Cast of naturals into fields

This file concerns the canonical homomorphism ℕ → F, where F is a field.

Main results

• Nat.cast_div: if n divides m, then ↑(m / n) = ↑m / ↑n
• Nat.cast_div_le: in all cases, ↑(m / n) ≤ ↑m / ↑ n

@[simp]

theorem Nat.cast_div {α : Type u_1} [DivisionSemiring α] {m : ℕ} {n : ℕ} (n_dvd : n ∣ m) (n_nonzero : ↑n ≠ 0) : ↑(m / n) = ↑m / ↑n

theorem Nat.cast_div_div_div_cancel_right {α : Type u_1} [DivisionSemiring α] [CharZero α] {m : ℕ} {n : ℕ} {d : ℕ} (hn : d ∣ n) (hm : d ∣ m) : ↑(m / d) / ↑(n / d) = ↑m / ↑n

theorem Nat.cast_inv_le_one {α : Type u_1} [LinearOrderedSemifield α] (n : ℕ) : (↑n)⁻¹ ≤ 1

theorem Nat.cast_div_le {α : Type u_1} [LinearOrderedSemifield α] {m : ℕ} {n : ℕ} : ↑(m / n) ≤ ↑m / ↑n

Natural division is always less than division in the field.

theorem Nat.inv_pos_of_nat {α : Type u_1} [LinearOrderedSemifield α] {n : ℕ} : 0 < (↑n + 1)⁻¹

theorem Nat.one_div_pos_of_nat {α : Type u_1} [LinearOrderedSemifield α] {n : ℕ} : 0 < 1 / (↑n + 1)

theorem Nat.one_div_le_one_div {α : Type u_1} [LinearOrderedSemifield α] {n : ℕ} {m : ℕ} (h : n ≤ m) : 1 / (↑m + 1) ≤ 1 / (↑n + 1)

theorem Nat.one_div_lt_one_div {α : Type u_1} [LinearOrderedSemifield α] {n : ℕ} {m : ℕ} (h : n < m) : 1 / (↑m + 1) < 1 / (↑n + 1)