ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- continuousOn_iff_lower_upperSemicontinuousOn
theorem continuousOn_iff_lower_upperSemicontinuousOn {f : α → γ} :
    ContinuousOn f s ↔ LowerSemicontinuousOn f s ∧ UpperSemicontinuousOn f s 
Arguments: {α : Type u_1} [TopologicalSpace α] {s : Set α} {γ : Type u_3} [LinearOrder γ] [TopologicalSpace γ] [OrderTopology γ] {f : α → γ}
Goal: ContinuousOn f s ↔ LowerSemicontinuousOn f s ∧ UpperSemicontinuousOn f s
-- continuous_iff_lower_upperSemicontinuous
theorem continuous_iff_lower_upperSemicontinuous {f : α → γ} :
    Continuous f ↔ LowerSemicontinuous f ∧ UpperSemicontinuous f 
Arguments: {α : Type u_1} [TopologicalSpace α] {γ : Type u_3} [LinearOrder γ] [TopologicalSpace γ] [OrderTopology γ] {f : α → γ}
Goal: Continuous f ↔ LowerSemicontinuous f ∧ UpperSemicontinuous f
-- IsClosed.mk_lt_continuum
theorem IsClosed.mk_lt_continuum [NormalSpace X] {s : Set X} (hs : IsClosed s)
    [DiscreteTopology s] : #s < 𝔠 
Arguments: {X : Type u} [TopologicalSpace X] [TopologicalSpace.SeparableSpace X] [NormalSpace X] {s : Set X} (hs : IsClosed s) [DiscreteTopology ↑s]
Goal: #↑s < 𝔠
-- IsClosed.not_normal_of_continuum_le_mk
theorem IsClosed.not_normal_of_continuum_le_mk {s : Set X} (hs : IsClosed s) [DiscreteTopology s]
    (hmk : 𝔠 ≤ #s) : ¬NormalSpace X 
Arguments: {X : Type u} [TopologicalSpace X] [TopologicalSpace.SeparableSpace X] {s : Set X} (hs : IsClosed s) [DiscreteTopology ↑s] (hmk : 𝔠 ≤ #↑s)
Goal: ¬NormalSpace X
-- polynomialFunctions_closure_eq_top'
theorem polynomialFunctions_closure_eq_top' : (polynomialFunctions I).topologicalClosure = ⊤ 
Goal: (polynomialFunctions I).topologicalClosure = ⊤
-- polynomialFunctions_closure_eq_top
theorem polynomialFunctions_closure_eq_top (a b : ℝ) :
    (polynomialFunctions (Set.Icc a b)).topologicalClosure = ⊤ 
Arguments: (a b : ℝ)
Goal: (polynomialFunctions (Set.Icc a b)).topologicalClosure = ⊤
-- continuousMap_mem_polynomialFunctions_closure
theorem continuousMap_mem_polynomialFunctions_closure (a b : ℝ) (f : C(Set.Icc a b, ℝ)) :
    f ∈ (polynomialFunctions (Set.Icc a b)).topologicalClosure 
Arguments: (a b : ℝ) (f : C(↑(Set.Icc a b), ℝ))
Goal: f ∈ (polynomialFunctions (Set.Icc a b)).topologicalClosure
-- exists_polynomial_near_continuousMap
theorem exists_polynomial_near_continuousMap (a b : ℝ) (f : C(Set.Icc a b, ℝ)) (ε : ℝ)
    (pos : 0 < ε) : ∃ p : ℝ[X], ‖p.toContinuousMapOn _ - f‖ < ε 
Arguments: (a b : ℝ) (f : C(↑(Set.Icc a b), ℝ)) (ε : ℝ) (pos : 0 < ε)
Goal: ∃ p, ‖p.toContinuousMapOn (Set.Icc a b) - f‖ < ε
-- exists_polynomial_near_of_continuousOn
theorem exists_polynomial_near_of_continuousOn (a b : ℝ) (f : ℝ → ℝ)
    (c : ContinuousOn f (Set.Icc a b)) (ε : ℝ) (pos : 0 < ε) :
    ∃ p : ℝ[X], ∀ x ∈ Set.Icc a b, |p.eval x - f x| < ε 
Arguments: (a b : ℝ) (f : ℝ → ℝ) (c : ContinuousOn f (Set.Icc a b)) (ε : ℝ) (pos : 0 < ε)
Goal: ∃ p, ∀ x ∈ Set.Icc a b, |Polynomial.eval x p - f x| < ε
-- ContinuousMap.starOrderedRing_of_sqrt
lemma starOrderedRing_of_sqrt {R : Type*} [PartialOrder R] [NonUnitalRing R] [StarRing R]
    [StarOrderedRing R] [TopologicalSpace R] [ContinuousStar R] [TopologicalRing R]
    (sqrt : R → R) (h_continuousOn : ContinuousOn sqrt {x : R | 0 ≤ x})
    (h_sqrt : ∀ x, 0 ≤ x → star (sqrt x) * sqrt x = x) : StarOrderedRing C(α, R) 
Arguments: {α : Type u_1} [TopologicalSpace α] {R : Type u_2} [PartialOrder R] [NonUnitalRing R] [StarRing R] [StarOrderedRing R] [TopologicalSpace R] [ContinuousStar R] [TopologicalRing R] (sqrt : R → R) (h_continuousOn : ContinuousOn sqrt {x | 0 ≤ x}) (h_sqrt : ∀ (x : R), 0 ≤ x → star (sqrt x) * sqrt x = x)
Goal: StarOrderedRing C(α, R)
-- TopologicalSpace.PseudoMetrizableSpace.exists_pseudo_metric
class PseudoMetrizableSpace (X : Type*) [t : TopologicalSpace X] : Prop where
  exists_pseudo_metric : ∃ m : PseudoMetricSpace X, m.toUniformSpace.toTopologicalSpace = t
Arguments: {X : Type u_5} [t : TopologicalSpace X] [self : TopologicalSpace.PseudoMetrizableSpace X]
Goal: ∃ m, UniformSpace.toTopologicalSpace = t
-- Inducing.pseudoMetrizableSpace
theorem _root_.Inducing.pseudoMetrizableSpace [PseudoMetrizableSpace Y] {f : X → Y}
    (hf : Inducing f) : PseudoMetrizableSpace X 
Arguments: {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace.PseudoMetrizableSpace Y] {f : X → Y} (hf : Inducing f)
Goal: TopologicalSpace.PseudoMetrizableSpace X
-- TopologicalSpace.MetrizableSpace.exists_metric
class MetrizableSpace (X : Type*) [t : TopologicalSpace X] : Prop where
  exists_metric : ∃ m : MetricSpace X, m.toUniformSpace.toTopologicalSpace = t
Arguments: {X : Type u_5} [t : TopologicalSpace X] [self : TopologicalSpace.MetrizableSpace X]
Goal: ∃ m, UniformSpace.toTopologicalSpace = t
-- Embedding.metrizableSpace
theorem _root_.Embedding.metrizableSpace [MetrizableSpace Y] {f : X → Y} (hf : Embedding f) :
    MetrizableSpace X 
Arguments: {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace.MetrizableSpace Y] {f : X → Y} (hf : Embedding f)
Goal: TopologicalSpace.MetrizableSpace X
-- TopologicalSpace.IsSeparable.secondCountableTopology
theorem IsSeparable.secondCountableTopology [PseudoMetrizableSpace X] {s : Set X}
    (hs : IsSeparable s) : SecondCountableTopology s 
Arguments: {X : Type u_2} [TopologicalSpace X] [TopologicalSpace.PseudoMetrizableSpace X] {s : Set X} (hs : TopologicalSpace.IsSeparable s)
Goal: SecondCountableTopology ↑s
-- ContinuousMap.equivBoundedOfCompact_apply
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) 
Arguments: (α : Type u_1) (β : Type u_2) [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
Goal: ⇑(ContinuousMap.equivBoundedOfCompact α β) = BoundedContinuousFunction.mkOfCompact
-- ContinuousMap.equivBoundedOfCompact_symm_apply
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) 
Arguments: (α : Type u_1) (β : Type u_2) [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
Goal: ⇑(ContinuousMap.equivBoundedOfCompact α β).symm = BoundedContinuousFunction.toContinuousMap
-- ContinuousMap.uniformInducing_equivBoundedOfCompact
theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) 
Arguments: (α : Type u_1) (β : Type u_2) [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
Goal: UniformInducing ⇑(ContinuousMap.equivBoundedOfCompact α β)
-- ContinuousMap.uniformEmbedding_equivBoundedOfCompact
theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) 
Arguments: (α : Type u_1) (β : Type u_2) [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
Goal: UniformEmbedding ⇑(ContinuousMap.equivBoundedOfCompact α β)
-- ContinuousMap.addEquivBoundedOfCompact_symm_apply
@[simp]
theorem addEquivBoundedOfCompact_symm_apply [AddMonoid β] [LipschitzAdd β] :
    ⇑((addEquivBoundedOfCompact α β).symm) = toContinuousMapAddHom α β 
Arguments: (α : Type u_1) (β : Type u_2) [TopologicalSpace α] [CompactSpace α] [MetricSpace β] [AddMonoid β] [LipschitzAdd β]
Goal: ⇑(ContinuousMap.addEquivBoundedOfCompact α β).symm = ⇑(BoundedContinuousFunction.toContinuousMapAddHom α β)
-- ContinuousMap.addEquivBoundedOfCompact_apply
@[simp]
theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
    ⇑(addEquivBoundedOfCompact α β) = mkOfCompact 
Arguments: (α : Type u_1) (β : Type u_2) [TopologicalSpace α] [CompactSpace α] [MetricSpace β] [AddMonoid β] [LipschitzAdd β]
Goal: ⇑(ContinuousMap.addEquivBoundedOfCompact α β) = BoundedContinuousFunction.mkOfCompact
-- ContinuousMap.isometryEquivBoundedOfCompact_symm_apply
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) 
Arguments: (α : Type u_1) (β : Type u_2) [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
Goal: ⇑(ContinuousMap.isometryEquivBoundedOfCompact α β).symm = BoundedContinuousFunction.toContinuousMap
-- ContinuousMap.isometryEquivBoundedOfCompact_toEquiv
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) 
Arguments: (α : Type u_1) (β : Type u_2) [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
Goal: (ContinuousMap.isometryEquivBoundedOfCompact α β).toEquiv = ContinuousMap.equivBoundedOfCompact α β
-- ContinuousMap.isometryEquivBoundedOfCompact_apply
@[simps! (config := .asFn) toEquiv apply symm_apply]
def isometryEquivBoundedOfCompact : C(α, β) ≃ᵢ (α →ᵇ β) 
Arguments: (α : Type u_1) (β : Type u_2) [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
Goal: ⇑(ContinuousMap.isometryEquivBoundedOfCompact α β) = BoundedContinuousFunction.mkOfCompact
-- BoundedContinuousFunction.dist_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.dist_mkOfCompact (f g : C(α, β)) :
    dist (mkOfCompact f) (mkOfCompact g) = dist f g 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [CompactSpace α] [MetricSpace β] (f g : C(α, β))
Goal: dist (BoundedContinuousFunction.mkOfCompact f) (BoundedContinuousFunction.mkOfCompact g) = dist f g
-- BoundedContinuousFunction.dist_toContinuousMap
@[simp]
theorem _root_.BoundedContinuousFunction.dist_toContinuousMap (f g : α →ᵇ β) :
    dist f.toContinuousMap g.toContinuousMap = dist f g 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [CompactSpace α] [MetricSpace β] (f g : α →ᵇ β)
Goal: dist f.toContinuousMap g.toContinuousMap = dist f g
-- ContinuousMap.dist_apply_le_dist
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [CompactSpace α] [MetricSpace β] {f g : C(α, β)} (x : α)
Goal: dist (f x) (g x) ≤ dist f g
-- ContinuousMap.dist_le
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [CompactSpace α] [MetricSpace β] {f g : C(α, β)} {C : ℝ} (C0 : 0 ≤ C)
Goal: dist f g ≤ C ↔ ∀ (x : α), dist (f x) (g x) ≤ C
-- ContinuousMap.dist_le_iff_of_nonempty
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [CompactSpace α] [MetricSpace β] {f g : C(α, β)} {C : ℝ} [Nonempty α]
Goal: dist f g ≤ C ↔ ∀ (x : α), dist (f x) (g x) ≤ C
-- ContinuousMap.dist_lt_iff_of_nonempty
theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [CompactSpace α] [MetricSpace β] {f g : C(α, β)} {C : ℝ} [Nonempty α]
Goal: dist f g < C ↔ ∀ (x : α), dist (f x) (g x) < C
-- ContinuousMap.dist_lt_of_nonempty
theorem dist_lt_of_nonempty [Nonempty α] (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [CompactSpace α] [MetricSpace β] {f g : C(α, β)} {C : ℝ} [Nonempty α] (w : ∀ (x : α), dist (f x) (g x) < C)
Goal: dist f g < C
-- ContinuousMap.dist_lt_iff
theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [CompactSpace α] [MetricSpace β] {f g : C(α, β)} {C : ℝ} (C0 : 0 < C)
Goal: dist f g < C ↔ ∀ (x : α), dist (f x) (g x) < C
-- BoundedContinuousFunction.norm_mkOfCompact
@[simp]
theorem _root_.BoundedContinuousFunction.norm_mkOfCompact (f : C(α, E)) : ‖mkOfCompact f‖ = ‖f‖ 
Arguments: {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [NormedAddCommGroup E] (f : C(α, E))
Goal: ‖BoundedContinuousFunction.mkOfCompact f‖ = ‖f‖
-- BoundedContinuousFunction.norm_toContinuousMap_eq
@[simp]
theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ E) :
    ‖f.toContinuousMap‖ = ‖f‖ 
Arguments: {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [NormedAddCommGroup E] (f : α →ᵇ E)
Goal: ‖f.toContinuousMap‖ = ‖f‖
-- ContinuousMap.norm_coe_le_norm
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ 
Arguments: {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [NormedAddCommGroup E] (f : C(α, E)) (x : α)
Goal: ‖f x‖ ≤ ‖f‖
-- ContinuousMap.dist_le_two_norm
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ 
Arguments: {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [NormedAddCommGroup E] (f : C(α, E)) (x y : α)
Goal: dist (f x) (f y) ≤ 2 * ‖f‖
-- ContinuousMap.norm_le
theorem norm_le {C : ℝ} (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C 
Arguments: {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [NormedAddCommGroup E] (f : C(α, E)) {C : ℝ} (C0 : 0 ≤ C)
Goal: ‖f‖ ≤ C ↔ ∀ (x : α), ‖f x‖ ≤ C
-- ContinuousMap.norm_le_of_nonempty
theorem norm_le_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M 
Arguments: {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [NormedAddCommGroup E] (f : C(α, E)) [Nonempty α] {M : ℝ}
Goal: ‖f‖ ≤ M ↔ ∀ (x : α), ‖f x‖ ≤ M
-- ContinuousMap.norm_lt_iff
theorem norm_lt_iff {M : ℝ} (M0 : 0 < M) : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M 
Arguments: {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [NormedAddCommGroup E] (f : C(α, E)) {M : ℝ} (M0 : 0 < M)
Goal: ‖f‖ < M ↔ ∀ (x : α), ‖f x‖ < M
-- ContinuousMap.nnnorm_lt_iff
theorem nnnorm_lt_iff {M : ℝ≥0} (M0 : 0 < M) : ‖f‖₊ < M ↔ ∀ x : α, ‖f x‖₊ < M 
Arguments: {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [NormedAddCommGroup E] (f : C(α, E)) {M : ℝ≥0} (M0 : 0 < M)
Goal: ‖f‖₊ < M ↔ ∀ (x : α), ‖f x‖₊ < M
-- ContinuousMap.norm_lt_iff_of_nonempty
theorem norm_lt_iff_of_nonempty [Nonempty α] {M : ℝ} : ‖f‖ < M ↔ ∀ x, ‖f x‖ < M 
Arguments: {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [NormedAddCommGroup E] (f : C(α, E)) [Nonempty α] {M : ℝ}
Goal: ‖f‖ < M ↔ ∀ (x : α), ‖f x‖ < M
-- ContinuousMap.nnnorm_lt_iff_of_nonempty
theorem nnnorm_lt_iff_of_nonempty [Nonempty α] {M : ℝ≥0} : ‖f‖₊ < M ↔ ∀ x, ‖f x‖₊ < M 
Arguments: {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [NormedAddCommGroup E] (f : C(α, E)) [Nonempty α] {M : ℝ≥0}
Goal: ‖f‖₊ < M ↔ ∀ (x : α), ‖f x‖₊ < M
-- ContinuousMap.apply_le_norm
theorem apply_le_norm (f : C(α, ℝ)) (x : α) : f x ≤ ‖f‖ 
Arguments: {α : Type u_1} [TopologicalSpace α] [CompactSpace α] (f : C(α, ℝ)) (x : α)
Goal: f x ≤ ‖f‖
-- ContinuousMap.neg_norm_le_apply
theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x 
Arguments: {α : Type u_1} [TopologicalSpace α] [CompactSpace α] (f : C(α, ℝ)) (x : α)
Goal: -‖f‖ ≤ f x
-- ContinuousMap.norm_eq_iSup_norm
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ 
Arguments: {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [NormedAddCommGroup E] (f : C(α, E))
Goal: ‖f‖ = ⨆ x, ‖f x‖
-- ContinuousMap.norm_restrict_mono_set
theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
    {K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {X : Type u_4} [TopologicalSpace X] (f : C(X, E)) {K L : TopologicalSpace.Compacts X} (hKL : K ≤ L)
Goal: ‖ContinuousMap.restrict (↑K) f‖ ≤ ‖ContinuousMap.restrict (↑L) f‖
-- ContinuousMap.linearIsometryBoundedOfCompact_symm_apply
@[simp]
theorem linearIsometryBoundedOfCompact_symm_apply (f : α →ᵇ E) :
    (linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap 
Arguments: {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [NormedAddCommGroup E] {𝕜 : Type u_4} [NormedField 𝕜] [NormedSpace 𝕜 E] (f : α →ᵇ E)
Goal: (ContinuousMap.linearIsometryBoundedOfCompact α E 𝕜).symm f = f.toContinuousMap
-- ContinuousMap.linearIsometryBoundedOfCompact_apply_apply
@[simp]
theorem linearIsometryBoundedOfCompact_apply_apply (f : C(α, E)) (a : α) :
    (linearIsometryBoundedOfCompact α E 𝕜 f) a = f a 
Arguments: {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [NormedAddCommGroup E] {𝕜 : Type u_4} [NormedField 𝕜] [NormedSpace 𝕜 E] (f : C(α, E)) (a : α)
Goal: ((ContinuousMap.linearIsometryBoundedOfCompact α E 𝕜) f) a = f a
-- ContinuousMap.linearIsometryBoundedOfCompact_toIsometryEquiv
@[simp]
theorem linearIsometryBoundedOfCompact_toIsometryEquiv :
    (linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = isometryEquivBoundedOfCompact α E 
Arguments: {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [NormedAddCommGroup E] {𝕜 : Type u_4} [NormedField 𝕜] [NormedSpace 𝕜 E]
Goal: (ContinuousMap.linearIsometryBoundedOfCompact α E 𝕜).toIsometryEquiv = ContinuousMap.isometryEquivBoundedOfCompact α E
-- ContinuousMap.linearIsometryBoundedOfCompact_toAddEquiv
@[simp] -- Porting note: adjusted LHS because `simpNF` complained it simplified.
theorem linearIsometryBoundedOfCompact_toAddEquiv :
    ((linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv : C(α, E) ≃+ (α →ᵇ E)) =
      addEquivBoundedOfCompact α E 
Arguments: {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [NormedAddCommGroup E] {𝕜 : Type u_4} [NormedField 𝕜] [NormedSpace 𝕜 E]
Goal: ↑(ContinuousMap.linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv = ContinuousMap.addEquivBoundedOfCompact α E
-- ContinuousMap.linearIsometryBoundedOfCompact_of_compact_toEquiv
@[simp]
theorem linearIsometryBoundedOfCompact_of_compact_toEquiv :
    (linearIsometryBoundedOfCompact α E 𝕜).toLinearEquiv.toEquiv = equivBoundedOfCompact α E 
Arguments: {α : Type u_1} {E : Type u_3} [TopologicalSpace α] [CompactSpace α] [NormedAddCommGroup E] {𝕜 : Type u_4} [NormedField 𝕜] [NormedSpace 𝕜 E]
Goal: (ContinuousMap.linearIsometryBoundedOfCompact α E 𝕜).toEquiv = ContinuousMap.equivBoundedOfCompact α E
-- ContinuousMap.uniform_continuity
theorem uniform_continuity (f : C(α, β)) (ε : ℝ) (h : 0 < ε) :
    ∃ δ > 0, ∀ {x y}, dist x y < δ → dist (f x) (f y) < ε 
Arguments: {α : Type u_1} {β : Type u_2} [MetricSpace α] [CompactSpace α] [MetricSpace β] (f : C(α, β)) (ε : ℝ) (h : 0 < ε)
Goal: ∃ δ > 0, ∀ {x y : α}, dist x y < δ → dist (f x) (f y) < ε
-- ContinuousMap.modulus_pos
theorem modulus_pos (f : C(α, β)) {ε : ℝ} {h : 0 < ε} : 0 < f.modulus ε h 
Arguments: {α : Type u_1} {β : Type u_2} [MetricSpace α] [CompactSpace α] [MetricSpace β] (f : C(α, β)) {ε : ℝ} {h : 0 < ε}
Goal: 0 < f.modulus ε h
-- ContinuousMap.dist_lt_of_dist_lt_modulus
theorem dist_lt_of_dist_lt_modulus (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α}
    (w : dist a b < f.modulus ε h) : dist (f a) (f b) < ε 
Arguments: {α : Type u_1} {β : Type u_2} [MetricSpace α] [CompactSpace α] [MetricSpace β] (f : C(α, β)) (ε : ℝ) (h : 0 < ε) {a b : α} (w : dist a b < f.modulus ε h)
Goal: dist (f a) (f b) < ε
-- ContinuousLinearMap.toLinear_compLeftContinuousCompact
@[simp]
theorem ContinuousLinearMap.toLinear_compLeftContinuousCompact (g : β →L[𝕜] γ) :
    (g.compLeftContinuousCompact X : C(X, β) →ₗ[𝕜] C(X, γ)) = g.compLeftContinuous 𝕜 X 
Arguments: (X : Type u_1) {𝕜 : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace X] [CompactSpace X] [NontriviallyNormedField 𝕜] [NormedAddCommGroup β] [NormedSpace 𝕜 β] [NormedAddCommGroup γ] [NormedSpace 𝕜 γ] (g : β →L[𝕜] γ)
Goal: ↑(ContinuousLinearMap.compLeftContinuousCompact X g) = ContinuousLinearMap.compLeftContinuous 𝕜 X g
-- ContinuousLinearMap.compLeftContinuousCompact_apply
@[simp]
theorem ContinuousLinearMap.compLeftContinuousCompact_apply (g : β →L[𝕜] γ) (f : C(X, β)) (x : X) :
    g.compLeftContinuousCompact X f x = g (f x) 
Arguments: (X : Type u_1) {𝕜 : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace X] [CompactSpace X] [NontriviallyNormedField 𝕜] [NormedAddCommGroup β] [NormedSpace 𝕜 β] [NormedAddCommGroup γ] [NormedSpace 𝕜 γ] (g : β →L[𝕜] γ) (f : C(X, β)) (x : X)
Goal: ((ContinuousLinearMap.compLeftContinuousCompact X g) f) x = g (f x)
-- ContinuousMap.compRightContinuousMap_apply
@[simp]
theorem compRightContinuousMap_apply {X Y : Type*} (T : Type*) [TopologicalSpace X]
    [CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y))
    (g : C(Y, T)) : (compRightContinuousMap T f) g = g.comp f 
Arguments: {X : Type u_1} {Y : Type u_2} (T : Type u_3) [TopologicalSpace X] [CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) (g : C(Y, T))
Goal: (ContinuousMap.compRightContinuousMap T f) g = g.comp f
-- ContinuousMap.compRightAlgHom_continuous
theorem compRightAlgHom_continuous {X Y : Type*} (R A : Type*) [TopologicalSpace X]
    [CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [CommSemiring R] [Semiring A]
    [MetricSpace A] [TopologicalSemiring A] [Algebra R A] (f : C(X, Y)) :
    Continuous (compRightAlgHom R A f) 
Arguments: {X : Type u_1} {Y : Type u_2} (R : Type u_3) (A : Type u_4) [TopologicalSpace X] [CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [CommSemiring R] [Semiring A] [MetricSpace A] [TopologicalSemiring A] [Algebra R A] (f : C(X, Y))
Goal: Continuous ⇑(ContinuousMap.compRightAlgHom R A f)
-- ContinuousMap.summable_of_locally_summable_norm
theorem summable_of_locally_summable_norm {ι : Type*} {F : ι → C(X, E)}
    (hF : ∀ K : Compacts X, Summable fun i => ‖(F i).restrict K‖) : Summable F 
Arguments: {X : Type u_1} [TopologicalSpace X] [LocallyCompactSpace X] {E : Type u_2} [NormedAddCommGroup E] [CompleteSpace E] {ι : Type u_3} {F : ι → C(X, E)} (hF : ∀ (K : TopologicalSpace.Compacts X), Summable fun i => ‖ContinuousMap.restrict (↑K) (F i)‖)
Goal: Summable F
-- BoundedContinuousFunction.mkOfCompact_star
theorem _root_.BoundedContinuousFunction.mkOfCompact_star [CompactSpace α] (f : C(α, β)) :
    mkOfCompact (star f) = star (mkOfCompact f) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [NormedAddCommGroup β] [StarAddMonoid β] [NormedStarGroup β] [CompactSpace α] (f : C(α, β))
Goal: BoundedContinuousFunction.mkOfCompact (star f) = star (BoundedContinuousFunction.mkOfCompact f)
-- Polynomial.toContinuousMap_apply
@[simps]
def toContinuousMap (p : R[X]) : C(R, R) 
Arguments: {R : Type u_1} [Semiring R] [TopologicalSpace R] [TopologicalSemiring R] (p : R[X]) (x : R)
Goal: p.toContinuousMap x = Polynomial.eval x p
-- Polynomial.toContinuousMap_X_eq_id
lemma toContinuousMap_X_eq_id : X.toContinuousMap = .id R 
Arguments: {R : Type u_1} [Semiring R] [TopologicalSpace R] [TopologicalSemiring R]
Goal: Polynomial.X.toContinuousMap = ContinuousMap.id R
-- Polynomial.toContinuousMapOn_apply
@[simps]
def toContinuousMapOn (p : R[X]) (X : Set R) : C(X, R) 
Arguments: {R : Type u_1} [Semiring R] [TopologicalSpace R] [TopologicalSemiring R] (p : R[X]) (X : Set R) (x : ↑X)
Goal: (p.toContinuousMapOn X) x = p.toContinuousMap ↑x
-- Polynomial.toContinuousMapOn_X_eq_restrict_id
lemma toContinuousMapOn_X_eq_restrict_id (s : Set R) :
    X.toContinuousMapOn s = restrict s (.id R) 
Arguments: {R : Type u_1} [Semiring R] [TopologicalSpace R] [TopologicalSemiring R] (s : Set R)
Goal: Polynomial.X.toContinuousMapOn s = ContinuousMap.restrict s (ContinuousMap.id R)
-- Polynomial.aeval_continuousMap_apply
@[simp]
theorem aeval_continuousMap_apply (g : R[X]) (f : C(α, R)) (x : α) :
    ((Polynomial.aeval f) g) x = g.eval (f x) 
Arguments: {R : Type u_1} {α : Type u_2} [TopologicalSpace α] [CommSemiring R] [TopologicalSpace R] [TopologicalSemiring R] (g : R[X]) (f : C(α, R)) (x : α)
Goal: ((Polynomial.aeval f) g) x = Polynomial.eval (f x) g
-- Polynomial.toContinuousMapAlgHom_apply
@[simps]
def toContinuousMapAlgHom : R[X] →ₐ[R] C(R, R) 
Arguments: {R : Type u_1} [CommSemiring R] [TopologicalSpace R] [TopologicalSemiring R] (p : R[X])
Goal: Polynomial.toContinuousMapAlgHom p = p.toContinuousMap
-- Polynomial.toContinuousMapOnAlgHom_apply
@[simps]
def toContinuousMapOnAlgHom (X : Set R) : R[X] →ₐ[R] C(X, R) 
Arguments: {R : Type u_1} [CommSemiring R] [TopologicalSpace R] [TopologicalSemiring R] (X : Set R) (p : R[X])
Goal: (Polynomial.toContinuousMapOnAlgHom X) p = p.toContinuousMapOn X
-- polynomialFunctions_coe
@[simp]
theorem polynomialFunctions_coe (X : Set R) :
    (polynomialFunctions X : Set C(X, R)) = Set.range (Polynomial.toContinuousMapOnAlgHom X) 
Arguments: {R : Type u_1} [CommSemiring R] [TopologicalSpace R] [TopologicalSemiring R] (X : Set R)
Goal: ↑(polynomialFunctions X) = Set.range ⇑(Polynomial.toContinuousMapOnAlgHom X)
-- polynomialFunctions_separatesPoints
theorem polynomialFunctions_separatesPoints (X : Set R) : (polynomialFunctions X).SeparatesPoints 
Arguments: {R : Type u_1} [CommSemiring R] [TopologicalSpace R] [TopologicalSemiring R] (X : Set R)
Goal: (polynomialFunctions X).SeparatesPoints