> **Puzzle 209.** Suppose you are trying to fry some eggs and also toast some slices of bread. Describe each process separately as a feasibility relation from \$$\mathbb{N}\$$ to \$$\mathbb{N}\$$ and then tensor these relations. What is the result?

Michael wrote:

> The exact requirements of frying and toasting is vague...

Yes - I wanted to make people think a little. If you have some number \$$x\$$ of eggs, and you want fried eggs, you can get at most \$$x\$$ fried eggs. Similarly with toasting slices of bread. I wanted people to translate these well-known facts into feasibility relations.

> Define \$$\Phi(x,x’)\$$ to be feasibility for frying where you get one fried egg (x) from one egg (x’). Similarly define \$$\Psi(y,y’)\$$ to be feasibility for toasting where you get one toast (y) from one slice of bread (y’).Then :

> $\Phi(x,x') = \begin{cases} \texttt{true} & \mbox{if } x \leq x' \\\\ \texttt{false} & \mbox{otherwise.} \end{cases}$

> $\Psi(y,y') = \begin{cases} \texttt{true} & \mbox{if } y \leq y' \\\\ \texttt{false} & \mbox{otherwise.} \end{cases}$

Right! And now we tensor them together...

> $(\Phi \otimes \Psi)((x,y),(x',y')) = \begin{cases} \texttt{true} & \mbox{if } (x \leq x') \mbox{ and } (y \leq y') \\\\ \texttt{false} & \mbox{otherwise.} \end{cases}$

Right!

Okay, on to the next puzzle.

> **Puzzle 211.** What general mathematical result is Puzzle 209 an example of?

Michael wrote:

> This is just the monoidal preorder law.

Hmm. First of all, the monoidal preorder law isn't a "result", it's just part of a definition. In a monoidal preorder we must have

$x \le x' \text{ and } y \le y \text{ implies } x \otimes x' \le y \otimes y'$

\$$\textbf{Bool}\$$ is a monoidal preorder, and the monoidal preorder law for \$$\textbf{Bool}\$$ says

$x \le x' \text{ and } y \le y \text{ implies } x \wedge x' \le y \wedge y'$

I believe this law is the key to solving Puzzle 210:

> **Puzzle 210.** Show that \$$\Phi \otimes \Psi\$$ is really a feasibility relation if \$$\Phi\$$ and \$$\Psi\$$ are feasibility relations.

But Puzzle 209 is about something more specific.

So, let me give some hints.

In Puzzle 209 the feasibility relation we're calling \$$\Phi\$$ is just any old feasibility relation from \$$\mathbb{N}\$$ to \$$\mathbb{N}\$$: it's a very famous one, with a name! Similarly for \$$\Psi\$$... and similarly for \$$\Phi \otimes \Psi\$$. So the answer to Puzzle 209 is a special case of a general result - a result I haven't stated yet. I'm trying to get people to guess this result based on this particular example.