-- Mathlib/GroupTheory/CosetCover.lean
theorem exists_finiteIndex_of_leftCoset_cover_aux [DecidableEq (Subgroup G)]
    (j : ι) (hj : j ∈ s) (hcovers' : ⋃ i ∈ s.filter (H · = H j), g i • (H i : Set G) ≠ Set.univ) :
    ∃ i ∈ s, H i ≠ H j ∧ (H i).FiniteIndex

Proof state:
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq (Subgroup G)
n : ℕ
ih :
  ∀ m < n,
    ∀ {ι : Type u_2} {H : ι → Subgroup G} {g : ι → G} {s : Finset ι},
      ⋃ i ∈ s, g i • ↑(H i) = Set.univ →
        ∀ j ∈ s,
          ⋃ i ∈ Finset.filter (fun x => H x = H j) s, g i • ↑(H i) ≠ Set.univ →
            m = (Finset.image H s).card → ∃ i ∈ s, H i ≠ H j ∧ (H i).FiniteIndex
ι : Type u_2
H : ι → Subgroup G
g : ι → G
s : Finset ι
j : ι
hcovers :
  (⋃ x ∈ Finset.filter (fun x => H x = H j) s, g x • ↑(H x)) ∪
      ⋃ x ∈ Finset.filter (fun a => ¬H a = H j) s, g x • ↑(H x) =
    Set.univ
hj : j ∈ s
hcovers' : ⋃ i ∈ Finset.filter (fun x => H x = H j) s, g i • ↑(H i) ≠ Set.univ
hn : n = (Finset.image H s).card
x : G
hx : ∀ (y : G), y • ↑(H j) ⊆ ⋃ i ∈ Finset.filter (fun x => H x ≠ H j) s, (y * x⁻¹ * g i) • ↑(H i)
κ : Type u_2 :=
  { x // x ∈ Finset.filter (fun x => H x ≠ H j) s } × Option { x // x ∈ Finset.filter (fun x => H x = H j) s }
f : κ → G :=
  fun x_1 =>
    match x_1 with
    | (k₁, some k₂) => g ↑k₂ * x⁻¹ * g ↑k₁
    | (k₁, none) => g ↑k₁
K : κ → Subgroup G := fun k => H ↑k.1
hK' : ∀ (k : κ), K k ∈ (Finset.image H s).erase (H j)
hK : ∀ (k : κ), K k ≠ H j
y : G
⊢ ∃ i, ∃ (_ : i ∈ Finset.univ), y ∈ f i • ↑(K i)

Tactic:
cases (Set.mem_union _ _ _).mp (hcovers.superset (Set.mem_univ y)) with
| inl hy =>
  have ⟨k, hk, hy⟩ := Set.mem_iUnion₂.mp hy
  have hk' : H k = H j := And.right <| by simpa using hk
  have ⟨i, hi, hy⟩ := Set.mem_iUnion₂.mp (hx (g k) (hk' ▸ hy))
  exact ⟨⟨⟨i, hi⟩, some ⟨k, hk⟩⟩, Finset.mem_univ _, hy⟩
| inr hy =>
  have ⟨i, hi, hy⟩ := Set.mem_iUnion₂.mp hy
  exact ⟨⟨⟨i, hi⟩, none⟩, Finset.mem_univ _, hy⟩

Result:
no goals

Proof state:
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq (Subgroup G)
n : ℕ
ih :
  ∀ m < n,
    ∀ {ι : Type u_2} {H : ι → Subgroup G} {g : ι → G} {s : Finset ι},
      ⋃ i ∈ s, g i • ↑(H i) = Set.univ →
        ∀ j ∈ s,
          ⋃ i ∈ Finset.filter (fun x => H x = H j) s, g i • ↑(H i) ≠ Set.univ →
            m = (Finset.image H s).card → ∃ i ∈ s, H i ≠ H j ∧ (H i).FiniteIndex
ι : Type u_2
H : ι → Subgroup G
g : ι → G
s : Finset ι
j : ι
hj : j ∈ s
hcovers'✝ : ⋃ i ∈ Finset.filter (fun x => H x = H j) s, g i • ↑(H i) ≠ Set.univ
hn : n = (Finset.image H s).card
x : G
hx : ∀ (y : G), y • ↑(H j) ⊆ ⋃ i ∈ Finset.filter (fun x => H x ≠ H j) s, (y * x⁻¹ * g i) • ↑(H i)
κ : Type u_2 :=
  { x // x ∈ Finset.filter (fun x => H x ≠ H j) s } × Option { x // x ∈ Finset.filter (fun x => H x = H j) s }
f : κ → G :=
  fun x_1 =>
    match x_1 with
    | (k₁, some k₂) => g ↑k₂ * x⁻¹ * g ↑k₁
    | (k₁, none) => g ↑k₁
K : κ → Subgroup G := fun k => H ↑k.1
hK' : ∀ (k : κ), K k ∈ (Finset.image H s).erase (H j)
hK : ∀ (k : κ), K k ≠ H j
hcovers : ⋃ k ∈ Finset.univ, f k • ↑(K k) = Set.univ
k : κ
hcovers' : ¬⋃ i ∈ Finset.filter (fun x => K x = K k) Finset.univ, f i • ↑(K i) = Set.univ
⊢ (Finset.image K Finset.univ).card < (Finset.image H s).card

Tactic:
refine ((Finset.card_le_card fun x => ?_).trans_lt <|
  Finset.card_erase_lt_of_mem (Finset.mem_image_of_mem H hj))

Result:
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq (Subgroup G)
n : ℕ
ih :
  ∀ m < n,
    ∀ {ι : Type u_2} {H : ι → Subgroup G} {g : ι → G} {s : Finset ι},
      ⋃ i ∈ s, g i • ↑(H i) = Set.univ →
        ∀ j ∈ s,
          ⋃ i ∈ Finset.filter (fun x => H x = H j) s, g i • ↑(H i) ≠ Set.univ →
            m = (Finset.image H s).card → ∃ i ∈ s, H i ≠ H j ∧ (H i).FiniteIndex
ι : Type u_2
H : ι → Subgroup G
g : ι → G
s : Finset ι
j : ι
hj : j ∈ s
hcovers'✝ : ⋃ i ∈ Finset.filter (fun x => H x = H j) s, g i • ↑(H i) ≠ Set.univ
hn : n = (Finset.image H s).card
x✝ : G
hx : ∀ (y : G), y • ↑(H j) ⊆ ⋃ i ∈ Finset.filter (fun x => H x ≠ H j) s, (y * x✝⁻¹ * g i) • ↑(H i)
κ : Type u_2 :=
  { x // x ∈ Finset.filter (fun x => H x ≠ H j) s } × Option { x // x ∈ Finset.filter (fun x => H x = H j) s }
f : κ → G :=
  fun x =>
    match x with
    | (k₁, some k₂) => g ↑k₂ * x✝⁻¹ * g ↑k₁
    | (k₁, none) => g ↑k₁
K : κ → Subgroup G := fun k => H ↑k.1
hK' : ∀ (k : κ), K k ∈ (Finset.image H s).erase (H j)
hK : ∀ (k : κ), K k ≠ H j
hcovers : ⋃ k ∈ Finset.univ, f k • ↑(K k) = Set.univ
k : κ
hcovers' : ¬⋃ i ∈ Finset.filter (fun x => K x = K k) Finset.univ, f i • ↑(K i) = Set.univ
x : Subgroup G
⊢ x ∈ Finset.image K Finset.univ → x ∈ (Finset.image H s).erase (H j)

Proof state:
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq (Subgroup G)
n : ℕ
ih :
  ∀ m < n,
    ∀ {ι : Type u_2} {H : ι → Subgroup G} {g : ι → G} {s : Finset ι},
      ⋃ i ∈ s, g i • ↑(H i) = Set.univ →
        ∀ j ∈ s,
          ⋃ i ∈ Finset.filter (fun x => H x = H j) s, g i • ↑(H i) ≠ Set.univ →
            m = (Finset.image H s).card → ∃ i ∈ s, H i ≠ H j ∧ (H i).FiniteIndex
ι : Type u_2
H : ι → Subgroup G
g : ι → G
s : Finset ι
j : ι
hj : j ∈ s
hcovers'✝ : ⋃ i ∈ Finset.filter (fun x => H x = H j) s, g i • ↑(H i) ≠ Set.univ
hn : n = (Finset.image H s).card
x✝ : G
hx : ∀ (y : G), y • ↑(H j) ⊆ ⋃ i ∈ Finset.filter (fun x => H x ≠ H j) s, (y * x✝⁻¹ * g i) • ↑(H i)
κ : Type u_2 :=
  { x // x ∈ Finset.filter (fun x => H x ≠ H j) s } × Option { x // x ∈ Finset.filter (fun x => H x = H j) s }
f : κ → G :=
  fun x =>
    match x with
    | (k₁, some k₂) => g ↑k₂ * x✝⁻¹ * g ↑k₁
    | (k₁, none) => g ↑k₁
K : κ → Subgroup G := fun k => H ↑k.1
hK' : ∀ (k : κ), K k ∈ (Finset.image H s).erase (H j)
hK : ∀ (k : κ), K k ≠ H j
hcovers : ⋃ k ∈ Finset.univ, f k • ↑(K k) = Set.univ
k : κ
hcovers' : ¬⋃ i ∈ Finset.filter (fun x => K x = K k) Finset.univ, f i • ↑(K i) = Set.univ
x : Subgroup G
⊢ (∃ y, K y = x) → x ∈ (Finset.image H s).erase (H j)

Tactic:
exact fun ⟨k, hk⟩ => hk ▸ hK' k

Result:
no goals

Proof state:
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq (Subgroup G)
n : ℕ
ih :
  ∀ m < n,
    ∀ {ι : Type u_2} {H : ι → Subgroup G} {g : ι → G} {s : Finset ι},
      ⋃ i ∈ s, g i • ↑(H i) = Set.univ →
        ∀ j ∈ s,
          ⋃ i ∈ Finset.filter (fun x => H x = H j) s, g i • ↑(H i) ≠ Set.univ →
            m = (Finset.image H s).card → ∃ i ∈ s, H i ≠ H j ∧ (H i).FiniteIndex
ι : Type u_2
H : ι → Subgroup G
g : ι → G
s : Finset ι
hcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ
j : ι
hj : j ∈ s
hcovers' : ⋃ i ∈ Finset.filter (fun x => H x = H j) s, g i • ↑(H i) ≠ Set.univ
hn : n = (Finset.image H s).card
x : G
hx : ∀ i ∈ s, H i = H j → ↑(g i) ≠ ↑x
y z : G
hz : z ∈ y • ↑(H j)
⊢ z ∈ ⋃ i ∈ Finset.filter (fun x => H x ≠ H j) s, (y * x⁻¹ * g i) • ↑(H i)

Tactic:
simp_rw [Finset.mem_filter, Set.mem_iUnion]

Result:
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq (Subgroup G)
n : ℕ
ih :
  ∀ m < n,
    ∀ {ι : Type u_2} {H : ι → Subgroup G} {g : ι → G} {s : Finset ι},
      ⋃ i ∈ s, g i • ↑(H i) = Set.univ →
        ∀ j ∈ s,
          ⋃ i ∈ Finset.filter (fun x => H x = H j) s, g i • ↑(H i) ≠ Set.univ →
            m = (Finset.image H s).card → ∃ i ∈ s, H i ≠ H j ∧ (H i).FiniteIndex
ι : Type u_2
H : ι → Subgroup G
g : ι → G
s : Finset ι
hcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ
j : ι
hj : j ∈ s
hcovers' : ⋃ i ∈ Finset.filter (fun x => H x = H j) s, g i • ↑(H i) ≠ Set.univ
hn : n = (Finset.image H s).card
x : G
hx : ∀ i ∈ s, H i = H j → ↑(g i) ≠ ↑x
y z : G
hz : z ∈ y • ↑(H j)
⊢ ∃ i, ∃ (_ : i ∈ s ∧ H i ≠ H j), z ∈ (y * x⁻¹ * g i) • ↑(H i)