awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral
Goal:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
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∞) → LE.le (α := ℝ) 0 (∫ (x : α) in s, f x ∂μ)
t : Set α
ht : MeasurableSet t
hμt : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) (∞ : ℝ≥0∞)
s : Set α
hs : MeasurableSet s
h : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s ∩ t) : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s ∩ t) : ℝ≥0∞) (∞ : ℝ≥0∞)

Tactic:
exact (measure_mono Set.inter_subset_right).trans_lt (lt_top_iff_ne_top.mpr hμt)

Premises used: Iff.mpr, lt_top_iff_ne_top, MeasureTheory.measure_mono, LE.le.trans_lt, Set.inter_subset_right
-- Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral
Goal:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
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 (α := ℝ) (∫ (x : α) in s, f x ∂μ) 0
t : Set α
ht : MeasurableSet t
hμt : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) (∞ : ℝ≥0∞)
suffices : f ≤ᶠ[ae (μ.restrict t)] 0 ∧ 0 ≤ᶠ[ae (μ.restrict t)] f
⊢ f =ᶠ[ae (μ.restrict t)] 0

Tactic:
suffices h_and : f ≤ᵐ[μ.restrict t] 0 ∧ 0 ≤ᵐ[μ.restrict t] f from
    h_and.1.mp (h_and.2.mono fun x hx1 hx2 => le_antisymm hx2 hx1)
-- Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral
Goal:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
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 (α := ℝ) (∫ (x : α) in s, f x ∂μ) 0
t : Set α
ht : MeasurableSet t
hμt : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ f ≤ᶠ[ae (μ.restrict t)] 0 ∧ 0 ≤ᶠ[ae (μ.restrict t)] f

Tactic:
⟨?_,
      ae_nonneg_restrict_of_forall_setIntegral_nonneg hf_int_finite
        (fun s hs hμs => (hf_zero s hs hμs).symm.le) ht hμt⟩

Premises used: MeasureTheory.ae_nonneg_restrict_of_forall_setIntegral_nonneg, Eq.le, And.intro, Eq.symm
-- Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral
Goal:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
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 (α := ℝ) (∫ (x : α) in s, f x ∂μ) 0
t : Set α
ht : MeasurableSet t
hμt : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) (∞ : ℝ≥0∞)
ae_nonneg_restrict_of_forall_setIntegral_nonneg : f ≤ᶠ[ae (μ.restrict t)] 0
⊢ f ≤ᶠ[ae (μ.restrict t)] 0 ∧ 0 ≤ᶠ[ae (μ.restrict t)] f

Tactic:
refine
    ⟨?_,
      ae_nonneg_restrict_of_forall_setIntegral_nonneg hf_int_finite
        (fun s hs hμs => (hf_zero s hs hμs).symm.le) ht hμt⟩
-- Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral
Goal:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
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 (α := ℝ) (∫ (x : α) in s, f x ∂μ) 0
t : Set α
ht : MeasurableSet t
hμt : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) (∞ : ℝ≥0∞)
h_neg : 0 ≤ᶠ[ae (μ.restrict t)] -f
⊢ f ≤ᶠ[ae (μ.restrict t)] 0

Tactic:
h_neg.mono fun x hx => ?_

Premises used: Filter.Eventually.mono, Filter.EventuallyLE
-- Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral
Goal:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
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 (α := ℝ) (∫ (x : α) in s, f x ∂μ) 0
t : Set α
ht : MeasurableSet t
hμt : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) (∞ : ℝ≥0∞)
h_neg : 0 ≤ᶠ[ae (μ.restrict t)] -f
refine : ∀ (x : α) (hx : 0 x ≤ (-f) x), f x ≤ 0 x
⊢ f ≤ᶠ[ae (μ.restrict t)] 0

Tactic:
refine h_neg.mono fun x hx => ?_

Premises used: Filter.Eventually.mono
-- Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral
Goal:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
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 (α := ℝ) (∫ (x : α) in s, f x ∂μ) 0
t : Set α
ht : MeasurableSet t
hμt : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) (∞ : ℝ≥0∞)
h_neg : 0 ≤ᶠ[ae (μ.restrict t)] -f
x : α
hx : 0 x ≤ (-f) x
⊢ 0 x ≤ -f x

Tactic:
rw [Pi.neg_apply] at hx

Premises used: Pi.neg_apply
-- Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral
Goal:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
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 (α := ℝ) (∫ (x : α) in s, f x ∂μ) 0
t : Set α
ht : MeasurableSet t
hμt : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) (∞ : ℝ≥0∞)
h_neg : 0 ≤ᶠ[ae (μ.restrict t)] -f
x : α
hx : 0 x ≤ -f x
⊢ f x ≤ 0 x

Tactic:
simpa using hx
-- Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral
Goal:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
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 (α := ℝ) (∫ (x : α) in s, f x ∂μ) 0
t : Set α
ht : MeasurableSet t
hμt : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) (∞ : ℝ≥0∞)
suffices : 0 ≤ᶠ[ae (μ.restrict t)] -f
⊢ f ≤ᶠ[ae (μ.restrict t)] 0

Tactic:
suffices h_neg : 0 ≤ᵐ[μ.restrict t] -f by
    refine h_neg.mono fun x hx => ?_
    rw [Pi.neg_apply] at hx
    simpa using hx
-- Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral
Goal:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
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 (α := ℝ) (∫ (x : α) in s, f x ∂μ) 0
t : Set α
ht : MeasurableSet t
hμt : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ 0 ≤ᶠ[ae (μ.restrict t)] -f

Tactic:
ae_nonneg_restrict_of_forall_setIntegral_nonneg (fun s hs hμs => (hf_int_finite s hs hμs).neg)
      (fun s hs hμs => ?_) ht hμt

Premises used: MeasureTheory.ae_nonneg_restrict_of_forall_setIntegral_nonneg, MeasureTheory.IntegrableOn.neg
-- Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral
Goal:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
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 (α := ℝ) (∫ (x : α) in s, f x ∂μ) 0
t : Set α
ht : MeasurableSet t
hμt : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) (∞ : ℝ≥0∞)
ae_nonneg_restrict_of_forall_setIntegral_nonneg :
  ∀ (s : Set α) (hs : MeasurableSet s) (hμs : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)),
    LE.le (α := ℝ) 0 (∫ (x : α) in s, (-f) x ∂μ)
⊢ 0 ≤ᶠ[ae (μ.restrict t)] -f

Tactic:
refine
    ae_nonneg_restrict_of_forall_setIntegral_nonneg (fun s hs hμs => (hf_int_finite s hs hμs).neg)
      (fun s hs hμs => ?_) ht hμt

Premises used: MeasureTheory.ae_nonneg_restrict_of_forall_setIntegral_nonneg, MeasureTheory.IntegrableOn.neg
-- Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral
Goal:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
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 (α := ℝ) (∫ (x : α) in s, f x ∂μ) 0
t : Set α
ht : MeasurableSet t
hμt : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) (∞ : ℝ≥0∞)
s : Set α
hs : MeasurableSet s
hμs : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
rw : LE.le (α := ℝ) (∫ (a : α) in s, f a ∂μ) 0
⊢ LE.le (α := ℝ) 0 (∫ (x : α) in s, -f x ∂μ)

Tactic:
rw [integral_neg, neg_nonneg]

Premises used: MeasureTheory.integral_neg, neg_nonneg
-- Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral
Goal:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
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 (α := ℝ) (∫ (x : α) in s, f x ∂μ) 0
t : Set α
ht : MeasurableSet t
hμt : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) (∞ : ℝ≥0∞)
s : Set α
hs : MeasurableSet s
hμs : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ LE.le (α := ℝ) (∫ (a : α) in s, f a ∂μ) 0

Tactic:
(hf_zero s hs hμs).le

Premises used: Eq.le
-- Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral
Goal:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
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 (α := ℝ) (∫ (x : α) in s, f x ∂μ) 0
t : Set α
ht : MeasurableSet t
hμt : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) (∞ : ℝ≥0∞)
s : Set α
hs : MeasurableSet s
hμs : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ LE.le (α := ℝ) (∫ (a : α) in s, f a ∂μ) 0

Tactic:
exact (hf_zero s hs hμs).le

Premises used: Eq.le
-- Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral
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
t : Set α
ht : MeasurableSet t
hμt : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) (∞ : ℝ≥0∞)
rcases : ∀ (u : Set E) (u_sep : IsSeparable u) (hu : ∀ᵐ (x : α) ∂μ.restrict t, f x ∈ u), f =ᶠ[ae (μ.restrict t)] 0
⊢ f =ᶠ[ae (μ.restrict t)] 0

Tactic:
rcases (hf_int_finite t ht hμt.lt_top).aestronglyMeasurable.isSeparable_ae_range with
    ⟨u, u_sep, hu⟩

Premises used: MeasureTheory.Integrable.aestronglyMeasurable, Ne.lt_top, MeasureTheory.AEStronglyMeasurable.isSeparable_ae_range
-- Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral
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
t : Set α
ht : MeasurableSet t
hμt : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) (∞ : ℝ≥0∞)
u : Set E
u_sep : IsSeparable u
hu : ∀ᵐ (x : α) ∂μ.restrict t, f x ∈ u
⊢ f =ᶠ[ae (μ.restrict t)] 0

Tactic:
ae_eq_zero_of_forall_dual_of_isSeparable ℝ u_sep (fun c => ?_) hu

Premises used: Real, MeasureTheory.ae_eq_zero_of_forall_dual_of_isSeparable
-- Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral
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
t : Set α
ht : MeasurableSet t
hμt : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) (∞ : ℝ≥0∞)
u : Set E
u_sep : IsSeparable u
hu : ∀ᵐ (x : α) ∂μ.restrict t, f x ∈ u
refine : ∀ (c : StrongDual ℝ E), (ae (μ.restrict t)).EventuallyEq (β := ℝ) (fun (x : α) ↦ (c : (a : E) → ℝ) (f x)) 0
⊢ f =ᶠ[ae (μ.restrict t)] 0

Tactic:
refine ae_eq_zero_of_forall_dual_of_isSeparable ℝ u_sep (fun c => ?_) hu

Premises used: Real, MeasureTheory.ae_eq_zero_of_forall_dual_of_isSeparable
-- Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral
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
t : Set α
ht : MeasurableSet t
hμt : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) (∞ : ℝ≥0∞)
u : Set E
u_sep : IsSeparable u
hu : ∀ᵐ (x : α) ∂μ.restrict t, f x ∈ u
c : StrongDual ℝ E
⊢ (ae (μ.restrict t)).EventuallyEq (β := ℝ) (fun (x : α) ↦ (c : (a : E) → ℝ) (f x)) 0

Tactic:
ae_eq_zero_restrict_of_forall_setIntegral_eq_zero_real ?_ ?_ ht hμt

Premises used: MeasureTheory.ae_eq_zero_restrict_of_forall_setIntegral_eq_zero_real
-- Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral
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
t : Set α
ht : MeasurableSet t
hμt : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) (∞ : ℝ≥0∞)
u : Set E
u_sep : IsSeparable u
hu : ∀ᵐ (x : α) ∂μ.restrict t, f x ∈ u
c : StrongDual ℝ E
refine_1 :
  ∀ (s : Set α),
    MeasurableSet s →
      LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) →
        IntegrableOn (ε := ℝ) (fun (x : α) ↦ (c : (a : E) → ℝ) (f x)) s μ
refine_2 :
  ∀ (s : Set α),
    MeasurableSet s →
      LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) →
        Eq (α := ℝ) (∫ (x : α) in s, (c : (a : E) → ℝ) (f x) ∂μ) 0
⊢ (ae (μ.restrict t)).EventuallyEq (β := ℝ) (fun (x : α) ↦ (c : (a : E) → ℝ) (f x)) 0

Tactic:
refine ae_eq_zero_restrict_of_forall_setIntegral_eq_zero_real ?_ ?_ ht hμt

Premises used: MeasureTheory.ae_eq_zero_restrict_of_forall_setIntegral_eq_zero_real
-- Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral
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
t : Set α
ht : MeasurableSet t
hμt : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) (∞ : ℝ≥0∞)
u : Set E
u_sep : IsSeparable u
hu : ∀ᵐ (x : α) ∂μ.restrict t, f x ∈ u
c : StrongDual ℝ E
s : Set α
hs : MeasurableSet s
hμs : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ IntegrableOn (ε := ℝ) (fun (x : α) ↦ (c : (a : E) → ℝ) (f x)) s μ

Tactic:
ContinuousLinearMap.integrable_comp c (hf_int_finite s hs hμs)

Premises used: MeasureTheory.IntegrableOn, ContinuousLinearMap.integrable_comp
-- Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral
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
t : Set α
ht : MeasurableSet t
hμt : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) (∞ : ℝ≥0∞)
u : Set E
u_sep : IsSeparable u
hu : ∀ᵐ (x : α) ∂μ.restrict t, f x ∈ u
c : StrongDual ℝ E
s : Set α
hs : MeasurableSet s
hμs : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ IntegrableOn (ε := ℝ) (fun (x : α) ↦ (c : (a : E) → ℝ) (f x)) s μ

Tactic:
exact ContinuousLinearMap.integrable_comp c (hf_int_finite s hs hμs)

Premises used: ContinuousLinearMap.integrable_comp
-- Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral
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
t : Set α
ht : MeasurableSet t
hμt : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) (∞ : ℝ≥0∞)
u : Set E
u_sep : IsSeparable u
hu : ∀ᵐ (x : α) ∂μ.restrict t, f x ∈ u
c : StrongDual ℝ E
s : Set α
hs : MeasurableSet s
hμs : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
refine_2 : Eq (α := ℝ) ((c : (a : E) → ℝ) 0 : ℝ) (0 : ℝ)
⊢ Eq (α := ℝ) (∫ (x : α) in s, (c : (a : E) → ℝ) (f x) ∂μ) 0

Tactic:
rw [ContinuousLinearMap.integral_comp_comm c (hf_int_finite s hs hμs), hf_zero s hs hμs]

Premises used: ContinuousLinearMap.integral_comp_comm
-- Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral
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
t : Set α
ht : MeasurableSet t
hμt : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) (∞ : ℝ≥0∞)
u : Set E
u_sep : IsSeparable u
hu : ∀ᵐ (x : α) ∂μ.restrict t, f x ∈ u
c : StrongDual ℝ E
s : Set α
hs : MeasurableSet s
hμs : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ Eq (α := ℝ) ((c : (a : E) → ℝ) 0 : ℝ) (0 : ℝ)

Tactic:
map_zero _

Premises used: map_zero
-- Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral
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
t : Set α
ht : MeasurableSet t
hμt : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) (∞ : ℝ≥0∞)
u : Set E
u_sep : IsSeparable u
hu : ∀ᵐ (x : α) ∂μ.restrict t, f x ∈ u
c : StrongDual ℝ E
s : Set α
hs : MeasurableSet s
hμs : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ Eq (α := ℝ) ((c : (a : E) → ℝ) 0 : ℝ) (0 : ℝ)

Tactic:
exact map_zero _

Premises used: map_zero
-- Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral
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_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) (∫ (x : α) in s, g x ∂μ)
t : Set α
ht : MeasurableSet t
hμt : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) (∞ : ℝ≥0∞)
rw : f - g =ᶠ[ae (μ.restrict t)] 0
⊢ f =ᶠ[ae (μ.restrict t)] g

Tactic:
rw [← sub_ae_eq_zero]

Premises used: MeasureTheory.sub_ae_eq_zero
-- Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral
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_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) (∫ (x : α) in s, g x ∂μ)
t : Set α
ht : MeasurableSet t
hμt : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) (∞ : ℝ≥0∞)
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

Tactic:
rw [integral_sub' (hf_int_finite s hs hμs) (hg_int_finite s hs hμs)]

Premises used: MeasureTheory.integral_sub'
-- Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral
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_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) (∫ (x : α) in s, g x ∂μ)
t : Set α
ht : MeasurableSet t
hμt : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) (∞ : ℝ≥0∞)
s : Set α
hs : MeasurableSet s
hμs : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ HSub.hSub (α := E) (∫ (a : α) in s, f a ∂μ) (∫ (a : α) in s, g a ∂μ) = 0

Tactic:
sub_eq_zero.mpr (hfg_zero s hs hμs)

Premises used: Iff.mpr, sub_eq_zero