| import OAI.Probability.RadialTransition.IdealSupports |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators ENNReal NNReal |
| open MeasureTheory Set Filter |
| namespace RadialTransition |
|
|
| def phaseSlope (i : Fin 3) : Parameter := parameter (phaseMass/2*(phaseDensity i)^2) (phaseDensity i) |
| def phaseCenter : Parameter := parameter 1 (-3/4) |
| def targetSlope : Parameter := parameter (17*packingDensity/4) (5*packingDensity/3) |
| def phaseWindow (r : ℝ) : Set Parameter := Metric.closedBall phaseCenter r |
|
|
| def HasPhaseGroups (Q : Pressure) (x : Parameter) (r : ℝ) : Prop := |
| supportSet (onParameters Q) positiveParameters x = closure (convexHull ℝ |
| {v : Parameter | v ∈ supportSet (onParameters Q) positiveParameters x ∧ |
| ∃ i : Fin 3, v ∈ Metric.closedBall (phaseSlope i) r}) |
|
|
| lemma support_energy_small {Q : Pressure} {β h e d σ ε : ℝ} |
| (hσ : 0 < σ) (hβ : σ < β) (hs : pressureSupport Q β h e d) |
| (happ : ∀ s ∈ ({-σ,0,σ} : Set ℝ), |Q (β+s) h-bareLimit h| ≤ ε) : |
| |e| ≤ 2*ε/σ := by |
| have hm := hs (β-σ) h (by linarith) |
| have hp := hs (β+σ) h (by linarith) |
| have ha0 := abs_le.mp (happ 0 (by simp)) |
| have ham := abs_le.mp (happ (-σ) (by simp)) |
| have hap := abs_le.mp (happ σ (by simp)) |
| simp only [sub_self, mul_zero, add_zero, sub_eq_add_neg] at hm hp ha0 ham hap |
| have heq := div_mul_cancel₀ (2*ε) hσ.ne' |
| rw [abs_le] |
| constructor <;> nlinarith |
|
|
| lemma phase_generator_distance {Q : Pressure} {β h U ε η σ u e : ℝ} |
| (hβ : 0 < β) (hσ : 0 < σ) (hβσ : σ < β) (hη : 0 ≤ η) |
| (hU : 4*packingDensity ≤ U) (hu : ActiveShift Q phaseMass β h u) (hub : u ∈ Icc (0:ℝ) U) |
| (happ : ∀ v ∈ Icc (0:ℝ) U, |Q β (h+β*phaseMass*v)-bareLimit (h+β*phaseMass*v)| ≤ ε) |
| (happe : ∀ s ∈ ({-σ,0,σ} : Set ℝ), |Q (β+s) (h+β*phaseMass*u)-bareLimit (h+β*phaseMass*u)| ≤ ε) |
| (hsmall : 2*ε ≤ β*phaseMass/2*η^2) |
| (hs : pressureSupport Q β (h+β*phaseMass*u) e u) : |
| ∃ i : Fin 3, ‖parameter (e+phaseMass*u^2/2) u-phaseSlope i‖ ≤ |
| 2*ε/σ+(1+phaseMass/2*(U+4*packingDensity))*η := by |
| have he := support_energy_small hσ hβσ hs happe |
| obtain ⟨i, hi⟩ := active_near_phase hβ.le hU hu hub happ |
| have hc : 0 < β*phaseMass/2 := by have := phaseMass_pos; positivity |
| have hdist : |u-phaseDensity i| ≤ η := by |
| have hsq : (u-phaseDensity i)^2 ≤ η^2 := by nlinarith |
| exact (sq_le_sq₀ (abs_nonneg _) hη).mp (by simpa only [sq_abs] using hsq) |
| have hsum : |u+phaseDensity i| ≤ U+4*packingDensity := by |
| rw [abs_of_nonneg (add_nonneg hub.1 (phaseDensity_bounds i).1)] |
| linarith [hub.2, (phaseDensity_bounds i).2] |
| have hdiff : |u^2-(phaseDensity i)^2| ≤ η*(U+4*packingDensity) := by |
| rw [show u^2-(phaseDensity i)^2=(u-phaseDensity i)*(u+phaseDensity i) by ring, abs_mul] |
| exact mul_le_mul hdist hsum (abs_nonneg _) hη |
| have hA := phaseMass_pos |
| have hshift : |e+phaseMass*u^2/2-phaseMass/2*(phaseDensity i)^2| ≤ |
| 2*ε/σ+phaseMass/2*η*(U+4*packingDensity) := by |
| calc |
| _ = |e+phaseMass/2*(u^2-(phaseDensity i)^2)| := by congr 1; ring |
| _ ≤ |e|+|phaseMass/2*(u^2-(phaseDensity i)^2)| := abs_add_le _ _ |
| _ = |e|+(phaseMass/2)*|u^2-(phaseDensity i)^2| := by rw [abs_mul, abs_of_pos (by positivity : 0 < phaseMass/2)] |
| _ ≤ _ := by nlinarith [mul_le_mul_of_nonneg_left hdiff (by positivity : 0 ≤ phaseMass/2)] |
| refine ⟨i, ?_⟩ |
| have hn := parameter_norm_le (parameter (e+phaseMass*u^2/2) u-phaseSlope i) |
| change ‖parameter (e+phaseMass*u^2/2) u-phaseSlope i‖ ≤ |
| |e+phaseMass*u^2/2-phaseMass/2*(phaseDensity i)^2|+|u-phaseDensity i| at hn |
| nlinarith |
|
|
| lemma phase_groups_of_generators {Q : Pressure} {x : Parameter} {G : Set Parameter} {r : ℝ} |
| (hgen : supportSet (onParameters Q) positiveParameters x=closure (convexHull ℝ G)) |
| (hG : ∀ v ∈ G, ∃ i : Fin 3, v ∈ Metric.closedBall (phaseSlope i) r) : |
| HasPhaseGroups Q x r := by |
| apply le_antisymm |
| · rw [hgen] |
| apply closure_mono |
| apply convexHull_mono |
| intro v hv |
| refine ⟨?_, hG v hv⟩ |
| exact subset_closure (subset_convexHull ℝ G hv) |
| · apply closure_minimal _ (isClosed_supportSet _ _ _) |
| exact convexHull_min (fun _ h => h.1) (convex_supportSet _ _ _) |
|
|
| end RadialTransition |
| end |
|
|
| end OAI |
|
|
