| {"name":"ProbabilityTheory.iIndepFun.reindex_four_abdc","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_abdc {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₁, Z₂, Z₄, Z₃] MeasureTheory.volume"} | |
| {"name":"ProbabilityTheory.iIndepFun.reindex_four_badc","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_badc {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₂, Z₁, Z₄, Z₃] MeasureTheory.volume"} | |
| {"name":"ProbabilityTheory.iIndepFun.reindex_four_bdca","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_bdca {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₂, Z₄, Z₃, Z₁] MeasureTheory.volume"} | |
| {"name":"ProbabilityTheory.iIndepFun.reindex_four_adcb","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_adcb {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₁, Z₄, Z₃, Z₂] MeasureTheory.volume"} | |
| {"name":"ProbabilityTheory.iIndepFun.reindex_four_dabc","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_dabc {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₄, Z₁, Z₂, Z₃] MeasureTheory.volume"} | |
| {"name":"ProbabilityTheory.iIndepFun.reindex_four_bcad","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_bcad {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₂, Z₃, Z₁, Z₄] MeasureTheory.volume"} | |
| {"name":"ProbabilityTheory.iIndepFun.reindex_four_bcda","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_bcda {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₂, Z₃, Z₄, Z₁] MeasureTheory.volume"} | |
| {"name":"ProbabilityTheory.iIndepFun.reindex_four_dbca","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_dbca {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₄, Z₂, Z₃, Z₁] MeasureTheory.volume"} | |
| {"name":"ProbabilityTheory.iIndepFun.κ","declaration":"def ProbabilityTheory.iIndepFun.κ : Fin 3 → Type"} | |
| {"name":"ProbabilityTheory.iIndepFun.reindex_four_acdb","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_acdb {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₁, Z₃, Z₄, Z₂] MeasureTheory.volume"} | |
| {"name":"ProbabilityTheory.iIndepFun.reindex_four_bacd","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_bacd {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₂, Z₁, Z₃, Z₄] MeasureTheory.volume"} | |
| {"name":"ProbabilityTheory.iIndepFun.reindex_four_acbd","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_acbd {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₁, Z₃, Z₂, Z₄] MeasureTheory.volume"} | |
| {"name":"ProbabilityTheory.iIndepFun.reindex_four_cadb","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_cadb {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₃, Z₁, Z₄, Z₂] MeasureTheory.volume"} | |
| {"name":"ProbabilityTheory.iIndepFun.reindex_four_cbad","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_cbad {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₃, Z₂, Z₁, Z₄] MeasureTheory.volume"} | |
| {"name":"ProbabilityTheory.iIndepFun.reindex_four_abcd","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_abcd {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume"} | |
| {"name":"ProbabilityTheory.iIndepFun.reindex_four_bdac","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_bdac {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₂, Z₄, Z₁, Z₃] MeasureTheory.volume"} | |
| {"name":"ProbabilityTheory.iIndepFun.reindex_four_adbc","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_adbc {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₁, Z₄, Z₂, Z₃] MeasureTheory.volume"} | |
| {"name":"ProbabilityTheory.iIndepFun.apply_two_last","declaration":"/-- If `(Z₁, Z₂, Z₃, Z₄)` are independent, so are `(Z₁, Z₂, φ Z₃ Z₄)` for any measurable `φ`. -/\ntheorem ProbabilityTheory.iIndepFun.apply_two_last {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) (hZ₁ : Measurable Z₁) (hZ₂ : Measurable Z₂) (hZ₃ : Measurable Z₃) (hZ₄ : Measurable Z₄) {phi : G → G → G} (hphi : Measurable (Function.uncurry phi)) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₁, Z₂, fun ω => phi (Z₃ ω) (Z₄ ω)] MeasureTheory.volume"} | |
| {"name":"ProbabilityTheory.iIndepFun.reindex_four_dbac","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_dbac {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₄, Z₂, Z₁, Z₃] MeasureTheory.volume"} | |