Theorem: is_ascending_rev_series_of_is_descending
File: Mathlib/GroupTheory/Nilpotent.lean

tactic: cases' hdesc with h0 hH
state_before: G : Type u_1
inst✝¹ : Group G
H✝ : Subgroup G
inst✝ : Normal H✝
H : ℕ → Subgroup G
n : ℕ
hn : H n = ⊥
hdesc : IsDescendingCentralSeries H
⊢ IsAscendingCentralSeries fun m => H (n - m)

tactic: refine' ⟨hn, fun x m hx g => _⟩
state_before: case intro
G : Type u_1
inst✝¹ : Group G
H✝ : Subgroup G
inst✝ : Normal H✝
H : ℕ → Subgroup G
n : ℕ
hn : H n = ⊥
h0 : H 0 = ⊤
hH : ∀ (x : G) (n : ℕ), x ∈ H n → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H (n + 1)
⊢ IsAscendingCentralSeries fun m => H (n - m)

tactic: dsimp only at hx⊢
state_before: case intro
G : Type u_1
inst✝¹ : Group G
H✝ : Subgroup G
inst✝ : Normal H✝
H : ℕ → Subgroup G
n : ℕ
hn : H n = ⊥
h0 : H 0 = ⊤
hH : ∀ (x : G) (n : ℕ), x ∈ H n → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H (n + 1)
x : G
m : ℕ
hx : x ∈ (fun m => H (n - m)) (m + 1)
g : G
⊢ x * g * x⁻¹ * g⁻¹ ∈ (fun m => H (n - m)) m

tactic: by_cases hm : n ≤ m
state_before: case intro
G : Type u_1
inst✝¹ : Group G
H✝ : Subgroup G
inst✝ : Normal H✝
H : ℕ → Subgroup G
n : ℕ
hn : H n = ⊥
h0 : H 0 = ⊤
hH : ∀ (x : G) (n : ℕ), x ∈ H n → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H (n + 1)
x : G
m : ℕ
hx : x ∈ H (n - (m + 1))
g : G
⊢ x * g * x⁻¹ * g⁻¹ ∈ H (n - m)

tactic: have hnm : n - m = 0 := tsub_eq_zero_iff_le.mpr hm
state_before: case pos
G : Type u_1
inst✝¹ : Group G
H✝ : Subgroup G
inst✝ : Normal H✝
H : ℕ → Subgroup G
n : ℕ
hn : H n = ⊥
h0 : H 0 = ⊤
hH : ∀ (x : G) (n : ℕ), x ∈ H n → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H (n + 1)
x : G
m : ℕ
hx : x ∈ H (n - (m + 1))
g : G
hm : n ≤ m
⊢ x * g * x⁻¹ * g⁻¹ ∈ H (n - m)

tactic: rw [hnm, h0]
state_before: case pos
G : Type u_1
inst✝¹ : Group G
H✝ : Subgroup G
inst✝ : Normal H✝
H : ℕ → Subgroup G
n : ℕ
hn : H n = ⊥
h0 : H 0 = ⊤
hH : ∀ (x : G) (n : ℕ), x ∈ H n → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H (n + 1)
x : G
m : ℕ
hx : x ∈ H (n - (m + 1))
g : G
hm : n ≤ m
hnm : n - m = 0
⊢ x * g * x⁻¹ * g⁻¹ ∈ H (n - m)

tactic: exact mem_top _
state_before: case pos
G : Type u_1
inst✝¹ : Group G
H✝ : Subgroup G
inst✝ : Normal H✝
H : ℕ → Subgroup G
n : ℕ
hn : H n = ⊥
h0 : H 0 = ⊤
hH : ∀ (x : G) (n : ℕ), x ∈ H n → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H (n + 1)
x : G
m : ℕ
hx : x ∈ H (n - (m + 1))
g : G
hm : n ≤ m
hnm : n - m = 0
⊢ x * g * x⁻¹ * g⁻¹ ∈ ⊤

tactic: push_neg  at hm
state_before: case neg
G : Type u_1
inst✝¹ : Group G
H✝ : Subgroup G
inst✝ : Normal H✝
H : ℕ → Subgroup G
n : ℕ
hn : H n = ⊥
h0 : H 0 = ⊤
hH : ∀ (x : G) (n : ℕ), x ∈ H n → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H (n + 1)
x : G
m : ℕ
hx : x ∈ H (n - (m + 1))
g : G
hm : ¬n ≤ m
⊢ x * g * x⁻¹ * g⁻¹ ∈ H (n - m)

tactic: convert hH x _ hx g using 1
state_before: case neg
G : Type u_1
inst✝¹ : Group G
H✝ : Subgroup G
inst✝ : Normal H✝
H : ℕ → Subgroup G
n : ℕ
hn : H n = ⊥
h0 : H 0 = ⊤
hH : ∀ (x : G) (n : ℕ), x ∈ H n → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H (n + 1)
x : G
m : ℕ
hx : x ∈ H (n - (m + 1))
g : G
hm : m < n
⊢ x * g * x⁻¹ * g⁻¹ ∈ H (n - m)

tactic: rw [tsub_add_eq_add_tsub (Nat.succ_le_of_lt hm), Nat.succ_sub_succ]
state_before: case h.e'_5
G : Type u_1
inst✝¹ : Group G
H✝ : Subgroup G
inst✝ : Normal H✝
H : ℕ → Subgroup G
n : ℕ
hn : H n = ⊥
h0 : H 0 = ⊤
hH : ∀ (x : G) (n : ℕ), x ∈ H n → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H (n + 1)
x : G
m : ℕ
hx : x ∈ H (n - (m + 1))
g : G
hm : m < n
⊢ H (n - m) = H (n - (m + 1) + 1)