mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Distribution.AEEqOfIntegralContDiff

Goal state before:
pp: E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : FiniteDimensional ℝ E
F : Type u_2
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
inst✝⁷ : CompleteSpace F
H : Type u_3
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_4
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : IsManifold I ∞ M
inst✝² : MeasurableSpace M
inst✝¹ : BorelSpace M
inst✝ : T2Space M
f : M → F
μ : Measure M
U✝ : Set M
hU : IsOpen[inst✝⁵] U✝
hSig : IsSigmaCompact U✝
hf : LocallyIntegrableOn f U✝ μ
meas_U : MeasurableSet U✝
U : Opens M := { carrier := U✝, is_open' := hU }
this : SigmaCompactSpace ↥U
g : ↥U → ℝ
h : ∫ (x : M), Function.extend Subtype.val g 0 x • f x ∂μ = 0
g_smth : ContMDiff I 𝓘(ℝ, ℝ) ∞ g
g_supp : HasCompactSupport g
x : M
hx : x ∉ ↑U
⊢ (∃ a, ↑a = x) → x ∈ ↑U

Tactic:
· rintro ⟨x, rfl⟩; exact x.2

Goal state after:
Module: Mathlib.Analysis.Calculus.Rademacher

Goal state before:
pp: case hF_meas
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 g : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
hg : Integrable g μ
v : E
⊢ ∀ᶠ (n : ℝ) in 𝓝[>] 0, AEStronglyMeasurable (fun a ↦ n⁻¹ • (f (a + n • v) - f a) * g a) μ

pp: case h_bound
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 g : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
hg : Integrable g μ
v : E
⊢ ∀ᶠ (n : ℝ) in 𝓝[>] 0, ∀ᵐ (a : E) ∂μ, ‖n⁻¹ • (f (a + n • v) - f a) * g a‖ ≤ ↑C * ‖v‖ * ‖g a‖

pp: case bound_integrable
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 g : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
hg : Integrable g μ
v : E
⊢ Integrable (fun x ↦ ↑C * ‖v‖ * ‖g x‖) μ

pp: case h_lim
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 g : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
hg : Integrable g μ
v : E
⊢ ∀ᵐ (a : E) ∂μ, Tendsto (fun n ↦ n⁻¹ • (f (a + n • v) - f a) * g a) (𝓝[>] 0) (𝓝 (lineDeriv ℝ f a v * g a))

Tactic:
· filter_upwards with t
    apply AEStronglyMeasurable.mul ?_ hg.aestronglyMeasurable
    apply aestronglyMeasurable_const.fun_smul
    apply AEStronglyMeasurable.sub _ hf.continuous.measurable.aestronglyMeasurable
    apply AEMeasurable.aestronglyMeasurable
    exact hf.continuous.measurable.comp_aemeasurable' (aemeasurable_id'.add_const _)

Goal state after:
case h_bound
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 g : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
hg : Integrable g μ
v : E
⊢ ∀ᶠ (n : ℝ) in 𝓝[>] 0, ∀ᵐ (a : E) ∂μ, ‖n⁻¹ • (f (a + n • v) - f a) * g a‖ ≤ ↑C * ‖v‖ * ‖g a‖

case bound_integrable
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 g : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
hg : Integrable g μ
v : E
⊢ Integrable (fun x ↦ ↑C * ‖v‖ * ‖g x‖) μ

case h_lim
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 g : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
hg : Integrable g μ
v : E
⊢ ∀ᵐ (a : E) ∂μ, Tendsto (fun n ↦ n⁻¹ • (f (a + n • v) - f a) * g a) (𝓝[>] 0) (𝓝 (lineDeriv ℝ f a v * g a))
Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral

Goal state before:
pp: E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
β : Type u_3
inst✝⁴ : TopologicalSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
inst✝¹ : SigmaCompactSpace β
inst✝ : R1Space β
μ : Measure β
f : β → E
hf : LocallyIntegrable f μ
h'f : ∀ (s : Set β), IsCompact s → ∫ (x : β) in s, f x ∂μ = 0
⊢ f =ᵐ[μ.restrict Set.univ] 0

Tactic:
← iUnion_closure_compactCovering

Goal state after:
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
β : Type u_3
inst✝⁴ : TopologicalSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
inst✝¹ : SigmaCompactSpace β
inst✝ : R1Space β
μ : Measure β
f : β → E
hf : LocallyIntegrable f μ
h'f : ∀ (s : Set β), IsCompact s → ∫ (x : β) in s, f x ∂μ = 0
⊢ f =ᵐ[μ.restrict (⋃ n, closure[inst✝⁴] (compactCovering β n))] 0
Module: Mathlib.Analysis.Calculus.TangentCone.Seq

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
x : E
hx : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
c : 𝕜
hc : 1 < ‖c‖
d : ℕ → E
hds : ∀ (n : ℕ), x + d n ∈ s
hd : ∀ (n : ℕ), ‖d n‖ < (1 / (2 * ‖c‖)) ^ n
N : ℕ
x✝ : True
n : ℕ
hn : N ≤ n
⊢ ‖c‖ ^ n * ‖d n‖ < 1 / 2 ^ N

Tactic:
rw [← lt_div_iff₀' (by positivity)]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
x : E
hx : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
c : 𝕜
hc : 1 < ‖c‖
d : ℕ → E
hds : ∀ (n : ℕ), x + d n ∈ s
hd : ∀ (n : ℕ), ‖d n‖ < (1 / (2 * ‖c‖)) ^ n
N : ℕ
x✝ : True
n : ℕ
hn : N ≤ n
⊢ ‖d n‖ < 1 / 2 ^ N / ‖c‖ ^ n
Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: F : Type u_2
inst✝ : SeminormedAddCommGroup F
a b : ℝ
f : ℝ → F
hf : AbsolutelyContinuousOnInterval f a b
⊢ Tendsto ?m.10 (totalLengthFilter ⊓ 𝓟 (disjWithin a b)) (𝓝 0)

Tactic:
by simpa using! hf

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

Goal state before:
pp: E✝ : Type u_1
inst✝⁴ : NormedAddCommGroup E✝
inst✝³ : NormedSpace ℝ E✝
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℝ → E
a b : ℝ
hf : IntervalIntegrable f volume a b
hab : a ≤ b
⊢ ∀ᵐ (x : ℝ), x ∈ Icc a b → ∀ c ∈ Icc a b, HasDerivAt (fun x ↦ ∫ (t : ℝ) in c..x, f t) (f x) x

Tactic:
have h₁ : ∀ᵐ x, x ≠ a := by simp [ae_iff, measure_singleton]

Goal state after:
E✝ : Type u_1
inst✝⁴ : NormedAddCommGroup E✝
inst✝³ : NormedSpace ℝ E✝
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℝ → E
a b : ℝ
hf : IntervalIntegrable f volume a b
hab : a ≤ b
h₁ : ∀ᵐ (x : ℝ), x ≠ a
⊢ ∀ᵐ (x : ℝ), x ∈ Icc a b → ∀ c ∈ Icc a b, HasDerivAt (fun x ↦ ∫ (t : ℝ) in c..x, f t) (f x) x
Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: F : Type u_2
inst✝ : SeminormedAddCommGroup F
a b : ℝ
f : ℝ → F
hf : AbsolutelyContinuousOnInterval f a b
⊢ comap (fun x ↦ (1, fun x_1 ↦ x)) totalLengthFilter ⊓ 𝓟 (uIcc a b ×ˢ uIcc a b) ≤
    comap (fun x ↦ (f x.1, f x.2)) (comap (fun x ↦ (1, fun x_1 ↦ x)) totalLengthFilter)

Tactic:
simp only [AbsolutelyContinuousOnInterval, Filter.tendsto_iff_comap] at hf

Goal state after:
F : Type u_2
inst✝ : SeminormedAddCommGroup F
a b : ℝ
f : ℝ → F
hf :
  totalLengthFilter ⊓ 𝓟 (disjWithin a b) ≤
    comap (fun E ↦ ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2)) (𝓝 0)
⊢ comap (fun x ↦ (1, fun x_1 ↦ x)) totalLengthFilter ⊓ 𝓟 (uIcc a b ×ˢ uIcc a b) ≤
    comap (fun x ↦ (f x.1, f x.2)) (comap (fun x ↦ (1, fun x_1 ↦ x)) totalLengthFilter)
Module: Mathlib.Analysis.Calculus.Rademacher

Goal state before:
pp: case h₂
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
⊢ ‖f (x + ρ • y) - f x - ρ • L y‖ ≤ δ * ρ

Tactic:
conv_rhs => rw [← norm_rho]

Goal state after:
case h₂
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
⊢ ‖f (x + ρ • y) - f x - ρ • L y‖ ≤ δ * ‖ρ‖
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun

Goal state before:
pp: case e'_3
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 ∈ Finset.range (T s).1, dist ((T s).2 i).1 ((T s).2 i).2 = b - d - ∑ x ∈ s, ((↑x).2 - (↑x).1)

Tactic:
Finset.sum_congr rfl (g := fun i ↦ ((T s).2 i).2 - ((T s).2 i).1)
            (fun i hi ↦ by rw [dist_comm, Real.dist_eq, abs_of_nonneg (by grind)])

Goal state after:
case e'_3
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 ∈ Finset.range (T s).1, (((T s).2 i).2 - ((T s).2 i).1) = b - d - ∑ x ∈ s, ((↑x).2 - (↑x).1)
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: M : ℝ
hM : M ≤ 1
z : ℂ
zn : 0 < 1 - ‖z‖
⊢ 1 * (1 - ‖z‖) = ‖1‖ - ‖z‖

Tactic:
rw [one_mul, norm_one]

Goal state after:
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
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

Tactic:
rw [interior_closedBall (0 : E) one_ne_zero, closedBall_sdiff_ball,
    mem_sphere_zero_iff_norm] at hx hy

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‖ = 1
y : E
hy : ‖y‖ = 1
hne : x ≠ y
⊢ ∃ c, (AffineMap.lineMap x y) c ∈ interior (closedBall 0 1)
Module: Mathlib.Analysis.Complex.AbsMax

Goal state before:
pp: F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → F
z w : ℂ
r : ℝ := dist w z
hd : DiffContOnCl ℂ f (ball z r)
hz : IsMaxOn (norm ∘ f) (closedBall z r) z
⊢ ‖f w‖ = ‖f z‖

Tactic:
have hw : w ∈ closedBall z r := mem_closedBall.2 le_rfl

Goal state after:
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → F
z w : ℂ
r : ℝ := dist w z
hd : DiffContOnCl ℂ f (ball z r)
hz : IsMaxOn (norm ∘ f) (closedBall z r) z
hw : w ∈ closedBall z r
⊢ ‖f w‖ = ‖f z‖
Module: Mathlib.Analysis.Convex.SpecificFunctions.Deriv

Goal state before:
pp: ⊢ StrictConcaveOn ℝ (Ioi 1) fun x ↦ √x * log x

Tactic:
apply strictConcaveOn_of_deriv2_neg' (convex_Ioi 1) _ fun x hx => ?_

Goal state after:
⊢ ContinuousOn (fun x ↦ √x * log x) (Ioi 1)

x : ℝ
hx : x ∈ Ioi 1
⊢ deriv^[2] (fun x ↦ √x * log x) x < 0
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : StrictMonoOn (deriv f) (interior D)
x y z : ℝ
hx : x ∈ D
hz : z ∈ D
hxy : x < y
hyz : y < z
hxzD : Icc x z ⊆ D
hxyD : Icc x y ⊆ D
hxyD' : Ioo x y ⊆ interior D
hyzD : Icc y z ⊆ D
hyzD' : Ioo y z ⊆ interior D
⊢ (f y - f x) / (y - x) < (f z - f y) / (z - y)

Tactic:
obtain ⟨a, ⟨hxa, hay⟩, ha⟩ : ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a :=
      StrictMonoOn.exists_slope_lt_deriv (hf.mono hxyD) hxy (hf'.mono hxyD')

Goal state after:
D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : StrictMonoOn (deriv f) (interior D)
x y z : ℝ
hx : x ∈ D
hz : z ∈ D
hxy : x < y
hyz : y < z
hxzD : Icc x z ⊆ D
hxyD : Icc x y ⊆ D
hxyD' : Ioo x y ⊆ interior D
hyzD : Icc y z ⊆ D
hyzD' : Ioo y z ⊆ interior D
a : ℝ
ha : (f y - f x) / (y - x) < deriv f a
hxa : x < a
hay : a < y
⊢ (f y - f x) / (y - x) < (f z - f y) / (z - y)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: case inr
x : ℝ
hx_gt : ¬x ≤ -(π / 2)
hx_lt : x < π / 2
r : ℤ
hxr_eq : x = (2 * ↑r + 1) * π / 2
h : r < 0
hr_le : r ≤ -1
⊢ ↑r ≤ (-1 - 1) / 2

pp: case inr
x : ℝ
hx_gt : ¬x ≤ -(π / 2)
hx_lt : x < π / 2
r : ℤ
hxr_eq : x = (2 * ↑r + 1) * π / 2
h : r < 0
hr_le : r ≤ -1
⊢ 0 < 2

Tactic:
· norm_num
      assumption_mod_cast

Goal state after:
case inr
x : ℝ
hx_gt : ¬x ≤ -(π / 2)
hx_lt : x < π / 2
r : ℤ
hxr_eq : x = (2 * ↑r + 1) * π / 2
h : r < 0
hr_le : r ≤ -1
⊢ 0 < 2
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: x : ℝ
h : 0 < x
⊢ arccos x = arctan (√(1 - x ^ 2) / x)

Tactic:
arccos,

Goal state after:
x : ℝ
h : 0 < x
⊢ π / 2 - arcsin x = arctan (√(1 - x ^ 2) / x)
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
⊢ IsMinOn f S x

Tactic:
by
  intro y hy
  rcases lt_trichotomy x y with hxy | h_eq | hyx
  · suffices 0 ≤ slope f x y by
      simp only [slope_def_field, div_nonneg_iff, sub_nonneg, tsub_le_iff_right, zero_add,
        not_le.mpr hxy, and_false, or_false] at this
      exact this.1
    exact hf_rd.trans <| rightDeriv_le_slope_of_mem_interior hf hx hy hxy
  · simp [h_eq]
  · suffices slope f x y ≤ 0 by
      simp only [slope_def_field, div_nonpos_iff, sub_nonneg, tsub_le_iff_right, zero_add,
        not_le.mpr hyx, and_false, or_false] at this
      exact this.1
    rw [slope_comm]
    exact (slope_le_leftDeriv_of_mem_interior hf hy hx hyx).trans hf_ld

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
hx : x ≠ 0
hy : y ≠ 0
h : ‖x - y‖ = ‖x‖ + ‖y‖
⊢ ⟪x, y⟫ = -(‖x‖ * ‖y‖)

Tactic:
obtain ⟨hxy₁, hxy₂⟩ := norm_nonneg (x - y), add_nonneg (norm_nonneg x) (norm_nonneg y)

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
h : ‖x - y‖ = ‖x‖ + ‖y‖
hxy₁ : 0 ≤ ‖x - y‖
hxy₂ : 0 ≤ ‖x‖ + ‖y‖
⊢ ⟪x, y⟫ = -(‖x‖ * ‖y‖)
Module: Mathlib.Analysis.Complex.CoveringMap

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : ProperSpace 𝕜
n : ℤ
hn : ↑n ≠ 0
⊢ IsCoveringMap fun x ↦ ⟨↑x ^ n, ⋯⟩

Tactic:
by
  obtain ⟨n, rfl | rfl⟩ := n.eq_nat_or_neg
  · convert! isCoveringMap_npow n _ <;> aesop
  · convert! (isCoveringMap_npow n _).comp_homeomorph (.inv₀ 𝕜)
    · simp [Homeomorph.inv₀]
    · simpa using hn

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

Goal state before:
pp: case insert
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
U : Set 𝕜
ι : Type u_3
f : ι → 𝕜 → 𝕜
a : ι
s : Finset ι
ha : a ∉ s
hs :
  (∀ i ∈ s, MeromorphicOn (f i) U) →
    (∀ i ∈ s, ∀ z ∈ U, meromorphicOrderAt (f i) z ≠ ⊤) → divisor (∏ i ∈ s, f i) U = ∑ i ∈ s, divisor (f i) U
h₁f : ∀ i ∈ insert a s, MeromorphicOn (f i) U
h₂f : ∀ i ∈ insert a s, ∀ z ∈ U, meromorphicOrderAt (f i) z ≠ ⊤
⊢ divisor (∏ i ∈ insert a s, f i) U = ∑ i ∈ insert a s, divisor (f i) U

Tactic:
have (z) (hz : z ∈ U) : meromorphicOrderAt (∏ i ∈ s, f i) z ≠ ⊤ := by
      simpa [meromorphicOrderAt_prod (fun i hi ↦ h₁f i (mem_insert_of_mem hi) z hz)]
        using fun i hi ↦ h₂f i (mem_insert_of_mem hi) z hz
    rw [prod_insert ha, sum_insert ha, divisor_mul (by aesop)
        (prod (fun i hi ↦ h₁f i (mem_insert_of_mem hi)))
        (h₂f a (mem_insert_self a s)) this,
      hs (fun i hi ↦ h₁f i (mem_insert_of_mem hi))
        (fun i hi ↦ h₂f i (mem_insert_of_mem hi))]

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

Goal state before:
pp: case neg.h₂g
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
ι : Type u_3
s : Finset ι
f : ι → 𝕜 → 𝕜
h₁f : ∀ i ∈ s, MeromorphicNFAt (f i) x
h₂f : {σ | σ ∈ s ∧ f σ x = 0}.Subsingleton
h₃f : ∀ {τ : ι}, τ ∈ s → τ ∉ {σ ∈ s | f σ x = 0} → AnalyticAt 𝕜 (f τ) x
τ : ι
h₁τ : τ ∈ s
h₂τ : f τ x = 0
this : ∀ {μ : ι}, μ ∈ s.erase τ → f μ x ≠ 0
⊢ ∏ c ∈ s.erase τ, f c x ≠ 0

Tactic:
Finset.prod_ne_zero_iff

Goal state after:
case neg.h₂g
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
ι : Type u_3
s : Finset ι
f : ι → 𝕜 → 𝕜
h₁f : ∀ i ∈ s, MeromorphicNFAt (f i) x
h₂f : {σ | σ ∈ s ∧ f σ x = 0}.Subsingleton
h₃f : ∀ {τ : ι}, τ ∈ s → τ ∉ {σ ∈ s | f σ x = 0} → AnalyticAt 𝕜 (f τ) x
τ : ι
h₁τ : τ ∈ s
h₂τ : f τ x = 0
this : ∀ {μ : ι}, μ ∈ s.erase τ → f μ x ≠ 0
⊢ ∀ a ∈ s.erase τ, f a x ≠ 0
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: case inl
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
g : 𝕜 → 𝕜
x : 𝕜
hg : AnalyticAt 𝕜 g x
hf : f =ᶠ[𝓝 (g x)] 0
⊢ MeromorphicNFAt (f ∘ g) x

Tactic:
exact Or.inl (hg.continuousAt.tendsto.eventually hf)

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
hf : MeromorphicOn f U
⊢ EqOn (toMeromorphicNFOn f U) f Uᶜ

Tactic:
by
  intro x hx
  simp_all [toMeromorphicNFOn]

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
ho : meromorphicOrderAt f x = 0
g : 𝕜 → E
g_an : AnalyticAt 𝕜 g x
gx : g x ≠ 0
hg : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ 0 • g z
y : 𝕜
hy : f y = (y - x) ^ 0 • g y
⊢ g y = f y

Tactic:
by simp [hy]

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
ho : 0 < meromorphicOrderAt f x
hf : MeromorphicAt f x
g : 𝕜 → E
g_an : AnalyticAt 𝕜 g x
gx : g x ≠ 0
n : ℕ
h'o : meromorphicOrderAt f x = ↑↑n
hg : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ ↑n • g z
this✝ : 0 = (x - x) ^ n • g x
this : ContinuousAt (fun z ↦ (z - x) ^ n • g z) x
y : 𝕜
hy : f y = (y - x) ^ ↑n • g y
⊢ (y - x) ^ n • g y = f y

Tactic:
by simp [hy]

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 (fun x ↦ ‖f x‖) (𝓝[≠] x) atTop

Tactic:
obtain ⟨c, hc⟩ := tendsto_nhds_of_meromorphicOrderAt_nonneg hf ho

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 : 0 ≤ meromorphicOrderAt f x
c : E
hc : Tendsto f (𝓝[≠] x) (𝓝 c)
⊢ ¬Tendsto (fun x ↦ ‖f x‖) (𝓝[≠] x) atTop
Module: Mathlib.Analysis.Meromorphic.FactorizedRational

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h : d x = 0
h₁ : HasFiniteMulSupport fun u ↦ (fun x ↦ x - u) ^ d u
⊢ ∏ c ∈ Finite.toFinset h₁, ((fun x ↦ x - c) ^ d c) x ≠ 0

Tactic:
Finset.prod_ne_zero_iff

Goal state after:
case pos
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
d : 𝕜 → ℤ
x : 𝕜
h : d x = 0
h₁ : HasFiniteMulSupport fun u ↦ (fun x ↦ x - u) ^ d u
⊢ ∀ a ∈ Finite.toFinset h₁, ((fun x ↦ x - a) ^ d a) x ≠ 0
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
⊢ ¬Tendsto f (𝓝[≠] x) (𝓝 0)

Tactic:
intro hc

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
hc : Tendsto f (𝓝[≠] x) (𝓝 0)
⊢ False
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
h : MeromorphicAt f x
h' : ContinuousAt f x
ho : meromorphicOrderAt f x = ⊤
this : AnalyticAt 𝕜 (fun x ↦ 0) x
y : 𝕜
hy : f y = 0
⊢ 0 = f y

Tactic:
by simp [hy]

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
h : MeromorphicAt f x
h' : ContinuousAt f x
n : ℤ
ho : meromorphicOrderAt f x = ↑n
g : 𝕜 → E
g_an : AnalyticAt 𝕜 g x
gx : g x ≠ 0
hg : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ n • g z
⊢ ∃ c, Tendsto f (𝓝[≠] x) (𝓝 c)

Tactic:
exact ⟨f x, h'.continuousWithinAt.tendsto⟩

Goal state after: