| import Mathlib | |
| namespace OAI | |
| open scoped BigOperators | |
| namespace TriangularLipschitz | |
| lemma radial_telescoping_sum (u : ℕ → ℝ) (m : ℕ) : | |
| ∑ k ∈ Finset.range m, (u (k+1)-u k)=u m-u 0 := by | |
| induction m with | |
| | zero => simp | |
| | succ m ih => rw [Finset.sum_range_succ,ih]; ring | |
| lemma radial_square_bound (u : ℕ → ℝ) (m : ℕ) {κ g : ℝ} (hκ : 0 ≤ κ) | |
| (hstep : ∀ k < m, κ*(u (k+1)-u k)^2 ≤ g) : | |
| κ*(u m-u 0)^2 ≤ (m:ℝ)^2*g := by | |
| have hc := sq_sum_le_card_mul_sum_sq (s := Finset.range m) | |
| (f := fun k => u (k+1)-u k) | |
| rw [radial_telescoping_sum,Finset.card_range] at hc | |
| have hs : κ*(∑ k ∈ Finset.range m, (u (k+1)-u k)^2) ≤ (m:ℝ)*g := by | |
| rw [Finset.mul_sum] | |
| simpa using Finset.sum_le_sum (fun k hk => hstep k (Finset.mem_range.mp hk)) | |
| calc | |
| _ ≤ κ*((m:ℝ)*∑ k ∈ Finset.range m, (u (k+1)-u k)^2) := mul_le_mul_of_nonneg_left hc hκ | |
| _ = (m:ℝ)*(κ*∑ k ∈ Finset.range m, (u (k+1)-u k)^2) := by ring | |
| _ ≤ (m:ℝ)*((m:ℝ)*g) := mul_le_mul_of_nonneg_left hs (Nat.cast_nonneg _) | |
| _ = _ := by ring | |
| lemma two_radial_square_bound (u v : ℕ → ℝ) (m : ℕ) {κ g : ℝ} (hκ : 0 ≤ κ) | |
| (hcenter : u m=v m) | |
| (hu : ∀ k < m, κ*(u (k+1)-u k)^2 ≤ g) | |
| (hv : ∀ k < m, κ*(v (k+1)-v k)^2 ≤ g) : | |
| κ*(u 0-v 0)^2 ≤ 4*(m:ℝ)^2*g := by | |
| have h₁ := radial_square_bound u m hκ hu | |
| have h₂ := radial_square_bound v m hκ hv | |
| have h₃ := mul_nonneg hκ (sq_nonneg ((u m-u 0)+(v m-v 0))) | |
| rw [hcenter] at h₁ h₃ | |
| nlinarith | |
| end TriangularLipschitz | |
| end OAI | |