| import Mathlib | |
| namespace OAI | |
| namespace Problem355.ZeroShell | |
| open scoped BigOperators | |
| theorem affine_shell_sum_le {α : Type*} (S : Finset α) | |
| (f g len J : α → ℝ) (R I A C : ℝ) | |
| (hR : 0 < R) (hI : 0 < I) (hA : 0 ≤ A) | |
| (hg : ∀ x ∈ S, 0 < g x) (hlen : ∀ x ∈ S, R ≤ len x) | |
| (hJ : ∀ x ∈ S, J x = I * len x / g x) | |
| (hf : ∀ x ∈ S, f x ≤ A / (J x) ^ 3) | |
| (hmoment : ∑ x ∈ S, (g x) ^ 3 ≤ C * R ^ 3 * I) : | |
| ∑ x ∈ S, f x ≤ A * C / I ^ 2 := by | |
| have hpoint : ∀ x ∈ S, f x ≤ (A / (I ^ 3 * R ^ 3)) * (g x) ^ 3 := by | |
| intro x hx | |
| have hgx := hg x hx | |
| have hlx : 0 < len x := lt_of_lt_of_le hR (hlen x hx) | |
| have hlen3 : R ^ 3 ≤ (len x) ^ 3 := pow_le_pow_left₀ hR.le (hlen x hx) 3 | |
| have hden : I ^ 3 * R ^ 3 ≤ I ^ 3 * (len x) ^ 3 := | |
| mul_le_mul_of_nonneg_left hlen3 (pow_nonneg hI.le 3) | |
| calc | |
| f x ≤ A / (J x) ^ 3 := hf x hx | |
| _ = (A * (g x) ^ 3) / (I ^ 3 * (len x) ^ 3) := by | |
| rw [hJ x hx] | |
| field_simp | |
| _ ≤ (A * (g x) ^ 3) / (I ^ 3 * R ^ 3) := | |
| div_le_div_of_nonneg_left (by positivity) (by positivity) hden | |
| _ = (A / (I ^ 3 * R ^ 3)) * (g x) ^ 3 := by ring | |
| calc | |
| (∑ x ∈ S, f x) ≤ ∑ x ∈ S, (A / (I ^ 3 * R ^ 3)) * (g x) ^ 3 := | |
| Finset.sum_le_sum hpoint | |
| _ = (A / (I ^ 3 * R ^ 3)) * ∑ x ∈ S, (g x) ^ 3 := by rw [Finset.mul_sum] | |
| _ ≤ (A / (I ^ 3 * R ^ 3)) * (C * R ^ 3 * I) := | |
| mul_le_mul_of_nonneg_left hmoment (by positivity) | |
| _ = A * C / I ^ 2 := by | |
| field_simp | |
| theorem uniform_shell_sum_le {α : Type*} (S : Finset α) | |
| (f g len J : α → ℝ) (R I A C H : ℝ) | |
| (hR : 0 < R) (hI : 0 < I) (hA : 0 ≤ A) (hH : 0 ≤ H) | |
| (hg : ∀ x ∈ S, 0 < g x) (hlen : ∀ x ∈ S, R ≤ len x) | |
| (hJ : ∀ x ∈ S, J x = I * len x / g x) | |
| (hf : ∀ x ∈ S, f x ≤ A / (J x) ^ 3) | |
| (hdivisor : ∀ x ∈ S, (g x) ^ 3 ≤ H * I) | |
| (hcard : (S.card : ℝ) ≤ C * R ^ 3) : | |
| ∑ x ∈ S, f x ≤ A * (C * H) / I ^ 2 := by | |
| apply affine_shell_sum_le S f g len J R I A (C * H) hR hI hA hg hlen hJ hf | |
| calc | |
| (∑ x ∈ S, (g x) ^ 3) ≤ ∑ _x ∈ S, H * I := Finset.sum_le_sum hdivisor | |
| _ = (S.card : ℝ) * (H * I) := by simp [nsmul_eq_mul] | |
| _ ≤ (C * R ^ 3) * (H * I) := mul_le_mul_of_nonneg_right hcard (mul_nonneg hH hI.le) | |
| _ = C * H * R ^ 3 * I := by ring | |
| end Problem355.ZeroShell | |
| end OAI | |