| import OAI.Algebra.FormulaHitting.Wronskian |
| import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas |
| import Mathlib.LinearAlgebra.LinearIndependent.Lemmas |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace NCHitting |
|
|
| open PowerSeries Submodule |
|
|
| variable {F : Type*} [Field F] [CharZero F] |
|
|
| def HasOrderBasis (V : Submodule F (PowerSeries F)) : Prop := |
| ∃ (k : ℕ) (b : Fin k → PowerSeries F) (a : Fin k → ℕ), |
| LinearIndependent F b ∧ Submodule.span F (Set.range b) = V ∧ |
| Function.Injective a ∧ |
| (∀ i, PowerSeries.coeff (a i) (b i) ≠ 0) ∧ |
| (∀ i r, r < a i → PowerSeries.coeff r (b i) = 0) |
|
|
| omit [CharZero F] in |
| theorem hasOrderBasis_bot : HasOrderBasis (⊥ : Submodule F (PowerSeries F)) := by |
| refine ⟨0, Fin.elim0, Fin.elim0, ?_, ?_, ?_, ?_, ?_⟩ |
| · exact linearIndependent_empty_type |
| · simp |
| · intro i; exact Fin.elim0 i |
| · intro i; exact Fin.elim0 i |
| · intro i; exact Fin.elim0 i |
|
|
| omit [CharZero F] in |
| theorem hasOrderBasis_of_finrank_le (N : ℕ) : |
| ∀ (V : Submodule F (PowerSeries F)) [FiniteDimensional F V], |
| Module.finrank F V ≤ N → HasOrderBasis V := by |
| classical |
| induction N with |
| | zero => |
| intro V _ hV |
| have he : V = ⊥ := Submodule.finrank_eq_zero.mp (Nat.eq_zero_of_le_zero hV) |
| rw [he] |
| exact hasOrderBasis_bot |
| | succ N ih => |
| intro V _ hV |
| by_cases he : V = ⊥ |
| · rw [he]; exact hasOrderBasis_bot |
| have hex : ∃ a : ℕ, ∃ v : PowerSeries F, v ∈ V ∧ PowerSeries.coeff a v ≠ 0 := by |
| obtain ⟨v, hv, hv0⟩ := Submodule.exists_mem_ne_zero_of_ne_bot he |
| have hc : ∃ a, PowerSeries.coeff a v ≠ 0 := by |
| by_contra h |
| push Not at h |
| apply hv0 |
| ext a |
| simpa using h a |
| obtain ⟨a, ha⟩ := hc |
| exact ⟨a, v, hv, ha⟩ |
| let a := Nat.find hex |
| obtain ⟨v, hv, hva⟩ := Nat.find_spec hex |
| have hlow : ∀ x ∈ V, ∀ r < a, PowerSeries.coeff r x = 0 := by |
| intro x hx r hr |
| by_contra hc |
| have hh := Nat.find_min' hex ⟨x, hx, hc⟩ |
| omega |
| let W : Submodule F (PowerSeries F) := V ⊓ LinearMap.ker (PowerSeries.coeff a) |
| have hWlt : W < V := by |
| refine lt_of_le_of_ne inf_le_left ?_ |
| intro h |
| have hvW : v ∈ W := h.symm ▸ hv |
| exact hva ((LinearMap.mem_ker.mp hvW.2)) |
| have hdim : Module.finrank F W ≤ N := by |
| have hlt := Submodule.finrank_lt_finrank_of_lt hWlt |
| omega |
| obtain ⟨k, b, aa, hli, hspan, hinj, hlead, hzero⟩ := ih W hdim |
| have hbW : ∀ i, b i ∈ W := by |
| intro i |
| rw [← hspan] |
| exact Submodule.subset_span ⟨i, rfl⟩ |
| have hvnot : v ∉ Submodule.span F (Set.range b) := by |
| rw [hspan] |
| intro h |
| exact hva (LinearMap.mem_ker.mp h.2) |
| have hneq : ∀ i, a ≠ aa i := by |
| intro i hi |
| exact hlead i (hi ▸ LinearMap.mem_ker.mp (hbW i).2) |
| refine ⟨k+1, Fin.cons v b, Fin.cons a aa, hli.finCons hvnot, ?_, ?_, ?_, ?_⟩ |
