In a military camp, there was sufficient food for $190$ days for a total of $4000$ men. After $30$ days, $800$ men left the camp. The remaining men will spend the remaining food provisions in how many days?

Let's use the man-days formula framework:
* Total original provision period = $190$ days.
* Provisions are consumed normally for the initial $30$ days by $4000$ men.
* Remaining days of food left for $4000$ men = $190-30=160$ days.

Thus, the total available remaining food volume can be quantified as:

$$\text{Remaining Food}=4000\times 160\text{man-days}$$

Now, $800$ men leave, so the remaining headcount of men is:

$$4000-800=3200\text{men}$$

Let $D$ be the number of days the remaining food will last for these $3200$ men. Equating the food volume:

$$3200\times D=4000\times 160$$

$$D=\frac{4000\times 160}{3200}$$

$$D=\frac{4000}{20}=200\text{days}$$