ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousOn.smul
@[to_additive (attr := fun_prop)]
theorem ContinuousOn.smul (hf : ContinuousOn f s) (hg : ContinuousOn g s) :
    ContinuousOn (fun x => f x • g x) s 
Arguments: {M : Type u_1} {X : Type u_2} {Y : Type u_3} [TopologicalSpace M] [TopologicalSpace X] [TopologicalSpace Y] [SMul M X] [ContinuousSMul M X] {f : Y → M} {g : Y → X} {s : Set Y} (hf : ContinuousOn f s) (hg : ContinuousOn g s)
Goal: ContinuousOn (fun x => f x • g x) s
-- Continuous.vadd
@[to_additive (attr := continuity, fun_prop)]
theorem Continuous.smul (hf : Continuous f) (hg : Continuous g) : Continuous fun x => f x • g x 
Arguments: {M : Type u_1} {X : Type u_2} {Y : Type u_3} [TopologicalSpace M] [TopologicalSpace X] [TopologicalSpace Y] [VAdd M X] [ContinuousVAdd M X] {f : Y → M} {g : Y → X} (hf : Continuous f) (hg : Continuous g)
Goal: Continuous fun x => f x +ᵥ g x
-- Continuous.smul
@[to_additive (attr := continuity, fun_prop)]
theorem Continuous.smul (hf : Continuous f) (hg : Continuous g) : Continuous fun x => f x • g x 
Arguments: {M : Type u_1} {X : Type u_2} {Y : Type u_3} [TopologicalSpace M] [TopologicalSpace X] [TopologicalSpace Y] [SMul M X] [ContinuousSMul M X] {f : Y → M} {g : Y → X} (hf : Continuous f) (hg : Continuous g)
Goal: Continuous fun x => f x • g x
-- Specializes.smul
@[to_additive]
protected theorem Specializes.smul {a b : M} {x y : X} (h₁ : a ⤳ b) (h₂ : x ⤳ y) :
    (a • x) ⤳ (b • y) 
Arguments: {M : Type u_1} {X : Type u_2} [TopologicalSpace M] [TopologicalSpace X] [SMul M X] [ContinuousSMul M X] {a b : M} {x y : X} (h₁ : a ⤳ b) (h₂ : x ⤳ y)
Goal: (a • x) ⤳ (b • y)
-- Specializes.vadd
@[to_additive]
protected theorem Specializes.smul {a b : M} {x y : X} (h₁ : a ⤳ b) (h₂ : x ⤳ y) :
    (a • x) ⤳ (b • y) 
Arguments: {M : Type u_1} {X : Type u_2} [TopologicalSpace M] [TopologicalSpace X] [VAdd M X] [ContinuousVAdd M X] {a b : M} {x y : X} (h₁ : a ⤳ b) (h₂ : x ⤳ y)
Goal: (a +ᵥ x) ⤳ (b +ᵥ y)
-- Inseparable.vadd
@[to_additive]
protected theorem Inseparable.smul {a b : M} {x y : X} (h₁ : Inseparable a b)
    (h₂ : Inseparable x y) : Inseparable (a • x) (b • y) 
Arguments: {M : Type u_1} {X : Type u_2} [TopologicalSpace M] [TopologicalSpace X] [VAdd M X] [ContinuousVAdd M X] {a b : M} {x y : X} (h₁ : Inseparable a b) (h₂ : Inseparable x y)
Goal: Inseparable (a +ᵥ x) (b +ᵥ y)
-- Inseparable.smul
@[to_additive]
protected theorem Inseparable.smul {a b : M} {x y : X} (h₁ : Inseparable a b)
    (h₂ : Inseparable x y) : Inseparable (a • x) (b • y) 
Arguments: {M : Type u_1} {X : Type u_2} [TopologicalSpace M] [TopologicalSpace X] [SMul M X] [ContinuousSMul M X] {a b : M} {x y : X} (h₁ : Inseparable a b) (h₂ : Inseparable x y)
Goal: Inseparable (a • x) (b • y)
-- IsCompact.vadd_set
@[to_additive]
lemma IsCompact.smul_set {k : Set M} {u : Set X} (hk : IsCompact k) (hu : IsCompact u) :
    IsCompact (k • u) 
Arguments: {M : Type u_1} {X : Type u_2} [TopologicalSpace M] [TopologicalSpace X] [VAdd M X] [ContinuousVAdd M X] {k : Set M} {u : Set X} (hk : IsCompact k) (hu : IsCompact u)
Goal: IsCompact (k +ᵥ u)
-- IsCompact.smul_set
@[to_additive]
lemma IsCompact.smul_set {k : Set M} {u : Set X} (hk : IsCompact k) (hu : IsCompact u) :
    IsCompact (k • u) 
Arguments: {M : Type u_1} {X : Type u_2} [TopologicalSpace M] [TopologicalSpace X] [SMul M X] [ContinuousSMul M X] {k : Set M} {u : Set X} (hk : IsCompact k) (hu : IsCompact u)
Goal: IsCompact (k • u)
-- vadd_set_closure_subset
@[to_additive]
lemma smul_set_closure_subset (K : Set M) (L : Set X) :
    closure K • closure L ⊆ closure (K • L) 
Arguments: {M : Type u_1} {X : Type u_2} [TopologicalSpace M] [TopologicalSpace X] [VAdd M X] [ContinuousVAdd M X] (K : Set M) (L : Set X)
Goal: closure K +ᵥ closure L ⊆ closure (K +ᵥ L)
-- smul_set_closure_subset
@[to_additive]
lemma smul_set_closure_subset (K : Set M) (L : Set X) :
    closure K • closure L ⊆ closure (K • L) 
Arguments: {M : Type u_1} {X : Type u_2} [TopologicalSpace M] [TopologicalSpace X] [SMul M X] [ContinuousSMul M X] (K : Set M) (L : Set X)
Goal: closure K • closure L ⊆ closure (K • L)
-- Inducing.continuousSMul
@[to_additive
  "Suppose that `N` additively acts on `X` and `M` continuously additively acts on `Y`.
Suppose that `g : Y → X` is an additive action homomorphism in the following sense:
there exists a continuous function `f : N → M` such that `g (c +ᵥ x) = f c +ᵥ g x`.
Then the action of `N` on `X` is continuous as well.

In many cases, `f = id` so that `g` is an action homomorphism in the sense of `AddActionHom`.
However, this version also works for `f = AddUnits.val`."]
lemma Inducing.continuousSMul {N : Type*} [SMul N Y] [TopologicalSpace N] {f : N → M}
    (hg : Inducing g) (hf : Continuous f) (hsmul : ∀ {c x}, g (c • x) = f c • g x) :
    ContinuousSMul N Y 
Arguments: {M : Type u_1} {X : Type u_2} {Y : Type u_3} [TopologicalSpace M] [TopologicalSpace X] [TopologicalSpace Y] [SMul M X] [ContinuousSMul M X] {g : Y → X} {N : Type u_5} [SMul N Y] [TopologicalSpace N] {f : N → M} (hg : Inducing g) (hf : Continuous f) (hsmul : ∀ {c : N} {x : Y}, g (c • x) = f c • g x)
Goal: ContinuousSMul N Y
-- Inducing.continuousVAdd
@[to_additive
  "Suppose that `N` additively acts on `X` and `M` continuously additively acts on `Y`.
Suppose that `g : Y → X` is an additive action homomorphism in the following sense:
there exists a continuous function `f : N → M` such that `g (c +ᵥ x) = f c +ᵥ g x`.
Then the action of `N` on `X` is continuous as well.

In many cases, `f = id` so that `g` is an action homomorphism in the sense of `AddActionHom`.
However, this version also works for `f = AddUnits.val`."]
lemma Inducing.continuousSMul {N : Type*} [SMul N Y] [TopologicalSpace N] {f : N → M}
    (hg : Inducing g) (hf : Continuous f) (hsmul : ∀ {c x}, g (c • x) = f c • g x) :
    ContinuousSMul N Y 
Arguments: {M : Type u_1} {X : Type u_2} {Y : Type u_3} [TopologicalSpace M] [TopologicalSpace X] [TopologicalSpace Y] [VAdd M X] [ContinuousVAdd M X] {g : Y → X} {N : Type u_5} [VAdd N Y] [TopologicalSpace N] {f : N → M} (hg : Inducing g) (hf : Continuous f) (hsmul : ∀ {c : N} {x : Y}, g (c +ᵥ x) = f c +ᵥ g x)
Goal: ContinuousVAdd N Y
-- AddAction.continuousVAdd_compHom
@[to_additive]
theorem MulAction.continuousSMul_compHom
    {N : Type*} [TopologicalSpace N] [Monoid N] {f : N →* M} (hf : Continuous f) :
    letI : MulAction N X := MulAction.compHom _ f
    ContinuousSMul N X 
Arguments: {M : Type u_1} {X : Type u_2} [TopologicalSpace M] [TopologicalSpace X] [AddMonoid M] [AddAction M X] [ContinuousVAdd M X] {N : Type u_5} [TopologicalSpace N] [AddMonoid N] {f : N →+ M} (hf : Continuous ⇑f)
Goal: ContinuousVAdd N X
-- MulAction.continuousSMul_compHom
@[to_additive]
theorem MulAction.continuousSMul_compHom
    {N : Type*} [TopologicalSpace N] [Monoid N] {f : N →* M} (hf : Continuous f) :
    letI : MulAction N X := MulAction.compHom _ f
    ContinuousSMul N X 
Arguments: {M : Type u_1} {X : Type u_2} [TopologicalSpace M] [TopologicalSpace X] [Monoid M] [MulAction M X] [ContinuousSMul M X] {N : Type u_5} [TopologicalSpace N] [Monoid N] {f : N →* M} (hf : Continuous ⇑f)
Goal: ContinuousSMul N X
-- continuousVAdd_sInf
@[to_additive]
theorem continuousSMul_sInf {ts : Set (TopologicalSpace X)}
    (h : ∀ t ∈ ts, @ContinuousSMul M X _ _ t) : @ContinuousSMul M X _ _ (sInf ts) 
Arguments: {M : Type u_2} {X : Type u_3} [TopologicalSpace M] [VAdd M X] {ts : Set (TopologicalSpace X)} (h : ∀ t ∈ ts, ContinuousVAdd M X)
Goal: ContinuousVAdd M X
-- continuousSMul_sInf
@[to_additive]
theorem continuousSMul_sInf {ts : Set (TopologicalSpace X)}
    (h : ∀ t ∈ ts, @ContinuousSMul M X _ _ t) : @ContinuousSMul M X _ _ (sInf ts) 
Arguments: {M : Type u_2} {X : Type u_3} [TopologicalSpace M] [SMul M X] {ts : Set (TopologicalSpace X)} (h : ∀ t ∈ ts, ContinuousSMul M X)
Goal: ContinuousSMul M X
-- continuousSMul_iInf
@[to_additive]
theorem continuousSMul_iInf {ts' : ι → TopologicalSpace X}
    (h : ∀ i, @ContinuousSMul M X _ _ (ts' i)) : @ContinuousSMul M X _ _ (⨅ i, ts' i) 
Arguments: {ι : Sort u_1} {M : Type u_2} {X : Type u_3} [TopologicalSpace M] [SMul M X] {ts' : ι → TopologicalSpace X} (h : ∀ (i : ι), ContinuousSMul M X)
Goal: ContinuousSMul M X
-- continuousVAdd_iInf
@[to_additive]
theorem continuousSMul_iInf {ts' : ι → TopologicalSpace X}
    (h : ∀ i, @ContinuousSMul M X _ _ (ts' i)) : @ContinuousSMul M X _ _ (⨅ i, ts' i) 
Arguments: {ι : Sort u_1} {M : Type u_2} {X : Type u_3} [TopologicalSpace M] [VAdd M X] {ts' : ι → TopologicalSpace X} (h : ∀ (i : ι), ContinuousVAdd M X)
Goal: ContinuousVAdd M X
-- continuousSMul_inf
@[to_additive]
theorem continuousSMul_inf {t₁ t₂ : TopologicalSpace X} [@ContinuousSMul M X _ _ t₁]
    [@ContinuousSMul M X _ _ t₂] : @ContinuousSMul M X _ _ (t₁ ⊓ t₂) 
Arguments: {M : Type u_2} {X : Type u_3} [TopologicalSpace M] [SMul M X] {t₁ t₂ : TopologicalSpace X} [ContinuousSMul M X] [ContinuousSMul M X]
Goal: ContinuousSMul M X
-- continuousVAdd_inf
@[to_additive]
theorem continuousSMul_inf {t₁ t₂ : TopologicalSpace X} [@ContinuousSMul M X _ _ t₁]
    [@ContinuousSMul M X _ _ t₂] : @ContinuousSMul M X _ _ (t₁ ⊓ t₂) 
Arguments: {M : Type u_2} {X : Type u_3} [TopologicalSpace M] [VAdd M X] {t₁ t₂ : TopologicalSpace X} [ContinuousVAdd M X] [ContinuousVAdd M X]
Goal: ContinuousVAdd M X
-- AddTorsor.connectedSpace
protected theorem AddTorsor.connectedSpace : ConnectedSpace P 
Arguments: (G : Type u_1) (P : Type u_2) [AddGroup G] [AddTorsor G P] [TopologicalSpace G] [PreconnectedSpace G] [TopologicalSpace P] [ContinuousVAdd G P]
Goal: ConnectedSpace P
-- HasUpperLowerClosure.isUpperSet_closure
class HasUpperLowerClosure (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where
  isUpperSet_closure : ∀ s : Set α, IsUpperSet s → IsUpperSet (closure s)
  isLowerSet_closure : ∀ s : Set α, IsLowerSet s → IsLowerSet (closure s)
  isOpen_upperClosure : ∀ s : Set α, IsOpen s → IsOpen (upperClosure s : Set α)
  isOpen_lowerClosure : ∀ s : Set α, IsOpen s → IsOpen (lowerClosure s : Set α)
Arguments: {α : Type u_1} [TopologicalSpace α] [Preorder α] [self : HasUpperLowerClosure α] (s : Set α)
Goal: IsUpperSet s → IsUpperSet (closure s)
-- HasUpperLowerClosure.isOpen_upperClosure
class HasUpperLowerClosure (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where
  isUpperSet_closure : ∀ s : Set α, IsUpperSet s → IsUpperSet (closure s)
  isLowerSet_closure : ∀ s : Set α, IsLowerSet s → IsLowerSet (closure s)
  isOpen_upperClosure : ∀ s : Set α, IsOpen s → IsOpen (upperClosure s : Set α)
  isOpen_lowerClosure : ∀ s : Set α, IsOpen s → IsOpen (lowerClosure s : Set α)
Arguments: {α : Type u_1} [TopologicalSpace α] [Preorder α] [self : HasUpperLowerClosure α] (s : Set α)
Goal: IsOpen s → IsOpen ↑(upperClosure s)
-- HasUpperLowerClosure.isOpen_lowerClosure
class HasUpperLowerClosure (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where
  isUpperSet_closure : ∀ s : Set α, IsUpperSet s → IsUpperSet (closure s)
  isLowerSet_closure : ∀ s : Set α, IsLowerSet s → IsLowerSet (closure s)
  isOpen_upperClosure : ∀ s : Set α, IsOpen s → IsOpen (upperClosure s : Set α)
  isOpen_lowerClosure : ∀ s : Set α, IsOpen s → IsOpen (lowerClosure s : Set α)
Arguments: {α : Type u_1} [TopologicalSpace α] [Preorder α] [self : HasUpperLowerClosure α] (s : Set α)
Goal: IsOpen s → IsOpen ↑(lowerClosure s)
-- HasUpperLowerClosure.isLowerSet_closure
class HasUpperLowerClosure (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where
  isUpperSet_closure : ∀ s : Set α, IsUpperSet s → IsUpperSet (closure s)
  isLowerSet_closure : ∀ s : Set α, IsLowerSet s → IsLowerSet (closure s)
  isOpen_upperClosure : ∀ s : Set α, IsOpen s → IsOpen (upperClosure s : Set α)
  isOpen_lowerClosure : ∀ s : Set α, IsOpen s → IsOpen (lowerClosure s : Set α)
Arguments: {α : Type u_1} [TopologicalSpace α] [Preorder α] [self : HasUpperLowerClosure α] (s : Set α)
Goal: IsLowerSet s → IsLowerSet (closure s)
-- IsUpperSet.closure
protected theorem IsUpperSet.closure : IsUpperSet s → IsUpperSet (closure s) 
Arguments: {α : Type u_1} [TopologicalSpace α] [Preorder α] [HasUpperLowerClosure α] {s : Set α}
Goal: IsUpperSet s → IsUpperSet (closure s)
-- IsLowerSet.closure
protected theorem IsLowerSet.closure : IsLowerSet s → IsLowerSet (closure s) 
Arguments: {α : Type u_1} [TopologicalSpace α] [Preorder α] [HasUpperLowerClosure α] {s : Set α}
Goal: IsLowerSet s → IsLowerSet (closure s)
-- IsOpen.upperClosure
protected theorem IsOpen.upperClosure : IsOpen s → IsOpen (upperClosure s : Set α) 
Arguments: {α : Type u_1} [TopologicalSpace α] [Preorder α] [HasUpperLowerClosure α] {s : Set α}
Goal: IsOpen s → IsOpen ↑(upperClosure s)
-- IsOpen.lowerClosure
protected theorem IsOpen.lowerClosure : IsOpen s → IsOpen (lowerClosure s : Set α) 
Arguments: {α : Type u_1} [TopologicalSpace α] [Preorder α] [HasUpperLowerClosure α] {s : Set α}
Goal: IsOpen s → IsOpen ↑(lowerClosure s)
-- IsUpperSet.interior
protected theorem IsUpperSet.interior (h : IsUpperSet s) : IsUpperSet (interior s) 
Arguments: {α : Type u_1} [TopologicalSpace α] [Preorder α] [HasUpperLowerClosure α] {s : Set α} (h : IsUpperSet s)
Goal: IsUpperSet (interior s)
-- IsLowerSet.interior
protected theorem IsLowerSet.interior (h : IsLowerSet s) : IsLowerSet (interior s) 
Arguments: {α : Type u_1} [TopologicalSpace α] [Preorder α] [HasUpperLowerClosure α] {s : Set α} (h : IsLowerSet s)
Goal: IsLowerSet (interior s)
-- Set.OrdConnected.interior
protected theorem Set.OrdConnected.interior (h : s.OrdConnected) : (interior s).OrdConnected 
Arguments: {α : Type u_1} [TopologicalSpace α] [Preorder α] [HasUpperLowerClosure α] {s : Set α} (h : s.OrdConnected)
Goal: (interior s).OrdConnected
-- ContinuousMapZero.map_zero'
structure ContinuousMapZero (X R : Type*) [Zero X] [Zero R] [TopologicalSpace X]
    [TopologicalSpace R] extends C(X, R) where
  map_zero' : toContinuousMap 0 = 0
Arguments: {X : Type u_1} {R : Type u_2} [Zero X] [Zero R] [TopologicalSpace X] [TopologicalSpace R] (self : ContinuousMapZero X R)
Goal: self.toContinuousMap 0 = 0
-- ContinuousMapZero.ext
@[ext]
lemma ext {f g : C(X, R)₀} (h : ∀ x, f x = g x) : f = g 
Arguments: {X : Type u_1} {R : Type u_3} [Zero X] [Zero R] [TopologicalSpace X] [TopologicalSpace R] {f g : ContinuousMapZero X R} (h : ∀ (x : X), f x = g x)
Goal: f = g
-- ContinuousMapZero.coe_mk
@[simp]
lemma coe_mk {f : C(X, R)} {h0 : f 0 = 0} : ⇑(mk f h0) = f 
Arguments: {X : Type u_1} {R : Type u_3} [Zero X] [Zero R] [TopologicalSpace X] [TopologicalSpace R] {f : C(X, R)} {h0 : f 0 = 0}
Goal: ⇑{ toContinuousMap := f, map_zero' := h0 } = ⇑f
-- ContinuousMapZero.toContinuousMap_injective
lemma toContinuousMap_injective [Zero R] : Injective ((↑) : C(X, R)₀ → C(X, R)) 
Arguments: {X : Type u_1} {R : Type u_3} [Zero X] [TopologicalSpace X] [TopologicalSpace R] [Zero R]
Goal: Function.Injective toContinuousMap
-- ContinuousMapZero.range_toContinuousMap
lemma range_toContinuousMap : range ((↑) : C(X, R)₀ → C(X, R)) = {f : C(X, R) | f 0 = 0} 
Arguments: {X : Type u_1} {R : Type u_3} [Zero X] [Zero R] [TopologicalSpace X] [TopologicalSpace R]
Goal: Set.range toContinuousMap = {f | f 0 = 0}
-- ContinuousMapZero.comp_apply
@[simp]
lemma comp_apply (g : C(Y, R)₀) (f : C(X, Y)₀) (x : X) : g.comp f x = g (f x) 
Arguments: {X : Type u_1} {Y : Type u_2} {R : Type u_3} [Zero X] [Zero Y] [Zero R] [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace R] (g : ContinuousMapZero Y R) (f : ContinuousMapZero X Y) (x : X)
Goal: (g.comp f) x = g (f x)
-- ContinuousMapZero.le_def
lemma le_def [PartialOrder R] (f g : C(X, R)₀) : f ≤ g ↔ ∀ x, f x ≤ g x 
Arguments: {X : Type u_1} {R : Type u_3} [Zero X] [Zero R] [TopologicalSpace X] [TopologicalSpace R] [PartialOrder R] (f g : ContinuousMapZero X R)
Goal: f ≤ g ↔ ∀ (x : X), f x ≤ g x
-- ContinuousMapZero.embedding_toContinuousMap
lemma embedding_toContinuousMap : Embedding ((↑) : C(X, R)₀ → C(X, R)) 
Arguments: {X : Type u_1} {R : Type u_3} [Zero X] [Zero R] [TopologicalSpace X] [TopologicalSpace R]
Goal: Embedding toContinuousMap
-- ContinuousMapZero.closedEmbedding_toContinuousMap
lemma closedEmbedding_toContinuousMap [T1Space R] :
    ClosedEmbedding ((↑) : C(X, R)₀ → C(X, R)) 
Arguments: {X : Type u_1} {R : Type u_3} [Zero X] [Zero R] [TopologicalSpace X] [TopologicalSpace R] [T1Space R]
Goal: ClosedEmbedding toContinuousMap
-- ContinuousMapZero.id_toFun
@[simps!]
protected def id {s : Set R} [Zero s] (h0 : ((0 : s) : R) = 0) : C(s, R)₀ 
Arguments: {R : Type u_3} [Zero R] [TopologicalSpace R] {s : Set R} [Zero ↑s] (h0 : ↑0 = 0)
Goal: ∀ (a : ↑s), (ContinuousMapZero.id h0) a = ↑a
-- ContinuousMapZero.toContinuousMap_id
@[simp]
lemma toContinuousMap_id {s : Set R} [Zero s] (h0 : ((0 : s) : R) = 0) :
    (ContinuousMapZero.id h0 : C(s, R)) = .restrict s (.id R) 
Arguments: {R : Type u_3} [Zero R] [TopologicalSpace R] {s : Set R} [Zero ↑s] (h0 : ↑0 = 0)
Goal: ↑(ContinuousMapZero.id h0) = ContinuousMap.restrict s (ContinuousMap.id R)
-- ContinuousMapZero.coe_zero
@[simp] lemma coe_zero [Zero R] : ⇑(0 : C(X, R)₀) = 0 
Arguments: {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] [Zero R]
Goal: ⇑0 = 0
-- ContinuousMapZero.coe_add
@[simp] lemma coe_add [AddZeroClass R] [ContinuousAdd R] (f g : C(X, R)₀) : ⇑(f + g) = f + g 
Arguments: {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] [AddZeroClass R] [ContinuousAdd R] (f g : ContinuousMapZero X R)
Goal: ⇑(f + g) = ⇑f + ⇑g
-- ContinuousMapZero.coe_mul
@[simp] lemma coe_mul [MulZeroClass R] [ContinuousMul R] (f g : C(X, R)₀) : ⇑(f * g) = f * g 
Arguments: {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] [MulZeroClass R] [ContinuousMul R] (f g : ContinuousMapZero X R)
Goal: ⇑(f * g) = ⇑f * ⇑g
-- ContinuousMapZero.coe_smul
@[simp] lemma coe_smul {M : Type*} [Zero R] [SMulZeroClass M R] [ContinuousConstSMul M R]
    (m : M) (f : C(X, R)₀) : ⇑(m • f) = m • f 
Arguments: {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] {M : Type u_3} [Zero R] [SMulZeroClass M R] [ContinuousConstSMul M R] (m : M) (f : ContinuousMapZero X R)
Goal: ⇑(m • f) = m • ⇑f
-- ContinuousMapZero.coe_star
@[simp] lemma coe_star [StarRing R] [ContinuousStar R] (f : C(X, R)₀) : ⇑(star f) = star ⇑f 
Arguments: {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] [CommSemiring R] [TopologicalSemiring R] [StarRing R] [ContinuousStar R] (f : ContinuousMapZero X R)
Goal: ⇑(star f) = star ⇑f
-- ContinuousMapZero.toContinuousMapHom_apply
@[simps]
def toContinuousMapHom [StarRing R] [ContinuousStar R] : C(X, R)₀ →⋆ₙₐ[R] C(X, R) 
Arguments: {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] [CommSemiring R] [TopologicalSemiring R] [StarRing R] [ContinuousStar R] (f : ContinuousMapZero X R)
Goal: ContinuousMapZero.toContinuousMapHom f = ↑f
-- ContinuousMapZero.coe_toContinuousMapHom
lemma coe_toContinuousMapHom [StarRing R] [ContinuousStar R] :
    ⇑(toContinuousMapHom (X := X) (R := R)) = (↑) 
Arguments: {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] [CommSemiring R] [TopologicalSemiring R] [StarRing R] [ContinuousStar R]
Goal: ⇑ContinuousMapZero.toContinuousMapHom = toContinuousMap
-- ContinuousMapZero.coe_sum
@[simp] lemma coe_sum {ι : Type*} (s : Finset ι)
    (f : ι → C(X, R)₀) : ⇑(s.sum f) = s.sum (fun i => ⇑(f i)) 
Arguments: {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [TopologicalSpace R] [CommSemiring R] [TopologicalSemiring R] {ι : Type u_3} (s : Finset ι) (f : ι → ContinuousMapZero X R)
Goal: ⇑(s.sum f) = ∑ i ∈ s, ⇑(f i)
-- ContinuousMapZero.coe_neg
@[simp]
lemma coe_neg (f : C(X, R)₀) : ⇑(-f) = -⇑f 
Arguments: {X : Type u_3} {R : Type u_4} [Zero X] [TopologicalSpace X] [CommRing R] [TopologicalSpace R] [TopologicalRing R] (f : ContinuousMapZero X R)
Goal: ⇑(-f) = -⇑f
-- ContinuousMapZero.uniformEmbedding_toContinuousMap
lemma uniformEmbedding_toContinuousMap :
    UniformEmbedding ((↑) : C(X, R)₀ → C(X, R)) 
Arguments: {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [Zero R] [UniformSpace R]
Goal: UniformEmbedding toContinuousMap
-- ContinuousMapZero.uniformEmbedding_comp
lemma uniformEmbedding_comp {Y : Type*} [UniformSpace Y] [Zero Y] (g : C(Y, R)₀)
    (hg : UniformEmbedding g) : UniformEmbedding (g.comp · : C(X, Y)₀ → C(X, R)₀) 
Arguments: {X : Type u_1} {R : Type u_2} [Zero X] [TopologicalSpace X] [Zero R] [UniformSpace R] {Y : Type u_3} [UniformSpace Y] [Zero Y] (g : ContinuousMapZero Y R) (hg : UniformEmbedding ⇑g)
Goal: UniformEmbedding fun x => g.comp x
-- ContinuousMapZero.nonUnitalStarAlgHom_precomp_apply
@[simps]
def nonUnitalStarAlgHom_precomp (f : C(X, Y)₀) : C(Y, R)₀ →⋆ₙₐ[R] C(X, R)₀ 
Arguments: {X : Type u_1} {Y : Type u_2} (R : Type u_4) [Zero X] [Zero Y] [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace R] [CommSemiring R] [StarRing R] [TopologicalSemiring R] [ContinuousStar R] (f : ContinuousMapZero X Y) (g : ContinuousMapZero Y R)
Goal: (ContinuousMapZero.nonUnitalStarAlgHom_precomp R f) g = g.comp f
-- ContinuousMapZero.nonUnitalStarAlgHom_postcomp_apply
@[simps apply]
def nonUnitalStarAlgHom_postcomp (φ : R →⋆ₙₐ[M] S) (hφ : Continuous φ) :
    C(X, R)₀ →⋆ₙₐ[M] C(X, S)₀ 
Arguments: (X : Type u_1) {M : Type u_3} {R : Type u_4} {S : Type u_5} [Zero X] [CommSemiring M] [TopologicalSpace X] [TopologicalSpace R] [TopologicalSpace S] [CommSemiring R] [StarRing R] [TopologicalSemiring R] [ContinuousStar R] [CommSemiring S] [StarRing S] [TopologicalSemiring S] [ContinuousStar S] [Module M R] [Module M S] [ContinuousConstSMul M R] [ContinuousConstSMul M S] (φ : R →⋆ₙₐ[M] S) (hφ : Continuous ⇑φ) (f : ContinuousMapZero X R)
Goal: (ContinuousMapZero.nonUnitalStarAlgHom_postcomp X φ hφ) f =
  { toFun := ⇑φ, continuous_toFun := hφ, map_zero' := ⋯ }.comp f
-- Valuation.subgroups_basis
theorem subgroups_basis : RingSubgroupsBasis fun γ : Γ₀ˣ => (v.ltAddSubgroup γ : AddSubgroup R) 
Arguments: {R : Type u} [Ring R] {Γ₀ : Type v} [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation R Γ₀)
Goal: RingSubgroupsBasis fun γ => v.ltAddSubgroup γ
-- Valued.is_topological_valuation
class Valued (R : Type u) [Ring R] (Γ₀ : outParam (Type v))
  [LinearOrderedCommGroupWithZero Γ₀] extends UniformSpace R, UniformAddGroup R where
  v : Valuation R Γ₀
  is_topological_valuation : ∀ s, s ∈ 𝓝 (0 : R) ↔ ∃ γ : Γ₀ˣ, { x : R | v x < γ } ⊆ s
Arguments: {R : Type u} [Ring R] {Γ₀ : outParam (Type v)} [LinearOrderedCommGroupWithZero Γ₀] [self : Valued R Γ₀] (s : Set R)
Goal: s ∈ 𝓝 0 ↔ ∃ γ, {x | Valued.v x < ↑γ} ⊆ s
-- Valued.hasBasis_nhds_zero
theorem hasBasis_nhds_zero :
    (𝓝 (0 : R)).HasBasis (fun _ => True) fun γ : Γ₀ˣ => { x | v x < (γ : Γ₀) } 
Arguments: (R : Type u) [Ring R] (Γ₀ : Type v) [LinearOrderedCommGroupWithZero Γ₀] [_i : Valued R Γ₀]
Goal: (𝓝 0).HasBasis (fun x => True) fun γ => {x | Valued.v x < ↑γ}
-- Valued.hasBasis_uniformity
theorem hasBasis_uniformity : (uniformity R).HasBasis (fun _ => True)
    fun γ : Γ₀ˣ => { p : R × R | v (p.2 - p.1) < (γ : Γ₀) } 
Arguments: (R : Type u) [Ring R] (Γ₀ : Type v) [LinearOrderedCommGroupWithZero Γ₀] [_i : Valued R Γ₀]
Goal: (uniformity R).HasBasis (fun x => True) fun γ => {p | Valued.v (p.2 - p.1) < ↑γ}
-- Valued.toUniformSpace_eq
theorem toUniformSpace_eq :
    toUniformSpace = @TopologicalAddGroup.toUniformSpace R _ v.subgroups_basis.topology _ 
Arguments: (R : Type u) [Ring R] (Γ₀ : Type v) [LinearOrderedCommGroupWithZero Γ₀] [_i : Valued R Γ₀]
Goal: Valued.toUniformSpace = TopologicalAddGroup.toUniformSpace R
-- Valued.mem_nhds
theorem mem_nhds {s : Set R} {x : R} : s ∈ 𝓝 x ↔ ∃ γ : Γ₀ˣ, { y | (v (y - x) : Γ₀) < γ } ⊆ s 
Arguments: {R : Type u} [Ring R] {Γ₀ : Type v} [LinearOrderedCommGroupWithZero Γ₀] [_i : Valued R Γ₀] {s : Set R} {x : R}
Goal: s ∈ 𝓝 x ↔ ∃ γ, {y | Valued.v (y - x) < ↑γ} ⊆ s
-- Valued.mem_nhds_zero
theorem mem_nhds_zero {s : Set R} : s ∈ 𝓝 (0 : R) ↔ ∃ γ : Γ₀ˣ, { x | v x < (γ : Γ₀) } ⊆ s 
Arguments: {R : Type u} [Ring R] {Γ₀ : Type v} [LinearOrderedCommGroupWithZero Γ₀] [_i : Valued R Γ₀] {s : Set R}
Goal: s ∈ 𝓝 0 ↔ ∃ γ, {x | Valued.v x < ↑γ} ⊆ s
-- Valued.loc_const
theorem loc_const {x : R} (h : (v x : Γ₀) ≠ 0) : { y : R | v y = v x } ∈ 𝓝 x 
Arguments: {R : Type u} [Ring R] {Γ₀ : Type v} [LinearOrderedCommGroupWithZero Γ₀] [_i : Valued R Γ₀] {x : R} (h : Valued.v x ≠ 0)
Goal: {y | Valued.v y = Valued.v x} ∈ 𝓝 x