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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 | import OAI.Probability.ContinuumTransition.ConstructionStep
namespace OAI
section
open scoped Topology
open Set Filter Metric
noncomputable section
namespace ContinuumTemperature
structure ConstructionData (p : ℝ) where
t : ℝ
τ : ℝ
γ : ℝ
τ_pos : 0 < τ
τ_small : τ ≤ p/100
γ_pos : 0 < γ
cap : ConstructionCap t (firstMass p+1)
geometry : ∀ w : Fin 3 → Parameter, (∀ i, dist (w i) (triangleVertex p i) ≤ τ) →
closedBall (triangleCenter p) γ ⊆ convexHull ℝ (range w)
stages : ∀ n, ConstructionStage t p τ n
ranges : ℕ → ℝ
initial_zero : (stages 0).σ = fun _ => 0
initial_mass : (stages 0).a = firstMass p
transitions : ∀ n, StageTransition (stages n) (stages (n+1)) (ranges n)
lemma sequence_of_initial {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))
(s₀ : ConstructionStage t p τ 0) (hzero : s₀.σ = fun _ => 0) (hfirst : s₀.a = firstMass p) :
Nonempty (ConstructionData p) := by
have hex := fun n (s : ConstructionStage t p τ n) => construction_step hp hτ hτp hγ hcap hgeom s
let next := fun n (s : ConstructionStage t p τ n) => Classical.choose (hex n s)
let length := fun n (s : ConstructionStage t p τ n) => Classical.choose (Classical.choose_spec (hex n s))
have hn (n) (s : ConstructionStage t p τ n) : StageTransition s (next n s) (length n s) :=
Classical.choose_spec (Classical.choose_spec (hex n s))
let seq : ∀ n, ConstructionStage t p τ n := Nat.rec s₀ (fun n s => next n s)
refine ⟨⟨t,τ,γ,hτ,hτp,hγ,hcap,hgeom,seq,(fun n => length n (seq n)),hzero,hfirst,?_⟩⟩
intro n
exact hn n (seq n)
lemma exists_constructionData : Nonempty (ConstructionData packingDensity) := by
have hp := packingDensity_pos
obtain ⟨τ₀,γ,hτ₀,hγ,hgeom₀⟩ := robust_vertex_triangle hp
let τ := min τ₀ (packingDensity/100)
have hτ : 0 < τ := lt_min hτ₀ (by positivity)
have hτp : τ ≤ packingDensity/100 := min_le_right _ _
have hgeom : ∀ w : Fin 3 → Parameter, (∀ i, dist (w i) (triangleVertex packingDensity i) ≤ τ) →
closedBall (triangleCenter packingDensity) γ ⊆ convexHull ℝ (range w) :=
fun w hw => hgeom₀ w (fun i => (hw i).trans (min_le_left _ _))
have ha : 0 < firstMass packingDensity := firstMass_pos hp
obtain ⟨T,hT⟩ := exists_constructionCap (show 0 < firstMass packingDensity+1 by linarith)
obtain ⟨t,ht,x,hx,htr⟩ := physical_initial_triple hτ hγ (show (0:ℝ) < 1/64 by norm_num)
(show (1:ℝ)/64 < 1/2 by norm_num) hT.t_large (hT.lt_gap (by linarith : firstMass packingDensity ≤ firstMass packingDensity+1))
(hgeom (triangleVertex packingDensity) (fun _ => by simp only [dist_self]; exact hτ.le))
let s₀ : ConstructionStage t packingDensity τ 0 := {
σ := fun _ => 0, R := fun _ => 1, a := firstMass packingDensity, x := x,
σ_nonneg := fun _ => le_refl _, R_one := fun _ => le_refl _, σ_summable := summable_zero,
a_pos := ha, mass := by simp [geometricBudget],
referenceBounded := ⟨1,by norm_num,fun _ _ => le_refl _⟩,
location := by
change dist x initialPoint < 1/64 at hx
norm_num [locationRadius,geometricBudget]
linarith,
triple := by
have he : groupRadius τ 0 = τ/2 := by norm_num [groupRadius,geometricBudget]; ring
rw [he,physicalProfile_zero]
exact htr }
exact sequence_of_initial hp hτ hτp hγ (hT.mono_t ht) hgeom s₀ rfl rfl
end ContinuumTemperature
end
end
end OAI
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