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

Goal state before:
pp: α : Type u_1
𝕜 : Type u_3
F : Type u_5
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : AddCommGroup F
inst✝⁵ : TopologicalSpace F
inst✝⁴ : Module 𝕜 F
l : Filter α
g : α → F
ι : Type u_7
E : ι → Type u_8
inst✝³ : (i : ι) → AddCommGroup (E i)
inst✝² : (i : ι) → Module 𝕜 (E i)
inst✝¹ : (i : ι) → TopologicalSpace (E i)
inst✝ : ∀ (i : ι), ContinuousSMul 𝕜 (E i)
f : (i : ι) → α → E i
h : ∀ (i : ι), f i =O[𝕜; l] g
this :
  (Filter.pi fun i ↦ 𝓝 0).HasBasis (fun If ↦ If.1.Finite ∧ ∀ i ∈ If.1, If.2 i ∈ 𝓝 0 ∧ Balanced 𝕜 (If.2 i)) fun If ↦
    If.1.pi fun i ↦ id (If.2 i)
⊢ (fun x i ↦ f i x) =O[𝕜; l] g

Tactic:
rw [← nhds_pi, ← Pi.zero_def] at this

Goal state after:
α : Type u_1
𝕜 : Type u_3
F : Type u_5
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : AddCommGroup F
inst✝⁵ : TopologicalSpace F
inst✝⁴ : Module 𝕜 F
l : Filter α
g : α → F
ι : Type u_7
E : ι → Type u_8
inst✝³ : (i : ι) → AddCommGroup (E i)
inst✝² : (i : ι) → Module 𝕜 (E i)
inst✝¹ : (i : ι) → TopologicalSpace (E i)
inst✝ : ∀ (i : ι), ContinuousSMul 𝕜 (E i)
f : (i : ι) → α → E i
h : ∀ (i : ι), f i =O[𝕜; l] g
this :
  (𝓝 0).HasBasis (fun If ↦ If.1.Finite ∧ ∀ i ∈ If.1, If.2 i ∈ 𝓝 0 ∧ Balanced 𝕜 (If.2 i)) fun If ↦
    If.1.pi fun i ↦ id (If.2 i)
⊢ (fun x i ↦ f i x) =O[𝕜; l] g
Module: Mathlib.Analysis.Calculus.FDeriv.Basic

Goal state before:
pp: case refine_1
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : TopologicalSpace E
F : Type u_3
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜 F
inst✝⁴ : TopologicalSpace F
f : E → F
f' : E →L[𝕜] F
x : E
inst✝³ : ContinuousAdd E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : ContinuousAdd F
inst✝ : ContinuousSMul 𝕜 F
hf : HasFDerivAt f f' x
v : E
α : Type u_4
c : α → 𝕜
l : Filter α
hc : Tendsto (fun n ↦ ‖c n‖) l atTop
⊢ Tendsto (fun n ↦ (c n)⁻¹ • v) l (𝓝 0)

Tactic:
tendsto_norm_atTop_iff_cobounded

Goal state after:
case refine_1
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : TopologicalSpace E
F : Type u_3
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜 F
inst✝⁴ : TopologicalSpace F
f : E → F
f' : E →L[𝕜] F
x : E
inst✝³ : ContinuousAdd E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : ContinuousAdd F
inst✝ : ContinuousSMul 𝕜 F
hf : HasFDerivAt f f' x
v : E
α : Type u_4
c : α → 𝕜
l : Filter α
hc : Tendsto c l (Bornology.cobounded 𝕜)
⊢ Tendsto (fun n ↦ (c n)⁻¹ • v) l (𝓝 0)
Module: Mathlib.Analysis.Calculus.FDeriv.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
x : E
s : Set E
inst✝² : ContinuousAdd E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : T2Space E
hxs : UniqueDiffWithinAt 𝕜 s x
⊢ fderivWithin 𝕜 id s x = ContinuousLinearMap.id 𝕜 E

Tactic:
rw [DifferentiableAt.fderivWithin differentiableAt_id hxs]

