| import Mathlib |
| import OAI.MathematicalPhysics.SheetFlows.Localization |
|
|
| namespace OAI |
|
|
| /-! SheetFlows shear core. -/ |
|
|
| section |
| noncomputable section |
| open Set MeasureTheory |
| open scoped BigOperators |
| namespace Solenoidal |
|
|
| variable (d : Collar) |
|
|
| def coreLift0 : StationaryPulse := |
| StationaryPulse.tensor 2 ![Dprofile (1/2) d.size 5, Cprofile (1/2) d.size 6, fun _ => 1] |
| (by intro k; fin_cases k |
| · exact Dprofile_contDiff _ _ d.positive |
| · exact Cprofile_contDiff _ _ d.positive |
| · exact contDiff_const) |
| (by intro k; fin_cases k |
| · exact Dprofile_periodic _ _ _ |
| · exact Cprofile_periodic _ _ _ |
| · exact fun _ => rfl) |
| (fun _ => rfl) 0 |
| (by apply Dprofile_integral_zero d.positive <;> linarith [d.small]) |
|
|
| def coreLift1 : StationaryPulse := |
| StationaryPulse.tensor 2 ![Cprofile (1/2) d.size 5, Dprofile (1/2) d.size 6, fun _ => 1] |
| (by intro k; fin_cases k |
| · exact Cprofile_contDiff _ _ d.positive |
| · exact Dprofile_contDiff _ _ d.positive |
| · exact contDiff_const) |
| (by intro k; fin_cases k |
| · exact Cprofile_periodic _ _ _ |
| · exact Dprofile_periodic _ _ _ |
| · exact fun _ => rfl) |
| (fun _ => rfl) 1 |
| (by apply Dprofile_integral_zero d.positive <;> linarith [d.small]) |
|
|
| def coreHorizontal (j : Fin 2) : StationaryPulse := |
| StationaryPulse.tensor j.castSucc ![fun _ => 1, fun _ => 1, Dprofile (1/2) d.size 5] |
| (by intro k; fin_cases k |
| · exact contDiff_const |
| · exact contDiff_const |
| · exact Dprofile_contDiff _ _ d.positive) |
| (by intro k; fin_cases k |
| · exact fun _ => rfl |
| · exact fun _ => rfl |
| · exact Dprofile_periodic _ _ _) |
| (by intro s; fin_cases j <;> rfl) 2 |
| (by apply Dprofile_integral_zero d.positive <;> linarith [d.small]) |
|
|
| theorem coreLift0_field (x : Space) : (coreLift0 d).field x = |
| (Dprofile (1/2) d.size 5 (x 0) * Cprofile (1/2) d.size 6 (x 1)) • basis 2 := by |
| simp [coreLift0, StationaryPulse.tensor, tensorVelocity, tensorScalar, Fin.prod_univ_succ] |
|
|
| theorem coreLift1_field (x : Space) : (coreLift1 d).field x = |
| (Cprofile (1/2) d.size 5 (x 0) * Dprofile (1/2) d.size 6 (x 1)) • basis 2 := by |
| simp [coreLift1, StationaryPulse.tensor, tensorVelocity, tensorScalar, Fin.prod_univ_succ] |
|
|
| theorem coreHorizontal_field (j : Fin 2) (x : Space) : (coreHorizontal d j).field x = |
| Dprofile (1/2) d.size 5 (x 2) • basis j.castSucc := by |
| simp [coreHorizontal, StationaryPulse.tensor, tensorVelocity, tensorScalar, Fin.prod_univ_succ] |
|
|
| theorem coreC_one {c r : ℝ} (hc : c ∈ Set.Icc 5 6) (hr : |r| ≤ 1/2) : |
| Cprofile (1/2) d.size c (c+r) = 1 := by |
| apply Cprofile_one d.positive |
| · linarith [d.small, hc.1] |
| · linarith [d.small, hc.2] |
| · have hh := abs_le.mp hr |
| constructor <;> linarith [hc.1, hc.2, hh.1, hh.2] |
| · rw [add_sub_cancel_left] |
| linarith [d.positive] |
|
|
| theorem coreD_linear {c r : ℝ} (hc : c ∈ Set.Icc 5 6) (hr : |r| ≤ 1/2) : |
| Dprofile (1/2) d.size c (c+r) = r := by |
| have hl : 1/2 + d.size ≤ c := by linarith [d.small, hc.1] |
| have hu : c + (1/2+d.size) ≤ 10 := by linarith [d.small, hc.2] |
| have hp : c+r ∈ Set.Icc 0 10 := by |
| have hh := abs_le.mp hr |
| constructor <;> linarith [hc.1, hc.2, hh.1, hh.2] |
| have hb : |c+r-c| ≤ 1/2+d.size/2 := by |
| rw [add_sub_cancel_left] |
| linarith [d.positive] |
| simpa only [add_sub_cancel_left] using Dprofile_linear d.positive hl hu hp hb |
|
|
| theorem coreC_off {c y : ℝ} (hc : c ∈ Set.Icc 5 6) (hy : y ∈ Set.Icc 2 3) : |
| Cprofile (1/2) d.size c y = 0 := by |
| apply Cprofile_zero_on_chart d.positive |
| · linarith [d.small, hc.1] |
| · linarith [d.small, hc.2] |
| · constructor <;> linarith [hy.1, hy.2] |
| · apply le_abs.mpr (Or.inr ?_) |
| linarith [d.small, hc.1, hy.2] |
|
|
| theorem coreD_off {c y : ℝ} (hc : c ∈ Set.Icc 5 6) (hy : y ∈ Set.Icc 2 3) : |
| Dprofile (1/2) d.size c y = 0 := by |
| apply Dprofile_zero_on_chart d.positive |
| · linarith [d.small, hc.1] |
| · linarith [d.small, hc.2] |
| · constructor <;> linarith [hy.1, hy.2] |
| · apply le_abs.mpr (Or.inr ?_) |
| linarith [d.small, hc.1, hy.2] |
|
|
| theorem coreLift0_on {r s : ℝ} (hr : |r| ≤ 1/2) (hs : |s| ≤ 1/2) (z : ℝ) : |
| (coreLift0 d).field ![5+r,6+s,z] = r • basis 2 := by |
| simp only [coreLift0_field, Matrix.cons_val_zero, Matrix.cons_val_one, |
| coreD_linear d (by norm_num : (5:ℝ) ∈ Set.Icc 5 6) hr, |
| coreC_one d (by norm_num : (6:ℝ) ∈ Set.Icc 5 6) hs, mul_one] |
|
|
| theorem coreLift1_on {r s : ℝ} (hr : |r| ≤ 1/2) (hs : |s| ≤ 1/2) (z : ℝ) : |
| (coreLift1 d).field ![5+r,6+s,z] = s • basis 2 := by |
| simp only [coreLift1_field, Matrix.cons_val_zero, Matrix.cons_val_one, |
| coreC_one d (by norm_num : (5:ℝ) ∈ Set.Icc 5 6) hr, |
| coreD_linear d (by norm_num : (6:ℝ) ∈ Set.Icc 5 6) hs, one_mul] |
|
|
| theorem coreLift0_off (x : Space) (hx : x 1 ∈ Set.Icc 2 3) : (coreLift0 d).field x = 0 := by |
| rw [coreLift0_field, coreC_off d (by norm_num : (6:ℝ) ∈ Set.Icc 5 6) hx, mul_zero, zero_smul] |
|
|
| theorem coreLift1_off (x : Space) (hx : x 1 ∈ Set.Icc 2 3) : (coreLift1 d).field x = 0 := by |
| rw [coreLift1_field, coreD_off d (by norm_num : (6:ℝ) ∈ Set.Icc 5 6) hx, mul_zero, zero_smul] |
|
|
| theorem coreHorizontal_on (j : Fin 2) (x y : ℝ) {r : ℝ} (hr : |r| ≤ 1/2) : |
| (coreHorizontal d j).field ![x,y,5+r] = r • basis j.castSucc := by |
| rw [coreHorizontal_field] |
| change Dprofile (1/2) d.size 5 (5+r) • basis j.castSucc = _ |
| rw [coreD_linear d (by norm_num : (5:ℝ) ∈ Set.Icc 5 6) hr] |
|
|
| theorem coreHorizontal_off (j : Fin 2) (x : Space) (hx : x 2 = 2) : |
| (coreHorizontal d j).field x = 0 := by |
| rw [coreHorizontal_field, hx, |
| coreD_off d (by norm_num : (5:ℝ) ∈ Set.Icc 5 6) (by norm_num : (2:ℝ) ∈ Set.Icc 2 3), zero_smul] |
|
|
| def tripleProgram0 (l : ℝ) : List StationaryPulse := |
| [coreLift0 d, (coreHorizontal d 0).scale (l-1), (coreLift0 d).scale (-1/l)] |
|
|
| def tripleProgram1 (l : ℝ) : List StationaryPulse := |
| [coreLift1 d, (coreHorizontal d 1).scale (l-1), (coreLift1 d).scale (-1/l)] |
|
|
| theorem tripleProgram0_on {l r s : ℝ} (hl : l ≠ 0) |
| (hr : |r| ≤ 1/2) (hs : |s| ≤ 1/2) (hlr : |l*r| ≤ 1/2) : |
| runProgram (tripleProgram0 d l) ![5+r,6+s,5] = ![5+l*r,6+s,5] := by |
| have h₁ : (coreLift0 d).apply ![5+r,6+s,5] = ![5+r,6+s,5+r] := by |
| rw [StationaryPulse.apply, coreLift0_on d hr hs] |
| ext i; fin_cases i <;> simp [basis] |
| have h₂ : ((coreHorizontal d 0).scale (l-1)).apply ![5+r,6+s,5+r] = |
| ![5+l*r,6+s,5+r] := by |
| rw [StationaryPulse.apply, StationaryPulse.scale_field, coreHorizontal_on d 0 _ _ hr] |
| ext i; fin_cases i <;> simp [basis] |
| ring |
| have h₃ : ((coreLift0 d).scale (-1/l)).apply ![5+l*r,6+s,5+r] = ![5+l*r,6+s,5] := by |
| rw [StationaryPulse.apply, StationaryPulse.scale_field, coreLift0_on d hlr hs] |
| ext i; fin_cases i <;> simp [basis] |
| field_simp [hl] |
| ring |
| change ((coreLift0 d).scale (-1/l)).apply |
| (((coreHorizontal d 0).scale (l-1)).apply ((coreLift0 d).apply ![5+r,6+s,5])) = _ |
| rw [h₁, h₂, h₃] |
|
|
| theorem tripleProgram1_on {l r s : ℝ} (hl : l ≠ 0) |
| (hr : |r| ≤ 1/2) (hs : |s| ≤ 1/2) (hls : |l*s| ≤ 1/2) : |
| runProgram (tripleProgram1 d l) ![5+r,6+s,5] = ![5+r,6+l*s,5] := by |
| have h₁ : (coreLift1 d).apply ![5+r,6+s,5] = ![5+r,6+s,5+s] := by |
| rw [StationaryPulse.apply, coreLift1_on d hr hs] |
| ext i; fin_cases i <;> simp [basis] |
| have h₂ : ((coreHorizontal d 1).scale (l-1)).apply ![5+r,6+s,5+s] = |
| ![5+r,6+l*s,5+s] := by |
| rw [StationaryPulse.apply, StationaryPulse.scale_field, coreHorizontal_on d 1 _ _ hs] |
| ext i; fin_cases i <;> simp [basis] |
| ring |
| have h₃ : ((coreLift1 d).scale (-1/l)).apply ![5+r,6+l*s,5+s] = ![5+r,6+l*s,5] := by |
| rw [StationaryPulse.apply, StationaryPulse.scale_field, coreLift1_on d hr hls] |
| ext i; fin_cases i <;> simp [basis] |
| field_simp [hl] |
| ring |
| change ((coreLift1 d).scale (-1/l)).apply |
| (((coreHorizontal d 1).scale (l-1)).apply ((coreLift1 d).apply ![5+r,6+s,5])) = _ |
| rw [h₁, h₂, h₃] |
|
|
| def processingProgram (l m : ℝ) : List StationaryPulse := |
| tripleProgram0 d l ++ tripleProgram1 d m |
|
|
| theorem processingProgram_on {l m r s : ℝ} (hl : l ≠ 0) (hm : m ≠ 0) |
| (hr : |r| ≤ 1/2) (hs : |s| ≤ 1/2) (hlr : |l*r| ≤ 1/2) (hms : |m*s| ≤ 1/2) : |
| runProgram (processingProgram d l m) ![5+r,6+s,5] = ![5+l*r,6+m*s,5] := by |
| rw [processingProgram, runProgram_append, tripleProgram0_on d hl hr hs hlr, |
| tripleProgram1_on d hm hlr hs hms] |
|
|
| theorem processingProgram_off (l m : ℝ) (x : Space) |
| (hx : x 1 ∈ Set.Icc 2 3) (hz : x 2 = 2) : |
| runProgram (processingProgram d l m) x = x := by |
| apply runProgram_fixed |
| intro p hp |
| simp only [processingProgram, tripleProgram0, tripleProgram1, List.mem_append, |
| List.mem_cons, List.not_mem_nil, or_false] at hp |
| rcases hp with (rfl | rfl | rfl) | (rfl | rfl | rfl) <;> |
| simp [coreLift0_off d x hx, coreLift1_off d x hx, coreHorizontal_off d _ x hz] |
|
|
| end Solenoidal |
| end |
| end |
|
|
| end OAI |
|
|
