I was going to link you to a page on Tae Kim, but considering you’ve used the exact same example, I’m assuming you’ve already seen it.

“One must not not go to school every day” – while you’re right that it’s a double negative, I think a better English transliteration would be something like “If you don’t go to school every day, it’s no good (i.e. you must go)”. It makes more sense when you consider the similar constructions using と and ば.

I think the positive equivalent of that example would be 勉強しちゃ. Here’s some example sentences using しちゃ with other words:
http://jisho.org/sentences?jap=shicha&eng=

The ～なくちゃ construction is somewhat feminine. It is a slur/contraction of ～なくては.

The ては goes to ちゃ. This is somewhat common with て form constructions. For example, 食べてしまった can be contracted to 食べちゃった. Which is another example of what Joel mentioned with しちゃう.

なきゃ short for なければ.

Both なくちゃ and なきゃ when used at the end of a sentence mean “must do”. They leave off the (implied) ending (ない、いけない、ならない、だめ).

The basic construction of all these types are [Don't do a thing] + [something bad].
That’s the basic construction (and the second part can be left off). If you remember that, the different forms should all make sense.

• This reply was modified 10 years, 3 months ago by  thisiskyle.

“I think the positive equivalent of that example would be 勉強しちゃ.”

Unlike しなくちゃ or しなきゃ, you cannot use しちゃ by itself. You have to add いけない or だめ.

毎日学校に行かなくてはいけない。
毎日学校に行かなければなりません。
毎日学校に行かないといけません。
毎日学校に行かなきゃいけない。
毎日学校に行かなくちゃだめ（だ）。
毎日学校に行かなけりゃいけない。
毎日学校に行かなくては。
毎日学校に行かなければ。
毎日学校に行かないと。
毎日学校に行かなきゃ。
毎日学校に行かなくちゃ。
毎日学校に行かなけりゃ。
毎日学校に行かねばならぬ。
毎日学校に行かなかったらあかん。
While having different nuances, all these mean “I have to go to school every day.”

毎日は学校に行ってはいけない。
毎日は学校に行っちゃだめ（だ）。
毎日は学校に行ったらあかん。
These three sentences mean “Every day I must not go to school.”