mathlib-initiative/mathlib-tactics huggingface dataset:
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
h₁ : MeromorphicAt f x
h₃ : f x = 0
n : ℤ
hn : ↑n = meromorphicOrderAt f x
meromorphicNFAt_iff_analyticAt_or : MeromorphicNFAt f x ↔ AnalyticAt 𝕜 f x ∨ MeromorphicAt f x ∧ ↑n < 0 ∧ f x = 0
h₂ : ↑n < 0
g : 𝕜 → E
h₁g : AnalyticAt 𝕜 g x
h₂g : g x ≠ 0
h₃g : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ n • g z
⊢ ∀ (a : 𝕜), (a ∈ {x}ᶜ → f a = (a - x) ^ n • g a) → f a = ((fun x_1 ↦ x_1 - x) ^ n • g) a

Tactic:
intro z

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
h₁ : MeromorphicAt f x
h₃ : f x = 0
n : ℤ
hn : ↑n = meromorphicOrderAt f x
meromorphicNFAt_iff_analyticAt_or : MeromorphicNFAt f x ↔ AnalyticAt 𝕜 f x ∨ MeromorphicAt f x ∧ ↑n < 0 ∧ f x = 0
h₂ : ↑n < 0
g : 𝕜 → E
h₁g : AnalyticAt 𝕜 g x
h₂g : g x ≠ 0
h₃g : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ n • g z
z : 𝕜
⊢ (z ∈ {x}ᶜ → f z = (z - x) ^ n • g z) → f z = ((fun x_1 ↦ x_1 - x) ^ n • g) z
Module: Mathlib.Analysis.Meromorphic.NormalForm

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
g : 𝕜 → E
hf : MeromorphicNFAt f x
hg : MeromorphicNFAt g x
h : f =ᶠ[𝓝[≠] x] g
t₀ : 0 = meromorphicOrderAt g x
cs : meromorphicOrderAt f x = 0
Z : ContinuousAt f x
W : ContinuousAt g x
⊢ f =ᶠ[𝓝 x] g

Tactic:
exact (Z.eventuallyEq_nhds_iff_eventuallyEq_nhdsNE W).1 h

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

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c x : 𝕜
f : 𝕜 → E
h : MeromorphicOn f (U + {c})
this : MeromorphicOn (f ∘ fun x ↦ x + c) U
h₁ : x ∉ U + {c}
⊢ 0 = (divisor f (U + {c})) x

pp: case pos.hz
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c x : 𝕜
f : 𝕜 → E
h : MeromorphicOn f (U + {c})
this : MeromorphicOn (f ∘ fun x ↦ x + c) U
h₁ : x ∉ U + {c}
⊢ x - c ∉ U

Tactic:
Function.locallyFinsuppWithin.apply_eq_zero_of_notMem

Goal state after:
case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c x : 𝕜
f : 𝕜 → E
h : MeromorphicOn f (U + {c})
this : MeromorphicOn (f ∘ fun x ↦ x + c) U
h₁ : x ∉ U + {c}
⊢ 0 = 0

case pos.hz
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c x : 𝕜
f : 𝕜 → E
h : MeromorphicOn f (U + {c})
this : MeromorphicOn (f ∘ fun x ↦ x + c) U
h₁ : x ∉ U + {c}
⊢ x ∉ U + {c}

case pos.hz
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c x : 𝕜
f : 𝕜 → E
h : MeromorphicOn f (U + {c})
this : MeromorphicOn (f ∘ fun x ↦ x + c) U
h₁ : x ∉ U + {c}
⊢ x - c ∉ U
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
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:
· filter_upwards [h₃g_f]
      intro y hy
      simp only [Pi.smul_apply', hy, Pi.pow_apply]
      rw [smul_comm]

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

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
hf : MeromorphicAt f x
h₂f : meromorphicOrderAt f x = ⊤
this : toMeromorphicNFAt f x =ᶠ[𝓝 x] 0
⊢ AnalyticAt 𝕜 0 x

Tactic:
exact analyticAt_const

Goal state after:
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 : toMeromorphicNFAt f x = f
⊢ MeromorphicNFAt f x

Tactic:
hf.symm

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 : toMeromorphicNFAt f x = f
⊢ MeromorphicNFAt (toMeromorphicNFAt f x) x
Module: Mathlib.Analysis.Meromorphic.NormalForm

