Absolute values in linear ordered rings.

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@[simp]

theorem abs_one {α : Type u_1} [inst : LinearOrderedRing α] : abs 1 = 1

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@[simp]

theorem abs_two {α : Type u_1} [inst : LinearOrderedRing α] : abs 2 = 2

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theorem abs_mul {α : Type u_1} [inst : LinearOrderedRing α] (a : α) (b : α) : abs (a * b) = abs a * abs b

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def absHom {α : Type u_1} [inst : LinearOrderedRing α] : α →*₀ α

abs as a MonoidWithZeroHom.

Equations

• absHom = { toZeroHom := { toFun := abs, map_zero' := (_ : abs 0 = 0) }, map_one' := (_ : abs 1 = 1), map_mul' := (_ : ∀ (a b : α), abs (a * b) = abs a * abs b) }

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@[simp]

theorem abs_mul_abs_self {α : Type u_1} [inst : LinearOrderedRing α] (a : α) : abs a * abs a = a * a

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theorem abs_mul_self {α : Type u_1} [inst : LinearOrderedRing α] (a : α) : abs (a * a) = a * a

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theorem abs_eq_self {α : Type u_1} [inst : LinearOrderedRing α] {a : α} : abs a = a ↔ 0 ≤ a

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theorem abs_eq_neg_self {α : Type u_1} [inst : LinearOrderedRing α] {a : α} : abs a = -a ↔ a ≤ 0

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theorem abs_cases {α : Type u_1} [inst : LinearOrderedRing α] (a : α) : abs a = a ∧ 0 ≤ a ∨ abs a = -a ∧ a < 0

For an element a of a linear ordered ring, either abs a = a and 0 ≤ a≤ a, or abs a = -a and a < 0. Use cases on this lemma to automate linarith in inequalities

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@[simp]

theorem max_zero_add_max_neg_zero_eq_abs_self {α : Type u_1} [inst : LinearOrderedRing α] (a : α) : max a 0 + max (-a) 0 = abs a

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theorem abs_eq_iff_mul_self_eq {α : Type u_1} [inst : LinearOrderedRing α] {a : α} {b : α} : abs a = abs b ↔ a * a = b * b

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theorem abs_lt_iff_mul_self_lt {α : Type u_1} [inst : LinearOrderedRing α] {a : α} {b : α} : abs a < abs b ↔ a * a < b * b

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theorem abs_le_iff_mul_self_le {α : Type u_1} [inst : LinearOrderedRing α] {a : α} {b : α} : abs a ≤ abs b ↔ a * a ≤ b * b

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theorem abs_le_one_iff_mul_self_le_one {α : Type u_1} [inst : LinearOrderedRing α] {a : α} : abs a ≤ 1 ↔ a * a ≤ 1

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theorem abs_sub_sq {α : Type u_1} [inst : LinearOrderedCommRing α] (a : α) (b : α) : abs (a - b) * abs (a - b) = a * a + b * b - (1 + 1) * a * b

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@[simp]

theorem abs_dvd {α : Type u_1} [inst : Ring α] [inst : LinearOrder α] (a : α) (b : α) : abs a ∣ b ↔ a ∣ b

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theorem abs_dvd_self {α : Type u_1} [inst : Ring α] [inst : LinearOrder α] (a : α) : abs a ∣ a

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@[simp]

theorem dvd_abs {α : Type u_1} [inst : Ring α] [inst : LinearOrder α] (a : α) (b : α) : a ∣ abs b ↔ a ∣ b

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theorem self_dvd_abs {α : Type u_1} [inst : Ring α] [inst : LinearOrder α] (a : α) : a ∣ abs a

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theorem abs_dvd_abs {α : Type u_1} [inst : Ring α] [inst : LinearOrder α] (a : α) (b : α) : abs a ∣ abs b ↔ a ∣ b