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

Goal state before:
pp: D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : StrictMonoOn (deriv f) (interior D)
x y z : ℝ
hx : x ∈ D
hz : z ∈ D
hxy : x < y
hyz : y < z
⊢ (f y - f x) / (y - x) < (f z - f y) / (z - y)

Tactic:
have hxzD : Icc x z ⊆ D := hD.ordConnected.out hx hz

Goal state after:
D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : StrictMonoOn (deriv f) (interior D)
x y z : ℝ
hx : x ∈ D
hz : z ∈ D
hxy : x < y
hyz : y < z
hxzD : Icc x z ⊆ D
⊢ (f y - f x) / (y - x) < (f z - f y) / (z - y)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: x : ℝ
h : 0 ≤ x
⊢ x / √(1 + x ^ 2) = √(1 - (√(1 + x ^ 2))⁻¹ ^ 2)

Tactic:
← sqrt_inv

Goal state after:
x : ℝ
h : 0 ≤ x
⊢ x / √(1 + x ^ 2) = √(1 - √(1 + x ^ 2)⁻¹ ^ 2)
Module: Mathlib.Analysis.Complex.Angle

Goal state before:
pp: a x y : ℂ
⊢ angle y (x * a) = angle (x * a) y

Tactic:
angle_comm

Goal state after:
a x y : ℂ
⊢ angle (x * a) y = angle (x * a) y
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
r : ℝ
hr : 0 < r
⊢ angle y x = angle x y

Tactic:
angle_comm

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
r : ℝ
hr : 0 < r
⊢ angle x y = angle x y
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
r : ℝ
hr : r < 0
⊢ angle y (-x) = angle (-x) y

Tactic:
angle_comm

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
r : ℝ
hr : r < 0
⊢ angle (-x) y = angle (-x) y
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: case inr.inl
S : Set ℝ
f : ℝ → ℝ
x : ℝ
hf : ConvexOn ℝ S f
hx : x ∈ interior S
hf_ld : derivWithin f (Iio x) x ≤ 0
hf_rd : 0 ≤ derivWithin f (Ioi x) x
y : ℝ
hy : y ∈ S
h_eq : x = y
⊢ y ∈ {x_1 | (fun x_2 ↦ f x ≤ f x_2) x_1}

pp: case inr.inr
S : Set ℝ
f : ℝ → ℝ
x : ℝ
hf : ConvexOn ℝ S f
hx : x ∈ interior S
hf_ld : derivWithin f (Iio x) x ≤ 0
hf_rd : 0 ≤ derivWithin f (Ioi x) x
y : ℝ
hy : y ∈ S
hyx : y < x
⊢ y ∈ {x_1 | (fun x_2 ↦ f x ≤ f x_2) x_1}

Tactic:
· simp [h_eq]

Goal state after:
case inr.inr
S : Set ℝ
f : ℝ → ℝ
x : ℝ
hf : ConvexOn ℝ S f
hx : x ∈ interior S
hf_ld : derivWithin f (Iio x) x ≤ 0
hf_rd : 0 ≤ derivWithin f (Ioi x) x
y : ℝ
hy : y ∈ S
hyx : y < x
⊢ y ∈ {x_1 | (fun x_2 ↦ f x ≤ f x_2) x_1}
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
⊢ angle y x = angle x y

Tactic:
angle_comm

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
⊢ angle x y = angle x y
Module: Mathlib.Analysis.Complex.BranchLogRoot

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : LocallyPathConnectedSpace X
U : Set X
hUc : IsSimplyConnected U
hUo : IsOpen U
g : X → ℂ
hgc : ContinuousOn g U
hU₀ : 0 ∉ g '' U
this✝ : SimplyConnectedSpace ↑U
this : LocallyPathConnectedSpace ↑U
x₀ : { x // x ∈ U }
hx₀ : g ↑x₀ ≠ 0
⊢ ∃ f, ContinuousOn f U ∧ EqOn (cexp ∘ f) g U

Tactic:
rcases isCoveringMapOn_exp.existsUnique_continuousMap_lifts
    ⟨U.domRestrict g, continuousOn_iff_continuous_domRestrict.mp hgc⟩ (exp_log hx₀)
    (fun x ↦ ne_of_mem_of_not_mem (mem_image_of_mem g x.2) hU₀) with ⟨f, ⟨-, hf⟩, -⟩

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : LocallyPathConnectedSpace X
U : Set X
hUc : IsSimplyConnected U
hUo : IsOpen U
g : X → ℂ
hgc : ContinuousOn g U
hU₀ : 0 ∉ g '' U
this✝ : SimplyConnectedSpace ↑U
this : LocallyPathConnectedSpace ↑U
x₀ : { x // x ∈ U }
hx₀ : g ↑x₀ ≠ 0
f : C(↑U, ℂ)
hf : cexp ∘ ⇑f = ⇑{ toFun := U.domRestrict g, continuous_toFun := ⋯ }
⊢ ∃ f, ContinuousOn f U ∧ EqOn (cexp ∘ f) g U
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: case refine_1
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
ta : ↑I × A
x✝ : ta ∈ Set.Icc 0 (t Nat.zero) ×ˢ Set.univ
ht : ta.1 = 0
right✝ : ta.2 ∈ Set.univ
⊢ g ta = ((fun x ↦ g (0, x)) ∘ Prod.snd) ta

Tactic:
← ta.eta,

Goal state after:
case refine_1
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
ta : ↑I × A
x✝ : ta ∈ Set.Icc 0 (t Nat.zero) ×ˢ Set.univ
ht : ta.1 = 0
right✝ : ta.2 ∈ Set.univ
⊢ g (ta.1, ta.2) = ((fun x ↦ g (0, x)) ∘ Prod.snd) (ta.1, ta.2)
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
f g : 𝕜 → 𝕜
x : 𝕜
hf : AnalyticAt 𝕜 f x
hg : MeromorphicNFAt g x
hor : g x ≠ 0 ∨ f x ≠ 0
⊢ MeromorphicNFAt (f * g⁻¹) x

Tactic:
rcases hor with hgne | hfne

Goal state after:
case inl
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
f g : 𝕜 → 𝕜
x : 𝕜
hf : AnalyticAt 𝕜 f x
hg : MeromorphicNFAt g x
hgne : g x ≠ 0
⊢ MeromorphicNFAt (f * g⁻¹) x

case inr
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
f g : 𝕜 → 𝕜
x : 𝕜
hf : AnalyticAt 𝕜 f x
hg : MeromorphicNFAt g x
hfne : f x ≠ 0
⊢ MeromorphicNFAt (f * g⁻¹) x
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
h₂ : ¬meromorphicOrderAt f x = ⊤
g : 𝕜 → E
h₁g : AnalyticAt 𝕜 g x
h₂g : g x ≠ 0
h₃g : f =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ (meromorphicOrderAt f x).untop₀ • g z
y : 𝕜
h₁y : f y = (y - x) ^ (meromorphicOrderAt f x).untop₀ • g y
h₂y : y ∈ {x}ᶜ
⊢ g y = (((y - x) ^ (meromorphicOrderAt f x).untop₀)⁻¹ • (y - x) ^ (meromorphicOrderAt f x).untop₀) • g y

Tactic:
smul_eq_mul,

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
h₂ : ¬meromorphicOrderAt f x = ⊤
g : 𝕜 → E
h₁g : AnalyticAt 𝕜 g x
h₂g : g x ≠ 0
h₃g : f =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ (meromorphicOrderAt f x).untop₀ • g z
y : 𝕜
h₁y : f y = (y - x) ^ (meromorphicOrderAt f x).untop₀ • g y
h₂y : y ∈ {x}ᶜ
⊢ g y = (((y - x) ^ (meromorphicOrderAt f x).untop₀)⁻¹ * (y - x) ^ (meromorphicOrderAt f x).untop₀) • g y
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: case mp
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
hf : MeromorphicAt f x
⊢ meromorphicOrderAt f x ≠ ⊤ → ∀ᶠ (x : 𝕜) in 𝓝[≠] x, f x ≠ 0

pp: case mpr
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
hf : MeromorphicAt f x
⊢ (∀ᶠ (x : 𝕜) in 𝓝[≠] x, f x ≠ 0) → meromorphicOrderAt f x ≠ ⊤

Tactic:
· intro h
    obtain ⟨g, h₁g, h₂g, h₃g⟩ := (meromorphicOrderAt_ne_top_iff hf).1 h
    filter_upwards [h₃g, self_mem_nhdsWithin, eventually_nhdsWithin_of_eventually_nhds
      ((h₁g.continuousAt.ne_iff_eventually_ne continuousAt_const).mp h₂g)]
    simp_all [zpow_ne_zero, sub_ne_zero]

Goal state after:
case mpr
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
hf : MeromorphicAt f x
⊢ (∀ᶠ (x : 𝕜) in 𝓝[≠] x, f x ≠ 0) → meromorphicOrderAt f x ≠ ⊤
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
U : Set 𝕜
ι : Type u_3
s : Finset ι
f : ι → 𝕜 → 𝕜
h₁f : ∀ i ∈ s, MeromorphicNFOn (f i) U
h₂f : ∀ x ∈ U, {σ | σ ∈ s ∧ f σ x = 0}.Subsingleton
⊢ MeromorphicNFOn (fun x ↦ ∏ i ∈ s, f i x) U

Tactic:
by
  convert! meromorphicNFOn_prod h₁f h₂f
  exact (Finset.prod_apply _ s f).symm

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
U : Set 𝕜
hf : ¬MeromorphicOn f U
⊢ toMeromorphicNFOn f U = 0

Tactic:
simp [toMeromorphicNFOn, hf]

Goal state after:
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
⊢ (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ d u) x ≠ 0

Tactic:
rw [finprod_eq_prod _ h₁, Finset.prod_apply, Finset.prod_ne_zero_iff]
    intro z hz
    simp only [Pi.pow_apply, ne_eq]
    by_cases h₂ : x = z <;> simp_all [zpow_ne_zero, sub_ne_zero]

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

Goal state before:
pp: case inl
R : ℝ
z : ℂ
hz : z ∈ univ
h : z ∈ ball 0 R
⊢ MeromorphicNFAt (canonicalFactor R z) z

Tactic:
rw [meromorphicNFAt_iff_analyticAt_or]

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.Order

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
n : ℤ
⊢ meromorphicOrderAt ((fun x_1 ↦ x_1 - x) ^ n) x = ↑n

Tactic:
rw [meromorphicOrderAt_eq_int_iff (by fun_prop)]

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.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:
rw [ne_eq, norm_eq_zero]; positivity

Goal state after:
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
a : 𝕜
h₂a : (divisor f U) a = 0 a
h₃a : MeromorphicNFAt (φ⁻¹ • f) a
h₄a : a ∈ U
⊢ f a = ((∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ (divisor f U) u) a • (φ a)⁻¹) • f a

Tactic:
smul_eq_mul,

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
a : 𝕜
h₂a : (divisor f U) a = 0 a
h₃a : MeromorphicNFAt (φ⁻¹ • f) a
h₄a : a ∈ U
⊢ f a = ((∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ (divisor f U) u) a * (φ a)⁻¹) • f a
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
inst✝¹ : PathConnectedSpace A
inst✝ : LocallyPathConnectedSpace A
f : C(A, X)
a₀ : A
e₀ : E
Γ : (γ : C(↑I, A)) → γ 0 = a₀ → C(↑I, E)
Γ_0 : ∀ (γ : C(↑I, A)) (a : γ 0 = a₀), (Γ γ a) 0 = e₀
F : A → E := fun a ↦ (Γ (somePath a₀ a).toContinuousMap ⋯) 1
a : A
p : OpenPartialHomeomorph E X
hep : F a ∈ p.source
homeo : IsLocalHomeomorph ↑p
he : ↑p e₀ = f a₀
uniq :
  ∀ (γ γ' : C(↑I, A)) (Γ Γ' : C(↑I, E)),
    γ 0 = a₀ →
      γ' 0 = a₀ → Γ 0 = e₀ → Γ' 0 = e₀ → ↑p ∘ ⇑Γ = ⇑(f.comp γ) → ↑p ∘ ⇑Γ' = ⇑(f.comp γ') → γ 1 = γ' 1 → Γ 1 = Γ' 1
Γ_lifts : ∀ (γ : C(↑I, A)) (a : γ 0 = a₀), ↑p ∘ ⇑(Γ γ a) = ⇑(f.comp γ)
this : ∀ (a : A), ↑p (F a) = f a
⊢ ↑p (F a) ∈ p.target

Tactic:
exact p.map_source hep

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

Goal state before:
pp: case neg.refine_2
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
hf₂ : MeromorphicAt f₂ x
hf₁ : MeromorphicAt f₁ x
n₂ : ℤ
hn₂ : ↑n₂ = meromorphicOrderAt f₂ x
n₁ : ℤ
hn₁ : ↑n₁ = meromorphicOrderAt f₁ x
meromorphicOrderAt_add_eq_left_of_lt : MeromorphicAt f₂ x → ↑n₁ < ↑n₂ → meromorphicOrderAt (f₁ + f₂) x = ↑n₁
h : ↑n₁ < ↑n₂
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
⊢ (g₁ + (fun x_1 ↦ x_1 - x) ^ (n₂ - n₁) • g₂) x ≠ 0

Tactic:
simpa [zero_zpow _ <| sub_ne_zero.mpr (WithTop.coe_lt_coe.1 h).ne']

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
a : ℂ
ha : a ∈ (-divisor f (ball 0 R)).supportᶜ
h₂a : a ∈ closedBall 0 R
x : ℂ
hx : x ∈ η₀.toFinset
⊢ canonicalFactor R x a ^ (-(divisor f (ball 0 R)) x) • canonicalFactor R x a ^ (divisor f (ball 0 R)) x = 1

Tactic:
smul_eq_mul,

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
x : ℂ
hx : x ∈ η₀.toFinset
⊢ canonicalFactor R x a ^ (-(divisor f (ball 0 R)) x) * canonicalFactor R x a ^ (divisor f (ball 0 R)) x = 1
Module: Mathlib.Analysis.Complex.Conformal

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℂ E
map : ℂ →L[ℂ] E
nonzero : map ≠ 0
minor₁ : ‖map 1‖ ≠ 0
x : ℂ
⊢ x = x • 1

Tactic:
smul_eq_mul,

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℂ E
map : ℂ →L[ℂ] E
nonzero : map ≠ 0
minor₁ : ‖map 1‖ ≠ 0
x : ℂ
⊢ x = x * 1
Module: Mathlib.Analysis.Complex.Conformal

Goal state before:
pp: case refine_1
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℂ E
map : ℂ →L[ℂ] E
nonzero : map ≠ 0
minor₁ : ‖map 1‖ ≠ 0
x : ℂ
this : x = x • 1
⊢ ‖‖map 1‖⁻¹ • (↑ℝ ↑map) x‖ = ‖x‖

Tactic:
nth_rw 1 [this]

Goal state after:
case refine_1
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℂ E
map : ℂ →L[ℂ] E
nonzero : map ≠ 0
minor₁ : ‖map 1‖ ≠ 0
x : ℂ
this : x = x • 1
⊢ ‖‖map 1‖⁻¹ • (↑ℝ ↑map) (x • 1)‖ = ‖x‖
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: case inl
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
g : 𝕜 → 𝕜
hf : MeromorphicAt f (g x)
hg : AnalyticAt 𝕜 g x
hg_nc : ¬EventuallyConst g (𝓝 x)
hf' : ∀ᶠ (z : 𝕜) in 𝓝[≠] g x, f z = 0
⊢ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, (f ∘ g) z = 0

Tactic:
rw [Function.comp_def, ← eventually_map (P := (f · = 0))]

Goal state after:
case inl
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
g : 𝕜 → 𝕜
hf : MeromorphicAt f (g x)
hg : AnalyticAt 𝕜 g x
hg_nc : ¬EventuallyConst g (𝓝 x)
hf' : ∀ᶠ (z : 𝕜) in 𝓝[≠] g x, f z = 0
⊢ ∀ᶠ (b : 𝕜) in map g (𝓝[≠] x), f b = 0
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: case mk
E : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace X
p : E → X
cov : IsCoveringMap p
x y z : X
γ✝ : Path.Homotopic.Quotient x y
γ' : Path.Homotopic.Quotient y z
e : ↑(p ⁻¹' {x})
γ : Path x y
⊢ cov.monodromy (Path.Homotopic.Quotient.trans (Quot.mk (⇑(Path.Homotopic.setoid x y)) γ) γ') e =
    cov.monodromy γ' (cov.monodromy (Quot.mk (⇑(Path.Homotopic.setoid x y)) γ) e)

Tactic:
obtain ⟨γ'⟩ := γ'

Goal state after:
case mk.mk
E : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace X
p : E → X
cov : IsCoveringMap p
x y z : X
γ✝ : Path.Homotopic.Quotient x y
γ'✝ : Path.Homotopic.Quotient y z
e : ↑(p ⁻¹' {x})
γ : Path x y
γ' : Path y z
⊢ cov.monodromy
      (Path.Homotopic.Quotient.trans (Quot.mk (⇑(Path.Homotopic.setoid x y)) γ)
        (Quot.mk (⇑(Path.Homotopic.setoid y z)) γ'))
      e =
    cov.monodromy (Quot.mk (⇑(Path.Homotopic.setoid y z)) γ')
      (cov.monodromy (Quot.mk (⇑(Path.Homotopic.setoid x y)) γ) e)
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
ε : ℝ
hε : ε > 0
x : ℝ
⊢ 0 < ε + sSupNormIm f x

Tactic:
linarith [sSupNormIm_nonneg f x]

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
⊢ (support fun i ↦ ↑((divisor f B₀R) i) * log ‖canonicalFactor R i w‖) ⊆ ↑t₂

Tactic:
intro _ _; simp_all [S₀R, B₀R]

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
⊢ (support fun i ↦ ↑((divisor f S₀R) i) * log ‖w - i‖) ⊆ ↑t₁

Tactic:
intro _ _; simp_all [S₀R, B₀R]

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

Goal state before:

Tactic:
_ = fun w ↦ (∫ y in z.im..w.im, f (w.re + y * I)) - (∫ _ in z.im..w.im, f z) := by simp
      _ =ᶠ[𝓝 z] fun w ↦ ∫ y in z.im..w.im, f (w.re + y * I) - f z := ?_
      _ =o[𝓝 z] fun w ↦ w - z := this

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
c : ℂ
r : ℝ
f : ℂ → E
inst✝ : CompleteSpace E
f_cont : ContinuousOn f (ball c r)
z : ℂ
hz : z ∈ ball c r
this : (fun w ↦ ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I) - f z) =o[𝓝 z] fun w ↦ w - z
⊢ (fun w ↦ (∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I)) - ∫ (x : ℝ) in z.im..w.im, f z) =ᶠ[𝓝 z] fun w ↦
    ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I) - f z
Module: Mathlib.Analysis.Complex.PhragmenLindelof

Goal state before:
pp: case refine_1
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b C : ℝ
f : ℂ → E
z : ℂ
hfd : DiffContOnCl ℂ f (re ⁻¹' Ioo a b)
hB : ∃ c < π / (b - a), ∃ B, f =O[comap (abs ∘ im) atTop ⊓ 𝓟 (re ⁻¹' Ioo a b)] fun z ↦ expR (B * expR (c * |z.im|))
hle_a : ∀ (z : ℂ), z.re = a → ‖f z‖ ≤ C
hle_b : ∀ (z : ℂ), z.re = b → ‖f z‖ ≤ C
hza : a ≤ z.re
hzb : z.re ≤ b
H : MapsTo (fun x ↦ x * -I) (im ⁻¹' Ioo a b) (re ⁻¹' Ioo a b)
⊢ ∃ c < π / (b - a),
    ∃ B, (fun z ↦ f (z * -I)) =O[comap (abs ∘ re) atTop ⊓ 𝓟 (im ⁻¹' Ioo a b)] fun z ↦ expR (B * expR (c * |z.re|))

Tactic:
rcases hB with ⟨c, hc, B, hO⟩
    refine ⟨c, hc, B, ?_⟩
    have : Tendsto (· * -I) (comap (|re ·|) atTop ⊓ 𝓟 (im ⁻¹' Ioo a b))
        (comap (|im ·|) atTop ⊓ 𝓟 (re ⁻¹' Ioo a b)) := by
      refine (tendsto_comap_iff.2 ?_).inf H.tendsto
      simpa [Function.comp_def] using tendsto_comap
    simpa [Function.comp_def] using hO.comp_tendsto this

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalAverage

Goal state before:
pp: f : ℝ → ℝ
a b : ℝ
⊢ (b - a)⁻¹ • ∫ (x : ℝ) in a..b, f x = (∫ (x : ℝ) in a..b, f x) / (b - a)

Tactic:
smul_eq_mul,

Goal state after:
f : ℝ → ℝ
a b : ℝ
⊢ (b - a)⁻¹ * ∫ (x : ℝ) in a..b, f x = (∫ (x : ℝ) in a..b, f x) / (b - a)
Module: Mathlib.Analysis.Complex.HasPrimitives

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
inst✝ : CompleteSpace E
h₁ : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
h₂ : IsConservativeOn f univ
⊢ IsExactOn f univ

Tactic:
by
  use (wedgeIntegral 0 · f)
  intro z _
  have h₃ : IsConservativeOn f (ball 0 (‖z‖ + 1)) := h₂.mono (subset_univ _)
  exact h₃.hasDerivAt_wedgeIntegral (by fun_prop) (by aesop)

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

Goal state before:
pp: case inr.refine_1
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f (Ioi 0 ×ℂ Ioi 0)
hre : ∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C
him : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C
ζ : ℂ
hζ : ζ.im ∈ Icc 0 (π / 2)
hz_re : 0 ≤ (cexp ζ).re
hz_im : 0 ≤ (cexp ζ).im
hzne : cexp ζ ≠ 0
H : MapsTo cexp (im ⁻¹' Ioo 0 (π / 2)) (Ioi 0 ×ℂ Ioi 0)
c : ℝ
hc : c < 2
B : ℝ
hO : f =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Ioi 0)] fun z ↦ expR (B * ‖z‖ ^ c)
⊢ (f ∘ cexp) =O[(comap re atBot ⊔ comap re atTop) ⊓ 𝓟 (im ⁻¹' Ioo 0 (π / 2))] fun z ↦ expR (max B 0 * expR (c * |z.re|))

Tactic:
inf_sup_right

Goal state after:
case inr.refine_1
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f (Ioi 0 ×ℂ Ioi 0)
hre : ∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C
him : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C
ζ : ℂ
hζ : ζ.im ∈ Icc 0 (π / 2)
hz_re : 0 ≤ (cexp ζ).re
hz_im : 0 ≤ (cexp ζ).im
hzne : cexp ζ ≠ 0
H : MapsTo cexp (im ⁻¹' Ioo 0 (π / 2)) (Ioi 0 ×ℂ Ioi 0)
c : ℝ
hc : c < 2
B : ℝ
hO : f =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Ioi 0)] fun z ↦ expR (B * ‖z‖ ^ c)
⊢ (f ∘ cexp) =O[comap re atBot ⊓ 𝓟 (im ⁻¹' Ioo 0 (π / 2)) ⊔ comap re atTop ⊓ 𝓟 (im ⁻¹' Ioo 0 (π / 2))] fun z ↦
    expR (max B 0 * expR (c * |z.re|))
Module: Mathlib.Analysis.Complex.Harmonic.Analytic

Goal state before:
pp: f : ℂ → ℝ
x : ℂ
hf : HarmonicAt f x
⊢ AnalyticAt ℝ f x

Tactic:
by
  obtain ⟨ε, h₁ε, h₂ε⟩ := isOpen_iff.1 (isOpen_setOfPred_harmonicAt (f := f)) x hf
  obtain ⟨F, h₁F, h₂F⟩ := InnerProductSpace.HarmonicOnNhd.exists_analyticOnNhd_ball_re_eq
    (fun _ hy ↦ h₂ε hy)
  rw [analyticAt_congr (Filter.eventually_of_mem (ball_mem_nhds x h₁ε) (fun y hy ↦ h₂F.symm hy))]
  exact (reCLM.analyticAt (F x)).comp (h₁F x (mem_ball_self h₁ε)).restrictScalars

Goal state after:
Module: Mathlib.MeasureTheory.Integral.CircleAverage

Goal state before:
pp: c : ℂ
R : ℝ
f : ℂ → ℝ
a : ℝ
hf : CircleIntegrable f c R
h₂f : ∀ x ∈ sphere c |R|, f x ≤ a
⊢ (2 * π)⁻¹ • ∫ (θ : ℝ) in 0..2 * π, f (circleMap c R θ) ≤ (2 * π)⁻¹ • ∫ (θ : ℝ) in 0..2 * π, a

Tactic:
smul_eq_mul,

Goal state after:
c : ℂ
R : ℝ
f : ℂ → ℝ
a : ℝ
hf : CircleIntegrable f c R
h₂f : ∀ x ∈ sphere c |R|, f x ≤ a
⊢ (2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, f (circleMap c R θ) ≤ (2 * π)⁻¹ • ∫ (θ : ℝ) in 0..2 * π, a