l3lab/ntp-mathlib huggingface dataset:
/-- The **measurable Schröder-Bernstein Theorem**: given measurable embeddings
`α → β` and `β → α`, we can find a measurable equivalence `α ≃ᵐ β`.-/
noncomputable def schroederBernstein {f : α → β} {g : β → α} (hf : MeasurableEmbedding f)
    (hg : MeasurableEmbedding g) : α ≃ᵐ β := by
  let F : Set α → Set α := fun A => (g '' (f '' A)ᶜ)ᶜ
  -- We follow the proof of the usual SB theorem in mathlib,
  -- the crux of which is finding a fixed point of this F.
  -- However, we must find this fixed point manually instead of invoking Knaster-Tarski
  -- in order to make sure it is measurable.
  suffices Σ'A : Set α, MeasurableSet A ∧ F A = A by
    

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
f✝ : α → β
g✝ : β → α
f : α → β
g : β → α
hf : MeasurableEmbedding f
hg : MeasurableEmbedding g
F : Set α → Set α := fun A => (g '' (f '' A)ᶜ)ᶜ
this : (A : Set α) ×' MeasurableSet A ∧ F A = A
⊢ α ≃ᵐ β

Next tactic:
rcases this with ⟨A, Ameas, Afp⟩
/-- The **measurable Schröder-Bernstein Theorem**: given measurable embeddings
`α → β` and `β → α`, we can find a measurable equivalence `α ≃ᵐ β`.-/
noncomputable def schroederBernstein {f : α → β} {g : β → α} (hf : MeasurableEmbedding f)
    (hg : MeasurableEmbedding g) : α ≃ᵐ β := by
  let F : Set α → Set α := fun A => (g '' (f '' A)ᶜ)ᶜ
  -- We follow the proof of the usual SB theorem in mathlib,
  -- the crux of which is finding a fixed point of this F.
  -- However, we must find this fixed point manually instead of invoking Knaster-Tarski
  -- in order to make sure it is measurable.
  suffices Σ'A : Set α, MeasurableSet A ∧ F A = A by
    rcases this with ⟨A, Ameas, Afp⟩
    

Proof state:
case mk.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
f✝ : α → β
g✝ : β → α
f : α → β
g : β → α
hf : MeasurableEmbedding f
hg : MeasurableEmbedding g
F : Set α → Set α := fun A => (g '' (f '' A)ᶜ)ᶜ
A : Set α
Ameas : MeasurableSet A
Afp : F A = A
⊢ α ≃ᵐ β

Next tactic:
let B := f '' A
/-- The **measurable Schröder-Bernstein Theorem**: given measurable embeddings
`α → β` and `β → α`, we can find a measurable equivalence `α ≃ᵐ β`.-/
noncomputable def schroederBernstein {f : α → β} {g : β → α} (hf : MeasurableEmbedding f)
    (hg : MeasurableEmbedding g) : α ≃ᵐ β := by
  let F : Set α → Set α := fun A => (g '' (f '' A)ᶜ)ᶜ
  -- We follow the proof of the usual SB theorem in mathlib,
  -- the crux of which is finding a fixed point of this F.
  -- However, we must find this fixed point manually instead of invoking Knaster-Tarski
  -- in order to make sure it is measurable.
  suffices Σ'A : Set α, MeasurableSet A ∧ F A = A by
    rcases this with ⟨A, Ameas, Afp⟩
    let B := f '' A
    

Proof state:
case mk.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
f✝ : α → β
g✝ : β → α
f : α → β
g : β → α
hf : MeasurableEmbedding f
hg : MeasurableEmbedding g
F : Set α → Set α := fun A => (g '' (f '' A)ᶜ)ᶜ
A : Set α
Ameas : MeasurableSet A
Afp : F A = A
B : Set β := f '' A
⊢ α ≃ᵐ β

Next tactic:
have Bmeas : MeasurableSet B := hf.measurableSet_image' Ameas
/-- The **measurable Schröder-Bernstein Theorem**: given measurable embeddings
`α → β` and `β → α`, we can find a measurable equivalence `α ≃ᵐ β`.-/
noncomputable def schroederBernstein {f : α → β} {g : β → α} (hf : MeasurableEmbedding f)
    (hg : MeasurableEmbedding g) : α ≃ᵐ β := by
  let F : Set α → Set α := fun A => (g '' (f '' A)ᶜ)ᶜ
  -- We follow the proof of the usual SB theorem in mathlib,
  -- the crux of which is finding a fixed point of this F.
  -- However, we must find this fixed point manually instead of invoking Knaster-Tarski
  -- in order to make sure it is measurable.
  suffices Σ'A : Set α, MeasurableSet A ∧ F A = A by
    rcases this with ⟨A, Ameas, Afp⟩
    let B := f '' A
    have Bmeas : MeasurableSet B := hf.measurableSet_image' Ameas
    

Proof state:
case mk.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
f✝ : α → β
g✝ : β → α
f : α → β
g : β → α
hf : MeasurableEmbedding f
hg : MeasurableEmbedding g
F : Set α → Set α := fun A => (g '' (f '' A)ᶜ)ᶜ
A : Set α
Ameas : MeasurableSet A
Afp : F A = A
B : Set β := f '' A
Bmeas : MeasurableSet B
⊢ α ≃ᵐ β

Next tactic:
refine'
      (MeasurableEquiv.sumCompl Ameas).symm.trans
        (MeasurableEquiv.trans _ (MeasurableEquiv.sumCompl Bmeas))
/-- The **measurable Schröder-Bernstein Theorem**: given measurable embeddings
`α → β` and `β → α`, we can find a measurable equivalence `α ≃ᵐ β`.-/
noncomputable def schroederBernstein {f : α → β} {g : β → α} (hf : MeasurableEmbedding f)
    (hg : MeasurableEmbedding g) : α ≃ᵐ β := by
  let F : Set α → Set α := fun A => (g '' (f '' A)ᶜ)ᶜ
  -- We follow the proof of the usual SB theorem in mathlib,
  -- the crux of which is finding a fixed point of this F.
  -- However, we must find this fixed point manually instead of invoking Knaster-Tarski
  -- in order to make sure it is measurable.
  suffices Σ'A : Set α, MeasurableSet A ∧ F A = A by
    rcases this with ⟨A, Ameas, Afp⟩
    let B := f '' A
    have Bmeas : MeasurableSet B := hf.measurableSet_image' Ameas
    refine'
      (MeasurableEquiv.sumCompl Ameas).symm.trans
        (MeasurableEquiv.trans _ (MeasurableEquiv.sumCompl Bmeas))
    

Proof state:
case mk.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
f✝ : α → β
g✝ : β → α
f : α → β
g : β → α
hf : MeasurableEmbedding f
hg : MeasurableEmbedding g
F : Set α → Set α := fun A => (g '' (f '' A)ᶜ)ᶜ
A : Set α
Ameas : MeasurableSet A
Afp : F A = A
B : Set β := f '' A
Bmeas : MeasurableSet B
⊢ ↑A ⊕ ↑Aᶜ ≃ᵐ ↑B ⊕ ↑Bᶜ

Next tactic:
apply MeasurableEquiv.sumCongr (hf.equivImage _)
/-- The **measurable Schröder-Bernstein Theorem**: given measurable embeddings
`α → β` and `β → α`, we can find a measurable equivalence `α ≃ᵐ β`.-/
noncomputable def schroederBernstein {f : α → β} {g : β → α} (hf : MeasurableEmbedding f)
    (hg : MeasurableEmbedding g) : α ≃ᵐ β := by
  let F : Set α → Set α := fun A => (g '' (f '' A)ᶜ)ᶜ
  -- We follow the proof of the usual SB theorem in mathlib,
  -- the crux of which is finding a fixed point of this F.
  -- However, we must find this fixed point manually instead of invoking Knaster-Tarski
  -- in order to make sure it is measurable.
  suffices Σ'A : Set α, MeasurableSet A ∧ F A = A by
    rcases this with ⟨A, Ameas, Afp⟩
    let B := f '' A
    have Bmeas : MeasurableSet B := hf.measurableSet_image' Ameas
    refine'
      (MeasurableEquiv.sumCompl Ameas).symm.trans
        (MeasurableEquiv.trans _ (MeasurableEquiv.sumCompl Bmeas))
    apply MeasurableEquiv.sumCongr (hf.equivImage _)
    

Proof state:
case mk.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
f✝ : α → β
g✝ : β → α
f : α → β
g : β → α
hf : MeasurableEmbedding f
hg : MeasurableEmbedding g
F : Set α → Set α := fun A => (g '' (f '' A)ᶜ)ᶜ
A : Set α
Ameas : MeasurableSet A
Afp : F A = A
B : Set β := f '' A
Bmeas : MeasurableSet B
⊢ ↑Aᶜ ≃ᵐ ↑Bᶜ

Next tactic:
have : Aᶜ = g '' Bᶜ := by
      apply compl_injective
      rw [← Afp]
      simp
/-- The **measurable Schröder-Bernstein Theorem**: given measurable embeddings
`α → β` and `β → α`, we can find a measurable equivalence `α ≃ᵐ β`.-/
noncomputable def schroederBernstein {f : α → β} {g : β → α} (hf : MeasurableEmbedding f)
    (hg : MeasurableEmbedding g) : α ≃ᵐ β := by
  let F : Set α → Set α := fun A => (g '' (f '' A)ᶜ)ᶜ
  -- We follow the proof of the usual SB theorem in mathlib,
  -- the crux of which is finding a fixed point of this F.
  -- However, we must find this fixed point manually instead of invoking Knaster-Tarski
  -- in order to make sure it is measurable.
  suffices Σ'A : Set α, MeasurableSet A ∧ F A = A by
    rcases this with ⟨A, Ameas, Afp⟩
    let B := f '' A
    have Bmeas : MeasurableSet B := hf.measurableSet_image' Ameas
    refine'
      (MeasurableEquiv.sumCompl Ameas).symm.trans
        (MeasurableEquiv.trans _ (MeasurableEquiv.sumCompl Bmeas))
    apply MeasurableEquiv.sumCongr (hf.equivImage _)
    have : Aᶜ = g '' Bᶜ := by
      

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
f✝ : α → β
g✝ : β → α
f : α → β
g : β → α
hf : MeasurableEmbedding f
hg : MeasurableEmbedding g
F : Set α → Set α := fun A => (g '' (f '' A)ᶜ)ᶜ
A : Set α
Ameas : MeasurableSet A
Afp : F A = A
B : Set β := f '' A
Bmeas : MeasurableSet B
⊢ Aᶜ = g '' Bᶜ

Next tactic:
apply compl_injective
/-- The **measurable Schröder-Bernstein Theorem**: given measurable embeddings
`α → β` and `β → α`, we can find a measurable equivalence `α ≃ᵐ β`.-/
noncomputable def schroederBernstein {f : α → β} {g : β → α} (hf : MeasurableEmbedding f)
    (hg : MeasurableEmbedding g) : α ≃ᵐ β := by
  let F : Set α → Set α := fun A => (g '' (f '' A)ᶜ)ᶜ
  -- We follow the proof of the usual SB theorem in mathlib,
  -- the crux of which is finding a fixed point of this F.
  -- However, we must find this fixed point manually instead of invoking Knaster-Tarski
  -- in order to make sure it is measurable.
  suffices Σ'A : Set α, MeasurableSet A ∧ F A = A by
    rcases this with ⟨A, Ameas, Afp⟩
    let B := f '' A
    have Bmeas : MeasurableSet B := hf.measurableSet_image' Ameas
    refine'
      (MeasurableEquiv.sumCompl Ameas).symm.trans
        (MeasurableEquiv.trans _ (MeasurableEquiv.sumCompl Bmeas))
    apply MeasurableEquiv.sumCongr (hf.equivImage _)
    have : Aᶜ = g '' Bᶜ := by
      apply compl_injective
      

Proof state:
case a
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
f✝ : α → β
g✝ : β → α
f : α → β
g : β → α
hf : MeasurableEmbedding f
hg : MeasurableEmbedding g
F : Set α → Set α := fun A => (g '' (f '' A)ᶜ)ᶜ
A : Set α
Ameas : MeasurableSet A
Afp : F A = A
B : Set β := f '' A
Bmeas : MeasurableSet B
⊢ Aᶜᶜ = (g '' Bᶜ)ᶜ

Next tactic:
rw [← Afp]
/-- The **measurable Schröder-Bernstein Theorem**: given measurable embeddings
`α → β` and `β → α`, we can find a measurable equivalence `α ≃ᵐ β`.-/
noncomputable def schroederBernstein {f : α → β} {g : β → α} (hf : MeasurableEmbedding f)
    (hg : MeasurableEmbedding g) : α ≃ᵐ β := by
  let F : Set α → Set α := fun A => (g '' (f '' A)ᶜ)ᶜ
  -- We follow the proof of the usual SB theorem in mathlib,
  -- the crux of which is finding a fixed point of this F.
  -- However, we must find this fixed point manually instead of invoking Knaster-Tarski
  -- in order to make sure it is measurable.
  suffices Σ'A : Set α, MeasurableSet A ∧ F A = A by
    rcases this with ⟨A, Ameas, Afp⟩
    let B := f '' A
    have Bmeas : MeasurableSet B := hf.measurableSet_image' Ameas
    refine'
      (MeasurableEquiv.sumCompl Ameas).symm.trans
        (MeasurableEquiv.trans _ (MeasurableEquiv.sumCompl Bmeas))
    apply MeasurableEquiv.sumCongr (hf.equivImage _)
    have : Aᶜ = g '' Bᶜ := by
      apply compl_injective
      rw [← Afp]
      

Proof state:
case a
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
f✝ : α → β
g✝ : β → α
f : α → β
g : β → α
hf : MeasurableEmbedding f
hg : MeasurableEmbedding g
F : Set α → Set α := fun A => (g '' (f '' A)ᶜ)ᶜ
A : Set α
Ameas : MeasurableSet A
Afp : F A = A
B : Set β := f '' A
Bmeas : MeasurableSet B
⊢ (F A)ᶜᶜ = (g '' Bᶜ)ᶜ

Next tactic:
simp
/-- The **measurable Schröder-Bernstein Theorem**: given measurable embeddings
`α → β` and `β → α`, we can find a measurable equivalence `α ≃ᵐ β`.-/
noncomputable def schroederBernstein {f : α → β} {g : β → α} (hf : MeasurableEmbedding f)
    (hg : MeasurableEmbedding g) : α ≃ᵐ β := by
  let F : Set α → Set α := fun A => (g '' (f '' A)ᶜ)ᶜ
  -- We follow the proof of the usual SB theorem in mathlib,
  -- the crux of which is finding a fixed point of this F.
  -- However, we must find this fixed point manually instead of invoking Knaster-Tarski
  -- in order to make sure it is measurable.
  suffices Σ'A : Set α, MeasurableSet A ∧ F A = A by
    rcases this with ⟨A, Ameas, Afp⟩
    let B := f '' A
    have Bmeas : MeasurableSet B := hf.measurableSet_image' Ameas
    refine'
      (MeasurableEquiv.sumCompl Ameas).symm.trans
        (MeasurableEquiv.trans _ (MeasurableEquiv.sumCompl Bmeas))
    apply MeasurableEquiv.sumCongr (hf.equivImage _)
    have : Aᶜ = g '' Bᶜ := by
      apply compl_injective
      rw [← Afp]
      simp
    

Proof state:
case mk.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
f✝ : α → β
g✝ : β → α
f : α → β
g : β → α
hf : MeasurableEmbedding f
hg : MeasurableEmbedding g
F : Set α → Set α := fun A => (g '' (f '' A)ᶜ)ᶜ
A : Set α
Ameas : MeasurableSet A
Afp : F A = A
B : Set β := f '' A
Bmeas : MeasurableSet B
this : Aᶜ = g '' Bᶜ
⊢ ↑Aᶜ ≃ᵐ ↑Bᶜ

