English
The decidable ≤ instance is determined by the inductive definition of ≤ on Localization and agrees with mk-based representations; it equates with the computed decision procedure on elements.
Русский
Разрешимое ≤ определяется через индуктивное определение ≤ на локализации и согласуется с представлениями через mk; совпадает с вычислимым алгоритмом сравнения элементов.
LaTeX
$$$(a \\le b) \\text{ is decidable and equals the mk-based decision}$$$
Lean4
/-- If `H` is a normal subgroup of `G`, then the set `{x : G | ∀ y : G, x*y*x⁻¹*y⁻¹ ∈ H}`
is a subgroup of `G` (because it is the preimage in `G` of the centre of the
quotient group `G/H`.)
-/
def upperCentralSeriesStep : Subgroup G
where
carrier := {x : G | ∀ y : G, x * y * x⁻¹ * y⁻¹ ∈ H}
one_mem' y := by simp
mul_mem' {a b} ha hb
y := by
convert Subgroup.mul_mem _ (ha (b * y * b⁻¹)) (hb y) using 1
group
inv_mem' {x} hx
y := by
specialize hx y⁻¹
rw [mul_assoc, inv_inv] at hx ⊢
exact Subgroup.Normal.mem_comm inferInstance hx
