ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- tendsto_const_mul_zpow_atTop_zero
theorem tendsto_const_mul_zpow_atTop_zero {n : ℤ} {c : 𝕜} (hn : n < 0) :
    Tendsto (fun x => c * x ^ n) atTop (𝓝 0) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {n : ℤ} {c : 𝕜} (hn : n < 0)
Goal: Filter.Tendsto (fun x => c * x ^ n) Filter.atTop (𝓝 0)
-- tendsto_const_mul_pow_nhds_iff'
theorem tendsto_const_mul_pow_nhds_iff' {n : ℕ} {c d : 𝕜} :
    Tendsto (fun x : 𝕜 => c * x ^ n) atTop (𝓝 d) ↔ (c = 0 ∨ n = 0) ∧ c = d 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {n : ℕ} {c d : 𝕜}
Goal: Filter.Tendsto (fun x => c * x ^ n) Filter.atTop (𝓝 d) ↔ (c = 0 ∨ n = 0) ∧ c = d
-- tendsto_const_mul_pow_nhds_iff
theorem tendsto_const_mul_pow_nhds_iff {n : ℕ} {c d : 𝕜} (hc : c ≠ 0) :
    Tendsto (fun x : 𝕜 => c * x ^ n) atTop (𝓝 d) ↔ n = 0 ∧ c = d 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {n : ℕ} {c d : 𝕜} (hc : c ≠ 0)
Goal: Filter.Tendsto (fun x => c * x ^ n) Filter.atTop (𝓝 d) ↔ n = 0 ∧ c = d
-- tendsto_const_mul_zpow_atTop_nhds_iff
theorem tendsto_const_mul_zpow_atTop_nhds_iff {n : ℤ} {c d : 𝕜} (hc : c ≠ 0) :
    Tendsto (fun x : 𝕜 => c * x ^ n) atTop (𝓝 d) ↔ n = 0 ∧ c = d ∨ n < 0 ∧ d = 0 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {n : ℤ} {c d : 𝕜} (hc : c ≠ 0)
Goal: Filter.Tendsto (fun x => c * x ^ n) Filter.atTop (𝓝 d) ↔ n = 0 ∧ c = d ∨ n < 0 ∧ d = 0
-- nhdsWithin_pos_comap_mul_left
theorem nhdsWithin_pos_comap_mul_left {x : 𝕜} (hx : 0 < x) :
    comap (x * ·) (𝓝[>] 0) = 𝓝[>] 0 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {x : 𝕜} (hx : 0 < x)
Goal: Filter.comap (fun x_1 => x * x_1) (𝓝[>] 0) = 𝓝[>] 0
-- eventually_nhdsWithin_pos_mul_left
theorem eventually_nhdsWithin_pos_mul_left {x : 𝕜} (hx : 0 < x) {p : 𝕜 → Prop}
    (h : ∀ᶠ ε in 𝓝[>] 0, p ε) : ∀ᶠ ε in 𝓝[>] 0, p (x * ε) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {x : 𝕜} (hx : 0 < x) {p : 𝕜 → Prop} (h : ∀ᶠ (ε : 𝕜) in 𝓝[>] 0, p ε)
Goal: ∀ᶠ (ε : 𝕜) in 𝓝[>] 0, p (x * ε)
-- NonarchimedeanGroup.cauchySeq_prod_of_tendsto_cofinite_one
@[to_additive "Let `G` be a nonarchimedean additive abelian group, and let `f : α → G` be a function
that tends to zero on the filter of cofinite sets. For each finite subset of `α`, consider the
partial sum of `f` on that subset. These partial sums form a Cauchy filter."]
theorem cauchySeq_prod_of_tendsto_cofinite_one {f : α → G} (hf : Tendsto f cofinite (𝓝 1)) :
    CauchySeq (fun s ↦ ∏ i ∈ s, f i) 
Arguments: {α : Type u_1} {G : Type u_2} [CommGroup G] [UniformSpace G] [UniformGroup G] [NonarchimedeanGroup G] {f : α → G} (hf : Filter.Tendsto f Filter.cofinite (𝓝 1))
Goal: CauchySeq fun s => ∏ i ∈ s, f i
-- NonarchimedeanAddGroup.cauchySeq_sum_of_tendsto_cofinite_zero
@[to_additive "Let `G` be a nonarchimedean additive abelian group, and let `f : α → G` be a function
that tends to zero on the filter of cofinite sets. For each finite subset of `α`, consider the
partial sum of `f` on that subset. These partial sums form a Cauchy filter."]
theorem cauchySeq_prod_of_tendsto_cofinite_one {f : α → G} (hf : Tendsto f cofinite (𝓝 1)) :
    CauchySeq (fun s ↦ ∏ i ∈ s, f i) 
Arguments: {α : Type u_1} {G : Type u_2} [AddCommGroup G] [UniformSpace G] [UniformAddGroup G] [NonarchimedeanAddGroup G] {f : α → G} (hf : Filter.Tendsto f Filter.cofinite (𝓝 0))
Goal: CauchySeq fun s => ∑ i ∈ s, f i
-- NonarchimedeanGroup.multipliable_of_tendsto_cofinite_one
@[to_additive "Let `G` be a complete nonarchimedean additive abelian group, and let `f : α → G` be a
function that tends to zero on the filter of cofinite sets. Then `f` is unconditionally summable."]
theorem multipliable_of_tendsto_cofinite_one [CompleteSpace G] {f : α → G}
    (hf : Tendsto f cofinite (𝓝 1)) : Multipliable f 
Arguments: {α : Type u_1} {G : Type u_2} [CommGroup G] [UniformSpace G] [UniformGroup G] [NonarchimedeanGroup G] [CompleteSpace G] {f : α → G} (hf : Filter.Tendsto f Filter.cofinite (𝓝 1))
Goal: Multipliable f
-- NonarchimedeanAddGroup.summable_of_tendsto_cofinite_zero
@[to_additive "Let `G` be a complete nonarchimedean additive abelian group, and let `f : α → G` be a
function that tends to zero on the filter of cofinite sets. Then `f` is unconditionally summable."]
theorem multipliable_of_tendsto_cofinite_one [CompleteSpace G] {f : α → G}
    (hf : Tendsto f cofinite (𝓝 1)) : Multipliable f 
Arguments: {α : Type u_1} {G : Type u_2} [AddCommGroup G] [UniformSpace G] [UniformAddGroup G] [NonarchimedeanAddGroup G] [CompleteSpace G] {f : α → G} (hf : Filter.Tendsto f Filter.cofinite (𝓝 0))
Goal: Summable f
-- NonarchimedeanGroup.multipliable_iff_tendsto_cofinite_one
@[to_additive "Let `G` be a complete nonarchimedean additive abelian group. Then a function
`f : α → G` is unconditionally summable if and only if it tends to zero on the filter of cofinite
sets."]
theorem multipliable_iff_tendsto_cofinite_one [CompleteSpace G] (f : α → G) :
    Multipliable f ↔ Tendsto f cofinite (𝓝 1) 
Arguments: {α : Type u_1} {G : Type u_2} [CommGroup G] [UniformSpace G] [UniformGroup G] [NonarchimedeanGroup G] [CompleteSpace G] (f : α → G)
Goal: Multipliable f ↔ Filter.Tendsto f Filter.cofinite (𝓝 1)
-- NonarchimedeanAddGroup.summable_iff_tendsto_cofinite_zero
@[to_additive "Let `G` be a complete nonarchimedean additive abelian group. Then a function
`f : α → G` is unconditionally summable if and only if it tends to zero on the filter of cofinite
sets."]
theorem multipliable_iff_tendsto_cofinite_one [CompleteSpace G] (f : α → G) :
    Multipliable f ↔ Tendsto f cofinite (𝓝 1) 
Arguments: {α : Type u_1} {G : Type u_2} [AddCommGroup G] [UniformSpace G] [UniformAddGroup G] [NonarchimedeanAddGroup G] [CompleteSpace G] (f : α → G)
Goal: Summable f ↔ Filter.Tendsto f Filter.cofinite (𝓝 0)
-- DomMulAct.continuous_mk
@[to_additive (attr := continuity, fun_prop)]
theorem continuous_mk : Continuous (@mk M) 
Arguments: {M : Type u_1} [TopologicalSpace M]
Goal: Continuous ⇑DomMulAct.mk
-- DomAddAct.continuous_mk
@[to_additive (attr := continuity, fun_prop)]
theorem continuous_mk : Continuous (@mk M) 
Arguments: {M : Type u_1} [TopologicalSpace M]
Goal: Continuous ⇑DomAddAct.mk
-- DomMulAct.continuous_mk_symm
@[to_additive (attr := continuity, fun_prop)]
theorem continuous_mk_symm : Continuous (@mk M).symm 
Arguments: {M : Type u_1} [TopologicalSpace M]
Goal: Continuous ⇑DomMulAct.mk.symm
-- DomAddAct.continuous_mk_symm
@[to_additive (attr := continuity, fun_prop)]
theorem continuous_mk_symm : Continuous (@mk M).symm 
Arguments: {M : Type u_1} [TopologicalSpace M]
Goal: Continuous ⇑DomAddAct.mk.symm
-- DomAddAct.mkHomeomorph_toEquiv
@[to_additive (attr := simps toEquiv) "`DomAddAct.mk` as a homeomorphism"]
def mkHomeomorph : M ≃ₜ Mᵈᵐᵃ 
Arguments: {M : Type u_1} [TopologicalSpace M]
Goal: DomAddAct.mkHomeomorph.toEquiv = DomAddAct.mk
-- DomMulAct.mkHomeomorph_toEquiv
@[to_additive (attr := simps toEquiv) "`DomAddAct.mk` as a homeomorphism"]
def mkHomeomorph : M ≃ₜ Mᵈᵐᵃ 
Arguments: {M : Type u_1} [TopologicalSpace M]
Goal: DomMulAct.mkHomeomorph.toEquiv = DomMulAct.mk
-- DomAddAct.coe_mkHomeomorph
@[to_additive (attr := simp)] theorem coe_mkHomeomorph : ⇑(mkHomeomorph : M ≃ₜ Mᵈᵐᵃ) = mk 
Arguments: {M : Type u_1} [TopologicalSpace M]
Goal: ⇑DomAddAct.mkHomeomorph = ⇑DomAddAct.mk
-- DomMulAct.coe_mkHomeomorph
@[to_additive (attr := simp)] theorem coe_mkHomeomorph : ⇑(mkHomeomorph : M ≃ₜ Mᵈᵐᵃ) = mk 
Arguments: {M : Type u_1} [TopologicalSpace M]
Goal: ⇑DomMulAct.mkHomeomorph = ⇑DomMulAct.mk
-- DomMulAct.coe_mkHomeomorph_symm
@[to_additive (attr := simp)]
theorem coe_mkHomeomorph_symm : ⇑(mkHomeomorph : M ≃ₜ Mᵈᵐᵃ).symm = mk.symm 
Arguments: {M : Type u_1} [TopologicalSpace M]
Goal: ⇑DomMulAct.mkHomeomorph.symm = ⇑DomMulAct.mk.symm
-- DomAddAct.coe_mkHomeomorph_symm
@[to_additive (attr := simp)]
theorem coe_mkHomeomorph_symm : ⇑(mkHomeomorph : M ≃ₜ Mᵈᵐᵃ).symm = mk.symm 
Arguments: {M : Type u_1} [TopologicalSpace M]
Goal: ⇑DomAddAct.mkHomeomorph.symm = ⇑DomAddAct.mk.symm
-- DomAddAct.inducing_mk
@[to_additive] theorem inducing_mk : Inducing (@mk M) 
Arguments: {M : Type u_1} [TopologicalSpace M]
Goal: Inducing ⇑DomAddAct.mk
-- DomMulAct.inducing_mk
@[to_additive] theorem inducing_mk : Inducing (@mk M) 
Arguments: {M : Type u_1} [TopologicalSpace M]
Goal: Inducing ⇑DomMulAct.mk
-- DomMulAct.embedding_mk
@[to_additive] theorem embedding_mk : Embedding (@mk M) 
Arguments: {M : Type u_1} [TopologicalSpace M]
Goal: Embedding ⇑DomMulAct.mk
-- DomAddAct.embedding_mk
@[to_additive] theorem embedding_mk : Embedding (@mk M) 
Arguments: {M : Type u_1} [TopologicalSpace M]
Goal: Embedding ⇑DomAddAct.mk
-- DomMulAct.openEmbedding_mk
@[to_additive] theorem openEmbedding_mk : OpenEmbedding (@mk M) 
Arguments: {M : Type u_1} [TopologicalSpace M]
Goal: OpenEmbedding ⇑DomMulAct.mk
-- DomAddAct.openEmbedding_mk
@[to_additive] theorem openEmbedding_mk : OpenEmbedding (@mk M) 
Arguments: {M : Type u_1} [TopologicalSpace M]
Goal: OpenEmbedding ⇑DomAddAct.mk
-- DomAddAct.closedEmbedding_mk
@[to_additive] theorem closedEmbedding_mk : ClosedEmbedding (@mk M) 
Arguments: {M : Type u_1} [TopologicalSpace M]
Goal: ClosedEmbedding ⇑DomAddAct.mk
-- DomMulAct.closedEmbedding_mk
@[to_additive] theorem closedEmbedding_mk : ClosedEmbedding (@mk M) 
Arguments: {M : Type u_1} [TopologicalSpace M]
Goal: ClosedEmbedding ⇑DomMulAct.mk
-- DomAddAct.quotientMap_mk
@[to_additive] theorem quotientMap_mk : QuotientMap (@mk M) 
Arguments: {M : Type u_1} [TopologicalSpace M]
Goal: QuotientMap ⇑DomAddAct.mk
-- DomMulAct.quotientMap_mk
@[to_additive] theorem quotientMap_mk : QuotientMap (@mk M) 
Arguments: {M : Type u_1} [TopologicalSpace M]
Goal: QuotientMap ⇑DomMulAct.mk
-- DomAddAct.inducing_mk_symm
@[to_additive] theorem inducing_mk_symm : Inducing (@mk M).symm 
Arguments: {M : Type u_1} [TopologicalSpace M]
Goal: Inducing ⇑DomAddAct.mk.symm
-- DomMulAct.inducing_mk_symm
@[to_additive] theorem inducing_mk_symm : Inducing (@mk M).symm 
Arguments: {M : Type u_1} [TopologicalSpace M]
Goal: Inducing ⇑DomMulAct.mk.symm
-- DomMulAct.embedding_mk_symm
@[to_additive] theorem embedding_mk_symm : Embedding (@mk M).symm 
Arguments: {M : Type u_1} [TopologicalSpace M]
Goal: Embedding ⇑DomMulAct.mk.symm
-- DomAddAct.embedding_mk_symm
@[to_additive] theorem embedding_mk_symm : Embedding (@mk M).symm 
Arguments: {M : Type u_1} [TopologicalSpace M]
Goal: Embedding ⇑DomAddAct.mk.symm
-- DomAddAct.openEmbedding_mk_symm
@[to_additive]
theorem openEmbedding_mk_symm : OpenEmbedding (@mk M).symm 
Arguments: {M : Type u_1} [TopologicalSpace M]
Goal: OpenEmbedding ⇑DomAddAct.mk.symm
-- DomMulAct.openEmbedding_mk_symm
@[to_additive]
theorem openEmbedding_mk_symm : OpenEmbedding (@mk M).symm 
Arguments: {M : Type u_1} [TopologicalSpace M]
Goal: OpenEmbedding ⇑DomMulAct.mk.symm
-- DomMulAct.closedEmbedding_mk_symm
@[to_additive]
theorem closedEmbedding_mk_symm : ClosedEmbedding (@mk M).symm 
Arguments: {M : Type u_1} [TopologicalSpace M]
Goal: ClosedEmbedding ⇑DomMulAct.mk.symm
-- DomAddAct.closedEmbedding_mk_symm
@[to_additive]
theorem closedEmbedding_mk_symm : ClosedEmbedding (@mk M).symm 
Arguments: {M : Type u_1} [TopologicalSpace M]
Goal: ClosedEmbedding ⇑DomAddAct.mk.symm
-- DomMulAct.quotientMap_mk_symm
@[to_additive]
theorem quotientMap_mk_symm : QuotientMap (@mk M).symm 
Arguments: {M : Type u_1} [TopologicalSpace M]
Goal: QuotientMap ⇑DomMulAct.mk.symm
-- DomAddAct.quotientMap_mk_symm
@[to_additive]
theorem quotientMap_mk_symm : QuotientMap (@mk M).symm 
Arguments: {M : Type u_1} [TopologicalSpace M]
Goal: QuotientMap ⇑DomAddAct.mk.symm
-- DomAddAct.map_mk_nhds
@[to_additive (attr := simp)]
theorem map_mk_nhds (x : M) : map (mk : M → Mᵈᵐᵃ) (𝓝 x) = 𝓝 (mk x) 
Arguments: {M : Type u_1} [TopologicalSpace M] (x : M)
Goal: Filter.map (⇑DomAddAct.mk) (𝓝 x) = 𝓝 (DomAddAct.mk x)
-- DomMulAct.map_mk_nhds
@[to_additive (attr := simp)]
theorem map_mk_nhds (x : M) : map (mk : M → Mᵈᵐᵃ) (𝓝 x) = 𝓝 (mk x) 
Arguments: {M : Type u_1} [TopologicalSpace M] (x : M)
Goal: Filter.map (⇑DomMulAct.mk) (𝓝 x) = 𝓝 (DomMulAct.mk x)
-- DomMulAct.map_mk_symm_nhds
@[to_additive (attr := simp)]
theorem map_mk_symm_nhds (x : Mᵈᵐᵃ) : map (mk.symm : Mᵈᵐᵃ → M) (𝓝 x) = 𝓝 (mk.symm x) 
Arguments: {M : Type u_1} [TopologicalSpace M] (x : Mᵈᵐᵃ)
Goal: Filter.map (⇑DomMulAct.mk.symm) (𝓝 x) = 𝓝 (DomMulAct.mk.symm x)
-- DomAddAct.map_mk_symm_nhds
@[to_additive (attr := simp)]
theorem map_mk_symm_nhds (x : Mᵈᵐᵃ) : map (mk.symm : Mᵈᵐᵃ → M) (𝓝 x) = 𝓝 (mk.symm x) 
Arguments: {M : Type u_1} [TopologicalSpace M] (x : Mᵈᵃᵃ)
Goal: Filter.map (⇑DomAddAct.mk.symm) (𝓝 x) = 𝓝 (DomAddAct.mk.symm x)
-- DomMulAct.comap_mk_nhds
@[to_additive (attr := simp)]
theorem comap_mk_nhds (x : Mᵈᵐᵃ) : comap (mk : M → Mᵈᵐᵃ) (𝓝 x) = 𝓝 (mk.symm x) 
Arguments: {M : Type u_1} [TopologicalSpace M] (x : Mᵈᵐᵃ)
Goal: Filter.comap (⇑DomMulAct.mk) (𝓝 x) = 𝓝 (DomMulAct.mk.symm x)
-- DomAddAct.comap_mk_nhds
@[to_additive (attr := simp)]
theorem comap_mk_nhds (x : Mᵈᵐᵃ) : comap (mk : M → Mᵈᵐᵃ) (𝓝 x) = 𝓝 (mk.symm x) 
Arguments: {M : Type u_1} [TopologicalSpace M] (x : Mᵈᵃᵃ)
Goal: Filter.comap (⇑DomAddAct.mk) (𝓝 x) = 𝓝 (DomAddAct.mk.symm x)
-- DomAddAct.comap_mk.symm_nhds
@[to_additive (attr := simp)]
theorem comap_mk.symm_nhds (x : M) : comap (mk.symm : Mᵈᵐᵃ → M) (𝓝 x) = 𝓝 (mk x) 
Arguments: {M : Type u_1} [TopologicalSpace M] (x : M)
Goal: Filter.comap (⇑DomAddAct.mk.symm) (𝓝 x) = 𝓝 (DomAddAct.mk x)
-- DomMulAct.comap_mk.symm_nhds
@[to_additive (attr := simp)]
theorem comap_mk.symm_nhds (x : M) : comap (mk.symm : Mᵈᵐᵃ → M) (𝓝 x) = 𝓝 (mk x) 
Arguments: {M : Type u_1} [TopologicalSpace M] (x : M)
Goal: Filter.comap (⇑DomMulAct.mk.symm) (𝓝 x) = 𝓝 (DomMulAct.mk x)
-- LocallyConstant.coe_zero
@[to_additive (attr := simp)]
theorem coe_one [One Y] : ⇑(1 : LocallyConstant X Y) = (1 : X → Y) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Zero Y]
Goal: ⇑0 = 0
-- LocallyConstant.coe_one
@[to_additive (attr := simp)]
theorem coe_one [One Y] : ⇑(1 : LocallyConstant X Y) = (1 : X → Y) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [One Y]
Goal: ⇑1 = 1
-- LocallyConstant.zero_apply
@[to_additive]
theorem one_apply [One Y] (x : X) : (1 : LocallyConstant X Y) x = 1 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Zero Y] (x : X)
Goal: 0 x = 0
-- LocallyConstant.one_apply
@[to_additive]
theorem one_apply [One Y] (x : X) : (1 : LocallyConstant X Y) x = 1 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [One Y] (x : X)
Goal: 1 x = 1
-- LocallyConstant.coe_neg
@[to_additive (attr := simp)]
theorem coe_inv [Inv Y] (f : LocallyConstant X Y) : ⇑(f⁻¹ : LocallyConstant X Y) = (f : X → Y)⁻¹ 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Neg Y] (f : LocallyConstant X Y)
Goal: ⇑(-f) = -⇑f
-- LocallyConstant.coe_inv
@[to_additive (attr := simp)]
theorem coe_inv [Inv Y] (f : LocallyConstant X Y) : ⇑(f⁻¹ : LocallyConstant X Y) = (f : X → Y)⁻¹ 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Inv Y] (f : LocallyConstant X Y)
Goal: ⇑f⁻¹ = (⇑f)⁻¹
-- LocallyConstant.neg_apply
@[to_additive]
theorem inv_apply [Inv Y] (f : LocallyConstant X Y) (x : X) : f⁻¹ x = (f x)⁻¹ 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Neg Y] (f : LocallyConstant X Y) (x : X)
Goal: (-f) x = -f x
-- LocallyConstant.inv_apply
@[to_additive]
theorem inv_apply [Inv Y] (f : LocallyConstant X Y) (x : X) : f⁻¹ x = (f x)⁻¹ 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Inv Y] (f : LocallyConstant X Y) (x : X)
Goal: f⁻¹ x = (f x)⁻¹
-- LocallyConstant.coe_add
@[to_additive (attr := simp)]
theorem coe_mul [Mul Y] (f g : LocallyConstant X Y) : ⇑(f * g) = f * g 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Add Y] (f g : LocallyConstant X Y)
Goal: ⇑(f + g) = ⇑f + ⇑g
-- LocallyConstant.coe_mul
@[to_additive (attr := simp)]
theorem coe_mul [Mul Y] (f g : LocallyConstant X Y) : ⇑(f * g) = f * g 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Mul Y] (f g : LocallyConstant X Y)
Goal: ⇑(f * g) = ⇑f * ⇑g
-- LocallyConstant.mul_apply
@[to_additive]
theorem mul_apply [Mul Y] (f g : LocallyConstant X Y) (x : X) : (f * g) x = f x * g x 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Mul Y] (f g : LocallyConstant X Y) (x : X)
Goal: (f * g) x = f x * g x
-- LocallyConstant.add_apply
@[to_additive]
theorem mul_apply [Mul Y] (f g : LocallyConstant X Y) (x : X) : (f * g) x = f x * g x 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Add Y] (f g : LocallyConstant X Y) (x : X)
Goal: (f + g) x = f x + g x
-- LocallyConstant.coeFnMonoidHom_apply
@[to_additive (attr := simps) "`DFunLike.coe` as an `AddMonoidHom`."]
def coeFnMonoidHom [MulOneClass Y] : LocallyConstant X Y →* X → Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [MulOneClass Y]
Goal: ∀ (a : LocallyConstant X Y) (a_1 : X), LocallyConstant.coeFnMonoidHom a a_1 = a a_1
-- LocallyConstant.coeFnAddMonoidHom_apply
@[to_additive (attr := simps) "`DFunLike.coe` as an `AddMonoidHom`."]
def coeFnMonoidHom [MulOneClass Y] : LocallyConstant X Y →* X → Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [AddZeroClass Y]
Goal: ∀ (a : LocallyConstant X Y) (a_1 : X), LocallyConstant.coeFnAddMonoidHom a a_1 = a a_1
-- LocallyConstant.constMonoidHom_apply
@[to_additive (attr := simps) "The constant-function embedding, as an additive monoid hom."]
def constMonoidHom [MulOneClass Y] : Y →* LocallyConstant X Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [MulOneClass Y] (y : Y)
Goal: LocallyConstant.constMonoidHom y = LocallyConstant.const X y
-- LocallyConstant.constAddMonoidHom_apply
@[to_additive (attr := simps) "The constant-function embedding, as an additive monoid hom."]
def constMonoidHom [MulOneClass Y] : Y →* LocallyConstant X Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [AddZeroClass Y] (y : Y)
Goal: LocallyConstant.constAddMonoidHom y = LocallyConstant.const X y
-- LocallyConstant.coe_charFn
theorem coe_charFn (hU : IsClopen U) : (charFn Y hU : X → Y) = Set.indicator U 1 
Arguments: {X : Type u_1} (Y : Type u_2) [TopologicalSpace X] [MulZeroOneClass Y] {U : Set X} (hU : IsClopen U)
Goal: ⇑(LocallyConstant.charFn Y hU) = U.indicator 1
-- LocallyConstant.charFn_eq_one
theorem charFn_eq_one [Nontrivial Y] (x : X) (hU : IsClopen U) : charFn Y hU x = (1 : Y) ↔ x ∈ U 
Arguments: {X : Type u_1} (Y : Type u_2) [TopologicalSpace X] [MulZeroOneClass Y] {U : Set X} [Nontrivial Y] (x : X) (hU : IsClopen U)
Goal: (LocallyConstant.charFn Y hU) x = 1 ↔ x ∈ U
-- LocallyConstant.charFn_eq_zero
theorem charFn_eq_zero [Nontrivial Y] (x : X) (hU : IsClopen U) : charFn Y hU x = (0 : Y) ↔ x ∉ U 
Arguments: {X : Type u_1} (Y : Type u_2) [TopologicalSpace X] [MulZeroOneClass Y] {U : Set X} [Nontrivial Y] (x : X) (hU : IsClopen U)
Goal: (LocallyConstant.charFn Y hU) x = 0 ↔ x ∉ U
-- LocallyConstant.charFn_inj
theorem charFn_inj [Nontrivial Y] (hU : IsClopen U) (hV : IsClopen V)
    (h : charFn Y hU = charFn Y hV) : U = V 
Arguments: {X : Type u_1} (Y : Type u_2) [TopologicalSpace X] [MulZeroOneClass Y] {U V : Set X} [Nontrivial Y] (hU : IsClopen U) (hV : IsClopen V) (h : LocallyConstant.charFn Y hU = LocallyConstant.charFn Y hV)
Goal: U = V
-- LocallyConstant.coe_div
@[to_additive]
theorem coe_div [Div Y] (f g : LocallyConstant X Y) : ⇑(f / g) = f / g 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Div Y] (f g : LocallyConstant X Y)
Goal: ⇑(f / g) = ⇑f / ⇑g
-- LocallyConstant.coe_sub
@[to_additive]
theorem coe_div [Div Y] (f g : LocallyConstant X Y) : ⇑(f / g) = f / g 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Sub Y] (f g : LocallyConstant X Y)
Goal: ⇑(f - g) = ⇑f - ⇑g
-- LocallyConstant.sub_apply
@[to_additive]
theorem div_apply [Div Y] (f g : LocallyConstant X Y) (x : X) : (f / g) x = f x / g x 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Sub Y] (f g : LocallyConstant X Y) (x : X)
Goal: (f - g) x = f x - g x
-- LocallyConstant.div_apply
@[to_additive]
theorem div_apply [Div Y] (f g : LocallyConstant X Y) (x : X) : (f / g) x = f x / g x 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Div Y] (f g : LocallyConstant X Y) (x : X)
Goal: (f / g) x = f x / g x
-- LocallyConstant.coe_smul
@[to_additive (attr := simp)]
theorem coe_smul [SMul R Y] (r : R) (f : LocallyConstant X Y) : ⇑(r • f) = r • (f : X → Y) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {R : Type u_4} [SMul R Y] (r : R) (f : LocallyConstant X Y)
Goal: ⇑(r • f) = r • ⇑f
-- LocallyConstant.coe_vadd
@[to_additive (attr := simp)]
theorem coe_smul [SMul R Y] (r : R) (f : LocallyConstant X Y) : ⇑(r • f) = r • (f : X → Y) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {R : Type u_4} [VAdd R Y] (r : R) (f : LocallyConstant X Y)
Goal: ⇑(r +ᵥ f) = r +ᵥ ⇑f
-- LocallyConstant.vadd_apply
@[to_additive]
theorem smul_apply [SMul R Y] (r : R) (f : LocallyConstant X Y) (x : X) : (r • f) x = r • f x 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {R : Type u_4} [VAdd R Y] (r : R) (f : LocallyConstant X Y) (x : X)
Goal: (r +ᵥ f) x = r +ᵥ f x
-- LocallyConstant.smul_apply
@[to_additive]
theorem smul_apply [SMul R Y] (r : R) (f : LocallyConstant X Y) (x : X) : (r • f) x = r • f x 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {R : Type u_4} [SMul R Y] (r : R) (f : LocallyConstant X Y) (x : X)
Goal: (r • f) x = r • f x
-- LocallyConstant.constRingHom_apply
@[simps]
def constRingHom [NonAssocSemiring Y] : Y →+* LocallyConstant X Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [NonAssocSemiring Y] (y : Y)
Goal: LocallyConstant.constRingHom y = LocallyConstant.const X y
-- LocallyConstant.coe_algebraMap
@[simp]
theorem coe_algebraMap (r : R) : ⇑(algebraMap R (LocallyConstant X Y) r) = algebraMap R (X → Y) r 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {R : Type u_5} [CommSemiring R] [Semiring Y] [Algebra R Y] (r : R)
Goal: ⇑((algebraMap R (LocallyConstant X Y)) r) = (algebraMap R (X → Y)) r
-- LocallyConstant.coeFnRingHom_apply
@[simps!] def coeFnRingHom [Semiring Y] : LocallyConstant X Y →+* X → Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Semiring Y]
Goal: ∀ (a : LocallyConstant X Y) (a_1 : X), LocallyConstant.coeFnRingHom a a_1 = a a_1
-- LocallyConstant.coeFnₗ_apply
@[simps!] def coeFnₗ (R : Type*) [Semiring R] [AddCommMonoid Y]
    [Module R Y] : LocallyConstant X Y →ₗ[R] X → Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] (R : Type u_6) [Semiring R] [AddCommMonoid Y] [Module R Y]
Goal: ∀ (a : LocallyConstant X Y) (a_1 : X), (LocallyConstant.coeFnₗ R) a a_1 = a a_1
-- LocallyConstant.coeFnAlgHom_apply
@[simps!] def coeFnAlgHom (R : Type*) [CommSemiring R] [Semiring Y] [Algebra R Y] :
    LocallyConstant X Y →ₐ[R] X → Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] (R : Type u_6) [CommSemiring R] [Semiring Y] [Algebra R Y]
Goal: ∀ (a : LocallyConstant X Y) (a_1 : X), (LocallyConstant.coeFnAlgHom R) a a_1 = a a_1
-- LocallyConstant.evalAddMonoidHom_apply
@[to_additive (attr := simps!) "Evaluation as an `AddMonoidHom`"]
def evalMonoidHom [MulOneClass Y] (x : X) : LocallyConstant X Y →* Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [AddZeroClass Y] (x : X)
Goal: ∀ (a : LocallyConstant X Y), (LocallyConstant.evalAddMonoidHom x) a = a x
-- LocallyConstant.evalMonoidHom_apply
@[to_additive (attr := simps!) "Evaluation as an `AddMonoidHom`"]
def evalMonoidHom [MulOneClass Y] (x : X) : LocallyConstant X Y →* Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [MulOneClass Y] (x : X)
Goal: ∀ (a : LocallyConstant X Y), (LocallyConstant.evalMonoidHom x) a = a x
-- LocallyConstant.evalₗ_apply
@[simps!] def evalₗ (R : Type*) [Semiring R] [AddCommMonoid Y]
    [Module R Y] (x : X) : LocallyConstant X Y →ₗ[R] Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] (R : Type u_6) [Semiring R] [AddCommMonoid Y] [Module R Y] (x : X)
Goal: ∀ (a : LocallyConstant X Y), (LocallyConstant.evalₗ R x) a = a x
-- LocallyConstant.evalRingHom_apply
@[simps!] def evalRingHom [Semiring Y] (x : X) : LocallyConstant X Y →+* Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Semiring Y] (x : X)
Goal: ∀ (a : LocallyConstant X Y), (LocallyConstant.evalRingHom x) a = a x