PFR-declarations/PFR.ForMathlib.Entropy.Kernel.Group.jsonl

{"name":"ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_add","declaration":"theorem ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_add {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableAdd₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.snd κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 + p.2) ⋯ , μ]"}
{"name":"ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_div","declaration":"theorem ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_div {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableDiv₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.fst κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 / p.2) ⋯ , μ]"}
{"name":"ProbabilityTheory.kernel.entropy_div_comm","declaration":"theorem ProbabilityTheory.kernel.entropy_div_comm {G : Type u_5} {T : Type u_6} [MeasurableSpace T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableDiv₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) (μ : MeasureTheory.Measure T) : Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 / p.2) ⋯ , μ] =\n  Hk[ProbabilityTheory.kernel.map κ (fun p => p.2 / p.1) ⋯ , μ]"}
{"name":"ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_div","declaration":"theorem ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_div {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableDiv₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : max (Hk[ProbabilityTheory.kernel.fst κ , μ]) (Hk[ProbabilityTheory.kernel.snd κ , μ]) - Ik[κ , μ] ≤\n  Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 / p.2) ⋯ , μ]"}
{"name":"measureEntropy_inv","declaration":"theorem measureEntropy_inv {G : Type u_5} [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [Countable G] (μ : MeasureTheory.Measure G) : Hm[MeasureTheory.Measure.map (fun x => x⁻¹) μ] = Hm[μ]"}
{"name":"ProbabilityTheory.kernel.max_entropy_le_entropy_div_prod","declaration":"theorem ProbabilityTheory.kernel.max_entropy_le_entropy_div_prod {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableDiv₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T G)) [ProbabilityTheory.IsMarkovKernel κ] (η : ↥(ProbabilityTheory.kernel T G)) [ProbabilityTheory.IsMarkovKernel η] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η μ) : max (Hk[κ , μ]) (Hk[η , μ]) ≤\n  Hk[ProbabilityTheory.kernel.map (ProbabilityTheory.kernel.prod κ η) (fun p => p.1 / p.2) ⋯ , μ]"}
{"name":"ProbabilityTheory.kernel.max_entropy_le_entropy_add_sum","declaration":"theorem ProbabilityTheory.kernel.max_entropy_le_entropy_add_sum {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableAdd₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T G)) [ProbabilityTheory.IsMarkovKernel κ] (η : ↥(ProbabilityTheory.kernel T G)) [ProbabilityTheory.IsMarkovKernel η] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η μ) : max (Hk[κ , μ]) (Hk[η , μ]) ≤\n  Hk[ProbabilityTheory.kernel.map (ProbabilityTheory.kernel.prod κ η) (fun p => p.1 + p.2) ⋯ , μ]"}
{"name":"ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_sub","declaration":"theorem ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_sub {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableSub₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.snd κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 - p.2) ⋯ , μ]"}
{"name":"ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_mul","declaration":"theorem ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_mul {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableMul₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.fst κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 * p.2) ⋯ , μ]"}
{"name":"measureEntropy_sub_comm","declaration":"theorem measureEntropy_sub_comm {G : Type u_5} [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableSub₂ G] [Countable G] (μ : MeasureTheory.Measure (G × G)) : Hm[MeasureTheory.Measure.map (fun p => p.1 - p.2) μ] = Hm[MeasureTheory.Measure.map (fun p => p.2 - p.1) μ]"}
{"name":"ProbabilityTheory.kernel.entropy_inv","declaration":"theorem ProbabilityTheory.kernel.entropy_inv {G : Type u_5} {T : Type u_6} [MeasurableSpace T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T G)) (μ : MeasureTheory.Measure T) : Hk[ProbabilityTheory.kernel.map κ (fun x => x⁻¹) ⋯ , μ] = Hk[κ , μ]"}
{"name":"ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_sub","declaration":"theorem ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_sub {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableSub₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.fst κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 - p.2) ⋯ , μ]"}
{"name":"ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_mul","declaration":"theorem ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_mul {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableMul₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.snd κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 * p.2) ⋯ , μ]"}
{"name":"measureEntropy_neg","declaration":"theorem measureEntropy_neg {G : Type u_5} [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [Countable G] (μ : MeasureTheory.Measure G) : Hm[MeasureTheory.Measure.map (fun x => -x) μ] = Hm[μ]"}
{"name":"ProbabilityTheory.kernel.max_entropy_le_entropy_sub_prod","declaration":"theorem ProbabilityTheory.kernel.max_entropy_le_entropy_sub_prod {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableSub₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T G)) [ProbabilityTheory.IsMarkovKernel κ] (η : ↥(ProbabilityTheory.kernel T G)) [ProbabilityTheory.IsMarkovKernel η] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η μ) : max (Hk[κ , μ]) (Hk[η , μ]) ≤\n  Hk[ProbabilityTheory.kernel.map (ProbabilityTheory.kernel.prod κ η) (fun p => p.1 - p.2) ⋯ , μ]"}
{"name":"ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_mul","declaration":"theorem ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_mul {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableMul₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : max (Hk[ProbabilityTheory.kernel.fst κ , μ]) (Hk[ProbabilityTheory.kernel.snd κ , μ]) - Ik[κ , μ] ≤\n  Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 * p.2) ⋯ , μ]"}
{"name":"ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_mul'","declaration":"theorem ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_mul' {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableMul₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : max (Hk[ProbabilityTheory.kernel.fst κ , μ]) (Hk[ProbabilityTheory.kernel.snd κ , μ]) - Ik[κ , μ] ≤\n  Hk[ProbabilityTheory.kernel.map κ (fun p => p.2 * p.1) ⋯ , μ]"}
{"name":"ProbabilityTheory.kernel.entropy_neg","declaration":"theorem ProbabilityTheory.kernel.entropy_neg {G : Type u_5} {T : Type u_6} [MeasurableSpace T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T G)) (μ : MeasureTheory.Measure T) : Hk[ProbabilityTheory.kernel.map κ (fun x => -x) ⋯ , μ] = Hk[κ , μ]"}
{"name":"ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_sub","declaration":"theorem ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_sub {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableSub₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : max (Hk[ProbabilityTheory.kernel.fst κ , μ]) (Hk[ProbabilityTheory.kernel.snd κ , μ]) - Ik[κ , μ] ≤\n  Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 - p.2) ⋯ , μ]"}
{"name":"ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_add","declaration":"theorem ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_add {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableAdd₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.fst κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 + p.2) ⋯ , μ]"}
{"name":"ProbabilityTheory.kernel.max_entropy_le_entropy_mul_prod","declaration":"theorem ProbabilityTheory.kernel.max_entropy_le_entropy_mul_prod {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableMul₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T G)) [ProbabilityTheory.IsMarkovKernel κ] (η : ↥(ProbabilityTheory.kernel T G)) [ProbabilityTheory.IsMarkovKernel η] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η μ) : max (Hk[κ , μ]) (Hk[η , μ]) ≤\n  Hk[ProbabilityTheory.kernel.map (ProbabilityTheory.kernel.prod κ η) (fun p => p.1 * p.2) ⋯ , μ]"}
{"name":"ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_add'","declaration":"theorem ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_add' {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableAdd₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.fst κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.2 + p.1) ⋯ , μ]"}
{"name":"ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_mul'","declaration":"theorem ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_mul' {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableMul₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.fst κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.2 * p.1) ⋯ , μ]"}
{"name":"ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_add'","declaration":"theorem ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_add' {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.snd κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.2 + p.1) ⋯ , μ]"}
{"name":"ProbabilityTheory.kernel.entropy_sub_comm","declaration":"theorem ProbabilityTheory.kernel.entropy_sub_comm {G : Type u_5} {T : Type u_6} [MeasurableSpace T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableSub₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) (μ : MeasureTheory.Measure T) : Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 - p.2) ⋯ , μ] =\n  Hk[ProbabilityTheory.kernel.map κ (fun p => p.2 - p.1) ⋯ , μ]"}
{"name":"ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_add","declaration":"theorem ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_add {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableAdd₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : max (Hk[ProbabilityTheory.kernel.fst κ , μ]) (Hk[ProbabilityTheory.kernel.snd κ , μ]) - Ik[κ , μ] ≤\n  Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 + p.2) ⋯ , μ]"}
{"name":"ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_add'","declaration":"theorem ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_add' {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableAdd₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : max (Hk[ProbabilityTheory.kernel.fst κ , μ]) (Hk[ProbabilityTheory.kernel.snd κ , μ]) - Ik[κ , μ] ≤\n  Hk[ProbabilityTheory.kernel.map κ (fun p => p.2 + p.1) ⋯ , μ]"}
{"name":"measureEntropy_div_comm","declaration":"theorem measureEntropy_div_comm {G : Type u_5} [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableDiv₂ G] [Countable G] (μ : MeasureTheory.Measure (G × G)) : Hm[MeasureTheory.Measure.map (fun p => p.1 / p.2) μ] = Hm[MeasureTheory.Measure.map (fun p => p.2 / p.1) μ]"}
{"name":"ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_mul'","declaration":"theorem ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_mul' {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.snd κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.2 * p.1) ⋯ , μ]"}
{"name":"ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_div","declaration":"theorem ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_div {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableDiv₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.snd κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 / p.2) ⋯ , μ]"}