feat(Data/Nat/Fib): formalize Lamé's theorem

[kennethgoodman]

Summary

Formalize Lamé's theorem (1844), the founding result of computational complexity theory.

Lamé's Theorem: If the Euclidean algorithm on inputs (a, b) with b ≤ a takes n + 1 division steps, then b ≥ fib(n + 1) and a ≥ fib(n + 2).

New definitions

• Nat.euclidSteps: counts the number of division steps in the Euclidean algorithm on natural number inputs.

New theorems

• Nat.fib_le_of_euclidSteps: the main Lamé bound — Fibonacci lower bound on inputs given a step count.
• Nat.euclidSteps_le_of_lt_fib: the contrapositive — step count upper bound given a Fibonacci bound on the smaller input.
• Nat.add_mod_le: helper lemma that b + a % b ≤ a when b ≤ a and 0 < b.

Proof strategy

Induction on n, tracking both elements of the pair. The key insight is that each Euclidean step replaces (a, b) with (b, a % b), and since a ≥ b + a % b (because a / b ≥ 1), the Fibonacci recurrence fib(n+3) = fib(n+2) + fib(n+1) matches the structure of the algorithm.

References

• Gabriel Lamé, Note sur la limite du nombre des divisions dans la recherche du plus grand commun diviseur entre deux nombres entiers, Comptes rendus de l'Académie des sciences, 1844.

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• builds cleanly (lake build Mathlib.Data.Nat.Fib.Lame)
• no sorry
• lines ≤ 100 characters, no trailing whitespace
• autoImplicit false
• docstrings on all public declarations

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