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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 | 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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