| import Mathlib |
| import OAI.MathematicalPhysics.SheetFlows.Routing |
| import OAI.MathematicalPhysics.SheetFlows.RationalFormula |
|
|
| namespace OAI |
|
|
| /-! SheetFlows formula profiles. -/ |
|
|
| noncomputable section |
| open Set MeasureTheory |
| open scoped BigOperators |
| namespace Solenoidal |
|
|
| def RationalParameter (x : ℝ) : Prop := ∃ q : ℚ, (q:ℝ) = x |
|
|
| namespace RationalParameter |
|
|
| theorem rat (q : ℚ) : RationalParameter (q:ℝ) := ⟨q,rfl⟩ |
| theorem ofNat (n : ℕ) : RationalParameter (n:ℝ) := ⟨n,by simp⟩ |
| theorem add {a b : ℝ} (ha : RationalParameter a) (hb : RationalParameter b) : |
| RationalParameter (a+b) := by |
| obtain ⟨a,rfl⟩ := ha; obtain ⟨b,rfl⟩ := hb |
| exact ⟨a+b,by simp⟩ |
| theorem sub {a b : ℝ} (ha : RationalParameter a) (hb : RationalParameter b) : |
| RationalParameter (a-b) := by |
| obtain ⟨a,rfl⟩ := ha; obtain ⟨b,rfl⟩ := hb |
| exact ⟨a-b,by simp⟩ |
| theorem mul {a b : ℝ} (ha : RationalParameter a) (hb : RationalParameter b) : |
| RationalParameter (a*b) := by |
| obtain ⟨a,rfl⟩ := ha; obtain ⟨b,rfl⟩ := hb |
| exact ⟨a*b,by simp⟩ |
| theorem div {a b : ℝ} (ha : RationalParameter a) (hb : RationalParameter b) : |
| RationalParameter (a/b) := by |
| obtain ⟨a,rfl⟩ := ha; obtain ⟨b,rfl⟩ := hb |
| exact ⟨a/b,by simp⟩ |
| theorem neg {a : ℝ} (ha : RationalParameter a) : RationalParameter (-a) := by |
| obtain ⟨a,rfl⟩ := ha |
| exact ⟨-a,by simp⟩ |
| end RationalParameter |
|
|
| theorem Rectangle.center_rational (R : Rectangle) (j : Fin 2) : |
| RationalParameter (R.center j) := |
| ((RationalParameter.rat _).add (.rat _)).div (.ofNat 2) |
|
|
| theorem Rectangle.halfWidth_rational (R : Rectangle) (j : Fin 2) : |
| RationalParameter (R.halfWidth j) := |
| ((RationalParameter.rat _).sub (.rat _)).div (.ofNat 2) |
|
|
| def PeriodicJet.sub {T : ℚ} {f g : ℝ → ℝ} (p : PeriodicJet T f) (q : PeriodicJet T g) : |
| PeriodicJet T (fun x => f x - g x) := by |
| simpa only [sub_eq_add_neg,neg_one_mul,Rat.cast_neg,Rat.cast_one] using |
| p.add ((PeriodicJet.const T p.positive (-1)).mul q) |
|
|
| theorem Cprofile_hasJet {h d c : ℝ} |
| (hh : RationalParameter h) (hd : RationalParameter d) (hc : RationalParameter c) |
| (hpos : 0 < d) (hl : h+d ≤ c) (hu : c+(h+d) ≤ 10) : |
| Nonempty (PeriodicJet 10 (Cprofile h d c)) := by |
| obtain ⟨h,rfl⟩ := hh; obtain ⟨d,rfl⟩ := hd; obtain ⟨c,rfl⟩ := hc |
| exact ⟨Cprofile_jet h d c (by exact_mod_cast hpos) (by exact_mod_cast hl) (by exact_mod_cast hu)⟩ |
|
|
| theorem Dprofile_hasJet {h d c : ℝ} |
| (hh : RationalParameter h) (hd : RationalParameter d) (hc : RationalParameter c) |
| (hpos : 0 < d) (hl : h+d ≤ c) (hu : c+(h+d) ≤ 10) : |
| Nonempty (PeriodicJet 10 (Dprofile h d c)) := by |
| obtain ⟨h,rfl⟩ := hh; obtain ⟨d,rfl⟩ := hd; obtain ⟨c,rfl⟩ := hc |
| exact ⟨Dprofile_jet h d c (by exact_mod_cast hpos) (by exact_mod_cast hl) (by exact_mod_cast hu)⟩ |
|
|
| namespace StationaryPulse |
|
|
| def RationalFormula (p : StationaryPulse) : Prop := HasRationalFormula (fun _ => p.field) |
|
|
| theorem RationalFormula.scale {p : StationaryPulse} {c : ℝ} |
| (hp : p.RationalFormula) (hc : RationalParameter c) : (p.scale c).RationalFormula := by |
| obtain ⟨q,rfl⟩ := hc |
| exact hp.smul q |
|
|
| theorem rationalFormula_tensor (i : Fin 3) (g : Fin 3 → ℝ → ℝ) |
| (hsm : ∀ j, ContDiff ℝ (⊤ : ℕ∞) (g j)) |
| (hp : ∀ j, Function.Periodic (g j) 10) |
| (hi : ∀ s, g i s = 1) (j : Fin 3) |
| (hz : (∫ s in Set.Ico (0 : ℝ) 10, g j s) = 0) |
| (hr : ∀ j, Nonempty (PeriodicJet 10 (g j))) : |
| (tensor i g hsm hp hi j hz).RationalFormula := tensorVelocity_formula i g hr |
|
|
| end StationaryPulse |
|
|
| variable (d : Collar) (hd : RationalParameter d.size) |
| include hd |
|
|
| theorem verticalPulse_formula (R : Rectangle) (hR : R.inCodingSquare) {s : ℝ} |
| (hs : s ∈ Set.Icc 0 4) (hsr : RationalParameter s) : |
| (verticalPulse d R hR s).RationalFormula := by |
| apply StationaryPulse.rationalFormula_tensor |
| have h (j : Fin 2) (v : ℝ) (hvr : RationalParameter v) (hv : v ∈ Set.Icc 0 4) : |
| Nonempty (PeriodicJet 10 (Cprofile (R.halfWidth j) d.size (R.center j+v))) := |
| Cprofile_hasJet (R.halfWidth_rational j) hd ((R.center_rational j).add hvr) |
| d.positive (rect_support_chart d R hR j hv).1 (rect_support_chart d R hR j hv).2 |
| intro j |
| fin_cases j |
| · exact h 0 s hsr hs |
| · obtain ⟨a⟩ := h 1 0 (by exact ⟨0, by norm_num⟩) (by norm_num) |
| obtain ⟨b⟩ := h 1 4 (by exact ⟨4, by norm_num⟩) (by norm_num) |
| exact ⟨by simpa using a.sub b⟩ |
| · exact ⟨by simpa using PeriodicJet.const 10 (by norm_num) 1⟩ |
|
|
| theorem horizontalPulse_formula (j : Fin 2) : (horizontalPulse d j).RationalFormula := by |
| apply StationaryPulse.rationalFormula_tensor |
| intro k |
| fin_cases k |
| · exact ⟨by simpa using PeriodicJet.const 10 (by norm_num) 1⟩ |
| · exact ⟨by simpa using PeriodicJet.const 10 (by norm_num) 1⟩ |
| · obtain ⟨a⟩ := Cprofile_hasJet (show RationalParameter 0 from ⟨0, by norm_num⟩) hd (show RationalParameter 5 from ⟨5, by norm_num⟩) d.positive |
| (show 0+d.size ≤ 5 by linarith [d.small]) |
| (show 5+(0+d.size) ≤ 10 by linarith [d.small]) |
| obtain ⟨b⟩ := Cprofile_hasJet (show RationalParameter 0 from ⟨0, by norm_num⟩) hd (show RationalParameter 8 from ⟨8, by norm_num⟩) d.positive |
| (show 0+d.size ≤ 8 by linarith [d.small]) |
| (show 8+(0+d.size) ≤ 10 by linarith [d.small]) |
| exact ⟨a.sub b⟩ |
|
|
| theorem coreLift0_formula : (coreLift0 d).RationalFormula := by |
| apply StationaryPulse.rationalFormula_tensor |
| intro k |
| fin_cases k |
| · exact Dprofile_hasJet (show RationalParameter (1/2) from ⟨1/2, by norm_num⟩) hd (show RationalParameter 5 from ⟨5, by norm_num⟩) |
| d.positive (by linarith [d.small]) (by linarith [d.small]) |
| · exact Cprofile_hasJet (show RationalParameter (1/2) from ⟨1/2, by norm_num⟩) hd (show RationalParameter 6 from ⟨6, by norm_num⟩) |
| d.positive (by linarith [d.small]) (by linarith [d.small]) |
| · exact ⟨by simpa using PeriodicJet.const 10 (by norm_num) 1⟩ |
|
|
| theorem coreLift1_formula : (coreLift1 d).RationalFormula := by |
| apply StationaryPulse.rationalFormula_tensor |
| intro k |
| fin_cases k |
| · exact Cprofile_hasJet (show RationalParameter (1/2) from ⟨1/2, by norm_num⟩) hd (show RationalParameter 5 from ⟨5, by norm_num⟩) |
| d.positive (by linarith [d.small]) (by linarith [d.small]) |
| · exact Dprofile_hasJet (show RationalParameter (1/2) from ⟨1/2, by norm_num⟩) hd (show RationalParameter 6 from ⟨6, by norm_num⟩) |
| d.positive (by linarith [d.small]) (by linarith [d.small]) |
| · exact ⟨by simpa using PeriodicJet.const 10 (by norm_num) 1⟩ |
|
|
| theorem coreHorizontal_formula (j : Fin 2) : (coreHorizontal d j).RationalFormula := by |
| apply StationaryPulse.rationalFormula_tensor |
| intro k |
| fin_cases k |
| · exact ⟨by simpa using PeriodicJet.const 10 (by norm_num) 1⟩ |
| · exact ⟨by simpa using PeriodicJet.const 10 (by norm_num) 1⟩ |
| · exact Dprofile_hasJet (show RationalParameter (1/2) from ⟨1/2, by norm_num⟩) hd (show RationalParameter 5 from ⟨5, by norm_num⟩) |
| d.positive (by linarith [d.small]) (by linarith [d.small]) |
|
|
| theorem travelProgram_formula (v : Plane) (hv : ∀ j, RationalParameter (v j)) : |
| ∀ p ∈ travelProgram d v, p.RationalFormula := by |
| intro p hp |
| simp only [travelProgram,List.mem_cons,List.not_mem_nil,or_false] at hp |
| rcases hp with rfl | rfl |
| · exact (horizontalPulse_formula d hd 0).scale (hv 0) |
| · exact (horizontalPulse_formula d hd 1).scale (hv 1) |
|
|
| theorem tripleProgram0_formula {l : ℝ} (hl : RationalParameter l) : |
| ∀ p ∈ tripleProgram0 d l, p.RationalFormula := by |
| have h1 : RationalParameter (1:ℝ) := ⟨1,by norm_num⟩ |
| intro p hp |
| simp only [tripleProgram0,List.mem_cons,List.not_mem_nil,or_false] at hp |
| rcases hp with rfl | rfl | rfl |
| · exact coreLift0_formula d hd |
| · exact (coreHorizontal_formula d hd 0).scale (hl.sub h1) |
| · exact (coreLift0_formula d hd).scale (h1.neg.div hl) |
|
|
| theorem tripleProgram1_formula {l : ℝ} (hl : RationalParameter l) : |
| ∀ p ∈ tripleProgram1 d l, p.RationalFormula := by |
| have h1 : RationalParameter (1:ℝ) := ⟨1,by norm_num⟩ |
| intro p hp |
| simp only [tripleProgram1,List.mem_cons,List.not_mem_nil,or_false] at hp |
| rcases hp with rfl | rfl | rfl |
| · exact coreLift1_formula d hd |
| · exact (coreHorizontal_formula d hd 1).scale (hl.sub h1) |
| · exact (coreLift1_formula d hd).scale (h1.neg.div hl) |
|
|
| theorem processingProgram_formula {l m : ℝ} (hl : RationalParameter l) |
| (hm : RationalParameter m) : ∀ p ∈ processingProgram d l m, p.RationalFormula := by |
| intro p hp |
| rcases List.mem_append.mp hp with hp | hp |
| · exact tripleProgram0_formula d hd hl p hp |
| · exact tripleProgram1_formula d hd hm p hp |
|
|
| theorem evacuateProgram_formula (R : Rectangle) (hR : R.inCodingSquare) : |
| ∀ p ∈ evacuateProgram d R hR, p.RationalFormula := by |
| intro p hp |
| simp only [evacuateProgram,List.mem_append,List.mem_singleton] at hp |
| rcases hp with (rfl | hp) | rfl |
| · exact (verticalPulse_formula d hd R hR (by norm_num) ⟨0,by norm_num⟩).scale ⟨3,by norm_num⟩ |
| · apply travelProgram_formula d hd _ _ _ hp |
| intro j; fin_cases j |
| · exact ⟨4,by norm_num⟩ |
| · exact ⟨0,by norm_num⟩ |
| · exact (verticalPulse_formula d hd R hR (by norm_num) ⟨4,by norm_num⟩).scale ⟨-3,by norm_num⟩ |
|
|
| theorem instructionProgram_formula {N : ℕ} (D : SheetData N) (i : Fin N) : |
| ∀ p ∈ instructionProgram d D i, p.RationalFormula := by |
| intro p hp |
| simp only [instructionProgram,List.mem_append,List.mem_singleton] at hp |
| rcases hp with (((rfl | hp) | hp) | hp) | rfl |
| · exact (verticalPulse_formula d hd _ (D.source_inside i) (by norm_num) |
| ⟨4,by norm_num⟩).scale ⟨3,by norm_num⟩ |
| · apply travelProgram_formula d hd _ _ _ hp |
| intro j; fin_cases j |
| · exact (show RationalParameter 1 from ⟨1,by norm_num⟩).sub ((D.source i).center_rational 0) |
| · exact (show RationalParameter 6 from ⟨6,by norm_num⟩).sub ((D.source i).center_rational 1) |
| · exact processingProgram_formula d hd (.rat _) (.rat _) _ hp |
| · apply travelProgram_formula d hd _ _ _ hp |
| intro j; fin_cases j |
| · exact ((D.target i).center_rational 0).sub ⟨5,by norm_num⟩ |
| · exact ((D.target i).center_rational 1).sub ⟨6,by norm_num⟩ |
| · exact (verticalPulse_formula d hd _ (D.target_inside i) (by norm_num) |
| ⟨0,by norm_num⟩).scale ⟨-3,by norm_num⟩ |
|
|
| theorem sheetProgram_formula {N : ℕ} (D : SheetData N) : |
| ∀ p ∈ sheetProgram d D, p.RationalFormula := by |
| intro p hp |
| rcases List.mem_append.mp hp with hp | hp |
| · obtain ⟨i,_,hi⟩ := List.mem_flatMap.mp hp |
| exact evacuateProgram_formula d hd _ (D.source_inside i) p hi |
| · obtain ⟨i,_,hi⟩ := List.mem_flatMap.mp hp |
| exact instructionProgram_formula d hd D i p hi |
|
|
| theorem sheetVelocity_formula {N : ℕ} (D : SheetData N) : |
| HasRationalFormula (scheduleVelocity |
| (fun i : Fin (sheetProgram d D).length => ((sheetProgram d D).get i).field)) := |
| scheduleVelocity_formula _ (fun index => sheetProgram_formula d hd D _ (List.get_mem _ index)) |
|
|
| end Solenoidal |
| end |
|
|
| end OAI |
|
|
