mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.Taylor

Goal state before:
pp: case inr
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℝ → F
x x₀ : ℝ
n : ℕ
hf : ContDiffOn ℝ (↑(n + 1)) f [[x₀, x]]
this✝ : x₀ ≠ x
this : UniqueDiffOn ℝ [[x₀, x]]
k : ℕ
⊢ k ≤ n →
    have u := fun t ↦ (x - t) ^ k / ↑k !;
    have v := fun t ↦ iteratedDerivWithin k f [[x₀, x]] t;
    ∫ (t : ℝ) in x₀..x, u t • deriv v t = u x • v x - u x₀ • v x₀ - ∫ (t : ℝ) in x₀..x, deriv u t • v t

Tactic:
hk

Goal state after:
case inr
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℝ → F
x x₀ : ℝ
n : ℕ
hf : ContDiffOn ℝ (↑(n + 1)) f [[x₀, x]]
this✝ : x₀ ≠ x
this : UniqueDiffOn ℝ [[x₀, x]]
k : ℕ
hk : k ≤ n
⊢ have u := fun t ↦ (x - t) ^ k / ↑k !;
  have v := fun t ↦ iteratedDerivWithin k f [[x₀, x]] t;
  ∫ (t : ℝ) in x₀..x, u t • deriv v t = u x • v x - u x₀ • v x₀ - ∫ (t : ℝ) in x₀..x, deriv u t • v t
Module: Mathlib.Analysis.Calculus.Taylor

Goal state before:
pp: case inr.huu'
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℝ → F
x x₀ : ℝ
n : ℕ
hf : ContDiffOn ℝ (↑(n + 1)) f [[x₀, x]]
this✝ : x₀ ≠ x
this : UniqueDiffOn ℝ [[x₀, x]]
k : ℕ
hk : k ≤ n
⊢ ∀ x_1 ∈ Ioo (min x₀ x) (max x₀ x),
    HasDerivAt (fun t ↦ (x - t) ^ k / ↑k !) (deriv (fun t ↦ (x - t) ^ k / ↑k !) x_1) x_1

Tactic:
intro t

Goal state after:
case inr.huu'
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℝ → F
x x₀ : ℝ
n : ℕ
hf : ContDiffOn ℝ (↑(n + 1)) f [[x₀, x]]
this✝ : x₀ ≠ x
this : UniqueDiffOn ℝ [[x₀, x]]
k : ℕ
hk : k ≤ n
t : ℝ
⊢ t ∈ Ioo (min x₀ x) (max x₀ x) → HasDerivAt (fun t ↦ (x - t) ^ k / ↑k !) (deriv (fun t ↦ (x - t) ^ k / ↑k !) t) t
Module: Mathlib.Analysis.Calculus.Taylor

Goal state before:
pp: case inr.hvv'
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℝ → F
x x₀ : ℝ
n : ℕ
hf : ContDiffOn ℝ (↑(n + 1)) f [[x₀, x]]
this✝ : x₀ ≠ x
this : UniqueDiffOn ℝ [[x₀, x]]
k : ℕ
hk : k ≤ n
⊢ ∀ x_1 ∈ Ioo (min x₀ x) (max x₀ x),
    HasDerivAt (fun t ↦ iteratedDerivWithin k f [[x₀, x]] t) (deriv (fun t ↦ iteratedDerivWithin k f [[x₀, x]] t) x_1)
      x_1

Tactic:
intro t

Goal state after:
case inr.hvv'
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℝ → F
x x₀ : ℝ
n : ℕ
hf : ContDiffOn ℝ (↑(n + 1)) f [[x₀, x]]
this✝ : x₀ ≠ x
this : UniqueDiffOn ℝ [[x₀, x]]
k : ℕ
hk : k ≤ n
t : ℝ
⊢ t ∈ Ioo (min x₀ x) (max x₀ x) →
    HasDerivAt (fun t ↦ iteratedDerivWithin k f [[x₀, x]] t) (deriv (fun t ↦ iteratedDerivWithin k f [[x₀, x]] t) t) t
Module: Mathlib.Analysis.Convex.Uniform

