| import OAI.Combinatorics.LowSpaceSubsetSum.Word.WordSeedPreparedBind |
| import OAI.Combinatorics.LowSpaceSubsetSum.Word.WordSeedDraws |
| import OAI.Combinatorics.LowSpaceSubsetSum.Word.WordSeedPermutationExecution |
| import OAI.Combinatorics.LowSpaceSubsetSum.Storage.FixedSizeSampleCells |
|
|
| namespace OAI |
|
|
| /-! # Building sampled subsets from stored permutation entries |
|
|
| The sampled permutation is materialized and its first selected entries are |
| scanned into a duplicate-free list. The Finset constructor uses that list |
| without a further deduplication pass. |
| -/ |
|
|
| noncomputable section |
| namespace LowSpaceSubsetSum.WordSeedMeasured |
|
|
| abbrev PermutationAccepted (n g : ℕ) := Accepted (permutationCoupling n g) |
|
|
| theorem permutation_materializes (n g : ℕ) (a : PermutationAccepted n g) : |
| ∃ v, (PermutationSamplerCells.materialize n g a.seed.val).1.1 = some v := by |
| have hv := PermutationDigitsExecution.value n g a.seed.val |
| rw [(SamplerExecution.permutation n g).value_eq,accepted_source a] at hv |
| cases hd : (PermutationDigitsExecution.run n g a.seed.val).value with |
| | none => simp only [hd,Option.map_none,reduceCtorEq] at hv |
| | some d => |
| refine ⟨(PermutationArray.build g d).1,?_⟩ |
| rw [PermutationSamplerCells.materialize_value] |
| simp only [PermutationDigitsExecution.materialize,hd] |
|
|
| theorem permutation_materialized_value (n g : ℕ) (a : PermutationAccepted n g) |
| (v : Vector (Fin g) g) |
| (hv : (PermutationSamplerCells.materialize n g a.seed.val).1.1 = some v) : |
| ∀ i : Fin g, v[i.val] = a.value i := by |
| rw [PermutationSamplerCells.materialize_value] at hv |
| obtain ⟨e,he,hv⟩ := PermutationDigitsExecution.materialize_value n g a.seed.val v hv |
| rw [(SamplerExecution.permutation n g).value_eq,accepted_source a] at he |
| cases Option.some.inj he |
| exact hv |
|
|
| abbrev Materialized (n g : ℕ) (a : PermutationAccepted n g) := |
| {v : Vector (Fin g) g // |
| (PermutationSamplerCells.materialize n g a.seed.val).1.1 = some v} |
|
|
| def materializeAccepted (n g : ℕ) (a : PermutationAccepted n g) : |
| (Materialized n g a × ℕ) × ℕ := |
| let r := PermutationSamplerCells.materialize n g a.seed.val |
| match h : r.1.1 with |
| | none => False.elim (by |
| obtain ⟨v,hv⟩ := permutation_materializes n g a |
| cases h.symm.trans hv) |
| | some v => ((⟨v,h⟩,g*n^3+r.1.2+1),r.2+8) |
|
|
| def selectedCompute (n g k : ℕ) (_hk : k ≤ g) (a : PermutationAccepted n g) : |
| (Finset (Fin g) × ℕ) × ℕ := |
| let material := materializeAccepted n g a |
| let v := material.1.1 |
| let xs := FixedSizeSampleCells.selected v.val k |
| let set : Finset (Fin g) := ⟨(xs.1 : Multiset (Fin g)),by |
| rw [FixedSizeSampleCells.prefix_value] |
| change (((List.finRange g).filter (fun i => decide (i.val < k))).map |
| (fun i => v.val[i.val])).Nodup |
| apply List.Nodup.map |
| · intro i j h |
| have hv := permutation_materialized_value n g a v.val v.property |
| exact a.value.injective ((hv i).symm.trans (h.trans (hv j))) |
| · exact (List.nodup_finRange g).filter _⟩ |
| ((set,material.1.2+4*g+3),material.2+xs.2+g+8) |
|
|
| theorem selectedCompute_value (n g k : ℕ) (hk : k ≤ g) (a : PermutationAccepted n g) : |
| (selectedCompute n g k hk a).1.1 = (permutationBlock (initialSubset g k hk) a.value).val := by |
| let v := (materializeAccepted n g a).1.1 |
| have hv := permutation_materialized_value n g a v.val v.property |
| have h := PermutationSubsetCode.prefix_value v.val a.value hv k hk |
| apply Finset.ext |
| intro i |
| change i ∈ (FixedSizeSampleCells.selected v.val k).1 ↔ _ |
| rw [FixedSizeSampleCells.prefix_value] |
| have hm := Finset.ext_iff.mp h i |
| simpa only [List.mem_toFinset,permutationBlock] using hm |
|
|
| theorem selectedCompute_work (n g k : ℕ) (hk : k ≤ g) (a : PermutationAccepted n g) : |
| (selectedCompute n g k hk a).1.2 ≤ g*n^3+10*(g+1)^2+4*g+5 := by |
| have hm := PermutationDigitsExecution.materialize_work n g a.seed.val |
| rw [←PermutationSamplerCells.materialize_value] at hm |
| obtain ⟨v,hv⟩ := permutation_materializes n g a |
| unfold selectedCompute materializeAccepted |
| dsimp only |
| split |
| next h => cases h.symm.trans hv |
| · dsimp only |
| omega |
|
|
| theorem selectedCompute_cells (n g k : ℕ) (hk : k ≤ g) (a : PermutationAccepted n g) : |
| (selectedCompute n g k hk a).2 ≤ (g+1)*(3*n^3+128)+6*g+32 := by |
| have hm := PermutationSamplerCells.materialize_cells n g a.seed.val |
| obtain ⟨v,hv⟩ := permutation_materializes n g a |
| unfold selectedCompute materializeAccepted |
| dsimp only |
| split |
| next h => cases h.symm.trans hv |
| next v' h => |
| have hx := FixedSizeSampleCells.prefix_cells v' k |
| dsimp only |
| omega |
|
|
| def selectedTyped (n g k : ℕ) (hk : k ≤ g) (a : PermutationAccepted n g) : |
| (↥((Finset.univ : Finset (Fin g)).powersetCard (initialSubset g k hk).card) × ℕ) × ℕ := |
| let r := selectedCompute n g k hk a |
| ((⟨r.1.1,by |
| rw [selectedCompute_value] |
| exact (permutationBlock (initialSubset g k hk) a.value).property⟩,r.1.2),r.2) |
|
|
| theorem selectedTyped_value (n g k : ℕ) (hk : k ≤ g) (a : PermutationAccepted n g) : |
| (selectedTyped n g k hk a).1.1 = permutationBlock (initialSubset g k hk) a.value := by |
| apply Subtype.ext |
| exact selectedCompute_value n g k hk a |
|
|
| def subsetInitial (w n g k : ℕ) (hk : k ≤ g) (hw : Nat.clog 2 g ≤ w) : |
| Execution (WordSeedRealization.subset w n g (initialSubset g k hk) hw) := |
| mapStored (permutation w n g hw) (permutationBlock (initialSubset g k hk)) |
| (selectedTyped n g k hk) (selectedTyped_value n g k hk) |
|
|
| def fixedSize (w n g k : ℕ) (hk : k ≤ g) (hw : Nat.clog 2 g ≤ w) : |
| Execution (WordSeedRealization.fixedSize w n g k hk hw) := |
| map (subsetInitial w n g k hk hw) (initialSubsetEquiv g k hk) |
| (fun a => (((initialSubsetEquiv g k hk) a,1),2)) (fun _ => rfl) |
|
|
| theorem fixedSize_bounded (w n g k : ℕ) (hk : k ≤ g) (hw : Nat.clog 2 g ≤ w) : |
| (fixedSize w n g k hk hw).Bounded |
| (g*(14*n^3+5*g+68)+10*(g+1)^2+11) |
| (10*(g+1)*(n^3+256)) (10*(g+1)*(n^3+256)) := by |
| have hs := mapStored_bounded (permutation w n g hw) |
| (permutationBlock (initialSubset g k hk)) (selectedTyped n g k hk) |
| (selectedTyped_value n g k hk) _ _ _ _ _ (permutation_bounded w n g hw) |
| (selectedCompute_work n g k hk) (selectedCompute_cells n g k hk) |
| have ht := map_bounded (subsetInitial w n g k hk hw) (initialSubsetEquiv g k hk) |
| (fun a => (((initialSubsetEquiv g k hk) a,1),2)) (fun _ => rfl) |
| _ _ _ 1 2 hs (fun _ => le_rfl) (fun _ => le_rfl) |
| intro stream cursor |
| have h := ht stream cursor |
| change _ ≤ _ ∧ _ ≤ _ ∧ _ ≤ _ |
| refine ⟨h.1.trans (by nlinarith),h.2.1.trans ?_,h.2.2.trans ?_⟩ |
| · apply max_le |
| · apply max_le <;> nlinarith |
| · nlinarith |
| · nlinarith |
|
|
| end LowSpaceSubsetSum.WordSeedMeasured |
|
|
| end |
|
|
| end OAI |
|
|
