-- Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean
theorem HasProd.nat_mul_neg {f : ℤ → M} (hf : HasProd f m) :
    HasProd (fun n : ℕ ↦ f n * f (-n)) (m * f 0)

Proof state:
case e_a.refine_1
M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
m : M
inst✝ : ContinuousMul M
f : ℤ → M
hf : HasProd f m
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 1) = 1

Tactic:
suffices x ≠ 0 by simp only [this, if_false]

Result:
case e_a.refine_1
M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
m : M
inst✝ : ContinuousMul M
f : ℤ → M
hf : HasProd f m
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

Proof state:
M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
m : M
inst✝ : ContinuousMul M
f : ℤ → M
hf : HasProd f m
⊢ HasProd (fun n => f ↑n * f (-↑n)) (m * f 0)

Tactic:
apply (hf.mul (hasProd_ite_eq (0 : ℤ) (f 0))).hasProd_of_prod_eq fun u ↦ ?_

Result:
M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
m : M
inst✝ : ContinuousMul M
f : ℤ → M
hf : HasProd f m
u : Finset ℤ
⊢ ∃ v, ∀ (v' : Finset ℕ), v ⊆ v' → ∃ u', u ⊆ u' ∧ (∏ x ∈ u', f x * if x = 0 then f 0 else 1) = ∏ b ∈ v', f ↑b * f (-↑b)

Proof state:
case e_a.refine_2
M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
m : M
inst✝ : ContinuousMul M
f : ℤ → M
hf : HasProd f m
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 1

Tactic:
intro x hx

Result:
case e_a.refine_2
M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
m : M
inst✝ : ContinuousMul M
f : ℤ → M
hf : HasProd f m
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 1

Proof state:
case e_a.refine_2
M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
m : M
inst✝ : ContinuousMul M
f : ℤ → M
hf : HasProd f m
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

Tactic:
omega

Result:
no goals

Proof state:
M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
m : M
inst✝ : ContinuousMul M
f : ℤ → M
hf : HasProd f m
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 1) = 1

Tactic:
simp only [this, if_false]

Result:
no goals

Proof state:
case e_a
M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
m : M
inst✝ : ContinuousMul M
f : ℤ → M
hf : HasProd f m
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, if x = 0 then f 0 else 1) = ∏ x ∈ u1 ∩ u2, f x

Tactic:
refine (prod_subset_one_on_sdiff inter_subset_union ?_ ?_).symm

Result:
case e_a.refine_1
M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
m : M
inst✝ : ContinuousMul M
f : ℤ → M
hf : HasProd f m
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 1) = 1

case e_a.refine_2
M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
m : M
inst✝ : ContinuousMul M
f : ℤ → M
hf : HasProd f m
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 1

Proof state:
case e_a.refine_1
M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
m : M
inst✝ : ContinuousMul M
f : ℤ → M
hf : HasProd f m
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 1) = 1

Tactic:
intro x hx

Result:
case e_a.refine_1
M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
m : M
inst✝ : ContinuousMul M
f : ℤ → M
hf : HasProd f m
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 1) = 1

Proof state:
M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
m : M
inst✝ : ContinuousMul M
f : ℤ → M
hf : HasProd f m
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'
⊢ u ⊆ u1 ∪ u2

Tactic:
intro x hx

Result:
M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
m : M
inst✝ : ContinuousMul M
f : ℤ → M
hf : HasProd f m
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'
x : ℤ
hx : x ∈ u
⊢ x ∈ u1 ∪ u2

Proof state:
M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
m : M
inst✝ : ContinuousMul M
f : ℤ → M
hf : HasProd f m
u : Finset ℤ
v' : Finset ℕ
hv' : image Int.natAbs u ⊆ v'
u1 : Finset ℤ := image (fun x => ↑x) v'
⊢ ∃ u', u ⊆ u' ∧ (∏ x ∈ u', f x * if x = 0 then f 0 else 1) = ∏ b ∈ v', f ↑b * f (-↑b)

Tactic:
let u2 := v'.image fun x : ℕ ↦ -(x : ℤ)

Result:
M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
m : M
inst✝ : ContinuousMul M
f : ℤ → M
hf : HasProd f m
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'
⊢ ∃ u', u ⊆ u' ∧ (∏ x ∈ u', f x * if x = 0 then f 0 else 1) = ∏ b ∈ v', f ↑b * f (-↑b)

Proof state:
case inr
M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
m : M
inst✝ : ContinuousMul M
f : ℤ → M
hf : HasProd f m
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'
x : ℤ
hx : x ∈ u
h'x : x ≤ 0
⊢ -↑x.natAbs = x

Tactic:
simp only [abs_of_nonpos h'x, Int.natCast_natAbs, neg_neg]

Result:
no goals

Proof state:
M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
m : M
inst✝ : ContinuousMul M
f : ℤ → M
hf : HasProd f m
u : Finset ℤ
v' : Finset ℕ
hv' : image Int.natAbs u ⊆ v'
⊢ ∃ u', u ⊆ u' ∧ (∏ x ∈ u', f x * if x = 0 then f 0 else 1) = ∏ b ∈ v', f ↑b * f (-↑b)

Tactic:
let u1 := v'.image fun x : ℕ ↦ (x : ℤ)

Result:
M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
m : M
inst✝ : ContinuousMul M
f : ℤ → M
hf : HasProd f m
u : Finset ℤ
v' : Finset ℕ
hv' : image Int.natAbs u ⊆ v'
u1 : Finset ℤ := image (fun x => ↑x) v'
⊢ ∃ u', u ⊆ u' ∧ (∏ x ∈ u', f x * if x = 0 then f 0 else 1) = ∏ b ∈ v', f ↑b * f (-↑b)