## How does progress fail in system $F_{\omega}$ when types $T_1 \to T_2$ and $T_2 \to T_1$ are equivalent?

Pierce’s TAPL book gives in exercise 30.3.17 the setting where $$T_1 \to T_2 \equiv T_2 \to T_1$$ (the function type are assumed to be equivalent). In the solutions, he claims that this assumption breaks the progress property.

It is easy to see that preservation fails. How can progress:

$$\vdash t:T \implies t \text{ is a value } \lor \exists t’. t \to t’$$

be wrong in this setting?

## JProve $T_1$ is finite if there exists bijection from $T_1$ to another finite set $T_2$

Assuming $$T_1$$ is non empty. Prove $$T_1$$ is finite if there exists bijection from $$T_1$$ to another finite set $$T_2$$

Now we have $$h : J_m \to T_1$$

Assume another finite set $$T_2$$, so we have $$h_1 : J_m \to T_2$$.

Since both $$h_1$$ and $$h_2$$ are bijections. so is $$h_1 \circ h^{-1}$$

$$J_m$$ = $${1,2,3,…,m$$}

Conversely, similar idea to that of this. But is this even correct ?

Thanks

## $T_1, … , T_n$ time of life of instruments with geometric distribution of parameter p independent

$$T_1, … , T_n$$ time of life of instruments with geometric distribution of parameter p independents. I define S as the first n instants of failure and U as the last n instant of failure. I want to find the law of S and the density of U.

This is what I did so far:

To find the law of S I start by calculating: $$\mathbb{P}(S = k) = \mathbb{P}(S \geq k) – \mathbb{P}(S \geq k + 1)$$ $$\mathbb{P}(S = k) = \mathbb{P}(S < k+1) – \mathbb{P}(S < k)$$ But here I am stuck on what I should do because I still do not know the distribution.

I do the same reasoning with U as follows: $$\mathbb{U}(S = k) = \mathbb{U}(U \leq k) – \mathbb{U}(S \leq k + 1)$$ But I still get nowhere.

Any suggestions?