In this session we'll come back to what we did in the last one.
Namely, proving equational properties of programs using structural induction.
We will practice the newly learned techniques in a somewhat larger proof.
For a more difficult example let's consider the reverse function.
We pick its inefficient definition here because it's more amenable to equational
proofs. So as defining process we would have nil
reverses nil and x followed by XS reverse is the reversal of the list xs followed by
the element X. These two equations are as we know equivalent to the more efficient
version of fork left and closer to what we want to prove.
So we pick them which of course does not prevent us from using at one time the more
efficient definition of reverse. So what we would like to do is prove the
proposition that XS reverse of reverse is XS. So to prove this it's an obvious
induction on the list excess, the base case is really easy. Nil reverse, reverse
is the same thing as nil reverse by the first law of reverse which says that nil
dot reverse is nil and then we again invoke the first law To show that, that
expression now is the same as nil, which establishes the proposition.
Let's look at the induction step so here we would have X followed by XS, and then a
double reverse. What can we do with that?
Well, we can apply the second clause of reverse, which would mean that this
expression here would, can be rewritten to the right-hand side, excess.reverse
followed by X. And then we have our reverse on both sides
here. What else can we do?
Well, That doesn't seem to be anything obvious.
So let's turn to the right hand side. The right hand side would read simply X
followed by XS. So what can we do that?
Well, one thing we could do is apply the induction hypothesis which says that the
list xs is the same as the list xs reverse, reverse.
So then we are left with x followed by xs reverse, reverse and again, there's not
much we can do anymore to this side here. So both sides, unfortunately, have
simplified to different expressions. This expression here and I've that
expression there. So we still need to show that the two
sides are the same and proving it directly by induction doesn't work as we have seen.
What we can try instead is, we can generalize this equation.
So the idea is that, instead of just saying excess to dot reverse here and
here. We replace that with.
An arbitrary list ys. So our new lemma that we want to prove is
for, that for any list ys, ys followed by x reverse is the same as x followed by
ys.reverse. And to prove that equation, we can use a second induction argument, this
time on the list bias so let's try that. So, let's look at the base case first.
So here, YS equals nil and the equation we want to show is that nil followed by
x.reverse is the same as x followed by nil.reverse.
That's the instansation of the lemma that we want to show here.
So, what could we do with the left hand side here?
Well, By the first close of plus, plus, nil is a
left unit, so this thing simply simplifies to list of x.reverse. reverse.
Then the next step would be to expand what list of X is.
So list of X, as we know by its definition, is X followed by nil.
In the next step then, we would involve the second clause of reverse to arrive at
this expression here. When actually there's one intermediate
step that we have to do here. So by the second clause of reverse, what
would we get? We would get the list that follows the
head element first so there would be nil.reverse Followed by, this suffix and
now we can simply find nil.reverse by the first clause of reverse is just nil.
Where as list of x expands to x followed by nil.
So, that's how we arrive, at this expression here.
Now what we can do here is we can again invoke the first clause of plus, plus to
say nil is a left unit and we're just left with x followed by nil.
There's one more step to do. Again by the first clause of reverse we
know that nil.reverse is nil. Or we've just now used this equation
backwards from going from nil to nil.reverse And that's the right hand side
if you want to show here. So we get equality here and the base case
is established. Let's look at the inductive step.
What we need to do here is show that y followed by ys, that's our list, followed
by x reverse is the same as x followed by then y followed by ys reversed.
So let's see how we would go about that let's work on the left hand side here.
First thing we can do is we can pull out the y from its binding with the list YS,
using the second list of comcat. So we have the Y as a head element here,
and then the list YS, followed by list of X.
The next thing we can do is we can invoke the law of reverse, which says well
reverse of a list that starts with Y is the same as reversal of the rest of the
list here. And the y becomes the last element of the
new list. The next thing we can do is apply the
induction hypothesis because we see here that the expression ys followed by list x
reverse, that's the left hand side of the equation with just the list ys and we can
assume that, that, that, that equation holds.
So we can rewrite it to the right hand side of the equation which will be x
followed by yes.reverse. Now we can apply the first clause of plus, plus to pull out
the accelerant. And we can apply the second clause of
reverse to establish that Y followed by ys reverse, is the same as ys reverse
followed by a list of y. And again we have equality here.
