| import OAI.Combinatorics.LowSpaceSubsetSum.Representation.RepresentationParentComplete |
|
|
| namespace OAI |
|
|
| /-! |
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|
| noncomputable section |
|
|
| namespace LowSpaceSubsetSum.RepresentationParentData |
|
|
| variable {n : ℕ} (D : RepresentationParentData n) |
|
|
| def entryUniverse : Finset (Entry (Fin n)) := by |
| classical |
| exact D.rawEntries.toFinset |
|
|
| theorem entries_nodup (B : ℕ) : (D.entries B).toList.Nodup := by |
| classical |
| have h := D.collect_invariant B |
| have hk : (((D.entries B).toList.map (WeightEncoding.entryKey D.input)).toFinset).card = |
| ((D.entries B).toList.map (WeightEncoding.entryKey D.input)).length := by |
| unfold entries |
| rw [← h.keys_eq, BinaryTrie.card_keySet, ← h.size_eq, List.length_map, Array.length_toList] |
| have hm : ((D.entries B).toList.map (WeightEncoding.entryKey D.input)).Nodup := |
| List.nodup_iff_length_dedup_eq.mpr (by rwa [← List.card_toFinset]) |
| exact List.Nodup.of_map (WeightEncoding.entryKey D.input) hm |
|
|
| theorem completed_size_le (B : ℕ) (hc : (D.collect B).state.user.keys.overflow = false) : |
| (D.entries B).size ≤ D.capacity := by |
| rw [entries, (D.collect_invariant B).size_eq] |
| exact (D.collect_invariant B).safe.complete_bound hc |
|
|
| theorem completed_size_eq_card (B : ℕ) |
| (hbounded : ∀ e ∈ D.rawEntries, WeightEncoding.BoundedWeight D.input e.value) |
| (hc : (D.collect B).state.user.keys.overflow = false) : |
| (D.entries B).size = D.entryUniverse.card := by |
| classical |
| have he : (D.entries B).toList.toFinset = D.entryUniverse := by |
| ext e |
| simp only [List.mem_toFinset, entryUniverse, D.entries_mem_iff_of_complete B hc hbounded] |
| rw [← he, List.toFinset_card_of_nodup (D.entries_nodup B), Array.length_toList] |
|
|
| theorem collect_clear_iff_card (B : ℕ) |
| (hbounded : ∀ e ∈ D.rawEntries, WeightEncoding.BoundedWeight D.input e.value) : |
| (D.collect B).state.user.keys.overflow = false ↔ D.entryUniverse.card ≤ D.capacity := by |
| classical |
| constructor |
| · intro hc |
| have hm := D.entries_mem_iff_of_complete B hc hbounded |
| have hsub : D.entryUniverse ⊆ (D.entries B).toList.toFinset := by |
| intro e he |
| exact List.mem_toFinset.mpr ((hm e).mpr (List.mem_toFinset.mp he)) |
| calc |
| D.entryUniverse.card ≤ (D.entries B).toList.toFinset.card := Finset.card_le_card hsub |
| _ ≤ (D.entries B).toList.length := List.toFinset_card_le _ |
| _ = (D.entries B).size := Array.length_toList |
| _ ≤ D.capacity := by |
| rw [entries, (D.collect_invariant B).size_eq] |
| exact (D.collect_invariant B).safe.complete_bound hc |
| · intro hc |
| exact (D.collect_fits B D.entryUniverse hc (fun e he => List.mem_toFinset.mpr he)).1 |
|
|
| theorem collect_overflow_iff_card (B : ℕ) |
| (hbounded : ∀ e ∈ D.rawEntries, WeightEncoding.BoundedWeight D.input e.value) : |
| (D.collect B).state.user.keys.overflow = true ↔ D.capacity < D.entryUniverse.card := by |
| rw [← not_le, ← D.collect_clear_iff_card B hbounded] |
| cases (D.collect B).state.user.keys.overflow <;> simp |
| |
| end LowSpaceSubsetSum.RepresentationParentData |
| |
| end |
| |
| end OAI |
| |
