mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f g : ↥(simpleFunc E p μ)
⊢ ⇑(toSimpleFunc (f - g)) =ᵐ[μ] ⇑(toSimpleFunc f) - ⇑(toSimpleFunc g)

Tactic:
filter_upwards [toSimpleFunc_eq_toFun (f - g), toSimpleFunc_eq_toFun f,
    toSimpleFunc_eq_toFun g, Lp.coeFn_sub (f : Lp E p μ) g] with _

Goal state after:
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f g : ↥(simpleFunc E p μ)
a✝ : α
⊢ (toSimpleFunc (f - g)) a✝ = ↑↑↑(f - g) a✝ →
    (toSimpleFunc f) a✝ = ↑↑↑f a✝ →
      (toSimpleFunc g) a✝ = ↑↑↑g a✝ →
        ↑↑(↑f - ↑g) a✝ = (↑↑↑f - ↑↑↑g) a✝ → (toSimpleFunc (f - g)) a✝ = (⇑(toSimpleFunc f) - ⇑(toSimpleFunc g)) a✝
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: case neg
α : Type u_1
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝² : PartialOrder F
inst✝¹ : IsOrderedAddMonoid F
inst✝ : IsOrderedModule ℝ F
ν : Measure α
f : α →ₛ F
hf : 0 ≤ᵐ[ν] ⇑f
hμν : μ ≤ ν
hfν : Integrable (⇑f) ν
x : α
hx : ¬0 ≤ f x
⊢ μ.real (⇑f ⁻¹' {f x}) • f x ≤ ν.real (⇑f ⁻¹' {f x}) • f x

Tactic:
suffices ν (f ⁻¹' {f x}) = 0 by
      have A : μ (f ⁻¹' {f x}) = 0 := by simpa using (hμν _ |>.trans_eq this)
      simp [measureReal_def, A, this]
    rw [← nonpos_iff_eq_zero, ← ae_iff.mp hf]
    refine measure_mono fun y hy ↦ ?_
    simp_all

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝² : NormedAddCommGroup E
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ↥(α →₁ₛ[μ] E)
⊢ L1.integral ↑f = integral f

Tactic:
simp only [integral, L1.integral]
  exact setToL1_eq_setToL1SCLM (dominatedFinMeasAdditive_weightedSMul μ) f

Goal state after:
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
F : Type u_3
F' : Type u_4
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace ℝ F'
m : MeasurableSpace α
μ : Measure α
T : Set α → F →L[ℝ] F'
C : ℝ
hT_norm : ∀ (s : Set α), MeasurableSet s → ‖T s‖ ≤ C * μ.real s
f : α →ₛ F
⊢ ∑ x ∈ f.range, C * μ.real (⇑f ⁻¹' {x}) * ‖x‖ ≤ C * ∑ x ∈ f.range, μ.real (⇑f ⁻¹' {x}) * ‖x‖

Tactic:
simp_rw [mul_sum, ← mul_assoc]

Goal state after:
α : Type u_1
F : Type u_3
F' : Type u_4
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace ℝ F'
m : MeasurableSpace α
μ : Measure α
T : Set α → F →L[ℝ] F'
C : ℝ
hT_norm : ∀ (s : Set α), MeasurableSet s → ‖T s‖ ≤ C * μ.real s
f : α →ₛ F
⊢ ∑ x ∈ f.range, C * μ.real (⇑f ⁻¹' {x}) * ‖x‖ ≤ ∑ x ∈ f.range, C * μ.real (⇑f ⁻¹' {x}) * ‖x‖
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
E : Type u_2
F' : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace ℝ F'
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F'
C : ℝ
hT_norm : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ‖T s‖ ≤ C * μ.real s
f : α →ₛ E
hf : Integrable (⇑f) μ
⊢ ∑ x ∈ f.range, C * μ.real (⇑f ⁻¹' {x}) * ‖x‖ ≤ C * ∑ x ∈ f.range, μ.real (⇑f ⁻¹' {x}) * ‖x‖

Tactic:
simp_rw [mul_sum, ← mul_assoc]

Goal state after:
α : Type u_1
E : Type u_2
F' : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace ℝ F'
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F'
C : ℝ
hT_norm : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ‖T s‖ ≤ C * μ.real s
f : α →ₛ E
hf : Integrable (⇑f) μ
⊢ ∑ x ∈ f.range, C * μ.real (⇑f ⁻¹' {x}) * ‖x‖ ≤ ∑ x ∈ f.range, C * μ.real (⇑f ⁻¹' {x}) * ‖x‖
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfi : Integrable f μ
⊢ (fun a ↦ ENNReal.ofReal (f a)) =ᵐ[μ] 0 ↔ f ≤ᵐ[μ] 0

Tactic:
Filter.EventuallyEq,

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfi : Integrable f μ
⊢ (∀ᵐ (x : α) ∂μ, ENNReal.ofReal (f x) = 0 x) ↔ f ≤ᵐ[μ] 0
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfi : Integrable f μ
⊢ ¬f =ᵐ[μ] 0 ↔ ¬μ (Function.support f) = 0

Tactic:
Filter.EventuallyEq,

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfi : Integrable f μ
⊢ (¬∀ᵐ (x : α) ∂μ, f x = 0 x) ↔ ¬μ (Function.support f) = 0
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable F μ
hf_tendsto : Tendsto (fun i ↦ ∫ (a : α), f i a ∂μ) atTop (𝓝 (∫ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f i a
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
f' : ℕ → α → ℝ≥0∞ := fun n a ↦ ENNReal.ofReal (f n a - f 0 a)
F' : α → ℝ≥0∞ := fun a ↦ ENNReal.ofReal (F a - f 0 a)
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun i ↦ f i a) atTop (𝓝 (F a))

Tactic:
have hf'_int_eq : ∀ i, ∫⁻ a, f' i a ∂μ = ENNReal.ofReal (∫ a, f i a ∂μ - ∫ a, f 0 a ∂μ) := by
    intro i
    unfold f'
    rw [← ofReal_integral_eq_lintegral_ofReal, integral_sub (hf_int i) (hf_int 0)]
    · exact (hf_int i).sub (hf_int 0)
    · filter_upwards [hf_mono] with a h_mono
      simp [h_mono zero_le]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable F μ
hf_tendsto : Tendsto (fun i ↦ ∫ (a : α), f i a ∂μ) atTop (𝓝 (∫ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f i a
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
f' : ℕ → α → ℝ≥0∞ := fun n a ↦ ENNReal.ofReal (f n a - f 0 a)
F' : α → ℝ≥0∞ := fun a ↦ ENNReal.ofReal (F a - f 0 a)
hf'_int_eq : ∀ (i : ℕ), ∫⁻ (a : α), f' i a ∂μ = ENNReal.ofReal (∫ (a : α), f i a ∂μ - ∫ (a : α), f 0 a ∂μ)
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun i ↦ f i a) atTop (𝓝 (F a))
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
h_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0
h_add : FinMeasAdditive μ T
f : ↥(α →₁ₛ[μ] E)
⊢ SimpleFunc.setToSimpleFunc T (toSimpleFunc (-f)) = -SimpleFunc.setToSimpleFunc T (toSimpleFunc f)

Tactic:
have : simpleFunc.toSimpleFunc (-f) =ᵐ[μ] ⇑(-simpleFunc.toSimpleFunc f) :=
    neg_toSimpleFunc f

Goal state after:
α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
h_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0
h_add : FinMeasAdditive μ T
f : ↥(α →₁ₛ[μ] E)
this : ⇑(toSimpleFunc (-f)) =ᵐ[μ] ⇑(-toSimpleFunc f)
⊢ SimpleFunc.setToSimpleFunc T (toSimpleFunc (-f)) = -SimpleFunc.setToSimpleFunc T (toSimpleFunc f)
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F : Type w
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : CompleteSpace 𝕜
PE : Type u_1
PF : Type u_2
inst✝⁴ : MetricSpace PE
inst✝³ : NormedAddTorsor E PE
inst✝² : MetricSpace PF
inst✝¹ : NormedAddTorsor F PF
inst✝ : FiniteDimensional 𝕜 E
f : PE →ᵃ[𝕜] PF
hf : Function.Injective ⇑f
K : ℝ≥0
hK : AntilipschitzWith K ⇑f.linear
x y : PE
⊢ ‖x -ᵥ y‖ ≤ ↑K * dist (f x) (f y)

Tactic:
dist_eq_norm_vsub F,

Goal state after:
𝕜 : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F : Type w
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : CompleteSpace 𝕜
PE : Type u_1
PF : Type u_2
inst✝⁴ : MetricSpace PE
inst✝³ : NormedAddTorsor E PE
inst✝² : MetricSpace PF
inst✝¹ : NormedAddTorsor F PF
inst✝ : FiniteDimensional 𝕜 E
f : PE →ᵃ[𝕜] PF
hf : Function.Injective ⇑f
K : ℝ≥0
hK : AntilipschitzWith K ⇑f.linear
x y : PE
⊢ ‖x -ᵥ y‖ ≤ ↑K * ‖f x -ᵥ f y‖
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type w
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : CompleteSpace 𝕜
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
⊢ Continuous[Pi.topologicalSpace, topologicalSpace]
    ⇑((LinearEquiv.piRing 𝕜 E ι 𝕜).symm ≪≫ₗ LinearMap.toContinuousLinearMap)

Tactic:
refine AddMonoidHomClass.continuous_of_bound
        (LinearMap.toContinuousLinearMap.toLinearMap.comp
            (LinearEquiv.piRing 𝕜 E ι 𝕜).symm.toLinearMap)
        (Fintype.card ι : ℝ) fun g ↦ ?_

Goal state after:
𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type w
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : CompleteSpace 𝕜
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
g : ι → E
⊢ ‖(↑LinearMap.toContinuousLinearMap ∘ₗ ↑(LinearEquiv.piRing 𝕜 E ι 𝕜).symm) g‖ ≤ ↑(Fintype.card ι) * ‖g‖
Module: Mathlib.Topology.ContinuousMap.Bounded.Normed

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : SeminormedAddCommGroup β
f g : α →ᵇ β
x : α
⊢ ‖f x - g x‖ ≤ dist f g

Tactic:
rw [dist_eq_norm]

Goal state after:
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : SeminormedAddCommGroup β
f g : α →ᵇ β
x : α
⊢ ‖f x - g x‖ ≤ ‖f - g‖
Module: Mathlib.MeasureTheory.Integral.Marginal

Goal state before:
pp: case insert
δ : Type u_1
δ' : Type u_2
X : δ → Type u_3
inst✝³ : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝² : DecidableEq δ
inst✝¹ : ∀ (i : δ), SigmaFinite (μ i)
inst✝ : DecidableEq δ'
e : δ' → δ
he : Injective e
f : ((i : δ') → X (e i)) → ℝ≥0∞
hf : Measurable f
h : Measurable fun x ↦ (fun {x_1} ↦ x) ∘' e
a✝ : δ'
s✝ : Finset δ'
hi : a✝ ∉ s✝
ih :
  ∀ (x : (i : δ) → X i),
    (∫⋯∫⁻_Finset.image e s✝, f ∘ fun x ↦ (fun {x_1} ↦ x) ∘' e ∂μ) x =
      (∫⋯∫⁻_s✝, f ∂(fun {x} ↦ μ) ∘' e) ((fun {x_1} ↦ x) ∘' e)
x : (i : δ) → X i
⊢ ∫⁻ (xᵢ : X (e a✝)), (∫⋯∫⁻_s✝, f ∂(fun {x} ↦ μ) ∘' e) ((fun {x_1} ↦ update x (e a✝) xᵢ) ∘' e) ∂μ (e a✝) =
    ∫⁻ (xᵢ : X (e a✝)), (∫⋯∫⁻_s✝, f ∂(fun {x} ↦ μ) ∘' e) (update ((fun {x_1} ↦ x) ∘' e) a✝ xᵢ) ∂((fun {x} ↦ μ) ∘' e) a✝

Tactic:
← update_comp_eq_of_injective' x he

Goal state after:
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : TopologicalSpace β
hα : TopologicalSpace α
h'α : OrderTopology α
f : α → β
a : α
h : 𝓝[<] a = ⊥
⊢ leftLim f a = f a

Tactic:
simp [leftLim, h]

Goal state after:
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : TopologicalSpace β
hα : TopologicalSpace α
h'α : OrderTopology α
f : α → β
a : α
h : ¬∃ y, Tendsto f (𝓝[<] a) (𝓝 y)
⊢ leftLim f a = f a

Tactic:
simp [leftLim, h]

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

Goal state before:
pp: case inr
R : Type u_1
inst✝³ : LinearOrder R
inst✝² : TopologicalSpace R
f : StieltjesFunction R
inst✝¹ : OrderTopology R
inst✝ : CompactIccSpace R
a : R
c d : ℕ → R
s✝ s : Finset ℕ
IH :
  ∀ t ⊂ s,
    ∀ (b : R), Icc a b ⊆ ⋃ i ∈ ↑t, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ t, ofReal (↑f (d i) - ↑f (c i))
b : R
ab : a ≤ b
i : ℕ
cv : Icc a b ⊆ Iotop (c i) (d i) ∪ ⋃ x ∈ ↑(s.erase i), Iotop (c x) (d x)
is : i ∈ s
bcd : b ∈ Iotop (c i) (d i)
⊢ ofReal (↑f b - ↑f a) ≤ ∑ i ∈ insert i (s.erase i), ofReal (↑f (d i) - ↑f (c i))

Tactic:
Finset.sum_insert (Finset.notMem_erase _ _)

Goal state after:
case inr
R : Type u_1
inst✝³ : LinearOrder R
inst✝² : TopologicalSpace R
f : StieltjesFunction R
inst✝¹ : OrderTopology R
inst✝ : CompactIccSpace R
a : R
c d : ℕ → R
s✝ s : Finset ℕ
IH :
  ∀ t ⊂ s,
    ∀ (b : R), Icc a b ⊆ ⋃ i ∈ ↑t, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ t, ofReal (↑f (d i) - ↑f (c i))
b : R
ab : a ≤ b
i : ℕ
cv : Icc a b ⊆ Iotop (c i) (d i) ∪ ⋃ x ∈ ↑(s.erase i), Iotop (c x) (d x)
is : i ∈ s
bcd : b ∈ Iotop (c i) (d i)
⊢ ofReal (↑f b - ↑f a) ≤ ofReal (↑f (d i) - ↑f (c i)) + ∑ x ∈ s.erase i, ofReal (↑f (d x) - ↑f (c x))
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
t : Finset ℕ
K : ℕ → Compacts G
⊢ μ (t.sup K) ≤ ∑ i ∈ t, μ (K i)

Tactic:
refine Finset.induction_on t ?_ ?_

Goal state after:
case refine_1
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
t : Finset ℕ
K : ℕ → Compacts G
⊢ μ (∅.sup K) ≤ ∑ i ∈ ∅, μ (K i)

case refine_2
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
t : Finset ℕ
K : ℕ → Compacts G
⊢ ∀ (a : ℕ) (s : Finset ℕ), a ∉ s → μ (s.sup K) ≤ ∑ i ∈ s, μ (K i) → μ ((insert a s).sup K) ≤ ∑ i ∈ insert a s, μ (K i)
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁴ : LinearOrder R
inst✝³ : TopologicalSpace R
f : StieltjesFunction R
inst✝² : OrderTopology R
inst✝¹ : CompactIccSpace R
inst✝ : DenselyOrdered R
a b : R
hab : a < b
s : ℕ → Set R
hs : Ioc a b ⊆ ⋃ i, s i
ε : ℝ≥0
εpos : 0 < ε
h : ∑' (i : ℕ), f.length (s i) < ∞
δ : ℝ≥0 := ε / 2
δpos : 0 < ↑δ
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑δ
a' : R
ha' : ↑f a' - ↑f a < ↑δ
aa' : a < a'
this : Nonempty R
i : ℕ
p q' : R
spq : s i \ botSet ⊆ Ioc p q'
hq' : ofReal (↑f q' - ↑f p) < f.length (s i) + ↑(ε' i)
⊢ Icc a' b ∩ s i ⊆ Ioc p q'

Tactic:
rintro x ⟨hx, h'x⟩

Goal state after:
R : Type u_1
inst✝⁴ : LinearOrder R
inst✝³ : TopologicalSpace R
f : StieltjesFunction R
inst✝² : OrderTopology R
inst✝¹ : CompactIccSpace R
inst✝ : DenselyOrdered R
a b : R
hab : a < b
s : ℕ → Set R
hs : Ioc a b ⊆ ⋃ i, s i
ε : ℝ≥0
εpos : 0 < ε
h : ∑' (i : ℕ), f.length (s i) < ∞
δ : ℝ≥0 := ε / 2
δpos : 0 < ↑δ
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑δ
a' : R
ha' : ↑f a' - ↑f a < ↑δ
aa' : a < a'
this : Nonempty R
i : ℕ
p q' : R
spq : s i \ botSet ⊆ Ioc p q'
hq' : ofReal (↑f q' - ↑f p) < f.length (s i) + ↑(ε' i)
x : R
hx : x ∈ Icc a' b
h'x : x ∈ s i
⊢ x ∈ Ioc p q'
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case neg
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s t : Set X
μ : Measure X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
ht : IntegrableOn f t μ
H : ¬IntegrableOn f (s ∪ t) μ
⊢ ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ

Tactic:
rw [integral_undef H, integral_undef]; simpa [integrableOn_union, ht] using! H

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

Goal state before:
pp: case inr
R : Type u_1
inst✝⁶ : LinearOrder R
inst✝⁵ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁴ : OrderTopology R
inst✝³ : CompactIccSpace R
inst✝² : MeasurableSpace R
inst✝¹ : BorelSpace R
inst✝ : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ i, t i
ε : ℝ≥0
ε0 : 0 < ε
h : ∑' (i : ℕ), f.length (t i) < ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε
i : ℕ
hR : Nonempty R
a b : R
h₁ : t i \ botSet ⊆ Ioc a b
h₂ : ofReal (↑f b - ↑f a) < f.length (t i) + ↑(ε' i)
⊢ ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ f.outer s ≤ f.length (t i) + ofReal ↑(ε' i)

Tactic:
rw [← f.outer_Ioc] at h₂

Goal state after:
case inr
R : Type u_1
inst✝⁶ : LinearOrder R
inst✝⁵ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁴ : OrderTopology R
inst✝³ : CompactIccSpace R
inst✝² : MeasurableSpace R
inst✝¹ : BorelSpace R
inst✝ : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ i, t i
ε : ℝ≥0
ε0 : 0 < ε
h : ∑' (i : ℕ), f.length (t i) < ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε
i : ℕ
hR : Nonempty R
a b : R
h₁ : t i \ botSet ⊆ Ioc a b
h₂ : f.outer (Ioc a b) < f.length (t i) + ↑(ε' i)
⊢ ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ f.outer s ≤ f.length (t i) + ofReal ↑(ε' i)
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝⁷ : Group G
inst✝⁶ : MulAction G α
inst✝⁵ : MeasurableSpace α
s t : Set α
μ : Measure α
inst✝⁴ : MeasurableConstSMul G α
inst✝³ : SMulInvariantMeasure G α μ
inst✝² : Countable G
β : Type u_6
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
hs : IsFundamentalDomain G s μ
ht : IsFundamentalDomain G t μ
f : α → β
hf : ∀ (g : G) (x : α), f (g • x) = f x
g : G
he : MeasurableEmbedding fun x ↦ g⁻¹ • x
⊢ AEStronglyMeasurable f (μ.restrict ((fun x ↦ g⁻¹ • x) '' (g • s ∩ t))) ↔
    AEStronglyMeasurable f (μ.restrict (g • s ∩ t))

Tactic:
← ((measurePreserving_smul g⁻¹ μ).restrict_image_emb he
        _).aestronglyMeasurable_comp_iff he

Goal state after:
G : Type u_1
α : Type u_3
inst✝⁷ : Group G
inst✝⁶ : MulAction G α
inst✝⁵ : MeasurableSpace α
s t : Set α
μ : Measure α
inst✝⁴ : MeasurableConstSMul G α
inst✝³ : SMulInvariantMeasure G α μ
inst✝² : Countable G
β : Type u_6
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
hs : IsFundamentalDomain G s μ
ht : IsFundamentalDomain G t μ
f : α → β
hf : ∀ (g : G) (x : α), f (g • x) = f x
g : G
he : MeasurableEmbedding fun x ↦ g⁻¹ • x
⊢ AEStronglyMeasurable (f ∘ fun x ↦ g⁻¹ • x) (μ.restrict (g • s ∩ t)) ↔ AEStronglyMeasurable f (μ.restrict (g • s ∩ t))
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case e_a
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
hfi : Integrable f μ
h_meas : NullMeasurableSet {x | 0 ≤ f x} μ
x : X
hx : x ∈ {x | 0 ≤ f x}ᶜ
⊢ ‖f x‖ = -f x

Tactic:
rw [Real.norm_eq_abs, abs_eq_neg_self.mpr _]

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
hfi : Integrable f μ
h_meas : NullMeasurableSet {x | 0 ≤ f x} μ
x : X
hx : x ∈ {x | 0 ≤ f x}ᶜ
⊢ f x ≤ 0
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure X
Y : Type u_5
inst✝ : MeasurableSpace Y
g : X → Y
f : Y → E
s : Set Y
hs : MeasurableSet s
hf : AEStronglyMeasurable f (Measure.map g μ)
hg : AEMeasurable g μ
⊢ ∫ (y : Y) in s, f y ∂Measure.map g μ = ∫ (x : X) in g ⁻¹' s, f (g x) ∂μ

Tactic:
rw [Measure.restrict_map_of_aemeasurable hg hs,
    integral_map (hg.mono_measure Measure.restrict_le_self) (hf.mono_measure _)]

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure X
Y : Type u_5
inst✝ : MeasurableSpace Y
g : X → Y
f : Y → E
s : Set Y
hs : MeasurableSet s
hf : AEStronglyMeasurable f (Measure.map g μ)
hg : AEMeasurable g μ
⊢ Measure.map g (μ.restrict (g ⁻¹' s)) ≤ Measure.map g μ
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝³ : Countable G
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : MeasurableSpace α
μ : Measure α
s : Set α
hs : IsFundamentalDomain G s μ
⊢ μ (fundamentalFrontier G s) = 0

Tactic:
by
  simpa only [fundamentalFrontier, iUnion₂_inter, one_smul, measure_iUnion_null_iff, inter_comm s,
    Function.onFun] using! fun g (hg : g ≠ 1) => hs.aedisjoint hg

Goal state after:
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: case refine_1
G : Type u_1
α : Type u_3
inst✝⁶ : Group G
inst✝⁵ : MulAction G α
inst✝⁴ : MeasurableSpace α
ν : Measure α
inst✝³ : SMulInvariantMeasure G α ν
inst✝² : Countable G
inst✝¹ : MeasurableConstSMul G α
i : SigmaFinite ν
μ : Measure (Quotient α_mod_G)
inst✝ : QuotientMeasureEqMeasurePreimage ν μ
A : ℕ → Set α
hA_meas : ∀ (i : ℕ), MeasurableSet (A i)
hA : ∀ (i : ℕ), ν (A i) < ∞
hA' : ⋃ i, A i = univ
n : ℕ
s : Set α
fund_dom_s : IsFundamentalDomain G s ν
this : Quotient.mk α_mod_G ⁻¹' Quotient.mk α_mod_G '' A n = ⋃ g, (fun x ↦ g • x) '' A n
measπAn : MeasurableSet (Quotient.mk α_mod_G '' A n)
⊢ ν ((⋃ g, (fun x ↦ g • x) '' A n) ∩ s) < ∞

Tactic:
iUnion_inter

Goal state after:
case refine_1
G : Type u_1
α : Type u_3
inst✝⁶ : Group G
inst✝⁵ : MulAction G α
inst✝⁴ : MeasurableSpace α
ν : Measure α
inst✝³ : SMulInvariantMeasure G α ν
inst✝² : Countable G
inst✝¹ : MeasurableConstSMul G α
i : SigmaFinite ν
μ : Measure (Quotient α_mod_G)
inst✝ : QuotientMeasureEqMeasurePreimage ν μ
A : ℕ → Set α
hA_meas : ∀ (i : ℕ), MeasurableSet (A i)
hA : ∀ (i : ℕ), ν (A i) < ∞
hA' : ⋃ i, A i = univ
n : ℕ
s : Set α
fund_dom_s : IsFundamentalDomain G s ν
this : Quotient.mk α_mod_G ⁻¹' Quotient.mk α_mod_G '' A n = ⋃ g, (fun x ↦ g • x) '' A n
measπAn : MeasurableSet (Quotient.mk α_mod_G '' A n)
⊢ ν (⋃ i, (fun x ↦ i • x) '' A n ∩ s) < ∞
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
⊢ ⟪-x, y⟫ = -⟪x, y⟫

Tactic:
← neg_one_smul 𝕜 x,

Goal state after:
𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
⊢ ⟪-1 • x, y⟫ = -⟪x, y⟫
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
g : StieltjesFunction R
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : R
⊢ ↑f x = ↑g x

Tactic:
have hf := measure_Iic f hfl x

Goal state after:
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
g : StieltjesFunction R
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : R
hf : f.measure (Iic x) = ofReal (↑f x - l)
⊢ ↑f x = ↑g x
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
mX : MeasurableSpace X
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_5
inst✝³ : NormedRing 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : Module 𝕜 F
inst✝ : IsBoundedSMul 𝕜 F
p✝ : ℝ≥0∞
μ✝ μ : Measure X
p : ℝ≥0∞
hp : Fact (1 ≤ p)
s : Set X
⊢ ∀ (x : ↥(Lp F p μ)), ‖{ toFun := fun f ↦ MemLp.toLp ↑↑f ⋯, map_add' := ⋯, map_smul' := ⋯ } x‖ ≤ 1 * ‖x‖

Tactic:
intro f; rw [one_mul]; exact norm_Lp_toLp_restrict_le s f

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
⊢ ⟪-x, y⟫ = -⟪x, y⟫

Tactic:
← neg_one_smul 𝕜 x,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
⊢ ⟪-1 • x, y⟫ = -⟪x, y⟫
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
h : ∀ (v : E), ⟪v, x⟫ = ⟪v, y⟫
⊢ x = y

Tactic:
← sub_eq_zero,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
h : ∀ (v : E), ⟪v, x⟫ = ⟪v, y⟫
⊢ x - y = 0
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
h : ∀ (v : E), ⟪x, v⟫ = ⟪y, v⟫
⊢ x = y

Tactic:
← sub_eq_zero,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
h : ∀ (v : E), ⟪x, v⟫ = ⟪y, v⟫
⊢ x - y = 0