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

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
E : Set α
⊢ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E = ⋂ δ, ⋂ (_ : 0 < δ), thickening δ E

Tactic:
rw [← cthickening_zero]
  exact cthickening_eq_iInter_thickening rfl.ge E

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

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
δ ε : ℝ
hε : 0 ≤ ε
hδ : 0 ≤ δ
s : Set α
⊢ cthickening ε (cthickening δ s) ⊆ cthickening (ε + δ) s

Tactic:
by
  intro x
  simp_rw [mem_cthickening_iff, ENNReal.ofReal_add hε hδ]
  exact fun hx => infEDist_le_infEDist_cthickening_add.trans (by grw [hx])

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 : AddLeftReflectLT α
inst✝ : IsLeftCancelAdd α
⊢ (∀ a < b + c, P a) ↔ (∀ a < b, P a) ∧ ∀ d < c, P (b + d)

Tactic:
lt_add_iff_lt_left_or_exists_lt

Goal state after:
α : Type u_1
inst✝⁴ : LinearOrder α
P : α → Prop
b c : α
inst✝³ : Add α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : AddLeftReflectLT α
inst✝ : IsLeftCancelAdd α
⊢ (∀ (a : α), (a < b ∨ ∃ d < c, a = b + d) → P a) ↔ (∀ a < b, P a) ∧ ∀ d < c, P (b + d)
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 : AddLeftReflectLT α
inst✝ : IsLeftCancelAdd α
⊢ (∃ a < b + c, P a) ↔ (∃ a < b, P a) ∨ ∃ d < c, P (b + d)

Tactic:
lt_add_iff_lt_left_or_exists_lt

Goal state after:
α : Type u_1
inst✝⁴ : LinearOrder α
P : α → Prop
b c : α
inst✝³ : Add α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : AddLeftReflectLT α
inst✝ : IsLeftCancelAdd α
⊢ (∃ a, (a < b ∨ ∃ d < c, a = b + d) ∧ P a) ↔ (∃ a < b, P a) ∨ ∃ d < c, P (b + d)
Module: Mathlib.Algebra.Order.Monoid.Canonical.Basic

Goal state before:
pp: α : Type u_1
inst✝⁴ : LinearOrder α
a b c : α
inst✝³ : AddCommMagma α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : AddLeftReflectLT α
inst✝ : IsLeftCancelAdd α
⊢ a < c + b ↔ a < c ∨ ∃ d < b, a = d + c

Tactic:
lt_add_iff_lt_left_or_exists_lt

Goal state after:
α : Type u_1
inst✝⁴ : LinearOrder α
a b c : α
inst✝³ : AddCommMagma α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : AddLeftReflectLT α
inst✝ : IsLeftCancelAdd α
⊢ (a < c ∨ ∃ d < b, a = c + d) ↔ a < c ∨ ∃ d < b, a = d + c
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case inr
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
this : {b | BddAbove (range fun i ↦ f i b)} = {x | ∃ n, ∀ (i : ι), f i x ≤ u n}
⊢ MeasurableSet {b | BddAbove (range fun i ↦ f i b)}

Tactic:
rw [this, ofPred_exists]

Goal state after:
case inr
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
this : {b | BddAbove (range fun i ↦ f i b)} = {x | ∃ n, ∀ (i : ι), f i x ≤ u n}
⊢ MeasurableSet (⋃ i, {x | ∀ (i_1 : ι), f i_1 x ≤ u i})
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), AEMeasurable (f i) μ
b : δ
hb : ∀ (i : ι), f i b = mk (f i) ⋯ b
⊢ ⨆ i, f i b = ⨆ i, mk (f i) ⋯ b

Tactic:
simp [hb]

Goal state after:
Module: Mathlib.Topology.IndicatorConstPointwise

Goal state before:
pp: case refine_3
α : Type u_1
A : Set α
β : Type u_2
inst✝¹ : Zero β
inst✝ : TopologicalSpace β
ι : Type u_3
L : Filter ι
As : ι → Set α
b : β
nhds_b : {0}ᶜ ∈ 𝓝 b
nhds_o : {b}ᶜ ∈ 𝓝 0
x : α
heart : Tendsto (fun i ↦ if x ∈ As i then b else 0) L (if x ∈ A then 𝓝 b else 𝓝 0) ↔ ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A
⊢ Tendsto (fun i ↦ (As i).indicator (fun x ↦ b) x) L (𝓝 (A.indicator (fun x ↦ b) x)) ↔ ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A

Tactic:
convert! heart
    by_cases hxA : x ∈ A <;> simp [hxA]

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ι : Type u_5
f : ι → α → ℝ≥0
g : α → ℝ≥0
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
hf : ∀ (i : ι), Measurable (f i)
lim : Tendsto f u (𝓝 g)
⊢ Measurable g

Tactic:
by
  simp_rw [← measurable_coe_nnreal_ennreal_iff] at hf ⊢
  refine ENNReal.measurable_of_tendsto' u hf ?_
  rw [tendsto_pi_nhds] at lim ⊢
  exact fun x => (ENNReal.continuous_coe.tendsto (g x)).comp (lim x)

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

Goal state before:
pp: α : Type u_1
m₀ : MeasurableSpace α
μ : Measure α
E : Type u_5
inst✝¹ : TopologicalSpace E
inst✝ : MetrizableSpace E
f g : α → E
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
x : α
hfx : f x = AEStronglyMeasurable.mk f hf x
hgx : g x = AEStronglyMeasurable.mk g hg x
⊢ (AEStronglyMeasurable.mk f hf x = AEStronglyMeasurable.mk g hg x) = (f x = g x)

Tactic:
simp only [hfx, hgx]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
inst✝² : Preorder β
inst✝¹ : OrderClosedTopology β
inst✝ : PseudoMetrizableSpace β
f g : α → β
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
x : α
hfx : f x = AEStronglyMeasurable.mk f hf x
hgx : g x = AEStronglyMeasurable.mk g hg x
⊢ (AEStronglyMeasurable.mk f hf x < AEStronglyMeasurable.mk g hg x) = (f x < g x)

Tactic:
simp only [hfx, hgx]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
inst✝² : Preorder β
inst✝¹ : OrderClosedTopology β
inst✝ : PseudoMetrizableSpace β
f g : α → β
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
x : α
hfx : f x = AEStronglyMeasurable.mk f hf x
hgx : g x = AEStronglyMeasurable.mk g hg x
⊢ (AEStronglyMeasurable.mk f hf x ≤ AEStronglyMeasurable.mk g hg x) = (f x ≤ g x)

Tactic:
simp only [hfx, hgx]

Goal state after:
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), Measurable (f n)
h_mono : Monotone f
c : ℝ≥0 → ℝ≥0∞ := ofNNReal
F : α → ℝ≥0∞ := fun a ↦ ⨆ n, f n a
⊢ ⨆ φ, ⨆ (_ : ∀ (x : α), ↑(φ x) ≤ ⨆ n, f n x), (SimpleFunc.map ofNNReal φ).lintegral μ ≤ ⨆ n, ∫⁻ (a : α), f n a ∂μ

Tactic:
refine iSup_le fun s => iSup_le fun hsf => ?_

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), Measurable (f n)
h_mono : Monotone f
c : ℝ≥0 → ℝ≥0∞ := ofNNReal
F : α → ℝ≥0∞ := fun a ↦ ⨆ n, f n a
s : α →ₛ ℝ≥0
hsf : ∀ (x : α), ↑(s x) ≤ ⨆ n, f n x
⊢ (SimpleFunc.map ofNNReal s).lintegral μ ≤ ⨆ n, ∫⁻ (a : α), f n a ∂μ
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
inst✝ : PseudoMetrizableSpace β
f : ℕ → α → β
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n ↦ f n x) atTop (𝓝 l)
⊢ ∃ f_lim, StronglyMeasurable f_lim ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (f_lim x))

Tactic:
borelize β
  obtain ⟨g, _, hg⟩ :
    ∃ g : α → β, Measurable g ∧ ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) :=
    measurable_limit_of_tendsto_metrizable_ae (fun n => (hf n).aemeasurable) h_ae_tendsto
  have Hg : AEStronglyMeasurable g μ := aestronglyMeasurable_of_tendsto_ae _ hf hg
  refine ⟨Hg.mk g, Hg.stronglyMeasurable_mk, ?_⟩
  filter_upwards [hg, Hg.ae_eq_mk] with x hx h'x
  rwa [h'x] at hx

Goal state after:
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 := ⋯
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
f_approx : ℕ → α →ₛ β := ⋯
fs : ℕ → α →ₛ β := ⋯
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, ∀ (m : ℕ), n ≤ m → x ∈ S m

Tactic:
rsuffices ⟨n, hn⟩ : ∃ n, x ∈ S n

Goal state after:
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 : x ∈ S n
⊢ ∃ n, ∀ (m : ℕ), n ≤ m → x ∈ S m

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 := ⋯
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
f_approx : ℕ → α →ₛ β := ⋯
fs : ℕ → α →ₛ β := ⋯
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, x ∈ S n
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
f g : α → β
mα : MeasurableSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : GroupWithZero β
inst✝² : ContinuousMul β
inst✝¹ : ContinuousInv₀ β
inst✝ : MetrizableSpace β
hf : StronglyMeasurable f
hg : StronglyMeasurable g
⊢ StronglyMeasurable (f / g)

Tactic:
borelize β
  refine ⟨fun n => hf.approx n / (hg.approx n).restrict {x | g x ≠ 0}, fun x => ?_⟩
  have : MeasurableSet {x | g x ≠ 0} := ((MeasurableSet.singleton 0).preimage hg.measurable).compl
  by_cases h : g x = 0
  · simp_all only [ne_eq, SimpleFunc.coe_div, SimpleFunc.coe_restrict, Pi.div_apply, mem_ofPred_eq,
      not_true_eq_false, not_false_eq_true, indicator_of_notMem, _root_.div_zero]
    exact tendsto_const_nhds
  · simp_all only [ne_eq, SimpleFunc.coe_div, SimpleFunc.coe_restrict,
      Pi.div_apply, mem_ofPred_eq, not_false_eq_true, indicator_of_mem]
    exact (hf.tendsto_approx x).div (hg.tendsto_approx x) h

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

Goal state before:
pp: α : Type u_5
E : Type u_6
x✝ : MeasurableSpace α
inst✝⁶ : AddGroup E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : ContinuousAdd E
inst✝¹ : ContinuousNeg E
inst✝ : PseudoMetrizableSpace E
g f : α → E
hg : Measurable g
hf : StronglyMeasurable f
⊢ Measurable (g + -f)

Tactic:
exact hg.add_stronglyMeasurable hf.neg

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

Goal state before:
pp: case insert
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
f : β → α → ℝ≥0∞
a : β
s : Finset β
has : a ∉ s
ih : (∀ b ∈ s, AEMeasurable (f b) μ) → ∫⁻ (a : α), ∑ b ∈ s, f b a ∂μ = ∑ b ∈ s, ∫⁻ (a : α), f b a ∂μ
hf : ∀ b ∈ insert a s, AEMeasurable (f b) μ
⊢ ∫⁻ (a_1 : α), ∑ b ∈ insert a s, f b a_1 ∂μ = ∑ b ∈ insert a s, ∫⁻ (a : α), f b a ∂μ

Tactic:
simp only [Finset.sum_insert has]
    rw [Finset.forall_mem_insert] at hf
    rw [lintegral_add_left' hf.1, ih hf.2]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
m : MeasurableSpace α
inst✝³ : TopologicalSpace β
inst✝² : PseudoMetrizableSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : PseudoMetrizableSpace γ
g : β → γ
f : α → β
hg : IsEmbedding g
⊢ (StronglyMeasurable fun x ↦ g (f x)) ↔ StronglyMeasurable f

Tactic:
by
  let := pseudoMetrizableSpacePseudoMetric γ
  borelize β γ
  refine
    ⟨fun H => stronglyMeasurable_iff_measurable_separable.2 ⟨?_, ?_⟩, fun H =>
      hg.continuous.comp_stronglyMeasurable H⟩
  · 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 : Measurable (G ∘ f) := Measurable.subtype_mk H.measurable
    exact hG.measurableEmbedding.measurable_comp_iff.1 this
  · have : IsSeparable (g ⁻¹' range (g ∘ f)) := hg.isSeparable_preimage H.isSeparable_range
    rwa [range_comp, hg.injective.preimage_image] at this

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✝ : IsFiniteMeasure ν
n : ℕ
hC_lt : ⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s ≤ 1 / ↑n
⊢ ∃ t,
    MeasurableSet t ∧
      SigmaFinite (μ.restrict t) ∧
        (⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s) - 1 / ↑n ≤ ν t

Tactic:
refine ⟨∅, MeasurableSet.empty, by rw [Measure.restrict_empty]; infer_instance, ?_⟩
    rw [tsub_eq_zero_of_le hC_lt]
    exact zero_le

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

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : Zero β
s : Set α
f : α → β
hs : MeasurableSet s
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
⊢ ∀ (x : α), Tendsto (fun n ↦ ((fun n ↦ (hf.approx n).restrict s) n) x) atTop (𝓝 (f x))

pp: case refine_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : Zero β
s : Set α
f : α → β
hs : MeasurableSet s
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
⊢ ∀ x ∉ s, ∀ (n : ℕ), ((fun n ↦ (hf.approx n).restrict s) n) x = 0

Tactic:
· intro x
    by_cases hx : x ∈ s
    · simpa [SimpleFunc.coe_restrict, hs, hx] using hf.tendsto_approx x
    · simpa [SimpleFunc.coe_restrict, hs, hx, hf_zero x hx] using tendsto_const_nhds

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : Zero β
s : Set α
f : α → β
hs : MeasurableSet s
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
⊢ ∀ x ∉ s, ∀ (n : ℕ), ((fun n ↦ (hf.approx n).restrict s) n) x = 0
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: case mpr
ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
s : Set ((i : ι) × E i)
H : ∀ x ∈ s, ∃ ε > 0, ∀ (y : (i : ι) × E i), dist x y < ε → y ∈ s
i : ι
x : E i
hx : x ∈ Sigma.mk i ⁻¹' s
ε : ℝ
εpos : ε > 0
hε : ∀ (y : (j : ι) × E j), dist ⟨i, x⟩ y < ε → y ∈ s
y : E i
hy : y ∈ ball x ε
⊢ dist ⟨i, x⟩ ⟨i, y⟩ < ε

Tactic:
Sigma.dist_same

Goal state after:
case mpr
ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
s : Set ((i : ι) × E i)
H : ∀ x ∈ s, ∃ ε > 0, ∀ (y : (i : ι) × E i), dist x y < ε → y ∈ s
i : ι
x : E i
hx : x ∈ Sigma.mk i ⁻¹' s
ε : ℝ
εpos : ε > 0
hε : ∀ (y : (j : ι) × E j), dist ⟨i, x⟩ y < ε → y ∈ s
y : E i
hy : y ∈ ball x ε
⊢ dist x y < ε
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: α : Type u_1
inst✝² : MeasurableSpace α
τ : TopologicalSpace α
inst✝¹ : BorelSpace α
inst✝ : PolishSpace α
⊢ StandardBorelSpace α

Tactic:
exists τ

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

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
g : α → β
hg : MeasurableEmbedding g
f : β → ℝ≥0∞
f₀ : SimpleFunc α ℝ≥0∞
hf₀ : ⇑f₀ ≤ fun a ↦ f (g a)
⊢ ∫⁻ (a : β), (f₀.extend g hg (const β 0)) a ∂Measure.map g μ ≤
    ⨆ g_1, ⨆ (_ : ⇑g_1 ≤ fun a ↦ f a), g_1.lintegral (Measure.map g μ)

Tactic:
← lintegral

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
g : α → β
hg : MeasurableEmbedding g
f : β → ℝ≥0∞
f₀ : SimpleFunc α ℝ≥0∞
hf₀ : ⇑f₀ ≤ fun a ↦ f (g a)
⊢ ∫⁻ (a : β), (f₀.extend g hg (const β 0)) a ∂Measure.map g μ ≤ ∫⁻ (a : β), f a ∂Measure.map g μ
Module: Mathlib.MeasureTheory.Measure.MutuallySingular

Goal state before:
pp: case refine_1
α : Type u_1
m0 : MeasurableSpace α
ν : Measure α
ι : Type u_2
inst✝ : Countable ι
μ : ι → Measure α
s : ι → Set α
hsm : ∀ (i : ι), MeasurableSet (s i)
hsμ : ∀ (i : ι), (μ i) (s i) = 0
hsν : ∀ (i : ι), ν (s i)ᶜ = 0
⊢ (sum μ) (⋂ i, s i) = 0

Tactic:
rw [sum_apply _ (MeasurableSet.iInter hsm), ENNReal.tsum_eq_zero]

Goal state after:
case refine_1
α : Type u_1
m0 : MeasurableSpace α
ν : Measure α
ι : Type u_2
inst✝ : Countable ι
μ : ι → Measure α
s : ι → Set α
hsm : ∀ (i : ι), MeasurableSet (s i)
hsμ : ∀ (i : ι), (μ i) (s i) = 0
hsν : ∀ (i : ι), ν (s i)ᶜ = 0
⊢ ∀ (i : ι), (μ i) (⋂ b, s b) = 0
Module: Mathlib.MeasureTheory.Measure.Count

Goal state before:
pp: 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) = ∞

Tactic:
rw [← finite_image_iff hf.injOn] at hs

Goal state after:
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 : ¬(f '' s).Finite
⊢ count (f '' s) = ∞
Module: Mathlib.MeasureTheory.Measure.Count

Goal state before:
pp: 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) = ∞

Tactic:
rw [← finite_image_iff hf.injOn] at hs

Goal state after:
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 : ¬(f '' s).Finite
⊢ count (f '' s) = ∞
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
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)
n : ℕ
⊢ (↑(p n)).1 = n

Tactic:
induction n with
    | zero => rfl
    | succ n IH => simp only [prec, hFn, IH]

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

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
μ✝ : Measure α
f : α → ℝ≥0∞
μ : Measure α
inst✝ : SFinite μ
h : IsFiniteMeasure μ
g : ℕ → α → ℝ≥0∞
hgm : ∀ (n : ℕ), Measurable (g n)
hgf : ∀ (n : ℕ), g n ≤ f
hgle : ∀ (n : ℕ), g n ≤ ↑n
hgint : ∀ (n : ℕ), ∫⁻ (a : α), min (f a) ↑n ∂μ = ∫⁻ (a : α), g n a ∂μ
φ : α → ℝ≥0∞ := fun x ↦ ⨆ n, g n x
hφm : Measurable φ
hφle : φ ≤ f
s : Set α
ψ : SimpleFunc α ℝ≥0
hψ : ∀ (x : α), ↑(ψ x) ≤ f x
n : ℕ
hn : ∀ (x : α), ψ x ≤ ↑n
this : ∫⁻ (a : α) in (toMeasurable μ s)ᶜ, min (f a) ↑n ∂μ ≠ ∞
hsm : MeasurableSet (toMeasurable μ s)
⊢ ∫⁻ (a : α) in s, min (f a) ↑n ∂μ ≤ ∫⁻ (a : α) in s, g n a ∂μ

Tactic:
apply ENNReal.le_of_add_le_add_right this

Goal state after:
α : Type u_1
inst✝¹ : MeasurableSpace α
μ✝ : Measure α
f : α → ℝ≥0∞
μ : Measure α
inst✝ : SFinite μ
h : IsFiniteMeasure μ
g : ℕ → α → ℝ≥0∞
hgm : ∀ (n : ℕ), Measurable (g n)
hgf : ∀ (n : ℕ), g n ≤ f
hgle : ∀ (n : ℕ), g n ≤ ↑n
hgint : ∀ (n : ℕ), ∫⁻ (a : α), min (f a) ↑n ∂μ = ∫⁻ (a : α), g n a ∂μ
φ : α → ℝ≥0∞ := ⋯
hφm : Measurable φ
hφle : φ ≤ f
s : Set α
ψ : SimpleFunc α ℝ≥0
hψ : ∀ (x : α), ↑(ψ x) ≤ f x
n : ℕ
hn : ∀ (x : α), ψ x ≤ ↑n
this : ∫⁻ (a : α) in (toMeasurable μ s)ᶜ, min (f a) ↑n ∂μ ≠ ∞
hsm : MeasurableSet (toMeasurable μ s)
⊢ ∫⁻ (a : α) in s, min (f a) ↑n ∂μ + ∫⁻ (a : α) in (toMeasurable μ s)ᶜ, min (f a) ↑n ∂μ ≤
    ∫⁻ (a : α) in s, g n a ∂μ + ∫⁻ (a : α) in (toMeasurable μ s)ᶜ, min (f a) ↑n ∂μ
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) → MetricSpace (F i)
⊢ ∀ (x y : (i : ι) → F i), edist x y = ENNReal.ofReal (dist x y)

Tactic:
simp [edist_dist]

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

Goal state before:
pp: E : ℕ → Type u_1
ι : Type u_2
X : Type u_3
Y : ι → Type u_4
f : (i : ι) → X → Y i
inst✝¹ : Encodable ι
inst✝ : (i : ι) → MetricSpace (Y i)
separating_f : Pairwise fun x y ↦ ∃ i, f i x ≠ f i y
x y : PiNatEmbed X Y f
⊢ edist x y ≠ ∞

Tactic:
simp [edist_dist]

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

Goal state before:
pp: X : Type u_3
inst✝¹ : MetricSpace X
inst✝ : SeparableSpace X
x : X
C : Set X
hxC : C ∈ 𝓝 x
⊢ ∃ n, C ∈ comap (distDenseSeq X n) (𝓝 (distDenseSeq X n x))

Tactic:
let ε : ℝ := min (infDist x (closure Cᶜ)) 1

Goal state after:
X : Type u_3
inst✝¹ : MetricSpace X
inst✝ : SeparableSpace X
x : X
C : Set X
hxC : C ∈ 𝓝 x
ε : ℝ := min (infDist x (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ)) 1
⊢ ∃ n, C ∈ comap (distDenseSeq X n) (𝓝 (distDenseSeq X n x))