openAI-math-dataset / lean /OAI /MathematicalPhysics /Boltzmann /Compactness /FiniteEntryPhaseBound.lean
| import OAI.MathematicalPhysics.Boltzmann.Branching.EntrySites | |
| import OAI.MathematicalPhysics.Boltzmann.Branching.ReferenceCollisionGeometry | |
| import OAI.MathematicalPhysics.Boltzmann.Branching.SeedEscapeMeasure | |
| import OAI.MathematicalPhysics.Boltzmann.Compactness.LocalizedDampedBalance | |
| import OAI.MathematicalPhysics.Boltzmann.Compactness.PhysicalEvolution | |
| import OAI.MathematicalPhysics.Boltzmann.Compactness.SelectedOriginDeletion | |
| namespace OAI | |
| section | |
| /-! # Reachable regions for a fixed finite number of physical gain collisions -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Set | |
| open scoped ENNReal ProbabilityTheory | |
| namespace BoltzmannNonuniqueness.ReferenceGeometry | |
| def finiteFlightCone (N n : ℕ) (h R : ℝ) (x : V) (p : PhysicalFlightState) : Prop := | |
| 0 ≤ p.1 ∧ p.1 ≤ h ∧ ‖p.2.2‖ ≤ R+2*((N:ℝ)-n) ∧ | |
| ‖p.2.1-x‖ ≤ (h-p.1)*(R+2*N) | |
| theorem finiteFlightCone_measurable (N n : ℕ) (h R : ℝ) (x : V) : | |
| MeasurableSet {p | finiteFlightCone N n h R x p} := by | |
| unfold finiteFlightCone | |
| measurability | |
| theorem finiteFlightCone_initial {N : ℕ} {h R : ℝ} (hh : 0 ≤ h) | |
| (x v : V) (hv : ‖v‖ ≤ R) : finiteFlightCone N N h R x (h,(x,v)) := by | |
| exact ⟨hh,le_rfl,by simpa using hv,by simp⟩ | |
| theorem markedChild_norm_le (m : HistoryMark) (v wp wm : V) (n : Sphere) | |
| (hp : ‖wp‖ ≤ 2) (hm : ‖wm‖ ≤ 2) : | |
| ‖markedChild m v wp wm n.1‖ ≤ ‖v‖+2 := by | |
| have hc (w : V) (hw : ‖w‖ ≤ 2) : ‖child v w n.1‖ ≤ ‖v‖+2 := by | |
| have hb := child_lipschitz v w 0 0 n.1 (mem_sphere_zero_iff_norm.mp n.2) | |
| have hz : child 0 0 n.1 = 0 := by simp [child] | |
| rw [hz,sub_zero,sub_zero,sub_zero] at hb | |
| exact hb.trans (add_le_add le_rfl hw) | |
| fin_cases m | |
| · change ‖v‖ ≤ ‖v‖+2; linarith | |
| · exact hc wp hp | |
| · exact hc wm hm | |
| theorem finiteFlightCone_parameter_step {N n : ℕ} {h R : ℝ} {x : V} | |
| (b : BeamCoefficients) (T : ℝ) {p : PhysicalFlightState} | |
| (hq : finiteFlightCone N (n+1) h R x p) (m : HistoryMark) | |
| {s : ℝ} (hs : s ∈ Icc 0 p.1) (z : Sphere) | |
| (hp : ‖b.plusPartner (mediumPoint T p s)‖ ≤ 2) | |
| (hm : ‖b.minusPartner (mediumPoint T p s)‖ ≤ 2) : | |
| finiteFlightCone N n h R x (mediumParameterState b T p (m,(s,z))) := by | |
| rcases hq with ⟨ht₀,ht,hvel,hpos⟩ | |
| push_cast at hvel | |
| have hv : ‖p.2.2‖ ≤ R+2*N := by | |
| have hn : 0 ≤ (n:ℝ) := Nat.cast_nonneg n | |
| linarith | |
| have hd : ‖p.2.1+s • p.2.2-x‖ ≤ (h-(p.1-s))*(R+2*N) := by | |
| calc | |
| _ = ‖(p.2.1-x)+s • p.2.2‖ := by congr 1; module | |
| _ ≤ ‖p.2.1-x‖+‖s • p.2.2‖ := norm_add_le _ _ | |
| _ = ‖p.2.1-x‖+s*‖p.2.2‖ := by rw [norm_smul,Real.norm_eq_abs,abs_of_nonneg hs.1] | |
| _ ≤ (h-p.1)*(R+2*N)+s*(R+2*N) := | |
| add_le_add hpos (mul_le_mul_of_nonneg_left hv hs.1) | |
| _ = _ := by ring | |
| refine ⟨sub_nonneg.mpr hs.2,?_,?_,hd⟩ | |
| · change p.1-s ≤ h | |
| linarith [hs.1] | |
| · change ‖markedChild m p.2.2 (b.plusPartner (mediumPoint T p s)) | |
