ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousAffineMap.neg_contLinear
@[simp]
theorem neg_contLinear (f : P →ᴬ[R] W) : (-f).contLinear = -f.contLinear 
Arguments: {R : Type u_2} {V : Type u_3} {W : Type u_4} {P : Type u_6} [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P] [NormedAddCommGroup W] [NormedField R] [NormedSpace R V] [NormedSpace R W] (f : P →ᴬ[R] W)
Goal: (-f).contLinear = -f.contLinear
-- ContinuousAffineMap.smul_contLinear
@[simp]
theorem smul_contLinear (t : R) (f : P →ᴬ[R] W) : (t • f).contLinear = t • f.contLinear 
Arguments: {R : Type u_2} {V : Type u_3} {W : Type u_4} {P : Type u_6} [NormedAddCommGroup V] [MetricSpace P] [NormedAddTorsor V P] [NormedAddCommGroup W] [NormedField R] [NormedSpace R V] [NormedSpace R W] (t : R) (f : P →ᴬ[R] W)
Goal: (t • f).contLinear = t • f.contLinear
-- ContinuousAffineMap.decomp
theorem decomp (f : V →ᴬ[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) 
Arguments: {R : Type u_2} {V : Type u_3} {W : Type u_4} [NormedAddCommGroup V] [NormedAddCommGroup W] [NormedField R] [NormedSpace R V] [NormedSpace R W] (f : V →ᴬ[R] W)
Goal: ⇑f = ⇑f.contLinear + Function.const V (f 0)
-- ContinuousAffineMap.norm_def
theorem norm_def : ‖f‖ = max ‖f 0‖ ‖f.contLinear‖ 
Arguments: {𝕜 : Type u_1} {V : Type u_3} {W : Type u_4} [NormedAddCommGroup V] [NormedAddCommGroup W] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] (f : V →ᴬ[𝕜] W)
Goal: ‖f‖ = max ‖f 0‖ ‖f.contLinear‖
-- ContinuousAffineMap.norm_contLinear_le
theorem norm_contLinear_le : ‖f.contLinear‖ ≤ ‖f‖ 
Arguments: {𝕜 : Type u_1} {V : Type u_3} {W : Type u_4} [NormedAddCommGroup V] [NormedAddCommGroup W] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] (f : V →ᴬ[𝕜] W)
Goal: ‖f.contLinear‖ ≤ ‖f‖
-- ContinuousAffineMap.norm_image_zero_le
theorem norm_image_zero_le : ‖f 0‖ ≤ ‖f‖ 
Arguments: {𝕜 : Type u_1} {V : Type u_3} {W : Type u_4} [NormedAddCommGroup V] [NormedAddCommGroup W] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] (f : V →ᴬ[𝕜] W)
Goal: ‖f 0‖ ≤ ‖f‖
-- ContinuousAffineMap.norm_eq
@[simp]
theorem norm_eq (h : f 0 = 0) : ‖f‖ = ‖f.contLinear‖ 
Arguments: {𝕜 : Type u_1} {V : Type u_3} {W : Type u_4} [NormedAddCommGroup V] [NormedAddCommGroup W] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] (f : V →ᴬ[𝕜] W) (h : f 0 = 0)
Goal: ‖f‖ = ‖f.contLinear‖
-- ContinuousAffineMap.norm_comp_le
theorem norm_comp_le (g : W₂ →ᴬ[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ 
Arguments: {𝕜 : Type u_1} {V : Type u_3} {W : Type u_4} {W₂ : Type u_5} [NormedAddCommGroup V] [NormedAddCommGroup W] [NormedAddCommGroup W₂] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [NormedSpace 𝕜 W₂] (f : V →ᴬ[𝕜] W) (g : W₂ →ᴬ[𝕜] V)
Goal: ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖
-- ContinuousAffineMap.toConstProdContinuousLinearMap_fst
@[simp]
theorem toConstProdContinuousLinearMap_fst (f : V →ᴬ[𝕜] W) :
    (toConstProdContinuousLinearMap 𝕜 V W f).fst = f 0 
Arguments: (𝕜 : Type u_1) (V : Type u_3) (W : Type u_4) [NormedAddCommGroup V] [NormedAddCommGroup W] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] (f : V →ᴬ[𝕜] W)
Goal: ((ContinuousAffineMap.toConstProdContinuousLinearMap 𝕜 V W) f).1 = f 0
-- ContinuousAffineMap.toConstProdContinuousLinearMap_snd
@[simp]
theorem toConstProdContinuousLinearMap_snd (f : V →ᴬ[𝕜] W) :
    (toConstProdContinuousLinearMap 𝕜 V W f).snd = f.contLinear 
Arguments: (𝕜 : Type u_1) (V : Type u_3) (W : Type u_4) [NormedAddCommGroup V] [NormedAddCommGroup W] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] (f : V →ᴬ[𝕜] W)
Goal: ((ContinuousAffineMap.toConstProdContinuousLinearMap 𝕜 V W) f).2 = f.contLinear
-- LinearMap.extendTo𝕜'_apply
theorem extendTo𝕜'_apply (fr : F →ₗ[ℝ] ℝ) (x : F) :
    fr.extendTo𝕜' x = (fr x : 𝕜) - (I : 𝕜) * (fr ((I : 𝕜) • x) : 𝕜) 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {F : Type u_2} [SeminormedAddCommGroup F] [NormedSpace 𝕜 F] [Module ℝ F] [IsScalarTower ℝ 𝕜 F] (fr : F →ₗ[ℝ] ℝ) (x : F)
Goal: fr.extendTo𝕜' x = ↑(fr x) - RCLike.I * ↑(fr (RCLike.I • x))
-- LinearMap.extendTo𝕜'_apply_re
@[simp]
theorem extendTo𝕜'_apply_re (fr : F →ₗ[ℝ] ℝ) (x : F) : re (fr.extendTo𝕜' x : 𝕜) = fr x 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {F : Type u_2} [SeminormedAddCommGroup F] [NormedSpace 𝕜 F] [Module ℝ F] [IsScalarTower ℝ 𝕜 F] (fr : F →ₗ[ℝ] ℝ) (x : F)
Goal: RCLike.re (fr.extendTo𝕜' x) = fr x
-- LinearMap.norm_extendTo𝕜'_apply_sq
theorem norm_extendTo𝕜'_apply_sq (fr : F →ₗ[ℝ] ℝ) (x : F) :
    ‖(fr.extendTo𝕜' x : 𝕜)‖ ^ 2 = fr (conj (fr.extendTo𝕜' x : 𝕜) • x) 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {F : Type u_2} [SeminormedAddCommGroup F] [NormedSpace 𝕜 F] [Module ℝ F] [IsScalarTower ℝ 𝕜 F] (fr : F →ₗ[ℝ] ℝ) (x : F)
Goal: ‖fr.extendTo𝕜' x‖ ^ 2 = fr ((starRingEnd 𝕜) (fr.extendTo𝕜' x) • x)
-- ContinuousLinearMap.norm_extendTo𝕜'_bound
theorem norm_extendTo𝕜'_bound (fr : F →L[ℝ] ℝ) (x : F) :
    ‖(fr.toLinearMap.extendTo𝕜' x : 𝕜)‖ ≤ ‖fr‖ * ‖x‖ 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {F : Type u_2} [SeminormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedSpace ℝ F] [IsScalarTower ℝ 𝕜 F] (fr : F →L[ℝ] ℝ) (x : F)
Goal: ‖(↑fr).extendTo𝕜' x‖ ≤ ‖fr‖ * ‖x‖
-- ContinuousLinearMap.extendTo𝕜'_apply
theorem extendTo𝕜'_apply (fr : F →L[ℝ] ℝ) (x : F) :
    fr.extendTo𝕜' x = (fr x : 𝕜) - (I : 𝕜) * (fr ((I : 𝕜) • x) : 𝕜) 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {F : Type u_2} [SeminormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedSpace ℝ F] [IsScalarTower ℝ 𝕜 F] (fr : F →L[ℝ] ℝ) (x : F)
Goal: fr.extendTo𝕜' x = ↑(fr x) - RCLike.I * ↑(fr (RCLike.I • x))
-- ContinuousLinearMap.norm_extendTo𝕜'
@[simp]
