| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.OrbitCount |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.Priority |
|
|
| namespace OAI |
|
|
| /-! Group assignments, orbit counts and extraction capacities. -/ |
|
|
| noncomputable section |
|
|
| namespace MatrixMultiplication.AllFieldGroupOrbitCount |
|
|
| open AllFieldHistory AllFieldHistorySupport PermutationMatching |
| open scoped BigOperators |
| attribute [local instance] Classical.propDecidable Classical.decEq |
|
|
| abbrev GroupClasses (K tick : ℕ) (sigma : Placement) := |
| ActiveOrder K tick sigma × JointPopulation.Shape |
|
|
| abbrev GroupPositions {K tick : ℕ} (allocation : Allocation) (m : ℕ) |
| {sigma : Placement} (c : GroupClasses K tick sigma) := |
| Fin (activeCounts allocation m c.1.val c.2) |
|
|
| abbrev GroupLetters {K tick : ℕ} {sigma : Placement} (c : GroupClasses K tick sigma) := |
| Fin (activeHalfLength c.1.val) → Fin 7 |
|
|
| abbrev GroupOrbit {K tick : ℕ} (allocation : Allocation) (m : ℕ) (sigma : Placement) := |
| PairOrbit (GroupPositions (K := K) (tick := tick) (sigma := sigma) allocation m) |
| (GroupLetters (K := K) (tick := tick) (sigma := sigma)) |
| (GroupLetters (K := K) (tick := tick) (sigma := sigma)) |
|
|
| noncomputable instance groupOrbitFintype {K tick : ℕ} |
| (allocation : Allocation) (m : ℕ) (sigma : Placement) : |
| Fintype (GroupOrbit (K := K) (tick := tick) allocation m sigma) := by |
| haveI : Finite (GroupOrbit (K := K) (tick := tick) allocation m sigma) := |
| Finite.of_injective pairOrbitProfile pairOrbitProfile_injective |
| exact Fintype.ofFinite _ |
|
|
| def groupDegree (K tick : ℕ) (sigma : Placement) : ℕ := |
| 2 * ∑ c : GroupClasses K tick sigma, 7 ^ activeHalfLength c.1.val |
|
|
| theorem card_groupLetters {K tick : ℕ} {sigma : Placement} |
| (c : GroupClasses K tick sigma) : |
| Fintype.card (GroupLetters c) = 7 ^ activeHalfLength c.1.val := by |
| simp only [GroupLetters, Fintype.card_fun, Fintype.card_fin] |
|
|
| theorem groupClass_card_le {K tick : ℕ} (allocation : Allocation) (m : ℕ) |
| {sigma : Placement} (c : GroupClasses K tick sigma) : |
| Fintype.card (GroupPositions allocation m c) ≤ |
| AllFieldHistoryOrbitCount.basePopulationBound (K := K) (tick := tick) allocation * m := |
| AllFieldHistoryOrbitCount.class_card_le allocation m (c.1.val, c.2) |
|
|
| theorem card_groupOrbit_le {K tick : ℕ} |
| (allocation : Allocation) (m : ℕ) (sigma : Placement) : |
| Nat.card (GroupOrbit (K := K) (tick := tick) allocation m sigma) ≤ |
| (AllFieldHistoryOrbitCount.basePopulationBound (K := K) (tick := tick) allocation * m + 1) ^ |
| groupDegree K tick sigma := by |
| simpa only [GroupOrbit, groupDegree, card_groupLetters, two_mul] using |
| (card_pairOrbits_le_common_bound |
| (P := GroupPositions (K := K) (tick := tick) (sigma := sigma) allocation m) |
| (X := GroupLetters (K := K) (tick := tick) (sigma := sigma)) |
| (Y := GroupLetters (K := K) (tick := tick) (sigma := sigma)) |
| (AllFieldHistoryOrbitCount.basePopulationBound (K := K) (tick := tick) allocation * m) |
| (groupClass_card_le allocation m)) |
|
|
| theorem card_groupOrbit_le_polynomial {K tick : ℕ} |
| (allocation : Allocation) (m : ℕ) (sigma : Placement) : |
| Nat.card (GroupOrbit (K := K) (tick := tick) allocation m sigma) ≤ |
| (AllFieldHistoryOrbitCount.basePopulationBound (K := K) (tick := tick) allocation + 1) ^ |
| groupDegree K tick sigma * (m + 1) ^ groupDegree K tick sigma := by |
| refine (card_groupOrbit_le allocation m sigma).trans ?_ |
| rw [← mul_pow] |
| apply Nat.pow_le_pow_left |
| nlinarith [Nat.zero_le |
| (AllFieldHistoryOrbitCount.basePopulationBound (K := K) (tick := tick) allocation)] |
|
|
| def groupOrbitPrefactor {K tick : ℕ} (allocation : Allocation) (sigma : Placement) : ℝ := |
| ((AllFieldHistoryOrbitCount.basePopulationBound (K := K) (tick := tick) allocation + 1 : ℕ) : ℝ) ^ |
| groupDegree K tick sigma |
|
|
| theorem groupOrbitPrefactor_pos {K tick : ℕ} (allocation : Allocation) (sigma : Placement) : |
| 0 < groupOrbitPrefactor (K := K) (tick := tick) allocation sigma := by |
| unfold groupOrbitPrefactor |
| positivity |
|
|
| theorem card_groupOrbit_le_polynomial_real {K tick : ℕ} |
| (allocation : Allocation) (m : ℕ) (sigma : Placement) : |
| (Nat.card (GroupOrbit (K := K) (tick := tick) allocation m sigma) : ℝ) ≤ |
| groupOrbitPrefactor (K := K) (tick := tick) allocation sigma * |
| ((m : ℝ) + 1) ^ groupDegree K tick sigma := by |
| unfold groupOrbitPrefactor |
| exact_mod_cast card_groupOrbit_le_polynomial (K := K) (tick := tick) allocation m sigma |
|
|
| theorem usedGroupOrbits_card_le_real {K tick : ℕ} (allocation : Allocation) (m : ℕ) |
| (sigma : Placement) |
| (s : Finset (GroupOrbit (K := K) (tick := tick) allocation m sigma)) : |
| (s.card : ℝ) ≤ groupOrbitPrefactor (K := K) (tick := tick) allocation sigma * |
| ((m : ℝ) + 1) ^ groupDegree K tick sigma := by |
| have hs : s.card ≤ Nat.card (GroupOrbit (K := K) (tick := tick) allocation m sigma) := by |
| rw [Nat.card_eq_fintype_card] |
| exact s.card_le_univ |
| have hsreal : (s.card : ℝ) ≤ |
| (Nat.card (GroupOrbit (K := K) (tick := tick) allocation m sigma) : ℝ) := by |
| exact_mod_cast hs |
| exact hsreal.trans (card_groupOrbit_le_polynomial_real allocation m sigma) |
|
|
| theorem taggedUsedGroupOrbits_card_le_real {K tick : ℕ} |
| (allocation : Allocation) (m : ℕ) (sigma : Placement) |
| (s : Finset (Fin 3 × GroupOrbit (K := K) (tick := tick) allocation m sigma)) : |
| (s.card : ℝ) ≤ (3 * groupOrbitPrefactor (K := K) (tick := tick) allocation sigma) * |
| ((m : ℝ) + 1) ^ groupDegree K tick sigma := by |
| have hs : s.card ≤ 3 * Nat.card |
| (GroupOrbit (K := K) (tick := tick) allocation m sigma) := 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 |
| (GroupOrbit (K := K) (tick := tick) allocation m sigma) : ℝ) := by |
| exact_mod_cast hs |
| calc |
| _ ≤ 3 * (Nat.card (GroupOrbit (K := K) (tick := tick) allocation m sigma) : ℝ) := hsreal |
| _ ≤ 3 * (groupOrbitPrefactor (K := K) (tick := tick) allocation sigma * |
| ((m : ℝ) + 1) ^ groupDegree K tick sigma) := |
| mul_le_mul_of_nonneg_left (card_groupOrbit_le_polynomial_real allocation m sigma) |
| (by norm_num) |
| _ = _ := (mul_assoc _ _ _).symm |
|
|
| end MatrixMultiplication.AllFieldGroupOrbitCount |
|
|
| end |
|
|
| end OAI |
|
|
