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

tactic: cases' hasc 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 = ⊤
hasc : IsAscendingCentralSeries H
⊢ IsDescendingCentralSeries 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 + 1) → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H n
⊢ IsDescendingCentralSeries fun m => H (n - m)

tactic: dsimp 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 + 1) → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H n
x : G
m : ℕ
hx : x ∈ (fun m => H (n - m)) m
g : G
⊢ x * g * x⁻¹ * g⁻¹ ∈ (fun m => H (n - m)) (m + 1)

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 + 1) → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H n
x : G
m : ℕ
hx : x ∈ H (n - m)
g : G
⊢ x * g * x⁻¹ * g⁻¹ ∈ (fun m => H (n - m)) (m + 1)

tactic: rw [tsub_eq_zero_of_le hm, h0, Subgroup.mem_bot] at hx
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 + 1) → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H n
x : G
m : ℕ
hx : x ∈ H (n - m)
g : G
hm : n ≤ m
⊢ x * g * x⁻¹ * g⁻¹ ∈ (fun m => H (n - m)) (m + 1)

tactic: subst hx
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 + 1) → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H n
x : G
m : ℕ
hx : x = 1
g : G
hm : n ≤ m
⊢ x * g * x⁻¹ * g⁻¹ ∈ (fun m => H (n - m)) (m + 1)

tactic: rw [show (1 : G) * g * (1⁻¹ : G) * g⁻¹ = 1 by group]
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 + 1) → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H n
m : ℕ
g : G
hm : n ≤ m
⊢ 1 * g * 1⁻¹ * g⁻¹ ∈ (fun m => H (n - m)) (m + 1)

tactic: exact Subgroup.one_mem _
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 + 1) → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H n
m : ℕ
g : G
hm : n ≤ m
⊢ 1 ∈ (fun m => H (n - m)) (m + 1)

tactic: group
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 = ⊤
h0 : H 0 = ⊥
hH : ∀ (x : G) (n : ℕ), x ∈ H (n + 1) → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H n
m : ℕ
g : G
hm : n ≤ m
⊢ 1 * g * 1⁻¹ * g⁻¹ = 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 + 1) → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H n
x : G
m : ℕ
hx : x ∈ H (n - m)
g : G
hm : ¬n ≤ m
⊢ x * g * x⁻¹ * g⁻¹ ∈ (fun m => H (n - m)) (m + 1)

tactic: apply hH
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 + 1) → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H n
x : G
m : ℕ
hx : x ∈ H (n - m)
g : G
hm : m < n
⊢ x * g * x⁻¹ * g⁻¹ ∈ (fun m => H (n - m)) (m + 1)

tactic: convert hx using 1
state_before: case neg.a
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 + 1) → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H n
x : G
m : ℕ
hx : x ∈ H (n - m)
g : G
hm : m < n
⊢ x ∈ H (n - (m + 1) + 1)

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 + 1) → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H n
x : G
m : ℕ
hx : x ∈ H (n - m)
g : G
hm : m < n
⊢ H (n - (m + 1) + 1) = H (n - m)