addition

multiplication

theorem Nat.eq_zero_of_mul_eq_zero {n : ℕ} {m : ℕ} : n * m = 0 → n = 0 ∨ m = 0

properties of inequality

theorem Nat.lt_of_le_and_ne {m : ℕ} {n : ℕ} (h1 : m ≤ n) : m ≠ n → m < n

instance Nat.linearOrder : LinearOrder ℕ

Equations

• Nat.linearOrder = LinearOrder.mk Nat.le_total inferInstance inferInstance inferInstance

def Nat.ltGeByCases {a : ℕ} {b : ℕ} {C : Sort u} (h₁ : a < b → C) (h₂ : b ≤ a → C) : C

Equations

• Nat.ltGeByCases h₁ h₂ = Decidable.byCases h₁ fun h => h₂ (_ : b ≤ a)

def Nat.ltByCases {a : ℕ} {b : ℕ} {C : Sort u} (h₁ : a < b → C) (h₂ : a = b → C) (h₃ : b < a → C) : C

Equations

• Nat.ltByCases h₁ h₂ h₃ = Nat.ltGeByCases h₁ fun h₁ => Nat.ltGeByCases h₃ fun h => h₂ (_ : a = b)

bit0/bit1 properties

theorem Nat.bit1_eq_succ_bit0 (n : ℕ) : bit1 n = Nat.succ (bit0 n)

theorem Nat.bit1_succ_eq (n : ℕ) : bit1 (Nat.succ n) = Nat.succ (Nat.succ (bit1 n))

theorem Nat.bit1_ne_one {n : ℕ} : n ≠ 0 → bit1 n ≠ 1

theorem Nat.bit0_ne_one (n : ℕ) : bit0 n ≠ 1

theorem Nat.bit1_ne_bit0 (n : ℕ) (m : ℕ) : bit1 n ≠ bit0 m

theorem Nat.bit0_ne_bit1 (n : ℕ) (m : ℕ) : bit0 n ≠ bit1 m

theorem Nat.bit0_inj {n : ℕ} {m : ℕ} : bit0 n = bit0 m → n = m

theorem Nat.bit1_inj {n : ℕ} {m : ℕ} : bit1 n = bit1 m → n = m

theorem Nat.bit0_ne {n : ℕ} {m : ℕ} : n ≠ m → bit0 n ≠ bit0 m

theorem Nat.bit1_ne {n : ℕ} {m : ℕ} : n ≠ m → bit1 n ≠ bit1 m

theorem Nat.zero_ne_bit0 {n : ℕ} : n ≠ 0 → 0 ≠ bit0 n

theorem Nat.zero_ne_bit1 (n : ℕ) : 0 ≠ bit1 n

theorem Nat.one_ne_bit0 (n : ℕ) : 1 ≠ bit0 n

theorem Nat.one_ne_bit1 {n : ℕ} : n ≠ 0 → 1 ≠ bit1 n

theorem Nat.one_lt_bit1 {n : ℕ} : n ≠ 0 → 1 < bit1 n

theorem Nat.one_lt_bit0 {n : ℕ} : n ≠ 0 → 1 < bit0 n

theorem Nat.bit0_lt {n : ℕ} {m : ℕ} (h : n < m) : bit0 n < bit0 m

theorem Nat.bit1_lt {n : ℕ} {m : ℕ} (h : n < m) : bit1 n < bit1 m

theorem Nat.bit0_lt_bit1 {n : ℕ} {m : ℕ} (h : n ≤ m) : bit0 n < bit1 m

theorem Nat.bit1_lt_bit0 {n : ℕ} {m : ℕ} : n < m → bit1 n < bit0 m

theorem Nat.one_le_bit1 (n : ℕ) : 1 ≤ bit1 n

theorem Nat.one_le_bit0 (n : ℕ) : n ≠ 0 → 1 ≤ bit0 n

successor and predecessor

def Nat.discriminate {B : Sort u} {n : ℕ} (H1 : n = 0 → B) (H2 : (m : ℕ) → n = Nat.succ m → B) : B

Equations

• Nat.discriminate H1 H2 = Nat.rec (fun x => n = x → B) (fun h => H1 h) (fun n n_ih h => H2 n h) n (_ : n = n)

theorem Nat.one_eq_succ_zero : 1 = Nat.succ 0

subtraction

Many lemmas are proven more generally in mathlib algebra/order/sub

theorem Nat.le_sub_iff_right {x : ℕ} {y : ℕ} {k : ℕ} (h : k ≤ y) : x ≤ y - k ↔ x + k ≤ y

theorem Nat.sub.right_comm (m : ℕ) (n : ℕ) (k : ℕ) : m - n - k = m - k - n

min

induction principles

def Nat.twoStepInduction {P : ℕ → Sort u} (H1 : P 0) (H2 : P 1) (H3 : (n : ℕ) → P n → P (Nat.succ n) → P (Nat.succ (Nat.succ n))) (a : ℕ) : P a

Equations

• Nat.twoStepInduction H1 H2 H3 0 = H1
• Nat.twoStepInduction H1 H2 H3 1 = H2
• Nat.twoStepInduction H1 H2 H3 (Nat.succ (Nat.succ _n)) = H3 _n (Nat.twoStepInduction H1 H2 H3 _n) (Nat.twoStepInduction H1 H2 H3 (Nat.succ _n))

def Nat.subInduction {P : ℕ → ℕ → Sort u} (H1 : (m : ℕ) → P 0 m) (H2 : (n : ℕ) → P (Nat.succ n) 0) (H3 : (n m : ℕ) → P n m → P (Nat.succ n) (Nat.succ m)) (n : ℕ) (m : ℕ) : P n m

Equations

• Nat.subInduction H1 H2 H3 0 x = H1 x
• Nat.subInduction H1 H2 H3 (Nat.succ _n) 0 = H2 _n
• Nat.subInduction H1 H2 H3 (Nat.succ n) (Nat.succ m) = H3 n m (Nat.subInduction H1 H2 H3 n m)

theorem Nat.strong_induction_on {p : ℕ → Prop} (n : ℕ) (h : (n : ℕ) → ((m : ℕ) → m < n → p m) → p n) : p n

theorem Nat.case_strong_induction_on {p : ℕ → Prop} (a : ℕ) (hz : p 0) (hi : (n : ℕ) → ((m : ℕ) → m ≤ n → p m) → p (Nat.succ n)) : p a

mod

theorem Nat.cond_decide_mod_two (x : ℕ) [d : Decidable (x % 2 = 1)] : (bif decide (x % 2 = 1) then 1 else 0) = x % 2

div

theorem Nat.mul_div_mul {m : ℕ} (n : ℕ) (k : ℕ) (H : 0 < m) : m * n / (m * k) = n / k

dvd

instance Nat.decidableDvd : DecidableRel fun x x_1 => x ∣ x_1

Equations

• Nat.decidableDvd _m _n = decidable_of_decidable_of_iff (_ : _n % _m = 0 ↔ _m ∣ _n)

find

def Nat.findX {p : ℕ → Prop} [DecidablePred p] (H : ∃ n, p n) : { n // p n ∧ ∀ (m : ℕ), m < n → ¬p m }

Equations

• One or more equations did not get rendered due to their size.

def Nat.find {p : ℕ → Prop} [DecidablePred p] (H : ∃ n, p n) : ℕ

If p is a (decidable) predicate on ℕ and hp : ∃ (n : ℕ), p n is a proof that there exists some natural number satisfying p, then nat.find hp is the smallest natural number satisfying p. Note that nat.find is protected, meaning that you can't just write find, even if the nat namespace is open.

The API for nat.find is:

• nat.find_spec is the proof that nat.find hp satisfies p.
• nat.find_min is the proof that if m < nat.find hp then m does not satisfy p.
• nat.find_min' is the proof that if m does satisfy p then nat.find hp ≤ m.

Equations

• Nat.find H = (Nat.findX H).val

theorem Nat.find_spec {p : ℕ → Prop} [DecidablePred p] (H : ∃ n, p n) : p (Nat.find H)

theorem Nat.find_min {p : ℕ → Prop} [DecidablePred p] (H : ∃ n, p n) {m : ℕ} : m < Nat.find H → ¬p m

theorem Nat.find_min' {p : ℕ → Prop} [DecidablePred p] (H : ∃ n, p n) {m : ℕ} (h : p m) : Nat.find H ≤ m

theorem Nat.to_digits_core_lens_eq_aux (b : ℕ) (f : ℕ) (n : ℕ) (l1 : List Char) (l2 : List Char) : List.length l1 = List.length l2 → List.length (Nat.toDigitsCore b f n l1) = List.length (Nat.toDigitsCore b f n l2)

theorem Nat.to_digits_core_lens_eq (b : ℕ) (f : ℕ) (n : ℕ) (c : Char) (tl : List Char) : List.length (Nat.toDigitsCore b f n (c :: tl)) = List.length (Nat.toDigitsCore b f n tl) + 1

theorem Nat.nat_repr_len_aux (n : ℕ) (b : ℕ) (e : ℕ) (h_b_pos : 0 < b) : n < b ^ Nat.succ e → n / b < b ^ e

theorem Nat.to_digits_core_length (b : ℕ) (h : 2 ≤ b) (f : ℕ) (n : ℕ) (e : ℕ) (hlt : n < b ^ e) (h_e_pos : 0 < e) : List.length (Nat.toDigitsCore b f n []) ≤ e

The String representation produced by toDigitsCore has the proper length relative to the number of digits in n < e for some base b. Since this works with any base greater than one, it can be used for binary, decimal, and hex.

theorem Nat.repr_length (n : ℕ) (e : ℕ) : 0 < e → n < 10 ^ e → String.length (Nat.repr n) ≤ e

The core implementation of Nat.repr returns a String with length less than or equal to the number of digits in the decimal number (represented by e). For example, the decimal string representation of any number less than 1000 (10 ^ 3) has a length less than or equal to 3.