# Left self-distributive

## Definition

$$\forall a,b,c \in S : a \circ (b \circ c) = (a \circ b) \circ (a \circ c)$$

**THEOREM**
Let \(\circ\) be idempotent, commutative, and associative. Then \(\circ\) is left self-distributive.

Proof available

**LEMMA**
Union is left self-distributive.

$$A \cup (B \cup C) = (A \cup B) \cup (A \cup C)$$

Proof available

**LEMMA**
Disjunction is left self-distributive.

$$P \vee (Q \vee R) \dashv\vdash (P \vee Q) \vee (P \vee R)$$

Proof available

**LEMMA**
Intersection is left self-distributive.

$$A \cap (B \cap C) = (A \cap B) \cap (A \cap C)$$

Proof available

**LEMMA**
Conjunction is left self-distributive.

$$P \wedge (Q \wedge R) \dashv\vdash (P \wedge Q) \wedge (P \wedge R)$$

Proof available