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

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
r : ℝ≥0∞
n : ℕ
x y : E
hf : HasFiniteFPowerSeriesOnBall f p x n r
h : y ∈ Metric.eball x r
⊢ CPolynomialAt 𝕜 f y

Tactic:
have : (‖y - x‖₊ : ℝ≥0∞) < r := by simpa [edist_eq_enorm_sub] using! h
  have := hf.changeOrigin this
  rw [add_sub_cancel] at this
  exact this.cpolynomialAt

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
m M N n : ℕ
f : Fin n → ℕ
hi : (m ≤ ⟨n, f⟩.fst ∧ ⟨n, f⟩.fst < M) ∧ ∀ (a : Fin ⟨n, f⟩.fst), 1 ≤ ⟨n, f⟩.snd a ∧ ⟨n, f⟩.snd a < N
⊢ (ofFn fun a ↦ f a).sum = ∑ j, f j

Tactic:
simp [sum_ofFn]

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
r'0 : 0 < ↑r'
h : ↑r' < r
⊢ (fun y ↦ f y.1 - f y.2 - (p 1) fun x ↦ y.1 - y.2) =O[𝓝[insert x s ×ˢ insert x s] (x, x)] fun y ↦
    ‖y - (x, x)‖ * ‖y.1 - y.2‖

Tactic:
refine (hf.isBigO_image_sub_image_sub_deriv_principal h).mono ?_

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
r'0 : 0 < ↑r'
h : ↑r' < r
⊢ 𝓝[insert x s ×ˢ insert x s] (x, x) ≤ 𝓟 (Metric.eball (x, x) ↑r' ∩ insert x s ×ˢ insert x s)
Module: Mathlib.Analysis.Analytic.Composition

Goal state before:
pp: m M N n : ℕ
f : Fin n → ℕ
hi : ⟨n, f⟩ ∈ compPartialSumSource m M N
j : Fin ⟨n, f⟩.fst
⊢ (compChangeOfVariables m M N ⟨n, f⟩ hi).snd.blocksFun ⟨↑j, ⋯⟩ = ⟨n, f⟩.snd j

Tactic:
dsimp [Composition.blocksFun, Composition.blocks, compChangeOfVariables]

Goal state after:
m M N n : ℕ
f : Fin n → ℕ
hi : ⟨n, f⟩ ∈ compPartialSumSource m M N
j : Fin ⟨n, f⟩.fst
⊢ (ofFn fun a ↦ f a)[↑j] = f j
Module: Mathlib.Analysis.Analytic.Inverse

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
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
⊢ S (n + 1) ≤ (I + 1) * a

Tactic:
have rSn : r * S n ≤ 1 / 2 :=
        calc
          r * S n ≤ r * ((I + 1) * a) := by gcongr
          _ ≤ 1 / 2 := by rwa [← mul_assoc]

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
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
⊢ S (n + 1) ≤ (I + 1) * a
Module: Mathlib.Analysis.Analytic.Within

Goal state before:
pp: case neg
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
g : E → F
hf : f =ᶠ[𝓝[insert x s] x] g
hg : AnalyticAt 𝕜 g x
u : Set E
u_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
xu : x ∈ u
hu : u ∩ insert x s ⊆ {x | (fun x ↦ f x = g x) x}
g' : E → F := u.piecewise g f
y : E
hy : y ∈ insert x s
h'y : y ∉ u
⊢ f y = g' y

Tactic:
· simp [g', h'y]

Goal state after:
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
⊢ (∃ g, f x = g x ∧ EqOn f g (insert x s) ∧ AnalyticAt 𝕜 g x) → ∃ g, f =ᶠ[𝓝[insert x s] x] g ∧ AnalyticAt 𝕜 g x

Tactic:
rintro ⟨g, -, hf, hg⟩

Goal state after:
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
g : E → F
hf : EqOn f g (insert x s)
hg : AnalyticAt 𝕜 g x
⊢ ∃ g, f =ᶠ[𝓝[insert x s] x] g ∧ AnalyticAt 𝕜 g x
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
a : Composition n
b : Composition a.length
i : Fin (a.gather b).length
⊢ (a.blocks.splitWrtComposition b)[↑i].sum = (a.gather b).blocksFun i

Tactic:
simp [Composition.blocksFun, getElem_map, Composition.gather]

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

Goal state before:
pp: case zero.succ
n : ℕ
a : Composition n
b : Composition a.length
i : ℕ
IH :
  ∀ (hi : i < b.length),
    0 < b.blocksFun ⟨i, hi⟩ →
      (take (take i b.blocks).sum a.blocks).sum = (take i (List.map sum (a.blocks.splitWrtComposition b))).sum
hi : i + 1 < b.length
hj : 0 < b.blocksFun ⟨i + 1, hi⟩
A : i < b.length
B : i < (List.map sum (a.blocks.splitWrtComposition b)).length
C : 0 < b.blocksFun ⟨i, A⟩
this : take (take i b.blocks).sum a.blocks = take (take i b.blocks).sum (take (take (i + 1) b.blocks).sum a.blocks)
⊢ (take (take (i + 1) b.blocks).sum a.blocks).sum =
    (take (take i b.blocks).sum (take (take (i + 1) b.blocks).sum a.blocks)).sum +
      (drop (b.sizeUpTo i) (take (b.sizeUpTo (i + 1)) a.blocks)).sum

Tactic:
←
        take_append_drop (sum (take i b.blocks)) (take (sum (take (Nat.succ i) b.blocks)) a.blocks),

Goal state after:
case zero.succ
n : ℕ
a : Composition n
b : Composition a.length
i : ℕ
IH :
  ∀ (hi : i < b.length),
    0 < b.blocksFun ⟨i, hi⟩ →
      (take (take i b.blocks).sum a.blocks).sum = (take i (List.map sum (a.blocks.splitWrtComposition b))).sum
hi : i + 1 < b.length
hj : 0 < b.blocksFun ⟨i + 1, hi⟩
A : i < b.length
B : i < (List.map sum (a.blocks.splitWrtComposition b)).length
C : 0 < b.blocksFun ⟨i, A⟩
this : take (take i b.blocks).sum a.blocks = take (take i b.blocks).sum (take (take (i + 1) b.blocks).sum a.blocks)
⊢ (take (take i b.blocks).sum (take (take i.succ b.blocks).sum a.blocks) ++
        drop (take i b.blocks).sum (take (take i.succ b.blocks).sum a.blocks)).sum =
    (take (take i b.blocks).sum
          (take (take i b.blocks).sum (take (take i.succ b.blocks).sum a.blocks) ++
            drop (take i b.blocks).sum (take (take i.succ b.blocks).sum a.blocks))).sum +
      (drop (b.sizeUpTo i) (take (b.sizeUpTo (i + 1)) a.blocks)).sum
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
i : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i))
⊢ (ofFn fun j ↦ (i.snd j).blocks.length).sum = (ofFn fun j ↦ (i.snd j).blocks).flatten.length

Tactic:
simp [sum_ofFn]

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

Goal state before:
pp: 𝕜 : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
𝕜' : Type u_9
inst✝⁵ : NontriviallyNormedField 𝕜'
inst✝⁴ : NormedAlgebra 𝕜 𝕜'
inst✝³ : NormedSpace 𝕜' E
inst✝² : IsScalarTower 𝕜 𝕜' E
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
f : E → F
x : E
s : Set E
hf : AnalyticWithinAt 𝕜' f s x
⊢ AnalyticWithinAt 𝕜 f s x

Tactic:
by
  rcases hf with ⟨p, hp⟩
  exact ⟨p.restrictScalars 𝕜, hp.restrictScalars⟩

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

Goal state before:
pp: 𝕜 : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
𝕜' : Type u_9
inst✝⁵ : NontriviallyNormedField 𝕜'
inst✝⁴ : NormedAlgebra 𝕜 𝕜'
inst✝³ : NormedSpace 𝕜' E
inst✝² : IsScalarTower 𝕜 𝕜' E
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
f : E → F
x : E
hf : AnalyticAt 𝕜' f x
⊢ AnalyticAt 𝕜 f x

Tactic:
by
  rcases hf with ⟨p, hp⟩
  exact ⟨p.restrictScalars 𝕜, hp.restrictScalars⟩

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:
rw [Set.preimage_compl, Set.mem_compl_iff, Set.mem_preimage, Set.mem_singleton_iff] at hy

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 : ¬f y = 0
⊢ g y = ((f⁻¹ • f) • g) y
Module: Mathlib.Analysis.Calculus.FDeriv.Add

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
f : E → F
x : E
s : Set E
R : Type u_4
inst✝⁴ : Monoid R
inst✝³ : DistribMulAction R F
inst✝² : SMulCommClass 𝕜 R F
inst✝¹ : ContinuousConstSMul R F
c : R
inst✝ : Invertible c
hs : UniqueDiffWithinAt 𝕜 s x
h : DifferentiableWithinAt 𝕜 (c • f) s x
⊢ DifferentiableWithinAt 𝕜 f s x

Tactic:
exact (differentiableWithinAt_smul_iff c).mp h

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

Goal state before:
pp: case inr
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
f : E → F
x : E
s : Set E
R : Type u_4
inst✝³ : DivisionSemiring R
inst✝² : Module R F
inst✝¹ : SMulCommClass 𝕜 R F
inst✝ : ContinuousConstSMul R F
c : R
hs : UniqueDiffWithinAt 𝕜 s x
ha : c ≠ 0
⊢ fderivWithin 𝕜 (c • f) s x = c • fderivWithin 𝕜 f s x

Tactic:
· have : Invertible c := invertibleOfNonzero ha
    simp [fderivWithin_const_smul_of_invertible c hs]

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

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
f' : E →L[𝕜] F
x : E
c : 𝕜ˣ
hc : ↑c ≠ 0
⊢ HasFDerivWithinAt (fun x ↦ f (↑c • x)) f' s x ↔ HasFDerivWithinAt (fun x ↦ ↑c • f (↑c • x)) (↑c • f') s x

Tactic:
exact (ContinuousLinearEquiv.smulLeft c).comp_hasFDerivWithinAt_iff.symm

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

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
n : ℕ
F' : Fin n.succ → Type u_6
inst✝¹ : (i : Fin n.succ) → NormedAddCommGroup (F' i)
inst✝ : (i : Fin n.succ) → NormedSpace 𝕜 (F' i)
φ : E → F' 0
φs : E → (i : Fin n) → F' i.succ
⊢ (Differentiable 𝕜 fun x ↦ Fin.cons (φ x) (φs x)) ↔ Differentiable 𝕜 φ ∧ Differentiable 𝕜 φs

Tactic:
rw [differentiable_pi, Fin.forall_fin_succ, differentiable_pi]

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
n : ℕ
F' : Fin n.succ → Type u_6
inst✝¹ : (i : Fin n.succ) → NormedAddCommGroup (F' i)
inst✝ : (i : Fin n.succ) → NormedSpace 𝕜 (F' i)
φ : E → F' 0
φs : E → (i : Fin n) → F' i.succ
⊢ ((Differentiable 𝕜 fun x ↦ Fin.cons (φ x) (φs x) 0) ∧
      ∀ (i : Fin n), Differentiable 𝕜 fun x ↦ Fin.cons (φ x) (φs x) i.succ) ↔
    Differentiable 𝕜 φ ∧ ∀ (i : Fin n), Differentiable 𝕜 fun x ↦ φs x i
Module: Mathlib.Analysis.Calculus.Deriv.Mul

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
𝔸 : Type u_3
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕜 𝔸
d : 𝕜 → 𝔸
d' c : 𝔸
hd : HasDerivAt d d' x
⊢ HasDerivAt (fun y ↦ c * d y) (c * d') x

Tactic:
rw [← hasDerivWithinAt_univ] at *
  exact hd.const_mul c

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

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

Tactic:
by
  rw [DifferentiableAt.derivWithin p.differentiableAt hxs]
  exact p.deriv

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

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
x : E
𝔸 : Type u_5
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕜 𝔸
a b : E → 𝔸
a' b' : E →L[𝕜] 𝔸
ha : HasFDerivAt a a' x
hb : HasFDerivAt b b' x
⊢ HasFDerivAt (a * b) (a x •> b' + a' <• b x) x

Tactic:
by
  -- `by exact` to solve unification issues.
  exact ((ContinuousLinearMap.mul 𝕜 𝔸).isBoundedBilinearMap.hasFDerivAt
    (a x, b x)).comp x (ha.prodMk hb)

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
ι : Type u_5
𝔸' : Type u_7
inst✝³ : NormedCommRing 𝔸'
inst✝² : NormedAlgebra 𝕜 𝔸'
inst✝¹ : DecidableEq ι
inst✝ : Finite ι
l : List ι
x : ι → 𝔸'
⊢ HasStrictFDerivAt (fun x ↦ (List.map x l).prod) (List.map (fun i ↦ (List.map x (l.erase i)).prod •> proj i) l).sum x

Tactic:
by
  have := Fintype.ofFinite ι
  refine hasStrictFDerivAt_list_prod'.congr_fderiv ?_
  conv_rhs => arg 1; arg 2; rw [← List.map_get_finRange l]
  simp only [List.map_map, ← List.sum_toFinset _ (List.nodup_finRange _), List.toFinset_finRange,
    Function.comp_def, ((List.erase_getElem _).map _).prod_eq, List.eraseIdx_eq_take_drop_succ,
    List.map_append, List.prod_append, List.get_eq_getElem, Fin.getElem_fin, Nat.succ_eq_add_one]
  exact Finset.sum_congr rfl fun i _ ↦ by
    ext; simp only [smul_apply, op_smul_eq_smul, smul_eq_mul]; ring

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
s : Set E
ι : Type u_5
𝔸' : Type u_7
inst✝² : NormedCommRing 𝔸'
inst✝¹ : NormedAlgebra 𝕜 𝔸'
u : Finset ι
g : ι → E → 𝔸'
g' : ι → E →L[𝕜] 𝔸'
inst✝ : DecidableEq ι
x : E
hg : ∀ i ∈ u, HasFDerivWithinAt (g i) (g' i) s x
⊢ HasFDerivWithinAt (fun x ↦ ∏ i ∈ u, g i x) (∑ i ∈ u, (∏ j ∈ u.erase i, g j x) •> g' i) s x

Tactic:
simpa [← Finset.prod_attach u] using .congr_fderiv
    (hasFDerivAt_finsetProd.comp_hasFDerivWithinAt x <|
      hasFDerivWithinAt_pi.mpr fun i ↦ hg (Subtype.val i) i.prop :)
    (by ext; simp [Finset.prod_erase_attach (g · x), ← u.sum_attach])

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
R : Type u_5
inst✝² : NormedRing R
inst✝¹ : HasSummableGeomSeries R
inst✝ : NormedAlgebra 𝕜 R
x : Rˣ
⊢ HasStrictFDerivAt Ring.inverse (-((mulLeftRight 𝕜 R) ↑x⁻¹) ↑x⁻¹) ↑x

Tactic:
convert! (analyticAt_inverse (𝕜 := 𝕜) x).hasStrictFDerivAt

Goal state after:
case e'_12
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
R : Type u_5
inst✝² : NormedRing R
inst✝¹ : HasSummableGeomSeries R
inst✝ : NormedAlgebra 𝕜 R
x : Rˣ
e_4✝ : inst✝².toAddCommGroup = NonUnitalNormedRing.toNormedAddCommGroup.toAddCommGroup
e_5✝ : (NormedAlgebra.toNormedSpace R).toModule ≍ (NormedAlgebra.toNormedSpace R).toModule
⊢ -((mulLeftRight 𝕜 R) ↑x⁻¹) ↑x⁻¹ = fderiv 𝕜 Ring.inverse ↑x
Module: Mathlib.Analysis.Calculus.Deriv.Comp

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
s' : Set 𝕜'
h : 𝕜 → 𝕜'
h₂ : 𝕜' → 𝕜'
y : 𝕜'
hh₂ : DifferentiableWithinAt 𝕜' h₂ s' y
hh : DifferentiableWithinAt 𝕜 h s x
hs : MapsTo h s s'
hy : h x = y
⊢ derivWithin (h₂ ∘ h) s x = derivWithin h₂ s' (h x) * derivWithin h s x

Tactic:
by
  subst hy; exact derivWithin_comp x hh₂ hh hs

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

Goal state before:
pp: case inr.inr
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
m : ℤ
x : 𝕜
h : x ≠ 0 ∨ 0 ≤ m
this : ∀ (m : ℤ), 0 < m → HasStrictDerivAt (fun x ↦ x ^ m) (↑m * x ^ (m - 1)) x
hm : 0 < m
⊢ HasStrictDerivAt (fun x ↦ x ^ m) (↑m * x ^ (m - 1)) x

Tactic:
· exact this m hm

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

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 f₁ : E → F
x : E
n : ℕ∞ω
h₁ : f₁ =ᶠ[𝓝[s] x] f
hx : f₁ x = f x
h : ContDiffWithinAt 𝕜 ω f s x
⊢ ContDiffWithinAt 𝕜 ω f₁ s x

Tactic:
obtain ⟨u, hu, p, H, H'⟩ := h

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 f₁ : E → F
x : E
n : ℕ∞ω
h₁ : f₁ =ᶠ[𝓝[s] x] f
hx : f₁ x = f x
u : Set E
hu : u ∈ 𝓝[insert x s] x
p : E → FormalMultilinearSeries 𝕜 E F
H : HasFTaylorSeriesUpToOn ω f p u
H' : ∀ (i : ℕ), AnalyticOn 𝕜 (fun x ↦ p x i) u
⊢ ContDiffWithinAt 𝕜 ω f₁ s x
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

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
x : E
n : ℕ∞ω
t : Set E
hst : s ∈ 𝓝[t] x
h : ContDiffWithinAt 𝕜 ω f s x
⊢ ContDiffWithinAt 𝕜 ω f t x

Tactic:
obtain ⟨u, hu, p, H, H'⟩ := h

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
x : E
n : ℕ∞ω
t : Set E
hst : s ∈ 𝓝[t] x
u : Set E
hu : u ∈ 𝓝[insert x s] x
p : E → FormalMultilinearSeries 𝕜 E F
H : HasFTaylorSeriesUpToOn ω f p u
H' : ∀ (i : ℕ), AnalyticOn 𝕜 (fun x ↦ p x i) u
⊢ ContDiffWithinAt 𝕜 ω f t x
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

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
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ c : OrderedFinpartition n
k m : Fin c.length
⊢ 0 < update c.partSize k (c.partSize k + 1) m

Tactic:
rcases eq_or_ne m k with rfl | hm
    · simp
    · simpa [hm] using c.partSize_pos m

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

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
h : ContDiffOn 𝕜 n f s
hs : UniqueDiffOn 𝕜 s
m : ℕ
hm : ↑m < n
x : E
hx : x ∈ s
this : ↑(m + 1) ≤ n
u : Set E
hu : u ∈ 𝓝[s] x
p : E → FormalMultilinearSeries 𝕜 E F
Hp : HasFTaylorSeriesUpToOn (↑(m + 1)) f p u
o : Set E
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
xo : x ∈ o
ho : s ∩ o ⊆ u
⊢ HasFDerivWithinAt (fun x ↦ ftaylorSeriesWithin 𝕜 f s x m) (ftaylorSeriesWithin 𝕜 f s x m.succ).curryLeft s x

Tactic:
have : p x m.succ = ftaylorSeriesWithin 𝕜 f s x m.succ := by
      change p x m.succ = iteratedFDerivWithin 𝕜 m.succ f s x
      rw [← iteratedFDerivWithin_inter_open o_open xo]
      exact (Hp.mono ho).eq_iteratedFDerivWithin_of_uniqueDiffOn le_rfl (hs.inter o_open) ⟨hx, xo⟩

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
n : ℕ∞ω
h : ContDiffOn 𝕜 n f s
hs : UniqueDiffOn 𝕜 s
m : ℕ
hm : ↑m < n
x : E
hx : x ∈ s
this✝ : ↑(m + 1) ≤ n
u : Set E
hu : u ∈ 𝓝[s] x
p : E → FormalMultilinearSeries 𝕜 E F
Hp : HasFTaylorSeriesUpToOn (↑(m + 1)) f p u
o : Set E
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
xo : x ∈ o
ho : s ∩ o ⊆ u
this : p x m.succ = ftaylorSeriesWithin 𝕜 f s x m.succ
⊢ HasFDerivWithinAt (fun x ↦ ftaylorSeriesWithin 𝕜 f s x m) (ftaylorSeriesWithin 𝕜 f s x m.succ).curryLeft s x
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: n : ℕ
c : OrderedFinpartition n
i : Fin c.length
⊢ (c.extendMiddle i).index 0 = i

Tactic:
have : (c.extendMiddle i).emb i 0 = 0 := by simp [extendMiddle]

Goal state after:
n : ℕ
c : OrderedFinpartition n
i : Fin c.length
this : (c.extendMiddle i).emb i 0 = 0
⊢ (c.extendMiddle i).index 0 = i
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

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∞ω
hs : UniqueDiffOn 𝕜 s
⊢ DifferentiableOn 𝕜 f s ∧ (n = ω → AnalyticOn 𝕜 f s) ∧ ContDiffOn 𝕜 n (fderivWithin 𝕜 f s) s ↔
    (n = ω → AnalyticOn 𝕜 f s) ∧ ∃ f', ContDiffOn 𝕜 n f' s ∧ ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x

Tactic:
refine ⟨fun h => ⟨h.2.1, fderivWithin 𝕜 f s, h.2.2,
    fun x hx => (h.1 x hx).hasFDerivWithinAt⟩, fun ⟨f_an, h⟩ => ?_⟩

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
n : ℕ∞ω
hs : UniqueDiffOn 𝕜 s
x✝ : (n = ω → AnalyticOn 𝕜 f s) ∧ ∃ f', ContDiffOn 𝕜 n f' s ∧ ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
f_an : n = ω → AnalyticOn 𝕜 f s
h : ∃ f', ContDiffOn 𝕜 n f' s ∧ ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
⊢ DifferentiableOn 𝕜 f s ∧ (n = ω → AnalyticOn 𝕜 f s) ∧ ContDiffOn 𝕜 n (fderivWithin 𝕜 f s) s