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

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
x : α
s t : Set α
ε : ℝ≥0
εpos : 0 < ε
h : infEDist x s + hausdorffEDist s t < ∞
⊢ infEDist x t ≤ infEDist x s + hausdorffEDist s t + ↑ε

Tactic:
by
    have ε0 : (ε / 2 : ℝ≥0∞) ≠ 0 := by simpa [pos_iff_ne_zero] using εpos
    have : infEDist x s < infEDist x s + ε / 2 :=
      ENNReal.lt_add_right (ENNReal.add_lt_top.1 h).1.ne ε0
    obtain ⟨y : α, ys : y ∈ s, dxy : edist x y < infEDist x s + ↑ε / 2⟩ := infEDist_lt_iff.mp this
    have : hausdorffEDist s t < hausdorffEDist s t + ε / 2 :=
      ENNReal.lt_add_right (ENNReal.add_lt_top.1 h).2.ne ε0
    obtain ⟨z : α, zt : z ∈ t, dyz : edist y z < hausdorffEDist s t + ↑ε / 2⟩ :=
      exists_edist_lt_of_hausdorffEDist_lt ys this
    calc
      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.HausdorffDistance

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
s t u : Set α
⊢ hausdorffEDist s u ≤ hausdorffEDist s t + hausdorffEDist t u

Tactic:
hausdorffEDist_def

Goal state after:
α : Type u
inst✝ : PseudoEMetricSpace α
s t u : Set α
⊢ max (⨆ x ∈ s, infEDist x u) (⨆ y ∈ u, infEDist y s) ≤ hausdorffEDist s t + hausdorffEDist t u
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case pos
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
a : α
ha : a ∈ t
b : α
hb : b ∈ t
⊢ MeasurableSet (⋃ (_ : a < b), ⋃ (_ : b ≤ a✝), Ico a b)

Tactic:
refine MeasurableSet.iUnion fun hab => MeasurableSet.iUnion fun _ => ?_

Goal state after:
case pos
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
a : α
ha : a ∈ t
b : α
hb : b ∈ t
hab : a < b
x✝ : b ≤ a✝
⊢ MeasurableSet (Ico a b)
Module: Mathlib.Topology.MetricSpace.Thickening

Goal state before:
pp: α : Type u_2
inst✝ : PseudoMetricSpace α
x : α
E : Set α
hx : x ∈ E
δ : ℝ
⊢ {x} ⊆ E

Tactic:
simpa using hx

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✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : β →ₛ γ
g : α → β
hgm : Measurable g
⊢ ↑(f.comp g hgm).range ⊆ ↑f.range

Tactic:
simp only [coe_range, coe_comp, Set.range_comp_subset_range]

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

Goal state before:
pp: α : Type u_2
inst✝ : PseudoMetricSpace α
x : α
ε δ : ℝ
⊢ thickening ε (thickening δ {x}) ⊆ thickening (ε + δ) {x}

Tactic:
apply thickening_thickening_subset

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

Goal state before:
pp: case a.refine_1
α : Type u_3
β : Type u_4
C : Set (Set α)
D : Set (Set β)
hC : IsCountablySpanning C
hD : IsCountablySpanning D
s : Set α
hs : s ∈ C
⊢ MeasurableSet (Prod.fst ⁻¹' s)

Tactic:
rcases hD with ⟨t, h1t, h2t⟩

Goal state after:
case a.refine_1
α : Type u_3
β : Type u_4
C : Set (Set α)
D : Set (Set β)
hC : IsCountablySpanning C
s : Set α
hs : s ∈ C
t : ℕ → Set β
h1t : ∀ (n : ℕ), t n ∈ D
h2t : ⋃ n, t n = univ
⊢ MeasurableSet (Prod.fst ⁻¹' s)
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 : Zero β
inst✝ : Zero γ
g : β → γ
hg : g 0 = 0
f : α →ₛ β
s : Set α
x : α
hs : ¬MeasurableSet s
⊢ (map g (f.restrict s)) x = ((map g f).restrict s) x

Tactic:
simp [restrict_of_not_measurable hs, hg]

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

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : CompleteLattice β
inst✝³ : OrderClosedTopology β
inst✝² : Zero β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
i : ℕ → β
f : α → β
a : α
hf : Measurable f
h_zero : 0 = ⊥
n : ℕ
⊢ ((Finset.range n).sup fun k ↦ if i k ≤ f a then i k else 0) ≤ ⨆ k, ⨆ (_ : i k ≤ f a), i k

