| import OAI.Geometry.IsometricImmersion.Metrics.MetricLocality |
| import OAI.Geometry.IsometricImmersion.Calculus.QuotientCalculus |
| import OAI.Geometry.IsometricImmersion.Calculus.HessianCommutator |
| import Mathlib.Tactic.Linarith |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter |
| open scoped ContDiff Topology |
|
|
| namespace SmoothLocal.Geometry |
| namespace LowQuotient |
|
|
| variable {U : Set Coord} {a b : Coord → ℝ} {p : Coord} |
|
|
| theorem iterated_contDiffOn (ha : ContDiffOn ℝ ∞ a U) (hU : IsOpen U) |
| (ds : List (Fin 2)) : ContDiffOn ℝ ∞ (iteratedCoordPartial ds a) U := by |
| induction ds with |
| | nil => exact ha |
| | cons i ds ih => exact partial_contDiffOn ih hU i |
|
|
| theorem iterated_differentiableAt (ha : ContDiffOn ℝ ∞ a U) (hU : IsOpen U) |
| (hp : p ∈ U) (ds : List (Fin 2)) : |
| DifferentiableAt ℝ (iteratedCoordPartial ds a) p := |
| ((iterated_contDiffOn ha hU ds).contDiffAt (hU.mem_nhds hp)).differentiableAt (by simp) |
|
|
| theorem product_one (ha : ContDiffOn ℝ ∞ a U) (hb : ContDiffOn ℝ ∞ b U) |
| (hU : IsOpen U) (hp : p ∈ U) (i : Fin 2) : |
| iteratedCoordPartial [i] (fun x => a x * b x) p = |
| iteratedCoordPartial [i] a p * b p + a p * iteratedCoordPartial [i] b p := |
| HessianCalculus.coordPartial_mul_at |
| (iterated_differentiableAt ha hU hp []) (iterated_differentiableAt hb hU hp []) i |
|
|
| theorem product_two (ha : ContDiffOn ℝ ∞ a U) (hb : ContDiffOn ℝ ∞ b U) |
| (hU : IsOpen U) (hp : p ∈ U) (i j : Fin 2) : |
| iteratedCoordPartial [i, j] (fun x => a x * b x) p = |
| iteratedCoordPartial [i, j] a p * b p + |
| iteratedCoordPartial [j] a p * iteratedCoordPartial [i] b p + |
| iteratedCoordPartial [i] a p * iteratedCoordPartial [j] b p + |
| a p * iteratedCoordPartial [i, j] b p := by |
| have he : coordPartial j (fun x => a x * b x) =ᶠ[𝓝 p] |
| (fun x => coordPartial j a x * b x + a x * coordPartial j b x) := by |
| filter_upwards [hU.mem_nhds hp] with x hx |
| exact product_one ha hb hU hx j |
| change coordPartial i (coordPartial j (fun x => a x * b x)) p = _ |
| rw [coordPartial_eq_of_eventuallyEq he i] |
| have ha0 := iterated_differentiableAt ha hU hp [] |
| have hb0 := iterated_differentiableAt hb hU hp [] |
| have haj := iterated_differentiableAt ha hU hp [j] |
| have hbj := iterated_differentiableAt hb hU hp [j] |
| have ht0 : DifferentiableAt ℝ (fun x => coordPartial j a x * b x) p := haj.mul hb0 |
| have ht1 : DifferentiableAt ℝ (fun x => a x * coordPartial j b x) p := ha0.mul hbj |
| rw [HessianCalculus.coordPartial_add_at |
| (f := fun x => coordPartial j a x * b x) (h := fun x => a x * coordPartial j b x) ht0 ht1 i, |
| HessianCalculus.coordPartial_mul_at (f := coordPartial j a) (h := b) haj hb0 i, |
| HessianCalculus.coordPartial_mul_at (f := a) (h := coordPartial j b) ha0 hbj i] |
| dsimp [iteratedCoordPartial] |
| ring |
|
|
| theorem product_three (ha : ContDiffOn ℝ ∞ a U) (hb : ContDiffOn ℝ ∞ b U) |
| (hU : IsOpen U) (hp : p ∈ U) (i j k : Fin 2) : |
| iteratedCoordPartial [i, j, k] (fun x => a x * b x) p = |
| iteratedCoordPartial [i, j, k] a p * b p + |
| iteratedCoordPartial [j, k] a p * iteratedCoordPartial [i] b p + |
| iteratedCoordPartial [i, k] a p * iteratedCoordPartial [j] b p + |
| iteratedCoordPartial [k] a p * iteratedCoordPartial [i, j] b p + |
| iteratedCoordPartial [i, j] a p * iteratedCoordPartial [k] b p + |
| iteratedCoordPartial [j] a p * iteratedCoordPartial [i, k] b p + |
| iteratedCoordPartial [i] a p * iteratedCoordPartial [j, k] b p + |
| a p * iteratedCoordPartial [i, j, k] b p := by |
| have he : iteratedCoordPartial [j, k] (fun x => a x * b x) =ᶠ[𝓝 p] |
| (fun x => iteratedCoordPartial [j, k] a x * b x + |
| iteratedCoordPartial [k] a x * iteratedCoordPartial [j] b x + |
| iteratedCoordPartial [j] a x * iteratedCoordPartial [k] b x + |
| a x * iteratedCoordPartial [j, k] b x) := by |
| filter_upwards [hU.mem_nhds hp] with x hx |
| exact product_two ha hb hU hx j k |
| change coordPartial i (iteratedCoordPartial [j, k] (fun x => a x * b x)) p = _ |
| rw [coordPartial_eq_of_eventuallyEq he i] |
| have ha0 := iterated_differentiableAt ha hU hp [] |
| have hb0 := iterated_differentiableAt hb hU hp [] |
| have haj := iterated_differentiableAt ha hU hp [j] |
| have hak := iterated_differentiableAt ha hU hp [k] |
| have hajk := iterated_differentiableAt ha hU hp [j, k] |
| have hbj := iterated_differentiableAt hb hU hp [j] |
| have hbk := iterated_differentiableAt hb hU hp [k] |
| have hbjk := iterated_differentiableAt hb hU hp [j, k] |
| have ht0 : DifferentiableAt ℝ (fun x => iteratedCoordPartial [j, k] a x * b x) p := hajk.mul hb0 |
| have ht1 : DifferentiableAt ℝ (fun x => iteratedCoordPartial [k] a x * iteratedCoordPartial [j] b x) p := hak.mul hbj |
| have ht2 : DifferentiableAt ℝ (fun x => iteratedCoordPartial [j] a x * iteratedCoordPartial [k] b x) p := haj.mul hbk |
| have ht3 : DifferentiableAt ℝ (fun x => a x * iteratedCoordPartial [j, k] b x) p := ha0.mul hbjk |
| have ht01 : DifferentiableAt ℝ (fun x => iteratedCoordPartial [j, k] a x * b x + |
| iteratedCoordPartial [k] a x * iteratedCoordPartial [j] b x) p := ht0.add ht1 |
| have ht012 : DifferentiableAt ℝ (fun x => iteratedCoordPartial [j, k] a x * b x + |
| iteratedCoordPartial [k] a x * iteratedCoordPartial [j] b x + |
| iteratedCoordPartial [j] a x * iteratedCoordPartial [k] b x) p := ht01.add ht2 |
| rw [HessianCalculus.coordPartial_add_at |
| ht012 ht3 i, |
| HessianCalculus.coordPartial_add_at ht01 ht2 i, |
| HessianCalculus.coordPartial_add_at ht0 ht1 i, |
| HessianCalculus.coordPartial_mul_at (f := iteratedCoordPartial [j, k] a) (h := b) hajk hb0 i, |
| HessianCalculus.coordPartial_mul_at (f := iteratedCoordPartial [k] a) (h := iteratedCoordPartial [j] b) hak hbj i, |
| HessianCalculus.coordPartial_mul_at (f := iteratedCoordPartial [j] a) (h := iteratedCoordPartial [k] b) haj hbk i, |
| HessianCalculus.coordPartial_mul_at (f := a) (h := iteratedCoordPartial [j, k] b) ha0 hbjk i] |
| dsimp [iteratedCoordPartial] |
| ring |
|
|
| theorem denominator_mul_quotient_word |
| (hU : IsOpen U) (hp : p ∈ U) (hne : ∀ x ∈ U, b x ≠ 0) (ds : List (Fin 2)) : |
| iteratedCoordPartial ds (fun x => b x * (a x / b x)) p = |
| iteratedCoordPartial ds a p := by |
| have he : (fun x => b x * (a x / b x)) =ᶠ[𝓝 p] a := by |
| filter_upwards [hU.mem_nhds hp] with x hx |
| field_simp [hne x hx] |
| exact (iteratedCoordPartial_eventuallyEq he ds).self_of_nhds |
|
|
| theorem quotient_one (ha : ContDiffOn ℝ ∞ a U) (hb : ContDiffOn ℝ ∞ b U) |
| (hU : IsOpen U) (hp : p ∈ U) (hne : ∀ x ∈ U, b x ≠ 0) (i : Fin 2) : |
| iteratedCoordPartial [i] (fun x => a x / b x) p = |
| (iteratedCoordPartial [i] a p - |
| iteratedCoordPartial [i] b p * (a p / b p)) / b p := by |
| have he := denominator_mul_quotient_word (a := a) hU hp hne [i] |
| rw [product_one hb (quotient_contDiffOn ha hb hne) hU hp i] at he |
| apply (eq_div_iff (hne p hp)).2 |
| nlinarith [he] |
|
|
| theorem quotient_two (ha : ContDiffOn ℝ ∞ a U) (hb : ContDiffOn ℝ ∞ b U) |
| (hU : IsOpen U) (hp : p ∈ U) (hne : ∀ x ∈ U, b x ≠ 0) (i j : Fin 2) : |
| iteratedCoordPartial [i, j] (fun x => a x / b x) p = |
| (iteratedCoordPartial [i, j] a p - |
| (iteratedCoordPartial [i, j] b p * (a p / b p) + |
| iteratedCoordPartial [j] b p * iteratedCoordPartial [i] (fun x => a x / b x) p + |
| iteratedCoordPartial [i] b p * iteratedCoordPartial [j] (fun x => a x / b x) p)) / b p := by |
| have he := denominator_mul_quotient_word (a := a) hU hp hne [i, j] |
| rw [product_two hb (quotient_contDiffOn ha hb hne) hU hp i j] at he |
| apply (eq_div_iff (hne p hp)).2 |
| nlinarith [he] |
|
|
| theorem quotient_three (ha : ContDiffOn ℝ ∞ a U) (hb : ContDiffOn ℝ ∞ b U) |
| (hU : IsOpen U) (hp : p ∈ U) (hne : ∀ x ∈ U, b x ≠ 0) (i j k : Fin 2) : |
| iteratedCoordPartial [i, j, k] (fun x => a x / b x) p = |
| (iteratedCoordPartial [i, j, k] a p - |
| (iteratedCoordPartial [i, j, k] b p * (a p / b p) + |
| iteratedCoordPartial [j, k] b p * iteratedCoordPartial [i] (fun x => a x / b x) p + |
| iteratedCoordPartial [i, k] b p * iteratedCoordPartial [j] (fun x => a x / b x) p + |
| iteratedCoordPartial [k] b p * iteratedCoordPartial [i, j] (fun x => a x / b x) p + |
| iteratedCoordPartial [i, j] b p * iteratedCoordPartial [k] (fun x => a x / b x) p + |
| iteratedCoordPartial [j] b p * iteratedCoordPartial [i, k] (fun x => a x / b x) p + |
| iteratedCoordPartial [i] b p * iteratedCoordPartial [j, k] (fun x => a x / b x) p)) / b p := by |
| have he := denominator_mul_quotient_word (a := a) hU hp hne [i, j, k] |
| rw [product_three hb (quotient_contDiffOn ha hb hne) hU hp i j k] at he |
| apply (eq_div_iff (hne p hp)).2 |
| nlinarith [he] |
|
|
| end LowQuotient |
| end SmoothLocal.Geometry |
|
|
| end |
|
|
| end OAI |
|
|
