awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Layercake (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioi 0))] 0
H2 :
  ∀ (s : ℝ) (_ : s > 0) (_ : LT.lt (α := ℝ) 0 (∫ (t : ℝ) in 0..s, g t)),
    Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) {a : α | s < f a} : ℝ≥0∞) (∞ : ℝ≥0∞)
M_bdd : BddAbove (α := ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
M : ℝ := sSup {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
zero_mem : Membership.mem (γ := Set ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0} 0
M_nonneg : 0 ≤ M
u : ℕ → ℝ
uM : ∀ (n : ℕ), u n < M
ulim : Tendsto u atTop (𝓝 M)
n : ℕ
⊢ g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 (u n)))] 0
have obtain : ∀ (s : ℝ) (hs : g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0) (uns : u n < s),
  g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 (u n)))] 0
Tactic: obtain ⟨s, hs, uns⟩ : ∃ s, g =ᶠ[ae (Measure.restrict volume (Ioc 0 s))] 0 ∧ u n < s :=
        exists_lt_of_lt_csSup (Set.nonempty_of_mem zero_mem) (uM n)
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Layercake (hypothesis)
Goal:
α : Type u_1
inst : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioi 0))] 0
H2 :
  ∀ (s : ℝ) (_ : s > 0) (_ : LT.lt (α := ℝ) 0 (∫ (t : ℝ) in 0..s, g t)),
    Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) {a : α | s < f a} : ℝ≥0∞) (∞ : ℝ≥0∞)
M_bdd : BddAbove (α := ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
M : ℝ := sSup {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
zero_mem : Membership.mem (γ := Set ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0} 0
M_nonneg : 0 ≤ M
u : ℕ → ℝ
uM : ∀ (n : ℕ), u n < M
ulim : Tendsto u atTop (𝓝 M)
⊢ g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioo 0 M))] 0
have I : ∀ (n : ℕ), g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 (u n)))] 0
Tactic: have I : ∀ n, g =ᵐ[volume.restrict (Ioc (0 : ℝ) (u n))] 0 := by
      intro n
      obtain ⟨s, hs, uns⟩ : ∃ s, g =ᶠ[ae (Measure.restrict volume (Ioc 0 s))] 0 ∧ u n < s :=
        exists_lt_of_lt_csSup (Set.nonempty_of_mem zero_mem) (uM n)
      exact ae_restrict_of_ae_restrict_of_subset (Ioc_subset_Ioc_right uns.le) hs
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Layercake (hypothesis)
Goal:
α : Type u_1
inst : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioi 0))] 0
H2 :
  ∀ (s : ℝ) (_ : s > 0) (_ : LT.lt (α := ℝ) 0 (∫ (t : ℝ) in 0..s, g t)),
    Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) {a : α | s < f a} : ℝ≥0∞) (∞ : ℝ≥0∞)
M_bdd : BddAbove (α := ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
M : ℝ := sSup {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
zero_mem : Membership.mem (γ := Set ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0} 0
M_nonneg : 0 ≤ M
u : ℕ → ℝ
uM : ∀ (n : ℕ), u n < M
ulim : Tendsto u atTop (𝓝 M)
I : ∀ (n : ℕ), g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 (u n)))] 0
⊢ g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioo 0 M))] 0
have this : g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (⋃ (n : ℕ), Ioc 0 (u n)))] 0
Tactic: have : g =ᵐ[volume.restrict (⋃ n, Ioc (0 : ℝ) (u n))] 0 := (ae_restrict_iUnion_iff _ _).2 I
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Layercake (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioi 0))] 0
H2 :
  ∀ (s : ℝ) (_ : s > 0) (_ : LT.lt (α := ℝ) 0 (∫ (t : ℝ) in 0..s, g t)),
    Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) {a : α | s < f a} : ℝ≥0∞) (∞ : ℝ≥0∞)
M_bdd : BddAbove (α := ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
M : ℝ := sSup {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
zero_mem : Membership.mem (γ := Set ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0} 0
M_nonneg : 0 ≤ M
u : ℕ → ℝ
uM : ∀ (n : ℕ), u n < M
ulim : Tendsto u atTop (𝓝 M)
I : ∀ (n : ℕ), g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 (u n)))] 0
this : g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (⋃ (n : ℕ), Ioc 0 (u n)))] 0
⊢ g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioo 0 M))] 0
have apply : Ioo 0 M ⊆ ⋃ (n : ℕ), Ioc 0 (u n)
Tactic: apply ae_restrict_of_ae_restrict_of_subset _ this
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Layercake (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioi 0))] 0
H2 :
  ∀ (s : ℝ) (_ : s > 0) (_ : LT.lt (α := ℝ) 0 (∫ (t : ℝ) in 0..s, g t)),
    Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) {a : α | s < f a} : ℝ≥0∞) (∞ : ℝ≥0∞)
M_bdd : BddAbove (α := ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
M : ℝ := sSup {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
zero_mem : Membership.mem (γ := Set ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0} 0
M_nonneg : 0 ≤ M
u : ℕ → ℝ
uM : ∀ (n : ℕ), u n < M
ulim : Tendsto u atTop (𝓝 M)
I : ∀ (n : ℕ), g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 (u n)))] 0
this : g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (⋃ (n : ℕ), Ioc 0 (u n)))] 0
x : ℝ
x_pos : 0 < x
xM : x < M
⊢ Membership.mem (γ := Set ℝ) (⋃ (n : ℕ), Ioc 0 (u n)) x
have obtain : ∀ (n : ℕ) (hn : x < u n), Membership.mem (γ := Set ℝ) (⋃ (n : ℕ), Ioc 0 (u n)) x
Tactic: obtain ⟨n, hn⟩ : ∃ n, x < u n := ((tendsto_order.1 ulim).1 _ xM).exists
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Layercake (hypothesis)
Goal:
α : Type u_1
inst : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioi 0))] 0
H2 :
  ∀ (s : ℝ) (_ : s > 0) (_ : LT.lt (α := ℝ) 0 (∫ (t : ℝ) in 0..s, g t)),
    Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) {a : α | s < f a} : ℝ≥0∞) (∞ : ℝ≥0∞)
M_bdd : BddAbove (α := ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
M : ℝ := sSup {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
zero_mem : Membership.mem (γ := Set ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0} 0
M_nonneg : 0 ≤ M
⊢ ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ =
    ∫⁻ (t : ℝ) in Ioi 0,
      HMul.hMul (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) {a : α | t ≤ f a} : ℝ≥0∞) (ENNReal.ofReal (g t))
have hgM : g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 M))] 0
Tactic: have hgM : g =ᵐ[volume.restrict (Ioc (0 : ℝ) M)] 0 := by
    rw [← restrict_Ioo_eq_restrict_Ioc]
    obtain ⟨u, -, uM, ulim⟩ : ∃ u, StrictMono u ∧ (∀ (n : ℕ), u n < M) ∧ Tendsto u atTop (𝓝 M) :=
      exists_seq_strictMono_tendsto M
    have I : ∀ n, g =ᵐ[volume.restrict (Ioc (0 : ℝ) (u n))] 0 := by
      intro n
      obtain ⟨s, hs, uns⟩ : ∃ s, g =ᶠ[ae (Measure.restrict volume (Ioc 0 s))] 0 ∧ u n < s :=
        exists_lt_of_lt_csSup (Set.nonempty_of_mem zero_mem) (uM n)
      exact ae_restrict_of_ae_restrict_of_subset (Ioc_subset_Ioc_right uns.le) hs
    have : g =ᵐ[volume.restrict (⋃ n, Ioc (0 : ℝ) (u n))] 0 := (ae_restrict_iUnion_iff _ _).2 I
    apply ae_restrict_of_ae_restrict_of_subset _ this
    rintro x ⟨x_pos, xM⟩
    obtain ⟨n, hn⟩ : ∃ n, x < u n := ((tendsto_order.1 ulim).1 _ xM).exists
    exact mem_iUnion.2 ⟨n, ⟨x_pos, hn.le⟩⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Layercake (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioi 0))] 0
H2 :
  ∀ (s : ℝ) (_ : s > 0) (_ : LT.lt (α := ℝ) 0 (∫ (t : ℝ) in 0..s, g t)),
    Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) {a : α | s < f a} : ℝ≥0∞) (∞ : ℝ≥0∞)
M_bdd : BddAbove (α := ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
M : ℝ := sSup {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
zero_mem : Membership.mem (γ := Set ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0} 0
M_nonneg : 0 ≤ M
hgM : g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 M))] 0
ν : Measure α := μ.restrict {a : α | M < f a}
⊢ SigmaFinite ν
have obtain : ∀ (u : ℕ → ℝ) (uM : ∀ (n : ℕ), M < u n) (ulim : Tendsto u atTop (𝓝 M)), SigmaFinite ν
Tactic: obtain ⟨u, -, uM, ulim⟩ : ∃ u, StrictAnti u ∧ (∀ (n : ℕ), M < u n) ∧ Tendsto u atTop (𝓝 M) :=
      exists_seq_strictAnti_tendsto M
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Layercake (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioi 0))] 0
H2 :
  ∀ (s : ℝ) (_ : s > 0) (_ : LT.lt (α := ℝ) 0 (∫ (t : ℝ) in 0..s, g t)),
    Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) {a : α | s < f a} : ℝ≥0∞) (∞ : ℝ≥0∞)
M_bdd : BddAbove (α := ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
M : ℝ := sSup {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
zero_mem : Membership.mem (γ := Set ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0} 0
M_nonneg : 0 ≤ M
hgM : g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 M))] 0
ν : Measure α := μ.restrict {a : α | M < f a}
u : ℕ → ℝ
uM : ∀ (n : ℕ), M < u n
ulim : Tendsto u atTop (𝓝 M)
n : ℕ
⊢ Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) {a : α | f a ≤ M} : ℝ≥0∞) (0 : ℝ≥0∞)
have rw : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) ({a : α | f a ≤ M} ∩ {a : α | M < f a}) : ℝ≥0∞) (0 : ℝ≥0∞)
Tactic: rw [Measure.restrict_apply (measurableSet_le f_mble measurable_const)]
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Layercake (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioi 0))] 0
H2 :
  ∀ (s : ℝ) (_ : s > 0) (_ : LT.lt (α := ℝ) 0 (∫ (t : ℝ) in 0..s, g t)),
    Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) {a : α | s < f a} : ℝ≥0∞) (∞ : ℝ≥0∞)
M_bdd : BddAbove (α := ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
M : ℝ := sSup {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
zero_mem : Membership.mem (γ := Set ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0} 0
M_nonneg : 0 ≤ M
hgM : g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 M))] 0
ν : Measure α := μ.restrict {a : α | M < f a}
u : ℕ → ℝ
uM : ∀ (n : ℕ), M < u n
ulim : Tendsto u atTop (𝓝 M)
n : ℕ
⊢ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) ({a : α | f a ≤ M} ∩ {a : α | M < f a}) : ℝ≥0∞) (0 : ℝ≥0∞)
have h : Eq.{u_1 + 1} (α := Set α) ({a : α | f a ≤ M} ∩ {a : α | M < f a}) ∅
Tactic: convert! measure_empty (μ := μ)
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Layercake (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioi 0))] 0
H2 :
  ∀ (s : ℝ) (_ : s > 0) (_ : LT.lt (α := ℝ) 0 (∫ (t : ℝ) in 0..s, g t)),
    Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) {a : α | s < f a} : ℝ≥0∞) (∞ : ℝ≥0∞)
M_bdd : BddAbove (α := ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
M : ℝ := sSup {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
zero_mem : Membership.mem (γ := Set ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0} 0
M_nonneg : 0 ≤ M
hgM : g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 M))] 0
ν : Measure α := μ.restrict {a : α | M < f a}
u : ℕ → ℝ
uM : ∀ (n : ℕ), M < u n
ulim : Tendsto u atTop (𝓝 M)
n : ℕ
⊢ Eq.{u_1 + 1} (α := Set α) ({a : α | f a ≤ M} ∩ {a : α | M < f a}) ∅
have h : Disjoint.{u_1} (α := Set α) {a : α | f a ≤ M} {a : α | M < f a}
Tactic: rw [← disjoint_iff_inter_eq_empty]
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Layercake (hypothesis)
Goal:
α : Type u_1
inst : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioi 0))] 0
H2 :
  ∀ (s : ℝ) (_ : s > 0) (_ : LT.lt (α := ℝ) 0 (∫ (t : ℝ) in 0..s, g t)),
    Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) {a : α | s < f a} : ℝ≥0∞) (∞ : ℝ≥0∞)
M_bdd : BddAbove (α := ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
M : ℝ := sSup {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
zero_mem : Membership.mem (γ := Set ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0} 0
M_nonneg : 0 ≤ M
hgM : g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 M))] 0
ν : Measure α := μ.restrict {a : α | M < f a}
u : ℕ → ℝ
uM : ∀ (n : ℕ), M < u n
ulim : Tendsto u atTop (𝓝 M)
n : ℕ
⊢ LT.lt (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) ({a : α | f a ≤ M} ∪ {a : α | u n < f a}) : ℝ≥0∞) (∞ : ℝ≥0∞)
have I : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) {a : α | f a ≤ M} : ℝ≥0∞) (0 : ℝ≥0∞)
Tactic: have I : ν {a | f a ≤ M} = 0 := by
          rw [Measure.restrict_apply (measurableSet_le f_mble measurable_const)]
          convert! measure_empty (μ := μ)
          rw [← disjoint_iff_inter_eq_empty]
          exact disjoint_left.mpr (fun a ha ↦ by simpa using ha)
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Layercake (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioi 0))] 0
H2 :
  ∀ (s : ℝ) (_ : s > 0) (_ : LT.lt (α := ℝ) 0 (∫ (t : ℝ) in 0..s, g t)),
    Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) {a : α | s < f a} : ℝ≥0∞) (∞ : ℝ≥0∞)
M_bdd : BddAbove (α := ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
M : ℝ := sSup {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
zero_mem : Membership.mem (γ := Set ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0} 0
M_nonneg : 0 ≤ M
hgM : g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 M))] 0
ν : Measure α := μ.restrict {a : α | M < f a}
u : ℕ → ℝ
uM : ∀ (n : ℕ), M < u n
ulim : Tendsto u atTop (𝓝 M)
n : ℕ
I : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) {a : α | f a ≤ M} : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) {a : α | u n < f a} : ℝ≥0∞) (∞ : ℝ≥0∞)
have rw : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) {a : α | u n < f a} : ℝ≥0∞) (∞ : ℝ≥0∞)
Tactic: rw [lt_top_iff_ne_top]
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Layercake (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioi 0))] 0
H2 :
  ∀ (s : ℝ) (_ : s > 0) (_ : LT.lt (α := ℝ) 0 (∫ (t : ℝ) in 0..s, g t)),
    Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) {a : α | s < f a} : ℝ≥0∞) (∞ : ℝ≥0∞)
M_bdd : BddAbove (α := ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
M : ℝ := sSup {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
zero_mem : Membership.mem (γ := Set ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0} 0
M_nonneg : 0 ≤ M
hgM : g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 M))] 0
ν : Measure α := μ.restrict {a : α | M < f a}
u : ℕ → ℝ
uM : ∀ (n : ℕ), M < u n
ulim : Tendsto u atTop (𝓝 M)
n : ℕ
I : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) {a : α | f a ≤ M} : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) {a : α | u n < f a} : ℝ≥0∞) (∞ : ℝ≥0∞)
have apply : LT.lt (α := ℝ) 0 (∫ (t : ℝ) in 0..u n, g t)
Tactic: apply H2 _ (M_nonneg.trans_lt (uM n))
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Layercake (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioi 0))] 0
H2 :
  ∀ (s : ℝ) (_ : s > 0) (_ : LT.lt (α := ℝ) 0 (∫ (t : ℝ) in 0..s, g t)),
    Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) {a : α | s < f a} : ℝ≥0∞) (∞ : ℝ≥0∞)
M_bdd : BddAbove (α := ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
M : ℝ := sSup {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
zero_mem : Membership.mem (γ := Set ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0} 0
M_nonneg : 0 ≤ M
hgM : g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 M))] 0
ν : Measure α := μ.restrict {a : α | M < f a}
u : ℕ → ℝ
uM : ∀ (n : ℕ), M < u n
ulim : Tendsto u atTop (𝓝 M)
n : ℕ
I : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) {a : α | f a ≤ M} : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ LT.lt (α := ℝ) 0 (∫ (t : ℝ) in 0..u n, g t)
have by_contra : ∀ (H3 : ¬LT.lt (α := ℝ) 0 (∫ (t : ℝ) in 0..u n, g t)), False
Tactic: by_contra H3
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Layercake (hypothesis)
Goal:
α : Type u_1
inst : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioi 0))] 0
H2 :
  ∀ (s : ℝ) (_ : s > 0) (_ : LT.lt (α := ℝ) 0 (∫ (t : ℝ) in 0..s, g t)),
    Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) {a : α | s < f a} : ℝ≥0∞) (∞ : ℝ≥0∞)
M_bdd : BddAbove (α := ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
M : ℝ := sSup {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
zero_mem : Membership.mem (γ := Set ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0} 0
M_nonneg : 0 ≤ M
hgM : g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 M))] 0
ν : Measure α := μ.restrict {a : α | M < f a}
u : ℕ → ℝ
uM : ∀ (n : ℕ), M < u n
ulim : Tendsto u atTop (𝓝 M)
n : ℕ
I : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) {a : α | f a ≤ M} : ℝ≥0∞) (0 : ℝ≥0∞)
H3 : ¬LT.lt (α := ℝ) 0 (∫ (t : ℝ) in 0..u n, g t)
⊢ False
have H3₁ : LE.le (α := ℝ) (@integral _ _ _ _ Real.measurableSpace (Measure.restrict volume (Ioc 0 (u n))) fun (x : ℝ) ↦ g x) 0
Tactic: rw [not_lt, intervalIntegral.integral_of_le (M_nonneg.trans (uM n).le)] at H3
Cleared hypotheses: H3
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Layercake (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioi 0))] 0
H2 :
  ∀ (s : ℝ) (_ : s > 0) (_ : LT.lt (α := ℝ) 0 (∫ (t : ℝ) in 0..s, g t)),
    Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) {a : α | s < f a} : ℝ≥0∞) (∞ : ℝ≥0∞)
M_bdd : BddAbove (α := ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
M : ℝ := sSup {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
zero_mem : Membership.mem (γ := Set ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0} 0
M_nonneg : 0 ≤ M
hgM : g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 M))] 0
ν : Measure α := μ.restrict {a : α | M < f a}
u : ℕ → ℝ
uM : ∀ (n : ℕ), M < u n
ulim : Tendsto u atTop (𝓝 M)
n : ℕ
I : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) {a : α | f a ≤ M} : ℝ≥0∞) (0 : ℝ≥0∞)
H3 : LE.le (α := ℝ) (@integral _ _ _ _ Real.measurableSpace (Measure.restrict volume (Ioc 0 (u n))) fun (x : ℝ) ↦ g x) 0
⊢ ∀ᵐ (t : ℝ) ∂Measure.restrict volume (Ioc 0 (u n)), 0 ≤ g t
have h : ∀ (a : ℝ),
  Membership.mem (γ := Set ℝ)
    {x : ℝ |
      Membership.mem (γ := Set ℝ) {x : ℝ | (fun (x : ℝ) ↦ x ∈ Ioc 0 (u n)) x} x →
        Membership.mem (γ := Set ℝ) {x : ℝ | (fun (t : ℝ) ↦ 0 ≤ g t) x} x}
    a
Tactic: filter_upwards [ae_restrict_mem measurableSet_Ioc] with s hs using g_nn _ hs.1
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.Layercake (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioi 0))] 0
H2 :
  ∀ (s : ℝ) (_ : s > 0) (_ : LT.lt (α := ℝ) 0 (∫ (t : ℝ) in 0..s, g t)),
    Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) {a : α | s < f a} : ℝ≥0∞) (∞ : ℝ≥0∞)
M_bdd : BddAbove (α := ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
M : ℝ := sSup {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0}
zero_mem : Membership.mem (γ := Set ℝ) {s : ℝ | g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 s))] 0} 0
M_nonneg : 0 ≤ M
hgM : g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 M))] 0
ν : Measure α := μ.restrict {a : α | M < f a}
u : ℕ → ℝ
uM : ∀ (n : ℕ), M < u n
ulim : Tendsto u atTop (𝓝 M)
n : ℕ
I : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) {a : α | f a ≤ M} : ℝ≥0∞) (0 : ℝ≥0∞)
H3 : LE.le (α := ℝ) (@integral _ _ _ _ Real.measurableSpace (Measure.restrict volume (Ioc 0 (u n))) fun (x : ℝ) ↦ g x) 0
h :
  ∀ (a : ℝ),
    Membership.mem (γ := Set ℝ)
      {x : ℝ |
        Membership.mem (γ := Set ℝ) {x : ℝ | (fun (x : ℝ) ↦ x ∈ Ioc 0 (u n)) x} x →
          Membership.mem (γ := Set ℝ) {x : ℝ | (fun (t : ℝ) ↦ 0 ≤ g t) x} x}
      a
⊢ ∀ᵐ (t : ℝ) ∂Measure.restrict volume (Ioc 0 (u n)), 0 ≤ g t
have h₁ : ∀ a ∈ Ioc 0 (u n), 0 ≤ g a
Tactic: filter_upwards [ae_restrict_mem measurableSet_Ioc] with s hs using g_nn _ hs.1