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f205175 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 | 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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