[GOAL]
a✝ : (Quotient.out (ord (aleph 1))).α
i : (Quotient.out (ord (aleph 1))).α := a✝
j : ↑(Iio i)
⊢ (invImage (fun a => a) (hasWellFoundedOut (ord (aleph 1)))).1 (↑j) a✝
[PROOFSTEP]
exact j.2
[GOAL]
α : Type u
s : Set (Set α)
i : (Quotient.out (ord (aleph 1))).α
⊢ s ⊆ generateMeasurableRec s i
[PROOFSTEP]
unfold generateMeasurableRec
[GOAL]
α : Type u
s : Set (Set α)
i : (Quotient.out (ord (aleph 1))).α
⊢ s ⊆
    let i := i;
    let S := ⋃ (j : ↑(Iio i)), generateMeasurableRec s ↑j;
    s ∪ {∅} ∪ compl '' S ∪ range fun f => ⋃ (n : ℕ), ↑(f n)
[PROOFSTEP]
apply_rules [subset_union_of_subset_left]
[GOAL]
case h.h.h
α : Type u
s : Set (Set α)
i : (Quotient.out (ord (aleph 1))).α
⊢ s ⊆ s
[PROOFSTEP]
exact subset_rfl
[GOAL]
α : Type u
s : Set (Set α)
i : (Quotient.out (ord (aleph 1))).α
⊢ ∅ ∈ generateMeasurableRec s i
[PROOFSTEP]
unfold generateMeasurableRec
[GOAL]
α : Type u
s : Set (Set α)
i : (Quotient.out (ord (aleph 1))).α
⊢ ∅ ∈
    let i := i;
    let S := ⋃ (j : ↑(Iio i)), generateMeasurableRec s ↑j;
    s ∪ {∅} ∪ compl '' S ∪ range fun f => ⋃ (n : ℕ), ↑(f n)
[PROOFSTEP]
exact mem_union_left _ (mem_union_left _ (mem_union_right _ (mem_singleton ∅)))
[GOAL]
α : Type u
s : Set (Set α)
i j : (Quotient.out (ord (aleph 1))).α
h : j < i
t : Set α
ht : t ∈ generateMeasurableRec s j
⊢ tᶜ ∈ generateMeasurableRec s i
[PROOFSTEP]
unfold generateMeasurableRec
[GOAL]
α : Type u
s : Set (Set α)
i j : (Quotient.out (ord (aleph 1))).α
h : j < i
t : Set α
ht : t ∈ generateMeasurableRec s j
⊢ tᶜ ∈
    let i := i;
    let S := ⋃ (j : ↑(Iio i)), generateMeasurableRec s ↑j;
    s ∪ {∅} ∪ compl '' S ∪ range fun f => ⋃ (n : ℕ), ↑(f n)
[PROOFSTEP]
exact mem_union_left _ (mem_union_right _ ⟨t, mem_iUnion.2 ⟨⟨j, h⟩, ht⟩, rfl⟩)
[GOAL]
α : Type u
s : Set (Set α)
i : (Quotient.out (ord (aleph 1))).α
f : ℕ → Set α
hf : ∀ (n : ℕ), ∃ j, j < i ∧ f n ∈ generateMeasurableRec s j
⊢ ⋃ (n : ℕ), f n ∈ generateMeasurableRec s i
[PROOFSTEP]
unfold generateMeasurableRec
[GOAL]
α : Type u
s : Set (Set α)
i : (Quotient.out (ord (aleph 1))).α
f : ℕ → Set α
hf : ∀ (n : ℕ), ∃ j, j < i ∧ f n ∈ generateMeasurableRec s j
⊢ ⋃ (n : ℕ), f n ∈
    let i := i;
    let S := ⋃ (j : ↑(Iio i)), generateMeasurableRec s ↑j;
    s ∪ {∅} ∪ compl '' S ∪ range fun f => ⋃ (n : ℕ), ↑(f n)
[PROOFSTEP]
exact
  mem_union_right _
    ⟨fun n =>
      ⟨f n,
        let ⟨j, hj, hf⟩ := hf n;
        mem_iUnion.2 ⟨⟨j, hj⟩, hf⟩⟩,
      rfl⟩
[GOAL]
α : Type u
s : Set (Set α)
i j : (Quotient.out (ord (aleph 1))).α
h : i ≤ j
x : Set α
hx : x ∈ generateMeasurableRec s i
⊢ x ∈ generateMeasurableRec s j
[PROOFSTEP]
rcases eq_or_lt_of_le h with (rfl | h)
[GOAL]
case inl
α : Type u
s : Set (Set α)
i : (Quotient.out (ord (aleph 1))).α
x : Set α
hx : x ∈ generateMeasurableRec s i
h : i ≤ i
⊢ x ∈ generateMeasurableRec s i
[PROOFSTEP]
exact hx
[GOAL]
case inr
α : Type u
s : Set (Set α)
i j : (Quotient.out (ord (aleph 1))).α
h✝ : i ≤ j
x : Set α
hx : x ∈ generateMeasurableRec s i
h : i < j
⊢ x ∈ generateMeasurableRec s j
[PROOFSTEP]
convert iUnion_mem_generateMeasurableRec fun _ => ⟨i, h, hx⟩
[GOAL]
case h.e'_4
α : Type u
s : Set (Set α)
i j : (Quotient.out (ord (aleph 1))).α
h✝ : i ≤ j
x : Set α
hx : x ∈ generateMeasurableRec s i
h : i < j
⊢ x = ⋃ (n : ℕ), x
[PROOFSTEP]
exact (iUnion_const x).symm
[GOAL]
α : Type u
s : Set (Set α)
i : (Quotient.out (ord (aleph 1))).α
⊢ #↑(generateMeasurableRec s i) ≤ max (#↑s) 2 ^ ℵ₀
[PROOFSTEP]
apply (aleph 1).ord.out.wo.wf.induction i
[GOAL]
α : Type u
s : Set (Set α)
i : (Quotient.out (ord (aleph 1))).α
⊢ ∀ (x : (Quotient.out (ord (aleph 1))).α),
    (∀ (y : (Quotient.out (ord (aleph 1))).α),
        WellOrder.r (Quotient.out (ord (aleph 1))) y x → #↑(generateMeasurableRec s y) ≤ max (#↑s) 2 ^ ℵ₀) →
      #↑(generateMeasurableRec s x) ≤ max (#↑s) 2 ^ ℵ₀
[PROOFSTEP]
intro i IH
[GOAL]
α : Type u
s : Set (Set α)
i✝ i : (Quotient.out (ord (aleph 1))).α
IH :
  ∀ (y : (Quotient.out (ord (aleph 1))).α),
    WellOrder.r (Quotient.out (ord (aleph 1))) y i → #↑(generateMeasurableRec s y) ≤ max (#↑s) 2 ^ ℵ₀
⊢ #↑(generateMeasurableRec s i) ≤ max (#↑s) 2 ^ ℵ₀
[PROOFSTEP]
have A := aleph0_le_aleph 1
[GOAL]
α : Type u
s : Set (Set α)
i✝ i : (Quotient.out (ord (aleph 1))).α
IH :
  ∀ (y : (Quotient.out (ord (aleph 1))).α),
    WellOrder.r (Quotient.out (ord (aleph 1))) y i → #↑(generateMeasurableRec s y) ≤ max (#↑s) 2 ^ ℵ₀
A : ℵ₀ ≤ aleph 1
⊢ #↑(generateMeasurableRec s i) ≤ max (#↑s) 2 ^ ℵ₀
[PROOFSTEP]
have B : aleph 1 ≤ max (#s) 2 ^ aleph0.{u} := aleph_one_le_continuum.trans (power_le_power_right (le_max_right _ _))
[GOAL]
α : Type u
s : Set (Set α)
i✝ i : (Quotient.out (ord (aleph 1))).α
IH :
  ∀ (y : (Quotient.out (ord (aleph 1))).α),
    WellOrder.r (Quotient.out (ord (aleph 1))) y i → #↑(generateMeasurableRec s y) ≤ max (#↑s) 2 ^ ℵ₀
A : ℵ₀ ≤ aleph 1
B : aleph 1 ≤ max (#↑s) 2 ^ ℵ₀
⊢ #↑(generateMeasurableRec s i) ≤ max (#↑s) 2 ^ ℵ₀
[PROOFSTEP]
have C : ℵ₀ ≤ max (#s) 2 ^ aleph0.{u} := A.trans B
[GOAL]
α : Type u
s : Set (Set α)
i✝ i : (Quotient.out (ord (aleph 1))).α
IH :
  ∀ (y : (Quotient.out (ord (aleph 1))).α),
    WellOrder.r (Quotient.out (ord (aleph 1))) y i → #↑(generateMeasurableRec s y) ≤ max (#↑s) 2 ^ ℵ₀
A : ℵ₀ ≤ aleph 1
B : aleph 1 ≤ max (#↑s) 2 ^ ℵ₀
C : ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀
⊢ #↑(generateMeasurableRec s i) ≤ max (#↑s) 2 ^ ℵ₀
[PROOFSTEP]
have J : #(⋃ j : Iio i, generateMeasurableRec s j.1) ≤ max (#s) 2 ^ aleph0.{u} :=
  by
  refine (mk_iUnion_le _).trans ?_
  have D : ⨆ j : Iio i, #(generateMeasurableRec s j) ≤ _ := ciSup_le' fun ⟨j, hj⟩ => IH j hj
  apply (mul_le_mul' ((mk_subtype_le _).trans (aleph 1).mk_ord_out.le) D).trans
  rw [mul_eq_max A C]
  exact max_le B le_rfl
[GOAL]
α : Type u
s : Set (Set α)
i✝ i : (Quotient.out (ord (aleph 1))).α
IH :
  ∀ (y : (Quotient.out (ord (aleph 1))).α),
    WellOrder.r (Quotient.out (ord (aleph 1))) y i → #↑(generateMeasurableRec s y) ≤ max (#↑s) 2 ^ ℵ₀
A : ℵ₀ ≤ aleph 1
B : aleph 1 ≤ max (#↑s) 2 ^ ℵ₀
C : ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀
⊢ #↑(⋃ (j : ↑(Iio i)), generateMeasurableRec s ↑j) ≤ max (#↑s) 2 ^ ℵ₀
[PROOFSTEP]
refine (mk_iUnion_le _).trans ?_
[GOAL]
α : Type u
s : Set (Set α)
i✝ i : (Quotient.out (ord (aleph 1))).α
IH :
  ∀ (y : (Quotient.out (ord (aleph 1))).α),
    WellOrder.r (Quotient.out (ord (aleph 1))) y i → #↑(generateMeasurableRec s y) ≤ max (#↑s) 2 ^ ℵ₀
A : ℵ₀ ≤ aleph 1
B : aleph 1 ≤ max (#↑s) 2 ^ ℵ₀
C : ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀
⊢ #↑(Iio i) * ⨆ (i_1 : ↑(Iio i)), #↑(generateMeasurableRec s ↑i_1) ≤ max (#↑s) 2 ^ ℵ₀
[PROOFSTEP]
have D : ⨆ j : Iio i, #(generateMeasurableRec s j) ≤ _ := ciSup_le' fun ⟨j, hj⟩ => IH j hj
[GOAL]
α : Type u
s : Set (Set α)
i✝ i : (Quotient.out (ord (aleph 1))).α
IH :
  ∀ (y : (Quotient.out (ord (aleph 1))).α),
    WellOrder.r (Quotient.out (ord (aleph 1))) y i → #↑(generateMeasurableRec s y) ≤ max (#↑s) 2 ^ ℵ₀
A : ℵ₀ ≤ aleph 1
B : aleph 1 ≤ max (#↑s) 2 ^ ℵ₀
C : ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀
D : ⨆ (j : ↑(Iio i)), #↑(generateMeasurableRec s ↑j) ≤ max (#↑s) 2 ^ ℵ₀
⊢ #↑(Iio i) * ⨆ (i_1 : ↑(Iio i)), #↑(generateMeasurableRec s ↑i_1) ≤ max (#↑s) 2 ^ ℵ₀
[PROOFSTEP]
apply (mul_le_mul' ((mk_subtype_le _).trans (aleph 1).mk_ord_out.le) D).trans
[GOAL]
α : Type u
s : Set (Set α)
i✝ i : (Quotient.out (ord (aleph 1))).α
IH :
  ∀ (y : (Quotient.out (ord (aleph 1))).α),
    WellOrder.r (Quotient.out (ord (aleph 1))) y i → #↑(generateMeasurableRec s y) ≤ max (#↑s) 2 ^ ℵ₀
A : ℵ₀ ≤ aleph 1
B : aleph 1 ≤ max (#↑s) 2 ^ ℵ₀
C : ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀
D : ⨆ (j : ↑(Iio i)), #↑(generateMeasurableRec s ↑j) ≤ max (#↑s) 2 ^ ℵ₀
⊢ aleph 1 * max (#↑s) 2 ^ ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀
[PROOFSTEP]
rw [mul_eq_max A C]
[GOAL]
α : Type u
s : Set (Set α)
i✝ i : (Quotient.out (ord (aleph 1))).α
IH :
  ∀ (y : (Quotient.out (ord (aleph 1))).α),
    WellOrder.r (Quotient.out (ord (aleph 1))) y i → #↑(generateMeasurableRec s y) ≤ max (#↑s) 2 ^ ℵ₀
A : ℵ₀ ≤ aleph 1
B : aleph 1 ≤ max (#↑s) 2 ^ ℵ₀
C : ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀
D : ⨆ (j : ↑(Iio i)), #↑(generateMeasurableRec s ↑j) ≤ max (#↑s) 2 ^ ℵ₀
⊢ max (aleph 1) (max (#↑s) 2 ^ ℵ₀) ≤ max (#↑s) 2 ^ ℵ₀
[PROOFSTEP]
exact max_le B le_rfl
[GOAL]
α : Type u
s : Set (Set α)
i✝ i : (Quotient.out (ord (aleph 1))).α
IH :
  ∀ (y : (Quotient.out (ord (aleph 1))).α),
    WellOrder.r (Quotient.out (ord (aleph 1))) y i → #↑(generateMeasurableRec s y) ≤ max (#↑s) 2 ^ ℵ₀
A : ℵ₀ ≤ aleph 1
B : aleph 1 ≤ max (#↑s) 2 ^ ℵ₀
C : ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀
J : #↑(⋃ (j : ↑(Iio i)), generateMeasurableRec s ↑j) ≤ max (#↑s) 2 ^ ℵ₀
⊢ #↑(generateMeasurableRec s i) ≤ max (#↑s) 2 ^ ℵ₀
[PROOFSTEP]
rw [generateMeasurableRec]
[GOAL]
α : Type u
s : Set (Set α)
i✝ i : (Quotient.out (ord (aleph 1))).α
IH :
  ∀ (y : (Quotient.out (ord (aleph 1))).α),
    WellOrder.r (Quotient.out (ord (aleph 1))) y i → #↑(generateMeasurableRec s y) ≤ max (#↑s) 2 ^ ℵ₀
A : ℵ₀ ≤ aleph 1
B : aleph 1 ≤ max (#↑s) 2 ^ ℵ₀
C : ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀
J : #↑(⋃ (j : ↑(Iio i)), generateMeasurableRec s ↑j) ≤ max (#↑s) 2 ^ ℵ₀
⊢ #↑((s ∪ {∅} ∪ compl '' ⋃ (j : ↑(Iio i)), generateMeasurableRec s ↑j) ∪ range fun f => ⋃ (n : ℕ), ↑(f n)) ≤
    max (#↑s) 2 ^ ℵ₀
[PROOFSTEP]
apply_rules [(mk_union_le _ _).trans, add_le_of_le C, mk_image_le.trans]
[GOAL]
case h1.h1.h1
α : Type u
s : Set (Set α)
i✝ i : (Quotient.out (ord (aleph 1))).α
IH :
  ∀ (y : (Quotient.out (ord (aleph 1))).α),
    WellOrder.r (Quotient.out (ord (aleph 1))) y i → #↑(generateMeasurableRec s y) ≤ max (#↑s) 2 ^ ℵ₀
A : ℵ₀ ≤ aleph 1
B : aleph 1 ≤ max (#↑s) 2 ^ ℵ₀
C : ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀
J : #↑(⋃ (j : ↑(Iio i)), generateMeasurableRec s ↑j) ≤ max (#↑s) 2 ^ ℵ₀
⊢ #↑s ≤ max (#↑s) 2 ^ ℵ₀
[PROOFSTEP]
exact (le_max_left _ _).trans (self_le_power _ one_lt_aleph0.le)
[GOAL]
case h1.h1.h2
α : Type u
s : Set (Set α)
i✝ i : (Quotient.out (ord (aleph 1))).α
IH :
  ∀ (y : (Quotient.out (ord (aleph 1))).α),
    WellOrder.r (Quotient.out (ord (aleph 1))) y i → #↑(generateMeasurableRec s y) ≤ max (#↑s) 2 ^ ℵ₀
A : ℵ₀ ≤ aleph 1
B : aleph 1 ≤ max (#↑s) 2 ^ ℵ₀
C : ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀
J : #↑(⋃ (j : ↑(Iio i)), generateMeasurableRec s ↑j) ≤ max (#↑s) 2 ^ ℵ₀
⊢ #↑{∅} ≤ max (#↑s) 2 ^ ℵ₀
[PROOFSTEP]
rw [mk_singleton]
[GOAL]
case h1.h1.h2
α : Type u
s : Set (Set α)
i✝ i : (Quotient.out (ord (aleph 1))).α
IH :
  ∀ (y : (Quotient.out (ord (aleph 1))).α),
    WellOrder.r (Quotient.out (ord (aleph 1))) y i → #↑(generateMeasurableRec s y) ≤ max (#↑s) 2 ^ ℵ₀
A : ℵ₀ ≤ aleph 1
B : aleph 1 ≤ max (#↑s) 2 ^ ℵ₀
C : ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀
J : #↑(⋃ (j : ↑(Iio i)), generateMeasurableRec s ↑j) ≤ max (#↑s) 2 ^ ℵ₀
⊢ 1 ≤ max (#↑s) 2 ^ ℵ₀
[PROOFSTEP]
exact one_lt_aleph0.le.trans C
[GOAL]
case h2
α : Type u
s : Set (Set α)
i✝ i : (Quotient.out (ord (aleph 1))).α
IH :
  ∀ (y : (Quotient.out (ord (aleph 1))).α),
    WellOrder.r (Quotient.out (ord (aleph 1))) y i → #↑(generateMeasurableRec s y) ≤ max (#↑s) 2 ^ ℵ₀
A : ℵ₀ ≤ aleph 1
B : aleph 1 ≤ max (#↑s) 2 ^ ℵ₀
C : ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀
J : #↑(⋃ (j : ↑(Iio i)), generateMeasurableRec s ↑j) ≤ max (#↑s) 2 ^ ℵ₀
⊢ #↑(range fun f => ⋃ (n : ℕ), ↑(f n)) ≤ max (#↑s) 2 ^ ℵ₀
[PROOFSTEP]
apply mk_range_le.trans
[GOAL]
case h2
α : Type u
s : Set (Set α)
i✝ i : (Quotient.out (ord (aleph 1))).α
IH :
  ∀ (y : (Quotient.out (ord (aleph 1))).α),
    WellOrder.r (Quotient.out (ord (aleph 1))) y i → #↑(generateMeasurableRec s y) ≤ max (#↑s) 2 ^ ℵ₀
A : ℵ₀ ≤ aleph 1
B : aleph 1 ≤ max (#↑s) 2 ^ ℵ₀
C : ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀
J : #↑(⋃ (j : ↑(Iio i)), generateMeasurableRec s ↑j) ≤ max (#↑s) 2 ^ ℵ₀
⊢ #(ℕ → ↑(⋃ (j : ↑(Iio i)), generateMeasurableRec s ↑j)) ≤ max (#↑s) 2 ^ ℵ₀
[PROOFSTEP]
simp only [mk_pi, prod_const, lift_uzero, mk_denumerable, lift_aleph0]
[GOAL]
case h2
α : Type u
s : Set (Set α)
i✝ i : (Quotient.out (ord (aleph 1))).α
IH :
  ∀ (y : (Quotient.out (ord (aleph 1))).α),
    WellOrder.r (Quotient.out (ord (aleph 1))) y i → #↑(generateMeasurableRec s y) ≤ max (#↑s) 2 ^ ℵ₀
A : ℵ₀ ≤ aleph 1
B : aleph 1 ≤ max (#↑s) 2 ^ ℵ₀
C : ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀
J : #↑(⋃ (j : ↑(Iio i)), generateMeasurableRec s ↑j) ≤ max (#↑s) 2 ^ ℵ₀
⊢ #↑(⋃ (j : ↑(Iio i)), generateMeasurableRec s ↑j) ^ ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀
[PROOFSTEP]
have := @power_le_power_right _ _ ℵ₀ J
[GOAL]
case h2
α : Type u
s : Set (Set α)
i✝ i : (Quotient.out (ord (aleph 1))).α
IH :
  ∀ (y : (Quotient.out (ord (aleph 1))).α),
    WellOrder.r (Quotient.out (ord (aleph 1))) y i → #↑(generateMeasurableRec s y) ≤ max (#↑s) 2 ^ ℵ₀
A : ℵ₀ ≤ aleph 1
B : aleph 1 ≤ max (#↑s) 2 ^ ℵ₀
C : ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀
J : #↑(⋃ (j : ↑(Iio i)), generateMeasurableRec s ↑j) ≤ max (#↑s) 2 ^ ℵ₀
this : #↑(⋃ (j : ↑(Iio i)), generateMeasurableRec s ↑j) ^ ℵ₀ ≤ (max (#↑s) 2 ^ ℵ₀) ^ ℵ₀
⊢ #↑(⋃ (j : ↑(Iio i)), generateMeasurableRec s ↑j) ^ ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀
[PROOFSTEP]
rwa [← power_mul, aleph0_mul_aleph0] at this 
[GOAL]
α : Type u
s : Set (Set α)
⊢ {t | GenerateMeasurable s t} = ⋃ (i : (Quotient.out (ord (aleph 1))).α), generateMeasurableRec s i
[PROOFSTEP]
ext t
[GOAL]
case h
α : Type u
s : Set (Set α)
t : Set α
⊢ t ∈ {t | GenerateMeasurable s t} ↔ t ∈ ⋃ (i : (Quotient.out (ord (aleph 1))).α), generateMeasurableRec s i
[PROOFSTEP]
refine' ⟨fun ht => _, fun ht => _⟩
[GOAL]
case h.refine'_1
α : Type u
s : Set (Set α)
t : Set α
ht : t ∈ {t | GenerateMeasurable s t}
⊢ t ∈ ⋃ (i : (Quotient.out (ord (aleph 1))).α), generateMeasurableRec s i
[PROOFSTEP]
inhabit ω₁
[GOAL]
case h.refine'_1
α : Type u
s : Set (Set α)
t : Set α
ht : t ∈ {t | GenerateMeasurable s t}
inhabited_h : Inhabited (Quotient.out (ord (aleph 1))).α
⊢ t ∈ ⋃ (i : (Quotient.out (ord (aleph 1))).α), generateMeasurableRec s i
[PROOFSTEP]
induction' ht with u hu u _ IH f _ IH
[GOAL]
case h.refine'_1.basic
α : Type u
s : Set (Set α)
t : Set α
inhabited_h : Inhabited (Quotient.out (ord (aleph 1))).α
u : Set α
hu : u ∈ s
⊢ u ∈ ⋃ (i : (Quotient.out (ord (aleph 1))).α), generateMeasurableRec s i
[PROOFSTEP]
exact mem_iUnion.2 ⟨default, self_subset_generateMeasurableRec s _ hu⟩
[GOAL]
case h.refine'_1.empty
α : Type u
s : Set (Set α)
t : Set α
inhabited_h : Inhabited (Quotient.out (ord (aleph 1))).α
⊢ ∅ ∈ ⋃ (i : (Quotient.out (ord (aleph 1))).α), generateMeasurableRec s i
[PROOFSTEP]
exact mem_iUnion.2 ⟨default, empty_mem_generateMeasurableRec s _⟩
[GOAL]
case h.refine'_1.compl
α : Type u
s : Set (Set α)
t : Set α
inhabited_h : Inhabited (Quotient.out (ord (aleph 1))).α
u : Set α
a✝ : GenerateMeasurable s u
IH : u ∈ ⋃ (i : (Quotient.out (ord (aleph 1))).α), generateMeasurableRec s i
⊢ uᶜ ∈ ⋃ (i : (Quotient.out (ord (aleph 1))).α), generateMeasurableRec s i
[PROOFSTEP]
rcases mem_iUnion.1 IH with ⟨i, hi⟩
[GOAL]
case h.refine'_1.compl.intro
α : Type u
s : Set (Set α)
t : Set α
inhabited_h : Inhabited (Quotient.out (ord (aleph 1))).α
u : Set α
a✝ : GenerateMeasurable s u
IH : u ∈ ⋃ (i : (Quotient.out (ord (aleph 1))).α), generateMeasurableRec s i
i : (Quotient.out (ord (aleph 1))).α
hi : u ∈ generateMeasurableRec s i
⊢ uᶜ ∈ ⋃ (i : (Quotient.out (ord (aleph 1))).α), generateMeasurableRec s i
[PROOFSTEP]
obtain ⟨j, hj⟩ := exists_gt i
[GOAL]
case h.refine'_1.compl.intro.intro
α : Type u
s : Set (Set α)
t : Set α
inhabited_h : Inhabited (Quotient.out (ord (aleph 1))).α
u : Set α
a✝ : GenerateMeasurable s u
IH : u ∈ ⋃ (i : (Quotient.out (ord (aleph 1))).α), generateMeasurableRec s i
i : (Quotient.out (ord (aleph 1))).α
hi : u ∈ generateMeasurableRec s i
j : (Quotient.out (ord (aleph 1))).α
hj : i < j
⊢ uᶜ ∈ ⋃ (i : (Quotient.out (ord (aleph 1))).α), generateMeasurableRec s i
[PROOFSTEP]
exact mem_iUnion.2 ⟨j, compl_mem_generateMeasurableRec hj hi⟩
[GOAL]
case h.refine'_1.iUnion
α : Type u
s : Set (Set α)
t : Set α
inhabited_h : Inhabited (Quotient.out (ord (aleph 1))).α
f : ℕ → Set α
a✝ : ∀ (n : ℕ), GenerateMeasurable s (f n)
IH : ∀ (n : ℕ), f n ∈ ⋃ (i : (Quotient.out (ord (aleph 1))).α), generateMeasurableRec s i
⊢ ⋃ (i : ℕ), f i ∈ ⋃ (i : (Quotient.out (ord (aleph 1))).α), generateMeasurableRec s i
[PROOFSTEP]
have : ∀ n, ∃ i, f n ∈ generateMeasurableRec s i := fun n => by simpa using IH n
[GOAL]
α : Type u
s : Set (Set α)
t : Set α
inhabited_h : Inhabited (Quotient.out (ord (aleph 1))).α
f : ℕ → Set α
a✝ : ∀ (n : ℕ), GenerateMeasurable s (f n)
IH : ∀ (n : ℕ), f n ∈ ⋃ (i : (Quotient.out (ord (aleph 1))).α), generateMeasurableRec s i
n : ℕ
⊢ ∃ i, f n ∈ generateMeasurableRec s i
[PROOFSTEP]
simpa using IH n
[GOAL]
case h.refine'_1.iUnion
α : Type u
s : Set (Set α)
t : Set α
inhabited_h : Inhabited (Quotient.out (ord (aleph 1))).α
f : ℕ → Set α
a✝ : ∀ (n : ℕ), GenerateMeasurable s (f n)
IH : ∀ (n : ℕ), f n ∈ ⋃ (i : (Quotient.out (ord (aleph 1))).α), generateMeasurableRec s i
this : ∀ (n : ℕ), ∃ i, f n ∈ generateMeasurableRec s i
⊢ ⋃ (i : ℕ), f i ∈ ⋃ (i : (Quotient.out (ord (aleph 1))).α), generateMeasurableRec s i
[PROOFSTEP]
choose I hI using this
[GOAL]
case h.refine'_1.iUnion
α : Type u
s : Set (Set α)
t : Set α
inhabited_h : Inhabited (Quotient.out (ord (aleph 1))).α
f : ℕ → Set α
a✝ : ∀ (n : ℕ), GenerateMeasurable s (f n)
IH : ∀ (n : ℕ), f n ∈ ⋃ (i : (Quotient.out (ord (aleph 1))).α), generateMeasurableRec s i
I : ℕ → (Quotient.out (ord (aleph 1))).α
hI : ∀ (n : ℕ), f n ∈ generateMeasurableRec s (I n)
⊢ ⋃ (i : ℕ), f i ∈ ⋃ (i : (Quotient.out (ord (aleph 1))).α), generateMeasurableRec s i
[PROOFSTEP]
have : IsWellOrder (ω₁ : Type u) (· < ·) := isWellOrder_out_lt _
[GOAL]
case h.refine'_1.iUnion
α : Type u
s : Set (Set α)
t : Set α
inhabited_h : Inhabited (Quotient.out (ord (aleph 1))).α
f : ℕ → Set α
a✝ : ∀ (n : ℕ), GenerateMeasurable s (f n)
IH : ∀ (n : ℕ), f n ∈ ⋃ (i : (Quotient.out (ord (aleph 1))).α), generateMeasurableRec s i
I : ℕ → (Quotient.out (ord (aleph 1))).α
hI : ∀ (n : ℕ), f n ∈ generateMeasurableRec s (I n)
this : IsWellOrder (Quotient.out (ord (aleph 1))).α fun x x_1 => x < x_1
⊢ ⋃ (i : ℕ), f i ∈ ⋃ (i : (Quotient.out (ord (aleph 1))).α), generateMeasurableRec s i
[PROOFSTEP]
refine'
  mem_iUnion.2
    ⟨Ordinal.enum (· < ·) (Ordinal.lsub fun n => Ordinal.typein.{u} (· < ·) (I n)) _,
      iUnion_mem_generateMeasurableRec fun n => ⟨I n, _, hI n⟩⟩
[GOAL]
case h.refine'_1.iUnion.refine'_1
α : Type u
s : Set (Set α)
t : Set α
inhabited_h : Inhabited (Quotient.out (ord (aleph 1))).α
f : ℕ → Set α
a✝ : ∀ (n : ℕ), GenerateMeasurable s (f n)
IH : ∀ (n : ℕ), f n ∈ ⋃ (i : (Quotient.out (ord (aleph 1))).α), generateMeasurableRec s i
I : ℕ → (Quotient.out (ord (aleph 1))).α
hI : ∀ (n : ℕ), f n ∈ generateMeasurableRec s (I n)
this : IsWellOrder (Quotient.out (ord (aleph 1))).α fun x x_1 => x < x_1
⊢ (Ordinal.lsub fun n => Ordinal.typein (fun x x_1 => x < x_1) (I n)) < Ordinal.type fun x x_1 => x < x_1
[PROOFSTEP]
rw [Ordinal.type_lt]
[GOAL]
case h.refine'_1.iUnion.refine'_1
α : Type u
s : Set (Set α)
t : Set α
inhabited_h : Inhabited (Quotient.out (ord (aleph 1))).α
f : ℕ → Set α
a✝ : ∀ (n : ℕ), GenerateMeasurable s (f n)
IH : ∀ (n : ℕ), f n ∈ ⋃ (i : (Quotient.out (ord (aleph 1))).α), generateMeasurableRec s i
I : ℕ → (Quotient.out (ord (aleph 1))).α
hI : ∀ (n : ℕ), f n ∈ generateMeasurableRec s (I n)
this : IsWellOrder (Quotient.out (ord (aleph 1))).α fun x x_1 => x < x_1
⊢ (Ordinal.lsub fun n => Ordinal.typein (fun x x_1 => x < x_1) (I n)) < ord (aleph 1)
[PROOFSTEP]
refine' Ordinal.lsub_lt_ord_lift _ fun i => Ordinal.typein_lt_self _
[GOAL]
case h.refine'_1.iUnion.refine'_1
α : Type u
s : Set (Set α)
t : Set α
inhabited_h : Inhabited (Quotient.out (ord (aleph 1))).α
f : ℕ → Set α
a✝ : ∀ (n : ℕ), GenerateMeasurable s (f n)
IH : ∀ (n : ℕ), f n ∈ ⋃ (i : (Quotient.out (ord (aleph 1))).α), generateMeasurableRec s i
I : ℕ → (Quotient.out (ord (aleph 1))).α
hI : ∀ (n : ℕ), f n ∈ generateMeasurableRec s (I n)
this : IsWellOrder (Quotient.out (ord (aleph 1))).α fun x x_1 => x < x_1
⊢ lift #ℕ < Ordinal.cof (ord (aleph 1))
[PROOFSTEP]
rw [mk_denumerable, lift_aleph0, isRegular_aleph_one.cof_eq]
[GOAL]
case h.refine'_1.iUnion.refine'_1
α : Type u
s : Set (Set α)
t : Set α
inhabited_h : Inhabited (Quotient.out (ord (aleph 1))).α
f : ℕ → Set α
a✝ : ∀ (n : ℕ), GenerateMeasurable s (f n)
IH : ∀ (n : ℕ), f n ∈ ⋃ (i : (Quotient.out (ord (aleph 1))).α), generateMeasurableRec s i
I : ℕ → (Quotient.out (ord (aleph 1))).α
hI : ∀ (n : ℕ), f n ∈ generateMeasurableRec s (I n)
this : IsWellOrder (Quotient.out (ord (aleph 1))).α fun x x_1 => x < x_1
⊢ ℵ₀ < aleph 1
[PROOFSTEP]
exact aleph0_lt_aleph_one
[GOAL]
case h.refine'_1.iUnion.refine'_2
α : Type u
s : Set (Set α)
t : Set α
inhabited_h : Inhabited (Quotient.out (ord (aleph 1))).α
f : ℕ → Set α
a✝ : ∀ (n : ℕ), GenerateMeasurable s (f n)
IH : ∀ (n : ℕ), f n ∈ ⋃ (i : (Quotient.out (ord (aleph 1))).α), generateMeasurableRec s i
I : ℕ → (Quotient.out (ord (aleph 1))).α
hI : ∀ (n : ℕ), f n ∈ generateMeasurableRec s (I n)
this : IsWellOrder (Quotient.out (ord (aleph 1))).α fun x x_1 => x < x_1
n : ℕ
⊢ I n <
    Ordinal.enum (fun x x_1 => x < x_1) (Ordinal.lsub fun n => Ordinal.typein (fun x x_1 => x < x_1) (I n))
      (_ : (Ordinal.lsub fun n => Ordinal.typein (fun x x_1 => x < x_1) (I n)) < Ordinal.type fun x x_1 => x < x_1)
[PROOFSTEP]
rw [← Ordinal.typein_lt_typein (· < ·), Ordinal.typein_enum]
[GOAL]
case h.refine'_1.iUnion.refine'_2
α : Type u
s : Set (Set α)
t : Set α
inhabited_h : Inhabited (Quotient.out (ord (aleph 1))).α
f : ℕ → Set α
a✝ : ∀ (n : ℕ), GenerateMeasurable s (f n)
IH : ∀ (n : ℕ), f n ∈ ⋃ (i : (Quotient.out (ord (aleph 1))).α), generateMeasurableRec s i
I : ℕ → (Quotient.out (ord (aleph 1))).α
hI : ∀ (n : ℕ), f n ∈ generateMeasurableRec s (I n)
this : IsWellOrder (Quotient.out (ord (aleph 1))).α fun x x_1 => x < x_1
n : ℕ
⊢ Ordinal.typein (fun x x_1 => x < x_1) (I n) < Ordinal.lsub fun n => Ordinal.typein (fun x x_1 => x < x_1) (I n)
[PROOFSTEP]
apply Ordinal.lt_lsub fun n : ℕ => _
[GOAL]
case h.refine'_2
α : Type u
s : Set (Set α)
t : Set α
ht : t ∈ ⋃ (i : (Quotient.out (ord (aleph 1))).α), generateMeasurableRec s i
⊢ t ∈ {t | GenerateMeasurable s t}
[PROOFSTEP]
rcases ht with ⟨t, ⟨i, rfl⟩, hx⟩
[GOAL]
case h.refine'_2.intro.intro.intro
α : Type u
s : Set (Set α)
t : Set α
i : (Quotient.out (ord (aleph 1))).α
hx : t ∈ (fun i => generateMeasurableRec s i) i
⊢ t ∈ {t | GenerateMeasurable s t}
[PROOFSTEP]
revert t
[GOAL]
case h.refine'_2.intro.intro.intro
α : Type u
s : Set (Set α)
i : (Quotient.out (ord (aleph 1))).α
⊢ ∀ (t : Set α), t ∈ (fun i => generateMeasurableRec s i) i → t ∈ {t | GenerateMeasurable s t}
[PROOFSTEP]
apply (aleph 1).ord.out.wo.wf.induction i
[GOAL]
case h.refine'_2.intro.intro.intro
α : Type u
s : Set (Set α)
i : (Quotient.out (ord (aleph 1))).α
⊢ ∀ (x : (Quotient.out (ord (aleph 1))).α),
    (∀ (y : (Quotient.out (ord (aleph 1))).α),
        WellOrder.r (Quotient.out (ord (aleph 1))) y x →
          ∀ (t : Set α), t ∈ (fun i => generateMeasurableRec s i) y → t ∈ {t | GenerateMeasurable s t}) →
      ∀ (t : Set α), t ∈ (fun i => generateMeasurableRec s i) x → t ∈ {t | GenerateMeasurable s t}
[PROOFSTEP]
intro j H t ht
[GOAL]
case h.refine'_2.intro.intro.intro
α : Type u
s : Set (Set α)
i j : (Quotient.out (ord (aleph 1))).α
H :
  ∀ (y : (Quotient.out (ord (aleph 1))).α),
    WellOrder.r (Quotient.out (ord (aleph 1))) y j →
      ∀ (t : Set α), t ∈ (fun i => generateMeasurableRec s i) y → t ∈ {t | GenerateMeasurable s t}
t : Set α
ht : t ∈ (fun i => generateMeasurableRec s i) j
⊢ t ∈ {t | GenerateMeasurable s t}
[PROOFSTEP]
unfold generateMeasurableRec at ht 
[GOAL]
case h.refine'_2.intro.intro.intro
α : Type u
s : Set (Set α)
i j : (Quotient.out (ord (aleph 1))).α
H :
  ∀ (y : (Quotient.out (ord (aleph 1))).α),
    WellOrder.r (Quotient.out (ord (aleph 1))) y j →
      ∀ (t : Set α), t ∈ (fun i => generateMeasurableRec s i) y → t ∈ {t | GenerateMeasurable s t}
t : Set α
ht :
  t ∈
    let i := j;
    let S := ⋃ (j : ↑(Iio i)), generateMeasurableRec s ↑j;
    s ∪ {∅} ∪ compl '' S ∪ range fun f => ⋃ (n : ℕ), ↑(f n)
⊢ t ∈ {t | GenerateMeasurable s t}
[PROOFSTEP]
rcases ht with (((h | (rfl : t = ∅)) | ⟨u, ⟨-, ⟨⟨k, hk⟩, rfl⟩, hu⟩, rfl⟩) | ⟨f, rfl⟩)
[GOAL]
case h.refine'_2.intro.intro.intro.inl.inl.inl
α : Type u
s : Set (Set α)
i j : (Quotient.out (ord (aleph 1))).α
H :
  ∀ (y : (Quotient.out (ord (aleph 1))).α),
    WellOrder.r (Quotient.out (ord (aleph 1))) y j →
      ∀ (t : Set α), t ∈ (fun i => generateMeasurableRec s i) y → t ∈ {t | GenerateMeasurable s t}
t : Set α
h : t ∈ s
⊢ t ∈ {t | GenerateMeasurable s t}
[PROOFSTEP]
exact .basic t h
[GOAL]
case h.refine'_2.intro.intro.intro.inl.inl.inr
α : Type u
s : Set (Set α)
i j : (Quotient.out (ord (aleph 1))).α
H :
  ∀ (y : (Quotient.out (ord (aleph 1))).α),
    WellOrder.r (Quotient.out (ord (aleph 1))) y j →
      ∀ (t : Set α), t ∈ (fun i => generateMeasurableRec s i) y → t ∈ {t | GenerateMeasurable s t}
⊢ ∅ ∈ {t | GenerateMeasurable s t}
[PROOFSTEP]
exact .empty
[GOAL]
case h.refine'_2.intro.intro.intro.inl.inr.intro.intro.intro.intro.intro.mk
α : Type u
s : Set (Set α)
i j : (Quotient.out (ord (aleph 1))).α
H :
  ∀ (y : (Quotient.out (ord (aleph 1))).α),
    WellOrder.r (Quotient.out (ord (aleph 1))) y j →
      ∀ (t : Set α), t ∈ (fun i => generateMeasurableRec s i) y → t ∈ {t | GenerateMeasurable s t}
u : Set α
k : (Quotient.out (ord (aleph 1))).α
hk : k ∈ Iio j
hu : u ∈ (fun j_1 => generateMeasurableRec s ↑j_1) { val := k, property := hk }
⊢ uᶜ ∈ {t | GenerateMeasurable s t}
[PROOFSTEP]
exact .compl u (H k hk u hu)
[GOAL]
case h.refine'_2.intro.intro.intro.inr.intro
α : Type u
s : Set (Set α)
i j : (Quotient.out (ord (aleph 1))).α
H :
  ∀ (y : (Quotient.out (ord (aleph 1))).α),
    WellOrder.r (Quotient.out (ord (aleph 1))) y j →
      ∀ (t : Set α), t ∈ (fun i => generateMeasurableRec s i) y → t ∈ {t | GenerateMeasurable s t}
f : ℕ → ↑(⋃ (j_1 : ↑(Iio j)), generateMeasurableRec s ↑j_1)
⊢ (fun f => ⋃ (n : ℕ), ↑(f n)) f ∈ {t | GenerateMeasurable s t}
[PROOFSTEP]
refine .iUnion _ @fun n => ?_
[GOAL]
case h.refine'_2.intro.intro.intro.inr.intro
α : Type u
s : Set (Set α)
i j : (Quotient.out (ord (aleph 1))).α
H :
  ∀ (y : (Quotient.out (ord (aleph 1))).α),
    WellOrder.r (Quotient.out (ord (aleph 1))) y j →
      ∀ (t : Set α), t ∈ (fun i => generateMeasurableRec s i) y → t ∈ {t | GenerateMeasurable s t}
f : ℕ → ↑(⋃ (j_1 : ↑(Iio j)), generateMeasurableRec s ↑j_1)
n : ℕ
⊢ GenerateMeasurable s ↑(f n)
[PROOFSTEP]
obtain ⟨-, ⟨⟨k, hk⟩, rfl⟩, hf⟩ := (f n).prop
[GOAL]
case h.refine'_2.intro.intro.intro.inr.intro.intro.intro.intro.mk
α : Type u
s : Set (Set α)
i j : (Quotient.out (ord (aleph 1))).α
H :
  ∀ (y : (Quotient.out (ord (aleph 1))).α),
    WellOrder.r (Quotient.out (ord (aleph 1))) y j →
      ∀ (t : Set α), t ∈ (fun i => generateMeasurableRec s i) y → t ∈ {t | GenerateMeasurable s t}
f : ℕ → ↑(⋃ (j_1 : ↑(Iio j)), generateMeasurableRec s ↑j_1)
n : ℕ
k : (Quotient.out (ord (aleph 1))).α
hk : k ∈ Iio j
hf : ↑(f n) ∈ (fun j_1 => generateMeasurableRec s ↑j_1) { val := k, property := hk }
⊢ GenerateMeasurable s ↑(f n)
[PROOFSTEP]
exact H k hk _ hf
[GOAL]
α : Type u
s : Set (Set α)
⊢ #↑{t | GenerateMeasurable s t} ≤ max (#↑s) 2 ^ ℵ₀
[PROOFSTEP]
rw [generateMeasurable_eq_rec]
[GOAL]
α : Type u
s : Set (Set α)
⊢ #↑(⋃ (i : (Quotient.out (ord (aleph 1))).α), generateMeasurableRec s i) ≤ max (#↑s) 2 ^ ℵ₀
[PROOFSTEP]
apply (mk_iUnion_le _).trans
[GOAL]
α : Type u
s : Set (Set α)
⊢ #(Quotient.out (ord (aleph 1))).α * ⨆ (i : (Quotient.out (ord (aleph 1))).α), #↑(generateMeasurableRec s i) ≤
    max (#↑s) 2 ^ ℵ₀
[PROOFSTEP]
rw [(aleph 1).mk_ord_out]
[GOAL]
α : Type u
s : Set (Set α)
⊢ aleph 1 * ⨆ (i : (Quotient.out (ord (aleph 1))).α), #↑(generateMeasurableRec s i) ≤ max (#↑s) 2 ^ ℵ₀
[PROOFSTEP]
refine le_trans (mul_le_mul' aleph_one_le_continuum (ciSup_le' fun i => cardinal_generateMeasurableRec_le s i)) ?_
[GOAL]
α : Type u
s : Set (Set α)
⊢ 𝔠 * max (#↑s) 2 ^ ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀
[PROOFSTEP]
refine (mul_le_max_of_aleph0_le_left aleph0_le_continuum).trans (max_le ?_ le_rfl)
[GOAL]
α : Type u
s : Set (Set α)
⊢ 𝔠 ≤ max (#↑s) 2 ^ ℵ₀
[PROOFSTEP]
exact power_le_power_right (le_max_right _ _)
[GOAL]
α : Type u
s : Set (Set α)
hs : #↑s ≤ 𝔠
⊢ max (#↑s) 2 ^ ℵ₀ ≤ 𝔠
[PROOFSTEP]
rw [← continuum_power_aleph0]
[GOAL]
α : Type u
s : Set (Set α)
hs : #↑s ≤ 𝔠
⊢ max (#↑s) 2 ^ ℵ₀ ≤ 𝔠 ^ ℵ₀
[PROOFSTEP]
exact_mod_cast power_le_power_right (max_le hs (nat_lt_continuum 2).le)
