| import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.AdmissibilityCore |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.Degrees |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.NativeLaws |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.MaskLaws |
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.PairClassWindows |
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.PairMixtureControls |
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.CompatibilityRateLimit |
|
|
| namespace OAI |
|
|
| /-! Group assignments, orbit counts and extraction capacities. -/ |
|
|
| noncomputable section |
|
|
| namespace MatrixMultiplication.AllFieldGroupPairWindows |
|
|
| open MatrixMultiplication.Foundation AllFieldHistory AllFieldHistorySupport |
| open AllFieldHistoryChildLaws AllFieldHistoryMasks AllFieldHistoryMaskLaws |
| open AllFieldHistoryGroupMasks AllFieldHistoryGroupedRecovery |
| open AllFieldGroupOrbitData AllFieldGroupDegrees AllFieldGroupNativeLaws |
| open JointCompatibilityScaling JointCompatibilityRateLimit |
| open scoped BigOperators |
| attribute [local instance] Classical.propDecidable |
|
|
| variable {K tick : ℕ} |
|
|
| def base (allocation : Allocation) (sigma : Placement) (side : Fin 3) := |
| JointPopulationCompatibility.classCounts |
| (Counts (K := K) (tick := tick) allocation 1 sigma) (sigma side) |
|
|
| def q (allocation : Allocation) (sigma : Placement) (side : Fin 3) |
| (c : Class (K := K) (tick := tick) sigma) : Symbol (K := K) (tick := tick) sigma c × Symbol (K := K) (tick := tick) sigma c → ℝ := |
| JointPairMixtureControls.pairMixture (base (K := K) (tick := tick) allocation sigma side) |
| (fun c u => supportedHalfLaw allocation false c.1.val u (sigma side)) |
| (fun c u => supportedHalfLaw allocation true c.1.val u (sigma side)) c |
|
|
| abbrev Positions (allocation : Allocation) (m : ℕ) (sigma : Placement) |
| (w : Raw (K := K) (tick := tick) allocation m sigma) := |
| JointPopulationCompatibility.ClassPos (Counts (K := K) (tick := tick) allocation m sigma) |
| (ownWord (K := K) (tick := tick) allocation m sigma w) |
|
|
| def pairCounts (allocation : Allocation) (m : ℕ) (sigma : Placement) |
| (w : Raw (K := K) (tick := tick) allocation m sigma) := |
| JointPopulationCompatibility.statisticCounts (Counts (K := K) (tick := tick) allocation m sigma) |
| (Letter (K := K) (tick := tick) sigma) (Letter (K := K) (tick := tick) sigma) (Symbol (K := K) (tick := tick) sigma) (Symbol (K := K) (tick := tick) sigma) |
| (projectedStatistic (K := K) (tick := tick) sigma) (projectedStatistic (K := K) (tick := tick) sigma) |
| (ownWord (K := K) (tick := tick) allocation m sigma w) w |
|
|
| def parentBase {sigma : Placement} (allocation : Allocation) (h : ActiveOrder K tick sigma) : ℕ := |
| population allocation 1 (h.val.val.1.source, h.val.val.2) |
|
|
| def parentPair (allocation : Allocation) (sigma : Placement) (side : Fin 3) |
| (h : ActiveOrder K tick sigma) (a : Statistic h.val × Statistic h.val) : ℝ := |
| (∑ u, (activeCounts allocation 1 h.val u : ℝ) * |
| supportedHalfLaw allocation false h.val u (sigma side) a.1 * |
| supportedHalfLaw allocation true h.val u (sigma side) a.2) / parentBase (K := K) (tick := tick) allocation h |
|
|
| def classMass (allocation : Allocation) (sigma : Placement) (side : Fin 3) |
| (c : Class (K := K) (tick := tick) sigma) : ℝ := |
| (baseSize (base (K := K) (tick := tick) allocation sigma side) c : ℝ) / parentBase (K := K) (tick := tick) allocation c.1 |
|
|
| def windowFactor (allocation : Allocation) (sigma : Placement) : ℝ := |
| 1 + ∑ h : ActiveOrder K tick sigma, (parentBase (K := K) (tick := tick) allocation h : ℝ) |
|
|
| theorem windowFactor_pos (allocation : Allocation) (sigma : Placement) : |
| 0 < windowFactor (K := K) (tick := tick) allocation sigma := by |
| unfold windowFactor |
| positivity |
|
|
| theorem parentBase_pos {sigma : Placement} (allocation : Allocation) (h : ActiveOrder K tick sigma) : |
| 0 < parentBase (K := K) (tick := tick) allocation h := |
| AllFieldHistorySupport.work_source_population_pos allocation (by decide : 0 < (1 : ℕ)) h.val.val |
|
|
| theorem parent_positions_card (allocation : Allocation) (m : ℕ) (sigma : Placement) |
| (h : ActiveOrder K tick sigma) : |
| Fintype.card (JointPopulation.Positions (Counts (K := K) (tick := tick) allocation m sigma) h) = |
| m * parentBase (K := K) (tick := tick) allocation h := by |
| change Fintype.card (Fin (∑ u, jointCounts allocation m h.val.val u)) = _ |
| rw [Fintype.card_fin, jointCounts_sum] |
| exact population_dilation allocation m _ |
|
|
| theorem positions_card (allocation : Allocation) (m : ℕ) (ε : ℝ) (sigma : Placement) |
| (side : Fin 3) (e : Targets (K := K) (tick := tick) allocation m sigma) |
| (w : Raw (K := K) (tick := tick) allocation m sigma) (hw : Admissible allocation m ε sigma side e w) |
| (c : Class (K := K) (tick := tick) sigma) : |
| Fintype.card (Positions (K := K) (tick := tick) allocation m sigma w c) = |
| m * baseSize (base (K := K) (tick := tick) allocation sigma side) c := by |
| have he : JointPopulationCompatibility.FixedSideTargets (Counts (K := K) (tick := tick) allocation m sigma) |
| (sigma side) (ownWord (K := K) (tick := tick) allocation m sigma w) := |
| ⟨e, fun h j => (hw.1.1 h j).symm⟩ |
| rw [JointPopulationCompatibility.classCounts_size |
| (Counts (K := K) (tick := tick) allocation m sigma) (sigma side) (ownWord (K := K) (tick := tick) allocation m sigma w) he c] |
| simp only [JointPopulationCompatibility.classCounts, Counts, groupCounts, |
| activeCounts_dilation allocation m, base, baseSize, Finset.mul_sum, mul_ite, mul_zero] |
|
|
| theorem parentPair_eq_parentCenter (allocation : Allocation) (sigma : Placement) |
| (side : Fin 3) (h : ActiveOrder K tick sigma) |
| (a : Statistic h.val × Statistic h.val) : |
| parentPair (K := K) (tick := tick) allocation sigma side h a = parentCenter allocation (sigma side) h.val a.1 a.2 := by |
| unfold parentPair parentCenter parentBase |
| rw [Finset.sum_div] |
| apply Finset.sum_congr rfl |
| intro u _ |
| by_cases hu : 0 < activeCounts allocation 1 h.val u |
| · simp only [supportedHalfLaw, ite_eq_left hu, halfLawAt, Bool.false_eq_true, |
| ite_false, ite_true] |
| ring |
| · have hz := Nat.eq_zero_of_not_pos hu |
| simp only [hz, Nat.cast_zero, zero_mul, zero_div] |
|
|
| theorem parentPair_eq_native (allocation : Allocation) (sigma : Placement) |
| (side : Fin 3) (h : ActiveOrder K tick sigma) |
| (a : Statistic h.val × Statistic h.val) : |
| parentPair (K := K) (tick := tick) allocation sigma side h a = orderPairMixture h side a := by |
| rw [parentPair_eq_parentCenter] |
| unfold parentCenter orderPairMixture |
| apply Finset.sum_congr rfl |
| intro u _ |
| rw [show (activeCounts allocation 1 h.val u : ℝ) / |
| population allocation 1 (h.val.val.1.source, h.val.val.2) = |
| (AllFieldActiveLaws.orderLaw h).mass u from |
| AllFieldActiveLaws.jointCounts_ratio allocation (by decide : 0 < (1 : ℕ)) h.val.val u] |
| ring |
|
|
| def parentStatistic (allocation : Allocation) (m : ℕ) (sigma : Placement) |
| (w : Raw (K := K) (tick := tick) allocation m sigma) (h : ActiveOrder K tick sigma) |
| (j : JointPopulation.Positions (Counts (K := K) (tick := tick) allocation m sigma) h) : |
| Statistic h.val × Statistic h.val := |
| (statistic h.val (w h j).1, statistic h.val (w h j).2) |
|
|
| theorem parent_window (allocation : Allocation) (m : ℕ) (ε : ℝ) (sigma : Placement) |
| (side : Fin 3) (w : Raw (K := K) (tick := tick) allocation m sigma) |
| (hw : AllFieldHistoryGroupMasks.rawPass allocation m ε (sigma side) sigma w) |
| (h : ActiveOrder K tick sigma) (a : Statistic h.val × Statistic h.val) : |
| |(wordPopulation (parentStatistic (K := K) (tick := tick) allocation m sigma w h) a : ℝ) / |
| Fintype.card (JointPopulation.Positions (Counts (K := K) (tick := tick) allocation m sigma) h) - |
| parentPair (K := K) (tick := tick) allocation sigma side h a| ≤ pairWidth ε h.val := by |
| let defaults : SymbolChoice K tick := fun h => statistic h (fun _ => 0) |
| let l : SymbolChoice K tick := Function.update defaults h.val a.1 |
| let r : SymbolChoice K tick := Function.update defaults h.val a.2 |
| have hp := hw (h.val, l, r) h.property |
| rw [center_eq_parentCenter] at hp |
| simp only [l, r, Function.update_self] at hp |
| rw [parentPair_eq_parentCenter] |
| have hpop : Fintype.card (JointPopulation.Positions (Counts (K := K) (tick := tick) allocation m sigma) h) = |
| population allocation m (h.val.val.1.source, h.val.val.2) := by |
| change Fintype.card (Fin (∑ u, jointCounts allocation m h.val.val u)) = _ |
| rw [Fintype.card_fin] |
| exact jointCounts_sum allocation m h.val.val |
| rw [hpop] |
| have hcount : (wordPopulation (parentStatistic (K := K) (tick := tick) allocation m sigma w h) a : ℝ) = |
| ∑ j : JointPopulation.Positions (activeCounts allocation m) h.val, |
| if statistic h.val (w h j).1 = a.1 ∧ statistic h.val (w h j).2 = a.2 |
| then (1 : ℝ) else 0 := by |
| rcases a with ⟨a₁, a₂⟩ |
| simp only [wordPopulation, Fintype.card_subtype, Finset.card_eq_sum_ones, |
| Finset.sum_filter, Nat.cast_sum, Nat.cast_ite, Nat.cast_one, Nat.cast_zero, |
| parentStatistic, Prod.mk.injEq] |
| apply Finset.sum_congr rfl |
| intro j _ |
| split_ifs <;> rfl |
| rw [hcount] |
| exact hp |
|
|
| def weightCode {K : ℕ} (w : Work K) : w.Statistic → Fin 17 := |
| match w with |
| | .stageA _ => AllFieldPairMarginals.halfWeightCode |
| | .stageB _ => AllFieldPairMarginals.statisticWeightCode |
| | .stageC _ => fun _ => 0 |
|
|
| @[simp] theorem weightCode_val {K : ℕ} (w : Work K) (a : w.Statistic) : |
| (weightCode w a).val = w.statisticWeight a := by |
| cases w <;> rfl |
|
|
| theorem weightCode_statistic {K : ℕ} (w : Work K) (hs : w.stage ≠ 2) |
| (x : CWWindowedLeaves.Raw w.halfLength) : |
| (weightCode w (w.statistic x)).val = CWStrands.weight x := |
| (weightCode_val w _).trans (w.statistic_weight hs x).symm |
|
|
| theorem left_supported_zero (allocation : Allocation) (sigma : Placement) (side : Fin 3) |
| (h : ActiveOrder K tick sigma) (hs : h.val.val.1.stage ≠ 2) |
| (u : JointPopulation.Shape) (a : Statistic h.val) |
| (ha : weightCode h.val.val.1 a ≠ JointPopulation.shapeSide (sigma side) u) : |
| supportedHalfLaw allocation false h.val u (sigma side) a = 0 := by |
| by_cases hu : 0 < activeCounts allocation 1 h.val u |
| · rw [supportedHalfLaw_eq_of_counts_pos allocation 1 false h.val u hu, halfLawAt_eq] |
| apply Rat.cast_eq_zero.mpr |
| apply h.val.val.1.childLaw_outside_support hs _ |
| (halfShape_mem_of_counts_pos allocation 1 false h.val u hu) _ a |
| rw [halfShape_childWeight] |
| simpa only [childWeight, Bool.false_eq_true, ite_false, ← weightCode_val, |
| ne_eq, Fin.ext_iff] using ha |
| · simp only [supportedHalfLaw, ite_eq_right hu] |
|
|
| theorem filtered_pair_sum (allocation : Allocation) (sigma : Placement) (side : Fin 3) |
| (c : Class (K := K) (tick := tick) sigma) (hs : c.1.val.val.1.stage ≠ 2) |
| (a : Symbol (K := K) (tick := tick) sigma c × Symbol (K := K) (tick := tick) sigma c) : |
| (∑ u, (base (K := K) (tick := tick) allocation sigma side c u : ℝ) * |
| supportedHalfLaw allocation false c.1.val u (sigma side) a.1 * |
| supportedHalfLaw allocation true c.1.val u (sigma side) a.2) = |
| if weightCode c.1.val.val.1 a.1 = c.2 then |
| ∑ u, (activeCounts allocation 1 c.1.val u : ℝ) * |
| supportedHalfLaw allocation false c.1.val u (sigma side) a.1 * |
| supportedHalfLaw allocation true c.1.val u (sigma side) a.2 |
| else 0 := by |
| have hterm (u : JointPopulation.Shape) : |
| (base (K := K) (tick := tick) allocation sigma side c u : ℝ) * |
| supportedHalfLaw allocation false c.1.val u (sigma side) a.1 * |
| supportedHalfLaw allocation true c.1.val u (sigma side) a.2 = |
| if weightCode c.1.val.val.1 a.1 = c.2 then |
| (activeCounts allocation 1 c.1.val u : ℝ) * |
| supportedHalfLaw allocation false c.1.val u (sigma side) a.1 * |
| supportedHalfLaw allocation true c.1.val u (sigma side) a.2 |
| else 0 := by |
| by_cases hu : JointPopulation.shapeSide (sigma side) u = c.2 |
| · simp only [base, JointPopulationCompatibility.classCounts, Counts, groupCounts, |
| hu, ite_true] |
| by_cases ha : weightCode c.1.val.val.1 a.1 = c.2 |
| · rw [ite_eq_left ha] |
| · have hz := left_supported_zero (K := K) (tick := tick) allocation sigma side c.1 hs u a.1 |
| (fun he => ha (he.trans hu)) |
| simp only [ha, ite_false, hz, mul_zero, zero_mul] |
| · simp only [base, JointPopulationCompatibility.classCounts, Counts, groupCounts, |
| hu, ite_false, Nat.cast_zero, zero_mul] |
| by_cases ha : weightCode c.1.val.val.1 a.1 = c.2 |
| · have hz := left_supported_zero (K := K) (tick := tick) allocation sigma side c.1 hs u a.1 |
| (fun he => hu (he.symm.trans ha)) |
| simp only [ha, ite_true, hz, mul_zero, zero_mul] |
| · rw [ite_eq_right ha] |
| simp_rw [hterm] |
| split_ifs <;> simp |
|
|
| theorem q_eq_conditionalCenter (allocation : Allocation) (sigma : Placement) (side : Fin 3) |
| (c : Class (K := K) (tick := tick) sigma) (hs : c.1.val.val.1.stage ≠ 2) : |
| q (K := K) (tick := tick) allocation sigma side c = JointPairClassWindows.conditionalCenter |
| (fun a : Symbol (K := K) (tick := tick) sigma c × Symbol (K := K) (tick := tick) sigma c => weightCode c.1.val.val.1 a.1) |
| c.2 (parentPair (K := K) (tick := tick) allocation sigma side c.1) (classMass (K := K) (tick := tick) allocation sigma side c) := by |
| funext a |
| rw [q, JointPairMixtureControls.pairMixture, filtered_pair_sum (K := K) (tick := tick) allocation sigma side c hs] |
| unfold JointPairClassWindows.conditionalCenter parentPair classMass |
| by_cases ha : weightCode c.1.val.val.1 a.1 = c.2 |
| · simp only [ha, ite_true] |
| have hp : (parentBase (K := K) (tick := tick) allocation c.1 : ℝ) ≠ 0 := |
| Nat.cast_ne_zero.mpr (Nat.ne_of_gt (parentBase_pos (K := K) (tick := tick) allocation c.1)) |
| rw [div_div_div_cancel_right₀ hp] |
| · simp only [ha, ite_false, zero_div] |
|
|
| theorem classMass_pos (allocation : Allocation) (sigma : Placement) (side : Fin 3) |
| (c : Class (K := K) (tick := tick) sigma) |
| (hc : 0 < baseSize (base (K := K) (tick := tick) allocation sigma side) c) : |
| 0 < classMass (K := K) (tick := tick) allocation sigma side c := |
| div_pos (Nat.cast_pos.mpr hc) (Nat.cast_pos.mpr (parentBase_pos (K := K) (tick := tick) allocation c.1)) |
|
|
| theorem fiber_mass (allocation : Allocation) (m : ℕ) (ε : ℝ) (sigma : Placement) |
| (side : Fin 3) (e : Targets (K := K) (tick := tick) allocation m sigma) |
| (w : Raw (K := K) (tick := tick) allocation m sigma) (hw : Admissible allocation m ε sigma side e w) |
| (c : Class (K := K) (tick := tick) sigma) : |
| (Fintype.card (Positions (K := K) (tick := tick) allocation m sigma w c) : ℝ) = |
| (Fintype.card (JointPopulation.Positions (Counts (K := K) (tick := tick) allocation m sigma) c.1) : ℝ) * |
| classMass (K := K) (tick := tick) allocation sigma side c := by |
| rw [positions_card (K := K) (tick := tick) allocation m ε sigma side e w hw, parent_positions_card] |
| simp only [Nat.cast_mul, classMass] |
| have hp : (parentBase (K := K) (tick := tick) allocation c.1 : ℝ) ≠ 0 := |
| Nat.cast_ne_zero.mpr (Nat.ne_of_gt (parentBase_pos (K := K) (tick := tick) allocation c.1)) |
| rw [mul_assoc, mul_div_cancel₀ _ hp] |
|
|
| theorem width_le (allocation : Allocation) {ε : ℝ} (hε : 0 ≤ ε) |
| (sigma : Placement) (side : Fin 3) (c : Class (K := K) (tick := tick) sigma) |
| (hc : 0 < baseSize (base (K := K) (tick := tick) allocation sigma side) c) : |
| pairWidth ε c.1.val / classMass (K := K) (tick := tick) allocation sigma side c ≤ |
| windowFactor (K := K) (tick := tick) allocation sigma * ε := by |
| have hpw : 0 ≤ pairWidth ε c.1.val := by unfold pairWidth; positivity |
| have hw : pairWidth ε c.1.val ≤ ε := |
| div_le_self hε (one_le_pow₀ (by norm_num)) |
| have hb : (1 : ℝ) ≤ baseSize (base (K := K) (tick := tick) allocation sigma side) c := by exact_mod_cast hc |
| have hn : (parentBase (K := K) (tick := tick) allocation c.1 : ℝ) ≤ windowFactor (K := K) (tick := tick) allocation sigma := by |
| unfold windowFactor |
| exact (Finset.single_le_sum (fun _ _ => Nat.cast_nonneg _) (Finset.mem_univ c.1)).trans |
| (le_add_of_nonneg_left zero_le_one) |
| unfold classMass |
| rw [div_div_eq_mul_div] |
| calc |
| pairWidth ε c.1.val * (parentBase (K := K) (tick := tick) allocation c.1 : ℝ) / |
| (baseSize (base (K := K) (tick := tick) allocation sigma side) c : ℝ) ≤ |
| pairWidth ε c.1.val * (parentBase (K := K) (tick := tick) allocation c.1 : ℝ) := |
| div_le_self (mul_nonneg hpw (Nat.cast_nonneg _)) hb |
| _ ≤ ε * windowFactor (K := K) (tick := tick) allocation sigma := mul_le_mul hw hn (Nat.cast_nonneg _) hε |
| _ = _ := mul_comm _ _ |
|
|
| theorem sharing_window (allocation : Allocation) (m : ℕ) {ε : ℝ} (hε : 0 ≤ ε) |
| (sigma : Placement) (side : Fin 3) |
| (e : Targets (K := K) (tick := tick) allocation m sigma) |
| (w : Raw (K := K) (tick := tick) allocation m sigma) (hw : Admissible allocation m ε sigma side e w) |
| (hp : AllFieldHistoryGroupMasks.rawPass allocation m ε (sigma side) sigma w) |
| (c : Class (K := K) (tick := tick) sigma) (hs : c.1.val.val.1.stage ≠ 2) |
| (hc : 0 < baseSize (base (K := K) (tick := tick) allocation sigma side) c) |
| (a : Symbol (K := K) (tick := tick) sigma c × Symbol (K := K) (tick := tick) sigma c) : |
| |(pairCounts (K := K) (tick := tick) allocation m sigma w c a : ℝ) / |
| Fintype.card (Positions (K := K) (tick := tick) allocation m sigma w c) - q (K := K) (tick := tick) allocation sigma side c a| ≤ |
| windowFactor (K := K) (tick := tick) allocation sigma * ε := by |
| let z := parentStatistic (K := K) (tick := tick) allocation m sigma w c.1 |
| let k : JointPopulation.Positions (Counts (K := K) (tick := tick) allocation m sigma) c.1 → Fin 17 := |
| fun j => ownWord (K := K) (tick := tick) allocation m sigma w ⟨c.1, j⟩ |
| let f : Symbol (K := K) (tick := tick) sigma c × Symbol (K := K) (tick := tick) sigma c → Fin 17 := |
| fun a => weightCode c.1.val.val.1 a.1 |
| have hf (j) : f (z j) = k j := by |
| apply Fin.ext |
| exact weightCode_statistic c.1.val.val.1 hs (w c.1 j).1 |
| have hη : 0 ≤ pairWidth ε c.1.val := by unfold pairWidth; positivity |
| have hh := JointPairClassWindows.fiber_window z k f hf c.2 |
| (parentPair (K := K) (tick := tick) allocation sigma side c.1) (classMass (K := K) (tick := tick) allocation sigma side c) |
| (pairWidth ε c.1.val) (classMass_pos (K := K) (tick := tick) allocation sigma side c hc) hη |
| (fiber_mass (K := K) (tick := tick) allocation m ε sigma side e w hw c) |
| (parent_window (K := K) (tick := tick) allocation m ε sigma side w hp c.1) a |
| rw [← q_eq_conditionalCenter (K := K) (tick := tick) allocation sigma side c hs] at hh |
| exact hh.trans (width_le (K := K) (tick := tick) allocation hε sigma side c hc) |
|
|
| theorem q_eq_one_of_subsingleton (allocation : Allocation) (sigma : Placement) (side : Fin 3) |
| (c : Class (K := K) (tick := tick) sigma) [Subsingleton (Symbol (K := K) (tick := tick) sigma c)] |
| (hc : 0 < baseSize (base (K := K) (tick := tick) allocation sigma side) c) |
| (a : Symbol (K := K) (tick := tick) sigma c × Symbol (K := K) (tick := tick) sigma c) : q (K := K) (tick := tick) allocation sigma side c a = 1 := by |
| have hl (right : Bool) (u : JointPopulation.Shape) |
| (hu : 0 < base (K := K) (tick := tick) allocation sigma side c u) : |
| supportedHalfLaw allocation right c.1.val u (sigma side) (if right then a.2 else a.1) = 1 := by |
| have hu' : 0 < activeCounts allocation 1 c.1.val u := by |
| unfold base JointPopulationCompatibility.classCounts Counts groupCounts at hu |
| split_ifs at hu with he |
| · exact hu |
| · omega |
| rw [supportedHalfLaw_eq_of_counts_pos allocation 1 right c.1.val u hu'] |
| have ht := halfLawAt_normalized allocation 1 right c.1.val u hu' (sigma side) |
| have he : (∑ b, halfLawAt right c.1.val u (sigma side) b) = |
| halfLawAt right c.1.val u (sigma side) (if right then a.2 else a.1) := |
| Fintype.sum_subsingleton _ _ |
| exact he.symm.trans ht |
| unfold q JointPairMixtureControls.pairMixture |
| have hn : (∑ u, (base (K := K) (tick := tick) allocation sigma side c u : ℝ) * |
| supportedHalfLaw allocation false c.1.val u (sigma side) a.1 * |
| supportedHalfLaw allocation true c.1.val u (sigma side) a.2) = |
| (baseSize (base (K := K) (tick := tick) allocation sigma side) c : ℝ) := by |
| rw [baseSize, Nat.cast_sum] |
| apply Finset.sum_congr rfl |
| intro u _ |
| by_cases hu : 0 < base (K := K) (tick := tick) allocation sigma side c u |
| · have hleft : supportedHalfLaw allocation false c.1.val u (sigma side) a.1 = 1 := by |
| simpa only [Bool.false_eq_true, ite_false] using hl false u hu |
| have hright : supportedHalfLaw allocation true c.1.val u (sigma side) a.2 = 1 := by |
| simpa only [ite_true] using hl true u hu |
| rw [hleft, hright, mul_one, mul_one] |
| · simp only [Nat.eq_zero_of_not_pos hu, Nat.cast_zero, zero_mul] |
| rw [hn] |
| exact div_self (Nat.cast_ne_zero.mpr (Nat.ne_of_gt hc)) |
| |
| theorem pairWindow_of_completeKeep (allocation : Allocation) {m : ℕ} (hm : 0 < m) |
| {ε : ℝ} (hε : 0 ≤ ε) (sigma : Placement) (side : Fin 3) |
| (e : Targets (K := K) (tick := tick) allocation m sigma) |
| (w : Raw (K := K) (tick := tick) allocation m sigma) (hw : Admissible allocation m ε sigma side e w) |
| (hp : completeKeep allocation m ε (sigma side) sigma w) : |
| PairWindow (base (K := K) (tick := tick) allocation sigma side) (q (K := K) (tick := tick) allocation sigma side) |
| (Positions (K := K) (tick := tick) allocation m sigma w) (pairCounts (K := K) (tick := tick) allocation m sigma w) |
| (windowFactor (K := K) (tick := tick) allocation sigma * ε) := by |
| intro c hc a |
| by_cases hs : c.1.val.val.1.stage ≠ 2 |
| · exact sharing_window (K := K) (tick := tick) allocation m hε sigma side e w hw hp.2.2 c hs hc a |
| · have hstage : c.1.val.val.1.stage = 2 := not_ne_iff.mp hs |
| have : Subsingleton (Symbol (K := K) (tick := tick) sigma c) := by |
| change Subsingleton c.1.val.val.1.Statistic |
| cases hh : c.1.val.val.1 with |
| | stageA h => simp [hh, Work.stage] at hstage |
| | stageB h => simp [hh, Work.stage] at hstage |
| | stageC h => dsimp [Work.Statistic]; infer_instance |
| have hpc : pairCounts (K := K) (tick := tick) allocation m sigma w c a = |
| Fintype.card (Positions (K := K) (tick := tick) allocation m sigma w c) := by |
| unfold pairCounts JointPopulationCompatibility.statisticCounts |
| have hall : ∀ i, JointPopulationCompatibility.statisticWord |
| (Counts (K := K) (tick := tick) allocation m sigma) (Letter (K := K) (tick := tick) sigma) (Letter (K := K) (tick := tick) sigma) (Symbol (K := K) (tick := tick) sigma) (Symbol (K := K) (tick := tick) sigma) |
| (projectedStatistic (K := K) (tick := tick) sigma) (projectedStatistic (K := K) (tick := tick) sigma) |
| (ownWord (K := K) (tick := tick) allocation m sigma w) w c i = a := fun _ => Subsingleton.elim _ _ |
| simp [wordPopulation, hall] |
| have hn : (Fintype.card (Positions (K := K) (tick := tick) allocation m sigma w c) : ℝ) ≠ 0 := by |
| rw [positions_card (K := K) (tick := tick) allocation m ε sigma side e w hw c] |
| exact Nat.cast_ne_zero.mpr (Nat.ne_of_gt (Nat.mul_pos hm hc)) |
| rw [hpc, div_self hn, q_eq_one_of_subsingleton (K := K) (tick := tick) allocation sigma side c hc a, |
| sub_self, abs_zero] |
| exact mul_nonneg (windowFactor_pos (K := K) (tick := tick) allocation sigma).le hε |
| |
| theorem passing_pairWindow (allocation : Allocation) {m : ℕ} (hm : 0 < m) |
| {ε : ℝ} (hε : 0 ≤ ε) (sigma : Placement) |
| (e : Targets (K := K) (tick := tick) allocation m sigma) |
| (o : Orbit (K := K) (tick := tick) allocation m sigma) (ho : o ∈ (data (K := K) (tick := tick) allocation m ε sigma).orbits e) |
| (v : Variable (K := K) (tick := tick) allocation m sigma) (hv : v ∈ (data (K := K) (tick := tick) allocation m ε sigma).passing e o) : |
| PairWindow (base (K := K) (tick := tick) allocation sigma v.1) (q (K := K) (tick := tick) allocation sigma v.1) |
| (Positions (K := K) (tick := tick) allocation m sigma v.2) (pairCounts (K := K) (tick := tick) allocation m sigma v.2) |
| (windowFactor (K := K) (tick := tick) allocation sigma * ε) := by |
| have hpass := (mem_passing allocation m ε sigma e o v).mp hv |
| exact pairWindow_of_completeKeep (K := K) (tick := tick) allocation hm hε sigma v.1 e v.2 |
| (AllFieldGroupAdmissible.full_admissible allocation m ε sigma e o ho v hpass.1) hpass.2 |
| |
| end MatrixMultiplication.AllFieldGroupPairWindows |
| |
| end |
| |
| end OAI |
| |
