mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Convex.EGauge

Goal state before:
pp: case inl
𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
x : E
hc : 1 < ‖c‖
this : NontriviallyNormedField 𝕜 := { toNormedField := inst✝², non_trivial := ⋯ }
h₀ : 0 ≠ 0 ∨ ‖x‖ ≠ 0
⊢ egauge 𝕜 (ball 0 ↑0) x ≤ ‖c‖ₑ * ‖x‖ₑ / ↑0

Tactic:
ENNReal.coe_zero,

Goal state after:
case inl
𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
x : E
hc : 1 < ‖c‖
this : NontriviallyNormedField 𝕜 := { toNormedField := inst✝², non_trivial := ⋯ }
h₀ : 0 ≠ 0 ∨ ‖x‖ ≠ 0
⊢ egauge 𝕜 (ball 0 ↑0) x ≤ ‖c‖ₑ * ‖x‖ₑ / 0
Module: Mathlib.Analysis.Calculus.TangentCone.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁶ : DivisionSemiring 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace 𝕜
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousSMul 𝕜 E
s : Set E
x y : E
α : Type u_3
l : Filter α
inst✝ : l.NeBot
c : α → 𝕜
hc₀ : Tendsto c l (𝓝 0) ∧ ∀ᶠ (n : α) in l, c n ∈ {0}ᶜ
hmem : ∀ᶠ (n : α) in l, x + c n • y ∈ s
⊢ y ∈ tangentConeAt 𝕜 s x

Tactic:
refine mem_tangentConeAt_of_seq l c⁻¹ (c · • y) ?_ hmem ?_

Goal state after:
case refine_1
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : DivisionSemiring 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace 𝕜
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousSMul 𝕜 E
s : Set E
x y : E
α : Type u_3
l : Filter α
inst✝ : l.NeBot
c : α → 𝕜
hc₀ : Tendsto c l (𝓝 0) ∧ ∀ᶠ (n : α) in l, c n ∈ {0}ᶜ
hmem : ∀ᶠ (n : α) in l, x + c n • y ∈ s
⊢ Tendsto (fun x ↦ c x • y) l (𝓝 0)

case refine_2
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : DivisionSemiring 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace 𝕜
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousSMul 𝕜 E
s : Set E
x y : E
α : Type u_3
l : Filter α
inst✝ : l.NeBot
c : α → 𝕜
hc₀ : Tendsto c l (𝓝 0) ∧ ∀ᶠ (n : α) in l, c n ∈ {0}ᶜ
hmem : ∀ᶠ (n : α) in l, x + c n • y ∈ s
⊢ Tendsto (fun n ↦ c⁻¹ n • c n • y) l (𝓝 y)
Module: Mathlib.Analysis.Calculus.TangentCone.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁷ : DivisionSemiring 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace 𝕜
inst✝² : (𝓝[≠] 0).NeBot
inst✝¹ : ContinuousSMul 𝕜 E
x : E
s : Set E
inst✝ : ContinuousAdd E
h : s ∈ 𝓝 x
⊢ UniqueDiffWithinAt 𝕜 s x

Tactic:
by
  simpa only [univ_inter] using uniqueDiffWithinAt_univ.inter h

Goal state after:
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:
rw [tendsto_norm_atTop_iff_cobounded] at hc
    simpa using (tendsto_inv₀_cobounded.comp hc).smul (tendsto_const_nhds (x := v))

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
x : E
s t : Set E
ht : t ∈ 𝓝[s] x
⊢ DifferentiableWithinAt 𝕜 f (s ∩ t) x ↔ DifferentiableWithinAt 𝕜 f s x

Tactic:
simp only [DifferentiableWithinAt, hasFDerivWithinAt_inter' ht]

Goal state after:
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
f' : E →L[𝕜] F
L : Filter (E × E)
⊢ HasFDerivAtFilter f f' L ↔ Tendsto (fun p ↦ ‖p.1 - p.2‖⁻¹ * ‖f p.1 - f p.2 - f' (p.1 - p.2)‖) L (𝓝 0)

