English
The iterate of a pointwise map equals the pointwise iterate of the maps: (Pi.map f)^{[n]} = Pi.map (i ↦ f(i)^{[n]}).
Русский
Итерация отображения по Карафонову типа равна точечной итерации: (Pi.map f)^{[n]} = Pi.map (i ↦ f(i)^{[n]}).
LaTeX
$$$$\forall ι,\alpha:\, ι\to \text{Type},\ f:\forall i,\alpha(i)\to \alpha(i),\ n:\mathbb{N},\ \mathrm{Nat.iterate}(\mathrm{Pi.map}\ f, n) = \mathrm{Pi.map}\ (i \mapsto \mathrm{Nat.iterate}(f(i), n)). $$$$
Lean4
@[simp]
theorem map_iterate {α : ι → Type*} (f : ∀ i, α i → α i) (n : ℕ) : (Pi.map f)^[n] = Pi.map fun i => (f i)^[n] := by
induction n <;> simp [*, map_comp_map]