In differential topology and geometry, the h-principle is a property enjoyed by certain construction problems. Roughly speaking, it states that the only obstructions to the existence of a solution come from algebraic topology. We describe a formalisation in Lean of the local h-principle for first-order, open, ample partial differential relations. This is a significant result in differential topology, originally proven by Gromov in 1973 as part of his sweeping effort which greatly generalised many previous flexibility results in topology and geometry. In particular it reproves Smale's celebrated sphere eversion theorem, a visually striking and counter-intuitive construction. Our formalisation uses Theilli\`ere's implementation of convex integration from 2018. This paper is the first part of the sphere eversion project, aiming to formalise the global version of the h-principle for open and ample first order differential relations, for maps between smooth manifolds. Our current local version for vector spaces is the main ingredient of this proof, and is sufficient to prove the titular corollary of the project. From a broader perspective, the goal of this project is to show that one can formalise advanced mathematics with a strongly geometric flavour and not only algebraically-flavoured
翻译:在不同的地形学和几何学中,h原则是某些建筑问题所享受的一种属性。粗略地说,它指出,阻碍解决方案存在的唯一障碍来自代数表层学。我们描述在利安将当地h原则正规化为一阶、开放和充分部分差异关系。这是不同地形学的一个重大结果,格罗莫夫最初于1973年证明了这一点,这是他大规模努力的一部分,极大地概括了许多先前在地形学和几何学方面的灵活性结果。特别是它反映了Smale所庆祝的球体在理论中反复出现,一种视觉惊人和反直观的构造。我们正规化利用了Theilli ⁇ ere从2018年起实施convex融合。本文是这个领域常态项目的第一个部分,旨在正式确定开放和充分第一阶差异关系原则的全球版本,用于平坦的马路图。我们目前对病媒空间的本地版本是这一证据的主要成分,足以证明该项目的顶端值。从更广的角度来看,这个工程的目标不是以高压的数学形式显示一个进步。