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9b6e014 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 | import OAI.Probability.ContinuumTransition.LowerLimit
namespace OAI
section
open scoped Topology
open Filter Set
namespace ContinuumTemperature
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)
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
end ContinuumTemperature
end
end OAI
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