Brunerie's 2016 PhD thesis contains the first synthetic proof in Homotopy Type Theory (HoTT) of the classical result that the fourth homotopy group of the 3-sphere is $\mathbb{Z}/2\mathbb{Z}$. The proof is one of the most impressive pieces of synthetic homotopy theory to date and uses a lot of advanced classical algebraic topology rephrased synthetically. Furthermore, Brunerie's proof is fully constructive and the main result can be reduced to the question of whether a particular ``Brunerie'' number $\beta$ can be normalized to $\pm 2$. The question of whether Brunerie's proof could be formalized in a proof assistant, either by computing this number or by formalizing the pen-and-paper proof, has since remained open. In this paper, we present a complete formalization in the Cubical Agda system, following Brunerie's pen-and-paper proof. We do this by modifying Brunerie's proof so that a key technical result, whose proof Brunerie only sketched in his thesis, can be avoided. We also present a formalization of a new and much simpler proof that $\beta$ is $\pm 2$. This formalization provides us with a sequence of simpler Brunerie numbers, one of which normalizes very quickly to $-2$ in Cubical Agda, resulting in a fully formalized computer assisted proof that $\pi_4(\mathbb{S}^3) \cong \mathbb{Z}/2\mathbb{Z}$.
翻译:2016 Brunerie 2016 年的 2016 年的 哲学博士论文 包含 经典结果中 智多基类型 { HATT (HotT) 的首个合成证明 经典结果中的第四组同质组是$\ m2\\ mathb ⁇ $。 该证明是迄今为止最令人印象深刻的合成同质同质理论之一, 并且使用了大量合成的先进古典代代代代数。 此外, 布鲁奈尔的证明完全具有建设性, 其主要结果可以降低到一个问题, 即 " Brunerie' $\ beta$( HATT) 的证明能否迅速正常化为$\ beta$( pm 2) 。 Brunerie' 的证明能否正式化, Bruse $( brunerie) 唯一的证明是C_ brentral_ brickrice.