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LeanCamCombi.lean

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import LeanCamCombi.ExtrProbCombi.BernoulliSeq
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import LeanCamCombi.ExtrProbCombi.BinomialRandomGraph
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import LeanCamCombi.ExtrProbCombi.BollobasContainment
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import LeanCamCombi.ExtrProbCombi.Connectivity
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import LeanCamCombi.ExtrProbCombi.GiantComponent
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import LeanCamCombi.Mathlib.Combinatorics.Additive.ApproximateSubgroup
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import LeanCamCombi.Mathlib.Combinatorics.SetFamily.LYM
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import LeanCamCombi.Mathlib.Combinatorics.SimpleGraph.Density
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import LeanCamCombi.Mathlib.Probability.Combinatorics.BinomialRandomGraph.Defs
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import LeanCamCombi.Mathlib.Probability.Distributions.Bernoulli
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import LeanCamCombi.Mathlib.Probability.HasLaw
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import LeanCamCombi.Mathlib.Probability.ProbabilityMassFunction.Constructions
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import LeanCamCombi.PlainCombi.KatonaCircle
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import LeanCamCombi.PlainCombi.LittlewoodOfford

LeanCamCombi/ExtrProbCombi/BernoulliSeq.lean

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LeanCamCombi/ExtrProbCombi/BinomialRandomGraph.lean

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LeanCamCombi/ExtrProbCombi/BollobasContainment.lean

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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Yaël Dillies
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-/
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import LeanCamCombi.ExtrProbCombi.BinomialRandomGraph
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import LeanCamCombi.Mathlib.Combinatorics.SimpleGraph.Density
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import LeanCamCombi.Mathlib.Probability.Combinatorics.BinomialRandomGraph.Defs
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import Mathlib.Analysis.SpecialFunctions.Pow.Real
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import Mathlib.Combinatorics.SimpleGraph.Copy
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import Mathlib.MeasureTheory.Measure.Typeclasses.Probability
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-/
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open Asymptotics Filter MeasureTheory
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open scoped MeasureTheory ENNReal NNReal SimpleGraph Topology
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variable {α β Ω : Type*} [Fintype β] {G : ℕ → Ω → SimpleGraph α} (H : SimpleGraph β)
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[Fintype H.edgeSet] [∀ n ω, DecidableRel (G n ω).Adj] [MeasurableSpace Ω] (μ : Measure Ω)
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[IsProbabilityMeasure μ] {p : ℕ → ℝ≥0} (hG : ∀ n, IsBinomialRandomGraph (G n) (p n) μ)
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open scoped MeasureTheory ENNReal NNReal SimpleGraph Topology unitInterval
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namespace SimpleGraph
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variable {V W Ω : Type*} [Fintype W] {G : ℕ → Ω → SimpleGraph V} (H : SimpleGraph W)
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[Fintype H.edgeSet] [∀ n ω, DecidableRel (G n ω).Adj] [MeasurableSpace Ω] (μ : Measure Ω)
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[IsProbabilityMeasure μ] {p : ℕ → I} (hG : ∀ n, IsBinomialRandom (G n) (p n) μ)
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/-- **Bollobás' Graph Containment Lemma, zero statement** -/
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/-- **Bollobás' Graph Containment Lemma**, zero statement -/
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lemma zero_statement
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(hp : (fun n ↦ p n : ℕ → ℝ) =o[atTop] (fun n ↦ n ^ (-H.maxEdgeSubdensity⁻¹ : ℝ) : ℕ → ℝ)) :
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Tendsto (fun n ↦ μ {ω | H ⊑ G n ω}) atTop (𝓝 0) :=
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sorry
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/-- **Bollobás' Graph Containment Lemma, one statement** -/
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/-- **Bollobás' Graph Containment Lemma**, one statement -/
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lemma one_statement
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(hp : (fun n ↦ n ^ (-H.maxEdgeSubdensity⁻¹ : ℝ) : ℕ → ℝ) =o[atTop] (fun n ↦ p n : ℕ → ℝ)) :
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Tendsto (fun n ↦ μ {ω | H ⊑ G n ω}) atTop (𝓝 1) :=
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sorry
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end SimpleGraph
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LeanCamCombi/ExtrProbCombi/Connectivity.lean

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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Yaël Dillies
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-/
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import LeanCamCombi.ExtrProbCombi.BinomialRandomGraph
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import LeanCamCombi.Mathlib.Probability.Combinatorics.BinomialRandomGraph.Defs
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/-!
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# Connectivity of the Erdős–Rényi model

LeanCamCombi/ExtrProbCombi/GiantComponent.lean

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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Yaël Dillies
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-/
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import LeanCamCombi.ExtrProbCombi.BinomialRandomGraph
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import LeanCamCombi.Mathlib.Probability.Combinatorics.BinomialRandomGraph.Defs
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/-!
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# The giant component

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