Rule of Association

Context
Natural deduction.

Definition
This rule is two-fold:


 * Conjunction is associative:
 * 1) $$p \land \left({q \land r}\right) \vdash \left({p \land q}\right) \land r$$
 * 2) $$\left({p \land q}\right) \land r \vdash p \land \left({q \land r}\right)$$


 * Disjunction is associative:
 * 1) $$p \lor \left({q \lor r}\right) \vdash \left({p \lor q}\right) \lor r$$
 * 2) $$\left({p \lor q}\right) \lor r \vdash p \lor \left({q \lor r}\right)$$

Its abbreviation in a tableau proof is $$\textrm{Assoc}$$.

Proof
These are proved by the tableau method:

$$p \land \left({q \land r}\right) \vdash \left({p \land q}\right) \land r$$:

$$\left({p \land q}\right) \land r \vdash p \land \left({q \land r}\right)$$

is proved similarly.

$$p \lor \left({q \lor r}\right) \vdash \left({p \lor q}\right) \lor r$$:

$$\left(p \lor q\right) \lor r \vdash p \lor \left(q \lor r\right)$$

is proved similarly.