l3lab/ntp-mathlib huggingface dataset:
theorem setToFun_finset_sum' (hT : DominatedFinMeasAdditive μ T C) {ι} (s : Finset ι)
    {f : ι → α → E} (hf : ∀ i ∈ s, Integrable (f i) μ) :
    setToFun μ T hT (∑ i in s, f i) = ∑ i in s, setToFun μ T hT (f i) := by
  revert hf
  refine' Finset.induction_on s _ _
  · intro _
    

Proof state:
case refine'_1
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
s : Finset ι
f : ι → α → E
hf✝ : ∀ i ∈ ∅, Integrable (f i)
⊢ setToFun μ T hT (∑ i in ∅, f i) = ∑ i in ∅, setToFun μ T hT (f i)

Next tactic:
simp only [setToFun_zero, Finset.sum_empty]
theorem setToFun_finset_sum' (hT : DominatedFinMeasAdditive μ T C) {ι} (s : Finset ι)
    {f : ι → α → E} (hf : ∀ i ∈ s, Integrable (f i) μ) :
    setToFun μ T hT (∑ i in s, f i) = ∑ i in s, setToFun μ T hT (f i) := by
  revert hf
  refine' Finset.induction_on s _ _
  · intro _
    simp only [setToFun_zero, Finset.sum_empty]
  · 

Proof state:
case refine'_2
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
s : Finset ι
f : ι → α → E
⊢ ∀ ⦃a : ι⦄ {s : Finset ι},
    a ∉ s →
      ((∀ i ∈ s, Integrable (f i)) → setToFun μ T hT (∑ i in s, f i) = ∑ i in s, setToFun μ T hT (f i)) →
        (∀ i ∈ insert a s, Integrable (f i)) →
          setToFun μ T hT (∑ i in insert a s, f i) = ∑ i in insert a s, setToFun μ T hT (f i)

Next tactic:
intro i s his ih hf
theorem setToFun_finset_sum' (hT : DominatedFinMeasAdditive μ T C) {ι} (s : Finset ι)
    {f : ι → α → E} (hf : ∀ i ∈ s, Integrable (f i) μ) :
    setToFun μ T hT (∑ i in s, f i) = ∑ i in s, setToFun μ T hT (f i) := by
  revert hf
  refine' Finset.induction_on s _ _
  · intro _
    simp only [setToFun_zero, Finset.sum_empty]
  · intro i s his ih hf
    

Proof state:
case refine'_2
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
s✝ : Finset ι
f : ι → α → E
i : ι
s : Finset ι
his : i ∉ s
ih : (∀ i ∈ s, Integrable (f i)) → setToFun μ T hT (∑ i in s, f i) = ∑ i in s, setToFun μ T hT (f i)
hf : ∀ i_1 ∈ insert i s, Integrable (f i_1)
⊢ setToFun μ T hT (∑ i in insert i s, f i) = ∑ i in insert i s, setToFun μ T hT (f i)

Next tactic:
simp only [his, Finset.sum_insert, not_false_iff]
theorem setToFun_finset_sum' (hT : DominatedFinMeasAdditive μ T C) {ι} (s : Finset ι)
    {f : ι → α → E} (hf : ∀ i ∈ s, Integrable (f i) μ) :
    setToFun μ T hT (∑ i in s, f i) = ∑ i in s, setToFun μ T hT (f i) := by
  revert hf
  refine' Finset.induction_on s _ _
  · intro _
    simp only [setToFun_zero, Finset.sum_empty]
  · intro i s his ih hf
    simp only [his, Finset.sum_insert, not_false_iff]
    

Proof state:
case refine'_2
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
s✝ : Finset ι
f : ι → α → E
i : ι
s : Finset ι
his : i ∉ s
ih : (∀ i ∈ s, Integrable (f i)) → setToFun μ T hT (∑ i in s, f i) = ∑ i in s, setToFun μ T hT (f i)
hf : ∀ i_1 ∈ insert i s, Integrable (f i_1)
⊢ setToFun μ T hT (f i + ∑ i in s, f i) = setToFun μ T hT (f i) + ∑ i in s, setToFun μ T hT (f i)

