openAI-math-dataset / lean /OAI /Analysis /Conductivity /Variational /CompactPhysicalCorrections.lean
| import Mathlib | |
| import OAI.Analysis.Conductivity.Sources.SymmetricCutoffSources | |
| import OAI.Analysis.Conductivity.Geometry.RegularPotentialCoordinates | |
| namespace OAI | |
| noncomputable section | |
| namespace ScalarConductivity | |
| open Set MeasureTheory Filter Topology Matrix | |
| open scoped Matrix.Norms.Elementwise | |
| abbrev PhysicalSourcePair := Fin 2 → Coord3 → ℝ | |
| def CompactSmoothPair (r : PhysicalSourcePair) : Prop := | |
| ∀ j,ContDiff ℝ (↑(⊤:ℕ∞)) (r j) ∧ HasCompactSupport (r j) | |
| def PairSupported (r : PhysicalSourcePair) (U : Set Coord3) : Prop := | |
| ∀ j,tsupport (r j)⊆U | |
| def physicalSourceMoment (u : Coord3 → Fin 2 → ℝ) (r : PhysicalSourcePair) : Coord3 := | |
| ![∫ x,r 0 x,∫ x,r 1 x,∫ x,u x 1*r 0 x-u x 0*r 1 x] | |
| namespace CompactSmoothPair | |
| variable {r s : PhysicalSourcePair} | |
| lemma zero : CompactSmoothPair 0 := fun _ => ⟨contDiff_const,HasCompactSupport.zero⟩ | |
| lemma add (hr : CompactSmoothPair r) (hs : CompactSmoothPair s) : CompactSmoothPair (r+s) := | |
| fun j => ⟨(hr j).1.add (hs j).1,(hr j).2.add (hs j).2⟩ | |
| lemma sub (hr : CompactSmoothPair r) (hs : CompactSmoothPair s) : CompactSmoothPair (r-s) := | |
| fun j => ⟨(hr j).1.sub (hs j).1,(hr j).2.sub (hs j).2⟩ | |
| lemma integrable (hr : CompactSmoothPair r) (j : Fin 2) : Integrable (r j) := | |
| (hr j).1.continuous.integrable_of_hasCompactSupport (hr j).2 | |
| lemma torque_integrable (hr : CompactSmoothPair r) {u : Coord3 → Fin 2 → ℝ} | |
| (hu : Continuous u) : Integrable (fun x => u x 1*r 0 x-u x 0*r 1 x) := | |
| ((continuous_apply 1 |>.comp hu).mul (hr 0).1.continuous |>.integrable_of_hasCompactSupport | |
| (hr 0).2.mul_left).sub | |
| ((continuous_apply 0 |>.comp hu).mul (hr 1).1.continuous |>.integrable_of_hasCompactSupport | |
| (hr 1).2.mul_left) | |
| end CompactSmoothPair | |
| namespace PairSupported | |
| variable {r s : PhysicalSourcePair} {U V : Set Coord3} | |
| lemma mono (hr : PairSupported r U) (hUV : U⊆V) : PairSupported r V := | |
| fun j => (hr j).trans hUV | |
| lemma zero : PairSupported 0 U := by intro j; simp [tsupport_zero] | |
| lemma add (hr : PairSupported r U) (hs : PairSupported s U) : PairSupported (r+s) U := | |
| fun j => (tsupport_add (r j) (s j)).trans (union_subset (hr j) (hs j)) | |
| lemma sub (hr : PairSupported r U) (hs : PairSupported s U) : PairSupported (r-s) U := | |
| fun j => (tsupport_sub (r j) (s j)).trans (union_subset (hr j) (hs j)) | |
| end PairSupported | |
| lemma physicalSourceMoment_zero (u : Coord3 → Fin 2 → ℝ) : physicalSourceMoment u 0=0 := by | |
| simp [physicalSourceMoment] | |
| lemma physicalSourceMoment_add {u : Coord3 → Fin 2 → ℝ} (hu : Continuous u) | |
| {r s : PhysicalSourcePair} (hr : CompactSmoothPair r) (hs : CompactSmoothPair s) : | |
| physicalSourceMoment u (r+s)=physicalSourceMoment u r+physicalSourceMoment u s := by | |
| have he : (fun x => u x 1*(r+s) 0 x-u x 0*(r+s) 1 x)= | |
| (fun x => (u x 1*r 0 x-u x 0*r 1 x)+(u x 1*s 0 x-u x 0*s 1 x)) := by | |
| funext x; change u x 1*(r 0 x+s 0 x)-u x 0*(r 1 x+s 1 x)=_; ring | |
| ext i | |
| fin_cases i | |
| · exact integral_add (hr.integrable 0) (hs.integrable 0) | |
| · exact integral_add (hr.integrable 1) (hs.integrable 1) | |
| · change (∫ x,u x 1*(r+s) 0 x-u x 0*(r+s) 1 x)=_ | |
| rw [he,integral_add (hr.torque_integrable hu) (hs.torque_integrable hu)] | |
| rfl | |
| lemma physicalSourceMoment_sub {u : Coord3 → Fin 2 → ℝ} (hu : Continuous u) | |
| {r s : PhysicalSourcePair} (hr : CompactSmoothPair r) (hs : CompactSmoothPair s) : | |
| physicalSourceMoment u (r-s)=physicalSourceMoment u r-physicalSourceMoment u s := by | |
| have he := physicalSourceMoment_add hu (hr.sub hs) hs | |
| rw [sub_add_cancel] at he | |
| exact eq_sub_iff_add_eq.mpr he.symm | |
| def PhysicallyCorrectable (u : Coord3 → Fin 2 → ℝ) (U : Set Coord3) | |
| (r : PhysicalSourcePair) : Prop := | |
| ∃ H : Coord3 → Mat3,ContDiff ℝ (↑(⊤:ℕ∞)) H ∧ HasCompactSupport H ∧ | |
| tsupport H⊆U ∧ (∀ x,(H x).IsSymm) ∧ ∀ j,symmetricSource H u j=r j | |
| lemma symmetricSource_eq_of_weak | |
| {u : Coord3 → Fin 2 → ℝ} (hu : ContDiff ℝ (↑(⊤:ℕ∞)) u) | |
| {H : Coord3 → Mat3} (hH : ContDiff ℝ (↑(⊤:ℕ∞)) H) (hc : HasCompactSupport H) | |
| {r : PhysicalSourcePair} (hr : CompactSmoothPair r) | |
| (hw : ∀ j (ψ : Coord3 → ℝ),ContDiff ℝ (↑(⊤:ℕ∞)) ψ → | |
| (∫ x,fderiv ℝ ψ x ((H x*gradientColumns (fderiv ℝ u x)).col j))=-(∫ x,ψ x*r j x)) : | |
| ∀ j,symmetricSource H u j=r j := by | |
| intro j | |
| have hs := coordinateDivergence_smooth (symmetricFlux_smooth hH hu j) | |
| apply Measure.eq_of_ae_eq (μ := volume) _ hs.continuous (hr j).1.continuous | |
| apply ae_eq_of_integral_contDiff_smul_eq hs.continuous.locallyIntegrable | |
| (hr j).1.continuous.locallyIntegrable | |
| intro ψ hψ _ | |
| have he := coordinateDivergence_weak (symmetricFlux_smooth hH hu j) | |
| (symmetricFlux_compact hc u j) ψ hψ | |
| have he' := hw j ψ hψ | |
| simp only [smul_eq_mul] | |
| change (∫ x,ψ x*coordinateDivergence (fun x => (H x*gradientColumns (fderiv ℝ u x)).col j) x)=_ | |
| linarith | |
| lemma coordinateDivergence_add {F G : Coord3 → Coord3} | |
| (hF : ContDiff ℝ (↑(⊤:ℕ∞)) F) (hG : ContDiff ℝ (↑(⊤:ℕ∞)) G) : | |
| coordinateDivergence (F+G)=coordinateDivergence F+coordinateDivergence G := by | |
| funext x | |
