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
hgM : g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 M))] 0
ν : Measure α := μ.restrict {a : α | M < f a}
this : SigmaFinite ν
meas : MeasurableSet.{u_1} (α := α) {a : α | M < f a}
s : α
hs : Membership.mem (γ := Set α) {a : α | M < f a}ᶜ s
⊢ Eq (α := ℝ) (∫ (t : ℝ) in 0..f s, g t) (∫ (t : ℝ) in 0..f s, 0)
have simp_rw : Eq (α := ℝ) (@integral _ _ _ _ Real.measurableSpace (Measure.restrict volume (Ioc 0 (f s))) fun (t : ℝ) ↦ g t)
  (@integral _ _ _ _ Real.measurableSpace (Measure.restrict volume (Ioc 0 (f s))) fun (t : ℝ) ↦ 0)
Tactic: simp_rw [intervalIntegral.integral_of_le (f_nonneg s)]
-- 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}
this : SigmaFinite ν
meas : MeasurableSet.{u_1} (α := α) {a : α | M < f a}
s : α
hs : Membership.mem (γ := Set α) {a : α | M < f a}ᶜ s
⊢ Eq (α := ℝ) (@integral _ _ _ _ Real.measurableSpace (Measure.restrict volume (Ioc 0 (f s))) fun (t : ℝ) ↦ g t)
    (@integral _ _ _ _ Real.measurableSpace (Measure.restrict volume (Ioc 0 (f s))) fun (t : ℝ) ↦ 0)
have h : g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 (f s)))] fun (a : ℝ) ↦ 0
Tactic: apply integral_congr_ae
-- 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}
this : SigmaFinite ν
meas : MeasurableSet.{u_1} (α := α) {a : α | M < f a}
s : α
hs : Membership.mem (γ := Set α) {a : α | M < f a}ᶜ s
⊢ g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 (f s)))] fun (a : ℝ) ↦ 0
have apply : Ioc 0 (f s) ⊆ Ioc 0 M
Tactic: apply ae_mono (restrict_mono ?_ le_rfl) hgM
-- 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}
this : SigmaFinite ν
meas : MeasurableSet.{u_1} (α := α) {a : α | M < f a}
s : α
hs : Membership.mem (γ := Set α) {a : α | M < f a}ᶜ s
⊢ Ioc 0 (f s) ⊆ Ioc 0 M
have h : f s ≤ M
Tactic: apply Ioc_subset_Ioc_right
-- 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}
this : SigmaFinite ν
meas : MeasurableSet.{u_1} (α := α) {a : α | M < f a}
s : α
hs : Membership.mem (γ := Set α) {a : α | M < f a}ᶜ s
⊢ ENNReal.ofReal (∫ (t : ℝ) in 0..f s, g t) = 0
have this₁ : Eq (α := ℝ) (∫ (t : ℝ) in 0..f s, g t) (∫ (t : ℝ) in 0..f s, 0)
Tactic: have : ∫ (t : ℝ) in 0..f s, g t = ∫ (t : ℝ) in 0..f s, 0 := by
        simp_rw [intervalIntegral.integral_of_le (f_nonneg s)]
        apply integral_congr_ae
        apply ae_mono (restrict_mono ?_ le_rfl) hgM
        apply Ioc_subset_Ioc_right
        simpa using 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
hgM : g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 M))] 0
ν : Measure α := μ.restrict {a : α | M < f a}
this : SigmaFinite ν
meas : MeasurableSet.{u_1} (α := α) {a : α | M < f a}
⊢ ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂ν
have I : ∫⁻ (ω : α) in {a : α | M < f a}ᶜ, ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = ∫⁻ (x : α) in {a : α | M < f a}ᶜ, 0 ∂μ
Tactic: have I : ∫⁻ ω in {a | M < f a}ᶜ, ENNReal.ofReal (∫ t in 0..f ω, g t) ∂μ
             = ∫⁻ _ in {a | M < f a}ᶜ, 0 ∂μ := by
      apply setLIntegral_congr_fun meas.compl (fun s hs ↦ ?_)
      have : ∫ (t : ℝ) in 0..f s, g t = ∫ (t : ℝ) in 0..f s, 0 := by
        simp_rw [intervalIntegral.integral_of_le (f_nonneg s)]
        apply integral_congr_ae
        apply ae_mono (restrict_mono ?_ le_rfl) hgM
        apply Ioc_subset_Ioc_right
        simpa using hs
      simp [this]
-- 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}
this : SigmaFinite ν
meas : MeasurableSet.{u_1} (α := α) {a : α | M < f a}
I :
  ∫⁻ (ω : α) in {a : α | M < f a}ᶜ, ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ =
    ∫⁻ (x : α) in {a : α | M < f a}ᶜ, 0 ∂μ
⊢ ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂ν
have I₁ : ∫⁻ (ω : α) in {a : α | M < f a}ᶜ, ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = 0
Tactic: simp only [lintegral_const, zero_mul] at I
Cleared hypotheses: 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
hgM : g =ᶠ[ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 M))] 0
ν : Measure α := μ.restrict {a : α | M < f a}
this : SigmaFinite ν
meas : MeasurableSet.{u_1} (α := α) {a : α | M < f a}
I : ∫⁻ (ω : α) in {a : α | M < f a}ᶜ, ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = 0
⊢ ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂ν
have rw : ∫⁻ (x : α) in {a : α | M < f a}, ENNReal.ofReal (∫ (t : ℝ) in 0..f x, g t) ∂μ +
    ∫⁻ (x : α) in {a : α | M < f a}ᶜ, ENNReal.ofReal (∫ (t : ℝ) in 0..f x, g t) ∂μ =
  ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂ν
Tactic: rw [← lintegral_add_compl _ meas, I, add_zero]
-- 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}
this : SigmaFinite ν
meas : MeasurableSet.{u_1} (α := α) {a : α | M < f a}
I : ∫⁻ (ω : α) in {a : α | M < f a}ᶜ, ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = 0
rw :
  ∫⁻ (x : α) in {a : α | M < f a}, ENNReal.ofReal (∫ (t : ℝ) in 0..f x, g t) ∂μ +
      ∫⁻ (x : α) in {a : α | M < f a}ᶜ, ENNReal.ofReal (∫ (t : ℝ) in 0..f x, g t) ∂μ =
    ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂ν
⊢ ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂ν
have rw₁ : ∫⁻ (x : α) in {a : α | M < f a}, ENNReal.ofReal (∫ (t : ℝ) in 0..f x, g t) ∂μ + 0 =
  ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂ν
Tactic: rw [← lintegral_add_compl _ meas, I, add_zero]
-- 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}
this : SigmaFinite ν
meas : MeasurableSet.{u_1} (α := α) {a : α | M < f a}
I : ∫⁻ (ω : α) in {a : α | M < f a}ᶜ, ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = 0
rw :
  ∫⁻ (x : α) in {a : α | M < f a}, ENNReal.ofReal (∫ (t : ℝ) in 0..f x, g t) ∂μ +
      ∫⁻ (x : α) in {a : α | M < f a}ᶜ, ENNReal.ofReal (∫ (t : ℝ) in 0..f x, g t) ∂μ =
    ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂ν
rw₁ :
  ∫⁻ (x : α) in {a : α | M < f a}, ENNReal.ofReal (∫ (t : ℝ) in 0..f x, g t) ∂μ + 0 =
    ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂ν
⊢ ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂ν
have rw₂ : ∫⁻ (x : α) in {a : α | M < f a}, ENNReal.ofReal (∫ (t : ℝ) in 0..f x, g t) ∂μ =
  ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂ν
Tactic: rw [← lintegral_add_compl _ meas, I, add_zero]
-- 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}
this : SigmaFinite ν
⊢ ∫⁻ (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 A : ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂ν
Tactic: have A : ∫⁻ ω, ENNReal.ofReal (∫ t in 0..f ω, g t) ∂μ
         = ∫⁻ ω, ENNReal.ofReal (∫ t in 0..f ω, g t) ∂ν := by
    have meas : MeasurableSet {a | M < f a} := measurableSet_lt measurable_const f_mble
    have I : ∫⁻ ω in {a | M < f a}ᶜ, ENNReal.ofReal (∫ t in 0..f ω, g t) ∂μ
             = ∫⁻ _ in {a | M < f a}ᶜ, 0 ∂μ := by
      apply setLIntegral_congr_fun meas.compl (fun s hs ↦ ?_)
      have : ∫ (t : ℝ) in 0..f s, g t = ∫ (t : ℝ) in 0..f s, 0 := by
        simp_rw [intervalIntegral.integral_of_le (f_nonneg s)]
        apply integral_congr_ae
        apply ae_mono (restrict_mono ?_ le_rfl) hgM
        apply Ioc_subset_Ioc_right
        simpa using hs
      simp [this]
    simp only [lintegral_const, zero_mul] at I
    rw [← lintegral_add_compl _ meas, I, add_zero]
-- 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}
this : SigmaFinite ν
A : ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂ν
⊢ ∫⁻ (t : ℝ) in Ioc 0 M,
      HMul.hMul (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) {a : α | t ≤ f a} : ℝ≥0∞) (ENNReal.ofReal (g t)) =
    ∫⁻ (t : ℝ) in Ioc 0 M,
      HMul.hMul (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) {a : α | t ≤ f a} : ℝ≥0∞) (ENNReal.ofReal (g t))
have h : (ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 M))).EventuallyEq (β := ℝ≥0∞)
  (fun (a : ℝ) ↦ HMul.hMul (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) {a_1 : α | a ≤ f a_1} : ℝ≥0∞) (ENNReal.ofReal (g a)))
  fun (a : ℝ) ↦ HMul.hMul (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) {a_1 : α | a ≤ f a_1} : ℝ≥0∞) (ENNReal.ofReal (g a))
Tactic: apply lintegral_congr_ae
-- 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}
this : SigmaFinite ν
A : ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂ν
⊢ (ae (F := Measure ℝ) (Measure.restrict volume (Ioc 0 M))).EventuallyEq (β := ℝ≥0∞)
    (fun (a : ℝ) ↦ HMul.hMul (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) {a_1 : α | a ≤ f a_1} : ℝ≥0∞) (ENNReal.ofReal (g a)))
    fun (a : ℝ) ↦ HMul.hMul (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) {a_1 : α | a ≤ f a_1} : ℝ≥0∞) (ENNReal.ofReal (g a))
have h : ∀ (t : ℝ) (ht : g t = 0 t),
  HMul.hMul (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) {a : α | t ≤ f a} : ℝ≥0∞) (ENNReal.ofReal (g t)) =
    HMul.hMul (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) {a : α | t ≤ f a} : ℝ≥0∞) (ENNReal.ofReal (g t))
Tactic: filter_upwards [hgM] with t ht
-- 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}
this : SigmaFinite ν
A : ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂ν
⊢ ∫⁻ (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)) =
    ∫⁻ (t : ℝ) in Ioi 0,
      HMul.hMul (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) {a : α | t ≤ f a} : ℝ≥0∞) (ENNReal.ofReal (g t))
have B1 : ∫⁻ (t : ℝ) in Ioc 0 M,
    HMul.hMul (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) {a : α | t ≤ f a} : ℝ≥0∞) (ENNReal.ofReal (g t)) =
  ∫⁻ (t : ℝ) in Ioc 0 M,
    HMul.hMul (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) {a : α | t ≤ f a} : ℝ≥0∞) (ENNReal.ofReal (g t))
Tactic: have B1 : ∫⁻ t in Ioc 0 M, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t)
         = ∫⁻ t in Ioc 0 M, ν {a : α | t ≤ f a} * ENNReal.ofReal (g t) := by
      apply lintegral_congr_ae
      filter_upwards [hgM] with t ht
      simp [ht]