Next tactic:
rw [this]
/-- The **measurable Schröder-Bernstein Theorem**: given measurable embeddings
`α → β` and `β → α`, we can find a measurable equivalence `α ≃ᵐ β`.-/
noncomputable def schroederBernstein {f : α → β} {g : β → α} (hf : MeasurableEmbedding f)
    (hg : MeasurableEmbedding g) : α ≃ᵐ β := by
  let F : Set α → Set α := fun A => (g '' (f '' A)ᶜ)ᶜ
  -- We follow the proof of the usual SB theorem in mathlib,
  -- the crux of which is finding a fixed point of this F.
  -- However, we must find this fixed point manually instead of invoking Knaster-Tarski
  -- in order to make sure it is measurable.
  suffices Σ'A : Set α, MeasurableSet A ∧ F A = A by
    rcases this with ⟨A, Ameas, Afp⟩
    let B := f '' A
    have Bmeas : MeasurableSet B := hf.measurableSet_image' Ameas
    refine'
      (MeasurableEquiv.sumCompl Ameas).symm.trans
        (MeasurableEquiv.trans _ (MeasurableEquiv.sumCompl Bmeas))
    apply MeasurableEquiv.sumCongr (hf.equivImage _)
    have : Aᶜ = g '' Bᶜ := by
      apply compl_injective
      rw [← Afp]
      simp
    rw [this]
    

Proof state:
case mk.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
f✝ : α → β
g✝ : β → α
f : α → β
g : β → α
hf : MeasurableEmbedding f
hg : MeasurableEmbedding g
F : Set α → Set α := fun A => (g '' (f '' A)ᶜ)ᶜ
A : Set α
Ameas : MeasurableSet A
Afp : F A = A
B : Set β := f '' A
Bmeas : MeasurableSet B
this : Aᶜ = g '' Bᶜ
⊢ ↑(g '' Bᶜ) ≃ᵐ ↑Bᶜ

Next tactic:
exact (hg.equivImage _).symm
/-- The **measurable Schröder-Bernstein Theorem**: given measurable embeddings
`α → β` and `β → α`, we can find a measurable equivalence `α ≃ᵐ β`.-/
noncomputable def schroederBernstein {f : α → β} {g : β → α} (hf : MeasurableEmbedding f)
    (hg : MeasurableEmbedding g) : α ≃ᵐ β := by
  let F : Set α → Set α := fun A => (g '' (f '' A)ᶜ)ᶜ
  -- We follow the proof of the usual SB theorem in mathlib,
  -- the crux of which is finding a fixed point of this F.
  -- However, we must find this fixed point manually instead of invoking Knaster-Tarski
  -- in order to make sure it is measurable.
  suffices Σ'A : Set α, MeasurableSet A ∧ F A = A by
    rcases this with ⟨A, Ameas, Afp⟩
    let B := f '' A
    have Bmeas : MeasurableSet B := hf.measurableSet_image' Ameas
    refine'
      (MeasurableEquiv.sumCompl Ameas).symm.trans
        (MeasurableEquiv.trans _ (MeasurableEquiv.sumCompl Bmeas))
    apply MeasurableEquiv.sumCongr (hf.equivImage _)
    have : Aᶜ = g '' Bᶜ := by
      apply compl_injective
      rw [← Afp]
      simp
    rw [this]
    exact (hg.equivImage _).symm
  

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
f✝ : α → β
g✝ : β → α
f : α → β
g : β → α
hf : MeasurableEmbedding f
hg : MeasurableEmbedding g
F : Set α → Set α := fun A => (g '' (f '' A)ᶜ)ᶜ
⊢ (A : Set α) ×' MeasurableSet A ∧ F A = A

Next tactic:
have Fmono : ∀ {A B}, A ⊆ B → F A ⊆ F B := fun h =>
    compl_subset_compl.mpr <| Set.image_subset _ <| compl_subset_compl.mpr <| Set.image_subset _ h
/-- The **measurable Schröder-Bernstein Theorem**: given measurable embeddings
`α → β` and `β → α`, we can find a measurable equivalence `α ≃ᵐ β`.-/
noncomputable def schroederBernstein {f : α → β} {g : β → α} (hf : MeasurableEmbedding f)
    (hg : MeasurableEmbedding g) : α ≃ᵐ β := by
  let F : Set α → Set α := fun A => (g '' (f '' A)ᶜ)ᶜ
  -- We follow the proof of the usual SB theorem in mathlib,
  -- the crux of which is finding a fixed point of this F.
  -- However, we must find this fixed point manually instead of invoking Knaster-Tarski
  -- in order to make sure it is measurable.
  suffices Σ'A : Set α, MeasurableSet A ∧ F A = A by
    rcases this with ⟨A, Ameas, Afp⟩
    let B := f '' A
    have Bmeas : MeasurableSet B := hf.measurableSet_image' Ameas
    refine'
      (MeasurableEquiv.sumCompl Ameas).symm.trans
        (MeasurableEquiv.trans _ (MeasurableEquiv.sumCompl Bmeas))
    apply MeasurableEquiv.sumCongr (hf.equivImage _)
    have : Aᶜ = g '' Bᶜ := by
      apply compl_injective
      rw [← Afp]
      simp
    rw [this]
    exact (hg.equivImage _).symm
  have Fmono : ∀ {A B}, A ⊆ B → F A ⊆ F B := fun h =>
    compl_subset_compl.mpr <| Set.image_subset _ <| compl_subset_compl.mpr <| Set.image_subset _ h
  

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
f✝ : α → β
g✝ : β → α
f : α → β
g : β → α
hf : MeasurableEmbedding f
hg : MeasurableEmbedding g
F : Set α → Set α := fun A => (g '' (f '' A)ᶜ)ᶜ
Fmono : ∀ {A B : Set α}, A ⊆ B → F A ⊆ F B
⊢ (A : Set α) ×' MeasurableSet A ∧ F A = A

Next tactic:
let X : ℕ → Set α := fun n => F^[n] univ
/-- The **measurable Schröder-Bernstein Theorem**: given measurable embeddings
`α → β` and `β → α`, we can find a measurable equivalence `α ≃ᵐ β`.-/
noncomputable def schroederBernstein {f : α → β} {g : β → α} (hf : MeasurableEmbedding f)
    (hg : MeasurableEmbedding g) : α ≃ᵐ β := by
  let F : Set α → Set α := fun A => (g '' (f '' A)ᶜ)ᶜ
  -- We follow the proof of the usual SB theorem in mathlib,
  -- the crux of which is finding a fixed point of this F.
  -- However, we must find this fixed point manually instead of invoking Knaster-Tarski
  -- in order to make sure it is measurable.
  suffices Σ'A : Set α, MeasurableSet A ∧ F A = A by
    rcases this with ⟨A, Ameas, Afp⟩
    let B := f '' A
    have Bmeas : MeasurableSet B := hf.measurableSet_image' Ameas
    refine'
      (MeasurableEquiv.sumCompl Ameas).symm.trans
        (MeasurableEquiv.trans _ (MeasurableEquiv.sumCompl Bmeas))
    apply MeasurableEquiv.sumCongr (hf.equivImage _)
    have : Aᶜ = g '' Bᶜ := by
      apply compl_injective
      rw [← Afp]
      simp
    rw [this]
    exact (hg.equivImage _).symm
  have Fmono : ∀ {A B}, A ⊆ B → F A ⊆ F B := fun h =>
    compl_subset_compl.mpr <| Set.image_subset _ <| compl_subset_compl.mpr <| Set.image_subset _ h
  let X : ℕ → Set α := fun n => F^[n] univ
  

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
f✝ : α → β
g✝ : β → α
f : α → β
g : β → α
hf : MeasurableEmbedding f
hg : MeasurableEmbedding g
F : Set α → Set α := fun A => (g '' (f '' A)ᶜ)ᶜ
Fmono : ∀ {A B : Set α}, A ⊆ B → F A ⊆ F B
X : ℕ → Set α := fun n => F^[n] univ
⊢ (A : Set α) ×' MeasurableSet A ∧ F A = A

Next tactic:
refine' ⟨iInter X, _, _⟩
/-- The **measurable Schröder-Bernstein Theorem**: given measurable embeddings
`α → β` and `β → α`, we can find a measurable equivalence `α ≃ᵐ β`.-/
noncomputable def schroederBernstein {f : α → β} {g : β → α} (hf : MeasurableEmbedding f)
    (hg : MeasurableEmbedding g) : α ≃ᵐ β := by
  let F : Set α → Set α := fun A => (g '' (f '' A)ᶜ)ᶜ
  -- We follow the proof of the usual SB theorem in mathlib,
  -- the crux of which is finding a fixed point of this F.
  -- However, we must find this fixed point manually instead of invoking Knaster-Tarski
  -- in order to make sure it is measurable.
  suffices Σ'A : Set α, MeasurableSet A ∧ F A = A by
    rcases this with ⟨A, Ameas, Afp⟩
    let B := f '' A
    have Bmeas : MeasurableSet B := hf.measurableSet_image' Ameas
    refine'
      (MeasurableEquiv.sumCompl Ameas).symm.trans
        (MeasurableEquiv.trans _ (MeasurableEquiv.sumCompl Bmeas))
    apply MeasurableEquiv.sumCongr (hf.equivImage _)
    have : Aᶜ = g '' Bᶜ := by
      apply compl_injective
      rw [← Afp]
      simp
    rw [this]
    exact (hg.equivImage _).symm
  have Fmono : ∀ {A B}, A ⊆ B → F A ⊆ F B := fun h =>
    compl_subset_compl.mpr <| Set.image_subset _ <| compl_subset_compl.mpr <| Set.image_subset _ h
  let X : ℕ → Set α := fun n => F^[n] univ
  refine' ⟨iInter X, _, _⟩
  · 

Proof state:
case refine'_1
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
f✝ : α → β
g✝ : β → α
f : α → β
g : β → α
hf : MeasurableEmbedding f
hg : MeasurableEmbedding g
F : Set α → Set α := fun A => (g '' (f '' A)ᶜ)ᶜ
Fmono : ∀ {A B : Set α}, A ⊆ B → F A ⊆ F B
X : ℕ → Set α := fun n => F^[n] univ
⊢ MeasurableSet (iInter X)

Next tactic:
apply MeasurableSet.iInter
/-- The **measurable Schröder-Bernstein Theorem**: given measurable embeddings
`α → β` and `β → α`, we can find a measurable equivalence `α ≃ᵐ β`.-/
noncomputable def schroederBernstein {f : α → β} {g : β → α} (hf : MeasurableEmbedding f)
    (hg : MeasurableEmbedding g) : α ≃ᵐ β := by
  let F : Set α → Set α := fun A => (g '' (f '' A)ᶜ)ᶜ
  -- We follow the proof of the usual SB theorem in mathlib,
  -- the crux of which is finding a fixed point of this F.
  -- However, we must find this fixed point manually instead of invoking Knaster-Tarski
  -- in order to make sure it is measurable.
  suffices Σ'A : Set α, MeasurableSet A ∧ F A = A by
    rcases this with ⟨A, Ameas, Afp⟩
    let B := f '' A
    have Bmeas : MeasurableSet B := hf.measurableSet_image' Ameas
    refine'
      (MeasurableEquiv.sumCompl Ameas).symm.trans
        (MeasurableEquiv.trans _ (MeasurableEquiv.sumCompl Bmeas))
    apply MeasurableEquiv.sumCongr (hf.equivImage _)
    have : Aᶜ = g '' Bᶜ := by
      apply compl_injective
      rw [← Afp]
      simp
    rw [this]
    exact (hg.equivImage _).symm
  have Fmono : ∀ {A B}, A ⊆ B → F A ⊆ F B := fun h =>
    compl_subset_compl.mpr <| Set.image_subset _ <| compl_subset_compl.mpr <| Set.image_subset _ h
  let X : ℕ → Set α := fun n => F^[n] univ
  refine' ⟨iInter X, _, _⟩
  · apply MeasurableSet.iInter
    

Proof state:
case refine'_1.h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
f✝ : α → β
g✝ : β → α
f : α → β
g : β → α
hf : MeasurableEmbedding f
hg : MeasurableEmbedding g
F : Set α → Set α := fun A => (g '' (f '' A)ᶜ)ᶜ
Fmono : ∀ {A B : Set α}, A ⊆ B → F A ⊆ F B
X : ℕ → Set α := fun n => F^[n] univ
⊢ ∀ (b : ℕ), MeasurableSet (F^[b] univ)

Next tactic:
intro n