ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Set.singleton_sub_singleton
@[to_additive]
theorem singleton_div_singleton : ({a} : Set α) / {b} = {a / b} 
Arguments: {α : Type u_2} [Sub α] {a b : α}
Goal: {a} - {b} = {a - b}
-- Set.sub_subset_sub
@[to_additive (attr := mono)]
theorem div_subset_div : s₁ ⊆ t₁ → s₂ ⊆ t₂ → s₁ / s₂ ⊆ t₁ / t₂ 
Arguments: {α : Type u_2} [Sub α] {s₁ s₂ t₁ t₂ : Set α}
Goal: s₁ ⊆ t₁ → s₂ ⊆ t₂ → s₁ - s₂ ⊆ t₁ - t₂
-- Set.div_subset_div
@[to_additive (attr := mono)]
theorem div_subset_div : s₁ ⊆ t₁ → s₂ ⊆ t₂ → s₁ / s₂ ⊆ t₁ / t₂ 
Arguments: {α : Type u_2} [Div α] {s₁ s₂ t₁ t₂ : Set α}
Goal: s₁ ⊆ t₁ → s₂ ⊆ t₂ → s₁ / s₂ ⊆ t₁ / t₂
-- Set.div_subset_div_left
@[to_additive]
theorem div_subset_div_left : t₁ ⊆ t₂ → s / t₁ ⊆ s / t₂ 
Arguments: {α : Type u_2} [Div α] {s t₁ t₂ : Set α}
Goal: t₁ ⊆ t₂ → s / t₁ ⊆ s / t₂
-- Set.sub_subset_sub_left
@[to_additive]
theorem div_subset_div_left : t₁ ⊆ t₂ → s / t₁ ⊆ s / t₂ 
Arguments: {α : Type u_2} [Sub α] {s t₁ t₂ : Set α}
Goal: t₁ ⊆ t₂ → s - t₁ ⊆ s - t₂
-- Set.sub_subset_sub_right
@[to_additive]
theorem div_subset_div_right : s₁ ⊆ s₂ → s₁ / t ⊆ s₂ / t 
Arguments: {α : Type u_2} [Sub α] {s₁ s₂ t : Set α}
Goal: s₁ ⊆ s₂ → s₁ - t ⊆ s₂ - t
-- Set.div_subset_div_right
@[to_additive]
theorem div_subset_div_right : s₁ ⊆ s₂ → s₁ / t ⊆ s₂ / t 
Arguments: {α : Type u_2} [Div α] {s₁ s₂ t : Set α}
Goal: s₁ ⊆ s₂ → s₁ / t ⊆ s₂ / t
-- Set.div_subset_iff
@[to_additive]
theorem div_subset_iff : s / t ⊆ u ↔ ∀ x ∈ s, ∀ y ∈ t, x / y ∈ u 
Arguments: {α : Type u_2} [Div α] {s t u : Set α}
Goal: s / t ⊆ u ↔ ∀ x ∈ s, ∀ y ∈ t, x / y ∈ u
-- Set.sub_subset_iff
@[to_additive]
theorem div_subset_iff : s / t ⊆ u ↔ ∀ x ∈ s, ∀ y ∈ t, x / y ∈ u 
Arguments: {α : Type u_2} [Sub α] {s t u : Set α}
Goal: s - t ⊆ u ↔ ∀ x ∈ s, ∀ y ∈ t, x - y ∈ u
-- Set.union_sub
@[to_additive]
theorem union_div : (s₁ ∪ s₂) / t = s₁ / t ∪ s₂ / t 
Arguments: {α : Type u_2} [Sub α] {s₁ s₂ t : Set α}
Goal: s₁ ∪ s₂ - t = s₁ - t ∪ (s₂ - t)
-- Set.union_div
@[to_additive]
theorem union_div : (s₁ ∪ s₂) / t = s₁ / t ∪ s₂ / t 
Arguments: {α : Type u_2} [Div α] {s₁ s₂ t : Set α}
Goal: (s₁ ∪ s₂) / t = s₁ / t ∪ s₂ / t
-- Set.div_union
@[to_additive]
theorem div_union : s / (t₁ ∪ t₂) = s / t₁ ∪ s / t₂ 
Arguments: {α : Type u_2} [Div α] {s t₁ t₂ : Set α}
Goal: s / (t₁ ∪ t₂) = s / t₁ ∪ s / t₂
-- Set.sub_union
@[to_additive]
theorem div_union : s / (t₁ ∪ t₂) = s / t₁ ∪ s / t₂ 
Arguments: {α : Type u_2} [Sub α] {s t₁ t₂ : Set α}
Goal: s - (t₁ ∪ t₂) = s - t₁ ∪ (s - t₂)
-- Set.inter_sub_subset
@[to_additive]
theorem inter_div_subset : s₁ ∩ s₂ / t ⊆ s₁ / t ∩ (s₂ / t) 
Arguments: {α : Type u_2} [Sub α] {s₁ s₂ t : Set α}
Goal: s₁ ∩ s₂ - t ⊆ (s₁ - t) ∩ (s₂ - t)
-- Set.inter_div_subset
@[to_additive]
theorem inter_div_subset : s₁ ∩ s₂ / t ⊆ s₁ / t ∩ (s₂ / t) 
Arguments: {α : Type u_2} [Div α] {s₁ s₂ t : Set α}
Goal: s₁ ∩ s₂ / t ⊆ s₁ / t ∩ (s₂ / t)
-- Set.sub_inter_subset
@[to_additive]
theorem div_inter_subset : s / (t₁ ∩ t₂) ⊆ s / t₁ ∩ (s / t₂) 
Arguments: {α : Type u_2} [Sub α] {s t₁ t₂ : Set α}
Goal: s - t₁ ∩ t₂ ⊆ (s - t₁) ∩ (s - t₂)
