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‖₊
⊢ f =O[𝕜; l] g → f =O[l] g

Tactic:
rw [nhds_basis_ball.isBigOTVS_iff nhds_basis_ball, 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.ConvergenceRadius

Goal state before:
pp: 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
r : ℝ≥0
h₀ : r ≠ 0
a : ℝ
ha : a ∈ Ioo (-1) 1
hp : (fun n ↦ ‖p n‖ * ↑r ^ n) =O[atTop] fun x ↦ a ^ x
⊢ ↑r < p.radius

Tactic:
have := ((TFAE_exists_lt_isLittleO_pow (fun n => ‖p n‖ * (r : ℝ) ^ n) 1).out 2 5)
  rcases this.mp ⟨a, ha, hp⟩ with ⟨a, ha, C, hC, hp⟩
  rw [← pos_iff_ne_zero, ← NNReal.coe_pos] at h₀
  lift a to ℝ≥0 using ha.1.le
  have : (r : ℝ) < r / a := by
    simpa only [div_one] using (div_lt_div_iff_of_pos_left h₀ zero_lt_one ha.1).2 ha.2
  norm_cast at this
  rw [← ENNReal.coe_lt_coe] at this
  refine this.trans_le (p.le_radius_of_bound C fun n => ?_)
  rw [NNReal.coe_div, div_pow, ← mul_div_assoc, div_le_iff₀ (pow_pos ha.1 n)]
  exact (le_abs_self _).trans (hp n)

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : Field 𝕜
inst✝⁴ : Ring E
inst✝³ : Algebra 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : IsTopologicalRing E
c : ℕ → 𝕜
inst✝ : Subsingleton E
⊢ ofScalars E c = 0

Tactic:
FormalMultilinearSeries.ext_iff,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : Field 𝕜
inst✝⁴ : Ring E
inst✝³ : Algebra 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : IsTopologicalRing E
c : ℕ → 𝕜
inst✝ : Subsingleton E
⊢ ∀ (n : ℕ), ofScalars E c n = 0 n
Module: Mathlib.Analysis.Analytic.Basic

Goal state before:
pp: case inr
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 g : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : EqOn g f (s ∩ Metric.eball x r)
h'' : g x = f x
y : E
h'y : y ∈ Metric.eball 0 r
hy : x + y ∈ s
⊢ x + y ∈ s ∩ Metric.eball x r

Tactic:
refine ⟨hy, ?_⟩

Goal state after:
case inr
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 g : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : EqOn g f (s ∩ Metric.eball x r)
h'' : g x = f x
y : E
h'y : y ∈ Metric.eball 0 r
hy : x + y ∈ s
⊢ x + y ∈ Metric.eball x r
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: case neg.refine_1
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing E
inst✝ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
hc : ∀ᶠ (n : ℕ) in atTop, c n ≠ 0
hc' : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 0)
r' : ℝ≥0
hrz : ¬r' = 0
⊢ Tendsto (fun n ↦ ‖‖‖c (n + 1)‖ * ↑r' ^ (n + 1)‖‖ / ‖‖‖c n‖ * ↑r' ^ n‖‖) atTop (𝓝 0)

pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing E
inst✝ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
hc : ∀ᶠ (n : ℕ) in atTop, c n ≠ 0
hc' : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 0)
r' : ℝ≥0
hrz : ¬r' = 0
⊢ ∀ (x : ℕ), c x ≠ 0 → ‖‖c x‖ * ↑r' ^ x‖ ≠ 0

Tactic:
· simp only [norm_norm]
        exact mul_zero (_ : ℝ) ▸ tendsto_succ_norm_div_norm _ hrz (NNReal.coe_zero ▸ hc')

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing E
inst✝ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
hc : ∀ᶠ (n : ℕ) in atTop, c n ≠ 0
hc' : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 0)
r' : ℝ≥0
hrz : ¬r' = 0
⊢ ∀ (x : ℕ), c x ≠ 0 → ‖‖c x‖ * ↑r' ^ x‖ ≠ 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
s : Set E
x : E
r : ℝ≥0∞
y : E
hf : HasFPowerSeriesWithinOnBall f p s x r
hy : y ∈ Metric.eball 0 r
h'y : x + y ∈ insert x s
A : Tendsto (fun z ↦ p.partialSum z.1 y) (atTop ×ˢ 𝓝 y) (𝓝 (f (x + y)))
ε : ℝ
εpos : ε > 0
r' : ℝ≥0
yr' : ‖y‖₊ < r'
r'r : ↑r' < r
⊢ ‖y‖ < ↑r'

Tactic:
simpa [← coe_nnnorm] using yr'

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

Goal state before:
pp: case hab
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∞
r' : ℝ≥0
hf : HasFPowerSeriesWithinOnBall f p s x r
h : ↑r' < r
a : ℝ
ha : a ∈ Ioo 0 1
C : ℝ
hC : C > 0
hp : ∀ y ∈ Metric.ball 0 ↑r', ∀ (n : ℕ), x + y ∈ insert x s → ‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / ↑r')) ^ n
y : E
hy : y ∈ Metric.ball 0 ↑r'
n : ℕ
ys : x + y ∈ insert x s
yr' : ‖y‖ < ↑r'
this : 0 ≤ a
⊢ a * (‖y‖ / ↑r') ≤ a

Tactic:
exact mul_le_of_le_one_right ha.1.le (div_le_one_of_le₀ yr'.le r'.coe_nonneg)

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

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
ι : Type u_5
Em : ι → Type u_6
inst✝² : (i : ι) → NormedAddCommGroup (Em i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (Em i)
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 Em F
y : (i : ι) → Em i
x✝ : y ∈ Metric.eball 0 ∞
⊢ ∀ b ∈ Finset.range (Fintype.card ι + 1), b ≠ Fintype.card ι → ((f.toFormalMultilinearSeries b) fun x ↦ y) = 0

Tactic:
· intro m _ ne; rw [toFormalMultilinearSeries, dif_neg ne.symm]; rfl

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

Goal state before:
pp: α : Type u_6
inst✝ : AddCommMonoid α
m M N : ℕ
f : (n : ℕ) × (Fin n → ℕ) → α
g : (n : ℕ) × Composition n → α
h : ∀ (e : (n : ℕ) × (Fin n → ℕ)) (he : e ∈ compPartialSumSource m M N), f e = g (compChangeOfVariables m M N e he)
k : ℕ
blocks_fun : Fin k → ℕ
H : ⟨k, blocks_fun⟩ ∈ compPartialSumSource m M N
k' : ℕ
blocks_fun' : Fin k' → ℕ
H' : ⟨k', blocks_fun'⟩ ∈ compPartialSumSource m M N
heq : compChangeOfVariables m M N ⟨k, blocks_fun⟩ H = compChangeOfVariables m M N ⟨k', blocks_fun'⟩ H'
this : ⟨k, blocks_fun⟩.fst = (compChangeOfVariables m M N ⟨k', blocks_fun'⟩ H').snd.length
⊢ k = k'

Tactic:
compChangeOfVariables_length

Goal state after:
α : Type u_6
inst✝ : AddCommMonoid α
m M N : ℕ
f : (n : ℕ) × (Fin n → ℕ) → α
g : (n : ℕ) × Composition n → α
h : ∀ (e : (n : ℕ) × (Fin n → ℕ)) (he : e ∈ compPartialSumSource m M N), f e = g (compChangeOfVariables m M N e he)
k : ℕ
blocks_fun : Fin k → ℕ
H : ⟨k, blocks_fun⟩ ∈ compPartialSumSource m M N
k' : ℕ
blocks_fun' : Fin k' → ℕ
H' : ⟨k', blocks_fun'⟩ ∈ compPartialSumSource m M N
heq : compChangeOfVariables m M N ⟨k, blocks_fun⟩ H = compChangeOfVariables m M N ⟨k', blocks_fun'⟩ H'
this : ⟨k, blocks_fun⟩.fst = ⟨k', blocks_fun'⟩.fst
⊢ k = k'
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
C r : ℝ
Cpos : 0 < C
rpos : 0 < r
ple : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ‖↑i.symm‖
a : ℝ
apos : 0 < a
ha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a ≤ 1
ha2 : r * (I + 1) * a ≤ 1 / 2
S : ℕ → ℝ := fun n ↦ ∑ k ∈ Ico 1 n, a ^ k * ‖p.rightInv i x k‖
n : ℕ
one_le_n : 1 ≤ n
hn : S n ≤ (I + 1) * a
In : 2 ≤ n + 1
rSn : r * S n ≤ 1 / 2
⊢ 1 / 2 < 1

Tactic:
by norm_num

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

Goal state before:
pp: case hac
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
C r : ℝ
Cpos : 0 < C
rpos : 0 < r
ple : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ⋯
a : ℝ
apos : 0 < a
ha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a ≤ 1
ha2 : r * (I + 1) * a ≤ 1 / 2
S : ℕ → ℝ := ⋯
n : ℕ
one_le_n : 1 ≤ n
hn : S n ≤ (I + 1) * a
In : 2 ≤ n + 1
rSn : r * S n ≤ 1 / 2
⊢ (r * S n) ^ 2 - (r * S n) ^ (n + 1) ≤ (r * S n) ^ 2

Tactic:
simp only [sub_le_self_iff]

Goal state after:
case hac
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
C r : ℝ
Cpos : 0 < C
rpos : 0 < r
ple : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ‖↑i.symm‖
a : ℝ
apos : 0 < a
ha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a ≤ 1
ha2 : r * (I + 1) * a ≤ 1 / 2
S : ℕ → ℝ := fun n ↦ ∑ k ∈ Ico 1 n, a ^ k * ‖p.rightInv i x k‖
n : ℕ
one_le_n : 1 ≤ n
hn : S n ≤ (I + 1) * a
In : 2 ≤ n + 1
rSn : r * S n ≤ 1 / 2
⊢ 0 ≤ (r * S n) ^ (n + 1)
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
H : Type u_5
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
n✝ n : ℕ
⊢ Function.LeftInverse
    (fun i ↦
      ⟨{ blocks := (ofFn fun j ↦ (i.snd j).blocks).flatten, blocks_pos := ⋯, blocks_sum := ⋯ },
        { blocks := ofFn fun j ↦ (i.snd j).length, blocks_pos := ⋯, blocks_sum := ⋯ }⟩)
    fun i ↦ ⟨i.fst.gather i.snd, i.fst.sigmaCompositionAux i.snd⟩

Tactic:
rintro ⟨a, b⟩
    rw [sigma_composition_eq_iff]
    dsimp
    constructor
    · conv_rhs =>
        rw [← flatten_splitWrtComposition a.blocks b, ← ofFn_get (splitWrtComposition a.blocks b)]
      have A : length (gather a b) = List.length (splitWrtComposition a.blocks b) := by
        simp only [length, gather, length_map, length_splitWrtComposition]
      congr! 2
      exact (Fin.heq_fun_iff A (α := List ℕ)).2 fun i => rfl
    · have B : Composition.length (Composition.gather a b) = List.length b.blocks :=
        Composition.length_gather _ _
      conv_rhs => rw [← ofFn_getElem (xs := b.blocks)]
      congr 1
      refine (Fin.heq_fun_iff B).2 fun i => ?_
      rw [sigmaCompositionAux, Composition.length, List.getElem_map_rev List.length,
        List.getElem_of_eq (map_length_splitWrtComposition _ _)]

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

Goal state before:
pp: 𝕜 : Type u_2
inst✝² : NontriviallyNormedField 𝕜
A : Type u_7
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
⊢ formalMultilinearSeries_geometric 𝕜 A = FormalMultilinearSeries.ofScalars A fun x ↦ 1

Tactic:
FormalMultilinearSeries.ext_iff,

Goal state after:
𝕜 : Type u_2
inst✝² : NontriviallyNormedField 𝕜
A : Type u_7
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
⊢ ∀ (n : ℕ), formalMultilinearSeries_geometric 𝕜 A n = FormalMultilinearSeries.ofScalars A (fun x ↦ 1) n
Module: Mathlib.Analysis.Analytic.Constructions

Goal state before:
pp: α : Type u_1
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : α → E → F
c : E
N : Finset α
h : ∀ n ∈ N, AnalyticAt 𝕜 (f n) c
⊢ AnalyticAt 𝕜 (∑ n ∈ N, f n) c

Tactic:
simp_rw [← analyticWithinAt_univ] at h ⊢
  exact N.analyticWithinAt_sum h

Goal state after:
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 : G → E
x : G
⊢ DifferentiableAt 𝕜 (⇑iso ∘ f) x ↔ DifferentiableAt 𝕜 f x

Tactic:
rw [← differentiableWithinAt_univ, ← differentiableWithinAt_univ,
    iso.comp_differentiableWithinAt_iff]

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, instTopologicalSpace] (compContinuousLinearMapₗ f).toFun

Tactic:
by
    rw [isEmbedding_toContinuousMultilinearMap.continuous_iff]
    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
f : E → F
x : E
n : ℕ
m : Fin (n + 1) → E
hf : DifferentiableAt 𝕜 (iteratedFDeriv 𝕜 n f) x
⊢ (iteratedFDeriv 𝕜 (n + 1) f x) m = (fderiv 𝕜 (fun y ↦ (iteratedFDeriv 𝕜 n f y) (tail m)) x) (m 0)

Tactic:
convert iteratedFDeriv_succ_apply_left m

Goal state after:
case e'_3
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
m : Fin (n + 1) → E
hf : DifferentiableAt 𝕜 (iteratedFDeriv 𝕜 n f) x
⊢ (fderiv 𝕜 (fun y ↦ (iteratedFDeriv 𝕜 n f y) (tail m)) x) (m 0) = ((fderiv 𝕜 (iteratedFDeriv 𝕜 n f) x) (m 0)) (tail m)
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: case inr
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
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : HasFPowerSeriesWithinOnBall f p s x r
y : E
hy✝ : y ∈ insert x s ∩ Metric.eball x r
Z : DifferentiableWithinAt 𝕜 f (insert y s) y
hy : y ≠ x
⊢ s ∈ 𝓝[insert x s ∩ Metric.eball x r] y

Tactic:
suffices s ∈ 𝓝[insert x s] y from nhdsWithin_mono _ inter_subset_left this

Goal state after:
case inr
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
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : HasFPowerSeriesWithinOnBall f p s x r
y : E
hy✝ : y ∈ insert x s ∩ Metric.eball x r
Z : DifferentiableWithinAt 𝕜 f (insert y s) y
hy : y ≠ x
⊢ s ∈ 𝓝[insert x s] y
Module: Mathlib.Analysis.Calculus.Deriv.Mul

Goal state before:
pp: 𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
f : 𝕜 → F
f' : F
x : 𝕜
s : Set 𝕜
𝕜' : Type u_2
inst✝⁴ : NormedRing 𝕜'
inst✝³ : NormedAlgebra 𝕜 𝕜'
inst✝² : Module 𝕜' F
inst✝¹ : IsBoundedSMul 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
c : 𝕜 → 𝕜'
c' : 𝕜'
hc : HasDerivWithinAt c c' s x
hf : HasDerivWithinAt f f' s x
⊢ HasDerivWithinAt (c • f) (c x • f' + c' • f x) s x

Tactic:
simpa using (HasFDerivWithinAt.smul hc hf).hasDerivWithinAt

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

Goal state before:
pp: 𝕜 : Type u_1
𝔸 : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedCommRing 𝔸
inst✝ : NormedAlgebra 𝕜 𝔸
x : 𝔸
n : ℕ
⊢ fderiv 𝕜 (fun x ↦ x ^ n) x = (n • x ^ (n - 1)) • ContinuousLinearMap.id 𝕜 𝔸

Tactic:
rw [fderiv_fun_pow n differentiableAt_fun_id, fderiv_fun_id]

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

Goal state before:
pp: case e'_12.hl.e_f.e_6
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
i j : ι
⊢ Function.update fa.2 i 0 j = (Function.update (fun x ↦ fa) (some i) (g, 0) (some j)).2 j

Tactic:
rcases eq_or_ne j i with rfl | hij <;> simp [*]

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 b : k
⊢ -((b - a)⁻¹ • (f a -ᵥ f b)) = (a - b)⁻¹ • (f a -ᵥ f b)

Tactic:
← neg_smul,

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 b : k
⊢ -(b - a)⁻¹ • (f a -ᵥ f b) = (a - b)⁻¹ • (f a -ᵥ f b)
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: case hr.e_f.e_6
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 : (i : ι) → E i
i j : ι
⊢ Function.update fa.2 i (g i) j = (Function.update (fun x ↦ fa) (some i) (0, g) (some j)).2 j

Tactic:
rcases eq_or_ne j i with rfl | hij <;> simp [*]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
ι : Type u_5
𝔸 : Type u_6
inst✝² : NormedRing 𝔸
inst✝¹ : NormedAlgebra 𝕜 𝔸
inst✝ : Finite ι
l : List ι
x : ι → 𝔸
this : Fintype ι
⊢ HasStrictFDerivAt (fun x ↦ (List.map x l).prod)
    (∑ i, (List.map x (List.take (↑i) l)).prod •> (proj l[i] <• (List.map x (List.drop (↑i).succ l)).prod)) x

Tactic:
induction l with
  | nil => simp [hasStrictFDerivAt_const]
  | cons a l IH =>
    simp only [List.map_cons, List.prod_cons, ← proj_apply (R := 𝕜) (φ := fun _ : ι ↦ 𝔸) a]
    exact .congr_fderiv (.mul' (ContinuousLinearMap.hasStrictFDerivAt _) IH)
      (by ext; simp [Fin.sum_univ_succ, Finset.mul_sum, mul_assoc, add_comm])

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

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → F
f' : F
x : 𝕜
s : Set 𝕜
l : F → E
l' : F →L[𝕜] E
y : F
t : Set F
hl : HasFDerivWithinAt l l' t (f x)
hf : HasDerivWithinAt f f' s x
hst : MapsTo f s t
hy : y = f x
⊢ HasDerivWithinAt (l ∘ f) (l' f') s x

Tactic:
exact hl.comp_hasDerivWithinAt x hf hst

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

Goal state before:
pp: 𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
m : ℤ
k : ℕ
⊢ (deriv^[k] fun x ↦ x ^ m) = fun x ↦ (∏ i ∈ Finset.range k, (↑m - ↑i)) * x ^ (m - ↑k)

Tactic:
induction k with

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

Goal state before:
pp: 𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
k : ℕ
c d : 𝕜
⊢ (deriv^[k] fun x ↦ (c * x + d)⁻¹) = fun x ↦ (-1) ^ k * ↑k ! * c ^ k * (c * x + d) ^ (-1 - ↑k)

Tactic:
induction k with

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

Goal state before:
pp: case cons.hdf
𝕜 : Type u_1
𝕜' : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
ι : Type u_3
f : ι → 𝕜 → 𝕜'
x : 𝕜
a : ι
s : Finset ι
ha : a ∉ s
ih :
  (∀ i ∈ s, f i x ≠ 0) →
    (∀ i ∈ s, DifferentiableAt 𝕜 (f i) x) → logDeriv (fun x ↦ ∏ i ∈ s, f i x) x = ∑ i ∈ s, logDeriv (f i) x
hf : f a x ≠ 0 ∧ ∀ x_1 ∈ s, f x_1 x ≠ 0
hd : DifferentiableAt 𝕜 (f a) x ∧ ∀ x_1 ∈ s, DifferentiableAt 𝕜 (f x_1) x
⊢ DifferentiableAt 𝕜 (f a) x

Tactic:
exact hd.1

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.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
⊢ (ftaylorSeriesWithin 𝕜 fun x ↦ 0) = 0

Tactic:
by
  apply ftaylorSeriesWithin_zero

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → E
x₀ : ℝ
n : ℕ
s : Set ℝ
hs : Convex ℝ s
hx₀s : x₀ ∈ s
hff' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
hf' : f' =o[𝓝[s] x₀] fun x ↦ (x - x₀) ^ n
⊢ (fun x ↦ f x - f x₀) =o[𝓝[s] x₀] fun x ↦ (x - x₀) ^ (n + 1)

Tactic:
by
  have h := hs.isLittleO_pow_succ hx₀s hff' ?_ (n := n)
  · rw [Asymptotics.isLittleO_iff] at h ⊢
    simpa using h
  · rw [Asymptotics.isLittleO_iff] at hf' ⊢
    convert! hf' using 4 with c hc x
    simp

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

Goal state before:
pp: 𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
H : Type u_5
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
f : G → H
f' : G → G →L[𝕜] H
x : G
hder : ∀ᶠ (y : G) in 𝓝 x, HasFDerivAt f (f' y) y
hcont : ContinuousAt f' x
c : ℝ
hc : 0 < c
ε : ℝ
ε0 : ε > 0
hε : ball x ε ⊆ {x | (fun y ↦ HasFDerivAt f (f' y) y) x} ∩ f' ⁻¹' ball (f' x) c
a b : G
h : (a, b) ∈ ball (x, x) ε
⊢ ‖f (a, b).1 - f (a, b).2 - (f' x) ((a, b).1 - (a, b).2)‖ ≤ c * ‖(a, b).1 - (a, b).2‖

Tactic:
← ball_prod_same,

Goal state after:
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
H : Type u_5
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
f : G → H
f' : G → G →L[𝕜] H
x : G
hder : ∀ᶠ (y : G) in 𝓝 x, HasFDerivAt f (f' y) y
hcont : ContinuousAt f' x
c : ℝ
hc : 0 < c
ε : ℝ
ε0 : ε > 0
hε : ball x ε ⊆ {x | (fun y ↦ HasFDerivAt f (f' y) y) x} ∩ f' ⁻¹' ball (f' x) c
a b : G
h : (a, b) ∈ ball x ε ×ˢ ball x ε
⊢ ‖f (a, b).1 - f (a, b).2 - (f' x) ((a, b).1 - (a, b).2)‖ ≤ c * ‖(a, b).1 - (a, b).2‖