ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- HasFiniteFPowerSeriesOnBall.differentiableOn
theorem HasFiniteFPowerSeriesOnBall.differentiableOn
    (h : HasFiniteFPowerSeriesOnBall f p x n r) : DifferentiableOn 𝕜 f (EMetric.ball x r) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {p : FormalMultilinearSeries 𝕜 E F} {r : ℝ≥0∞} {n : ℕ} {f : E → F} {x : E} (h : HasFiniteFPowerSeriesOnBall f p x n r)
Goal: DifferentiableOn 𝕜 f (EMetric.ball x r)
-- HasFiniteFPowerSeriesOnBall.hasFDerivAt
theorem HasFiniteFPowerSeriesOnBall.hasFDerivAt (h : HasFiniteFPowerSeriesOnBall f p x n r)
    {y : E} (hy : (‖y‖₊ : ℝ≥0∞) < r) :
    HasFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p.changeOrigin y 1)) (x + y) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {p : FormalMultilinearSeries 𝕜 E F} {r : ℝ≥0∞} {n : ℕ} {f : E → F} {x : E} (h : HasFiniteFPowerSeriesOnBall f p x n r) {y : E} (hy : ↑‖y‖₊ < r)
Goal: HasFDerivAt f ((continuousMultilinearCurryFin1 𝕜 E F) (p.changeOrigin y 1)) (x + y)
-- HasFiniteFPowerSeriesOnBall.fderiv_eq
theorem HasFiniteFPowerSeriesOnBall.fderiv_eq (h : HasFiniteFPowerSeriesOnBall f p x n r)
    {y : E} (hy : (‖y‖₊ : ℝ≥0∞) < r) :
    fderiv 𝕜 f (x + y) = continuousMultilinearCurryFin1 𝕜 E F (p.changeOrigin y 1) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {p : FormalMultilinearSeries 𝕜 E F} {r : ℝ≥0∞} {n : ℕ} {f : E → F} {x : E} (h : HasFiniteFPowerSeriesOnBall f p x n r) {y : E} (hy : ↑‖y‖₊ < r)
Goal: fderiv 𝕜 f (x + y) = (continuousMultilinearCurryFin1 𝕜 E F) (p.changeOrigin y 1)
-- HasFiniteFPowerSeriesOnBall.fderiv
protected theorem HasFiniteFPowerSeriesOnBall.fderiv
    (h : HasFiniteFPowerSeriesOnBall f p x (n + 1) r) :
    HasFiniteFPowerSeriesOnBall (fderiv 𝕜 f) p.derivSeries x n r 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {p : FormalMultilinearSeries 𝕜 E F} {r : ℝ≥0∞} {n : ℕ} {f : E → F} {x : E} (h : HasFiniteFPowerSeriesOnBall f p x (n + 1) r)
Goal: HasFiniteFPowerSeriesOnBall (fderiv 𝕜 f) p.derivSeries x n r
-- HasFiniteFPowerSeriesOnBall.fderiv'
theorem HasFiniteFPowerSeriesOnBall.fderiv' (h : HasFiniteFPowerSeriesOnBall f p x n r) :
    HasFiniteFPowerSeriesOnBall (fderiv 𝕜 f) p.derivSeries x (n - 1) r 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {p : FormalMultilinearSeries 𝕜 E F} {r : ℝ≥0∞} {n : ℕ} {f : E → F} {x : E} (h : HasFiniteFPowerSeriesOnBall f p x n r)
Goal: HasFiniteFPowerSeriesOnBall (fderiv 𝕜 f) p.derivSeries x (n - 1) r
-- CPolynomialOn.fderiv
theorem CPolynomialOn.fderiv (h : CPolynomialOn 𝕜 f s) :
    CPolynomialOn 𝕜 (fderiv 𝕜 f) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} (h : CPolynomialOn 𝕜 f s)
Goal: CPolynomialOn 𝕜 (fderiv 𝕜 f) s
-- CPolynomialOn.iteratedFDeriv
theorem CPolynomialOn.iteratedFDeriv (h : CPolynomialOn 𝕜 f s) (n : ℕ) :
    CPolynomialOn 𝕜 (iteratedFDeriv 𝕜 n f) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} (h : CPolynomialOn 𝕜 f s) (n : ℕ)
Goal: CPolynomialOn 𝕜 (iteratedFDeriv 𝕜 n f) s
-- CPolynomialOn.contDiffOn
theorem CPolynomialOn.contDiffOn (h : CPolynomialOn 𝕜 f s) {n : ℕ∞} :
    ContDiffOn 𝕜 n f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} (h : CPolynomialOn 𝕜 f s) {n : ℕ∞}
Goal: ContDiffOn 𝕜 n f s
-- CPolynomialAt.contDiffAt
theorem CPolynomialAt.contDiffAt (h : CPolynomialAt 𝕜 f x) {n : ℕ∞} :
    ContDiffAt 𝕜 n f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E} (h : CPolynomialAt 𝕜 f x) {n : ℕ∞}
Goal: ContDiffAt 𝕜 n f x
-- CPolynomialOn.deriv
protected theorem CPolynomialOn.deriv (h : CPolynomialOn 𝕜 f s) : CPolynomialOn 𝕜 (deriv f) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {s : Set 𝕜} (h : CPolynomialOn 𝕜 f s)
Goal: CPolynomialOn 𝕜 (deriv f) s
-- CPolynomialOn.iterated_deriv
theorem CPolynomialOn.iterated_deriv (h : CPolynomialOn 𝕜 f s) (n : ℕ) :
    CPolynomialOn 𝕜 (deriv^[n] f) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {s : Set 𝕜} (h : CPolynomialOn 𝕜 f s) (n : ℕ)
Goal: CPolynomialOn 𝕜 (deriv^[n] f) s
-- ContinuousMultilinearMap.hasFiniteFPowerSeriesOnBall
protected theorem hasFiniteFPowerSeriesOnBall :
    HasFiniteFPowerSeriesOnBall f f.toFormalMultilinearSeries 0 (Fintype.card ι + 1) ⊤ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {ι : Type u_2} {E : ι → Type u_3} [(i : ι) → NormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [Fintype ι] (f : ContinuousMultilinearMap 𝕜 E F)
Goal: HasFiniteFPowerSeriesOnBall (⇑f) f.toFormalMultilinearSeries 0 (Fintype.card ι + 1) ⊤
-- ContinuousMultilinearMap.changeOriginSeries_support
theorem changeOriginSeries_support {k l : ℕ} (h : k + l ≠ Fintype.card ι) :
    f.toFormalMultilinearSeries.changeOriginSeries k l = 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {ι : Type u_2} {E : ι → Type u_3} [(i : ι) → NormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [Fintype ι] (f : ContinuousMultilinearMap 𝕜 E F) {k l : ℕ} (h : k + l ≠ Fintype.card ι)
Goal: f.toFormalMultilinearSeries.changeOriginSeries k l = 0
-- ContinuousMultilinearMap.changeOrigin_toFormalMultilinearSeries
theorem changeOrigin_toFormalMultilinearSeries [DecidableEq ι] :
    continuousMultilinearCurryFin1 𝕜 (∀ i, E i) F (f.toFormalMultilinearSeries.changeOrigin x 1) =
    f.linearDeriv x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {ι : Type u_2} {E : ι → Type u_3} [(i : ι) → NormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [Fintype ι] (f : ContinuousMultilinearMap 𝕜 E F) (x : (i : ι) → E i) [DecidableEq ι]
Goal: (continuousMultilinearCurryFin1 𝕜 ((i : ι) → E i) F) (f.toFormalMultilinearSeries.changeOrigin x 1) = f.linearDeriv x
-- ContinuousMultilinearMap.hasFDerivAt
protected theorem hasFDerivAt [DecidableEq ι] : HasFDerivAt f (f.linearDeriv x) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {ι : Type u_2} {E : ι → Type u_3} [(i : ι) → NormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [Fintype ι] (f : ContinuousMultilinearMap 𝕜 E F) (x : (i : ι) → E i) [DecidableEq ι]
Goal: HasFDerivAt (⇑f) (f.linearDeriv x) x
-- _private.Mathlib.Analysis.Calculus.FDeriv.Analytic.0.Equiv.succ_embeddingFinSucc_fst_symm_apply
private lemma _root_.Equiv.succ_embeddingFinSucc_fst_symm_apply {ι : Type*} [DecidableEq ι]
    {n : ℕ} (e : Fin (n+1) ↪ ι) {k : ι}
    (h'k : k ∈ Set.range (Equiv.embeddingFinSucc n ι e).1) (hk : k ∈ Set.range e) :
    Fin.succ ((Equiv.embeddingFinSucc n ι e).1.toEquivRange.symm ⟨k, h'k⟩)
      = e.toEquivRange.symm ⟨k, hk⟩ 
Arguments: {ι : Type u_4} [DecidableEq ι] {n : ℕ} (e : Fin (n + 1) ↪ ι) {k : ι} (h'k : k ∈ Set.range ⇑((Equiv.embeddingFinSucc n ι) e).fst) (hk : k ∈ Set.range ⇑e)
Goal: (((Equiv.embeddingFinSucc n ι) e).fst.toEquivRange.symm ⟨k, h'k⟩).succ = e.toEquivRange.symm ⟨k, hk⟩
-- ContinuousMultilinearMap.hasFTaylorSeriesUpTo_iteratedFDeriv
theorem hasFTaylorSeriesUpTo_iteratedFDeriv :
    HasFTaylorSeriesUpTo ⊤ f (fun v n ↦ f.iteratedFDeriv n v) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {ι : Type u_2} {E : ι → Type u_3} [(i : ι) → NormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [Fintype ι] (f : ContinuousMultilinearMap 𝕜 E F)
Goal: HasFTaylorSeriesUpTo ⊤ ⇑f fun v n => f.iteratedFDeriv n v
-- ContinuousMultilinearMap.iteratedFDeriv_eq
theorem iteratedFDeriv_eq (n : ℕ) :
    iteratedFDeriv 𝕜 n f = f.iteratedFDeriv n 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {ι : Type u_2} {E : ι → Type u_3} [(i : ι) → NormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [Fintype ι] (f : ContinuousMultilinearMap 𝕜 E F) (n : ℕ)
Goal: iteratedFDeriv 𝕜 n ⇑f = f.iteratedFDeriv n
-- ContinuousMultilinearMap.norm_iteratedFDeriv_le
theorem norm_iteratedFDeriv_le (n : ℕ) (x : (i : ι) → E i) :
    ‖iteratedFDeriv 𝕜 n f x‖
      ≤ Nat.descFactorial (Fintype.card ι) n * ‖f‖ * ‖x‖ ^ (Fintype.card ι - n) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {ι : Type u_2} {E : ι → Type u_3} [(i : ι) → NormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [Fintype ι] (f : ContinuousMultilinearMap 𝕜 E F) (n : ℕ) (x : (i : ι) → E i)
Goal: ‖iteratedFDeriv 𝕜 n (⇑f) x‖ ≤ ↑((Fintype.card ι).descFactorial n) * ‖f‖ * ‖x‖ ^ (Fintype.card ι - n)
-- ContinuousMultilinearMap.cPolynomialAt
lemma cPolynomialAt : CPolynomialAt 𝕜 f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {ι : Type u_2} {E : ι → Type u_3} [(i : ι) → NormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [Fintype ι] (f : ContinuousMultilinearMap 𝕜 E F) (x : (i : ι) → E i)
Goal: CPolynomialAt 𝕜 (⇑f) x
-- ContinuousMultilinearMap.cPolyomialOn
lemma cPolyomialOn : CPolynomialOn 𝕜 f ⊤ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {ι : Type u_2} {E : ι → Type u_3} [(i : ι) → NormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [Fintype ι] (f : ContinuousMultilinearMap 𝕜 E F)
Goal: CPolynomialOn 𝕜 ⇑f ⊤
-- ContinuousMultilinearMap.contDiffAt
lemma contDiffAt : ContDiffAt 𝕜 n f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {ι : Type u_2} {E : ι → Type u_3} [(i : ι) → NormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [Fintype ι] (f : ContinuousMultilinearMap 𝕜 E F) {n : ℕ∞} (x : (i : ι) → E i)
Goal: ContDiffAt 𝕜 n (⇑f) x
-- ContinuousMultilinearMap.contDiff
lemma contDiff : ContDiff 𝕜 n f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {ι : Type u_2} {E : ι → Type u_3} [(i : ι) → NormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [Fintype ι] (f : ContinuousMultilinearMap 𝕜 E F) {n : ℕ∞}
Goal: ContDiff 𝕜 n ⇑f
-- FormalMultilinearSeries.derivSeries_apply_diag
theorem derivSeries_apply_diag (n : ℕ) (x : E) :
    derivSeries p n (fun _ ↦ x) x = (n + 1) • p (n + 1) fun _ ↦ x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) (x : E)
Goal: ((p.derivSeries n) fun x_1 => x) x = (n + 1) • (p (n + 1)) fun x_1 => x
-- HasFPowerSeriesOnBall.iteratedFDeriv_zero_apply_diag
theorem iteratedFDeriv_zero_apply_diag : iteratedFDeriv 𝕜 0 f x = p 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {p : FormalMultilinearSeries 𝕜 E F} {f : E → F} {x : E} {r : ℝ≥0∞} (h : HasFPowerSeriesOnBall f p x r)
Goal: iteratedFDeriv 𝕜 0 f x = p 0
-- _private.Mathlib.Analysis.Calculus.FDeriv.Analytic.0.HasFPowerSeriesOnBall.factorial_smul'
private theorem factorial_smul' {n : ℕ} : ∀ {F : Type max u v} [NormedAddCommGroup F]
    [NormedSpace 𝕜 F] [CompleteSpace F] {p : FormalMultilinearSeries 𝕜 E F}
    {f : E → F}, HasFPowerSeriesOnBall f p x r →
    n ! • p n (fun _ ↦ y) = iteratedFDeriv 𝕜 n f x (fun _ ↦ y) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {r : ℝ≥0∞} (y : E) {n : ℕ} {F : Type (max u v)} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {p : FormalMultilinearSeries 𝕜 E F} {f : E → F}
Goal: HasFPowerSeriesOnBall f p x r → (n.factorial • (p n) fun x => y) = (iteratedFDeriv 𝕜 n f x) fun x => y
-- HasFPowerSeriesOnBall.factorial_smul
theorem factorial_smul (n : ℕ) :
    n ! • p n (fun _ ↦ y) = iteratedFDeriv 𝕜 n f x (fun _ ↦ y) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {p : FormalMultilinearSeries 𝕜 E F} {f : E → F} {x : E} {r : ℝ≥0∞} (h : HasFPowerSeriesOnBall f p x r) (y : E) [CompleteSpace F] (n : ℕ)
Goal: (n.factorial • (p n) fun x => y) = (iteratedFDeriv 𝕜 n f x) fun x => y
-- HasFPowerSeriesOnBall.hasSum_iteratedFDeriv
theorem hasSum_iteratedFDeriv [CharZero 𝕜] {y : E} (hy : y ∈ EMetric.ball 0 r) :
    HasSum (fun n ↦ (n ! : 𝕜)⁻¹ • iteratedFDeriv 𝕜 n f x fun _ ↦ y) (f (x + y)) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {p : FormalMultilinearSeries 𝕜 E F} {f : E → F} {x : E} {r : ℝ≥0∞} (h : HasFPowerSeriesOnBall f p x r) [CompleteSpace F] [CharZero 𝕜] {y : E} (hy : y ∈ EMetric.ball 0 r)
Goal: HasSum (fun n => (↑n.factorial)⁻¹ • (iteratedFDeriv 𝕜 n f x) fun x => y) (f (x + y))
-- uniformCauchySeqOnFilter_of_fderiv
theorem uniformCauchySeqOnFilter_of_fderiv (hf' : UniformCauchySeqOnFilter f' l (𝓝 x))
    (hf : ∀ᶠ n : ι × E in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2)
    (hfg : Cauchy (map (fun n => f n x) l)) : UniformCauchySeqOnFilter f l (𝓝 x) 
Arguments: {ι : Type u_1} {l : Filter ι} {E : Type u_2} [NormedAddCommGroup E] {𝕜 : Type u_3} [RCLike 𝕜] [NormedSpace 𝕜 E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : ι → E → G} {f' : ι → E → E →L[𝕜] G} {x : E} (hf' : UniformCauchySeqOnFilter f' l (𝓝 x)) (hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2) (hfg : Cauchy (Filter.map (fun n => f n x) l))
Goal: UniformCauchySeqOnFilter f l (𝓝 x)
-- uniformCauchySeqOn_ball_of_fderiv
theorem uniformCauchySeqOn_ball_of_fderiv {r : ℝ} (hf' : UniformCauchySeqOn f' l (Metric.ball x r))
    (hf : ∀ n : ι, ∀ y : E, y ∈ Metric.ball x r → HasFDerivAt (f n) (f' n y) y)
    (hfg : Cauchy (map (fun n => f n x) l)) : UniformCauchySeqOn f l (Metric.ball x r) 
Arguments: {ι : Type u_1} {l : Filter ι} {E : Type u_2} [NormedAddCommGroup E] {𝕜 : Type u_3} [RCLike 𝕜] [NormedSpace 𝕜 E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : ι → E → G} {f' : ι → E → E →L[𝕜] G} {x : E} {r : ℝ} (hf' : UniformCauchySeqOn f' l (Metric.ball x r)) (hf : ∀ (n : ι), ∀ y ∈ Metric.ball x r, HasFDerivAt (f n) (f' n y) y) (hfg : Cauchy (Filter.map (fun n => f n x) l))
Goal: UniformCauchySeqOn f l (Metric.ball x r)
-- cauchy_map_of_uniformCauchySeqOn_fderiv
theorem cauchy_map_of_uniformCauchySeqOn_fderiv {s : Set E} (hs : IsOpen s) (h's : IsPreconnected s)
    (hf' : UniformCauchySeqOn f' l s) (hf : ∀ n : ι, ∀ y : E, y ∈ s → HasFDerivAt (f n) (f' n y) y)
    {x₀ x : E} (hx₀ : x₀ ∈ s) (hx : x ∈ s) (hfg : Cauchy (map (fun n => f n x₀) l)) :
    Cauchy (map (fun n => f n x) l) 
Arguments: {ι : Type u_1} {l : Filter ι} {E : Type u_2} [NormedAddCommGroup E] {𝕜 : Type u_3} [RCLike 𝕜] [NormedSpace 𝕜 E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : ι → E → G} {f' : ι → E → E →L[𝕜] G} {s : Set E} (hs : IsOpen s) (h's : IsPreconnected s) (hf' : UniformCauchySeqOn f' l s) (hf : ∀ (n : ι), ∀ y ∈ s, HasFDerivAt (f n) (f' n y) y) {x₀ x : E} (hx₀ : x₀ ∈ s) (hx : x ∈ s) (hfg : Cauchy (Filter.map (fun n => f n x₀) l))
Goal: Cauchy (Filter.map (fun n => f n x) l)
-- difference_quotients_converge_uniformly
theorem difference_quotients_converge_uniformly (hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x))
    (hf : ∀ᶠ n : ι × E in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2)
    (hfg : ∀ᶠ y : E in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))) :
    TendstoUniformlyOnFilter (fun n : ι => fun y : E => (‖y - x‖⁻¹ : 𝕜) • (f n y - f n x))
      (fun y : E => (‖y - x‖⁻¹ : 𝕜) • (g y - g x)) l (𝓝 x) 
Arguments: {ι : Type u_1} {l : Filter ι} {E : Type u_2} [NormedAddCommGroup E] {𝕜 : Type u_3} [RCLike 𝕜] [NormedSpace 𝕜 E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : ι → E → G} {g : E → G} {f' : ι → E → E →L[𝕜] G} {g' : E → E →L[𝕜] G} {x : E} (hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)) (hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2) (hfg : ∀ᶠ (y : E) in 𝓝 x, Filter.Tendsto (fun n => f n y) l (𝓝 (g y)))
Goal: TendstoUniformlyOnFilter (fun n y => (↑‖y - x‖)⁻¹ • (f n y - f n x)) (fun y => (↑‖y - x‖)⁻¹ • (g y - g x)) l (𝓝 x)
-- hasFDerivAt_of_tendstoUniformlyOnFilter
theorem hasFDerivAt_of_tendstoUniformlyOnFilter [NeBot l]
    (hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x))
    (hf : ∀ᶠ n : ι × E in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2)
    (hfg : ∀ᶠ y in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))) : HasFDerivAt g (g' x) x 
Arguments: {ι : Type u_1} {l : Filter ι} {E : Type u_2} [NormedAddCommGroup E] {𝕜 : Type u_3} [RCLike 𝕜] [NormedSpace 𝕜 E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : ι → E → G} {g : E → G} {f' : ι → E → E →L[𝕜] G} {g' : E → E →L[𝕜] G} {x : E} [l.NeBot] (hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)) (hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2) (hfg : ∀ᶠ (y : E) in 𝓝 x, Filter.Tendsto (fun n => f n y) l (𝓝 (g y)))
Goal: HasFDerivAt g (g' x) x
-- hasFDerivAt_of_tendstoLocallyUniformlyOn
theorem hasFDerivAt_of_tendstoLocallyUniformlyOn [NeBot l] {s : Set E} (hs : IsOpen s)
    (hf' : TendstoLocallyUniformlyOn f' g' l s) (hf : ∀ n, ∀ x ∈ s, HasFDerivAt (f n) (f' n x) x)
    (hfg : ∀ x ∈ s, Tendsto (fun n => f n x) l (𝓝 (g x))) (hx : x ∈ s) :
    HasFDerivAt g (g' x) x 
Arguments: {ι : Type u_1} {l : Filter ι} {E : Type u_2} [NormedAddCommGroup E] {𝕜 : Type u_3} [RCLike 𝕜] [NormedSpace 𝕜 E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : ι → E → G} {g : E → G} {f' : ι → E → E →L[𝕜] G} {g' : E → E →L[𝕜] G} {x : E} [l.NeBot] {s : Set E} (hs : IsOpen s) (hf' : TendstoLocallyUniformlyOn f' g' l s) (hf : ∀ (n : ι), ∀ x ∈ s, HasFDerivAt (f n) (f' n x) x) (hfg : ∀ x ∈ s, Filter.Tendsto (fun n => f n x) l (𝓝 (g x))) (hx : x ∈ s)
Goal: HasFDerivAt g (g' x) x
-- hasFDerivAt_of_tendsto_locally_uniformly_on'
theorem hasFDerivAt_of_tendsto_locally_uniformly_on' [NeBot l] {s : Set E} (hs : IsOpen s)
    (hf' : TendstoLocallyUniformlyOn (fderiv 𝕜 ∘ f) g' l s) (hf : ∀ n, DifferentiableOn 𝕜 (f n) s)
    (hfg : ∀ x ∈ s, Tendsto (fun n => f n x) l (𝓝 (g x))) (hx : x ∈ s) :
    HasFDerivAt g (g' x) x 
Arguments: {ι : Type u_1} {l : Filter ι} {E : Type u_2} [NormedAddCommGroup E] {𝕜 : Type u_3} [RCLike 𝕜] [NormedSpace 𝕜 E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : ι → E → G} {g : E → G} {g' : E → E →L[𝕜] G} {x : E} [l.NeBot] {s : Set E} (hs : IsOpen s) (hf' : TendstoLocallyUniformlyOn (fderiv 𝕜 ∘ f) g' l s) (hf : ∀ (n : ι), DifferentiableOn 𝕜 (f n) s) (hfg : ∀ x ∈ s, Filter.Tendsto (fun n => f n x) l (𝓝 (g x))) (hx : x ∈ s)
Goal: HasFDerivAt g (g' x) x
-- hasFDerivAt_of_tendstoUniformlyOn
theorem hasFDerivAt_of_tendstoUniformlyOn [NeBot l] {s : Set E} (hs : IsOpen s)
    (hf' : TendstoUniformlyOn f' g' l s)
    (hf : ∀ n : ι, ∀ x : E, x ∈ s → HasFDerivAt (f n) (f' n x) x)
    (hfg : ∀ x : E, x ∈ s → Tendsto (fun n => f n x) l (𝓝 (g x))) :
    ∀ x : E, x ∈ s → HasFDerivAt g (g' x) x 
Arguments: {ι : Type u_1} {l : Filter ι} {E : Type u_2} [NormedAddCommGroup E] {𝕜 : Type u_3} [RCLike 𝕜] [NormedSpace 𝕜 E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : ι → E → G} {g : E → G} {f' : ι → E → E →L[𝕜] G} {g' : E → E →L[𝕜] G} [l.NeBot] {s : Set E} (hs : IsOpen s) (hf' : TendstoUniformlyOn f' g' l s) (hf : ∀ (n : ι), ∀ x ∈ s, HasFDerivAt (f n) (f' n x) x) (hfg : ∀ x ∈ s, Filter.Tendsto (fun n => f n x) l (𝓝 (g x))) (x : E)
Goal: x ∈ s → HasFDerivAt g (g' x) x
-- hasFDerivAt_of_tendstoUniformly
theorem hasFDerivAt_of_tendstoUniformly [NeBot l] (hf' : TendstoUniformly f' g' l)
    (hf : ∀ n : ι, ∀ x : E, HasFDerivAt (f n) (f' n x) x)
    (hfg : ∀ x : E, Tendsto (fun n => f n x) l (𝓝 (g x))) : ∀ x : E, HasFDerivAt g (g' x) x 
Arguments: {ι : Type u_1} {l : Filter ι} {E : Type u_2} [NormedAddCommGroup E] {𝕜 : Type u_3} [RCLike 𝕜] [NormedSpace 𝕜 E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : ι → E → G} {g : E → G} {f' : ι → E → E →L[𝕜] G} {g' : E → E →L[𝕜] G} [l.NeBot] (hf' : TendstoUniformly f' g' l) (hf : ∀ (n : ι) (x : E), HasFDerivAt (f n) (f' n x) x) (hfg : ∀ (x : E), Filter.Tendsto (fun n => f n x) l (𝓝 (g x))) (x : E)
Goal: HasFDerivAt g (g' x) x
-- UniformCauchySeqOnFilter.one_smulRight
theorem UniformCauchySeqOnFilter.one_smulRight {l' : Filter 𝕜}
    (hf' : UniformCauchySeqOnFilter f' l l') :
    UniformCauchySeqOnFilter (fun n => fun z => (1 : 𝕜 →L[𝕜] 𝕜).smulRight (f' n z)) l l' 
Arguments: {ι : Type u_1} {l : Filter ι} {𝕜 : Type u_2} [RCLike 𝕜] {G : Type u_3} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f' : ι → 𝕜 → G} {l' : Filter 𝕜} (hf' : UniformCauchySeqOnFilter f' l l')
Goal: UniformCauchySeqOnFilter (fun n z => ContinuousLinearMap.smulRight 1 (f' n z)) l l'
-- uniformCauchySeqOnFilter_of_deriv
theorem uniformCauchySeqOnFilter_of_deriv (hf' : UniformCauchySeqOnFilter f' l (𝓝 x))
    (hf : ∀ᶠ n : ι × 𝕜 in l ×ˢ 𝓝 x, HasDerivAt (f n.1) (f' n.1 n.2) n.2)
    (hfg : Cauchy (map (fun n => f n x) l)) : UniformCauchySeqOnFilter f l (𝓝 x) 
Arguments: {ι : Type u_1} {l : Filter ι} {𝕜 : Type u_2} [RCLike 𝕜] {G : Type u_3} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f f' : ι → 𝕜 → G} {x : 𝕜} (hf' : UniformCauchySeqOnFilter f' l (𝓝 x)) (hf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasDerivAt (f n.1) (f' n.1 n.2) n.2) (hfg : Cauchy (Filter.map (fun n => f n x) l))
Goal: UniformCauchySeqOnFilter f l (𝓝 x)
-- uniformCauchySeqOn_ball_of_deriv
theorem uniformCauchySeqOn_ball_of_deriv {r : ℝ} (hf' : UniformCauchySeqOn f' l (Metric.ball x r))
    (hf : ∀ n : ι, ∀ y : 𝕜, y ∈ Metric.ball x r → HasDerivAt (f n) (f' n y) y)
    (hfg : Cauchy (map (fun n => f n x) l)) : UniformCauchySeqOn f l (Metric.ball x r) 
Arguments: {ι : Type u_1} {l : Filter ι} {𝕜 : Type u_2} [RCLike 𝕜] {G : Type u_3} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f f' : ι → 𝕜 → G} {x : 𝕜} {r : ℝ} (hf' : UniformCauchySeqOn f' l (Metric.ball x r)) (hf : ∀ (n : ι), ∀ y ∈ Metric.ball x r, HasDerivAt (f n) (f' n y) y) (hfg : Cauchy (Filter.map (fun n => f n x) l))
Goal: UniformCauchySeqOn f l (Metric.ball x r)
-- hasDerivAt_of_tendstoUniformlyOnFilter
theorem hasDerivAt_of_tendstoUniformlyOnFilter [NeBot l]
    (hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x))
    (hf : ∀ᶠ n : ι × 𝕜 in l ×ˢ 𝓝 x, HasDerivAt (f n.1) (f' n.1 n.2) n.2)
    (hfg : ∀ᶠ y in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))) : HasDerivAt g (g' x) x 
Arguments: {ι : Type u_1} {l : Filter ι} {𝕜 : Type u_2} [RCLike 𝕜] {G : Type u_3} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : ι → 𝕜 → G} {g : 𝕜 → G} {f' : ι → 𝕜 → G} {g' : 𝕜 → G} {x : 𝕜} [l.NeBot] (hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)) (hf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasDerivAt (f n.1) (f' n.1 n.2) n.2) (hfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Filter.Tendsto (fun n => f n y) l (𝓝 (g y)))
Goal: HasDerivAt g (g' x) x