| simp only [coordinateDivergence,Pi.add_apply,←Finset.sum_add_distrib] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| have he := (((contDiff_pi.mp hF i).differentiable (by simp) x).hasFDerivAt.add | |
| ((contDiff_pi.mp hG i).differentiable (by simp) x).hasFDerivAt).fderiv | |
| exact congrArg (fun D : Coord3 →L[ℝ] ℝ => D (Pi.single i 1)) he | |
| lemma symmetricSource_add {H K : Coord3 → Mat3} {u : Coord3 → Fin 2 → ℝ} | |
| (hH : ContDiff ℝ (↑(⊤:ℕ∞)) H) (hK : ContDiff ℝ (↑(⊤:ℕ∞)) K) | |
| (hu : ContDiff ℝ (↑(⊤:ℕ∞)) u) (j : Fin 2) : | |
| symmetricSource (H+K) u j=symmetricSource H u j+symmetricSource K u j := by | |
| have he : (fun x => ((H+K) x*gradientColumns (fderiv ℝ u x)).col j)= | |
| (fun x => (H x*gradientColumns (fderiv ℝ u x)).col j)+ | |
| (fun x => (K x*gradientColumns (fderiv ℝ u x)).col j) := by | |
| funext x i | |
| simp [Matrix.add_mul] | |
| unfold symmetricSource | |
| rw [he,coordinateDivergence_add (symmetricFlux_smooth hH hu j) (symmetricFlux_smooth hK hu j)] | |
| namespace PhysicallyCorrectable | |
| variable {u : Coord3 → Fin 2 → ℝ} {U : Set Coord3} {r s : PhysicalSourcePair} | |
| lemma zero : PhysicallyCorrectable u U 0 := by | |
| refine ⟨0,contDiff_const,HasCompactSupport.zero,by simp [tsupport_zero],fun _ => Matrix.isSymm_zero,?_⟩ | |
| intro j | |
| funext x | |
| simp [symmetricSource,coordinateDivergence,Matrix.col] | |
| lemma add (hu : ContDiff ℝ (↑(⊤:ℕ∞)) u) | |
| (hr : PhysicallyCorrectable u U r) (hs : PhysicallyCorrectable u U s) : | |
| PhysicallyCorrectable u U (r+s) := by | |
| obtain ⟨H,hH,hHc,hHs,hHsy,hHr⟩ := hr | |
| obtain ⟨K,hK,hKc,hKs,hKsy,hKr⟩ := hs | |
| refine ⟨H+K,hH.add hK,hHc.add hKc,(tsupport_add H K).trans (union_subset hHs hKs), | |
| fun x => (hHsy x).add (hKsy x),?_⟩ | |
| intro j | |
| rw [symmetricSource_add hH hK hu j,hHr j,hKr j] | |
| rfl | |
| end PhysicallyCorrectable | |
| lemma RegularCorrectionBox.correct | |
| {u : Coord3 → Fin 2 → ℝ} (hu : ContDiff ℝ (↑(⊤:ℕ∞)) u) {U : Set Coord3} | |
| (B : RegularCorrectionBox u U) {r : PhysicalSourcePair} | |
| (hr : CompactSmoothPair r) (hs : PairSupported r B.region) | |
| (hm : physicalSourceMoment u r=0) : PhysicallyCorrectable u U r := by | |
| have hz (j : Fin 2) : (∫ x,r j x)=0 := by | |
| fin_cases j | |
| · exact congrFun hm 0 | |
| · exact congrFun hm 1 | |
| have ht : (∫ x,u x 1*r 0 x-u x 0*r 1 x)=0 := congrFun hm 2 | |
| obtain ⟨H,hH,hc,hsub,hsy,hw⟩ := local_physical_symmetric_correction B.chart B.smooth | |
| B.inverse_smooth B.widths B.subset_source u B.potentials r | |
| (fun j => (hr j).1) (fun j => (hr j).2) hs hz ht | |
| exact ⟨H,hH,hc,hsub.trans B.target_subset,hsy,symmetricSource_eq_of_weak hu hH hc hr hw⟩ | |
| end ScalarConductivity | |
| end | |
| end OAI | |