| import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.Degrees |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.NativeLaws |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.AmbientDegree |
|
|
| namespace OAI |
|
|
| /-! Group assignments, orbit counts and extraction capacities. -/ |
|
|
| noncomputable section |
|
|
| namespace MatrixMultiplication.AllFieldGroupDegreeLaws |
|
|
| open AllFieldHistory AllFieldHistorySupport AllFieldHistoryChildLaws |
| open AllFieldGroupOrbitData AllFieldGroupDegrees AllFieldGroupNativeLaws |
| open JointPopulation JointCanonicalization MatrixMultiplication.Foundation Filter |
| open scoped BigOperators Topology |
| attribute [local instance] Classical.propDecidable Classical.decEq |
|
|
| variable {K tick : ℕ} |
|
|
| def nativeLaw (allocation : Allocation) (right : Bool) (side : Fin 3) |
| (sigma : Placement) (c : Class (K := K) (tick := tick) sigma) |
| (u : JointPopulation.Shape) : Symbol (K := K) (tick := tick) sigma c → ℝ := |
| supportedHalfLaw allocation right c.1.val u (sigma side) |
|
|
| theorem nativeLaw_bounds (allocation : Allocation) (right : Bool) (side : Fin 3) |
| (sigma : Placement) (c : Class (K := K) (tick := tick) sigma) |
| (u : JointPopulation.Shape) (a : Symbol (K := K) (tick := tick) sigma c) : |
| 0 ≤ nativeLaw allocation right side sigma c u a ∧ |
| nativeLaw allocation right side sigma c u a ≤ 1 := |
| supportedHalfLaw_bounds allocation right c.1.val u (sigma side) a |
|
|
| theorem weighted_classLaw_eq_native (allocation : Allocation) (m : ℕ) |
| (right : Bool) (side : Fin 3) (sigma : Placement) |
| (h : ActiveOrder K tick sigma) (u : JointPopulation.Shape) |
| (hd : designated right side sigma h u) (a : Statistic h.val) : |
| (Counts (K := K) (tick := tick) allocation m sigma h u : ℝ) * compatibilityLaw right side sigma h u a = |
| (Counts (K := K) (tick := tick) allocation m sigma h u : ℝ) * |
| supportedHalfLaw allocation right h.val u (sigma side) a := by |
| by_cases hp : 0 < Counts (K := K) (tick := tick) allocation m sigma h u |
| · rw [compatibilityLaw_eq allocation m sigma right side h u hp hd, |
| supportedHalfLaw_eq_of_counts_pos allocation m right h.val u hp (sigma side)] |
| rfl |
| · have hz : Counts (K := K) (tick := tick) allocation m sigma h u = 0 := Nat.eq_zero_of_not_pos hp |
| simp only [hz, Nat.cast_zero, zero_mul] |
|
|
| theorem compatible_iff_native (allocation : Allocation) (m : ℕ) (χ : ℝ) |
| (sigma : Placement) (side : Fin 3) |
| (e : Targets (K := K) (tick := tick) allocation m sigma) |
| (w : Raw (K := K) (tick := tick) allocation m sigma) : |
| JointPopulationCompatibility.Compatible (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) |
| (classDesignated false side sigma) (classDesignated true side sigma) |
| (classLaw false side sigma) (classLaw true side sigma) χ (sigma side) e w ↔ |
| JointPopulationCompatibility.Compatible (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) |
| (classDesignated false side sigma) (classDesignated true side sigma) |
| (nativeLaw allocation false side sigma) (nativeLaw allocation true side sigma) |
| χ (sigma side) e w := by |
| constructor <;> rintro ⟨hL, hR⟩ <;> constructor |
| · intro h u hd a |
| have hh := hL h u hd a |
| change |_ - (Counts (K := K) (tick := tick) allocation m sigma h u : ℝ) * |
| supportedHalfLaw allocation false h.val u (sigma side) a| ≤ _ |
| rw [← weighted_classLaw_eq_native allocation m false side sigma h u hd a] |
| exact hh |
| · intro h u hd a |
| have hh := hR h u hd a |
| change |_ - (Counts (K := K) (tick := tick) allocation m sigma h u : ℝ) * |
| supportedHalfLaw allocation true h.val u (sigma side) a| ≤ _ |
| rw [← weighted_classLaw_eq_native allocation m true side sigma h u hd a] |
| exact hh |
| · intro h u hd a |
| have hh := hL h u hd a |
| change |_ - (Counts (K := K) (tick := tick) allocation m sigma h u : ℝ) * |
| compatibilityLaw false side sigma h u a| ≤ _ |
| rw [weighted_classLaw_eq_native allocation m false side sigma h u hd a] |
| exact hh |
| · intro h u hd a |
| have hh := hR h u hd a |
| change |_ - (Counts (K := K) (tick := tick) allocation m sigma h u : ℝ) * |
| compatibilityLaw true side sigma h u a| ≤ _ |
| rw [weighted_classLaw_eq_native allocation m true side sigma h u hd a] |
| exact hh |
|
|
| def nativeCount (allocation : Allocation) (m : ℕ) (χ : ℝ) |
| (sigma : Placement) (side : Fin 3) (w : Raw (K := K) (tick := tick) allocation m sigma) : ℕ := |
| JointPopulationCompatibility.compatibleTargetCount (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) |
| (classDesignated false side sigma) (classDesignated true side sigma) |
| (nativeLaw allocation false side sigma) (nativeLaw allocation true side sigma) χ |
| (sigma side) (ownWord allocation m sigma w) w |
|
|
| theorem conditionalCount_eq_native (allocation : Allocation) (m : ℕ) (χ : ℝ) |
| (sigma : Placement) (side : Fin 3) (w : Raw (K := K) (tick := tick) allocation m sigma) : |
| conditionalCount allocation m χ sigma side w = nativeCount allocation m χ sigma side w := by |
| unfold conditionalCount nativeCount JointPopulationCompatibility.compatibleTargetCount |
| exact Fintype.card_congr (Equiv.subtypeEquivRight |
| (fun e : JointPopulationCompatibility.FixedSideTargets |
| (Counts (K := K) (tick := tick) allocation m sigma) (sigma side) |
| (ownWord allocation m sigma w) => |
| compatible_iff_native allocation m χ sigma side e.val w)) |
|
|
| theorem competitors_card_le_nativeCount (allocation : Allocation) (m : ℕ) |
| {ε χ : ℝ} (hε : 0 ≤ ε) (hχ : ε ≤ χ) (sigma : Placement) |
| (e : Targets (K := K) (tick := tick) allocation m sigma) |
| (o : Orbit (K := K) (tick := tick) allocation m sigma) (v : Variable (K := K) (tick := tick) allocation m sigma) : |
| (competitors allocation m ε sigma e o v).card ≤ nativeCount allocation m χ sigma v.1 v.2 := by |
| rw [← conditionalCount_eq_native] |
| exact competitors_card_le_conditionalCount allocation m hε hχ sigma e o v |
|
|
| theorem competitors_subset_eligibility_of_first (allocation : Allocation) (m : ℕ) |
| (width : ℝ) (sigma : Placement) |
| (e : Targets (K := K) (tick := tick) allocation m sigma) |
| (o : Orbit (K := K) (tick := tick) allocation m sigma) (v : Variable (K := K) (tick := tick) allocation m sigma) (hv : v.1 = 0) : |
| competitors allocation m width sigma e o v ⊆ |
| (data allocation m width sigma).eligibilityCompetitors e := by |
| intro t ht |
| apply ((data allocation m width sigma).mem_eligibilityCompetitors e t).2 |
| refine ⟨competitor_mem allocation m width sigma e o v t ht, |
| (Finset.mem_filter.mp ht).2.1, ?_⟩ |
| have hh := (Finset.mem_filter.mp ht).2.2.1 |
| simpa only [hv, tripleSide, Matrix.cons_val_zero] using! hh |
|
|
| theorem eventually_eligibility_card_le (allocation : Allocation) (sigma : Placement) |
| (width : ℝ) {δ : ℝ} (hδ : 0 < δ) : |
| ∀ᶠ m : ℕ in atTop, ∀ e : Targets (K := K) (tick := tick) allocation m sigma, |
| (((data allocation m width sigma).eligibilityCompetitors e).card : ℝ) ≤ |
| Real.exp ((m : ℝ) * |
| (AllFieldActiveCapacity.orderNativeDegree (K := K) (tick := tick) allocation sigma 0 + δ)) := by |
| filter_upwards [AllFieldGroupAmbientDegree.eventually_sharedAmbient_first_card_le |
| (K := K) (tick := tick) allocation sigma hδ] with m hm |
| intro e |
| exact (Nat.cast_le.mpr (eligibility_card_le_sharedAmbient allocation m width sigma e)).trans (hm e) |
|
|
| theorem eventually_first_competitors_card_le (allocation : Allocation) (sigma : Placement) |
| (width : ℝ) {δ : ℝ} (hδ : 0 < δ) : |
| ∀ᶠ m : ℕ in atTop, ∀ (e : Targets (K := K) (tick := tick) allocation m sigma) |
| (o : Orbit (K := K) (tick := tick) allocation m sigma) (v : Variable (K := K) (tick := tick) allocation m sigma), v.1 = 0 → |
| ((competitors allocation m width sigma e o v).card : ℝ) ≤ |
| Real.exp ((m : ℝ) * |
| (AllFieldActiveCapacity.orderNativeDegree (K := K) (tick := tick) allocation sigma 0 + δ)) := by |
| filter_upwards [eventually_eligibility_card_le (K := K) (tick := tick) |
| allocation sigma width hδ] with m hm |
| intro e o v hv |
| exact (Nat.cast_le.mpr (Finset.card_le_card |
| (competitors_subset_eligibility_of_first allocation m width sigma e o v hv))).trans (hm e) |
|
|
| end MatrixMultiplication.AllFieldGroupDegreeLaws |
|
|
| end |
|
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| end OAI |
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|
