mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z ↦ ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
⊢ ‖1 - z‖ * ∑ i ∈ Ico B₁ (max B₁ B₂), ε / 4 / M * ‖z‖ ^ i = ‖1 - z‖ * (ε / 4 / M) * ∑ i ∈ Ico B₁ (max B₁ B₂), ‖z‖ ^ i

Tactic:
by
        rw [← mul_sum, ← mul_assoc]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex

Goal state before:
pp: case inl
x y : ℂ
h1 : ∀ (k : ℤ), x ≠ (2 * ↑k + 1) * ↑π / 2
h2 : ∀ (l : ℤ), y ≠ (2 * ↑l + 1) * ↑π / 2
⊢ tan (x + y) = (tan x + tan y) / (1 - tan x * tan y)

Tactic:
rw [tan, sin_add, cos_add, ←
      div_div_div_cancel_right₀ (mul_ne_zero (cos_ne_zero_iff.mpr h1) (cos_ne_zero_iff.mpr h2)),
      add_div, sub_div]

Goal state after:
case inl
x y : ℂ
h1 : ∀ (k : ℤ), x ≠ (2 * ↑k + 1) * ↑π / 2
h2 : ∀ (l : ℤ), y ≠ (2 * ↑l + 1) * ↑π / 2
⊢ (sin x * cos y / (cos x * cos y) + cos x * sin y / (cos x * cos y)) /
      (cos x * cos y / (cos x * cos y) - sin x * sin y / (cos x * cos y)) =
    (tan x + tan y) / (1 - tan x * tan y)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex

Goal state before:
pp: n : ℕ
a : ℂ
h : ∀ (k : ℤ), a ≠ ↑k * (2 * ↑π)
b : ℂ
⊢ ∑ i ∈ Finset.range n, cos (a * ↑i + b) = sin (↑n * a / 2) * cos ((↑n - 1) * a / 2 + b) / sin (a / 2)

Tactic:
rw [← sin_mul_sum_cos]

Goal state after:
n : ℕ
a : ℂ
h : ∀ (k : ℤ), a ≠ ↑k * (2 * ↑π)
b : ℂ
⊢ ∑ i ∈ Finset.range n, cos (a * ↑i + b) = (sin (a / 2) * ∑ i ∈ Finset.range n, cos (a * ↑i + b)) / sin (a / 2)
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: case refine_1
D : Set ℝ
hD : Convex ℝ D
f f' f'' : ℝ → ℝ
hf : ContinuousOn f D
hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x
hf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x
hf''₀ : ∀ x ∈ interior D, 0 ≤ f'' x
this : EqOn (deriv f) f' (interior D)
⊢ DifferentiableOn ℝ (deriv f) (interior D)

Tactic:
rw [differentiableOn_congr this]
    exact fun x hx ↦ (hf'' _ hx).differentiableWithinAt

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: case inr
x : ℝ
hx_gt : ¬x ≤ -(π / 2)
hx_lt : x < π / 2
r : ℤ
hxr_eq : x = (2 * ↑r + 1) * π / 2
h : r < 0
⊢ x ≤ -(π / 2)

Tactic:
rw [hxr_eq, ← one_mul (π / 2), mul_div_assoc, neg_mul_eq_neg_mul,
      mul_le_mul_iff_left₀ (half_pos pi_pos)]

Goal state after:
case inr
x : ℝ
hx_gt : ¬x ≤ -(π / 2)
hx_lt : x < π / 2
r : ℤ
hxr_eq : x = (2 * ↑r + 1) * π / 2
h : r < 0
⊢ 2 * ↑r + 1 ≤ -1
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: case refine_1
D : Set ℝ
hD : Convex ℝ D
f f' f'' : ℝ → ℝ
hf : ContinuousOn f D
hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x
hf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x
hf''₀ : ∀ x ∈ interior D, f'' x ≤ 0
this : EqOn (deriv f) f' (interior D)
⊢ DifferentiableOn ℝ (deriv f) (interior D)

Tactic:
rw [differentiableOn_congr this]
    exact fun x hx ↦ (hf'' _ hx).differentiableWithinAt

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: x y : ℝ
⊢ |sin x - sin y| ≤ |x - y|

Tactic:
by
  simpa [edist_dist] using! lipschitzWith_sin x y

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
⊢ √(1 - (⟪x, y⟫ / (‖x‖ * ‖y‖)) ^ 2) * (‖x‖ * ‖y‖) = √(⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ * ⟪x, y⟫)

Tactic:
nth_rw 2 [← Real.sqrt_sq (mul_nonneg (norm_nonneg x) (norm_nonneg y))]

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
⊢ √(1 - (⟪x, y⟫ / (‖x‖ * ‖y‖)) ^ 2) * √((‖x‖ * ‖y‖) ^ 2) = √(⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ * ⟪x, y⟫)
Module: Mathlib.Analysis.Meromorphic.IsolatedZeros

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f g : 𝕜 → E
hf : MeromorphicOn f U
hg : MeromorphicOn g U
hU : Preperfect U
h : f =ᶠ[codiscreteWithin U] g
⊢ ∀ᶠ (x : 𝕜) in 𝓝ˢ U, f =ᶠ[𝓝[≠] x] g

Tactic:
rw [eventually_nhdsSet_iff_exists]
  use {x | f =ᶠ[𝓝[≠] x] g}
  simp only [Set.mem_ofPred_eq, imp_self, implies_true, and_true]
  constructor
  · apply isOpen_setOfPred_eventually_nhdsWithin
  · intro x hx
    rw [Set.mem_ofPred]
    exact eventuallyEq_nhdsNE_of_eventuallyEq_codiscreteWithin (hf x hx) (hg x hx) hx (hU x hx) h

Goal state after:
Module: Mathlib.Analysis.Meromorphic.TrailingCoefficient

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
h : ¬MeromorphicAt f x
⊢ meromorphicTrailingCoeffAt f x = 0

Tactic:
by simp [meromorphicTrailingCoeffAt, h]

Goal state after:
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: case neg.h₁g
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
ι : Type u_3
s : Finset ι
f : ι → 𝕜 → 𝕜
h₁f : ∀ i ∈ s, MeromorphicNFAt (f i) x
h₂f : {σ | σ ∈ s ∧ f σ x = 0}.Subsingleton
h₃f : ∀ {τ : ι}, τ ∈ s → τ ∉ {σ ∈ s | f σ x = 0} → AnalyticAt 𝕜 (f τ) x
τ : ι
h₁τ : τ ∈ s
h₂τ : f τ x = 0
this : ∀ {μ : ι}, μ ∈ s.erase τ → f μ x ≠ 0
⊢ AnalyticAt 𝕜 (∏ x ∈ s.erase τ, f x) x

pp: case neg.h₂g
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
ι : Type u_3
s : Finset ι
f : ι → 𝕜 → 𝕜
h₁f : ∀ i ∈ s, MeromorphicNFAt (f i) x
h₂f : {σ | σ ∈ s ∧ f σ x = 0}.Subsingleton
h₃f : ∀ {τ : ι}, τ ∈ s → τ ∉ {σ ∈ s | f σ x = 0} → AnalyticAt 𝕜 (f τ) x
τ : ι
h₁τ : τ ∈ s
h₂τ : f τ x = 0
this : ∀ {μ : ι}, μ ∈ s.erase τ → f μ x ≠ 0
⊢ (∏ x ∈ s.erase τ, f x) x ≠ 0

Tactic:
· apply Finset.analyticAt_prod _ (fun μ hμ ↦ h₃f (Finset.mem_of_mem_erase hμ) (by aesop))

Goal state after:
case neg.h₂g
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
ι : Type u_3
s : Finset ι
f : ι → 𝕜 → 𝕜
h₁f : ∀ i ∈ s, MeromorphicNFAt (f i) x
h₂f : {σ | σ ∈ s ∧ f σ x = 0}.Subsingleton
h₃f : ∀ {τ : ι}, τ ∈ s → τ ∉ {σ ∈ s | f σ x = 0} → AnalyticAt 𝕜 (f τ) x
τ : ι
h₁τ : τ ∈ s
h₂τ : f τ x = 0
this : ∀ {μ : ι}, μ ∈ s.erase τ → f μ x ≠ 0
⊢ (∏ x ∈ s.erase τ, f x) x ≠ 0
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : ¬MeromorphicAt f x
⊢ meromorphicOrderAt f x = ⊤ ↔ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = 0

pp: case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : MeromorphicAt f x
⊢ meromorphicOrderAt f x = ⊤ ↔ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = 0

Tactic:
· simp only [hf, not_false_eq_true, meromorphicOrderAt_of_not_meromorphicAt, WithTop.zero_ne_top,
      false_iff]
    contrapose hf
    exact (MeromorphicAt.const 0 x).congr (EventuallyEq.symm hf)

Goal state after:
case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : MeromorphicAt f x
⊢ meromorphicOrderAt f x = ⊤ ↔ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = 0
Module: Mathlib.Analysis.Meromorphic.TrailingCoefficient

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : 𝕜
f₁ f₂ : 𝕜 → E
hf₁ : MeromorphicAt f₁ x
hf₂ : MeromorphicAt f₂ x
h₂ : meromorphicTrailingCoeffAt f₁ x + meromorphicTrailingCoeffAt f₂ x ≠ 0
n₁ : ℤ
hn₁ : ↑n₁ = meromorphicOrderAt f₁ x
h₁ : ↑n₁ = meromorphicOrderAt f₂ x
h₁f₁ : ¬↑n₁ = ⊤
g₁ : 𝕜 → E
h₁g₁ : AnalyticAt 𝕜 g₁ x
h₂g₁ : g₁ x ≠ 0
h₃g₁ : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f₁ z = (z - x) ^ n₁ • g₁ z
⊢ meromorphicTrailingCoeffAt (f₁ + f₂) x = meromorphicTrailingCoeffAt f₁ x + meromorphicTrailingCoeffAt f₂ x

Tactic:
lift meromorphicOrderAt f₂ x to ℤ using (by lia) with n₂ hn₂

Goal state after:
case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : 𝕜
f₁ f₂ : 𝕜 → E
hf₁ : MeromorphicAt f₁ x
hf₂ : MeromorphicAt f₂ x
h₂ : meromorphicTrailingCoeffAt f₁ x + meromorphicTrailingCoeffAt f₂ x ≠ 0
n₁ : ℤ
hn₁ : ↑n₁ = meromorphicOrderAt f₁ x
h₁f₁ : ¬↑n₁ = ⊤
g₁ : 𝕜 → E
h₁g₁ : AnalyticAt 𝕜 g₁ x
h₂g₁ : g₁ x ≠ 0
h₃g₁ : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f₁ z = (z - x) ^ n₁ • g₁ z
n₂ : ℤ
hn₂ : ↑n₂ = meromorphicOrderAt f₂ x
h₁ : ↑n₁ = ↑n₂
⊢ meromorphicTrailingCoeffAt (f₁ + f₂) x = meromorphicTrailingCoeffAt f₁ x + meromorphicTrailingCoeffAt f₂ x
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
⊢ EqOn (toMeromorphicNFOn f U) f Uᶜ

Tactic:
intro x hx
  simp_all [toMeromorphicNFOn]

Goal state after:
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: case top
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
h : MeromorphicAt f x
h' : ContinuousAt f x
ho : meromorphicOrderAt f x = ⊤
⊢ AnalyticAt 𝕜 f x

