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a458723 | 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 | 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
end
end OAI
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