ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Finset.inter_add_singleton
@[to_additive]
theorem inter_mul_singleton : s ∩ t * {a} = s * {a} ∩ (t * {a}) 
Arguments: {α : Type u_2} [Add α] [IsRightCancelAdd α] [DecidableEq α] (s t : Finset α) (a : α)
Goal: s ∩ t + {a} = (s + {a}) ∩ (t + {a})
-- Finset.card_le_card_mul_right
@[to_additive]
theorem card_le_card_mul_right {t : Finset α} (ht : t.Nonempty) : s.card ≤ (s * t).card 
Arguments: {α : Type u_2} [Mul α] [IsRightCancelMul α] [DecidableEq α] (s : Finset α) {t : Finset α} (ht : t.Nonempty)
Goal: s.card ≤ (s * t).card
-- Finset.card_le_card_add_right
@[to_additive]
theorem card_le_card_mul_right {t : Finset α} (ht : t.Nonempty) : s.card ≤ (s * t).card 
Arguments: {α : Type u_2} [Add α] [IsRightCancelAdd α] [DecidableEq α] (s : Finset α) {t : Finset α} (ht : t.Nonempty)
Goal: s.card ≤ (s + t).card
-- Finset.card_le_card_div_left
@[to_additive] lemma card_le_card_div_left (hs : s.Nonempty) : t.card ≤ (s / t).card 
Arguments: {α : Type u_2} [Group α] [DecidableEq α] {s t : Finset α} (hs : s.Nonempty)
Goal: t.card ≤ (s / t).card
-- Finset.card_le_card_sub_left
@[to_additive] lemma card_le_card_div_left (hs : s.Nonempty) : t.card ≤ (s / t).card 
Arguments: {α : Type u_2} [AddGroup α] [DecidableEq α] {s t : Finset α} (hs : s.Nonempty)
Goal: t.card ≤ (s - t).card
-- Finset.card_le_card_div_right
@[to_additive] lemma card_le_card_div_right (ht : t.Nonempty) : s.card ≤ (s / t).card 
Arguments: {α : Type u_2} [Group α] [DecidableEq α] {s t : Finset α} (ht : t.Nonempty)
Goal: s.card ≤ (s / t).card
-- Finset.card_le_card_sub_right
@[to_additive] lemma card_le_card_div_right (ht : t.Nonempty) : s.card ≤ (s / t).card 
Arguments: {α : Type u_2} [AddGroup α] [DecidableEq α] {s t : Finset α} (ht : t.Nonempty)
Goal: s.card ≤ (s - t).card
-- Finset.image_smul_comm
@[to_additive]
theorem image_smul_comm [DecidableEq β] [DecidableEq γ] [SMul α β] [SMul α γ] (f : β → γ) (a : α)
    (s : Finset β) : (∀ b, f (a • b) = a • f b) → (a • s).image f = a • s.image f 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [DecidableEq β] [DecidableEq γ] [SMul α β] [SMul α γ] (f : β → γ) (a : α) (s : Finset β)
Goal: (∀ (b : β), f (a • b) = a • f b) → Finset.image f (a • s) = a • Finset.image f s
-- Finset.image_vadd_comm
@[to_additive]
theorem image_smul_comm [DecidableEq β] [DecidableEq γ] [SMul α β] [SMul α γ] (f : β → γ) (a : α)
    (s : Finset β) : (∀ b, f (a • b) = a • f b) → (a • s).image f = a • s.image f 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [DecidableEq β] [DecidableEq γ] [VAdd α β] [VAdd α γ] (f : β → γ) (a : α) (s : Finset β)
Goal: (∀ (b : β), f (a +ᵥ b) = a +ᵥ f b) → Finset.image f (a +ᵥ s) = a +ᵥ Finset.image f s
-- Finset.image_vadd_distrib
@[to_additive]
theorem image_smul_distrib [DecidableEq α] [DecidableEq β] [Monoid α] [Monoid β] [FunLike F α β]
    [MonoidHomClass F α β] (f : F) (a : α) (s : Finset α) : (a • s).image f = f a • s.image f 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} [DecidableEq α] [DecidableEq β] [AddMonoid α] [AddMonoid β] [FunLike F α β] [AddMonoidHomClass F α β] (f : F) (a : α) (s : Finset α)
Goal: Finset.image (⇑f) (a +ᵥ s) = f a +ᵥ Finset.image (⇑f) s
-- Finset.image_smul_distrib
@[to_additive]
theorem image_smul_distrib [DecidableEq α] [DecidableEq β] [Monoid α] [Monoid β] [FunLike F α β]
    [MonoidHomClass F α β] (f : F) (a : α) (s : Finset α) : (a • s).image f = f a • s.image f 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} [DecidableEq α] [DecidableEq β] [Monoid α] [Monoid β] [FunLike F α β] [MonoidHomClass F α β] (f : F) (a : α) (s : Finset α)
Goal: Finset.image (⇑f) (a • s) = f a • Finset.image (⇑f) s
-- Finset.smul_mem_smul_finset_iff
@[to_additive (attr := simp)]
theorem smul_mem_smul_finset_iff (a : α) : a • b ∈ a • s ↔ b ∈ s 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [Group α] [MulAction α β] {s : Finset β} {b : β} (a : α)
Goal: a • b ∈ a • s ↔ b ∈ s
-- Finset.vadd_mem_vadd_finset_iff
@[to_additive (attr := simp)]
theorem smul_mem_smul_finset_iff (a : α) : a • b ∈ a • s ↔ b ∈ s 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [AddGroup α] [AddAction α β] {s : Finset β} {b : β} (a : α)
Goal: a +ᵥ b ∈ a +ᵥ s ↔ b ∈ s
-- Finset.inv_smul_mem_iff
@[to_additive]
theorem inv_smul_mem_iff : a⁻¹ • b ∈ s ↔ b ∈ a • s 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [Group α] [MulAction α β] {s : Finset β} {a : α} {b : β}
Goal: a⁻¹ • b ∈ s ↔ b ∈ a • s
-- Finset.neg_vadd_mem_iff
@[to_additive]
theorem inv_smul_mem_iff : a⁻¹ • b ∈ s ↔ b ∈ a • s 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [AddGroup α] [AddAction α β] {s : Finset β} {a : α} {b : β}
Goal: -a +ᵥ b ∈ s ↔ b ∈ a +ᵥ s
-- Finset.mem_neg_vadd_finset_iff
@[to_additive]
theorem mem_inv_smul_finset_iff : b ∈ a⁻¹ • s ↔ a • b ∈ s 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [AddGroup α] [AddAction α β] {s : Finset β} {a : α} {b : β}
Goal: b ∈ -a +ᵥ s ↔ a +ᵥ b ∈ s
-- Finset.mem_inv_smul_finset_iff
@[to_additive]
theorem mem_inv_smul_finset_iff : b ∈ a⁻¹ • s ↔ a • b ∈ s 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [Group α] [MulAction α β] {s : Finset β} {a : α} {b : β}
Goal: b ∈ a⁻¹ • s ↔ a • b ∈ s
-- Finset.vadd_finset_subset_vadd_finset_iff
@[to_additive (attr := simp)]
theorem smul_finset_subset_smul_finset_iff : a • s ⊆ a • t ↔ s ⊆ t 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [AddGroup α] [AddAction α β] {s t : Finset β} {a : α}
Goal: a +ᵥ s ⊆ a +ᵥ t ↔ s ⊆ t
-- Finset.smul_finset_subset_smul_finset_iff
@[to_additive (attr := simp)]
theorem smul_finset_subset_smul_finset_iff : a • s ⊆ a • t ↔ s ⊆ t 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [Group α] [MulAction α β] {s t : Finset β} {a : α}
Goal: a • s ⊆ a • t ↔ s ⊆ t
-- Finset.vadd_finset_subset_iff
@[to_additive]
theorem smul_finset_subset_iff : a • s ⊆ t ↔ s ⊆ a⁻¹ • t 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [AddGroup α] [AddAction α β] {s t : Finset β} {a : α}
Goal: a +ᵥ s ⊆ t ↔ s ⊆ -a +ᵥ t
-- Finset.smul_finset_subset_iff
@[to_additive]
theorem smul_finset_subset_iff : a • s ⊆ t ↔ s ⊆ a⁻¹ • t 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [Group α] [MulAction α β] {s t : Finset β} {a : α}
Goal: a • s ⊆ t ↔ s ⊆ a⁻¹ • t
-- Finset.subset_vadd_finset_iff
@[to_additive]
theorem subset_smul_finset_iff : s ⊆ a • t ↔ a⁻¹ • s ⊆ t 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [AddGroup α] [AddAction α β] {s t : Finset β} {a : α}
Goal: s ⊆ a +ᵥ t ↔ -a +ᵥ s ⊆ t
-- Finset.subset_smul_finset_iff
@[to_additive]
theorem subset_smul_finset_iff : s ⊆ a • t ↔ a⁻¹ • s ⊆ t 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [Group α] [MulAction α β] {s t : Finset β} {a : α}
Goal: s ⊆ a • t ↔ a⁻¹ • s ⊆ t
-- Finset.vadd_finset_inter
@[to_additive]
theorem smul_finset_inter : a • (s ∩ t) = a • s ∩ a • t 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [AddGroup α] [AddAction α β] {s t : Finset β} {a : α}
Goal: a +ᵥ s ∩ t = (a +ᵥ s) ∩ (a +ᵥ t)
-- Finset.smul_finset_inter
@[to_additive]
theorem smul_finset_inter : a • (s ∩ t) = a • s ∩ a • t 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [Group α] [MulAction α β] {s t : Finset β} {a : α}
Goal: a • (s ∩ t) = a • s ∩ a • t
-- Finset.vadd_finset_sdiff
@[to_additive]
theorem smul_finset_sdiff : a • (s \ t) = a • s \ a • t 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [AddGroup α] [AddAction α β] {s t : Finset β} {a : α}
Goal: a +ᵥ s \ t = (a +ᵥ s) \ (a +ᵥ t)
-- Finset.smul_finset_sdiff
@[to_additive]
theorem smul_finset_sdiff : a • (s \ t) = a • s \ a • t 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [Group α] [MulAction α β] {s t : Finset β} {a : α}
Goal: a • (s \ t) = a • s \ a • t
-- Finset.smul_finset_symmDiff
@[to_additive]
theorem smul_finset_symmDiff : a • s ∆ t = (a • s) ∆ (a • t) 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [Group α] [MulAction α β] {s t : Finset β} {a : α}
Goal: a • symmDiff s t = symmDiff (a • s) (a • t)
-- Finset.vadd_finset_symmDiff
@[to_additive]
theorem smul_finset_symmDiff : a • s ∆ t = (a • s) ∆ (a • t) 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [AddGroup α] [AddAction α β] {s t : Finset β} {a : α}
Goal: a +ᵥ symmDiff s t = symmDiff (a +ᵥ s) (a +ᵥ t)
-- Finset.smul_finset_univ
@[to_additive (attr := simp)]
theorem smul_finset_univ [Fintype β] : a • (univ : Finset β) = univ 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [Group α] [MulAction α β] {a : α} [Fintype β]
Goal: a • Finset.univ = Finset.univ
-- Finset.vadd_finset_univ
@[to_additive (attr := simp)]
theorem smul_finset_univ [Fintype β] : a • (univ : Finset β) = univ 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [AddGroup α] [AddAction α β] {a : α} [Fintype β]
Goal: a +ᵥ Finset.univ = Finset.univ
-- Finset.vadd_univ
@[to_additive (attr := simp)]
theorem smul_univ [Fintype β] {s : Finset α} (hs : s.Nonempty) : s • (univ : Finset β) = univ 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [AddGroup α] [AddAction α β] [Fintype β] {s : Finset α} (hs : s.Nonempty)
Goal: s +ᵥ Finset.univ = Finset.univ
-- Finset.smul_univ
@[to_additive (attr := simp)]
theorem smul_univ [Fintype β] {s : Finset α} (hs : s.Nonempty) : s • (univ : Finset β) = univ 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [Group α] [MulAction α β] [Fintype β] {s : Finset α} (hs : s.Nonempty)
Goal: s • Finset.univ = Finset.univ
-- Finset.card_smul_finset
@[to_additive (attr := simp)]
theorem card_smul_finset (a : α) (s : Finset β) : (a • s).card = s.card 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [Group α] [MulAction α β] (a : α) (s : Finset β)
Goal: (a • s).card = s.card
-- Finset.card_vadd_finset
@[to_additive (attr := simp)]
theorem card_smul_finset (a : α) (s : Finset β) : (a • s).card = s.card 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [AddGroup α] [AddAction α β] (a : α) (s : Finset β)
Goal: (a +ᵥ s).card = s.card
-- Finset.card_dvd_card_smul_right
@[to_additive "If the left cosets of `t` by elements of `s` are disjoint (but not necessarily
distinct!), then the size of `t` divides the size of `s +ᵥ t`."]
theorem card_dvd_card_smul_right {s : Finset α} :
    ((· • t) '' (s : Set α)).PairwiseDisjoint id → t.card ∣ (s • t).card 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [Group α] [MulAction α β] {t : Finset β} {s : Finset α}
Goal: ((fun x => x • t) '' ↑s).PairwiseDisjoint id → t.card ∣ (s • t).card
-- Finset.card_dvd_card_vadd_right
@[to_additive "If the left cosets of `t` by elements of `s` are disjoint (but not necessarily
distinct!), then the size of `t` divides the size of `s +ᵥ t`."]
theorem card_dvd_card_smul_right {s : Finset α} :
    ((· • t) '' (s : Set α)).PairwiseDisjoint id → t.card ∣ (s • t).card 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [AddGroup α] [AddAction α β] {t : Finset β} {s : Finset α}
Goal: ((fun x => x +ᵥ t) '' ↑s).PairwiseDisjoint id → t.card ∣ (s +ᵥ t).card
-- Finset.card_dvd_card_add_left
@[to_additive "If the right cosets of `s` by elements of `t` are disjoint (but not necessarily
distinct!), then the size of `s` divides the size of `s + t`."]
theorem card_dvd_card_mul_left {s t : Finset α} :
    ((fun b => s.image fun a => a * b) '' (t : Set α)).PairwiseDisjoint id →
      s.card ∣ (s * t).card 
Arguments: {α : Type u_2} [AddGroup α] [DecidableEq α] {s t : Finset α}
Goal: ((fun b => Finset.image (fun a => a + b) s) '' ↑t).PairwiseDisjoint id → s.card ∣ (s + t).card
-- Finset.card_dvd_card_mul_left
@[to_additive "If the right cosets of `s` by elements of `t` are disjoint (but not necessarily
distinct!), then the size of `s` divides the size of `s + t`."]
theorem card_dvd_card_mul_left {s t : Finset α} :
    ((fun b => s.image fun a => a * b) '' (t : Set α)).PairwiseDisjoint id →
      s.card ∣ (s * t).card 
Arguments: {α : Type u_2} [Group α] [DecidableEq α] {s t : Finset α}
Goal: ((fun b => Finset.image (fun a => a * b) s) '' ↑t).PairwiseDisjoint id → s.card ∣ (s * t).card
-- Finset.card_dvd_card_mul_right
@[to_additive "If the left cosets of `t` by elements of `s` are disjoint (but not necessarily
distinct!), then the size of `t` divides the size of `s + t`."]
theorem card_dvd_card_mul_right {s t : Finset α} :
    ((· • t) '' (s : Set α)).PairwiseDisjoint id → t.card ∣ (s * t).card 
Arguments: {α : Type u_2} [Group α] [DecidableEq α] {s t : Finset α}
Goal: ((fun x => x • t) '' ↑s).PairwiseDisjoint id → t.card ∣ (s * t).card
-- Finset.card_dvd_card_add_right
@[to_additive "If the left cosets of `t` by elements of `s` are disjoint (but not necessarily
distinct!), then the size of `t` divides the size of `s + t`."]
theorem card_dvd_card_mul_right {s t : Finset α} :
    ((· • t) '' (s : Set α)).PairwiseDisjoint id → t.card ∣ (s * t).card 
Arguments: {α : Type u_2} [AddGroup α] [DecidableEq α] {s t : Finset α}
Goal: ((fun x => x +ᵥ t) '' ↑s).PairwiseDisjoint id → t.card ∣ (s + t).card
-- Finset.inv_smul_finset_distrib
@[to_additive (attr := simp)]
lemma inv_smul_finset_distrib (a : α) (s : Finset α) : (a • s)⁻¹ = op a⁻¹ • s⁻¹ 
Arguments: {α : Type u_2} [Group α] [DecidableEq α] (a : α) (s : Finset α)
Goal: (a • s)⁻¹ = MulOpposite.op a⁻¹ • s⁻¹
-- Finset.neg_vadd_finset_distrib
@[to_additive (attr := simp)]
lemma inv_smul_finset_distrib (a : α) (s : Finset α) : (a • s)⁻¹ = op a⁻¹ • s⁻¹ 
Arguments: {α : Type u_2} [AddGroup α] [DecidableEq α] (a : α) (s : Finset α)
Goal: -(a +ᵥ s) = AddOpposite.op (-a) +ᵥ -s
-- Finset.inv_op_smul_finset_distrib
@[to_additive (attr := simp)]
lemma inv_op_smul_finset_distrib (a : α) (s : Finset α) : (op a • s)⁻¹ = a⁻¹ • s⁻¹ 
Arguments: {α : Type u_2} [Group α] [DecidableEq α] (a : α) (s : Finset α)
Goal: (MulOpposite.op a • s)⁻¹ = a⁻¹ • s⁻¹
-- Finset.neg_op_vadd_finset_distrib
@[to_additive (attr := simp)]
lemma inv_op_smul_finset_distrib (a : α) (s : Finset α) : (op a • s)⁻¹ = a⁻¹ • s⁻¹ 
Arguments: {α : Type u_2} [AddGroup α] [DecidableEq α] (a : α) (s : Finset α)
Goal: -(AddOpposite.op a +ᵥ s) = -a +ᵥ -s
-- Finset.smul_zero_subset
theorem smul_zero_subset (s : Finset α) : s • (0 : Finset β) ⊆ 0 
Arguments: {α : Type u_2} {β : Type u_3} [Zero β] [SMulZeroClass α β] [DecidableEq β] (s : Finset α)
Goal: s • 0 ⊆ 0
-- Finset.Nonempty.smul_zero
theorem Nonempty.smul_zero (hs : s.Nonempty) : s • (0 : Finset β) = 0 
Arguments: {α : Type u_2} {β : Type u_3} [Zero β] [SMulZeroClass α β] [DecidableEq β] {s : Finset α} (hs : s.Nonempty)
Goal: s • 0 = 0
-- Finset.zero_mem_smul_finset
theorem zero_mem_smul_finset (h : (0 : β) ∈ t) : (0 : β) ∈ a • t 
Arguments: {α : Type u_2} {β : Type u_3} [Zero β] [SMulZeroClass α β] [DecidableEq β] {t : Finset β} {a : α} (h : 0 ∈ t)
Goal: 0 ∈ a • t
-- Finset.zero_mem_smul_finset_iff
theorem zero_mem_smul_finset_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] [DecidableEq β] {t : Finset β} {a : α} [Zero α] [NoZeroSMulDivisors α β] (ha : a ≠ 0)
Goal: 0 ∈ a • t ↔ 0 ∈ t
-- Finset.zero_smul_subset
lemma zero_smul_subset (t : Finset β) : (0 : Finset α) • t ⊆ 0 
Arguments: {α : Type u_2} {β : Type u_3} [Zero α] [Zero β] [SMulWithZero α β] [DecidableEq β] (t : Finset β)
Goal: 0 • t ⊆ 0
-- Finset.Nonempty.zero_smul
lemma Nonempty.zero_smul (ht : t.Nonempty) : (0 : Finset α) • t = 0 
Arguments: {α : Type u_2} {β : Type u_3} [Zero α] [Zero β] [SMulWithZero α β] [DecidableEq β] {t : Finset β} (ht : t.Nonempty)
Goal: 0 • t = 0
-- Finset.zero_smul_finset
@[simp] lemma zero_smul_finset {s : Finset β} (h : s.Nonempty) : (0 : α) • s = (0 : Finset β) 
Arguments: {α : Type u_2} {β : Type u_3} [Zero α] [Zero β] [SMulWithZero α β] [DecidableEq β] {s : Finset β} (h : s.Nonempty)
Goal: 0 • s = 0
-- Finset.zero_smul_finset_subset
lemma zero_smul_finset_subset (s : Finset β) : (0 : α) • s ⊆ 0 
Arguments: {α : Type u_2} {β : Type u_3} [Zero α] [Zero β] [SMulWithZero α β] [DecidableEq β] (s : Finset β)
Goal: 0 • s ⊆ 0
-- Finset.zero_mem_smul_iff
lemma 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] [DecidableEq β] {s : Finset α} {t : Finset β} [NoZeroSMulDivisors α β]
Goal: 0 ∈ s • t ↔ 0 ∈ s ∧ t.Nonempty ∨ 0 ∈ t ∧ s.Nonempty
-- Finset.smul_mem_smul_finset_iff₀
@[simp]
theorem smul_mem_smul_finset_iff₀ (ha : a ≠ 0) : a • b ∈ a • s ↔ b ∈ s 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [GroupWithZero α] [MulAction α β] {s : Finset β} {a : α} {b : β} (ha : a ≠ 0)
Goal: a • b ∈ a • s ↔ b ∈ s
-- Finset.inv_smul_mem_iff₀
theorem inv_smul_mem_iff₀ (ha : a ≠ 0) : a⁻¹ • b ∈ s ↔ b ∈ a • s 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [GroupWithZero α] [MulAction α β] {s : Finset β} {a : α} {b : β} (ha : a ≠ 0)
Goal: a⁻¹ • b ∈ s ↔ b ∈ a • s
-- Finset.mem_inv_smul_finset_iff₀
theorem mem_inv_smul_finset_iff₀ (ha : a ≠ 0) : b ∈ a⁻¹ • s ↔ a • b ∈ s 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [GroupWithZero α] [MulAction α β] {s : Finset β} {a : α} {b : β} (ha : a ≠ 0)
Goal: b ∈ a⁻¹ • s ↔ a • b ∈ s
-- Finset.smul_finset_subset_smul_finset_iff₀
@[simp]
theorem smul_finset_subset_smul_finset_iff₀ (ha : a ≠ 0) : a • s ⊆ a • t ↔ s ⊆ t 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [GroupWithZero α] [MulAction α β] {s t : Finset β} {a : α} (ha : a ≠ 0)
Goal: a • s ⊆ a • t ↔ s ⊆ t
-- Finset.smul_finset_subset_iff₀
theorem smul_finset_subset_iff₀ (ha : a ≠ 0) : a • s ⊆ t ↔ s ⊆ a⁻¹ • t 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [GroupWithZero α] [MulAction α β] {s t : Finset β} {a : α} (ha : a ≠ 0)
Goal: a • s ⊆ t ↔ s ⊆ a⁻¹ • t
-- Finset.subset_smul_finset_iff₀
theorem subset_smul_finset_iff₀ (ha : a ≠ 0) : s ⊆ a • t ↔ a⁻¹ • s ⊆ t 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [GroupWithZero α] [MulAction α β] {s t : Finset β} {a : α} (ha : a ≠ 0)
Goal: s ⊆ a • t ↔ a⁻¹ • s ⊆ t
-- Finset.smul_finset_inter₀
theorem smul_finset_inter₀ (ha : a ≠ 0) : a • (s ∩ t) = a • s ∩ a • t 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [GroupWithZero α] [MulAction α β] {s t : Finset β} {a : α} (ha : a ≠ 0)
Goal: a • (s ∩ t) = a • s ∩ a • t
-- Finset.smul_finset_sdiff₀
theorem smul_finset_sdiff₀ (ha : a ≠ 0) : a • (s \ t) = a • s \ a • t 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [GroupWithZero α] [MulAction α β] {s t : Finset β} {a : α} (ha : a ≠ 0)
Goal: a • (s \ t) = a • s \ a • t
-- Finset.smul_finset_symmDiff₀
theorem smul_finset_symmDiff₀ (ha : a ≠ 0) : a • s ∆ t = (a • s) ∆ (a • t) 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [GroupWithZero α] [MulAction α β] {s t : Finset β} {a : α} (ha : a ≠ 0)
Goal: a • symmDiff s t = symmDiff (a • s) (a • t)
-- Finset.smul_finset_univ₀
lemma smul_finset_univ₀ [Fintype β] (ha : a ≠ 0) : a • (univ : Finset β) = univ 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [GroupWithZero α] [MulAction α β] {a : α} [Fintype β] (ha : a ≠ 0)
Goal: a • Finset.univ = Finset.univ
-- Finset.smul_univ₀
theorem smul_univ₀ [Fintype β] {s : Finset α} (hs : ¬s ⊆ 0) : s • (univ : Finset β) = univ 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [GroupWithZero α] [MulAction α β] [Fintype β] {s : Finset α} (hs : ¬s ⊆ 0)
Goal: s • Finset.univ = Finset.univ
-- Finset.smul_univ₀'
lemma smul_univ₀' [Fintype β] {s : Finset α} (hs : s.Nontrivial) : s • (univ : Finset β) = univ 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [GroupWithZero α] [MulAction α β] [Fintype β] {s : Finset α} (hs : s.Nontrivial)
Goal: s • Finset.univ = Finset.univ
-- Finset.inv_zero
@[simp] protected lemma inv_zero : (0 : Finset α)⁻¹ = 0 
Arguments: {α : Type u_2} [GroupWithZero α] [DecidableEq α]
Goal: 0⁻¹ = 0
-- Finset.inv_smul_finset_distrib₀
@[simp] lemma inv_smul_finset_distrib₀ (a : α) (s : Finset α) : (a • s)⁻¹ = op a⁻¹ • s⁻¹ 
Arguments: {α : Type u_2} [GroupWithZero α] [DecidableEq α] (a : α) (s : Finset α)
Goal: (a • s)⁻¹ = MulOpposite.op a⁻¹ • s⁻¹
-- Finset.inv_op_smul_finset_distrib₀
@[simp] lemma inv_op_smul_finset_distrib₀ (a : α) (s : Finset α) : (op a • s)⁻¹ = a⁻¹ • s⁻¹ 
Arguments: {α : Type u_2} [GroupWithZero α] [DecidableEq α] (a : α) (s : Finset α)
Goal: (MulOpposite.op a • s)⁻¹ = a⁻¹ • s⁻¹
-- Finset.smul_finset_neg
@[simp]
theorem smul_finset_neg : a • -t = -(a • t) 
Arguments: {α : Type u_2} {β : Type u_3} [Monoid α] [AddGroup β] [DistribMulAction α β] [DecidableEq β] (a : α) (t : Finset β)
Goal: a • -t = -(a • t)
-- Finset.smul_neg
@[simp]
protected theorem smul_neg : s • -t = -(s • t) 
Arguments: {α : Type u_2} {β : Type u_3} [Monoid α] [AddGroup β] [DistribMulAction α β] [DecidableEq β] (s : Finset α) (t : Finset β)
Goal: s • -t = -(s • t)
-- Finset.neg_smul_finset
@[simp]
theorem neg_smul_finset : -a • t = -(a • t) 
Arguments: {α : Type u_2} {β : Type u_3} [Ring α] [AddCommGroup β] [Module α β] [DecidableEq β] {t : Finset β} {a : α}
Goal: -a • t = -(a • t)
-- Finset.neg_smul
@[simp]
protected theorem neg_smul [DecidableEq α] : -s • t = -(s • t) 
Arguments: {α : Type u_2} {β : Type u_3} [Ring α] [AddCommGroup β] [Module α β] [DecidableEq β] {s : Finset α} {t : Finset β} [DecidableEq α]
Goal: -s • t = -(s • t)
-- Finset.sum_neg_index
@[to_additive (attr := simp)] lemma prod_inv_index [InvolutiveInv ι] (s : Finset ι) (f : ι → α) :
    ∏ i ∈ s⁻¹, f i = ∏ i ∈ s, f i⁻¹ 
Arguments: {α : Type u_2} [AddCommMonoid α] {ι : Type u_5} [DecidableEq ι] [InvolutiveNeg ι] (s : Finset ι) (f : ι → α)
Goal: ∑ i ∈ -s, f i = ∑ i ∈ s, f (-i)
-- Finset.prod_inv_index
@[to_additive (attr := simp)] lemma prod_inv_index [InvolutiveInv ι] (s : Finset ι) (f : ι → α) :
    ∏ i ∈ s⁻¹, f i = ∏ i ∈ s, f i⁻¹ 
Arguments: {α : Type u_2} [CommMonoid α] {ι : Type u_5} [DecidableEq ι] [InvolutiveInv ι] (s : Finset ι) (f : ι → α)
Goal: ∏ i ∈ s⁻¹, f i = ∏ i ∈ s, f i⁻¹
-- Finset.prod_neg_index
@[to_additive existing, simp] lemma prod_neg_index [InvolutiveNeg ι] (s : Finset ι) (f : ι → α) :
    ∏ i ∈ -s, f i = ∏ i ∈ s, f (-i) 
Arguments: {α : Type u_2} [CommMonoid α] {ι : Type u_5} [DecidableEq ι] [InvolutiveNeg ι] (s : Finset ι) (f : ι → α)
Goal: ∏ i ∈ -s, f i = ∏ i ∈ s, f (-i)
-- Finset.sum_inv_index
@[to_additive existing, simp] lemma sum_inv_index [InvolutiveInv ι] (s : Finset ι) (f : ι → α) :
    ∑ i ∈ s⁻¹, f i = ∑ i ∈ s, f i⁻¹ 
Arguments: {α : Type u_2} [AddCommMonoid α] {ι : Type u_5} [DecidableEq ι] [InvolutiveInv ι] (s : Finset ι) (f : ι → α)
Goal: ∑ i ∈ s⁻¹, f i = ∑ i ∈ s, f i⁻¹
-- Fintype.piFinset_mul
@[to_additive]
lemma piFinset_mul [∀ i, Mul (α i)] (s t : ∀ i, Finset (α i)) :
    piFinset (fun i ↦ s i * t i) = piFinset s * piFinset t 
Arguments: {ι : Type u_5} {α : ι → Type u_6} [Fintype ι] [DecidableEq ι] [(i : ι) → DecidableEq (α i)] [(i : ι) → Mul (α i)] (s t : (i : ι) → Finset (α i))
Goal: (Fintype.piFinset fun i => s i * t i) = Fintype.piFinset s * Fintype.piFinset t
-- Fintype.piFinset_add
@[to_additive]
lemma piFinset_mul [∀ i, Mul (α i)] (s t : ∀ i, Finset (α i)) :
    piFinset (fun i ↦ s i * t i) = piFinset s * piFinset t 
Arguments: {ι : Type u_5} {α : ι → Type u_6} [Fintype ι] [DecidableEq ι] [(i : ι) → DecidableEq (α i)] [(i : ι) → Add (α i)] (s t : (i : ι) → Finset (α i))
Goal: (Fintype.piFinset fun i => s i + t i) = Fintype.piFinset s + Fintype.piFinset t
-- Fintype.piFinset_sub
@[to_additive]
lemma piFinset_div [∀ i, Div (α i)] (s t : ∀ i, Finset (α i)) :
    piFinset (fun i ↦ s i / t i) = piFinset s / piFinset t 
Arguments: {ι : Type u_5} {α : ι → Type u_6} [Fintype ι] [DecidableEq ι] [(i : ι) → DecidableEq (α i)] [(i : ι) → Sub (α i)] (s t : (i : ι) → Finset (α i))
Goal: (Fintype.piFinset fun i => s i - t i) = Fintype.piFinset s - Fintype.piFinset t
-- Fintype.piFinset_div
@[to_additive]
lemma piFinset_div [∀ i, Div (α i)] (s t : ∀ i, Finset (α i)) :
    piFinset (fun i ↦ s i / t i) = piFinset s / piFinset t 
Arguments: {ι : Type u_5} {α : ι → Type u_6} [Fintype ι] [DecidableEq ι] [(i : ι) → DecidableEq (α i)] [(i : ι) → Div (α i)] (s t : (i : ι) → Finset (α i))
Goal: (Fintype.piFinset fun i => s i / t i) = Fintype.piFinset s / Fintype.piFinset t
-- Fintype.piFinset_neg
@[to_additive (attr := simp)]
lemma piFinset_inv [∀ i, Inv (α i)] (s : ∀ i, Finset (α i)) :
    piFinset (fun i ↦ (s i)⁻¹) = (piFinset s)⁻¹ 
Arguments: {ι : Type u_5} {α : ι → Type u_6} [Fintype ι] [DecidableEq ι] [(i : ι) → DecidableEq (α i)] [(i : ι) → Neg (α i)] (s : (i : ι) → Finset (α i))
Goal: (Fintype.piFinset fun i => -s i) = -Fintype.piFinset s