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

Goal state before:
pp: 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'
⊢ if h : IsRCLikeNormedField 𝕜 then Convex ℝ (range ↑(I.toPartialEquiv ≫ (↑e).toEquiv.toPartialEquiv))
  else range ↑(I.toPartialEquiv ≫ (↑e).toEquiv.toPartialEquiv) = univ

Tactic:
by
    split_ifs with h
    · simp only [PartialEquiv.coe_trans, Equiv.toPartialEquiv_apply, LinearEquiv.coe_toEquiv,
      ContinuousLinearEquiv.coe_toLinearEquiv, toPartialEquiv_coe]
      rw [range_comp]
      let := h.rclike
      let := NormedSpace.restrictScalars ℝ 𝕜 E
      let := NormedSpace.restrictScalars ℝ 𝕜 E'
      let eR : E →L[ℝ] E' := ContinuousLinearMap.restrictScalars ℝ (e : E →L[𝕜] E')
      change Convex ℝ (⇑eR '' range ↑I)
      apply I.convex_range.linear_image
    · simp [range_eq_univ_of_not_isRCLikeNormedField I h, range_comp]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
F : Type u_5
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
n : ℕ∞ω
V : M → Type u_6
inst✝⁵ : TopologicalSpace (TotalSpace F V)
inst✝⁴ : (x : M) → TopologicalSpace (V x)
inst✝³ : FiberBundle F V
inst✝² : (x : M) → AddCommGroup (V x)
inst✝¹ : (x : M) → Module 𝕜 (V x)
inst✝ : VectorBundle 𝕜 F V
u : Set M
x₀ : M
ι : Type u_7
t : ι → (x : M) → V x
ht : LocallyFinite fun i ↦ {x | t i x ≠ 0}
ht' : ∀ (i : ι), ContMDiffWithinAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, t i x⟩) u x₀
y : M
hy : y ∈ Set.univ
⊢ HasFiniteSupport fun i ↦ t i y

Tactic:
choose U hu hfin using ht y

Goal state after:
𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
F : Type u_5
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
n : ℕ∞ω
V : M → Type u_6
inst✝⁵ : TopologicalSpace (TotalSpace F V)
inst✝⁴ : (x : M) → TopologicalSpace (V x)
inst✝³ : FiberBundle F V
inst✝² : (x : M) → AddCommGroup (V x)
inst✝¹ : (x : M) → Module 𝕜 (V x)
inst✝ : VectorBundle 𝕜 F V
u : Set M
x₀ : M
ι : Type u_7
t : ι → (x : M) → V x
ht : LocallyFinite fun i ↦ {x | t i x ≠ 0}
ht' : ∀ (i : ι), ContMDiffWithinAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, t i x⟩) u x₀
y : M
hy : y ∈ Set.univ
U : Set M
hu : U ∈ nhds y
hfin : {i | ((fun i ↦ {x | t i x ≠ 0}) i ∩ U).Nonempty}.Finite
⊢ HasFiniteSupport fun i ↦ t i y
Module: Mathlib.Analysis.Distribution.AEEqOfIntegralContDiff

Goal state before:
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
meas_U : MeasurableSet U✝
U : Opens M := { carrier := U✝, is_open' := hU }
this : SigmaCompactSpace ↥U
g : ↥U → ℝ
h : ∫ (x : ↑↑U), Function.extend Subtype.val g 0 ↑x • f ↑x ∂Measure.comap Subtype.val μ = 0
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:
Subtype.val_injective.extend_apply

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
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
h : ∫ (x : ↑↑U), g x • f ↑x ∂Measure.comap Subtype.val μ = 0
⊢ ∫ (x : ↥U), g x • f ↑x ∂Measure.comap Subtype.val μ = 0
Module: Mathlib.Analysis.Distribution.AEEqOfIntegralContDiff

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

Tactic:
rintro ⟨x, rfl⟩

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

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : SigmaFinite μ
f : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = 0
S : ℕ → Set α := spanningSets μ
⊢ μ ({a | ¬f a = 0 a} ∩ ⋃ b, spanningSets μ b) = 0

Tactic:
Set.inter_iUnion,

Goal state after:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : SigmaFinite μ
f : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = 0
S : ℕ → Set α := spanningSets μ
⊢ μ (⋃ i, {a | ¬f a = 0 a} ∩ spanningSets μ i) = 0
Module: Mathlib.MeasureTheory.Measure.Haar.Disintegration

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹³ : NontriviallyNormedField 𝕜
inst✝¹² : CompleteSpace 𝕜
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : MeasurableSpace E
inst✝⁹ : BorelSpace E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : MeasurableSpace F
inst✝⁵ : BorelSpace F
inst✝⁴ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝³ : μ.IsAddHaarMeasure
inst✝² : ν.IsAddHaarMeasure
inst✝¹ : LocallyCompactSpace E
inst✝ : LocallyCompactSpace F
s : Set F
hs : MeasurableSet s
this : FiniteDimensional 𝕜 E
⊢ (∀ᵐ (y : F) ∂ν, ∀ᵐ (x : E) ∂μ, y + L x ∈ s) ↔ ∀ᵐ (y : F) ∂ν, y ∈ s

Tactic:
have : FiniteDimensional 𝕜 F := .of_locallyCompactSpace 𝕜

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹³ : NontriviallyNormedField 𝕜
inst✝¹² : CompleteSpace 𝕜
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : MeasurableSpace E
inst✝⁹ : BorelSpace E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : MeasurableSpace F
inst✝⁵ : BorelSpace F
inst✝⁴ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝³ : μ.IsAddHaarMeasure
inst✝² : ν.IsAddHaarMeasure
inst✝¹ : LocallyCompactSpace E
inst✝ : LocallyCompactSpace F
s : Set F
hs : MeasurableSet s
this✝ : FiniteDimensional 𝕜 E
this : FiniteDimensional 𝕜 F
⊢ (∀ᵐ (y : F) ∂ν, ∀ᵐ (x : E) ∂μ, y + L x ∈ s) ↔ ∀ᵐ (y : F) ∂ν, y ∈ s
Module: Mathlib.Analysis.Calculus.TangentCone.Pi

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : Semiring 𝕜
ι : Type u_2
E : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommGroup (E i)
inst✝³ : (i : ι) → Module 𝕜 (E i)
inst✝² : (i : ι) → TopologicalSpace (E i)
inst✝¹ : ∀ (i : ι), ContinuousAdd (E i)
inst✝ : ∀ (i : ι), ContinuousConstSMul 𝕜 (E i)
s : (i : ι) → Set (E i)
x : (i : ι) → E i
h : ∀ (i : ι), UniqueDiffWithinAt 𝕜 (s i) (x i)
⊢ UniqueDiffWithinAt 𝕜 (univ.pi s) x

Tactic:
by
  classical
  simp only [uniqueDiffWithinAt_iff, closure_pi_set] at h ⊢
  refine ⟨.of_closure <| (dense_pi univ fun i _ ↦ (h i).1).closure.mono ?_, fun i _ => (h i).2⟩
  simp only [closure_pi_set, ← Submodule.closure_coe_iSup_map_single, Submodule.map_span]
  gcongr
  refine iSup_le fun i ↦ ?_
  gcongr
  exact mapsTo_tangentConeAt_pi (fun j _ ↦ (h j).2) |>.image_subset

Goal state after:
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 c l (Bornology.cobounded 𝕜)
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)

Tactic:
simpa using Tendsto.congr' this <| (tendsto_inv₀_cobounded.comp hc).smul hd

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

Goal state before:
pp: E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
β : Type u_3
inst✝⁴ : TopologicalSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
inst✝¹ : SigmaCompactSpace β
inst✝ : R1Space β
μ : Measure β
f : β → E
hf : Integrable f μ
h'f : ∀ (s : Set β), IsCompact s → ∫ (x : β) in s, f x ∂μ = 0
s : Set β
hs : IsClosed[inst✝⁴] s
t : ℕ → Set β := fun n ↦ closure[inst✝⁴] (compactCovering β n) ∩ s
B : s = ⋃ n, t n
⊢ Tendsto (fun n ↦ ∫ (x : β) in t n, f x ∂μ) atTop (𝓝 (∫ (x : β) in s, f x ∂μ))

Tactic:
B

Goal state after:
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
β : Type u_3
inst✝⁴ : TopologicalSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
inst✝¹ : SigmaCompactSpace β
inst✝ : R1Space β
μ : Measure β
f : β → E
hf : Integrable f μ
h'f : ∀ (s : Set β), IsCompact s → ∫ (x : β) in s, f x ∂μ = 0
s : Set β
hs : IsClosed[inst✝⁴] s
t : ℕ → Set β := fun n ↦ closure[inst✝⁴] (compactCovering β n) ∩ s
B : s = ⋃ n, t n
⊢ Tendsto (fun n ↦ ∫ (x : β) in t n, f x ∂μ) atTop (𝓝 (∫ (x : β) in ⋃ n, t n, f x ∂μ))
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.DerivIntegrable

Goal state before:
pp: case e'_3
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 a..b, ‖deriv f x‖ ≤ f b - f a
⊢ EqOn (deriv f) (fun x ↦ ‖deriv f x‖) (uIoo a b)

Tactic:
uIoo_of_le hab

Goal state after:
case e'_3
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 a..b, ‖deriv f x‖ ≤ f b - f a
⊢ EqOn (deriv f) (fun x ↦ ‖deriv f x‖) (Ioo a b)
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
⊢ ∃ᶠ (ε : ℝ) in 𝓝[>] 0, ∃ a ∈ t, (Prod.snd - Prod.fst) a = ε ∧ a.1 = x

Tactic:
apply Eventually.frequently

Goal state after:
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) := ⋯
s : Set ℝ := ⋯
x : ℝ
hx : x ∈ s
⊢ ∀ᶠ (x_1 : ℝ) in 𝓝[>] 0, ∃ a ∈ t, (Prod.snd - Prod.fst) a = x_1 ∧ 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}
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
⊢ ε < η → ε < δ → ε < (b - x) / 2 → ∃ a ∈ t, (Prod.snd - Prod.fst) a = ε ∧ a.1 = x

Tactic:
hε₂

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
⊢ ε < δ → ε < (b - x) / 2 → ∃ a ∈ t, (Prod.snd - Prod.fst) a = ε ∧ a.1 = x
Module: Mathlib.Analysis.Calculus.Taylor

Goal state before:
pp: E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
⊢ -((↑k !)⁻¹ * (x - y) ^ k) = ((↑k + 1) * ↑k !)⁻¹ * (-(↑k + 1) * (x - y) ^ k)

Tactic:
field

Goal state after:
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)
⊢ (↑K + 1) * (ε / (↑K + 1)) = ε

Tactic:
field

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

Goal state before:
pp: case e'_3
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)
δ' : ℝ := (b - a) / (↑n + 1)
⊢ (b - a) / (↑n + 1) = 1 / (↑n + 1) * (b - a)

Tactic:
field

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

Goal state before:
pp: case e'_4
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)
δ' : ℝ := (b - a) / (↑n + 1)
⊢ δ = δ / (b - a) * (b - a)

Tactic:
field

Goal state after:
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
n : ℕ
hn : 1 / (↑n + 1) < δ / (b - a)
δ' : ℝ := (b - a) / (↑n + 1)
hδ₃ : δ' < δ
⊢ BoundedVariationOn f (Icc a b)

Tactic:
have h_mono : Monotone fun (i : ℕ) ↦ a + ↑i * δ' := by
    apply Monotone.const_add
    apply Monotone.mul_const Nat.mono_cast
    simp only [δ']
    refine div_nonneg ?_ ?_ <;> 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
hab₁ : 0 < b - a
n : ℕ
hn : 1 / (↑n + 1) < δ / (b - a)
δ' : ℝ := (b - a) / (↑n + 1)
hδ₃ : δ' < δ
h_mono : Monotone fun i ↦ a + ↑i * δ'
⊢ BoundedVariationOn f (Icc a b)
Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: case e'_2.e'_4
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)
δ' : ℝ := (b - a) / (↑n + 1)
hδ₃ : δ' < δ
h_mono : Monotone fun i ↦ a + ↑i * δ'
⊢ b = a + (↑n + 1) * ((b - a) / (↑n + 1))

Tactic:
field

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

Goal state before:
pp: f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n + 1) f [[x₀, x]]
hu : UniqueDiffOn ℝ [[x₀, x]]
⊢ ∃ x' ∈ uIoo x₀ x,
    f x - taylorWithinEval f n [[x₀, x]] x₀ x = iteratedDeriv (n + 1) f x' * (x - x₀) ^ (n + 1) / ↑(n + 1)!

Tactic:
have hd : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIcc x₀ x) := by
    refine hf.differentiableOn_iteratedDerivWithin ?_ hu
    norm_cast
    simp

Goal state after:
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n + 1) f [[x₀, x]]
hu : UniqueDiffOn ℝ [[x₀, x]]
hd : DifferentiableOn ℝ (iteratedDerivWithin n f [[x₀, x]]) [[x₀, x]]
⊢ ∃ x' ∈ uIoo x₀ x,
    f x - taylorWithinEval f n [[x₀, x]] x₀ x = iteratedDeriv (n + 1) f x' * (x - x₀) ^ (n + 1) / ↑(n + 1)!
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun

Goal state before:
pp: X : Type u_1
inst✝ : PseudoMetricSpace X
f : ℝ → X
d b y : ℝ
hdb : d ≤ b
hf : AbsolutelyContinuousOnInterval f d b
u : Set (ℝ × ℝ)
hu₃ : HasSum (fun z ↦ (↑z).2 - (↑z).1) (b - d)
hu₄ : HasSum (fun z ↦ dist (f (↑z).1) (f (↑z).2)) y
u_coe : Finset ↑u → Finset (ℝ × ℝ) := fun s ↦ Finset.image Subtype.val s
hu₁ : ∀ (s : Finset ↑u) ⦃z : ℝ × ℝ⦄, z ∈ u_coe s → d ≤ z.1 ∧ z.1 ≤ z.2 ∧ z.2 ≤ b
hu₂ : ∀ (s : Finset ↑u), (↑(u_coe s)).PairwiseDisjoint fun z ↦ Icc z.1 z.2
T : Finset ↑u → ℕ × (ℕ → ℝ × ℝ) := fun s ↦ ((u_coe s).card + 1, fun i ↦ (u_coe s).intervalGapsWithin ⋯ d b ↑i)
hT₁ : ∀ (s : Finset ↑u) (i : ℕ), d ≤ ((u_coe s).intervalGapsWithin ⋯ d b ↑i).1
hT₂ : ∀ (s : Finset ↑u) (i : ℕ), ((u_coe s).intervalGapsWithin ⋯ d b ↑i).1 ≤ ((u_coe s).intervalGapsWithin ⋯ d b ↑i).2
hT₃ : ∀ (s : Finset ↑u) (i : ℕ), ((u_coe s).intervalGapsWithin ⋯ d b ↑i).2 ≤ b
hT₄ :
  ∀ (s : Finset ↑u),
    (Iio ((u_coe s).card + 1)).PairwiseDisjoint fun j ↦
      Ioc ((u_coe s).intervalGapsWithin ⋯ d b ↑j).1 ((u_coe s).intervalGapsWithin ⋯ d b ↑j).2
hT : MapsTo T univ (disjWithin d b)
s : Finset ↑u
g : ℝ → ℝ → ℝ
⊢ SurjOn Subtype.val ↑s ↑(u_coe s)

Tactic:
simp only [Finset.coe_image, u_coe]

Goal state after:
X : Type u_1
inst✝ : PseudoMetricSpace X
f : ℝ → X
d b y : ℝ
hdb : d ≤ b
hf : AbsolutelyContinuousOnInterval f d b
u : Set (ℝ × ℝ)
hu₃ : HasSum (fun z ↦ (↑z).2 - (↑z).1) (b - d)
hu₄ : HasSum (fun z ↦ dist (f (↑z).1) (f (↑z).2)) y
u_coe : Finset ↑u → Finset (ℝ × ℝ) := fun s ↦ Finset.image Subtype.val s
hu₁ : ∀ (s : Finset ↑u) ⦃z : ℝ × ℝ⦄, z ∈ u_coe s → d ≤ z.1 ∧ z.1 ≤ z.2 ∧ z.2 ≤ b
hu₂ : ∀ (s : Finset ↑u), (↑(u_coe s)).PairwiseDisjoint fun z ↦ Icc z.1 z.2
T : Finset ↑u → ℕ × (ℕ → ℝ × ℝ) := fun s ↦ ((u_coe s).card + 1, fun i ↦ (u_coe s).intervalGapsWithin ⋯ d b ↑i)
hT₁ : ∀ (s : Finset ↑u) (i : ℕ), d ≤ ((u_coe s).intervalGapsWithin ⋯ d b ↑i).1
hT₂ : ∀ (s : Finset ↑u) (i : ℕ), ((u_coe s).intervalGapsWithin ⋯ d b ↑i).1 ≤ ((u_coe s).intervalGapsWithin ⋯ d b ↑i).2
hT₃ : ∀ (s : Finset ↑u) (i : ℕ), ((u_coe s).intervalGapsWithin ⋯ d b ↑i).2 ≤ b
hT₄ :
  ∀ (s : Finset ↑u),
    (Iio ((u_coe s).card + 1)).PairwiseDisjoint fun j ↦
      Ioc ((u_coe s).intervalGapsWithin ⋯ d b ↑j).1 ((u_coe s).intervalGapsWithin ⋯ d b ↑j).2
hT : MapsTo T univ (disjWithin d b)
s : Finset ↑u
g : ℝ → ℝ → ℝ
⊢ SurjOn Subtype.val (↑s) (Subtype.val '' ↑s)