l3lab/ntp-mathlib huggingface dataset:
theorem condexpIndL1Fin_disjoint_union (hs : MeasurableSet s) (ht : MeasurableSet t) (hμs : μ s ≠ ∞)
    (hμt : μ t ≠ ∞) (hst : s ∩ t = ∅) (x : G) :
    condexpIndL1Fin hm (hs.union ht) ((measure_union_le s t).trans_lt
      (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne x =
    condexpIndL1Fin hm hs hμs x + condexpIndL1Fin hm ht hμt x := by
  ext1
  have hμst :=
    ((measure_union_le s t).trans_lt (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne
  refine' (condexpIndL1Fin_ae_eq_condexpIndSMul hm (hs.union ht) hμst x).trans _
  refine' EventuallyEq.trans _ (Lp.coeFn_add _ _).symm
  have hs_eq := condexpIndL1Fin_ae_eq_condexpIndSMul hm hs hμs x
  have ht_eq := condexpIndL1Fin_ae_eq_condexpIndSMul hm ht hμt x
  refine' EventuallyEq.trans _ (EventuallyEq.add hs_eq.symm ht_eq.symm)
  

Proof state:
case h
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
hs : MeasurableSet s
ht : MeasurableSet t
hμs : ↑↑μ s ≠ ⊤
hμt : ↑↑μ t ≠ ⊤
hst : s ∩ t = ∅
x : G
hμst : ↑↑μ (s ∪ t) ≠ ⊤
hs_eq : ↑↑(condexpIndL1Fin hm hs hμs x) =ᵐ[μ] ↑↑(condexpIndSMul hm hs hμs x)
ht_eq : ↑↑(condexpIndL1Fin hm ht hμt x) =ᵐ[μ] ↑↑(condexpIndSMul hm ht hμt x)
⊢ ↑↑(condexpIndSMul hm (_ : MeasurableSet (s ∪ t)) hμst x) =ᵐ[μ] fun x_1 =>
    ↑↑(condexpIndSMul hm hs hμs x) x_1 + ↑↑(condexpIndSMul hm ht hμt x) x_1

Next tactic:
rw [condexpIndSMul]
theorem condexpIndL1Fin_disjoint_union (hs : MeasurableSet s) (ht : MeasurableSet t) (hμs : μ s ≠ ∞)
    (hμt : μ t ≠ ∞) (hst : s ∩ t = ∅) (x : G) :
    condexpIndL1Fin hm (hs.union ht) ((measure_union_le s t).trans_lt
      (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne x =
    condexpIndL1Fin hm hs hμs x + condexpIndL1Fin hm ht hμt x := by
  ext1
  have hμst :=
    ((measure_union_le s t).trans_lt (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne
  refine' (condexpIndL1Fin_ae_eq_condexpIndSMul hm (hs.union ht) hμst x).trans _
  refine' EventuallyEq.trans _ (Lp.coeFn_add _ _).symm
  have hs_eq := condexpIndL1Fin_ae_eq_condexpIndSMul hm hs hμs x
  have ht_eq := condexpIndL1Fin_ae_eq_condexpIndSMul hm ht hμt x
  refine' EventuallyEq.trans _ (EventuallyEq.add hs_eq.symm ht_eq.symm)
  rw [condexpIndSMul]
  

Proof state:
case h
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
hs : MeasurableSet s
ht : MeasurableSet t
hμs : ↑↑μ s ≠ ⊤
hμt : ↑↑μ t ≠ ⊤
hst : s ∩ t = ∅
x : G
hμst : ↑↑μ (s ∪ t) ≠ ⊤
hs_eq : ↑↑(condexpIndL1Fin hm hs hμs x) =ᵐ[μ] ↑↑(condexpIndSMul hm hs hμs x)
ht_eq : ↑↑(condexpIndL1Fin hm ht hμt x) =ᵐ[μ] ↑↑(condexpIndSMul hm ht hμt x)
⊢ ↑↑((compLpL 2 μ (toSpanSingleton ℝ x))
          ↑((condexpL2 ℝ ℝ hm) (indicatorConstLp 2 (_ : MeasurableSet (s ∪ t)) hμst 1))) =ᵐ[μ]
    fun x_1 => ↑↑(condexpIndSMul hm hs hμs x) x_1 + ↑↑(condexpIndSMul hm ht hμt x) x_1

Next tactic:
rw [indicatorConstLp_disjoint_union hs ht hμs hμt hst (1 : ℝ)]
theorem condexpIndL1Fin_disjoint_union (hs : MeasurableSet s) (ht : MeasurableSet t) (hμs : μ s ≠ ∞)
    (hμt : μ t ≠ ∞) (hst : s ∩ t = ∅) (x : G) :
    condexpIndL1Fin hm (hs.union ht) ((measure_union_le s t).trans_lt
      (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne x =
    condexpIndL1Fin hm hs hμs x + condexpIndL1Fin hm ht hμt x := by
  ext1
  have hμst :=
    ((measure_union_le s t).trans_lt (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne
  refine' (condexpIndL1Fin_ae_eq_condexpIndSMul hm (hs.union ht) hμst x).trans _
  refine' EventuallyEq.trans _ (Lp.coeFn_add _ _).symm
  have hs_eq := condexpIndL1Fin_ae_eq_condexpIndSMul hm hs hμs x
  have ht_eq := condexpIndL1Fin_ae_eq_condexpIndSMul hm ht hμt x
  refine' EventuallyEq.trans _ (EventuallyEq.add hs_eq.symm ht_eq.symm)
  rw [condexpIndSMul]
  rw [indicatorConstLp_disjoint_union hs ht hμs hμt hst (1 : ℝ)]
  

Proof state:
case h
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
hs : MeasurableSet s
ht : MeasurableSet t
hμs : ↑↑μ s ≠ ⊤
hμt : ↑↑μ t ≠ ⊤
hst : s ∩ t = ∅
x : G
hμst : ↑↑μ (s ∪ t) ≠ ⊤
hs_eq : ↑↑(condexpIndL1Fin hm hs hμs x) =ᵐ[μ] ↑↑(condexpIndSMul hm hs hμs x)
ht_eq : ↑↑(condexpIndL1Fin hm ht hμt x) =ᵐ[μ] ↑↑(condexpIndSMul hm ht hμt x)
⊢ ↑↑((compLpL 2 μ (toSpanSingleton ℝ x))
          ↑((condexpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1 + indicatorConstLp 2 ht hμt 1))) =ᵐ[μ]
    fun x_1 => ↑↑(condexpIndSMul hm hs hμs x) x_1 + ↑↑(condexpIndSMul hm ht hμt x) x_1

Next tactic:
rw [(condexpL2 ℝ ℝ hm).map_add]
theorem condexpIndL1Fin_disjoint_union (hs : MeasurableSet s) (ht : MeasurableSet t) (hμs : μ s ≠ ∞)
    (hμt : μ t ≠ ∞) (hst : s ∩ t = ∅) (x : G) :
    condexpIndL1Fin hm (hs.union ht) ((measure_union_le s t).trans_lt
      (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne x =
    condexpIndL1Fin hm hs hμs x + condexpIndL1Fin hm ht hμt x := by
  ext1
  have hμst :=
    ((measure_union_le s t).trans_lt (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne
  refine' (condexpIndL1Fin_ae_eq_condexpIndSMul hm (hs.union ht) hμst x).trans _
  refine' EventuallyEq.trans _ (Lp.coeFn_add _ _).symm
  have hs_eq := condexpIndL1Fin_ae_eq_condexpIndSMul hm hs hμs x
  have ht_eq := condexpIndL1Fin_ae_eq_condexpIndSMul hm ht hμt x
  refine' EventuallyEq.trans _ (EventuallyEq.add hs_eq.symm ht_eq.symm)
  rw [condexpIndSMul]
  rw [indicatorConstLp_disjoint_union hs ht hμs hμt hst (1 : ℝ)]
  rw [(condexpL2 ℝ ℝ hm).map_add]
  

Proof state:
case h
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
hs : MeasurableSet s
ht : MeasurableSet t
hμs : ↑↑μ s ≠ ⊤
hμt : ↑↑μ t ≠ ⊤
hst : s ∩ t = ∅
x : G
hμst : ↑↑μ (s ∪ t) ≠ ⊤
hs_eq : ↑↑(condexpIndL1Fin hm hs hμs x) =ᵐ[μ] ↑↑(condexpIndSMul hm hs hμs x)
ht_eq : ↑↑(condexpIndL1Fin hm ht hμt x) =ᵐ[μ] ↑↑(condexpIndSMul hm ht hμt x)
⊢ ↑↑((compLpL 2 μ (toSpanSingleton ℝ x))
          ↑((condexpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1) + (condexpL2 ℝ ℝ hm) (indicatorConstLp 2 ht hμt 1))) =ᵐ[μ]
    fun x_1 => ↑↑(condexpIndSMul hm hs hμs x) x_1 + ↑↑(condexpIndSMul hm ht hμt x) x_1

Next tactic:
push_cast
theorem condexpIndL1Fin_disjoint_union (hs : MeasurableSet s) (ht : MeasurableSet t) (hμs : μ s ≠ ∞)
    (hμt : μ t ≠ ∞) (hst : s ∩ t = ∅) (x : G) :
    condexpIndL1Fin hm (hs.union ht) ((measure_union_le s t).trans_lt
      (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne x =
    condexpIndL1Fin hm hs hμs x + condexpIndL1Fin hm ht hμt x := by
  ext1
  have hμst :=
    ((measure_union_le s t).trans_lt (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne
  refine' (condexpIndL1Fin_ae_eq_condexpIndSMul hm (hs.union ht) hμst x).trans _
  refine' EventuallyEq.trans _ (Lp.coeFn_add _ _).symm
  have hs_eq := condexpIndL1Fin_ae_eq_condexpIndSMul hm hs hμs x
  have ht_eq := condexpIndL1Fin_ae_eq_condexpIndSMul hm ht hμt x
  refine' EventuallyEq.trans _ (EventuallyEq.add hs_eq.symm ht_eq.symm)
  rw [condexpIndSMul]
  rw [indicatorConstLp_disjoint_union hs ht hμs hμt hst (1 : ℝ)]
  rw [(condexpL2 ℝ ℝ hm).map_add]
  push_cast
  

Proof state:
case h
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
hs : MeasurableSet s
ht : MeasurableSet t
hμs : ↑↑μ s ≠ ⊤
hμt : ↑↑μ t ≠ ⊤
hst : s ∩ t = ∅
x : G
hμst : ↑↑μ (s ∪ t) ≠ ⊤
hs_eq : ↑↑(condexpIndL1Fin hm hs hμs x) =ᵐ[μ] ↑↑(condexpIndSMul hm hs hμs x)
ht_eq : ↑↑(condexpIndL1Fin hm ht hμt x) =ᵐ[μ] ↑↑(condexpIndSMul hm ht hμt x)
⊢ ↑↑((compLpL 2 μ (toSpanSingleton ℝ x))
          (↑((condexpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) +
            ↑((condexpL2 ℝ ℝ hm) (indicatorConstLp 2 ht hμt 1)))) =ᵐ[μ]
    fun x_1 => ↑↑(condexpIndSMul hm hs hμs x) x_1 + ↑↑(condexpIndSMul hm ht hμt x) x_1

Next tactic:
rw [((toSpanSingleton ℝ x).compLpL 2 μ).map_add]
theorem condexpIndL1Fin_disjoint_union (hs : MeasurableSet s) (ht : MeasurableSet t) (hμs : μ s ≠ ∞)
    (hμt : μ t ≠ ∞) (hst : s ∩ t = ∅) (x : G) :
    condexpIndL1Fin hm (hs.union ht) ((measure_union_le s t).trans_lt
      (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne x =
    condexpIndL1Fin hm hs hμs x + condexpIndL1Fin hm ht hμt x := by
  ext1
  have hμst :=
    ((measure_union_le s t).trans_lt (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne
  refine' (condexpIndL1Fin_ae_eq_condexpIndSMul hm (hs.union ht) hμst x).trans _
  refine' EventuallyEq.trans _ (Lp.coeFn_add _ _).symm
  have hs_eq := condexpIndL1Fin_ae_eq_condexpIndSMul hm hs hμs x
  have ht_eq := condexpIndL1Fin_ae_eq_condexpIndSMul hm ht hμt x
  refine' EventuallyEq.trans _ (EventuallyEq.add hs_eq.symm ht_eq.symm)
  rw [condexpIndSMul]
  rw [indicatorConstLp_disjoint_union hs ht hμs hμt hst (1 : ℝ)]
  rw [(condexpL2 ℝ ℝ hm).map_add]
  push_cast
  rw [((toSpanSingleton ℝ x).compLpL 2 μ).map_add]
  

Proof state:
case h
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
hs : MeasurableSet s
ht : MeasurableSet t
hμs : ↑↑μ s ≠ ⊤
hμt : ↑↑μ t ≠ ⊤
hst : s ∩ t = ∅
x : G
hμst : ↑↑μ (s ∪ t) ≠ ⊤
hs_eq : ↑↑(condexpIndL1Fin hm hs hμs x) =ᵐ[μ] ↑↑(condexpIndSMul hm hs hμs x)
ht_eq : ↑↑(condexpIndL1Fin hm ht hμt x) =ᵐ[μ] ↑↑(condexpIndSMul hm ht hμt x)
⊢ ↑↑((compLpL 2 μ (toSpanSingleton ℝ x)) ↑((condexpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) +
          (compLpL 2 μ (toSpanSingleton ℝ x)) ↑((condexpL2 ℝ ℝ hm) (indicatorConstLp 2 ht hμt 1))) =ᵐ[μ]
    fun x_1 => ↑↑(condexpIndSMul hm hs hμs x) x_1 + ↑↑(condexpIndSMul hm ht hμt x) x_1

Next tactic:
refine' (Lp.coeFn_add _ _).trans _
theorem condexpIndL1Fin_disjoint_union (hs : MeasurableSet s) (ht : MeasurableSet t) (hμs : μ s ≠ ∞)
    (hμt : μ t ≠ ∞) (hst : s ∩ t = ∅) (x : G) :
    condexpIndL1Fin hm (hs.union ht) ((measure_union_le s t).trans_lt
      (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne x =
    condexpIndL1Fin hm hs hμs x + condexpIndL1Fin hm ht hμt x := by
  ext1
  have hμst :=
    ((measure_union_le s t).trans_lt (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne
  refine' (condexpIndL1Fin_ae_eq_condexpIndSMul hm (hs.union ht) hμst x).trans _
  refine' EventuallyEq.trans _ (Lp.coeFn_add _ _).symm
  have hs_eq := condexpIndL1Fin_ae_eq_condexpIndSMul hm hs hμs x
  have ht_eq := condexpIndL1Fin_ae_eq_condexpIndSMul hm ht hμt x
  refine' EventuallyEq.trans _ (EventuallyEq.add hs_eq.symm ht_eq.symm)
  rw [condexpIndSMul]
  rw [indicatorConstLp_disjoint_union hs ht hμs hμt hst (1 : ℝ)]
  rw [(condexpL2 ℝ ℝ hm).map_add]
  push_cast
  rw [((toSpanSingleton ℝ x).compLpL 2 μ).map_add]
  refine' (Lp.coeFn_add _ _).trans _
  

Proof state:
case h
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
hs : MeasurableSet s
ht : MeasurableSet t
hμs : ↑↑μ s ≠ ⊤
hμt : ↑↑μ t ≠ ⊤
hst : s ∩ t = ∅
x : G
hμst : ↑↑μ (s ∪ t) ≠ ⊤
hs_eq : ↑↑(condexpIndL1Fin hm hs hμs x) =ᵐ[μ] ↑↑(condexpIndSMul hm hs hμs x)
ht_eq : ↑↑(condexpIndL1Fin hm ht hμt x) =ᵐ[μ] ↑↑(condexpIndSMul hm ht hμt x)
⊢ ↑↑((compLpL 2 μ (toSpanSingleton ℝ x)) ↑((condexpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1))) +
      ↑↑((compLpL 2 μ (toSpanSingleton ℝ x)) ↑((condexpL2 ℝ ℝ hm) (indicatorConstLp 2 ht hμt 1))) =ᵐ[μ]
    fun x_1 => ↑↑(condexpIndSMul hm hs hμs x) x_1 + ↑↑(condexpIndSMul hm ht hμt x) x_1

Next tactic:
refine' eventually_of_forall fun y => _
theorem condexpIndL1Fin_disjoint_union (hs : MeasurableSet s) (ht : MeasurableSet t) (hμs : μ s ≠ ∞)
    (hμt : μ t ≠ ∞) (hst : s ∩ t = ∅) (x : G) :
    condexpIndL1Fin hm (hs.union ht) ((measure_union_le s t).trans_lt
      (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne x =
    condexpIndL1Fin hm hs hμs x + condexpIndL1Fin hm ht hμt x := by
  ext1
  have hμst :=
    ((measure_union_le s t).trans_lt (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne
  refine' (condexpIndL1Fin_ae_eq_condexpIndSMul hm (hs.union ht) hμst x).trans _
  refine' EventuallyEq.trans _ (Lp.coeFn_add _ _).symm
  have hs_eq := condexpIndL1Fin_ae_eq_condexpIndSMul hm hs hμs x
  have ht_eq := condexpIndL1Fin_ae_eq_condexpIndSMul hm ht hμt x
  refine' EventuallyEq.trans _ (EventuallyEq.add hs_eq.symm ht_eq.symm)
  rw [condexpIndSMul]
  rw [indicatorConstLp_disjoint_union hs ht hμs hμt hst (1 : ℝ)]
  rw [(condexpL2 ℝ ℝ hm).map_add]
  push_cast
  rw [((toSpanSingleton ℝ x).compLpL 2 μ).map_add]
  refine' (Lp.coeFn_add _ _).trans _
  refine' eventually_of_forall fun y => _
  

Proof state:
case h
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
hs : MeasurableSet s
ht : MeasurableSet t
hμs : ↑↑μ s ≠ ⊤
hμt : ↑↑μ t ≠ ⊤
hst : s ∩ t = ∅
x : G
hμst : ↑↑μ (s ∪ t) ≠ ⊤
hs_eq : ↑↑(condexpIndL1Fin hm hs hμs x) =ᵐ[μ] ↑↑(condexpIndSMul hm hs hμs x)
ht_eq : ↑↑(condexpIndL1Fin hm ht hμt x) =ᵐ[μ] ↑↑(condexpIndSMul hm ht hμt x)
y : α
⊢ (↑↑((compLpL 2 μ (toSpanSingleton ℝ x)) ↑((condexpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1))) +
        ↑↑((compLpL 2 μ (toSpanSingleton ℝ x)) ↑((condexpL2 ℝ ℝ hm) (indicatorConstLp 2 ht hμt 1))))
      y =
    (fun x_1 => ↑↑(condexpIndSMul hm hs hμs x) x_1 + ↑↑(condexpIndSMul hm ht hμt x) x_1) y

Next tactic:
rfl
theorem condexpIndL1_of_measurableSet_of_measure_ne_top (hs : MeasurableSet s) (hμs : μ s ≠ ∞)
    (x : G) : condexpIndL1 hm μ s x = condexpIndL1Fin hm hs hμs x := by
  

Proof state:
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
hs : MeasurableSet s
hμs : ↑↑μ s ≠ ⊤
x : G
⊢ condexpIndL1 hm μ s x = condexpIndL1Fin hm hs hμs x

Next tactic:
simp only [condexpIndL1, And.intro hs hμs, dif_pos, Ne.def, not_false_iff, and_self_iff]
theorem condexpIndL1_of_measure_eq_top (hμs : μ s = ∞) (x : G) : condexpIndL1 hm μ s x = 0 := by
  

Proof state:
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
hμs : ↑↑μ s = ⊤
x : G
⊢ condexpIndL1 hm μ s x = 0

Next tactic:
simp only [condexpIndL1, hμs, eq_self_iff_true, not_true, Ne.def, dif_neg, not_false_iff,
    and_false_iff]
theorem condexpIndL1_of_not_measurableSet (hs : ¬MeasurableSet s) (x : G) :
    condexpIndL1 hm μ s x = 0 := by
  

Proof state:
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
hs : ¬MeasurableSet s
x : G
⊢ condexpIndL1 hm μ s x = 0

Next tactic:
simp only [condexpIndL1, hs, dif_neg, not_false_iff, false_and_iff]
theorem condexpIndL1_add (x y : G) :
    condexpIndL1 hm μ s (x + y) = condexpIndL1 hm μ s x + condexpIndL1 hm μ s y := by
  

Proof state:
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
x y : G
⊢ condexpIndL1 hm μ s (x + y) = condexpIndL1 hm μ s x + condexpIndL1 hm μ s y

Next tactic:
by_cases hs : MeasurableSet s
theorem condexpIndL1_add (x y : G) :
    condexpIndL1 hm μ s (x + y) = condexpIndL1 hm μ s x + condexpIndL1 hm μ s y := by
  by_cases hs : MeasurableSet s
  

Proof state:
case pos
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
x y : G
hs : MeasurableSet s
⊢ condexpIndL1 hm μ s (x + y) = condexpIndL1 hm μ s x + condexpIndL1 hm μ s y
case neg
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
x y : G
hs : ¬MeasurableSet s
⊢ condexpIndL1 hm μ s (x + y) = condexpIndL1 hm μ s x + condexpIndL1 hm μ s y

Next tactic:
swap
theorem condexpIndL1_add (x y : G) :
    condexpIndL1 hm μ s (x + y) = condexpIndL1 hm μ s x + condexpIndL1 hm μ s y := by
  by_cases hs : MeasurableSet s
  swap; · 

Proof state:
case neg
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
x y : G
hs : ¬MeasurableSet s
⊢ condexpIndL1 hm μ s (x + y) = condexpIndL1 hm μ s x + condexpIndL1 hm μ s y

Next tactic:
simp_rw [condexpIndL1_of_not_measurableSet hs]
theorem condexpIndL1_add (x y : G) :
    condexpIndL1 hm μ s (x + y) = condexpIndL1 hm μ s x + condexpIndL1 hm μ s y := by
  by_cases hs : MeasurableSet s
  swap; · simp_rw [condexpIndL1_of_not_measurableSet hs]; 

Proof state:
case neg
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
x y : G
hs : ¬MeasurableSet s
⊢ 0 = 0 + 0

Next tactic:
rw [zero_add]
theorem condexpIndL1_add (x y : G) :
    condexpIndL1 hm μ s (x + y) = condexpIndL1 hm μ s x + condexpIndL1 hm μ s y := by
  by_cases hs : MeasurableSet s
  swap; · simp_rw [condexpIndL1_of_not_measurableSet hs]; rw [zero_add]
  

Proof state:
case pos
α : Type u_1
β : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
G' : Type u_6
𝕜 : Type u_7
p : ℝ≥0∞
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup F'
inst✝⁸ : NormedSpace 𝕜 F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : CompleteSpace F'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace ℝ G'
inst✝² : CompleteSpace G'
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
x y : G
hs : MeasurableSet s
⊢ condexpIndL1 hm μ s (x + y) = condexpIndL1 hm μ s x + condexpIndL1 hm μ s y

Next tactic:
by_cases hμs : μ s = ∞