diff --git a/lean4/src/putnam_1994_b3.lean b/lean4/src/putnam_1994_b3.lean index ec69120f..be9f6068 100644 --- a/lean4/src/putnam_1994_b3.lean +++ b/lean4/src/putnam_1994_b3.lean @@ -8,9 +8,7 @@ abbrev putnam_1994_b3_solution : Set ℝ := sorry /-- Find the set of all real numbers $k$ with the following property: For any positive, differentiable function $f$ that satisfies $f'(x)>f(x)$ for all $x$, there is some number $N$ such that $f(x)>e^{kx}$ for all $x>N$. -/ -theorem putnam_1994_b3 -(k : ℝ) -(allfexN : Prop) -(hallfexN : allfexN = ∀ f : ℝ → ℝ, (f > 0 ∧ Differentiable ℝ f ∧ ∀ x : ℝ, deriv f x > f x) → (∃ N : ℝ, ∀ x > N, f x > Real.exp (k * x))) -: allfexN ↔ k ∈ putnam_1994_b3_solution := -sorry +theorem putnam_1994_b3 : + {k | ∀ f (hf : (∀ x, 0 < f x ∧ f x < deriv f x) ∧ Differentiable ℝ f), + ∃ N, ∀ x > N, Real.exp (k * x) < f x} = putnam_1994_b3_solution := + sorry