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Write the two formal pseudodifferential operators in normal order aswhere $p={\alpha }_1$ and $q={\alpha }_2$. The formal pseudodifferential composition rule givesThus $\boxed{\mathrm{ord}(P_1{P}_2)=p+q}$ whenever $ab$ is not identically zero, in particular when both leading coefficients are nowhere zero. In general the rigorous statement over the specified coefficient algebra is $\mathrm{ord}(P_1{P}_2)\le p+q$.

The leading scalar coefficients commute. Subtracting the reversed composition also cancels the terms involving $a_1,b_1$, leavingTherefore the order filtration of formal pseudodifferential operators giveswith equality precisely when the displayed coefficient is not identically zero. The bound can be strict: constant-coefficient powers of $\partial$ commute, giving the zero operator, whose order can be assigned $-\infty$.

There is also a genuine qualification to product-order addition in $C^{\infty }(\mathbb{R})$. Choose nonzero smooth functions $a,b$ with disjoint compact supports and take $P_1=m_a,P_2=m_b$. Each has order zero, but their product is zero. Nonzero coefficients as elements of this algebra need not have a nonzero product. Thus an unqualified assertion of exact product order requires an additional leading-symbol hypothesis.