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

Goal state before:
pp: case mp
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
f : α → E
g : α → F
l : Filter α
c : 𝕜
hc : 1 < ‖c‖₊
⊢ (∀ (i : ℝ), 0 < i → ∃ j, 0 < j ∧ ∀ᶠ (x : α) in l, egauge 𝕜 (ball 0 i) (f x) ≤ egauge 𝕜 (ball 0 j) (g x)) → f =O[l] g

Tactic:
isBigO_iff

Goal state after:
case mp
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
f : α → E
g : α → F
l : Filter α
c : 𝕜
hc : 1 < ‖c‖₊
⊢ (∀ (i : ℝ), 0 < i → ∃ j, 0 < j ∧ ∀ᶠ (x : α) in l, egauge 𝕜 (ball 0 i) (f x) ≤ egauge 𝕜 (ball 0 j) (g x)) →
    ∃ c, ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g x‖
Module: Mathlib.Analysis.Analytic.ChangeOrigin

Goal state before:
pp: case e'_5
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
p : FormalMultilinearSeries 𝕜 E F
r r' : ℝ≥0
hr : ↑r + ↑r' < p.radius
n : ℕ
s : Finset (Fin n)
⊢ ‖p (n - s.card + s.card)‖₊ * r ^ s.card * r' ^ (n - s.card) = ‖p n‖₊ * (r ^ s.card * r' ^ (n - s.card))

Tactic:
tsub_add_cancel_of_le (card_finset_fin_le _),

Goal state after:
case e'_5
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
p : FormalMultilinearSeries 𝕜 E F
r r' : ℝ≥0
hr : ↑r + ↑r' < p.radius
n : ℕ
s : Finset (Fin n)
⊢ ‖p n‖₊ * r ^ s.card * r' ^ (n - s.card) = ‖p n‖₊ * (r ^ s.card * r' ^ (n - s.card))
Module: Mathlib.Analysis.Analytic.ConvergenceRadius

Goal state before:
pp: case mp
𝕜 : Type u_1
E : Type u_3
F : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
h : p.radius = ∞
r : ℝ≥0
a : ℝ
ha : a ∈ Ioo 0 1
C : ℝ
hp : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C * a ^ n
⊢ Summable fun n ↦ ‖p n‖ * ↑r ^ n

Tactic:
refine .of_norm_bounded
      (g := fun n ↦ (C : ℝ) * a ^ n) ((summable_geometric_of_lt_one ha.1.le ha.2).mul_left _)
      fun n ↦ ?_

Goal state after:
case mp
𝕜 : Type u_1
E : Type u_3
F : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
h : p.radius = ∞
r : ℝ≥0
a : ℝ
ha : a ∈ Ioo 0 1
C : ℝ
hp : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C * a ^ n
n : ℕ
⊢ ‖‖p n‖ * ↑r ^ n‖ ≤ C * a ^ n
Module: Mathlib.Analysis.Analytic.ConvergenceRadius

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
F : Type u_4
G : Type u_5
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
p : FormalMultilinearSeries 𝕜 F G
u : E →ₗᵢ[𝕜] F
⊢ p.radius ≤ (p.compContinuousLinearMap u.toContinuousLinearMap).radius

Tactic:
obtain (h | h) := subsingleton_or_nontrivial E
  · simp [Subsingleton.elim u.toContinuousLinearMap 0]
  · simpa [u.norm_toContinuousLinearMap]
      using div_le_radius_compContinuousLinearMap p u.toContinuousLinearMap

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

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 : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
p : FormalMultilinearSeries 𝕜 E F
x y : E
h : ↑‖x‖₊ + ↑‖y‖₊ < p.radius
x_mem_ball : x ∈ Metric.eball 0 p.radius
y_mem_ball : y ∈ Metric.eball 0 (p.changeOrigin x).radius
x_add_y_mem_ball : x + y ∈ Metric.eball 0 p.radius
f : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F :=
  fun s ↦ ((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 ↦ x) fun x ↦ y
hsf : Summable f
hf : HasSum f ((p.changeOrigin x).sum y)
⊢ HasSum (f ∘ ⇑changeOriginIndexEquiv.symm) (p.sum (x + y))

Tactic:
refine HasSum.sigma_of_hasSum
    (p.hasSum x_add_y_mem_ball) (fun n => ?_) (changeOriginIndexEquiv.symm.summable_iff.2 hsf)

Goal state after:
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
inst✝ : CompleteSpace F
p : FormalMultilinearSeries 𝕜 E F
x y : E
h : ↑‖x‖₊ + ↑‖y‖₊ < p.radius
x_mem_ball : x ∈ Metric.eball 0 p.radius
y_mem_ball : y ∈ Metric.eball 0 (p.changeOrigin x).radius
x_add_y_mem_ball : x + y ∈ Metric.eball 0 p.radius
f : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F :=
  fun s ↦ ((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 ↦ x) fun x ↦ y
hsf : Summable f
hf : HasSum f ((p.changeOrigin x).sum y)
n : ℕ
⊢ HasSum (fun c ↦ (f ∘ ⇑changeOriginIndexEquiv.symm) ⟨n, c⟩) ((p n) fun x_1 ↦ x + y)
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : Ring E
inst✝² : Algebra 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : IsTopologicalRing E
c : ℕ → 𝕜
n : ℕ
hc : c n = 0
⊢ 0 • ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 n E = 0

Tactic:
zero_smul 𝕜 (ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 n E)

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : Ring E
inst✝² : Algebra 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : IsTopologicalRing E
c : ℕ → 𝕜
n : ℕ
hc : c n = 0
⊢ 0 = 0
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
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesOnBall f p x r
y z : E
hz : z ∈ Metric.eball 0 r
⊢ HasSum (fun n ↦ (p n) fun x ↦ z) (f (x + y + z - y))

Tactic:
convert hf.hasSum hz
      abel

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
r' : ℝ≥0
hr : ↑r' < r
hr'0 : 0 < r'
a : ℝ
ha : a ∈ Ioo 0 1
C : ℝ
hC : 0 < C
hp : ∀ (n : ℕ), ‖p n‖ ≤ C * a ^ n / ↑r' ^ n
L : E × E → ℝ := fun y ↦ C * (a / ↑r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * (a / (1 - a) ^ 2 + 2 / (1 - a))
y : E × E
hy' : y ∈ Metric.eball (x, x) ↑r'
ys : y ∈ insert x s ×ˢ insert x s
hy : y ∈ Metric.eball x r ×ˢ Metric.eball x r
A : ℕ → F := fun n ↦ ((p n) fun x_1 ↦ y.1 - x) - (p n) fun x_1 ↦ y.2 - x
⊢ ((p 1) fun x ↦ y.1 - y.2) = f x - f x + (((p 1) fun x_1 ↦ y.1 - x) - (p 1) fun x_1 ↦ y.2 - x)

Tactic:
sub_self,

Goal state after:
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
r' : ℝ≥0
hr : ↑r' < r
hr'0 : 0 < r'
a : ℝ
ha : a ∈ Ioo 0 1
C : ℝ
hC : 0 < C
hp : ∀ (n : ℕ), ‖p n‖ ≤ C * a ^ n / ↑r' ^ n
L : E × E → ℝ := fun y ↦ C * (a / ↑r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * (a / (1 - a) ^ 2 + 2 / (1 - a))
y : E × E
hy' : y ∈ Metric.eball (x, x) ↑r'
ys : y ∈ insert x s ×ˢ insert x s
hy : y ∈ Metric.eball x r ×ˢ Metric.eball x r
A : ℕ → F := fun n ↦ ((p n) fun x_1 ↦ y.1 - x) - (p n) fun x_1 ↦ y.2 - x
⊢ ((p 1) fun x ↦ y.1 - y.2) = 0 + (((p 1) fun x_1 ↦ y.1 - x) - (p 1) fun x_1 ↦ y.2 - x)
Module: Mathlib.Analysis.Analytic.Inverse

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
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i
⊢ p.comp (p.rightInv i x) = id 𝕜 F ((p 0) 0)

Tactic:
by
  ext (n v)
  match n with
  | 0 =>
    simp only [comp_coeff_zero', Matrix.zero_empty, id_apply_zero]
    congr
    ext i
    exact i.elim0
  | 1 =>
    simp only [comp_coeff_one, h, rightInv_coeff_one, ContinuousLinearEquiv.apply_symm_apply,
      id_apply_one, ContinuousLinearEquiv.coe_apply, continuousMultilinearCurryFin1_symm_apply]
  | n + 2 =>
    have N : 0 < n + 2 := by simp
    simp [comp_rightInv_aux1 N, h, rightInv, comp_rightInv_aux2, -Set.toFinset_ofPred]

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

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
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
hp : 0 < p.radius
h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i
⊢ 0 < (p.rightInv i x).radius

Tactic:
exact radius_rightInv_pos_of_radius_pos hp

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module 𝕜 E
inst✝⁸ : TopologicalSpace E
F : Type u_3
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace F
f : E →L[𝕜] F
x : E
s : Set E
inst✝⁴ : ContinuousAdd E
inst✝³ : ContinuousSMul 𝕜 E
inst✝² : ContinuousAdd F
inst✝¹ : ContinuousSMul 𝕜 F
inst✝ : T2Space F
hxs : UniqueDiffWithinAt 𝕜 s x
⊢ fderivWithin 𝕜 (⇑f) s x = f

Tactic:
by
  rw [DifferentiableAt.fderivWithin f.differentiableAt hxs]
  exact f.fderiv

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

Goal state before:
pp: 𝕜 : Type u_2
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
𝕝 : Type u_8
inst✝⁴ : NormedDivisionRing 𝕝
inst✝³ : NormedAlgebra 𝕜 𝕝
inst✝² : Module 𝕝 F
inst✝¹ : IsBoundedSMul 𝕝 F
inst✝ : IsScalarTower 𝕜 𝕝 F
f : E → 𝕝
g : E → F
z : E
h₁f : AnalyticAt 𝕜 f z
h₂f : f z ≠ 0
hprod : AnalyticAt 𝕜 (fun z ↦ f z • g z) z
y : E
hy : y ∈ f ⁻¹' {0}ᶜ
⊢ g y = ((f⁻¹ • f) • g) y

Tactic:
Set.preimage_compl,

Goal state after:
𝕜 : Type u_2
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
𝕝 : Type u_8
inst✝⁴ : NormedDivisionRing 𝕝
inst✝³ : NormedAlgebra 𝕜 𝕝
inst✝² : Module 𝕝 F
inst✝¹ : IsBoundedSMul 𝕝 F
inst✝ : IsScalarTower 𝕜 𝕝 F
f : E → 𝕝
g : E → F
z : E
h₁f : AnalyticAt 𝕜 f z
h₂f : f z ≠ 0
hprod : AnalyticAt 𝕜 (fun z ↦ f z • g z) z
y : E
hy : y ∈ (f ⁻¹' {0})ᶜ
⊢ g y = ((f⁻¹ • f) • g) y
Module: Mathlib.Analysis.Calculus.FDeriv.Add

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 : DifferentiableWithinAt 𝕜 (-f) s x
⊢ DifferentiableWithinAt 𝕜 f s x

Tactic:
by simpa only [neg_neg] using h.neg

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

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
h : DifferentiableAt 𝕜 (-f) x
⊢ DifferentiableAt 𝕜 f x

Tactic:
by simpa only [neg_neg] using h.neg

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

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
s : Set E
h : DifferentiableOn 𝕜 (-f) s
⊢ DifferentiableOn 𝕜 f s

Tactic:
by simpa only [neg_neg] using h.neg

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

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
h : Differentiable 𝕜 (-f)
⊢ Differentiable 𝕜 f

Tactic:
by simpa only [neg_neg] using h.neg

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
ι : Type u_4
inst✝¹⁴ : NormedField 𝕜
inst✝¹³ : AddCommGroup E
inst✝¹² : Module 𝕜 E
inst✝¹¹ : TopologicalSpace E
inst✝¹⁰ : AddCommGroup F
inst✝⁹ : Module 𝕜 F
inst✝⁸ : TopologicalSpace F
inst✝⁷ : IsTopologicalAddGroup F
G : Type u_5
inst✝⁶ : AddCommGroup G
inst✝⁵ : Module 𝕜 G
inst✝⁴ : TopologicalSpace G
inst✝³ : ContinuousConstSMul 𝕜 F
E' : Type u_6
inst✝² : AddCommGroup E'
inst✝¹ : Module 𝕜 E'
inst✝ : TopologicalSpace E'
f : E →L[𝕜] E'
⊢ Continuous[instTopologicalSpace, ContinuousMultilinearMap.instTopologicalSpace]
    (toContinuousMultilinearMap ∘ (compContinuousLinearMapₗ f).toFun)

Tactic:
exact (map_continuous <| ContinuousMultilinearMap.compContinuousLinearMapL fun _ ↦ f).comp
      continuous_toContinuousMultilinearMap

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
hs : UniqueDiffOn 𝕜 s
n : ℕ
IH :
  ∀ {x : E},
    x ∈ s →
      ∀ (m : Fin (n + 1) → E),
        (iteratedFDerivWithin 𝕜 (n + 1) f s x) m =
          ((iteratedFDerivWithin 𝕜 n (fun y ↦ fderivWithin 𝕜 f s y) s x) (init m)) (m (last n))
x : E
hx : x ∈ s
m : Fin (n + 1 + 1) → E
I : (E [×n]→L[𝕜] E →L[𝕜] F) ≃ₗᵢ[𝕜] E [×n.succ]→L[𝕜] F := (continuousMultilinearCurryRightEquiv' 𝕜 n E F).symm
A : ∀ y ∈ s, iteratedFDerivWithin 𝕜 n.succ f s y = (⇑I ∘ iteratedFDerivWithin 𝕜 n (fun y ↦ fderivWithin 𝕜 f s y) s) y
⊢ ((⇑I ∘ ⇑(fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n (fderivWithin 𝕜 f s) s) s x)) (m 0)) (tail m) =
    (((fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n (fun y ↦ fderivWithin 𝕜 f s y) s) s x) (m 0)) (init (tail m)))
      (tail m (last n))

Tactic:
by
        simp [I]

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
f : E → F
x : E
s : Set E
h : HasFPowerSeriesWithinAt f p s x
⊢ ‖(x, x) - (x, x)‖ = 0

Tactic:
sub_self,

Goal state after:
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
⊢ ‖0‖ = 0
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
f : E → F
x : E
n : ℕ
⊢ ‖fderiv 𝕜 (iteratedFDeriv 𝕜 n f) x‖ = ‖iteratedFDeriv 𝕜 (n + 1) f x‖

Tactic:
rw [iteratedFDeriv_succ_eq_comp_left, comp_apply, LinearIsometryEquiv.norm_map]

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

Goal state before:
pp: case right
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
⊢ ∃ a, (fun x ↦ ⟨{x}ᶜ, ⋯⟩) a = ⟨s, hs⟩

Tactic:
have h : #sᶜ = 1 := by rw [card_compl, hs, Fintype.card_fin, Nat.add_sub_cancel]

Goal state after:
case right
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
h : #sᶜ = 1
⊢ ∃ a, (fun x ↦ ⟨{x}ᶜ, ⋯⟩) a = ⟨s, hs⟩
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
⊢ (a - a)⁻¹ • (f a -ᵥ f a) = 0

Tactic:
sub_self,

Goal state after:
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
⊢ 0⁻¹ • (f a -ᵥ f a) = 0
Module: Mathlib.LinearAlgebra.AffineSpace.Slope

Goal state before:
pp: case inl
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
⊢ (a - a) • slope f a a = f a -ᵥ f a

Tactic:
sub_self,

Goal state after:
case inl
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
⊢ 0 • slope f a a = f a -ᵥ f a
Module: Mathlib.LinearAlgebra.AffineSpace.Slope

Goal state before:
pp: case pos
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 c : k
⊢ ((a - a) / (c - a)) • slope f a a + ((c - a) / (c - a)) • slope f a c = slope f a c

Tactic:
sub_self,

Goal state after:
case pos
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 c : k
⊢ (0 / (c - a)) • slope f a a + ((c - a) / (c - a)) • slope f a c = slope f a c
Module: Mathlib.LinearAlgebra.AffineSpace.Slope

Goal state before:
pp: case neg
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 c : k
hac : ¬a = c
⊢ ((c - a) / (c - a)) • slope f a c = slope f a c

Tactic:
div_self (sub_ne_zero.2 <| Ne.symm hac),

Goal state after:
case neg
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 c : k
hac : ¬a = c
⊢ 1 • slope f a c = slope f a c
Module: Mathlib.Analysis.Analytic.IsolatedZeros

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : E
n : ℕ
z : 𝕜
a : ℕ → E
hs : HasSum (fun m ↦ z ^ m • a m) s
ha : ∀ k < n, a k = 0
hn : n > 0
h : ¬z = 0
⊢ z ^ n • (z ^ n)⁻¹ • s = s

Tactic:
by match_scalars; field

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
m n : ℤ
hm : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = (z - z₀) ^ m • g z
hn : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = (z - z₀) ^ n • g z
h_le : n ≤ m
g : 𝕜 → E
hg_an : AnalyticAt 𝕜 g z₀
left✝ : g z₀ ≠ 0
hg_eq : ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = (z - z₀) ^ m • g z
j : 𝕜 → E
hj_an : AnalyticAt 𝕜 j z₀
hj_eq : ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = (z - z₀) ^ n • j z
hj_ne : m ≠ n
this : ∀ᶠ (z : 𝕜) in 𝓝 z₀, j z = (z - z₀) ^ (m - n) • g z
⊢ (z₀ - z₀) ^ (m - n) • g z₀ = 0

Tactic:
sub_self,

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
m n : ℤ
hm : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = (z - z₀) ^ m • g z
hn : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = (z - z₀) ^ n • g z
h_le : n ≤ m
g : 𝕜 → E
hg_an : AnalyticAt 𝕜 g z₀
left✝ : g z₀ ≠ 0
hg_eq : ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = (z - z₀) ^ m • g z
j : 𝕜 → E
hj_an : AnalyticAt 𝕜 j z₀
hj_eq : ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = (z - z₀) ^ n • j z
hj_ne : m ≠ n
this : ∀ᶠ (z : 𝕜) in 𝓝 z₀, j z = (z - z₀) ^ (m - n) • g z
⊢ 0 ^ (m - n) • g z₀ = 0
Module: Mathlib.Analysis.Calculus.FDeriv.Affine

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 →ᴬ[𝕜] F
x : E
s : Set E
hxs : UniqueDiffWithinAt 𝕜 s x
⊢ fderivWithin 𝕜 (⇑f) s x = f.contLinear

Tactic:
by
  rw [DifferentiableAt.fderivWithin f.differentiableAt hxs]
  exact f.fderiv

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
f g : 𝕜 → 𝕜'
U : Set 𝕜
hU : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
h : f =ᶠ[codiscreteWithin U] g
x : 𝕜
hx : x ∈ U
z : 𝕜
hz : z ∈ {x | f x = g x} ∪ Uᶜ
⊢ z ∈ U → f z = g z

Tactic:
hzU

Goal state after:
𝕜 : Type u_1
𝕜' : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
f g : 𝕜 → 𝕜'
U : Set 𝕜
hU : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
h : f =ᶠ[codiscreteWithin U] g
x : 𝕜
hx : x ∈ U
z : 𝕜
hz : z ∈ {x | f x = g x} ∪ Uᶜ
hzU : z ∈ U
⊢ f z = g z
Module: Mathlib.Analysis.Calculus.MeanValue

Goal state before:
pp: case ha
f : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x
bound : ∀ x ∈ Ico a b, ∀ (r : ℝ), B' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
x : ℝ
hx : x ∈ Icc a b
r : ℝ
hr : r > 0
⊢ f a ≤ B a + r * (a - a)

Tactic:
sub_self,

Goal state after:
case ha
f : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x
bound : ∀ x ∈ Ico a b, ∀ (r : ℝ), B' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
x : ℝ
hx : x ∈ Icc a b
r : ℝ
hr : r > 0
⊢ f a ≤ B a + r * 0
Module: Mathlib.Analysis.Calculus.MeanValue

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
f' : ℝ → E
C : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x
bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C
g : ℝ → E := fun x ↦ f x - f a
hg : ContinuousOn g (Icc a b)
hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x
B : ℝ → ℝ := fun x ↦ C * (x - a)
hB : ∀ (x : ℝ), HasDerivAt B C x
⊢ ‖f a - f a‖ ≤ C * (a - a)

Tactic:
sub_self,

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
f' : ℝ → E
C : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x
bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C
g : ℝ → E := fun x ↦ f x - f a
hg : ContinuousOn g (Icc a b)
hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x
B : ℝ → ℝ := fun x ↦ C * (x - a)
hB : ∀ (x : ℝ), HasDerivAt B C x
⊢ ‖0‖ ≤ C * (a - a)
Module: Mathlib.Analysis.Calculus.MeanValue

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
f' : ℝ → E
C : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x
bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C
g : ℝ → E := fun x ↦ f x - f a
hg : ContinuousOn g (Icc a b)
hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x
B : ℝ → ℝ := fun x ↦ C * (x - a)
hB : ∀ (x : ℝ), HasDerivAt B C x
⊢ 0 ≤ C * (a - a)

Tactic:
sub_self,

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
f' : ℝ → E
C : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x
bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C
g : ℝ → E := fun x ↦ f x - f a
hg : ContinuousOn g (Icc a b)
hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x
B : ℝ → ℝ := fun x ↦ C * (x - a)
hB : ∀ (x : ℝ), HasDerivAt B C x
⊢ 0 ≤ C * 0