# Left-distributive

## Definition

Let $$(S, *, +)$$ be an algebraic structure. Then $$*$$ is left-distributive over $$+$$ if and only if

$$\forall x,y,z \in S : x * (y + z) = (x * y) + (x * z)$$
THEOREM

Minimum is left-distributive over maximum.

$$\forall a,b,c \in \mathbb{R} : \text{min}(a,\text{max}(b,c))=\text{max}(\text{min}(a,b),\text{min}(a,c))$$

Proof available

THEOREM

Scalar multiplication is left-distributive over vector addition.

$$c(\mathbf{u} + \mathbf{v}) = c\mathbf{u} + c\mathbf{v}$$

Proof available

LEMMA    $$P \wedge (Q \vee R) \vdash P \wedge Q \vee P \wedge R$$

Proof available

LEMMA    $$P \wedge Q \vee P \wedge R \vdash P \wedge (Q \vee R)$$

Proof available

THEOREM

Conjunction is left-distributive over disjunction.

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

Proof available

LEMMA    $$(P \vee Q) \wedge (P \vee R) \vdash P \vee Q \wedge R$$

Proof available

THEOREM

Disjunction is left-distributive over conjunction.

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

Proof available