ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Set.vadd_union
@[to_additive]
theorem smul_union : s • (t₁ ∪ t₂) = s • t₁ ∪ s • t₂ 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {s : Set α} {t₁ t₂ : Set β}
Goal: s +ᵥ (t₁ ∪ t₂) = s +ᵥ t₁ ∪ (s +ᵥ t₂)
-- Set.smul_union
@[to_additive]
theorem smul_union : s • (t₁ ∪ t₂) = s • t₁ ∪ s • t₂ 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {s : Set α} {t₁ t₂ : Set β}
Goal: s • (t₁ ∪ t₂) = s • t₁ ∪ s • t₂
-- Set.inter_vadd_subset
@[to_additive]
theorem inter_smul_subset : (s₁ ∩ s₂) • t ⊆ s₁ • t ∩ s₂ • t 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {s₁ s₂ : Set α} {t : Set β}
Goal: s₁ ∩ s₂ +ᵥ t ⊆ (s₁ +ᵥ t) ∩ (s₂ +ᵥ t)
-- Set.inter_smul_subset
@[to_additive]
theorem inter_smul_subset : (s₁ ∩ s₂) • t ⊆ s₁ • t ∩ s₂ • t 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {s₁ s₂ : Set α} {t : Set β}
Goal: (s₁ ∩ s₂) • t ⊆ s₁ • t ∩ s₂ • t
-- Set.vadd_inter_subset
@[to_additive]
theorem smul_inter_subset : s • (t₁ ∩ t₂) ⊆ s • t₁ ∩ s • t₂ 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {s : Set α} {t₁ t₂ : Set β}
Goal: s +ᵥ t₁ ∩ t₂ ⊆ (s +ᵥ t₁) ∩ (s +ᵥ t₂)
-- Set.smul_inter_subset
@[to_additive]
theorem smul_inter_subset : s • (t₁ ∩ t₂) ⊆ s • t₁ ∩ s • t₂ 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {s : Set α} {t₁ t₂ : Set β}
Goal: s • (t₁ ∩ t₂) ⊆ s • t₁ ∩ s • t₂
-- Set.inter_smul_union_subset_union
@[to_additive]
theorem inter_smul_union_subset_union : (s₁ ∩ s₂) • (t₁ ∪ t₂) ⊆ s₁ • t₁ ∪ s₂ • t₂ 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {s₁ s₂ : Set α} {t₁ t₂ : Set β}
Goal: (s₁ ∩ s₂) • (t₁ ∪ t₂) ⊆ s₁ • t₁ ∪ s₂ • t₂
-- Set.inter_vadd_union_subset_union
@[to_additive]
theorem inter_smul_union_subset_union : (s₁ ∩ s₂) • (t₁ ∪ t₂) ⊆ s₁ • t₁ ∪ s₂ • t₂ 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {s₁ s₂ : Set α} {t₁ t₂ : Set β}
Goal: s₁ ∩ s₂ +ᵥ (t₁ ∪ t₂) ⊆ s₁ +ᵥ t₁ ∪ (s₂ +ᵥ t₂)
-- Set.union_vadd_inter_subset_union
@[to_additive]
theorem union_smul_inter_subset_union : (s₁ ∪ s₂) • (t₁ ∩ t₂) ⊆ s₁ • t₁ ∪ s₂ • t₂ 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {s₁ s₂ : Set α} {t₁ t₂ : Set β}
Goal: s₁ ∪ s₂ +ᵥ t₁ ∩ t₂ ⊆ s₁ +ᵥ t₁ ∪ (s₂ +ᵥ t₂)
-- Set.union_smul_inter_subset_union
@[to_additive]
theorem union_smul_inter_subset_union : (s₁ ∪ s₂) • (t₁ ∩ t₂) ⊆ s₁ • t₁ ∪ s₂ • t₂ 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {s₁ s₂ : Set α} {t₁ t₂ : Set β}
Goal: (s₁ ∪ s₂) • (t₁ ∩ t₂) ⊆ s₁ • t₁ ∪ s₂ • t₂
-- Set.iUnion_smul_left_image
@[to_additive]
theorem iUnion_smul_left_image : ⋃ a ∈ s, a • t = s • t 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {s : Set α} {t : Set β}
Goal: ⋃ a ∈ s, a • t = s • t
-- Set.iUnion_vadd_left_image
@[to_additive]
theorem iUnion_smul_left_image : ⋃ a ∈ s, a • t = s • t 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {s : Set α} {t : Set β}
Goal: ⋃ a ∈ s, a +ᵥ t = s +ᵥ t
-- Set.iUnion_smul_right_image
@[to_additive]
theorem iUnion_smul_right_image : ⋃ a ∈ t, (· • a) '' s = s • t 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {s : Set α} {t : Set β}
