What type of conditional is this

Question

Well, I was reading a book when I read this sentence

Should you fail to do so, we would have no choice but to take further action.

I want to know what type of conditional is this? The if statement part is in present (First conditional) and the result part is using would (Second conditional). Is it a inverted conditional or a mixed conditional or something totally different?

Should we win your prize we would like to build a new school close to our production center.

To me it seems like a structure to show politeness.

Answer 1

"Should" emphasises the hypothetical nature of the condition and makes it seem less likely or more remote.

Should you fail to do so, we would have no choice but to take further action.

This is the inverted form of "If you should fail...".

It can equally be expressed as "If you failed..." or "If you were to fail...".

In the above example, it is similar in function to a so-called "second conditional", but there are other cases where it is close in function to the "first conditional":

Should your child become anxious or nervous about any activity, it is a good idea to inform the team-leader. ( Cambridge )

This could be expressed as "If your child should become..." or "If your child becomes" (present tense). Similarly:

Should we win your prize we would like to build a new school close to our production center.

This could be expressed as "If we should win..." or "If we win". ("If we won" wouldn't work as well here.)