Tactic:
have : AnalyticAt 𝕜 (fun _ ↦ (0 : E)) x := analyticAt_const

Goal state after:
case top
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
h : MeromorphicAt f x
h' : ContinuousAt f x
ho : meromorphicOrderAt f x = ⊤
this : AnalyticAt 𝕜 (fun x ↦ 0) x
⊢ AnalyticAt 𝕜 f x
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
h : MeromorphicAt f x
h' : ContinuousAt f x
g : 𝕜 → E
g_an : AnalyticAt 𝕜 g x
gx : g x ≠ 0
this : 0 ≤ meromorphicOrderAt f x
n : ℕ
ho : meromorphicOrderAt f x = ↑↑n
hg : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ ↑n • g z
⊢ ∀ᶠ (z : 𝕜) in 𝓝 x, (z - x) ^ n • g z = f z

Tactic:
apply (ContinuousAt.eventuallyEq_nhds_iff_eventuallyEq_nhdsNE (by fun_prop) h').1

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
h : MeromorphicAt f x
h' : ContinuousAt f x
g : 𝕜 → E
g_an : AnalyticAt 𝕜 g x
gx : g x ≠ 0
this : 0 ≤ meromorphicOrderAt f x
n : ℕ
ho : meromorphicOrderAt f x = ↑↑n
hg : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ ↑n • g z
⊢ (fun x_1 ↦ (x_1 - x) ^ n • g x_1) =ᶠ[𝓝[≠] x] f
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: case mp.inr.inl
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f g : 𝕜 → E
D : locallyFinsuppWithin U ℤ
hg : ∀ (u : ↑U), g ↑u ≠ 0
h✝ : f =ᶠ[codiscreteWithin U] (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ D u) • g
u : 𝕜
hx : (fun x ↦ ↑(D u) * log ‖x - u‖) = 0
y : 𝕜
hy : 1 < ‖y‖
h : y = 0
⊢ D u = 0

Tactic:
simp only [h, norm_zero] at hy
        linarith

Goal state after:
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
f : ℂ → E
h₁f : MeromorphicOn f (closedBall 0 R)
h₂f : ∀ (u : ↑(closedBall 0 R)), meromorphicOrderAt f ↑u ≠ ⊤
hR : 0 < R
φ : ℂ → E := (∏ᶠ (c : ℂ), canonicalFactor R c ^ (divisor f (ball 0 R)) c) • f
hφ : MeromorphicOn φ (closedBall 0 R)
g : ℂ → E := toMeromorphicNFOn φ (closedBall 0 R)
h₃g : divisor g (ball 0 R) = 0
h₂g : MeromorphicNFOn g (closedBall 0 R)
h₄g : ∀ {z : ℂ}, z ∈ closedBall 0 R → meromorphicOrderAt g z ≠ ⊤
η₀ : (-divisor f (ball 0 R)).support.Finite
⊢ f =ᶠ[codiscreteWithin (closedBall 0 R)]
    (∏ i ∈ η₀.toFinset, canonicalFactor R i ^ (-(divisor f (ball 0 R)) i)) •
      (∏ i ∈ η₀.toFinset, canonicalFactor R i ^ (divisor f (ball 0 R)) i) • f

Tactic:
filter_upwards [codiscreteWithin_mono (by tauto) η₀.compl_mem_codiscrete,
            self_mem_codiscreteWithin (closedBall 0 R)] with a ha h₂a

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
f : ℂ → E
h₁f : MeromorphicOn f (closedBall 0 R)
h₂f : ∀ (u : ↑(closedBall 0 R)), meromorphicOrderAt f ↑u ≠ ⊤
hR : 0 < R
φ : ℂ → E := (∏ᶠ (c : ℂ), canonicalFactor R c ^ (divisor f (ball 0 R)) c) • f
hφ : MeromorphicOn φ (closedBall 0 R)
g : ℂ → E := toMeromorphicNFOn φ (closedBall 0 R)
h₃g : divisor g (ball 0 R) = 0
h₂g : MeromorphicNFOn g (closedBall 0 R)
h₄g : ∀ {z : ℂ}, z ∈ closedBall 0 R → meromorphicOrderAt g z ≠ ⊤
η₀ : (-divisor f (ball 0 R)).support.Finite
a : ℂ
ha : a ∈ (-divisor f (ball 0 R)).supportᶜ
h₂a : a ∈ closedBall 0 R
⊢ f a =
    ((∏ i ∈ η₀.toFinset, canonicalFactor R i ^ (-(divisor f (ball 0 R)) i)) •
        (∏ i ∈ η₀.toFinset, canonicalFactor R i ^ (divisor f (ball 0 R)) i) • f)
      a
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: case refine_2
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
U : Set 𝕜
h : MeromorphicOn f U
hx : x ∈ U
this : {x}ᶜ = U \ {x} ∪ Uᶜ
⊢ ∀ᶠ (y : 𝕜) in 𝓝[Uᶜ] x, AnalyticAt 𝕜 f y ∨ y ∉ U

Tactic:
· filter_upwards [self_mem_nhdsWithin] with y hy using Or.inr hy

Goal state after:
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace X
p : E → X
cov : IsCoveringMap p
γ₀ γ₁ : C(↑I, X)
h : γ₀.HomotopicRel γ₁ {0, 1}
e : E
h₀ : γ₀ 0 = p e
h₁ : γ₁ 0 = p e
⊢ (cov.liftPath γ₀ e h₀) 1 = (cov.liftPath γ₁ e h₁) 1

Tactic:
have := (cov.homotopicRel_liftPath h e h₀ h₁).some
  rw [← this.eq_fst 0 (.inr rfl), ← this.eq_snd 0 (.inr rfl)]

Goal state after:
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
z : ℂ
ε : ℝ
hε : ε > 0
⊢ ‖↑(ε + sSupNormIm f 0) ^ (z - 1)‖ * ‖↑(ε + sSupNormIm f 1) ^ (-z)‖ =
    (ε + sSupNormIm f 0) ^ (z.re - 1) * (ε + sSupNormIm f 1) ^ (-z.re)

Tactic:
norm_cpow_eq_rpow_re_of_pos (sSupNormIm_eps_pos f hε _) _

Goal state after:
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
z : ℂ
ε : ℝ
hε : ε > 0
⊢ (ε + sSupNormIm f 0) ^ (z - 1).re * ‖↑(ε + sSupNormIm f 1) ^ (-z)‖ =
    (ε + sSupNormIm f 0) ^ (z.re - 1) * (ε + sSupNormIm f 1) ^ (-z.re)
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
z : ℂ
ε : ℝ
hε : ε > 0
⊢ (ε + sSupNormIm f 0) ^ (z - 1).re * ‖↑(ε + sSupNormIm f 1) ^ (-z)‖ =
    (ε + sSupNormIm f 0) ^ (z.re - 1) * (ε + sSupNormIm f 1) ^ (-z.re)

Tactic:
norm_cpow_eq_rpow_re_of_pos (sSupNormIm_eps_pos f hε _) _

Goal state after:
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
z : ℂ
ε : ℝ
hε : ε > 0
⊢ (ε + sSupNormIm f 0) ^ (z - 1).re * (ε + sSupNormIm f 1) ^ (-z).re =
    (ε + sSupNormIm f 0) ^ (z.re - 1) * (ε + sSupNormIm f 1) ^ (-z.re)
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
z : ℂ
ε : ℝ
hε : ε > 0
⊢ (ε + sSupNormIm f 0) ^ (z - 1).re * (ε + sSupNormIm f 1) ^ (-z).re =
    (ε + sSupNormIm f 0) ^ (z.re - 1) * (ε + sSupNormIm f 1) ^ (-z.re)

Tactic:
norm_cpow_eq_rpow_re_of_pos (sSupNormIm_eps_pos f hε _) _

Goal state after:
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
z : ℂ
ε : ℝ
hε : ε > 0
⊢ (ε + sSupNormIm f 0) ^ (z - 1).re * (ε + sSupNormIm f 1) ^ (-z).re =
    (ε + sSupNormIm f 0) ^ (z.re - 1) * (ε + sSupNormIm f 1) ^ (-z.re)
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: E : Type u_1
X : Type u_2
A : Type u_3
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace A
p : E → X
G : Type u_4
inst✝¹ : Group G
inst✝ : MulAction G E
hp : IsQuotientCoveringMap p G
g : G
x : X
e : ↑(p ⁻¹' {x})
γ✝ γ γ' : FundamentalGroup X x
⊢ op ((hp.fiberEquivGroup e) (⋯.monodromy (Path.Homotopic.Quotient.trans γ' γ) e)) =
    op ((hp.fiberEquivGroup e) (⋯.monodromy γ e)) * op ((hp.fiberEquivGroup e) (⋯.monodromy γ' e))

Tactic:
IsCoveringMap.monodromy_trans_apply,

Goal state after:
E : Type u_1
X : Type u_2
A : Type u_3
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace A
p : E → X
G : Type u_4
inst✝¹ : Group G
inst✝ : MulAction G E
hp : IsQuotientCoveringMap p G
g : G
x : X
e : ↑(p ⁻¹' {x})
γ✝ γ γ' : FundamentalGroup X x
⊢ op ((hp.fiberEquivGroup e) (⋯.monodromy γ (⋯.monodromy γ' e))) =
    op ((hp.fiberEquivGroup e) (⋯.monodromy γ e)) * op ((hp.fiberEquivGroup e) (⋯.monodromy γ' e))
Module: Mathlib.MeasureTheory.Integral.CircleAverage

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℂ → E
⊢ ∫ (θ : ℝ) in 0..2 * π, f (circleMap 0 1 (-θ)) = ∫ (θ : ℝ) in 0..2 * π, f (circleMap 0 1 θ)

Tactic:
by
    rw [intervalIntegral.integral_comp_neg (fun w ↦ f (circleMap 0 1 w))]
    have t₀ : Function.Periodic (fun w ↦ f (circleMap 0 1 w)) (2 * π) :=
      fun x ↦ by simp [periodic_circleMap 0 1 x]
    simpa using (t₀.intervalIntegral_add_eq (-(2 * π)) 0)

Goal state after:
Module: Mathlib.Analysis.Complex.Harmonic.Poisson

Goal state before:
pp: f : ℂ → ℝ
c w : ℂ
R : ℝ
hf : HarmonicOnNhd f (closedBall c R)
hw : w ∈ ball c R
e : ℝ
h₁e : 0 < e
h₂e : ball c (e + R) ⊆ {x | HarmonicAt f x}
F : ℂ → ℂ
h₁F : AnalyticOnNhd ℂ F (ball c (e + R))
h₂F : Set.EqOn (fun z ↦ (F z).re) f (ball c (e + R))
h₃F : DifferentiableOn ℂ F (closure (ball c R))
h₄F :
  Set.EqOn (re ∘ herglotzRieszKernel c w • f) (⇑reCLM ∘ fun z ↦ ((z - c + (w - c)) / (z - c - (w - c))).re • F z)
    (sphere c |R|)
⊢ circleAverage (re ∘ herglotzRieszKernel c w • f) c R = f w

Tactic:
circleAverage_congr_sphere h₄F,

Goal state after:
f : ℂ → ℝ
c w : ℂ
R : ℝ
hf : HarmonicOnNhd f (closedBall c R)
hw : w ∈ ball c R
e : ℝ
h₁e : 0 < e
h₂e : ball c (e + R) ⊆ {x | HarmonicAt f x}
F : ℂ → ℂ
h₁F : AnalyticOnNhd ℂ F (ball c (e + R))
h₂F : Set.EqOn (fun z ↦ (F z).re) f (ball c (e + R))
h₃F : DifferentiableOn ℂ F (closure (ball c R))
h₄F :
  Set.EqOn (re ∘ herglotzRieszKernel c w • f) (⇑reCLM ∘ fun z ↦ ((z - c + (w - c)) / (z - c - (w - c))).re • F z)
    (sphere c |R|)
⊢ circleAverage (⇑reCLM ∘ fun z ↦ ((z - c + (w - c)) / (z - c - (w - c))).re • F z) c R = f w
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
ha : 1 < a
hb : 1 < b
this : ∫ (t : ℝ) in a..b, deriv (fun t ↦ (log t)⁻¹) t = (log b)⁻¹ - (log a)⁻¹
⊢ ∫ (t : ℝ) in a..b, t⁻¹ / log t ^ 2 = (log a)⁻¹ - (log b)⁻¹

Tactic:
simp_rw [deriv_inv_log, neg_div, intervalIntegral.integral_neg] at this

Goal state after:
a b : ℝ
ha : 1 < a
hb : 1 < b
this : -∫ (x : ℝ) in a..b, x⁻¹ / log x ^ 2 = (log b)⁻¹ - (log a)⁻¹
⊢ ∫ (t : ℝ) in a..b, t⁻¹ / log t ^ 2 = (log a)⁻¹ - (log b)⁻¹
Module: Mathlib.Order.Filter.ZeroAndBoundedAtFilter

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝ : SeminormedRing β
l : Filter α
f g : α → β
hf : l.ZeroAtFilter f
hg : l.BoundedAtFilter g
⊢ l.ZeroAtFilter (f * g)

Tactic:
ZeroAtFilter,

Goal state after:
α : Type u_2
β : Type u_3
inst✝ : SeminormedRing β
l : Filter α
f g : α → β
hf : Tendsto f l (𝓝 0)
hg : l.BoundedAtFilter g
⊢ Tendsto (f * g) l (𝓝 0)
Module: Mathlib.Order.Filter.ZeroAndBoundedAtFilter

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝ : SeminormedRing β
l : Filter α
f g : α → β
hf : l.BoundedAtFilter f
hg : l.ZeroAtFilter g
⊢ l.ZeroAtFilter (f * g)

Tactic:
ZeroAtFilter,

Goal state after:
α : Type u_2
β : Type u_3
inst✝ : SeminormedRing β
l : Filter α
f g : α → β
hf : l.BoundedAtFilter f
hg : Tendsto g l (𝓝 0)
⊢ Tendsto (f * g) l (𝓝 0)
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: w ρ : ℂ
R r₀ r : ℝ
hR : 0 < R
hρ : ‖ρ‖ = R
hr₀ : 0 < r₀
hw : ‖w‖ < r₀
hr₀r : r₀ ≤ r
hrR : r ≤ R
θ : ℝ
hdR : 0 < ‖circleMap 0 R θ - ρ‖
⊢ ‖herglotzLogIntegrand w ρ (circleMap 0 r θ)‖ ≤
    (R + ‖w‖) / (r₀ - ‖w‖) * (|log (2 * R)| + |log √(r₀ / R)| + |log ‖circleMap 0 R θ - ρ‖|)

Tactic:
by
  simp only [herglotzLogIntegrand, Pi.smul_apply', Function.comp_apply, smul_eq_mul, norm_mul,
    norm_eq_abs]
  have ⟨hrw, hr⟩ : 0 < r₀ - ‖w‖ ∧ 0 < r := by grind
  have h_norm_sub₁ := const_mul_norm_sub_circleMap_le_norm_sub_circleMap hρ hr₀ hR hr₀r hrR θ
  have h_norm_sub₂ : 0 < ‖circleMap 0 r θ - ρ‖ := lt_of_lt_of_le (by positivity) h_norm_sub₁
  gcongr
  · simp only [herglotzRieszKernel_def, sub_zero]
    calc
     |((circleMap 0 r θ + w) / (circleMap 0 r θ - w)).re|
     _ ≤ ‖circleMap 0 r θ + w‖ / ‖circleMap 0 r θ - w‖ := by grw [Complex.abs_re_le_norm, norm_div]
     _ ≤ (r + ‖w‖) / (r - ‖w‖) := by
        grw [norm_add_le, ← norm_sub_norm_le]
        all_goals simp only [norm_circleMap_zero, abs_of_pos hr]; grind
      _ ≤ (R + ‖w‖) / (r₀ - ‖w‖) := by gcongr
  · apply abs_le.mpr ⟨_, _⟩
    · have h_log_lower_bound :
          log ‖circleMap 0 r θ - ρ‖ ≥ log (sqrt (r₀ / R)) + log ‖circleMap 0 R θ - ρ‖ := by
        rw [← log_mul (by positivity) (by positivity)]
        gcongr
      grind
    · calc log ‖circleMap 0 r θ - ρ‖
      _ ≤ |log (2 * R)| + 0 + 0 := by
        grw [← le_abs_self, norm_sub_le, hρ, two_mul, norm_circleMap_zero, abs_of_pos hr, hrR]
        simp
      _ ≤ |log (2 * R)| + |log √(r₀ / R)| + |log ‖circleMap 0 R θ - ρ‖| := by
        gcongr <;> positivity

Goal state after:
Module: Mathlib.Analysis.Complex.Periodic

Goal state before:
pp: h : ℝ
f : ℂ → ℂ
hh : 0 < h
hf : Periodic f ↑h
h_hol : ∀ᶠ (z : ℂ) in I∞, DifferentiableAt ℂ f z
h_bd : I∞.BoundedAtFilter f
⊢ DifferentiableAt ℂ (cuspFunction h f) 0

Tactic:
by
  obtain ⟨c, t⟩ := (boundedAtFilter_cuspFunction hh h_bd).bound
  replace t := (eventually_differentiableAt_cuspFunction_nhds_ne_zero hh hf h_hol).and t
  simp only [norm_one, Pi.one_apply, mul_one] at t
  obtain ⟨S, hS1, hS2, hS3⟩ := eventually_nhds_iff.mp (eventually_nhdsWithin_iff.mp t)
  have h_diff : DifferentiableOn ℂ (cuspFunction h f) (S \ {0}) :=
    fun x hx ↦ (hS1 x hx.1 hx.2).1.differentiableWithinAt
  have hF_bd : BddAbove (norm ∘ cuspFunction h f '' (S \ {0})) := by
    use c
    simp only [mem_upperBounds, Set.mem_image, Set.mem_sdiff, forall_exists_index, and_imp]
    intro y q hq hq2 hy
    simpa only [← hy, norm_one, mul_one] using! (hS1 q hq hq2).2
  have := differentiableOn_update_limUnder_of_bddAbove (IsOpen.mem_nhds hS2 hS3) h_diff hF_bd
  rw [← cuspFunction_zero_eq_limUnder_nhds_ne, update_eq_self] at this
  exact this.differentiableAt (IsOpen.mem_nhds hS2 hS3)

Goal state after:
Module: Mathlib.Analysis.Complex.Periodic

Goal state before:
pp: case inr
h : ℝ
f g : ℂ → ℂ
hfcts : ContinuousAt (cuspFunction h f) 0
hgcts : ContinuousAt (cuspFunction h g) 0
⊢ update ((f + g) ∘ invQParam h) 0 ((𝓝[≠] 0).limUnder ((f + g) ∘ invQParam h)) 0 =
    (update (f ∘ invQParam h) 0 ((𝓝[≠] 0).limUnder (f ∘ invQParam h)) +
        update (g ∘ invQParam h) 0 ((𝓝[≠] 0).limUnder (g ∘ invQParam h)))
      0

Tactic:
simpa using! (tendsto_nhds_limUnder ⟨_, tendsto_nhds_zero hfcts⟩).add
      (tendsto_nhds_limUnder ⟨_, tendsto_nhds_zero hgcts⟩) |>.limUnder_eq

Goal state after: