| import OAI.Combinatorics.MatrixRemoval.HostSampling |
| import OAI.Combinatorics.MatrixRemoval.GuardedPath |
|
|
| namespace OAI |
|
|
| /-! |
| Containment of guarded-path cells in the protected sampling set. |
| The geometric validity of a selection and the cellwise counting bound |
| are separate requirements. |
| -/ |
|
|
| namespace Problem348.Construction.HostSampling |
|
|
| open Problem348.TreeSampling Problem348.GuardedPath |
|
|
| variable {h n : ℕ} |
|
|
| / |
| def RowChosen (row : Position h ≃ Fin n) (s : Seeds (2 ^ h)) (r : Fin n) : Prop := |
| coordinate s.1 s.2.1 (row.symm r) = offset (row.symm r) |
|
|
| / |
| def ColChosen (col : Position h ≃ Fin n) (s : Seeds (2 ^ h)) (c : Fin n) : Prop := |
| coordinate s.1 s.2.2 (col.symm c) = offset (col.symm c) |
|
|
| / |
| def GoodCell (row col : Position h ≃ Fin n) (s : Seeds (2 ^ h)) |
| (rc : Fin n × Fin n) : Prop := |
| Protected (row.symm rc.1) (col.symm rc.2) ∧ |
| Selected s (row.symm rc.1) (col.symm rc.2) |
|
|
| theorem goodCell_of_chosen {row col : Position h ≃ Fin n} {s : Seeds (2 ^ h)} |
| {r c : Fin n} (hp : Protected (row.symm r) (col.symm c)) |
| (hr : RowChosen row s r) (hc : ColChosen col s c) : |
| GoodCell row col s (r, c) := |
| ⟨hp, Prod.ext hr hc⟩ |
|
|
| / |
| structure FrameChosen (row col : Position h ≃ Fin n) (s : Seeds (2 ^ h)) |
| (F : AnchorFrame n) : Prop where |
| row_independent : ∀ u, sampleDepth (row.symm (F.rows.val u)) = none |
| col_independent : ∀ v, sampleDepth (col.symm (F.cols.val v)) = none |
| row_chosen : ∀ u, RowChosen row s (F.rows.val u) |
| col_chosen : ∀ v, ColChosen col s (F.cols.val v) |
|
|
| / |
| theorem FrameChosen.cell_good {row col : Position h ≃ Fin n} {s : Seeds (2 ^ h)} |
| {F : AnchorFrame n} (hF : FrameChosen row col s F) |
| {r₀ r₁ c₀ c₁ : Fin n} |
| (hr₀ : RowChosen row s r₀) (hr₁ : RowChosen row s r₁) |
| (hc₀ : ColChosen col s c₀) (hc₁ : ColChosen col s c₁) |
| {rc : Fin n × Fin n} (hcell : F.Cell r₀ r₁ c₀ c₁ rc) : |
| GoodCell row col s rc := by |
| rcases hcell with ⟨u, v, rfl⟩ | ⟨v, rfl | rfl⟩ | ⟨u, rfl | rfl⟩ |
| · exact goodCell_of_chosen (Or.inl (hF.row_independent u)) |
| (hF.row_chosen u) (hF.col_chosen v) |
| · exact goodCell_of_chosen (Or.inr (Or.inl (hF.col_independent v))) |
| hr₀ (hF.col_chosen v) |
| · exact goodCell_of_chosen (Or.inr (Or.inl (hF.col_independent v))) |
| hr₁ (hF.col_chosen v) |
| · exact goodCell_of_chosen (Or.inl (hF.row_independent u)) |
| (hF.row_chosen u) hc₀ |
| · exact goodCell_of_chosen (Or.inl (hF.row_independent u)) |
| (hF.row_chosen u) hc₁ |
|
|
| / |
| structure PathChosen (row col : Position h ≃ Fin n) (s : Seeds (2 ^ h)) |
| (P : Selection n) : Prop where |
| xp_chosen : ∀ i, i ≤ h → RowChosen row s (P.xp i) |
| xm_chosen : ∀ i, i ≤ h → RowChosen row s (P.xm i) |
| yp_chosen : ∀ i, i ≤ h → ColChosen col s (P.yp i) |
| ym_chosen : ∀ i, i ≤ h → ColChosen col s (P.ym i) |
| xd_chosen : RowChosen row s P.xd |
| yd_chosen : ColChosen col s P.yd |
| xd_independent : sampleDepth (row.symm P.xd) = none |
| yd_independent : sampleDepth (col.symm P.yd) = none |
