In differential topology and geometry, the h-principle is a property exploited in certain construction problems. Roughly speaking, he states that the only obstacle to the existence of a solution lies in algebraic topology.

We present a lean formalization of the local h-principle for first-order, open, and sufficient partial differential relations. This is an important result in differential topology, which he first proved by Gromov in 1973. This was done as part of his radical effort which greatly generalized many previous results of flexibility in topology and geometry. In particular, we denounce Smeer’s famous sphere reversal theorem, a visually striking and counterintuitive construction. Our formalization uses his implementation of Theilli\`ere’s convex integral from 2018.

This paper is the first part of the sphere inversion project and aims to formalize a global version of the h-principle of open and sufficient first-order differential relations for maps between smooth manifolds. The current local version of the vector space is the main component of this proof, and is sufficient to prove the project’s nominal consequences. From a broader perspective, the goal of this project is to show that advanced mathematics can be formalized not only with an algebraic flavor, but with a strong geometric flavor.

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