mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Asymptotics.TVS

Goal state before:
pp: α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : Module 𝕜 E
inst✝³ : AddCommGroup F
inst✝² : TopologicalSpace F
inst✝¹ : Module 𝕜 F
l : Filter α
f : α → E
g : α → F
inst✝ : ContinuousConstSMul 𝕜 F
ι : Sort u_7
p : ι → Prop
U : ι → Set E
hb : (𝓝 0).HasBasis p U
h : ∀ (i : ι), p i → ∃ C, ∃ V ∈ 𝓝 0, (fun x ↦ egauge 𝕜 (U i) (f x)) ≤ᶠ[l] fun x ↦ ↑C * egauge 𝕜 V (g x)
i : ι
hi : p i
C : ℝ≥0
V : Set F
hV₀ : V ∈ 𝓝 0
hV : (fun x ↦ egauge 𝕜 (U i) (f x)) ≤ᶠ[l] fun x ↦ ↑C * egauge 𝕜 V (g x)
c : 𝕜
hc : C < ‖c‖₊
hc₀ : c ≠ 0
⊢ ∃ j ∈ 𝓝 0, ∀ᶠ (x : α) in l, egauge 𝕜 (U i) (f x) ≤ egauge 𝕜 (id j) (g x)

Tactic:
refine ⟨c⁻¹ • V, (set_smul_mem_nhds_zero_iff <| inv_ne_zero hc₀).mpr hV₀, ?_⟩

Goal state after:
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : Module 𝕜 E
inst✝³ : AddCommGroup F
inst✝² : TopologicalSpace F
inst✝¹ : Module 𝕜 F
l : Filter α
f : α → E
g : α → F
inst✝ : ContinuousConstSMul 𝕜 F
ι : Sort u_7
p : ι → Prop
U : ι → Set E
hb : (𝓝 0).HasBasis p U
h : ∀ (i : ι), p i → ∃ C, ∃ V ∈ 𝓝 0, (fun x ↦ egauge 𝕜 (U i) (f x)) ≤ᶠ[l] fun x ↦ ↑C * egauge 𝕜 V (g x)
i : ι
hi : p i
C : ℝ≥0
V : Set F
hV₀ : V ∈ 𝓝 0
hV : (fun x ↦ egauge 𝕜 (U i) (f x)) ≤ᶠ[l] fun x ↦ ↑C * egauge 𝕜 V (g x)
c : 𝕜
hc : C < ‖c‖₊
hc₀ : c ≠ 0
⊢ ∀ᶠ (x : α) in l, egauge 𝕜 (U i) (f x) ≤ egauge 𝕜 (id (c⁻¹ • V)) (g x)
Module: Mathlib.Analysis.Calculus.FDeriv.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
F : Type u_3
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 F
inst✝ : TopologicalSpace F
f : E → F
x : E
s : Set E
h : DifferentiableWithinAt 𝕜 f s x
⊢ HasFDerivWithinAt f (fderivWithin 𝕜 f s x) s x

Tactic:
by
  simp only [fderivWithin, dif_pos h]
  split_ifs with h₀
  exacts [h₀, Classical.choose_spec h]

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Multilinear.Curry

Goal state before:
pp: 𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
Ei : Fin n.succ → Type wEi
G : Type wG
G' : Type wG'
inst✝¹¹ : Fintype ι
inst✝¹⁰ : Fintype ι'
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁶ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)
inst✝⁵ : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : NormedAddCommGroup G'
inst✝¹ : NormedSpace 𝕜 G'
f : ContinuousMultilinearMap 𝕜 Ei G
inst✝ : DecidableEq (Fin n)
m : (i : Fin n) → Ei i.castSucc
i : Fin n
x y : Ei i.castSucc
⊢ (f.curryRight (update m i (x + y))).mkContinuous (‖f‖ * ∏ i_1, ‖update m i (x + y) i_1‖) ⋯ =
    (f.curryRight (update m i x)).mkContinuous (‖f‖ * ∏ i_1, ‖update m i x i_1‖) ⋯ +
      (f.curryRight (update m i y)).mkContinuous (‖f‖ * ∏ i_1, ‖update m i y i_1‖) ⋯

