| import OAI.Geometry.IsometricImmersion.Assembly.AccumulatingDisks |
| import Mathlib.Analysis.Calculus.SmoothSeries |
| import Mathlib.Analysis.Normed.Group.Bounded |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped ContDiff Topology BigOperators Matrix |
| open Filter Set |
|
|
| namespace SmoothLocal.Geometry |
|
|
| def patchSeriesWeight (ε : ℝ) (n : ℕ) : ℝ := ε / 2 / (2 : ℝ) ^ n |
|
|
| theorem patchSeriesWeight_hasSum (ε : ℝ) : HasSum (patchSeriesWeight ε) ε := |
| hasSum_geometric_two' ε |
|
|
| theorem patchSeriesWeight_pos {ε : ℝ} (hε : 0 < ε) (n : ℕ) : |
| 0 < patchSeriesWeight ε n := by |
| unfold patchSeriesWeight |
| positivity |
|
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| variable {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] [CompleteSpace V] |
|
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| def flatPatchSum (f : ℕ → Coord → V) (p : Coord) : V := ∑' n, f n p |
|
|
| omit [CompleteSpace V] in |
| theorem compactSmooth_has_derivative_bounds (f : Coord → V) |
| (hf : ContDiff ℝ ∞ f) (hc : HasCompactSupport f) (k : ℕ) : |
| ∃ C : ℝ, ∀ p, ‖iteratedFDeriv ℝ k f p‖ ≤ C := |
| (hf.continuous_iteratedFDeriv (WithTop.coe_le_coe.mpr le_top)).bounded_above_of_compact_support |
| (hc.iteratedFDeriv k) |
|
|
| omit [CompleteSpace V] in |
| theorem exists_summable_derivative_majorant (f : ℕ → Coord → V) (ε : ℝ) |
| (hf : ∀ n, ContDiff ℝ ∞ (f n)) (hc : ∀ n, HasCompactSupport (f n)) |
| (hbound : ∀ n k : ℕ, k ≤ n → ∀ p : Coord, |
| ‖iteratedFDeriv ℝ k (f n) p‖ ≤ patchSeriesWeight ε n) : |
| ∃ v : ℕ → ℕ → ℝ, (∀ k, Summable (v k)) ∧ |
| ∀ k n p, ‖iteratedFDeriv ℝ k (f n) p‖ ≤ v k n := by |
| classical |
| have hC (k n : ℕ) : ∃ C : ℝ, ∀ p, ‖iteratedFDeriv ℝ k (f n) p‖ ≤ C := |
| compactSmooth_has_derivative_bounds (f n) (hf n) (hc n) k |
| choose C hC using hC |
| let v : ℕ → ℕ → ℝ := fun k n => if k ≤ n then patchSeriesWeight ε n else C k n |
| refine ⟨v, ?_, ?_⟩ |
| · intro k |
| apply (patchSeriesWeight_hasSum ε).summable.congr_atTop |
| filter_upwards [eventually_ge_atTop k] with n hn |
| simp only [v, ite_eq_left hn] |
| · intro k n p |
| by_cases hkn : k ≤ n |
| · simpa only [v, ite_eq_left hkn] using hbound n k hkn p |
| · simpa only [v, ite_eq_right hkn] using hC k n p |
|
|
| theorem flatPatchSum_summable (f : ℕ → Coord → V) (ε : ℝ) |
| (hbound : ∀ n k : ℕ, k ≤ n → ∀ p : Coord, |
| ‖iteratedFDeriv ℝ k (f n) p‖ ≤ patchSeriesWeight ε n) (p : Coord) : |
| Summable (fun n => f n p) := by |
| apply Summable.of_norm_bounded (patchSeriesWeight_hasSum ε).summable |
| intro n |
| simpa only [norm_iteratedFDeriv_zero] using hbound n 0 (Nat.zero_le n) p |
|
|
| omit [CompleteSpace V] in |
| theorem flatPatchSum_norm_le (f : ℕ → Coord → V) (ε : ℝ) |
| (hbound : ∀ n k : ℕ, k ≤ n → ∀ p : Coord, |
| ‖iteratedFDeriv ℝ k (f n) p‖ ≤ patchSeriesWeight ε n) (p : Coord) : |
| ‖flatPatchSum f p‖ ≤ ε := by |
| apply tsum_of_norm_bounded (patchSeriesWeight_hasSum ε) |
| intro n |
| simpa only [norm_iteratedFDeriv_zero] using hbound n 0 (Nat.zero_le n) p |
|
|
| theorem flatPatchSum_contDiff (f : ℕ → Coord → V) (ε : ℝ) |
| (hf : ∀ n, ContDiff ℝ ∞ (f n)) (hc : ∀ n, HasCompactSupport (f n)) |
| (hbound : ∀ n k : ℕ, k ≤ n → ∀ p : Coord, |
| ‖iteratedFDeriv ℝ k (f n) p‖ ≤ patchSeriesWeight ε n) : |
| ContDiff ℝ ∞ (flatPatchSum f) := by |
| obtain ⟨v, hv, hfv⟩ := exists_summable_derivative_majorant f ε hf hc hbound |
| exact contDiff_tsum (N := ⊤) hf (fun k _ => hv k) (fun k n p _ => hfv k n p) |
|
|
| theorem flatPatchSum_iteratedFDeriv (f : ℕ → Coord → V) (ε : ℝ) |
| (hf : ∀ n, ContDiff ℝ ∞ (f n)) (hc : ∀ n, HasCompactSupport (f n)) |
| (hbound : ∀ n k : ℕ, k ≤ n → ∀ p : Coord, |
| ‖iteratedFDeriv ℝ k (f n) p‖ ≤ patchSeriesWeight ε n) |
| (k : ℕ) (p : Coord) : |
| iteratedFDeriv ℝ k (flatPatchSum f) p = ∑' n, iteratedFDeriv ℝ k (f n) p := by |
| obtain ⟨v, hv, hfv⟩ := exists_summable_derivative_majorant f ε hf hc hbound |
| exact iteratedFDeriv_tsum_apply (N := ⊤) hf (fun k _ => hv k) |
| (fun k n p _ => hfv k n p) (by simp) p |
|
|
| theorem flatPatchSum_derivatives_tendstoUniformly (f : ℕ → Coord → V) (ε : ℝ) |
| (hf : ∀ n, ContDiff ℝ ∞ (f n)) (hc : ∀ n, HasCompactSupport (f n)) |
| (hbound : ∀ n k : ℕ, k ≤ n → ∀ p : Coord, |
| ‖iteratedFDeriv ℝ k (f n) p‖ ≤ patchSeriesWeight ε n) (k : ℕ) : |
| TendstoUniformly (fun N : ℕ => fun p => |
| ∑ n ∈ Finset.range N, iteratedFDeriv ℝ k (f n) p) |
| (iteratedFDeriv ℝ k (flatPatchSum f)) atTop := by |
| obtain ⟨v, hv, hfv⟩ := exists_summable_derivative_majorant f ε hf hc hbound |
| have heq : iteratedFDeriv ℝ k (flatPatchSum f) = |
| (fun p => ∑' n, iteratedFDeriv ℝ k (f n) p) := by |
| funext p |
| exact flatPatchSum_iteratedFDeriv f ε hf hc hbound k p |
| rw [heq] |
| exact tendstoUniformly_tsum_nat (hv k) (fun n p => hfv k n p) |
|
|
| theorem flatPatchSum_all_jets_zero (f : ℕ → Coord → V) (ε : ℝ) |
| (hf : ∀ n, ContDiff ℝ ∞ (f n)) (hc : ∀ n, HasCompactSupport (f n)) |
| (hbound : ∀ n k : ℕ, k ≤ n → ∀ p : Coord, |
| ‖iteratedFDeriv ℝ k (f n) p‖ ≤ patchSeriesWeight ε n) |
| {p : Coord} (hp : ∀ n, p ∉ tsupport (f n)) (k : ℕ) : |
| iteratedFDeriv ℝ k (flatPatchSum f) p = 0 := by |
| rw [flatPatchSum_iteratedFDeriv f ε hf hc hbound k p] |
| have hz (n : ℕ) : iteratedFDeriv ℝ k (f n) p = 0 := by |
| by_contra hne |
| exact hp n ((support_iteratedFDeriv_subset (𝕜 := ℝ) k) hne) |
| simp only [hz, tsum_zero] |
|
|
| omit [NormedSpace ℝ V] [CompleteSpace V] in |
| theorem hasCompactSupport_of_accumulatingDisk_support (f : ℕ → Coord → V) |
| (hsupp : ∀ n, tsupport (f n) ⊆ enlargedAccumulatingDisk n) (n : ℕ) : |
| HasCompactSupport (f n) := |
| (enlargedAccumulatingDisk_isCompact n).of_isClosed_subset |
| (isClosed_tsupport (f n)) (hsupp n) |
|
|
| theorem accumulatingDisk_flat_sum (f : ℕ → Coord → V) (ε : ℝ) |
| (hf : ∀ n, ContDiff ℝ ∞ (f n)) |
| (hsupp : ∀ n, tsupport (f n) ⊆ enlargedAccumulatingDisk n) |
| (hbound : ∀ n k : ℕ, k ≤ n → ∀ p : Coord, |
| ‖iteratedFDeriv ℝ k (f n) p‖ ≤ patchSeriesWeight ε n) : |
| (∀ p, Summable (fun n => f n p)) ∧ ContDiff ℝ ∞ (flatPatchSum f) ∧ |
| (∀ p, ‖flatPatchSum f p‖ ≤ ε) ∧ |
| (∀ k : ℕ, iteratedFDeriv ℝ k (flatPatchSum f) (0 : Coord) = 0) := by |
| have hc := hasCompactSupport_of_accumulatingDisk_support f hsupp |
| refine ⟨flatPatchSum_summable f ε hbound, |
| flatPatchSum_contDiff f ε hf hc hbound, |
| flatPatchSum_norm_le f ε hbound, ?_⟩ |
| intro k |
| apply flatPatchSum_all_jets_zero f ε hf hc hbound _ k |
| intro n hn |
| exact zero_not_mem_enlargedAccumulatingDisk n (hsupp n hn) |
|
|
| end SmoothLocal.Geometry |
|
|
| end |
|
|
| end OAI |
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|
