| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.MaskCore |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.ChildLaws |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.Support |
| import OAI.LinearAlgebra.MatrixMultiplication.Recovery.HistorySymmetry |
| import OAI.LinearAlgebra.MatrixMultiplication.Recovery.InheritedMaskAction |
| import OAI.LinearAlgebra.MatrixMultiplication.Recovery.InheritedMaskOrbitFilter |
|
|
| namespace OAI |
|
|
| /-! Finite extraction histories, inherited masks and recovery bounds. -/ |
|
|
| noncomputable section |
|
|
| namespace MatrixMultiplication.AllFieldHistoryMasks |
|
|
| open AllFieldHistory AllFieldHistoryChildLaws AllFieldHistorySupport |
| open JointCanonicalization InheritedMasks PermutationMatching HistorySymmetry |
| open scoped BigOperators |
|
|
| attribute [local instance] Classical.propDecidable Classical.decEq |
|
|
| variable {K tick : ℕ} |
|
|
| theorem testedClasses_parent (allocation : Allocation) (i : MaskIndex K tick) |
| (c : Classes K tick) (hc : c ∈ testedClasses allocation i) : c.1 = i.1 := |
| (Finset.mem_filter.mp hc).2.1 |
|
|
| theorem rightProbability_le_one (allocation : Allocation) (side : Fin 3) |
| (i : MaskIndex K tick) (c : Classes K tick) (hc : c ∈ testedClasses allocation i) : |
| rightProbability side i c ≤ 1 := by |
| have hsum := rightLaw_normalized allocation 1 c.1 c.2 |
| (positiveClasses_positive allocation i.1 c hc) side |
| rw [← hsum] |
| exact Finset.single_le_sum (fun a _ => rightLaw_nonnegative c.1 c.2 side a) |
| (Finset.mem_univ _) |
|
|
| theorem left_density_le (allocation : Allocation) {m : ℕ} (hm : 0 < m) |
| (ε : ℝ) (side : Fin 3) (w : Words (K := K) (tick := tick) allocation m) |
| (hw : childWindows allocation m ε side w) (i : MaskIndex K tick) |
| (c : Classes K tick) (hc : c ∈ testedClasses allocation i) : |
| |classDensity (wordPairLeftPositions w classStatistic (leftSymbol i)) c - |
| leftProbability side i c| ≤ childWidth ε i.1 := by |
| have hp : 0 < activeCounts allocation m c.1 c.2 := |
| (activeCounts_pos_iff allocation hm c.1 c.2).2 |
| (positiveClasses_positive allocation i.1 c hc) |
| have ht := (hw c).1 (leftSymbol i c) |
| have hh := testedClasses_parent allocation i c hc |
| have ht' : |empiricalLaw (classStatistic c ∘ w.left c) (leftSymbol i c) - |
| leftProbability side i c| ≤ childWidth ε c.1 := by |
| simpa only [activeLaw, activeWidth, ite_eq_left hp, leftProbability] using ht |
| have hd := classDensity_statisticPositions w.left classStatistic (leftSymbol i) c |
| change classDensity (wordPairLeftPositions w classStatistic (leftSymbol i)) c = _ at hd |
| rw [hd] |
| exact ht'.trans_eq (congrArg (childWidth ε) hh) |
|
|
| theorem right_density_le (allocation : Allocation) {m : ℕ} (hm : 0 < m) |
| (ε : ℝ) (side : Fin 3) (w : Words (K := K) (tick := tick) allocation m) |
| (hw : childWindows allocation m ε side w) (i : MaskIndex K tick) |
| (c : Classes K tick) (hc : c ∈ testedClasses allocation i) : |
| |classDensity (wordPairRightPositions w classStatistic (rightSymbol i)) c - |
| rightProbability side i c| ≤ childWidth ε i.1 := by |
| have hp : 0 < activeCounts allocation m c.1 c.2 := |
| (activeCounts_pos_iff allocation hm c.1 c.2).2 |
| (positiveClasses_positive allocation i.1 c hc) |
| have ht := (hw c).2 (rightSymbol i c) |
| have hh := testedClasses_parent allocation i c hc |
| have ht' : |empiricalLaw (classStatistic c ∘ w.right c) (rightSymbol i c) - |
| rightProbability side i c| ≤ childWidth ε c.1 := by |
| simpa only [activeLaw, activeWidth, ite_eq_left hp, rightProbability] using ht |
| have hd := classDensity_statisticPositions w.right classStatistic (rightSymbol i) c |
| change classDensity (wordPairRightPositions w classStatistic (rightSymbol i)) c = _ at hd |
| rw [hd] |
| exact ht'.trans_eq (congrArg (childWidth ε) hh) |
|
|
| theorem mixture_error_zero (allocation : Allocation) {m : ℕ} (hm : 0 < m) |
| (side : Fin 3) (i : MaskIndex K tick) : |
| |classProductMixture (fun c : testedClasses allocation i => Positions allocation m c.val) |
| (fun c => leftProbability side i c.val) (fun c => rightProbability side i c.val) - |
| center allocation side i| ≤ (0 : ℝ) := by |
| rw [selected_classProductMixture_eq_one allocation hm] |
| simp only [center, sub_self, abs_zero, le_refl] |
|
|
| theorem orbitSet_as_filter {G V : Type*} [Group G] [Fintype G] [MulAction G V] |
| [Fintype V] [DecidableEq V] (v : V) : |
| OrbitCounting.orbitSet (G := G) v = |
| Finset.univ.filter (fun w => ∃ g : G, g • v = w) := by |
| ext w |
| simp only [OrbitCounting.orbitSet, Finset.mem_image, Finset.mem_univ, |
| true_and, Finset.mem_filter] |
|
|
| attribute [local irreducible] activeCounts |
|
|
| theorem full_filter_loss_le (allocation : Allocation) {m : ℕ} (hm : 2 ≤ m) |
| {ε : ℝ} (hε : 0 < ε) (side : Fin 3) |
| (w : Words (K := K) (tick := tick) allocation m) |
| (hw : childWindows allocation m ε side w) : |
| ((rejectedFilter |
| (fullOrbitFilter (HalfClassPermutations (Positions (K := K) (tick := tick) allocation m)) w) |
| (fun v => ∃ i : MaskIndex K tick, |
| selectedWordPairMaskBad (testedClasses allocation i) classStatistic classStatistic |
| (leftSymbol i) (rightSymbol i) (pairWidth ε i.1) (center allocation side i) v)).card : ℝ) / |
| (fullOrbitFilter |
| (HalfClassPermutations (Positions (K := K) (tick := tick) allocation m)) w).card ≤ |
| lossConstant (K := K) (tick := tick) allocation ε / m := by |
| have hmpos : 0 < m := lt_of_lt_of_le (by decide : 0 < 2) hm |
| have hsize (i : MaskIndex K tick) : |
| populationSize (fun c : testedClasses allocation i => Positions allocation m c.val) = |
| selectedBasePopulation allocation (testedClasses allocation i) * m := |
| selected_populationSize allocation m (testedClasses allocation i) |
| have hn (i : MaskIndex K tick) (c : Classes K tick) |
| (hc : c ∈ testedClasses allocation i) : 2 ≤ Fintype.card (Positions allocation m c) := by |
| exact selected_two_le allocation hm (testedClasses allocation i) |
| (positiveClasses_positive allocation i.1) ⟨c, hc⟩ |
| have hmass (i : MaskIndex K tick) : |
| 0 < selectedBasePopulation allocation (testedClasses allocation i) := |
| positiveClasses_basePopulation_pos allocation i.1 |
| have hright0 (i : MaskIndex K tick) (c : Classes K tick) |
| (_hc : c ∈ testedClasses allocation i) : 0 ≤ rightProbability side i c := |
| rightLaw_nonnegative c.1 c.2 side _ |
| have h := finite_selectedWordPair_full_filter_inverse_linear_of_windows |
| (C := Classes K tick) (P := Positions (K := K) (tick := tick) allocation m) |
| (X := Letters) (Y := Letters) (I := MaskIndex K tick) |
| (SL := fun _ c => Statistic c.1) (SR := fun _ c => Statistic c.1) |
| (testedClasses allocation) w (fun _ => classStatistic) (fun _ => classStatistic) |
| leftSymbol rightSymbol hn |
| (fun i => selectedBasePopulation allocation (testedClasses allocation i)) (m : ℝ) |
| (by exact_mod_cast hmpos) hmass hsize |
| (leftProbability side) (rightProbability side) |
| (fun i => pairWidth ε i.1) (fun i => childWidth ε i.1) (fun _ => 0) |
| (center allocation side) (fun i => pairWidth_pos hε i.1) |
| hright0 (rightProbability_le_one allocation side) |
| (left_density_le allocation hmpos ε side w hw) |
| (right_density_le allocation hmpos ε side w hw) |
| (mixture_error_zero allocation hmpos side) (fun _ => le_rfl) |
| (fun i => (div_pos (pairWidth_pos hε i.1) (by norm_num : (0 : ℝ) < 4)).le) |
| simpa only [lossConstant] using h |
|
|
| theorem orbit_loss_le (allocation : Allocation) {m : ℕ} (hm : 2 ≤ m) |
| {ε : ℝ} (hε : 0 < ε) (side : Fin 3) |
| (w : Words (K := K) (tick := tick) allocation m) |
| (hw : childWindows allocation m ε side w) : |
| (((OrbitCounting.orbitSet |
| (G := HalfClassPermutations (Positions (K := K) (tick := tick) allocation m)) w).filter |
| (fun v => ¬passes allocation m ε side v)).card : ℝ) / |
| (OrbitCounting.orbitSet |
| (G := HalfClassPermutations (Positions (K := K) (tick := tick) allocation m)) w).card ≤ |
| lossConstant (K := K) (tick := tick) allocation ε / m := by |
| have heq := orbit_bad_fraction_eq_fullOrbitFilter_of_iff |
| (G := HalfClassPermutations (Positions (K := K) (tick := tick) allocation m)) w |
| (fun v => ¬passes allocation m ε side v) |
| (fun v => ∃ i : MaskIndex K tick, |
| selectedWordPairMaskBad (testedClasses allocation i) classStatistic classStatistic |
| (leftSymbol i) (rightSymbol i) (pairWidth ε i.1) (center allocation side i) v) |
| (fun v => by simp only [passes, not_forall, not_not]) |
| exact heq.trans_le (full_filter_loss_le allocation hm hε side w hw) |
|
|
| end MatrixMultiplication.AllFieldHistoryMasks |
|
|
| end |
|
|
| end OAI |
|
|
