TOPIC: Another suggestion is to not smoke cigarettes

I 'm confused about this phrase " Another suggestion is to not smoke cigarettes."

I think " to no smoke cigarettes" is a negative of " to-infinitive". In my opinion, the negative of " to-infinitive" should be " not + to + verb". So I re-write the sentence here.

I 'm sure my re-writing sentence can't work. I don't know what is wrong with me??? Maybe we can use the pattern here.

".... is + to + not + verb +...". I follow the rule and make the following sentence below.

The important thing is to not tell a lie
Could you please tell me if I 'm right?
Thanks!

You are right. The NORMAL way of forming the negative of "to-infinitive" is"not +to+verb". This means that the normal order in the above sentence would be "Another suggestion is not to smoke cigarettes."

However, as you can also see in the example you gave, sometimes people move"not" to follow the "to". Some grammarians would fault that structure as split infinitive. Other grammarians may argue, though, that this is not really a problem. As you can see, English language grammarians don't quite agree on this issue.
You would be wise to follow the normal standard usage, as"not + to + verb" will always be acceptable in standard English

Thanks !

You are welcome