{"name":"ProbabilityTheory.kernel.rdist_zero_kernel_left","declaration":"theorem ProbabilityTheory.kernel.rdist_zero_kernel_left {T : Type u_1} {T' : Type u_2} {G : Type u_4} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable T'] [MeasurableSpace T'] [MeasurableSingletonClass T'] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] {η : ↥(ProbabilityTheory.kernel T' G)} [ProbabilityTheory.IsFiniteKernel η] {μ : MeasureTheory.Measure T} {ν : MeasureTheory.Measure T'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] : dk[0 ; μ # η ; ν] = -Hk[η , ν] / 2"}
{"name":"ProbabilityTheory.kernel.rdist","declaration":"/-- The Rusza distance between two kernels taking values in the same space, defined as the average\nRusza distance between the image measures. -/\ndef ProbabilityTheory.kernel.rdist {T : Type u_1} {T' : Type u_2} {G : Type u_4} [MeasurableSpace T] [MeasurableSpace T'] [MeasurableSpace G] [AddCommGroup G] (κ : ↥(ProbabilityTheory.kernel T G)) (η : ↥(ProbabilityTheory.kernel T' G)) (μ : MeasureTheory.Measure T) (ν : MeasureTheory.Measure T') : ℝ"}
{"name":"ProbabilityTheory.kernel.rdist_zero_left","declaration":"theorem ProbabilityTheory.kernel.rdist_zero_left {T : Type u_1} {T' : Type u_2} {G : Type u_4} [MeasurableSpace T] [MeasurableSpace T'] [MeasurableSpace G] [AddCommGroup G] (κ : ↥(ProbabilityTheory.kernel T G)) (η : ↥(ProbabilityTheory.kernel T' G)) (ν' : MeasureTheory.Measure T') : dk[κ ; 0 # η ; ν'] = 0"}
{"name":"ProbabilityTheory.kernel.rdist_symm","declaration":"theorem ProbabilityTheory.kernel.rdist_symm {T : Type u_1} {T' : Type u_2} {G : Type u_4} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable T'] [MeasurableSpace T'] [MeasurableSingletonClass T'] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] {κ : ↥(ProbabilityTheory.kernel T G)} {η : ↥(ProbabilityTheory.kernel T' G)} [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η] {μ : MeasureTheory.Measure T} {ν : MeasureTheory.Measure T'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] : dk[κ ; μ # η ; ν] = dk[η ; ν # κ ; μ]"}
{"name":"ProbabilityTheory.kernel.rdist_dirac_zero_left","declaration":"theorem ProbabilityTheory.kernel.rdist_dirac_zero_left {T : Type u_1} {T' : Type u_2} {G : Type u_4} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable T'] [MeasurableSpace T'] [MeasurableSingletonClass T'] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] {η : ↥(ProbabilityTheory.kernel T' G)} [ProbabilityTheory.IsFiniteKernel η] {μ : MeasureTheory.Measure T} {ν : MeasureTheory.Measure T'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] : dk[ProbabilityTheory.kernel.const T (MeasureTheory.Measure.dirac 0) ; μ # η ; ν] = Hk[η , ν] / 2"}
{"name":"ProbabilityTheory.kernel.abs_sub_entropy_le_rdist","declaration":"theorem ProbabilityTheory.kernel.abs_sub_entropy_le_rdist {T : Type u_1} {T' : Type u_2} {G : Type u_4} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable T'] [Nonempty T'] [MeasurableSpace T'] [MeasurableSingletonClass T'] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] {κ : ↥(ProbabilityTheory.kernel T G)} {η : ↥(ProbabilityTheory.kernel T' G)} [ProbabilityTheory.IsMarkovKernel κ] [ProbabilityTheory.IsMarkovKernel η] {μ : MeasureTheory.Measure T} {ν : MeasureTheory.Measure T'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η ν) : |Hk[κ , μ] - Hk[η , ν]| ≤ 2 * dk[κ ; μ # η ; ν]"}
