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7d9acbd | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 | 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
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