ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Set.to_countable
theorem to_countable (s : Set α) [Countable s] : s.Countable 
Arguments: {α : Type u} (s : Set α) [Countable ↑s]
Goal: s.Countable
-- Set.countable_iff_exists_injective
protected theorem countable_iff_exists_injective {s : Set α} :
    s.Countable ↔ ∃ f : s → ℕ, Injective f 
Arguments: {α : Type u} {s : Set α}
Goal: s.Countable ↔ ∃ f, Function.Injective f
-- Set.countable_iff_exists_injOn
theorem countable_iff_exists_injOn {s : Set α} : s.Countable ↔ ∃ f : α → ℕ, InjOn f s 
Arguments: {α : Type u} {s : Set α}
Goal: s.Countable ↔ ∃ f, Set.InjOn f s
-- Set.countable_iff_nonempty_encodable
theorem countable_iff_nonempty_encodable {s : Set α} : s.Countable ↔ Nonempty (Encodable s) 
Arguments: {α : Type u} {s : Set α}
Goal: s.Countable ↔ Nonempty (Encodable ↑s)
-- Set.subset_range_enumerate
theorem subset_range_enumerate {s : Set α} (h : s.Countable) (default : α) :
    s ⊆ range (enumerateCountable h default) 
Arguments: {α : Type u} {s : Set α} (h : s.Countable) (default : α)
Goal: s ⊆ Set.range (Set.enumerateCountable h default)
-- Set.range_enumerateCountable_subset
lemma range_enumerateCountable_subset {s : Set α} (h : s.Countable) (default : α) :
    range (enumerateCountable h default) ⊆ insert default s 
Arguments: {α : Type u} {s : Set α} (h : s.Countable) (default : α)
Goal: Set.range (Set.enumerateCountable h default) ⊆ insert default s
-- Set.range_enumerateCountable_of_mem
lemma range_enumerateCountable_of_mem {s : Set α} (h : s.Countable) {default : α}
    (h_mem : default ∈ s) :
    range (enumerateCountable h default) = s 
Arguments: {α : Type u} {s : Set α} (h : s.Countable) {default : α} (h_mem : default ∈ s)
Goal: Set.range (Set.enumerateCountable h default) = s
-- Set.enumerateCountable_mem
lemma enumerateCountable_mem {s : Set α} (h : s.Countable) {default : α} (h_mem : default ∈ s)
    (n : ℕ) :
    enumerateCountable h default n ∈ s 
Arguments: {α : Type u} {s : Set α} (h : s.Countable) {default : α} (h_mem : default ∈ s) (n : ℕ)
Goal: Set.enumerateCountable h default n ∈ s
-- Set.Countable.mono
theorem Countable.mono {s₁ s₂ : Set α} (h : s₁ ⊆ s₂) (hs : s₂.Countable) : s₁.Countable 
Arguments: {α : Type u} {s₁ s₂ : Set α} (h : s₁ ⊆ s₂) (hs : s₂.Countable)
Goal: s₁.Countable
-- Set.countable_range
theorem countable_range [Countable ι] (f : ι → β) : (range f).Countable 
Arguments: {β : Type v} {ι : Sort x} [Countable ι] (f : ι → β)
Goal: (Set.range f).Countable
-- Set.countable_iff_exists_subset_range
theorem countable_iff_exists_subset_range [Nonempty α] {s : Set α} :
    s.Countable ↔ ∃ f : ℕ → α, s ⊆ range f 
Arguments: {α : Type u} [Nonempty α] {s : Set α}
Goal: s.Countable ↔ ∃ f, s ⊆ Set.range f
-- Set.countable_iff_exists_surjective
protected theorem countable_iff_exists_surjective {s : Set α} (hs : s.Nonempty) :
    s.Countable ↔ ∃ f : ℕ → s, Surjective f 
Arguments: {α : Type u} {s : Set α} (hs : s.Nonempty)
Goal: s.Countable ↔ ∃ f, Function.Surjective f
-- Set.countable_univ
theorem countable_univ [Countable α] : (univ : Set α).Countable 
Arguments: {α : Type u} [Countable α]
Goal: Set.univ.Countable
-- Set.countable_univ_iff
theorem countable_univ_iff : (univ : Set α).Countable ↔ Countable α 
Arguments: {α : Type u}
Goal: Set.univ.Countable ↔ Countable α
-- Set.Countable.exists_eq_range
theorem Countable.exists_eq_range {s : Set α} (hc : s.Countable) (hs : s.Nonempty) :
    ∃ f : ℕ → α, s = range f 
Arguments: {α : Type u} {s : Set α} (hc : s.Countable) (hs : s.Nonempty)
Goal: ∃ f, s = Set.range f
-- Set.countable_empty
@[simp] theorem countable_empty : (∅ : Set α).Countable 
Arguments: {α : Type u}
Goal: ∅.Countable
-- Set.countable_singleton
@[simp] theorem countable_singleton (a : α) : ({a} : Set α).Countable 
Arguments: {α : Type u} (a : α)
Goal: {a}.Countable
-- Set.Countable.image
theorem Countable.image {s : Set α} (hs : s.Countable) (f : α → β) : (f '' s).Countable 
Arguments: {α : Type u} {β : Type v} {s : Set α} (hs : s.Countable) (f : α → β)
Goal: (f '' s).Countable
-- Set.MapsTo.countable_of_injOn
theorem MapsTo.countable_of_injOn {s : Set α} {t : Set β} {f : α → β} (hf : MapsTo f s t)
    (hf' : InjOn f s) (ht : t.Countable) : s.Countable 
Arguments: {α : Type u} {β : Type v} {s : Set α} {t : Set β} {f : α → β} (hf : Set.MapsTo f s t) (hf' : Set.InjOn f s) (ht : t.Countable)
Goal: s.Countable
-- Set.Countable.preimage_of_injOn
theorem Countable.preimage_of_injOn {s : Set β} (hs : s.Countable) {f : α → β}
    (hf : InjOn f (f ⁻¹' s)) : (f ⁻¹' s).Countable 
Arguments: {α : Type u} {β : Type v} {s : Set β} (hs : s.Countable) {f : α → β} (hf : Set.InjOn f (f ⁻¹' s))
Goal: (f ⁻¹' s).Countable
-- Set.Countable.preimage
protected theorem Countable.preimage {s : Set β} (hs : s.Countable) {f : α → β} (hf : Injective f) :
    (f ⁻¹' s).Countable 
Arguments: {α : Type u} {β : Type v} {s : Set β} (hs : s.Countable) {f : α → β} (hf : Function.Injective f)
Goal: (f ⁻¹' s).Countable
-- Set.exists_seq_iSup_eq_top_iff_countable
theorem exists_seq_iSup_eq_top_iff_countable [CompleteLattice α] {p : α → Prop} (h : ∃ x, p x) :
    (∃ s : ℕ → α, (∀ n, p (s n)) ∧ ⨆ n, s n = ⊤) ↔
      ∃ S : Set α, S.Countable ∧ (∀ s ∈ S, p s) ∧ sSup S = ⊤ 
Arguments: {α : Type u} [CompleteLattice α] {p : α → Prop} (h : ∃ x, p x)
Goal: (∃ s, (∀ (n : ℕ), p (s n)) ∧ ⨆ n, s n = ⊤) ↔ ∃ S, S.Countable ∧ (∀ s ∈ S, p s) ∧ sSup S = ⊤
-- Set.exists_seq_cover_iff_countable
theorem exists_seq_cover_iff_countable {p : Set α → Prop} (h : ∃ s, p s) :
    (∃ s : ℕ → Set α, (∀ n, p (s n)) ∧ ⋃ n, s n = univ) ↔
      ∃ S : Set (Set α), S.Countable ∧ (∀ s ∈ S, p s) ∧ ⋃₀ S = univ 
Arguments: {α : Type u} {p : Set α → Prop} (h : ∃ s, p s)
Goal: (∃ s, (∀ (n : ℕ), p (s n)) ∧ ⋃ n, s n = Set.univ) ↔ ∃ S, S.Countable ∧ (∀ s ∈ S, p s) ∧ ⋃₀ S = Set.univ
-- Set.countable_of_injective_of_countable_image
theorem countable_of_injective_of_countable_image {s : Set α} {f : α → β} (hf : InjOn f s)
    (hs : (f '' s).Countable) : s.Countable 
Arguments: {α : Type u} {β : Type v} {s : Set α} {f : α → β} (hf : Set.InjOn f s) (hs : (f '' s).Countable)
Goal: s.Countable
-- Set.countable_iUnion
theorem countable_iUnion {t : ι → Set α} [Countable ι] (ht : ∀ i, (t i).Countable) :
    (⋃ i, t i).Countable 
Arguments: {α : Type u} {ι : Sort x} {t : ι → Set α} [Countable ι] (ht : ∀ (i : ι), (t i).Countable)
Goal: (⋃ i, t i).Countable
-- Set.countable_iUnion_iff
@[simp]
theorem countable_iUnion_iff [Countable ι] {t : ι → Set α} :
    (⋃ i, t i).Countable ↔ ∀ i, (t i).Countable 
Arguments: {α : Type u} {ι : Sort x} [Countable ι] {t : ι → Set α}
Goal: (⋃ i, t i).Countable ↔ ∀ (i : ι), (t i).Countable
-- Set.Countable.biUnion_iff
theorem Countable.biUnion_iff {s : Set α} {t : ∀ a ∈ s, Set β} (hs : s.Countable) :
    (⋃ a ∈ s, t a ‹_›).Countable ↔ ∀ a (ha : a ∈ s), (t a ha).Countable 
Arguments: {α : Type u} {β : Type v} {s : Set α} {t : (a : α) → a ∈ s → Set β} (hs : s.Countable)
Goal: (⋃ a, ⋃ (h : a ∈ s), t a h).Countable ↔ ∀ (a : α) (ha : a ∈ s), (t a ha).Countable
-- Set.Countable.sUnion_iff
theorem Countable.sUnion_iff {s : Set (Set α)} (hs : s.Countable) :
    (⋃₀ s).Countable ↔ ∀ a ∈ s, a.Countable 
Arguments: {α : Type u} {s : Set (Set α)} (hs : s.Countable)
Goal: (⋃₀ s).Countable ↔ ∀ a ∈ s, a.Countable
-- Set.countable_union
@[simp]
theorem countable_union {s t : Set α} : (s ∪ t).Countable ↔ s.Countable ∧ t.Countable 
Arguments: {α : Type u} {s t : Set α}
Goal: (s ∪ t).Countable ↔ s.Countable ∧ t.Countable
-- Set.Countable.union
theorem Countable.union {s t : Set α} (hs : s.Countable) (ht : t.Countable) : (s ∪ t).Countable 
Arguments: {α : Type u} {s t : Set α} (hs : s.Countable) (ht : t.Countable)
Goal: (s ∪ t).Countable
-- Set.Countable.of_diff
theorem Countable.of_diff {s t : Set α} (h : (s \ t).Countable) (ht : t.Countable) : s.Countable 
Arguments: {α : Type u} {s t : Set α} (h : (s \ t).Countable) (ht : t.Countable)
Goal: s.Countable
-- Set.countable_insert
@[simp]
theorem countable_insert {s : Set α} {a : α} : (insert a s).Countable ↔ s.Countable 
Arguments: {α : Type u} {s : Set α} {a : α}
Goal: (insert a s).Countable ↔ s.Countable
-- Set.Countable.insert
protected theorem Countable.insert {s : Set α} (a : α) (h : s.Countable) : (insert a s).Countable 
Arguments: {α : Type u} {s : Set α} (a : α) (h : s.Countable)
Goal: (insert a s).Countable
-- Set.Finite.countable
theorem Finite.countable {s : Set α} (hs : s.Finite) : s.Countable 
Arguments: {α : Type u} {s : Set α} (hs : s.Finite)
Goal: s.Countable
-- Set.Countable.of_subsingleton
@[nontriviality]
theorem Countable.of_subsingleton [Subsingleton α] (s : Set α) : s.Countable 
Arguments: {α : Type u} [Subsingleton α] (s : Set α)
Goal: s.Countable
-- Set.Subsingleton.countable
theorem Subsingleton.countable {s : Set α} (hs : s.Subsingleton) : s.Countable 
Arguments: {α : Type u} {s : Set α} (hs : s.Subsingleton)
Goal: s.Countable
-- Set.countable_isTop
theorem countable_isTop (α : Type*) [PartialOrder α] : { x : α | IsTop x }.Countable 
Arguments: (α : Type u_1) [PartialOrder α]
Goal: {x | IsTop x}.Countable
-- Set.countable_isBot
theorem countable_isBot (α : Type*) [PartialOrder α] : { x : α | IsBot x }.Countable 
Arguments: (α : Type u_1) [PartialOrder α]
Goal: {x | IsBot x}.Countable
-- Set.countable_setOf_finite_subset
theorem countable_setOf_finite_subset {s : Set α} (hs : s.Countable) :
    { t | Set.Finite t ∧ t ⊆ s }.Countable 
Arguments: {α : Type u} {s : Set α} (hs : s.Countable)
Goal: {t | t.Finite ∧ t ⊆ s}.Countable
-- Set.countable_univ_pi
theorem countable_univ_pi {π : α → Type*} [Finite α] {s : ∀ a, Set (π a)}
    (hs : ∀ a, (s a).Countable) : (pi univ s).Countable 
Arguments: {α : Type u} {π : α → Type u_1} [Finite α] {s : (a : α) → Set (π a)} (hs : ∀ (a : α), (s a).Countable)
Goal: (Set.univ.pi s).Countable
-- Set.countable_pi
theorem countable_pi {π : α → Type*} [Finite α] {s : ∀ a, Set (π a)} (hs : ∀ a, (s a).Countable) :
    { f : ∀ a, π a | ∀ a, f a ∈ s a }.Countable 
Arguments: {α : Type u} {π : α → Type u_1} [Finite α] {s : (a : α) → Set (π a)} (hs : ∀ (a : α), (s a).Countable)
Goal: {f | ∀ (a : α), f a ∈ s a}.Countable
-- Set.Countable.prod
protected theorem Countable.prod {s : Set α} {t : Set β} (hs : s.Countable) (ht : t.Countable) :
    Set.Countable (s ×ˢ t) 
Arguments: {α : Type u} {β : Type v} {s : Set α} {t : Set β} (hs : s.Countable) (ht : t.Countable)
Goal: (s ×ˢ t).Countable
-- Set.Countable.image2
theorem Countable.image2 {s : Set α} {t : Set β} (hs : s.Countable) (ht : t.Countable)
    (f : α → β → γ) : (image2 f s t).Countable 
Arguments: {α : Type u} {β : Type v} {γ : Type w} {s : Set α} {t : Set β} (hs : s.Countable) (ht : t.Countable) (f : α → β → γ)
Goal: (Set.image2 f s t).Countable
-- Set.countable_setOf_nonempty_of_disjoint
theorem countable_setOf_nonempty_of_disjoint {f : β → Set α}
    (hf : Pairwise (Disjoint on f)) {s : Set α} (h'f : ∀ t, f t ⊆ s) (hs : s.Countable) :
    Set.Countable {t | (f t).Nonempty} 
Arguments: {α : Type u} {β : Type v} {f : β → Set α} (hf : Pairwise (Disjoint on f)) {s : Set α} (h'f : ∀ (t : β), f t ⊆ s) (hs : s.Countable)
Goal: {t | (f t).Nonempty}.Countable
-- Finset.countable_toSet
theorem Finset.countable_toSet (s : Finset α) : Set.Countable (↑s : Set α) 
Arguments: {α : Type u} (s : Finset α)
Goal: (↑s).Countable
-- Set.pairwise_iUnion
theorem pairwise_iUnion {f : κ → Set α} (h : Directed (· ⊆ ·) f) :
    (⋃ n, f n).Pairwise r ↔ ∀ n, (f n).Pairwise r 
Arguments: {α : Type u_1} {κ : Sort u_6} {r : α → α → Prop} {f : κ → Set α} (h : Directed (fun x x_1 => x ⊆ x_1) f)
Goal: (⋃ n, f n).Pairwise r ↔ ∀ (n : κ), (f n).Pairwise r
-- Set.pairwise_sUnion
theorem pairwise_sUnion {r : α → α → Prop} {s : Set (Set α)} (h : DirectedOn (· ⊆ ·) s) :
    (⋃₀ s).Pairwise r ↔ ∀ a ∈ s, Set.Pairwise a r 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set (Set α)} (h : DirectedOn (fun x x_1 => x ⊆ x_1) s)
Goal: (⋃₀ s).Pairwise r ↔ ∀ a ∈ s, a.Pairwise r
-- Set.pairwiseDisjoint_iUnion
theorem pairwiseDisjoint_iUnion {g : ι' → Set ι} (h : Directed (· ⊆ ·) g) :
    (⋃ n, g n).PairwiseDisjoint f ↔ ∀ ⦃n⦄, (g n).PairwiseDisjoint f 
Arguments: {α : Type u_1} {ι : Type u_4} {ι' : Type u_5} [PartialOrder α] [OrderBot α] {f : ι → α} {g : ι' → Set ι} (h : Directed (fun x x_1 => x ⊆ x_1) g)
Goal: (⋃ n, g n).PairwiseDisjoint f ↔ ∀ ⦃n : ι'⦄, (g n).PairwiseDisjoint f
-- Set.pairwiseDisjoint_sUnion
theorem pairwiseDisjoint_sUnion {s : Set (Set ι)} (h : DirectedOn (· ⊆ ·) s) :
    (⋃₀ s).PairwiseDisjoint f ↔ ∀ ⦃a⦄, a ∈ s → Set.PairwiseDisjoint a f 
Arguments: {α : Type u_1} {ι : Type u_4} [PartialOrder α] [OrderBot α] {f : ι → α} {s : Set (Set ι)} (h : DirectedOn (fun x x_1 => x ⊆ x_1) s)
Goal: (⋃₀ s).PairwiseDisjoint f ↔ ∀ ⦃a : Set ι⦄, a ∈ s → a.PairwiseDisjoint f
-- Set.PairwiseDisjoint.biUnion
theorem PairwiseDisjoint.biUnion {s : Set ι'} {g : ι' → Set ι} {f : ι → α}
    (hs : s.PairwiseDisjoint fun i' : ι' => ⨆ i ∈ g i', f i)
    (hg : ∀ i ∈ s, (g i).PairwiseDisjoint f) : (⋃ i ∈ s, g i).PairwiseDisjoint f 
Arguments: {α : Type u_1} {ι : Type u_4} {ι' : Type u_5} [CompleteLattice α] {s : Set ι'} {g : ι' → Set ι} {f : ι → α} (hs : s.PairwiseDisjoint fun i' => ⨆ i ∈ g i', f i) (hg : ∀ i ∈ s, (g i).PairwiseDisjoint f)
Goal: (⋃ i ∈ s, g i).PairwiseDisjoint f
-- Set.PairwiseDisjoint.prod_left
theorem PairwiseDisjoint.prod_left {f : ι × ι' → α}
    (hs : s.PairwiseDisjoint fun i => ⨆ i' ∈ t, f (i, i'))
    (ht : t.PairwiseDisjoint fun i' => ⨆ i ∈ s, f (i, i')) :
    (s ×ˢ t : Set (ι × ι')).PairwiseDisjoint f 
Arguments: {α : Type u_1} {ι : Type u_4} {ι' : Type u_5} [CompleteLattice α] {s : Set ι} {t : Set ι'} {f : ι × ι' → α} (hs : s.PairwiseDisjoint fun i => ⨆ i' ∈ t, f (i, i')) (ht : t.PairwiseDisjoint fun i' => ⨆ i ∈ s, f (i, i'))
Goal: (s ×ˢ t).PairwiseDisjoint f
-- Set.pairwiseDisjoint_prod_left
theorem pairwiseDisjoint_prod_left {s : Set ι} {t : Set ι'} {f : ι × ι' → α} :
    (s ×ˢ t : Set (ι × ι')).PairwiseDisjoint f ↔
      (s.PairwiseDisjoint fun i => ⨆ i' ∈ t, f (i, i')) ∧
        t.PairwiseDisjoint fun i' => ⨆ i ∈ s, f (i, i') 
Arguments: {α : Type u_1} {ι : Type u_4} {ι' : Type u_5} [Order.Frame α] {s : Set ι} {t : Set ι'} {f : ι × ι' → α}
Goal: (s ×ˢ t).PairwiseDisjoint f ↔
  (s.PairwiseDisjoint fun i => ⨆ i' ∈ t, f (i, i')) ∧ t.PairwiseDisjoint fun i' => ⨆ i ∈ s, f (i, i')
-- Set.biUnion_diff_biUnion_eq
theorem biUnion_diff_biUnion_eq {s t : Set ι} {f : ι → Set α} (h : (s ∪ t).PairwiseDisjoint f) :
    ((⋃ i ∈ s, f i) \ ⋃ i ∈ t, f i) = ⋃ i ∈ s \ t, f i 
Arguments: {α : Type u_1} {ι : Type u_4} {s t : Set ι} {f : ι → Set α} (h : (s ∪ t).PairwiseDisjoint f)
Goal: (⋃ i ∈ s, f i) \ ⋃ i ∈ t, f i = ⋃ i ∈ s \ t, f i
-- Set.PairwiseDisjoint.subset_of_biUnion_subset_biUnion
theorem Set.PairwiseDisjoint.subset_of_biUnion_subset_biUnion (h₀ : (s ∪ t).PairwiseDisjoint f)
    (h₁ : ∀ i ∈ s, (f i).Nonempty) (h : ⋃ i ∈ s, f i ⊆ ⋃ i ∈ t, f i) : s ⊆ t 
Arguments: {α : Type u_1} {ι : Type u_4} {f : ι → Set α} {s t : Set ι} (h₀ : (s ∪ t).PairwiseDisjoint f) (h₁ : ∀ i ∈ s, (f i).Nonempty) (h : ⋃ i ∈ s, f i ⊆ ⋃ i ∈ t, f i)
Goal: s ⊆ t
-- Pairwise.subset_of_biUnion_subset_biUnion
theorem Pairwise.subset_of_biUnion_subset_biUnion (h₀ : Pairwise (Disjoint on f))
    (h₁ : ∀ i ∈ s, (f i).Nonempty) (h : ⋃ i ∈ s, f i ⊆ ⋃ i ∈ t, f i) : s ⊆ t 
Arguments: {α : Type u_1} {ι : Type u_4} {f : ι → Set α} {s t : Set ι} (h₀ : Pairwise (Disjoint on f)) (h₁ : ∀ i ∈ s, (f i).Nonempty) (h : ⋃ i ∈ s, f i ⊆ ⋃ i ∈ t, f i)
Goal: s ⊆ t
-- Pairwise.biUnion_injective
theorem Pairwise.biUnion_injective (h₀ : Pairwise (Disjoint on f)) (h₁ : ∀ i, (f i).Nonempty) :
    Injective fun s : Set ι => ⋃ i ∈ s, f i 
Arguments: {α : Type u_1} {ι : Type u_4} {f : ι → Set α} (h₀ : Pairwise (Disjoint on f)) (h₁ : ∀ (i : ι), (f i).Nonempty)
Goal: Function.Injective fun s => ⋃ i ∈ s, f i
-- pairwiseDisjoint_unique
theorem pairwiseDisjoint_unique {y : α}
    (h_disjoint : PairwiseDisjoint s f)
    (hy : y ∈ (⋃ i ∈ s, f i)) : ∃! i, i ∈ s ∧ y ∈ f i 
Arguments: {α : Type u_1} {ι : Type u_4} {f : ι → Set α} {s : Set ι} {y : α} (h_disjoint : s.PairwiseDisjoint f) (hy : y ∈ ⋃ i ∈ s, f i)
Goal: ∃! i, i ∈ s ∧ y ∈ f i
-- Set.image2_smul
@[to_additive (attr := simp)]
theorem image2_smul : image2 SMul.smul s t = s • t 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {s : Set α} {t : Set β}
Goal: Set.image2 SMul.smul s t = s • t
-- Set.image2_vadd
@[to_additive (attr := simp)]
theorem image2_smul : image2 SMul.smul s t = s • t 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {s : Set α} {t : Set β}
Goal: Set.image2 VAdd.vadd s t = s +ᵥ t
-- Set.image_smul_prod
@[to_additive vadd_image_prod]
theorem image_smul_prod : (fun x : α × β ↦ x.fst • x.snd) '' s ×ˢ t = s • t 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {s : Set α} {t : Set β}
Goal: (fun x => x.1 • x.2) '' s ×ˢ t = s • t
-- Set.vadd_image_prod
@[to_additive vadd_image_prod]
theorem image_smul_prod : (fun x : α × β ↦ x.fst • x.snd) '' s ×ˢ t = s • t 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {s : Set α} {t : Set β}
Goal: (fun x => x.1 +ᵥ x.2) '' s ×ˢ t = s +ᵥ t
-- Set.mem_vadd
@[to_additive]
theorem mem_smul : b ∈ s • t ↔ ∃ x ∈ s, ∃ y ∈ t, x • y = b 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {s : Set α} {t : Set β} {b : β}
Goal: b ∈ s +ᵥ t ↔ ∃ x ∈ s, ∃ y ∈ t, x +ᵥ y = b
-- Set.mem_smul
@[to_additive]
theorem mem_smul : b ∈ s • t ↔ ∃ x ∈ s, ∃ y ∈ t, x • y = b 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {s : Set α} {t : Set β} {b : β}
Goal: b ∈ s • t ↔ ∃ x ∈ s, ∃ y ∈ t, x • y = b
-- Set.vadd_mem_vadd
@[to_additive]
theorem smul_mem_smul : a ∈ s → b ∈ t → a • b ∈ s • t 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {s : Set α} {t : Set β} {a : α} {b : β}
Goal: a ∈ s → b ∈ t → a +ᵥ b ∈ s +ᵥ t
-- Set.smul_mem_smul
@[to_additive]
theorem smul_mem_smul : a ∈ s → b ∈ t → a • b ∈ s • t 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {s : Set α} {t : Set β} {a : α} {b : β}
Goal: a ∈ s → b ∈ t → a • b ∈ s • t
-- Set.empty_vadd
@[to_additive (attr := simp)]
theorem empty_smul : (∅ : Set α) • t = ∅ 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {t : Set β}
Goal: ∅ +ᵥ t = ∅
-- Set.empty_smul
@[to_additive (attr := simp)]
theorem empty_smul : (∅ : Set α) • t = ∅ 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {t : Set β}
Goal: ∅ • t = ∅
-- Set.smul_empty
@[to_additive (attr := simp)]
theorem smul_empty : s • (∅ : Set β) = ∅ 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {s : Set α}
Goal: s • ∅ = ∅
-- Set.vadd_empty
@[to_additive (attr := simp)]
theorem smul_empty : s • (∅ : Set β) = ∅ 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {s : Set α}
Goal: s +ᵥ ∅ = ∅
-- Set.smul_eq_empty
@[to_additive (attr := simp)]
theorem smul_eq_empty : s • t = ∅ ↔ s = ∅ ∨ t = ∅ 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {s : Set α} {t : Set β}
Goal: s • t = ∅ ↔ s = ∅ ∨ t = ∅
-- Set.vadd_eq_empty
@[to_additive (attr := simp)]
theorem smul_eq_empty : s • t = ∅ ↔ s = ∅ ∨ t = ∅ 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {s : Set α} {t : Set β}
Goal: s +ᵥ t = ∅ ↔ s = ∅ ∨ t = ∅
-- Set.smul_nonempty
@[to_additive (attr := simp)]
theorem smul_nonempty : (s • t).Nonempty ↔ s.Nonempty ∧ t.Nonempty 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {s : Set α} {t : Set β}
Goal: (s • t).Nonempty ↔ s.Nonempty ∧ t.Nonempty
-- Set.vadd_nonempty
@[to_additive (attr := simp)]
theorem smul_nonempty : (s • t).Nonempty ↔ s.Nonempty ∧ t.Nonempty 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {s : Set α} {t : Set β}
Goal: (s +ᵥ t).Nonempty ↔ s.Nonempty ∧ t.Nonempty
-- Set.Nonempty.vadd
@[to_additive]
theorem Nonempty.smul : s.Nonempty → t.Nonempty → (s • t).Nonempty 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {s : Set α} {t : Set β}
Goal: s.Nonempty → t.Nonempty → (s +ᵥ t).Nonempty
-- Set.Nonempty.smul
@[to_additive]
theorem Nonempty.smul : s.Nonempty → t.Nonempty → (s • t).Nonempty 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {s : Set α} {t : Set β}
Goal: s.Nonempty → t.Nonempty → (s • t).Nonempty
-- Set.Nonempty.of_vadd_left
@[to_additive]
theorem Nonempty.of_smul_left : (s • t).Nonempty → s.Nonempty 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {s : Set α} {t : Set β}
Goal: (s +ᵥ t).Nonempty → s.Nonempty
-- Set.Nonempty.of_smul_left
@[to_additive]
theorem Nonempty.of_smul_left : (s • t).Nonempty → s.Nonempty 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {s : Set α} {t : Set β}
Goal: (s • t).Nonempty → s.Nonempty
-- Set.Nonempty.of_vadd_right
@[to_additive]
theorem Nonempty.of_smul_right : (s • t).Nonempty → t.Nonempty 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {s : Set α} {t : Set β}
Goal: (s +ᵥ t).Nonempty → t.Nonempty
-- Set.Nonempty.of_smul_right
@[to_additive]
theorem Nonempty.of_smul_right : (s • t).Nonempty → t.Nonempty 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {s : Set α} {t : Set β}
Goal: (s • t).Nonempty → t.Nonempty
-- Set.smul_singleton
@[to_additive (attr := simp low+1)]
theorem smul_singleton : s • ({b} : Set β) = (· • b) '' s 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {s : Set α} {b : β}
Goal: s • {b} = (fun x => x • b) '' s
-- Set.vadd_singleton
@[to_additive (attr := simp low+1)]
theorem smul_singleton : s • ({b} : Set β) = (· • b) '' s 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {s : Set α} {b : β}
Goal: s +ᵥ {b} = (fun x => x +ᵥ b) '' s
-- Set.singleton_smul
@[to_additive (attr := simp low+1)]
theorem singleton_smul : ({a} : Set α) • t = a • t 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {t : Set β} {a : α}
Goal: {a} • t = a • t
-- Set.singleton_vadd
@[to_additive (attr := simp low+1)]
theorem singleton_smul : ({a} : Set α) • t = a • t 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {t : Set β} {a : α}
Goal: {a} +ᵥ t = a +ᵥ t
-- Set.singleton_smul_singleton
@[to_additive (attr := simp high)]
theorem singleton_smul_singleton : ({a} : Set α) • ({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.singleton_vadd_singleton
@[to_additive (attr := simp high)]
theorem singleton_smul_singleton : ({a} : Set α) • ({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_subset_vadd
@[to_additive (attr := mono)]
theorem smul_subset_smul : 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 → t₁ ⊆ t₂ → s₁ +ᵥ t₁ ⊆ s₂ +ᵥ t₂
-- Set.smul_subset_smul
@[to_additive (attr := mono)]
theorem smul_subset_smul : 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.smul_subset_smul_left
@[to_additive]
theorem smul_subset_smul_left : 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: t₁ ⊆ t₂ → s • t₁ ⊆ s • t₂
-- Set.vadd_subset_vadd_left
@[to_additive]
theorem smul_subset_smul_left : 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: t₁ ⊆ t₂ → s +ᵥ t₁ ⊆ s +ᵥ t₂
-- Set.vadd_subset_vadd_right
@[to_additive]
theorem smul_subset_smul_right : s₁ ⊆ s₂ → 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 → s₁ +ᵥ t ⊆ s₂ +ᵥ t
-- Set.smul_subset_smul_right
@[to_additive]
theorem smul_subset_smul_right : s₁ ⊆ s₂ → 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 → s₁ • t ⊆ s₂ • t
-- Set.vadd_subset_iff
@[to_additive]
theorem smul_subset_iff : s • t ⊆ u ↔ ∀ a ∈ s, ∀ b ∈ t, a • b ∈ u 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {s : Set α} {t u : Set β}
Goal: s +ᵥ t ⊆ u ↔ ∀ a ∈ s, ∀ b ∈ t, a +ᵥ b ∈ u
-- Set.smul_subset_iff
@[to_additive]
theorem smul_subset_iff : s • t ⊆ u ↔ ∀ a ∈ s, ∀ b ∈ t, a • b ∈ u 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {s : Set α} {t u : Set β}
Goal: s • t ⊆ u ↔ ∀ a ∈ s, ∀ b ∈ t, a • b ∈ u
-- Set.union_vadd
@[to_additive]
theorem union_smul : (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.union_smul
@[to_additive]
theorem union_smul : (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