File: Mathlib/Analysis/Calculus/FDeriv/Analytic.lean

full_name: HasFPowerSeriesAt.hasStrictFDerivAt
code: theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
    HasStrictFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p 1)) x 
goal: HasStrictFDerivAt f ((continuousMultilinearCurryFin1 𝕜 E F) (p 1)) x

full_name: HasFPowerSeriesAt.hasFDerivAt
code: theorem HasFPowerSeriesAt.hasFDerivAt (h : HasFPowerSeriesAt f p x) :
    HasFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p 1)) x 
goal: HasFDerivAt f ((continuousMultilinearCurryFin1 𝕜 E F) (p 1)) x

full_name: HasFPowerSeriesAt.differentiableAt
code: theorem HasFPowerSeriesAt.differentiableAt (h : HasFPowerSeriesAt f p x) : DifferentiableAt 𝕜 f x 
goal: DifferentiableAt 𝕜 f x

full_name: AnalyticAt.differentiableAt
code: theorem AnalyticAt.differentiableAt : AnalyticAt 𝕜 f x → DifferentiableAt 𝕜 f x
  
goal: AnalyticAt 𝕜 f x → DifferentiableAt 𝕜 f x

full_name: AnalyticAt.differentiableWithinAt
code: theorem AnalyticAt.differentiableWithinAt (h : AnalyticAt 𝕜 f x) : DifferentiableWithinAt 𝕜 f s x 
goal: DifferentiableWithinAt 𝕜 f s x

full_name: HasFPowerSeriesAt.fderiv_eq
code: theorem HasFPowerSeriesAt.fderiv_eq (h : HasFPowerSeriesAt f p x) :
    fderiv 𝕜 f x = continuousMultilinearCurryFin1 𝕜 E F (p 1) 
goal: fderiv 𝕜 f x = (continuousMultilinearCurryFin1 𝕜 E F) (p 1)

full_name: HasFPowerSeriesOnBall.differentiableOn
code: theorem HasFPowerSeriesOnBall.differentiableOn [CompleteSpace F]
    (h : HasFPowerSeriesOnBall f p x r) : DifferentiableOn 𝕜 f (EMetric.ball x r) 
goal: DifferentiableOn 𝕜 f (EMetric.ball x r)

full_name: AnalyticOn.differentiableOn
code: theorem AnalyticOn.differentiableOn (h : AnalyticOn 𝕜 f s) : DifferentiableOn 𝕜 f s 
goal: DifferentiableOn 𝕜 f s

full_name: HasFPowerSeriesOnBall.hasFDerivAt
code: theorem HasFPowerSeriesOnBall.hasFDerivAt [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
    {y : E} (hy : (‖y‖₊ : ℝ≥0∞) < r) :
    HasFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p.changeOrigin y 1)) (x + y) 
goal: HasFDerivAt f ((continuousMultilinearCurryFin1 𝕜 E F) (p.changeOrigin y 1)) (x + y)

full_name: HasFPowerSeriesOnBall.fderiv_eq
code: theorem HasFPowerSeriesOnBall.fderiv_eq [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
    {y : E} (hy : (‖y‖₊ : ℝ≥0∞) < r) :
    fderiv 𝕜 f (x + y) = continuousMultilinearCurryFin1 𝕜 E F (p.changeOrigin y 1) 
goal: fderiv 𝕜 f (x + y) = (continuousMultilinearCurryFin1 𝕜 E F) (p.changeOrigin y 1)

full_name: HasFPowerSeriesOnBall.fderiv
code: /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
    HasFPowerSeriesOnBall (fderiv 𝕜 f) p.derivSeries x r 
goal: HasFPowerSeriesOnBall (fderiv 𝕜 f) p.derivSeries x r

full_name: AnalyticOn.fderiv
code: /-- If a function is analytic on a set `s`, so is its Fréchet derivative. -/
theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) :
    AnalyticOn 𝕜 (fderiv 𝕜 f) s 
goal: AnalyticOn 𝕜 (fderiv 𝕜 f) s

