mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.PartitionOfUnity

Goal state before:
pp: ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
f : BumpCovering ι X s
i : ι
x : X
h : (f i) x = 0
⊢ f.toPOUFun i x = 0

Tactic:
toPOUFun,

Goal state after:
ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
f : BumpCovering ι X s
i : ι
x : X
h : (f i) x = 0
⊢ (f i) x * ∏ᶠ (j : ι) (_ : WellOrderingRel j i), (1 - (f j) x) = 0
Module: Mathlib.Geometry.Manifold.PartitionOfUnity

Goal state before:
pp: ι : Type uι
E : Type uE
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
H : Type uH
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
inst✝⁴ : FiniteDimensional ℝ E
n : ℕ∞
M : Type u_1
inst✝³ : MetricSpace M
inst✝² : ChartedSpace H M
inst✝¹ : IsManifold I ∞ M
inst✝ : SigmaCompactSpace M
K U : ι → Set M
hK : ∀ (i : ι), IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (K i)
hU : ∀ (i : ι), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)
hKU : ∀ (i : ι), K i ⊆ U i
hfin : LocallyFinite K
⊢ ∃ δ, (∀ (x : M), 0 < δ x) ∧ ∀ (i : ι), ∀ x ∈ K i, closedBall x (δ x) ⊆ U i

Tactic:
by
  rcases Metric.exists_contMDiffMap_forall_closedEBall_subset I hK hU hKU hfin with ⟨δ, hδ0, hδ⟩
  refine ⟨δ, hδ0, fun i x hx => ?_⟩
  rw [← Metric.closedEBall_ofReal (hδ0 _).le]
  exact hδ i x hx

Goal state after:
Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = 0
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ∞
h_neg : 0 ≤ᵐ[μ.restrict t] -f
⊢ f ≤ᵐ[μ.restrict t] 0

Tactic:
by
    refine h_neg.mono fun x hx => ?_
    rw [Pi.neg_apply] at hx
    simpa using hx

Goal state after:
Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f g : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn g s μ
hfg_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ∞
⊢ f - g =ᵐ[μ.restrict t] 0

Tactic:
have hfg' : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, (f - g) x ∂μ) = 0 := by
    intro s hs hμs
    rw [integral_sub' (hf_int_finite s hs hμs) (hg_int_finite s hs hμs)]
    exact sub_eq_zero.mpr (hfg_zero s hs hμs)

Goal state after:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f g : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn g s μ
hfg_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ∞
hfg' : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, (f - g) x ∂μ = 0
⊢ f - g =ᵐ[μ.restrict t] 0
Module: Mathlib.MeasureTheory.Measure.Haar.Disintegration

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹³ : NontriviallyNormedField 𝕜
inst✝¹² : CompleteSpace 𝕜
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : MeasurableSpace E
inst✝⁹ : BorelSpace E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : MeasurableSpace F
inst✝⁵ : BorelSpace F
inst✝⁴ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝³ : μ.IsAddHaarMeasure
inst✝² : ν.IsAddHaarMeasure
inst✝¹ : LocallyCompactSpace E
inst✝ : LocallyCompactSpace F
s : Set F
hs : MeasurableSet s
this✝² : FiniteDimensional 𝕜 E
this✝¹ : FiniteDimensional 𝕜 F
this✝ : ProperSpace E
this : ProperSpace F
M : F × E →ₗ[𝕜] F := LinearMap.id.coprod L
M_cont : Continuous ⇑M
hM : Function.Surjective ⇑M
A : ∀ (x : F × E), M x ∈ s ↔ x ∈ ⇑M ⁻¹' s
⊢ (∀ᵐ (y : F) ∂ν, ∀ᵐ (x : E) ∂μ, y + L x ∈ s) ↔ ∀ᵐ (x : F) ∂ν, ∀ᵐ (y : E) ∂μ, (x, y) ∈ ⇑M ⁻¹' s

Tactic:
simp only [M, mem_preimage, LinearMap.coprod_apply, LinearMap.id_coe, id_eq]

Goal state after:
Module: Mathlib.Analysis.Calculus.TangentCone.Seq

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NormedDivisionRing 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul 𝕜 E
s : Set E
x y : E
r : 𝕜
hr₀ : r ≠ 0
hr : ‖r‖ < 1
hs : ∀ᶠ (n : ℕ) in atTop, x + r ^ n • y ∈ s
⊢ y ∈ tangentConeAt 𝕜 s x

Tactic:
refine mem_tangentConeAt_of_add_smul_mem
    (tendsto_nhdsWithin_iff.mpr ⟨tendsto_pow_atTop_nhds_zero_of_norm_lt_one hr, ?_⟩) hs

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NormedDivisionRing 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul 𝕜 E
s : Set E
x y : E
r : 𝕜
hr₀ : r ≠ 0
hr : ‖r‖ < 1
hs : ∀ᶠ (n : ℕ) in atTop, x + r ^ n • y ∈ s
⊢ ∀ᶠ (n : ℕ) in atTop, r ^ n ∈ {0}ᶜ
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Slope

Goal state before:
pp: f : ℝ → ℝ
a b c : ℝ
hf : IntervalIntegrable f volume a (b + c)
hab : a ≤ b
hc : 0 ≤ c
⊢ IntervalIntegrable (fun x ↦ slope f x (x + c)) volume a b

Tactic:
simp only [slope, add_sub_cancel_left, vsub_eq_sub, smul_eq_mul]
  exact hf.comp_add_right c |>.mono_set (by grind [uIcc]) |>.sub (hf.mono_set (by grind [uIcc]))
    |>.const_mul (c := c⁻¹)

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Slope

Goal state before:
pp: case inr
f : ℝ → ℝ
a b c : ℝ
hf : MonotoneOn f (uIcc a (b + c))
hab : a ≤ b
hc✝ : 0 ≤ c
hc : 0 < c
hf' : IntervalIntegrable f volume a (b + c)
fU : ∫ (x : ℝ) in b..b + c, f x ≤ c * f (b + c)
fL : c * f a ≤ ∫ (x : ℝ) in a..a + c, f x
⊢ c⁻¹ * ((∫ (x : ℝ) in b..b + c, f x) - ∫ (x : ℝ) in a..a + c, f x) ≤ f (b + c) - f a

Tactic:
fU,

Goal state after:
case inr
f : ℝ → ℝ
a b c : ℝ
hf : MonotoneOn f (uIcc a (b + c))
hab : a ≤ b
hc✝ : 0 ≤ c
hc : 0 < c
hf' : IntervalIntegrable f volume a (b + c)
fU : ∫ (x : ℝ) in b..b + c, f x ≤ c * f (b + c)
fL : c * f a ≤ ∫ (x : ℝ) in a..a + c, f x
⊢ c⁻¹ * (c * f (b + c) - ∫ (x : ℝ) in a..a + c, f x) ≤ f (b + c) - f a
Module: Mathlib.Analysis.Calculus.Taylor

Goal state before:
pp: case e'_9.e'_5
t x : ℝ
n : ℕ
⊢ ↑n + 1 = ↑(n + 1)

Tactic:
simp only [Nat.cast_add, Nat.cast_one]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun

Goal state before:
pp: case inr
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → F
d b η : ℝ
hdb✝ : d ≤ b
hf : ∀ᵐ (x : ℝ), x ∈ Ioo d b → HasDerivAt f (f' x) x
hη : 0 < η
hdb : d < b
t : Set (ℝ × ℝ) := {z | (d < z.1 ∧ z.1 < z.2 ∧ z.2 < b) ∧ dist (slope f z.1 z.2) (f' z.1) < η}
s : Set ℝ := {x | x ∈ Ioo d b ∧ HasDerivAt f (f' x) x}
u : Set (ℝ × ℝ)
hu₁ : ∀ x ∈ u, (d < x.1 ∧ x.1 < x.2 ∧ x.2 < b) ∧ dist (slope f x.1 x.2) (f' x.1) < η
hu₂ : u.Countable
hu₃ : u.PairwiseDisjoint fun z ↦ Icc z.1 z.2
hu₄ : volume (s \ ⋃ z ∈ u, Icc z.1 z.2) = 0
this : Countable ↑u
⊢ HasSum (fun z ↦ (↑z).2 - (↑z).1) (b - d)

Tactic:
have : Pairwise (Disjoint on fun (z : u) ↦ Icc z.val.1 z.val.2) :=
    fun z₁ z₂ hz₁z₂ ↦ hu₃ z₁.prop z₂.prop (Subtype.coe_ne_coe.mpr hz₁z₂)

Goal state after:
case inr
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → F
d b η : ℝ
hdb✝ : d ≤ b
hf : ∀ᵐ (x : ℝ), x ∈ Ioo d b → HasDerivAt f (f' x) x
hη : 0 < η
hdb : d < b
t : Set (ℝ × ℝ) := {z | (d < z.1 ∧ z.1 < z.2 ∧ z.2 < b) ∧ dist (slope f z.1 z.2) (f' z.1) < η}
s : Set ℝ := {x | x ∈ Ioo d b ∧ HasDerivAt f (f' x) x}
u : Set (ℝ × ℝ)
hu₁ : ∀ x ∈ u, (d < x.1 ∧ x.1 < x.2 ∧ x.2 < b) ∧ dist (slope f x.1 x.2) (f' x.1) < η
hu₂ : u.Countable
hu₃ : u.PairwiseDisjoint fun z ↦ Icc z.1 z.2
hu₄ : volume (s \ ⋃ z ∈ u, Icc z.1 z.2) = 0
this✝ : Countable ↑u
this : Pairwise (Disjoint on fun z ↦ Icc (↑z).1 (↑z).2)
⊢ HasSum (fun z ↦ (↑z).2 - (↑z).1) (b - d)
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: case h.h.h.h
s : ℝ
hs : 0 < s
M ε : ℝ
hM : 0 < M
hε : 0 < ε
H : ∀ (x y : ℝ), 0 < x → x < ε → |y| < s * x → √(x ^ 2 + y ^ 2) < M * (1 - √((1 - x) ^ 2 + y ^ 2))
z : ℂ
hzl : (1 - z).re < ε
hzr : z ∈ stolzCone s
⊢ z ∈ stolzSet M

Tactic:
rw [stolzCone, Set.mem_ofPred_eq, ← one_re, ← sub_re] at hzr

Goal state after:
case h.h.h.h
s : ℝ
hs : 0 < s
M ε : ℝ
hM : 0 < M
hε : 0 < ε
H : ∀ (x y : ℝ), 0 < x → x < ε → |y| < s * x → √(x ^ 2 + y ^ 2) < M * (1 - √((1 - x) ^ 2 + y ^ 2))
z : ℂ
hzl : (1 - z).re < ε
hzr : |z.im| < s * (1 - z).re
⊢ z ∈ stolzSet M
Module: Mathlib.Analysis.Normed.Module.Ray

Goal state before:
pp: case inr
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
u : E
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
h : SameRay ℝ (a • u) (b • u)
H : ∀ (a b : ℝ), 0 ≤ a → 0 ≤ b → SameRay ℝ (a • u) (b • u) → b ≤ a → ‖a • u - b • u‖ = |‖a • u‖ - ‖b • u‖|
hab : ¬b ≤ a
⊢ ‖a • u - b • u‖ = |‖a • u‖ - ‖b • u‖|

pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
u : E
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
h : SameRay ℝ (a • u) (b • u)
hab : b ≤ a
⊢ ‖a • u - b • u‖ = |‖a • u‖ - ‖b • u‖|

Tactic:
· rw [SameRay.sameRay_comm] at h
    rw [norm_sub_rev, abs_sub_comm]
    exact H b a hb ha h (le_of_not_ge hab)

Goal state after:
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
u : E
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
h : SameRay ℝ (a • u) (b • u)
hab : b ≤ a
⊢ ‖a • u - b • u‖ = |‖a • u‖ - ‖b • u‖|
Module: Mathlib.Analysis.Normed.Module.Ray

Goal state before:
pp: F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
x y : F
hx : x ≠ 0
hy : y ≠ 0
⊢ SameRay ℝ x y ↔ ‖x‖⁻¹ • x = ‖y‖⁻¹ • y

Tactic:
rw [inv_smul_eq_iff₀, smul_comm, eq_comm, inv_smul_eq_iff₀, sameRay_iff_norm_smul_eq] <;>
    rwa [norm_ne_zero_iff]

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

Goal state before:
pp: E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : Nontrivial E
f g : E → F
U : Set E
hU : Bornology.IsBounded U
hf : DiffContOnCl ℂ f U
hg : DiffContOnCl ℂ g U
hfg : EqOn f g (frontier U)
H : ∀ z ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U, ‖(f - g) z‖ ≤ 0
⊢ EqOn f g (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U)

Tactic:
by simpa [sub_eq_zero] using! H

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: x : ℝ
hx : 0 ≤ x
hx' : x ≤ 1
⊢ x ≤ sin (π / 2 * x)

Tactic:
by
  simpa [mul_comm x] using
    strictConcaveOn_sin_Icc.concaveOn.2 ⟨le_rfl, pi_pos.le⟩
      ⟨pi_div_two_pos.le, half_le_self pi_pos.le⟩ (sub_nonneg.2 hx') hx

Goal state after:
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: case inr
S : Set ℝ
f : ℝ → ℝ
hfc : ConcaveOn ℝ S f
hfd : DifferentiableOn ℝ f S
x : ℝ
hx : x ∈ S
y : ℝ
hy : y ∈ S
hxy : x ≤ y
hxy' : x < y
⊢ derivWithin f S y ≤ derivWithin f S x

Tactic:
exact (hfc.derivWithin_le_slope hx hy hxy' (hfd y hy)).trans
    (hfc.slope_le_derivWithin hx hy hxy' (hfd x hx))

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

Goal state before:
pp: f : ℂ → ℂ
M R : ℝ
z : ℂ
hM : 0 < M
hf : DifferentiableOn ℂ f (ball 0 R)
hf₁ : Set.MapsTo f (ball 0 R) {z | z.re ≤ M}
hR : 0 < R
hz : z ∈ ball 0 R
hf₂ : f 0 = 0
w : ℂ := f z / (2 * ↑M - f z)
hzR : ‖z‖ < R
hwR : ‖f z / (2 * ↑M - f z)‖ ≤ ‖z‖ / R
h_denom : 2 * ↑M - f z ≠ 0
⊢ 2 * M * (‖z‖ / R / (1 - ‖z‖ / R)) = 2 * M * ‖z‖ / (R - ‖z‖)

Tactic:
by field_simp

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R₁ R₂ : ℝ
f : ℂ → E
c : ℂ
hd : DifferentiableOn ℂ f (ball c R₁)
h_maps : MapsTo f (ball c R₁) (closedBall (f c) R₂)
h₀ : 0 < R₁
⊢ ‖deriv f c‖ ≤ R₂ / R₁

Tactic:
by
  rw [norm_deriv_eq_norm_fderiv]
  exact norm_fderiv_le_div_of_mapsTo_ball hd h_maps h₀

Goal state after:
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: case refine_2.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
n : ℕ
x✝ :
  ∃ N,
    a ∈ N ∧
      IsOpen[inst✝] N ∧
        ∃ g',
          ContinuousOn g' (Set.Icc 0 (t n) ×ˢ N) ∧
            p ∘ g' = ⇑f ∧ (∀ (a : A), g' (0, a) = g (0, a)) ∧ ∀ t' ≤ t n, g' (t', a) = g (t', a)
N : Set A
haN : a ∈ N
N_open : IsOpen[inst✝] N
g' : ↑I × A → E
cont_g' : ContinuousOn g' (Set.Icc 0 (t n) ×ˢ N)
g'_lifts : p ∘ g' = ⇑f
g'_0 : ∀ (a : A), g' (0, a) = g (0, a)
g'_a : ∀ t' ≤ t n, g' (t', a) = g (t', a)
e : E
h_sub : Set.Icc (t n) (t (n + 1)) ⊆ (fun x ↦ g (x, a)) ⁻¹' (q e).source
this : Set.Icc (t n) (t (n + 1)) ×ˢ {a} ⊆ ⇑f ⁻¹' (q e).target
u : Set ↑I
v : Set A
v_open : IsOpen[inst✝] v
hu : Set.Icc (t n) (t (n + 1)) ⊆ u
hav : {a} ⊆ v
huv : u ×ˢ v ⊆ ⇑f ⁻¹' (q e).target
⊢ a ∈ v ∩ (N ∩ (g' ∘ fun y ↦ (t n, y)) ⁻¹' (q e).source)

Tactic:
refine ⟨Set.singleton_subset_iff.mp hav, haN, ?_⟩
    change g' (t n, a) ∈ (q e).source; rw [g'_a _ le_rfl]
    exact h_sub ⟨le_rfl, t_mono n.le_succ⟩

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

Goal state before:
pp: 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
hf₁ : MeromorphicOn f₁ U
hf₂ : AnalyticOnNhd 𝕜 f₂ U
x : 𝕜
hx : x ∈ U
this : max (-(meromorphicOrderAt (f₁ + f₂) x).untop₀) 0 = max (-(meromorphicOrderAt f₁ x).untop₀) 0
⊢ (divisor (f₁ + f₂) U)⁻ x = (divisor f₁ U)⁻ x

Tactic:
simpa [negPart_def, hx, hf₁, hf₁.add hf₂.meromorphicOn]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
f : 𝕜 → E
inst✝¹ : CompleteSpace 𝕜
inst✝ : CharZero 𝕜
x : 𝕜
g : 𝕜 → 𝕜
hg : AnalyticAt 𝕜 g x
hg' : deriv g x ≠ 0
⊢ MeromorphicNFAt (f ∘ g) x ↔ MeromorphicNFAt f (g x)

Tactic:
simp [meromorphicNFAt_iff_analyticAt_or, analyticAt_comp_iff_of_deriv_ne_zero hg hg',
    meromorphicAt_comp_iff_of_deriv_ne_zero hg hg',
    meromorphicOrderAt_comp_of_deriv_ne_zero hg hg']

Goal state after:
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
x : 𝕜
f₁ f₂ : 𝕜 → E
h : f₁ =ᶠ[𝓝[≠] x] f₂
⊢ meromorphicTrailingCoeffAt f₁ x = meromorphicTrailingCoeffAt f₂ x

Tactic:
by_cases h₁ : ¬MeromorphicAt f₁ x
  · simp [h₁, (MeromorphicAt.meromorphicAt_congr h).not.1 h₁]
  rw [not_not] at h₁
  by_cases h₂ : meromorphicOrderAt f₁ x = ⊤
  · simp_all [meromorphicOrderAt_congr h]
  obtain ⟨g, h₁g, h₂g, h₃g⟩ := (meromorphicOrderAt_ne_top_iff h₁).1 h₂
  rw [h₁g.meromorphicTrailingCoeffAt_of_ne_zero_of_eq_nhdsNE h₂g h₃g,
    h₁g.meromorphicTrailingCoeffAt_of_ne_zero_of_eq_nhdsNE h₂g (h.symm.trans h₃g)]

Goal state after:
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
hf : MeromorphicAt f x
⊢ (∃ c, Tendsto f (𝓝[≠] x) (𝓝 c)) ↔ 0 ≤ meromorphicOrderAt f x

Tactic:
rcases lt_or_ge (meromorphicOrderAt f x) 0 with ho | ho
  · simp only [← not_lt, ho, not_true_eq_false, iff_false, not_exists]
    intro c hc
    apply not_tendsto_atTop_of_tendsto_nhds hc.norm
    rw [tendsto_norm_atTop_iff_cobounded]
    exact tendsto_cobounded_of_meromorphicOrderAt_neg ho
  · simp [ho, tendsto_nhds_of_meromorphicOrderAt_nonneg hf ho]

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

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

Tactic:
by
  convert! meromorphicOrderAt_zpow_id_sub_const
  simp only [zpow_natCast]

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

Goal state before:
pp: z : ℂ
R : ℝ
w : ℂ
hR : 0 < R
⊢ meromorphicOrderAt (canonicalFactor R w) z ≠ ⊤

Tactic:
by
  apply (meromorphic_canonicalFactor R w).exists_meromorphicOrderAt_ne_top_iff_forall.1
  use 0
  by_cases hw : w = 0
  · simp_all [meromorphicOrderAt_canonicalFactor (mem_ball_self hR)]
  suffices meromorphicOrderAt (canonicalFactor R w) 0 = 0 by simp_all
  rw [MeromorphicNFAt.meromorphicOrderAt_eq_zero_iff]
  · simp_all [canonicalFactor, ne_of_gt hR]
  · apply AnalyticAt.meromorphicNFAt
    apply analyticOnNhd_canonicalFactor
    grind

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

Goal state before:
pp: case refine_1
R : ℝ
F : locallyFinsuppWithin (ball 0 R) ℤ
w : ℂ
⊢ MeromorphicNFOn (canonicalFactor R w ^ F w) (ball 0 R)

Tactic:
by_cases hw : w ∈ ball 0 R

Goal state after:
case pos
R : ℝ
F : locallyFinsuppWithin (ball 0 R) ℤ
w : ℂ
hw : w ∈ ball 0 R
⊢ MeromorphicNFOn (canonicalFactor R w ^ F w) (ball 0 R)

case neg
R : ℝ
F : locallyFinsuppWithin (ball 0 R) ℤ
w : ℂ
hw : w ∉ ball 0 R
⊢ MeromorphicNFOn (canonicalFactor R w ^ F w) (ball 0 R)
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: case coe
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
𝕜' : Type u_4
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
f : 𝕜 → 𝕜'
x : 𝕜
hf : MeromorphicAt f x
n : ℕ
hn : meromorphicOrderAt (f ^ n) x = ↑n * meromorphicOrderAt f x
a✝ : ℤ
⊢ ↑n * a✝ + a✝ = ↑(n + 1) * a✝

Tactic:
simp only [Nat.cast_add, Nat.cast_one]

Goal state after:
case coe
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
𝕜' : Type u_4
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
f : 𝕜 → 𝕜'
x : 𝕜
hf : MeromorphicAt f x
n : ℕ
hn : meromorphicOrderAt (f ^ n) x = ↑n * meromorphicOrderAt f x
a✝ : ℤ
⊢ ↑n * a✝ + a✝ = (↑n + 1) * a✝
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
f g : 𝕜 → 𝕜'
hf : MeromorphicAt f x
hg : MeromorphicAt g x
⊢ meromorphicOrderAt (f / g) x = meromorphicOrderAt f x - meromorphicOrderAt g x

Tactic:
rw [div_eq_mul_inv, meromorphicOrderAt_mul hf hg.inv, meromorphicOrderAt_inv, sub_eq_add_neg]

Goal state after:
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: case refine_4
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
this : Continuous[_, inst✝¹] fun st ↦ (Γ st.2) st.1
t t' : ↑I
⊢ p (((fun st ↦ (Γ st.2) st.1) ∘ fun y ↦ (1, y)) t) = p (((fun st ↦ (Γ st.2) st.1) ∘ fun y ↦ (1, y)) t')

Tactic:
change p (Γ _ _) = p (Γ _ _)

Goal state after:
case refine_4
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
this : Continuous[_, inst✝¹] fun st ↦ (Γ st.2) st.1
t t' : ↑I
⊢ p ((Γ ((fun y ↦ (1, y)) t).2) ((fun y ↦ (1, y)) t).1) = p ((Γ ((fun y ↦ (1, y)) t').2) ((fun y ↦ (1, y)) t').1)
Module: Mathlib.Analysis.Complex.Conformal

Goal state before:
pp: case inl
c : ℝ
a : Circle
⊢ (∃ map, restrictScalars ℝ map = c • (rotation a).toLinearIsometry.toContinuousLinearMap) ∨
    ∃ map, restrictScalars ℝ map = (c • (rotation a).toLinearIsometry.toContinuousLinearMap) ∘SL ↑conjCLE

Tactic:
refine Or.inl ⟨c • (a : ℂ) • ContinuousLinearMap.id ℂ ℂ, ?_⟩
    ext1
    simp

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
z : ℂ
f : ℂ → E
h : DifferentiableAt ℂ f z
hf' : deriv f z ≠ 0
⊢ ConformalAt f z

Tactic:
conformalAt_iff_isConformalMap_fderiv,

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
z : ℂ
f : ℂ → E
h : DifferentiableAt ℂ f z
hf' : deriv f z ≠ 0
⊢ IsConformalMap (fderiv ℝ f z)
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
g : G
x : X
γ : FundamentalGroup X x
x✝ : ↑(p ⁻¹' {x})
⊢ ↑(((⋯.monodromyPerm x) γ * (hp.toPermFiber x) g) x✝) = ↑(((hp.toPermFiber x) g * (⋯.monodromyPerm x) γ) x✝)

Tactic:
exact congr($hp.monodromy_toPermFiber)

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
f : E → F
x : E
h : HarmonicAt f x
⊢ ContDiffAt ℝ 2 (-f) x

Tactic:
simpa using! h.1.neg

Goal state after: