l3lab/ntp-mathlib huggingface dataset:
/-- Consider a series of functions `∑' i, f i x`. Assume that each individual function `f i` is of
class `C^N`, and moreover there is a uniform summable upper bound on the `k`-th derivative
for each `k ≤ N` (except maybe for finitely many `i`s). Then the series is also `C^N`. -/
theorem contDiff_tsum_of_eventually (hf : ∀ i, ContDiff 𝕜 N (f i))
    (hv : ∀ k : ℕ, (k : ℕ∞) ≤ N → Summable (v k))
    (h'f :
      ∀ k : ℕ,
        (k : ℕ∞) ≤ N →
          ∀ᶠ i in (Filter.cofinite : Filter α), ∀ x : E, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i) :
    ContDiff 𝕜 N fun x => ∑' i, f i x := by
  classical
    refine contDiff_iff_forall_nat_le.2 fun m hm => ?_
    let t : Set α :=
      { i : α | ¬∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i }
    have ht : Set.Finite t :=
      haveI A :
        ∀ᶠ i in (Filter.cofinite : Filter α),
          ∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x : E, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i := by
        

Proof state:
α : Type u_1
β : Type u_2
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
v : ℕ → α → ℝ
s : Set E
x₀ x : E
N : ℕ∞
hf : ∀ (i : α), ContDiff 𝕜 N (f i)
hv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)
h'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i
m : ℕ
hm : ↑m ≤ N
t : Set α := {i | ¬∀ k ∈ Finset.range (m + 1), ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}
⊢ ∀ᶠ (i : α) in cofinite, ∀ k ∈ Finset.range (m + 1), ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i

Next tactic:
rw [eventually_all_finset]
/-- Consider a series of functions `∑' i, f i x`. Assume that each individual function `f i` is of
class `C^N`, and moreover there is a uniform summable upper bound on the `k`-th derivative
for each `k ≤ N` (except maybe for finitely many `i`s). Then the series is also `C^N`. -/
theorem contDiff_tsum_of_eventually (hf : ∀ i, ContDiff 𝕜 N (f i))
    (hv : ∀ k : ℕ, (k : ℕ∞) ≤ N → Summable (v k))
    (h'f :
      ∀ k : ℕ,
        (k : ℕ∞) ≤ N →
          ∀ᶠ i in (Filter.cofinite : Filter α), ∀ x : E, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i) :
    ContDiff 𝕜 N fun x => ∑' i, f i x := by
  classical
    refine contDiff_iff_forall_nat_le.2 fun m hm => ?_
    let t : Set α :=
      { i : α | ¬∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i }
    have ht : Set.Finite t :=
      haveI A :
        ∀ᶠ i in (Filter.cofinite : Filter α),
          ∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x : E, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i := by
        rw [eventually_all_finset]
        

Proof state:
α : Type u_1
β : Type u_2
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
v : ℕ → α → ℝ
s : Set E
x₀ x : E
N : ℕ∞
hf : ∀ (i : α), ContDiff 𝕜 N (f i)
hv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)
h'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i
m : ℕ
hm : ↑m ≤ N
t : Set α := {i | ¬∀ k ∈ Finset.range (m + 1), ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}
⊢ ∀ i ∈ Finset.range (m + 1), ∀ᶠ (x : α) in cofinite, ∀ (x_1 : E), ‖iteratedFDeriv 𝕜 i (f x) x_1‖ ≤ v i x

Next tactic:
intro i hi
/-- Consider a series of functions `∑' i, f i x`. Assume that each individual function `f i` is of
class `C^N`, and moreover there is a uniform summable upper bound on the `k`-th derivative
for each `k ≤ N` (except maybe for finitely many `i`s). Then the series is also `C^N`. -/
theorem contDiff_tsum_of_eventually (hf : ∀ i, ContDiff 𝕜 N (f i))
    (hv : ∀ k : ℕ, (k : ℕ∞) ≤ N → Summable (v k))
    (h'f :
      ∀ k : ℕ,
        (k : ℕ∞) ≤ N →
          ∀ᶠ i in (Filter.cofinite : Filter α), ∀ x : E, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i) :
    ContDiff 𝕜 N fun x => ∑' i, f i x := by
  classical
    refine contDiff_iff_forall_nat_le.2 fun m hm => ?_
    let t : Set α :=
      { i : α | ¬∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i }
    have ht : Set.Finite t :=
      haveI A :
        ∀ᶠ i in (Filter.cofinite : Filter α),
          ∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x : E, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i := by
        rw [eventually_all_finset]
        intro i hi
        

Proof state:
α : Type u_1
β : Type u_2
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
v : ℕ → α → ℝ
s : Set E
x₀ x : E
N : ℕ∞
hf : ∀ (i : α), ContDiff 𝕜 N (f i)
hv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)
h'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i
m : ℕ
hm : ↑m ≤ N
t : Set α := {i | ¬∀ k ∈ Finset.range (m + 1), ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}
i : ℕ
hi : i ∈ Finset.range (m + 1)
⊢ ∀ᶠ (x : α) in cofinite, ∀ (x_1 : E), ‖iteratedFDeriv 𝕜 i (f x) x_1‖ ≤ v i x

Next tactic:
apply h'f
/-- Consider a series of functions `∑' i, f i x`. Assume that each individual function `f i` is of
class `C^N`, and moreover there is a uniform summable upper bound on the `k`-th derivative
for each `k ≤ N` (except maybe for finitely many `i`s). Then the series is also `C^N`. -/
theorem contDiff_tsum_of_eventually (hf : ∀ i, ContDiff 𝕜 N (f i))
    (hv : ∀ k : ℕ, (k : ℕ∞) ≤ N → Summable (v k))
    (h'f :
      ∀ k : ℕ,
        (k : ℕ∞) ≤ N →
          ∀ᶠ i in (Filter.cofinite : Filter α), ∀ x : E, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i) :
    ContDiff 𝕜 N fun x => ∑' i, f i x := by
  classical
    refine contDiff_iff_forall_nat_le.2 fun m hm => ?_
    let t : Set α :=
      { i : α | ¬∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i }
    have ht : Set.Finite t :=
      haveI A :
        ∀ᶠ i in (Filter.cofinite : Filter α),
          ∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x : E, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i := by
        rw [eventually_all_finset]
        intro i hi
        apply h'f
        

Proof state:
case a
α : Type u_1
β : Type u_2
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
v : ℕ → α → ℝ
s : Set E
x₀ x : E
N : ℕ∞
hf : ∀ (i : α), ContDiff 𝕜 N (f i)
hv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)
h'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i
m : ℕ
hm : ↑m ≤ N
t : Set α := {i | ¬∀ k ∈ Finset.range (m + 1), ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}
i : ℕ
hi : i ∈ Finset.range (m + 1)
⊢ ↑i ≤ N

Next tactic:
simp only [Finset.mem_range_succ_iff] at hi
/-- Consider a series of functions `∑' i, f i x`. Assume that each individual function `f i` is of
class `C^N`, and moreover there is a uniform summable upper bound on the `k`-th derivative
for each `k ≤ N` (except maybe for finitely many `i`s). Then the series is also `C^N`. -/
theorem contDiff_tsum_of_eventually (hf : ∀ i, ContDiff 𝕜 N (f i))
    (hv : ∀ k : ℕ, (k : ℕ∞) ≤ N → Summable (v k))
    (h'f :
      ∀ k : ℕ,
        (k : ℕ∞) ≤ N →
          ∀ᶠ i in (Filter.cofinite : Filter α), ∀ x : E, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i) :
    ContDiff 𝕜 N fun x => ∑' i, f i x := by
  classical
    refine contDiff_iff_forall_nat_le.2 fun m hm => ?_
    let t : Set α :=
      { i : α | ¬∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i }
    have ht : Set.Finite t :=
      haveI A :
        ∀ᶠ i in (Filter.cofinite : Filter α),
          ∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x : E, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i := by
        rw [eventually_all_finset]
        intro i hi
        apply h'f
        simp only [Finset.mem_range_succ_iff] at hi
        

Proof state:
case a
α : Type u_1
β : Type u_2
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
v : ℕ → α → ℝ
s : Set E
x₀ x : E
N : ℕ∞
hf : ∀ (i : α), ContDiff 𝕜 N (f i)
hv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)
h'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i
m : ℕ
hm : ↑m ≤ N
t : Set α := {i | ¬∀ k ∈ Finset.range (m + 1), ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}
i : ℕ
hi : i ≤ m
⊢ ↑i ≤ N

Next tactic:
exact (WithTop.coe_le_coe.2 hi).trans hm
/-- Consider a series of functions `∑' i, f i x`. Assume that each individual function `f i` is of
class `C^N`, and moreover there is a uniform summable upper bound on the `k`-th derivative
for each `k ≤ N` (except maybe for finitely many `i`s). Then the series is also `C^N`. -/
theorem contDiff_tsum_of_eventually (hf : ∀ i, ContDiff 𝕜 N (f i))
    (hv : ∀ k : ℕ, (k : ℕ∞) ≤ N → Summable (v k))
    (h'f :
      ∀ k : ℕ,
        (k : ℕ∞) ≤ N →
          ∀ᶠ i in (Filter.cofinite : Filter α), ∀ x : E, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i) :
    ContDiff 𝕜 N fun x => ∑' i, f i x := by
  classical
    refine contDiff_iff_forall_nat_le.2 fun m hm => ?_
    let t : Set α :=
      { i : α | ¬∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i }
    have ht : Set.Finite t :=
      haveI A :
        ∀ᶠ i in (Filter.cofinite : Filter α),
          ∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x : E, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i := by
        rw [eventually_all_finset]
        intro i hi
        apply h'f
        simp only [Finset.mem_range_succ_iff] at hi
        exact (WithTop.coe_le_coe.2 hi).trans hm
      eventually_cofinite.2 A
    

Proof state:
α : Type u_1
β : Type u_2
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
v : ℕ → α → ℝ
s : Set E
x₀ x : E
N : ℕ∞
hf : ∀ (i : α), ContDiff 𝕜 N (f i)
hv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)
h'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i
m : ℕ
hm : ↑m ≤ N
t : Set α := {i | ¬∀ k ∈ Finset.range (m + 1), ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}
ht : Set.Finite t
⊢ ContDiff 𝕜 ↑m fun x => ∑' (i : α), f i x

Next tactic:
let T : Finset α := ht.toFinset
/-- Consider a series of functions `∑' i, f i x`. Assume that each individual function `f i` is of
class `C^N`, and moreover there is a uniform summable upper bound on the `k`-th derivative
for each `k ≤ N` (except maybe for finitely many `i`s). Then the series is also `C^N`. -/
theorem contDiff_tsum_of_eventually (hf : ∀ i, ContDiff 𝕜 N (f i))
    (hv : ∀ k : ℕ, (k : ℕ∞) ≤ N → Summable (v k))
    (h'f :
      ∀ k : ℕ,
        (k : ℕ∞) ≤ N →
          ∀ᶠ i in (Filter.cofinite : Filter α), ∀ x : E, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i) :
    ContDiff 𝕜 N fun x => ∑' i, f i x := by
  classical
    refine contDiff_iff_forall_nat_le.2 fun m hm => ?_
    let t : Set α :=
      { i : α | ¬∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i }
    have ht : Set.Finite t :=
      haveI A :
        ∀ᶠ i in (Filter.cofinite : Filter α),
          ∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x : E, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i := by
        rw [eventually_all_finset]
        intro i hi
        apply h'f
        simp only [Finset.mem_range_succ_iff] at hi
        exact (WithTop.coe_le_coe.2 hi).trans hm
      eventually_cofinite.2 A
    let T : Finset α := ht.toFinset
    

Proof state:
α : Type u_1
β : Type u_2
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
v : ℕ → α → ℝ
s : Set E
x₀ x : E
N : ℕ∞
hf : ∀ (i : α), ContDiff 𝕜 N (f i)
hv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)
h'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i
m : ℕ
hm : ↑m ≤ N
t : Set α := {i | ¬∀ k ∈ Finset.range (m + 1), ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}
ht : Set.Finite t
T : Finset α := Finite.toFinset ht
⊢ ContDiff 𝕜 ↑m fun x => ∑' (i : α), f i x

Next tactic:
have : (fun x => ∑' i, f i x) = (fun x => ∑ i in T, f i x) +
        fun x => ∑' i : { i // i ∉ T }, f i x := by
      ext1 x
      refine' (sum_add_tsum_subtype_compl _ T).symm
      refine' .of_norm_bounded_eventually _ (hv 0 (zero_le _)) _
      filter_upwards [h'f 0 (zero_le _)] with i hi
      simpa only [norm_iteratedFDeriv_zero] using hi x
/-- Consider a series of functions `∑' i, f i x`. Assume that each individual function `f i` is of
class `C^N`, and moreover there is a uniform summable upper bound on the `k`-th derivative
for each `k ≤ N` (except maybe for finitely many `i`s). Then the series is also `C^N`. -/
theorem contDiff_tsum_of_eventually (hf : ∀ i, ContDiff 𝕜 N (f i))
    (hv : ∀ k : ℕ, (k : ℕ∞) ≤ N → Summable (v k))
    (h'f :
      ∀ k : ℕ,
        (k : ℕ∞) ≤ N →
          ∀ᶠ i in (Filter.cofinite : Filter α), ∀ x : E, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i) :
    ContDiff 𝕜 N fun x => ∑' i, f i x := by
  classical
    refine contDiff_iff_forall_nat_le.2 fun m hm => ?_
    let t : Set α :=
      { i : α | ¬∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i }
    have ht : Set.Finite t :=
      haveI A :
        ∀ᶠ i in (Filter.cofinite : Filter α),
          ∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x : E, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i := by
        rw [eventually_all_finset]
        intro i hi
        apply h'f
        simp only [Finset.mem_range_succ_iff] at hi
        exact (WithTop.coe_le_coe.2 hi).trans hm
      eventually_cofinite.2 A
    let T : Finset α := ht.toFinset
    have : (fun x => ∑' i, f i x) = (fun x => ∑ i in T, f i x) +
        fun x => ∑' i : { i // i ∉ T }, f i x := by
      

Proof state:
α : Type u_1
β : Type u_2
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
v : ℕ → α → ℝ
s : Set E
x₀ x : E
N : ℕ∞
hf : ∀ (i : α), ContDiff 𝕜 N (f i)
hv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)
h'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i
m : ℕ
hm : ↑m ≤ N
t : Set α := {i | ¬∀ k ∈ Finset.range (m + 1), ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}
ht : Set.Finite t
T : Finset α := Finite.toFinset ht
⊢ (fun x => ∑' (i : α), f i x) = (fun x => ∑ i in T, f i x) + fun x => ∑' (i : { i // i ∉ T }), f (↑i) x

Next tactic:
ext1 x
/-- Consider a series of functions `∑' i, f i x`. Assume that each individual function `f i` is of
class `C^N`, and moreover there is a uniform summable upper bound on the `k`-th derivative
for each `k ≤ N` (except maybe for finitely many `i`s). Then the series is also `C^N`. -/
theorem contDiff_tsum_of_eventually (hf : ∀ i, ContDiff 𝕜 N (f i))
    (hv : ∀ k : ℕ, (k : ℕ∞) ≤ N → Summable (v k))
    (h'f :
      ∀ k : ℕ,
        (k : ℕ∞) ≤ N →
          ∀ᶠ i in (Filter.cofinite : Filter α), ∀ x : E, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i) :
    ContDiff 𝕜 N fun x => ∑' i, f i x := by
  classical
    refine contDiff_iff_forall_nat_le.2 fun m hm => ?_
    let t : Set α :=
      { i : α | ¬∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i }
    have ht : Set.Finite t :=
      haveI A :
        ∀ᶠ i in (Filter.cofinite : Filter α),
          ∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x : E, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i := by
        rw [eventually_all_finset]
        intro i hi
        apply h'f
        simp only [Finset.mem_range_succ_iff] at hi
        exact (WithTop.coe_le_coe.2 hi).trans hm
      eventually_cofinite.2 A
    let T : Finset α := ht.toFinset
    have : (fun x => ∑' i, f i x) = (fun x => ∑ i in T, f i x) +
        fun x => ∑' i : { i // i ∉ T }, f i x := by
      ext1 x
      

Proof state:
case h
α : Type u_1
β : Type u_2
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
v : ℕ → α → ℝ
s : Set E
x₀ x✝ : E
N : ℕ∞
hf : ∀ (i : α), ContDiff 𝕜 N (f i)
hv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)
h'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i
m : ℕ
hm : ↑m ≤ N
t : Set α := {i | ¬∀ k ∈ Finset.range (m + 1), ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}
ht : Set.Finite t
T : Finset α := Finite.toFinset ht
x : E
⊢ ∑' (i : α), f i x = ((fun x => ∑ i in T, f i x) + fun x => ∑' (i : { i // i ∉ T }), f (↑i) x) x

Next tactic:
refine' (sum_add_tsum_subtype_compl _ T).symm
/-- Consider a series of functions `∑' i, f i x`. Assume that each individual function `f i` is of
class `C^N`, and moreover there is a uniform summable upper bound on the `k`-th derivative
for each `k ≤ N` (except maybe for finitely many `i`s). Then the series is also `C^N`. -/
theorem contDiff_tsum_of_eventually (hf : ∀ i, ContDiff 𝕜 N (f i))
    (hv : ∀ k : ℕ, (k : ℕ∞) ≤ N → Summable (v k))
    (h'f :
      ∀ k : ℕ,
        (k : ℕ∞) ≤ N →
          ∀ᶠ i in (Filter.cofinite : Filter α), ∀ x : E, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i) :
    ContDiff 𝕜 N fun x => ∑' i, f i x := by
  classical
    refine contDiff_iff_forall_nat_le.2 fun m hm => ?_
    let t : Set α :=
      { i : α | ¬∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i }
    have ht : Set.Finite t :=
      haveI A :
        ∀ᶠ i in (Filter.cofinite : Filter α),
          ∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x : E, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i := by
        rw [eventually_all_finset]
        intro i hi
        apply h'f
        simp only [Finset.mem_range_succ_iff] at hi
        exact (WithTop.coe_le_coe.2 hi).trans hm
      eventually_cofinite.2 A
    let T : Finset α := ht.toFinset
    have : (fun x => ∑' i, f i x) = (fun x => ∑ i in T, f i x) +
        fun x => ∑' i : { i // i ∉ T }, f i x := by
      ext1 x
      refine' (sum_add_tsum_subtype_compl _ T).symm
      

Proof state:
case h
α : Type u_1
β : Type u_2
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
v : ℕ → α → ℝ
s : Set E
x₀ x✝ : E
N : ℕ∞
hf : ∀ (i : α), ContDiff 𝕜 N (f i)
hv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)
h'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i
m : ℕ
hm : ↑m ≤ N
t : Set α := {i | ¬∀ k ∈ Finset.range (m + 1), ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}
ht : Set.Finite t
T : Finset α := Finite.toFinset ht
x : E
⊢ Summable fun i => f i x

Next tactic:
refine' .of_norm_bounded_eventually _ (hv 0 (zero_le _)) _
/-- Consider a series of functions `∑' i, f i x`. Assume that each individual function `f i` is of
class `C^N`, and moreover there is a uniform summable upper bound on the `k`-th derivative
for each `k ≤ N` (except maybe for finitely many `i`s). Then the series is also `C^N`. -/
theorem contDiff_tsum_of_eventually (hf : ∀ i, ContDiff 𝕜 N (f i))
    (hv : ∀ k : ℕ, (k : ℕ∞) ≤ N → Summable (v k))
    (h'f :
      ∀ k : ℕ,
        (k : ℕ∞) ≤ N →
          ∀ᶠ i in (Filter.cofinite : Filter α), ∀ x : E, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i) :
    ContDiff 𝕜 N fun x => ∑' i, f i x := by
  classical
    refine contDiff_iff_forall_nat_le.2 fun m hm => ?_
    let t : Set α :=
      { i : α | ¬∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i }
    have ht : Set.Finite t :=
      haveI A :
        ∀ᶠ i in (Filter.cofinite : Filter α),
          ∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x : E, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i := by
        rw [eventually_all_finset]
        intro i hi
        apply h'f
        simp only [Finset.mem_range_succ_iff] at hi
        exact (WithTop.coe_le_coe.2 hi).trans hm
      eventually_cofinite.2 A
    let T : Finset α := ht.toFinset
    have : (fun x => ∑' i, f i x) = (fun x => ∑ i in T, f i x) +
        fun x => ∑' i : { i // i ∉ T }, f i x := by
      ext1 x
      refine' (sum_add_tsum_subtype_compl _ T).symm
      refine' .of_norm_bounded_eventually _ (hv 0 (zero_le _)) _
      

Proof state:
case h
α : Type u_1
β : Type u_2
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
v : ℕ → α → ℝ
s : Set E
x₀ x✝ : E
N : ℕ∞
hf : ∀ (i : α), ContDiff 𝕜 N (f i)
hv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)
h'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i
m : ℕ
hm : ↑m ≤ N
t : Set α := {i | ¬∀ k ∈ Finset.range (m + 1), ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}
ht : Set.Finite t
T : Finset α := Finite.toFinset ht
x : E
⊢ ∀ᶠ (i : α) in cofinite, ‖f i x‖ ≤ v 0 i

Next tactic:
filter_upwards [h'f 0 (zero_le _)] with i hi
/-- Consider a series of functions `∑' i, f i x`. Assume that each individual function `f i` is of
class `C^N`, and moreover there is a uniform summable upper bound on the `k`-th derivative
for each `k ≤ N` (except maybe for finitely many `i`s). Then the series is also `C^N`. -/
theorem contDiff_tsum_of_eventually (hf : ∀ i, ContDiff 𝕜 N (f i))
    (hv : ∀ k : ℕ, (k : ℕ∞) ≤ N → Summable (v k))
    (h'f :
      ∀ k : ℕ,
        (k : ℕ∞) ≤ N →
          ∀ᶠ i in (Filter.cofinite : Filter α), ∀ x : E, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i) :
    ContDiff 𝕜 N fun x => ∑' i, f i x := by
  classical
    refine contDiff_iff_forall_nat_le.2 fun m hm => ?_
    let t : Set α :=
      { i : α | ¬∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i }
    have ht : Set.Finite t :=
      haveI A :
        ∀ᶠ i in (Filter.cofinite : Filter α),
          ∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x : E, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i := by
        rw [eventually_all_finset]
        intro i hi
        apply h'f
        simp only [Finset.mem_range_succ_iff] at hi
        exact (WithTop.coe_le_coe.2 hi).trans hm
      eventually_cofinite.2 A
    let T : Finset α := ht.toFinset
    have : (fun x => ∑' i, f i x) = (fun x => ∑ i in T, f i x) +
        fun x => ∑' i : { i // i ∉ T }, f i x := by
      ext1 x
      refine' (sum_add_tsum_subtype_compl _ T).symm
      refine' .of_norm_bounded_eventually _ (hv 0 (zero_le _)) _
      filter_upwards [h'f 0 (zero_le _)] with i hi
      

Proof state:
case h
α : Type u_1
β : Type u_2
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
v : ℕ → α → ℝ
s : Set E
x₀ x✝ : E
N : ℕ∞
hf : ∀ (i : α), ContDiff 𝕜 N (f i)
hv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)
h'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i
m : ℕ
hm : ↑m ≤ N
t : Set α := {i | ¬∀ k ∈ Finset.range (m + 1), ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}
ht : Set.Finite t
T : Finset α := Finite.toFinset ht
x : E
i : α
hi : ∀ (x : E), ‖iteratedFDeriv 𝕜 0 (f i) x‖ ≤ v 0 i
⊢ ‖f i x‖ ≤ v 0 i

Next tactic:
simpa only [norm_iteratedFDeriv_zero] using hi x