Is everybody happy with the current exercise sheet?

We designed them to make you happy obviously.

We did not. So what problems are there?

There are some bangs missing.

Some bangs missing? Oh God. So where are the bangs missing?

The next exercise sheet.

There's...oh I...yes, yes.

Yes, yes, obviously, yeah, obviously. I didn't notice that. Didn't we correct it?

Where is it?

Didn't we notice at some point that there was a bang missing there?

Well, you said you weren't sure if it was missing or not.

Ah, it seems fairly obvious that it's missing, yeah.

Did we know...I think we managed to know about...

I don't think we can see.

We did the party exercise on Friday and I think you managed to show without the bang.

This looks weird. I mean how would one show it without the bang?

As I mean Bang says I can move the formula either zero or one time, zero times one time or two times.

How would I get to three times from there?

Cutting with what?

Yeah, yeah.

Yeah, obviously, yeah.

Well, actually I mean...

I would expect this, right?

So you said you managed to prove it this way.

Yeah, but you used promotion.

Okay, so wait, if you use promotion then what?

Okay, why do you cut with this and then...

It is of course different.

Wait, this is one with bangs in front of the...

Okay, out of which the intermediate one would be superfluous, right?

I mean this one you could derive because this is additive conjunction.

So in this proof here there is apparently bangs floating around.

This is an axiom. Oh, actually you could...

Right, you can apply it to a formula that already has a bang in it.

Yeah, that looks much more reasonable.

And how did Ido write it?

Did you say Ido wrote it? It's different than what?

Ah, than what we wrote. Okay, so yeah, maybe.

But you're right, if you say that...

I mean we can promote this with another bang in front.

And we haven't removed the promotion rule and then...

Yes, that should work.

Okay, so for the moment apparently we claim that the exercise isn't wrongly posed.

And we recall that you have the promotion rule.

So we were removing the weakening, contraction and dereliction rule, but not the promotion rule.

So as a tiny hint, you can add another bang to that thing in front.

So you can turn the single bang into a double bang.

And then it should be clear.

So this is in the one-sided or two-sided system actually.

Well, this is the two-sided system.