| import Mathlib | |
| namespace OAI | |
| open scoped BigOperators | |
| namespace PiExponent.FiniteSpacing | |
| theorem lower_bound_of_spacing {N : ℕ} (x : Fin N → ℝ) {d : ℝ} | |
| (hmin : ∀ i, d ≤ x i) | |
| (hsep : ∀ i j, i < j → d ≤ x j - x i) : | |
| ∀ i, ((i.val : ℝ) + 1) * d ≤ x i := by | |
| have aux : ∀ k (hk : k < N), ((k : ℝ) + 1) * d ≤ x ⟨k, hk⟩ := by | |
| intro k | |
| induction k with | |
| | zero => | |
| intro hk | |
| simpa using hmin ⟨0, hk⟩ | |
| | succ k ih => | |
| intro hk | |
| have hk' : k < N := Nat.lt_trans (Nat.lt_succ_self k) hk | |
| have hp := ih hk' | |
| have hs := hsep ⟨k, hk'⟩ ⟨k + 1, hk⟩ (by simp) | |
| push_cast | |
| linarith | |
| intro i | |
| exact aux i.val i.isLt | |
| theorem sum_inv_succ_sq_le_two_sub (N : ℕ) : | |
| (∑ k ∈ Finset.range N, 1 / ((k : ℝ) + 1) ^ 2) ≤ | |
| 2 - 2 / ((N : ℝ) + 1) := by | |
| induction N with | |
| | zero => norm_num | |
| | succ N ih => | |
| rw [Finset.sum_range_succ] | |
| have hN : (0 : ℝ) ≤ N := Nat.cast_nonneg _ | |
| have hN1 : (0 : ℝ) < (N : ℝ) + 1 := by positivity | |
| have hN2 : (0 : ℝ) < (N : ℝ) + 2 := by positivity | |
| have hterm : 1 / ((N : ℝ) + 1) ^ 2 ≤ | |
| 2 / ((N : ℝ) + 1) - 2 / ((N : ℝ) + 2) := by | |
| field_simp | |
| nlinarith | |
| push_cast | |
| convert add_le_add ih hterm using 1 | |
| ring | |
| theorem sum_inv_succ_sq_le_two (N : ℕ) : | |
| (∑ k ∈ Finset.range N, 1 / ((k : ℝ) + 1) ^ 2) ≤ 2 := by | |
| have h := sum_inv_succ_sq_le_two_sub N | |
| have hnonneg : 0 ≤ 2 / ((N : ℝ) + 1) := by positivity | |
| linarith | |
| theorem sum_inv_sq_le_two_div_sq_of_spacing {N : ℕ} | |
| (x : Fin N → ℝ) {d : ℝ} (hd : 0 < d) | |
| (hmin : ∀ i, d ≤ x i) | |
| (hsep : ∀ i j, i < j → d ≤ x j - x i) : | |
| (∑ i, 1 / (x i) ^ 2) ≤ 2 / d ^ 2 := by | |
| have hlower := lower_bound_of_spacing x hmin hsep | |
| calc | |
| (∑ i, 1 / (x i) ^ 2) ≤ | |
| ∑ i : Fin N, 1 / ((((i.val : ℝ) + 1) * d) ^ 2) := by | |
| apply Finset.sum_le_sum | |
| intro i _ | |
| have hpos : 0 < ((i.val : ℝ) + 1) * d := by positivity | |
| exact one_div_le_one_div_of_le (sq_pos_of_pos hpos) | |
| (pow_le_pow_left₀ hpos.le (hlower i) 2) | |
| _ = (1 / d ^ 2) * ∑ i : Fin N, 1 / ((i.val : ℝ) + 1) ^ 2 := by | |
| rw [Finset.mul_sum] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| rw [mul_pow, one_div_mul_one_div] | |
| ring | |
| _ ≤ (1 / d ^ 2) * 2 := by | |
| apply mul_le_mul_of_nonneg_left _ (by positivity) | |
| rw [Fin.sum_univ_eq_sum_range (fun k : ℕ => 1 / ((k : ℝ) + 1) ^ 2)] | |
| exact sum_inv_succ_sq_le_two N | |
| _ = 2 / d ^ 2 := by ring | |
| theorem sum_inv_sq_le_two_div_sq (s : Finset ℝ) {d : ℝ} (hd : 0 < d) | |
| (hmin : ∀ x ∈ s, d ≤ x) | |
| (hsep : ∀ x ∈ s, ∀ y ∈ s, x ≠ y → d ≤ |x - y|) : | |
| (∑ x ∈ s, 1 / x ^ 2) ≤ 2 / d ^ 2 := by | |
| classical | |
| let e := s.orderEmbOfFin rfl | |
| have hmem : ∀ i, e i ∈ s := fun i => s.orderEmbOfFin_mem rfl i | |
| have hstep : ∀ i j, i < j → d ≤ e j - e i := by | |
| intro i j hij | |
| have hij' : e i < e j := e.strictMono hij | |
| have h := hsep (e j) (hmem j) (e i) (hmem i) (ne_of_gt hij') | |
| rwa [abs_of_pos (sub_pos.mpr hij')] at h | |
| have hbound := sum_inv_sq_le_two_div_sq_of_spacing e hd | |
| (fun i => hmin (e i) (hmem i)) hstep | |
| have hsum : (∑ i : Fin s.card, 1 / (e i) ^ 2) = ∑ x ∈ s, 1 / x ^ 2 := by | |
| have himage : Finset.univ.image e = s := s.image_orderEmbOfFin_univ rfl | |
| calc | |
| _ = ∑ y ∈ Finset.univ.image e, 1 / y ^ 2 := | |
| (Finset.sum_image e.injective.injOn).symm | |
| _ = _ := congrArg (fun t : Finset ℝ => ∑ y ∈ t, 1 / y ^ 2) himage | |
| rwa [hsum] at hbound | |
| theorem sum_inv_sq_le_two_div_sq_indexed {ι : Type*} | |
| (rows : Finset ι) (x : ι → ℝ) {d : ℝ} (hd : 0 < d) | |
| (hmin : ∀ i ∈ rows, d ≤ x i) | |
| (hsep : ∀ i ∈ rows, ∀ j ∈ rows, i ≠ j → d ≤ |x i - x j|) : | |
| (∑ i ∈ rows, 1 / (x i) ^ 2) ≤ 2 / d ^ 2 := by | |
| classical | |
| have hinj : Set.InjOn x (rows : Set ι) := by | |
| intro i hi j hj h | |
| by_contra hij | |
| have hs := hsep i hi j hj hij | |
| rw [h, sub_self, abs_zero] at hs | |
| exact (not_le_of_gt hd) hs | |
| have h := sum_inv_sq_le_two_div_sq (rows.image x) hd | |
| (by intro a ha; obtain ⟨i, hi, rfl⟩ := Finset.mem_image.mp ha; exact hmin i hi) | |
| (by | |
| intro a ha b hb hab | |
| obtain ⟨i, hi, rfl⟩ := Finset.mem_image.mp ha | |
| obtain ⟨j, hj, rfl⟩ := Finset.mem_image.mp hb | |
| exact hsep i hi j hj (fun hij => hab (congrArg x hij))) | |
| rwa [Finset.sum_image hinj] at h | |
| end PiExponent.FiniteSpacing | |
| end OAI | |