File size: 5,387 Bytes
9b6e014 | 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 | import OAI.Probability.ContinuumTransition.ConstructionStage
namespace OAI
section
open scoped Topology
open Set Filter Metric
noncomputable section
namespace ContinuumTemperature
structure StageTransition {t p τ : ℝ} {n : ℕ} (s : ConstructionStage t p τ n)
(s' : ConstructionStage t p τ (n+1)) (Rnew : ℝ) : Prop where
range_large : 4 < Rnew
mass_eq : s'.σ = prepend s.a s.σ
range_eq : s'.R = prepend Rnew s.R
next_small : s'.a < geometricBudget n/2
displacement : dist s'.x s.x < geometricBudget n/32
approximation : ∀ y ∈ closedBall initialPoint (1/4), |s'.pressure y-s.ideal y| ≤ geometricBudget n
lemma construction_step {t p τ γ : ℝ} (hp : 0 < p) (hτ : 0 < τ) (hτp : τ ≤ p/100)
(hγ : 0 < γ) (hcap : ConstructionCap t (firstMass p+1))
(hgeom : ∀ w : Fin 3 → Parameter, (∀ i, dist (w i) (triangleVertex p i) ≤ τ) →
closedBall (triangleCenter p) γ ⊆ convexHull ℝ (range w))
{n : ℕ} (s : ConstructionStage t p τ n) :
∃ s' : ConstructionStage t p τ (n+1), ∃ Rnew, StageTransition s s' Rnew := by
have hWmod : closedBall initialPoint (1/4) ⊆ moderateBox :=
(closedBall_subset_closedBall (by norm_num : (1:ℝ)/4 ≤ 1/2)).trans initial_ball_moderate
have hball := s.near_moderate.trans (hWmod.trans moderateBox_subset)
have hxmod : s.x ∈ moderateBox := hWmod (s.x_near.le.trans (by norm_num : (1:ℝ)/8 ≤ 1/4))
have hxdom := moderateBox_subset hxmod
have hsl := s.triple.support_ball hp (groupRadius_le hτ.le n) hτp hgeom
have hb : groupRadius τ n ≤ p/100 := (groupRadius_le hτ.le n).trans hτp
let ε := τ*geometricBudget n/4
have hε : 0 < ε := by dsimp [ε]; positivity [geometricBudget_pos n]
obtain ⟨S,hS,δ,hδ,hpersist⟩ := three_group_persistence (s.convex hcap) hxdom
(show (0:ℝ) < 1/8 by norm_num) hγ hball hsl
(slopeGroups_convex s.ideal s.x (triangleVertex p) (groupRadius τ n))
(fun i j hij => HasVertexTriple.separated hp hb i j hij) s.triple.hull hε
(fun _ => inter_subset_right) (1048576^2) 1048576
let ell := geometricBudget n/32
have hell : 0 < ell := by dsimp [ell]; positivity [geometricBudget_pos n]
let a := min δ (min (ell/(S+1)) (geometricBudget n/2))/2
have ha : 0 < a := by dsimp [a]; positivity [geometricBudget_pos n]
have ham : a < min δ (min (ell/(S+1)) (geometricBudget n/2)) := by dsimp [a] at ha ⊢; linarith
have haδ : a < δ := ham.trans_le (min_le_left _ _)
have haell : a < ell/(S+1) := ham.trans_le ((min_le_right _ _).trans (min_le_left _ _))
have hapow : a < geometricBudget n/2 := ham.trans_le ((min_le_right _ _).trans (min_le_right _ _))
have hSa : S*a < ell := by
have hh := (lt_div_iff₀ (show 0 < S+1 by linarith)).mp haell
nlinarith only [hh,ha]
obtain ⟨B,hB,hbound⟩ := s.referenceBounded
have hprec : 0 < min (a^2) (geometricBudget n) := lt_min (sq_pos_of_pos ha) (geometricBudget_pos n)
obtain ⟨Rnew,hRnew,hRbig,happ⟩ := physical_kac_approximate hcap s.a_pos s.σ_nonneg s.R_one s.σ_summable
s.mass_total hB hbound hprec (B+1)
let σ' := prepend s.a s.σ
let R' := prepend Rnew s.R
have hσ' : ∀ k, 0 ≤ σ' k := prepend_nonneg s.a_pos.le s.σ_nonneg
have hR' : ∀ k, 1 ≤ R' k := prepend_ge_one (by linarith) s.R_one
have hs' : Summable σ' := summable_prepend s.a s.σ_summable
have hmass' : ∑' k, σ' k = s.a+∑' k, s.σ k := tsum_prepend s.a s.σ_summable
have hm' : ∑' k, σ' k ≤ firstMass p+1 := hmass' ▸ s.mass_total
have htot : a+∑' k, σ' k ≤ firstMass p+1-geometricBudget (n+1) := by
rw [hmass',geometricBudget_succ]
linarith only [s.mass,hapow]
have haM : a ≤ firstMass p+1 := by
have hh : 0 ≤ ∑' k, σ' k := tsum_nonneg hσ'
have hh' := geometricBudget_pos (n+1)
linarith only [htot,hh,hh']
let Q := physicalPressure (physicalProfile t σ' R') t
let P := meanField Q a
have hQ := physical_convex_bounded hcap.t_large hσ' hR' hs' (hcap.weak hm')
have hMF := physical_meanField_bounded hcap.t_large ha hσ' hR' hs' (hcap.weak hm') (hcap.lt_gap haM)
obtain ⟨z,hz,hzr,hg,hull⟩ := hpersist a ha haδ Q P hQ hMF.1 hMF.2
(fun y hy => (happ y (s.near_moderate hy)).trans (min_le_left _ _))
(physical_transform_gain hcap.t_large ha hσ' hR' hs' (hcap.weak hm') (hcap.lt_gap haM)
hxmod.1 hxmod.2.1 hxmod.2.2)
(fun y hy w hw => by
have hh := hWmod (s.near_moderate hy)
have hd := physical_transform_active_bound hcap.t_large ha hσ' hR' hs' (hcap.weak hm') (hcap.lt_gap haM)
hh.1 hh.2.1 hh.2.2 hw
simpa only [abs_of_nonneg hd.1] using hd.2)
have hdisp : dist z s.x < geometricBudget n/32 := hz.trans hSa
let s' : ConstructionStage t p τ (n+1) := {
σ := σ', R := R', a := a, x := z,
σ_nonneg := hσ', R_one := hR', σ_summable := hs', a_pos := ha, mass := htot,
referenceBounded := ⟨Rnew,by linarith,by
intro k hk
cases k with
| zero => exact le_refl _
| succ k => exact (hbound k hk).trans (by linarith)⟩,
location := by
rw [locationRadius_succ]
have hh := dist_triangle z s.x initialPoint
linarith only [hh,hdisp,s.location],
triple := ⟨hg,by rw [groupRadius_succ]; exact hull⟩ }
refine ⟨s',Rnew,hRnew,rfl,rfl,hapow,hdisp,?_⟩
intro y hy
exact (happ y hy).trans (min_le_right _ _)
end ContinuumTemperature
end
end
end OAI
|