awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.AEEqOfIntegral (hypothesis)
Goal:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : CompleteSpace E
inst_3 : SigmaFinite μ
f : α → E
hf_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn f s μ
hf_zero :
  ∀ (s : Set α),
    MeasurableSet s →
      LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) →
        Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, f x ∂μ) 0
S : (i : ℕ) → Set α := spanningSets μ
n : ℕ
h_meas_n : MeasurableSet (S n)
⊢ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) ({a : α | ¬f a = 0 a} ∩ spanningSets μ n) : ℝ≥0∞) (0 : ℝ≥0∞)
have hμn : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (S n) : ℝ≥0∞) (∞ : ℝ≥0∞)
Tactic: have hμn : μ (S n) < ∞ := measure_spanningSets_lt_top μ n
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.AEEqOfIntegral (goal)
Goal:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : CompleteSpace E
inst_3 : SigmaFinite μ
f : α → E
hf_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn f s μ
hf_zero :
  ∀ (s : Set α),
    MeasurableSet s →
      LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) →
        Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, f x ∂μ) 0
S : (i : ℕ) → Set α := spanningSets μ
n : ℕ
h_meas_n : MeasurableSet (S n)
hμn : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (S n) : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) ({a : α | ¬f a = 0 a} ∩ spanningSets μ n) : ℝ≥0∞) (0 : ℝ≥0∞)
have rw : Eq (α := ℝ≥0∞) ((μ.restrict (S n) : (a : Set α) → ℝ≥0∞) {a : α | ¬f a = 0 a} : ℝ≥0∞) (0 : ℝ≥0∞)
Tactic: rw [← Measure.restrict_apply' h_meas_n]
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.AEEqOfIntegral (goal)
Goal:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : CompleteSpace E
inst_3 : SigmaFinite μ
f g : α → E
hf_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn f s μ
hg_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn g s μ
hfg_eq :
  ∀ (s : Set α),
    MeasurableSet s →
      LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) →
        Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, f x ∂μ) (∫ (x : α) in s, g x ∂μ)
⊢ f =ᶠ[ae μ] g
have rw : f - g =ᶠ[ae μ] 0
Tactic: rw [← sub_ae_eq_zero]
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.AEEqOfIntegral (goal)
Goal:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : CompleteSpace E
inst_3 : SigmaFinite μ
f g : α → E
hf_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn f s μ
hg_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn g s μ
hfg_eq :
  ∀ (s : Set α),
    MeasurableSet s →
      LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) →
        Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, f x ∂μ) (∫ (x : α) in s, g x ∂μ)
s : Set α
hs : MeasurableSet s
hμs : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, (f - g) x ∂μ) 0
have rw : HSub.hSub (α := E) (∫ (a : α) in s, f a ∂μ) (∫ (a : α) in s, g a ∂μ) = 0
Tactic: rw [integral_sub' (hf_int_finite s hs hμs) (hg_int_finite s hs hμs),
      sub_eq_zero.mpr (hfg_eq s hs hμs)]
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.AEEqOfIntegral (goal)
Goal:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : CompleteSpace E
inst_3 : SigmaFinite μ
f g : α → E
hf_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn f s μ
hg_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn g s μ
hfg_eq :
  ∀ (s : Set α),
    MeasurableSet s →
      LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) →
        Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, f x ∂μ) (∫ (x : α) in s, g x ∂μ)
s : Set α
hs : MeasurableSet s
hμs : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
rw : HSub.hSub (α := E) (∫ (a : α) in s, f a ∂μ) (∫ (a : α) in s, g a ∂μ) = 0
⊢ Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, (f - g) x ∂μ) 0
have rw₁ : Eq.{u_2 + 1} (α := E) 0 0
Tactic: rw [integral_sub' (hf_int_finite s hs hμs) (hg_int_finite s hs hμs),
      sub_eq_zero.mpr (hfg_eq s hs hμs)]
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.AEEqOfIntegral (hypothesis)
Goal:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : CompleteSpace E
inst_3 : SigmaFinite μ
f g : α → E
hf_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn f s μ
hg_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn g s μ
hfg_eq :
  ∀ (s : Set α),
    MeasurableSet s →
      LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) →
        Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, f x ∂μ) (∫ (x : α) in s, g x ∂μ)
⊢ f - g =ᶠ[ae μ] 0
have hfg : ∀ (s : Set α),
  MeasurableSet s →
    LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) →
      Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, (f - g) x ∂μ) 0
Tactic: have hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, (f - g) x ∂μ) = 0 := by
    intro s hs hμs
    rw [integral_sub' (hf_int_finite s hs hμs) (hg_int_finite s hs hμs),
      sub_eq_zero.mpr (hfg_eq s hs hμs)]
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.AEEqOfIntegral (hypothesis)
Goal:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : CompleteSpace E
inst_3 : SigmaFinite μ
f g : α → E
hf_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn f s μ
hg_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn g s μ
hfg_eq :
  ∀ (s : Set α),
    MeasurableSet s →
      LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) →
        Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, f x ∂μ) (∫ (x : α) in s, g x ∂μ)
hfg :
  ∀ (s : Set α),
    MeasurableSet s →
      LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) →
        Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, (f - g) x ∂μ) 0
⊢ f - g =ᶠ[ae μ] 0
have hfg_int : ∀ (s : Set α),
  MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn (f - g) s μ
Tactic: have hfg_int : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn (f - g) s μ := fun s hs hμs =>
    (hf_int_finite s hs hμs).sub (hg_int_finite s hs hμs)
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.AEEqOfIntegral (goal)
Goal:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : CompleteSpace E
f : α → E
hf_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn f s μ
hf_zero :
  ∀ (s : Set α),
    MeasurableSet s →
      LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) →
        Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, f x ∂μ) 0
hf : AEFinStronglyMeasurable f μ
t : Set α := hf.sigmaFiniteSet
⊢ f =ᶠ[ae μ] 0
have suffices : f =ᶠ[ae (μ.restrict t)] 0
Tactic: suffices f =ᵐ[μ.restrict t] 0 from
    ae_of_ae_restrict_of_ae_restrict_compl _ this hf.ae_eq_zero_compl
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.AEEqOfIntegral (hypothesis)
Goal:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : CompleteSpace E
f : α → E
hf_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn f s μ
hf_zero :
  ∀ (s : Set α),
    MeasurableSet s →
      LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) →
        Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, f x ∂μ) 0
hf : AEFinStronglyMeasurable f μ
t : Set α := hf.sigmaFiniteSet
⊢ f =ᶠ[ae (μ.restrict t)] 0
have this : SigmaFinite (μ.restrict t)
Tactic: haveI : SigmaFinite (μ.restrict t) := hf.sigmaFinite_restrict
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.AEEqOfIntegral (goal)
Goal:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : CompleteSpace E
f : α → E
hf_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn f s μ
hf_zero :
  ∀ (s : Set α),
    MeasurableSet s →
      LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) →
        Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, f x ∂μ) 0
hf : AEFinStronglyMeasurable f μ
t : Set α := hf.sigmaFiniteSet
this : SigmaFinite (μ.restrict t)
⊢ f =ᶠ[ae (μ.restrict t)] 0
have refine_1 : ∀ (s : Set α),
  MeasurableSet s →
    LT.lt (α := ℝ≥0∞) ((μ.restrict t : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn f s (μ.restrict t)
Tactic: refine ae_eq_zero_of_forall_setIntegral_eq_of_sigmaFinite ?_ ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.AEEqOfIntegral (goal)
Goal:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : CompleteSpace E
f : α → E
hf_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn f s μ
hf_zero :
  ∀ (s : Set α),
    MeasurableSet s →
      LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) →
        Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, f x ∂μ) 0
hf : AEFinStronglyMeasurable f μ
t : Set α := hf.sigmaFiniteSet
this : SigmaFinite (μ.restrict t)
refine_1 :
  ∀ (s : Set α),
    MeasurableSet s →
      LT.lt (α := ℝ≥0∞) ((μ.restrict t : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn f s (μ.restrict t)
⊢ f =ᶠ[ae (μ.restrict t)] 0
have refine_2 : ∀ (s : Set α),
  MeasurableSet s →
    LT.lt (α := ℝ≥0∞) ((μ.restrict t : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) →
      Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, f x ∂μ.restrict t) 0
Tactic: refine ae_eq_zero_of_forall_setIntegral_eq_of_sigmaFinite ?_ ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.AEEqOfIntegral (goal)
Goal:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : CompleteSpace E
f : α → E
hf_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn f s μ
hf_zero :
  ∀ (s : Set α),
    MeasurableSet s →
      LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) →
        Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, f x ∂μ) 0
hf : AEFinStronglyMeasurable f μ
t : Set α := hf.sigmaFiniteSet
this : SigmaFinite (μ.restrict t)
s : Set α
hs : MeasurableSet s
hμs : LT.lt (α := ℝ≥0∞) ((μ.restrict t : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ IntegrableOn f s (μ.restrict t)
have refine_1 : Integrable f (μ.restrict (s ∩ t))
Tactic: rw [IntegrableOn, Measure.restrict_restrict hs]
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.AEEqOfIntegral (hypothesis)
Goal:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : CompleteSpace E
f : α → E
hf_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn f s μ
hf_zero :
  ∀ (s : Set α),
    MeasurableSet s →
      LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) →
        Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, f x ∂μ) 0
hf : AEFinStronglyMeasurable f μ
t : Set α := hf.sigmaFiniteSet
this : SigmaFinite (μ.restrict t)
s : Set α
hs : MeasurableSet s
hμs : LT.lt (α := ℝ≥0∞) ((μ.restrict t : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ Integrable f (μ.restrict (s ∩ t))
have hμs₁ : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s ∩ t) : ℝ≥0∞) (∞ : ℝ≥0∞)
Tactic: rw [Measure.restrict_apply hs] at hμs
Cleared hypotheses: hμs
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.AEEqOfIntegral (goal)
Goal:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : CompleteSpace E
f : α → E
hf_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn f s μ
hf_zero :
  ∀ (s : Set α),
    MeasurableSet s →
      LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) →
        Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, f x ∂μ) 0
hf : AEFinStronglyMeasurable f μ
t : Set α := hf.sigmaFiniteSet
this : SigmaFinite (μ.restrict t)
s : Set α
hs : MeasurableSet s
hμs : LT.lt (α := ℝ≥0∞) ((μ.restrict t : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, f x ∂μ.restrict t) 0
have refine_2 : Eq.{u_2 + 1} (α := E) (∫ (x : α) in s ∩ t, f x ∂μ) 0
Tactic: rw [Measure.restrict_restrict hs]
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.AEEqOfIntegral (hypothesis)
Goal:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : CompleteSpace E
f : α → E
hf_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn f s μ
hf_zero :
  ∀ (s : Set α),
    MeasurableSet s →
      LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) →
        Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, f x ∂μ) 0
hf : AEFinStronglyMeasurable f μ
t : Set α := hf.sigmaFiniteSet
this : SigmaFinite (μ.restrict t)
s : Set α
hs : MeasurableSet s
hμs : LT.lt (α := ℝ≥0∞) ((μ.restrict t : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ Eq.{u_2 + 1} (α := E) (∫ (x : α) in s ∩ t, f x ∂μ) 0
have hμs₁ : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s ∩ t) : ℝ≥0∞) (∞ : ℝ≥0∞)
Tactic: rw [Measure.restrict_apply hs] at hμs
Cleared hypotheses: hμs
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.AEEqOfIntegral (goal)
Goal:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : CompleteSpace E
f g : α → E
hf_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn f s μ
hg_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn g s μ
hfg_eq :
  ∀ (s : Set α),
    MeasurableSet s →
      LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) →
        Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, f x ∂μ) (∫ (x : α) in s, g x ∂μ)
hf : AEFinStronglyMeasurable f μ
hg : AEFinStronglyMeasurable g μ
⊢ f =ᶠ[ae μ] g
have rw : f - g =ᶠ[ae μ] 0
Tactic: rw [← sub_ae_eq_zero]
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.AEEqOfIntegral (goal)
Goal:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : CompleteSpace E
f g : α → E
hf_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn f s μ
hg_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn g s μ
hfg_eq :
  ∀ (s : Set α),
    MeasurableSet s →
      LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) →
        Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, f x ∂μ) (∫ (x : α) in s, g x ∂μ)
hf : AEFinStronglyMeasurable f μ
hg : AEFinStronglyMeasurable g μ
s : Set α
hs : MeasurableSet s
hμs : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, (f - g) x ∂μ) 0
have rw : HSub.hSub (α := E) (∫ (a : α) in s, f a ∂μ) (∫ (a : α) in s, g a ∂μ) = 0
Tactic: rw [integral_sub' (hf_int_finite s hs hμs) (hg_int_finite s hs hμs),
      sub_eq_zero.mpr (hfg_eq s hs hμs)]
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.AEEqOfIntegral (goal)
Goal:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : CompleteSpace E
f g : α → E
hf_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn f s μ
hg_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn g s μ
hfg_eq :
  ∀ (s : Set α),
    MeasurableSet s →
      LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) →
        Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, f x ∂μ) (∫ (x : α) in s, g x ∂μ)
hf : AEFinStronglyMeasurable f μ
hg : AEFinStronglyMeasurable g μ
s : Set α
hs : MeasurableSet s
hμs : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
rw : HSub.hSub (α := E) (∫ (a : α) in s, f a ∂μ) (∫ (a : α) in s, g a ∂μ) = 0
⊢ Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, (f - g) x ∂μ) 0
have rw₁ : Eq.{u_2 + 1} (α := E) 0 0
Tactic: rw [integral_sub' (hf_int_finite s hs hμs) (hg_int_finite s hs hμs),
      sub_eq_zero.mpr (hfg_eq s hs hμs)]
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.AEEqOfIntegral (hypothesis)
Goal:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : CompleteSpace E
f g : α → E
hf_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn f s μ
hg_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn g s μ
hfg_eq :
  ∀ (s : Set α),
    MeasurableSet s →
      LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) →
        Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, f x ∂μ) (∫ (x : α) in s, g x ∂μ)
hf : AEFinStronglyMeasurable f μ
hg : AEFinStronglyMeasurable g μ
⊢ f - g =ᶠ[ae μ] 0
have hfg : ∀ (s : Set α),
  MeasurableSet s →
    LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) →
      Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, (f - g) x ∂μ) 0
Tactic: have hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, (f - g) x ∂μ) = 0 := by
    intro s hs hμs
    rw [integral_sub' (hf_int_finite s hs hμs) (hg_int_finite s hs hμs),
      sub_eq_zero.mpr (hfg_eq s hs hμs)]
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.AEEqOfIntegral (hypothesis)
Goal:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : CompleteSpace E
f g : α → E
hf_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn f s μ
hg_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn g s μ
hfg_eq :
  ∀ (s : Set α),
    MeasurableSet s →
      LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) →
        Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, f x ∂μ) (∫ (x : α) in s, g x ∂μ)
hf : AEFinStronglyMeasurable f μ
hg : AEFinStronglyMeasurable g μ
hfg :
  ∀ (s : Set α),
    MeasurableSet s →
      LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) →
        Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, (f - g) x ∂μ) 0
⊢ f - g =ᶠ[ae μ] 0
have hfg_int : ∀ (s : Set α),
  MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn (f - g) s μ
Tactic: have hfg_int : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn (f - g) s μ := fun s hs hμs =>
    (hf_int_finite s hs hμs).sub (hg_int_finite s hs hμs)
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.AEEqOfIntegral (goal)
Goal:
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : CompleteSpace E
hm : m ≤ m0
f : α → E
hf_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn f s μ
hf_zero :
  ∀ (s : Set α),
    MeasurableSet s →
      LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) →
        Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, f x ∂μ) 0
hf : FinStronglyMeasurable f (μ.trim hm)
⊢ f =ᶠ[ae μ] 0
have obtain : ∀ (t : Set α) (ht_meas : MeasurableSet t) (htf_zero : ∀ x ∈ tᶜ, f x = 0) (htμ : SigmaFinite ((μ.trim hm).restrict t)),
  f =ᶠ[ae μ] 0
Tactic: obtain ⟨t, ht_meas, htf_zero, htμ⟩ := hf.exists_set_sigmaFinite
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.AEEqOfIntegral (goal)
Goal:
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : CompleteSpace E
hm : m ≤ m0
f : α → E
hf_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn f s μ
hf_zero :
  ∀ (s : Set α),
    MeasurableSet s →
      LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) →
        Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, f x ∂μ) 0
hf : FinStronglyMeasurable f (μ.trim hm)
t : Set α
ht_meas : MeasurableSet t
htf_zero : ∀ x ∈ tᶜ, f x = 0
htμ : SigmaFinite ((μ.trim hm).restrict t)
⊢ SigmaFinite ((μ.restrict t).trim hm)
have rwa : ∀ (htμ : SigmaFinite ((μ.restrict t).trim hm)), SigmaFinite ((μ.restrict t).trim hm)
Tactic: rwa [restrict_trim hm μ ht_meas] at htμ
Cleared hypotheses: htμ
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.AEEqOfIntegral (hypothesis)
Goal:
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : CompleteSpace E
hm : m ≤ m0
f : α → E
hf_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn f s μ
hf_zero :
  ∀ (s : Set α),
    MeasurableSet s →
      LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) →
        Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, f x ∂μ) 0
hf : FinStronglyMeasurable f (μ.trim hm)
t : Set α
ht_meas : MeasurableSet t
htf_zero : ∀ x ∈ tᶜ, f x = 0
htμ : SigmaFinite ((μ.trim hm).restrict t)
⊢ f =ᶠ[ae μ] 0
have this : SigmaFinite ((μ.restrict t).trim hm)
Tactic: haveI : SigmaFinite ((μ.restrict t).trim hm) := by rwa [restrict_trim hm μ ht_meas] at htμ
-- Lean 4 proof step from Mathlib.MeasureTheory.Function.AEEqOfIntegral (goal)
Goal:
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : CompleteSpace E
hm : m ≤ m0
f : α → E
hf_int_finite :
  ∀ (s : Set α), MeasurableSet s → LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) → IntegrableOn f s μ
hf_zero :
  ∀ (s : Set α),
    MeasurableSet s →
      LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) →
        Eq.{u_2 + 1} (α := E) (∫ (x : α) in s, f x ∂μ) 0
hf : FinStronglyMeasurable f (μ.trim hm)
t : Set α
ht_meas : MeasurableSet t
htf_zero : ∀ x ∈ tᶜ, f x = 0
htμ : SigmaFinite ((μ.trim hm).restrict t)
this : SigmaFinite ((μ.restrict t).trim hm)
⊢ f =ᶠ[ae (μ.restrict tᶜ)] 0
have rw : ∀ᵐ (x : α) ∂μ, x ∈ tᶜ → f x = 0 x
Tactic: rw [EventuallyEq, ae_restrict_iff' (MeasurableSet.compl (hm _ ht_meas))]