openAI-math-dataset / lean /OAI /Analysis /Conductivity /Variational /CompletePhysicalAttachment.lean
| import Mathlib | |
| import OAI.Analysis.Conductivity.Branching.PhysicalJoin | |
| import OAI.Analysis.Conductivity.Branching.PhysicalAttachedAssembly | |
| namespace OAI | |
| section | |
| noncomputable section | |
| namespace ScalarConductivity | |
| open Set MeasureTheory Filter Topology | |
| def attachedPhysicalJet (s a κ : Fin 3 → ℝ) (p : centralEnergySpace s) | |
| (i : Fin 3) (x : R3) : JetFiber := | |
| Fin.cases | |
| (physicalJet (fullAttachedEndValue s (centralT s 0 p) (a 0) centralThickness | |
| (κ 0) (WithLp.ofLp x)) | |
| (fullAttachedEndGradient s (centralT s 0 p) (a 0) centralThickness | |
| (κ 0) (WithLp.ofLp x))) | |
| (fun k => physicalJet | |
| (fullAttachedEndValue s (centralT s k.succ p) (a k.succ) (-centralThickness) | |
| (κ k.succ) ((sourceChildHomeomorph (actualChildSign k)).symm (WithLp.ofLp x))) | |
| (fun j => sourceScale⁻¹*fullAttachedEndGradient s (centralT s k.succ p) | |
| (a k.succ) (-centralThickness) (κ k.succ) | |
| ((sourceChildHomeomorph (actualChildSign k)).symm (WithLp.ofLp x)) (childAxis j))) i | |
| theorem complete_physical_attachment (s : Fin 3 → ℝ) | |
| (hs : ∀ u v : ℝ,(1/2)*(u^2+v^2) ≤ s 0*u^2+2*s 1*u*v+s 2*v^2) | |
| (a κ : Fin 3 → ℝ) (ha₀ : a 0<0) (ha : ∀ k : Fin 2,0<a k.succ) | |
| (p : centralEnergySpace s) : | |
| ∃ w : H1,w∈H10 ∧ | |
| H1JetOn w centralPhysical (centralFullJetCLM s p.val) ∧ | |
| ∀ i : Fin 3,H1JetOn w (physicalEndRegion i) (attachedPhysicalJet s a κ p i) := by | |
| let zi := centralInteriorCompletion s (centralJoinPartition_smooth 0) | |
| centralJoinPartition_interior_compact centralJoinPartition_interior_support | |
| (fun _ hy => centralPhysical_subset_ball (centralJoinPartition_interior_support hy)) p | |
| let zp := parentCompletionJoin s hs ha₀ (centralJoinPartition_smooth 1) | |
| centralJoinPartition_parent_compact (centralJoinPartition_bound 1) | |
| centralJoinPartition_parent_support p | |
| let zc (k : Fin 2) := physicalChildCompletionJoin s hs (ha k) k (centralJoinChildCutoff_smooth k) | |
| (centralJoinChildCutoff_compact k) (centralJoinChildCutoff_bound k) | |
| (centralJoinChildCutoff_support k) p | |
| obtain ⟨dp,hdp,hds,hdpZ,hdpE⟩ := parentAttachedCorrection_supported s hs ha₀ (κ 0) p | |
| have hdcEx (k : Fin 2) := physicalChildAttachedCorrection_supported s hs (ha k) k (κ k.succ) p | |
| choose dc hdc hdcZ hdcE using hdcEx | |
| let w : H1 := zi+(zp+dp)+((zc 0+dc 0)+(zc 1+dc 1)) | |
| have hw : w∈H10 := H10.add_mem | |
| (H10.add_mem (centralInteriorCompletion_mem_H10 s _ _ _ _ p) | |
| (H10.add_mem (parentCompletionJoin_mem_H10 s hs ha₀ _ _ _ _ p) hdp)) | |
| (H10.add_mem | |
| (H10.add_mem (physicalChildCompletionJoin_mem_H10 s hs (ha 0) 0 _ _ _ _ p) (hdc 0)) | |
| (H10.add_mem (physicalChildCompletionJoin_mem_H10 s hs (ha 1) 1 _ _ _ _ p) (hdc 1))) | |
| have hpC : H1JetOn dp centralPhysical (fun _ => 0) := | |
| (H1JetOn.zero_of_weak (D:={y | centralThickness≤ sourceCollarTime y}) hdpZ).mono | |
| (fun y hy => ((centralPhysical_time_iff y).mp hy).1) | |
| have hcC (k : Fin 2) : H1JetOn (dc k) centralPhysical (fun _ => 0) := | |
| (H1JetOn.zero_of_weak (D:={y | sourceCollarTime | |
| ((sourceChildHomeomorph (actualChildSign k)).symm y)≤-centralThickness}) (hdcZ k)).mono | |
| (fun y hy => ((centralPhysical_time_iff y).mp hy).2 k) | |
| have hpO (k : Fin 2) : H1JetOn (zp+dp) (physicalEndRegion k.succ) (fun _ => 0) := by | |
| apply H1JetOn.add_zero | |
| · exact H1JetOn.zero_of_weak (parent_join_zero_on_children s hs ha₀ p k) | |
| · apply (H1JetOn.zero_of_weak (D:={y | centralThickness≤ sourceCollarTime y}) hdpZ).mono | |
| intro y hy | |
| exact (show centralThickness≤2*centralThickness by norm_num [centralThickness]).trans | |
| (physicalChildEnd_parent_time k hy) | |
| have hcO (k : Fin 2) (i : Fin 3) (hik : i≠k.succ) : | |
| H1JetOn (zc k+dc k) (physicalEndRegion i) (fun _ => 0) := by | |
| apply H1JetOn.add_zero | |
| · exact H1JetOn.zero_of_weak (physical_child_join_zero_on_other_ends s hs (ha k) p k i hik) | |
| · apply (H1JetOn.zero_of_weak (D:={y | sourceCollarTime | |
| ((sourceChildHomeomorph (actualChildSign k)).symm y)≤-centralThickness}) (hdcZ k)).mono | |
| intro y hy | |
| exact (physicalOtherEnd_child_time k i hik hy).le.trans | |
| (by norm_num [centralThickness]) | |
| have hiO (i : Fin 3) : H1JetOn zi (physicalEndRegion i) (fun _ => 0) := | |
| H1JetOn.zero_of_weak (central_interior_join_zero_on_ends s p i) | |
| have hpE : H1JetOn (zp+dp) (physicalEndRegion 0) (attachedPhysicalJet s a κ p 0) := | |
| H1JetOn.of_weak hdpE | |
| have hcE (k : Fin 2) : H1JetOn (zc k+dc k) (physicalEndRegion k.succ) | |
| (attachedPhysicalJet s a κ p k.succ) := H1JetOn.of_weak (hdcE k) | |
| refine ⟨w,hw,?_,?_⟩ | |
| · have he : w=centralNeighborhoodCompletion s hs ha₀ ha p+(dp+(dc 0+dc 1)) := by | |
| simp only [w,centralNeighborhoodCompletion,Fin.sum_univ_two,add_apply] | |
| change zi+(zp+dp)+((zc 0+dc 0)+(zc 1+dc 1))=zi+zp+(zc 0+zc 1)+(dp+(dc 0+dc 1)) | |
| abel | |
| rw [he] | |
| have hc : H1JetOn (centralNeighborhoodCompletion s hs ha₀ ha p) centralPhysical | |
| (centralFullJetCLM s p.val) := centralNeighborhoodCompletion_central_ae s hs ha₀ ha p | |
| exact hc.add_zero (hpC.add_zero ((hcC 0).add_zero (hcC 1))) | |
| · intro i | |
| refine Fin.cases ?_ (fun k => ?_) i | |
| · exact ((hiO 0).zero_add hpE).add_zero | |
| ((hcO 0 0 (by decide)).add_zero (hcO 1 0 (by decide))) | |
| · fin_cases k | |
| · exact ((hiO 1).add_zero (hpO 0)).zero_add | |
| ((hcE 0).add_zero (hcO 1 1 (by decide))) | |
| · exact ((hiO 2).add_zero (hpO 1)).zero_add | |
| ((hcO 0 2 (by decide)).zero_add (hcE 1)) | |
| end ScalarConductivity | |
| end | |
| end | |
| end OAI | |