mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.PowerSeries.Substitution

Goal state before:
pp: case succ.succ
R : Type u_2
inst✝¹ : CommRing R
P : R⟦X⟧
hP : constantCoeff P = 0
inst✝ : Invertible ((coeff 1) P)
n : ℕ
B : R⟦X⟧
hB : ∑ i, C (P.substInvFun ↑i) * X ^ ↑i = B
hB' : constantCoeff B = 0
⊢ ∑ᶠ (d : ℕ),
      (coeff d) P •
        (coeff (n + 1 + 1)) ((B + -(C ⅟((coeff 1) P) * C ((coeff (n + 1 + 1)) (subst B P)) * X ^ (n + 2))) ^ d) =
    0

pp: case succ.succ.hb
R : Type u_2
inst✝¹ : CommRing R
P : R⟦X⟧
hP : constantCoeff P = 0
inst✝ : Invertible ((coeff 1) P)
n : ℕ
B : R⟦X⟧
hB : ∑ i, C (P.substInvFun ↑i) * X ^ ↑i = B
hB' : constantCoeff B = 0
⊢ HasSubst (B + -(C ⅟((coeff 1) P) * C ((coeff (n + 1 + 1)) (subst B P)) * X ^ (n + 2)))

Tactic:
· simp only [smul_eq_mul, ← map_mul]
      generalize hk : ⅟(P.coeff 1) * coeff (n + 1 + 1) (subst B P) = k
      trans ∑ᶠ d, P.coeff d * (coeff (n + 1 + 1) (B ^ d) - if d = 1 then k else 0)
      · refine finsum_congr fun i ↦ ?_
        · congr 1
          obtain (_ | _ | i) := i
          · simp
          · simp [← sub_eq_add_neg]
          · simp only [add_assoc, Nat.reduceAdd]
            rw [add_comm B, add_pow, map_sum, Finset.sum_eq_single (a := 0)]
            · simp
            · rintro (_ | _ | j) hj hj'
              · simp at hj'
              · simp [mul_comm (C k), hB', mul_assoc, coeff_X_pow_mul']
              · rw [← neg_mul, mul_pow, ← pow_mul, mul_comm (_ ^ _)]
                simp [mul_assoc, coeff_X_pow_mul']
            · simp
      · simp_rw [mul_sub]
        rw [finsum_sub_distrib]
        · simp only [mul_ite, mul_zero]
          nth_rw 2 [finsum_eq_single (a := 1)]
          · simp only [↓reduceIte, ← hk, mul_invOf_cancel_left', sub_eq_zero]
            rw [coeff_subst']
            · rfl
            · simp [HasSubst, ← PowerSeries.constantCoeff.eq_def, hB']
          · simp +contextual
        · refine .subset (Set.finite_Iio (n + 3)) fun i ↦ ?_
          obtain ⟨B, rfl⟩ : X ∣ B := by rwa [X_dvd_iff]
          simp +contextual [mul_pow, coeff_X_pow_mul', Nat.lt_succ_iff]
        · exact .subset (Set.finite_singleton 1) (fun _ ↦ by simp +contextual)

Goal state after:
case succ.succ.hb
R : Type u_2
inst✝¹ : CommRing R
P : R⟦X⟧
hP : constantCoeff P = 0
inst✝ : Invertible ((coeff 1) P)
n : ℕ
B : R⟦X⟧
hB : ∑ i, C (P.substInvFun ↑i) * X ^ ↑i = B
hB' : constantCoeff B = 0
⊢ HasSubst (B + -(C ⅟((coeff 1) P) * C ((coeff (n + 1 + 1)) (subst B P)) * X ^ (n + 2)))
Module: Mathlib.Analysis.Calculus.FDeriv.Const

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
F : Type u_3
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 F
inst✝ : TopologicalSpace F
f : E → F
x : E
h : x ∉ tsupport f
⊢ HasStrictFDerivAt f 0 x

Tactic:
by
  rw [notMem_tsupport_iff_eventuallyEq] at h
  exact (hasStrictFDerivAt_const (0 : F) x).congr_of_eventuallyEq h.symm

Goal state after:
Module: Mathlib.Analysis.Asymptotics.TVS

Goal state before:
pp: α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : TopologicalSpace E
inst✝³ : Module 𝕜 E
inst✝² : AddCommGroup F
inst✝¹ : TopologicalSpace F
inst✝ : Module 𝕜 F
f₁ f₂ : α → E
g₁ g₂ : α → F
l : Filter α
hf : f₁ =ᶠ[l] f₂
hg : g₁ =ᶠ[l] g₂
U : Set E
hU : U ∈ 𝓝 0
V : Set F
hV : V ∈ 𝓝 0
⊢ ((fun x ↦ egauge 𝕜 U (f₁ x)) ≤ᶠ[l] fun x ↦ egauge 𝕜 V (g₁ x)) ↔
    (fun x ↦ egauge 𝕜 U (f₂ x)) ≤ᶠ[l] fun x ↦ egauge 𝕜 V (g₂ x)

Tactic:
exact eventuallyLE_congr (hf.fun_comp (egauge 𝕜 U)) (hg.fun_comp (egauge 𝕜 V))

Goal state after:
Module: Mathlib.Analysis.Asymptotics.TVS

Goal state before:
pp: α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : TopologicalSpace E
inst✝³ : Module 𝕜 E
inst✝² : AddCommGroup F
inst✝¹ : TopologicalSpace F
inst✝ : Module 𝕜 F
f₁ f₂ : α → E
g₁ g₂ : α → F
l : Filter α
hf : f₁ =ᶠ[l] f₂
hg : g₁ =ᶠ[l] g₂
⊢ f₁ =O[𝕜; l] g₁ ↔ f₂ =O[𝕜; l] g₂

Tactic:
by
  simp only [isBigOTVS_iff]
  peel with U hU V hV
  exact eventuallyLE_congr (hf.fun_comp (egauge 𝕜 U)) (hg.fun_comp (egauge 𝕜 V))

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

Goal state before:
pp: 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
inst✝⁶ : ContinuousAdd E
inst✝⁵ : ContinuousSMul 𝕜 E
F : Type u_3
inst✝⁴ : AddCommGroup F
inst✝³ : Module 𝕜 F
inst✝² : TopologicalSpace F
inst✝¹ : ContinuousAdd F
inst✝ : ContinuousSMul 𝕜 F
f : E → F
f' : E →L[𝕜] F
x : E
s : Set E
h : HasFDerivWithinAt f f' s x
α : Type u_4
l : Filter α
c : α → 𝕜
d : α → E
v : E
dlim : Tendsto d l (𝓝 0)
dtop : ∀ᶠ (n : α) in l, x + d n ∈ s
cdlim : Tendsto (fun n ↦ c n • d n) l (𝓝 v)
⊢ Tendsto (fun n ↦ x + d n) l (𝓝 x)

Tactic:
by simpa using tendsto_const_nhds.add dlim

Goal state after:
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
F : Type u_3
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 F
inst✝ : TopologicalSpace F
f : E → F
f' : E →L[𝕜] F
x : E
⊢ HasFDerivWithinAt f f' univ x ↔ HasFDerivAt f f' x

Tactic:
by
  simp only [HasFDerivWithinAt, nhdsWithin_univ, HasFDerivAt]

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

Goal state before:
pp: case inr
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
F : Type u_3
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 F
inst✝ : TopologicalSpace F
f : E → F
s t : Set E
hf : DifferentiableOn 𝕜 f s
hf' : DifferentiableOn 𝕜 f t
hs : IsOpen[inst✝³] s
ht : IsOpen[inst✝³] t
x : E
hx : x ∈ t
⊢ DifferentiableWithinAt 𝕜 f (s ∪ t) x

Tactic:
· exact (hf' x hx).differentiableAt (ht.mem_nhds hx) |>.differentiableWithinAt

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

Goal state before:
pp: 𝕜 : Type u
E : Type v
F : Type w
G : Type x
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 F G
n : ℕ
v : Fin n → E
⊢ (p.compContinuousLinearMap 0 n) v = (constFormalMultilinearSeries 𝕜 E ((p 0) 0) n) v

Tactic:
cases n with

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
x y : E
r : ℝ≥0
⊢ Function.LeftInverse (fun s ↦ ⟨s.fst - s.snd.card, ⟨s.snd.card, ⟨Finset.map (finCongr ⋯).toEmbedding s.snd, ⋯⟩⟩⟩)
    fun s ↦ ⟨s.fst + s.snd.fst, ↑s.snd.snd⟩

Tactic:
rintro ⟨k, l, ⟨s : Finset (Fin <| k + l), hs : s.card = l⟩⟩
    dsimp only [Subtype.coe_mk]
    -- Lean can't automatically generalize `k' = k + l - s.card`, `l' = s.card`, so we explicitly
    -- formulate the generalized goal
    suffices ∀ k' l', k' = k → l' = l → ∀ (hkl : k + l = k' + l') (hs'),
        (⟨k', l', ⟨s.map (finCongr hkl).toEmbedding, hs'⟩⟩ :
          Σ k l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) = ⟨k, l, ⟨s, hs⟩⟩ by
      apply this <;> simp only [hs, add_tsub_cancel_right]
    simp

Goal state after:
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
x : E
hx : x ∈ Metric.eball 0 p.radius
⊢ Summable fun n ↦ ‖(p n) fun x_1 ↦ x‖

Tactic:
rw [mem_eball_zero_iff] at hx
  refine .of_nonneg_of_le
    (fun _ ↦ norm_nonneg _) (fun n ↦ ((p n).le_opNorm _).trans_eq ?_) (p.summable_norm_mul_pow hx)
  simp

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
f : E → F
pf : FormalMultilinearSeries 𝕜 E F
x : E
r : ℝ≥0∞
hf : HasFiniteFPowerSeriesOnBall f pf x 0 r
⊢ ∀ y ∈ Metric.eball x r, f y = 0

Tactic:
by
  intro y hy
  rw [hf.eq_partialSum' y hy 0 le_rfl, FormalMultilinearSeries.partialSum]
  simp only [Finset.range_zero, Finset.sum_empty]

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
x : E
f : E → F
p : FormalMultilinearSeries 𝕜 E F
U : Set E
hU : U ∈ 𝓝 x
⊢ HasFPowerSeriesWithinAt f p U x ↔ HasFPowerSeriesAt f p x

Tactic:
rw [← hasFPowerSeriesWithinAt_univ]

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
x : E
f : E → F
p : FormalMultilinearSeries 𝕜 E F
U : Set E
hU : U ∈ 𝓝 x
⊢ HasFPowerSeriesWithinAt f p U x ↔ HasFPowerSeriesWithinAt f p univ x
Module: Mathlib.Analysis.Analytic.OfScalars

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing E
inst✝¹ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
inst✝ : NormOneClass E
r : ℝ≥0∞
hc' : Tendsto (fun n ↦ ENNReal.ofReal ‖c n.succ‖ / ENNReal.ofReal ‖c n‖) atTop (𝓝 r)
⊢ (ofScalars E c).radius = r⁻¹

Tactic:
rcases ENNReal.trichotomy r with (hr | hr | hr)

Goal state after:
case inl
𝕜 : Type u_1
E : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing E
inst✝¹ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
inst✝ : NormOneClass E
r : ℝ≥0∞
hc' : Tendsto (fun n ↦ ENNReal.ofReal ‖c n.succ‖ / ENNReal.ofReal ‖c n‖) atTop (𝓝 r)
hr : r = 0
⊢ (ofScalars E c).radius = r⁻¹

case inr.inl
𝕜 : Type u_1
E : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing E
inst✝¹ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
inst✝ : NormOneClass E
r : ℝ≥0∞
hc' : Tendsto (fun n ↦ ENNReal.ofReal ‖c n.succ‖ / ENNReal.ofReal ‖c n‖) atTop (𝓝 r)
hr : r = ∞
⊢ (ofScalars E c).radius = r⁻¹

case inr.inr
𝕜 : Type u_1
E : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing E
inst✝¹ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
inst✝ : NormOneClass E
r : ℝ≥0∞
hc' : Tendsto (fun n ↦ ENNReal.ofReal ‖c n.succ‖ / ENNReal.ofReal ‖c n‖) atTop (𝓝 r)
hr : 0 < r.toReal
⊢ (ofScalars E c).radius = r⁻¹
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
w : E
hw : ‖w‖₊ < r'
⊢ ∑ i ∈ Finset.range (n - k), ‖(p (i + k)) fun x ↦ w‖ ≤ ∑ i ∈ Finset.range (n - k), ‖p (i + k)‖ * ‖w‖ ^ (i + k)

Tactic:
by
        gcongr with i _hi; exact ((p (i + k)).le_opNorm _).trans_eq (by simp)

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
n : ℕ
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
c : Composition n
⊢ ‖q.compAlongComposition p c‖₊ ≤ ‖q c.length‖₊ * ∏ i, ‖p (c.blocksFun i)‖₊

Tactic:
rw [← NNReal.coe_le_coe]

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
n : ℕ
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
c : Composition n
⊢ ↑‖q.compAlongComposition p c‖₊ ≤ ↑(‖q c.length‖₊ * ∏ i, ‖p (c.blocksFun i)‖₊)
Module: Mathlib.Analysis.Analytic.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesWithinOnBall f p s x r
r' : ℝ≥0
hr : ↑r' < r
hr'0 : 0 < r'
a : ℝ
ha : a ∈ Ioo 0 1
C : ℝ
hC : 0 < C
hp : ∀ (n : ℕ), ‖p n‖ ≤ C * a ^ n / ↑r' ^ n
L : E × E → ℝ := fun y ↦ C * (a / ↑r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * (a / (1 - a) ^ 2 + 2 / (1 - a))
y : E × E
hy' : y ∈ Metric.eball (x, x) ↑r'
ys : y ∈ insert x s ×ˢ insert x s
⊢ y ∈ Metric.eball x r ×ˢ Metric.eball x r

Tactic:
rw [Metric.eball_prod_same]

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesWithinOnBall f p s x r
r' : ℝ≥0
hr : ↑r' < r
hr'0 : 0 < r'
a : ℝ
ha : a ∈ Ioo 0 1
C : ℝ
hC : 0 < C
hp : ∀ (n : ℕ), ‖p n‖ ≤ C * a ^ n / ↑r' ^ n
L : E × E → ℝ := fun y ↦ C * (a / ↑r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * (a / (1 - a) ^ 2 + 2 / (1 - a))
y : E × E
hy' : y ∈ Metric.eball (x, x) ↑r'
ys : y ∈ insert x s ×ˢ insert x s
⊢ y ∈ Metric.eball (x, x) r
Module: Mathlib.Analysis.Analytic.Composition

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
x : E
n : ℕ
v : Fin n → E
i✝ : ℕ
him✝ : i✝ < (Composition.ones n).length
hin✝ : i✝ < n
⊢ (id 𝕜 E x).applyComposition (Composition.ones n) v ⟨i✝, him✝⟩ = v ⟨i✝, hin✝⟩

Tactic:
applyComposition_ones

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
x : E
n : ℕ
v : Fin n → E
i✝ : ℕ
him✝ : i✝ < (Composition.ones n).length
hin✝ : i✝ < n
⊢ (fun v i ↦ (id 𝕜 E x 1) fun x ↦ v (Fin.castLE ⋯ i)) v ⟨i✝, him✝⟩ = v ⟨i✝, hin✝⟩
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
M N : ℕ
z : E
⊢ q.partialSum M (∑ i ∈ Finset.Ico 1 N, (p i) fun _j ↦ z) =
    ∑ i ∈ compPartialSumTarget 0 M N, (q.compAlongComposition p i.snd) fun _j ↦ z

Tactic:
suffices H :
    (∑ n ∈ Finset.range M,
        ∑ r ∈ Fintype.piFinset fun i : Fin n => Finset.Ico 1 N,
          q n fun i : Fin n => p (r i) fun _j => z) =
      ∑ i ∈ compPartialSumTarget 0 M N, q.compAlongComposition p i.2 fun _j => z by
    simpa only [FormalMultilinearSeries.partialSum, ContinuousMultilinearMap.map_sum_finset] using H

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
M N : ℕ
z : E
⊢ (∑ n ∈ Finset.range M, ∑ r ∈ Fintype.piFinset fun i ↦ Finset.Ico 1 N, (q n) fun i ↦ (p (r i)) fun _j ↦ z) =
    ∑ i ∈ compPartialSumTarget 0 M N, (q.compAlongComposition p i.snd) fun _j ↦ z
Module: Mathlib.Analysis.Analytic.Basic

Goal state before:
pp: case refine_1
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
r : ℝ
r_pos : r > 0
h : ∀ ⦃y : 𝕜⦄, dist y 0 < r → HasSum (fun n ↦ y ^ n • p.coeff n) (f (z₀ + y))
z : 𝕜
z_pos : 0 < ‖z‖
le_z : ‖z‖ < r
this : ↑‖z‖₊ ≤ p.radius
⊢ 0 < ‖z‖₊

Tactic:
exact zero_lt_iff.mpr (nnnorm_ne_zero_iff.mpr (norm_pos_iff.mp z_pos))

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‖
n : ℕ
one_le_n : 1 ≤ n
hn : S n ≤ (I + 1) * a
In : 2 ≤ n + 1
rSn : r * S n ≤ 1 / 2
⊢ I * a + I * C * ∑ k ∈ Ico 2 (n + 1), (r * S n) ^ k =
    I * a + I * C * (((r * S n) ^ 2 - (r * S n) ^ (n + 1)) / (1 - r * S n))

Tactic:
rw [geom_sum_Ico' _ In]

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 := ‖↑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
⊢ r * S n ≠ 1
Module: Mathlib.Analysis.Analytic.Within

Goal state before:
pp: case 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
s : Set E
x : E
g : E → F
hf : f =ᶠ[𝓝[insert x s] x] g
hg : AnalyticAt 𝕜 g x
⊢ ∃ g, f x = g x ∧ EqOn f g (insert x s) ∧ AnalyticAt 𝕜 g x

Tactic:
rcases mem_nhdsWithin.1 hf with ⟨u, u_open, xu, hu⟩

Goal state after:
case 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
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, f x = g x ∧ EqOn f g (insert x s) ∧ 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
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 : 0 < c.length
⊢ ∑ j, c.blocksFun j < ∑ j, n

Tactic:
by
        apply Finset.sum_lt_sum (fun j hj ↦ (hi j).le)
        exact ⟨⟨0, h'c⟩, Finset.mem_univ _, hi _⟩

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:
← add_assoc,

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.gather b).sizeUpTo i + (a.sigmaCompositionAux b ⟨i, hi2⟩).sizeUpTo j + k
Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
hs : UniqueDiffOn 𝕜 s
n : ℕ
IH :
  ∀ {x : E},
    x ∈ s →
      ∀ (m : Fin (n + 1) → E),
        (iteratedFDerivWithin 𝕜 (n + 1) f s x) m =
          ((iteratedFDerivWithin 𝕜 n (fun y ↦ fderivWithin 𝕜 f s y) s x) (init m)) (m (last n))
x : E
hx : x ∈ s
m : Fin (n + 1 + 1) → E
I : (E [×n]→L[𝕜] E →L[𝕜] F) ≃ₗᵢ[𝕜] E [×n.succ]→L[𝕜] F := (continuousMultilinearCurryRightEquiv' 𝕜 n E F).symm
y : E
hy : y ∈ s
⊢ iteratedFDerivWithin 𝕜 n.succ f s y = (⇑I ∘ iteratedFDerivWithin 𝕜 n (fun y ↦ fderivWithin 𝕜 f s y) s) y

Tactic:
by
      ext m
      simp [IH hy m, I]

Goal state after: