Absolute values in ordered groups.

instance Neg.toHasAbs {α : Type u_1} [Neg α] [Sup α] : Abs α

abs a is the absolute value of a

instance Inv.toHasAbs {α : Type u_1} [Inv α] [Sup α] : Abs α

abs a is the absolute value of a.

theorem abs_eq_sup_neg {α : Type u_1} [Neg α] [Sup α] (a : α) : |a| = a ⊔ -a

theorem abs_eq_sup_inv {α : Type u_1} [Inv α] [Sup α] (a : α) : |a| = a ⊔ a⁻¹

theorem abs_eq_max_neg {α : Type u_1} [Neg α] [LinearOrder α] {a : α} : |a| = max a (-a)

theorem abs_choice {α : Type u_1} [Neg α] [LinearOrder α] (x : α) : |x| = x ∨ |x| = -x

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

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

theorem le_abs_self {α : Type u_1} [Neg α] [LinearOrder α] (a : α) : a ≤ |a|

theorem neg_le_abs_self {α : Type u_1} [Neg α] [LinearOrder α] (a : α) : -a ≤ |a|

theorem lt_abs {α : Type u_1} [Neg α] [LinearOrder α] {a : α} {b : α} : a < |b| ↔ a < b ∨ a < -b

theorem abs_le_abs {α : Type u_1} [Neg α] [LinearOrder α] {a : α} {b : α} (h₀ : a ≤ b) (h₁ : -a ≤ b) : |a| ≤ |b|

theorem abs_by_cases {α : Type u_1} [Neg α] [LinearOrder α] (P : α → Prop) {a : α} (h1 : P a) (h2 : P (-a)) : P |a|

@[simp]

theorem abs_neg {α : Type u_1} [AddGroup α] [LinearOrder α] (a : α) : |(-a)| = |a|

theorem eq_or_eq_neg_of_abs_eq {α : Type u_1} [AddGroup α] [LinearOrder α] {a : α} {b : α} (h : |a| = b) : a = b ∨ a = -b

theorem abs_eq_abs {α : Type u_1} [AddGroup α] [LinearOrder α] {a : α} {b : α} : |a| = |b| ↔ a = b ∨ a = -b

theorem abs_sub_comm {α : Type u_1} [AddGroup α] [LinearOrder α] (a : α) (b : α) : |a - b| = |b - a|

theorem abs_of_nonneg {α : Type u_1} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a : α} (h : 0 ≤ a) : |a| = a

theorem abs_of_pos {α : Type u_1} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a : α} (h : 0 < a) : |a| = a

theorem abs_of_nonpos {α : Type u_1} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a : α} (h : a ≤ 0) : |a| = -a

theorem abs_of_neg {α : Type u_1} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a : α} (h : a < 0) : |a| = -a

theorem abs_le_abs_of_nonneg {α : Type u_1} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a : α} {b : α} (ha : 0 ≤ a) (hab : a ≤ b) : |a| ≤ |b|

@[simp]

theorem abs_zero {α : Type u_1} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] : |0| = 0

@[simp]

theorem abs_pos {α : Type u_1} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a : α} : 0 < |a| ↔ a ≠ 0

theorem abs_pos_of_pos {α : Type u_1} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a : α} (h : 0 < a) : 0 < |a|

theorem abs_pos_of_neg {α : Type u_1} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a : α} (h : a < 0) : 0 < |a|

theorem neg_abs_le_self {α : Type u_1} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a : α) : -|a| ≤ a

theorem add_abs_nonneg {α : Type u_1} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a : α) : 0 ≤ a + |a|

theorem neg_abs_le_neg {α : Type u_1} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a : α) : -|a| ≤ -a

@[simp]

theorem abs_nonneg {α : Type u_1} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a : α) : 0 ≤ |a|

@[simp]

theorem abs_abs {α : Type u_1} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a : α) : |(|a|)| = |a|

@[simp]

theorem abs_eq_zero {α : Type u_1} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a : α} : |a| = 0 ↔ a = 0

@[simp]

theorem abs_nonpos_iff {α : Type u_1} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a : α} : |a| ≤ 0 ↔ a = 0

theorem abs_le_abs_of_nonpos {α : Type u_1} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a : α} {b : α} [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (ha : a ≤ 0) (hab : b ≤ a) : |a| ≤ |b|

theorem abs_lt {α : Type u_1} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a : α} {b : α} [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] : |a| < b ↔ -b < a ∧ a < b

theorem neg_lt_of_abs_lt {α : Type u_1} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a : α} {b : α} [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : |a| < b) : -b < a

theorem lt_of_abs_lt {α : Type u_1} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a : α} {b : α} [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : |a| < b) : a < b

theorem max_sub_min_eq_abs' {α : Type u_1} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a : α) (b : α) : max a b - min a b = |a - b|

theorem max_sub_min_eq_abs {α : Type u_1} [AddGroup α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a : α) (b : α) : max a b - min a b = |b - a|

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 x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass β β (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {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 x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass β β (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {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 x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass β β (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {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 x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass β β (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {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_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