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

Goal state before:
pp: E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
inst✝¹¹ : FiniteDimensional ℝ E
F : Type u_2
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace ℝ F
inst✝⁸ : CompleteSpace F
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : IsManifold I ∞ M
inst✝³ : MeasurableSpace M
inst✝² : BorelSpace M
inst✝¹ : T2Space M
f : M → F
μ : Measure M
inst✝ : SigmaCompactSpace M
hf : LocallyIntegrable f μ
h : ∀ (g : M → ℝ), ContMDiff I 𝓘(ℝ, ℝ) ∞ g → HasCompactSupport g → ∫ (x : M), g x • f x ∂μ = 0
this✝² : LocallyCompactSpace H
this✝¹ : LocallyCompactSpace M
this✝ : SecondCountableTopology H
this : SecondCountableTopology M
x✝ : MetricSpace M := metrizableSpaceMetric M
s : Set M
hs : IsCompact s
δ : ℝ
δpos : 0 < δ
hδ : IsCompact (cthickening δ s)
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), u n ∈ Ioo 0 δ
u_lim : Tendsto u atTop (𝓝 0)
v : ℕ → Set M := fun n ↦ thickening (u n) s
K : Set M
K_compact : IsCompact K
vK : ∀ (n : ℕ), v n ⊆ K
g : ℕ → M → ℝ
g_supp : ∀ (n : ℕ), support (g n) = v n
g_diff : ∀ (n : ℕ), ContMDiff I 𝓘(ℝ, ℝ) ∞ (g n)
g_range : ∀ (n : ℕ), range (g n) ⊆ Icc 0 1
hg : ∀ (n : ℕ), ∀ x ∈ s, g n x = 1
bound : M → ℝ := K.indicator fun x ↦ ‖f x‖
A : ∀ (n : ℕ), AEStronglyMeasurable (fun x ↦ g n x • f x) μ
B : Integrable bound μ
C : ∀ (n : ℕ), ∀ᵐ (x : M) ∂μ, ‖g n x • f x‖ ≤ bound x
x : M
hxs : x ∉ s
⊢ ∃ ε, 0 < ε ∧ x ∉ thickening ε s

Tactic:
rw [← hs.isClosed.closure_eq, closure_eq_iInter_thickening s] at hxs

Goal state after:
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
inst✝¹¹ : FiniteDimensional ℝ E
F : Type u_2
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace ℝ F
inst✝⁸ : CompleteSpace F
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : IsManifold I ∞ M
inst✝³ : MeasurableSpace M
inst✝² : BorelSpace M
inst✝¹ : T2Space M
f : M → F
μ : Measure M
inst✝ : SigmaCompactSpace M
hf : LocallyIntegrable f μ
h : ∀ (g : M → ℝ), ContMDiff I 𝓘(ℝ, ℝ) ∞ g → HasCompactSupport g → ∫ (x : M), g x • f x ∂μ = 0
this✝² : LocallyCompactSpace H
this✝¹ : LocallyCompactSpace M
this✝ : SecondCountableTopology H
this : SecondCountableTopology M
x✝ : MetricSpace M := metrizableSpaceMetric M
s : Set M
hs : IsCompact s
δ : ℝ
δpos : 0 < δ
hδ : IsCompact (cthickening δ s)
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), u n ∈ Ioo 0 δ
u_lim : Tendsto u atTop (𝓝 0)
v : ℕ → Set M := fun n ↦ thickening (u n) s
K : Set M
K_compact : IsCompact K
vK : ∀ (n : ℕ), v n ⊆ K
g : ℕ → M → ℝ
g_supp : ∀ (n : ℕ), support (g n) = v n
g_diff : ∀ (n : ℕ), ContMDiff I 𝓘(ℝ, ℝ) ∞ (g n)
g_range : ∀ (n : ℕ), range (g n) ⊆ Icc 0 1
hg : ∀ (n : ℕ), ∀ x ∈ s, g n x = 1
bound : M → ℝ := K.indicator fun x ↦ ‖f x‖
A : ∀ (n : ℕ), AEStronglyMeasurable (fun x ↦ g n x • f x) μ
B : Integrable bound μ
C : ∀ (n : ℕ), ∀ᵐ (x : M) ∂μ, ‖g n x • f x‖ ≤ bound x
x : M
hxs : x ∉ ⋂ δ, ⋂ (_ : 0 < δ), thickening δ s
⊢ ∃ ε, 0 < ε ∧ x ∉ thickening ε s
Module: Mathlib.Geometry.Manifold.PartitionOfUnity

Goal state before:
pp: E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type uH
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : FiniteDimensional ℝ E
inst✝² : IsManifold I ∞ M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
n : ℕ∞
s t : Set M
hs : IsOpen[inst✝⁵] s
ht : IsClosed[inst✝⁵] t
h : t ⊆ s
f : M → ℝ
f_supp : support f = s
f_diff : ContMDiff I 𝓘(ℝ, ℝ) (↑?m.68) f
f_pos : ∀ (x : M), 0 ≤ f x
g : M → ℝ
g_supp : support g = tᶜ
g_diff : ContMDiff I 𝓘(ℝ, ℝ) (↑?m.109) g
g_pos : ∀ (x : M), 0 ≤ g x
x : M
xs : x ∉ tᶜ
⊢ x ∈ support f

Tactic:
exact h.trans f_supp.symm.subset (by simpa using xs)

Goal state after:
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 g : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
h'f : HasCompactSupport f
hg : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] g
v : E
K : Set E := cthickening ‖v‖ (tsupport f)
K_compact : IsCompact K
t : ℝ
ht : t ∈ Ioc 0 1
t_pos : 0 < t
x : E
hx : x ∈ Function.support f
⊢ x ∈ K

Tactic:
exact self_subset_cthickening _ (subset_tsupport _ hx)

Goal state after:
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 D : ℝ≥0
f g : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
hg : LipschitzWith D g
h'g : HasCompactSupport g
v : E
A :
  Tendsto (fun t ↦ ∫ (x : E), t⁻¹ • (f (x + t • v) - f x) * g x ∂μ) (𝓝[>] 0) (𝓝 (∫ (x : E), lineDeriv ℝ f x v * g x ∂μ))
B :
  Tendsto (fun t ↦ ∫ (x : E), t⁻¹ • (g (x + t • -v) - g x) * f x ∂μ) (𝓝[>] 0)
    (𝓝 (∫ (x : E), lineDeriv ℝ g x (-v) * f x ∂μ))
t : ℝ
S3 : ∫ (x : E), f (x + t • v) * g x ∂μ = ∫ (x : E), f x * g (x + t • -v) ∂μ
⊢ ∫ (x : E), f (x + t • v) * g x - f x * g x ∂μ = ∫ (x : E), f x * g (x + t • -v) - f x * g x ∂μ

Tactic:
integral_sub,

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 D : ℝ≥0
f g : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
hg : LipschitzWith D g
h'g : HasCompactSupport g
v : E
A :
  Tendsto (fun t ↦ ∫ (x : E), t⁻¹ • (f (x + t • v) - f x) * g x ∂μ) (𝓝[>] 0) (𝓝 (∫ (x : E), lineDeriv ℝ f x v * g x ∂μ))
B :
  Tendsto (fun t ↦ ∫ (x : E), t⁻¹ • (g (x + t • -v) - g x) * f x ∂μ) (𝓝[>] 0)
    (𝓝 (∫ (x : E), lineDeriv ℝ g x (-v) * f x ∂μ))
t : ℝ
S3 : ∫ (x : E), f (x + t • v) * g x ∂μ = ∫ (x : E), f x * g (x + t • -v) ∂μ
⊢ ∫ (a : E), f (a + t • v) * g a ∂μ - ∫ (a : E), f a * g a ∂μ = ∫ (x : E), f x * g (x + t • -v) - f x * g x ∂μ

case hf
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C D : ℝ≥0
f g : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
hg : LipschitzWith D g
h'g : HasCompactSupport g
v : E
A :
  Tendsto (fun t ↦ ∫ (x : E), t⁻¹ • (f (x + t • v) - f x) * g x ∂μ) (𝓝[>] 0) (𝓝 (∫ (x : E), lineDeriv ℝ f x v * g x ∂μ))
B :
  Tendsto (fun t ↦ ∫ (x : E), t⁻¹ • (g (x + t • -v) - g x) * f x ∂μ) (𝓝[>] 0)
    (𝓝 (∫ (x : E), lineDeriv ℝ g x (-v) * f x ∂μ))
t : ℝ
S3 : ∫ (x : E), f (x + t • v) * g x ∂μ = ∫ (x : E), f x * g (x + t • -v) ∂μ
⊢ Integrable (fun x ↦ f (x + t • v) * g x) μ

case hg
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C D : ℝ≥0
f g : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
hg : LipschitzWith D g
h'g : HasCompactSupport g
v : E
A :
  Tendsto (fun t ↦ ∫ (x : E), t⁻¹ • (f (x + t • v) - f x) * g x ∂μ) (𝓝[>] 0) (𝓝 (∫ (x : E), lineDeriv ℝ f x v * g x ∂μ))
B :
  Tendsto (fun t ↦ ∫ (x : E), t⁻¹ • (g (x + t • -v) - g x) * f x ∂μ) (𝓝[>] 0)
    (𝓝 (∫ (x : E), lineDeriv ℝ g x (-v) * f x ∂μ))
t : ℝ
S3 : ∫ (x : E), f (x + t • v) * g x ∂μ = ∫ (x : E), f x * g (x + t • -v) ∂μ
⊢ Integrable (fun x ↦ f x * g x) μ
Module: Mathlib.Analysis.Calculus.Rademacher

Goal state before:
pp: case hf.hf
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C D : ℝ≥0
f g : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
hg : LipschitzWith D g
h'g : HasCompactSupport g
v : E
A :
  Tendsto (fun t ↦ ∫ (x : E), t⁻¹ • (f (x + t • v) - f x) * g x ∂μ) (𝓝[>] 0) (𝓝 (∫ (x : E), lineDeriv ℝ f x v * g x ∂μ))
B :
  Tendsto (fun t ↦ ∫ (x : E), t⁻¹ • (g (x + t • -v) - g x) * f x ∂μ) (𝓝[>] 0)
    (𝓝 (∫ (x : E), lineDeriv ℝ g x (-v) * f x ∂μ))
t : ℝ
S3 : ∫ (x : E), f (x + t • v) * g x ∂μ = ∫ (x : E), f x * g (x + t • -v) ∂μ
⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] fun x ↦ f (x + t • v) * g x

Tactic:
exact (hf.continuous.comp (continuous_add_const _)).mul hg.continuous

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

Goal state before:
pp: F : Type u_2
inst✝³ : SeminormedAddCommGroup F
a b : ℝ
M : Type u_3
inst✝² : SeminormedRing M
inst✝¹ : Module M F
inst✝ : NormSMulClass M F
f : ℝ → M
g : ℝ → F
hf :
  Tendsto (fun E ↦ ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2)) (totalLengthFilter ⊓ 𝓟 (disjWithin a b))
    (𝓝 0)
hg :
  Tendsto (fun E ↦ ∑ i ∈ Finset.range E.1, dist (g (E.2 i).1) (g (E.2 i).2)) (totalLengthFilter ⊓ 𝓟 (disjWithin a b))
    (𝓝 0)
C : ℝ
hC : ∀ x ∈ uIcc a b, ‖f x‖ ≤ C
D : ℝ
hD : ∀ x ∈ uIcc a b, ‖g x‖ ≤ D
n : ℕ
I : ℕ → ℝ × ℝ
⊢ (n, I) ∈ disjWithin a b →
    ∑ i ∈ Finset.range (n, I).1, dist ((f • g) ((n, I).2 i).1) ((f • g) ((n, I).2 i).2) ≤
      C * ∑ i ∈ Finset.range (n, I).1, dist (g ((n, I).2 i).1) (g ((n, I).2 i).2) +
        D * ∑ i ∈ Finset.range (n, I).1, dist (f ((n, I).2 i).1) (f ((n, I).2 i).2)

Tactic:
hnI

Goal state after:
F : Type u_2
inst✝³ : SeminormedAddCommGroup F
a b : ℝ
M : Type u_3
inst✝² : SeminormedRing M
inst✝¹ : Module M F
inst✝ : NormSMulClass M F
f : ℝ → M
g : ℝ → F
hf :
  Tendsto (fun E ↦ ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2)) (totalLengthFilter ⊓ 𝓟 (disjWithin a b))
    (𝓝 0)
hg :
  Tendsto (fun E ↦ ∑ i ∈ Finset.range E.1, dist (g (E.2 i).1) (g (E.2 i).2)) (totalLengthFilter ⊓ 𝓟 (disjWithin a b))
    (𝓝 0)
C : ℝ
hC : ∀ x ∈ uIcc a b, ‖f x‖ ≤ C
D : ℝ
hD : ∀ x ∈ uIcc a b, ‖g x‖ ≤ D
n : ℕ
I : ℕ → ℝ × ℝ
hnI : (n, I) ∈ disjWithin a b
⊢ ∑ i ∈ Finset.range (n, I).1, dist ((f • g) ((n, I).2 i).1) ((f • g) ((n, I).2 i).2) ≤
    C * ∑ i ∈ Finset.range (n, I).1, dist (g ((n, I).2 i).1) (g ((n, I).2 i).2) +
      D * ∑ i ∈ Finset.range (n, I).1, dist (f ((n, I).2 i).1) (f ((n, I).2 i).2)
Module: Mathlib.Analysis.Calculus.Taylor

Goal state before:
pp: case e_a
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
n : ℕ
s : Set ℝ
x₀ x : ℝ
⊢ (((↑n + ↑1) * ↑n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ =
    ((↑n !)⁻¹ * (↑n + 1)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀

Tactic:
Nat.cast_one,

Goal state after:
case e_a
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
n : ℕ
s : Set ℝ
x₀ x : ℝ
⊢ (((↑n + 1) * ↑n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ =
    ((↑n !)⁻¹ * (↑n + 1)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀
Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: case inr
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
⊢ BoundedVariationOn f (Icc a b)

Tactic:
obtain ⟨δ, hδ₁, hδ₂⟩ := hf 1 (by linarith)

Goal state after:
case inr
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
⊢ BoundedVariationOn f (Icc a b)
Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: case inr
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
⊢ BoundedVariationOn f (Icc a b)

Tactic:
obtain ⟨n, hn⟩ := exists_nat_one_div_lt (div_pos hδ₁ hab₁)

Goal state after:
case inr
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)
⊢ BoundedVariationOn f (Icc a b)
Module: Mathlib.Analysis.Calculus.Taylor

Goal state before:
pp: case inr.hvv'
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℝ → F
x x₀ : ℝ
n : ℕ
hf : ContDiffOn ℝ (↑(n + 1)) f [[x₀, x]]
this✝ : x₀ ≠ x
this : UniqueDiffOn ℝ [[x₀, x]]
k : ℕ
hk : k ≤ n
⊢ ∀ x_1 ∈ Ioo (min x₀ x) (max x₀ x),
    HasDerivAt (fun t ↦ iteratedDerivWithin k f [[x₀, x]] t) (deriv (fun t ↦ iteratedDerivWithin k f [[x₀, x]] t) x_1)
      x_1

Tactic:
intro t ht
    refine DifferentiableOn.hasDerivAt (s := uIoo x₀ x) ?_ (by grind [Ioo_mem_nhds, uIoo])
    exact hf.differentiableOn_iteratedDerivWithin (by norm_cast; omega) this
      |>.mono (by grind [uIoo, uIcc])

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Ray

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

Tactic:
← sub_smul,

Goal state after:
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
u : E
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
h : SameRay ℝ (a • u) (b • u)
hab : 0 ≤ a - b
⊢ ‖(a - b) • u‖ = |‖a • u‖ - ‖b • u‖|
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
z : E
hz : ‖z‖ ≤ 1
hδz : 1 - δ' ≤ ‖z‖
⊢ ‖‖z‖⁻¹ • z - 1 • z‖ ≤ δ'

Tactic:
← sub_smul,

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
z : E
hz : ‖z‖ ≤ 1
hδz : 1 - δ' ≤ ‖z‖
⊢ ‖(‖z‖⁻¹ - 1) • z‖ ≤ δ'
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
⊢ (n, I) ∈ disjWithin a b →
    ∑ x ∈ Finset.range (n, I).1, dist (∫ (v : ℝ) in c..((n, I).2 x).1, f v) (∫ (v : ℝ) in c..((n, I).2 x).2, f v) ≤
      (∫⁻ (x : ℝ) in ⋃ i ∈ Finset.range (n, I).1, uIoc ((n, I).2 i).1 ((n, I).2 i).2,
          ‖f x‖ₑ ∂volume.restrict (uIoc a b)).toReal

Tactic:
hnI

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
⊢ ∑ x ∈ Finset.range (n, I).1, dist (∫ (v : ℝ) in c..((n, I).2 x).1, f v) (∫ (v : ℝ) in c..((n, I).2 x).2, f v) ≤
    (∫⁻ (x : ℝ) in ⋃ i ∈ Finset.range (n, I).1, uIoc ((n, I).2 i).1 ((n, I).2 i).2,
        ‖f x‖ₑ ∂volume.restrict (uIoc a b)).toReal
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: case hbc
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 := ⋯
g : ℂ → ℂ := ⋯
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ⋯
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
⊢ ‖∑ i ∈ range B₁, (l - s (i + 1)) * z ^ i‖ ≤ ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i

Tactic:
← norm_pow,

Goal state after:
case hbc
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
⊢ ‖∑ i ∈ range B₁, (l - s (i + 1)) * z ^ i‖ ≤ ∑ x ∈ range B₁, ‖l - s (x + 1)‖ * ‖z ^ x‖
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: case hbc
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 := ⋯
g : ℂ → ℂ := ⋯
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ⋯
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
⊢ ‖∑ i ∈ Ico B₁ (max B₁ B₂), (l - s (i + 1)) * z ^ i‖ ≤ ∑ i ∈ Ico B₁ (max B₁ B₂), ‖l - s (i + 1)‖ * ‖z‖ ^ i

Tactic:
← norm_pow,

Goal state after:
case hbc
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
⊢ ‖∑ i ∈ Ico B₁ (max B₁ B₂), (l - s (i + 1)) * z ^ i‖ ≤ ∑ x ∈ Ico B₁ (max B₁ B₂), ‖l - s (x + 1)‖ * ‖z ^ x‖
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : DifferentiableOn ℝ f (interior D)
hf'_mono : MonotoneOn (deriv f) (interior D)
⊢ ∀ {x y z : ℝ}, x ∈ D → z ∈ D → x < y → y < z → (f y - f x) / (y - x) ≤ (f z - f y) / (z - y)

Tactic:
intro x y z hx hz hxy hyz

Goal state after:
D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : DifferentiableOn ℝ f (interior D)
hf'_mono : MonotoneOn (deriv f) (interior D)
x y z : ℝ
hx : x ∈ D
hz : z ∈ D
hxy : x < y
hyz : y < z
⊢ (f y - f x) / (y - x) ≤ (f z - f y) / (z - y)
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x : V
⊢ arccos (⟪0, x⟫ / (‖0‖ * ‖x‖)) = π / 2

Tactic:
inner_zero_left,

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x : V
⊢ arccos (0 / (‖0‖ * ‖x‖)) = π / 2
Module: Mathlib.Analysis.Complex.BranchLogRoot

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : LocallyPathConnectedSpace X
U : Set X
hUc : IsSimplyConnected U
hUo : IsOpen U
g : X → 𝔻
hgc : ContinuousOn g U
hU₀ : 0 ∉ g '' U
n : ℕ+
f : X → ℂ
hfc : ContinuousOn f U
hf : ∀ (x : X), f x ^ ↑n = (UnitDisc.coe ∘ g) x
x : X
⊢ ‖f x‖ ^ ↑n < 1

Tactic:
← norm_pow,

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : LocallyPathConnectedSpace X
U : Set X
hUc : IsSimplyConnected U
hUo : IsOpen U
g : X → 𝔻
hgc : ContinuousOn g U
hU₀ : 0 ∉ g '' U
n : ℕ+
f : X → ℂ
hfc : ContinuousOn f U
hf : ∀ (x : X), f x ^ ↑n = (UnitDisc.coe ∘ g) x
x : X
⊢ ‖f x ^ ↑n‖ < 1
Module: Mathlib.Analysis.Meromorphic.Divisor

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
U : Set 𝕜
z₀ x : 𝕜
hx : x ≠ z₀
hu : x ∈ U
⊢ (divisor (fun x ↦ x - z₀) U) x = 0

Tactic:
divisor_apply (show MeromorphicOn (· - z₀) U from fun_sub id <| const z₀) hu,

Goal state after:
case pos
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
U : Set 𝕜
z₀ x : 𝕜
hx : x ≠ z₀
hu : x ∈ U
⊢ (meromorphicOrderAt (fun x ↦ x - z₀) x).untop₀ = 0
Module: Mathlib.Analysis.Meromorphic.TrailingCoefficient

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
x : 𝕜
f₁ f₂ : 𝕜 → E
h : f₁ =ᶠ[𝓝[≠] x] f₂
h₁ : MeromorphicAt f₁ x
h₂ : ¬meromorphicOrderAt f₁ x = ⊤
g : 𝕜 → E
h₁g : AnalyticAt 𝕜 g x
h₂g : g x ≠ 0
h₃g : f₁ =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ (meromorphicOrderAt f₁ x).untop₀ • g z
⊢ meromorphicTrailingCoeffAt f₁ x = meromorphicTrailingCoeffAt f₂ x

Tactic:
h₁g.meromorphicTrailingCoeffAt_of_ne_zero_of_eq_nhdsNE h₂g h₃g,

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
x : 𝕜
f₁ f₂ : 𝕜 → E
h : f₁ =ᶠ[𝓝[≠] x] f₂
h₁ : MeromorphicAt f₁ x
h₂ : ¬meromorphicOrderAt f₁ x = ⊤
g : 𝕜 → E
h₁g : AnalyticAt 𝕜 g x
h₂g : g x ≠ 0
h₃g : f₁ =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ (meromorphicOrderAt f₁ x).untop₀ • g z
⊢ g x = meromorphicTrailingCoeffAt f₂ x
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: case pos.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 z : 𝕜
hz : z = x
h₀f : MeromorphicNFAt f x
n : ℤ
g : 𝕜 → E
h₁g : AnalyticAt 𝕜 g x
h₂g : g x ≠ 0
h₃g : f =ᶠ[𝓝 x] (fun x_1 ↦ x_1 - x) ^ n • g
⊢ (if h : True then Function.update f x (if h_1 : meromorphicOrderAt f x = 0 then Classical.choose ⋯ x else 0) else 0)
      x =
    ((fun x_1 ↦ x_1 - x) ^ n • g) x

Tactic:
have : meromorphicOrderAt f x = n := by
          rw [meromorphicOrderAt_eq_int_iff h₀f.meromorphicAt]
          use g, h₁g, h₂g
          exact eventually_nhdsWithin_of_eventually_nhds h₃g

Goal state after:
case pos.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 z : 𝕜
hz : z = x
h₀f : MeromorphicNFAt f x
n : ℤ
g : 𝕜 → E
h₁g : AnalyticAt 𝕜 g x
h₂g : g x ≠ 0
h₃g : f =ᶠ[𝓝 x] (fun x_1 ↦ x_1 - x) ^ n • g
this : meromorphicOrderAt f x = ↑n
⊢ (if h : True then Function.update f x (if h_1 : meromorphicOrderAt f x = 0 then Classical.choose ⋯ x else 0) else 0)
      x =
    ((fun x_1 ↦ x_1 - x) ^ n • g) x
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: case refine_1.coe
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
nneg : 0 ≤ meromorphicOrderAt f x
n : ℤ
h₀ : meromorphicOrderAt f x = ↑n
⊢ ∃ g, AnalyticAt 𝕜 g x ∧ f =ᶠ[𝓝[≠] x] g

Tactic:
obtain ⟨g, hg, -, hfg⟩ := (meromorphicOrderAt_eq_int_iff hf).mp h₀
      refine ⟨fun z ↦ (z - x) ^ n • g z, ?_, hfg⟩
      exact (AnalyticAt.zpow_nonneg (by fun_prop) (by simpa [h₀] using nneg)).smul hg

Goal state after: