Definitions of basic functions

theorem Int.subNatNat_of_sub_eq_zero {m n : Nat} (h : n - m = 0) : subNatNat m n = ↑(m - n)

theorem Int.subNatNat_of_sub_eq_succ {m n k : Nat} (h : n - m = k.succ) : subNatNat m n = negSucc k

@[simp]

theorem Int.neg_zero : -0 = 0

@[simp]

theorem Int.natCast_add (n m : Nat) : ↑(n + m) = ↑n + ↑m

@[simp]

theorem Int.natCast_mul (n m : Nat) : ↑(n * m) = ↑n * ↑m

theorem Int.natCast_succ (n : Nat) : ↑n.succ = ↑n + 1

theorem Int.natCast_add_one (n : Nat) : ↑(n + 1) = ↑n + 1

@[deprecated Int.natCast_add (since := "2025-04-17")]

theorem Int.ofNat_add (n m : Nat) : ↑(n + m) = ↑n + ↑m

@[deprecated Int.natCast_mul (since := "2025-04-17")]

theorem Int.ofNat_mul (n m : Nat) : ↑(n * m) = ↑n * ↑m

@[deprecated Int.natCast_succ (since := "2025-04-17")]

theorem Int.ofNat_succ (n : Nat) : ↑n.succ = ↑n + 1

theorem Int.neg_ofNat_zero : -↑0 = 0

theorem Int.neg_ofNat_succ (n : Nat) : -↑n.succ = negSucc n

@[simp]

theorem Int.neg_negSucc (n : Nat) : -negSucc n = ↑(n + 1)

theorem Int.negOfNat_eq {n : Nat} : negOfNat n = -ofNat n

@[simp]

theorem Int.add_def {a b : Int} : a.add b = a + b

@[simp]

theorem Int.mul_def {a b : Int} : a.mul b = a * b

These are only for internal use

Ideally these could all be made private, but they are used in downstream libraries.

theorem Int.ofNat_add_ofNat (m n : Nat) : ↑m + ↑n = ↑(m + n)

theorem Int.ofNat_add_negSucc (m n : Nat) : ↑m + negSucc n = subNatNat m n.succ

theorem Int.negSucc_add_ofNat (m n : Nat) : negSucc m + ↑n = subNatNat n m.succ

theorem Int.negSucc_add_negSucc (m n : Nat) : negSucc m + negSucc n = negSucc (m + n).succ

theorem Int.ofNat_mul_ofNat (m n : Nat) : ↑m * ↑n = ↑(m * n)

theorem Int.ofNat_inj {m n : Nat} : ↑m = ↑n ↔ m = n

theorem Int.ofNat_eq_zero {n : Nat} : ↑n = 0 ↔ n = 0

theorem Int.ofNat_ne_zero {n : Nat} : ↑n ≠ 0 ↔ n ≠ 0

theorem Int.negSucc_inj {m n : Nat} : negSucc m = negSucc n ↔ m = n

theorem Int.negSucc_eq (n : Nat) : negSucc n = -(↑n + 1)

@[deprecated Int.negSucc_eq (since := "2025-03-11")]

theorem Int.negSucc_coe (n : Nat) : negSucc n = -↑(n + 1)

@[simp]

theorem Int.negSucc_ne_zero (n : Nat) : negSucc n ≠ 0

@[simp]

theorem Int.zero_ne_negSucc (n : Nat) : 0 ≠ negSucc n

@[simp]

theorem Int.cast_ofNat_Int {n : Nat} : ↑(OfNat.ofNat n) = OfNat.ofNat n

@[simp]

theorem Int.beq'_eq (a b : Int) : (a.beq' b = true) = (a = b)

@[simp]

theorem Int.beq'_ne (a b : Int) : (a.beq' b = false) = (a ≠ b)

theorem Int.beq'_eq_beq (a b : Int) : a.beq' b = (a == b)

@[simp]

theorem Int.neg_neg (a : Int) : - -a = a

@[simp]

theorem Int.neg_inj {a b : Int} : -a = -b ↔ a = b

@[simp]

theorem Int.neg_eq_zero {a : Int} : -a = 0 ↔ a = 0

theorem Int.neg_ne_zero {a : Int} : -a ≠ 0 ↔ a ≠ 0

theorem Int.sub_eq_add_neg {a b : Int} : a - b = a + -b

theorem Int.add_neg_eq_sub {a b : Int} : a + -b = a - b

theorem Int.add_neg_one (i : Int) : i + -1 = i - 1

theorem Int.subNatNat_elim (m n : Nat) (motive : Nat → Nat → Int → Prop) (hp : ∀ (i n : Nat), motive (n + i) n ↑i) (hn : ∀ (i m : Nat), motive m (m + i + 1) (negSucc i)) : motive m n (subNatNat m n)

theorem Int.subNatNat_add_left {m n : Nat} : subNatNat (m + n) m = ↑n

theorem Int.subNatNat_add_right {m n : Nat} : subNatNat m (m + n + 1) = negSucc n

theorem Int.subNatNat_add_add (m n k : Nat) : subNatNat (m + k) (n + k) = subNatNat m n

theorem Int.subNatNat_of_le {m n : Nat} (h : n ≤ m) : subNatNat m n = ↑(m - n)

theorem Int.subNatNat_of_lt {m n : Nat} (h : m < n) : subNatNat m n = negSucc (n - m).pred

@[simp]

theorem Int.subNat_eq_zero_iff {a b : Nat} : subNatNat a b = 0 ↔ a = b

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

theorem Int.subNatNat_add (m n k : Nat) : subNatNat (m + n) k = ↑m + subNatNat n k

theorem Int.subNatNat_add_negSucc (m n k : Nat) : subNatNat m n + negSucc k = subNatNat m (n + k.succ)

theorem Int.subNatNat_self (n : Nat) : subNatNat n n = 0

theorem Int.add_comm (a b : Int) : a + b = b + a

instance Int.instCommutativeHAdd : Std.Commutative fun (x1 x2 : Int) => x1 + x2

@[simp]

theorem Int.add_zero (a : Int) : a + 0 = a

@[simp]

theorem Int.zero_add (a : Int) : 0 + a = a

instance Int.instLawfulIdentityHAddOfNat : Std.LawfulIdentity (fun (x1 x2 : Int) => x1 + x2) 0

theorem Int.ofNat_add_negSucc_of_lt {m n : Nat} (h : m < n.succ) : ofNat m + negSucc n = negSucc (n - m)

theorem Int.add_assoc (a b c : Int) : a + b + c = a + (b + c)

instance Int.instAssociativeHAdd : Std.Associative fun (x1 x2 : Int) => x1 + x2

theorem Int.add_left_comm (a b c : Int) : a + (b + c) = b + (a + c)

theorem Int.add_right_comm (a b c : Int) : a + b + c = a + c + b

theorem Int.add_left_neg (a : Int) : -a + a = 0

theorem Int.add_right_neg (a : Int) : a + -a = 0

theorem Int.neg_eq_of_add_eq_zero {a b : Int} (h : a + b = 0) : -a = b

theorem Int.eq_neg_of_eq_neg {a b : Int} (h : a = -b) : b = -a

theorem Int.eq_neg_comm {a b : Int} : a = -b ↔ b = -a

theorem Int.neg_eq_comm {a b : Int} : -a = b ↔ -b = a

theorem Int.neg_add_cancel_left (a b : Int) : -a + (a + b) = b

theorem Int.add_neg_cancel_left (a b : Int) : a + (-a + b) = b

theorem Int.add_neg_cancel_right (a b : Int) : a + b + -b = a

theorem Int.neg_add_cancel_right (a b : Int) : a + -b + b = a

theorem Int.add_left_cancel {a b c : Int} (h : a + b = a + c) : b = c

theorem Int.neg_add {a b : Int} : -(a + b) = -a + -b

theorem Int.wlog_sign {P : Int → Prop} (inv : ∀ (i : Int), P i ↔ P (-i)) (w : ∀ (n : Nat), P ↑n) (i : Int) : P i

If a predicate on the integers is invariant under negation, then it is sufficient to prove it for the nonnegative integers.

@[simp]

theorem Int.negSucc_sub_one (n : Nat) : negSucc n - 1 = negSucc (n + 1)

@[simp]

theorem Int.sub_self (a : Int) : a - a = 0

@[simp]

theorem Int.sub_zero (a : Int) : a - 0 = a

@[simp]

theorem Int.zero_sub (a : Int) : 0 - a = -a

theorem Int.sub_eq_zero_of_eq {a b : Int} (h : a = b) : a - b = 0

theorem Int.eq_of_sub_eq_zero {a b : Int} (h : a - b = 0) : a = b

theorem Int.sub_eq_zero {a b : Int} : a - b = 0 ↔ a = b

theorem Int.sub_sub (a b c : Int) : a - b - c = a - (b + c)

theorem Int.neg_sub (a b : Int) : -(a - b) = b - a

theorem Int.sub_sub_self (a b : Int) : a - (a - b) = b

@[simp]

theorem Int.sub_neg (a b : Int) : a - -b = a + b

@[simp]

theorem Int.sub_add_cancel (a b : Int) : a - b + b = a

@[simp]

theorem Int.add_sub_cancel (a b : Int) : a + b - b = a

theorem Int.add_sub_assoc (a b c : Int) : a + b - c = a + (b - c)

theorem Int.ofNat_sub {m n : Nat} (h : m ≤ n) : ↑(n - m) = ↑n - ↑m

@[deprecated Int.negSucc_eq (since := "2025-03-11")]

theorem Int.negSucc_coe' (n : Nat) : negSucc n = -↑n - 1

theorem Int.subNatNat_eq_coe {m n : Nat} : subNatNat m n = ↑m - ↑n

@[simp]

theorem Int.toNat_sub (m n : Nat) : (↑m - ↑n).toNat = m - n

theorem Int.toNat_of_nonpos {z : Int} : z ≤ 0 → z.toNat = 0

@[simp]

theorem Int.neg_ofNat_eq_negSucc_iff {a b : Nat} : -↑a = negSucc b ↔ a = b + 1

@[simp]

theorem Int.negSucc_eq_neg_ofNat_iff {a b : Nat} : negSucc a = -↑b ↔ a + 1 = b

@[simp]

theorem Int.neg_ofNat_eq_negSucc_add_one_iff {a b : Nat} : -↑a = negSucc b + 1 ↔ a = b

@[simp]

theorem Int.negSucc_add_one_eq_neg_ofNat_iff {a b : Nat} : negSucc a + 1 = -↑b ↔ a = b

theorem Int.sub_eq_iff_eq_add {b a c : Int} : a - b = c ↔ a = c + b

theorem Int.sub_eq_iff_eq_add' {b a c : Int} : a - b = c ↔ a = b + c

@[simp]

theorem Int.add_left_inj {i j : Int} (k : Int) : i + k = j + k ↔ i = j

@[simp]

theorem Int.add_right_inj {i j : Int} (k : Int) : k + i = k + j ↔ i = j

@[simp]

theorem Int.sub_right_inj {i j : Int} (k : Int) : k - i = k - j ↔ i = j

@[simp]

theorem Int.sub_left_inj {i j : Int} (k : Int) : i - k = j - k ↔ i = j

theorem Int.ofNat_mul_negSucc (m n : Nat) : ↑m * negSucc n = -↑(m * n.succ)

theorem Int.negSucc_mul_ofNat (m n : Nat) : negSucc m * ↑n = -↑(m.succ * n)

theorem Int.negSucc_mul_negSucc (m n : Nat) : negSucc m * negSucc n = ↑m.succ * ↑n.succ

theorem Int.mul_comm (a b : Int) : a * b = b * a

instance Int.instCommutativeHMul : Std.Commutative fun (x1 x2 : Int) => x1 * x2

theorem Int.ofNat_mul_negOfNat (m n : Nat) : ↑m * negOfNat n = negOfNat (m * n)

theorem Int.negOfNat_mul_ofNat (m n : Nat) : negOfNat m * ↑n = negOfNat (m * n)

theorem Int.negSucc_mul_negOfNat (m n : Nat) : negSucc m * negOfNat n = ofNat (m.succ * n)

theorem Int.negOfNat_mul_negSucc (m n : Nat) : negOfNat n * negSucc m = ofNat (n * m.succ)

theorem Int.mul_assoc (a b c : Int) : a * b * c = a * (b * c)

instance Int.instAssociativeHMul : Std.Associative fun (x1 x2 : Int) => x1 * x2

theorem Int.mul_left_comm (a b c : Int) : a * (b * c) = b * (a * c)

theorem Int.mul_right_comm (a b c : Int) : a * b * c = a * c * b

@[simp]

theorem Int.mul_zero (a : Int) : a * 0 = 0

@[simp]

theorem Int.zero_mul (a : Int) : 0 * a = 0

theorem Int.negOfNat_eq_subNatNat_zero (n : Nat) : negOfNat n = subNatNat 0 n

theorem Int.ofNat_mul_subNatNat (m n k : Nat) : ↑m * subNatNat n k = subNatNat (m * n) (m * k)

theorem Int.negOfNat_add (m n : Nat) : negOfNat m + negOfNat n = negOfNat (m + n)

theorem Int.negSucc_mul_subNatNat (m n k : Nat) : negSucc m * subNatNat n k = subNatNat (m.succ * k) (m.succ * n)

theorem Int.mul_add (a b c : Int) : a * (b + c) = a * b + a * c

theorem Int.add_mul (a b c : Int) : (a + b) * c = a * c + b * c

theorem Int.neg_mul_eq_neg_mul (a b : Int) : -(a * b) = -a * b

theorem Int.neg_mul_eq_mul_neg (a b : Int) : -(a * b) = a * -b

theorem Int.neg_mul (a b : Int) : -a * b = -(a * b)

theorem Int.mul_neg (a b : Int) : a * -b = -(a * b)

theorem Int.neg_mul_neg (a b : Int) : -a * -b = a * b

theorem Int.neg_mul_comm (a b : Int) : -a * b = a * -b

theorem Int.mul_sub (a b c : Int) : a * (b - c) = a * b - a * c

theorem Int.sub_mul (a b c : Int) : (a - b) * c = a * c - b * c

@[simp]

theorem Int.one_mul (a : Int) : 1 * a = a

@[simp]

theorem Int.mul_one (a : Int) : a * 1 = a

instance Int.instLawfulIdentityHMulOfNat : Std.LawfulIdentity (fun (x1 x2 : Int) => x1 * x2) 1

@[simp]

theorem Int.mul_neg_one (a : Int) : a * -1 = -a

@[simp]

theorem Int.neg_one_mul (a : Int) : -1 * a = -a

theorem Int.neg_eq_neg_one_mul (a : Int) : -a = -1 * a

theorem Int.mul_eq_zero {a b : Int} : a * b = 0 ↔ a = 0 ∨ b = 0

theorem Int.mul_ne_zero {a b : Int} (a0 : a ≠ 0) (b0 : b ≠ 0) : a * b ≠ 0

instance Int.instNeZeroHMul {a b : Int} [NeZero a] [NeZero b] : NeZero (a * b)

@[simp]

theorem Int.mul_ne_zero_iff {a b : Int} : a * b ≠ 0 ↔ a ≠ 0 ∧ b ≠ 0

theorem Int.eq_of_mul_eq_mul_right {a b c : Int} (ha : a ≠ 0) (h : b * a = c * a) : b = c

theorem Int.eq_of_mul_eq_mul_left {a b c : Int} (ha : a ≠ 0) (h : a * b = a * c) : b = c

theorem Int.mul_eq_mul_left_iff {a b c : Int} (h : c ≠ 0) : c * a = c * b ↔ a = b

theorem Int.mul_eq_mul_right_iff {a b c : Int} (h : c ≠ 0) : a * c = b * c ↔ a = b

theorem Int.eq_one_of_mul_eq_self_left {a b : Int} (Hpos : a ≠ 0) (H : b * a = a) : b = 1

theorem Int.eq_one_of_mul_eq_self_right {a b : Int} (Hpos : b ≠ 0) (H : b * a = b) : a = 1

theorem Int.two_mul (n : Int) : 2 * n = n + n

NatCast lemmas

The following lemmas are later subsumed by e.g. Nat.cast_add and Nat.cast_mul in Mathlib but it is convenient to have these earlier, for users who only need Nat and Int.

theorem Int.natCast_zero : ↑0 = 0

theorem Int.natCast_one : ↑1 = 1

@[simp]

theorem Int.natAbs_cast (n : Nat) : (↑n).natAbs = n