full_name: AnalyticOn.iteratedFDeriv
code: /-- If a function is analytic on a set `s`, so are its successive Fréchet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) (n : ℕ) :
    AnalyticOn 𝕜 (iteratedFDeriv 𝕜 n f) s 
goal: AnalyticOn 𝕜 (iteratedFDeriv 𝕜 n f) s

full_name: AnalyticOn.contDiffOn
code: /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn 𝕜 f s) {n : ℕ∞} :
    ContDiffOn 𝕜 n f s 
goal: ContDiffOn 𝕜 n f s

full_name: AnalyticAt.contDiffAt
code: theorem AnalyticAt.contDiffAt [CompleteSpace F] (h : AnalyticAt 𝕜 f x) {n : ℕ∞} :
    ContDiffAt 𝕜 n f x 
goal: ContDiffAt 𝕜 n f x

full_name: HasFPowerSeriesAt.hasStrictDerivAt
code: protected theorem HasFPowerSeriesAt.hasStrictDerivAt (h : HasFPowerSeriesAt f p x) :
    HasStrictDerivAt f (p 1 fun _ => 1) x 
goal: HasStrictDerivAt f ((p 1) fun x => 1) x

full_name: HasFPowerSeriesAt.hasDerivAt
code: protected theorem HasFPowerSeriesAt.hasDerivAt (h : HasFPowerSeriesAt f p x) :
    HasDerivAt f (p 1 fun _ => 1) x 
goal: HasDerivAt f ((p 1) fun x => 1) x

full_name: HasFPowerSeriesAt.deriv
code: protected theorem HasFPowerSeriesAt.deriv (h : HasFPowerSeriesAt f p x) :
    deriv f x = p 1 fun _ => 1 
goal: deriv f x = (p 1) fun x => 1

full_name: AnalyticOn.deriv
code: /-- If a function is analytic on a set `s`, so is its derivative. -/
theorem AnalyticOn.deriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) : AnalyticOn 𝕜 (deriv f) s 
goal: AnalyticOn 𝕜 (deriv f) s

full_name: AnalyticOn.iterated_deriv
code: /-- If a function is analytic on a set `s`, so are its successive derivatives. -/
theorem AnalyticOn.iterated_deriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) (n : ℕ) :
    AnalyticOn 𝕜 (_root_.deriv^[n] f) s 
goal: AnalyticOn 𝕜 (deriv^[n] f) s

full_name: HasFiniteFPowerSeriesOnBall.differentiableOn
code: theorem HasFiniteFPowerSeriesOnBall.differentiableOn
    (h : HasFiniteFPowerSeriesOnBall f p x n r) : DifferentiableOn 𝕜 f (EMetric.ball x r) 
goal: DifferentiableOn 𝕜 f (EMetric.ball x r)

full_name: HasFiniteFPowerSeriesOnBall.hasFDerivAt
code: 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) 
goal: HasFDerivAt f ((continuousMultilinearCurryFin1 𝕜 E F) (p.changeOrigin y 1)) (x + y)

full_name: HasFiniteFPowerSeriesOnBall.fderiv_eq
code: 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) 
goal: fderiv 𝕜 f (x + y) = (continuousMultilinearCurryFin1 𝕜 E F) (p.changeOrigin y 1)

full_name: HasFiniteFPowerSeriesOnBall.fderiv
code: /-- If a function has a finite power series on a ball, then so does its derivative. -/
protected theorem HasFiniteFPowerSeriesOnBall.fderiv
    (h : HasFiniteFPowerSeriesOnBall f p x (n + 1) r) :
    HasFiniteFPowerSeriesOnBall (fderiv 𝕜 f) p.derivSeries x n r 
goal: HasFiniteFPowerSeriesOnBall (fderiv 𝕜 f) p.derivSeries x n r

full_name: HasFiniteFPowerSeriesOnBall.fderiv'
code: /-- If a function has a finite power series on a ball, then so does its derivative.
This is a variant of `HasFiniteFPowerSeriesOnBall.fderiv` where the degree of `f` is `< n`
and not `< n + 1`. -/
theorem HasFiniteFPowerSeriesOnBall.fderiv' (h : HasFiniteFPowerSeriesOnBall f p x n r) :
    HasFiniteFPowerSeriesOnBall (fderiv 𝕜 f) p.derivSeries x (n - 1) r 
goal: HasFiniteFPowerSeriesOnBall (fderiv 𝕜 f) p.derivSeries x (n - 1) r

full_name: CPolynomialOn.fderiv
code: /-- If a function is polynomial on a set `s`, so is its Fréchet derivative. -/
theorem CPolynomialOn.fderiv (h : CPolynomialOn 𝕜 f s) :
    CPolynomialOn 𝕜 (fderiv 𝕜 f) s 
goal: CPolynomialOn 𝕜 (fderiv 𝕜 f) s

full_name: CPolynomialOn.iteratedFDeriv
code: /-- If a function is polynomial on a set `s`, so are its successive Fréchet derivative. -/
theorem CPolynomialOn.iteratedFDeriv (h : CPolynomialOn 𝕜 f s) (n : ℕ) :
    CPolynomialOn 𝕜 (iteratedFDeriv 𝕜 n f) s 
goal: CPolynomialOn 𝕜 (iteratedFDeriv 𝕜 n f) s

full_name: CPolynomialOn.contDiffOn
code: /-- A polynomial function is infinitely differentiable. -/
theorem CPolynomialOn.contDiffOn (h : CPolynomialOn 𝕜 f s) {n : ℕ∞} :
    ContDiffOn 𝕜 n f s 
goal: ContDiffOn 𝕜 n f s

full_name: CPolynomialAt.contDiffAt
code: theorem CPolynomialAt.contDiffAt (h : CPolynomialAt 𝕜 f x) {n : ℕ∞} :
    ContDiffAt 𝕜 n f x 
goal: ContDiffAt 𝕜 n f x

full_name: CPolynomialOn.deriv
code: /-- If a function is polynomial on a set `s`, so is its derivative. -/
protected theorem CPolynomialOn.deriv (h : CPolynomialOn 𝕜 f s) : CPolynomialOn 𝕜 (deriv f) s 
goal: CPolynomialOn 𝕜 (deriv f) s

full_name: CPolynomialOn.iterated_deriv
code: /-- If a function is polynomial on a set `s`, so are its successive derivatives. -/
theorem CPolynomialOn.iterated_deriv (h : CPolynomialOn 𝕜 f s) (n : ℕ) :
    CPolynomialOn 𝕜 (deriv^[n] f) s 
goal: CPolynomialOn 𝕜 (deriv^[n] f) s

full_name: ContinuousMultilinearMap.hasFiniteFPowerSeriesOnBall
code: protected theorem hasFiniteFPowerSeriesOnBall :
    HasFiniteFPowerSeriesOnBall f f.toFormalMultilinearSeries 0 (Fintype.card ι + 1) ⊤ 
goal: HasFiniteFPowerSeriesOnBall (⇑f) f.toFormalMultilinearSeries 0 (Fintype.card ι + 1) ⊤

full_name: ContinuousMultilinearMap.changeOriginSeries_support
code: theorem changeOriginSeries_support {k l : ℕ} (h : k + l ≠ Fintype.card ι) :
    f.toFormalMultilinearSeries.changeOriginSeries k l = 0 
goal: f.toFormalMultilinearSeries.changeOriginSeries k l = 0

full_name: ContinuousMultilinearMap.changeOrigin_toFormalMultilinearSeries
code: theorem changeOrigin_toFormalMultilinearSeries [DecidableEq ι] :
    continuousMultilinearCurryFin1 𝕜 (∀ i, E i) F (f.toFormalMultilinearSeries.changeOrigin x 1) =
    f.linearDeriv x 
goal: (continuousMultilinearCurryFin1 𝕜 ((i : ι) → E i) F) (f.toFormalMultilinearSeries.changeOrigin x 1) = f.linearDeriv x

full_name: ContinuousMultilinearMap.hasFDerivAt
code: protected theorem hasFDerivAt [DecidableEq ι] : HasFDerivAt f (f.linearDeriv x) x 
goal: HasFDerivAt (⇑f) (f.linearDeriv x) x

full_name: _private.Mathlib.Analysis.Calculus.FDeriv.Analytic.0.Equiv.succ_embeddingFinSucc_fst_symm_apply
code: /-- Technical lemma used in the proof of `hasFTaylorSeriesUpTo_iteratedFDeriv`, to compare sums
over embedding of `Fin k` and `Fin (k + 1)`. -/
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⟩ 
goal: (((Equiv.embeddingFinSucc n ι) e).fst.toEquivRange.symm ⟨k, h'k⟩).succ = e.toEquivRange.symm ⟨k, hk⟩

full_name: ContinuousMultilinearMap.hasFTaylorSeriesUpTo_iteratedFDeriv
code: /-- A continuous multilinear function `f` admits a Taylor series, whose successive terms are given
by `f.iteratedFDeriv n`. This is the point of the definition of `f.iteratedFDeriv`. -/
theorem hasFTaylorSeriesUpTo_iteratedFDeriv :
    HasFTaylorSeriesUpTo ⊤ f (fun v n ↦ f.iteratedFDeriv n v) 
goal: HasFTaylorSeriesUpTo ⊤ ⇑f fun v n => f.iteratedFDeriv n v

full_name: ContinuousMultilinearMap.iteratedFDeriv_eq
code: theorem iteratedFDeriv_eq (n : ℕ) :
    iteratedFDeriv 𝕜 n f = f.iteratedFDeriv n 
goal: iteratedFDeriv 𝕜 n ⇑f = f.iteratedFDeriv n

full_name: ContinuousMultilinearMap.norm_iteratedFDeriv_le
code: 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) 
goal: ‖iteratedFDeriv 𝕜 n (⇑f) x‖ ≤ ↑((Fintype.card ι).descFactorial n) * ‖f‖ * ‖x‖ ^ (Fintype.card ι - n)

full_name: ContinuousMultilinearMap.cPolynomialAt
code: lemma cPolynomialAt : CPolynomialAt 𝕜 f x 
goal: CPolynomialAt 𝕜 (⇑f) x

full_name: ContinuousMultilinearMap.cPolyomialOn
code: lemma cPolyomialOn : CPolynomialOn 𝕜 f ⊤ 
goal: CPolynomialOn 𝕜 ⇑f ⊤

full_name: ContinuousMultilinearMap.contDiffAt
code: lemma contDiffAt : ContDiffAt 𝕜 n f x 
goal: ContDiffAt 𝕜 n (⇑f) x

full_name: ContinuousMultilinearMap.contDiff
code: lemma contDiff : ContDiff 𝕜 n f 
goal: ContDiff 𝕜 n ⇑f

full_name: FormalMultilinearSeries.derivSeries_apply_diag
code: theorem derivSeries_apply_diag (n : ℕ) (x : E) :
    derivSeries p n (fun _ ↦ x) x = (n + 1) • p (n + 1) fun _ ↦ x 
goal: ((p.derivSeries n) fun x_1 => x) x = (n + 1) • (p (n + 1)) fun x_1 => x

full_name: HasFPowerSeriesOnBall.iteratedFDeriv_zero_apply_diag
code: theorem iteratedFDeriv_zero_apply_diag : iteratedFDeriv 𝕜 0 f x = p 0 
goal: iteratedFDeriv 𝕜 0 f x = p 0

full_name: _private.Mathlib.Analysis.Calculus.FDeriv.Analytic.0.HasFPowerSeriesOnBall.factorial_smul'
code: 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) 
goal: HasFPowerSeriesOnBall f p x r → (n.factorial • (p n) fun x => y) = (iteratedFDeriv 𝕜 n f x) fun x => y

full_name: HasFPowerSeriesOnBall.factorial_smul
code: theorem factorial_smul (n : ℕ) :
    n ! • p n (fun _ ↦ y) = iteratedFDeriv 𝕜 n f x (fun _ ↦ y) 
goal: (n.factorial • (p n) fun x => y) = (iteratedFDeriv 𝕜 n f x) fun x => y

full_name: HasFPowerSeriesOnBall.hasSum_iteratedFDeriv
code: 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)) 
goal: HasSum (fun n => (↑n.factorial)⁻¹ • (iteratedFDeriv 𝕜 n f x) fun x => y) (f (x + y))