ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Finset.smul_union
@[to_additive]
theorem smul_union : s • (t₁ ∪ t₂) = s • t₁ ∪ s • t₂ 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [SMul α β] {s : Finset α} {t₁ t₂ : Finset β}
Goal: s • (t₁ ∪ t₂) = s • t₁ ∪ s • t₂
-- Finset.inter_vadd_subset
@[to_additive]
theorem inter_smul_subset [DecidableEq α] : (s₁ ∩ s₂) • t ⊆ s₁ • t ∩ s₂ • t 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [VAdd α β] {s₁ s₂ : Finset α} {t : Finset β} [DecidableEq α]
Goal: s₁ ∩ s₂ +ᵥ t ⊆ (s₁ +ᵥ t) ∩ (s₂ +ᵥ t)
-- Finset.inter_smul_subset
@[to_additive]
theorem inter_smul_subset [DecidableEq α] : (s₁ ∩ s₂) • t ⊆ s₁ • t ∩ s₂ • t 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [SMul α β] {s₁ s₂ : Finset α} {t : Finset β} [DecidableEq α]
Goal: (s₁ ∩ s₂) • t ⊆ s₁ • t ∩ s₂ • t
-- Finset.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} [DecidableEq β] [VAdd α β] {s : Finset α} {t₁ t₂ : Finset β}
Goal: s +ᵥ t₁ ∩ t₂ ⊆ (s +ᵥ t₁) ∩ (s +ᵥ t₂)
-- Finset.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} [DecidableEq β] [SMul α β] {s : Finset α} {t₁ t₂ : Finset β}
Goal: s • (t₁ ∩ t₂) ⊆ s • t₁ ∩ s • t₂
-- Finset.inter_vadd_union_subset_union
@[to_additive]
theorem inter_smul_union_subset_union [DecidableEq α] : (s₁ ∩ s₂) • (t₁ ∪ t₂) ⊆ s₁ • t₁ ∪ s₂ • t₂ 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [VAdd α β] {s₁ s₂ : Finset α} {t₁ t₂ : Finset β} [DecidableEq α]
Goal: s₁ ∩ s₂ +ᵥ (t₁ ∪ t₂) ⊆ s₁ +ᵥ t₁ ∪ (s₂ +ᵥ t₂)
-- Finset.inter_smul_union_subset_union
@[to_additive]
theorem inter_smul_union_subset_union [DecidableEq α] : (s₁ ∩ s₂) • (t₁ ∪ t₂) ⊆ s₁ • t₁ ∪ s₂ • t₂ 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [SMul α β] {s₁ s₂ : Finset α} {t₁ t₂ : Finset β} [DecidableEq α]
Goal: (s₁ ∩ s₂) • (t₁ ∪ t₂) ⊆ s₁ • t₁ ∪ s₂ • t₂
-- Finset.union_vadd_inter_subset_union
@[to_additive]
theorem union_smul_inter_subset_union [DecidableEq α] : (s₁ ∪ s₂) • (t₁ ∩ t₂) ⊆ s₁ • t₁ ∪ s₂ • t₂ 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [VAdd α β] {s₁ s₂ : Finset α} {t₁ t₂ : Finset β} [DecidableEq α]
Goal: s₁ ∪ s₂ +ᵥ t₁ ∩ t₂ ⊆ s₁ +ᵥ t₁ ∪ (s₂ +ᵥ t₂)
-- Finset.union_smul_inter_subset_union
@[to_additive]
theorem union_smul_inter_subset_union [DecidableEq α] : (s₁ ∪ s₂) • (t₁ ∩ t₂) ⊆ s₁ • t₁ ∪ s₂ • t₂ 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [SMul α β] {s₁ s₂ : Finset α} {t₁ t₂ : Finset β} [DecidableEq α]
Goal: (s₁ ∪ s₂) • (t₁ ∩ t₂) ⊆ s₁ • t₁ ∪ s₂ • t₂
-- Finset.subset_vadd
@[to_additive
      "If a finset `u` is contained in the scalar sum of two sets `s +ᵥ t`, we can find two
      finsets `s'`, `t'` such that `s' ⊆ s`, `t' ⊆ t` and `u ⊆ s' +ᵥ t'`."]
theorem subset_smul {s : Set α} {t : Set β} :
    ↑u ⊆ s • t → ∃ (s' : Finset α) (t' : Finset β), ↑s' ⊆ s ∧ ↑t' ⊆ t ∧ u ⊆ s' • t' 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [VAdd α β] {u : Finset β} {s : Set α} {t : Set β}
Goal: ↑u ⊆ s +ᵥ t → ∃ s' t', ↑s' ⊆ s ∧ ↑t' ⊆ t ∧ u ⊆ s' +ᵥ t'
-- Finset.subset_smul
@[to_additive
      "If a finset `u` is contained in the scalar sum of two sets `s +ᵥ t`, we can find two
      finsets `s'`, `t'` such that `s' ⊆ s`, `t' ⊆ t` and `u ⊆ s' +ᵥ t'`."]
theorem subset_smul {s : Set α} {t : Set β} :
    ↑u ⊆ s • t → ∃ (s' : Finset α) (t' : Finset β), ↑s' ⊆ s ∧ ↑t' ⊆ t ∧ u ⊆ s' • t' 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [SMul α β] {u : Finset β} {s : Set α} {t : Set β}
Goal: ↑u ⊆ s • t → ∃ s' t', ↑s' ⊆ s ∧ ↑t' ⊆ t ∧ u ⊆ s' • t'
-- Finset.vsub_def
theorem vsub_def : s -ᵥ t = image₂ (· -ᵥ ·) s t 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] [DecidableEq α] {s t : Finset β}
Goal: s -ᵥ t = Finset.image₂ (fun x x_1 => x -ᵥ x_1) s t
-- Finset.image_vsub_product
@[simp]
theorem image_vsub_product : image₂ (· -ᵥ ·) s t = s -ᵥ t 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] [DecidableEq α] {s t : Finset β}
Goal: Finset.image₂ (fun x x_1 => x -ᵥ x_1) s t = s -ᵥ t
-- Finset.mem_vsub
theorem mem_vsub : a ∈ s -ᵥ t ↔ ∃ b ∈ s, ∃ c ∈ t, b -ᵥ c = a 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] [DecidableEq α] {s t : Finset β} {a : α}
Goal: a ∈ s -ᵥ t ↔ ∃ b ∈ s, ∃ c ∈ t, b -ᵥ c = a
-- Finset.coe_vsub
@[simp, norm_cast]
theorem coe_vsub (s t : Finset β) : (↑(s -ᵥ t) : Set α) = (s : Set β) -ᵥ t 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] [DecidableEq α] (s t : Finset β)
Goal: ↑(s -ᵥ t) = ↑s -ᵥ ↑t
-- Finset.vsub_mem_vsub
theorem vsub_mem_vsub : b ∈ s → c ∈ t → b -ᵥ c ∈ s -ᵥ t 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] [DecidableEq α] {s t : Finset β} {b c : β}
Goal: b ∈ s → c ∈ t → b -ᵥ c ∈ s -ᵥ t
-- Finset.vsub_card_le
theorem vsub_card_le : (s -ᵥ t : Finset α).card ≤ s.card * t.card 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] [DecidableEq α] {s t : Finset β}
Goal: (s -ᵥ t).card ≤ s.card * t.card
-- Finset.empty_vsub
@[simp]
theorem empty_vsub (t : Finset β) : (∅ : Finset β) -ᵥ t = ∅ 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] [DecidableEq α] (t : Finset β)
Goal: ∅ -ᵥ t = ∅
-- Finset.vsub_empty
@[simp]
theorem vsub_empty (s : Finset β) : s -ᵥ (∅ : Finset β) = ∅ 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] [DecidableEq α] (s : Finset β)
Goal: s -ᵥ ∅ = ∅
-- Finset.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] [DecidableEq α] {s t : Finset β}
Goal: s -ᵥ t = ∅ ↔ s = ∅ ∨ t = ∅
-- Finset.vsub_nonempty
@[simp, aesop safe apply (rule_sets := [finsetNonempty])]
theorem vsub_nonempty : (s -ᵥ t : Finset α).Nonempty ↔ s.Nonempty ∧ t.Nonempty 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] [DecidableEq α] {s t : Finset β}
Goal: (s -ᵥ t).Nonempty ↔ s.Nonempty ∧ t.Nonempty
-- Finset.Nonempty.vsub
theorem Nonempty.vsub : s.Nonempty → t.Nonempty → (s -ᵥ t : Finset α).Nonempty 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] [DecidableEq α] {s t : Finset β}
Goal: s.Nonempty → t.Nonempty → (s -ᵥ t).Nonempty
-- Finset.Nonempty.of_vsub_left
theorem Nonempty.of_vsub_left : (s -ᵥ t : Finset α).Nonempty → s.Nonempty 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] [DecidableEq α] {s t : Finset β}
Goal: (s -ᵥ t).Nonempty → s.Nonempty
-- Finset.Nonempty.of_vsub_right
theorem Nonempty.of_vsub_right : (s -ᵥ t : Finset α).Nonempty → t.Nonempty 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] [DecidableEq α] {s t : Finset β}
Goal: (s -ᵥ t).Nonempty → t.Nonempty
-- Finset.vsub_singleton
@[simp]
theorem vsub_singleton (b : β) : s -ᵥ ({b} : Finset β) = s.image (· -ᵥ b) 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] [DecidableEq α] {s : Finset β} (b : β)
Goal: s -ᵥ {b} = Finset.image (fun x => x -ᵥ b) s
-- Finset.singleton_vsub
theorem singleton_vsub (a : β) : ({a} : Finset β) -ᵥ t = t.image (a -ᵥ ·) 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] [DecidableEq α] {t : Finset β} (a : β)
Goal: {a} -ᵥ t = Finset.image (fun x => a -ᵥ x) t
-- Finset.singleton_vsub_singleton
theorem singleton_vsub_singleton (a b : β) : ({a} : Finset β) -ᵥ {b} = {a -ᵥ b} 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] [DecidableEq α] (a b : β)
Goal: {a} -ᵥ {b} = {a -ᵥ b}
-- Finset.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] [DecidableEq α] {s₁ s₂ t₁ t₂ : Finset β}
Goal: s₁ ⊆ s₂ → t₁ ⊆ t₂ → s₁ -ᵥ t₁ ⊆ s₂ -ᵥ t₂
-- Finset.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] [DecidableEq α] {s t₁ t₂ : Finset β}
Goal: t₁ ⊆ t₂ → s -ᵥ t₁ ⊆ s -ᵥ t₂
-- Finset.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] [DecidableEq α] {s₁ s₂ t : Finset β}
Goal: s₁ ⊆ s₂ → s₁ -ᵥ t ⊆ s₂ -ᵥ t
-- Finset.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] [DecidableEq α] {s t : Finset β} {u : Finset α}
Goal: s -ᵥ t ⊆ u ↔ ∀ x ∈ s, ∀ y ∈ t, x -ᵥ y ∈ u
-- Finset.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] [DecidableEq α] {s₁ s₂ t : Finset β} [DecidableEq β]
Goal: s₁ ∪ s₂ -ᵥ t = s₁ -ᵥ t ∪ (s₂ -ᵥ t)
-- Finset.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] [DecidableEq α] {s t₁ t₂ : Finset β} [DecidableEq β]
Goal: s -ᵥ (t₁ ∪ t₂) = s -ᵥ t₁ ∪ (s -ᵥ t₂)
-- Finset.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] [DecidableEq α] {s₁ s₂ t : Finset β} [DecidableEq β]
Goal: s₁ ∩ s₂ -ᵥ t ⊆ (s₁ -ᵥ t) ∩ (s₂ -ᵥ t)
-- Finset.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] [DecidableEq α] {s t₁ t₂ : Finset β} [DecidableEq β]
Goal: s -ᵥ t₁ ∩ t₂ ⊆ (s -ᵥ t₁) ∩ (s -ᵥ t₂)
-- Finset.subset_vsub
theorem subset_vsub {s t : Set β} :
    ↑u ⊆ s -ᵥ t → ∃ s' t' : Finset β, ↑s' ⊆ s ∧ ↑t' ⊆ t ∧ u ⊆ s' -ᵥ t' 
Arguments: {α : Type u_2} {β : Type u_3} [VSub α β] [DecidableEq α] {u : Finset α} {s t : Set β}
Goal: ↑u ⊆ s -ᵥ t → ∃ s' t', ↑s' ⊆ s ∧ ↑t' ⊆ t ∧ u ⊆ s' -ᵥ t'
-- Finset.smul_finset_def
@[to_additive]
theorem smul_finset_def : a • s = s.image (a • ·) 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [SMul α β] {s : Finset β} {a : α}
Goal: a • s = Finset.image (fun x => a • x) s
-- Finset.vadd_finset_def
@[to_additive]
theorem smul_finset_def : a • s = s.image (a • ·) 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [VAdd α β] {s : Finset β} {a : α}
Goal: a +ᵥ s = Finset.image (fun x => a +ᵥ x) s
-- Finset.image_vadd
@[to_additive]
theorem image_smul : (s.image fun x => a • x) = a • s 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [VAdd α β] {s : Finset β} {a : α}
Goal: Finset.image (fun x => a +ᵥ x) s = a +ᵥ s
-- Finset.image_smul
@[to_additive]
theorem image_smul : (s.image fun x => a • x) = a • s 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [SMul α β] {s : Finset β} {a : α}
Goal: Finset.image (fun x => a • x) s = a • s
-- Finset.mem_vadd_finset
@[to_additive]
theorem mem_smul_finset {x : β} : x ∈ a • s ↔ ∃ y, y ∈ s ∧ a • y = x 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [VAdd α β] {s : Finset β} {a : α} {x : β}
Goal: x ∈ a +ᵥ s ↔ ∃ y ∈ s, a +ᵥ y = x
-- Finset.mem_smul_finset
@[to_additive]
theorem mem_smul_finset {x : β} : x ∈ a • s ↔ ∃ y, y ∈ s ∧ a • y = x 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [SMul α β] {s : Finset β} {a : α} {x : β}
Goal: x ∈ a • s ↔ ∃ y ∈ s, a • y = x
-- Finset.coe_smul_finset
@[to_additive (attr := simp, norm_cast)]
theorem coe_smul_finset (a : α) (s : Finset β) : ↑(a • s) = a • (↑s : Set β) 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [SMul α β] (a : α) (s : Finset β)
Goal: ↑(a • s) = a • ↑s
-- Finset.coe_vadd_finset
@[to_additive (attr := simp, norm_cast)]
theorem coe_smul_finset (a : α) (s : Finset β) : ↑(a • s) = a • (↑s : Set β) 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [VAdd α β] (a : α) (s : Finset β)
Goal: ↑(a +ᵥ s) = a +ᵥ ↑s
-- Finset.smul_mem_smul_finset
@[to_additive]
theorem smul_mem_smul_finset : b ∈ s → a • b ∈ a • s 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [SMul α β] {s : Finset β} {a : α} {b : β}
Goal: b ∈ s → a • b ∈ a • s
-- Finset.vadd_mem_vadd_finset
@[to_additive]
theorem smul_mem_smul_finset : b ∈ s → a • b ∈ a • s 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [VAdd α β] {s : Finset β} {a : α} {b : β}
Goal: b ∈ s → a +ᵥ b ∈ a +ᵥ s
-- Finset.smul_finset_card_le
@[to_additive]
theorem smul_finset_card_le : (a • s).card ≤ s.card 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [SMul α β] {s : Finset β} {a : α}
Goal: (a • s).card ≤ s.card
-- Finset.vadd_finset_card_le
@[to_additive]
theorem smul_finset_card_le : (a • s).card ≤ s.card 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [VAdd α β] {s : Finset β} {a : α}
Goal: (a +ᵥ s).card ≤ s.card
-- Finset.smul_finset_empty
@[to_additive (attr := simp)]
theorem smul_finset_empty (a : α) : a • (∅ : Finset β) = ∅ 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [SMul α β] (a : α)
Goal: a • ∅ = ∅
-- Finset.vadd_finset_empty
@[to_additive (attr := simp)]
theorem smul_finset_empty (a : α) : a • (∅ : Finset β) = ∅ 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [VAdd α β] (a : α)
Goal: a +ᵥ ∅ = ∅
-- Finset.vadd_finset_eq_empty
@[to_additive (attr := simp)]
theorem smul_finset_eq_empty : a • s = ∅ ↔ s = ∅ 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [VAdd α β] {s : Finset β} {a : α}
Goal: a +ᵥ s = ∅ ↔ s = ∅
-- Finset.smul_finset_eq_empty
@[to_additive (attr := simp)]
theorem smul_finset_eq_empty : a • s = ∅ ↔ s = ∅ 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [SMul α β] {s : Finset β} {a : α}
Goal: a • s = ∅ ↔ s = ∅
-- Finset.smul_finset_nonempty
@[to_additive (attr := simp, aesop safe apply (rule_sets := [finsetNonempty]))]
theorem smul_finset_nonempty : (a • s).Nonempty ↔ s.Nonempty 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [SMul α β] {s : Finset β} {a : α}
Goal: (a • s).Nonempty ↔ s.Nonempty
-- Finset.vadd_finset_nonempty
@[to_additive (attr := simp, aesop safe apply (rule_sets := [finsetNonempty]))]
theorem smul_finset_nonempty : (a • s).Nonempty ↔ s.Nonempty 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [VAdd α β] {s : Finset β} {a : α}
Goal: (a +ᵥ s).Nonempty ↔ s.Nonempty
-- Finset.Nonempty.vadd_finset
@[to_additive]
theorem Nonempty.smul_finset (hs : s.Nonempty) : (a • s).Nonempty 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [VAdd α β] {s : Finset β} {a : α} (hs : s.Nonempty)
Goal: (a +ᵥ s).Nonempty
-- Finset.Nonempty.smul_finset
@[to_additive]
theorem Nonempty.smul_finset (hs : s.Nonempty) : (a • s).Nonempty 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [SMul α β] {s : Finset β} {a : α} (hs : s.Nonempty)
Goal: (a • s).Nonempty
-- Finset.singleton_smul
@[to_additive (attr := simp)]
theorem singleton_smul (a : α) : ({a} : Finset α) • t = a • t 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [SMul α β] {t : Finset β} (a : α)
Goal: {a} • t = a • t
-- Finset.singleton_vadd
@[to_additive (attr := simp)]
theorem singleton_smul (a : α) : ({a} : Finset α) • t = a • t 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [VAdd α β] {t : Finset β} (a : α)
Goal: {a} +ᵥ t = a +ᵥ t
-- Finset.vadd_finset_subset_vadd_finset
@[to_additive (attr := mono)]
theorem smul_finset_subset_smul_finset : s ⊆ t → a • s ⊆ a • t 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [VAdd α β] {s t : Finset β} {a : α}
Goal: s ⊆ t → a +ᵥ s ⊆ a +ᵥ t
-- Finset.smul_finset_subset_smul_finset
@[to_additive (attr := mono)]
theorem smul_finset_subset_smul_finset : s ⊆ t → a • s ⊆ a • t 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [SMul α β] {s t : Finset β} {a : α}
Goal: s ⊆ t → a • s ⊆ a • t
-- Finset.smul_finset_singleton
@[to_additive (attr := simp)]
theorem smul_finset_singleton (b : β) : a • ({b} : Finset β) = {a • b} 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [SMul α β] {a : α} (b : β)
Goal: a • {b} = {a • b}
-- Finset.vadd_finset_singleton
@[to_additive (attr := simp)]
theorem smul_finset_singleton (b : β) : a • ({b} : Finset β) = {a • b} 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [VAdd α β] {a : α} (b : β)
Goal: a +ᵥ {b} = {a +ᵥ b}
-- Finset.vadd_finset_union
@[to_additive]
theorem smul_finset_union : a • (s₁ ∪ s₂) = a • s₁ ∪ a • s₂ 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [VAdd α β] {s₁ s₂ : Finset β} {a : α}
Goal: a +ᵥ (s₁ ∪ s₂) = a +ᵥ s₁ ∪ (a +ᵥ s₂)
-- Finset.smul_finset_union
@[to_additive]
theorem smul_finset_union : a • (s₁ ∪ s₂) = a • s₁ ∪ a • s₂ 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [SMul α β] {s₁ s₂ : Finset β} {a : α}
Goal: a • (s₁ ∪ s₂) = a • s₁ ∪ a • s₂
-- Finset.vadd_finset_inter_subset
@[to_additive]
theorem smul_finset_inter_subset : a • (s₁ ∩ s₂) ⊆ a • s₁ ∩ a • s₂ 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [VAdd α β] {s₁ s₂ : Finset β} {a : α}
Goal: a +ᵥ s₁ ∩ s₂ ⊆ (a +ᵥ s₁) ∩ (a +ᵥ s₂)
-- Finset.smul_finset_inter_subset
@[to_additive]
theorem smul_finset_inter_subset : a • (s₁ ∩ s₂) ⊆ a • s₁ ∩ a • s₂ 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [SMul α β] {s₁ s₂ : Finset β} {a : α}
Goal: a • (s₁ ∩ s₂) ⊆ a • s₁ ∩ a • s₂
-- Finset.vadd_finset_subset_vadd
@[to_additive]
theorem smul_finset_subset_smul {s : Finset α} : a ∈ s → a • t ⊆ s • t 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [VAdd α β] {t : Finset β} {a : α} {s : Finset α}
Goal: a ∈ s → a +ᵥ t ⊆ s +ᵥ t
-- Finset.smul_finset_subset_smul
@[to_additive]
theorem smul_finset_subset_smul {s : Finset α} : a ∈ s → a • t ⊆ s • t 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [SMul α β] {t : Finset β} {a : α} {s : Finset α}
Goal: a ∈ s → a • t ⊆ s • t
-- Finset.biUnion_vadd_finset
@[to_additive (attr := simp)]
theorem biUnion_smul_finset (s : Finset α) (t : Finset β) : s.biUnion (· • t) = s • t 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [VAdd α β] (s : Finset α) (t : Finset β)
Goal: (s.biUnion fun x => x +ᵥ t) = s +ᵥ t
-- Finset.biUnion_smul_finset
@[to_additive (attr := simp)]
theorem biUnion_smul_finset (s : Finset α) (t : Finset β) : s.biUnion (· • t) = s • t 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] [SMul α β] (s : Finset α) (t : Finset β)
Goal: (s.biUnion fun x => x • t) = s • t
-- Finset.op_vadd_finset_vadd_eq_vadd_vadd_finset
@[to_additive]
theorem op_smul_finset_smul_eq_smul_smul_finset (a : α) (s : Finset β) (t : Finset 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} [DecidableEq β] [DecidableEq γ] [VAdd αᵃᵒᵖ β] [VAdd β γ] [VAdd α γ] (a : α) (s : Finset β) (t : Finset 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)
-- Finset.op_smul_finset_smul_eq_smul_smul_finset
@[to_additive]
theorem op_smul_finset_smul_eq_smul_smul_finset (a : α) (s : Finset β) (t : Finset 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} [DecidableEq β] [DecidableEq γ] [SMul αᵐᵒᵖ β] [SMul β γ] [SMul α γ] (a : α) (s : Finset β) (t : Finset 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
-- Finset.op_vadd_finset_subset_add
@[to_additive]
theorem op_smul_finset_subset_mul : a ∈ t → op a • s ⊆ s * t 
Arguments: {α : Type u_2} [Add α] [DecidableEq α] {s t : Finset α} {a : α}
Goal: a ∈ t → AddOpposite.op a +ᵥ s ⊆ s + t
-- Finset.op_smul_finset_subset_mul
@[to_additive]
theorem op_smul_finset_subset_mul : a ∈ t → op a • s ⊆ s * t 
Arguments: {α : Type u_2} [Mul α] [DecidableEq α] {s t : Finset α} {a : α}
Goal: a ∈ t → MulOpposite.op a • s ⊆ s * t
-- Finset.biUnion_op_smul_finset
@[to_additive (attr := simp)]
theorem biUnion_op_smul_finset (s t : Finset α) : (t.biUnion fun a => op a • s) = s * t 
Arguments: {α : Type u_2} [Mul α] [DecidableEq α] (s t : Finset α)
Goal: (t.biUnion fun a => MulOpposite.op a • s) = s * t
-- Finset.biUnion_op_vadd_finset
@[to_additive (attr := simp)]
theorem biUnion_op_smul_finset (s t : Finset α) : (t.biUnion fun a => op a • s) = s * t 
Arguments: {α : Type u_2} [Add α] [DecidableEq α] (s t : Finset α)
Goal: (t.biUnion fun a => AddOpposite.op a +ᵥ s) = s + t
-- Finset.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] [DecidableEq α] {s t u : Finset α}
Goal: s + t ⊆ u ↔ ∀ a ∈ s, a +ᵥ t ⊆ u
-- Finset.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] [DecidableEq α] {s t u : Finset α}
Goal: s * t ⊆ u ↔ ∀ a ∈ s, a • t ⊆ u
-- Finset.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] [DecidableEq α] {s t u : Finset α}
Goal: s * t ⊆ u ↔ ∀ b ∈ t, MulOpposite.op b • s ⊆ u
-- Finset.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] [DecidableEq α] {s t u : Finset α}
Goal: s + t ⊆ u ↔ ∀ b ∈ t, AddOpposite.op b +ᵥ s ⊆ u
-- Finset.op_smul_finset_mul_eq_mul_smul_finset
@[to_additive]
theorem op_smul_finset_mul_eq_mul_smul_finset (a : α) (s : Finset α) (t : Finset α) :
    op a • s * t = s * a • t 
Arguments: {α : Type u_2} [Semigroup α] [DecidableEq α] (a : α) (s t : Finset α)
Goal: MulOpposite.op a • s * t = s * a • t
-- Finset.op_vadd_finset_add_eq_add_vadd_finset
@[to_additive]
theorem op_smul_finset_mul_eq_mul_smul_finset (a : α) (s : Finset α) (t : Finset α) :
    op a • s * t = s * a • t 
Arguments: {α : Type u_2} [AddSemigroup α] [DecidableEq α] (a : α) (s t : Finset α)
Goal: AddOpposite.op a +ᵥ s + t = s + (a +ᵥ t)
-- Finset.pairwiseDisjoint_vadd_iff
@[to_additive]
theorem pairwiseDisjoint_smul_iff {s : Set α} {t : Finset α} :
    s.PairwiseDisjoint (· • t) ↔ (s ×ˢ t : Set (α × α)).InjOn fun p => p.1 * p.2 
Arguments: {α : Type u_2} [Add α] [IsLeftCancelAdd α] [DecidableEq α] {s : Set α} {t : Finset α}
Goal: (s.PairwiseDisjoint fun x => x +ᵥ t) ↔ Set.InjOn (fun p => p.1 + p.2) (s ×ˢ ↑t)
-- Finset.pairwiseDisjoint_smul_iff
@[to_additive]
theorem pairwiseDisjoint_smul_iff {s : Set α} {t : Finset α} :
    s.PairwiseDisjoint (· • t) ↔ (s ×ˢ t : Set (α × α)).InjOn fun p => p.1 * p.2 
Arguments: {α : Type u_2} [Mul α] [IsLeftCancelMul α] [DecidableEq α] {s : Set α} {t : Finset α}
Goal: (s.PairwiseDisjoint fun x => x • t) ↔ Set.InjOn (fun p => p.1 * p.2) (s ×ˢ ↑t)
-- Finset.card_singleton_mul
@[to_additive (attr := simp)]
theorem card_singleton_mul : ({a} * t).card = t.card 
Arguments: {α : Type u_2} [Mul α] [IsLeftCancelMul α] [DecidableEq α] (t : Finset α) (a : α)
Goal: ({a} * t).card = t.card
-- Finset.card_singleton_add
@[to_additive (attr := simp)]
theorem card_singleton_mul : ({a} * t).card = t.card 
Arguments: {α : Type u_2} [Add α] [IsLeftCancelAdd α] [DecidableEq α] (t : Finset α) (a : α)
Goal: ({a} + t).card = t.card
-- Finset.singleton_add_inter
@[to_additive]
theorem singleton_mul_inter : {a} * (s ∩ t) = {a} * s ∩ ({a} * t) 
Arguments: {α : Type u_2} [Add α] [IsLeftCancelAdd α] [DecidableEq α] (s t : Finset α) (a : α)
Goal: {a} + s ∩ t = ({a} + s) ∩ ({a} + t)
-- Finset.singleton_mul_inter
@[to_additive]
theorem singleton_mul_inter : {a} * (s ∩ t) = {a} * s ∩ ({a} * t) 
Arguments: {α : Type u_2} [Mul α] [IsLeftCancelMul α] [DecidableEq α] (s t : Finset α) (a : α)
Goal: {a} * (s ∩ t) = {a} * s ∩ ({a} * t)
-- Finset.card_le_card_add_left
@[to_additive]
theorem card_le_card_mul_left {s : Finset α} (hs : s.Nonempty) : t.card ≤ (s * t).card 
Arguments: {α : Type u_2} [Add α] [IsLeftCancelAdd α] [DecidableEq α] (t : Finset α) {s : Finset α} (hs : s.Nonempty)
Goal: t.card ≤ (s + t).card
-- Finset.card_le_card_mul_left
@[to_additive]
theorem card_le_card_mul_left {s : Finset α} (hs : s.Nonempty) : t.card ≤ (s * t).card 
Arguments: {α : Type u_2} [Mul α] [IsLeftCancelMul α] [DecidableEq α] (t : Finset α) {s : Finset α} (hs : s.Nonempty)
Goal: t.card ≤ (s * t).card
-- Finset.card_add_singleton
@[to_additive (attr := simp)]
theorem card_mul_singleton : (s * {a}).card = s.card 
Arguments: {α : Type u_2} [Add α] [IsRightCancelAdd α] [DecidableEq α] (s : Finset α) (a : α)
Goal: (s + {a}).card = s.card
-- Finset.card_mul_singleton
@[to_additive (attr := simp)]
theorem card_mul_singleton : (s * {a}).card = s.card 
Arguments: {α : Type u_2} [Mul α] [IsRightCancelMul α] [DecidableEq α] (s : Finset α) (a : α)
Goal: (s * {a}).card = s.card
-- Finset.inter_mul_singleton
@[to_additive]
theorem inter_mul_singleton : s ∩ t * {a} = s * {a} ∩ (t * {a}) 
Arguments: {α : Type u_2} [Mul α] [IsRightCancelMul α] [DecidableEq α] (s t : Finset α) (a : α)
Goal: s ∩ t * {a} = s * {a} ∩ (t * {a})