# Regular formula

From LLWiki

A *regular formula* is a formula *R* such that .

A formula *L* is *co-regular* if its dual is regular, that is if .

## Alternative characterization

*R* is regular if and only if it is equivalent to a formula of the shape for some positive formula *P*.

*Proof.*
If *R* is regular then with positive. If with *P* positive then *R* is regular since .

## Regular connectives

A connective *c* of arity *n* is *regular* if for any regular formulas *R*_{1},...,*R*_{n}, is regular.

**Proposition** (Regular connectives)

, and define regular connectives.

*Proof.*
If *R* and *S* are regular, thus it is regular since is positive.

thus it is regular since is positive.

If *R* is regular then is regular, since .

More generally, is regular for any formula *A*.