{"name":"entropic_PFR_conjecture","declaration":"/-- `entropic_PFR_conjecture`: For two $G$-valued random variables $X^0_1, X^0_2$, there is some\nsubgroup $H \\leq G$ such that $d[X^0_1;U_H] + d[X^0_2;U_H] \\le 11 d[X^0_1;X^0_2]$. -/\ntheorem entropic_PFR_conjecture {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {G : Type uG} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (p : refPackage Ω₀₁ Ω₀₂ G) (hpη : p.η = 1 / 9) : ∃ H Ω mΩ U,\n  MeasureTheory.IsProbabilityMeasure MeasureTheory.volume ∧\n    Measurable U ∧\n      ProbabilityTheory.IsUniform (↑H) U MeasureTheory.volume ∧ d[p.X₀₁ # U] + d[p.X₀₂ # U] ≤ 11 * d[p.X₀₁ # p.X₀₂]"}
{"name":"tau_strictly_decreases","declaration":"/-- If $d[X_1;X_2] > 0$ then there are $G$-valued random variables $X'_1, X'_2$ such that $\\tau[X'_1;X'_2] < \\tau[X_1;X_2]$.\nPhrased in the contrapositive form for convenience of proof. -/\ntheorem tau_strictly_decreases {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] {Ω : Type u_3} [mΩ : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {G : Type uG} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (p : refPackage Ω₀₁ Ω₀₂ G) {X₁ : Ω → G} {X₂ : Ω → G} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (h_min : tau_minimizes p X₁ X₂) (hpη : p.η = 1 / 9) : d[X₁ # X₂] = 0"}
{"name":"entropic_PFR_conjecture'","declaration":"theorem entropic_PFR_conjecture' {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {G : Type uG} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (p : refPackage Ω₀₁ Ω₀₂ G) (hpη : p.η = 1 / 9) : ∃ H Ω mΩ U,\n  ProbabilityTheory.IsUniform (↑H) U MeasureTheory.volume ∧\n    d[p.X₀₁ # U] ≤ 6 * d[p.X₀₁ # p.X₀₂] ∧ d[p.X₀₂ # U] ≤ 6 * d[p.X₀₁ # p.X₀₂]"}