mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case refine_2
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
u : α
⊢ MeasurableSet (Iic u)

Tactic:
← compl_Ioi

Goal state after:
case refine_2
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
u : α
⊢ MeasurableSet (Ioi u)ᶜ
Module: Mathlib.Topology.MetricSpace.HausdorffDistance

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
ι : Sort u_2
s t : ι → Set α
⊢ ⨆ x ∈ ⋃ i, s i, infEDist x (⋃ i, t i) ≤ ⨆ i, max (⨆ x ∈ s i, infEDist x (t i)) (⨆ y ∈ t i, infEDist y (s i)) ∧
    ⨆ y ∈ ⋃ i, t i, infEDist y (⋃ i, s i) ≤ ⨆ i, max (⨆ x ∈ s i, infEDist x (t i)) (⨆ y ∈ t i, infEDist y (s i))

Tactic:
iSup_iUnion,

Goal state after:
α : Type u
inst✝ : PseudoEMetricSpace α
ι : Sort u_2
s t : ι → Set α
⊢ ⨆ i, ⨆ a ∈ s i, infEDist a (⋃ i, t i) ≤ ⨆ i, max (⨆ x ∈ s i, infEDist x (t i)) (⨆ y ∈ t i, infEDist y (s i)) ∧
    ⨆ i, ⨆ a ∈ t i, infEDist a (⋃ i, s i) ≤ ⨆ i, max (⨆ x ∈ s i, infEDist x (t i)) (⨆ y ∈ t i, infEDist y (s i))
Module: Mathlib.Algebra.Order.Monoid.Canonical.Basic

Goal state before:
pp: α : Type u_1
inst✝⁴ : LinearOrder α
P : α → Prop
b c : α
inst✝³ : Add α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : AddLeftMono α
inst✝ : IsLeftCancelAdd α
⊢ (∀ a ≤ b + c, P a) ↔ (∀ a < b, P a) ∧ ∀ d ≤ c, P (b + d)

Tactic:
simp_rw [le_add_iff_lt_left_or_exists_le]
  aesop

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.Canonical.Basic

Goal state before:
pp: α : Type u_1
inst✝⁴ : LinearOrder α
P : α → Prop
b c : α
inst✝³ : Add α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : AddLeftMono α
inst✝ : IsLeftCancelAdd α
⊢ (∃ a ≤ b + c, P a) ↔ (∃ a < b, P a) ∨ ∃ d ≤ c, P (b + d)

Tactic:
simp_rw [le_add_iff_lt_left_or_exists_le]
  aesop

Goal state after:
Module: Mathlib.MeasureTheory.Function.SimpleFunc

Goal state before:
pp: case refine_3
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α →ₛ ℝ≥0∞
s : Finset ℝ≥0∞
hs : ∀ (x : α), f x ≠ 0 → μ (⇑f ⁻¹' {f x}) ≠ 0 → f x ∈ s
b : ℝ≥0∞
a✝ : b ∈ s
hb : b * μ (⇑f ⁻¹' {b}) ≠ 0
⊢ b ∈ f.range

Tactic:
rw [mem_range, ← preimage_singleton_nonempty]

Goal state after:
case refine_3
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α →ₛ ℝ≥0∞
s : Finset ℝ≥0∞
hs : ∀ (x : α), f x ≠ 0 → μ (⇑f ⁻¹' {f x}) ≠ 0 → f x ∈ s
b : ℝ≥0∞
a✝ : b ∈ s
hb : b * μ (⇑f ⁻¹' {b}) ≠ 0
⊢ (⇑f ⁻¹' {b}).Nonempty
Module: Mathlib.Topology.Instances.Real.Lemmas

Goal state before:
pp: case refine_1
s : Set ℝ
conn : s.OrdConnected
nt : s.Nontrivial
x : ℝ
hx : x ∈ s
ε : ℝ
ε_pos : ε > 0
z : ℝ
hz : z ∈ s
ne : z ≠ x
lt : z < x
q : ℚ
h₁ : max z (x - ε) < ↑q
h₂ : ↑q < x
⊢ ∃ y ∈ s ∩ range Rat.cast, |y - x| < ε

Tactic:
rw [max_lt_iff] at h₁

Goal state after:
case refine_1
s : Set ℝ
conn : s.OrdConnected
nt : s.Nontrivial
x : ℝ
hx : x ∈ s
ε : ℝ
ε_pos : ε > 0
z : ℝ
hz : z ∈ s
ne : z ≠ x
lt : z < x
q : ℚ
h₁ : z < ↑q ∧ x - ε < ↑q
h₂ : ↑q < x
⊢ ∃ y ∈ s ∩ range Rat.cast, |y - x| < ε
Module: Mathlib.Topology.IndicatorConstPointwise

Goal state before:
pp: case neg
ι : Type u_3
L : Filter ι
β : Type u_4
p : ι → Prop
inst✝¹ : DecidablePred p
q : Prop
inst✝ : Decidable q
a b : β
F G : Filter β
haG : {a}ᶜ ∈ G
hbF : {b}ᶜ ∈ F
haF : 𝓟 {a} ≤ F
hbG : 𝓟 {b} ≤ G
h : ∀ᶠ (i : ι) in L, p i ↔ q
obs : (fun x ↦ if q then a else b) =ᶠ[L] fun i ↦ if p i then a else b
hq : ¬q
⊢ Tendsto (fun x ↦ if q then a else b) L (if q then F else G)

Tactic:
· simp only [hq, ite_false]
      apply le_trans _ hbG
      simp only [principal_singleton, le_pure_iff, mem_map, Set.mem_singleton_iff,
        Set.preimage_const_of_mem, univ_mem]

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Real

Goal state before:
pp: case refine_2
q : ℚ
⊢ MeasurableSet (Iic ↑q)

Tactic:
← compl_Ioi

Goal state after:
case refine_2
q : ℚ
⊢ MeasurableSet (Ioi ↑q)ᶜ
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Real

Goal state before:
pp: case refine_2
α : Type u_1
mα : MeasurableSpace α
ι : Type u_5
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
μ : Measure α
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
hf : ∀ (i : ι), AEMeasurable (f i) μ
hlim : ∀ᵐ (a : α) ∂μ, Tendsto (fun i ↦ f i a) u (𝓝 (g a))
v : ℕ → ι
hv : Tendsto v atTop u
h'f : ∀ (n : ℕ), AEMeasurable (f (v n)) μ
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' ↦ Tendsto f' atTop (𝓝 (g x))
hp : ∀ᵐ (x : α) ∂μ, p x fun n ↦ f (v n) x
aeSeqLim : α → ℝ≥0∞ := fun x ↦ if x ∈ aeSeqSet h'f p then g x else ⋯.some
⊢ g =ᵐ[μ] aeSeqLim

Tactic:
exact (ite_ae_eq_of_measure_compl_zero g (fun x ↦ (⟨f (v 0) x⟩ : Nonempty ℝ≥0∞).some)
      (aeSeqSet h'f p) (aeSeq.measure_compl_aeSeqSet_eq_zero h'f hp)).symm

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_5
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
⊢ μ s = μ (s ∩ univ)

Tactic:
inter_univ

Goal state after:
α : Type u_5
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
⊢ μ s = μ s
Module: Mathlib.MeasureTheory.Function.SimpleFuncDense

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : PseudoEMetricSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : MeasurableSpace β
f : β → α
hf : Measurable f
s : Set α
y₀ : α
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
x : β
m n : ℕ
h : m ≤ n
⊢ edist ((approxOn f hf s y₀ h₀ n) x) (f x) ≤ edist ((approxOn f hf s y₀ h₀ m) x) (f x)

Tactic:
dsimp only [approxOn, coe_comp, Function.comp_def]

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : PseudoEMetricSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : MeasurableSpace β
f : β → α
hf : Measurable f
s : Set α
y₀ : α
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
x : β
m n : ℕ
h : m ≤ n
⊢ edist ((nearestPt (fun k ↦ Nat.rec y₀ (fun n n_ih ↦ ↑(denseSeq (↑s) n)) k) n) (f x)) (f x) ≤
    edist ((nearestPt (fun k ↦ Nat.rec y₀ (fun n n_ih ↦ ↑(denseSeq (↑s) n)) k) m) (f x)) (f x)
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_5
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : μ s = μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' Ioi 0)
B : μ (s ∩ f ⁻¹' Ioi 0) = μ (s ∩ f ⁻¹' {∞}) + μ (s ∩ f ⁻¹' Ioo 0 ∞)
C : μ (s ∩ f ⁻¹' Ioo 0 ∞) = ∑' (n : ℤ), μ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))
⊢ μ (s ∩ f ⁻¹' {0}) + (μ (s ∩ f ⁻¹' {∞}) + ∑' (n : ℤ), μ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))) =
    μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' {∞}) + ∑' (n : ℤ), μ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))

Tactic:
add_assoc

Goal state after:
α : Type u_5
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : μ s = μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' Ioi 0)
B : μ (s ∩ f ⁻¹' Ioi 0) = μ (s ∩ f ⁻¹' {∞}) + μ (s ∩ f ⁻¹' Ioo 0 ∞)
C : μ (s ∩ f ⁻¹' Ioo 0 ∞) = ∑' (n : ℤ), μ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))
⊢ μ (s ∩ f ⁻¹' {0}) + (μ (s ∩ f ⁻¹' {∞}) + ∑' (n : ℤ), μ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))) =
    μ (s ∩ f ⁻¹' {0}) + (μ (s ∩ f ⁻¹' {∞}) + ∑' (n : ℤ), μ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))))
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: case inr.refine_1
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
h₀ : ∫⁻ (a : α), f a ∂μ ≠ 0
L : ℕ → ℝ≥0∞
left✝ : StrictMono L
hLf : ∀ (n : ℕ), L n ∈ Ioo ⊥ (∫⁻ (a : α), f a ∂μ)
hL_tendsto : Tendsto L atTop (𝓝 (∫⁻ (a : α), f a ∂μ))
g : ℕ → α → ℝ≥0∞
hgm : ∀ (n : ℕ), Measurable (g n)
hgf : ∀ (n : ℕ), g n ≤ f
hLg : ∀ (n : ℕ), L n < ∫⁻ (a : α), g n a ∂μ
⊢ ∫⁻ (a : α), f a ∂μ ≤ ∫⁻ (a : α), (fun x ↦ ⨆ n, g n x) a ∂μ

pp: case inr.refine_2
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
h₀ : ∫⁻ (a : α), f a ∂μ ≠ 0
L : ℕ → ℝ≥0∞
left✝ : StrictMono L
hLf : ∀ (n : ℕ), L n ∈ Ioo ⊥ (∫⁻ (a : α), f a ∂μ)
hL_tendsto : Tendsto L atTop (𝓝 (∫⁻ (a : α), f a ∂μ))
g : ℕ → α → ℝ≥0∞
hgm : ∀ (n : ℕ), Measurable (g n)
hgf : ∀ (n : ℕ), g n ≤ f
hLg : ∀ (n : ℕ), L n < ∫⁻ (a : α), g n a ∂μ
⊢ ∫⁻ (a : α), (fun x ↦ ⨆ n, g n x) a ∂μ ≤ ∫⁻ (a : α), f a ∂μ

Tactic:
· refine le_of_tendsto' hL_tendsto fun n => (hLg n).le.trans <| lintegral_mono fun x => ?_
    exact le_iSup (fun n => g n x) n

Goal state after:
case inr.refine_2
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
h₀ : ∫⁻ (a : α), f a ∂μ ≠ 0
L : ℕ → ℝ≥0∞
left✝ : StrictMono L
hLf : ∀ (n : ℕ), L n ∈ Ioo ⊥ (∫⁻ (a : α), f a ∂μ)
hL_tendsto : Tendsto L atTop (𝓝 (∫⁻ (a : α), f a ∂μ))
g : ℕ → α → ℝ≥0∞
hgm : ∀ (n : ℕ), Measurable (g n)
hgf : ∀ (n : ℕ), g n ≤ f
hLg : ∀ (n : ℕ), L n < ∫⁻ (a : α), g n a ∂μ
⊢ ∫⁻ (a : α), (fun x ↦ ⨆ n, g n x) a ∂μ ≤ ∫⁻ (a : α), f a ∂μ
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: case pos
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : NormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
hf : StronglyMeasurable f
c : ℝ
x : α
hfx : ‖f x‖ ≤ c
h_tendsto : Tendsto (fun n ↦ (hf.approx n) x) atTop (𝓝 (f x))
hfx0 : ‖f x‖ = 0
⊢ Tendsto (fun n ↦ min 1 (c / ‖(hf.approx n) x‖) • (hf.approx n) x) atTop (𝓝 (f x))

Tactic:
norm_eq_zero

Goal state after:
case pos
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : NormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
hf : StronglyMeasurable f
c : ℝ
x : α
hfx : ‖f x‖ ≤ c
h_tendsto : Tendsto (fun n ↦ (hf.approx n) x) atTop (𝓝 (f x))
hfx0 : f x = 0
⊢ Tendsto (fun n ↦ min 1 (c / ‖(hf.approx n) x‖) • (hf.approx n) x) atTop (𝓝 (f x))
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: α : Type u_1
m✝ : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), AEMeasurable (f n) μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n ↦ f n x
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' ↦ Monotone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
n m : ℕ
⊢ n ≤ m → aeSeq hf p n ≤ aeSeq hf p m

Tactic:
hnm

Goal state after:
α : Type u_1
m✝ : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), AEMeasurable (f n) μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n ↦ f n x
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' ↦ Monotone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
n m : ℕ
hnm : n ≤ m
⊢ aeSeq hf p n ≤ aeSeq hf p m
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
h : ∫⁻ (x : α), f x ∂μ ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
ε₂ : ℝ≥0∞
hε₂0 : 0 < ε₂
hε₂ε : ε₂ < ε
ε₁ : ℝ≥0∞
hε₁0 : 0 < ε₁
hε₁₂ : ε₁ < ε₂
φ : α →ₛ ℝ≥0
left✝ : ∀ (x : α), ↑(φ x) ≤ f x
hφ : ∀ (ψ : α →ₛ ℝ≥0), (∀ (x : α), ↑(ψ x) ≤ f x) → (SimpleFunc.map ofNNReal (ψ - φ)).lintegral μ < ε₁
C : ℝ≥0
hC : ∀ (x : α), φ x ≤ C
s : Set α
hs : μ s < (ε₂ - ε₁) / ↑C
ψ : α →ₛ ℝ≥0
hψ : ∀ (x : α), ↑(ψ x) ≤ f x
x : α
⊢ (SimpleFunc.map ofNNReal ψ) x ≤ (SimpleFunc.map ofNNReal (φ + (ψ - φ))) x

Tactic:
simp only [add_tsub_eq_max, le_max_right, coe_map, Function.comp_apply, SimpleFunc.coe_add,
        SimpleFunc.coe_sub, Pi.add_apply, Pi.sub_apply, ENNReal.coe_max (φ x) (ψ x)]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Sub

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i ↦ f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ∞
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' ↦ Antitone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
n m : ℕ
⊢ n ≤ m → aeSeq h_meas p m ≤ aeSeq h_meas p n

Tactic:
hnm

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i ↦ f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ∞
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' ↦ Antitone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
n m : ℕ
hnm : n ≤ m
⊢ aeSeq h_meas p m ≤ aeSeq h_meas p n
Module: Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence

Goal state before:
pp: case refine_2
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
g : α → ℝ≥0∞
hf_meas : ∀ (n : ℕ), Measurable (f n)
h_bound : ∀ (n : ℕ), f n ≤ᵐ[μ] g
h_fin : ∫⁻ (a : α), g a ∂μ ≠ ∞
n m : ℕ
⊢ n ≤ m → (fun n a ↦ ⨆ i, ⨆ (_ : i ≥ n), f i a) m ≤ (fun n a ↦ ⨆ i, ⨆ (_ : i ≥ n), f i a) n

Tactic:
hnm

Goal state after:
case refine_2
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
g : α → ℝ≥0∞
hf_meas : ∀ (n : ℕ), Measurable (f n)
h_bound : ∀ (n : ℕ), f n ≤ᵐ[μ] g
h_fin : ∫⁻ (a : α), g a ∂μ ≠ ∞
n m : ℕ
hnm : n ≤ m
⊢ (fun n a ↦ ⨆ i, ⨆ (_ : i ≥ n), f i a) m ≤ (fun n a ↦ ⨆ i, ⨆ (_ : i ≥ n), f i a) n
Module: Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
ι : Type u_2
l : Filter ι
inst✝ : l.IsCountablyGenerated
F : ι → α → ℝ≥0∞
f bound : α → ℝ≥0∞
hF_meas : ∀ᶠ (n : ι) in l, Measurable (F n)
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a
h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ∞
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F n a) l (𝓝 (f a))
⊢ ∀ᶠ (n : ι) in l, AEMeasurable (F n) μ

Tactic:
filter_upwards [hF_meas] using by fun_prop

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion

Goal state before:
pp: case neg
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : SFinite ν
hμν : μ ≪ ν
s : Set α
a✝ : MeasurableSet s
hνs : ¬ν (s ∩ (μ.sigmaFiniteSetWRT ν)ᶜ) = 0
⊢ μ (s ∩ (μ.sigmaFiniteSetWRT ν)ᶜ) = ∞ * ν (s ∩ (μ.sigmaFiniteSetWRT ν)ᶜ)

Tactic:
rw [ENNReal.top_mul hνs, measure_eq_top_of_subset_compl_sigmaFiniteSetWRT
      Set.inter_subset_right hνs]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
s : Set α
⊢ (μ.restrict μ.sigmaFiniteSetᶜ) s = 0 ∨ (μ.restrict μ.sigmaFiniteSetᶜ) s = ∞

Tactic:
rw [Measure.restrict_apply' measurableSet_sigmaFiniteSet.compl]

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
s : Set α
⊢ μ (s ∩ μ.sigmaFiniteSetᶜ) = 0 ∨ μ (s ∩ μ.sigmaFiniteSetᶜ) = ∞
Module: Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence

Goal state before:
pp: case a
α : Type u_2
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable 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
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ∞
g : ℕ → α → ℝ≥0∞
gmeas : ∀ (n : ℕ), Measurable (g n)
gf : ∀ (n : ℕ), g n ≤ f n
hg : ∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g n a ∂μ
g' : ℕ → α → ℝ≥0∞ := fun t ↦ Nat.rec (motive := fun x ↦ α → ℝ≥0∞) (g 0) (fun n I x ↦ max (g (n + 1) x) (I x)) t
M : ∀ (n : ℕ), Measurable (g' n)
I : ∀ (n : ℕ) (x : α), g n x ≤ g' n x
I' : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), g' n x ≤ f n x
n : ℕ
⊢ ∫⁻ (a : α), g n a ∂μ ≤ ∫⁻ (x : α), g' n x ∂μ

Tactic:
exact lintegral_mono (I n)

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: x : ℝ
⊢ σ (projIcc 0 1 ⋯ x) = projIcc 0 1 ⋯ (1 - x)

Tactic:
ext
  rcases le_total x 0 with h₀ | h₀
  · simp [projIcc_of_le_left, projIcc_of_right_le, h₀]
  · rcases le_total x 1 with h₁ | h₁
    · lift x to I using ⟨h₀, h₁⟩
      simp_rw [← coe_symm_eq, projIcc_val]
    · simp [projIcc_of_le_left, projIcc_of_right_le, h₁]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence

Goal state before:
pp: case inl
α : Type u_2
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hf_meas : ∀ (n : ℕ), AEMeasurable (f n) 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
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), F a ≤ f i a
h0 : ∫⁻ (a : α), f 0 a ∂μ ≠ ∞
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ∞
a : α
h_mono : Antitone fun i ↦ f i a
h : Tendsto (fun i ↦ f i a) atTop atBot
⊢ atBot ≤ 𝓝 0

Tactic:
OrderBot.atBot_eq

Goal state after:
case inl
α : Type u_2
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hf_meas : ∀ (n : ℕ), AEMeasurable (f n) 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
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), F a ≤ f i a
h0 : ∫⁻ (a : α), f 0 a ∂μ ≠ ∞
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ∞
a : α
h_mono : Antitone fun i ↦ f i a
h : Tendsto (fun i ↦ f i a) atTop atBot
⊢ pure ⊥ ≤ 𝓝 0
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: X : Type u
Y : Type v
Z : Type w
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : Nonempty Z
Φ : Z → X
Ψ : Z → Y
ε : ℝ
H : ∀ (p q : Z), |dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)| ≤ 2 * ε
x : X
y : Y
z : X
⊢ dist x z ≤ ((⨅ p, dist x (Φ p) + dist y (Ψ p)) + ⨅ p, dist z (Φ p) + dist y (Ψ p)) + (ε + ε)

Tactic:
add_assoc

Goal state after:
X : Type u
Y : Type v
Z : Type w
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : Nonempty Z
Φ : Z → X
Ψ : Z → Y
ε : ℝ
H : ∀ (p q : Z), |dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)| ≤ 2 * ε
x : X
y : Y
z : X
⊢ dist x z ≤ (⨅ p, dist x (Φ p) + dist y (Ψ p)) + ((⨅ p, dist z (Φ p) + dist y (Ψ p)) + (ε + ε))
Module: Mathlib.Topology.MetricSpace.Polish

Goal state before:
pp: case refine_4.refine_2
α : Type u_1
β : Type u_2
inst✝ : MetricSpace α
s : Opens α
t : Set s.CompleteCopy
h : ∀ x ∈ t, ∃ ε > 0, ∀ (y : s.CompleteCopy), dist x y < ε → y ∈ t
x : s.CompleteCopy
hx : x ∈ t
⊢ t ∈ 𝓝 x

Tactic:
rcases h x hx with ⟨ε, ε0, hε⟩

Goal state after:
case refine_4.refine_2
α : Type u_1
β : Type u_2
inst✝ : MetricSpace α
s : Opens α
t : Set s.CompleteCopy
h : ∀ x ∈ t, ∃ ε > 0, ∀ (y : s.CompleteCopy), dist x y < ε → y ∈ t
x : s.CompleteCopy
hx : x ∈ t
ε : ℝ
ε0 : ε > 0
hε : ∀ (y : s.CompleteCopy), dist x y < ε → y ∈ t
⊢ t ∈ 𝓝 x
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: case inr.inl
X : Type u
Y : Type v
Z : Type w
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
p : Y
q : X
h : Sum.dist (Sum.inr p) (Sum.inl q) = 0
⊢ Sum.inr p = Sum.inl q

pp: case inr.inr
X : Type u
Y : Type v
Z : Type w
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
p q : Y
h : Sum.dist (Sum.inr p) (Sum.inr q) = 0
⊢ Sum.inr p = Sum.inr q

Tactic:
· exact eq_of_glueDist_eq_zero _ _ _ one_pos _ _ ((Sum.dist_eq_glueDist q p).symm.trans h)

Goal state after:
case inr.inr
X : Type u
Y : Type v
Z : Type w
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
p q : Y
h : Sum.dist (Sum.inr p) (Sum.inr q) = 0
⊢ Sum.inr p = Sum.inr q
Module: Mathlib.Topology.MetricSpace.Polish

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : MetricSpace α
s : Opens α
t : Set s.CompleteCopy
h : ∀ x ∈ t, ∃ ε > 0, ∀ (y : s.CompleteCopy), dist x y < ε → y ∈ t
x : s.CompleteCopy
hx : x ∈ t
ε : ℝ
ε0 : ε > 0
hε : ∀ (y : s.CompleteCopy), dist ↑x ↑y + |(infDist (↑x) (↑s)ᶜ)⁻¹ - (infDist (↑y) (↑s)ᶜ)⁻¹| < ε → y ∈ t
⊢ ↑x ∉ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (↑s)ᶜ

Tactic:
rw [s.isOpen.isClosed_compl.closure_eq, mem_compl_iff, not_not]

Goal state after:
α : Type u_1
β : Type u_2
inst✝ : MetricSpace α
s : Opens α
t : Set s.CompleteCopy
h : ∀ x ∈ t, ∃ ε > 0, ∀ (y : s.CompleteCopy), dist x y < ε → y ∈ t
x : s.CompleteCopy
hx : x ∈ t
ε : ℝ
ε0 : ε > 0
hε : ∀ (y : s.CompleteCopy), dist ↑x ↑y + |(infDist (↑x) (↑s)ᶜ)⁻¹ - (infDist (↑y) (↑s)ᶜ)⁻¹| < ε → y ∈ t
⊢ ↑x ∈ ↑s
Module: Mathlib.Topology.MetricSpace.CantorScheme

Goal state before:
pp: β : Type u_1
α : Type u_2
A : List β → Set α
inst✝¹ : PseudoMetricSpace α
inst✝ : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : ℕ → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
this : Antitone fun n ↦ A (res x n)
n m : ℕ
⊢ n ≤ m → u m ∈ A (res x n)

Tactic:
hnm

Goal state after:
β : Type u_1
α : Type u_2
A : List β → Set α
inst✝¹ : PseudoMetricSpace α
inst✝ : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : ℕ → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
this : Antitone fun n ↦ A (res x n)
n m : ℕ
hnm : n ≤ m
⊢ u m ∈ A (res x n)
Module: Mathlib.Topology.MetricSpace.Polish

Goal state before:
pp: α : Type u_1
t : TopologicalSpace α
inst✝ : PolishSpace α
s : ℕ → Set α
m : ℕ → TopologicalSpace α
mt : ∀ (n : ℕ), m n ≤ t
m_polish : ∀ (n : ℕ), PolishSpace α
h✝ : ∀ (n : ℕ), IsClosed[m n] (s n)
m_open : ∀ (n : ℕ), IsOpen[m n] (s n)
t' : TopologicalSpace α
t'm : ∀ (n : ℕ), t' ≤ m n
t'_polish : PolishSpace α
A : IsOpen[t'] (⋃ n, s n)
⊢ IsClopenable (⋃ n, s n)

Tactic:
obtain ⟨t'', t''_le, t''_polish, h1, h2⟩ : ∃ t'' : TopologicalSpace α,
      t'' ≤ t' ∧ @PolishSpace α t'' ∧ IsClosed[t''] (⋃ n, s n) ∧ IsOpen[t''] (⋃ n, s n) :=
    @IsOpen.isClopenable α t' t'_polish _ A

Goal state after:
α : Type u_1
t : TopologicalSpace α
inst✝ : PolishSpace α
s : ℕ → Set α
m : ℕ → TopologicalSpace α
mt : ∀ (n : ℕ), m n ≤ t
m_polish : ∀ (n : ℕ), PolishSpace α
h✝ : ∀ (n : ℕ), IsClosed[m n] (s n)
m_open : ∀ (n : ℕ), IsOpen[m n] (s n)
t' : TopologicalSpace α
t'm : ∀ (n : ℕ), t' ≤ m n
t'_polish : PolishSpace α
A : IsOpen[t'] (⋃ n, s n)
t'' : TopologicalSpace α
t''_le : t'' ≤ t'
t''_polish : PolishSpace α
h1 : IsClosed[t''] (⋃ n, s n)
h2 : IsOpen[t''] (⋃ n, s n)
⊢ IsClopenable (⋃ n, s n)
Module: Mathlib.Topology.MetricSpace.Perfect

Goal state before:
pp: case false.true
α : Type u_1
inst✝¹ : MetricSpace α
C : Set α
hC : Perfect C
hnonempty : C.Nonempty
inst✝ : CompleteSpace α
u : ℕ → ℝ≥0∞
upos' : ∀ (n : ℕ), u n ∈ Ioo 0 1
hu : Tendsto u atTop (nhds 0)
upos : ∀ (n : ℕ), 0 < u n
P : Type (max 0 u_1) := { E // Perfect E ∧ E.Nonempty }
C0 C1 : {C : Set α} → Perfect C → C.Nonempty → {ε : ℝ≥0∞} → 0 < ε → Set α
h0 :
  ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
    Perfect (C0 hC hnonempty hε) ∧
      (C0 hC hnonempty hε).Nonempty ∧ C0 hC hnonempty hε ⊆ C ∧ ediam (C0 hC hnonempty hε) ≤ ε
h1 :
  ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
    Perfect (C1 hC hnonempty hε) ∧
      (C1 hC hnonempty hε).Nonempty ∧ C1 hC hnonempty hε ⊆ C ∧ ediam (C1 hC hnonempty hε) ≤ ε
hdisj :
  ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
    Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)
DP : List Bool → P :=
  fun l ↦
    List.rec ⟨C, ⋯⟩
      (fun a l ih ↦ Bool.casesOn (motive := fun t ↦ a = t → P) a (fun h ↦ ⟨C0 ⋯ ⋯ ⋯, ⋯⟩) (fun h ↦ ⟨C1 ⋯ ⋯ ⋯, ⋯⟩) ⋯) l
D : List Bool → Set α := fun l ↦ ↑(DP l)
hanti : ClosureAntitone D
hdiam : VanishingDiam D
l : List Bool
hab : false ≠ true
⊢ Disjoint (D (false :: l)) (D (true :: l))

Tactic:
exact hdisj _ _ _

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
h : Disjoint μ ν
ε : ℝ≥0
hε : 0 < ε
h₁ : sInf {m | ∃ t, m = μ (t ∩ univ) + ν (tᶜ ∩ univ)} = 0
⊢ ∃ s, μ s = 0 ∧ ν sᶜ ≤ 2 * ↑ε

Tactic:
inter_univ

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
h : Disjoint μ ν
ε : ℝ≥0
hε : 0 < ε
h₁ : sInf {m | ∃ t, m = μ t + ν tᶜ} = 0
⊢ ∃ s, μ s = 0 ∧ ν sᶜ ≤ 2 * ↑ε
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Probability

Goal state before:
pp: case inl
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsZeroOrProbabilityMeasure μ
h : μ univ = 0
⊢ μ = 0 ∨ IsProbabilityMeasure μ

Tactic:
apply Or.inl (measure_univ_eq_zero.mp h)

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Probability

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s : Set α
inst✝ : IsZeroOrProbabilityMeasure μ
p : α → Prop
f✝ : β → α
f : α → β
hf : ¬AEMeasurable f μ
⊢ IsZeroOrProbabilityMeasure (Measure.map f μ)

Tactic:
simp [isZeroOrProbabilityMeasure_iff, hf]

Goal state after: