mathlib-initiative/mathlib-tactics huggingface dataset:
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
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0
hr : Summable fun i ↦ ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst
n : ℕ
⊢ ‖q.comp p n‖₊ * r ^ n ≤ ∑' (i : (n : ℕ) × Composition n), ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst

Tactic:
calc
    ‖FormalMultilinearSeries.comp q p n‖₊ * r ^ n ≤
        ∑' c : Composition n, ‖compAlongComposition q p c‖₊ * r ^ n := by
      rw [tsum_fintype, ← Finset.sum_mul]
      exact mul_le_mul' (nnnorm_sum_le _ _) le_rfl
    _ ≤ ∑' i : Σ n : ℕ, Composition n, ‖compAlongComposition q p i.snd‖₊ * r ^ i.fst :=
      NNReal.tsum_comp_le_tsum_of_inj hr sigma_mk_injective

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∞
r' : ℝ≥0
hf : HasFPowerSeriesWithinOnBall f p s x r
h : ↑r' < r
a : ℝ
ha : a ∈ Ioo 0 1
C : ℝ
hp : ∀ y ∈ Metric.ball 0 ↑r', ∀ (n : ℕ), x + y ∈ insert x s → ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n
ε : ℝ
εpos : ε > 0
L : Tendsto (fun n ↦ C * a ^ n) atTop (𝓝 (C * 0))
⊢ ∀ᶠ (n : ℕ) in atTop,
    ∀ x_1 ∈ (fun x_2 ↦ x + x_2) ⁻¹' insert x s ∩ Metric.ball 0 ↑r', dist (f (x + x_1)) (p.partialSum n x_1) < ε

Tactic:
mul_zero

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∞
r' : ℝ≥0
hf : HasFPowerSeriesWithinOnBall f p s x r
h : ↑r' < r
a : ℝ
ha : a ∈ Ioo 0 1
C : ℝ
hp : ∀ y ∈ Metric.ball 0 ↑r', ∀ (n : ℕ), x + y ∈ insert x s → ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n
ε : ℝ
εpos : ε > 0
L : Tendsto (fun n ↦ C * a ^ n) atTop (𝓝 0)
⊢ ∀ᶠ (n : ℕ) in atTop,
    ∀ x_1 ∈ (fun x_2 ↦ x + x_2) ⁻¹' insert x s ∩ Metric.ball 0 ↑r', dist (f (x + x_1)) (p.partialSum n x_1) < ε
Module: Mathlib.Analysis.Analytic.Inverse

Goal state before:
pp: n : ℕ
p : ℕ → ℝ
hp : ∀ (k : ℕ), 0 ≤ p k
r a : ℝ
hr : 0 ≤ r
ha : 0 ≤ a
⊢ ∑ e ∈ compPartialSumSource 2 (n + 1) n, ∏ j, r * (a ^ e.snd j * p (e.snd j)) =
    ∑ j ∈ Ico 2 (n + 1), r ^ j * (∑ k ∈ Ico 1 n, a ^ k * p k) ^ j

Tactic:
by
      rw [compPartialSumSource,
        ← sum_sigma' (Ico 2 (n + 1))
          (fun k : ℕ => (Fintype.piFinset fun _ : Fin k => Ico 1 n : Finset (Fin k → ℕ)))
          (fun n e => ∏ j : Fin n, r * (a ^ e j * p (e j)))]
      congr! with j
      simp only [← @MultilinearMap.mkPiAlgebra_apply ℝ (Fin j) _ ℝ]
      simp only [←
        MultilinearMap.map_sum_finset (MultilinearMap.mkPiAlgebra ℝ (Fin j) ℝ) fun _ (m : ℕ) =>
          r * (a ^ m * p m)]
      simp only [MultilinearMap.mkPiAlgebra_apply]
      simp [prod_const, ← mul_sum, mul_pow]

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
⊢ ContinuousWithinAt f (insert x s) x

Tactic:
apply (hf.continuousOn.continuousWithinAt (x := x) (by simp [hf.r_pos])).mono_of_mem_nhdsWithin

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
⊢ insert x s ∩ Metric.eball x r ∈ 𝓝[insert x s] x
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
fa : AnalyticOnNhd 𝕜 f univ
⊢ ContinuousOn f univ

Tactic:
exact fa.continuousOn

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
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‖
IRec : ∀ (n : ℕ), 1 ≤ n → S n ≤ (I + 1) * a
a' : NNReal := ⟨a, ⋯⟩
⊢ ↑a' ≤ (p.rightInv i x).radius

Tactic:
apply le_radius_of_eventually_le _ ((I + 1) * a)

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
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 := ⋯
IRec : ∀ (n : ℕ), 1 ≤ n → S n ≤ (I + 1) * a
a' : NNReal := ⋯
⊢ ∀ᶠ (n : ℕ) in atTop, ‖p.rightInv i x n‖ * ↑a' ^ n ≤ (I + 1) * a
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:
simp_rw [FormalMultilinearSeries.ext_iff, FormalMultilinearSeries.ofScalars,
    formalMultilinearSeries_geometric, one_smul, implies_true]

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

Goal state before:
pp: case hasSum
𝕜 : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
A : Type u_7
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : HasSummableGeomSeries A
⊢ ∀ {y : A},
    y ∈ Metric.eball 0 1 → HasSum (fun n ↦ (formalMultilinearSeries_geometric 𝕜 A n) fun x ↦ y) (1 - (0 + y))⁻¹ʳ

Tactic:
intro y hy
    simp only [Metric.mem_eball, edist_dist, dist_zero_right, ofReal_lt_one] at hy
    simp only [zero_add, NormedRing.inverse_one_sub _ hy, Units.oneSub, Units.inv_mk,
      formalMultilinearSeries_geometric, ContinuousMultilinearMap.mkPiAlgebraFin_apply,
      List.ofFn_const, List.prod_replicate]
    exact (summable_geometric_of_norm_lt_one hy).hasSum

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
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
r : FormalMultilinearSeries 𝕜 G H
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
v : Fin n → E
f : (a : Composition n) × Composition a.length → H :=
  fun c ↦ (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))
g : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=
  fun c ↦ (r c.fst.length) fun i ↦ (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))
a : Composition n
b : Composition a.length
a✝ : ⟨a, b⟩ ∈ Finset.univ
i : ℕ
hi1 : i < b.length
hi2 : i < (a.gather b).length
j : ℕ
hj1 : j < b.blocksFun ⟨i, hi1⟩
hj2 : j < (a.sigmaCompositionAux b ⟨↑⟨i, hi1⟩, ⋯⟩).length
k : ℕ
hk1 : k < a.blocksFun ((b.embedding ⟨i, hi1⟩) ⟨j, hj1⟩)
hk2 : k < (a.sigmaCompositionAux b ⟨↑⟨i, hi1⟩, ⋯⟩).blocksFun ⟨↑⟨j, hj1⟩, ⋯⟩
⊢ a.sizeUpTo (b.sizeUpTo i + j) + k = (a.gather b).sizeUpTo i + (a.sigmaCompositionAux b ⟨i, hi2⟩).sizeUpTo j + k

Tactic:
← sizeUpTo_sizeUpTo_add _ _ hi1 hj1

Goal state after:
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
r : FormalMultilinearSeries 𝕜 G H
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
v : Fin n → E
f : (a : Composition n) × Composition a.length → H :=
  fun c ↦ (r c.snd.length) (q.applyComposition c.snd (p.applyComposition c.fst v))
g : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)) → H :=
  fun c ↦ (r c.fst.length) fun i ↦ (q (c.snd i).length) (p.applyComposition (c.snd i) (v ∘ ⇑(c.fst.embedding i)))
a : Composition n
b : Composition a.length
a✝ : ⟨a, b⟩ ∈ Finset.univ
i : ℕ
hi1 : i < b.length
hi2 : i < (a.gather b).length
j : ℕ
hj1 : j < b.blocksFun ⟨i, hi1⟩
hj2 : j < (a.sigmaCompositionAux b ⟨↑⟨i, hi1⟩, ⋯⟩).length
k : ℕ
hk1 : k < a.blocksFun ((b.embedding ⟨i, hi1⟩) ⟨j, hj1⟩)
hk2 : k < (a.sigmaCompositionAux b ⟨↑⟨i, hi1⟩, ⋯⟩).blocksFun ⟨↑⟨j, hj1⟩, ⋯⟩
⊢ a.sizeUpTo (b.sizeUpTo i + j) + k = a.sizeUpTo (b.sizeUpTo i + j) + k
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
⊢ fderiv 𝕜 (fun y ↦ -f y) x = -fderiv 𝕜 f x

Tactic:
simp only [← fderivWithin_univ, fderivWithin_fun_neg uniqueDiffWithinAt_univ]

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Alternating.Basic

Goal state before:
pp: 𝕜 : Type u
n : ℕ
E : Type wE
F : Type wF
G : Type wG
ι : Type v
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : SeminormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f : F [⋀^ι]→L[𝕜] G
g dg : E →L[𝕜] F
v : ι → E
a b : ι
heq : v a = v b
hne : a ≠ b
⊢ ∀ x ∈ {a, b}ᶜ, (f fun j ↦ (Function.update (fun x ↦ g) x dg j) (v j)) = 0

Tactic:
· simp only [mem_compl, mem_insert, mem_singleton, not_or, and_imp]
        intro i hia hib
        apply f.map_eq_zero_of_eq _ _ hne
        simp [*, Ne.symm]

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

Goal state before:
pp: case fderivWithin
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
N : ℕ∞ω
p : E → FormalMultilinearSeries 𝕜 E F
hN : ∞ ≤ N
k : ℕ
H : ∀ (n : ℕ), HasFTaylorSeriesUpToOn (↑(n + k)) f p s
⊢ ∀ (m : ℕ), ↑m < N → ∀ x ∈ s, HasFDerivWithinAt (fun x ↦ p x m) (p x m.succ).curryLeft s x

pp: case mpr.cont
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
N : ℕ∞ω
p : E → FormalMultilinearSeries 𝕜 E F
hN : ∞ ≤ N
k : ℕ
H : ∀ (n : ℕ), HasFTaylorSeriesUpToOn (↑(n + k)) f p s
⊢ ∀ (m : ℕ), ↑m ≤ N → ContinuousOn (fun x ↦ p x m) s

Tactic:
· intro m _
      apply (H m.succ).fderivWithin m (by norm_cast; lia)

Goal state after:
case mpr.cont
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
N : ℕ∞ω
p : E → FormalMultilinearSeries 𝕜 E F
hN : ∞ ≤ N
k : ℕ
H : ∀ (n : ℕ), HasFTaylorSeriesUpToOn (↑(n + k)) f p s
⊢ ∀ (m : ℕ), ↑m ≤ N → ContinuousOn (fun x ↦ p x m) s
Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries

Goal state before:
pp: case mpr.zero_eq
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
n : ℕ
h :
  HasFTaylorSeriesUpToOn (↑n) f p s ∧
    (∀ x ∈ s, HasFDerivWithinAt (fun y ↦ p y n) (p x n.succ).curryLeft s x) ∧ ContinuousOn (fun x ↦ p x (n + 1)) s
⊢ ∀ x ∈ s, (p x 0).curry0 = f x

pp: case fderivWithin
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
n : ℕ
h :
  HasFTaylorSeriesUpToOn (↑n) f p s ∧
    (∀ x ∈ s, HasFDerivWithinAt (fun y ↦ p y n) (p x n.succ).curryLeft s x) ∧ ContinuousOn (fun x ↦ p x (n + 1)) s
⊢ ∀ (m : ℕ), ↑m < ↑n + 1 → ∀ x ∈ s, HasFDerivWithinAt (fun x ↦ p x m) (p x m.succ).curryLeft s x

pp: case mpr.cont
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
n : ℕ
h :
  HasFTaylorSeriesUpToOn (↑n) f p s ∧
    (∀ x ∈ s, HasFDerivWithinAt (fun y ↦ p y n) (p x n.succ).curryLeft s x) ∧ ContinuousOn (fun x ↦ p x (n + 1)) s
⊢ ∀ (m : ℕ), ↑m ≤ ↑n + 1 → ContinuousOn (fun x ↦ p x m) s

Tactic:
· exact h.1.zero_eq

Goal state after:
case fderivWithin
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
n : ℕ
h :
  HasFTaylorSeriesUpToOn (↑n) f p s ∧
    (∀ x ∈ s, HasFDerivWithinAt (fun y ↦ p y n) (p x n.succ).curryLeft s x) ∧ ContinuousOn (fun x ↦ p x (n + 1)) s
⊢ ∀ (m : ℕ), ↑m < ↑n + 1 → ∀ x ∈ s, HasFDerivWithinAt (fun x ↦ p x m) (p x m.succ).curryLeft s x

case mpr.cont
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
n : ℕ
h :
  HasFTaylorSeriesUpToOn (↑n) f p s ∧
    (∀ x ∈ s, HasFDerivWithinAt (fun y ↦ p y n) (p x n.succ).curryLeft s x) ∧ ContinuousOn (fun x ↦ p x (n + 1)) s
⊢ ∀ (m : ℕ), ↑m ≤ ↑n + 1 → ContinuousOn (fun x ↦ p x m) s
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
n : ℕ
a : E
⊢ iteratedFDerivWithin 𝕜 n (fun z ↦ f (z + a)) s = fun x ↦ iteratedFDerivWithin 𝕜 n f (a +ᵥ s) (x + a)

Tactic:
simpa [add_comm a] using iteratedFDerivWithin_comp_add_left' n a

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
⊢ ∀ ⦃c : ℝ⦄,
    0 < c →
      ∀ᶠ (x_1 : E) in 𝓝[insert x s] x,
        ‖f x_1 - f x - ((continuousMultilinearCurryFin1 𝕜 E F) (p 1)) (x_1 - x)‖ ≤ c * ‖x_1 - x‖

Tactic:
intro c

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
c : ℝ
⊢ 0 < c →
    ∀ᶠ (x_1 : E) in 𝓝[insert x s] x,
      ‖f x_1 - f x - ((continuousMultilinearCurryFin1 𝕜 E F) (p 1)) (x_1 - x)‖ ≤ c * ‖x_1 - x‖
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'
v : 𝕜 → 𝕜'
u : 𝕜'
⊢ derivWithin (fun y ↦ v y * u) s x = derivWithin v s x * u

Tactic:
by_cases hv : DifferentiableWithinAt 𝕜 v s x
  · rw [derivWithin_mul_const hv u]
  by_cases hu : u = 0
  · simp [hu]
  rw [derivWithin_zero_of_not_differentiableWithinAt hv, zero_mul,
      derivWithin_zero_of_not_differentiableWithinAt]
  have : v = fun x ↦ (v x * u) * u⁻¹ := by ext; simp [hu]
  exact fun h_diff ↦ hv <| this ▸ h_diff.mul_const _

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:
rw [deriv_const_sub c, deriv_id'']

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

Goal state before:
pp: case inr
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
⊢ (∑ i, ((f.toFormalMultilinearSeries.changeOriginSeriesTerm 1 (Fintype.card ι - 1) ↑i ⋯) fun x_1 ↦ x) fun x ↦ y) =
    ∑ i, f (Function.update x i (y i))

Tactic:
simp_rw [changeOriginSeriesTerm_apply]

Goal state after:
case inr
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_1, (f.toFormalMultilinearSeries (1 + (Fintype.card ι - 1))) ((↑x_1).piecewise (fun x_2 ↦ x) fun x ↦ y) =
    ∑ i, f (Function.update x i (y i))
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: case succ
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
x : E
r : ℝ≥0∞
y : E
n : ℕ
ih :
  ∀ {F : Type (max u v)} [inst : NormedAddCommGroup F] [inst_1 : NormedSpace 𝕜 F] [CompleteSpace F]
    {p : FormalMultilinearSeries 𝕜 E F} {f : E → F},
    HasFPowerSeriesOnBall f p x r → (n ! • (p n) fun x ↦ y) = (iteratedFDeriv 𝕜 n f x) fun x ↦ y
F : Type (max u v)
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
p : FormalMultilinearSeries 𝕜 E F
f : E → F
h : HasFPowerSeriesOnBall f p x r
⊢ (n ! • (n + 1) • (p (n + 1)) fun x ↦ y) = (iteratedFDeriv 𝕜 (n + 1) f x) fun x ↦ y

Tactic:
← derivSeries_apply_diag,

Goal state after:
case succ
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
x : E
r : ℝ≥0∞
y : E
n : ℕ
ih :
  ∀ {F : Type (max u v)} [inst : NormedAddCommGroup F] [inst_1 : NormedSpace 𝕜 F] [CompleteSpace F]
    {p : FormalMultilinearSeries 𝕜 E F} {f : E → F},
    HasFPowerSeriesOnBall f p x r → (n ! • (p n) fun x ↦ y) = (iteratedFDeriv 𝕜 n f x) fun x ↦ y
F : Type (max u v)
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
p : FormalMultilinearSeries 𝕜 E F
f : E → F
h : HasFPowerSeriesOnBall f p x r
⊢ n ! • ((p.derivSeries n) fun x ↦ y) y = (iteratedFDeriv 𝕜 (n + 1) f x) fun x ↦ y
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: case succ
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
f : E → F
x : E
r : ℝ≥0∞
h : HasFPowerSeriesOnBall f p x r
y : E
inst✝ : CompleteSpace F
n✝ : ℕ
⊢ (n✝! • (n✝ + 1) • (p (n✝ + 1)) fun x ↦ y) = (iteratedFDeriv 𝕜 (n✝ + 1) f x) fun x ↦ y

Tactic:
← derivSeries_apply_diag,

Goal state after:
case succ
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
f : E → F
x : E
r : ℝ≥0∞
h : HasFPowerSeriesOnBall f p x r
y : E
inst✝ : CompleteSpace F
n✝ : ℕ
⊢ n✝! • ((p.derivSeries n✝) fun x ↦ y) y = (iteratedFDeriv 𝕜 (n✝ + 1) f x) fun x ↦ y
Module: Mathlib.Analysis.Complex.ReImTopology

Goal state before:
pp: α : Type u_1
ι : Type u_2
f : ι → α → ℂ
p : Filter ι
g : α → ℂ
K : Set α
hf : TendstoUniformlyOn f g p K
⊢ TendstoUniformlyOn (fun n x ↦ (f n x).im) (fun y ↦ (g y).im) p K

Tactic:
apply UniformContinuous.comp_tendstoUniformlyOn uniformContinuous_im hf

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

Goal state before:
pp: case inr
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
hne : x ≠ 0
⊢ deriv (fun x ↦ x⁻¹) x = -(x ^ 2)⁻¹

Tactic:
exact (hasDerivAt_inv hne).deriv

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

Goal state before:
pp: case pos
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c : 𝕜 → 𝕜'
hc : DifferentiableWithinAt 𝕜 c s x
hx : c x ≠ 0
hsx : UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (fun x ↦ (c x)⁻¹) s x = -derivWithin c s x / c x ^ 2

pp: case neg
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c : 𝕜 → 𝕜'
hc : DifferentiableWithinAt 𝕜 c s x
hx : c x ≠ 0
hsx : ¬UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (fun x ↦ (c x)⁻¹) s x = -derivWithin c s x / c x ^ 2

Tactic:
· exact (hc.hasDerivWithinAt.inv hx).derivWithin hsx

Goal state after:
case neg
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c : 𝕜 → 𝕜'
hc : DifferentiableWithinAt 𝕜 c s x
hx : c x ≠ 0
hsx : ¬UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (fun x ↦ (c x)⁻¹) s x = -derivWithin c s x / c x ^ 2
Module: Mathlib.Analysis.Calculus.Deriv.ZPow

Goal state before:
pp: case neg
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
m : ℤ
x : 𝕜
H : ¬(x ≠ 0 ∨ 0 ≤ m)
⊢ deriv (fun x ↦ x ^ m) x = ↑m * x ^ (m - 1)

Tactic:
deriv_zero_of_not_differentiableAt (mt differentiableAt_zpow.1 H)

Goal state after:
case neg
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
m : ℤ
x : 𝕜
H : ¬(x ≠ 0 ∨ 0 ≤ m)
⊢ 0 = ↑m * x ^ (m - 1)
Module: Mathlib.Analysis.Calculus.Deriv.ZPow

Goal state before:
pp: case neg
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
m : ℤ
hm : m < 0
⊢ 0 = ↑m * 0

Tactic:
mul_zero

Goal state after:
case neg
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
m : ℤ
hm : m < 0
⊢ 0 = 0
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
x : 𝕜
s : Set 𝕜
l : F → E
y : F
t : Set F
hl : DifferentiableWithinAt 𝕜 l t (f x)
hf : DifferentiableWithinAt 𝕜 f s x
hs : MapsTo f s t
hy : y = f x
⊢ derivWithin (l ∘ f) s x = (fderivWithin 𝕜 l t (f x)) (derivWithin f s x)

Tactic:
exact fderivWithin_comp_derivWithin x hl hf hs

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

Goal state before:
pp: case inr.inr
𝕜 : Type u_1
𝕜' : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
f : 𝕜 → 𝕜'
x : 𝕜
hdf : DifferentiableAt 𝕜 f x
n : ℤ
hn : n ≠ 0
hf : f x ≠ 0
⊢ logDeriv (fun x ↦ f x ^ n) x = ↑n * logDeriv f x

Tactic:
rw [logDeriv_apply, ← comp_def (· ^ n), deriv_comp _ (differentiableAt_zpow.2 <| .inl hf) hdf,
      deriv_zpow, logDeriv_apply]

Goal state after:
case inr.inr
𝕜 : Type u_1
𝕜' : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
f : 𝕜 → 𝕜'
x : 𝕜
hdf : DifferentiableAt 𝕜 f x
n : ℤ
hn : n ≠ 0
hf : f x ≠ 0
⊢ ↑n * f x ^ (n - 1) * deriv f x / f x ^ n = ↑n * (deriv f x / f x)