ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- inv_le_inv_of_neg
theorem inv_le_inv_of_neg (ha : a < 0) (hb : b < 0) : a⁻¹ ≤ b⁻¹ ↔ b ≤ a 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α} (ha : a < 0) (hb : b < 0)
Goal: a⁻¹ ≤ b⁻¹ ↔ b ≤ a
-- inv_le_of_neg
theorem inv_le_of_neg (ha : a < 0) (hb : b < 0) : a⁻¹ ≤ b ↔ b⁻¹ ≤ a 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α} (ha : a < 0) (hb : b < 0)
Goal: a⁻¹ ≤ b ↔ b⁻¹ ≤ a
-- le_inv_of_neg
theorem le_inv_of_neg (ha : a < 0) (hb : b < 0) : a ≤ b⁻¹ ↔ b ≤ a⁻¹ 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α} (ha : a < 0) (hb : b < 0)
Goal: a ≤ b⁻¹ ↔ b ≤ a⁻¹
-- inv_lt_inv_of_neg
theorem inv_lt_inv_of_neg (ha : a < 0) (hb : b < 0) : a⁻¹ < b⁻¹ ↔ b < a 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α} (ha : a < 0) (hb : b < 0)
Goal: a⁻¹ < b⁻¹ ↔ b < a
-- inv_lt_of_neg
theorem inv_lt_of_neg (ha : a < 0) (hb : b < 0) : a⁻¹ < b ↔ b⁻¹ < a 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α} (ha : a < 0) (hb : b < 0)
Goal: a⁻¹ < b ↔ b⁻¹ < a
-- lt_inv_of_neg
theorem lt_inv_of_neg (ha : a < 0) (hb : b < 0) : a < b⁻¹ ↔ b < a⁻¹ 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α} (ha : a < 0) (hb : b < 0)
Goal: a < b⁻¹ ↔ b < a⁻¹
-- sub_inv_antitoneOn_Ioi
theorem sub_inv_antitoneOn_Ioi :
    AntitoneOn (fun x ↦ (x-c)⁻¹) (Set.Ioi c) 
Arguments: {α : Type u_2} [LinearOrderedField α] {c : α}
Goal: AntitoneOn (fun x => (x - c)⁻¹) (Set.Ioi c)
-- sub_inv_antitoneOn_Iio
theorem sub_inv_antitoneOn_Iio :
    AntitoneOn (fun x ↦ (x-c)⁻¹) (Set.Iio c) 
Arguments: {α : Type u_2} [LinearOrderedField α] {c : α}
Goal: AntitoneOn (fun x => (x - c)⁻¹) (Set.Iio c)
-- sub_inv_antitoneOn_Icc_right
theorem sub_inv_antitoneOn_Icc_right (ha : c < a) :
    AntitoneOn (fun x ↦ (x-c)⁻¹) (Set.Icc a b) 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b c : α} (ha : c < a)
Goal: AntitoneOn (fun x => (x - c)⁻¹) (Set.Icc a b)
-- sub_inv_antitoneOn_Icc_left
theorem sub_inv_antitoneOn_Icc_left (ha : b < c) :
    AntitoneOn (fun x ↦ (x-c)⁻¹) (Set.Icc a b) 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b c : α} (ha : b < c)
Goal: AntitoneOn (fun x => (x - c)⁻¹) (Set.Icc a b)
-- inv_antitoneOn_Ioi
theorem inv_antitoneOn_Ioi :
    AntitoneOn (fun x : α ↦ x⁻¹) (Set.Ioi 0) 
Arguments: {α : Type u_2} [LinearOrderedField α]
Goal: AntitoneOn (fun x => x⁻¹) (Set.Ioi 0)
-- inv_antitoneOn_Iio
theorem inv_antitoneOn_Iio :
    AntitoneOn (fun x : α ↦ x⁻¹) (Set.Iio 0) 
Arguments: {α : Type u_2} [LinearOrderedField α]
Goal: AntitoneOn (fun x => x⁻¹) (Set.Iio 0)
-- inv_antitoneOn_Icc_right
theorem inv_antitoneOn_Icc_right (ha : 0 < a) :
    AntitoneOn (fun x : α ↦ x⁻¹) (Set.Icc a b) 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α} (ha : 0 < a)
Goal: AntitoneOn (fun x => x⁻¹) (Set.Icc a b)
-- inv_antitoneOn_Icc_left
theorem inv_antitoneOn_Icc_left (hb : b < 0) :
    AntitoneOn (fun x : α ↦ x⁻¹) (Set.Icc a b) 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α} (hb : b < 0)
Goal: AntitoneOn (fun x => x⁻¹) (Set.Icc a b)
-- div_le_div_of_nonpos_of_le
