ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousMap.nsmul_apply
@[to_additive (attr := norm_cast)]
theorem pow_apply [Monoid β] [ContinuousMul β] (f : C(α, β)) (n : ℕ) (x : α) :
    (f ^ n) x = f x ^ n 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [AddMonoid β] [ContinuousAdd β] (f : C(α, β)) (n : ℕ) (x : α)
Goal: (n • f) x = n • f x
-- ContinuousMap.pow_apply
@[to_additive (attr := norm_cast)]
theorem pow_apply [Monoid β] [ContinuousMul β] (f : C(α, β)) (n : ℕ) (x : α) :
    (f ^ n) x = f x ^ n 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [Monoid β] [ContinuousMul β] (f : C(α, β)) (n : ℕ) (x : α)
Goal: (f ^ n) x = f x ^ n
-- ContinuousMap.nsmul_comp
@[to_additive]
theorem pow_comp [Monoid γ] [ContinuousMul γ] (f : C(β, γ)) (n : ℕ) (g : C(α, β)) :
    (f ^ n).comp g = f.comp g ^ n 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [AddMonoid γ] [ContinuousAdd γ] (f : C(β, γ)) (n : ℕ) (g : C(α, β))
Goal: (n • f).comp g = n • f.comp g
-- ContinuousMap.pow_comp
@[to_additive]
theorem pow_comp [Monoid γ] [ContinuousMul γ] (f : C(β, γ)) (n : ℕ) (g : C(α, β)) :
    (f ^ n).comp g = f.comp g ^ n 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [Monoid γ] [ContinuousMul γ] (f : C(β, γ)) (n : ℕ) (g : C(α, β))
Goal: (f ^ n).comp g = f.comp g ^ n
-- ContinuousMap.coe_neg
@[to_additive (attr := simp)]
theorem coe_inv [Inv β] [ContinuousInv β] (f : C(α, β)) : ⇑f⁻¹ = (⇑f)⁻¹ 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [Neg β] [ContinuousNeg β] (f : C(α, β))
Goal: ⇑(-f) = -⇑f
-- ContinuousMap.coe_inv
@[to_additive (attr := simp)]
theorem coe_inv [Inv β] [ContinuousInv β] (f : C(α, β)) : ⇑f⁻¹ = (⇑f)⁻¹ 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [Inv β] [ContinuousInv β] (f : C(α, β))
Goal: ⇑f⁻¹ = (⇑f)⁻¹
-- ContinuousMap.neg_apply
@[to_additive (attr := simp)]
theorem inv_apply [Inv β] [ContinuousInv β] (f : C(α, β)) (x : α) : f⁻¹ x = (f x)⁻¹ 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [Neg β] [ContinuousNeg β] (f : C(α, β)) (x : α)
Goal: (-f) x = -f x
-- ContinuousMap.inv_apply
@[to_additive (attr := simp)]
theorem inv_apply [Inv β] [ContinuousInv β] (f : C(α, β)) (x : α) : f⁻¹ x = (f x)⁻¹ 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [Inv β] [ContinuousInv β] (f : C(α, β)) (x : α)
Goal: f⁻¹ x = (f x)⁻¹
-- ContinuousMap.inv_comp
@[to_additive (attr := simp)]
theorem inv_comp [Inv γ] [ContinuousInv γ] (f : C(β, γ)) (g : C(α, β)) :
    f⁻¹.comp g = (f.comp g)⁻¹ 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [Inv γ] [ContinuousInv γ] (f : C(β, γ)) (g : C(α, β))
Goal: f⁻¹.comp g = (f.comp g)⁻¹
-- ContinuousMap.neg_comp
@[to_additive (attr := simp)]
theorem inv_comp [Inv γ] [ContinuousInv γ] (f : C(β, γ)) (g : C(α, β)) :
    f⁻¹.comp g = (f.comp g)⁻¹ 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [Neg γ] [ContinuousNeg γ] (f : C(β, γ)) (g : C(α, β))
Goal: (-f).comp g = -f.comp g
-- ContinuousMap.coe_div
@[to_additive (attr := norm_cast, simp)]
theorem coe_div [Div β] [ContinuousDiv β] (f g : C(α, β)) : ⇑(f / g) = f / g 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [Div β] [ContinuousDiv β] (f g : C(α, β))
Goal: ⇑(f / g) = ⇑f / ⇑g
-- ContinuousMap.coe_sub
@[to_additive (attr := norm_cast, simp)]
theorem coe_div [Div β] [ContinuousDiv β] (f g : C(α, β)) : ⇑(f / g) = f / g 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [Sub β] [ContinuousSub β] (f g : C(α, β))
Goal: ⇑(f - g) = ⇑f - ⇑g
-- ContinuousMap.div_apply
@[to_additive (attr := simp)]
theorem div_apply [Div β] [ContinuousDiv β] (f g : C(α, β)) (x : α) : (f / g) x = f x / g x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [Div β] [ContinuousDiv β] (f g : C(α, β)) (x : α)
Goal: (f / g) x = f x / g x
-- ContinuousMap.sub_apply
@[to_additive (attr := simp)]
theorem div_apply [Div β] [ContinuousDiv β] (f g : C(α, β)) (x : α) : (f / g) x = f x / g x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [Sub β] [ContinuousSub β] (f g : C(α, β)) (x : α)
Goal: (f - g) x = f x - g x
-- ContinuousMap.sub_comp
@[to_additive (attr := simp)]
theorem div_comp [Div γ] [ContinuousDiv γ] (f g : C(β, γ)) (h : C(α, β)) :
    (f / g).comp h = f.comp h / g.comp h 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [Sub γ] [ContinuousSub γ] (f g : C(β, γ)) (h : C(α, β))
Goal: (f - g).comp h = f.comp h - g.comp h
-- ContinuousMap.div_comp
@[to_additive (attr := simp)]
theorem div_comp [Div γ] [ContinuousDiv γ] (f g : C(β, γ)) (h : C(α, β)) :
    (f / g).comp h = f.comp h / g.comp h 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [Div γ] [ContinuousDiv γ] (f g : C(β, γ)) (h : C(α, β))
Goal: (f / g).comp h = f.comp h / g.comp h
-- ContinuousMap.coe_zsmul
@[to_additive (attr := norm_cast) (reorder := 7 8)]
theorem coe_zpow [Group β] [TopologicalGroup β] (f : C(α, β)) (z : ℤ) : ⇑(f ^ z) = (⇑f) ^ z 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [AddGroup β] [TopologicalAddGroup β] (z : ℤ) (f : C(α, β))
Goal: ⇑(z • f) = z • ⇑f
-- ContinuousMap.coe_zpow
@[to_additive (attr := norm_cast) (reorder := 7 8)]
theorem coe_zpow [Group β] [TopologicalGroup β] (f : C(α, β)) (z : ℤ) : ⇑(f ^ z) = (⇑f) ^ z 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [Group β] [TopologicalGroup β] (f : C(α, β)) (z : ℤ)
Goal: ⇑(f ^ z) = ⇑f ^ z
-- ContinuousMap.zsmul_apply
@[to_additive]
theorem zpow_apply [Group β] [TopologicalGroup β] (f : C(α, β)) (z : ℤ) (x : α) :
    (f ^ z) x = f x ^ z 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [AddGroup β] [TopologicalAddGroup β] (f : C(α, β)) (z : ℤ) (x : α)
Goal: (z • f) x = z • f x
-- ContinuousMap.zpow_apply
@[to_additive]
theorem zpow_apply [Group β] [TopologicalGroup β] (f : C(α, β)) (z : ℤ) (x : α) :
    (f ^ z) x = f x ^ z 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [Group β] [TopologicalGroup β] (f : C(α, β)) (z : ℤ) (x : α)
Goal: (f ^ z) x = f x ^ z
-- ContinuousMap.zpow_comp
@[to_additive]
theorem zpow_comp [Group γ] [TopologicalGroup γ] (f : C(β, γ)) (z : ℤ) (g : C(α, β)) :
    (f ^ z).comp g = f.comp g ^ z 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [Group γ] [TopologicalGroup γ] (f : C(β, γ)) (z : ℤ) (g : C(α, β))
Goal: (f ^ z).comp g = f.comp g ^ z
-- ContinuousMap.zsmul_comp
@[to_additive]
theorem zpow_comp [Group γ] [TopologicalGroup γ] (f : C(β, γ)) (z : ℤ) (g : C(α, β)) :
    (f ^ z).comp g = f.comp g ^ z 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [AddGroup γ] [TopologicalAddGroup γ] (f : C(β, γ)) (z : ℤ) (g : C(α, β))
Goal: (z • f).comp g = z • f.comp g
-- ContinuousMap.coeFnMonoidHom_apply
@[to_additive (attr := simps)
  "Coercion to a function as an `AddMonoidHom`. Similar to `AddMonoidHom.coeFn`." ]
def coeFnMonoidHom [Monoid β] [ContinuousMul β] : C(α, β) →* α → β 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [Monoid β] [ContinuousMul β] (f : C(α, β)) (a : α)
Goal: ContinuousMap.coeFnMonoidHom f a = f a
-- ContinuousMap.coeFnAddMonoidHom_apply
@[to_additive (attr := simps)
  "Coercion to a function as an `AddMonoidHom`. Similar to `AddMonoidHom.coeFn`." ]
def coeFnMonoidHom [Monoid β] [ContinuousMul β] : C(α, β) →* α → β 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [AddMonoid β] [ContinuousAdd β] (f : C(α, β)) (a : α)
Goal: ContinuousMap.coeFnAddMonoidHom f a = f a
-- MonoidHom.compLeftContinuous_apply
@[to_additive (attr := simps)
"Composition on the left by a (continuous) homomorphism of topological `AddMonoid`s, as an
`AddMonoidHom`. Similar to `AddMonoidHom.comp_left`."]
protected def _root_.MonoidHom.compLeftContinuous {γ : Type*} [Monoid β] [ContinuousMul β]
    [TopologicalSpace γ] [Monoid γ] [ContinuousMul γ] (g : β →* γ) (hg : Continuous g) :
    C(α, β) →* C(α, γ) 
Arguments: (α : Type u_1) {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {γ : Type u_3} [Monoid β] [ContinuousMul β] [TopologicalSpace γ] [Monoid γ] [ContinuousMul γ] (g : β →* γ) (hg : Continuous ⇑g) (f : C(α, β))
Goal: (MonoidHom.compLeftContinuous α g hg) f = { toFun := ⇑g, continuous_toFun := hg }.comp f
-- AddMonoidHom.compLeftContinuous_apply
@[to_additive (attr := simps)
"Composition on the left by a (continuous) homomorphism of topological `AddMonoid`s, as an
`AddMonoidHom`. Similar to `AddMonoidHom.comp_left`."]
protected def _root_.MonoidHom.compLeftContinuous {γ : Type*} [Monoid β] [ContinuousMul β]
    [TopologicalSpace γ] [Monoid γ] [ContinuousMul γ] (g : β →* γ) (hg : Continuous g) :
    C(α, β) →* C(α, γ) 
Arguments: (α : Type u_1) {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {γ : Type u_3} [AddMonoid β] [ContinuousAdd β] [TopologicalSpace γ] [AddMonoid γ] [ContinuousAdd γ] (g : β →+ γ) (hg : Continuous ⇑g) (f : C(α, β))
Goal: (AddMonoidHom.compLeftContinuous α g hg) f = { toFun := ⇑g, continuous_toFun := hg }.comp f
-- ContinuousMap.compAddMonoidHom'_apply
@[to_additive (attr := simps)
      "Composition on the right as an `AddMonoidHom`. Similar to `AddMonoidHom.compHom'`."]
def compMonoidHom' {γ : Type*} [TopologicalSpace γ] [MulOneClass γ] [ContinuousMul γ]
    (g : C(α, β)) : C(β, γ) →* C(α, γ) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {γ : Type u_3} [TopologicalSpace γ] [AddZeroClass γ] [ContinuousAdd γ] (g : C(α, β)) (f : C(β, γ))
Goal: g.compAddMonoidHom' f = f.comp g
-- ContinuousMap.compMonoidHom'_apply
@[to_additive (attr := simps)
      "Composition on the right as an `AddMonoidHom`. Similar to `AddMonoidHom.compHom'`."]
def compMonoidHom' {γ : Type*} [TopologicalSpace γ] [MulOneClass γ] [ContinuousMul γ]
    (g : C(α, β)) : C(β, γ) →* C(α, γ) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {γ : Type u_3} [TopologicalSpace γ] [MulOneClass γ] [ContinuousMul γ] (g : C(α, β)) (f : C(β, γ))
Goal: g.compMonoidHom' f = f.comp g
-- ContinuousMap.coe_prod
@[to_additive (attr := simp)]
theorem coe_prod [CommMonoid β] [ContinuousMul β] {ι : Type*} (s : Finset ι) (f : ι → C(α, β)) :
    ⇑(∏ i ∈ s, f i) = ∏ i ∈ s, (f i : α → β) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [CommMonoid β] [ContinuousMul β] {ι : Type u_3} (s : Finset ι) (f : ι → C(α, β))
Goal: ⇑(∏ i ∈ s, f i) = ∏ i ∈ s, ⇑(f i)
-- ContinuousMap.coe_sum
@[to_additive (attr := simp)]
theorem coe_prod [CommMonoid β] [ContinuousMul β] {ι : Type*} (s : Finset ι) (f : ι → C(α, β)) :
    ⇑(∏ i ∈ s, f i) = ∏ i ∈ s, (f i : α → β) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [AddCommMonoid β] [ContinuousAdd β] {ι : Type u_3} (s : Finset ι) (f : ι → C(α, β))
Goal: ⇑(∑ i ∈ s, f i) = ∑ i ∈ s, ⇑(f i)
-- ContinuousMap.prod_apply
@[to_additive]
theorem prod_apply [CommMonoid β] [ContinuousMul β] {ι : Type*} (s : Finset ι) (f : ι → C(α, β))
    (a : α) : (∏ i ∈ s, f i) a = ∏ i ∈ s, f i a 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [CommMonoid β] [ContinuousMul β] {ι : Type u_3} (s : Finset ι) (f : ι → C(α, β)) (a : α)
Goal: (∏ i ∈ s, f i) a = ∏ i ∈ s, (f i) a
-- ContinuousMap.sum_apply
@[to_additive]
theorem prod_apply [CommMonoid β] [ContinuousMul β] {ι : Type*} (s : Finset ι) (f : ι → C(α, β))
    (a : α) : (∏ i ∈ s, f i) a = ∏ i ∈ s, f i a 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [AddCommMonoid β] [ContinuousAdd β] {ι : Type u_3} (s : Finset ι) (f : ι → C(α, β)) (a : α)
Goal: (∑ i ∈ s, f i) a = ∑ i ∈ s, (f i) a
-- ContinuousMap.hasSum_apply
theorem hasSum_apply {γ : Type*} [AddCommMonoid β] [ContinuousAdd β]
    {f : γ → C(α, β)} {g : C(α, β)} (hf : HasSum f g) (x : α) :
    HasSum (fun i : γ => f i x) (g x) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {γ : Type u_3} [AddCommMonoid β] [ContinuousAdd β] {f : γ → C(α, β)} {g : C(α, β)} (hf : HasSum f g) (x : α)
Goal: HasSum (fun i => (f i) x) (g x)
-- ContinuousMap.summable_apply
theorem summable_apply [AddCommMonoid β] [ContinuousAdd β] {γ : Type*} {f : γ → C(α, β)}
    (hf : Summable f) (x : α) : Summable fun i : γ => f i x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [AddCommMonoid β] [ContinuousAdd β] {γ : Type u_3} {f : γ → C(α, β)} (hf : Summable f) (x : α)
Goal: Summable fun i => (f i) x
-- ContinuousMap.tsum_apply
theorem tsum_apply [T2Space β] [AddCommMonoid β] [ContinuousAdd β] {γ : Type*} {f : γ → C(α, β)}
    (hf : Summable f) (x : α) :
    ∑' i : γ, f i x = (∑' i : γ, f i) x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [T2Space β] [AddCommMonoid β] [ContinuousAdd β] {γ : Type u_3} {f : γ → C(α, β)} (hf : Summable f) (x : α)
Goal: ∑' (i : γ), (f i) x = (∑' (i : γ), f i) x
-- RingHom.compLeftContinuous_apply_apply
@[simps!]
protected def _root_.RingHom.compLeftContinuous (α : Type*) {β : Type*} {γ : Type*}
    [TopologicalSpace α]
    [TopologicalSpace β] [Semiring β] [TopologicalSemiring β] [TopologicalSpace γ] [Semiring γ]
    [TopologicalSemiring γ] (g : β →+* γ) (hg : Continuous g) : C(α, β) →+* C(α, γ) 
Arguments: (α : Type u_1) {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [Semiring β] [TopologicalSemiring β] [TopologicalSpace γ] [Semiring γ] [TopologicalSemiring γ] (g : β →+* γ) (hg : Continuous ⇑g) (f : C(α, β))
Goal: ∀ (a : α), ((RingHom.compLeftContinuous α g hg) f) a = g (f a)
-- ContinuousMap.coeFnRingHom_apply
@[simps!]
def coeFnRingHom {α : Type*} {β : Type*} [TopologicalSpace α] [TopologicalSpace β] [Semiring β]
    [TopologicalSemiring β] : C(α, β) →+* α → β 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [Semiring β] [TopologicalSemiring β] (f : C(α, β)) (a : α)
Goal: ContinuousMap.coeFnRingHom f a = f a
-- ContinuousMap.coe_smul
@[to_additive (attr := simp, norm_cast)]
theorem coe_smul [SMul R M] [ContinuousConstSMul R M] (c : R) (f : C(α, M)) : ⇑(c • f) = c • ⇑f 
Arguments: {α : Type u_1} [TopologicalSpace α] {R : Type u_3} {M : Type u_5} [TopologicalSpace M] [SMul R M] [ContinuousConstSMul R M] (c : R) (f : C(α, M))
Goal: ⇑(c • f) = c • ⇑f
-- ContinuousMap.coe_vadd
@[to_additive (attr := simp, norm_cast)]
theorem coe_smul [SMul R M] [ContinuousConstSMul R M] (c : R) (f : C(α, M)) : ⇑(c • f) = c • ⇑f 
Arguments: {α : Type u_1} [TopologicalSpace α] {R : Type u_3} {M : Type u_5} [TopologicalSpace M] [VAdd R M] [ContinuousConstVAdd R M] (c : R) (f : C(α, M))
Goal: ⇑(c +ᵥ f) = c +ᵥ ⇑f
-- ContinuousMap.vadd_apply
@[to_additive]
theorem smul_apply [SMul R M] [ContinuousConstSMul R M] (c : R) (f : C(α, M)) (a : α) :
    (c • f) a = c • f a 
Arguments: {α : Type u_1} [TopologicalSpace α] {R : Type u_3} {M : Type u_5} [TopologicalSpace M] [VAdd R M] [ContinuousConstVAdd R M] (c : R) (f : C(α, M)) (a : α)
Goal: (c +ᵥ f) a = c +ᵥ f a
-- ContinuousMap.smul_apply
@[to_additive]
theorem smul_apply [SMul R M] [ContinuousConstSMul R M] (c : R) (f : C(α, M)) (a : α) :
    (c • f) a = c • f a 
Arguments: {α : Type u_1} [TopologicalSpace α] {R : Type u_3} {M : Type u_5} [TopologicalSpace M] [SMul R M] [ContinuousConstSMul R M] (c : R) (f : C(α, M)) (a : α)
Goal: (c • f) a = c • f a
-- ContinuousMap.smul_comp
@[to_additive (attr := simp)]
theorem smul_comp [SMul R M] [ContinuousConstSMul R M] (r : R) (f : C(β, M)) (g : C(α, β)) :
    (r • f).comp g = r • f.comp g 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {R : Type u_3} {M : Type u_5} [TopologicalSpace M] [SMul R M] [ContinuousConstSMul R M] (r : R) (f : C(β, M)) (g : C(α, β))
Goal: (r • f).comp g = r • f.comp g
-- ContinuousMap.vadd_comp
@[to_additive (attr := simp)]
theorem smul_comp [SMul R M] [ContinuousConstSMul R M] (r : R) (f : C(β, M)) (g : C(α, β)) :
    (r • f).comp g = r • f.comp g 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {R : Type u_3} {M : Type u_5} [TopologicalSpace M] [VAdd R M] [ContinuousConstVAdd R M] (r : R) (f : C(β, M)) (g : C(α, β))
Goal: (r +ᵥ f).comp g = r +ᵥ f.comp g
-- ContinuousLinearMap.compLeftContinuous_apply
@[simps]
protected def _root_.ContinuousLinearMap.compLeftContinuous (α : Type*) [TopologicalSpace α]
    (g : M →L[R] M₂) : C(α, M) →ₗ[R] C(α, M₂) 
Arguments: (R : Type u_3) {M : Type u_5} [TopologicalSpace M] {M₂ : Type u_6} [TopologicalSpace M₂] [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [ContinuousAdd M] [Module R M] [ContinuousConstSMul R M] [ContinuousAdd M₂] [Module R M₂] [ContinuousConstSMul R M₂] (α : Type u_7) [TopologicalSpace α] (g : M →L[R] M₂)
Goal: ∀ (a : C(α, M)),
  (ContinuousLinearMap.compLeftContinuous R α g) a =
    (↑(AddMonoidHom.compLeftContinuous α (↑g).toAddMonoidHom ⋯)).toFun a
-- ContinuousMap.coeFnLinearMap_apply
@[simps]
def coeFnLinearMap : C(α, M) →ₗ[R] α → M 
Arguments: {α : Type u_1} [TopologicalSpace α] (R : Type u_3) {M : Type u_5} [TopologicalSpace M] [Semiring R] [AddCommMonoid M] [ContinuousAdd M] [Module R M] [ContinuousConstSMul R M]
Goal: ∀ (a : C(α, M)) (a_1 : α), (ContinuousMap.coeFnLinearMap R) a a_1 = (↑ContinuousMap.coeFnAddMonoidHom).toFun a a_1
-- ContinuousMap.C_apply
@[simp]
theorem ContinuousMap.C_apply (r : R) (a : α) : ContinuousMap.C r a = algebraMap R A r 
Arguments: {α : Type u_1} [TopologicalSpace α] {R : Type u_2} [CommSemiring R] {A : Type u_3} [TopologicalSpace A] [Semiring A] [Algebra R A] [TopologicalSemiring A] (r : R) (a : α)
Goal: (ContinuousMap.C r) a = (algebraMap R A) r
-- AlgHom.compLeftContinuous_apply_apply
@[simps!]
protected def AlgHom.compLeftContinuous {α : Type*} [TopologicalSpace α] (g : A →ₐ[R] A₂)
    (hg : Continuous g) : C(α, A) →ₐ[R] C(α, A₂) 
Arguments: (R : Type u_2) [CommSemiring R] {A : Type u_3} [TopologicalSpace A] [Semiring A] [Algebra R A] [TopologicalSemiring A] {A₂ : Type u_4} [TopologicalSpace A₂] [Semiring A₂] [Algebra R A₂] [TopologicalSemiring A₂] {α : Type u_5} [TopologicalSpace α] (g : A →ₐ[R] A₂) (hg : Continuous ⇑g) (f : C(α, A))
Goal: ∀ (a : α), ((AlgHom.compLeftContinuous R g hg) f) a = g (f a)
-- ContinuousMap.compRightAlgHom_apply
@[simps]
def ContinuousMap.compRightAlgHom {α β : Type*} [TopologicalSpace α] [TopologicalSpace β]
    (f : C(α, β)) : C(β, A) →ₐ[R] C(α, A) 
Arguments: (R : Type u_2) [CommSemiring R] (A : Type u_3) [TopologicalSpace A] [Semiring A] [Algebra R A] [TopologicalSemiring A] {α : Type u_5} {β : Type u_6} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β)) (g : C(β, A))
Goal: (ContinuousMap.compRightAlgHom R A f) g = g.comp f
-- ContinuousMap.coeFnAlgHom_apply
@[simps!]
def ContinuousMap.coeFnAlgHom : C(α, A) →ₐ[R] α → A 
Arguments: {α : Type u_1} [TopologicalSpace α] (R : Type u_2) [CommSemiring R] {A : Type u_3} [TopologicalSpace A] [Semiring A] [Algebra R A] [TopologicalSemiring A] (f : C(α, A)) (a : α)
Goal: (ContinuousMap.coeFnAlgHom R) f a = f a
-- Subalgebra.separatesPoints_monotone
theorem Subalgebra.separatesPoints_monotone :
    Monotone fun s : Subalgebra R C(α, A) => s.SeparatesPoints 
Arguments: {α : Type u_1} [TopologicalSpace α] {R : Type u_2} [CommSemiring R] {A : Type u_3} [TopologicalSpace A] [Semiring A] [Algebra R A] [TopologicalSemiring A]
Goal: Monotone fun s => s.SeparatesPoints
-- algebraMap_apply
@[simp]
theorem algebraMap_apply (k : R) (a : α) : algebraMap R C(α, A) k a = k • (1 : A) 
Arguments: {α : Type u_1} [TopologicalSpace α] {R : Type u_2} [CommSemiring R] {A : Type u_3} [TopologicalSpace A] [Semiring A] [Algebra R A] [TopologicalSemiring A] (k : R) (a : α)
Goal: ((algebraMap R C(α, A)) k) a = k • 1
-- Subalgebra.SeparatesPoints.strongly
theorem Subalgebra.SeparatesPoints.strongly {s : Subalgebra 𝕜 C(α, 𝕜)} (h : s.SeparatesPoints) :
    (s : Set C(α, 𝕜)).SeparatesPointsStrongly 
Arguments: {α : Type u_1} [TopologicalSpace α] {𝕜 : Type u_5} [TopologicalSpace 𝕜] [Field 𝕜] [TopologicalRing 𝕜] {s : Subalgebra 𝕜 C(α, 𝕜)} (h : s.SeparatesPoints)
Goal: (↑s).SeparatesPointsStrongly
-- ContinuousMap.coe_mabs
@[to_additive (attr := simp, norm_cast)]
lemma coe_mabs (f : C(α, β)) : ⇑|f|ₘ = |⇑f|ₘ 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [TopologicalSpace β] [Group β] [TopologicalGroup β] [Lattice β] [TopologicalLattice β] (f : C(α, β))
Goal: ⇑(mabs f) = mabs ⇑f
-- ContinuousMap.coe_abs
@[to_additive (attr := simp, norm_cast)]
lemma coe_mabs (f : C(α, β)) : ⇑|f|ₘ = |⇑f|ₘ 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [TopologicalSpace β] [AddGroup β] [TopologicalAddGroup β] [Lattice β] [TopologicalLattice β] (f : C(α, β))
Goal: ⇑|f| = |⇑f|
-- ContinuousMap.abs_apply
@[to_additive (attr := simp)]
lemma mabs_apply (f : C(α, β)) (x : α) : |f|ₘ x = |f x|ₘ 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [TopologicalSpace β] [AddGroup β] [TopologicalAddGroup β] [Lattice β] [TopologicalLattice β] (f : C(α, β)) (x : α)
Goal: |f| x = |f x|
-- ContinuousMap.mabs_apply
@[to_additive (attr := simp)]
lemma mabs_apply (f : C(α, β)) (x : α) : |f|ₘ x = |f x|ₘ 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [TopologicalSpace β] [Group β] [TopologicalGroup β] [Lattice β] [TopologicalLattice β] (f : C(α, β)) (x : α)
Goal: (mabs f) x = mabs (f x)
-- ContinuousMap.coe_star
@[simp]
theorem coe_star (f : C(α, β)) : ⇑(star f) = star (⇑f) 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [Star β] [ContinuousStar β] (f : C(α, β))
Goal: ⇑(star f) = star ⇑f
-- ContinuousMap.star_apply
@[simp]
theorem star_apply (f : C(α, β)) (x : α) : star f x = star (f x) 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [Star β] [ContinuousStar β] (f : C(α, β)) (x : α)
Goal: (star f) x = star (f x)
-- ContinuousMap.compStarAlgHom'_apply
@[simps]
def compStarAlgHom' (f : C(X, Y)) : C(Y, A) →⋆ₐ[𝕜] C(X, A) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (𝕜 : Type u_4) [CommSemiring 𝕜] (A : Type u_5) [TopologicalSpace A] [Semiring A] [TopologicalSemiring A] [Star A] [ContinuousStar A] [Algebra 𝕜 A] (f : C(X, Y)) (g : C(Y, A))
Goal: (ContinuousMap.compStarAlgHom' 𝕜 A f) g = g.comp f
-- ContinuousMap.compStarAlgHom'_id
theorem compStarAlgHom'_id : compStarAlgHom' 𝕜 A (ContinuousMap.id X) = StarAlgHom.id 𝕜 C(X, A) 
Arguments: {X : Type u_1} [TopologicalSpace X] (𝕜 : Type u_4) [CommSemiring 𝕜] (A : Type u_5) [TopologicalSpace A] [Semiring A] [TopologicalSemiring A] [Star A] [ContinuousStar A] [Algebra 𝕜 A]
Goal: ContinuousMap.compStarAlgHom' 𝕜 A (ContinuousMap.id X) = StarAlgHom.id 𝕜 C(X, A)
-- ContinuousMap.compStarAlgHom'_comp
theorem compStarAlgHom'_comp (g : C(Y, Z)) (f : C(X, Y)) :
    compStarAlgHom' 𝕜 A (g.comp f) = (compStarAlgHom' 𝕜 A f).comp (compStarAlgHom' 𝕜 A g) 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (𝕜 : Type u_4) [CommSemiring 𝕜] (A : Type u_5) [TopologicalSpace A] [Semiring A] [TopologicalSemiring A] [Star A] [ContinuousStar A] [Algebra 𝕜 A] (g : C(Y, Z)) (f : C(X, Y))
Goal: ContinuousMap.compStarAlgHom' 𝕜 A (g.comp f) =
  (ContinuousMap.compStarAlgHom' 𝕜 A f).comp (ContinuousMap.compStarAlgHom' 𝕜 A g)