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59d1c80 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 | 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
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