openAI-math-dataset / lean /OAI /Analysis /Conductivity /Variational /CentralCorrectedRepresentative.lean
| import Mathlib | |
| import OAI.Analysis.Conductivity.Branching.PhysicalFiniteEndingData | |
| import OAI.Analysis.Conductivity.Variational.CentralHarmonicOpen | |
| import OAI.Analysis.Conductivity.Flux.VariableBlockTensor | |
| import OAI.Analysis.Conductivity.Fourier.SpectralAffineCovector | |
| import OAI.Analysis.Conductivity.Sobolev.WeylPairGluing | |
| namespace OAI | |
| section | |
| noncomputable section | |
| namespace ScalarConductivity | |
| open Set MeasureTheory Filter Topology Matrix | |
| open scoped Matrix.Norms.Elementwise | |
| lemma H1JetOn.value_cartesian {w : H1} {D : Set Coord3} {f : R3 → JetFiber} | |
| (hw : H1JetOn w D f) (hDb : ∀ y∈D,WithLp.toLp 2 y∈ball) : | |
| ∀ᵐ y : Coord3,y∈D → weakValue w (WithLp.toLp 2 y)=f (WithLp.toLp 2 y) 0 := by | |
| have hh := (PiLp.volume_preserving_toLp (Fin 3)).quasiMeasurePreserving.ae | |
| ((ae_restrict_iff' (show MeasurableSet ball from Metric.isOpen_ball.measurableSet)).mp hw) | |
| filter_upwards [hh] with y hy hyD | |
| exact congrArg (fun z : JetFiber => z 0) (hy (hDb y hyD) hyD) | |
| lemma centralFullJet_value_ae (s : Fin 3 → ℝ) (p : CentralAmbient s) : | |
| ∀ᵐ y : Coord3,y∈centralPhysical → centralFullJetCLM s p (WithLp.toLp 2 y) 0= | |
| centralWholeL2 (centralAmbientComponent s 0 p) (WithLp.toLp 2 y) := by | |
| have hh := physicalFourJetCLM_ae (fun i => centralPhysicalWholeCLM (centralAmbientComponent s i p)) | |
| have hh' := (PiLp.volume_preserving_toLp (Fin 3)).quasiMeasurePreserving.ae | |
| ((ae_restrict_iff' (show MeasurableSet ball from Metric.isOpen_ball.measurableSet)).mp hh) | |
| have hr := (PiLp.volume_preserving_toLp (Fin 3)).quasiMeasurePreserving.ae | |
| (centralWholeL2_ae (centralAmbientComponent s 0 p)) | |
| filter_upwards [hh',hr,centralPhysicalWholeCLM_ae (centralAmbientComponent s 0 p)] with y hy hr hc hyC | |
| change physicalFourJetCLM (fun i => centralPhysicalWholeCLM (centralAmbientComponent s i p)) | |
| (WithLp.toLp 2 y) 0=_ | |
| rw [hy (centralPhysical_subset_ball hyC),hr] | |
| change centralPhysicalWholeCLM (centralAmbientComponent s 0 p) y=_ | |
| rw [hc] | |
| change centralPhysical.indicator (fun y => centralAmbientComponent s 0 p (sourcePairCoordinates y)) y= | |
| centralClosed.indicator (centralAmbientComponent s 0 p) (centralEuclideanCLE (WithLp.toLp 2 y)) | |
| rw [indicator_of_mem hyC,indicator_of_mem (show centralEuclideanCLE (WithLp.toLp 2 y)∈centralClosed from hyC)] | |
| rfl | |
| namespace PhysicalFiniteEndingData | |
| variable {s : Fin 3 → ℝ} (z : PhysicalFiniteEndingData s) | |
| lemma central_representative {w : Fin 2 → H1} | |
| (hw : ∀ j,H1JetOn (w j) centralPhysical (centralFullJetCLM s (z.p j).val)) : | |
| ∃ u : Coord3 → Fin 2 → ℝ, | |
| ContDiffOn ℝ (↑(⊤:ℕ∞)) u centralStrict ∧ | |
| (∀ᵐ y : Coord3,y∈centralStrict → u y=fun j => weakValue (w j) (WithLp.toLp 2 y)) ∧ | |
| ∀ᵐ y : Coord3,y∈centralStrict → y∈regularRegion u | |
| (fun _ => ⟨1,Matrix.transpose_one⟩) centralStrict := by | |
| let U : Set R3 := (WithLp.ofLp : R3 → Coord3) ⁻¹' centralStrict | |
| have hU : IsOpen U := centralStrict_open.preimage | |
| (PiLp.continuousLinearEquiv 2 ℝ (fun _ : Fin 3 => ℝ)).continuous | |
| obtain ⟨u,hs,he,hr⟩ := weakWeyl_pair_cartesian_representative | |
| (fun j => centralWholeL2 (centralAmbientComponent s 0 (z.p j).val)) hU | |
| (fun j => (z.harmonic j).mono centralStrict_harmonic) | |
| refine ⟨u,hs,?_,hr⟩ | |
| have hj (j : Fin 2) : ∀ᵐ y : Coord3,y∈centralPhysical → | |
| weakValue (w j) (WithLp.toLp 2 y)=centralWholeL2 (centralAmbientComponent s 0 (z.p j).val) (WithLp.toLp 2 y) := by | |
| filter_upwards [(hw j).value_cartesian (fun _ => centralPhysical_subset_ball), | |
| centralFullJet_value_ae s (z.p j).val] with y hw hv hy | |
| exact (hw hy).trans (hv hy) | |
| filter_upwards [he,ae_all_iff.mpr hj] with y he hw hy | |
| rw [he hy] | |
| funext j | |
| exact (hw j (centralStrict_subset hy)).symm | |
| end PhysicalFiniteEndingData | |
| end ScalarConductivity | |
| end | |
| end | |
| section | |
| noncomputable section | |
| namespace ScalarConductivity | |
| open Set MeasureTheory Filter Topology Matrix | |
| open scoped Matrix.Norms.Elementwise | |
| lemma correctedEndValue_face {s : Fin 3 → ℝ} | |
| (hs : ∀ x y : ℝ,(1/2)*(x^2+y^2)≤ s 0*x^2+2*s 1*x*y+s 2*y^2) | |
| (f : Fin 2 → spectralTraceGraph (torusRate s)) (i : Fin 3) | |
| (z : TorusEndingData s (branchNormalize i (fun j => torusSpectralSynthesis s (f j)))) | |
| (j : Fin 2) (a b : ℝ) (m n : Fin 4) {x : Coord3} | |
| (hx : x∈sourceCollarOpenBox) (ht : 0<a*(x 0-b)) : | |
| fullAttachedEndValue s (f j) a b (centralBasisSlopes j i) (sourceCollarPiece m n x)+ | |
| attachedPeriodicField (branchCorrection i z j) a b (sourceCollarPiece m n x)= | |
| variableCollarField (fun y j => branchSplicedField i z j y) a b (sourceCollarPiece m n x) j := by | |
| rw [variableCollarField_open _ a b m n hx] | |
| change (attachedEndPoissonField s (f j) a b 0 (sourceCollarPiece m n x)).re+ | |
| (centralBasisSlopes j i)*(a*(sourceCollarTime (sourceCollarPiece m n x)-b))+ | |
| physicalPeriodicField (fun y => branchCorrection i z j (flatEndCoordinates a b y)) (sourceCollarPiece m n x)=_ | |
| rw [attachedEndPoissonField_piece _ _ _ _ _ _ (sourceCollarOpenBox_subset hx), | |
| physicalPeriodicField_piece ((branchCorrection_periodic i z j).flatEnd a b) m n | |
| (sourceCollarOpenBox_subset hx),sourceCollarPiece_time_open hx] | |
| unfold branchSplicedField torusAffineField | |
| rw [(torusRealContinuation_spectral_eventually hs (f j) | |
| (show 0<(flatEndCoordinates a b (sourceFaceAngles m n x)) 0 from ht)).eq_of_nhds] | |
| change _+_+_= (centralBasisSlopes j i)*(a*(x 0-b))+ | |
| (endFlatPoisson s (f j) (endAxialAffine a b (sourceFaceAngles m n x))).re+_ | |
| ring | |
| lemma correctedEndValue_band {s : Fin 3 → ℝ} | |
| (hs : ∀ x y : ℝ,(1/2)*(x^2+y^2)≤ s 0*x^2+2*s 1*x*y+s 2*y^2) | |
| (f : Fin 2 → spectralTraceGraph (torusRate s)) (i : Fin 3) | |
| (z : TorusEndingData s (branchNormalize i (fun j => torusSpectralSynthesis s (f j)))) | |
| (j : Fin 2) (a b : ℝ) {l r : ℝ} (hl : -(1:ℝ)/100≤l) (hr : r≤1/100) | |
| (hT : ∀ t∈Ioo l r,0<a*(t-b)) : | |
| ∀ᵐ y∂volume.restrict (sourceClosedCollarBand l r), | |
| fullAttachedEndValue s (f j) a b (centralBasisSlopes j i) y+ | |
| attachedPeriodicField (branchCorrection i z j) a b y= | |
| variableCollarField (fun y j => branchSplicedField i z j y) a b y j := by | |
| apply sourceClosedCollarBand_ae_faces hl hr | |
| intro m n | |
| filter_upwards [sourceExtendedBox_open_ae hl hr,sourceExtendedBox_axial_strict_ae l r] | |
| with x hxo hxt | |
| exact correctedEndValue_face hs f i z j a b m n hxo (hT _ hxt) | |
| namespace PhysicalFiniteEndingData | |
| variable {s : Fin 3 → ℝ} (z : PhysicalFiniteEndingData s) | |
| def endRepresentative (i : Fin 3) : Coord3 → Fin 2 → ℝ := | |
| Fin.cases (variableCollarField (fun y j => branchSplicedField 0 (z.ending 0) j y) | |
| (z.compression 0) centralThickness) | |
| (fun k => fun y => variableCollarField (fun y j => branchSplicedField k.succ (z.ending k.succ) j y) | |
| (z.compression k.succ) (-centralThickness) ((sourceChildHomeomorph (actualChildSign k)).symm y)) i | |
| lemma parent_open_time {t : ℝ} (ht : t∈Ioo 0 centralThickness) : | |
| 0<z.compression 0*(t-centralThickness) := mul_pos_of_neg_of_neg z.compression_parent (sub_neg.mpr ht.2) | |
| lemma child_open_time (k : Fin 2) {t : ℝ} (ht : t∈Ioo (-centralThickness) 0) : | |
| 0<z.compression k.succ*(t-(-centralThickness)) := mul_pos (z.compression_child k) (sub_pos.mpr ht.1) | |
| lemma endRepresentative_value | |
| (hs : ∀ x y : ℝ,(1/2)*(x^2+y^2)≤ s 0*x^2+2*s 1*x*y+s 2*y^2) (i : Fin 3) (j : Fin 2) : | |
| ∀ᵐ y : Coord3,y∈physicalEndRegion i → z.correctedEndJet j i (WithLp.toLp 2 y) 0=z.endRepresentative i y j := by | |
| refine Fin.cases ?_ (fun k => ?_) i | |
| · have hh := correctedEndValue_band hs (fun j => centralT s 0 (z.p j)) 0 (z.ending 0) j | |
| (z.compression 0) centralThickness (by norm_num : -(1:ℝ)/100≤0) | |
| (by norm_num [centralThickness] : centralThickness≤(1:ℝ)/100) (fun t => z.parent_open_time) | |
| exact (ae_restrict_iff' (physicalEndRegion_compact 0).measurableSet).mp hh | |
| · have hh := correctedEndValue_band hs (fun j => centralT s k.succ (z.p j)) k.succ (z.ending k.succ) j | |
| (z.compression k.succ) (-centralThickness) (by norm_num [centralThickness] : -(1:ℝ)/100≤-centralThickness) | |
| (by norm_num : (0:ℝ)≤1/100) (fun t => z.child_open_time k) | |
| exact (sourceChildInverse_quasi (actualChildSign k)).ae | |
| ((ae_restrict_iff' childEndBand_measurable).mp hh) | |
| lemma endRepresentative_regular | |
| (hs : ∀ x y : ℝ,(1/2)*(x^2+y^2)≤ s 0*x^2+2*s 1*x*y+s 2*y^2) | |
| (hS : ∀ i y,(variableEndTensor (fun i => branchSplicedTensor i (z.ending i)) z.compression i y).IsSymm) | |
| (i : Fin 3) : ∀ᵐ y : Coord3,y∈physicalEndRegion i → | |
| y∈regularRegion (z.endRepresentative i) | |
| (fun y => ⟨variableEndTensor (fun i => branchSplicedTensor i (z.ending i)) z.compression i y,hS i y⟩) univ := by | |
| refine Fin.cases ?_ (fun k => ?_) i | |
| · exact (ae_restrict_iff' (physicalEndRegion_compact 0).measurableSet).mp | |
| (branchCollar_regular_ae hs 0 (z.ending 0) (z.compression_ne 0) | |
| (by norm_num : -(1:ℝ)/100≤0) (by norm_num [centralThickness] : centralThickness≤(1:ℝ)/100) | |
| (fun t => z.parent_open_time)) | |
| · have hh := branchCollar_regular_ae hs k.succ (z.ending k.succ) (z.compression_ne k.succ) | |
| (by norm_num [centralThickness] : -(1:ℝ)/100≤-centralThickness) (by norm_num : (0:ℝ)≤1/100) | |
| (fun t => z.child_open_time k) | |
| have hh' := (sourceChildInverse_quasi (actualChildSign k)).ae | |
| ((ae_restrict_iff' childEndBand_measurable).mp hh) | |
| filter_upwards [hh'] with y hy hyE | |
| have hr := sourceChild_regular_at k | |
| (variableCollarTensor_symm (branchSplicedTensor_symm k.succ (z.ending k.succ)) | |
| (z.compression k.succ) (-centralThickness)) (hy hyE) | |
| simpa only [preimage_univ, endRepresentative, variableEndTensor, Fin.cases_succ] using hr | |
| end PhysicalFiniteEndingData | |
| end ScalarConductivity | |
| end | |
| end | |
| end OAI | |