| import OAI.Probability.ContinuumTransition.LowerLimit |
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| namespace OAI |
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| section |
| open scoped Topology |
| open Filter Set |
| namespace ContinuumTemperature |
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| lemma core_pressure_nonneg {t β : ℝ} (ht : 1 < t) (hβ : 0 ≤ β) (h : ℝ) : |
| 0 ≤ physicalPressure (coreProfile t) t (β,h) := by |
| exact div_nonneg (canonicalPressure_nonneg |
| (measurable_radialPotential (measurable_coreProfile t)) ht hβ |
| (coreProfile_stable_zero t) (fun _ _ => le_refl _)) (by linarith) |
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| theorem core_pressure_tendsto {β : ℝ} (hβ : 0 < β) (h : ℝ) : |
| Tendsto (fun t : ℝ => physicalPressure (coreProfile t) t (β,h)) |
| atTop (𝓝 (idealCorePressure h)) := by |
| refine tendsto_order.2 ⟨?_,?_⟩ |
| · intro c hc |
| unfold idealCorePressure at hc |
| rcases lt_max_iff.mp hc with hc | hc |
| · filter_upwards [eventually_gt_atTop (1 : ℝ)] with t ht |
| exact hc.trans_le (core_pressure_nonneg ht hβ.le h) |
| · rcases lt_max_iff.mp hc with hc | hc |
| · obtain ⟨η,hη,he⟩ := exists_packing_dilate_lt hc |
| exact core_one_lower_eventually hβ.le hη h he |
| · have hc' : c < packingDensity*(4*h-9) := by nlinarith only [hc] |
| obtain ⟨η,hη,he⟩ := exists_packing_dilate_lt hc' |
| exact core_tetra_lower_eventually hβ.le hη h he |
| · intro c hc |
| have hsep : ∀ᶠ t : ℝ in atTop, 2*(1+shellDelta)*Real.exp (-t) < 1 := by |
| have hh := (Real.tendsto_exp_atBot.comp tendsto_neg_atTop_atBot).const_mul (2*(1+shellDelta)) |
| simp only [mul_zero] at hh |
| exact hh.eventually_lt_const zero_lt_one |
| filter_upwards [(core_upper_bound_tendsto hβ h).eventually_lt_const hc, |
| eventually_gt_atTop (1 : ℝ),hsep] with t ht h1 hs |
| exact (core_pressure_upper h1 hβ.le hs h).trans_lt ht |
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| end ContinuumTemperature |
| end |
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| end OAI |
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