| import OAI.Probability.RadialTransition.InitialSeparation |
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| namespace OAI |
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| noncomputable section |
| open scoped BigOperators NNReal |
| open Set Filter |
| namespace RadialTransition |
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| lemma exists_family_pressure_ball : |
| ∃ t b : ℝ, 1 ≤ t ∧ 2 ≤ b ∧ |
| 2197*t*bumpIntegral⁻¹*iterationBudget ≤ (t^2/8)/8 ∧ |
| ∃ x ∈ phaseWindow (1/8), Metric.closedBall targetSlope pressureRadius ⊆ |
| supportSet (onParameters (grandPressure (familyPotential t (iterationMass b) (scaleRange b)) t)) positiveParameters x := by |
| have hinitE := initial_separation phaseRadius_pos (show 0<pressureRadius/64 by have := pressureRadius_pos; positivity) |
| obtain ⟨t,⟨⟨⟨hinit,ht⟩,htA⟩,htrow⟩⟩ := (hinitE.and (eventually_ge_atTop (1:ℝ)) |>.and |
| (eventually_ge_atTop (6912*iterationBudget)) |>.and |
| (eventually_ge_atTop (140608*bumpIntegral⁻¹*iterationBudget))).exists |
| have ht0 : 0 < t := by linarith |
| have hrow : 2197*t*bumpIntegral⁻¹*iterationBudget ≤ (t^2/8)/8 := by |
| have hh := mul_le_mul_of_nonneg_left htrow ht0.le |
| nlinarith only [hh] |
| obtain ⟨U,C,C₀,hU,hC,hC₀,hcontrol⟩ := family_controls ht iterationBudget_pos.le htA hrow |
| obtain ⟨D,K,hD,hK,hkac⟩ := family_kac_uniform ht (by norm_num : (0:ℝ)<1/2) |
| (by norm_num : (1/2:ℝ)≤3/2) (by norm_num : (0:ℝ)≤2) iterationBudget_pos.le htA |
| obtain ⟨b,k,l,hb,hbK,hk,hl,hdom,hrad,hband,hpress⟩ := |
| scale_budgets (K := K) (by linarith : 0≤U) (by linarith : 0≤C) hC₀ (show 0<2*D by positivity) |
| have had := iteration_admissible hb t |
| have hpre (n : ℕ) := reversePrefix_admissible (t := t) had.1 had.2.1 had.2.2.1 had.2.2.2 n |
| have hmass (n : ℕ) : iterationMass b n ∈ Ioc (0:ℝ) iterationBudget := |
| ⟨iterationMass_pos (by linarith) n,iterationMass_le hb n⟩ |
| have hctrl (n : ℕ) := hcontrol _ _ (hpre n) _ (hmass n) |
| have hP (n : ℕ) : ConvexOn ℝ positiveParameters (onParameters (iterationP t b n)) := |
| convexOn_idealAttraction (admissible_pressure_convex ht (hpre n) hrow) (hctrl n).1 |
| have hQ (n : ℕ) : ConvexOn ℝ positiveParameters (onParameters (iterationQ t b n)) := |
| admissible_pressure_convex ht (hpre (n+1)) hrow |
| have happ := iteration_kac ht hD hK hb hbK hrow hkac |
| have hlip (n : ℕ) (x : Parameter) (hx : x ∈ phaseWindow (1/4)) |
| (y : Parameter) (hy : y ∈ phaseWindow (1/4)) : |
| |onParameters (iterationP t b n) x-onParameters (iterationP t b n) y| ≤ C*dist x y := |
| (hctrl n).2.2.1 x hx y hy |
| have hinc (n : ℕ) (x : Parameter) (hx : x ∈ phaseWindow (1/4)) : |
| |onParameters (iterationP t b (n+1)) x-onParameters (iterationQ t b n) x| ≤ C₀*scaleTail b n := by |
| have hh := (hctrl (n+1)).2.2.2 x hx |
| change 0 ≤ onParameters (iterationP t b (n+1)) x-onParameters (iterationQ t b n) x ∧ |
| onParameters (iterationP t b (n+1)) x-onParameters (iterationQ t b n) x ≤ C₀*scaleTail b n at hh |
| rw [abs_of_nonneg hh.1] |
| exact hh.2 |
| have hball : ∀ n, ∃ x ∈ Metric.ball phaseCenter (1/8), Metric.closedBall targetSlope pressureRadius ⊆ |
| supportSet (onParameters (iterationP t b n)) positiveParameters x := by |
| apply multiscale_pressure_balls (fun n => scaleTail_pos (by linarith) n) |
| (fun n => show 0≤2*D*(scaleTail b n)^2 by positivity) (by linarith : 0≤U) |
| (by linarith : 0≤C) hk hl hP hQ (iterationP_succ t b) happ hlip hinc |
| (fun n => (hctrl (n+1)).1) (fun n => (hctrl (n+1)).2.1) ?_ hdom hrad hband hpress |
| intro x hx |
| rw [iterationP_zero] |
| exact ⟨(hinit x hx).2,(hinit x hx).1⟩ |
| have hlim (x : Parameter) (hx : x ∈ phaseWindow (1/4)) : |
| Tendsto (fun n => onParameters (iterationP t b n) x) atTop |
| (nhds (onParameters (grandPressure (familyPotential t (iterationMass b) (scaleRange b)) t) x)) := by |
| have hβ : 0 < x 0 := phaseWindow_positive (by norm_num) hx |
| have hlimQ := (grandPressure_reversePrefix_tendsto (h := x 1) ht hβ had hrow).comp (tendsto_add_atTop_nat 1) |
| have hdiff : Tendsto (fun n => onParameters (iterationP t b n) x-onParameters (iterationQ t b n) x) atTop (nhds 0) := by |
| apply squeeze_zero_norm (a := fun n => 2*D*(scaleTail b n)^2) |
| · intro n |
| simpa only [Real.norm_eq_abs,abs_sub_comm] using happ n x hx |
| · simpa using ((summable_scaleTail hb).tendsto_atTop_zero.pow 2).const_mul (2*D) |
| have hsum := hdiff.add hlimQ |
| simpa only [Function.comp_apply,onParameters,iterationQ,sub_add_cancel,zero_add] using hsum |
| exact ⟨t,b,ht,hb,hrow,pressure_ball_limit (admissible_pressure_convex ht had hrow) hball hlip hlim⟩ |
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| end RadialTransition |
| end |
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| end OAI |
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