| import OAI.Probability.RadialTransition.CoreLocalLower |
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| namespace OAI |
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| noncomputable section |
| open scoped BigOperators ENNReal NNReal |
| open MeasureTheory Set Filter |
| namespace RadialTransition |
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| 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))) |
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| 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 |
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| 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 _ _) |
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| lemma tendsto_core_micro : Tendsto microLength atTop (nhds 0) := by |
| exact Real.tendsto_exp_atBot.comp tendsto_neg_atTop_atBot |
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| 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 |
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| 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'⟩ |
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| end RadialTransition |
| end |
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| end OAI |
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