"In constructive mathematics, proof by contradiction, while not universally rejected, is treated with caution and often replaced with direct or constructive proofs."
(gemini llm answer to google query: constructive math contradiction)
"Wiles proved the modularity theorem for semistable elliptic curves, from which Fermat’s last theorem follows using proof by contradiction."
https://en.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem
So, will the Lean formalization of FLT involve translation to a direct or constructive proof? It seems not, I gather the proof will rely on classical not constructive logic.
"3. Proof by Contradiction:
The core of the formal proof involves assuming ¬Fermat_Last_Theorem and deriving a contradiction. This contradiction usually arises from building a mathematical structure (like an elliptic curve) based on the assumed solution and then demonstrating that this structure must possess contradictory properties, violating established theorems.
4. Formalizing Contradiction:
The contradiction is formalized in Lean by deriving two conflicting statements, often denoted as Q and ¬Q, within the context of the assumed ¬Fermat_Last_Theorem. Since Lean adheres to classical logic, the existence of these conflicting statements implies that the initial assumption (¬Fermat_Last_Theorem) must be false."
(gemini llm answer to google query: Lean formalization of fermat's last theorem "proof by contradiction")
"3. Proof by Contradiction: The core of the formal proof involves assuming ¬Fermat_Last_Theorem and deriving a contradiction. This contradiction usually arises from building a mathematical structure (like an elliptic curve) based on the assumed solution and then demonstrating that this structure must possess contradictory properties, violating established theorems. 4. Formalizing Contradiction: The contradiction is formalized in Lean by deriving two conflicting statements, often denoted as Q and ¬Q, within the context of the assumed ¬Fermat_Last_Theorem. Since Lean adheres to classical logic, the existence of these conflicting statements implies that the initial assumption (¬Fermat_Last_Theorem) must be false."
(gemini llm answer to google query: Lean formalization of fermat's last theorem "proof by contradiction")