| import Mathlib.Algebra.Polynomial.Derivative |
| import Mathlib.Algebra.Polynomial.BigOperators |
| import Mathlib.Data.Matrix.Basic |
| import Lean.Elab.Tactic.Omega |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace NCHitting |
|
|
| open Polynomial |
|
|
| variable {F : Type*} [Field F] |
|
|
| def numeratorRow {I : Type*} [Fintype I] (q : F[X]) (P : Matrix I I F[X]) |
| (u : I → F) : ℕ → I → F[X] |
| | 0, l => C (u l) |
| | j + 1, l => q * (numeratorRow q P u j l).derivative - |
| C (j : F) * q.derivative * numeratorRow q P u j l + |
| ∑ t : I, numeratorRow q P u j t * P t l |
|
|
| private theorem degree_q_derivative {q p : F[X]} {n b : ℕ} |
| (hn : 0 < n) (hq : q.natDegree ≤ n) (hp : p.natDegree ≤ b) : |
| (q * p.derivative).natDegree ≤ b + (n-1) := by |
| by_cases hzero : p.natDegree = 0 |
| · simp [Polynomial.derivative_of_natDegree_zero hzero] |
| · have hb : 0 < b := by omega |
| have hd := p.natDegree_derivative_le |
| have hm : (q * p.derivative).natDegree ≤ q.natDegree + p.derivative.natDegree := |
| Polynomial.natDegree_mul_le |
| omega |
|
|
| theorem numeratorRow_degree {I : Type*} [Fintype I] {n : ℕ} (hn : 0 < n) |
| (q : F[X]) (P : Matrix I I F[X]) (u : I → F) |
| (hq : q.natDegree ≤ n) (hP : ∀ t l, (P t l).natDegree ≤ n-1) |
| (j : ℕ) (l : I) : |
| (numeratorRow q P u j l).natDegree ≤ j * (n-1) := by |
| induction j generalizing l with |
| | zero => simp [numeratorRow] |
| | succ j ih => |
| have ihall : ∀ l, (numeratorRow q P u j l).natDegree ≤ j * (n-1) := by |
| intro l' |
| exact ih l' |
| have ha := degree_q_derivative hn hq (ihall l) |
| have hb : (C (j : F) * q.derivative * numeratorRow q P u j l).natDegree ≤ |
| j * (n-1) + (n-1) := by |
| have hq' : q.derivative.natDegree ≤ n-1 := |
| q.natDegree_derivative_le.trans (Nat.sub_le_sub_right hq 1) |
| have hC := Polynomial.natDegree_C_mul_le (j : F) q.derivative |
| have hmul : (C (j : F) * q.derivative * numeratorRow q P u j l).natDegree ≤ |
| (C (j : F) * q.derivative).natDegree + (numeratorRow q P u j l).natDegree := |
| Polynomial.natDegree_mul_le |
| have hp := ihall l |
| omega |
| have hc : (∑ t : I, numeratorRow q P u j t * P t l).natDegree ≤ |
| j * (n-1) + (n-1) := by |
| apply Polynomial.natDegree_sum_le_of_forall_le |
| intro t _ |
| exact Polynomial.natDegree_mul_le.trans (Nat.add_le_add (ihall t) (hP t l)) |
| rw [numeratorRow, Nat.add_mul, one_mul] |
| apply Polynomial.natDegree_add_le_of_degree_le _ hc |
| exact (Polynomial.natDegree_sub_le _ _).trans (max_le ha hb) |
|
|
| end NCHitting |
|
|
| end |
|
|
| end OAI |
|
|
