ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ENorm.coe_max
@[simp, norm_cast]
theorem coe_max (e₁ e₂ : ENorm 𝕜 V) : ⇑(e₁ ⊔ e₂) = fun x => max (e₁ x) (e₂ x) 
Arguments: {𝕜 : Type u_1} {V : Type u_2} [NormedField 𝕜] [AddCommGroup V] [Module 𝕜 V] (e₁ e₂ : ENorm 𝕜 V)
Goal: ↑(e₁ ⊔ e₂) = fun x => max (↑e₁ x) (↑e₂ x)
-- ENorm.max_map
@[norm_cast]
theorem max_map (e₁ e₂ : ENorm 𝕜 V) (x : V) : (e₁ ⊔ e₂) x = max (e₁ x) (e₂ x) 
Arguments: {𝕜 : Type u_1} {V : Type u_2} [NormedField 𝕜] [AddCommGroup V] [Module 𝕜 V] (e₁ e₂ : ENorm 𝕜 V) (x : V)
Goal: ↑(e₁ ⊔ e₂) x = max (↑e₁ x) (↑e₂ x)
-- ENorm.finite_dist_eq
theorem finite_dist_eq (x y : e.finiteSubspace) : dist x y = (e (x - y)).toReal 
Arguments: {𝕜 : Type u_1} {V : Type u_2} [NormedField 𝕜] [AddCommGroup V] [Module 𝕜 V] (e : ENorm 𝕜 V) (x y : ↥e.finiteSubspace)
Goal: dist x y = (↑e (↑x - ↑y)).toReal
-- ENorm.finite_edist_eq
theorem finite_edist_eq (x y : e.finiteSubspace) : edist x y = e (x - y) 
Arguments: {𝕜 : Type u_1} {V : Type u_2} [NormedField 𝕜] [AddCommGroup V] [Module 𝕜 V] (e : ENorm 𝕜 V) (x y : ↥e.finiteSubspace)
Goal: edist x y = ↑e (↑x - ↑y)
-- ENorm.finite_norm_eq
theorem finite_norm_eq (x : e.finiteSubspace) : ‖x‖ = (e x).toReal 
Arguments: {𝕜 : Type u_1} {V : Type u_2} [NormedField 𝕜] [AddCommGroup V] [Module 𝕜 V] (e : ENorm 𝕜 V) (x : ↥e.finiteSubspace)
Goal: ‖x‖ = (↑e ↑x).toReal
-- LinearMap.bound_of_shell
theorem bound_of_shell [RingHomIsometric σ₁₂] (f : E →ₛₗ[σ₁₂] F) {ε C : ℝ} (ε_pos : 0 < ε) {c : 𝕜}
    (hc : 1 < ‖c‖) (hf : ∀ x, ε / ‖c‖ ≤ ‖x‖ → ‖x‖ < ε → ‖f x‖ ≤ C * ‖x‖) (x : E) :
    ‖f x‖ ≤ C * ‖x‖ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [NormedAddCommGroup E] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (f : E →ₛₗ[σ₁₂] F) {ε C : ℝ} (ε_pos : 0 < ε) {c : 𝕜} (hc : 1 < ‖c‖) (hf : ∀ (x : E), ε / ‖c‖ ≤ ‖x‖ → ‖x‖ < ε → ‖f x‖ ≤ C * ‖x‖) (x : E)
Goal: ‖f x‖ ≤ C * ‖x‖
-- LinearMap.bound_of_ball_bound
theorem bound_of_ball_bound {r : ℝ} (r_pos : 0 < r) (c : ℝ) (f : E →ₗ[𝕜] Fₗ)
    (h : ∀ z ∈ Metric.ball (0 : E) r, ‖f z‖ ≤ c) : ∃ C, ∀ z : E, ‖f z‖ ≤ C * ‖z‖ 
Arguments: {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} [NormedAddCommGroup E] [NormedAddCommGroup Fₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] {r : ℝ} (r_pos : 0 < r) (c : ℝ) (f : E →ₗ[𝕜] Fₗ) (h : ∀ z ∈ Metric.ball 0 r, ‖f z‖ ≤ c)
Goal: ∃ C, ∀ (z : E), ‖f z‖ ≤ C * ‖z‖
-- LinearMap.antilipschitz_of_comap_nhds_le
theorem antilipschitz_of_comap_nhds_le [h : RingHomIsometric σ₁₂] (f : E →ₛₗ[σ₁₂] F)
    (hf : (𝓝 0).comap f ≤ 𝓝 0) : ∃ K, AntilipschitzWith K f 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [NormedAddCommGroup E] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [h : RingHomIsometric σ₁₂] (f : E →ₛₗ[σ₁₂] F) (hf : Filter.comap (⇑f) (𝓝 0) ≤ 𝓝 0)
Goal: ∃ K, AntilipschitzWith K ⇑f
-- ContinuousLinearMap.opNorm_zero_iff
theorem opNorm_zero_iff [RingHomIsometric σ₁₂] : ‖f‖ = 0 ↔ f = 0 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [NormedAddCommGroup E] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →SL[σ₁₂] F) [RingHomIsometric σ₁₂]
Goal: ‖f‖ = 0 ↔ f = 0
-- ContinuousLinearMap.norm_id
@[simp]
theorem norm_id [Nontrivial E] : ‖id 𝕜 E‖ = 1 
Arguments: {𝕜 : Type u_1} {E : Type u_4} [NormedAddCommGroup E] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [Nontrivial E]
Goal: ‖ContinuousLinearMap.id 𝕜 E‖ = 1
-- ContinuousLinearMap.nnnorm_id
@[simp]
lemma nnnorm_id [Nontrivial E] : ‖id 𝕜 E‖₊ = 1 
Arguments: {𝕜 : Type u_1} {E : Type u_4} [NormedAddCommGroup E] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [Nontrivial E]
Goal: ‖ContinuousLinearMap.id 𝕜 E‖₊ = 1
-- ContinuousLinearMap.homothety_norm
theorem homothety_norm [RingHomIsometric σ₁₂] [Nontrivial E] (f : E →SL[σ₁₂] F) {a : ℝ}
    (hf : ∀ x, ‖f x‖ = a * ‖x‖) : ‖f‖ = a 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [NormedAddCommGroup E] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] [Nontrivial E] (f : E →SL[σ₁₂] F) {a : ℝ} (hf : ∀ (x : E), ‖f x‖ = a * ‖x‖)
Goal: ‖f‖ = a
-- ContinuousLinearMap.antilipschitz_of_embedding
theorem antilipschitz_of_embedding (f : E →L[𝕜] Fₗ) (hf : Embedding f) :
    ∃ K, AntilipschitzWith K f 
Arguments: {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} [NormedAddCommGroup E] [NormedAddCommGroup Fₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] (f : E →L[𝕜] Fₗ) (hf : Embedding ⇑f)
Goal: ∃ K, AntilipschitzWith K ⇑f
-- LinearIsometry.norm_toContinuousLinearMap
@[simp]
theorem norm_toContinuousLinearMap [Nontrivial E] [RingHomIsometric σ₁₂] (f : E →ₛₗᵢ[σ₁₂] F) :
    ‖f.toContinuousLinearMap‖ = 1 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [NormedAddCommGroup E] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [Nontrivial E] [RingHomIsometric σ₁₂] (f : E →ₛₗᵢ[σ₁₂] F)
Goal: ‖f.toContinuousLinearMap‖ = 1
-- LinearIsometry.norm_toContinuousLinearMap_comp
theorem norm_toContinuousLinearMap_comp [RingHomIsometric σ₁₂] (f : F →ₛₗᵢ[σ₂₃] G)
    {g : E →SL[σ₁₂] F} : ‖f.toContinuousLinearMap.comp g‖ = ‖g‖ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomIsometric σ₁₂] (f : F →ₛₗᵢ[σ₂₃] G) {g : E →SL[σ₁₂] F}
Goal: ‖f.toContinuousLinearMap.comp g‖ = ‖g‖
-- ContinuousLinearMap.opNorm_comp_linearIsometryEquiv
theorem opNorm_comp_linearIsometryEquiv (f : F →SL[σ₂₃] G) (g : F' ≃ₛₗᵢ[σ₂'] F) :
    ‖f.comp g.toLinearIsometry.toContinuousLinearMap‖ = ‖f‖ 
Arguments: {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {F : Type u_6} {G : Type u_8} [NormedAddCommGroup F] [NormedAddCommGroup G] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {𝕜₂' : Type u_11} [NontriviallyNormedField 𝕜₂'] {F' : Type u_12} [NormedAddCommGroup F'] [NormedSpace 𝕜₂' F'] {σ₂' : 𝕜₂' →+* 𝕜₂} {σ₂'' : 𝕜₂ →+* 𝕜₂'} {σ₂₃' : 𝕜₂' →+* 𝕜₃} [RingHomInvPair σ₂' σ₂''] [RingHomInvPair σ₂'' σ₂'] [RingHomCompTriple σ₂' σ₂₃ σ₂₃'] [RingHomCompTriple σ₂'' σ₂₃' σ₂₃] [RingHomIsometric σ₂₃] [RingHomIsometric σ₂'] [RingHomIsometric σ₂''] [RingHomIsometric σ₂₃'] (f : F →SL[σ₂₃] G) (g : F' ≃ₛₗᵢ[σ₂'] F)
Goal: ‖f.comp g.toLinearIsometry.toContinuousLinearMap‖ = ‖f‖
-- ContinuousLinearMap.norm_smulRight_apply
@[simp]
theorem norm_smulRight_apply (c : E →L[𝕜] 𝕜) (f : Fₗ) : ‖smulRight c f‖ = ‖c‖ * ‖f‖ 
Arguments: {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} [NormedAddCommGroup E] [NormedAddCommGroup Fₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] (c : E →L[𝕜] 𝕜) (f : Fₗ)
Goal: ‖c.smulRight f‖ = ‖c‖ * ‖f‖
-- ContinuousLinearMap.nnnorm_smulRight_apply
@[simp]
theorem nnnorm_smulRight_apply (c : E →L[𝕜] 𝕜) (f : Fₗ) : ‖smulRight c f‖₊ = ‖c‖₊ * ‖f‖₊ 
Arguments: {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} [NormedAddCommGroup E] [NormedAddCommGroup Fₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] (c : E →L[𝕜] 𝕜) (f : Fₗ)
Goal: ‖c.smulRight f‖₊ = ‖c‖₊ * ‖f‖₊
-- ContinuousLinearMap.norm_smulRightL_apply
@[simp]
theorem norm_smulRightL_apply (c : E →L[𝕜] 𝕜) (f : Fₗ) : ‖smulRightL 𝕜 E Fₗ c f‖ = ‖c‖ * ‖f‖ 
Arguments: {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} [NormedAddCommGroup E] [NormedAddCommGroup Fₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] (c : E →L[𝕜] 𝕜) (f : Fₗ)
Goal: ‖((ContinuousLinearMap.smulRightL 𝕜 E Fₗ) c) f‖ = ‖c‖ * ‖f‖
-- ContinuousLinearMap.norm_smulRightL
@[simp]
theorem norm_smulRightL (c : E →L[𝕜] 𝕜) [Nontrivial Fₗ] : ‖smulRightL 𝕜 E Fₗ c‖ = ‖c‖ 
Arguments: {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} [NormedAddCommGroup E] [NormedAddCommGroup Fₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] (c : E →L[𝕜] 𝕜) [Nontrivial Fₗ]
Goal: ‖(ContinuousLinearMap.smulRightL 𝕜 E Fₗ) c‖ = ‖c‖
-- ContinuousLinearMap.norm_smulRightL_le
lemma norm_smulRightL_le : ‖smulRightL 𝕜 E Fₗ‖ ≤ 1 
Arguments: {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} [NormedAddCommGroup E] [NormedAddCommGroup Fₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ]
Goal: ‖ContinuousLinearMap.smulRightL 𝕜 E Fₗ‖ ≤ 1
-- Submodule.norm_subtypeL
theorem norm_subtypeL (K : Submodule 𝕜 E) [Nontrivial K] : ‖K.subtypeL‖ = 1 
Arguments: {𝕜 : Type u_1} {E : Type u_4} [NormedAddCommGroup E] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] (K : Submodule 𝕜 E) [Nontrivial ↥K]
Goal: ‖K.subtypeL‖ = 1
-- ContinuousLinearEquiv.antilipschitz
protected theorem antilipschitz (e : E ≃SL[σ₁₂] F) :
    AntilipschitzWith ‖(e.symm : F →SL[σ₂₁] E)‖₊ e 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [NormedAddCommGroup E] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [RingHomIsometric σ₂₁] (e : E ≃SL[σ₁₂] F)
Goal: AntilipschitzWith ‖↑e.symm‖₊ ⇑e
-- ContinuousLinearEquiv.one_le_norm_mul_norm_symm
theorem one_le_norm_mul_norm_symm [RingHomIsometric σ₁₂] [Nontrivial E] (e : E ≃SL[σ₁₂] F) :
    1 ≤ ‖(e : E →SL[σ₁₂] F)‖ * ‖(e.symm : F →SL[σ₂₁] E)‖ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [NormedAddCommGroup E] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [RingHomIsometric σ₂₁] [RingHomIsometric σ₁₂] [Nontrivial E] (e : E ≃SL[σ₁₂] F)
Goal: 1 ≤ ‖↑e‖ * ‖↑e.symm‖
-- ContinuousLinearEquiv.norm_pos
theorem norm_pos [RingHomIsometric σ₁₂] [Nontrivial E] (e : E ≃SL[σ₁₂] F) :
    0 < ‖(e : E →SL[σ₁₂] F)‖ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [NormedAddCommGroup E] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [RingHomIsometric σ₂₁] [RingHomIsometric σ₁₂] [Nontrivial E] (e : E ≃SL[σ₁₂] F)
Goal: 0 < ‖↑e‖
-- ContinuousLinearEquiv.norm_symm_pos
theorem norm_symm_pos [RingHomIsometric σ₁₂] [Nontrivial E] (e : E ≃SL[σ₁₂] F) :
    0 < ‖(e.symm : F →SL[σ₂₁] E)‖ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [NormedAddCommGroup E] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [RingHomIsometric σ₂₁] [RingHomIsometric σ₁₂] [Nontrivial E] (e : E ≃SL[σ₁₂] F)
Goal: 0 < ‖↑e.symm‖
-- ContinuousLinearEquiv.nnnorm_symm_pos
theorem nnnorm_symm_pos [RingHomIsometric σ₁₂] [Nontrivial E] (e : E ≃SL[σ₁₂] F) :
    0 < ‖(e.symm : F →SL[σ₂₁] E)‖₊ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [NormedAddCommGroup E] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [RingHomIsometric σ₂₁] [RingHomIsometric σ₁₂] [Nontrivial E] (e : E ≃SL[σ₁₂] F)
Goal: 0 < ‖↑e.symm‖₊
-- ContinuousLinearEquiv.subsingleton_or_norm_symm_pos
theorem subsingleton_or_norm_symm_pos [RingHomIsometric σ₁₂] (e : E ≃SL[σ₁₂] F) :
    Subsingleton E ∨ 0 < ‖(e.symm : F →SL[σ₂₁] E)‖ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [NormedAddCommGroup E] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [RingHomIsometric σ₂₁] [RingHomIsometric σ₁₂] (e : E ≃SL[σ₁₂] F)
Goal: Subsingleton E ∨ 0 < ‖↑e.symm‖
-- ContinuousLinearEquiv.subsingleton_or_nnnorm_symm_pos
theorem subsingleton_or_nnnorm_symm_pos [RingHomIsometric σ₁₂] (e : E ≃SL[σ₁₂] F) :
    Subsingleton E ∨ 0 < ‖(e.symm : F →SL[σ₂₁] E)‖₊ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [NormedAddCommGroup E] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] [RingHomIsometric σ₂₁] [RingHomIsometric σ₁₂] (e : E ≃SL[σ₁₂] F)
Goal: Subsingleton E ∨ 0 < ‖↑e.symm‖₊
-- ContinuousLinearEquiv.coord_norm
@[simp]
theorem coord_norm (x : E) (h : x ≠ 0) : ‖coord 𝕜 x h‖ = ‖x‖⁻¹ 
Arguments: (𝕜 : Type u_1) {E : Type u_4} [NormedAddCommGroup E] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] (x : E) (h : x ≠ 0)
Goal: ‖ContinuousLinearEquiv.coord 𝕜 x h‖ = ‖x‖⁻¹
-- NormedSpace.equicontinuous_TFAE
protected theorem NormedSpace.equicontinuous_TFAE : List.TFAE
    [ EquicontinuousAt ((↑) ∘ f) 0,
      Equicontinuous ((↑) ∘ f),
      UniformEquicontinuous ((↑) ∘ f),
      ∃ C, ∀ i x, ‖f i x‖ ≤ C * ‖x‖,
      ∃ C ≥ 0, ∀ i x, ‖f i x‖ ≤ C * ‖x‖,
      ∃ C, ∀ i, ‖f i‖ ≤ C,
      ∃ C ≥ 0, ∀ i, ‖f i‖ ≤ C,
      BddAbove (Set.range (‖f ·‖)),
      (⨆ i, (‖f i‖₊ : ENNReal)) < ⊤ ] 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} {ι : Type u_11} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] (f : ι → E →SL[σ₁₂] F)
Goal: [EquicontinuousAt (DFunLike.coe ∘ f) 0, Equicontinuous (DFunLike.coe ∘ f), UniformEquicontinuous (DFunLike.coe ∘ f),
    ∃ C, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖, ∃ C ≥ 0, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖,
    ∃ C, ∀ (i : ι), ‖f i‖ ≤ C, ∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C, BddAbove (Set.range fun x => ‖f x‖),
    ⨆ i, ↑‖f i‖₊ < ⊤].TFAE
-- riesz_lemma
theorem riesz_lemma {F : Subspace 𝕜 E} (hFc : IsClosed (F : Set E)) (hF : ∃ x : E, x ∉ F) {r : ℝ}
    (hr : r < 1) : ∃ x₀ : E, x₀ ∉ F ∧ ∀ y ∈ F, r * ‖x₀‖ ≤ ‖x₀ - y‖ 
Arguments: {𝕜 : Type u_1} [NormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Subspace 𝕜 E} (hFc : IsClosed ↑F) (hF : ∃ x, x ∉ F) {r : ℝ} (hr : r < 1)
Goal: ∃ x₀ ∉ F, ∀ y ∈ F, r * ‖x₀‖ ≤ ‖x₀ - y‖
-- riesz_lemma_of_norm_lt
theorem riesz_lemma_of_norm_lt {c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖c‖ < R) {F : Subspace 𝕜 E}
    (hFc : IsClosed (F : Set E)) (hF : ∃ x : E, x ∉ F) :
    ∃ x₀ : E, ‖x₀‖ ≤ R ∧ ∀ y ∈ F, 1 ≤ ‖x₀ - y‖ 
Arguments: {𝕜 : Type u_1} [NormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖c‖ < R) {F : Subspace 𝕜 E} (hFc : IsClosed ↑F) (hF : ∃ x, x ∉ F)
Goal: ∃ x₀, ‖x₀‖ ≤ R ∧ ∀ y ∈ F, 1 ≤ ‖x₀ - y‖
-- Metric.closedBall_infDist_compl_subset_closure
theorem Metric.closedBall_infDist_compl_subset_closure {x : F} {s : Set F} (hx : x ∈ s) :
    closedBall x (infDist x sᶜ) ⊆ closure s 
Arguments: {F : Type u_3} [SeminormedAddCommGroup F] [NormedSpace ℝ F] {x : F} {s : Set F} (hx : x ∈ s)
Goal: Metric.closedBall x (Metric.infDist x sᶜ) ⊆ closure s
-- Finset.sum_indicator_mod
lemma Finset.sum_indicator_mod {R : Type*} [AddCommMonoid R] (m : ℕ) [NeZero m] (f : ℕ → R) :
    f = ∑ a : ZMod m, {n : ℕ | (n : ZMod m) = a}.indicator f 
Arguments: {R : Type u_1} [AddCommMonoid R] (m : ℕ) [NeZero m] (f : ℕ → R)
Goal: f = ∑ a : ZMod m, {n | ↑n = a}.indicator f
-- summable_indicator_mod_iff_summable
lemma summable_indicator_mod_iff_summable {R : Type*} [AddCommGroup R] [TopologicalSpace R]
    [TopologicalAddGroup R] (m : ℕ) [hm : NeZero m] (k : ℕ) (f : ℕ → R) :
    Summable ({n : ℕ | (n : ZMod m) = k}.indicator f) ↔ Summable fun n ↦ f (m * n + k) 
Arguments: {R : Type u_1} [AddCommGroup R] [TopologicalSpace R] [TopologicalAddGroup R] (m : ℕ) [hm : NeZero m] (k : ℕ) (f : ℕ → R)
Goal: Summable ({n | ↑n = ↑k}.indicator f) ↔ Summable fun n => f (m * n + k)
-- not_summable_of_antitone_of_neg
lemma not_summable_of_antitone_of_neg {f : ℕ → ℝ} (hf : Antitone f) {n : ℕ} (hn : f n < 0) :
    ¬ Summable f 
Arguments: {f : ℕ → ℝ} (hf : Antitone f) {n : ℕ} (hn : f n < 0)
Goal: ¬Summable f
-- not_summable_indicator_mod_of_antitone_of_neg
lemma not_summable_indicator_mod_of_antitone_of_neg {m : ℕ} [hm : NeZero m] {f : ℕ → ℝ}
    (hf : Antitone f) {n : ℕ} (hn : f n < 0) (k : ZMod m) :
    ¬ Summable ({n : ℕ | (n : ZMod m) = k}.indicator f) 
Arguments: {m : ℕ} [hm : NeZero m] {f : ℕ → ℝ} (hf : Antitone f) {n : ℕ} (hn : f n < 0) (k : ZMod m)
Goal: ¬Summable ({n | ↑n = k}.indicator f)
-- summable_indicator_mod_iff_summable_indicator_mod
lemma summable_indicator_mod_iff_summable_indicator_mod {m : ℕ} [NeZero m] {f : ℕ → ℝ}
    (hf : Antitone f) {k : ZMod m} (l : ZMod m)
    (hs : Summable ({n : ℕ | (n : ZMod m) = k}.indicator f)) :
    Summable ({n : ℕ | (n : ZMod m) = l}.indicator f) 
Arguments: {m : ℕ} [NeZero m] {f : ℕ → ℝ} (hf : Antitone f) {k : ZMod m} (l : ZMod m) (hs : Summable ({n | ↑n = k}.indicator f))
Goal: Summable ({n | ↑n = l}.indicator f)
-- summable_indicator_mod_iff
lemma summable_indicator_mod_iff {m : ℕ} [NeZero m] {f : ℕ → ℝ} (hf : Antitone f) (k : ZMod m) :
    Summable ({n : ℕ | (n : ZMod m) = k}.indicator f) ↔ Summable f 
Arguments: {m : ℕ} [NeZero m] {f : ℕ → ℝ} (hf : Antitone f) (k : ZMod m)
Goal: Summable ({n | ↑n = k}.indicator f) ↔ Summable f
-- intervalIntegral.intervalIntegrable_pow
@[simp]
theorem intervalIntegrable_pow : IntervalIntegrable (fun x => x ^ n) μ a b 
Arguments: {a b : ℝ} (n : ℕ) {μ : MeasureTheory.Measure ℝ} [MeasureTheory.IsLocallyFiniteMeasure μ]
Goal: IntervalIntegrable (fun x => x ^ n) μ a b
-- intervalIntegral.intervalIntegrable_zpow
theorem intervalIntegrable_zpow {n : ℤ} (h : 0 ≤ n ∨ (0 : ℝ) ∉ [[a, b]]) :
    IntervalIntegrable (fun x => x ^ n) μ a b 
Arguments: {a b : ℝ} {μ : MeasureTheory.Measure ℝ} [MeasureTheory.IsLocallyFiniteMeasure μ] {n : ℤ} (h : 0 ≤ n ∨ 0 ∉ [[a, b]])
Goal: IntervalIntegrable (fun x => x ^ n) μ a b
-- intervalIntegral.intervalIntegrable_rpow
theorem intervalIntegrable_rpow {r : ℝ} (h : 0 ≤ r ∨ (0 : ℝ) ∉ [[a, b]]) :
    IntervalIntegrable (fun x => x ^ r) μ a b 
Arguments: {a b : ℝ} {μ : MeasureTheory.Measure ℝ} [MeasureTheory.IsLocallyFiniteMeasure μ] {r : ℝ} (h : 0 ≤ r ∨ 0 ∉ [[a, b]])
Goal: IntervalIntegrable (fun x => x ^ r) μ a b
-- intervalIntegral.intervalIntegrable_rpow'
theorem intervalIntegrable_rpow' {r : ℝ} (h : -1 < r) :
    IntervalIntegrable (fun x => x ^ r) volume a b 
Arguments: {a b r : ℝ} (h : -1 < r)
Goal: IntervalIntegrable (fun x => x ^ r) MeasureTheory.volume a b
-- intervalIntegral.integrableOn_Ioo_rpow_iff
lemma integrableOn_Ioo_rpow_iff {s t : ℝ} (ht : 0 < t) :
    IntegrableOn (fun x ↦ x ^ s) (Ioo (0 : ℝ) t) ↔ -1 < s 
Arguments: {s t : ℝ} (ht : 0 < t)
Goal: MeasureTheory.IntegrableOn (fun x => x ^ s) (Set.Ioo 0 t) MeasureTheory.volume ↔ -1 < s
-- intervalIntegral.intervalIntegrable_cpow
theorem intervalIntegrable_cpow {r : ℂ} (h : 0 ≤ r.re ∨ (0 : ℝ) ∉ [[a, b]]) :
    IntervalIntegrable (fun x : ℝ => (x : ℂ) ^ r) μ a b 
Arguments: {a b : ℝ} {μ : MeasureTheory.Measure ℝ} [MeasureTheory.IsLocallyFiniteMeasure μ] {r : ℂ} (h : 0 ≤ r.re ∨ 0 ∉ [[a, b]])
Goal: IntervalIntegrable (fun x => ↑x ^ r) μ a b
-- intervalIntegral.intervalIntegrable_cpow'
theorem intervalIntegrable_cpow' {r : ℂ} (h : -1 < r.re) :
    IntervalIntegrable (fun x : ℝ => (x : ℂ) ^ r) volume a b 
Arguments: {a b : ℝ} {r : ℂ} (h : -1 < r.re)
Goal: IntervalIntegrable (fun x => ↑x ^ r) MeasureTheory.volume a b
-- intervalIntegral.integrableOn_Ioo_cpow_iff
theorem integrableOn_Ioo_cpow_iff {s : ℂ} {t : ℝ} (ht : 0 < t) :
    IntegrableOn (fun x : ℝ ↦ (x : ℂ) ^ s) (Ioo (0 : ℝ) t) ↔ -1 < s.re 
Arguments: {s : ℂ} {t : ℝ} (ht : 0 < t)
Goal: MeasureTheory.IntegrableOn (fun x => ↑x ^ s) (Set.Ioo 0 t) MeasureTheory.volume ↔ -1 < s.re
-- intervalIntegral.intervalIntegrable_id
@[simp]
theorem intervalIntegrable_id : IntervalIntegrable (fun x => x) μ a b 
Arguments: {a b : ℝ} {μ : MeasureTheory.Measure ℝ} [MeasureTheory.IsLocallyFiniteMeasure μ]
Goal: IntervalIntegrable (fun x => x) μ a b
-- intervalIntegral.intervalIntegrable_const
theorem intervalIntegrable_const : IntervalIntegrable (fun _ => c) μ a b 
Arguments: {a b : ℝ} {μ : MeasureTheory.Measure ℝ} [MeasureTheory.IsLocallyFiniteMeasure μ] (c : ℝ)
Goal: IntervalIntegrable (fun x => c) μ a b
-- intervalIntegral.intervalIntegrable_one_div
theorem intervalIntegrable_one_div (h : ∀ x : ℝ, x ∈ [[a, b]] → f x ≠ 0)
    (hf : ContinuousOn f [[a, b]]) : IntervalIntegrable (fun x => 1 / f x) μ a b 
Arguments: {a b : ℝ} {f : ℝ → ℝ} {μ : MeasureTheory.Measure ℝ} [MeasureTheory.IsLocallyFiniteMeasure μ] (h : ∀ x ∈ [[a, b]], f x ≠ 0) (hf : ContinuousOn f [[a, b]])
Goal: IntervalIntegrable (fun x => 1 / f x) μ a b
-- intervalIntegral.intervalIntegrable_inv
@[simp]
theorem intervalIntegrable_inv (h : ∀ x : ℝ, x ∈ [[a, b]] → f x ≠ 0)
    (hf : ContinuousOn f [[a, b]]) : IntervalIntegrable (fun x => (f x)⁻¹) μ a b 
Arguments: {a b : ℝ} {f : ℝ → ℝ} {μ : MeasureTheory.Measure ℝ} [MeasureTheory.IsLocallyFiniteMeasure μ] (h : ∀ x ∈ [[a, b]], f x ≠ 0) (hf : ContinuousOn f [[a, b]])
Goal: IntervalIntegrable (fun x => (f x)⁻¹) μ a b
-- intervalIntegral.intervalIntegrable_exp
@[simp]
theorem intervalIntegrable_exp : IntervalIntegrable exp μ a b 
Arguments: {a b : ℝ} {μ : MeasureTheory.Measure ℝ} [MeasureTheory.IsLocallyFiniteMeasure μ]
Goal: IntervalIntegrable rexp μ a b
-- IntervalIntegrable.log
@[simp]
theorem _root_.IntervalIntegrable.log (hf : ContinuousOn f [[a, b]])
    (h : ∀ x : ℝ, x ∈ [[a, b]] → f x ≠ 0) :
    IntervalIntegrable (fun x => log (f x)) μ a b 
Arguments: {a b : ℝ} {f : ℝ → ℝ} {μ : MeasureTheory.Measure ℝ} [MeasureTheory.IsLocallyFiniteMeasure μ] (hf : ContinuousOn f [[a, b]]) (h : ∀ x ∈ [[a, b]], f x ≠ 0)
Goal: IntervalIntegrable (fun x => Real.log (f x)) μ a b
-- intervalIntegral.intervalIntegrable_log
@[simp]
theorem intervalIntegrable_log (h : (0 : ℝ) ∉ [[a, b]]) : IntervalIntegrable log μ a b 
Arguments: {a b : ℝ} {μ : MeasureTheory.Measure ℝ} [MeasureTheory.IsLocallyFiniteMeasure μ] (h : 0 ∉ [[a, b]])
Goal: IntervalIntegrable Real.log μ a b
-- intervalIntegral.intervalIntegrable_sin
@[simp]
theorem intervalIntegrable_sin : IntervalIntegrable sin μ a b 
Arguments: {a b : ℝ} {μ : MeasureTheory.Measure ℝ} [MeasureTheory.IsLocallyFiniteMeasure μ]
Goal: IntervalIntegrable Real.sin μ a b
-- intervalIntegral.intervalIntegrable_cos
@[simp]
theorem intervalIntegrable_cos : IntervalIntegrable cos μ a b 
Arguments: {a b : ℝ} {μ : MeasureTheory.Measure ℝ} [MeasureTheory.IsLocallyFiniteMeasure μ]
Goal: IntervalIntegrable Real.cos μ a b
-- intervalIntegral.intervalIntegrable_one_div_one_add_sq
theorem intervalIntegrable_one_div_one_add_sq :
    IntervalIntegrable (fun x : ℝ => 1 / (↑1 + x ^ 2)) μ a b 
Arguments: {a b : ℝ} {μ : MeasureTheory.Measure ℝ} [MeasureTheory.IsLocallyFiniteMeasure μ]
Goal: IntervalIntegrable (fun x => 1 / (1 + x ^ 2)) μ a b
-- intervalIntegral.intervalIntegrable_inv_one_add_sq
@[simp]
theorem intervalIntegrable_inv_one_add_sq :
    IntervalIntegrable (fun x : ℝ => (↑1 + x ^ 2)⁻¹) μ a b 
Arguments: {a b : ℝ} {μ : MeasureTheory.Measure ℝ} [MeasureTheory.IsLocallyFiniteMeasure μ]
Goal: IntervalIntegrable (fun x => (1 + x ^ 2)⁻¹) μ a b
-- intervalIntegral.mul_integral_comp_mul_right
@[simp]
theorem mul_integral_comp_mul_right : (c * ∫ x in a..b, f (x * c)) = ∫ x in a * c..b * c, f x 
Arguments: {a b : ℝ} {f : ℝ → ℝ} (c : ℝ)
Goal: c * ∫ (x : ℝ) in a..b, f (x * c) = ∫ (x : ℝ) in a * c..b * c, f x
-- intervalIntegral.mul_integral_comp_mul_left
@[simp]
theorem mul_integral_comp_mul_left : (c * ∫ x in a..b, f (c * x)) = ∫ x in c * a..c * b, f x 
Arguments: {a b : ℝ} {f : ℝ → ℝ} (c : ℝ)
Goal: c * ∫ (x : ℝ) in a..b, f (c * x) = ∫ (x : ℝ) in c * a..c * b, f x