ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Set.vsub_union
theorem vsub_union : s -ᵥ (t₁ ∪ t₂) = s -ᵥ t₁ ∪ (s -ᵥ t₂) 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] {s t₁ t₂ : Set β}
Goal: s -ᵥ (t₁ ∪ t₂) = s -ᵥ t₁ ∪ (s -ᵥ t₂)
-- Set.inter_vsub_subset
theorem inter_vsub_subset : s₁ ∩ s₂ -ᵥ t ⊆ (s₁ -ᵥ t) ∩ (s₂ -ᵥ t) 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] {s₁ s₂ t : Set β}
Goal: s₁ ∩ s₂ -ᵥ t ⊆ (s₁ -ᵥ t) ∩ (s₂ -ᵥ t)
-- Set.vsub_inter_subset
theorem vsub_inter_subset : s -ᵥ t₁ ∩ t₂ ⊆ (s -ᵥ t₁) ∩ (s -ᵥ t₂) 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] {s t₁ t₂ : Set β}
Goal: s -ᵥ t₁ ∩ t₂ ⊆ (s -ᵥ t₁) ∩ (s -ᵥ t₂)
-- Set.inter_vsub_union_subset_union
theorem inter_vsub_union_subset_union : s₁ ∩ s₂ -ᵥ (t₁ ∪ t₂) ⊆ s₁ -ᵥ t₁ ∪ (s₂ -ᵥ t₂) 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] {s₁ s₂ t₁ t₂ : Set β}
Goal: s₁ ∩ s₂ -ᵥ (t₁ ∪ t₂) ⊆ s₁ -ᵥ t₁ ∪ (s₂ -ᵥ t₂)
-- Set.union_vsub_inter_subset_union
theorem union_vsub_inter_subset_union : s₁ ∪ s₂ -ᵥ t₁ ∩ t₂ ⊆ s₁ -ᵥ t₁ ∪ (s₂ -ᵥ t₂) 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] {s₁ s₂ t₁ t₂ : Set β}
Goal: s₁ ∪ s₂ -ᵥ t₁ ∩ t₂ ⊆ s₁ -ᵥ t₁ ∪ (s₂ -ᵥ t₂)
-- Set.iUnion_vsub_left_image
theorem iUnion_vsub_left_image : ⋃ a ∈ s, (a -ᵥ ·) '' t = s -ᵥ t 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] {s t : Set β}
Goal: ⋃ a ∈ s, (fun x => a -ᵥ x) '' t = s -ᵥ t
-- Set.iUnion_vsub_right_image
theorem iUnion_vsub_right_image : ⋃ a ∈ t, (· -ᵥ a) '' s = s -ᵥ t 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] {s t : Set β}
Goal: ⋃ a ∈ t, (fun x => x -ᵥ a) '' s = s -ᵥ t
-- Set.iUnion_vsub
theorem iUnion_vsub (s : ι → Set β) (t : Set β) : (⋃ i, s i) -ᵥ t = ⋃ i, s i -ᵥ t 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Sort u_5} [VSub α β] (s : ι → Set β) (t : Set β)
Goal: (⋃ i, s i) -ᵥ t = ⋃ i, s i -ᵥ t
-- Set.vsub_iUnion
theorem vsub_iUnion (s : Set β) (t : ι → Set β) : (s -ᵥ ⋃ i, t i) = ⋃ i, s -ᵥ t i 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Sort u_5} [VSub α β] (s : Set β) (t : ι → Set β)
Goal: s -ᵥ ⋃ i, t i = ⋃ i, s -ᵥ t i
-- Set.iUnion₂_vsub
theorem iUnion₂_vsub (s : ∀ i, κ i → Set β) (t : Set β) :
    (⋃ (i) (j), s i j) -ᵥ t = ⋃ (i) (j), s i j -ᵥ t 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Sort u_5} {κ : ι → Sort u_6} [VSub α β] (s : (i : ι) → κ i → Set β) (t : Set β)
Goal: (⋃ i, ⋃ j, s i j) -ᵥ t = ⋃ i, ⋃ j, s i j -ᵥ t
-- Set.vsub_iUnion₂
theorem vsub_iUnion₂ (s : Set β) (t : ∀ i, κ i → Set β) :
    (s -ᵥ ⋃ (i) (j), t i j) = ⋃ (i) (j), s -ᵥ t i j 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Sort u_5} {κ : ι → Sort u_6} [VSub α β] (s : Set β) (t : (i : ι) → κ i → Set β)
Goal: s -ᵥ ⋃ i, ⋃ j, t i j = ⋃ i, ⋃ j, s -ᵥ t i j
-- Set.iInter_vsub_subset
theorem iInter_vsub_subset (s : ι → Set β) (t : Set β) : (⋂ i, s i) -ᵥ t ⊆ ⋂ i, s i -ᵥ t 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Sort u_5} [VSub α β] (s : ι → Set β) (t : Set β)
Goal: (⋂ i, s i) -ᵥ t ⊆ ⋂ i, s i -ᵥ t
-- Set.vsub_iInter_subset
theorem vsub_iInter_subset (s : Set β) (t : ι → Set β) : (s -ᵥ ⋂ i, t i) ⊆ ⋂ i, s -ᵥ t i 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Sort u_5} [VSub α β] (s : Set β) (t : ι → Set β)
