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a2ee19a | 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 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 | import OAI.Probability.RadialTransition.PackingBlob
namespace OAI
noncomputable section
open scoped BigOperators ENNReal NNReal
open MeasureTheory Set Filter
namespace RadialTransition
lemma core_nonpos_outer {t : ℝ} (ht : 1 ≤ t) :
∀ r : ℝ≥0, 2 ≤ (r:ℝ) → corePotential t r ≤ 0 := by
intro r hr
have ht0 : 0 < t := by linarith
have hinv : 1/t ≤ 1 := (div_le_one ht0).mpr ht
rw [corePotential, coreReal_zero_outer ht0 (by linarith)]
lemma core_finitePressure_upper {t α L β h : ℝ} (hL : 1 ≤ L)
(ht0 : 0 < t) (ht : 2*((1+(1:ℝ)/1000)*microLength t) < 1)
(hβ : 0 ≤ β) (hα : 0 ≤ α) (h1 : h ≤ α) (h4 : 4*h-9 ≤ α) :
finiteGrandPressure (corePotential t) t L β h ≤
α*((packingNumber L:ℝ)/L^3)+(216*Real.log 15001+Real.exp (t*h-β*t^2/2))/t := by
have hL0 : 0 < L := by linarith
have hh := core_grand_upper hL0.le ht0.le ht hβ (mul_nonneg ht0.le hα) h1 h4
have hp := (grandPartition_bounds (continuous_corePotential t) (bounded_corePotential t)
(stable_corePotential t) hL0.le hβ t h).1
have hreal := ENNReal.toReal_mono (by finiteness :
ENNReal.ofReal (Real.exp (t*α*packingNumber L)) *
((15001:ℝ≥0∞)^(cubeCells L).card * ENNReal.ofReal (Real.exp (L^3*Real.exp (t*h-β*t^2/2)))) ≠ ⊤) hh
simp only [ENNReal.toReal_ofReal (show 0 ≤ grandPartition (corePotential t) t L β h by linarith),
ENNReal.toReal_mul, ENNReal.toReal_pow, ENNReal.toReal_ofReal (Real.exp_pos _).le,
ENNReal.toReal_ofNat] at hreal
have hlog := Real.log_le_log (show 0 < grandPartition (corePotential t) t L β h by linarith) hreal
rw [Real.log_mul (Real.exp_ne_zero _) (by positivity), Real.log_exp,
Real.log_mul (by positivity) (Real.exp_ne_zero _), Real.log_pow, Real.log_exp] at hlog
have hc := mul_le_mul_of_nonneg_right (card_cubeCells hL) (show 0 ≤ Real.log (15001:ℝ) from Real.log_nonneg (by norm_num))
have hden : 0 < t*L^3 := by positivity
dsimp only [finiteGrandPressure]
apply (div_le_iff₀ hden).mpr
have heq : (α*((packingNumber L:ℝ)/L^3)+(216*Real.log 15001+Real.exp (t*h-β*t^2/2))/t)*(t*L^3) =
t*α*packingNumber L+216*L^3*Real.log 15001+L^3*Real.exp (t*h-β*t^2/2) := by
field_simp
ring
rw [heq]
linarith
lemma corePressure_upper {t α β h : ℝ} (ht0 : 1 ≤ t)
(ht : 2*((1+(1:ℝ)/1000)*microLength t) < 1)
(hβ : 0 ≤ β) (hα : 0 ≤ α) (h1 : h ≤ α) (h4 : 4*h-9 ≤ α) :
grandPressure (corePotential t) t β h ≤
α*packingDensity+(216*Real.log 15001+Real.exp (t*h-β*t^2/2))/t := by
have htpos : 0 < t := by linarith
have hl := (grandPressure_spec (continuous_corePotential t) (bounded_corePotential t)
(stable_corePotential t) htpos hβ (core_nonpos_outer ht0) h).1
have hr := packingDensity_limit.const_mul α |>.add_const ((216*Real.log 15001+Real.exp (t*h-β*t^2/2))/t)
apply le_of_tendsto_of_tendsto hl hr
filter_upwards [eventually_ge_atTop (1:ℝ)] with L hL
exact core_finitePressure_upper hL htpos ht hβ hα h1 h4
end RadialTransition
end
end OAI
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