Tactic:
rw [hasFDerivAtFilter_iff_isLittleO, ← isLittleO_norm_left, ← isLittleO_norm_right,
    isLittleO_iff_tendsto]
  · simp [div_eq_inv_mul]
  · simp +contextual [sub_eq_zero]

Goal state after:
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
f' : E →L[𝕜] F
x : E
⊢ ((fun x' ↦ f x' - f x - f' (x' - x)) ∘ fun x_1 ↦ x + x_1) =o[𝓝 0] ((fun x' ↦ x' - x) ∘ fun x_1 ↦ x + x_1) ↔
    (fun h ↦ f (x + h) - f x - f' h) =o[𝓝 0] fun h ↦ h

Tactic:
simp [Function.comp_def]

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

Goal state before:
pp: 𝕜 : Type u
E : Type v
F : Type w
inst✝¹⁰ : Semiring 𝕜
inst✝⁹ : AddCommMonoid E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : TopologicalSpace E
inst✝⁶ : ContinuousAdd E
inst✝⁵ : ContinuousConstSMul 𝕜 E
inst✝⁴ : AddCommMonoid F
inst✝³ : Module 𝕜 F
inst✝² : TopologicalSpace F
inst✝¹ : ContinuousAdd F
inst✝ : ContinuousConstSMul 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
hp : p.order ≠ 0
h : p = 0
⊢ False

Tactic:
simp [h] at hp

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✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
𝕜' : Type u_6
c : 𝕜'
inst✝³ : NormedRing 𝕜'
inst✝² : Module 𝕜' F
inst✝¹ : SMulCommClass 𝕜 𝕜' F
inst✝ : IsBoundedSMul 𝕜' F
r : ℝ≥0
C : ℝ
h : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C
n : ℕ
⊢ ‖c‖ * ‖p n‖ * ↑r ^ n ≤ ‖c‖ * C

Tactic:
mul_assoc,

Goal state after:
𝕜 : Type u_1
E : Type u_3
F : Type u_4
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
𝕜' : Type u_6
c : 𝕜'
inst✝³ : NormedRing 𝕜'
inst✝² : Module 𝕜' F
inst✝¹ : SMulCommClass 𝕜 𝕜' F
inst✝ : IsBoundedSMul 𝕜' F
r : ℝ≥0
C : ℝ
h : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C
n : ℕ
⊢ ‖c‖ * (‖p n‖ * ↑r ^ n) ≤ ‖c‖ * C
Module: Mathlib.Analysis.Analytic.Linear

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 →L[𝕜] F
x : E
m n : ℕ
hm : 2 ≤ n + 2
⊢ f.fpowerSeries x (n + 2) = 0

Tactic:
simp [fpowerSeries]

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

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

Tactic:
by
    rw [Finset.sum_eq_single_of_mem _ (Finset.self_mem_range_succ _), zero_add]
    · rw [toFormalMultilinearSeries, dif_pos rfl]; rfl
    · intro m _ ne; rw [toFormalMultilinearSeries, dif_neg ne.symm]; rfl

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
⊢ p.removeZero.leftInv i x = p.leftInv i x

Tactic:
ext1 n
  induction n using Nat.strong_induction_on with | _ n IH
  match n with
  | 0 => simp -- if one replaces `simp` with `refl`, the proof times out in the kernel.
  | 1 => simp -- TODO: why?
  | n + 2 =>
    simp only [leftInv, neg_inj]
    refine Finset.sum_congr rfl fun c cuniv => ?_
    rcases c with ⟨c, hc⟩
    ext v
    simp [IH _ hc]

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
⊢ ∀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:
have L : Tendsto (fun n => (C : ℝ) * a ^ n) atTop (𝓝 ((C : ℝ) * 0)) :=
    tendsto_const_nhds.mul (tendsto_pow_atTop_nhds_zero_of_lt_one ha.1.le ha.2)

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 (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
Module: Mathlib.Analysis.Analytic.Composition

Goal state before:
pp: case i_surj
α : Type u_6
inst✝ : AddCommMonoid α
m M N : ℕ
f : (n : ℕ) × (Fin n → ℕ) → α
g : (n : ℕ) × Composition n → α
h : ∀ (e : (n : ℕ) × (Fin n → ℕ)) (he : e ∈ compPartialSumSource m M N), f e = g (compChangeOfVariables m M N e he)
⊢ ∀ b ∈ compPartialSumTarget m M N, ∃ a, ∃ (ha : a ∈ compPartialSumSource m M N), compChangeOfVariables m M N a ha = b

pp: case h
α : Type u_6
inst✝ : AddCommMonoid α
m M N : ℕ
f : (n : ℕ) × (Fin n → ℕ) → α
g : (n : ℕ) × Composition n → α
h : ∀ (e : (n : ℕ) × (Fin n → ℕ)) (he : e ∈ compPartialSumSource m M N), f e = g (compChangeOfVariables m M N e he)
⊢ ∀ (a : (n : ℕ) × (Fin n → ℕ)) (ha : a ∈ compPartialSumSource m M N), f a = g (compChangeOfVariables m M N a ha)

Tactic:
· intro i hi
    apply compPartialSumTargetSet_image_compPartialSumSource m M N i
    simpa [compPartialSumTarget] using hi

Goal state after:
case h
α : Type u_6
inst✝ : AddCommMonoid α
m M N : ℕ
f : (n : ℕ) × (Fin n → ℕ) → α
g : (n : ℕ) × Composition n → α
h : ∀ (e : (n : ℕ) × (Fin n → ℕ)) (he : e ∈ compPartialSumSource m M N), f e = g (compChangeOfVariables m M N e he)
⊢ ∀ (a : (n : ℕ) × (Fin n → ℕ)) (ha : a ∈ compPartialSumSource m M N), f a = g (compChangeOfVariables m M N a ha)
Module: Mathlib.Analysis.Analytic.Basic

Goal state before:
pp: case e'_6
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
⊢ f = (fun y ↦ f (x + y)) ∘ fun y ↦ y - x

Tactic:
simp [Function.comp_def]

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
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_assoc,

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 +
      ∑ 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)‖)
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 +
      ‖↑i.symm‖ *
        ∑ x_1 ∈ Ico 2 (n + 1),
          C * a ^ x_1 * ∑ i_1 ∈ {c | 1 < c.length}.toFinset, r ^ i_1.length * ∏ j, ‖p.rightInv i x (i_1.blocksFun j)‖ =
    ‖↑i.symm‖ * a +
      C * ‖↑i.symm‖ *
        ∑ 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_assoc,

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‖ *
        ∑ x_1 ∈ Ico 2 (n + 1),
          C *
            (a ^ x_1 * ∑ i_1 ∈ {c | 1 < c.length}.toFinset, r ^ i_1.length * ∏ j, ‖p.rightInv i x (i_1.blocksFun j)‖) =
    ‖↑i.symm‖ * a +
      C *
        (‖↑i.symm‖ *
          ∑ 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
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
⊢ 0 < (p.rightInv i x).radius

Tactic:
let a' : NNReal := ⟨a, apos.le⟩

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‖
IRec : ∀ (n : ℕ), 1 ≤ n → S n ≤ (I + 1) * a
a' : NNReal := ⟨a, ⋯⟩
⊢ 0 < (p.rightInv i x).radius
Module: Mathlib.Analysis.Analytic.Inverse

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
f : E → G
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
x : E
hq : 0 < q.radius
hp : 0 < p.radius
r0 : ℝ≥0∞
h0 : HasFPowerSeriesOnBall f (q.comp p) x r0
⊢ ∀ᶠ (y : E) in 𝓝 0, Tendsto (fun a ↦ q.partialSum a.1 (p.partialSum a.2 y - (p 0) fun x ↦ 0)) atTop (𝓝 (f (x + y)))

Tactic:
rcases q.comp_summable_nnreal p hq hp with ⟨r1, r1_pos : 0 < r1, hr1⟩

Goal state after:
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
f : E → G
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
x : E
hq : 0 < q.radius
hp : 0 < p.radius
r0 : ℝ≥0∞
h0 : HasFPowerSeriesOnBall f (q.comp p) x r0
r1 : NNReal
r1_pos : 0 < r1
hr1 : Summable fun i ↦ ‖q.compAlongComposition p i.snd‖₊ * r1 ^ i.fst
⊢ ∀ᶠ (y : E) in 𝓝 0, Tendsto (fun a ↦ q.partialSum a.1 (p.partialSum a.2 y - (p 0) fun x ↦ 0)) atTop (𝓝 (f (x + y)))
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
a : Composition n
b : Composition a.length
⊢ (fun i ↦
        ⟨{ blocks := (ofFn fun j ↦ (i.snd j).blocks).flatten, blocks_pos := ⋯, blocks_sum := ⋯ },
          { blocks := ofFn fun j ↦ (i.snd j).length, blocks_pos := ⋯, blocks_sum := ⋯ }⟩)
      ((fun i ↦ ⟨i.fst.gather i.snd, i.fst.sigmaCompositionAux i.snd⟩) ⟨a, b⟩) =
    ⟨a, b⟩

Tactic:
sigma_composition_eq_iff

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
n✝ n : ℕ
a : Composition n
b : Composition a.length
⊢ ((fun i ↦
              ⟨{ blocks := (ofFn fun j ↦ (i.snd j).blocks).flatten, blocks_pos := ⋯, blocks_sum := ⋯ },
                { blocks := ofFn fun j ↦ (i.snd j).length, blocks_pos := ⋯, blocks_sum := ⋯ }⟩)
            ((fun i ↦ ⟨i.fst.gather i.snd, i.fst.sigmaCompositionAux i.snd⟩) ⟨a, b⟩)).fst.blocks =
      ⟨a, b⟩.fst.blocks ∧
    ((fun i ↦
              ⟨{ blocks := (ofFn fun j ↦ (i.snd j).blocks).flatten, blocks_pos := ⋯, blocks_sum := ⋯ },
                { blocks := ofFn fun j ↦ (i.snd j).length, blocks_pos := ⋯, blocks_sum := ⋯ }⟩)
            ((fun i ↦ ⟨i.fst.gather i.snd, i.fst.sigmaCompositionAux i.snd⟩) ⟨a, b⟩)).snd.blocks =
      ⟨a, b⟩.snd.blocks
Module: Mathlib.Analysis.Analytic.Constructions

Goal state before:
pp: case empty
α : Type u_1
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : α → E → F
c : E
s : Set E
h : ∀ n ∈ ∅, AnalyticWithinAt 𝕜 (f n) s c
⊢ AnalyticWithinAt 𝕜 0 s c

Tactic:
exact analyticWithinAt_const

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

Goal state before:
pp: case empty
α : Type u_1
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
A : Type u_9
inst✝¹ : NormedCommRing A
inst✝ : NormedAlgebra 𝕜 A
f : α → E → A
c : E
s : Set E
h : ∀ n ∈ ∅, AnalyticWithinAt 𝕜 (f n) s c
⊢ AnalyticWithinAt 𝕜 (fun z ↦ 1) s c

Tactic:
exact analyticWithinAt_const

Goal state after:
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
f' : E →L[𝕜] F
x a : E
⊢ HasFDerivAt (fun x ↦ f (x - a)) f' x ↔ HasFDerivAt f f' (x - a)

Tactic:
by
  simp [← hasFDerivWithinAt_univ, hasFDerivWithinAt_comp_sub]

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Alternating.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
M' : Type u_3
N : Type u_4
N' : Type u_5
ι : Type u_6
inst✝¹² : Semiring R
inst✝¹¹ : AddCommMonoid M
inst✝¹⁰ : Module R M
inst✝⁹ : TopologicalSpace M
inst✝⁸ : AddCommMonoid M'
inst✝⁷ : Module R M'
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : TopologicalSpace N
inst✝² : AddCommMonoid N'
inst✝¹ : Module R N'
inst✝ : TopologicalSpace N'
n : ℕ
f✝ g : M [⋀^ι]→L[R] N
e : M ≃L[R] M'
f : M [⋀^ι]→L[R] N
x✝ : ι → M
⊢ ((fun f ↦ f.compContinuousLinearMap ↑e) ((fun f ↦ f.compContinuousLinearMap ↑e.symm) f)) x✝ = f x✝

Tactic:
simp [Function.comp_def]

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Alternating.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
M' : Type u_3
N : Type u_4
N' : Type u_5
ι : Type u_6
inst✝¹² : Semiring R
inst✝¹¹ : AddCommMonoid M
inst✝¹⁰ : Module R M
inst✝⁹ : TopologicalSpace M
inst✝⁸ : AddCommMonoid M'
inst✝⁷ : Module R M'
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : TopologicalSpace N
inst✝² : AddCommMonoid N'
inst✝¹ : Module R N'
inst✝ : TopologicalSpace N'
n : ℕ
f✝ g : M [⋀^ι]→L[R] N
e : M ≃L[R] M'
f : M' [⋀^ι]→L[R] N
x✝ : ι → M'
⊢ ((fun f ↦ f.compContinuousLinearMap ↑e.symm) ((fun f ↦ f.compContinuousLinearMap ↑e) f)) x✝ = f x✝

Tactic:
simp [Function.comp_def]

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Alternating.Topology

Goal state before:
pp: case inr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
ι : Type u_4
inst✝¹⁰ : NormedField 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : TopologicalSpace E
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜 F
inst✝⁴ : UniformSpace F
inst✝³ : IsUniformAddGroup F
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : ContinuousConstSMul 𝕜 F
inst✝ : CompleteSpace F
h : IsCoherentWith {s | IsVonNBounded 𝕜 s}
this :
  ∀ {F : Type u_3} [inst : AddCommGroup F] [inst_1 : Module 𝕜 F] [inst_2 : UniformSpace F]
    [inst_3 : IsUniformAddGroup F] [ContinuousConstSMul 𝕜 F] [CompleteSpace F],
    T2Space F → CompleteSpace (E [⋀^ι]→L[𝕜] F)
hF : ¬T2Space F
⊢ CompleteSpace (E [⋀^ι]→L[𝕜] F)

Tactic:
(isUniformInducing_postcomp (SeparationQuotient.mkCLM _ _)
      SeparationQuotient.isUniformInducing_mk).completeSpace_congr

Goal state after:
case inr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
ι : Type u_4
inst✝¹⁰ : NormedField 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : TopologicalSpace E
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜 F
inst✝⁴ : UniformSpace F
inst✝³ : IsUniformAddGroup F
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : ContinuousConstSMul 𝕜 F
inst✝ : CompleteSpace F
h : IsCoherentWith {s | IsVonNBounded 𝕜 s}
this :
  ∀ {F : Type u_3} [inst : AddCommGroup F] [inst_1 : Module 𝕜 F] [inst_2 : UniformSpace F]
    [inst_3 : IsUniformAddGroup F] [ContinuousConstSMul 𝕜 F] [CompleteSpace F],
    T2Space F → CompleteSpace (E [⋀^ι]→L[𝕜] F)
hF : ¬T2Space F
⊢ CompleteSpace (E [⋀^ι]→L[𝕜] SeparationQuotient F)

case inr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
ι : Type u_4
inst✝¹⁰ : NormedField 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : TopologicalSpace E
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜 F
inst✝⁴ : UniformSpace F
inst✝³ : IsUniformAddGroup F
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : ContinuousConstSMul 𝕜 F
inst✝ : CompleteSpace F
h : IsCoherentWith {s | IsVonNBounded 𝕜 s}
this :
  ∀ {F : Type u_3} [inst : AddCommGroup F] [inst_1 : Module 𝕜 F] [inst_2 : UniformSpace F]
    [inst_3 : IsUniformAddGroup F] [ContinuousConstSMul 𝕜 F] [CompleteSpace F],
    T2Space F → CompleteSpace (E [⋀^ι]→L[𝕜] F)
hF : ¬T2Space F
⊢ Surjective (SeparationQuotient.mkCLM 𝕜 F).compContinuousAlternatingMap