awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Probability.Kernel.Basic (goal)
Goal:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
μ : Measure α
h : ∀ᵐ (a : α) ∂μ, IsProbabilityMeasure.{u_2} (α := β) (m0 := mβ) ((κ : (a : α) → Measure β) a)
h' : μ ≠ 0
obtain :
  ∃ (s : Set α),
    MeasurableSet s ∧
      Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (0 : ℝ≥0∞) ∧
        ∀ a ∉ s, IsProbabilityMeasure.{u_2} (α := β) (m0 := mβ) ((κ : (a : α) → Measure β) a)
obtain₁ :
  ∀ (s : Set α) (s_meas : MeasurableSet s) (μs : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (0 : ℝ≥0∞))
    (hs : ∀ a ∉ s, IsProbabilityMeasure.{u_2} (α := β) (m0 := mβ) ((κ : (a : α) → Measure β) a)),
    ∃ (η : Kernel α β),
      Filter.EventuallyEq.{u_1, u_2} (β := Measure β) (ae μ) (⇑κ : (a : α) → Measure β) (⇑η : (a : α) → Measure β) ∧
        IsMarkovKernel η
obtain₂ :
  ∀ (s : Set α) (s_meas : MeasurableSet s) (μs : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (0 : ℝ≥0∞))
    (hs : ∀ a ∉ s, IsProbabilityMeasure.{u_2} (α := β) (m0 := mβ) ((κ : (a : α) → Measure β) a)) (a : α) (ha : a ∈ sᶜ),
    ∃ (η : Kernel α β),
      Filter.EventuallyEq.{u_1, u_2} (β := Measure β) (ae μ) (⇑κ : (a : α) → Measure β) (⇑η : (a : α) → Measure β) ∧
        IsMarkovKernel η
refine_1 :
  ∀ (s : Set α) (s_meas : MeasurableSet s) (μs : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (0 : ℝ≥0∞))
    (hs : ∀ a ∉ s, IsProbabilityMeasure.{u_2} (α := β) (m0 := mβ) ((κ : (a : α) → Measure β) a)) (a : α) (ha : a ∈ sᶜ),
    Filter.EventuallyEq.{u_1, u_2} (β := Measure β) (ae μ) (⇑κ : (a : α) → Measure β)
      (⇑(piecewise (s := s) s_meas (const α ((κ : (a : α) → Measure β) a)) κ) : (a : α) → Measure β)
⊢ ∃ (η : Kernel α β),
    Filter.EventuallyEq.{u_1, u_2} (β := Measure β) (ae μ) (⇑κ : (a : α) → Measure β) (⇑η : (a : α) → Measure β) ∧
      IsMarkovKernel η
have refine_2 : ∀ (s : Set α) (s_meas : MeasurableSet s) (μs : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (0 : ℝ≥0∞))
  (hs : ∀ a ∉ s, IsProbabilityMeasure.{u_2} (α := β) (m0 := mβ) ((κ : (a : α) → Measure β) a)) (a : α) (ha : a ∈ sᶜ),
  IsMarkovKernel (piecewise (s := s) s_meas (const α ((κ : (a : α) → Measure β) a)) κ)
Tactic: classical
  obtain ⟨s, s_meas, μs, hs⟩ : ∃ s, MeasurableSet s ∧ μ s = 0
      ∧ ∀ a ∉ s, IsProbabilityMeasure (κ a) := by
    refine ⟨toMeasurable μ {a | ¬ IsProbabilityMeasure (κ a)}, measurableSet_toMeasurable _ _,
      by simpa [measure_toMeasurable] using h, ?_⟩
    intro a ha
    contrapose ha
    exact subset_toMeasurable _ _ ha
  obtain ⟨a, ha⟩ : sᶜ.Nonempty := by
    contrapose! h'; simpa [μs, h'] using measure_univ_le_add_compl s (μ := μ)
  refine ⟨Kernel.piecewise s_meas (Kernel.const _ (κ a)) κ, ?_, ?_⟩
  · filter_upwards [measure_eq_zero_iff_ae_notMem.1 μs] with b hb
    simp [hb, piecewise]
  · refine ⟨fun b ↦ ?_⟩
    by_cases hb : b ∈ s
    · simpa [hb, piecewise] using hs _ ha
    · simpa [hb, piecewise] using hs _ hb
-- Lean 4 proof step from Mathlib.Probability.Kernel.Basic (goal)
Goal:
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
μ ν : Measure α
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
b : Bool
⊢ LE.le (α := ℝ≥0∞) ((((boolKernel μ ν : (a : Bool) → Measure α) b : Measure α) : (a : Set α) → ℝ≥0∞) Set.univ : ℝ≥0∞)
    (max ((μ : (a : Set α) → ℝ≥0∞) Set.univ) ((ν : (a : Set α) → ℝ≥0∞) Set.univ) : ℝ≥0∞)
have false : LE.le (α := ℝ≥0∞) ((((boolKernel μ ν : (a : Bool) → Measure α) false : Measure α) : (a : Set α) → ℝ≥0∞) Set.univ : ℝ≥0∞)
  (max ((μ : (a : Set α) → ℝ≥0∞) Set.univ) ((ν : (a : Set α) → ℝ≥0∞) Set.univ) : ℝ≥0∞)
Tactic: cases b
Cleared hypotheses: b
-- Lean 4 proof step from Mathlib.Probability.Kernel.Basic (goal)
Goal:
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
μ ν : Measure α
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
b : Bool
false :
  LE.le (α := ℝ≥0∞)
    ((((boolKernel μ ν : (a : Bool) → Measure α) Bool.false : Measure α) : (a : Set α) → ℝ≥0∞) Set.univ : ℝ≥0∞)
    (max ((μ : (a : Set α) → ℝ≥0∞) Set.univ) ((ν : (a : Set α) → ℝ≥0∞) Set.univ) : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞) ((((boolKernel μ ν : (a : Bool) → Measure α) b : Measure α) : (a : Set α) → ℝ≥0∞) Set.univ : ℝ≥0∞)
    (max ((μ : (a : Set α) → ℝ≥0∞) Set.univ) ((ν : (a : Set α) → ℝ≥0∞) Set.univ) : ℝ≥0∞)
have true : LE.le (α := ℝ≥0∞) ((((boolKernel μ ν : (a : Bool) → Measure α) true : Measure α) : (a : Set α) → ℝ≥0∞) Set.univ : ℝ≥0∞)
  (max ((μ : (a : Set α) → ℝ≥0∞) Set.univ) ((ν : (a : Set α) → ℝ≥0∞) Set.univ) : ℝ≥0∞)
Tactic: cases b
Cleared hypotheses: b
-- Lean 4 proof step from Mathlib.Probability.Kernel.Basic (goal)
Goal:
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
μ ν : Measure α
inst : IsProbabilityMeasure μ
inst_1 : IsProbabilityMeasure ν
b : Bool
⊢ IsProbabilityMeasure.{u_1} (α := α) (m0 := mα) ((boolKernel μ ν : (a : Bool) → Measure α) b)
have false : IsProbabilityMeasure.{u_1} (α := α) (m0 := mα) ((boolKernel μ ν : (a : Bool) → Measure α) false)
Tactic: cases b
Cleared hypotheses: b
-- Lean 4 proof step from Mathlib.Probability.Kernel.Basic (goal)
Goal:
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
μ ν : Measure α
inst : IsProbabilityMeasure μ
inst_1 : IsProbabilityMeasure ν
b : Bool
false : IsProbabilityMeasure.{u_1} (α := α) (m0 := mα) ((boolKernel μ ν : (a : Bool) → Measure α) Bool.false)
⊢ IsProbabilityMeasure.{u_1} (α := α) (m0 := mα) ((boolKernel μ ν : (a : Bool) → Measure α) b)
have true : IsProbabilityMeasure.{u_1} (α := α) (m0 := mα) ((boolKernel μ ν : (a : Bool) → Measure α) true)
Tactic: cases b
Cleared hypotheses: b
-- Lean 4 proof step from Mathlib.Probability.Kernel.Basic (goal)
Goal:
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
μ ν : Measure α
inst : SFinite μ
inst_1 : SFinite ν
⊢ ∃ (κs : ℕ → Kernel Bool α), (∀ (n : ℕ), IsFiniteKernel (κs n)) ∧ boolKernel μ ν = Kernel.sum κs
have refine : boolKernel μ ν = Kernel.sum fun (n : ℕ) ↦ boolKernel (sfiniteSeq μ n) (sfiniteSeq ν n)
Tactic: refine ⟨fun n ↦ boolKernel (sfiniteSeq μ n) (sfiniteSeq ν n), fun n ↦ inferInstance, ?_⟩
-- Lean 4 proof step from Mathlib.Probability.Kernel.Basic (goal)
Goal:
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
μ ν : Measure α
inst : SFinite μ
inst_1 : SFinite ν
⊢ boolKernel μ ν = Kernel.sum fun (n : ℕ) ↦ boolKernel (sfiniteSeq μ n) (sfiniteSeq ν n)
have h : ∀ (b : Bool) (s : Set α),
  MeasurableSet s →
    Eq (α := ℝ≥0∞) ((((boolKernel μ ν : (a : Bool) → Measure α) b : Measure α) : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
      ((((Kernel.sum fun (n : ℕ) ↦ boolKernel (sfiniteSeq μ n) (sfiniteSeq ν n) : (a : Bool) → Measure α) b :
              Measure α) :
            (a : Set α) → ℝ≥0∞)
          s :
        ℝ≥0∞)
Tactic: ext b
-- Lean 4 proof step from Mathlib.Probability.Kernel.Basic (goal)
Goal:
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
μ ν : Measure α
inst : SFinite μ
inst_1 : SFinite ν
b : Bool
s : Set α
h : MeasurableSet s
⊢ Eq (α := ℝ≥0∞) ((((boolKernel μ ν : (a : Bool) → Measure α) b : Measure α) : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
    ((((Kernel.sum fun (n : ℕ) ↦ boolKernel (sfiniteSeq μ n) (sfiniteSeq ν n) : (a : Bool) → Measure α) b : Measure α) :
          (a : Set α) → ℝ≥0∞)
        s :
      ℝ≥0∞)
have h : Eq (α := ℝ≥0∞) ((((boolKernel μ ν : (a : Bool) → Measure α) b : Measure α) : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
  (((Measure.sum fun (n : ℕ) ↦ (boolKernel (sfiniteSeq μ n) (sfiniteSeq ν n) : (a : Bool) → Measure α) b : Measure α) :
        (a : Set α) → ℝ≥0∞)
      s :
    ℝ≥0∞)
Tactic: rw [Kernel.sum_apply]
-- Lean 4 proof step from Mathlib.Probability.Kernel.Basic (goal)
Goal:
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
μ ν : Measure α
inst : SFinite μ
inst_1 : SFinite ν
b : Bool
s : Set α
h : MeasurableSet s
⊢ Eq (α := ℝ≥0∞) ((((boolKernel μ ν : (a : Bool) → Measure α) b : Measure α) : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
    (((Measure.sum fun (n : ℕ) ↦ (boolKernel (sfiniteSeq μ n) (sfiniteSeq ν n) : (a : Bool) → Measure α) b :
            Measure α) :
          (a : Set α) → ℝ≥0∞)
        s :
      ℝ≥0∞)
have h : Eq (α := ℝ≥0∞) ((((boolKernel μ ν : (a : Bool) → Measure α) false : Measure α) : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
  (((Measure.sum fun (n : ℕ) ↦ (boolKernel (sfiniteSeq μ n) (sfiniteSeq ν n) : (a : Bool) → Measure α) false :
          Measure α) :
        (a : Set α) → ℝ≥0∞)
      s :
    ℝ≥0∞)
Tactic: cases b
Cleared hypotheses: b
-- Lean 4 proof step from Mathlib.Probability.Kernel.Basic (goal)
Goal:
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
μ ν : Measure α
inst : SFinite μ
inst_1 : SFinite ν
b : Bool
s : Set α
h_1 : MeasurableSet s
h :
  Eq (α := ℝ≥0∞) ((((boolKernel μ ν : (a : Bool) → Measure α) false : Measure α) : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
    (((Measure.sum fun (n : ℕ) ↦ (boolKernel (sfiniteSeq μ n) (sfiniteSeq ν n) : (a : Bool) → Measure α) false :
            Measure α) :
          (a : Set α) → ℝ≥0∞)
        s :
      ℝ≥0∞)
⊢ Eq (α := ℝ≥0∞) ((((boolKernel μ ν : (a : Bool) → Measure α) b : Measure α) : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
    (((Measure.sum fun (n : ℕ) ↦ (boolKernel (sfiniteSeq μ n) (sfiniteSeq ν n) : (a : Bool) → Measure α) b :
            Measure α) :
          (a : Set α) → ℝ≥0∞)
        s :
      ℝ≥0∞)
have h₁ : Eq (α := ℝ≥0∞) ((((boolKernel μ ν : (a : Bool) → Measure α) true : Measure α) : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
  (((Measure.sum fun (n : ℕ) ↦ (boolKernel (sfiniteSeq μ n) (sfiniteSeq ν n) : (a : Bool) → Measure α) true :
          Measure α) :
        (a : Set α) → ℝ≥0∞)
      s :
    ℝ≥0∞)
Tactic: cases b
Cleared hypotheses: b
-- Lean 4 proof step from Mathlib.Probability.Kernel.Basic (goal)
Goal:
α : Type u_1
mα : MeasurableSpace α
κ : Kernel Bool α
⊢ κ = boolKernel ((κ : (a : Bool) → Measure α) false) ((κ : (a : Bool) → Measure α) true)
have h : ∀ (s : Set α),
  MeasurableSet s →
    Eq (α := ℝ≥0∞) ((((κ : (a : Bool) → Measure α) false : Measure α) : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
      ((((boolKernel ((κ : (a : Bool) → Measure α) false) ((κ : (a : Bool) → Measure α) true) : (a : Bool) → Measure α)
                false :
              Measure α) :
            (a : Set α) → ℝ≥0∞)
          s :
        ℝ≥0∞)
Tactic: ext (_ | _)
-- Lean 4 proof step from Mathlib.Probability.Kernel.Basic (goal)
Goal:
α : Type u_1
mα : MeasurableSpace α
κ : Kernel Bool α
h :
  ∀ (s : Set α),
    MeasurableSet s →
      Eq (α := ℝ≥0∞) ((((κ : (a : Bool) → Measure α) false : Measure α) : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
        ((((boolKernel ((κ : (a : Bool) → Measure α) false) ((κ : (a : Bool) → Measure α) true) :
                    (a : Bool) → Measure α)
                  false :
                Measure α) :
              (a : Set α) → ℝ≥0∞)
            s :
          ℝ≥0∞)
⊢ κ = boolKernel ((κ : (a : Bool) → Measure α) false) ((κ : (a : Bool) → Measure α) true)
have h₁ : ∀ (s : Set α),
  MeasurableSet s →
    Eq (α := ℝ≥0∞) ((((κ : (a : Bool) → Measure α) true : Measure α) : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
      ((((boolKernel ((κ : (a : Bool) → Measure α) false) ((κ : (a : Bool) → Measure α) true) : (a : Bool) → Measure α)
                true :
              Measure α) :
            (a : Set α) → ℝ≥0∞)
          s :
        ℝ≥0∞)
Tactic: ext (_ | _)
-- Lean 4 proof step from Mathlib.Probability.Kernel.Category.SFinKer (goal)
Goal:
X Y : SFinKer
⊢ MonoidalCategoryStruct.tensorObj (𝒞 := instLargeCategory) X Y ≅
    MonoidalCategoryStruct.tensorObj (𝒞 := instLargeCategory) Y X
have refine_1 : CategoryStruct.comp (Y := MonoidalCategoryStruct.tensorObj (𝒞 := instLargeCategory) Y X)
    { hom := Kernel.swap X.carrier Y.carrier, property := sorry }
    { hom := Kernel.swap Y.carrier X.carrier, property := sorry } =
  𝟙 (MonoidalCategoryStruct.tensorObj (𝒞 := instLargeCategory) X Y)
Tactic: refine ⟨⟨Kernel.swap _ _, by rw [Kernel.swap]; infer_instance⟩,
      ⟨Kernel.swap _ _, by rw [Kernel.swap]; infer_instance⟩, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Probability.Kernel.Category.SFinKer (goal)
Goal:
X Y : SFinKer
refine_1 :
  CategoryStruct.comp (Y := MonoidalCategoryStruct.tensorObj (𝒞 := instLargeCategory) Y X)
      { hom := Kernel.swap X.carrier Y.carrier, property := ⋯ }
      { hom := Kernel.swap Y.carrier X.carrier, property := ⋯ } =
    𝟙 (MonoidalCategoryStruct.tensorObj (𝒞 := instLargeCategory) X Y)
⊢ MonoidalCategoryStruct.tensorObj (𝒞 := instLargeCategory) X Y ≅
    MonoidalCategoryStruct.tensorObj (𝒞 := instLargeCategory) Y X
have refine_2 : CategoryStruct.comp (Y := MonoidalCategoryStruct.tensorObj (𝒞 := instLargeCategory) X Y)
    { hom := Kernel.swap Y.carrier X.carrier, property := sorry }
    { hom := Kernel.swap X.carrier Y.carrier, property := sorry } =
  𝟙 (MonoidalCategoryStruct.tensorObj (𝒞 := instLargeCategory) Y X)
Tactic: refine ⟨⟨Kernel.swap _ _, by rw [Kernel.swap]; infer_instance⟩,
      ⟨Kernel.swap _ _, by rw [Kernel.swap]; infer_instance⟩, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Probability.Kernel.Category.SFinKer (goal)
Goal:
X Y : SFinKer
⊢ CategoryStruct.comp (Y := MonoidalCategoryStruct.tensorObj (𝒞 := instLargeCategory) Y X)
      { hom := Kernel.swap X.carrier Y.carrier, property := ⋯ }
      { hom := Kernel.swap Y.carrier X.carrier, property := ⋯ } =
    𝟙 (MonoidalCategoryStruct.tensorObj (𝒞 := instLargeCategory) X Y)
have refine_1 : Hom.hom
    (CategoryStruct.comp (Y := MonoidalCategoryStruct.tensorObj (𝒞 := instLargeCategory) Y X)
      { hom := Kernel.swap X.carrier Y.carrier, property := sorry }
      { hom := Kernel.swap Y.carrier X.carrier, property := sorry }) =
  Hom.hom (𝟙 (MonoidalCategoryStruct.tensorObj (𝒞 := instLargeCategory) X Y))
Tactic: ext : 1
-- Lean 4 proof step from Mathlib.Probability.Kernel.Category.SFinKer (goal)
Goal:
X Y : SFinKer
⊢ CategoryStruct.comp (Y := MonoidalCategoryStruct.tensorObj (𝒞 := instLargeCategory) X Y)
      { hom := Kernel.swap Y.carrier X.carrier, property := ⋯ }
      { hom := Kernel.swap X.carrier Y.carrier, property := ⋯ } =
    𝟙 (MonoidalCategoryStruct.tensorObj (𝒞 := instLargeCategory) Y X)
have refine_2 : Hom.hom
    (CategoryStruct.comp (Y := MonoidalCategoryStruct.tensorObj (𝒞 := instLargeCategory) X Y)
      { hom := Kernel.swap Y.carrier X.carrier, property := sorry }
      { hom := Kernel.swap X.carrier Y.carrier, property := sorry }) =
  Hom.hom (𝟙 (MonoidalCategoryStruct.tensorObj (𝒞 := instLargeCategory) Y X))
Tactic: ext : 1
-- Lean 4 proof step from Mathlib.Probability.Kernel.Category.SFinKer (goal)
Goal:
X Y Z : SFinKer
κ : Y ⟶ Z
⊢ Eq (α :=
    MonoidalCategoryStruct.tensorObj (𝒞 := instLargeCategory) X Y ⟶
      MonoidalCategoryStruct.tensorObj (𝒞 := instLargeCategory) Z X)
    (X ◁ κ ≫
      Iso.hom
        { hom := { hom := Kernel.swap X.carrier Z.carrier, property := ⋯ },
          inv := { hom := Kernel.swap Z.carrier X.carrier, property := ⋯ }, hom_inv_id := ⋯, inv_hom_id := ⋯ })
    (Iso.hom
        { hom := { hom := Kernel.swap X.carrier Y.carrier, property := ⋯ },
          inv := { hom := Kernel.swap Y.carrier X.carrier, property := ⋯ }, hom_inv_id := ⋯, inv_hom_id := ⋯ } ≫
      κ ▷ X)
have h : Eq (α :=
  Kernel (MonoidalCategoryStruct.tensorObj (𝒞 := instLargeCategory) X Y).carrier
    (MonoidalCategoryStruct.tensorObj (𝒞 := instLargeCategory) Z X).carrier)
  (Hom.hom
    (X ◁ κ ≫
      Iso.hom
        { hom := { hom := Kernel.swap X.carrier Z.carrier, property := sorry },
          inv := { hom := Kernel.swap Z.carrier X.carrier, property := sorry }, hom_inv_id := sorry, inv_hom_id := sorry }))
  (Hom.hom
    (Iso.hom
        { hom := { hom := Kernel.swap X.carrier Y.carrier, property := sorry },
          inv := { hom := Kernel.swap Y.carrier X.carrier, property := sorry }, hom_inv_id := sorry, inv_hom_id := sorry } ≫
      κ ▷ X))
Tactic: ext : 1
-- Lean 4 proof step from Mathlib.Probability.Kernel.Category.SFinKer (goal)
Goal:
X_1 Y : SFinKer
κ : X_1 ⟶ Y
X : SFinKer
⊢ Eq (α :=
    MonoidalCategoryStruct.tensorObj (𝒞 := instLargeCategory) X_1 X ⟶
      MonoidalCategoryStruct.tensorObj (𝒞 := instLargeCategory) X Y)
    (κ ▷ X ≫
      Iso.hom
        { hom := { hom := Kernel.swap Y.carrier X.carrier, property := ⋯ },
          inv := { hom := Kernel.swap X.carrier Y.carrier, property := ⋯ }, hom_inv_id := ⋯, inv_hom_id := ⋯ })
    (Iso.hom
        { hom := { hom := Kernel.swap X_1.carrier X.carrier, property := ⋯ },
          inv := { hom := Kernel.swap X.carrier X_1.carrier, property := ⋯ }, hom_inv_id := ⋯, inv_hom_id := ⋯ } ≫
      X ◁ κ)
have h : Eq (α :=
  Kernel (MonoidalCategoryStruct.tensorObj (𝒞 := instLargeCategory) X_1 X).carrier
    (MonoidalCategoryStruct.tensorObj (𝒞 := instLargeCategory) X Y).carrier)
  (Hom.hom
    (κ ▷ X ≫
      Iso.hom
        { hom := { hom := Kernel.swap Y.carrier X.carrier, property := sorry },
          inv := { hom := Kernel.swap X.carrier Y.carrier, property := sorry }, hom_inv_id := sorry, inv_hom_id := sorry }))
  (Hom.hom
    (Iso.hom
        { hom := { hom := Kernel.swap X_1.carrier X.carrier, property := sorry },
          inv := { hom := Kernel.swap X.carrier X_1.carrier, property := sorry }, hom_inv_id := sorry, inv_hom_id := sorry } ≫
      X ◁ κ))
Tactic: ext : 1
-- Lean 4 proof step from Mathlib.Probability.Kernel.Category.SFinKer (goal)
Goal:
X Y Z : SFinKer
⊢ Kernel.deterministic
          (⇑MeasurableEquiv.prodAssoc : (a : (Y.carrier × Z.carrier) × X.carrier) → Y.carrier × Z.carrier × X.carrier)
          ⋯ ∘ₖ
        Kernel.deterministic Prod.swap ⋯ ∘ₖ
      Kernel.deterministic
        (⇑MeasurableEquiv.prodAssoc : (a : (X.carrier × Y.carrier) × Z.carrier) → X.carrier × Y.carrier × Z.carrier) ⋯ =
    Kernel.deterministic id ⋯ ∥ₖ Kernel.deterministic Prod.swap ⋯ ∘ₖ
        Kernel.deterministic
          (⇑MeasurableEquiv.prodAssoc : (a : (Y.carrier × X.carrier) × Z.carrier) → Y.carrier × X.carrier × Z.carrier)
          ⋯ ∘ₖ
      (Kernel.deterministic Prod.swap ⋯ ∥ₖ Kernel.deterministic id ⋯)
have h : Kernel.deterministic
        (⇑MeasurableEquiv.prodAssoc : (a : (Y.carrier × Z.carrier) × X.carrier) → Y.carrier × Z.carrier × X.carrier)
        sorry ∘ₖ
      Kernel.deterministic Prod.swap sorry ∘ₖ
    Kernel.deterministic
      (⇑MeasurableEquiv.prodAssoc : (a : (X.carrier × Y.carrier) × Z.carrier) → X.carrier × Y.carrier × Z.carrier) sorry =
  Kernel.deterministic (Prod.map id Prod.swap) sorry ∘ₖ
      Kernel.deterministic
        (⇑MeasurableEquiv.prodAssoc : (a : (Y.carrier × X.carrier) × Z.carrier) → Y.carrier × X.carrier × Z.carrier)
        sorry ∘ₖ
    (Kernel.deterministic Prod.swap sorry ∥ₖ Kernel.deterministic id sorry)
Tactic: rw [Kernel.deterministic_parallelComp_deterministic]
-- Lean 4 proof step from Mathlib.Probability.Kernel.Category.SFinKer (goal)
Goal:
X Y Z : SFinKer
⊢ Kernel.deterministic
          (⇑MeasurableEquiv.prodAssoc : (a : (Y.carrier × Z.carrier) × X.carrier) → Y.carrier × Z.carrier × X.carrier)
          ⋯ ∘ₖ
        Kernel.deterministic Prod.swap ⋯ ∘ₖ
      Kernel.deterministic
        (⇑MeasurableEquiv.prodAssoc : (a : (X.carrier × Y.carrier) × Z.carrier) → X.carrier × Y.carrier × Z.carrier) ⋯ =
    Kernel.deterministic (Prod.map id Prod.swap) ⋯ ∘ₖ
        Kernel.deterministic
          (⇑MeasurableEquiv.prodAssoc : (a : (Y.carrier × X.carrier) × Z.carrier) → Y.carrier × X.carrier × Z.carrier)
          ⋯ ∘ₖ
      (Kernel.deterministic Prod.swap ⋯ ∥ₖ Kernel.deterministic id ⋯)
have h : Kernel.deterministic
        (⇑MeasurableEquiv.prodAssoc : (a : (Y.carrier × Z.carrier) × X.carrier) → Y.carrier × Z.carrier × X.carrier)
        sorry ∘ₖ
      Kernel.deterministic Prod.swap sorry ∘ₖ
    Kernel.deterministic
      (⇑MeasurableEquiv.prodAssoc : (a : (X.carrier × Y.carrier) × Z.carrier) → X.carrier × Y.carrier × Z.carrier) sorry =
  Kernel.deterministic (Prod.map id Prod.swap) sorry ∘ₖ
      Kernel.deterministic
        (⇑MeasurableEquiv.prodAssoc : (a : (Y.carrier × X.carrier) × Z.carrier) → Y.carrier × X.carrier × Z.carrier)
        sorry ∘ₖ
    Kernel.deterministic (Prod.map Prod.swap id) sorry
Tactic: rw [Kernel.deterministic_parallelComp_deterministic]
-- Lean 4 proof step from Mathlib.Probability.Kernel.Category.SFinKer (goal)
Goal:
X Y Z : SFinKer
⊢ Kernel.deterministic
          (⇑MeasurableEquiv.prodAssoc : (a : (Y.carrier × Z.carrier) × X.carrier) → Y.carrier × Z.carrier × X.carrier)
          ⋯ ∘ₖ
        Kernel.deterministic Prod.swap ⋯ ∘ₖ
      Kernel.deterministic
        (⇑MeasurableEquiv.prodAssoc : (a : (X.carrier × Y.carrier) × Z.carrier) → X.carrier × Y.carrier × Z.carrier) ⋯ =
    Kernel.deterministic id ⋯ ∥ₖ Kernel.deterministic Prod.swap ⋯ ∘ₖ
        Kernel.deterministic
          (⇑MeasurableEquiv.prodAssoc : (a : (Y.carrier × X.carrier) × Z.carrier) → Y.carrier × X.carrier × Z.carrier)
          ⋯ ∘ₖ
      (Kernel.deterministic Prod.swap ⋯ ∥ₖ Kernel.deterministic id ⋯)
have h : Kernel.deterministic
        (⇑MeasurableEquiv.prodAssoc : (a : (Y.carrier × Z.carrier) × X.carrier) → Y.carrier × Z.carrier × X.carrier)
        sorry ∘ₖ
      Kernel.deterministic Prod.swap sorry ∘ₖ
    Kernel.deterministic
      (⇑MeasurableEquiv.prodAssoc : (a : (X.carrier × Y.carrier) × Z.carrier) → X.carrier × Y.carrier × Z.carrier) sorry =
  Kernel.deterministic (Prod.map id Prod.swap) sorry ∘ₖ
      Kernel.deterministic
        (⇑MeasurableEquiv.prodAssoc : (a : (Y.carrier × X.carrier) × Z.carrier) → Y.carrier × X.carrier × Z.carrier)
        sorry ∘ₖ
    Kernel.deterministic (Prod.map Prod.swap id) sorry
Tactic: repeat rw [Kernel.deterministic_parallelComp_deterministic]
-- Lean 4 proof step from Mathlib.Probability.Kernel.Category.SFinKer (goal)
Goal:
X Y Z : SFinKer
⊢ Kernel.deterministic
          (⇑MeasurableEquiv.prodAssoc : (a : (Y.carrier × Z.carrier) × X.carrier) → Y.carrier × Z.carrier × X.carrier)
          ⋯ ∘ₖ
        Kernel.deterministic Prod.swap ⋯ ∘ₖ
      Kernel.deterministic
        (⇑MeasurableEquiv.prodAssoc : (a : (X.carrier × Y.carrier) × Z.carrier) → X.carrier × Y.carrier × Z.carrier) ⋯ =
    Kernel.deterministic (Prod.map id Prod.swap) ⋯ ∘ₖ
        Kernel.deterministic
          (⇑MeasurableEquiv.prodAssoc : (a : (Y.carrier × X.carrier) × Z.carrier) → Y.carrier × X.carrier × Z.carrier)
          ⋯ ∘ₖ
      Kernel.deterministic (Prod.map Prod.swap id) ⋯
have h : Kernel.deterministic
      ((⇑MeasurableEquiv.prodAssoc : (a : (Y.carrier × Z.carrier) × X.carrier) → Y.carrier × Z.carrier × X.carrier) ∘
        Prod.swap)
      sorry ∘ₖ
    Kernel.deterministic
      (⇑MeasurableEquiv.prodAssoc : (a : (X.carrier × Y.carrier) × Z.carrier) → X.carrier × Y.carrier × Z.carrier) sorry =
  Kernel.deterministic (Prod.map id Prod.swap) sorry ∘ₖ
      Kernel.deterministic
        (⇑MeasurableEquiv.prodAssoc : (a : (Y.carrier × X.carrier) × Z.carrier) → Y.carrier × X.carrier × Z.carrier)
        sorry ∘ₖ
    Kernel.deterministic (Prod.map Prod.swap id) sorry
Tactic: rw [Kernel.deterministic_comp_deterministic]