| import OAI.Probability.RadialTransition.PhysicalIteration |
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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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| lemma core_ideal_convex_active {t : ℝ} (ht : 1 ≤ t) (htA : 6912*phaseMass ≤ t) : |
| ConvexOn ℝ positiveParameters (onParameters (grandPressure (corePotential t) t)) ∧ |
| (∀ β h, 0 < β → ∃ u, ActiveShift (grandPressure (corePotential t) t) phaseMass β h u) := by |
| have hb : ∀ n : ℕ, 0 ≤ (0:ℝ) := fun _ => le_rfl |
| have hS : ∀ n : ℕ, 1 ≤ (1:ℝ) := fun _ => le_rfl |
| have hs : Summable (fun _ : ℕ => (0:ℝ)) := summable_zero |
| have hrow : 2197*t*bumpIntegral⁻¹*(∑' _ : ℕ, (0:ℝ)) ≤ (t^2/8)/8 := by simp; positivity |
| have hact := family_ideal_global_active ht phaseMass_pos.le htA hb hS hs hrow |
| rw [familyPotential_zero] at hact |
| refine ⟨?_, hact⟩ |
| have ht0 : 0 < t := by linarith |
| have hst := stable_family_of_master ht0.le hb hS hs hs (fun _ => le_rfl) hrow |
| rw [familyPotential_zero] at hst |
| exact convexOn_grandPressure (continuous_corePotential t) (bounded_corePotential t) hst ht0 |
| (fun r hr => by simpa only [familyPotential_zero] using familyPotential_nonpos_outer hb hS ht (r := r) hr) |
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| lemma initial_separation {τ ε₀ : ℝ} (hτ : 0 < τ) (hε₀ : 0 < ε₀) : |
| ∀ᶠ t : ℝ in atTop, ∀ x ∈ phaseWindow (1/4), |
| |onParameters (idealAttraction (grandPressure (corePotential t) t) phaseMass) x- |
| onParameters threePlane x| ≤ ε₀ ∧ |
| HasPhaseGroups (idealAttraction (grandPressure (corePotential t) t) phaseMass) x (τ/2) := by |
| obtain ⟨U₀, hU₀, hcut⟩ := uniform_ideal_cutoff (by norm_num : (0:ℝ)<1/2) |
| (by norm_num : (1/2:ℝ)≤3/2) (by norm_num : (0:ℝ)≤2) phaseMass_pos.le |
| let U := max U₀ (4*packingDensity) |
| have hUU : U₀ ≤ U := le_max_left _ _ |
| have hU : 1 ≤ U := hU₀.trans hUU |
| have hUp : 4*packingDensity ≤ U := le_max_right _ _ |
| let K := 1+phaseMass/2*(U+4*packingDensity) |
| have hK : 0 < K := by have := phaseMass_pos; have := packingDensity_pos; dsimp [K]; positivity |
| let η := τ/(4*K) |
| have hη : 0 < η := div_pos hτ (by positivity) |
| let ε := min ε₀ (min (τ/32) (phaseMass*η^2/16)) |
| have hε : 0 < ε := lt_min hε₀ (lt_min (by positivity) (by have := phaseMass_pos; positivity)) |
| have he₀ : ε ≤ ε₀ := min_le_left _ _ |
| have heτ : ε ≤ τ/32 := (min_le_right _ _).trans (min_le_left _ _) |
| have heη : ε ≤ phaseMass*η^2/16 := (min_le_right _ _).trans (min_le_right _ _) |
| let H := 2+2*phaseMass*U |
| have hH : 0 ≤ H := by have := phaseMass_pos; dsimp [H]; positivity |
| filter_upwards [corePressure_uniform_bare (by norm_num : (0:ℝ)<1/4) hH hε, |
| eventually_ge_atTop (1:ℝ), eventually_ge_atTop (6912*phaseMass)] with t happ ht htA |
| let Q := grandPressure (corePotential t) t |
| obtain ⟨hQ, hact⟩ := core_ideal_convex_active ht htA |
| have hb : ∀ n : ℕ, 0 ≤ (0:ℝ) := fun _ => le_rfl |
| have hS : ∀ n : ℕ, 1 ≤ (1:ℝ) := fun _ => le_rfl |
| have hs : Summable (fun _ : ℕ => (0:ℝ)) := summable_zero |
| have hrow : 2197*t*bumpIntegral⁻¹*(∑' _ : ℕ, (0:ℝ)) ≤ (t^2/8)/8 := by simp; positivity |
| have hactive (β h : ℝ) (hβ : β ∈ Icc (1/2:ℝ) (3/2)) (hh : |h| ≤ 2) : |
| ∃ u ∈ Icc (0:ℝ) U, ActiveShift Q phaseMass β h u := by |
| obtain ⟨κ, hκ, hc, hbound⟩ := hcut t ht htA β hβ h (abs_le.mp hh) phaseMass ⟨phaseMass_pos.le,le_rfl⟩ |
| have hbound' := hbound.trans (mul_le_mul_of_nonneg_left hUU hκ.le) |
| obtain ⟨u, hu, ha⟩ := family_ideal_exists_active ht (by linarith [hβ.1]) phaseMass_pos.le |
| hb hS hs hrow hκ hc hU hbound' |
| rw [familyPotential_zero] at ha |
| exact ⟨u, hu, ha⟩ |
| have hshift (β h u : ℝ) (hβ : β ∈ Icc (1/2:ℝ) (3/2)) (hh : |h| ≤ 2) |
| (hu : u ∈ Icc (0:ℝ) U) : |h+β*phaseMass*u| ≤ H := by |
| have hb0 : 0 ≤ β := by linarith [hβ.1] |
| have ha0 := phaseMass_pos.le |
| have hu0 := hu.1 |
| have hmul : β*phaseMass*u ≤ 2*phaseMass*U := |
| mul_le_mul (mul_le_mul_of_nonneg_right (show β ≤ 2 by linarith [hβ.2]) phaseMass_pos.le) |
| hu.2 hu.1 (by have := phaseMass_pos; positivity) |
| calc |
| _ ≤ |h|+|β*phaseMass*u| := abs_add_le _ _ |
| _ ≤ H := by rw [abs_of_nonneg (by positivity : 0 ≤ β*phaseMass*u)]; dsimp [H]; linarith |
| intro x hx |
| obtain ⟨hβ, hh⟩ := phaseWindow_quarter_bounds hx |
| have hβ' : x 0 ∈ Icc (1/2:ℝ) (3/2) := ⟨by linarith [hβ.1], by linarith [hβ.2]⟩ |
| have hβ0 : 0 < x 0 := by linarith [hβ.1] |
| have happx : ∀ u ∈ Icc (0:ℝ) U, |Q (x 0) (x 1+x 0*phaseMass*u)-bareLimit (x 1+x 0*phaseMass*u)| ≤ ε := by |
| intro u hu |
| exact happ _ _ (by linarith [hβ.1]) (hshift _ _ _ hβ' hh hu) |
| constructor |
| · obtain ⟨u, hub, hu⟩ := hactive _ _ hβ' hh |
| exact (ideal_approx_threePlane hβ0.le hUp hu hub happx).trans he₀ |
| · have hlocal : ∀ y ∈ Metric.closedBall x (1/16), ∃ u ∈ Icc (0:ℝ) U, ActiveShift Q phaseMass (y 0) (y 1) u := by |
| intro y hy |
| have hy' : y ∈ phaseWindow (1/2) := phaseWindow_mono (by norm_num : (1/4:ℝ)+1/16≤1/2) (phaseWindow_near hx hy) |
| obtain ⟨hyβ, hyh⟩ := phaseWindow_half_bounds hy' |
| exact hactive _ _ hyβ hyh |
| have hgen := ideal_supportSet_generated hQ phaseMass_pos hact (by norm_num : (0:ℝ)<1/16) |
| (by linarith [hβ.1] : 2*(1/16:ℝ)<x 0) hlocal |
| apply phase_groups_of_generators hgen |
| rintro v ⟨u, hub, hu, e, hes, rfl⟩ |
| have happe : ∀ s ∈ ({-(1/4:ℝ),0,1/4} : Set ℝ), |
| |Q (x 0+s) (x 1+x 0*phaseMass*u)-bareLimit (x 1+x 0*phaseMass*u)| ≤ ε := by |
| intro s hs |
| have hs' : -(1/4:ℝ) ≤ s := by |
| simp only [mem_insert_iff,mem_singleton_iff] at hs |
| rcases hs with rfl | rfl | rfl <;> norm_num |
| exact happ _ _ (by linarith [hβ.1]) (hshift _ _ _ hβ' hh hub) |
| have hsmall : 2*ε ≤ x 0*phaseMass/2*η^2 := by |
| have hm := mul_le_mul_of_nonneg_right hβ.1 (mul_nonneg phaseMass_pos.le (sq_nonneg η)) |
| nlinarith |
| obtain ⟨i, hi⟩ := phase_generator_distance hβ0 (by norm_num : (0:ℝ)<1/4) |
| (by linarith [hβ.1]) hη.le hUp hu hub happx happe hsmall hes |
| refine ⟨i, Metric.mem_closedBall.mpr ?_⟩ |
| rw [dist_eq_norm] |
| apply hi.trans |
| have heq : K*η=τ/4 := by dsimp [η]; field_simp |
| change 2*ε/(1/4)+K*η ≤ τ/2 |
| rw [heq] |
| linarith |
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| end RadialTransition |
| end |
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| end OAI |
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