Periodic Functions on ℕ

This file identifies a few functions on ℕ which are periodic, and also proves a lemma about periodic predicates which helps determine their cardinality when filtering intervals over them.

theorem Nat.periodic_gcd (a : ℕ) : Function.Periodic (Nat.gcd a) a

theorem Nat.periodic_coprime (a : ℕ) : Function.Periodic (Nat.Coprime a) a

theorem Nat.periodic_mod (a : ℕ) : Function.Periodic (fun (n : ℕ) => n % a) a

theorem Function.Periodic.map_mod_nat {α : Type u_1} {f : ℕ → α} {a : ℕ} (hf : Function.Periodic f a) (n : ℕ) : f (n % a) = f n

theorem Nat.filter_multiset_Ico_card_eq_of_periodic (n : ℕ) (a : ℕ) (p : ℕ → Prop) [DecidablePred p] (pp : Function.Periodic p a) : Multiset.card (Multiset.filter p (Multiset.Ico n (n + a))) = Nat.count p a

An interval of length a filtered over a periodic predicate of period a has cardinality equal to the number naturals below a for which p a is true.

theorem Nat.filter_Ico_card_eq_of_periodic (n : ℕ) (a : ℕ) (p : ℕ → Prop) [DecidablePred p] (pp : Function.Periodic p a) : (Finset.filter p (Finset.Ico n (n + a))).card = Nat.count p a

An interval of length a filtered over a periodic predicate of period a has cardinality equal to the number naturals below a for which p a is true.