Formalising a proof of Bézout's Lemma

In Section 8.7, we considered a well-known theorem from number theory, and we have given a mathematical proof of it in Section 8.8. We now revisit this theorem and its proof, which are reproduced below, and translate it into the formal λD-format.

A thorough inspection of what we need for the formalisation of the proof in its entirety will take up the space of a full chapter: the present one. It acts as a final exercise, showing several important aspects of λD.

In the process, we will encounter various questions and problems. We'll try to foresee some of these questions and solve them before we start the actual proof. Other problems we solve ‘on the fly’. On some occasions, we come across situations of missing foreknowledge that is either too laborious or too uninspiring to be dealt with in this book; in those cases we resort to only summarising what is lacking. Hence, we decide neither to fill every gap, nor to always supply the relevant details.

The mentioned theorem reads as follows:

'Theorem (“Bézout's Lemma”, restricted version)

Let m, n ∈ ℕ+ be coprime. Then ∃x,y∈ℤ(mx + ny = 1).’

Remark 15.1.1The lemma has been attributed to the French mathematician É. Bézout (1730–1783), although it already appeared in earlier work of others.