Goal state before:
pp: case inr.inr
E : Type u_1
inst✝² : SeminormedAddCommGroup E
inst✝¹ : UniformConvexSpace E
ε : ℝ
inst✝ : NormedSpace ℝ E
hε : 0 < ε
hε' : 0 < ε / 3
δ : ℝ
hδ : 0 < δ
h : ∀ ⦃x : E⦄, ‖x‖ = 1 → ∀ ⦃y : E⦄, ‖y‖ = 1 → ε / 3 ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 - δ
δ' : ℝ := min (1 / 2) (min (ε / 3) (δ / 3))
x : E
hx : ‖x‖ ≤ 1
y : E
hy : ‖y‖ ≤ 1
hxy : ε ≤ ‖x - y‖
hx' : 1 - δ' < ‖x‖
hy' : 1 - δ' < ‖y‖
hδ' : 0 < 1 - δ'
h₁ : ∀ (z : E), 1 - δ' < ‖z‖ → ‖‖z‖⁻¹ • z‖ = 1
⊢ ‖x + y‖ ≤ 2 - δ'

Tactic:
have h₂ : ∀ z : E, ‖z‖ ≤ 1 → 1 - δ' ≤ ‖z‖ → ‖‖z‖⁻¹ • z - z‖ ≤ δ' := by
    rintro z hz hδz
    nth_rw 3 [← one_smul ℝ z]
    rwa [← sub_smul,
      norm_smul_of_nonneg (sub_nonneg_of_le <| (one_le_inv₀ (hδ'.trans_le hδz)).2 hz),
      sub_mul, inv_mul_cancel₀ (hδ'.trans_le hδz).ne', one_mul, sub_le_comm]

Goal state after:
case inr.inr
E : Type u_1
inst✝² : SeminormedAddCommGroup E
inst✝¹ : UniformConvexSpace E
ε : ℝ
inst✝ : NormedSpace ℝ E
hε : 0 < ε
hε' : 0 < ε / 3
δ : ℝ
hδ : 0 < δ
h : ∀ ⦃x : E⦄, ‖x‖ = 1 → ∀ ⦃y : E⦄, ‖y‖ = 1 → ε / 3 ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 - δ
δ' : ℝ := min (1 / 2) (min (ε / 3) (δ / 3))
x : E
hx : ‖x‖ ≤ 1
y : E
hy : ‖y‖ ≤ 1
hxy : ε ≤ ‖x - y‖
hx' : 1 - δ' < ‖x‖
hy' : 1 - δ' < ‖y‖
hδ' : 0 < 1 - δ'
h₁ : ∀ (z : E), 1 - δ' < ‖z‖ → ‖‖z‖⁻¹ • z‖ = 1
h₂ : ∀ (z : E), ‖z‖ ≤ 1 → 1 - δ' ≤ ‖z‖ → ‖‖z‖⁻¹ • z - z‖ ≤ δ'
⊢ ‖x + y‖ ≤ 2 - δ'
Module: Mathlib.Analysis.Convex.SpecificFunctions.Deriv

Goal state before:
pp: case inl
x : ℝ
hx : 0 < x
⊢ deriv (fun x ↦ √x * log x) x = (2 + log x) / (2 * √x)

pp: case inr
x : ℝ
hx : x ≤ 0
⊢ deriv (fun x ↦ √x * log x) x = (2 + log x) / (2 * √x)

Tactic:
· exact (hasDerivAt_sqrt_mul_log hx.ne').deriv

Goal state after:
case inr
x : ℝ
hx : x ≤ 0
⊢ deriv (fun x ↦ √x * log x) x = (2 + log x) / (2 * √x)
Module: Mathlib.Analysis.Complex.AbsMax

Goal state before:
pp: F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → F
z w : ℂ
r : ℝ := dist w z
hd : DiffContOnCl ℂ f (ball z r)
hz : IsMaxOn (norm ∘ f) (closedBall z r) z
hw : w ∈ closedBall z r
hw_lt : ‖f w‖ < ‖f z‖
hr : 0 < r
this : ‖∮ (ζ : ℂ) in C(z, r), (ζ - z)⁻¹ • f ζ‖ < 2 * π * r * (‖f z‖ / r)
⊢ ‖∮ (ζ : ℂ) in C(z, r), (ζ - z)⁻¹ • f ζ‖ < 2 * π * ‖f z‖

