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cfa2fd2 | 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 | import Mathlib
import OAI.Algebra.FrobeniusObstruction.Obstruction
import OAI.Algebra.AlgebraicObstruction.SeriesJets
namespace OAI
noncomputable section
open scoped BigOperators
namespace BoundaryOnly.FormalObstruction.FormalCorrection
open MvPowerSeries
variable {R σ τ ι : Type*} [CommRing R]
theorem Jet.sum {n : ℕ} (s : Finset ι) {f g : ι → MvPowerSeries σ R}
(h : ∀ i ∈ s, Jet n (f i) (g i)) : Jet n (∑ i ∈ s, f i) (∑ i ∈ s, g i) := by
apply Jet.iff_coeff.mpr
intro e he
simp only [map_sum]
exact Finset.sum_congr rfl fun i hi => Jet.iff_coeff.mp (h i hi) e he
theorem positively_weighted_triple {n : ℕ} {f f' h h' g g' : MvPowerSeries σ R}
(hf : Jet n f f') (hg : Jet n g g') (h0 : Vanishes 1 h) (h0' : Vanishes 1 h')
(hh : Jet (n+1) h h') : Jet (n+1) (f*h*g) (f'*h'*g') := by
change Vanishes (n+1) (f*h*g - f'*h'*g')
have heq : f*h*g - f'*h'*g' =
(f-f')*h*g + f'*(h-h')*g + f'*h'*(g-g') := by ring
rw [heq]
apply Vanishes.add
· exact ((Vanishes.mul hf h0).mul_right g).add
((Vanishes.mul_left hh f').mul_right g)
· simpa only [Nat.add_comm, mul_assoc] using (h0'.mul hg).mul_left f'
variable [Fintype ι]
def correctionArguments (x : τ → MvPowerSeries σ R) (v : ι → MvPowerSeries σ R)
(B : Matrix ι ι (MvPowerSeries σ R)) : τ ⊕ ι → MvPowerSeries σ R
| .inl t => x t
| .inr i => ∑ j, B i j * v j
def remainderValue (H : Matrix ι ι (MvPowerSeries (τ ⊕ ι) R))
(x : τ → MvPowerSeries σ R) (v : ι → MvPowerSeries σ R)
(B : Matrix ι ι (MvPowerSeries σ R)) : Matrix ι ι (MvPowerSeries σ R) :=
fun i j => subst (correctionArguments x v B) (H i j)
def correctionOperator (b : Matrix ι ι (MvPowerSeries σ R))
(H : Matrix ι ι (MvPowerSeries (τ ⊕ ι) R))
(x : τ → MvPowerSeries σ R) (v : ι → MvPowerSeries σ R)
(B : Matrix ι ι (MvPowerSeries σ R)) : Matrix ι ι (MvPowerSeries σ R) :=
-b - B.transpose * remainderValue H x v B * B
theorem correctionArguments_centered (x : τ → MvPowerSeries σ R)
(v : ι → MvPowerSeries σ R) (hx : ∀ t, constantCoeff (x t) = 0)
(hv : ∀ i, Vanishes 2 (v i)) (B : Matrix ι ι (MvPowerSeries σ R)) :
∀ z, constantCoeff (correctionArguments x v B z) = 0 := by
intro z
cases z with
| inl t => exact hx t
| inr i =>
apply Vanishes.one_iff.mp
exact (Vanishes.sum Finset.univ (fun j => B i j * v j)
(fun j _ => (hv j).mul_left (B i j))).mono (by omega)
theorem correctionArguments_jet (x : τ → MvPowerSeries σ R)
(v : ι → MvPowerSeries σ R) (hv : ∀ i, Vanishes 2 (v i))
{n : ℕ} {B B' : Matrix ι ι (MvPowerSeries σ R)}
(hB : ∀ i j, Jet n (B i j) (B' i j)) :
∀ z, Jet (n+2) (correctionArguments x v B z) (correctionArguments x v B' z) := by
intro z
cases z with
| inl t => exact Jet.refl _ _
| inr i =>
apply Jet.sum
intro j _
change Vanishes (n+2) (B i j * v j - B' i j * v j)
rw [← sub_mul]
exact (hB i j).mul (hv j)
variable [Finite σ] [Finite τ]
omit [Finite σ] in
theorem remainderValue_positive (H : Matrix ι ι (MvPowerSeries (τ ⊕ ι) R))
(x : τ → MvPowerSeries σ R) (v : ι → MvPowerSeries σ R)
(hH : ∀ i j, Vanishes 1 (H i j)) (hx : ∀ t, constantCoeff (x t) = 0)
(hv : ∀ i, Vanishes 2 (v i)) (B : Matrix ι ι (MvPowerSeries σ R)) (i j : ι) :
Vanishes 1 (remainderValue H x v B i j) := by
apply Vanishes.one_iff.mpr
have ha := correctionArguments_centered x v hx hv B
exact constantCoeff_subst_eq_zero (hasSubst_of_constantCoeff_zero ha) ha
(Vanishes.one_iff.mp (hH i j))
theorem remainderValue_jet (H : Matrix ι ι (MvPowerSeries (τ ⊕ ι) R))
(x : τ → MvPowerSeries σ R) (v : ι → MvPowerSeries σ R)
(hx : ∀ t, constantCoeff (x t) = 0) (hv : ∀ i, Vanishes 2 (v i))
{n : ℕ} {B B' : Matrix ι ι (MvPowerSeries σ R)}
(hB : ∀ i j, Jet n (B i j) (B' i j)) (i j : ι) :
Jet (n+2) (remainderValue H x v B i j) (remainderValue H x v B' i j) := by
exact Jet.subst (correctionArguments_centered x v hx hv B)
(correctionArguments_centered x v hx hv B') (correctionArguments_jet x v hv hB) (H i j)
theorem correctionOperator_contracts (b : Matrix ι ι (MvPowerSeries σ R))
(H : Matrix ι ι (MvPowerSeries (τ ⊕ ι) R))
(x : τ → MvPowerSeries σ R) (v : ι → MvPowerSeries σ R)
(hH : ∀ i j, Vanishes 1 (H i j)) (hx : ∀ t, constantCoeff (x t) = 0)
(hv : ∀ i, Vanishes 2 (v i))
{n : ℕ} {B B' : Matrix ι ι (MvPowerSeries σ R)}
(hB : ∀ i j, Jet n (B i j) (B' i j)) (i j : ι) :
Jet (n+1) (correctionOperator b H x v B i j) (correctionOperator b H x v B' i j) := by
classical
simp only [correctionOperator, Matrix.sub_apply, Matrix.neg_apply, Matrix.mul_apply,
Matrix.transpose_apply, Finset.sum_mul]
apply Jet.sub (Jet.refl _ _)
apply Jet.sum
intro l _
apply Jet.sum
intro k _
exact positively_weighted_triple (hB k i) (hB l j)
(remainderValue_positive H x v hH hx hv B k l)
(remainderValue_positive H x v hH hx hv B' k l)
((remainderValue_jet H x v hx hv hB k l).mono (by omega))
theorem exists_correction_matrix (b : Matrix ι ι (MvPowerSeries σ R))
(H : Matrix ι ι (MvPowerSeries (τ ⊕ ι) R))
(x : τ → MvPowerSeries σ R) (v : ι → MvPowerSeries σ R)
(hH : ∀ i j, Vanishes 1 (H i j)) (hx : ∀ t, constantCoeff (x t) = 0)
(hv : ∀ i, Vanishes 2 (v i)) :
∃ B : Matrix ι ι (MvPowerSeries σ R), B = -b - B.transpose * remainderValue H x v B * B := by
let T : ((ι × ι) → MvPowerSeries σ R) → ((ι × ι) → MvPowerSeries σ R) :=
fun B ij => correctionOperator b H x v (fun i j => B (i,j)) ij.1 ij.2
have hT : ∀ n f g, Close n f g → Close (n+1) (T f) (T g) := by
intro n f g hfg ij e he
exact Jet.iff_coeff.mp (correctionOperator_contracts b H x v hH hx hv
(fun i j => Jet.iff_coeff.mpr (hfg (i,j))) ij.1 ij.2) e he
refine ⟨fun i j => fixedPoint T (i,j), ?_⟩
funext i j
exact (congrFun (fixedPoint_eq T hT) (i,j)).symm
end BoundaryOnly.FormalObstruction.FormalCorrection
end
end OAI
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