| import Mathlib | |
| import OAI.Analysis.HardSphere.Geometry.GridAlignment | |
| import OAI.Analysis.HardSphere.PathBounds.VisitCounts | |
| import OAI.Analysis.HardSphere.Thermodynamics.ShiftCloseTail | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Set Filter | |
| open scoped ENNReal NNReal Topology BigOperators | |
| namespace HardSphere.Groups | |
| open Wiener PathCutoff CellOrbital BathSlice HardSphere.Lattice | |
| lemma joint_event_image {W V U : Type*} [MeasurableSpace W] [DecidableEq U] | |
| (μ : Measure W) (f : V→U) (P : U→W→Prop) (A : Finset V) : | |
| μ {ω | ∀v∈A,P (f v) ω}=μ {ω | ∀u∈A.image f,P u ω} := by | |
| congr 1 | |
| ext ω | |
| simp only [Set.mem_ofPred_eq,Finset.mem_image,forall_exists_index,and_imp,forall_apply_eq_imp_iff₂] | |
| lemma grid_visitors_eq {n K : ℕ} [NeZero K] (L : Length) | |
| (v : Fin 3→Fin K) (ω : ConfigurationPath n L) : | |
| ShiftVisits.visitorCount L (gridShift L) Set.univ (gridSite v) ω=(visitors (gridCenter L v) ω).card := by | |
| classical | |
| rw [grid_visitorCount] | |
| apply congrArg Finset.card | |
| ext i | |
| simp [visitors] | |
| lemma grid_longVisitors_eq {n K : ℕ} [NeZero K] (L : Length) | |
| (v : Fin 3→Fin K) (ω : ConfigurationPath n L) : | |
| ShiftVisits.visitorCount L (gridShift L) {p | 1 < intervalDiameter 0 1 p} (gridSite v) ω= | |
| (longVisitors (gridCenter L v) ω).card := by | |
| classical | |
| rw [grid_visitorCount] | |
| apply congrArg Finset.card | |
| ext i | |
| simp [longVisitors,and_comm] | |
| lemma grid_nonregular_iff {n K : ℕ} [NeZero K] (L : Length) | |
| (m : ℕ) (r R C S : ℝ) (u : Time) (v : Fin 3→Fin K) (ω : ConfigurationPath n L) : | |
| (¬∀i,pathVisitsBall L 12 (gridCenter L v) i ω→Regular m r R C u S (onePathAt i ω)) ↔ | |
| 1≤(ShiftVisits.visitorCount L (gridShift L) {p | ¬Regular m r R C u S p} (gridSite v) ω:ℝ) := by | |
| classical | |
| rw [grid_visitorCount] | |
| have hcast (k : ℕ) : (1:ℝ)≤(k:ℝ) ↔ 1≤k := by exact_mod_cast Iff.rfl | |
| rw [hcast,Finset.one_le_card] | |
| change (¬∀i, _ → _) ↔ ∃i, i∈_ | |
| simp only [Finset.mem_filter,Finset.mem_univ,true_and,Set.mem_ofPred_eq] | |
| push Not | |
| exact exists_congr (fun i ↦ and_comm) | |
| lemma ofReal_power_tail {x q : ℝ} (hx : 0≤x) (hq : 0≤q) (h : x≤q^6) (n : ℕ) : | |
| ENNReal.ofReal (x^n)≤(ENNReal.ofReal q)^(6*n) := by | |
| rw [pow_mul,←ENNReal.ofReal_pow hq,←ENNReal.ofReal_pow (pow_nonneg hq _)] | |
| exact ENNReal.ofReal_le_ofReal (pow_le_pow_left₀ hx h n) | |
| lemma exp_power_tail {x q : ℝ} (hq : 0≤q) (h : Real.exp x≤q^6) (n : ℕ) : | |
| ENNReal.ofReal (Real.exp (x*(n:ℝ)))≤(ENNReal.ofReal q)^(6*n) := by | |
| rw [mul_comm x (n:ℝ),Real.exp_nat_mul] | |
| exact ofReal_power_tail (Real.exp_pos x).le hq h n | |
| lemma grid_close_tail : ∃ Q : ℕ, 0<Q ∧ ∀ θ : ℝ, 0<θ → | |
| ∃ b Gθ : ℝ, 0<b ∧ b≤1/4 ∧ 0<Gθ ∧ | |
| Tendsto (fun D : ℝ ↦ ((Gθ/b^2)*D^(-(4/3):ℝ))^(1/(Q:ℝ))) atTop (𝓝 0) ∧ | |
| ∀ αmax : ℝ, ∃ D₀ : ℝ, 1≤D₀ ∧ | |
| ∀ (n K : ℕ) [Nontrivial (Fin n)] (L : Length) (_hL : L.val=(K:ℝ)) (D α : ℝ), | |
| 100≤L.val → D=(n:ℝ)/L.val^3 → D₀≤D → 1≤α → α≤αmax → | |
| ∀ (g : Jet n L), IsGroundVector n L (α/D) g → | |
| ∀ (T : ℝ), 0≤T → ∀ (q : Time) (A : Finset (Fin 3→Fin K)), | |
| groundPathLaw n L (α/D) T g {ω | ∀ v∈A, | |
| θ*D≤((localCloseLabels L (spatialCloseScale D b) (gridCenter L v) (ω q)).card:ℝ)}≤ | |
| ENNReal.ofReal (((((Gθ/b^2)*D^(-(4/3):ℝ))^(1/(Q:ℝ)))^A.card)) := by | |
| obtain ⟨Q,hQ,htail⟩ := spatial_translated_stationary_close_tail | |
| refine ⟨Q,hQ,?_⟩ | |
| intro θ hθ | |
| obtain ⟨b,G,hb,hb1,hG,hl,ht⟩ := htail θ hθ | |
| refine ⟨b,G,hb,hb1,hG,hl,?_⟩ | |
| intro αmax | |
| obtain ⟨D₀,hD₀,hh⟩ := ht αmax | |
| refine ⟨D₀,hD₀,?_⟩ | |
| intro n K inst L hL D α hL100 hDeq hD hα hαmax g hg T hT q A | |
| have h := hh n K L hL D α hL100 hDeq hD hα hαmax g hg T hT q (gridShift L) (A.image gridCube) | |
| rw [Finset.card_image_of_injective _ gridCube_injective] at h | |
| convert h using 1 | |
| congr 1 | |
| ext ω | |
| simp only [Set.mem_ofPred_eq,Finset.mem_image,forall_exists_index,and_imp,forall_apply_eq_imp_iff₂,shifted_unitCenter] | |
| end HardSphere.Groups | |
| end | |
| end | |
| end | |
| end OAI | |