mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Group.AddTorsor

Goal state before:
pp: α : Type u_1
V✝ : Type u_2
P✝ : Type u_3
W : Type u_4
Q : Type u_5
inst✝⁷ : SeminormedAddCommGroup V✝
inst✝⁶ : PseudoMetricSpace P✝
inst✝⁵ : NormedAddTorsor V✝ P✝
inst✝⁴ : SeminormedAddCommGroup W
inst✝³ : PseudoMetricSpace Q
inst✝² : NormedAddTorsor W Q
V : Type u_6
P : Type u_7
inst✝¹ : SeminormedAddCommGroup V
inst✝ : AddTorsor V P
x y z : P
⊢ ‖x -ᵥ ?p₂ + (?p₂ -ᵥ z)‖ ≤ ‖x -ᵥ y‖ + ‖y -ᵥ z‖

pp: case p₂
α : Type u_1
V✝ : Type u_2
P✝ : Type u_3
W : Type u_4
Q : Type u_5
inst✝⁷ : SeminormedAddCommGroup V✝
inst✝⁶ : PseudoMetricSpace P✝
inst✝⁵ : NormedAddTorsor V✝ P✝
inst✝⁴ : SeminormedAddCommGroup W
inst✝³ : PseudoMetricSpace Q
inst✝² : NormedAddTorsor W Q
V : Type u_6
P : Type u_7
inst✝¹ : SeminormedAddCommGroup V
inst✝ : AddTorsor V P
x y z : P
⊢ P

Tactic:
apply norm_add_le

Goal state after:
Module: Mathlib.Analysis.Normed.Group.AddTorsor

Goal state before:
pp: α : Type u_1
V✝ : Type u_2
P✝ : Type u_3
W : Type u_4
Q : Type u_5
inst✝⁷ : SeminormedAddCommGroup V✝
inst✝⁶ : PseudoMetricSpace P✝
inst✝⁵ : NormedAddTorsor V✝ P✝
inst✝⁴ : SeminormedAddCommGroup W
inst✝³ : PseudoMetricSpace Q
inst✝² : NormedAddTorsor W Q
V : Type u_6
P : Type u_7
inst✝¹ : NormedAddCommGroup V
inst✝ : AddTorsor V P
x y z : P
⊢ ‖x -ᵥ ?p₂ + (?p₂ -ᵥ z)‖ ≤ ‖x -ᵥ y‖ + ‖y -ᵥ z‖

pp: case p₂
α : Type u_1
V✝ : Type u_2
P✝ : Type u_3
W : Type u_4
Q : Type u_5
inst✝⁷ : SeminormedAddCommGroup V✝
inst✝⁶ : PseudoMetricSpace P✝
inst✝⁵ : NormedAddTorsor V✝ P✝
inst✝⁴ : SeminormedAddCommGroup W
inst✝³ : PseudoMetricSpace Q
inst✝² : NormedAddTorsor W Q
V : Type u_6
P : Type u_7
inst✝¹ : NormedAddCommGroup V
inst✝ : AddTorsor V P
x y z : P
⊢ P

Tactic:
apply norm_add_le

Goal state after:
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
r : ℝ
r_pos : 0 < r
c : ℝ
f : E →ₗ[𝕜] 𝕜
h : ∀ z ∈ sphere 0 r, ‖f z‖ ≤ c
z : E
⊢ ‖f z‖ ≤ c / r * ‖z‖

Tactic:
by
  by_cases z_zero : z = 0
  · rw [z_zero]
    simp only [map_zero, norm_zero, mul_zero]
    exact le_rfl
  set z₁ := ((r : 𝕜) * (‖z‖ : 𝕜)⁻¹) • z with hz₁
  have norm_f_z₁ : ‖f z₁‖ ≤ c := by
    apply h
    rw [mem_sphere_zero_iff_norm]
    exact norm_smul_inv_norm' r_pos.le z_zero
  have r_ne_zero : (r : 𝕜) ≠ 0 := RCLike.ofReal_ne_zero.mpr r_pos.ne'
  have eq : f z = ‖z‖ / r * f z₁ := by
    rw [hz₁, map_smul, smul_eq_mul]
    rw [← mul_assoc, ← mul_assoc, div_mul_cancel₀ _ r_ne_zero, mul_inv_cancel₀, one_mul]
    simp only [z_zero, RCLike.ofReal_eq_zero, norm_eq_zero, Ne, not_false_iff]
  rw [eq, norm_mul, norm_div, RCLike.norm_coe_norm, RCLike.norm_of_nonneg r_pos.le,
    div_mul_eq_mul_div, div_mul_eq_mul_div, mul_comm]
  apply div_le_div₀ _ _ r_pos rfl.ge
  · exact mul_nonneg ((norm_nonneg _).trans norm_f_z₁) (norm_nonneg z)
  apply mul_le_mul norm_f_z₁ rfl.le (norm_nonneg z) ((norm_nonneg _).trans norm_f_z₁)

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

Goal state before:
pp: case refine_2
α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝⁸ : NormedAddCommGroup G''
inst✝⁷ : PartialOrder G''
inst✝⁶ : IsOrderedAddMonoid G''
inst✝⁵ : NormedSpace ℝ G''
inst✝⁴ : CompleteSpace G''
inst✝³ : NormedAddCommGroup G'
inst✝² : PartialOrder G'
inst✝¹ : NormedSpace ℝ G'
inst✝ : ClosedIciTopology 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 : ↥(Lp G' 1 μ)
hf : 0 ≤ f
g : { g // 0 ≤ g }
⊢ ∀ (a : { g // 0 ≤ g }), 0 ≤ (setToL1 hT) ↑(coeSimpleFuncNonnegToLpNonneg 1 μ G' a)

Tactic:
intro g
    have : (coeSimpleFuncNonnegToLpNonneg 1 μ G' g : α →₁[μ] G') = (g : α →₁ₛ[μ] G') := rfl
    rw [this, setToL1_eq_setToL1SCLM]
    exact setToL1S_nonneg (fun s => hT.eq_zero_of_measure_zero) hT.1 hT_nonneg g.2

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 < ∞
⊢ ∀ᶠ (y : ℝ≥0∞) in 𝓝 (∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ), ∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ < y + ↑ε

Tactic:
refine tendsto_id.add tendsto_const_nhds (lt_mem_nhds (α := ℝ≥0∞) <| ENNReal.lt_add_right ?_ ?_)

Goal state after:
case refine_1
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 < ∞
⊢ ∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ ≠ ∞

case refine_2
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 < ∞
⊢ ↑ε ≠ 0
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
hT : DominatedFinMeasAdditive μ T C
c : ℝ
f : α → E
⊢ setToFun μ (fun s ↦ c • T s) ⋯ f = c • setToFun μ T hT f

Tactic:
by_cases hF : CompleteSpace F; swap
  · simp [setToFun, hF]
  by_cases hf : Integrable f μ
  · simp_rw [setToFun_eq _ hf, L1.setToL1_smul_left hT c]
  · simp_rw [setToFun_undef _ hf, smul_zero]

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

Goal state before:
pp: case refine_1
α : Type u_1
inst✝³ : PseudoMetricSpace α
E' : Type u_2
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : FiniteDimensional ℝ E'
s : Set α
f : α → E'
K : ℝ≥0
hf : LipschitzOnWith K f s
ι : Type u_2 := ⋯
A : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ := ⋯
LA : LipschitzWith ‖↑A‖₊ ⇑A
L : LipschitzOnWith (‖↑A‖₊ * K) (⇑A ∘ f) s
g : α → ι → ℝ
hg : LipschitzWith (‖↑A‖₊ * K) g
gs : EqOn (⇑A ∘ f) g s
LAsymm : LipschitzWith ‖↑A.symm‖₊ ⇑A.symm
⊢ ‖↑A.symm‖₊ * (‖↑A‖₊ * K) ≤ lipschitzExtensionConstant E' * K

Tactic:
lipschitzExtensionConstant,

Goal state after:
case refine_1
α : Type u_1
inst✝³ : PseudoMetricSpace α
E' : Type u_2
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : FiniteDimensional ℝ E'
s : Set α
f : α → E'
K : ℝ≥0
hf : LipschitzOnWith K f s
ι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')
A : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ := (Basis.ofVectorSpace ℝ E').equivFun.toContinuousLinearEquiv
LA : LipschitzWith ‖↑A‖₊ ⇑A
L : LipschitzOnWith (‖↑A‖₊ * K) (⇑A ∘ f) s
g : α → ι → ℝ
hg : LipschitzWith (‖↑A‖₊ * K) g
gs : EqOn (⇑A ∘ f) g s
LAsymm : LipschitzWith ‖↑A.symm‖₊ ⇑A.symm
⊢ ‖↑A.symm‖₊ * (‖↑A‖₊ * K) ≤
    (have A := (Basis.ofVectorSpace ℝ E').equivFun.toContinuousLinearEquiv;
      max (‖↑A.symm‖₊ * ‖↑A‖₊) 1) *
      K
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
ι : Type u_6
m : MeasurableSpace α
f : α → G
μ : ι → Measure α
s : Finset ι
hf : ∀ i ∈ s, Integrable f (μ i)
⊢ ∫ (a : α), f a ∂∑ i ∈ s, μ i = ∑ i ∈ s, ∫ (a : α), f a ∂μ i

Tactic:
by
  induction s using Finset.cons_induction_on with
  | empty => simp
  | cons _ _ h ih =>
    rw [Finset.forall_mem_cons] at hf
    rw [Finset.sum_cons, Finset.sum_cons, ← ih hf.2]
    exact integral_add_measure hf.1 (integrable_finsetSum_measure.2 hf.2)

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

Goal state before:
pp: case inr
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : PseudoMetricSpace β
f g : α →ᵇ β
h✝ : Nonempty α
⊢ dist f g = ⨆ x, dist (f x) (g x)

Tactic:
refine (dist_le_iff_of_nonempty.mpr <| le_ciSup ?_).antisymm (ciSup_le dist_coe_le_dist)

Goal state after:
case inr
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : PseudoMetricSpace β
f g : α →ᵇ β
h✝ : Nonempty α
⊢ BddAbove (range fun x ↦ dist (f x) (g x))
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: case refine_1
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
x : δ
⊢ dist ((extend f g₁ h₁) x) ((extend f g₂ h₂) x) ≤
    max (dist g₁ g₂) (dist (h₁.domRestrict (range ⇑f)ᶜ) (h₂.domRestrict (range ⇑f)ᶜ))

Tactic:
rcases em (∃ y, f y = x) with (⟨x, rfl⟩ | hx)

Goal state after:
case refine_1.inl
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
x : α
⊢ dist ((extend f g₁ h₁) (f x)) ((extend f g₂ h₂) (f x)) ≤
    max (dist g₁ g₂) (dist (h₁.domRestrict (range ⇑f)ᶜ) (h₂.domRestrict (range ⇑f)ᶜ))

case refine_1.inr
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
x : δ
hx : ¬∃ y, f y = x
⊢ dist ((extend f g₁ h₁) x) ((extend f g₂ h₂) x) ≤
    max (dist g₁ g₂) (dist (h₁.domRestrict (range ⇑f)ᶜ) (h₂.domRestrict (range ⇑f)ᶜ))
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
E : Type v
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace 𝕜
h : ¬FiniteDimensional 𝕜 E
⊢ ∃ R f, 1 < R ∧ (∀ (n : ℕ), ‖f n‖ ≤ R) ∧ Pairwise fun m n ↦ 1 ≤ ‖f m - f n‖

Tactic:
by
  obtain ⟨c, hc⟩ : ∃ c : 𝕜, 1 < ‖c‖ := NormedField.exists_one_lt_norm 𝕜
  have A : ‖c‖ < ‖c‖ + 1 := by linarith
  rcases exists_seq_norm_le_one_le_norm_sub' hc A h with ⟨f, hf⟩
  exact ⟨‖c‖ + 1, f, hc.trans A, hf.1, hf.2⟩

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✝¹ : 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 : StronglyMeasurable f
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
hF : CompleteSpace F
hfi' : Integrable f μ'
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : SeparableSpace ↑(range f ∪ {0})
i : ℕ
y : E
x✝ : y ∈ (SimpleFunc.approxOn f ⋯ (insert 0 (range f)) 0 ⋯ i).range
hy : ⇑((SimpleFunc.approxOn f ⋯ (insert 0 (range f)) 0 ⋯ i).comp φ hφ) ⁻¹' {y} = ∅
⊢ (T (⇑(SimpleFunc.approxOn f ⋯ (insert 0 (range f)) 0 ⋯ i) ∘ φ ⁻¹' {y})) y = 0

Tactic:
SimpleFunc.coe_comp

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
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
β : Type u_7
x✝¹ : 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 : StronglyMeasurable f
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
hF : CompleteSpace F
hfi' : Integrable f μ'
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : SeparableSpace ↑(range f ∪ {0})
i : ℕ
y : E
x✝ : y ∈ (SimpleFunc.approxOn f ⋯ (insert 0 (range f)) 0 ⋯ i).range
hy : ⇑(SimpleFunc.approxOn f ⋯ (insert 0 (range f)) 0 ⋯ i) ∘ φ ⁻¹' {y} = ∅
⊢ (T (⇑(SimpleFunc.approxOn f ⋯ (insert 0 (range f)) 0 ⋯ i) ∘ φ ⁻¹' {y})) y = 0
Module: Mathlib.Topology.ContinuousMap.Bounded.Star

Goal state before:
pp: F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : NonUnitalNormedRing β
inst✝¹ : StarRing β
inst✝ : CStarRing β
f : α →ᵇ β
x : α
⊢ ‖f x‖ ^ 2 ≤ ‖star f * f‖

Tactic:
sq,

Goal state after:
F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : NonUnitalNormedRing β
inst✝¹ : StarRing β
inst✝ : CStarRing β
f : α →ᵇ β
x : α
⊢ ‖f x‖ * ‖f x‖ ≤ ‖star f * f‖
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : CompactSpace α
inst✝² : NonUnitalNormedRing β
inst✝¹ : StarRing β
inst✝ : CStarRing β
f : C(α, β)
x : α
⊢ ‖f x‖ ^ 2 ≤ ‖star f * f‖

Tactic:
sq,

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : CompactSpace α
inst✝² : NonUnitalNormedRing β
inst✝¹ : StarRing β
inst✝ : CStarRing β
f : C(α, β)
x : α
⊢ ‖f x‖ * ‖f x‖ ≤ ‖star f * f‖
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: case e_f.e_a
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
β : Type u_6
m m0 : MeasurableSpace β
μ : Measure β
hm : m ≤ m0
f : β →ₛ F
hf_int : Integrable (⇑f) μ
hf : StronglyMeasurable ⇑f
hf_int_m : Integrable (⇑f) (μ.trim hm)
x : F
⊢ μ (⇑f ⁻¹' {x}) = (μ.trim hm) (⇑f ⁻¹' {x})

Tactic:
exact (trim_measurableSet_eq hm (@SimpleFunc.measurableSet_fiber β F m f x)).symm

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

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x : α
⊢ (thickenedIndicator δ_pos E) x ≤ 1

Tactic:
thickenedIndicator.coeFn_eq_comp

Goal state after:
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x : α
⊢ (ENNReal.toNNReal ∘ thickenedIndicatorAux δ E) x ≤ 1
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case neg.refine_2
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
c : ℝ≥0∞
hc_ne_top : c ≠ ∞
hT : DominatedFinMeasAdditive μ T C
hT_smul : DominatedFinMeasAdditive (c • μ) T C'
f : α → E
hc0 : ¬c = 0
⊢ μ = (c⁻¹ * c) • μ

Tactic:
ENNReal.inv_mul_cancel hc0 hc_ne_top,

Goal state after:
case neg.refine_2
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
c : ℝ≥0∞
hc_ne_top : c ≠ ∞
hT : DominatedFinMeasAdditive μ T C
hT_smul : DominatedFinMeasAdditive (c • μ) T C'
f : α → E
hc0 : ¬c = 0
⊢ μ = 1 • μ
Module: Mathlib.MeasureTheory.Integral.Marginal

Goal state before:
pp: δ : Type u_1
X : δ → Type u_3
inst✝² : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝¹ : DecidableEq δ
s t : Finset δ
inst✝ : ∀ (i : δ), SigmaFinite (μ i)
f g : ((i : δ) → X i) → ℝ≥0∞
hst : s ⊆ t
hf : Measurable f
hg : Measurable g
hfg : ∫⋯∫⁻_s, f ∂μ = ∫⋯∫⁻_s, g ∂μ
⊢ ∫⋯∫⁻_t, f ∂μ = ∫⋯∫⁻_t, g ∂μ

Tactic:
← union_sdiff_of_subset hst,

Goal state after:
δ : Type u_1
X : δ → Type u_3
inst✝² : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝¹ : DecidableEq δ
s t : Finset δ
inst✝ : ∀ (i : δ), SigmaFinite (μ i)
f g : ((i : δ) → X i) → ℝ≥0∞
hst : s ⊆ t
hf : Measurable f
hg : Measurable g
hfg : ∫⋯∫⁻_s, f ∂μ = ∫⋯∫⁻_s, g ∂μ
⊢ ∫⋯∫⁻_s ∪ t \ s, f ∂μ = ∫⋯∫⁻_s ∪ t \ s, g ∂μ
Module: Mathlib.MeasureTheory.Integral.Marginal

Goal state before:
pp: δ : Type u_1
X : δ → Type u_3
inst✝² : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝¹ : DecidableEq δ
s t : Finset δ
inst✝ : ∀ (i : δ), SigmaFinite (μ i)
f g : ((i : δ) → X i) → ℝ≥0∞
hst : s ⊆ t
hf : Measurable f
hg : Measurable g
hfg : ∫⋯∫⁻_s, f ∂μ ≤ ∫⋯∫⁻_s, g ∂μ
⊢ ∫⋯∫⁻_t, f ∂μ ≤ ∫⋯∫⁻_t, g ∂μ

Tactic:
← union_sdiff_of_subset hst,

Goal state after:
δ : Type u_1
X : δ → Type u_3
inst✝² : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝¹ : DecidableEq δ
s t : Finset δ
inst✝ : ∀ (i : δ), SigmaFinite (μ i)
f g : ((i : δ) → X i) → ℝ≥0∞
hst : s ⊆ t
hf : Measurable f
hg : Measurable g
hfg : ∫⋯∫⁻_s, f ∂μ ≤ ∫⋯∫⁻_s, g ∂μ
⊢ ∫⋯∫⁻_s ∪ t \ s, f ∂μ ≤ ∫⋯∫⁻_s ∪ t \ s, g ∂μ
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
⊢ ∀ᵐ (a : α) ∂μ, Summable fun n ↦ ↑‖f n a‖₊

Tactic:
← lintegral_tsum hf''

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' : ∫⁻ (a : α), ∑' (i : ι), ‖f i a‖ₑ ∂μ ≠ ∞
hF : CompleteSpace F
hf'' : ∀ (i : ι), AEMeasurable (fun x ↦ ‖f i x‖ₑ) μ
⊢ ∀ᵐ (a : α) ∂μ, Summable fun n ↦ ↑‖f n a‖₊
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inr.inl
R : Type u_1
inst✝² : LinearOrder R
inst✝¹ : TopologicalSpace R
f✝ : StieltjesFunction R
inst✝ : OrderTopology R
f : R → ℝ
hf : Monotone f
x : R
s : Set ℝ
hs : s ∈ 𝓝 (rightLim f x)
l u : ℝ
hlu : rightLim f x ∈ Ioo l u
lus : Ioo l u ⊆ s
y₀ : R
hy₀ : x < y₀
y : R
xy : x < y
h'y : Ioo x y ⊆ f ⁻¹' Ioo l u
z : R
hz : z ∈ Ico x y
h''z : x < z
h'z : 𝓝[>] z = ⊥
⊢ rightLim f z < u

Tactic:
rw [rightLim_eq_of_eq_bot _ h'z]
      have : z ∈ Ioo x y := ⟨h''z, hz.2⟩
      exact (h'y this).2

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✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
β : Type u_7
mβ : MeasurableSpace β
inst✝ : SFinite μ
hT : DominatedFinMeasAdditive μ T C
h'T : ∀ (s : Set (β × α)), MeasurableSet s → StronglyMeasurable fun x ↦ T (Prod.mk x ⁻¹' s)
f : β → α → E
hf : StronglyMeasurable (Function.uncurry f)
hF : CompleteSpace F
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : SeparableSpace ↑(range (Function.uncurry f) ∪ {0})
s : ℕ → SimpleFunc (β × α) E := SimpleFunc.approxOn (Function.uncurry f) ⋯ (range (Function.uncurry f) ∪ {0}) 0 ⋯
s' : ℕ → β → SimpleFunc α E := fun n x ↦ (s n).comp (Prod.mk x) ⋯
f' : ℕ → β → F := fun n ↦ {x | Integrable (f x) μ}.indicator fun x ↦ SimpleFunc.setToSimpleFunc T (s' n x)
hf' : ∀ (n : ℕ), StronglyMeasurable (f' n)
x : β
hfx : Integrable (f x) μ
n : ℕ
y : α
⊢ ‖((s n).comp (Prod.mk x) ⋯) y‖ ≤ ((fun a ↦ ‖f x a‖) + fun a ↦ ‖f x a‖) y

Tactic:
SimpleFunc.coe_comp

Goal state after:
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
β : Type u_7
mβ : MeasurableSpace β
inst✝ : SFinite μ
hT : DominatedFinMeasAdditive μ T C
h'T : ∀ (s : Set (β × α)), MeasurableSet s → StronglyMeasurable fun x ↦ T (Prod.mk x ⁻¹' s)
f : β → α → E
hf : StronglyMeasurable (Function.uncurry f)
hF : CompleteSpace F
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : SeparableSpace ↑(range (Function.uncurry f) ∪ {0})
s : ℕ → SimpleFunc (β × α) E := SimpleFunc.approxOn (Function.uncurry f) ⋯ (range (Function.uncurry f) ∪ {0}) 0 ⋯
s' : ℕ → β → SimpleFunc α E := fun n x ↦ (s n).comp (Prod.mk x) ⋯
f' : ℕ → β → F := fun n ↦ {x | Integrable (f x) μ}.indicator fun x ↦ SimpleFunc.setToSimpleFunc T (s' n x)
hf' : ∀ (n : ℕ), StronglyMeasurable (f' n)
x : β
hfx : Integrable (f x) μ
n : ℕ
y : α
⊢ ‖(⇑(s n) ∘ Prod.mk x) y‖ ≤ ((fun a ↦ ‖f x a‖) + fun a ↦ ‖f x a‖) y
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:

Tactic:
f.outer (botSet ∪ Ioc a b)
      _ ≤ f.outer botSet + f.outer (Ioc a b) := measure_union_le _ _
      _ ≤ f.length botSet + f.outer (Ioc a b) := by gcongr; apply outer_le_length
      _ = 0 + f.outer (Ioc a b) := by
        simp only [← length_sdiff_botSet, sdiff_self, empty_sdiff, outer_Ioc, zero_add]
        simp [empty_sdiff]
      _ = f.outer (Ioc a b) := by simp
      _ < f.length (t i) + ofReal ↑(ε' i) := by simpa using h₂

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

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

Tactic:
by
  rw [hf.restrict_map, hf.integral_map]

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

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K : Set G
hK : IsCompact K
g : G
V : Set G
hV : (interior V).Nonempty
⊢ index K V ≤ index ((fun h ↦ g * h) '' K) V

Tactic:
convert! mul_left_index_le (hK.image <| continuous_const_mul g) hV g⁻¹

Goal state after:
case e'_3
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K : Set G
hK : IsCompact K
g : G
V : Set G
hV : (interior V).Nonempty
⊢ K = (fun h ↦ g⁻¹ * h) '' (fun x ↦ g * x) '' K
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 : F
⊢ im ⟪x, x⟫ = 0

Tactic:
← @ofReal_inj 𝕜,

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 : F
⊢ ↑(im ⟪x, x⟫) = ↑0
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
C : ℝ
h : ∀ (x : R), |↑f x| ≤ C
h✝ : Nonempty R
⊢ ∃ u, Tendsto (↑f) atTop (𝓝 u)

Tactic:
rcases tendsto_atTop_of_monotone f.mono with H | H

Goal state after:
case inl
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
C : ℝ
h : ∀ (x : R), |↑f x| ≤ C
h✝ : Nonempty R
H : Tendsto (↑f) atTop atTop
⊢ ∃ u, Tendsto (↑f) atTop (𝓝 u)

case inr
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
C : ℝ
h : ∀ (x : R), |↑f x| ≤ C
h✝ : Nonempty R
H : ∃ l, Tendsto (↑f) atTop (𝓝 l)
⊢ ∃ u, Tendsto (↑f) atTop (𝓝 u)
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f ⊥
this : Continuous eval
⊢ prehaar ↑K₀ '' {U | U ⊆ ↑⊤.toOpens ∧ IsOpen[inst✝¹] U ∧ 1 ∈ U} ⊆ eval ⁻¹' {0}

pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f ⊥
this : Continuous eval
⊢ IsClosed (eval ⁻¹' {0})

Tactic:
· rintro _ ⟨U, _, rfl⟩; apply prehaar_empty

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f ⊥
this : Continuous eval
⊢ IsClosed (eval ⁻¹' {0})