mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.DerivIntegrable

Goal state before:
pp: case left
f : ℝ → ℝ
a b : ℝ
hf : MonotoneOn f (Icc a b)
hab : a ≤ b
G : ℕ → ℝ → ℝ
hGf : ∀ᵐ (x : ℝ), x ∈ uIcc a b → Tendsto (fun n ↦ G n x) atTop (𝓝 (deriv f x))
hG : ∀ (n : ℕ), AEStronglyMeasurable (G n) (volume.restrict (uIcc a b))
hG' : liminf (fun n ↦ ∫⁻ (x : ℝ) in uIcc a b, ‖G n x‖ₑ) atTop ≤ ENNReal.ofReal (f b - f a)
hG'₀ : liminf (fun n ↦ ∫⁻ (x : ℝ) in Icc a b, ‖G n x‖ₑ) atTop ≠ ⊤
integrable_f_deriv : Integrable (deriv f) (volume.restrict (Icc a b))
this : f a ≤ f b
f_deriv_nonneg : ∀ {x : ℝ}, x ∈ Ioo a b → 0 ≤ deriv f x
⊢ 0 ≤ᵐ[volume.restrict (Icc a b)] deriv f

Tactic:
rw [Filter.EventuallyLE, ← restrict_Ioo_eq_restrict_Icc,
      MeasureTheory.ae_restrict_iff' measurableSet_Ioo]

Goal state after:
case left
f : ℝ → ℝ
a b : ℝ
hf : MonotoneOn f (Icc a b)
hab : a ≤ b
G : ℕ → ℝ → ℝ
hGf : ∀ᵐ (x : ℝ), x ∈ uIcc a b → Tendsto (fun n ↦ G n x) atTop (𝓝 (deriv f x))
hG : ∀ (n : ℕ), AEStronglyMeasurable (G n) (volume.restrict (uIcc a b))
hG' : liminf (fun n ↦ ∫⁻ (x : ℝ) in uIcc a b, ‖G n x‖ₑ) atTop ≤ ENNReal.ofReal (f b - f a)
hG'₀ : liminf (fun n ↦ ∫⁻ (x : ℝ) in Icc a b, ‖G n x‖ₑ) atTop ≠ ⊤
integrable_f_deriv : Integrable (deriv f) (volume.restrict (Icc a b))
this : f a ≤ f b
f_deriv_nonneg : ∀ {x : ℝ}, x ∈ Ioo a b → 0 ≤ deriv f x
⊢ ∀ᵐ (x : ℝ), x ∈ Ioo a b → 0 x ≤ deriv f x
Module: Mathlib.Analysis.Calculus.Rademacher

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
C : ℝ≥0
inst✝ : FiniteDimensional ℝ E
f : E → F
hf : LipschitzWith C f
s : Set E
hs : sphere 0 1 ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
L : E →L[ℝ] F
x : E
hL : ∀ v ∈ s, HasLineDerivAt ℝ f (L v) x v
ε : ℝ
εpos : 0 < ε
δ : ℝ
δpos : 0 < δ
hδ : (↑C + ‖L‖ + 1) * δ = ε
q : Set E
hqs : q ⊆ s
q_fin : q.Finite
hq : sphere 0 1 ⊆ ⋃ y ∈ q, ball y δ
I : ∀ᶠ (t : ℝ) in 𝓝 0, ∀ v ∈ q, ‖f (x + t • v) - f x - t • L v‖ ≤ δ * ‖t‖
r : ℝ
r_pos : 0 < r
hr : ∀ (t : ℝ), ‖t‖ < r → ∀ v ∈ q, ‖f (x + t • v) - f x - t • L v‖ ≤ δ * ‖t‖
v : E
hv : v ∈ ball 0 r
v_ne : v ≠ 0
w : E
ρ : ℝ
w_mem : w ∈ sphere 0 1
hvw : v = ρ • w
hρ : ρ = ‖v‖
norm_rho : ‖ρ‖ = ρ
rho_pos : 0 ≤ ρ
y : E
yq : y ∈ q
hy : ‖w - y‖ < δ
this : ‖y - w‖ < δ
| δ * ρ

Tactic:
← norm_rho

Goal state after:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
C : ℝ≥0
inst✝ : FiniteDimensional ℝ E
f : E → F
hf : LipschitzWith C f
s : Set E
hs : sphere 0 1 ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
L : E →L[ℝ] F
x : E
hL : ∀ v ∈ s, HasLineDerivAt ℝ f (L v) x v
ε : ℝ
εpos : 0 < ε
δ : ℝ
δpos : 0 < δ
hδ : (↑C + ‖L‖ + 1) * δ = ε
q : Set E
hqs : q ⊆ s
q_fin : q.Finite
hq : sphere 0 1 ⊆ ⋃ y ∈ q, ball y δ
I : ∀ᶠ (t : ℝ) in 𝓝 0, ∀ v ∈ q, ‖f (x + t • v) - f x - t • L v‖ ≤ δ * ‖t‖
r : ℝ
r_pos : 0 < r
hr : ∀ (t : ℝ), ‖t‖ < r → ∀ v ∈ q, ‖f (x + t • v) - f x - t • L v‖ ≤ δ * ‖t‖
v : E
hv : v ∈ ball 0 r
v_ne : v ≠ 0
w : E
ρ : ℝ
w_mem : w ∈ sphere 0 1
hvw : v = ρ • w
hρ : ρ = ‖v‖
norm_rho : ‖ρ‖ = ρ
rho_pos : 0 ≤ ρ
y : E
yq : y ∈ q
hy : ‖w - y‖ < δ
this : ‖y - w‖ < δ
| δ * ‖ρ‖
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: s x y : ℝ
hx₀ : 0 < x
hx₁ : x < 1 / (1 + s ^ 2)
hy : |y| < s * x
⊢ (1 - x) ^ 2 + y ^ 2 ≤ (1 - x) ^ 2 + (s * x) ^ 2

Tactic:
← sq_abs y

Goal state after:
s x y : ℝ
hx₀ : 0 < x
hx₁ : x < 1 / (1 + s ^ 2)
hy : |y| < s * x
⊢ (1 - x) ^ 2 + |y| ^ 2 ≤ (1 - x) ^ 2 + (s * x) ^ 2
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: s x y : ℝ
hx₀ : 0 < x
hx₁ : x < 1 / (1 + s ^ 2)
hy : |y| < s * x
H : √((1 - x) ^ 2 + y ^ 2) ≤ 1 - x / 2
⊢ √(x ^ 2 + y ^ 2) ≤ √(x ^ 2 + (s * x) ^ 2)

Tactic:
← sq_abs y

Goal state after:
s x y : ℝ
hx₀ : 0 < x
hx₁ : x < 1 / (1 + s ^ 2)
hy : |y| < s * x
H : √((1 - x) ^ 2 + y ^ 2) ≤ 1 - x / 2
⊢ √(x ^ 2 + |y| ^ 2) ≤ √(x ^ 2 + (s * x) ^ 2)
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: s x y : ℝ
hx₀ : 0 < x
hx₁ : x < 1 / (1 + s ^ 2)
hy : |y| < s * x
H : √((1 - x) ^ 2 + y ^ 2) ≤ 1 - x / 2
⊢ √(1 + s ^ 2) * √(x ^ 2) = √(1 + s ^ 2) * x

Tactic:
sqrt_sq hx₀.le

Goal state after:
s x y : ℝ
hx₀ : 0 < x
hx₁ : x < 1 / (1 + s ^ 2)
hy : |y| < s * x
H : √((1 - x) ^ 2 + y ^ 2) ≤ 1 - x / 2
⊢ √(1 + s ^ 2) * x = √(1 + s ^ 2) * x
Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: case inr.x
F : Type u_2
inst✝ : SeminormedAddCommGroup F
a✝ b✝ : ℝ
f : ℝ → F
a b : ℝ
hf :
  ∀ ε > 0,
    ∃ δ > 0,
      ∀ E ∈ disjWithin a b,
        ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2 < δ →
          ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2) < ε
hab₀ : a ≤ b
hab : a < b
δ : ℝ
hδ₁ : δ > 0
hδ₂ :
  ∀ E ∈ disjWithin a b,
    ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2 < δ → ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2) < 1
hab₁ : 0 < b - a
n : ℕ
hn : 1 / (↑n + 1) < δ / (b - a)
δ' : ℝ := ⋯
hδ₃ : δ' < δ
h_mono : Monotone fun i ↦ a + ↑i * δ'
v_sum : eVariationOn f (Icc a b) = ∑ i ∈ Finset.range (n + 1), eVariationOn f (Icc (a + ↑i * δ') (a + (↑i + 1) * δ'))
v_each : ∀ (x y : ℝ), a ≤ x → x ≤ y → y < x + δ → y ≤ b → eVariationOn f (Icc x y) ≤ 1
i : ℕ
hi : i < n + 1
⊢ a + (↑i + 1) * δ' ≤ b

Tactic:
convert! h_mono (show i + 1 ≤ n + 1 by lia)

Goal state after:
case e'_3.e'_6
F : Type u_2
inst✝ : SeminormedAddCommGroup F
a✝ b✝ : ℝ
f : ℝ → F
a b : ℝ
hf :
  ∀ ε > 0,
    ∃ δ > 0,
      ∀ E ∈ disjWithin a b,
        ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2 < δ →
          ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2) < ε
hab₀ : a ≤ b
hab : a < b
δ : ℝ
hδ₁ : δ > 0
hδ₂ :
  ∀ E ∈ disjWithin a b,
    ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2 < δ → ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2) < 1
hab₁ : 0 < b - a
n : ℕ
hn : 1 / (↑n + 1) < δ / (b - a)
δ' : ℝ := (b - a) / (↑n + 1)
hδ₃ : δ' < δ
h_mono : Monotone fun i ↦ a + ↑i * δ'
v_sum : eVariationOn f (Icc a b) = ∑ i ∈ Finset.range (n + 1), eVariationOn f (Icc (a + ↑i * δ') (a + (↑i + 1) * δ'))
v_each : ∀ (x y : ℝ), a ≤ x → x ≤ y → y < x + δ → y ≤ b → eVariationOn f (Icc x y) ≤ 1
i : ℕ
hi : i < n + 1
⊢ ↑i + 1 = ↑(i + 1)

case e'_4
F : Type u_2
inst✝ : SeminormedAddCommGroup F
a✝ b✝ : ℝ
f : ℝ → F
a b : ℝ
hf :
  ∀ ε > 0,
    ∃ δ > 0,
      ∀ E ∈ disjWithin a b,
        ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2 < δ →
          ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2) < ε
hab₀ : a ≤ b
hab : a < b
δ : ℝ
hδ₁ : δ > 0
hδ₂ :
  ∀ E ∈ disjWithin a b,
    ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2 < δ → ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2) < 1
hab₁ : 0 < b - a
n : ℕ
hn : 1 / (↑n + 1) < δ / (b - a)
δ' : ℝ := (b - a) / (↑n + 1)
hδ₃ : δ' < δ
h_mono : Monotone fun i ↦ a + ↑i * δ'
v_sum : eVariationOn f (Icc a b) = ∑ i ∈ Finset.range (n + 1), eVariationOn f (Icc (a + ↑i * δ') (a + (↑i + 1) * δ'))
v_each : ∀ (x y : ℝ), a ≤ x → x ≤ y → y < x + δ → y ≤ b → eVariationOn f (Icc x y) ≤ 1
i : ℕ
hi : i < n + 1
⊢ b = (fun i ↦ a + ↑i * δ') (n + 1)
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 := fun E ↦ ⋃ i ∈ Finset.range E.1, uIoc (E.2 i).1 (E.2 i).2
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
⊢ ∑ x ∈ Finset.range n, dist (∫ (v : ℝ) in c..(I x).1, f v) (∫ (v : ℝ) in c..(I x).2, f v) ≤
    ∑ i ∈ Finset.range (n, I).1, ∫ (x : ℝ) in uIoc ((n, I).2 i).1 ((n, I).2 i).2, ‖f x‖ ∂volume.restrict (uIoc a b)

Tactic:
refine Finset.sum_le_sum (fun i hi ↦ ?_)

Goal state after:
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 := fun E ↦ ⋃ i ∈ Finset.range E.1, uIoc (E.2 i).1 (E.2 i).2
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
⊢ dist (∫ (v : ℝ) in c..(I i).1, f v) (∫ (v : ℝ) in c..(I i).2, f v) ≤
    ∫ (x : ℝ) in uIoc ((n, I).2 i).1 ((n, I).2 i).2, ‖f x‖ ∂volume.restrict (uIoc a b)
Module: Mathlib.Analysis.Convex.SpecificFunctions.Deriv

Goal state before:
pp: m : ℤ
n : ℕ
hn : Even n
hm : m ∉ Ico 0 ↑n
h : ∏ k ∈ Finset.range n, (m - ↑k) = 0
⊢ m ∈ Ico 0 ↑n

Tactic:
Finset.prod_eq_zero_iff

Goal state after:
m : ℤ
n : ℕ
hn : Even n
hm : m ∉ Ico 0 ↑n
h : ∃ a ∈ Finset.range n, m - ↑a = 0
⊢ m ∈ Ico 0 ↑n
Module: Mathlib.Analysis.Complex.AbsMax

Goal state before:
pp: F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : ℂ → F
z w : ℂ
hd : DiffContOnCl ℂ f (ball z (dist w z))
hz : IsMaxOn (norm ∘ f) (ball z (dist w z)) z
⊢ ‖f w‖ = ‖f z‖

Tactic:
rcases eq_or_ne w z with (rfl | hne)

Goal state after:
case inl
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : ℂ → F
w : ℂ
hd : DiffContOnCl ℂ f (ball w (dist w w))
hz : IsMaxOn (norm ∘ f) (ball w (dist w w)) w
⊢ ‖f w‖ = ‖f w‖

case inr
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : ℂ → F
z w : ℂ
hd : DiffContOnCl ℂ f (ball z (dist w z))
hz : IsMaxOn (norm ∘ f) (ball z (dist w z)) z
hne : w ≠ z
⊢ ‖f w‖ = ‖f z‖
Module: Mathlib.Analysis.Complex.AbsMax

Goal state before:
pp: E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : E → ℂ
c : E
hf : ∀ᶠ (z : E) in 𝓝 c, DifferentiableAt ℂ f z
hc : IsLocalMin (norm ∘ f) c
h : ¬f c = 0
h1 : ∀ᶠ (z : E) in 𝓝 c, f z ≠ 0
⊢ ∀ᶠ (z : E) in 𝓝 c, f z = f c

Tactic:
have h2 : IsLocalMax (norm ∘ f)⁻¹ c := hc.inv (h1.mono fun z => norm_pos_iff.mpr)

Goal state after:
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : E → ℂ
c : E
hf : ∀ᶠ (z : E) in 𝓝 c, DifferentiableAt ℂ f z
hc : IsLocalMin (norm ∘ f) c
h : ¬f c = 0
h1 : ∀ᶠ (z : E) in 𝓝 c, f z ≠ 0
h2 : IsLocalMax (norm ∘ f)⁻¹ c
⊢ ∀ᶠ (z : E) in 𝓝 c, f z = f c
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: S : Set ℝ
f : ℝ → ℝ
x : ℝ
hf : ConvexOn ℝ S f
hx : x ∈ interior S
hf_ld : derivWithin f (Iio x) x ≤ 0
hf_rd : 0 ≤ derivWithin f (Ioi x) x
y : ℝ
hy : y ∈ S
⊢ y ∈ {x_1 | (fun x_2 ↦ f x ≤ f x_2) x_1}

Tactic:
rcases lt_trichotomy x y with hxy | h_eq | hyx

Goal state after:
case inl
S : Set ℝ
f : ℝ → ℝ
x : ℝ
hf : ConvexOn ℝ S f
hx : x ∈ interior S
hf_ld : derivWithin f (Iio x) x ≤ 0
hf_rd : 0 ≤ derivWithin f (Ioi x) x
y : ℝ
hy : y ∈ S
hxy : x < y
⊢ y ∈ {x_1 | (fun x_2 ↦ f x ≤ f x_2) x_1}

case inr.inl
S : Set ℝ
f : ℝ → ℝ
x : ℝ
hf : ConvexOn ℝ S f
hx : x ∈ interior S
hf_ld : derivWithin f (Iio x) x ≤ 0
hf_rd : 0 ≤ derivWithin f (Ioi x) x
y : ℝ
hy : y ∈ S
h_eq : x = y
⊢ y ∈ {x_1 | (fun x_2 ↦ f x ≤ f x_2) x_1}

case inr.inr
S : Set ℝ
f : ℝ → ℝ
x : ℝ
hf : ConvexOn ℝ S f
hx : x ∈ interior S
hf_ld : derivWithin f (Iio x) x ≤ 0
hf_rd : 0 ≤ derivWithin f (Ioi x) x
y : ℝ
hy : y ∈ S
hyx : y < x
⊢ y ∈ {x_1 | (fun x_2 ↦ f x ≤ f x_2) x_1}
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
⊢ ?m.101

Tactic:
simpa only [dist_zero_right, div_one, mul_comm (1 / R), mul_one_div]
    using schwarz_applied hM hf hf₁ hz hf₂

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

Goal state before:
pp: f : ℂ → ℂ
c z : ℂ
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 - c‖ < R₁
hR₁ : 0 < R₁
this :
  ∀ {R₁ : ℝ},
    DifferentiableOn ℂ f (ball c R₁) →
      MapsTo f (ball c R₁) (closedBall (f c) R₂) →
        z ∈ ball c R₁ →
          0 < R₁ →
            DifferentiableOn ℂ f (closedBall c R₁) ∧ MapsTo f (closedBall c R₁) (closedBall (f c) R₂) →
              ‖f z - f c‖ ≤ R₂ * (‖z - c‖ / R₁) ^ (n + 1)
hd' : ¬(DifferentiableOn ℂ f (closedBall c R₁) ∧ MapsTo f (closedBall c R₁) (closedBall (f c) R₂))
r : ℝ
⊢ r ∈ Ioo ‖z - c‖ R₁ → ‖f z - f c‖ ≤ R₂ * (‖z - c‖ / r) ^ (n + 1)

Tactic:
⟨hzr, hrR₁⟩

Goal state after:
f : ℂ → ℂ
c z : ℂ
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 - c‖ < R₁
hR₁ : 0 < R₁
this :
  ∀ {R₁ : ℝ},
    DifferentiableOn ℂ f (ball c R₁) →
      MapsTo f (ball c R₁) (closedBall (f c) R₂) →
        z ∈ ball c R₁ →
          0 < R₁ →
            DifferentiableOn ℂ f (closedBall c R₁) ∧ MapsTo f (closedBall c R₁) (closedBall (f c) R₂) →
              ‖f z - f c‖ ≤ R₂ * (‖z - c‖ / R₁) ^ (n + 1)
hd' : ¬(DifferentiableOn ℂ f (closedBall c R₁) ∧ MapsTo f (closedBall c R₁) (closedBall (f c) R₂))
r : ℝ
hzr : ‖z - c‖ < r
hrR₁ : r < R₁
⊢ ‖f z - f c‖ ≤ R₂ * (‖z - c‖ / r) ^ (n + 1)
Module: Mathlib.Analysis.Complex.Schwarz

Goal state before:
pp: f : ℂ → ℂ
c z : ℂ
R₂ : ℝ
n : ℕ
hn : (fun x ↦ f x - f c) =o[𝓝 c] fun w ↦ (w - c) ^ n
R₁ : ℝ
hz : z ∈ ball c R₁
hR₁ : 0 < R₁
hd : DifferentiableOn ℂ f (closedBall c R₁)
h_maps : MapsTo f (closedBall c R₁) (closedBall (f c) R₂)
hne : z ≠ c
g : ℂ → ℂ := fun w ↦ ((w - c) ^ (n + 1))⁻¹ * (f w - f c)
g' : ℂ → ℂ := update g c ((𝓝[≠] c).limUnder g)
hdg' : DifferentiableOn ℂ g' (closedBall c R₁)
this : ‖g' z‖ ≤ R₂ / R₁ ^ (n + 1)
⊢ ‖f z - f c‖ ≤ R₂ * (‖z - c‖ / R₁) ^ (n + 1)

Tactic:
by
    have hz' : ‖z - c‖ ≠ 0 := by simpa [sub_eq_zero] using hne
    simpa [g', hne, g, div_pow, mul_comm, field] using this

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

Goal state before:
pp: case refine_2
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : LocallyPathConnectedSpace X
U : Set X
hUc : IsSimplyConnected U
hUo : IsOpen U
g : X → ℂ
hgc : ContinuousOn g U
hU₀ : 0 ∉ g '' U
n : ℕ
hn : n ≠ 0
f : X → ℂ
hfc : ContinuousOn f U
hf : EqOn (cexp ∘ f) g U
z : X
⊢ U.piecewise (fun x ↦ cexp (f x / ↑n)) (fun x ↦ g x ^ (1 / ↑n)) z ^ n = g z

Tactic:
by_cases hz : z ∈ U

Goal state after:
case pos
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : LocallyPathConnectedSpace X
U : Set X
hUc : IsSimplyConnected U
hUo : IsOpen U
g : X → ℂ
hgc : ContinuousOn g U
hU₀ : 0 ∉ g '' U
n : ℕ
hn : n ≠ 0
f : X → ℂ
hfc : ContinuousOn f U
hf : EqOn (cexp ∘ f) g U
z : X
hz : z ∈ U
⊢ U.piecewise (fun x ↦ cexp (f x / ↑n)) (fun x ↦ g x ^ (1 / ↑n)) z ^ n = g z

case neg
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : LocallyPathConnectedSpace X
U : Set X
hUc : IsSimplyConnected U
hUo : IsOpen U
g : X → ℂ
hgc : ContinuousOn g U
hU₀ : 0 ∉ g '' U
n : ℕ
hn : n ≠ 0
f : X → ℂ
hfc : ContinuousOn f U
hf : EqOn (cexp ∘ f) g U
z : X
hz : z ∉ U
⊢ U.piecewise (fun x ↦ cexp (f x / ↑n)) (fun x ↦ g x ^ (1 / ↑n)) z ^ n = g z
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
x : 𝕜
hx : x ∈ U
⊢ (divisor f₁ U) x = (divisor f₂ U) x

Tactic:
simp only [hf₁, hx, divisor_apply, hf₁.congr_codiscreteWithin h₁ h₂]

Goal state after:
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
x : 𝕜
hx : x ∈ U
⊢ (meromorphicOrderAt f₁ x).untop₀ = (meromorphicOrderAt f₂ x).untop₀
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: 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
h : f =ᶠ[𝓝 x] 0
⊢ AnalyticAt 𝕜 f x ∨ MeromorphicAt f x ∧ meromorphicOrderAt f x < 0 ∧ f x = 0

Tactic:
simp [(analyticAt_congr h).2 analyticAt_const]

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 : 𝕜 → 𝕜
f₂ : 𝕜 → E
h₁f₁ : MeromorphicOn f₁ U
h₁f₂ : MeromorphicOn f₂ U
h₂f₁ : ∀ z ∈ U, meromorphicOrderAt f₁ z ≠ ⊤
h₂f₂ : ∀ z ∈ U, meromorphicOrderAt f₂ z ≠ ⊤
z : 𝕜
⊢ (divisor (f₁ • f₂) U) z = (divisor f₁ U + divisor f₂ U) z

Tactic:
by_cases hz : z ∈ U

Goal state after:
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 : 𝕜 → 𝕜
f₂ : 𝕜 → E
h₁f₁ : MeromorphicOn f₁ U
h₁f₂ : MeromorphicOn f₂ U
h₂f₁ : ∀ z ∈ U, meromorphicOrderAt f₁ z ≠ ⊤
h₂f₂ : ∀ z ∈ U, meromorphicOrderAt f₂ z ≠ ⊤
z : 𝕜
hz : z ∈ U
⊢ (divisor (f₁ • f₂) U) z = (divisor f₁ U + divisor f₂ U) z

case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f₁ : 𝕜 → 𝕜
f₂ : 𝕜 → E
h₁f₁ : MeromorphicOn f₁ U
h₁f₂ : MeromorphicOn f₂ U
h₂f₁ : ∀ z ∈ U, meromorphicOrderAt f₁ z ≠ ⊤
h₂f₂ : ∀ z ∈ U, meromorphicOrderAt f₂ z ≠ ⊤
z : 𝕜
hz : z ∉ U
⊢ (divisor (f₁ • f₂) U) z = (divisor f₁ U + divisor f₂ U) z
Module: Mathlib.Analysis.Meromorphic.Divisor

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
U : Set 𝕜
f : 𝕜 → 𝕜
hf : MeromorphicOn f U
n : ℕ
z : 𝕜
hn : ¬n = 0
⊢ (divisor (f ^ n) U) z = (n • divisor f U) z

Tactic:
by_cases hz : z ∈ U

Goal state after:
case pos
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
U : Set 𝕜
f : 𝕜 → 𝕜
hf : MeromorphicOn f U
n : ℕ
z : 𝕜
hn : ¬n = 0
hz : z ∈ U
⊢ (divisor (f ^ n) U) z = (n • divisor f U) z

case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
U : Set 𝕜
f : 𝕜 → 𝕜
hf : MeromorphicOn f U
n : ℕ
z : 𝕜
hn : ¬n = 0
hz : z ∉ U
⊢ (divisor (f ^ n) U) z = (n • divisor f U) z
Module: Mathlib.Analysis.Meromorphic.Divisor

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
U : Set 𝕜
f : 𝕜 → 𝕜
hf : MeromorphicOn f U
n : ℤ
z : 𝕜
hn : ¬n = 0
⊢ (divisor (f ^ n) U) z = (n • divisor f U) z

Tactic:
by_cases hz : z ∈ U

Goal state after:
case pos
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
U : Set 𝕜
f : 𝕜 → 𝕜
hf : MeromorphicOn f U
n : ℤ
z : 𝕜
hn : ¬n = 0
hz : z ∈ U
⊢ (divisor (f ^ n) U) z = (n • divisor f U) z

case neg
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
U : Set 𝕜
f : 𝕜 → 𝕜
hf : MeromorphicOn f U
n : ℤ
z : 𝕜
hn : ¬n = 0
hz : z ∉ U
⊢ (divisor (f ^ n) U) z = (n • divisor f U) z
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
f : 𝕜 → E
x : 𝕜
h : MeromorphicAt f x
⊢ Tendsto ((fun x_1 ↦ x_1 - x) ^ (-(meromorphicOrderAt f x).untop₀) • f) (𝓝[≠] x) (𝓝 (meromorphicTrailingCoeffAt f x))

Tactic:
by_cases h₂ : meromorphicOrderAt f x = ⊤
  · 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
  obtain ⟨g, h₁g, h₂g, h₃g⟩ := (meromorphicOrderAt_ne_top_iff h).1 h₂
  apply Tendsto.congr' (f₁ := g)
  · filter_upwards [h₃g, self_mem_nhdsWithin] with y h₁y h₂y
    rw [zpow_neg, Pi.smul_apply', Pi.inv_apply, Pi.pow_apply, h₁y, ← smul_assoc, smul_eq_mul,
      ← zpow_neg, ← zpow_add', neg_add_cancel, zpow_zero, one_smul]
    left
    simp_all [sub_ne_zero]
  · rw [h₁g.meromorphicTrailingCoeffAt_of_eq_nhdsNE h₃g]
    apply h₁g.continuousAt.continuousWithinAt

Goal state after:
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
h₁f : MeromorphicNFOn f U
⊢ 0 ≤ MeromorphicOn.divisor f U ↔ AnalyticOnNhd 𝕜 f U

Tactic:
constructor <;> intro h x

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
h₁f : MeromorphicNFOn f U
h : 0 ≤ MeromorphicOn.divisor f U
x : 𝕜
⊢ x ∈ U → AnalyticAt 𝕜 f x

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
h₁f : MeromorphicNFOn f U
h : AnalyticOnNhd 𝕜 f U
x : 𝕜
⊢ 0 x ≤ (MeromorphicOn.divisor f U) x
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
h₁f : MeromorphicOn f U
z : 𝕜
⊢ E

Tactic:
by_cases hz : z ∈ U

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
f : 𝕜 → E
g : 𝕜 → 𝕜
x : 𝕜
U : Set 𝕜
h₁f : MeromorphicOn f U
z : 𝕜
hz : z ∈ U
⊢ E

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
U : Set 𝕜
h₁f : MeromorphicOn f U
z : 𝕜
hz : z ∉ U
⊢ E
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
z : 𝕜
⊢ (divisor (toMeromorphicNFOn f U) U) z = (divisor f U) z

Tactic:
by_cases hz : z ∈ U

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
f : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
z : 𝕜
hz : z ∈ U
⊢ (divisor (toMeromorphicNFOn f U) U) z = (divisor f U) z

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
U : Set 𝕜
hf : MeromorphicOn f U
z : 𝕜
hz : z ∉ U
⊢ (divisor (toMeromorphicNFOn f U) U) z = (divisor f U) z
Module: Mathlib.Analysis.Meromorphic.TrailingCoefficient

Goal state before:
pp: case insert
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
ι : Type u_3
f : ι → 𝕜 → 𝕜
x : 𝕜
σ : ι
s₁ : Finset ι
hσ : σ ∉ s₁
hind :
  (∀ σ ∈ s₁, MeromorphicAt (f σ) x) →
    meromorphicTrailingCoeffAt (∏ n ∈ s₁, f n) x = ∏ n ∈ s₁, meromorphicTrailingCoeffAt (f n) x
h : ∀ σ_1 ∈ insert σ s₁, MeromorphicAt (f σ_1) x
this : ∀ σ₀ ∈ s₁, MeromorphicAt (f σ₀) x
⊢ meromorphicTrailingCoeffAt (∏ n ∈ insert σ s₁, f n) x = ∏ n ∈ insert σ s₁, meromorphicTrailingCoeffAt (f n) x

Tactic:
rw [Finset.prod_insert hσ, Finset.prod_insert hσ,
      (h σ (Finset.mem_insert_self σ s₁)).meromorphicTrailingCoeffAt_mul
      (MeromorphicAt.prod this), hind this]

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 : meromorphicOrderAt f x ≠ 0
⊢ (∃ c, c ≠ 0 ∧ Tendsto f (𝓝[≠] x) (𝓝 c)) ↔ meromorphicOrderAt f x = 0

Tactic:
simp only [ne_eq, ho, iff_false, not_exists, not_and]

Goal state after:
case inr
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : MeromorphicAt f x
ho : meromorphicOrderAt f x ≠ 0
⊢ ∀ (x_1 : E), ¬x_1 = 0 → ¬Tendsto f (𝓝[≠] x) (𝓝 x_1)