| import OAI.Algebra.FormulaHitting.Multiplicity |
| import OAI.Algebra.FormulaHitting.DimensionBound |
| import OAI.Algebra.FormulaHitting.SystemOutput |
| import OAI.Algebra.FormulaHitting.SeriesFaithfulness |
| import OAI.Algebra.FormulaHitting.JetMatrix |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace NCHitting |
|
|
| open Matrix PowerSeries |
|
|
| variable {F : Type*} [Field F] [CharZero F] |
|
|
| omit [CharZero F] in |
| theorem AcyclicRepresentation.scalarOutput_eq_contract {n : ℕ} |
| (R : AcyclicRepresentation F n) : |
| systemScalarOutput R.u R.v R.fundamentalMatrix = |
| R.contract R.fundamentalMatrix := by |
| simp only [systemScalarOutput, AcyclicRepresentation.contract, |
| LinearMap.coe_mk, AddHom.coe_mk, dotProduct, Matrix.mulVec, |
| Finset.sum_mul, Finset.mul_sum] |
| rw [Finset.sum_comm] |
| simp only [mul_assoc] |
|
|
| theorem Formula.series_order_lt {n : ℕ} (f : Formula F n) (hn : 0 < n) |
| (hf : f.coeff ≠ 0) : |
| f.series.order.toNat < |
| (n-1) * (Fintype.card f.representation.State).choose 2 + |
| Fintype.card f.representation.State := by |
| let R := f.representation |
| have hq : AcyclicRepresentation.denominator (F := F) n ≠ 0 := by |
| intro hz |
| have he := AcyclicRepresentation.denominator_eval_zero_ne (F := F) (n := n) |
| exact he (by rw [hz, Polynomial.eval_zero]) |
| have ho : systemScalarOutput R.u R.v R.fundamentalMatrix = f.series := |
| R.scalarOutput_eq_contract.trans f.system_output_eq_series |
| have hbound := system_multiplicity n hn |
| (AcyclicRepresentation.denominator (F := F) n) R.systemNumerator R.u R.v |
| R.fundamentalMatrix (AcyclicRepresentation.denominator_degree (F := F) (n := n)) |
| hq R.systemNumerator_degree |
| (R.fundamentalMatrix_isUnit f.representation_card_pos) |
| R.fundamentalMatrix_polynomial_system |
| (by rw [ho]; exact f.series_ne_zero hf) |
| rwa [ho] at hbound |
| |
| theorem Formula.eval_generator_one_ne_zero (f : Formula F 1) |
| (hs : f.size ≤ 1) (hf : f.coeff ≠ 0) : |
| f.eval (Matrix.scalar (Fin (dimension 1 1))) |
| (fun i r q => (generator 1 1 i r q : F)) ≠ 0 := by |
| let r : Fin (dimension 1 1) := ⟨0, by decide⟩ |
| rcases (Formula.size_le_one_iff f).mp hs with ⟨a, rfl⟩ | ⟨i, rfl⟩ |
| · intro hz |
| have ha : a = 0 := by |
| have he := congrFun (congrFun hz r) r |
| simpa [Formula.eval, Matrix.scalar_apply] using he |
| apply hf |
| funext w |
| simp [Formula.coeff, ha] |
| · intro hz |
| have he := congrFun (congrFun hz r) r |
| simp [Formula.eval, generator_one] at he |
|
|
| theorem Formula.eval_generator_ne_zero {n s : ℕ} (f : Formula F n) |
| (hn : 0 < n) (hs : 0 < s) (hsize : f.size ≤ s) (hf : f.coeff ≠ 0) : |
| f.eval (Matrix.scalar (Fin (dimension n s))) |
| (fun i r q => (generator n s i r q : F)) ≠ 0 := by |
| by_cases he : n = 1 ∧ s = 1 |
| · rcases he with ⟨rfl, rfl⟩ |
| exact f.eval_generator_one_ne_zero hsize hf |
| · have hns : 2 ≤ n*s := by |
| have hnle : n ≤ n*s := by simpa using Nat.mul_le_mul_left n hs |
| have hsle : s ≤ n*s := by simpa using Nat.mul_le_mul_right s hn |
| omega |
| have hm : Fintype.card f.representation.State ≤ 2*s := |
| f.representation_card_le.trans (Nat.mul_le_mul_left 2 hsize) |
| have hbound : f.series.order.toNat < dimension n s := by |
| rw [dimension_eq hns] |
| exact (f.series_order_lt hn hf).trans_le (multiplicity_dimension_bound hn hs hm) |
| apply f.eval_generator_ne_zero_of_jet_series_ne_zero hns |
| apply jet_ne_zero_of_order_lt |
| rw [← PowerSeries.coe_toNat_order (f.series_ne_zero hf), ENat.natCast_lt_natCast] |
| exact hbound |
|
|
| theorem universal_hitting (n s : ℕ) (hn : 1 ≤ n) (hs : 1 ≤ s) |
| (F : Type*) [Field F] [CharZero F] (f : Formula F n) |
| (hsize : f.size ≤ s) (hf : ∃ w, f.coeff w ≠ 0) : |
| f.eval (Matrix.scalar (Fin (dimension n s))) |
| (fun i r q => (generator n s i r q : F)) ≠ 0 := by |
| apply f.eval_generator_ne_zero hn hs hsize |
| intro hz |
| obtain ⟨w, hw⟩ := hf |
| exact hw (congrFun hz w) |
|
|
| end NCHitting |
|
|
| end |
|
|
| end OAI |
|
|
