ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- HNNExtension.NormalWord.unitsSMulEquiv_apply
@[simps]
noncomputable def unitsSMulEquiv : NormalWord d ≃ NormalWord d 
Arguments: {G : Type u_1} [Group G] {A B : Subgroup G} (φ : ↥A ≃* ↥B) {d : HNNExtension.NormalWord.TransversalPair G A B} (w : HNNExtension.NormalWord d)
Goal: (HNNExtension.NormalWord.unitsSMulEquiv φ) w = HNNExtension.NormalWord.unitsSMul φ 1 w
-- HNNExtension.NormalWord.unitsSMulEquiv_symm_apply
@[simps]
noncomputable def unitsSMulEquiv : NormalWord d ≃ NormalWord d 
Arguments: {G : Type u_1} [Group G] {A B : Subgroup G} (φ : ↥A ≃* ↥B) {d : HNNExtension.NormalWord.TransversalPair G A B} (w : HNNExtension.NormalWord d)
Goal: (HNNExtension.NormalWord.unitsSMulEquiv φ).symm w = HNNExtension.NormalWord.unitsSMul φ (-1) w
-- HNNExtension.NormalWord.unitsSMul_one_group_smul
theorem unitsSMul_one_group_smul (g : A) (w : NormalWord d) :
    unitsSMul φ 1 ((g : G) • w) = (φ g : G) • (unitsSMul φ 1 w) 
Arguments: {G : Type u_1} [Group G] {A B : Subgroup G} (φ : ↥A ≃* ↥B) {d : HNNExtension.NormalWord.TransversalPair G A B} (g : ↥A) (w : HNNExtension.NormalWord d)
Goal: HNNExtension.NormalWord.unitsSMul φ 1 (↑g • w) = ↑(φ g) • HNNExtension.NormalWord.unitsSMul φ 1 w
-- HNNExtension.NormalWord.prod_group_smul
@[simp]
theorem prod_group_smul (g : G) (w : NormalWord d) :
    (g • w).prod φ = of g * (w.prod φ) 
Arguments: {G : Type u_1} [Group G] {A B : Subgroup G} (φ : ↥A ≃* ↥B) {d : HNNExtension.NormalWord.TransversalPair G A B} (g : G) (w : HNNExtension.NormalWord d)
Goal: HNNExtension.NormalWord.ReducedWord.prod φ (g • w).toReducedWord =
  HNNExtension.of g * HNNExtension.NormalWord.ReducedWord.prod φ w.toReducedWord
-- HNNExtension.NormalWord.of_smul_eq_smul
theorem of_smul_eq_smul (g : G) (w : NormalWord d) :
    (of g : HNNExtension G A B φ) • w = g • w 
Arguments: {G : Type u_1} [Group G] {A B : Subgroup G} (φ : ↥A ≃* ↥B) {d : HNNExtension.NormalWord.TransversalPair G A B} (g : G) (w : HNNExtension.NormalWord d)
Goal: HNNExtension.of g • w = g • w
-- HNNExtension.NormalWord.t_smul_eq_unitsSMul
theorem t_smul_eq_unitsSMul (w : NormalWord d) :
    (t : HNNExtension G A B φ) • w = unitsSMul φ 1 w 
Arguments: {G : Type u_1} [Group G] {A B : Subgroup G} (φ : ↥A ≃* ↥B) {d : HNNExtension.NormalWord.TransversalPair G A B} (w : HNNExtension.NormalWord d)
Goal: HNNExtension.t • w = HNNExtension.NormalWord.unitsSMul φ 1 w
-- HNNExtension.NormalWord.t_pow_smul_eq_unitsSMul
theorem t_pow_smul_eq_unitsSMul (u : ℤˣ) (w : NormalWord d) :
    (t ^ (u : ℤ) : HNNExtension G A B φ) • w = unitsSMul φ u w 
Arguments: {G : Type u_1} [Group G] {A B : Subgroup G} (φ : ↥A ≃* ↥B) {d : HNNExtension.NormalWord.TransversalPair G A B} (u : ℤˣ) (w : HNNExtension.NormalWord d)
Goal: HNNExtension.t ^ ↑u • w = HNNExtension.NormalWord.unitsSMul φ u w
-- HNNExtension.NormalWord.prod_cons
@[simp]
theorem prod_cons (g : G) (u : ℤˣ) (w : NormalWord d) (h1 : w.head ∈ d.set u)
    (h2 : ∀ u' ∈ Option.map Prod.fst w.toList.head?,
      w.head ∈ toSubgroup A B u → u = u') :
    (cons g u w h1 h2).prod φ = of g * (t ^ (u : ℤ) * w.prod φ) 
Arguments: {G : Type u_1} [Group G] {A B : Subgroup G} (φ : ↥A ≃* ↥B) {d : HNNExtension.NormalWord.TransversalPair G A B} (g : G) (u : ℤˣ) (w : HNNExtension.NormalWord d) (h1 : w.head ∈ d.set u) (h2 : ∀ u' ∈ Option.map Prod.fst w.toList.head?, w.head ∈ HNNExtension.toSubgroup A B u → u = u')
Goal: HNNExtension.NormalWord.ReducedWord.prod φ (HNNExtension.NormalWord.cons g u w h1 h2).toReducedWord =
  HNNExtension.of g * (HNNExtension.t ^ ↑u * HNNExtension.NormalWord.ReducedWord.prod φ w.toReducedWord)
-- HNNExtension.NormalWord.prod_unitsSMul
theorem prod_unitsSMul (u : ℤˣ) (w : NormalWord d) :
    (unitsSMul φ u w).prod φ = (t^(u : ℤ) * w.prod φ : HNNExtension G A B φ) 
Arguments: {G : Type u_1} [Group G] {A B : Subgroup G} (φ : ↥A ≃* ↥B) {d : HNNExtension.NormalWord.TransversalPair G A B} (u : ℤˣ) (w : HNNExtension.NormalWord d)
Goal: HNNExtension.NormalWord.ReducedWord.prod φ (HNNExtension.NormalWord.unitsSMul φ u w).toReducedWord =
  HNNExtension.t ^ ↑u * HNNExtension.NormalWord.ReducedWord.prod φ w.toReducedWord
-- HNNExtension.NormalWord.prod_empty
@[simp]
theorem prod_empty : (empty : NormalWord d).prod φ = 1 
Arguments: {G : Type u_1} [Group G] {A B : Subgroup G} (φ : ↥A ≃* ↥B) {d : HNNExtension.NormalWord.TransversalPair G A B}
Goal: HNNExtension.NormalWord.ReducedWord.prod φ HNNExtension.NormalWord.empty.toReducedWord = 1
-- HNNExtension.NormalWord.prod_smul
@[simp]
theorem prod_smul (g : HNNExtension G A B φ) (w : NormalWord d) :
    (g • w).prod φ = g * w.prod φ 
Arguments: {G : Type u_1} [Group G] {A B : Subgroup G} (φ : ↥A ≃* ↥B) {d : HNNExtension.NormalWord.TransversalPair G A B} (g : HNNExtension G A B φ) (w : HNNExtension.NormalWord d)
Goal: HNNExtension.NormalWord.ReducedWord.prod φ (g • w).toReducedWord =
  g * HNNExtension.NormalWord.ReducedWord.prod φ w.toReducedWord
-- HNNExtension.NormalWord.prod_smul_empty
@[simp]
theorem prod_smul_empty (w : NormalWord d) :
    (w.prod φ) • empty = w 
Arguments: {G : Type u_1} [Group G] {A B : Subgroup G} (φ : ↥A ≃* ↥B) {d : HNNExtension.NormalWord.TransversalPair G A B} (w : HNNExtension.NormalWord d)
Goal: HNNExtension.NormalWord.ReducedWord.prod φ w.toReducedWord • HNNExtension.NormalWord.empty = w
-- HNNExtension.NormalWord.prod_injective
theorem prod_injective : Injective
    (fun w => w.prod φ : NormalWord d → HNNExtension G A B φ) 
Arguments: {G : Type u_1} [Group G] {A B : Subgroup G} (φ : ↥A ≃* ↥B) (d : HNNExtension.NormalWord.TransversalPair G A B)
Goal: Function.Injective fun w => HNNExtension.NormalWord.ReducedWord.prod φ w.toReducedWord
-- HNNExtension.of_injective
theorem of_injective : Function.Injective (of : G → HNNExtension G A B φ) 
Arguments: {G : Type u_1} [Group G] {A B : Subgroup G} (φ : ↥A ≃* ↥B)
Goal: Function.Injective ⇑HNNExtension.of
-- HNNExtension.ReducedWord.exists_normalWord_prod_eq
theorem exists_normalWord_prod_eq
    (d : TransversalPair G A B) (w : ReducedWord G A B) :
    ∃ w' : NormalWord d, w'.prod φ = w.prod φ ∧
      w'.toList.map Prod.fst = w.toList.map Prod.fst ∧
      ∀ u ∈ w.toList.head?.map Prod.fst,
      w'.head⁻¹ * w.head ∈ toSubgroup A B (-u) 
Arguments: {G : Type u_1} [Group G] {A B : Subgroup G} (φ : ↥A ≃* ↥B) (d : HNNExtension.NormalWord.TransversalPair G A B) (w : HNNExtension.NormalWord.ReducedWord G A B)
Goal: ∃ w',
  HNNExtension.NormalWord.ReducedWord.prod φ w'.toReducedWord = HNNExtension.NormalWord.ReducedWord.prod φ w ∧
    List.map Prod.fst w'.toList = List.map Prod.fst w.toList ∧
      ∀ u ∈ Option.map Prod.fst w.toList.head?, w'.head⁻¹ * w.head ∈ HNNExtension.toSubgroup A B (-u)
-- HNNExtension.ReducedWord.map_fst_eq_and_of_prod_eq
theorem map_fst_eq_and_of_prod_eq {w₁ w₂ : ReducedWord G A B}
    (hprod : w₁.prod φ = w₂.prod φ) :
    w₁.toList.map Prod.fst = w₂.toList.map Prod.fst ∧
     ∀ u ∈ w₁.toList.head?.map Prod.fst,
      w₁.head⁻¹ * w₂.head ∈ toSubgroup A B (-u) 
Arguments: {G : Type u_1} [Group G] {A B : Subgroup G} (φ : ↥A ≃* ↥B) {w₁ w₂ : HNNExtension.NormalWord.ReducedWord G A B} (hprod : HNNExtension.NormalWord.ReducedWord.prod φ w₁ = HNNExtension.NormalWord.ReducedWord.prod φ w₂)
Goal: List.map Prod.fst w₁.toList = List.map Prod.fst w₂.toList ∧
  ∀ u ∈ Option.map Prod.fst w₁.toList.head?, w₁.head⁻¹ * w₂.head ∈ HNNExtension.toSubgroup A B (-u)
-- HNNExtension.ReducedWord.toList_eq_nil_of_mem_of_range
theorem toList_eq_nil_of_mem_of_range (w : ReducedWord G A B)
    (hw : w.prod φ ∈ (of.range : Subgroup (HNNExtension G A B φ))) :
    w.toList = [] 
Arguments: {G : Type u_1} [Group G] {A B : Subgroup G} (φ : ↥A ≃* ↥B) (w : HNNExtension.NormalWord.ReducedWord G A B) (hw : HNNExtension.NormalWord.ReducedWord.prod φ w ∈ HNNExtension.of.range)
Goal: w.toList = []
-- mulIndicator_eventuallyEq
@[to_additive]
theorem mulIndicator_eventuallyEq (hf : f =ᶠ[l ⊓ 𝓟 s] g) (hs : s =ᶠ[l] t) :
    mulIndicator s f =ᶠ[l] mulIndicator t g 
Arguments: {α : Type u_1} {M : Type u_3} [One M] {s t : Set α} {f g : α → M} {l : Filter α} (hf : f =ᶠ[l ⊓ 𝓟 s] g) (hs : s =ᶠ[l] t)
Goal: s.mulIndicator f =ᶠ[l] t.mulIndicator g
-- indicator_eventuallyEq
@[to_additive]
theorem mulIndicator_eventuallyEq (hf : f =ᶠ[l ⊓ 𝓟 s] g) (hs : s =ᶠ[l] t) :
    mulIndicator s f =ᶠ[l] mulIndicator t g 
Arguments: {α : Type u_1} {M : Type u_3} [Zero M] {s t : Set α} {f g : α → M} {l : Filter α} (hf : f =ᶠ[l ⊓ 𝓟 s] g) (hs : s =ᶠ[l] t)
Goal: s.indicator f =ᶠ[l] t.indicator g
-- mulIndicator_union_eventuallyEq
@[to_additive]
theorem mulIndicator_union_eventuallyEq (h : ∀ᶠ a in l, a ∉ s ∩ t) :
    mulIndicator (s ∪ t) f =ᶠ[l] mulIndicator s f * mulIndicator t f 
Arguments: {α : Type u_1} {M : Type u_3} [Monoid M] {s t : Set α} {f : α → M} {l : Filter α} (h : ∀ᶠ (a : α) in l, a ∉ s ∩ t)
Goal: (s ∪ t).mulIndicator f =ᶠ[l] s.mulIndicator f * t.mulIndicator f
-- indicator_union_eventuallyEq
@[to_additive]
theorem mulIndicator_union_eventuallyEq (h : ∀ᶠ a in l, a ∉ s ∩ t) :
    mulIndicator (s ∪ t) f =ᶠ[l] mulIndicator s f * mulIndicator t f 
Arguments: {α : Type u_1} {M : Type u_3} [AddMonoid M] {s t : Set α} {f : α → M} {l : Filter α} (h : ∀ᶠ (a : α) in l, a ∉ s ∩ t)
Goal: (s ∪ t).indicator f =ᶠ[l] s.indicator f + t.indicator f
-- mulIndicator_eventuallyLE_mulIndicator
@[to_additive]
theorem mulIndicator_eventuallyLE_mulIndicator (h : f ≤ᶠ[l ⊓ 𝓟 s] g) :
    mulIndicator s f ≤ᶠ[l] mulIndicator s g 
Arguments: {α : Type u_1} {β : Type u_2} [One β] [Preorder β] {s : Set α} {f g : α → β} {l : Filter α} (h : f ≤ᶠ[l ⊓ 𝓟 s] g)
Goal: s.mulIndicator f ≤ᶠ[l] s.mulIndicator g
-- indicator_eventuallyLE_indicator
@[to_additive]
theorem mulIndicator_eventuallyLE_mulIndicator (h : f ≤ᶠ[l ⊓ 𝓟 s] g) :
    mulIndicator s f ≤ᶠ[l] mulIndicator s g 
Arguments: {α : Type u_1} {β : Type u_2} [Zero β] [Preorder β] {s : Set α} {f g : α → β} {l : Filter α} (h : f ≤ᶠ[l ⊓ 𝓟 s] g)
Goal: s.indicator f ≤ᶠ[l] s.indicator g
-- Monotone.indicator_eventuallyEq_iUnion
@[to_additive]
theorem Monotone.mulIndicator_eventuallyEq_iUnion {ι} [Preorder ι] [One β] (s : ι → Set α)
    (hs : Monotone s) (f : α → β) (a : α) :
    (fun i => mulIndicator (s i) f a) =ᶠ[atTop] fun _ ↦ mulIndicator (⋃ i, s i) f a 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_5} [Preorder ι] [Zero β] (s : ι → Set α) (hs : Monotone s) (f : α → β) (a : α)
Goal: (fun i => (s i).indicator f a) =ᶠ[Filter.atTop] fun x => (⋃ i, s i).indicator f a
-- Monotone.mulIndicator_eventuallyEq_iUnion
@[to_additive]
theorem Monotone.mulIndicator_eventuallyEq_iUnion {ι} [Preorder ι] [One β] (s : ι → Set α)
    (hs : Monotone s) (f : α → β) (a : α) :
    (fun i => mulIndicator (s i) f a) =ᶠ[atTop] fun _ ↦ mulIndicator (⋃ i, s i) f a 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_5} [Preorder ι] [One β] (s : ι → Set α) (hs : Monotone s) (f : α → β) (a : α)
Goal: (fun i => (s i).mulIndicator f a) =ᶠ[Filter.atTop] fun x => (⋃ i, s i).mulIndicator f a
-- Monotone.tendsto_indicator
@[to_additive]
theorem Monotone.tendsto_mulIndicator {ι} [Preorder ι] [One β] (s : ι → Set α) (hs : Monotone s)
    (f : α → β) (a : α) :
    Tendsto (fun i => mulIndicator (s i) f a) atTop (pure <| mulIndicator (⋃ i, s i) f a) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_5} [Preorder ι] [Zero β] (s : ι → Set α) (hs : Monotone s) (f : α → β) (a : α)
Goal: Filter.Tendsto (fun i => (s i).indicator f a) Filter.atTop (pure ((⋃ i, s i).indicator f a))
-- Monotone.tendsto_mulIndicator
@[to_additive]
theorem Monotone.tendsto_mulIndicator {ι} [Preorder ι] [One β] (s : ι → Set α) (hs : Monotone s)
    (f : α → β) (a : α) :
    Tendsto (fun i => mulIndicator (s i) f a) atTop (pure <| mulIndicator (⋃ i, s i) f a) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_5} [Preorder ι] [One β] (s : ι → Set α) (hs : Monotone s) (f : α → β) (a : α)
Goal: Filter.Tendsto (fun i => (s i).mulIndicator f a) Filter.atTop (pure ((⋃ i, s i).mulIndicator f a))
-- Antitone.mulIndicator_eventuallyEq_iInter
@[to_additive]
theorem Antitone.mulIndicator_eventuallyEq_iInter {ι} [Preorder ι] [One β] (s : ι → Set α)
    (hs : Antitone s) (f : α → β) (a : α) :
    (fun i => mulIndicator (s i) f a) =ᶠ[atTop] fun _ ↦ mulIndicator (⋂ i, s i) f a 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_5} [Preorder ι] [One β] (s : ι → Set α) (hs : Antitone s) (f : α → β) (a : α)
Goal: (fun i => (s i).mulIndicator f a) =ᶠ[Filter.atTop] fun x => (⋂ i, s i).mulIndicator f a
-- Antitone.indicator_eventuallyEq_iInter
@[to_additive]
theorem Antitone.mulIndicator_eventuallyEq_iInter {ι} [Preorder ι] [One β] (s : ι → Set α)
    (hs : Antitone s) (f : α → β) (a : α) :
    (fun i => mulIndicator (s i) f a) =ᶠ[atTop] fun _ ↦ mulIndicator (⋂ i, s i) f a 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_5} [Preorder ι] [Zero β] (s : ι → Set α) (hs : Antitone s) (f : α → β) (a : α)
Goal: (fun i => (s i).indicator f a) =ᶠ[Filter.atTop] fun x => (⋂ i, s i).indicator f a
-- Antitone.tendsto_mulIndicator
@[to_additive]
theorem Antitone.tendsto_mulIndicator {ι} [Preorder ι] [One β] (s : ι → Set α) (hs : Antitone s)
    (f : α → β) (a : α) :
    Tendsto (fun i => mulIndicator (s i) f a) atTop (pure <| mulIndicator (⋂ i, s i) f a) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_5} [Preorder ι] [One β] (s : ι → Set α) (hs : Antitone s) (f : α → β) (a : α)
Goal: Filter.Tendsto (fun i => (s i).mulIndicator f a) Filter.atTop (pure ((⋂ i, s i).mulIndicator f a))
-- Antitone.tendsto_indicator
@[to_additive]
theorem Antitone.tendsto_mulIndicator {ι} [Preorder ι] [One β] (s : ι → Set α) (hs : Antitone s)
    (f : α → β) (a : α) :
    Tendsto (fun i => mulIndicator (s i) f a) atTop (pure <| mulIndicator (⋂ i, s i) f a) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_5} [Preorder ι] [Zero β] (s : ι → Set α) (hs : Antitone s) (f : α → β) (a : α)
Goal: Filter.Tendsto (fun i => (s i).indicator f a) Filter.atTop (pure ((⋂ i, s i).indicator f a))
-- mulIndicator_biUnion_finset_eventuallyEq
@[to_additive]
theorem mulIndicator_biUnion_finset_eventuallyEq {ι} [One β] (s : ι → Set α) (f : α → β) (a : α) :
    (fun n : Finset ι => mulIndicator (⋃ i ∈ n, s i) f a) =ᶠ[atTop]
      fun _ ↦ mulIndicator (iUnion s) f a 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_5} [One β] (s : ι → Set α) (f : α → β) (a : α)
Goal: (fun n => (⋃ i ∈ n, s i).mulIndicator f a) =ᶠ[Filter.atTop] fun x => (Set.iUnion s).mulIndicator f a
-- indicator_biUnion_finset_eventuallyEq
@[to_additive]
theorem mulIndicator_biUnion_finset_eventuallyEq {ι} [One β] (s : ι → Set α) (f : α → β) (a : α) :
    (fun n : Finset ι => mulIndicator (⋃ i ∈ n, s i) f a) =ᶠ[atTop]
      fun _ ↦ mulIndicator (iUnion s) f a 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_5} [Zero β] (s : ι → Set α) (f : α → β) (a : α)
Goal: (fun n => (⋃ i ∈ n, s i).indicator f a) =ᶠ[Filter.atTop] fun x => (Set.iUnion s).indicator f a
-- tendsto_mulIndicator_biUnion_finset
@[to_additive]
theorem tendsto_mulIndicator_biUnion_finset {ι} [One β] (s : ι → Set α) (f : α → β) (a : α) :
    Tendsto (fun n : Finset ι => mulIndicator (⋃ i ∈ n, s i) f a) atTop
      (pure <| mulIndicator (iUnion s) f a) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_5} [One β] (s : ι → Set α) (f : α → β) (a : α)
Goal: Filter.Tendsto (fun n => (⋃ i ∈ n, s i).mulIndicator f a) Filter.atTop (pure ((Set.iUnion s).mulIndicator f a))
-- tendsto_indicator_biUnion_finset
@[to_additive]
theorem tendsto_mulIndicator_biUnion_finset {ι} [One β] (s : ι → Set α) (f : α → β) (a : α) :
    Tendsto (fun n : Finset ι => mulIndicator (⋃ i ∈ n, s i) f a) atTop
      (pure <| mulIndicator (iUnion s) f a) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_5} [Zero β] (s : ι → Set α) (f : α → β) (a : α)
Goal: Filter.Tendsto (fun n => (⋃ i ∈ n, s i).indicator f a) Filter.atTop (pure ((Set.iUnion s).indicator f a))
-- Filter.EventuallyEq.support
@[to_additive]
protected theorem Filter.EventuallyEq.mulSupport [One β] {f g : α → β} {l : Filter α}
    (h : f =ᶠ[l] g) :
    Function.mulSupport f =ᶠ[l] Function.mulSupport g 
Arguments: {α : Type u_1} {β : Type u_2} [Zero β] {f g : α → β} {l : Filter α} (h : f =ᶠ[l] g)
Goal: Function.support f =ᶠ[l] Function.support g
-- Filter.EventuallyEq.mulSupport
@[to_additive]
protected theorem Filter.EventuallyEq.mulSupport [One β] {f g : α → β} {l : Filter α}
    (h : f =ᶠ[l] g) :
    Function.mulSupport f =ᶠ[l] Function.mulSupport g 
Arguments: {α : Type u_1} {β : Type u_2} [One β] {f g : α → β} {l : Filter α} (h : f =ᶠ[l] g)
Goal: Function.mulSupport f =ᶠ[l] Function.mulSupport g
-- Filter.EventuallyEq.mulIndicator
@[to_additive]
protected theorem Filter.EventuallyEq.mulIndicator [One β] {l : Filter α} {f g : α → β} {s : Set α}
    (hfg : f =ᶠ[l] g) : s.mulIndicator f =ᶠ[l] s.mulIndicator g 
Arguments: {α : Type u_1} {β : Type u_2} [One β] {l : Filter α} {f g : α → β} {s : Set α} (hfg : f =ᶠ[l] g)
Goal: s.mulIndicator f =ᶠ[l] s.mulIndicator g
-- Filter.EventuallyEq.indicator
@[to_additive]
protected theorem Filter.EventuallyEq.mulIndicator [One β] {l : Filter α} {f g : α → β} {s : Set α}
    (hfg : f =ᶠ[l] g) : s.mulIndicator f =ᶠ[l] s.mulIndicator g 
Arguments: {α : Type u_1} {β : Type u_2} [Zero β] {l : Filter α} {f g : α → β} {s : Set α} (hfg : f =ᶠ[l] g)
Goal: s.indicator f =ᶠ[l] s.indicator g
-- Filter.EventuallyEq.indicator_zero
@[to_additive]
theorem Filter.EventuallyEq.mulIndicator_one [One β] {l : Filter α} {f : α → β} {s : Set α}
    (hf : f =ᶠ[l] 1) : s.mulIndicator f =ᶠ[l] 1 
Arguments: {α : Type u_1} {β : Type u_2} [Zero β] {l : Filter α} {f : α → β} {s : Set α} (hf : f =ᶠ[l] 0)
Goal: s.indicator f =ᶠ[l] 0
-- Filter.EventuallyEq.mulIndicator_one
@[to_additive]
theorem Filter.EventuallyEq.mulIndicator_one [One β] {l : Filter α} {f : α → β} {s : Set α}
    (hf : f =ᶠ[l] 1) : s.mulIndicator f =ᶠ[l] 1 
Arguments: {α : Type u_1} {β : Type u_2} [One β] {l : Filter α} {f : α → β} {s : Set α} (hf : f =ᶠ[l] 1)
Goal: s.mulIndicator f =ᶠ[l] 1
-- Filter.EventuallyEq.of_mulIndicator
@[to_additive]
theorem Filter.EventuallyEq.of_mulIndicator [One β] {l : Filter α} {f : α → β}
    (hf : ∀ᶠ x in l, f x ≠ 1) {s t : Set α} (h : s.mulIndicator f =ᶠ[l] t.mulIndicator f) :
    s =ᶠ[l] t 
Arguments: {α : Type u_1} {β : Type u_2} [One β] {l : Filter α} {f : α → β} (hf : ∀ᶠ (x : α) in l, f x ≠ 1) {s t : Set α} (h : s.mulIndicator f =ᶠ[l] t.mulIndicator f)
Goal: s =ᶠ[l] t
-- Filter.EventuallyEq.of_indicator
@[to_additive]
theorem Filter.EventuallyEq.of_mulIndicator [One β] {l : Filter α} {f : α → β}
    (hf : ∀ᶠ x in l, f x ≠ 1) {s t : Set α} (h : s.mulIndicator f =ᶠ[l] t.mulIndicator f) :
    s =ᶠ[l] t 
Arguments: {α : Type u_1} {β : Type u_2} [Zero β] {l : Filter α} {f : α → β} (hf : ∀ᶠ (x : α) in l, f x ≠ 0) {s t : Set α} (h : s.indicator f =ᶠ[l] t.indicator f)
Goal: s =ᶠ[l] t
-- Filter.EventuallyEq.of_indicator_const
@[to_additive]
theorem Filter.EventuallyEq.of_mulIndicator_const [One β] {l : Filter α} {c : β} (hc : c ≠ 1)
    {s t : Set α} (h : s.mulIndicator (fun _ ↦ c) =ᶠ[l] t.mulIndicator fun _ ↦ c) : s =ᶠ[l] t 
Arguments: {α : Type u_1} {β : Type u_2} [Zero β] {l : Filter α} {c : β} (hc : c ≠ 0) {s t : Set α} (h : (s.indicator fun x => c) =ᶠ[l] t.indicator fun x => c)
Goal: s =ᶠ[l] t
-- Filter.EventuallyEq.of_mulIndicator_const
@[to_additive]
theorem Filter.EventuallyEq.of_mulIndicator_const [One β] {l : Filter α} {c : β} (hc : c ≠ 1)
    {s t : Set α} (h : s.mulIndicator (fun _ ↦ c) =ᶠ[l] t.mulIndicator fun _ ↦ c) : s =ᶠ[l] t 
Arguments: {α : Type u_1} {β : Type u_2} [One β] {l : Filter α} {c : β} (hc : c ≠ 1) {s t : Set α} (h : (s.mulIndicator fun x => c) =ᶠ[l] t.mulIndicator fun x => c)
Goal: s =ᶠ[l] t
-- Filter.mulIndicator_const_eventuallyEq
@[to_additive]
theorem Filter.mulIndicator_const_eventuallyEq [One β] {l : Filter α} {c : β} (hc : c ≠ 1)
    {s t : Set α} : s.mulIndicator (fun _ ↦ c) =ᶠ[l] t.mulIndicator (fun _ ↦ c) ↔ s =ᶠ[l] t 
Arguments: {α : Type u_1} {β : Type u_2} [One β] {l : Filter α} {c : β} (hc : c ≠ 1) {s t : Set α}
Goal: ((s.mulIndicator fun x => c) =ᶠ[l] t.mulIndicator fun x => c) ↔ s =ᶠ[l] t
-- Filter.indicator_const_eventuallyEq
@[to_additive]
theorem Filter.mulIndicator_const_eventuallyEq [One β] {l : Filter α} {c : β} (hc : c ≠ 1)
    {s t : Set α} : s.mulIndicator (fun _ ↦ c) =ᶠ[l] t.mulIndicator (fun _ ↦ c) ↔ s =ᶠ[l] t 
Arguments: {α : Type u_1} {β : Type u_2} [Zero β] {l : Filter α} {c : β} (hc : c ≠ 0) {s t : Set α}
Goal: ((s.indicator fun x => c) =ᶠ[l] t.indicator fun x => c) ↔ s =ᶠ[l] t
-- sSupHom.map_sSup'
structure sSupHom (α β : Type*) [SupSet α] [SupSet β] where
    toFun : α → β
    map_sSup' (s : Set α) : toFun (sSup s) = sSup (toFun '' s)
Arguments: {α : Type u_8} {β : Type u_9} [SupSet α] [SupSet β] (self : sSupHom α β) (s : Set α)
Goal: self.toFun (sSup s) = sSup (self.toFun '' s)
-- sInfHom.map_sInf'
structure sInfHom (α β : Type*) [InfSet α] [InfSet β] where
    toFun : α → β
    map_sInf' (s : Set α) : toFun (sInf s) = sInf (toFun '' s)
Arguments: {α : Type u_8} {β : Type u_9} [InfSet α] [InfSet β] (self : sInfHom α β) (s : Set α)
Goal: self.toFun (sInf s) = sInf (self.toFun '' s)
-- FrameHom.map_sSup'
structure FrameHom (α β : Type*) [CompleteLattice α] [CompleteLattice β] extends
  InfTopHom α β where
    map_sSup' (s : Set α) : toFun (sSup s) = sSup (toFun '' s)
Arguments: {α : Type u_8} {β : Type u_9} [CompleteLattice α] [CompleteLattice β] (self : FrameHom α β) (s : Set α)
Goal: self.toFun (sSup s) = sSup (self.toFun '' s)
-- CompleteLatticeHom.map_sSup'
structure CompleteLatticeHom (α β : Type*) [CompleteLattice α] [CompleteLattice β] extends
  sInfHom α β where
    map_sSup' (s : Set α) : toFun (sSup s) = sSup (toFun '' s)
Arguments: {α : Type u_8} {β : Type u_9} [CompleteLattice α] [CompleteLattice β] (self : CompleteLatticeHom α β) (s : Set α)
Goal: self.toFun (sSup s) = sSup (self.toFun '' s)
-- sSupHomClass.map_sSup
class sSupHomClass (F α β : Type*) [SupSet α] [SupSet β] [FunLike F α β] : Prop where
    map_sSup (f : F) (s : Set α) : f (sSup s) = sSup (f '' s)
Arguments: {F : Type u_8} {α : Type u_9} {β : Type u_10} [SupSet α] [SupSet β] [FunLike F α β] [self : sSupHomClass F α β] (f : F) (s : Set α)
Goal: f (sSup s) = sSup (⇑f '' s)
-- sInfHomClass.map_sInf
class sInfHomClass (F α β : Type*) [InfSet α] [InfSet β] [FunLike F α β] : Prop where
    map_sInf (f : F) (s : Set α) : f (sInf s) = sInf (f '' s)
Arguments: {F : Type u_8} {α : Type u_9} {β : Type u_10} [InfSet α] [InfSet β] [FunLike F α β] [self : sInfHomClass F α β] (f : F) (s : Set α)
Goal: f (sInf s) = sInf (⇑f '' s)
-- FrameHomClass.map_sSup
class FrameHomClass (F α β : Type*) [CompleteLattice α] [CompleteLattice β] [FunLike F α β]
  extends InfTopHomClass F α β : Prop where
    map_sSup (f : F) (s : Set α) : f (sSup s) = sSup (f '' s)
Arguments: {F : Type u_8} {α : Type u_9} {β : Type u_10} [CompleteLattice α] [CompleteLattice β] [FunLike F α β] [self : FrameHomClass F α β] (f : F) (s : Set α)
Goal: f (sSup s) = sSup (⇑f '' s)
-- CompleteLatticeHomClass.map_sSup
class CompleteLatticeHomClass (F α β : Type*) [CompleteLattice α] [CompleteLattice β]
  [FunLike F α β] extends sInfHomClass F α β : Prop where
    map_sSup (f : F) (s : Set α) : f (sSup s) = sSup (f '' s)
Arguments: {F : Type u_8} {α : Type u_9} {β : Type u_10} [CompleteLattice α] [CompleteLattice β] [FunLike F α β] [self : CompleteLatticeHomClass F α β] (f : F) (s : Set α)
Goal: f (sSup s) = sSup (⇑f '' s)
-- map_iSup
@[simp] theorem map_iSup [SupSet α] [SupSet β] [sSupHomClass F α β] (f : F) (g : ι → α) :
    f (⨆ i, g i) = ⨆ i, f (g i) 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} {ι : Sort u_6} [FunLike F α β] [SupSet α] [SupSet β] [sSupHomClass F α β] (f : F) (g : ι → α)
Goal: f (⨆ i, g i) = ⨆ i, f (g i)
-- map_iSup₂
theorem map_iSup₂ [SupSet α] [SupSet β] [sSupHomClass F α β] (f : F) (g : ∀ i, κ i → α) :
    f (⨆ (i) (j), g i j) = ⨆ (i) (j), f (g i j) 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} {ι : Sort u_6} {κ : ι → Sort u_7} [FunLike F α β] [SupSet α] [SupSet β] [sSupHomClass F α β] (f : F) (g : (i : ι) → κ i → α)
Goal: f (⨆ i, ⨆ j, g i j) = ⨆ i, ⨆ j, f (g i j)
-- map_iInf
@[simp] theorem map_iInf [InfSet α] [InfSet β] [sInfHomClass F α β] (f : F) (g : ι → α) :
    f (⨅ i, g i) = ⨅ i, f (g i) 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} {ι : Sort u_6} [FunLike F α β] [InfSet α] [InfSet β] [sInfHomClass F α β] (f : F) (g : ι → α)
Goal: f (⨅ i, g i) = ⨅ i, f (g i)