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

Tactic:
apply eventuallyEq_nhds_of_eventuallyEq_nhdsNE

Goal state after:
case h₁
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 = ⊤
⊢ toMeromorphicNFAt f x =ᶠ[𝓝[≠] x] 0

case h₂
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 = ⊤
⊢ toMeromorphicNFAt f x x = 0 x
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
c : 𝕜
R : ℝ
⊢ MeromorphicNFOn f (sphere c R - {c}) ↔ MeromorphicNFOn f (sphere 0 R)

Tactic:
sphere_sub_singleton,

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
c : 𝕜
R : ℝ
⊢ MeromorphicNFOn f (sphere (c - c) R) ↔ MeromorphicNFOn f (sphere 0 R)
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
this : U ∈ Filter.codiscreteWithin U
⊢ f =ᶠ[Filter.codiscreteWithin U] toMeromorphicNFOn f U

Tactic:
filter_upwards [hf.analyticAt_mem_codiscreteWithin, this] with a h₁a h₂a

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
U : Set 𝕜
hf : MeromorphicOn f U
this : U ∈ Filter.codiscreteWithin U
a : 𝕜
h₁a : AnalyticAt 𝕜 f a
h₂a : a ∈ U
⊢ f a = toMeromorphicNFOn f U a
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
U : Set 𝕜
hf : MeromorphicOn f U
hx : x ∈ U
⊢ toMeromorphicNFOn f U =ᶠ[𝓝 x] toMeromorphicNFAt f x

Tactic:
apply eventuallyEq_nhds_of_eventuallyEq_nhdsNE

Goal state after:
case h₁
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
U : Set 𝕜
hf : MeromorphicOn f U
hx : x ∈ U
⊢ toMeromorphicNFOn f U =ᶠ[𝓝[≠] x] toMeromorphicNFAt f x

case h₂
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
U : Set 𝕜
hf : MeromorphicOn f U
hx : x ∈ U
⊢ toMeromorphicNFOn f U x = toMeromorphicNFAt f x x
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
𝕜' : Type u_4
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
x : 𝕜
ι : Type u_5
s : Finset ι
f : ι → 𝕜 → 𝕜'
hf : ∀ i ∈ s, MeromorphicAt (f i) x
⊢ meromorphicOrderAt (∏ i ∈ s, f i) x = ∑ i ∈ s, meromorphicOrderAt (f i) x

Tactic:
induction s using Finset.induction with
  | empty =>
    rw [Finset.prod_empty, Finset.sum_empty, ← WithTop.coe_zero, meromorphicOrderAt_eq_int_iff]
    · exact ⟨1, analyticAt_const, by simp⟩
    · apply MeromorphicAt.const
  | insert a s ha hs =>
    rw [Finset.sum_insert ha, Finset.prod_insert ha, meromorphicOrderAt_mul
      (hf a (Finset.mem_insert_self a s))
      (MeromorphicAt.prod (fun i hi ↦ hf i (Finset.mem_insert_of_mem hi)))]
    congr
    rw [hs (fun i hi ↦ hf i (Finset.mem_insert_of_mem hi))]

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

Goal state before:
pp: case refine_1
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
⊢ 0 ≤ -divisor f U + divisor f U

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:
neg_add_cancel

Goal state after:
case refine_1
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
⊢ 0 ≤ 0

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 ≠ ⊤
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: case refine_2
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
⊢ Continuous[_, inst✝¹] fun x ↦ (Γ (0, x).2) (0, x).1

Tactic:
simp_rw [Γ_0]; exact continuous_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
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)
⊢ -divisor f (ball 0 R) + divisor f (ball 0 R) = 0

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)
⊢ MeromorphicOn (∏ᶠ (c : ℂ), canonicalFactor R c ^ (divisor f (ball 0 R)) c) (ball 0 R)

Tactic:
neg_add_cancel

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)
⊢ 0 = 0

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)
⊢ MeromorphicOn (∏ᶠ (c : ℂ), canonicalFactor R c ^ (divisor f (ball 0 R)) c) (ball 0 R)
Module: Mathlib.Analysis.Complex.Isometry

Goal state before:
pp: case refine_1
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : f 1 = 1
h0 : f I = I ∨ f I = -I
h' : f I = I
⊢ f = LinearIsometryEquiv.refl ℝ ℂ

Tactic:
apply LinearIsometryEquiv.toLinearEquiv_injective
      apply Complex.basisOneI.ext'
      intro i
      fin_cases i <;> simp [h, h']

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

Goal state before:
pp: case refine_2
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : f 1 = 1
h0 : f I = I ∨ f I = -I
h' : f I = -I
⊢ f = conjLIE

Tactic:
apply LinearIsometryEquiv.toLinearEquiv_injective
      apply Complex.basisOneI.ext'
      intro i
      fin_cases i <;> simp [h, h']

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

Goal state before:
pp: case refine_2
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
⊢ restrictScalars ℝ map = ‖map 1‖ • { toLinearMap := ‖map 1‖⁻¹ • ↑ℝ ↑map, norm_map' := ⋯ }.toContinuousLinearMap

Tactic:
· ext1
    simp [minor₁]

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 g : ℂ → E
x : ℂ
D : CanonicalDecomp f g R
hR : 0 < R
h : ‖x‖ = R
η₁ : AnalyticAt ℂ (∏ᶠ (u : ℂ), canonicalFactor R u ^ (-(divisor f (ball 0 R)) u)) x
⊢ Preperfect (closure (ball 0 R))

Tactic:
exact isOpen_ball.perfect_closure.2

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
⊢ f =ᶠ[codiscreteWithin {0}]
    ((∏ᶠ (u : ℂ), canonicalFactor 0 u ^ (-(divisor f (ball 0 R)) u)) *
        ∏ᶠ (v : ℂ), (fun x ↦ x - v) ^ (divisor f (sphere 0 R)) v) •
      fun x ↦ meromorphicTrailingCoeffAt f 0

Tactic:
apply subsingleton_singleton.mem_codiscreteWithin

Goal state after:
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
t₀ : (↑a.im • I) • ↑b.re = ↑(a.im * b.re) • I
t₁ : (↑a.im • I) • ↑b.im • I = ↑(-a.im * b.im) • 1
⊢ ℓ (↑a.re • ↑b.re) + ℓ (↑(a.im * b.re) • I) + (ℓ (↑a.re • ↑b.im • I) + ℓ (↑(-a.im * b.im) • 1)) =
    ↑a.re • ℓ ↑b.re + (↑a.im • I) • ℓ ↑b.re + (↑a.re • ℓ (↑b.im • I) + (↑a.im • I) • ℓ (↑b.im • I))

Tactic:
repeat rw [Complex.coe_smul, ℓ.map_smul]

Goal state after:
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
t₀ : (↑a.im • I) • ↑b.re = ↑(a.im * b.re) • I
t₁ : (↑a.im • I) • ↑b.im • I = ↑(-a.im * b.im) • 1
⊢ a.re • ℓ ↑b.re + (a.im * b.re) • ℓ I + (a.re • b.im • ℓ I + (-a.im * b.im) • ℓ 1) =
    ↑a.re • ℓ ↑b.re + (↑a.im • I) • ℓ ↑b.re + (↑a.re • b.im • ℓ I + (↑a.im • I) • b.im • ℓ I)
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
hw : w ∈ closedBall 0 R
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
this : MeromorphicAt h w
⊢ Preperfect (closure (ball 0 R))

