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data.nat.periodic

Periodic Functions on ℕ #

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

theorem nat.periodic_mod (a : ) :
function.periodic (λ (n : ), n % a) a
theorem function.periodic.map_mod_nat {α : Type u_1} {f : α} {a : } (hf : function.periodic f a) (n : ) :
f (n % a) = f n

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 a : ) (p : Prop) [decidable_pred p] (pp : function.periodic 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.