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159357f | 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 | import Mathlib
import OAI.Analysis.Conductivity.Branching.PhysicalBlockEnergySum
namespace OAI
section
noncomputable section
namespace ScalarConductivity
open Set MeasureTheory Filter Topology
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
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
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]
end ScalarConductivity
end
end
end OAI
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