ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Set.Icc_mul_Ico_subset'
@[to_additive Icc_add_Ico_subset]
theorem Icc_mul_Ico_subset' (a b c d : α) : Icc a b * Ico c d ⊆ Ico (a * c) (b * d) 
Arguments: {α : Type u_1} [Mul α] [PartialOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap HMul.hMul) LT.lt] (a b c d : α)
Goal: Set.Icc a b * Set.Ico c d ⊆ Set.Ico (a * c) (b * d)
-- Set.Icc_add_Ico_subset
@[to_additive Icc_add_Ico_subset]
theorem Icc_mul_Ico_subset' (a b c d : α) : Icc a b * Ico c d ⊆ Ico (a * c) (b * d) 
Arguments: {α : Type u_1} [Add α] [PartialOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap HAdd.hAdd) LT.lt] (a b c d : α)
Goal: Set.Icc a b + Set.Ico c d ⊆ Set.Ico (a + c) (b + d)
-- Set.Ico_add_Icc_subset
@[to_additive Ico_add_Icc_subset]
theorem Ico_mul_Icc_subset' (a b c d : α) : Ico a b * Icc c d ⊆ Ico (a * c) (b * d) 
Arguments: {α : Type u_1} [Add α] [PartialOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap HAdd.hAdd) LT.lt] (a b c d : α)
Goal: Set.Ico a b + Set.Icc c d ⊆ Set.Ico (a + c) (b + d)
-- Set.Ico_mul_Icc_subset'
@[to_additive Ico_add_Icc_subset]
theorem Ico_mul_Icc_subset' (a b c d : α) : Ico a b * Icc c d ⊆ Ico (a * c) (b * d) 
Arguments: {α : Type u_1} [Mul α] [PartialOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap HMul.hMul) LT.lt] (a b c d : α)
Goal: Set.Ico a b * Set.Icc c d ⊆ Set.Ico (a * c) (b * d)
-- Set.Ioc_add_Ico_subset
@[to_additive Ioc_add_Ico_subset]
theorem Ioc_mul_Ico_subset' (a b c d : α) : Ioc a b * Ico c d ⊆ Ioo (a * c) (b * d) 
Arguments: {α : Type u_1} [Add α] [PartialOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap HAdd.hAdd) LT.lt] (a b c d : α)
Goal: Set.Ioc a b + Set.Ico c d ⊆ Set.Ioo (a + c) (b + d)
-- Set.Ioc_mul_Ico_subset'
@[to_additive Ioc_add_Ico_subset]
theorem Ioc_mul_Ico_subset' (a b c d : α) : Ioc a b * Ico c d ⊆ Ioo (a * c) (b * d) 
Arguments: {α : Type u_1} [Mul α] [PartialOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap HMul.hMul) LT.lt] (a b c d : α)
Goal: Set.Ioc a b * Set.Ico c d ⊆ Set.Ioo (a * c) (b * d)
-- Set.Ico_mul_Ioc_subset'
@[to_additive Ico_add_Ioc_subset]
theorem Ico_mul_Ioc_subset' (a b c d : α) : Ico a b * Ioc c d ⊆ Ioo (a * c) (b * d) 
Arguments: {α : Type u_1} [Mul α] [PartialOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap HMul.hMul) LT.lt] (a b c d : α)
Goal: Set.Ico a b * Set.Ioc c d ⊆ Set.Ioo (a * c) (b * d)
-- Set.Ico_add_Ioc_subset
@[to_additive Ico_add_Ioc_subset]
theorem Ico_mul_Ioc_subset' (a b c d : α) : Ico a b * Ioc c d ⊆ Ioo (a * c) (b * d) 
Arguments: {α : Type u_1} [Add α] [PartialOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap HAdd.hAdd) LT.lt] (a b c d : α)
Goal: Set.Ico a b + Set.Ioc c d ⊆ Set.Ioo (a + c) (b + d)
-- Set.Iic_mul_Iio_subset'
@[to_additive Iic_add_Iio_subset]
theorem Iic_mul_Iio_subset' (a b : α) : Iic a * Iio b ⊆ Iio (a * b) 
Arguments: {α : Type u_1} [Mul α] [PartialOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap HMul.hMul) LT.lt] (a b : α)
Goal: Set.Iic a * Set.Iio b ⊆ Set.Iio (a * b)
-- Set.Iic_add_Iio_subset
@[to_additive Iic_add_Iio_subset]
theorem Iic_mul_Iio_subset' (a b : α) : Iic a * Iio b ⊆ Iio (a * b) 
Arguments: {α : Type u_1} [Add α] [PartialOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap HAdd.hAdd) LT.lt] (a b : α)
