| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.Support |
| import OAI.LinearAlgebra.MatrixMultiplication.Recovery.InheritedOrbitProfiles |
|
|
| namespace OAI |
|
|
| /-! Finite extraction histories, inherited masks and recovery bounds. -/ |
|
|
| noncomputable section |
|
|
| namespace MatrixMultiplication.AllFieldHistoryOrbitCount |
|
|
| open AllFieldHistory AllFieldHistorySupport PermutationMatching |
| open scoped BigOperators |
| attribute [local instance] Classical.propDecidable Classical.decEq |
|
|
| abbrev Classes (K tick : ℕ) := ActiveClass K tick |
|
|
| abbrev Positions {K tick : ℕ} (allocation : Allocation) (m : ℕ) |
| (c : Classes K tick) := |
| JointCanonicalization.ClassPositions (activeCounts allocation m) c |
|
|
| abbrev Letters {K tick : ℕ} (c : Classes K tick) := |
| Fin (activeHalfLength c.1) → Fin 7 |
|
|
| abbrev ActualOrbit {K tick : ℕ} (allocation : Allocation) (m : ℕ) := |
| PairOrbit (Positions (K := K) (tick := tick) allocation m) |
| (Letters (K := K) (tick := tick)) (Letters (K := K) (tick := tick)) |
|
|
| noncomputable instance actualOrbitFintype {K tick : ℕ} |
| (allocation : Allocation) (m : ℕ) : |
| Fintype (ActualOrbit (K := K) (tick := tick) allocation m) := by |
| haveI : Finite (ActualOrbit (K := K) (tick := tick) allocation m) := |
| Finite.of_injective pairOrbitProfile pairOrbitProfile_injective |
| exact Fintype.ofFinite _ |
|
|
| def basePopulationBound {K tick : ℕ} (allocation : Allocation) : ℕ := |
| ∑ c : Classes K tick, activeCounts allocation 1 c.1 c.2 |
|
|
| def orbitDegree (K tick : ℕ) : ℕ := |
| 2 * ∑ c : Classes K tick, 7 ^ activeHalfLength c.1 |
|
|
| theorem card_letters {K tick : ℕ} (c : Classes K tick) : |
| Fintype.card (Letters c) = 7 ^ activeHalfLength c.1 := by |
| simp only [Letters, Fintype.card_fun, Fintype.card_fin] |
|
|
| theorem class_card_le {K tick : ℕ} (allocation : Allocation) (m : ℕ) |
| (c : Classes K tick) : |
| Fintype.card (Positions allocation m c) ≤ |
| basePopulationBound (K := K) (tick := tick) allocation * m := by |
| have hc : activeCounts allocation 1 c.1 c.2 ≤ |
| basePopulationBound (K := K) (tick := tick) allocation := by |
| unfold basePopulationBound |
| exact Finset.single_le_sum |
| (fun (d : Classes K tick) _ => Nat.zero_le (activeCounts allocation 1 d.1 d.2)) |
| (Finset.mem_univ c) |
| simp only [Positions, JointCanonicalization.ClassPositions, Fintype.card_fin, |
| activeCounts_dilation allocation m] |
| simpa only [Nat.mul_comm] using Nat.mul_le_mul_left m hc |
|
|
| theorem card_actualOrbit_le {K tick : ℕ} (allocation : Allocation) (m : ℕ) : |
| Nat.card (ActualOrbit (K := K) (tick := tick) allocation m) ≤ |
| (basePopulationBound (K := K) (tick := tick) allocation * m + 1) ^ |
| orbitDegree K tick := by |
| simpa only [ActualOrbit, orbitDegree, card_letters, two_mul] using |
| (card_pairOrbits_le_common_bound |
| (P := Positions (K := K) (tick := tick) allocation m) |
| (X := Letters (K := K) (tick := tick)) |
| (Y := Letters (K := K) (tick := tick)) |
| (basePopulationBound (K := K) (tick := tick) allocation * m) |
| (class_card_le allocation m)) |
|
|
| theorem card_actualOrbit_le_polynomial {K tick : ℕ} |
| (allocation : Allocation) (m : ℕ) : |
| Nat.card (ActualOrbit (K := K) (tick := tick) allocation m) ≤ |
| (basePopulationBound (K := K) (tick := tick) allocation + 1) ^ orbitDegree K tick * |
| (m + 1) ^ orbitDegree K tick := by |
| refine (card_actualOrbit_le allocation m).trans ?_ |
| rw [← mul_pow] |
| apply Nat.pow_le_pow_left |
| nlinarith [Nat.zero_le (basePopulationBound (K := K) (tick := tick) allocation)] |
|
|
| def orbitPrefactor {K tick : ℕ} (allocation : Allocation) : ℝ := |
| ((basePopulationBound (K := K) (tick := tick) allocation + 1 : ℕ) : ℝ) ^ |
| orbitDegree K tick |
|
|
| theorem orbitPrefactor_pos {K tick : ℕ} (allocation : Allocation) : |
| 0 < orbitPrefactor (K := K) (tick := tick) allocation := by |
| unfold orbitPrefactor |
| positivity |
|
|
| theorem card_actualOrbit_le_polynomial_real {K tick : ℕ} |
| (allocation : Allocation) (m : ℕ) : |
| (Nat.card (ActualOrbit (K := K) (tick := tick) allocation m) : ℝ) ≤ |
| orbitPrefactor (K := K) (tick := tick) allocation * |
| ((m : ℝ) + 1) ^ orbitDegree K tick := by |
| unfold orbitPrefactor |
| exact_mod_cast card_actualOrbit_le_polynomial (K := K) (tick := tick) allocation m |
|
|
| theorem usedOrbits_card_le {K tick : ℕ} (allocation : Allocation) (m : ℕ) |
| (s : Finset (ActualOrbit (K := K) (tick := tick) allocation m)) : |
| s.card ≤ (basePopulationBound (K := K) (tick := tick) allocation + 1) ^ |
| orbitDegree K tick * (m + 1) ^ orbitDegree K tick := by |
| have hs : s.card ≤ Nat.card (ActualOrbit (K := K) (tick := tick) allocation m) := by |
| rw [Nat.card_eq_fintype_card] |
| exact s.card_le_univ |
| exact hs.trans (card_actualOrbit_le_polynomial allocation m) |
|
|
| theorem usedOrbits_card_le_real {K tick : ℕ} (allocation : Allocation) (m : ℕ) |
| (s : Finset (ActualOrbit (K := K) (tick := tick) allocation m)) : |
| (s.card : ℝ) ≤ orbitPrefactor (K := K) (tick := tick) allocation * |
| ((m : ℝ) + 1) ^ orbitDegree K tick := by |
| unfold orbitPrefactor |
| exact_mod_cast usedOrbits_card_le allocation m s |
|
|
| theorem taggedUsedOrbits_card_le_real {K tick : ℕ} (allocation : Allocation) (m : ℕ) |
| (s : Finset (Fin 3 × ActualOrbit (K := K) (tick := tick) allocation m)) : |
| (s.card : ℝ) ≤ (3 * orbitPrefactor (K := K) (tick := tick) allocation) * |
| ((m : ℝ) + 1) ^ orbitDegree K tick := by |
| have hs : s.card ≤ 3 * Nat.card |
| (ActualOrbit (K := K) (tick := tick) allocation m) := by |
| simpa only [Fintype.card_prod, Fintype.card_fin, Nat.card_eq_fintype_card] using |
| s.card_le_univ |
| have hsreal : (s.card : ℝ) ≤ 3 * (Nat.card |
| (ActualOrbit (K := K) (tick := tick) allocation m) : ℝ) := by |
| exact_mod_cast hs |
| calc |
| _ ≤ 3 * (Nat.card (ActualOrbit (K := K) (tick := tick) allocation m) : ℝ) := hsreal |
| _ ≤ 3 * (orbitPrefactor (K := K) (tick := tick) allocation * |
| ((m : ℝ) + 1) ^ orbitDegree K tick) := |
| mul_le_mul_of_nonneg_left (card_actualOrbit_le_polynomial_real allocation m) |
| (by norm_num) |
| _ = _ := (mul_assoc _ _ _).symm |
|
|
| end MatrixMultiplication.AllFieldHistoryOrbitCount |
|
|
| end |
|
|
| end OAI |
|
|
