awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.MeasureTheory.Constructions.Polish.Basic
Goal:
ι : Type u_2
inst : Countable ι
α : Type u_3
inst_1 : MeasurableSpace α
s t : ι → Set α
u : (m n : ι) → Set α
hsu : ∀ (m n : ι), s m ⊆ u m n
htu : ∀ (m n : ι), Disjoint (t n) (u m n)
hu : ∀ (m n : ι), MeasurableSet (u m n)
⊢ Subset.{u_3} (α := Set α) (⋃ (n : ι), s n) (⋃ (m : ι), ⋂ (n : ι), u m n)

Tactic:
iUnion_subset fun m => subset_iUnion_of_subset m ?_

Premises used: Set.iUnion_subset, Set.subset_iUnion_of_subset
-- Module: Mathlib.MeasureTheory.Constructions.Polish.Basic
Goal:
ι : Type u_2
inst : Countable ι
α : Type u_3
inst_1 : MeasurableSpace α
s t : ι → Set α
u : (m n : ι) → Set α
hsu : ∀ (m n : ι), s m ⊆ u m n
htu : ∀ (m n : ι), Disjoint (t n) (u m n)
hu : ∀ (m n : ι), MeasurableSet (u m n)
refine_1 : ∀ (m : ι), s m ⊆ ⋂ (n : ι), u m n
⊢ Subset.{u_3} (α := Set α) (⋃ (n : ι), s n) (⋃ (m : ι), ⋂ (n : ι), u m n)

Tactic:
refine iUnion_subset fun m => subset_iUnion_of_subset m ?_

Premises used: Set.iUnion_subset, Set.subset_iUnion_of_subset
-- Module: Mathlib.MeasureTheory.Constructions.Polish.Basic
Goal:
ι : Type u_2
inst : Countable ι
α : Type u_3
inst_1 : MeasurableSpace α
s t : ι → Set α
u : (m n : ι) → Set α
hsu : ∀ (m n : ι), s m ⊆ u m n
htu : ∀ (m n : ι), Disjoint (t n) (u m n)
hu : ∀ (m n : ι), MeasurableSet (u m n)
m : ι
⊢ s m ⊆ ⋂ (n : ι), u m n

Tactic:
subset_iInter fun n => hsu m n

Premises used: Set.subset_iInter
-- Module: Mathlib.MeasureTheory.Constructions.Polish.Basic
Goal:
ι : Type u_2
inst : Countable ι
α : Type u_3
inst_1 : MeasurableSpace α
s t : ι → Set α
u : (m n : ι) → Set α
hsu : ∀ (m n : ι), s m ⊆ u m n
htu : ∀ (m n : ι), Disjoint (t n) (u m n)
hu : ∀ (m n : ι), MeasurableSet (u m n)
m : ι
⊢ s m ⊆ ⋂ (n : ι), u m n

Tactic:
exact subset_iInter fun n => hsu m n

Premises used: Set.subset_iInter
-- Module: Mathlib.MeasureTheory.Constructions.Polish.Basic
Goal:
ι : Type u_2
inst : Countable ι
α : Type u_3
inst_1 : MeasurableSpace α
s t : ι → Set α
u : (m n : ι) → Set α
hsu : ∀ (m n : ι), s m ⊆ u m n
htu : ∀ (m n : ι), Disjoint (t n) (u m n)
hu : ∀ (m n : ι), MeasurableSet (u m n)
refine_2 : ∀ (i i_1 : ι), Disjoint (t i) (⋂ (n : ι), u i_1 n)
⊢ Disjoint.{u_3} (α := Set α) (⋃ (m : ι), t m) (⋃ (m : ι), ⋂ (n : ι), u m n)

Tactic:
simp_rw [disjoint_iUnion_left, disjoint_iUnion_right]

Premises used: Set.disjoint_iUnion_left, Set.disjoint_iUnion_right
-- Module: Mathlib.MeasureTheory.Constructions.Polish.Basic
Goal:
ι : Type u_2
inst : Countable ι
α : Type u_3
inst_1 : MeasurableSpace α
s t : ι → Set α
u : (m n : ι) → Set α
hsu : ∀ (m n : ι), s m ⊆ u m n
htu : ∀ (m n : ι), Disjoint (t n) (u m n)
hu : ∀ (m n : ι), MeasurableSet (u m n)
n m : ι
apply : ⋂ (n : ι), u m n ≤ u m n
⊢ Disjoint (t n) (⋂ (n : ι), u m n)

Tactic:
apply Disjoint.mono_right _ (htu m n)

Premises used: Disjoint.mono_right
-- Module: Mathlib.MeasureTheory.Constructions.Polish.Basic
Goal:
ι : Type u_2
inst : Countable ι
α : Type u_3
inst_1 : MeasurableSpace α
s t : ι → Set α
u : (m n : ι) → Set α
hsu : ∀ (m n : ι), s m ⊆ u m n
htu : ∀ (m n : ι), Disjoint (t n) (u m n)
hu : ∀ (m n : ι), MeasurableSet (u m n)
n m : ι
⊢ ⋂ (n : ι), u m n ≤ u m n

Tactic:
apply iInter_subset

Premises used: Set.iInter_subset
-- Module: Mathlib.MeasureTheory.Constructions.Polish.Basic
Goal:
ι : Type u_2
inst : Countable ι
α : Type u_3
inst_1 : MeasurableSpace α
s t : ι → Set α
u : (m n : ι) → Set α
hsu : ∀ (m n : ι), s m ⊆ u m n
htu : ∀ (m n : ι), Disjoint (t n) (u m n)
hu : ∀ (m n : ι), MeasurableSet (u m n)
⊢ MeasurableSet.{u_3} (α := α) (⋃ (m : ι), ⋂ (n : ι), u m n)

Tactic:
MeasurableSet.iUnion fun m => ?_

Premises used: MeasurableSet.iUnion
-- Module: Mathlib.MeasureTheory.Constructions.Polish.Basic
Goal:
ι : Type u_2
inst : Countable ι
α : Type u_3
inst_1 : MeasurableSpace α
s t : ι → Set α
u : (m n : ι) → Set α
hsu : ∀ (m n : ι), s m ⊆ u m n
htu : ∀ (m n : ι), Disjoint (t n) (u m n)
hu : ∀ (m n : ι), MeasurableSet (u m n)
refine_3 : ∀ (m : ι), MeasurableSet.{u_3} (α := α) (⋂ (n : ι), u m n)
⊢ MeasurableSet.{u_3} (α := α) (⋃ (m : ι), ⋂ (n : ι), u m n)

Tactic:
refine MeasurableSet.iUnion fun m => ?_

Premises used: MeasurableSet.iUnion
-- Module: Mathlib.MeasureTheory.Constructions.Polish.Basic
Goal:
ι : Type u_2
inst : Countable ι
α : Type u_3
inst_1 : MeasurableSpace α
s t : ι → Set α
u : (m n : ι) → Set α
hsu : ∀ (m n : ι), s m ⊆ u m n
htu : ∀ (m n : ι), Disjoint (t n) (u m n)
hu : ∀ (m n : ι), MeasurableSet (u m n)
m : ι
⊢ MeasurableSet.{u_3} (α := α) (⋂ (n : ι), u m n)

Tactic:
MeasurableSet.iInter fun n => hu m n

Premises used: MeasurableSet.iInter
-- Module: Mathlib.MeasureTheory.Constructions.Polish.Basic
Goal:
ι : Type u_2
inst : Countable ι
α : Type u_3
inst_1 : MeasurableSpace α
s t : ι → Set α
u : (m n : ι) → Set α
hsu : ∀ (m n : ι), s m ⊆ u m n
htu : ∀ (m n : ι), Disjoint (t n) (u m n)
hu : ∀ (m n : ι), MeasurableSet (u m n)
m : ι
⊢ MeasurableSet.{u_3} (α := α) (⋂ (n : ι), u m n)

Tactic:
exact MeasurableSet.iInter fun n => hu m n

Premises used: MeasurableSet.iInter
-- Module: Mathlib.MeasureTheory.Constructions.Polish.Basic
Goal:
α : Type u_1
inst : TopologicalSpace α
inst_1 : T2Space α
inst_2 : MeasurableSpace α
inst_3 : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous[_, inst] f
hg : Continuous[_, inst] g
h : Disjoint (range f) (range g)
by_contra : ∀ (hfg : ¬MeasurablySeparable (range f) (range g)), False
⊢ MeasurablySeparable (range f) (range g)

Tactic:
by_contra hfg
-- Module: Mathlib.MeasureTheory.Constructions.Polish.Basic
Goal:
α : Type u_1
inst : TopologicalSpace α
inst_1 : T2Space α
inst_2 : MeasurableSpace α
inst_3 : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous[_, inst] f
hg : Continuous[_, inst] g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
n : ℕ
x y : (n : ℕ) → ℕ
contrapose! :
  (∀ (x' y' : (n : ℕ) → ℕ) (_ : x' ∈ cylinder (E := fun (n : ℕ) ↦ ℕ) x n)
      (_ : y' ∈ cylinder (E := fun (n : ℕ) ↦ ℕ) y n),
      MeasurablySeparable (f '' cylinder (E := fun (a : ℕ) ↦ ℕ) x' (n + 1))
        (g '' cylinder (E := fun (a : ℕ) ↦ ℕ) y' (n + 1))) →
    MeasurablySeparable (f '' cylinder (E := fun (n : ℕ) ↦ ℕ) x n) (g '' cylinder (E := fun (n : ℕ) ↦ ℕ) y n)
⊢ ¬MeasurablySeparable (f '' cylinder (E := fun (a : ℕ) ↦ ℕ) x n) (g '' cylinder (E := fun (a : ℕ) ↦ ℕ) y n) →
    ∃ (x' : (n : ℕ) → ℕ) (y' : (n : ℕ) → ℕ),
      x' ∈ cylinder (E := fun (n : ℕ) ↦ ℕ) x n ∧
        y' ∈ cylinder (E := fun (n : ℕ) ↦ ℕ) y n ∧
          ¬MeasurablySeparable (f '' cylinder (E := fun (a : ℕ) ↦ ℕ) x' (n + 1))
              (g '' cylinder (E := fun (a : ℕ) ↦ ℕ) y' (n + 1))

Tactic:
contrapose!
-- Module: Mathlib.MeasureTheory.Constructions.Polish.Basic
Goal:
α : Type u_1
inst : TopologicalSpace α
inst_1 : T2Space α
inst_2 : MeasurableSpace α
inst_3 : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous[_, inst] f
hg : Continuous[_, inst] g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
n : ℕ
x y : (n : ℕ) → ℕ
H :
  ∀ (x' y' : (n : ℕ) → ℕ) (_ : x' ∈ cylinder (E := fun (n : ℕ) ↦ ℕ) x n) (_ : y' ∈ cylinder (E := fun (n : ℕ) ↦ ℕ) y n),
    MeasurablySeparable (f '' cylinder (E := fun (a : ℕ) ↦ ℕ) x' (n + 1))
      (g '' cylinder (E := fun (a : ℕ) ↦ ℕ) y' (n + 1))
rw :
  MeasurablySeparable.{u_1} (α := α)
    (⋃ (i : ℕ), f '' cylinder (E := fun (n : ℕ) ↦ ℕ) (update (β := fun (n : ℕ) ↦ ℕ) x n i) (n + 1))
    (⋃ (i : ℕ), g '' cylinder (E := fun (n : ℕ) ↦ ℕ) (update (β := fun (n : ℕ) ↦ ℕ) y n i) (n + 1))
⊢ MeasurablySeparable (f '' cylinder (E := fun (n : ℕ) ↦ ℕ) x n) (g '' cylinder (E := fun (n : ℕ) ↦ ℕ) y n)

Tactic:
rw [← iUnion_cylinder_update x n, ← iUnion_cylinder_update y n, image_iUnion, image_iUnion]

Premises used: PiNat.iUnion_cylinder_update, Set.image_iUnion
-- Module: Mathlib.MeasureTheory.Constructions.Polish.Basic
Goal:
α : Type u_1
inst : TopologicalSpace α
inst_1 : T2Space α
inst_2 : MeasurableSpace α
inst_3 : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous[_, inst] f
hg : Continuous[_, inst] g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
n : ℕ
x y : (n : ℕ) → ℕ
H :
  ∀ (x' y' : (n : ℕ) → ℕ) (_ : x' ∈ cylinder (E := fun (n : ℕ) ↦ ℕ) x n) (_ : y' ∈ cylinder (E := fun (n : ℕ) ↦ ℕ) y n),
    MeasurablySeparable (f '' cylinder (E := fun (a : ℕ) ↦ ℕ) x' (n + 1))
      (g '' cylinder (E := fun (a : ℕ) ↦ ℕ) y' (n + 1))
⊢ MeasurablySeparable.{u_1} (α := α)
    (⋃ (i : ℕ), f '' cylinder (E := fun (n : ℕ) ↦ ℕ) (update (β := fun (n : ℕ) ↦ ℕ) x n i) (n + 1))
    (⋃ (i : ℕ), g '' cylinder (E := fun (n : ℕ) ↦ ℕ) (update (β := fun (n : ℕ) ↦ ℕ) y n i) (n + 1))

Tactic:
MeasurablySeparable.iUnion fun i j => ?_

Premises used: MeasureTheory.MeasurablySeparable.iUnion
-- Module: Mathlib.MeasureTheory.Constructions.Polish.Basic
Goal:
α : Type u_1
inst : TopologicalSpace α
inst_1 : T2Space α
inst_2 : MeasurableSpace α
inst_3 : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous[_, inst] f
hg : Continuous[_, inst] g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
n : ℕ
x y : (n : ℕ) → ℕ
H :
  ∀ (x' y' : (n : ℕ) → ℕ) (_ : x' ∈ cylinder (E := fun (n : ℕ) ↦ ℕ) x n) (_ : y' ∈ cylinder (E := fun (n : ℕ) ↦ ℕ) y n),
    MeasurablySeparable (f '' cylinder (E := fun (a : ℕ) ↦ ℕ) x' (n + 1))
      (g '' cylinder (E := fun (a : ℕ) ↦ ℕ) y' (n + 1))
refine :
  ∀ (i j : ℕ),
    MeasurablySeparable (f '' cylinder (E := fun (n : ℕ) ↦ ℕ) (update (β := fun (n : ℕ) ↦ ℕ) x n i) (n + 1))
      (g '' cylinder (E := fun (n : ℕ) ↦ ℕ) (update (β := fun (n : ℕ) ↦ ℕ) y n j) (n + 1))
⊢ MeasurablySeparable.{u_1} (α := α)
    (⋃ (i : ℕ), f '' cylinder (E := fun (n : ℕ) ↦ ℕ) (update (β := fun (n : ℕ) ↦ ℕ) x n i) (n + 1))
    (⋃ (i : ℕ), g '' cylinder (E := fun (n : ℕ) ↦ ℕ) (update (β := fun (n : ℕ) ↦ ℕ) y n i) (n + 1))

Tactic:
refine MeasurablySeparable.iUnion fun i j => ?_

Premises used: MeasureTheory.MeasurablySeparable.iUnion
-- Module: Mathlib.MeasureTheory.Constructions.Polish.Basic
Goal:
α : Type u_1
inst : TopologicalSpace α
inst_1 : T2Space α
inst_2 : MeasurableSpace α
inst_3 : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous[_, inst] f
hg : Continuous[_, inst] g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
n : ℕ
x y : (n : ℕ) → ℕ
H :
  ∀ (x' y' : (n : ℕ) → ℕ) (_ : x' ∈ cylinder (E := fun (n : ℕ) ↦ ℕ) x n) (_ : y' ∈ cylinder (E := fun (n : ℕ) ↦ ℕ) y n),
    MeasurablySeparable (f '' cylinder (E := fun (a : ℕ) ↦ ℕ) x' (n + 1))
      (g '' cylinder (E := fun (a : ℕ) ↦ ℕ) y' (n + 1))
i j : ℕ
⊢ MeasurablySeparable (f '' cylinder (E := fun (n : ℕ) ↦ ℕ) (update (β := fun (n : ℕ) ↦ ℕ) x n i) (n + 1))
    (g '' cylinder (E := fun (n : ℕ) ↦ ℕ) (update (β := fun (n : ℕ) ↦ ℕ) y n j) (n + 1))

Tactic:
H _ _ (update_mem_cylinder _ _ _) (update_mem_cylinder _ _ _)

Premises used: PiNat.update_mem_cylinder
-- Module: Mathlib.MeasureTheory.Constructions.Polish.Basic
Goal:
α : Type u_1
inst : TopologicalSpace α
inst_1 : T2Space α
inst_2 : MeasurableSpace α
inst_3 : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous[_, inst] f
hg : Continuous[_, inst] g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
n : ℕ
x y : (n : ℕ) → ℕ
H :
  ∀ (x' y' : (n : ℕ) → ℕ) (_ : x' ∈ cylinder (E := fun (n : ℕ) ↦ ℕ) x n) (_ : y' ∈ cylinder (E := fun (n : ℕ) ↦ ℕ) y n),
    MeasurablySeparable (f '' cylinder (E := fun (a : ℕ) ↦ ℕ) x' (n + 1))
      (g '' cylinder (E := fun (a : ℕ) ↦ ℕ) y' (n + 1))
i j : ℕ
⊢ MeasurablySeparable (f '' cylinder (E := fun (n : ℕ) ↦ ℕ) (update (β := fun (n : ℕ) ↦ ℕ) x n i) (n + 1))
    (g '' cylinder (E := fun (n : ℕ) ↦ ℕ) (update (β := fun (n : ℕ) ↦ ℕ) y n j) (n + 1))

Tactic:
exact H _ _ (update_mem_cylinder _ _ _) (update_mem_cylinder _ _ _)

Premises used: PiNat.update_mem_cylinder
-- Module: Mathlib.MeasureTheory.Constructions.Polish.Basic
Goal:
α : Type u_1
inst : TopologicalSpace α
inst_1 : T2Space α
inst_2 : MeasurableSpace α
inst_3 : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous[_, inst] f
hg : Continuous[_, inst] g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
⊢ ∀ (n : ℕ) (x y : (n : ℕ) → ℕ),
    ¬MeasurablySeparable (f '' cylinder (E := fun (a : ℕ) ↦ ℕ) x n) (g '' cylinder (E := fun (a : ℕ) ↦ ℕ) y n) →
      ∃ (x' : (n : ℕ) → ℕ) (y' : (n : ℕ) → ℕ),
        x' ∈ cylinder (E := fun (n : ℕ) ↦ ℕ) x n ∧
          y' ∈ cylinder (E := fun (n : ℕ) ↦ ℕ) y n ∧
            ¬MeasurablySeparable (f '' cylinder (E := fun (a : ℕ) ↦ ℕ) x' (n + 1))
                (g '' cylinder (E := fun (a : ℕ) ↦ ℕ) y' (n + 1))

Tactic:
have I : ∀ n x y, ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →
      ∃ x' y', x' ∈ cylinder x n ∧ y' ∈ cylinder y n ∧
      ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) := by
    intro n x y
    contrapose!
    intro H
    rw [← iUnion_cylinder_update x n, ← iUnion_cylinder_update y n, image_iUnion, image_iUnion]
    refine MeasurablySeparable.iUnion fun i j => ?_
    exact H _ _ (update_mem_cylinder _ _ _) (update_mem_cylinder _ _ _)
-- Module: Mathlib.MeasureTheory.Constructions.Polish.Basic
Goal:
α : Type u_1
inst : TopologicalSpace α
inst_1 : T2Space α
inst_2 : MeasurableSpace α
inst_3 : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous[_, inst] f
hg : Continuous[_, inst] g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
I :
  ∀ (n : ℕ) (x y : (n : ℕ) → ℕ),
    ¬MeasurablySeparable (f '' cylinder (E := fun (a : ℕ) ↦ ℕ) x n) (g '' cylinder (E := fun (a : ℕ) ↦ ℕ) y n) →
      ∃ (x' : (n : ℕ) → ℕ) (y' : (n : ℕ) → ℕ),
        x' ∈ cylinder (E := fun (n : ℕ) ↦ ℕ) x n ∧
          y' ∈ cylinder (E := fun (n : ℕ) ↦ ℕ) y n ∧
            ¬MeasurablySeparable (f '' cylinder (E := fun (a : ℕ) ↦ ℕ) x' (n + 1))
                (g '' cylinder (E := fun (a : ℕ) ↦ ℕ) y' (n + 1))
A : Type :=
  { p : ℕ × (ℕ → ℕ) × (ℕ → ℕ) //
    ¬MeasurablySeparable (f '' cylinder (E := fun (a : ℕ) ↦ ℕ) p.2.1 p.1)
        (g '' cylinder (E := fun (a : ℕ) ↦ ℕ) p.2.2 p.1) }
⊢ ∀ (p : A),
    ∃ (q : A),
      (↑q : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).1 = (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).1 + 1 ∧
        (↑q : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).2.1 ∈
            cylinder (E := fun (n : ℕ) ↦ ℕ) (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).2.1 (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).1 ∧
          (↑q : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).2.2 ∈
            cylinder (E := fun (n : ℕ) ↦ ℕ) (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).2.2 (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).1

Tactic:
have : ∀ p : A, ∃ q : A,
      q.1.1 = p.1.1 + 1 ∧ q.1.2.1 ∈ cylinder p.1.2.1 p.1.1 ∧ q.1.2.2 ∈ cylinder p.1.2.2 p.1.1 := by
    rintro ⟨⟨n, x, y⟩, hp⟩
    rcases I n x y hp with ⟨x', y', hx', hy', h'⟩
    exact ⟨⟨⟨n + 1, x', y'⟩, h'⟩, rfl, hx', hy'⟩
-- Module: Mathlib.MeasureTheory.Constructions.Polish.Basic
Goal:
α : Type u_1
inst : TopologicalSpace α
inst_1 : T2Space α
inst_2 : MeasurableSpace α
inst_3 : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous[_, inst] f
hg : Continuous[_, inst] g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
I :
  ∀ (n : ℕ) (x y : (n : ℕ) → ℕ),
    ¬MeasurablySeparable (f '' cylinder (E := fun (a : ℕ) ↦ ℕ) x n) (g '' cylinder (E := fun (a : ℕ) ↦ ℕ) y n) →
      ∃ (x' : (n : ℕ) → ℕ) (y' : (n : ℕ) → ℕ),
        x' ∈ cylinder (E := fun (n : ℕ) ↦ ℕ) x n ∧
          y' ∈ cylinder (E := fun (n : ℕ) ↦ ℕ) y n ∧
            ¬MeasurablySeparable (f '' cylinder (E := fun (a : ℕ) ↦ ℕ) x' (n + 1))
                (g '' cylinder (E := fun (a : ℕ) ↦ ℕ) y' (n + 1))
A : Type :=
  { p : ℕ × (ℕ → ℕ) × (ℕ → ℕ) //
    ¬MeasurablySeparable (f '' cylinder (E := fun (a : ℕ) ↦ ℕ) p.2.1 p.1)
        (g '' cylinder (E := fun (a : ℕ) ↦ ℕ) p.2.2 p.1) }
this :
  ∀ (p : A),
    ∃ (q : A),
      (↑q : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).1 = (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).1 + 1 ∧
        (↑q : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).2.1 ∈
            cylinder (E := fun (n : ℕ) ↦ ℕ) (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).2.1 (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).1 ∧
          (↑q : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).2.2 ∈
            cylinder (E := fun (n : ℕ) ↦ ℕ) (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).2.2 (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).1
choose :
  ∀ (F : (p : A) → A) (hFn : ∀ (p : A), (↑(F p) : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).1 = (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).1 + 1)
    (hFx :
      ∀ (p : A),
        (↑(F p) : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).2.1 ∈
          cylinder (E := fun (n : ℕ) ↦ ℕ) (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).2.1 (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).1)
    (hFy :
      ∀ (p : A),
        (↑(F p) : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).2.2 ∈
          cylinder (E := fun (n : ℕ) ↦ ℕ) (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).2.2 (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).1),
    False
⊢ False

Tactic:
choose F hFn hFx hFy using this
-- Module: Mathlib.MeasureTheory.Constructions.Polish.Basic
Goal:
α : Type u_1
inst : TopologicalSpace α
inst_1 : T2Space α
inst_2 : MeasurableSpace α
inst_3 : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous[_, inst] f
hg : Continuous[_, inst] g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
I :
  ∀ (n : ℕ) (x y : (n : ℕ) → ℕ),
    ¬MeasurablySeparable (f '' cylinder (E := fun (a : ℕ) ↦ ℕ) x n) (g '' cylinder (E := fun (a : ℕ) ↦ ℕ) y n) →
      ∃ (x' : (n : ℕ) → ℕ) (y' : (n : ℕ) → ℕ),
        x' ∈ cylinder (E := fun (n : ℕ) ↦ ℕ) x n ∧
          y' ∈ cylinder (E := fun (n : ℕ) ↦ ℕ) y n ∧
            ¬MeasurablySeparable (f '' cylinder (E := fun (a : ℕ) ↦ ℕ) x' (n + 1))
                (g '' cylinder (E := fun (a : ℕ) ↦ ℕ) y' (n + 1))
A : Type :=
  { p : ℕ × (ℕ → ℕ) × (ℕ → ℕ) //
    ¬MeasurablySeparable (f '' cylinder (E := fun (a : ℕ) ↦ ℕ) p.2.1 p.1)
        (g '' cylinder (E := fun (a : ℕ) ↦ ℕ) p.2.2 p.1) }
F : (p : A) → A
hFn : ∀ (p : A), (↑(F p) : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).1 = (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).1 + 1
hFx :
  ∀ (p : A),
    (↑(F p) : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).2.1 ∈
      cylinder (E := fun (n : ℕ) ↦ ℕ) (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).2.1 (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).1
hFy :
  ∀ (p : A),
    (↑(F p) : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).2.2 ∈
      cylinder (E := fun (n : ℕ) ↦ ℕ) (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).2.2 (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).1
⊢ ¬MeasurablySeparable
      (f ''
        cylinder (E := fun (a : ℕ) ↦ ℕ) (0, fun (x : ℕ) ↦ 0, fun (x : ℕ) ↦ 0).2.1
          (0, fun (x : ℕ) ↦ 0, fun (x : ℕ) ↦ 0).1)
      (g ''
        cylinder (E := fun (a : ℕ) ↦ ℕ) (0, fun (x : ℕ) ↦ 0, fun (x : ℕ) ↦ 0).2.2
          (0, fun (x : ℕ) ↦ 0, fun (x : ℕ) ↦ 0).1)

Tactic:
simp [hfg]
-- Module: Mathlib.MeasureTheory.Constructions.Polish.Basic
Goal:
α : Type u_1
inst : TopologicalSpace α
inst_1 : T2Space α
inst_2 : MeasurableSpace α
inst_3 : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous[_, inst] f
hg : Continuous[_, inst] g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
I :
  ∀ (n : ℕ) (x y : (n : ℕ) → ℕ),
    ¬MeasurablySeparable (f '' cylinder (E := fun (a : ℕ) ↦ ℕ) x n) (g '' cylinder (E := fun (a : ℕ) ↦ ℕ) y n) →
      ∃ (x' : (n : ℕ) → ℕ) (y' : (n : ℕ) → ℕ),
        x' ∈ cylinder (E := fun (n : ℕ) ↦ ℕ) x n ∧
          y' ∈ cylinder (E := fun (n : ℕ) ↦ ℕ) y n ∧
            ¬MeasurablySeparable (f '' cylinder (E := fun (a : ℕ) ↦ ℕ) x' (n + 1))
                (g '' cylinder (E := fun (a : ℕ) ↦ ℕ) y' (n + 1))
A : Type :=
  { p : ℕ × (ℕ → ℕ) × (ℕ → ℕ) //
    ¬MeasurablySeparable (f '' cylinder (E := fun (a : ℕ) ↦ ℕ) p.2.1 p.1)
        (g '' cylinder (E := fun (a : ℕ) ↦ ℕ) p.2.2 p.1) }
F : (p : A) → A
hFn : ∀ (p : A), (↑(F p) : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).1 = (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).1 + 1
hFx :
  ∀ (p : A),
    (↑(F p) : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).2.1 ∈
      cylinder (E := fun (n : ℕ) ↦ ℕ) (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).2.1 (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).1
hFy :
  ∀ (p : A),
    (↑(F p) : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).2.2 ∈
      cylinder (E := fun (n : ℕ) ↦ ℕ) (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).2.2 (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).1
p0 : A :=
  Subtype.mk (p := fun (p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)) ↦
    ¬MeasurablySeparable (f '' cylinder (E := fun (a : ℕ) ↦ ℕ) p.2.1 p.1)
        (g '' cylinder (E := fun (a : ℕ) ↦ ℕ) p.2.2 p.1))
    (0, fun (x : ℕ) ↦ 0, fun (x : ℕ) ↦ 0) ⋯
p : ℕ → A := fun (n : ℕ) ↦ F^[n] p0
n : ℕ
⊢ p (n + 1) = F (p n)

Tactic:
simp only [p, iterate_succ', Function.comp]

Premises used: Function.iterate_succ', Function.comp
-- Module: Mathlib.MeasureTheory.Constructions.Polish.Basic
Goal:
α : Type u_1
inst : TopologicalSpace α
inst_1 : T2Space α
inst_2 : MeasurableSpace α
inst_3 : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous[_, inst] f
hg : Continuous[_, inst] g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
I :
  ∀ (n : ℕ) (x y : (n : ℕ) → ℕ),
    ¬MeasurablySeparable (f '' cylinder (E := fun (a : ℕ) ↦ ℕ) x n) (g '' cylinder (E := fun (a : ℕ) ↦ ℕ) y n) →
      ∃ (x' : (n : ℕ) → ℕ) (y' : (n : ℕ) → ℕ),
        x' ∈ cylinder (E := fun (n : ℕ) ↦ ℕ) x n ∧
          y' ∈ cylinder (E := fun (n : ℕ) ↦ ℕ) y n ∧
            ¬MeasurablySeparable (f '' cylinder (E := fun (a : ℕ) ↦ ℕ) x' (n + 1))
                (g '' cylinder (E := fun (a : ℕ) ↦ ℕ) y' (n + 1))
A : Type :=
  { p : ℕ × (ℕ → ℕ) × (ℕ → ℕ) //
    ¬MeasurablySeparable (f '' cylinder (E := fun (a : ℕ) ↦ ℕ) p.2.1 p.1)
        (g '' cylinder (E := fun (a : ℕ) ↦ ℕ) p.2.2 p.1) }
F : (p : A) → A
hFn : ∀ (p : A), (↑(F p) : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).1 = (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).1 + 1
hFx :
  ∀ (p : A),
    (↑(F p) : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).2.1 ∈
      cylinder (E := fun (n : ℕ) ↦ ℕ) (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).2.1 (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).1
hFy :
  ∀ (p : A),
    (↑(F p) : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).2.2 ∈
      cylinder (E := fun (n : ℕ) ↦ ℕ) (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).2.2 (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).1
p0 : A :=
  Subtype.mk (p := fun (p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)) ↦
    ¬MeasurablySeparable (f '' cylinder (E := fun (a : ℕ) ↦ ℕ) p.2.1 p.1)
        (g '' cylinder (E := fun (a : ℕ) ↦ ℕ) p.2.2 p.1))
    (0, fun (x : ℕ) ↦ 0, fun (x : ℕ) ↦ 0) ⋯
p : ℕ → A := fun (n : ℕ) ↦ F^[n] p0
⊢ ∀ (n : ℕ), p (n + 1) = F (p n)

Tactic:
have prec : ∀ n, p (n + 1) = F (p n) := fun n => by simp only [p, iterate_succ', Function.comp]
-- Module: Mathlib.MeasureTheory.Constructions.Polish.Basic
Goal:
α : Type u_1
inst : TopologicalSpace α
inst_1 : T2Space α
inst_2 : MeasurableSpace α
inst_3 : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous[_, inst] f
hg : Continuous[_, inst] g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
I :
  ∀ (n : ℕ) (x y : (n : ℕ) → ℕ),
    ¬MeasurablySeparable (f '' cylinder (E := fun (a : ℕ) ↦ ℕ) x n) (g '' cylinder (E := fun (a : ℕ) ↦ ℕ) y n) →
      ∃ (x' : (n : ℕ) → ℕ) (y' : (n : ℕ) → ℕ),
        x' ∈ cylinder (E := fun (n : ℕ) ↦ ℕ) x n ∧
          y' ∈ cylinder (E := fun (n : ℕ) ↦ ℕ) y n ∧
            ¬MeasurablySeparable (f '' cylinder (E := fun (a : ℕ) ↦ ℕ) x' (n + 1))
                (g '' cylinder (E := fun (a : ℕ) ↦ ℕ) y' (n + 1))
A : Type :=
  { p : ℕ × (ℕ → ℕ) × (ℕ → ℕ) //
    ¬MeasurablySeparable (f '' cylinder (E := fun (a : ℕ) ↦ ℕ) p.2.1 p.1)
        (g '' cylinder (E := fun (a : ℕ) ↦ ℕ) p.2.2 p.1) }
F : (p : A) → A
hFn : ∀ (p : A), (↑(F p) : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).1 = (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).1 + 1
hFx :
  ∀ (p : A),
    (↑(F p) : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).2.1 ∈
      cylinder (E := fun (n : ℕ) ↦ ℕ) (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).2.1 (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).1
hFy :
  ∀ (p : A),
    (↑(F p) : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).2.2 ∈
      cylinder (E := fun (n : ℕ) ↦ ℕ) (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).2.2 (↑p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).1
p0 : A :=
  Subtype.mk (p := fun (p : ℕ × (ℕ → ℕ) × (ℕ → ℕ)) ↦
    ¬MeasurablySeparable (f '' cylinder (E := fun (a : ℕ) ↦ ℕ) p.2.1 p.1)
        (g '' cylinder (E := fun (a : ℕ) ↦ ℕ) p.2.2 p.1))
    (0, fun (x : ℕ) ↦ 0, fun (x : ℕ) ↦ 0) ⋯
p : ℕ → A := fun (n : ℕ) ↦ F^[n] p0
prec : ∀ (n : ℕ), p (n + 1) = F (p n)
⊢ (↑(p 0) : ℕ × (ℕ → ℕ) × (ℕ → ℕ)).1 = 0

Tactic:
rfl