| import Mathlib | |
| import OAI.Analysis.HardSphere.Geometry.DeficiencyPenalty | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter Set | |
| open scoped ENNReal Topology BigOperators | |
| open scoped BigOperators ComplexConjugate | |
| open MeasureTheory ProbabilityTheory Filter Set | |
| open scoped ENNReal NNReal Topology BigOperators | |
| open MeasureTheory ProbabilityTheory Real Complex | |
| open scoped ENNReal ComplexConjugate | |
| open ProbabilityTheory | |
| namespace HardSphere | |
| open MeasureTheory Set Filter PeriodicCutoff TightCutoff | |
| open scoped ComplexConjugate | |
| lemma exists_balanced_insertion (S : Type) [Fintype S] [Nonempty S] [DecidableEq S] (m : ℕ) : | |
| ∃ site : ℕ → S, ∀ z, insertionCount site (m*Fintype.card S) z=m := by | |
| classical | |
| let T := m*Fintype.card S | |
| let e : Fin T ≃ Fin m × S := Fintype.equivOfCardEq (by simp [T]) | |
| let site : ℕ → S := fun i ↦ if h : i<T then (e ⟨i,h⟩).2 else Classical.choice ‹Nonempty S› | |
| refine ⟨site,?_⟩ | |
| intro z | |
| let A := (Finset.range T).filter (fun i ↦ site i=z) | |
| have hl (i : A) : i.val<T := Finset.mem_range.mp (Finset.mem_filter.mp i.property).1 | |
| have hz (i : A) : (e ⟨i.val,hl i⟩).2=z := by | |
| have h := (Finset.mem_filter.mp i.property).2 | |
| simpa only [site,dite_eq_left (hl i)] using h | |
| let f : A → Fin m := fun i ↦ (e ⟨i.val,hl i⟩).1 | |
| let g : Fin m → A := fun j ↦ ⟨(e.symm (j,z)).val,by | |
| apply Finset.mem_filter.mpr | |
| refine ⟨Finset.mem_range.mpr (e.symm (j,z)).isLt,?_⟩ | |
| simp only [site,dite_eq_left (e.symm (j,z)).isLt,e.apply_symm_apply]⟩ | |
| have hg (j : Fin m) : f (g j)=j := by | |
| change (e (e.symm (j,z))).1=j | |
| rw [e.apply_symm_apply] | |
| have hf (i : A) : g (f i)=i := by | |
| apply Subtype.ext | |
| change (e.symm ((e ⟨i.val,hl i⟩).1,z)).val=i.val | |
| rw [←hz i] | |
| simp only [Prod.mk.eta,e.symm_apply_apply] | |
| let q : A ≃ Fin m := ⟨f,g,hf,hg⟩ | |
| have hc := Fintype.card_congr q | |
| simpa only [Fintype.card_coe,Fintype.card_fin,insertionCount,A,T] using hc | |
| lemma deficiency_penalty_cells : ∃ C : ℝ, 0<C ∧ | |
| ∀ (S : Type) [Fintype S] [Nonempty S] [DecidableEq S] | |
| (n : ℕ) (L : Length) (r : ℝ), 0<r → 0<n → | |
| ∀ (g : Jet n L), IsGroundVector n L r g → | |
| ∀ (w : S → SmoothMultiplier 1 L), | |
| (∀ z X, conj ((w z).toFun X)=(w z).toFun X) → | |
| (∀ z, (∫ y : Torus L, ‖(w z).orbital y‖^2)=1) → | |
| (∀ z v : S, z≠v → ∀ x : Torus L, (w v).orbital x≠0 → | |
| ¬ ∃ y∈tsupport (w z).orbital, torusDistance L y x ≤ 2*r) → | |
| ∀ (k m : ℕ) (B E : ℝ), | |
| (∀ z, (w z).orbitalBound ≤ B) → (∀ z, (w z).orbitalEnergy ≤ E) → | |
| C*B*(k+m)*r^3 ≤ 1/4 → | |
| ∀ u : Jet n L, u∈dirichletDomain n L r → | |
| (∀ᵐ Y ∂volume, u.1 Y≠0 → ∀ z, (closeLabels r (w z) Y).card ≤ k) → | |
| m*Fintype.card S*(energy g/(n:ℝ)-C*(E+B*(k+m)*r))*‖u.1‖^2 ≤ | |
| energy u-energy g*‖u.1‖^2 := by | |
| obtain ⟨C,hC,hdef⟩ := deficiency_penalty | |
| refine ⟨C,hC,?_⟩ | |
| intro S _ _ _ n L r hr hn g hg w hw hwN hsep k m B E hB hE hs u hu hk | |
| obtain ⟨site,hsite⟩ := exists_balanced_insertion S m | |
| simpa only [Nat.cast_mul] using hdef S n L r hr hn g hg w hw hwN hsep (m*Fintype.card S) k m site B E hB hE | |
| (fun z ↦ le_of_eq (hsite z)) hs u hu hk | |
| end HardSphere | |
| end | |
| end OAI | |