Goal: s -ᵥ ⋂ i, t i ⊆ ⋂ i, s -ᵥ t i
-- Set.iInter₂_vsub_subset
theorem iInter₂_vsub_subset (s : ∀ i, κ i → Set β) (t : Set β) :
    (⋂ (i) (j), s i j) -ᵥ t ⊆ ⋂ (i) (j), s i j -ᵥ t 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Sort u_5} {κ : ι → Sort u_6} [VSub α β] (s : (i : ι) → κ i → Set β) (t : Set β)
Goal: (⋂ i, ⋂ j, s i j) -ᵥ t ⊆ ⋂ i, ⋂ j, s i j -ᵥ t
-- Set.vsub_iInter₂_subset
theorem vsub_iInter₂_subset (s : Set β) (t : ∀ i, κ i → Set β) :
    (s -ᵥ ⋂ (i) (j), t i j) ⊆ ⋂ (i) (j), s -ᵥ t i j 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Sort u_5} {κ : ι → Sort u_6} [VSub α β] (s : Set β) (t : (i : ι) → κ i → Set β)
Goal: s -ᵥ ⋂ i, ⋂ j, t i j ⊆ ⋂ i, ⋂ j, s -ᵥ t i j
-- Set.image_vadd_comm
@[to_additive]
theorem image_smul_comm [SMul α β] [SMul α γ] (f : β → γ) (a : α) (s : Set β) :
    (∀ b, f (a • b) = a • f b) → f '' (a • s) = a • f '' s 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [VAdd α β] [VAdd α γ] (f : β → γ) (a : α) (s : Set β)
Goal: (∀ (b : β), f (a +ᵥ b) = a +ᵥ f b) → f '' (a +ᵥ s) = a +ᵥ f '' s
-- Set.image_smul_comm
@[to_additive]
theorem image_smul_comm [SMul α β] [SMul α γ] (f : β → γ) (a : α) (s : Set β) :
    (∀ b, f (a • b) = a • f b) → f '' (a • s) = a • f '' s 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [SMul α β] [SMul α γ] (f : β → γ) (a : α) (s : Set β)
Goal: (∀ (b : β), f (a • b) = a • f b) → f '' (a • s) = a • f '' s
-- Set.image_vadd_distrib
@[to_additive]
theorem image_smul_distrib [MulOneClass α] [MulOneClass β] [FunLike F α β] [MonoidHomClass F α β]
    (f : F) (a : α) (s : Set α) :
    f '' (a • s) = f a • f '' s 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} [AddZeroClass α] [AddZeroClass β] [FunLike F α β] [AddMonoidHomClass F α β] (f : F) (a : α) (s : Set α)
Goal: ⇑f '' (a +ᵥ s) = f a +ᵥ ⇑f '' s
-- Set.image_smul_distrib
@[to_additive]
theorem image_smul_distrib [MulOneClass α] [MulOneClass β] [FunLike F α β] [MonoidHomClass F α β]
    (f : F) (a : α) (s : Set α) :
    f '' (a • s) = f a • f '' s 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} [MulOneClass α] [MulOneClass β] [FunLike F α β] [MonoidHomClass F α β] (f : F) (a : α) (s : Set α)
Goal: ⇑f '' (a • s) = f a • ⇑f '' s
-- Set.op_vadd_set_vadd_eq_vadd_vadd_set
@[to_additive]
theorem op_smul_set_smul_eq_smul_smul_set (a : α) (s : Set β) (t : Set γ)
    (h : ∀ (a : α) (b : β) (c : γ), (op a • b) • c = b • a • c) : (op a • s) • t = s • a • t 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [VAdd αᵃᵒᵖ β] [VAdd β γ] [VAdd α γ] (a : α) (s : Set β) (t : Set γ) (h : ∀ (a : α) (b : β) (c : γ), AddOpposite.op a +ᵥ b +ᵥ c = b +ᵥ (a +ᵥ c))
Goal: AddOpposite.op a +ᵥ s +ᵥ t = s +ᵥ (a +ᵥ t)
-- Set.op_smul_set_smul_eq_smul_smul_set
@[to_additive]
theorem op_smul_set_smul_eq_smul_smul_set (a : α) (s : Set β) (t : Set γ)
    (h : ∀ (a : α) (b : β) (c : γ), (op a • b) • c = b • a • c) : (op a • s) • t = s • a • t 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [SMul αᵐᵒᵖ β] [SMul β γ] [SMul α γ] (a : α) (s : Set β) (t : Set γ) (h : ∀ (a : α) (b : β) (c : γ), (MulOpposite.op a • b) • c = b • a • c)
Goal: (MulOpposite.op a • s) • t = s • a • t
-- Set.smul_zero_subset
theorem smul_zero_subset (s : Set α) : s • (0 : Set β) ⊆ 0 
Arguments: {α : Type u_2} {β : Type u_3} [Zero β] [SMulZeroClass α β] (s : Set α)
Goal: s • 0 ⊆ 0
-- Set.Nonempty.smul_zero
theorem Nonempty.smul_zero (hs : s.Nonempty) : s • (0 : Set β) = 0 
Arguments: {α : Type u_2} {β : Type u_3} [Zero β] [SMulZeroClass α β] {s : Set α} (hs : s.Nonempty)
Goal: s • 0 = 0
-- Set.zero_mem_smul_set
theorem zero_mem_smul_set (h : (0 : β) ∈ t) : (0 : β) ∈ a • t 
Arguments: {α : Type u_2} {β : Type u_3} [Zero β] [SMulZeroClass α β] {t : Set β} {a : α} (h : 0 ∈ t)
Goal: 0 ∈ a • t
-- Set.zero_mem_smul_set_iff
theorem zero_mem_smul_set_iff (ha : a ≠ 0) : (0 : β) ∈ a • t ↔ (0 : β) ∈ t 
Arguments: {α : Type u_2} {β : Type u_3} [Zero β] [SMulZeroClass α β] {t : Set β} {a : α} [Zero α] [NoZeroSMulDivisors α β] (ha : a ≠ 0)
Goal: 0 ∈ a • t ↔ 0 ∈ t
-- Set.zero_smul_subset
theorem zero_smul_subset (t : Set β) : (0 : Set α) • t ⊆ 0 
Arguments: {α : Type u_2} {β : Type u_3} [Zero α] [Zero β] [SMulWithZero α β] (t : Set β)
Goal: 0 • t ⊆ 0
-- Set.Nonempty.zero_smul
theorem Nonempty.zero_smul (ht : t.Nonempty) : (0 : Set α) • t = 0 
Arguments: {α : Type u_2} {β : Type u_3} [Zero α] [Zero β] [SMulWithZero α β] {t : Set β} (ht : t.Nonempty)
Goal: 0 • t = 0
-- Set.zero_smul_set
@[simp] theorem zero_smul_set {s : Set β} (h : s.Nonempty) : (0 : α) • s = (0 : Set β) 
Arguments: {α : Type u_2} {β : Type u_3} [Zero α] [Zero β] [SMulWithZero α β] {s : Set β} (h : s.Nonempty)
Goal: 0 • s = 0
-- Set.zero_smul_set_subset
theorem zero_smul_set_subset (s : Set β) : (0 : α) • s ⊆ 0 
Arguments: {α : Type u_2} {β : Type u_3} [Zero α] [Zero β] [SMulWithZero α β] (s : Set β)
Goal: 0 • s ⊆ 0
-- Set.subsingleton_zero_smul_set
theorem subsingleton_zero_smul_set (s : Set β) : ((0 : α) • s).Subsingleton 
Arguments: {α : Type u_2} {β : Type u_3} [Zero α] [Zero β] [SMulWithZero α β] (s : Set β)
Goal: (0 • s).Subsingleton
-- Set.zero_mem_smul_iff
theorem zero_mem_smul_iff :
    (0 : β) ∈ s • t ↔ (0 : α) ∈ s ∧ t.Nonempty ∨ (0 : β) ∈ t ∧ s.Nonempty 
Arguments: {α : Type u_2} {β : Type u_3} [Zero α] [Zero β] [SMulWithZero α β] {s : Set α} {t : Set β} [NoZeroSMulDivisors α β]
Goal: 0 ∈ s • t ↔ 0 ∈ s ∧ t.Nonempty ∨ 0 ∈ t ∧ s.Nonempty
-- Set.op_vadd_set_add_eq_add_vadd_set
@[to_additive]
theorem op_smul_set_mul_eq_mul_smul_set (a : α) (s : Set α) (t : Set α) :
    op a • s * t = s * a • t 
Arguments: {α : Type u_2} [AddSemigroup α] (a : α) (s t : Set α)
Goal: AddOpposite.op a +ᵥ s + t = s + (a +ᵥ t)
-- Set.op_smul_set_mul_eq_mul_smul_set
@[to_additive]
theorem op_smul_set_mul_eq_mul_smul_set (a : α) (s : Set α) (t : Set α) :
    op a • s * t = s * a • t 
Arguments: {α : Type u_2} [Semigroup α] (a : α) (s t : Set α)
Goal: MulOpposite.op a • s * t = s * a • t
-- Set.pairwiseDisjoint_vadd_iff
@[to_additive]
theorem pairwiseDisjoint_smul_iff :
    s.PairwiseDisjoint (· • t) ↔ (s ×ˢ t).InjOn fun p ↦ p.1 * p.2 
Arguments: {α : Type u_2} [Add α] [IsLeftCancelAdd α] {s t : Set α}
Goal: (s.PairwiseDisjoint fun x => x +ᵥ t) ↔ Set.InjOn (fun p => p.1 + p.2) (s ×ˢ t)
-- Set.pairwiseDisjoint_smul_iff
@[to_additive]
theorem pairwiseDisjoint_smul_iff :
    s.PairwiseDisjoint (· • t) ↔ (s ×ˢ t).InjOn fun p ↦ p.1 * p.2 
Arguments: {α : Type u_2} [Mul α] [IsLeftCancelMul α] {s t : Set α}
Goal: (s.PairwiseDisjoint fun x => x • t) ↔ Set.InjOn (fun p => p.1 * p.2) (s ×ˢ t)
-- Set.smul_mem_smul_set_iff
@[to_additive (attr := simp)]
theorem smul_mem_smul_set_iff : a • x ∈ a • s ↔ x ∈ s 
Arguments: {α : Type u_2} {β : Type u_3} [Group α] [MulAction α β] {s : Set β} {a : α} {x : β}
Goal: a • x ∈ a • s ↔ x ∈ s
-- Set.vadd_mem_vadd_set_iff
@[to_additive (attr := simp)]
theorem smul_mem_smul_set_iff : a • x ∈ a • s ↔ x ∈ s 
Arguments: {α : Type u_2} {β : Type u_3} [AddGroup α] [AddAction α β] {s : Set β} {a : α} {x : β}
Goal: a +ᵥ x ∈ a +ᵥ s ↔ x ∈ s
-- Set.mem_vadd_set_iff_neg_vadd_mem
@[to_additive]
theorem mem_smul_set_iff_inv_smul_mem : x ∈ a • A ↔ a⁻¹ • x ∈ A 
Arguments: {α : Type u_2} {β : Type u_3} [AddGroup α] [AddAction α β] {A : Set β} {a : α} {x : β}
Goal: x ∈ a +ᵥ A ↔ -a +ᵥ x ∈ A
-- Set.mem_smul_set_iff_inv_smul_mem
@[to_additive]
theorem mem_smul_set_iff_inv_smul_mem : x ∈ a • A ↔ a⁻¹ • x ∈ A 
Arguments: {α : Type u_2} {β : Type u_3} [Group α] [MulAction α β] {A : Set β} {a : α} {x : β}
Goal: x ∈ a • A ↔ a⁻¹ • x ∈ A
-- Set.mem_neg_vadd_set_iff
@[to_additive]
theorem mem_inv_smul_set_iff : x ∈ a⁻¹ • A ↔ a • x ∈ A 
Arguments: {α : Type u_2} {β : Type u_3} [AddGroup α] [AddAction α β] {A : Set β} {a : α} {x : β}
Goal: x ∈ -a +ᵥ A ↔ a +ᵥ x ∈ A
-- Set.mem_inv_smul_set_iff
@[to_additive]
theorem mem_inv_smul_set_iff : x ∈ a⁻¹ • A ↔ a • x ∈ A 
Arguments: {α : Type u_2} {β : Type u_3} [Group α] [MulAction α β] {A : Set β} {a : α} {x : β}
Goal: x ∈ a⁻¹ • A ↔ a • x ∈ A
-- Set.preimage_smul
@[to_additive]
theorem preimage_smul (a : α) (t : Set β) : (fun x ↦ a • x) ⁻¹' t = a⁻¹ • t 
Arguments: {α : Type u_2} {β : Type u_3} [Group α] [MulAction α β] (a : α) (t : Set β)
Goal: (fun x => a • x) ⁻¹' t = a⁻¹ • t
-- Set.preimage_vadd
@[to_additive]
theorem preimage_smul (a : α) (t : Set β) : (fun x ↦ a • x) ⁻¹' t = a⁻¹ • t 
Arguments: {α : Type u_2} {β : Type u_3} [AddGroup α] [AddAction α β] (a : α) (t : Set β)
Goal: (fun x => a +ᵥ x) ⁻¹' t = -a +ᵥ t
-- Set.preimage_smul_inv
@[to_additive]
theorem preimage_smul_inv (a : α) (t : Set β) : (fun x ↦ a⁻¹ • x) ⁻¹' t = a • t 
Arguments: {α : Type u_2} {β : Type u_3} [Group α] [MulAction α β] (a : α) (t : Set β)
Goal: (fun x => a⁻¹ • x) ⁻¹' t = a • t
-- Set.preimage_vadd_neg
@[to_additive]
theorem preimage_smul_inv (a : α) (t : Set β) : (fun x ↦ a⁻¹ • x) ⁻¹' t = a • t 
Arguments: {α : Type u_2} {β : Type u_3} [AddGroup α] [AddAction α β] (a : α) (t : Set β)
Goal: (fun x => -a +ᵥ x) ⁻¹' t = a +ᵥ t
-- Set.set_smul_subset_set_smul_iff
@[to_additive (attr := simp)]
theorem set_smul_subset_set_smul_iff : a • A ⊆ a • B ↔ A ⊆ B 
Arguments: {α : Type u_2} {β : Type u_3} [Group α] [MulAction α β] {A B : Set β} {a : α}
Goal: a • A ⊆ a • B ↔ A ⊆ B
-- Set.set_vadd_subset_set_vadd_iff
@[to_additive (attr := simp)]
theorem set_smul_subset_set_smul_iff : a • A ⊆ a • B ↔ A ⊆ B 
Arguments: {α : Type u_2} {β : Type u_3} [AddGroup α] [AddAction α β] {A B : Set β} {a : α}
Goal: a +ᵥ A ⊆ a +ᵥ B ↔ A ⊆ B
-- Set.set_smul_subset_iff
@[to_additive]
theorem set_smul_subset_iff : a • A ⊆ B ↔ A ⊆ a⁻¹ • B 
Arguments: {α : Type u_2} {β : Type u_3} [Group α] [MulAction α β] {A B : Set β} {a : α}
Goal: a • A ⊆ B ↔ A ⊆ a⁻¹ • B
-- Set.set_vadd_subset_iff
@[to_additive]
theorem set_smul_subset_iff : a • A ⊆ B ↔ A ⊆ a⁻¹ • B 
Arguments: {α : Type u_2} {β : Type u_3} [AddGroup α] [AddAction α β] {A B : Set β} {a : α}
Goal: a +ᵥ A ⊆ B ↔ A ⊆ -a +ᵥ B
-- Set.subset_set_vadd_iff
@[to_additive]
theorem subset_set_smul_iff : A ⊆ a • B ↔ a⁻¹ • A ⊆ B 
Arguments: {α : Type u_2} {β : Type u_3} [AddGroup α] [AddAction α β] {A B : Set β} {a : α}
Goal: A ⊆ a +ᵥ B ↔ -a +ᵥ A ⊆ B
-- Set.subset_set_smul_iff
@[to_additive]
theorem subset_set_smul_iff : A ⊆ a • B ↔ a⁻¹ • A ⊆ B 
Arguments: {α : Type u_2} {β : Type u_3} [Group α] [MulAction α β] {A B : Set β} {a : α}
Goal: A ⊆ a • B ↔ a⁻¹ • A ⊆ B
-- Set.vadd_set_inter
@[to_additive]
theorem smul_set_inter : a • (s ∩ t) = a • s ∩ a • t 
Arguments: {α : Type u_2} {β : Type u_3} [AddGroup α] [AddAction α β] {s t : Set β} {a : α}
Goal: a +ᵥ s ∩ t = (a +ᵥ s) ∩ (a +ᵥ t)
-- Set.smul_set_inter
@[to_additive]
theorem smul_set_inter : a • (s ∩ t) = a • s ∩ a • t 
Arguments: {α : Type u_2} {β : Type u_3} [Group α] [MulAction α β] {s t : Set β} {a : α}
Goal: a • (s ∩ t) = a • s ∩ a • t
-- Set.smul_set_iInter
@[to_additive]
theorem smul_set_iInter {ι : Type*}
    (a : α) (t : ι → Set β) : (a • ⋂ i, t i) = ⋂ i, a • t i 
Arguments: {α : Type u_2} {β : Type u_3} [Group α] [MulAction α β] {ι : Type u_5} (a : α) (t : ι → Set β)
Goal: a • ⋂ i, t i = ⋂ i, a • t i
-- Set.vadd_set_iInter
@[to_additive]
theorem smul_set_iInter {ι : Type*}
    (a : α) (t : ι → Set β) : (a • ⋂ i, t i) = ⋂ i, a • t i 
Arguments: {α : Type u_2} {β : Type u_3} [AddGroup α] [AddAction α β] {ι : Type u_5} (a : α) (t : ι → Set β)
Goal: a +ᵥ ⋂ i, t i = ⋂ i, a +ᵥ t i
-- Set.vadd_set_sdiff
@[to_additive]
theorem smul_set_sdiff : a • (s \ t) = a • s \ a • t 
Arguments: {α : Type u_2} {β : Type u_3} [AddGroup α] [AddAction α β] {s t : Set β} {a : α}
Goal: a +ᵥ s \ t = (a +ᵥ s) \ (a +ᵥ t)
-- Set.smul_set_sdiff
@[to_additive]
theorem smul_set_sdiff : a • (s \ t) = a • s \ a • t 
Arguments: {α : Type u_2} {β : Type u_3} [Group α] [MulAction α β] {s t : Set β} {a : α}
Goal: a • (s \ t) = a • s \ a • t
-- Set.smul_set_symmDiff
@[to_additive]
theorem smul_set_symmDiff : a • s ∆ t = (a • s) ∆ (a • t) 
Arguments: {α : Type u_2} {β : Type u_3} [Group α] [MulAction α β] {s t : Set β} {a : α}
Goal: a • symmDiff s t = symmDiff (a • s) (a • t)
-- Set.vadd_set_symmDiff
@[to_additive]
theorem smul_set_symmDiff : a • s ∆ t = (a • s) ∆ (a • t) 
Arguments: {α : Type u_2} {β : Type u_3} [AddGroup α] [AddAction α β] {s t : Set β} {a : α}
Goal: a +ᵥ symmDiff s t = symmDiff (a +ᵥ s) (a +ᵥ t)
-- Set.smul_set_univ
@[to_additive (attr := simp)]
theorem smul_set_univ : a • (univ : Set β) = univ 
Arguments: {α : Type u_2} {β : Type u_3} [Group α] [MulAction α β] {a : α}
Goal: a • Set.univ = Set.univ
-- Set.vadd_set_univ
@[to_additive (attr := simp)]
theorem smul_set_univ : a • (univ : Set β) = univ 
Arguments: {α : Type u_2} {β : Type u_3} [AddGroup α] [AddAction α β] {a : α}
Goal: a +ᵥ Set.univ = Set.univ
-- Set.smul_univ
@[to_additive (attr := simp)]
theorem smul_univ {s : Set α} (hs : s.Nonempty) : s • (univ : Set β) = univ 
Arguments: {α : Type u_2} {β : Type u_3} [Group α] [MulAction α β] {s : Set α} (hs : s.Nonempty)
Goal: s • Set.univ = Set.univ
-- Set.vadd_univ
@[to_additive (attr := simp)]
theorem smul_univ {s : Set α} (hs : s.Nonempty) : s • (univ : Set β) = univ 
Arguments: {α : Type u_2} {β : Type u_3} [AddGroup α] [AddAction α β] {s : Set α} (hs : s.Nonempty)
Goal: s +ᵥ Set.univ = Set.univ
-- Set.vadd_set_compl
@[to_additive]
theorem smul_set_compl : a • sᶜ = (a • s)ᶜ 
Arguments: {α : Type u_2} {β : Type u_3} [AddGroup α] [AddAction α β] {s : Set β} {a : α}
Goal: a +ᵥ sᶜ = (a +ᵥ s)ᶜ
-- Set.smul_set_compl
@[to_additive]
theorem smul_set_compl : a • sᶜ = (a • s)ᶜ 
Arguments: {α : Type u_2} {β : Type u_3} [Group α] [MulAction α β] {s : Set β} {a : α}
Goal: a • sᶜ = (a • s)ᶜ
-- Set.smul_inter_ne_empty_iff
@[to_additive]
theorem smul_inter_ne_empty_iff {s t : Set α} {x : α} :
    x • s ∩ t ≠ ∅ ↔ ∃ a b, (a ∈ t ∧ b ∈ s) ∧ a * b⁻¹ = x 
Arguments: {α : Type u_2} [Group α] {s t : Set α} {x : α}
Goal: x • s ∩ t ≠ ∅ ↔ ∃ a b, (a ∈ t ∧ b ∈ s) ∧ a * b⁻¹ = x
-- Set.vadd_inter_ne_empty_iff
@[to_additive]
theorem smul_inter_ne_empty_iff {s t : Set α} {x : α} :
    x • s ∩ t ≠ ∅ ↔ ∃ a b, (a ∈ t ∧ b ∈ s) ∧ a * b⁻¹ = x 
Arguments: {α : Type u_2} [AddGroup α] {s t : Set α} {x : α}
Goal: (x +ᵥ s) ∩ t ≠ ∅ ↔ ∃ a b, (a ∈ t ∧ b ∈ s) ∧ a + -b = x
-- Set.vadd_inter_ne_empty_iff'
@[to_additive]
theorem smul_inter_ne_empty_iff' {s t : Set α} {x : α} :
    x • s ∩ t ≠ ∅ ↔ ∃ a b, (a ∈ t ∧ b ∈ s) ∧ a / b = x 
Arguments: {α : Type u_2} [AddGroup α] {s t : Set α} {x : α}
Goal: (x +ᵥ s) ∩ t ≠ ∅ ↔ ∃ a b, (a ∈ t ∧ b ∈ s) ∧ a - b = x
-- Set.smul_inter_ne_empty_iff'
@[to_additive]
theorem smul_inter_ne_empty_iff' {s t : Set α} {x : α} :
    x • s ∩ t ≠ ∅ ↔ ∃ a b, (a ∈ t ∧ b ∈ s) ∧ a / b = x 
Arguments: {α : Type u_2} [Group α] {s t : Set α} {x : α}
Goal: x • s ∩ t ≠ ∅ ↔ ∃ a b, (a ∈ t ∧ b ∈ s) ∧ a / b = x
-- Set.op_vadd_inter_ne_empty_iff
@[to_additive]
theorem op_smul_inter_ne_empty_iff {s t : Set α} {x : αᵐᵒᵖ} :
    x • s ∩ t ≠ ∅ ↔ ∃ a b, (a ∈ s ∧ b ∈ t) ∧ a⁻¹ * b = MulOpposite.unop x 
Arguments: {α : Type u_2} [AddGroup α] {s t : Set α} {x : αᵃᵒᵖ}
Goal: (x +ᵥ s) ∩ t ≠ ∅ ↔ ∃ a b, (a ∈ s ∧ b ∈ t) ∧ -a + b = AddOpposite.unop x
-- Set.op_smul_inter_ne_empty_iff
@[to_additive]
theorem op_smul_inter_ne_empty_iff {s t : Set α} {x : αᵐᵒᵖ} :
    x • s ∩ t ≠ ∅ ↔ ∃ a b, (a ∈ s ∧ b ∈ t) ∧ a⁻¹ * b = MulOpposite.unop x 
Arguments: {α : Type u_2} [Group α] {s t : Set α} {x : αᵐᵒᵖ}
Goal: x • s ∩ t ≠ ∅ ↔ ∃ a b, (a ∈ s ∧ b ∈ t) ∧ a⁻¹ * b = MulOpposite.unop x
-- Set.iUnion_neg_vadd
@[to_additive (attr := simp)]
theorem iUnion_inv_smul : ⋃ g : α, g⁻¹ • s = ⋃ g : α, g • s 
Arguments: {α : Type u_2} {β : Type u_3} [AddGroup α] [AddAction α β] {s : Set β}
Goal: ⋃ g, -g +ᵥ s = ⋃ g, g +ᵥ s
-- Set.iUnion_inv_smul
@[to_additive (attr := simp)]
theorem iUnion_inv_smul : ⋃ g : α, g⁻¹ • s = ⋃ g : α, g • s 
Arguments: {α : Type u_2} {β : Type u_3} [Group α] [MulAction α β] {s : Set β}
Goal: ⋃ g, g⁻¹ • s = ⋃ g, g • s
-- Set.iUnion_smul_eq_setOf_exists
@[to_additive]
theorem iUnion_smul_eq_setOf_exists {s : Set β} : ⋃ g : α, g • s = { a | ∃ g : α, g • a ∈ s } 
Arguments: {α : Type u_2} {β : Type u_3} [Group α] [MulAction α β] {s : Set β}
Goal: ⋃ g, g • s = {a | ∃ g, g • a ∈ s}
-- Set.iUnion_vadd_eq_setOf_exists
@[to_additive]
theorem iUnion_smul_eq_setOf_exists {s : Set β} : ⋃ g : α, g • s = { a | ∃ g : α, g • a ∈ s } 
Arguments: {α : Type u_2} {β : Type u_3} [AddGroup α] [AddAction α β] {s : Set β}
Goal: ⋃ g, g +ᵥ s = {a | ∃ g, g +ᵥ a ∈ s}
-- Set.neg_vadd_set_distrib
@[to_additive (attr := simp)]
lemma inv_smul_set_distrib (a : α) (s : Set α) : (a • s)⁻¹ = op a⁻¹ • s⁻¹ 
Arguments: {α : Type u_2} [AddGroup α] (a : α) (s : Set α)
Goal: -(a +ᵥ s) = AddOpposite.op (-a) +ᵥ -s
-- Set.inv_smul_set_distrib
@[to_additive (attr := simp)]
lemma inv_smul_set_distrib (a : α) (s : Set α) : (a • s)⁻¹ = op a⁻¹ • s⁻¹ 
Arguments: {α : Type u_2} [Group α] (a : α) (s : Set α)
Goal: (a • s)⁻¹ = MulOpposite.op a⁻¹ • s⁻¹
-- Set.neg_op_vadd_set_distrib
@[to_additive (attr := simp)]
lemma inv_op_smul_set_distrib (a : α) (s : Set α) : (op a • s)⁻¹ = a⁻¹ • s⁻¹ 
Arguments: {α : Type u_2} [AddGroup α] (a : α) (s : Set α)
Goal: -(AddOpposite.op a +ᵥ s) = -a +ᵥ -s
-- Set.inv_op_smul_set_distrib
@[to_additive (attr := simp)]
lemma inv_op_smul_set_distrib (a : α) (s : Set α) : (op a • s)⁻¹ = a⁻¹ • s⁻¹ 
Arguments: {α : Type u_2} [Group α] (a : α) (s : Set α)
Goal: (MulOpposite.op a • s)⁻¹ = a⁻¹ • s⁻¹
-- Set.smul_set_disjoint_iff
@[to_additive (attr := simp)]
lemma smul_set_disjoint_iff : Disjoint (a • s) (a • t) ↔ Disjoint s t 
Arguments: {α : Type u_2} {β : Type u_3} [Group α] [MulAction α β] {s t : Set β} {a : α}
Goal: Disjoint (a • s) (a • t) ↔ Disjoint s t
-- Set.vadd_set_disjoint_iff
@[to_additive (attr := simp)]
lemma smul_set_disjoint_iff : Disjoint (a • s) (a • t) ↔ Disjoint s t 
Arguments: {α : Type u_2} {β : Type u_3} [AddGroup α] [AddAction α β] {s t : Set β} {a : α}
Goal: Disjoint (a +ᵥ s) (a +ᵥ t) ↔ Disjoint s t
-- Set.smul_mem_smul_set_iff₀
@[simp]
theorem smul_mem_smul_set_iff₀ (ha : a ≠ 0) (A : Set β) (x : β) : a • x ∈ a • A ↔ x ∈ A 
Arguments: {α : Type u_2} {β : Type u_3} [GroupWithZero α] [MulAction α β] {a : α} (ha : a ≠ 0) (A : Set β) (x : β)
Goal: a • x ∈ a • A ↔ x ∈ A
-- Set.mem_smul_set_iff_inv_smul_mem₀
theorem mem_smul_set_iff_inv_smul_mem₀ (ha : a ≠ 0) (A : Set β) (x : β) : x ∈ a • A ↔ a⁻¹ • x ∈ A 
Arguments: {α : Type u_2} {β : Type u_3} [GroupWithZero α] [MulAction α β] {a : α} (ha : a ≠ 0) (A : Set β) (x : β)
Goal: x ∈ a • A ↔ a⁻¹ • x ∈ A
-- Set.mem_inv_smul_set_iff₀
theorem mem_inv_smul_set_iff₀ (ha : a ≠ 0) (A : Set β) (x : β) : x ∈ a⁻¹ • A ↔ a • x ∈ A 
Arguments: {α : Type u_2} {β : Type u_3} [GroupWithZero α] [MulAction α β] {a : α} (ha : a ≠ 0) (A : Set β) (x : β)
Goal: x ∈ a⁻¹ • A ↔ a • x ∈ A
-- Set.preimage_smul₀
theorem preimage_smul₀ (ha : a ≠ 0) (t : Set β) : (fun x ↦ a • x) ⁻¹' t = a⁻¹ • t 
Arguments: {α : Type u_2} {β : Type u_3} [GroupWithZero α] [MulAction α β] {a : α} (ha : a ≠ 0) (t : Set β)
Goal: (fun x => a • x) ⁻¹' t = a⁻¹ • t
-- Set.preimage_smul_inv₀
theorem preimage_smul_inv₀ (ha : a ≠ 0) (t : Set β) : (fun x ↦ a⁻¹ • x) ⁻¹' t = a • t 
Arguments: {α : Type u_2} {β : Type u_3} [GroupWithZero α] [MulAction α β] {a : α} (ha : a ≠ 0) (t : Set β)
Goal: (fun x => a⁻¹ • x) ⁻¹' t = a • t
-- Set.set_smul_subset_set_smul_iff₀
@[simp]
theorem set_smul_subset_set_smul_iff₀ (ha : a ≠ 0) {A B : Set β} : a • A ⊆ a • B ↔ A ⊆ B 
Arguments: {α : Type u_2} {β : Type u_3} [GroupWithZero α] [MulAction α β] {a : α} (ha : a ≠ 0) {A B : Set β}
Goal: a • A ⊆ a • B ↔ A ⊆ B
-- Set.set_smul_subset_iff₀
theorem set_smul_subset_iff₀ (ha : a ≠ 0) {A B : Set β} : a • A ⊆ B ↔ A ⊆ a⁻¹ • B 
Arguments: {α : Type u_2} {β : Type u_3} [GroupWithZero α] [MulAction α β] {a : α} (ha : a ≠ 0) {A B : Set β}
Goal: a • A ⊆ B ↔ A ⊆ a⁻¹ • B
-- Set.subset_set_smul_iff₀
theorem subset_set_smul_iff₀ (ha : a ≠ 0) {A B : Set β} : A ⊆ a • B ↔ a⁻¹ • A ⊆ B 
Arguments: {α : Type u_2} {β : Type u_3} [GroupWithZero α] [MulAction α β] {a : α} (ha : a ≠ 0) {A B : Set β}
Goal: A ⊆ a • B ↔ a⁻¹ • A ⊆ B
-- Set.smul_set_inter₀
theorem smul_set_inter₀ (ha : a ≠ 0) : a • (s ∩ t) = a • s ∩ a • t 
Arguments: {α : Type u_2} {β : Type u_3} [GroupWithZero α] [MulAction α β] {s t : Set β} {a : α} (ha : a ≠ 0)
Goal: a • (s ∩ t) = a • s ∩ a • t
-- Set.smul_set_sdiff₀
theorem smul_set_sdiff₀ (ha : a ≠ 0) : a • (s \ t) = a • s \ a • t 
Arguments: {α : Type u_2} {β : Type u_3} [GroupWithZero α] [MulAction α β] {s t : Set β} {a : α} (ha : a ≠ 0)
Goal: a • (s \ t) = a • s \ a • t
-- Set.smul_set_symmDiff₀
theorem smul_set_symmDiff₀ (ha : a ≠ 0) : a • s ∆ t = (a • s) ∆ (a • t) 
Arguments: {α : Type u_2} {β : Type u_3} [GroupWithZero α] [MulAction α β] {s t : Set β} {a : α} (ha : a ≠ 0)
Goal: a • symmDiff s t = symmDiff (a • s) (a • t)
-- Set.smul_set_univ₀
theorem smul_set_univ₀ (ha : a ≠ 0) : a • (univ : Set β) = univ 
Arguments: {α : Type u_2} {β : Type u_3} [GroupWithZero α] [MulAction α β] {a : α} (ha : a ≠ 0)
Goal: a • Set.univ = Set.univ
-- Set.smul_univ₀
theorem smul_univ₀ {s : Set α} (hs : ¬s ⊆ 0) : s • (univ : Set β) = univ 
Arguments: {α : Type u_2} {β : Type u_3} [GroupWithZero α] [MulAction α β] {s : Set α} (hs : ¬s ⊆ 0)
Goal: s • Set.univ = Set.univ
-- Set.smul_univ₀'
theorem smul_univ₀' {s : Set α} (hs : s.Nontrivial) : s • (univ : Set β) = univ 
Arguments: {α : Type u_2} {β : Type u_3} [GroupWithZero α] [MulAction α β] {s : Set α} (hs : s.Nontrivial)
Goal: s • Set.univ = Set.univ
-- Set.inv_zero
@[simp] protected lemma inv_zero : (0 : Set α)⁻¹ = 0 
Arguments: {α : Type u_2} [GroupWithZero α]
Goal: 0⁻¹ = 0