| import OAI.Probability.ContinuumTransition.ConstructionStep |
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| namespace OAI |
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| section |
| open scoped Topology |
| open Set Filter Metric |
| noncomputable section |
| namespace ContinuumTemperature |
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| 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) |
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| 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) |
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| 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 |
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| end ContinuumTemperature |
| end |
| end |
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| end OAI |
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