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.norm_coe_units (e : ˣ) :
e = 1
theorem Int.nnnorm_natCast (n : ) :
n‖₊ = n
@[deprecated Int.nnnorm_natCast]
theorem Int.nnnorm_coe_nat (n : ) :
n‖₊ = n

Alias of Int.nnnorm_natCast.

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