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
x : E
s : Set E
inst✝² : ContinuousAdd E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : T2Space E
hxs : UniqueDiffWithinAt 𝕜 s x
⊢ fderiv 𝕜 id x = ContinuousLinearMap.id 𝕜 E
Module: Mathlib.Analysis.Calculus.FDeriv.Basic

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
x₀ : E
n : ℕ
h : f =O[𝓝 x₀] fun x ↦ ‖x - x₀‖ ^ n
hn : 1 < n
⊢ HasFDerivAt f 0 x₀

Tactic:
by
  rw [← nhdsWithin_univ] at h
  exact (h.hasFDerivWithinAt (mem_univ _) hn).hasFDerivAt_of_univ

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✝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
n : ℕ
hp : p (n + 1) = 0
s : { s // s.card = n }
hs : s ∈ Finset.univ
⊢ n + 1 = 1 + n

Tactic:
abel

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

Goal state before:
pp: 𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
Ei : Fin n.succ → Type wEi
G : Type wG
G' : Type wG'
inst✝¹⁰ : Fintype ι
inst✝⁹ : Fintype ι'
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁵ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)
inst✝⁴ : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : ContinuousMultilinearMap 𝕜 Ei G
⊢ ‖{ toFun := curryLeft, map_add' := ⋯, map_smul' := ⋯, invFun := ContinuousLinearMap.uncurryLeft, left_inv := ⋯,
          right_inv := ⋯ }
        f‖ ≤
    ‖f‖

Tactic:
by dsimp; exact MultilinearMap.mkContinuousLinear_norm_le _ (norm_nonneg f) _

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

Goal state before:
pp: 𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
Ei : Fin n.succ → Type wEi
G : Type wG
G' : Type wG'
inst✝¹⁰ : Fintype ι
inst✝⁹ : Fintype ι'
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁵ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)
inst✝⁴ : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
p : Fin (n + 1)
f : ContinuousMultilinearMap 𝕜 Ei G
⊢ ‖{ toFun := curryMid p, map_add' := ⋯, map_smul' := ⋯, invFun := ContinuousLinearMap.uncurryMid p, left_inv := ⋯,
          right_inv := ⋯ }
        f‖ ≤
    ‖f‖

Tactic:
by dsimp; exact MultilinearMap.mkContinuousLinear_norm_le _ (norm_nonneg f) _

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
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesOnBall f p x r
y z : E
hz : z ∈ Metric.eball 0 r
⊢ x + y + z - y = x + z

Tactic:
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
y z : E
hz1 : x + y + z ∈ insert (x + y) (s + {y})
hz2 : z ∈ Metric.eball 0 r
⊢ x + z ∈ insert x s

Tactic:
simp only [add_singleton, image_add_right, mem_insert_iff, add_eq_left, mem_preimage] at hz1 ⊢

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
y z : E
hz2 : z ∈ Metric.eball 0 r
hz1 : z = 0 ∨ x + y + z + -y ∈ s
⊢ z = 0 ∨ x + z ∈ s
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
y z : E
hz1 : x + y + z ∈ insert (x + y) (s + {y})
hz2 : z ∈ Metric.eball 0 r
this : x + z ∈ insert x s
⊢ x + y + z - y = x + z

Tactic:
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
s : Set E
hf : AnalyticOnNhd 𝕜 f s
y x : E
hx : x + -y ∈ s
⊢ x = x - y + y

Tactic:
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
s : Set E
hf : AnalyticOnNhd 𝕜 f s
y x : E
hx : x + -y ∈ s
⊢ x - y = x + -y

Tactic:
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
s : Set E
hf : AnalyticOn 𝕜 f s
y x : E
hx : x + -y ∈ s
⊢ x = x - y + y

Tactic:
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
s : Set E
hf : AnalyticOn 𝕜 f s
y x : E
hx : x + -y ∈ s
⊢ x - y = x + -y

Tactic:
abel

Goal state after:
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
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → p m = 0
x y : E
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
⊢ (p.changeOrigin x).sum y = p.sum (x + y)

Tactic:
have finsupp : f.support.Finite := by
    apply Set.Finite.subset (s := changeOriginIndexEquiv ⁻¹' Sigma.fst ⁻¹' {m | m < n})
    · apply Set.Finite.preimage (Equiv.injective _).injOn
      simp_rw [← {m | m < n}.iUnion_of_singleton_coe, preimage_iUnion, ← range_sigmaMk]
      exact finite_iUnion fun _ ↦ finite_range _
    · refine fun s ↦ Not.imp_symm fun hs ↦ ?_
      simp only [preimage_ofPred_eq, changeOriginIndexEquiv_apply_fst, mem_ofPred, not_lt] at hs
      dsimp only [f]
      rw [changeOriginSeriesTerm_bound p hn _ _ _ hs, _root_.zero_apply, _root_.zero_apply]

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
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → p m = 0
x y : E
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
finsupp : (Function.support f).Finite
⊢ (p.changeOrigin x).sum y = p.sum (x + y)
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
s t : Set E
x : E
h : AnalyticWithinAt 𝕜 f s x
hst : s ∈ 𝓝[t] x
⊢ AnalyticWithinAt 𝕜 f t x

Tactic:
by
  rcases h with ⟨p, hp⟩
  exact ⟨p, hp.mono_of_mem_nhdsWithin hst⟩

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∞
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
yr'_2 : ‖y‖ < ↑r'
S : Summable fun n ↦ ‖p n‖ * ↑r' ^ n
k : ℕ
hk : ∑' (n : ℕ), ‖p (n + k)‖ * ↑r' ^ (n + k) < ε / 4
A : ∀ᶠ (z : ℕ × E) in atTop ×ˢ 𝓝 y, dist (p.partialSum k z.2) (p.partialSum k y) < ε / 4
B : ∀ᶠ (z : ℕ × E) in atTop ×ˢ 𝓝 y, ‖z.2‖₊ < r'
C : ∀ᶠ (z : ℕ × E) in atTop ×ˢ 𝓝 y, k ≤ z.1
n : ℕ
z : E
h'z : ‖(n, z).2‖₊ < r'
hkn : k ≤ (n, z).1
hz : ‖p.partialSum k z - p.partialSum k y‖ < ε / 4
I : ∀ (w : E), ‖w‖₊ < r' → ‖∑ i ∈ Finset.Ico k n, (p i) fun x ↦ w‖ ≤ ε / 4
⊢ ((∑ i ∈ Finset.range k, (p i) fun x ↦ z) + ∑ i ∈ Finset.Ico k n, (p i) fun x ↦ z) -
      ((∑ i ∈ Finset.range k, (p i) fun x ↦ y) + ∑ i ∈ Finset.Ico k n, (p i) fun x ↦ y) =
    (((∑ i ∈ Finset.range k, (p i) fun x ↦ z) - ∑ i ∈ Finset.range k, (p i) fun x ↦ y) +
        ∑ i ∈ Finset.Ico k n, (p i) fun x ↦ z) +
      -∑ i ∈ Finset.Ico k n, (p i) fun x ↦ y

Tactic:
abel

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
hq : 0 < q.radius
hp : 0 < p.radius
rq rp : ℝ≥0
hrp : rp < 1 ∧ ↑rp < p.radius
hrq : rq < 1 ∧ ↑rq < q.radius
rp_pos : 0 < rp
rq_pos : 0 < rq
⊢ ∃ r > 0, Summable fun i ↦ ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst

Tactic:
obtain ⟨Cq, _hCq0, hCq⟩ : ∃ Cq > 0, ∀ n, ‖q n‖₊ * rq ^ n ≤ Cq :=
    q.nnnorm_mul_pow_le_of_lt_radius hrq.2

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
hq : 0 < q.radius
hp : 0 < p.radius
rq rp : ℝ≥0
hrp : rp < 1 ∧ ↑rp < p.radius
hrq : rq < 1 ∧ ↑rq < q.radius
rp_pos : 0 < rp
rq_pos : 0 < rq
Cq : ℝ≥0
_hCq0 : Cq > 0
hCq : ∀ (n : ℕ), ‖q n‖₊ * rq ^ n ≤ Cq
⊢ ∃ r > 0, Summable fun i ↦ ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst
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
g : F → G
f : E → F
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
x : E
t : Set F
s : Set E
hs : Set.MapsTo f s t
rg : ℝ≥0∞
Hg : HasFPowerSeriesWithinOnBall g q t (f x) rg
rf : ℝ≥0∞
Hf : HasFPowerSeriesWithinOnBall f p s x rf
r : ℝ≥0
r_pos : 0 < r
hr : Summable fun i ↦ ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst
⊢ ∃ δ, 0 < δ ∧ ∀ {z : E}, z ∈ insert x s ∩ Metric.eball x δ → f z ∈ insert (f x) t ∩ Metric.eball (f x) rg

Tactic:
have : insert (f x) t ∩ Metric.eball (f x) rg ∈ 𝓝[insert (f x) t] (f x) := by
      apply inter_mem_nhdsWithin
      exact Metric.eball_mem_nhds _ Hg.r_pos

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
g : F → G
f : E → F
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
x : E
t : Set F
s : Set E
hs : Set.MapsTo f s t
rg : ℝ≥0∞
Hg : HasFPowerSeriesWithinOnBall g q t (f x) rg
rf : ℝ≥0∞
Hf : HasFPowerSeriesWithinOnBall f p s x rf
r : ℝ≥0
r_pos : 0 < r
hr : Summable fun i ↦ ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst
this : insert (f x) t ∩ Metric.eball (f x) rg ∈ 𝓝[insert (f x) t] f x
⊢ ∃ δ, 0 < δ ∧ ∀ {z : E}, z ∈ insert x s ∩ Metric.eball x δ → f z ∈ insert (f x) t ∩ Metric.eball (f x) rg
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:
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)))]

Goal state after:
n : ℕ
p : ℕ → ℝ
hp : ∀ (k : ℕ), 0 ≤ p k
r a : ℝ
hr : 0 ≤ r
ha : 0 ≤ a
⊢ ∑ a_1 ∈ Ico 2 (n + 1), ∑ s ∈ Fintype.piFinset fun x ↦ Ico 1 n, ∏ j, r * (a ^ s j * p (s j)) =
    ∑ j ∈ Ico 2 (n + 1), r ^ j * (∑ k ∈ Ico 1 n, a ^ k * p k) ^ j
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
x : E
n : ℕ
hn : 2 ≤ n + 1
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
r a C : ℝ
hr : 0 ≤ r
ha : 0 ≤ a
hC : 0 ≤ C
hp : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ‖↑i.symm‖
⊢ a * ‖↑i.symm‖ +
      ∑ x_1 ∈ Ico 2 (n + 1),
        a ^ x_1 *
          (‖↑i.symm‖ *
            ∑ x_2 ∈ {c | 1 < c.length}.toFinset, C * (r ^ x_2.length * ∏ j, ‖p.rightInv i x (x_2.blocksFun j)‖)) =
    ‖↑i.symm‖ * a +
      ‖↑i.symm‖ * C *
        ∑ k ∈ Ico 2 (n + 1),
          a ^ k * ∑ c ∈ {c | 1 < c.length}.toFinset, r ^ c.length * ∏ j, ‖p.rightInv i x (c.blocksFun j)‖

Tactic:
← mul_sum,

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
x : E
n : ℕ
hn : 2 ≤ n + 1
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
r a C : ℝ
hr : 0 ≤ r
ha : 0 ≤ a
hC : 0 ≤ C
hp : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ‖↑i.symm‖
⊢ a * ‖↑i.symm‖ +
      ∑ x_1 ∈ Ico 2 (n + 1),
        a ^ x_1 *
          (‖↑i.symm‖ *
            (C * ∑ i_1 ∈ {c | 1 < c.length}.toFinset, r ^ i_1.length * ∏ j, ‖p.rightInv i x (i_1.blocksFun j)‖)) =
    ‖↑i.symm‖ * a +
      ‖↑i.symm‖ * C *
        ∑ k ∈ Ico 2 (n + 1),
          a ^ k * ∑ c ∈ {c | 1 < c.length}.toFinset, r ^ c.length * ∏ j, ‖p.rightInv i x (c.blocksFun j)‖
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
x : E
n : ℕ
hn : 2 ≤ n + 1
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
r a C : ℝ
hr : 0 ≤ r
ha : 0 ≤ a
hC : 0 ≤ C
hp : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ‖↑i.symm‖
⊢ ‖↑i.symm‖ * a +
      ∑ x_1 ∈ Ico 2 (n + 1),
        ‖↑i.symm‖ *
          (a ^ x_1 *
            (C * ∑ i_1 ∈ {c | 1 < c.length}.toFinset, r ^ i_1.length * ∏ j, ‖p.rightInv i x (i_1.blocksFun j)‖)) =
    ‖↑i.symm‖ * a +
      ‖↑i.symm‖ *
        (C *
          ∑ k ∈ Ico 2 (n + 1),
            a ^ k * ∑ c ∈ {c | 1 < c.length}.toFinset, r ^ c.length * ∏ j, ‖p.rightInv i x (c.blocksFun j)‖)

Tactic:
← mul_sum,

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
x : E
n : ℕ
hn : 2 ≤ n + 1
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
r a C : ℝ
hr : 0 ≤ r
ha : 0 ≤ a
hC : 0 ≤ C
hp : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ‖↑i.symm‖
⊢ ‖↑i.symm‖ * a +
      ‖↑i.symm‖ *
        ∑ i_1 ∈ Ico 2 (n + 1),
          a ^ i_1 *
            (C * ∑ i_2 ∈ {c | 1 < c.length}.toFinset, r ^ i_2.length * ∏ j, ‖p.rightInv i x (i_2.blocksFun j)‖) =
    ‖↑i.symm‖ * a +
      ‖↑i.symm‖ *
        (C *
          ∑ k ∈ Ico 2 (n + 1),
            a ^ k * ∑ c ∈ {c | 1 < c.length}.toFinset, r ^ c.length * ∏ j, ‖p.rightInv i x (c.blocksFun j)‖)
Module: Mathlib.Analysis.Analytic.Basic

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
h : 0 < p.radius
y✝ : E
hy : y✝ ∈ Metric.eball 0 p.radius
⊢ HasSum (fun n ↦ (p n) fun x ↦ y✝) (p.sum (0 + y✝))

Tactic:
by
      rw [zero_add]
      exact p.hasSum hy

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, ⋯⟩
H : ↑a' ≤ (p.rightInv i x).radius
⊢ 0 < (p.rightInv i x).radius

Tactic:
apply lt_of_lt_of_le _ H
    -- Prior to https://github.com/leanprover/lean4/pull/2734, this was `exact_mod_cast apos`.
    simpa only [ENNReal.coe_pos]

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

Goal state before:
pp: case mpr.refine_1
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
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
g : E → F
r : ℝ≥0∞
hg : HasFPowerSeriesOnBall g p x r
e : ℝ
e0 : e > 0
hfg : ∀ ⦃y : E⦄, dist y x < e → y ∈ insert x s → f y = g y
y : E
ys : y ∈ insert x s
xy : y ∈ eball x (min r (ENNReal.ofReal e))
⊢ f y = g y

Tactic:
refine hfg ?_ ys

Goal state after:
case mpr.refine_1
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
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
g : E → F
r : ℝ≥0∞
hg : HasFPowerSeriesOnBall g p x r
e : ℝ
e0 : e > 0
hfg : ∀ ⦃y : E⦄, dist y x < e → y ∈ insert x s → f y = g y
y : E
ys : y ∈ insert x s
xy : y ∈ eball x (min r (ENNReal.ofReal e))
⊢ dist y x < e
Module: Mathlib.Analysis.Analytic.Composition

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

Tactic:
have := j.2

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

Goal state before:
pp: case e'_2.h
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
c : Composition n
d : (i : Fin c.length) → Composition (c.blocksFun i)
this : List.map sum (ofFn fun i ↦ (d i).blocks) = c.blocks
⊢ List.map List.length (ofFn fun j ↦ (d j).blocks) =
    { blocks := ofFn fun j ↦ (d j).length, blocks_pos := ⋯, blocks_sum := ⋯ }.blocks

Tactic:
simp only [map_ofFn, Function.comp_def]

Goal state after: