| import Mathlib |
| import OAI.Analysis.Conductivity.Fourier.RectangularTorusInverse |
| import OAI.Analysis.Conductivity.Geometry.BoxEnergyPoincare |
|
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| namespace OAI |
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| noncomputable section |
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| namespace ScalarConductivity |
| open Set MeasureTheory |
|
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| lemma sourceCollarPiece_radial_pos (i j : Fin 4) {x : Fin 3 → ℝ} |
| (hr : 0≤ sourceCollarRadius j (x 0) (x 2)) (ha : |x 1|≤1) : |
| sourceRadial (sourceCollarPiece i j x)=sourceCollarRadius j (x 0) (x 2) := by |
| have hL : 0<sourceLength := by norm_num [sourceLength] |
| simp only [sourceRadial,sourceCollarPiece,Matrix.cons_val_zero,Matrix.cons_val_one, |
| abs_mul,abs_of_pos hL,abs_of_nonneg hr] |
| have he : sourceLength*sourceCollarRadius j (x 0) (x 2)*|squareFace i (x 1) 0|/sourceLength = |
| sourceCollarRadius j (x 0) (x 2)*|squareFace i (x 1) 0| := by field_simp |
| rw [he,←mul_max_of_nonneg _ _ hr,squareFace_max i ha,mul_one] |
|
|
| lemma sourceCollarPiece_cross_pos (i j : Fin 4) {x : Fin 3 → ℝ} |
| (hr : 0≤ sourceCollarRadius j (x 0) (x 2)) (ha : |x 1|≤1) : |
| sourceCrossCoordinates (sourceCollarPiece i j x)=(1-x 0) • squareFace j (x 2) := by |
| ext k; fin_cases k |
| · change (sourceRadial (sourceCollarPiece i j x)-sourceRadialCenter)/sourceRadialWidth= |
| (1-x 0)*squareFace j (x 2) 0 |
| rw [sourceCollarPiece_radial_pos i j hr ha,sourceCollarRadius] |
| simp only [add_sub_cancel_left] |
| have hw : sourceRadialWidth≠0 := by norm_num [sourceRadialWidth,sourceHole] |
| field_simp |
| · simp [sourceCrossCoordinates,sourceCollarPiece] |
|
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| lemma sourceCollarPiece_time_pos (i j : Fin 4) {x : Fin 3 → ℝ} |
| (hr : 0≤ sourceCollarRadius j (x 0) (x 2)) (ht : x 0≤1) |
| (ha : |x 1|≤1) (hb : |x 2|≤1) : |
| sourceCollarTime (sourceCollarPiece i j x)=x 0 := by |
| rw [sourceCollarTime,sourceCollarPiece_cross_pos i j hr ha] |
| simp only [Pi.smul_apply,smul_eq_mul,abs_mul,abs_of_nonneg (sub_nonneg.mpr ht)] |
| rw [←mul_max_of_nonneg _ _ (sub_nonneg.mpr ht),squareFace_max j hb] |
| ring |
|
|
| lemma sourceCollarInverse_left_pos (i j : Fin 4) {x : Fin 3 → ℝ} |
| (hr : 0<sourceCollarRadius j (x 0) (x 2)) (ht : x 0<1) |
| (ha : |x 1|≤1) (hb : |x 2|≤1) : |
| sourceCollarInverse i j (sourceCollarPiece i j x)=x := by |
| have hq : 1-x 0≠0 := (sub_pos.mpr ht).ne' |
| ext k; fin_cases k |
| · exact sourceCollarPiece_time_pos i j hr.le ht.le ha hb |
| · change squareFaceParameter i ![sourceCollarPiece i j x 0/sourceLength, |
| sourceCollarPiece i j x 1]/sourceRadial (sourceCollarPiece i j x)=x 1 |
| rw [sourceCollarPiece_angular,squareFace_parameter,sourceCollarPiece_radial_pos i j hr.le ha] |
| exact mul_div_cancel_left₀ _ hr.ne' |
| · change squareFaceParameter j (sourceCrossCoordinates (sourceCollarPiece i j x))/ |
| (1-sourceCollarTime (sourceCollarPiece i j x))=x 2 |
| rw [sourceCollarPiece_cross_pos i j hr.le ha,squareFace_parameter, |
| sourceCollarPiece_time_pos i j hr.le ht.le ha hb] |
| exact mul_div_cancel_left₀ _ hq |
|
|
| def sourceExtendedBox (l r : ℝ) : Set (Fin 3 → ℝ) := Icc ![l,-1,-1] ![r,1,1] |
|
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| lemma mem_sourceExtendedBox {l r : ℝ} {x : Fin 3 → ℝ} : |
| x∈sourceExtendedBox l r ↔ x 0∈Icc l r ∧ |x 1| ≤ 1 ∧ |x 2| ≤ 1 := by |
| simp [sourceExtendedBox,mem_Icc,Pi.le_def,Fin.forall_fin_succ,abs_le] |
| tauto |
|
|
| lemma sourceCollarRadius_extended (j : Fin 4) {t b : ℝ} |
| (ht : t∈Icc (-(1:ℝ)/100) (1/100)) (hb : |b|≤1) : |
| (1:ℝ)/50≤ sourceCollarRadius j t b ∧ sourceCollarRadius j t b≤2 := by |
| have hq : |squareFace j b 0|≤1 := (le_max_left _ _).trans (squareFace_max j hb).le |
| have hh := abs_le.mp hq |
| have hprod : |(1-t)*squareFace j b 0|≤101/100 := by |
| rw [abs_mul,abs_of_nonneg (by linarith [ht.2] : 0≤1-t)] |
| nlinarith [ht.1,ht.2] |
| have hh' := abs_le.mp hprod |
| dsimp [sourceCollarRadius,sourceRadialCenter,sourceRadialWidth,sourceHole] |
| constructor <;> linarith |
|
|
| lemma sourceExtendedBox_injOn (i j : Fin 4) {l r : ℝ} |
| (hl : -(1:ℝ)/100≤l) (hr : r≤1/100) : |
| InjOn (sourceCollarPiece i j) (sourceExtendedBox l r) := by |
| have hi (x : Fin 3 → ℝ) (hx : x∈sourceExtendedBox l r) : |
| sourceCollarInverse i j (sourceCollarPiece i j x)=x := by |
| obtain ⟨ht,ha,hb⟩ := mem_sourceExtendedBox.mp hx |
| exact sourceCollarInverse_left_pos i j |
| (lt_of_lt_of_le (by norm_num : (0:ℝ)<1/50) |
| (sourceCollarRadius_extended j ⟨hl.trans ht.1,ht.2.trans hr⟩ hb).1) |
| (by linarith [ht.2]) ha hb |
| intro x hx y hy he |
| apply_fun sourceCollarInverse i j at he |
| rwa [hi x hx,hi y hy] at he |
|
|
| lemma sourceExtended_jacobian_lower (i j : Fin 4) {l r : ℝ} |
| (hl : -(1:ℝ)/100≤l) (hr : r≤1/100) |
| {x : Fin 3 → ℝ} (hx : x∈sourceExtendedBox l r) : |
| (1:ℝ)/100≤abs ((sourceCollarDerivative i j x).det) := by |
| obtain ⟨ht,ha,hb⟩ := mem_sourceExtendedBox.mp hx |
| have ht' : x 0∈Icc (-(1:ℝ)/100) (1/100) := ⟨hl.trans ht.1,ht.2.trans hr⟩ |
| have hR := (sourceCollarRadius_extended j ht' hb).1 |
| rw [sourceCollarDerivative_det] |
| have he : -sourceLength*sourceCollarRadius j (x 0) (x 2)*sourceRadialWidth*(1-x 0)= |
| -(sourceLength*sourceCollarRadius j (x 0) (x 2)*sourceRadialWidth*(1-x 0)) := by ring |
| rw [he,abs_neg,abs_of_nonneg (by |
| have : 0≤ sourceCollarRadius j (x 0) (x 2) := by linarith |
| have : 0≤1-x 0 := by linarith [ht'.2] |
| norm_num [sourceLength,sourceRadialWidth,sourceHole]; positivity)] |
| have hh := mul_le_mul hR (show (99:ℝ)/100≤1-x 0 by linarith [ht'.2]) |
| (by norm_num : (0:ℝ)≤99/100) (by linarith : 0≤ sourceCollarRadius j (x 0) (x 2)) |
| norm_num [sourceLength,sourceRadialWidth,sourceHole] |
| nlinarith |
|
|
| theorem sourceExtended_integral {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] |
| (i j : Fin 4) {l r : ℝ} (hl : -(1:ℝ)/100≤l) (hr : r≤1/100) |
| (g : (Fin 3 → ℝ) → F) : |
| (∫ y in sourceCollarPiece i j '' sourceExtendedBox l r,g y)= |
| ∫ x in sourceExtendedBox l r,abs ((sourceCollarDerivative i j x).det) • |
| g (sourceCollarPiece i j x) := by |
| exact integral_image_eq_integral_abs_det_fderiv_smul volume measurableSet_Icc |
| (fun x _ => (sourceCollarPiece_hasFDeriv i j x).hasFDerivWithinAt) |
| (sourceExtendedBox_injOn i j hl hr) g |
|
|
| lemma squareFace_abs_le (i : Fin 4) {a : ℝ} (ha : |a|≤1) (k : Fin 2) : |
| |squareFace i a k|≤1 := by |
| fin_cases k |
| · exact (le_max_left _ _).trans (squareFace_max i ha).le |
| · exact (le_max_right _ _).trans (squareFace_max i ha).le |
|
|
| lemma sourceRadius_trim_bounds (j : Fin 4) {t b : ℝ} |
| (ht : centralThickness≤t ∧ t≤2*centralThickness) (hb : |b|≤1) : |
| centralInnerY≤ sourceCollarRadius j t b ∧ sourceCollarRadius j t b≤centralOuterY := by |
| have hq := squareFace_abs_le j hb 0 |
| have hqt : |(1-t)*squareFace j b 0|≤1-centralThickness := by |
| have ht0 : 0≤1-t := by norm_num [centralThickness] at ht; linarith |
| rw [abs_mul,abs_of_nonneg ht0] |
| have hh := mul_le_mul_of_nonneg_left hq ht0 |
| linarith [ht.1] |
| have hh := abs_le.mp hqt |
| dsimp [sourceCollarRadius,centralInnerY,centralOuterY,sourceRadialCenter,sourceRadialWidth,sourceHole,centralThickness] at * |
| constructor <;> linarith |
|
|
| def sourcePairCoordinates (y : Fin 3 → ℝ) : Box3 := ((y 0,y 1),y 2) |
|
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| lemma sourceParentCollar_parent (i j : Fin 4) {x : Fin 3 → ℝ} |
| (hx : x∈sourceExtendedBox centralThickness (2*centralThickness)) : |
| centralParentFootprint (sourcePairCoordinates (sourceCollarPiece i j x)).1 ∧ |
| |(sourcePairCoordinates (sourceCollarPiece i j x)).2|≤centralHeight := by |
| obtain ⟨ht,ha,hb⟩ := mem_sourceExtendedBox.mp hx |
| have hr := sourceRadius_trim_bounds j ht hb |
| have hr0 : 0≤ sourceCollarRadius j (x 0) (x 2) := by |
| have hh : 0<centralInnerY := by norm_num [centralInnerY,sourceHole,sourceRadialWidth,centralThickness] |
| linarith |
| have hL : 0<sourceLength := by norm_num [sourceLength] |
| have ht0 : 0≤1-x 0 := by norm_num [centralThickness] at ht; linarith |
| have hq0 := squareFace_abs_le i ha 0 |
| have hq1 := squareFace_abs_le i ha 1 |
| have hq2 := squareFace_abs_le j hb 1 |
| have hx0 : |sourceLength*sourceCollarRadius j (x 0) (x 2)*squareFace i (x 1) 0|≤centralOuterX := by |
| rw [abs_mul,abs_mul,abs_of_pos hL,abs_of_nonneg hr0] |
| have hh := mul_le_mul_of_nonneg_left hq0 (mul_nonneg hL.le hr0) |
| dsimp [centralOuterX,centralOuterY] at * |
| nlinarith |
| have hx1 : |sourceCollarRadius j (x 0) (x 2)*squareFace i (x 1) 1|≤centralOuterY := by |
| rw [abs_mul,abs_of_nonneg hr0] |
| nlinarith |
| have hin : centralInnerX≤|sourceLength*sourceCollarRadius j (x 0) (x 2)*squareFace i (x 1) 0| ∨ |
| centralInnerY≤|sourceCollarRadius j (x 0) (x 2)*squareFace i (x 1) 1| := by |
| have hm := squareFace_max i ha |
| rcases max_cases |squareFace i (x 1) 0| |squareFace i (x 1) 1| with h|h |
| · left |
| rw [abs_mul,abs_mul,abs_of_pos hL,abs_of_nonneg hr0,←h.1,hm,mul_one] |
| dsimp [centralInnerX,centralInnerY] at * |
| nlinarith |
| · right |
| rw [abs_mul,abs_of_nonneg hr0,←h.1,hm,mul_one] |
| exact hr.1 |
| have ht1 : |(1-x 0)*squareFace j (x 2) 1|≤centralHeight := by |
| rw [abs_mul,abs_of_nonneg ht0] |
| dsimp [centralHeight] |
| nlinarith [ht.1] |
| exact ⟨⟨hx0,hx1,hin⟩,ht1⟩ |
|
|
| lemma sourceParentCollar_avoids (i j : Fin 4) {x : Fin 3 → ℝ} |
| (hx : x∈sourceExtendedBox centralThickness (2*centralThickness)) : |
| centralChildHeight≤|(sourceCollarPiece i j x) 2| ∨ |
| (centralAvoidChild 1 ((sourceCollarPiece i j x) 0,(sourceCollarPiece i j x) 1) ∧ |
| centralAvoidChild (-1) ((sourceCollarPiece i j x) 0,(sourceCollarPiece i j x) 1)) := by |
| obtain ⟨ht,ha,hb⟩ := mem_sourceExtendedBox.mp hx |
| have hr := sourceRadius_trim_bounds j ht hb |
| have hr0 : 0≤ sourceCollarRadius j (x 0) (x 2) := by |
| have hh : 0<centralInnerY := by norm_num [centralInnerY,sourceHole,sourceRadialWidth,centralThickness] |
| linarith |
| have hL : 0<sourceLength := by norm_num [sourceLength] |
| fin_cases j |
| · right |
| change 0≤ sourceCollarRadius 0 (x 0) (x 2) at hr0 |
| have hlarge : (999:ℝ)/1000≤ sourceCollarRadius 0 (x 0) (x 2) := by |
| norm_num [sourceCollarRadius,squareFace,squareFaceBase,squareFaceDirection,Matrix.cons_val_succ,sourceRadialCenter,sourceRadialWidth,sourceHole,centralThickness] at ht ⊢ |
| linarith |
| have hm := squareFace_max i ha |
| rcases max_cases |squareFace i (x 1) 0| |squareFace i (x 1) 1| with h|h |
| · have he : |(sourceCollarPiece i 0 x) 0|=sourceLength*sourceCollarRadius 0 (x 0) (x 2) := by |
| simp only [sourceCollarPiece,Matrix.cons_val_zero,abs_mul,abs_of_pos hL,abs_of_nonneg hr0] |
| rw [←h.1,hm,mul_one] |
| have hbig : (1598:ℝ)/1000≤|(sourceCollarPiece i 0 x) 0| := by rw [he]; norm_num [sourceLength]; linarith |
| have hb1 := abs_sub_le ((sourceCollarPiece i 0 x) 0) sourceOffset |
| have hb2 := abs_sub_le ((sourceCollarPiece i 0 x) 0) (-sourceOffset) |
| have hlow1 := abs_add_le ((sourceCollarPiece i 0 x) 0-sourceOffset) sourceOffset |
| have hlow2 := abs_add_le ((sourceCollarPiece i 0 x) 0+sourceOffset) (-sourceOffset) |
| simp only [sub_add_cancel,add_neg_cancel_right] at hlow1 hlow2 |
| constructor <;> apply Or.inl |
| · norm_num [centralChildOuterX,sourceScale,sourceRadialWidth,sourceHole,centralThickness,sourceOffset] at * |
| linarith |
| · norm_num [centralChildOuterX,sourceScale,sourceRadialWidth,sourceHole,centralThickness,sourceOffset] at * |
| linarith |
| · have he : |(sourceCollarPiece i 0 x) 1|=sourceCollarRadius 0 (x 0) (x 2) := by |
| simp only [sourceCollarPiece,Matrix.cons_val_one,Matrix.cons_val_zero,abs_mul,abs_of_nonneg hr0] |
| rw [←h.1,hm,mul_one] |
| have hh : centralChildOuterY≤|(sourceCollarPiece i 0 x) 1| := by |
| rw [he] |
| norm_num [centralChildOuterY,sourceScale,sourceLength,sourceRadialWidth,sourceHole,centralThickness] |
| linarith |
| exact ⟨Or.inr (Or.inl hh),Or.inr (Or.inl hh)⟩ |
| · left |
| simp only [sourceCollarPiece,Matrix.cons_val_two,squareFace,squareFaceBase,squareFaceDirection, |
| Pi.add_apply,Pi.smul_apply,smul_eq_mul] |
| norm_num [centralChildHeight,sourceScale,centralThickness] at ht ⊢ |
| linarith [le_abs_self (1-x 0),neg_le_abs (x 0-1)] |
| · right |
| change 0≤ sourceCollarRadius 2 (x 0) (x 2) at hr0 |
| have hsmall : sourceCollarRadius 2 (x 0) (x 2)≤(31:ℝ)/1000 := by |
| change sourceRadialCenter+sourceRadialWidth*(1-x 0)*((-1:ℝ)+x 2*0)≤31/1000 |
| norm_num [sourceRadialCenter,sourceRadialWidth,sourceHole,centralThickness] at ht ⊢ |
| linarith |
| have hq := squareFace_abs_le i ha 0 |
| have hsmall' : |(sourceCollarPiece i 2 x) 0|≤(1:ℝ)/20 := by |
| simp only [sourceCollarPiece,Matrix.cons_val_zero,abs_mul,abs_of_pos hL,abs_of_nonneg hr0] |
| norm_num [sourceLength] |
| nlinarith |
| have hlow1 := abs_sub_le sourceOffset ((sourceCollarPiece i 2 x) 0) |
| have hlow2 := abs_sub_le (-sourceOffset) ((sourceCollarPiece i 2 x) 0) |
| have htri1 := abs_add_le (sourceOffset-(sourceCollarPiece i 2 x) 0) ((sourceCollarPiece i 2 x) 0) |
| have htri2 := abs_add_le (-sourceOffset-(sourceCollarPiece i 2 x) 0) ((sourceCollarPiece i 2 x) 0) |
| simp only [sub_add_cancel,abs_sub_comm sourceOffset,abs_sub_comm (-sourceOffset)] at htri1 htri2 |
| have hc1 : centralChildOuterX≤|(sourceCollarPiece i 2 x) 0-sourceOffset| := by |
| norm_num [centralChildOuterX,sourceScale,sourceRadialWidth,sourceHole,centralThickness,sourceOffset] at * |
| linarith |
| have hc2 : centralChildOuterX≤|(sourceCollarPiece i 2 x) 0+sourceOffset| := by |
| norm_num [centralChildOuterX,sourceScale,sourceRadialWidth,sourceHole,centralThickness,sourceOffset] at * |
| linarith |
| exact ⟨Or.inl (by simpa using hc1),Or.inl (by simpa using hc2)⟩ |
| · left |
| simp only [sourceCollarPiece,Matrix.cons_val_two,squareFace,squareFaceBase,squareFaceDirection, |
| Pi.add_apply,Pi.smul_apply,smul_eq_mul] |
| norm_num [centralChildHeight,sourceScale,centralThickness] at ht ⊢ |
| linarith [le_abs_self (1-x 0),neg_le_abs (x 0-1)] |
|
|
| theorem sourceParentCollar_central (i j : Fin 4) {x : Fin 3 → ℝ} |
| (hx : x∈sourceExtendedBox centralThickness (2*centralThickness)) : |
| sourcePairCoordinates (sourceCollarPiece i j x)∈centralClosed := |
| ⟨(sourceParentCollar_parent i j hx).1,(sourceParentCollar_parent i j hx).2, |
| sourceParentCollar_avoids i j hx⟩ |
|
|
| end ScalarConductivity |
|
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| end |
|
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| end OAI |
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