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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 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 | import OAI.Probability.RadialTransition.CoreLocalLower
namespace OAI
noncomputable section
open scoped BigOperators ENNReal NNReal
open MeasureTheory Set Filter
namespace RadialTransition
def bareAlpha (h : ℝ) : ℝ := max 0 (max h (4*h-9))
def bareLimit (h : ℝ) : ℝ := max 0 (max (packingDensity*h) (packingDensity*(4*h-9)))
lemma bareLimit_eq (h : ℝ) : bareLimit h = bareAlpha h*packingDensity := by
have hp := packingDensity_pos.le
simp only [bareLimit, bareAlpha, max_mul_of_nonneg _ _ hp, zero_mul]
congr 2 <;> ring
lemma packing_coefficient_approx {ε : ℝ} (hε : 0 < ε) :
∃ ν : ℝ, 0 < ν ∧ ν ≤ 1 ∧ |packingDensity/(1+ν)^3-packingDensity| < ε := by
have hh : Tendsto (fun x : ℝ => packingDensity/(1+x⁻¹)^3) atTop (nhds packingDensity) := by
have hh : Tendsto (fun x : ℝ => (1+x⁻¹)^3) atTop (nhds (1:ℝ)) := by
simpa using ((tendsto_const_nhds (x := (1:ℝ))).add tendsto_inv_atTop_zero).pow 3
simpa [div_eq_mul_inv] using (hh.inv₀ (by norm_num : (1:ℝ) ≠ 0)).const_mul packingDensity
have he := (Metric.tendsto_atTop.mp hh ε hε)
obtain ⟨x, hx⟩ := he
let y := max x 1
have hy : 1 ≤ y := le_max_right _ _
refine ⟨y⁻¹, inv_pos.mpr (by linarith), inv_le_one_of_one_le₀ hy, ?_⟩
simpa only [Real.dist_eq] using hx y (le_max_left _ _)
lemma tendsto_core_micro : Tendsto microLength atTop (nhds 0) := by
exact Real.tendsto_exp_atBot.comp tendsto_neg_atTop_atBot
lemma corePressure_lower_branch {ν b t β h : ℝ} (ht : 1 ≤ t) (hb : 0 < b)
(hν : 0 ≤ ν) (hsep : 1/t+4*b ≤ ν) (hβ : 0 ≤ β)
(htb : 3*microLength t ≤ b/2) :
packingDensity/(1+ν)^3*h-packingDensity*|Real.log (b^3)|/t ≤
grandPressure (corePotential t) t β h ∧
packingDensity/(1+ν)^3*(4*h-9)-packingDensity*|Real.log (blobTetraConstant b)|/t ≤
grandPressure (corePotential t) t β h := by
let c := packingDensity/(1+ν)^3
have hp := packingDensity_pos
have hc0 : 0 ≤ c := by dsimp [c]; positivity
have hcp : c ≤ packingDensity := by
dsimp [c]
exact div_le_self packingDensity_pos.le (one_le_pow₀ (by linarith))
have ht0 : 0 < t := by linarith
have hl := corePressure_packing_lower ht hb.le hν hsep hβ (h := h)
have hh (z : ℝ) : -packingDensity*|z|/t ≤ c*(z/t) := by
have hz := neg_abs_le z
have h1 := mul_le_mul_of_nonneg_left hz hc0
have h2 := mul_le_mul_of_nonneg_right hcp (abs_nonneg z)
have h3 : -packingDensity*|z| ≤ c*z := by nlinarith
have h4 := div_le_div_of_nonneg_right h3 ht0.le
simpa only [mul_div_assoc] using h4
constructor
· have hm := mul_le_mul_of_nonneg_left (core_local_log_monomer ht0 hb hβ (h := h)) hc0
have he := hh (Real.log (b^3))
dsimp only [c] at hm he
rw [mul_add] at hm
simp only [neg_mul, neg_div] at he
linarith
· have hm := mul_le_mul_of_nonneg_left (core_local_log_tetra ht0 hb hβ htb (h := h)) hc0
have he := hh (Real.log (blobTetraConstant b))
dsimp only [c] at hm he
rw [mul_add] at hm
simp only [neg_mul, neg_div] at he
linarith
lemma corePressure_uniform_bare {β₀ H ε : ℝ} (hβ₀ : 0 < β₀) (hH : 0 ≤ H) (hε : 0 < ε) :
∀ᶠ t : ℝ in atTop, ∀ β h : ℝ, β₀ ≤ β → |h| ≤ H →
|grandPressure (corePotential t) t β h-bareLimit h| ≤ ε := by
let D : ℝ := 4*H+10
have hD : 0 < D := by dsimp [D]; positivity
obtain ⟨ν, hν, _, hc⟩ := packing_coefficient_approx (div_pos hε (mul_pos (by norm_num : (0:ℝ)<2) hD))
let b := ν/10
let c := packingDensity/(1+ν)^3
let K := packingDensity*(|Real.log (b^3)|+|Real.log (blobTetraConstant b)|)
have hb : 0 < b := by dsimp [b]; positivity
have hp := packingDensity_pos
have hK : 0 ≤ K := by dsimp [K]; positivity
have hlim1 : Tendsto (fun t : ℝ => 1/t) atTop (nhds 0) := by simpa using tendsto_inv_atTop_zero
have hsep : ∀ᶠ t : ℝ in atTop, 1/t+4*b ≤ ν := by
have hh := ((hlim1.add_const (4*b)).eventually (eventually_le_nhds (by dsimp [b]; linarith : 0+4*b < ν)))
exact hh
have hmic : ∀ᶠ t : ℝ in atTop, 3*microLength t ≤ b/2 :=
(tendsto_core_micro.const_mul 3).eventually (eventually_le_nhds (by linarith : (3:ℝ)*0 < b/2))
have hsmall : ∀ᶠ t : ℝ in atTop, 2*((1+(1:ℝ)/1000)*microLength t) < 1 := by
have hh := (tendsto_core_micro.const_mul (2*(1+(1:ℝ)/1000))).eventually
(eventually_lt_nhds (by norm_num : 2*(1+(1:ℝ)/1000)*0 < 1))
simpa only [mul_assoc] using hh
have herror : ∀ᶠ t : ℝ in atTop, K/t ≤ ε/2 := by
have hh : Tendsto (fun t : ℝ => K/t) atTop (nhds 0) := by simpa [div_eq_mul_inv] using hlim1.const_mul K
exact hh.eventually (eventually_le_nhds (by positivity : (0:ℝ)<ε/2))
have hupper : ∀ᶠ t : ℝ in atTop, (216*Real.log 15001+1)/t ≤ ε := by
have hh : Tendsto (fun t : ℝ => (216*Real.log 15001+1)/t) atTop (nhds 0) := by
simpa [div_eq_mul_inv] using hlim1.const_mul (216*Real.log 15001+1)
exact hh.eventually (eventually_le_nhds hε)
filter_upwards [eventually_ge_atTop (max 1 (2*H/β₀)), hsep, hmic, hsmall, herror, hupper]
with t ht hsep hmic hsmall herror hupper
intro β h hβ hh
have ht1 : 1 ≤ t := (le_max_left _ _).trans ht
have ht0 : 0 < t := by linarith
have hβpos : 0 < β := hβ₀.trans_le hβ
have hhH := abs_le.mp hh
have hα0 : 0 ≤ bareAlpha h := le_max_left _ _
have hα1 : h ≤ bareAlpha h := (le_max_left _ _).trans (le_max_right _ _)
have hα4 : 4*h-9 ≤ bareAlpha h := (le_max_right _ _).trans (le_max_right _ _)
have hup := corePressure_upper ht1 hsmall hβpos.le hα0 hα1 hα4
have hexp : Real.exp (t*h-β*t^2/2) ≤ 1 := by
apply Real.exp_le_one_iff.mpr
have hth : 2*H ≤ t*β₀ := (div_le_iff₀ hβ₀).mp ((le_max_right _ _).trans ht)
have hh1 := mul_le_mul_of_nonneg_right hth ht0.le
have hh2 := mul_le_mul_of_nonneg_right hβ (sq_nonneg t)
have hh3 := mul_le_mul_of_nonneg_left hhH.2 ht0.le
nlinarith
have hup' : grandPressure (corePotential t) t β h-bareLimit h ≤ ε := by
rw [bareLimit_eq]
have hdiv := div_le_div_of_nonneg_right (show 216*Real.log 15001+Real.exp (t*h-β*t^2/2) ≤ 216*Real.log 15001+1 by linarith) ht0.le
linarith
have hlo := corePressure_lower_branch ht1 hb hν.le hsep hβpos.le hmic (h := h)
have hz := (grandPressure_spec (continuous_corePotential t) (bounded_corePotential t)
(stable_corePotential t) ht0 hβpos.le (core_nonpos_outer ht1) h).2.1
have hcoeff (x : ℝ) (hx : |x| ≤ D) : |packingDensity*x-c*x| ≤ ε/2 := by
have he : packingDensity*x-c*x = (packingDensity-c)*x := by ring
rw [he, abs_mul]
have hcx : |packingDensity-c| ≤ ε/(2*D) := by
dsimp [c]
rw [abs_sub_comm]
exact hc.le
calc
_ ≤ (ε/(2*D))*D := mul_le_mul hcx hx (abs_nonneg _) (by positivity)
_ = ε/2 := by field_simp
have he1 := hcoeff h (hh.trans (by dsimp [D]; linarith))
have he4 := hcoeff (4*h-9) (by rw [abs_le]; dsimp [D]; constructor <;> linarith)
have hlog1 : packingDensity*|Real.log (b^3)|/t ≤ ε/2 := by
apply le_trans _ herror
apply div_le_div_of_nonneg_right _ ht0.le
dsimp [K]
nlinarith [packingDensity_pos, abs_nonneg (Real.log (blobTetraConstant b))]
have hlog4 : packingDensity*|Real.log (blobTetraConstant b)|/t ≤ ε/2 := by
apply le_trans _ herror
apply div_le_div_of_nonneg_right _ ht0.le
dsimp [K]
nlinarith [packingDensity_pos, abs_nonneg (Real.log (b^3))]
have hlo' : bareLimit h-grandPressure (corePotential t) t β h ≤ ε := by
have h1 : packingDensity*h ≤ grandPressure (corePotential t) t β h+ε := by
have hh := (abs_le.mp he1).2
dsimp only [c] at hh
linarith [hlo.1]
have h4 : packingDensity*(4*h-9) ≤ grandPressure (corePotential t) t β h+ε := by
have hh := (abs_le.mp he4).2
dsimp only [c] at hh
linarith [hlo.2]
have hm : bareLimit h ≤ grandPressure (corePotential t) t β h+ε :=
max_le (by linarith) (max_le h1 h4)
linarith
exact abs_le.mpr ⟨by linarith, hup'⟩
end RadialTransition
end
end OAI
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