mathlib documentation

data.nat.dist

Distance function on ℕ #

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

def nat.dist (n m : ℕ) :

Distance (absolute value of difference) between natural numbers.

Equations

n.dist m = n - m + (m - n)

theorem nat.dist.def (n m : ℕ) :

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) :

0.dist 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_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