| import Mathlib |
| import OAI.Analysis.HardSphere.Geometry.FrozenChecks |
| import OAI.Analysis.HardSphere.Conditioning.BathPhase |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped ENNReal |
| namespace HardSphere |
| lemma insertFirst_preserving {N : ℕ} (hN : 0<N) (L : Length) : |
| MeasurePreserving (fun p : Torus L×Configuration (N-1) L ↦ insertFirst hN p.1 p.2) |
| ((volume : Measure (Torus L)).prod volume) volume := by |
| cases N with |
| | zero => omega |
| | succ n => |
| unfold insertFirst |
| have h := (cons_preserving (n:=n) L).comp |
| (Measure.measurePreserving_swap (μ:=(volume : Measure (Torus L))) (ν:=volume)) |
| convert h using 1 <;> simp [Function.comp_def] |
|
|
| lemma bathFirst_preserving {N : ℕ} (hN : 0<N) (L : Length) : |
| MeasurePreserving (fun p : Configuration (N-1) L×Torus L ↦ insertFirst hN p.2 p.1) |
| ((volume : Measure (Configuration (N-1) L)).prod volume) volume := |
| (insertFirst_preserving hN L).comp (Measure.measurePreserving_swap) |
|
|
| lemma ground_slice_memLp {N : ℕ} (hN : 0<N) {L : Length} (ψ : ScalarL2 N L) : |
| MemLp (fun p : Configuration (N-1) L×Torus L ↦ ψ (insertFirst hN p.2 p.1)) 2 |
| ((volume : Measure (Configuration (N-1) L)).prod volume) := |
| (Lp.memLp ψ).comp_measurePreserving (bathFirst_preserving hN L) |
|
|
| lemma occupation_nonneg {N : ℕ} (hN : 0<N) {L : Length} (ψ : ScalarL2 N L) : |
| 0≤occupation hN ψ := by |
| unfold occupation |
| exact mul_nonneg (inv_nonneg.mpr (pow_nonneg L.property.le _)) |
| (integral_nonneg fun _ ↦ sq_nonneg _) |
|
|
| end HardSphere |
|
|
| end |
| end |
|
|
| section |
| noncomputable section |
| open MeasureTheory Set |
| open scoped ENNReal |
| namespace HardSphere.Comparison |
| variable {I C : Type*} [Fintype I] [DecidableEq I] [Fintype C] |
| {X : I → Type*} [∀i,MeasurableSpace (X i)] |
| omit [Fintype C] in |
| lemma frozen_uniformInsertion_on (checks : C → LocalCheck X) (U S : Finset I) |
| (hSU : S⊆U) (x : ∀i,X i) (μ : ∀i,Measure (X i)) {a : ℝ≥0∞} |
| (h : UniformInsertionOn checks μ a S) : |
| UniformInsertionOn (frozenChecks checks U x) μ a S := by |
| intro i hi s his y |
| have hin : i∉s∪(Finset.univ\U) := by simp [Finset.mem_union,his,hSU hi] |
| apply (h i hi (s∪(Finset.univ\U)) hin (freezeOutside U x y)).trans |
| apply lintegral_mono |
| intro v |
| dsimp only |
| rw [←freezeOutside_update U x y (hSU hi) v] |
| by_cases hp : freezeOutside U x (Function.update y i v)∈insertionEvent checks (s∪(Finset.univ\U)) i |
| · rw [Set.indicator_of_mem hp,Set.indicator_of_mem (frozen_insertion checks U s x (Function.update y i v) i hp)] |
| · rw [Set.indicator_of_notMem hp] |
| exact bot_le |
| end HardSphere.Comparison |
|
|
| end |
| end |
|
|
| end |
|
|
| end OAI |
|
|
