## Section: New Results

### Formalizing Bourbaki-style mathematics

Participant : José Grimm.

Most of the work described here is inspired by the experiment of giving formal proofs in Coq of the exercises found in Bourbaki's exposition of set theory. However, some of the results go beyond what can be found in Bourbaki.

We studied order relations by proving several properties about the *length* and *width* of order
relations, for instance showing that when a set has $nm+1$ elements, the lengh or the with of any order on this set
is larger than either $n$ or $m$. We then considered similar theorems on the set of all parts of a given
set, ordered by inclusion. In particular, this gives formal proofs of results by Dilworth and Erdös and Zserkeres.

We also studied ordinal addition, which is non-commutative. Given a finite sequence of ordinals, one can compute the number of different results of the sum of these elements, depending on the order in which this sequence is taken. There is an explicit formula for this number, with a proof that we formalized.

Last, we studied a footnote from Bourbaki, that indicates that 1 is a notation for a term whose normal form has several tens of thousands of signs. We compute this size (about ${10}^{13}$ or ${10}^{60}$ depending on whether some constructs are given by axioms or by definitions) and provide statistics on the distributions of signs in the normal form.