Tactic:
exact isOpen_ball.perfect_closure.2

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
hw : w ∈ closedBall 0 R
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
this : MeromorphicAt h w
⊢ ((∏ᶠ (i : ℂ), meromorphicTrailingCoeffAt (canonicalFactor R i) w ^ (divisor f B₀R) i) *
        ∏ᶠ (i : ℂ), meromorphicTrailingCoeffAt (fun x ↦ x - i) w ^ (-divisor f S₀R) i) •
      meromorphicTrailingCoeffAt f w =
    ((∏ᶠ (i : ℂ), meromorphicTrailingCoeffAt (canonicalFactor R i) w ^ (divisor f B₀R) i) *
        ∏ᶠ (i : ℂ), meromorphicTrailingCoeffAt (fun x ↦ x - i) w ^ (-divisor f S₀R) i) •
      meromorphicTrailingCoeffAt
        (((∏ᶠ (u : ℂ), canonicalFactor R u ^ (-(divisor f B₀R) u)) * ∏ᶠ (v : ℂ), (fun x ↦ x - v) ^ (divisor f S₀R) v) •
          h)
        w

Tactic:
by
      rw [meromorphicTrailingCoeffAt_congr_nhdsNE
        ((D.meromorphicOn w hw).eventuallyEq_nhdsNE_of_eventuallyEq_codiscreteWithin_preperfect
        (by fun_prop) hw ?η₁ D.eventuallyEq)]
      case η₁ =>
        rw [← closure_ball _ hR.ne']
        exact isOpen_ball.perfect_closure.2

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

Goal state before:
pp: case inr
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)
r : ℤ := (meromorphicOrderAt f (g x)).untop₀
hr : meromorphicOrderAt f (g x) = ↑r
F : 𝕜 → E
hFan : AnalyticAt 𝕜 F (g x)
hFne : F (g x) ≠ 0
hFev : f =ᶠ[𝓝[≠] g x] fun z ↦ (z - g x) ^ r • F z
aux1 : f ∘ g =ᶠ[𝓝[≠] x] (fun x_1 ↦ g x_1 - g x) ^ r • F ∘ g
aux2 : meromorphicOrderAt (F ∘ g) x = 0
⊢ meromorphicOrderAt (f ∘ g) x = ↑r * ENat.map Nat.cast (analyticOrderAt (fun x_1 ↦ g x_1 - g x) x)

Tactic:
rw [meromorphicOrderAt_congr aux1, meromorphicOrderAt_smul ?_ (AnalyticAt.meromorphicAt ?_),
    aux2, add_zero, meromorphicOrderAt_zpow, AnalyticAt.meromorphicOrderAt_eq]

Goal state after:
case inr
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)
r : ℤ := (meromorphicOrderAt f (g x)).untop₀
hr : meromorphicOrderAt f (g x) = ↑r
F : 𝕜 → E
hFan : AnalyticAt 𝕜 F (g x)
hFne : F (g x) ≠ 0
hFev : f =ᶠ[𝓝[≠] g x] fun z ↦ (z - g x) ^ r • F z
aux1 : f ∘ g =ᶠ[𝓝[≠] x] (fun x_1 ↦ g x_1 - g x) ^ r • F ∘ g
aux2 : meromorphicOrderAt (F ∘ g) x = 0
⊢ AnalyticAt 𝕜 (fun x_1 ↦ g x_1 - g x) x

case inr.hf
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)
r : ℤ := (meromorphicOrderAt f (g x)).untop₀
hr : meromorphicOrderAt f (g x) = ↑r
F : 𝕜 → E
hFan : AnalyticAt 𝕜 F (g x)
hFne : F (g x) ≠ 0
hFev : f =ᶠ[𝓝[≠] g x] fun z ↦ (z - g x) ^ r • F z
aux1 : f ∘ g =ᶠ[𝓝[≠] x] (fun x_1 ↦ g x_1 - g x) ^ r • F ∘ g
aux2 : meromorphicOrderAt (F ∘ g) x = 0
⊢ MeromorphicAt (fun x_1 ↦ g x_1 - g x) x

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)
r : ℤ := (meromorphicOrderAt f (g x)).untop₀
hr : meromorphicOrderAt f (g x) = ↑r
F : 𝕜 → E
hFan : AnalyticAt 𝕜 F (g x)
hFne : F (g x) ≠ 0
hFev : f =ᶠ[𝓝[≠] g x] fun z ↦ (z - g x) ^ r • F z
aux1 : f ∘ g =ᶠ[𝓝[≠] x] (fun x_1 ↦ g x_1 - g x) ^ r • F ∘ g
aux2 : meromorphicOrderAt (F ∘ g) x = 0
⊢ MeromorphicAt ((fun x_1 ↦ g x_1 - g x) ^ r) x

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)
r : ℤ := (meromorphicOrderAt f (g x)).untop₀
hr : meromorphicOrderAt f (g x) = ↑r
F : 𝕜 → E
hFan : AnalyticAt 𝕜 F (g x)
hFne : F (g x) ≠ 0
hFev : f =ᶠ[𝓝[≠] g x] fun z ↦ (z - g x) ^ r • F z
aux1 : f ∘ g =ᶠ[𝓝[≠] x] (fun x_1 ↦ g x_1 - g x) ^ r • F ∘ g
aux2 : meromorphicOrderAt (F ∘ g) x = 0
⊢ AnalyticAt 𝕜 (F ∘ g) x
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
ε : ℝ
hε : ε > 0
B : ℝ
z : ℂ
hB : ‖f z‖ ≤ B
hset : z ∈ verticalClosedStrip 0 1
hM1_one : ε + sSupNormIm f 1 < 1
⊢ ‖↑(ε + sSupNormIm f 1) ^ (-z)‖ ≤ (ε + sSupNormIm f 1) ^ (-1)

Tactic:
simp only [norm_cpow_eq_rpow_re_of_pos (sSupNormIm_eps_pos f hε 1),
        neg_re, Real.rpow_le_rpow_of_exponent_ge (sSupNormIm_eps_pos f hε 1)
        (le_of_lt hM1_one) (neg_le_neg_iff.mpr hset.2)]

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

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
l u : ℝ
hul : l < u
⊢ sSupNormIm (scale f l u) 0 = sSupNormIm f l

Tactic:
sSupNormIm,

Goal state after:
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
l u : ℝ
hul : l < u
⊢ sSup (norm ∘ scale f l u '' re ⁻¹' {0}) = sSup (norm ∘ f '' re ⁻¹' {l})
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
l u : ℝ
hul : l < u
⊢ sSupNormIm (scale f l u) 1 = sSupNormIm f u

Tactic:
sSupNormIm,

Goal state after:
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
l u : ℝ
hul : l < u
⊢ sSup (norm ∘ scale f l u '' re ⁻¹' {1}) = sSup (norm ∘ f '' re ⁻¹' {u})
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: case h₁
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
this : ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re
⊢ sSupNormIm f 0 ≤ a

Tactic:
rw [sSupNormIm]

Goal state after:
case h₁
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
this : ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re
⊢ sSup (norm ∘ f '' re ⁻¹' {0}) ≤ a
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
this : ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re
⊢ sSupNormIm f 1 ≤ b

Tactic:
rw [sSupNormIm]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
z : ℂ
a b : ℝ
hz : z ∈ verticalClosedStrip 0 1
hd : DiffContOnCl ℂ f (verticalStrip 0 1)
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a
hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b
this : ‖interpStrip f z‖ ≤ sSupNormIm f 0 ^ (1 - z.re) * sSupNormIm f 1 ^ z.re
⊢ sSup (norm ∘ f '' re ⁻¹' {1}) ≤ b
Module: Mathlib.Analysis.Complex.PhragmenLindelof

Goal state before:
pp: case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C✝ : ℝ
f : ℂ → E
z : ℂ
C : ℝ
hC₀ : 0 < C
a b : ℝ
hza : a - b < z.im
hle_a : ∀ (z : ℂ), z.im = a - b → ‖f z‖ ≤ C
hzb : z.im < a + b
hle_b : ∀ (z : ℂ), z.im = a + b → ‖f z‖ ≤ C
hfd : DiffContOnCl ℂ f (im ⁻¹' Ioo (a - b) (a + b))
hab : a - b < a + b
hb : 0 < b
hπb : 0 < π / 2 / b
c : ℝ
hc : c < π / 2 / b
B : ℝ
hO : f =O[comap (abs ∘ re) atTop ⊓ 𝓟 (im ⁻¹' Ioo (a - b) (a + b))] fun z ↦ expR (B * expR (c * |z.re|))
d : ℝ
hcd : c < d
hd₀ : 0 < d
hb' : d * b < π / 2
aff : ℂ → ℂ := fun w ↦ ↑d * (w - ↑a * I)
g : ℝ → ℂ → ℂ := fun ε w ↦ cexp (↑ε * (cexp (aff w) + cexp (-aff w)))
ε : ℝ
ε₀ : ε < 0
δ : ℝ
δ₀ : δ < 0
hδ : ∀ ⦃w : ℂ⦄, w.im ∈ Icc (a - b) (a + b) → ‖g ε w‖ ≤ expR (δ * expR (d * |w.re|))
hg₁ : ∀ (w : ℂ), w.im = a - b ∨ w.im = a + b → ‖g ε w‖ ≤ 1
R : ℝ
hzR : |z.re| < R
hR : ∀ (w : ℂ), |w.re| = R → w.im ∈ Ioo (a - b) (a + b) → ‖g ε w • f w‖ ≤ C
hR₀ : 0 < R
hgd : Differentiable ℂ (g ε)
hd : DiffContOnCl ℂ (fun w ↦ g ε w • f w) (Ioo (-R) R ×ℂ Ioo (a - b) (a + b))
w : ℂ
him : ¬(w.im = a - b ∨ w.im = a + b)
hw : w ∈ {-R, R} ×ℂ Icc (a - b) (a + b)
hw' : w.im = a - b ∨ w.im = a + b ∨ w.im ∈ Ioo (a - b) (a + b)
⊢ ‖g ε w • f w‖ ≤ C

Tactic:
rw [← or_assoc] at hw'

Goal state after:
case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C✝ : ℝ
f : ℂ → E
z : ℂ
C : ℝ
hC₀ : 0 < C
a b : ℝ
hza : a - b < z.im
hle_a : ∀ (z : ℂ), z.im = a - b → ‖f z‖ ≤ C
hzb : z.im < a + b
hle_b : ∀ (z : ℂ), z.im = a + b → ‖f z‖ ≤ C
hfd : DiffContOnCl ℂ f (im ⁻¹' Ioo (a - b) (a + b))
hab : a - b < a + b
hb : 0 < b
hπb : 0 < π / 2 / b
c : ℝ
hc : c < π / 2 / b
B : ℝ
hO : f =O[comap (abs ∘ re) atTop ⊓ 𝓟 (im ⁻¹' Ioo (a - b) (a + b))] fun z ↦ expR (B * expR (c * |z.re|))
d : ℝ
hcd : c < d
hd₀ : 0 < d
hb' : d * b < π / 2
aff : ℂ → ℂ := fun w ↦ ↑d * (w - ↑a * I)
g : ℝ → ℂ → ℂ := fun ε w ↦ cexp (↑ε * (cexp (aff w) + cexp (-aff w)))
ε : ℝ
ε₀ : ε < 0
δ : ℝ
δ₀ : δ < 0
hδ : ∀ ⦃w : ℂ⦄, w.im ∈ Icc (a - b) (a + b) → ‖g ε w‖ ≤ expR (δ * expR (d * |w.re|))
hg₁ : ∀ (w : ℂ), w.im = a - b ∨ w.im = a + b → ‖g ε w‖ ≤ 1
R : ℝ
hzR : |z.re| < R
hR : ∀ (w : ℂ), |w.re| = R → w.im ∈ Ioo (a - b) (a + b) → ‖g ε w • f w‖ ≤ C
hR₀ : 0 < R
hgd : Differentiable ℂ (g ε)
hd : DiffContOnCl ℂ (fun w ↦ g ε w • f w) (Ioo (-R) R ×ℂ Ioo (a - b) (a + b))
w : ℂ
him : ¬(w.im = a - b ∨ w.im = a + b)
hw : w ∈ {-R, R} ×ℂ Icc (a - b) (a + b)
hw' : (w.im = a - b ∨ w.im = a + b) ∨ w.im ∈ Ioo (a - b) (a + b)
⊢ ‖g ε w • f w‖ ≤ C
Module: Mathlib.Analysis.InnerProductSpace.Harmonic.Basic

Goal state before:
pp: case right
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : FiniteDimensional ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
x : E
c : ℝ
h : HarmonicAt f x
⊢ Δ (c • f) =ᶠ[𝓝 x] 0

Tactic:
filter_upwards [laplacian_smul_nhds c h.1, h.2]

Goal state after:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : FiniteDimensional ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
x : E
c : ℝ
h : HarmonicAt f x
⊢ ∀ (a : E), Δ (c • f) a = (c • Δ f) a → Δ f a = 0 a → Δ (c • f) a = 0 a
Module: Mathlib.Analysis.InnerProductSpace.Laplacian

Goal state before:
pp: E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : FiniteDimensional ℝ E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f₁ f₂ : E → F
x : E
s : Set E
h₁ : ContDiffWithinAt ℝ 2 f₁ s x
h₂ : ContDiffWithinAt ℝ 2 f₂ s x
hs : UniqueDiffOn ℝ s
hx : x ∈ s
y : E
h₁y : ContDiffWithinAt ℝ 2 f₁ s y
h₂y : ContDiffWithinAt ℝ 2 f₂ s y
h₃y : y ∈ insert x s
⊢ (Δ[s] f₁ + f₂) y = ((Δ[s] f₁) + Δ[s] f₂) y

Tactic:
rw [s.insert_eq_of_mem hx] at h₃y

Goal state after:
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : FiniteDimensional ℝ E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f₁ f₂ : E → F
x : E
s : Set E
h₁ : ContDiffWithinAt ℝ 2 f₁ s x
h₂ : ContDiffWithinAt ℝ 2 f₂ s x
hs : UniqueDiffOn ℝ s
hx : x ∈ s
y : E
h₁y : ContDiffWithinAt ℝ 2 f₁ s y
h₂y : ContDiffWithinAt ℝ 2 f₂ s y
h₃y : y ∈ s
⊢ (Δ[s] f₁ + f₂) y = ((Δ[s] f₁) + Δ[s] f₂) y
Module: Mathlib.Analysis.InnerProductSpace.Laplacian

Goal state before:
pp: E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : FiniteDimensional ℝ E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f₁ f₂ : E → F
x : E
s : Set E
h₁ : ContDiffWithinAt ℝ 2 f₁ s x
h₂ : ContDiffWithinAt ℝ 2 f₂ s x
hs : UniqueDiffOn ℝ s
hx : x ∈ s
y : E
h₁y : ContDiffWithinAt ℝ 2 f₁ s y
h₂y : ContDiffWithinAt ℝ 2 f₂ s y
h₃y : y ∈ insert x s
⊢ (Δ[s] f₁ - f₂) y = ((Δ[s] f₁) - Δ[s] f₂) y

Tactic:
rw [s.insert_eq_of_mem hx] at h₃y

Goal state after:
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : FiniteDimensional ℝ E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f₁ f₂ : E → F
x : E
s : Set E
h₁ : ContDiffWithinAt ℝ 2 f₁ s x
h₂ : ContDiffWithinAt ℝ 2 f₂ s x
hs : UniqueDiffOn ℝ s
hx : x ∈ s
y : E
h₁y : ContDiffWithinAt ℝ 2 f₁ s y
h₂y : ContDiffWithinAt ℝ 2 f₂ s y
h₃y : y ∈ s
⊢ (Δ[s] f₁ - f₂) y = ((Δ[s] f₁) - Δ[s] f₂) y