| import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.DegreeLaws |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.PairWindows |
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.CompatibilityCountProducer |
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.PairMixtureControls |
|
|
| namespace OAI |
|
|
| /-! Group assignments, orbit counts and extraction capacities. -/ |
|
|
| noncomputable section |
|
|
| namespace MatrixMultiplication.AllFieldGroupCountRate |
|
|
| open AllFieldHistory AllFieldHistorySupport AllFieldHistoryChildLaws |
| open AllFieldGroupOrbitData AllFieldGroupDegrees AllFieldGroupDegreeLaws |
| open AllFieldGroupNativeLaws AllFieldGroupPairWindows |
| open JointCompatibilityControls JointCompatibilityScaling JointCompatibilityRateLimit |
| open JointPopulationCompatibility JointTypeCounts MatrixMultiplication.Foundation Filter |
| open scoped BigOperators Topology |
| attribute [local instance] Classical.propDecidable Classical.decEq |
|
|
| variable {K tick : ℕ} |
|
|
| def nativeCompatibilityRate (allocation : Allocation) (sigma : Placement) (side : Fin 3) : ℝ := |
| compatibilityRate (base (K := K) (tick := tick) allocation sigma side) |
| (q allocation sigma side) |
| (classDesignated false side sigma) (classDesignated true side sigma) |
| (groupCap (base allocation sigma side) (classDesignated false side sigma) |
| (nativeLaw allocation false side sigma) 0) |
| (groupCap (base allocation sigma side) (classDesignated true side sigma) |
| (nativeLaw allocation true side sigma) 0) |
|
|
| theorem base_positive_counts (allocation : Allocation) (sigma : Placement) (side : Fin 3) |
| (c : Class (K := K) (tick := tick) sigma) (u : JointPopulation.Shape) |
| (hu : 0 < base allocation sigma side c u) : |
| 0 < Counts (K := K) (tick := tick) allocation 1 sigma c.1 u := by |
| unfold base classCounts at hu |
| split_ifs at hu with hs |
| · exact hu |
| · exact (Nat.lt_irrefl 0 hu).elim |
|
|
| theorem nativeLaw_normalized (allocation : Allocation) (right : Bool) |
| (sigma : Placement) (side : Fin 3) |
| (c : Class (K := K) (tick := tick) sigma) (u : JointPopulation.Shape) |
| (hu : 0 < base allocation sigma side c u) : |
| ∑ a, nativeLaw allocation right side sigma c u a = 1 := by |
| have hp := base_positive_counts allocation sigma side c u hu |
| simp only [nativeLaw, |
| supportedHalfLaw_eq_of_counts_pos allocation 1 right c.1.val u hp (sigma side)] |
| exact halfLawAt_normalized allocation 1 right c.1.val u hp (sigma side) |
|
|
| theorem classCounts_dilation (allocation : Allocation) (m : ℕ) |
| (sigma : Placement) (side : Fin 3) : |
| classCounts (Counts (K := K) (tick := tick) allocation m sigma) (sigma side) = |
| fun c u => m * base allocation sigma side c u := by |
| funext c u |
| change (if JointPopulation.shapeSide (sigma side) u = c.2 then |
| activeCounts allocation m c.1.val u else 0) = |
| m * (if JointPopulation.shapeSide (sigma side) u = c.2 then |
| activeCounts allocation 1 c.1.val u else 0) |
| rw [activeCounts_dilation] |
| split_ifs <;> simp |
|
|
| theorem exists_widths_eventually_nativeCount_le (allocation : Allocation) (sigma : Placement) |
| (side : Fin 3) {δ χMax ηMax : ℝ} |
| (hδ : 0 < δ) (hχMax : 0 < χMax) (hηMax : 0 < ηMax) : |
| ∃ χ η : ℝ, 0 < χ ∧ χ ≤ χMax ∧ 0 < η ∧ η ≤ ηMax ∧ |
| ∀ᶠ m : ℕ in atTop, |
| ∀ (e : Targets (K := K) (tick := tick) allocation m sigma) |
| (w : Raw (K := K) (tick := tick) allocation m sigma), |
| (∀ h j, JointPopulation.shapeSide (sigma side) ((e h).val j) = |
| ownWord allocation m sigma w ⟨h, j⟩) → |
| PairWindow (base allocation sigma side) (q allocation sigma side) |
| (Positions (K := K) (tick := tick) allocation m sigma w) (pairCounts allocation m sigma w) η → |
| (nativeCount allocation m χ sigma side w : ℝ) ≤ |
| Real.exp ((m : ℝ) * |
| (nativeCompatibilityRate (K := K) (tick := tick) allocation sigma side + δ)) := by |
| have hparts : ∀ c : Class (K := K) (tick := tick) sigma, |
| Function.Injective (fun a : Symbol (K := K) (tick := tick) sigma c × Symbol (K := K) (tick := tick) sigma c => (a.1, a.2)) := |
| fun _ => fun _ _ h => h |
| have hleft : ∀ (c : Class (K := K) (tick := tick) sigma) (a : Symbol (K := K) (tick := tick) sigma c), |
| (baseSize (base allocation sigma side) c : ℝ) * |
| (∑ b : {b : Symbol (K := K) (tick := tick) sigma c × Symbol (K := K) (tick := tick) sigma c // b.1 = a}, q allocation sigma side c b.val) = |
| totalCenterCount (base allocation sigma side) (nativeLaw allocation false side sigma) c a := by |
| intro c a |
| exact JointPairMixtureControls.pairMixture_left_balance |
| (base allocation sigma side) (nativeLaw allocation false side sigma) |
| (nativeLaw allocation true side sigma) |
| (nativeLaw_normalized allocation true sigma side) c a |
| have hright : ∀ (c : Class (K := K) (tick := tick) sigma) (a : Symbol (K := K) (tick := tick) sigma c), |
| (baseSize (base allocation sigma side) c : ℝ) * |
| (∑ b : {b : Symbol (K := K) (tick := tick) sigma c × Symbol (K := K) (tick := tick) sigma c // b.2 = a}, q allocation sigma side c b.val) = |
| totalCenterCount (base allocation sigma side) (nativeLaw allocation true side sigma) c a := by |
| intro c a |
| exact JointPairMixtureControls.pairMixture_right_balance |
| (base allocation sigma side) (nativeLaw allocation false side sigma) |
| (nativeLaw allocation true side sigma) |
| (nativeLaw_normalized allocation false sigma side) c a |
| obtain ⟨χ, η, hχ, hχMax', hη, hηMax', hbound⟩ := |
| JointCompatibilityCountProducer.exists_widths_eventually_count_le |
| (base allocation sigma side) (q allocation sigma side) |
| (fun _ p => p.1) (fun _ p => p.2) |
| (classDesignated false side sigma) (classDesignated true side sigma) |
| (nativeLaw allocation false side sigma) (nativeLaw allocation true side sigma) |
| hparts |
| (fun c u a => (nativeLaw_bounds allocation false side sigma c u a).1) |
| (fun c u a => (nativeLaw_bounds allocation false side sigma c u a).2) |
| (fun c u a => (nativeLaw_bounds allocation true side sigma c u a).1) |
| (fun c u a => (nativeLaw_bounds allocation true side sigma c u a).2) |
| hleft hright hδ hχMax hηMax |
| refine ⟨χ, η, hχ, hχMax', hη, hηMax', ?_⟩ |
| filter_upwards [hbound] with m hm |
| intro e w hown hwindow |
| rw [← classCounts_dilation allocation m sigma side] at hm |
| let target : FixedSideTargets (Counts (K := K) (tick := tick) allocation m sigma) (sigma side) |
| (ownWord allocation m sigma w) := ⟨e, hown⟩ |
| have hb := (hm (Positions (K := K) (tick := tick) allocation m sigma w) (pairCounts allocation m sigma w) |
| (candidateEquiv (Counts (K := K) (tick := tick) allocation m sigma) (sigma side) |
| (ownWord allocation m sigma w) target) |
| (statisticElement (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 sigma) (projectedStatistic sigma) |
| (ownWord allocation m sigma w) w) hwindow).1 |
| rw [nativeCount, compatibleTargetCount_eq_kernel] |
| exact hb |
|
|
| theorem exists_threshold_eventually_competitors_le (allocation : Allocation) |
| (sigma : Placement) (side : Fin 3) {δ widthMax : ℝ} |
| (hδ : 0 < δ) (hwidthMax : 0 < widthMax) : |
| ∃ threshold : ℝ, 0 < threshold ∧ threshold ≤ widthMax ∧ |
| ∀ width : ℝ, 0 < width → width ≤ threshold → |
| ∀ᶠ m : ℕ in atTop, |
| ∀ (e : Targets (K := K) (tick := tick) allocation m sigma) |
| (o : Orbit (K := K) (tick := tick) allocation m sigma), o ∈ (data allocation m width sigma).orbits e → |
| ∀ w : Raw (K := K) (tick := tick) allocation m sigma, |
| (side, w) ∈ (data allocation m width sigma).passing e o → |
| ((competitors allocation m width sigma e o (side, w)).card : ℝ) ≤ |
| Real.exp ((m : ℝ) * |
| (nativeCompatibilityRate (K := K) (tick := tick) allocation sigma side + δ)) := by |
| obtain ⟨χ, η, hχ, hχMax, hη, _, hbound⟩ := |
| exists_widths_eventually_nativeCount_le (K := K) (tick := tick) |
| allocation sigma side (ηMax := 1) hδ hwidthMax zero_lt_one |
| let threshold := min χ (η / windowFactor (K := K) (tick := tick) allocation sigma) |
| have hf := windowFactor_pos (K := K) (tick := tick) allocation sigma |
| have ht : 0 < threshold := lt_min hχ (div_pos hη hf) |
| refine ⟨threshold, ht, (min_le_left _ _).trans hχMax, ?_⟩ |
| intro width hw hwt |
| have hwχ : width ≤ χ := hwt.trans (min_le_left _ _) |
| have hww : windowFactor (K := K) (tick := tick) allocation sigma * width ≤ η := by |
| have hh := (le_div_iff₀ hf).mp (hwt.trans (min_le_right _ _)) |
| simpa only [mul_comm] using hh |
| filter_upwards [hbound, eventually_gt_atTop (0 : ℕ)] with m hm hmpos |
| intro e o ho w hpass |
| have ha := AllFieldGroupAdmissible.full_admissible allocation m width sigma |
| e o ho (side, w) ((data allocation m width sigma).passing_subset e o ho hpass) |
| have hp := passing_pairWindow allocation hmpos hw.le sigma e o ho (side, w) hpass |
| have hpair : PairWindow (base allocation sigma side) (q allocation sigma side) |
| (Positions (K := K) (tick := tick) allocation m sigma w) (pairCounts allocation m sigma w) η := |
| fun c hc a => (hp c hc a).trans hww |
| have hc := hm e w (fun h j => (ha.1.1 h j).symm) hpair |
| exact (Nat.cast_le.mpr (competitors_card_le_nativeCount allocation m hw.le hwχ sigma |
| e o (side, w))).trans hc |
|
|
| end MatrixMultiplication.AllFieldGroupCountRate |
|
|
| end |
|
|
| end OAI |
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|
