| import Mathlib |
| import OAI.Analysis.Conductivity.Variational.PhysicalFaceUniqueness |
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| namespace OAI |
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| section |
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| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Matrix |
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| variable {E : Type*} |
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| def sourceFacePasteList (B : Fin 4 → Fin 4 → Coord3 → E) (v : E) |
| (ks : List (Fin 4 × Fin 4)) (y : Coord3) : E := by |
| classical |
| exact ks.foldr (fun k z => if y∈sourceFlatPatch k.1 k.2 then B k.1 k.2 y else z) v |
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| lemma sourceFacePasteList_cons (B : Fin 4 → Fin 4 → Coord3 → E) (v : E) |
| (k : Fin 4 × Fin 4) (ks : List (Fin 4 × Fin 4)) (y : Coord3) : |
| sourceFacePasteList B v (k::ks) y= |
| (by classical exact if y∈sourceFlatPatch k.1 k.2 then B k.1 k.2 y else sourceFacePasteList B v ks y) := rfl |
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| lemma sourceFacePasteList_measurable [MeasurableSpace E] |
| {B : Fin 4 → Fin 4 → Coord3 → E} (hB : ∀ i j,Measurable (B i j)) |
| (v : E) (ks : List (Fin 4 × Fin 4)) : Measurable (sourceFacePasteList B v ks) := by |
| induction ks with |
| | nil => exact measurable_const |
| | cons k ks ih => |
| exact Measurable.ite (isCompact_sourceFlatPatch k.1 k.2).measurableSet (hB k.1 k.2) ih |
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| lemma sourceFacePasteList_local (B : Fin 4 → Fin 4 → Coord3 → E) (v : E) |
| (ks : List (Fin 4 × Fin 4)) (y : Coord3) |
| (hy : ∃ k∈ks,y∈sourceFlatPatch k.1 k.2) : |
| ∃ k∈ks,y∈sourceFlatPatch k.1 k.2 ∧ sourceFacePasteList B v ks y=B k.1 k.2 y := by |
| classical |
| induction ks with |
| | nil => simp at hy |
| | cons k ks ih => |
| by_cases hk : y∈sourceFlatPatch k.1 k.2 |
| · exact ⟨k,List.mem_cons_self,hk,by rw [sourceFacePasteList_cons,ite_eq_left hk]⟩ |
| · obtain ⟨l,hl,hly⟩ := hy |
| have hl' : l∈ks := (List.mem_cons.mp hl).resolve_left (by rintro rfl; exact hk hly) |
| obtain ⟨l,hl,hly,he⟩ := ih ⟨l,hl',hly⟩ |
| exact ⟨l,List.mem_cons_of_mem _ hl,hly,by rw [sourceFacePasteList_cons,ite_eq_right hk,he]⟩ |
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| lemma sourceFacePasteList_open (B : Fin 4 → Fin 4 → Coord3 → E) (v : E) |
| (ks : List (Fin 4 × Fin 4)) (i j : Fin 4) {x : Coord3} |
| (hx : x∈sourceCollarOpenBox) (hk : (i,j)∈ks) : |
| sourceFacePasteList B v ks (sourceCollarPiece i j x)=B i j (sourceCollarPiece i j x) := by |
| obtain ⟨k,_,hy,he⟩ := sourceFacePasteList_local B v ks _ |
| ⟨(i,j),hk,⟨x,sourceCollarOpenBox_subset hx,rfl⟩⟩ |
| obtain ⟨z,hz,hze⟩ := hy |
| obtain ⟨hi,hj⟩ := sourceCollarPiece_open_unique i j k.1 k.2 hx hz hze.symm |
| simpa only [←hi,←hj] using he |
|
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| def sourceFacePaste (B : Fin 4 → Fin 4 → Coord3 → E) (v : E) : Coord3 → E := |
| sourceFacePasteList B v (Finset.univ : Finset (Fin 4 × Fin 4)).toList |
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| lemma sourceFacePaste_measurable [MeasurableSpace E] |
| {B : Fin 4 → Fin 4 → Coord3 → E} (hB : ∀ i j,Measurable (B i j)) (v : E) : |
| Measurable (sourceFacePaste B v) := sourceFacePasteList_measurable hB v _ |
|
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| lemma sourceFacePaste_open (B : Fin 4 → Fin 4 → Coord3 → E) (v : E) |
| (i j : Fin 4) {x : Coord3} (hx : x∈sourceCollarOpenBox) : |
| sourceFacePaste B v (sourceCollarPiece i j x)=B i j (sourceCollarPiece i j x) := |
| sourceFacePasteList_open B v _ i j hx (by simp) |
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| lemma measurable_sourceCollarInverse (i j : Fin 4) : Measurable (sourceCollarInverse i j) := by |
| unfold sourceCollarInverse squareFaceParameter sourceCollarTime sourceCrossCoordinates sourceRadial |
| fun_prop |
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| def flatEndCoordinates (a b : ℝ) (x : Coord3) : Coord3 := ![a*(x 0-b),x 1,x 2] |
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| lemma contDiff_flatEndCoordinates (a b : ℝ) : ContDiff ℝ (↑(⊤:ℕ∞)) (flatEndCoordinates a b) := by |
| apply contDiff_pi.mpr |
| intro i |
| fin_cases i <;> dsimp [flatEndCoordinates] <;> fun_prop |
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| lemma flatEndCoordinates_angles (a b : ℝ) (x : Coord3) : |
| torusAngles (flatEndCoordinates a b x)=torusAngles x := rfl |
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| lemma flatEndCoordinates_axial (a b : ℝ) (x : Coord3) : |
| flatEndCoordinates a b x 0=a*(x 0-b) := rfl |
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| end ScalarConductivity |
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| end |
| end |
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| end OAI |
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