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Fix a misformalization of putnam_2022_b1 #227

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Oct 15, 2024
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Fix a misformalization of putnam_2022_b1 #227

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b377431
Fix a misformalization of putnam_2022_b1
eric-wieser Oct 7, 2024
fca6ef3
Drop a useless assumption
eric-wieser Oct 14, 2024
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18 changes: 7 additions & 11 deletions lean4/src/putnam_2022_b1.lean
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Expand Up @@ -7,14 +7,10 @@ open Polynomial
Suppose that $P(x)=a_1x+a_2x^2+\cdots+a_nx^n$ is a polynomial with integer coefficients, with $a_1$ odd. Suppose that $e^{P(x)}=b_0+b_1x+b_2x^2+\dots$ for all $x$. Prove that $b_k$ is nonzero for all $k \geq 0$.
-/
theorem putnam_2022_b1
(n : ℕ)
(P : Polynomial ℝ)
(B : Polynomial ℝ)
(npos : n ≥ 1)
(Pconst : P.coeff 0 = 0)
(Pdegree : P.degree = n)
(Pint : ∀ k : Set.Icc 1 n, P.coeff k = round (P.coeff k))
(Podd : Odd (round (P.coeff 1)))
(hB : ∀ x : ℝ, Real.exp (P.eval x) = B.eval x)
: ∀ k : ℕ, B.coeff k ≠ 0 :=
sorry
(P : Polynomial ℤ)
(b : ℕ → ℝ)
(Pconst : P.coeff 0 = 0)
(Podd : Odd (P.coeff 1))
(hB : ∀ x : ℝ, HasSum (fun i => b i * x ^ i) (Real.exp (aeval x P))) :
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I think there might be a preferred spelling of this with PowerSeries or FormalMultilinearSeries, but I couldn't work it out.

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@ocfnash, do you think HasSum or tsum is better here?

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I have a mild preference for HasSum here but either is fine.

∀ k : ℕ, b k ≠ 0 := by
sorry
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