Formalizing Computational Paths and Fundamental Groups in Lean

Abstract

Computational paths treat propositional equality as explicit paths built from labelled deduction steps and rewrite rules. This view originates in work by de Queiroz and collaborators [1] and yields a weak groupoid structure for equality, together with a computational account of homotopy inspired by homotopy type theory. In this paper we present a complete mechanization of this framework in Lean 4 and show how it supports concrete homotopy theoretic computations. Our contributions are threefold. First, we formalize the theory of computational paths in Lean, including path formation, composition, inverses, and a rewrite system that identifies redundant or trivial paths. We prove that equality types with computational paths carry a weak groupoid structure in the sense of the original theory. Second, we organize this material into a reusable Lean library, ComputationalPathsLean, which exposes an interface for paths, rewrites, and loop spaces. This library allows later developments to treat computational paths as a drop-in replacement for propositional equality when reasoning about homotopical structure. Third, we apply the library to six canonical examples in algebraic topology. We give Lean proofs that the fundamental group of the circle is isomorphic to the integers, the cylinder and Mobius band also have fundamental group isomorphic to the integers (via retraction to the circle), the fundamental group of the torus is isomorphic to the product of two copies of the integers, the fundamental group of the Klein bottle is isomorphic to the semidirect product Z cross Z, and the fundamental group of the real projective plane is isomorphic to Z_2. These case studies demonstrate that the computational paths approach scales to nontrivial homotopical computations in a modern proof assistant. All the definitions and proofs described here are available in an open-source Lean 4 repository.