mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Process.Filtration

Goal state before:
pp: case refine_2.inl
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
β : ι → Type u_3
inst✝⁵ : (i : ι) → TopologicalSpace (β i)
inst✝⁴ : ∀ (i : ι), MetrizableSpace (β i)
mβ : (i : ι) → MeasurableSpace (β i)
inst✝³ : ∀ (i : ι), BorelSpace (β i)
inst✝² : Preorder ι
inst✝¹ : (i : ι) → MulZeroOneClass (β i)
inst✝ : ∀ (i : ι), Nontrivial (β i)
s : ι → Set Ω
hsm : ∀ (i : ι), MeasurableSet (s i)
i : ι
t✝ : Set Ω
n : ι
ht : MeasurableSet t✝
t : Set Ω
hn : n ≤ i
u : Set (β n)
left✝ : MeasurableSet u
hu' : Set.univ = t
heq : ((s n).indicator fun x ↦ 1) ⁻¹' u = Set.univ
⊢ MeasurableSet Set.univ

pp: case refine_2.inr.inl
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
β : ι → Type u_3
inst✝⁵ : (i : ι) → TopologicalSpace (β i)
inst✝⁴ : ∀ (i : ι), MetrizableSpace (β i)
mβ : (i : ι) → MeasurableSpace (β i)
inst✝³ : ∀ (i : ι), BorelSpace (β i)
inst✝² : Preorder ι
inst✝¹ : (i : ι) → MulZeroOneClass (β i)
inst✝ : ∀ (i : ι), Nontrivial (β i)
s : ι → Set Ω
hsm : ∀ (i : ι), MeasurableSet (s i)
i : ι
t✝ : Set Ω
n : ι
ht : MeasurableSet t✝
t : Set Ω
hn : n ≤ i
u : Set (β n)
left✝ : MeasurableSet u
hu' : s n = t
heq : ((s n).indicator fun x ↦ 1) ⁻¹' u = s n
⊢ MeasurableSet (s n)

pp: case refine_2.inr.inr.inl
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
β : ι → Type u_3
inst✝⁵ : (i : ι) → TopologicalSpace (β i)
inst✝⁴ : ∀ (i : ι), MetrizableSpace (β i)
mβ : (i : ι) → MeasurableSpace (β i)
inst✝³ : ∀ (i : ι), BorelSpace (β i)
inst✝² : Preorder ι
inst✝¹ : (i : ι) → MulZeroOneClass (β i)
inst✝ : ∀ (i : ι), Nontrivial (β i)
s : ι → Set Ω
hsm : ∀ (i : ι), MeasurableSet (s i)
i : ι
t✝ : Set Ω
n : ι
ht : MeasurableSet t✝
t : Set Ω
hn : n ≤ i
u : Set (β n)
left✝ : MeasurableSet u
hu' : (s n)ᶜ = t
heq : ((s n).indicator fun x ↦ 1) ⁻¹' u = (s n)ᶜ
⊢ MeasurableSet (s n)ᶜ

pp: case refine_2.inr.inr.inr
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
β : ι → Type u_3
inst✝⁵ : (i : ι) → TopologicalSpace (β i)
inst✝⁴ : ∀ (i : ι), MetrizableSpace (β i)
mβ : (i : ι) → MeasurableSpace (β i)
inst✝³ : ∀ (i : ι), BorelSpace (β i)
inst✝² : Preorder ι
inst✝¹ : (i : ι) → MulZeroOneClass (β i)
inst✝ : ∀ (i : ι), Nontrivial (β i)
s : ι → Set Ω
hsm : ∀ (i : ι), MeasurableSet (s i)
i : ι
t✝ : Set Ω
n : ι
ht : MeasurableSet t✝
t : Set Ω
hn : n ≤ i
u : Set (β n)
left✝ : MeasurableSet u
hu' : ∅ = t
heq : ((s n).indicator fun x ↦ 1) ⁻¹' u = ∅
⊢ MeasurableSet ∅

Tactic:
exacts [MeasurableSet.univ, measurableSet_generateFrom ⟨n, hn, rfl⟩,
      MeasurableSet.compl (measurableSet_generateFrom ⟨n, hn, rfl⟩), measurableSet_empty _]

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSpace.DomAct.Basic

Goal state before:
pp: M : Type u_1
N : Type u_2
α : Type u_3
E : Type u_4
inst✝⁴ : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
inst✝² : SMul M α
inst✝¹ : SMulInvariantMeasure M α μ
inst✝ : MeasurableConstSMul M α
⊢ ∀ (a : Mᵈᵐᵃ) (x y : ↥(Lp E p μ)), a • (x + y) = a • x + a • y

Tactic:
rintro _ ⟨⟨⟩, _⟩ ⟨⟨⟩, _⟩

Goal state after:
case mk.mk
M : Type u_1
N : Type u_2
α : Type u_3
E : Type u_4
inst✝⁴ : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
inst✝² : SMul M α
inst✝¹ : SMulInvariantMeasure M α μ
inst✝ : MeasurableConstSMul M α
a✝² : Mᵈᵐᵃ
val✝¹ : α →ₘ[μ] E
a✝¹ : { f // AEStronglyMeasurable f μ }
property✝¹ : Quot.mk (⇑(Measure.aeEqSetoid E μ)) a✝¹ ∈ Lp E p μ
val✝ : α →ₘ[μ] E
a✝ : { f // AEStronglyMeasurable f μ }
property✝ : Quot.mk (⇑(Measure.aeEqSetoid E μ)) a✝ ∈ Lp E p μ
⊢ a✝² • (⟨Quot.mk (⇑(Measure.aeEqSetoid E μ)) a✝¹, property✝¹⟩ + ⟨Quot.mk (⇑(Measure.aeEqSetoid E μ)) a✝, property✝⟩) =
    a✝² • ⟨Quot.mk (⇑(Measure.aeEqSetoid E μ)) a✝¹, property✝¹⟩ +
      a✝² • ⟨Quot.mk (⇑(Measure.aeEqSetoid E μ)) a✝, property✝⟩
Module: Mathlib.MeasureTheory.Function.Intersectivity

Goal state before:
pp: case refine_1.inl
α : Type u_2
inst✝¹ : MeasurableSpace α
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
f : ℕ → α → ℝ≥0∞ := fun n ↦ (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1
hfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a
hf : ∀ (n : ℕ), Measurable (f n)
hf₁ : ∀ (n : ℕ), f n ≤ 1
x : α
n : ℕ
inst✝ : IsFiniteMeasure 0
hr : ∀ (n : ℕ), r ≤ 0 (s n)
M : (α → ℝ) → Set α := fun f ↦ {x | eLpNormEssSup f 0 < ↑‖f x‖₊}
N : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)
hN₀ : 0 N = 0
hN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \ N).Nonempty → 0 < 0 (⋂ n ∈ u, s n)
hrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂0
hμ : 0 ≠ 0
this : ∫⁻ (x : α), limsup (fun x_1 ↦ f x_1 x) atTop ∂0 ≤ 0 univ
hxN : x ∉ N
hx : ⨍⁻ (a : α), limsup (fun x ↦ f x a) atTop ∂0 ≤ limsup (fun x_1 ↦ f x_1 x) atTop
⊢ ⨍⁻ (x : α), f n x ∂0 ≤ 1

pp: case refine_1.inr
α : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ μ (s n)
M : (α → ℝ) → Set α := fun f ↦ {x | eLpNormEssSup f μ < ↑‖f x‖₊}
N : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)
hN₀ : μ N = 0
hN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)
f : ℕ → α → ℝ≥0∞ := fun n ↦ (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1
hfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a
hf : ∀ (n : ℕ), Measurable (f n)
hf₁ : ∀ (n : ℕ), f n ≤ 1
hrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ
hμ : μ ≠ 0
this : ∫⁻ (x : α), limsup (fun x_1 ↦ f x_1 x) atTop ∂μ ≤ μ univ
x : α
hxN : x ∉ N
hx : ⨍⁻ (a : α), limsup (fun x ↦ f x a) atTop ∂μ ≤ limsup (fun x_1 ↦ f x_1 x) atTop
n : ℕ
h✝ : NeZero μ
⊢ ⨍⁻ (x : α), f n x ∂μ ≤ 1

Tactic:
· simp

Goal state after:
case refine_1.inr
α : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ μ (s n)
M : (α → ℝ) → Set α := fun f ↦ {x | eLpNormEssSup f μ < ↑‖f x‖₊}
N : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)
hN₀ : μ N = 0
hN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)
f : ℕ → α → ℝ≥0∞ := fun n ↦ (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1
hfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a
hf : ∀ (n : ℕ), Measurable (f n)
hf₁ : ∀ (n : ℕ), f n ≤ 1
hrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ
hμ : μ ≠ 0
this : ∫⁻ (x : α), limsup (fun x_1 ↦ f x_1 x) atTop ∂μ ≤ μ univ
x : α
hxN : x ∉ N
hx : ⨍⁻ (a : α), limsup (fun x ↦ f x a) atTop ∂μ ≤ limsup (fun x_1 ↦ f x_1 x) atTop
n : ℕ
h✝ : NeZero μ
⊢ ⨍⁻ (x : α), f n x ∂μ ≤ 1
Module: Mathlib.MeasureTheory.Function.Piecewise

Goal state before:
pp: ι : Type u_1
α : Type u_2
β : Type u_3
inst✝² : MeasurableSpace α
s : ι → Set α
f : ι → α → β
inst✝¹ : Countable ι
hs : IndexedPartition s
hm : ∀ (i : ι), MeasurableSet (s i)
inst✝ : TopologicalSpace β
hf : ∀ (i : ι), StronglyMeasurable (f i)
Fi : Infinite ι
e : ℕ ≃ ι
n : ℕ
i : ι
hi : ∃ m < n, i = e m
⊢ hi.choose < n + 1

Tactic:
by grind

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

Goal state before:
pp: ι : Type u_1
α : Type u_2
β : Type u_3
inst✝² : MeasurableSpace α
s : ι → Set α
f : ι → α → β
inst✝¹ : Countable ι
hs : IndexedPartition s
hm : ∀ (i : ι), MeasurableSet (s i)
inst✝ : TopologicalSpace β
hf : ∀ (i : ι), StronglyMeasurable (f i)
Fi : Infinite ι
e : ℕ ≃ ι
g : (n : ℕ) → ι → Fin (n + 1) := fun n i ↦ if hi : ∃ m < n, i = e m then ⟨hi.choose, ⋯⟩ else Fin.last n
sg : ∀ (n : ℕ), Function.Surjective (g n)
G : (n : ℕ) → IndexedPartition fun k ↦ ⋃ i ∈ g n ⁻¹' {k}, s i
x : α
this : ∀ᶠ (n : ℕ) in atTop, e ↑((G n).index x) = hs.index x
n : ℕ
hn : e ↑((G n).index x) = hs.index x
⊢ (⋯.approx n) x = (⋯.approx n) x

Tactic:
by rw [hn]

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

Goal state before:
pp: case pos.inl
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
ε : ℝ≥0∞
hε : ε > 0
hfinε : ε ≠ ∞
hg' : MemLp g p 0
hui : UnifIntegrable f p 0
hut : UnifTight f p 0
hfg : ∀ᵐ (x : α) ∂0, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
⊢ ∃ N, ∀ n ≥ N, eLpNorm (f n - g) p 0 ≤ ε

pp: case pos.inr
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hut : UnifTight f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
hfinε : ε ≠ ∞
hμ : μ ≠ 0
⊢ ∃ N, ∀ n ≥ N, eLpNorm (f n - g) p μ ≤ ε

Tactic:
· simp

Goal state after:
case pos.inr
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hut : UnifTight f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
hfinε : ε ≠ ∞
hμ : μ ≠ 0
⊢ ∃ N, ∀ n ≥ N, eLpNorm (f n - g) p μ ≤ ε
Module: Mathlib.MeasureTheory.Group.ModularCharacter

Goal state before:
pp: G : Type u_1
inst✝³ : TopologicalSpace G
inst✝² : Group G
inst✝¹ : IsTopologicalGroup G
inst✝ : LocallyCompactSpace G
g h : G
this✝¹ : MeasurableSpace G := borel G
this✝ : BorelSpace G
mul_g_meas : Measurable fun x ↦ x * g
mul_h_meas : Measurable fun x ↦ x * h
ν : Measure G := haar
⊢ modularCharacterFun h * modularCharacterFun g =
    (map (fun x ↦ x * h) (map (fun x ↦ x * g) ν)).haarScalarFactor (map (fun x ↦ x * g) ν) * modularCharacterFun g

Tactic:
modularCharacterFun_eq_haarScalarFactor (map (· * g) ν) _

Goal state after:
G : Type u_1
inst✝³ : TopologicalSpace G
inst✝² : Group G
inst✝¹ : IsTopologicalGroup G
inst✝ : LocallyCompactSpace G
g h : G
this✝¹ : MeasurableSpace G := borel G
this✝ : BorelSpace G
mul_g_meas : Measurable fun x ↦ x * g
mul_h_meas : Measurable fun x ↦ x * h
ν : Measure G := haar
⊢ (map (fun x ↦ x * h) (map (fun x ↦ x * g) ν)).haarScalarFactor (map (fun x ↦ x * g) ν) * modularCharacterFun g =
    (map (fun x ↦ x * h) (map (fun x ↦ x * g) ν)).haarScalarFactor (map (fun x ↦ x * g) ν) * modularCharacterFun g
Module: Mathlib.MeasureTheory.Group.GeometryOfNumbers

Goal state before:
pp: case bc
E : Type u_1
inst✝⁸ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : Nontrivial E
inst✝² : μ.IsAddHaarMeasure
L : AddSubgroup E
inst✝¹ : Countable ↥L
inst✝ : DiscreteTopology ↥L
fund : IsAddFundamentalDomain (↥L) F μ
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h_cpt : IsCompact s
h : μ F * 2 ^ finrank ℝ E ≤ μ s
h_mes : μ s ≠ 0
h_nemp : s.Nonempty
u : ℕ → ℝ≥0 := ⋯.choose
K : ConvexBody E := { carrier := s, convex' := h_conv, isCompact' := h_cpt, nonempty' := h_nemp }
S : ℕ → ConvexBody E := fun n ↦ (1 + u n) • K
Z : ℕ → Set E := fun n ↦ ↑(S n) ∩ (↑L \ {0})
h_zero : 0 ∈ K
h_clos : IsClosed (↑L \ {0})
n : ℕ
⊢ 1 < ENNReal.ofReal (1 + ↑(u n))

Tactic:
simp [u, (exists_seq_strictAnti_tendsto (0 : ℝ≥0)).choose_spec.2.1 n]

Goal state after:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β✝ : Type u_3
γ✝ : Type u_4
inst✝⁶ : TopologicalSpace α
inst✝⁵ : TopologicalSpace β✝
inst✝⁴ : Zero β✝
γ : Type u_5
inst✝³ : TopologicalSpace γ
inst✝² : Zero γ
β : Type u_6
inst✝¹ : PseudoMetricSpace β
inst✝ : Zero β
f : α →C_c β
this : IsCompact (Set.range ⇑f)
C : ℝ
hC : ∀ ⦃x : β⦄, x ∈ Set.range ⇑f → ∀ ⦃y : β⦄, y ∈ Set.range ⇑f → dist x y ≤ C
⊢ ∀ (x y : α), dist (f x) (f y) ≤ C

Tactic:
by grind

Goal state after:
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Basic

Goal state before:
pp: case inr.inl
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
a b c : E
ω : E → E →L[𝕜] F
γab : Path a b
γbc : Path b c
instE : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
ht : 0 < 1 / 2
H₁ : ⇑(γab.trans γbc).extend =ᶠ[𝓝 0] fun s ↦ γab.extend (2 * s)
⊢ 𝓝[Icc 0 2] (2 * 0) = 𝓝[I] (2 * 0)

pp: case inr.inr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
a b c : E
t : ℝ
ω : E → E →L[𝕜] F
γab : Path a b
γbc : Path b c
ht : t < 1 / 2
instE : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
H₁ : ⇑(γab.trans γbc).extend =ᶠ[𝓝 t] fun s ↦ γab.extend (2 * s)
ht₀ : 0 < t
⊢ 𝓝[Icc 0 2] (2 * t) = 𝓝[I] (2 * t)

Tactic:
· simp

Goal state after:
case inr.inr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
a b c : E
t : ℝ
ω : E → E →L[𝕜] F
γab : Path a b
γbc : Path b c
ht : t < 1 / 2
instE : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
H₁ : ⇑(γab.trans γbc).extend =ᶠ[𝓝 t] fun s ↦ γab.extend (2 * s)
ht₀ : 0 < t
⊢ 𝓝[Icc 0 2] (2 * t) = 𝓝[I] (2 * t)
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
a b c : E
t : ℝ
ω : E → E →L[𝕜] F
γab : Path a b
γbc : Path b c
ht : t < 1 / 2
⊢ curveIntegralFun ω (γab.trans γbc) t = 2 • curveIntegralFun ω γab (2 * t)

Tactic:
by
  let instE := NormedSpace.restrictScalars ℝ 𝕜 E
  have H₁ : (γab.trans γbc).extend =ᶠ[𝓝 t] (fun s ↦ γab.extend (2 * s)) :=
    (eventually_le_nhds ht).mono fun _ ↦ Path.extend_trans_of_le_half _ _
  have H₂ : (2 : ℝ) • I =ᶠ[𝓝 (2 * t)] I := by
    rw [LinearOrderedField.smul_Icc two_pos, mul_zero, mul_one, ← nhdsWithin_eq_iff_eventuallyEq]
    rcases lt_trichotomy t 0 with ht₀ | rfl | ht₀
    · rw [notMem_closure_iff_nhdsWithin_eq_bot.mp, notMem_closure_iff_nhdsWithin_eq_bot.mp] <;>
        simp_intro h <;> linarith
    · simp
    · rw [nhdsWithin_eq_nhds.2, nhdsWithin_eq_nhds.2] <;> simp [*] <;> linarith
  rw [curveIntegralFun_def, H₁.self_of_nhds, H₁.derivWithin_eq_of_nhds, curveIntegralFun_def,
    derivWithin_comp_mul_left, ofNat_smul_eq_nsmul, map_nsmul, derivWithin_congr_set H₂]

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

Goal state before:
pp: case e_a.e_a
f : ℝ → ℝ
N : ℕ
N_nonzero : 0 < N
a b : ℝ
⊢ ∀ x ∈ Finset.range (N - 1), f (a + (↑(N - 1 - 1 - x) + 1) * (b - a) / ↑N) = f (b + (↑x + 1) * (a - b) / ↑N)

Tactic:
intro k hk

Goal state after:
case e_a.e_a
f : ℝ → ℝ
N : ℕ
N_nonzero : 0 < N
a b : ℝ
k : ℕ
hk : k ∈ Finset.range (N - 1)
⊢ f (a + (↑(N - 1 - 1 - k) + 1) * (b - a) / ↑N) = f (b + (↑k + 1) * (a - b) / ↑N)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule

Goal state before:
pp: case e_a.e_a
f : ℝ → ℝ
N : ℕ
N_nonzero : 0 < N
a b : ℝ
k : ℕ
hk : k ∈ Finset.range (N - 1)
⊢ f (a + ↑(N - 1 - (k + 1) + 1) * (b - a) / ↑N) = f (b + ↑(k + 1) * (a - b) / ↑N)

Tactic:
Nat.cast_add,

Goal state after:
case e_a.e_a
f : ℝ → ℝ
N : ℕ
N_nonzero : 0 < N
a b : ℝ
k : ℕ
hk : k ∈ Finset.range (N - 1)
⊢ f (a + (↑(N - 1 - (k + 1)) + ↑1) * (b - a) / ↑N) = f (b + ↑(k + 1) * (a - b) / ↑N)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral

Goal state before:
pp: case xs
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
f : E → F
a : E
r : ℝ
hr : 0 ≤ r
hdf : ∀ᶠ (x : E) in 𝓝 a, DifferentiableAt ℝ f x
hderiv : fderiv ℝ f =O[𝓝 a] fun x ↦ ‖x - a‖ ^ r
C : ℝ
hC₀ : C > 0
hC : ∀ᶠ (x : E) in 𝓝 a, ‖fderiv ℝ f x‖ ≤ C * ‖‖x - a‖ ^ r‖
ε : ℝ
hε₀ : ε > 0
hε : ∀ y ∈ ball a ε, DifferentiableAt ℝ f y ∧ ‖fderiv ℝ f y‖ ≤ C * ‖‖y - a‖ ^ r‖
y : E
hy : y ∈ ball a ε
hsub : closedBall a ‖y - a‖ ⊆ ball a ε
⊢ a ∈ closedBall a ‖y - a‖

pp: case ys
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
f : E → F
a : E
r : ℝ
hr : 0 ≤ r
hdf : ∀ᶠ (x : E) in 𝓝 a, DifferentiableAt ℝ f x
hderiv : fderiv ℝ f =O[𝓝 a] fun x ↦ ‖x - a‖ ^ r
C : ℝ
hC₀ : C > 0
hC : ∀ᶠ (x : E) in 𝓝 a, ‖fderiv ℝ f x‖ ≤ C * ‖‖x - a‖ ^ r‖
ε : ℝ
hε₀ : ε > 0
hε : ∀ y ∈ ball a ε, DifferentiableAt ℝ f y ∧ ‖fderiv ℝ f y‖ ≤ C * ‖‖y - a‖ ^ r‖
y : E
hy : y ∈ ball a ε
hsub : closedBall a ‖y - a‖ ⊆ ball a ε
⊢ y ∈ closedBall a ‖y - a‖

Tactic:
· simp

Goal state after:
case ys
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
f : E → F
a : E
r : ℝ
hr : 0 ≤ r
hdf : ∀ᶠ (x : E) in 𝓝 a, DifferentiableAt ℝ f x
hderiv : fderiv ℝ f =O[𝓝 a] fun x ↦ ‖x - a‖ ^ r
C : ℝ
hC₀ : C > 0
hC : ∀ᶠ (x : E) in 𝓝 a, ‖fderiv ℝ f x‖ ≤ C * ‖‖x - a‖ ^ r‖
ε : ℝ
hε₀ : ε > 0
hε : ∀ y ∈ ball a ε, DifferentiableAt ℝ f y ∧ ‖fderiv ℝ f y‖ ≤ C * ‖‖y - a‖ ^ r‖
y : E
hy : y ∈ ball a ε
hsub : closedBall a ‖y - a‖ ⊆ ball a ε
⊢ y ∈ closedBall a ‖y - a‖
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule

Goal state before:
pp: f : ℝ → ℝ
N : ℕ
a h : ℝ
N_nonzero : 0 < N
⊢ h / 2 *
      (f (a + ↑0 * h) + f (a + (↑N - ↑(Nat.succ 0) + 1) * h) +
        ∑ x ∈ Finset.range (N - 1), (f (a + ↑(x + 1) * h) + f (a + (↑x + 1) * h))) =
    h * ((f a + f (a + ↑N * h)) / 2 + ∑ x ∈ Finset.range (N - 1), f (a + (↑x + 1) * h))

Tactic:
Nat.cast_add,

Goal state after:
f : ℝ → ℝ
N : ℕ
a h : ℝ
N_nonzero : 0 < N
⊢ h / 2 *
      (f (a + ↑0 * h) + f (a + (↑N - ↑(Nat.succ 0) + 1) * h) +
        ∑ x ∈ Finset.range (N - 1), (f (a + (↑x + ↑1) * h) + f (a + (↑x + 1) * h))) =
    h * ((f a + f (a + ↑N * h)) / 2 + ∑ x ∈ Finset.range (N - 1), f (a + (↑x + 1) * h))
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule

Goal state before:
pp: f : ℝ → ℝ
N : ℕ
a h : ℝ
N_nonzero : 0 < N
h_f_int : IntervalIntegrable f volume a (a + ↑N * h)
⊢ trapezoidal_integral f N a (a + ↑N * h) - ∫ (x : ℝ) in a + ↑0 * h..a + ↑N * h, f x =
    trapezoidal_integral f N a (a + ↑N * h) - ∫ (x : ℝ) in a..a + ↑N * h, f x

pp: f : ℝ → ℝ
N : ℕ
a h : ℝ
N_nonzero : 0 < N
h_f_int : IntervalIntegrable f volume a (a + ↑N * h)
⊢ ∀ k < N, IntervalIntegrable f volume (a + ↑k * h) (a + ↑(k + 1) * h)

Tactic:
· simp

Goal state after:
f : ℝ → ℝ
N : ℕ
a h : ℝ
N_nonzero : 0 < N
h_f_int : IntervalIntegrable f volume a (a + ↑N * h)
⊢ ∀ k < N, IntervalIntegrable f volume (a + ↑k * h) (a + ↑(k + 1) * h)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule

Goal state before:
pp: f : ℝ → ℝ
N : ℕ
a h : ℝ
N_nonzero : 0 < N
h_f_int : IntervalIntegrable f volume a (a + ↑N * h)
⊢ ∀ k < N, IntervalIntegrable f volume (a + ↑k * h) (a + ↑(k + 1) * h)

Tactic:
intro k hk

Goal state after:
f : ℝ → ℝ
N : ℕ
a h : ℝ
N_nonzero : 0 < N
h_f_int : IntervalIntegrable f volume a (a + ↑N * h)
k : ℕ
hk : k < N
⊢ IntervalIntegrable f volume (a + ↑k * h) (a + ↑(k + 1) * h)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule

Goal state before:
pp: case inl
f : ℝ → ℝ
N : ℕ
a h : ℝ
N_nonzero : 0 < N
h_f_int : IntervalIntegrable f volume a (a + ↑N * h)
k : ℕ
hk : k < N
h_neg : h ≤ 0
⊢ ∀ {k : ℕ}, k ≤ N → a + ↑k * h ∈ [[a, a + ↑N * h]]

Tactic:
intro k hk

Goal state after:
case inl
f : ℝ → ℝ
N : ℕ
a h : ℝ
N_nonzero : 0 < N
h_f_int : IntervalIntegrable f volume a (a + ↑N * h)
k✝ : ℕ
hk✝ : k✝ < N
h_neg : h ≤ 0
k : ℕ
hk : k ≤ N
⊢ a + ↑k * h ∈ [[a, a + ↑N * h]]
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule

Goal state before:
pp: case inr
f : ℝ → ℝ
N : ℕ
a h : ℝ
N_nonzero : 0 < N
h_f_int : IntervalIntegrable f volume a (a + ↑N * h)
k : ℕ
hk : k < N
h_pos : 0 ≤ h
⊢ ∀ {k : ℕ}, k ≤ N → a + ↑k * h ∈ [[a, a + ↑N * h]]

Tactic:
intro k hk

Goal state after:
case inr
f : ℝ → ℝ
N : ℕ
a h : ℝ
N_nonzero : 0 < N
h_f_int : IntervalIntegrable f volume a (a + ↑N * h)
k✝ : ℕ
hk✝ : k✝ < N
h_pos : 0 ≤ h
k : ℕ
hk : k ≤ N
⊢ a + ↑k * h ∈ [[a, a + ↑N * h]]
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule

Goal state before:
pp: f : ℝ → ℝ
ζ a b : ℝ
a_lt_b : a < b
h_df : DifferentiableOn ℝ f (Set.Icc a b)
h_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)
fpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ
N : ℕ
N_nonzero : 0 < N
h : ℝ := (b - a) / ↑N
ak : ℕ → ℝ := fun k ↦ a + ↑k * h
h0 : ∀ (k : ℕ), ak (k + 1) - ak k = h
hab : 0 < b - a
hpos : 0 < h
hb : b = a + ↑N * h
⊢ ∀ k ∈ Finset.range N, |trapezoidal_error f 1 (ak k) (ak (k + 1))| ≤ ζ / 12 * h ^ 3

Tactic:
intro k hk

Goal state after:
f : ℝ → ℝ
ζ a b : ℝ
a_lt_b : a < b
h_df : DifferentiableOn ℝ f (Set.Icc a b)
h_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)
fpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ
N : ℕ
N_nonzero : 0 < N
h : ℝ := (b - a) / ↑N
ak : ℕ → ℝ := fun k ↦ a + ↑k * h
h0 : ∀ (k : ℕ), ak (k + 1) - ak k = h
hab : 0 < b - a
hpos : 0 < h
hb : b = a + ↑N * h
k : ℕ
hk : k ∈ Finset.range N
⊢ |trapezoidal_error f 1 (ak k) (ak (k + 1))| ≤ ζ / 12 * h ^ 3
Module: Mathlib.MeasureTheory.Integral.TorusIntegral

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : (Fin 1 → ℂ) → E
c : Fin 1 → ℂ
R : Fin 1 → ℝ
H₁ : (⇑(MeasurableEquiv.funUnique (Fin 1) ℝ).symm ⁻¹' Icc 0 fun x ↦ 2 * π) = Icc 0 (2 * π)
⊢ torusMap c R = fun θ x ↦ circleMap (c 0) (R 0) (θ 0)

Tactic:
ext θ i : 2

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : (Fin 1 → ℂ) → E
c : Fin 1 → ℂ
R : Fin 1 → ℝ
H₁ : (⇑(MeasurableEquiv.funUnique (Fin 1) ℝ).symm ⁻¹' Icc 0 fun x ↦ 2 * π) = Icc 0 (2 * π)
θ : Fin 1 → ℝ
i : Fin 1
⊢ torusMap c R θ i = circleMap (c 0) (R 0) (θ 0)
Module: Mathlib.MeasureTheory.Integral.TorusIntegral

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : (Fin 1 → ℂ) → E
c : Fin 1 → ℂ
R : Fin 1 → ℝ
⊢ (∯ (x : Fin 1 → ℂ) in T(c, R), f x) = ∮ (z : ℂ) in C(c 0, R 0), f fun x ↦ z

Tactic:
have H₁ : (((MeasurableEquiv.funUnique _ _).symm) ⁻¹' Icc 0 fun _ => 2 * π) = Icc 0 (2 * π) :=
    (OrderIso.funUnique (Fin 1) ℝ).symm.preimage_Icc _ _
  have H₂ : torusMap c R = fun θ _ ↦ circleMap (c 0) (R 0) (θ 0) := by
    ext θ i : 2
    rw [Subsingleton.elim i 0]; rfl
  rw [torusIntegral, circleIntegral, intervalIntegral.integral_of_le Real.two_pi_pos.le,
    Measure.restrict_congr_set Ioc_ae_eq_Icc,
    ← ((volume_preserving_funUnique (Fin 1) ℝ).symm _).setIntegral_preimage_emb
      (MeasurableEquiv.measurableEmbedding _), H₁, H₂]
  simp [circleMap_zero]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.FiniteMeasureExt

Goal state before:
pp: E : Type u_1
𝕜 : Type u_2
inst✝⁶ : RCLike 𝕜
inst✝⁵ : MeasurableSpace E
inst✝⁴ : TopologicalSpace E
inst✝³ : PolishSpace E
inst✝² : BorelSpace E
P P' : Measure E
inst✝¹ : IsFiniteMeasure P
inst✝ : IsFiniteMeasure P'
A : StarSubalgebra 𝕜 (E →ᵇ 𝕜)
hA : (StarSubalgebra.map (toContinuousMapStarₐ 𝕜) A).SeparatesPoints
heq : ∀ g ∈ A, ∫ (x : E), g x ∂P = ∫ (x : E), g x ∂P'
⊢ P = P'

Tactic:
let := TopologicalSpace.upgradeIsCompletelyMetrizable E
  exact ext_of_forall_mem_subalgebra_integral_eq_of_pseudoEMetric_complete_countable hA heq

Goal state after:
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.TaylorExpansion

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝ : IsProbabilityMeasure P
X : Ω → ℝ
hX : AEMeasurable X P
h0 : ∫ (x : Ω), X x ∂P = 0
h1 : ∫ (x : Ω), (X ^ 2) x ∂P = 1
t : ℝ
⊢ 1 + ↑0 * ↑t * I - ↑1 * ↑t ^ 2 / 2 = 1 - ↑t ^ 2 / 2

pp: case hint
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝ : IsProbabilityMeasure P
X : Ω → ℝ
hX : AEMeasurable X P
h0 : ∫ (x : Ω), X x ∂P = 0
h1 : ∫ (x : Ω), (X ^ 2) x ∂P = 1
t : ℝ
⊢ MemLp id 2 (Measure.map X P)

Tactic:
· simp

Goal state after:
case hint
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝ : IsProbabilityMeasure P
X : Ω → ℝ
hX : AEMeasurable X P
h0 : ∫ (x : Ω), X x ∂P = 0
h1 : ∫ (x : Ω), (X ^ 2) x ∂P = 1
t : ℝ
⊢ MemLp id 2 (Measure.map X P)