| import OAI.Geometry.RuledCovering.SingularOverlap |
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| namespace OAI |
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| noncomputable section |
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| open Set MeasureTheory Filter Metric |
| open scoped Topology ContDiff Pointwise |
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| namespace RuledApproximation |
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| lemma singular_geometric_cover {p gamma : ℝ} (hp : 1 < p) (hp2 : p < 2) |
| (hpg : 0 < 1+gamma*(p-2)) : |
| ∀ᶠ c : ℝ in 𝓝[>] 0, |
| tetrahedron ⊆ singularLower p gamma c ∪ singularUpper p gamma c ∪ |
| lowerTip (c^gamma) ∪ upperTip (c^gamma) := by |
| obtain ⟨C,_,hinv⟩ := singular_inverse_small hp2 |
| obtain ⟨K,_,hover⟩ := singular_overlap_small (by linarith : 0 ≤ p) hp2.le |
| have h0 : Tendsto (fun c : ℝ => c) (𝓝[>] 0) (𝓝 0) := tendsto_id.mono_left nhdsWithin_le_nhds |
| have hcut : Tendsto (fun c : ℝ => c*C*(c^gamma)^(p-2)) (𝓝[>] 0) (𝓝 0) := by |
| simpa only [mul_assoc,mul_left_comm,mul_zero] using (singular_cutoff_small hpg).const_mul C |
| have hKlim : Tendsto (fun c : ℝ => c*K) (𝓝[>] 0) (𝓝 0) := by simpa using h0.mul_const K |
| filter_upwards [self_mem_nhdsWithin,h0.eventually (gt_mem_nhds (by norm_num : (0:ℝ)<1)), |
| hcut.eventually (gt_mem_nhds (by norm_num : (0:ℝ)<1/8)), |
| hKlim.eventually (gt_mem_nhds (by norm_num : (0:ℝ)<1/2)),overlap_error_eventually K] |
| with c hc hc1 hcut hcK herr |
| have hc : 0 < c := hc |
| intro z hz |
| obtain ⟨t,ht0,ht1,hx,hy,hz2⟩ := hz |
| by_cases htip : t ≤ 2*c^gamma |
| · exact Or.inl (Or.inr ⟨t,ht0,htip,hy,hz2⟩) |
| by_cases htip' : 1-t ≤ 2*c^gamma |
| · exact Or.inr ⟨1-t,by linarith,htip',hx,by rw [hz2]; congr 1; ring⟩ |
| have hx1 : |z 0| ≤ 1 := hx.trans (by linarith) |
| have hy1 : |z 1| ≤ 1 := hy.trans ht1 |
| have hkx : |c*z 0| ≤ c := by simpa only [abs_mul,abs_of_pos hc,mul_one] using mul_le_mul_of_nonneg_left hx1 hc.le |
| have hky : |-c*z 1| ≤ c := by simpa only [abs_mul,abs_neg,abs_of_pos hc,mul_one] using mul_le_mul_of_nonneg_left hy1 hc.le |
| obtain ⟨zm,hm0,hm2,hmb,hem⟩ := hinv (c^gamma) c (z 1) t (c*z 0) |
| (Real.rpow_pos_of_pos hc gamma) hc.le hcut.le (by linarith) ht1 hy hkx |
| obtain ⟨zp,hp0,hp2',hpb,hep⟩ := hinv (c^gamma) c (z 0) (1-t) (-c*z 1) |
| (Real.rpow_pos_of_pos hc gamma) hc.le hcut.le (by linarith) (by linarith) hx hky |
| have hem0 := congrArg (fun w : Plane => w 0) hem |
| have hem1 := congrArg (fun w : Plane => w 1) hem |
| have hep0 := congrArg (fun w : Plane => w 0) hep |
| have hep1 := congrArg (fun w : Plane => w 1) hep |
| change z 1 = zm 0+(c*z 0)*singularField p zm 0 at hem0 |
| change t = zm 1+(c*z 0)*singularField p zm 1 at hem1 |
| change z 0 = zp 0+(-c*z 1)*singularField p zp 0 at hep0 |
| change 1-t = zp 1+(-c*z 1)*singularField p zp 1 at hep1 |
| have hceil := hover c (z 0) (z 1) zm zp hc.le hc1.le hcK.le herr hx1 hy1 hm2 hp2' hmb hpb |
| (by linarith only [hem0]) (by linarith only [hep0]) (by nlinarith only [hem1,hep1]) |
| rcases hceil with hm | hp' |
| · apply Or.inl ∘ Or.inl ∘ Or.inl |
| refine ⟨zm,⟨hm0,hm2,hmb,hm⟩,z 0,hx1,?_⟩ |
| ext i |
| fin_cases i |
| · rfl |
| · change z 1 = zm 0+z 0*(c*singularField p zm 0) |
| nlinarith only [hem0] |
| · change z 2 = Real.sqrt 2*(zm 1+z 0*(c*singularField p zm 1)) |
| rw [hz2] |
| congr 1 |
| nlinarith only [hem1] |
| · apply Or.inl ∘ Or.inl ∘ Or.inr |
| refine ⟨zp,⟨hp0,hp2',hpb,hp'⟩,z 1,hy1,?_⟩ |
| ext i |
| fin_cases i |
| · change z 0 = zp 0-c*z 1*singularField p zp 0 |
| nlinarith only [hep0] |
| · rfl |
| · change z 2 = Real.sqrt 2*(1-(zp 1-c*z 1*singularField p zp 1)) |
| rw [hz2] |
| congr 1 |
| nlinarith only [hep1] |
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| end RuledApproximation |
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| end |
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| end OAI |
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