ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- LocallyFinite.exists_finset_nhd_mulSupport_subset
@[to_additive " If a family of functions `f` has locally-finite support, subordinate to a family of
open sets, then for any point we can find a neighbourhood on which only finitely-many members of `f`
are non-zero. "]
theorem LocallyFinite.exists_finset_nhd_mulSupport_subset {U : ι → Set X} [One R] {f : ι → X → R}
    (hlf : LocallyFinite fun i => mulSupport (f i)) (hso : ∀ i, mulTSupport (f i) ⊆ U i)
    (ho : ∀ i, IsOpen (U i)) (x : X) :
    ∃ (is : Finset ι), ∃ n, n ∈ 𝓝 x ∧ (n ⊆ ⋂ i ∈ is, U i) ∧
      ∀ z ∈ n, (mulSupport fun i => f i z) ⊆ is 
Arguments: {X : Type u_1} {R : Type u_9} {ι : Type u_10} [TopologicalSpace X] {U : ι → Set X} [One R] {f : ι → X → R} (hlf : LocallyFinite fun i => Function.mulSupport (f i)) (hso : ∀ (i : ι), mulTSupport (f i) ⊆ U i) (ho : ∀ (i : ι), IsOpen (U i)) (x : X)
Goal: ∃ is, ∃ n ∈ 𝓝 x, n ⊆ ⋂ i ∈ is, U i ∧ ∀ z ∈ n, (Function.mulSupport fun i => f i z) ⊆ ↑is
-- locallyFinite_support_iff
@[to_additive]
theorem locallyFinite_mulSupport_iff [CommMonoid M] {f : ι → X → M} :
    (LocallyFinite fun i ↦ mulSupport <| f i) ↔ LocallyFinite fun i ↦ mulTSupport <| f i 
Arguments: {X : Type u_1} {M : Type u_7} {ι : Type u_10} [TopologicalSpace X] [AddCommMonoid M] {f : ι → X → M}
Goal: (LocallyFinite fun i => Function.support (f i)) ↔ LocallyFinite fun i => tsupport (f i)
-- locallyFinite_mulSupport_iff
@[to_additive]
theorem locallyFinite_mulSupport_iff [CommMonoid M] {f : ι → X → M} :
    (LocallyFinite fun i ↦ mulSupport <| f i) ↔ LocallyFinite fun i ↦ mulTSupport <| f i 
Arguments: {X : Type u_1} {M : Type u_7} {ι : Type u_10} [TopologicalSpace X] [CommMonoid M] {f : ι → X → M}
Goal: (LocallyFinite fun i => Function.mulSupport (f i)) ↔ LocallyFinite fun i => mulTSupport (f i)
-- LocallyFinite.smul_left
theorem LocallyFinite.smul_left [Zero R] [Zero M] [SMulWithZero R M]
    {s : ι → X → R} (h : LocallyFinite fun i ↦ support <| s i) (f : ι → X → M) :
    LocallyFinite fun i ↦ support <| s i • f i 
Arguments: {X : Type u_1} {M : Type u_7} {R : Type u_9} {ι : Type u_10} [TopologicalSpace X] [Zero R] [Zero M] [SMulWithZero R M] {s : ι → X → R} (h : LocallyFinite fun i => Function.support (s i)) (f : ι → X → M)
Goal: LocallyFinite fun i => Function.support (s i • f i)
-- LocallyFinite.smul_right
theorem LocallyFinite.smul_right [Zero M] [SMulZeroClass R M]
    {f : ι → X → M} (h : LocallyFinite fun i ↦ support <| f i) (s : ι → X → R) :
    LocallyFinite fun i ↦ support <| s i • f i 
Arguments: {X : Type u_1} {M : Type u_7} {R : Type u_9} {ι : Type u_10} [TopologicalSpace X] [Zero M] [SMulZeroClass R M] {f : ι → X → M} (h : LocallyFinite fun i => Function.support (f i)) (s : ι → X → R)
Goal: LocallyFinite fun i => Function.support (s i • f i)
-- PseudoEpimorphism.exists_map_eq_of_map_le'
structure PseudoEpimorphism (α β : Type*) [Preorder α] [Preorder β] extends α →o β where
  exists_map_eq_of_map_le' ⦃a : α⦄ ⦃b : β⦄ : toFun a ≤ b → ∃ c, a ≤ c ∧ toFun c = b
Arguments: {α : Type u_6} {β : Type u_7} [Preorder α] [Preorder β] (self : PseudoEpimorphism α β) ⦃a : α⦄ ⦃b : β⦄
Goal: self.toFun a ≤ b → ∃ c, a ≤ c ∧ self.toFun c = b
-- EsakiaHom.exists_map_eq_of_map_le'
structure EsakiaHom (α β : Type*) [TopologicalSpace α] [Preorder α] [TopologicalSpace β]
  [Preorder β] extends α →Co β where
  exists_map_eq_of_map_le' ⦃a : α⦄ ⦃b : β⦄ : toFun a ≤ b → ∃ c, a ≤ c ∧ toFun c = b
Arguments: {α : Type u_6} {β : Type u_7} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] (self : EsakiaHom α β) ⦃a : α⦄ ⦃b : β⦄
Goal: self.toFun a ≤ b → ∃ c, a ≤ c ∧ self.toFun c = b
-- PseudoEpimorphismClass.exists_map_eq_of_map_le
class PseudoEpimorphismClass (F : Type*) (α β : outParam Type*)
    [Preorder α] [Preorder β] [FunLike F α β]
    extends RelHomClass F ((· ≤ ·) : α → α → Prop) ((· ≤ ·) : β → β → Prop) : Prop where
  exists_map_eq_of_map_le (f : F) ⦃a : α⦄ ⦃b : β⦄ : f a ≤ b → ∃ c, a ≤ c ∧ f c = b
Arguments: {F : Type u_6} {α : outParam (Type u_7)} {β : outParam (Type u_8)} [Preorder α] [Preorder β] [FunLike F α β] [self : PseudoEpimorphismClass F α β] (f : F) ⦃a : α⦄ ⦃b : β⦄
Goal: f a ≤ b → ∃ c, a ≤ c ∧ f c = b
-- EsakiaHomClass.exists_map_eq_of_map_le
class EsakiaHomClass (F : Type*) (α β : outParam Type*) [TopologicalSpace α] [Preorder α]
    [TopologicalSpace β] [Preorder β] [FunLike F α β]
    extends ContinuousOrderHomClass F α β : Prop where
  exists_map_eq_of_map_le (f : F) ⦃a : α⦄ ⦃b : β⦄ : f a ≤ b → ∃ c, a ≤ c ∧ f c = b
Arguments: {F : Type u_6} {α : outParam (Type u_7)} {β : outParam (Type u_8)} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] [FunLike F α β] [self : EsakiaHomClass F α β] (f : F) ⦃a : α⦄ ⦃b : β⦄
Goal: f a ≤ b → ∃ c, a ≤ c ∧ f c = b
-- PseudoEpimorphism.toOrderHom_eq_coe
@[simp]
theorem toOrderHom_eq_coe (f : PseudoEpimorphism α β) : ⇑f.toOrderHom = f 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (f : PseudoEpimorphism α β)
Goal: ⇑f.toOrderHom = ⇑f
-- PseudoEpimorphism.toFun_eq_coe
theorem toFun_eq_coe {f : PseudoEpimorphism α β} : f.toFun = (f : α → β) 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] {f : PseudoEpimorphism α β}
Goal: f.toFun = ⇑f
-- PseudoEpimorphism.ext
@[ext]
theorem ext {f g : PseudoEpimorphism α β} (h : ∀ a, f a = g a) : f = g 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] {f g : PseudoEpimorphism α β} (h : ∀ (a : α), f a = g a)
Goal: f = g
-- PseudoEpimorphism.coe_copy
@[simp]
theorem coe_copy (f : PseudoEpimorphism α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (f : PseudoEpimorphism α β) (f' : α → β) (h : f' = ⇑f)
Goal: ⇑(f.copy f' h) = f'
-- PseudoEpimorphism.copy_eq
theorem copy_eq (f : PseudoEpimorphism α β) (f' : α → β) (h : f' = f) : f.copy f' h = f 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (f : PseudoEpimorphism α β) (f' : α → β) (h : f' = ⇑f)
Goal: f.copy f' h = f
-- PseudoEpimorphism.coe_id
@[simp]
theorem coe_id : ⇑(PseudoEpimorphism.id α) = id 
Arguments: (α : Type u_2) [Preorder α]
Goal: ⇑(PseudoEpimorphism.id α) = id
-- PseudoEpimorphism.coe_id_orderHom
@[simp]
theorem coe_id_orderHom : (PseudoEpimorphism.id α : α →o α) = OrderHom.id 
Arguments: (α : Type u_2) [Preorder α]
Goal: ↑(PseudoEpimorphism.id α) = OrderHom.id
-- PseudoEpimorphism.id_apply
@[simp]
theorem id_apply (a : α) : PseudoEpimorphism.id α a = a 
Arguments: {α : Type u_2} [Preorder α] (a : α)
Goal: (PseudoEpimorphism.id α) a = a
-- PseudoEpimorphism.coe_comp
@[simp]
theorem coe_comp (g : PseudoEpimorphism β γ) (f : PseudoEpimorphism α β) :
    (g.comp f : α → γ) = g ∘ f 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] (g : PseudoEpimorphism β γ) (f : PseudoEpimorphism α β)
Goal: ⇑(g.comp f) = ⇑g ∘ ⇑f
-- PseudoEpimorphism.coe_comp_orderHom
@[simp]
theorem coe_comp_orderHom (g : PseudoEpimorphism β γ) (f : PseudoEpimorphism α β) :
    (g.comp f : α →o γ) = (g : β →o γ).comp f 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] (g : PseudoEpimorphism β γ) (f : PseudoEpimorphism α β)
Goal: ↑(g.comp f) = (↑g).comp ↑f
-- PseudoEpimorphism.comp_apply
@[simp]
theorem comp_apply (g : PseudoEpimorphism β γ) (f : PseudoEpimorphism α β) (a : α) :
    (g.comp f) a = g (f a) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] (g : PseudoEpimorphism β γ) (f : PseudoEpimorphism α β) (a : α)
Goal: (g.comp f) a = g (f a)
-- PseudoEpimorphism.comp_assoc
@[simp]
theorem comp_assoc (h : PseudoEpimorphism γ δ) (g : PseudoEpimorphism β γ)
    (f : PseudoEpimorphism α β) : (h.comp g).comp f = h.comp (g.comp f) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} {δ : Type u_5} [Preorder α] [Preorder β] [Preorder γ] [Preorder δ] (h : PseudoEpimorphism γ δ) (g : PseudoEpimorphism β γ) (f : PseudoEpimorphism α β)
Goal: (h.comp g).comp f = h.comp (g.comp f)
-- PseudoEpimorphism.comp_id
@[simp]
theorem comp_id (f : PseudoEpimorphism α β) : f.comp (PseudoEpimorphism.id α) = f 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (f : PseudoEpimorphism α β)
Goal: f.comp (PseudoEpimorphism.id α) = f
-- PseudoEpimorphism.id_comp
@[simp]
theorem id_comp (f : PseudoEpimorphism α β) : (PseudoEpimorphism.id β).comp f = f 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (f : PseudoEpimorphism α β)
Goal: (PseudoEpimorphism.id β).comp f = f
-- PseudoEpimorphism.cancel_right
@[simp]
theorem cancel_right {g₁ g₂ : PseudoEpimorphism β γ} {f : PseudoEpimorphism α β}
    (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] {g₁ g₂ : PseudoEpimorphism β γ} {f : PseudoEpimorphism α β} (hf : Function.Surjective ⇑f)
Goal: g₁.comp f = g₂.comp f ↔ g₁ = g₂
-- PseudoEpimorphism.cancel_left
@[simp]
theorem cancel_left {g : PseudoEpimorphism β γ} {f₁ f₂ : PseudoEpimorphism α β} (hg : Injective g) :
    g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Preorder α] [Preorder β] [Preorder γ] {g : PseudoEpimorphism β γ} {f₁ f₂ : PseudoEpimorphism α β} (hg : Function.Injective ⇑g)
Goal: g.comp f₁ = g.comp f₂ ↔ f₁ = f₂
-- EsakiaHom.toContinuousOrderHom_coe
@[simp]
theorem toContinuousOrderHom_coe {f : EsakiaHom α β} :
    f.toContinuousOrderHom = (f : α → β) 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] {f : EsakiaHom α β}
Goal: ⇑f.toContinuousOrderHom = ⇑f
-- EsakiaHom.toFun_eq_coe
theorem toFun_eq_coe {f : EsakiaHom α β} : f.toFun = (f : α → β) 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] {f : EsakiaHom α β}
Goal: f.toFun = ⇑f
-- EsakiaHom.ext
@[ext]
theorem ext {f g : EsakiaHom α β} (h : ∀ a, f a = g a) : f = g 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] {f g : EsakiaHom α β} (h : ∀ (a : α), f a = g a)
Goal: f = g
-- EsakiaHom.coe_copy
@[simp]
theorem coe_copy (f : EsakiaHom α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] (f : EsakiaHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: ⇑(f.copy f' h) = f'
-- EsakiaHom.copy_eq
theorem copy_eq (f : EsakiaHom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] (f : EsakiaHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: f.copy f' h = f
-- EsakiaHom.coe_id
@[simp]
theorem coe_id : ⇑(EsakiaHom.id α) = id 
Arguments: (α : Type u_2) [TopologicalSpace α] [Preorder α]
Goal: ⇑(EsakiaHom.id α) = id
-- EsakiaHom.coe_id_pseudoEpimorphism
@[simp]
theorem coe_id_pseudoEpimorphism :
    (EsakiaHom.id α : PseudoEpimorphism α α) = PseudoEpimorphism.id α 
Arguments: (α : Type u_2) [TopologicalSpace α] [Preorder α]
Goal: { toOrderHom := ↑(EsakiaHom.id α), exists_map_eq_of_map_le' := ⋯ } = PseudoEpimorphism.id α
-- EsakiaHom.id_apply
@[simp]
theorem id_apply (a : α) : EsakiaHom.id α a = a 
Arguments: {α : Type u_2} [TopologicalSpace α] [Preorder α] (a : α)
Goal: (EsakiaHom.id α) a = a
-- EsakiaHom.coe_id_continuousOrderHom
@[simp]
theorem coe_id_continuousOrderHom : (EsakiaHom.id α : α →Co α) = ContinuousOrderHom.id α 
Arguments: {α : Type u_2} [TopologicalSpace α] [Preorder α]
Goal: ↑(EsakiaHom.id α) = ContinuousOrderHom.id α
-- EsakiaHom.coe_comp_continuousOrderHom
@[simp]
theorem coe_comp_continuousOrderHom (g : EsakiaHom β γ) (f : EsakiaHom α β) :
    (g.comp f : α →Co γ) = (g : β →Co γ).comp f 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] [TopologicalSpace γ] [Preorder γ] (g : EsakiaHom β γ) (f : EsakiaHom α β)
Goal: ↑(g.comp f) = (↑g).comp ↑f
-- EsakiaHom.coe_comp_pseudoEpimorphism
@[simp]
theorem coe_comp_pseudoEpimorphism (g : EsakiaHom β γ) (f : EsakiaHom α β) :
    (g.comp f : PseudoEpimorphism α γ) = (g : PseudoEpimorphism β γ).comp f 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] [TopologicalSpace γ] [Preorder γ] (g : EsakiaHom β γ) (f : EsakiaHom α β)
Goal: { toOrderHom := ↑(g.comp f), exists_map_eq_of_map_le' := ⋯ } =
  { toOrderHom := ↑g, exists_map_eq_of_map_le' := ⋯ }.comp { toOrderHom := ↑f, exists_map_eq_of_map_le' := ⋯ }
-- EsakiaHom.coe_comp
@[simp]
theorem coe_comp (g : EsakiaHom β γ) (f : EsakiaHom α β) : (g.comp f : α → γ) = g ∘ f 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] [TopologicalSpace γ] [Preorder γ] (g : EsakiaHom β γ) (f : EsakiaHom α β)
Goal: ⇑(g.comp f) = ⇑g ∘ ⇑f
-- EsakiaHom.comp_apply
@[simp]
theorem comp_apply (g : EsakiaHom β γ) (f : EsakiaHom α β) (a : α) : (g.comp f) a = g (f a) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] [TopologicalSpace γ] [Preorder γ] (g : EsakiaHom β γ) (f : EsakiaHom α β) (a : α)
Goal: (g.comp f) a = g (f a)
-- EsakiaHom.comp_assoc
@[simp]
theorem comp_assoc (h : EsakiaHom γ δ) (g : EsakiaHom β γ) (f : EsakiaHom α β) :
    (h.comp g).comp f = h.comp (g.comp f) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} {δ : Type u_5} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] [TopologicalSpace γ] [Preorder γ] [TopologicalSpace δ] [Preorder δ] (h : EsakiaHom γ δ) (g : EsakiaHom β γ) (f : EsakiaHom α β)
Goal: (h.comp g).comp f = h.comp (g.comp f)
-- EsakiaHom.comp_id
@[simp]
theorem comp_id (f : EsakiaHom α β) : f.comp (EsakiaHom.id α) = f 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] (f : EsakiaHom α β)
Goal: f.comp (EsakiaHom.id α) = f
-- EsakiaHom.id_comp
@[simp]
theorem id_comp (f : EsakiaHom α β) : (EsakiaHom.id β).comp f = f 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] (f : EsakiaHom α β)
Goal: (EsakiaHom.id β).comp f = f
-- EsakiaHom.cancel_right
@[simp]
theorem cancel_right {g₁ g₂ : EsakiaHom β γ} {f : EsakiaHom α β} (hf : Surjective f) :
    g₁.comp f = g₂.comp f ↔ g₁ = g₂ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] [TopologicalSpace γ] [Preorder γ] {g₁ g₂ : EsakiaHom β γ} {f : EsakiaHom α β} (hf : Function.Surjective ⇑f)
Goal: g₁.comp f = g₂.comp f ↔ g₁ = g₂
-- EsakiaHom.cancel_left
@[simp]
theorem cancel_left {g : EsakiaHom β γ} {f₁ f₂ : EsakiaHom α β} (hg : Injective g) :
    g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] [TopologicalSpace γ] [Preorder γ] {g : EsakiaHom β γ} {f₁ f₂ : EsakiaHom α β} (hg : Function.Injective ⇑g)
Goal: g.comp f₁ = g.comp f₂ ↔ f₁ = f₂
-- Filter.Tendsto.partialSups
protected lemma partialSups (hf : ∀ k ≤ n, Tendsto (f k) l (𝓝 (g k))) :
    Tendsto (partialSups f n) l (𝓝 (partialSups g n)) 
Arguments: {L : Type u_1} [SemilatticeSup L] [TopologicalSpace L] [ContinuousSup L] {α : Type u_2} {l : Filter α} {f : ℕ → α → L} {g : ℕ → L} {n : ℕ} (hf : ∀ k ≤ n, Filter.Tendsto (f k) l (𝓝 (g k)))
Goal: Filter.Tendsto ((partialSups f) n) l (𝓝 ((partialSups g) n))
-- Filter.Tendsto.partialSups_apply
protected lemma partialSups_apply (hf : ∀ k ≤ n, Tendsto (f k) l (𝓝 (g k))) :
    Tendsto (fun a ↦ partialSups (f · a) n) l (𝓝 (partialSups g n)) 
Arguments: {L : Type u_1} [SemilatticeSup L] [TopologicalSpace L] [ContinuousSup L] {α : Type u_2} {l : Filter α} {f : ℕ → α → L} {g : ℕ → L} {n : ℕ} (hf : ∀ k ≤ n, Filter.Tendsto (f k) l (𝓝 (g k)))
Goal: Filter.Tendsto (fun a => (partialSups fun x => f x a) n) l (𝓝 ((partialSups g) n))
-- ContinuousAt.partialSups_apply
protected lemma ContinuousAt.partialSups_apply (hf : ∀ k ≤ n, ContinuousAt (f k) x) :
    ContinuousAt (fun a ↦ partialSups (f · a) n) x 
Arguments: {L : Type u_1} [SemilatticeSup L] [TopologicalSpace L] [ContinuousSup L] {X : Type u_2} [TopologicalSpace X] {f : ℕ → X → L} {n : ℕ} {x : X} (hf : ∀ k ≤ n, ContinuousAt (f k) x)
Goal: ContinuousAt (fun a => (partialSups fun x => f x a) n) x
-- ContinuousAt.partialSups
protected lemma ContinuousAt.partialSups (hf : ∀ k ≤ n, ContinuousAt (f k) x) :
    ContinuousAt (partialSups f n) x 
Arguments: {L : Type u_1} [SemilatticeSup L] [TopologicalSpace L] [ContinuousSup L] {X : Type u_2} [TopologicalSpace X] {f : ℕ → X → L} {n : ℕ} {x : X} (hf : ∀ k ≤ n, ContinuousAt (f k) x)
Goal: ContinuousAt ((partialSups f) n) x
-- ContinuousWithinAt.partialSups_apply
protected lemma ContinuousWithinAt.partialSups_apply (hf : ∀ k ≤ n, ContinuousWithinAt (f k) s x) :
    ContinuousWithinAt (fun a ↦ partialSups (f · a) n) s x 
Arguments: {L : Type u_1} [SemilatticeSup L] [TopologicalSpace L] [ContinuousSup L] {X : Type u_2} [TopologicalSpace X] {f : ℕ → X → L} {n : ℕ} {s : Set X} {x : X} (hf : ∀ k ≤ n, ContinuousWithinAt (f k) s x)
Goal: ContinuousWithinAt (fun a => (partialSups fun x => f x a) n) s x
-- ContinuousWithinAt.partialSups
protected lemma ContinuousWithinAt.partialSups (hf : ∀ k ≤ n, ContinuousWithinAt (f k) s x) :
    ContinuousWithinAt (partialSups f n) s x 
Arguments: {L : Type u_1} [SemilatticeSup L] [TopologicalSpace L] [ContinuousSup L] {X : Type u_2} [TopologicalSpace X] {f : ℕ → X → L} {n : ℕ} {s : Set X} {x : X} (hf : ∀ k ≤ n, ContinuousWithinAt (f k) s x)
Goal: ContinuousWithinAt ((partialSups f) n) s x
-- ContinuousOn.partialSups_apply
protected lemma ContinuousOn.partialSups_apply (hf : ∀ k ≤ n, ContinuousOn (f k) s) :
    ContinuousOn (fun a ↦ partialSups (f · a) n) s 
Arguments: {L : Type u_1} [SemilatticeSup L] [TopologicalSpace L] [ContinuousSup L] {X : Type u_2} [TopologicalSpace X] {f : ℕ → X → L} {n : ℕ} {s : Set X} (hf : ∀ k ≤ n, ContinuousOn (f k) s)
Goal: ContinuousOn (fun a => (partialSups fun x => f x a) n) s
-- ContinuousOn.partialSups
protected lemma ContinuousOn.partialSups (hf : ∀ k ≤ n, ContinuousOn (f k) s) :
    ContinuousOn (partialSups f n) s 
Arguments: {L : Type u_1} [SemilatticeSup L] [TopologicalSpace L] [ContinuousSup L] {X : Type u_2} [TopologicalSpace X] {f : ℕ → X → L} {n : ℕ} {s : Set X} (hf : ∀ k ≤ n, ContinuousOn (f k) s)
Goal: ContinuousOn ((partialSups f) n) s
-- Continuous.partialSups_apply
protected lemma Continuous.partialSups_apply (hf : ∀ k ≤ n, Continuous (f k)) :
    Continuous (fun a ↦ partialSups (f · a) n) 
Arguments: {L : Type u_1} [SemilatticeSup L] [TopologicalSpace L] [ContinuousSup L] {X : Type u_2} [TopologicalSpace X] {f : ℕ → X → L} {n : ℕ} (hf : ∀ k ≤ n, Continuous (f k))
Goal: Continuous fun a => (partialSups fun x => f x a) n
-- Continuous.partialSups
protected lemma Continuous.partialSups (hf : ∀ k ≤ n, Continuous (f k)) :
    Continuous (partialSups f n) 
Arguments: {L : Type u_1} [SemilatticeSup L] [TopologicalSpace L] [ContinuousSup L] {X : Type u_2} [TopologicalSpace X] {f : ℕ → X → L} {n : ℕ} (hf : ∀ k ≤ n, Continuous (f k))
Goal: Continuous ((partialSups f) n)
-- TopologicalSpace.NoetherianSpace.wellFounded_opens
@[mk_iff]
class NoetherianSpace : Prop where
  wellFounded_opens : WellFounded ((· > ·) : Opens α → Opens α → Prop)
Arguments: {α : Type u_1} [TopologicalSpace α] [self : TopologicalSpace.NoetherianSpace α]
Goal: WellFounded fun x x_1 => x > x_1
-- TopologicalSpace.noetherianSpace_iff
@[mk_iff]
class NoetherianSpace : Prop where
  wellFounded_opens : WellFounded ((· > ·) : Opens α → Opens α → Prop)
Arguments: (α : Type u_1) [TopologicalSpace α]
Goal: TopologicalSpace.NoetherianSpace α ↔ WellFounded fun x x_1 => x > x_1
-- TopologicalSpace.noetherianSpace_iff_opens
theorem noetherianSpace_iff_opens : NoetherianSpace α ↔ ∀ s : Opens α, IsCompact (s : Set α) 
Arguments: (α : Type u_1) [TopologicalSpace α]
Goal: TopologicalSpace.NoetherianSpace α ↔ ∀ (s : TopologicalSpace.Opens α), IsCompact ↑s
-- TopologicalSpace.NoetherianSpace.isCompact
protected theorem NoetherianSpace.isCompact [NoetherianSpace α] (s : Set α) : IsCompact s 
Arguments: {α : Type u_1} [TopologicalSpace α] [TopologicalSpace.NoetherianSpace α] (s : Set α)
Goal: IsCompact s
-- Inducing.noetherianSpace
protected theorem _root_.Inducing.noetherianSpace [NoetherianSpace α] {i : β → α}
    (hi : Inducing i) : NoetherianSpace β 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace.NoetherianSpace α] {i : β → α} (hi : Inducing i)
Goal: TopologicalSpace.NoetherianSpace β
-- TopologicalSpace.noetherianSpace_TFAE
theorem noetherianSpace_TFAE :
    TFAE [NoetherianSpace α,
      WellFounded fun s t : Closeds α => s < t,
      ∀ s : Set α, IsCompact s,
      ∀ s : Opens α, IsCompact (s : Set α)] 
Arguments: (α : Type u_1) [TopologicalSpace α]
Goal: [TopologicalSpace.NoetherianSpace α, WellFounded fun s t => s < t, ∀ (s : Set α), IsCompact s,
    ∀ (s : TopologicalSpace.Opens α), IsCompact ↑s].TFAE
-- TopologicalSpace.noetherianSpace_iff_isCompact
theorem noetherianSpace_iff_isCompact : NoetherianSpace α ↔ ∀ s : Set α, IsCompact s 
Arguments: {α : Type u_1} [TopologicalSpace α]
Goal: TopologicalSpace.NoetherianSpace α ↔ ∀ (s : Set α), IsCompact s
-- TopologicalSpace.NoetherianSpace.wellFounded_closeds
theorem NoetherianSpace.wellFounded_closeds [NoetherianSpace α] :
    WellFounded fun s t : Closeds α => s < t 
Arguments: {α : Type u_1} [TopologicalSpace α] [TopologicalSpace.NoetherianSpace α]
Goal: WellFounded fun s t => s < t
-- TopologicalSpace.noetherianSpace_of_surjective
theorem noetherianSpace_of_surjective [NoetherianSpace α] (f : α → β) (hf : Continuous f)
    (hf' : Function.Surjective f) : NoetherianSpace β 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace.NoetherianSpace α] (f : α → β) (hf : Continuous f) (hf' : Function.Surjective f)
Goal: TopologicalSpace.NoetherianSpace β
-- TopologicalSpace.noetherianSpace_iff_of_homeomorph
theorem noetherianSpace_iff_of_homeomorph (f : α ≃ₜ β) : NoetherianSpace α ↔ NoetherianSpace β 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α ≃ₜ β)
Goal: TopologicalSpace.NoetherianSpace α ↔ TopologicalSpace.NoetherianSpace β
-- TopologicalSpace.NoetherianSpace.range
theorem NoetherianSpace.range [NoetherianSpace α] (f : α → β) (hf : Continuous f) :
    NoetherianSpace (Set.range f) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace.NoetherianSpace α] (f : α → β) (hf : Continuous f)
Goal: TopologicalSpace.NoetherianSpace ↑(Set.range f)
-- TopologicalSpace.noetherianSpace_set_iff
theorem noetherianSpace_set_iff (s : Set α) :
    NoetherianSpace s ↔ ∀ t, t ⊆ s → IsCompact t 
Arguments: {α : Type u_1} [TopologicalSpace α] (s : Set α)
Goal: TopologicalSpace.NoetherianSpace ↑s ↔ ∀ t ⊆ s, IsCompact t
-- TopologicalSpace.noetherian_univ_iff
@[simp]
theorem noetherian_univ_iff : NoetherianSpace (Set.univ : Set α) ↔ NoetherianSpace α 
Arguments: {α : Type u_1} [TopologicalSpace α]
Goal: TopologicalSpace.NoetherianSpace ↑Set.univ ↔ TopologicalSpace.NoetherianSpace α
-- TopologicalSpace.NoetherianSpace.iUnion
theorem NoetherianSpace.iUnion {ι : Type*} (f : ι → Set α) [Finite ι]
    [hf : ∀ i, NoetherianSpace (f i)] : NoetherianSpace (⋃ i, f i) 
Arguments: {α : Type u_1} [TopologicalSpace α] {ι : Type u_3} (f : ι → Set α) [Finite ι] [hf : ∀ (i : ι), TopologicalSpace.NoetherianSpace ↑(f i)]
Goal: TopologicalSpace.NoetherianSpace ↑(⋃ i, f i)
-- TopologicalSpace.NoetherianSpace.discrete
theorem NoetherianSpace.discrete [NoetherianSpace α] [T2Space α] : DiscreteTopology α 
Arguments: {α : Type u_1} [TopologicalSpace α] [TopologicalSpace.NoetherianSpace α] [T2Space α]
Goal: DiscreteTopology α
-- TopologicalSpace.NoetherianSpace.finite
theorem NoetherianSpace.finite [NoetherianSpace α] [T2Space α] : Finite α 
Arguments: {α : Type u_1} [TopologicalSpace α] [TopologicalSpace.NoetherianSpace α] [T2Space α]
Goal: Finite α