mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.MetricSpace.Thickening

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
E : Set α
x : α
h : x ∉ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
ε : ℝ
ε_pos : 0 < ε
ε_lt : ENNReal.ofReal ε < infEDist x E
δ : ℝ
hδ : δ < ε
⊢ x ∉ cthickening δ E

Tactic:
simp only [cthickening, mem_ofPred_eq, not_le]

Goal state after:
α : Type u
inst✝ : PseudoEMetricSpace α
E : Set α
x : α
h : x ∉ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
ε : ℝ
ε_pos : 0 < ε
ε_lt : ENNReal.ofReal ε < infEDist x E
δ : ℝ
hδ : δ < ε
⊢ ENNReal.ofReal δ < infEDist x E
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case h₂
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hα : Nonempty α
A : ∀ (i : ι) (c : α), MeasurableSet {x | f i x ≤ c}
B : ∀ (c : α), MeasurableSet {x | ∀ (i : ι), f i x ≤ c}
u : ℕ → α
hu : Tendsto u atTop atTop
x : δ
n : ℕ
hn : ∀ (i : ι), f i x ≤ u n
i : ι
⊢ (fun i ↦ f i x) i ≤ u n

Tactic:
exact hn i

Goal state after:
Module: Mathlib.Topology.Instances.Real.Lemmas

Goal state before:
pp: q : ℚ
⊢ closure (Rat.cast '' {x | q < x}) = {r | ↑q ≤ r}

Tactic:
convert! closure_ordConnected_inter_rat (ordConnected_Ioi (a := (q : ℝ))) _ using 1
  · congr!; aesop
  · exact (closure_Ioi _).symm
  · exact ⟨q + 1, show (q : ℝ) < _ by linarith, q + 2, show (q : ℝ) < _ by linarith, by simp⟩

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
f : α → ℝ≥0
hf : Measurable f
μ : Measure α
inst✝ : SigmaFinite μ
sigma_finite_sets : ℕ → Set α := spanningSets μ
norm_sets : ℕ → Set α := fun n ↦ {x | f x ≤ ↑n}
norm_sets_spanning : ⋃ n, norm_sets n = univ
sets : ℕ → Set α := fun n ↦ sigma_finite_sets n ∩ norm_sets n
h_meas : ∀ (n : ℕ), MeasurableSet (sets n)
n : ℕ
⊢ μ (sigma_finite_sets n) < ∞

Tactic:
exact measure_spanningSets_lt_top μ n

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
x : β
hx : f x ∈ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s
this : Nonempty ↑s
⊢ Tendsto (fun n ↦ (approxOn f hf s y₀ h₀ n) x) atTop (𝓝 (f x))

Tactic:
rw [← @Subtype.range_coe _ s, ← image_univ, ← (denseRange_denseSeq s).closure_eq] at hx

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
this : Nonempty ↑s
hx :
  f x ∈
    closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace]
      (Subtype.val '' closure[instTopologicalSpaceSubtype] (range (denseSeq ↑s)))
⊢ Tendsto (fun n ↦ (approxOn f hf s y₀ h₀ n) x) atTop (𝓝 (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)
⊢ μ (s ∩ f ⁻¹' Ioi 0) = μ (s ∩ f ⁻¹' {∞} ∪ s ∩ f ⁻¹' Ioo 0 ∞)

Tactic:
rw [← inter_union_distrib_left]

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)
⊢ μ (s ∩ f ⁻¹' Ioi 0) = μ (s ∩ (f ⁻¹' {∞} ∪ f ⁻¹' Ioo 0 ∞))
Module: Mathlib.MeasureTheory.Function.SimpleFuncDense

Goal state before:
pp: X : Type u_3
Y : Type u_4
α : Type u_5
inst✝⁷ : Zero α
inst✝⁶ : TopologicalSpace X
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : MeasurableSpace X
inst✝³ : MeasurableSpace Y
inst✝² : OpensMeasurableSpace X
inst✝¹ : OpensMeasurableSpace Y
inst✝ : PseudoMetricSpace α
f : X × Y → α
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
h'f : HasCompactSupport f
ε : ℝ
hε : 0 < ε
M : ∀ (K : Set (X × Y)), IsCompact K → ∃ g s, MeasurableSet s ∧ K ⊆ s ∧ ∀ x ∈ s, dist (f x) (g x) < ε
g : SimpleFunc (X × Y) α
s : Set (X × Y)
s_meas : MeasurableSet s
fs : tsupport f ⊆ s
hg : ∀ x ∈ s, dist (f x) (g x) < ε
p : X × Y
H : p ∈ support f
⊢ p ∈ s

Tactic:
exact fs (subset_tsupport _ H)

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

Goal state before:
pp: 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))
⊢ Tendsto (fun n ↦ min 1 (c / ‖(hf.approx n) x‖) • (hf.approx n) x) atTop (𝓝 (f x))

Tactic:
by_cases hfx0 : ‖f x‖ = 0

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))

case neg
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.Function.StronglyMeasurable.AEStronglyMeasurable

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
m₀ : MeasurableSpace α
μ : Measure α
inst✝¹ : PseudoMetrizableSpace β
inst✝ : PseudoMetrizableSpace γ
g : β → γ
f : α → β
hg : IsEmbedding g
this : PseudoMetricSpace γ := pseudoMetrizableSpacePseudoMetric γ
this✝³ : MeasurableSpace β := borel β
this✝² : BorelSpace β
this✝¹ : MeasurableSpace γ := borel γ
this✝ : BorelSpace γ
H : AEStronglyMeasurable (fun x ↦ g (f x)) μ
⊢ AEMeasurable f μ

Tactic:
let G : β → range g := rangeFactorization g
    have hG : IsClosedEmbedding G :=
      { hg.codRestrict _ _ with
        isClosed_range := by rw [rangeFactorization_surjective.range_eq]; exact isClosed_univ }
    have : AEMeasurable (G ∘ f) μ := AEMeasurable.subtype_mk H.aemeasurable
    exact hG.measurableEmbedding.aemeasurable_comp_iff.1 this

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

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : Nonempty β
inst✝ : T2Space β
hα : IsEmpty α
⊢ StronglyMeasurable f ↔ ∃ c, f = fun x ↦ c

pp: case inr
α : Type u_1
β : Type u_2
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : Nonempty β
inst✝ : T2Space β
hα : Nonempty α
⊢ StronglyMeasurable f ↔ ∃ c, f = fun x ↦ c

Tactic:
· simp [eq_iff_true_of_subsingleton]

Goal state after:
case inr
α : Type u_1
β : Type u_2
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : Nonempty β
inst✝ : T2Space β
hα : Nonempty α
⊢ StronglyMeasurable f ↔ ∃ c, f = fun x ↦ c
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
f : α → β
inst✝¹ : TopologicalSpace β
inst✝ : Zero β
m✝ : MeasurableSpace α
μ : Measure α
hf_meas : StronglyMeasurable f
t : Set α
ht : MeasurableSet t
hft_zero : ∀ x ∈ tᶜ, f x = 0
htμ this : SigmaFinite (μ.restrict t)
S : ℕ → Set α := spanningSets (μ.restrict t)
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
f_approx : ℕ → α →ₛ β := hf_meas.approx
fs : ℕ → α →ₛ β := fun n ↦ (f_approx n).restrict (S n ∩ t)
h_fs_t_compl : ∀ (n : ℕ), ∀ x ∉ t, (fs n) x = 0
x : α
hxt : x ∈ t
h : Tendsto (fun n ↦ (f_approx n) x) atTop (𝓝 (f x))
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → x ∈ S m ∩ t
m : ℕ
hnm : n ≤ m
⊢ (S m ∩ t).indicator (⇑(f_approx m)) x = (f_approx m) x

Tactic:
Set.indicator_of_mem (hn m hnm)

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
⊢ μ = ν ↔ ∀ (f : α → ℝ≥0∞), Measurable f → ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), f a ∂ν

Tactic:
refine ⟨fun h _ _ ↦ by rw [h], ?_⟩

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
⊢ (∀ (f : α → ℝ≥0∞), Measurable f → ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), f a ∂ν) → μ = ν
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
A : Set α
hA : MeasurableSet A
⊢ ∫⁻ (x : α) in A, f x ∂μ + ∫⁻ (x : α) in Aᶜ, f x ∂μ = ∫⁻ (x : α), f x ∂μ

Tactic:
by
  rw [← lintegral_add_measure, Measure.restrict_add_restrict_compl hA]

Goal state after:
Module: Mathlib.Topology.Perfect

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
C : Set α
x : α
U : Set α
h_acc : AccPt x (𝓟 C)
hU : U ∈ 𝓝 x
this : 𝓝[≠] x ≤ 𝓟 U
⊢ (𝓝[≠] x ⊓ 𝓟 C).NeBot

Tactic:
exact h_acc

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
t : Set α
inst✝ : SFinite μ
ht_subset : t ⊆ μ.sigmaFiniteSetᶜ
⊢ ¬μ t = 0 → μ t = ∞

Tactic:
exact measure_eq_top_of_subset_compl_sigmaFiniteSetWRT ht_subset

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

Goal state before:
pp: case mp.h₂
E : ℕ → Type u_1
x y : (n : ℕ) → E n
n : ℕ
hy : y ∈ cylinder x n
z : (n : ℕ) → E n
hz : z ∈ cylinder x n
i : ℕ
hi : i < n
⊢ z i = y i

Tactic:
rw [hy i hi]

Goal state after:
case mp.h₂
E : ℕ → Type u_1
x y : (n : ℕ) → E n
n : ℕ
hy : y ∈ cylinder x n
z : (n : ℕ) → E n
hz : z ∈ cylinder x n
i : ℕ
hi : i < n
⊢ z i = x i
Module: Mathlib.Topology.MetricSpace.Polish

Goal state before:
pp: α : Type u_1
ι : Type u_3
inst✝ : Countable ι
t : ι → TopologicalSpace α
ht : ∀ (i : ι), PolishSpace α
i₀ : ι
hi₀ : ∀ (i : ι), t i ≤ t i₀
u : UniformSpace α
hcomp : CompleteSpace α
hcount : (𝓤 α).IsCountablyGenerated
htop : u.toTopologicalSpace = ⨅ i, t i
this : SecondCountableTopology α
⊢ T1Space α

Tactic:
let := t i₀; have := ht i₀; infer_instance

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

Goal state before:
pp: ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
i : ι
x : E i
j : ι
y : E j
k : ι
z : E k
hik : i ≠ k
hij : i ≠ j
hjk : j ≠ k
⊢ dist x ⋯.some + 1 + dist ⋯.some z = dist x ⋯.some + 1 + 0 + (0 + 0 + dist ⋯.some z)

Tactic:
simp only [add_zero, zero_add]

Goal state after:
Module: Mathlib.Dynamics.Ergodic.MeasurePreserving

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μa : Measure α
μb : Measure β
μc : Measure γ
g : α → β
e : β ≃ᵐ γ
h : MeasurePreserving (⇑e) μb μc
⊢ MeasurePreserving (⇑e ∘ g) μa μc ↔ MeasurePreserving g μa μb

Tactic:
refine ⟨fun hg => ?_, fun hg => h.comp hg⟩
  convert! (MeasurePreserving.symm e h).comp hg
  simp [← Function.comp_assoc e.symm e g]

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

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
s : Set α
s_mble : MeasurableSet s
⊢ count s = ∞ ↔ s.Infinite

Tactic:
by_cases hs : s.Finite

Goal state after:
case pos
α : Type u_1
inst✝ : MeasurableSpace α
s : Set α
s_mble : MeasurableSet s
hs : s.Finite
⊢ count s = ∞ ↔ s.Infinite

case neg
α : Type u_1
inst✝ : MeasurableSpace α
s : Set α
s_mble : MeasurableSet s
hs : ¬s.Finite
⊢ count s = ∞ ↔ s.Infinite
Module: Mathlib.MeasureTheory.Measure.Count

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
s : Set α
inst✝ : MeasurableSingletonClass α
⊢ count s = ∞ ↔ s.Infinite

Tactic:
by_cases hs : s.Finite

Goal state after:
case pos
α : Type u_1
inst✝¹ : MeasurableSpace α
s : Set α
inst✝ : MeasurableSingletonClass α
hs : s.Finite
⊢ count s = ∞ ↔ s.Infinite

case neg
α : Type u_1
inst✝¹ : MeasurableSpace α
s : Set α
inst✝ : MeasurableSingletonClass α
hs : ¬s.Finite
⊢ count s = ∞ ↔ s.Infinite
Module: Mathlib.MeasureTheory.Measure.Count

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
s : Set α
h : count s = 0
x : α
hx : x ∈ s
⊢ False

Tactic:
simpa [hx] using ((ENNReal.le_tsum x).trans <| le_sum_apply _ _).trans_eq h

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : β → α
hf : Function.Injective f
s : Set β
s_mble : MeasurableSet s
fs_mble : MeasurableSet (f '' s)
⊢ count (f '' s) = count s

Tactic:
by_cases hs : s.Finite

Goal state after:
case pos
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : β → α
hf : Function.Injective f
s : Set β
s_mble : MeasurableSet s
fs_mble : MeasurableSet (f '' s)
hs : s.Finite
⊢ count (f '' s) = count s

case neg
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : β → α
hf : Function.Injective f
s : Set β
s_mble : MeasurableSet s
fs_mble : MeasurableSet (f '' s)
hs : ¬s.Finite
⊢ count (f '' s) = count s
Module: Mathlib.MeasureTheory.Measure.Count

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : β → α
hf : Function.Injective f
s : Finset β
s_mble : MeasurableSet ↑s
fs_mble : MeasurableSet (f '' ↑s)
⊢ count (f '' ↑s) = count ↑s

Tactic:
← Finset.coe_image,

Goal state after:
case pos
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : β → α
hf : Function.Injective f
s : Finset β
s_mble : MeasurableSet ↑s
fs_mble : MeasurableSet (f '' ↑s)
⊢ count ↑(Finset.image f s) = count ↑s
Module: Mathlib.MeasureTheory.Measure.Count

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSingletonClass α
inst✝ : MeasurableSingletonClass β
f : β → α
hf : Function.Injective f
s : Set β
⊢ count (f '' s) = count s

Tactic:
by_cases hs : s.Finite

Goal state after:
case pos
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSingletonClass α
inst✝ : MeasurableSingletonClass β
f : β → α
hf : Function.Injective f
s : Set β
hs : s.Finite
⊢ count (f '' s) = count s

case neg
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSingletonClass α
inst✝ : MeasurableSingletonClass β
f : β → α
hf : Function.Injective f
s : Set β
hs : ¬s.Finite
⊢ count (f '' s) = count s
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: 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 ∈ K, edist (x i) (y i) + ∑' (i : ↑(↑K)ᶜ), 2⁻¹ ^ encode ↑i < ∑ _i ∈ K, δ + ε / 2

Tactic:
by
            refine ENNReal.add_lt_add_of_le_of_lt (by simpa using fun i hi ↦ (hxy i hi).ne_top) ?_
              hK
            gcongr with i hi
            exact (hxy i hi).le

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

Goal state before:
pp: α : Type u_1
inst✝² : PseudoMetricSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsUnifLocDoublingMeasure μ
K : ℝ
C : ℝ≥0 := doublingConstant μ
n : ℕ
ihn : ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), μ (closedBall x (2 ^ n * (2 * ε))) ≤ ↑C ^ n * μ (closedBall x (2 * ε))
ε : ℝ
hεn : ∀ (x : α), μ (closedBall x (2 ^ n * (2 * ε))) ≤ ↑C ^ n * μ (closedBall x (2 * ε))
hε : ∀ (x : α), μ (closedBall x (2 * ε)) ≤ ↑(doublingConstant μ) * μ (closedBall x ε)
x : α
⊢ μ (closedBall x (2 ^ n * (2 * ε))) ≤ ↑C ^ (n + 1) * μ (closedBall x ε)

Tactic:
hεn,

Goal state after:
α : Type u_1
inst✝² : PseudoMetricSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsUnifLocDoublingMeasure μ
K : ℝ
C : ℝ≥0 := doublingConstant μ
n : ℕ
ihn : ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), μ (closedBall x (2 ^ n * (2 * ε))) ≤ ↑C ^ n * μ (closedBall x (2 * ε))
ε : ℝ
hεn : ∀ (x : α), μ (closedBall x (2 ^ n * (2 * ε))) ≤ ↑C ^ n * μ (closedBall x (2 * ε))
hε : ∀ (x : α), μ (closedBall x (2 * ε)) ≤ ↑(doublingConstant μ) * μ (closedBall x ε)
x : α
⊢ ↑C ^ n * μ (closedBall x (2 * ε)) ≤ ↑C ^ (n + 1) * μ (closedBall x ε)
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
ν : Measure β
inst✝ : IsFiniteMeasure ν
s : Set (α × β)
hs : MeasurableSet s
⊢ Measurable fun x ↦ ν (Prod.mk x ⁻¹' s)

Tactic:
induction s, hs using induction_on_inter generateFrom_prod.symm isPiSystem_prod with

Goal state after:
Module: Mathlib.Dynamics.Minimal

Goal state before:
pp: case mp
M : Type u_1
α : Type u_3
inst✝³ : Monoid M
inst✝² : TopologicalSpace α
inst✝¹ : MulAction M α
inst✝ : ContinuousConstSMul M α
⊢ IsMinimal M α → ∀ (s : Set α), IsClosed s → (∀ (c : M), c • s ⊆ s) → s = ∅ ∨ s = univ

Tactic:
intro _ _

Goal state after:
case mp
M : Type u_1
α : Type u_3
inst✝³ : Monoid M
inst✝² : TopologicalSpace α
inst✝¹ : MulAction M α
inst✝ : ContinuousConstSMul M α
a✝ : IsMinimal M α
s✝ : Set α
⊢ IsClosed s✝ → (∀ (c : M), c • s✝ ⊆ s✝) → s✝ = ∅ ∨ s✝ = univ
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : SFinite ν
inst✝ : IsProbabilityMeasure μ
⊢ map Prod.snd (μ.prod ν) = ν

Tactic:
rw [map_snd_prod, measure_univ, one_smul]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝ : SFinite ν
s : Set (α × β)
hs : MeasurableSet s
h2s : (μ.prod ν) s ≠ ∞
⊢ ∀ᵐ (x : α) ∂μ, ν (Prod.mk x ⁻¹' s) < ∞

Tactic:
rw [prod_apply hs] at h2s

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝ : SFinite ν
s : Set (α × β)
hs : MeasurableSet s
h2s : ∫⁻ (x : α), ν (Prod.mk x ⁻¹' s) ∂μ ≠ ∞
⊢ ∀ᵐ (x : α) ∂μ, ν (Prod.mk x ⁻¹' s) < ∞
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝ : SFinite ν
s : Set (α × β)
hs : MeasurableSet s
⊢ (μ.prod ν) s = 0 ↔ (fun x ↦ ν (Prod.mk x ⁻¹' s)) =ᵐ[μ] 0

Tactic:
prod_apply hs,

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝ : SFinite ν
s : Set (α × β)
hs : MeasurableSet s
⊢ ∫⁻ (x : α), ν (Prod.mk x ⁻¹' s) ∂μ = 0 ↔ (fun x ↦ ν (Prod.mk x ⁻¹' s)) =ᵐ[μ] 0
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : T2Space α
inst✝¹ : MeasurableSpace α
inst✝ : OpensMeasurableSpace α
s : Set α
hs : AnalyticSet s
hsc : AnalyticSet sᶜ
⊢ MeasurableSet s

Tactic:
rcases hs.measurablySeparable hsc disjoint_compl_right with ⟨u, hsu, hdu, hmu⟩
  obtain rfl : s = u := hsu.antisymm (disjoint_compl_left_iff_subset.1 hdu)
  exact hmu

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

Goal state before:
pp: case refine_1
α : Type u_4
β : Type u_5
γ : Type u_6
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ ν : Measure ((α × β) × γ)
inst✝ : IsFiniteMeasure μ
h_eq :
  ∀ (ν : Measure ((α × β) × γ)) {s : Set α} {t : Set β} {u : Set γ},
    MeasurableSet s →
      MeasurableSet t → MeasurableSet u → (map (⇑MeasurableEquiv.prodAssoc) ν) (s ×ˢ t ×ˢ u) = ν ((s ×ˢ t) ×ˢ u)
s : Set α
t : Set β
u : Set γ
h : (map (⇑MeasurableEquiv.prodAssoc) μ) (s ×ˢ t ×ˢ u) = (map (⇑MeasurableEquiv.prodAssoc) ν) (s ×ˢ t ×ˢ u)
hs : MeasurableSet s
ht : MeasurableSet t
hu : MeasurableSet u
⊢ μ ((s ×ˢ t) ×ˢ u) = ν ((s ×ˢ t) ×ˢ u)

pp: case refine_2
α : Type u_4
β : Type u_5
γ : Type u_6
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ ν : Measure ((α × β) × γ)
inst✝ : IsFiniteMeasure μ
h_eq :
  ∀ (ν : Measure ((α × β) × γ)) {s : Set α} {t : Set β} {u : Set γ},
    MeasurableSet s →
      MeasurableSet t → MeasurableSet u → (map (⇑MeasurableEquiv.prodAssoc) ν) (s ×ˢ t ×ˢ u) = ν ((s ×ˢ t) ×ˢ u)
s : Set α
t : Set β
u : Set γ
h : μ ((s ×ˢ t) ×ˢ u) = ν ((s ×ˢ t) ×ˢ u)
hs : MeasurableSet s
ht : MeasurableSet t
hu : MeasurableSet u
⊢ (map (⇑MeasurableEquiv.prodAssoc) μ) (s ×ˢ t ×ˢ u) = (map (⇑MeasurableEquiv.prodAssoc) ν) (s ×ˢ t ×ˢ u)

Tactic:
· rwa [h_eq μ hs ht hu, h_eq ν hs ht hu] at h

Goal state after:
case refine_2
α : Type u_4
β : Type u_5
γ : Type u_6
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ ν : Measure ((α × β) × γ)
inst✝ : IsFiniteMeasure μ
h_eq :
  ∀ (ν : Measure ((α × β) × γ)) {s : Set α} {t : Set β} {u : Set γ},
    MeasurableSet s →
      MeasurableSet t → MeasurableSet u → (map (⇑MeasurableEquiv.prodAssoc) ν) (s ×ˢ t ×ˢ u) = ν ((s ×ˢ t) ×ˢ u)
s : Set α
t : Set β
u : Set γ
h : μ ((s ×ˢ t) ×ˢ u) = ν ((s ×ˢ t) ×ˢ u)
hs : MeasurableSet s
ht : MeasurableSet t
hu : MeasurableSet u
⊢ (map (⇑MeasurableEquiv.prodAssoc) μ) (s ×ˢ t ×ˢ u) = (map (⇑MeasurableEquiv.prodAssoc) ν) (s ×ˢ t ×ˢ u)
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
p : Set α → Prop
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).InnerRegularWRT p MeasurableSet
hs : univ ⊆ ⋃ n, s n
hmono : Monotone s
⊢ μ.InnerRegularWRT p MeasurableSet

Tactic:
intro F hF r hr

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
p : Set α → Prop
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).InnerRegularWRT p MeasurableSet
hs : univ ⊆ ⋃ n, s n
hmono : Monotone s
F : Set α
hF : MeasurableSet F
r : ℝ≥0∞
hr : r < μ F
⊢ ∃ K ⊆ F, p K ∧ r < μ K
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
ν : Measure β
inst✝² : SFinite ν
μ : Measure α
R : Type u_4
inst✝¹ : SMul R ℝ≥0∞
inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
c : R
s : Set (α × β)
hs : MeasurableSet s
⊢ ((c • μ).prod ν) s = (c • μ.prod ν) s

Tactic:
prod_apply hs,

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
ν : Measure β
inst✝² : SFinite ν
μ : Measure α
R : Type u_4
inst✝¹ : SMul R ℝ≥0∞
inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
c : R
s : Set (α × β)
hs : MeasurableSet s
⊢ ∫⁻ (x : α), ν (Prod.mk x ⁻¹' s) ∂c • μ = (c • μ.prod ν) s
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: X : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : SigmaCompactSpace X
inst✝ : MeasurableSpace X
μ : Measure X
⊢ μ.InnerRegularWRT IsCompact IsClosed[inst✝²]

Tactic:
intro F hF r hr

Goal state after:
X : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : SigmaCompactSpace X
inst✝ : MeasurableSpace X
μ : Measure X
F : Set X
hF : IsClosed[inst✝²] F
r : ℝ≥0∞
hr : r < μ F
⊢ ∃ K ⊆ F, IsCompact K ∧ r < μ K
Module: Mathlib.MeasureTheory.Measure.Prod

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
X : α → β
Y : α → γ
μ : Measure α
hX : AEMeasurable X μ
hY : ¬AEMeasurable Y μ
this : ¬AEMeasurable (fun x ↦ (X x, Y x)) μ
⊢ (map (fun a ↦ (X a, Y a)) μ).snd = map Y μ

Tactic:
simp [map_of_not_aemeasurable, hY, this]

Goal state after: