# The order relation on the integers #

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

Alias of the reverse direction of `Int.ofNat_le`.

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

Alias of the forward direction of `Int.ofNat_le`.

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

Alias of the reverse direction of `Int.ofNat_lt`.

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

Alias of the forward direction of `Int.ofNat_lt`.

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instance Int.instLinearOrder :
LinearOrder ℤ
Equations

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

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theorem Int.eq_zero_or_eq_zero_of_mul_eq_zero {a b : ℤ} (h : a * b = 0) :
a = 0 ∨ b = 0
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theorem Int.nonneg_or_nonpos_of_mul_nonneg {a b : ℤ} :
0 ≤ a * b → 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0
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theorem Int.mul_nonneg_of_nonneg_or_nonpos {a b : ℤ} :
0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0 → 0 ≤ a * b
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theorem Int.mul_nonneg_iff {a b : ℤ} :
0 ≤ a * b ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0