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_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0
h_add : FinMeasAdditive μ T
f g : α →ₛ E
hf : Integrable (⇑f) μ
h✝ : μ {a | ¬f a = g a} = 0
x y : E
hxy : x ≠ y
z : α
h : f z = x ∧ g z = y
⊢ ¬x = g z

Tactic:
h.2

Goal state after:
α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
h_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0
h_add : FinMeasAdditive μ T
f g : α →ₛ E
hf : Integrable (⇑f) μ
h✝ : μ {a | ¬f a = g a} = 0
x y : E
hxy : x ≠ y
z : α
h : f z = x ∧ g z = y
⊢ ¬x = y
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: case mp
α : Type u_1
β : Type u_2
inst✝ : NormedAddCommGroup β
u : α → β
l : Filter α
c : β
h : c ≠ 0
⊢ Tendsto (u - fun x ↦ c) l (𝓝 0) → Tendsto u l (𝓝 c)

Tactic:
intro h

Goal state after:
case mp
α : Type u_1
β : Type u_2
inst✝ : NormedAddCommGroup β
u : α → β
l : Filter α
c : β
h✝ : c ≠ 0
h : Tendsto (u - fun x ↦ c) l (𝓝 0)
⊢ Tendsto u l (𝓝 c)
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: case mpr
α : Type u_1
β : Type u_2
inst✝ : NormedAddCommGroup β
u : α → β
l : Filter α
c : β
h : c ≠ 0
⊢ Tendsto u l (𝓝 c) → Tendsto (u - fun x ↦ c) l (𝓝 0)

Tactic:
intro h

Goal state after:
case mpr
α : Type u_1
β : Type u_2
inst✝ : NormedAddCommGroup β
u : α → β
l : Filter α
c : β
h✝ : c ≠ 0
h : Tendsto u l (𝓝 c)
⊢ Tendsto (u - fun x ↦ c) l (𝓝 0)
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: case pos
α : Type u_1
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
inst✝ : OrderClosedTopology E
f : α → E
ν : Measure α
hle : μ ≤ ν
hf : 0 ≤ᵐ[ν] f
hE : CompleteSpace E
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
g : α → E
hfi : Integrable g ν
hg : StronglyMeasurable g
hg_nonneg : ∀ (x : α), g x ∈ Ici 0
hfg : f =ᵐ[ν] g
x✝ : SeparableSpace ↑(range g ∪ {0})
h₁ : Tendsto (fun n ↦ SimpleFunc.integral ν (approxOn g ⋯ (range g ∪ {0}) 0 ⋯ n)) atTop (𝓝 (∫ (x : α), g x ∂ν))
⊢ ∫ (a : α), g a ∂μ ≤ ∫ (a : α), g a ∂ν

Tactic:
have h₂ := tendsto_integral_approxOn_of_measurable_of_range_subset hg.measurable
    (hfi.mono_measure hle) _ le_rfl

Goal state after:
case pos
α : Type u_1
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
inst✝ : OrderClosedTopology E
f : α → E
ν : Measure α
hle : μ ≤ ν
hf : 0 ≤ᵐ[ν] f
hE : CompleteSpace E
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
g : α → E
hfi : Integrable g ν
hg : StronglyMeasurable g
hg_nonneg : ∀ (x : α), g x ∈ Ici 0
hfg : f =ᵐ[ν] g
x✝ : SeparableSpace ↑(range g ∪ {0})
h₁ : Tendsto (fun n ↦ SimpleFunc.integral ν (approxOn g ⋯ (range g ∪ {0}) 0 ⋯ n)) atTop (𝓝 (∫ (x : α), g x ∂ν))
h₂ : Tendsto (fun n ↦ SimpleFunc.integral μ (approxOn g ⋯ (range g ∪ {0}) 0 ⋯ n)) atTop (𝓝 (∫ (x : α), g x ∂μ))
⊢ ∫ (a : α), g a ∂μ ≤ ∫ (a : α), g a ∂ν
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, v t < 0
φ : α → β
hφ : ∀ᶠ (x : α) in l, 0 < φ x
h_eq : u =ᶠ[l] φ * v
⊢ ∀ᶠ (x : α) in l, u x < 0

Tactic:
exact (hφ.and (hv.and h_eq)).mono (fun x ⟨hφ, hv, h_eq⟩ ↦ h_eq ▸ mul_neg_of_pos_of_neg hφ hv)

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 : SeminormedAddCommGroup V
inst✝ : AddTorsor V P
x y z : P
⊢ ‖x -ᵥ z‖ ≤ ‖x -ᵥ y‖ + ‖y -ᵥ z‖

Tactic:
← vsub_add_vsub_cancel

Goal state after:
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‖

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
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 z‖ ≤ ‖x -ᵥ y‖ + ‖y -ᵥ z‖

Tactic:
← vsub_add_vsub_cancel

Goal state after:
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‖

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
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
P : (α → E) → Prop
h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → P (s.indicator fun x ↦ c)
h_closed : IsClosed {f | P ↑↑f}
h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → MemLp f 1 μ → MemLp g 1 μ → P f → P g → P (f + g)
h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → MemLp f 1 μ → P f → P g
⊢ ∀ ⦃f : α → E⦄, MemLp f 1 μ → P f

Tactic:
exact MemLp.induction one_ne_top (motive := P) h_ind h_add h_closed h_ae

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))
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun i ↦ f i a) atTop (𝓝 (F a))

Tactic:
have h_mono : ∀ᵐ a ∂μ, Monotone (fun i ↦ f' i a) := by
    filter_upwards [hf_mono] with a ha_mono i j hij
    refine ENNReal.ofReal_le_ofReal ?_
    simp [ha_mono hij]

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
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun i ↦ f i a) atTop (𝓝 (F a))
Module: Mathlib.Analysis.Normed.Affine.Isometry

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
V₁ : Type u_3
V₁' : Type u_4
V₂ : Type u_5
V₃ : Type u_6
V₄ : Type u_7
P₁ : Type u_8
P₁' : Type u_9
P : Type u_10
P₂ : Type u_11
P₃ : Type u_12
P₄ : Type u_13
inst✝²⁴ : NormedField 𝕜
inst✝²³ : SeminormedAddCommGroup V
inst✝²² : NormedSpace 𝕜 V
inst✝²¹ : PseudoMetricSpace P
inst✝²⁰ : NormedAddTorsor V P
inst✝¹⁹ : SeminormedAddCommGroup V₁
inst✝¹⁸ : NormedSpace 𝕜 V₁
inst✝¹⁷ : PseudoMetricSpace P₁
inst✝¹⁶ : NormedAddTorsor V₁ P₁
inst✝¹⁵ : SeminormedAddCommGroup V₁'
inst✝¹⁴ : NormedSpace 𝕜 V₁'
inst✝¹³ : MetricSpace P₁'
inst✝¹² : NormedAddTorsor V₁' P₁'
inst✝¹¹ : SeminormedAddCommGroup V₂
inst✝¹⁰ : NormedSpace 𝕜 V₂
inst✝⁹ : PseudoMetricSpace P₂
inst✝⁸ : NormedAddTorsor V₂ P₂
inst✝⁷ : SeminormedAddCommGroup V₃
inst✝⁶ : NormedSpace 𝕜 V₃
inst✝⁵ : PseudoMetricSpace P₃
inst✝⁴ : NormedAddTorsor V₃ P₃
inst✝³ : SeminormedAddCommGroup V₄
inst✝² : NormedSpace 𝕜 V₄
inst✝¹ : PseudoMetricSpace P₄
inst✝ : NormedAddTorsor V₄ P₄
f✝ f g : P →ᵃⁱ[𝕜] P₂
⊢ (fun f ↦ f.toFun) f = (fun f ↦ f.toFun) g → f = g

Tactic:
cases f; cases g; simp

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 ν : Measure α
f : α → G
hμ : Integrable f μ
hν : Integrable f ν
s : Set α
hs : MeasurableSet s
h's : (μ + ν) s < ∞
⊢ weightedSMul (μ + ν) s = weightedSMul μ s + weightedSMul ν s

Tactic:
simp only [Measure.coe_add, Pi.add_apply, add_lt_top] at h's

Goal state after:
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
m : MeasurableSpace α
μ ν : Measure α
f : α → G
hμ : Integrable f μ
hν : Integrable f ν
s : Set α
hs : MeasurableSet s
h's : μ s < ∞ ∧ ν s < ∞
⊢ weightedSMul (μ + ν) s = weightedSMul μ s + weightedSMul ν s
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: case neg
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
f : α → β
hf : MeasurableEmbedding f
g : β → G
hgm : ¬AEStronglyMeasurable g (Measure.map f μ)
⊢ ¬AEStronglyMeasurable (fun x ↦ g (f x)) μ

Tactic:
exact fun hgf => hgm (hf.aestronglyMeasurable_map_iff.2 hgf)

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

Goal state before:
pp: case neg
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
⊢ 0 ≤ 0

Tactic:
le_rfl

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

Goal state before:
pp: α : Type u
β : Type v
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
f g : α →ᵇ β
C : ℝ
inst✝ : CompactSpace α
C0 : 0 < C
h : ¬Nonempty α
⊢ dist f g ≤ 0

Tactic:
dist_eq

Goal state after:
α : Type u
β : Type v
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
f g : α →ᵇ β
C : ℝ
inst✝ : CompactSpace α
C0 : 0 < C
h : ¬Nonempty α
⊢ sInf {C | 0 ≤ C ∧ ∀ (x : α), dist (f x) (g x) ≤ C} ≤ 0
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case neg
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 : 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 : α → G'
hf : 0 ≤ᵐ[μ] f
hG'' : CompleteSpace G''
hfi : ¬Integrable f μ
⊢ 0 ≤ 0

Tactic:
le_rfl

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

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
R : Type u_4
inst✝¹ : NonUnitalSeminormedRing R
inst✝ : IsCancelMulZero R
f g : C(α, R)
h : f * g = 0
x : α
⊢ (mkOfCompact f * mkOfCompact g) x = 0 x

Tactic:
simpa using! congr($h x)

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

Goal state before:
pp: F : Type u_1
inst✝² : NormedRing F
inst✝¹ : NormOneClass F
inst✝ : NormMulClass F
k : ℕ
r : F
hr : ‖r‖ < 1
u : ℕ → F
hu : u =O[atTop] fun n ↦ ↑(n ^ k)
r' : ℝ
hrr' : ‖r‖ < r'
h : r' < 1
⊢ (fun n ↦ ‖u n‖ * ‖r‖ ^ n) =O[atTop] fun n ↦ ‖↑n ^ k‖ * ‖r‖ ^ n

Tactic:
by
      simpa [Nat.cast_pow] using
      (isBigO_norm_left.mpr (isBigO_norm_right.mpr hu)).mul (isBigO_refl (fun n ↦ (‖r‖ ^ n)) atTop)

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

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
E : Set α
x : α
⊢ thickenedIndicatorAux δ E x ≤ 1

Tactic:
apply tsub_le_self (α := ℝ≥0∞)

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

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δseq : ℕ → ℝ
δseq_pos : ∀ (n : ℕ), 0 < δseq n
δseq_lim : Tendsto δseq atTop (𝓝 0)
E : Set α
⊢ Tendsto (fun n ↦ ⇑(thickenedIndicator ⋯ E)) atTop
    (𝓝 ((closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).indicator fun x ↦ 1))

Tactic:
have key := thickenedIndicatorAux_tendsto_indicator_closure δseq_lim E

Goal state after:
α : Type u_1
inst✝ : PseudoEMetricSpace α
δseq : ℕ → ℝ
δseq_pos : ∀ (n : ℕ), 0 < δseq n
δseq_lim : Tendsto δseq atTop (𝓝 0)
E : Set α
key :
  Tendsto (fun n ↦ thickenedIndicatorAux (δseq n) E) atTop
    (𝓝 ((closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).indicator fun x ↦ 1))
⊢ Tendsto (fun n ↦ ⇑(thickenedIndicator ⋯ E)) atTop
    (𝓝 ((closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).indicator fun x ↦ 1))
Module: Mathlib.MeasureTheory.Integral.Marginal

Goal state before:
pp: case inl
δ : Type u_1
X : δ → Type u_3
inst✝³ : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝² : DecidableEq δ
inst✝¹ : ∀ (i : δ), SigmaFinite (μ i)
inst✝ : Fintype δ
s : Finset δ
f g : ((i : δ) → X i) → ℝ≥0∞
hf : Measurable f
hg : Measurable g
hfg : ∫⋯∫⁻_s, f ∂μ ≤ ∫⋯∫⁻_s, g ∂μ
h : IsEmpty ((i : δ) → X i)
⊢ 0 ≤ 0

Tactic:
le_rfl

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
f : StieltjesFunction R
x : R
hx : x ∈ {x | leftLim (↑f) x ≠ ↑f x}
h'x : ContinuousAt (↑f) x
⊢ leftLim (↑f) x = ↑f x

Tactic:
exact (Monotone.continuousWithinAt_Iio_iff_leftLim_eq f.mono).1 h'x.continuousWithinAt

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✝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
hfi : IntegrableOn f (⋃ n, s n) μ
hne : atTop.NeBot
⊢ Tendsto (fun i ↦ ∫ (x : X) in s i, f x ∂μ) atTop (𝓝 (∫ (x : X) in ⋃ n, s n, f x ∂μ))

Tactic:
have := (atTop_neBot_iff.mp hne).2

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
hfi : IntegrableOn f (⋃ n, s n) μ
hne : atTop.NeBot
this : IsDirectedOrder ι
⊢ Tendsto (fun i ↦ ∫ (x : X) in s i, f x ∂μ) atTop (𝓝 (∫ (x : X) in ⋃ n, s n, f x ∂μ))
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), MeasurableSet (s i)
h_anti : Antitone s
hfi : ∃ i, IntegrableOn f (s i) μ
hne : atTop.NeBot
⊢ Tendsto (fun i ↦ ∫ (x : X) in s i, f x ∂μ) atTop (𝓝 (∫ (x : X) in ⋂ n, s n, f x ∂μ))

Tactic:
have := (atTop_neBot_iff.mp hne).2

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), MeasurableSet (s i)
h_anti : Antitone s
hfi : ∃ i, IntegrableOn f (s i) μ
hne : atTop.NeBot
this : IsDirectedOrder ι
⊢ Tendsto (fun i ↦ ∫ (x : X) in s i, f x ∂μ) atTop (𝓝 (∫ (x : X) in ⋂ n, s n, f x ∂μ))
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : R1Space G
S : MeasurableSpace G
inst✝¹ : BorelSpace G
inst✝ : WeaklyLocallyCompactSpace G
this : IsFiniteMeasureOnCompacts μ.measure
U : Set G
hU : IsOpen U
r : ℝ≥0∞
hr : r < μ.innerContent { carrier := U, is_open' := hU }
⊢ ∃ K ⊆ U, IsCompact K ∧ r < μ.measure K

Tactic:
simp only [innerContent, lt_iSup_iff] at hr

Goal state after:
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : R1Space G
S : MeasurableSpace G
inst✝¹ : BorelSpace G
inst✝ : WeaklyLocallyCompactSpace G
this : IsFiniteMeasureOnCompacts μ.measure
U : Set G
hU : IsOpen U
r : ℝ≥0∞
hr : ∃ i, ∃ (_ : ↑i ⊆ ↑{ carrier := U, is_open' := hU }), r < μ i
⊢ ∃ K ⊆ U, IsCompact K ∧ r < μ.measure K
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

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

Tactic:
by
      simp only [smul_set_inter, inter_comm, smul_inv_smul, aestronglyMeasurable_sum_measure_iff]

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

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

Tactic:
abs_eq_self.mpr _

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

X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
hfi : Integrable f μ
h_meas : NullMeasurableSet {x | 0 ≤ f x} μ
x : X
hx : x ∈ {x | 0 ≤ f x}
⊢ 0 ≤ f x
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝⁵ : Group G
inst✝⁴ : MulAction G α
inst✝³ : MeasurableSpace α
s : Set α
μ : Measure α
inst✝² : MeasurableConstSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
hs : IsFundamentalDomain G s μ
f : α → ℝ≥0∞
hf : ∀ (γ : G) (x : α), f (γ • x) = f x
⊢ essSup f (μ.restrict s) = essSup f μ

Tactic:
refine le_antisymm (essSup_mono_measure' Measure.restrict_le_self) ?_
  rw [essSup_eq_sInf (μ.restrict s) f, essSup_eq_sInf μ f]
  refine sInf_le_sInf ?_
  rintro a (ha : (μ.restrict s) {x : α | a < f x} = 0)
  rw [Measure.restrict_apply₀' hs.nullMeasurableSet] at ha
  refine measure_zero_of_invariant hs _ ?_ ha
  intro γ
  ext x
  rw [mem_smul_set_iff_inv_smul_mem]
  simp only [mem_ofPred_eq, hf γ⁻¹ x]

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

Goal state before:
pp: case left.refine_2
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : (interior V).Nonempty
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V
h2s : s.card = index (K₁.carrier ∪ K₂.carrier) V
this : ∀ K ⊆ ⋃ g ∈ s, (fun x ↦ g * x) ⁻¹' V, index K V ≤ {g ∈ s | ((fun x ↦ g * x) ⁻¹' V ∩ K).Nonempty}.card
g₁ : G
a✝ : g₁ ∈ s
g₂ : G
h2g₂ : g₂ ∈ K₁.carrier
g₃ : G
h2g₃ : g₃ ∈ K₂.carrier
h1g₃ : g₁ * g₃ ∈ V
h1g₂ : g₁ * g₂ ∈ V
⊢ g₂ * (g₁ * g₂)⁻¹ = g₁⁻¹

Tactic:
· simp only [mul_inv_rev, mul_inv_cancel_left]

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

Goal state before:
pp: case right.refine_2
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : (interior V).Nonempty
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V
h2s : s.card = index (K₁.carrier ∪ K₂.carrier) V
this : ∀ K ⊆ ⋃ g ∈ s, (fun x ↦ g * x) ⁻¹' V, index K V ≤ {g ∈ s | ((fun x ↦ g * x) ⁻¹' V ∩ K).Nonempty}.card
g₁ : G
a✝ : g₁ ∈ s
g₂ : G
h2g₂ : g₂ ∈ K₁.carrier
g₃ : G
h2g₃ : g₃ ∈ K₂.carrier
h1g₃ : g₁ * g₃ ∈ V
h1g₂ : g₁ * g₂ ∈ V
⊢ g₃ * (g₁ * g₃)⁻¹ = g₁⁻¹

Tactic:
· simp only [mul_inv_rev, mul_inv_cancel_left]

Goal state after:
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
⊢ ↑(normSq x) = ⟪x, x⟫

Tactic:
ext_iff

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
⊢ re ↑(normSq x) = re ⟪x, x⟫ ∧ im ↑(normSq x) = im ⟪x, x⟫
Module: Mathlib.Analysis.InnerProductSpace.Defs

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

Tactic:
inner_re_symm

Goal state after:
𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
t : ℝ
⊢ normSq x * t * t + re ⟪y, x⟫ * t + re ⟪y, x⟫ * t + normSq y =
    normSq x * t * t + re ⟪y, x⟫ * t + re ⟪y, x⟫ * t + normSq y
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
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := ⋯
this : Continuous eval
⊢ IsClosed (Ici 0)

Tactic:
exact isClosed_Ici

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✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure X
s : Set X
f : ↥(Lp E p μ)
⊢ eLpNorm (↑↑f) p (μ.restrict s) ≤ eLpNorm (↑↑f) p μ

Tactic:
exact eLpNorm_mono_measure _ Measure.restrict_le_self

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
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
eval : (Compacts G → ℝ) → ℝ := ⋯
this : Continuous eval
⊢ IsClosed (Ici 0)

Tactic:
exact isClosed_Ici

Goal state after: