proof of this

Theorem: Let $$$p=\lceil\frac km\times2^{64}\rceil$$$, when $$$a_i\le \frac{2^{64}}m$$$, the computation of the lower rounding is exact.

Proof: Let $$$\frac p{2^{64}}=\frac km+\epsilon$$$, where $$$\epsilon\in[0,\frac1{2^{64}})$$$.

Here if $$$\lfloor a_i\times\frac km+a_i\epsilon\rfloor>\lfloor a_i\times \frac km\rfloor$$$ must be wrong, $$$a_i\times \frac km$$$ is at least $$$\frac1m$$$ away from $$$\lfloor a_i\times \frac km\rfloor+1$$$, so as long as $$$a_i\epsilon<\frac1m$$$ it's OK, let's continue the derivation.

Since $$$\{a_i\times\frac km\}\times m$$$ is an integer, the result is exact as long as $$$a_im\epsilon<1$$$.

The $$$a_i\epsilon<\frac1m$$$ and $$$a_im\epsilon<1$$$ are the same, and then the condition can be rewritten as $$$a_i\le\frac{2^{64}}m$$$ according to $$$\epsilon<\frac1{2^{64}}$$$.