| import Mathlib |
| import OAI.Analysis.Conductivity.Branching.PhysicalBlockEnergySum |
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| namespace OAI |
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| section |
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| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology |
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| lemma sourceCollarTime_pos_iff_domain (y : Fin 3 → ℝ) : |
| 0<sourceCollarTime y ↔ WithLp.toLp 2 y∈sourceDomain := by |
| change 0<sourceCollarTime y ↔ sourceHole<sourceRadial y ∧ sourceRadial y<1 ∧ |y 2|<1 |
| rw [←not_le,sourceCollarTime_le_iff] |
| simp only [sub_zero,mul_zero,add_zero,not_or,not_le] |
| tauto |
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| lemma sourceDomain_closure_time (y : Fin 3 → ℝ) : |
| WithLp.toLp 2 y∈closure sourceDomain ↔ 0≤ sourceCollarTime y := by |
| have hcoord : Continuous (fun x : R3 => (fun i : Fin 3 => x i)) := |
| continuous_pi (fun i => (EuclideanSpace.proj i).continuous) |
| have hclosed : IsClosed {x : R3 | 0≤ sourceCollarTime (fun i => x i)} := |
| isClosed_le continuous_const (locallyLipschitz_sourceCollarTime.continuous.comp hcoord) |
| constructor |
| · intro hy |
| exact closure_minimal (fun x hx => ((sourceCollarTime_pos_iff_domain _).mpr hx).le) hclosed hy |
| · intro hy |
| by_cases hp : 0<sourceCollarTime y |
| · exact subset_closure ((sourceCollarTime_pos_iff_domain y).mp hp) |
| · have he : sourceCollarTime y=0 := le_antisymm (le_of_not_gt hp) hy |
| let g : ℝ → R3 := fun t => WithLp.toLp 2 (sourceAngularCollar t (sourcePhysicalAngles y)) |
| have hg : Continuous g := (PiLp.continuous_toLp 2 (fun _ : Fin 3 => ℝ)).comp |
| (continuous_uncurry_sourceAngularCollar.comp (continuous_id.prodMk continuous_const)) |
| have hz : (0:ℝ)∈closure (Ioo 0 centralThickness) := by |
| rw [closure_Ioo (show (0:ℝ)≠centralThickness by norm_num [centralThickness])] |
| exact ⟨le_rfl,by norm_num [centralThickness]⟩ |
| have hin : g '' Ioo 0 centralThickness ⊆ sourceDomain := by |
| rintro _ ⟨t,ht,rfl⟩ |
| apply (sourceCollarTime_pos_iff_domain _).mp |
| rw [sourceAngular_time (show t∈Icc (-(1:ℝ)/100) (1/100) from |
| ⟨by linarith [ht.1],ht.2.le.trans (by norm_num [centralThickness])⟩)] |
| exact ht.1 |
| have hmem := closure_mono hin (image_closure_subset_closure_image hg ⟨0,hz,rfl⟩) |
| have hright := sourcePhysicalCoordinates_right |
| (show sourceCollarTime y∈Icc (-(1:ℝ)/100) (1/100) by rw [he]; norm_num) |
| rw [he] at hright |
| simpa only [g,hright] using hmem |
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| theorem physicalBlockRegion_sourceDomain (y : Fin 3 → ℝ) : |
| y∈physicalBlockRegion ↔ WithLp.toLp 2 y∈closure sourceDomain ∧ |
| ∀ k : Fin 2,WithLp.toLp 2 ((sourceChildHomeomorph (actualChildSign k)).symm y)∉sourceDomain := by |
| rw [physicalBlockRegion_time_iff,sourceDomain_closure_time] |
| simp only [←sourceCollarTime_pos_iff_domain,not_lt] |
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| end ScalarConductivity |
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| end |
| end |
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| end OAI |
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