mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
u₀ : 𝕜
hd : HasFiniteSupport d
h₁d : ¬d u₀ = 0
this✝ : (Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d u) ⊆ ↑(Finite.toFinset hd)
this : u₀ ∈ Finite.toFinset hd
⊢ support (update d u₀ 0) ⊆ ↑((Finite.toFinset hd).erase u₀)

Tactic:
]

Goal state after:
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
u₀ : 𝕜
hd : HasFiniteSupport d
h₁d : ¬d u₀ = 0
this✝ : (Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d u) ⊆ ↑(Finite.toFinset hd)
this : u₀ ∈ Finite.toFinset hd
⊢ support (update d u₀ 0) ⊆ ↑((Finite.toFinset hd).erase u₀)
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
u₀ : 𝕜
hd : HasFiniteSupport d
h₁d : ¬d u₀ = 0
this✝ : (Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d u) ⊆ ↑(Finite.toFinset hd)
this : u₀ ∈ Finite.toFinset hd
⊢ (Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ update d u₀ 0 u) ⊆ ↑((Finite.toFinset hd).erase u₀)

Tactic:
rw [mulSupport]

Goal state after:
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
u₀ : 𝕜
hd : HasFiniteSupport d
h₁d : ¬d u₀ = 0
this✝ : (Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d u) ⊆ ↑(Finite.toFinset hd)
this : u₀ ∈ Finite.toFinset hd
⊢ support (update d u₀ 0) ⊆ ↑((Finite.toFinset hd).erase u₀)
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
g : 𝕜 → 𝕜
hf : MeromorphicAt f (g x)
hg : AnalyticAt 𝕜 g x
hg_nc : ¬EventuallyConst g (𝓝 x)
h : ∀ᶠ (z : 𝕜) in 𝓝[≠] g x, f z = 0
⊢ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, (f ∘ g) z = 0

Tactic:
]

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
g : 𝕜 → 𝕜
hf : MeromorphicAt f (g x)
hg : AnalyticAt 𝕜 g x
hg_nc : ¬EventuallyConst g (𝓝 x)
h : ∀ᶠ (z : 𝕜) in 𝓝[≠] g x, f z = 0
⊢ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, (f ∘ g) z = 0
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: case pos.coe
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
hf₁₂ : f₁ =ᶠ[𝓝[≠] x] f₂
hf₁ : MeromorphicAt f₁ x
n : ℤ
h₁f₁ : meromorphicOrderAt f₁ x = ↑n
g : 𝕜 → E
h₁g : AnalyticAt 𝕜 g x
h₂g : g x ≠ 0
h₃g : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f₁ z = (z - x) ^ n • g z
⊢ ∃ g, AnalyticAt 𝕜 g x ∧ g x ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f₂ z = (z - x) ^ n • g z

Tactic:
]

Goal state after:
case pos.coe
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
hf₁₂ : f₁ =ᶠ[𝓝[≠] x] f₂
hf₁ : MeromorphicAt f₁ x
n : ℤ
h₁f₁ : meromorphicOrderAt f₁ x = ↑n
g : 𝕜 → E
h₁g : AnalyticAt 𝕜 g x
h₂g : g x ≠ 0
h₃g : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f₁ z = (z - x) ^ n • g z
⊢ ∃ g, AnalyticAt 𝕜 g x ∧ g x ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f₂ z = (z - x) ^ n • g z
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
hd : (support d).Finite
z : 𝕜
hz : z ∈ univ
⊢ MeromorphicNFAt ((fun x ↦ x - z) ^ d z * ∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ update d z 0 u) z

Tactic:
]

Goal state after:
case pos
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
hd : (support d).Finite
z : 𝕜
hz : z ∈ univ
⊢ MeromorphicNFAt ((fun x ↦ x - z) ^ d z * ∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ update d z 0 u) z
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
hf : AnalyticAt 𝕜 f x
hn : analyticOrderAt f x = ⊤
⊢ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = 0

Tactic:
]

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
hf : AnalyticAt 𝕜 f x
hn : analyticOrderAt 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
f : 𝕜 → E
x : 𝕜
g : 𝕜 → 𝕜
hf : MeromorphicAt f (g x)
hg : AnalyticAt 𝕜 g x
hg_nc : ¬EventuallyConst g (𝓝 x)
h : ¬meromorphicOrderAt f (g x) = ⊤
r : ℤ := (meromorphicOrderAt f (g x)).untop₀
F : 𝕜 → E
h₁F : AnalyticAt 𝕜 F (g x)
h₂F : F (g x) ≠ 0
h₃F : f =ᶠ[𝓝[≠] g x] fun z ↦ (z - g x) ^ (meromorphicOrderAt f (g x)).untop₀ • F z
h₁ : meromorphicTrailingCoeffAt (f ∘ g) x = meromorphicTrailingCoeffAt ((fun x_1 ↦ g x_1 - g x) ^ r • F ∘ g) x
⊢ meromorphicTrailingCoeffAt ((fun x_1 ↦ g x_1 - g x) ^ r) x • (F ∘ g) x =
    meromorphicTrailingCoeffAt (fun x_1 ↦ g x_1 - g x) x ^ r • F (g x)

Tactic:
]

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
f : 𝕜 → E
x : 𝕜
g : 𝕜 → 𝕜
hf : MeromorphicAt f (g x)
hg : AnalyticAt 𝕜 g x
hg_nc : ¬EventuallyConst g (𝓝 x)
h : ¬meromorphicOrderAt f (g x) = ⊤
r : ℤ := (meromorphicOrderAt f (g x)).untop₀
F : 𝕜 → E
h₁F : AnalyticAt 𝕜 F (g x)
h₂F : F (g x) ≠ 0
h₃F : f =ᶠ[𝓝[≠] g x] fun z ↦ (z - g x) ^ (meromorphicOrderAt f (g x)).untop₀ • F z
h₁ : meromorphicTrailingCoeffAt (f ∘ g) x = meromorphicTrailingCoeffAt ((fun x_1 ↦ g x_1 - g x) ^ r • F ∘ g) x
⊢ meromorphicTrailingCoeffAt ((fun x_1 ↦ g x_1 - g x) ^ r) x • (F ∘ g) x =
    meromorphicTrailingCoeffAt (fun x_1 ↦ g x_1 - g x) x ^ r • F (g x)
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
hd : ¬(Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d u).Finite
⊢ MeromorphicNFOn (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ d u) univ

Tactic:
]

Goal state after:
case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
hd : ¬(Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d u).Finite
⊢ MeromorphicNFOn (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ d u) univ
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
hd : ¬(Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d u).Finite
⊢ MeromorphicNFOn 1 univ

Tactic:
]

Goal state after:
case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
hd : ¬(Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d u).Finite
⊢ MeromorphicNFOn 1 univ
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
z : 𝕜
d : 𝕜 → ℤ
h₁d : HasFiniteSupport d
⊢ ∃ g,
    AnalyticAt 𝕜 g z ∧
      g z ≠ 0 ∧ ∀ᶠ (z_1 : 𝕜) in nhdsWithin z {z}ᶜ, (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ d u) z_1 = (z_1 - z) ^ d z • g z_1

Tactic:
]

Goal state after:
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
z : 𝕜
d : 𝕜 → ℤ
h₁d : HasFiniteSupport d
⊢ ∃ g,
    AnalyticAt 𝕜 g z ∧
      g z ≠ 0 ∧ ∀ᶠ (z_1 : 𝕜) in nhdsWithin z {z}ᶜ, (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ d u) z_1 = (z_1 - z) ^ d z • g z_1
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
z : 𝕜
d : 𝕜 → ℤ
hd : ¬(Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d u).Finite
⊢ meromorphicOrderAt (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ d u) z ≠ ⊤

Tactic:
]

Goal state after:
case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
z : 𝕜
d : 𝕜 → ℤ
hd : ¬(Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d u).Finite
⊢ meromorphicOrderAt (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ d u) z ≠ ⊤
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
z : 𝕜
d : 𝕜 → ℤ
hd : ¬(support d).Finite
⊢ meromorphicOrderAt (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ d u) z ≠ ⊤

Tactic:
rw [← mulSupport] at hd

Goal state after:
case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
z : 𝕜
d : 𝕜 → ℤ
hd : ¬(Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d u).Finite
⊢ meromorphicOrderAt (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ d u) z ≠ ⊤
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
f : 𝕜 → E
x c : 𝕜
h : ¬¬MeromorphicAt f (x + c)
⊢ meromorphicTrailingCoeffAt (fun x_1 ↦ x_1 + c - (x + c)) x ^ (meromorphicOrderAt f (x + c)).untop₀ •
      meromorphicTrailingCoeffAt f (x + c) =
    meromorphicTrailingCoeffAt f (x + c)

Tactic:
]

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
f : 𝕜 → E
x c : 𝕜
h : ¬¬MeromorphicAt f (x + c)
⊢ meromorphicTrailingCoeffAt (fun x_1 ↦ x_1 + c - (x + c)) x ^ (meromorphicOrderAt f (x + c)).untop₀ •
      meromorphicTrailingCoeffAt f (x + c) =
    meromorphicTrailingCoeffAt f (x + c)
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h : (support d).Finite
u : 𝕜
h₁ : u = x
⊢ x ∉ ↑h.toFinset → (x - x) ^ update d x 0 x = 1

Tactic:
]

Goal state after:
case pos
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h : (support d).Finite
u : 𝕜
h₁ : u = x
⊢ x ∉ ↑h.toFinset → (x - x) ^ update d x 0 x = 1
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: R : ℝ
w x : ℂ
hx : x ∈ {w}ᶜ
⊢ AnalyticAt ℂ (fun z ↦ (↑R ^ 2 - (starRingEnd ℂ) w * z) / (↑R * (z - w))) x

Tactic:
]

Goal state after:
R : ℝ
w x : ℂ
hx : x ∈ {w}ᶜ
⊢ AnalyticAt ℂ (fun z ↦ (↑R ^ 2 - (starRingEnd ℂ) w * z) / (↑R * (z - w))) x
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: 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)
⊢ ∏ n ∈ Finite.toFinset h, meromorphicTrailingCoeffAt ((fun x ↦ x - n) ^ d n) x =
    ∏ i ∈ Finite.toFinset h, (x - i) ^ update d x 0 i

Tactic:
]

Goal state after:
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)
⊢ ∏ n ∈ Finite.toFinset h, meromorphicTrailingCoeffAt ((fun x ↦ x - n) ^ d n) x =
    ∏ i ∈ Finite.toFinset h, (x - i) ^ update d x 0 i
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: R : ℝ
w : ℂ
h : w ∈ ball 0 R
⊢ meromorphicOrderAt (fun z ↦ ↑R ^ 2 - (starRingEnd ℂ) w * z) w -
      (meromorphicOrderAt (fun z ↦ ↑R) w + meromorphicOrderAt (fun z ↦ z - w) w) =
    -1

Tactic:
]

Goal state after:
R : ℝ
w : ℂ
h : w ∈ ball 0 R
⊢ meromorphicOrderAt (fun z ↦ ↑R ^ 2 - (starRingEnd ℂ) w * z) w -
      (meromorphicOrderAt (fun z ↦ ↑R) w + meromorphicOrderAt (fun z ↦ z - w) w) =
    -1
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 = ⊤
this : AnalyticAt 𝕜 (fun x ↦ 0) x
⊢ (fun x ↦ 0) =ᶠ[𝓝[≠] x] f

Tactic:
]

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
⊢ (fun x ↦ 0) =ᶠ[𝓝[≠] x] f
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: case refine_2.refine_7
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
f : C(↑I × A, X)
g : ↑I × A → E
g_lifts : p ∘ g = ⇑f
cont_0 : Continuous[inst✝, inst✝²] fun x ↦ g (0, x)
a✝ : A
cont_a : Continuous[_, inst✝²] fun x ↦ g (x, a✝)
q : E → OpenPartialHomeomorph E X
mem_source : ∀ (x : E), x ∈ (q x).source
hpq : ∀ (x : E), p = ↑(q x)
t : ℕ → ↑I
t_0 : t 0 = 0
t_mono : Monotone t
t_sub : ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ (fun x ↦ g (x, a✝)) ⁻¹' (q i).source
n_max : ℕ
h_max : ∀ m ≥ n_max, t m = 1
n : ℕ
x✝ :
  ∃ N,
    a✝ ∈ N ∧
      IsOpen[inst✝] N ∧
        ∃ g',
          ContinuousOn g' (Set.Icc 0 (t n) ×ˢ N) ∧
            p ∘ g' = ⇑f ∧ (∀ (a : A), g' (0, a) = g (0, a)) ∧ ∀ t' ≤ t n, g' (t', a✝) = g (t', a✝)
N : Set A
haN : a✝ ∈ N
N_open : IsOpen[inst✝] N
g' : ↑I × A → E
cont_g' : ContinuousOn g' (Set.Icc 0 (t n) ×ˢ N)
g'_lifts : p ∘ g' = ⇑f
g'_0 : ∀ (a : A), g' (0, a) = g (0, a)
g'_a : ∀ t' ≤ t n, g' (t', a✝) = g (t', a✝)
e : E
h_sub : Set.Icc (t n) (t (n + 1)) ⊆ (fun x ↦ g (x, a✝)) ⁻¹' (q e).source
this : Set.Icc (t n) (t (n + 1)) ×ˢ {a✝} ⊆ ⇑f ⁻¹' (q e).target
u : Set ↑I
v : Set A
v_open : IsOpen[inst✝] v
hu : Set.Icc (t n) (t (n + 1)) ⊆ u
hav : {a✝} ⊆ v
huv : u ×ˢ v ⊆ ⇑f ⁻¹' (q e).target
a : A
⊢ (fun ta ↦ if ta.1 ≤ t n then g' ta else if f ta ∈ (q e).target then ↑(q e).symm (f ta) else g ta) (0, a) = g' (0, a)

Tactic:
]

Goal state after:
case refine_2.refine_7
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
f : C(↑I × A, X)
g : ↑I × A → E
g_lifts : p ∘ g = ⇑f
cont_0 : Continuous[inst✝, inst✝²] fun x ↦ g (0, x)
a✝ : A
cont_a : Continuous[_, inst✝²] fun x ↦ g (x, a✝)
q : E → OpenPartialHomeomorph E X
mem_source : ∀ (x : E), x ∈ (q x).source
hpq : ∀ (x : E), p = ↑(q x)
t : ℕ → ↑I
t_0 : t 0 = 0
t_mono : Monotone t
t_sub : ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ (fun x ↦ g (x, a✝)) ⁻¹' (q i).source
n_max : ℕ
h_max : ∀ m ≥ n_max, t m = 1
n : ℕ
x✝ :
  ∃ N,
    a✝ ∈ N ∧
      IsOpen[inst✝] N ∧
        ∃ g',
          ContinuousOn g' (Set.Icc 0 (t n) ×ˢ N) ∧
            p ∘ g' = ⇑f ∧ (∀ (a : A), g' (0, a) = g (0, a)) ∧ ∀ t' ≤ t n, g' (t', a✝) = g (t', a✝)
N : Set A
haN : a✝ ∈ N
N_open : IsOpen[inst✝] N
g' : ↑I × A → E
cont_g' : ContinuousOn g' (Set.Icc 0 (t n) ×ˢ N)
g'_lifts : p ∘ g' = ⇑f
g'_0 : ∀ (a : A), g' (0, a) = g (0, a)
g'_a : ∀ t' ≤ t n, g' (t', a✝) = g (t', a✝)
e : E
h_sub : Set.Icc (t n) (t (n + 1)) ⊆ (fun x ↦ g (x, a✝)) ⁻¹' (q e).source
this : Set.Icc (t n) (t (n + 1)) ×ˢ {a✝} ⊆ ⇑f ⁻¹' (q e).target
u : Set ↑I
v : Set A
v_open : IsOpen[inst✝] v
hu : Set.Icc (t n) (t (n + 1)) ⊆ u
hav : {a✝} ⊆ v
huv : u ×ˢ v ⊆ ⇑f ⁻¹' (q e).target
a : A
⊢ (fun ta ↦ if ta.1 ≤ t n then g' ta else if f ta ∈ (q e).target then ↑(q e).symm (f ta) else g ta) (0, a) = g' (0, a)
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: 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
⊢ meromorphicTrailingCoeffAt (fun x ↦ x - y) x ^ d y = (x - y) ^ update d x 0 y

Tactic:
]

Goal state after:
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
⊢ meromorphicTrailingCoeffAt (fun x ↦ x - y) x ^ d y = (x - y) ^ update d x 0 y
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: case refine_2
R : ℝ
w : ℂ
h : w ∈ ball 0 R
⊢ ¬R = ‖w‖

Tactic:
]

Goal state after:
case refine_2
R : ℝ
w : ℂ
h : w ∈ ball 0 R
⊢ ¬R = ‖w‖
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: case refine_2
R : ℝ
w : ℂ
h : ‖w‖ < R
⊢ ¬R = ‖w‖

Tactic:
]

Goal state after:
case refine_2
R : ℝ
w : ℂ
h : ‖w‖ < R
⊢ ¬R = ‖w‖
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: case pos
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
⊢ (if y = y then 1 else y - y) ^ d y = (y - y) ^ update d y 0 y

Tactic:
]

Goal state after:
case pos
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
⊢ (if y = y then 1 else y - y) ^ d y = (y - y) ^ update d y 0 y
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: case inl
R : ℝ
z : ℂ
hz : z ∈ univ
h : z ∈ ball 0 R
⊢ AnalyticAt ℂ (canonicalFactor R z) z ∨
    MeromorphicAt (canonicalFactor R z) z ∧ meromorphicOrderAt (canonicalFactor R z) z < 0 ∧ canonicalFactor R z z = 0

Tactic:
]

Goal state after:
case inl
R : ℝ
z : ℂ
hz : z ∈ univ
h : z ∈ ball 0 R
⊢ AnalyticAt ℂ (canonicalFactor R z) z ∨
    MeromorphicAt (canonicalFactor R z) z ∧ meromorphicOrderAt (canonicalFactor R z) z < 0 ∧ canonicalFactor R z z = 0
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
⊢ ∏ i ∈ Finite.toFinset h₁, (x - i) ^ update d x 0 i = ∏ᶠ (u : 𝕜), (x - u) ^ d u

Tactic:
]

Goal state after:
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h₁ : HasFiniteSupport d
h₂ : x ∉ support d
⊢ ∏ i ∈ Finite.toFinset h₁, (x - i) ^ update d x 0 i = ∏ᶠ (u : 𝕜), (x - u) ^ d u
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: R : ℝ
w z : ℂ
h₂z : z ≠ w
hR : 0 < R
hzw : z - w ≠ 0
hw : ‖w‖ < R
h₁z : ‖z‖ ≤ R
⊢ ‖w‖ * R < R * R

Tactic:
]

Goal state after:
R : ℝ
w z : ℂ
h₂z : z ≠ w
hR : 0 < R
hzw : z - w ≠ 0
hw : ‖w‖ < R
h₁z : ‖z‖ ≤ R
⊢ ‖w‖ * R < R * R
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
this : (Function.mulSupport fun u ↦ (x - u) ^ d u) ⊆ ↑(Finite.toFinset h₁)
⊢ (Finite.toFinset h₁).prod ?m.107 = ∏ i ∈ Finite.toFinset h₁, (x - i) ^ d i

pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h₁ : HasFiniteSupport d
h₂ : x ∉ support d
this : (Function.mulSupport fun u ↦ (x - u) ^ d u) ⊆ ↑(Finite.toFinset h₁)
⊢ ∀ x_1 ∈ Finite.toFinset h₁, (x - x_1) ^ update d x 0 x_1 = ?m.107 x_1

pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h₁ : HasFiniteSupport d
h₂ : x ∉ support d
this : (Function.mulSupport fun u ↦ (x - u) ^ d u) ⊆ ↑(Finite.toFinset h₁)
⊢ 𝕜 → 𝕜

Tactic:
]

Goal state after:
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h₁ : HasFiniteSupport d
h₂ : x ∉ support d
this : (Function.mulSupport fun u ↦ (x - u) ^ d u) ⊆ ↑(Finite.toFinset h₁)
⊢ ∀ x_1 ∈ Finite.toFinset h₁, (x - x_1) ^ update d x 0 x_1 = (x - x_1) ^ d x_1
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 : 0 < meromorphicOrderAt f x
hf : f x = 0
⊢ Tendsto f (𝓝[≠] x) (𝓝 0)

Tactic:
]

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 : 0 < meromorphicOrderAt f x
hf : f x = 0
⊢ Tendsto f (𝓝[≠] x) (𝓝 0)
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: R : ℝ
w z : ℂ
h₂z : z ≠ w
hR : 0 < R
hzw : z - w ≠ 0
hw : ‖w‖ < R
h₁z : ‖z‖ ≤ R
h_num_ne_zero : ↑R ^ 2 - (starRingEnd ℂ) w * z ≠ 0
⊢ (↑R ^ 2 - (starRingEnd ℂ) w * z) / (↑R * (z - w)) ≠ 0

Tactic:
]

Goal state after:
R : ℝ
w z : ℂ
h₂z : z ≠ w
hR : 0 < R
hzw : z - w ≠ 0
hw : ‖w‖ < R
h₁z : ‖z‖ ≤ R
h_num_ne_zero : ↑R ^ 2 - (starRingEnd ℂ) w * z ≠ 0
⊢ (↑R ^ 2 - (starRingEnd ℂ) w * z) / (↑R * (z - w)) ≠ 0
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h : HasFiniteSupport d
⊢ log ‖∏ i ∈ Finite.toFinset h, (x - i) ^ update d x 0 i‖ = ∑ᶠ (u : 𝕜), ↑(d u) * log ‖x - u‖

Tactic:
]

Goal state after:
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h : HasFiniteSupport d
⊢ log ‖∏ i ∈ Finite.toFinset h, (x - i) ^ update d x 0 i‖ = ∑ᶠ (u : 𝕜), ↑(d u) * log ‖x - u‖
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: R : ℝ
w z : ℂ
hw : w ∈ ball 0 R
hz : z ∈ sphere 0 R
hR : 0 < R
hzw : z - w ≠ 0
⊢ ¬↑R * (z - w) = 0

Tactic:
]

Goal state after:
R : ℝ
w z : ℂ
hw : w ∈ ball 0 R
hz : z ∈ sphere 0 R
hR : 0 < R
hzw : z - w ≠ 0
⊢ ¬↑R * (z - w) = 0
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
n : ℤ
⊢ ∃ g, AnalyticAt 𝕜 g x ∧ g x ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, ((fun x_1 ↦ x_1 - x) ^ n) z = (z - x) ^ n • g z

Tactic:
]

Goal state after:
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
n : ℤ
⊢ ∃ g, AnalyticAt 𝕜 g x ∧ g x ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, ((fun x_1 ↦ x_1 - x) ^ n) z = (z - x) ^ n • g z
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h✝ : HasFiniteSupport d
y : 𝕜
a✝ : y ∈ Finite.toFinset h✝
h : x = y
⊢ ‖(y - y) ^ update d y 0 y‖ ≠ 0

Tactic:
]

Goal state after:
case pos
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h✝ : HasFiniteSupport d
y : 𝕜
a✝ : y ∈ Finite.toFinset h✝
h : x = y
⊢ ‖(y - y) ^ update d y 0 y‖ ≠ 0
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: R : ℝ
w z : ℂ
hw : w ∈ ball 0 R
hz : z ∈ sphere 0 R
hR : 0 < R
hzw : z - w ≠ 0
⊢ ‖↑R ^ 2 - (starRingEnd ℂ) w * z‖ = ‖↑R * (z - w)‖

Tactic:
]

Goal state after:
R : ℝ
w z : ℂ
hw : w ∈ ball 0 R
hz : z ∈ sphere 0 R
hR : 0 < R
hzw : z - w ≠ 0
⊢ ‖↑R ^ 2 - (starRingEnd ℂ) w * z‖ = ‖↑R * (z - w)‖
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
⊢ meromorphicOrderAt (fun x_1 ↦ x_1 - x) x = meromorphicOrderAt (fun x ↦ x - ?m.35) ?m.35

pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
⊢ 𝕜

Tactic:
]

Goal state after:
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
x : 𝕜
f : 𝕜 → E
h₁ : MeromorphicAt f x
⊢ meromorphicOrderAt f x = meromorphicOrderAt (-f) x

Tactic:
]

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
f : 𝕜 → E
h₁ : MeromorphicAt f x
⊢ meromorphicOrderAt f x = meromorphicOrderAt (-f) x
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
x : 𝕜
f : 𝕜 → E
h₁ : MeromorphicAt f x
h₂ : meromorphicOrderAt f x = ⊤
⊢ meromorphicOrderAt (-f) x = ⊤

Tactic:
]

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
x : 𝕜
f : 𝕜 → E
h₁ : MeromorphicAt f x
h₂ : meromorphicOrderAt f x = ⊤
⊢ meromorphicOrderAt (-f) x = ⊤
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: 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
⊢ ∑ i ∈ Finite.toFinset h, log ‖(x - i) ^ update d x 0 i‖ = ∑ᶠ (u : 𝕜), ↑(d u) * log ‖x - u‖

Tactic:
]

Goal state after:
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
⊢ ∑ i ∈ Finite.toFinset h, log ‖(x - i) ^ update d x 0 i‖ = ∑ᶠ (u : 𝕜), ↑(d u) * log ‖x - u‖
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
x : 𝕜
f : 𝕜 → E
h₁ : MeromorphicAt f x
n : ℤ
hn : ∃ g, AnalyticAt 𝕜 g x ∧ g x ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ n • g z
⊢ ∃ g, AnalyticAt 𝕜 g x ∧ g x ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, (-f) z = (z - x) ^ n • g z

Tactic:
]

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
f : 𝕜 → E
h₁ : MeromorphicAt f x
n : ℤ
hn : ∃ g, AnalyticAt 𝕜 g x ∧ g x ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ n • g z
⊢ ∃ g, AnalyticAt 𝕜 g x ∧ g x ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, (-f) z = (z - x) ^ n • g z