l3lab/ntp-mathlib huggingface dataset:
/-- Don't use this lemma. Use `ae_nonneg_of_forall_set_integral_nonneg`. -/
theorem ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable (hfm : StronglyMeasurable f)
    (hf : Integrable f μ) (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) :
    0 ≤ᵐ[μ] f := by
  simp_rw [EventuallyLE, Pi.zero_apply]
  rw [ae_const_le_iff_forall_lt_measure_zero]
  intro b hb_neg
  let s := {x | f x ≤ b}
  have hs : MeasurableSet s := hfm.measurableSet_le stronglyMeasurable_const
  have mus : μ s < ∞ := by
    let c : ℝ≥0 := ⟨|b|, abs_nonneg _⟩
    have c_pos : (c : ℝ≥0∞) ≠ 0 := by simpa [← NNReal.coe_eq_zero] using hb_neg.ne
    calc
      μ s ≤ μ {x | (c : ℝ≥0∞) ≤ ‖f x‖₊} := by
        apply measure_mono
        intro x hx
        simp only [Set.mem_setOf_eq] at hx
        simpa only [nnnorm, abs_of_neg hb_neg, abs_of_neg (hx.trans_lt hb_neg), Real.norm_eq_abs,
          Subtype.mk_le_mk, neg_le_neg_iff, Set.mem_setOf_eq, ENNReal.coe_le_coe, NNReal] using hx
      _ ≤ (∫⁻ x, ‖f x‖₊ ∂μ) / c :=
        (meas_ge_le_lintegral_div hfm.aemeasurable.ennnorm c_pos ENNReal.coe_ne_top)
      _ < ∞ := ENNReal.div_lt_top (ne_of_lt hf.2) c_pos
  have h_int_gt : (∫ x in s, f x ∂μ) ≤ b * (μ s).toReal := by
    

Proof state:
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ≥0∞
f : α → ℝ
hfm : StronglyMeasurable f
hf : Integrable f
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
b : ℝ
hb_neg : b < 0
s : Set α := {x | f x ≤ b}
hs : MeasurableSet s
mus : ↑↑μ s < ⊤
⊢ ∫ (x : α) in s, f x ∂μ ≤ b * ENNReal.toReal (↑↑μ s)

Next tactic:
have h_const_le : (∫ x in s, f x ∂μ) ≤ ∫ _ in s, b ∂μ := by
      refine'
        set_integral_mono_ae_restrict hf.integrableOn (integrableOn_const.mpr (Or.inr mus)) _
      rw [EventuallyLE, ae_restrict_iff hs]
      exact eventually_of_forall fun x hxs => hxs
/-- Don't use this lemma. Use `ae_nonneg_of_forall_set_integral_nonneg`. -/
theorem ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable (hfm : StronglyMeasurable f)
    (hf : Integrable f μ) (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) :
    0 ≤ᵐ[μ] f := by
  simp_rw [EventuallyLE, Pi.zero_apply]
  rw [ae_const_le_iff_forall_lt_measure_zero]
  intro b hb_neg
  let s := {x | f x ≤ b}
  have hs : MeasurableSet s := hfm.measurableSet_le stronglyMeasurable_const
  have mus : μ s < ∞ := by
    let c : ℝ≥0 := ⟨|b|, abs_nonneg _⟩
    have c_pos : (c : ℝ≥0∞) ≠ 0 := by simpa [← NNReal.coe_eq_zero] using hb_neg.ne
    calc
      μ s ≤ μ {x | (c : ℝ≥0∞) ≤ ‖f x‖₊} := by
        apply measure_mono
        intro x hx
        simp only [Set.mem_setOf_eq] at hx
        simpa only [nnnorm, abs_of_neg hb_neg, abs_of_neg (hx.trans_lt hb_neg), Real.norm_eq_abs,
          Subtype.mk_le_mk, neg_le_neg_iff, Set.mem_setOf_eq, ENNReal.coe_le_coe, NNReal] using hx
      _ ≤ (∫⁻ x, ‖f x‖₊ ∂μ) / c :=
        (meas_ge_le_lintegral_div hfm.aemeasurable.ennnorm c_pos ENNReal.coe_ne_top)
      _ < ∞ := ENNReal.div_lt_top (ne_of_lt hf.2) c_pos
  have h_int_gt : (∫ x in s, f x ∂μ) ≤ b * (μ s).toReal := by
    have h_const_le : (∫ x in s, f x ∂μ) ≤ ∫ _ in s, b ∂μ := by
      

Proof state:
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ≥0∞
f : α → ℝ
hfm : StronglyMeasurable f
hf : Integrable f
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
b : ℝ
hb_neg : b < 0
s : Set α := {x | f x ≤ b}
hs : MeasurableSet s
mus : ↑↑μ s < ⊤
⊢ ∫ (x : α) in s, f x ∂μ ≤ ∫ (x : α) in s, b ∂μ

Next tactic:
refine'
        set_integral_mono_ae_restrict hf.integrableOn (integrableOn_const.mpr (Or.inr mus)) _
/-- Don't use this lemma. Use `ae_nonneg_of_forall_set_integral_nonneg`. -/
theorem ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable (hfm : StronglyMeasurable f)
    (hf : Integrable f μ) (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) :
    0 ≤ᵐ[μ] f := by
  simp_rw [EventuallyLE, Pi.zero_apply]
  rw [ae_const_le_iff_forall_lt_measure_zero]
  intro b hb_neg
  let s := {x | f x ≤ b}
  have hs : MeasurableSet s := hfm.measurableSet_le stronglyMeasurable_const
  have mus : μ s < ∞ := by
    let c : ℝ≥0 := ⟨|b|, abs_nonneg _⟩
    have c_pos : (c : ℝ≥0∞) ≠ 0 := by simpa [← NNReal.coe_eq_zero] using hb_neg.ne
    calc
      μ s ≤ μ {x | (c : ℝ≥0∞) ≤ ‖f x‖₊} := by
        apply measure_mono
        intro x hx
        simp only [Set.mem_setOf_eq] at hx
        simpa only [nnnorm, abs_of_neg hb_neg, abs_of_neg (hx.trans_lt hb_neg), Real.norm_eq_abs,
          Subtype.mk_le_mk, neg_le_neg_iff, Set.mem_setOf_eq, ENNReal.coe_le_coe, NNReal] using hx
      _ ≤ (∫⁻ x, ‖f x‖₊ ∂μ) / c :=
        (meas_ge_le_lintegral_div hfm.aemeasurable.ennnorm c_pos ENNReal.coe_ne_top)
      _ < ∞ := ENNReal.div_lt_top (ne_of_lt hf.2) c_pos
  have h_int_gt : (∫ x in s, f x ∂μ) ≤ b * (μ s).toReal := by
    have h_const_le : (∫ x in s, f x ∂μ) ≤ ∫ _ in s, b ∂μ := by
      refine'
        set_integral_mono_ae_restrict hf.integrableOn (integrableOn_const.mpr (Or.inr mus)) _
      

Proof state:
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ≥0∞
f : α → ℝ
hfm : StronglyMeasurable f
hf : Integrable f
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
b : ℝ
hb_neg : b < 0
s : Set α := {x | f x ≤ b}
hs : MeasurableSet s
mus : ↑↑μ s < ⊤
⊢ (fun x => f x) ≤ᵐ[Measure.restrict μ s] fun x => b

Next tactic:
rw [EventuallyLE, ae_restrict_iff hs]
/-- Don't use this lemma. Use `ae_nonneg_of_forall_set_integral_nonneg`. -/
theorem ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable (hfm : StronglyMeasurable f)
    (hf : Integrable f μ) (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) :
    0 ≤ᵐ[μ] f := by
  simp_rw [EventuallyLE, Pi.zero_apply]
  rw [ae_const_le_iff_forall_lt_measure_zero]
  intro b hb_neg
  let s := {x | f x ≤ b}
  have hs : MeasurableSet s := hfm.measurableSet_le stronglyMeasurable_const
  have mus : μ s < ∞ := by
    let c : ℝ≥0 := ⟨|b|, abs_nonneg _⟩
    have c_pos : (c : ℝ≥0∞) ≠ 0 := by simpa [← NNReal.coe_eq_zero] using hb_neg.ne
    calc
      μ s ≤ μ {x | (c : ℝ≥0∞) ≤ ‖f x‖₊} := by
        apply measure_mono
        intro x hx
        simp only [Set.mem_setOf_eq] at hx
        simpa only [nnnorm, abs_of_neg hb_neg, abs_of_neg (hx.trans_lt hb_neg), Real.norm_eq_abs,
          Subtype.mk_le_mk, neg_le_neg_iff, Set.mem_setOf_eq, ENNReal.coe_le_coe, NNReal] using hx
      _ ≤ (∫⁻ x, ‖f x‖₊ ∂μ) / c :=
        (meas_ge_le_lintegral_div hfm.aemeasurable.ennnorm c_pos ENNReal.coe_ne_top)
      _ < ∞ := ENNReal.div_lt_top (ne_of_lt hf.2) c_pos
  have h_int_gt : (∫ x in s, f x ∂μ) ≤ b * (μ s).toReal := by
    have h_const_le : (∫ x in s, f x ∂μ) ≤ ∫ _ in s, b ∂μ := by
      refine'
        set_integral_mono_ae_restrict hf.integrableOn (integrableOn_const.mpr (Or.inr mus)) _
      rw [EventuallyLE, ae_restrict_iff hs]
      

Proof state:
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ≥0∞
f : α → ℝ
hfm : StronglyMeasurable f
hf : Integrable f
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
b : ℝ
hb_neg : b < 0
s : Set α := {x | f x ≤ b}
hs : MeasurableSet s
mus : ↑↑μ s < ⊤
⊢ ∀ᵐ (x : α) ∂μ, x ∈ s → f x ≤ b

Next tactic:
exact eventually_of_forall fun x hxs => hxs
/-- Don't use this lemma. Use `ae_nonneg_of_forall_set_integral_nonneg`. -/
theorem ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable (hfm : StronglyMeasurable f)
    (hf : Integrable f μ) (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) :
    0 ≤ᵐ[μ] f := by
  simp_rw [EventuallyLE, Pi.zero_apply]
  rw [ae_const_le_iff_forall_lt_measure_zero]
  intro b hb_neg
  let s := {x | f x ≤ b}
  have hs : MeasurableSet s := hfm.measurableSet_le stronglyMeasurable_const
  have mus : μ s < ∞ := by
    let c : ℝ≥0 := ⟨|b|, abs_nonneg _⟩
    have c_pos : (c : ℝ≥0∞) ≠ 0 := by simpa [← NNReal.coe_eq_zero] using hb_neg.ne
    calc
      μ s ≤ μ {x | (c : ℝ≥0∞) ≤ ‖f x‖₊} := by
        apply measure_mono
        intro x hx
        simp only [Set.mem_setOf_eq] at hx
        simpa only [nnnorm, abs_of_neg hb_neg, abs_of_neg (hx.trans_lt hb_neg), Real.norm_eq_abs,
          Subtype.mk_le_mk, neg_le_neg_iff, Set.mem_setOf_eq, ENNReal.coe_le_coe, NNReal] using hx
      _ ≤ (∫⁻ x, ‖f x‖₊ ∂μ) / c :=
        (meas_ge_le_lintegral_div hfm.aemeasurable.ennnorm c_pos ENNReal.coe_ne_top)
      _ < ∞ := ENNReal.div_lt_top (ne_of_lt hf.2) c_pos
  have h_int_gt : (∫ x in s, f x ∂μ) ≤ b * (μ s).toReal := by
    have h_const_le : (∫ x in s, f x ∂μ) ≤ ∫ _ in s, b ∂μ := by
      refine'
        set_integral_mono_ae_restrict hf.integrableOn (integrableOn_const.mpr (Or.inr mus)) _
      rw [EventuallyLE, ae_restrict_iff hs]
      exact eventually_of_forall fun x hxs => hxs
    

Proof state:
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ≥0∞
f : α → ℝ
hfm : StronglyMeasurable f
hf : Integrable f
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
b : ℝ
hb_neg : b < 0
s : Set α := {x | f x ≤ b}
hs : MeasurableSet s
mus : ↑↑μ s < ⊤
h_const_le : ∫ (x : α) in s, f x ∂μ ≤ ∫ (x : α) in s, b ∂μ
⊢ ∫ (x : α) in s, f x ∂μ ≤ b * ENNReal.toReal (↑↑μ s)

Next tactic:
rwa [set_integral_const, smul_eq_mul, mul_comm] at h_const_le
/-- Don't use this lemma. Use `ae_nonneg_of_forall_set_integral_nonneg`. -/
theorem ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable (hfm : StronglyMeasurable f)
    (hf : Integrable f μ) (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) :
    0 ≤ᵐ[μ] f := by
  simp_rw [EventuallyLE, Pi.zero_apply]
  rw [ae_const_le_iff_forall_lt_measure_zero]
  intro b hb_neg
  let s := {x | f x ≤ b}
  have hs : MeasurableSet s := hfm.measurableSet_le stronglyMeasurable_const
  have mus : μ s < ∞ := by
    let c : ℝ≥0 := ⟨|b|, abs_nonneg _⟩
    have c_pos : (c : ℝ≥0∞) ≠ 0 := by simpa [← NNReal.coe_eq_zero] using hb_neg.ne
    calc
      μ s ≤ μ {x | (c : ℝ≥0∞) ≤ ‖f x‖₊} := by
        apply measure_mono
        intro x hx
        simp only [Set.mem_setOf_eq] at hx
        simpa only [nnnorm, abs_of_neg hb_neg, abs_of_neg (hx.trans_lt hb_neg), Real.norm_eq_abs,
          Subtype.mk_le_mk, neg_le_neg_iff, Set.mem_setOf_eq, ENNReal.coe_le_coe, NNReal] using hx
      _ ≤ (∫⁻ x, ‖f x‖₊ ∂μ) / c :=
        (meas_ge_le_lintegral_div hfm.aemeasurable.ennnorm c_pos ENNReal.coe_ne_top)
      _ < ∞ := ENNReal.div_lt_top (ne_of_lt hf.2) c_pos
  have h_int_gt : (∫ x in s, f x ∂μ) ≤ b * (μ s).toReal := by
    have h_const_le : (∫ x in s, f x ∂μ) ≤ ∫ _ in s, b ∂μ := by
      refine'
        set_integral_mono_ae_restrict hf.integrableOn (integrableOn_const.mpr (Or.inr mus)) _
      rw [EventuallyLE, ae_restrict_iff hs]
      exact eventually_of_forall fun x hxs => hxs
    rwa [set_integral_const, smul_eq_mul, mul_comm] at h_const_le
  

Proof state:
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ≥0∞
f : α → ℝ
hfm : StronglyMeasurable f
hf : Integrable f
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
b : ℝ
hb_neg : b < 0
s : Set α := {x | f x ≤ b}
hs : MeasurableSet s
mus : ↑↑μ s < ⊤
h_int_gt : ∫ (x : α) in s, f x ∂μ ≤ b * ENNReal.toReal (↑↑μ s)
⊢ ↑↑μ {x | f x ≤ b} = 0

Next tactic:
by_contra h
/-- Don't use this lemma. Use `ae_nonneg_of_forall_set_integral_nonneg`. -/
theorem ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable (hfm : StronglyMeasurable f)
    (hf : Integrable f μ) (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) :
    0 ≤ᵐ[μ] f := by
  simp_rw [EventuallyLE, Pi.zero_apply]
  rw [ae_const_le_iff_forall_lt_measure_zero]
  intro b hb_neg
  let s := {x | f x ≤ b}
  have hs : MeasurableSet s := hfm.measurableSet_le stronglyMeasurable_const
  have mus : μ s < ∞ := by
    let c : ℝ≥0 := ⟨|b|, abs_nonneg _⟩
    have c_pos : (c : ℝ≥0∞) ≠ 0 := by simpa [← NNReal.coe_eq_zero] using hb_neg.ne
    calc
      μ s ≤ μ {x | (c : ℝ≥0∞) ≤ ‖f x‖₊} := by
        apply measure_mono
        intro x hx
        simp only [Set.mem_setOf_eq] at hx
        simpa only [nnnorm, abs_of_neg hb_neg, abs_of_neg (hx.trans_lt hb_neg), Real.norm_eq_abs,
          Subtype.mk_le_mk, neg_le_neg_iff, Set.mem_setOf_eq, ENNReal.coe_le_coe, NNReal] using hx
      _ ≤ (∫⁻ x, ‖f x‖₊ ∂μ) / c :=
        (meas_ge_le_lintegral_div hfm.aemeasurable.ennnorm c_pos ENNReal.coe_ne_top)
      _ < ∞ := ENNReal.div_lt_top (ne_of_lt hf.2) c_pos
  have h_int_gt : (∫ x in s, f x ∂μ) ≤ b * (μ s).toReal := by
    have h_const_le : (∫ x in s, f x ∂μ) ≤ ∫ _ in s, b ∂μ := by
      refine'
        set_integral_mono_ae_restrict hf.integrableOn (integrableOn_const.mpr (Or.inr mus)) _
      rw [EventuallyLE, ae_restrict_iff hs]
      exact eventually_of_forall fun x hxs => hxs
    rwa [set_integral_const, smul_eq_mul, mul_comm] at h_const_le
  by_contra h
  

Proof state:
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ≥0∞
f : α → ℝ
hfm : StronglyMeasurable f
hf : Integrable f
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
b : ℝ
hb_neg : b < 0
s : Set α := {x | f x ≤ b}
hs : MeasurableSet s
mus : ↑↑μ s < ⊤
h_int_gt : ∫ (x : α) in s, f x ∂μ ≤ b * ENNReal.toReal (↑↑μ s)
h : ¬↑↑μ {x | f x ≤ b} = 0
⊢ False

Next tactic:
refine' (lt_self_iff_false (∫ x in s, f x ∂μ)).mp (h_int_gt.trans_lt _)
/-- Don't use this lemma. Use `ae_nonneg_of_forall_set_integral_nonneg`. -/
theorem ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable (hfm : StronglyMeasurable f)
    (hf : Integrable f μ) (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) :
    0 ≤ᵐ[μ] f := by
  simp_rw [EventuallyLE, Pi.zero_apply]
  rw [ae_const_le_iff_forall_lt_measure_zero]
  intro b hb_neg
  let s := {x | f x ≤ b}
  have hs : MeasurableSet s := hfm.measurableSet_le stronglyMeasurable_const
  have mus : μ s < ∞ := by
    let c : ℝ≥0 := ⟨|b|, abs_nonneg _⟩
    have c_pos : (c : ℝ≥0∞) ≠ 0 := by simpa [← NNReal.coe_eq_zero] using hb_neg.ne
    calc
      μ s ≤ μ {x | (c : ℝ≥0∞) ≤ ‖f x‖₊} := by
        apply measure_mono
        intro x hx
        simp only [Set.mem_setOf_eq] at hx
        simpa only [nnnorm, abs_of_neg hb_neg, abs_of_neg (hx.trans_lt hb_neg), Real.norm_eq_abs,
          Subtype.mk_le_mk, neg_le_neg_iff, Set.mem_setOf_eq, ENNReal.coe_le_coe, NNReal] using hx
      _ ≤ (∫⁻ x, ‖f x‖₊ ∂μ) / c :=
        (meas_ge_le_lintegral_div hfm.aemeasurable.ennnorm c_pos ENNReal.coe_ne_top)
      _ < ∞ := ENNReal.div_lt_top (ne_of_lt hf.2) c_pos
  have h_int_gt : (∫ x in s, f x ∂μ) ≤ b * (μ s).toReal := by
    have h_const_le : (∫ x in s, f x ∂μ) ≤ ∫ _ in s, b ∂μ := by
      refine'
        set_integral_mono_ae_restrict hf.integrableOn (integrableOn_const.mpr (Or.inr mus)) _
      rw [EventuallyLE, ae_restrict_iff hs]
      exact eventually_of_forall fun x hxs => hxs
    rwa [set_integral_const, smul_eq_mul, mul_comm] at h_const_le
  by_contra h
  refine' (lt_self_iff_false (∫ x in s, f x ∂μ)).mp (h_int_gt.trans_lt _)
  

Proof state:
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ≥0∞
f : α → ℝ
hfm : StronglyMeasurable f
hf : Integrable f
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
b : ℝ
hb_neg : b < 0
s : Set α := {x | f x ≤ b}
hs : MeasurableSet s
mus : ↑↑μ s < ⊤
h_int_gt : ∫ (x : α) in s, f x ∂μ ≤ b * ENNReal.toReal (↑↑μ s)
h : ¬↑↑μ {x | f x ≤ b} = 0
⊢ b * ENNReal.toReal (↑↑μ s) < ∫ (x : α) in s, f x ∂μ

Next tactic:
refine' (mul_neg_iff.mpr (Or.inr ⟨hb_neg, _⟩)).trans_le _
/-- Don't use this lemma. Use `ae_nonneg_of_forall_set_integral_nonneg`. -/
theorem ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable (hfm : StronglyMeasurable f)
    (hf : Integrable f μ) (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) :
    0 ≤ᵐ[μ] f := by
  simp_rw [EventuallyLE, Pi.zero_apply]
  rw [ae_const_le_iff_forall_lt_measure_zero]
  intro b hb_neg
  let s := {x | f x ≤ b}
  have hs : MeasurableSet s := hfm.measurableSet_le stronglyMeasurable_const
  have mus : μ s < ∞ := by
    let c : ℝ≥0 := ⟨|b|, abs_nonneg _⟩
    have c_pos : (c : ℝ≥0∞) ≠ 0 := by simpa [← NNReal.coe_eq_zero] using hb_neg.ne
    calc
      μ s ≤ μ {x | (c : ℝ≥0∞) ≤ ‖f x‖₊} := by
        apply measure_mono
        intro x hx
        simp only [Set.mem_setOf_eq] at hx
        simpa only [nnnorm, abs_of_neg hb_neg, abs_of_neg (hx.trans_lt hb_neg), Real.norm_eq_abs,
          Subtype.mk_le_mk, neg_le_neg_iff, Set.mem_setOf_eq, ENNReal.coe_le_coe, NNReal] using hx
      _ ≤ (∫⁻ x, ‖f x‖₊ ∂μ) / c :=
        (meas_ge_le_lintegral_div hfm.aemeasurable.ennnorm c_pos ENNReal.coe_ne_top)
      _ < ∞ := ENNReal.div_lt_top (ne_of_lt hf.2) c_pos
  have h_int_gt : (∫ x in s, f x ∂μ) ≤ b * (μ s).toReal := by
    have h_const_le : (∫ x in s, f x ∂μ) ≤ ∫ _ in s, b ∂μ := by
      refine'
        set_integral_mono_ae_restrict hf.integrableOn (integrableOn_const.mpr (Or.inr mus)) _
      rw [EventuallyLE, ae_restrict_iff hs]
      exact eventually_of_forall fun x hxs => hxs
    rwa [set_integral_const, smul_eq_mul, mul_comm] at h_const_le
  by_contra h
  refine' (lt_self_iff_false (∫ x in s, f x ∂μ)).mp (h_int_gt.trans_lt _)
  refine' (mul_neg_iff.mpr (Or.inr ⟨hb_neg, _⟩)).trans_le _
  

Proof state:
case refine'_1
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ≥0∞
f : α → ℝ
hfm : StronglyMeasurable f
hf : Integrable f
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
b : ℝ
hb_neg : b < 0
s : Set α := {x | f x ≤ b}
hs : MeasurableSet s
mus : ↑↑μ s < ⊤
h_int_gt : ∫ (x : α) in s, f x ∂μ ≤ b * ENNReal.toReal (↑↑μ s)
h : ¬↑↑μ {x | f x ≤ b} = 0
⊢ 0 < ENNReal.toReal (↑↑μ s)
case refine'_2
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ≥0∞
f : α → ℝ
hfm : StronglyMeasurable f
hf : Integrable f
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
b : ℝ
hb_neg : b < 0
s : Set α := {x | f x ≤ b}
hs : MeasurableSet s
mus : ↑↑μ s < ⊤
h_int_gt : ∫ (x : α) in s, f x ∂μ ≤ b * ENNReal.toReal (↑↑μ s)
h : ¬↑↑μ {x | f x ≤ b} = 0
⊢ 0 ≤ ∫ (x : α) in s, f x ∂μ

Next tactic:
swap
/-- Don't use this lemma. Use `ae_nonneg_of_forall_set_integral_nonneg`. -/
theorem ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable (hfm : StronglyMeasurable f)
    (hf : Integrable f μ) (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) :
    0 ≤ᵐ[μ] f := by
  simp_rw [EventuallyLE, Pi.zero_apply]
  rw [ae_const_le_iff_forall_lt_measure_zero]
  intro b hb_neg
  let s := {x | f x ≤ b}
  have hs : MeasurableSet s := hfm.measurableSet_le stronglyMeasurable_const
  have mus : μ s < ∞ := by
    let c : ℝ≥0 := ⟨|b|, abs_nonneg _⟩
    have c_pos : (c : ℝ≥0∞) ≠ 0 := by simpa [← NNReal.coe_eq_zero] using hb_neg.ne
    calc
      μ s ≤ μ {x | (c : ℝ≥0∞) ≤ ‖f x‖₊} := by
        apply measure_mono
        intro x hx
        simp only [Set.mem_setOf_eq] at hx
        simpa only [nnnorm, abs_of_neg hb_neg, abs_of_neg (hx.trans_lt hb_neg), Real.norm_eq_abs,
          Subtype.mk_le_mk, neg_le_neg_iff, Set.mem_setOf_eq, ENNReal.coe_le_coe, NNReal] using hx
      _ ≤ (∫⁻ x, ‖f x‖₊ ∂μ) / c :=
        (meas_ge_le_lintegral_div hfm.aemeasurable.ennnorm c_pos ENNReal.coe_ne_top)
      _ < ∞ := ENNReal.div_lt_top (ne_of_lt hf.2) c_pos
  have h_int_gt : (∫ x in s, f x ∂μ) ≤ b * (μ s).toReal := by
    have h_const_le : (∫ x in s, f x ∂μ) ≤ ∫ _ in s, b ∂μ := by
      refine'
        set_integral_mono_ae_restrict hf.integrableOn (integrableOn_const.mpr (Or.inr mus)) _
      rw [EventuallyLE, ae_restrict_iff hs]
      exact eventually_of_forall fun x hxs => hxs
    rwa [set_integral_const, smul_eq_mul, mul_comm] at h_const_le
  by_contra h
  refine' (lt_self_iff_false (∫ x in s, f x ∂μ)).mp (h_int_gt.trans_lt _)
  refine' (mul_neg_iff.mpr (Or.inr ⟨hb_neg, _⟩)).trans_le _
  swap
  · 

Proof state:
case refine'_2
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ≥0∞
f : α → ℝ
hfm : StronglyMeasurable f
hf : Integrable f
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
b : ℝ
hb_neg : b < 0
s : Set α := {x | f x ≤ b}
hs : MeasurableSet s
mus : ↑↑μ s < ⊤
h_int_gt : ∫ (x : α) in s, f x ∂μ ≤ b * ENNReal.toReal (↑↑μ s)
h : ¬↑↑μ {x | f x ≤ b} = 0
⊢ 0 ≤ ∫ (x : α) in s, f x ∂μ

Next tactic:
exact hf_zero s hs mus
/-- Don't use this lemma. Use `ae_nonneg_of_forall_set_integral_nonneg`. -/
theorem ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable (hfm : StronglyMeasurable f)
    (hf : Integrable f μ) (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) :
    0 ≤ᵐ[μ] f := by
  simp_rw [EventuallyLE, Pi.zero_apply]
  rw [ae_const_le_iff_forall_lt_measure_zero]
  intro b hb_neg
  let s := {x | f x ≤ b}
  have hs : MeasurableSet s := hfm.measurableSet_le stronglyMeasurable_const
  have mus : μ s < ∞ := by
    let c : ℝ≥0 := ⟨|b|, abs_nonneg _⟩
    have c_pos : (c : ℝ≥0∞) ≠ 0 := by simpa [← NNReal.coe_eq_zero] using hb_neg.ne
    calc
      μ s ≤ μ {x | (c : ℝ≥0∞) ≤ ‖f x‖₊} := by
        apply measure_mono
        intro x hx
        simp only [Set.mem_setOf_eq] at hx
        simpa only [nnnorm, abs_of_neg hb_neg, abs_of_neg (hx.trans_lt hb_neg), Real.norm_eq_abs,
          Subtype.mk_le_mk, neg_le_neg_iff, Set.mem_setOf_eq, ENNReal.coe_le_coe, NNReal] using hx
      _ ≤ (∫⁻ x, ‖f x‖₊ ∂μ) / c :=
        (meas_ge_le_lintegral_div hfm.aemeasurable.ennnorm c_pos ENNReal.coe_ne_top)
      _ < ∞ := ENNReal.div_lt_top (ne_of_lt hf.2) c_pos
  have h_int_gt : (∫ x in s, f x ∂μ) ≤ b * (μ s).toReal := by
    have h_const_le : (∫ x in s, f x ∂μ) ≤ ∫ _ in s, b ∂μ := by
      refine'
        set_integral_mono_ae_restrict hf.integrableOn (integrableOn_const.mpr (Or.inr mus)) _
      rw [EventuallyLE, ae_restrict_iff hs]
      exact eventually_of_forall fun x hxs => hxs
    rwa [set_integral_const, smul_eq_mul, mul_comm] at h_const_le
  by_contra h
  refine' (lt_self_iff_false (∫ x in s, f x ∂μ)).mp (h_int_gt.trans_lt _)
  refine' (mul_neg_iff.mpr (Or.inr ⟨hb_neg, _⟩)).trans_le _
  swap
  · exact hf_zero s hs mus
  

Proof state:
case refine'_1
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s✝ t : Set α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ≥0∞
f : α → ℝ
hfm : StronglyMeasurable f
hf : Integrable f
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
b : ℝ
hb_neg : b < 0
s : Set α := {x | f x ≤ b}
hs : MeasurableSet s
mus : ↑↑μ s < ⊤
h_int_gt : ∫ (x : α) in s, f x ∂μ ≤ b * ENNReal.toReal (↑↑μ s)
h : ¬↑↑μ {x | f x ≤ b} = 0
⊢ 0 < ENNReal.toReal (↑↑μ s)

Next tactic:
refine' ENNReal.toReal_nonneg.lt_of_ne fun h_eq => h _