Skip to content

[Merged by Bors] - feat: lift (ℶ_ o) = ℶ_ (lift o) #33059

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Closed
vihdzp wants to merge 2 commits into from
+19 −0
Closed

[Merged by Bors] - feat: lift (ℶ_ o) = ℶ_ (lift o) #33059

Changes from all commits
Commits
Show all changes
2 commits
Select commit Hold shift + click to select a range
5f16b52
proof
vihdzp Dec 18, 2025
077c737
Update Aleph.lean
vihdzp Dec 18, 2025
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
19 changes: 19 additions & 0 deletions Mathlib/SetTheory/Cardinal/Aleph.lean
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -587,6 +587,21 @@ theorem isStrongLimit_preBeth {o : Ordinal} : IsStrongLimit (preBeth o) ↔ IsSu
· intro h
simpa using h.two_power_lt (preBeth_strictMono (lt_succ a))

@[simp]
theorem lift_preBeth (o : Ordinal) : lift.{v} (preBeth o) = preBeth (Ordinal.lift.{v} o) := by
induction o using SuccOrder.prelimitRecOn with
| succ o _ IH => simp [IH]
| isSuccPrelimit o ho IH =>
rw [preBeth_limit ho, preBeth_limit (isSuccPrelimit_lift.2 ho), lift_iSup (bddAbove_of_small _)]
apply congrArg sSup
ext x
constructor <;> rintro ⟨⟨i, hi⟩, rfl⟩
· refine ⟨⟨i.lift, ?_⟩, (IH _ hi).symm⟩
simpa
· obtain ⟨i, rfl⟩ := Ordinal.mem_range_lift_of_le hi.le
rw [mem_Iio, Ordinal.lift_lt] at hi
exact ⟨⟨i, hi⟩, IH _ hi⟩

/-- The Beth function is defined so that `beth 0 = ℵ₀'`, `beth (succ o) = 2 ^ beth o`, and that for
a limit ordinal `o`, `beth o` is the supremum of `beth a` for `a < o`.

Expand Down Expand Up @@ -649,4 +664,8 @@ theorem beth_ne_zero (o : Ordinal) : ℶ_ o ≠ 0 :=
theorem isStrongLimit_beth {o : Ordinal} : IsStrongLimit (ℶ_ o) ↔ IsSuccPrelimit o := by
rw [beth_eq_preBeth, isStrongLimit_preBeth, isSuccLimit_add_iff_of_isSuccLimit isSuccLimit_omega0]

@[simp]
theorem lift_beth (o : Ordinal) : lift.{v} (ℶ_ o) = ℶ_ (Ordinal.lift.{v} o) := by
rw [beth_eq_preBeth, beth_eq_preBeth, lift_preBeth, Ordinal.lift_add, lift_omega0]

end Cardinal
⚡