ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- WithZeroTopology.isOpen_Iio
theorem isOpen_Iio {a : Γ₀} : IsOpen (Iio a) 
Arguments: {Γ₀ : Type u_2} [LinearOrderedCommGroupWithZero Γ₀] {a : Γ₀}
Goal: IsOpen (Set.Iio a)
-- LinearPMap.IsClosed.isClosable
theorem IsClosed.isClosable {f : E →ₗ.[R] F} (hf : f.IsClosed) : f.IsClosable 
Arguments: {R : Type u_1} {E : Type u_2} {F : Type u_3} [CommRing R] [AddCommGroup E] [AddCommGroup F] [Module R E] [Module R F] [TopologicalSpace E] [TopologicalSpace F] [ContinuousAdd E] [ContinuousAdd F] [TopologicalSpace R] [ContinuousSMul R E] [ContinuousSMul R F] {f : E →ₗ.[R] F} (hf : f.IsClosed)
Goal: f.IsClosable
-- LinearPMap.IsClosable.leIsClosable
theorem IsClosable.leIsClosable {f g : E →ₗ.[R] F} (hf : f.IsClosable) (hfg : g ≤ f) :
    g.IsClosable 
Arguments: {R : Type u_1} {E : Type u_2} {F : Type u_3} [CommRing R] [AddCommGroup E] [AddCommGroup F] [Module R E] [Module R F] [TopologicalSpace E] [TopologicalSpace F] [ContinuousAdd E] [ContinuousAdd F] [TopologicalSpace R] [ContinuousSMul R E] [ContinuousSMul R F] {f g : E →ₗ.[R] F} (hf : f.IsClosable) (hfg : g ≤ f)
Goal: g.IsClosable
-- LinearPMap.IsClosable.existsUnique
theorem IsClosable.existsUnique {f : E →ₗ.[R] F} (hf : f.IsClosable) :
    ∃! f' : E →ₗ.[R] F, f.graph.topologicalClosure = f'.graph 
Arguments: {R : Type u_1} {E : Type u_2} {F : Type u_3} [CommRing R] [AddCommGroup E] [AddCommGroup F] [Module R E] [Module R F] [TopologicalSpace E] [TopologicalSpace F] [ContinuousAdd E] [ContinuousAdd F] [TopologicalSpace R] [ContinuousSMul R E] [ContinuousSMul R F] {f : E →ₗ.[R] F} (hf : f.IsClosable)
Goal: ∃! f', f.graph.topologicalClosure = f'.graph
-- LinearPMap.closure_def
theorem closure_def {f : E →ₗ.[R] F} (hf : f.IsClosable) : f.closure = hf.choose 
Arguments: {R : Type u_1} {E : Type u_2} {F : Type u_3} [CommRing R] [AddCommGroup E] [AddCommGroup F] [Module R E] [Module R F] [TopologicalSpace E] [TopologicalSpace F] [ContinuousAdd E] [ContinuousAdd F] [TopologicalSpace R] [ContinuousSMul R E] [ContinuousSMul R F] {f : E →ₗ.[R] F} (hf : f.IsClosable)
Goal: f.closure = Exists.choose hf
-- LinearPMap.closure_def'
theorem closure_def' {f : E →ₗ.[R] F} (hf : ¬f.IsClosable) : f.closure = f 
Arguments: {R : Type u_1} {E : Type u_2} {F : Type u_3} [CommRing R] [AddCommGroup E] [AddCommGroup F] [Module R E] [Module R F] [TopologicalSpace E] [TopologicalSpace F] [ContinuousAdd E] [ContinuousAdd F] [TopologicalSpace R] [ContinuousSMul R E] [ContinuousSMul R F] {f : E →ₗ.[R] F} (hf : ¬f.IsClosable)
Goal: f.closure = f
-- LinearPMap.IsClosable.graph_closure_eq_closure_graph
theorem IsClosable.graph_closure_eq_closure_graph {f : E →ₗ.[R] F} (hf : f.IsClosable) :
    f.graph.topologicalClosure = f.closure.graph 
Arguments: {R : Type u_1} {E : Type u_2} {F : Type u_3} [CommRing R] [AddCommGroup E] [AddCommGroup F] [Module R E] [Module R F] [TopologicalSpace E] [TopologicalSpace F] [ContinuousAdd E] [ContinuousAdd F] [TopologicalSpace R] [ContinuousSMul R E] [ContinuousSMul R F] {f : E →ₗ.[R] F} (hf : f.IsClosable)
Goal: f.graph.topologicalClosure = f.closure.graph
-- LinearPMap.le_closure
theorem le_closure (f : E →ₗ.[R] F) : f ≤ f.closure 
Arguments: {R : Type u_1} {E : Type u_2} {F : Type u_3} [CommRing R] [AddCommGroup E] [AddCommGroup F] [Module R E] [Module R F] [TopologicalSpace E] [TopologicalSpace F] [ContinuousAdd E] [ContinuousAdd F] [TopologicalSpace R] [ContinuousSMul R E] [ContinuousSMul R F] (f : E →ₗ.[R] F)
Goal: f ≤ f.closure
-- LinearPMap.IsClosable.closure_mono
theorem IsClosable.closure_mono {f g : E →ₗ.[R] F} (hg : g.IsClosable) (h : f ≤ g) :
    f.closure ≤ g.closure 
Arguments: {R : Type u_1} {E : Type u_2} {F : Type u_3} [CommRing R] [AddCommGroup E] [AddCommGroup F] [Module R E] [Module R F] [TopologicalSpace E] [TopologicalSpace F] [ContinuousAdd E] [ContinuousAdd F] [TopologicalSpace R] [ContinuousSMul R E] [ContinuousSMul R F] {f g : E →ₗ.[R] F} (hg : g.IsClosable) (h : f ≤ g)
Goal: f.closure ≤ g.closure
-- LinearPMap.IsClosable.closure_isClosed
theorem IsClosable.closure_isClosed {f : E →ₗ.[R] F} (hf : f.IsClosable) : f.closure.IsClosed 
Arguments: {R : Type u_1} {E : Type u_2} {F : Type u_3} [CommRing R] [AddCommGroup E] [AddCommGroup F] [Module R E] [Module R F] [TopologicalSpace E] [TopologicalSpace F] [ContinuousAdd E] [ContinuousAdd F] [TopologicalSpace R] [ContinuousSMul R E] [ContinuousSMul R F] {f : E →ₗ.[R] F} (hf : f.IsClosable)
Goal: f.closure.IsClosed
-- LinearPMap.IsClosable.closureIsClosable
theorem IsClosable.closureIsClosable {f : E →ₗ.[R] F} (hf : f.IsClosable) : f.closure.IsClosable 
Arguments: {R : Type u_1} {E : Type u_2} {F : Type u_3} [CommRing R] [AddCommGroup E] [AddCommGroup F] [Module R E] [Module R F] [TopologicalSpace E] [TopologicalSpace F] [ContinuousAdd E] [ContinuousAdd F] [TopologicalSpace R] [ContinuousSMul R E] [ContinuousSMul R F] {f : E →ₗ.[R] F} (hf : f.IsClosable)
Goal: f.closure.IsClosable
-- LinearPMap.isClosable_iff_exists_closed_extension
theorem isClosable_iff_exists_closed_extension {f : E →ₗ.[R] F} :
    f.IsClosable ↔ ∃ g : E →ₗ.[R] F, g.IsClosed ∧ f ≤ g 
Arguments: {R : Type u_1} {E : Type u_2} {F : Type u_3} [CommRing R] [AddCommGroup E] [AddCommGroup F] [Module R E] [Module R F] [TopologicalSpace E] [TopologicalSpace F] [ContinuousAdd E] [ContinuousAdd F] [TopologicalSpace R] [ContinuousSMul R E] [ContinuousSMul R F] {f : E →ₗ.[R] F}
Goal: f.IsClosable ↔ ∃ g, g.IsClosed ∧ f ≤ g
-- LinearPMap.HasCore.le_domain
structure HasCore (f : E →ₗ.[R] F) (S : Submodule R E) : Prop where
  le_domain : S ≤ f.domain
  closure_eq : (f.domRestrict S).closure = f
Arguments: {R : Type u_1} {E : Type u_2} {F : Type u_3} [CommRing R] [AddCommGroup E] [AddCommGroup F] [Module R E] [Module R F] [TopologicalSpace E] [TopologicalSpace F] [ContinuousAdd E] [ContinuousAdd F] [TopologicalSpace R] [ContinuousSMul R E] [ContinuousSMul R F] {f : E →ₗ.[R] F} {S : Submodule R E} (self : f.HasCore S)
Goal: S ≤ f.domain
-- LinearPMap.HasCore.closure_eq
structure HasCore (f : E →ₗ.[R] F) (S : Submodule R E) : Prop where
  le_domain : S ≤ f.domain
  closure_eq : (f.domRestrict S).closure = f
Arguments: {R : Type u_1} {E : Type u_2} {F : Type u_3} [CommRing R] [AddCommGroup E] [AddCommGroup F] [Module R E] [Module R F] [TopologicalSpace E] [TopologicalSpace F] [ContinuousAdd E] [ContinuousAdd F] [TopologicalSpace R] [ContinuousSMul R E] [ContinuousSMul R F] {f : E →ₗ.[R] F} {S : Submodule R E} (self : f.HasCore S)
Goal: (f.domRestrict S).closure = f
-- LinearPMap.hasCore_def
theorem hasCore_def {f : E →ₗ.[R] F} {S : Submodule R E} (h : f.HasCore S) :
    (f.domRestrict S).closure = f 
Arguments: {R : Type u_1} {E : Type u_2} {F : Type u_3} [CommRing R] [AddCommGroup E] [AddCommGroup F] [Module R E] [Module R F] [TopologicalSpace E] [TopologicalSpace F] [ContinuousAdd E] [ContinuousAdd F] [TopologicalSpace R] [ContinuousSMul R E] [ContinuousSMul R F] {f : E →ₗ.[R] F} {S : Submodule R E} (h : f.HasCore S)
Goal: (f.domRestrict S).closure = f
-- LinearPMap.closureHasCore
theorem closureHasCore (f : E →ₗ.[R] F) : f.closure.HasCore f.domain 
Arguments: {R : Type u_1} {E : Type u_2} {F : Type u_3} [CommRing R] [AddCommGroup E] [AddCommGroup F] [Module R E] [Module R F] [TopologicalSpace E] [TopologicalSpace F] [ContinuousAdd E] [ContinuousAdd F] [TopologicalSpace R] [ContinuousSMul R E] [ContinuousSMul R F] (f : E →ₗ.[R] F)
Goal: f.closure.HasCore f.domain
-- LinearPMap.closure_inverse_graph
theorem closure_inverse_graph (hf : LinearMap.ker f.toFun = ⊥) (hf' : f.IsClosable)
    (hcf : LinearMap.ker f.closure.toFun = ⊥) :
    f.closure.inverse.graph = f.inverse.graph.topologicalClosure 
Arguments: {R : Type u_1} {E : Type u_2} {F : Type u_3} [CommRing R] [AddCommGroup E] [AddCommGroup F] [Module R E] [Module R F] [TopologicalSpace E] [TopologicalSpace F] [ContinuousAdd E] [ContinuousAdd F] [TopologicalSpace R] [ContinuousSMul R E] [ContinuousSMul R F] {f : E →ₗ.[R] F} (hf : LinearMap.ker f.toFun = ⊥) (hf' : f.IsClosable) (hcf : LinearMap.ker f.closure.toFun = ⊥)
Goal: f.closure.inverse.graph = f.inverse.graph.topologicalClosure
-- LinearPMap.inverse_isClosable_iff
theorem inverse_isClosable_iff (hf : LinearMap.ker f.toFun = ⊥) (hf' : f.IsClosable) :
    f.inverse.IsClosable ↔ LinearMap.ker f.closure.toFun = ⊥ 
Arguments: {R : Type u_1} {E : Type u_2} {F : Type u_3} [CommRing R] [AddCommGroup E] [AddCommGroup F] [Module R E] [Module R F] [TopologicalSpace E] [TopologicalSpace F] [ContinuousAdd E] [ContinuousAdd F] [TopologicalSpace R] [ContinuousSMul R E] [ContinuousSMul R F] {f : E →ₗ.[R] F} (hf : LinearMap.ker f.toFun = ⊥) (hf' : f.IsClosable)
Goal: f.inverse.IsClosable ↔ LinearMap.ker f.closure.toFun = ⊥
-- LinearPMap.inverse_closure
theorem inverse_closure (hf : LinearMap.ker f.toFun = ⊥) (hf' : f.IsClosable)
    (hcf : LinearMap.ker f.closure.toFun = ⊥) :
    f.inverse.closure = f.closure.inverse 
Arguments: {R : Type u_1} {E : Type u_2} {F : Type u_3} [CommRing R] [AddCommGroup E] [AddCommGroup F] [Module R E] [Module R F] [TopologicalSpace E] [TopologicalSpace F] [ContinuousAdd E] [ContinuousAdd F] [TopologicalSpace R] [ContinuousSMul R E] [ContinuousSMul R F] {f : E →ₗ.[R] F} (hf : LinearMap.ker f.toFun = ⊥) (hf' : f.IsClosable) (hcf : LinearMap.ker f.closure.toFun = ⊥)
Goal: f.inverse.closure = f.closure.inverse
-- exists_idempotent_of_compact_t2_of_continuous_mul_left
@[to_additive
      "Any nonempty compact Hausdorff additive semigroup where right-addition is continuous
      contains an idempotent, i.e. an `m` such that `m + m = m`"]
theorem exists_idempotent_of_compact_t2_of_continuous_mul_left {M} [Nonempty M] [Semigroup M]
    [TopologicalSpace M] [CompactSpace M] [T2Space M]
    (continuous_mul_left : ∀ r : M, Continuous (· * r)) : ∃ m : M, m * m = m 
Arguments: {M : Type u_1} [Nonempty M] [Semigroup M] [TopologicalSpace M] [CompactSpace M] [T2Space M] (continuous_mul_left : ∀ (r : M), Continuous fun x => x * r)
Goal: ∃ m, m * m = m
-- exists_idempotent_of_compact_t2_of_continuous_add_left
@[to_additive
      "Any nonempty compact Hausdorff additive semigroup where right-addition is continuous
      contains an idempotent, i.e. an `m` such that `m + m = m`"]
theorem exists_idempotent_of_compact_t2_of_continuous_mul_left {M} [Nonempty M] [Semigroup M]
    [TopologicalSpace M] [CompactSpace M] [T2Space M]
    (continuous_mul_left : ∀ r : M, Continuous (· * r)) : ∃ m : M, m * m = m 
Arguments: {M : Type u_1} [Nonempty M] [AddSemigroup M] [TopologicalSpace M] [CompactSpace M] [T2Space M] (continuous_mul_left : ∀ (r : M), Continuous fun x => x + r)
Goal: ∃ m, m + m = m
-- exists_idempotent_in_compact_subsemigroup
@[to_additive exists_idempotent_in_compact_add_subsemigroup
      "A version of
      `exists_idempotent_of_compact_t2_of_continuous_add_left` where the idempotent lies in
      some specified nonempty compact additive subsemigroup."]
theorem exists_idempotent_in_compact_subsemigroup {M} [Semigroup M] [TopologicalSpace M] [T2Space M]
    (continuous_mul_left : ∀ r : M, Continuous (· * r)) (s : Set M) (snemp : s.Nonempty)
    (s_compact : IsCompact s) (s_add : ∀ᵉ (x ∈ s) (y ∈ s), x * y ∈ s) :
    ∃ m ∈ s, m * m = m 
Arguments: {M : Type u_1} [Semigroup M] [TopologicalSpace M] [T2Space M] (continuous_mul_left : ∀ (r : M), Continuous fun x => x * r) (s : Set M) (snemp : s.Nonempty) (s_compact : IsCompact s) (s_add : ∀ x ∈ s, ∀ y ∈ s, x * y ∈ s)
Goal: ∃ m ∈ s, m * m = m
-- exists_idempotent_in_compact_add_subsemigroup
@[to_additive exists_idempotent_in_compact_add_subsemigroup
      "A version of
      `exists_idempotent_of_compact_t2_of_continuous_add_left` where the idempotent lies in
      some specified nonempty compact additive subsemigroup."]
theorem exists_idempotent_in_compact_subsemigroup {M} [Semigroup M] [TopologicalSpace M] [T2Space M]
    (continuous_mul_left : ∀ r : M, Continuous (· * r)) (s : Set M) (snemp : s.Nonempty)
    (s_compact : IsCompact s) (s_add : ∀ᵉ (x ∈ s) (y ∈ s), x * y ∈ s) :
    ∃ m ∈ s, m * m = m 
Arguments: {M : Type u_1} [AddSemigroup M] [TopologicalSpace M] [T2Space M] (continuous_mul_left : ∀ (r : M), Continuous fun x => x + r) (s : Set M) (snemp : s.Nonempty) (s_compact : IsCompact s) (s_add : ∀ x ∈ s, ∀ y ∈ s, x + y ∈ s)
Goal: ∃ m ∈ s, m + m = m
-- Ideal.adic_basis
theorem adic_basis (I : Ideal R) : SubmodulesRingBasis fun n : ℕ => (I ^ n • ⊤ : Ideal R) 
Arguments: {R : Type u_1} [CommRing R] (I : Ideal R)
Goal: SubmodulesRingBasis fun n => I ^ n • ⊤
-- Ideal.nonarchimedean
theorem nonarchimedean (I : Ideal R) : @NonarchimedeanRing R _ I.adicTopology 
Arguments: {R : Type u_1} [CommRing R] (I : Ideal R)
Goal: NonarchimedeanRing R
-- Ideal.hasBasis_nhds_zero_adic
theorem hasBasis_nhds_zero_adic (I : Ideal R) :
    HasBasis (@nhds R I.adicTopology (0 : R)) (fun _n : ℕ => True) fun n =>
      ((I ^ n : Ideal R) : Set R) 
Arguments: {R : Type u_1} [CommRing R] (I : Ideal R)
Goal: (𝓝 0).HasBasis (fun _n => True) fun n => ↑(I ^ n)
-- Ideal.hasBasis_nhds_adic
theorem hasBasis_nhds_adic (I : Ideal R) (x : R) :
    HasBasis (@nhds R I.adicTopology x) (fun _n : ℕ => True) fun n =>
      (fun y => x + y) '' (I ^ n : Ideal R) 
Arguments: {R : Type u_1} [CommRing R] (I : Ideal R) (x : R)
Goal: (𝓝 x).HasBasis (fun _n => True) fun n => (fun y => x + y) '' ↑(I ^ n)
-- Ideal.adic_module_basis
theorem adic_module_basis :
    I.ringFilterBasis.SubmodulesBasis fun n : ℕ => I ^ n • (⊤ : Submodule R M) 
Arguments: {R : Type u_1} [CommRing R] (I : Ideal R) (M : Type u_2) [AddCommGroup M] [Module R M]
Goal: I.ringFilterBasis.SubmodulesBasis fun n => I ^ n • ⊤
-- isAdic_iff
theorem isAdic_iff [top : TopologicalSpace R] [TopologicalRing R] {J : Ideal R} :
    IsAdic J ↔
      (∀ n : ℕ, IsOpen ((J ^ n : Ideal R) : Set R)) ∧
        ∀ s ∈ 𝓝 (0 : R), ∃ n : ℕ, ((J ^ n : Ideal R) : Set R) ⊆ s 
Arguments: {R : Type u_1} [CommRing R] [top : TopologicalSpace R] [TopologicalRing R] {J : Ideal R}
Goal: IsAdic J ↔ (∀ (n : ℕ), IsOpen ↑(J ^ n)) ∧ ∀ s ∈ 𝓝 0, ∃ n, ↑(J ^ n) ⊆ s
-- is_ideal_adic_pow
theorem is_ideal_adic_pow {J : Ideal R} (h : IsAdic J) {n : ℕ} (hn : 0 < n) : IsAdic (J ^ n) 
Arguments: {R : Type u_1} [CommRing R] [TopologicalSpace R] [TopologicalRing R] {J : Ideal R} (h : IsAdic J) {n : ℕ} (hn : 0 < n)
Goal: IsAdic (J ^ n)
-- is_bot_adic_iff
theorem is_bot_adic_iff {A : Type*} [CommRing A] [TopologicalSpace A] [TopologicalRing A] :
    IsAdic (⊥ : Ideal A) ↔ DiscreteTopology A 
Arguments: {A : Type u_2} [CommRing A] [TopologicalSpace A] [TopologicalRing A]
Goal: IsAdic ⊥ ↔ DiscreteTopology A
-- ContinuousMultilinearMap.range_toUniformOnFun
lemma range_toUniformOnFun [DecidableEq ι] [TopologicalSpace F] :
    range toUniformOnFun =
      {f : (Π i, E i) →ᵤ[{s | IsVonNBounded 𝕜 s}] F |
        Continuous (toFun _ f) ∧
        (∀ (m : Π i, E i) i x y,
          toFun _ f (update m i (x + y)) = toFun _ f (update m i x) + toFun _ f (update m i y)) ∧
        (∀ (m : Π i, E i) i (c : 𝕜) x,
          toFun _ f (update m i (c • x)) = c • toFun _ f (update m i x))} 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} {E : ι → Type u_3} {F : Type u_4} [NormedField 𝕜] [(i : ι) → TopologicalSpace (E i)] [(i : ι) → AddCommGroup (E i)] [(i : ι) → Module 𝕜 (E i)] [AddCommGroup F] [Module 𝕜 F] [DecidableEq ι] [TopologicalSpace F]
Goal: Set.range ContinuousMultilinearMap.toUniformOnFun =
  {f |
    Continuous ((UniformOnFun.toFun {s | Bornology.IsVonNBounded 𝕜 s}) f) ∧
      (∀ (m : (i : ι) → E i) (i : ι) (x y : E i),
          (UniformOnFun.toFun {s | Bornology.IsVonNBounded 𝕜 s}) f (Function.update m i (x + y)) =
            (UniformOnFun.toFun {s | Bornology.IsVonNBounded 𝕜 s}) f (Function.update m i x) +
              (UniformOnFun.toFun {s | Bornology.IsVonNBounded 𝕜 s}) f (Function.update m i y)) ∧
        ∀ (m : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i),
          (UniformOnFun.toFun {s | Bornology.IsVonNBounded 𝕜 s}) f (Function.update m i (c • x)) =
            c • (UniformOnFun.toFun {s | Bornology.IsVonNBounded 𝕜 s}) f (Function.update m i x)}
-- ContinuousMultilinearMap.toUniformOnFun_toFun
@[simp]
lemma toUniformOnFun_toFun [TopologicalSpace F] (f : ContinuousMultilinearMap 𝕜 E F) :
    UniformOnFun.toFun _ f.toUniformOnFun = f 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} {E : ι → Type u_3} {F : Type u_4} [NormedField 𝕜] [(i : ι) → TopologicalSpace (E i)] [(i : ι) → AddCommGroup (E i)] [(i : ι) → Module 𝕜 (E i)] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] (f : ContinuousMultilinearMap 𝕜 E F)
Goal: (UniformOnFun.toFun {s | Bornology.IsVonNBounded 𝕜 s}) f.toUniformOnFun = ⇑f
-- ContinuousMultilinearMap.uniformEmbedding_toUniformOnFun
lemma uniformEmbedding_toUniformOnFun :
    UniformEmbedding (toUniformOnFun : ContinuousMultilinearMap 𝕜 E F → _) 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} {E : ι → Type u_3} {F : Type u_4} [NormedField 𝕜] [(i : ι) → TopologicalSpace (E i)] [(i : ι) → AddCommGroup (E i)] [(i : ι) → Module 𝕜 (E i)] [AddCommGroup F] [Module 𝕜 F] [UniformSpace F] [UniformAddGroup F]
Goal: UniformEmbedding ContinuousMultilinearMap.toUniformOnFun
-- ContinuousMultilinearMap.embedding_toUniformOnFun
lemma embedding_toUniformOnFun : Embedding (toUniformOnFun : ContinuousMultilinearMap 𝕜 E F → _) 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} {E : ι → Type u_3} {F : Type u_4} [NormedField 𝕜] [(i : ι) → TopologicalSpace (E i)] [(i : ι) → AddCommGroup (E i)] [(i : ι) → Module 𝕜 (E i)] [AddCommGroup F] [Module 𝕜 F] [UniformSpace F] [UniformAddGroup F]
Goal: Embedding ContinuousMultilinearMap.toUniformOnFun
-- ContinuousMultilinearMap.uniformContinuous_coe_fun
theorem uniformContinuous_coe_fun [∀ i, ContinuousSMul 𝕜 (E i)] :
    UniformContinuous (DFunLike.coe : ContinuousMultilinearMap 𝕜 E F → (Π i, E i) → F) 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} {E : ι → Type u_3} {F : Type u_4} [NormedField 𝕜] [(i : ι) → TopologicalSpace (E i)] [(i : ι) → AddCommGroup (E i)] [(i : ι) → Module 𝕜 (E i)] [AddCommGroup F] [Module 𝕜 F] [UniformSpace F] [UniformAddGroup F] [∀ (i : ι), ContinuousSMul 𝕜 (E i)]
Goal: UniformContinuous DFunLike.coe
-- ContinuousMultilinearMap.uniformContinuous_eval_const
theorem uniformContinuous_eval_const [∀ i, ContinuousSMul 𝕜 (E i)] (x : Π i, E i) :
    UniformContinuous fun f : ContinuousMultilinearMap 𝕜 E F ↦ f x 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} {E : ι → Type u_3} {F : Type u_4} [NormedField 𝕜] [(i : ι) → TopologicalSpace (E i)] [(i : ι) → AddCommGroup (E i)] [(i : ι) → Module 𝕜 (E i)] [AddCommGroup F] [Module 𝕜 F] [UniformSpace F] [UniformAddGroup F] [∀ (i : ι), ContinuousSMul 𝕜 (E i)] (x : (i : ι) → E i)
Goal: UniformContinuous fun f => f x
-- ContinuousMultilinearMap.completeSpace
theorem completeSpace (h : RestrictGenTopology {s : Set (Π i, E i) | IsVonNBounded 𝕜 s}) :
    CompleteSpace (ContinuousMultilinearMap 𝕜 E F) 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} {E : ι → Type u_3} {F : Type u_4} [NormedField 𝕜] [(i : ι) → TopologicalSpace (E i)] [(i : ι) → AddCommGroup (E i)] [(i : ι) → Module 𝕜 (E i)] [AddCommGroup F] [Module 𝕜 F] [UniformSpace F] [UniformAddGroup F] [∀ (i : ι), ContinuousSMul 𝕜 (E i)] [ContinuousConstSMul 𝕜 F] [CompleteSpace F] [T2Space F] (h : RestrictGenTopology {s | Bornology.IsVonNBounded 𝕜 s})
Goal: CompleteSpace (ContinuousMultilinearMap 𝕜 E F)
-- ContinuousMultilinearMap.hasBasis_nhds_zero_of_basis
theorem hasBasis_nhds_zero_of_basis {ι : Type*} {p : ι → Prop} {b : ι → Set F}
    (h : (𝓝 (0 : F)).HasBasis p b) :
    (𝓝 (0 : ContinuousMultilinearMap 𝕜 E F)).HasBasis
      (fun Si : Set (Π i, E i) × ι => IsVonNBounded 𝕜 Si.1 ∧ p Si.2)
      fun Si => { f | MapsTo f Si.1 (b Si.2) } 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} {E : ι → Type u_3} {F : Type u_4} [NormedField 𝕜] [(i : ι) → TopologicalSpace (E i)] [(i : ι) → AddCommGroup (E i)] [(i : ι) → Module 𝕜 (E i)] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F]
Goal: ∀ {ι_1 : Type u_5} {p : ι_1 → Prop} {b : ι_1 → Set F},
  (𝓝 0).HasBasis p b →
    (𝓝 0).HasBasis (fun Si => Bornology.IsVonNBounded 𝕜 Si.1 ∧ p Si.2) fun Si => {f | Set.MapsTo (⇑f) Si.1 (b Si.2)}
-- ContinuousMultilinearMap.hasBasis_nhds_zero
theorem hasBasis_nhds_zero :
    (𝓝 (0 : ContinuousMultilinearMap 𝕜 E F)).HasBasis
      (fun SV : Set (Π i, E i) × Set F => IsVonNBounded 𝕜 SV.1 ∧ SV.2 ∈ 𝓝 0) fun SV =>
      { f | MapsTo f SV.1 SV.2 } 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} {E : ι → Type u_3} {F : Type u_4} [NormedField 𝕜] [(i : ι) → TopologicalSpace (E i)] [(i : ι) → AddCommGroup (E i)] [(i : ι) → Module 𝕜 (E i)] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F]
Goal: (𝓝 0).HasBasis (fun SV => Bornology.IsVonNBounded 𝕜 SV.1 ∧ SV.2 ∈ 𝓝 0) fun SV => {f | Set.MapsTo (⇑f) SV.1 SV.2}
-- ContinuousMultilinearMap.continuous_eval_const
theorem continuous_eval_const (x : Π i, E i) :
    Continuous fun p : ContinuousMultilinearMap 𝕜 E F ↦ p x 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} {E : ι → Type u_3} {F : Type u_4} [NormedField 𝕜] [(i : ι) → TopologicalSpace (E i)] [(i : ι) → AddCommGroup (E i)] [(i : ι) → Module 𝕜 (E i)] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [∀ (i : ι), ContinuousSMul 𝕜 (E i)] (x : (i : ι) → E i)
Goal: Continuous fun p => p x
-- ContinuousMultilinearMap.continuous_coe_fun
theorem continuous_coe_fun :
    Continuous (DFunLike.coe : ContinuousMultilinearMap 𝕜 E F → (Π i, E i) → F) 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} {E : ι → Type u_3} {F : Type u_4} [NormedField 𝕜] [(i : ι) → TopologicalSpace (E i)] [(i : ι) → AddCommGroup (E i)] [(i : ι) → Module 𝕜 (E i)] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [∀ (i : ι), ContinuousSMul 𝕜 (E i)]
Goal: Continuous DFunLike.coe
-- ContinuousMultilinearMap.apply_apply
@[simp]
lemma apply_apply [ContinuousConstSMul 𝕜 F] {m : Π i, E i} {c : ContinuousMultilinearMap 𝕜 E F} :
    apply 𝕜 E F m c = c m 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} {E : ι → Type u_3} {F : Type u_4} [NormedField 𝕜] [(i : ι) → TopologicalSpace (E i)] [(i : ι) → AddCommGroup (E i)] [(i : ι) → Module 𝕜 (E i)] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [∀ (i : ι), ContinuousSMul 𝕜 (E i)] [ContinuousConstSMul 𝕜 F] {m : (i : ι) → E i} {c : ContinuousMultilinearMap 𝕜 E F}
Goal: (ContinuousMultilinearMap.apply 𝕜 E F m) c = c m
-- ContinuousMultilinearMap.hasSum_eval
theorem hasSum_eval {α : Type*} {p : α → ContinuousMultilinearMap 𝕜 E F}
    {q : ContinuousMultilinearMap 𝕜 E F} (h : HasSum p q) (m : Π i, E i) :
    HasSum (fun a => p a m) (q m) 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} {E : ι → Type u_3} {F : Type u_4} [NormedField 𝕜] [(i : ι) → TopologicalSpace (E i)] [(i : ι) → AddCommGroup (E i)] [(i : ι) → Module 𝕜 (E i)] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [∀ (i : ι), ContinuousSMul 𝕜 (E i)] {α : Type u_5} {p : α → ContinuousMultilinearMap 𝕜 E F} {q : ContinuousMultilinearMap 𝕜 E F} (h : HasSum p q) (m : (i : ι) → E i)
Goal: HasSum (fun a => (p a) m) (q m)
-- ContinuousMultilinearMap.tsum_eval
theorem tsum_eval [T2Space F] {α : Type*} {p : α → ContinuousMultilinearMap 𝕜 E F} (hp : Summable p)
    (m : Π i, E i) : (∑' a, p a) m = ∑' a, p a m 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} {E : ι → Type u_3} {F : Type u_4} [NormedField 𝕜] [(i : ι) → TopologicalSpace (E i)] [(i : ι) → AddCommGroup (E i)] [(i : ι) → Module 𝕜 (E i)] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [∀ (i : ι), ContinuousSMul 𝕜 (E i)] [T2Space F] {α : Type u_5} {p : α → ContinuousMultilinearMap 𝕜 E F} (hp : Summable p) (m : (i : ι) → E i)
Goal: (∑' (a : α), p a) m = ∑' (a : α), (p a) m
-- CompletableTopField.nice
class CompletableTopField extends T0Space K : Prop where
  nice : ∀ F : Filter K, Cauchy F → 𝓝 0 ⊓ F = ⊥ → Cauchy (map (fun x => x⁻¹) F)
Arguments: {K : Type u_1} [Field K] [UniformSpace K] [self : CompletableTopField K] (F : Filter K)
Goal: Cauchy F → 𝓝 0 ⊓ F = ⊥ → Cauchy (Filter.map (fun x => x⁻¹) F)
-- UniformSpace.Completion.continuous_hatInv
theorem continuous_hatInv [CompletableTopField K] {x : hat K} (h : x ≠ 0) :
    ContinuousAt hatInv x 
Arguments: {K : Type u_1} [Field K] [UniformSpace K] [CompletableTopField K] {x : hat K} (h : x ≠ 0)
Goal: ContinuousAt UniformSpace.Completion.hatInv x
-- UniformSpace.Completion.hatInv_extends
theorem hatInv_extends {x : K} (h : x ≠ 0) : hatInv (x : hat K) = ↑(x⁻¹ : K) 
Arguments: {K : Type u_1} [Field K] [UniformSpace K] [TopologicalDivisionRing K] {x : K} (h : x ≠ 0)
Goal: (↑K x).hatInv = ↑K x⁻¹