PFR-declarations/PFR.Endgame.jsonl

{"name":"hV","declaration":"theorem hV {G : Type u_1} [AddCommGroup G] [hG : MeasurableSpace G] [MeasurableAdd₂ G] {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (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, X₁', X₂'] MeasureTheory.volume) : H[X₁' + X₂] = H[X₁ + X₂']"}
{"name":"independenceCondition5","declaration":"theorem independenceCondition5 {G : Type u_1} [AddCommGroup G] [hG : MeasurableSpace G] [MeasurableAdd₂ G] {Ω : Type u_4} [mΩ : 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_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![X₁, X₁', X₂ + X₂'] MeasureTheory.volume"}
{"name":"independenceCondition3","declaration":"theorem independenceCondition3 {G : Type u_1} [AddCommGroup G] [hG : MeasurableSpace G] [MeasurableAdd₂ G] {Ω : Type u_4} [mΩ : 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_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![X₁', X₂, X₁ + X₂'] MeasureTheory.volume"}
{"name":"independenceCondition4","declaration":"theorem independenceCondition4 {G : Type u_1} [AddCommGroup G] [hG : MeasurableSpace G] [MeasurableAdd₂ G] {Ω : Type u_4} [mΩ : 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_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![X₂, X₁', X₁ + X₂'] MeasureTheory.volume"}
{"name":"sum_dist_diff_le","declaration":"/-- $$ \\sum_{i=1}^2 \\sum_{A\\in\\{U,V,W\\}} \\big(d[X^0_i;A|S] - d[X^0_i;X_i]\\big)$$\nis less than or equal to\n$$ \\leq (6 - 3\\eta) k + 3(2 \\eta k - I_1).$$\n-/\ntheorem sum_dist_diff_le {G : Type u_1} [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} [mΩ : 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, X₁', X₂'] MeasureTheory.volume) (h_min : tau_minimizes p X₁ X₂) : d[p.X₀₁ # X₁ + X₂ | X₁ + X₂ + X₁' + X₂'] - d[p.X₀₁ # X₁] + (d[p.X₀₂ # X₁ + X₂ | X₁ + X₂ + X₁' + X₂'] - d[p.X₀₂ # X₂]) +\n      (d[p.X₀₁ # X₁' + X₂ | X₁ + X₂ + X₁' + X₂'] - d[p.X₀₁ # X₁] +\n        (d[p.X₀₂ # X₁' + X₂ | X₁ + X₂ + X₁' + X₂'] - d[p.X₀₂ # X₂])) +\n    (d[p.X₀₁ # X₁' + X₁ | X₁ + X₂ + X₁' + X₂'] - d[p.X₀₁ # X₁] +\n      (d[p.X₀₂ # X₁' + X₁ | X₁ + X₂ + X₁' + X₂'] - d[p.X₀₂ # X₂])) ≤\n  (6 - 3 * p.η) * d[X₁ # X₂] + 3 * (2 * p.η * d[X₁ # X₂] - I[X₁ + X₂ : X₁' + X₂|X₁ + X₂ + X₁' + X₂'])"}
{"name":"independenceCondition2","declaration":"theorem independenceCondition2 {G : Type u_1} [AddCommGroup G] [hG : MeasurableSpace G] [MeasurableAdd₂ G] {Ω : Type u_4} [mΩ : 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_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![X₂, X₁, X₁' + X₂'] MeasureTheory.volume"}
{"name":"construct_good'","declaration":"theorem construct_good' {G : Type u_1} [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} [mΩ : MeasureTheory.MeasureSpace Ω] (X₁ : Ω → G) (X₂ : Ω → G) (h_min : tau_minimizes p X₁ X₂) {Ω' : Type u_5} [MeasureTheory.MeasureSpace Ω'] {T₁ : Ω' → G} {T₂ : Ω' → G} {T₃ : Ω' → G} (hT : T₁ + T₂ + T₃ = 0) (hT₁ : Measurable T₁) (hT₂ : Measurable T₂) (hT₃ : Measurable T₃) (μ : MeasureTheory.Measure Ω') [MeasureTheory.IsProbabilityMeasure μ] : d[X₁ # X₂] ≤\n  I[T₁ : T₂ ; μ] + I[T₂ : T₃ ; μ] + I[T₃ : T₁ ; μ] +\n    p.η / 3 *\n      (I[T₁ : T₂ ; μ] + I[T₂ : T₃ ; μ] + I[T₃ : T₁ ; μ] +\n            (d[p.X₀₁ ; MeasureTheory.volume # T₁ ; μ] - d[p.X₀₁ # X₁] +\n              (d[p.X₀₂ ; MeasureTheory.volume # T₁ ; μ] - d[p.X₀₂ # X₂])) +\n          (d[p.X₀₁ ; MeasureTheory.volume # T₂ ; μ] - d[p.X₀₁ # X₁] +\n            (d[p.X₀₂ ; MeasureTheory.volume # T₂ ; μ] - d[p.X₀₂ # X₂])) +\n        (d[p.X₀₁ ; MeasureTheory.volume # T₃ ; μ] - d[p.X₀₁ # X₁] +\n          (d[p.X₀₂ ; MeasureTheory.volume # T₃ ; μ] - d[p.X₀₂ # X₂])))"}
{"name":"I₃_eq","declaration":"/-- The quantity $I_3 = I[V:W|S]$ is equal to $I_2$. -/\ntheorem I₃_eq {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [MeasurableAdd₂ G] {Ω : Type u_4} [mΩ : 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) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) : I[X₁' + X₂ : X₁' + X₁|X₁ + X₂ + X₁' + X₂'] = I[X₁ + X₂ : X₁' + X₁|X₁ + X₂ + X₁' + X₂']"}
{"name":"cond_c_eq_integral","declaration":"theorem cond_c_eq_integral {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] (X₁ : Ω → G) (X₂ : Ω → G) {Ω' : Type u_5} [MeasureTheory.MeasureSpace Ω'] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {Y : Ω' → G} {Z : Ω' → G} (hY : Measurable Y) (hZ : Measurable Z) : d[p.X₀₁ # Y | Z] - d[p.X₀₁ # X₁] + (d[p.X₀₂ # Y | Z] - d[p.X₀₂ # X₂]) =\n  ∫ (x : G),\n    (fun z =>\n        d[p.X₀₁ ; MeasureTheory.volume # Y ; ProbabilityTheory.cond MeasureTheory.volume (Z ⁻¹' {z})] - d[p.X₀₁ # X₁] +\n          (d[p.X₀₂ ; MeasureTheory.volume # Y ; ProbabilityTheory.cond MeasureTheory.volume (Z ⁻¹' {z})] -\n            d[p.X₀₂ # X₂]))\n      x ∂MeasureTheory.Measure.map Z MeasureTheory.volume"}
{"name":"independenceCondition1","declaration":"theorem independenceCondition1 {G : Type u_1} [AddCommGroup G] [hG : MeasurableSpace G] [MeasurableAdd₂ G] {Ω : Type u_4} [mΩ : 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_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![X₁, X₂, X₁' + X₂'] MeasureTheory.volume"}
{"name":"construct_good","declaration":"/-- If $T_1, T_2, T_3$ are $G$-valued random variables with $T_1+T_2+T_3=0$ holds identically and\n-\n$$ \\delta := \\sum_{1 \\leq i < j \\leq 3} I[T_i;T_j]$$\n\nThen there exist random variables $T'_1, T'_2$ such that\n\n$$ d[T'_1;T'_2] + \\eta (d[X_1^0;T'_1] - d[X_1^0;X _1]) + \\eta(d[X_2^0;T'_2] - d[X_2^0;X_2])$$\n\nis at most\n\n$$\\delta + \\frac{\\eta}{3} \\biggl( \\delta + \\sum_{i=1}^2 \\sum_{j = 1}^3\n    (d[X^0_i;T_j] - d[X^0_i; X_i]) \\biggr).$$\n-/\ntheorem construct_good {G : Type u_1} [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} [mΩ : MeasureTheory.MeasureSpace Ω] (X₁ : Ω → G) (X₂ : Ω → G) (h_min : tau_minimizes p X₁ X₂) {Ω' : Type u_5} [MeasureTheory.MeasureSpace Ω'] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {T₁ : Ω' → G} {T₂ : Ω' → G} {T₃ : Ω' → G} (hT : T₁ + T₂ + T₃ = 0) (hT₁ : Measurable T₁) (hT₂ : Measurable T₂) (hT₃ : Measurable T₃) : d[X₁ # X₂] ≤\n  I[T₁ : T₂] + I[T₂ : T₃] + I[T₃ : T₁] +\n    p.η / 3 *\n      (I[T₁ : T₂] + I[T₂ : T₃] + I[T₃ : T₁] + (d[p.X₀₁ # T₁] - d[p.X₀₁ # X₁] + (d[p.X₀₂ # T₁] - d[p.X₀₂ # X₂])) +\n          (d[p.X₀₁ # T₂] - d[p.X₀₁ # X₁] + (d[p.X₀₂ # T₂] - d[p.X₀₂ # X₂])) +\n        (d[p.X₀₁ # T₃] - d[p.X₀₁ # X₁] + (d[p.X₀₂ # T₃] - d[p.X₀₂ # X₂])))"}
{"name":"independenceCondition6","declaration":"theorem independenceCondition6 {G : Type u_1} [AddCommGroup G] [hG : MeasurableSpace G] [MeasurableAdd₂ G] {Ω : Type u_4} [mΩ : 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_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![X₂, X₂', X₁' + X₁] MeasureTheory.volume"}
{"name":"sum_condMutual_le","declaration":"/-- $$ I(U : V | S) + I(V : W | S) + I(W : U | S) $$\nis less than or equal to\n$$ 6 \\eta k - \\frac{1 - 5 \\eta}{1-\\eta} (2 \\eta k - I_1).$$\n-/\ntheorem sum_condMutual_le {G : Type u_1} [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} [mΩ : 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, X₁', X₂'] MeasureTheory.volume) (h_min : tau_minimizes p X₁ X₂) : I[X₁ + X₂ : X₁' + X₂|X₁ + X₂ + X₁' + X₂'] + I[X₁' + X₂ : X₁' + X₁|X₁ + X₂ + X₁' + X₂'] +\n    I[X₁' + X₁ : X₁ + X₂|X₁ + X₂ + X₁' + X₂'] ≤\n  6 * p.η * d[X₁ # X₂] - (1 - 5 * p.η) / (1 - p.η) * (2 * p.η * d[X₁ # X₂] - I[X₁ + X₂ : X₁' + X₂|X₁ + X₂ + X₁' + X₂'])"}
{"name":"cond_construct_good","declaration":"theorem cond_construct_good {G : Type u_1} [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} [mΩ : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (h_min : tau_minimizes p X₁ X₂) {Ω' : Type u_5} [MeasureTheory.MeasureSpace Ω'] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {T₁ : Ω' → G} {T₂ : Ω' → G} {T₃ : Ω' → G} (hT : T₁ + T₂ + T₃ = 0) (hT₁ : Measurable T₁) (hT₂ : Measurable T₂) (hT₃ : Measurable T₃) {R : Ω' → G} (hR : Measurable R) : d[X₁ # X₂] ≤\n  I[T₁ : T₂|R] + I[T₂ : T₃|R] + I[T₃ : T₁|R] +\n    p.η / 3 *\n      (I[T₁ : T₂|R] + I[T₂ : T₃|R] + I[T₃ : T₁|R] +\n            (d[p.X₀₁ # T₁ | R] - d[p.X₀₁ # X₁] + (d[p.X₀₂ # T₁ | R] - d[p.X₀₂ # X₂])) +\n          (d[p.X₀₁ # T₂ | R] - d[p.X₀₁ # X₁] + (d[p.X₀₂ # T₂ | R] - d[p.X₀₂ # X₂])) +\n        (d[p.X₀₁ # T₃ | R] - d[p.X₀₁ # X₁] + (d[p.X₀₂ # T₃ | R] - d[p.X₀₂ # X₂])))"}
{"name":"tau_strictly_decreases_aux","declaration":"/-- If $d[X_1;X_2] > 0$ then there are $G$-valued random variables $X'_1, X'_2$ such that\nPhrased in the contrapositive form for convenience of proof. -/\ntheorem tau_strictly_decreases_aux {G : Type u_1} [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} [mΩ : 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, X₁', X₂'] MeasureTheory.volume) (h_min : tau_minimizes p X₁ X₂) (hpη : p.η = 1 / 9) : d[X₁ # X₂] = 0"}
{"name":"sum_uvw_eq_zero","declaration":"/-- $U+V+W=0$. -/\ntheorem sum_uvw_eq_zero {G : Type u_1} [AddCommGroup G] [elem : ElementaryAddCommGroup G 2] {Ω : Type u_4} (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) : X₁ + X₂ + (X₁' + X₂) + (X₁' + X₁) = 0"}
{"name":"delta'_eq_integral","declaration":"theorem delta'_eq_integral {G : Type u_1} [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] {Ω' : Type u_5} [MeasureTheory.MeasureSpace Ω'] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {T₁ : Ω' → G} {T₂ : Ω' → G} {T₃ : Ω' → G} {R : Ω' → G} : I[T₁ : T₂|R] + I[T₂ : T₃|R] + I[T₃ : T₁|R] =\n  ∫ (x : G),\n    (fun r =>\n        I[T₁ : T₂ ; ProbabilityTheory.cond MeasureTheory.volume (R ⁻¹' {r})] +\n            I[T₂ : T₃ ; ProbabilityTheory.cond MeasureTheory.volume (R ⁻¹' {r})] +\n          I[T₃ : T₁ ; ProbabilityTheory.cond MeasureTheory.volume (R ⁻¹' {r})])\n      x ∂MeasureTheory.Measure.map R MeasureTheory.volume"}
{"name":"construct_good_prelim","declaration":"/-- If $T_1, T_2, T_3$ are $G$-valued random variables with $T_1+T_2+T_3=0$ holds identically and\n$$ \\delta := \\sum_{1 \\leq i < j \\leq 3} I[T_i;T_j]$$\nThen there exist random variables $T'_1, T'_2$ such that\n$$ d[T'_1;T'_2] + \\eta (d[X_1^0;T'_1] - d[X_1^0;X_1]) + \\eta(d[X_2^0;T'_2] - d[X_2^0;X_2]) $$\nis at most\n$$ \\delta + \\eta ( d[X^0_1;T_1]-d[X^0_1;X_1]) + \\eta (d[X^0_2;T_2]-d[X^0_2;X_2]) $$\n$$ + \\tfrac12 \\eta I[T_1: T_3] + \\tfrac12 \\eta I[T_2: T_3].$$\n-/\ntheorem construct_good_prelim {G : Type u_1} [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} [mΩ : MeasureTheory.MeasureSpace Ω] (X₁ : Ω → G) (X₂ : Ω → G) (h_min : tau_minimizes p X₁ X₂) {Ω' : Type u_5} [MeasureTheory.MeasureSpace Ω'] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {T₁ : Ω' → G} {T₂ : Ω' → G} {T₃ : Ω' → G} (hT : T₁ + T₂ + T₃ = 0) (hT₁ : Measurable T₁) (hT₂ : Measurable T₂) (hT₃ : Measurable T₃) : d[X₁ # X₂] ≤\n  I[T₁ : T₂] + I[T₂ : T₃] + I[T₃ : T₁] + p.η * (d[p.X₀₁ # T₁] - d[p.X₀₁ # X₁] + (d[p.X₀₂ # T₂] - d[p.X₀₂ # X₂])) +\n    p.η * (I[T₁ : T₃] + I[T₂ : T₃]) / 2"}
{"name":"hU","declaration":"theorem hU {G : Type u_1} [AddCommGroup G] [hG : MeasurableSpace G] [MeasurableAdd₂ G] {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (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, X₁', X₂'] MeasureTheory.volume) : H[X₁ + X₂] = H[X₁' + X₂']"}