###### tags: `TAS`
@mamy @theOnlyAlex @wlin @nlejeune @gnouf

# TAS - TD 10

## Question 1

1. $[0, 4] \leq^\# [1, 3] = [0, 3], [1, 3]$
2. $[1, 3] \leq^\# [0, 4] = [1, 3], [1, 4]$
3. $[0, 2] \leq^\# [3, 5] = [0, 2], [3, 5]$
4. $[3, 5] \leq^\# [0, 2] = \bot, \bot$

## Question 2

$$
X \leq_g Y = \{ x \in X\ |\ \exists y \in Y; x \leq y \} \\
X \leq_d Y = \{ y \in Y\ |\ \exists y \in Y; x \leq y \} \\
$$
$$
\mathbb{C} [[e_1 \leq e_2]] R = \{ \rho \in R\ |\ \exists (v_1, v_2) \in \mathbb{Z}^2, v_1 \in \mathbb{E} [[e]] \rho, v_2 \in \mathbb{E} [[e_2]] \rho, v_1 \leq v_2 \} \\
= \{ \rho \in R\ |\ \exists (v_1, v_2) \in \mathbb{Z}^2, v_1 \in \mathbb{E} [[e]] \rho \leq_g \mathbb{E} [[e_2]] \rho, v_2 \in \mathbb{E} [[e_2]] \rho \leq_d \mathbb{E} [[e_2]] \rho \}
$$

## Question 3

$$
[a, b] \leq^\#_g [c, d] = \alpha (\gamma ([a, b]) \leq_g \gamma([c, d])) \\
= \alpha (\{ x \in \gamma ([a, b])\ |\ \exists y \in \gamma ([c, d]), x \leq y \}) \\
= \alpha (\{ x\ |\ a \leq x \leq b, \exists y, c \leq y \leq d\ et\ x \leq y \}) \\
= \alpha (\{ x\ |\ a \leq x \leq b\ et\ x \leq d \}) \\
= \Bigg\{ \begin{matrix} \bot\ si\ a \geq d \\ [a, min(b, d)]\ si\ a \leq d \end{matrix}
$$

## Question 4

$$
[a, b] \leq^\# [c ,d] = \Bigg\{ \begin{matrix} [a, min(b, d)], [max(a, c), d]\ si\ a \leq d \\ \bot, \bot\ sinon \end{matrix}
$$

## Question 5

$$
x \leq y \Longleftrightarrow x + 1 \leq y
$$