mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.Rademacher

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C : ℝ≥0
f : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
ι : Type u_3
s : Finset ι
a : ι → ℝ
v : ι → E
g : E → ℝ
g_smooth : ContDiff ℝ (↑⊤) g
g_comp : HasCompactSupport g
⊢ ∫ (x : E), g x • lineDeriv ℝ f x (∑ i ∈ s, a i • v i) ∂μ =
    ∫ (x : E), ∑ x_1 ∈ s, g x • (a x_1 • fun x ↦ lineDeriv ℝ f x (v x_1)) x ∂μ

Tactic:
have A : ∀ i ∈ s, Integrable (fun x ↦ g x • (a i • fun x ↦ lineDeriv ℝ f x (v i)) x) μ :=
    fun i hi ↦ (g_smooth.continuous.integrable_of_hasCompactSupport g_comp).smul_of_top_left
      ((hf.memLp_lineDeriv (v i)).const_smul (a i))

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C : ℝ≥0
f : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
ι : Type u_3
s : Finset ι
a : ι → ℝ
v : ι → E
g : E → ℝ
g_smooth : ContDiff ℝ (↑⊤) g
g_comp : HasCompactSupport g
A : ∀ i ∈ s, Integrable (fun x ↦ g x • (a i • fun x ↦ lineDeriv ℝ f x (v i)) x) μ
⊢ ∫ (x : E), g x • lineDeriv ℝ f x (∑ i ∈ s, a i • v i) ∂μ =
    ∫ (x : E), ∑ x_1 ∈ s, g x • (a x_1 • fun x ↦ lineDeriv ℝ f x (v x_1)) x ∂μ
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.DerivIntegrable

Goal state before:
pp: f : ℝ → ℝ
a b : ℝ
hab : a ≤ b
hf : MonotoneOn f (Icc a b)
g : ℝ → ℝ := fun x ↦ f (max a (min x b))
⊢ MapsTo (fun x ↦ max a (min x b)) univ (Icc a b)

Tactic:
by simpa

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

Goal state before:
pp: X : Type u_1
inst✝ : PseudoMetricSpace X
⊢ totalLengthFilter.HasBasis (fun ε ↦ 0 < ε) fun ε ↦ {E | ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2 < ε}

Tactic:
convert! Filter.HasBasis.comap (α := ℝ) _ (nhds_basis_Ioo_pos _) using 1

Goal state after:
case e'_5
X : Type u_1
inst✝ : PseudoMetricSpace X
⊢ (fun ε ↦ {E | ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2 < ε}) = fun i ↦
    (fun E ↦ ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2) ⁻¹' Ioo (0 - i) (0 + i)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun

Goal state before:
pp: X : Type u_1
inst✝ : PseudoMetricSpace X
f : ℝ → X
d b y : ℝ
hdb : d ≤ b
hf : AbsolutelyContinuousOnInterval f d b
u : Set (ℝ × ℝ)
hu₃ : HasSum (fun z ↦ (↑z).2 - (↑z).1) (b - d)
hu₄ : HasSum (fun z ↦ dist (f (↑z).1) (f (↑z).2)) y
u_coe : Finset ↑u → Finset (ℝ × ℝ) := fun s ↦ Finset.image Subtype.val s
hu₁ : ∀ (s : Finset ↑u) ⦃z : ℝ × ℝ⦄, z ∈ u_coe s → d ≤ z.1 ∧ z.1 ≤ z.2 ∧ z.2 ≤ b
hu₂ : ∀ (s : Finset ↑u), (↑(u_coe s)).PairwiseDisjoint fun z ↦ Icc z.1 z.2
T : Finset ↑u → ℕ × (ℕ → ℝ × ℝ) := fun s ↦ ((u_coe s).card + 1, fun i ↦ (u_coe s).intervalGapsWithin ⋯ d b ↑i)
hT₁ : ∀ (s : Finset ↑u) (i : ℕ), d ≤ ((u_coe s).intervalGapsWithin ⋯ d b ↑i).1
hT₂ : ∀ (s : Finset ↑u) (i : ℕ), ((u_coe s).intervalGapsWithin ⋯ d b ↑i).1 ≤ ((u_coe s).intervalGapsWithin ⋯ d b ↑i).2
hT₃ : ∀ (s : Finset ↑u) (i : ℕ), ((u_coe s).intervalGapsWithin ⋯ d b ↑i).2 ≤ b
hT₄ :
  ∀ (s : Finset ↑u),
    (Iio ((u_coe s).card + 1)).PairwiseDisjoint fun j ↦
      Ioc ((u_coe s).intervalGapsWithin ⋯ d b ↑j).1 ((u_coe s).intervalGapsWithin ⋯ d b ↑j).2
hT : MapsTo T univ (disjWithin d b)
u_coe_sum : ∀ (s : Finset ↑u) (g : ℝ → ℝ → ℝ), ∑ b ∈ s, g (↑b).1 (↑b).2 = ∑ z ∈ u_coe s, g z.1 z.2
s : Finset ↑u
i : ℕ
hi : i ∈ Finset.range (T s).1
⊢ |((T s).2 i).2 - ((T s).2 i).1| = ((T s).2 i).2 - ((T s).2 i).1

Tactic:
abs_of_nonneg (by grind)

Goal state after:
X : Type u_1
inst✝ : PseudoMetricSpace X
f : ℝ → X
d b y : ℝ
hdb : d ≤ b
hf : AbsolutelyContinuousOnInterval f d b
u : Set (ℝ × ℝ)
hu₃ : HasSum (fun z ↦ (↑z).2 - (↑z).1) (b - d)
hu₄ : HasSum (fun z ↦ dist (f (↑z).1) (f (↑z).2)) y
u_coe : Finset ↑u → Finset (ℝ × ℝ) := fun s ↦ Finset.image Subtype.val s
hu₁ : ∀ (s : Finset ↑u) ⦃z : ℝ × ℝ⦄, z ∈ u_coe s → d ≤ z.1 ∧ z.1 ≤ z.2 ∧ z.2 ≤ b
hu₂ : ∀ (s : Finset ↑u), (↑(u_coe s)).PairwiseDisjoint fun z ↦ Icc z.1 z.2
T : Finset ↑u → ℕ × (ℕ → ℝ × ℝ) := fun s ↦ ((u_coe s).card + 1, fun i ↦ (u_coe s).intervalGapsWithin ⋯ d b ↑i)
hT₁ : ∀ (s : Finset ↑u) (i : ℕ), d ≤ ((u_coe s).intervalGapsWithin ⋯ d b ↑i).1
hT₂ : ∀ (s : Finset ↑u) (i : ℕ), ((u_coe s).intervalGapsWithin ⋯ d b ↑i).1 ≤ ((u_coe s).intervalGapsWithin ⋯ d b ↑i).2
hT₃ : ∀ (s : Finset ↑u) (i : ℕ), ((u_coe s).intervalGapsWithin ⋯ d b ↑i).2 ≤ b
hT₄ :
  ∀ (s : Finset ↑u),
    (Iio ((u_coe s).card + 1)).PairwiseDisjoint fun j ↦
      Ioc ((u_coe s).intervalGapsWithin ⋯ d b ↑j).1 ((u_coe s).intervalGapsWithin ⋯ d b ↑j).2
hT : MapsTo T univ (disjWithin d b)
u_coe_sum : ∀ (s : Finset ↑u) (g : ℝ → ℝ → ℝ), ∑ b ∈ s, g (↑b).1 (↑b).2 = ∑ z ∈ u_coe s, g z.1 z.2
s : Finset ↑u
i : ℕ
hi : i ∈ Finset.range (T s).1
⊢ ((T s).2 i).2 - ((T s).2 i).1 = ((T s).2 i).2 - ((T s).2 i).1
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)
z : ℂ
hz : ‖z‖ < 1
s : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i
k : Tendsto (fun x ↦ (1 - z) * ∑ i ∈ range x, f i * ∑ i ∈ range i, z ^ i) atTop (𝓝 (l - ∑' (i : ℕ), f i * z ^ i))
⊢ Tendsto (fun n ↦ (1 - z) * ∑ i ∈ range n, (l - ∑ j ∈ range (i + 1), f j) * z ^ i) atTop
    (𝓝 (l - ∑' (n : ℕ), f n * z ^ n))

Tactic:
mul_sum _ _ (f _),

Goal state after:
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)
z : ℂ
hz : ‖z‖ < 1
s : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i
k : Tendsto (fun x ↦ (1 - z) * ∑ x ∈ range x, ∑ i ∈ range x, f x * z ^ i) atTop (𝓝 (l - ∑' (i : ℕ), f i * z ^ i))
⊢ Tendsto (fun n ↦ (1 - z) * ∑ i ∈ range n, (l - ∑ j ∈ range (i + 1), f j) * z ^ i) atTop
    (𝓝 (l - ∑' (n : ℕ), f n * z ^ n))
Module: Mathlib.Analysis.Convex.StrictConvexSpace

Goal state before:
pp: E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
h : ∀ (x y : E), ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b, a + b = 1 ∧ ‖a • x + b • y‖ < 1
⊢ StrictConvexSpace ℝ E

Tactic:
refine
    StrictConvexSpace.of_strictConvex_unitClosedBall ℝ
      ((convex_closedBall _ _).strictConvex' fun x hx y hy hne => ?_)

Goal state after:
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
h : ∀ (x y : E), ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b, a + b = 1 ∧ ‖a • x + b • y‖ < 1
x : E
hx : x ∈ closedBall 0 1 \ interior (closedBall 0 1)
y : E
hy : y ∈ closedBall 0 1 \ interior (closedBall 0 1)
hne : x ≠ y
⊢ (fun x y ↦ ∃ c, (AffineMap.lineMap x y) c ∈ interior (closedBall 0 1)) x y
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)
z : ℂ
hz : ‖z‖ < 1
s : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i
k :
  Tendsto (fun n ↦ (1 - z) * ∑ j ∈ range n, (∑ k ∈ range n, f k - ∑ k ∈ range (j + 1), f k) * z ^ j) atTop
    (𝓝 (l - ∑' (i : ℕ), f i * z ^ i))
⊢ Tendsto (fun n ↦ (1 - z) * ∑ i ∈ range n, (l - ∑ j ∈ range (i + 1), f j) * z ^ i) atTop
    (𝓝 (l - ∑' (n : ℕ), f n * z ^ n))

Tactic:
suffices Tendsto (fun n ↦ (l - s n) * ∑ i ∈ range n, z ^ i) atTop (𝓝 0) by
    simp_rw [mul_sum] at this
    replace this := (this.const_mul (1 - z)).add k
    conv at this =>
      enter [1, n]
      rw [← mul_add, ← sum_add_distrib]
      enter [2, 2, i]
      rw [← add_mul, sub_add_sub_cancel]
    rwa [mul_zero, zero_add] at this

Goal state after:
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)
z : ℂ
hz : ‖z‖ < 1
s : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i
k :
  Tendsto (fun n ↦ (1 - z) * ∑ j ∈ range n, (∑ k ∈ range n, f k - ∑ k ∈ range (j + 1), f k) * z ^ j) atTop
    (𝓝 (l - ∑' (i : ℕ), f i * z ^ i))
⊢ Tendsto (fun n ↦ (l - s n) * ∑ i ∈ range n, z ^ i) atTop (𝓝 0)
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z ↦ ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
i : ℕ
hi : i ∈ Ico B₁ (max B₁ B₂)
⊢ i + 1 ≥ B₁

Tactic:
linarith only [(mem_Ico.mp hi).1]

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

Goal state before:
pp: x : ℝ
hx : cos x = 0
⊢ Tendsto (fun x ↦ |tan x|) (𝓝[≠] x) atTop

Tactic:
have hx : Complex.cos x = 0 := mod_cast hx
  simp only [← Real.norm_eq_abs, ← Complex.norm_real, Complex.ofReal_tan]
  refine (Complex.tendsto_norm_tan_of_cos_eq_zero hx).comp ?_
  refine Tendsto.inf Complex.continuous_ofReal.continuousAt ?_
  exact tendsto_principal_principal.2 fun y => mt Complex.ofReal_inj.1

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

Goal state before:
pp: x : ℝ
A : cos (arctan x) ≠ 0
⊢ 1 / cos (↑tanPartialHomeomorph.symm x) ^ 2 ≠ 0

Tactic:
by simpa

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

Goal state before:
pp: n : WithTop ℕ∞
x : ℝ
this : cos (arctan x) ≠ 0
⊢ 1 / cos (↑tanPartialHomeomorph.symm x) ^ 2 ≠ 0

Tactic:
by simpa

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

Goal state before:
pp: x : ℝ
⊢ sin x ^ 2 ≤ x ^ 2

Tactic:
rcases eq_or_ne x 0 with rfl | hx
  case inl => simp
  case inr => exact (sin_sq_lt_sq hx).le

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

Goal state before:
pp: x : ℝ
hx₀ : 0 ≤ x
hx : x ≤ π / 2
⊢ 0 ≤ π / 2 - x

Tactic:
by simpa

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

Goal state before:
pp: x : ℝ
hx₀ : 0 ≤ x
hx : x ≤ π / 2
⊢ π / 2 - x ≤ π / 2

Tactic:
by simpa

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

Goal state before:
pp: x : ℝ
hx₀ : 0 ≤ x
hx : x ≤ π / 2
⊢ 1 - 2 / π * x ≤ cos x

Tactic:
by
  simpa [sin_pi_div_two_sub, mul_sub, div_mul_div_comm, mul_comm π, pi_pos.ne']
    using mul_le_sin (x := π / 2 - x) (by simpa) (by simpa)

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
⊢ arccos (⟪x, y⟫ / (‖x‖ * ‖y‖)) = π ↔ x ≠ 0 ∧ ∃ r < 0, y = r • x

Tactic:
← real_inner_div_norm_mul_norm_eq_neg_one_iff,

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
⊢ arccos (⟪x, y⟫ / (‖x‖ * ‖y‖)) = π ↔ ⟪x, y⟫ / (‖x‖ * ‖y‖) = -1
Module: Mathlib.Analysis.Complex.Schwarz

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
c z : E
R₁ R₂ : ℝ
n : ℕ
hd : DifferentiableOn ℂ f (ball c R₁)
h_maps : MapsTo f (ball c R₁) (closedBall (f c) R₂)
hn : (fun x ↦ f x - f c) =o[𝓝 c] fun w ↦ ‖w - c‖ ^ n
hz : z ∈ ball c R₁
hR₁ : 0 < R₁
hR₂ : 0 ≤ R₂
hfne : f z ≠ f c
hne : z ≠ c
g : StrongDual ℂ F
hg : ‖g‖ = 1
hgf : g (f z - f c) = ↑‖f z - f c‖
h : ℂ → ℂ := ⇑g ∘ f ∘ ⇑(lineMap c z)
w : ℂ
hw : w ∈ ball 0 (R₁ / dist z c)
⊢ (lineMap c z) w ∈ ball c R₁

Tactic:
simpa [lt_div_iff₀, hne, dist_comm c] using hw

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
hf : MeromorphicNFAt f x
h₁f : meromorphicOrderAt f x = 0
h₂f : AnalyticAt 𝕜 f x
⊢ f x ≠ 0

Tactic:
rw [← h₂f.analyticOrderAt_eq_zero, ← ENat.map_natCast_eq_zero (α := 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
x : 𝕜
hf : MeromorphicNFAt f x
h₁f : meromorphicOrderAt f x = 0
h₂f : AnalyticAt 𝕜 f x
⊢ ENat.map Nat.cast (analyticOrderAt f x) = 0
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
f g : 𝕜 → 𝕜
x : 𝕜
hf✝ : AnalyticAt 𝕜 f x
hg : MeromorphicNFAt g x
hgne : g x ≠ 0
hf : MeromorphicNFAt f x
⊢ AnalyticAt 𝕜 g x

Tactic:
grind [meromorphicNFAt_iff_analyticAt_or]

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
U : Set 𝕜
hf : MeromorphicAt f x
h₁f : ∃ g, AnalyticAt 𝕜 g x ∧ g x ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ 0 • g z
⊢ E

Tactic:
exact (Classical.choose h₁f) x

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
f : 𝕜 → E
x : 𝕜
hf : MeromorphicAt f x
ho : 0 ≤ meromorphicOrderAt f x
⊢ Tendsto f (𝓝[≠] x) (Bornology.cobounded E) ↔ meromorphicOrderAt f x < 0

Tactic:
simp only [lt_iff_not_ge, ho, not_true_eq_false, iff_false, ← tendsto_norm_atTop_iff_cobounded]
    obtain ⟨c, hc⟩ := tendsto_nhds_of_meromorphicOrderAt_nonneg hf ho
    exact not_tendsto_atTop_of_tendsto_nhds hc.norm

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
this : (Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d u) ⊆ ↑(Finite.toFinset h)
y : 𝕜
hy : y ∈ Finite.toFinset h
hxy : x = y
⊢ meromorphicTrailingCoeffAt (fun x ↦ x - y) y ^ d y = (y - y) ^ update d y 0 y

Tactic:
meromorphicTrailingCoeffAt_id_sub_const

Goal state after:
case pos
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
hxy : x = y
⊢ (if y = y then 1 else y - y) ^ d y = (y - y) ^ update d y 0 y
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
z₀ : 𝕜
e : E
inst✝ : Decidable (e = 0)
he : ¬e = 0
⊢ (fun x ↦ e) z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, e = (z - z₀) ^ 0 • (fun x ↦ e) z

Tactic:
by simpa

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h : HasFiniteSupport d
⊢ log ‖∏ i ∈ Finite.toFinset h, (x - i) ^ update d x 0 i‖ = ∑ᶠ (u : 𝕜), ↑(d u) * log ‖x - u‖

Tactic:
have : ∀ y ∈ h.toFinset, ‖(x - y) ^ update d x 0 y‖ ≠ 0 := by
    intro y _
    by_cases h : x = y
    · rw [h]
      simp_all
    · simp_all [zpow_ne_zero, sub_ne_zero]

Goal state after:
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h : HasFiniteSupport d
this : ∀ y ∈ Finite.toFinset h, ‖(x - y) ^ update d x 0 y‖ ≠ 0
⊢ log ‖∏ i ∈ Finite.toFinset h, (x - i) ^ update d x 0 i‖ = ∑ᶠ (u : 𝕜), ↑(d u) * log ‖x - u‖
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
⊢ MeromorphicOn φ (closedBall 0 R)

Tactic:
apply smul (MeromorphicOn.finprod _) h₁f

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 := ⋯
⊢ ∀ (σ : ℂ), MeromorphicOn (canonicalFactor R σ ^ (divisor f (ball 0 R)) σ) (closedBall 0 R)
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)
z : ℂ
hz : z ∈ closedBall 0 R
⊢ MeromorphicAt (∏ᶠ (c : ℂ), canonicalFactor R c ^ (divisor f (ball 0 R)) c) z

Tactic:
· apply MeromorphicAt.finprod (fun x ↦ (meromorphic_canonicalFactor R x z).zpow _)

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

Goal state before:
pp: case neg.refine_1
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f₁ f₂ : 𝕜 → E
x : 𝕜
hf₂ : MeromorphicAt f₂ x
hf₁ : MeromorphicAt f₁ x
n₂ : ℤ
hn₂ : ↑n₂ = meromorphicOrderAt f₂ x
n₁ : ℤ
hn₁ : ↑n₁ = meromorphicOrderAt f₁ x
meromorphicOrderAt_add_eq_left_of_lt : MeromorphicAt f₂ x → ↑n₁ < ↑n₂ → meromorphicOrderAt (f₁ + f₂) x = ↑n₁
h : ↑n₁ < ↑n₂
g₁ : 𝕜 → E
h₁g₁ : AnalyticAt 𝕜 g₁ x
h₂g₁ : g₁ x ≠ 0
h₃g₁ : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f₁ z = (z - x) ^ n₁ • g₁ z
g₂ : 𝕜 → E
h₁g₂ : AnalyticAt 𝕜 g₂ x
h₂g₂ : g₂ x ≠ 0
h₃g₂ : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f₂ z = (z - x) ^ n₂ • g₂ z
⊢ AnalyticAt 𝕜 (g₁ + (fun x_1 ↦ x_1 - x) ^ (n₂ - n₁) • g₂) x

Tactic:
apply h₁g₁.add (AnalyticAt.smul _ h₁g₂)
    apply AnalyticAt.zpow_nonneg (by fun_prop)
      (sub_nonneg.2 (le_of_lt (WithTop.coe_lt_coe.1 h)))

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
f : ℂ → E
h₁f : MeromorphicOn f (closedBall 0 R)
h₂f : ∀ (u : ↑(closedBall 0 R)), meromorphicOrderAt f ↑u ≠ ⊤
hR : 0 < R
φ : ℂ → E := (∏ᶠ (c : ℂ), canonicalFactor R c ^ (divisor f (ball 0 R)) c) • f
hφ : MeromorphicOn φ (closedBall 0 R)
g : ℂ → E := toMeromorphicNFOn φ (closedBall 0 R)
h₃g : divisor g (ball 0 R) = 0
h₂g : MeromorphicNFOn g (closedBall 0 R)
h₄g : ∀ {z : ℂ}, z ∈ closedBall 0 R → meromorphicOrderAt g z ≠ ⊤
η₀ : (-divisor f (ball 0 R)).support.Finite
⊢ f =ᶠ[codiscreteWithin (closedBall 0 R)]
    (∏ i ∈ η₀.toFinset, canonicalFactor R i ^ (-(divisor f (ball 0 R)) i)) •
      (∏ᶠ (c : ℂ), canonicalFactor R c ^ (divisor f (ball 0 R)) c) • f

Tactic:
rw [finprod_eq_prod_of_mulSupport_subset_of_finite _ (by aesop) η₀]

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
⊢ f =ᶠ[codiscreteWithin (closedBall 0 R)]
    (∏ 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
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: E : Type u_1
X : Type u_2
A : Type u_3
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace A
p : E → X
homeo : IsLocalHomeomorph p
inst✝¹ : PathConnectedSpace A
inst✝ : LocallyPathConnectedSpace A
f : C(A, X)
a₀ : A
e₀ : E
he : p e₀ = f a₀
uniq :
  ∀ (γ γ' : C(↑I, A)) (Γ Γ' : C(↑I, E)),
    γ 0 = a₀ →
      γ' 0 = a₀ → Γ 0 = e₀ → Γ' 0 = e₀ → p ∘ ⇑Γ = ⇑(f.comp γ) → p ∘ ⇑Γ' = ⇑(f.comp γ') → γ 1 = γ' 1 → Γ 1 = Γ' 1
Γ : (γ : C(↑I, A)) → γ 0 = a₀ → C(↑I, E)
Γ_0 : ∀ (γ : C(↑I, A)) (a : γ 0 = a₀), (Γ γ a) 0 = e₀
Γ_lifts : ∀ (γ : C(↑I, A)) (a : γ 0 = a₀), p ∘ ⇑(Γ γ a) = ⇑(f.comp γ)
F : A → E := fun a ↦ (Γ (somePath a₀ a).toContinuousMap ⋯) 1
this : ∀ (a : A), p (F a) = f a
⊢ p ∘ ⇑(ContinuousMap.const (↑I) e₀) = ⇑(f.comp (ContinuousMap.const (↑I) a₀))

Tactic:
by simpa

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

Goal state before:
pp: E : Type u_1
X : Type u_2
A : Type u_3
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace A
p : E → X
homeo : IsLocalHomeomorph p
inst✝¹ : PathConnectedSpace A
inst✝ : LocallyPathConnectedSpace A
f : C(A, X)
a₀ : A
e₀ : E
he : p e₀ = f a₀
uniq :
  ∀ (γ γ' : C(↑I, A)) (Γ Γ' : C(↑I, E)),
    γ 0 = a₀ →
      γ' 0 = a₀ → Γ 0 = e₀ → Γ' 0 = e₀ → p ∘ ⇑Γ = ⇑(f.comp γ) → p ∘ ⇑Γ' = ⇑(f.comp γ') → γ 1 = γ' 1 → Γ 1 = Γ' 1
Γ : (γ : C(↑I, A)) → γ 0 = a₀ → C(↑I, E)
Γ_0 : ∀ (γ : C(↑I, A)) (a : γ 0 = a₀), (Γ γ a) 0 = e₀
Γ_lifts : ∀ (γ : C(↑I, A)) (a : γ 0 = a₀), p ∘ ⇑(Γ γ a) = ⇑(f.comp γ)
F : A → E := fun a ↦ (Γ (somePath a₀ a).toContinuousMap ⋯) 1
this : ∀ (a : A), p (F a) = f a
F' : C(A, E)
x✝ : (fun F ↦ F a₀ = e₀ ∧ p ∘ ⇑F = ⇑f) F'
F'_0 : F' a₀ = e₀
hpF' : p ∘ ⇑F' = ⇑f
a : A
γ : Path a₀ a := somePath a₀ a
⊢ (F'.comp ↑γ) 0 = e₀

Tactic:
by simpa

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 := ⋯
hφ : MeromorphicOn φ (closedBall 0 R)
g : ℂ → E := ⋯
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
⊢ (∏ i ∈ η₀.toFinset, canonicalFactor R i ^ (-(divisor f (ball 0 R)) i)) • φ =ᶠ[codiscreteWithin (closedBall 0 R)]
    (∏ᶠ (u : ℂ), canonicalFactor R u ^ (-(divisor f (ball 0 R)) u)) • g

Tactic:
rw [finprod_eq_prod_of_mulSupport_subset_of_finite _ (by aesop) η₀]

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
⊢ (∏ i ∈ η₀.toFinset, canonicalFactor R i ^ (-(divisor f (ball 0 R)) i)) • φ =ᶠ[codiscreteWithin (closedBall 0 R)]
    (∏ i ∈ η₀.toFinset, canonicalFactor R i ^ (-(divisor f (ball 0 R)) i)) • g
Module: Mathlib.Analysis.Complex.Conformal

Goal state before:
pp: g : ℂ →L[ℝ] ℂ
h : IsConformalMap g
⊢ (∃ map, restrictScalars ℝ map = g) ∨ ∃ map, restrictScalars ℝ map = g ∘SL ↑conjCLE

Tactic:
by
  rcases h with ⟨c, -, li, rfl⟩
  obtain ⟨li, rfl⟩ : ∃ li' : ℂ ≃ₗᵢ[ℝ] ℂ, li'.toLinearIsometry = li :=
    ⟨li.toLinearIsometryEquiv rfl, by ext1; rfl⟩
  rcases linear_isometry_complex li with ⟨a, rfl | rfl⟩
  -- let rot := c • (a : ℂ) • ContinuousLinearMap.id ℂ ℂ,
  · refine Or.inl ⟨c • (a : ℂ) • ContinuousLinearMap.id ℂ ℂ, ?_⟩
    ext1
    simp
  · refine Or.inr ⟨c • (a : ℂ) • ContinuousLinearMap.id ℂ ℂ, ?_⟩
    ext1
    simp

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

Goal state before:
pp: 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
cov : IsCoveringMap p
γ✝ : C(↑I, X)
e✝ : E
γ_0 : γ✝ 0 = p e✝
x y z : X
e : E
hpe : x = p e
γ : Path x y
γ' : Path y z
Γ : C(↑I, E) := cov.liftPath (↑γ) e ⋯
⊢ ↑(γ.trans γ') 0 = p e

Tactic:
by simpa

Goal state after: