{"name":"first_estimate","declaration":"/-- We have $I_1 \\leq 2 \\eta k$ -/\ntheorem first_estimate {G : Type u_1} [addgroup : AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₂', X₁'] MeasureTheory.volume) (h_min : tau_minimizes p X₁ X₂) : I[X₁ + X₂ : X₁' + X₂|X₁ + X₂ + X₁' + X₂'] ≤ 2 * p.η * d[X₁ # X₂]"}
{"name":"diff_rdist_le_1","declaration":"/-- $$d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1] \\leq \\tfrac{1}{2} k + \\tfrac{1}{4} \\bbH[X_2] - \\tfrac{1}{4} \\bbH[X_1].$$ -/\ntheorem diff_rdist_le_1 {G : Type u_1} [addgroup : AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₁ : Measurable X₁) (hX₂' : Measurable X₂') (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₂', X₁'] MeasureTheory.volume) : d[p.X₀₁ # X₁ + X₂'] - d[p.X₀₁ # X₁] ≤ d[X₁ # X₂] / 2 + H[X₂] / 4 - H[X₁] / 4"}
{"name":"condRuzsaDist_of_sums_ge","declaration":"/-- The distance $d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1 | X_1 + \\tilde X_2] - d[X^0_1; X_1]) - \\eta(d[X^0_2; X_2 | X_2 + \\tilde X_1] - d[X^0_2; X_2]).$$\n-/\ntheorem condRuzsaDist_of_sums_ge {G : Type u_1} [addgroup : AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h_min : tau_minimizes p X₁ X₂) : d[X₁ | X₁ + X₂' # X₂ | X₂ + X₁'] ≥\n  d[X₁ # X₂] - p.η * (d[p.X₀₁ # X₁ | X₁ + X₂'] - d[p.X₀₁ # X₁]) - p.η * (d[p.X₀₂ # X₂ | X₂ + X₁'] - d[p.X₀₂ # X₂])"}
{"name":"diff_rdist_le_2","declaration":"/-- $$ d[X^0_2;X_2+\\tilde X_1] - d[X^0_2; X_2] \\leq \\tfrac{1}{2} k + \\tfrac{1}{4} \\mathbb{H}[X_1] - \\tfrac{1}{4} \\mathbb{H}[X_2].$$ -/\ntheorem diff_rdist_le_2 {G : Type u_1} [addgroup : AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₂', X₁'] MeasureTheory.volume) : d[p.X₀₂ # X₂ + X₁'] - d[p.X₀₂ # X₂] ≤ d[X₁ # X₂] / 2 + H[X₁] / 4 - H[X₂] / 4"}
{"name":"diff_rdist_le_3","declaration":"/-- $$ d[X_1^0;X_1|X_1+\\tilde X_2] - d[X_1^0;X_1] \\leq\n\\tfrac{1}{2} k + \\tfrac{1}{4} \\mathbb{H}[X_1] - \\tfrac{1}{4} \\mathbb{H}[X_2].$$ -/\ntheorem diff_rdist_le_3 {G : Type u_1} [addgroup : AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₁ : Measurable X₁) (hX₂' : Measurable X₂') (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₂', X₁'] MeasureTheory.volume) : d[p.X₀₁ # X₁ | X₁ + X₂'] - d[p.X₀₁ # X₁] ≤ d[X₁ # X₂] / 2 + H[X₁] / 4 - H[X₂] / 4"}
{"name":"rdist_of_sums_ge","declaration":"/-- The distance $d[X_1+\\tilde X_2; X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1]) - \\eta (d[X^0_2; X_2+\\tilde X_1] - d[X^0_2; X_2]).$$ -/\ntheorem rdist_of_sums_ge {G : Type u_1} [addgroup : AddCommGroup G] [hG : MeasurableSpace G] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h_min : tau_minimizes p X₁ X₂) : d[X₁ + X₂' # X₂ + X₁'] ≥\n  d[X₁ # X₂] - p.η * (d[p.X₀₁ # X₁ + X₂'] - d[p.X₀₁ # X₁]) - p.η * (d[p.X₀₂ # X₂ + X₁'] - d[p.X₀₂ # X₂])"}
{"name":"rdist_add_rdist_add_condMutual_eq","declaration":"/-- The sum of\n$$ d[X_1+\\tilde X_2;X_2+\\tilde X_1] + d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1] $$\nand\n$$ I[X_1+ X_2 : \\tilde X_1 + X_2 \\,|\\, X_1 + X_2 + \\tilde X_1 + \\tilde X_2] $$\nis equal to $2k$. -/\ntheorem rdist_add_rdist_add_condMutual_eq {G : Type u_1} [addgroup : AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₂', X₁'] MeasureTheory.volume) : d[X₁ + X₂' # X₂ + X₁'] + d[X₁ | X₁ + X₂' # X₂ | X₂ + X₁'] + I[X₁ + X₂ : X₁' + X₂|X₁ + X₂ + X₁' + X₂'] = 2 * d[X₁ # X₂]"}
{"name":"ent_ofsum_le","declaration":"/-- $$\\mathbb{H}[X_1+X_2+\\tilde X_1+\\tilde X_2] \\le \\tfrac{1}{2} \\mathbb{H}[X_1]+\\tfrac{1}{2} \\mathbb{H}[X_2] + (2 + \\eta) k - I_1.$$\n-/\ntheorem ent_ofsum_le {G : Type u_1} [addgroup : AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₂', X₁'] MeasureTheory.volume) (h_min : tau_minimizes p X₁ X₂) : H[X₁ + X₂ + X₁' + X₂'] ≤ H[X₁] / 2 + H[X₂] / 2 + (2 + p.η) * d[X₁ # X₂] - I[X₁ + X₂ : X₁' + X₂|X₁ + X₂ + X₁' + X₂']"}
{"name":"diff_rdist_le_4","declaration":"/-- $$ d[X_2^0; X_2|X_2+\\tilde X_1] - d[X_2^0; X_2] \\leq\n\\tfrac{1}{2}k + \\tfrac{1}{4} \\mathbb{H}[X_2] - \\tfrac{1}{4} \\mathbb{H}[X_1].$$ -/\ntheorem diff_rdist_le_4 {G : Type u_1} [addgroup : AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₂', X₁'] MeasureTheory.volume) : d[p.X₀₂ # X₂ | X₂ + X₁'] - d[p.X₀₂ # X₂] ≤ d[X₁ # X₂] / 2 + H[X₂] / 4 - H[X₁] / 4"}