Proof exercises Propositional natural deduction

The following sequents provide practice in the art of constructing proofs. Answers are given, but of course the idea is to come up with proofs of your own before looking them up. As ever, the answers are not unique: in some cases, alternative proofs are just as good as the ones given.

(p AND q) AND r ⊢ s OR (q OR t)	Try it	Answer
(p IMP q) IMP p ⊢ (p IMP q) IMP q	Try it	Answer
(p AND q) IMP NOTr ⊢ (p AND r) IMP NOTq	Try it	Answer
p IMP q ⊢ (q IMP r) IMP ((r IMP s) IMP (NOTs IMP NOTp))	Try it	Answer
p AND q ⊢ (p IMP q) AND (q IMP p)	Try it	Answer
NOT(NOTp AND NOTq) ⊢ p OR q	Try it	Answer
p IMP NOTq ⊢ (p AND q) IMP r	Try it	Answer
p OR q ⊢ NOTp IMP q	Try it	Answer
NOTp IMP q ⊢ p OR q	Try it	Answer
NOT(p IMP q) ⊢ p AND NOTq	Try it	Answer
(p IMP q) IMP q ⊢ (q IMP p) IMP p	Try it	Answer
p IMP q, r IMP s ⊢ (p OR r) IMP (q OR s)	Try it	Answer
p OR q, p OR r, NOT(q AND r) ⊢ p	Try it	Answer
p IMP (q OR r) ⊢ NOTq IMP (p IMP r)	Try it	Answer
(p AND q) IMP (r OR s) ⊢ (p IMP r) OR (q IMP s)	Try it	Answer