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
g : 𝕜 → 𝕜
x : 𝕜
U : Set 𝕜
h₁f : MeromorphicOn f U
⊢ 𝕜 → E

Tactic:
intro z

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
g : 𝕜 → 𝕜
x : 𝕜
U : Set 𝕜
h₁f : MeromorphicOn f U
z : 𝕜
⊢ E
Module: Mathlib.Analysis.Meromorphic.NormalForm

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
U : Set 𝕜
hf : MeromorphicOn f U
⊢ MeromorphicNFOn (toMeromorphicNFOn f U) U

Tactic:
intro z

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
U : Set 𝕜
hf : MeromorphicOn f U
z : 𝕜
⊢ z ∈ U → MeromorphicNFAt (toMeromorphicNFOn f U) z
Module: Mathlib.Analysis.Meromorphic.NormalForm

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
U : Set 𝕜
hf : MeromorphicOn f U
z : 𝕜
hz : z ∈ U
⊢ MeromorphicNFAt (toMeromorphicNFOn f U) z

Tactic:
meromorphicNFAt_congr (toMeromorphicNFOn_eq_toMeromorphicNFAt_on_nhds hf hz)

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
U : Set 𝕜
hf : MeromorphicOn f U
z : 𝕜
hz : z ∈ U
⊢ MeromorphicNFAt (toMeromorphicNFAt f z) 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 : d x = 0
h₁ : HasFiniteMulSupport fun u ↦ (fun x ↦ x - u) ^ d u
⊢ ∀ a ∈ Finite.toFinset h₁, ((fun x ↦ x - a) ^ d a) x ≠ 0

Tactic:
intro z

Goal state after:
case pos
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h : d x = 0
h₁ : HasFiniteMulSupport fun u ↦ (fun x ↦ x - u) ^ d u
z : 𝕜
⊢ z ∈ Finite.toFinset h₁ → ((fun x ↦ x - z) ^ d z) x ≠ 0
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
⊢ MeromorphicNFOn (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ d u) univ

Tactic:
intro z

Goal state after:
case pos
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
hd : (support d).Finite
z : 𝕜
⊢ z ∈ univ → MeromorphicNFAt (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ d u) z
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: case refine_2
R : ℝ
w : ℂ
h : w ∈ ball 0 R
⊢ ¬R ^ 2 = ‖w‖ ^ 2

Tactic:
sq_eq_sq₀ (pos_of_mem_ball h).le (norm_nonneg w)

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: R : ℝ
w : ℂ
h : w ∈ ball 0 R
⊢ MeromorphicNFOn (canonicalFactor R w) univ

Tactic:
intro z

Goal state after:
R : ℝ
w : ℂ
h : w ∈ ball 0 R
z : ℂ
⊢ z ∈ univ → MeromorphicNFAt (canonicalFactor R w) z
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

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 : 𝕜 → E
h₁f : MeromorphicOn f U
h₂f : ∀ (u : ↑U), meromorphicOrderAt f ↑u ≠ ⊤
h₃f : (divisor f U).support.Finite
φ : 𝕜 → 𝕜 := ∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ (divisor f U) u
hφ : MeromorphicOn φ U
g : 𝕜 → E := toMeromorphicNFOn (φ⁻¹ • f) U
hg : MeromorphicNFOn g U
⊢ ∀ z ∈ U, meromorphicOrderAt φ⁻¹ z ≠ ⊤

Tactic:
intro z

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f : 𝕜 → E
h₁f : MeromorphicOn f U
h₂f : ∀ (u : ↑U), meromorphicOrderAt f ↑u ≠ ⊤
h₃f : (divisor f U).support.Finite
φ : 𝕜 → 𝕜 := ∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ (divisor f U) u
hφ : MeromorphicOn φ U
g : 𝕜 → E := toMeromorphicNFOn (φ⁻¹ • f) U
hg : MeromorphicNFOn g U
z : 𝕜
⊢ z ∈ U → meromorphicOrderAt φ⁻¹ z ≠ ⊤
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
𝕜' : Type u_4
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
x : 𝕜
f : 𝕜 → 𝕜'
hf : MeromorphicAt f x
a : ℤ
ha : ↑a = meromorphicOrderAt f x
g : 𝕜 → 𝕜'
h₁g : AnalyticAt 𝕜 g x
h₂g : g x ≠ 0
h₃g : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ a • g z
⊢ ∃ g, AnalyticAt 𝕜 g x ∧ g x ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f⁻¹ z = (z - x) ^ (fun a ↦ -a) a • g z

Tactic:
use g⁻¹, h₁g.inv h₂g, inv_eq_zero.not.2 h₂g

Goal state after:
case right
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
𝕜' : Type u_4
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
x : 𝕜
f : 𝕜 → 𝕜'
hf : MeromorphicAt f x
a : ℤ
ha : ↑a = meromorphicOrderAt f x
g : 𝕜 → 𝕜'
h₁g : AnalyticAt 𝕜 g x
h₂g : g x ≠ 0
h₃g : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ a • g z
⊢ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f⁻¹ z = (z - x) ^ (fun a ↦ -a) a • g⁻¹ z
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: case h₁f
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
f : ℂ → E
h₁f : MeromorphicOn f (closedBall 0 R)
η₀ : (-divisor f (ball 0 R)).support.Finite
⊢ ∀ i ∈ η₀.toFinset, MeromorphicOn (canonicalFactor R i ^ (divisor f (ball 0 R)) i) (ball 0 R)

Tactic:
intro z

Goal state after:
case h₁f
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
f : ℂ → E
h₁f : MeromorphicOn f (closedBall 0 R)
η₀ : (-divisor f (ball 0 R)).support.Finite
z : ℂ
⊢ z ∈ η₀.toFinset → MeromorphicOn (canonicalFactor R z ^ (divisor f (ball 0 R)) z) (ball 0 R)
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: case h₂f
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
f : ℂ → E
h₁f : MeromorphicOn f (closedBall 0 R)
η₀ : (-divisor f (ball 0 R)).support.Finite
⊢ ∀ i ∈ η₀.toFinset, ∀ z ∈ ball 0 R, meromorphicOrderAt (canonicalFactor R i ^ (divisor f (ball 0 R)) i) z ≠ ⊤

Tactic:
intro z

Goal state after:
case h₂f
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
f : ℂ → E
h₁f : MeromorphicOn f (closedBall 0 R)
η₀ : (-divisor f (ball 0 R)).support.Finite
z : ℂ
⊢ z ∈ η₀.toFinset → ∀ z_1 ∈ ball 0 R, meromorphicOrderAt (canonicalFactor R z ^ (divisor f (ball 0 R)) z) z_1 ≠ ⊤
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
⊢ ∀ u ∈ ball 0 R, g u ≠ 0

Tactic:
intro z

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
z : ℂ
⊢ z ∈ ball 0 R → g z ≠ 0
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: case 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
⊢ ∀ x ∈ {u | meromorphicOrderAt f ↑u = ⊤}ᶜ,
    ∃ t ⊆ {u | meromorphicOrderAt f ↑u = ⊤}ᶜ, IsOpen[instTopologicalSpaceSubtype] t ∧ x ∈ t

Tactic:
intro z

Goal state after:
case 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
⊢ z ∈ {u | meromorphicOrderAt f ↑u = ⊤}ᶜ →
    ∃ t ⊆ {u | meromorphicOrderAt f ↑u = ⊤}ᶜ, IsOpen[instTopologicalSpaceSubtype] t ∧ z ∈ t
Module: Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Basic

Goal state before:
pp: R : Type u_1
n : Type u_2
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommRing R
g : GL n R
⊢ g ∈ Subgroup.center (GL n R) ↔ ↑g ∈ Set.range ⇑(Matrix.scalar n)

Tactic:
constructor
  · intro hg
    refine Matrix.mem_range_scalar_of_commute_transvectionStruct fun t ↦ ?_
    simpa [Units.ext_iff] using! Subgroup.mem_center_iff.mp hg (.mk _ _ t.mul_inv t.inv_mul)
  · refine fun ⟨a, ha⟩ ↦ Subgroup.mem_center_iff.mpr fun h ↦ ?_
    simp [-scalar_apply, Units.ext_iff, ← ha, Matrix.scalar_comm a (Commute.all _)]

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Basic

Goal state before:
pp: R : Type u_1
n : Type u_2
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommRing R
⊢ Subgroup.center (GL n R) = (scalar n).range

Tactic:
ext g

Goal state after:
R : Type u_1
n : Type u_2
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommRing R
g : GL n R
⊢ g ∈ Subgroup.center (GL n R) ↔ g ∈ (scalar n).range
Module: Mathlib.Analysis.Complex.Isometry

Goal state before:
pp: case right
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : f 1 = 1
⊢ ∀ (z : ℂ), (f.toLinearIsometry z).re = z.re

Tactic:
intro z

Goal state after:
case right
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : f 1 = 1
z : ℂ
⊢ (f.toLinearIsometry z).re = z.re
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
f g : ℂ → E
x : ℂ
D : CanonicalDecomp f g R
hR : 0 < R
h : ‖x‖ = R
a : ℂ
ha : R ≤ ‖a‖
⊢ AnalyticAt ℂ 1 x

Tactic:
exact analyticAt_const

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

Goal state before:
pp: case right
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
⊢ ∀ x ∈ {u | meromorphicOrderAt f ↑u = ⊤},
    ∃ t ⊆ {u | meromorphicOrderAt f ↑u = ⊤}, IsOpen[instTopologicalSpaceSubtype] t ∧ x ∈ t

Tactic:
intro z

Goal state after:
case right
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
⊢ z ∈ {u | meromorphicOrderAt f ↑u = ⊤} →
    ∃ t ⊆ {u | meromorphicOrderAt f ↑u = ⊤}, IsOpen[instTopologicalSpaceSubtype] t ∧ z ∈ t
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
g : ℂ → E
D : CanonicalDecomp f g R
s₁ : 0 ∈ closedBall 0 R
⊢ ∃ u, meromorphicOrderAt g ↑u ≠ ⊤

Tactic:
use ⟨0, s₁⟩

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 : 0 < R
g : ℂ → E
D : CanonicalDecomp f g R
s₁ : 0 ∈ closedBall 0 R
⊢ meromorphicOrderAt g ↑⟨0, s₁⟩ ≠ ⊤
Module: Mathlib.Analysis.Complex.Conformal

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
ℓ : ℂ →ₗ[ℝ] E
h : ℓ I = I • ℓ 1
a b : ℂ
⊢ ℓ (a • b) = a • ℓ b

Tactic:
by
  rw [← re_add_im a, ← re_add_im b, ← smul_eq_mul _ I, ← smul_eq_mul _ I]
  have t₀ : ((a.im : ℂ) • I) • (b.re : ℂ) = (↑(a.im * b.re) : ℂ) • I := by
    simp only [smul_eq_mul, ofReal_mul, ← mul_assoc, mul_comm _ I]
  have t₁ : ((a.im : ℂ) • I) • (b.im : ℂ) • I = (↑(- a.im * b.im) : ℂ) • (1 : ℂ) := by
    simp [mul_mul_mul_comm _ I]
  simp only [add_smul, smul_add, ℓ.map_add, t₀, t₁]
  repeat rw [Complex.coe_smul, ℓ.map_smul]
  have t₂ {r : ℝ} : ℓ (r : ℂ) = r • ℓ (1 : ℂ) := by simp [← ℓ.map_smul]
  simp only [t₂, h]
  match_scalars
  simp [mul_mul_mul_comm _ I]
  ring

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

Goal state before:
pp: case h
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
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' ⊆
      {x |
        ∃ t, (∀ y ∈ t, y ∈ {↑x}ᶜ → f y = 0) ∧ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t ∧ ↑x ∈ t} ∧
    IsOpen[instTopologicalSpaceSubtype] (Subtype.val ⁻¹' t') ∧ z ∈ Subtype.val ⁻¹' t'

Tactic:
simp only [mem_compl_iff, mem_singleton_iff, isOpen_induced h₂t', mem_preimage,
      h₃t', and_self, and_true]

Goal state after:
case h
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
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' ⊆
    {x | ∃ t, (∀ y ∈ t, ¬y = ↑x → f y = 0) ∧ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t ∧ ↑x ∈ t}
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: case mp.inl
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
hU : IsConnected U
h₂f : ∃ u, meromorphicOrderAt f ↑u ≠ ⊤
this✝ : PreconnectedSpace ↑U
h : {u | meromorphicOrderAt f ↑u = ⊤} = ∅
u : ↑U
this : u ∉ ∅
⊢ meromorphicOrderAt f ↑u ≠ ⊤

Tactic:
← h

Goal state after:
case mp.inl
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
hU : IsConnected U
h₂f : ∃ u, meromorphicOrderAt f ↑u ≠ ⊤
this✝ : PreconnectedSpace ↑U
h : {u | meromorphicOrderAt f ↑u = ⊤} = ∅
u : ↑U
this : u ∉ {u | meromorphicOrderAt f ↑u = ⊤}
⊢ meromorphicOrderAt f ↑u ≠ ⊤
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: case mp.inr
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
hU✝ : IsConnected U
this✝ : PreconnectedSpace ↑U
h : {u | meromorphicOrderAt f ↑u = ⊤} = univ
u : ↑U
hU : meromorphicOrderAt f ↑u ≠ ⊤
this : u ∈ univ
⊢ ∀ (u : ↑U), meromorphicOrderAt f ↑u ≠ ⊤

Tactic:
← h

Goal state after:
case mp.inr
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
hU✝ : IsConnected U
this✝ : PreconnectedSpace ↑U
h : {u | meromorphicOrderAt f ↑u = ⊤} = univ
u : ↑U
hU : meromorphicOrderAt f ↑u ≠ ⊤
this : u ∈ {u | meromorphicOrderAt f ↑u = ⊤}
⊢ ∀ (u : ↑U), meromorphicOrderAt f ↑u ≠ ⊤
Module: Mathlib.Analysis.Complex.ExponentialBounds

Goal state before:
pp: t : |2⁻¹| = 2⁻¹
z : |429895798851992810179897 / 620208537104218167705600 + log 2⁻¹| ≤ 1 / 17179869184
⊢ |log 2 - 287209 / 414355| ≤ 1 / 17179869184 + (1 / 10 ^ 10 - 1 / 2 ^ 34)

Tactic:
log_inv,

Goal state after:
t : |2⁻¹| = 2⁻¹
z : |429895798851992810179897 / 620208537104218167705600 + -log 2| ≤ 1 / 17179869184
⊢ |log 2 - 287209 / 414355| ≤ 1 / 17179869184 + (1 / 10 ^ 10 - 1 / 2 ^ 34)
Module: Mathlib.Analysis.Complex.ExponentialBounds

Goal state before:
pp: t : |2 / 3| = 2 / 3
z : |∑ i ∈ range 70, (2 / 3) ^ (i + 1) / (↑i + 1) + log 3⁻¹| ≤ (2 / 3) ^ (70 + 1) / 3⁻¹
⊢ |log 3 - 109861228867 / 100000000000| ≤ (2 / 3) ^ 71 / 3⁻¹ + (1 / 10 ^ 10 - (2 / 3) ^ 71 / 3⁻¹)

Tactic:
log_inv,

Goal state after:
t : |2 / 3| = 2 / 3
z : |∑ i ∈ range 70, (2 / 3) ^ (i + 1) / (↑i + 1) + -log 3| ≤ (2 / 3) ^ (70 + 1) / 3⁻¹
⊢ |log 3 - 109861228867 / 100000000000| ≤ (2 / 3) ^ 71 / 3⁻¹ + (1 / 10 ^ 10 - (2 / 3) ^ 71 / 3⁻¹)
Module: Mathlib.Analysis.Complex.ExponentialBounds

Goal state before:
pp: t : |4 / 5| = 4 / 5
z : |∑ i ∈ range 130, (4 / 5) ^ (i + 1) / (↑i + 1) + log 5⁻¹| ≤ (4 / 5) ^ (130 + 1) / 5⁻¹
⊢ |log 5 - 160943791243 / 100000000000| ≤ (4 / 5) ^ 131 / 5⁻¹ + (1 / 10 ^ 10 - (4 / 5) ^ 131 / 5⁻¹)

Tactic:
log_inv,

Goal state after:
t : |4 / 5| = 4 / 5
z : |∑ i ∈ range 130, (4 / 5) ^ (i + 1) / (↑i + 1) + -log 5| ≤ (4 / 5) ^ (130 + 1) / 5⁻¹
⊢ |log 5 - 160943791243 / 100000000000| ≤ (4 / 5) ^ 131 / 5⁻¹ + (1 / 10 ^ 10 - (4 / 5) ^ 131 / 5⁻¹)
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: case refine_2
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
cov : IsCoveringMap p
inst✝¹ : PathConnectedSpace A
inst✝ : LocallyPathConnectedSpace A
f : C(A, X)
a₀ : A
e₀ : E
he : p e₀ = f a₀
le : (FundamentalGroup.map f a₀).range ≤ (mapOfEq { toFun := p, continuous_toFun := ⋯ } he).range
γ γ' : C(↑I, A)
Γ Γ' : C(↑I, E)
γ_0 : γ 0 = a₀
γ'_0 : γ' 0 = a₀
Γ_0 : Γ 0 = e₀
Γ'_0 : Γ' 0 = e₀
Γ_lifts : p ∘ ⇑Γ = ⇑(f.comp γ)
Γ'_lifts : p ∘ ⇑Γ' = ⇑(f.comp γ')
γγ'1 : γ 1 = γ' 1
pγ : Path a₀ (γ 1) := { toContinuousMap := γ, source' := γ_0, target' := ⋯ }
pγ' : Path a₀ (γ 1) := { toContinuousMap := γ', source' := γ'_0, target' := ⋯ }
⊢ ↑(cov.monodromy (Path.Homotopic.Quotient.mk (pγ.map ⋯)) ⟨e₀, he⟩) =
    ↑(cov.monodromy (Path.Homotopic.Quotient.mk (pγ'.map ⋯)) ⟨e₀, he⟩)

Tactic:
← Subtype.ext_iff

Goal state after:
case refine_2
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
cov : IsCoveringMap p
inst✝¹ : PathConnectedSpace A
inst✝ : LocallyPathConnectedSpace A
f : C(A, X)
a₀ : A
e₀ : E
he : p e₀ = f a₀
le : (FundamentalGroup.map f a₀).range ≤ (mapOfEq { toFun := p, continuous_toFun := ⋯ } he).range
γ γ' : C(↑I, A)
Γ Γ' : C(↑I, E)
γ_0 : γ 0 = a₀
γ'_0 : γ' 0 = a₀
Γ_0 : Γ 0 = e₀
Γ'_0 : Γ' 0 = e₀
Γ_lifts : p ∘ ⇑Γ = ⇑(f.comp γ)
Γ'_lifts : p ∘ ⇑Γ' = ⇑(f.comp γ')
γγ'1 : γ 1 = γ' 1
pγ : Path a₀ (γ 1) := { toContinuousMap := γ, source' := γ_0, target' := ⋯ }
pγ' : Path a₀ (γ 1) := { toContinuousMap := γ', source' := γ'_0, target' := ⋯ }
⊢ cov.monodromy (Path.Homotopic.Quotient.mk (pγ.map ⋯)) ⟨e₀, he⟩ =
    cov.monodromy (Path.Homotopic.Quotient.mk (pγ'.map ⋯)) ⟨e₀, he⟩
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: case h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
ε : ℝ
hε : ε > 0
B : ℝ
hB : ∀ y ∈ norm ∘ f '' verticalClosedStrip 0 1, y ≤ B
⊢ ∀ a ∈ verticalClosedStrip 0 1,
    ‖((↑ε + ↑(sSupNormIm f 0)) ^ (a - 1) * (↑ε + ↑(sSupNormIm f 1)) ^ (-a)) • f a‖ ≤
      max 1 ((ε + sSupNormIm f 0) ^ (-1)) * max 1 ((ε + sSupNormIm f 1) ^ (-1)) * B

Tactic:
intro z

Goal state after:
case h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
ε : ℝ
hε : ε > 0
B : ℝ
hB : ∀ y ∈ norm ∘ f '' verticalClosedStrip 0 1, y ≤ B
z : ℂ
⊢ z ∈ verticalClosedStrip 0 1 →
    ‖((↑ε + ↑(sSupNormIm f 0)) ^ (z - 1) * (↑ε + ↑(sSupNormIm f 1)) ^ (-z)) • f z‖ ≤
      max 1 ((ε + sSupNormIm f 0) ^ (-1)) * max 1 ((ε + sSupNormIm f 1) ^ (-1)) * B