| 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 |
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| end |
|
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| end OAI |
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