Good.

All right.

Hello, everyone.

Welcome to this

I think

second SMI lecture.

Today I'm substituting for Professor Kohlhaase because he's at a workshop.

I'm one of his PhD students.

Okay.

The quiz is over.

It seems to have worked

which is very good because yesterday we had technical problems.

So let me open the slides.

Good.

So what you've done so far in the lecture is some introductory admin stuff, and you

started with unary natural numbers

which is defining natural numbers by having a zero

and a successor function.

All the way it was visualized in the slides was initially just with slashes

and that's

the basic way of counting.

And the last thing you did

I think

was a bit of doing proofs by induction where you

do a proof that some property holds for all natural numbers.

And if I understand correctly

I'll start exactly at this slide where we have another

example of a proof by induction

in this case about something more physical.

So the idea is we have a sequence of dominoes

and we say basically if we push the first

one, then all of them will fall.

And to prove that

basically

we need to know a few things about the setup.

So we need to know that they're all placed next to each other in the sequence.

We need to know that they are high enough that if one falls, the next one will fall.

We need to know that the weights are so that they will actually fall over and have enough

energy to knock over the next one.

So we say they have to have the same height.

And then we say, and we push the first one.

And then we say in that case

all of them will fall

which is obviously true.

But the question is how do you prove that?

And the basic idea is we do a proof by induction because proofs by induction exactly follow

this kind of domino pattern.

So the proof is shown here.

Proof by induction is always two cases.

We have the base case where we say it starts.