theorem div_le_div_of_nonpos_of_le (hc : c ≤ 0) (h : b ≤ a) : a / c ≤ b / c 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b c : α} (hc : c ≤ 0) (h : b ≤ a)
Goal: a / c ≤ b / c
-- div_lt_div_of_neg_of_lt
theorem div_lt_div_of_neg_of_lt (hc : c < 0) (h : b < a) : a / c < b / c 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b c : α} (hc : c < 0) (h : b < a)
Goal: a / c < b / c
-- div_le_div_right_of_neg
theorem div_le_div_right_of_neg (hc : c < 0) : a / c ≤ b / c ↔ b ≤ a 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b c : α} (hc : c < 0)
Goal: a / c ≤ b / c ↔ b ≤ a
-- div_lt_div_right_of_neg
theorem div_lt_div_right_of_neg (hc : c < 0) : a / c < b / c ↔ b < a 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b c : α} (hc : c < 0)
Goal: a / c < b / c ↔ b < a
-- one_le_div_of_neg
theorem one_le_div_of_neg (hb : b < 0) : 1 ≤ a / b ↔ a ≤ b 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α} (hb : b < 0)
Goal: 1 ≤ a / b ↔ a ≤ b
-- div_le_one_of_neg
theorem div_le_one_of_neg (hb : b < 0) : a / b ≤ 1 ↔ b ≤ a 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α} (hb : b < 0)
Goal: a / b ≤ 1 ↔ b ≤ a
-- one_lt_div_of_neg
theorem one_lt_div_of_neg (hb : b < 0) : 1 < a / b ↔ a < b 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α} (hb : b < 0)
Goal: 1 < a / b ↔ a < b
-- div_lt_one_of_neg
theorem div_lt_one_of_neg (hb : b < 0) : a / b < 1 ↔ b < a 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α} (hb : b < 0)
Goal: a / b < 1 ↔ b < a
-- one_div_le_of_neg
theorem one_div_le_of_neg (ha : a < 0) (hb : b < 0) : 1 / a ≤ b ↔ 1 / b ≤ a 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α} (ha : a < 0) (hb : b < 0)
Goal: 1 / a ≤ b ↔ 1 / b ≤ a
-- one_div_lt_of_neg
theorem one_div_lt_of_neg (ha : a < 0) (hb : b < 0) : 1 / a < b ↔ 1 / b < a 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α} (ha : a < 0) (hb : b < 0)
Goal: 1 / a < b ↔ 1 / b < a
-- le_one_div_of_neg
theorem le_one_div_of_neg (ha : a < 0) (hb : b < 0) : a ≤ 1 / b ↔ b ≤ 1 / a 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α} (ha : a < 0) (hb : b < 0)
Goal: a ≤ 1 / b ↔ b ≤ 1 / a
-- lt_one_div_of_neg
theorem lt_one_div_of_neg (ha : a < 0) (hb : b < 0) : a < 1 / b ↔ b < 1 / a 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α} (ha : a < 0) (hb : b < 0)
Goal: a < 1 / b ↔ b < 1 / a
-- one_lt_div_iff
theorem one_lt_div_iff : 1 < a / b ↔ 0 < b ∧ b < a ∨ b < 0 ∧ a < b 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α}
Goal: 1 < a / b ↔ 0 < b ∧ b < a ∨ b < 0 ∧ a < b
-- one_le_div_iff
theorem one_le_div_iff : 1 ≤ a / b ↔ 0 < b ∧ b ≤ a ∨ b < 0 ∧ a ≤ b 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α}
Goal: 1 ≤ a / b ↔ 0 < b ∧ b ≤ a ∨ b < 0 ∧ a ≤ b
-- div_lt_one_iff
theorem div_lt_one_iff : a / b < 1 ↔ 0 < b ∧ a < b ∨ b = 0 ∨ b < 0 ∧ b < a 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α}
Goal: a / b < 1 ↔ 0 < b ∧ a < b ∨ b = 0 ∨ b < 0 ∧ b < a
-- div_le_one_iff
theorem div_le_one_iff : a / b ≤ 1 ↔ 0 < b ∧ a ≤ b ∨ b = 0 ∨ b < 0 ∧ b ≤ a 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α}
Goal: a / b ≤ 1 ↔ 0 < b ∧ a ≤ b ∨ b = 0 ∨ b < 0 ∧ b ≤ a
-- one_div_le_one_div_of_neg_of_le
theorem one_div_le_one_div_of_neg_of_le (hb : b < 0) (h : a ≤ b) : 1 / b ≤ 1 / a 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α} (hb : b < 0) (h : a ≤ b)
Goal: 1 / b ≤ 1 / a
-- one_div_lt_one_div_of_neg_of_lt
theorem one_div_lt_one_div_of_neg_of_lt (hb : b < 0) (h : a < b) : 1 / b < 1 / a 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α} (hb : b < 0) (h : a < b)
Goal: 1 / b < 1 / a
-- le_of_neg_of_one_div_le_one_div
theorem le_of_neg_of_one_div_le_one_div (hb : b < 0) (h : 1 / a ≤ 1 / b) : b ≤ a 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α} (hb : b < 0) (h : 1 / a ≤ 1 / b)
Goal: b ≤ a
-- lt_of_neg_of_one_div_lt_one_div
theorem lt_of_neg_of_one_div_lt_one_div (hb : b < 0) (h : 1 / a < 1 / b) : b < a 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α} (hb : b < 0) (h : 1 / a < 1 / b)
Goal: b < a
-- one_div_le_one_div_of_neg
theorem one_div_le_one_div_of_neg (ha : a < 0) (hb : b < 0) : 1 / a ≤ 1 / b ↔ b ≤ a 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α} (ha : a < 0) (hb : b < 0)
Goal: 1 / a ≤ 1 / b ↔ b ≤ a
-- one_div_lt_one_div_of_neg
theorem one_div_lt_one_div_of_neg (ha : a < 0) (hb : b < 0) : 1 / a < 1 / b ↔ b < a 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α} (ha : a < 0) (hb : b < 0)
Goal: 1 / a < 1 / b ↔ b < a
-- one_div_lt_neg_one
theorem one_div_lt_neg_one (h1 : a < 0) (h2 : -1 < a) : 1 / a < -1 
Arguments: {α : Type u_2} [LinearOrderedField α] {a : α} (h1 : a < 0) (h2 : -1 < a)
Goal: 1 / a < -1
-- one_div_le_neg_one
theorem one_div_le_neg_one (h1 : a < 0) (h2 : -1 ≤ a) : 1 / a ≤ -1 
Arguments: {α : Type u_2} [LinearOrderedField α] {a : α} (h1 : a < 0) (h2 : -1 ≤ a)
Goal: 1 / a ≤ -1
-- sub_self_div_two
theorem sub_self_div_two (a : α) : a - a / 2 = a / 2 
Arguments: {α : Type u_2} [LinearOrderedField α] (a : α)
Goal: a - a / 2 = a / 2
-- div_two_sub_self
theorem div_two_sub_self (a : α) : a / 2 - a = -(a / 2) 
Arguments: {α : Type u_2} [LinearOrderedField α] (a : α)
Goal: a / 2 - a = -(a / 2)
-- add_sub_div_two_lt
theorem add_sub_div_two_lt (h : a < b) : a + (b - a) / 2 < b 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α} (h : a < b)
Goal: a + (b - a) / 2 < b
-- sub_one_div_inv_le_two
theorem sub_one_div_inv_le_two (a2 : 2 ≤ a) : (1 - 1 / a)⁻¹ ≤ 2 
Arguments: {α : Type u_2} [LinearOrderedField α] {a : α} (a2 : 2 ≤ a)
Goal: (1 - 1 / a)⁻¹ ≤ 2
-- IsLUB.mul_left
theorem IsLUB.mul_left {s : Set α} (ha : 0 ≤ a) (hs : IsLUB s b) :
    IsLUB ((fun b => a * b) '' s) (a * b) 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α} {s : Set α} (ha : 0 ≤ a) (hs : IsLUB s b)
Goal: IsLUB ((fun b => a * b) '' s) (a * b)
-- IsLUB.mul_right
theorem IsLUB.mul_right {s : Set α} (ha : 0 ≤ a) (hs : IsLUB s b) :
    IsLUB ((fun b => b * a) '' s) (b * a) 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α} {s : Set α} (ha : 0 ≤ a) (hs : IsLUB s b)
Goal: IsLUB ((fun b => b * a) '' s) (b * a)
-- mul_sub_mul_div_mul_neg_iff