| · apply le_antisymm |
| · apply Submodule.span_le.mpr |
| rintro x ⟨i, rfl⟩ |
| refine Fin.cases ?_ (fun j => ?_) i |
| · simpa using hv |
| · simpa using (hbW j).1 |
| · intro x hx |
| let c : F := PowerSeries.coeff a x / PowerSeries.coeff a v |
| have hy : x - c • v ∈ W := by |
| constructor |
| · exact V.sub_mem hx (V.smul_mem c hv) |
| · apply LinearMap.mem_ker.mpr |
| simp only [map_sub, map_smul, smul_eq_mul, c] |
| rw [div_mul_cancel₀ _ hva, sub_self] |
| have hvm : v ∈ Submodule.span F (Set.range (Fin.cons v b)) := |
| Submodule.subset_span ⟨0, rfl⟩ |
| have hym : x - c • v ∈ Submodule.span F (Set.range (Fin.cons v b)) := by |
| apply (Submodule.span_mono (s := Set.range b) ?_) |
| · rwa [hspan] |
| · rintro y ⟨j, rfl⟩ |
| exact ⟨j.succ, rfl⟩ |
| have := Submodule.add_mem (Submodule.span F (Set.range (Fin.cons v b))) |
| hym (Submodule.smul_mem _ c hvm) |
| simpa using this |
| · intro i j |
| refine Fin.cases ?_ (fun i' => ?_) i |
| · refine Fin.cases ?_ (fun j' => ?_) j |
| · intro _; rfl |
| · intro hij; exact (hneq j' (by simpa using hij)).elim |
| · refine Fin.cases ?_ (fun j' => ?_) j |
| · intro hij; exact (hneq i' (by simpa using hij.symm)).elim |
| · intro hij |
| have hij' : aa i' = aa j' := by simpa using hij |
| exact congrArg Fin.succ (hinj hij') |
| · intro i |
| exact Fin.cases hva hlead i |
| · intro i |
| exact Fin.cases (hlow v hv) hzero i |
|
|
| omit [CharZero F] in |
| theorem hasOrderBasis (V : Submodule F (PowerSeries F)) [FiniteDimensional F V] : |
| HasOrderBasis V := hasOrderBasis_of_finrank_le (Module.finrank F V) V le_rfl |
|
|
| theorem wronskian_ne_zero_of_linearIndependent {k : ℕ} |
| (f : Fin k → PowerSeries F) (hf : LinearIndependent F f) : wronskian f ≠ 0 := by |
| classical |
| let := FiniteDimensional.span_of_finite F (Set.finite_range f) |
| obtain ⟨l, b, a, hb, hspan, ha, hlead, hzero⟩ := hasOrderBasis (Submodule.span F (Set.range f)) |
| have hdimf : Module.finrank F (Submodule.span F (Set.range f)) = k := by |
| simpa using finrank_span_eq_card hf |
| have hdimb : Module.finrank F (Submodule.span F (Set.range b)) = l := by |
| simpa using finrank_span_eq_card hb |
| rw [hspan, hdimf] at hdimb |
| subst l |
| have hm : ∀ j, ∃ c : Fin k → F, ∑ i : Fin k, c i • f i = b j := by |
| intro j |
| apply (Submodule.mem_span_range_iff_exists_fun F).mp |
| rw [← hspan] |
| exact Submodule.subset_span ⟨j, rfl⟩ |
| choose c hc using hm |
| have hfn : (fun j => ∑ i : Fin k, c j i • f i) = b := funext hc |
| have he := wronskian_linearCombination f (fun i j => c j i) |
| rw [hfn] at he |
| have hbn := (wronskian_order_sum b a ha hlead hzero).1 |
| intro h |
| rw [h, zero_mul] at he |
| exact hbn he |
|
|
| end NCHitting |
|
|
| end |
|
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| end OAI |
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|
