l3lab/ntp-mathlib huggingface dataset:
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
    [OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
    μ[f|m] ≤ᵐ[μ] μ[g|m] := by
  by_cases hm : m ≤ m0
  swap; · simp_rw [condexp_of_not_le hm]; rfl
  by_cases hμm : SigmaFinite (μ.trim hm)
  swap; · 

Proof state:
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
E : Type u_5
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f g : α → E
hf : Integrable f
hg : Integrable g
hfg : f ≤ᵐ[μ] g
hm : m ≤ m0
hμm : ¬SigmaFinite (Measure.trim μ hm)
⊢ μ[f|m] ≤ᵐ[μ] μ[g|m]

Next tactic:
simp_rw [condexp_of_not_sigmaFinite hm hμm]
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
    [OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
    μ[f|m] ≤ᵐ[μ] μ[g|m] := by
  by_cases hm : m ≤ m0
  swap; · simp_rw [condexp_of_not_le hm]; rfl
  by_cases hμm : SigmaFinite (μ.trim hm)
  swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; 

Proof state:
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
E : Type u_5
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f g : α → E
hf : Integrable f
hg : Integrable g
hfg : f ≤ᵐ[μ] g
hm : m ≤ m0
hμm : ¬SigmaFinite (Measure.trim μ hm)
⊢ 0 ≤ᵐ[μ] 0

Next tactic:
rfl
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
    [OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
    μ[f|m] ≤ᵐ[μ] μ[g|m] := by
  by_cases hm : m ≤ m0
  swap; · simp_rw [condexp_of_not_le hm]; rfl
  by_cases hμm : SigmaFinite (μ.trim hm)
  swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
  

Proof state:
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
E : Type u_5
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f g : α → E
hf : Integrable f
hg : Integrable g
hfg : f ≤ᵐ[μ] g
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
⊢ μ[f|m] ≤ᵐ[μ] μ[g|m]

Next tactic:
haveI : SigmaFinite (μ.trim hm) := hμm
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
    [OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
    μ[f|m] ≤ᵐ[μ] μ[g|m] := by
  by_cases hm : m ≤ m0
  swap; · simp_rw [condexp_of_not_le hm]; rfl
  by_cases hμm : SigmaFinite (μ.trim hm)
  swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
  haveI : SigmaFinite (μ.trim hm) := hμm
  

Proof state:
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
E : Type u_5
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f g : α → E
hf : Integrable f
hg : Integrable g
hfg : f ≤ᵐ[μ] g
hm : m ≤ m0
hμm this : SigmaFinite (Measure.trim μ hm)
⊢ μ[f|m] ≤ᵐ[μ] μ[g|m]

Next tactic:
exact (condexp_ae_eq_condexpL1 hm _).trans_le
    ((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_condexpL1 hm _).symm)
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
    [OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
  

Proof state:
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
E : Type u_5
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f : α → E
hf : 0 ≤ᵐ[μ] f
⊢ 0 ≤ᵐ[μ] μ[f|m]

Next tactic:
by_cases hfint : Integrable f μ
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
    [OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
  by_cases hfint : Integrable f μ
  · 

Proof state:
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
E : Type u_5
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f : α → E
hf : 0 ≤ᵐ[μ] f
hfint : Integrable f
⊢ 0 ≤ᵐ[μ] μ[f|m]

Next tactic:
rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
    [OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
  by_cases hfint : Integrable f μ
  · rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
    

Proof state:
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
E : Type u_5
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f : α → E
hf : 0 ≤ᵐ[μ] f
hfint : Integrable f
⊢ μ[0|m] ≤ᵐ[μ] μ[f|m]

Next tactic:
exact condexp_mono (integrable_zero _ _ _) hfint hf
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
    [OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] := by
  by_cases hfint : Integrable f μ
  · rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
    exact condexp_mono (integrable_zero _ _ _) hfint hf
  · 

Proof state:
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
E : Type u_5
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f : α → E
hf : 0 ≤ᵐ[μ] f
hfint : ¬Integrable f
⊢ 0 ≤ᵐ[μ] μ[f|m]

Next tactic:
rw [condexp_undef hfint]
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
    [OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0 := by
  

Proof state:
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
E : Type u_5
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f : α → E
hf : f ≤ᵐ[μ] 0
⊢ μ[f|m] ≤ᵐ[μ] 0

Next tactic:
by_cases hfint : Integrable f μ
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
    [OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0 := by
  by_cases hfint : Integrable f μ
  · 

Proof state:
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
E : Type u_5
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f : α → E
hf : f ≤ᵐ[μ] 0
hfint : Integrable f
⊢ μ[f|m] ≤ᵐ[μ] 0

Next tactic:
rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
    [OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0 := by
  by_cases hfint : Integrable f μ
  · rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
    

Proof state:
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
E : Type u_5
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f : α → E
hf : f ≤ᵐ[μ] 0
hfint : Integrable f
⊢ μ[f|m] ≤ᵐ[μ] μ[0|m]

Next tactic:
exact condexp_mono hfint (integrable_zero _ _ _) hf
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
    [OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0 := by
  by_cases hfint : Integrable f μ
  · rw [(condexp_zero.symm : (0 : α → E) = μ[0|m])]
    exact condexp_mono hfint (integrable_zero _ _ _) hf
  · 

Proof state:
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
inst✝⁵ : NormedSpace ℝ F'
inst✝⁴ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
E : Type u_5
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
f : α → E
hf : f ≤ᵐ[μ] 0
hfint : ¬Integrable f
⊢ μ[f|m] ≤ᵐ[μ] 0

Next tactic:
rw [condexp_undef hfint]
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
    (hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
    (hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
    (hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
    (h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
    (h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
    (hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
    μ[f|m] =ᵐ[μ] μ[g|m] := by
  

Proof state:
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
fs gs : ℕ → α → F'
f g : α → F'
hfs_int : ∀ (n : ℕ), Integrable (fs n)
hgs_int : ∀ (n : ℕ), Integrable (gs n)
hfs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))
hgs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))
bound_fs : α → ℝ
h_int_bound_fs : Integrable bound_fs
bound_gs : α → ℝ
h_int_bound_gs : Integrable bound_gs
hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x
hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x
hfg : ∀ (n : ℕ), μ[fs n|m] =ᵐ[μ] μ[gs n|m]
⊢ μ[f|m] =ᵐ[μ] μ[g|m]

Next tactic:
by_cases hm : m ≤ m0
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
    (hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
    (hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
    (hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
    (h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
    (h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
    (hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
    μ[f|m] =ᵐ[μ] μ[g|m] := by
  by_cases hm : m ≤ m0; 

Proof state:
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
fs gs : ℕ → α → F'
f g : α → F'
hfs_int : ∀ (n : ℕ), Integrable (fs n)
hgs_int : ∀ (n : ℕ), Integrable (gs n)
hfs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))
hgs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))
bound_fs : α → ℝ
h_int_bound_fs : Integrable bound_fs
bound_gs : α → ℝ
h_int_bound_gs : Integrable bound_gs
hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x
hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x
hfg : ∀ (n : ℕ), μ[fs n|m] =ᵐ[μ] μ[gs n|m]
hm : m ≤ m0
⊢ μ[f|m] =ᵐ[μ] μ[g|m]
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
fs gs : ℕ → α → F'
f g : α → F'
hfs_int : ∀ (n : ℕ), Integrable (fs n)
hgs_int : ∀ (n : ℕ), Integrable (gs n)
hfs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))
hgs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))
bound_fs : α → ℝ
h_int_bound_fs : Integrable bound_fs
bound_gs : α → ℝ
h_int_bound_gs : Integrable bound_gs
hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x
hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x
hfg : ∀ (n : ℕ), μ[fs n|m] =ᵐ[μ] μ[gs n|m]
hm : ¬m ≤ m0
⊢ μ[f|m] =ᵐ[μ] μ[g|m]

Next tactic:
swap
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
    (hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
    (hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
    (hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
    (h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
    (h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
    (hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
    μ[f|m] =ᵐ[μ] μ[g|m] := by
  by_cases hm : m ≤ m0; swap; · 

Proof state:
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
fs gs : ℕ → α → F'
f g : α → F'
hfs_int : ∀ (n : ℕ), Integrable (fs n)
hgs_int : ∀ (n : ℕ), Integrable (gs n)
hfs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))
hgs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))
bound_fs : α → ℝ
h_int_bound_fs : Integrable bound_fs
bound_gs : α → ℝ
h_int_bound_gs : Integrable bound_gs
hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x
hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x
hfg : ∀ (n : ℕ), μ[fs n|m] =ᵐ[μ] μ[gs n|m]
hm : ¬m ≤ m0
⊢ μ[f|m] =ᵐ[μ] μ[g|m]

Next tactic:
simp_rw [condexp_of_not_le hm]
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
    (hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
    (hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
    (hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
    (h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
    (h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
    (hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
    μ[f|m] =ᵐ[μ] μ[g|m] := by
  by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; 

Proof state:
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
fs gs : ℕ → α → F'
f g : α → F'
hfs_int : ∀ (n : ℕ), Integrable (fs n)
hgs_int : ∀ (n : ℕ), Integrable (gs n)
hfs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))
hgs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))
bound_fs : α → ℝ
h_int_bound_fs : Integrable bound_fs
bound_gs : α → ℝ
h_int_bound_gs : Integrable bound_gs
hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x
hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x
hfg : ∀ (n : ℕ), μ[fs n|m] =ᵐ[μ] μ[gs n|m]
hm : ¬m ≤ m0
⊢ 0 =ᵐ[μ] 0

Next tactic:
rfl
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
    (hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
    (hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
    (hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
    (h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
    (h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
    (hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
    μ[f|m] =ᵐ[μ] μ[g|m] := by
  by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; rfl
  

Proof state:
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
fs gs : ℕ → α → F'
f g : α → F'
hfs_int : ∀ (n : ℕ), Integrable (fs n)
hgs_int : ∀ (n : ℕ), Integrable (gs n)
hfs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))
hgs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))
bound_fs : α → ℝ
h_int_bound_fs : Integrable bound_fs
bound_gs : α → ℝ
h_int_bound_gs : Integrable bound_gs
hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x
hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x
hfg : ∀ (n : ℕ), μ[fs n|m] =ᵐ[μ] μ[gs n|m]
hm : m ≤ m0
⊢ μ[f|m] =ᵐ[μ] μ[g|m]

Next tactic:
by_cases hμm : SigmaFinite (μ.trim hm)
/-- If two sequences of functions have a.e. equal conditional expectations at each step, converge
and verify dominated convergence hypotheses, then the conditional expectations of their limits are
a.e. equal. -/
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
    (hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
    (hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
    (hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
    (h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
    (h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
    (hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
    μ[f|m] =ᵐ[μ] μ[g|m] := by
  by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; rfl
  by_cases hμm : SigmaFinite (μ.trim hm); 

Proof state:
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
fs gs : ℕ → α → F'
f g : α → F'
hfs_int : ∀ (n : ℕ), Integrable (fs n)
hgs_int : ∀ (n : ℕ), Integrable (gs n)
hfs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))
hgs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))
bound_fs : α → ℝ
h_int_bound_fs : Integrable bound_fs
bound_gs : α → ℝ
h_int_bound_gs : Integrable bound_gs
hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x
hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x
hfg : ∀ (n : ℕ), μ[fs n|m] =ᵐ[μ] μ[gs n|m]
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
⊢ μ[f|m] =ᵐ[μ] μ[g|m]
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g✝ : α → F'
s : Set α
fs gs : ℕ → α → F'
f g : α → F'
hfs_int : ∀ (n : ℕ), Integrable (fs n)
hgs_int : ∀ (n : ℕ), Integrable (gs n)
hfs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))
hgs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))
bound_fs : α → ℝ
h_int_bound_fs : Integrable bound_fs
bound_gs : α → ℝ
h_int_bound_gs : Integrable bound_gs
hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x
hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x
hfg : ∀ (n : ℕ), μ[fs n|m] =ᵐ[μ] μ[gs n|m]
hm : m ≤ m0
hμm : ¬SigmaFinite (Measure.trim μ hm)
⊢ μ[f|m] =ᵐ[μ] μ[g|m]

Next tactic:
swap