| import Mathlib |
| import OAI.Algebra.FrobeniusObstruction.Obstruction |
| import OAI.Algebra.AlgebraicObstruction.GraphCorrection |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators |
|
|
| namespace BoundaryOnly.FormalObstruction.FormalCorrection |
| open MvPowerSeries Matrix |
| open scoped Classical |
| variable {R α ι : Type*} [CommRing R] [Fintype ι] |
|
|
| def centeredPart (f : MvPowerSeries α R) : MvPowerSeries α R := f - C (constantCoeff f) |
|
|
| theorem centeredPart_centered [Fintype α] (f : MvPowerSeries α R) : |
| constantCoeff (centeredPart f) = 0 := by |
| simp [centeredPart] |
|
|
| theorem centeredPart_eq [Fintype α] {f : MvPowerSeries α R} (hf : constantCoeff f = 0) : |
| centeredPart f = f := by simp [centeredPart,hf] |
|
|
| theorem centeredPart_jet [Fintype α] {n : ℕ} {f g : MvPowerSeries α R} (h : Jet n f g) : |
| Jet n (centeredPart f) (centeredPart g) := by |
| by_cases hn : n = 0 |
| · subst n; exact Vanishes.all _ |
| · have hc : constantCoeff f = constantCoeff g := |
| Jet.iff_coeff.mp h 0 (by simpa using Nat.pos_of_ne_zero hn) |
| rw [centeredPart,centeredPart,hc] |
| exact h.sub (Jet.refl _ _) |
|
|
| variable [Fintype α] |
|
|
| theorem quadratic_subst_contracts (H : MvPowerSeries (α ⊕ ι) R) (hH : Vanishes 2 H) |
| (f g : ι → MvPowerSeries α R) {n : ℕ} |
| (hfg : ∀ i, Jet n (f i) (g i)) : |
| Jet (n+1) (subst (Sum.elim X (fun i => centeredPart (f i))) H) |
| (subst (Sum.elim X (fun i => centeredPart (g i))) H) := by |
| let x : (α ⊕ ι) → MvPowerSeries α R := Sum.elim X (fun i => centeredPart (g i)) |
| let δ : ι → MvPowerSeries α R := fun i => centeredPart (f i)-centeredPart (g i) |
| have hx : ∀ j, constantCoeff (x j) = 0 := by |
| intro j; cases j <;> simp [x,centeredPart_centered] |
| have hδ : ∀ j, constantCoeff (δ j) = 0 := by |
| intro j; simp [δ,centeredPart_centered] |
| have hext := extended_centered x δ hx hδ |
| let hs := hasSubst_of_constantCoeff_zero hext |
| obtain ⟨q,hq,hfact⟩ := exists_first_remainder H |
| have hq' : ∀ i, Vanishes 1 (subst (Sum.elim x δ) (q i)) := |
| fun i => (hq 1 hH i).subst _ hext |
| have hfact' := congrArg (substAlgHom hs) hfact |
| simp only [map_sub,map_sum,map_mul,substAlgHom_apply,subst_X hs,Sum.elim_inr, |
| subst_extended_shifted x δ hx hδ,subst_extended_base x δ hx hδ] at hfact' |
| have harg : shiftedGraph x δ = Sum.elim X (fun i => centeredPart (f i)) := by |
| funext j |
| cases j <;> simp [shiftedGraph,x,δ] |
| change Vanishes (n+1) _ |
| rw [← harg] |
| change Vanishes (n+1) (subst (shiftedGraph x δ) H-subst x H) |
| rw [hfact'] |
| exact Vanishes.sum _ _ fun i _ => by |
| simpa only [Nat.add_comm 1 n] using (hq' i).mul (centeredPart_jet (hfg i)) |
|
|
| def implicitOperator (L : ι → MvPowerSeries α R) |
| (H : ι → MvPowerSeries (α ⊕ ι) R) (w : ι → MvPowerSeries α R) : |
| ι → MvPowerSeries α R := fun i => |
| L i + subst (Sum.elim X (fun j => centeredPart (w j))) (H i) |
|
|
| theorem implicitOperator_contracts (L : ι → MvPowerSeries α R) |
| (H : ι → MvPowerSeries (α ⊕ ι) R) (hH : ∀ i, Vanishes 2 (H i)) : |
| ∀ n f g, Close n f g → Close (n+1) (implicitOperator L H f) (implicitOperator L H g) := by |
| intro n f g h i e he |
| have hj := quadratic_subst_contracts (H i) (hH i) f g |
| (fun j => Jet.iff_coeff.mpr (h j)) |
| simpa only [implicitOperator,map_add] using congrArg (fun z => coeff e (L i)+z) |
| (Jet.iff_coeff.mp hj e he) |
|
|
| theorem implicitOperator_centered (L : ι → MvPowerSeries α R) |
| (H : ι → MvPowerSeries (α ⊕ ι) R) |
| (hL : ∀ i, constantCoeff (L i) = 0) (hH : ∀ i, Vanishes 2 (H i)) |
| (w : ι → MvPowerSeries α R) : ∀ i, constantCoeff (implicitOperator L H w i) = 0 := by |
| intro i |
| have hargs : ∀ j, constantCoeff (Sum.elim X (fun j => centeredPart (w j)) j) = 0 := by |
| intro j; cases j <;> simp [centeredPart_centered] |
| have hz := Vanishes.one_iff.mp (((hH i).subst _ hargs).mono (by omega : 1 ≤ 2)) |
| simpa only [implicitOperator,map_add,hL,zero_add] using hz |
|
|
| theorem centered_implicit_solution (L : ι → MvPowerSeries α R) |
| (H : ι → MvPowerSeries (α ⊕ ι) R) |
| (hL : ∀ i, constantCoeff (L i) = 0) (hH : ∀ i, Vanishes 2 (H i)) : |
| ∃! w : ι → MvPowerSeries α R, |
| (∀ i, constantCoeff (w i) = 0) ∧ |
| (∀ i, w i = L i + subst (Sum.elim X w) (H i)) := by |
| let T := implicitOperator L H |
| have hT := implicitOperator_contracts L H hH |
| have hw : T (fixedPoint T) = fixedPoint T := fixedPoint_eq T hT |
| have hcenter : ∀ i, constantCoeff (fixedPoint T i) = 0 := by |
| rw [← hw]; exact implicitOperator_centered L H hL hH _ |
| refine ⟨fixedPoint T,⟨hcenter,?_⟩,?_⟩ |
| · intro i |
| have he := congrFun hw i |
| simpa only [T,implicitOperator,centeredPart_eq (hcenter _)] using he.symm |
| · intro w ⟨hc,hw⟩ |
| apply fixedPoint_unique T hT |
| funext i |
| simpa only [T,implicitOperator,centeredPart_eq (hc _)] using (hw i).symm |
|
|
| end BoundaryOnly.FormalObstruction.FormalCorrection |
|
|
| end |
|
|
| end OAI |
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