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

Goal state before:
pp: case refine_1
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✝¹⁵ : NontriviallyNormedField 𝕜
inst✝¹⁴ : NormedAddCommGroup F
inst✝¹³ : NormedSpace 𝕜 F
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
EB : Type u_7
inst✝¹⁰ : NormedAddCommGroup EB
inst✝⁹ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝⁸ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
EM : Type u_10
inst✝⁷ : NormedAddCommGroup EM
inst✝⁶ : NormedSpace 𝕜 EM
HM : Type u_11
inst✝⁵ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace HM M
inst✝² : TopologicalSpace B
inst✝¹ : ChartedSpace HB B
inst✝ : FiberBundle F E
f : M → TotalSpace F E
s : Set M
x₀ : M
hf : ContinuousWithinAt f s x₀
h1 : (fun x ↦ (f x).proj) ⁻¹' (trivializationAt F E (f x₀).proj).baseSet ∈ 𝓝[s] x₀
x : M
hx : x ∈ (fun x ↦ (f x).proj) ⁻¹' (trivializationAt F E (f x₀).proj).baseSet
⊢ (Prod.fst ∘ fun x ↦
        (↑(extChartAt IB (f x₀).proj) (↑(trivializationAt F E (f x₀).proj).toPartialEquiv (f x)).1,
          (↑(trivializationAt F E (f x₀).proj).toPartialEquiv (f x)).2))
      x =
    ↑(extChartAt IB (f x₀).proj) (f x).proj

Tactic:
Function.comp,

Goal state after:
case refine_1
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✝¹⁵ : NontriviallyNormedField 𝕜
inst✝¹⁴ : NormedAddCommGroup F
inst✝¹³ : NormedSpace 𝕜 F
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
EB : Type u_7
inst✝¹⁰ : NormedAddCommGroup EB
inst✝⁹ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝⁸ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
EM : Type u_10
inst✝⁷ : NormedAddCommGroup EM
inst✝⁶ : NormedSpace 𝕜 EM
HM : Type u_11
inst✝⁵ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace HM M
inst✝² : TopologicalSpace B
inst✝¹ : ChartedSpace HB B
inst✝ : FiberBundle F E
f : M → TotalSpace F E
s : Set M
x₀ : M
hf : ContinuousWithinAt f s x₀
h1 : (fun x ↦ (f x).proj) ⁻¹' (trivializationAt F E (f x₀).proj).baseSet ∈ 𝓝[s] x₀
x : M
hx : x ∈ (fun x ↦ (f x).proj) ⁻¹' (trivializationAt F E (f x₀).proj).baseSet
⊢ ↑(extChartAt IB (f x₀).proj) (↑(trivializationAt F E (f x₀).proj).toPartialEquiv (f x)).1 =
    ↑(extChartAt IB (f x₀).proj) (f x).proj
Module: Mathlib.Topology.VectorBundle.Basic

Goal state before:
pp: R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁶ : NontriviallyNormedField R
inst✝⁵ : (x : B) → AddCommMonoid (E x)
inst✝⁴ : (x : B) → Module R (E x)
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace R F
inst✝¹ : TopologicalSpace B
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle R F E
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.pretrivializationAtlas
this : ContinuousOn (fun x ↦ (a.coordChange he' he x.1) x.2) ((e'.baseSet ∩ e.baseSet) ×ˢ univ)
b : B
f : F
hb : (b, f).1 ∈ e'.baseSet ∩ e.baseSet
⊢ ↑e (↑e'.symm (b, f)) = (b, (a.coordChange he' he b) f)

Tactic:
a.mk_coordChange _ _ hb,

Goal state after:
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁶ : NontriviallyNormedField R
inst✝⁵ : (x : B) → AddCommMonoid (E x)
inst✝⁴ : (x : B) → Module R (E x)
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace R F
inst✝¹ : TopologicalSpace B
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle R F E
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.pretrivializationAtlas
this : ContinuousOn (fun x ↦ (a.coordChange he' he x.1) x.2) ((e'.baseSet ∩ e.baseSet) ×ˢ univ)
b : B
f : F
hb : (b, f).1 ∈ e'.baseSet ∩ e.baseSet
⊢ ↑e (↑e'.symm (b, f)) = ↑e { proj := (b, f).1, snd := e'.symm (b, f).1 f }
Module: Mathlib.Geometry.Manifold.VectorBundle.Basic

Goal state before:
pp: case refine_2
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
f : M → B
g : M → F
x : M
hf : ContMDiffWithinAt IM IB n f s x
hg : ContMDiffWithinAt IM 𝓘(𝕜, F) n g s x
he : f x ∈ e.baseSet
he' : f x ∈ e'.baseSet
⊢ e.coordChange e' (f x) (g x) = ↑(Trivialization.coordChangeL 𝕜 e e' (f x)) (g x)

Tactic:
· exact (Trivialization.coordChangeL_apply' e e' ⟨he, he'⟩ (g x)).symm

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
n : ℕ∞ω
E : Type u_2
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace 𝕜 E
E' : Type u_3
inst✝¹¹ : NormedAddCommGroup E'
inst✝¹⁰ : NormedSpace 𝕜 E'
H : Type u_4
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
H' : Type u_5
inst✝⁸ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M : Type u_6
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
M' : Type u_7
inst✝⁵ : TopologicalSpace M'
inst✝⁴ : ChartedSpace H' M'
F : Type u_8
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : IsManifold I 1 M
inst✝ : IsManifold I' 1 M'
h : IsManifold I 2 M
⊢ ContMDiffVectorBundle 1 E (TangentSpace I) I

Tactic:
have : IsManifold I (1 + 1) M := h
  exact TangentBundle.contMDiffVectorBundle

Goal state after:
Module: Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u_21
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H M'
q' : M'
t : Set M'
⊢ HasMFDerivAt[t] Sum.inr q' (ContinuousLinearMap.id 𝕜 (TangentSpace I q'))

Tactic:
by
  refine ⟨by fun_prop, ?_⟩
  have : (writtenInExtChartAt I I q' (@Sum.inr M M'))
      =ᶠ[𝓝[(extChartAt I q').symm ⁻¹' t ∩ Set.range I] (extChartAt I q' q')] id :=
    writtenInExtChartAt_sumInr_eventuallyEq_id.filter_mono (nhdsWithin_mono _ (fun _y hy ↦ hy.2))
  exact (hasFDerivWithinAt_id (extChartAt I q' q') _).congr_of_eventuallyEq this
    (by simp [writtenInExtChartAt, extChartAt])

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

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝¹⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace 𝕜 E
E' : Type u_3
inst✝¹⁵ : NormedAddCommGroup E'
inst✝¹⁴ : NormedSpace 𝕜 E'
F : Type u_4
inst✝¹³ : NormedAddCommGroup F
inst✝¹² : NormedSpace 𝕜 F
H : Type u_5
inst✝¹¹ : TopologicalSpace H
H' : Type u_6
inst✝¹⁰ : TopologicalSpace H'
G : Type u_7
inst✝⁹ : TopologicalSpace G
G' : Type u_8
inst✝⁸ : TopologicalSpace G'
I : ModelWithCorners 𝕜 E H
I' : ModelWithCorners 𝕜 E' H'
J : ModelWithCorners 𝕜 F G
J' : ModelWithCorners 𝕜 F G'
M : Type u_9
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
M' : Type u_10
inst✝⁵ : TopologicalSpace M'
inst✝⁴ : ChartedSpace H' M'
N : Type u_11
inst✝³ : TopologicalSpace N
inst✝² : ChartedSpace G N
N' : Type u_12
inst✝¹ : TopologicalSpace N'
inst✝ : ChartedSpace G' N'
n : ℕ∞ω
e : E ≃L[𝕜] E'
h : IsRCLikeNormedField 𝕜
this✝¹ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this✝ : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
this : NormedSpace ℝ E' := NormedSpace.restrictScalars ℝ 𝕜 E'
eR : E →L[ℝ] E' := ContinuousLinearMap.restrictScalars ℝ ↑e
⊢ Convex ℝ (⇑eR '' range ↑I)

Tactic:
apply I.convex_range.linear_image

Goal state after:
Module: Mathlib.Topology.PartitionOfUnity

Goal state before:
pp: ι : Type u
X : Type v
inst✝² : TopologicalSpace X
s : Set X
ρ : PartitionOfUnity ι X s
x₀ : X
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module ℝ M
φ : ι → X → M
this : (fun i ↦ (ρ i) x₀ • φ i x₀) = (fun i ↦ (ρ i) x₀) • fun i ↦ φ i x₀
⊢ (support fun i ↦ (ρ i) x₀ • φ i x₀) ⊆ ↑(ρ.finsupport x₀)

Tactic:
rw [ρ.coe_finsupport x₀, this, support_smul]

Goal state after:
ι : Type u
X : Type v
inst✝² : TopologicalSpace X
s : Set X
ρ : PartitionOfUnity ι X s
x₀ : X
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module ℝ M
φ : ι → X → M
this : (fun i ↦ (ρ i) x₀ • φ i x₀) = (fun i ↦ (ρ i) x₀) • fun i ↦ φ i x₀
⊢ ((support fun i ↦ (ρ i) x₀) ∩ support fun i ↦ φ i x₀) ⊆ support fun i ↦ (ρ i) x₀
Module: Mathlib.Geometry.Manifold.PartitionOfUnity

Goal state before:
pp: ι : Type uι
E : Type uE
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
H : Type uH
inst✝² : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
s : Set M
f : SmoothPartitionOfUnity ι I M s
x : M
hx : x ∈ s
⊢ ∃ i, 0 < (f i) x

Tactic:
by
  by_contra! h
  have H : ∀ i, f i x = 0 := fun i ↦ le_antisymm (h i) (f.nonneg i x)
  have := f.sum_eq_one hx
  simp_rw [H] at this
  simpa

Goal state after:
Module: Mathlib.Analysis.Distribution.AEEqOfIntegralContDiff

Goal state before:
pp: case refine_1
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
h : ∀ (g : M → ℝ), ContMDiff I 𝓘(ℝ, ℝ) ∞ g → HasCompactSupport g → tsupport g ⊆ U✝ → ∫ (x : M), g x • f x ∂μ = 0
meas_U : MeasurableSet U✝
U : Opens M := { carrier := U✝, is_open' := hU }
this : SigmaCompactSpace ↥U
⊢ LocallyIntegrable (fun x ↦ f ↑x) (Measure.comap Subtype.val μ)

pp: case refine_2
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
h : ∀ (g : M → ℝ), ContMDiff I 𝓘(ℝ, ℝ) ∞ g → HasCompactSupport g → tsupport g ⊆ U✝ → ∫ (x : M), g x • f x ∂μ = 0
meas_U : MeasurableSet U✝
U : Opens M := { carrier := U✝, is_open' := hU }
this : SigmaCompactSpace ↥U
g : ↥U → ℝ
g_smth : ContMDiff I 𝓘(ℝ, ℝ) ∞ g
g_supp : HasCompactSupport g
⊢ ∫ (x : ↥U), g x • f ↑x ∂Measure.comap Subtype.val μ = 0

Tactic:
· exact (locallyIntegrable_comap meas_U).mpr hf

Goal state after:
case refine_2
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
h : ∀ (g : M → ℝ), ContMDiff I 𝓘(ℝ, ℝ) ∞ g → HasCompactSupport g → tsupport g ⊆ U✝ → ∫ (x : M), g x • f x ∂μ = 0
meas_U : MeasurableSet U✝
U : Opens M := { carrier := U✝, is_open' := hU }
this : SigmaCompactSpace ↥U
g : ↥U → ℝ
g_smth : ContMDiff I 𝓘(ℝ, ℝ) ∞ g
g_supp : HasCompactSupport g
⊢ ∫ (x : ↥U), g x • f ↑x ∂Measure.comap Subtype.val μ = 0
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 : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
v : E
L : ℝ →L[ℝ] E := ContinuousLinearMap.smulRight 1 v
p : E
this✝ : ∀ᵐ (s : ℝ), DifferentiableAt ℝ (fun t ↦ f (p + t • v)) s
s : ℝ
hs : DifferentiableAt ℝ (fun t ↦ f (p + t • v)) s
h's : DifferentiableAt ℝ (fun t ↦ f (p + t • v)) (s + 0)
this : DifferentiableAt ℝ (fun t ↦ s + t) 0
⊢ DifferentiableAt ℝ (fun t ↦ f (p + s • v + t • v)) 0

Tactic:
convert! h's.comp 0 this with _ t

Goal state after:
case e'_11
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 : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
v : E
L : ℝ →L[ℝ] E := ContinuousLinearMap.smulRight 1 v
p : E
this✝ : ∀ᵐ (s : ℝ), DifferentiableAt ℝ (fun t ↦ f (p + t • v)) s
s : ℝ
hs : DifferentiableAt ℝ (fun t ↦ f (p + t • v)) s
h's : DifferentiableAt ℝ (fun t ↦ f (p + t • v)) (s + 0)
this : DifferentiableAt ℝ (fun t ↦ s + t) 0
t : ℝ
⊢ f (p + s • v + t • v) = ((fun t ↦ f (p + t • v)) ∘ HAdd.hAdd s) t
Module: Mathlib.Analysis.Calculus.TangentCone.Seq

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NormedDivisionRing 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul 𝕜 E
α : Type u_3
l : Filter α
c : α → 𝕜
d : α → E
y : E
hc : Tendsto (fun n ↦ ‖c n‖) l atTop
hd : Tendsto (fun n ↦ c n • d n) l (𝓝 y)
⊢ Tendsto d l (𝓝 0)

Tactic:
have : ∀ᶠ n in l, (c n)⁻¹ • c n • d n = d n :=
    (eventually_ne_of_tendsto_norm_atTop hc 0).mono fun n hn ↦ inv_smul_smul₀ hn (d n)

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NormedDivisionRing 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul 𝕜 E
α : Type u_3
l : Filter α
c : α → 𝕜
d : α → E
y : E
hc : Tendsto (fun n ↦ ‖c n‖) l atTop
hd : Tendsto (fun n ↦ c n • d n) l (𝓝 y)
this : ∀ᶠ (n : α) in l, (c n)⁻¹ • c n • d n = d n
⊢ Tendsto d l (𝓝 0)
Module: Mathlib.Order.SuccPred.IntervalSucc

Goal state before:
pp: case h
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : SuccOrder α
inst✝¹ : IsSuccArchimedean α
inst✝ : LinearOrder β
f : α → β
a : α
hf : ∀ i ∈ Ici a, f a ≤ f i
b : β
hb : b ∈ Ici (f a)
h2f : b ∉ ⋃ i ∈ Ici a, Ico (f i) (f (succ i))
⊢ b ∈ {x | ∀ a_1 ∈ Ici a, f a_1 ≤ x}

Tactic:
exact Succ.rec (P := fun i _ ↦ f i ≤ b) hb (by simp_all)

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

Goal state before:
pp: case hf
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → F
d b η : ℝ
hdb✝ : d ≤ b
hf : ∀ᵐ (x : ℝ), x ∈ Ioo d b → HasDerivAt f (f' x) x
hη : 0 < η
hdb : d < b
t : Set (ℝ × ℝ) := {z | (d < z.1 ∧ z.1 < z.2 ∧ z.2 < b) ∧ dist (slope f z.1 z.2) (f' z.1) < η}
s : Set ℝ := {x | x ∈ Ioo d b ∧ HasDerivAt f (f' x) x}
x : ℝ
hx : x ∈ s
this : Tendsto (slope f x) (𝓝[≠] x) (𝓝 (f' x))
δ : ℝ
hδ₁ : δ > 0
hδ₂ : ∀ ⦃x_1 : ℝ⦄, x_1 ∈ {x}ᶜ → dist x_1 x < δ → dist (slope f x x_1) (f' x) < η
evn_bound : ∀ {α : ℝ}, 0 < α → ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ε < α
evn_pos : ∀ᶠ (ε : ℝ) in 𝓝[>] 0, 0 < ε
⊢ ∀ᶠ (x_1 : ℝ) in 𝓝[>] 0, ∃ a ∈ t, (Prod.snd - Prod.fst) a = x_1 ∧ a.1 = x

Tactic:
filter_upwards [evn_pos, evn_bound hη, evn_bound hδ₁,
                      evn_bound (α := (b - x) / 2) (by simp [hx.left.right])]
        with ε hε₁ hε₂ hε₃ hε₄

Goal state after:
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → F
d b η : ℝ
hdb✝ : d ≤ b
hf : ∀ᵐ (x : ℝ), x ∈ Ioo d b → HasDerivAt f (f' x) x
hη : 0 < η
hdb : d < b
t : Set (ℝ × ℝ) := {z | (d < z.1 ∧ z.1 < z.2 ∧ z.2 < b) ∧ dist (slope f z.1 z.2) (f' z.1) < η}
s : Set ℝ := {x | x ∈ Ioo d b ∧ HasDerivAt f (f' x) x}
x : ℝ
hx : x ∈ s
this : Tendsto (slope f x) (𝓝[≠] x) (𝓝 (f' x))
δ : ℝ
hδ₁ : δ > 0
hδ₂ : ∀ ⦃x_1 : ℝ⦄, x_1 ∈ {x}ᶜ → dist x_1 x < δ → dist (slope f x x_1) (f' x) < η
evn_bound : ∀ {α : ℝ}, 0 < α → ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ε < α
evn_pos : ∀ᶠ (ε : ℝ) in 𝓝[>] 0, 0 < ε
ε : ℝ
hε₁ : 0 < ε
hε₂ : ε < η
hε₃ : ε < δ
hε₄ : ε < (b - x) / 2
⊢ ∃ a ∈ t, (Prod.snd - Prod.fst) a = ε ∧ a.1 = x
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun

Goal state before:
pp: F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → F
d b η : ℝ
hdb✝ : d ≤ b
hf : ∀ᵐ (x : ℝ), x ∈ Ioo d b → HasDerivAt f (f' x) x
hη : 0 < η
hdb : d < b
t : Set (ℝ × ℝ) := {z | (d < z.1 ∧ z.1 < z.2 ∧ z.2 < b) ∧ dist (slope f z.1 z.2) (f' z.1) < η}
s : Set ℝ := {x | x ∈ Ioo d b ∧ HasDerivAt f (f' x) x}
u : Set (ℝ × ℝ)
hu₁ : ∀ x ∈ u, (d < x.1 ∧ x.1 < x.2 ∧ x.2 < b) ∧ dist (slope f x.1 x.2) (f' x.1) < η
hu₂ : u.Countable
hu₃ : u.PairwiseDisjoint fun z ↦ Icc z.1 z.2
hu₄ : volume (s \ ⋃ z ∈ u, Icc z.1 z.2) = 0
this✝ : Countable ↑u
this : Pairwise (Disjoint on fun z ↦ Icc (↑z).1 (↑z).2)
⊢ volume (Ioo d b \ ⋃ z ∈ u, Icc z.1 z.2) = 0

Tactic:
measure_eq_zero_iff_ae_notMem

Goal state after:
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → F
d b η : ℝ
hdb✝ : d ≤ b
hf : ∀ᵐ (x : ℝ), x ∈ Ioo d b → HasDerivAt f (f' x) x
hη : 0 < η
hdb : d < b
t : Set (ℝ × ℝ) := {z | (d < z.1 ∧ z.1 < z.2 ∧ z.2 < b) ∧ dist (slope f z.1 z.2) (f' z.1) < η}
s : Set ℝ := {x | x ∈ Ioo d b ∧ HasDerivAt f (f' x) x}
u : Set (ℝ × ℝ)
hu₁ : ∀ x ∈ u, (d < x.1 ∧ x.1 < x.2 ∧ x.2 < b) ∧ dist (slope f x.1 x.2) (f' x.1) < η
hu₂ : u.Countable
hu₃ : u.PairwiseDisjoint fun z ↦ Icc z.1 z.2
hu₄ : ∀ᵐ (a : ℝ), a ∉ s \ ⋃ z ∈ u, Icc z.1 z.2
this✝ : Countable ↑u
this : Pairwise (Disjoint on fun z ↦ Icc (↑z).1 (↑z).2)
⊢ ∀ᵐ (a : ℝ), a ∉ Ioo d b \ ⋃ z ∈ u, Icc z.1 z.2
Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: X : Type u_1
inst✝ : PseudoMetricSpace X
a b : ℝ
f : ℝ → X
K : ℝ≥0
hfK : LipschitzOnWith K f (uIcc a b)
ε : ℝ
hε : ε > 0
x✝ : ℕ × (ℕ → ℝ × ℝ)
n : ℕ
I : ℕ → ℝ × ℝ
hnI₁ : (n, I) ∈ disjWithin a b
hnI₂ : ∑ i ∈ Finset.range (n, I).1, dist ((n, I).2 i).1 ((n, I).2 i).2 < ε / (↑K + 1)
⊢ ∑ i ∈ Finset.range (n, I).1, dist (f ((n, I).2 i).1) (f ((n, I).2 i).2) < ε

Tactic:
calc
    _ ≤ ∑ i ∈ Finset.range n, K * dist (I i).1 (I i).2 := by
      apply Finset.sum_le_sum
      intro i hi
      have := hfK (hnI₁.left i hi).left (hnI₁.left i hi).right
      apply ENNReal.toReal_mono (Ne.symm (not_eq_of_beq_eq_false rfl)) at this
      rwa [ENNReal.toReal_mul, ← dist_edist, ← dist_edist] at this
    _ = K * ∑ i ∈ Finset.range n, dist (I i).1 (I i).2 := by symm; exact Finset.mul_sum _ _ _
    _ ≤ K * (ε / (K + 1)) := by gcongr
    _ < (K + 1) * (ε / (K + 1)) := by gcongr; linarith
    _ = ε := by field

Goal state after:
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
⊢ ‖ρ • L y - ρ • L w‖ ≤ ρ * (‖L‖ * ‖y - w‖)

Tactic:
rw [← smul_sub, norm_smul, norm_rho]
        gcongr
        exact L.lipschitz.norm_sub_le _ _

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

Goal state before:
pp: case inl
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
x : F
h : ‖x‖ = ‖0‖
⊢ SameRay ℝ x 0 ↔ x = 0

pp: case inr
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
x y : F
h : ‖x‖ = ‖y‖
hy : y ≠ 0
⊢ SameRay ℝ x y ↔ x = y

Tactic:
· rw [norm_zero, norm_eq_zero] at h
    exact iff_of_true (SameRay.zero_right _) h

Goal state after:
case inr
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
x y : F
h : ‖x‖ = ‖y‖
hy : y ≠ 0
⊢ SameRay ℝ x y ↔ x = y
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun

Goal state before:
pp: f g : ℝ → ℝ
a b : ℝ
hf : AbsolutelyContinuousOnInterval f a b
hg : AbsolutelyContinuousOnInterval g a b
⊢ ∫ (x : ℝ) in a..b, deriv f x * g x + f x * deriv g x = f b * g b - f a * g a

Tactic:
← (hf.fun_mul hg).integral_deriv_eq_sub

Goal state after:
f g : ℝ → ℝ
a b : ℝ
hf : AbsolutelyContinuousOnInterval f a b
hg : AbsolutelyContinuousOnInterval g a b
⊢ ∫ (x : ℝ) in a..b, deriv f x * g x + f x * deriv g x = ∫ (x : ℝ) in a..b, deriv (fun i ↦ f i * g i) x
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
hw : w ∈ closedBall z r
hw_lt : ‖f w‖ < ‖f z‖
⊢ False

Tactic:
have hr : 0 < r := dist_pos.2 (ne_of_apply_ne (norm ∘ f) hw_lt.ne)

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
hw_lt : ‖f w‖ < ‖f z‖
hr : 0 < r
⊢ False
Module: Mathlib.Analysis.Convex.SpecificFunctions.Deriv

Goal state before:
pp: case hf''
m : ℤ
hm₀ : m ≠ 0
hm₁ : m ≠ 1
x : ℝ
hx : 0 < x
⊢ 0 < (∏ i ∈ Finset.range 2, (↑m - ↑i)) * x ^ (m - ↑2)

Tactic:
refine mul_pos ?_ (zpow_pos hx _)

Goal state after:
case hf''
m : ℤ
hm₀ : m ≠ 0
hm₁ : m ≠ 1
x : ℝ
hx : 0 < x
⊢ 0 < ∏ i ∈ Finset.range 2, (↑m - ↑i)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex

Goal state before:
pp: θ : ℂ
⊢ sin θ = 0 ∨ cos θ = 0 ↔ ∃ k, ↑k * ↑π / 2 = θ

Tactic:
← mul_eq_zero,

Goal state after:
θ : ℂ
⊢ sin θ * cos θ = 0 ↔ ∃ k, ↑k * ↑π / 2 = θ
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex

Goal state before:
pp: x y : ℂ
h :
  ((∀ (k : ℤ), x ≠ (2 * ↑k + 1) * ↑π / 2) ∧ ∀ (l : ℤ), y ≠ (2 * ↑l + 1) * ↑π / 2) ∨
    (∃ k, x = (2 * ↑k + 1) * ↑π / 2) ∧ ∃ l, y = (2 * ↑l + 1) * ↑π / 2
this : tan (x + -y) = (tan x + -tan y) / (1 - tan x * -tan y)
⊢ tan (x - y) = (tan x - tan y) / (1 + tan x * tan y)

Tactic:
convert! this using 2

Goal state after:
case e'_3.e'_6
x y : ℂ
h :
  ((∀ (k : ℤ), x ≠ (2 * ↑k + 1) * ↑π / 2) ∧ ∀ (l : ℤ), y ≠ (2 * ↑l + 1) * ↑π / 2) ∨
    (∃ k, x = (2 * ↑k + 1) * ↑π / 2) ∧ ∃ l, y = (2 * ↑l + 1) * ↑π / 2
this : tan (x + -y) = (tan x + -tan y) / (1 - tan x * -tan y)
⊢ 1 + tan x * tan y = 1 - tan x * -tan y
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: x✝ x : ℝ
hx : x ≠ 0
hx₀ : 0 < x
hxπ : 1 < x
⊢ 1 < x ^ 2

Tactic:
one_lt_sq_iff₀ hx₀.le

Goal state after:
x✝ x : ℝ
hx : x ≠ 0
hx₀ : 0 < x
hxπ : 1 < x
⊢ 1 < x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: x : ℝ
h : x ∈ Ioo (-1) 1
⊢ arcsin x = arctan (x / √(1 - x ^ 2))

Tactic:
by
  rw_mod_cast [arctan_eq_arcsin, div_pow, sq_sqrt, one_add_div, div_div, ← sqrt_mul,
    mul_div_cancel₀, sub_add_cancel, sqrt_one, div_one] <;> simp at h <;> nlinarith [h.1, h.2]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: x y : ℝ
h : x * y < 1
| (x + y) / (1 - x * y)

Tactic:
rw [← tan_arctan x, ← tan_arctan y]

Goal state after:
x y : ℝ
h : x * y < 1
| (tan (arctan x) + tan (arctan y)) / (1 - tan (arctan x) * tan (arctan y))