mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Module.Ball.Action

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
inst✝³ : NormedField 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CharZero 𝕜
r : ℝ
hr : r ≠ 0
x : ↑(sphere 0 r)
this : IsAddTorsionFree E
h : x = -x
| ↑x

Tactic:
h

Goal state after:
𝕜 : Type u_1
E : Type u_3
inst✝³ : NormedField 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CharZero 𝕜
r : ℝ
hr : r ≠ 0
x : ↑(sphere 0 r)
this : IsAddTorsionFree E
h : x = -x
| ↑(-x)
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 → 𝔻
hgc : ContinuousOn g U
hU₀ : 0 ∉ g '' U
n : ℕ+
⊢ ∃ f, ContinuousOn f U ∧ ∀ (x : X), f x ^ n = g x

Tactic:
by
  rcases exists_continuousOn_pow_eq hUc hUo
    (continuous_coe.comp_continuousOn hgc)
    (by simpa using hU₀) n.ne_zero with ⟨f, hfc, hf⟩
  suffices ∀ x, ‖f x‖ < 1 by
    lift f to X → 𝔻 using this
    refine ⟨f, isEmbedding_coe.continuousOn_iff.mpr hfc, fun x ↦ ?_⟩
    simpa only [← coe_pow, Function.comp_apply, coe_inj] using hf x
  intro x
  rw [← pow_lt_one_iff_of_nonneg (norm_nonneg _) n.ne_zero, ← norm_pow, hf]
  exact (g x).norm_lt_one

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
f₁ f₂ : 𝕜 → E
h₁ : f₁ =ᶠ[codiscreteWithin U] f₂
h₂ : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
hf₁ : MeromorphicOn f₁ U
⊢ divisor f₁ U = divisor f₂ U

Tactic:
ext x
    by_cases hx : x ∈ U
    · simp only [hf₁, hx, divisor_apply, hf₁.congr_codiscreteWithin h₁ h₂]
      congr 1
      apply meromorphicOrderAt_congr
      simp_rw [EventuallyEq, Filter.Eventually, mem_codiscreteWithin,
        disjoint_principal_right] at h₁
      have : U ∈ 𝓝[≠] x := by
        apply mem_nhdsWithin.mpr
        use U, h₂, hx, Set.inter_subset_left
      filter_upwards [this, h₁ x hx] with a h₁a h₂a
      simp only [Set.mem_compl_iff, Set.mem_sdiff, Set.mem_ofPred_eq, not_and] at h₂a
      tauto
    · simp [hx]

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

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
f : 𝕜 → E
x : 𝕜
g : 𝕜 → E
hfg : f =ᶠ[𝓝 x] g
⊢ MeromorphicNFAt f x → MeromorphicNFAt g x

pp: case mpr
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
g : 𝕜 → E
hfg : f =ᶠ[𝓝 x] g
⊢ MeromorphicNFAt g x → MeromorphicNFAt f x

Tactic:
· rintro (h | ⟨n, h, h₁h, h₂h, h₃h⟩)
    · exact .inl (hfg.symm.trans h)
    · exact .inr ⟨n, h, h₁h, h₂h, hfg.symm.trans h₃h⟩

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
f : 𝕜 → E
x : 𝕜
g : 𝕜 → E
hfg : f =ᶠ[𝓝 x] g
⊢ MeromorphicNFAt g x → MeromorphicNFAt f 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
g : 𝕜 → 𝕜
x : 𝕜
h₁g : AnalyticAt 𝕜 g x
h₂g : g x ≠ 0
hprod : MeromorphicNFAt (g • f) x
⊢ MeromorphicNFAt f x

Tactic:
have : f =ᶠ[𝓝 x] g⁻¹ • g • f := by
      filter_upwards [h₁g.continuousAt.preimage_mem_nhds (compl_singleton_mem_nhds_iff.mpr h₂g)]
      intro y hy
      rw [Set.preimage_compl, Set.mem_compl_iff, Set.mem_preimage,
        Set.mem_singleton_iff] at hy
      simp [hy]

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
g : 𝕜 → 𝕜
x : 𝕜
h₁g : AnalyticAt 𝕜 g x
h₂g : g x ≠ 0
hprod : MeromorphicNFAt (g • f) x
this : f =ᶠ[𝓝 x] g⁻¹ • g • f
⊢ MeromorphicNFAt f x
Module: Mathlib.Analysis.Meromorphic.TrailingCoefficient

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
h : MeromorphicAt f x
h₂ : meromorphicOrderAt f x = ⊤
⊢ Tendsto ((fun x_1 ↦ x_1 - x) ^ (-(meromorphicOrderAt f x).untop₀) • f) (𝓝[≠] x) (𝓝 (meromorphicTrailingCoeffAt f x))

Tactic:
simp_all only [WithTop.untop₀_top, neg_zero, zpow_zero, one_smul,
      meromorphicTrailingCoeffAt_of_order_eq_top]
    apply Tendsto.congr' (f₁ := 0)
    · filter_upwards [meromorphicOrderAt_eq_top_iff.1 h₂] with y hy
      simp_all
    · apply Tendsto.congr' (f₁ := 0) (by rfl) continuousWithinAt_const.tendsto

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
g : 𝕜 → 𝕜
x : 𝕜
hg : AnalyticAt 𝕜 g x
n : ℤ
q : 𝕜 → E
hq_an : AnalyticAt 𝕜 q (g x)
hq_ne : q (g x) ≠ 0
hf : f =ᶠ[𝓝 (g x)] (fun x_1 ↦ x_1 - g x) ^ n • q
hord : ∀ᶠ (z : 𝕜) in 𝓝 x, g z - g x = 0
h : f (g x) = 0
⊢ f ∘ g =ᶠ[𝓝 x] 0

Tactic:
filter_upwards [hord, hg.continuousAt.preimage_mem_nhds (hf.filter_mono (by simp))]
        using by simp_all [sub_eq_zero]

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

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
g : 𝕜 → 𝕜
x : 𝕜
hg : AnalyticAt 𝕜 g x
n : ℤ
q : 𝕜 → E
hq_an : AnalyticAt 𝕜 q (g x)
hq_ne : q (g x) ≠ 0
hf : f =ᶠ[𝓝 (g x)] (fun x_1 ↦ x_1 - g x) ^ n • q
hord : ∀ᶠ (z : 𝕜) in 𝓝 x, g z - g x = 0
h : ¬f (g x) = 0
⊢ f ∘ g =ᶠ[𝓝 x] (fun x_1 ↦ x_1 - x) ^ 0 • fun x_1 ↦ f (g x)

Tactic:
filter_upwards [hord, hg.continuousAt.preimage_mem_nhds (hf.filter_mono (by simp))]
        using by simp_all [sub_eq_zero]

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

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
f : 𝕜 → E
U : Set 𝕜
g : 𝕜 → 𝕜
h₁g : AnalyticOnNhd 𝕜 g U
h₂g : ∀ u ∈ U, g u ≠ 0
h : MeromorphicNFOn (g • f) U
⊢ MeromorphicNFOn f U

Tactic:
z

Goal state after:
case mp
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
g : 𝕜 → 𝕜
h₁g : AnalyticOnNhd 𝕜 g U
h₂g : ∀ u ∈ U, g u ≠ 0
h : MeromorphicNFOn (g • f) U
z : 𝕜
⊢ z ∈ U → MeromorphicNFAt f z
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: case mpr
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
U : Set 𝕜
g : 𝕜 → 𝕜
h₁g : AnalyticOnNhd 𝕜 g U
h₂g : ∀ u ∈ U, g u ≠ 0
h : MeromorphicNFOn f U
⊢ MeromorphicNFOn (g • f) U

Tactic:
z

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
f : 𝕜 → E
U : Set 𝕜
g : 𝕜 → 𝕜
h₁g : AnalyticOnNhd 𝕜 g U
h₂g : ∀ u ∈ U, g u ≠ 0
h : MeromorphicNFOn f U
z : 𝕜
⊢ z ∈ U → MeromorphicNFAt (g • f) z
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 : MeromorphicAt f x
h' : ContinuousAt f x
n : ℤ
ho : meromorphicOrderAt f x = ↑n
g : 𝕜 → E
g_an : AnalyticAt 𝕜 g x
gx : g x ≠ 0
hg : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ n • g z
this : 0 ≤ meromorphicOrderAt f x
⊢ 0 ≤ n

Tactic:
simpa [ho] using this

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]
    right
    refine ⟨meromorphic_canonicalFactor R z z, ?_, by simp⟩
    simpa [meromorphicOrderAt_canonicalFactor h] using WithTop.coe_lt_zero.mpr (by lia : -1 < 0)

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✝ : HasFiniteSupport d
y : 𝕜
a✝ : y ∈ Finite.toFinset h✝
h : x = y
⊢ ‖(x - y) ^ update d x 0 y‖ ≠ 0

Tactic:
h

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.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

pp: case refine_3
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
⊢ ∀ (a : ↑I), Continuous[_, inst✝¹] fun x ↦ (Γ (x, a).2) (x, a).1

Tactic:
· simp_rw [Γ_0]; exact continuous_const

Goal state after:
case refine_3
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
⊢ ∀ (a : ↑I), Continuous[_, inst✝¹] fun x ↦ (Γ (x, a).2) (x, a).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
a : ℂ
ha : a ∈ (-divisor f (ball 0 R)).supportᶜ
h₂a : a ∈ closedBall 0 R
⊢ f a =
    ((∏ i ∈ η₀.toFinset, canonicalFactor R i ^ (-(divisor f (ball 0 R)) i)) •
        (∏ i ∈ η₀.toFinset, canonicalFactor R i ^ (divisor f (ball 0 R)) i) • f)
      a

Tactic:
simp only [Pi.smul_apply', Finset.prod_apply, Pi.pow_apply]

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
⊢ f a =
    (∏ x ∈ η₀.toFinset, canonicalFactor R x a ^ (-(divisor f (ball 0 R)) x)) •
      (∏ x ∈ η₀.toFinset, canonicalFactor R x a ^ (divisor f (ball 0 R)) x) • f a
Module: Mathlib.Analysis.Complex.Isometry

Goal state before:
pp: a : Circle
h : rotation a = conjLIE
h1 : (rotation a) 1 = (starRingEnd ℂ) 1
⊢ False

Tactic:
have hI : rotation a I = conj I := LinearIsometryEquiv.congr_fun h I

Goal state after:
a : Circle
h : rotation a = conjLIE
h1 : (rotation a) 1 = (starRingEnd ℂ) 1
hI : (rotation a) I = (starRingEnd ℂ) I
⊢ False
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: case succ.refine_3
E : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace X
p : E → X
cov : IsCoveringMap p
γ : C(↑I, X)
e : E
γ_0 : γ 0 = p e
U : X → Set X := fun x ↦ ⋯.choose
mem_base : ∀ (x : X), x ∈ ⋯.choose
U_open : ∀ (x : X), IsOpen[inst✝] ⋯.choose
h✝ : ∀ (x : X), IsOpen[inst✝¹] (p ⁻¹' ⋯.choose)
H : (x : X) → ↑(p ⁻¹' ⋯.choose) ≃ₜ ↑⋯.choose × ↑(p ⁻¹' {x})
x✝ : ∀ (x : X) (x_1 : ↑(p ⁻¹' ⋯.choose)), ↑((H x) x_1).1 = p ↑x_1
t : ℕ → ↑I
t_0 : t 0 = 0
t_mono : Monotone t
t_sub✝ : ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ ⇑γ ⁻¹' ⋯.choose
n_max : ℕ
h_max : ∀ m ≥ n_max, t m = 1
n : ℕ
Γ : ↑I → E
cont : ContinuousOn Γ (Set.Icc 0 (t n))
eqOn : Set.EqOn (p ∘ Γ) (⇑γ) (Set.Icc 0 (t n))
Γ_0 : Γ 0 = e
x : X
t_sub : Set.Icc (t n) (t (n + 1)) ⊆ ⇑γ ⁻¹' ⋯.choose
pΓtn : p (Γ (t n)) = γ (t n)
this : Nonempty ↑(p ⁻¹' {x})
q : Bundle.Trivialization (↑(p ⁻¹' {x})) p := ⋯.toTrivialization
s : ↑I
⊢ s ∈ Set.Icc 0 (t (n + 1)) ∩ closure {a | t n < a} →
    ((fun x_1 ↦ (x_1, (↑q (Γ (t n))).2)) ∘ γ.toFun) s ∈ q.baseSet ×ˢ Set.univ

Tactic:
h

Goal state after:
case succ.refine_3
E : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace X
p : E → X
cov : IsCoveringMap p
γ : C(↑I, X)
e : E
γ_0 : γ 0 = p e
U : X → Set X := fun x ↦ ⋯.choose
mem_base : ∀ (x : X), x ∈ ⋯.choose
U_open : ∀ (x : X), IsOpen[inst✝] ⋯.choose
h✝ : ∀ (x : X), IsOpen[inst✝¹] (p ⁻¹' ⋯.choose)
H : (x : X) → ↑(p ⁻¹' ⋯.choose) ≃ₜ ↑⋯.choose × ↑(p ⁻¹' {x})
x✝ : ∀ (x : X) (x_1 : ↑(p ⁻¹' ⋯.choose)), ↑((H x) x_1).1 = p ↑x_1
t : ℕ → ↑I
t_0 : t 0 = 0
t_mono : Monotone t
t_sub✝ : ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ ⇑γ ⁻¹' ⋯.choose
n_max : ℕ
h_max : ∀ m ≥ n_max, t m = 1
n : ℕ
Γ : ↑I → E
cont : ContinuousOn Γ (Set.Icc 0 (t n))
eqOn : Set.EqOn (p ∘ Γ) (⇑γ) (Set.Icc 0 (t n))
Γ_0 : Γ 0 = e
x : X
t_sub : Set.Icc (t n) (t (n + 1)) ⊆ ⇑γ ⁻¹' ⋯.choose
pΓtn : p (Γ (t n)) = γ (t n)
this : Nonempty ↑(p ⁻¹' {x})
q : Bundle.Trivialization (↑(p ⁻¹' {x})) p := ⋯.toTrivialization
s : ↑I
h : s ∈ Set.Icc 0 (t (n + 1)) ∩ closure {a | t n < a}
⊢ ((fun x_1 ↦ (x_1, (↑q (Γ (t n))).2)) ∘ γ.toFun) s ∈ q.baseSet ×ˢ Set.univ
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace X
p : E → X
G : Type u_4
inst✝¹ : Group G
inst✝ : MulAction G E
hp : IsQuotientCoveringMap p G
x : X
e : ↑(p ⁻¹' {x})
γ : FundamentalGroup X x
h : (hp.fundamentalGroupToMulOpposite e) γ = 1
⊢ MulOpposite.unop ((hp.fundamentalGroupToMulOpposite e) γ) • ↑e = ↑e

Tactic:
h

Goal state after:
E : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace X
p : E → X
G : Type u_4
inst✝¹ : Group G
inst✝ : MulAction G E
hp : IsQuotientCoveringMap p G
x : X
e : ↑(p ⁻¹' {x})
γ : FundamentalGroup X x
h : (hp.fundamentalGroupToMulOpposite e) γ = 1
⊢ MulOpposite.unop 1 • ↑e = ↑e
Module: Mathlib.Analysis.Complex.HasPrimitives

Goal state before:
pp: c z : ℂ
r : ℝ
hz : z ∈ ball c r
⊢ ↑z.re + ↑c.im * I ∈ ball c r

Tactic:
suffices dist (z.re + c.im * I) c ≤ dist z c from lt_of_le_of_lt this hz

Goal state after:
c z : ℂ
r : ℝ
hz : z ∈ ball c r
⊢ dist (↑z.re + ↑c.im * I) c ≤ dist z c
Module: Mathlib.Analysis.Complex.Hadamard

Goal state before:
pp: case mp
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
l u : ℝ
hul : l < u
e : E
⊢ (∃ x, x.re = 0 ∧ f (↑l + x * (↑u - ↑l)) = e) → ∃ x, x.re = l ∧ f x = e

Tactic:
intro h
      obtain ⟨z, hz₁, hz₂⟩ := h
      use ↑l + z * (↑u - ↑l)
      simp [hz₁, hz₂]

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

Goal state before:
pp: case hbc.ha₂
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
c : ℂ
r : ℝ
f : ℂ → E
f_cont : ContinuousOn f (ball c r)
z : ℂ
hz : z ∈ ball c r
hf : IsConservativeOn f (ball c r)
w : ℂ
w_in_z_ball : w ∈ ball z (r - dist z c)
I₁ : E := ⋯
I₂ : E := ⋯
I₃ : E := ⋯
I₄ : E := ⋯
I₅ : E := ⋯
I₆ : E := ⋯
I₇ : E := ⋯
I₈ : E := ⋯
z_ball : ball z (r - dist z c) ⊆ ball c r
w_mem : w ∈ ball c r
integrableHoriz :
  ∀ (a₁ a₂ b : ℝ),
    ↑a₁ + ↑b * I ∈ ball c r → ↑a₂ + ↑b * I ∈ ball c r → IntervalIntegrable (fun x ↦ f (↑x + ↑b * I)) volume a₁ a₂
integrableVert :
  ∀ (a b₁ b₂ : ℝ),
    ↑a + ↑b₁ * I ∈ ball c r → ↑a + ↑b₂ * I ∈ ball c r → IntervalIntegrable (fun y ↦ f (↑a + ↑y * I)) volume b₁ b₂
⊢ ↑w.re + ↑c.im * I ∈ ball c r

Tactic:
exact re_add_im_mul_mem_ball w_mem

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

Goal state before:
pp: case hab.hb₁
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
c : ℂ
r : ℝ
f : ℂ → E
f_cont : ContinuousOn f (ball c r)
z : ℂ
hz : z ∈ ball c r
hf : IsConservativeOn f (ball c r)
w : ℂ
w_in_z_ball : w ∈ ball z (r - dist z c)
I₁ : E := ⋯
I₂ : E := ⋯
I₃ : E := ⋯
I₄ : E := ⋯
I₅ : E := ⋯
I₆ : E := ⋯
I₇ : E := ⋯
I₈ : E := ⋯
z_ball : ball z (r - dist z c) ⊆ ball c r
w_mem : w ∈ ball c r
integrableHoriz :
  ∀ (a₁ a₂ b : ℝ),
    ↑a₁ + ↑b * I ∈ ball c r → ↑a₂ + ↑b * I ∈ ball c r → IntervalIntegrable (fun x ↦ f (↑x + ↑b * I)) volume a₁ a₂
integrableVert :
  ∀ (a b₁ b₂ : ℝ),
    ↑a + ↑b₁ * I ∈ ball c r → ↑a + ↑b₂ * I ∈ ball c r → IntervalIntegrable (fun y ↦ f (↑a + ↑y * I)) volume b₁ b₂
hI₁ : I₁ = I₃ + I₇
⊢ ↑w.re + ↑c.im * I ∈ ball c r

Tactic:
exact re_add_im_mul_mem_ball w_mem

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Harmonic.Basic

Goal state before:
pp: case left
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
f₁ f₂ : E → F
x : E
h₁ : HarmonicAt f₁ x
h₂ : HarmonicAt f₂ x
⊢ ContDiffAt ℝ 2 (f₁ - f₂) x

Tactic:
exact h₁.1.sub h₂.1

Goal state after:
Module: Mathlib.Analysis.Complex.Harmonic.Liouville

Goal state before:
pp: case hb
f : ℂ → ℝ
h_harm : HarmonicOnNhd f univ
h_bound : IsBounded (range f)
F : ℂ → ℂ
hF_diff : AnalyticOnNhd ℂ F univ
hF_re : (fun z ↦ (F z).re) = f
⊢ IsBounded (range fun z ↦ cexp (F z))

Tactic:
isBounded_iff_forall_norm_le

Goal state after:
case hb
f : ℂ → ℝ
h_harm : HarmonicOnNhd f univ
h_bound : ∃ C, ∀ x ∈ range f, ‖x‖ ≤ C
F : ℂ → ℂ
hF_diff : AnalyticOnNhd ℂ F univ
hF_re : (fun z ↦ (F z).re) = f
⊢ ∃ C, ∀ x ∈ range fun z ↦ cexp (F z), ‖x‖ ≤ C
Module: Mathlib.Analysis.Complex.PhragmenLindelof

Goal state before:
pp: 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)
x : ℝ
hx : 0 ≤ x
⊢ ∀ (a : ℂ), a.re = x → a ∈ im ⁻¹' Ioo 0 (π / 2) → B * expR (a.re * c) ≤ max B 0 * expR (c * |a.re|)

Tactic:
z

Goal state after:
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)
x : ℝ
hx : 0 ≤ x
z : ℂ
⊢ z.re = x → z ∈ im ⁻¹' Ioo 0 (π / 2) → B * expR (z.re * c) ≤ max B 0 * expR (c * |z.re|)
Module: Mathlib.Analysis.InnerProductSpace.Harmonic.HarmonicContOnCl

Goal state before:
pp: case inl
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
h : HarmonicContOnCl f (ball x 0)
⊢ ContinuousOn f (closedBall x 0)

pp: case inr
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
r : ℝ
h : HarmonicContOnCl f (ball x r)
hr : r ≠ 0
⊢ ContinuousOn f (closedBall x r)

Tactic:
· rw [closedBall_zero]
    exact continuousOn_singleton f x

Goal state after:
case inr
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
r : ℝ
h : HarmonicContOnCl f (ball x r)
hr : r ≠ 0
⊢ ContinuousOn f (closedBall x r)
Module: Mathlib.MeasureTheory.Integral.CircleAverage

Goal state before:
pp: c : ℂ
R : ℝ
f : ℂ → ℝ
⊢ |(2 * π)⁻¹| * |∫ (θ : ℝ) in 0..2 * π, f (circleMap c R θ)| ≤ (2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, |f| (circleMap c R θ)

