The integers as normed ring

This file contains basic facts about the integers as normed ring.

Recall that ‖n‖ denotes the norm of n as real number. This norm is always nonnegative, so we can bundle the norm together with this fact, to obtain a term of type NNReal (the nonnegative real numbers). The resulting nonnegative real number is denoted by ‖n‖₊.

theorem Int.nnnorm_coe_units (e : ℤˣ) : ‖↑e‖₊ = 1

theorem Int.norm_coe_units (e : ℤˣ) : ‖↑e‖ = 1

@[simp]

theorem Int.nnnorm_coe_nat (n : ℕ) : ‖↑n‖₊ = ↑n

@[simp]

theorem Int.toNat_add_toNat_neg_eq_nnnorm (n : ℤ) : ↑(Int.toNat n) + ↑(Int.toNat (-n)) = ‖n‖₊

@[simp]

theorem Int.toNat_add_toNat_neg_eq_norm (n : ℤ) : ↑(Int.toNat n) + ↑(Int.toNat (-n)) = ‖n‖