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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 | import OAI.Probability.RadialTransition.PhysicalIteration
namespace OAI
noncomputable section
open scoped BigOperators ENNReal NNReal
open MeasureTheory Set Filter
namespace RadialTransition
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)
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
end RadialTransition
end
end OAI
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