| import OAI.Geometry.Arnold.Coordinates |
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| namespace OAI |
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| universe uE |
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| noncomputable section |
| open scoped BigOperators ContDiff |
| open MeasureTheory Set Filter |
| open scoped Matrix.Norms.Operator |
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| open scoped ContDiff Topology |
| open Set Filter |
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| namespace ArnoldCounterexample.Extension |
| variable {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] |
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| omit [FiniteDimensional ℝ E] in |
| lemma contDiff_mul_of_tsupport_subset {b g : E → ℝ} {U : Set E} |
| (hb : ContDiff ℝ ∞ b) (hU : IsOpen U) (hg : ContDiffOn ℝ ∞ g U) |
| (hs : tsupport b ⊆ U) : ContDiff ℝ ∞ (fun x ↦ b x * g x) := by |
| rw [contDiff_iff_contDiffAt] |
| intro x |
| by_cases hx : x ∈ U |
| · exact hb.contDiffAt.mul ((hg x hx).contDiffAt (hU.mem_nhds hx)) |
| · have hz : x ∉ tsupport b := fun h ↦ hx (hs h) |
| have heq : (fun y ↦ b y * g y) =ᶠ[𝓝 x] fun _ ↦ (0 : ℝ) := by |
| filter_upwards [(isClosed_tsupport b).isOpen_compl.mem_nhds hz] with y hy |
| rw [image_eq_zero_of_notMem_tsupport hy, zero_mul] |
| exact contDiffAt_const.congr_of_eventuallyEq heq |
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| /-- Local ambient smooth extensions glue globally on a compact subset of a |
| finite-dimensional real vector space. No extension theorem is assumed. -/ |
| theorem global_extension {K : Set E} (hK : IsCompact K) (f : K → ℝ) |
| (hlocal : ∀ z : K, ∃ U : Set E, ∃ g : E → ℝ, |
| IsOpen U ∧ ↑z ∈ U ∧ ContDiffOn ℝ ∞ g U ∧ ∀ y : K, ↑y ∈ U → g y = f y) : |
| ∃ G : E → ℝ, ContDiff ℝ ∞ G ∧ ∀ z : K, G z = f z := by |
| classical |
| choose U g hU hz hg heq using hlocal |
| have hb : ∀ z : K, ∃ b : E → ℝ, |
| tsupport b ⊆ U z ∧ HasCompactSupport b ∧ ContDiff ℝ ∞ b ∧ |
| range b ⊆ Icc 0 1 ∧ b z = 1 := fun z ↦ |
| exists_contDiff_tsupport_subset ((hU z).mem_nhds (hz z)) |
| choose b hbs hbc hbd hbr hbz using hb |
| obtain ⟨T, hT⟩ := hK.elim_finite_subcover (fun z : K ↦ Function.support (b z)) |
| (fun z ↦ (hbd z).continuous.isOpen_support) (by |
| intro x hx |
| exact mem_iUnion.mpr ⟨⟨x, hx⟩, by simp [Function.mem_support, hbz]⟩) |
| obtain ⟨δ, hδs, hδd, hδr⟩ := hK.isClosed.isOpen_compl.exists_contDiff_support_eq (n := ⊤) |
| let S : E → ℝ := fun x ↦ ∑ z ∈ T, b z x |
| let N : E → ℝ := fun x ↦ ∑ z ∈ T, b z x * g z x |
| have hS : ContDiff ℝ ∞ S := ContDiff.sum (fun z _ ↦ hbd z) |
| have hN : ContDiff ℝ ∞ N := ContDiff.sum (fun z _ ↦ |
| contDiff_mul_of_tsupport_subset (hbd z) (hU z) (hg z) (hbs z)) |
| have hSnon (x : E) : 0 ≤ S x := Finset.sum_nonneg (fun z _ ↦ (hbr z ⟨x, rfl⟩).1) |
| have hpositive (x : E) : 0 < S x + δ x := by |
| by_cases hx : x ∈ K |
| · obtain ⟨z, hzT, hzbx⟩ := mem_iUnion₂.mp (hT hx) |
| have hbpos : 0 < b z x := lt_of_le_of_ne (hbr z ⟨x, rfl⟩).1 (Ne.symm hzbx) |
| have hlt : 0 < S x := lt_of_lt_of_le hbpos |
| (Finset.single_le_sum (fun i _ ↦ (hbr i ⟨x, rfl⟩).1) hzT) |
| exact add_pos_of_pos_of_nonneg hlt (hδr ⟨x, rfl⟩).1 |
| · have hδx : δ x ≠ 0 := by |
| change x ∈ Function.support δ |
| rw [hδs] |
| exact hx |
| exact add_pos_of_nonneg_of_pos (hSnon x) |
| (lt_of_le_of_ne (hδr ⟨x, rfl⟩).1 hδx.symm) |
| refine ⟨fun x ↦ N x / (S x + δ x), hN.div (hS.add hδd) (fun x ↦ (hpositive x).ne'), ?_⟩ |
| intro z |
| have hδz : δ z = 0 := by |
| by_contra hne |
| have hz : ↑z ∈ Function.support δ := hne |
| rw [hδs] at hz |
| exact hz z.property |
| have hNz : N z = S z * f z := by |
| dsimp [N, S] |
| rw [Finset.sum_mul] |
| apply Finset.sum_congr rfl |
| intro y _ |
| by_cases hby : b y z = 0 |
| · simp [hby] |
| · rw [heq y z (hbs y (subset_tsupport (b y) hby))] |
| change N z / (S z + δ z) = f z |
| rw [hNz, hδz, add_zero] |
| exact mul_div_cancel_left₀ (f z) (by simpa [hδz] using (hpositive z).ne') |
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| end ArnoldCounterexample.Extension |
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| end |
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| end OAI |
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