l3lab/ntp-mathlib huggingface dataset:
theorem HasSum.sum_nat_of_sum_int {α : Type*} [AddCommMonoid α] [TopologicalSpace α]
    [ContinuousAdd α] {a : α} {f : ℤ → α} (hf : HasSum f a) :
    HasSum (fun n : ℕ => f n + f (-n)) (a + f 0) := by
  apply (hf.add (hasSum_ite_eq (0 : ℤ) (f 0))).hasSum_of_sum_eq fun u => ?_
  refine' ⟨u.image Int.natAbs, fun v' hv' => _⟩
  let u1 := v'.image fun x : ℕ => (x : ℤ)
  let u2 := v'.image fun x : ℕ => -(x : ℤ)
  have A : u ⊆ u1 ∪ u2 := by
    intro x hx
    simp only [mem_union, mem_image, exists_prop]
    rcases le_total 0 x with (h'x | h'x)
    · left
      refine' ⟨Int.natAbs x, hv' _, _⟩
      · simp only [mem_image, exists_prop]
        exact ⟨x, hx, rfl⟩
      · simp only [h'x, Int.coe_natAbs, abs_eq_self]
    · right
      refine' ⟨Int.natAbs x, hv' _, _⟩
      · simp only [mem_image, exists_prop]
        exact ⟨x, hx, rfl⟩
      · simp only [abs_of_nonpos h'x, Int.coe_natAbs, neg_neg]
  refine' ⟨u1 ∪ u2, A, _⟩
  

Proof state:
α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : AddCommGroup α✝
inst✝⁴ : TopologicalSpace α✝
inst✝³ : TopologicalAddGroup α✝
f✝ g : β → α✝
a✝ a₁ a₂ : α✝
α : Type u_5
inst✝² : AddCommMonoid α
inst✝¹ : TopologicalSpace α
inst✝ : ContinuousAdd α
a : α
f : ℤ → α
hf : HasSum f a
u : Finset ℤ
v' : Finset ℕ
hv' : image Int.natAbs u ⊆ v'
u1 : Finset ℤ := image (fun x => ↑x) v'
u2 : Finset ℤ := image (fun x => -↑x) v'
A : u ⊆ u1 ∪ u2
⊢ ∑ x in u1 ∪ u2, (f x + if x = 0 then f 0 else 0) = ∑ b in v', (f ↑b + f (-↑b))

Next tactic:
calc
    (∑ x in u1 ∪ u2, (f x + ite (x = 0) (f 0) 0)) =
        (∑ x in u1 ∪ u2, f x) + ∑ x in u1 ∩ u2, f x := by
      rw [sum_add_distrib]
      congr 1
      refine' (sum_subset_zero_on_sdiff inter_subset_union _ _).symm
      · intro x hx
        suffices x ≠ 0 by simp only [this, if_false]
        rintro rfl
        simp at hx
      · intro x hx
        simp only [mem_inter, mem_image, exists_prop] at hx
        have : x = 0 := by
          apply le_antisymm
          · rcases hx.2 with ⟨a, _, rfl⟩
            simp only [Right.neg_nonpos_iff, Nat.cast_nonneg]
          · rcases hx.1 with ⟨a, _, rfl⟩
            simp only [Nat.cast_nonneg]
        simp only [this, eq_self_iff_true, if_true]
    _ = (∑ x in u1, f x) + ∑ x in u2, f x := sum_union_inter
    _ = (∑ b in v', f b) + ∑ b in v', f (-b) := by simp
    _ = ∑ b in v', (f b + f (-b)) := sum_add_distrib.symm
theorem HasSum.sum_nat_of_sum_int {α : Type*} [AddCommMonoid α] [TopologicalSpace α]
    [ContinuousAdd α] {a : α} {f : ℤ → α} (hf : HasSum f a) :
    HasSum (fun n : ℕ => f n + f (-n)) (a + f 0) := by
  apply (hf.add (hasSum_ite_eq (0 : ℤ) (f 0))).hasSum_of_sum_eq fun u => ?_
  refine' ⟨u.image Int.natAbs, fun v' hv' => _⟩
  let u1 := v'.image fun x : ℕ => (x : ℤ)
  let u2 := v'.image fun x : ℕ => -(x : ℤ)
  have A : u ⊆ u1 ∪ u2 := by
    intro x hx
    simp only [mem_union, mem_image, exists_prop]
    rcases le_total 0 x with (h'x | h'x)
    · left
      refine' ⟨Int.natAbs x, hv' _, _⟩
      · simp only [mem_image, exists_prop]
        exact ⟨x, hx, rfl⟩
      · simp only [h'x, Int.coe_natAbs, abs_eq_self]
    · right
      refine' ⟨Int.natAbs x, hv' _, _⟩
      · simp only [mem_image, exists_prop]
        exact ⟨x, hx, rfl⟩
      · simp only [abs_of_nonpos h'x, Int.coe_natAbs, neg_neg]
  refine' ⟨u1 ∪ u2, A, _⟩
  calc
    (∑ x in u1 ∪ u2, (f x + ite (x = 0) (f 0) 0)) =
        (∑ x in u1 ∪ u2, f x) + ∑ x in u1 ∩ u2, f x := by
      

Proof state:
α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : AddCommGroup α✝
inst✝⁴ : TopologicalSpace α✝
inst✝³ : TopologicalAddGroup α✝
f✝ g : β → α✝
a✝ a₁ a₂ : α✝
α : Type u_5
inst✝² : AddCommMonoid α
inst✝¹ : TopologicalSpace α
inst✝ : ContinuousAdd α
a : α
f : ℤ → α
hf : HasSum f a
u : Finset ℤ
v' : Finset ℕ
hv' : image Int.natAbs u ⊆ v'
u1 : Finset ℤ := image (fun x => ↑x) v'
u2 : Finset ℤ := image (fun x => -↑x) v'
A : u ⊆ u1 ∪ u2
⊢ ∑ x in u1 ∪ u2, (f x + if x = 0 then f 0 else 0) = ∑ x in u1 ∪ u2, f x + ∑ x in u1 ∩ u2, f x

Next tactic:
rw [sum_add_distrib]
theorem HasSum.sum_nat_of_sum_int {α : Type*} [AddCommMonoid α] [TopologicalSpace α]
    [ContinuousAdd α] {a : α} {f : ℤ → α} (hf : HasSum f a) :
    HasSum (fun n : ℕ => f n + f (-n)) (a + f 0) := by
  apply (hf.add (hasSum_ite_eq (0 : ℤ) (f 0))).hasSum_of_sum_eq fun u => ?_
  refine' ⟨u.image Int.natAbs, fun v' hv' => _⟩
  let u1 := v'.image fun x : ℕ => (x : ℤ)
  let u2 := v'.image fun x : ℕ => -(x : ℤ)
  have A : u ⊆ u1 ∪ u2 := by
    intro x hx
    simp only [mem_union, mem_image, exists_prop]
    rcases le_total 0 x with (h'x | h'x)
    · left
      refine' ⟨Int.natAbs x, hv' _, _⟩
      · simp only [mem_image, exists_prop]
        exact ⟨x, hx, rfl⟩
      · simp only [h'x, Int.coe_natAbs, abs_eq_self]
    · right
      refine' ⟨Int.natAbs x, hv' _, _⟩
      · simp only [mem_image, exists_prop]
        exact ⟨x, hx, rfl⟩
      · simp only [abs_of_nonpos h'x, Int.coe_natAbs, neg_neg]
  refine' ⟨u1 ∪ u2, A, _⟩
  calc
    (∑ x in u1 ∪ u2, (f x + ite (x = 0) (f 0) 0)) =
        (∑ x in u1 ∪ u2, f x) + ∑ x in u1 ∩ u2, f x := by
      rw [sum_add_distrib]
      

Proof state:
α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : AddCommGroup α✝
inst✝⁴ : TopologicalSpace α✝
inst✝³ : TopologicalAddGroup α✝
f✝ g : β → α✝
a✝ a₁ a₂ : α✝
α : Type u_5
inst✝² : AddCommMonoid α
inst✝¹ : TopologicalSpace α
inst✝ : ContinuousAdd α
a : α
f : ℤ → α
hf : HasSum f a
u : Finset ℤ
v' : Finset ℕ
hv' : image Int.natAbs u ⊆ v'
u1 : Finset ℤ := image (fun x => ↑x) v'
u2 : Finset ℤ := image (fun x => -↑x) v'
A : u ⊆ u1 ∪ u2
⊢ (∑ x in u1 ∪ u2, f x + ∑ x in u1 ∪ u2, if x = 0 then f 0 else 0) = ∑ x in u1 ∪ u2, f x + ∑ x in u1 ∩ u2, f x

Next tactic:
congr 1
theorem HasSum.sum_nat_of_sum_int {α : Type*} [AddCommMonoid α] [TopologicalSpace α]
    [ContinuousAdd α] {a : α} {f : ℤ → α} (hf : HasSum f a) :
    HasSum (fun n : ℕ => f n + f (-n)) (a + f 0) := by
  apply (hf.add (hasSum_ite_eq (0 : ℤ) (f 0))).hasSum_of_sum_eq fun u => ?_
  refine' ⟨u.image Int.natAbs, fun v' hv' => _⟩
  let u1 := v'.image fun x : ℕ => (x : ℤ)
  let u2 := v'.image fun x : ℕ => -(x : ℤ)
  have A : u ⊆ u1 ∪ u2 := by
    intro x hx
    simp only [mem_union, mem_image, exists_prop]
    rcases le_total 0 x with (h'x | h'x)
    · left
      refine' ⟨Int.natAbs x, hv' _, _⟩
      · simp only [mem_image, exists_prop]
        exact ⟨x, hx, rfl⟩
      · simp only [h'x, Int.coe_natAbs, abs_eq_self]
    · right
      refine' ⟨Int.natAbs x, hv' _, _⟩
      · simp only [mem_image, exists_prop]
        exact ⟨x, hx, rfl⟩
      · simp only [abs_of_nonpos h'x, Int.coe_natAbs, neg_neg]
  refine' ⟨u1 ∪ u2, A, _⟩
  calc
    (∑ x in u1 ∪ u2, (f x + ite (x = 0) (f 0) 0)) =
        (∑ x in u1 ∪ u2, f x) + ∑ x in u1 ∩ u2, f x := by
      rw [sum_add_distrib]
      congr 1
      

Proof state:
case e_a
α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : AddCommGroup α✝
inst✝⁴ : TopologicalSpace α✝
inst✝³ : TopologicalAddGroup α✝
f✝ g : β → α✝
a✝ a₁ a₂ : α✝
α : Type u_5
inst✝² : AddCommMonoid α
inst✝¹ : TopologicalSpace α
inst✝ : ContinuousAdd α
a : α
f : ℤ → α
hf : HasSum f a
u : Finset ℤ
v' : Finset ℕ
hv' : image Int.natAbs u ⊆ v'
u1 : Finset ℤ := image (fun x => ↑x) v'
u2 : Finset ℤ := image (fun x => -↑x) v'
A : u ⊆ u1 ∪ u2
⊢ (∑ x in u1 ∪ u2, if x = 0 then f 0 else 0) = ∑ x in u1 ∩ u2, f x

Next tactic:
refine' (sum_subset_zero_on_sdiff inter_subset_union _ _).symm
theorem HasSum.sum_nat_of_sum_int {α : Type*} [AddCommMonoid α] [TopologicalSpace α]
    [ContinuousAdd α] {a : α} {f : ℤ → α} (hf : HasSum f a) :
    HasSum (fun n : ℕ => f n + f (-n)) (a + f 0) := by
  apply (hf.add (hasSum_ite_eq (0 : ℤ) (f 0))).hasSum_of_sum_eq fun u => ?_
  refine' ⟨u.image Int.natAbs, fun v' hv' => _⟩
  let u1 := v'.image fun x : ℕ => (x : ℤ)
  let u2 := v'.image fun x : ℕ => -(x : ℤ)
  have A : u ⊆ u1 ∪ u2 := by
    intro x hx
    simp only [mem_union, mem_image, exists_prop]
    rcases le_total 0 x with (h'x | h'x)
    · left
      refine' ⟨Int.natAbs x, hv' _, _⟩
      · simp only [mem_image, exists_prop]
        exact ⟨x, hx, rfl⟩
      · simp only [h'x, Int.coe_natAbs, abs_eq_self]
    · right
      refine' ⟨Int.natAbs x, hv' _, _⟩
      · simp only [mem_image, exists_prop]
        exact ⟨x, hx, rfl⟩
      · simp only [abs_of_nonpos h'x, Int.coe_natAbs, neg_neg]
  refine' ⟨u1 ∪ u2, A, _⟩
  calc
    (∑ x in u1 ∪ u2, (f x + ite (x = 0) (f 0) 0)) =
        (∑ x in u1 ∪ u2, f x) + ∑ x in u1 ∩ u2, f x := by
      rw [sum_add_distrib]
      congr 1
      refine' (sum_subset_zero_on_sdiff inter_subset_union _ _).symm
      · 

Proof state:
case e_a.refine'_1
α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : AddCommGroup α✝
inst✝⁴ : TopologicalSpace α✝
inst✝³ : TopologicalAddGroup α✝
f✝ g : β → α✝
a✝ a₁ a₂ : α✝
α : Type u_5
inst✝² : AddCommMonoid α
inst✝¹ : TopologicalSpace α
inst✝ : ContinuousAdd α
a : α
f : ℤ → α
hf : HasSum f a
u : Finset ℤ
v' : Finset ℕ
hv' : image Int.natAbs u ⊆ v'
u1 : Finset ℤ := image (fun x => ↑x) v'
u2 : Finset ℤ := image (fun x => -↑x) v'
A : u ⊆ u1 ∪ u2
⊢ ∀ x ∈ (u1 ∪ u2) \ (u1 ∩ u2), (if x = 0 then f 0 else 0) = 0

Next tactic:
intro x hx
theorem HasSum.sum_nat_of_sum_int {α : Type*} [AddCommMonoid α] [TopologicalSpace α]
    [ContinuousAdd α] {a : α} {f : ℤ → α} (hf : HasSum f a) :
    HasSum (fun n : ℕ => f n + f (-n)) (a + f 0) := by
  apply (hf.add (hasSum_ite_eq (0 : ℤ) (f 0))).hasSum_of_sum_eq fun u => ?_
  refine' ⟨u.image Int.natAbs, fun v' hv' => _⟩
  let u1 := v'.image fun x : ℕ => (x : ℤ)
  let u2 := v'.image fun x : ℕ => -(x : ℤ)
  have A : u ⊆ u1 ∪ u2 := by
    intro x hx
    simp only [mem_union, mem_image, exists_prop]
    rcases le_total 0 x with (h'x | h'x)
    · left
      refine' ⟨Int.natAbs x, hv' _, _⟩
      · simp only [mem_image, exists_prop]
        exact ⟨x, hx, rfl⟩
      · simp only [h'x, Int.coe_natAbs, abs_eq_self]
    · right
      refine' ⟨Int.natAbs x, hv' _, _⟩
      · simp only [mem_image, exists_prop]
        exact ⟨x, hx, rfl⟩
      · simp only [abs_of_nonpos h'x, Int.coe_natAbs, neg_neg]
  refine' ⟨u1 ∪ u2, A, _⟩
  calc
    (∑ x in u1 ∪ u2, (f x + ite (x = 0) (f 0) 0)) =
        (∑ x in u1 ∪ u2, f x) + ∑ x in u1 ∩ u2, f x := by
      rw [sum_add_distrib]
      congr 1
      refine' (sum_subset_zero_on_sdiff inter_subset_union _ _).symm
      · intro x hx
        

Proof state:
case e_a.refine'_1
α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : AddCommGroup α✝
inst✝⁴ : TopologicalSpace α✝
inst✝³ : TopologicalAddGroup α✝
f✝ g : β → α✝
a✝ a₁ a₂ : α✝
α : Type u_5
inst✝² : AddCommMonoid α
inst✝¹ : TopologicalSpace α
inst✝ : ContinuousAdd α
a : α
f : ℤ → α
hf : HasSum f a
u : Finset ℤ
v' : Finset ℕ
hv' : image Int.natAbs u ⊆ v'
u1 : Finset ℤ := image (fun x => ↑x) v'
u2 : Finset ℤ := image (fun x => -↑x) v'
A : u ⊆ u1 ∪ u2
x : ℤ
hx : x ∈ (u1 ∪ u2) \ (u1 ∩ u2)
⊢ (if x = 0 then f 0 else 0) = 0

Next tactic:
suffices x ≠ 0 by simp only [this, if_false]
theorem HasSum.sum_nat_of_sum_int {α : Type*} [AddCommMonoid α] [TopologicalSpace α]
    [ContinuousAdd α] {a : α} {f : ℤ → α} (hf : HasSum f a) :
    HasSum (fun n : ℕ => f n + f (-n)) (a + f 0) := by
  apply (hf.add (hasSum_ite_eq (0 : ℤ) (f 0))).hasSum_of_sum_eq fun u => ?_
  refine' ⟨u.image Int.natAbs, fun v' hv' => _⟩
  let u1 := v'.image fun x : ℕ => (x : ℤ)
  let u2 := v'.image fun x : ℕ => -(x : ℤ)
  have A : u ⊆ u1 ∪ u2 := by
    intro x hx
    simp only [mem_union, mem_image, exists_prop]
    rcases le_total 0 x with (h'x | h'x)
    · left
      refine' ⟨Int.natAbs x, hv' _, _⟩
      · simp only [mem_image, exists_prop]
        exact ⟨x, hx, rfl⟩
      · simp only [h'x, Int.coe_natAbs, abs_eq_self]
    · right
      refine' ⟨Int.natAbs x, hv' _, _⟩
      · simp only [mem_image, exists_prop]
        exact ⟨x, hx, rfl⟩
      · simp only [abs_of_nonpos h'x, Int.coe_natAbs, neg_neg]
  refine' ⟨u1 ∪ u2, A, _⟩
  calc
    (∑ x in u1 ∪ u2, (f x + ite (x = 0) (f 0) 0)) =
        (∑ x in u1 ∪ u2, f x) + ∑ x in u1 ∩ u2, f x := by
      rw [sum_add_distrib]
      congr 1
      refine' (sum_subset_zero_on_sdiff inter_subset_union _ _).symm
      · intro x hx
        suffices x ≠ 0 by 

Proof state:
α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : AddCommGroup α✝
inst✝⁴ : TopologicalSpace α✝
inst✝³ : TopologicalAddGroup α✝
f✝ g : β → α✝
a✝ a₁ a₂ : α✝
α : Type u_5
inst✝² : AddCommMonoid α
inst✝¹ : TopologicalSpace α
inst✝ : ContinuousAdd α
a : α
f : ℤ → α
hf : HasSum f a
u : Finset ℤ
v' : Finset ℕ
hv' : image Int.natAbs u ⊆ v'
u1 : Finset ℤ := image (fun x => ↑x) v'
u2 : Finset ℤ := image (fun x => -↑x) v'
A : u ⊆ u1 ∪ u2
x : ℤ
hx : x ∈ (u1 ∪ u2) \ (u1 ∩ u2)
this : x ≠ 0
⊢ (if x = 0 then f 0 else 0) = 0

Next tactic:
simp only [this, if_false]
theorem HasSum.sum_nat_of_sum_int {α : Type*} [AddCommMonoid α] [TopologicalSpace α]
    [ContinuousAdd α] {a : α} {f : ℤ → α} (hf : HasSum f a) :
    HasSum (fun n : ℕ => f n + f (-n)) (a + f 0) := by
  apply (hf.add (hasSum_ite_eq (0 : ℤ) (f 0))).hasSum_of_sum_eq fun u => ?_
  refine' ⟨u.image Int.natAbs, fun v' hv' => _⟩
  let u1 := v'.image fun x : ℕ => (x : ℤ)
  let u2 := v'.image fun x : ℕ => -(x : ℤ)
  have A : u ⊆ u1 ∪ u2 := by
    intro x hx
    simp only [mem_union, mem_image, exists_prop]
    rcases le_total 0 x with (h'x | h'x)
    · left
      refine' ⟨Int.natAbs x, hv' _, _⟩
      · simp only [mem_image, exists_prop]
        exact ⟨x, hx, rfl⟩
      · simp only [h'x, Int.coe_natAbs, abs_eq_self]
    · right
      refine' ⟨Int.natAbs x, hv' _, _⟩
      · simp only [mem_image, exists_prop]
        exact ⟨x, hx, rfl⟩
      · simp only [abs_of_nonpos h'x, Int.coe_natAbs, neg_neg]
  refine' ⟨u1 ∪ u2, A, _⟩
  calc
    (∑ x in u1 ∪ u2, (f x + ite (x = 0) (f 0) 0)) =
        (∑ x in u1 ∪ u2, f x) + ∑ x in u1 ∩ u2, f x := by
      rw [sum_add_distrib]
      congr 1
      refine' (sum_subset_zero_on_sdiff inter_subset_union _ _).symm
      · intro x hx
        suffices x ≠ 0 by simp only [this, if_false]
        

Proof state:
case e_a.refine'_1
α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : AddCommGroup α✝
inst✝⁴ : TopologicalSpace α✝
inst✝³ : TopologicalAddGroup α✝
f✝ g : β → α✝
a✝ a₁ a₂ : α✝
α : Type u_5
inst✝² : AddCommMonoid α
inst✝¹ : TopologicalSpace α
inst✝ : ContinuousAdd α
a : α
f : ℤ → α
hf : HasSum f a
u : Finset ℤ
v' : Finset ℕ
hv' : image Int.natAbs u ⊆ v'
u1 : Finset ℤ := image (fun x => ↑x) v'
u2 : Finset ℤ := image (fun x => -↑x) v'
A : u ⊆ u1 ∪ u2
x : ℤ
hx : x ∈ (u1 ∪ u2) \ (u1 ∩ u2)
⊢ x ≠ 0

Next tactic:
rintro rfl
theorem HasSum.sum_nat_of_sum_int {α : Type*} [AddCommMonoid α] [TopologicalSpace α]
    [ContinuousAdd α] {a : α} {f : ℤ → α} (hf : HasSum f a) :
    HasSum (fun n : ℕ => f n + f (-n)) (a + f 0) := by
  apply (hf.add (hasSum_ite_eq (0 : ℤ) (f 0))).hasSum_of_sum_eq fun u => ?_
  refine' ⟨u.image Int.natAbs, fun v' hv' => _⟩
  let u1 := v'.image fun x : ℕ => (x : ℤ)
  let u2 := v'.image fun x : ℕ => -(x : ℤ)
  have A : u ⊆ u1 ∪ u2 := by
    intro x hx
    simp only [mem_union, mem_image, exists_prop]
    rcases le_total 0 x with (h'x | h'x)
    · left
      refine' ⟨Int.natAbs x, hv' _, _⟩
      · simp only [mem_image, exists_prop]
        exact ⟨x, hx, rfl⟩
      · simp only [h'x, Int.coe_natAbs, abs_eq_self]
    · right
      refine' ⟨Int.natAbs x, hv' _, _⟩
      · simp only [mem_image, exists_prop]
        exact ⟨x, hx, rfl⟩
      · simp only [abs_of_nonpos h'x, Int.coe_natAbs, neg_neg]
  refine' ⟨u1 ∪ u2, A, _⟩
  calc
    (∑ x in u1 ∪ u2, (f x + ite (x = 0) (f 0) 0)) =
        (∑ x in u1 ∪ u2, f x) + ∑ x in u1 ∩ u2, f x := by
      rw [sum_add_distrib]
      congr 1
      refine' (sum_subset_zero_on_sdiff inter_subset_union _ _).symm
      · intro x hx
        suffices x ≠ 0 by simp only [this, if_false]
        rintro rfl
        

Proof state:
case e_a.refine'_1
α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : AddCommGroup α✝
inst✝⁴ : TopologicalSpace α✝
inst✝³ : TopologicalAddGroup α✝
f✝ g : β → α✝
a✝ a₁ a₂ : α✝
α : Type u_5
inst✝² : AddCommMonoid α
inst✝¹ : TopologicalSpace α
inst✝ : ContinuousAdd α
a : α
f : ℤ → α
hf : HasSum f a
u : Finset ℤ
v' : Finset ℕ
hv' : image Int.natAbs u ⊆ v'
u1 : Finset ℤ := image (fun x => ↑x) v'
u2 : Finset ℤ := image (fun x => -↑x) v'
A : u ⊆ u1 ∪ u2
hx : 0 ∈ (u1 ∪ u2) \ (u1 ∩ u2)
⊢ False

Next tactic:
simp at hx
theorem HasSum.sum_nat_of_sum_int {α : Type*} [AddCommMonoid α] [TopologicalSpace α]
    [ContinuousAdd α] {a : α} {f : ℤ → α} (hf : HasSum f a) :
    HasSum (fun n : ℕ => f n + f (-n)) (a + f 0) := by
  apply (hf.add (hasSum_ite_eq (0 : ℤ) (f 0))).hasSum_of_sum_eq fun u => ?_
  refine' ⟨u.image Int.natAbs, fun v' hv' => _⟩
  let u1 := v'.image fun x : ℕ => (x : ℤ)
  let u2 := v'.image fun x : ℕ => -(x : ℤ)
  have A : u ⊆ u1 ∪ u2 := by
    intro x hx
    simp only [mem_union, mem_image, exists_prop]
    rcases le_total 0 x with (h'x | h'x)
    · left
      refine' ⟨Int.natAbs x, hv' _, _⟩
      · simp only [mem_image, exists_prop]
        exact ⟨x, hx, rfl⟩
      · simp only [h'x, Int.coe_natAbs, abs_eq_self]
    · right
      refine' ⟨Int.natAbs x, hv' _, _⟩
      · simp only [mem_image, exists_prop]
        exact ⟨x, hx, rfl⟩
      · simp only [abs_of_nonpos h'x, Int.coe_natAbs, neg_neg]
  refine' ⟨u1 ∪ u2, A, _⟩
  calc
    (∑ x in u1 ∪ u2, (f x + ite (x = 0) (f 0) 0)) =
        (∑ x in u1 ∪ u2, f x) + ∑ x in u1 ∩ u2, f x := by
      rw [sum_add_distrib]
      congr 1
      refine' (sum_subset_zero_on_sdiff inter_subset_union _ _).symm
      · intro x hx
        suffices x ≠ 0 by simp only [this, if_false]
        rintro rfl
        simp at hx
      · 

Proof state:
case e_a.refine'_2
α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : AddCommGroup α✝
inst✝⁴ : TopologicalSpace α✝
inst✝³ : TopologicalAddGroup α✝
f✝ g : β → α✝
a✝ a₁ a₂ : α✝
α : Type u_5
inst✝² : AddCommMonoid α
inst✝¹ : TopologicalSpace α
inst✝ : ContinuousAdd α
a : α
f : ℤ → α
hf : HasSum f a
u : Finset ℤ
v' : Finset ℕ
hv' : image Int.natAbs u ⊆ v'
u1 : Finset ℤ := image (fun x => ↑x) v'
u2 : Finset ℤ := image (fun x => -↑x) v'
A : u ⊆ u1 ∪ u2
⊢ ∀ x ∈ u1 ∩ u2, f x = if x = 0 then f 0 else 0

Next tactic:
intro x hx
theorem HasSum.sum_nat_of_sum_int {α : Type*} [AddCommMonoid α] [TopologicalSpace α]
    [ContinuousAdd α] {a : α} {f : ℤ → α} (hf : HasSum f a) :
    HasSum (fun n : ℕ => f n + f (-n)) (a + f 0) := by
  apply (hf.add (hasSum_ite_eq (0 : ℤ) (f 0))).hasSum_of_sum_eq fun u => ?_
  refine' ⟨u.image Int.natAbs, fun v' hv' => _⟩
  let u1 := v'.image fun x : ℕ => (x : ℤ)
  let u2 := v'.image fun x : ℕ => -(x : ℤ)
  have A : u ⊆ u1 ∪ u2 := by
    intro x hx
    simp only [mem_union, mem_image, exists_prop]
    rcases le_total 0 x with (h'x | h'x)
    · left
      refine' ⟨Int.natAbs x, hv' _, _⟩
      · simp only [mem_image, exists_prop]
        exact ⟨x, hx, rfl⟩
      · simp only [h'x, Int.coe_natAbs, abs_eq_self]
    · right
      refine' ⟨Int.natAbs x, hv' _, _⟩
      · simp only [mem_image, exists_prop]
        exact ⟨x, hx, rfl⟩
      · simp only [abs_of_nonpos h'x, Int.coe_natAbs, neg_neg]
  refine' ⟨u1 ∪ u2, A, _⟩
  calc
    (∑ x in u1 ∪ u2, (f x + ite (x = 0) (f 0) 0)) =
        (∑ x in u1 ∪ u2, f x) + ∑ x in u1 ∩ u2, f x := by
      rw [sum_add_distrib]
      congr 1
      refine' (sum_subset_zero_on_sdiff inter_subset_union _ _).symm
      · intro x hx
        suffices x ≠ 0 by simp only [this, if_false]
        rintro rfl
        simp at hx
      · intro x hx
        

Proof state:
case e_a.refine'_2
α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : AddCommGroup α✝
inst✝⁴ : TopologicalSpace α✝
inst✝³ : TopologicalAddGroup α✝
f✝ g : β → α✝
a✝ a₁ a₂ : α✝
α : Type u_5
inst✝² : AddCommMonoid α
inst✝¹ : TopologicalSpace α
inst✝ : ContinuousAdd α
a : α
f : ℤ → α
hf : HasSum f a
u : Finset ℤ
v' : Finset ℕ
hv' : image Int.natAbs u ⊆ v'
u1 : Finset ℤ := image (fun x => ↑x) v'
u2 : Finset ℤ := image (fun x => -↑x) v'
A : u ⊆ u1 ∪ u2
x : ℤ
hx : x ∈ u1 ∩ u2
⊢ f x = if x = 0 then f 0 else 0

Next tactic:
simp only [mem_inter, mem_image, exists_prop] at hx
theorem HasSum.sum_nat_of_sum_int {α : Type*} [AddCommMonoid α] [TopologicalSpace α]
    [ContinuousAdd α] {a : α} {f : ℤ → α} (hf : HasSum f a) :
    HasSum (fun n : ℕ => f n + f (-n)) (a + f 0) := by
  apply (hf.add (hasSum_ite_eq (0 : ℤ) (f 0))).hasSum_of_sum_eq fun u => ?_
  refine' ⟨u.image Int.natAbs, fun v' hv' => _⟩
  let u1 := v'.image fun x : ℕ => (x : ℤ)
  let u2 := v'.image fun x : ℕ => -(x : ℤ)
  have A : u ⊆ u1 ∪ u2 := by
    intro x hx
    simp only [mem_union, mem_image, exists_prop]
    rcases le_total 0 x with (h'x | h'x)
    · left
      refine' ⟨Int.natAbs x, hv' _, _⟩
      · simp only [mem_image, exists_prop]
        exact ⟨x, hx, rfl⟩
      · simp only [h'x, Int.coe_natAbs, abs_eq_self]
    · right
      refine' ⟨Int.natAbs x, hv' _, _⟩
      · simp only [mem_image, exists_prop]
        exact ⟨x, hx, rfl⟩
      · simp only [abs_of_nonpos h'x, Int.coe_natAbs, neg_neg]
  refine' ⟨u1 ∪ u2, A, _⟩
  calc
    (∑ x in u1 ∪ u2, (f x + ite (x = 0) (f 0) 0)) =
        (∑ x in u1 ∪ u2, f x) + ∑ x in u1 ∩ u2, f x := by
      rw [sum_add_distrib]
      congr 1
      refine' (sum_subset_zero_on_sdiff inter_subset_union _ _).symm
      · intro x hx
        suffices x ≠ 0 by simp only [this, if_false]
        rintro rfl
        simp at hx
      · intro x hx
        simp only [mem_inter, mem_image, exists_prop] at hx
        

Proof state:
case e_a.refine'_2
α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : AddCommGroup α✝
inst✝⁴ : TopologicalSpace α✝
inst✝³ : TopologicalAddGroup α✝
f✝ g : β → α✝
a✝ a₁ a₂ : α✝
α : Type u_5
inst✝² : AddCommMonoid α
inst✝¹ : TopologicalSpace α
inst✝ : ContinuousAdd α
a : α
f : ℤ → α
hf : HasSum f a
u : Finset ℤ
v' : Finset ℕ
hv' : image Int.natAbs u ⊆ v'
u1 : Finset ℤ := image (fun x => ↑x) v'
u2 : Finset ℤ := image (fun x => -↑x) v'
A : u ⊆ u1 ∪ u2
x : ℤ
hx : (∃ a ∈ v', ↑a = x) ∧ ∃ a ∈ v', -↑a = x
⊢ f x = if x = 0 then f 0 else 0

Next tactic:
have : x = 0 := by
          apply le_antisymm
          · rcases hx.2 with ⟨a, _, rfl⟩
            simp only [Right.neg_nonpos_iff, Nat.cast_nonneg]
          · rcases hx.1 with ⟨a, _, rfl⟩
            simp only [Nat.cast_nonneg]
theorem HasSum.sum_nat_of_sum_int {α : Type*} [AddCommMonoid α] [TopologicalSpace α]
    [ContinuousAdd α] {a : α} {f : ℤ → α} (hf : HasSum f a) :
    HasSum (fun n : ℕ => f n + f (-n)) (a + f 0) := by
  apply (hf.add (hasSum_ite_eq (0 : ℤ) (f 0))).hasSum_of_sum_eq fun u => ?_
  refine' ⟨u.image Int.natAbs, fun v' hv' => _⟩
  let u1 := v'.image fun x : ℕ => (x : ℤ)
  let u2 := v'.image fun x : ℕ => -(x : ℤ)
  have A : u ⊆ u1 ∪ u2 := by
    intro x hx
    simp only [mem_union, mem_image, exists_prop]
    rcases le_total 0 x with (h'x | h'x)
    · left
      refine' ⟨Int.natAbs x, hv' _, _⟩
      · simp only [mem_image, exists_prop]
        exact ⟨x, hx, rfl⟩
      · simp only [h'x, Int.coe_natAbs, abs_eq_self]
    · right
      refine' ⟨Int.natAbs x, hv' _, _⟩
      · simp only [mem_image, exists_prop]
        exact ⟨x, hx, rfl⟩
      · simp only [abs_of_nonpos h'x, Int.coe_natAbs, neg_neg]
  refine' ⟨u1 ∪ u2, A, _⟩
  calc
    (∑ x in u1 ∪ u2, (f x + ite (x = 0) (f 0) 0)) =
        (∑ x in u1 ∪ u2, f x) + ∑ x in u1 ∩ u2, f x := by
      rw [sum_add_distrib]
      congr 1
      refine' (sum_subset_zero_on_sdiff inter_subset_union _ _).symm
      · intro x hx
        suffices x ≠ 0 by simp only [this, if_false]
        rintro rfl
        simp at hx
      · intro x hx
        simp only [mem_inter, mem_image, exists_prop] at hx
        have : x = 0 := by
          

Proof state:
α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : AddCommGroup α✝
inst✝⁴ : TopologicalSpace α✝
inst✝³ : TopologicalAddGroup α✝
f✝ g : β → α✝
a✝ a₁ a₂ : α✝
α : Type u_5
inst✝² : AddCommMonoid α
inst✝¹ : TopologicalSpace α
inst✝ : ContinuousAdd α
a : α
f : ℤ → α
hf : HasSum f a
u : Finset ℤ
v' : Finset ℕ
hv' : image Int.natAbs u ⊆ v'
u1 : Finset ℤ := image (fun x => ↑x) v'
u2 : Finset ℤ := image (fun x => -↑x) v'
A : u ⊆ u1 ∪ u2
x : ℤ
hx : (∃ a ∈ v', ↑a = x) ∧ ∃ a ∈ v', -↑a = x
⊢ x = 0

Next tactic:
apply le_antisymm