Goal: Set.Iic a + Set.Iio b ⊆ Set.Iio (a + b)
-- Set.Iio_add_Iic_subset
@[to_additive Iio_add_Iic_subset]
theorem Iio_mul_Iic_subset' (a b : α) : Iio a * Iic b ⊆ Iio (a * b) 
Arguments: {α : Type u_1} [Add α] [PartialOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap HAdd.hAdd) LT.lt] (a b : α)
Goal: Set.Iio a + Set.Iic b ⊆ Set.Iio (a + b)
-- Set.Iio_mul_Iic_subset'
@[to_additive Iio_add_Iic_subset]
theorem Iio_mul_Iic_subset' (a b : α) : Iio a * Iic b ⊆ Iio (a * b) 
Arguments: {α : Type u_1} [Mul α] [PartialOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap HMul.hMul) LT.lt] (a b : α)
Goal: Set.Iio a * Set.Iic b ⊆ Set.Iio (a * b)
-- Set.Ioi_add_Ici_subset
@[to_additive Ioi_add_Ici_subset]
theorem Ioi_mul_Ici_subset' (a b : α) : Ioi a * Ici b ⊆ Ioi (a * b) 
Arguments: {α : Type u_1} [Add α] [PartialOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap HAdd.hAdd) LT.lt] (a b : α)
Goal: Set.Ioi a + Set.Ici b ⊆ Set.Ioi (a + b)
-- Set.Ioi_mul_Ici_subset'
@[to_additive Ioi_add_Ici_subset]
theorem Ioi_mul_Ici_subset' (a b : α) : Ioi a * Ici b ⊆ Ioi (a * b) 
Arguments: {α : Type u_1} [Mul α] [PartialOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap HMul.hMul) LT.lt] (a b : α)
Goal: Set.Ioi a * Set.Ici b ⊆ Set.Ioi (a * b)
-- Set.Ici_mul_Ioi_subset'
@[to_additive Ici_add_Ioi_subset]
theorem Ici_mul_Ioi_subset' (a b : α) : Ici a * Ioi b ⊆ Ioi (a * b) 
Arguments: {α : Type u_1} [Mul α] [PartialOrder α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap HMul.hMul) LT.lt] (a b : α)
Goal: Set.Ici a * Set.Ioi b ⊆ Set.Ioi (a * b)
-- Set.Ici_add_Ioi_subset
@[to_additive Ici_add_Ioi_subset]
theorem Ici_mul_Ioi_subset' (a b : α) : Ici a * Ioi b ⊆ Ioi (a * b) 
Arguments: {α : Type u_1} [Add α] [PartialOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] [CovariantClass α α (Function.swap HAdd.hAdd) LT.lt] (a b : α)
Goal: Set.Ici a + Set.Ioi b ⊆ Set.Ioi (a + b)
-- Set.preimage_const_add_Ici
@[simp]
theorem preimage_const_add_Ici : (fun x => a + x) ⁻¹' Ici b = Ici (b - a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: (fun x => a + x) ⁻¹' Set.Ici b = Set.Ici (b - a)
-- Set.preimage_const_add_Ioi
@[simp]
theorem preimage_const_add_Ioi : (fun x => a + x) ⁻¹' Ioi b = Ioi (b - a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: (fun x => a + x) ⁻¹' Set.Ioi b = Set.Ioi (b - a)
-- Set.preimage_const_add_Iic
@[simp]
theorem preimage_const_add_Iic : (fun x => a + x) ⁻¹' Iic b = Iic (b - a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: (fun x => a + x) ⁻¹' Set.Iic b = Set.Iic (b - a)
-- Set.preimage_const_add_Iio
@[simp]
theorem preimage_const_add_Iio : (fun x => a + x) ⁻¹' Iio b = Iio (b - a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: (fun x => a + x) ⁻¹' Set.Iio b = Set.Iio (b - a)
-- Set.preimage_const_add_Icc
@[simp]
theorem preimage_const_add_Icc : (fun x => a + x) ⁻¹' Icc b c = Icc (b - a) (c - a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b c : α)
Goal: (fun x => a + x) ⁻¹' Set.Icc b c = Set.Icc (b - a) (c - a)
-- Set.preimage_const_add_Ico
@[simp]
theorem preimage_const_add_Ico : (fun x => a + x) ⁻¹' Ico b c = Ico (b - a) (c - a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b c : α)
Goal: (fun x => a + x) ⁻¹' Set.Ico b c = Set.Ico (b - a) (c - a)
-- Set.preimage_const_add_Ioc
@[simp]
theorem preimage_const_add_Ioc : (fun x => a + x) ⁻¹' Ioc b c = Ioc (b - a) (c - a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b c : α)
Goal: (fun x => a + x) ⁻¹' Set.Ioc b c = Set.Ioc (b - a) (c - a)
-- Set.preimage_const_add_Ioo
@[simp]
theorem preimage_const_add_Ioo : (fun x => a + x) ⁻¹' Ioo b c = Ioo (b - a) (c - a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b c : α)
Goal: (fun x => a + x) ⁻¹' Set.Ioo b c = Set.Ioo (b - a) (c - a)
-- Set.preimage_add_const_Ici
@[simp]
theorem preimage_add_const_Ici : (fun x => x + a) ⁻¹' Ici b = Ici (b - a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: (fun x => x + a) ⁻¹' Set.Ici b = Set.Ici (b - a)
-- Set.preimage_add_const_Ioi
@[simp]
theorem preimage_add_const_Ioi : (fun x => x + a) ⁻¹' Ioi b = Ioi (b - a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: (fun x => x + a) ⁻¹' Set.Ioi b = Set.Ioi (b - a)
-- Set.preimage_add_const_Iic
@[simp]
theorem preimage_add_const_Iic : (fun x => x + a) ⁻¹' Iic b = Iic (b - a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: (fun x => x + a) ⁻¹' Set.Iic b = Set.Iic (b - a)
-- Set.preimage_add_const_Iio
@[simp]
theorem preimage_add_const_Iio : (fun x => x + a) ⁻¹' Iio b = Iio (b - a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: (fun x => x + a) ⁻¹' Set.Iio b = Set.Iio (b - a)
-- Set.preimage_add_const_Icc
@[simp]
theorem preimage_add_const_Icc : (fun x => x + a) ⁻¹' Icc b c = Icc (b - a) (c - a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b c : α)
Goal: (fun x => x + a) ⁻¹' Set.Icc b c = Set.Icc (b - a) (c - a)
-- Set.preimage_add_const_Ico
@[simp]
theorem preimage_add_const_Ico : (fun x => x + a) ⁻¹' Ico b c = Ico (b - a) (c - a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b c : α)
Goal: (fun x => x + a) ⁻¹' Set.Ico b c = Set.Ico (b - a) (c - a)
-- Set.preimage_add_const_Ioc
@[simp]
theorem preimage_add_const_Ioc : (fun x => x + a) ⁻¹' Ioc b c = Ioc (b - a) (c - a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b c : α)
Goal: (fun x => x + a) ⁻¹' Set.Ioc b c = Set.Ioc (b - a) (c - a)
-- Set.preimage_add_const_Ioo
@[simp]
theorem preimage_add_const_Ioo : (fun x => x + a) ⁻¹' Ioo b c = Ioo (b - a) (c - a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b c : α)
Goal: (fun x => x + a) ⁻¹' Set.Ioo b c = Set.Ioo (b - a) (c - a)
-- Set.preimage_neg_Ici
@[simp]
theorem preimage_neg_Ici : -Ici a = Iic (-a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a : α)
Goal: -Set.Ici a = Set.Iic (-a)
-- Set.preimage_neg_Iic
@[simp]
theorem preimage_neg_Iic : -Iic a = Ici (-a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a : α)
Goal: -Set.Iic a = Set.Ici (-a)
-- Set.preimage_neg_Ioi
@[simp]
theorem preimage_neg_Ioi : -Ioi a = Iio (-a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a : α)
Goal: -Set.Ioi a = Set.Iio (-a)
-- Set.preimage_neg_Iio
@[simp]
theorem preimage_neg_Iio : -Iio a = Ioi (-a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a : α)
Goal: -Set.Iio a = Set.Ioi (-a)
-- Set.preimage_neg_Icc
@[simp]
theorem preimage_neg_Icc : -Icc a b = Icc (-b) (-a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: -Set.Icc a b = Set.Icc (-b) (-a)
-- Set.preimage_neg_Ico
@[simp]
theorem preimage_neg_Ico : -Ico a b = Ioc (-b) (-a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: -Set.Ico a b = Set.Ioc (-b) (-a)
-- Set.preimage_neg_Ioc
@[simp]
theorem preimage_neg_Ioc : -Ioc a b = Ico (-b) (-a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: -Set.Ioc a b = Set.Ico (-b) (-a)
-- Set.preimage_neg_Ioo
@[simp]
theorem preimage_neg_Ioo : -Ioo a b = Ioo (-b) (-a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: -Set.Ioo a b = Set.Ioo (-b) (-a)
-- Set.preimage_sub_const_Ici
@[simp]
theorem preimage_sub_const_Ici : (fun x => x - a) ⁻¹' Ici b = Ici (b + a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: (fun x => x - a) ⁻¹' Set.Ici b = Set.Ici (b + a)
-- Set.preimage_sub_const_Ioi
@[simp]
theorem preimage_sub_const_Ioi : (fun x => x - a) ⁻¹' Ioi b = Ioi (b + a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: (fun x => x - a) ⁻¹' Set.Ioi b = Set.Ioi (b + a)
-- Set.preimage_sub_const_Iic
@[simp]
theorem preimage_sub_const_Iic : (fun x => x - a) ⁻¹' Iic b = Iic (b + a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: (fun x => x - a) ⁻¹' Set.Iic b = Set.Iic (b + a)
-- Set.preimage_sub_const_Iio
@[simp]
theorem preimage_sub_const_Iio : (fun x => x - a) ⁻¹' Iio b = Iio (b + a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: (fun x => x - a) ⁻¹' Set.Iio b = Set.Iio (b + a)
-- Set.preimage_sub_const_Icc
@[simp]
theorem preimage_sub_const_Icc : (fun x => x - a) ⁻¹' Icc b c = Icc (b + a) (c + a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b c : α)
Goal: (fun x => x - a) ⁻¹' Set.Icc b c = Set.Icc (b + a) (c + a)
-- Set.preimage_sub_const_Ico
@[simp]
theorem preimage_sub_const_Ico : (fun x => x - a) ⁻¹' Ico b c = Ico (b + a) (c + a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b c : α)
Goal: (fun x => x - a) ⁻¹' Set.Ico b c = Set.Ico (b + a) (c + a)
-- Set.preimage_sub_const_Ioc
@[simp]
theorem preimage_sub_const_Ioc : (fun x => x - a) ⁻¹' Ioc b c = Ioc (b + a) (c + a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b c : α)
Goal: (fun x => x - a) ⁻¹' Set.Ioc b c = Set.Ioc (b + a) (c + a)
-- Set.preimage_sub_const_Ioo
@[simp]
theorem preimage_sub_const_Ioo : (fun x => x - a) ⁻¹' Ioo b c = Ioo (b + a) (c + a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b c : α)
Goal: (fun x => x - a) ⁻¹' Set.Ioo b c = Set.Ioo (b + a) (c + a)
-- Set.preimage_const_sub_Ici
@[simp]
theorem preimage_const_sub_Ici : (fun x => a - x) ⁻¹' Ici b = Iic (a - b) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: (fun x => a - x) ⁻¹' Set.Ici b = Set.Iic (a - b)
-- Set.preimage_const_sub_Iic
@[simp]
theorem preimage_const_sub_Iic : (fun x => a - x) ⁻¹' Iic b = Ici (a - b) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: (fun x => a - x) ⁻¹' Set.Iic b = Set.Ici (a - b)
-- Set.preimage_const_sub_Ioi
@[simp]
theorem preimage_const_sub_Ioi : (fun x => a - x) ⁻¹' Ioi b = Iio (a - b) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: (fun x => a - x) ⁻¹' Set.Ioi b = Set.Iio (a - b)
-- Set.preimage_const_sub_Iio
@[simp]
theorem preimage_const_sub_Iio : (fun x => a - x) ⁻¹' Iio b = Ioi (a - b) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: (fun x => a - x) ⁻¹' Set.Iio b = Set.Ioi (a - b)
-- Set.preimage_const_sub_Icc
@[simp]
theorem preimage_const_sub_Icc : (fun x => a - x) ⁻¹' Icc b c = Icc (a - c) (a - b) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b c : α)
Goal: (fun x => a - x) ⁻¹' Set.Icc b c = Set.Icc (a - c) (a - b)
-- Set.preimage_const_sub_Ico
@[simp]
theorem preimage_const_sub_Ico : (fun x => a - x) ⁻¹' Ico b c = Ioc (a - c) (a - b) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b c : α)
Goal: (fun x => a - x) ⁻¹' Set.Ico b c = Set.Ioc (a - c) (a - b)
-- Set.preimage_const_sub_Ioc
@[simp]
theorem preimage_const_sub_Ioc : (fun x => a - x) ⁻¹' Ioc b c = Ico (a - c) (a - b) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b c : α)
Goal: (fun x => a - x) ⁻¹' Set.Ioc b c = Set.Ico (a - c) (a - b)
-- Set.preimage_const_sub_Ioo
@[simp]
theorem preimage_const_sub_Ioo : (fun x => a - x) ⁻¹' Ioo b c = Ioo (a - c) (a - b) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b c : α)
Goal: (fun x => a - x) ⁻¹' Set.Ioo b c = Set.Ioo (a - c) (a - b)
-- Set.image_const_add_Iic
theorem image_const_add_Iic : (fun x => a + x) '' Iic b = Iic (a + b) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: (fun x => a + x) '' Set.Iic b = Set.Iic (a + b)
-- Set.image_const_add_Iio
theorem image_const_add_Iio : (fun x => a + x) '' Iio b = Iio (a + b) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: (fun x => a + x) '' Set.Iio b = Set.Iio (a + b)
-- Set.image_add_const_Iic
theorem image_add_const_Iic : (fun x => x + a) '' Iic b = Iic (b + a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: (fun x => x + a) '' Set.Iic b = Set.Iic (b + a)
-- Set.image_add_const_Iio
theorem image_add_const_Iio : (fun x => x + a) '' Iio b = Iio (b + a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: (fun x => x + a) '' Set.Iio b = Set.Iio (b + a)
-- Set.image_neg_Ici
theorem image_neg_Ici : Neg.neg '' Ici a = Iic (-a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a : α)
Goal: Neg.neg '' Set.Ici a = Set.Iic (-a)
-- Set.image_neg_Iic
theorem image_neg_Iic : Neg.neg '' Iic a = Ici (-a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a : α)
Goal: Neg.neg '' Set.Iic a = Set.Ici (-a)
-- Set.image_neg_Ioi
theorem image_neg_Ioi : Neg.neg '' Ioi a = Iio (-a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a : α)
Goal: Neg.neg '' Set.Ioi a = Set.Iio (-a)
-- Set.image_neg_Iio
theorem image_neg_Iio : Neg.neg '' Iio a = Ioi (-a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a : α)
Goal: Neg.neg '' Set.Iio a = Set.Ioi (-a)
-- Set.image_neg_Icc
theorem image_neg_Icc : Neg.neg '' Icc a b = Icc (-b) (-a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: Neg.neg '' Set.Icc a b = Set.Icc (-b) (-a)
-- Set.image_neg_Ico
theorem image_neg_Ico : Neg.neg '' Ico a b = Ioc (-b) (-a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: Neg.neg '' Set.Ico a b = Set.Ioc (-b) (-a)
-- Set.image_neg_Ioc
theorem image_neg_Ioc : Neg.neg '' Ioc a b = Ico (-b) (-a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: Neg.neg '' Set.Ioc a b = Set.Ico (-b) (-a)
-- Set.image_neg_Ioo
theorem image_neg_Ioo : Neg.neg '' Ioo a b = Ioo (-b) (-a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: Neg.neg '' Set.Ioo a b = Set.Ioo (-b) (-a)
-- Set.image_const_sub_Ici
@[simp]
theorem image_const_sub_Ici : (fun x => a - x) '' Ici b = Iic (a - b) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: (fun x => a - x) '' Set.Ici b = Set.Iic (a - b)
-- Set.image_const_sub_Iic
@[simp]
theorem image_const_sub_Iic : (fun x => a - x) '' Iic b = Ici (a - b) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: (fun x => a - x) '' Set.Iic b = Set.Ici (a - b)
-- Set.image_const_sub_Ioi
@[simp]
theorem image_const_sub_Ioi : (fun x => a - x) '' Ioi b = Iio (a - b) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: (fun x => a - x) '' Set.Ioi b = Set.Iio (a - b)
-- Set.image_const_sub_Iio
@[simp]
theorem image_const_sub_Iio : (fun x => a - x) '' Iio b = Ioi (a - b) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: (fun x => a - x) '' Set.Iio b = Set.Ioi (a - b)
-- Set.image_const_sub_Icc
@[simp]
theorem image_const_sub_Icc : (fun x => a - x) '' Icc b c = Icc (a - c) (a - b) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b c : α)
Goal: (fun x => a - x) '' Set.Icc b c = Set.Icc (a - c) (a - b)
-- Set.image_const_sub_Ico
@[simp]
theorem image_const_sub_Ico : (fun x => a - x) '' Ico b c = Ioc (a - c) (a - b) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b c : α)
Goal: (fun x => a - x) '' Set.Ico b c = Set.Ioc (a - c) (a - b)
-- Set.image_const_sub_Ioc
@[simp]
theorem image_const_sub_Ioc : (fun x => a - x) '' Ioc b c = Ico (a - c) (a - b) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b c : α)
Goal: (fun x => a - x) '' Set.Ioc b c = Set.Ico (a - c) (a - b)
-- Set.image_const_sub_Ioo
@[simp]
theorem image_const_sub_Ioo : (fun x => a - x) '' Ioo b c = Ioo (a - c) (a - b) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b c : α)
Goal: (fun x => a - x) '' Set.Ioo b c = Set.Ioo (a - c) (a - b)
-- Set.image_sub_const_Ici
@[simp]
theorem image_sub_const_Ici : (fun x => x - a) '' Ici b = Ici (b - a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: (fun x => x - a) '' Set.Ici b = Set.Ici (b - a)
-- Set.image_sub_const_Iic
@[simp]
theorem image_sub_const_Iic : (fun x => x - a) '' Iic b = Iic (b - a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: (fun x => x - a) '' Set.Iic b = Set.Iic (b - a)
-- Set.image_sub_const_Ioi
@[simp]
theorem image_sub_const_Ioi : (fun x => x - a) '' Ioi b = Ioi (b - a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: (fun x => x - a) '' Set.Ioi b = Set.Ioi (b - a)
-- Set.image_sub_const_Iio
@[simp]
theorem image_sub_const_Iio : (fun x => x - a) '' Iio b = Iio (b - a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: (fun x => x - a) '' Set.Iio b = Set.Iio (b - a)
-- Set.image_sub_const_Icc
@[simp]
theorem image_sub_const_Icc : (fun x => x - a) '' Icc b c = Icc (b - a) (c - a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b c : α)
Goal: (fun x => x - a) '' Set.Icc b c = Set.Icc (b - a) (c - a)
-- Set.image_sub_const_Ico
@[simp]
theorem image_sub_const_Ico : (fun x => x - a) '' Ico b c = Ico (b - a) (c - a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b c : α)
Goal: (fun x => x - a) '' Set.Ico b c = Set.Ico (b - a) (c - a)
-- Set.image_sub_const_Ioc
@[simp]
theorem image_sub_const_Ioc : (fun x => x - a) '' Ioc b c = Ioc (b - a) (c - a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b c : α)
Goal: (fun x => x - a) '' Set.Ioc b c = Set.Ioc (b - a) (c - a)
-- Set.image_sub_const_Ioo
@[simp]
theorem image_sub_const_Ioo : (fun x => x - a) '' Ioo b c = Ioo (b - a) (c - a) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b c : α)
Goal: (fun x => x - a) '' Set.Ioo b c = Set.Ioo (b - a) (c - a)
-- Set.Iic_add_bij
theorem Iic_add_bij : BijOn (· + a) (Iic b) (Iic (b + a)) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: Set.BijOn (fun x => x + a) (Set.Iic b) (Set.Iic (b + a))
-- Set.Iio_add_bij
theorem Iio_add_bij : BijOn (· + a) (Iio b) (Iio (b + a)) 
Arguments: {α : Type u_1} [OrderedAddCommGroup α] (a b : α)
Goal: Set.BijOn (fun x => x + a) (Set.Iio b) (Set.Iio (b + a))
-- Set.preimage_const_add_uIcc
@[simp]
theorem preimage_const_add_uIcc : (fun x => a + x) ⁻¹' [[b, c]] = [[b - a, c - a]] 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] (a b c : α)
Goal: (fun x => a + x) ⁻¹' [[b, c]] = [[b - a, c - a]]
-- Set.preimage_add_const_uIcc
@[simp]
theorem preimage_add_const_uIcc : (fun x => x + a) ⁻¹' [[b, c]] = [[b - a, c - a]] 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] (a b c : α)
Goal: (fun x => x + a) ⁻¹' [[b, c]] = [[b - a, c - a]]
-- Set.preimage_neg_uIcc
@[simp]
theorem preimage_neg_uIcc : @Neg.neg (Set α) Set.neg [[a, b]] = [[-a, -b]] 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] (a b : α)
Goal: -[[a, b]] = [[-a, -b]]
-- Set.preimage_sub_const_uIcc
@[simp]
theorem preimage_sub_const_uIcc : (fun x => x - a) ⁻¹' [[b, c]] = [[b + a, c + a]] 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] (a b c : α)
Goal: (fun x => x - a) ⁻¹' [[b, c]] = [[b + a, c + a]]
-- Set.preimage_const_sub_uIcc
@[simp]
theorem preimage_const_sub_uIcc : (fun x => a - x) ⁻¹' [[b, c]] = [[a - b, a - c]] 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] (a b c : α)
Goal: (fun x => a - x) ⁻¹' [[b, c]] = [[a - b, a - c]]
-- Set.image_const_add_uIcc
theorem image_const_add_uIcc : (fun x => a + x) '' [[b, c]] = [[a + b, a + c]] 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] (a b c : α)
Goal: (fun x => a + x) '' [[b, c]] = [[a + b, a + c]]
-- Set.image_add_const_uIcc
theorem image_add_const_uIcc : (fun x => x + a) '' [[b, c]] = [[b + a, c + a]] 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] (a b c : α)
Goal: (fun x => x + a) '' [[b, c]] = [[b + a, c + a]]
-- Set.image_const_sub_uIcc
@[simp]
theorem image_const_sub_uIcc : (fun x => a - x) '' [[b, c]] = [[a - b, a - c]] 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] (a b c : α)
Goal: (fun x => a - x) '' [[b, c]] = [[a - b, a - c]]
-- Set.image_sub_const_uIcc
@[simp]
theorem image_sub_const_uIcc : (fun x => x - a) '' [[b, c]] = [[b - a, c - a]] 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] (a b c : α)
Goal: (fun x => x - a) '' [[b, c]] = [[b - a, c - a]]
-- Set.image_neg_uIcc
theorem image_neg_uIcc : Neg.neg '' [[a, b]] = [[-a, -b]] 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] (a b : α)
Goal: Neg.neg '' [[a, b]] = [[-a, -b]]
-- Set.abs_sub_le_of_uIcc_subset_uIcc
theorem abs_sub_le_of_uIcc_subset_uIcc (h : [[c, d]] ⊆ [[a, b]]) : |d - c| ≤ |b - a| 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a b c d : α} (h : [[c, d]] ⊆ [[a, b]])
Goal: |d - c| ≤ |b - a|
-- Set.abs_sub_left_of_mem_uIcc
theorem abs_sub_left_of_mem_uIcc (h : c ∈ [[a, b]]) : |c - a| ≤ |b - a| 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a b c : α} (h : c ∈ [[a, b]])
Goal: |c - a| ≤ |b - a|
-- Set.abs_sub_right_of_mem_uIcc
theorem abs_sub_right_of_mem_uIcc (h : c ∈ [[a, b]]) : |b - c| ≤ |b - a| 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] {a b c : α} (h : c ∈ [[a, b]])
Goal: |b - c| ≤ |b - a|