| import OAI.Geometry.Immersion.ClosedSurface.GridPartition | |
| namespace OAI | |
| noncomputable section | |
| open Set Complex Bundle Manifold | |
| open scoped ContDiff Matrix Topology Manifold BigOperators | |
| namespace ClosedSurfaceR4.PhaseGrid | |
| open Set | |
| lemma isOpen_cell (h R : ℝ) (a : Index) : IsOpen (cell h R a) := | |
| (isOpen_lt ((continuous_fst.sub continuous_const).abs) continuous_const).inter | |
| (isOpen_lt ((continuous_snd.sub continuous_const).abs) continuous_const) | |
| def coverRegion (s : Finset Index) (h : ℝ) : Set Base := ⋃ a ∈ s, cell h 1 a | |
| lemma coverRegion_open (s : Finset Index) (h : ℝ) : IsOpen (coverRegion s h) := | |
| isOpen_iUnion (fun a => isOpen_iUnion (fun _ => isOpen_cell h 1 a)) | |
| lemma coverRegion_covers (s : Finset Index) (h : ℝ) {x : Base} | |
| (hx : x ∈ coverRegion s h) : ∃ a ∈ s, x ∈ cell h 1 a := by | |
| simpa only [coverRegion,mem_iUnion,exists_prop] using hx | |
| theorem compact_polynomial_square_partition {K U : Set Base} | |
| (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U) : | |
| ∃ z₀ D : ℝ, ∃ C : ℕ → ℝ, 0 < z₀ ∧ z₀ ≤ 1 ∧ 0 < D ∧ (∀ j, 1 ≤ C j) ∧ | |
| ∀ z : ℝ, 0 < z → z ≤ z₀ → ∃ s : Finset Index, | |
| (s.card : ℝ) ≤ D/z^12 ∧ IsOpen (coverRegion s (z^6)) ∧ | |
| K ⊆ coverRegion s (z^6) ∧ coverRegion s (z^6) ⊆ U ∧ | |
| (∀ a ∈ s, z^6 • center a ∈ U ∧ tsupport (cutoff (z^6) a) ⊆ U) ∧ | |
| (∀ a, ContDiffOn ℝ ∞ (normalizedCutoff s (z^6) a) (coverRegion s (z^6))) ∧ | |
| (∀ a j x, x ∈ coverRegion s (z^6) → | |
| ‖iteratedFDeriv ℝ j (normalizedCutoff s (z^6) a) x‖ ≤ C j / z^(6*j)) ∧ | |
| (∀ x ∈ coverRegion s (z^6), ∑ a ∈ s, (normalizedCutoff s (z^6) a x)^2 = 1) ∧ | |
| (∀ a ∈ s, tsupport (normalizedCutoff s (z^6) a) ⊆ U) := by | |
| obtain ⟨z₀,D,hz₀,hz₀1,hD,hgrid⟩ := compact_subpatch_polynomial_grid hK hU hKU | |
| obtain ⟨C,hC,hderiv⟩ := normalizedCutoff_polynomial_derivatives | |
| refine ⟨z₀,D,C,hz₀,hz₀1,hD,hC,fun z hz hzsmall => ?_⟩ | |
| obtain ⟨s,hcard,hcover,hsub⟩ := hgrid z hz hzsmall | |
| have hopen := coverRegion_open s (z^6) | |
| have hinner : K ⊆ coverRegion s (z^6) := by | |
| intro x hx | |
| simpa only [coverRegion,mem_iUnion,exists_prop] using hcover x hx | |
| have houter : coverRegion s (z^6) ⊆ U := by | |
| intro x hx | |
| obtain ⟨a,ha,hax⟩ := coverRegion_covers s (z^6) hx | |
| apply (hsub a ha).2 | |
| apply subset_closure | |
| change cutoff (z^6) a x ≠ 0 | |
| rw [cutoff_one_on_cell (pow_pos hz _) a hax] | |
| norm_num | |
| refine ⟨s,hcard,hopen,hinner,houter,hsub,?_,?_,?_,?_⟩ | |
| · exact fun a => normalizedCutoff_smooth (pow_pos hz _) (fun x hx => coverRegion_covers s (z^6) hx) a | |
| · exact fun a j x hx => hderiv z hz s _ hopen (fun x hx => coverRegion_covers s (z^6) hx) a j x hx | |
| · intro x hx | |
| apply normalizedCutoff_squares | |
| exact zero_lt_one.trans_le (squareSum_ge_one (pow_pos hz _) (coverRegion_covers s (z^6) hx)) | |
| · exact fun a ha => (normalizedCutoff_tsupport s (z^6) a).trans (hsub a ha).2 | |
| end ClosedSurfaceR4.PhaseGrid | |
| end | |
| end OAI | |