Basic lemmas about natural numbers

The primary purpose of the lemmas in this file is to assist with reasoning about sizes of objects, array indices and such.

This file was upstreamed from Std, and later these lemmas should be organised into other files more systematically.

@[reducible, inline, deprecated Nat.and_forall_add_one (since := "2024-07-30")]

abbrev Nat.and_forall_succ {p : Nat → Prop} : (p 0 ∧ ∀ (n : Nat), p (n + 1)) ↔ ∀ (n : Nat), p n

Equations

• @Nat.and_forall_succ = @Nat.and_forall_add_one

@[reducible, inline, deprecated Nat.or_exists_add_one (since := "2024-07-30")]

abbrev Nat.or_exists_succ {p : Nat → Prop} : (p 0 ∨ ∃ (n : Nat), p (n + 1)) ↔ Exists p

Equations

• @Nat.or_exists_succ = @Nat.or_exists_add_one

@[simp]

theorem Nat.exists_ne_zero {P : Nat → Prop} : (∃ (n : Nat), ¬n = 0 ∧ P n) ↔ ∃ (n : Nat), P (n + 1)

@[simp]

theorem Nat.exists_eq_add_one {a : Nat} : (∃ (n : Nat), a = n + 1) ↔ 0 < a

@[simp]

theorem Nat.exists_add_one_eq {a : Nat} : (∃ (n : Nat), n + 1 = a) ↔ 0 < a

theorem Nat.forall_lt_succ_right' {n : Nat} {p : (m : Nat) → m < n + 1 → Prop} : (∀ (m : Nat) (h : m < n + 1), p m h) ↔ (∀ (m : Nat) (h : m < n), p m ⋯) ∧ p n ⋯

Dependent variant of forall_lt_succ_right.

theorem Nat.forall_lt_succ_right {n : Nat} {p : Nat → Prop} : (∀ (m : Nat), m < n + 1 → p m) ↔ (∀ (m : Nat), m < n → p m) ∧ p n

See forall_lt_succ_right' for a variant where p takes the bound as an argument.

theorem Nat.forall_lt_succ_left' {n : Nat} {p : (m : Nat) → m < n + 1 → Prop} : (∀ (m : Nat) (h : m < n + 1), p m h) ↔ p 0 ⋯ ∧ ∀ (m : Nat) (h : m < n), p (m + 1) ⋯

Dependent variant of forall_lt_succ_left.

theorem Nat.forall_lt_succ_left {n : Nat} {p : Nat → Prop} : (∀ (m : Nat), m < n + 1 → p m) ↔ p 0 ∧ ∀ (m : Nat), m < n → p (m + 1)

See forall_lt_succ_left' for a variant where p takes the bound as an argument.

theorem Nat.exists_lt_succ_right' {n : Nat} {p : (m : Nat) → m < n + 1 → Prop} : (∃ (m : Nat), ∃ (h : m < n + 1), p m h) ↔ (∃ (m : Nat), ∃ (h : m < n), p m ⋯) ∨ p n ⋯

Dependent variant of exists_lt_succ_right.

theorem Nat.exists_lt_succ_right {n : Nat} {p : Nat → Prop} : (∃ (m : Nat), m < n + 1 ∧ p m) ↔ (∃ (m : Nat), m < n ∧ p m) ∨ p n

See exists_lt_succ_right' for a variant where p takes the bound as an argument.

theorem Nat.exists_lt_succ_left' {n : Nat} {p : (m : Nat) → m < n + 1 → Prop} : (∃ (m : Nat), ∃ (h : m < n + 1), p m h) ↔ p 0 ⋯ ∨ ∃ (m : Nat), ∃ (h : m < n), p (m + 1) ⋯

Dependent variant of exists_lt_succ_left.

theorem Nat.exists_lt_succ_left {n : Nat} {p : Nat → Prop} : (∃ (m : Nat), m < n + 1 ∧ p m) ↔ p 0 ∨ ∃ (m : Nat), m < n ∧ p (m + 1)

See exists_lt_succ_left' for a variant where p takes the bound as an argument.

add

theorem Nat.add_add_add_comm (a b c d : Nat) : a + b + (c + d) = a + c + (b + d)

theorem Nat.one_add (n : Nat) : 1 + n = n.succ

theorem Nat.succ_eq_one_add (n : Nat) : n.succ = 1 + n

theorem Nat.succ_add_eq_add_succ (a b : Nat) : a.succ + b = a + b.succ

theorem Nat.eq_zero_of_add_eq_zero_right {n m : Nat} (h : n + m = 0) : n = 0

@[simp]

theorem Nat.add_eq_zero_iff {n m : Nat} : n + m = 0 ↔ n = 0 ∧ m = 0

@[simp]

theorem Nat.add_left_cancel_iff {m k n : Nat} : n + m = n + k ↔ m = k

@[simp]

theorem Nat.add_right_cancel_iff {m k n : Nat} : m + n = k + n ↔ m = k

@[simp]

theorem Nat.add_left_eq_self {a b : Nat} : a + b = b ↔ a = 0

@[simp]

theorem Nat.add_right_eq_self {a b : Nat} : a + b = a ↔ b = 0

@[simp]

theorem Nat.self_eq_add_right {a b : Nat} : a = a + b ↔ b = 0

@[simp]

theorem Nat.self_eq_add_left {a b : Nat} : a = b + a ↔ b = 0

@[simp]

theorem Nat.add_le_add_iff_left {m k n : Nat} : n + m ≤ n + k ↔ m ≤ k

theorem Nat.lt_of_add_lt_add_right {k m n : Nat} : k + n < m + n → k < m

theorem Nat.lt_of_add_lt_add_left {k m n : Nat} : n + k < n + m → k < m

@[simp]

theorem Nat.add_lt_add_iff_left {k n m : Nat} : k + n < k + m ↔ n < m

@[simp]

theorem Nat.add_lt_add_iff_right {k n m : Nat} : n + k < m + k ↔ n < m

theorem Nat.add_lt_add_of_le_of_lt {a b c d : Nat} (hle : a ≤ b) (hlt : c < d) : a + c < b + d

theorem Nat.add_lt_add_of_lt_of_le {a b c d : Nat} (hlt : a < b) (hle : c ≤ d) : a + c < b + d

theorem Nat.pos_of_lt_add_right {n k : Nat} (h : n < n + k) : 0 < k

theorem Nat.pos_of_lt_add_left {n k : Nat} : n < k + n → 0 < k

@[simp]

theorem Nat.lt_add_right_iff_pos {n k : Nat} : n < n + k ↔ 0 < k

@[simp]

theorem Nat.lt_add_left_iff_pos {n k : Nat} : n < k + n ↔ 0 < k

theorem Nat.add_pos_left {m : Nat} (h : 0 < m) (n : Nat) : 0 < m + n

theorem Nat.add_pos_right {n : Nat} (m : Nat) (h : 0 < n) : 0 < m + n

theorem Nat.add_self_ne_one (n : Nat) : n + n ≠ 1

theorem Nat.le_iff_lt_add_one {x y : Nat} : x ≤ y ↔ x < y + 1

sub

theorem Nat.sub_one (n : Nat) : n - 1 = n.pred

theorem Nat.one_sub (n : Nat) : 1 - n = if n = 0 then 1 else 0

theorem Nat.succ_sub_sub_succ (n m k : Nat) : n.succ - m - k.succ = n - m - k

theorem Nat.add_sub_sub_add_right (n m k l : Nat) : n + l - m - (k + l) = n - m - k

theorem Nat.sub_right_comm (m n k : Nat) : m - n - k = m - k - n

theorem Nat.add_sub_cancel_right (n m : Nat) : n + m - m = n

@[simp]

theorem Nat.add_sub_cancel' {n m : Nat} (h : m ≤ n) : m + (n - m) = n

theorem Nat.succ_sub_one (n : Nat) : n.succ - 1 = n

theorem Nat.one_add_sub_one (n : Nat) : 1 + n - 1 = n

theorem Nat.sub_sub_self {n m : Nat} (h : m ≤ n) : n - (n - m) = m

theorem Nat.sub_add_comm {n m k : Nat} (h : k ≤ n) : n + m - k = n - k + m

theorem Nat.sub_eq_zero_iff_le {n m : Nat} : n - m = 0 ↔ n ≤ m

theorem Nat.sub_pos_iff_lt {n m : Nat} : 0 < n - m ↔ m < n

theorem Nat.sub_le_iff_le_add {a b c : Nat} : a - b ≤ c ↔ a ≤ c + b

theorem Nat.sub_le_iff_le_add' {a b c : Nat} : a - b ≤ c ↔ a ≤ b + c

theorem Nat.le_sub_iff_add_le {k m n : Nat} (h : k ≤ m) : n ≤ m - k ↔ n + k ≤ m

theorem Nat.add_lt_iff_lt_sub_right {a b c : Nat} : a + b < c ↔ a < c - b

theorem Nat.add_le_of_le_sub' {n k m : Nat} (h : m ≤ k) : n ≤ k - m → m + n ≤ k

theorem Nat.le_sub_of_add_le' {n k m : Nat} : m + n ≤ k → n ≤ k - m

theorem Nat.le_sub_iff_add_le' {k m n : Nat} (h : k ≤ m) : n ≤ m - k ↔ k + n ≤ m

theorem Nat.le_of_sub_le_sub_left {n k m : Nat} : n ≤ k → k - m ≤ k - n → n ≤ m

theorem Nat.sub_le_sub_iff_left {n m k : Nat} (h : n ≤ k) : k - m ≤ k - n ↔ n ≤ m

theorem Nat.sub_lt_of_pos_le {a b : Nat} (h₀ : 0 < a) (h₁ : a ≤ b) : b - a < b

@[reducible, inline]

abbrev Nat.sub_lt_self {a b : Nat} (h₀ : 0 < a) (h₁ : a ≤ b) : b - a < b

Equations

• @Nat.sub_lt_self = @Nat.sub_lt_of_pos_le

theorem Nat.add_lt_of_lt_sub' {a b c : Nat} : b < c - a → a + b < c

theorem Nat.sub_add_lt_sub {m k n : Nat} (h₁ : m + k ≤ n) (h₂ : 0 < k) : n - (m + k) < n - m

theorem Nat.sub_one_lt_of_le {a b : Nat} (h₀ : 0 < a) (h₁ : a ≤ b) : a - 1 < b

theorem Nat.sub_lt_succ (a b : Nat) : a - b < a.succ

theorem Nat.sub_lt_add_one (a b : Nat) : a - b < a + 1

theorem Nat.sub_one_sub_lt {i n : Nat} (h : i < n) : n - 1 - i < n

theorem Nat.exists_eq_add_of_le {m n : Nat} (h : m ≤ n) : ∃ (k : Nat), n = m + k

theorem Nat.exists_eq_add_of_le' {m n : Nat} (h : m ≤ n) : ∃ (k : Nat), n = k + m

theorem Nat.exists_eq_add_of_lt {m n : Nat} (h : m < n) : ∃ (k : Nat), n = m + k + 1

min/max

theorem Nat.succ_min_succ (x y : Nat) : min x.succ y.succ = (min x y).succ

@[simp]

theorem Nat.min_self (a : Nat) : min a a = a

instance Nat.instIdempotentOpMin : Std.IdempotentOp min

@[simp]

theorem Nat.zero_min (a : Nat) : min 0 a = 0

@[simp]

theorem Nat.min_zero (a : Nat) : min a 0 = 0

@[simp]

theorem Nat.min_assoc (a b c : Nat) : min (min a b) c = min a (min b c)

instance Nat.instAssociativeMin : Std.Associative min

@[simp]

theorem Nat.min_self_assoc {m n : Nat} : min m (min m n) = min m n

@[simp]

theorem Nat.min_self_assoc' {m n : Nat} : min n (min m n) = min n m

@[simp]

theorem Nat.min_add_left {a b : Nat} : min a (b + a) = a

@[simp]

theorem Nat.min_add_right {a b : Nat} : min a (a + b) = a

@[simp]

theorem Nat.add_left_min {a b : Nat} : min (b + a) a = a

@[simp]

theorem Nat.add_right_min {a b : Nat} : min (a + b) a = a

theorem Nat.sub_sub_eq_min (a b : Nat) : a - (a - b) = min a b

theorem Nat.sub_eq_sub_min (n m : Nat) : n - m = n - min n m

@[simp]

theorem Nat.sub_add_min_cancel (n m : Nat) : n - m + min n m = n

theorem Nat.max_eq_right {a b : Nat} (h : a ≤ b) : max a b = b

theorem Nat.max_eq_left {a b : Nat} (h : b ≤ a) : max a b = a

theorem Nat.succ_max_succ (x y : Nat) : max x.succ y.succ = (max x y).succ

theorem Nat.max_le_of_le_of_le {a b c : Nat} : a ≤ c → b ≤ c → max a b ≤ c

theorem Nat.max_le {a b c : Nat} : max a b ≤ c ↔ a ≤ c ∧ b ≤ c

theorem Nat.max_lt {a b c : Nat} : max a b < c ↔ a < c ∧ b < c

@[simp]

theorem Nat.max_self (a : Nat) : max a a = a

instance Nat.instIdempotentOpMax : Std.IdempotentOp max

@[simp]

theorem Nat.zero_max (a : Nat) : max 0 a = a

@[simp]

theorem Nat.max_zero (a : Nat) : max a 0 = a

instance Nat.instLawfulIdentityMaxOfNat : Std.LawfulIdentity max 0

theorem Nat.max_assoc (a b c : Nat) : max (max a b) c = max a (max b c)

instance Nat.instAssociativeMax : Std.Associative max

@[simp]

theorem Nat.max_add_left {a b : Nat} : max a (b + a) = b + a

@[simp]

theorem Nat.max_add_right {a b : Nat} : max a (a + b) = a + b

@[simp]

theorem Nat.add_left_max {a b : Nat} : max (b + a) a = b + a

@[simp]

theorem Nat.add_right_max {a b : Nat} : max (a + b) a = a + b

theorem Nat.sub_add_eq_max (a b : Nat) : a - b + b = max a b

theorem Nat.sub_eq_max_sub (n m : Nat) : n - m = max n m - m

theorem Nat.max_min_distrib_left (a b c : Nat) : max a (min b c) = min (max a b) (max a c)

theorem Nat.min_max_distrib_left (a b c : Nat) : min a (max b c) = max (min a b) (min a c)

theorem Nat.max_min_distrib_right (a b c : Nat) : max (min a b) c = min (max a c) (max b c)

theorem Nat.min_max_distrib_right (a b c : Nat) : min (max a b) c = max (min a c) (min b c)

theorem Nat.add_max_add_right (a b c : Nat) : max (a + c) (b + c) = max a b + c

theorem Nat.add_min_add_right (a b c : Nat) : min (a + c) (b + c) = min a b + c

theorem Nat.add_max_add_left (a b c : Nat) : max (a + b) (a + c) = a + max b c

theorem Nat.add_min_add_left (a b c : Nat) : min (a + b) (a + c) = a + min b c

theorem Nat.pred_min_pred (x y : Nat) : min x.pred y.pred = (min x y).pred

theorem Nat.pred_max_pred (x y : Nat) : max x.pred y.pred = (max x y).pred

theorem Nat.sub_min_sub_right (a b c : Nat) : min (a - c) (b - c) = min a b - c

theorem Nat.sub_max_sub_right (a b c : Nat) : max (a - c) (b - c) = max a b - c

theorem Nat.sub_min_sub_left (a b c : Nat) : min (a - b) (a - c) = a - max b c

theorem Nat.sub_max_sub_left (a b c : Nat) : max (a - b) (a - c) = a - min b c

theorem Nat.mul_max_mul_right (a b c : Nat) : max (a * c) (b * c) = max a b * c

theorem Nat.mul_min_mul_right (a b c : Nat) : min (a * c) (b * c) = min a b * c

theorem Nat.mul_max_mul_left (a b c : Nat) : max (a * b) (a * c) = a * max b c

theorem Nat.mul_min_mul_left (a b c : Nat) : min (a * b) (a * c) = a * min b c

mul

theorem Nat.mul_right_comm (n m k : Nat) : n * m * k = n * k * m

theorem Nat.mul_mul_mul_comm (a b c d : Nat) : a * b * (c * d) = a * c * (b * d)

theorem Nat.mul_eq_zero {m n : Nat} : n * m = 0 ↔ n = 0 ∨ m = 0

theorem Nat.mul_ne_zero_iff {n m : Nat} : n * m ≠ 0 ↔ n ≠ 0 ∧ m ≠ 0

theorem Nat.mul_ne_zero {n m : Nat} : n ≠ 0 → m ≠ 0 → n * m ≠ 0

theorem Nat.ne_zero_of_mul_ne_zero_left {n m : Nat} (h : n * m ≠ 0) : n ≠ 0

theorem Nat.mul_left_cancel {n m k : Nat} (np : 0 < n) (h : n * m = n * k) : m = k

theorem Nat.mul_right_cancel {n m k : Nat} (mp : 0 < m) (h : n * m = k * m) : n = k

theorem Nat.mul_left_cancel_iff {n : Nat} (p : 0 < n) {m k : Nat} : n * m = n * k ↔ m = k

theorem Nat.mul_right_cancel_iff {m : Nat} (p : 0 < m) {n k : Nat} : n * m = k * m ↔ n = k

theorem Nat.ne_zero_of_mul_ne_zero_right {n m : Nat} (h : n * m ≠ 0) : m ≠ 0

theorem Nat.le_mul_of_pos_left {n : Nat} (m : Nat) (h : 0 < n) : m ≤ n * m

theorem Nat.le_mul_of_pos_right {m : Nat} (n : Nat) (h : 0 < m) : n ≤ n * m

theorem Nat.mul_lt_mul_of_lt_of_le {a c b d : Nat} (hac : a < c) (hbd : b ≤ d) (hd : 0 < d) : a * b < c * d

theorem Nat.mul_lt_mul_of_lt_of_le' {a c b d : Nat} (hac : a < c) (hbd : b ≤ d) (hb : 0 < b) : a * b < c * d

theorem Nat.mul_lt_mul_of_le_of_lt {a c b d : Nat} (hac : a ≤ c) (hbd : b < d) (hc : 0 < c) : a * b < c * d

theorem Nat.mul_lt_mul_of_le_of_lt' {a c b d : Nat} (hac : a ≤ c) (hbd : b < d) (ha : 0 < a) : a * b < c * d

theorem Nat.mul_lt_mul_of_lt_of_lt {a b c d : Nat} (hac : a < c) (hbd : b < d) : a * b < c * d

theorem Nat.succ_mul_succ (a b : Nat) : a.succ * b.succ = a * b + a + b + 1

theorem Nat.add_one_mul_add_one (a b : Nat) : (a + 1) * (b + 1) = a * b + a + b + 1

theorem Nat.mul_le_add_right {m k n : Nat} : k * m ≤ m + n ↔ (k - 1) * m ≤ n

theorem Nat.succ_mul_succ_eq (a b : Nat) : a.succ * b.succ = a * b + a + b + 1

theorem Nat.mul_self_sub_mul_self_eq (a b : Nat) : a * a - b * b = (a + b) * (a - b)

theorem Nat.pos_of_mul_pos_left {a b : Nat} (h : 0 < a * b) : 0 < b

theorem Nat.pos_of_mul_pos_right {a b : Nat} (h : 0 < a * b) : 0 < a

@[simp]

theorem Nat.mul_pos_iff_of_pos_left {a b : Nat} (h : 0 < a) : 0 < a * b ↔ 0 < b

@[simp]

theorem Nat.mul_pos_iff_of_pos_right {a b : Nat} (h : 0 < b) : 0 < a * b ↔ 0 < a

theorem Nat.pos_of_lt_mul_left {a b c : Nat} (h : a < b * c) : 0 < c

theorem Nat.pos_of_lt_mul_right {a b c : Nat} (h : a < b * c) : 0 < b

div/mod

theorem Nat.mod_two_eq_zero_or_one (n : Nat) : n % 2 = 0 ∨ n % 2 = 1

@[simp]

theorem Nat.mod_two_bne_zero {a : Nat} : (a % 2 != 0) = (a % 2 == 1)

@[simp]

theorem Nat.mod_two_bne_one {a : Nat} : (a % 2 != 1) = (a % 2 == 0)

theorem Nat.le_of_mod_lt {a b : Nat} (h : a % b < a) : b ≤ a

theorem Nat.mul_mod_mul_right (z x y : Nat) : x * z % (y * z) = x % y * z

theorem Nat.sub_mul_mod {x k n : Nat} (h₁ : n * k ≤ x) : (x - n * k) % n = x % n

@[simp]

theorem Nat.mod_mod (a n : Nat) : a % n % n = a % n

theorem Nat.mul_mod (a b n : Nat) : a * b % n = a % n * (b % n) % n

@[simp]

theorem Nat.mod_add_mod (m n k : Nat) : (m % n + k) % n = (m + k) % n

@[simp]

