| import Mathlib |
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| namespace OAI |
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| noncomputable section |
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| section |
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| noncomputable section |
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| namespace Kadison.FinitePatching |
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| variable (I : Type*) [Fintype I] |
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| def listPool (N : ℕ) : Finset (List I) := by |
| classical |
| exact Finset.univ.biUnion (fun n : Fin (N+1) => |
| Finset.univ.image (fun f : Fin n → I => List.ofFn f)) |
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| lemma mem_listPool (N : ℕ) (l : List I) : l∈listPool I N ↔ l.length ≤ N := by |
| classical |
| constructor |
| · intro h |
| simp only [listPool,Finset.mem_biUnion,Finset.mem_univ,Finset.mem_image,true_and] at h |
| obtain ⟨n,f,rfl⟩ := h |
| simpa only [List.length_ofFn] using Nat.le_of_lt_succ n.isLt |
| · intro h |
| simp only [listPool,Finset.mem_biUnion,Finset.mem_univ,Finset.mem_image,true_and] |
| exact ⟨⟨l.length,Nat.lt_succ_of_le h⟩,l.get,List.ofFn_get l⟩ |
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| abbrev SmallList (N : ℕ) := ↥(listPool I N) |
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| def smallList {N : ℕ} (l : List I) (h : l.length ≤ N) : SmallList I N := |
| ⟨l,(mem_listPool I N l).mpr h⟩ |
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| lemma SmallList.length_le {N : ℕ} (l : SmallList I N) : l.val.length ≤ N := |
| (mem_listPool I N l.val).mp l.property |
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| end Kadison.FinitePatching |
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| end |
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| end |
| end |
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| end OAI |
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