{"name":"ProbabilityTheory.kernel.rdist_zero_kernel_right","declaration":"theorem ProbabilityTheory.kernel.rdist_zero_kernel_right {T : Type u_1} {T' : Type u_2} {G : Type u_4} [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable T'] [MeasurableSpace T'] [MeasurableSingletonClass T'] [MeasurableSpace G] [AddCommGroup G] [MeasurableSub₂ G] {κ : ↥(ProbabilityTheory.kernel T G)} [ProbabilityTheory.IsFiniteKernel κ] {μ : MeasureTheory.Measure T} {ν : MeasureTheory.Measure T'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] : dk[κ ; μ # 0 ; ν] = -Hk[κ , μ] / 2"}
{"name":"ProbabilityTheory.kernel.rdist_zero_right","declaration":"theorem ProbabilityTheory.kernel.rdist_zero_right {T : Type u_1} {T' : Type u_2} {G : Type u_4} [MeasurableSpace T] [MeasurableSpace T'] [MeasurableSpace G] [AddCommGroup G] (κ : ↥(ProbabilityTheory.kernel T G)) (η : ↥(ProbabilityTheory.kernel T' G)) (μ : MeasureTheory.Measure T) : dk[κ ; μ # η ; 0] = 0"}
{"name":"ProbabilityTheory.kernel.rdist_nonneg","declaration":"theorem ProbabilityTheory.kernel.rdist_nonneg {T : Type u_1} {T' : Type u_2} {G : Type u_4} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable T'] [Nonempty T'] [MeasurableSpace T'] [MeasurableSingletonClass T'] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] {κ : ↥(ProbabilityTheory.kernel T G)} {η : ↥(ProbabilityTheory.kernel T' G)} [ProbabilityTheory.IsMarkovKernel κ] [ProbabilityTheory.IsMarkovKernel η] {μ : MeasureTheory.Measure T} {ν : MeasureTheory.Measure T'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η ν) : 0 ≤ dk[κ ; μ # η ; ν]"}
{"name":"ProbabilityTheory.kernel.ent_of_diff_le","declaration":"/-- The **improved entropic Ruzsa triangle inequality**. -/\ntheorem ProbabilityTheory.kernel.ent_of_diff_le {T : Type u_1} {G : Type u_4} [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) (η : ↥(ProbabilityTheory.kernel T G)) [ProbabilityTheory.IsMarkovKernel κ] [ProbabilityTheory.IsMarkovKernel η] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.FiniteKernelSupport κ) (hη : ProbabilityTheory.kernel.FiniteKernelSupport η) : Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 - p.2) ⋯ , μ] ≤\n  Hk[ProbabilityTheory.kernel.map (ProbabilityTheory.kernel.prod (ProbabilityTheory.kernel.fst κ) η)\n          (fun p => p.1 - p.2) ⋯ ,\n        μ] +\n      Hk[ProbabilityTheory.kernel.map (ProbabilityTheory.kernel.prod η (ProbabilityTheory.kernel.snd κ))\n          (fun p => p.1 - p.2) ⋯ ,\n        μ] -\n    Hk[η , μ]"}
{"name":"ProbabilityTheory.kernel.«termDk[_;_#_;_]»","declaration":"/-- The Rusza distance between two kernels taking values in the same space, defined as the average\nRusza distance between the image measures. -/\ndef ProbabilityTheory.kernel.«termDk[_;_#_;_]»  : Lean.ParserDescr"}
{"name":"ProbabilityTheory.kernel.«termDk[_;_#_;_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef ProbabilityTheory.kernel.«termDk[_;_#_;_]».delab  : Lean.PrettyPrinter.Delaborator.Delab"}
{"name":"ProbabilityTheory.kernel.ruzsa_triangle_aux","declaration":"theorem ProbabilityTheory.kernel.ruzsa_triangle_aux {T : Type u_1} {G : Type u_4} [MeasurableSpace T] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) (η : ↥(ProbabilityTheory.kernel T G)) [ProbabilityTheory.IsMarkovKernel κ] [ProbabilityTheory.IsMarkovKernel η] : ProbabilityTheory.kernel.map (ProbabilityTheory.kernel.prod κ η) (fun p => p.2 - p.1.2) ⋯ =\n  ProbabilityTheory.kernel.map (ProbabilityTheory.kernel.prod η (ProbabilityTheory.kernel.snd κ)) (fun p => p.1 - p.2) ⋯"}
{"name":"ProbabilityTheory.kernel.rdist_eq","declaration":"theorem ProbabilityTheory.kernel.rdist_eq {T : Type u_1} {T' : Type u_2} {G : Type u_4} [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable T'] [MeasurableSpace T'] [MeasurableSingletonClass T'] [MeasurableSpace G] [AddCommGroup G] {κ : ↥(ProbabilityTheory.kernel T G)} {η : ↥(ProbabilityTheory.kernel T' G)} {μ : MeasureTheory.Measure T} {ν : MeasureTheory.Measure T'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] : dk[κ ; μ # η ; ν] =\n  ∫ (x : T × T'),\n        (fun p => Hm[MeasureTheory.Measure.map (fun x => x.1 - x.2) (MeasureTheory.Measure.prod (κ p.1) (η p.2))])\n          x ∂MeasureTheory.Measure.prod μ ν -\n      Hk[κ , μ] / 2 -\n    Hk[η , ν] / 2"}
{"name":"ProbabilityTheory.kernel.rdist_eq'","declaration":"theorem ProbabilityTheory.kernel.rdist_eq' {T : Type u_1} {T' : Type u_2} {G : Type u_4} [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable T'] [MeasurableSpace T'] [MeasurableSingletonClass T'] [MeasurableSpace G] [AddCommGroup G] [MeasurableSub₂ G] {κ : ↥(ProbabilityTheory.kernel T G)} {η : ↥(ProbabilityTheory.kernel T' G)} [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η] {μ : MeasureTheory.Measure T} {ν : MeasureTheory.Measure T'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] : dk[κ ; μ # η ; ν] =\n  Hk[ProbabilityTheory.kernel.map\n          (ProbabilityTheory.kernel.prod (ProbabilityTheory.kernel.prodMkRight T' κ)\n            (ProbabilityTheory.kernel.prodMkLeft T η))\n          (fun x => x.1 - x.2) ⋯ ,\n        MeasureTheory.Measure.prod μ ν] -\n      Hk[κ , μ] / 2 -\n    Hk[η , ν] / 2"}
{"name":"ProbabilityTheory.kernel.rdist_dirac_zero_right","declaration":"theorem ProbabilityTheory.kernel.rdist_dirac_zero_right {T : Type u_1} {T' : Type u_2} {G : Type u_4} [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable T'] [Nonempty T'] [MeasurableSpace T'] [MeasurableSingletonClass T'] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] {κ : ↥(ProbabilityTheory.kernel T G)} [ProbabilityTheory.IsFiniteKernel κ] {μ : MeasureTheory.Measure T} {ν : MeasureTheory.Measure T'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] : dk[κ ; μ # ProbabilityTheory.kernel.const T' (MeasureTheory.Measure.dirac 0) ; ν] = Hk[κ , μ] / 2"}
{"name":"ProbabilityTheory.kernel.rdistm","declaration":"/-- The Rusza distance between two measures, defined as `H[X - Y] - H[X]/2 - H[Y]/2` where `X`\nand `Y` are independent variables distributed according to the two measures. -/\ndef ProbabilityTheory.kernel.rdistm {G : Type u_4} [MeasurableSpace G] [AddCommGroup G] (μ : MeasureTheory.Measure G) (ν : MeasureTheory.Measure G) : ℝ"}
{"name":"ProbabilityTheory.kernel.rdist_triangle_aux1","declaration":"theorem ProbabilityTheory.kernel.rdist_triangle_aux1 {T : Type u_1} {T' : Type u_2} {T'' : Type u_3} {G : Type u_4} [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace T'] [MeasurableSingletonClass T'] [MeasurableSpace T''] [MeasurableSingletonClass T''] [MeasurableSpace G] [AddCommGroup G] [MeasurableSub₂ G] (κ : ↥(ProbabilityTheory.kernel T G)) (η : ↥(ProbabilityTheory.kernel T' G)) [ProbabilityTheory.IsMarkovKernel κ] [ProbabilityTheory.IsMarkovKernel η] (μ : MeasureTheory.Measure T) (μ' : MeasureTheory.Measure T') (μ'' : MeasureTheory.Measure T'') [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] [MeasureTheory.IsProbabilityMeasure μ''] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport μ'] [ProbabilityTheory.FiniteSupport μ''] : Hk[ProbabilityTheory.kernel.map\n      (ProbabilityTheory.kernel.prod\n        (ProbabilityTheory.kernel.prodMkRight T' (ProbabilityTheory.kernel.prodMkRight T'' κ))\n        (ProbabilityTheory.kernel.prodMkLeft (T × T'') η))\n      (fun p => p.1 - p.2) ⋯ ,\n    MeasureTheory.Measure.prod (MeasureTheory.Measure.prod μ μ'') μ'] =\n  Hk[ProbabilityTheory.kernel.map\n      (ProbabilityTheory.kernel.prod (ProbabilityTheory.kernel.prodMkRight T' κ)\n        (ProbabilityTheory.kernel.prodMkLeft T η))\n      (fun x => x.1 - x.2) ⋯ ,\n    MeasureTheory.Measure.prod μ μ']"}
{"name":"ProbabilityTheory.kernel.rdist_triangle_aux2","declaration":"theorem ProbabilityTheory.kernel.rdist_triangle_aux2 {T : Type u_1} {T' : Type u_2} {T'' : Type u_3} {G : Type u_4} [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace T'] [MeasurableSingletonClass T'] [MeasurableSpace T''] [MeasurableSingletonClass T''] [MeasurableSpace G] [AddCommGroup G] [MeasurableSub₂ G] (η : ↥(ProbabilityTheory.kernel T' G)) (ξ : ↥(ProbabilityTheory.kernel T'' G)) [ProbabilityTheory.IsMarkovKernel η] [ProbabilityTheory.IsMarkovKernel ξ] (μ : MeasureTheory.Measure T) (μ' : MeasureTheory.Measure T') (μ'' : MeasureTheory.Measure T'') [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] [MeasureTheory.IsProbabilityMeasure μ''] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport μ'] [ProbabilityTheory.FiniteSupport μ''] : Hk[ProbabilityTheory.kernel.map\n      (ProbabilityTheory.kernel.prod (ProbabilityTheory.kernel.prodMkLeft (T × T'') η)\n        (ProbabilityTheory.kernel.prodMkRight T' (ProbabilityTheory.kernel.prodMkLeft T ξ)))\n      (fun p => p.1 - p.2) ⋯ ,\n    MeasureTheory.Measure.prod (MeasureTheory.Measure.prod μ μ'') μ'] =\n  Hk[ProbabilityTheory.kernel.map\n      (ProbabilityTheory.kernel.prod (ProbabilityTheory.kernel.prodMkRight T'' η)\n        (ProbabilityTheory.kernel.prodMkLeft T' ξ))\n      (fun x => x.1 - x.2) ⋯ ,\n    MeasureTheory.Measure.prod μ' μ'']"}
{"name":"ProbabilityTheory.kernel.rdist_triangle","declaration":"theorem ProbabilityTheory.kernel.rdist_triangle {T : Type u_1} {T' : Type u_2} {T'' : Type u_3} {G : Type u_4} [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable T'] [Nonempty T'] [MeasurableSpace T'] [MeasurableSingletonClass T'] [Countable T''] [MeasurableSpace T''] [MeasurableSingletonClass T''] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] (κ : ↥(ProbabilityTheory.kernel T G)) (η : ↥(ProbabilityTheory.kernel T' G)) (ξ : ↥(ProbabilityTheory.kernel T'' G)) [ProbabilityTheory.IsMarkovKernel κ] [ProbabilityTheory.IsMarkovKernel η] [ProbabilityTheory.IsMarkovKernel ξ] (μ : MeasureTheory.Measure T) (μ' : MeasureTheory.Measure T') (μ'' : MeasureTheory.Measure T'') [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] [MeasureTheory.IsProbabilityMeasure μ''] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport μ'] [ProbabilityTheory.FiniteSupport μ''] (hκ : ProbabilityTheory.kernel.FiniteKernelSupport κ) (hη : ProbabilityTheory.kernel.FiniteKernelSupport η) (hξ : ProbabilityTheory.kernel.FiniteKernelSupport ξ) : dk[κ ; μ # ξ ; μ''] ≤ dk[κ ; μ # η ; μ'] + dk[η ; μ' # ξ ; μ'']"}