Left-distributive


Definition

\(*\) 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

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

Proof available

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

Proof available