a proof of Theorem (9.21):
(9.21) Provided that there are no free occurrences of x in P,
Hint: Use Generalized De Morgan to translate existential quantification into universal, do some manipulations, and then translate back to existential quantification.
6. Prove (∃x |: P ==> Q) ≡ (∃x |: ¬P) ∨ (∃x |: Q)