local_to_global_hasse_principle

The local-global principle determines rational solvability by verifying a polynomial system has solutions over ℝ and over ℚ_p for all primes p, then lifting local data to a global rational witness using Hasse-Minkowski theorem applications.

You apply Hensel lifting in Lean 4 by using provided formalization templates to encode local checks over ℚ_p, lifting approximate solutions to higher p-adic precision, and constructing a global rational solution from the local data.

The Hasse principle applies directly to quadratic forms over ℚ via the Hasse-Minkowski theorem, but analogous local-global principles and descent methods can extend the reasoning to related algebraic structures where local data governs global solvability.

You can use Lean 4 Mathlib to formalize local-global reasoning by encoding real and p-adic solvability checks, applying Hensel lifting steps, and constructing a global solution witness within Lean 4 templates.

The best way to verify real and p-adic solvability is to check the polynomial system over ℝ and ℚ_p for all primes p individually, then apply the Hasse-Minkowski theorem to deduce and construct a global rational witness.