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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 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 | import Mathlib
import OAI.Probability.CriticalPercolation.GraphGeometry.HitEvent
import OAI.Probability.CriticalPercolation.ProductMeasure.FreshRegionLaw
import OAI.Probability.CriticalPercolation.Gluing.CollarErrorBudget
import OAI.Probability.CriticalPercolation.Gluing.SourceCollarTransfer
namespace OAI
section
section
noncomputable section
open Set Function MeasureTheory ProbabilityTheory
open scoped Classical BigOperators ENNReal
namespace CriticalPercolation.BondGraph
open BernoulliProduct
variable {V E : Type*} [Countable V] [Countable E] (G : BondGraph V E)
theorem fresh_collar_transfer (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(A Input T : Set V) (hA : A.Finite) (q : (E⊕A) → unitInterval)
(m R : ℕ) (hcR : m<R) (Dstar : Set E)
(houtside : ∀ a∈A, a∉G.vertexCollar Input R)
(hregion : G.inducedEdges (G.vertexCollar Input R)⊆Dstar)
(pc p' : unitInterval) (hpc₀ : 0<(pc : ℝ)) (hpc : (pc : ℝ)<1)
(hpp : (pc : ℝ)/2 ≤ p') (hppc : (p' : ℝ) ≤ pc)
(hq : ∀ e∈Dstar, q (Sum.inl e)=p')
(D k M VM : ℕ) (hD : ∀ v, (G.incidentEdges {v}).ncard ≤ D)
(hM : 0 < M) (hMbound : ∀ v, (G.vertexBall v (4*m+2)).ncard ≤ M)
(hVM : ∀ v, (G.vertexBall v m).ncard ≤ VM)
(κ η : ℝ) (hκ : 0<κ) (hκ1 : κ≤1) (hη₀ : 0≤η)
(hη : η ≤ κ*(κ/4)/(4*(Real.log (4/κ)+1)))
(hk : M*⌈Real.log (4/κ)/((pc : ℝ)/2)^(m+1+VM)⌉₊ ≤ k)
(hR : 4/(κ*(1-(pc : ℝ))^(D*k))<(R : ℝ)-m)
(hmean : ∀ y∈G.vertexCollar Input R,
1-η ≤ (BernoulliProduct.law (fun _ : E => p')).real
((forceOn Dstarᶜ ∅) ⁻¹' G.seedHitEvent (G.vertexBall y m) T)) :
(BernoulliProduct.law q).real ((G.withSource A).hitEvent none (some '' Input)\
(G.withSource A).hitEvent none (some '' T)) ≤ κ := by
by_cases hI : Input.Nonempty
· let p₀ : unitInterval := ⟨(pc : ℝ)/2,by constructor <;> linarith [pc.2.1,pc.2.2]⟩
let J : Finset ℤ := Finset.Icc ((m : ℤ)+1) (R : ℤ)
have hJ : J.Nonempty := Finset.nonempty_Icc.mpr (by omega)
have hJcard : (J.card : ℝ)=(R : ℝ)-m := by
have hjc : J.card=R-m := by
change (Finset.Icc ((m : ℤ)+1) (R : ℤ)).card=R-m
rw [Int.card_Icc]
omega
rw [hjc,Nat.cast_sub hcR.le]
have hK : 0 ≤ Real.log (4/κ) := Real.log_nonneg ((le_div_iff₀ hκ).mpr (by linarith))
have hs : 0<((pc : ℝ)/2)^(m+1+VM) := positive_trial_power _ hpc₀ _ _
have hh := G.source_distance_collar_transfer hc hl A Input T hA q hI m R J hJ
(by intro j hj; have hh := Finset.mem_Icc.mp hj; omega) houtside p₀ pc hpc
(by
intro e he
rw [hq e (hregion he)]
exact ⟨hpp,hppc⟩)
D k M ⌈Real.log (4/κ)/((pc : ℝ)/2)^(m+1+VM)⌉₊ VM hD hM hk hMbound hVM
(Real.log (4/κ)) η (κ/4) hK hη₀ (by linarith)
(trial_ceiling_budget _ _ hs)
(by
intro y hy
exact (hmean y hy).trans (fresh_region_lower (fun e => q (Sum.inl e))
(fun _ => p') Dstar hq _ (G.measurableSet_seedHitEvent _ _)
(G.seedHitEvent_upper _ _)))
rw [hJcard] at hh
exact hh.trans (collar_error_budget κ η ((R : ℝ)-m) _ hκ hκ1 hη
(pow_pos (by linarith) _) hR)
· have he : Input=∅ := Set.not_nonempty_iff_eq_empty.mp hI
subst Input
have he : (G.withSource A).hitEvent none (some '' (∅ : Set V))=∅ := by
ext ω
simp [hitEvent]
rw [he,Set.empty_sdiff,measureReal_empty]
exact hκ.le
end CriticalPercolation.BondGraph
end
end
end
end OAI
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