Module: Mathlib.GroupTheory.CosetCover

Goal state before:
pp: case pos
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 ∈ {x ∈ s | H x = H j}, 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 ∈ {x ∈ s | H x = H j}, g i • ↑(H i) ≠ Set.univ
hn : n = (Finset.image H s).card
x : G
hx : ∀ (y : G), y • ↑(H j) ⊆ ⋃ i ∈ {x ∈ s | H x ≠ H j}, (y * x⁻¹ * g i) • ↑(H i)
κ : Type u_2 := ↥({x ∈ s | H x ≠ H j}) × Option ↥({x ∈ s | H x = H j})
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 ∈ {x | K x = K k}, f i • ↑(K i) = Set.univ
⊢ ∃ i ∈ s, H i ≠ H j ∧ (H i).FiniteIndex

pp: case neg
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 ∈ {x ∈ s | H x = H j}, 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 ∈ {x ∈ s | H x = H j}, g i • ↑(H i) ≠ Set.univ
hn : n = (Finset.image H s).card
x : G
hx : ∀ (y : G), y • ↑(H j) ⊆ ⋃ i ∈ {x ∈ s | H x ≠ H j}, (y * x⁻¹ * g i) • ↑(H i)
κ : Type u_2 := ↥({x ∈ s | H x ≠ H j}) × Option ↥({x ∈ s | H x = H j})
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 ∈ {x | K x = K k}, f i • ↑(K i) = Set.univ
⊢ ∃ i ∈ s, H i ≠ H j ∧ (H i).FiniteIndex

Tactic:
· rw [Set.iUnion₂_congr fun i hi => by rw [(Finset.mem_filter.mp hi).right]] at hcovers'
      exact ⟨k.1, Finset.mem_of_mem_filter k.1.1 k.1.2, hK k,
        finiteIndex_of_leftCoset_cover_const hcovers'⟩

Goal state after:
case neg
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 ∈ {x ∈ s | H x = H j}, 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 ∈ {x ∈ s | H x = H j}, g i • ↑(H i) ≠ Set.univ
hn : n = (Finset.image H s).card
x : G
hx : ∀ (y : G), y • ↑(H j) ⊆ ⋃ i ∈ {x ∈ s | H x ≠ H j}, (y * x⁻¹ * g i) • ↑(H i)
κ : Type u_2 := ↥({x ∈ s | H x ≠ H j}) × Option ↥({x ∈ s | H x = H j})
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 ∈ {x | K x = K k}, f i • ↑(K i) = Set.univ
⊢ ∃ i ∈ s, H i ≠ H j ∧ (H i).FiniteIndex