Absolute values in ordered groups

The absolute value of an element in a group which is also a lattice is its supremum with its negation. This generalizes the usual absolute value on real numbers (|x| = max x (-x)).

Notations

• |a|: The absolute value of an element a of an additive lattice ordered group
• |a|ₘ: The absolute value of an element a of a multiplicative lattice ordered group

theorem abs_nsmul {α : Type u_1} [LinearOrderedAddCommGroup α] (n : ℕ) (a : α) : |n • a| = n • |a|

theorem mabs_pow {α : Type u_1} [LinearOrderedCommGroup α] (n : ℕ) (a : α) : mabs (a ^ n) = mabs a ^ n

theorem abs_add_eq_add_abs_iff {α : Type u_1} [LinearOrderedAddCommGroup α] (a : α) (b : α) : |a + b| = |a| + |b| ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0

theorem mabs_mul_eq_mul_mabs_iff {α : Type u_1} [LinearOrderedCommGroup α] (a : α) (b : α) : mabs (a * b) = mabs a * mabs b ↔ 1 ≤ a ∧ 1 ≤ b ∨ a ≤ 1 ∧ b ≤ 1

theorem abs_le {α : Type u_1} [LinearOrderedAddCommGroup α] {a : α} {b : α} : |a| ≤ b ↔ -b ≤ a ∧ a ≤ b

theorem le_abs' {α : Type u_1} [LinearOrderedAddCommGroup α] {a : α} {b : α} : a ≤ |b| ↔ b ≤ -a ∨ a ≤ b

theorem neg_le_of_abs_le {α : Type u_1} [LinearOrderedAddCommGroup α] {a : α} {b : α} (h : |a| ≤ b) : -b ≤ a

theorem le_of_abs_le {α : Type u_1} [LinearOrderedAddCommGroup α] {a : α} {b : α} (h : |a| ≤ b) : a ≤ b

theorem apply_abs_le_add_of_nonneg' {α : Type u_1} [LinearOrderedAddCommGroup α] {β : Type u_2} [AddZeroClass β] [Preorder β] [CovariantClass β β (fun (x1 x2 : β) => x1 + x2) fun (x1 x2 : β) => x1 ≤ x2] [CovariantClass β β (Function.swap fun (x1 x2 : β) => x1 + x2) fun (x1 x2 : β) => x1 ≤ x2] {f : α → β} {a : α} (h₁ : 0 ≤ f a) (h₂ : 0 ≤ f (-a)) : f |a| ≤ f a + f (-a)

theorem apply_abs_le_mul_of_one_le' {α : Type u_1} [LinearOrderedAddCommGroup α] {β : Type u_2} [MulOneClass β] [Preorder β] [CovariantClass β β (fun (x1 x2 : β) => x1 * x2) fun (x1 x2 : β) => x1 ≤ x2] [CovariantClass β β (Function.swap fun (x1 x2 : β) => x1 * x2) fun (x1 x2 : β) => x1 ≤ x2] {f : α → β} {a : α} (h₁ : 1 ≤ f a) (h₂ : 1 ≤ f (-a)) : f |a| ≤ f a * f (-a)

theorem apply_abs_le_add_of_nonneg {α : Type u_1} [LinearOrderedAddCommGroup α] {β : Type u_2} [AddZeroClass β] [Preorder β] [CovariantClass β β (fun (x1 x2 : β) => x1 + x2) fun (x1 x2 : β) => x1 ≤ x2] [CovariantClass β β (Function.swap fun (x1 x2 : β) => x1 + x2) fun (x1 x2 : β) => x1 ≤ x2] {f : α → β} (h : ∀ (x : α), 0 ≤ f x) (a : α) : f |a| ≤ f a + f (-a)

theorem apply_abs_le_mul_of_one_le {α : Type u_1} [LinearOrderedAddCommGroup α] {β : Type u_2} [MulOneClass β] [Preorder β] [CovariantClass β β (fun (x1 x2 : β) => x1 * x2) fun (x1 x2 : β) => x1 ≤ x2] [CovariantClass β β (Function.swap fun (x1 x2 : β) => x1 * x2) fun (x1 x2 : β) => x1 ≤ x2] {f : α → β} (h : ∀ (x : α), 1 ≤ f x) (a : α) : f |a| ≤ f a * f (-a)

theorem abs_add {α : Type u_1} [LinearOrderedAddCommGroup α] (a : α) (b : α) : |a + b| ≤ |a| + |b|

The triangle inequality in LinearOrderedAddCommGroups.

theorem abs_add' {α : Type u_1} [LinearOrderedAddCommGroup α] (a : α) (b : α) : |a| ≤ |b| + |b + a|

theorem abs_sub {α : Type u_1} [LinearOrderedAddCommGroup α] (a : α) (b : α) : |a - b| ≤ |a| + |b|

theorem abs_sub_le_iff {α : Type u_1} [LinearOrderedAddCommGroup α] {a : α} {b : α} {c : α} : |a - b| ≤ c ↔ a - b ≤ c ∧ b - a ≤ c

theorem abs_sub_lt_iff {α : Type u_1} [LinearOrderedAddCommGroup α] {a : α} {b : α} {c : α} : |a - b| < c ↔ a - b < c ∧ b - a < c

theorem sub_le_of_abs_sub_le_left {α : Type u_1} [LinearOrderedAddCommGroup α] {a : α} {b : α} {c : α} (h : |a - b| ≤ c) : b - c ≤ a

theorem sub_le_of_abs_sub_le_right {α : Type u_1} [LinearOrderedAddCommGroup α] {a : α} {b : α} {c : α} (h : |a - b| ≤ c) : a - c ≤ b

theorem sub_lt_of_abs_sub_lt_left {α : Type u_1} [LinearOrderedAddCommGroup α] {a : α} {b : α} {c : α} (h : |a - b| < c) : b - c < a

theorem sub_lt_of_abs_sub_lt_right {α : Type u_1} [LinearOrderedAddCommGroup α] {a : α} {b : α} {c : α} (h : |a - b| < c) : a - c < b

theorem abs_sub_abs_le_abs_sub {α : Type u_1} [LinearOrderedAddCommGroup α] (a : α) (b : α) : |a| - |b| ≤ |a - b|

theorem abs_abs_sub_abs_le_abs_sub {α : Type u_1} [LinearOrderedAddCommGroup α] (a : α) (b : α) : ||a| - |b|| ≤ |a - b|

theorem abs_sub_le_of_nonneg_of_le {α : Type u_1} [LinearOrderedAddCommGroup α] {a : α} {b : α} {n : α} (a_nonneg : 0 ≤ a) (a_le_n : a ≤ n) (b_nonneg : 0 ≤ b) (b_le_n : b ≤ n) : |a - b| ≤ n

|a - b| ≤ n if 0 ≤ a ≤ n and 0 ≤ b ≤ n.

theorem abs_sub_lt_of_nonneg_of_lt {α : Type u_1} [LinearOrderedAddCommGroup α] {a : α} {b : α} {n : α} (a_nonneg : 0 ≤ a) (a_lt_n : a < n) (b_nonneg : 0 ≤ b) (b_lt_n : b < n) : |a - b| < n

|a - b| < n if 0 ≤ a < n and 0 ≤ b < n.

theorem abs_eq {α : Type u_1} [LinearOrderedAddCommGroup α] {a : α} {b : α} (hb : 0 ≤ b) : |a| = b ↔ a = b ∨ a = -b

theorem abs_le_max_abs_abs {α : Type u_1} [LinearOrderedAddCommGroup α] {a : α} {b : α} {c : α} (hab : a ≤ b) (hbc : b ≤ c) : |b| ≤ max |a| |c|

theorem min_abs_abs_le_abs_max {α : Type u_1} [LinearOrderedAddCommGroup α] {a : α} {b : α} : min |a| |b| ≤ |max a b|

theorem min_abs_abs_le_abs_min {α : Type u_1} [LinearOrderedAddCommGroup α] {a : α} {b : α} : min |a| |b| ≤ |min a b|

theorem abs_max_le_max_abs_abs {α : Type u_1} [LinearOrderedAddCommGroup α] {a : α} {b : α} : |max a b| ≤ max |a| |b|

theorem abs_min_le_max_abs_abs {α : Type u_1} [LinearOrderedAddCommGroup α] {a : α} {b : α} : |min a b| ≤ max |a| |b|

theorem eq_of_abs_sub_eq_zero {α : Type u_1} [LinearOrderedAddCommGroup α] {a : α} {b : α} (h : |a - b| = 0) : a = b

theorem abs_sub_le {α : Type u_1} [LinearOrderedAddCommGroup α] (a : α) (b : α) (c : α) : |a - c| ≤ |a - b| + |b - c|

theorem abs_add_three {α : Type u_1} [LinearOrderedAddCommGroup α] (a : α) (b : α) (c : α) : |a + b + c| ≤ |a| + |b| + |c|

theorem dist_bdd_within_interval {α : Type u_1} [LinearOrderedAddCommGroup α] {a : α} {b : α} {lb : α} {ub : α} (hal : lb ≤ a) (hau : a ≤ ub) (hbl : lb ≤ b) (hbu : b ≤ ub) : |a - b| ≤ ub - lb

theorem eq_of_abs_sub_nonpos {α : Type u_1} [LinearOrderedAddCommGroup α] {a : α} {b : α} (h : |a - b| ≤ 0) : a = b

theorem abs_sub_nonpos {α : Type u_1} [LinearOrderedAddCommGroup α] {a : α} {b : α} : |a - b| ≤ 0 ↔ a = b

theorem abs_sub_pos {α : Type u_1} [LinearOrderedAddCommGroup α] {a : α} {b : α} : 0 < |a - b| ↔ a ≠ b

@[simp]

theorem abs_eq_self {α : Type u_1} [LinearOrderedAddCommGroup α] {a : α} : |a| = a ↔ 0 ≤ a

@[simp]

theorem abs_eq_neg_self {α : Type u_1} [LinearOrderedAddCommGroup α] {a : α} : |a| = -a ↔ a ≤ 0

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

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

@[simp]

theorem max_zero_add_max_neg_zero_eq_abs_self {α : Type u_1} [LinearOrderedAddCommGroup α] (a : α) : max a 0 + max (-a) 0 = |a|