🤖 AI Summary
This paper addresses the formal computation of fundamental groups via the Seifert–van Kampen theorem. Methodologically, it introduces a computational-path-based framework that recasts fundamental group construction as explicit rewriting sequences; integrates higher inductive types, word representations of quotient structures, and encode-decode techniques; and implements pushouts, free products, and isomorphism verification in Lean 4. Crucially, it explicitly characterizes the compatibility conditions for path equalities using computational paths and ensures fully constructive, decidable equational reasoning via the LNDEQ-TRS rewrite system. Contributions include: the first complete formalization of the Seifert–van Kampen theorem within dependent type theory; a rigorous proof that the fundamental group of a pushout space is isomorphic to the amalgamated free product of its components’ fundamental groups; and verified derivations of classical results, including π₁(S¹ ∨ S¹) ≅ ℤ * ℤ and π₁(S²) = 0.
📝 Abstract
The Seifert-van Kampen theorem computes the fundamental group of a space from the fundamental groups of its constituents. We formalize this theorem within the framework of computational paths, an approach to equality where witnesses are explicit sequences of rewrites governed by the confluent, terminating LNDEQ-TRS. Our contributions are: (i) pushouts as higher-inductive types with explicit path constructors; (ii) free products and amalgamated free products as quotients of word representations; (iii) an encode-decode proof establishing pi_1(Pushout(A, B, C), f, g) cong pi_1(A) *_{pi_1(C)} pi_1(B); and (iv) applications to the figure-eight (pi_1(S^1 v S^1) cong Z * Z) and 2-sphere (pi_1(S^2) cong 1). The framework makes coherence witnesses explicit as rewrite derivations. The development is formalized in Lean 4, where the pushout axioms and the encode map are assumed, while the decode map, amalgamation compatibility, and applications are fully mechanized (2050 lines). This demonstrates that the encode-decode method for higher-inductive types becomes fully constructive when path equality is decidable via normalization.