mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: case refine_2
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
ι : Type u_6
f : α → G
hfi : AEStronglyMeasurable f μ
F : ι → α → G
l : Filter ι
hFi : ∀ᶠ (i : ι) in l, Integrable (F i) μ
s : Set α
hF : Tendsto (fun i ↦ eLpNorm (fun x ↦ F i x - f x) 1 μ) l (𝓝 0)
⊢ Tendsto (fun i ↦ eLpNorm (fun x ↦ F i x - f x) 1 (μ.restrict s)) l (𝓝 0)

Tactic:
exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hF (fun _ ↦ zero_le)
      (fun _ ↦ eLpNorm_mono_measure _ Measure.restrict_le_self)

Goal state after:
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : NormedAddCommGroup β
u v w : α → β
l : Filter α
huv : u ~[l] v
hwv : w =o[l] v
⊢ u - w ~[l] v

Tactic:
simpa only [sub_eq_add_neg] using! huv.add_isLittleO hwv.neg_left

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

Goal state before:
pp: case neg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfm : AEStronglyMeasurable f μ
hfi : ¬Integrable f μ
⊢ ∫ (a : α), f a ∂μ = (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal

Tactic:
rw [integral_undef hfi]
    simp_rw [Integrable, hfm, hasFiniteIntegral_iff_norm, lt_top_iff_ne_top, Ne, true_and,
      Classical.not_not] at hfi
    have : ∫⁻ a : α, ENNReal.ofReal (f a) ∂μ = ∫⁻ a, ENNReal.ofReal ‖f a‖ ∂μ := by
      refine lintegral_congr_ae (hf.mono fun a h => ?_)
      dsimp only
      rw [Real.norm_eq_abs, abs_of_nonneg h]
    rw [this, hfi, toReal_top]

Goal state after:
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
_i : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
motive : (α → E) → Prop
indicator : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → motive (s.indicator fun x ↦ c)
add :
  ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → MemLp f p μ → MemLp g p μ → motive f → motive g → motive (f + g)
closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {f | motive ↑↑f}
ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → MemLp f p μ → motive f → motive g
this✝¹ : ∀ (f : α →ₛ E), MemLp (⇑f) p μ → motive ⇑f
this✝ : ∀ (f : ↥(Lp.simpleFunc E p μ)), motive ↑↑↑f
this : ∀ (f : ↥(Lp E p μ)), motive ↑↑f
⊢ ∀ ⦃f : α → E⦄, MemLp f p μ → motive f

Tactic:
exact fun f hf => ae hf.coeFn_toLp (Lp.memLp _) (this (hf.toLp f))

Goal state after:
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: α : Type u_1
β : Type u_3
inst✝ : NormedField β
t u : α → β
l : Filter α
h : t ~[l] u
z : ℤ
a✝ : ℕ
⊢ t ^ Int.negSucc a✝ ~[l] u ^ Int.negSucc a✝

Tactic:
simpa using (h.pow _).inv

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

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
inst✝¹ : ClosedIciTopology E
β : Type u_6
inst✝ : Preorder β
f : α → β → E
s : Set β
hf_mono : ∀ᵐ (x : α) ∂μ, MonotoneOn (f x) s
hf_int : ∀ a ∈ s, Integrable (fun x ↦ f x a) μ
a : β
ha : a ∈ s
⊢ ∀ ⦃b : β⦄, b ∈ s → a ≤ b → (fun b ↦ ∫ (x : α), f x b ∂μ) a ≤ (fun b ↦ ∫ (x : α), f x b ∂μ) b

Tactic:
b

Goal state after:
α : Type u_1
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
inst✝¹ : ClosedIciTopology E
β : Type u_6
inst✝ : Preorder β
f : α → β → E
s : Set β
hf_mono : ∀ᵐ (x : α) ∂μ, MonotoneOn (f x) s
hf_int : ∀ a ∈ s, Integrable (fun x ↦ f x a) μ
a : β
ha : a ∈ s
b : β
⊢ b ∈ s → a ≤ b → (fun b ↦ ∫ (x : α), f x b ∂μ) a ≤ (fun b ↦ ∫ (x : α), f x b ∂μ) b
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
inst✝¹ : ClosedIciTopology E
β : Type u_6
inst✝ : Preorder β
f : α → β → E
s : Set β
hf_anti : ∀ᵐ (x : α) ∂μ, AntitoneOn (f x) s
hf_int : ∀ a ∈ s, Integrable (fun x ↦ f x a) μ
a : β
ha : a ∈ s
⊢ ∀ ⦃b : β⦄, b ∈ s → a ≤ b → (fun b ↦ ∫ (x : α), f x b ∂μ) b ≤ (fun b ↦ ∫ (x : α), f x b ∂μ) a

Tactic:
b

Goal state after:
α : Type u_1
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
inst✝¹ : ClosedIciTopology E
β : Type u_6
inst✝ : Preorder β
f : α → β → E
s : Set β
hf_anti : ∀ᵐ (x : α) ∂μ, AntitoneOn (f x) s
hf_int : ∀ a ∈ s, Integrable (fun x ↦ f x a) μ
a : β
ha : a ∈ s
b : β
⊢ b ∈ s → a ≤ b → (fun b ↦ ∫ (x : α), f x b ∂μ) b ≤ (fun b ↦ ∫ (x : α), f x b ∂μ) a
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝⁵ : PartialOrder E
inst✝⁴ : IsOrderedAddMonoid E
inst✝³ : IsOrderedModule ℝ E
inst✝² : ClosedIciTopology E
β : Type u_6
inst✝¹ : AddCommMonoid β
inst✝ : Module ℝ β
f : α → β → E
s : Set β
hs : Convex ℝ s
hf_conv : ∀ᵐ (x : α) ∂μ, ConvexOn ℝ s (f x)
hf_int : ∀ a ∈ s, Integrable (fun x ↦ f x a) μ
a : β
ha : a ∈ s
⊢ ∀ ⦃y : β⦄,
    y ∈ s →
      ∀ ⦃a_2 b : ℝ⦄,
        0 ≤ a_2 →
          0 ≤ b →
            a_2 + b = 1 →
              (fun b ↦ ∫ (x : α), f x b ∂μ) (a_2 • a + b • y) ≤
                a_2 • (fun b ↦ ∫ (x : α), f x b ∂μ) a + b • (fun b ↦ ∫ (x : α), f x b ∂μ) y

Tactic:
b

Goal state after:
α : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝⁵ : PartialOrder E
inst✝⁴ : IsOrderedAddMonoid E
inst✝³ : IsOrderedModule ℝ E
inst✝² : ClosedIciTopology E
β : Type u_6
inst✝¹ : AddCommMonoid β
inst✝ : Module ℝ β
f : α → β → E
s : Set β
hs : Convex ℝ s
hf_conv : ∀ᵐ (x : α) ∂μ, ConvexOn ℝ s (f x)
hf_int : ∀ a ∈ s, Integrable (fun x ↦ f x a) μ
a : β
ha : a ∈ s
b : β
⊢ b ∈ s →
    ∀ ⦃a_2 b_1 : ℝ⦄,
      0 ≤ a_2 →
        0 ≤ b_1 →
          a_2 + b_1 = 1 →
            (fun b ↦ ∫ (x : α), f x b ∂μ) (a_2 • a + b_1 • b) ≤
              a_2 • (fun b ↦ ∫ (x : α), f x b ∂μ) a + b_1 • (fun b ↦ ∫ (x : α), f x b ∂μ) b
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)
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))

Tactic:
have hF'_int_eq : ∫⁻ a, F' a ∂μ = ENNReal.ofReal (∫ a, F a ∂μ - ∫ a, f 0 a ∂μ) := by
    unfold F'
    rw [← ofReal_integral_eq_lintegral_ofReal, integral_sub hF_int (hf_int 0)]
    · exact hF_int.sub (hf_int 0)
    · filter_upwards [hf_bound] with a h_bound
      simp [h_bound 0]

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)
hF'_int_eq : ∫⁻ (a : α), F' a ∂μ = ENNReal.ofReal (∫ (a : α), F 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: case neg
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
C : ℝ
hT : DominatedFinMeasAdditive μ T C
c : ℝ
f : α → E
hF : CompleteSpace F
hf : ¬Integrable f μ
⊢ setToFun μ (fun s ↦ c • T s) ⋯ f = c • setToFun μ T hT f

Tactic:
· simp_rw [setToFun_undef _ hf, smul_zero]

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

Goal state before:
pp: X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝⁴ : NormedAddCommGroup E
f : X → E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSingletonClass X
μ : Measure X
inst✝ : Countable X
hf : Integrable f (Measure.sum fun a ↦ μ {a} • Measure.dirac a)
⊢ ∑' (i : X), ∫ (x : X), f x ∂μ {i} • Measure.dirac i =
    ∑' (x : X), (Measure.sum fun a ↦ μ {a} • Measure.dirac a).real {x} • f x

Tactic:
congr 1 with a : 1

Goal state after:
case e_f
X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝⁴ : NormedAddCommGroup E
f : X → E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSingletonClass X
μ : Measure X
inst✝ : Countable X
hf : Integrable f (Measure.sum fun a ↦ μ {a} • Measure.dirac a)
a : X
⊢ ∫ (x : X), f x ∂μ {a} • Measure.dirac a = (Measure.sum fun a ↦ μ {a} • Measure.dirac a).real {a} • f a
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case neg
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 T' : Set α → E →L[ℝ] F
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
c : ℝ
h_smul : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T' s = c • T s
f : α → E
hF : CompleteSpace F
hf : ¬Integrable f μ
⊢ setToFun μ T' hT' f = c • setToFun μ T hT f

Tactic:
· simp_rw [setToFun_undef _ hf, smul_zero]

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

Goal state before:
pp: X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : MeasurableSingletonClass X
μ : Measure X
f : X → E
s : Set X
hs : s.Countable
hf : IntegrableOn f s μ
hi : Countable { x // x ∈ s }
hf' : Integrable (fun x ↦ f ↑x) (Measure.comap Subtype.val μ)
⊢ ∑' (x : ↑s), (Measure.comap Subtype.val μ).real {x} • f ↑x = ∑' (x : ↑s), μ.real {↑x} • f ↑x

Tactic:
congr 1 with a : 1

Goal state after:
case e_f
X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : MeasurableSingletonClass X
μ : Measure X
f : X → E
s : Set X
hs : s.Countable
hf : IntegrableOn f s μ
hi : Countable { x // x ∈ s }
hf' : Integrable (fun x ↦ f ↑x) (Measure.comap Subtype.val μ)
a : ↑s
⊢ (Measure.comap Subtype.val μ).real {a} • f ↑a = μ.real {↑a} • f ↑a
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case pos
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
C : ℝ
f g : α → E
hT : DominatedFinMeasAdditive μ T C
hf : Integrable f μ
hg : Integrable g μ
hF : CompleteSpace F
⊢ (L1.setToL1 hT) (Integrable.toL1 (f + g) ⋯) = setToFun μ T hT f + setToFun μ T hT g

Tactic:
setToFun_eq hT hf,

Goal state after:
case pos
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
C : ℝ
f g : α → E
hT : DominatedFinMeasAdditive μ T C
hf : Integrable f μ
hg : Integrable g μ
hF : CompleteSpace F
⊢ (L1.setToL1 hT) (Integrable.toL1 (f + g) ⋯) = (L1.setToL1 hT) (Integrable.toL1 f hf) + setToFun μ T hT g
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case pos
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
C : ℝ
hT : DominatedFinMeasAdditive μ T C
f : α → E
hF : CompleteSpace F
hf : Integrable f μ
⊢ setToFun μ T hT (-f) = -setToFun μ T hT f

Tactic:
setToFun_eq hT hf,

Goal state after:
case pos
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
C : ℝ
hT : DominatedFinMeasAdditive μ T C
f : α → E
hF : CompleteSpace F
hf : Integrable f μ
⊢ setToFun μ T hT (-f) = -(L1.setToL1 hT) (Integrable.toL1 f hf)
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case pos
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_6
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
inst✝⁴ : NormedDivisionRing 𝕜
inst✝³ : Module 𝕜 E
inst✝² : NormSMulClass 𝕜 E
inst✝¹ : Module 𝕜 F
inst✝ : NormSMulClass 𝕜 F
hT : DominatedFinMeasAdditive μ T C
h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c • x) = c • (T s) x
c : 𝕜
f : α → E
hF : CompleteSpace F
hf : Integrable f μ
⊢ setToFun μ T hT (c • f) = c • setToFun μ T hT f

Tactic:
setToFun_eq hT hf,

Goal state after:
case pos
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_6
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
inst✝⁴ : NormedDivisionRing 𝕜
inst✝³ : Module 𝕜 E
inst✝² : NormSMulClass 𝕜 E
inst✝¹ : Module 𝕜 F
inst✝ : NormSMulClass 𝕜 F
hT : DominatedFinMeasAdditive μ T C
h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c • x) = c • (T s) x
c : 𝕜
f : α → E
hF : CompleteSpace F
hf : Integrable f μ
⊢ setToFun μ T hT (c • f) = c • (L1.setToL1 hT) (Integrable.toL1 f hf)
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case pos
α : Type u_1
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
G'' : Type u_8
inst✝⁴ : NormedAddCommGroup G''
inst✝³ : PartialOrder G''
inst✝² : IsOrderedAddMonoid G''
inst✝¹ : NormedSpace ℝ G''
inst✝ : OrderClosedTopology G''
T T' : Set α → E →L[ℝ] G''
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
hTT' : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∀ (x : E), (T s) x ≤ (T' s) x
f : α → E
hG'' : CompleteSpace G''
hf : Integrable f μ
⊢ setToFun μ T hT f ≤ setToFun μ T' hT' f

Tactic:
setToFun_eq _ hf

Goal state after:
case pos
α : Type u_1
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
G'' : Type u_8
inst✝⁴ : NormedAddCommGroup G''
inst✝³ : PartialOrder G''
inst✝² : IsOrderedAddMonoid G''
inst✝¹ : NormedSpace ℝ G''
inst✝ : OrderClosedTopology G''
T T' : Set α → E →L[ℝ] G''
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
hTT' : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∀ (x : E), (T s) x ≤ (T' s) x
f : α → E
hG'' : CompleteSpace G''
hf : Integrable f μ
⊢ (L1.setToL1 hT) (Integrable.toL1 f hf) ≤ (L1.setToL1 hT') (Integrable.toL1 f hf)
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: case refine_3
α : Type u
β : Type v
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetricSpace β
δ : Type u_2
inst✝¹ : TopologicalSpace δ
inst✝ : DiscreteTopology δ
f : α ↪ δ
g₁ g₂ : α →ᵇ β
h₁ h₂ : δ →ᵇ β
⊢ dist (h₁.domRestrict (range ⇑f)ᶜ) (h₂.domRestrict (range ⇑f)ᶜ) ≤ dist (extend f g₁ h₁) (extend f g₂ h₂)

Tactic:
refine (dist_le dist_nonneg).2 fun x => ?_
    calc
      dist (h₁ x) (h₂ x) = dist (extend f g₁ h₁ x) (extend f g₂ h₂ x) := by
        rw [extend_apply' x.coe_prop, extend_apply' x.coe_prop]
      _ ≤ _ := dist_coe_le_dist _

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: α : Type u_2
inst✝ : TopologicalSpace α
f : α →ᵇ ℝ≥0
x : α
key : nndist (f x) (0 x) ≤ nndist f 0
⊢ f x ≤ nndist f 0

Tactic:
simp only [coe_zero, Pi.zero_apply] at key

Goal state after:
α : Type u_2
inst✝ : TopologicalSpace α
f : α →ᵇ ℝ≥0
x : α
key : nndist (f x) 0 ≤ nndist f 0
⊢ f x ≤ nndist f 0
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_6
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : p.HolderConjugate q
hf : MemLp f (ENNReal.ofReal p) μ
hg : MemLp g (ENNReal.ofReal q) μ
h_left : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ * ‖g a‖) ∂μ = ∫⁻ (a : α), ((fun x ↦ ‖f x‖ₑ) * fun x ↦ ‖g x‖ₑ) a ∂μ
h_right_f : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ ^ p) ∂μ = ∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ
h_right_g : ∫⁻ (a : α), ENNReal.ofReal (‖g a‖ ^ q) ∂μ = ∫⁻ (a : α), ‖g a‖ₑ ^ q ∂μ
⊢ (∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ) ^ (1 / p) =
    (∫⁻ (x : α), ‖f x‖ₑ ^ (ENNReal.ofReal p).toReal ∂μ) ^ (1 / (ENNReal.ofReal p).toReal)

Tactic:
rw [ENNReal.toReal_ofReal hpq.nonneg]

Goal state after:
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : CompleteSpace 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : LocallyCompactSpace E
S : Submodule 𝕜 E
a✝ : Nontrivial E
this : ProperSpace 𝕜
⊢ ProperSpace ↥S

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

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : CompleteSpace 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : LocallyCompactSpace E
S : Submodule 𝕜 E
a✝ : Nontrivial E
this✝ : ProperSpace 𝕜
this : FiniteDimensional 𝕜 E
⊢ ProperSpace ↥S
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: case inl
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : Nontrivial E
x : E
K : Set E
hK : IsCompact K
hx : x ∈ K
hx' : K ∈ 𝓝 x
⊢ ∃ y ∈ frontier K, infDist x Kᶜ = dist x y

Tactic:
Metric.nhds_basis_closedBall.mem_iff

Goal state after:
case inl
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : Nontrivial E
x : E
K : Set E
hK : IsCompact K
hx : x ∈ K
hx' : ∃ i, 0 < i ∧ closedBall x i ⊆ K
⊢ ∃ y ∈ frontier K, infDist x Kᶜ = dist x y
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: case neg
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
r : ℝ≥0
hfint : Integrable f μ
hfint' : 0 ≤ ∫ (x : α), f x ∂μ
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ ↑r
hr : ¬r = 0
hμ : ¬IsFiniteMeasure μ
this : μ univ = ∞
⊢ eLpNorm f 1 μ ≤ if ↑r = 0 then 0 else ∞

pp: case neg.hnc
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
r : ℝ≥0
hfint : Integrable f μ
hfint' : 0 ≤ ∫ (x : α), f x ∂μ
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ ↑r
hr : ¬r = 0
hμ : ¬IsFiniteMeasure μ
this : μ univ = ∞
⊢ ¬2 = 0

Tactic:
if_neg

Goal state after:
case neg
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
r : ℝ≥0
hfint : Integrable f μ
hfint' : 0 ≤ ∫ (x : α), f x ∂μ
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ ↑r
hr : ¬r = 0
hμ : ¬IsFiniteMeasure μ
this : μ univ = ∞
⊢ eLpNorm f 1 μ ≤ ∞

case neg.hnc
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
r : ℝ≥0
hfint : Integrable f μ
hfint' : 0 ≤ ∫ (x : α), f x ∂μ
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ ↑r
hr : ¬r = 0
hμ : ¬IsFiniteMeasure μ
this : μ univ = ∞
⊢ ¬↑r = 0

case neg.hnc
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
r : ℝ≥0
hfint : Integrable f μ
hfint' : 0 ≤ ∫ (x : α), f x ∂μ
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ ↑r
hr : ¬r = 0
hμ : ¬IsFiniteMeasure μ
this : μ univ = ∞
⊢ ¬2 = 0
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case neg
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
C : ℝ
f : α → E
hT : DominatedFinMeasAdditive μ T C
hC : 0 ≤ C
hF : CompleteSpace F
hf : ¬Integrable f μ
⊢ ‖setToFun μ T hT f‖ₑ ≤ ↑(NNReal.mk C hC) * ∫⁻ (x : α), ‖f x‖ₑ ∂μ

Tactic:
simp [setToFun_undef _ hf]

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

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : T3Space β
f : α → β
b : β
val✝ : OrderTop α
h : Tendsto f (pure ⊤) (𝓝 b)
⊢ Tendsto (rightLim f) (pure ⊤) (𝓝 b)

Tactic:
have : f.rightLim ⊤ = f ⊤ := rightLim_eq_of_isTop isTop_top

Goal state after:
case inl
α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : T3Space β
f : α → β
b : β
val✝ : OrderTop α
h : Tendsto f (pure ⊤) (𝓝 b)
this : rightLim f ⊤ = f ⊤
⊢ Tendsto (rightLim f) (pure ⊤) (𝓝 b)
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
f : α → β
hf : Monotone f
x : α
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
⊢ ContinuousWithinAt f (Iio x) x ↔ leftLim f x = f x

Tactic:
by
  rcases eq_or_neBot (𝓝[<] x) with h' | h'
  · simp [leftLim_eq_of_eq_bot f h', ContinuousWithinAt, h']
  refine ⟨fun h => tendsto_nhds_unique (hf.tendsto_leftLim x) h.tendsto, fun h => ?_⟩
  have := hf.tendsto_leftLim x
  rwa [h] at this

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

Goal state before:
pp: R : Type u_1
inst✝² : LinearOrder R
inst✝¹ : TopologicalSpace R
inst✝ : OrderTopology R
a b : R
h : IsTop b
x : R
hx : x ∈ Ioi a
⊢ x ∈ Ioc a b

Tactic:
exact ⟨hx, h _⟩

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

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
ι : Type u_5
t : Finset ι
s : ι → Set X
hs : ∀ i ∈ t, MeasurableSet (s i)
hf : ∀ i ∈ t, μ (s i) ≠ ∞
⊢ μ.real (⋃ i ∈ t, s i) = ∑ u ∈ t.powerset with u.Nonempty, (-1) ^ (#u + 1) * μ.real (⋂ i ∈ u, s i)

Tactic:
← setIntegral_one_eq_measureReal

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
ι : Type u_5
t : Finset ι
s : ι → Set X
hs : ∀ i ∈ t, MeasurableSet (s i)
hf : ∀ i ∈ t, μ (s i) ≠ ∞
⊢ ∫ (x : X) in ⋃ i ∈ t, s i, 1 ∂μ = ∑ x ∈ t.powerset with x.Nonempty, (-1) ^ (#x + 1) * ∫ (x : X) in ⋂ i ∈ x, s i, 1 ∂μ