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ae00908 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 | 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
end OAI
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