ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- seminormFromConst_isLimit
theorem seminormFromConst_isLimit (x : R) :
    Tendsto (seminormFromConst_seq c f x) atTop (𝓝 (seminormFromConst' hf1 hc hpm x)) 
Arguments: {R : Type u_1} [CommRing R] {c : R} {f : RingSeminorm R} (hf1 : f 1 ≤ 1) (hc : f c ≠ 0) (hpm : IsPowMul ⇑f) (x : R)
Goal: Filter.Tendsto (seminormFromConst_seq c f x) Filter.atTop (𝓝 (seminormFromConst' hf1 hc hpm x))
-- seminormFromConst_one
theorem seminormFromConst_one : seminormFromConst' hf1 hc hpm 1 = 1 
Arguments: {R : Type u_1} [CommRing R] {c : R} {f : RingSeminorm R} (hf1 : f 1 ≤ 1) (hc : f c ≠ 0) (hpm : IsPowMul ⇑f)
Goal: seminormFromConst' hf1 hc hpm 1 = 1
-- seminormFromConst_def
theorem seminormFromConst_def (x : R) :
    seminormFromConst hf1 hc hpm x = seminormFromConst' hf1 hc hpm x 
Arguments: {R : Type u_1} [CommRing R] {c : R} {f : RingSeminorm R} (hf1 : f 1 ≤ 1) (hc : f c ≠ 0) (hpm : IsPowMul ⇑f) (x : R)
Goal: (seminormFromConst hf1 hc hpm) x = seminormFromConst' hf1 hc hpm x
-- seminormFromConst_one_le
theorem seminormFromConst_one_le : seminormFromConst' hf1 hc hpm 1 ≤ 1 
Arguments: {R : Type u_1} [CommRing R] {c : R} {f : RingSeminorm R} (hf1 : f 1 ≤ 1) (hc : f c ≠ 0) (hpm : IsPowMul ⇑f)
Goal: seminormFromConst' hf1 hc hpm 1 ≤ 1
-- seminormFromConst_isNonarchimedean
theorem seminormFromConst_isNonarchimedean (hna : IsNonarchimedean f) :
    IsNonarchimedean (seminormFromConst' hf1 hc hpm) 
Arguments: {R : Type u_1} [CommRing R] {c : R} {f : RingSeminorm R} (hf1 : f 1 ≤ 1) (hc : f c ≠ 0) (hpm : IsPowMul ⇑f) (hna : IsNonarchimedean ⇑f)
Goal: IsNonarchimedean (seminormFromConst' hf1 hc hpm)
-- seminormFromConst_isPowMul
theorem seminormFromConst_isPowMul : IsPowMul (seminormFromConst' hf1 hc hpm) 
Arguments: {R : Type u_1} [CommRing R] {c : R} {f : RingSeminorm R} (hf1 : f 1 ≤ 1) (hc : f c ≠ 0) (hpm : IsPowMul ⇑f)
Goal: IsPowMul (seminormFromConst' hf1 hc hpm)
-- seminormFromConst_le_seminorm
theorem seminormFromConst_le_seminorm (x : R) : seminormFromConst' hf1 hc hpm x ≤ f x 
Arguments: {R : Type u_1} [CommRing R] {c : R} {f : RingSeminorm R} (hf1 : f 1 ≤ 1) (hc : f c ≠ 0) (hpm : IsPowMul ⇑f) (x : R)
Goal: seminormFromConst' hf1 hc hpm x ≤ f x
-- seminormFromConst_apply_of_isMul
theorem seminormFromConst_apply_of_isMul {x : R} (hx : ∀ y : R, f (x * y) = f x * f y) :
    seminormFromConst' hf1 hc hpm x = f x 
Arguments: {R : Type u_1} [CommRing R] {c : R} {f : RingSeminorm R} (hf1 : f 1 ≤ 1) (hc : f c ≠ 0) (hpm : IsPowMul ⇑f) {x : R} (hx : ∀ (y : R), f (x * y) = f x * f y)
Goal: seminormFromConst' hf1 hc hpm x = f x
-- seminormFromConst_isMul_of_isMul
theorem seminormFromConst_isMul_of_isMul {x : R} (hx : ∀ y : R, f (x * y) = f x * f y) (y : R) :
    seminormFromConst' hf1 hc hpm (x * y) =
      seminormFromConst' hf1 hc hpm x * seminormFromConst' hf1 hc hpm y 
Arguments: {R : Type u_1} [CommRing R] {c : R} {f : RingSeminorm R} (hf1 : f 1 ≤ 1) (hc : f c ≠ 0) (hpm : IsPowMul ⇑f) {x : R} (hx : ∀ (y : R), f (x * y) = f x * f y) (y : R)
Goal: seminormFromConst' hf1 hc hpm (x * y) = seminormFromConst' hf1 hc hpm x * seminormFromConst' hf1 hc hpm y
-- seminormFromConst_apply_c
theorem seminormFromConst_apply_c : seminormFromConst' hf1 hc hpm c = f c 
Arguments: {R : Type u_1} [CommRing R] {c : R} {f : RingSeminorm R} (hf1 : f 1 ≤ 1) (hc : f c ≠ 0) (hpm : IsPowMul ⇑f)
Goal: seminormFromConst' hf1 hc hpm c = f c
-- seminormFromConst_const_mul
theorem seminormFromConst_const_mul (x : R) :
    seminormFromConst' hf1 hc hpm (c * x) =
      seminormFromConst' hf1 hc hpm c * seminormFromConst' hf1 hc hpm x 
Arguments: {R : Type u_1} [CommRing R] {c : R} {f : RingSeminorm R} (hf1 : f 1 ≤ 1) (hc : f c ≠ 0) (hpm : IsPowMul ⇑f) (x : R)
Goal: seminormFromConst' hf1 hc hpm (c * x) = seminormFromConst' hf1 hc hpm c * seminormFromConst' hf1 hc hpm x
-- seminormFromConstRingNormOfField_def
theorem seminormFromConstRingNormOfField_def {k : K} {g : RingSeminorm K} (hg1 : g 1 ≤ 1)
    (hg_k : g k ≠ 0) (hg_pm : IsPowMul g) (x : K) :
    normFromConst hg1 hg_k hg_pm x = seminormFromConst' hg1 hg_k hg_pm x 
Arguments: {K : Type u_1} [Field K] {k : K} {g : RingSeminorm K} (hg1 : g 1 ≤ 1) (hg_k : g k ≠ 0) (hg_pm : IsPowMul ⇑g) (x : K)
Goal: (normFromConst hg1 hg_k hg_pm) x = seminormFromConst' hg1 hg_k hg_pm x
-- LinearIsometry.coe_toLinearIsometryEquiv
@[simp]
theorem coe_toLinearIsometryEquiv (li : E₁ →ₗᵢ[R₁] F) (h : finrank R₁ E₁ = finrank R₁ F) :
    (li.toLinearIsometryEquiv h : E₁ → F) = li 
Arguments: {F : Type u_2} {E₁ : Type u_3} [SeminormedAddCommGroup F] [NormedAddCommGroup E₁] {R₁ : Type u_4} [Field R₁] [Module R₁ E₁] [Module R₁ F] [FiniteDimensional R₁ E₁] [FiniteDimensional R₁ F] (li : E₁ →ₗᵢ[R₁] F) (h : FiniteDimensional.finrank R₁ E₁ = FiniteDimensional.finrank R₁ F)
Goal: ⇑(li.toLinearIsometryEquiv h) = ⇑li
-- LinearIsometry.toLinearIsometryEquiv_apply
@[simp]
theorem toLinearIsometryEquiv_apply (li : E₁ →ₗᵢ[R₁] F) (h : finrank R₁ E₁ = finrank R₁ F)
    (x : E₁) : (li.toLinearIsometryEquiv h) x = li x 
Arguments: {F : Type u_2} {E₁ : Type u_3} [SeminormedAddCommGroup F] [NormedAddCommGroup E₁] {R₁ : Type u_4} [Field R₁] [Module R₁ E₁] [Module R₁ F] [FiniteDimensional R₁ E₁] [FiniteDimensional R₁ F] (li : E₁ →ₗᵢ[R₁] F) (h : FiniteDimensional.finrank R₁ E₁ = FiniteDimensional.finrank R₁ F) (x : E₁)
Goal: (li.toLinearIsometryEquiv h) x = li x
-- AffineIsometry.coe_toAffineIsometryEquiv
@[simp]
theorem coe_toAffineIsometryEquiv [Inhabited P₁] (li : P₁ →ᵃⁱ[𝕜] P₂)
    (h : finrank 𝕜 V₁ = finrank 𝕜 V₂) : (li.toAffineIsometryEquiv h : P₁ → P₂) = li 
Arguments: {𝕜 : Type u_1} {V₁ : Type u_2} {V₂ : Type u_3} {P₁ : Type u_4} {P₂ : Type u_5} [NormedField 𝕜] [NormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₁] [NormedSpace 𝕜 V₂] [MetricSpace P₁] [PseudoMetricSpace P₂] [NormedAddTorsor V₁ P₁] [NormedAddTorsor V₂ P₂] [FiniteDimensional 𝕜 V₁] [FiniteDimensional 𝕜 V₂] [Inhabited P₁] (li : P₁ →ᵃⁱ[𝕜] P₂) (h : FiniteDimensional.finrank 𝕜 V₁ = FiniteDimensional.finrank 𝕜 V₂)
Goal: ⇑(li.toAffineIsometryEquiv h) = ⇑li
-- AffineIsometry.toAffineIsometryEquiv_apply
@[simp]
theorem toAffineIsometryEquiv_apply [Inhabited P₁] (li : P₁ →ᵃⁱ[𝕜] P₂)
    (h : finrank 𝕜 V₁ = finrank 𝕜 V₂) (x : P₁) : (li.toAffineIsometryEquiv h) x = li x 
Arguments: {𝕜 : Type u_1} {V₁ : Type u_2} {V₂ : Type u_3} {P₁ : Type u_4} {P₂ : Type u_5} [NormedField 𝕜] [NormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₁] [NormedSpace 𝕜 V₂] [MetricSpace P₁] [PseudoMetricSpace P₂] [NormedAddTorsor V₁ P₁] [NormedAddTorsor V₂ P₂] [FiniteDimensional 𝕜 V₁] [FiniteDimensional 𝕜 V₂] [Inhabited P₁] (li : P₁ →ᵃⁱ[𝕜] P₂) (h : FiniteDimensional.finrank 𝕜 V₁ = FiniteDimensional.finrank 𝕜 V₂) (x : P₁)
Goal: (li.toAffineIsometryEquiv h) x = li x
-- AffineMap.continuous_of_finiteDimensional
theorem AffineMap.continuous_of_finiteDimensional (f : PE →ᵃ[𝕜] PF) : Continuous f 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type w} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace 𝕜] {PE : Type u_1} {PF : Type u_2} [MetricSpace PE] [NormedAddTorsor E PE] [MetricSpace PF] [NormedAddTorsor F PF] [FiniteDimensional 𝕜 E] (f : PE →ᵃ[𝕜] PF)
Goal: Continuous ⇑f
-- AffineEquiv.continuous_of_finiteDimensional
theorem AffineEquiv.continuous_of_finiteDimensional (f : PE ≃ᵃ[𝕜] PF) : Continuous f 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type w} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace 𝕜] {PE : Type u_1} {PF : Type u_2} [MetricSpace PE] [NormedAddTorsor E PE] [MetricSpace PF] [NormedAddTorsor F PF] [FiniteDimensional 𝕜 E] (f : PE ≃ᵃ[𝕜] PF)
Goal: Continuous ⇑f
-- AffineEquiv.coe_toHomeomorphOfFiniteDimensional
@[simp]
theorem AffineEquiv.coe_toHomeomorphOfFiniteDimensional (f : PE ≃ᵃ[𝕜] PF) :
    ⇑f.toHomeomorphOfFiniteDimensional = f 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type w} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace 𝕜] {PE : Type u_1} {PF : Type u_2} [MetricSpace PE] [NormedAddTorsor E PE] [MetricSpace PF] [NormedAddTorsor F PF] [FiniteDimensional 𝕜 E] (f : PE ≃ᵃ[𝕜] PF)
Goal: ⇑f.toHomeomorphOfFiniteDimensional = ⇑f
-- AffineEquiv.coe_toHomeomorphOfFiniteDimensional_symm
@[simp]
theorem AffineEquiv.coe_toHomeomorphOfFiniteDimensional_symm (f : PE ≃ᵃ[𝕜] PF) :
    ⇑f.toHomeomorphOfFiniteDimensional.symm = f.symm 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type w} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace 𝕜] {PE : Type u_1} {PF : Type u_2} [MetricSpace PE] [NormedAddTorsor E PE] [MetricSpace PF] [NormedAddTorsor F PF] [FiniteDimensional 𝕜 E] (f : PE ≃ᵃ[𝕜] PF)
Goal: ⇑f.toHomeomorphOfFiniteDimensional.symm = ⇑f.symm
-- ContinuousLinearMap.continuous_det
theorem ContinuousLinearMap.continuous_det : Continuous fun f : E →L[𝕜] E => f.det 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace 𝕜]
Goal: Continuous fun f => f.det
-- lipschitzExtensionConstant_def
irreducible_def lipschitzExtensionConstant (E' : Type*) [NormedAddCommGroup E'] [NormedSpace ℝ E']
  [FiniteDimensional ℝ E'] : ℝ≥0 
Arguments: (E' : Type u_1) [NormedAddCommGroup E'] [NormedSpace ℝ E'] [FiniteDimensional ℝ E']
Goal: lipschitzExtensionConstant E' =
  let A := (Basis.ofVectorSpace ℝ E').equivFun.toContinuousLinearEquiv;
  max (‖↑A.symm‖₊ * ‖↑A‖₊) 1
-- lipschitzExtensionConstant_pos
theorem lipschitzExtensionConstant_pos (E' : Type*) [NormedAddCommGroup E'] [NormedSpace ℝ E']
    [FiniteDimensional ℝ E'] : 0 < lipschitzExtensionConstant E' 
Arguments: (E' : Type u_1) [NormedAddCommGroup E'] [NormedSpace ℝ E'] [FiniteDimensional ℝ E']
Goal: 0 < lipschitzExtensionConstant E'
-- LipschitzOnWith.extend_finite_dimension
theorem LipschitzOnWith.extend_finite_dimension {α : Type*} [PseudoMetricSpace α] {E' : Type*}
    [NormedAddCommGroup E'] [NormedSpace ℝ E'] [FiniteDimensional ℝ E'] {s : Set α} {f : α → E'}
    {K : ℝ≥0} (hf : LipschitzOnWith K f s) :
    ∃ g : α → E', LipschitzWith (lipschitzExtensionConstant E' * K) g ∧ EqOn f g s 
Arguments: {α : Type u_1} [PseudoMetricSpace α] {E' : Type u_2} [NormedAddCommGroup E'] [NormedSpace ℝ E'] [FiniteDimensional ℝ E'] {s : Set α} {f : α → E'} {K : ℝ≥0} (hf : LipschitzOnWith K f s)
Goal: ∃ g, LipschitzWith (lipschitzExtensionConstant E' * K) g ∧ Set.EqOn f g s
-- LinearMap.exists_antilipschitzWith
theorem LinearMap.exists_antilipschitzWith [FiniteDimensional 𝕜 E] (f : E →ₗ[𝕜] F)
    (hf : LinearMap.ker f = ⊥) : ∃ K > 0, AntilipschitzWith K f 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type w} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace 𝕜] [FiniteDimensional 𝕜 E] (f : E →ₗ[𝕜] F) (hf : LinearMap.ker f = ⊥)
Goal: ∃ K > 0, AntilipschitzWith K ⇑f
-- LinearMap.injective_iff_antilipschitz
theorem LinearMap.injective_iff_antilipschitz [FiniteDimensional 𝕜 E] (f : E →ₗ[𝕜] F) :
    Injective f ↔ ∃ K > 0, AntilipschitzWith K f 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type w} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace 𝕜] [FiniteDimensional 𝕜 E] (f : E →ₗ[𝕜] F)
Goal: Function.Injective ⇑f ↔ ∃ K > 0, AntilipschitzWith K ⇑f
-- ContinuousLinearMap.isOpen_injective
theorem ContinuousLinearMap.isOpen_injective [FiniteDimensional 𝕜 E] :
    IsOpen { L : E →L[𝕜] F | Injective L } 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type w} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace 𝕜] [FiniteDimensional 𝕜 E]
Goal: IsOpen {L | Function.Injective ⇑L}
-- LinearIndependent.eventually
protected theorem LinearIndependent.eventually {ι} [Finite ι] {f : ι → E}
    (hf : LinearIndependent 𝕜 f) : ∀ᶠ g in 𝓝 f, LinearIndependent 𝕜 g 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace 𝕜] {ι : Type u_1} [Finite ι] {f : ι → E} (hf : LinearIndependent 𝕜 f)
Goal: ∀ᶠ (g : ι → E) in 𝓝 f, LinearIndependent 𝕜 g
-- isOpen_setOf_linearIndependent
theorem isOpen_setOf_linearIndependent {ι : Type*} [Finite ι] :
    IsOpen { f : ι → E | LinearIndependent 𝕜 f } 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace 𝕜] {ι : Type u_1} [Finite ι]
Goal: IsOpen {f | LinearIndependent 𝕜 f}
-- isOpen_setOf_nat_le_rank
theorem isOpen_setOf_nat_le_rank (n : ℕ) :
    IsOpen { f : E →L[𝕜] F | ↑n ≤ (f : E →ₗ[𝕜] F).rank } 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type w} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace 𝕜] (n : ℕ)
Goal: IsOpen {f | ↑n ≤ (↑f).rank}
-- Basis.opNNNorm_le
theorem Basis.opNNNorm_le {ι : Type*} [Fintype ι] (v : Basis ι 𝕜 E) {u : E →L[𝕜] F} (M : ℝ≥0)
    (hu : ∀ i, ‖u (v i)‖₊ ≤ M) : ‖u‖₊ ≤ Fintype.card ι • ‖v.equivFunL.toContinuousLinearMap‖₊ * M 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type w} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace 𝕜] {ι : Type u_1} [Fintype ι] (v : Basis ι 𝕜 E) {u : E →L[𝕜] F} (M : ℝ≥0) (hu : ∀ (i : ι), ‖u (v i)‖₊ ≤ M)
Goal: ‖u‖₊ ≤ Fintype.card ι • ‖↑v.equivFunL‖₊ * M
-- Basis.opNorm_le
theorem Basis.opNorm_le {ι : Type*} [Fintype ι] (v : Basis ι 𝕜 E) {u : E →L[𝕜] F} {M : ℝ}
    (hM : 0 ≤ M) (hu : ∀ i, ‖u (v i)‖ ≤ M) :
    ‖u‖ ≤ Fintype.card ι • ‖v.equivFunL.toContinuousLinearMap‖ * M 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type w} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace 𝕜] {ι : Type u_1} [Fintype ι] (v : Basis ι 𝕜 E) {u : E →L[𝕜] F} {M : ℝ} (hM : 0 ≤ M) (hu : ∀ (i : ι), ‖u (v i)‖ ≤ M)
Goal: ‖u‖ ≤ Fintype.card ι • ‖↑v.equivFunL‖ * M
-- Basis.exists_opNNNorm_le
theorem Basis.exists_opNNNorm_le {ι : Type*} [Finite ι] (v : Basis ι 𝕜 E) :
    ∃ C > (0 : ℝ≥0), ∀ {u : E →L[𝕜] F} (M : ℝ≥0), (∀ i, ‖u (v i)‖₊ ≤ M) → ‖u‖₊ ≤ C * M 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type w} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace 𝕜] {ι : Type u_1} [Finite ι] (v : Basis ι 𝕜 E)
Goal: ∃ C > 0, ∀ {u : E →L[𝕜] F} (M : ℝ≥0), (∀ (i : ι), ‖u (v i)‖₊ ≤ M) → ‖u‖₊ ≤ C * M
-- Basis.exists_opNorm_le
theorem Basis.exists_opNorm_le {ι : Type*} [Finite ι] (v : Basis ι 𝕜 E) :
    ∃ C > (0 : ℝ), ∀ {u : E →L[𝕜] F} {M : ℝ}, 0 ≤ M → (∀ i, ‖u (v i)‖ ≤ M) → ‖u‖ ≤ C * M 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type w} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace 𝕜] {ι : Type u_1} [Finite ι] (v : Basis ι 𝕜 E)
Goal: ∃ C > 0, ∀ {u : E →L[𝕜] F} {M : ℝ}, 0 ≤ M → (∀ (i : ι), ‖u (v i)‖ ≤ M) → ‖u‖ ≤ C * M
-- AffineSubspace.closed_of_finiteDimensional
theorem AffineSubspace.closed_of_finiteDimensional {P : Type*} [MetricSpace P]
    [NormedAddTorsor E P] (s : AffineSubspace 𝕜 P) [FiniteDimensional 𝕜 s.direction] :
    IsClosed (s : Set P) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace 𝕜] {P : Type u_1} [MetricSpace P] [NormedAddTorsor E P] (s : AffineSubspace 𝕜 P) [FiniteDimensional 𝕜 ↥s.direction]
Goal: IsClosed ↑s
-- exists_norm_le_le_norm_sub_of_finset
theorem exists_norm_le_le_norm_sub_of_finset {c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖c‖ < R)
    (h : ¬FiniteDimensional 𝕜 E) (s : Finset E) : ∃ x : E, ‖x‖ ≤ R ∧ ∀ y ∈ s, 1 ≤ ‖y - x‖ 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace 𝕜] {c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖c‖ < R) (h : ¬FiniteDimensional 𝕜 E) (s : Finset E)
Goal: ∃ x, ‖x‖ ≤ R ∧ ∀ y ∈ s, 1 ≤ ‖y - x‖
-- exists_seq_norm_le_one_le_norm_sub'
theorem exists_seq_norm_le_one_le_norm_sub' {c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖c‖ < R)
    (h : ¬FiniteDimensional 𝕜 E) :
    ∃ f : ℕ → E, (∀ n, ‖f n‖ ≤ R) ∧ Pairwise fun m n => 1 ≤ ‖f m - f n‖ 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace 𝕜] {c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖c‖ < R) (h : ¬FiniteDimensional 𝕜 E)
Goal: ∃ f, (∀ (n : ℕ), ‖f n‖ ≤ R) ∧ Pairwise fun m n => 1 ≤ ‖f m - f n‖
-- exists_seq_norm_le_one_le_norm_sub
theorem exists_seq_norm_le_one_le_norm_sub (h : ¬FiniteDimensional 𝕜 E) :
    ∃ (R : ℝ) (f : ℕ → E), 1 < R ∧ (∀ n, ‖f n‖ ≤ R) ∧ Pairwise fun m n => 1 ≤ ‖f m - f n‖ 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace 𝕜] (h : ¬FiniteDimensional 𝕜 E)
Goal: ∃ R f, 1 < R ∧ (∀ (n : ℕ), ‖f n‖ ≤ R) ∧ Pairwise fun m n => 1 ≤ ‖f m - f n‖
-- FiniteDimensional.of_isCompact_closedBall₀
theorem FiniteDimensional.of_isCompact_closedBall₀ {r : ℝ} (rpos : 0 < r)
    (h : IsCompact (Metric.closedBall (0 : E) r)) : FiniteDimensional 𝕜 E 
Arguments: (𝕜 : Type u) [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace 𝕜] {r : ℝ} (rpos : 0 < r) (h : IsCompact (Metric.closedBall 0 r))
Goal: FiniteDimensional 𝕜 E
-- FiniteDimensional.of_isCompact_closedBall
theorem FiniteDimensional.of_isCompact_closedBall {r : ℝ} (rpos : 0 < r) {c : E}
    (h : IsCompact (Metric.closedBall c r)) : FiniteDimensional 𝕜 E 
Arguments: (𝕜 : Type u) [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace 𝕜] {r : ℝ} (rpos : 0 < r) {c : E} (h : IsCompact (Metric.closedBall c r))
Goal: FiniteDimensional 𝕜 E
-- FiniteDimensional.of_locallyCompactSpace
theorem FiniteDimensional.of_locallyCompactSpace [LocallyCompactSpace E] :
    FiniteDimensional 𝕜 E 
Arguments: (𝕜 : Type u) [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace 𝕜] [LocallyCompactSpace E]
Goal: FiniteDimensional 𝕜 E
-- HasCompactSupport.eq_zero_or_finiteDimensional
theorem HasCompactSupport.eq_zero_or_finiteDimensional {X : Type*} [TopologicalSpace X] [Zero X]
    [T1Space X] {f : E → X} (hf : HasCompactSupport f) (h'f : Continuous f) :
    f = 0 ∨ FiniteDimensional 𝕜 E 
Arguments: (𝕜 : Type u) [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace 𝕜] {X : Type u_1} [TopologicalSpace X] [Zero X] [T1Space X] {f : E → X} (hf : HasCompactSupport f) (h'f : Continuous f)
Goal: f = 0 ∨ FiniteDimensional 𝕜 E
-- HasCompactMulSupport.eq_one_or_finiteDimensional
@[to_additive existing]
theorem HasCompactMulSupport.eq_one_or_finiteDimensional {X : Type*} [TopologicalSpace X] [One X]
    [T1Space X] {f : E → X} (hf : HasCompactMulSupport f) (h'f : Continuous f) :
    f = 1 ∨ FiniteDimensional 𝕜 E 
Arguments: (𝕜 : Type u) [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace 𝕜] {X : Type u_1} [TopologicalSpace X] [One X] [T1Space X] {f : E → X} (hf : HasCompactMulSupport f) (h'f : Continuous f)
Goal: f = 1 ∨ FiniteDimensional 𝕜 E
-- ProperSpace.of_locallyCompactSpace
lemma ProperSpace.of_locallyCompactSpace (𝕜 : Type*) [NontriviallyNormedField 𝕜]
    {E : Type*} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] [LocallyCompactSpace E] :
    ProperSpace E 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] [LocallyCompactSpace E]
Goal: ProperSpace E
-- ProperSpace.of_locallyCompact_module
lemma ProperSpace.of_locallyCompact_module [Nontrivial E] [LocallyCompactSpace E] :
    ProperSpace 𝕜 
Arguments: (𝕜 : Type u) [NontriviallyNormedField 𝕜] (E : Type v) [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace 𝕜] [Nontrivial E] [LocallyCompactSpace E]
Goal: ProperSpace 𝕜
-- continuousOn_clm_apply
theorem continuousOn_clm_apply {X : Type*} [TopologicalSpace X] [FiniteDimensional 𝕜 E]
    {f : X → E →L[𝕜] F} {s : Set X} : ContinuousOn f s ↔ ∀ y, ContinuousOn (fun x => f x y) s 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type w} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace 𝕜] {X : Type u_1} [TopologicalSpace X] [FiniteDimensional 𝕜 E] {f : X → E →L[𝕜] F} {s : Set X}
Goal: ContinuousOn f s ↔ ∀ (y : E), ContinuousOn (fun x => (f x) y) s
-- continuous_clm_apply
theorem continuous_clm_apply {X : Type*} [TopologicalSpace X] [FiniteDimensional 𝕜 E]
    {f : X → E →L[𝕜] F} : Continuous f ↔ ∀ y, Continuous (f · y) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type w} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace 𝕜] {X : Type u_1} [TopologicalSpace X] [FiniteDimensional 𝕜 E] {f : X → E →L[𝕜] F}
Goal: Continuous f ↔ ∀ (y : E), Continuous fun x => (f x) y
-- FiniteDimensional.proper
theorem FiniteDimensional.proper [FiniteDimensional 𝕜 E] : ProperSpace E 
Arguments: (𝕜 : Type u) [NontriviallyNormedField 𝕜] (E : Type v) [NormedAddCommGroup E] [NormedSpace 𝕜 E] [LocallyCompactSpace 𝕜] [FiniteDimensional 𝕜 E]
Goal: ProperSpace E
-- exists_mem_frontier_infDist_compl_eq_dist
theorem exists_mem_frontier_infDist_compl_eq_dist {E : Type*} [NormedAddCommGroup E]
    [NormedSpace ℝ E] [FiniteDimensional ℝ E] {x : E} {s : Set E} (hx : x ∈ s) (hs : s ≠ univ) :
    ∃ y ∈ frontier s, Metric.infDist x sᶜ = dist x y 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {x : E} {s : Set E} (hx : x ∈ s) (hs : s ≠ Set.univ)
Goal: ∃ y ∈ frontier s, Metric.infDist x sᶜ = dist x y
-- IsCompact.exists_mem_frontier_infDist_compl_eq_dist
nonrec theorem IsCompact.exists_mem_frontier_infDist_compl_eq_dist {E : Type*}
    [NormedAddCommGroup E] [NormedSpace ℝ E] [Nontrivial E] {x : E} {K : Set E} (hK : IsCompact K)
    (hx : x ∈ K) :
    ∃ y ∈ frontier K, Metric.infDist x Kᶜ = dist x y 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [Nontrivial E] {x : E} {K : Set E} (hK : IsCompact K) (hx : x ∈ K)
Goal: ∃ y ∈ frontier K, Metric.infDist x Kᶜ = dist x y
-- summable_norm_iff
theorem summable_norm_iff {α E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    [FiniteDimensional ℝ E] {f : α → E} : (Summable fun x => ‖f x‖) ↔ Summable f 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {f : α → E}
Goal: (Summable fun x => ‖f x‖) ↔ Summable f
-- summable_of_isBigO'
theorem summable_of_isBigO' {ι E F : Type*} [NormedAddCommGroup E] [CompleteSpace E]
    [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F] {f : ι → E} {g : ι → F}
    (hg : Summable g) (h : f =O[cofinite] g) : Summable f 
Arguments: {ι : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [CompleteSpace E] [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F] {f : ι → E} {g : ι → F} (hg : Summable g) (h : f =O[Filter.cofinite] g)
Goal: Summable f
-- Asymptotics.IsBigO.comp_summable
lemma Asymptotics.IsBigO.comp_summable {ι E F : Type*}
    [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
    [NormedAddCommGroup F] [CompleteSpace F]
    {f : E → F} (hf : f =O[𝓝 0] id) {g : ι → E} (hg : Summable g) : Summable (f ∘ g) 
Arguments: {ι : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [NormedAddCommGroup F] [CompleteSpace F] {f : E → F} (hf : f =O[𝓝 0] id) {g : ι → E} (hg : Summable g)
Goal: Summable (f ∘ g)
-- summable_of_isBigO_nat'
theorem summable_of_isBigO_nat' {E F : Type*} [NormedAddCommGroup E] [CompleteSpace E]
    [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F] {f : ℕ → E} {g : ℕ → F}
    (hg : Summable g) (h : f =O[atTop] g) : Summable f 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [CompleteSpace E] [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F] {f : ℕ → E} {g : ℕ → F} (hg : Summable g) (h : f =O[Filter.atTop] g)
Goal: Summable f