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

Goal state before:
pp: γ : Type u_3
inst✝³ : TopologicalSpace γ
inst✝² : MeasurableSpace γ
inst✝¹ : BorelSpace γ
inst✝ : R1Space γ
K s : Set γ
hK : IsCompact K
hs : MeasurableSet s
hKs : K ⊆ s
⊢ ⋃ x ∈ K, {y | Inseparable x y} ⊆ s

Tactic:
iUnion₂_subset_iff

Goal state after:
γ : Type u_3
inst✝³ : TopologicalSpace γ
inst✝² : MeasurableSpace γ
inst✝¹ : BorelSpace γ
inst✝ : R1Space γ
K s : Set γ
hK : IsCompact K
hs : MeasurableSet s
hKs : K ⊆ s
⊢ ∀ i ∈ K, {y | Inseparable i y} ⊆ s
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_5
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → x ∈ s
S : Set (Set α) := {S | ∃ l ∈ s, ∃ u ∈ s, l ≤ u ∧ Icc l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : t.Countable
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : ∀ b < a, (Ioo b a).Nonempty
⊢ Iio a = ⋃ l ∈ t, ⋃ u ∈ t, ⋃ (_ : l < u), ⋃ (_ : u < a), Icc l u

Tactic:
ext y

Goal state after:
α : Type u_5
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → x ∈ s
S : Set (Set α) := {S | ∃ l ∈ s, ∃ u ∈ s, l ≤ u ∧ Icc l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : t.Countable
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : ∀ b < a, (Ioo b a).Nonempty
y : α
⊢ y ∈ Iio a ↔ y ∈ ⋃ l ∈ t, ⋃ u ∈ t, ⋃ (_ : l < u), ⋃ (_ : u < a), Icc l u
Module: Mathlib.Topology.MetricSpace.HausdorffDistance

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
U : Set α
hU : IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] U
a : ℝ≥0∞
a_pos : 0 < a
a_lt_one : a < 1
F : ℕ → Set α := fun n ↦ (fun x ↦ infEDist x Uᶜ) ⁻¹' Ici (a ^ n)
F_subset : ∀ (n : ℕ), F n ⊆ U
⊢ ⋃ n, F n ⊆ U

Tactic:
by simp only [iUnion_subset_iff, F_subset, forall_const]

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

Goal state before:
pp: α : Type u_5
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l ∈ s, ∃ u ∈ s, l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : t.Countable
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : ∀ b < a, (Ioo b a).Nonempty
⊢ Iio a = ⋃ l ∈ t, ⋃ u ∈ t, ⋃ (_ : l < u), ⋃ (_ : u ≤ a), Ico l u

Tactic:
ext y

Goal state after:
α : Type u_5
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l ∈ s, ∃ u ∈ s, l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : t.Countable
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : ∀ b < a, (Ioo b a).Nonempty
y : α
⊢ y ∈ Iio a ↔ y ∈ ⋃ l ∈ t, ⋃ u ∈ t, ⋃ (_ : l < u), ⋃ (_ : u ≤ a), Ico l u
Module: Mathlib.Topology.MetricSpace.HausdorffDistance

Goal state before:

Tactic:
infEDist x t ≤ edist x z := infEDist_le_edist_of_mem zt
      _ ≤ edist x y + edist y z := edist_triangle _ _ _
      _ ≤ infEDist x s + ε / 2 + (hausdorffEDist s t + ε / 2) := add_le_add dxy.le dyz.le
      _ = infEDist x s + hausdorffEDist s t + ε := by
        rw [add_add_add_comm, ENNReal.add_halves]

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

Goal state before:
pp: α : Type u_2
inst✝ : PseudoMetricSpace α
x : α
δ : ℝ
hδ : 0 ≤ δ
⊢ cthickening δ {x} = closedBall x δ

Tactic:
ext y

Goal state after:
α : Type u_2
inst✝ : PseudoMetricSpace α
x : α
δ : ℝ
hδ : 0 ≤ δ
y : α
⊢ y ∈ cthickening δ {x} ↔ y ∈ closedBall x δ
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case refine_2
α : Type u_5
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → μ (Ico a b) ≠ ∞
h : ∀ ⦃a b : α⦄, a < b → μ (Ico a b) = ν (Ico a b)
s : Set α
hsc : s.Countable
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
this : {x | ∃ i, ∃ (_ : i ∈ s), ∃ i_2, ∃ (_ : i_2 ∈ s) (_ : i < i_2), x = Ico i i_2}.Countable
⊢ ⋃₀ {x | ∃ i, ∃ (_ : i ∈ s), ∃ i_2, ∃ (_ : i_2 ∈ s) (_ : i < i_2), x = Ico i i_2} = univ

Tactic:
refine sUnion_eq_univ_iff.2 fun x => ?_
    rcases hsd.exists_le' hsb x with ⟨l, hls, hlx⟩
    rcases hsd.exists_gt x with ⟨u, hus, hxu⟩
    exact ⟨_, ⟨l, hls, u, hus, hlx.trans_lt hxu, rfl⟩, hlx, hxu⟩

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : MeasurableSpace α
inst✝¹ : SemilatticeSup β
inst✝ : OrderBot β
f : γ → α →ₛ β
s : Finset γ
a : α
⊢ (s.sup f) a = s.sup fun c ↦ (f c) a

Tactic:
refine Finset.induction_on s rfl ?_

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : MeasurableSpace α
inst✝¹ : SemilatticeSup β
inst✝ : OrderBot β
f : γ → α →ₛ β
s : Finset γ
a : α
⊢ ∀ (a_1 : γ) (s : Finset γ),
    a_1 ∉ s → ((s.sup f) a = s.sup fun c ↦ (f c) a) → ((insert a_1 s).sup f) a = (insert a_1 s).sup fun c ↦ (f c) a
Module: Mathlib.MeasureTheory.Function.SimpleFunc

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁸ : MeasurableSpace α
inst✝⁷ : SemilatticeSup β
inst✝⁶ : OrderBot β
inst✝⁵ : Zero β
inst✝⁴ : TopologicalSpace β
inst✝³ : OrderClosedTopology β
inst✝² : MeasurableSpace β
inst✝¹ : OpensMeasurableSpace β
inst✝ : MeasurableSpace γ
i : ℕ → β
f : γ → β
g : α → γ
n : ℕ
a : α
hf : Measurable f
hg : Measurable g
⊢ (approx i (f ∘ g) n) a = (approx i f n) (g a)

Tactic:
rw [approx_apply _ hf, approx_apply _ (hf.comp hg), Function.comp_apply]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
g : β → ℝ≥0∞
f : α →ₛ β
a : α
hb : f a ∈ f.range
⊢ g (f a) * μ (⇑f ⁻¹' ↑({b ∈ f.range | g b = g (f a)})) = ∑ j ∈ f.range with g j = g (f a), g j * μ (⇑f ⁻¹' {j})

Tactic:
← f.sum_measure_preimage_singleton,

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
g : β → ℝ≥0∞
f : α →ₛ β
a : α
hb : f a ∈ f.range
⊢ g (f a) * ∑ y ∈ f.range with g y = g (f a), μ (⇑f ⁻¹' {y}) = ∑ j ∈ f.range with g j = g (f a), g j * μ (⇑f ⁻¹' {j})
Module: Mathlib.MeasureTheory.Function.SimpleFunc

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
c : ℝ≥0∞
s : Set α
hs : MeasurableSet s
⊢ (const α c).lintegral (μ.restrict s) = c * μ s

Tactic:
const_lintegral_restrict

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
c : ℝ≥0∞
s : Set α
hs : MeasurableSet s
⊢ c * μ s = c * μ s
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Real

Goal state before:
pp: case refine_1
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
x : α
⊢ Tendsto (fun i ↦ aeSeq h'f p i x) atTop (𝓝 (aeSeqLim x))

Tactic:
unfold aeSeqLim
    simp_rw [aeSeq]
    split_ifs with hx
    · simp_rw [aeSeq.mk_eq_fun_of_mem_aeSeqSet h'f hx]
      exact aeSeq.fun_prop_of_mem_aeSeqSet h'f hx
    · exact tendsto_const_nhds

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

Goal state before:
pp: β : Type u_6
mβ : MeasurableSpace β
f : EReal × EReal → β
h_real : Measurable fun p ↦ f (↑p.1, ↑p.2)
h_bot_left : Measurable fun r ↦ f (⊥, ↑r)
h_top_left : Measurable fun r ↦ f (⊤, ↑r)
h_bot_right : Measurable fun r ↦ f (↑r, ⊥)
h_top_right : Measurable fun r ↦ f (↑r, ⊤)
⊢ Measurable f

Tactic:
refine measurable_of_real_prod ?_ ?_ ?_
  · refine measurable_swap_iff.mp <| measurable_of_real_prod ?_ h_bot_right h_top_right
    exact h_real.comp measurable_swap
  · exact measurable_of_measurable_real h_bot_left
  · exact measurable_of_measurable_real h_top_left

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

Goal state before:
pp: case neg
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : PseudoEMetricSpace α
inst✝ : OpensMeasurableSpace α
e : ℕ → α
x : α
N : ℕ
ihN : ∀ {k : ℕ}, k ≤ N → edist ((nearestPt e N) x) x ≤ edist (e k) x
k : ℕ
hk : k ≤ N + 1
h : ∃ k ≤ N, edist (e k) x ≤ edist (e (N + 1)) x
⊢ edist (e ((nearestPtInd e N) x)) x ≤ edist (e k) x

Tactic:
rcases h with ⟨l, hlN, hxl⟩

Goal state after:
case neg
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : PseudoEMetricSpace α
inst✝ : OpensMeasurableSpace α
e : ℕ → α
x : α
N : ℕ
ihN : ∀ {k : ℕ}, k ≤ N → edist ((nearestPt e N) x) x ≤ edist (e k) x
k : ℕ
hk : k ≤ N + 1
l : ℕ
hlN : l ≤ N
hxl : edist (e l) x ≤ edist (e (N + 1)) x
⊢ edist (e ((nearestPtInd e N) x)) x ≤ edist (e k) 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
⊢ μ s = μ (s ∩ f ⁻¹' {0} ∪ s ∩ f ⁻¹' Ioi 0)

Tactic:
rw [← inter_union_distrib_left, ← preimage_union, singleton_union, Ioi_insert,
        ← _root_.bot_eq_zero, Ici_bot, preimage_univ, inter_univ]

Goal state after:
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
n : ℕ
x : β
this : Nonempty ↑s
⊢ ∀ (n : ℕ), Nat.casesOn n y₀ (Subtype.val ∘ denseSeq ↑s) ∈ s

Tactic:
rintro (_ | n)

Goal state after:
case zero
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
n : ℕ
x : β
this : Nonempty ↑s
⊢ Nat.casesOn 0 y₀ (Subtype.val ∘ denseSeq ↑s) ∈ s

case succ
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
n✝ : ℕ
x : β
this : Nonempty ↑s
n : ℕ
⊢ Nat.casesOn (n + 1) y₀ (Subtype.val ∘ denseSeq ↑s) ∈ s
Module: Mathlib.MeasureTheory.Function.SimpleFunc

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
motive : (α → ℝ≥0∞) → Prop
indicator : ∀ (c : ℝ≥0∞) ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → motive (s.indicator fun x ↦ c)
add :
  ∀ ⦃f g : α → ℝ≥0∞⦄,
    Disjoint (support f) (support g) → Measurable f → Measurable g → motive f → motive g → motive (f + g)
iSup :
  ∀ ⦃f : ℕ → α → ℝ≥0∞⦄,
    (∀ (n : ℕ), Measurable (f n)) → Monotone f → (∀ (n : ℕ), motive (f n)) → motive fun x ↦ ⨆ n, f n x
f : α → ℝ≥0∞
hf : Measurable f
c : ℝ≥0∞
s : Set α
hs : MeasurableSet s
m n : ℕ
hmn : m ≤ n
a : α
⊢ (s ∩ spanningSets μ m).indicator (fun x ↦ c) a ≤ (s ∩ spanningSets μ n).indicator (fun x ↦ c) a

Tactic:
grw [hmn]

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

Goal state before:
pp: case refine_2
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
h : ∀ᵐ (a : α) ∂μ, f a ≤ g a
t : Set α
hts : {x | (fun a ↦ f a ≤ g a) x}ᶜ ⊆ t
ht : MeasurableSet t
ht0 : μ t = 0
this : ∀ᵐ (x : α) ∂μ, x ∉ t
s : α →ₛ ℝ≥0∞
hfs : ⇑s ≤ fun a ↦ f a
⊢ s.lintegral μ ≤ (s.restrict tᶜ).lintegral μ

Tactic:
exact le_of_eq <| SimpleFunc.lintegral_congr <| this.mono fun a hnt => by
      simp [restrict_apply s ht.compl, hnt]

Goal state after:
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
f : α → β
inst✝ : PseudoMetrizableSpace β
ν : Measure α
⊢ AEStronglyMeasurable f (μ + ν) ↔ AEStronglyMeasurable f μ ∧ AEStronglyMeasurable f ν

Tactic:
rw [← sum_cond, aestronglyMeasurable_sum_measure_iff, Bool.forall_bool, and_comm]

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
f : α → β
inst✝ : PseudoMetrizableSpace β
ν : Measure α
⊢ AEStronglyMeasurable f (bif true then μ else ν) ∧ AEStronglyMeasurable f (bif false then μ else ν) ↔
    AEStronglyMeasurable f μ ∧ AEStronglyMeasurable f ν
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Markov

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
h2f : ∫⁻ (x : α), f x ∂μ ≠ ∞
⊢ ∀ᵐ (x : α) ∂μ, f x < ∞

Tactic:
ae_iff,

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
h2f : ∫⁻ (x : α), f x ∂μ ≠ ∞
⊢ μ {a | ¬f a < ∞} = 0
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_1
E : Type u_5
m m₀ : MeasurableSpace α
μ : Measure α
f g : α → E
inst✝³ : TopologicalSpace E
inst✝² : Preorder E
inst✝¹ : OrderClosedTopology E
inst✝ : PseudoMetrizableSpace E
hm : m ≤ m₀
hf : StronglyMeasurable f
hg : StronglyMeasurable g
hfg : f ≤ᵐ[μ] g
⊢ ∀ᵐ (x : α) ∂μ.trim hm, f x ≤ g x

Tactic:
ae_iff,

Goal state after:
α : Type u_1
E : Type u_5
m m₀ : MeasurableSpace α
μ : Measure α
f g : α → E
inst✝³ : TopologicalSpace E
inst✝² : Preorder E
inst✝¹ : OrderClosedTopology E
inst✝ : PseudoMetrizableSpace E
hm : m ≤ m₀
hf : StronglyMeasurable f
hg : StronglyMeasurable g
hfg : f ≤ᵐ[μ] g
⊢ (μ.trim hm) {a | ¬f a ≤ g a} = 0
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_1
E : Type u_5
m m₀ : MeasurableSpace α
μ : Measure α
f g : α → E
inst✝¹ : TopologicalSpace E
inst✝ : MetrizableSpace E
hm : m ≤ m₀
hf : StronglyMeasurable f
hg : StronglyMeasurable g
hfg : f =ᵐ[μ] g
⊢ ∀ᵐ (x : α) ∂μ.trim hm, f x = g x

Tactic:
ae_iff,

Goal state after:
α : Type u_1
E : Type u_5
m m₀ : MeasurableSpace α
μ : Measure α
f g : α → E
inst✝¹ : TopologicalSpace E
inst✝ : MetrizableSpace E
hm : m ≤ m₀
hf : StronglyMeasurable f
hg : StronglyMeasurable g
hfg : f =ᵐ[μ] g
⊢ (μ.trim hm) {a | ¬f a = g a} = 0
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
μ : Measure α
f : α → β
inst✝² : Zero β
inst✝¹ : TopologicalSpace β
inst✝ : T2Space β
fs : ℕ → α →ₛ β
hT_lt_top : ∀ (n : ℕ), μ (support ⇑(fs n)) < ∞
h_approx : ∀ (x : α), Tendsto (fun n ↦ (fs n) x) atTop (𝓝 (f x))
T : ℕ → Set α := fun n ↦ support ⇑(fs n)
hT_meas : ∀ (n : ℕ), MeasurableSet (T n)
t : Set α := ⋃ n, T n
⊢ SigmaFinite (μ.restrict t)

Tactic:
refine ⟨⟨⟨fun n => tᶜ ∪ T n, fun _ => trivial, fun n => ?_, ?_⟩⟩⟩
    · rw [Measure.restrict_apply' (MeasurableSet.iUnion hT_meas), Set.union_inter_distrib_right,
        Set.compl_inter_self t, Set.empty_union]
      exact (measure_mono Set.inter_subset_left).trans_lt (hT_lt_top n)
    · rw [← Set.union_iUnion tᶜ T]
      exact Set.compl_union_self _

Goal state after:
Module: Mathlib.Topology.Metrizable.CompletelyMetrizable

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace.IsCompletelyPseudoMetrizableSpace Y
f : X → Y
hf : IsClosedEmbedding f
this : UpgradedIsCompletelyPseudoMetrizableSpace Y := upgradeIsCompletelyPseudoMetrizable Y
⊢ TopologicalSpace.IsCompletelyPseudoMetrizableSpace X

Tactic:
let : PseudoMetricSpace X := hf.isEmbedding.comapPseudoMetricSpace

Goal state after:
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace.IsCompletelyPseudoMetrizableSpace Y
f : X → Y
hf : IsClosedEmbedding f
this✝ : UpgradedIsCompletelyPseudoMetrizableSpace Y := upgradeIsCompletelyPseudoMetrizable Y
this : PseudoMetricSpace X := ⋯.comapPseudoMetricSpace
⊢ TopologicalSpace.IsCompletelyPseudoMetrizableSpace X
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: E : ℕ → Type u_1
x : (n : ℕ) → E n
n : ℕ
⊢ cylinder x n = (↑(Finset.range n)).pi fun i ↦ {x i}

Tactic:
ext y

Goal state after:
E : ℕ → Type u_1
x : (n : ℕ) → E n
n : ℕ
y : (n : ℕ) → E n
⊢ y ∈ cylinder x n ↔ y ∈ (↑(Finset.range n)).pi fun i ↦ {x i}
Module: Mathlib.Topology.Metrizable.CompletelyMetrizable

Goal state before:
pp: case pos
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
x y z : X
h✝ : x = y
⊢ (if x = z then 0 else 1) ≤ (if x = y then 0 else 1) + if y = z then 0 else 1

Tactic:
by_cases x = z

Goal state after:
case pos
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
x y z : X
h✝¹ : x = y
h✝ : x = z
⊢ (if x = z then 0 else 1) ≤ (if x = y then 0 else 1) + if y = z then 0 else 1

case neg
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
x y z : X
h✝¹ : x = y
h✝ : ¬x = z
⊢ (if x = z then 0 else 1) ≤ (if x = y then 0 else 1) + if y = z then 0 else 1
Module: Mathlib.Topology.Metrizable.CompletelyMetrizable

Goal state before:
pp: case neg
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
x y z : X
h✝ : ¬x = y
⊢ (if x = z then 0 else 1) ≤ (if x = y then 0 else 1) + if y = z then 0 else 1

Tactic:
by_cases x = z

Goal state after:
case pos
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
x y z : X
h✝¹ : ¬x = y
h✝ : x = z
⊢ (if x = z then 0 else 1) ≤ (if x = y then 0 else 1) + if y = z then 0 else 1

case neg
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
x y z : X
h✝¹ : ¬x = y
h✝ : ¬x = z
⊢ (if x = z then 0 else 1) ≤ (if x = y then 0 else 1) + if y = z then 0 else 1
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: E : ℕ → Type u_1
x : (n : ℕ) → E n
n : ℕ
⊢ ⋃ k, cylinder (update x n k) (n + 1) = cylinder x n

Tactic:
ext y

Goal state after:
E : ℕ → Type u_1
x : (n : ℕ) → E n
n : ℕ
y : (n : ℕ) → E n
⊢ y ∈ ⋃ k, cylinder (update x n k) (n + 1) ↔ y ∈ cylinder x n
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: α : Type u_2
x : ℕ → α
n : ℕ
⊢ cylinder x n = {y | res y n = res x n}

Tactic:
ext y

Goal state after:
α : Type u_2
x : ℕ → α
n : ℕ
y : ℕ → α
⊢ y ∈ cylinder x n ↔ y ∈ {y | res y n = res x n}
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
x y : (i : ι) × E i
h : dist x y < 1
⊢ x.fst = y.fst

Tactic:
by
  cases x; cases y
  contrapose! h
  apply one_le_dist_of_ne h

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

Goal state before:
pp: case refine_2
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : PolishSpace α
s : Set α
hs : IsClosed[inst✝¹] s
this✝ : PolishSpace ↑s
t : Set α := sᶜ
this : PolishSpace ↑t
f : ↑s ⊕ ↑t ≃ α := Equiv.Set.sumCompl s
hle : coinduced (⇑f) instTopologicalSpaceSum ≤ inst✝¹
⊢ IsOpen[coinduced (⇑f) instTopologicalSpaceSum] s

Tactic:
rw [isOpen_coinduced, isOpen_sum_iff]

Goal state after:
case refine_2
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : PolishSpace α
s : Set α
hs : IsClosed[inst✝¹] s
this✝ : PolishSpace ↑s
t : Set α := sᶜ
this : PolishSpace ↑t
f : ↑s ⊕ ↑t ≃ α := Equiv.Set.sumCompl s
hle : coinduced (⇑f) instTopologicalSpaceSum ≤ inst✝¹
⊢ IsOpen[instTopologicalSpaceSubtype] (Sum.inl ⁻¹' ⇑f ⁻¹' s) ∧
    IsOpen[instTopologicalSpaceSubtype] (Sum.inr ⁻¹' ⇑f ⁻¹' s)
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: case inl
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
x : (n : ℕ) → E n
hs : IsClosed[Pi.topologicalSpace] ∅
hx : x ∉ ∅
⊢ ∃ n, Disjoint ∅ (cylinder x n)

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
x : (n : ℕ) → E n
hx : x ∉ s
hne : s.Nonempty
⊢ ∃ n, Disjoint s (cylinder x n)

Tactic:
· exact ⟨0, by simp⟩

Goal state after:
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
x : (n : ℕ) → E n
hx : x ∉ s
hne : s.Nonempty
⊢ ∃ n, Disjoint s (cylinder x n)
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: X : ℕ → Type u
inst✝ : (n : ℕ) → MetricSpace (X n)
f : (n : ℕ) → X n → X (n + 1)
I : ∀ (n : ℕ), Isometry (f n)
x : (n : ℕ) × X n
⊢ inductiveLimitDist f x x = 0

Tactic:
by simp [inductiveLimitDist]

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

Goal state before:
pp: case refine_1.h₁
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
y : (n : ℕ) → E n
⊢ f y ∈ s

Tactic:
by_cases hy : y ∈ s

Goal state after:
case pos
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
y : (n : ℕ) → E n
hy : y ∈ s
⊢ f y ∈ s

case neg
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
y : (n : ℕ) → E n
hy : y ∉ s
⊢ f y ∈ s
Module: Mathlib.MeasureTheory.Measure.Count

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
p : α → Prop
h : ∀ᵐ (x : α) ∂count, p x
x : α
⊢ p x

Tactic:
ae_iff,

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
p : α → Prop
h : count {a | ¬p a} = 0
x : α
⊢ p x
Module: Mathlib.MeasureTheory.Measure.Dirac

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSingletonClass α
a : α
⊢ ae (dirac a) = pure a

Tactic:
ext s
  simp [mem_ae_iff, imp_false]

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

Goal state before:
pp: case h
E : ℕ → Type u_1
ι : Type u_2
inst✝¹ : Encodable ι
F : ι → Type u_3
inst✝ : (i : ι) → PseudoEMetricSpace (F i)
ε : ℝ≥0∞
hε : 0 < ε
K : Finset ι
hK : ∑' (i : { j // j ∉ K }), 2⁻¹ ^ encode ↑i < ε / 2
δ : ℝ≥0∞
δpos : 0 < δ
hδ : δ * ↑K.card < ε / 2
x y : (i : ι) → F i
hxy : ∀ (x_1 : ι) (h : x_1 ∈ K), edist (x x_1) (y x_1) < δ
i : ι
a✝ : i ∈ K
⊢ min (2⁻¹ ^ encode i) (edist (x i) (y i)) ≤ edist (x i) (y i)

Tactic:
apply min_le_right

Goal state after: