mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Manifold.MFDeriv.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
s : Set M
e : OpenPartialHomeomorph M H
e' : OpenPartialHomeomorph M' H'
he : e ∈ maximalAtlas I 1 M
he' : e' ∈ maximalAtlas I' 1 M'
hs : s ⊆ e.source
h2s : MapsTo f s e'.source
⊢ MDiff[s] f ↔
    (∀ x ∈ s, ContinuousWithinAt f s x) ∧
      ∀ ⦃x : M⦄,
        x ∈ s →
          DifferentiableWithinAt 𝕜 (↑(e'.extend I') ∘ f ∘ ↑(e.extend I).symm) (↑(e.extend I) '' s) (↑(e.extend I) x)

Tactic:
← forall_and,

Goal state after:
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
s : Set M
e : OpenPartialHomeomorph M H
e' : OpenPartialHomeomorph M' H'
he : e ∈ maximalAtlas I 1 M
he' : e' ∈ maximalAtlas I' 1 M'
hs : s ⊆ e.source
h2s : MapsTo f s e'.source
⊢ MDiff[s] f ↔
    ∀ x ∈ s,
      ContinuousWithinAt f s x ∧
        DifferentiableWithinAt 𝕜 (↑(e'.extend I') ∘ f ∘ ↑(e.extend I).symm) (↑(e.extend I) '' s) (↑(e.extend I) x)
Module: Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace B
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
EB : Type u_4
inst✝³ : NormedAddCommGroup EB
inst✝² : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹ : TopologicalSpace HB
inst✝ : ChartedSpace HB B
IB : ModelWithCorners 𝕜 EB HB
n : ℕ∞ω
e : OpenPartialHomeomorph (B × F) (B × F)
U : Set B
hU : e.source = U ×ˢ univ
φ : ↑U → B → F ≃L[𝕜] F
u : ↑U → Set B
hu : ∀ (x : ↑U), IsOpen[inst✝⁷] (u x)
hUu : ∀ (x : ↑U), u x ⊆ U
hux : ∀ (x : ↑U), ↑x ∈ u x
hφ : ∀ (x : ↑U), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 ↦ ↑(φ x x_1)) (u x)
h2φ : ∀ (x : ↑U), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 ↦ ↑(φ x x_1).symm) (u x)
heφ : ∀ (x : ↑U), (e.restr (u x ×ˢ univ)).EqOnSource (FiberwiseLinear.openPartialHomeomorph (φ x) ⋯ ⋯ ⋯)
heuφ : ∀ (x : ↑U), EqOn (↑e) (fun q ↦ (q.1, (φ x q.1) q.2)) (u x ×ˢ univ)
huφ : ∀ (x x' : ↑U), ∀ y ∈ u x, y ∈ u x' → φ x y = φ x' y
hUu' : U = ⋃ i, u i
hU' : IsOpen[inst✝⁷] U
Φ₀ : ↑U → F ≃L[𝕜] F := iUnionLift u (fun x ↦ φ x ∘ Subtype.val) huφ U ⋯
Φ : B → F ≃L[𝕜] F := fun y ↦ if hy : y ∈ U then Φ₀ ⟨y, hy⟩ else ContinuousLinearEquiv.refl 𝕜 F
hΦ✝ : ∀ (y : B) (hy : y ∈ U), Φ y = Φ₀ ⟨y, hy⟩
hΦφ : ∀ (x : ↑U), ∀ y ∈ u x, Φ y = φ x y
hΦ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun y ↦ ↑(Φ y)) U
x : B
hx : x ∈ U
⊢ ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun y ↦ ↑(Φ y).symm) (U ∩ u ⟨x, hx⟩)

Tactic:
refine (ContMDiffOn.congr (h2φ ⟨x, hx⟩) ?_).mono inter_subset_right

Goal state after:
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace B
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
EB : Type u_4
inst✝³ : NormedAddCommGroup EB
inst✝² : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹ : TopologicalSpace HB
inst✝ : ChartedSpace HB B
IB : ModelWithCorners 𝕜 EB HB
n : ℕ∞ω
e : OpenPartialHomeomorph (B × F) (B × F)
U : Set B
hU : e.source = U ×ˢ univ
φ : ↑U → B → F ≃L[𝕜] F
u : ↑U → Set B
hu : ∀ (x : ↑U), IsOpen[inst✝⁷] (u x)
hUu : ∀ (x : ↑U), u x ⊆ U
hux : ∀ (x : ↑U), ↑x ∈ u x
hφ : ∀ (x : ↑U), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 ↦ ↑(φ x x_1)) (u x)
h2φ : ∀ (x : ↑U), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 ↦ ↑(φ x x_1).symm) (u x)
heφ : ∀ (x : ↑U), (e.restr (u x ×ˢ univ)).EqOnSource (FiberwiseLinear.openPartialHomeomorph (φ x) ⋯ ⋯ ⋯)
heuφ : ∀ (x : ↑U), EqOn (↑e) (fun q ↦ (q.1, (φ x q.1) q.2)) (u x ×ˢ univ)
huφ : ∀ (x x' : ↑U), ∀ y ∈ u x, y ∈ u x' → φ x y = φ x' y
hUu' : U = ⋃ i, u i
hU' : IsOpen[inst✝⁷] U
Φ₀ : ↑U → F ≃L[𝕜] F := iUnionLift u (fun x ↦ φ x ∘ Subtype.val) huφ U ⋯
Φ : B → F ≃L[𝕜] F := fun y ↦ if hy : y ∈ U then Φ₀ ⟨y, hy⟩ else ContinuousLinearEquiv.refl 𝕜 F
hΦ✝ : ∀ (y : B) (hy : y ∈ U), Φ y = Φ₀ ⟨y, hy⟩
hΦφ : ∀ (x : ↑U), ∀ y ∈ u x, Φ y = φ x y
hΦ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun y ↦ ↑(Φ y)) U
x : B
hx : x ∈ U
⊢ ∀ y ∈ u ⟨x, hx⟩, ↑(Φ y).symm = ↑(φ ⟨x, hx⟩ y).symm
Module: Mathlib.Topology.FiberBundle.Constructions

Goal state before:
pp: B : Type u
F : Type v
E : B → Type w₁
B' : Type w₂
f✝ : B' → B
inst✝⁶ : TopologicalSpace B'
inst✝⁵ : TopologicalSpace (TotalSpace F E)
inst✝⁴ : TopologicalSpace F
inst✝³ : TopologicalSpace B
inst✝² : ∀ (_b : B), Nonempty (E _b)
K : Type U
inst✝¹ : FunLike K B' B
inst✝ : ContinuousMapClass K B' B
e : Trivialization F TotalSpace.proj
f : K
x : TotalSpace F (⇑f *ᵖ E)
h : x ∈ Pullback.lift ⇑f ⁻¹' e.source
⊢ (x.proj, (↑e (Pullback.lift (⇑f) x)).2) ∈ (⇑f ⁻¹' e.baseSet) ×ˢ univ

Tactic:
by
    simp_rw [e.source_eq, mem_preimage, Pullback.lift_proj] at h
    simp_rw [prodMk_mem_set_prod_eq, mem_univ, and_true, mem_preimage, h]

Goal state after:
Module: Mathlib.Topology.VectorBundle.Constructions

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
inst✝¹⁵ : NontriviallyNormedField 𝕜
inst✝¹⁴ : TopologicalSpace B
F₁ : Type u_3
inst✝¹³ : NormedAddCommGroup F₁
inst✝¹² : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹¹ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹⁰ : NormedAddCommGroup F₂
inst✝⁹ : NormedSpace 𝕜 F₂
E₂ : B → Type u_6
inst✝⁸ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁷ : (x : B) → AddCommMonoid (E₁ x)
inst✝⁶ : (x : B) → Module 𝕜 (E₁ x)
inst✝⁵ : (x : B) → AddCommMonoid (E₂ x)
inst✝⁴ : (x : B) → Module 𝕜 (E₂ x)
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝³ : Trivialization.IsLinear 𝕜 e₁
inst✝² : Trivialization.IsLinear 𝕜 e₁'
inst✝¹ : Trivialization.IsLinear 𝕜 e₂
inst✝ : Trivialization.IsLinear 𝕜 e₂'
b : B
hb : (b ∈ e₁.baseSet ∧ b ∈ e₂.baseSet) ∧ b ∈ e₁'.baseSet ∧ b ∈ e₂'.baseSet
⊢ ↑(coordChangeL 𝕜 (e₁.prod e₂) (e₁'.prod e₂') b) = (↑(coordChangeL 𝕜 e₁ e₁' b)).prodMap ↑(coordChangeL 𝕜 e₂ e₂' b)

Tactic:
ContinuousLinearMap.ext_iff,

Goal state after:
𝕜 : Type u_1
B : Type u_2
inst✝¹⁵ : NontriviallyNormedField 𝕜
inst✝¹⁴ : TopologicalSpace B
F₁ : Type u_3
inst✝¹³ : NormedAddCommGroup F₁
inst✝¹² : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹¹ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹⁰ : NormedAddCommGroup F₂
inst✝⁹ : NormedSpace 𝕜 F₂
E₂ : B → Type u_6
inst✝⁸ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁷ : (x : B) → AddCommMonoid (E₁ x)
inst✝⁶ : (x : B) → Module 𝕜 (E₁ x)
inst✝⁵ : (x : B) → AddCommMonoid (E₂ x)
inst✝⁴ : (x : B) → Module 𝕜 (E₂ x)
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝³ : Trivialization.IsLinear 𝕜 e₁
inst✝² : Trivialization.IsLinear 𝕜 e₁'
inst✝¹ : Trivialization.IsLinear 𝕜 e₂
inst✝ : Trivialization.IsLinear 𝕜 e₂'
b : B
hb : (b ∈ e₁.baseSet ∧ b ∈ e₂.baseSet) ∧ b ∈ e₁'.baseSet ∧ b ∈ e₂'.baseSet
⊢ ∀ (x : F₁ × F₂),
    ↑(coordChangeL 𝕜 (e₁.prod e₂) (e₁'.prod e₂') b) x =
      ((↑(coordChangeL 𝕜 e₁ e₁' b)).prodMap ↑(coordChangeL 𝕜 e₂ e₂' b)) x
Module: Mathlib.Geometry.Manifold.VectorBundle.Basic

Goal state before:
pp: n : ℕ∞ω
𝕜 : Type u_1
B : Type u_2
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝²¹ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝²⁰ : NormedAddCommGroup EB
inst✝¹⁹ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹⁸ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹⁷ : TopologicalSpace B
inst✝¹⁶ : ChartedSpace HB B
EM : Type u_9
inst✝¹⁵ : NormedAddCommGroup EM
inst✝¹⁴ : NormedSpace 𝕜 EM
HM : Type u_10
inst✝¹³ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝¹² : TopologicalSpace M
inst✝¹¹ : ChartedSpace HM M
inst✝¹⁰ : (x : B) → AddCommMonoid (E x)
inst✝⁹ : (x : B) → Module 𝕜 (E x)
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : TopologicalSpace (TotalSpace F E)
inst✝⁵ : (x : B) → TopologicalSpace (E x)
inst✝⁴ : FiberBundle F E
inst✝³ : VectorBundle 𝕜 F E
inst✝² : ContMDiffVectorBundle n F E IB
e e' : Trivialization F TotalSpace.proj
inst✝¹ : MemTrivializationAtlas e
inst✝ : MemTrivializationAtlas e'
s : Set M
x : M
f : M → TotalSpace F E
hp : ContMDiffWithinAt IM IB n (TotalSpace.proj ∘ f) s x
hf : ContMDiffWithinAt IM 𝓘(𝕜, F) n (fun y ↦ (↑e (f y)).2) s x
he : (f x).proj ∈ e.baseSet
he' : (f x).proj ∈ e'.baseSet
⊢ (↑e' (f x)).2 = e.coordChange e' ((TotalSpace.proj ∘ f) x) (↑e (f x)).2

Tactic:
by simp [he]

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorBundle.Basic

Goal state before:
pp: case fst
n : ℕ∞ω
𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹⁰ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝⁹ : NormedAddCommGroup EB
inst✝⁸ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝⁷ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝⁶ : TopologicalSpace B
inst✝⁵ : ChartedSpace HB B
inst✝⁴ : (x : B) → AddCommMonoid (E x)
inst✝³ : (x : B) → Module 𝕜 (E x)
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle 𝕜 F E
ha : IsContMDiff IB a n
e e' : Pretrivialization F TotalSpace.proj
he : e ∈ a.pretrivializationAtlas
he' : e' ∈ a.pretrivializationAtlas
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
v : F
⊢ (b, (contMDiffCoordChange n IB a he he' b) v).1 = (↑e' { proj := b, snd := e.symm b v }).1

Tactic:
rw [e.mk_symm hb.1 v, e'.coe_fst', e.proj_symm_apply' hb.1]
    rw [e.proj_symm_apply' hb.1]; exact hb.2

Goal state after:
Module: Mathlib.Geometry.Manifold.BumpFunction

Goal state before:
pp: E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
x : M
inst✝ : FiniteDimensional ℝ E
hs : x ∈ (chartAt H c).source
hd : dist (↑(extChartAt I c) x) (↑(extChartAt I c) c) ≤ f.rIn
⊢ ↑f x = 1

Tactic:
by
  simp only [f.eqOn_source hs, (· ∘ ·), f.one_of_mem_closedBall hd]

Goal state after:
Module: Mathlib.Topology.PartitionOfUnity

Goal state before:
pp: ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
ρ : PartitionOfUnity ι X s
x₀ : X
⊢ ∀ᶠ (y : X) in 𝓝 x₀, ρ.fintsupport y ⊆ ρ.fintsupport x₀

Tactic:
by
  apply (ρ.locallyFinite.closure.eventually_subset (fun _ ↦ isClosed_closure) x₀).mono
  intro y hy z hz
  rw [PartitionOfUnity.mem_fintsupport_iff] at *
  exact hy hz

Goal state after:
Module: Mathlib.Geometry.Manifold.PartitionOfUnity

Goal state before:
pp: case refine_2
E : Type uE
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
H : Type uH
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : IsManifold I ∞ M
n : ℕ∞
inst✝² : T2Space M
inst✝¹ : NormalSpace M
inst✝ : SigmaCompactSpace M
s t : Set M
hs : IsClosed[inst✝⁶] s
ht : IsClosed[inst✝⁶] t
hd : Disjoint s t
u : Set M
u_op : IsOpen[inst✝⁶] u
hsu : s ⊆ u
hut : closure[inst✝⁶] u ⊆ tᶜ
v : Set M
v_op : IsOpen[inst✝⁶] v
htv : t ⊆ v
hvu : closure[inst✝⁶] v ⊆ (closure[inst✝⁶] u)ᶜ
f : C^↑n⟮I, M; 𝓘(ℝ, ℝ), ℝ⟯
hfu : EqOn (⇑f) 0 (closure[inst✝⁶] u)
hfv : EqOn (⇑f) 1 (closure[inst✝⁶] v)
hf : ∀ (x : M), f x ∈ Icc 0 1
⊢ ∀ᶠ (x : M) in 𝓝ˢ t, f x = 1

Tactic:
· exact eventually_of_mem (mem_of_superset (v_op.mem_nhdsSet.mpr htv) subset_closure) hfv

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

Goal state before:
pp: case refine_1
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : AEFinStronglyMeasurable f μ
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → 0 ≤ ∫ (x : α) in s, f x ∂μ
t : Set α := hf.sigmaFiniteSet
this : SigmaFinite (μ.restrict t)
s : Set α
hs : MeasurableSet s
hμts : (μ.restrict t) s < ∞
⊢ Integrable f (μ.restrict (s ∩ t))

Tactic:
rw [Measure.restrict_apply hs] at hμts

Goal state after:
case refine_1
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : AEFinStronglyMeasurable f μ
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → 0 ≤ ∫ (x : α) in s, f x ∂μ
t : Set α := hf.sigmaFiniteSet
this : SigmaFinite (μ.restrict t)
s : Set α
hs : MeasurableSet s
hμts : μ (s ∩ t) < ∞
⊢ Integrable f (μ.restrict (s ∩ t))
Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral

Goal state before:
pp: case refine_2
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : AEFinStronglyMeasurable f μ
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → 0 ≤ ∫ (x : α) in s, f x ∂μ
t : Set α := hf.sigmaFiniteSet
this : SigmaFinite (μ.restrict t)
s : Set α
hs : MeasurableSet s
hμts : (μ.restrict t) s < ∞
⊢ 0 ≤ ∫ (x : α) in s ∩ t, f x ∂μ

Tactic:
rw [Measure.restrict_apply hs] at hμts

Goal state after:
case refine_2
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : AEFinStronglyMeasurable f μ
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → 0 ≤ ∫ (x : α) in s, f x ∂μ
t : Set α := hf.sigmaFiniteSet
this : SigmaFinite (μ.restrict t)
s : Set α
hs : MeasurableSet s
hμts : μ (s ∩ t) < ∞
⊢ 0 ≤ ∫ (x : α) in s ∩ t, f x ∂μ
Module: Mathlib.Analysis.Calculus.Rademacher

Goal state before:
pp: case neg
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 ∉ K
A : f x = 0
⊢ ‖t⁻¹ • (f (x + t • v) - f x) * g x‖ ≤ K.indicator (fun x ↦ ↑C * ‖v‖ * ‖g x‖) x

Tactic:
have B : f (x + t • v) = 0 := by
        rw [← Function.notMem_support]
        contrapose hx
        apply mem_cthickening_of_dist_le _ _ (‖v‖) (tsupport f) (subset_tsupport _ hx)
        simp only [dist_eq_norm, sub_add_cancel_left, norm_neg, norm_smul, Real.norm_eq_abs,
          abs_of_nonneg t_pos.le]
        exact mul_le_of_le_one_left (norm_nonneg v) ht.2

Goal state after:
case neg
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 ∉ K
A : f x = 0
B : f (x + t • v) = 0
⊢ ‖t⁻¹ • (f (x + t • v) - f x) * g x‖ ≤ K.indicator (fun x ↦ ↑C * ‖v‖ * ‖g x‖) x
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.DerivIntegrable

Goal state before:
pp: case right
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
ebound : ∫⁻ (x : ℝ) in Icc a b, ‖deriv f x‖ₑ ∂volume ≤ ENNReal.ofReal (f b - f a)
⊢ ∫ (x : ℝ) in a..b, deriv f x ≤ f b - f a

Tactic:
← MeasureTheory.ofReal_integral_norm_eq_lintegral_enorm integrable_f_deriv,

Goal state after:
case right
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
ebound : ENNReal.ofReal (∫ (x : ℝ) in Icc a b, ‖deriv f x‖ ∂volume) ≤ ENNReal.ofReal (f b - f a)
⊢ ∫ (x : ℝ) in a..b, deriv f x ≤ f b - f a
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.LebesgueDifferentiationThm

Goal state before:
pp: case refine_1
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
h₂ : ∀ᵐ (x : ℝ), x ≠ b
g : ℝ → E := fun x ↦ if x ∈ Ioc a b then f x else 0
hg : LocallyIntegrable g volume
x : ℝ
hx : ∀ (c : ℝ), HasDerivAt (fun x ↦ ∫ (t : ℝ) in c..x, g t) (g x) x
a✝² : x ≠ a
a✝¹ : x ≠ b
a✝ : x ∈ Icc a b
c : ℝ
hc : c ∈ Icc a b
y : ℝ
hy : y ∈ Ioo a b
⊢ ∫ (t : ℝ) in c..y, f t = ∫ (t : ℝ) in c..y, g t

Tactic:
apply intervalIntegral.integral_congr_ae'

Goal state after:
case refine_1.h
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
h₂ : ∀ᵐ (x : ℝ), x ≠ b
g : ℝ → E := ⋯
hg : LocallyIntegrable g volume
x : ℝ
hx : ∀ (c : ℝ), HasDerivAt (fun x ↦ ∫ (t : ℝ) in c..x, g t) (g x) x
a✝² : x ≠ a
a✝¹ : x ≠ b
a✝ : x ∈ Icc a b
c : ℝ
hc : c ∈ Icc a b
y : ℝ
hy : y ∈ Ioo a b
⊢ ∀ᵐ (x : ℝ), x ∈ Ioc c y → f x = g x

case h'
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
h₂ : ∀ᵐ (x : ℝ), x ≠ b
g : ℝ → E := ⋯
hg : LocallyIntegrable g volume
x : ℝ
hx : ∀ (c : ℝ), HasDerivAt (fun x ↦ ∫ (t : ℝ) in c..x, g t) (g x) x
a✝² : x ≠ a
a✝¹ : x ≠ b
a✝ : x ∈ Icc a b
c : ℝ
hc : c ∈ Icc a b
y : ℝ
hy : y ∈ Ioo a b
⊢ ∀ᵐ (x : ℝ), x ∈ Ioc y c → f x = g x
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.LebesgueDifferentiationThm

Goal state before:
pp: case refine_2
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
h₂ : ∀ᵐ (x : ℝ), x ≠ b
g : ℝ → E := fun x ↦ if x ∈ Ioc a b then f x else 0
hg : LocallyIntegrable g volume
x : ℝ
hx : ∀ (c : ℝ), HasDerivAt (fun x ↦ ∫ (t : ℝ) in c..x, g t) (g x) x
a✝² : x ≠ a
a✝¹ : x ≠ b
a✝ : x ∈ Icc a b
c : ℝ
hc : c ∈ Icc a b
⊢ ∫ (t : ℝ) in c..x, f t = ∫ (t : ℝ) in c..x, g t

Tactic:
apply intervalIntegral.integral_congr_ae'

Goal state after:
case refine_2.h
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
h₂ : ∀ᵐ (x : ℝ), x ≠ b
g : ℝ → E := ⋯
hg : LocallyIntegrable g volume
x : ℝ
hx : ∀ (c : ℝ), HasDerivAt (fun x ↦ ∫ (t : ℝ) in c..x, g t) (g x) x
a✝² : x ≠ a
a✝¹ : x ≠ b
a✝ : x ∈ Icc a b
c : ℝ
hc : c ∈ Icc a b
⊢ ∀ᵐ (x_1 : ℝ), x_1 ∈ Ioc c x → f x_1 = g x_1

case h'
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
h₂ : ∀ᵐ (x : ℝ), x ≠ b
g : ℝ → E := ⋯
hg : LocallyIntegrable g volume
x : ℝ
hx : ∀ (c : ℝ), HasDerivAt (fun x ↦ ∫ (t : ℝ) in c..x, g t) (g x) x
a✝² : x ≠ a
a✝¹ : x ≠ b
a✝ : x ∈ Icc a b
c : ℝ
hc : c ∈ Icc a b
⊢ ∀ᵐ (x_1 : ℝ), x_1 ∈ Ioc x c → f x_1 = g x_1
Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: case h₁
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 :
  (∀ i ∈ Finset.range n, (I i).1 ∈ uIcc a b ∧ (I i).2 ∈ uIcc a b) ∧
    (↑(Finset.range n)).PairwiseDisjoint fun i ↦ uIoc (I i).1 (I i).2
i : ℕ
hi : i ∈ Finset.range n
⊢ dist (f (I i).1 • g (I i).1) (f (I i).1 • g (I i).2) ≤ C * dist (g (I i).1) (g (I i).2)

Tactic:
dist_smul₀

Goal state after:
case h₁
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 :
  (∀ i ∈ Finset.range n, (I i).1 ∈ uIcc a b ∧ (I i).2 ∈ uIcc a b) ∧
    (↑(Finset.range n)).PairwiseDisjoint fun i ↦ uIoc (I i).1 (I i).2
i : ℕ
hi : i ∈ Finset.range n
⊢ ‖f (I i).1‖ * dist (g (I i).1) (g (I i).2) ≤ C * dist (g (I i).1) (g (I i).2)
Module: Mathlib.Analysis.Calculus.Rademacher

Goal state before:
pp: E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
F : Type u_2
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
C : ℝ≥0
s : Set E
μ : Measure E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : FiniteDimensional ℝ F
inst✝ : μ.IsAddHaarMeasure
f : E → F
hf : LipschitzOnWith C f s
hs : MeasurableSet s
⊢ ∀ᵐ (x : E) ∂μ.restrict s, DifferentiableWithinAt ℝ f s x

Tactic:
rw [ae_restrict_iff' hs]
  exact hf.ae_differentiableWithinAt_of_mem

Goal state after: