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

Goal state before:
pp: E : Type u_1
inst✝² : SeminormedAddCommGroup E
inst✝¹ : UniformConvexSpace E
ε : ℝ
inst✝ : NormedSpace ℝ E
hε : 0 < ε
hε' : 0 < ε / 3
δ : ℝ
hδ : 0 < δ
h : ∀ ⦃x : E⦄, ‖x‖ = 1 → ∀ ⦃y : E⦄, ‖y‖ = 1 → ε / 3 ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 - δ
δ' : ℝ := min (1 / 2) (min (ε / 3) (δ / 3))
x : E
hx : ‖x‖ ≤ 1
y : E
hy : ‖y‖ ≤ 1
hxy : ε ≤ ‖x - y‖
hx' : 1 - δ' < ‖x‖
hy' : 1 - δ' < ‖y‖
hδ' : 0 < 1 - δ'
h₁ : ∀ (z : E), 1 - δ' < ‖z‖ → ‖‖z‖⁻¹ • z‖ = 1
h₂ : ∀ (z : E), ‖z‖ ≤ 1 → 1 - δ' ≤ ‖z‖ → ‖‖z‖⁻¹ • z - z‖ ≤ δ'
x' : E := ‖x‖⁻¹ • x
y' : E := ‖y‖⁻¹ • y
hxy' : ε / 3 ≤ ‖x' - y'‖
this : ∀ (x' y' : E), x + y = x' + y' + (x - x') + (y - y')
⊢ ‖x + y‖ ≤ ‖x' + y'‖ + ‖x' - x‖ + ‖y' - y‖

Tactic:
rw [norm_sub_rev, norm_sub_rev y', this]

Goal state after:
E : Type u_1
inst✝² : SeminormedAddCommGroup E
inst✝¹ : UniformConvexSpace E
ε : ℝ
inst✝ : NormedSpace ℝ E
hε : 0 < ε
hε' : 0 < ε / 3
δ : ℝ
hδ : 0 < δ
h : ∀ ⦃x : E⦄, ‖x‖ = 1 → ∀ ⦃y : E⦄, ‖y‖ = 1 → ε / 3 ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 - δ
δ' : ℝ := min (1 / 2) (min (ε / 3) (δ / 3))
x : E
hx : ‖x‖ ≤ 1
y : E
hy : ‖y‖ ≤ 1
hxy : ε ≤ ‖x - y‖
hx' : 1 - δ' < ‖x‖
hy' : 1 - δ' < ‖y‖
hδ' : 0 < 1 - δ'
h₁ : ∀ (z : E), 1 - δ' < ‖z‖ → ‖‖z‖⁻¹ • z‖ = 1
h₂ : ∀ (z : E), ‖z‖ ≤ 1 → 1 - δ' ≤ ‖z‖ → ‖‖z‖⁻¹ • z - z‖ ≤ δ'
x' : E := ‖x‖⁻¹ • x
y' : E := ‖y‖⁻¹ • y
hxy' : ε / 3 ≤ ‖x' - y'‖
this : ∀ (x' y' : E), x + y = x' + y' + (x - x') + (y - y')
⊢ ‖?x' + ?y' + (x - ?x') + (y - ?y')‖ ≤ ‖x' + y'‖ + ‖x - x'‖ + ‖y - y'‖

case x'
E : Type u_1
inst✝² : SeminormedAddCommGroup E
inst✝¹ : UniformConvexSpace E
ε : ℝ
inst✝ : NormedSpace ℝ E
hε : 0 < ε
hε' : 0 < ε / 3
δ : ℝ
hδ : 0 < δ
h : ∀ ⦃x : E⦄, ‖x‖ = 1 → ∀ ⦃y : E⦄, ‖y‖ = 1 → ε / 3 ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 - δ
δ' : ℝ := ⋯
x : E
hx : ‖x‖ ≤ 1
y : E
hy : ‖y‖ ≤ 1
hxy : ε ≤ ‖x - y‖
hx' : 1 - δ' < ‖x‖
hy' : 1 - δ' < ‖y‖
hδ' : 0 < 1 - δ'
h₁ : ∀ (z : E), 1 - δ' < ‖z‖ → ‖‖z‖⁻¹ • z‖ = 1
h₂ : ∀ (z : E), ‖z‖ ≤ 1 → 1 - δ' ≤ ‖z‖ → ‖‖z‖⁻¹ • z - z‖ ≤ δ'
x' : E := ⋯
y' : E := ⋯
hxy' : ε / 3 ≤ ‖x' - y'‖
this : ∀ (x' y' : E), x + y = x' + y' + (x - x') + (y - y')
⊢ E

case y'
E : Type u_1
inst✝² : SeminormedAddCommGroup E
inst✝¹ : UniformConvexSpace E
ε : ℝ
inst✝ : NormedSpace ℝ E
hε : 0 < ε
hε' : 0 < ε / 3
δ : ℝ
hδ : 0 < δ
h : ∀ ⦃x : E⦄, ‖x‖ = 1 → ∀ ⦃y : E⦄, ‖y‖ = 1 → ε / 3 ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 - δ
δ' : ℝ := ⋯
x : E
hx : ‖x‖ ≤ 1
y : E
hy : ‖y‖ ≤ 1
hxy : ε ≤ ‖x - y‖
hx' : 1 - δ' < ‖x‖
hy' : 1 - δ' < ‖y‖
hδ' : 0 < 1 - δ'
h₁ : ∀ (z : E), 1 - δ' < ‖z‖ → ‖‖z‖⁻¹ • z‖ = 1
h₂ : ∀ (z : E), ‖z‖ ≤ 1 → 1 - δ' ≤ ‖z‖ → ‖‖z‖⁻¹ • z - z‖ ≤ δ'
x' : E := ⋯
y' : E := ⋯
hxy' : ε / 3 ≤ ‖x' - y'‖
this : ∀ (x' y' : E), x + y = x' + y' + (x - x') + (y - y')
⊢ E
Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: f : ℝ → ℝ
a b c : ℝ
h : IntervalIntegrable f volume a b
hc : c ∈ uIcc a b
s : ℕ × (ℕ → ℝ × ℝ) → Set ℝ := ⋯
this✝¹ :
  Tendsto (fun i ↦ ∫⁻ (x : ℝ) in s i, ‖f x‖ₑ ∂volume.restrict (uIoc a b)) (totalLengthFilter ⊓ 𝓟 (disjWithin a b)) (𝓝 0)
this✝ :
  Tendsto (ENNReal.toReal ∘ fun i ↦ ∫⁻ (x : ℝ) in s i, ‖f x‖ₑ ∂volume.restrict (uIoc a b))
    (totalLengthFilter ⊓ 𝓟 (disjWithin a b)) (𝓝 0)
this : ∀ᶠ (E : ℕ × (ℕ → ℝ × ℝ)) in totalLengthFilter ⊓ 𝓟 (disjWithin a b), E ∈ disjWithin a b
n : ℕ
I : ℕ → ℝ × ℝ
hnI : (n, I) ∈ disjWithin a b
hnI1 : ∀ i ∈ Finset.range (n, I).1, ((n, I).2 i).1 ∈ uIcc a b ∧ ((n, I).2 i).2 ∈ uIcc a b
hnI2 : (↑(Finset.range (n, I).1)).PairwiseDisjoint fun i ↦ uIoc ((n, I).2 i).1 ((n, I).2 i).2
i : ℕ
hi : i ∈ Finset.range n
⊢ uIoc c (I i).1 ⊆ uIoc a b

Tactic:
grind [uIoc, uIcc]

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

Goal state before:
pp: f : ℝ → ℝ
a b c : ℝ
h : IntervalIntegrable f volume a b
hc : c ∈ uIcc a b
s : ℕ × (ℕ → ℝ × ℝ) → Set ℝ := ⋯
this✝¹ :
  Tendsto (fun i ↦ ∫⁻ (x : ℝ) in s i, ‖f x‖ₑ ∂volume.restrict (uIoc a b)) (totalLengthFilter ⊓ 𝓟 (disjWithin a b)) (𝓝 0)
this✝ :
  Tendsto (ENNReal.toReal ∘ fun i ↦ ∫⁻ (x : ℝ) in s i, ‖f x‖ₑ ∂volume.restrict (uIoc a b))
    (totalLengthFilter ⊓ 𝓟 (disjWithin a b)) (𝓝 0)
this : ∀ᶠ (E : ℕ × (ℕ → ℝ × ℝ)) in totalLengthFilter ⊓ 𝓟 (disjWithin a b), E ∈ disjWithin a b
n : ℕ
I : ℕ → ℝ × ℝ
hnI : (n, I) ∈ disjWithin a b
hnI1 : ∀ i ∈ Finset.range (n, I).1, ((n, I).2 i).1 ∈ uIcc a b ∧ ((n, I).2 i).2 ∈ uIcc a b
hnI2 : (↑(Finset.range (n, I).1)).PairwiseDisjoint fun i ↦ uIoc ((n, I).2 i).1 ((n, I).2 i).2
i : ℕ
hi : i ∈ Finset.range n
⊢ uIoc c (I i).2 ⊆ uIoc a b

Tactic:
grind [uIoc, uIcc]

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

Goal state before:
pp: E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
c : E
hd : ∀ᶠ (z : E) in 𝓝 c, DifferentiableAt ℂ f z
hc : IsLocalMax (norm ∘ f) c
⊢ ∀ᶠ (y : E) in 𝓝 c, ‖f y‖ = ‖f c‖

Tactic:
rcases nhds_basis_closedBall.eventually_iff.1 (hd.and hc) with ⟨r, hr₀, hr⟩

Goal state after:
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
c : E
hd : ∀ᶠ (z : E) in 𝓝 c, DifferentiableAt ℂ f z
hc : IsLocalMax (norm ∘ f) c
r : ℝ
hr₀ : 0 < r
hr : ∀ ⦃x : E⦄, x ∈ closedBall c r → DifferentiableAt ℂ f x ∧ (norm ∘ f) x ≤ (norm ∘ f) c
⊢ ∀ᶠ (y : E) in 𝓝 c, ‖f y‖ = ‖f c‖
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
a : ℝ
ha : (f w - f x) / (w - x) < deriv f a
hxa : x < a
haw : a < w
⊢ ∃ b ∈ Ioo w y, (f y - f w) / (y - w) < deriv f b

Tactic:
by
      apply StrictMonoOn.exists_slope_lt_deriv_aux _ hwy _ _
      · refine hf.mono (Icc_subset_Icc hxw.le le_rfl)
      · exact hf'_mono.mono (Ioo_subset_Ioo hxw.le le_rfl)
      · intro z hz
        rw [← hw]
        apply ne_of_gt
        exact hf'_mono ⟨hxw, hwy⟩ ⟨hxw.trans hz.1, hz.2⟩ hz.1

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✝ : StrictConvexSpace ℝ F
f : E → F
c : E
hd : ∀ᶠ (z : E) in 𝓝 c, DifferentiableAt ℂ f z
hc : IsLocalMax (norm ∘ f) c
⊢ ∀ᶠ (y : E) in 𝓝 c, f y = f c

Tactic:
rcases nhds_basis_closedBall.eventually_iff.1 (hd.and hc) with ⟨r, hr₀, hr⟩

Goal state after:
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✝ : StrictConvexSpace ℝ F
f : E → F
c : E
hd : ∀ᶠ (z : E) in 𝓝 c, DifferentiableAt ℂ f z
hc : IsLocalMax (norm ∘ f) c
r : ℝ
hr₀ : 0 < r
hr : ∀ ⦃x : E⦄, x ∈ closedBall c r → DifferentiableAt ℂ f x ∧ (norm ∘ f) x ≤ (norm ∘ f) c
⊢ ∀ᶠ (y : E) in 𝓝 c, f y = f c
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv

Goal state before:
pp: x : ℂ
h : cos x ≠ 0
⊢ HasStrictDerivAt tan (1 / cos x ^ 2) x

Tactic:
convert! (hasStrictDerivAt_sin x).div (hasStrictDerivAt_cos x) h using 1
  rw_mod_cast [← sin_sq_add_cos_sq x]
  ring

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

Goal state before:
pp: x : ℝ
⊢ sin x = 1 ∨ sin x = -1 ↔ ∃ k, π / 2 + ↑k * π = x

Tactic:
sin_eq_one_iff,

Goal state after:
x : ℝ
⊢ (∃ k, π / 2 + ↑k * (2 * π) = x) ∨ sin x = -1 ↔ ∃ k, π / 2 + ↑k * π = x
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: case inr.refine_2
S : Set ℝ
f : ℝ → ℝ
hfc : ConvexOn ℝ S f
x : ℝ
hxs : x ∈ interior S
y : ℝ
hys : y ∈ interior S
hxy✝ : x ≤ y
hxy : x < y
⊢ ∀ b ∈ slope f y '' {y_1 | y_1 ∈ S ∧ y < y_1}, slope f x y ≤ b

Tactic:
· rintro _ ⟨z, ⟨hzs, hyz : y < z⟩, rfl⟩
    rw [slope_comm]
    exact slope_mono hfc (interior_subset hys) ⟨interior_subset hxs, hxy.ne⟩ ⟨hzs, hyz.ne'⟩
      (hxy.trans hyz).le

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

Goal state before:
pp: S : Set ℝ
f : ℝ → ℝ
x y f' : ℝ
hfc : ConvexOn ℝ S f
hx : x ∈ S
hy : y ∈ S
hxy : x < y
hf' : HasDerivWithinAt f f' (Ioi x) x
⊢ f' ≤ slope f x y

Tactic:
apply le_of_tendsto <| (hasDerivWithinAt_iff_tendsto_slope' self_notMem_Ioi).mp hf'
  simp_rw [eventually_nhdsWithin_iff, slope_def_field]
  filter_upwards [eventually_lt_nhds hxy] with t ht (ht' : x < t)
  refine hfc.secant_mono hx (?_ : t ∈ S) hy ht'.ne' hxy.ne' ht.le
  exact hfc.1.ordConnected.out hx hy ⟨ht'.le, ht.le⟩

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
x : ℂ
hx : x ∈ ball 0 R
⊢ (f x).re - (f 0).re ≤ M + ‖f 0‖

Tactic:
calc (f x).re - (f 0).re ≤ M - (f 0).re := by gcongr; exact hf₁ hx
      _ ≤ M + ‖f 0‖ := by linarith [neg_le_abs (f 0).re, abs_re_le_norm (f 0)]

Goal state after:
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
h : sin (angle x y) = 1
⊢ angle x y = π / 2

Tactic:
← cos_eq_zero_iff_angle_eq_pi_div_two,

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : sin (angle x y) = 1
⊢ cos (angle x y) = 0
Module: Mathlib.Analysis.Complex.UnitDisc.Basic

Goal state before:
pp: z : 𝔻
⊢ ‖↑z‖ ^ 2 < 1

Tactic:
sq_lt_one_iff_abs_lt_one,

Goal state after:
z : 𝔻
⊢ |‖↑z‖| < 1
Module: Mathlib.Analysis.Complex.UnitDisc.Basic

Goal state before:
pp: z : 𝕔𝔻
⊢ ‖↑z‖ ^ 2 ≤ 1

Tactic:
rw [sq_le_one_iff_abs_le_one, abs_norm]

Goal state after:
z : 𝕔𝔻
⊢ ‖↑z‖ ≤ 1
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
hf : AnalyticAt 𝕜 f x ∨ MeromorphicAt f x ∧ meromorphicOrderAt f x < 0 ∧ f x = 0
h₂f : 0 ≤ meromorphicOrderAt f x
⊢ AnalyticAt 𝕜 f x

Tactic:
rcases hf with h | ⟨_, h₃f, _⟩

Goal state after:
case mp.inl
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
h₂f : 0 ≤ meromorphicOrderAt f x
h : AnalyticAt 𝕜 f x
⊢ AnalyticAt 𝕜 f x

case mp.inr
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
h₂f : 0 ≤ meromorphicOrderAt f x
left✝ : MeromorphicAt f x
h₃f : meromorphicOrderAt f x < 0
right✝ : f x = 0
⊢ AnalyticAt 𝕜 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
U : Set 𝕜
hf : MeromorphicOn f U
⊢ {x | MeromorphicNFAt f x} ∈ Filter.codiscreteWithin U

Tactic:
by
  filter_upwards [hf.analyticAt_mem_codiscreteWithin] with _ ha
  exact ha.meromorphicNFAt

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

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
x : 𝕜
hf : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = 0
⊢ MeromorphicAt f x

Tactic:
exact (MeromorphicAt.const 0 x).congr (EventuallyEq.symm hf)

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

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : MeromorphicAt f x
h : analyticOrderAt (fun z ↦ (z - x) ^ Exists.choose hf • f z) x = ⊤
⊢ analyticOrderAt (fun z ↦ (z - x) ^ Exists.choose ⋯ • f z) x = ⊤ ↔ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = 0

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
x : 𝕜
hf : MeromorphicAt f x
h : ¬analyticOrderAt (fun z ↦ (z - x) ^ Exists.choose hf • f z) x = ⊤
⊢ analyticOrderAt (fun z ↦ (z - x) ^ Exists.choose ⋯ • f z) x = ⊤ ↔ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = 0

Tactic:
· simp only [h, eventually_nhdsWithin_iff, mem_compl_iff, mem_singleton_iff, true_iff]
    rw [analyticOrderAt_eq_top] at h
    filter_upwards [h] with z hf hz
    rwa [smul_eq_zero_iff_right <| pow_ne_zero _ (sub_ne_zero.mpr hz)] at hf

Goal state after:
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
x : 𝕜
hf : MeromorphicAt f x
h : ¬analyticOrderAt (fun z ↦ (z - x) ^ Exists.choose hf • f z) x = ⊤
⊢ analyticOrderAt (fun z ↦ (z - x) ^ Exists.choose ⋯ • f z) x = ⊤ ↔ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = 0
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h : HasFiniteSupport d
this : (Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d u) ⊆ ↑(Finite.toFinset h)
y : 𝕜
hy : y ∈ Finite.toFinset h
⊢ meromorphicTrailingCoeffAt ((fun x ↦ x - y) ^ d y) x = (x - y) ^ update d x 0 y

Tactic:
rw [MeromorphicAt.meromorphicTrailingCoeffAt_zpow (by fun_prop)]

Goal state after:
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h : HasFiniteSupport d
this : (Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d u) ⊆ ↑(Finite.toFinset h)
y : 𝕜
hy : y ∈ Finite.toFinset h
⊢ meromorphicTrailingCoeffAt (fun x ↦ x - y) x ^ d y = (x - y) ^ update d x 0 y
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)
η₀ : (-divisor f (ball 0 R)).support.Finite
x : ℂ
hx : x ∈ η₀.toFinset
⊢ -((divisor f (ball 0 R)) x • locallyFinsuppWithin.restrict (locallyFinsuppWithin.single x 1) ⋯) =
    (-⇑(divisor f (ball 0 R))) x • locallyFinsuppWithin.restrict (locallyFinsuppWithin.single x 1) ⋯

pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
f : ℂ → E
h₁f : MeromorphicOn f (closedBall 0 R)
η₀ : (-divisor f (ball 0 R)).support.Finite
x : ℂ
hx : x ∈ η₀.toFinset
⊢ x ∈ ball 0 R

Tactic:
Pi.neg_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)
η₀ : (-divisor f (ball 0 R)).support.Finite
x : ℂ
hx : x ∈ η₀.toFinset
⊢ -((divisor f (ball 0 R)) x • locallyFinsuppWithin.restrict (locallyFinsuppWithin.single x 1) ⋯) =
    -(divisor f (ball 0 R)) x • locallyFinsuppWithin.restrict (locallyFinsuppWithin.single x 1) ⋯

E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
f : ℂ → E
h₁f : MeromorphicOn f (closedBall 0 R)
η₀ : (-divisor f (ball 0 R)).support.Finite
x : ℂ
hx : x ∈ η₀.toFinset
⊢ x ∈ ball 0 R
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 g : 𝕜 → E
D : locallyFinsuppWithin U ℤ
hg : ∀ (u : ↑U), g ↑u ≠ 0
h : f =ᶠ[codiscreteWithin U] (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ D u) • g
t₁ : (support fun u x ↦ ↑(D u) * log ‖x - u‖) = D.support
⊢ (fun x ↦ log ‖f x‖) =ᶠ[codiscreteWithin U] (∑ᶠ (u : 𝕜), fun x ↦ ↑(D u) * log ‖x - u‖) + fun x ↦ log ‖g x‖

Tactic:
by_cases h₃f : D.support.Finite

Goal state after:
case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f g : 𝕜 → E
D : locallyFinsuppWithin U ℤ
hg : ∀ (u : ↑U), g ↑u ≠ 0
h : f =ᶠ[codiscreteWithin U] (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ D u) • g
t₁ : (support fun u x ↦ ↑(D u) * log ‖x - u‖) = D.support
h₃f : D.support.Finite
⊢ (fun x ↦ log ‖f x‖) =ᶠ[codiscreteWithin U] (∑ᶠ (u : 𝕜), fun x ↦ ↑(D u) * log ‖x - u‖) + fun x ↦ log ‖g x‖

case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f g : 𝕜 → E
D : locallyFinsuppWithin U ℤ
hg : ∀ (u : ↑U), g ↑u ≠ 0
h : f =ᶠ[codiscreteWithin U] (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ D u) • g
t₁ : (support fun u x ↦ ↑(D u) * log ‖x - u‖) = D.support
h₃f : ¬D.support.Finite
⊢ (fun x ↦ log ‖f x‖) =ᶠ[codiscreteWithin U] (∑ᶠ (u : 𝕜), fun x ↦ ↑(D u) * log ‖x - u‖) + fun x ↦ log ‖g x‖
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
x : 𝕜
f g : 𝕜 → E
D : 𝕜 → ℤ
hD : HasFiniteSupport D
h₁x : x ∈ U
h₂x : AccPt x (𝓟 U)
hf : MeromorphicAt f x
h₁g : AnalyticAt 𝕜 g x
h₂g : g x ≠ 0
h : f =ᶠ[codiscreteWithin U] (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ D u) • g
t₀ : MeromorphicAt (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ D u) x
⊢ meromorphicTrailingCoeffAt f x = (∏ᶠ (u : 𝕜), (x - u) ^ update D x 0 u) • g x

Tactic:
rw [meromorphicTrailingCoeffAt_congr_nhdsNE
      (hf.eventuallyEq_nhdsNE_of_eventuallyEq_codiscreteWithin (by fun_prop) h₁x h₂x h),
    t₀.meromorphicTrailingCoeffAt_smul h₁g.meromorphicAt,
    h₁g.meromorphicTrailingCoeffAt_of_ne_zero h₂g]

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
x : 𝕜
f g : 𝕜 → E
D : 𝕜 → ℤ
hD : HasFiniteSupport D
h₁x : x ∈ U
h₂x : AccPt x (𝓟 U)
hf : MeromorphicAt f x
h₁g : AnalyticAt 𝕜 g x
h₂g : g x ≠ 0
h : f =ᶠ[codiscreteWithin U] (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ D u) • g
t₀ : MeromorphicAt (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ D u) x
⊢ meromorphicTrailingCoeffAt (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ D u) x • g x = (∏ᶠ (u : 𝕜), (x - u) ^ update D x 0 u) • g x
Module: Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Basic

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

Tactic:
refine fun ⟨a, ha⟩ ↦ Subgroup.mem_center_iff.mpr fun h ↦ ?_
    simp [-scalar_apply, Units.ext_iff, ← ha, Matrix.scalar_comm a (Commute.all _)]

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

Goal state before:
pp: f : ℂ → ℂ
z : ℂ
⊢ ConformalAt f z ↔
    (DifferentiableAt ℂ f z ∨ DifferentiableAt ℂ (f ∘ ⇑(starRingEnd ℂ)) ((starRingEnd ℂ) z)) ∧ fderiv ℝ f z ≠ 0

Tactic:
rw [conformalAt_iff_isConformalMap_fderiv]
  rw [isConformalMap_iff_is_complex_or_conj_linear]
  apply and_congr_left
  intro h
  have h_diff := h.imp_symm fderiv_zero_of_not_differentiableAt
  apply or_congr
  · rw [differentiableAt_iff_restrictScalars ℝ h_diff]
  rw [← conj_conj z] at h_diff
  rw [differentiableAt_iff_restrictScalars ℝ (h_diff.comp _ conjCLE.differentiableAt)]
  refine exists_congr fun g => rfl.congr ?_
  have : fderiv ℝ conj (conj z) = _ := conjCLE.fderiv
  simp [fderiv_comp _ h_diff conjCLE.differentiableAt, this]

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

Goal state before:
pp: case h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
f : ℂ → E
h₁f : MeromorphicOn f (closedBall 0 R)
h₂f : ∀ (u : ↑(closedBall 0 R)), meromorphicOrderAt f ↑u ≠ ⊤
hR : 0 < R
g : ℂ → E
D : CanonicalDecomp f g R
s₁ : 0 ∈ closedBall 0 R
⊢ meromorphicOrderAt g ↑⟨0, s₁⟩ ≠ ⊤

Tactic:
simp [(D.meromorphicNFOn s₁).meromorphicOrderAt_eq_zero_iff.2 (D.ne_zero 0 (by simp [hR]))]

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
x : ℂ
hx : x ∈ t₁
⊢ meromorphicTrailingCoeffAt (fun x_1 ↦ x_1 - x) w ^ ((-⇑(divisor f S₀R)) x + (divisor f S₀R) x) = 1

pp: case ha
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
x : ℂ
hx : x ∈ t₁
⊢ meromorphicTrailingCoeffAt (fun x_1 ↦ x_1 - x) w ≠ 0

Tactic:
Pi.neg_apply,

Goal state after:
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
x : ℂ
hx : x ∈ t₁
⊢ meromorphicTrailingCoeffAt (fun x_1 ↦ x_1 - x) w ^ (-(divisor f S₀R) x + (divisor f S₀R) x) = 1

case ha
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
x : ℂ
hx : x ∈ t₁
⊢ meromorphicTrailingCoeffAt (fun x_1 ↦ x_1 - x) w ≠ 0
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
x : 𝕜
f : 𝕜 → E
g : 𝕜 → 𝕜
hg : AnalyticAt 𝕜 g x
hg' : deriv g x ≠ 0
inst✝¹ : CompleteSpace 𝕜
inst✝ : CharZero 𝕜
hf : ¬MeromorphicAt f (g x)
⊢ meromorphicOrderAt (f ∘ g) x = 0

Tactic:
meromorphicOrderAt_of_not_meromorphicAt

Goal state after:
case neg
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
x : 𝕜
f : 𝕜 → E
g : 𝕜 → 𝕜
hg : AnalyticAt 𝕜 g x
hg' : deriv g x ≠ 0
inst✝¹ : CompleteSpace 𝕜
inst✝ : CharZero 𝕜
hf : ¬MeromorphicAt f (g x)
⊢ 0 = 0

case neg
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
x : 𝕜
f : 𝕜 → E
g : 𝕜 → 𝕜
hg : AnalyticAt 𝕜 g x
hg' : deriv g x ≠ 0
inst✝¹ : CompleteSpace 𝕜
inst✝ : CharZero 𝕜
hf : ¬MeromorphicAt f (g x)
⊢ ¬MeromorphicAt (f ∘ g) x
Module: Mathlib.Analysis.Meromorphic.Order

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
x : 𝕜
g : 𝕜 → 𝕜
hg : AnalyticAt 𝕜 g x
hg' : g x ≠ 0
hf : ¬MeromorphicAt f x
⊢ meromorphicOrderAt (g • f) x = 0

Tactic:
meromorphicOrderAt_of_not_meromorphicAt

Goal state after:
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
x : 𝕜
g : 𝕜 → 𝕜
hg : AnalyticAt 𝕜 g x
hg' : g x ≠ 0
hf : ¬MeromorphicAt f x
⊢ 0 = 0

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
x : 𝕜
g : 𝕜 → 𝕜
hg : AnalyticAt 𝕜 g x
hg' : g x ≠ 0
hf : ¬MeromorphicAt f x
⊢ ¬MeromorphicAt (g • f) x
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
p : E → X
G : Type u_4
inst✝² : Group G
inst✝¹ : MulAction G E
hp : IsQuotientCoveringMap p G
x : X
inst✝ : SimplyConnectedSpace E
⊢ Injective ⇑(⋯.monodromyPerm x)

Tactic:
let e : p⁻¹' {x} := ⟨(hp.surjective x).choose, (hp.surjective x).choose_spec⟩

Goal state after:
E : Type u_1
X : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
p : E → X
G : Type u_4
inst✝² : Group G
inst✝¹ : MulAction G E
hp : IsQuotientCoveringMap p G
x : X
inst✝ : SimplyConnectedSpace E
e : ↑(p ⁻¹' {x}) := ⟨⋯.choose, ⋯⟩
⊢ Injective ⇑(⋯.monodromyPerm x)
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
U : Set ℂ
hU : IsOpen U
g : ℂ → E
hg : ∀ z ∈ U, HasDerivAt g (f z) z
hg' : DifferentiableOn ℂ g U
⊢ DifferentiableOn ℂ f U

Tactic:
exact (differentiableOn_congr <| fun z hz ↦ (hg z hz).deriv).mp <| hg'.deriv hU

Goal state after: