| import Lean.Elab.Tactic.Omega |
| import Mathlib.Algebra.BigOperators.Fin |
| import Mathlib.Tactic.Linarith |
| import Mathlib.Tactic.Positivity |
| import Mathlib.Tactic.Ring |
|
|
| namespace OAI |
|
|
| /-! Multiplicity sequences and their arithmetic identities. -/ |
|
|
|
|
| namespace Kaplansky |
| open scoped BigOperators |
|
|
| def testMultiplicity (m R j : ℕ) (i : Fin (j+1)) : ℕ := |
| if i.val=0 then 64*m*4^(R-j) else 960*m*4^(R+2*i.val-j-2) |
|
|
| def testRank (m R j : ℕ) : ℕ := 4^j*(64*m*4^R) |
|
|
| theorem testRank_eq (m R j : ℕ) : testRank m R j = 64*m*4^(R+j) := by |
| simp only [testRank,pow_add] |
| ring |
|
|
| theorem testMultiplicity_pos {m : ℕ} (hm : 0<m) (R j : ℕ) (i : Fin (j+1)) : |
| 0<testMultiplicity m R j i := by |
| unfold testMultiplicity |
| split <;> positivity |
|
|
| theorem testMultiplicity_quarter {m R j : ℕ} (hj : j<R) (i : Fin (j+1)) : |
| 4*testMultiplicity m R (j+1) i.castSucc = testMultiplicity m R j i := by |
| unfold testMultiplicity |
| simp only [Fin.val_castSucc] |
| split_ifs with hi |
| · have he : R-j=(R-(j+1))+1 := by omega |
| rw [he,pow_succ] |
| ring |
| · have he : R+2*i.val-j-2=(R+2*i.val-(j+1)-2)+1 := by omega |
| rw [he,pow_succ] |
| ring |
|
|
| theorem testMultiplicity_fresh {m R j : ℕ} (hj : j<R) : |
| 4*testMultiplicity m R (j+1) (Fin.last (j+1)) = 15*testRank m R j := by |
| have hi : (Fin.last (j+1)).val ≠ 0 := by simp |
| rw [testMultiplicity,ite_eq_right hi,testRank_eq] |
| have he : R+j=(R+2*(Fin.last (j+1)).val-(j+1)-2)+1 := by |
| simp only [Fin.val_last] |
| omega |
| rw [he,pow_succ] |
| ring |
|
|
| theorem sum_testMultiplicity {m R j : ℕ} (hj : j≤R) : |
| ∑ i, testMultiplicity m R j i = testRank m R j := by |
| induction j with |
| | zero => simp [testMultiplicity,testRank] |
| | succ j ih => |
| have hlt : j<R := by omega |
| have hprev := ih (by omega) |
| have hq : 4*(∑ i : Fin (j+1), testMultiplicity m R (j+1) i.castSucc) = |
| testRank m R j := by |
| rw [Finset.mul_sum] |
| simpa only [testMultiplicity_quarter hlt] using hprev |
| rw [Fin.sum_univ_castSucc] |
| have hf := testMultiplicity_fresh (m := m) hlt |
| have hr : testRank m R (j+1)=4*testRank m R j := by |
| simp only [testRank,pow_succ] |
| ring |
| rw [hr] |
| omega |
|
|
| theorem testMultiplicity_divisible {m R j : ℕ} (hj : j<R) (i : Fin (j+1)) : |
| 4 ∣ testMultiplicity m R j i := ⟨_,(testMultiplicity_quarter hj i).symm⟩ |
|
|
| theorem testMultiplicity_lower_bound {m R j : ℕ} (hj : j<R) (i : Fin (j+1)) : |
| 256*m ≤ testMultiplicity m R j i := by |
| rw [← testMultiplicity_quarter hj i] |
| have h : 64*m ≤ testMultiplicity m R (j+1) i.castSucc := by |
| unfold testMultiplicity |
| split |
| · have hp : 1≤4^(R-(j+1)) := Nat.one_le_pow _ _ (by omega) |
| nlinarith |
| · have hp : 1≤4^(R+2*(i.castSucc).val-(j+1)-2) := Nat.one_le_pow _ _ (by omega) |
| nlinarith |
| omega |
|
|
| end Kaplansky |
|
|
| end OAI |
|
|