Next tactic:
rw [setToFun_add hT (hf i (Finset.mem_insert_self i s)) _]
theorem setToFun_finset_sum' (hT : DominatedFinMeasAdditive μ T C) {ι} (s : Finset ι)
    {f : ι → α → E} (hf : ∀ i ∈ s, Integrable (f i) μ) :
    setToFun μ T hT (∑ i in s, f i) = ∑ i in s, setToFun μ T hT (f i) := by
  revert hf
  refine' Finset.induction_on s _ _
  · intro _
    simp only [setToFun_zero, Finset.sum_empty]
  · intro i s his ih hf
    simp only [his, Finset.sum_insert, not_false_iff]
    rw [setToFun_add hT (hf i (Finset.mem_insert_self i s)) _]
    · 

Proof state:
case refine'_2
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
s✝ : Finset ι
f : ι → α → E
i : ι
s : Finset ι
his : i ∉ s
ih : (∀ i ∈ s, Integrable (f i)) → setToFun μ T hT (∑ i in s, f i) = ∑ i in s, setToFun μ T hT (f i)
hf : ∀ i_1 ∈ insert i s, Integrable (f i_1)
⊢ setToFun μ T hT (f i) + setToFun μ T hT (∑ i in s, f i) = setToFun μ T hT (f i) + ∑ i in s, setToFun μ T hT (f i)

Next tactic:
rw [ih fun i hi => hf i (Finset.mem_insert_of_mem hi)]
theorem setToFun_finset_sum' (hT : DominatedFinMeasAdditive μ T C) {ι} (s : Finset ι)
    {f : ι → α → E} (hf : ∀ i ∈ s, Integrable (f i) μ) :
    setToFun μ T hT (∑ i in s, f i) = ∑ i in s, setToFun μ T hT (f i) := by
  revert hf
  refine' Finset.induction_on s _ _
  · intro _
    simp only [setToFun_zero, Finset.sum_empty]
  · intro i s his ih hf
    simp only [his, Finset.sum_insert, not_false_iff]
    rw [setToFun_add hT (hf i (Finset.mem_insert_self i s)) _]
    · rw [ih fun i hi => hf i (Finset.mem_insert_of_mem hi)]
    · 

Proof state:
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
s✝ : Finset ι
f : ι → α → E
i : ι
s : Finset ι
his : i ∉ s
ih : (∀ i ∈ s, Integrable (f i)) → setToFun μ T hT (∑ i in s, f i) = ∑ i in s, setToFun μ T hT (f i)
hf : ∀ i_1 ∈ insert i s, Integrable (f i_1)
⊢ Integrable (∑ i in s, f i)

Next tactic:
convert integrable_finset_sum s fun i hi => hf i (Finset.mem_insert_of_mem hi) with x
theorem setToFun_finset_sum' (hT : DominatedFinMeasAdditive μ T C) {ι} (s : Finset ι)
    {f : ι → α → E} (hf : ∀ i ∈ s, Integrable (f i) μ) :
    setToFun μ T hT (∑ i in s, f i) = ∑ i in s, setToFun μ T hT (f i) := by
  revert hf
  refine' Finset.induction_on s _ _
  · intro _
    simp only [setToFun_zero, Finset.sum_empty]
  · intro i s his ih hf
    simp only [his, Finset.sum_insert, not_false_iff]
    rw [setToFun_add hT (hf i (Finset.mem_insert_self i s)) _]
    · rw [ih fun i hi => hf i (Finset.mem_insert_of_mem hi)]
    · convert integrable_finset_sum s fun i hi => hf i (Finset.mem_insert_of_mem hi) with x
      

Proof state:
case h.e'_5.h
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
s✝ : Finset ι
f : ι → α → E
i : ι
s : Finset ι
his : i ∉ s
ih : (∀ i ∈ s, Integrable (f i)) → setToFun μ T hT (∑ i in s, f i) = ∑ i in s, setToFun μ T hT (f i)
hf : ∀ i_1 ∈ insert i s, Integrable (f i_1)
x : α
⊢ Finset.sum s (fun i => f i) x = ∑ i in s, f i x

Next tactic:
simp
theorem setToFun_finset_sum (hT : DominatedFinMeasAdditive μ T C) {ι} (s : Finset ι) {f : ι → α → E}
    (hf : ∀ i ∈ s, Integrable (f i) μ) :
    (setToFun μ T hT fun a => ∑ i in s, f i a) = ∑ i in s, setToFun μ T hT (f i) := by
  

Proof state:
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
s : Finset ι
f : ι → α → E
hf : ∀ i ∈ s, Integrable (f i)
⊢ (setToFun μ T hT fun a => ∑ i in s, f i a) = ∑ i in s, setToFun μ T hT (f i)

Next tactic:
convert setToFun_finset_sum' hT s hf with a
theorem setToFun_finset_sum (hT : DominatedFinMeasAdditive μ T C) {ι} (s : Finset ι) {f : ι → α → E}
    (hf : ∀ i ∈ s, Integrable (f i) μ) :
    (setToFun μ T hT fun a => ∑ i in s, f i a) = ∑ i in s, setToFun μ T hT (f i) := by
  convert setToFun_finset_sum' hT s hf with a; 

Proof state:
case h.e'_2.h.e'_14.h
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
s : Finset ι
f : ι → α → E
hf : ∀ i ∈ s, Integrable (f i)
a : α
⊢ ∑ i in s, f i a = Finset.sum s (fun i => f i) a

Next tactic:
simp
theorem setToFun_neg (hT : DominatedFinMeasAdditive μ T C) (f : α → E) :
    setToFun μ T hT (-f) = -setToFun μ T hT f := by
  

Proof state:
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
f : α → E
⊢ setToFun μ T hT (-f) = -setToFun μ T hT f

Next tactic:
by_cases hf : Integrable f μ
theorem setToFun_neg (hT : DominatedFinMeasAdditive μ T C) (f : α → E) :
    setToFun μ T hT (-f) = -setToFun μ T hT f := by
  by_cases hf : Integrable f μ
  · 

Proof state:
case pos
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
f : α → E
hf : Integrable f
⊢ setToFun μ T hT (-f) = -setToFun μ T hT f

Next tactic:
rw [setToFun_eq hT hf, setToFun_eq hT hf.neg, Integrable.toL1_neg,
      (L1.setToL1 hT).map_neg]
theorem setToFun_neg (hT : DominatedFinMeasAdditive μ T C) (f : α → E) :
    setToFun μ T hT (-f) = -setToFun μ T hT f := by
  by_cases hf : Integrable f μ
  · rw [setToFun_eq hT hf, setToFun_eq hT hf.neg, Integrable.toL1_neg,
      (L1.setToL1 hT).map_neg]
  · 

Proof state:
case neg
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
f : α → E
hf : ¬Integrable f
⊢ setToFun μ T hT (-f) = -setToFun μ T hT f

Next tactic:
rw [setToFun_undef hT hf, setToFun_undef hT, neg_zero]
theorem setToFun_neg (hT : DominatedFinMeasAdditive μ T C) (f : α → E) :
    setToFun μ T hT (-f) = -setToFun μ T hT f := by
  by_cases hf : Integrable f μ
  · rw [setToFun_eq hT hf, setToFun_eq hT hf.neg, Integrable.toL1_neg,
      (L1.setToL1 hT).map_neg]
  · rw [setToFun_undef hT hf, setToFun_undef hT, neg_zero]
    

Proof state:
case neg
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
hT : DominatedFinMeasAdditive μ T C
f : α → E
hf : ¬Integrable f
⊢ ¬Integrable (-f)

Next tactic:
rwa [← integrable_neg_iff] at hf
theorem setToFun_sub (hT : DominatedFinMeasAdditive μ T C) (hf : Integrable f μ)
    (hg : Integrable g μ) : setToFun μ T hT (f - g) = setToFun μ T hT f - setToFun μ T hT g := by
  

Proof state:
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f g : α → E
hT : DominatedFinMeasAdditive μ T C
hf : Integrable f
hg : Integrable g
⊢ setToFun μ T hT (f - g) = setToFun μ T hT f - setToFun μ T hT g

Next tactic:
rw [sub_eq_add_neg, sub_eq_add_neg, setToFun_add hT hf hg.neg, setToFun_neg hT g]
theorem setToFun_smul [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F]
    (hT : DominatedFinMeasAdditive μ T C) (h_smul : ∀ c : 𝕜, ∀ s x, T s (c • x) = c • T s x) (c : 𝕜)
    (f : α → E) : setToFun μ T hT (c • f) = c • setToFun μ T hT f := by
  

Proof state:
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedAddCommGroup F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝³ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
hT : DominatedFinMeasAdditive μ T C
h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c • x) = c • (T s) x
c : 𝕜
f : α → E
⊢ setToFun μ T hT (c • f) = c • setToFun μ T hT f

Next tactic:
by_cases hf : Integrable f μ
theorem setToFun_smul [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F]
    (hT : DominatedFinMeasAdditive μ T C) (h_smul : ∀ c : 𝕜, ∀ s x, T s (c • x) = c • T s x) (c : 𝕜)
    (f : α → E) : setToFun μ T hT (c • f) = c • setToFun μ T hT f := by
  by_cases hf : Integrable f μ
  · 

Proof state:
case pos
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedAddCommGroup F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝³ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
hT : DominatedFinMeasAdditive μ T C
h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c • x) = c • (T s) x
c : 𝕜
f : α → E
hf : Integrable f
⊢ setToFun μ T hT (c • f) = c • setToFun μ T hT f

Next tactic:
rw [setToFun_eq hT hf, setToFun_eq hT, Integrable.toL1_smul',
      L1.setToL1_smul hT h_smul c _]
theorem setToFun_smul [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F]
    (hT : DominatedFinMeasAdditive μ T C) (h_smul : ∀ c : 𝕜, ∀ s x, T s (c • x) = c • T s x) (c : 𝕜)
    (f : α → E) : setToFun μ T hT (c • f) = c • setToFun μ T hT f := by
  by_cases hf : Integrable f μ
  · rw [setToFun_eq hT hf, setToFun_eq hT, Integrable.toL1_smul',
      L1.setToL1_smul hT h_smul c _]
  · 

Proof state:
case neg
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedAddCommGroup F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝³ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
hT : DominatedFinMeasAdditive μ T C
h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c • x) = c • (T s) x
c : 𝕜
f : α → E
hf : ¬Integrable f
⊢ setToFun μ T hT (c • f) = c • setToFun μ T hT f

Next tactic:
by_cases hr : c = 0
theorem setToFun_smul [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F]
    (hT : DominatedFinMeasAdditive μ T C) (h_smul : ∀ c : 𝕜, ∀ s x, T s (c • x) = c • T s x) (c : 𝕜)
    (f : α → E) : setToFun μ T hT (c • f) = c • setToFun μ T hT f := by
  by_cases hf : Integrable f μ
  · rw [setToFun_eq hT hf, setToFun_eq hT, Integrable.toL1_smul',
      L1.setToL1_smul hT h_smul c _]
  · by_cases hr : c = 0
    · 

Proof state:
case pos
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedAddCommGroup F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝³ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
hT : DominatedFinMeasAdditive μ T C
h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c • x) = c • (T s) x
c : 𝕜
f : α → E
hf : ¬Integrable f
hr : c = 0
⊢ setToFun μ T hT (c • f) = c • setToFun μ T hT f

Next tactic:
rw [hr]
theorem setToFun_smul [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F]
    (hT : DominatedFinMeasAdditive μ T C) (h_smul : ∀ c : 𝕜, ∀ s x, T s (c • x) = c • T s x) (c : 𝕜)
    (f : α → E) : setToFun μ T hT (c • f) = c • setToFun μ T hT f := by
  by_cases hf : Integrable f μ
  · rw [setToFun_eq hT hf, setToFun_eq hT, Integrable.toL1_smul',
      L1.setToL1_smul hT h_smul c _]
  · by_cases hr : c = 0
    · rw [hr]; 

Proof state:
case pos
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedAddCommGroup F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝³ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
hT : DominatedFinMeasAdditive μ T C
h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c • x) = c • (T s) x
c : 𝕜
f : α → E
hf : ¬Integrable f
hr : c = 0
⊢ setToFun μ T hT (0 • f) = 0 • setToFun μ T hT f

Next tactic:
simp
theorem setToFun_smul [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F]
    (hT : DominatedFinMeasAdditive μ T C) (h_smul : ∀ c : 𝕜, ∀ s x, T s (c • x) = c • T s x) (c : 𝕜)
    (f : α → E) : setToFun μ T hT (c • f) = c • setToFun μ T hT f := by
  by_cases hf : Integrable f μ
  · rw [setToFun_eq hT hf, setToFun_eq hT, Integrable.toL1_smul',
      L1.setToL1_smul hT h_smul c _]
  · by_cases hr : c = 0
    · rw [hr]; simp
    · 

Proof state:
case neg
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedAddCommGroup F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝³ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
hT : DominatedFinMeasAdditive μ T C
h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c • x) = c • (T s) x
c : 𝕜
f : α → E
hf : ¬Integrable f
hr : ¬c = 0
⊢ setToFun μ T hT (c • f) = c • setToFun μ T hT f

Next tactic:
have hf' : ¬Integrable (c • f) μ := by rwa [integrable_smul_iff hr f]
theorem setToFun_smul [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F]
    (hT : DominatedFinMeasAdditive μ T C) (h_smul : ∀ c : 𝕜, ∀ s x, T s (c • x) = c • T s x) (c : 𝕜)
    (f : α → E) : setToFun μ T hT (c • f) = c • setToFun μ T hT f := by
  by_cases hf : Integrable f μ
  · rw [setToFun_eq hT hf, setToFun_eq hT, Integrable.toL1_smul',
      L1.setToL1_smul hT h_smul c _]
  · by_cases hr : c = 0
    · rw [hr]; simp
    · have hf' : ¬Integrable (c • f) μ := by 

Proof state:
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedAddCommGroup F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝³ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
hT : DominatedFinMeasAdditive μ T C
h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c • x) = c • (T s) x
c : 𝕜
f : α → E
hf : ¬Integrable f
hr : ¬c = 0
⊢ ¬Integrable (c • f)

Next tactic:
rwa [integrable_smul_iff hr f]
theorem setToFun_smul [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F]
    (hT : DominatedFinMeasAdditive μ T C) (h_smul : ∀ c : 𝕜, ∀ s x, T s (c • x) = c • T s x) (c : 𝕜)
    (f : α → E) : setToFun μ T hT (c • f) = c • setToFun μ T hT f := by
  by_cases hf : Integrable f μ
  · rw [setToFun_eq hT hf, setToFun_eq hT, Integrable.toL1_smul',
      L1.setToL1_smul hT h_smul c _]
  · by_cases hr : c = 0
    · rw [hr]; simp
    · have hf' : ¬Integrable (c • f) μ := by rwa [integrable_smul_iff hr f]
      

Proof state:
case neg
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedAddCommGroup F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝³ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f✝ g : α → E
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
hT : DominatedFinMeasAdditive μ T C
h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c • x) = c • (T s) x
c : 𝕜
f : α → E
hf : ¬Integrable f
hr : ¬c = 0
hf' : ¬Integrable (c • f)
⊢ setToFun μ T hT (c • f) = c • setToFun μ T hT f

Next tactic:
rw [setToFun_undef hT hf, setToFun_undef hT hf', smul_zero]
theorem setToFun_congr_ae (hT : DominatedFinMeasAdditive μ T C) (h : f =ᵐ[μ] g) :
    setToFun μ T hT f = setToFun μ T hT g := by
  

Proof state:
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f g : α → E
hT : DominatedFinMeasAdditive μ T C
h : f =ᵐ[μ] g
⊢ setToFun μ T hT f = setToFun μ T hT g

Next tactic:
by_cases hfi : Integrable f μ