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

Goal state before:
pp: E : Type uE
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type uH
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
c : M
inst✝¹ : FiniteDimensional ℝ E
inst✝ : T2Space M
⊢ (𝓝 c).HasBasis (fun x ↦ True) fun f ↦ tsupport ↑f

Tactic:
by
  have :
    (𝓝 c).HasBasis (fun _ : SmoothBumpFunction I c => True) fun f =>
      (extChartAt I c).symm '' (closedBall (extChartAt I c c) f.rOut ∩ range I) := by
    rw [← map_extChartAt_symm_nhdsWithin_range (I := I) c]
    exact nhdsWithin_range_basis.map _
  exact this.to_hasBasis' (fun f _ => ⟨f, trivial, f.tsupport_subset_symm_image_closedBall⟩)
    fun f _ => f.tsupport_mem_nhds

Goal state after:
Module: Mathlib.Topology.ShrinkingLemma

Goal state before:
pp: case h
ι : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
u : ι → Set X
s : Set X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
v : PartialRefinement u s fun w ↦ IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier
si : Set X := ⋯
hsi : si = s ∩ ⋂ i_1, ⋂ (_ : ¬i_1 = i), (v.toFun i_1)ᶜ
hsic : IsCompact si
this : si ⊆ v.toFun i
vi : Set X
hvi : IsOpen vi ∧ si ⊆ vi ∧ closure vi ⊆ v.toFun i ∧ IsCompact (closure vi)
x : X
hx : x ∈ s
h : ∀ (j : ι), j ≠ i → x ∉ v.toFun j
⊢ x ∈ si

Tactic:
rw [hsi]

Goal state after:
case h
ι : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
u : ι → Set X
s : Set X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
v : PartialRefinement u s fun w ↦ IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier
si : Set X := s ∩ (⋃ j, ⋃ (_ : j ≠ i), v.toFun j)ᶜ
hsi : si = s ∩ ⋂ i_1, ⋂ (_ : ¬i_1 = i), (v.toFun i_1)ᶜ
hsic : IsCompact si
this : si ⊆ v.toFun i
vi : Set X
hvi : IsOpen vi ∧ si ⊆ vi ∧ closure vi ⊆ v.toFun i ∧ IsCompact (closure vi)
x : X
hx : x ∈ s
h : ∀ (j : ι), j ≠ i → x ∉ v.toFun j
⊢ x ∈ s ∩ ⋂ i_1, ⋂ (_ : ¬i_1 = i), (v.toFun i_1)ᶜ
Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
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
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ∞
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ ∫ (a : α) in s, f a ∂μ ≤ 0

Tactic:
exact (hf_zero s hs hμs).le

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.Disintegration

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := L.ker
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I✝ : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I✝
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
I : map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑M.symm) μ))
μS : Measure ↥S := addHaar
μT : Measure ↥T := addHaar
c₀ : ℝ≥0∞
c₀_pos : c₀ ≠ 0
c₀_fin : c₀ ≠ ∞
h₀ : map (⇑M.symm) μ = c₀ • μS.prod μT
J : map (⇑P) (μS.prod μT) = μS univ • μT
c₁ : ℝ≥0∞
c₁_pos : c₁ ≠ 0
c₁_fin : c₁ ≠ ∞
h₁ : map (⇑L') μT = c₁ • ν
⊢ (c₀ * μS univ * c₁) • ν = (c₀ * c₁ * addHaar univ) • ν

Tactic:
mul_assoc,

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := L.ker
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I✝ : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I✝
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
I : map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑M.symm) μ))
μS : Measure ↥S := addHaar
μT : Measure ↥T := addHaar
c₀ : ℝ≥0∞
c₀_pos : c₀ ≠ 0
c₀_fin : c₀ ≠ ∞
h₀ : map (⇑M.symm) μ = c₀ • μS.prod μT
J : map (⇑P) (μS.prod μT) = μS univ • μT
c₁ : ℝ≥0∞
c₁_pos : c₁ ≠ 0
c₁_fin : c₁ ≠ ∞
h₁ : map (⇑L') μT = c₁ • ν
⊢ (c₀ * (μS univ * c₁)) • ν = (c₀ * c₁ * addHaar univ) • ν
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
ι : Type u_3
s : Finset ι
a : ι → ℝ
v : ι → E
g : E → ℝ
g_smooth : ContDiff ℝ (↑⊤) g
g_comp : HasCompactSupport g
A : ∀ i ∈ s, Integrable (fun x ↦ g x • (a i • fun x ↦ lineDeriv ℝ f x (v i)) x) μ
S1 : ∫ (x : E), lineDeriv ℝ f x (∑ i ∈ s, a i • v i) * g x ∂μ = ∑ i ∈ s, a i * ∫ (x : E), lineDeriv ℝ f x (v i) * g x ∂μ
⊢ ∫ (x : E), g x * lineDeriv ℝ f x (∑ i ∈ s, a i • v i) ∂μ =
    ∑ x ∈ s, ∫ (a_1 : E), a x * g a_1 * lineDeriv ℝ f a_1 (v x) ∂μ

Tactic:
mul_assoc,

Goal state after:
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
ι : Type u_3
s : Finset ι
a : ι → ℝ
v : ι → E
g : E → ℝ
g_smooth : ContDiff ℝ (↑⊤) g
g_comp : HasCompactSupport g
A : ∀ i ∈ s, Integrable (fun x ↦ g x • (a i • fun x ↦ lineDeriv ℝ f x (v i)) x) μ
S1 : ∫ (x : E), lineDeriv ℝ f x (∑ i ∈ s, a i • v i) * g x ∂μ = ∑ i ∈ s, a i * ∫ (x : E), lineDeriv ℝ f x (v i) * g x ∂μ
⊢ ∫ (x : E), g x * lineDeriv ℝ f x (∑ i ∈ s, a i • v i) ∂μ =
    ∑ x ∈ s, ∫ (a_1 : E), a x * (g a_1 * lineDeriv ℝ f a_1 (v x)) ∂μ
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
ι : Type u_3
s : Finset ι
a : ι → ℝ
v : ι → E
g : E → ℝ
g_smooth : ContDiff ℝ (↑⊤) g
g_comp : HasCompactSupport g
A : ∀ i ∈ s, Integrable (fun x ↦ g x • (a i • fun x ↦ lineDeriv ℝ f x (v i)) x) μ
B : ∀ i ∈ s, Integrable (fun x ↦ a i * ((fderiv ℝ g x) (v i) * f x)) μ
⊢ ∫ (x : E), ∑ i ∈ s, a i * (fderiv ℝ g x) (v i) * f x ∂μ = ∑ i ∈ s, a i * ∫ (x : E), (fderiv ℝ g x) (v i) * f x ∂μ

Tactic:
mul_assoc,

Goal state after:
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
ι : Type u_3
s : Finset ι
a : ι → ℝ
v : ι → E
g : E → ℝ
g_smooth : ContDiff ℝ (↑⊤) g
g_comp : HasCompactSupport g
A : ∀ i ∈ s, Integrable (fun x ↦ g x • (a i • fun x ↦ lineDeriv ℝ f x (v i)) x) μ
B : ∀ i ∈ s, Integrable (fun x ↦ a i * ((fderiv ℝ g x) (v i) * f x)) μ
⊢ ∫ (x : E), ∑ x_1 ∈ s, a x_1 * ((fderiv ℝ g x) (v x_1) * f x) ∂μ =
    ∑ i ∈ s, a i * ∫ (x : E), (fderiv ℝ g x) (v i) * f x ∂μ
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
ι : Type u_3
s : Finset ι
a : ι → ℝ
v : ι → E
g : E → ℝ
g_smooth : ContDiff ℝ (↑⊤) g
g_comp : HasCompactSupport g
A : ∀ i ∈ s, Integrable (fun x ↦ g x • (a i • fun x ↦ lineDeriv ℝ f x (v i)) x) μ
i : ι
⊢ i ∈ s → Integrable (fun x ↦ a i * ((fderiv ℝ g x) (v i) * f x)) μ

Tactic:
_hi

Goal state after:
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
ι : Type u_3
s : Finset ι
a : ι → ℝ
v : ι → E
g : E → ℝ
g_smooth : ContDiff ℝ (↑⊤) g
g_comp : HasCompactSupport g
A : ∀ i ∈ s, Integrable (fun x ↦ g x • (a i • fun x ↦ lineDeriv ℝ f x (v i)) x) μ
i : ι
_hi : i ∈ s
⊢ Integrable (fun x ↦ a i * ((fderiv ℝ g x) (v i) * f x)) μ
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Slope

Goal state before:
pp: case inr
f : ℝ → ℝ
a b c : ℝ
hf : MonotoneOn f (uIcc a (b + c))
hab : a ≤ b
hc✝ : 0 ≤ c
hc : 0 < c
hf' : IntervalIntegrable f volume a (b + c)
⊢ c⁻¹ * ((∫ (x : ℝ) in a + c..b + c, f x) - ∫ (x : ℝ) in a..b, f x) ≤ f (b + c) - f a

Tactic:
intervalIntegral.integral_interval_sub_interval_comm'
        (hf'.mono_set (by grind [uIcc]))
        (hf'.mono_set (by grind [uIcc]))
        (hf'.mono_set (by grind [uIcc]))

Goal state after:
case inr
f : ℝ → ℝ
a b c : ℝ
hf : MonotoneOn f (uIcc a (b + c))
hab : a ≤ b
hc✝ : 0 ≤ c
hc : 0 < c
hf' : IntervalIntegrable f volume a (b + c)
⊢ c⁻¹ * ((∫ (x : ℝ) in b..b + c, f x) - ∫ (x : ℝ) in a..a + c, f x) ≤ f (b + c) - f a
Module: Mathlib.Analysis.Calculus.Taylor

Goal state before:
pp: t x : ℝ
n : ℕ
⊢ HasDerivAt (fun y ↦ (-y + x) ^ (n + 1)) ((↑n + 1) * -1 * (-t + x) ^ n) t

Tactic:
mul_assoc,

Goal state after:
t x : ℝ
n : ℕ
⊢ HasDerivAt (fun y ↦ (-y + x) ^ (n + 1)) ((↑n + 1) * (-1 * (-t + x) ^ n)) t
Module: Mathlib.Analysis.Calculus.Taylor

Goal state before:
pp: case succ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x₀ : ℝ
s : Set ℝ
hs : Convex ℝ s
hx₀s : x₀ ∈ s
n : ℕ
h : ∀ {f : ℝ → E}, ContDiffOn ℝ (↑n) f s → (fun x ↦ f x - taylorWithinEval f n s x₀ x) =o[𝓝[s] x₀] fun x ↦ (x - x₀) ^ n
f : ℝ → E
hf : ContDiffOn ℝ (↑(n + 1)) f s
⊢ (fun x ↦ f x - taylorWithinEval f (n + 1) s x₀ x) =o[𝓝[s] x₀] fun x ↦ (x - x₀) ^ (n + 1)

Tactic:
rcases s.eq_singleton_or_nontrivial hx₀s with rfl | hs'

Goal state after:
case succ.inl
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x₀ : ℝ
n : ℕ
f : ℝ → E
hs : Convex ℝ {x₀}
hx₀s : x₀ ∈ {x₀}
h :
  ∀ {f : ℝ → E},
    ContDiffOn ℝ (↑n) f {x₀} → (fun x ↦ f x - taylorWithinEval f n {x₀} x₀ x) =o[𝓝[{x₀}] x₀] fun x ↦ (x - x₀) ^ n
hf : ContDiffOn ℝ (↑(n + 1)) f {x₀}
⊢ (fun x ↦ f x - taylorWithinEval f (n + 1) {x₀} x₀ x) =o[𝓝[{x₀}] x₀] fun x ↦ (x - x₀) ^ (n + 1)

case succ.inr
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x₀ : ℝ
s : Set ℝ
hs : Convex ℝ s
hx₀s : x₀ ∈ s
n : ℕ
h : ∀ {f : ℝ → E}, ContDiffOn ℝ (↑n) f s → (fun x ↦ f x - taylorWithinEval f n s x₀ x) =o[𝓝[s] x₀] fun x ↦ (x - x₀) ^ n
f : ℝ → E
hf : ContDiffOn ℝ (↑(n + 1)) f s
hs' : s.Nontrivial
⊢ (fun x ↦ f x - taylorWithinEval f (n + 1) s x₀ x) =o[𝓝[s] x₀] fun x ↦ (x - x₀) ^ (n + 1)
Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: 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'
v_sum : eVariationOn f (Icc a b) = ∑ i ∈ Finset.range (n + 1), eVariationOn f (Icc (a + ↑i * δ') (a + (↑i + 1) * δ'))
x y : ℝ
x✝⁴ : a ≤ x
x✝³ : x ≤ y
x✝² : y < x + δ
x✝¹ : y ≤ b
p : ℕ × { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ Icc x y }
hp₁ : Monotone ↑p.2
hp₂ : ∀ (i : ℕ), ↑p.2 i ∈ Icc x y
x✝ : ℕ
⊢ uIoc ((p.1, fun i ↦ (↑p.2 i, ↑p.2 (i + 1))).2 x✝).1 ((p.1, fun i ↦ (↑p.2 i, ↑p.2 (i + 1))).2 x✝).2 =
    Ioc (↑p.2 x✝) (↑p.2 (Order.succ x✝))

Tactic:
uIoc_of_le (hp₁ (by lia)),

Goal state after:
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'
v_sum : eVariationOn f (Icc a b) = ∑ i ∈ Finset.range (n + 1), eVariationOn f (Icc (a + ↑i * δ') (a + (↑i + 1) * δ'))
x y : ℝ
x✝⁴ : a ≤ x
x✝³ : x ≤ y
x✝² : y < x + δ
x✝¹ : y ≤ b
p : ℕ × { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ Icc x y }
hp₁ : Monotone ↑p.2
hp₂ : ∀ (i : ℕ), ↑p.2 i ∈ Icc x y
x✝ : ℕ
⊢ Ioc (↑p.2 x✝) (↑p.2 (x✝ + 1)) = Ioc (↑p.2 x✝) (↑p.2 (Order.succ x✝))
Module: Mathlib.Analysis.Calculus.Taylor

Goal state before:
pp: case inr.hg
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hf₁ : ContDiffOn ℝ (↑n) f [[x₀, x]]
hf₂ : AbsolutelyContinuousOnInterval (iteratedDerivWithin n f [[x₀, x]]) x₀ x
this : x₀ ≠ x
k : ℕ
hk : k ≤ n
⊢ AbsolutelyContinuousOnInterval (fun t ↦ iteratedDerivWithin k f [[x₀, x]] t) x₀ x

Tactic:
rcases hk.eq_or_lt with rfl | hk

Goal state after:
case inr.hg.inl
f : ℝ → ℝ
x x₀ : ℝ
this : x₀ ≠ x
k : ℕ
hf₁ : ContDiffOn ℝ (↑k) f [[x₀, x]]
hf₂ : AbsolutelyContinuousOnInterval (iteratedDerivWithin k f [[x₀, x]]) x₀ x
hk : k ≤ k
⊢ AbsolutelyContinuousOnInterval (fun t ↦ iteratedDerivWithin k f [[x₀, x]] t) x₀ x

case inr.hg.inr
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hf₁ : ContDiffOn ℝ (↑n) f [[x₀, x]]
hf₂ : AbsolutelyContinuousOnInterval (iteratedDerivWithin n f [[x₀, x]]) x₀ x
this : x₀ ≠ x
k : ℕ
hk✝ : k ≤ n
hk : k < n
⊢ AbsolutelyContinuousOnInterval (fun t ↦ iteratedDerivWithin k f [[x₀, x]] t) x₀ x
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)
z : ℂ
hz : ‖z‖ < 1
s : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i
k :
  Tendsto (fun x ↦ (1 - z) * ∑ x_1 ∈ range x, (∑ i ∈ Ico (x_1 + 1) x, f i) * z ^ x_1) atTop
    (𝓝 (l - ∑' (i : ℕ), f i * z ^ i))
| Tendsto (fun x ↦ (1 - z) * ∑ x_1 ∈ range x, (∑ i ∈ Ico (x_1 + 1) x, f i) * z ^ x_1) atTop
    (𝓝 (l - ∑' (i : ℕ), f i * z ^ i))

Tactic:
enter [1, n]
    rw [sum_congr (g := fun j ↦ (∑ k ∈ range n, f k - ∑ k ∈ range (j + 1), f k) * z ^ j)
      rfl (fun j hj ↦ by congr 1; exact sum_Ico_eq_sub _ (mem_range.mp hj))]

Goal state after:
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)
z : ℂ
hz : ‖z‖ < 1
s : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i
k :
  Tendsto (fun x ↦ (1 - z) * ∑ x_1 ∈ range x, (∑ i ∈ Ico (x_1 + 1) x, f i) * z ^ x_1) atTop
    (𝓝 (l - ∑' (i : ℕ), f i * z ^ i))
n : ℕ
| (1 - z) * ∑ j ∈ range n, (∑ k ∈ range n, f k - ∑ k ∈ range (j + 1), f k) * z ^ j
Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: case inr.x
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 := ⋯
hδ₃ : δ' < δ
h_mono : Monotone fun i ↦ a + ↑i * δ'
v_sum : eVariationOn f (Icc a b) = ∑ i ∈ Finset.range (n + 1), eVariationOn f (Icc (a + ↑i * δ') (a + (↑i + 1) * δ'))
v_each : ∀ (x y : ℝ), a ≤ x → x ≤ y → y < x + δ → y ≤ b → eVariationOn f (Icc x y) ≤ 1
i : ℕ
hi : i < n + 1
⊢ a + (↑i + 1) * δ' < a + ↑i * δ' + δ

pp: case inr.x
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 := ⋯
hδ₃ : δ' < δ
h_mono : Monotone fun i ↦ a + ↑i * δ'
v_sum : eVariationOn f (Icc a b) = ∑ i ∈ Finset.range (n + 1), eVariationOn f (Icc (a + ↑i * δ') (a + (↑i + 1) * δ'))
v_each : ∀ (x y : ℝ), a ≤ x → x ≤ y → y < x + δ → y ≤ b → eVariationOn f (Icc x y) ≤ 1
i : ℕ
hi : i < n + 1
⊢ a + (↑i + 1) * δ' ≤ b

Tactic:
· rw [add_mul, ← add_assoc]; simpa

Goal state after:
case inr.x
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 := ⋯
hδ₃ : δ' < δ
h_mono : Monotone fun i ↦ a + ↑i * δ'
v_sum : eVariationOn f (Icc a b) = ∑ i ∈ Finset.range (n + 1), eVariationOn f (Icc (a + ↑i * δ') (a + (↑i + 1) * δ'))
v_each : ∀ (x y : ℝ), a ≤ x → x ≤ y → y < x + δ → y ≤ b → eVariationOn f (Icc x y) ≤ 1
i : ℕ
hi : i < n + 1
⊢ a + (↑i + 1) * δ' ≤ b
Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: f : ℝ → ℝ
a b c : ℝ
h : IntervalIntegrable f volume a b
hc : c ∈ uIcc a b
s : ℕ × (ℕ → ℝ × ℝ) → Set ℝ := fun E ↦ ⋃ i ∈ Finset.range E.1, uIoc (E.2 i).1 (E.2 i).2
this✝¹ :
  Tendsto (fun i ↦ ∫⁻ (x : ℝ) in s i, ‖f x‖ₑ ∂volume.restrict (uIoc a b)) (totalLengthFilter ⊓ 𝓟 (disjWithin a b)) (𝓝 0)
this✝ :
  Tendsto (ENNReal.toReal ∘ fun i ↦ ∫⁻ (x : ℝ) in s i, ‖f x‖ₑ ∂volume.restrict (uIoc a b))
    (totalLengthFilter ⊓ 𝓟 (disjWithin a b)) (𝓝 0)
this : ∀ᶠ (E : ℕ × (ℕ → ℝ × ℝ)) in totalLengthFilter ⊓ 𝓟 (disjWithin a b), E ∈ disjWithin a b
n : ℕ
I : ℕ → ℝ × ℝ
hnI : (n, I) ∈ disjWithin a b
hnI1 : ∀ i ∈ Finset.range (n, I).1, ((n, I).2 i).1 ∈ uIcc a b ∧ ((n, I).2 i).2 ∈ uIcc a b
hnI2 : (↑(Finset.range (n, I).1)).PairwiseDisjoint fun i ↦ uIoc ((n, I).2 i).1 ((n, I).2 i).2
⊢ ∑ x ∈ Finset.range n, dist (∫ (v : ℝ) in c..(I x).1, f v) (∫ (v : ℝ) in c..(I x).2, f v) ≤
    ∑ i ∈ Finset.range (n, I).1, ∫ (x : ℝ) in uIoc ((n, I).2 i).1 ((n, I).2 i).2, ‖f x‖ ∂volume.restrict (uIoc a b)

Tactic:
refine Finset.sum_le_sum (fun i hi ↦ ?_)
    rw [Real.dist_eq,
        intervalIntegral.integral_interval_sub_left
          (by apply IntervalIntegrable.mono_set' h; grind [uIoc, uIcc])
          (by apply IntervalIntegrable.mono_set' h; grind [uIoc, uIcc]),
        Measure.restrict_restrict_of_subset
          (uIoc_subset_of_mem_disjWithin hnI (Finset.mem_range.mp hi)),
        intervalIntegral.integral_symm, abs_neg,
        intervalIntegral.abs_intervalIntegral_eq]
    exact abs_integral_le_integral_abs

Goal state after:
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‖
hr : 0 < r
this : ‖∮ (ζ : ℂ) in C(z, r), (ζ - z)⁻¹ • f ζ‖ < 2 * π * r * (‖f z‖ / r)
⊢ ‖∮ (ζ : ℂ) in C(z, r), (ζ - z)⁻¹ • f ζ‖ < 2 * π * ‖f z‖

Tactic:
mul_assoc,

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
this : ‖∮ (ζ : ℂ) in C(z, r), (ζ - z)⁻¹ • f ζ‖ < 2 * π * (r * (‖f z‖ / r))
⊢ ‖∮ (ζ : ℂ) in C(z, r), (ζ - z)⁻¹ • f ζ‖ < 2 * π * ‖f z‖
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: case neg
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
h : ∃ w ∈ Ioo x y, deriv f w = 0
⊢ ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a

Tactic:
· rcases h with ⟨w, ⟨hxw, hwy⟩, hw⟩
    obtain ⟨a, ⟨hxa, haw⟩, ha⟩ : ∃ a ∈ Ioo x w, (f w - f x) / (w - x) < deriv f a := by
      apply StrictMonoOn.exists_slope_lt_deriv_aux _ hxw _ _
      · exact hf.mono (Icc_subset_Icc le_rfl hwy.le)
      · exact hf'_mono.mono (Ioo_subset_Ioo le_rfl hwy.le)
      · intro z hz
        rw [← hw]
        apply ne_of_lt
        exact hf'_mono ⟨hz.1, hz.2.trans hwy⟩ ⟨hxw, hwy⟩ hz.2
    obtain ⟨b, ⟨hwb, hby⟩, hb⟩ : ∃ b ∈ Ioo w y, (f y - f w) / (y - w) < deriv f b := by
      apply StrictMonoOn.exists_slope_lt_deriv_aux _ hwy _ _
      · refine hf.mono (Icc_subset_Icc hxw.le le_rfl)
      · exact hf'_mono.mono (Ioo_subset_Ioo hxw.le le_rfl)
      · intro z hz
        rw [← hw]
        apply ne_of_gt
        exact hf'_mono ⟨hxw, hwy⟩ ⟨hxw.trans hz.1, hz.2⟩ hz.1
    refine ⟨b, ⟨hxw.trans hwb, hby⟩, ?_⟩
    simp only [div_lt_iff₀, hxy, hxw, hwy, sub_pos] at ha hb ⊢
    have : deriv f a * (w - x) < deriv f b * (w - x) := by
      apply mul_lt_mul _ le_rfl (sub_pos.2 hxw) _
      · exact hf'_mono ⟨hxa, haw.trans hwy⟩ ⟨hxw.trans hwb, hby⟩ (haw.trans hwb)
      · rw [← hw]
        exact (hf'_mono ⟨hxw, hwy⟩ ⟨hxw.trans hwb, hby⟩ hwb).le
    linarith

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

Goal state before:
pp: x : ℝ
h : 0 < x
⊢ sin x < x

Tactic:
by
  rcases lt_or_ge 1 x with h' | h'
  · exact (sin_le_one x).trans_lt h'
  have hx : |x| = x := abs_of_nonneg h.le
  have := le_of_abs_le (sin_bound <| show |x| ≤ 1 by rwa [hx])
  rw [sub_le_iff_le_add', hx] at this
  apply this.trans_lt
  rw [sub_add, sub_lt_self_iff, sub_pos, div_eq_mul_inv (x ^ 3)]
  refine mul_lt_mul' ?_ (by norm_num) (by norm_num) (pow_pos h 3)
  apply pow_le_pow_of_le_one h.le h'
  simp

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

Goal state before:
pp: x : ℝ
k : ℤ
h : arctan x = (2 * ↑k + 1) * π / 2
lb : -1 < 2 * ↑k + 1
ub : 2 * ↑k + 1 < 1
⊢ False

Tactic:
norm_cast at lb ub

Goal state after:
x : ℝ
k : ℤ
h : arctan x = (2 * ↑k + 1) * π / 2
lb : Int.negSucc 0 < 2 * k + 1
ub : 2 * k + 1 < 1
⊢ False
Module: Mathlib.Analysis.Complex.Angle

Goal state before:
pp: x✝ y x : ℂ
hx : ∃ θ ∈ Ioc (-π) π, cexp (↑θ * I) = x
hy : ‖1‖ = 1
⊢ ‖x - 1‖ ∈ Icc (2 / π * angle x 1) (angle x 1)

Tactic:
obtain ⟨θ, hθ, rfl⟩ := hx

Goal state after:
x y : ℂ
hy : ‖1‖ = 1
θ : ℝ
hθ : θ ∈ Ioc (-π) π
⊢ ‖cexp (↑θ * I) - 1‖ ∈ Icc (2 / π * angle (cexp (↑θ * I)) 1) (angle (cexp (↑θ * I)) 1)
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
r : ℝ
hr : 0 < r
⊢ arccos (r * ⟪x, y⟫ / (r * ‖x‖ * ‖y‖)) = arccos (⟪x, y⟫ / (‖x‖ * ‖y‖))

Tactic:
mul_assoc,

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
r : ℝ
hr : 0 < r
⊢ arccos (r * ⟪x, y⟫ / (r * (‖x‖ * ‖y‖))) = arccos (⟪x, y⟫ / (‖x‖ * ‖y‖))
Module: Mathlib.Analysis.Complex.Angle

Goal state before:
pp: x y : ℂ
hy : ‖1‖ = 1
θ : ℝ
hθ : θ ∈ Ioc (-π) π
⊢ 2 * (1 - Real.cos θ) = (Real.cos θ - 1) ^ 2 + Real.sin θ ^ 2

Tactic:
by linear_combination -θ.cos_sq_add_sin_sq

Goal state after: