ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- finsum_eventually_eq_sum
@[to_additive]
theorem finprod_eventually_eq_prod {M : Type*} [CommMonoid M] {f : ι → X → M}
    (hf : LocallyFinite fun i => mulSupport (f i)) (x : X) :
    ∃ s : Finset ι, ∀ᶠ y in 𝓝 x, ∏ᶠ i, f i y = ∏ i ∈ s, f i y 
Arguments: {ι : Type u_1} {X : Type u_5} [TopologicalSpace X] {M : Type u_6} [AddCommMonoid M] {f : ι → X → M} (hf : LocallyFinite fun i => Function.support (f i)) (x : X)
Goal: ∃ s, ∀ᶠ (y : X) in 𝓝 x, ∑ᶠ (i : ι), f i y = ∑ i ∈ s, f i y
-- continuous_finprod
@[to_additive]
theorem continuous_finprod {f : ι → X → M} (hc : ∀ i, Continuous (f i))
    (hf : LocallyFinite fun i => mulSupport (f i)) : Continuous fun x => ∏ᶠ i, f i x 
Arguments: {ι : Type u_1} {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [CommMonoid M] [ContinuousMul M] {f : ι → X → M} (hc : ∀ (i : ι), Continuous (f i)) (hf : LocallyFinite fun i => Function.mulSupport (f i))
Goal: Continuous fun x => ∏ᶠ (i : ι), f i x
-- continuous_finsum
@[to_additive]
theorem continuous_finprod {f : ι → X → M} (hc : ∀ i, Continuous (f i))
    (hf : LocallyFinite fun i => mulSupport (f i)) : Continuous fun x => ∏ᶠ i, f i x 
Arguments: {ι : Type u_1} {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddCommMonoid M] [ContinuousAdd M] {f : ι → X → M} (hc : ∀ (i : ι), Continuous (f i)) (hf : LocallyFinite fun i => Function.support (f i))
Goal: Continuous fun x => ∑ᶠ (i : ι), f i x
-- continuous_finprod_cond
@[to_additive]
theorem continuous_finprod_cond {f : ι → X → M} {p : ι → Prop} (hc : ∀ i, p i → Continuous (f i))
    (hf : LocallyFinite fun i => mulSupport (f i)) :
    Continuous fun x => ∏ᶠ (i) (_ : p i), f i x 
Arguments: {ι : Type u_1} {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [CommMonoid M] [ContinuousMul M] {f : ι → X → M} {p : ι → Prop} (hc : ∀ (i : ι), p i → Continuous (f i)) (hf : LocallyFinite fun i => Function.mulSupport (f i))
Goal: Continuous fun x => ∏ᶠ (i : ι) (_ : p i), f i x
-- continuous_finsum_cond
@[to_additive]
theorem continuous_finprod_cond {f : ι → X → M} {p : ι → Prop} (hc : ∀ i, p i → Continuous (f i))
    (hf : LocallyFinite fun i => mulSupport (f i)) :
    Continuous fun x => ∏ᶠ (i) (_ : p i), f i x 
Arguments: {ι : Type u_1} {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddCommMonoid M] [ContinuousAdd M] {f : ι → X → M} {p : ι → Prop} (hc : ∀ (i : ι), p i → Continuous (f i)) (hf : LocallyFinite fun i => Function.support (f i))
Goal: Continuous fun x => ∑ᶠ (i : ι) (_ : p i), f i x
-- continuousAdd_sInf
@[to_additive]
theorem continuousMul_sInf {ts : Set (TopologicalSpace M)}
    (h : ∀ t ∈ ts, @ContinuousMul M t _) : @ContinuousMul M (sInf ts) _ 
Arguments: {M : Type u_3} [Add M] {ts : Set (TopologicalSpace M)} (h : ∀ t ∈ ts, ContinuousAdd M)
Goal: ContinuousAdd M
-- continuousMul_sInf
@[to_additive]
theorem continuousMul_sInf {ts : Set (TopologicalSpace M)}
    (h : ∀ t ∈ ts, @ContinuousMul M t _) : @ContinuousMul M (sInf ts) _ 
Arguments: {M : Type u_3} [Mul M] {ts : Set (TopologicalSpace M)} (h : ∀ t ∈ ts, ContinuousMul M)
Goal: ContinuousMul M
-- continuousMul_iInf
@[to_additive]
theorem continuousMul_iInf {ts : ι' → TopologicalSpace M}
    (h' : ∀ i, @ContinuousMul M (ts i) _) : @ContinuousMul M (⨅ i, ts i) _ 
Arguments: {M : Type u_3} {ι' : Sort u_6} [Mul M] {ts : ι' → TopologicalSpace M} (h' : ∀ (i : ι'), ContinuousMul M)
Goal: ContinuousMul M
-- continuousAdd_iInf
@[to_additive]
theorem continuousMul_iInf {ts : ι' → TopologicalSpace M}
    (h' : ∀ i, @ContinuousMul M (ts i) _) : @ContinuousMul M (⨅ i, ts i) _ 
Arguments: {M : Type u_3} {ι' : Sort u_6} [Add M] {ts : ι' → TopologicalSpace M} (h' : ∀ (i : ι'), ContinuousAdd M)
Goal: ContinuousAdd M
-- continuousAdd_inf
@[to_additive]
theorem continuousMul_inf {t₁ t₂ : TopologicalSpace M} (h₁ : @ContinuousMul M t₁ _)
    (h₂ : @ContinuousMul M t₂ _) : @ContinuousMul M (t₁ ⊓ t₂) _ 
Arguments: {M : Type u_3} [Add M] {t₁ t₂ : TopologicalSpace M} (h₁ : ContinuousAdd M) (h₂ : ContinuousAdd M)
Goal: ContinuousAdd M
-- continuousMul_inf
@[to_additive]
theorem continuousMul_inf {t₁ t₂ : TopologicalSpace M} (h₁ : @ContinuousMul M t₁ _)
    (h₂ : @ContinuousMul M t₂ _) : @ContinuousMul M (t₁ ⊓ t₂) _ 
Arguments: {M : Type u_3} [Mul M] {t₁ t₂ : TopologicalSpace M} (h₁ : ContinuousMul M) (h₂ : ContinuousMul M)
Goal: ContinuousMul M
-- ContinuousMap.coe_mulRight
@[to_additive (attr := simp)]
theorem coe_mulRight (x : X) : ⇑(ContinuousMap.mulRight x) = fun y => y * x 
Arguments: {X : Type u_5} [TopologicalSpace X] [Mul X] [ContinuousMul X] (x : X)
Goal: ⇑(ContinuousMap.mulRight x) = fun y => y * x
-- ContinuousMap.coe_addRight
@[to_additive (attr := simp)]
theorem coe_mulRight (x : X) : ⇑(ContinuousMap.mulRight x) = fun y => y * x 
Arguments: {X : Type u_5} [TopologicalSpace X] [Add X] [ContinuousAdd X] (x : X)
Goal: ⇑(ContinuousMap.addRight x) = fun y => y + x
-- ContinuousMap.coe_addLeft
@[to_additive (attr := simp)]
theorem coe_mulLeft (x : X) : ⇑(ContinuousMap.mulLeft x) = fun y => x * y 
Arguments: {X : Type u_5} [TopologicalSpace X] [Add X] [ContinuousAdd X] (x : X)
Goal: ⇑(ContinuousMap.addLeft x) = fun y => x + y
-- ContinuousMap.coe_mulLeft
@[to_additive (attr := simp)]
theorem coe_mulLeft (x : X) : ⇑(ContinuousMap.mulLeft x) = fun y => x * y 
Arguments: {X : Type u_5} [TopologicalSpace X] [Mul X] [ContinuousMul X] (x : X)
Goal: ⇑(ContinuousMap.mulLeft x) = fun y => x * y
-- AffineMap.continuous_iff
theorem continuous_iff {f : E →ᵃ[R] F} : Continuous f ↔ Continuous f.linear 
Arguments: {R : Type u_1} {E : Type u_2} {F : Type u_3} [AddCommGroup E] [TopologicalSpace E] [AddCommGroup F] [TopologicalSpace F] [TopologicalAddGroup F] [Ring R] [Module R E] [Module R F] {f : E →ᵃ[R] F}
Goal: Continuous ⇑f ↔ Continuous ⇑f.linear
-- AffineMap.lineMap_continuous
@[continuity]
theorem lineMap_continuous [TopologicalSpace R] [ContinuousSMul R F] {p v : F} :
    Continuous (lineMap p v : R →ᵃ[R] F) 
Arguments: {R : Type u_1} {F : Type u_3} [AddCommGroup F] [TopologicalSpace F] [TopologicalAddGroup F] [Ring R] [Module R F] [TopologicalSpace R] [ContinuousSMul R F] {p v : F}
Goal: Continuous ⇑(AffineMap.lineMap p v)
-- AffineMap.homothety_continuous
@[continuity]
theorem homothety_continuous (x : F) (t : R) : Continuous <| homothety x t 
Arguments: {R : Type u_1} {F : Type u_3} [AddCommGroup F] [TopologicalSpace F] [TopologicalAddGroup F] [CommRing R] [Module R F] [ContinuousConstSMul R F] (x : F) (t : R)
Goal: Continuous ⇑(AffineMap.homothety x t)
-- AffineMap.homothety_isOpenMap
theorem homothety_isOpenMap (x : F) (t : R) (ht : t ≠ 0) : IsOpenMap <| homothety x t 
Arguments: {R : Type u_1} {F : Type u_3} [AddCommGroup F] [TopologicalSpace F] [TopologicalAddGroup F] [Field R] [Module R F] [ContinuousConstSMul R F] (x : F) (t : R) (ht : t ≠ 0)
Goal: IsOpenMap ⇑(AffineMap.homothety x t)
-- TopologicalSemiring.continuousNeg_of_mul
theorem TopologicalSemiring.continuousNeg_of_mul [TopologicalSpace α] [NonAssocRing α]
    [ContinuousMul α] : ContinuousNeg α 
Arguments: {α : Type u_1} [TopologicalSpace α] [NonAssocRing α] [ContinuousMul α]
Goal: ContinuousNeg α
-- TopologicalSemiring.toTopologicalRing
theorem TopologicalSemiring.toTopologicalRing [TopologicalSpace α] [NonAssocRing α]
    (_ : TopologicalSemiring α) : TopologicalRing α 
Arguments: {α : Type u_1} [TopologicalSpace α] [NonAssocRing α]
Goal: TopologicalSemiring α → TopologicalRing α
-- Subsemiring.topologicalClosure_coe
@[simp]
theorem Subsemiring.topologicalClosure_coe (s : Subsemiring α) :
    (s.topologicalClosure : Set α) = _root_.closure (s : Set α) 
Arguments: {α : Type u_1} [TopologicalSpace α] [Semiring α] [TopologicalSemiring α] (s : Subsemiring α)
Goal: ↑s.topologicalClosure = closure ↑s
-- Subsemiring.le_topologicalClosure
theorem Subsemiring.le_topologicalClosure (s : Subsemiring α) : s ≤ s.topologicalClosure 
Arguments: {α : Type u_1} [TopologicalSpace α] [Semiring α] [TopologicalSemiring α] (s : Subsemiring α)
Goal: s ≤ s.topologicalClosure
-- Subsemiring.isClosed_topologicalClosure
theorem Subsemiring.isClosed_topologicalClosure (s : Subsemiring α) :
    IsClosed (s.topologicalClosure : Set α) 
Arguments: {α : Type u_1} [TopologicalSpace α] [Semiring α] [TopologicalSemiring α] (s : Subsemiring α)
Goal: IsClosed ↑s.topologicalClosure
-- Subsemiring.topologicalClosure_minimal
theorem Subsemiring.topologicalClosure_minimal (s : Subsemiring α) {t : Subsemiring α} (h : s ≤ t)
    (ht : IsClosed (t : Set α)) : s.topologicalClosure ≤ t 
Arguments: {α : Type u_1} [TopologicalSpace α] [Semiring α] [TopologicalSemiring α] (s : Subsemiring α) {t : Subsemiring α} (h : s ≤ t) (ht : IsClosed ↑t)
Goal: s.topologicalClosure ≤ t
-- TopologicalRing.of_addGroup_of_nhds_zero
theorem TopologicalRing.of_addGroup_of_nhds_zero [TopologicalAddGroup R]
    (hmul : Tendsto (uncurry ((· * ·) : R → R → R)) (𝓝 0 ×ˢ 𝓝 0) <| 𝓝 0)
    (hmul_left : ∀ x₀ : R, Tendsto (fun x : R => x₀ * x) (𝓝 0) <| 𝓝 0)
    (hmul_right : ∀ x₀ : R, Tendsto (fun x : R => x * x₀) (𝓝 0) <| 𝓝 0) : TopologicalRing R 
Arguments: {R : Type u_2} [NonUnitalNonAssocRing R] [TopologicalSpace R] [TopologicalAddGroup R] (hmul : Filter.Tendsto (Function.uncurry fun x x_1 => x * x_1) (𝓝 0 ×ˢ 𝓝 0) (𝓝 0)) (hmul_left : ∀ (x₀ : R), Filter.Tendsto (fun x => x₀ * x) (𝓝 0) (𝓝 0)) (hmul_right : ∀ (x₀ : R), Filter.Tendsto (fun x => x * x₀) (𝓝 0) (𝓝 0))
Goal: TopologicalRing R
-- TopologicalRing.of_nhds_zero
theorem TopologicalRing.of_nhds_zero
    (hadd : Tendsto (uncurry ((· + ·) : R → R → R)) (𝓝 0 ×ˢ 𝓝 0) <| 𝓝 0)
    (hneg : Tendsto (fun x => -x : R → R) (𝓝 0) (𝓝 0))
    (hmul : Tendsto (uncurry ((· * ·) : R → R → R)) (𝓝 0 ×ˢ 𝓝 0) <| 𝓝 0)
    (hmul_left : ∀ x₀ : R, Tendsto (fun x : R => x₀ * x) (𝓝 0) <| 𝓝 0)
    (hmul_right : ∀ x₀ : R, Tendsto (fun x : R => x * x₀) (𝓝 0) <| 𝓝 0)
    (hleft : ∀ x₀ : R, 𝓝 x₀ = map (fun x => x₀ + x) (𝓝 0)) : TopologicalRing R 
Arguments: {R : Type u_2} [NonUnitalNonAssocRing R] [TopologicalSpace R] (hadd : Filter.Tendsto (Function.uncurry fun x x_1 => x + x_1) (𝓝 0 ×ˢ 𝓝 0) (𝓝 0)) (hneg : Filter.Tendsto (fun x => -x) (𝓝 0) (𝓝 0)) (hmul : Filter.Tendsto (Function.uncurry fun x x_1 => x * x_1) (𝓝 0 ×ˢ 𝓝 0) (𝓝 0)) (hmul_left : ∀ (x₀ : R), Filter.Tendsto (fun x => x₀ * x) (𝓝 0) (𝓝 0)) (hmul_right : ∀ (x₀ : R), Filter.Tendsto (fun x => x * x₀) (𝓝 0) (𝓝 0)) (hleft : ∀ (x₀ : R), 𝓝 x₀ = Filter.map (fun x => x₀ + x) (𝓝 0))
Goal: TopologicalRing R
-- mulLeft_continuous
theorem mulLeft_continuous (x : α) : Continuous (AddMonoidHom.mulLeft x) 
Arguments: {α : Type u_1} [TopologicalSpace α] [NonUnitalNonAssocRing α] [TopologicalRing α] (x : α)
Goal: Continuous ⇑(AddMonoidHom.mulLeft x)
-- mulRight_continuous
theorem mulRight_continuous (x : α) : Continuous (AddMonoidHom.mulRight x) 
Arguments: {α : Type u_1} [TopologicalSpace α] [NonUnitalNonAssocRing α] [TopologicalRing α] (x : α)
Goal: Continuous ⇑(AddMonoidHom.mulRight x)
-- Subring.le_topologicalClosure
theorem Subring.le_topologicalClosure (s : Subring α) : s ≤ s.topologicalClosure 
Arguments: {α : Type u_1} [TopologicalSpace α] [Ring α] [TopologicalRing α] (s : Subring α)
Goal: s ≤ s.topologicalClosure
-- Subring.isClosed_topologicalClosure
theorem Subring.isClosed_topologicalClosure (s : Subring α) :
    IsClosed (s.topologicalClosure : Set α) 
Arguments: {α : Type u_1} [TopologicalSpace α] [Ring α] [TopologicalRing α] (s : Subring α)
Goal: IsClosed ↑s.topologicalClosure
-- Subring.topologicalClosure_minimal
theorem Subring.topologicalClosure_minimal (s : Subring α) {t : Subring α} (h : s ≤ t)
    (ht : IsClosed (t : Set α)) : s.topologicalClosure ≤ t 
Arguments: {α : Type u_1} [TopologicalSpace α] [Ring α] [TopologicalRing α] (s : Subring α) {t : Subring α} (h : s ≤ t) (ht : IsClosed ↑t)
Goal: s.topologicalClosure ≤ t
-- RingTopology.toTopologicalSpace_injective
theorem toTopologicalSpace_injective :
    Injective (toTopologicalSpace : RingTopology α → TopologicalSpace α) 
Arguments: {α : Type u_1} [Ring α]
Goal: Function.Injective RingTopology.toTopologicalSpace
-- RingTopology.ext
@[ext]
theorem ext {f g : RingTopology α} (h : f.IsOpen = g.IsOpen) : f = g 
Arguments: {α : Type u_1} [Ring α] {f g : RingTopology α} (h : TopologicalSpace.IsOpen = TopologicalSpace.IsOpen)
Goal: f = g
-- RingTopology.coinduced_continuous
theorem coinduced_continuous {α β : Type*} [t : TopologicalSpace α] [Ring β] (f : α → β) :
    Continuous[t, (coinduced f).toTopologicalSpace] f 
Arguments: {α : Type u_2} {β : Type u_3} [t : TopologicalSpace α] [Ring β] (f : α → β)
Goal: Continuous f
-- continuous_algebraMap
@[continuity, fun_prop]
theorem continuous_algebraMap [ContinuousSMul R A] : Continuous (algebraMap R A) 
Arguments: (R : Type u_1) (A : Type u) [CommSemiring R] [Semiring A] [Algebra R A] [TopologicalSpace R] [TopologicalSpace A] [ContinuousSMul R A]
Goal: Continuous ⇑(algebraMap R A)
-- continuous_algebraMap_iff_smul
theorem continuous_algebraMap_iff_smul [TopologicalSemiring A] :
    Continuous (algebraMap R A) ↔ Continuous fun p : R × A => p.1 • p.2 
Arguments: (R : Type u_1) (A : Type u) [CommSemiring R] [Semiring A] [Algebra R A] [TopologicalSpace R] [TopologicalSpace A] [TopologicalSemiring A]
Goal: Continuous ⇑(algebraMap R A) ↔ Continuous fun p => p.1 • p.2
-- continuousSMul_of_algebraMap
theorem continuousSMul_of_algebraMap [TopologicalSemiring A] (h : Continuous (algebraMap R A)) :
    ContinuousSMul R A 
Arguments: (R : Type u_1) (A : Type u) [CommSemiring R] [Semiring A] [Algebra R A] [TopologicalSpace R] [TopologicalSpace A] [TopologicalSemiring A] (h : Continuous ⇑(algebraMap R A))
Goal: ContinuousSMul R A
-- algebraMapCLM_toFun
@[simps]
def algebraMapCLM : R →L[R] A 
Arguments: (R : Type u_1) (A : Type u) [CommSemiring R] [Semiring A] [Algebra R A] [TopologicalSpace R] [TopologicalSpace A] [ContinuousSMul R A] (a : R)
Goal: (algebraMapCLM R A) a = (algebraMap R A) a
-- algebraMapCLM_apply
@[simps]
def algebraMapCLM : R →L[R] A 
Arguments: (R : Type u_1) (A : Type u) [CommSemiring R] [Semiring A] [Algebra R A] [TopologicalSpace R] [TopologicalSpace A] [ContinuousSMul R A] (a : R)
Goal: (algebraMapCLM R A) a = (algebraMap R A) a
-- algebraMapCLM_coe
theorem algebraMapCLM_coe : ⇑(algebraMapCLM R A) = algebraMap R A 
Arguments: (R : Type u_1) (A : Type u) [CommSemiring R] [Semiring A] [Algebra R A] [TopologicalSpace R] [TopologicalSpace A] [ContinuousSMul R A]
Goal: ⇑(algebraMapCLM R A) = ⇑(algebraMap R A)
-- algebraMapCLM_toLinearMap
theorem algebraMapCLM_toLinearMap : (algebraMapCLM R A).toLinearMap = Algebra.linearMap R A 
Arguments: (R : Type u_1) (A : Type u) [CommSemiring R] [Semiring A] [Algebra R A] [TopologicalSpace R] [TopologicalSpace A] [ContinuousSMul R A]
Goal: ↑(algebraMapCLM R A) = Algebra.linearMap R A
-- ContinuousAlgHom.cont
structure ContinuousAlgHom (R : Type*) [CommSemiring R] (A : Type*) [Semiring A]
    [TopologicalSpace A] (B : Type*) [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B]
    extends A →ₐ[R] B where
-- TODO: replace with `fun_prop` when that is stable
  cont : Continuous toFun := by continuity
Arguments: {R : Type u_3} [CommSemiring R] {A : Type u_4} [Semiring A] [TopologicalSpace A] {B : Type u_5} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] (self : A →A[R] B)
Goal: Continuous (↑↑self.toRingHom).toFun
-- ContinuousAlgHom.toAlgHom_eq_coe
@[simp]
theorem toAlgHom_eq_coe (f : A →A[R] B) : f.toAlgHom = f 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] (f : A →A[R] B)
Goal: f.toAlgHom = ↑f
-- ContinuousAlgHom.coe_inj
@[simp, norm_cast]
theorem coe_inj {f g : A →A[R] B} : (f : A →ₐ[R] B) = g ↔ f = g 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] {f g : A →A[R] B}
Goal: ↑f = ↑g ↔ f = g
-- ContinuousAlgHom.coe_mk
@[simp]
theorem coe_mk (f : A →ₐ[R] B) (h) : (mk f h : A →ₐ[R] B) = f 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) (h : Continuous (↑↑f.toRingHom).toFun)
Goal: ↑{ toAlgHom := f, cont := h } = f
-- ContinuousAlgHom.coe_mk'
@[simp]
theorem coe_mk' (f : A →ₐ[R] B) (h) : (mk f h : A → B) = f 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) (h : Continuous (↑↑f.toRingHom).toFun)
Goal: ⇑{ toAlgHom := f, cont := h } = ⇑f
-- ContinuousAlgHom.coe_coe
@[simp, norm_cast]
theorem coe_coe (f : A →A[R] B) : ⇑(f : A →ₐ[R] B) = f 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] (f : A →A[R] B)
Goal: ⇑↑f = ⇑f
-- ContinuousAlgHom.continuous
@[fun_prop]
protected theorem continuous (f : A →A[R] B) : Continuous f 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] (f : A →A[R] B)
Goal: Continuous ⇑f
-- ContinuousAlgHom.uniformContinuous
protected theorem uniformContinuous {E₁ E₂ : Type*} [UniformSpace E₁] [UniformSpace E₂]
    [Ring E₁] [Ring E₂] [Algebra R E₁] [Algebra R E₂] [UniformAddGroup E₁]
    [UniformAddGroup E₂] (f : E₁ →A[R] E₂) : UniformContinuous f 
Arguments: {R : Type u_1} [CommSemiring R] {E₁ : Type u_4} {E₂ : Type u_5} [UniformSpace E₁] [UniformSpace E₂] [Ring E₁] [Ring E₂] [Algebra R E₁] [Algebra R E₂] [UniformAddGroup E₁] [UniformAddGroup E₂] (f : E₁ →A[R] E₂)
Goal: UniformContinuous ⇑f
-- ContinuousAlgHom.ext
@[ext]
theorem ext {f g : A →A[R] B} (h : ∀ x, f x = g x) : f = g 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] {f g : A →A[R] B} (h : ∀ (x : A), f x = g x)
Goal: f = g
-- ContinuousAlgHom.ext_iff
theorem ext_iff {f g : A →A[R] B} : f = g ↔ ∀ x, f x = g x 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] {f g : A →A[R] B}
Goal: f = g ↔ ∀ (x : A), f x = g x
-- ContinuousAlgHom.coe_copy
@[simp]
theorem coe_copy (f : A →A[R] B) (f' : A → B) (h : f' = ⇑f) : ⇑(f.copy f' h) = f' 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] (f : A →A[R] B) (f' : A → B) (h : f' = ⇑f)
Goal: ⇑(f.copy f' h) = f'
-- ContinuousAlgHom.copy_eq
theorem copy_eq (f : A →A[R] B) (f' : A → B) (h : f' = ⇑f) : f.copy f' h = f 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] (f : A →A[R] B) (f' : A → B) (h : f' = ⇑f)
Goal: f.copy f' h = f
-- ContinuousAlgHom.map_zero
protected theorem map_zero (f : A →A[R] B) : f (0 : A) = 0 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] (f : A →A[R] B)
Goal: f 0 = 0
-- ContinuousAlgHom.map_add
protected theorem map_add (f : A →A[R] B) (x y : A) : f (x + y) = f x + f y 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] (f : A →A[R] B) (x y : A)
Goal: f (x + y) = f x + f y
-- ContinuousAlgHom.map_smul
protected theorem map_smul (f : A →A[R] B) (c : R) (x : A) :
    f (c • x) = c • f x 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] (f : A →A[R] B) (c : R) (x : A)
Goal: f (c • x) = c • f x
-- ContinuousAlgHom.map_smul_of_tower
theorem map_smul_of_tower {R S : Type*} [CommSemiring S] [SMul R A] [Algebra S A] [SMul R B]
    [Algebra S B] [MulActionHomClass (A →A[S] B) R A B] (f : A →A[S] B) (c : R) (x : A) :
    f (c • x) = c • f x 
Arguments: {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] {R : Type u_4} {S : Type u_5} [CommSemiring S] [SMul R A] [Algebra S A] [SMul R B] [Algebra S B] [MulActionHomClass (A →A[S] B) R A B] (f : A →A[S] B) (c : R) (x : A)
Goal: f (c • x) = c • f x
-- ContinuousAlgHom.map_sum
protected theorem map_sum {ι : Type*} (f : A →A[R] B) (s : Finset ι) (g : ι → A) :
    f (∑ i in s, g i) = ∑ i in s, f (g i) 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] {ι : Type u_4} (f : A →A[R] B) (s : Finset ι) (g : ι → A)
Goal: f (∑ i ∈ s, g i) = ∑ i ∈ s, f (g i)
-- ContinuousAlgHom.ext_ring
@[ext]
theorem ext_ring [TopologicalSpace R] {f g : R →A[R] A} : f = g 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] [Algebra R A] [TopologicalSpace R] {f g : R →A[R] A}
Goal: f = g
-- ContinuousAlgHom.ext_ring_iff
theorem ext_ring_iff [TopologicalSpace R] {f g : R →A[R] A} : f = g ↔ f 1 = g 1 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] [Algebra R A] [TopologicalSpace R] {f g : R →A[R] A}
Goal: f = g ↔ f 1 = g 1
-- ContinuousAlgHom.eqOn_closure_adjoin
theorem eqOn_closure_adjoin [T2Space B] {s : Set A} {f g : A →A[R] B} (h : Set.EqOn f g s) :
    Set.EqOn f g (closure (Algebra.adjoin R s : Set A)) 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] [T2Space B] {s : Set A} {f g : A →A[R] B} (h : Set.EqOn (⇑f) (⇑g) s)
Goal: Set.EqOn (⇑f) (⇑g) (closure ↑(Algebra.adjoin R s))
-- ContinuousAlgHom.ext_on
theorem ext_on [T2Space B] {s : Set A} (hs : Dense (Algebra.adjoin R s : Set A))
    {f g : A →A[R] B} (h : Set.EqOn f g s) : f = g 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] [T2Space B] {s : Set A} (hs : Dense ↑(Algebra.adjoin R s)) {f g : A →A[R] B} (h : Set.EqOn (⇑f) (⇑g) s)
Goal: f = g
-- Subalgebra.topologicalClosure_map
theorem _root_.Subalgebra.topologicalClosure_map
    [TopologicalSemiring B] (f : A →A[R] B) (s : Subalgebra R A) :
    s.topologicalClosure.map f ≤ (s.map f.toAlgHom).topologicalClosure 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] [TopologicalSemiring A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] [TopologicalSemiring B] (f : A →A[R] B) (s : Subalgebra R A)
Goal: Subalgebra.map (↑f) s.topologicalClosure ≤ (Subalgebra.map f.toAlgHom s).topologicalClosure
-- Subalgebra.topologicalClosure_coe
@[simp]
theorem _root_.Subalgebra.topologicalClosure_coe
    (s : Subalgebra R A) :
  (s.topologicalClosure : Set A) = closure ↑s 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] [TopologicalSemiring A] [Algebra R A] (s : Subalgebra R A)
Goal: ↑s.topologicalClosure = closure ↑s
-- DenseRange.topologicalClosure_map_subalgebra
theorem _root_.DenseRange.topologicalClosure_map_subalgebra
    [TopologicalSemiring B] {f : A →A[R] B}  (hf' : DenseRange f) {s : Subalgebra R A}
    (hs : s.topologicalClosure = ⊤) : (s.map (f : A →ₐ[R] B)).topologicalClosure = ⊤ 
Arguments: {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] [TopologicalSemiring A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] [TopologicalSemiring B] {f : A →A[R] B} (hf' : DenseRange ⇑f) {s : Subalgebra R A} (hs : s.topologicalClosure = ⊤)
Goal: (Subalgebra.map (↑f) s).topologicalClosure = ⊤
-- ContinuousAlgHom.one_def
theorem one_def : (1 : A →A[R] A) = ContinuousAlgHom.id R A 
Arguments: (R : Type u_1) [CommSemiring R] (A : Type u_2) [Semiring A] [TopologicalSpace A] [Algebra R A]
Goal: 1 = ContinuousAlgHom.id R A