| import Mathlib.Data.Nat.Choose.Basic |
| import Mathlib.Tactic.Ring |
| import Lean.Elab.Tactic.Omega |
|
|
| namespace OAI |
|
|
| namespace NCHitting |
|
|
| theorem choose_twice_add {s : ℕ} (hs : 0 < s) : |
| (2*s).choose 2 + s = 2*s^2 := by |
| rw [Nat.choose_two_right] |
| have he : (2*s)*(2*s-1) = (s*(2*s-1))*2 := by ring |
| rw [he, Nat.mul_div_left _ (by decide)] |
| calc |
| s*(2*s-1)+s = s*((2*s-1)+1) := by ring |
| _ = s*(2*s) := by rw [Nat.sub_add_cancel (by omega : 1 ≤ 2*s)] |
| _ = 2*s^2 := by ring |
|
|
| theorem multiplicity_dimension_bound {n s m : ℕ} |
| (hn : 0 < n) (hs : 0 < s) (hm : m ≤ 2*s) : |
| (n-1)*m.choose 2 + m ≤ 2*n*s^2 := by |
| by_cases h1 : n = 1 |
| · subst n |
| simp only [Nat.sub_self, zero_mul, zero_add, mul_one] |
| have hsq : s ≤ s*s := Nat.le_mul_of_pos_right s hs |
| calc |
| m ≤ 2*s := hm |
| _ ≤ 2*(s*s) := Nat.mul_le_mul_left 2 hsq |
| _ = 2*s^2 := by ring |
| · have hn2 : 2 ≤ n := by omega |
| have hc : m.choose 2 ≤ (2*s).choose 2 := Nat.choose_le_choose 2 hm |
| calc |
| (n-1)*m.choose 2 + m ≤ n*((2*s).choose 2)+2*s := |
| Nat.add_le_add (Nat.mul_le_mul (Nat.sub_le _ _) hc) hm |
| _ ≤ n*((2*s).choose 2)+n*s := |
| Nat.add_le_add_left (Nat.mul_le_mul_right s hn2) _ |
| _ = n*((2*s).choose 2+s) := by ring |
| _ = n*(2*s^2) := by rw [choose_twice_add hs] |
| _ = 2*n*s^2 := by ring |
|
|
| end NCHitting |
|
|
| end OAI |
|
|
