| import Mathlib |
| import OAI.Analysis.Conductivity.Geometry.ExtendedCollar |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory |
|
|
| def threeArrow : (Fin 3 → ℝ) ≃ᵐ ℝ×(ℝ×ℝ) := |
| (MeasurableEquiv.piFinSuccAbove (fun _ : Fin 3 => ℝ) 0).trans |
| ((MeasurableEquiv.refl ℝ).prodCongr (MeasurableEquiv.finTwoArrow)) |
|
|
| lemma threeArrow_apply (x : Fin 3 → ℝ) : threeArrow x=(x 0,x 1,x 2) := by rfl |
| lemma threeArrow_symm_apply (x : ℝ×(ℝ×ℝ)) : threeArrow.symm x=![x.1,x.2.1,x.2.2] := by |
| ext i; fin_cases i <;> rfl |
|
|
| lemma threeArrow_volume : MeasurePreserving threeArrow := by |
| exact ((MeasurePreserving.id volume).prod (volume_preserving_finTwoArrow ℝ)).comp |
| (volume_preserving_piFinSuccAbove (fun _ : Fin 3 => ℝ) 0) |
|
|
| lemma sourceExtendedBox_image (l r : ℝ) : |
| threeArrow '' sourceExtendedBox l r=Icc l r ×ˢ (Icc (-1:ℝ) 1 ×ˢ Icc (-1:ℝ) 1) := by |
| ext z |
| rw [MeasurableEquiv.image_eq_preimage_symm] |
| simp [threeArrow_symm_apply,mem_sourceExtendedBox,abs_le,Prod.le_def] |
| tauto |
|
|
| lemma integral_sourceExtendedBox {g : (Fin 3 → ℝ) → ℝ} (hg : Continuous g) |
| {l r : ℝ} (hlr : l≤r) : |
| (∫ x in sourceExtendedBox l r,g x)= |
| ∫ t in l..r,∫ a in (-1:ℝ)..1,∫ b in (-1:ℝ)..1,g ![t,a,b] := by |
| have hc : Continuous (fun z : ℝ×(ℝ×ℝ) => g ![z.1,z.2.1,z.2.2]) := hg.comp (by fun_prop) |
| have hi := hc.continuousOn.integrableOn_compact (μ:=volume) |
| (isCompact_Icc.prod (isCompact_Icc.prod isCompact_Icc) : |
| IsCompact (Icc l r ×ˢ (Icc (-1:ℝ) 1 ×ˢ Icc (-1:ℝ) 1))) |
| have he := threeArrow_volume.setIntegral_image_emb threeArrow.measurableEmbedding |
| (fun z : ℝ×(ℝ×ℝ) => g ![z.1,z.2.1,z.2.2]) (sourceExtendedBox l r) |
| simp only [threeArrow_apply] at he |
| have hid (x : Fin 3 → ℝ) : ![x 0,x 1,x 2]=x := by ext i; fin_cases i <;> rfl |
| simp_rw [hid] at he |
| have him := sourceExtendedBox_image l r |
| simp only [threeArrow_apply] at him |
| rw [←he,him] |
| change (∫ y in Icc l r ×ˢ (Icc (-1:ℝ) 1 ×ˢ Icc (-1:ℝ) 1), |
| g ![y.1,y.2.1,y.2.2] ∂volume.prod volume)=_ |
| rw [setIntegral_prod _ hi] |
| simp_rw [intervalIntegral.integral_of_le hlr, |
| intervalIntegral.integral_of_le (by norm_num : (-1:ℝ)≤1),←integral_Icc_eq_integral_Ioc] |
| apply setIntegral_congr_fun measurableSet_Icc |
| intro t ht |
| exact setIntegral_prod _ ((hc.comp (continuous_const.prodMk continuous_id)).continuousOn.integrableOn_compact |
| (isCompact_Icc.prod isCompact_Icc)) |
|
|
| end ScalarConductivity |
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|
| end |
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| end OAI |
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