Tactic:
by
    rwa [mul_assoc, mul_div_cancel₀ _ hr.ne'] at this

Goal state after:
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
A : DifferentiableOn ℝ f (Ioo x y)
a : ℝ
ha : deriv f a = (f y - f x) / (y - x)
hxa : x < a
hay : a < y
⊢ ∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x)

Tactic:
rcases nonempty_Ioo.2 hxa with ⟨b, ⟨hxb, hba⟩⟩

Goal state after:
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
A : DifferentiableOn ℝ f (Ioo x y)
a : ℝ
ha : deriv f a = (f y - f x) / (y - x)
hxa : x < a
hay : a < y
b : ℝ
hxb : x < b
hba : b < a
⊢ ∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: x : ℝ
hx : x < 0
⊢ x < sin x

Tactic:
by simpa using sin_lt <| neg_pos.2 hx

Goal state after:
Module: Mathlib.Analysis.Complex.UnitDisc.Basic

Goal state before:
pp: z : 𝔻
n : ℕ+
⊢ ↑(z ^ n) = ↑0 ↔ z = 0

Tactic:
coe_pow

Goal state after:
z : 𝔻
n : ℕ+
⊢ ↑z ^ ↑n = ↑0 ↔ z = 0
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: case right.left
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
g : 𝕜 → 𝕜
x : 𝕜
h₁g : AnalyticAt 𝕜 g x
h₂g : g x ≠ 0
n : ℤ
g_f : 𝕜 → E
h₁g_f : AnalyticAt 𝕜 g_f x
h₂g_f : g_f x ≠ 0
h₃g_f : f =ᶠ[𝓝 x] (fun x_1 ↦ x_1 - x) ^ n • g_f
⊢ (g • g_f) x ≠ 0

pp: case right.right
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
g : 𝕜 → 𝕜
x : 𝕜
h₁g : AnalyticAt 𝕜 g x
h₂g : g x ≠ 0
n : ℤ
g_f : 𝕜 → E
h₁g_f : AnalyticAt 𝕜 g_f x
h₂g_f : g_f x ≠ 0
h₃g_f : f =ᶠ[𝓝 x] (fun x_1 ↦ x_1 - x) ^ n • g_f
⊢ g • f =ᶠ[𝓝 x] (fun x_1 ↦ x_1 - x) ^ n • g • g_f

Tactic:
· simp [smul_ne_zero h₂g h₂g_f]

Goal state after:
case right.right
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
g : 𝕜 → 𝕜
x : 𝕜
h₁g : AnalyticAt 𝕜 g x
h₂g : g x ≠ 0
n : ℤ
g_f : 𝕜 → E
h₁g_f : AnalyticAt 𝕜 g_f x
h₂g_f : g_f x ≠ 0
h₃g_f : f =ᶠ[𝓝 x] (fun x_1 ↦ x_1 - x) ^ n • g_f
⊢ g • f =ᶠ[𝓝 x] (fun x_1 ↦ x_1 - x) ^ n • g • g_f
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
hf : MeromorphicAt f x
m : ℕ
hm : ↑m = analyticOrderAt (fun z ↦ (z - x) ^ Exists.choose hf • f z) x
h : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = 0
z : 𝕜
hfz : f z = 0
⊢ (z - x) ^ Exists.choose hf • f z = 0

Tactic:
hfz,

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
hf : MeromorphicAt f x
m : ℕ
hm : ↑m = analyticOrderAt (fun z ↦ (z - x) ^ Exists.choose hf • f z) x
h : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = 0
z : 𝕜
hfz : f z = 0
⊢ (z - x) ^ Exists.choose hf • 0 = 0
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
x : 𝕜
hf : MeromorphicNFAt f x
z : 𝕜
⊢ toMeromorphicNFAt f x z = f z

Tactic:
by_cases hz : z = x

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 : MeromorphicNFAt f x
z : 𝕜
hz : z = x
⊢ toMeromorphicNFAt f x z = f z

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 : MeromorphicNFAt f x
z : 𝕜
hz : ¬z = x
⊢ toMeromorphicNFAt f x z = f z
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
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
n : ℤ
hf : MeromorphicAt f x
h : ∀ᶠ (z : 𝕜) in 𝓝 x, (z - x) ^ Exists.choose hf • f z = 0
x✝ : ∃ g, AnalyticAt 𝕜 g x ∧ g x ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ n • g z
g : 𝕜 → E
hg_an : AnalyticAt 𝕜 g x
hg_ne : g x ≠ 0
hg_eq : ∀ᶠ (x_1 : 𝕜) in 𝓝 x, x_1 ∈ {x}ᶜ → f x_1 = (x_1 - x) ^ n • g x_1
⊢ ∀ᶠ (x_1 : 𝕜) in 𝓝 x, x_1 ∈ {x}ᶜ → g x_1 = 0

Tactic:
filter_upwards [h, hg_eq] with z hfz hfz_eq hz

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
n : ℤ
hf : MeromorphicAt f x
h : ∀ᶠ (z : 𝕜) in 𝓝 x, (z - x) ^ Exists.choose hf • f z = 0
x✝ : ∃ g, AnalyticAt 𝕜 g x ∧ g x ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ n • g z
g : 𝕜 → E
hg_an : AnalyticAt 𝕜 g x
hg_ne : g x ≠ 0
hg_eq : ∀ᶠ (x_1 : 𝕜) in 𝓝 x, x_1 ∈ {x}ᶜ → f x_1 = (x_1 - x) ^ n • g x_1
z : 𝕜
hfz : (z - x) ^ Exists.choose hf • f z = 0
hfz_eq : z ∈ {x}ᶜ → f z = (z - x) ^ n • g z
hz : z ∈ {x}ᶜ
⊢ g z = 0
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: case mpr
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
h : MeromorphicNFOn f U
⊢ toMeromorphicNFOn f U = f

Tactic:
ext x
    by_cases hx : x ∈ U
    · simp only [toMeromorphicNFOn, h.meromorphicOn, ↓reduceDIte, hx]
      rw [toMeromorphicNFAt_eq_self.2 (h hx)]
    · simp [toMeromorphicNFOn, h.meromorphicOn, hx]

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

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
n : ℤ
f : 𝕜 → 𝕜
h₁ : MeromorphicAt f x
h₂ : meromorphicOrderAt f x = ⊤
h₃ : n = 0
⊢ meromorphicTrailingCoeffAt (f ^ n) x = meromorphicTrailingCoeffAt f x ^ n

Tactic:
simp only [h₃, zpow_zero]
      apply analyticAt_const.meromorphicTrailingCoeffAt_of_ne_zero (ne_zero_of_eq_one rfl)

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

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h : HasFiniteSupport d
this : (Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d u) ⊆ ↑(Finite.toFinset h)
y : 𝕜
hy : y ∈ Finite.toFinset h
hxy : ¬x = y
⊢ meromorphicTrailingCoeffAt (fun x ↦ x - y) x ^ d y = (x - y) ^ update d x 0 y

Tactic:
grind [meromorphicTrailingCoeffAt_id_sub_const]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h₁ : HasFiniteSupport d
h₂ : x ∉ support d
⊢ meromorphicTrailingCoeffAt (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ d u) x = ∏ᶠ (u : 𝕜), (x - u) ^ d u

Tactic:
rw [meromorphicTrailingCoeffAt_factorizedRational h₁,
    finprod_eq_prod_of_mulSupport_subset _ (mulSupport_update h₁)]
  have : (fun u ↦ (x - u) ^ d u).mulSupport ⊆ h₁.toFinset := by
    intro u
    contrapose
    simp_all
  rw [finprod_eq_prod_of_mulSupport_subset _ this, Finset.prod_congr rfl]
  intro y hy
  congr
  apply Function.update_of_ne
  by_contra hCon
  simp_all

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

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h✝ : HasFiniteSupport d
this✝ : ∀ y ∈ Finite.toFinset h✝, ‖(x - y) ^ update d x 0 y‖ ≠ 0
this : (support fun u ↦ ↑(d u) * log ‖x - u‖) ⊆ ↑(Finite.toFinset h✝)
y : 𝕜
hy : y ∈ Finite.toFinset h✝
h : ¬x = y
⊢ ↑(update d x 0 y) * log ‖x - y‖ = ↑(d y) * log ‖x - y‖

Tactic:
Function.update_of_ne (by tauto)

Goal state after:
case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h✝ : HasFiniteSupport d
this✝ : ∀ y ∈ Finite.toFinset h✝, ‖(x - y) ^ update d x 0 y‖ ≠ 0
this : (support fun u ↦ ↑(d u) * log ‖x - u‖) ⊆ ↑(Finite.toFinset h✝)
y : 𝕜
hy : y ∈ Finite.toFinset h✝
h : ¬x = y
⊢ ↑(d y) * log ‖x - y‖ = ↑(d y) * log ‖x - y‖
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: case refine_4
E : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace X
p : E → X
homeo : IsLocalHomeomorph p
sep : IsSeparatedMap p
γ₀ γ₁ : C(↑I, X)
γ : γ₀.HomotopyRel γ₁ {0, 1}
Γ : ↑I → C(↑I, E)
Γ_lifts : ∀ (t s : ↑I), p ((Γ t) s) = γ (t, s)
Γ_0 : ∀ (t : ↑I), (Γ t) 0 = (Γ 0) 0
t : ↑I
this : Continuous[_, inst✝¹] fun st ↦ (Γ st.2) st.1
⊢ ∀ (a a' : ↑I), p (((fun st ↦ (Γ st.2) st.1) ∘ fun y ↦ (1, y)) a) = p (((fun st ↦ (Γ st.2) st.1) ∘ fun y ↦ (1, y)) a')

Tactic:
intro t

Goal state after:
case refine_4
E : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace X
p : E → X
homeo : IsLocalHomeomorph p
sep : IsSeparatedMap p
γ₀ γ₁ : C(↑I, X)
γ : γ₀.HomotopyRel γ₁ {0, 1}
Γ : ↑I → C(↑I, E)
Γ_lifts : ∀ (t s : ↑I), p ((Γ t) s) = γ (t, s)
Γ_0 : ∀ (t : ↑I), (Γ t) 0 = (Γ 0) 0
t✝ : ↑I
this : Continuous[_, inst✝¹] fun st ↦ (Γ st.2) st.1
t : ↑I
⊢ ∀ (a' : ↑I), p (((fun st ↦ (Γ st.2) st.1) ∘ fun y ↦ (1, y)) t) = p (((fun st ↦ (Γ st.2) st.1) ∘ fun y ↦ (1, y)) a')
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: case pos
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 : R ≤ 0
⊢ ∃ g, CanonicalDecomp f g R

Tactic:
use fun _ ↦ f 0

Goal state after:
case h
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 : R ≤ 0
⊢ CanonicalDecomp f (fun x ↦ f 0) R
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: case hf
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f₁ f₂ : 𝕜 → E
x : 𝕜
hf₁ : MeromorphicAt f₁ x
hf₂ : MeromorphicAt f₂ x
n₁ : ℤ
hn₁ : ↑n₁ = meromorphicOrderAt f₁ x
n₂ : ℤ
hn₂ : ↑n₂ = meromorphicOrderAt f₂ x
meromorphicOrderAt_add : MeromorphicAt f₁ x → MeromorphicAt f₂ x → min ↑n₁ ↑n₂ ≤ meromorphicOrderAt (f₁ + f₂) x
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
g₂ : 𝕜 → E
h₁g₂ : AnalyticAt 𝕜 g₂ x
h₂g₂ : g₂ x ≠ 0
h₃g₂ : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f₂ z = (z - x) ^ n₂ • g₂ z
n : ℤ := ⋯
g : 𝕜 → E := ⋯
⊢ AnalyticAt 𝕜 ((fun z ↦ (z - x) ^ (n₁ - n)) • g₁) x

Tactic:
apply (AnalyticAt.zpow_nonneg (by fun_prop) (sub_nonneg.2 (min_le_left n₁ n₂))).smul h₁g₁

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

Goal state before:
pp: case h.left
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
z : ↑U
hz : z ∈ {u | meromorphicOrderAt f ↑u = ⊤}ᶜ
h : ∀ᶠ (z : 𝕜) in 𝓝[≠] ↑z, f z ≠ 0
t' : Set 𝕜
h₁t' : ∀ y ∈ t', y ∈ {↑z}ᶜ → f y ≠ 0
h₂t' : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t'
h₃t' : ↑z ∈ t'
⊢ Subtype.val ⁻¹' t' ⊆ {u | meromorphicOrderAt f ↑u = ⊤}ᶜ

Tactic:
intro w hw
        push _ ∈ _
        by_cases h₁w : w = z
        · rwa [h₁w]
        · rw [meromorphicOrderAt_eq_top_iff, not_eventually]
          apply Filter.Eventually.frequently
          rw [eventually_nhdsWithin_iff, eventually_nhds_iff]
          use t' \ {z.1}, fun y h₁y h₂y ↦ h₁t' y h₁y.1 h₁y.2, h₂t'.sdiff isClosed_singleton, hw,
            mem_singleton_iff.not.2 (Subtype.coe_ne_coe.mpr h₁w)

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

Goal state before:
pp: case inl
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 : R < 0
⊢ ∃ h, ECanonicalDecomp f h R

Tactic:
use fun _ ↦ f 0

Goal state after:
case h
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 : R < 0
⊢ ECanonicalDecomp f (fun x ↦ f 0) R
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: E : Type u_1
X : Type u_2
A : Type u_3
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
p : E → X
cov : IsCoveringMap p
H : C(↑I × A, X)
f : C(A, E)
H_0 : ∀ (a : A), H (0, a) = p (f a)
⊢ (p ∘ fun ta ↦
      (cov.liftPath (H.comp ((ContinuousMap.id ↑I).prodMk (ContinuousMap.const (↑I) ta.2))) (f ta.2) ⋯) ta.1) =
    ⇑H

Tactic:
ext ⟨t, a⟩

Goal state after:
E : Type u_1
X : Type u_2
A : Type u_3
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
p : E → X
cov : IsCoveringMap p
H : C(↑I × A, X)
f : C(A, E)
H_0 : ∀ (a : A), H (0, a) = p (f a)
t : ↑I
a : A
⊢ (p ∘ fun ta ↦ (cov.liftPath (H.comp ((ContinuousMap.id ↑I).prodMk (ContinuousMap.const (↑I) ta.2))) (f ta.2) ⋯) ta.1)
      (t, a) =
    H (t, a)
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: E : Type u_1
X : Type u_2
A : Type u_3
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
p : E → X
cov : IsCoveringMap p
H : C(↑I × A, X)
f : C(A, E)
H_0 : ∀ (a : A), H (0, a) = p (f a)
H' : ↑I × A → E
x✝¹ : (∀ (a : A), Continuous[_, inst✝²] fun x ↦ H' (x, a)) ∧ p ∘ H' = ⇑H ∧ ∀ (a : A), H' (0, a) = f a
H'_cont : ∀ (a : A), Continuous[_, inst✝²] fun x ↦ H' (x, a)
H'_lifts : p ∘ H' = ⇑H
H'_0 : ∀ (a : A), H' (0, a) = f a
x✝ : ↑I × A
t : ↑I
a : A
⊢ (p ∘ fun x ↦ H' (x, a)) = ⇑(H.comp ((ContinuousMap.id ↑I).prodMk (ContinuousMap.const ↑I (t, a).2)))

Tactic:
ext ⟨t, a⟩

Goal state after:
E : Type u_1
X : Type u_2
A : Type u_3
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
p : E → X
cov : IsCoveringMap p
H : C(↑I × A, X)
f : C(A, E)
H_0 : ∀ (a : A), H (0, a) = p (f a)
H' : ↑I × A → E
x✝¹ : (∀ (a : A), Continuous[_, inst✝²] fun x ↦ H' (x, a)) ∧ p ∘ H' = ⇑H ∧ ∀ (a : A), H' (0, a) = f a
H'_cont : ∀ (a : A), Continuous[_, inst✝²] fun x ↦ H' (x, a)
H'_lifts : p ∘ H' = ⇑H
H'_0 : ∀ (a : A), H' (0, a) = f a
x✝ : ↑I × A
t✝ : ↑I
a✝ : A
t : ℝ
a : t ∈ I
⊢ (p ∘ fun x ↦ H' (x, a✝)) ⟨t, a⟩ = (H.comp ((ContinuousMap.id ↑I).prodMk (ContinuousMap.const ↑I (t✝, a✝).2))) ⟨t, a⟩
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: E : Type u_1
X : Type u_2
A : Type u_3
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace A
p : E → X
cov : IsCoveringMap p
H : C(↑I × A, X)
f : C(A, E)
H_0 : ∀ (a : A), H (0, a) = p (f a)
f₀ f₁ : C(A, X)
S : Set A
F : f₀.HomotopyRel f₁ S
inst✝ : PreconnectedSpace A
f₀' f₁' : C(A, E)
he : ∃ a ∈ S, f₀' a = f₁' a
h₀ : p ∘ ⇑f₀' = ⇑f₀
h₁ : p ∘ ⇑f₁' = ⇑f₁
F_0 : ∀ (a : A), F (0, a) = p (f₀' a)
t : ↑I
a : A
ha : a ∈ S
⊢ ∀ (a_1 a' : ↑I),
    p
        ((cov.liftPath ((↑F).comp ((ContinuousMap.id ↑I).prodMk (ContinuousMap.const ↑I (t, a).2))) (f₀' (t, a).2) ⋯)
          a_1) =
      p
        ((cov.liftPath ((↑F).comp ((ContinuousMap.id ↑I).prodMk (ContinuousMap.const ↑I (t, a).2))) (f₀' (t, a).2) ⋯)
          a')

Tactic:
intro t

Goal state after:
E : Type u_1
X : Type u_2
A : Type u_3
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace A
p : E → X
cov : IsCoveringMap p
H : C(↑I × A, X)
f : C(A, E)
H_0 : ∀ (a : A), H (0, a) = p (f a)
f₀ f₁ : C(A, X)
S : Set A
F : f₀.HomotopyRel f₁ S
inst✝ : PreconnectedSpace A
f₀' f₁' : C(A, E)
he : ∃ a ∈ S, f₀' a = f₁' a
h₀ : p ∘ ⇑f₀' = ⇑f₀
h₁ : p ∘ ⇑f₁' = ⇑f₁
F_0 : ∀ (a : A), F (0, a) = p (f₀' a)
t✝ : ↑I
a : A
ha : a ∈ S
t : ↑I
⊢ ∀ (a' : ↑I),
    p
        ((cov.liftPath ((↑F).comp ((ContinuousMap.id ↑I).prodMk (ContinuousMap.const ↑I (t✝, a).2))) (f₀' (t✝, a).2) ⋯)
          t) =
      p
        ((cov.liftPath ((↑F).comp ((ContinuousMap.id ↑I).prodMk (ContinuousMap.const ↑I (t✝, a).2))) (f₀' (t✝, a).2) ⋯)
          a')
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
w : ℂ
f h✝ : ℂ → E
D : ECanonicalDecomp f h✝ R
h₁w : w ∈ closedBall 0 R
h₂w : meromorphicOrderAt f w = 0
hR : 0 < R
x : ℂ
h : ¬x = w
⊢ meromorphicTrailingCoeffAt (fun x_1 ↦ x_1 - x) w ^ (-divisor f (sphere 0 R)) x = (w - x) ^ (-divisor f (sphere 0 R)) x

Tactic:
grind [meromorphicTrailingCoeffAt_id_sub_const]

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
w : ℂ
f h : ℂ → E
D : ECanonicalDecomp f h R
h₁w : w ∈ closedBall 0 R
h₂w : meromorphicOrderAt f w = 0
hR : 0 < R
B₀R : Set ℂ := ball 0 R
S₀R : Set ℂ := sphere 0 R
t₁ : Finset ℂ
ht₁ : ↑t₁ = (divisor f S₀R).support
t₂ : Finset ℂ
ht₂ : ↑t₂ = (divisor f B₀R).support
⊢ log
      ‖((∏ i ∈ t₂, canonicalFactor R i w ^ (divisor f B₀R) i) * ∏ᶠ (i : ℂ), (w - i) ^ (-divisor f S₀R) i) •
          meromorphicTrailingCoeffAt f w‖ =
    log
      ‖((∏ i ∈ t₂, canonicalFactor R i w ^ (divisor f B₀R) i) * ∏ i ∈ t₁, (w - i) ^ (-divisor f S₀R) i) •
          meromorphicTrailingCoeffAt f w‖

pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
w : ℂ
f h : ℂ → E
D : ECanonicalDecomp f h R
h₁w : w ∈ closedBall 0 R
h₂w : meromorphicOrderAt f w = 0
hR : 0 < R
B₀R : Set ℂ := ball 0 R
S₀R : Set ℂ := sphere 0 R
t₁ : Finset ℂ
ht₁ : ↑t₁ = (divisor f S₀R).support
t₂ : Finset ℂ
ht₂ : ↑t₂ = (divisor f B₀R).support
⊢ (mulSupport fun i ↦ canonicalFactor R i w ^ (divisor f B₀R) i) ⊆ ↑t₂

Tactic:
finprod_eq_prod_of_mulSupport_subset (s := t₁) _ _

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
w : ℂ
f h : ℂ → E
D : ECanonicalDecomp f h R
h₁w : w ∈ closedBall 0 R
h₂w : meromorphicOrderAt f w = 0
hR : 0 < R
B₀R : Set ℂ := ball 0 R
S₀R : Set ℂ := sphere 0 R
t₁ : Finset ℂ
ht₁ : ↑t₁ = (divisor f S₀R).support
t₂ : Finset ℂ
ht₂ : ↑t₂ = (divisor f B₀R).support
⊢ log
      ‖((∏ i ∈ t₂, canonicalFactor R i w ^ (divisor f B₀R) i) * ∏ i ∈ t₁, (w - i) ^ (-divisor f S₀R) i) •
          meromorphicTrailingCoeffAt f w‖ =
    log
      ‖((∏ i ∈ t₂, canonicalFactor R i w ^ (divisor f B₀R) i) * ∏ i ∈ t₁, (w - i) ^ (-divisor f S₀R) i) •
          meromorphicTrailingCoeffAt f w‖

E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
w : ℂ
f h : ℂ → E
D : ECanonicalDecomp f h R
h₁w : w ∈ closedBall 0 R
h₂w : meromorphicOrderAt f w = 0
hR : 0 < R
B₀R : Set ℂ := ball 0 R
S₀R : Set ℂ := sphere 0 R
t₁ : Finset ℂ
ht₁ : ↑t₁ = (divisor f S₀R).support
t₂ : Finset ℂ
ht₂ : ↑t₂ = (divisor f B₀R).support
⊢ (mulSupport fun i ↦ (w - i) ^ (-divisor f S₀R) i) ⊆ ↑t₁

E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
w : ℂ
f h : ℂ → E
D : ECanonicalDecomp f h R
h₁w : w ∈ closedBall 0 R
h₂w : meromorphicOrderAt f w = 0
hR : 0 < R
B₀R : Set ℂ := ball 0 R
S₀R : Set ℂ := sphere 0 R
t₁ : Finset ℂ
ht₁ : ↑t₁ = (divisor f S₀R).support
t₂ : Finset ℂ
ht₂ : ↑t₂ = (divisor f B₀R).support
⊢ (mulSupport fun i ↦ canonicalFactor R i w ^ (divisor f B₀R) i) ⊆ ↑t₂
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: l u : ℝ
hul : l < u
z : ℂ
⊢ (z / (↑u - ↑l)).re - (↑l / (↑u - ↑l)).re = (z.re - l) / (u - l)

Tactic:
norm_cast
  rw [Complex.div_re, Complex.normSq_ofReal, Complex.ofReal_re, Complex.ofReal_im, mul_div_assoc,
    div_mul_eq_div_div_swap, div_self (by norm_cast; linarith),
    ← div_eq_mul_one_div]
  simp only [mul_zero, zero_div, add_zero]
  rw [← div_sub_div_same]

Goal state after: