Consider the environment $Δ ≡ D_1,D_2, D_3, D_ 4$ of Section 9.6. Describe the dependencies between the four definitions and give all possible linearisations of the corresponding partial order.
Let's remind ourselves of the environment $\Delta$.
$ (D_1) \quad x : \mathbb{Z},y : \mathbb{Z} \quad \triangleright \quad a(x,y) := x^2 +y^2 \quad : \quad \mathbb{Z} $
$ (D_2) \quad x : \mathbb{Z},y : \mathbb{Z} \mathbb{Z} \quad \triangleright \quad b(x,y) := 2·(x·y) \quad : \quad \mathbb{Z} $
$ (D_3) \quad x : \mathbb{Z},y : \mathbb{Z} \mathbb{Z} \quad \triangleright \quad c(x,y) := a(x,y)+b(x,y) \quad : \quad \mathbb{Z} $
$ (D_4) \quad x : \mathbb{Z},y : \mathbb{Z} \mathbb{Z} \quad \triangleright \quad \textit{lemma}(x,y) := c(x,y) = (x+y)^2 \quad : \quad ∗_p $
Here, $D_3$ depends on $D_1$ and also $D_2$. There is no relation between $D_1$ and $D_2$. Furthermore, $D_4$ depends only on $D_3$.
There are two possible linearisation of the partial order:
$$ D_1 \leftarrow D_2 \leftarrow D_3 \leftarrow D_4 $$
$$ D_2 \leftarrow D_1 \leftarrow D_3 \leftarrow D_4 $$
