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.

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theorem Nat.periodic_gcd (a : ℕ) :

Function.Periodic (Nat.gcd a) a

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theorem Nat.periodic_coprime (a : ℕ) :

Function.Periodic (Nat.Coprime a) a

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theorem Nat.periodic_mod (a : ℕ) :

Function.Periodic (fun n => n % a) a

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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

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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.

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theorem Nat.filter_Ico_card_eq_of_periodic (n : ℕ) (a : ℕ) (p : ℕ → Prop) [DecidablePred p] (pp : Function.Periodic p a) :

Finset.card (Finset.filter p (Finset.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.