| import OAI.Combinatorics.MatrixRemoval.HostSampledSelection |
| import OAI.Combinatorics.MatrixRemoval.HostSamplingCompatibility |
|
|
| namespace OAI |
|
|
| /-! |
| # Arbitrary-repair lower bound for the actual canonical host |
|
|
| The concrete common-leaf selection satisfies both independent certificates: |
| its guarded path is valid, and its chosen coordinates bound each guard cell's |
| sampling multiplicity. |
| Together these prove the integer repair lower bound. No assumption |
| about preservation of arbitrary variable-variable cells is made. |
| -/ |
|
|
| noncomputable section |
| namespace Problem348.Construction.SampledSelection |
|
|
| open HostSampledPath HostSampling TreeSampling |
|
|
| / |
| theorem frame_chosen {h n : ℕ} (er ec : Position h ≃ Fin n) |
| (ha : AxisConditions er ec) (s : Seeds (2 ^ h)) (t : Mode h) : |
| FrameChosen er ec s (frame er ec ha t s.2.1 s.2.2) := by |
| constructor |
| · intro u |
| simp [frame, SelectedAnchor.frame, sampleDepth] |
| · intro v |
| simp [frame, SelectedAnchor.frame, sampleDepth] |
| · intro u |
| exact rowChosen_path_anchor er s t u |
| · intro v |
| exact colChosen_path_anchor ec s t v |
|
|
| / |
| coordinate equations. This is separate from all matrix-entry claims. -/ |
| theorem selection_chosen {h n : ℕ} (hh : 1 ≤ h) (er ec : Position h ≃ Fin n) |
| (ha : AxisConditions er ec) (s : Seeds (2 ^ h)) : |
| PathChosen er ec s (selection hh er ec ha s.1 s.2.1 s.2.2) := by |
| constructor |
| · intro i hi |
| simpa only [selection, path_of_le hi] using rowChosen_path er i hi true s |
| · intro i hi |
| simpa only [selection, path_of_le hi] using rowChosen_path er i hi false s |
| · intro i hi |
| simpa only [selection, path_of_le hi] using colChosen_path ec i hi true s |
| · intro i hi |
| simpa only [selection, path_of_le hi] using colChosen_path ec i hi false s |
| · exact rowChosen_path_dummy er s |
| · exact colChosen_path_dummy ec s |
| · simp [selection, HostSampledPath.dummy, sampleDepth] |
| · simp [selection, HostSampledPath.dummy, sampleDepth] |
| · simpa only [selection, Equiv.symm_apply_apply, path_of_le (Nat.zero_le h)] using |
| protected_path_roots h true true s.1 s.2.1 s.2.2 |
| · simpa only [selection, Equiv.symm_apply_apply, path_of_le (Nat.zero_le h)] using |
| protected_path_roots h false false s.1 s.2.1 s.2.2 |
| · intro i hi |
| exact frame_chosen er ec ha s _ |
| · intro i hi |
| exact frame_chosen er ec ha s _ |
| · intro i hi |
| exact frame_chosen er ec ha s _ |
| · intro i hi |
| exact frame_chosen er ec ha s _ |
|
|
| / |
| of the three-seed sample space, for the actual selected paths. -/ |
| theorem selection_guard_frequency {h n : ℕ} (hh : 1 ≤ h) |
| (er ec : Position h ≃ Fin n) (ha : AxisConditions er ec) |
| (rc : Fin n × Fin n) : |
| (Finset.univ.filter fun s : Seeds (2 ^ h) => |
| rc ∈ (selection hh er ec ha s.1 s.2.1 s.2.2).guards h).card * (2 ^ h) ^ 2 ≤ |
| Fintype.card (Seeds (2 ^ h)) := |
| path_guard_frequency er ec |
| (fun s => selection hh er ec ha s.1 s.2.1 s.2.2) |
| (fun s => selection_chosen hh er ec ha s) rc |
|
|
| / |
| The independent axis enumerations satisfy `AxisConditions`: increasing anchor |
| groups precede all nonanchors, variable rows precede dummy rows, dummy columns |
| precede variable columns, and sampled parent/child pairs obey the four signed |
| tree-order inequalities. No entry-preservation condition is imposed on repairs. -/ |
| theorem hammingDistance_ge {h n : ℕ} (hh : 1 ≤ h) |
| (er ec : Position h ≃ Fin n) (ha : AxisConditions er ec) |
| (B : BinaryMatrix n) (hfree : HFree fixedH B) : |
| (2 ^ h) ^ 2 ≤ hammingDistance (SelectedAnchor.matrix er ec) B := by |
| classical |
| apply Sampling.hammingDistance_ge_of_guard_sampling fixedH |
| (SelectedAnchor.matrix er ec) B |
| (fun s : Seeds (2 ^ h) => (selection hh er ec ha s.1 s.2.1 s.2.2).guards h) |
| (2 ^ h) |
| · rw [card_seeds] |
| positivity |
| · exact selection_guard_frequency hh er ec ha |
| · intro s hag |
| exact GuardedPath.Selection.not_HFree_of_agree |
| (selection_valid hh er ec ha s.1 s.2.1 s.2.2) hag |
| · exact hfree |
|
|
| end Problem348.Construction.SampledSelection |
|
|
| end |
|
|
| end OAI |
|
|
