File size: 3,076 Bytes
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 | 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
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