mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Operator.Extend

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
Eₗ : Type u_4
F : Type u_5
inst✝¹⁰ : NormedDivisionRing 𝕜
inst✝⁹ : NormedDivisionRing 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝⁸ : AddCommGroup E
inst✝⁷ : SeminormedAddCommGroup Eₗ
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : Module 𝕜 E
inst✝⁴ : Module 𝕜₂ F
inst✝³ : IsBoundedSMul 𝕜₂ F
inst✝² : Module 𝕜 Eₗ
inst✝¹ : IsBoundedSMul 𝕜 Eₗ
inst✝ : CompleteSpace F
f : E →ₛₗ[σ₁₂] F
e : E →ₗ[𝕜] Eₗ
h_dense : DenseRange ⇑e
h_norm : ∃ C, ∀ (x : E), ‖f x‖ ≤ C * ‖e x‖
x : E
⊢ (f.extendOfNorm e) (e x) = f x

Tactic:
have := (f.compLeftInverse e).extend_eq (e := (LinearMap.range e).subtypeL)
    (by simpa using! h_dense) isUniformEmbedding_subtype_val.isUniformInducing

Goal state after:
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
Eₗ : Type u_4
F : Type u_5
inst✝¹⁰ : NormedDivisionRing 𝕜
inst✝⁹ : NormedDivisionRing 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝⁸ : AddCommGroup E
inst✝⁷ : SeminormedAddCommGroup Eₗ
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : Module 𝕜 E
inst✝⁴ : Module 𝕜₂ F
inst✝³ : IsBoundedSMul 𝕜₂ F
inst✝² : Module 𝕜 Eₗ
inst✝¹ : IsBoundedSMul 𝕜 Eₗ
inst✝ : CompleteSpace F
f : E →ₛₗ[σ₁₂] F
e : E →ₗ[𝕜] Eₗ
h_dense : DenseRange ⇑e
h_norm : ∃ C, ∀ (x : E), ‖f x‖ ≤ C * ‖e x‖
x : E
this : ∀ (x : ↥e.range), ((f.compLeftInverse e).extend e.range.subtypeL) (e.range.subtypeL x) = (f.compLeftInverse e) x
⊢ (f.extendOfNorm e) (e x) = f x
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: case inr
α : Type u_1
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝ : DecidablePred fun x ↦ x ≠ 0
f : α →ₛ F
s : Finset F
hs : {x ∈ f.range | x ≠ 0} ⊆ s
x : F
hx✝ : x ∉ f.range ∨ x = 0
hx : x ∉ Set.range ⇑f
⊢ μ.real ∅ • x = 0

Tactic:
simp [Set.disjoint_singleton_left, hx]

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

Goal state before:
pp: case inr
α : Type u_1
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝ : DecidablePred fun x ↦ x ≠ 0
f : α →ₛ F
s : Finset F
hs : {x ∈ f.range | x ≠ 0} ⊆ s
x : F
hx✝ : x ∉ f.range ∨ x = 0
hx : x ∉ Set.range ⇑f
⊢ Disjoint {x} (Set.range ⇑f)

Tactic:
simp [Set.disjoint_singleton_left, hx]

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

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
β : Type u_6
x✝ : MeasurableSpace β
ν : Measure β
f g : α → β → G
h : ∀ᵐ (a : α) ∂μ, f a =ᵐ[ν] g a
a✝ : α
⊢ f a✝ =ᵐ[ν] g a✝ → ∫ (b : β), f a✝ b ∂ν = ∫ (b : β), g a✝ b ∂ν

Tactic:
ha

Goal state after:
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
β : Type u_6
x✝ : MeasurableSpace β
ν : Measure β
f g : α → β → G
h : ∀ᵐ (a : α) ∂μ, f a =ᵐ[ν] g a
a✝ : α
ha : f a✝ =ᵐ[ν] g a✝
⊢ ∫ (b : β), f a✝ b ∂ν = ∫ (b : β), g a✝ b ∂ν
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: case refine_1
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
hF : Tendsto (fun i ↦ ∫⁻ (x : α), ‖F i x - f x‖ₑ ∂μ) l (𝓝 0)
s : Set α
⊢ ∀ᶠ (i : ι) in l, Integrable (F i) (μ.restrict s)

Tactic:
filter_upwards [hFi] with i hi using hi.restrict

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

Goal state before:
pp: case mpr
α : Type u_1
β : Type u_2
inst✝ : NormedField β
u v : α → β
l : Filter α
hz : ∀ᶠ (x : α) in l, v x ≠ 0
⊢ Tendsto (u / v) l (𝓝 1) → u ~[l] v

Tactic:
exact isEquivalent_of_tendsto_one

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

Goal state before:
pp: case h₂
α : Type u_1
E : Type u_2
𝕜 : Type u_3
inst✝² : NormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
a b : α → 𝕜
u v : α → E
l : Filter α
hab : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : α) in l, ‖(a - b) x‖ ≤ c * ‖b x‖
φ : α → 𝕜
habφ : a =ᶠ[l] φ * b
this : ((fun x ↦ a x • u x) - fun x ↦ b x • v x) =ᶠ[l] fun x ↦ b x • (φ x • u x - v x)
C : ℝ
hC : C > 0
hCuv : ∀ᶠ (x : α) in l, ‖u x‖ ≤ C * ‖v x‖
c : ℝ
huv : ∀ᶠ (x : α) in l, ‖(u - v) x‖ ≤ c / 2 * ‖v x‖
hφ : ∀ᶠ (x : α) in l, ‖φ x - 1‖ < c / 2 / C
hc : 0 < c
x : α
hCuvx : ‖u x‖ ≤ C * ‖v x‖
huvx : ‖(u - v) x‖ ≤ c / 2 * ‖v x‖
hφx : ‖φ x - 1‖ < c / 2 / C
key : ‖φ x - 1‖ * ‖u x‖ ≤ c / 2 * ‖v x‖
⊢ ‖u x - v x‖ ≤ c / 2 * ‖v x‖

Tactic:
exact huvx

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
⊢ 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:
ha

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.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : NormedField β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
u v : α → β
l : Filter α
inst✝ : ClosedIicTopology β
h : u ~[l] v
hv : ∀ᶠ (t : α) in l, 0 < v t
⊢ ∀ᶠ (x : α) in l, 0 < u x

Tactic:
by
  obtain ⟨φ, hφ, h_eq⟩ := h.exists_pos_eq_mul
  exact (hφ.and (hv.and h_eq)).mono (fun x ⟨hφ, hv, h_eq⟩ ↦ h_eq ▸ mul_pos hφ hv)

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_anti : ∀ᵐ (x : α) ∂μ, AntitoneOn (f x) s
hf_int : ∀ a ∈ s, Integrable (fun x ↦ f x a) μ
a : β
⊢ 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:
ha

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
⊢ a ∈ s →
    ∀ ⦃y : β⦄,
      y ∈ s →
        ∀ ⦃a_3 b : ℝ⦄,
          0 ≤ a_3 →
            0 ≤ b →
              a_3 + b = 1 →
                (fun b ↦ ∫ (x : α), f x b ∂μ) (a_3 • a + b • y) ≤
                  a_3 • (fun b ↦ ∫ (x : α), f x b ∂μ) a + b • (fun b ↦ ∫ (x : α), f x b ∂μ) y

Tactic:
ha

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
⊢ ∀ ⦃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
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 : ∀ (n : ℕ), Integrable (f n) μ
hF : Integrable F μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n ↦ f n x
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))
f' : ℕ → α → ℝ := fun n x ↦ f n x - f 0 x
a : α
⊢ (Monotone fun n ↦ f n a) → ∀ (n : ℕ), 0 ≤ f' n a

Tactic:
ha

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf : ∀ (n : ℕ), Integrable (f n) μ
hF : Integrable F μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n ↦ f n x
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))
f' : ℕ → α → ℝ := fun n x ↦ f n x - f 0 x
a : α
ha : Monotone fun n ↦ f n a
⊢ ∀ (n : ℕ), 0 ≤ f' n a
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_6
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝⁵ : NormedRing 𝕜
inst✝⁴ : Module 𝕜 E
inst✝³ : Module 𝕜 F
inst✝² : IsBoundedSMul 𝕜 E
inst✝¹ : IsBoundedSMul 𝕜 F
inst✝ : CompleteSpace F
T : Set α → E →L[ℝ] F
C : ℝ
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 : ↥(Lp E 1 μ)
⊢ (setToL1 hT) (c • f) = c • (setToL1 hT) f

Tactic:
rw [setToL1_eq_setToL1' hT h_smul, setToL1_eq_setToL1' hT h_smul]
  exact map_smul _ _ _

Goal state after:
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)
hF'_int_eq : ∫⁻ (a : α), F' a ∂μ = ENNReal.ofReal (∫ (a : α), F a ∂μ - ∫ (a : α), f 0 a ∂μ)
h_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f' i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F' a ∂μ))
h_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f' i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f' i a ≤ F' a
h : ∀ᵐ (a : α) ∂μ, Tendsto (fun i ↦ f' i a) atTop (𝓝 (F' a))
a : α
⊢ Tendsto (fun i ↦ f' i a) atTop (𝓝 (F' a)) →
    (Monotone fun i ↦ f i a) → (∀ (i : ℕ), f i a ≤ F a) → Tendsto (fun i ↦ f i a) atTop (𝓝 (F a))

Tactic:
ha

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)
h_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f' i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F' a ∂μ))
h_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f' i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f' i a ≤ F' a
h : ∀ᵐ (a : α) ∂μ, Tendsto (fun i ↦ f' i a) atTop (𝓝 (F' a))
a : α
ha : Tendsto (fun i ↦ f' i a) atTop (𝓝 (F' a))
⊢ (Monotone fun i ↦ f i a) → (∀ (i : ℕ), f i a ≤ F a) → 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
C : ℝ
f g : α → E
hT : DominatedFinMeasAdditive μ T C
hf : Integrable f μ
hg : Integrable g μ
⊢ setToFun μ T hT (f + g) = setToFun μ T hT f + setToFun μ T hT g

Tactic:
by
  by_cases hF : CompleteSpace F; swap
  · simp [setToFun, hF]
  rw [setToFun_eq hT (hf.add hg), setToFun_eq hT hf, setToFun_eq hT hg, Integrable.toL1_add,
    (L1.setToL1 hT).map_add]

Goal state after:
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, Antitone fun i ↦ f i a
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), F a ≤ f i a
f' : ℕ → α → ℝ := fun i a ↦ -f i a
F' : α → ℝ := fun a ↦ -F a
a : α
⊢ (Antitone fun i ↦ f i a) → Monotone fun i ↦ f' i a

Tactic:
ha

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, Antitone fun i ↦ f i a
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), F a ≤ f i a
f' : ℕ → α → ℝ := fun i a ↦ -f i a
F' : α → ℝ := fun a ↦ -F a
a : α
ha : Antitone fun i ↦ f i a
⊢ Monotone fun i ↦ f' i a
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, Antitone fun i ↦ f i a
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), F a ≤ f i a
f' : ℕ → α → ℝ := fun i a ↦ -f i a
F' : α → ℝ := fun a ↦ -F a
a : α
⊢ (∀ (i : ℕ), F a ≤ f i a) → ∀ (i : ℕ), f' i a ≤ F' a

Tactic:
ha

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, Antitone fun i ↦ f i a
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), F a ≤ f i a
f' : ℕ → α → ℝ := fun i a ↦ -f i a
F' : α → ℝ := fun a ↦ -F a
a : α
ha : ∀ (i : ℕ), F a ≤ f i a
⊢ ∀ (i : ℕ), f' i a ≤ F' a
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetricSpace β
f g : α →ᵇ β
C : ℝ
inst✝¹ : Nonempty α
inst✝ : CompactSpace α
w : ∀ (x : α), dist (f x) (g x) < C
c : Continuous[inst✝³, _] fun x ↦ dist (f x) (g x)
x : α
le : IsMaxOn (fun x ↦ dist (f x) (g x)) univ x
⊢ dist f g < C

Tactic:
exact lt_of_le_of_lt (dist_le_iff_of_nonempty.mpr fun y => le trivial) (w x)

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
E : Set α
⊢ (E.indicator fun x ↦ 1) ≤ thickenedIndicatorAux δ E

Tactic:
intro a
  by_cases h : a ∈ E
  · simp only [h, indicator_of_mem, thickenedIndicatorAux_one δ E h, le_refl]
  · simp only [h, indicator_of_notMem, not_false_iff, zero_le]

Goal state after:
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
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
x : α
⊢ Tendsto f (𝓝[<] x) (𝓝[≤] leftLim f x)

Tactic:
apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within f (hf.tendsto_leftLim x)

Goal state after:
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
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
x : α
⊢ ∀ᶠ (x_1 : α) in 𝓝[<] x, f x_1 ∈ Iic (leftLim f x)
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 : Set X
μ : Measure X
hs : MeasurableSet s
hfi : ¬IntegrableOn f s μ
⊢ ¬Integrable (s.indicator f) μ

Tactic:
rwa [integrable_indicator_iff hs]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
inst✝¹ : CompleteSpace E
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
inst✝ : Countable ι
f : ι → α → E
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hf' : ∑' (i : ι), ∫⁻ (a : α), ‖f i a‖ₑ ∂μ ≠ ∞
hF : CompleteSpace F
hf'' : ∀ (i : ι), AEMeasurable (fun x ↦ ‖f i x‖ₑ) μ
hhh : ∀ᵐ (a : α) ∂μ, Summable fun n ↦ ↑‖f n a‖₊
this : ∫⁻ (a : α), ∑' (n : ι), ‖f n a‖ₑ ∂μ < ∞
a : α
⊢ (Summable fun n ↦ ↑‖f n a‖₊) → ↑(∑' (a_2 : ι), ‖f a_2 a‖₊) = ∑' (n : ι), ↑‖f n a‖₊

Tactic:
ha

Goal state after:
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
inst✝¹ : CompleteSpace E
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
inst✝ : Countable ι
f : ι → α → E
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hf' : ∑' (i : ι), ∫⁻ (a : α), ‖f i a‖ₑ ∂μ ≠ ∞
hF : CompleteSpace F
hf'' : ∀ (i : ι), AEMeasurable (fun x ↦ ‖f i x‖ₑ) μ
hhh : ∀ᵐ (a : α) ∂μ, Summable fun n ↦ ↑‖f n a‖₊
this : ∫⁻ (a : α), ∑' (n : ι), ‖f n a‖ₑ ∂μ < ∞
a : α
ha : Summable fun n ↦ ↑‖f n a‖₊
⊢ ↑(∑' (a_1 : ι), ‖f a_1 a‖₊) = ∑' (n : ι), ↑‖f n a‖₊
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set X
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
h_mono : Monotone s
hne : atTop.NeBot
this✝ : IsDirectedOrder ι
S : Set X := ⋃ i, s i
hfi : IntegrableOn f S μ
hfi' : (μ.withDensity fun x ↦ ↑‖f x‖₊) S < ∞
hSm : NullMeasurableSet S μ
hsub : ∀ {i : ι}, s i ⊆ S
ν : Measure X := μ.withDensity fun x ↦ ‖f x‖ₑ
hν : ν = μ.withDensity fun x ↦ ‖f x‖ₑ
ε : ℝ≥0
ε0 : 0 < ↑ε
this : ∀ᶠ (i : ι) in atTop, ν (s i) ∈ Icc (ν S - ↑ε) (ν S + ↑ε)
i : ι
hi : ν (s i) ∈ Icc (ν S - ↑ε) (ν S + ↑ε)
hsm' : NullMeasurableSet (s i) ν
⊢ ∫⁻ (a : X) in S \ s i, ‖f a‖ₑ ∂μ ≤ ↑ε

Tactic:
rw [← withDensity_apply₀ _ (hSm.diff (hsm _)), ← hν, measure_sdiff hsub hsm']

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set X
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
h_mono : Monotone s
hne : atTop.NeBot
this✝ : IsDirectedOrder ι
S : Set X := ⋃ i, s i
hfi : IntegrableOn f S μ
hfi' : (μ.withDensity fun x ↦ ↑‖f x‖₊) S < ∞
hSm : NullMeasurableSet S μ
hsub : ∀ {i : ι}, s i ⊆ S
ν : Measure X := μ.withDensity fun x ↦ ‖f x‖ₑ
hν : ν = μ.withDensity fun x ↦ ‖f x‖ₑ
ε : ℝ≥0
ε0 : 0 < ↑ε
this : ∀ᶠ (i : ι) in atTop, ν (s i) ∈ Icc (ν S - ↑ε) (ν S + ↑ε)
i : ι
hi : ν (s i) ∈ Icc (ν S - ↑ε) (ν S + ↑ε)
hsm' : NullMeasurableSet (s i) ν
⊢ ν S - ν (s i) ≤ ↑ε

X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set X
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
h_mono : Monotone s
hne : atTop.NeBot
this✝ : IsDirectedOrder ι
S : Set X := ⋃ i, s i
hfi : IntegrableOn f S μ
hfi' : (μ.withDensity fun x ↦ ↑‖f x‖₊) S < ∞
hSm : NullMeasurableSet S μ
hsub : ∀ {i : ι}, s i ⊆ S
ν : Measure X := μ.withDensity fun x ↦ ‖f x‖ₑ
hν : ν = μ.withDensity fun x ↦ ‖f x‖ₑ
ε : ℝ≥0
ε0 : 0 < ↑ε
this : ∀ᶠ (i : ι) in atTop, ν (s i) ∈ Icc (ν S - ↑ε) (ν S + ↑ε)
i : ι
hi : ν (s i) ∈ Icc (ν S - ↑ε) (ν S + ↑ε)
hsm' : NullMeasurableSet (s i) ν
⊢ ν (s i) ≠ ∞
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
X : Type u_7
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
hT : DominatedFinMeasAdditive μ T C
fs : X → α → E
bound : α → ℝ
s : Set X
hfs_meas : ∀ x ∈ s, AEStronglyMeasurable (fs x) μ
h_bound : ∀ x ∈ s, ∀ᵐ (a : α) ∂μ, ‖fs x a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_cont : ∀ᵐ (a : α) ∂μ, ContinuousOn (fun x ↦ fs x a) s
x : X
hx : x ∈ s
a : α
⊢ ContinuousOn (fun x ↦ fs x a) s → ContinuousWithinAt (fun x ↦ fs x a) s x

Tactic:
ha

Goal state after:
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
X : Type u_7
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
hT : DominatedFinMeasAdditive μ T C
fs : X → α → E
bound : α → ℝ
s : Set X
hfs_meas : ∀ x ∈ s, AEStronglyMeasurable (fs x) μ
h_bound : ∀ x ∈ s, ∀ᵐ (a : α) ∂μ, ‖fs x a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_cont : ∀ᵐ (a : α) ∂μ, ContinuousOn (fun x ↦ fs x a) s
x : X
hx : x ∈ s
a : α
ha : ContinuousOn (fun x ↦ fs x a) s
⊢ ContinuousWithinAt (fun x ↦ fs x a) s x
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
⊢ (⨆ x, μ ↑x) + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'

Tactic:
have : Nonempty { L : Compacts G // (L : Set G) ⊆ U' ∩ U } := ⟨⟨⊥, empty_subset _⟩⟩

Goal state after:
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
⊢ (⨆ x, μ ↑x) + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
α : Type u_3
E : Type u_5
inst✝⁶ : Group G
inst✝⁵ : MulAction G α
inst✝⁴ : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
s t : Set α
μ : Measure α
inst✝² : MeasurableConstSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
hs : IsFundamentalDomain G s μ
ht : IsFundamentalDomain G t μ
f : α → E
hf : ∀ (g : G) (x : α), f (g • x) = f x
⊢ ∫⁻ (a : α) in s, ‖f a‖ₑ ∂μ < ∞ ↔ ∫⁻ (a : α) in t, ‖f a‖ₑ ∂μ < ∞

Tactic:
rw [hs.setLIntegral_eq ht]

Goal state after:
case hf
G : Type u_1
α : Type u_3
E : Type u_5
inst✝⁶ : Group G
inst✝⁵ : MulAction G α
inst✝⁴ : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
s t : Set α
μ : Measure α
inst✝² : MeasurableConstSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
hs : IsFundamentalDomain G s μ
ht : IsFundamentalDomain G t μ
f : α → E
hf : ∀ (g : G) (x : α), f (g • x) = f x
⊢ ∀ (g : G) (x : α), ‖f (g • x)‖ₑ = ‖f x‖ₑ