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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
⊢ ∫⁻ (a : α), ⨆ b, f b a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ

Tactic:
← iSup_apply

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
⊢ ∫⁻ (a : α), (⨆ i, f i) a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: case e'_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' ↦ Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
h_ae_seq_directed : Directed LE.le (aeSeq hf p)
⊢ ∫⁻ (a : α), (⨆ i, f i) a ∂μ = ∫⁻ (a : α), ⨆ b, aeSeq hf p b a ∂?m.244

Tactic:
← iSup_apply

Goal state after:
case e'_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' ↦ Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
h_ae_seq_directed : Directed LE.le (aeSeq hf p)
⊢ ∫⁻ (a : α), (⨆ i, f i) a ∂μ = ∫⁻ (a : α), (⨆ i, aeSeq hf p i) a ∂?m.244
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: case h_mono
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
i j : ℕ
h : i ≤ j
a : α
⊢ (fun n ↦ ⇑(eapprox f n) + ⇑(eapprox g n)) i a ≤ (fun n ↦ ⇑(eapprox f n) + ⇑(eapprox g n)) j a

Tactic:
dsimp

Goal state after:
case h_mono
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
i j : ℕ
h : i ≤ j
a : α
⊢ (eapprox f i) a + (eapprox g i) a ≤ (eapprox f j) a + (eapprox g j) a
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: case h_mono
α : Type u_1
m : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hf : Measurable f
i j : ℕ
h : i ≤ j
a : α
⊢ (fun n ↦ ⇑(const α r * eapprox f n)) i a ≤ (fun n ↦ ⇑(const α r * eapprox f n)) j a

Tactic:
dsimp

Goal state after:
case h_mono
α : Type u_1
m : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hf : Measurable f
i j : ℕ
h : i ≤ j
a : α
⊢ r * (eapprox f i) a ≤ r * (eapprox f j) a
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
s : α →ₛ ℝ≥0∞
hs : ⇑s ≤ fun a ↦ f a
x : α
⊢ (const α r * s) x ≤ (fun a ↦ r * f a) x

Tactic:
dsimp

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
s : α →ₛ ℝ≥0∞
hs : ⇑s ≤ fun a ↦ f a
x : α
⊢ r * s x ≤ r * f x
Module: Mathlib.Analysis.Normed.Group.InfiniteSum

Goal state before:
pp: ι : Type u_1
ε : Type u_5
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddCommMonoid ε
f : ι → ε
g : ι → ℝ≥0∞
a : ε
b : ℝ≥0∞
hf : HasSum f a
hg : HasSum g b
h : ∀ (i : ι), ‖f i‖ₑ ≤ g i
⊢ ‖a‖ₑ ≤ b

Tactic:
by
  exact le_of_tendsto_of_tendsto' hf.enorm hg fun _s ↦ enorm_sum_le_of_le _ fun i _hi ↦ h i

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

Goal state before:
pp: ι : Type u_1
ε : Type u_5
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddCommMonoid ε
f : ι → ε
g : ι → ℝ≥0∞
a : ℝ≥0∞
hg : HasSum g a
h : ∀ (i : ι), ‖f i‖ₑ ≤ g i
⊢ ‖∑' (i : ι), f i‖ₑ ≤ a

Tactic:
by_cases hf : Summable f

Goal state after:
case pos
ι : Type u_1
ε : Type u_5
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddCommMonoid ε
f : ι → ε
g : ι → ℝ≥0∞
a : ℝ≥0∞
hg : HasSum g a
h : ∀ (i : ι), ‖f i‖ₑ ≤ g i
hf : Summable f
⊢ ‖∑' (i : ι), f i‖ₑ ≤ a

case neg
ι : Type u_1
ε : Type u_5
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddCommMonoid ε
f : ι → ε
g : ι → ℝ≥0∞
a : ℝ≥0∞
hg : HasSum g a
h : ∀ (i : ι), ‖f i‖ₑ ≤ g i
hf : ¬Summable f
⊢ ‖∑' (i : ι), f i‖ₑ ≤ a
Module: Mathlib.Analysis.Normed.Group.InfiniteSum

Goal state before:
pp: ι : Type u_1
E : Type u_3
inst✝ : SeminormedAddCommGroup E
f : ι → E
g : ι → ℝ
a : ℝ
hg : HasSum g a
h : ∀ (i : ι), ‖f i‖ ≤ g i
⊢ ‖∑' (i : ι), f i‖ ≤ a

Tactic:
by_cases hf : Summable f

Goal state after:
case pos
ι : Type u_1
E : Type u_3
inst✝ : SeminormedAddCommGroup E
f : ι → E
g : ι → ℝ
a : ℝ
hg : HasSum g a
h : ∀ (i : ι), ‖f i‖ ≤ g i
hf : Summable f
⊢ ‖∑' (i : ι), f i‖ ≤ a

case neg
ι : Type u_1
E : Type u_3
inst✝ : SeminormedAddCommGroup E
f : ι → E
g : ι → ℝ
a : ℝ
hg : HasSum g a
h : ∀ (i : ι), ‖f i‖ ≤ g i
hf : ¬Summable f
⊢ ‖∑' (i : ι), f i‖ ≤ a
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: E✝ : ℕ → Type u_1
inst✝² : (n : ℕ) → TopologicalSpace (E✝ n)
inst✝¹ : ∀ (n : ℕ), DiscreteTopology (E✝ n)
E : ℕ → Type u_2
inst✝ : (n : ℕ) → UniformSpace (E n)
h : ∀ (n : ℕ), uniformity (E n) = 𝓟 SetRel.id
this : ∀ (n : ℕ), DiscreteTopology (E n)
ε : ℝ
εpos : ε > 0
n : ℕ
hn : (1 / 2) ^ n < ε
x y : (n : ℕ) → E n
hxy : ∀ i < n, x i = y i
⊢ (x, y).1 ∈ cylinder (x, y).2 n

Tactic:
mem_cylinder_iff

Goal state after:
E✝ : ℕ → Type u_1
inst✝² : (n : ℕ) → TopologicalSpace (E✝ n)
inst✝¹ : ∀ (n : ℕ), DiscreteTopology (E✝ n)
E : ℕ → Type u_2
inst✝ : (n : ℕ) → UniformSpace (E n)
h : ∀ (n : ℕ), uniformity (E n) = 𝓟 SetRel.id
this : ∀ (n : ℕ), DiscreteTopology (E n)
ε : ℝ
εpos : ε > 0
n : ℕ
hn : (1 / 2) ^ n < ε
x y : (n : ℕ) → E n
hxy : ∀ i < n, x i = y i
⊢ ∀ i < n, (x, y).1 i = (x, y).2 i
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
x : (n : ℕ) → E n
hx : x ∉ s
y : (n : ℕ) → E n
hy : y ∈ s
⊢ 0 < shortestPrefixDiff x s

Tactic:
exact (firstDiff_lt_shortestPrefixDiff hs hx hy).pos

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

Goal state before:
pp: case hfh
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
x : ℕ → Bool
n : ℕ
hm : n.succ ≠ 0
⊢ ediam (D (PiNat.res x n.succ)) ≤ u n.succ

Tactic:
dsimp

Goal state after:
case hfh
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
x : ℕ → Bool
n : ℕ
hm : n.succ ≠ 0
⊢ ediam (D (x n :: PiNat.res x n)) ≤ u (n + 1)
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Map

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
ν : Measure β
g : α → β
hg : MeasurePreserving g μ ν
hge : MeasurableEmbedding g
f : β → ℝ≥0∞
⊢ ∫⁻ (a : α), f (g a) ∂μ = ∫⁻ (b : β), f b ∂ν

Tactic:
by rw [← hg.map_eq, hge.lintegral_map]

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

Goal state before:
pp: case inr
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x ↦ if x ∈ s then x else ⋯.some
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)
xs : x ∉ s
ys : y ∉ s
Ax : (s ∩ cylinder x (longestPrefix x s)).Nonempty
fx : f x = Ax.some
Ay : (s ∩ cylinder y (longestPrefix y s)).Nonempty
fy : f y = Ay.some
H : longestPrefix y s < firstDiff x y
⊢ cylinder x (longestPrefix x s) = cylinder y (longestPrefix y s)

Tactic:
· symm
              rw [firstDiff_comm] at H
              exact cylinder_longestPrefix_eq_of_longestPrefix_lt_firstDiff hs hne H ys xs

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: ι : Type u_2
inst✝¹ : Countable ι
α : Type u_3
inst✝ : MeasurableSpace α
s t : ι → Set α
h : ∀ (m n : ι), MeasurablySeparable (s m) (t n)
⊢ MeasurablySeparable (⋃ n, s n) (⋃ m, t m)

Tactic:
choose u hsu htu hu using h

Goal state after:
ι : Type u_2
inst✝¹ : Countable ι
α : Type u_3
inst✝ : MeasurableSpace α
s t : ι → Set α
u : ι → ι → Set α
hsu : ∀ (m n : ι), s m ⊆ u m n
htu : ∀ (m n : ι), Disjoint (t n) (u m n)
hu : ∀ (m n : ι), MeasurableSet (u m n)
⊢ MeasurablySeparable (⋃ n, s n) (⋃ m, t m)
Module: Mathlib.MeasureTheory.Measure.Dirac

Goal state before:
pp: α : Type u_4
mα : MeasurableSpace α
inst✝ : MeasurableSingletonClass α
μ : Measure α
s : Finset α
hμ : ∀ᵐ (a : α) ∂μ, a ∈ s
t : Set α
ht : MeasurableSet t
⊢ μ (t \ {x | (fun a ↦ a ∈ s) x}ᶜ) = (∑ a ∈ s, μ {a} • dirac a) t

Tactic:
dsimp

Goal state after:
α : Type u_4
mα : MeasurableSpace α
inst✝ : MeasurableSingletonClass α
μ : Measure α
s : Finset α
hμ : ∀ᵐ (a : α) ∂μ, a ∈ s
t : Set α
ht : MeasurableSet t
⊢ μ (t \ (↑s)ᶜ) = (∑ a ∈ s, μ {a} • dirac a) t
Module: Mathlib.MeasureTheory.Measure.GiryMonad

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
m : Measure α
f : α → Measure β
s : Set β
hs : MeasurableSet s
⊢ (m.bind f) s ≤ ∫⁻ (a : α), (f a) s ∂m

Tactic:
by
  rw [bind, join_apply hs]
  apply lintegral_map_le

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

Goal state before:
pp: case right
E : ℕ → Type u_1
ι : Type u_2
inst✝¹ : Encodable ι
F : ι → Type u_3
inst✝ : (i : ι) → PseudoEMetricSpace (F i)
⊢ ∀ (i : ι) (i_1 : ℝ≥0∞), 0 < i_1 → {a | edist (a.1 i) (a.2 i) < i_1} ∈ ⨅ ε, ⨅ (_ : 0 < ε), 𝓟 {p | edist p.1 p.2 < ε}

Tactic:
· intro i ε hε₀
      have : (0 : ℝ≥0∞) < 2⁻¹ ^ encode i := ENNReal.pow_pos (by norm_num) _
      refine mem_iInf_of_mem (min (2⁻¹ ^ encode i) ε) <| mem_iInf_of_mem (by positivity) ?_
      simp only [and_imp, Prod.forall, ofPred_subset_ofPred, lt_min_iff, mem_principal]
      intro x y hn
      exact (edist_le_edist_pi_of_edist_lt hn).trans_lt

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

Goal state before:
pp: ι : Type u_2
inst✝¹ : Encodable ι
F : ι → Type u_3
inst✝ : (i : ι) → PseudoMetricSpace (F i)
x y : (i : ι) → F i
i : ι
h : dist x y < 2⁻¹ ^ encode i
⊢ dist (x i) (y i) ≤ dist x y

Tactic:
simpa only [not_le.2 h, false_or] using min_le_iff.1 (min_dist_le_dist_pi x y i)

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: case e'_3
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : T2Space α
inst✝¹ : MeasurableSpace α
inst✝ : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous[_, inst✝³] f
hg : Continuous[_, inst✝³] g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
I :
  ∀ (n : ℕ) (x y : ℕ → ℕ),
    ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →
      ∃ x' y',
        x' ∈ cylinder x n ∧
          y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))
A : Type := { p // ¬MeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }
F : A → A
hFn : ∀ (p : A), (↑(F p)).1 = (↑p).1 + 1
hFx : ∀ (p : A), (↑(F p)).2.1 ∈ cylinder (↑p).2.1 (↑p).1
hFy : ∀ (p : A), (↑(F p)).2.2 ∈ cylinder (↑p).2.2 (↑p).1
p0 : A := ⟨(0, fun x ↦ 0, fun x ↦ 0), ⋯⟩
p : ℕ → A := fun n ↦ F^[n] p0
prec : ∀ (n : ℕ), p (n + 1) = F (p n)
pn_fst : ∀ (n : ℕ), (↑(p n)).1 = n
Ix : ∀ (m n : ℕ), m + 1 ≤ n → (↑(p n)).2.1 m = (↑(p (m + 1))).2.1 m
Iy : ∀ (m n : ℕ), m + 1 ≤ n → (↑(p n)).2.2 m = (↑(p (m + 1))).2.2 m
x : ℕ → ℕ := fun n ↦ (↑(p (n + 1))).2.1 n
hx : x = fun n ↦ (↑(p (n + 1))).2.1 n
y : ℕ → ℕ := fun n ↦ (↑(p (n + 1))).2.2 n
hy : y = fun n ↦ (↑(p (n + 1))).2.2 n
n : ℕ
⊢ x ∈ cylinder (↑(p n)).2.1 n

Tactic:
mem_cylinder_iff

Goal state after:
case e'_3
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : T2Space α
inst✝¹ : MeasurableSpace α
inst✝ : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous[_, inst✝³] f
hg : Continuous[_, inst✝³] g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
I :
  ∀ (n : ℕ) (x y : ℕ → ℕ),
    ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →
      ∃ x' y',
        x' ∈ cylinder x n ∧
          y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))
A : Type := { p // ¬MeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }
F : A → A
hFn : ∀ (p : A), (↑(F p)).1 = (↑p).1 + 1
hFx : ∀ (p : A), (↑(F p)).2.1 ∈ cylinder (↑p).2.1 (↑p).1
hFy : ∀ (p : A), (↑(F p)).2.2 ∈ cylinder (↑p).2.2 (↑p).1
p0 : A := ⟨(0, fun x ↦ 0, fun x ↦ 0), ⋯⟩
p : ℕ → A := fun n ↦ F^[n] p0
prec : ∀ (n : ℕ), p (n + 1) = F (p n)
pn_fst : ∀ (n : ℕ), (↑(p n)).1 = n
Ix : ∀ (m n : ℕ), m + 1 ≤ n → (↑(p n)).2.1 m = (↑(p (m + 1))).2.1 m
Iy : ∀ (m n : ℕ), m + 1 ≤ n → (↑(p n)).2.2 m = (↑(p (m + 1))).2.2 m
x : ℕ → ℕ := fun n ↦ (↑(p (n + 1))).2.1 n
hx : x = fun n ↦ (↑(p (n + 1))).2.1 n
y : ℕ → ℕ := fun n ↦ (↑(p (n + 1))).2.2 n
hy : y = fun n ↦ (↑(p (n + 1))).2.2 n
n : ℕ
⊢ ∀ i < n, x i = (↑(p n)).2.1 i
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: case e'_4
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : T2Space α
inst✝¹ : MeasurableSpace α
inst✝ : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous[_, inst✝³] f
hg : Continuous[_, inst✝³] g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
I :
  ∀ (n : ℕ) (x y : ℕ → ℕ),
    ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →
      ∃ x' y',
        x' ∈ cylinder x n ∧
          y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))
A : Type := { p // ¬MeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }
F : A → A
hFn : ∀ (p : A), (↑(F p)).1 = (↑p).1 + 1
hFx : ∀ (p : A), (↑(F p)).2.1 ∈ cylinder (↑p).2.1 (↑p).1
hFy : ∀ (p : A), (↑(F p)).2.2 ∈ cylinder (↑p).2.2 (↑p).1
p0 : A := ⟨(0, fun x ↦ 0, fun x ↦ 0), ⋯⟩
p : ℕ → A := fun n ↦ F^[n] p0
prec : ∀ (n : ℕ), p (n + 1) = F (p n)
pn_fst : ∀ (n : ℕ), (↑(p n)).1 = n
Ix : ∀ (m n : ℕ), m + 1 ≤ n → (↑(p n)).2.1 m = (↑(p (m + 1))).2.1 m
Iy : ∀ (m n : ℕ), m + 1 ≤ n → (↑(p n)).2.2 m = (↑(p (m + 1))).2.2 m
x : ℕ → ℕ := fun n ↦ (↑(p (n + 1))).2.1 n
hx : x = fun n ↦ (↑(p (n + 1))).2.1 n
y : ℕ → ℕ := fun n ↦ (↑(p (n + 1))).2.2 n
hy : y = fun n ↦ (↑(p (n + 1))).2.2 n
n : ℕ
⊢ y ∈ cylinder (↑(p n)).2.2 n

Tactic:
mem_cylinder_iff

Goal state after:
case e'_4
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : T2Space α
inst✝¹ : MeasurableSpace α
inst✝ : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous[_, inst✝³] f
hg : Continuous[_, inst✝³] g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
I :
  ∀ (n : ℕ) (x y : ℕ → ℕ),
    ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →
      ∃ x' y',
        x' ∈ cylinder x n ∧
          y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))
A : Type := { p // ¬MeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }
F : A → A
hFn : ∀ (p : A), (↑(F p)).1 = (↑p).1 + 1
hFx : ∀ (p : A), (↑(F p)).2.1 ∈ cylinder (↑p).2.1 (↑p).1
hFy : ∀ (p : A), (↑(F p)).2.2 ∈ cylinder (↑p).2.2 (↑p).1
p0 : A := ⟨(0, fun x ↦ 0, fun x ↦ 0), ⋯⟩
p : ℕ → A := fun n ↦ F^[n] p0
prec : ∀ (n : ℕ), p (n + 1) = F (p n)
pn_fst : ∀ (n : ℕ), (↑(p n)).1 = n
Ix : ∀ (m n : ℕ), m + 1 ≤ n → (↑(p n)).2.1 m = (↑(p (m + 1))).2.1 m
Iy : ∀ (m n : ℕ), m + 1 ≤ n → (↑(p n)).2.2 m = (↑(p (m + 1))).2.2 m
x : ℕ → ℕ := fun n ↦ (↑(p (n + 1))).2.1 n
hx : x = fun n ↦ (↑(p (n + 1))).2.1 n
y : ℕ → ℕ := fun n ↦ (↑(p (n + 1))).2.2 n
hy : y = fun n ↦ (↑(p (n + 1))).2.2 n
n : ℕ
⊢ ∀ i < n, y i = (↑(p n)).2.2 i
Module: Mathlib.MeasureTheory.Group.Convolution

Goal state before:
pp: M : Type u_1
inst✝³ : Monoid M
inst✝² : MeasurableSpace M
inst✝¹ : MeasurableMul₂ M
x : M
μ : Measure M
inst✝ : SFinite μ
⊢ map (fun x ↦ x.1 * x.2) (map (Prod.mk x) μ) = map (fun y ↦ x * y) μ

Tactic:
map_map (by fun_prop) (by fun_prop)

Goal state after:
M : Type u_1
inst✝³ : Monoid M
inst✝² : MeasurableSpace M
inst✝¹ : MeasurableMul₂ M
x : M
μ : Measure M
inst✝ : SFinite μ
⊢ map ((fun x ↦ x.1 * x.2) ∘ Prod.mk x) μ = map (fun y ↦ x * y) μ
Module: Mathlib.MeasureTheory.Group.Convolution

Goal state before:
pp: M : Type u_1
inst✝³ : Monoid M
inst✝² : MeasurableSpace M
inst✝¹ : MeasurableMul₂ M
μ : Measure M
inst✝ : SFinite μ
x : M
⊢ map (fun x ↦ x.1 * x.2) (map (fun x_1 ↦ (x_1, x)) μ) = map (fun y ↦ y * x) μ

Tactic:
map_map (by fun_prop) (by fun_prop)

Goal state after:
M : Type u_1
inst✝³ : Monoid M
inst✝² : MeasurableSpace M
inst✝¹ : MeasurableMul₂ M
μ : Measure M
inst✝ : SFinite μ
x : M
⊢ map ((fun x ↦ x.1 * x.2) ∘ fun x_1 ↦ (x_1, x)) μ = map (fun y ↦ y * x) μ
Module: Mathlib.MeasureTheory.Group.Convolution

Goal state before:
pp: M : Type u_1
inst✝⁴ : Monoid M
inst✝³ : MeasurableSpace M
inst✝² : MeasurableMul₂ M
μ ν ρ : Measure M
inst✝¹ : ρ.IsMulLeftInvariant
inst✝ : SFinite ν
hν : ν ≪ ρ
⊢ μ ∗ₘ ν ≪ ρ

Tactic:
refine AbsolutelyContinuous.mk (fun s hs h ↦ ?_)

Goal state after:
M : Type u_1
inst✝⁴ : Monoid M
inst✝³ : MeasurableSpace M
inst✝² : MeasurableMul₂ M
μ ν ρ : Measure M
inst✝¹ : ρ.IsMulLeftInvariant
inst✝ : SFinite ν
hν : ν ≪ ρ
s : Set M
hs : MeasurableSet s
h : ρ s = 0
⊢ (μ ∗ₘ ν) s = 0
Module: Mathlib.MeasureTheory.Group.Convolution

Goal state before:
pp: M : Type u_2
M' : Type u_3
mM : MeasurableSpace M
inst✝⁵ : Monoid M
inst✝⁴ : MeasurableMul₂ M
mM' : MeasurableSpace M'
inst✝³ : Monoid M'
inst✝² : MeasurableMul₂ M'
μ ν : Measure M
inst✝¹ : SFinite μ
inst✝ : SFinite ν
L : M →* M'
hL : Measurable ⇑L
⊢ map (⇑L) (map (fun x ↦ x.1 * x.2) (μ.prod ν)) = map (fun x ↦ x.1 * x.2) ((map (⇑L) μ).prod (map (⇑L) ν))

Tactic:
map_map (by fun_prop) (by fun_prop)

Goal state after:
M : Type u_2
M' : Type u_3
mM : MeasurableSpace M
inst✝⁵ : Monoid M
inst✝⁴ : MeasurableMul₂ M
mM' : MeasurableSpace M'
inst✝³ : Monoid M'
inst✝² : MeasurableMul₂ M'
μ ν : Measure M
inst✝¹ : SFinite μ
inst✝ : SFinite ν
L : M →* M'
hL : Measurable ⇑L
⊢ map (⇑L ∘ fun x ↦ x.1 * x.2) (μ.prod ν) = map (fun x ↦ x.1 * x.2) ((map (⇑L) μ).prod (map (⇑L) ν))
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace β
inst✝⁵ : MeasurableSpace γ
μ✝ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝⁴ : SFinite ν✝
X : Type u_4
Y : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
m : MeasurableSpace X
μ : Measure X
inst✝¹ : IsLocallyFiniteMeasure μ
m' : MeasurableSpace Y
ν : Measure Y
inst✝ : IsLocallyFiniteMeasure ν
x : X × Y
⊢ (μ.prod ν).FiniteAtFilter (𝓝 x)

Tactic:
rw [nhds_prod_eq]
    exact μ.finiteAt_nhds _ |>.prod <| ν.finiteAt_nhds _

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
ν : Measure β
s : Set (α × β)
hsm : MeasurableSet s
hs : (fun x ↦ ν (Prod.mk x ⁻¹' s)) =ᵐ[μ] 0
⊢ (μ.prod ν) s = 0

Tactic:
rw [← nonpos_iff_eq_zero]
  calc
    μ.prod ν s ≤ ∫⁻ x, ν (Prod.mk x ⁻¹' s) ∂μ := prod_apply_le hsm
    _ = 0 := by simp [lintegral_congr_ae hs]

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

Goal state before:
pp: case e_6
G : Type u_1
A : Type u_2
inst✝³ : Group G
inst✝² : MulAction G A
inst✝¹ : MeasurableSpace A
inst✝ : MeasurableConstSMul G A
μ : Measure A
g : Gᵈᵐᵃ
s : Set A
⊢ ⇑(MeasurableEquiv.smul (DomMulAct.mk.symm g)⁻¹) ⁻¹' s = DomMulAct.mk.symm g • s

Tactic:
exact Set.preimage_smul_inv (DomMulAct.mk.symm g) s

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace γ
μ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝ : SFinite ν✝
ν : Measure β
C : Set (Set α)
D : Set (Set β)
hμ : μ.FiniteSpanningSetsIn C
hν : ν.FiniteSpanningSetsIn D
⊢ (μ.prod ν).FiniteSpanningSetsIn (image2 (fun x1 x2 ↦ x1 ×ˢ x2) C D)

Tactic:
haveI := hν.sigmaFinite

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace γ
μ μ' : Measure α
ν✝ ν' : Measure β
τ : Measure γ
inst✝ : SFinite ν✝
ν : Measure β
C : Set (Set α)
D : Set (Set β)
hμ : μ.FiniteSpanningSetsIn C
hν : ν.FiniteSpanningSetsIn D
this : SigmaFinite ν
⊢ (μ.prod ν).FiniteSpanningSetsIn (image2 (fun x1 x2 ↦ x1 ×ˢ x2) C D)
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
mβ : MeasurableSpace β
inst✝¹ : TopologicalSpace β
μ : Measure β
inst✝ : μ.OuterRegular
f : α → β
f_cont : Continuous[inst✝², inst✝¹] f
f_me : MeasurableEmbedding f
A : Set α
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > μ (f '' A)
⊢ ∃ U ⊇ A, IsOpen[inst✝²] U ∧ (comap f μ) U < r

Tactic:
obtain ⟨U, hUA, Uopen, hμU⟩ := OuterRegular.outerRegular (f_me.measurableSet_image' hA) r hr

Goal state after:
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
mβ : MeasurableSpace β
inst✝¹ : TopologicalSpace β
μ : Measure β
inst✝ : μ.OuterRegular
f : α → β
f_cont : Continuous[inst✝², inst✝¹] f
f_me : MeasurableEmbedding f
A : Set α
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > μ (f '' A)
U : Set β
hUA : U ⊇ f '' A
Uopen : IsOpen[inst✝¹] U
hμU : μ U < r
⊢ ∃ U ⊇ A, IsOpen[inst✝²] U ∧ (comap f μ) U < r
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen[inst✝¹] (s n)
h'' : univ ⊆ ⋃ n, s n
A : Set α
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > μ A
HA : μ A < ∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
⊢ A = ⋃ n, (fun n ↦ A ∩ disjointed s n) n

Tactic:
← inter_iUnion,

Goal state after:
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen[inst✝¹] (s n)
h'' : univ ⊆ ⋃ n, s n
A : Set α
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > μ A
HA : μ A < ∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
⊢ A = A ∩ ⋃ i, disjointed s i