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

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_add : FinMeasAdditive μ T
f g : α →ₛ E
hf : Integrable (⇑f) μ
hg : Integrable (⇑g) μ
h : Pairwise fun x y ↦ T (⇑f ⁻¹' {x} ∩ ⇑g ⁻¹' {y}) = 0
h_pair : Integrable (⇑(f.pair g)) μ
a : α
hp : (f.pair g) a ∈ (f.pair g).range
⊢ (T (⇑(f.pair g) ⁻¹' {(f.pair g) a})) ((f.pair g) a).1 = (T (⇑(f.pair g) ⁻¹' {(f.pair g) a})) ((f.pair g) a).2

Tactic:
by_cases eq : f a = g a

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
h_add : FinMeasAdditive μ T
f g : α →ₛ E
hf : Integrable (⇑f) μ
hg : Integrable (⇑g) μ
h : Pairwise fun x y ↦ T (⇑f ⁻¹' {x} ∩ ⇑g ⁻¹' {y}) = 0
h_pair : Integrable (⇑(f.pair g)) μ
a : α
hp : (f.pair g) a ∈ (f.pair g).range
eq : f a = g a
⊢ (T (⇑(f.pair g) ⁻¹' {(f.pair g) a})) ((f.pair g) a).1 = (T (⇑(f.pair g) ⁻¹' {(f.pair g) a})) ((f.pair g) a).2

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
h_add : FinMeasAdditive μ T
f g : α →ₛ E
hf : Integrable (⇑f) μ
hg : Integrable (⇑g) μ
h : Pairwise fun x y ↦ T (⇑f ⁻¹' {x} ∩ ⇑g ⁻¹' {y}) = 0
h_pair : Integrable (⇑(f.pair g)) μ
a : α
hp : (f.pair g) a ∈ (f.pair g).range
eq : ¬f a = g a
⊢ (T (⇑(f.pair g) ⁻¹' {(f.pair g) a})) ((f.pair g) a).1 = (T (⇑(f.pair g) ⁻¹' {(f.pair g) a})) ((f.pair g) a).2
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f μ
f₁ : ↥(Lp ℝ 1 μ) := Integrable.toL1 f hf
eq₁ : (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal = ‖Lp.posPart f₁‖
a✝ : α
h₁ : ↑↑(Lp.negPart f₁) a✝ = -min (↑↑f₁ a✝) 0
h₂ : ↑↑(Integrable.toL1 f hf) a✝ = f a✝
⊢ max (-f a✝) 0 = ‖min (f a✝) 0‖

Tactic:
Real.norm_of_nonpos (min_le_right _ _),

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f μ
f₁ : ↥(Lp ℝ 1 μ) := Integrable.toL1 f hf
eq₁ : (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal = ‖Lp.posPart f₁‖
a✝ : α
h₁ : ↑↑(Lp.negPart f₁) a✝ = -min (↑↑f₁ a✝) 0
h₂ : ↑↑(Integrable.toL1 f hf) a✝ = f a✝
⊢ max (-f a✝) 0 = -min (f a✝) 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)
i : ℕ
⊢ ∫⁻ (a : α), f' i a ∂μ = ENNReal.ofReal (∫ (a : α), f i a ∂μ - ∫ (a : α), f 0 a ∂μ)

Tactic:
unfold f'

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)
i : ℕ
⊢ ∫⁻ (a : α), ENNReal.ofReal (f i a - f 0 a) ∂μ = ENNReal.ofReal (∫ (a : α), f i a ∂μ - ∫ (a : α), f 0 a ∂μ)
Module: Mathlib.Analysis.Normed.Module.RCLike.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
hx : x ≠ 0
this : ‖x‖ ≠ 0
⊢ ‖(↑‖x‖)⁻¹ • x‖ = 1

Tactic:
simp [field, norm_smul]

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

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
f : ↥(α →₁ₛ[μ] G)
h_eq : ∀ (a : α), (SimpleFunc.map (fun x ↦ ‖x‖ₑ) (toSimpleFunc f)) a = ‖(toSimpleFunc f) a‖ₑ
⊢ ∀ a ∈ (toSimpleFunc f).range, ‖a‖ₑ * μ (⇑(toSimpleFunc f) ⁻¹' {a}) ≠ ∞

Tactic:
intro x _
    by_cases hx0 : x = 0
    · rw [hx0]; simp
    · finiteness [SimpleFunc.measure_preimage_lt_top_of_integrable _ (SimpleFunc.integrable f) hx0]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
G'' : Type u_7
G' : Type u_8
inst✝⁷ : NormedAddCommGroup G'
inst✝⁶ : PartialOrder G'
inst✝⁵ : IsOrderedAddMonoid G'
inst✝⁴ : NormedSpace ℝ G'
inst✝³ : NormedAddCommGroup G''
inst✝² : PartialOrder G''
inst✝¹ : IsOrderedAddMonoid G''
inst✝ : NormedSpace ℝ G''
T : Set α → G'' →L[ℝ] G'
h_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0
h_add : FinMeasAdditive μ T
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∀ (x : G''), 0 ≤ x → 0 ≤ (T s) x
f g : ↥(α →₁ₛ[μ] G'')
hfg : f ≤ g
⊢ setToL1S T f ≤ setToL1S T g

Tactic:
← sub_nonneg

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
G'' : Type u_7
G' : Type u_8
inst✝⁷ : NormedAddCommGroup G'
inst✝⁶ : PartialOrder G'
inst✝⁵ : IsOrderedAddMonoid G'
inst✝⁴ : NormedSpace ℝ G'
inst✝³ : NormedAddCommGroup G''
inst✝² : PartialOrder G''
inst✝¹ : IsOrderedAddMonoid G''
inst✝ : NormedSpace ℝ G''
T : Set α → G'' →L[ℝ] G'
h_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0
h_add : FinMeasAdditive μ T
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∀ (x : G''), 0 ≤ x → 0 ≤ (T s) x
f g : ↥(α →₁ₛ[μ] G'')
hfg : 0 ≤ g - f
⊢ 0 ≤ setToL1S T g - setToL1S T f
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
ha : Tendsto (fun i ↦ f' i a) atTop (𝓝 (F' a))
ha_mono : Monotone fun i ↦ f i a
ha_bound : ∀ (i : ℕ), f i a ≤ F a
⊢ (fun i ↦ f i a) = fun i ↦ (f' i a).toReal + f 0 a

Tactic:
unfold f'

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))
ha_mono : Monotone fun i ↦ f i a
ha_bound : ∀ (i : ℕ), f i a ≤ F a
⊢ (fun i ↦ f i a) = fun i ↦ (ENNReal.ofReal (f i a - f 0 a)).toReal + f 0 a
Module: Mathlib.Topology.Instances.Matrix

Goal state before:
pp: m : Type u_4
R : Type u_8
inst✝³ : TopologicalSpace R
α : Type u_11
b : m → α
inst✝² : LinearOrder α
inst✝¹ : Zero R
inst✝ : T2Space R
⊢ IsClosed {M | M.BlockTriangular b}

Tactic:
simp only [BlockTriangular, Set.ofPred_forall]
  refine isClosed_iInter fun i => isClosed_iInter fun j => isClosed_iInter fun _ => ?_
  exact isClosed_eq (continuous_id.matrix_elem i j) continuous_const

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝⁹ : NormedAddCommGroup G''
inst✝⁸ : PartialOrder G''
inst✝⁷ : IsOrderedAddMonoid G''
inst✝⁶ : NormedSpace ℝ G''
inst✝⁵ : CompleteSpace G''
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : PartialOrder G'
inst✝² : NormedSpace ℝ G'
inst✝¹ : ClosedIciTopology G''
inst✝ : IsOrderedAddMonoid G'
T : Set α → G' →L[ℝ] G''
C : ℝ
hT : DominatedFinMeasAdditive μ T C
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f g : ↥(Lp G' 1 μ)
hfg : f ≤ g
⊢ (setToL1 hT) f ≤ (setToL1 hT) g

Tactic:
← sub_nonneg

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝⁹ : NormedAddCommGroup G''
inst✝⁸ : PartialOrder G''
inst✝⁷ : IsOrderedAddMonoid G''
inst✝⁶ : NormedSpace ℝ G''
inst✝⁵ : CompleteSpace G''
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : PartialOrder G'
inst✝² : NormedSpace ℝ G'
inst✝¹ : ClosedIciTopology G''
inst✝ : IsOrderedAddMonoid G'
T : Set α → G' →L[ℝ] G''
C : ℝ
hT : DominatedFinMeasAdditive μ T C
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f g : ↥(Lp G' 1 μ)
hfg : 0 ≤ g - f
⊢ 0 ≤ (setToL1 hT) g - (setToL1 hT) f
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'
f : α → E
hF : CompleteSpace F
hf : ¬Integrable f μ
⊢ setToFun μ (T + T') ⋯ f = setToFun μ T hT f + setToFun μ T' hT' f

Tactic:
simp_rw [setToFun_undef _ hf, add_zero]

Goal state after:
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' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
hT'' : DominatedFinMeasAdditive μ T'' C''
h_add : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T'' s = T s + T' s
f : α → E
hF : CompleteSpace F
hf : ¬Integrable f μ
⊢ setToFun μ T'' hT'' f = setToFun μ T hT f + setToFun μ T' hT' f

Tactic:
simp_rw [setToFun_undef _ hf, add_zero]

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

Goal state before:
pp: ι : Type u_1
X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
μ : ι → Measure X
f : X → E
inst✝ : NormedSpace ℝ E
hf : Integrable f (Measure.sum μ)
hfi : ∀ (i : ι), Integrable f (μ i)
ε : ℝ≥0
ε0 : 0 < ↑ε
hf_lt : ∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ < ∞
hmem : ∀ᶠ (y : ℝ≥0∞) in 𝓝 (∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ), ∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ < y + ↑ε
s : Finset ι
hs : ∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ < ∑ b ∈ s, (fun i ↦ ∫⁻ (a : X), ‖f a‖ₑ ∂μ i) b + ↑ε
ν : Measure X
hν : ∑ i ∈ s, μ i + ν = Measure.sum μ
⊢ ↑(nndist (∫ (x : X), f x ∂∑ i ∈ s, μ i) (∫ (x : X), f x ∂(∑ i ∈ s, μ i + ν))) < ↑ε

Tactic:
rw [← hν, integrable_add_measure] at hf

Goal state after:
ι : Type u_1
X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
μ : ι → Measure X
f : X → E
inst✝ : NormedSpace ℝ E
hfi : ∀ (i : ι), Integrable f (μ i)
ε : ℝ≥0
ε0 : 0 < ↑ε
hf_lt : ∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ < ∞
hmem : ∀ᶠ (y : ℝ≥0∞) in 𝓝 (∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ), ∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ < y + ↑ε
s : Finset ι
hs : ∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ < ∑ b ∈ s, (fun i ↦ ∫⁻ (a : X), ‖f a‖ₑ ∂μ i) b + ↑ε
ν : Measure X
hf : Integrable f (∑ i ∈ s, μ i) ∧ Integrable f ν
hν : ∑ i ∈ s, μ i + ν = Measure.sum μ
⊢ ↑(nndist (∫ (x : X), f x ∂∑ i ∈ s, μ i) (∫ (x : X), f x ∂(∑ i ∈ s, μ i + ν))) < ↑ε
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
ha : Tendsto (fun i ↦ f' i a) atTop (𝓝 (F' a))
ha_mono : Monotone fun i ↦ f i a
ha_bound : ∀ (i : ℕ), f i a ≤ F a
h1 : (fun i ↦ f i a) = fun i ↦ (f' i a).toReal + f 0 a
h2 : F a = (F' a).toReal + f 0 a
⊢ Tendsto (fun i ↦ (f' i a).toReal) atTop (𝓝 (F' a).toReal)

Tactic:
exact (ENNReal.continuousAt_toReal (by finiteness)).tendsto.comp ha

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

Goal state before:
pp: 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
⊢ (μ {a}).toReal • ∫ (x : X), f x ∂Measure.dirac a = (Measure.sum fun a ↦ μ {a} • Measure.dirac a).real {a} • f a

Tactic:
integral_dirac,

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
⊢ (μ {a}).toReal • f a = (Measure.sum fun a ↦ μ {a} • Measure.dirac a).real {a} • f a
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: case refine_2
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
⊢ ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f' i a

Tactic:
filter_upwards [hf_mono] with a ha i j hij
    simp [f', ha hij]

Goal state after:
Module: Mathlib.Topology.Bornology.BoundedOperation

Goal state before:

Tactic:
‖x₁ * y₁ - x₂ * y₂‖
     _ ≤ ‖x₁ * y₁‖ + ‖x₂ * y₂‖ := norm_sub_le _ _
     _ ≤ Af * Ag + Af * Ag     := add_le_add (aux hx₁ hy₁) (aux hx₂ hy₂)
     _ = 2 * Af * Ag           := by simp [← two_mul, mul_assoc]

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : CompactSpace α
inst✝ : PseudoMetricSpace β
⊢ ∀ (s : Set (C(α, β) × C(α, β))),
    s ∈ uniformity C(α, β) ↔
      ∃ t ∈ uniformity (α →ᵇ β),
        ∀ (x y : C(α, β)), ((equivBoundedOfCompact α β) x, (equivBoundedOfCompact α β) y) ∈ t → (x, y) ∈ s

Tactic:
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
      exact fun s =>
        ⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
          ⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
            fun f g h => hs fun x _ => h x⟩,
          fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
          ⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
            fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩

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✝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 C' : ℝ
hT : DominatedFinMeasAdditive μ T C
β : Type u_7
x✝ : MeasurableSpace β
μ' : Measure β
φ : α → β
T' : Set β → E →L[ℝ] F
hT' : DominatedFinMeasAdditive μ' T' C'
f : β → E
hf : Integrable (f ∘ φ) μ
hfm : AEStronglyMeasurable f (Measure.map φ μ)
hφ : Measurable φ
hμ' : μ' ≤ Measure.map φ μ
h : ∀ (s : Set β) (x : E), MeasurableSet s → (T' s) x = (T (φ ⁻¹' s)) x
g : β → E := AEStronglyMeasurable.mk f hfm
A : setToFun μ' T' hT' f = setToFun μ' T' hT' g
B : setToFun μ T hT (f ∘ φ) = setToFun μ T hT (g ∘ φ)
⊢ setToFun μ' T' hT' g = setToFun μ T hT (g ∘ φ)

Tactic:
exact setToFun_of_le_map_of_stronglyMeasurable _ _
    (hf.congr (ae_of_ae_map hφ.aemeasurable hfm.ae_eq_mk)) hfm.stronglyMeasurable_mk hφ hμ' h

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : SeminormedAddCommGroup β
inst✝² : StarAddMonoid β
inst✝¹ : NormedStarGroup β
inst✝ : CompactSpace α
f : C(α, β)
⊢ ‖mkOfCompact f‖ ≤ ‖f‖

Tactic:
BoundedContinuousFunction.norm_mkOfCompact

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : SeminormedAddCommGroup β
inst✝² : StarAddMonoid β
inst✝¹ : NormedStarGroup β
inst✝ : CompactSpace α
f : C(α, β)
⊢ ‖f‖ ≤ ‖f‖
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case cons
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
f : α → E
ι : Type u_7
s✝ : Finset ι
μ : ι → Measure α
T : ι → Set α → E →L[ℝ] F
C : ι → ℝ
hTs : ∀ (i : ι), DominatedFinMeasAdditive (μ i) (T i) (C i)
i : ι
s : Finset ι
his : i ∉ s
hs' : s.Nonempty
ih : (∀ i ∈ s, Integrable f (μ i)) → setToFun (∑ i ∈ s, μ i) (∑ i ∈ s, T i) ⋯ f = ∑ i ∈ s, setToFun (μ i) (T i) ⋯ f
hf : ∀ i_1 ∈ Finset.cons i s his, Integrable f (μ i_1)
⊢ setToFun (∑ i ∈ Finset.cons i s his, μ i) (∑ i ∈ Finset.cons i s his, T i) ⋯ f =
    ∑ i ∈ Finset.cons i s his, setToFun (μ i) (T i) ⋯ f

Tactic:
simpa [his, ih fun j hj => hf j (Finset.mem_cons_of_mem hj)] using!
      setToFun_add_measure (hTs i) (DominatedFinMeasAdditive.finsetSum_measure hs' μ T C hTs)
      (hf i (Finset.mem_cons_self i s))
      (integrable_finsetSum_measure.2 fun j hj => hf j (Finset.mem_cons_of_mem hj))

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✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
f : α → E
ι : Type u_7
s : Finset ι
hs : s.Nonempty
μ : ι → Measure α
T : ι → Set α → E →L[ℝ] F
C : ι → ℝ
hTs : ∀ (i : ι), DominatedFinMeasAdditive (μ i) (T i) (C i)
hf : ∀ i ∈ s, Integrable f (μ i)
⊢ setToFun (∑ i ∈ s, μ i) (∑ i ∈ s, T i) ⋯ f = ∑ i ∈ s, setToFun (μ i) (T i) ⋯ f

Tactic:
induction hs using Finset.Nonempty.cons_induction with

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✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ μ' μ'' : Measure α
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f : α → E
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ' T' C'
hT'' : DominatedFinMeasAdditive μ'' T'' C''
h : ∀ (s : Set α), MeasurableSet s → (μ + μ') s < ∞ → T'' s = T s + T' s
hf : Integrable f μ
hf' : Integrable f μ'
hμ : μ'' ≤ μ + μ'
hC : 0 ≤ C
hC' : 0 ≤ C'
hC'' : 0 ≤ C''
I : DominatedFinMeasAdditive (μ + μ') T C
A : setToFun (μ + μ') T I f = setToFun μ T hT f
I' : DominatedFinMeasAdditive (μ + μ') T' C'
A' : setToFun (μ + μ') T' I' f = setToFun μ' T' hT' f
I'' : DominatedFinMeasAdditive (μ + μ') T'' C''
⊢ setToFun (μ + μ') T'' I'' f = setToFun μ'' T'' hT'' f

Tactic:
apply setToFun_congr_measure_of_integrable (c' := 1) (by simp) (by simpa using hμ)

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 μ' μ'' : Measure α
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f : α → E
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ' T' C'
hT'' : DominatedFinMeasAdditive μ'' T'' C''
h : ∀ (s : Set α), MeasurableSet s → (μ + μ') s < ∞ → T'' s = T s + T' s
hf : Integrable f μ
hf' : Integrable f μ'
hμ : μ'' ≤ μ + μ'
hC : 0 ≤ C
hC' : 0 ≤ C'
hC'' : 0 ≤ C''
I : DominatedFinMeasAdditive (μ + μ') T C
A : setToFun (μ + μ') T I f = setToFun μ T hT f
I' : DominatedFinMeasAdditive (μ + μ') T' C'
A' : setToFun (μ + μ') T' I' f = setToFun μ' T' hT' f
I'' : DominatedFinMeasAdditive (μ + μ') T'' C''
⊢ Integrable f (μ + μ')
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝¹ : LinearOrder R
inst✝ : TopologicalSpace R
f : StieltjesFunction R
s₁ s₂ : Set R
h : s₁ ⊆ s₂
⊢ f.length s₁ ≤ f.length s₂

Tactic:
rcases isEmpty_or_nonempty R with hR | hR
  · simp [length_eq_of_isEmpty]
  simp only [length_eq]
  exact iInf_mono fun a => biInf_mono fun b => by gcongr

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

Goal state before:
pp: R : Type u_1
inst✝³ : LinearOrder R
inst✝² : TopologicalSpace R
f : StieltjesFunction R
inst✝¹ : OrderTopology R
inst✝ : CompactIccSpace R
a b : R
c d : ℕ → R
ss : Icc a b ⊆ ⋃ i, Iotop (c i) (d i)
this :
  ∀ (s : Finset ℕ) (b : R),
    Icc a b ⊆ ⋃ i ∈ ↑s, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ s, ofReal (↑f (d i) - ↑f (c i))
s : Set ℕ
left✝ : s ⊆ univ
hf : s.Finite
hs : Icc a b ⊆ ⋃ i ∈ s, Iotop (c i) (d i)
⊢ ⋃ i ∈ ↑hf.toFinset, Iotop (c i) (d i) = ⋃ i ∈ s, Iotop (c i) (d i)

Tactic:
simp only [Finset.set_biUnion_coe,
        Finite.mem_toFinset]

Goal state after:
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
a : R
ha : ¬IsBot a
b : R
hb : b < a
u : ℕ → R
u_mono : StrictMono u
u_lim : Tendsto u atTop (𝓝 a)
u_lt_a : ∀ (n : ℕ), u n < a
A : {a} = ⋂ n, Ioc (u n) a
⊢ f.measure (Ioc (u 0) a) ≠ ∞

Tactic:
simpa only [measure_Ioc] using ENNReal.ofReal_ne_top

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

Goal state before:
pp: case pos
G : Type u_1
α : Type u_3
inst✝⁷ : Group G
inst✝⁶ : MulAction G α
inst✝⁵ : MeasurableSpace α
ν : Measure α
inst✝⁴ : SMulInvariantMeasure G α ν
inst✝³ : Countable G
inst✝² : MeasurableConstSMul G α
μ : Measure (Quotient α_mod_G)
inst✝¹ : QuotientMeasureEqMeasurePreimage ν μ
inst✝ : IsFiniteMeasure μ
hasFun : HasFundamentalDomain G α ν
𝓕 : Set α
h𝓕 : IsFundamentalDomain G 𝓕 ν
H : μ univ < ∞
⊢ covolume G α ν < ∞

Tactic:
h𝓕.projection_respects_measure_apply (μ := μ) MeasurableSet.univ

Goal state after:
case pos
G : Type u_1
α : Type u_3
inst✝⁷ : Group G
inst✝⁶ : MulAction G α
inst✝⁵ : MeasurableSpace α
ν : Measure α
inst✝⁴ : SMulInvariantMeasure G α ν
inst✝³ : Countable G
inst✝² : MeasurableConstSMul G α
μ : Measure (Quotient α_mod_G)
inst✝¹ : QuotientMeasureEqMeasurePreimage ν μ
inst✝ : IsFiniteMeasure μ
hasFun : HasFundamentalDomain G α ν
𝓕 : Set α
h𝓕 : IsFundamentalDomain G 𝓕 ν
H : ν (Quotient.mk α_mod_G ⁻¹' univ ∩ 𝓕) < ∞
⊢ covolume G α ν < ∞