theorem norm_extendTo𝕜' (fr : F →L[ℝ] ℝ) : ‖(fr.extendTo𝕜' : F →L[𝕜] 𝕜)‖ = ‖fr‖ 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {F : Type u_2} [SeminormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedSpace ℝ F] [IsScalarTower ℝ 𝕜 F] (fr : F →L[ℝ] ℝ)
Goal: ‖fr.extendTo𝕜'‖ = ‖fr‖
-- LinearMap.extendTo𝕜_apply
theorem LinearMap.extendTo𝕜_apply (fr : RestrictScalars ℝ 𝕜 F →ₗ[ℝ] ℝ) (x : F) :
    fr.extendTo𝕜 x = (fr x : 𝕜) - (I : 𝕜) * (fr ((I : 𝕜) • x) : 𝕜) 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {F : Type u_2} [SeminormedAddCommGroup F] [NormedSpace 𝕜 F] (fr : RestrictScalars ℝ 𝕜 F →ₗ[ℝ] ℝ) (x : F)
Goal: fr.extendTo𝕜 x = ↑(fr x) - RCLike.I * ↑(fr (RCLike.I • x))
-- ContinuousLinearMap.extendTo𝕜_apply
theorem ContinuousLinearMap.extendTo𝕜_apply (fr : RestrictScalars ℝ 𝕜 F →L[ℝ] ℝ) (x : F) :
    fr.extendTo𝕜 x = (fr x : 𝕜) - (I : 𝕜) * (fr ((I : 𝕜) • x) : 𝕜) 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {F : Type u_2} [SeminormedAddCommGroup F] [NormedSpace 𝕜 F] (fr : RestrictScalars ℝ 𝕜 F →L[ℝ] ℝ) (x : F)
Goal: fr.extendTo𝕜 x = ↑(fr x) - RCLike.I * ↑(fr (RCLike.I • x))
-- ContinuousLinearMap.norm_extendTo𝕜
@[simp]
theorem ContinuousLinearMap.norm_extendTo𝕜 (fr : RestrictScalars ℝ 𝕜 F →L[ℝ] ℝ) :
    ‖fr.extendTo𝕜‖ = ‖fr‖ 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {F : Type u_2} [SeminormedAddCommGroup F] [NormedSpace 𝕜 F] (fr : RestrictScalars ℝ 𝕜 F →L[ℝ] ℝ)
Goal: ‖fr.extendTo𝕜‖ = ‖fr‖
-- ZMod.dft_apply
lemma dft_apply (Φ : ZMod N → ℂ) (k : ZMod N) :
    𝓕 Φ k = ∑ j : ZMod N, toCircle (-(j * k)) • Φ j 
Arguments: {N : ℕ} [NeZero N] (Φ : ZMod N → ℂ) (k : ZMod N)
Goal: 𝓕 Φ k = ∑ j : ZMod N, ZMod.toCircle (-(j * k)) • Φ j
-- ZMod.dft_def
lemma dft_def (Φ : ZMod N → ℂ) : 𝓕 Φ = fun k ↦ ∑ j : ZMod N, toCircle (-(j * k)) • Φ j 
Arguments: {N : ℕ} [NeZero N] (Φ : ZMod N → ℂ)
Goal: 𝓕 Φ = fun k => ∑ j : ZMod N, ZMod.toCircle (-(j * k)) • Φ j
-- DirichletCharacter.fourierTransform_eq_gaussSum_mulShift
lemma fourierTransform_eq_gaussSum_mulShift (k : ZMod N) :
    𝓕 χ k = gaussSum χ (stdAddChar.mulShift (-k)) 
Arguments: {N : ℕ} [NeZero N] (χ : DirichletCharacter ℂ N) (k : ZMod N)
Goal: 𝓕 (⇑χ) k = gaussSum χ (ZMod.stdAddChar.mulShift (-k))
-- DirichletCharacter.fourierTransform_eq_inv_mul_gaussSum
lemma fourierTransform_eq_inv_mul_gaussSum (k : ZMod N) (hχ : IsPrimitive χ) :
    𝓕 χ k = χ⁻¹ (-k) * gaussSum χ stdAddChar 
Arguments: {N : ℕ} [NeZero N] (χ : DirichletCharacter ℂ N) (k : ZMod N) (hχ : χ.IsPrimitive)
Goal: 𝓕 (⇑χ) k = χ⁻¹ (-k) * gaussSum χ ZMod.stdAddChar
-- Quaternion.exp_coe
@[simp, norm_cast]
theorem exp_coe (r : ℝ) : exp ℝ (r : ℍ[ℝ]) = ↑(exp ℝ r) 
Arguments: (r : ℝ)
Goal: NormedSpace.exp ℝ ↑r = ↑(NormedSpace.exp ℝ r)
-- Quaternion.expSeries_even_of_imaginary
theorem expSeries_even_of_imaginary {q : Quaternion ℝ} (hq : q.re = 0) (n : ℕ) :
    expSeries ℝ (Quaternion ℝ) (2 * n) (fun _ => q) =
      ↑((-1 : ℝ) ^ n * ‖q‖ ^ (2 * n) / (2 * n)!) 
Arguments: {q : ℍ} (hq : q.re = 0) (n : ℕ)
Goal: ((NormedSpace.expSeries ℝ ℍ (2 * n)) fun x => q) = ↑((-1) ^ n * ‖q‖ ^ (2 * n) / ↑(2 * n)!)
-- Quaternion.expSeries_odd_of_imaginary
theorem expSeries_odd_of_imaginary {q : Quaternion ℝ} (hq : q.re = 0) (n : ℕ) :
    expSeries ℝ (Quaternion ℝ) (2 * n + 1) (fun _ => q) =
      (((-1 : ℝ) ^ n * ‖q‖ ^ (2 * n + 1) / (2 * n + 1)!) / ‖q‖) • q 
Arguments: {q : ℍ} (hq : q.re = 0) (n : ℕ)
Goal: ((NormedSpace.expSeries ℝ ℍ (2 * n + 1)) fun x => q) = ((-1) ^ n * ‖q‖ ^ (2 * n + 1) / ↑(2 * n + 1)! / ‖q‖) • q
-- Quaternion.hasSum_expSeries_of_imaginary
theorem hasSum_expSeries_of_imaginary {q : Quaternion ℝ} (hq : q.re = 0) {c s : ℝ}
    (hc : HasSum (fun n => (-1 : ℝ) ^ n * ‖q‖ ^ (2 * n) / (2 * n)!) c)
    (hs : HasSum (fun n => (-1 : ℝ) ^ n * ‖q‖ ^ (2 * n + 1) / (2 * n + 1)!) s) :
    HasSum (fun n => expSeries ℝ (Quaternion ℝ) n fun _ => q) (↑c + (s / ‖q‖) • q) 
Arguments: {q : ℍ} (hq : q.re = 0) {c s : ℝ} (hc : HasSum (fun n => (-1) ^ n * ‖q‖ ^ (2 * n) / ↑(2 * n)!) c) (hs : HasSum (fun n => (-1) ^ n * ‖q‖ ^ (2 * n + 1) / ↑(2 * n + 1)!) s)
Goal: HasSum (fun n => (NormedSpace.expSeries ℝ ℍ n) fun x => q) (↑c + (s / ‖q‖) • q)
-- Quaternion.exp_of_re_eq_zero
theorem exp_of_re_eq_zero (q : Quaternion ℝ) (hq : q.re = 0) :
    exp ℝ q = ↑(Real.cos ‖q‖) + (Real.sin ‖q‖ / ‖q‖) • q 
Arguments: (q : ℍ) (hq : q.re = 0)
Goal: NormedSpace.exp ℝ q = ↑(Real.cos ‖q‖) + (Real.sin ‖q‖ / ‖q‖) • q
-- Quaternion.exp_eq
theorem exp_eq (q : Quaternion ℝ) :
    exp ℝ q = exp ℝ q.re • (↑(Real.cos ‖q.im‖) + (Real.sin ‖q.im‖ / ‖q.im‖) • q.im) 
Arguments: (q : ℍ)
Goal: NormedSpace.exp ℝ q = NormedSpace.exp ℝ q.re • (↑(Real.cos ‖q.im‖) + (Real.sin ‖q.im‖ / ‖q.im‖) • q.im)
-- Quaternion.re_exp
theorem re_exp (q : ℍ[ℝ]) : (exp ℝ q).re = exp ℝ q.re * Real.cos ‖q - q.re‖ 
Arguments: (q : ℍ)
Goal: (NormedSpace.exp ℝ q).re = NormedSpace.exp ℝ q.re * Real.cos ‖q - ↑q.re‖
-- Quaternion.im_exp
theorem im_exp (q : ℍ[ℝ]) : (exp ℝ q).im = (exp ℝ q.re * (Real.sin ‖q.im‖ / ‖q.im‖)) • q.im 
Arguments: (q : ℍ)
Goal: (NormedSpace.exp ℝ q).im = (NormedSpace.exp ℝ q.re * (Real.sin ‖q.im‖ / ‖q.im‖)) • q.im
-- Quaternion.normSq_exp
theorem normSq_exp (q : ℍ[ℝ]) : normSq (exp ℝ q) = exp ℝ q.re ^ 2 
Arguments: (q : ℍ)
Goal: Quaternion.normSq (NormedSpace.exp ℝ q) = NormedSpace.exp ℝ q.re ^ 2
-- Quaternion.norm_exp
@[simp]
theorem norm_exp (q : ℍ[ℝ]) : ‖exp ℝ q‖ = ‖exp ℝ q.re‖ 
Arguments: (q : ℍ)
Goal: ‖NormedSpace.exp ℝ q‖ = ‖NormedSpace.exp ℝ q.re‖
-- UniformConvergenceCLM.locallyConvexSpace
theorem locallyConvexSpace (𝔖 : Set (Set E)) (h𝔖₁ : 𝔖.Nonempty)
    (h𝔖₂ : DirectedOn (· ⊆ ·) 𝔖) :
    LocallyConvexSpace R (UniformConvergenceCLM σ F 𝔖) 
Arguments: (R : Type u_1) {𝕜₁ : Type u_2} {𝕜₂ : Type u_3} {E : Type u_4} {F : Type u_5} [AddCommGroup E] [TopologicalSpace E] [AddCommGroup F] [TopologicalSpace F] [TopologicalAddGroup F] [OrderedSemiring R] [NormedField 𝕜₁] [NormedField 𝕜₂] [Module 𝕜₁ E] [Module 𝕜₂ F] {σ : 𝕜₁ →+* 𝕜₂} [Module R F] [ContinuousConstSMul R F] [LocallyConvexSpace R F] [SMulCommClass 𝕜₂ R F] (𝔖 : Set (Set E)) (h𝔖₁ : 𝔖.Nonempty) (h𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖)
Goal: LocallyConvexSpace R (UniformConvergenceCLM σ F 𝔖)
-- oscillationWithin_nhd_eq_oscillation
theorem oscillationWithin_nhd_eq_oscillation [TopologicalSpace E] (f : E → F) (D : Set E) (x : E)
    (hD : D ∈ 𝓝 x) : oscillationWithin f D x = oscillation f x 
Arguments: {E : Type u} {F : Type v} [PseudoEMetricSpace F] [TopologicalSpace E] (f : E → F) (D : Set E) (x : E) (hD : D ∈ 𝓝 x)
Goal: oscillationWithin f D x = oscillation f x
-- oscillationWithin_univ_eq_oscillation
theorem oscillationWithin_univ_eq_oscillation [TopologicalSpace E] (f : E → F) (x : E) :
    oscillationWithin f univ x = oscillation f x 
Arguments: {E : Type u} {F : Type v} [PseudoEMetricSpace F] [TopologicalSpace E] (f : E → F) (x : E)
Goal: oscillationWithin f Set.univ x = oscillation f x
-- ContinuousWithinAt.oscillationWithin_eq_zero
theorem oscillationWithin_eq_zero [TopologicalSpace E] {f : E → F} {D : Set E}
    {x : E} (hf : ContinuousWithinAt f D x) : oscillationWithin f D x = 0 
Arguments: {E : Type u} {F : Type v} [PseudoEMetricSpace F] [TopologicalSpace E] {f : E → F} {D : Set E} {x : E} (hf : ContinuousWithinAt f D x)
Goal: oscillationWithin f D x = 0
-- ContinuousAt.oscillation_eq_zero
theorem oscillation_eq_zero [TopologicalSpace E] {f : E → F} {x : E} (hf : ContinuousAt f x) :
    oscillation f x = 0 
Arguments: {E : Type u} {F : Type v} [PseudoEMetricSpace F] [TopologicalSpace E] {f : E → F} {x : E} (hf : ContinuousAt f x)
Goal: oscillation f x = 0
-- OscillationWithin.eq_zero_iff_continuousWithinAt
theorem eq_zero_iff_continuousWithinAt [TopologicalSpace E] (f : E → F) {D : Set E}
    {x : E} (xD : x ∈ D) : oscillationWithin f D x = 0 ↔ ContinuousWithinAt f D x 
Arguments: {E : Type u} {F : Type v} [PseudoEMetricSpace F] [TopologicalSpace E] (f : E → F) {D : Set E} {x : E} (xD : x ∈ D)
Goal: oscillationWithin f D x = 0 ↔ ContinuousWithinAt f D x
-- Oscillation.eq_zero_iff_continuousAt
theorem eq_zero_iff_continuousAt [TopologicalSpace E] (f : E → F) (x : E) :
    oscillation f x = 0 ↔ ContinuousAt f x 
Arguments: {E : Type u} {F : Type v} [PseudoEMetricSpace F] [TopologicalSpace E] (f : E → F) (x : E)
Goal: oscillation f x = 0 ↔ ContinuousAt f x
-- IsCompact.uniform_oscillationWithin
theorem uniform_oscillationWithin (hK : ∀ x ∈ K, oscillationWithin f D x < ε) :
    ∃ δ > 0, ∀ x ∈ K, diam (f '' (ball x (ENNReal.ofReal δ) ∩ D)) ≤ ε 
Arguments: {E : Type u} {F : Type v} [PseudoEMetricSpace F] [PseudoEMetricSpace E] {K : Set E} (comp : IsCompact K) {f : E → F} {D : Set E} {ε : ℝ≥0∞} (hK : ∀ x ∈ K, oscillationWithin f D x < ε)
Goal: ∃ δ > 0, ∀ x ∈ K, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal δ) ∩ D)) ≤ ε
-- IsCompact.uniform_oscillation
theorem uniform_oscillation [PseudoEMetricSpace E] {K : Set E} (comp : IsCompact K)
    {f : E → F} {ε : ENNReal} (hK : ∀ x ∈ K, oscillation f x < ε) :
    ∃ δ > 0, ∀ x ∈ K, diam (f '' (ball x (ENNReal.ofReal δ))) ≤ ε 
Arguments: {E : Type u} {F : Type v} [PseudoEMetricSpace F] [PseudoEMetricSpace E] {K : Set E} (comp : IsCompact K) {f : E → F} {ε : ℝ≥0∞} (hK : ∀ x ∈ K, oscillation f x < ε)
Goal: ∃ δ > 0, ∀ x ∈ K, EMetric.diam (f '' EMetric.ball x (ENNReal.ofReal δ)) ≤ ε
-- BarrelledSpace.continuous_of_lowerSemicontinuous
class BarrelledSpace (𝕜 E : Type*) [SeminormedRing 𝕜] [AddGroup E] [SMul 𝕜 E]
    [TopologicalSpace E] : Prop where
    continuous_of_lowerSemicontinuous : ∀ p : Seminorm 𝕜 E, LowerSemicontinuous p → Continuous p
Arguments: {𝕜 : Type u_1} {E : Type u_2} [SeminormedRing 𝕜] [AddGroup E] [SMul 𝕜 E] [TopologicalSpace E] [self : BarrelledSpace 𝕜 E] (p : Seminorm 𝕜 E)
Goal: LowerSemicontinuous ⇑p → Continuous ⇑p
-- Seminorm.continuous_of_lowerSemicontinuous
theorem Seminorm.continuous_of_lowerSemicontinuous {𝕜 E : Type*} [AddGroup E] [SMul 𝕜 E]
    [SeminormedRing 𝕜] [TopologicalSpace E] [BarrelledSpace 𝕜 E] (p : Seminorm 𝕜 E)
    (hp : LowerSemicontinuous p) : Continuous p 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [AddGroup E] [SMul 𝕜 E] [SeminormedRing 𝕜] [TopologicalSpace E] [BarrelledSpace 𝕜 E] (p : Seminorm 𝕜 E) (hp : LowerSemicontinuous ⇑p)
Goal: Continuous ⇑p
-- Seminorm.continuous_iSup
theorem Seminorm.continuous_iSup
    {ι : Sort*} {𝕜 E : Type*} [NormedField 𝕜]  [AddCommGroup E] [Module 𝕜 E]
    [TopologicalSpace E] [BarrelledSpace 𝕜 E] (p : ι → Seminorm 𝕜 E)
    (hp : ∀ i, Continuous (p i)) (bdd : BddAbove (range p)) :
    Continuous (⨆ i, p i) 
Arguments: {ι : Sort u_1} {𝕜 : Type u_2} {E : Type u_3} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [BarrelledSpace 𝕜 E] (p : ι → Seminorm 𝕜 E) (hp : ∀ (i : ι), Continuous ⇑(p i)) (bdd : BddAbove (Set.range p))
Goal: Continuous (⨆ i, ⇑(p i))
-- WithSeminorms.banach_steinhaus
protected theorem banach_steinhaus (H : ∀ k x, BddAbove (range fun i ↦ q k (𝓕 i x))) :
    UniformEquicontinuous ((↑) ∘ 𝓕) 
Arguments: {ι : Type u_2} {κ : Type u_3} {𝕜₁ : Type u_4} {𝕜₂ : Type u_5} {E : Type u_6} {F : Type u_7} [Nonempty κ] [NontriviallyNormedField 𝕜₁] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} [RingHomIsometric σ₁₂] [AddCommGroup E] [AddCommGroup F] [Module 𝕜₁ E] [Module 𝕜₂ F] [UniformSpace E] [UniformSpace F] [UniformAddGroup E] [UniformAddGroup F] [ContinuousSMul 𝕜₁ E] [BarrelledSpace 𝕜₁ E] {𝓕 : ι → E →SL[σ₁₂] F} {q : SeminormFamily 𝕜₂ F κ} (hq : WithSeminorms q) (H : ∀ (k : κ) (x : E), BddAbove (Set.range fun i => (q k) ((𝓕 i) x)))
Goal: UniformEquicontinuous (DFunLike.coe ∘ 𝓕)
-- norm_image_of_norm_zero
theorem norm_image_of_norm_zero [SemilinearMapClass 𝓕 σ₁₂ E F] (f : 𝓕) (hf : Continuous f) {x : E}
    (hx : ‖x‖ = 0) : ‖f x‖ = 0 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} {𝓕 : Type u_10} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [FunLike 𝓕 E F] [SemilinearMapClass 𝓕 σ₁₂ E F] (f : 𝓕) (hf : Continuous ⇑f) {x : E} (hx : ‖x‖ = 0)
Goal: ‖f x‖ = 0
-- SemilinearMapClass.bound_of_shell_semi_normed
theorem SemilinearMapClass.bound_of_shell_semi_normed [SemilinearMapClass 𝓕 σ₁₂ E F] (f : 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} (hx : ‖x‖ ≠ 0) :
    ‖f x‖ ≤ C * ‖x‖ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} {𝓕 : Type u_10} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [FunLike 𝓕 E F] [RingHomIsometric σ₁₂] [SemilinearMapClass 𝓕 σ₁₂ E F] (f : 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} (hx : ‖x‖ ≠ 0)
Goal: ‖f x‖ ≤ C * ‖x‖
-- SemilinearMapClass.bound_of_continuous
theorem SemilinearMapClass.bound_of_continuous [SemilinearMapClass 𝓕 σ₁₂ E F] (f : 𝓕)
    (hf : Continuous f) : ∃ C, 0 < C ∧ ∀ x : E, ‖f x‖ ≤ C * ‖x‖ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} {𝓕 : Type u_10} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [FunLike 𝓕 E F] [RingHomIsometric σ₁₂] [SemilinearMapClass 𝓕 σ₁₂ E F] (f : 𝓕) (hf : Continuous ⇑f)
Goal: ∃ C, 0 < C ∧ ∀ (x : E), ‖f x‖ ≤ C * ‖x‖
-- ContinuousLinearMap.bound
theorem bound [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) : ∃ C, 0 < C ∧ ∀ x : E, ‖f x‖ ≤ C * ‖x‖ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F)
Goal: ∃ C, 0 < C ∧ ∀ (x : E), ‖f x‖ ≤ C * ‖x‖
-- LinearIsometry.toSpanSingleton_apply
@[simp]
theorem _root_.LinearIsometry.toSpanSingleton_apply {v : E} (hv : ‖v‖ = 1) (a : 𝕜) :
    LinearIsometry.toSpanSingleton 𝕜 E hv a = a • v 
Arguments: {𝕜 : Type u_1} {E : Type u_4} [SeminormedAddCommGroup E] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] {v : E} (hv : ‖v‖ = 1) (a : 𝕜)
Goal: (LinearIsometry.toSpanSingleton 𝕜 E hv) a = a • v
-- LinearIsometry.coe_toSpanSingleton
@[simp]
theorem _root_.LinearIsometry.coe_toSpanSingleton {v : E} (hv : ‖v‖ = 1) :
    (LinearIsometry.toSpanSingleton 𝕜 E hv).toLinearMap = LinearMap.toSpanSingleton 𝕜 E v 
Arguments: {𝕜 : Type u_1} {E : Type u_4} [SeminormedAddCommGroup E] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] {v : E} (hv : ‖v‖ = 1)
Goal: (LinearIsometry.toSpanSingleton 𝕜 E hv).toLinearMap = LinearMap.toSpanSingleton 𝕜 E v
-- ContinuousLinearMap.norm_def
theorem norm_def (f : E →SL[σ₁₂] F) : ‖f‖ = sInf { c | 0 ≤ c ∧ ∀ x, ‖f x‖ ≤ c * ‖x‖ } 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →SL[σ₁₂] F)
Goal: ‖f‖ = sInf {c | 0 ≤ c ∧ ∀ (x : E), ‖f x‖ ≤ c * ‖x‖}
-- ContinuousLinearMap.bounds_nonempty
theorem bounds_nonempty [RingHomIsometric σ₁₂] {f : E →SL[σ₁₂] F} :
    ∃ c, c ∈ { c | 0 ≤ c ∧ ∀ x, ‖f x‖ ≤ c * ‖x‖ } 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] {f : E →SL[σ₁₂] F}
Goal: ∃ c, c ∈ {c | 0 ≤ c ∧ ∀ (x : E), ‖f x‖ ≤ c * ‖x‖}
-- ContinuousLinearMap.bounds_bddBelow
theorem bounds_bddBelow {f : E →SL[σ₁₂] F} : BddBelow { c | 0 ≤ c ∧ ∀ x, ‖f x‖ ≤ c * ‖x‖ } 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {f : E →SL[σ₁₂] F}
Goal: BddBelow {c | 0 ≤ c ∧ ∀ (x : E), ‖f x‖ ≤ c * ‖x‖}
-- ContinuousLinearMap.isLeast_opNorm
theorem isLeast_opNorm [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) :
    IsLeast {c | 0 ≤ c ∧ ∀ x, ‖f x‖ ≤ c * ‖x‖} ‖f‖ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F)
Goal: IsLeast {c | 0 ≤ c ∧ ∀ (x : E), ‖f x‖ ≤ c * ‖x‖} ‖f‖
-- ContinuousLinearMap.opNorm_le_bound
theorem opNorm_le_bound (f : E →SL[σ₁₂] F) {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ x, ‖f x‖ ≤ M * ‖x‖) :
    ‖f‖ ≤ M 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →SL[σ₁₂] F) {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ (x : E), ‖f x‖ ≤ M * ‖x‖)
Goal: ‖f‖ ≤ M
-- ContinuousLinearMap.opNorm_le_bound'
theorem opNorm_le_bound' (f : E →SL[σ₁₂] F) {M : ℝ} (hMp : 0 ≤ M)
    (hM : ∀ x, ‖x‖ ≠ 0 → ‖f x‖ ≤ M * ‖x‖) : ‖f‖ ≤ M 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →SL[σ₁₂] F) {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ (x : E), ‖x‖ ≠ 0 → ‖f x‖ ≤ M * ‖x‖)
Goal: ‖f‖ ≤ M
-- ContinuousLinearMap.opNorm_le_of_lipschitz
theorem opNorm_le_of_lipschitz {f : E →SL[σ₁₂] F} {K : ℝ≥0} (hf : LipschitzWith K f) : ‖f‖ ≤ K 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {f : E →SL[σ₁₂] F} {K : ℝ≥0} (hf : LipschitzWith K ⇑f)
Goal: ‖f‖ ≤ ↑K
-- ContinuousLinearMap.opNorm_eq_of_bounds
theorem opNorm_eq_of_bounds {φ : E →SL[σ₁₂] F} {M : ℝ} (M_nonneg : 0 ≤ M)
    (h_above : ∀ x, ‖φ x‖ ≤ M * ‖x‖) (h_below : ∀ N ≥ 0, (∀ x, ‖φ x‖ ≤ N * ‖x‖) → M ≤ N) :
    ‖φ‖ = M 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {φ : E →SL[σ₁₂] F} {M : ℝ} (M_nonneg : 0 ≤ M) (h_above : ∀ (x : E), ‖φ x‖ ≤ M * ‖x‖) (h_below : ∀ N ≥ 0, (∀ (x : E), ‖φ x‖ ≤ N * ‖x‖) → M ≤ N)
Goal: ‖φ‖ = M
-- ContinuousLinearMap.opNorm_neg
theorem opNorm_neg (f : E →SL[σ₁₂] F) : ‖-f‖ = ‖f‖ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →SL[σ₁₂] F)
Goal: ‖-f‖ = ‖f‖