Tactic:
ext
        simp

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Multilinear.Curry

Goal state before:
pp: 𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
Ei : Fin n.succ → Type wEi
G : Type wG
G' : Type wG'
inst✝¹¹ : Fintype ι
inst✝¹⁰ : Fintype ι'
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁶ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)
inst✝⁵ : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : NormedAddCommGroup G'
inst✝¹ : NormedSpace 𝕜 G'
f : ContinuousMultilinearMap 𝕜 Ei G
inst✝ : DecidableEq (Fin n)
m : (i : Fin n) → Ei i.castSucc
i : Fin n
c : 𝕜
x : Ei i.castSucc
⊢ (f.curryRight (update m i (c • x))).mkContinuous (‖f‖ * ∏ i_1, ‖update m i (c • x) i_1‖) ⋯ =
    c • (f.curryRight (update m i x)).mkContinuous (‖f‖ * ∏ i_1, ‖update m i x i_1‖) ⋯

Tactic:
ext
        simp

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

Goal state before:
pp: case refine_2
𝕜 : Type u_1
E : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing E
inst✝¹ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
inst✝ : NormOneClass E
hc : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop atTop
r : ℝ≥0
hr : ↑r < (ofScalars E c).radius
this : 0 < r
⊢ ∀ᶠ (n : ℕ) in atTop, 2 * ‖‖ofScalars E c n‖ * ↑r ^ n‖ ≤ ‖‖ofScalars E c (n + 1)‖ * ↑r ^ (n + 1)‖

Tactic:
filter_upwards [hc.eventually_ge_atTop (2 * r⁻¹), eventually_ne_atTop 0] with n hc hn

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing E
inst✝¹ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
inst✝ : NormOneClass E
hc✝ : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop atTop
r : ℝ≥0
hr : ↑r < (ofScalars E c).radius
this : 0 < r
n : ℕ
hc : 2 * ↑r⁻¹ ≤ ‖c n.succ‖ / ‖c n‖
hn : n ≠ 0
⊢ 2 * ‖‖ofScalars E c n‖ * ↑r ^ n‖ ≤ ‖‖ofScalars E c (n + 1)‖ * ↑r ^ (n + 1)‖
Module: Mathlib.Analysis.Analytic.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x y : E
⊢ HasFPowerSeriesWithinAt f p (insert y s) x ↔ HasFPowerSeriesWithinAt f p s x

Tactic:
rcases eq_or_ne x y with rfl | hy
  · simp [HasFPowerSeriesWithinAt]
  · refine ⟨fun h ↦ h.mono (subset_insert _ _), fun h ↦ ?_⟩
    apply HasFPowerSeriesWithinAt.mono_of_mem_nhdsWithin h
    rw [nhdsWithin_insert_of_ne hy]
    exact self_mem_nhdsWithin

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesWithinOnBall f p s x r
⊢ TendstoLocallyUniformlyOn (fun n y ↦ p.partialSum n (y - x)) f atTop (insert x s ∩ Metric.eball x r)

Tactic:
have A : ContinuousOn (fun y : E => y - x) (insert x s ∩ Metric.eball (x : E) r) := by fun_prop
  convert! hf.tendstoLocallyUniformlyOn.comp (fun y : E => y - x) _ A using 1
  · ext z
    simp
  · intro z
    simp [edist_eq_enorm_sub]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesWithinOnBall f p s x r
⊢ TendstoLocallyUniformlyOn (fun n y ↦ p.partialSum n (y - x)) f atTop (insert x s ∩ Metric.eball x r)

Tactic:
by
  have A : ContinuousOn (fun y : E => y - x) (insert x s ∩ Metric.eball (x : E) r) := by fun_prop
  convert! hf.tendstoLocallyUniformlyOn.comp (fun y : E => y - x) _ A using 1
  · ext z
    simp
  · intro z
    simp [edist_eq_enorm_sub]

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

Goal state before:
pp: case inl
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
inst✝ : CompleteSpace F
h : p.radius = 0
⊢ ContinuousOn p.sum (Metric.eball 0 p.radius)

Tactic:
simp [h, continuousOn_empty]

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

Goal state before:
pp: case inr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
inst✝ : CompleteSpace F
h : 0 < p.radius
⊢ ContinuousOn p.sum (Metric.eball 0 p.radius)

Tactic:
· exact (p.hasFPowerSeriesOnBall h).continuousOn

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
m n : ℕ
g : F → G
f : E → F
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
x : E
hg : HasFiniteFPowerSeriesAt g q (f x) m
hf : HasFiniteFPowerSeriesAt f p x n
hn : 0 < n
r : ℝ≥0∞
hr : HasFPowerSeriesOnBall (g ∘ f) (q.comp p) x r
i : ℕ
hi : m * n ≤ i
c : Composition i
v : Fin i → E
hc : c.length < m
⊢ ∃ j, n ≤ c.blocksFun j

Tactic:
contrapose! hi

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
m n : ℕ
g : F → G
f : E → F
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
x : E
hg : HasFiniteFPowerSeriesAt g q (f x) m
hf : HasFiniteFPowerSeriesAt f p x n
hn : 0 < n
r : ℝ≥0∞
hr : HasFPowerSeriesOnBall (g ∘ f) (q.comp p) x r
i : ℕ
c : Composition i
v : Fin i → E
hc : c.length < m
hi : ∀ (j : Fin c.length), c.blocksFun j < n
⊢ i < m * n
Module: Mathlib.Analysis.Analytic.Within

Goal state before:
pp: case refine_2
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
f : E → F
s : Set E
x : E
h : AnalyticWithinAt 𝕜 f s x
g : E → F
h'g : EqOn f g (insert x s)
hg : AnalyticAt 𝕜 g x
u : Set E := insert x s ∩ {y | AnalyticAt 𝕜 g y}
y : E
hy : y ∈ u
this : AnalyticWithinAt 𝕜 g u y
⊢ AnalyticWithinAt 𝕜 f u y

Tactic:
exact this.congr (h'g.mono (inter_subset_left)) (h'g (inter_subset_left hy))

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

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
f : E → F
x : E
s : Set E
h : IsBoundedLinearMap 𝕜 f
hxs : UniqueDiffWithinAt 𝕜 s x
⊢ fderivWithin 𝕜 f s x = toContinuousLinearMap f h

Tactic:
DifferentiableAt.fderivWithin h.differentiableAt hxs

Goal state after:
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
f : E → F
x : E
s : Set E
h : IsBoundedLinearMap 𝕜 f
hxs : UniqueDiffWithinAt 𝕜 s x
⊢ fderiv 𝕜 f x = toContinuousLinearMap f h
Module: Mathlib.Analysis.Calculus.FDeriv.Comp

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E → F
f' : E →L[𝕜] F
x : E
g : F → G
g' : F →L[𝕜] G
t : Set F
y : F
hg : HasFDerivWithinAt g g' t y
hf : HasFDerivAt f f' x
ht : ∀ᶠ (x' : E) in 𝓝 x, f x' ∈ t
hy : y = f x
⊢ HasFDerivAt (g ∘ f) (g' ∘SL f') x

Tactic:
subst y

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E → F
f' : E →L[𝕜] F
x : E
g : F → G
g' : F →L[𝕜] G
t : Set F
hf : HasFDerivAt f f' x
ht : ∀ᶠ (x' : E) in 𝓝 x, f x' ∈ t
hg : HasFDerivWithinAt g g' t (f x)
⊢ HasFDerivAt (g ∘ f) (g' ∘SL f') x
Module: Mathlib.Analysis.Calculus.FDeriv.Comp

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
g : F → G
f : E → F
x : E
y : F
s : Set E
t : Set F
hg : DifferentiableWithinAt 𝕜 g t y
hf : DifferentiableWithinAt 𝕜 f s x
h : MapsTo f s t
hxs : UniqueDiffWithinAt 𝕜 s x
hy : f x = y
v : E
⊢ (fderivWithin 𝕜 g t y) ((fderivWithin 𝕜 f s x) v) = (fderivWithin 𝕜 (g ∘ f) s x) v

Tactic:
subst y

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
g : F → G
f : E → F
x : E
s : Set E
t : Set F
hf : DifferentiableWithinAt 𝕜 f s x
h : MapsTo f s t
hxs : UniqueDiffWithinAt 𝕜 s x
v : E
hg : DifferentiableWithinAt 𝕜 g t (f x)
⊢ (fderivWithin 𝕜 g t (f x)) ((fderivWithin 𝕜 f s x) v) = (fderivWithin 𝕜 (g ∘ f) s x) v
Module: Mathlib.Analysis.Analytic.Constructions

Goal state before:
pp: 𝕜 : Type u_2
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
ι : Type u_9
inst✝² : Fintype ι
Fm : ι → Type u_10
inst✝¹ : (i : ι) → NormedAddCommGroup (Fm i)
inst✝ : (i : ι) → NormedSpace 𝕜 (Fm i)
p : (i : ι) → FormalMultilinearSeries 𝕜 E (Fm i)
⊢ (pi p).radius = ⨅ i, (p i).radius

Tactic:
refine le_antisymm (by simp [radius_pi_le]) ?_

Goal state after:
𝕜 : Type u_2
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
ι : Type u_9
inst✝² : Fintype ι
Fm : ι → Type u_10
inst✝¹ : (i : ι) → NormedAddCommGroup (Fm i)
inst✝ : (i : ι) → NormedSpace 𝕜 (Fm i)
p : (i : ι) → FormalMultilinearSeries 𝕜 E (Fm i)
⊢ ⨅ i, (p i).radius ≤ (pi p).radius
Module: Mathlib.Analysis.Calculus.FDeriv.Equiv

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
iso : E ≃L[𝕜] F
f : F → G
x : E
f' : F →L[𝕜] G
⊢ HasFDerivAt (f ∘ ⇑iso) (f' ∘SL ↑iso) x ↔ HasFDerivAt f f' (iso x)

Tactic:
by
  simp only [← hasFDerivWithinAt_univ, ← comp_right_hasFDerivWithinAt_iff, preimage_univ]

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Alternating.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
ι : Type u_4
inst✝⁹ : Semiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : Module R M
inst✝⁶ : TopologicalSpace M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : TopologicalSpace N
inst✝² : IsTopologicalAddGroup N
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f✝ f : ContinuousMultilinearMap R (fun x ↦ M) N
v : ι → M
i j : ι
hv : v i = v j
hne : i ≠ j
⊢ (∑ σ, Equiv.Perm.sign σ • domDomCongr σ f).toFun v = 0

Tactic:
by
        simpa [MultilinearMap.alternatization_apply]
          using f.1.alternatization.map_eq_zero_of_eq' v i j hv hne

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
p : E → FormalMultilinearSeries 𝕜 E F
H : ContinuousOn f s ∧ ∀ x ∈ s, (p x 0).curry0 = f x
hm : ↑0 ≤ 0
this : EqOn (fun x ↦ p x 0) (⇑(continuousMultilinearCurryFin0 𝕜 E F).symm ∘ f) s
⊢ ContinuousOn (fun x ↦ p x 0) s

Tactic:
continuousOn_congr this,

Goal state after:
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
p : E → FormalMultilinearSeries 𝕜 E F
H : ContinuousOn f s ∧ ∀ x ∈ s, (p x 0).curry0 = f x
hm : ↑0 ≤ 0
this : EqOn (fun x ↦ p x 0) (⇑(continuousMultilinearCurryFin0 𝕜 E F).symm ∘ f) s
⊢ ContinuousOn (⇑(continuousMultilinearCurryFin0 𝕜 E F).symm ∘ f) s
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
f : E → F
x : E
s : Set E
h : HasFPowerSeriesWithinAt f p s x
⊢ (fun y ↦
      f y.1 - f y.2 - ((continuousMultilinearCurryFin1 𝕜 E F) (p 1)) (y.1 - y.2)) =o[𝓝[insert x s ×ˢ insert x s] (x, x)]
    fun y ↦ y.1 - y.2

Tactic:
refine h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right ?_)
  refine isLittleO_iff_exists_eq_mul.2 ⟨fun y => ‖y - (x, x)‖, ?_, EventuallyEq.rfl⟩
  apply Tendsto.mono_left _ nhdsWithin_le_nhds
  refine (continuous_id.fun_sub continuous_const).norm.tendsto' _ _ ?_
  rw [_root_.id, sub_self, norm_zero]

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries

Goal state before:
pp: case zero
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
⊢ tsupport (iteratedFDeriv 𝕜 0 f) ⊆ tsupport f

Tactic:
rw [iteratedFDeriv_zero_eq_comp]

Goal state after:
case zero
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
⊢ tsupport (⇑(continuousMultilinearCurryFin0 𝕜 E F).symm ∘ f) ⊆ tsupport f
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
G : Type u_1
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
x : 𝕜
B : E →L[𝕜] F →L[𝕜] G
u : 𝕜 → E
v : 𝕜 → F
u' : E
v' : F
hu : x ∈ tsupport v → HasDerivAt u u' x
hv : x ∈ tsupport u → HasDerivAt v v' x
⊢ HasDerivAt (fun x ↦ (B (u x)) (v x)) ((B (u x)) v' + (B u') (v x)) x

Tactic:
by_cases hxu : x ∈ tsupport u

Goal state after:
case pos
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
G : Type u_1
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
x : 𝕜
B : E →L[𝕜] F →L[𝕜] G
u : 𝕜 → E
v : 𝕜 → F
u' : E
v' : F
hu : x ∈ tsupport v → HasDerivAt u u' x
hv : x ∈ tsupport u → HasDerivAt v v' x
hxu : x ∈ tsupport u
⊢ HasDerivAt (fun x ↦ (B (u x)) (v x)) ((B (u x)) v' + (B u') (v x)) x

case neg
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
G : Type u_1
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
x : 𝕜
B : E →L[𝕜] F →L[𝕜] G
u : 𝕜 → E
v : 𝕜 → F
u' : E
v' : F
hu : x ∈ tsupport v → HasDerivAt u u' x
hv : x ∈ tsupport u → HasDerivAt v v' x
hxu : x ∉ tsupport u
⊢ HasDerivAt (fun x ↦ (B (u x)) (v x)) ((B (u x)) v' + (B u') (v x)) x
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: case zero
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
s : Set E
inst✝ : CompleteSpace F
h : AnalyticOnNhd 𝕜 f s
⊢ AnalyticOnNhd 𝕜 (iteratedFDeriv 𝕜 0 f) s

Tactic:
rw [iteratedFDeriv_zero_eq_comp]

Goal state after:
case zero
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
s : Set E
inst✝ : CompleteSpace F
h : AnalyticOnNhd 𝕜 f s
⊢ AnalyticOnNhd 𝕜 (⇑(continuousMultilinearCurryFin0 𝕜 E F).symm ∘ f) s
Module: Mathlib.Analysis.Calculus.Deriv.Mul

Goal state before:
pp: case neg
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
𝕜' : Type u_2
inst✝¹ : NormedDivisionRing 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
u : 𝕜 → 𝕜'
v : 𝕜'
hu : ¬DifferentiableAt 𝕜 u x
⊢ deriv (fun y ↦ u y * v) x = 0

Tactic:
rcases eq_or_ne v 0 with (rfl | hd)

Goal state after:
case neg.inl
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
𝕜' : Type u_2
inst✝¹ : NormedDivisionRing 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
u : 𝕜 → 𝕜'
hu : ¬DifferentiableAt 𝕜 u x
⊢ deriv (fun y ↦ u y * 0) x = 0

case neg.inr
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
𝕜' : Type u_2
inst✝¹ : NormedDivisionRing 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
u : 𝕜 → 𝕜'
v : 𝕜'
hu : ¬DifferentiableAt 𝕜 u x
hd : v ≠ 0
⊢ deriv (fun y ↦ u y * v) x = 0
Module: Mathlib.Analysis.Calculus.Deriv.Mul

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

Tactic:
by
  simp only [deriv_const_mul c differentiableAt_fun_id, deriv_id'', mul_one]

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

Goal state before:
pp: case zero
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_.iteratedFDeriv 𝕜 0 f) s

Tactic:
rw [iteratedFDeriv_zero_eq_comp]

Goal state after:
case zero
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 (⇑(continuousMultilinearCurryFin0 𝕜 E F).symm ∘ f) s
Module: Mathlib.Analysis.Analytic.Uniqueness

Goal state before:
pp: case succ
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 : 0 < ‖y‖
⊢ (p fun x ↦ y) = 0

Tactic:
refine norm_eq_zero.mp (le_antisymm (le_of_forall_pos_le_add fun ε ε_pos => ?_) (norm_nonneg _))

Goal state after:
case succ
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 : 0 < ‖y‖
ε : ℝ
ε_pos : 0 < ε
⊢ ‖p fun x ↦ y‖ ≤ 0 + ε
Module: Mathlib.Analysis.Calculus.Deriv.Slope

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
x : 𝕜
L : Filter 𝕜
⊢ HasDerivAtFilter f f' (L ×ˢ pure x) ↔ Tendsto (fun y ↦ slope f x y - (y - x)⁻¹ • (y - x) • f') L (𝓝 0)

Tactic:
simp only [hasDerivAtFilter_iff_tendsto, prod_pure, tendsto_map'_iff, Function.comp_def,
        ← norm_inv, ← norm_smul, ← tendsto_zero_iff_norm_tendsto_zero, slope_def_module, smul_sub]

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

Goal state before:
pp: case e'_2.a
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
n : ℕ
x : E
x✝ : { s // s.card = n }
a✝ : x✝ ∈ Finset.univ
⊢ ((p.changeOriginSeriesTerm 1 n ↑x✝ ⋯) fun x_1 ↦ x) (Fin.snoc ![] x) = (p (n + 1)) fun x_1 ↦ x

pp: case e'_3.e'_5
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
n : ℕ
x : E
⊢ n + 1 = Finset.univ.card

Tactic:
· rw [Fin.snoc_zero, changeOriginSeriesTerm_apply, Finset.piecewise_same, add_comm]

Goal state after:
case e'_3.e'_5
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
n : ℕ
x : E
⊢ n + 1 = Finset.univ.card