mathlib3 documentation


Admissible absolute value on the integers #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4. This file defines an admissible absolute value absolute_value.abs_is_admissible which we use to show the class number of the ring of integers of a number field is finite.

Main results #

theorem absolute_value.exists_partition_int (n : ) {ε : } (hε : 0 < ε) {b : } (hb : b 0) (A : fin n ) :
(t : fin n fin 1 / ε⌉₊), (i₀ i₁ : fin n), t i₀ = t i₁ |A i₁ % b - A i₀ % b| < |b| ε

We can partition a finite family into partition_card ε sets, such that the remainders in each set are close together.

abs : ℤ → ℤ is an admissible absolute value