mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Covering.Besicovitch

Goal state before:
pp: α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : HasBesicovitchCovering α
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun t ↦ ((↑t).PairwiseDisjoint fun p ↦ closedBall p.1 p.2) ∧ (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
t : Finset (α × ℝ)
ht : P t
B : Set α := ⋃ p ∈ t, closedBall p.1 p.2
hB : B = ⋃ p ∈ t, closedBall p.1 p.2
B_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] B
s' : Set α := s \ B
x : α
hx : x ∈ s'
xs : x ∈ s
⊢ ∃ r ∈ f x ∩ Ioo 0 1, Disjoint B (closedBall x r)

Tactic:
rcases eq_empty_or_nonempty B with (hB | hB)

Goal state after:
case inl
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : HasBesicovitchCovering α
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun t ↦ ((↑t).PairwiseDisjoint fun p ↦ closedBall p.1 p.2) ∧ (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
t : Finset (α × ℝ)
ht : P t
B : Set α := ⋃ p ∈ t, closedBall p.1 p.2
hB✝ : B = ⋃ p ∈ t, closedBall p.1 p.2
B_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] B
s' : Set α := s \ B
x : α
hx : x ∈ s'
xs : x ∈ s
hB : B = ∅
⊢ ∃ r ∈ f x ∩ Ioo 0 1, Disjoint B (closedBall x r)

case inr
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : HasBesicovitchCovering α
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun t ↦ ((↑t).PairwiseDisjoint fun p ↦ closedBall p.1 p.2) ∧ (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
t : Finset (α × ℝ)
ht : P t
B : Set α := ⋃ p ∈ t, closedBall p.1 p.2
hB✝ : B = ⋃ p ∈ t, closedBall p.1 p.2
B_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] B
s' : Set α := s \ B
x : α
hx : x ∈ s'
xs : x ∈ s
hB : B.Nonempty
⊢ ∃ r ∈ f x ∩ Ioo 0 1, Disjoint B (closedBall x r)
Module: Mathlib.MeasureTheory.Covering.Besicovitch

Goal state before:
pp: case inr
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : HasBesicovitchCovering α
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun t ↦ ((↑t).PairwiseDisjoint fun p ↦ closedBall p.1 p.2) ∧ (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
t : Finset (α × ℝ)
ht : P t
B : Set α := ⋃ p ∈ t, closedBall p.1 p.2
hB✝ : B = ⋃ p ∈ t, closedBall p.1 p.2
B_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] B
s' : Set α := s \ B
x : α
hx : x ∈ s'
xs : x ∈ s
hB : B.Nonempty
r : ℝ := infDist x B
this : 0 < min r 1
⊢ ∃ r ∈ f x ∩ Ioo 0 1, Disjoint B (closedBall x r)

Tactic:
rcases hf x xs _ this with ⟨r, hr, h'r⟩

Goal state after:
case inr
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : HasBesicovitchCovering α
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun t ↦ ((↑t).PairwiseDisjoint fun p ↦ closedBall p.1 p.2) ∧ (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
t : Finset (α × ℝ)
ht : P t
B : Set α := ⋃ p ∈ t, closedBall p.1 p.2
hB✝ : B = ⋃ p ∈ t, closedBall p.1 p.2
B_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] B
s' : Set α := s \ B
x : α
hx : x ∈ s'
xs : x ∈ s
hB : B.Nonempty
r✝ : ℝ := infDist x B
this : 0 < min r✝ 1
r : ℝ
hr : r ∈ f x
h'r : r ∈ Ioo 0 (min r✝ 1)
⊢ ∃ r ∈ f x ∩ Ioo 0 1, Disjoint B (closedBall x r)
Module: Mathlib.MeasureTheory.Covering.Besicovitch

Goal state before:
pp: case refine_1.left
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : HasBesicovitchCovering α
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun t ↦ ((↑t).PairwiseDisjoint fun p ↦ closedBall p.1 p.2) ∧ (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
t : Finset (α × ℝ)
ht : P t
B : Set α := ⋃ p ∈ t, closedBall p.1 p.2
hB : B = ⋃ p ∈ t, closedBall p.1 p.2
B_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] B
s' : Set α := s \ B
r : α → ℝ
hr : ∀ x ∈ s', r x ∈ f x ∩ Ioo 0 1 ∧ Disjoint B (closedBall x (r x))
v : Finset α
vs' : ↑v ⊆ s'
hμv : μ (s' \ ⋃ x ∈ v, closedBall x (r x)) ≤ ↑N / (↑N + 1) * μ s'
hv : (↑v).PairwiseDisjoint fun x ↦ closedBall x (r x)
p' : α
p'v : p' ∈ ↑v
hp : (p', r p') ∈ (fun x ↦ (x, r x)) '' ↑v
q' : α
q'v : q' ∈ ↑v
hq : (q', r q') ∈ (fun x ↦ (x, r x)) '' ↑v
hpq : (p', r p') ≠ (q', r q')
hp'q' : p' = (q', r q').1
⊢ False

Tactic:
rw [hp'q'] at hpq

Goal state after:
case refine_1.left
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : HasBesicovitchCovering α
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun t ↦ ((↑t).PairwiseDisjoint fun p ↦ closedBall p.1 p.2) ∧ (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
t : Finset (α × ℝ)
ht : P t
B : Set α := ⋃ p ∈ t, closedBall p.1 p.2
hB : B = ⋃ p ∈ t, closedBall p.1 p.2
B_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] B
s' : Set α := s \ B
r : α → ℝ
hr : ∀ x ∈ s', r x ∈ f x ∩ Ioo 0 1 ∧ Disjoint B (closedBall x (r x))
v : Finset α
vs' : ↑v ⊆ s'
hμv : μ (s' \ ⋃ x ∈ v, closedBall x (r x)) ≤ ↑N / (↑N + 1) * μ s'
hv : (↑v).PairwiseDisjoint fun x ↦ closedBall x (r x)
p' : α
p'v : p' ∈ ↑v
hp : (p', r p') ∈ (fun x ↦ (x, r x)) '' ↑v
q' : α
q'v : q' ∈ ↑v
hq : (q', r q') ∈ (fun x ↦ (x, r x)) '' ↑v
hpq : ((q', r q').1, r (q', r q').1) ≠ (q', r q')
hp'q' : p' = (q', r q').1
⊢ False
Module: Mathlib.Analysis.Calculus.Deriv.Star

Goal state before:
pp: 𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : StarRing 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : StarAddMonoid F
inst✝² : StarModule 𝕜 F
inst✝¹ : ContinuousStar F
inst✝ : NormedStarGroup 𝕜
f : 𝕜 → F
x : 𝕜
f' : F
hf : HasDerivAt (star ∘ f ∘ ⇑(starRingEnd 𝕜)) f' x
⊢ HasDerivAt f (star f') ((starRingEnd 𝕜) x)

Tactic:
convert! hf.star_conj

Goal state after:
𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : StarRing 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : StarAddMonoid F
inst✝² : StarModule 𝕜 F
inst✝¹ : ContinuousStar F
inst✝ : NormedStarGroup 𝕜
f : 𝕜 → F
x : 𝕜
f' : F
hf : HasDerivAt (star ∘ f ∘ ⇑(starRingEnd 𝕜)) f' x
⊢ f = star ∘ (star ∘ f ∘ ⇑(starRingEnd 𝕜)) ∘ ⇑(starRingEnd 𝕜)
Module: Mathlib.Analysis.Calculus.Deriv.Star

Goal state before:
pp: 𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : StarRing 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : StarAddMonoid F
inst✝² : StarModule 𝕜 F
inst✝¹ : ContinuousStar F
inst✝ : NormedStarGroup 𝕜
f : 𝕜 → F
x : 𝕜
f' : F
hf : HasDerivAt f (star f') ((starRingEnd 𝕜) x)
⊢ HasDerivAt (star ∘ f ∘ ⇑(starRingEnd 𝕜)) f' x

Tactic:
convert! hf.star_conj

Goal state after:
case e'_9
𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : StarRing 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : StarAddMonoid F
inst✝² : StarModule 𝕜 F
inst✝¹ : ContinuousStar F
inst✝ : NormedStarGroup 𝕜
f : 𝕜 → F
x : 𝕜
f' : F
hf : HasDerivAt f (star f') ((starRingEnd 𝕜) x)
⊢ f' = star (star f')

case e'_10
𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : StarRing 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : StarAddMonoid F
inst✝² : StarModule 𝕜 F
inst✝¹ : ContinuousStar F
inst✝ : NormedStarGroup 𝕜
f : 𝕜 → F
x : 𝕜
f' : F
hf : HasDerivAt f (star f') ((starRingEnd 𝕜) x)
⊢ x = (starRingEnd 𝕜) ((starRingEnd 𝕜) x)
Module: Mathlib.Analysis.Calculus.Deriv.Star

Goal state before:
pp: 𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : StarRing 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : StarAddMonoid F
inst✝² : StarModule 𝕜 F
inst✝¹ : ContinuousStar F
x : 𝕜
inst✝ : NormedStarGroup 𝕜
f : 𝕜 → F
hf : DifferentiableAt 𝕜 (star ∘ f ∘ ⇑(starRingEnd 𝕜)) x
⊢ DifferentiableAt 𝕜 f ((starRingEnd 𝕜) x)

Tactic:
convert! hf.star_conj

Goal state after:
𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : StarRing 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : StarAddMonoid F
inst✝² : StarModule 𝕜 F
inst✝¹ : ContinuousStar F
x : 𝕜
inst✝ : NormedStarGroup 𝕜
f : 𝕜 → F
hf : DifferentiableAt 𝕜 (star ∘ f ∘ ⇑(starRingEnd 𝕜)) x
⊢ f = star ∘ (star ∘ f ∘ ⇑(starRingEnd 𝕜)) ∘ ⇑(starRingEnd 𝕜)
Module: Mathlib.Analysis.Calculus.Deriv.Star

Goal state before:
pp: 𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : StarRing 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : StarAddMonoid F
inst✝² : StarModule 𝕜 F
inst✝¹ : ContinuousStar F
x : 𝕜
inst✝ : NormedStarGroup 𝕜
f : 𝕜 → F
hf : DifferentiableAt 𝕜 f ((starRingEnd 𝕜) x)
⊢ DifferentiableAt 𝕜 (star ∘ f ∘ ⇑(starRingEnd 𝕜)) x

Tactic:
convert! hf.star_conj

Goal state after:
𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : StarRing 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : StarAddMonoid F
inst✝² : StarModule 𝕜 F
inst✝¹ : ContinuousStar F
x : 𝕜
inst✝ : NormedStarGroup 𝕜
f : 𝕜 → F
hf : DifferentiableAt 𝕜 f ((starRingEnd 𝕜) x)
⊢ x = (starRingEnd 𝕜) ((starRingEnd 𝕜) x)
Module: Mathlib.Analysis.Calculus.DerivativeTest

Goal state before:
pp: f : ℝ → ℝ
a b c : ℝ
hb : ContinuousAt f b
hc : ContinuousAt f c
hd₀ : DifferentiableOn ℝ f (Ioo a b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x
⊢ IsMinOn f (Ioc a c) b

Tactic:
refine isMinOn_Ioc_of_anti_mono ?_ ?_

Goal state after:
case refine_1
f : ℝ → ℝ
a b c : ℝ
hb : ContinuousAt f b
hc : ContinuousAt f c
hd₀ : DifferentiableOn ℝ f (Ioo a b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x
⊢ AntitoneOn f (Ioc a b)

case refine_2
f : ℝ → ℝ
a b c : ℝ
hb : ContinuousAt f b
hc : ContinuousAt f c
hd₀ : DifferentiableOn ℝ f (Ioo a b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x
⊢ MonotoneOn f (Icc b c)
Module: Mathlib.Analysis.Calculus.DerivativeTest

Goal state before:
pp: f : ℝ → ℝ
x₀ : ℝ
hf : deriv f x₀ > 0
hx : f x₀ = 0
h_tendsto : Tendsto (slope f x₀) (𝓝[≠] x₀) (𝓝 (deriv f x₀))
x : ℝ
hx₀ : 0 < slope f x₀ x
hx₁ : x ≠ x₀
⊢ sign (f x) = sign (x - x₀)

Tactic:
obtain (hx' | hx') := hx₁.lt_or_gt

Goal state after:
case inl
f : ℝ → ℝ
x₀ : ℝ
hf : deriv f x₀ > 0
hx : f x₀ = 0
h_tendsto : Tendsto (slope f x₀) (𝓝[≠] x₀) (𝓝 (deriv f x₀))
x : ℝ
hx₀ : 0 < slope f x₀ x
hx₁ : x ≠ x₀
hx' : x < x₀
⊢ sign (f x) = sign (x - x₀)

case inr
f : ℝ → ℝ
x₀ : ℝ
hf : deriv f x₀ > 0
hx : f x₀ = 0
h_tendsto : Tendsto (slope f x₀) (𝓝[≠] x₀) (𝓝 (deriv f x₀))
x : ℝ
hx₀ : 0 < slope f x₀ x
hx₁ : x ≠ x₀
hx' : x₀ < x
⊢ sign (f x) = sign (x - x₀)
Module: Mathlib.Analysis.Calculus.FDeriv.Symmetric

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set E
s_conv : Convex ℝ s
f : E → F
f' : E → E →L[ℝ] F
f'' : E →L[ℝ] E →L[ℝ] F
hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x
x : E
xs : x ∈ s
hx : HasFDerivWithinAt f' f'' (interior s) x
v w : E
h4v : x + 4 • v ∈ interior s
h4w : x + 4 • w ∈ interior s
A : 1 / 2 ∈ Ioc 0 1
B : 1 / 2 ∈ Icc 0 1
⊢ x + 2 • v + 2 • w ∈ interior s

Tactic:
convert! s_conv.interior.add_smul_sub_mem h4v h4w B using 1

Goal state after:
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set E
s_conv : Convex ℝ s
f : E → F
f' : E → E →L[ℝ] F
f'' : E →L[ℝ] E →L[ℝ] F
hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x
x : E
xs : x ∈ s
hx : HasFDerivWithinAt f' f'' (interior s) x
v w : E
h4v : x + 4 • v ∈ interior s
h4w : x + 4 • w ∈ interior s
A : 1 / 2 ∈ Ioc 0 1
B : 1 / 2 ∈ Icc 0 1
⊢ x + 2 • v + 2 • w = x + 4 • v + (1 / 2) • (x + 4 • w - (x + 4 • v))
Module: Mathlib.Analysis.Calculus.Gradient.Basic

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : CompleteSpace F
f : F → 𝕜
f' x y : F
s : Set F
h : HasGradientWithinAt f f' s x
hs : UniqueDiffWithinAt 𝕜 s x
⊢ (fderivWithin 𝕜 f s x) y = ⟪f', y⟫

Tactic:
rw [h.hasFDerivWithinAt.fderivWithin hs, toDual_apply_apply]

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

Goal state before:
pp: a b : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hab : a < b
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Ioo a b, g' x ≠ 0
hfa : Tendsto f (𝓝[>] a) (𝓝 0)
hga : Tendsto g (𝓝[>] a) (𝓝 0)
hdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝[>] a) l
sub : ∀ x ∈ Ioo a b, Ioo a x ⊆ Ioo a b
⊢ ∀ x ∈ Ioo a b, g x ≠ 0

Tactic:
intro x hx h

Goal state after:
a b : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hab : a < b
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Ioo a b, g' x ≠ 0
hfa : Tendsto f (𝓝[>] a) (𝓝 0)
hga : Tendsto g (𝓝[>] a) (𝓝 0)
hdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝[>] a) l
sub : ∀ x ∈ Ioo a b, Ioo a x ⊆ Ioo a b
x : ℝ
hx : x ∈ Ioo a b
h : g x = 0
⊢ False
Module: Mathlib.Analysis.Calculus.LagrangeMultipliers

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
φ : E → ℝ
x₀ : E
φ' : StrongDual ℝ E
ι : Type u_3
inst✝ : Fintype ι
f : ι → E → ℝ
f' : ι → StrongDual ℝ E
hextr : IsLocalExtrOn φ {x | ∀ (i : ι), f i x = f i x₀} x₀
hf' : ∀ (i : ι), HasStrictFDerivAt (f i) (f' i) x₀
hφ' : HasStrictFDerivAt φ φ' x₀
this : DecidableEq ι := Classical.decEq ι
⊢ ∃ Λ Λ₀, (Λ, Λ₀) ≠ 0 ∧ ∑ i, Λ i • f' i + Λ₀ • φ' = 0

Tactic:
replace hextr : IsLocalExtrOn φ {x | (fun i => f i x) = fun i => f i x₀} x₀ := by
    simpa only [funext_iff] using hextr

Goal state after:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
φ : E → ℝ
x₀ : E
φ' : StrongDual ℝ E
ι : Type u_3
inst✝ : Fintype ι
f : ι → E → ℝ
f' : ι → StrongDual ℝ E
hf' : ∀ (i : ι), HasStrictFDerivAt (f i) (f' i) x₀
hφ' : HasStrictFDerivAt φ φ' x₀
this : DecidableEq ι := Classical.decEq ι
hextr : IsLocalExtrOn φ {x | (fun i ↦ f i x) = fun i ↦ f i x₀} x₀
⊢ ∃ Λ Λ₀, (Λ, Λ₀) ≠ 0 ∧ ∑ i, Λ i • f' i + Λ₀ • φ' = 0
Module: Mathlib.Analysis.Calculus.LHopital

Goal state before:
pp: a b : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hab : a < b
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Ioo a b, g' x ≠ 0
hfb : Tendsto f (𝓝[<] b) (𝓝 0)
hgb : Tendsto g (𝓝[<] b) (𝓝 0)
hdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝[<] b) l
hdnf : ∀ x ∈ Ioo (-b) (-a), HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x
hdng : ∀ x ∈ Ioo (-b) (-a), HasDerivAt (g ∘ Neg.neg) (g' (-x) * -1) x
this : Tendsto (fun x ↦ (f ∘ Neg.neg) x / (g ∘ Neg.neg) x) (𝓝[>] (-b)) l
⊢ Tendsto (fun x ↦ f x / g x) (𝓝[<] b) l

Tactic:
have := this.comp tendsto_neg_nhdsLT

Goal state after:
a b : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hab : a < b
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Ioo a b, g' x ≠ 0
hfb : Tendsto f (𝓝[<] b) (𝓝 0)
hgb : Tendsto g (𝓝[<] b) (𝓝 0)
hdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝[<] b) l
hdnf : ∀ x ∈ Ioo (-b) (-a), HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x
hdng : ∀ x ∈ Ioo (-b) (-a), HasDerivAt (g ∘ Neg.neg) (g' (-x) * -1) x
this✝ : Tendsto (fun x ↦ (f ∘ Neg.neg) x / (g ∘ Neg.neg) x) (𝓝[>] (-b)) l
this : Tendsto ((fun x ↦ (f ∘ Neg.neg) x / (g ∘ Neg.neg) x) ∘ Neg.neg) (𝓝[<] b) l
⊢ Tendsto (fun x ↦ f x / g x) (𝓝[<] b) l
Module: Mathlib.Analysis.Calculus.LHopital

Goal state before:
pp: a : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Ioi a, g' x ≠ 0
hftop : Tendsto f atTop (𝓝 0)
hgtop : Tendsto g atTop (𝓝 0)
hdiv : Tendsto (fun x ↦ f' x / g' x) atTop l
a' : ℝ
haa' : a < a'
ha' : 0 < a'
fact1 : ∀ x ∈ Ioo 0 a'⁻¹, x ≠ 0
fact2 : ∀ x ∈ Ioo 0 a'⁻¹, a < x⁻¹
hdnf : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (f ∘ Inv.inv) (f' x⁻¹ * -(x ^ 2)⁻¹) x
hdng : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (g ∘ Inv.inv) (g' x⁻¹ * -(x ^ 2)⁻¹) x
x : ℝ
hx : x ∈ Ioo 0 a'⁻¹
⊢ g' x⁻¹ ≠ 0

Tactic:
exact hg' _ (fact2 x hx)

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
m : ℝ≥0
hm : ENNReal.ofReal |A.det| < ↑m
d : ℝ≥0∞ := ENNReal.ofReal |A.det|
ε : ℝ
hε : μ (closedBall 0 ε + ⇑A '' closedBall 0 1) < ↑m * μ (closedBall 0 1)
εpos : 0 < ε
this : Iio (NNReal.mk ε ⋯) ∈ 𝓝 0
δ : ℝ≥0
s : Set E
f : E → E
hf : ApproximatesLinearOn f A s δ
hδ : ↑δ < ε
x : E
r : ℝ
xs : x ∈ s
r0 : 0 ≤ r
z : E
zs : z ∈ s
zr : ‖z - x‖ ≤ r
⊢ ‖f z - f x - A (z - x)‖ ≤ ε * r

Tactic:
calc
          ‖f z - f x - A (z - x)‖ ≤ δ * ‖z - x‖ := hf _ zs _ xs
          _ ≤ ε * r := by gcongr

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

Goal state before:
pp: a b : ℝ
f f' g : ℝ → ℝ
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x
hf' : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ f' x
⊢ IntervalIntegrable (fun x ↦ (g ∘ f) x * f' x) volume a b ↔ IntervalIntegrable g volume (f a) (f b)

Tactic:
simpa [mul_comm] using! integrable_deriv_smul_comp_iff_of_deriv_nonneg hf hff' hf'

Goal state after:
Module: Mathlib.Analysis.Calculus.LineDeriv.Measurable

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : LocallyCompactSpace 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : OpensMeasurableSpace E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace F
f : E → F
v : E
inst✝ : SecondCountableTopologyEither E F
hf : Continuous f
⊢ StronglyMeasurable fun x ↦ lineDeriv 𝕜 f x v

Tactic:
borelize 𝕜
  let g : E → 𝕜 → F := fun x t ↦ f (x + t • v)
  have hg : Continuous g.uncurry := by fun_prop
  exact (stronglyMeasurable_deriv_with_param hg).comp_measurable measurable_prodMk_right

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

Goal state before:
pp: E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
U : Set ℂ
hU : IsOpen U
c w₀ : ℂ
R : ℝ
f : ℂ → E
hc : closedBall c R ⊆ U
hf : DifferentiableOn ℂ f U
hw₀ : w₀ ∈ ball c R
hf' : DifferentiableOn ℂ (dslope f w₀) U
⊢ (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • f z = deriv f w₀

Tactic:
have h0 := (hf'.diffContOnCl_ball hc).two_pi_i_inv_smul_circleIntegral_sub_inv_smul hw₀

Goal state after:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
U : Set ℂ
hU : IsOpen U
c w₀ : ℂ
R : ℝ
f : ℂ → E
hc : closedBall c R ⊆ U
hf : DifferentiableOn ℂ f U
hw₀ : w₀ ∈ ball c R
hf' : DifferentiableOn ℂ (dslope f w₀) U
h0 : (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z = dslope f w₀ w₀
⊢ (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • f z = deriv f w₀
Module: Mathlib.Topology.Algebra.InfiniteSum.UniformOn

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
inst✝³ : CommMonoid α
f : ι → β → α
g : β → α
x : β
s : Set β
inst✝² : UniformSpace α
inst✝¹ : TopologicalSpace β
inst✝ : LocallyCompactSpace β
h : HasProdLocallyUniformlyOn f g s
hx : s ∈ 𝓝 x
K : Set β
hK3 : K ⊆ s
hK1 : K ∈ 𝓝 x
hK2 : IsCompact K
⊢ ∃ t ∈ 𝓝[s] x, HasProdUniformlyOn f g t

Tactic:
exact ⟨K, nhdsWithin_le_nhds hK1,
    HasProdLocallyUniformlyOn.hasProdUniformlyOn_of_isCompact (h.mono hK3) hK2⟩

Goal state after:
Module: Mathlib.Analysis.RCLike.TangentCone

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
h𝕜 : IsRCLikeNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace ℝ E
s : Set E
x : E
⊢ tangentConeAt ℝ s x ⊆ tangentConeAt 𝕜 s x

Tactic:
let := h𝕜.rclike

Goal state after:
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
h𝕜 : IsRCLikeNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace ℝ E
s : Set E
x : E
this : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
⊢ tangentConeAt ℝ s x ⊆ tangentConeAt 𝕜 s x
Module: Mathlib.Analysis.RCLike.TangentCone

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
h𝕜 : IsRCLikeNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace ℝ E
s : Set E
x : E
hs : UniqueDiffWithinAt ℝ s x
⊢ UniqueDiffWithinAt 𝕜 s x

Tactic:
let := h𝕜.rclike

Goal state after:
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
h𝕜 : IsRCLikeNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace ℝ E
s : Set E
x : E
hs : UniqueDiffWithinAt ℝ s x
this : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
⊢ UniqueDiffWithinAt 𝕜 s x
Module: Mathlib.Geometry.Manifold.StructureGroupoid

Goal state before:
pp: case left
H : Type u_1
inst✝ : TopologicalSpace H
PG : Pregroupoid H
e e' : OpenPartialHomeomorph H H
he : e ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
ee' : e' ≈ e
⊢ PG.property (↑e') e'.source

Tactic:
apply PG.congr e'.open_source ee'.2

Goal state after:
case left
H : Type u_1
inst✝ : TopologicalSpace H
PG : Pregroupoid H
e e' : OpenPartialHomeomorph H H
he : e ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}
ee' : e' ≈ e
⊢ PG.property (↑e) e'.source
Module: Mathlib.Geometry.Manifold.HasGroupoid

Goal state before:
pp: H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝⁶ : TopologicalSpace H
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : TopologicalSpace M'
inst✝² : TopologicalSpace M''
G : StructureGroupoid H
inst✝¹ : ChartedSpace H M'
inst✝ : ChartedSpace H M''
e : Structomorph G M M'
e' : Structomorph G M' M''
c : OpenPartialHomeomorph M H
c' : OpenPartialHomeomorph M'' H
hc : c ∈ atlas H M
hc' : c' ∈ atlas H M''
x : H
hx : x ∈ (c.symm ≫ₕ (e.trans e'.toHomeomorph).toOpenPartialHomeomorph ≫ₕ c').source
f₁ : OpenPartialHomeomorph M M' := e.toOpenPartialHomeomorph
f₂ : OpenPartialHomeomorph M' M'' := e'.toOpenPartialHomeomorph
f : OpenPartialHomeomorph M M'' := (e.trans e'.toHomeomorph).toOpenPartialHomeomorph
feq : f = f₁ ≫ₕ f₂
y : M' := ↑(c.symm ≫ₕ f₁) x
g : OpenPartialHomeomorph M' H := chartAt H y
hg₁ : chartAt H y ∈ atlas H M'
hg₂ : y ∈ (chartAt H y).source
s : Set H := (c.symm ≫ₕ f₁).source ∩ ↑(c.symm ≫ₕ f₁) ⁻¹' g.source
open_s : IsOpen[inst✝⁶] s
this : x ∈ s
F₁ : OpenPartialHomeomorph H H := (c.symm ≫ₕ f₁ ≫ₕ g) ≫ₕ g.symm ≫ₕ f₂ ≫ₕ c'
A : F₁ ∈ G
F₂ : OpenPartialHomeomorph H H := (c.symm ≫ₕ f ≫ₕ c').restr s
⊢ F₁ ≈ c.symm ≫ₕ f₁ ≫ₕ (g ≫ₕ g.symm) ≫ₕ f₂ ≫ₕ c'

Tactic:
simp only [F₁, trans_assoc, _root_.refl]

Goal state after: