mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: R : Type u_2
G : Type u_3
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommGroup G
inst✝ : SMul R G
C : ConvexCone R G
x : G
hx : x ∈ C
left✝ : x ≠ 0
hxneg : -x ∈ C
⊢ C.Pointed

Tactic:
rw [Pointed, ← add_neg_cancel x]

Goal state after:
R : Type u_2
G : Type u_3
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommGroup G
inst✝ : SMul R G
C : ConvexCone R G
x : G
hx : x ∈ C
left✝ : x ≠ 0
hxneg : -x ∈ C
⊢ x + -x ∈ C
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: 𝕜 : Type u_1
M : Type u_4
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup M
inst✝ : Module 𝕜 M
C : ConvexCone 𝕜 M
s : Set M
hs : Convex 𝕜 s
⊢ Disjoint C (hull 𝕜 s) ↔ Disjoint (↑C) s

Tactic:
by
  rw [disjoint_comm, disjoint_hull_left_of_convex hs, disjoint_comm]

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: case smul_mem'
𝕜 : Type u_1
R : Type u_2
G : Type u_3
M : Type u_4
N : Type u_5
O : Type u_6
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup M
inst✝ : Module 𝕜 M
s : Set M
hs : Convex 𝕜 s
⊢ ∀ ⦃c : 𝕜⦄, 0 < c → ∀ ⦃x : M⦄, (∃ i, ∃ (_ : 0 < i), ∃ y ∈ s, i • y = x) → ∃ i, ∃ (_ : 0 < i), ∃ y ∈ s, i • y = c • x

Tactic:
rintro c c_pos _ ⟨c', c'_pos, x, hx, rfl⟩

Goal state after:
case smul_mem'
𝕜 : Type u_1
R : Type u_2
G : Type u_3
M : Type u_4
N : Type u_5
O : Type u_6
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup M
inst✝ : Module 𝕜 M
s : Set M
hs : Convex 𝕜 s
c : 𝕜
c_pos : 0 < c
c' : 𝕜
c'_pos : 0 < c'
x : M
hx : x ∈ s
⊢ ∃ i, ∃ (_ : 0 < i), ∃ y ∈ s, i • y = c • c' • x
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: case refine_3
x : ℝ
h : |x| < 1
n : ℕ
F : ℝ → ℝ := fun x ↦ ∑ i ∈ Finset.range n, x ^ (i + 1) / (↑i + 1) + log (1 - x)
F' : ℝ → ℝ := fun x ↦ -x ^ n / (1 - x)
A : ∀ y ∈ Set.Ioo (-1) 1, HasDerivAt F (F' y) y
B : ∀ y ∈ Set.Icc (-|x|) |x|, |F' y| ≤ |x| ^ n / (1 - |x|)
⊢ x ∈ Set.Icc (-|x|) |x|

Tactic:
simp [le_abs_self x, neg_le.mp (neg_le_abs x)]

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Pointed

Goal state before:
pp: case neg
R : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁴ : Ring R
inst✝³ : LinearOrder R
inst✝² : IsOrderedRing R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
C : PointedCone R E
r : R
x✝ : E
hx : x✝ ∈ C.support.carrier
hr : ¬0 ≤ r
⊢ r • x✝ ∈ C.support.carrier

Tactic:
have hr := le_of_lt <| neg_pos_of_neg <| lt_of_not_ge hr

Goal state after:
case neg
R : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁴ : Ring R
inst✝³ : LinearOrder R
inst✝² : IsOrderedRing R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
C : PointedCone R E
r : R
x✝ : E
hx : x✝ ∈ C.support.carrier
hr✝ : ¬0 ≤ r
hr : 0 ≤ -r
⊢ r • x✝ ∈ C.support.carrier
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: a : ℝ
h : 0 < a
h₁ : |1 / (2 * a + 1)| < 1
h₂ : 2 * a + 1 ≠ 0
h₃ : a ≠ 0
⊢ log (1 + a⁻¹) = log ((1 + 1 / (2 * a + 1)) / (1 - 1 / (2 * a + 1)))

Tactic:
congr
    simp [field]
    ring

Goal state after:
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: case refine_1
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
a : ℝ
hs₁ : Convex ℝ s
hs₀ : 0 ∈ s
hs₂ : Absorbent ℝ s
ha : 0 ≤ a
x : E
h : gauge s x ≤ a
r : ℝ
hr : a < r
hr' : 0 < r
⊢ x ∈ r • s

Tactic:
mem_smul_set_iff_inv_smul_mem₀ hr'.ne'

Goal state after:
case refine_1
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
a : ℝ
hs₁ : Convex ℝ s
hs₀ : 0 ∈ s
hs₂ : Absorbent ℝ s
ha : 0 ≤ a
x : E
h : gauge s x ≤ a
r : ℝ
hr : a < r
hr' : 0 < r
⊢ r⁻¹ • x ∈ s
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: E : Type u_2
inst✝² : AddCommGroup E
inst✝¹ : Module ℝ E
s : Set E
inst✝ : TopologicalSpace E
ha : Absorbent ℝ s
hb : Bornology.IsVonNBounded ℝ s
u : Set E
hu : u ∈ 𝓝 0
⊢ u ∈ comap (gauge s) (𝓝 0)

Tactic:
rcases (hb hu).exists_pos with ⟨r, hr₀, hr⟩
  filter_upwards [preimage_mem_comap (gt_mem_nhds (inv_pos.2 hr₀))] with x (hx : gauge s x < r⁻¹)
  rcases exists_lt_of_gauge_lt ha hx with ⟨c, hc₀, hcr, y, hy, rfl⟩
  have hrc := (lt_inv_comm₀ hr₀ hc₀).2 hcr
  rcases hr c⁻¹ (hrc.le.trans (le_abs_self _)) hy with ⟨z, hz, rfl⟩
  simpa only [smul_inv_smul₀ hc₀.ne']

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.SeparatingDual

Goal state before:
pp: R : Type u_1
V : Type u_2
inst✝⁵ : Field R
inst✝⁴ : AddCommGroup V
inst✝³ : TopologicalSpace R
inst✝² : TopologicalSpace V
inst✝¹ : IsTopologicalRing R
inst✝ : Module R V
⊢ SeparatingDual R V ↔ Function.Injective ⇑(ContinuousLinearMap.coeLM R).flip

Tactic:
separatingDual_def,

Goal state after:
R : Type u_1
V : Type u_2
inst✝⁵ : Field R
inst✝⁴ : AddCommGroup V
inst✝³ : TopologicalSpace R
inst✝² : TopologicalSpace V
inst✝¹ : IsTopologicalRing R
inst✝ : Module R V
⊢ (∀ (x : V), x ≠ 0 → ∃ f, f x ≠ 0) ↔ Function.Injective ⇑(ContinuousLinearMap.coeLM R).flip
Module: Mathlib.Analysis.Normed.Operator.CompleteCodomain

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : SeparatingDual 𝕜 E
inst✝¹ : Nontrivial E
inst✝ : CompleteSpace (E →L[𝕜] F)
f : ℕ → F
hf : CauchySeq f
⊢ ∃ a, Tendsto f atTop (𝓝 a)

Tactic:
obtain ⟨v, hv⟩ : ∃ (v : E), v ≠ 0 := exists_ne 0

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : SeparatingDual 𝕜 E
inst✝¹ : Nontrivial E
inst✝ : CompleteSpace (E →L[𝕜] F)
f : ℕ → F
hf : CauchySeq f
v : E
hv : v ≠ 0
⊢ ∃ a, Tendsto f atTop (𝓝 a)
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: case refine_2
α : Type u_1
E : Type u_2
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : ℕ → Set α
f : α → E
hsm : ∀ (i : ℕ), MeasurableSet (s i)
h_anti : Antitone s
hfi : IntegrableOn f (s 0) μ
bound : α → ℝ := (s 0).indicator fun a ↦ ‖f a‖
h_int_eq : (fun i ↦ ∫ (a : α) in s i, f a ∂μ) = fun i ↦ ∫ (a : α), (s i).indicator f a ∂μ
⊢ IntegrableOn (fun a ↦ ‖f a‖) (s 0) μ

Tactic:
exact hfi.norm

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

Goal state before:
pp: case e_μ
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
hf : AEMeasurable f μ
g : X → E
f' : X → ℝ≥0 := ⋯
⊢ (fun x ↦ ↑(f x)) =ᵐ[μ] fun x ↦ ↑(f' x)

Tactic:
filter_upwards [hf.ae_eq_mk] with x hx

Goal state after:
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
hf : AEMeasurable f μ
g : X → E
f' : X → ℝ≥0 := AEMeasurable.mk f hf
x : X
hx : f x = AEMeasurable.mk f hf x
⊢ ↑(f x) = ↑(f' x)
Module: Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

Goal state before:
pp: X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
hf : AEMeasurable f μ
g : X → E
f' : X → ℝ≥0 := ⋯
⊢ (fun a ↦ f' a • g a) =ᵐ[μ] fun a ↦ f a • g a

Tactic:
filter_upwards [hf.ae_eq_mk] with x hx

Goal state after:
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
hf : AEMeasurable f μ
g : X → E
f' : X → ℝ≥0 := AEMeasurable.mk f hf
x : X
hx : f x = AEMeasurable.mk f hf x
⊢ f' x • g x = f x • g x
Module: Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

Goal state before:
pp: X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0∞
f_meas : AEMeasurable f μ
hf_lt_top : ∀ᵐ (x : X) ∂μ, f x < ∞
g : X → E
⊢ ∫ (x : X), g x ∂μ.withDensity f = ∫ (x : X), g x ∂μ.withDensity fun x ↦ ↑(f x).toNNReal

Tactic:
withDensity_congr_ae (coe_toNNReal_ae_eq hf_lt_top)

Goal state after:
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0∞
f_meas : AEMeasurable f μ
hf_lt_top : ∀ᵐ (x : X) ∂μ, f x < ∞
g : X → E
⊢ ∫ (x : X), g x ∂μ.withDensity f = ∫ (x : X), g x ∂μ.withDensity f
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b₀ b₁ b₂ : ℝ
μ : Measure ℝ
f : ℝ → E
hb₀ : μ {b₀} = 0
h_int : IntervalIntegrable f μ (min a b₁) (max a b₂)
h₀ : b₀ ∈ Icc b₁ b₂
h₁₂ : b₁ ≤ b₂
min₁₂ : min b₁ b₂ = b₁
h_int' : ∀ {x : ℝ}, x ∈ Icc b₁ b₂ → IntervalIntegrable f μ b₁ x
this : ∀ b ∈ Icc b₁ b₂, ∫ (x : ℝ) in a..b, f x ∂μ = ∫ (x : ℝ) in a..b₁, f x ∂μ + ∫ (x : ℝ) in b₁..b, f x ∂μ
⊢ ContinuousWithinAt (fun y ↦ ∫ (x : ℝ) in a..b₁, f x ∂μ + ∫ (x : ℝ) in b₁..y, f x ∂μ) (Icc b₁ b₂) b₀

Tactic:
clear this

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b₀ b₁ b₂ : ℝ
μ : Measure ℝ
f : ℝ → E
hb₀ : μ {b₀} = 0
h_int : IntervalIntegrable f μ (min a b₁) (max a b₂)
h₀ : b₀ ∈ Icc b₁ b₂
h₁₂ : b₁ ≤ b₂
min₁₂ : min b₁ b₂ = b₁
h_int' : ∀ {x : ℝ}, x ∈ Icc b₁ b₂ → IntervalIntegrable f μ b₁ x
⊢ ContinuousWithinAt (fun y ↦ ∫ (x : ℝ) in a..b₁, f x ∂μ + ∫ (x : ℝ) in b₁..y, f x ∂μ) (Icc b₁ b₂) b₀
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: case refine_1
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b₀ b₁ b₂ : ℝ
μ : Measure ℝ
f : ℝ → E
hb₀ : μ {b₀} = 0
h_int : IntervalIntegrable f μ (min a b₁) (max a b₂)
h₀ : b₀ ∈ Icc b₁ b₂
h₁₂ : b₁ ≤ b₂
min₁₂ : min b₁ b₂ = b₁
h_int' : ∀ {x : ℝ}, x ∈ Icc b₁ b₂ → IntervalIntegrable f μ b₁ x
this✝ :
  (fun b ↦ ∫ (x : ℝ) in b₁..b, f x ∂μ) =ᶠ[𝓝[Icc b₁ b₂] b₀] fun b ↦ ∫ (x : ℝ) in b₁..b₂, {x | x ≤ b}.indicator f x ∂μ
this : IntervalIntegrable (fun x ↦ ‖f x‖) μ b₁ b₂
⊢ ∀ᶠ (x : ℝ) in 𝓝[Icc b₁ b₂] b₀, AEStronglyMeasurable ({x_1 | x_1 ≤ x}.indicator f) (μ.restrict (Ι b₁ b₂))

Tactic:
clear this

Goal state after:
case refine_1
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b₀ b₁ b₂ : ℝ
μ : Measure ℝ
f : ℝ → E
hb₀ : μ {b₀} = 0
h_int : IntervalIntegrable f μ (min a b₁) (max a b₂)
h₀ : b₀ ∈ Icc b₁ b₂
h₁₂ : b₁ ≤ b₂
min₁₂ : min b₁ b₂ = b₁
h_int' : ∀ {x : ℝ}, x ∈ Icc b₁ b₂ → IntervalIntegrable f μ b₁ x
this :
  (fun b ↦ ∫ (x : ℝ) in b₁..b, f x ∂μ) =ᶠ[𝓝[Icc b₁ b₂] b₀] fun b ↦ ∫ (x : ℝ) in b₁..b₂, {x | x ≤ b}.indicator f x ∂μ
⊢ ∀ᶠ (x : ℝ) in 𝓝[Icc b₁ b₂] b₀, AEStronglyMeasurable ({x_1 | x_1 ≤ x}.indicator f) (μ.restrict (Ι b₁ b₂))
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: case refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b₀ b₁ b₂ : ℝ
μ : Measure ℝ
f : ℝ → E
hb₀ : μ {b₀} = 0
h_int : IntervalIntegrable f μ (min a b₁) (max a b₂)
h₀ : b₀ ∈ Icc b₁ b₂
h₁₂ : b₁ ≤ b₂
min₁₂ : min b₁ b₂ = b₁
h_int' : ∀ {x : ℝ}, x ∈ Icc b₁ b₂ → IntervalIntegrable f μ b₁ x
this✝ :
  (fun b ↦ ∫ (x : ℝ) in b₁..b, f x ∂μ) =ᶠ[𝓝[Icc b₁ b₂] b₀] fun b ↦ ∫ (x : ℝ) in b₁..b₂, {x | x ≤ b}.indicator f x ∂μ
this : IntervalIntegrable (fun x ↦ ‖f x‖) μ b₁ b₂
⊢ ∀ᶠ (x : ℝ) in 𝓝[Icc b₁ b₂] b₀, ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι b₁ b₂ → ‖{x_1 | x_1 ≤ x}.indicator f t‖ ≤ ‖f t‖

Tactic:
clear this

Goal state after:
case refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b₀ b₁ b₂ : ℝ
μ : Measure ℝ
f : ℝ → E
hb₀ : μ {b₀} = 0
h_int : IntervalIntegrable f μ (min a b₁) (max a b₂)
h₀ : b₀ ∈ Icc b₁ b₂
h₁₂ : b₁ ≤ b₂
min₁₂ : min b₁ b₂ = b₁
h_int' : ∀ {x : ℝ}, x ∈ Icc b₁ b₂ → IntervalIntegrable f μ b₁ x
this :
  (fun b ↦ ∫ (x : ℝ) in b₁..b, f x ∂μ) =ᶠ[𝓝[Icc b₁ b₂] b₀] fun b ↦ ∫ (x : ℝ) in b₁..b₂, {x | x ≤ b}.indicator f x ∂μ
⊢ ∀ᶠ (x : ℝ) in 𝓝[Icc b₁ b₂] b₀, ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι b₁ b₂ → ‖{x_1 | x_1 ≤ x}.indicator f t‖ ≤ ‖f t‖
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: case refine_3
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b₀ b₁ b₂ : ℝ
μ : Measure ℝ
f : ℝ → E
hb₀ : μ {b₀} = 0
h_int : IntervalIntegrable f μ (min a b₁) (max a b₂)
h₀ : b₀ ∈ Icc b₁ b₂
h₁₂ : b₁ ≤ b₂
min₁₂ : min b₁ b₂ = b₁
h_int' : ∀ {x : ℝ}, x ∈ Icc b₁ b₂ → IntervalIntegrable f μ b₁ x
this✝ :
  (fun b ↦ ∫ (x : ℝ) in b₁..b, f x ∂μ) =ᶠ[𝓝[Icc b₁ b₂] b₀] fun b ↦ ∫ (x : ℝ) in b₁..b₂, {x | x ≤ b}.indicator f x ∂μ
this : IntervalIntegrable (fun x ↦ ‖f x‖) μ b₁ b₂
⊢ ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι b₁ b₂ → ContinuousWithinAt (fun x ↦ {x_1 | x_1 ≤ x}.indicator f t) (Icc b₁ b₂) b₀

Tactic:
clear this

Goal state after:
case refine_3
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b₀ b₁ b₂ : ℝ
μ : Measure ℝ
f : ℝ → E
hb₀ : μ {b₀} = 0
h_int : IntervalIntegrable f μ (min a b₁) (max a b₂)
h₀ : b₀ ∈ Icc b₁ b₂
h₁₂ : b₁ ≤ b₂
min₁₂ : min b₁ b₂ = b₁
h_int' : ∀ {x : ℝ}, x ∈ Icc b₁ b₂ → IntervalIntegrable f μ b₁ x
this :
  (fun b ↦ ∫ (x : ℝ) in b₁..b, f x ∂μ) =ᶠ[𝓝[Icc b₁ b₂] b₀] fun b ↦ ∫ (x : ℝ) in b₁..b₂, {x | x ≤ b}.indicator f x ∂μ
⊢ ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι b₁ b₂ → ContinuousWithinAt (fun x ↦ {x_1 | x_1 ≤ x}.indicator f t) (Icc b₁ b₂) b₀
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: case hf
E : Type u_1
X : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : TopologicalSpace X
μ : Measure ℝ
inst✝ : FirstCountableTopology X
F : X → ℝ → E
bound : ℝ → ℝ
a b a₀ b₀ : ℝ
x₀ : X
hF_meas : ∀ (x : X), AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
h_bound : ∀ᶠ (x : X) in 𝓝 x₀, ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x t‖ ≤ bound t
bound_integrable : IntervalIntegrable bound μ a b
h_cont : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ContinuousAt (fun x ↦ F x t) x₀
ha₀ : a₀ ∈ Ioo a b
hb₀ : b₀ ∈ Ioo a b
hμb₀ : μ {b₀} = 0
hsub : ∀ {a₀ b₀ : ℝ}, a₀ ∈ Ioo a b → b₀ ∈ Ioo a b → Ι a₀ b₀ ⊆ Ι a b
Ioo_nhds : Ioo a b ∈ 𝓝 b₀
Icc_nhds : Icc a b ∈ 𝓝 b₀
hx₀ : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x₀ t‖ ≤ bound t
x : X
t : ℝ
hx : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x t‖ ≤ bound t
ht : t ∈ Ioo a b
hiF :
  ∀ {x : X} {a₀ b₀ : ℝ},
    (∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x t‖ ≤ bound t) → a₀ ∈ Ioo a b → b₀ ∈ Ioo a b → IntervalIntegrable (F x) μ a₀ b₀
⊢ IntervalIntegrable (F x) μ b₀ t

pp: case hg
E : Type u_1
X : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : TopologicalSpace X
μ : Measure ℝ
inst✝ : FirstCountableTopology X
F : X → ℝ → E
bound : ℝ → ℝ
a b a₀ b₀ : ℝ
x₀ : X
hF_meas : ∀ (x : X), AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
h_bound : ∀ᶠ (x : X) in 𝓝 x₀, ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x t‖ ≤ bound t
bound_integrable : IntervalIntegrable bound μ a b
h_cont : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ContinuousAt (fun x ↦ F x t) x₀
ha₀ : a₀ ∈ Ioo a b
hb₀ : b₀ ∈ Ioo a b
hμb₀ : μ {b₀} = 0
hsub : ∀ {a₀ b₀ : ℝ}, a₀ ∈ Ioo a b → b₀ ∈ Ioo a b → Ι a₀ b₀ ⊆ Ι a b
Ioo_nhds : Ioo a b ∈ 𝓝 b₀
Icc_nhds : Icc a b ∈ 𝓝 b₀
hx₀ : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x₀ t‖ ≤ bound t
x : X
t : ℝ
hx : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x t‖ ≤ bound t
ht : t ∈ Ioo a b
hiF :
  ∀ {x : X} {a₀ b₀ : ℝ},
    (∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x t‖ ≤ bound t) → a₀ ∈ Ioo a b → b₀ ∈ Ioo a b → IntervalIntegrable (F x) μ a₀ b₀
⊢ IntervalIntegrable (F x₀) μ b₀ t

Tactic:
· exact hiF hx hb₀ ht

Goal state after:
case hg
E : Type u_1
X : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : TopologicalSpace X
μ : Measure ℝ
inst✝ : FirstCountableTopology X
F : X → ℝ → E
bound : ℝ → ℝ
a b a₀ b₀ : ℝ
x₀ : X
hF_meas : ∀ (x : X), AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
h_bound : ∀ᶠ (x : X) in 𝓝 x₀, ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x t‖ ≤ bound t
bound_integrable : IntervalIntegrable bound μ a b
h_cont : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ContinuousAt (fun x ↦ F x t) x₀
ha₀ : a₀ ∈ Ioo a b
hb₀ : b₀ ∈ Ioo a b
hμb₀ : μ {b₀} = 0
hsub : ∀ {a₀ b₀ : ℝ}, a₀ ∈ Ioo a b → b₀ ∈ Ioo a b → Ι a₀ b₀ ⊆ Ι a b
Ioo_nhds : Ioo a b ∈ 𝓝 b₀
Icc_nhds : Icc a b ∈ 𝓝 b₀
hx₀ : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x₀ t‖ ≤ bound t
x : X
t : ℝ
hx : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x t‖ ≤ bound t
ht : t ∈ Ioo a b
hiF :
  ∀ {x : X} {a₀ b₀ : ℝ},
    (∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x t‖ ≤ bound t) → a₀ ∈ Ioo a b → b₀ ∈ Ioo a b → IntervalIntegrable (F x) μ a₀ b₀
⊢ IntervalIntegrable (F x₀) μ b₀ t
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : TopologicalSpace X
μ : Measure ℝ
inst✝ : FirstCountableTopology X
F : X → ℝ → E
bound : ℝ → ℝ
a b a₀ b₀ : ℝ
x₀ : X
hF_meas : ∀ (x : X), AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
h_bound : ∀ᶠ (x : X) in 𝓝 x₀, ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x t‖ ≤ bound t
bound_integrable : IntervalIntegrable bound μ a b
h_cont : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ContinuousAt (fun x ↦ F x t) x₀
ha₀ : a₀ ∈ Ioo a b
hb₀ : b₀ ∈ Ioo a b
hμb₀ : μ {b₀} = 0
hsub : ∀ {a₀ b₀ : ℝ}, a₀ ∈ Ioo a b → b₀ ∈ Ioo a b → Ι a₀ b₀ ⊆ Ι a b
Ioo_nhds : Ioo a b ∈ 𝓝 b₀
Icc_nhds : Icc a b ∈ 𝓝 b₀
hx₀ : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x₀ t‖ ≤ bound t
this :
  ∀ᶠ (p : X × ℝ) in 𝓝 (x₀, b₀),
    ∫ (s : ℝ) in a₀..p.2, F p.1 s ∂μ =
      ∫ (s : ℝ) in a₀..b₀, F p.1 s ∂μ + ∫ (s : ℝ) in b₀..p.2, F x₀ s ∂μ + ∫ (s : ℝ) in b₀..p.2, F p.1 s - F x₀ s ∂μ
⊢ ContinuousAt
    (fun x ↦
      ∫ (s : ℝ) in a₀..b₀, F x.1 s ∂μ + ∫ (s : ℝ) in b₀..x.2, F x₀ s ∂μ + ∫ (s : ℝ) in b₀..x.2, F x.1 s - F x₀ s ∂μ)
    (x₀, b₀)

Tactic:
clear this

Goal state after:
E : Type u_1
X : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : TopologicalSpace X
μ : Measure ℝ
inst✝ : FirstCountableTopology X
F : X → ℝ → E
bound : ℝ → ℝ
a b a₀ b₀ : ℝ
x₀ : X
hF_meas : ∀ (x : X), AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
h_bound : ∀ᶠ (x : X) in 𝓝 x₀, ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x t‖ ≤ bound t
bound_integrable : IntervalIntegrable bound μ a b
h_cont : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ContinuousAt (fun x ↦ F x t) x₀
ha₀ : a₀ ∈ Ioo a b
hb₀ : b₀ ∈ Ioo a b
hμb₀ : μ {b₀} = 0
hsub : ∀ {a₀ b₀ : ℝ}, a₀ ∈ Ioo a b → b₀ ∈ Ioo a b → Ι a₀ b₀ ⊆ Ι a b
Ioo_nhds : Ioo a b ∈ 𝓝 b₀
Icc_nhds : Icc a b ∈ 𝓝 b₀
hx₀ : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x₀ t‖ ≤ bound t
⊢ ContinuousAt
    (fun x ↦
      ∫ (s : ℝ) in a₀..b₀, F x.1 s ∂μ + ∫ (s : ℝ) in b₀..x.2, F x₀ s ∂μ + ∫ (s : ℝ) in b₀..x.2, F x.1 s - F x₀ s ∂μ)
    (x₀, b₀)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: f : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : 0 ≤ᵐ[μ.restrict (Icc a b)] f
⊢ 0 ≤ ∫ (u : ℝ) in a..b, f u ∂μ

Tactic:
let H := ae_restrict_of_ae_restrict_of_subset Ioc_subset_Icc_self hf
  simpa only [integral_of_le hab] using setIntegral_nonneg_of_ae_restrict H

Goal state after:
Module: Mathlib.Topology.Algebra.Order.Floor

Goal state before:
pp: α : Type u_1
inst✝⁴ : Ring α
inst✝³ : LinearOrder α
inst✝² : FloorRing α
inst✝¹ : TopologicalSpace α
inst✝ : OrderClosedTopology α
x : α
⊢ ↑⌊x⌋ + 1 - 1 ≤ x

Tactic:
add_sub_cancel_right

Goal state after:
α : Type u_1
inst✝⁴ : Ring α
inst✝³ : LinearOrder α
inst✝² : FloorRing α
inst✝¹ : TopologicalSpace α
inst✝ : OrderClosedTopology α
x : α
⊢ ↑⌊x⌋ ≤ x
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
⊢ AddQuotientMeasureEqMeasurePreimage volume volume

Tactic:
by
  apply MeasureTheory.leftInvariantIsAddQuotientMeasureEqMeasurePreimage
  simp [(isAddFundamentalDomain_Ioc' hT.out 0).covolume_eq_volume, AddCircle.measure_univ]

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

Goal state before:
pp: case convert_2
G : Type u_1
inst✝¹⁴ : Group G
inst✝¹³ : MeasurableSpace G
inst✝¹² : TopologicalSpace G
inst✝¹¹ : IsTopologicalGroup G
inst✝¹⁰ : BorelSpace G
inst✝⁹ : PolishSpace G
Γ : Subgroup G
inst✝⁸ : Γ.Normal
inst✝⁷ : T2Space (G ⧸ Γ)
inst✝⁶ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝⁵ : ν.IsMulLeftInvariant
inst✝⁴ : Countable ↥Γ
inst✝³ : ν.IsMulRightInvariant
inst✝² : SigmaFinite ν
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : SigmaFinite μ
s : Set G
fund_dom_s : IsFundamentalDomain (↥Γ.op) s ν
V : Set (G ⧸ Γ)
meas_V : MeasurableSet V
neZeroV : μ V ≠ 0
hV : μ V = ν (QuotientGroup.mk ⁻¹' V ∩ s)
neTopV : μ V ≠ ∞
U : Set (G ⧸ Γ)
a✝ : MeasurableSet U
meas_π : Measurable QuotientGroup.mk
μ' : Measure (G ⧸ Γ) := ⋯
has_fund : HasFundamentalDomain (↥Γ.op) G ν
i : QuotientMeasureEqMeasurePreimage ν μ'
this✝ : μ'.IsMulLeftInvariant
this : SigmaFinite μ'
⊢ ν (QuotientGroup.mk ⁻¹' V ∩ s) ≠ 0

pp: case convert_3
G : Type u_1
inst✝¹⁴ : Group G
inst✝¹³ : MeasurableSpace G
inst✝¹² : TopologicalSpace G
inst✝¹¹ : IsTopologicalGroup G
inst✝¹⁰ : BorelSpace G
inst✝⁹ : PolishSpace G
Γ : Subgroup G
inst✝⁸ : Γ.Normal
inst✝⁷ : T2Space (G ⧸ Γ)
inst✝⁶ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝⁵ : ν.IsMulLeftInvariant
inst✝⁴ : Countable ↥Γ
inst✝³ : ν.IsMulRightInvariant
inst✝² : SigmaFinite ν
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : SigmaFinite μ
s : Set G
fund_dom_s : IsFundamentalDomain (↥Γ.op) s ν
V : Set (G ⧸ Γ)
meas_V : MeasurableSet V
neZeroV : μ V ≠ 0
hV : μ V = ν (QuotientGroup.mk ⁻¹' V ∩ s)
neTopV : μ V ≠ ∞
U : Set (G ⧸ Γ)
a✝ : MeasurableSet U
meas_π : Measurable QuotientGroup.mk
μ' : Measure (G ⧸ Γ) := ⋯
has_fund : HasFundamentalDomain (↥Γ.op) G ν
i : QuotientMeasureEqMeasurePreimage ν μ'
this✝ : μ'.IsMulLeftInvariant
this : SigmaFinite μ'
⊢ ν (QuotientGroup.mk ⁻¹' V ∩ s) ≠ ∞

Tactic:
· exact trans hV.symm neZeroV

Goal state after:
case convert_3
G : Type u_1
inst✝¹⁴ : Group G
inst✝¹³ : MeasurableSpace G
inst✝¹² : TopologicalSpace G
inst✝¹¹ : IsTopologicalGroup G
inst✝¹⁰ : BorelSpace G
inst✝⁹ : PolishSpace G
Γ : Subgroup G
inst✝⁸ : Γ.Normal
inst✝⁷ : T2Space (G ⧸ Γ)
inst✝⁶ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝⁵ : ν.IsMulLeftInvariant
inst✝⁴ : Countable ↥Γ
inst✝³ : ν.IsMulRightInvariant
inst✝² : SigmaFinite ν
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : SigmaFinite μ
s : Set G
fund_dom_s : IsFundamentalDomain (↥Γ.op) s ν
V : Set (G ⧸ Γ)
meas_V : MeasurableSet V
neZeroV : μ V ≠ 0
hV : μ V = ν (QuotientGroup.mk ⁻¹' V ∩ s)
neTopV : μ V ≠ ∞
U : Set (G ⧸ Γ)
a✝ : MeasurableSet U
meas_π : Measurable QuotientGroup.mk
μ' : Measure (G ⧸ Γ) := ⋯
has_fund : HasFundamentalDomain (↥Γ.op) G ν
i : QuotientMeasureEqMeasurePreimage ν μ'
this✝ : μ'.IsMulLeftInvariant
this : SigmaFinite μ'
⊢ ν (QuotientGroup.mk ⁻¹' V ∩ s) ≠ ∞
Module: Mathlib.MeasureTheory.Measure.Haar.Quotient

Goal state before:
pp: case hV
G : Type u_1
inst✝¹⁴ : Group G
inst✝¹³ : MeasurableSpace G
inst✝¹² : TopologicalSpace G
inst✝¹¹ : IsTopologicalGroup G
inst✝¹⁰ : BorelSpace G
inst✝⁹ : PolishSpace G
Γ : Subgroup G
inst✝⁸ : Γ.Normal
inst✝⁷ : T2Space (G ⧸ Γ)
inst✝⁶ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
inst✝⁵ : Countable ↥Γ
ν : Measure G
inst✝⁴ : ν.IsHaarMeasure
inst✝³ : ν.IsMulRightInvariant
inst✝² : SigmaFinite ν
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 ν
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : SigmaFinite μ
V : Set (G ⧸ Γ)
hV : (interior V).Nonempty
meas_V : MeasurableSet V
hμK : μ V = ν (QuotientGroup.mk ⁻¹' V ∩ 𝓕)
neTopV : μ V ≠ ∞
⊢ μ V = ν (QuotientGroup.mk ⁻¹' V ∩ 𝓕)

Tactic:
exact hμK

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

Goal state before:
pp: case hμK
G : Type u_1
inst✝¹² : Group G
inst✝¹¹ : MeasurableSpace G
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : IsTopologicalGroup G
inst✝⁸ : BorelSpace G
inst✝⁷ : PolishSpace G
Γ : Subgroup G
inst✝⁶ : Γ.Normal
inst✝⁵ : T2Space (G ⧸ Γ)
inst✝⁴ : SecondCountableTopology (G ⧸ Γ)
inst✝³ : Countable ↥Γ
ν : Measure G
inst✝² : ν.IsHaarMeasure
inst✝¹ : ν.IsMulRightInvariant
inst✝ : SigmaFinite ν
K : PositiveCompacts (G ⧸ Γ)
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 ν
h𝓕_finite : ν 𝓕 ≠ ∞
c : ℝ≥0∞ := ⋯
c_ne_top : c ≠ ∞
μ : Measure (G ⧸ Γ) := ⋯
hμK : μ ↑K = c
this : SigmaFinite μ
⊢ μ ↑K = ν (QuotientGroup.mk ⁻¹' ↑K ∩ 𝓕)

Tactic:
exact hμK

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

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
t : ℝ
f : ℝ → ℂ
p : ℝ≥0∞
hLp : MemLp f p (volume.restrict (Ioc t (t + T)))
⊢ MemLp (AddCircle.liftIoc T t f) p volume

Tactic:
simp only [AddCircle.liftIoc, Set.domRestrict_def, Function.comp_def]

Goal state after:
T : ℝ
hT : Fact (0 < T)
t : ℝ
f : ℝ → ℂ
p : ℝ≥0∞
hLp : MemLp f p (volume.restrict (Ioc t (t + T)))
⊢ MemLp (fun x ↦ f ↑((AddCircle.equivIoc T t) x)) p volume