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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 | import OAI.Geometry.Kahler.BaseChartJets
namespace OAI
universe uKahler10013_1 uKahler10013_2 uKahler10035_1 uKahler10035_2 uKahler9999_1 uKahler9999_2
open Complex
open scoped ContDiff Matrix Matrix.Norms.Elementwise
open scoped ContDiff Matrix Matrix.Norms.Elementwise ComplexOrder
open scoped ContDiff ComplexOrder
open scoped ContDiff ENNReal
open Set Filter Topology MeasureTheory
open scoped ContDiff ENNReal Pointwise
open Set Filter Topology
open scoped ContDiff
noncomputable section
open Set Filter Topology
open scoped ContDiff
namespace PinchedHartogs.BaseConstruction
lemma local_jet_series_derivative {E : Type uKahler9999_1} {F : Type uKahler9999_2} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F]
{f : ℕ → E → F} (hf : ∀ j, ContDiff ℝ ∞ (f j))
{s : Set E} (hs : IsOpen s) (hc : IsPreconnected s) {x : E} (hx : x ∈ s)
{v : ℕ → ℕ → ℝ} (hv : ∀ n, Summable (v n))
(hb : ∀ n j y, y ∈ s → ‖iteratedFDeriv ℝ n (f j) y‖ ≤ v n j) (n : ℕ) :
HasFDerivAt (fun y => ∑' j, iteratedFDeriv ℝ n (f j) y)
(∑' j, fderiv ℝ (iteratedFDeriv ℝ n (f j)) x) x := by
apply hasFDerivAt_tsum_of_isPreconnected (hv (n+1)) hs hc
(fun j y _ => ((hf j).differentiable_iteratedFDeriv (by exact_mod_cast (ENat.natCast_lt_top n))).differentiableAt.hasFDerivAt)
(fun j y hy => ?_) hx
(Summable.of_norm_bounded (hv n) (fun j => hb n j x hx)) hx
simpa only [norm_fderiv_iteratedFDeriv] using hb (n+1) j y hy
lemma local_iteratedFDeriv_tsum {E : Type uKahler10013_1} {F : Type uKahler10013_2} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F]
{f : ℕ → E → F} (hf : ∀ j, ContDiff ℝ ∞ (f j))
{s : Set E} (hs : IsOpen s) (hc : IsPreconnected s)
{v : ℕ → ℕ → ℝ} (hv : ∀ n, Summable (v n))
(hb : ∀ n j y, y ∈ s → ‖iteratedFDeriv ℝ n (f j) y‖ ≤ v n j) (n : ℕ) :
EqOn (iteratedFDeriv ℝ n (fun y => ∑' j, f j y))
(fun y => ∑' j, iteratedFDeriv ℝ n (f j) y) s := by
induction n with
| zero =>
intro x hx
simp_rw [iteratedFDeriv_zero_eq_comp]
exact (continuousMultilinearCurryFin0 ℝ E F).symm.toContinuousLinearEquiv.map_tsum
| succ n ih =>
intro x hx
have he : iteratedFDeriv ℝ n (fun y => ∑' j, f j y) =ᶠ[𝓝 x]
(fun y => ∑' j, iteratedFDeriv ℝ n (f j) y) :=
Filter.eventuallyEq_of_mem (hs.mem_nhds hx) (fun y hy => ih hy)
simp_rw [iteratedFDeriv_succ_eq_comp_left,Function.comp_apply]
rw [he.fderiv_eq,(local_jet_series_derivative hf hs hc hx hv hb n).fderiv]
exact (continuousMultilinearCurryLeftEquiv ℝ (fun _ : Fin (n+1) => E) F).symm.toContinuousLinearEquiv.map_tsum
lemma local_contDiff_tsum {E : Type uKahler10035_1} {F : Type uKahler10035_2} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F]
{f : ℕ → E → F} (hf : ∀ j, ContDiff ℝ ∞ (f j))
{s : Set E} (hs : IsOpen s) (hc : IsPreconnected s)
{v : ℕ → ℕ → ℝ} (hv : ∀ n, Summable (v n))
(hb : ∀ n j y, y ∈ s → ‖iteratedFDeriv ℝ n (f j) y‖ ≤ v n j) :
ContDiffOn ℝ ∞ (fun y => ∑' j, f j y) s := by
apply contDiffOn_of_differentiableOn
intro n hn x hx
have he : iteratedFDerivWithin ℝ n (fun y => ∑' j, f j y) s =ᶠ[𝓝 x]
(fun y => ∑' j, iteratedFDeriv ℝ n (f j) y) := by
filter_upwards [hs.mem_nhds hx] with y hy
rw [iteratedFDerivWithin_of_isOpen n hs hy]
exact local_iteratedFDeriv_tsum hf hs hc hv hb n hy
exact ((local_jet_series_derivative hf hs hc hx hv hb n).differentiableAt.congr_of_eventuallyEq he).differentiableWithinAt
end PinchedHartogs.BaseConstruction
end
end OAI
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