l3lab/ntp-mathlib huggingface dataset:
theorem stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable {α β ι : Type*}
    [TopologicalSpace ι] [MetrizableSpace ι] [MeasurableSpace ι] [SecondCountableTopology ι]
    [OpensMeasurableSpace ι] [TopologicalSpace β] [PseudoMetrizableSpace β] [MeasurableSpace α]
    {u : ι → α → β} (hu_cont : ∀ x, Continuous fun i => u i x) (h : ∀ i, StronglyMeasurable (u i)) :
    StronglyMeasurable (Function.uncurry u) := by
  

Proof state:
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
ι✝ : Type u_4
inst✝⁸ : Countable ι✝
α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous fun i => u i x
h : ∀ (i : ι), StronglyMeasurable (u i)
⊢ StronglyMeasurable (uncurry u)

Next tactic:
borelize β
theorem stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable {α β ι : Type*}
    [TopologicalSpace ι] [MetrizableSpace ι] [MeasurableSpace ι] [SecondCountableTopology ι]
    [OpensMeasurableSpace ι] [TopologicalSpace β] [PseudoMetrizableSpace β] [MeasurableSpace α]
    {u : ι → α → β} (hu_cont : ∀ x, Continuous fun i => u i x) (h : ∀ i, StronglyMeasurable (u i)) :
    StronglyMeasurable (Function.uncurry u) := by
  borelize β
  

Proof state:
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
ι✝ : Type u_4
inst✝⁸ : Countable ι✝
α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous fun i => u i x
h : ∀ (i : ι), StronglyMeasurable (u i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
⊢ StronglyMeasurable (uncurry u)

Next tactic:
obtain ⟨t_sf, ht_sf⟩ :
    ∃ t : ℕ → SimpleFunc ι ι, ∀ j x, Tendsto (fun n => u (t n j) x) atTop (𝓝 <| u j x) := by
    have h_str_meas : StronglyMeasurable (id : ι → ι) := stronglyMeasurable_id
    refine' ⟨h_str_meas.approx, fun j x => _⟩
    exact ((hu_cont x).tendsto j).comp (h_str_meas.tendsto_approx j)
theorem stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable {α β ι : Type*}
    [TopologicalSpace ι] [MetrizableSpace ι] [MeasurableSpace ι] [SecondCountableTopology ι]
    [OpensMeasurableSpace ι] [TopologicalSpace β] [PseudoMetrizableSpace β] [MeasurableSpace α]
    {u : ι → α → β} (hu_cont : ∀ x, Continuous fun i => u i x) (h : ∀ i, StronglyMeasurable (u i)) :
    StronglyMeasurable (Function.uncurry u) := by
  borelize β
  obtain ⟨t_sf, ht_sf⟩ :
    ∃ t : ℕ → SimpleFunc ι ι, ∀ j x, Tendsto (fun n => u (t n j) x) atTop (𝓝 <| u j x) := by
    

Proof state:
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
ι✝ : Type u_4
inst✝⁸ : Countable ι✝
α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous fun i => u i x
h : ∀ (i : ι), StronglyMeasurable (u i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
⊢ ∃ t, ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t n) j) x) atTop (𝓝 (u j x))

Next tactic:
have h_str_meas : StronglyMeasurable (id : ι → ι) := stronglyMeasurable_id
theorem stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable {α β ι : Type*}
    [TopologicalSpace ι] [MetrizableSpace ι] [MeasurableSpace ι] [SecondCountableTopology ι]
    [OpensMeasurableSpace ι] [TopologicalSpace β] [PseudoMetrizableSpace β] [MeasurableSpace α]
    {u : ι → α → β} (hu_cont : ∀ x, Continuous fun i => u i x) (h : ∀ i, StronglyMeasurable (u i)) :
    StronglyMeasurable (Function.uncurry u) := by
  borelize β
  obtain ⟨t_sf, ht_sf⟩ :
    ∃ t : ℕ → SimpleFunc ι ι, ∀ j x, Tendsto (fun n => u (t n j) x) atTop (𝓝 <| u j x) := by
    have h_str_meas : StronglyMeasurable (id : ι → ι) := stronglyMeasurable_id
    

Proof state:
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
ι✝ : Type u_4
inst✝⁸ : Countable ι✝
α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous fun i => u i x
h : ∀ (i : ι), StronglyMeasurable (u i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
h_str_meas : StronglyMeasurable id
⊢ ∃ t, ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t n) j) x) atTop (𝓝 (u j x))

Next tactic:
refine' ⟨h_str_meas.approx, fun j x => _⟩
theorem stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable {α β ι : Type*}
    [TopologicalSpace ι] [MetrizableSpace ι] [MeasurableSpace ι] [SecondCountableTopology ι]
    [OpensMeasurableSpace ι] [TopologicalSpace β] [PseudoMetrizableSpace β] [MeasurableSpace α]
    {u : ι → α → β} (hu_cont : ∀ x, Continuous fun i => u i x) (h : ∀ i, StronglyMeasurable (u i)) :
    StronglyMeasurable (Function.uncurry u) := by
  borelize β
  obtain ⟨t_sf, ht_sf⟩ :
    ∃ t : ℕ → SimpleFunc ι ι, ∀ j x, Tendsto (fun n => u (t n j) x) atTop (𝓝 <| u j x) := by
    have h_str_meas : StronglyMeasurable (id : ι → ι) := stronglyMeasurable_id
    refine' ⟨h_str_meas.approx, fun j x => _⟩
    

Proof state:
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
ι✝ : Type u_4
inst✝⁸ : Countable ι✝
α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous fun i => u i x
h : ∀ (i : ι), StronglyMeasurable (u i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
h_str_meas : StronglyMeasurable id
j : ι
x : α
⊢ Tendsto (fun n => u (↑(StronglyMeasurable.approx h_str_meas n) j) x) atTop (𝓝 (u j x))

Next tactic:
exact ((hu_cont x).tendsto j).comp (h_str_meas.tendsto_approx j)
theorem stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable {α β ι : Type*}
    [TopologicalSpace ι] [MetrizableSpace ι] [MeasurableSpace ι] [SecondCountableTopology ι]
    [OpensMeasurableSpace ι] [TopologicalSpace β] [PseudoMetrizableSpace β] [MeasurableSpace α]
    {u : ι → α → β} (hu_cont : ∀ x, Continuous fun i => u i x) (h : ∀ i, StronglyMeasurable (u i)) :
    StronglyMeasurable (Function.uncurry u) := by
  borelize β
  obtain ⟨t_sf, ht_sf⟩ :
    ∃ t : ℕ → SimpleFunc ι ι, ∀ j x, Tendsto (fun n => u (t n j) x) atTop (𝓝 <| u j x) := by
    have h_str_meas : StronglyMeasurable (id : ι → ι) := stronglyMeasurable_id
    refine' ⟨h_str_meas.approx, fun j x => _⟩
    exact ((hu_cont x).tendsto j).comp (h_str_meas.tendsto_approx j)
  

Proof state:
case intro
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
ι✝ : Type u_4
inst✝⁸ : Countable ι✝
α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous fun i => u i x
h : ∀ (i : ι), StronglyMeasurable (u i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
t_sf : ℕ → ι →ₛ ι
ht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))
⊢ StronglyMeasurable (uncurry u)

Next tactic:
let U (n : ℕ) (p : ι × α) := u (t_sf n p.fst) p.snd
theorem stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable {α β ι : Type*}
    [TopologicalSpace ι] [MetrizableSpace ι] [MeasurableSpace ι] [SecondCountableTopology ι]
    [OpensMeasurableSpace ι] [TopologicalSpace β] [PseudoMetrizableSpace β] [MeasurableSpace α]
    {u : ι → α → β} (hu_cont : ∀ x, Continuous fun i => u i x) (h : ∀ i, StronglyMeasurable (u i)) :
    StronglyMeasurable (Function.uncurry u) := by
  borelize β
  obtain ⟨t_sf, ht_sf⟩ :
    ∃ t : ℕ → SimpleFunc ι ι, ∀ j x, Tendsto (fun n => u (t n j) x) atTop (𝓝 <| u j x) := by
    have h_str_meas : StronglyMeasurable (id : ι → ι) := stronglyMeasurable_id
    refine' ⟨h_str_meas.approx, fun j x => _⟩
    exact ((hu_cont x).tendsto j).comp (h_str_meas.tendsto_approx j)
  let U (n : ℕ) (p : ι × α) := u (t_sf n p.fst) p.snd
  

Proof state:
case intro
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
ι✝ : Type u_4
inst✝⁸ : Countable ι✝
α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous fun i => u i x
h : ∀ (i : ι), StronglyMeasurable (u i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
t_sf : ℕ → ι →ₛ ι
ht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))
U : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.1) p.2
⊢ StronglyMeasurable (uncurry u)

Next tactic:
have h_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd) := by
    rw [tendsto_pi_nhds]
    exact fun p => ht_sf p.fst p.snd
theorem stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable {α β ι : Type*}
    [TopologicalSpace ι] [MetrizableSpace ι] [MeasurableSpace ι] [SecondCountableTopology ι]
    [OpensMeasurableSpace ι] [TopologicalSpace β] [PseudoMetrizableSpace β] [MeasurableSpace α]
    {u : ι → α → β} (hu_cont : ∀ x, Continuous fun i => u i x) (h : ∀ i, StronglyMeasurable (u i)) :
    StronglyMeasurable (Function.uncurry u) := by
  borelize β
  obtain ⟨t_sf, ht_sf⟩ :
    ∃ t : ℕ → SimpleFunc ι ι, ∀ j x, Tendsto (fun n => u (t n j) x) atTop (𝓝 <| u j x) := by
    have h_str_meas : StronglyMeasurable (id : ι → ι) := stronglyMeasurable_id
    refine' ⟨h_str_meas.approx, fun j x => _⟩
    exact ((hu_cont x).tendsto j).comp (h_str_meas.tendsto_approx j)
  let U (n : ℕ) (p : ι × α) := u (t_sf n p.fst) p.snd
  have h_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd) := by
    

Proof state:
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
ι✝ : Type u_4
inst✝⁸ : Countable ι✝
α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous fun i => u i x
h : ∀ (i : ι), StronglyMeasurable (u i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
t_sf : ℕ → ι →ₛ ι
ht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))
U : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.1) p.2
⊢ Tendsto U atTop (𝓝 fun p => u p.1 p.2)

Next tactic:
rw [tendsto_pi_nhds]
theorem stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable {α β ι : Type*}
    [TopologicalSpace ι] [MetrizableSpace ι] [MeasurableSpace ι] [SecondCountableTopology ι]
    [OpensMeasurableSpace ι] [TopologicalSpace β] [PseudoMetrizableSpace β] [MeasurableSpace α]
    {u : ι → α → β} (hu_cont : ∀ x, Continuous fun i => u i x) (h : ∀ i, StronglyMeasurable (u i)) :
    StronglyMeasurable (Function.uncurry u) := by
  borelize β
  obtain ⟨t_sf, ht_sf⟩ :
    ∃ t : ℕ → SimpleFunc ι ι, ∀ j x, Tendsto (fun n => u (t n j) x) atTop (𝓝 <| u j x) := by
    have h_str_meas : StronglyMeasurable (id : ι → ι) := stronglyMeasurable_id
    refine' ⟨h_str_meas.approx, fun j x => _⟩
    exact ((hu_cont x).tendsto j).comp (h_str_meas.tendsto_approx j)
  let U (n : ℕ) (p : ι × α) := u (t_sf n p.fst) p.snd
  have h_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd) := by
    rw [tendsto_pi_nhds]
    

Proof state:
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
ι✝ : Type u_4
inst✝⁸ : Countable ι✝
α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous fun i => u i x
h : ∀ (i : ι), StronglyMeasurable (u i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
t_sf : ℕ → ι →ₛ ι
ht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))
U : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.1) p.2
⊢ ∀ (x : ι × α), Tendsto (fun i => U i x) atTop (𝓝 (u x.1 x.2))

Next tactic:
exact fun p => ht_sf p.fst p.snd
theorem stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable {α β ι : Type*}
    [TopologicalSpace ι] [MetrizableSpace ι] [MeasurableSpace ι] [SecondCountableTopology ι]
    [OpensMeasurableSpace ι] [TopologicalSpace β] [PseudoMetrizableSpace β] [MeasurableSpace α]
    {u : ι → α → β} (hu_cont : ∀ x, Continuous fun i => u i x) (h : ∀ i, StronglyMeasurable (u i)) :
    StronglyMeasurable (Function.uncurry u) := by
  borelize β
  obtain ⟨t_sf, ht_sf⟩ :
    ∃ t : ℕ → SimpleFunc ι ι, ∀ j x, Tendsto (fun n => u (t n j) x) atTop (𝓝 <| u j x) := by
    have h_str_meas : StronglyMeasurable (id : ι → ι) := stronglyMeasurable_id
    refine' ⟨h_str_meas.approx, fun j x => _⟩
    exact ((hu_cont x).tendsto j).comp (h_str_meas.tendsto_approx j)
  let U (n : ℕ) (p : ι × α) := u (t_sf n p.fst) p.snd
  have h_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd) := by
    rw [tendsto_pi_nhds]
    exact fun p => ht_sf p.fst p.snd
  

Proof state:
case intro
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
ι✝ : Type u_4
inst✝⁸ : Countable ι✝
α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous fun i => u i x
h : ∀ (i : ι), StronglyMeasurable (u i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
t_sf : ℕ → ι →ₛ ι
ht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))
U : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.1) p.2
h_tendsto : Tendsto U atTop (𝓝 fun p => u p.1 p.2)
⊢ StronglyMeasurable (uncurry u)

Next tactic:
refine' stronglyMeasurable_of_tendsto _ (fun n => _) h_tendsto
theorem stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable {α β ι : Type*}
    [TopologicalSpace ι] [MetrizableSpace ι] [MeasurableSpace ι] [SecondCountableTopology ι]
    [OpensMeasurableSpace ι] [TopologicalSpace β] [PseudoMetrizableSpace β] [MeasurableSpace α]
    {u : ι → α → β} (hu_cont : ∀ x, Continuous fun i => u i x) (h : ∀ i, StronglyMeasurable (u i)) :
    StronglyMeasurable (Function.uncurry u) := by
  borelize β
  obtain ⟨t_sf, ht_sf⟩ :
    ∃ t : ℕ → SimpleFunc ι ι, ∀ j x, Tendsto (fun n => u (t n j) x) atTop (𝓝 <| u j x) := by
    have h_str_meas : StronglyMeasurable (id : ι → ι) := stronglyMeasurable_id
    refine' ⟨h_str_meas.approx, fun j x => _⟩
    exact ((hu_cont x).tendsto j).comp (h_str_meas.tendsto_approx j)
  let U (n : ℕ) (p : ι × α) := u (t_sf n p.fst) p.snd
  have h_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd) := by
    rw [tendsto_pi_nhds]
    exact fun p => ht_sf p.fst p.snd
  refine' stronglyMeasurable_of_tendsto _ (fun n => _) h_tendsto
  

Proof state:
case intro
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
ι✝ : Type u_4
inst✝⁸ : Countable ι✝
α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous fun i => u i x
h : ∀ (i : ι), StronglyMeasurable (u i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
t_sf : ℕ → ι →ₛ ι
ht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))
U : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.1) p.2
h_tendsto : Tendsto U atTop (𝓝 fun p => u p.1 p.2)
n : ℕ
⊢ StronglyMeasurable (U n)

Next tactic:
have h_str_meas : StronglyMeasurable fun p : (t_sf n).range × α => u (↑p.fst) p.snd := by
    refine' stronglyMeasurable_iff_measurable_separable.2 ⟨_, _⟩
    · have :
        (fun p : ↥(t_sf n).range × α => u (↑p.fst) p.snd) =
          (fun p : α × (t_sf n).range => u (↑p.snd) p.fst) ∘ Prod.swap :=
        rfl
      rw [this, measurable_swap_iff]
      exact measurable_from_prod_countable fun j => (h j).measurable
    · have : IsSeparable (⋃ i : (t_sf n).range, range (u i)) :=
        isSeparable_iUnion fun i => (h i).isSeparable_range
      apply this.mono
      rintro _ ⟨⟨i, x⟩, rfl⟩
      simp only [mem_iUnion, mem_range]
      exact ⟨i, x, rfl⟩
theorem stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable {α β ι : Type*}
    [TopologicalSpace ι] [MetrizableSpace ι] [MeasurableSpace ι] [SecondCountableTopology ι]
    [OpensMeasurableSpace ι] [TopologicalSpace β] [PseudoMetrizableSpace β] [MeasurableSpace α]
    {u : ι → α → β} (hu_cont : ∀ x, Continuous fun i => u i x) (h : ∀ i, StronglyMeasurable (u i)) :
    StronglyMeasurable (Function.uncurry u) := by
  borelize β
  obtain ⟨t_sf, ht_sf⟩ :
    ∃ t : ℕ → SimpleFunc ι ι, ∀ j x, Tendsto (fun n => u (t n j) x) atTop (𝓝 <| u j x) := by
    have h_str_meas : StronglyMeasurable (id : ι → ι) := stronglyMeasurable_id
    refine' ⟨h_str_meas.approx, fun j x => _⟩
    exact ((hu_cont x).tendsto j).comp (h_str_meas.tendsto_approx j)
  let U (n : ℕ) (p : ι × α) := u (t_sf n p.fst) p.snd
  have h_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd) := by
    rw [tendsto_pi_nhds]
    exact fun p => ht_sf p.fst p.snd
  refine' stronglyMeasurable_of_tendsto _ (fun n => _) h_tendsto
  have h_str_meas : StronglyMeasurable fun p : (t_sf n).range × α => u (↑p.fst) p.snd := by
    

Proof state:
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
ι✝ : Type u_4
inst✝⁸ : Countable ι✝
α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous fun i => u i x
h : ∀ (i : ι), StronglyMeasurable (u i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
t_sf : ℕ → ι →ₛ ι
ht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))
U : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.1) p.2
h_tendsto : Tendsto U atTop (𝓝 fun p => u p.1 p.2)
n : ℕ
⊢ StronglyMeasurable fun p => u (↑p.1) p.2

Next tactic:
refine' stronglyMeasurable_iff_measurable_separable.2 ⟨_, _⟩
theorem stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable {α β ι : Type*}
    [TopologicalSpace ι] [MetrizableSpace ι] [MeasurableSpace ι] [SecondCountableTopology ι]
    [OpensMeasurableSpace ι] [TopologicalSpace β] [PseudoMetrizableSpace β] [MeasurableSpace α]
    {u : ι → α → β} (hu_cont : ∀ x, Continuous fun i => u i x) (h : ∀ i, StronglyMeasurable (u i)) :
    StronglyMeasurable (Function.uncurry u) := by
  borelize β
  obtain ⟨t_sf, ht_sf⟩ :
    ∃ t : ℕ → SimpleFunc ι ι, ∀ j x, Tendsto (fun n => u (t n j) x) atTop (𝓝 <| u j x) := by
    have h_str_meas : StronglyMeasurable (id : ι → ι) := stronglyMeasurable_id
    refine' ⟨h_str_meas.approx, fun j x => _⟩
    exact ((hu_cont x).tendsto j).comp (h_str_meas.tendsto_approx j)
  let U (n : ℕ) (p : ι × α) := u (t_sf n p.fst) p.snd
  have h_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd) := by
    rw [tendsto_pi_nhds]
    exact fun p => ht_sf p.fst p.snd
  refine' stronglyMeasurable_of_tendsto _ (fun n => _) h_tendsto
  have h_str_meas : StronglyMeasurable fun p : (t_sf n).range × α => u (↑p.fst) p.snd := by
    refine' stronglyMeasurable_iff_measurable_separable.2 ⟨_, _⟩
    · 

Proof state:
case refine'_1
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
ι✝ : Type u_4
inst✝⁸ : Countable ι✝
α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous fun i => u i x
h : ∀ (i : ι), StronglyMeasurable (u i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
t_sf : ℕ → ι →ₛ ι
ht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))
U : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.1) p.2
h_tendsto : Tendsto U atTop (𝓝 fun p => u p.1 p.2)
n : ℕ
⊢ Measurable fun p => u (↑p.1) p.2

Next tactic:
have :
        (fun p : ↥(t_sf n).range × α => u (↑p.fst) p.snd) =
          (fun p : α × (t_sf n).range => u (↑p.snd) p.fst) ∘ Prod.swap :=
        rfl
theorem stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable {α β ι : Type*}
    [TopologicalSpace ι] [MetrizableSpace ι] [MeasurableSpace ι] [SecondCountableTopology ι]
    [OpensMeasurableSpace ι] [TopologicalSpace β] [PseudoMetrizableSpace β] [MeasurableSpace α]
    {u : ι → α → β} (hu_cont : ∀ x, Continuous fun i => u i x) (h : ∀ i, StronglyMeasurable (u i)) :
    StronglyMeasurable (Function.uncurry u) := by
  borelize β
  obtain ⟨t_sf, ht_sf⟩ :
    ∃ t : ℕ → SimpleFunc ι ι, ∀ j x, Tendsto (fun n => u (t n j) x) atTop (𝓝 <| u j x) := by
    have h_str_meas : StronglyMeasurable (id : ι → ι) := stronglyMeasurable_id
    refine' ⟨h_str_meas.approx, fun j x => _⟩
    exact ((hu_cont x).tendsto j).comp (h_str_meas.tendsto_approx j)
  let U (n : ℕ) (p : ι × α) := u (t_sf n p.fst) p.snd
  have h_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd) := by
    rw [tendsto_pi_nhds]
    exact fun p => ht_sf p.fst p.snd
  refine' stronglyMeasurable_of_tendsto _ (fun n => _) h_tendsto
  have h_str_meas : StronglyMeasurable fun p : (t_sf n).range × α => u (↑p.fst) p.snd := by
    refine' stronglyMeasurable_iff_measurable_separable.2 ⟨_, _⟩
    · have :
        (fun p : ↥(t_sf n).range × α => u (↑p.fst) p.snd) =
          (fun p : α × (t_sf n).range => u (↑p.snd) p.fst) ∘ Prod.swap :=
        rfl
      

Proof state:
case refine'_1
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
ι✝ : Type u_4
inst✝⁸ : Countable ι✝
α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous fun i => u i x
h : ∀ (i : ι), StronglyMeasurable (u i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
t_sf : ℕ → ι →ₛ ι
ht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))
U : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.1) p.2
h_tendsto : Tendsto U atTop (𝓝 fun p => u p.1 p.2)
n : ℕ
this : (fun p => u (↑p.1) p.2) = (fun p => u (↑p.2) p.1) ∘ Prod.swap
⊢ Measurable fun p => u (↑p.1) p.2

Next tactic:
rw [this, measurable_swap_iff]