ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- inf_sq_eq_mul_div_mabs_div
@[to_additive two_nsmul_inf_eq_add_sub_abs_sub]
lemma inf_sq_eq_mul_div_mabs_div (a b : α) : (a ⊓ b) ^ 2 = a * b / |b / a|ₘ 
Arguments: {α : Type u_1} [Lattice α] [CommGroup α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a b : α)
Goal: (a ⊓ b) ^ 2 = a * b / mabs (b / a)
-- two_nsmul_inf_eq_add_sub_abs_sub
@[to_additive two_nsmul_inf_eq_add_sub_abs_sub]
lemma inf_sq_eq_mul_div_mabs_div (a b : α) : (a ⊓ b) ^ 2 = a * b / |b / a|ₘ 
Arguments: {α : Type u_1} [Lattice α] [AddCommGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a b : α)
Goal: 2 • (a ⊓ b) = a + b - |b - a|
-- abs_sub_sup_add_abs_sub_inf
@[to_additive]
lemma mabs_div_sup_mul_mabs_div_inf (a b c : α) :
    |(a ⊔ c) / (b ⊔ c)|ₘ * |(a ⊓ c) / (b ⊓ c)|ₘ = |a / b|ₘ 
Arguments: {α : Type u_1} [Lattice α] [AddCommGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a b c : α)
Goal: |a ⊔ c - b ⊔ c| + |a ⊓ c - b ⊓ c| = |a - b|
-- mabs_div_sup_mul_mabs_div_inf
@[to_additive]
lemma mabs_div_sup_mul_mabs_div_inf (a b c : α) :
    |(a ⊔ c) / (b ⊔ c)|ₘ * |(a ⊓ c) / (b ⊓ c)|ₘ = |a / b|ₘ 
Arguments: {α : Type u_1} [Lattice α] [CommGroup α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a b c : α)
Goal: mabs ((a ⊔ c) / (b ⊔ c)) * mabs ((a ⊓ c) / (b ⊓ c)) = mabs (a / b)
-- abs_sup_sub_sup_le_abs
@[to_additive] lemma mabs_sup_div_sup_le_mabs (a b c : α) : |(a ⊔ c) / (b ⊔ c)|ₘ ≤ |a / b|ₘ 
Arguments: {α : Type u_1} [Lattice α] [AddCommGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a b c : α)
Goal: |a ⊔ c - b ⊔ c| ≤ |a - b|
-- mabs_sup_div_sup_le_mabs
@[to_additive] lemma mabs_sup_div_sup_le_mabs (a b c : α) : |(a ⊔ c) / (b ⊔ c)|ₘ ≤ |a / b|ₘ 
Arguments: {α : Type u_1} [Lattice α] [CommGroup α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a b c : α)
Goal: mabs ((a ⊔ c) / (b ⊔ c)) ≤ mabs (a / b)
-- abs_inf_sub_inf_le_abs
@[to_additive] lemma mabs_inf_div_inf_le_mabs (a b c : α) : |(a ⊓ c) / (b ⊓ c)|ₘ ≤ |a / b|ₘ 
Arguments: {α : Type u_1} [Lattice α] [AddCommGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a b c : α)
Goal: |a ⊓ c - b ⊓ c| ≤ |a - b|
-- mabs_inf_div_inf_le_mabs
@[to_additive] lemma mabs_inf_div_inf_le_mabs (a b c : α) : |(a ⊓ c) / (b ⊓ c)|ₘ ≤ |a / b|ₘ 
Arguments: {α : Type u_1} [Lattice α] [CommGroup α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a b c : α)
Goal: mabs ((a ⊓ c) / (b ⊓ c)) ≤ mabs (a / b)
-- m_Birkhoff_inequalities
@[to_additive Birkhoff_inequalities]
lemma m_Birkhoff_inequalities (a b c : α) :
    |(a ⊔ c) / (b ⊔ c)|ₘ ⊔ |(a ⊓ c) / (b ⊓ c)|ₘ ≤ |a / b|ₘ 
Arguments: {α : Type u_1} [Lattice α] [CommGroup α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a b c : α)
Goal: mabs ((a ⊔ c) / (b ⊔ c)) ⊔ mabs ((a ⊓ c) / (b ⊓ c)) ≤ mabs (a / b)
-- Birkhoff_inequalities
@[to_additive Birkhoff_inequalities]
lemma m_Birkhoff_inequalities (a b c : α) :
    |(a ⊔ c) / (b ⊔ c)|ₘ ⊔ |(a ⊓ c) / (b ⊓ c)|ₘ ≤ |a / b|ₘ 
Arguments: {α : Type u_1} [Lattice α] [AddCommGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a b c : α)
Goal: |a ⊔ c - b ⊔ c| ⊔ |a ⊓ c - b ⊓ c| ≤ |a - b|
-- mabs_choice
@[to_additive] lemma mabs_choice (x : α) : |x|ₘ = x ∨ |x|ₘ = x⁻¹ 
Arguments: {α : Type u_1} [Group α] [LinearOrder α] (x : α)
Goal: mabs x = x ∨ mabs x = x⁻¹
-- abs_choice
@[to_additive] lemma mabs_choice (x : α) : |x|ₘ = x ∨ |x|ₘ = x⁻¹ 
Arguments: {α : Type u_1} [AddGroup α] [LinearOrder α] (x : α)
Goal: |x| = x ∨ |x| = -x
-- le_abs
@[to_additive] lemma le_mabs : a ≤ |b|ₘ ↔ a ≤ b ∨ a ≤ b⁻¹ 
Arguments: {α : Type u_1} [AddGroup α] [LinearOrder α] {a b : α}
Goal: a ≤ |b| ↔ a ≤ b ∨ a ≤ -b
-- le_mabs
@[to_additive] lemma le_mabs : a ≤ |b|ₘ ↔ a ≤ b ∨ a ≤ b⁻¹ 
Arguments: {α : Type u_1} [Group α] [LinearOrder α] {a b : α}
Goal: a ≤ mabs b ↔ a ≤ b ∨ a ≤ b⁻¹
-- mabs_eq_max_inv
@[to_additive] lemma mabs_eq_max_inv : |a|ₘ = max a a⁻¹ 
Arguments: {α : Type u_1} [Group α] [LinearOrder α] {a : α}
Goal: mabs a = max a a⁻¹
-- abs_eq_max_neg
@[to_additive] lemma mabs_eq_max_inv : |a|ₘ = max a a⁻¹ 
Arguments: {α : Type u_1} [AddGroup α] [LinearOrder α] {a : α}
Goal: |a| = max a (-a)
-- lt_abs
@[to_additive] lemma lt_mabs : a < |b|ₘ ↔ a < b ∨ a < b⁻¹ 
Arguments: {α : Type u_1} [AddGroup α] [LinearOrder α] {a b : α}
Goal: a < |b| ↔ a < b ∨ a < -b
-- lt_mabs
@[to_additive] lemma lt_mabs : a < |b|ₘ ↔ a < b ∨ a < b⁻¹ 
Arguments: {α : Type u_1} [Group α] [LinearOrder α] {a b : α}
Goal: a < mabs b ↔ a < b ∨ a < b⁻¹
-- mabs_by_cases
@[to_additive] lemma mabs_by_cases (P : α → Prop) (h1 : P a) (h2 : P a⁻¹) : P |a|ₘ 
Arguments: {α : Type u_1} [Group α] [LinearOrder α] {a : α} (P : α → Prop) (h1 : P a) (h2 : P a⁻¹)
Goal: P (mabs a)
-- abs_by_cases
@[to_additive] lemma mabs_by_cases (P : α → Prop) (h1 : P a) (h2 : P a⁻¹) : P |a|ₘ 
Arguments: {α : Type u_1} [AddGroup α] [LinearOrder α] {a : α} (P : α → Prop) (h1 : P a) (h2 : P (-a))
Goal: P |a|
-- eq_or_eq_neg_of_abs_eq
@[to_additive] lemma eq_or_eq_inv_of_mabs_eq (h : |a|ₘ = b) : a = b ∨ a = b⁻¹ 
Arguments: {α : Type u_1} [AddGroup α] [LinearOrder α] {a b : α} (h : |a| = b)
Goal: a = b ∨ a = -b
-- eq_or_eq_inv_of_mabs_eq
@[to_additive] lemma eq_or_eq_inv_of_mabs_eq (h : |a|ₘ = b) : a = b ∨ a = b⁻¹ 
Arguments: {α : Type u_1} [Group α] [LinearOrder α] {a b : α} (h : mabs a = b)
Goal: a = b ∨ a = b⁻¹
-- mabs_eq_mabs
@[to_additive] lemma mabs_eq_mabs : |a|ₘ = |b|ₘ ↔ a = b ∨ a = b⁻¹ 
Arguments: {α : Type u_1} [Group α] [LinearOrder α] {a b : α}
Goal: mabs a = mabs b ↔ a = b ∨ a = b⁻¹
-- abs_eq_abs
@[to_additive] lemma mabs_eq_mabs : |a|ₘ = |b|ₘ ↔ a = b ∨ a = b⁻¹ 
Arguments: {α : Type u_1} [AddGroup α] [LinearOrder α] {a b : α}
Goal: |a| = |b| ↔ a = b ∨ a = -b
-- even_abs
@[to_additive] lemma isSquare_mabs : IsSquare |a|ₘ ↔ IsSquare a 
Arguments: {α : Type u_1} [AddGroup α] [LinearOrder α] {a : α}
Goal: Even |a| ↔ Even a
-- isSquare_mabs
@[to_additive] lemma isSquare_mabs : IsSquare |a|ₘ ↔ IsSquare a 
Arguments: {α : Type u_1} [Group α] [LinearOrder α] {a : α}
Goal: IsSquare (mabs a) ↔ IsSquare a
-- one_lt_mabs
@[to_additive (attr := simp) abs_pos] lemma one_lt_mabs : 1 < |a|ₘ ↔ a ≠ 1 
Arguments: {α : Type u_1} [Group α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a : α}
Goal: 1 < mabs a ↔ a ≠ 1
-- abs_pos
@[to_additive (attr := simp) abs_pos] lemma one_lt_mabs : 1 < |a|ₘ ↔ a ≠ 1 
Arguments: {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}
Goal: 0 < |a| ↔ a ≠ 0
-- one_lt_mabs_pos_of_one_lt
@[to_additive abs_pos_of_pos] lemma one_lt_mabs_pos_of_one_lt (h : 1 < a) : 1 < |a|ₘ 
Arguments: {α : Type u_1} [Group α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a : α} (h : 1 < a)
Goal: 1 < mabs a
-- abs_pos_of_pos
@[to_additive abs_pos_of_pos] lemma one_lt_mabs_pos_of_one_lt (h : 1 < a) : 1 < |a|ₘ 
Arguments: {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)
Goal: 0 < |a|
-- abs_pos_of_neg
@[to_additive abs_pos_of_neg] lemma one_lt_mabs_of_lt_one (h : a < 1) : 1 < |a|ₘ 
Arguments: {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)
Goal: 0 < |a|
-- one_lt_mabs_of_lt_one
@[to_additive abs_pos_of_neg] lemma one_lt_mabs_of_lt_one (h : a < 1) : 1 < |a|ₘ 
Arguments: {α : Type u_1} [Group α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a : α} (h : a < 1)
Goal: 1 < mabs a
-- neg_abs_le
@[to_additive] lemma inv_mabs_le (a : α) : |a|ₘ⁻¹ ≤ a 
Arguments: {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)
Goal: -|a| ≤ a
-- inv_mabs_le
@[to_additive] lemma inv_mabs_le (a : α) : |a|ₘ⁻¹ ≤ a 
Arguments: {α : Type u_1} [Group α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a : α)
Goal: (mabs a)⁻¹ ≤ a
-- one_le_mul_mabs
@[to_additive add_abs_nonneg] lemma one_le_mul_mabs (a : α) : 1 ≤ a * |a|ₘ 
Arguments: {α : Type u_1} [Group α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a : α)
Goal: 1 ≤ a * mabs a
-- add_abs_nonneg
@[to_additive add_abs_nonneg] lemma one_le_mul_mabs (a : α) : 1 ≤ a * |a|ₘ 
Arguments: {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)
Goal: 0 ≤ a + |a|
-- inv_mabs_le_inv
@[to_additive] lemma inv_mabs_le_inv (a : α) : |a|ₘ⁻¹ ≤ a⁻¹ 
Arguments: {α : Type u_1} [Group α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a : α)
Goal: (mabs a)⁻¹ ≤ a⁻¹
-- neg_abs_le_neg
@[to_additive] lemma inv_mabs_le_inv (a : α) : |a|ₘ⁻¹ ≤ a⁻¹ 
Arguments: {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)
Goal: -|a| ≤ -a
-- abs_ne_zero
@[to_additive] lemma mabs_ne_one : |a|ₘ ≠ 1 ↔ a ≠ 1 
Arguments: {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} [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: |a| ≠ 0 ↔ a ≠ 0
-- mabs_ne_one
@[to_additive] lemma mabs_ne_one : |a|ₘ ≠ 1 ↔ a ≠ 1 
Arguments: {α : Type u_1} [Group α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a : α} [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1]
Goal: mabs a ≠ 1 ↔ a ≠ 1
-- abs_eq_zero
@[to_additive (attr := simp)] lemma mabs_eq_one : |a|ₘ = 1 ↔ a = 1 
Arguments: {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} [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: |a| = 0 ↔ a = 0
-- mabs_eq_one
@[to_additive (attr := simp)] lemma mabs_eq_one : |a|ₘ = 1 ↔ a = 1 
Arguments: {α : Type u_1} [Group α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a : α} [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1]
Goal: mabs a = 1 ↔ a = 1
-- abs_nonpos_iff
@[to_additive (attr := simp) abs_nonpos_iff] lemma mabs_le_one : |a|ₘ ≤ 1 ↔ a = 1 
Arguments: {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} [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: |a| ≤ 0 ↔ a = 0
-- mabs_le_one
@[to_additive (attr := simp) abs_nonpos_iff] lemma mabs_le_one : |a|ₘ ≤ 1 ↔ a = 1 
Arguments: {α : Type u_1} [Group α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a : α} [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1]
Goal: mabs a ≤ 1 ↔ a = 1
-- mabs_le_mabs_of_le_one
@[to_additive] lemma mabs_le_mabs_of_le_one (ha : a ≤ 1) (hab : b ≤ a) : |a|ₘ ≤ |b|ₘ 
Arguments: {α : Type u_1} [Group α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α} [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (ha : a ≤ 1) (hab : b ≤ a)
Goal: mabs a ≤ mabs b
-- abs_le_abs_of_nonpos
@[to_additive] lemma mabs_le_mabs_of_le_one (ha : a ≤ 1) (hab : b ≤ a) : |a|ₘ ≤ |b|ₘ 
Arguments: {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 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)
Goal: |a| ≤ |b|
-- abs_lt
@[to_additive] lemma mabs_lt : |a|ₘ < b ↔ b⁻¹ < a ∧ a < b 
Arguments: {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 b : α} [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: |a| < b ↔ -b < a ∧ a < b
-- mabs_lt
@[to_additive] lemma mabs_lt : |a|ₘ < b ↔ b⁻¹ < a ∧ a < b 
Arguments: {α : Type u_1} [Group α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α} [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1]
Goal: mabs a < b ↔ b⁻¹ < a ∧ a < b
-- inv_lt_of_mabs_lt
@[to_additive] lemma inv_lt_of_mabs_lt (h : |a|ₘ < b) : b⁻¹ < a 
Arguments: {α : Type u_1} [Group α] [LinearOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α} [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (h : mabs a < b)
Goal: b⁻¹ < a
-- neg_lt_of_abs_lt
@[to_additive] lemma inv_lt_of_mabs_lt (h : |a|ₘ < b) : b⁻¹ < a 
Arguments: {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 b : α} [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : |a| < b)
Goal: -b < a
-- lt_of_mabs_lt
@[to_additive] lemma lt_of_mabs_lt : |a|ₘ < b → a < b 
Arguments: {α : Type u_1} [Group α] [LinearOrder α] {a b : α}
Goal: mabs a < b → a < b
-- lt_of_abs_lt
@[to_additive] lemma lt_of_mabs_lt : |a|ₘ < b → a < b 
Arguments: {α : Type u_1} [AddGroup α] [LinearOrder α] {a b : α}
Goal: |a| < b → a < b
-- max_div_min_eq_mabs'
@[to_additive] lemma max_div_min_eq_mabs' (a b : α) : max a b / min a b = |a / b|ₘ 
Arguments: {α : Type u_1} [Group 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 b : α)
Goal: max a b / min a b = mabs (a / b)
-- max_sub_min_eq_abs'
@[to_additive] lemma max_div_min_eq_mabs' (a b : α) : max a b / min a b = |a / b|ₘ 
Arguments: {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 b : α)
Goal: max a b - min a b = |a - b|
-- max_sub_min_eq_abs
@[to_additive] lemma max_div_min_eq_mabs (a b : α) : max a b / min a b = |b / a|ₘ 
Arguments: {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 b : α)
Goal: max a b - min a b = |b - a|
-- max_div_min_eq_mabs
@[to_additive] lemma max_div_min_eq_mabs (a b : α) : max a b / min a b = |b / a|ₘ 
Arguments: {α : Type u_1} [Group 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 b : α)
Goal: max a b / min a b = mabs (b / a)
-- LatticeOrderedAddCommGroup.isSolid_solidClosure
lemma isSolid_solidClosure (s : Set α) : IsSolid (solidClosure s) 
Arguments: {α : Type u_1} [Lattice α] [AddCommGroup α] (s : Set α)
Goal: LatticeOrderedAddCommGroup.IsSolid (LatticeOrderedAddCommGroup.solidClosure s)
-- LatticeOrderedAddCommGroup.solidClosure_min
lemma solidClosure_min (hst : s ⊆ t) (ht : IsSolid t) : solidClosure s ⊆ t 
Arguments: {α : Type u_1} [Lattice α] [AddCommGroup α] {s t : Set α} (hst : s ⊆ t) (ht : LatticeOrderedAddCommGroup.IsSolid t)
Goal: LatticeOrderedAddCommGroup.solidClosure s ⊆ t
-- Pi.abs_apply
@[simp] lemma abs_apply (f : ∀ i, α i) (i : ι) : |f| i = |f i| 
Arguments: {ι : Type u_2} {α : ι → Type u_3} [(i : ι) → AddGroup (α i)] [(i : ι) → Lattice (α i)] (f : (i : ι) → α i) (i : ι)
Goal: |f| i = |f i|
-- Pi.abs_def
lemma abs_def (f : ∀ i, α i) : |f| = fun i ↦ |f i| 
Arguments: {ι : Type u_2} {α : ι → Type u_3} [(i : ι) → AddGroup (α i)] [(i : ι) → Lattice (α i)] (f : (i : ι) → α i)
Goal: |f| = fun i => |f i|
-- abs_nsmul
@[to_additive] lemma mabs_pow (n : ℕ) (a : α) : |a ^ n|ₘ = |a|ₘ ^ n 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] (n : ℕ) (a : α)
Goal: |n • a| = n • |a|
-- mabs_pow
@[to_additive] lemma mabs_pow (n : ℕ) (a : α) : |a ^ n|ₘ = |a|ₘ ^ n 
Arguments: {α : Type u_1} [LinearOrderedCommGroup α] (n : ℕ) (a : α)
Goal: mabs (a ^ n) = mabs a ^ n
-- _private.Mathlib.Algebra.Order.Group.Abs.0.abs_add_eq_add_abs_le
@[to_additive] private lemma mabs_mul_eq_mul_mabs_le (hab : a ≤ b) :
    |a * b|ₘ = |a|ₘ * |b|ₘ ↔ 1 ≤ a ∧ 1 ≤ b ∨ a ≤ 1 ∧ b ≤ 1 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a b : α} (hab : a ≤ b)
Goal: |a + b| = |a| + |b| ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0
-- _private.Mathlib.Algebra.Order.Group.Abs.0.mabs_mul_eq_mul_mabs_le
@[to_additive] private lemma mabs_mul_eq_mul_mabs_le (hab : a ≤ b) :
    |a * b|ₘ = |a|ₘ * |b|ₘ ↔ 1 ≤ a ∧ 1 ≤ b ∨ a ≤ 1 ∧ b ≤ 1 
Arguments: {α : Type u_1} [LinearOrderedCommGroup α] {a b : α} (hab : a ≤ b)
Goal: mabs (a * b) = mabs a * mabs b ↔ 1 ≤ a ∧ 1 ≤ b ∨ a ≤ 1 ∧ b ≤ 1
-- mabs_mul_eq_mul_mabs_iff
@[to_additive] lemma mabs_mul_eq_mul_mabs_iff (a b : α) :
    |a * b|ₘ = |a|ₘ * |b|ₘ ↔ 1 ≤ a ∧ 1 ≤ b ∨ a ≤ 1 ∧ b ≤ 1 
Arguments: {α : Type u_1} [LinearOrderedCommGroup α] (a b : α)
Goal: mabs (a * b) = mabs a * mabs b ↔ 1 ≤ a ∧ 1 ≤ b ∨ a ≤ 1 ∧ b ≤ 1
-- abs_add_eq_add_abs_iff
@[to_additive] lemma mabs_mul_eq_mul_mabs_iff (a b : α) :
    |a * b|ₘ = |a|ₘ * |b|ₘ ↔ 1 ≤ a ∧ 1 ≤ b ∨ a ≤ 1 ∧ b ≤ 1 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] (a b : α)
Goal: |a + b| = |a| + |b| ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0
-- abs_le
theorem abs_le : |a| ≤ b ↔ -b ≤ a ∧ a ≤ b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a b : α}
Goal: |a| ≤ b ↔ -b ≤ a ∧ a ≤ b
-- le_abs'
theorem le_abs' : a ≤ |b| ↔ b ≤ -a ∨ a ≤ b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a b : α}
Goal: a ≤ |b| ↔ b ≤ -a ∨ a ≤ b
-- neg_le_of_abs_le
theorem neg_le_of_abs_le (h : |a| ≤ b) : -b ≤ a 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a b : α} (h : |a| ≤ b)
Goal: -b ≤ a
-- le_of_abs_le
theorem le_of_abs_le (h : |a| ≤ b) : a ≤ b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a b : α} (h : |a| ≤ b)
Goal: a ≤ b
-- apply_abs_le_mul_of_one_le'
@[to_additive]
theorem apply_abs_le_mul_of_one_le' {β : Type*} [MulOneClass β] [Preorder β]
    [CovariantClass β β (· * ·) (· ≤ ·)] [CovariantClass β β (swap (· * ·)) (· ≤ ·)] {f : α → β}
    {a : α} (h₁ : 1 ≤ f a) (h₂ : 1 ≤ f (-a)) : f |a| ≤ f a * f (-a) 
Arguments: {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))
Goal: f |a| ≤ f a * f (-a)
-- apply_abs_le_add_of_nonneg'
@[to_additive]
theorem apply_abs_le_mul_of_one_le' {β : Type*} [MulOneClass β] [Preorder β]
    [CovariantClass β β (· * ·) (· ≤ ·)] [CovariantClass β β (swap (· * ·)) (· ≤ ·)] {f : α → β}
    {a : α} (h₁ : 1 ≤ f a) (h₂ : 1 ≤ f (-a)) : f |a| ≤ f a * f (-a) 
Arguments: {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))
Goal: f |a| ≤ f a + f (-a)
-- apply_abs_le_mul_of_one_le
@[to_additive]
theorem apply_abs_le_mul_of_one_le {β : Type*} [MulOneClass β] [Preorder β]
    [CovariantClass β β (· * ·) (· ≤ ·)] [CovariantClass β β (swap (· * ·)) (· ≤ ·)] {f : α → β}
    (h : ∀ x, 1 ≤ f x) (a : α) : f |a| ≤ f a * f (-a) 
Arguments: {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)
Goal: f |a| ≤ f a * f (-a)
-- apply_abs_le_add_of_nonneg
@[to_additive]
theorem apply_abs_le_mul_of_one_le {β : Type*} [MulOneClass β] [Preorder β]
    [CovariantClass β β (· * ·) (· ≤ ·)] [CovariantClass β β (swap (· * ·)) (· ≤ ·)] {f : α → β}
    (h : ∀ x, 1 ≤ f x) (a : α) : f |a| ≤ f a * f (-a) 
Arguments: {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)
Goal: f |a| ≤ f a + f (-a)
-- abs_add
theorem abs_add (a b : α) : |a + b| ≤ |a| + |b| 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] (a b : α)
Goal: |a + b| ≤ |a| + |b|
-- abs_add'
theorem abs_add' (a b : α) : |a| ≤ |b| + |b + a| 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] (a b : α)
Goal: |a| ≤ |b| + |b + a|
-- abs_sub
theorem abs_sub (a b : α) : |a - b| ≤ |a| + |b| 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] (a b : α)
Goal: |a - b| ≤ |a| + |b|
-- abs_sub_le_iff
theorem abs_sub_le_iff : |a - b| ≤ c ↔ a - b ≤ c ∧ b - a ≤ c 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a b c : α}
Goal: |a - b| ≤ c ↔ a - b ≤ c ∧ b - a ≤ c
-- abs_sub_lt_iff
theorem abs_sub_lt_iff : |a - b| < c ↔ a - b < c ∧ b - a < c 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a b c : α}
Goal: |a - b| < c ↔ a - b < c ∧ b - a < c
-- sub_le_of_abs_sub_le_left
theorem sub_le_of_abs_sub_le_left (h : |a - b| ≤ c) : b - c ≤ a 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a b c : α} (h : |a - b| ≤ c)
Goal: b - c ≤ a
-- sub_le_of_abs_sub_le_right
theorem sub_le_of_abs_sub_le_right (h : |a - b| ≤ c) : a - c ≤ b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a b c : α} (h : |a - b| ≤ c)
Goal: a - c ≤ b
-- sub_lt_of_abs_sub_lt_left
theorem sub_lt_of_abs_sub_lt_left (h : |a - b| < c) : b - c < a 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a b c : α} (h : |a - b| < c)
Goal: b - c < a
-- sub_lt_of_abs_sub_lt_right
theorem sub_lt_of_abs_sub_lt_right (h : |a - b| < c) : a - c < b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a b c : α} (h : |a - b| < c)
Goal: a - c < b
-- abs_sub_abs_le_abs_sub
theorem abs_sub_abs_le_abs_sub (a b : α) : |a| - |b| ≤ |a - b| 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] (a b : α)
Goal: |a| - |b| ≤ |a - b|
-- abs_abs_sub_abs_le_abs_sub
theorem abs_abs_sub_abs_le_abs_sub (a b : α) : |(|a| - |b|)| ≤ |a - b| 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] (a b : α)
Goal: ||a| - |b|| ≤ |a - b|
-- abs_sub_le_of_nonneg_of_le
theorem abs_sub_le_of_nonneg_of_le {a b n : α} (a_nonneg : 0 ≤ a) (a_le_n : a ≤ n)
    (b_nonneg : 0 ≤ b) (b_le_n : b ≤ n) : |a - b| ≤ n 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a b n : α} (a_nonneg : 0 ≤ a) (a_le_n : a ≤ n) (b_nonneg : 0 ≤ b) (b_le_n : b ≤ n)
Goal: |a - b| ≤ n
-- abs_sub_lt_of_nonneg_of_lt
theorem abs_sub_lt_of_nonneg_of_lt {a b n : α} (a_nonneg : 0 ≤ a) (a_lt_n : a < n)
    (b_nonneg : 0 ≤ b) (b_lt_n : b < n) : |a - b| < n 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a b n : α} (a_nonneg : 0 ≤ a) (a_lt_n : a < n) (b_nonneg : 0 ≤ b) (b_lt_n : b < n)
Goal: |a - b| < n
-- abs_eq
theorem abs_eq (hb : 0 ≤ b) : |a| = b ↔ a = b ∨ a = -b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a b : α} (hb : 0 ≤ b)
Goal: |a| = b ↔ a = b ∨ a = -b
-- abs_le_max_abs_abs
theorem abs_le_max_abs_abs (hab : a ≤ b) (hbc : b ≤ c) : |b| ≤ max |a| |c| 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a b c : α} (hab : a ≤ b) (hbc : b ≤ c)
Goal: |b| ≤ max |a| |c|
-- min_abs_abs_le_abs_max
theorem min_abs_abs_le_abs_max : min |a| |b| ≤ |max a b| 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a b : α}
Goal: min |a| |b| ≤ |max a b|
-- min_abs_abs_le_abs_min
theorem min_abs_abs_le_abs_min : min |a| |b| ≤ |min a b| 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a b : α}
Goal: min |a| |b| ≤ |min a b|
-- abs_max_le_max_abs_abs
theorem abs_max_le_max_abs_abs : |max a b| ≤ max |a| |b| 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a b : α}
Goal: |max a b| ≤ max |a| |b|
-- abs_min_le_max_abs_abs
theorem abs_min_le_max_abs_abs : |min a b| ≤ max |a| |b| 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a b : α}
Goal: |min a b| ≤ max |a| |b|
-- eq_of_abs_sub_eq_zero
theorem eq_of_abs_sub_eq_zero {a b : α} (h : |a - b| = 0) : a = b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a b : α} (h : |a - b| = 0)
Goal: a = b
-- abs_sub_le
theorem abs_sub_le (a b c : α) : |a - c| ≤ |a - b| + |b - c| 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] (a b c : α)
Goal: |a - c| ≤ |a - b| + |b - c|