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

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

Tactic:
inhabit β

Goal state after:
case intro.inr
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), Measurable (f b)
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
val✝ : Encodable β
h✝ : Nonempty β
inhabited_h : Inhabited β
⊢ ∫⁻ (a : α), ⨆ b, f b a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f : α → ℝ≥0∞
⊢ ∫⁻ (a : α), s.indicator f a ∂μ = ∫⁻ (a : α) in s, f a ∂μ

Tactic:
apply le_antisymm (lintegral_indicator_le f s)
  simp only [lintegral, ← restrict_lintegral_eq_lintegral_restrict _ hs, iSup_subtype']
  refine iSup_mono' (Subtype.forall.2 fun φ hφ => ?_)
  refine ⟨⟨φ.restrict s, fun x => ?_⟩, le_rfl⟩
  simp [hφ x, hs, indicator_le_indicator]

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✝² : TopologicalSpace β
inst✝¹ : Zero β
m0 : MeasurableSpace α
hf : StronglyMeasurable f
μ : Measure α
inst✝ : SigmaFinite μ
⊢ ∀ x ∈ univᶜ, f x = 0

Tactic:
simp

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

Goal state before:
pp: case intro.inr
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : Countable β
f : β → α → ℝ≥0∞
μ : Measure α
hμ : μ ≠ 0
hf : ∀ (b : β), Measurable (f b)
hf_int : ∀ (b : β), ∫⁻ (a : α), f b a ∂μ ≠ ∞
h_directed : Directed (fun x1 x2 ↦ x1 ≥ x2) f
val✝ : Encodable β
h✝ : Nonempty β
⊢ ∫⁻ (a : α), ⨅ b, f b a ∂μ = ⨅ b, ∫⁻ (a : α), f b a ∂μ

Tactic:
inhabit β

Goal state after:
case intro.inr
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : Countable β
f : β → α → ℝ≥0∞
μ : Measure α
hμ : μ ≠ 0
hf : ∀ (b : β), Measurable (f b)
hf_int : ∀ (b : β), ∫⁻ (a : α), f b a ∂μ ≠ ∞
h_directed : Directed (fun x1 x2 ↦ x1 ≥ x2) f
val✝ : Encodable β
h✝ : Nonempty β
inhabited_h : Inhabited β
⊢ ∫⁻ (a : α), ⨅ b, f b a ∂μ = ⨅ b, ∫⁻ (a : α), f b a ∂μ
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Sub

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∫⁻ (a : α), f a ∂μ ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
hf₀ : ∫⁻ (a : α), f a ∂μ = 0
⊢ μ ∅ < ∞

Tactic:
simp

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

Goal state before:
pp: α : Type u_1
β : Type u_2
f : α → β
mα : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : TopologicalSpace β
inst✝² : PseudoMetrizableSpace β
inst✝¹ : SecondCountableTopology β
inst✝ : OpensMeasurableSpace β
hf : Measurable f
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
a✝ : Nontrivial β
⊢ StronglyMeasurable f

Tactic:
inhabit β

Goal state after:
α : Type u_1
β : Type u_2
f : α → β
mα : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : TopologicalSpace β
inst✝² : PseudoMetrizableSpace β
inst✝¹ : SecondCountableTopology β
inst✝ : OpensMeasurableSpace β
hf : Measurable f
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
a✝ : Nontrivial β
inhabited_h : Inhabited β
⊢ StronglyMeasurable f
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
f : α → β
mα : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : TopologicalSpace β
inst✝² : PseudoMetrizableSpace β
inst✝¹ : SecondCountableTopology β
inst✝ : OpensMeasurableSpace β
hf : Measurable f
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
a✝ : Nontrivial β
inhabited_h : Inhabited β
x : α
⊢ f x ∈ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] univ

Tactic:
simp

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

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

Tactic:
simp

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

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
h_nonneg : 0 ≤ᵐ[μ] f
⊢ ∫⁻ (x : α), ‖f x‖ₑ ∂μ = ∫⁻ (x : α), ENNReal.ofReal (f x) ∂μ

Tactic:
by
  apply lintegral_congr_ae
  filter_upwards [h_nonneg] with x hx
  rw [Real.enorm_eq_ofReal hx]

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
hC_lt : 1 / ↑n < ⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s
h_lt_top : ⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s < ∞
⊢ 1 / ↑n ≠ 0

Tactic:
simp

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

Goal state before:
pp: case e_f
α : Type u_1
m : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hf : Measurable f
a : α
⊢ ⨆ i, r * (eapprox f i) a = ⨆ n, (const α r * eapprox f n) a

Tactic:
simp

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

Goal state before:
pp: α : Type u_1
β : Type u_2
f g : α → β
m : MeasurableSpace α
inst✝ : TopologicalSpace β
s : Set α
x✝ : DecidablePred fun x ↦ x ∈ s
hs : MeasurableSet s
hf : StronglyMeasurable f
hg : StronglyMeasurable g
⊢ StronglyMeasurable (s.piecewise f g)

Tactic:
refine ⟨fun n => SimpleFunc.piecewise s hs (hf.approx n) (hg.approx n), fun x => ?_⟩

Goal state after:
α : Type u_1
β : Type u_2
f g : α → β
m : MeasurableSpace α
inst✝ : TopologicalSpace β
s : Set α
x✝ : DecidablePred fun x ↦ x ∈ s
hs : MeasurableSet s
hf : StronglyMeasurable f
hg : StronglyMeasurable g
x : α
⊢ Tendsto (fun n ↦ ((fun n ↦ SimpleFunc.piecewise s hs (hf.approx n) (hg.approx n)) n) x) atTop (𝓝 (s.piecewise f g x))
Module: Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure ν
this : ∀ (n : ℕ), SigmaFinite (μ.restrict (μ.sigmaFiniteSetGE ν n))
f : ℕ × ℕ → Set α :=
  fun p ↦
    (μ.sigmaFiniteSetWRT' ν)ᶜ ∪ spanningSets (μ.restrict (μ.sigmaFiniteSetGE ν p.1)) p.2 ∩ μ.sigmaFiniteSetGE ν p.1
e : ℕ ≃ ℕ × ℕ := Nat.pairEquiv.symm
x✝ : ℕ
⊢ f (e x✝) ∈ Set.range f

Tactic:
simp

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

Goal state before:
pp: α : Type u_1
β : Type u_2
s : Set α
m : MeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : (x : α) → Decidable (x ∈ s)
f : ↑s → β
hf : StronglyMeasurable f
g : ↑sᶜ → β
hg : StronglyMeasurable g
hs : MeasurableSet s
⊢ StronglyMeasurable fun x ↦ if hx : x ∈ s then f ⟨x, hx⟩ else g ⟨x, hx⟩

Tactic:
refine ⟨fun n ↦ SimpleFunc.dite s hs (hf.approx n) (hg.approx n), fun x ↦ ?_⟩
  by_cases hx : x ∈ s
  · simpa [hx] using hf.tendsto_approx ⟨x, hx⟩
  · simpa [hx] using hg.tendsto_approx ⟨x, hx⟩

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace α
inst✝³ : OpensMeasurableSpace α
inst✝² : MeasurableSingletonClass α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
h : SecondCountableTopologyEither α β
f : α → β
s : Set α
hf : ContinuousOn f s
hs : sᶜ.Countable
h's : MeasurableSet s
⊢ f = fun x ↦ if hx : x ∈ s then f ↑⟨x, hx⟩ else f ↑⟨x, hx⟩

Tactic:
simp

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

Goal state before:
pp: case refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f✝ : α → ℝ≥0∞
hf : Measurable f✝
f g : α → ℝ≥0∞
a✝ : Disjoint (Function.support f) (Function.support g)
⊢ Measurable f →
    Measurable g →
      ∫⁻ (a : α), f a ∂μ.trim hm = ∫⁻ (a : α), f a ∂μ →
        ∫⁻ (a : α), g a ∂μ.trim hm = ∫⁻ (a : α), g a ∂μ → ∫⁻ (a : α), (f + g) a ∂μ.trim hm = ∫⁻ (a : α), (f + g) a ∂μ

Tactic:
hf

Goal state after:
case refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f✝ : α → ℝ≥0∞
hf✝ : Measurable f✝
f g : α → ℝ≥0∞
a✝ : Disjoint (Function.support f) (Function.support g)
hf : Measurable f
⊢ Measurable g →
    ∫⁻ (a : α), f a ∂μ.trim hm = ∫⁻ (a : α), f a ∂μ →
      ∫⁻ (a : α), g a ∂μ.trim hm = ∫⁻ (a : α), g a ∂μ → ∫⁻ (a : α), (f + g) a ∂μ.trim hm = ∫⁻ (a : α), (f + g) a ∂μ
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: case refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f✝ : α → ℝ≥0∞
hf✝ : Measurable f✝
f g : α → ℝ≥0∞
a✝¹ : Disjoint (Function.support f) (Function.support g)
hf : Measurable f
a✝ : Measurable g
hf_prop : ∫⁻ (a : α), f a ∂μ.trim hm = ∫⁻ (a : α), f a ∂μ
hg_prop : ∫⁻ (a : α), g a ∂μ.trim hm = ∫⁻ (a : α), g a ∂μ
h_m : ∫⁻ (a : α), f a + g a ∂μ.trim hm = ∫⁻ (a : α), f a ∂μ.trim hm + ∫⁻ (a : α), g a ∂μ.trim hm
⊢ ∫⁻ (a : α), (f + g) a ∂μ.trim hm = ∫⁻ (a : α), (f + g) a ∂μ

Tactic:
have h_m0 := lintegral_add_left (μ := μ) (Measurable.mono hf hm le_rfl) g

Goal state after:
case refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f✝ : α → ℝ≥0∞
hf✝ : Measurable f✝
f g : α → ℝ≥0∞
a✝¹ : Disjoint (Function.support f) (Function.support g)
hf : Measurable f
a✝ : Measurable g
hf_prop : ∫⁻ (a : α), f a ∂μ.trim hm = ∫⁻ (a : α), f a ∂μ
hg_prop : ∫⁻ (a : α), g a ∂μ.trim hm = ∫⁻ (a : α), g a ∂μ
h_m : ∫⁻ (a : α), f a + g a ∂μ.trim hm = ∫⁻ (a : α), f a ∂μ.trim hm + ∫⁻ (a : α), g a ∂μ.trim hm
h_m0 : ∫⁻ (a : α), f a + g a ∂μ = ∫⁻ (a : α), f a ∂μ + ∫⁻ (a : α), g a ∂μ
⊢ ∫⁻ (a : α), (f + g) a ∂μ.trim hm = ∫⁻ (a : α), (f + g) a ∂μ
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : TopologicalSpace β
f : α → β
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
h : univ ⊆ s ∪ t
hc : StronglyMeasurable fun a ↦ f ↑a
hd : StronglyMeasurable fun a ↦ f ↑a
a✝ : Nontrivial β
⊢ StronglyMeasurable f

Tactic:
inhabit β

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : TopologicalSpace β
f : α → β
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
h : univ ⊆ s ∪ t
hc : StronglyMeasurable fun a ↦ f ↑a
hd : StronglyMeasurable fun a ↦ f ↑a
a✝ : Nontrivial β
inhabited_h : Inhabited β
⊢ StronglyMeasurable f
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: case refine_3
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f✝ : α → ℝ≥0∞
hf : Measurable f✝
f : ℕ → α → ℝ≥0∞
⊢ (∀ (n : ℕ), Measurable (f n)) →
    Monotone f →
      (∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ.trim hm = ∫⁻ (a : α), f n a ∂μ) →
        ∫⁻ (a : α), (fun x ↦ ⨆ n, f n x) a ∂μ.trim hm = ∫⁻ (a : α), (fun x ↦ ⨆ n, f n x) a ∂μ

Tactic:
hf

Goal state after:
case refine_3
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f✝ : α → ℝ≥0∞
hf✝ : Measurable f✝
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), Measurable (f n)
⊢ Monotone f →
    (∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ.trim hm = ∫⁻ (a : α), f n a ∂μ) →
      ∫⁻ (a : α), (fun x ↦ ⨆ n, f n x) a ∂μ.trim hm = ∫⁻ (a : α), (fun x ↦ ⨆ n, f n x) a ∂μ
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : TopologicalSpace β
f : α → β
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
h : univ ⊆ s ∪ t
hc : StronglyMeasurable fun a ↦ f ↑a
hd : StronglyMeasurable fun a ↦ f ↑a
a✝ : Nontrivial β
inhabited_h : Inhabited β
x : { x // x ∈ s }
⊢ Function.extend Subtype.val (fun x ↦ f ↑x) (Function.extend Subtype.val (fun x ↦ f ↑x) fun x ↦ default) ↑x = f ↑x

Tactic:
simp

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

Goal state before:
pp: case pcw
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : Nonempty β
inst✝ : TopologicalSpace β
P : (f : α → β) → StronglyMeasurable f → Prop
const : ∀ (c : β), P (fun x ↦ c) ⋯
pcw :
  ∀ ⦃f g : α → β⦄ {s : Set α} (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) (hs : MeasurableSet s),
    P f hf → P g hg → P (s.piecewise f g) ⋯
lim :
  ∀ ⦃f : ℕ → α → β⦄ ⦃g : α → β⦄ (hf : ∀ (n : ℕ), StronglyMeasurable (f n)) (hg : StronglyMeasurable g),
    (∀ (n : ℕ), P (f n) ⋯) → (∀ (x : α), Tendsto (fun x_1 ↦ f x_1 x) atTop (𝓝 (g x))) → P g hg
f✝ : α → β
hf : StronglyMeasurable f✝
s✝ : ℕ → α →ₛ β := hf.approx
n : ℕ
f g : α →ₛ β
s : Set α
hs : MeasurableSet s
Pf : P ⇑f ⋯
Pg : P ⇑g ⋯
⊢ P ⇑(SimpleFunc.piecewise s hs f g) ⋯

Tactic:
SimpleFunc.coe_piecewise

Goal state after:
case pcw
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : Nonempty β
inst✝ : TopologicalSpace β
P : (f : α → β) → StronglyMeasurable f → Prop
const : ∀ (c : β), P (fun x ↦ c) ⋯
pcw :
  ∀ ⦃f g : α → β⦄ {s : Set α} (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) (hs : MeasurableSet s),
    P f hf → P g hg → P (s.piecewise f g) ⋯
lim :
  ∀ ⦃f : ℕ → α → β⦄ ⦃g : α → β⦄ (hf : ∀ (n : ℕ), StronglyMeasurable (f n)) (hg : StronglyMeasurable g),
    (∀ (n : ℕ), P (f n) ⋯) → (∀ (x : α), Tendsto (fun x_1 ↦ f x_1 x) atTop (𝓝 (g x))) → P g hg
f✝ : α → β
hf : StronglyMeasurable f✝
s✝ : ℕ → α →ₛ β := hf.approx
n : ℕ
f g : α →ₛ β
s : Set α
hs : MeasurableSet s
Pf : P ⇑f ⋯
Pg : P ⇑g ⋯
⊢ P (s.piecewise ⇑f ⇑g) ⋯
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
u : Set α
hu : MeasurableSet u
⊢ Subtype.val ⁻¹' (s ∩ u) = Subtype.val ⁻¹' u

Tactic:
simp

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

Goal state before:
pp: α : Type u_2
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hf_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ∞
h_bound_finite : ∀ᵐ (a : α) ∂μ, F a ≠ ∞
h_exists : ∀ᵐ (a : α) ∂μ, ∃ l, Tendsto (fun i ↦ f i a) atTop (𝓝 l)
F' : α → ℝ≥0∞ := fun a ↦ if h : ∃ l, Tendsto (fun i ↦ f i a) atTop (𝓝 l) then h.choose else ∞
hF'_tendsto : ∀ᵐ (a : α) ∂μ, Tendsto (fun i ↦ f i a) atTop (𝓝 (F' a))
hF'_le : F' ≤ᵐ[μ] F
⊢ Tendsto (fun i ↦ ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F' a ∂μ))

Tactic:
exact lintegral_tendsto_of_tendsto_of_monotone hf_meas hf_mono hF'_tendsto

Goal state after:
Module: Mathlib.Topology.Perfect

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : T1Space α
U : Set α
hu : U.Nontrivial
h :
  ∀ (t t' : Set α),
    IsClosed[inst✝¹] t →
      IsClosed[inst✝¹] t' → U ⊆ t ∪ t' → (U ∩ t).Nonempty → (U ∩ t').Nonempty → (U ∩ (t ∩ t')).Nonempty
x : α
hx : x ∈ U
⊢ U ⊆ {x} ∪ U \ {x}

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.Order.ProjIcc

Goal state before:
pp: α : Type u_1
inst✝² : LinearOrder α
a b : α
h : a ≤ b
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
s : Set ↑(Icc a b)
hs : IsOpen[inst✝¹] (projIcc a b h ⁻¹' s)
x✝ : { x // x ∈ Icc a b }
⊢ x✝ ∈ Subtype.val ⁻¹' projIcc a b h ⁻¹' s ↔ x✝ ∈ s

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.Algebra.MetricSpace.Lipschitz

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
x : α
r : ℝ
hr : 0 < r
K : ℝ
h : ∀ (y : α), dist y x < r → dist (f y) (f x) ≤ K * dist y x
⊢ K * dist x x = 0

Tactic:
simp

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

Goal state before:
pp: α : Type u_2
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hf_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
hf_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Antitone fun i ↦ f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), F a ≤ f i a
h0 : ∫⁻ (a : α), f 0 a ∂μ ≠ ∞
⊢ ∫⁻ (a : α), F a ∂μ ≠ ∞

Tactic:
refine ((lintegral_mono_ae ?_).trans_lt h0.lt_top).ne

Goal state after:
α : Type u_2
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hf_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
hf_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Antitone fun i ↦ f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), F a ≤ f i a
h0 : ∫⁻ (a : α), f 0 a ∂μ ≠ ∞
⊢ ∀ᵐ (a : α) ∂μ, F a ≤ f 0 a
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: X : Type u
Y : Type v
Z : Type w
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : Nonempty Z
Φ : Z → X
Ψ : Z → Y
ε : ℝ
p : Z
A : ∀ (q : Z), 0 ≤ dist (Φ p) (Φ q) + dist (Ψ p) (Ψ q)
⊢ 0 = dist (Φ p) (Φ p) + dist (Ψ p) (Ψ p)

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: x : ↑I
⊢ Subtype.val ⁻¹' Ioi ↑x = Ioc x 1

Tactic:
rw [preimage_subtype_val_Ioi]

Goal state after:
x : ↑I
⊢ Ioi x = Ioc x 1
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: a b : ℝ
x y : ↑(Icc a b)
h : ↑x ≤ ↑y
t : ↑unitInterval
⊢ ↑x ≤ ↑(convexComb x y t)

Tactic:
simp

Goal state after:
a b : ℝ
x y : ↑(Icc a b)
h : ↑x ≤ ↑y
t : ↑unitInterval
⊢ ↑x ≤ (1 - ↑t) * ↑x + ↑t * ↑y
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: a b : ℝ
x y : ↑(Icc a b)
h : ↑x ≤ ↑y
t : ↑unitInterval
⊢ ↑(convexComb x y t) ≤ ↑y

Tactic:
simp

Goal state after:
a b : ℝ
x y : ↑(Icc a b)
h : ↑x ≤ ↑y
t : ↑unitInterval
⊢ (1 - ↑t) * ↑x + ↑t * ↑y ≤ ↑y
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: case neg
a b : ℝ
x y z : ↑(Icc a b)
s t : ↑unitInterval
hs : ↑s = 1
ht : ¬↑t = 1
this : 1 - ↑t ≠ 0
⊢ (1 - 1) * ↑x + ((1 - ↑t) * ↑y + ↑t * ↑z) = (1 - ↑t) * ((1 - 1) * ↑x + ↑y) + ↑t * ↑z

Tactic:
simp

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

Goal state before:
pp: E : ℕ → Type u_1
x : (n : ℕ) → E n
n : ℕ
⊢ x ∈ cylinder x n

Tactic:
simp

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
x : X
⊢ (if x = x then 0 else 1) = 0

Tactic:
simp

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
⊢ ∀ {x y : X}, (if x = y then 0 else 1) = 0 → x = y

Tactic:
simp

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
m : MetricSpace X :=
  { dist := fun x y ↦ if x = y then 0 else 1, dist_self := ⋯, dist_comm := ⋯, dist_triangle := ⋯, edist_dist := ⋯,
    uniformity_dist := ⋯, cobounded_sets := ⋯, eq_of_dist_eq_zero := ⋯ }
x : X
⊢ 0 < 1

Tactic:
simp

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
m : MetricSpace X :=
  { dist := fun x y ↦ if x = y then 0 else 1, dist_self := ⋯, dist_comm := ⋯, dist_triangle := ⋯, edist_dist := ⋯,
    uniformity_dist := ⋯, cobounded_sets := ⋯, eq_of_dist_eq_zero := ⋯ }
u : ℕ → X
hu : ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε
⊢ 1 > 0

Tactic:
simp

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

Goal state before:
pp: case mpr.inr
E : ℕ → Type u_1
x y : (n : ℕ) → E n
i : ℕ
H : ∀ i_1 < i, y i_1 = x i_1
hi : i < i + 1
⊢ y i = update x i (y i) i

Tactic:
simp

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

Goal state before:
pp: case mp.zero
α : Type u_2
x y : ℕ → α
h : res x 0 = res y 0
⊢ ∀ ⦃m : ℕ⦄, m < 0 → x m = y m

Tactic:
simp

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

Goal state before:
pp: case mpr.zero
α : Type u_2
x y : ℕ → α
h : ∀ ⦃m : ℕ⦄, m < 0 → x m = y m
⊢ res x 0 = res y 0

Tactic:
simp

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

Goal state before:
pp: E : ℕ → Type u_1
x y : (n : ℕ) → E n
⊢ dist x y ≤ 1

Tactic:
by
  rcases eq_or_ne x y with (rfl | h)
  · simp [dist]
  · simp only [dist, ne_eq, h, not_false_eq_true, ↓reduceIte, one_div, inv_pow]
    bound

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

Goal state before:
pp: case inr.inl
E : ℕ → Type u_1
x z : (n : ℕ) → E n
hxz : x ≠ z
⊢ dist x z ≤ max (dist x x) (dist x z)

Tactic:
simp

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

Goal state before:
pp: case inr.inr.inl
E : ℕ → Type u_1
x y : (n : ℕ) → E n
hxy hxz : x ≠ y
⊢ dist x y ≤ max (dist x y) (dist y y)

Tactic:
simp

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

Goal state before:
pp: X : Type u
Y : Type v
Z : Type w
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
p : X
q : Y
x✝ : X ⊕ Y
⊢ ∀ (p_1 q_1 : Unit), |dist ⋯.some ⋯.some - dist ⋯.some ⋯.some| ≤ 2 * 1

Tactic:
simp

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

Goal state before:
pp: X : Type u
Y : Type v
Z : Type w
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
x✝ : X ⊕ Y
q : X
r : Y
⊢ ∀ (p q_1 : Unit), |dist ⋯.some ⋯.some - dist ⋯.some ⋯.some| ≤ 2 * 1

Tactic:
simp

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

Goal state before:
pp: X : Type u
Y : Type v
Z : Type w
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
x✝ : X ⊕ Y
q : X
r : Y
⊢ Sum.dist x✝ (Sum.inr r) ≤ Sum.dist x✝ (Sum.inl q) + Sum.dist (Sum.inl q) (Sum.inr r)

Tactic:
simp only [Sum.dist_eq_glueDist q r]
      exact glueDist_triangle _ _ _ (by simp) _ _ _

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

Goal state before:
pp: X : Type u
Y : Type v
Z : Type w
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
p : Y
x✝ : X ⊕ Y
r : X
⊢ ∀ (p_1 q : Unit), |dist ⋯.some ⋯.some - dist ⋯.some ⋯.some| ≤ 2 * 1

Tactic:
simp

Goal state after:
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
⊢ ∀ (n : ℕ), 0 < (1 / 2) ^ n

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : IsTopologicalRing 𝕜
a b : 𝕜
h : a < b
e : ↑(Icc 0 1) ≃ₜ ↑(⇑(affineHomeomorph (b - a) a ⋯) '' Icc 0 1) := (affineHomeomorph (b - a) a ⋯).image (Icc 0 1)
⊢ Icc a (b - a + a) = Icc a b

Tactic:
simp

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)
x : (n : ℕ) → E n
n : ℕ
⊢ IsOpen[Pi.topologicalSpace] (cylinder x n)

Tactic:
rw [PiNat.cylinder_eq_pi]

Goal state after:
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
x : (n : ℕ) → E n
n : ℕ
⊢ IsOpen[Pi.topologicalSpace] ((↑(Finset.range n)).pi fun i ↦ {x i})
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
⊢ 0 ≤ 1 / 2

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: x : ↑I
⊢ ↑(toNNReal x + toNNReal (σ x)) = ↑1

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: x : ↑I
⊢ ↑(toNNReal (σ x) + toNNReal x) = ↑1

Tactic:
simp

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

Goal state before:
pp: case zero
β : Type u_1
α : Type u_2
A : List β → Set α
hA : CantorScheme.Disjoint A
x y : ↑(inducedMap A).fst
hxy : (inducedMap A).snd x = (inducedMap A).snd y
⊢ res (↑x) 0 = res (↑y) 0

Tactic:
simp

Goal state after: