| import Mathlib |
| import OAI.Probability.CriticalPercolation.Criticality.BlockIndependence |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open Set MeasureTheory ProbabilityTheory |
| open scoped Classical ENNReal |
|
|
| namespace CriticalPercolation.BernoulliProduct |
| variable {E : Type*} |
|
|
| lemma fewOpenBonds_probability_le (p : unitInterval) (S : Set E) (hS : S.Finite) |
| (N L : ℕ) (hcard : (N+1)*L ≤ S.ncard) : |
| (law (fun _ : E => p)).real {ω | (S∩ω).ncard ≤ N} ≤ |
| (N+1 : ℝ)*(1-(p:ℝ))^L := by |
| let : Fintype S := hS.fintype |
| obtain ⟨e⟩ := Function.Embedding.nonempty_of_card_le |
| (α := Fin (N+1) × Fin L) (β := S) |
| (show Fintype.card (Fin (N+1) × Fin L) ≤ Fintype.card S by |
| rw [Fintype.card_prod,Fintype.card_fin,Fintype.card_fin] |
| simpa only [← Nat.card_coe_set_eq, Nat.card_eq_fintype_card] using hcard) |
| let f : Fin (N+1) × Fin L → E := fun x => (e x).val |
| have hf : Function.Injective f := fun x y h => e.injective (Subtype.ext h) |
| let B : Fin (N+1) → Set E := fun i => Set.range (fun j : Fin L => f (i,j)) |
| have hBfinite (i : Fin (N+1)) : (B i).Finite := Set.finite_range _ |
| have hBcard (i : Fin (N+1)) : (B i).ncard = L := by |
| rw [Set.ncard_range_of_injective (fun a b h => (Prod.mk.inj (hf h)).2)] |
| simp |
| have hcover : {ω : Set E | (S∩ω).ncard ≤ N} ⊆ ⋃ i, closedBlock (B i) := by |
| intro ω hω |
| by_contra hn |
| have hopen : ∀ i : Fin (N+1), ∃ j : Fin L, f (i,j)∈ω := by |
| intro i |
| have hi : ¬ Disjoint (B i) ω := fun hh => hn (Set.mem_iUnion.mpr ⟨i,hh⟩) |
| obtain ⟨b,hb,hbω⟩ := Set.not_disjoint_iff.mp hi |
| obtain ⟨j,rfl⟩ := hb |
| exact ⟨j,hbω⟩ |
| choose j hj using hopen |
| have hinj : Function.Injective (fun i : Fin (N+1) => f (i,j i)) := |
| fun a b h => (Prod.mk.inj (hf h)).1 |
| have hsub : Set.range (fun i : Fin (N+1) => f (i,j i)) ⊆ S∩ω := by |
| rintro x ⟨i,rfl⟩ |
| exact ⟨(e (i,j i)).property,hj i⟩ |
| have hh := Set.ncard_le_ncard hsub (hS.subset Set.inter_subset_left) |
| rw [Set.ncard_range_of_injective hinj] at hh |
| simp only [Nat.card_eq_fintype_card,Fintype.card_fin] at hh |
| exact Nat.not_succ_le_self N (hh.trans hω) |
| have hreal : (bitMeasure p {False}).toReal = 1-(p:ℝ) := by |
| simpa only [Measure.real,bitWeight,ite_false] using bitMeasure_real_singleton p False |
| calc |
| _ ≤ (law (fun _ : E => p)).real (⋃ i, closedBlock (B i)) := measureReal_mono hcover |
| _ ≤ ∑ i, (law (fun _ : E => p)).real (closedBlock (B i)) := |
| measureReal_iUnion_fintype_le _ |
| _ = ∑ _i : Fin (N+1), (1-(p:ℝ))^L := by |
| apply Finset.sum_congr rfl |
| intro i _ |
| rw [measureReal_def,law_closedBlock_const p _ (hBfinite i),hBcard, |
| ENNReal.toReal_pow,hreal] |
| _ = _ := by simp [Nat.cast_add,Nat.cast_one] |
|
|
| end CriticalPercolation.BernoulliProduct |
| end |
| end |
|
|
| end OAI |
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|
