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159357f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 | import Mathlib
import OAI.Analysis.Conductivity.Variational.PhysicalFaceUniqueness
namespace OAI
section
noncomputable section
namespace ScalarConductivity
open Set MeasureTheory Matrix
variable {E : Type*}
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
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
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
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]⟩
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
def sourceFacePaste (B : Fin 4 → Fin 4 → Coord3 → E) (v : E) : Coord3 → E :=
sourceFacePasteList B v (Finset.univ : Finset (Fin 4 × Fin 4)).toList
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 _
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)
lemma measurable_sourceCollarInverse (i j : Fin 4) : Measurable (sourceCollarInverse i j) := by
unfold sourceCollarInverse squareFaceParameter sourceCollarTime sourceCrossCoordinates sourceRadial
fun_prop
def flatEndCoordinates (a b : ℝ) (x : Coord3) : Coord3 := ![a*(x 0-b),x 1,x 2]
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
lemma flatEndCoordinates_angles (a b : ℝ) (x : Coord3) :
torusAngles (flatEndCoordinates a b x)=torusAngles x := rfl
lemma flatEndCoordinates_axial (a b : ℝ) (x : Coord3) :
flatEndCoordinates a b x 0=a*(x 0-b) := rfl
end ScalarConductivity
end
end
end OAI
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