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- P ^ (Q ^ R) <=> P ^ Q ^ R
- P v (Q v R) <=> P v Q v R

A note to users, typically association is used as P ^ (Q ^ R) <=> (P ^ Q) ^ R. While Aris will allow you to prove that this is equivalent, association allows the removal of one pair of parentheses at a time. (P ^ Q) ^ (R ^ S) <=> P ^ Q ^ R ^ S is also valid in Aris, because association allows recursion, but only when removing several sets of parentheses or adding several sets of parentheses.