File size: 3,129 Bytes
5d5048e | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 | import OAI.Combinatorics.LowSpaceSubsetSum.Representation.RepresentationParentComplete
namespace OAI
/-! # Exact overflow decisions for the distinct-entry analysis universe -/
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
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