theorem mul_sub_mul_div_mul_neg_iff (hc : c ≠ 0) (hd : d ≠ 0) :
    (a * d - b * c) / (c * d) < 0 ↔ a / c < b / d 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b c d : α} (hc : c ≠ 0) (hd : d ≠ 0)
Goal: (a * d - b * c) / (c * d) < 0 ↔ a / c < b / d
-- mul_sub_mul_div_mul_nonpos_iff
theorem mul_sub_mul_div_mul_nonpos_iff (hc : c ≠ 0) (hd : d ≠ 0) :
    (a * d - b * c) / (c * d) ≤ 0 ↔ a / c ≤ b / d 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b c d : α} (hc : c ≠ 0) (hd : d ≠ 0)
Goal: (a * d - b * c) / (c * d) ≤ 0 ↔ a / c ≤ b / d
-- exists_add_lt_and_pos_of_lt
theorem exists_add_lt_and_pos_of_lt (h : b < a) : ∃ c, b + c < a ∧ 0 < c 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α} (h : b < a)
Goal: ∃ c, b + c < a ∧ 0 < c
-- le_of_forall_sub_le
theorem le_of_forall_sub_le (h : ∀ ε > 0, b - ε ≤ a) : b ≤ a 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α} (h : ∀ ε > 0, b - ε ≤ a)
Goal: b ≤ a
-- mul_self_inj_of_nonneg
theorem mul_self_inj_of_nonneg (a0 : 0 ≤ a) (b0 : 0 ≤ b) : a * a = b * b ↔ a = b 
Arguments: {α : Type u_2} [LinearOrderedField α] {a b : α} (a0 : 0 ≤ a) (b0 : 0 ≤ b)
Goal: a * a = b * b ↔ a = b
-- min_div_div_right_of_nonpos
theorem min_div_div_right_of_nonpos (hc : c ≤ 0) (a b : α) : min (a / c) (b / c) = max a b / c 
Arguments: {α : Type u_2} [LinearOrderedField α] {c : α} (hc : c ≤ 0) (a b : α)
Goal: min (a / c) (b / c) = max a b / c
-- max_div_div_right_of_nonpos
theorem max_div_div_right_of_nonpos (hc : c ≤ 0) (a b : α) : max (a / c) (b / c) = min a b / c 
Arguments: {α : Type u_2} [LinearOrderedField α] {c : α} (hc : c ≤ 0) (a b : α)
Goal: max (a / c) (b / c) = min a b / c
-- abs_inv
theorem abs_inv (a : α) : |a⁻¹| = |a|⁻¹ 
Arguments: {α : Type u_2} [LinearOrderedField α] (a : α)
Goal: |a⁻¹| = |a|⁻¹
-- abs_div
theorem abs_div (a b : α) : |a / b| = |a| / |b| 
Arguments: {α : Type u_2} [LinearOrderedField α] (a b : α)
Goal: |a / b| = |a| / |b|
-- abs_one_div
theorem abs_one_div (a : α) : |1 / a| = 1 / |a| 
Arguments: {α : Type u_2} [LinearOrderedField α] (a : α)
Goal: |1 / a| = 1 / |a|
-- _private.Mathlib.Algebra.Order.Field.Basic.0.Mathlib.Meta.Positivity.div_nonneg_of_pos_of_nonneg
private lemma div_nonneg_of_pos_of_nonneg (ha : 0 < a) (hb : 0 ≤ b) : 0 ≤ a / b 
Arguments: {α : Type u_4} [LinearOrderedSemifield α] {a b : α} (ha : 0 < a) (hb : 0 ≤ b)
Goal: 0 ≤ a / b
-- _private.Mathlib.Algebra.Order.Field.Basic.0.Mathlib.Meta.Positivity.div_nonneg_of_nonneg_of_pos
private lemma div_nonneg_of_nonneg_of_pos (ha : 0 ≤ a) (hb : 0 < b) : 0 ≤ a / b 
Arguments: {α : Type u_4} [LinearOrderedSemifield α] {a b : α} (ha : 0 ≤ a) (hb : 0 < b)
Goal: 0 ≤ a / b
-- _private.Mathlib.Algebra.Order.Field.Basic.0.Mathlib.Meta.Positivity.div_ne_zero_of_pos_of_ne_zero
private lemma div_ne_zero_of_pos_of_ne_zero (ha : 0 < a) (hb : b ≠ 0) : a / b ≠ 0 
Arguments: {α : Type u_4} [LinearOrderedSemifield α] {a b : α} (ha : 0 < a) (hb : b ≠ 0)
Goal: a / b ≠ 0
-- _private.Mathlib.Algebra.Order.Field.Basic.0.Mathlib.Meta.Positivity.div_ne_zero_of_ne_zero_of_pos
private lemma div_ne_zero_of_ne_zero_of_pos (ha : a ≠ 0) (hb : 0 < b) : a / b ≠ 0 
Arguments: {α : Type u_4} [LinearOrderedSemifield α] {a b : α} (ha : a ≠ 0) (hb : 0 < b)
Goal: a / b ≠ 0
-- _private.Mathlib.Algebra.Order.Field.Basic.0.Mathlib.Meta.Positivity.zpow_zero_pos
private lemma zpow_zero_pos (a : α) : 0 < a ^ (0 : ℤ) 
Arguments: {α : Type u_4} [LinearOrderedSemifield α] (a : α)
Goal: 0 < a ^ 0
-- Set.Ici_add_bij
theorem Ici_add_bij : BijOn (· + d) (Ici a) (Ici (a + d)) 
Arguments: {M : Type u_1} [OrderedCancelAddCommMonoid M] [ExistsAddOfLE M] (a d : M)
Goal: Set.BijOn (fun x => x + d) (Set.Ici a) (Set.Ici (a + d))
-- Set.Ioi_add_bij
theorem Ioi_add_bij : BijOn (· + d) (Ioi a) (Ioi (a + d)) 
Arguments: {M : Type u_1} [OrderedCancelAddCommMonoid M] [ExistsAddOfLE M] (a d : M)
Goal: Set.BijOn (fun x => x + d) (Set.Ioi a) (Set.Ioi (a + d))
-- Set.Icc_add_bij
theorem Icc_add_bij : BijOn (· + d) (Icc a b) (Icc (a + d) (b + d)) 
Arguments: {M : Type u_1} [OrderedCancelAddCommMonoid M] [ExistsAddOfLE M] (a b d : M)
Goal: Set.BijOn (fun x => x + d) (Set.Icc a b) (Set.Icc (a + d) (b + d))
-- Set.Ioo_add_bij
theorem Ioo_add_bij : BijOn (· + d) (Ioo a b) (Ioo (a + d) (b + d)) 
Arguments: {M : Type u_1} [OrderedCancelAddCommMonoid M] [ExistsAddOfLE M] (a b d : M)
Goal: Set.BijOn (fun x => x + d) (Set.Ioo a b) (Set.Ioo (a + d) (b + d))
-- Set.Ioc_add_bij
theorem Ioc_add_bij : BijOn (· + d) (Ioc a b) (Ioc (a + d) (b + d)) 
Arguments: {M : Type u_1} [OrderedCancelAddCommMonoid M] [ExistsAddOfLE M] (a b d : M)
Goal: Set.BijOn (fun x => x + d) (Set.Ioc a b) (Set.Ioc (a + d) (b + d))
-- Set.Ico_add_bij
theorem Ico_add_bij : BijOn (· + d) (Ico a b) (Ico (a + d) (b + d)) 
Arguments: {M : Type u_1} [OrderedCancelAddCommMonoid M] [ExistsAddOfLE M] (a b d : M)
Goal: Set.BijOn (fun x => x + d) (Set.Ico a b) (Set.Ico (a + d) (b + d))
-- Set.image_add_const_Ici
@[simp]
theorem image_add_const_Ici : (fun x => x + a) '' Ici b = Ici (b + a) 
Arguments: {M : Type u_1} [OrderedCancelAddCommMonoid M] [ExistsAddOfLE M] (a b : M)
Goal: (fun x => x + a) '' Set.Ici b = Set.Ici (b + a)
-- Set.image_add_const_Ioi
@[simp]
theorem image_add_const_Ioi : (fun x => x + a) '' Ioi b = Ioi (b + a) 
Arguments: {M : Type u_1} [OrderedCancelAddCommMonoid M] [ExistsAddOfLE M] (a b : M)
Goal: (fun x => x + a) '' Set.Ioi b = Set.Ioi (b + a)
-- Set.image_add_const_Icc
@[simp]
theorem image_add_const_Icc : (fun x => x + a) '' Icc b c = Icc (b + a) (c + a) 
Arguments: {M : Type u_1} [OrderedCancelAddCommMonoid M] [ExistsAddOfLE M] (a b c : M)
Goal: (fun x => x + a) '' Set.Icc b c = Set.Icc (b + a) (c + a)
-- Set.image_add_const_Ico
@[simp]
theorem image_add_const_Ico : (fun x => x + a) '' Ico b c = Ico (b + a) (c + a) 
Arguments: {M : Type u_1} [OrderedCancelAddCommMonoid M] [ExistsAddOfLE M] (a b c : M)
Goal: (fun x => x + a) '' Set.Ico b c = Set.Ico (b + a) (c + a)
-- Set.image_add_const_Ioc
@[simp]
theorem image_add_const_Ioc : (fun x => x + a) '' Ioc b c = Ioc (b + a) (c + a) 
Arguments: {M : Type u_1} [OrderedCancelAddCommMonoid M] [ExistsAddOfLE M] (a b c : M)
Goal: (fun x => x + a) '' Set.Ioc b c = Set.Ioc (b + a) (c + a)
-- Set.image_add_const_Ioo
@[simp]
theorem image_add_const_Ioo : (fun x => x + a) '' Ioo b c = Ioo (b + a) (c + a) 
Arguments: {M : Type u_1} [OrderedCancelAddCommMonoid M] [ExistsAddOfLE M] (a b c : M)
Goal: (fun x => x + a) '' Set.Ioo b c = Set.Ioo (b + a) (c + a)
-- Set.image_const_add_Ici
@[simp]
theorem image_const_add_Ici : (fun x => a + x) '' Ici b = Ici (a + b) 
Arguments: {M : Type u_1} [OrderedCancelAddCommMonoid M] [ExistsAddOfLE M] (a b : M)
Goal: (fun x => a + x) '' Set.Ici b = Set.Ici (a + b)
-- Set.image_const_add_Ioi
@[simp]
theorem image_const_add_Ioi : (fun x => a + x) '' Ioi b = Ioi (a + b) 
Arguments: {M : Type u_1} [OrderedCancelAddCommMonoid M] [ExistsAddOfLE M] (a b : M)
Goal: (fun x => a + x) '' Set.Ioi b = Set.Ioi (a + b)
-- Set.image_const_add_Icc
@[simp]
theorem image_const_add_Icc : (fun x => a + x) '' Icc b c = Icc (a + b) (a + c) 
Arguments: {M : Type u_1} [OrderedCancelAddCommMonoid M] [ExistsAddOfLE M] (a b c : M)
Goal: (fun x => a + x) '' Set.Icc b c = Set.Icc (a + b) (a + c)
-- Set.image_const_add_Ico
@[simp]
theorem image_const_add_Ico : (fun x => a + x) '' Ico b c = Ico (a + b) (a + c) 
Arguments: {M : Type u_1} [OrderedCancelAddCommMonoid M] [ExistsAddOfLE M] (a b c : M)
Goal: (fun x => a + x) '' Set.Ico b c = Set.Ico (a + b) (a + c)
-- Set.image_const_add_Ioc
@[simp]
theorem image_const_add_Ioc : (fun x => a + x) '' Ioc b c = Ioc (a + b) (a + c) 
Arguments: {M : Type u_1} [OrderedCancelAddCommMonoid M] [ExistsAddOfLE M] (a b c : M)
Goal: (fun x => a + x) '' Set.Ioc b c = Set.Ioc (a + b) (a + c)
-- Set.image_const_add_Ioo
@[simp]
theorem image_const_add_Ioo : (fun x => a + x) '' Ioo b c = Ioo (a + b) (a + c) 
Arguments: {M : Type u_1} [OrderedCancelAddCommMonoid M] [ExistsAddOfLE M] (a b c : M)
Goal: (fun x => a + x) '' Set.Ioo b c = Set.Ioo (a + b) (a + c)
-- Multiset.map_add_left_Icc
lemma map_add_left_Icc (a b c : α) : (Icc a b).map (c + ·) = Icc (c + a) (c + b) 
Arguments: {α : Type u_1} [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] (a b c : α)
Goal: Multiset.map (fun x => c + x) (Multiset.Icc a b) = Multiset.Icc (c + a) (c + b)
-- Multiset.map_add_left_Ico
lemma map_add_left_Ico (a b c : α) : (Ico a b).map (c + ·) = Ico (c + a) (c + b) 
Arguments: {α : Type u_1} [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] (a b c : α)
Goal: Multiset.map (fun x => c + x) (Multiset.Ico a b) = Multiset.Ico (c + a) (c + b)
-- Multiset.map_add_left_Ioc
lemma map_add_left_Ioc (a b c : α) : (Ioc a b).map (c + ·) = Ioc (c + a) (c + b) 
Arguments: {α : Type u_1} [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] (a b c : α)
Goal: Multiset.map (fun x => c + x) (Multiset.Ioc a b) = Multiset.Ioc (c + a) (c + b)
-- Multiset.map_add_left_Ioo
lemma map_add_left_Ioo (a b c : α) : (Ioo a b).map (c + ·) = Ioo (c + a) (c + b) 
Arguments: {α : Type u_1} [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] (a b c : α)
Goal: Multiset.map (fun x => c + x) (Multiset.Ioo a b) = Multiset.Ioo (c + a) (c + b)
-- Multiset.map_add_right_Icc
lemma map_add_right_Icc (a b c : α) : ((Icc a b).map fun x => x + c) = Icc (a + c) (b + c) 
Arguments: {α : Type u_1} [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] (a b c : α)
Goal: Multiset.map (fun x => x + c) (Multiset.Icc a b) = Multiset.Icc (a + c) (b + c)
-- Multiset.map_add_right_Ico
lemma map_add_right_Ico (a b c : α) : ((Ico a b).map fun x => x + c) = Ico (a + c) (b + c) 
Arguments: {α : Type u_1} [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] (a b c : α)
Goal: Multiset.map (fun x => x + c) (Multiset.Ico a b) = Multiset.Ico (a + c) (b + c)
-- Multiset.map_add_right_Ioc
lemma map_add_right_Ioc (a b c : α) : ((Ioc a b).map fun x => x + c) = Ioc (a + c) (b + c) 
Arguments: {α : Type u_1} [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] (a b c : α)
Goal: Multiset.map (fun x => x + c) (Multiset.Ioc a b) = Multiset.Ioc (a + c) (b + c)
-- Multiset.map_add_right_Ioo
lemma map_add_right_Ioo (a b c : α) : ((Ioo a b).map fun x => x + c) = Ioo (a + c) (b + c) 
Arguments: {α : Type u_1} [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] (a b c : α)
Goal: Multiset.map (fun x => x + c) (Multiset.Ioo a b) = Multiset.Ioo (a + c) (b + c)
-- MonoidHom.map_neg_one
theorem map_neg_one : f (-1) = 1 
Arguments: {M : Type u_2} {R : Type u_3} [Ring R] [Monoid M] [LinearOrder M] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (f : R →* M)
Goal: f (-1) = 1
-- MonoidHom.map_neg
@[simp]
theorem map_neg (x : R) : f (-x) = f x 
Arguments: {M : Type u_2} {R : Type u_3} [Ring R] [Monoid M] [LinearOrder M] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (f : R →* M) (x : R)
Goal: f (-x) = f x
-- MonoidHom.map_sub_swap
theorem map_sub_swap (x y : R) : f (x - y) = f (y - x) 
Arguments: {M : Type u_2} {R : Type u_3} [Ring R] [Monoid M] [LinearOrder M] [CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (f : R →* M) (x y : R)
Goal: f (x - y) = f (y - x)
-- zero_pow_le_one
theorem zero_pow_le_one : ∀ n : ℕ, (0 : R) ^ n ≤ 1
  
Arguments: {R : Type u_3} [OrderedSemiring R] (n : ℕ)
Goal: 0 ^ n ≤ 1
-- pow_add_pow_le
theorem pow_add_pow_le (hx : 0 ≤ x) (hy : 0 ≤ y) (hn : n ≠ 0) : x ^ n + y ^ n ≤ (x + y) ^ n 
Arguments: {R : Type u_3} [OrderedSemiring R] {x y : R} {n : ℕ} (hx : 0 ≤ x) (hy : 0 ≤ y) (hn : n ≠ 0)
Goal: x ^ n + y ^ n ≤ (x + y) ^ n
-- pow_le_one
theorem pow_le_one : ∀ n : ℕ, 0 ≤ a → a ≤ 1 → a ^ n ≤ 1
  
Arguments: {R : Type u_3} [OrderedSemiring R] {a : R} (n : ℕ)
Goal: 0 ≤ a → a ≤ 1 → a ^ n ≤ 1
-- pow_lt_one
theorem pow_lt_one (h₀ : 0 ≤ a) (h₁ : a < 1) : ∀ {n : ℕ}, n ≠ 0 → a ^ n < 1
  
Arguments: {R : Type u_3} [OrderedSemiring R] {a : R} (h₀ : 0 ≤ a) (h₁ : a < 1) {n : ℕ}
Goal: n ≠ 0 → a ^ n < 1
-- one_le_pow_of_one_le
theorem one_le_pow_of_one_le (H : 1 ≤ a) : ∀ n : ℕ, 1 ≤ a ^ n
  
Arguments: {R : Type u_3} [OrderedSemiring R] {a : R} (H : 1 ≤ a) (n : ℕ)
Goal: 1 ≤ a ^ n
-- pow_right_mono
theorem pow_right_mono (h : 1 ≤ a) : Monotone (a ^ ·) 
Arguments: {R : Type u_3} [OrderedSemiring R] {a : R} (h : 1 ≤ a)
Goal: Monotone fun x => a ^ x
-- pow_le_pow_right
@[gcongr]
theorem pow_le_pow_right (ha : 1 ≤ a) (h : n ≤ m) : a ^ n ≤ a ^ m 
Arguments: {R : Type u_3} [OrderedSemiring R] {a : R} {n m : ℕ} (ha : 1 ≤ a) (h : n ≤ m)
Goal: a ^ n ≤ a ^ m
-- le_self_pow
theorem le_self_pow (ha : 1 ≤ a) (h : m ≠ 0) : a ≤ a ^ m 
Arguments: {R : Type u_3} [OrderedSemiring R] {a : R} {m : ℕ} (ha : 1 ≤ a) (h : m ≠ 0)
Goal: a ≤ a ^ m
-- pow_le_pow_left
@[mono, gcongr]
theorem pow_le_pow_left {a b : R} (ha : 0 ≤ a) (hab : a ≤ b) : ∀ n, a ^ n ≤ b ^ n
  
Arguments: {R : Type u_3} [OrderedSemiring R] {a b : R} (ha : 0 ≤ a) (hab : a ≤ b) (n : ℕ)
Goal: a ^ n ≤ b ^ n
-- one_lt_pow
theorem one_lt_pow (ha : 1 < a) : ∀ {n : ℕ} (_ : n ≠ 0), 1 < a ^ n
  
Arguments: {R : Type u_3} [OrderedSemiring R] {a : R} (ha : 1 < a) {n : ℕ}
Goal: n ≠ 0 → 1 < a ^ n
-- pow_add_pow_le'
lemma pow_add_pow_le' (ha : 0 ≤ a) (hb : 0 ≤ b) : a ^ n + b ^ n ≤ 2 * (a + b) ^ n 
Arguments: {R : Type u_3} [OrderedSemiring R] {a b : R} {n : ℕ} (ha : 0 ≤ a) (hb : 0 ≤ b)
Goal: a ^ n + b ^ n ≤ 2 * (a + b) ^ n
-- pow_lt_pow_left
@[gcongr]
theorem pow_lt_pow_left (h : x < y) (hx : 0 ≤ x) : ∀ {n : ℕ}, n ≠ 0 → x ^ n < y ^ n
  
Arguments: {R : Type u_3} [StrictOrderedSemiring R] {x y : R} (h : x < y) (hx : 0 ≤ x) {n : ℕ}
Goal: n ≠ 0 → x ^ n < y ^ n
-- pow_left_strictMonoOn
lemma pow_left_strictMonoOn (hn : n ≠ 0) : StrictMonoOn (· ^ n : R → R) {a | 0 ≤ a} 
Arguments: {R : Type u_3} [StrictOrderedSemiring R] {n : ℕ} (hn : n ≠ 0)
Goal: StrictMonoOn (fun x => x ^ n) {a | 0 ≤ a}
-- pow_right_strictMono
lemma pow_right_strictMono (h : 1 < a) : StrictMono (a ^ ·) 
Arguments: {R : Type u_3} [StrictOrderedSemiring R] {a : R} (h : 1 < a)
Goal: StrictMono fun x => a ^ x
-- pow_lt_pow_right
@[gcongr]
theorem pow_lt_pow_right (h : 1 < a) (hmn : m < n) : a ^ m < a ^ n 
Arguments: {R : Type u_3} [StrictOrderedSemiring R] {a : R} {n m : ℕ} (h : 1 < a) (hmn : m < n)
Goal: a ^ m < a ^ n
-- pow_lt_pow_iff_right
lemma pow_lt_pow_iff_right (h : 1 < a) : a ^ n < a ^ m ↔ n < m 
Arguments: {R : Type u_3} [StrictOrderedSemiring R] {a : R} {n m : ℕ} (h : 1 < a)
Goal: a ^ n < a ^ m ↔ n < m
-- pow_le_pow_iff_right
lemma pow_le_pow_iff_right (h : 1 < a) : a ^ n ≤ a ^ m ↔ n ≤ m 
Arguments: {R : Type u_3} [StrictOrderedSemiring R] {a : R} {n m : ℕ} (h : 1 < a)
Goal: a ^ n ≤ a ^ m ↔ n ≤ m
-- lt_self_pow
theorem lt_self_pow (h : 1 < a) (hm : 1 < m) : a < a ^ m 
Arguments: {R : Type u_3} [StrictOrderedSemiring R] {a : R} {m : ℕ} (h : 1 < a) (hm : 1 < m)
Goal: a < a ^ m
-- pow_right_strictAnti
theorem pow_right_strictAnti (h₀ : 0 < a) (h₁ : a < 1) : StrictAnti (a ^ ·) 
Arguments: {R : Type u_3} [StrictOrderedSemiring R] {a : R} (h₀ : 0 < a) (h₁ : a < 1)
Goal: StrictAnti fun x => a ^ x
-- pow_lt_pow_iff_right_of_lt_one
theorem pow_lt_pow_iff_right_of_lt_one (h₀ : 0 < a) (h₁ : a < 1) : a ^ m < a ^ n ↔ n < m 
Arguments: {R : Type u_3} [StrictOrderedSemiring R] {a : R} {n m : ℕ} (h₀ : 0 < a) (h₁ : a < 1)
Goal: a ^ m < a ^ n ↔ n < m