Theorem: MeasureTheory.Memℒp.memℒp_of_exponent_le
File: Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean

tactic: cases' hfq with hfq_m hfq_lt_top
state_before: α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hfq : Memℒp f q μ
hpq : p ≤ q
⊢ Memℒp f p μ
state_after: case intro
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
⊢ Memℒp f p μ

tactic: by_cases hp0 : p = 0
state_before: case intro
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
⊢ Memℒp f p μ
state_after: case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p = 0
⊢ Memℒp f p μ

case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : ¬p = 0
⊢ Memℒp f p μ

tactic: rw [← Ne] at hp0
state_before: case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : ¬p = 0
⊢ Memℒp f p μ
state_after: case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
⊢ Memℒp f p μ

tactic: refine ⟨hfq_m, ?_⟩
state_before: case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
⊢ Memℒp f p μ
state_after: case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
⊢ snorm f p μ < ⊤

tactic: by_cases hp_top : p = ∞
state_before: case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
⊢ snorm f p μ < ⊤
state_after: case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : p = ⊤
⊢ snorm f p μ < ⊤

case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
⊢ snorm f p μ < ⊤

tactic: have hp_pos : 0 < p.toReal := ENNReal.toReal_pos hp0 hp_top
state_before: case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
⊢ snorm f p μ < ⊤
state_after: case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
⊢ snorm f p μ < ⊤

tactic: by_cases hq_top : q = ∞
state_before: case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
⊢ snorm f p μ < ⊤
state_after: case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : q = ⊤
⊢ snorm f p μ < ⊤

case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : ¬q = ⊤
⊢ snorm f p μ < ⊤

tactic: have hq0 : q ≠ 0 := by
  by_contra hq_eq_zero
  have hp_eq_zero : p = 0 := le_antisymm (by rwa [hq_eq_zero] at hpq) (zero_le _)
  rw [hp_eq_zero, ENNReal.zero_toReal] at hp_pos
  exact (lt_irrefl _) hp_pos
state_before: case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : ¬q = ⊤
⊢ snorm f p μ < ⊤
state_after: case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : ¬q = ⊤
hq0 : q ≠ 0
⊢ snorm f p μ < ⊤

tactic: have hpq_real : p.toReal ≤ q.toReal := by rwa [ENNReal.toReal_le_toReal hp_top hq_top]
state_before: case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : ¬q = ⊤
hq0 : q ≠ 0
⊢ snorm f p μ < ⊤
state_after: case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : ¬q = ⊤
hq0 : q ≠ 0
hpq_real : p.toReal ≤ q.toReal
⊢ snorm f p μ < ⊤

tactic: rw [snorm_eq_snorm' hp0 hp_top]
state_before: case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : ¬q = ⊤
hq0 : q ≠ 0
hpq_real : p.toReal ≤ q.toReal
⊢ snorm f p μ < ⊤
state_after: case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : ¬q = ⊤
hq0 : q ≠ 0
hpq_real : p.toReal ≤ q.toReal
⊢ snorm' f p.toReal μ < ⊤

tactic: rw [snorm_eq_snorm' hq0 hq_top] at hfq_lt_top
state_before: case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : ¬q = ⊤
hq0 : q ≠ 0
hpq_real : p.toReal ≤ q.toReal
⊢ snorm' f p.toReal μ < ⊤
state_after: case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm' f q.toReal μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : ¬q = ⊤
hq0 : q ≠ 0
hpq_real : p.toReal ≤ q.toReal
⊢ snorm' f p.toReal μ < ⊤

tactic: exact snorm'_lt_top_of_snorm'_lt_top_of_exponent_le hfq_m hfq_lt_top (le_of_lt hp_pos) hpq_real
state_before: case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm' f q.toReal μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : ¬q = ⊤
hq0 : q ≠ 0
hpq_real : p.toReal ≤ q.toReal
⊢ snorm' f p.toReal μ < ⊤
state_after: no goals

tactic: rwa [hp0, memℒp_zero_iff_aestronglyMeasurable]
state_before: case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p = 0
⊢ Memℒp f p μ
state_after: no goals

tactic: have hq_top : q = ∞ := by rwa [hp_top, top_le_iff] at hpq
state_before: case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : p = ⊤
⊢ snorm f p μ < ⊤
state_after: case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : p = ⊤
hq_top : q = ⊤
⊢ snorm f p μ < ⊤

tactic: rw [hp_top]
state_before: case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : p = ⊤
hq_top : q = ⊤
⊢ snorm f p μ < ⊤
state_after: case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : p = ⊤
hq_top : q = ⊤
⊢ snorm f ⊤ μ < ⊤

tactic: rwa [hq_top] at hfq_lt_top
state_before: case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : p = ⊤
hq_top : q = ⊤
⊢ snorm f ⊤ μ < ⊤
state_after: no goals

tactic: rwa [hp_top, top_le_iff] at hpq
state_before: α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : p = ⊤
⊢ q = ⊤
state_after: no goals

tactic: rw [snorm_eq_snorm' hp0 hp_top]
state_before: case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : q = ⊤
⊢ snorm f p μ < ⊤
state_after: case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : q = ⊤
⊢ snorm' f p.toReal μ < ⊤

tactic: rw [hq_top, snorm_exponent_top] at hfq_lt_top
state_before: case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : q = ⊤
⊢ snorm' f p.toReal μ < ⊤
state_after: case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snormEssSup f μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : q = ⊤
⊢ snorm' f p.toReal μ < ⊤

tactic: refine lt_of_le_of_lt (snorm'_le_snormEssSup_mul_rpow_measure_univ hp_pos) ?_
state_before: case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snormEssSup f μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : q = ⊤
⊢ snorm' f p.toReal μ < ⊤
state_after: case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snormEssSup f μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : q = ⊤
⊢ snormEssSup f μ * μ Set.univ ^ (1 / p.toReal) < ⊤

tactic: refine ENNReal.mul_lt_top hfq_lt_top.ne ?_
state_before: case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snormEssSup f μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : q = ⊤
⊢ snormEssSup f μ * μ Set.univ ^ (1 / p.toReal) < ⊤
state_after: case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snormEssSup f μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : q = ⊤
⊢ μ Set.univ ^ (1 / p.toReal) ≠ ⊤

tactic: exact (ENNReal.rpow_lt_top_of_nonneg (by simp [hp_pos.le]) (measure_ne_top μ Set.univ)).ne
state_before: case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snormEssSup f μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : q = ⊤
⊢ μ Set.univ ^ (1 / p.toReal) ≠ ⊤
state_after: no goals

tactic: simp [hp_pos.le]
state_before: α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snormEssSup f μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : q = ⊤
⊢ 0 ≤ 1 / p.toReal
state_after: no goals

tactic: by_contra hq_eq_zero
state_before: α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : ¬q = ⊤
⊢ q ≠ 0
state_after: α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : ¬q = ⊤
hq_eq_zero : q = 0
⊢ False

tactic: have hp_eq_zero : p = 0 := le_antisymm (by rwa [hq_eq_zero] at hpq) (zero_le _)
state_before: α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : ¬q = ⊤
hq_eq_zero : q = 0
⊢ False
state_after: α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : ¬q = ⊤
hq_eq_zero : q = 0
hp_eq_zero : p = 0
⊢ False

tactic: rw [hp_eq_zero, ENNReal.zero_toReal] at hp_pos
state_before: α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : ¬q = ⊤
hq_eq_zero : q = 0
hp_eq_zero : p = 0
⊢ False
state_after: α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < 0
hq_top : ¬q = ⊤
hq_eq_zero : q = 0
hp_eq_zero : p = 0
⊢ False

tactic: exact (lt_irrefl _) hp_pos
state_before: α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < 0
hq_top : ¬q = ⊤
hq_eq_zero : q = 0
hp_eq_zero : p = 0
⊢ False
state_after: no goals

tactic: rwa [hq_eq_zero] at hpq
state_before: α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : ¬q = ⊤
hq_eq_zero : q = 0
⊢ p ≤ 0
state_after: no goals

tactic: rwa [ENNReal.toReal_le_toReal hp_top hq_top]
state_before: α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
f✝ : α → E
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : snorm f q μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : ¬q = ⊤
hq0 : q ≠ 0
⊢ p.toReal ≤ q.toReal
state_after: no goals