Exercise 2: Show that the union of C1 and C2 is consistent.

[dag7dev]

Exercise 2: Let C1 and C2 be two consistent cuts. Show that the union of C1 and C2 is consistent.

[rotiroti]

The previous solutions contain some errors. Following is an updated version of it.

We aim to prove the consistency of the union, $C_1 \cup C_2$, and this requires us to show that:

$$(e' \in C_1 \cup C_2) \land (e \to e') \Rightarrow e \in C_1 \cup C_2.$$

Let's break down the proof step by step:

• $(e' \in C_1 \cup C_2) \land (e \to e')$, as per the precondition for consistency.
• $(e' \in C_1 \lor e' \in C_2) \land (e \to e')$, by the definition of set union.
• $(e' \in C_1 \land e \to e') \lor (e' \in C_2 \land e \to e')$, since $C_1$ and $C_2$ are consistent cuts.
• This implies that either $(e \in C_1) \lor (e \in C_2)$ or both.
• By the definition of set union, we can conclude that $e \in C_1 \cup C_2$.