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

Goal state before:
pp: case hnhds
α : Type u
inst✝¹ : PseudoEMetricSpace α
β : Type u_2
inst✝ : PseudoEMetricSpace β
f : α → β
K : Set α
U : Set β
hK : IsCompact K
ho : IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] U
hf : ∀ x ∈ K, ContinuousAt f x
hKU : MapsTo f K U
x : α
⊢ x ∈ K → ∃ t ∈ 𝓝[K] x, ∃ ε > 0, ∃ V ∈ 𝓝ˢ t, thickening ε (f '' V) ⊆ U

Tactic:
hx

Goal state after:
case hnhds
α : Type u
inst✝¹ : PseudoEMetricSpace α
β : Type u_2
inst✝ : PseudoEMetricSpace β
f : α → β
K : Set α
U : Set β
hK : IsCompact K
ho : IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] U
hf : ∀ x ∈ K, ContinuousAt f x
hKU : MapsTo f K U
x : α
hx : x ∈ K
⊢ ∃ t ∈ 𝓝[K] x, ∃ ε > 0, ∃ V ∈ 𝓝ˢ t, thickening ε (f '' V) ⊆ U
Module: Mathlib.MeasureTheory.Function.SimpleFunc

Goal state before:
pp: case refine_2
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 = ⊥
k : ℕ
hk : i k ≤ f a
⊢ i k ≤ (approx i f (k + 1)) a

Tactic:
rw [approx_apply a hf]

Goal state after:
case refine_2
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 = ⊥
k : ℕ
hk : i k ≤ f a
⊢ i k ≤ (Finset.range (k + 1)).sup fun k ↦ if i k ≤ f a then i k else 0
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case pos
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
g g' : δ → α
hf : ∀ (i : ι), Measurable (f i)
s : Set δ
hs : MeasurableSet s
hg : ∀ b ∈ s, IsLUB {a | ∃ i, f i b = a} (g b)
hg' : EqOn g g' sᶜ
g'_meas : Measurable g'
i : ι
this : Nonempty ι
f' : ι → δ → α := fun i ↦ s.piecewise (f i) g'
b : δ
hb : b ∈ s
⊢ IsLUB {a | ∃ i, f' i b = a} (g b)

Tactic:
have A : ∀ i, f' i b = f i b := fun i ↦ by simp [f', hb]

Goal state after:
case pos
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
g g' : δ → α
hf : ∀ (i : ι), Measurable (f i)
s : Set δ
hs : MeasurableSet s
hg : ∀ b ∈ s, IsLUB {a | ∃ i, f i b = a} (g b)
hg' : EqOn g g' sᶜ
g'_meas : Measurable g'
i : ι
this : Nonempty ι
f' : ι → δ → α := fun i ↦ s.piecewise (f i) g'
b : δ
hb : b ∈ s
A : ∀ (i : ι), f' i b = f i b
⊢ IsLUB {a | ∃ i, f' i b = a} (g b)
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_1
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ x ∈ s, s ∈ 𝓝[>] x
M : α → α
hM : ∀ x ∈ s, x < M x
y : α → α
hy : ∀ x ∈ s, y x ∈ Ioi x
h'y : ∀ x ∈ s, Ioo x (y x) ⊆ s
x : α
⊢ x ∈ s \ interior s → ∀ ⦃y_1 : α⦄, y_1 ∈ s \ interior s → x ≠ y_1 → Function.onFun Disjoint (fun x ↦ Ioo x (y x)) x y_1

Tactic:
hx

Goal state after:
α : Type u_1
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ x ∈ s, s ∈ 𝓝[>] x
M : α → α
hM : ∀ x ∈ s, x < M x
y : α → α
hy : ∀ x ∈ s, y x ∈ Ioi x
h'y : ∀ x ∈ s, Ioo x (y x) ⊆ s
x : α
hx : x ∈ s \ interior s
⊢ ∀ ⦃y_1 : α⦄, y_1 ∈ s \ interior s → x ≠ y_1 → Function.onFun Disjoint (fun x ↦ Ioo x (y x)) x y_1
Module: Mathlib.Topology.Instances.Real.Lemmas

