l3lab/ntp-mathlib huggingface dataset:
/-- Assume that a function `f` has a derivative at every point of a set `s`. Then one may cover `s`
with countably many closed sets `t n` on which `f` is well approximated by linear maps `A n`. -/
theorem exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAt [SecondCountableTopology F]
    (f : E → F) (s : Set E) (f' : E → E →L[ℝ] F) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x)
    (r : (E →L[ℝ] F) → ℝ≥0) (rpos : ∀ A, r A ≠ 0) :
    ∃ (t : ℕ → Set E) (A : ℕ → E →L[ℝ] F),
      (∀ n, IsClosed (t n)) ∧
        (s ⊆ ⋃ n, t n) ∧
          (∀ n, ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧
            (s.Nonempty → ∀ n, ∃ y ∈ s, A n = f' y) := by
  /- Choose countably many linear maps `f' z`. For every such map, if `f` has a derivative at `x`
    close enough to `f' z`, then `f y - f x` is well approximated by `f' z (y - x)` for `y` close
    enough to `x`, say on a ball of radius `r` (or even `u n` for some `n`, where `u` is a fixed
    sequence tending to `0`).
    Let `M n z` be the points where this happens. Then this set is relatively closed inside `s`,
    and moreover in every closed ball of radius `u n / 3` inside it the map is well approximated by
    `f' z`. Using countably many closed balls to split `M n z` into small diameter subsets
    `K n z p`, one obtains the desired sets `t q` after reindexing.
    -/
  -- exclude the trivial case where `s` is empty
  

Proof state:
E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
s✝ : Set E
f✝ : E → E
f'✝ : E → E →L[ℝ] E
inst✝ : SecondCountableTopology F
f : E → F
s : Set E
f' : E → E →L[ℝ] F
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
r : (E →L[ℝ] F) → ℝ≥0
rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0
⊢ ∃ t A,
    (∀ (n : ℕ), IsClosed (t n)) ∧
      s ⊆ ⋃ n, t n ∧
        (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧
          (Set.Nonempty s → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y)

Next tactic:
rcases eq_empty_or_nonempty s with (rfl | hs)
/-- Assume that a function `f` has a derivative at every point of a set `s`. Then one may cover `s`
with countably many closed sets `t n` on which `f` is well approximated by linear maps `A n`. -/
theorem exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAt [SecondCountableTopology F]
    (f : E → F) (s : Set E) (f' : E → E →L[ℝ] F) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x)
    (r : (E →L[ℝ] F) → ℝ≥0) (rpos : ∀ A, r A ≠ 0) :
    ∃ (t : ℕ → Set E) (A : ℕ → E →L[ℝ] F),
      (∀ n, IsClosed (t n)) ∧
        (s ⊆ ⋃ n, t n) ∧
          (∀ n, ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧
            (s.Nonempty → ∀ n, ∃ y ∈ s, A n = f' y) := by
  /- Choose countably many linear maps `f' z`. For every such map, if `f` has a derivative at `x`
    close enough to `f' z`, then `f y - f x` is well approximated by `f' z (y - x)` for `y` close
    enough to `x`, say on a ball of radius `r` (or even `u n` for some `n`, where `u` is a fixed
    sequence tending to `0`).
    Let `M n z` be the points where this happens. Then this set is relatively closed inside `s`,
    and moreover in every closed ball of radius `u n / 3` inside it the map is well approximated by
    `f' z`. Using countably many closed balls to split `M n z` into small diameter subsets
    `K n z p`, one obtains the desired sets `t q` after reindexing.
    -/
  -- exclude the trivial case where `s` is empty
  rcases eq_empty_or_nonempty s with (rfl | hs)
  · 

Proof state:
case inl
E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
s : Set E
f✝ : E → E
f'✝ : E → E →L[ℝ] E
inst✝ : SecondCountableTopology F
f : E → F
f' : E → E →L[ℝ] F
r : (E →L[ℝ] F) → ℝ≥0
rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0
hf' : ∀ x ∈ ∅, HasFDerivWithinAt f (f' x) ∅ x
⊢ ∃ t A,
    (∀ (n : ℕ), IsClosed (t n)) ∧
      ∅ ⊆ ⋃ n, t n ∧
        (∀ (n : ℕ), ApproximatesLinearOn f (A n) (∅ ∩ t n) (r (A n))) ∧
          (Set.Nonempty ∅ → ∀ (n : ℕ), ∃ y ∈ ∅, A n = f' y)

Next tactic:
refine' ⟨fun _ => ∅, fun _ => 0, _, _, _, _⟩
/-- Assume that a function `f` has a derivative at every point of a set `s`. Then one may cover `s`
with countably many closed sets `t n` on which `f` is well approximated by linear maps `A n`. -/
theorem exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAt [SecondCountableTopology F]
    (f : E → F) (s : Set E) (f' : E → E →L[ℝ] F) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x)
    (r : (E →L[ℝ] F) → ℝ≥0) (rpos : ∀ A, r A ≠ 0) :
    ∃ (t : ℕ → Set E) (A : ℕ → E →L[ℝ] F),
      (∀ n, IsClosed (t n)) ∧
        (s ⊆ ⋃ n, t n) ∧
          (∀ n, ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧
            (s.Nonempty → ∀ n, ∃ y ∈ s, A n = f' y) := by
  /- Choose countably many linear maps `f' z`. For every such map, if `f` has a derivative at `x`
    close enough to `f' z`, then `f y - f x` is well approximated by `f' z (y - x)` for `y` close
    enough to `x`, say on a ball of radius `r` (or even `u n` for some `n`, where `u` is a fixed
    sequence tending to `0`).
    Let `M n z` be the points where this happens. Then this set is relatively closed inside `s`,
    and moreover in every closed ball of radius `u n / 3` inside it the map is well approximated by
    `f' z`. Using countably many closed balls to split `M n z` into small diameter subsets
    `K n z p`, one obtains the desired sets `t q` after reindexing.
    -/
  -- exclude the trivial case where `s` is empty
  rcases eq_empty_or_nonempty s with (rfl | hs)
  · refine' ⟨fun _ => ∅, fun _ => 0, _, _, _, _⟩ <;> 

Proof state:
case inl.refine'_1
E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
s : Set E
f✝ : E → E
f'✝ : E → E →L[ℝ] E
inst✝ : SecondCountableTopology F
f : E → F
f' : E → E →L[ℝ] F
r : (E →L[ℝ] F) → ℝ≥0
rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0
hf' : ∀ x ∈ ∅, HasFDerivWithinAt f (f' x) ∅ x
⊢ ∀ (n : ℕ), IsClosed ((fun x => ∅) n)

Next tactic:
simp
/-- Assume that a function `f` has a derivative at every point of a set `s`. Then one may cover `s`
with countably many closed sets `t n` on which `f` is well approximated by linear maps `A n`. -/
theorem exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAt [SecondCountableTopology F]
    (f : E → F) (s : Set E) (f' : E → E →L[ℝ] F) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x)
    (r : (E →L[ℝ] F) → ℝ≥0) (rpos : ∀ A, r A ≠ 0) :
    ∃ (t : ℕ → Set E) (A : ℕ → E →L[ℝ] F),
      (∀ n, IsClosed (t n)) ∧
        (s ⊆ ⋃ n, t n) ∧
          (∀ n, ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧
            (s.Nonempty → ∀ n, ∃ y ∈ s, A n = f' y) := by
  /- Choose countably many linear maps `f' z`. For every such map, if `f` has a derivative at `x`
    close enough to `f' z`, then `f y - f x` is well approximated by `f' z (y - x)` for `y` close
    enough to `x`, say on a ball of radius `r` (or even `u n` for some `n`, where `u` is a fixed
    sequence tending to `0`).
    Let `M n z` be the points where this happens. Then this set is relatively closed inside `s`,
    and moreover in every closed ball of radius `u n / 3` inside it the map is well approximated by
    `f' z`. Using countably many closed balls to split `M n z` into small diameter subsets
    `K n z p`, one obtains the desired sets `t q` after reindexing.
    -/
  -- exclude the trivial case where `s` is empty
  rcases eq_empty_or_nonempty s with (rfl | hs)
  · refine' ⟨fun _ => ∅, fun _ => 0, _, _, _, _⟩ <;> 

Proof state:
case inl.refine'_2
E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
s : Set E
f✝ : E → E
f'✝ : E → E →L[ℝ] E
inst✝ : SecondCountableTopology F
f : E → F
f' : E → E →L[ℝ] F
r : (E →L[ℝ] F) → ℝ≥0
rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0
hf' : ∀ x ∈ ∅, HasFDerivWithinAt f (f' x) ∅ x
⊢ ∅ ⊆ ⋃ n, (fun x => ∅) n

Next tactic:
simp
/-- Assume that a function `f` has a derivative at every point of a set `s`. Then one may cover `s`
with countably many closed sets `t n` on which `f` is well approximated by linear maps `A n`. -/
theorem exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAt [SecondCountableTopology F]
    (f : E → F) (s : Set E) (f' : E → E →L[ℝ] F) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x)
    (r : (E →L[ℝ] F) → ℝ≥0) (rpos : ∀ A, r A ≠ 0) :
    ∃ (t : ℕ → Set E) (A : ℕ → E →L[ℝ] F),
      (∀ n, IsClosed (t n)) ∧
        (s ⊆ ⋃ n, t n) ∧
          (∀ n, ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧
            (s.Nonempty → ∀ n, ∃ y ∈ s, A n = f' y) := by
  /- Choose countably many linear maps `f' z`. For every such map, if `f` has a derivative at `x`
    close enough to `f' z`, then `f y - f x` is well approximated by `f' z (y - x)` for `y` close
    enough to `x`, say on a ball of radius `r` (or even `u n` for some `n`, where `u` is a fixed
    sequence tending to `0`).
    Let `M n z` be the points where this happens. Then this set is relatively closed inside `s`,
    and moreover in every closed ball of radius `u n / 3` inside it the map is well approximated by
    `f' z`. Using countably many closed balls to split `M n z` into small diameter subsets
    `K n z p`, one obtains the desired sets `t q` after reindexing.
    -/
  -- exclude the trivial case where `s` is empty
  rcases eq_empty_or_nonempty s with (rfl | hs)
  · refine' ⟨fun _ => ∅, fun _ => 0, _, _, _, _⟩ <;> 

Proof state:
case inl.refine'_3
E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
s : Set E
f✝ : E → E
f'✝ : E → E →L[ℝ] E
inst✝ : SecondCountableTopology F
f : E → F
f' : E → E →L[ℝ] F
r : (E →L[ℝ] F) → ℝ≥0
rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0
hf' : ∀ x ∈ ∅, HasFDerivWithinAt f (f' x) ∅ x
⊢ ∀ (n : ℕ), ApproximatesLinearOn f ((fun x => 0) n) (∅ ∩ (fun x => ∅) n) (r ((fun x => 0) n))

Next tactic:
simp
/-- Assume that a function `f` has a derivative at every point of a set `s`. Then one may cover `s`
with countably many closed sets `t n` on which `f` is well approximated by linear maps `A n`. -/
theorem exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAt [SecondCountableTopology F]
    (f : E → F) (s : Set E) (f' : E → E →L[ℝ] F) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x)
    (r : (E →L[ℝ] F) → ℝ≥0) (rpos : ∀ A, r A ≠ 0) :
    ∃ (t : ℕ → Set E) (A : ℕ → E →L[ℝ] F),
      (∀ n, IsClosed (t n)) ∧
        (s ⊆ ⋃ n, t n) ∧
          (∀ n, ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧
            (s.Nonempty → ∀ n, ∃ y ∈ s, A n = f' y) := by
  /- Choose countably many linear maps `f' z`. For every such map, if `f` has a derivative at `x`
    close enough to `f' z`, then `f y - f x` is well approximated by `f' z (y - x)` for `y` close
    enough to `x`, say on a ball of radius `r` (or even `u n` for some `n`, where `u` is a fixed
    sequence tending to `0`).
    Let `M n z` be the points where this happens. Then this set is relatively closed inside `s`,
    and moreover in every closed ball of radius `u n / 3` inside it the map is well approximated by
    `f' z`. Using countably many closed balls to split `M n z` into small diameter subsets
    `K n z p`, one obtains the desired sets `t q` after reindexing.
    -/
  -- exclude the trivial case where `s` is empty
  rcases eq_empty_or_nonempty s with (rfl | hs)
  · refine' ⟨fun _ => ∅, fun _ => 0, _, _, _, _⟩ <;> 

Proof state:
case inl.refine'_4
E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
s : Set E
f✝ : E → E
f'✝ : E → E →L[ℝ] E
inst✝ : SecondCountableTopology F
f : E → F
f' : E → E →L[ℝ] F
r : (E →L[ℝ] F) → ℝ≥0
rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0
hf' : ∀ x ∈ ∅, HasFDerivWithinAt f (f' x) ∅ x
⊢ Set.Nonempty ∅ → ∀ (n : ℕ), ∃ y ∈ ∅, (fun x => 0) n = f' y

Next tactic:
simp
/-- Assume that a function `f` has a derivative at every point of a set `s`. Then one may cover `s`
with countably many closed sets `t n` on which `f` is well approximated by linear maps `A n`. -/
theorem exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAt [SecondCountableTopology F]
    (f : E → F) (s : Set E) (f' : E → E →L[ℝ] F) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x)
    (r : (E →L[ℝ] F) → ℝ≥0) (rpos : ∀ A, r A ≠ 0) :
    ∃ (t : ℕ → Set E) (A : ℕ → E →L[ℝ] F),
      (∀ n, IsClosed (t n)) ∧
        (s ⊆ ⋃ n, t n) ∧
          (∀ n, ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧
            (s.Nonempty → ∀ n, ∃ y ∈ s, A n = f' y) := by
  /- Choose countably many linear maps `f' z`. For every such map, if `f` has a derivative at `x`
    close enough to `f' z`, then `f y - f x` is well approximated by `f' z (y - x)` for `y` close
    enough to `x`, say on a ball of radius `r` (or even `u n` for some `n`, where `u` is a fixed
    sequence tending to `0`).
    Let `M n z` be the points where this happens. Then this set is relatively closed inside `s`,
    and moreover in every closed ball of radius `u n / 3` inside it the map is well approximated by
    `f' z`. Using countably many closed balls to split `M n z` into small diameter subsets
    `K n z p`, one obtains the desired sets `t q` after reindexing.
    -/
  -- exclude the trivial case where `s` is empty
  rcases eq_empty_or_nonempty s with (rfl | hs)
  · refine' ⟨fun _ => ∅, fun _ => 0, _, _, _, _⟩ <;> simp
  -- we will use countably many linear maps. Select these from all the derivatives since the
  -- space of linear maps is second-countable
  

Proof state:
case inr
E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
s✝ : Set E
f✝ : E → E
f'✝ : E → E →L[ℝ] E
inst✝ : SecondCountableTopology F
f : E → F
s : Set E
f' : E → E →L[ℝ] F
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
r : (E →L[ℝ] F) → ℝ≥0
rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0
hs : Set.Nonempty s
⊢ ∃ t A,
    (∀ (n : ℕ), IsClosed (t n)) ∧
      s ⊆ ⋃ n, t n ∧
        (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧
          (Set.Nonempty s → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y)

Next tactic:
obtain ⟨T, T_count, hT⟩ :
    ∃ T : Set s,
      T.Countable ∧ ⋃ x ∈ T, ball (f' (x : E)) (r (f' x)) = ⋃ x : s, ball (f' x) (r (f' x)) :=
    TopologicalSpace.isOpen_iUnion_countable _ fun x => isOpen_ball
/-- Assume that a function `f` has a derivative at every point of a set `s`. Then one may cover `s`
with countably many closed sets `t n` on which `f` is well approximated by linear maps `A n`. -/
theorem exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAt [SecondCountableTopology F]
    (f : E → F) (s : Set E) (f' : E → E →L[ℝ] F) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x)
    (r : (E →L[ℝ] F) → ℝ≥0) (rpos : ∀ A, r A ≠ 0) :
    ∃ (t : ℕ → Set E) (A : ℕ → E →L[ℝ] F),
      (∀ n, IsClosed (t n)) ∧
        (s ⊆ ⋃ n, t n) ∧
          (∀ n, ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧
            (s.Nonempty → ∀ n, ∃ y ∈ s, A n = f' y) := by
  /- Choose countably many linear maps `f' z`. For every such map, if `f` has a derivative at `x`
    close enough to `f' z`, then `f y - f x` is well approximated by `f' z (y - x)` for `y` close
    enough to `x`, say on a ball of radius `r` (or even `u n` for some `n`, where `u` is a fixed
    sequence tending to `0`).
    Let `M n z` be the points where this happens. Then this set is relatively closed inside `s`,
    and moreover in every closed ball of radius `u n / 3` inside it the map is well approximated by
    `f' z`. Using countably many closed balls to split `M n z` into small diameter subsets
    `K n z p`, one obtains the desired sets `t q` after reindexing.
    -/
  -- exclude the trivial case where `s` is empty
  rcases eq_empty_or_nonempty s with (rfl | hs)
  · refine' ⟨fun _ => ∅, fun _ => 0, _, _, _, _⟩ <;> simp
  -- we will use countably many linear maps. Select these from all the derivatives since the
  -- space of linear maps is second-countable
  obtain ⟨T, T_count, hT⟩ :
    ∃ T : Set s,
      T.Countable ∧ ⋃ x ∈ T, ball (f' (x : E)) (r (f' x)) = ⋃ x : s, ball (f' x) (r (f' x)) :=
    TopologicalSpace.isOpen_iUnion_countable _ fun x => isOpen_ball
  -- fix a sequence `u` of positive reals tending to zero.
  

Proof state:
case inr.intro.intro
E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
s✝ : Set E
f✝ : E → E
f'✝ : E → E →L[ℝ] E
inst✝ : SecondCountableTopology F
f : E → F
s : Set E
f' : E → E →L[ℝ] F
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
r : (E →L[ℝ] F) → ℝ≥0
rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0
hs : Set.Nonempty s
T : Set ↑s
T_count : Set.Countable T
hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x))
⊢ ∃ t A,
    (∀ (n : ℕ), IsClosed (t n)) ∧
      s ⊆ ⋃ n, t n ∧
        (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧
          (Set.Nonempty s → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y)

Next tactic:
obtain ⟨u, _, u_pos, u_lim⟩ :
    ∃ u : ℕ → ℝ, StrictAnti u ∧ (∀ n : ℕ, 0 < u n) ∧ Tendsto u atTop (𝓝 0) :=
    exists_seq_strictAnti_tendsto (0 : ℝ)
/-- Assume that a function `f` has a derivative at every point of a set `s`. Then one may
partition `s` into countably many disjoint relatively measurable sets (i.e., intersections
of `s` with measurable sets `t n`) on which `f` is well approximated by linear maps `A n`. -/
theorem exists_partition_approximatesLinearOn_of_hasFDerivWithinAt [SecondCountableTopology F]
    (f : E → F) (s : Set E) (f' : E → E →L[ℝ] F) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x)
    (r : (E →L[ℝ] F) → ℝ≥0) (rpos : ∀ A, r A ≠ 0) :
    ∃ (t : ℕ → Set E) (A : ℕ → E →L[ℝ] F),
      Pairwise (Disjoint on t) ∧
        (∀ n, MeasurableSet (t n)) ∧
          (s ⊆ ⋃ n, t n) ∧
            (∀ n, ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧
              (s.Nonempty → ∀ n, ∃ y ∈ s, A n = f' y) := by
  

Proof state:
E : Type u_1
F : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
s✝ : Set E
f✝ : E → E
f'✝ : E → E →L[ℝ] E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
μ : Measure E
inst✝¹ : IsAddHaarMeasure μ
inst✝ : SecondCountableTopology F
f : E → F
s : Set E
f' : E → E →L[ℝ] F
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
r : (E →L[ℝ] F) → ℝ≥0
rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0
⊢ ∃ t A,
    Pairwise (Disjoint on t) ∧
      (∀ (n : ℕ), MeasurableSet (t n)) ∧
        s ⊆ ⋃ n, t n ∧
          (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧
            (Set.Nonempty s → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y)

Next tactic:
rcases exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAt f s f' hf' r rpos with
    ⟨t, A, t_closed, st, t_approx, ht⟩
/-- Assume that a function `f` has a derivative at every point of a set `s`. Then one may
partition `s` into countably many disjoint relatively measurable sets (i.e., intersections
of `s` with measurable sets `t n`) on which `f` is well approximated by linear maps `A n`. -/
theorem exists_partition_approximatesLinearOn_of_hasFDerivWithinAt [SecondCountableTopology F]
    (f : E → F) (s : Set E) (f' : E → E →L[ℝ] F) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x)
    (r : (E →L[ℝ] F) → ℝ≥0) (rpos : ∀ A, r A ≠ 0) :
    ∃ (t : ℕ → Set E) (A : ℕ → E →L[ℝ] F),
      Pairwise (Disjoint on t) ∧
        (∀ n, MeasurableSet (t n)) ∧
          (s ⊆ ⋃ n, t n) ∧
            (∀ n, ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧
              (s.Nonempty → ∀ n, ∃ y ∈ s, A n = f' y) := by
  rcases exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAt f s f' hf' r rpos with
    ⟨t, A, t_closed, st, t_approx, ht⟩
  

Proof state:
case intro.intro.intro.intro.intro
E : Type u_1
F : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
s✝ : Set E
f✝ : E → E
f'✝ : E → E →L[ℝ] E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
μ : Measure E
inst✝¹ : IsAddHaarMeasure μ
inst✝ : SecondCountableTopology F
f : E → F
s : Set E
f' : E → E →L[ℝ] F
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
r : (E →L[ℝ] F) → ℝ≥0
rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0
t : ℕ → Set E
A : ℕ → E →L[ℝ] F
t_closed : ∀ (n : ℕ), IsClosed (t n)
st : s ⊆ ⋃ n, t n
t_approx : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))
ht : Set.Nonempty s → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y
⊢ ∃ t A,
    Pairwise (Disjoint on t) ∧
      (∀ (n : ℕ), MeasurableSet (t n)) ∧
        s ⊆ ⋃ n, t n ∧
          (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧
            (Set.Nonempty s → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y)

Next tactic:
refine'
    ⟨disjointed t, A, disjoint_disjointed _,
      MeasurableSet.disjointed fun n => (t_closed n).measurableSet, _, _, ht⟩
/-- Assume that a function `f` has a derivative at every point of a set `s`. Then one may
partition `s` into countably many disjoint relatively measurable sets (i.e., intersections
of `s` with measurable sets `t n`) on which `f` is well approximated by linear maps `A n`. -/
theorem exists_partition_approximatesLinearOn_of_hasFDerivWithinAt [SecondCountableTopology F]
    (f : E → F) (s : Set E) (f' : E → E →L[ℝ] F) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x)
    (r : (E →L[ℝ] F) → ℝ≥0) (rpos : ∀ A, r A ≠ 0) :
    ∃ (t : ℕ → Set E) (A : ℕ → E →L[ℝ] F),
      Pairwise (Disjoint on t) ∧
        (∀ n, MeasurableSet (t n)) ∧
          (s ⊆ ⋃ n, t n) ∧
            (∀ n, ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧
              (s.Nonempty → ∀ n, ∃ y ∈ s, A n = f' y) := by
  rcases exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAt f s f' hf' r rpos with
    ⟨t, A, t_closed, st, t_approx, ht⟩
  refine'
    ⟨disjointed t, A, disjoint_disjointed _,
      MeasurableSet.disjointed fun n => (t_closed n).measurableSet, _, _, ht⟩
  · 

Proof state:
case intro.intro.intro.intro.intro.refine'_1
E : Type u_1
F : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
s✝ : Set E
f✝ : E → E
f'✝ : E → E →L[ℝ] E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
μ : Measure E
inst✝¹ : IsAddHaarMeasure μ
inst✝ : SecondCountableTopology F
f : E → F
s : Set E
f' : E → E →L[ℝ] F
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
r : (E →L[ℝ] F) → ℝ≥0
rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0
t : ℕ → Set E
A : ℕ → E →L[ℝ] F
t_closed : ∀ (n : ℕ), IsClosed (t n)
st : s ⊆ ⋃ n, t n
t_approx : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))
ht : Set.Nonempty s → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y
⊢ s ⊆ ⋃ n, disjointed t n

Next tactic:
rw [iUnion_disjointed]