Formal mathematical reasoning with Lean 4 proof verification
Mathematics forms the foundation of trustworthy AI. Using the Lean 4 theorem prover and mathlib, we ensure that every mathematical computation is formally verified and mathematically rigorous.
This eliminates the risk of subtle mathematical errors that can propagate through complex calculations, ensuring results are not just numerically accurate but provably correct.
We use Lean 4 to write mathematical proofs that computers can check. This helps verify logical soundness and catch errors.
theorem convergence_of_series (a : ℕ → ℝ) (h : ∀ n, |a n| ≤ (1/2)^n) : ∃ L, tendsto (λ n, ∑ i in range n, a i) atTop (𝓝 L) := by -- Proof verified by Lean sorry
We build on mathlib's collection of verified mathematical theorems in algebra, analysis, topology, and number theory.
We explore using AI to suggest proof strategies that Lean 4 can verify. This helps bridge mathematical intuition and formal verification.
We use symbolic manipulation to avoid numerical approximation errors when working with algebraic expressions and differential equations.
Verify that numerical methods satisfy conservation laws and boundary conditions
Prove convergence properties of optimization algorithms
Formally verify stability and robustness properties
Verify Fourier transforms and filter properties
Modern theorem prover with dependent type theory for mathematical verification
Comprehensive library of verified mathematical theorems
Symbolic mathematics for algebraic manipulation
Here's how we verify a convergence property in Lean 4:
import Mathlib.Analysis.SpecificLimits theorem geometric_series_convergence (r : ℝ) (hr : |r| < 1) : ∃ L : ℝ, tendsto (λ n, ∑ i in range n, r^i) atTop (𝓝 L) := by use (1 - r)⁻¹ -- Lean verifies this converges to 1/(1-r) apply tendsto_geometric_sum hr