Goal: ⋃ a ∈ t, (fun x => x • a) '' s = s • t
-- Set.iUnion_vadd_right_image
@[to_additive]
theorem iUnion_smul_right_image : ⋃ a ∈ t, (· • a) '' s = s • t 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {s : Set α} {t : Set β}
Goal: ⋃ a ∈ t, (fun x => x +ᵥ a) '' s = s +ᵥ t
-- Set.iUnion_vadd
@[to_additive]
theorem iUnion_smul (s : ι → Set α) (t : Set β) : (⋃ i, s i) • t = ⋃ i, s i • t 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Sort u_5} [VAdd α β] (s : ι → Set α) (t : Set β)
Goal: (⋃ i, s i) +ᵥ t = ⋃ i, s i +ᵥ t
-- Set.iUnion_smul
@[to_additive]
theorem iUnion_smul (s : ι → Set α) (t : Set β) : (⋃ i, s i) • t = ⋃ i, s i • t 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Sort u_5} [SMul α β] (s : ι → Set α) (t : Set β)
Goal: (⋃ i, s i) • t = ⋃ i, s i • t
-- Set.vadd_iUnion
@[to_additive]
theorem smul_iUnion (s : Set α) (t : ι → Set β) : (s • ⋃ i, t i) = ⋃ i, s • t i 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Sort u_5} [VAdd α β] (s : Set α) (t : ι → Set β)
Goal: s +ᵥ ⋃ i, t i = ⋃ i, s +ᵥ t i
-- Set.smul_iUnion
@[to_additive]
theorem smul_iUnion (s : Set α) (t : ι → Set β) : (s • ⋃ i, t i) = ⋃ i, s • t i 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Sort u_5} [SMul α β] (s : Set α) (t : ι → Set β)
Goal: s • ⋃ i, t i = ⋃ i, s • t i
-- Set.iUnion₂_smul
@[to_additive]
theorem iUnion₂_smul (s : ∀ i, κ i → Set α) (t : Set β) :
    (⋃ (i) (j), s i j) • t = ⋃ (i) (j), s i j • t 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Sort u_5} {κ : ι → Sort u_6} [SMul α β] (s : (i : ι) → κ i → Set α) (t : Set β)
Goal: (⋃ i, ⋃ j, s i j) • t = ⋃ i, ⋃ j, s i j • t
-- Set.iUnion₂_vadd
@[to_additive]
theorem iUnion₂_smul (s : ∀ i, κ i → Set α) (t : Set β) :
    (⋃ (i) (j), s i j) • t = ⋃ (i) (j), s i j • t 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Sort u_5} {κ : ι → Sort u_6} [VAdd α β] (s : (i : ι) → κ i → Set α) (t : Set β)
Goal: (⋃ i, ⋃ j, s i j) +ᵥ t = ⋃ i, ⋃ j, s i j +ᵥ t
-- Set.smul_iUnion₂
@[to_additive]
theorem smul_iUnion₂ (s : Set α) (t : ∀ i, κ i → Set β) :
    (s • ⋃ (i) (j), t i j) = ⋃ (i) (j), s • t i j 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Sort u_5} {κ : ι → Sort u_6} [SMul α β] (s : Set α) (t : (i : ι) → κ i → Set β)
Goal: s • ⋃ i, ⋃ j, t i j = ⋃ i, ⋃ j, s • t i j
-- Set.vadd_iUnion₂
@[to_additive]
theorem smul_iUnion₂ (s : Set α) (t : ∀ i, κ i → Set β) :
    (s • ⋃ (i) (j), t i j) = ⋃ (i) (j), s • t i j 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Sort u_5} {κ : ι → Sort u_6} [VAdd α β] (s : Set α) (t : (i : ι) → κ i → Set β)
Goal: s +ᵥ ⋃ i, ⋃ j, t i j = ⋃ i, ⋃ j, s +ᵥ t i j
-- Set.iInter_vadd_subset
@[to_additive]
theorem iInter_smul_subset (s : ι → Set α) (t : Set β) : (⋂ i, s i) • t ⊆ ⋂ i, s i • t 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Sort u_5} [VAdd α β] (s : ι → Set α) (t : Set β)
Goal: (⋂ i, s i) +ᵥ t ⊆ ⋂ i, s i +ᵥ t
-- Set.iInter_smul_subset
@[to_additive]
theorem iInter_smul_subset (s : ι → Set α) (t : Set β) : (⋂ i, s i) • t ⊆ ⋂ i, s i • t 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Sort u_5} [SMul α β] (s : ι → Set α) (t : Set β)
Goal: (⋂ i, s i) • t ⊆ ⋂ i, s i • t
-- Set.smul_iInter_subset
@[to_additive]
theorem smul_iInter_subset (s : Set α) (t : ι → Set β) : (s • ⋂ i, t i) ⊆ ⋂ i, s • t i 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Sort u_5} [SMul α β] (s : Set α) (t : ι → Set β)
Goal: s • ⋂ i, t i ⊆ ⋂ i, s • t i
-- Set.vadd_iInter_subset
@[to_additive]
theorem smul_iInter_subset (s : Set α) (t : ι → Set β) : (s • ⋂ i, t i) ⊆ ⋂ i, s • t i 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Sort u_5} [VAdd α β] (s : Set α) (t : ι → Set β)
Goal: s +ᵥ ⋂ i, t i ⊆ ⋂ i, s +ᵥ t i
-- Set.iInter₂_smul_subset
@[to_additive]
theorem iInter₂_smul_subset (s : ∀ i, κ i → Set α) (t : Set β) :
    (⋂ (i) (j), s i j) • t ⊆ ⋂ (i) (j), s i j • t 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Sort u_5} {κ : ι → Sort u_6} [SMul α β] (s : (i : ι) → κ i → Set α) (t : Set β)
Goal: (⋂ i, ⋂ j, s i j) • t ⊆ ⋂ i, ⋂ j, s i j • t
-- Set.iInter₂_vadd_subset
@[to_additive]
theorem iInter₂_smul_subset (s : ∀ i, κ i → Set α) (t : Set β) :
    (⋂ (i) (j), s i j) • t ⊆ ⋂ (i) (j), s i j • t 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Sort u_5} {κ : ι → Sort u_6} [VAdd α β] (s : (i : ι) → κ i → Set α) (t : Set β)
Goal: (⋂ i, ⋂ j, s i j) +ᵥ t ⊆ ⋂ i, ⋂ j, s i j +ᵥ t
-- Set.vadd_iInter₂_subset
@[to_additive]
theorem smul_iInter₂_subset (s : Set α) (t : ∀ i, κ i → Set β) :
    (s • ⋂ (i) (j), t i j) ⊆ ⋂ (i) (j), s • t i j 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Sort u_5} {κ : ι → Sort u_6} [VAdd α β] (s : Set α) (t : (i : ι) → κ i → Set β)
Goal: s +ᵥ ⋂ i, ⋂ j, t i j ⊆ ⋂ i, ⋂ j, s +ᵥ t i j
-- Set.smul_iInter₂_subset
@[to_additive]
theorem smul_iInter₂_subset (s : Set α) (t : ∀ i, κ i → Set β) :
    (s • ⋂ (i) (j), t i j) ⊆ ⋂ (i) (j), s • t i j 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Sort u_5} {κ : ι → Sort u_6} [SMul α β] (s : Set α) (t : (i : ι) → κ i → Set β)
Goal: s • ⋂ i, ⋂ j, t i j ⊆ ⋂ i, ⋂ j, s • t i j
-- Set.vadd_set_subset_vadd
@[to_additive]
theorem smul_set_subset_smul {s : Set α} : a ∈ s → a • t ⊆ s • t 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {t : Set β} {a : α} {s : Set α}
Goal: a ∈ s → a +ᵥ t ⊆ s +ᵥ t
-- Set.smul_set_subset_smul
@[to_additive]
theorem smul_set_subset_smul {s : Set α} : a ∈ s → a • t ⊆ s • t 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {t : Set β} {a : α} {s : Set α}
Goal: a ∈ s → a • t ⊆ s • t
-- Set.iUnion_smul_set
@[to_additive (attr := simp)]
theorem iUnion_smul_set (s : Set α) (t : Set β) : ⋃ a ∈ s, a • t = s • t 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] (s : Set α) (t : Set β)
Goal: ⋃ a ∈ s, a • t = s • t
-- Set.iUnion_vadd_set
@[to_additive (attr := simp)]
theorem iUnion_smul_set (s : Set α) (t : Set β) : ⋃ a ∈ s, a • t = s • t 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] (s : Set α) (t : Set β)
Goal: ⋃ a ∈ s, a +ᵥ t = s +ᵥ t
-- Set.image_smul
@[to_additive]
theorem image_smul : (fun x ↦ a • x) '' t = a • t 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {t : Set β} {a : α}
Goal: (fun x => a • x) '' t = a • t
-- Set.image_vadd
@[to_additive]
theorem image_smul : (fun x ↦ a • x) '' t = a • t 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {t : Set β} {a : α}
Goal: (fun x => a +ᵥ x) '' t = a +ᵥ t
-- Set.mem_smul_set
@[to_additive]
theorem mem_smul_set : x ∈ a • t ↔ ∃ y, y ∈ t ∧ a • y = x 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {t : Set β} {a : α} {x : β}
Goal: x ∈ a • t ↔ ∃ y ∈ t, a • y = x
-- Set.mem_vadd_set
@[to_additive]
theorem mem_smul_set : x ∈ a • t ↔ ∃ y, y ∈ t ∧ a • y = x 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {t : Set β} {a : α} {x : β}
Goal: x ∈ a +ᵥ t ↔ ∃ y ∈ t, a +ᵥ y = x
-- Set.vadd_mem_vadd_set
@[to_additive]
theorem smul_mem_smul_set : b ∈ s → a • b ∈ a • s 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {s : Set β} {a : α} {b : β}
Goal: b ∈ s → a +ᵥ b ∈ a +ᵥ s
-- Set.smul_mem_smul_set
@[to_additive]
theorem smul_mem_smul_set : b ∈ s → a • b ∈ a • s 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {s : Set β} {a : α} {b : β}
Goal: b ∈ s → a • b ∈ a • s
-- Set.vadd_set_empty
@[to_additive (attr := simp)]
theorem smul_set_empty : a • (∅ : Set β) = ∅ 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {a : α}
Goal: a +ᵥ ∅ = ∅
-- Set.smul_set_empty
@[to_additive (attr := simp)]
theorem smul_set_empty : a • (∅ : Set β) = ∅ 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {a : α}
Goal: a • ∅ = ∅
-- Set.smul_set_eq_empty
@[to_additive (attr := simp)]
theorem smul_set_eq_empty : a • s = ∅ ↔ s = ∅ 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {s : Set β} {a : α}
Goal: a • s = ∅ ↔ s = ∅
-- Set.vadd_set_eq_empty
@[to_additive (attr := simp)]
theorem smul_set_eq_empty : a • s = ∅ ↔ s = ∅ 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {s : Set β} {a : α}
Goal: a +ᵥ s = ∅ ↔ s = ∅
-- Set.vadd_set_nonempty
@[to_additive (attr := simp)]
theorem smul_set_nonempty : (a • s).Nonempty ↔ s.Nonempty 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {s : Set β} {a : α}
Goal: (a +ᵥ s).Nonempty ↔ s.Nonempty
-- Set.smul_set_nonempty
@[to_additive (attr := simp)]
theorem smul_set_nonempty : (a • s).Nonempty ↔ s.Nonempty 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {s : Set β} {a : α}
Goal: (a • s).Nonempty ↔ s.Nonempty
-- Set.smul_set_singleton
@[to_additive (attr := simp)]
theorem smul_set_singleton : a • ({b} : Set β) = {a • b} 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {a : α} {b : β}
Goal: a • {b} = {a • b}
-- Set.vadd_set_singleton
@[to_additive (attr := simp)]
theorem smul_set_singleton : a • ({b} : Set β) = {a • b} 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {a : α} {b : β}
Goal: a +ᵥ {b} = {a +ᵥ b}
-- Set.vadd_set_mono
@[to_additive]
theorem smul_set_mono : s ⊆ t → a • s ⊆ a • t 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {s t : Set β} {a : α}
Goal: s ⊆ t → a +ᵥ s ⊆ a +ᵥ t
-- Set.smul_set_mono
@[to_additive]
theorem smul_set_mono : s ⊆ t → a • s ⊆ a • t 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {s t : Set β} {a : α}
Goal: s ⊆ t → a • s ⊆ a • t
-- Set.smul_set_subset_iff
@[to_additive]
theorem smul_set_subset_iff : a • s ⊆ t ↔ ∀ ⦃b⦄, b ∈ s → a • b ∈ t 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {s t : Set β} {a : α}
Goal: a • s ⊆ t ↔ ∀ ⦃b : β⦄, b ∈ s → a • b ∈ t
-- Set.vadd_set_subset_iff
@[to_additive]
theorem smul_set_subset_iff : a • s ⊆ t ↔ ∀ ⦃b⦄, b ∈ s → a • b ∈ t 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {s t : Set β} {a : α}
Goal: a +ᵥ s ⊆ t ↔ ∀ ⦃b : β⦄, b ∈ s → a +ᵥ b ∈ t
-- Set.smul_set_union
@[to_additive]
theorem smul_set_union : a • (t₁ ∪ t₂) = a • t₁ ∪ a • t₂ 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {t₁ t₂ : Set β} {a : α}
Goal: a • (t₁ ∪ t₂) = a • t₁ ∪ a • t₂
-- Set.vadd_set_union
@[to_additive]
theorem smul_set_union : a • (t₁ ∪ t₂) = a • t₁ ∪ a • t₂ 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {t₁ t₂ : Set β} {a : α}
Goal: a +ᵥ (t₁ ∪ t₂) = a +ᵥ t₁ ∪ (a +ᵥ t₂)
-- Set.smul_set_inter_subset
@[to_additive]
theorem smul_set_inter_subset : a • (t₁ ∩ t₂) ⊆ a • t₁ ∩ a • t₂ 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {t₁ t₂ : Set β} {a : α}
Goal: a • (t₁ ∩ t₂) ⊆ a • t₁ ∩ a • t₂
-- Set.vadd_set_inter_subset
@[to_additive]
theorem smul_set_inter_subset : a • (t₁ ∩ t₂) ⊆ a • t₁ ∩ a • t₂ 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {t₁ t₂ : Set β} {a : α}
Goal: a +ᵥ t₁ ∩ t₂ ⊆ (a +ᵥ t₁) ∩ (a +ᵥ t₂)
-- Set.smul_set_iUnion
@[to_additive]
theorem smul_set_iUnion (a : α) (s : ι → Set β) : (a • ⋃ i, s i) = ⋃ i, a • s i 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Sort u_5} [SMul α β] (a : α) (s : ι → Set β)
Goal: a • ⋃ i, s i = ⋃ i, a • s i
-- Set.vadd_set_iUnion
@[to_additive]
theorem smul_set_iUnion (a : α) (s : ι → Set β) : (a • ⋃ i, s i) = ⋃ i, a • s i 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Sort u_5} [VAdd α β] (a : α) (s : ι → Set β)
Goal: a +ᵥ ⋃ i, s i = ⋃ i, a +ᵥ s i
-- Set.vadd_set_iUnion₂
@[to_additive]
theorem smul_set_iUnion₂ (a : α) (s : ∀ i, κ i → Set β) :
    (a • ⋃ (i) (j), s i j) = ⋃ (i) (j), a • s i j 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Sort u_5} {κ : ι → Sort u_6} [VAdd α β] (a : α) (s : (i : ι) → κ i → Set β)
Goal: a +ᵥ ⋃ i, ⋃ j, s i j = ⋃ i, ⋃ j, a +ᵥ s i j
-- Set.smul_set_iUnion₂
@[to_additive]
theorem smul_set_iUnion₂ (a : α) (s : ∀ i, κ i → Set β) :
    (a • ⋃ (i) (j), s i j) = ⋃ (i) (j), a • s i j 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Sort u_5} {κ : ι → Sort u_6} [SMul α β] (a : α) (s : (i : ι) → κ i → Set β)
Goal: a • ⋃ i, ⋃ j, s i j = ⋃ i, ⋃ j, a • s i j
-- Set.smul_set_iInter_subset
@[to_additive]
theorem smul_set_iInter_subset (a : α) (t : ι → Set β) : (a • ⋂ i, t i) ⊆ ⋂ i, a • t i 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Sort u_5} [SMul α β] (a : α) (t : ι → Set β)
Goal: a • ⋂ i, t i ⊆ ⋂ i, a • t i
-- Set.vadd_set_iInter_subset
@[to_additive]
theorem smul_set_iInter_subset (a : α) (t : ι → Set β) : (a • ⋂ i, t i) ⊆ ⋂ i, a • t i 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Sort u_5} [VAdd α β] (a : α) (t : ι → Set β)
Goal: a +ᵥ ⋂ i, t i ⊆ ⋂ i, a +ᵥ t i
-- Set.smul_set_iInter₂_subset
@[to_additive]
theorem smul_set_iInter₂_subset (a : α) (t : ∀ i, κ i → Set β) :
    (a • ⋂ (i) (j), t i j) ⊆ ⋂ (i) (j), a • t i j 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Sort u_5} {κ : ι → Sort u_6} [SMul α β] (a : α) (t : (i : ι) → κ i → Set β)
Goal: a • ⋂ i, ⋂ j, t i j ⊆ ⋂ i, ⋂ j, a • t i j
-- Set.vadd_set_iInter₂_subset
@[to_additive]
theorem smul_set_iInter₂_subset (a : α) (t : ∀ i, κ i → Set β) :
    (a • ⋂ (i) (j), t i j) ⊆ ⋂ (i) (j), a • t i j 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Sort u_5} {κ : ι → Sort u_6} [VAdd α β] (a : α) (t : (i : ι) → κ i → Set β)
Goal: a +ᵥ ⋂ i, ⋂ j, t i j ⊆ ⋂ i, ⋂ j, a +ᵥ t i j
-- Set.Nonempty.smul_set
@[to_additive]
theorem Nonempty.smul_set : s.Nonempty → (a • s).Nonempty 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {s : Set β} {a : α}
Goal: s.Nonempty → (a • s).Nonempty
-- Set.Nonempty.vadd_set
@[to_additive]
theorem Nonempty.smul_set : s.Nonempty → (a • s).Nonempty 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {s : Set β} {a : α}
Goal: s.Nonempty → (a +ᵥ s).Nonempty
-- Set.op_vadd_set_subset_add
@[to_additive]
theorem op_smul_set_subset_mul : a ∈ t → op a • s ⊆ s * t 
Arguments: {α : Type u_2} [Add α] {s t : Set α} {a : α}
Goal: a ∈ t → AddOpposite.op a +ᵥ s ⊆ s + t
-- Set.op_smul_set_subset_mul
@[to_additive]
theorem op_smul_set_subset_mul : a ∈ t → op a • s ⊆ s * t 
Arguments: {α : Type u_2} [Mul α] {s t : Set α} {a : α}
Goal: a ∈ t → MulOpposite.op a • s ⊆ s * t
-- Set.image_op_smul
@[to_additive]
theorem image_op_smul : (op '' s) • t = t * s 
Arguments: {α : Type u_2} [Mul α] {s t : Set α}
Goal: MulOpposite.op '' s • t = t * s
-- Set.image_op_vadd
@[to_additive]
theorem image_op_smul : (op '' s) • t = t * s 
Arguments: {α : Type u_2} [Add α] {s t : Set α}
Goal: AddOpposite.op '' s +ᵥ t = t + s
-- Set.iUnion_op_smul_set
@[to_additive (attr := simp)]
theorem iUnion_op_smul_set (s t : Set α) : ⋃ a ∈ t, MulOpposite.op a • s = s * t 
Arguments: {α : Type u_2} [Mul α] (s t : Set α)
Goal: ⋃ a ∈ t, MulOpposite.op a • s = s * t
-- Set.iUnion_op_vadd_set
@[to_additive (attr := simp)]
theorem iUnion_op_smul_set (s t : Set α) : ⋃ a ∈ t, MulOpposite.op a • s = s * t 
Arguments: {α : Type u_2} [Add α] (s t : Set α)
Goal: ⋃ a ∈ t, AddOpposite.op a +ᵥ s = s + t
-- Set.mul_subset_iff_left
@[to_additive]
theorem mul_subset_iff_left : s * t ⊆ u ↔ ∀ a ∈ s, a • t ⊆ u 
Arguments: {α : Type u_2} [Mul α] {s t u : Set α}
Goal: s * t ⊆ u ↔ ∀ a ∈ s, a • t ⊆ u
-- Set.add_subset_iff_left
@[to_additive]
theorem mul_subset_iff_left : s * t ⊆ u ↔ ∀ a ∈ s, a • t ⊆ u 
Arguments: {α : Type u_2} [Add α] {s t u : Set α}
Goal: s + t ⊆ u ↔ ∀ a ∈ s, a +ᵥ t ⊆ u
-- Set.mul_subset_iff_right
@[to_additive]
theorem mul_subset_iff_right : s * t ⊆ u ↔ ∀ b ∈ t, op b • s ⊆ u 
Arguments: {α : Type u_2} [Mul α] {s t u : Set α}
Goal: s * t ⊆ u ↔ ∀ b ∈ t, MulOpposite.op b • s ⊆ u
-- Set.add_subset_iff_right
@[to_additive]
theorem mul_subset_iff_right : s * t ⊆ u ↔ ∀ b ∈ t, op b • s ⊆ u 
Arguments: {α : Type u_2} [Add α] {s t u : Set α}
Goal: s + t ⊆ u ↔ ∀ b ∈ t, AddOpposite.op b +ᵥ s ⊆ u
-- Set.range_smul_range
@[to_additive]
theorem range_smul_range {ι κ : Type*} [SMul α β] (b : ι → α) (c : κ → β) :
    range b • range c = range fun p : ι × κ ↦ b p.1 • c p.2 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Type u_5} {κ : Type u_6} [SMul α β] (b : ι → α) (c : κ → β)
Goal: Set.range b • Set.range c = Set.range fun p => b p.1 • c p.2
-- Set.range_vadd_range
@[to_additive]
theorem range_smul_range {ι κ : Type*} [SMul α β] (b : ι → α) (c : κ → β) :
    range b • range c = range fun p : ι × κ ↦ b p.1 • c p.2 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Type u_5} {κ : Type u_6} [VAdd α β] (b : ι → α) (c : κ → β)
Goal: Set.range b +ᵥ Set.range c = Set.range fun p => b p.1 +ᵥ c p.2
-- Set.vadd_set_range
@[to_additive]
theorem smul_set_range [SMul α β] {ι : Sort*} (a : α) (f : ι → β) :
    a • range f = range fun i ↦ a • f i 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {ι : Sort u_5} (a : α) (f : ι → β)
Goal: a +ᵥ Set.range f = Set.range fun i => a +ᵥ f i
-- Set.smul_set_range
@[to_additive]
theorem smul_set_range [SMul α β] {ι : Sort*} (a : α) (f : ι → β) :
    a • range f = range fun i ↦ a • f i 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {ι : Sort u_5} (a : α) (f : ι → β)
Goal: a • Set.range f = Set.range fun i => a • f i
-- Set.range_vadd
@[to_additive] lemma range_smul [SMul α β] {ι : Sort*} (a : α) (f : ι → β) :
    range (fun i ↦ a • f i) = a • range f 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {ι : Sort u_5} (a : α) (f : ι → β)
Goal: (Set.range fun i => a +ᵥ f i) = a +ᵥ Set.range f
-- Set.range_smul
@[to_additive] lemma range_smul [SMul α β] {ι : Sort*} (a : α) (f : ι → β) :
    range (fun i ↦ a • f i) = a • range f 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {ι : Sort u_5} (a : α) (f : ι → β)
Goal: (Set.range fun i => a • f i) = a • Set.range f
-- Set.range_add
@[to_additive] lemma range_mul [Mul α] {ι : Sort*} (a : α) (f : ι → α) :
    range (fun i ↦ a * f i) = a • range f 
Arguments: {α : Type u_2} [Add α] {ι : Sort u_5} (a : α) (f : ι → α)
Goal: (Set.range fun i => a + f i) = a +ᵥ Set.range f
-- Set.range_mul
@[to_additive] lemma range_mul [Mul α] {ι : Sort*} (a : α) (f : ι → α) :
    range (fun i ↦ a * f i) = a • range f 
Arguments: {α : Type u_2} [Mul α] {ι : Sort u_5} (a : α) (f : ι → α)
Goal: (Set.range fun i => a * f i) = a • Set.range f
-- Set.image2_vsub
@[simp]
theorem image2_vsub : (image2 VSub.vsub s t : Set α) = s -ᵥ t 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] {s t : Set β}
Goal: Set.image2 VSub.vsub s t = s -ᵥ t
-- Set.image_vsub_prod
theorem image_vsub_prod : (fun x : β × β ↦ x.fst -ᵥ x.snd) '' s ×ˢ t = s -ᵥ t 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] {s t : Set β}
Goal: (fun x => x.1 -ᵥ x.2) '' s ×ˢ t = s -ᵥ t
-- Set.mem_vsub
theorem mem_vsub : a ∈ s -ᵥ t ↔ ∃ x ∈ s, ∃ y ∈ t, x -ᵥ y = a 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] {s t : Set β} {a : α}
Goal: a ∈ s -ᵥ t ↔ ∃ x ∈ s, ∃ y ∈ t, x -ᵥ y = a
-- Set.vsub_mem_vsub
theorem vsub_mem_vsub (hb : b ∈ s) (hc : c ∈ t) : b -ᵥ c ∈ s -ᵥ t 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] {s t : Set β} {b c : β} (hb : b ∈ s) (hc : c ∈ t)
Goal: b -ᵥ c ∈ s -ᵥ t
-- Set.empty_vsub
@[simp]
theorem empty_vsub (t : Set β) : ∅ -ᵥ t = ∅ 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] (t : Set β)
Goal: ∅ -ᵥ t = ∅
-- Set.vsub_empty
@[simp]
theorem vsub_empty (s : Set β) : s -ᵥ ∅ = ∅ 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] (s : Set β)
Goal: s -ᵥ ∅ = ∅
-- Set.vsub_eq_empty
@[simp]
theorem vsub_eq_empty : s -ᵥ t = ∅ ↔ s = ∅ ∨ t = ∅ 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] {s t : Set β}
Goal: s -ᵥ t = ∅ ↔ s = ∅ ∨ t = ∅
-- Set.vsub_nonempty
@[simp]
theorem vsub_nonempty : (s -ᵥ t : Set α).Nonempty ↔ s.Nonempty ∧ t.Nonempty 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] {s t : Set β}
Goal: (s -ᵥ t).Nonempty ↔ s.Nonempty ∧ t.Nonempty
-- Set.Nonempty.vsub
theorem Nonempty.vsub : s.Nonempty → t.Nonempty → (s -ᵥ t : Set α).Nonempty 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] {s t : Set β}
Goal: s.Nonempty → t.Nonempty → (s -ᵥ t).Nonempty
-- Set.Nonempty.of_vsub_left
theorem Nonempty.of_vsub_left : (s -ᵥ t : Set α).Nonempty → s.Nonempty 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] {s t : Set β}
Goal: (s -ᵥ t).Nonempty → s.Nonempty
-- Set.Nonempty.of_vsub_right
theorem Nonempty.of_vsub_right : (s -ᵥ t : Set α).Nonempty → t.Nonempty 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] {s t : Set β}
Goal: (s -ᵥ t).Nonempty → t.Nonempty
-- Set.vsub_singleton
@[simp low+1]
theorem vsub_singleton (s : Set β) (b : β) : s -ᵥ {b} = (· -ᵥ b) '' s 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] (s : Set β) (b : β)
Goal: s -ᵥ {b} = (fun x => x -ᵥ b) '' s
-- Set.singleton_vsub
@[simp low+1]
theorem singleton_vsub (t : Set β) (b : β) : {b} -ᵥ t = (b -ᵥ ·) '' t 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] (t : Set β) (b : β)
Goal: {b} -ᵥ t = (fun x => b -ᵥ x) '' t
-- Set.singleton_vsub_singleton
@[simp high]
theorem singleton_vsub_singleton : ({b} : Set β) -ᵥ {c} = {b -ᵥ c} 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] {b c : β}
Goal: {b} -ᵥ {c} = {b -ᵥ c}
-- Set.vsub_subset_vsub
@[mono]
theorem vsub_subset_vsub : s₁ ⊆ s₂ → 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 → t₁ ⊆ t₂ → s₁ -ᵥ t₁ ⊆ s₂ -ᵥ t₂
-- Set.vsub_subset_vsub_left
theorem vsub_subset_vsub_left : 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: t₁ ⊆ t₂ → s -ᵥ t₁ ⊆ s -ᵥ t₂
-- Set.vsub_subset_vsub_right
theorem vsub_subset_vsub_right : s₁ ⊆ s₂ → 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 → s₁ -ᵥ t ⊆ s₂ -ᵥ t
-- Set.vsub_subset_iff
theorem vsub_subset_iff : s -ᵥ t ⊆ u ↔ ∀ x ∈ s, ∀ y ∈ t, x -ᵥ y ∈ u 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] {s t : Set β} {u : Set α}
Goal: s -ᵥ t ⊆ u ↔ ∀ x ∈ s, ∀ y ∈ t, x -ᵥ y ∈ u
-- Set.vsub_self_mono
theorem vsub_self_mono (h : s ⊆ t) : s -ᵥ s ⊆ t -ᵥ t 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] {s t : Set β} (h : s ⊆ t)
Goal: s -ᵥ s ⊆ t -ᵥ t
-- Set.union_vsub
theorem union_vsub : 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)