-- Set.div_inter_subset
@[to_additive]
theorem div_inter_subset : s / (t₁ ∩ t₂) ⊆ s / t₁ ∩ (s / t₂) 
Arguments: {α : Type u_2} [Div α] {s t₁ t₂ : Set α}
Goal: s / (t₁ ∩ t₂) ⊆ s / t₁ ∩ (s / t₂)
-- Set.inter_sub_union_subset_union
@[to_additive]
theorem inter_div_union_subset_union : s₁ ∩ s₂ / (t₁ ∪ t₂) ⊆ s₁ / t₁ ∪ s₂ / t₂ 
Arguments: {α : Type u_2} [Sub α] {s₁ s₂ t₁ t₂ : Set α}
Goal: s₁ ∩ s₂ - (t₁ ∪ t₂) ⊆ s₁ - t₁ ∪ (s₂ - t₂)
-- Set.inter_div_union_subset_union
@[to_additive]
theorem inter_div_union_subset_union : s₁ ∩ s₂ / (t₁ ∪ t₂) ⊆ s₁ / t₁ ∪ s₂ / t₂ 
Arguments: {α : Type u_2} [Div α] {s₁ s₂ t₁ t₂ : Set α}
Goal: s₁ ∩ s₂ / (t₁ ∪ t₂) ⊆ s₁ / t₁ ∪ s₂ / t₂
-- Set.union_div_inter_subset_union
@[to_additive]
theorem union_div_inter_subset_union : (s₁ ∪ s₂) / (t₁ ∩ t₂) ⊆ s₁ / t₁ ∪ s₂ / t₂ 
Arguments: {α : Type u_2} [Div α] {s₁ s₂ t₁ t₂ : Set α}
Goal: (s₁ ∪ s₂) / (t₁ ∩ t₂) ⊆ s₁ / t₁ ∪ s₂ / t₂
-- Set.union_sub_inter_subset_union
@[to_additive]
theorem union_div_inter_subset_union : (s₁ ∪ s₂) / (t₁ ∩ t₂) ⊆ s₁ / t₁ ∪ s₂ / t₂ 
Arguments: {α : Type u_2} [Sub α] {s₁ s₂ t₁ t₂ : Set α}
Goal: s₁ ∪ s₂ - t₁ ∩ t₂ ⊆ s₁ - t₁ ∪ (s₂ - t₂)
-- Set.iUnion_sub_left_image
@[to_additive]
theorem iUnion_div_left_image : ⋃ a ∈ s, (a / ·) '' t = s / t 
Arguments: {α : Type u_2} [Sub α] {s t : Set α}
Goal: ⋃ a ∈ s, (fun x => a - x) '' t = s - t
-- Set.iUnion_div_left_image
@[to_additive]
theorem iUnion_div_left_image : ⋃ a ∈ s, (a / ·) '' t = s / t 
Arguments: {α : Type u_2} [Div α] {s t : Set α}
Goal: ⋃ a ∈ s, (fun x => a / x) '' t = s / t
-- Set.iUnion_sub_right_image
@[to_additive]
theorem iUnion_div_right_image : ⋃ a ∈ t, (· / a) '' s = s / t 
Arguments: {α : Type u_2} [Sub α] {s t : Set α}
Goal: ⋃ a ∈ t, (fun x => x - a) '' s = s - t
-- Set.iUnion_div_right_image
@[to_additive]
theorem iUnion_div_right_image : ⋃ a ∈ t, (· / a) '' s = s / t 
Arguments: {α : Type u_2} [Div α] {s t : Set α}
Goal: ⋃ a ∈ t, (fun x => x / a) '' s = s / t
-- Set.iUnion_div
@[to_additive]
theorem iUnion_div (s : ι → Set α) (t : Set α) : (⋃ i, s i) / t = ⋃ i, s i / t 
Arguments: {α : Type u_2} {ι : Sort u_5} [Div α] (s : ι → Set α) (t : Set α)
Goal: (⋃ i, s i) / t = ⋃ i, s i / t
-- Set.iUnion_sub
@[to_additive]
theorem iUnion_div (s : ι → Set α) (t : Set α) : (⋃ i, s i) / t = ⋃ i, s i / t 
Arguments: {α : Type u_2} {ι : Sort u_5} [Sub α] (s : ι → Set α) (t : Set α)
Goal: (⋃ i, s i) - t = ⋃ i, s i - t
-- Set.sub_iUnion
@[to_additive]
theorem div_iUnion (s : Set α) (t : ι → Set α) : (s / ⋃ i, t i) = ⋃ i, s / t i 
Arguments: {α : Type u_2} {ι : Sort u_5} [Sub α] (s : Set α) (t : ι → Set α)
Goal: s - ⋃ i, t i = ⋃ i, s - t i
-- Set.div_iUnion
@[to_additive]
theorem div_iUnion (s : Set α) (t : ι → Set α) : (s / ⋃ i, t i) = ⋃ i, s / t i 
Arguments: {α : Type u_2} {ι : Sort u_5} [Div α] (s : Set α) (t : ι → Set α)
Goal: s / ⋃ i, t i = ⋃ i, s / t i
-- Set.iUnion₂_div
@[to_additive]
theorem iUnion₂_div (s : ∀ i, κ i → Set α) (t : Set α) :
    (⋃ (i) (j), s i j) / t = ⋃ (i) (j), s i j / t 
Arguments: {α : Type u_2} {ι : Sort u_5} {κ : ι → Sort u_6} [Div α] (s : (i : ι) → κ i → Set α) (t : Set α)
Goal: (⋃ i, ⋃ j, s i j) / t = ⋃ i, ⋃ j, s i j / t
-- Set.iUnion₂_sub
@[to_additive]
theorem iUnion₂_div (s : ∀ i, κ i → Set α) (t : Set α) :
    (⋃ (i) (j), s i j) / t = ⋃ (i) (j), s i j / t 
Arguments: {α : Type u_2} {ι : Sort u_5} {κ : ι → Sort u_6} [Sub α] (s : (i : ι) → κ i → Set α) (t : Set α)
Goal: (⋃ i, ⋃ j, s i j) - t = ⋃ i, ⋃ j, s i j - t
-- Set.div_iUnion₂
@[to_additive]
theorem div_iUnion₂ (s : Set α) (t : ∀ i, κ i → Set α) :
    (s / ⋃ (i) (j), t i j) = ⋃ (i) (j), s / t i j 
Arguments: {α : Type u_2} {ι : Sort u_5} {κ : ι → Sort u_6} [Div α] (s : Set α) (t : (i : ι) → κ i → Set α)
Goal: s / ⋃ i, ⋃ j, t i j = ⋃ i, ⋃ j, s / t i j
-- Set.sub_iUnion₂
@[to_additive]
theorem div_iUnion₂ (s : Set α) (t : ∀ i, κ i → Set α) :
    (s / ⋃ (i) (j), t i j) = ⋃ (i) (j), s / t i j 
Arguments: {α : Type u_2} {ι : Sort u_5} {κ : ι → Sort u_6} [Sub α] (s : Set α) (t : (i : ι) → κ i → Set α)
Goal: s - ⋃ i, ⋃ j, t i j = ⋃ i, ⋃ j, s - t i j
-- Set.iInter_div_subset
@[to_additive]
theorem iInter_div_subset (s : ι → Set α) (t : Set α) : (⋂ i, s i) / t ⊆ ⋂ i, s i / t 
Arguments: {α : Type u_2} {ι : Sort u_5} [Div α] (s : ι → Set α) (t : Set α)
Goal: (⋂ i, s i) / t ⊆ ⋂ i, s i / t
-- Set.iInter_sub_subset
@[to_additive]
theorem iInter_div_subset (s : ι → Set α) (t : Set α) : (⋂ i, s i) / t ⊆ ⋂ i, s i / t 
Arguments: {α : Type u_2} {ι : Sort u_5} [Sub α] (s : ι → Set α) (t : Set α)
Goal: (⋂ i, s i) - t ⊆ ⋂ i, s i - t
-- Set.sub_iInter_subset
@[to_additive]
theorem div_iInter_subset (s : Set α) (t : ι → Set α) : (s / ⋂ i, t i) ⊆ ⋂ i, s / t i 
Arguments: {α : Type u_2} {ι : Sort u_5} [Sub α] (s : Set α) (t : ι → Set α)
Goal: s - ⋂ i, t i ⊆ ⋂ i, s - t i
-- Set.div_iInter_subset
@[to_additive]
theorem div_iInter_subset (s : Set α) (t : ι → Set α) : (s / ⋂ i, t i) ⊆ ⋂ i, s / t i 
Arguments: {α : Type u_2} {ι : Sort u_5} [Div α] (s : Set α) (t : ι → Set α)
Goal: s / ⋂ i, t i ⊆ ⋂ i, s / t i
-- Set.iInter₂_sub_subset
@[to_additive]
theorem iInter₂_div_subset (s : ∀ i, κ i → Set α) (t : Set α) :
    (⋂ (i) (j), s i j) / t ⊆ ⋂ (i) (j), s i j / t 
Arguments: {α : Type u_2} {ι : Sort u_5} {κ : ι → Sort u_6} [Sub α] (s : (i : ι) → κ i → Set α) (t : Set α)
Goal: (⋂ i, ⋂ j, s i j) - t ⊆ ⋂ i, ⋂ j, s i j - t
-- Set.iInter₂_div_subset
@[to_additive]
theorem iInter₂_div_subset (s : ∀ i, κ i → Set α) (t : Set α) :
    (⋂ (i) (j), s i j) / t ⊆ ⋂ (i) (j), s i j / t 
Arguments: {α : Type u_2} {ι : Sort u_5} {κ : ι → Sort u_6} [Div α] (s : (i : ι) → κ i → Set α) (t : Set α)
Goal: (⋂ i, ⋂ j, s i j) / t ⊆ ⋂ i, ⋂ j, s i j / t
-- Set.div_iInter₂_subset
@[to_additive]
theorem div_iInter₂_subset (s : Set α) (t : ∀ i, κ i → Set α) :
    (s / ⋂ (i) (j), t i j) ⊆ ⋂ (i) (j), s / t i j 
Arguments: {α : Type u_2} {ι : Sort u_5} {κ : ι → Sort u_6} [Div α] (s : Set α) (t : (i : ι) → κ i → Set α)
Goal: s / ⋂ i, ⋂ j, t i j ⊆ ⋂ i, ⋂ j, s / t i j
-- Set.sub_iInter₂_subset
@[to_additive]
theorem div_iInter₂_subset (s : Set α) (t : ∀ i, κ i → Set α) :
    (s / ⋂ (i) (j), t i j) ⊆ ⋂ (i) (j), s / t i j 
Arguments: {α : Type u_2} {ι : Sort u_5} {κ : ι → Sort u_6} [Sub α] (s : Set α) (t : (i : ι) → κ i → Set α)
Goal: s - ⋂ i, ⋂ j, t i j ⊆ ⋂ i, ⋂ j, s - t i j
-- Set.inter_mul_union_subset
@[to_additive]
theorem inter_mul_union_subset : s ∩ t * (s ∪ t) ⊆ s * t 
Arguments: {α : Type u_2} [CommSemigroup α] {s t : Set α}
Goal: s ∩ t * (s ∪ t) ⊆ s * t
-- Set.inter_add_union_subset
@[to_additive]
theorem inter_mul_union_subset : s ∩ t * (s ∪ t) ⊆ s * t 
Arguments: {α : Type u_2} [AddCommSemigroup α] {s t : Set α}
Goal: s ∩ t + (s ∪ t) ⊆ s + t
-- Set.union_mul_inter_subset
@[to_additive]
theorem union_mul_inter_subset : (s ∪ t) * (s ∩ t) ⊆ s * t 
Arguments: {α : Type u_2} [CommSemigroup α] {s t : Set α}
Goal: (s ∪ t) * (s ∩ t) ⊆ s * t
-- Set.union_add_inter_subset
@[to_additive]
theorem union_mul_inter_subset : (s ∪ t) * (s ∩ t) ⊆ s * t 
Arguments: {α : Type u_2} [AddCommSemigroup α] {s t : Set α}
Goal: s ∪ t + s ∩ t ⊆ s + t
-- Set.subset_add_left
@[to_additive]
theorem subset_mul_left (s : Set α) {t : Set α} (ht : (1 : α) ∈ t) : s ⊆ s * t 
Arguments: {α : Type u_2} [AddZeroClass α] (s : Set α) {t : Set α} (ht : 0 ∈ t)
Goal: s ⊆ s + t
-- Set.subset_mul_left
@[to_additive]
theorem subset_mul_left (s : Set α) {t : Set α} (ht : (1 : α) ∈ t) : s ⊆ s * t 
Arguments: {α : Type u_2} [MulOneClass α] (s : Set α) {t : Set α} (ht : 1 ∈ t)
Goal: s ⊆ s * t
-- Set.subset_mul_right
@[to_additive]
theorem subset_mul_right {s : Set α} (t : Set α) (hs : (1 : α) ∈ s) : t ⊆ s * t 
Arguments: {α : Type u_2} [MulOneClass α] {s : Set α} (t : Set α) (hs : 1 ∈ s)
Goal: t ⊆ s * t
-- Set.subset_add_right
@[to_additive]
theorem subset_mul_right {s : Set α} (t : Set α) (hs : (1 : α) ∈ s) : t ⊆ s * t 
Arguments: {α : Type u_2} [AddZeroClass α] {s : Set α} (t : Set α) (hs : 0 ∈ s)
Goal: t ⊆ s + t
-- Set.coe_singletonMonoidHom
@[to_additive (attr := simp)]
theorem coe_singletonMonoidHom : (singletonMonoidHom : α → Set α) = singleton 
Arguments: {α : Type u_2} [MulOneClass α]
Goal: ⇑Set.singletonMonoidHom = singleton
-- Set.coe_singletonAddMonoidHom
@[to_additive (attr := simp)]
theorem coe_singletonMonoidHom : (singletonMonoidHom : α → Set α) = singleton 
Arguments: {α : Type u_2} [AddZeroClass α]
Goal: ⇑Set.singletonAddMonoidHom = singleton
-- Set.singletonAddMonoidHom_apply
@[to_additive (attr := simp)]
theorem singletonMonoidHom_apply (a : α) : singletonMonoidHom a = {a} 
Arguments: {α : Type u_2} [AddZeroClass α] (a : α)
Goal: Set.singletonAddMonoidHom a = {a}
-- Set.singletonMonoidHom_apply
@[to_additive (attr := simp)]
theorem singletonMonoidHom_apply (a : α) : singletonMonoidHom a = {a} 
Arguments: {α : Type u_2} [MulOneClass α] (a : α)
Goal: Set.singletonMonoidHom a = {a}
-- Set.pow_mem_pow
@[to_additive]
theorem pow_mem_pow (ha : a ∈ s) : ∀ n : ℕ, a ^ n ∈ s ^ n
  
Arguments: {α : Type u_2} [Monoid α] {s : Set α} {a : α} (ha : a ∈ s) (n : ℕ)
Goal: a ^ n ∈ s ^ n
-- Set.nsmul_mem_nsmul
@[to_additive]
theorem pow_mem_pow (ha : a ∈ s) : ∀ n : ℕ, a ^ n ∈ s ^ n
  
Arguments: {α : Type u_2} [AddMonoid α] {s : Set α} {a : α} (ha : a ∈ s) (n : ℕ)
Goal: n • a ∈ n • s
-- Set.pow_subset_pow
@[to_additive]
theorem pow_subset_pow (hst : s ⊆ t) : ∀ n : ℕ, s ^ n ⊆ t ^ n
  
Arguments: {α : Type u_2} [Monoid α] {s t : Set α} (hst : s ⊆ t) (n : ℕ)
Goal: s ^ n ⊆ t ^ n
-- Set.nsmul_subset_nsmul
@[to_additive]
theorem pow_subset_pow (hst : s ⊆ t) : ∀ n : ℕ, s ^ n ⊆ t ^ n
  
Arguments: {α : Type u_2} [AddMonoid α] {s t : Set α} (hst : s ⊆ t) (n : ℕ)
Goal: n • s ⊆ n • t
-- Set.nsmul_subset_nsmul_of_zero_mem
@[to_additive]
theorem pow_subset_pow_of_one_mem (hs : (1 : α) ∈ s) (hn : m ≤ n) : s ^ m ⊆ s ^ n 
Arguments: {α : Type u_2} [AddMonoid α] {s : Set α} {m n : ℕ} (hs : 0 ∈ s) (hn : m ≤ n)
Goal: m • s ⊆ n • s
-- Set.pow_subset_pow_of_one_mem
@[to_additive]
theorem pow_subset_pow_of_one_mem (hs : (1 : α) ∈ s) (hn : m ≤ n) : s ^ m ⊆ s ^ n 
Arguments: {α : Type u_2} [Monoid α] {s : Set α} {m n : ℕ} (hs : 1 ∈ s) (hn : m ≤ n)
Goal: s ^ m ⊆ s ^ n
-- Set.empty_nsmul
@[to_additive (attr := simp)]
theorem empty_pow {n : ℕ} (hn : n ≠ 0) : (∅ : Set α) ^ n = ∅ 
Arguments: {α : Type u_2} [AddMonoid α] {n : ℕ} (hn : n ≠ 0)
Goal: n • ∅ = ∅
-- Set.empty_pow
@[to_additive (attr := simp)]
theorem empty_pow {n : ℕ} (hn : n ≠ 0) : (∅ : Set α) ^ n = ∅ 
Arguments: {α : Type u_2} [Monoid α] {n : ℕ} (hn : n ≠ 0)
Goal: ∅ ^ n = ∅
-- Set.mul_univ_of_one_mem
@[to_additive]
theorem mul_univ_of_one_mem (hs : (1 : α) ∈ s) : s * univ = univ 
Arguments: {α : Type u_2} [Monoid α] {s : Set α} (hs : 1 ∈ s)
Goal: s * Set.univ = Set.univ
-- Set.add_univ_of_zero_mem
@[to_additive]
theorem mul_univ_of_one_mem (hs : (1 : α) ∈ s) : s * univ = univ 
Arguments: {α : Type u_2} [AddMonoid α] {s : Set α} (hs : 0 ∈ s)
Goal: s + Set.univ = Set.univ
-- Set.univ_mul_of_one_mem
@[to_additive]
theorem univ_mul_of_one_mem (ht : (1 : α) ∈ t) : univ * t = univ 
Arguments: {α : Type u_2} [Monoid α] {t : Set α} (ht : 1 ∈ t)
Goal: Set.univ * t = Set.univ
-- Set.univ_add_of_zero_mem
@[to_additive]
theorem univ_mul_of_one_mem (ht : (1 : α) ∈ t) : univ * t = univ 
Arguments: {α : Type u_2} [AddMonoid α] {t : Set α} (ht : 0 ∈ t)
Goal: Set.univ + t = Set.univ
-- Set.univ_add_univ
@[to_additive (attr := simp)]
theorem univ_mul_univ : (univ : Set α) * univ = univ 
Arguments: {α : Type u_2} [AddMonoid α]
Goal: Set.univ + Set.univ = Set.univ
-- Set.univ_mul_univ
@[to_additive (attr := simp)]
theorem univ_mul_univ : (univ : Set α) * univ = univ 
Arguments: {α : Type u_2} [Monoid α]
Goal: Set.univ * Set.univ = Set.univ
-- Set.nsmul_univ
@[simp]
theorem nsmul_univ {α : Type*} [AddMonoid α] : ∀ {n : ℕ}, n ≠ 0 → n • (univ : Set α) = univ
  
Arguments: {α : Type u_5} [AddMonoid α] {n : ℕ}
Goal: n ≠ 0 → n • Set.univ = Set.univ
-- Set.univ_pow
@[to_additive existing (attr := simp) nsmul_univ]
theorem univ_pow : ∀ {n : ℕ}, n ≠ 0 → (univ : Set α) ^ n = univ
  
Arguments: {α : Type u_2} [Monoid α] {n : ℕ}
Goal: n ≠ 0 → Set.univ ^ n = Set.univ
-- IsUnit.set
@[to_additive]
protected theorem _root_.IsUnit.set : IsUnit a → IsUnit ({a} : Set α) 
Arguments: {α : Type u_2} [Monoid α] {a : α}
Goal: IsUnit a → IsUnit {a}
-- IsAddUnit.set
@[to_additive]
protected theorem _root_.IsUnit.set : IsUnit a → IsUnit ({a} : Set α) 
Arguments: {α : Type u_2} [AddMonoid α] {a : α}
Goal: IsAddUnit a → IsAddUnit {a}
-- Set.mul_eq_one_iff
@[to_additive]
protected theorem mul_eq_one_iff : s * t = 1 ↔ ∃ a b, s = {a} ∧ t = {b} ∧ a * b = 1 
Arguments: {α : Type u_2} [DivisionMonoid α] {s t : Set α}
Goal: s * t = 1 ↔ ∃ a b, s = {a} ∧ t = {b} ∧ a * b = 1
-- Set.add_eq_zero_iff
@[to_additive]
protected theorem mul_eq_one_iff : s * t = 1 ↔ ∃ a b, s = {a} ∧ t = {b} ∧ a * b = 1 
Arguments: {α : Type u_2} [SubtractionMonoid α] {s t : Set α}
Goal: s + t = 0 ↔ ∃ a b, s = {a} ∧ t = {b} ∧ a + b = 0
-- Set.isAddUnit_iff
@[to_additive (attr := simp 500)]
theorem isUnit_iff : IsUnit s ↔ ∃ a, s = {a} ∧ IsUnit a 
Arguments: {α : Type u_2} [SubtractionMonoid α] {s : Set α}
Goal: IsAddUnit s ↔ ∃ a, s = {a} ∧ IsAddUnit a
-- Set.isUnit_iff
@[to_additive (attr := simp 500)]
theorem isUnit_iff : IsUnit s ↔ ∃ a, s = {a} ∧ IsUnit a 
Arguments: {α : Type u_2} [DivisionMonoid α] {s : Set α}
Goal: IsUnit s ↔ ∃ a, s = {a} ∧ IsUnit a
-- Set.univ_div_univ
@[to_additive (attr := simp)]
lemma univ_div_univ : (univ / univ : Set α) = univ 
Arguments: {α : Type u_2} [DivisionMonoid α]
Goal: Set.univ / Set.univ = Set.univ
-- Set.univ_sub_univ
@[to_additive (attr := simp)]
lemma univ_div_univ : (univ / univ : Set α) = univ 
Arguments: {α : Type u_2} [SubtractionMonoid α]
Goal: Set.univ - Set.univ = Set.univ
-- Set.mul_add_subset
theorem mul_add_subset : s * (t + u) ⊆ s * t + s * u 
Arguments: {α : Type u_2} [Distrib α] (s t u : Set α)
Goal: s * (t + u) ⊆ s * t + s * u
-- Set.add_mul_subset
theorem add_mul_subset : (s + t) * u ⊆ s * u + t * u 
Arguments: {α : Type u_2} [Distrib α] (s t u : Set α)
Goal: (s + t) * u ⊆ s * u + t * u
-- Set.mul_zero_subset
theorem mul_zero_subset (s : Set α) : s * 0 ⊆ 0 
Arguments: {α : Type u_2} [MulZeroClass α] (s : Set α)
Goal: s * 0 ⊆ 0
-- Set.zero_mul_subset
theorem zero_mul_subset (s : Set α) : 0 * s ⊆ 0 
Arguments: {α : Type u_2} [MulZeroClass α] (s : Set α)
Goal: 0 * s ⊆ 0
-- Set.Nonempty.mul_zero
theorem Nonempty.mul_zero (hs : s.Nonempty) : s * 0 = 0 
Arguments: {α : Type u_2} [MulZeroClass α] {s : Set α} (hs : s.Nonempty)
Goal: s * 0 = 0
-- Set.Nonempty.zero_mul
theorem Nonempty.zero_mul (hs : s.Nonempty) : 0 * s = 0 
Arguments: {α : Type u_2} [MulZeroClass α] {s : Set α} (hs : s.Nonempty)
Goal: 0 * s = 0
-- Set.one_mem_div_iff
@[to_additive (attr := simp)]
theorem one_mem_div_iff : (1 : α) ∈ s / t ↔ ¬Disjoint s t 
Arguments: {α : Type u_2} [Group α] {s t : Set α}
Goal: 1 ∈ s / t ↔ ¬Disjoint s t
-- Set.zero_mem_sub_iff
@[to_additive (attr := simp)]
theorem one_mem_div_iff : (1 : α) ∈ s / t ↔ ¬Disjoint s t 
Arguments: {α : Type u_2} [AddGroup α] {s t : Set α}
Goal: 0 ∈ s - t ↔ ¬Disjoint s t
-- Set.not_one_mem_div_iff
@[to_additive]
theorem not_one_mem_div_iff : (1 : α) ∉ s / t ↔ Disjoint s t 
Arguments: {α : Type u_2} [Group α] {s t : Set α}
Goal: 1 ∉ s / t ↔ Disjoint s t
-- Set.not_zero_mem_sub_iff
@[to_additive]
theorem not_one_mem_div_iff : (1 : α) ∉ s / t ↔ Disjoint s t 
Arguments: {α : Type u_2} [AddGroup α] {s t : Set α}
Goal: 0 ∉ s - t ↔ Disjoint s t
-- Set.Nonempty.one_mem_div
@[to_additive]
theorem Nonempty.one_mem_div (h : s.Nonempty) : (1 : α) ∈ s / s 
Arguments: {α : Type u_2} [Group α] {s : Set α} (h : s.Nonempty)
Goal: 1 ∈ s / s
-- Set.Nonempty.zero_mem_sub
@[to_additive]
theorem Nonempty.one_mem_div (h : s.Nonempty) : (1 : α) ∈ s / s 
Arguments: {α : Type u_2} [AddGroup α] {s : Set α} (h : s.Nonempty)
Goal: 0 ∈ s - s
-- Set.isUnit_singleton
@[to_additive]
theorem isUnit_singleton (a : α) : IsUnit ({a} : Set α) 
Arguments: {α : Type u_2} [Group α] (a : α)
Goal: IsUnit {a}
-- Set.isAddUnit_singleton
@[to_additive]
theorem isUnit_singleton (a : α) : IsUnit ({a} : Set α) 
Arguments: {α : Type u_2} [AddGroup α] (a : α)
Goal: IsAddUnit {a}
-- Set.isUnit_iff_singleton
@[to_additive (attr := simp)]
theorem isUnit_iff_singleton : IsUnit s ↔ ∃ a, s = {a} 
Arguments: {α : Type u_2} [Group α] {s : Set α}
Goal: IsUnit s ↔ ∃ a, s = {a}
-- Set.isAddUnit_iff_singleton
@[to_additive (attr := simp)]
theorem isUnit_iff_singleton : IsUnit s ↔ ∃ a, s = {a} 
Arguments: {α : Type u_2} [AddGroup α] {s : Set α}
Goal: IsAddUnit s ↔ ∃ a, s = {a}
-- Set.image_add_left
@[to_additive (attr := simp)]
theorem image_mul_left : (a * ·) '' t = (a⁻¹ * ·) ⁻¹' t 
Arguments: {α : Type u_2} [AddGroup α] {t : Set α} {a : α}
Goal: (fun x => a + x) '' t = (fun x => -a + x) ⁻¹' t
-- Set.image_mul_left
@[to_additive (attr := simp)]
theorem image_mul_left : (a * ·) '' t = (a⁻¹ * ·) ⁻¹' t 
Arguments: {α : Type u_2} [Group α] {t : Set α} {a : α}
Goal: (fun x => a * x) '' t = (fun x => a⁻¹ * x) ⁻¹' t
-- Set.image_add_right
@[to_additive (attr := simp)]
theorem image_mul_right : (· * b) '' t = (· * b⁻¹) ⁻¹' t 
Arguments: {α : Type u_2} [AddGroup α] {t : Set α} {b : α}
Goal: (fun x => x + b) '' t = (fun x => x + -b) ⁻¹' t
-- Set.image_mul_right
@[to_additive (attr := simp)]
theorem image_mul_right : (· * b) '' t = (· * b⁻¹) ⁻¹' t 
Arguments: {α : Type u_2} [Group α] {t : Set α} {b : α}
Goal: (fun x => x * b) '' t = (fun x => x * b⁻¹) ⁻¹' t
-- Set.image_mul_left'
@[to_additive]
theorem image_mul_left' : (a⁻¹ * ·) '' t = (a * ·) ⁻¹' t 
Arguments: {α : Type u_2} [Group α] {t : Set α} {a : α}
Goal: (fun x => a⁻¹ * x) '' t = (fun x => a * x) ⁻¹' t
-- Set.image_add_left'
@[to_additive]
theorem image_mul_left' : (a⁻¹ * ·) '' t = (a * ·) ⁻¹' t 
Arguments: {α : Type u_2} [AddGroup α] {t : Set α} {a : α}
Goal: (fun x => -a + x) '' t = (fun x => a + x) ⁻¹' t
-- Set.image_add_right'
@[to_additive]
theorem image_mul_right' : (· * b⁻¹) '' t = (· * b) ⁻¹' t 
Arguments: {α : Type u_2} [AddGroup α] {t : Set α} {b : α}
Goal: (fun x => x + -b) '' t = (fun x => x + b) ⁻¹' t
-- Set.image_mul_right'
@[to_additive]
theorem image_mul_right' : (· * b⁻¹) '' t = (· * b) ⁻¹' t 
Arguments: {α : Type u_2} [Group α] {t : Set α} {b : α}
Goal: (fun x => x * b⁻¹) '' t = (fun x => x * b) ⁻¹' t
-- Set.preimage_add_left_singleton
@[to_additive (attr := simp)]
theorem preimage_mul_left_singleton : (a * ·) ⁻¹' {b} = {a⁻¹ * b} 
Arguments: {α : Type u_2} [AddGroup α] {a b : α}
Goal: (fun x => a + x) ⁻¹' {b} = {-a + b}
-- Set.preimage_mul_left_singleton
@[to_additive (attr := simp)]
theorem preimage_mul_left_singleton : (a * ·) ⁻¹' {b} = {a⁻¹ * b} 
Arguments: {α : Type u_2} [Group α] {a b : α}
Goal: (fun x => a * x) ⁻¹' {b} = {a⁻¹ * b}
-- Set.preimage_mul_right_singleton
@[to_additive (attr := simp)]
theorem preimage_mul_right_singleton : (· * a) ⁻¹' {b} = {b * a⁻¹} 
Arguments: {α : Type u_2} [Group α] {a b : α}
Goal: (fun x => x * a) ⁻¹' {b} = {b * a⁻¹}
-- Set.preimage_add_right_singleton
@[to_additive (attr := simp)]
theorem preimage_mul_right_singleton : (· * a) ⁻¹' {b} = {b * a⁻¹} 
Arguments: {α : Type u_2} [AddGroup α] {a b : α}
Goal: (fun x => x + a) ⁻¹' {b} = {b + -a}
-- Set.preimage_mul_left_one
@[to_additive (attr := simp)]
theorem preimage_mul_left_one : (a * ·) ⁻¹' 1 = {a⁻¹} 
Arguments: {α : Type u_2} [Group α] {a : α}
Goal: (fun x => a * x) ⁻¹' 1 = {a⁻¹}
-- Set.preimage_add_left_zero
@[to_additive (attr := simp)]
theorem preimage_mul_left_one : (a * ·) ⁻¹' 1 = {a⁻¹} 
Arguments: {α : Type u_2} [AddGroup α] {a : α}
Goal: (fun x => a + x) ⁻¹' 0 = {-a}
-- Set.preimage_mul_right_one
@[to_additive (attr := simp)]
theorem preimage_mul_right_one : (· * b) ⁻¹' 1 = {b⁻¹} 
Arguments: {α : Type u_2} [Group α] {b : α}
Goal: (fun x => x * b) ⁻¹' 1 = {b⁻¹}
-- Set.preimage_add_right_zero
@[to_additive (attr := simp)]
theorem preimage_mul_right_one : (· * b) ⁻¹' 1 = {b⁻¹} 
Arguments: {α : Type u_2} [AddGroup α] {b : α}
Goal: (fun x => x + b) ⁻¹' 0 = {-b}
-- Set.preimage_mul_left_one'
@[to_additive]
theorem preimage_mul_left_one' : (a⁻¹ * ·) ⁻¹' 1 = {a} 
Arguments: {α : Type u_2} [Group α] {a : α}
Goal: (fun x => a⁻¹ * x) ⁻¹' 1 = {a}
-- Set.preimage_add_left_zero'
@[to_additive]
theorem preimage_mul_left_one' : (a⁻¹ * ·) ⁻¹' 1 = {a} 
Arguments: {α : Type u_2} [AddGroup α] {a : α}
Goal: (fun x => -a + x) ⁻¹' 0 = {a}
-- Set.preimage_mul_right_one'
@[to_additive]
theorem preimage_mul_right_one' : (· * b⁻¹) ⁻¹' 1 = {b} 
Arguments: {α : Type u_2} [Group α] {b : α}
Goal: (fun x => x * b⁻¹) ⁻¹' 1 = {b}
-- Set.preimage_add_right_zero'
@[to_additive]
theorem preimage_mul_right_one' : (· * b⁻¹) ⁻¹' 1 = {b} 
Arguments: {α : Type u_2} [AddGroup α] {b : α}
Goal: (fun x => x + -b) ⁻¹' 0 = {b}
-- Set.add_univ
@[to_additive (attr := simp)]
theorem mul_univ (hs : s.Nonempty) : s * (univ : Set α) = univ 
Arguments: {α : Type u_2} [AddGroup α] {s : Set α} (hs : s.Nonempty)
Goal: s + Set.univ = Set.univ
-- Set.mul_univ
@[to_additive (attr := simp)]
theorem mul_univ (hs : s.Nonempty) : s * (univ : Set α) = univ 
Arguments: {α : Type u_2} [Group α] {s : Set α} (hs : s.Nonempty)
Goal: s * Set.univ = Set.univ
-- Set.univ_mul
@[to_additive (attr := simp)]
theorem univ_mul (ht : t.Nonempty) : (univ : Set α) * t = univ 
Arguments: {α : Type u_2} [Group α] {t : Set α} (ht : t.Nonempty)
Goal: Set.univ * t = Set.univ
-- Set.univ_add
@[to_additive (attr := simp)]
theorem univ_mul (ht : t.Nonempty) : (univ : Set α) * t = univ 
Arguments: {α : Type u_2} [AddGroup α] {t : Set α} (ht : t.Nonempty)
Goal: Set.univ + t = Set.univ