| import OAI.Combinatorics.MatrixRemoval.Host |
| import OAI.Combinatorics.MatrixRemoval.ThreeSeedSampling |
|
|
| namespace OAI |
|
|
| /-! |
| The canonical host projections and protected-cell sampling bridge. |
| Three seeds give the cellwise sampling bound. |
| -/ |
|
|
| namespace Problem348.Construction.HostSampling |
|
|
| open Problem348.TreeSampling |
|
|
| / |
| def sampleDepth {h : ℕ} : Position h → Option (Fin (h + 1)) |
| | Sum.inl _ => none |
| | Sum.inr (Sum.inl v) => some ⟨depth v, Nat.lt_succ_of_le (depth_le v)⟩ |
| | Sum.inr (Sum.inr _) => none |
|
|
| / |
| def offset {h : ℕ} : Position h → Fin (2 ^ h) |
| | Sum.inl (_, _, x) => x |
| | Sum.inr (Sum.inl v) => v.2 |
| | Sum.inr (Sum.inr x) => x |
|
|
| / |
| def coordinate {h : ℕ} (z u : Fin (2 ^ h)) : Position h → Fin (2 ^ h) |
| | Sum.inl _ => u |
| | Sum.inr (Sum.inl v) => atLevel h (depth v) (depth_le v) z u |
| | Sum.inr (Sum.inr _) => u |
|
|
| / |
| def Selected {h : ℕ} (s : Seeds (2 ^ h)) (r c : Position h) : Prop := |
| (coordinate s.1 s.2.1 r, coordinate s.1 s.2.2 c) = (offset r, offset c) |
|
|
| instance {h : ℕ} (s : Seeds (2 ^ h)) (r c : Position h) : |
| Decidable (Selected s r c) := inferInstanceAs (Decidable (_ = _)) |
|
|
| / |
| def Protected {h : ℕ} (r c : Position h) : Prop := |
| sampleDepth r = none ∨ sampleDepth c = none ∨ |
| sampleDepth r = some ⟨0, by omega⟩ ∧ sampleDepth c = some ⟨0, by omega⟩ |
|
|
| @[simp] theorem protected_anchor_left {h : ℕ} (a : AnchorPosition h) |
| (c : Position h) : Protected (Sum.inl a) c := by |
| simp [Protected, sampleDepth] |
|
|
| @[simp] theorem protected_anchor_right {h : ℕ} (r : Position h) |
| (a : AnchorPosition h) : Protected r (Sum.inl a) := by |
| simp [Protected, sampleDepth] |
|
|
| @[simp] theorem protected_dummy_left {h : ℕ} (x : Fin (2 ^ h)) |
| (c : Position h) : Protected (Sum.inr (Sum.inr x)) c := by |
| simp [Protected, sampleDepth] |
|
|
| @[simp] theorem protected_dummy_right {h : ℕ} (r : Position h) |
| (x : Fin (2 ^ h)) : Protected r (Sum.inr (Sum.inr x)) := by |
| simp [Protected, sampleDepth] |
|
|
| theorem protected_roots {h : ℕ} (v w : VariablePosition h) |
| (hv : depth v = 0) (hw : depth w = 0) : |
| Protected (Sum.inr (Sum.inl v)) (Sum.inr (Sum.inl w)) := by |
| simp [Protected, sampleDepth, hv, hw] |
|
|
| / |
| theorem selected_fiber_card {h : ℕ} (r c : Position h) (hp : Protected r c) : |
| Fintype.card {s : Seeds (2 ^ h) // Selected s r c} = 2 ^ h := by |
| rcases r with ⟨t, u, x⟩ | (v | x) <;> |
| rcases c with ⟨t', u', y⟩ | (w | y) |
| · exact card_both_fiber x y |
| · exact card_atLevel_column_fiber h (depth w) (depth_le w) x w.2 |
| · exact card_both_fiber x y |
| · exact card_atLevel_row_fiber h (depth v) (depth_le v) v.2 y |
| · have hd : depth v = 0 ∧ depth w = 0 := by |
| simpa [Protected, sampleDepth, Fin.ext_iff] using hp |
| have he : ∀ s : Seeds (2 ^ h), Selected s (Sum.inr (Sum.inl v)) |
| (Sum.inr (Sum.inl w)) ↔ s.2 = (v.2, w.2) := by |
| intro s |
| simp [Selected, coordinate, offset, hd.1, hd.2] |
| calc |
| _ = Fintype.card {s : Seeds (2 ^ h) // s.2 = (v.2, w.2)} := |
| Fintype.card_congr (Equiv.subtypeEquivRight he) |
| _ = _ := card_both_fiber v.2 w.2 |
| · exact card_atLevel_row_fiber h (depth v) (depth_le v) v.2 y |
| · exact card_both_fiber x y |
| · exact card_atLevel_column_fiber h (depth w) (depth_le w) x w.2 |
| · exact card_both_fiber x y |
|
|
| / |
| theorem selected_fiber_card_mul {h : ℕ} (r c : Position h) (hp : Protected r c) : |
| Fintype.card {s : Seeds (2 ^ h) // Selected s r c} * (2 ^ h) ^ 2 = |
| Fintype.card (Seeds (2 ^ h)) := by |
| rw [selected_fiber_card r c hp, card_seeds] |
| ring |
|
|
| / |
| theorem selected_filter_card {h : ℕ} (r c : Position h) (hp : Protected r c) : |
| (Finset.univ.filter fun s : Seeds (2 ^ h) => Selected s r c).card = 2 ^ h := by |
| simpa only [Fintype.card_subtype] using selected_fiber_card r c hp |
|
|
| / |
| theorem protected_event_card_le {h : ℕ} (r c : Position h) |
| (event : Finset (Seeds (2 ^ h))) |
| (hevent : ∀ s ∈ event, Protected r c ∧ Selected s r c) : |
| event.card ≤ 2 ^ h := by |
| classical |
| rcases event.eq_empty_or_nonempty with hempty | ⟨s, hs⟩ |
| · simp [hempty] |
| · have hp := (hevent s hs).1 |
| calc |
| event.card ≤ (Finset.univ.filter fun s => Selected s r c).card := by |
| apply Finset.card_le_card |
| intro t ht |
| simp only [Finset.mem_filter, Finset.mem_univ, true_and] |
| exact (hevent t ht).2 |
| _ = _ := selected_filter_card r c hp |
|
|
| / |
| the union-bound frequency estimate for each fixed raw cell. -/ |
| theorem guard_frequency {h : ℕ} |
| (guards : Seeds (2 ^ h) → Finset (Position h × Position h)) |
| (hguards : ∀ s rc, rc ∈ guards s → |
| Protected rc.1 rc.2 ∧ Selected s rc.1 rc.2) |
| (rc : Position h × Position h) : |
| (Finset.univ.filter fun s => rc ∈ guards s).card * (2 ^ h) ^ 2 ≤ |
| Fintype.card (Seeds (2 ^ h)) := by |
| classical |
| have hb := protected_event_card_le rc.1 rc.2 |
| (Finset.univ.filter fun s => rc ∈ guards s) |
| (fun s hs => hguards s rc (Finset.mem_filter.mp hs).2) |
| rw [card_seeds] |
| calc |
| _ ≤ 2 ^ h * (2 ^ h) ^ 2 := Nat.mul_le_mul_right _ hb |
| _ = (2 ^ h) ^ 3 := by ring |
|
|
| / |
| theorem transported_guard_frequency {h n : ℕ} |
| (row col : Position h ≃ Fin n) |
| (guards : Seeds (2 ^ h) → Finset (Fin n × Fin n)) |
| (hguards : ∀ s rc, rc ∈ guards s → |
| Protected (row.symm rc.1) (col.symm rc.2) ∧ |
| Selected s (row.symm rc.1) (col.symm rc.2)) |
| (rc : Fin n × Fin n) : |
| (Finset.univ.filter fun s => rc ∈ guards s).card * (2 ^ h) ^ 2 ≤ |
| Fintype.card (Seeds (2 ^ h)) := by |
| classical |
| have hb := protected_event_card_le (row.symm rc.1) (col.symm rc.2) |
| (Finset.univ.filter fun s => rc ∈ guards s) |
| (fun s hs => hguards s rc (Finset.mem_filter.mp hs).2) |
| rw [card_seeds] |
| calc |
| _ ≤ 2 ^ h * (2 ^ h) ^ 2 := Nat.mul_le_mul_right _ hb |
| _ = (2 ^ h) ^ 3 := by ring |
|
|
| end Problem348.Construction.HostSampling |
|
|
| end OAI |
|
|
