mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : 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
⊢ HasFiniteFPowerSeriesOnBall (fderiv 𝕜 f) p.derivSeries x n r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Add

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f g : 𝕜 → F
f' g' : F
L : Filter (𝕜 × 𝕜)
hf : HasDerivAtFilter f f' L
hg : HasDerivAtFilter g g' L
⊢ HasDerivAtFilter (f - g) (f' - g') L

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Mul

Goal state before:
pp: 𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
ι : Type u_2
inst✝² : DecidableEq ι
𝔸' : Type u_3
inst✝¹ : NormedCommRing 𝔸'
inst✝ : NormedAlgebra 𝕜 𝔸'
u : Finset ι
f : ι → 𝕜 → 𝔸'
hf : ∀ i ∈ u, DifferentiableWithinAt 𝕜 (f i) s x
⊢ derivWithin (fun x ↦ ∏ i ∈ u, f i x) s x = ∑ i ∈ u, (∏ j ∈ u.erase i, f j x) • derivWithin (f i) s x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Add

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f g : 𝕜 → F
x : 𝕜
s : Set 𝕜
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

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Add

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
x : 𝕜
s : Set 𝕜
c : F
⊢ derivWithin (fun y ↦ f y - c) s x = derivWithin f s x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Mul

Goal state before:
pp: 𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
ι : Type u_2
inst✝² : DecidableEq ι
𝔸' : Type u_3
inst✝¹ : NormedCommRing 𝔸'
inst✝ : NormedAlgebra 𝕜 𝔸'
u : Finset ι
f : ι → 𝕜 → 𝔸'
hf : ∀ i ∈ u, DifferentiableWithinAt 𝕜 (f i) s x
⊢ derivWithin (∏ i ∈ u, f i) s x = ∑ i ∈ u, (∏ j ∈ u.erase i, f j x) • derivWithin (f i) s x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Add

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
c : F
⊢ (derivWithin fun x ↦ f x - c) = derivWithin f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Add

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
x : 𝕜
c : F
⊢ deriv (fun y ↦ f y - c) x = deriv f x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Mul

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
ι : Type u_2
𝔸' : Type u_3
inst✝¹ : NormedCommRing 𝔸'
inst✝ : NormedAlgebra 𝕜 𝔸'
u : Finset ι
f : ι → 𝕜 → 𝔸'
hd : ∀ i ∈ u, DifferentiableAt 𝕜 (f i) x
⊢ DifferentiableAt 𝕜 (fun x ↦ ∏ i ∈ u, f i x) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Add

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
c : F
⊢ (deriv fun x ↦ f x - c) = deriv f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
h : HasFiniteFPowerSeriesOnBall f p x 0 r
y : E
hy : y ∈ Metric.eball x r
z : E
hz : z ∈ Metric.eball x r
⊢ f z = (fun x ↦ 0) z

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Add

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
f' : F
L : Filter (𝕜 × 𝕜)
c : F
hf : HasDerivAtFilter f f' L
⊢ HasDerivAtFilter (fun x ↦ c - f x) (-f') L

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Mul

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
ι : Type u_2
𝔸' : Type u_3
inst✝¹ : NormedCommRing 𝔸'
inst✝ : NormedAlgebra 𝕜 𝔸'
u : Finset ι
f : ι → 𝕜 → 𝔸'
hd : ∀ i ∈ u, DifferentiableAt 𝕜 (f i) x
⊢ DifferentiableAt 𝕜 (∏ i ∈ u, f i) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Add

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
x : 𝕜
s : Set 𝕜
c : F
⊢ derivWithin (fun y ↦ c - f y) s x = -derivWithin f s x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Add

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
x : 𝕜
c : F
⊢ deriv (fun x ↦ c - f x) x = -deriv f x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Mul

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
ι : Type u_2
𝔸' : Type u_3
inst✝¹ : NormedCommRing 𝔸'
inst✝ : NormedAlgebra 𝕜 𝔸'
u : Finset ι
f : ι → 𝕜 → 𝔸'
hd : ∀ i ∈ u, DifferentiableWithinAt 𝕜 (f i) s x
⊢ DifferentiableWithinAt 𝕜 (fun x ↦ ∏ i ∈ u, f i x) s x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Add

Goal state before:
pp: 𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
x c : 𝕜
⊢ deriv (fun x ↦ c - x) x = -1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Add

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
a b : 𝕜
⊢ DifferentiableAt 𝕜 (fun x ↦ f (x - b)) a ↔ DifferentiableAt 𝕜 f (a - b)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Add

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
a b : 𝕜
⊢ DifferentiableAt 𝕜 (fun x ↦ f (b - x)) a ↔ DifferentiableAt 𝕜 f (b - a)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Mul

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
ι : Type u_2
𝔸' : Type u_3
inst✝¹ : NormedCommRing 𝔸'
inst✝ : NormedAlgebra 𝕜 𝔸'
u : Finset ι
f : ι → 𝕜 → 𝔸'
hd : ∀ i ∈ u, DifferentiableWithinAt 𝕜 (f i) s x
⊢ DifferentiableWithinAt 𝕜 (∏ i ∈ u, f i) s x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Add

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
a b : 𝕜
⊢ DifferentiableAt 𝕜 f a ↔ DifferentiableAt 𝕜 (fun x ↦ f (x - b)) (a + b)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Add

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
a b : 𝕜
⊢ DifferentiableAt 𝕜 f a ↔ DifferentiableAt 𝕜 (fun x ↦ f (b - x)) (b - a)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Mul

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
𝕜' : Type u_2
inst✝¹ : NormedDivisionRing 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c : 𝕜 → 𝕜'
c' : 𝕜'
hc : HasDerivAt c c' x
d : 𝕜'
⊢ HasDerivAt (fun x ↦ c x / d) (c' / d) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Mul

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
𝕜' : Type u_2
inst✝¹ : NormedDivisionRing 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c : 𝕜 → 𝕜'
c' : 𝕜'
hc : HasDerivAt c c' x
d : 𝕜'
⊢ HasDerivAt (fun x ↦ c x / d) (c' / d) x

Tactic:
by
  simpa only [div_eq_mul_inv] using hc.mul_const d⁻¹

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Mul

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
𝕜' : Type u_2
inst✝¹ : NormedDivisionRing 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c : 𝕜 → 𝕜'
c' : 𝕜'
hc : HasDerivWithinAt c c' s x
d : 𝕜'
⊢ HasDerivWithinAt (fun x ↦ c x / d) (c' / d) s x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Mul

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
𝕜' : Type u_2
inst✝¹ : NormedDivisionRing 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c : 𝕜 → 𝕜'
c' : 𝕜'
hc : HasDerivWithinAt c c' s x
d : 𝕜'
⊢ HasDerivWithinAt (fun x ↦ c x / d) (c' / d) s x

Tactic:
by
  simpa only [div_eq_mul_inv] using hc.mul_const d⁻¹

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Mul

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
𝕜' : Type u_2
inst✝¹ : NormedDivisionRing 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c : 𝕜 → 𝕜'
c' : 𝕜'
hc : HasStrictDerivAt c c' x
d : 𝕜'
⊢ HasStrictDerivAt (fun x ↦ c x / d) (c' / d) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Mul

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
𝕜' : Type u_2
inst✝¹ : NormedDivisionRing 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c : 𝕜 → 𝕜'
c' : 𝕜'
hc : HasStrictDerivAt c c' x
d : 𝕜'
⊢ HasStrictDerivAt (fun x ↦ c x / d) (c' / d) x

Tactic:
by
  simpa only [div_eq_mul_inv] using hc.mul_const d⁻¹

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Mul

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
𝕜' : Type u_2
inst✝¹ : NormedDivisionRing 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c : 𝕜 → 𝕜'
d : 𝕜'
⊢ derivWithin (fun x ↦ c x / d) s x = derivWithin c s x / d

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Mul

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
𝕜' : Type u_2
inst✝¹ : NormedDivisionRing 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c : 𝕜 → 𝕜'
d : 𝕜'
⊢ deriv (fun x ↦ c x / d) x = deriv c x / d

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Polynomial

Goal state before:
pp: 𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
p : 𝕜[X]
x : 𝕜
⊢ HasStrictDerivAt (fun x ↦ eval x p) (eval x (derivative p)) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Polynomial

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : Algebra R 𝕜
q : R[X]
x : 𝕜
⊢ HasStrictDerivAt (fun x ↦ (aeval x) q) ((aeval x) (derivative q)) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : 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
⊢ HasFiniteFPowerSeriesOnBall (fderiv 𝕜 f) p.derivSeries x (n - 1) r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Mul

Goal state before:
pp: 𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
E : Type w
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
x : 𝕜
G : Type u_2
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
c : 𝕜 → F →L[𝕜] G
c' : F →L[𝕜] G
d : 𝕜 → E →L[𝕜] F
d' : E →L[𝕜] F
hc : HasStrictDerivAt c c' x
hd : HasStrictDerivAt d d' x
⊢ HasStrictDerivAt (fun y ↦ c y ∘SL d y) (c' ∘SL d x + c x ∘SL d') x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Polynomial

Goal state before:
pp: 𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
p : 𝕜[X]
hxs : UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (fun x ↦ eval x p) s x = eval x (derivative p)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Polynomial

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : Algebra R 𝕜
q : R[X]
hxs : UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (fun x ↦ (aeval x) q) s x = (aeval x) (derivative q)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Mul

Goal state before:
pp: 𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
E : Type w
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
x : 𝕜
s : Set 𝕜
G : Type u_2
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
c : 𝕜 → F →L[𝕜] G
c' : F →L[𝕜] G
d : 𝕜 → E →L[𝕜] F
d' : E →L[𝕜] F
hc : HasDerivWithinAt c c' s x
hd : HasDerivWithinAt d d' s x
⊢ HasDerivWithinAt (fun y ↦ c y ∘SL d y) (c' ∘SL d x + c x ∘SL d') s x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
s : Set E
h : CPolynomialOn 𝕜 f s
⊢ CPolynomialOn 𝕜 (_root_.fderiv 𝕜 f) s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
s : Set E
h : CPolynomialOn 𝕜 f s
n : ℕ
IH : CPolynomialOn 𝕜 (_root_.iteratedFDeriv 𝕜 n f) s
⊢ (E →L[𝕜] E [×n]→L[𝕜] F) →L[𝕜] E [×n + 1]→L[𝕜] F

Tactic:
exact ↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun _ : Fin (n + 1) ↦ E) F).symm

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Mul

Goal state before:
pp: 𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
E : Type w
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
x : 𝕜
G : Type u_2
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
c : 𝕜 → F →L[𝕜] G
c' : F →L[𝕜] G
d : 𝕜 → E →L[𝕜] F
d' : E →L[𝕜] F
hc : HasDerivAt c c' x
hd : HasDerivAt d d' x
⊢ HasDerivAt (fun y ↦ c y ∘SL d y) (c' ∘SL d x + c x ∘SL d') x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
s : Set E
h : CPolynomialOn 𝕜 f s
n : ℕ
⊢ CPolynomialOn 𝕜 (_root_.iteratedFDeriv 𝕜 n f) s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Mul

Goal state before:
pp: 𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
E : Type w
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
x : 𝕜
s : Set 𝕜
G : Type u_2
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
c : 𝕜 → F →L[𝕜] G
d : 𝕜 → E →L[𝕜] F
hc : DifferentiableWithinAt 𝕜 c s x
hd : DifferentiableWithinAt 𝕜 d s x
⊢ derivWithin (fun y ↦ c y ∘SL d y) s x = derivWithin c s x ∘SL d x + c x ∘SL derivWithin d s x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
s : Set 𝕜
h : CPolynomialOn 𝕜 f s
n : ℕ
⊢ CPolynomialOn 𝕜 (deriv^[n] f) s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Mul

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
x : 𝕜
G : Type u_2
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
c : 𝕜 → F →L[𝕜] G
c' : F →L[𝕜] G
u : 𝕜 → F
u' : F
hc : HasStrictDerivAt c c' x
hu : HasStrictDerivAt u u' x
⊢ HasStrictDerivAt (fun y ↦ (c y) (u y)) (c' (u x) + (c x) u') x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Mul

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
x : 𝕜
s : Set 𝕜
G : Type u_2
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
c : 𝕜 → F →L[𝕜] G
c' : F →L[𝕜] G
u : 𝕜 → F
u' : F
hc : HasDerivWithinAt c c' s x
hu : HasDerivWithinAt u u' s x
⊢ HasDerivWithinAt (fun y ↦ (c y) (u y)) (c' (u x) + (c x) u') s x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
k l : ℕ
h : k + l ≠ Fintype.card ι
x✝¹ : { s // s.card = l }
x✝ : x✝¹ ∈ Finset.univ
⊢ f.toFormalMultilinearSeries.changeOriginSeriesTerm k l ↑x✝¹ ⋯ = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Mul

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
x : 𝕜
G : Type u_2
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
c : 𝕜 → F →L[𝕜] G
c' : F →L[𝕜] G
u : 𝕜 → F
u' : F
hc : HasDerivAt c c' x
hu : HasDerivAt u u' x
⊢ HasDerivAt (fun y ↦ (c y) (u y)) (c' (u x) + (c x) u') x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Mul

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
x : 𝕜
s : Set 𝕜
G : Type u_2
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
c : 𝕜 → F →L[𝕜] G
u : 𝕜 → F
hc : DifferentiableWithinAt 𝕜 c s x
hu : DifferentiableWithinAt 𝕜 u s x
⊢ derivWithin (fun y ↦ (c y) (u y)) s x = (derivWithin c s x) (u x) + (c x) (derivWithin u s x)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝³ : (i : ι) → NormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
x : (i : ι) → E i
inst✝ : DecidableEq ι
y : (i : ι) → E i
h✝ : IsEmpty ι
l : ℕ
⊢ 1 + l ≠ Fintype.card ι

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝³ : (i : ι) → NormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
x : (i : ι) → E i
inst✝ : DecidableEq ι
y : (i : ι) → E i
h✝ : Nonempty ι
m : ℕ
hm : ¬m = Fintype.card ι - 1
⊢ 1 ≤ Fintype.card ι

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝³ : (i : ι) → NormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
x : (i : ι) → E i
inst✝ : DecidableEq ι
y : (i : ι) → E i
h✝ : Nonempty ι
x✝ : Fin (Nat.succ 0 + (Fintype.card ι - Nat.succ 0))
⊢ #{x✝}ᶜ = Fintype.card ι - 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝³ : (i : ι) → NormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
x : (i : ι) → E i
inst✝ : DecidableEq ι
y : (i : ι) → E i
h✝ : Nonempty ι
s : Finset (Fin (1 + (Fintype.card ι - 1)))
hs : #s = Fintype.card ι - 1
⊢ #sᶜ = 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝³ : (i : ι) → NormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
x : (i : ι) → E i
inst✝ : DecidableEq ι
⊢ (continuousMultilinearCurryFin1 𝕜 ((i : ι) → E i) F) (f.toFormalMultilinearSeries.changeOrigin x 1) = f.linearDeriv x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝³ : (i : ι) → NormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
x : (i : ι) → E i
inst✝ : DecidableEq ι
⊢ HasStrictFDerivAt (⇑f) (f.linearDeriv x) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝³ : (i : ι) → NormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
fa : ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i)
f : ContinuousMultilinearMap 𝕜 (fun x ↦ ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i)) F :=
  (ContinuousLinearMap.id 𝕜 (ContinuousMultilinearMap 𝕜 E F)).continuousMultilinearMapOption
Hf :
  HasStrictFDerivAt (fun x ↦ f fun x_1 ↦ x)
    ((f.linearDeriv fun x ↦ fa) ∘SL
      ContinuousLinearMap.pi fun x ↦ ContinuousLinearMap.id 𝕜 (ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i)))
    fa
e_4✝ : Prod.instAddCommGroup = Prod.normedAddCommGroup.toAddCommGroup
e_5✝ : Prod.instModule ≍ Prod.normedSpace.toModule
e_6✝ : instTopologicalSpaceProd = PseudoMetricSpace.toUniformSpace.toTopologicalSpace
g : ContinuousMultilinearMap 𝕜 E F
this :
  ∑ i, fa.1 (Function.update fa.2 i 0) = ∑ i, fa.1 fun j ↦ (Function.update (fun x ↦ fa) (some i) (g, 0) (some j)).2 j
⊢ ((apply 𝕜 E F fa.2 ∘SL ContinuousLinearMap.fst 𝕜 (ContinuousMultilinearMap 𝕜 E F) ((i : ι) → E i) +
          fa.1.linearDeriv fa.2 ∘SL ContinuousLinearMap.snd 𝕜 (ContinuousMultilinearMap 𝕜 E F) ((i : ι) → E i)) ∘SL
        ContinuousLinearMap.inl 𝕜 (ContinuousMultilinearMap 𝕜 E F) ((i : ι) → E i))
      g =
    (((f.linearDeriv fun x ↦ fa) ∘SL
          ContinuousLinearMap.pi fun x ↦
            ContinuousLinearMap.id 𝕜 (ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i))) ∘SL
        ContinuousLinearMap.inl 𝕜 (ContinuousMultilinearMap 𝕜 E F) ((i : ι) → E i))
      g

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Uniqueness

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
y : E
c : ℝ
c_pos : c > 0
t : Set E
t_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
δ : ℝ
δ_pos : δ > 0
δε : Metric.ball 0 δ ⊆ t
p : E [×0]→L[𝕜] F
ht : ∀ y ∈ t, ‖p fun x ↦ y‖ ≤ c * ‖‖y‖ ^ (0 + 1)‖
⊢ ‖p fun x ↦ y‖ = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Analytic.Uniqueness

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
y : E
c : ℝ
c_pos : c > 0
t : Set E
t_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
δ : ℝ
δ_pos : δ > 0
δε : Metric.ball 0 δ ⊆ t
n : ℕ
p : E [×n + 1]→L[𝕜] F
ht : ∀ y ∈ t, ‖p fun x ↦ y‖ ≤ c * ‖‖y‖ ^ (n + 1 + 1)‖
hy : y = 0
⊢ (p fun x ↦ y) = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.DiffContOnCl

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜 F
f : E → F
inst✝ : NormedSpace ℝ E
x : E
r : ℝ
h : DiffContOnCl 𝕜 f (ball x r)
⊢ ContinuousOn f (closedBall x r)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Slope

Goal state before:
pp: k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → PE
a : k
⊢ slope f a a = 0

Tactic:
by

Goal state after: