Created
May 27, 2025 12:03
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Lean proof that higher universes don't embed into lower ones.
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| inductive Tree : Sort (max (u + 1) v) | |
| | node {A : Sort u} (c : A → Tree) | |
| theorem no_embedding | |
| (Lower : Sort (max (u + 1) v) → Sort u) | |
| (down : ∀ {α}, α → Lower α) | |
| (up : ∀ {α}, Lower α → α) | |
| (up_down : ∀ {α} (x : α), x = up (down x)) | |
| : False := by | |
| induction h : Tree.node up | |
| case node ih => | |
| cases h | |
| exact ih _ (up_down _) |
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