Tactic:
abs_of_pos (inv_pos.2 two_pi_pos),

Goal state after:
c : ℂ
R : ℝ
f : ℂ → ℝ
⊢ (2 * π)⁻¹ * |∫ (θ : ℝ) in 0..2 * π, f (circleMap c R θ)| ≤ (2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, |f| (circleMap c R θ)
Module: Mathlib.Analysis.Complex.PhragmenLindelof

Goal state before:
pp: case inr.refine_3
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)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Ioi 0)] fun z ↦ expR (B * ‖z‖ ^ c)
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)
ζ : ℂ
hζ : ζ.im = π / 2
⊢ ‖f (cexp ↑ζ.re * (↑(Real.cos (π / 2)) + sin ↑(π / 2) * I))‖ ≤ C

Tactic:
← ofReal_sin,

Goal state after:
case inr.refine_3
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)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Ioi 0)] fun z ↦ expR (B * ‖z‖ ^ c)
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)
ζ : ℂ
hζ : ζ.im = π / 2
⊢ ‖f (cexp ↑ζ.re * (↑(Real.cos (π / 2)) + ↑(Real.sin (π / 2)) * I))‖ ≤ C
Module: Mathlib.Analysis.SpecialFunctions.Log.NegMulLog

Goal state before:
pp: x : ℝ
h : ∀ (x : ℝ), deriv (deriv fun x ↦ x * log x) x = x⁻¹
⊢ -deriv (deriv fun i ↦ i * log i) x = -x⁻¹

Tactic:
h

Goal state after:
x : ℝ
h : ∀ (x : ℝ), deriv (deriv fun x ↦ x * log x) x = x⁻¹
⊢ -x⁻¹ = -x⁻¹
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: case refine_2
s t : ℝ
ht : 0 < t
h : -1 < s
⊢ IntegrableOn (fun x ↦ x ^ s) (Ioo 0 t) volume

Tactic:
rw [← intervalIntegrable_iff_integrableOn_Ioo_of_le ht.le]

Goal state after:
case refine_2
s t : ℝ
ht : 0 < t
h : -1 < s
⊢ IntervalIntegrable (fun x ↦ x ^ s) volume 0 t
Module: Mathlib.Analysis.Complex.Poisson

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
inst✝ : CompleteSpace E
hf : DiffContOnCl ℂ f (ball 0 R)
hw : w ∈ ball 0 R
hR : 0 < R
h₁w : w ≠ 0
W : ℂ := ↑R * cexp (↑w.arg * I)
q : ℝ := ‖w‖ / R
h₁q : 0 < q
h₂q : q < 1
η₀ : ∀ {x : ℂ}, ‖x‖ ≤ R → ↑q * x - W ≠ 0
⊢ DifferentiableOn ℂ (fun z ↦ ↑q / (↑q * z - W)) (closure (ball 0 R))

Tactic:
rw [closure_ball 0 hR.ne']

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
inst✝ : CompleteSpace E
hf : DiffContOnCl ℂ f (ball 0 R)
hw : w ∈ ball 0 R
hR : 0 < R
h₁w : w ≠ 0
W : ℂ := ↑R * cexp (↑w.arg * I)
q : ℝ := ‖w‖ / R
h₁q : 0 < q
h₂q : q < 1
η₀ : ∀ {x : ℂ}, ‖x‖ ≤ R → ↑q * x - W ≠ 0
⊢ DifferentiableOn ℂ (fun z ↦ ↑q / (↑q * z - W)) (closedBall 0 R)
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: case neg.inr.refine_2.inl
a b : ℝ
μ : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
r : ℂ
h : 0 ≤ r.re
h2 : ¬0 ∉ [[a, b]]
h' : 0 = r.re
c : ℝ
hc : 0 ≤ c
⊢ IntervalIntegrable (fun x ↦ ‖↑x ^ r‖) μ 0 c

Tactic:
have : IntervalIntegrable (fun _ => 1 : ℝ → ℝ) μ 0 c := intervalIntegrable_const

Goal state after:
case neg.inr.refine_2.inl
a b : ℝ
μ : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
r : ℂ
h : 0 ≤ r.re
h2 : ¬0 ∉ [[a, b]]
h' : 0 = r.re
c : ℝ
hc : 0 ≤ c
this : IntervalIntegrable (fun x ↦ 1) μ 0 c
⊢ IntervalIntegrable (fun x ↦ ‖↑x ^ r‖) μ 0 c