English
If x is periodic with period n, then f^{m mod n} (x) = f^{m}(x) for all m.
Русский
Если x имеет период n, то f^{m mod n}(x) = f^{m}(x) для всех m.
LaTeX
$$$IsPeriodicPt f n x \to \forall m, f^{m \% n}(x) = f^{m}(x)$$$
Lean4
protected theorem gcd (hm : IsPeriodicPt f m x) (hn : IsPeriodicPt f n x) : IsPeriodicPt f (m.gcd n) x :=
by
revert hm hn
refine Nat.gcd.induction m n (fun n _ hn => ?_) fun m n _ ih hm hn => ?_
· rwa [Nat.gcd_zero_left]
· rw [Nat.gcd_rec]
exact ih (hn.mod hm) hm