Distance function on ℕ

This file defines a simple distance function on naturals from truncated subtraction.

def Nat.dist (n m : ℕ) : ℕ

Distance (absolute value of difference) between natural numbers.

Equations

• n.dist m = n - m + (m - n)

theorem Nat.dist_comm (n m : ℕ) : n.dist m = m.dist n

@[simp]

theorem Nat.dist_self (n : ℕ) : n.dist n = 0

theorem Nat.eq_of_dist_eq_zero {n m : ℕ} (h : n.dist m = 0) : n = m

theorem Nat.dist_eq_zero {n m : ℕ} (h : n = m) : n.dist m = 0

theorem Nat.dist_eq_sub_of_le {n m : ℕ} (h : n ≤ m) : n.dist m = m - n

theorem Nat.dist_eq_sub_of_le_right {n m : ℕ} (h : m ≤ n) : n.dist m = n - m

theorem Nat.dist_tri_left (n m : ℕ) : m ≤ n.dist m + n

theorem Nat.dist_tri_right (n m : ℕ) : m ≤ n + n.dist m

theorem Nat.dist_tri_left' (n m : ℕ) : n ≤ n.dist m + m

theorem Nat.dist_tri_right' (n m : ℕ) : n ≤ m + n.dist m

theorem Nat.dist_zero_right (n : ℕ) : n.dist 0 = n

theorem Nat.dist_zero_left (n : ℕ) : Nat.dist 0 n = n

theorem Nat.dist_add_add_right (n k m : ℕ) : (n + k).dist (m + k) = n.dist m

theorem Nat.dist_add_add_left (k n m : ℕ) : (k + n).dist (k + m) = n.dist m

theorem Nat.dist_eq_intro {n m k l : ℕ} (h : n + m = k + l) : n.dist k = l.dist m

theorem Nat.dist.triangle_inequality (n m k : ℕ) : n.dist k ≤ n.dist m + m.dist k

theorem Nat.dist_mul_right (n k m : ℕ) : (n * k).dist (m * k) = n.dist m * k

theorem Nat.dist_mul_left (k n m : ℕ) : (k * n).dist (k * m) = k * n.dist m

theorem Nat.dist_eq_max_sub_min {i j : ℕ} : i.dist j = i ⊔ j - i ⊓ j

theorem Nat.dist_succ_succ {i j : ℕ} : i.succ.dist j.succ = i.dist j

theorem Nat.dist_pos_of_ne {i j : ℕ} : i ≠ j → 0 < i.dist j