## TL;DR
AI is learning to prove mathematical theorems -- from International Mathematical Olympiad geometry problems (AlphaGeometry) to formal proofs in Lean 4 (AlphaProof). The combination of LLM reasoning, reinforcement learning, and formal verification is creating AI systems that not only solve problems but produce human-verifiable, logically rigorous proofs.

## Core Explanation
Theorem proving approaches: (1) Natural language -- LLMs (Minerva, GPT-f) generate proof text in human-readable form. Flexible but unverifiable; (2) Formal -- proofs written in interactive theorem provers (Lean 4, Coq, Isabelle). Every step is machine-verified. The proof is a mathematical guarantee, not a probabilistic guess; (3) Neuro-symbolic -- AlphaGeometry combines a neural language model (generating auxiliary constructions) with a symbolic deduction engine (Deductive Database) that exhaustively applies geometry rules. The neural component suggests creative steps; the symbolic component verifies them. Key insight: formal mathematics is amenable to RL because proof states are fully observable and reward (proof complete or not) is clear. AlphaProof uses RL to train a policy that selects proof tactics given the current proof state.

## Detailed Analysis
AlphaGeometry (Nature 2024): trained on 100M synthetic geometry theorems generated by randomly constructing points, lines, and circles with random intersections. The Deductive Database computes all derivable facts. Without the language model (pure symbolic): solves 10/30 IMO problems. With it: 25/30. AlphaProof (2025): three training stages -- pretrain on formal math corpus (Mathlib), fine-tune via supervised learning on human-written Lean proofs, RL via self-play where the model generates proof attempts and receives reward for successful proofs. Nature 2025 report notes it proved IMO-level problems spanning algebra, number theory, and combinatorics. LeanDojo, LLMLean, and COPRA provide the tooling. Key open problem: generating novel mathematical knowledge -- can AI discover new lemmas, definitions, or conjectures? Math libraries (Mathlib, 130K+ theorems) provide the training corpus.