Note about Mathlib/Init/

The files in Mathlib/Init are leftovers from the port from Mathlib3. (They contain content moved from lean3 itself that Mathlib needed but was not moved to lean4.)

We intend to move all the content of these files out into the main Mathlib directory structure. Contributions assisting with this are appreciated.

The order relation on the integers

theorem Int.le.elim {a : ℤ} {b : ℤ} (h : a ≤ b) {P : Prop} (h' : ∀ (n : ℕ), a + ↑n = b → P) : P

theorem Int.ofNat_le_ofNat_of_le {m : ℕ} {n : ℕ} : m ≤ n → ↑m ≤ ↑n

Alias of the reverse direction of Int.ofNat_le.

theorem Int.le_of_ofNat_le_ofNat {m : ℕ} {n : ℕ} : ↑m ≤ ↑n → m ≤ n

Alias of the forward direction of Int.ofNat_le.

theorem Int.lt.elim {a : ℤ} {b : ℤ} (h : a < b) {P : Prop} (h' : ∀ (n : ℕ), a + ↑n.succ = b → P) : P

theorem Int.lt_of_ofNat_lt_ofNat {n : ℕ} {m : ℕ} : ↑n < ↑m → n < m

Alias of the forward direction of Int.ofNat_lt.

theorem Int.ofNat_lt_ofNat_of_lt {n : ℕ} {m : ℕ} : n < m → ↑n < ↑m

Alias of the reverse direction of Int.ofNat_lt.

instance Int.instLinearOrder : LinearOrder ℤ

Equations

• Int.instLinearOrder = LinearOrder.mk Int.le_total inferInstance inferInstance inferInstance Int.instLinearOrder.proof_1 Int.instLinearOrder.proof_2 Int.instLinearOrder.proof_3

theorem Int.neg_mul_eq_neg_mul_symm (a : ℤ) (b : ℤ) : -a * b = -(a * b)

theorem Int.mul_neg_eq_neg_mul_symm (a : ℤ) (b : ℤ) : a * -b = -(a * b)

theorem Int.eq_zero_or_eq_zero_of_mul_eq_zero {a : ℤ} {b : ℤ} (h : a * b = 0) : a = 0 ∨ b = 0