-- Mathlib/MeasureTheory/Group/AddCircle.lean
theorem isAddFundamentalDomain_of_ae_ball (I : Set <| AddCircle T) (u x : AddCircle T)
    (hu : IsOfFinAddOrder u) (hI : I =ᵐ[volume] ball x (T / (2 * addOrderOf u))) :
    IsAddFundamentalDomain (AddSubgroup.zmultiples u) I

Proof state:
T : ℝ
hT : Fact (0 < T)
I : Set (AddCircle T)
u x : AddCircle T
hu : IsOfFinAddOrder u
G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u
n : ℕ := addOrderOf u
B : Set (AddCircle T) := ball x (T / (2 * ↑n))
hI : I =ᶠ[ae volume] B
⊢ 1 ≤ ↑n

Tactic:
norm_cast

Result:
T : ℝ
hT : Fact (0 < T)
I : Set (AddCircle T)
u x : AddCircle T
hu : IsOfFinAddOrder u
G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u
n : ℕ := addOrderOf u
B : Set (AddCircle T) := ball x (T / (2 * ↑n))
hI : I =ᶠ[ae volume] B
⊢ 1 ≤ n

Proof state:
T : ℝ
hT : Fact (0 < T)
I : Set (AddCircle T)
u x : AddCircle T
hu : IsOfFinAddOrder u
G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u
n : ℕ := addOrderOf u
B : Set (AddCircle T) := ball x (T / (2 * ↑n))
hI : I =ᶠ[ae volume] B
hn : 1 ≤ ↑n
⊢ IsAddFundamentalDomain (↥G) I volume

Tactic:
refine IsAddFundamentalDomain.mk_of_measure_univ_le ?_ ?_ ?_ ?_

Result:
case refine_1
T : ℝ
hT : Fact (0 < T)
I : Set (AddCircle T)
u x : AddCircle T
hu : IsOfFinAddOrder u
G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u
n : ℕ := addOrderOf u
B : Set (AddCircle T) := ball x (T / (2 * ↑n))
hI : I =ᶠ[ae volume] B
hn : 1 ≤ ↑n
⊢ NullMeasurableSet I volume

case refine_2
T : ℝ
hT : Fact (0 < T)
I : Set (AddCircle T)
u x : AddCircle T
hu : IsOfFinAddOrder u
G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u
n : ℕ := addOrderOf u
B : Set (AddCircle T) := ball x (T / (2 * ↑n))
hI : I =ᶠ[ae volume] B
hn : 1 ≤ ↑n
⊢ ∀ (g : ↥G), g ≠ 0 → AEDisjoint volume (g +ᵥ I) I

case refine_3
T : ℝ
hT : Fact (0 < T)
I : Set (AddCircle T)
u x : AddCircle T
hu : IsOfFinAddOrder u
G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u
n : ℕ := addOrderOf u
B : Set (AddCircle T) := ball x (T / (2 * ↑n))
hI : I =ᶠ[ae volume] B
hn : 1 ≤ ↑n
⊢ ∀ (g : ↥G), QuasiMeasurePreserving (fun x => g +ᵥ x) volume volume

case refine_4
T : ℝ
hT : Fact (0 < T)
I : Set (AddCircle T)
u x : AddCircle T
hu : IsOfFinAddOrder u
G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u
n : ℕ := addOrderOf u
B : Set (AddCircle T) := ball x (T / (2 * ↑n))
hI : I =ᶠ[ae volume] B
hn : 1 ≤ ↑n
⊢ volume univ ≤ ∑' (g : ↥G), volume (g +ᵥ I)

Proof state:
case refine_2.mk
T : ℝ
hT : Fact (0 < T)
I : Set (AddCircle T)
u x : AddCircle T
hu : IsOfFinAddOrder u
G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u
n : ℕ := addOrderOf u
B : Set (AddCircle T) := ball x (T / (2 * ↑n))
hI : I =ᶠ[ae volume] B
hn : 1 ≤ ↑n
g : AddCircle T
hg : g ∈ G
hg' : g ≠ 0
⊢ AEDisjoint volume (g +ᵥ I) I

Tactic:
refine AEDisjoint.congr (Disjoint.aedisjoint ?_)
  ((quasiMeasurePreserving_add_left volume (-g)).vadd_ae_eq_of_ae_eq g hI) hI

Result:
case refine_2.mk
T : ℝ
hT : Fact (0 < T)
I : Set (AddCircle T)
u x : AddCircle T
hu : IsOfFinAddOrder u
G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u
n : ℕ := addOrderOf u
B : Set (AddCircle T) := ball x (T / (2 * ↑n))
hI : I =ᶠ[ae volume] B
hn : 1 ≤ ↑n
g : AddCircle T
hg : g ∈ G
hg' : g ≠ 0
⊢ Disjoint (g +ᵥ B) B

Proof state:
case refine_2.mk.h
T : ℝ
hT : Fact (0 < T)
I : Set (AddCircle T)
u x : AddCircle T
hu : IsOfFinAddOrder u
G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u
n : ℕ := addOrderOf u
B : Set (AddCircle T) := ball x (T / (2 * ↑n))
hI : I =ᶠ[ae volume] B
hn : 1 ≤ ↑n
g : AddCircle T
hg : g ∈ G
hg' : g ≠ 0
hBg : g +ᵥ B = ball (g + x) (T / (2 * ↑n))
⊢ addOrderOf g ≤ n

Tactic:
exact Nat.le_of_dvd (addOrderOf_pos_iff.mpr hu) (addOrderOf_dvd_of_mem_zmultiples hg)

Result:
no goals

Proof state:
T : ℝ
hT : Fact (0 < T)
I : Set (AddCircle T)
u x : AddCircle T
hu : IsOfFinAddOrder u
hI : I =ᶠ[ae volume] ball x (T / (2 * ↑(addOrderOf u)))
⊢ IsAddFundamentalDomain (↥(AddSubgroup.zmultiples u)) I volume

Tactic:
set G := AddSubgroup.zmultiples u

Result:
T : ℝ
hT : Fact (0 < T)
I : Set (AddCircle T)
u x : AddCircle T
hu : IsOfFinAddOrder u
hI : I =ᶠ[ae volume] ball x (T / (2 * ↑(addOrderOf u)))
G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u
⊢ IsAddFundamentalDomain (↥G) I volume

Proof state:
T : ℝ
hT : Fact (0 < T)
I : Set (AddCircle T)
u x : AddCircle T
hu : IsOfFinAddOrder u
G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u
n : ℕ := addOrderOf u
B : Set (AddCircle T) := ball x (T / (2 * ↑n))
hI : I =ᶠ[ae volume] B
hn : 1 ≤ ↑n
g : AddCircle T
hg : g ∈ G
hg' : g ≠ 0
⊢ g +ᵥ B = ball (g + x) (T / (2 * ↑n))

Tactic:
rw [add_comm g x, ← singleton_add_ball _ x g, add_ball, thickening_singleton]

Result:
no goals

Proof state:
case refine_2.mk
T : ℝ
hT : Fact (0 < T)
I : Set (AddCircle T)
u x : AddCircle T
hu : IsOfFinAddOrder u
G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u
n : ℕ := addOrderOf u
B : Set (AddCircle T) := ball x (T / (2 * ↑n))
hI : I =ᶠ[ae volume] B
hn : 1 ≤ ↑n
g : AddCircle T
hg : g ∈ G
hg' : g ≠ 0
⊢ Disjoint (g +ᵥ B) B

Tactic:
have hBg : g +ᵥ B = ball (g + x) (T / (2 * n)) := by
  rw [add_comm g x, ← singleton_add_ball _ x g, add_ball, thickening_singleton]

Result:
case refine_2.mk
T : ℝ
hT : Fact (0 < T)
I : Set (AddCircle T)
u x : AddCircle T
hu : IsOfFinAddOrder u
G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u
n : ℕ := addOrderOf u
B : Set (AddCircle T) := ball x (T / (2 * ↑n))
hI : I =ᶠ[ae volume] B
hn : 1 ≤ ↑n
g : AddCircle T
hg : g ∈ G
hg' : g ≠ 0
hBg : g +ᵥ B = ball (g + x) (T / (2 * ↑n))
⊢ Disjoint (g +ᵥ B) B

Proof state:
T : ℝ
hT : Fact (0 < T)
I : Set (AddCircle T)
u x : AddCircle T
hu : IsOfFinAddOrder u
G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u
n : ℕ := addOrderOf u
B : Set (AddCircle T) := ball x (T / (2 * ↑n))
hn : 1 ≤ ↑n
hI : I =ᶠ[ae volume] closedBall x (T / (2 * ↑n))
this : Fintype ↥G
⊢ Finset.univ.card = Fintype.card ↥(AddSubgroup.zmultiples u)

Tactic:
rfl

Result:
no goals

Proof state:
case refine_4
T : ℝ
hT : Fact (0 < T)
I : Set (AddCircle T)
u x : AddCircle T
hu : IsOfFinAddOrder u
G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u
n : ℕ := addOrderOf u
B : Set (AddCircle T) := ball x (T / (2 * ↑n))
hI : I =ᶠ[ae volume] B
hn : 1 ≤ ↑n
⊢ volume univ ≤ ∑' (g : ↥G), volume (g +ᵥ I)

Tactic:
replace hI := hI.trans closedBall_ae_eq_ball.symm

Result:
case refine_4
T : ℝ
hT : Fact (0 < T)
I : Set (AddCircle T)
u x : AddCircle T
hu : IsOfFinAddOrder u
G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u
n : ℕ := addOrderOf u
B : Set (AddCircle T) := ball x (T / (2 * ↑n))
hn : 1 ≤ ↑n
hI : I =ᶠ[ae volume] closedBall x (T / (2 * ↑n))
⊢ volume univ ≤ ∑' (g : ↥G), volume (g +ᵥ I)

Proof state:
case refine_2.mk.h
T : ℝ
hT : Fact (0 < T)
I : Set (AddCircle T)
u x : AddCircle T
hu : IsOfFinAddOrder u
G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u
n : ℕ := addOrderOf u
B : Set (AddCircle T) := ball x (T / (2 * ↑n))
hI : I =ᶠ[ae volume] B
hn : 1 ≤ ↑n
g : AddCircle T
hg : g ∈ G
hg' : g ≠ 0
hBg : g +ᵥ B = ball (g + x) (T / (2 * ↑n))
⊢ T / (2 * ↑n) + T / (2 * ↑n) ≤ dist (g + x) x

Tactic:
rw [dist_eq_norm, add_sub_cancel_right, div_mul_eq_div_div, ← add_div, ← add_div,
  add_self_div_two, div_le_iff₀' (by positivity : 0 < (n : ℝ)), ← nsmul_eq_mul]

Result:
case refine_2.mk.h
T : ℝ
hT : Fact (0 < T)
I : Set (AddCircle T)
u x : AddCircle T
hu : IsOfFinAddOrder u
G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u
n : ℕ := addOrderOf u
B : Set (AddCircle T) := ball x (T / (2 * ↑n))
hI : I =ᶠ[ae volume] B
hn : 1 ≤ ↑n
g : AddCircle T
hg : g ∈ G
hg' : g ≠ 0
hBg : g +ᵥ B = ball (g + x) (T / (2 * ↑n))
⊢ T ≤ n • ‖g‖