Tactic:
h_zero

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : CompleteLattice β
inst✝³ : OrderClosedTopology β
inst✝² : Zero β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
i : ℕ → β
f : α → β
a : α
hf : Measurable f
h_zero : 0 = ⊥
n : ℕ
⊢ ((Finset.range n).sup fun k ↦ if i k ≤ f a then i k else ⊥) ≤ ⨆ k, ⨆ (_ : i k ≤ f a), i k
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
⊢ NullMeasurableSet {a | f a ≤ g a} μ

Tactic:
by
  apply
    (hf.stronglyMeasurable_mk.measurableSet_le hg.stronglyMeasurable_mk).nullMeasurableSet.congr
  filter_upwards [hf.ae_eq_mk, hg.ae_eq_mk] with x hfx hgx
  change (hf.mk f x ≤ hg.mk g x) = (f x ≤ g x)
  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✝² : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
mβ : MeasurableSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : BorelSpace β
f : α → β
hf : AEStronglyMeasurable f μ
⊢ AEStronglyMeasurable id (Measure.map f μ)

Tactic:
obtain ⟨t, ht1, ht2⟩ := hf.isSeparable_ae_range

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
mβ : MeasurableSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : BorelSpace β
f : α → β
hf : AEStronglyMeasurable f μ
t : Set β
ht1 : IsSeparable t
ht2 : ∀ᵐ (x : α) ∂μ, f x ∈ t
⊢ AEStronglyMeasurable id (Measure.map f μ)
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: case refine_1
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
n : ℕ
y : β
hy : y ∈ ↑({y ∈ (fs n).range | y ≠ 0})
⊢ y ≠ 0

Tactic:
let : (y : β) → Decidable (y = 0) := fun y => Classical.propDecidable _
      rw [Finset.mem_coe, Finset.mem_filter] at hy
      exact hy.2

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 := 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, ∀ (m : ℕ), n ≤ m → x ∈ S m ∩ t

Tactic:
rsuffices ⟨n, hn⟩ : ∃ n, ∀ m, n ≤ m → x ∈ S m
        · exact ⟨n, fun m hnm => Set.mem_inter (hn m hnm) hxt⟩
        rsuffices ⟨n, hn⟩ : ∃ n, x ∈ S n
        · exact ⟨n, fun m hnm => monotone_spanningSets (μ.restrict t) hnm hn⟩
        rw [← Set.mem_iUnion, iUnion_spanningSets (μ.restrict t)]
        trivial

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

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

Tactic:
rw [Measure.restrict_empty, lintegral_zero_measure]

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

Goal state before:
pp: case pos
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 : ∀ s ∈ 𝓝 (f x), ∃ a, ∀ (b : ℕ), a ≤ b → (f_approx b) x ∈ s
n₁ : ℕ
hn₁ : ∀ (m_1 : ℕ), n₁ ≤ m_1 → (fs m_1) x = (f_approx m_1) x
s : Set β
hs : s ∈ 𝓝 (f x)
⊢ ∃ a, ∀ (b : ℕ), a ≤ b → (fs b) x ∈ s

Tactic:
obtain ⟨n₂, hn₂⟩ := h s hs

Goal state after:
case pos
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 : ∀ s ∈ 𝓝 (f x), ∃ a, ∀ (b : ℕ), a ≤ b → (f_approx b) x ∈ s
n₁ : ℕ
hn₁ : ∀ (m_1 : ℕ), n₁ ≤ m_1 → (fs m_1) x = (f_approx m_1) x
s : Set β
hs : s ∈ 𝓝 (f x)
n₂ : ℕ
hn₂ : ∀ (b : ℕ), n₂ ≤ b → (f_approx b) x ∈ s
⊢ ∃ a, ∀ (b : ℕ), a ≤ b → (fs b) x ∈ s
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s t : Set α
⊢ ∫⁻ (a : α) in s ∪ t, f a ∂μ ≤ ∫⁻ (a : α) in s, f a ∂μ + ∫⁻ (a : α) in t, f a ∂μ

Tactic:
by
  rw [← lintegral_add_measure]
  exact lintegral_mono' (restrict_union_le _ _) le_rfl

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

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

Tactic:
intro n m hnm a

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

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
⊢ ∫⁻ (x : α), ENNReal.ofReal (f x) ∂μ ≤ ∫⁻ (x : α), ‖f x‖ₑ ∂μ

Tactic:
by
  simp_rw [← ofReal_norm]
  refine lintegral_mono fun x => ENNReal.ofReal_le_ofReal ?_
  rw [Real.norm_eq_abs]
  exact le_abs_self (f x)

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
⊢ 0 ≤ ν ∅

Tactic:
exact zero_le

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure ν
n : ℕ
⊢ ν (μ.sigmaFiniteSetGE ν n) ≤ ⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s

Tactic:
by
  refine (le_iSup (f := fun s ↦ _)
    (sigmaFinite_restrict_sigmaFiniteSetGE μ ν n)).trans ?_
  exact le_iSup₂ (f := fun s _ ↦ ⨆ (_ : SigmaFinite (μ.restrict s)), ν s) (μ.sigmaFiniteSetGE ν n)
    (measurableSet_sigmaFiniteSetGE n)

Goal state after:
Module: Mathlib.Topology.Perfect

Goal state before:
pp: case h.left
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : T25Space α
hC : Perfect C
y : α
yC : y ∈ C
x : α
xC : x ∈ C
hxy : x ≠ y
U : Set α
xU : x ∈ U
Uop : IsOpen[inst✝¹] U
V : Set α
yV : y ∈ V
Vop : IsOpen[inst✝¹] V
hUV : Disjoint (closure[inst✝¹] U) (closure[inst✝¹] V)
⊢ closure[inst✝¹] (U ∩ C) ⊆ C

Tactic:
hC.closed.closure_subset_iff

Goal state after:
case h.left
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : T25Space α
hC : Perfect C
y : α
yC : y ∈ C
x : α
xC : x ∈ C
hxy : x ≠ y
U : Set α
xU : x ∈ U
Uop : IsOpen[inst✝¹] U
V : Set α
yV : y ∈ V
Vop : IsOpen[inst✝¹] V
hUV : Disjoint (closure[inst✝¹] U) (closure[inst✝¹] V)
⊢ U ∩ C ⊆ C
Module: Mathlib.Topology.Perfect

Goal state before:
pp: case h.right.left
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : T25Space α
hC : Perfect C
y : α
yC : y ∈ C
x : α
xC : x ∈ C
hxy : x ≠ y
U : Set α
xU : x ∈ U
Uop : IsOpen[inst✝¹] U
V : Set α
yV : y ∈ V
Vop : IsOpen[inst✝¹] V
hUV : Disjoint (closure[inst✝¹] U) (closure[inst✝¹] V)
⊢ closure[inst✝¹] (V ∩ C) ⊆ C

Tactic:
hC.closed.closure_subset_iff

Goal state after:
case h.right.left
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : T25Space α
hC : Perfect C
y : α
yC : y ∈ C
x : α
xC : x ∈ C
hxy : x ≠ y
U : Set α
xU : x ∈ U
Uop : IsOpen[inst✝¹] U
V : Set α
yV : y ∈ V
Vop : IsOpen[inst✝¹] V
hUV : Disjoint (closure[inst✝¹] U) (closure[inst✝¹] V)
⊢ V ∩ C ⊆ C
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: a b : ℝ
x y z : ↑(Icc a b)
s t : ↑unitInterval
⊢ convexComb (convexComb z y (convexComb_assoc_coeff₁ (unitInterval.symm t) (unitInterval.symm s))) x
      (convexComb_assoc_coeff₂ (unitInterval.symm t) (unitInterval.symm s)) =
    convexComb x (convexComb y z (convexComb_assoc_coeff₂' s t)) (convexComb_assoc_coeff₁' s t)

Tactic:
← convexComb_symm x,

Goal state after:
a b : ℝ
x y z : ↑(Icc a b)
s t : ↑unitInterval
⊢ convexComb x (convexComb z y (convexComb_assoc_coeff₁ (unitInterval.symm t) (unitInterval.symm s)))
      (unitInterval.symm (convexComb_assoc_coeff₂ (unitInterval.symm t) (unitInterval.symm s))) =
    convexComb x (convexComb y z (convexComb_assoc_coeff₂' s t)) (convexComb_assoc_coeff₁' s t)
Module: Mathlib.Topology.MetricSpace.Polish

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Opens α
inst✝ : CompleteSpace α
u : ℕ → s.CompleteCopy
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < (1 / 2) ^ N
A : CauchySeq fun n ↦ ↑(u n)
x : α
xlim : Tendsto (fun n ↦ ↑(u n)) atTop (𝓝 x)
⊢ ∃ x, Tendsto u atTop (𝓝 x)

Tactic:
by_cases xs : x ∈ s

Goal state after:
case pos
α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Opens α
inst✝ : CompleteSpace α
u : ℕ → s.CompleteCopy
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < (1 / 2) ^ N
A : CauchySeq fun n ↦ ↑(u n)
x : α
xlim : Tendsto (fun n ↦ ↑(u n)) atTop (𝓝 x)
xs : x ∈ s
⊢ ∃ x, Tendsto u atTop (𝓝 x)

case neg
α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Opens α
inst✝ : CompleteSpace α
u : ℕ → s.CompleteCopy
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < (1 / 2) ^ N
A : CauchySeq fun n ↦ ↑(u n)
x : α
xlim : Tendsto (fun n ↦ ↑(u n)) atTop (𝓝 x)
xs : x ∉ s
⊢ ∃ x, Tendsto u atTop (𝓝 x)
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: X : Type u
Y : Type v
Z : Type w
inst✝³ : Nonempty Z
inst✝² : MetricSpace Z
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
Φ : Z → X
Ψ : Z → Y
hΦ : Isometry Φ
hΨ : Isometry Ψ
i : PseudoMetricSpace (X ⊕ Y) := gluePremetric hΦ hΨ
⊢ toGlueL hΦ hΨ ∘ Φ = toGlueR hΦ hΨ ∘ Ψ

Tactic:
let _ := i.toUniformSpace.toTopologicalSpace

Goal state after:
X : Type u
Y : Type v
Z : Type w
inst✝³ : Nonempty Z
inst✝² : MetricSpace Z
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
Φ : Z → X
Ψ : Z → Y
hΦ : Isometry Φ
hΨ : Isometry Ψ
i : PseudoMetricSpace (X ⊕ Y) := gluePremetric hΦ hΨ
x✝ : TopologicalSpace (X ⊕ Y) := PseudoMetricSpace.toUniformSpace.toTopologicalSpace
⊢ toGlueL hΦ hΨ ∘ Φ = toGlueR hΦ hΨ ∘ Ψ
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: case inr.inr
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
x y : (n : ℕ) → E n
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
H : longestPrefix x s < firstDiff x y
xs : x ∉ s
ys : y ∉ s
L : longestPrefix x s < longestPrefix y s
Ay : (s ∩ cylinder y (longestPrefix y s)).Nonempty
A'y : (s ∩ cylinder y (longestPrefix x s).succ).Nonempty
⊢ longestPrefix y s = longestPrefix x s

Tactic:
have Z := disjoint_cylinder_of_longestPrefix_lt hs xs (Nat.lt_succ_self _)

Goal state after:
case inr.inr
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
x y : (n : ℕ) → E n
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
H : longestPrefix x s < firstDiff x y
xs : x ∉ s
ys : y ∉ s
L : longestPrefix x s < longestPrefix y s
Ay : (s ∩ cylinder y (longestPrefix y s)).Nonempty
A'y : (s ∩ cylinder y (longestPrefix x s).succ).Nonempty
Z : Disjoint s (cylinder x (longestPrefix x s).succ)
⊢ longestPrefix y s = longestPrefix x s
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)
x y : (n : ℕ) → E n
s : Set ((n : ℕ) → E n)
hs : IsClosed[Pi.topologicalSpace] s
hne : s.Nonempty
H : longestPrefix x s < firstDiff x y
xs : x ∉ s
ys : y ∉ s
⊢ cylinder x (longestPrefix x s) = cylinder y (longestPrefix y s)

Tactic:
have l_eq : longestPrefix y s = longestPrefix x s := by
    rcases lt_trichotomy (longestPrefix y s) (longestPrefix x s) with (L | L | L)
    · have Ax : (s ∩ cylinder x (longestPrefix x s)).Nonempty :=
        inter_cylinder_longestPrefix_nonempty hs hne x
      have Z := disjoint_cylinder_of_longestPrefix_lt hs ys L
      rw [firstDiff_comm] at H
      rw [cylinder_eq_cylinder_of_le_firstDiff _ _ H.le] at Z
      exact (Ax.not_disjoint Z).elim
    · exact L
    · have Ay : (s ∩ cylinder y (longestPrefix y s)).Nonempty :=
        inter_cylinder_longestPrefix_nonempty hs hne y
      have A'y : (s ∩ cylinder y (longestPrefix x s).succ).Nonempty :=
        Ay.mono (inter_subset_inter_right s (cylinder_anti _ L))
      have Z := disjoint_cylinder_of_longestPrefix_lt hs xs (Nat.lt_succ_self _)
      rw [cylinder_eq_cylinder_of_le_firstDiff _ _ H] at Z
      exact (A'y.not_disjoint Z).elim
  rw [l_eq, ← mem_cylinder_iff_eq]
  exact cylinder_anti y H.le (mem_cylinder_firstDiff x y)

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

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
f : β → ℝ≥0∞
g : α → β
hg : AEMeasurable g μ
i : β → ℝ≥0∞
hi : Measurable i
h'i : i ≤ f
⊢ ∫⁻ (a : β), i a ∂Measure.map g μ ≤ ∫⁻ (a : α), f (g a) ∂μ

Tactic:
lintegral_map' hi.aemeasurable hg

Goal state after:
case pos
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
f : β → ℝ≥0∞
g : α → β
hg : AEMeasurable g μ
i : β → ℝ≥0∞
hi : Measurable i
h'i : i ≤ f
⊢ ∫⁻ (a : α), i (g a) ∂μ ≤ ∫⁻ (a : α), f (g a) ∂μ
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: case refine_2.inr.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)
⊢ firstDiff x y ≤ firstDiff (f x) (f y)

Tactic:
by_cases xs : x ∈ 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
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
⊢ firstDiff x y ≤ firstDiff (f x) (f y)

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
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
⊢ firstDiff x y ≤ firstDiff (f x) (f y)
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Probability

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsProbabilityMeasure μ
⊢ Nonempty α

Tactic:
by_contra! maybe_empty
  have : μ Set.univ = 0 := by
    rw [Set.univ_eq_empty_iff.mpr maybe_empty, measure_empty]
  simp at this

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

Goal state before:
pp: case inr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : IsZeroOrProbabilityMeasure μ
p : ℝ≥0∞
hμs : p < μ s
s_mble : MeasurableSet s
h : IsProbabilityMeasure μ
⊢ μ sᶜ < 1 - p

Tactic:
rw [prob_compl_eq_one_sub s_mble]

Goal state after:
case inr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : IsZeroOrProbabilityMeasure μ
p : ℝ≥0∞
hμs : p < μ s
s_mble : MeasurableSet s
h : IsProbabilityMeasure μ
⊢ 1 - μ s < 1 - p
Module: Mathlib.MeasureTheory.Measure.Dirac

Goal state before:
pp: δ : Type u_3
ι : Type u_4
mδ : MeasurableSpace δ
c : ι → ℝ
d : ι → δ
h1 : ∀ (i : ι), 0 ≤ c i
h2 : HasSum c 1
⊢ HasSum (fun i ↦ (c i).toNNReal) 1

Tactic:
by simpa using h2.toNNReal h1

Goal state after:
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
hne : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n
fs : ∀ x ∈ s, f x = x
frange : range f = s
hf : LipschitzWith 1 f
⊢ ∃ f, (∀ x ∈ s, f x = x) ∧ range f = s ∧ Continuous[Pi.topologicalSpace, Pi.topologicalSpace] f

Tactic:
exact ⟨f, fs, frange, hf.continuous⟩

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

Goal state before:
pp: case mp
α : Type u_1
inst✝ : MeasurableSpace α
x y : α
h : dirac x = dirac y
A : Set α
A_mble : MeasurableSet A
obs : (dirac x) A = (dirac y) A
⊢ x ∈ A ↔ y ∈ A

Tactic:
simp only [Measure.dirac_apply' _ A_mble] at obs

Goal state after:
case mp
α : Type u_1
inst✝ : MeasurableSpace α
x y : α
h : dirac x = dirac y
A : Set α
A_mble : MeasurableSet A
obs : A.indicator 1 x = A.indicator 1 y
⊢ x ∈ A ↔ y ∈ A
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
⊢ ∑ p ∈ s, μ (t ∩ {p}) = (∑ a ∈ s, μ {a} • dirac a) t

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
i : α
hi : i ∈ ↑s
j : α
hj : j ∈ ↑s
hij : i ≠ j
⊢ Disjoint {i} {j}

Tactic:
· simp only [coe_finsetSum, Finset.sum_apply, smul_apply]
      congr with a
      by_cases ha : a ∈ t <;> simp [*]

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
i : α
hi : i ∈ ↑s
j : α
hj : j ∈ ↑s
hij : i ≠ j
⊢ Disjoint {i} {j}