theorem Nat.add_mod_mod (m n k : Nat) : (m + n % k) % k = (m + n) % k

theorem Nat.add_mod (a b n : Nat) : (a + b) % n = (a % n + b % n) % n

@[simp]

theorem Nat.self_sub_mod (n k : Nat) [NeZero k] : (n - k) % n = n - k

@[simp]

theorem Nat.mod_mul_mod {a b c : Nat} : a % c * b % c = a * b % c

theorem Nat.mod_eq_sub (x w : Nat) : x % w = x - w * (x / w)

pow

theorem Nat.pow_succ' {m n : Nat} : m ^ n.succ = m * m ^ n

theorem Nat.pow_add_one' {m n : Nat} : m ^ (n + 1) = m * m ^ n

@[simp]

theorem Nat.pow_eq {m n : Nat} : m.pow n = m ^ n

theorem Nat.one_shiftLeft (n : Nat) : 1 <<< n = 2 ^ n

theorem Nat.zero_pow {n : Nat} (H : 0 < n) : 0 ^ n = 0

@[simp]

theorem Nat.one_pow (n : Nat) : 1 ^ n = 1

@[simp]

theorem Nat.pow_one (a : Nat) : a ^ 1 = a

theorem Nat.pow_two (a : Nat) : a ^ 2 = a * a

theorem Nat.pow_add (a m n : Nat) : a ^ (m + n) = a ^ m * a ^ n

theorem Nat.pow_add' (a m n : Nat) : a ^ (m + n) = a ^ n * a ^ m

theorem Nat.pow_mul (a m n : Nat) : a ^ (m * n) = (a ^ m) ^ n

theorem Nat.pow_mul' (a m n : Nat) : a ^ (m * n) = (a ^ n) ^ m

theorem Nat.pow_right_comm (a m n : Nat) : (a ^ m) ^ n = (a ^ n) ^ m

theorem Nat.mul_pow (a b n : Nat) : (a * b) ^ n = a ^ n * b ^ n

@[reducible, inline]

abbrev Nat.pow_le_pow_left {n m : Nat} (h : n ≤ m) (i : Nat) : n ^ i ≤ m ^ i

Equations

• @Nat.pow_le_pow_left = @Nat.pow_le_pow_of_le_left

@[reducible, inline]

abbrev Nat.pow_le_pow_right {n : Nat} (hx : n > 0) {i j : Nat} : i ≤ j → n ^ i ≤ n ^ j

Equations

• @Nat.pow_le_pow_right = @Nat.pow_le_pow_of_le_right

theorem Nat.one_lt_two_pow {n : Nat} (h : n ≠ 0) : 1 < 2 ^ n

@[simp]

theorem Nat.one_lt_two_pow_iff {n : Nat} : 1 < 2 ^ n ↔ n ≠ 0

theorem Nat.one_le_two_pow {n : Nat} : 1 ≤ 2 ^ n

@[simp]

theorem Nat.one_mod_two_pow_eq_one {n : Nat} : 1 % 2 ^ n = 1 ↔ 0 < n

@[simp]

theorem Nat.one_mod_two_pow {n : Nat} (h : 0 < n) : 1 % 2 ^ n = 1

theorem Nat.pow_pos {a n : Nat} (h : 0 < a) : 0 < a ^ n

theorem Nat.pow_lt_pow_succ {a n : Nat} (h : 1 < a) : a ^ n < a ^ (n + 1)

theorem Nat.pow_lt_pow_of_lt {a n m : Nat} (h : 1 < a) (w : n < m) : a ^ n < a ^ m

theorem Nat.pow_le_pow_of_le {a n m : Nat} (h : 1 < a) (w : n ≤ m) : a ^ n ≤ a ^ m

theorem Nat.pow_le_pow_iff_right {a n m : Nat} (h : 1 < a) : a ^ n ≤ a ^ m ↔ n ≤ m

theorem Nat.pow_lt_pow_iff_right {a n m : Nat} (h : 1 < a) : a ^ n < a ^ m ↔ n < m

@[simp]

theorem Nat.pow_pred_mul {x w : Nat} (h : 0 < w) : x ^ (w - 1) * x = x ^ w

theorem Nat.pow_pred_lt_pow {x w : Nat} (h₁ : 1 < x) (h₂ : 0 < w) : x ^ (w - 1) < x ^ w

theorem Nat.two_pow_pred_lt_two_pow {w : Nat} (h : 0 < w) : 2 ^ (w - 1) < 2 ^ w

@[simp]

theorem Nat.two_pow_pred_add_two_pow_pred {w : Nat} (h : 0 < w) : 2 ^ (w - 1) + 2 ^ (w - 1) = 2 ^ w

@[simp]

theorem Nat.two_pow_sub_two_pow_pred {w : Nat} (h : 0 < w) : 2 ^ w - 2 ^ (w - 1) = 2 ^ (w - 1)

@[simp]

theorem Nat.two_pow_pred_mod_two_pow {w : Nat} (h : 0 < w) : 2 ^ (w - 1) % 2 ^ w = 2 ^ (w - 1)

theorem Nat.pow_lt_pow_iff_pow_mul_le_pow {a n m : Nat} (h : 1 < a) : a ^ n < a ^ m ↔ a ^ n * a ≤ a ^ m

theorem Nat.lt_pow_self {n a : Nat} (h : 1 < a) : n < a ^ n

theorem Nat.lt_two_pow_self {n : Nat} : n < 2 ^ n

@[simp]

theorem Nat.mod_two_pow_self {n : Nat} : n % 2 ^ n = n

@[simp]

theorem Nat.two_pow_pred_mul_two {w : Nat} (h : 0 < w) : 2 ^ (w - 1) * 2 = 2 ^ w

log2

@[simp]

theorem Nat.log2_zero : log2 0 = 0

theorem Nat.le_log2 {n k : Nat} (h : n ≠ 0) : k ≤ n.log2 ↔ 2 ^ k ≤ n

theorem Nat.log2_lt {n k : Nat} (h : n ≠ 0) : n.log2 < k ↔ n < 2 ^ k

@[simp]

theorem Nat.log2_two_pow {n : Nat} : (2 ^ n).log2 = n

theorem Nat.log2_self_le {n : Nat} (h : n ≠ 0) : 2 ^ n.log2 ≤ n

theorem Nat.lt_log2_self {n : Nat} : n < 2 ^ (n.log2 + 1)

dvd

theorem Nat.eq_mul_of_div_eq_right {a b c : Nat} (H1 : b ∣ a) (H2 : a / b = c) : a = b * c

theorem Nat.div_eq_iff_eq_mul_right {a b c : Nat} (H : 0 < b) (H' : b ∣ a) : a / b = c ↔ a = b * c

theorem Nat.div_eq_iff_eq_mul_left {a b c : Nat} (H : 0 < b) (H' : b ∣ a) : a / b = c ↔ a = c * b

theorem Nat.pow_dvd_pow_iff_pow_le_pow {k l x : Nat} : 0 < x → (x ^ k ∣ x ^ l ↔ x ^ k ≤ x ^ l)

theorem Nat.pow_dvd_pow_iff_le_right {x k l : Nat} (w : 1 < x) : x ^ k ∣ x ^ l ↔ k ≤ l

If 1 < x, then x^k divides x^l if and only if k is at most l.

theorem Nat.pow_dvd_pow_iff_le_right' {b k l : Nat} : (b + 2) ^ k ∣ (b + 2) ^ l ↔ k ≤ l

theorem Nat.pow_dvd_pow {m n : Nat} (a : Nat) (h : m ≤ n) : a ^ m ∣ a ^ n

theorem Nat.pow_sub_mul_pow (a : Nat) {m n : Nat} (h : m ≤ n) : a ^ (n - m) * a ^ m = a ^ n

theorem Nat.pow_dvd_of_le_of_pow_dvd {p m n k : Nat} (hmn : m ≤ n) (hdiv : p ^ n ∣ k) : p ^ m ∣ k

theorem Nat.dvd_of_pow_dvd {p k m : Nat} (hk : 1 ≤ k) (hpk : p ^ k ∣ m) : p ∣ m

theorem Nat.pow_div {x m n : Nat} (h : n ≤ m) (hx : 0 < x) : x ^ m / x ^ n = x ^ (m - n)

shiftLeft and shiftRight

@[simp]

theorem Nat.shiftLeft_zero {n : Nat} : n <<< 0 = n

theorem Nat.shiftLeft_succ_inside (m n : Nat) : m <<< (n + 1) = (2 * m) <<< n

Shiftleft on successor with multiple moved inside.

theorem Nat.shiftLeft_succ (m n : Nat) : m <<< (n + 1) = 2 * m <<< n

Shiftleft on successor with multiple moved to outside.

theorem Nat.shiftRight_succ_inside (m n : Nat) : m >>> (n + 1) = (m / 2) >>> n

Shiftright on successor with division moved inside.

@[simp]

theorem Nat.zero_shiftLeft (n : Nat) : 0 <<< n = 0

@[simp]

theorem Nat.zero_shiftRight (n : Nat) : 0 >>> n = 0

theorem Nat.shiftLeft_add (m n k : Nat) : m <<< (n + k) = m <<< n <<< k

@[simp]

theorem Nat.shiftLeft_shiftRight (x n : Nat) : x <<< n >>> n = x

theorem Nat.mul_add_div {m : Nat} (m_pos : m > 0) (x y : Nat) : (m * x + y) / m = x + y / m

theorem Nat.mul_add_mod (m x y : Nat) : (m * x + y) % m = y % m

@[simp]

theorem Nat.mod_div_self (m n : Nat) : m % n / n = 0

Decidability of predicates

instance Nat.decidableBallLT (n : Nat) (P : (k : Nat) → k < n → Prop) [(n_1 : Nat) → (h : n_1 < n) → Decidable (P n_1 h)] : Decidable (∀ (n_1 : Nat) (h : n_1 < n), P n_1 h)

Equations

• One or more equations did not get rendered due to their size.
• Nat.decidableBallLT 0 x_3 = isTrue ⋯

instance Nat.decidableForallFin {n : Nat} (P : Fin n → Prop) [DecidablePred P] : Decidable (∀ (i : Fin n), P i)

Equations

• Nat.decidableForallFin P = decidable_of_iff (∀ (k : Nat) (h : k < n), P ⟨k, h⟩) ⋯

instance Nat.decidableBallLE (n : Nat) (P : (k : Nat) → k ≤ n → Prop) [(n_1 : Nat) → (h : n_1 ≤ n) → Decidable (P n_1 h)] : Decidable (∀ (n_1 : Nat) (h : n_1 ≤ n), P n_1 h)

Equations

• n.decidableBallLE P = decidable_of_iff (∀ (k : Nat) (h : k < n.succ), P k ⋯) ⋯

instance Nat.decidableExistsLT {p : Nat → Prop} [h : DecidablePred p] : DecidablePred fun (n : Nat) => ∃ (m : Nat), m < n ∧ p m

Equations

• Nat.decidableExistsLT 0 = isFalse ⋯
• n.succ.decidableExistsLT = decidable_of_decidable_of_iff ⋯

instance Nat.decidableExistsLE {p : Nat → Prop} [DecidablePred p] : DecidablePred fun (n : Nat) => ∃ (m : Nat), m ≤ n ∧ p m

Equations

• n.decidableExistsLE = decidable_of_iff (∃ (m : Nat), m < n + 1 ∧ p m) ⋯

instance Nat.decidableExistsLT' {k : Nat} {p : (m : Nat) → m < k → Prop} [I : (m : Nat) → (h : m < k) → Decidable (p m h)] : Decidable (∃ (m : Nat), ∃ (h : m < k), p m h)

Dependent version of decidableExistsLT.

Equations

• Nat.decidableExistsLT' = isFalse ⋯
• Nat.decidableExistsLT' = decidable_of_iff ((∃ (m : Nat), ∃ (h : m < n), P m ⋯) ∨ P n ⋯) ⋯

instance Nat.decidableExistsLE' {k : Nat} {p : (m : Nat) → m ≤ k → Prop} [I : (m : Nat) → (h : m ≤ k) → Decidable (p m h)] : Decidable (∃ (m : Nat), ∃ (h : m ≤ k), p m h)

Dependent version of decidableExistsLE.

Equations

• Nat.decidableExistsLE' = decidable_of_iff (∃ (m : Nat), ∃ (h : m < k + 1), p m ⋯) ⋯

Results about List.sum specialized to Nat

theorem Nat.sum_pos_iff_exists_pos {l : List Nat} : 0 < l.sum ↔ ∃ (x : Nat), x ∈ l ∧ 0 < x