| root_plus : Protected (row.symm (P.xp 0)) (col.symm (P.yp 0)) |
| root_minus : Protected (row.symm (P.xm 0)) (col.symm (P.ym 0)) |
| vertical_plus : ∀ i, i < h → FrameChosen row col s (P.vPlus i) |
| vertical_minus : ∀ i, i < h → FrameChosen row col s (P.vMinus i) |
| horizontal_plus : ∀ i, i < h → FrameChosen row col s (P.wPlus i) |
| horizontal_minus : ∀ i, i < h → FrameChosen row col s (P.wMinus i) |
|
|
| theorem PathChosen.core_good {row col : Position h ≃ Fin n} {s : Seeds (2 ^ h)} |
| {P : Selection n} (hP : PathChosen row col s P) |
| {rc : Fin n × Fin n} (hc : P.CoreCell h rc) : GoodCell row col s rc := by |
| rcases hc with rfl | rfl | ⟨i, hi, rfl | rfl | rfl | rfl⟩ |
| · exact goodCell_of_chosen hP.root_plus |
| (hP.xp_chosen 0 (Nat.zero_le _)) (hP.yp_chosen 0 (Nat.zero_le _)) |
| · exact goodCell_of_chosen hP.root_minus |
| (hP.xm_chosen 0 (Nat.zero_le _)) (hP.ym_chosen 0 (Nat.zero_le _)) |
| · exact goodCell_of_chosen (Or.inr (Or.inl hP.yd_independent)) |
| (hP.xp_chosen i hi) hP.yd_chosen |
| · exact goodCell_of_chosen (Or.inr (Or.inl hP.yd_independent)) |
| (hP.xm_chosen i hi) hP.yd_chosen |
| · exact goodCell_of_chosen (Or.inl hP.xd_independent) |
| hP.xd_chosen (hP.yp_chosen i hi) |
| · exact goodCell_of_chosen (Or.inl hP.xd_independent) |
| hP.xd_chosen (hP.ym_chosen i hi) |
|
|
| theorem PathChosen.edge_good {row col : Position h ≃ Fin n} {s : Seeds (2 ^ h)} |
| {P : Selection n} (hP : PathChosen row col s P) {i : ℕ} (hi : i < h) |
| {rc : Fin n × Fin n} (hc : P.EdgeCell i rc) : GoodCell row col s rc := by |
| have hil : i ≤ h := Nat.le_of_lt hi |
| have his : i + 1 ≤ h := hi |
| rcases hc with hc | hc | hc | hc |
| · exact (hP.vertical_plus i hi).cell_good |
| (hP.xp_chosen (i + 1) his) (hP.xp_chosen i hil) |
| hP.yd_chosen (hP.yp_chosen i hil) hc |
| · exact (hP.vertical_minus i hi).cell_good |
| (hP.xm_chosen i hil) (hP.xm_chosen (i + 1) his) |
| hP.yd_chosen (hP.ym_chosen i hil) hc |
| · exact (hP.horizontal_plus i hi).cell_good |
| (hP.xp_chosen (i + 1) his) hP.xd_chosen |
| (hP.yp_chosen i hil) (hP.yp_chosen (i + 1) his) hc |
| · exact (hP.horizontal_minus i hi).cell_good |
| (hP.xm_chosen (i + 1) his) hP.xd_chosen |
| (hP.ym_chosen (i + 1) his) (hP.ym_chosen i hil) hc |
|
|
| / |
| theorem PathChosen.guard_good {row col : Position h ≃ Fin n} {s : Seeds (2 ^ h)} |
| {P : Selection n} (hP : PathChosen row col s P) |
| {rc : Fin n × Fin n} (hc : rc ∈ P.guards h) : GoodCell row col s rc := by |
| rw [Selection.mem_guards] at hc |
| rcases hc with hc | ⟨i, hi, hc⟩ |
| · exact hP.core_good hc |
| · exact hP.edge_good hi hc |
|
|
| / |
| theorem path_guard_frequency (row col : Position h ≃ Fin n) |
| (P : Seeds (2 ^ h) → Selection n) |
| (hP : ∀ s, PathChosen row col s (P s)) (rc : Fin n × Fin n) : |
| (Finset.univ.filter fun s => rc ∈ (P s).guards h).card * (2 ^ h) ^ 2 ≤ |
| Fintype.card (Seeds (2 ^ h)) := by |
| classical |
| exact transported_guard_frequency row col (fun s => (P s).guards h) |
| (fun s _ hc => (hP s).guard_good hc) rc |
|
|
| end Problem348.Construction.HostSampling |
|
|
| end OAI |
|
|