Goal state before:
pp: f : ℕ → ℝ
k : ℕ
h_mon : MonotoneOn f (Ici k)
h_bdd : BddAbove (f '' Ici k)
⊢ Tendsto f atTop (𝓝 (sSup (f '' Ici k)))

Tactic:
by
  rw [← range_add_eq_image_Ici] at h_bdd
  rw [Ici, ← monotone_add_nat_iff_monotoneOn_nat_Ici] at h_mon
  rw [← tendsto_add_atTop_iff_nat k, ← range_add_eq_image_Ici, sSup_range]
  exact tendsto_atTop_ciSup h_mon h_bdd

Goal state after:
Module: Mathlib.Topology.IndicatorConstPointwise

Goal state before:
pp: ι : Type u_3
L : Filter ι
β : Type u_4
p : ι → Prop
inst✝¹ : DecidablePred p
q : Prop
inst✝ : Decidable q
a b : β
F G : Filter β
haG : {a}ᶜ ∈ G
hbF : {b}ᶜ ∈ F
haF : 𝓟 {a} ≤ F
hbG : 𝓟 {b} ≤ G
h : ∀ᶠ (i : ι) in L, p i ↔ q
⊢ (fun x ↦ if q then a else b) =ᶠ[L] fun i ↦ if p i then a else b

Tactic:
filter_upwards [h] with i hi

Goal state after:
ι : Type u_3
L : Filter ι
β : Type u_4
p : ι → Prop
inst✝¹ : DecidablePred p
q : Prop
inst✝ : Decidable q
a b : β
F G : Filter β
haG : {a}ᶜ ∈ G
hbF : {b}ᶜ ∈ F
haF : 𝓟 {a} ≤ F
hbG : 𝓟 {b} ≤ G
h : ∀ᶠ (i : ι) in L, p i ↔ q
i : ι
hi : p i ↔ q
⊢ (if q then a else b) = if p i then a else b
Module: Mathlib.MeasureTheory.Function.SimpleFunc

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : Zero β
μ : Measure α
f : α →ₛ β
hf : ∀ (y : β), y ≠ 0 → μ (⇑f ⁻¹' {y}) < ∞
y : β
hy : y ∈ {y ∈ f.range | y ≠ 0}
⊢ μ (⇑f ⁻¹' {y}) < ∞

Tactic:
rw [Finset.mem_filter] at hy

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : Zero β
μ : Measure α
f : α →ₛ β
hf : ∀ (y : β), y ≠ 0 → μ (⇑f ⁻¹' {y}) < ∞
y : β
hy : y ∈ f.range ∧ y ≠ 0
⊢ μ (⇑f ⁻¹' {y}) < ∞
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Real

Goal state before:
pp: ⊢ borel ℝ = generateFrom (⋃ a, {Iio ↑a})

Tactic:
by
  rw [borel_eq_generateFrom_Iio]
  refine le_antisymm
    (generateFrom_le ?_)
    (generateFrom_mono <| iUnion_subset fun q ↦ singleton_subset_iff.mpr <| mem_range_self _)
  rintro _ ⟨a, rfl⟩
  have : IsLUB (range ((↑) : ℚ → ℝ) ∩ Iio a) a := by
    simp [isLUB_iff_le_iff, mem_upperBounds, ← le_iff_forall_rat_lt_imp_le]
  rw [← this.biUnion_Iio_eq, ← image_univ, ← image_inter_preimage, univ_inter, biUnion_image]
  exact MeasurableSet.biUnion (to_countable _)
    fun b _ => GenerateMeasurable.basic (Iio (b : ℝ)) (by simp)

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : PseudoMetrizableSpace β
inst✝² : MeasurableSpace β
inst✝¹ : BorelSpace β
ι : Type u_3
μ : Measure α
f : ι → α → β
g : α → β
u : Filter ι
hu : u.NeBot
inst✝ : u.IsCountablyGenerated
hf : ∀ (n : ι), AEMeasurable (f n) μ
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) u (𝓝 (g x))
v : ℕ → ι
hv : Tendsto v atTop u
h'f : ∀ (n : ℕ), AEMeasurable (f (v n)) μ
p : α → (ℕ → β) → Prop := fun x f' ↦ Tendsto (fun n ↦ f' n) atTop (𝓝 (g x))
x : α
⊢ Tendsto (fun n ↦ f n x) u (𝓝 (g x)) → p x fun n ↦ f (v n) x

Tactic:
hx

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : PseudoMetrizableSpace β
inst✝² : MeasurableSpace β
inst✝¹ : BorelSpace β
ι : Type u_3
μ : Measure α
f : ι → α → β
g : α → β
u : Filter ι
hu : u.NeBot
inst✝ : u.IsCountablyGenerated
hf : ∀ (n : ι), AEMeasurable (f n) μ
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) u (𝓝 (g x))
v : ℕ → ι
hv : Tendsto v atTop u
h'f : ∀ (n : ℕ), AEMeasurable (f (v n)) μ
p : α → (ℕ → β) → Prop := fun x f' ↦ Tendsto (fun n ↦ f' n) atTop (𝓝 (g x))
x : α
hx : Tendsto (fun n ↦ f n x) u (𝓝 (g x))
⊢ p x fun n ↦ f (v n) x
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∞
μ : Measure α
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
hf : ∀ (i : ι), AEMeasurable (f i) μ
hlim : ∀ᵐ (a : α) ∂μ, Tendsto (fun i ↦ f i a) u (𝓝 (g a))
v : ℕ → ι
hv : Tendsto v atTop u
h'f : ∀ (n : ℕ), AEMeasurable (f (v n)) μ
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' ↦ Tendsto f' atTop (𝓝 (g x))
x : α
⊢ Tendsto (fun i ↦ f i x) u (𝓝 (g x)) → p x fun n ↦ f (v n) x

Tactic:
hx

Goal state after:
α : Type u_1
mα : MeasurableSpace α
ι : Type u_5
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
μ : Measure α
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
hf : ∀ (i : ι), AEMeasurable (f i) μ
hlim : ∀ᵐ (a : α) ∂μ, Tendsto (fun i ↦ f i a) u (𝓝 (g a))
v : ℕ → ι
hv : Tendsto v atTop u
h'f : ∀ (n : ℕ), AEMeasurable (f (v n)) μ
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' ↦ Tendsto f' atTop (𝓝 (g x))
x : α
hx : Tendsto (fun i ↦ f i x) u (𝓝 (g x))
⊢ p x fun n ↦ f (v n) x
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable

Goal state before:
pp: α : Type u_1
inst✝³ : MeasurableSpace α
β : Type u_3
inst✝² : MeasurableSpace β
inst✝¹ : PseudoMetricSpace β
inst✝ : BorelSpace β
μ : Measure α
g : α → β
hf : ∀ ε > 0, ∃ f, AEMeasurable f μ ∧ ∀ᵐ (x : α) ∂μ, dist (f x) (g x) ≤ ε
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
f : ℕ → α → β
Hf : ∀ (n : ℕ), AEMeasurable (f n) μ ∧ ∀ᵐ (x : α) ∂μ, dist (f n x) (g x) ≤ u n
this : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), dist (f n x) (g x) ≤ u n
x : α
⊢ (∀ (n : ℕ), dist (f n x) (g x) ≤ u n) → Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))

Tactic:
hx

Goal state after:
α : Type u_1
inst✝³ : MeasurableSpace α
β : Type u_3
inst✝² : MeasurableSpace β
inst✝¹ : PseudoMetricSpace β
inst✝ : BorelSpace β
μ : Measure α
g : α → β
hf : ∀ ε > 0, ∃ f, AEMeasurable f μ ∧ ∀ᵐ (x : α) ∂μ, dist (f x) (g x) ≤ ε
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
f : ℕ → α → β
Hf : ∀ (n : ℕ), AEMeasurable (f n) μ ∧ ∀ᵐ (x : α) ∂μ, dist (f n x) (g x) ≤ u n
this : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), dist (f n x) (g x) ≤ u n
x : α
hx : ∀ (n : ℕ), dist (f n x) (g x) ≤ u n
⊢ Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : PseudoMetrizableSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
ι : Type u_3
inst✝¹ : Nonempty ι
μ : Measure α
f : ι → α → β
L : Filter ι
inst✝ : L.IsCountablyGenerated
hf : ∀ (n : ι), AEMeasurable (f n) μ
h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n ↦ f n x) L (𝓝 l)
inhabited_h : Inhabited ι
hL : L.NeBot
f_lim : α → β := fun x ↦ if h : ∃ l, Tendsto (fun n ↦ f n x) L (𝓝 l) then h.choose else ⋯.some
x : α
⊢ (∃ l, Tendsto (fun n ↦ f n x) L (𝓝 l)) → Tendsto (fun n ↦ f n x) L (𝓝 (f_lim x))

Tactic:
hx

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : PseudoMetrizableSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
ι : Type u_3
inst✝¹ : Nonempty ι
μ : Measure α
f : ι → α → β
L : Filter ι
inst✝ : L.IsCountablyGenerated
hf : ∀ (n : ι), AEMeasurable (f n) μ
h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n ↦ f n x) L (𝓝 l)
inhabited_h : Inhabited ι
hL : L.NeBot
f_lim : α → β := fun x ↦ if h : ∃ l, Tendsto (fun n ↦ f n x) L (𝓝 l) then h.choose else ⋯.some
x : α
hx : ∃ l, Tendsto (fun n ↦ f n x) L (𝓝 l)
⊢ Tendsto (fun n ↦ f n x) L (𝓝 (f_lim x))
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : PseudoMetrizableSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
ι : Type u_3
inst✝¹ : Nonempty ι
μ : Measure α
f : ι → α → β
L : Filter ι
inst✝ : L.IsCountablyGenerated
hf : ∀ (n : ι), AEMeasurable (f n) μ
h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n ↦ f n x) L (𝓝 l)
inhabited_h : Inhabited ι
hL : L.NeBot
f_lim : α → β := fun x ↦ if h : ∃ l, Tendsto (fun n ↦ f n x) L (𝓝 l) then h.choose else ⋯.some
h_ae_tendsto_f_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) L (𝓝 (f_lim x))
hf_lim : AEMeasurable f_lim μ
x : α
⊢ Tendsto (fun n ↦ f n x) L (𝓝 (f_lim x)) →
    f_lim x = AEMeasurable.mk f_lim hf_lim x → Tendsto (fun n ↦ f n x) L (𝓝 (AEMeasurable.mk f_lim hf_lim x))

Tactic:
hx

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : PseudoMetrizableSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
ι : Type u_3
inst✝¹ : Nonempty ι
μ : Measure α
f : ι → α → β
L : Filter ι
inst✝ : L.IsCountablyGenerated
hf : ∀ (n : ι), AEMeasurable (f n) μ
h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n ↦ f n x) L (𝓝 l)
inhabited_h : Inhabited ι
hL : L.NeBot
f_lim : α → β := fun x ↦ if h : ∃ l, Tendsto (fun n ↦ f n x) L (𝓝 l) then h.choose else ⋯.some
h_ae_tendsto_f_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) L (𝓝 (f_lim x))
hf_lim : AEMeasurable f_lim μ
x : α
hx : Tendsto (fun n ↦ f n x) L (𝓝 (f_lim x))
⊢ f_lim x = AEMeasurable.mk f_lim hf_lim x → Tendsto (fun n ↦ f n x) L (𝓝 (AEMeasurable.mk f_lim hf_lim x))
Module: Mathlib.MeasureTheory.Function.SimpleFuncDense

Goal state before:
pp: α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : PseudoEMetricSpace α
inst✝ : OpensMeasurableSpace α
e : ℕ → α
x : α
hx : x ∈ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (range e)
ε : ℝ≥0∞
hε : 0 < ε
N : ℕ
hN : edist x (e N) < ε
⊢ ∃ ia, True ∧ ∀ x_1 ∈ Ici ia, (nearestPt e x_1) x ∈ eball x ε

Tactic:
edist_comm

Goal state after:
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : PseudoEMetricSpace α
inst✝ : OpensMeasurableSpace α
e : ℕ → α
x : α
hx : x ∈ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (range e)
ε : ℝ≥0∞
hε : 0 < ε
N : ℕ
hN : edist (e N) x < ε
⊢ ∃ ia, True ∧ ∀ x_1 ∈ Ici ia, (nearestPt e x_1) x ∈ eball x ε
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace β
m : MeasurableSpace α
f : α → β
m' m₀ : MeasurableSpace α
μ : Measure α
inst✝ : Zero β
hm : m ≤ m₀
s : Set α
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : AEStronglyMeasurable f μ
hf_zero : f =ᵐ[μ.restrict sᶜ] 0
x : α
⊢ f x = AEStronglyMeasurable.mk f hf x → s.indicator (AEStronglyMeasurable.mk f hf) x = s.indicator f x

Tactic:
hx

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace β
m : MeasurableSpace α
f : α → β
m' m₀ : MeasurableSpace α
μ : Measure α
inst✝ : Zero β
hm : m ≤ m₀
s : Set α
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : AEStronglyMeasurable f μ
hf_zero : f =ᵐ[μ.restrict sᶜ] 0
x : α
hx : f x = AEStronglyMeasurable.mk f hf x
⊢ s.indicator (AEStronglyMeasurable.mk f hf) x = s.indicator f x
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
f : α → β
hf : AEStronglyMeasurable f μ
x : α
⊢ f x = AEStronglyMeasurable.mk f hf x → f x ∈ range (AEStronglyMeasurable.mk f hf)

Tactic:
hx

Goal state after:
α : Type u_1
β : Type u_2
inst✝ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
f : α → β
hf : AEStronglyMeasurable f μ
x : α
hx : f x = AEStronglyMeasurable.mk f hf x
⊢ f x ∈ range (AEStronglyMeasurable.mk f hf)
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
s : α →ₛ ℝ≥0
hsf : ∀ (x : α), ↑(s x) ≤ ⨆ n, f n x
r✝ : ℝ≥0
right✝ ha✝ : ↑r✝ < 1
ha : r✝ < 1
rs : α →ₛ ℝ≥0 := SimpleFunc.map (fun a ↦ r✝ * a) s
eq_rs : SimpleFunc.map c rs = const α ↑r✝ * SimpleFunc.map c s
eq : ∀ (p : ℝ≥0∞), ⇑(SimpleFunc.map c rs) ⁻¹' {p} = ⋃ n, ⇑(SimpleFunc.map c rs) ⁻¹' {p} ∩ {a | p ≤ f n a}
r : ℝ≥0∞
i j : ℕ
h : i ≤ j
x : α
⊢ r ≤ f i x → r ≤ f j x

Tactic:
hx

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
r✝ : ℝ≥0
right✝ ha✝ : ↑r✝ < 1
ha : r✝ < 1
rs : α →ₛ ℝ≥0 := SimpleFunc.map (fun a ↦ r✝ * a) s
eq_rs : SimpleFunc.map c rs = const α ↑r✝ * SimpleFunc.map c s
eq : ∀ (p : ℝ≥0∞), ⇑(SimpleFunc.map c rs) ⁻¹' {p} = ⋃ n, ⇑(SimpleFunc.map c rs) ⁻¹' {p} ∩ {a | p ≤ f n a}
r : ℝ≥0∞
i j : ℕ
h : i ≤ j
x : α
hx : r ≤ f i x
⊢ r ≤ f j x
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
ι : Type u_5
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
f : ι → α → β
g : α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) u (𝓝 (g x))
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
v : ℕ → ι
hv : Tendsto v atTop u
t : ℕ → Set β
t_sep : ∀ (n : ℕ), IsSeparable (t n)
ht : ∀ (n : ℕ), f (v n) ⁻¹' t n ∈ ae μ
x : α
⊢ (∀ (i : ℕ), f (v i) x ∈ t i) → Tendsto (fun n ↦ f n x) u (𝓝 (g x)) → g x ∈ closure[inst✝³] (⋃ i, t i)

Tactic:
hx

Goal state after:
α : Type u_1
β : Type u_2
inst✝³ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
ι : Type u_5
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
f : ι → α → β
g : α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) u (𝓝 (g x))
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
v : ℕ → ι
hv : Tendsto v atTop u
t : ℕ → Set β
t_sep : ∀ (n : ℕ), IsSeparable (t n)
ht : ∀ (n : ℕ), f (v n) ⁻¹' t n ∈ ae μ
x : α
hx : ∀ (i : ℕ), f (v i) x ∈ t i
⊢ Tendsto (fun n ↦ f n x) u (𝓝 (g x)) → g x ∈ closure[inst✝³] (⋃ i, t i)
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)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
g : α → β
left✝ : Measurable g
hg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
Hg : AEStronglyMeasurable g μ
x : α
⊢ Tendsto (fun n ↦ f n x) atTop (𝓝 (g x)) →
    g x = AEStronglyMeasurable.mk g Hg x → Tendsto (fun n ↦ f n x) atTop (𝓝 (AEStronglyMeasurable.mk g Hg x))

Tactic:
hx

Goal state after:
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)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
g : α → β
left✝ : Measurable g
hg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
Hg : AEStronglyMeasurable g μ
x : α
hx : Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
⊢ g x = AEStronglyMeasurable.mk g Hg x → Tendsto (fun n ↦ f n x) atTop (𝓝 (AEStronglyMeasurable.mk g Hg x))
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable

Goal state before:
pp: α : Type u_5
β : Type u_6
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
mb : MeasurableSpace β
inst✝ : BorelSpace β
m : MeasurableSpace α
μ : Measure α
f : α → β
s : Set β
hs : MeasurableSet s
h_nonempty : s.Nonempty
h_mem : ∀ᵐ (x : α) ∂μ, f x ∈ s
f' : α → β
hf' : StronglyMeasurable f'
hff' : f =ᵐ[μ] f'
x : α
⊢ f x ∈ s → f x = f' x → f' x = (f' ⁻¹' s).piecewise f' (fun x ↦ h_nonempty.some) x

Tactic:
hx

Goal state after:
α : Type u_5
β : Type u_6
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
mb : MeasurableSpace β
inst✝ : BorelSpace β
m : MeasurableSpace α
μ : Measure α
f : α → β
s : Set β
hs : MeasurableSet s
h_nonempty : s.Nonempty
h_mem : ∀ᵐ (x : α) ∂μ, f x ∈ s
f' : α → β
hf' : StronglyMeasurable f'
hff' : f =ᵐ[μ] f'
x : α
hx : f x ∈ s
⊢ f x = f' x → f' x = (f' ⁻¹' s).piecewise f' (fun x ↦ h_nonempty.some) x
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
s : Set α
hs : MeasurableSet s
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → f x = g x
⊢ ∀ᵐ (x : α) ∂μ.restrict s, f x = g x

Tactic:
rwa [ae_restrict_iff' hs]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
f g : α → β
s : Set α
inst✝ : DecidablePred fun x ↦ x ∈ s
hs : MeasurableSet s
hf : AEStronglyMeasurable f (μ.restrict s)
hg : AEStronglyMeasurable g (μ.restrict sᶜ)
h : ∀ᵐ (x : α) ∂μ, x ∈ s → f x = AEStronglyMeasurable.mk f hf x
x : α
⊢ (x ∈ s → f x = AEStronglyMeasurable.mk f hf x) →
    x ∈ s → s.piecewise f g x = s.piecewise (AEStronglyMeasurable.mk f hf) (AEStronglyMeasurable.mk g hg) x

Tactic:
hx

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
f g : α → β
s : Set α
inst✝ : DecidablePred fun x ↦ x ∈ s
hs : MeasurableSet s
hf : AEStronglyMeasurable f (μ.restrict s)
hg : AEStronglyMeasurable g (μ.restrict sᶜ)
h : ∀ᵐ (x : α) ∂μ, x ∈ s → f x = AEStronglyMeasurable.mk f hf x
x : α
hx : x ∈ s → f x = AEStronglyMeasurable.mk f hf x
⊢ x ∈ s → s.piecewise f g x = s.piecewise (AEStronglyMeasurable.mk f hf) (AEStronglyMeasurable.mk g hg) x
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
f g : α → β
s : Set α
inst✝ : DecidablePred fun x ↦ x ∈ s
hs : MeasurableSet s
hf : AEStronglyMeasurable f (μ.restrict s)
hg : AEStronglyMeasurable g (μ.restrict sᶜ)
h : ∀ᵐ (x : α) ∂μ, x ∈ sᶜ → g x = AEStronglyMeasurable.mk g hg x
x : α
⊢ (x ∈ sᶜ → g x = AEStronglyMeasurable.mk g hg x) →
    x ∈ sᶜ → s.piecewise f g x = s.piecewise (AEStronglyMeasurable.mk f hf) (AEStronglyMeasurable.mk g hg) x

Tactic:
hx

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
f g : α → β
s : Set α
inst✝ : DecidablePred fun x ↦ x ∈ s
hs : MeasurableSet s
hf : AEStronglyMeasurable f (μ.restrict s)
hg : AEStronglyMeasurable g (μ.restrict sᶜ)
h : ∀ᵐ (x : α) ∂μ, x ∈ sᶜ → g x = AEStronglyMeasurable.mk g hg x
x : α
hx : x ∈ sᶜ → g x = AEStronglyMeasurable.mk g hg x
⊢ x ∈ sᶜ → s.piecewise f g x = s.piecewise (AEStronglyMeasurable.mk f hf) (AEStronglyMeasurable.mk g hg) x
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_4
inst✝² : Countable ι
inst✝¹ : TopologicalSpace β
f : α → β
inst✝ : PseudoMetrizableSpace β
m : MeasurableSpace α
μ : ι → Measure α
h : ∀ (i : ι), AEStronglyMeasurable f (μ i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
t : ι → Set β
t_sep : ∀ (i : ι), IsSeparable (t i)
ht : ∀ (i : ι), f ⁻¹' t i ∈ ae (μ i)
i : ι
x : α
⊢ x ∈ f ⁻¹' t i → ∃ i, f x ∈ t i

Tactic:
hx

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_4
inst✝² : Countable ι
inst✝¹ : TopologicalSpace β
f : α → β
inst✝ : PseudoMetrizableSpace β
m : MeasurableSpace α
μ : ι → Measure α
h : ∀ (i : ι), AEStronglyMeasurable f (μ i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
t : ι → Set β
t_sep : ∀ (i : ι), IsSeparable (t i)
ht : ∀ (i : ι), f ⁻¹' t i ∈ ae (μ i)
i : ι
x : α
hx : x ∈ f ⁻¹' t i
⊢ ∃ i, f x ∈ t i
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁶ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
f : α → β
G₀ : Type u_7
inst✝⁵ : GroupWithZero G₀
inst✝⁴ : MulAction G₀ β
inst✝³ : TopologicalSpace G₀
inst✝² : ContinuousInv₀ G₀
inst✝¹ : MetrizableSpace G₀
inst✝ : ContinuousSMul G₀ β
c : α → G₀
hc : AEStronglyMeasurable c μ
hc0 : ∀ᵐ (x : α) ∂μ, c x ≠ 0
h : AEStronglyMeasurable (fun x ↦ c x • f x) μ
x : α
⊢ c x ≠ 0 → (c x)⁻¹ • c x • f x = f x

Tactic:
hx

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁶ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
f : α → β
G₀ : Type u_7
inst✝⁵ : GroupWithZero G₀
inst✝⁴ : MulAction G₀ β
inst✝³ : TopologicalSpace G₀
inst✝² : ContinuousInv₀ G₀
inst✝¹ : MetrizableSpace G₀
inst✝ : ContinuousSMul G₀ β
c : α → G₀
hc : AEStronglyMeasurable c μ
hc0 : ∀ᵐ (x : α) ∂μ, c x ≠ 0
h : AEStronglyMeasurable (fun x ↦ c x • f x) μ
x : α
hx : c x ≠ 0
⊢ (c x)⁻¹ • c x • f x = f x