ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- hasDerivAt_neg'
theorem hasDerivAt_neg' : HasDerivAt (fun x => -x) (-1) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (x : 𝕜)
Goal: HasDerivAt (fun x => -x) (-1) x
-- hasStrictDerivAt_neg
theorem hasStrictDerivAt_neg : HasStrictDerivAt Neg.neg (-1) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (x : 𝕜)
Goal: HasStrictDerivAt Neg.neg (-1) x
-- deriv_neg
theorem deriv_neg : deriv Neg.neg x = -1 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (x : 𝕜)
Goal: deriv Neg.neg x = -1
-- deriv_neg'
@[simp]
theorem deriv_neg' : deriv (Neg.neg : 𝕜 → 𝕜) = fun _ => -1 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜]
Goal: deriv Neg.neg = fun x => -1
-- deriv_neg''
@[simp]
theorem deriv_neg'' : deriv (fun x : 𝕜 => -x) x = -1 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (x : 𝕜)
Goal: deriv (fun x => -x) x = -1
-- derivWithin_neg
theorem derivWithin_neg (hxs : UniqueDiffWithinAt 𝕜 s x) : derivWithin Neg.neg s x = -1 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (x : 𝕜) (s : Set 𝕜) (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: derivWithin Neg.neg s x = -1
-- differentiable_neg
theorem differentiable_neg : Differentiable 𝕜 (Neg.neg : 𝕜 → 𝕜) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜]
Goal: Differentiable 𝕜 Neg.neg
-- differentiableOn_neg
theorem differentiableOn_neg : DifferentiableOn 𝕜 (Neg.neg : 𝕜 → 𝕜) s 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (s : Set 𝕜)
Goal: DifferentiableOn 𝕜 Neg.neg s
-- not_differentiableAt_abs_zero
theorem not_differentiableAt_abs_zero : ¬ DifferentiableAt ℝ (abs : ℝ → ℝ) 0 
Goal: ¬DifferentiableAt ℝ abs 0
-- differentiableAt_comp_neg_iff
lemma differentiableAt_comp_neg_iff {a : 𝕜} :
    DifferentiableAt 𝕜 f (-a) ↔ DifferentiableAt 𝕜 (fun x ↦ f (-x)) a 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {a : 𝕜}
Goal: DifferentiableAt 𝕜 f (-a) ↔ DifferentiableAt 𝕜 (fun x => f (-x)) a
-- HasDerivAtFilter.sub
theorem HasDerivAtFilter.sub (hf : HasDerivAtFilter f f' x L) (hg : HasDerivAtFilter g g' x L) :
    HasDerivAtFilter (fun x => f x - g x) (f' - g') x L 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : 𝕜 → F} {f' g' : F} {x : 𝕜} {L : Filter 𝕜} (hf : HasDerivAtFilter f f' x L) (hg : HasDerivAtFilter g g' x L)
Goal: HasDerivAtFilter (fun x => f x - g x) (f' - g') x L
-- HasDerivWithinAt.sub
nonrec theorem HasDerivWithinAt.sub (hf : HasDerivWithinAt f f' s x)
    (hg : HasDerivWithinAt g g' s x) : HasDerivWithinAt (fun x => f x - g x) (f' - g') s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : 𝕜 → F} {f' g' : F} {x : 𝕜} {s : Set 𝕜} (hf : HasDerivWithinAt f f' s x) (hg : HasDerivWithinAt g g' s x)
Goal: HasDerivWithinAt (fun x => f x - g x) (f' - g') s x
-- HasDerivAt.sub
nonrec theorem HasDerivAt.sub (hf : HasDerivAt f f' x) (hg : HasDerivAt g g' x) :
    HasDerivAt (fun x => f x - g x) (f' - g') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : 𝕜 → F} {f' g' : F} {x : 𝕜} (hf : HasDerivAt f f' x) (hg : HasDerivAt g g' x)
Goal: HasDerivAt (fun x => f x - g x) (f' - g') x
-- HasStrictDerivAt.sub
theorem HasStrictDerivAt.sub (hf : HasStrictDerivAt f f' x) (hg : HasStrictDerivAt g g' x) :
    HasStrictDerivAt (fun x => f x - g x) (f' - g') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : 𝕜 → F} {f' g' : F} {x : 𝕜} (hf : HasStrictDerivAt f f' x) (hg : HasStrictDerivAt g g' x)
Goal: HasStrictDerivAt (fun x => f x - g x) (f' - g') x
-- derivWithin_sub
theorem derivWithin_sub (hxs : UniqueDiffWithinAt 𝕜 s x) (hf : DifferentiableWithinAt 𝕜 f s x)
    (hg : DifferentiableWithinAt 𝕜 g s x) :
    derivWithin (fun y => f y - g y) s x = derivWithin f s x - derivWithin g s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : 𝕜 → F} {x : 𝕜} {s : Set 𝕜} (hxs : UniqueDiffWithinAt 𝕜 s x) (hf : DifferentiableWithinAt 𝕜 f s x) (hg : DifferentiableWithinAt 𝕜 g s x)
Goal: derivWithin (fun y => f y - g y) s x = derivWithin f s x - derivWithin g s x
-- deriv_sub
@[simp]
theorem deriv_sub (hf : DifferentiableAt 𝕜 f x) (hg : DifferentiableAt 𝕜 g x) :
    deriv (fun y => f y - g y) x = deriv f x - deriv g x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : 𝕜 → F} {x : 𝕜} (hf : DifferentiableAt 𝕜 f x) (hg : DifferentiableAt 𝕜 g x)
Goal: deriv (fun y => f y - g y) x = deriv f x - deriv g x
-- HasDerivAtFilter.sub_const
theorem HasDerivAtFilter.sub_const (hf : HasDerivAtFilter f f' x L) (c : F) :
    HasDerivAtFilter (fun x => f x - c) f' x L 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {L : Filter 𝕜} (hf : HasDerivAtFilter f f' x L) (c : F)
Goal: HasDerivAtFilter (fun x => f x - c) f' x L
-- HasDerivWithinAt.sub_const
nonrec theorem HasDerivWithinAt.sub_const (hf : HasDerivWithinAt f f' s x) (c : F) :
    HasDerivWithinAt (fun x => f x - c) f' s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {s : Set 𝕜} (hf : HasDerivWithinAt f f' s x) (c : F)
Goal: HasDerivWithinAt (fun x => f x - c) f' s x
-- HasDerivAt.sub_const
nonrec theorem HasDerivAt.sub_const (hf : HasDerivAt f f' x) (c : F) :
    HasDerivAt (fun x => f x - c) f' x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} (hf : HasDerivAt f f' x) (c : F)
Goal: HasDerivAt (fun x => f x - c) f' x
-- derivWithin_sub_const
theorem derivWithin_sub_const (hxs : UniqueDiffWithinAt 𝕜 s x) (c : F) :
    derivWithin (fun y => f y - c) s x = derivWithin f s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {s : Set 𝕜} (hxs : UniqueDiffWithinAt 𝕜 s x) (c : F)
Goal: derivWithin (fun y => f y - c) s x = derivWithin f s x
-- deriv_sub_const
theorem deriv_sub_const (c : F) : deriv (fun y => f y - c) x = deriv f x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} (c : F)
Goal: deriv (fun y => f y - c) x = deriv f x
-- HasDerivAtFilter.const_sub
theorem HasDerivAtFilter.const_sub (c : F) (hf : HasDerivAtFilter f f' x L) :
    HasDerivAtFilter (fun x => c - f x) (-f') x L 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {L : Filter 𝕜} (c : F) (hf : HasDerivAtFilter f f' x L)
Goal: HasDerivAtFilter (fun x => c - f x) (-f') x L
-- HasDerivWithinAt.const_sub
nonrec theorem HasDerivWithinAt.const_sub (c : F) (hf : HasDerivWithinAt f f' s x) :
    HasDerivWithinAt (fun x => c - f x) (-f') s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {s : Set 𝕜} (c : F) (hf : HasDerivWithinAt f f' s x)
Goal: HasDerivWithinAt (fun x => c - f x) (-f') s x
-- HasStrictDerivAt.const_sub
theorem HasStrictDerivAt.const_sub (c : F) (hf : HasStrictDerivAt f f' x) :
    HasStrictDerivAt (fun x => c - f x) (-f') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} (c : F) (hf : HasStrictDerivAt f f' x)
Goal: HasStrictDerivAt (fun x => c - f x) (-f') x
-- HasDerivAt.const_sub
nonrec theorem HasDerivAt.const_sub (c : F) (hf : HasDerivAt f f' x) :
    HasDerivAt (fun x => c - f x) (-f') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} (c : F) (hf : HasDerivAt f f' x)
Goal: HasDerivAt (fun x => c - f x) (-f') x
-- derivWithin_const_sub
theorem derivWithin_const_sub (hxs : UniqueDiffWithinAt 𝕜 s x) (c : F) :
    derivWithin (fun y => c - f y) s x = -derivWithin f s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {s : Set 𝕜} (hxs : UniqueDiffWithinAt 𝕜 s x) (c : F)
Goal: derivWithin (fun y => c - f y) s x = -derivWithin f s x
-- deriv_const_sub
theorem deriv_const_sub (c : F) : deriv (fun y => c - f y) x = -deriv f x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} (c : F)
Goal: deriv (fun y => c - f y) x = -deriv f x
-- ContDiffBump.convolution_eq_right
theorem convolution_eq_right {x₀ : G} (hg : ∀ x ∈ ball x₀ φ.rOut, g x = g x₀) :
    (φ ⋆[lsmul ℝ ℝ, μ] g : G → E') x₀ = integral μ φ • g x₀ 
Arguments: {G : Type uG} {E' : Type uE'} [NormedAddCommGroup E'] {g : G → E'} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ E'] [NormedAddCommGroup G] [NormedSpace ℝ G] [HasContDiffBump G] [CompleteSpace E'] {φ : ContDiffBump 0} {x₀ : G} (hg : ∀ x ∈ Metric.ball x₀ φ.rOut, g x = g x₀)
Goal: (↑φ ⋆[ContinuousLinearMap.lsmul ℝ ℝ, μ] g) x₀ = MeasureTheory.integral μ ↑φ • g x₀
-- ContDiffBump.normed_convolution_eq_right
theorem normed_convolution_eq_right {x₀ : G} (hg : ∀ x ∈ ball x₀ φ.rOut, g x = g x₀) :
    (φ.normed μ ⋆[lsmul ℝ ℝ, μ] g : G → E') x₀ = g x₀ 
Arguments: {G : Type uG} {E' : Type uE'} [NormedAddCommGroup E'] {g : G → E'} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ E'] [NormedAddCommGroup G] [NormedSpace ℝ G] [HasContDiffBump G] [CompleteSpace E'] {φ : ContDiffBump 0} [BorelSpace G] [MeasureTheory.IsLocallyFiniteMeasure μ] [μ.IsOpenPosMeasure] [FiniteDimensional ℝ G] {x₀ : G} (hg : ∀ x ∈ Metric.ball x₀ φ.rOut, g x = g x₀)
Goal: (φ.normed μ ⋆[ContinuousLinearMap.lsmul ℝ ℝ, μ] g) x₀ = g x₀
-- ContDiffBump.dist_normed_convolution_le
theorem dist_normed_convolution_le {x₀ : G} {ε : ℝ} (hmg : AEStronglyMeasurable g μ)
    (hg : ∀ x ∈ ball x₀ φ.rOut, dist (g x) (g x₀) ≤ ε) :
    dist ((φ.normed μ ⋆[lsmul ℝ ℝ, μ] g : G → E') x₀) (g x₀) ≤ ε 
Arguments: {G : Type uG} {E' : Type uE'} [NormedAddCommGroup E'] {g : G → E'} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ E'] [NormedAddCommGroup G] [NormedSpace ℝ G] [HasContDiffBump G] [CompleteSpace E'] {φ : ContDiffBump 0} [BorelSpace G] [MeasureTheory.IsLocallyFiniteMeasure μ] [μ.IsOpenPosMeasure] [FiniteDimensional ℝ G] [μ.IsAddLeftInvariant] {x₀ : G} {ε : ℝ} (hmg : MeasureTheory.AEStronglyMeasurable g μ) (hg : ∀ x ∈ Metric.ball x₀ φ.rOut, dist (g x) (g x₀) ≤ ε)
Goal: dist ((φ.normed μ ⋆[ContinuousLinearMap.lsmul ℝ ℝ, μ] g) x₀) (g x₀) ≤ ε
-- ContDiffBump.convolution_tendsto_right
nonrec theorem convolution_tendsto_right {ι} {φ : ι → ContDiffBump (0 : G)} {g : ι → G → E'}
    {k : ι → G} {x₀ : G} {z₀ : E'} {l : Filter ι} (hφ : Tendsto (fun i => (φ i).rOut) l (𝓝 0))
    (hig : ∀ᶠ i in l, AEStronglyMeasurable (g i) μ) (hcg : Tendsto (uncurry g) (l ×ˢ 𝓝 x₀) (𝓝 z₀))
    (hk : Tendsto k l (𝓝 x₀)) :
    Tendsto (fun i => ((φ i).normed μ ⋆[lsmul ℝ ℝ, μ] g i) (k i)) l (𝓝 z₀) 
Arguments: {G : Type uG} {E' : Type uE'} [NormedAddCommGroup E'] [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ E'] [NormedAddCommGroup G] [NormedSpace ℝ G] [HasContDiffBump G] [CompleteSpace E'] [BorelSpace G] [MeasureTheory.IsLocallyFiniteMeasure μ] [μ.IsOpenPosMeasure] [FiniteDimensional ℝ G] [μ.IsAddLeftInvariant] {ι : Type u_1} {φ : ι → ContDiffBump 0} {g : ι → G → E'} {k : ι → G} {x₀ : G} {z₀ : E'} {l : Filter ι} (hφ : Filter.Tendsto (fun i => (φ i).rOut) l (𝓝 0)) (hig : ∀ᶠ (i : ι) in l, MeasureTheory.AEStronglyMeasurable (g i) μ) (hcg : Filter.Tendsto (Function.uncurry g) (l ×ˢ 𝓝 x₀) (𝓝 z₀)) (hk : Filter.Tendsto k l (𝓝 x₀))
Goal: Filter.Tendsto (fun i => ((φ i).normed μ ⋆[ContinuousLinearMap.lsmul ℝ ℝ, μ] g i) (k i)) l (𝓝 z₀)
-- ContDiffBump.convolution_tendsto_right_of_continuous
theorem convolution_tendsto_right_of_continuous {ι} {φ : ι → ContDiffBump (0 : G)} {l : Filter ι}
    (hφ : Tendsto (fun i => (φ i).rOut) l (𝓝 0)) (hg : Continuous g) (x₀ : G) :
    Tendsto (fun i => ((φ i).normed μ ⋆[lsmul ℝ ℝ, μ] g) x₀) l (𝓝 (g x₀)) 
Arguments: {G : Type uG} {E' : Type uE'} [NormedAddCommGroup E'] {g : G → E'} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ E'] [NormedAddCommGroup G] [NormedSpace ℝ G] [HasContDiffBump G] [CompleteSpace E'] [BorelSpace G] [MeasureTheory.IsLocallyFiniteMeasure μ] [μ.IsOpenPosMeasure] [FiniteDimensional ℝ G] [μ.IsAddLeftInvariant] {ι : Type u_1} {φ : ι → ContDiffBump 0} {l : Filter ι} (hφ : Filter.Tendsto (fun i => (φ i).rOut) l (𝓝 0)) (hg : Continuous g) (x₀ : G)
Goal: Filter.Tendsto (fun i => ((φ i).normed μ ⋆[ContinuousLinearMap.lsmul ℝ ℝ, μ] g) x₀) l (𝓝 (g x₀))
-- ContDiffBump.ae_convolution_tendsto_right_of_locallyIntegrable
theorem ae_convolution_tendsto_right_of_locallyIntegrable
    {ι} {φ : ι → ContDiffBump (0 : G)} {l : Filter ι} {K : ℝ}
    (hφ : Tendsto (fun i ↦ (φ i).rOut) l (𝓝 0))
    (h'φ : ∀ᶠ i in l, (φ i).rOut ≤ K * (φ i).rIn) (hg : LocallyIntegrable g μ) : ∀ᵐ x₀ ∂μ,
    Tendsto (fun i ↦ ((φ i).normed μ ⋆[lsmul ℝ ℝ, μ] g) x₀) l (𝓝 (g x₀)) 
Arguments: {G : Type uG} {E' : Type uE'} [NormedAddCommGroup E'] {g : G → E'} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ E'] [NormedAddCommGroup G] [NormedSpace ℝ G] [HasContDiffBump G] [CompleteSpace E'] [BorelSpace G] [MeasureTheory.IsLocallyFiniteMeasure μ] [μ.IsOpenPosMeasure] [FiniteDimensional ℝ G] [μ.IsAddLeftInvariant] {ι : Type u_1} {φ : ι → ContDiffBump 0} {l : Filter ι} {K : ℝ} (hφ : Filter.Tendsto (fun i => (φ i).rOut) l (𝓝 0)) (h'φ : ∀ᶠ (i : ι) in l, (φ i).rOut ≤ K * (φ i).rIn) (hg : MeasureTheory.LocallyIntegrable g μ)
Goal: ∀ᵐ (x₀ : G) ∂μ, Filter.Tendsto (fun i => ((φ i).normed μ ⋆[ContinuousLinearMap.lsmul ℝ ℝ, μ] g) x₀) l (𝓝 (g x₀))
-- smooth_barycentric_coord
theorem smooth_barycentric_coord (b : AffineBasis ι 𝕜 E) (i : ι) : ContDiff 𝕜 ⊤ (b.coord i) 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} {E : Type u_3} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [FiniteDimensional 𝕜 E] (b : AffineBasis ι 𝕜 E) (i : ι)
Goal: ContDiff 𝕜 ⊤ ⇑(b.coord i)
-- ContinuousAffineMap.contDiff
theorem contDiff {n : ℕ∞} (f : V →ᴬ[𝕜] W) : ContDiff 𝕜 n f 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {W : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup V] [NormedSpace 𝕜 V] [NormedAddCommGroup W] [NormedSpace 𝕜 W] {n : ℕ∞} (f : V →ᴬ[𝕜] W)
Goal: ContDiff 𝕜 n ⇑f
-- QuadraticMap.hasLineDerivAt
theorem hasLineDerivAt (f : QuadraticMap 𝕜 E F) (a b : E) :
    HasLineDerivAt 𝕜 f (polar f a b) a b 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] (f : QuadraticMap 𝕜 E F) (a b : E)
Goal: HasLineDerivAt 𝕜 (⇑f) (QuadraticMap.polar (⇑f) a b) a b
-- QuadraticMap.lineDifferentiableAt
theorem lineDifferentiableAt (f : QuadraticMap 𝕜 E F) (a b : E) : LineDifferentiableAt 𝕜 f a b 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] (f : QuadraticMap 𝕜 E F) (a b : E)
Goal: LineDifferentiableAt 𝕜 (⇑f) a b
-- QuadraticMap.lineDeriv
@[simp]
protected theorem lineDeriv (f : QuadraticMap 𝕜 E F) : lineDeriv 𝕜 f = polar f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] (f : QuadraticMap 𝕜 E F)
Goal: lineDeriv 𝕜 ⇑f = QuadraticMap.polar ⇑f
-- ContDiffBump.normed_def
theorem normed_def {μ : Measure E} (x : E) : f.normed μ x = f x / ∫ x, f x ∂μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] [MeasurableSpace E] {c : E} (f : ContDiffBump c) {μ : MeasureTheory.Measure E} (x : E)
Goal: f.normed μ x = ↑f x / ∫ (x : E), ↑f x ∂μ
-- ContDiffBump.nonneg_normed
theorem nonneg_normed (x : E) : 0 ≤ f.normed μ x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] [MeasurableSpace E] {c : E} (f : ContDiffBump c) {μ : MeasureTheory.Measure E} (x : E)
Goal: 0 ≤ f.normed μ x
-- ContDiffBump.contDiff_normed
theorem contDiff_normed {n : ℕ∞} : ContDiff ℝ n (f.normed μ) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] [MeasurableSpace E] {c : E} (f : ContDiffBump c) {μ : MeasureTheory.Measure E} {n : ℕ∞}
Goal: ContDiff ℝ n (f.normed μ)
-- ContDiffBump.continuous_normed
theorem continuous_normed : Continuous (f.normed μ) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] [MeasurableSpace E] {c : E} (f : ContDiffBump c) {μ : MeasureTheory.Measure E}
Goal: Continuous (f.normed μ)
-- ContDiffBump.normed_sub
theorem normed_sub (x : E) : f.normed μ (c - x) = f.normed μ (c + x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] [MeasurableSpace E] {c : E} (f : ContDiffBump c) {μ : MeasureTheory.Measure E} (x : E)
Goal: f.normed μ (c - x) = f.normed μ (c + x)
-- ContDiffBump.normed_neg
theorem normed_neg (f : ContDiffBump (0 : E)) (x : E) : f.normed μ (-x) = f.normed μ x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] [MeasurableSpace E] {μ : MeasureTheory.Measure E} (f : ContDiffBump 0) (x : E)
Goal: f.normed μ (-x) = f.normed μ x
-- ContDiffBump.integrable
protected theorem integrable : Integrable f μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] [MeasurableSpace E] {c : E} (f : ContDiffBump c) {μ : MeasureTheory.Measure E} [BorelSpace E] [FiniteDimensional ℝ E] [MeasureTheory.IsLocallyFiniteMeasure μ]
Goal: MeasureTheory.Integrable (↑f) μ
-- ContDiffBump.integrable_normed
protected theorem integrable_normed : Integrable (f.normed μ) μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] [MeasurableSpace E] {c : E} (f : ContDiffBump c) {μ : MeasureTheory.Measure E} [BorelSpace E] [FiniteDimensional ℝ E] [MeasureTheory.IsLocallyFiniteMeasure μ]
Goal: MeasureTheory.Integrable (f.normed μ) μ
-- ContDiffBump.integral_pos
theorem integral_pos : 0 < ∫ x, f x ∂μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] [MeasurableSpace E] {c : E} (f : ContDiffBump c) {μ : MeasureTheory.Measure E} [BorelSpace E] [FiniteDimensional ℝ E] [MeasureTheory.IsLocallyFiniteMeasure μ] [μ.IsOpenPosMeasure]
Goal: 0 < ∫ (x : E), ↑f x ∂μ
-- ContDiffBump.integral_normed
theorem integral_normed : ∫ x, f.normed μ x ∂μ = 1 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] [MeasurableSpace E] {c : E} (f : ContDiffBump c) {μ : MeasureTheory.Measure E} [BorelSpace E] [FiniteDimensional ℝ E] [MeasureTheory.IsLocallyFiniteMeasure μ] [μ.IsOpenPosMeasure]
Goal: ∫ (x : E), f.normed μ x ∂μ = 1
-- ContDiffBump.support_normed_eq
theorem support_normed_eq : Function.support (f.normed μ) = Metric.ball c f.rOut 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] [MeasurableSpace E] {c : E} (f : ContDiffBump c) {μ : MeasureTheory.Measure E} [BorelSpace E] [FiniteDimensional ℝ E] [MeasureTheory.IsLocallyFiniteMeasure μ] [μ.IsOpenPosMeasure]
Goal: Function.support (f.normed μ) = Metric.ball c f.rOut
-- ContDiffBump.tsupport_normed_eq
theorem tsupport_normed_eq : tsupport (f.normed μ) = Metric.closedBall c f.rOut 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] [MeasurableSpace E] {c : E} (f : ContDiffBump c) {μ : MeasureTheory.Measure E} [BorelSpace E] [FiniteDimensional ℝ E] [MeasureTheory.IsLocallyFiniteMeasure μ] [μ.IsOpenPosMeasure]
Goal: tsupport (f.normed μ) = Metric.closedBall c f.rOut
-- ContDiffBump.hasCompactSupport_normed
theorem hasCompactSupport_normed : HasCompactSupport (f.normed μ) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] [MeasurableSpace E] {c : E} (f : ContDiffBump c) {μ : MeasureTheory.Measure E} [BorelSpace E] [FiniteDimensional ℝ E] [MeasureTheory.IsLocallyFiniteMeasure μ] [μ.IsOpenPosMeasure]
Goal: HasCompactSupport (f.normed μ)
-- ContDiffBump.tendsto_support_normed_smallSets
theorem tendsto_support_normed_smallSets {ι} {φ : ι → ContDiffBump c} {l : Filter ι}
    (hφ : Tendsto (fun i => (φ i).rOut) l (𝓝 0)) :
    Tendsto (fun i => Function.support fun x => (φ i).normed μ x) l (𝓝 c).smallSets 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] [MeasurableSpace E] {c : E} {μ : MeasureTheory.Measure E} [BorelSpace E] [FiniteDimensional ℝ E] [MeasureTheory.IsLocallyFiniteMeasure μ] [μ.IsOpenPosMeasure] {ι : Type u_2} {φ : ι → ContDiffBump c} {l : Filter ι} (hφ : Filter.Tendsto (fun i => (φ i).rOut) l (𝓝 0))
Goal: Filter.Tendsto (fun i => Function.support fun x => (φ i).normed μ x) l (𝓝 c).smallSets
-- ContDiffBump.integral_normed_smul
theorem integral_normed_smul {X} [NormedAddCommGroup X] [NormedSpace ℝ X]
    [CompleteSpace X] (z : X) : ∫ x, f.normed μ x • z ∂μ = z 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] [MeasurableSpace E] {c : E} (f : ContDiffBump c) (μ : MeasureTheory.Measure E) [BorelSpace E] [FiniteDimensional ℝ E] [MeasureTheory.IsLocallyFiniteMeasure μ] [μ.IsOpenPosMeasure] {X : Type u_2} [NormedAddCommGroup X] [NormedSpace ℝ X] [CompleteSpace X] (z : X)
Goal: ∫ (x : E), f.normed μ x • z ∂μ = z
-- ContDiffBump.measure_closedBall_le_integral
theorem measure_closedBall_le_integral : (μ (closedBall c f.rIn)).toReal ≤ ∫ x, f x ∂μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] [MeasurableSpace E] {c : E} (f : ContDiffBump c) (μ : MeasureTheory.Measure E) [BorelSpace E] [FiniteDimensional ℝ E] [MeasureTheory.IsLocallyFiniteMeasure μ]
Goal: (μ (Metric.closedBall c f.rIn)).toReal ≤ ∫ (x : E), ↑f x ∂μ
-- ContDiffBump.normed_le_div_measure_closedBall_rIn
theorem normed_le_div_measure_closedBall_rIn (x : E) :
    f.normed μ x ≤ 1 / (μ (closedBall c f.rIn)).toReal 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] [MeasurableSpace E] {c : E} (f : ContDiffBump c) (μ : MeasureTheory.Measure E) [BorelSpace E] [FiniteDimensional ℝ E] [MeasureTheory.IsLocallyFiniteMeasure μ] [μ.IsOpenPosMeasure] (x : E)
Goal: f.normed μ x ≤ 1 / (μ (Metric.closedBall c f.rIn)).toReal
-- ContDiffBump.integral_le_measure_closedBall
theorem integral_le_measure_closedBall : ∫ x, f x ∂μ ≤ (μ (closedBall c f.rOut)).toReal 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] [MeasurableSpace E] {c : E} (f : ContDiffBump c) (μ : MeasureTheory.Measure E) [BorelSpace E] [FiniteDimensional ℝ E] [MeasureTheory.IsLocallyFiniteMeasure μ]
Goal: ∫ (x : E), ↑f x ∂μ ≤ (μ (Metric.closedBall c f.rOut)).toReal
-- ContDiffBump.measure_closedBall_div_le_integral
theorem measure_closedBall_div_le_integral [IsAddHaarMeasure μ] (K : ℝ) (h : f.rOut ≤ K * f.rIn) :
    (μ (closedBall c f.rOut)).toReal / K ^ finrank ℝ E ≤ ∫ x, f x ∂μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] [MeasurableSpace E] {c : E} (f : ContDiffBump c) (μ : MeasureTheory.Measure E) [BorelSpace E] [FiniteDimensional ℝ E] [MeasureTheory.IsLocallyFiniteMeasure μ] [μ.IsAddHaarMeasure] (K : ℝ) (h : f.rOut ≤ K * f.rIn)
Goal: (μ (Metric.closedBall c f.rOut)).toReal / K ^ FiniteDimensional.finrank ℝ E ≤ ∫ (x : E), ↑f x ∂μ
-- ContDiffBump.normed_le_div_measure_closedBall_rOut
theorem normed_le_div_measure_closedBall_rOut [IsAddHaarMeasure μ] (K : ℝ) (h : f.rOut ≤ K * f.rIn)
    (x : E) :
    f.normed μ x ≤ K ^ finrank ℝ E / (μ (closedBall c f.rOut)).toReal 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [HasContDiffBump E] [MeasurableSpace E] {c : E} (f : ContDiffBump c) (μ : MeasureTheory.Measure E) [BorelSpace E] [FiniteDimensional ℝ E] [MeasureTheory.IsLocallyFiniteMeasure μ] [μ.IsOpenPosMeasure] [μ.IsAddHaarMeasure] (K : ℝ) (h : f.rOut ≤ K * f.rIn) (x : E)
Goal: f.normed μ x ≤ K ^ FiniteDimensional.finrank ℝ E / (μ (Metric.closedBall c f.rOut)).toReal