| (b.minusPartner (mediumPoint T p s)) z.1‖ ≤ _ | |
| have hc := markedChild_norm_le m p.2.2 _ _ z hp hm | |
| linarith | |
| end BoltzmannNonuniqueness.ReferenceGeometry | |
| end | |
| end | |
| section | |
| /-! # A bounded finite collision cone stays above the cold cutoff and in a compact phase set -/ | |
| noncomputable section | |
| open Set | |
| open BoltzmannNonuniqueness.EuclideanFlight BoltzmannNonuniqueness.VelocityWeight | |
| namespace BoltzmannNonuniqueness.ReferenceGeometry | |
| theorem finiteFlightCone_displacement {N n : ℕ} {h R : ℝ} {x : V} | |
| {q : PhysicalFlightState} (hq : finiteFlightCone N n h R x q) : | |
| ‖q.2.1-x‖ ≤ h*(R+2*N) := by | |
| have hB : 0 ≤ R+2*N := by | |
| have hn : 0 ≤ (n:ℝ) := Nat.cast_nonneg n | |
| linarith [hq.2.2.1,norm_nonneg q.2.2] | |
| exact hq.2.2.2.trans (mul_le_mul_of_nonneg_right (by linarith [hq.1]) hB) | |
| theorem finiteFlightCone_cold_domain {N n : ℕ} {t h R a η l : ℝ} {x : V} | |
| (ha : a ≤ t) (hl : 0 ≤ l) (hshort : h*(R+2*N) ≤ 1) | |
| (hx : Real.exp η*l+1 < ‖transverse x‖) | |
| {q : PhysicalFlightState} (hq : finiteFlightCone N n h R x q) : | |
| flightCharacteristicEquiv (t+h) q ∈ coldCharacteristicDomain a η l := by | |
| have hd := (finiteFlightCone_displacement hq).trans hshort | |
| have ht := norm_sub_norm_le (transverse x) (transverse q.2.1) | |
| have he : ‖transverse x-transverse q.2.1‖ ≤ 1 := by | |
| rw [← PlaneChart.transverse_sub] | |
| exact (LargeOutput.transverse_norm_le (x-q.2.1)).trans (by rwa [norm_sub_rev]) | |
| have hy : Real.exp η*l < ‖transverse q.2.1‖ := by linarith | |
| change a ≤ t+h-q.1 ∧ 0 < ‖transverse (q.2.1-(t+h-q.1) • q.2.2+ | |
| (t+h-q.1) • q.2.2)‖ ∧ Real.exp η*l < ‖transverse (q.2.1-(t+h-q.1) • q.2.2+ | |
| (t+h-q.1) • q.2.2)‖ | |
| rw [sub_add_cancel] | |
| exact ⟨by linarith [hq.2.1],(mul_nonneg (Real.exp_pos η).le hl).trans_lt hy,hy⟩ | |
| theorem finiteFlightCone_phase_bound {N n : ℕ} {t h R X : ℝ} {x : V} | |
| (ht : 0 ≤ t) (hR : 0 ≤ R) (hX : 0 ≤ X) (hx : ‖x‖ ≤ X) | |
| {q : PhysicalFlightState} (hq : finiteFlightCone N n h R x q) : | |
| ‖(flightCharacteristicEquiv (t+h) q).2‖ ≤ X+(t+2*h+1)*(R+2*N) := by | |
| have hh : 0 ≤ h := hq.1.trans hq.2.1 | |
| have hB : 0 ≤ R+2*N := by positivity | |
| have hv : ‖q.2.2‖ ≤ R+2*N := by | |
| have hn : 0 ≤ (n:ℝ) := Nat.cast_nonneg n | |
| linarith [hq.2.2.1] | |
| have hpos : ‖q.2.1‖ ≤ X+h*(R+2*N) := by | |
| calc | |
| _ ≤ ‖q.2.1-x‖+‖x‖ := by simpa only [sub_add_cancel] using norm_add_le (q.2.1-x) x | |
| _ ≤ _ := by simpa only [add_comm] using add_le_add (finiteFlightCone_displacement hq) hx | |
| have htime : 0 ≤ t+h-q.1 := by linarith [hq.2.1] | |
| have hphase : ‖q.2.1-(t+h-q.1) • q.2.2‖ ≤ X+(t+2*h)*(R+2*N) := by | |
| apply (norm_sub_le _ _).trans | |
| rw [norm_smul,Real.norm_eq_abs,abs_of_nonneg htime] | |
| have hm := mul_le_mul_of_nonneg_left hv htime | |
| have hm' := mul_le_mul_of_nonneg_right (show t+h-q.1 ≤ t+h by linarith [hq.1]) hB | |
| nlinarith | |
| rw [Prod.norm_def,max_le_iff] | |
| exact ⟨by change ‖q.2.1-(t+h-q.1) • q.2.2‖ ≤ _; nlinarith, | |
| hv.trans (by nlinarith [mul_nonneg (show 0 ≤ t+2*h by positivity) hB])⟩ | |
| end BoltzmannNonuniqueness.ReferenceGeometry | |
| end | |
| end | |
| section | |
| /-! # Uniform upper and positive transverse bounds for finitely many successive windows -/ | |
| noncomputable section | |
| open MeasureTheory Set | |
| open BoltzmannNonuniqueness.VelocityWeight | |
| namespace BoltzmannNonuniqueness.JetCopyDatum | |
| theorem finite_velocity_window_bounds (R b : ℕ → ℝ) | |
| (hR : ∀ i, 0 < R i) (hb : ∀ i, 0 < b i) (m : ℕ) : | |
| ∃ V β : ℝ, 0 < V ∧ 0 < β ∧ ∀ i v, | |
| v ∈ velocityWindow (R (min i m)) (b (min i m)) → | |
| ‖v‖ ≤ V ∧ β ≤ ‖transverse v‖ := by | |
| let I := Finset.range (m+1) | |
| have hI : I.Nonempty := Finset.nonempty_range_iff.mpr (Nat.succ_ne_zero m) | |
| let V := I.sup' hI R | |
| let β := I.inf' hI b | |
| have hzero : 0 ∈ I := Finset.mem_range.mpr (Nat.zero_lt_succ m) | |
| have hV : 0 < V := (hR 0).trans_le (Finset.le_sup' R hzero) | |
| have hβ : 0 < β := (Finset.lt_inf'_iff hI).mpr (fun i _ => hb i) | |
| refine ⟨V,β,hV,hβ,?_⟩ | |
| intro i v hv | |
| have hi : min i m ∈ I := Finset.mem_range.mpr (lt_of_le_of_lt (min_le_right i m) | |
| (Nat.lt_succ_self m)) | |
| exact ⟨hv.1.trans (Finset.le_sup' R hi),(Finset.inf'_le b hi).trans hv.2⟩ | |
| end BoltzmannNonuniqueness.JetCopyDatum | |
| end | |
| end | |
| section | |
| /-! # One compact phase cutoff contains all short entry histories in a fixed time window -/ | |
| noncomputable section | |
| open MeasureTheory Set | |
| open BoltzmannNonuniqueness.EuclideanFlight | |
| namespace BoltzmannNonuniqueness.ReferenceGeometry | |
| def phaseBallCore (M : ℝ) : Set CharacteristicPhase := {z | ‖z.2‖ ≤ M} | |
| theorem phaseBallCore_measurable (M : ℝ) : MeasurableSet (phaseBallCore M) := | |
| measurableSet_le measurable_snd.norm measurable_const | |
| theorem finiteFlightCone_window_core {N n : ℕ} {s t U R X : ℝ} {x : V} | |
| (hs : 0 ≤ s) (ht : t ≤ U) (hR : 0 ≤ R) (hX : 0 ≤ X) (hx : ‖x‖ ≤ X) | |
| {q : PhysicalFlightState} (hq : finiteFlightCone N n (t-s) R x q) : | |
| flightCharacteristicEquiv t q ∈ phaseBallCore (X+(2*U+1)*(R+2*N)) := by | |
| have hb := finiteFlightCone_phase_bound hs hR hX hx hq | |
| have he : s+(t-s) = t := by ring | |
| rw [he] at hb | |
| change ‖(flightCharacteristicEquiv t q).2‖ ≤ _ | |
| apply hb.trans | |
| have hB : 0 ≤ R+2*N := by positivity | |
| nlinarith [mul_le_mul_of_nonneg_right ht hB,mul_nonneg hs hB] | |
| theorem exists_phaseBall_cutoff (M : ℝ) : | |
| ∃ β : WeakTime.Phase → ℝ, ContDiff ℝ (⊤ : ℕ∞) β ∧ HasCompactSupport β ∧ | |
| (∀ p, 0 ≤ β p) ∧ (∀ p, β p ≤ 1) ∧ (∀ p, ‖p‖ ≤ M → β p = 1) := by | |
| obtain ⟨n,hn⟩ := exists_nat_gt M | |
| refine ⟨seedPhaseCutoff n,seedPhaseCutoff_smooth n,seedPhaseCutoff_compact n, | |
| (fun p => (seedPhaseCutoff_mem n p).1),(fun p => (seedPhaseCutoff_mem n p).2),?_⟩ | |
| intro p hp | |
| apply (seedPhaseBump n).one_of_mem_closedBall | |
| rw [Metric.mem_closedBall,dist_zero_right] | |
| dsimp [seedPhaseBump] | |
| linarith | |
| end BoltzmannNonuniqueness.ReferenceGeometry | |
| end | |
| end | |
| end OAI | |