Set containment of MinimumBadPrefix and BadPrefix of a Safety Property This question is related to formal model checking theory, but I cannot find a tag for it.

Let $P_{safe}$ be a safety property.

$BadPref(P_{safe})$ is the set of bad prefixes for $P_{safe}$

A Safety property $P_{safe}$ over Atomic Proposition $AP$ is regular
if its set of bad prefixes is a regular language over $2^{AP}$

Is the following true? 


If $L$ is a regular language with $MinBadPref(P_{safe}) \subset L \subset BadPref(P_{safe})$,
then $P_{safe}$ is regular. 
If $P_{safe}$ is regular, then any L for which
$MinBadPref(P_{safe}) \subset L \subset BadPref(P_{safe})$ is regular.


My attempt to 2. is that it's true

$P_{safe}$ is regular then $MinBadPref(P_{safe})$ is regular and so is $BadPref(P_{safe})$. A NFA $N$ recognizing $BadPref(P_{safe})$ can be obtained by adding self loops on the final states of the NFA $M$ recognizing $MinBadPref(P_{safe})$. And $L$ is recognized by some NFA containing a subset of the self loops added to $M$. Therefore $L$ is regular.

And what about 1.?
For (1), I will only give a sketch at first, in case you want to work it out yourself; I can add some details later if you want. Try proving the following:

$w\in BadPref$ if and only if $w$ has a prefix $u$ which is in $L$.

(2) is already covered quite well by David's comments.