openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldGroups /NativeDegreeBound.lean
| import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.CountRate | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.PairEntropy | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.ProjectedEntropy | |
| namespace OAI | |
| /-! Group assignments, orbit counts and extraction capacities. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.AllFieldGroupNativeDegreeBound | |
| open AllFieldHistory AllFieldActiveLaws AllFieldActiveCapacity | |
| open AllFieldGroupOrbitData AllFieldGroupDegrees AllFieldGroupDegreeLaws | |
| open AllFieldGroupCountRate AllFieldGroupPairWindows AllFieldGroupPairEntropy | |
| open AllFieldGroupProjectedEntropy JointCompatibilityRateLimit | |
| open MatrixMultiplication.Foundation Filter | |
| open scoped BigOperators Topology | |
| attribute [local instance] Classical.propDecidable Classical.decEq | |
| variable {K tick : ℕ} | |
| def maxNativeDegree (allocation : Allocation) (sigma : Placement) : ℝ := | |
| max (orderNativeDegree (K := K) (tick := tick) allocation sigma 0) | |
| (max (orderNativeDegree (K := K) (tick := tick) allocation sigma 1) | |
| (orderNativeDegree (K := K) (tick := tick) allocation sigma 2)) | |
| theorem orderNativeDegree_le_max (allocation : Allocation) (sigma : Placement) (side : Fin 3) : | |
| orderNativeDegree (K := K) (tick := tick) allocation sigma side ≤ | |
| maxNativeDegree (K := K) (tick := tick) allocation sigma := by | |
| fin_cases side | |
| · exact le_max_left _ _ | |
| · exact (le_max_left _ _).trans (le_max_right _ _) | |
| · exact (le_max_right _ _).trans (le_max_right _ _) | |
| theorem nativeCompatibilityRate_eq (allocation : Allocation) (sigma : Placement) | |
| (side : Fin 3) (hside : side ≠ 0) : | |
| nativeCompatibilityRate (K := K) (tick := tick) allocation sigma side = | |
| orderNativeDegree (K := K) (tick := tick) allocation sigma side := by | |
| have hp := projected_native_entropy_eq (K := K) (tick := tick) allocation sigma side hside | |
| have hb := baseEntropy_sub_pairReferenceEntropy (K := K) (tick := tick) allocation sigma side | |
| unfold nativeCompatibilityRate compatibilityRate | |
| rw [show ∀ a b c d : ℝ, a - b + c + d = (a - b) + (c + d) by intros; ring, | |
| hb, hp, ← Finset.sum_add_distrib] | |
| unfold orderNativeDegree | |
| apply Finset.sum_congr rfl | |
| intro h _ | |
| have hs : (parentBase allocation h : ℝ) = | |
| (populationLength (K := K) allocation 1 : ℝ) * orderMass allocation h := | |
| population_cast_real allocation 1 (h.val.val.1.source, h.val.val.2) | |
| rw [hs] | |
| ring | |
| theorem exists_threshold_eventually_counts_le (allocation : Allocation) (sigma : Placement) | |
| {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, | |
| (((data allocation m width sigma).eligibilityCompetitors e).card : ℝ) ≤ | |
| Real.exp ((m : ℝ) * (maxNativeDegree (K := K) (tick := tick) allocation sigma + δ))) ∧ | |
| (∀ (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 → | |
| ∀ v : Variable (K := K) (tick := tick) allocation m sigma, | |
| v ∈ (data allocation m width sigma).passing e o → | |
| (((data allocation m width sigma).competitors e o v).card : ℝ) ≤ | |
| Real.exp ((m : ℝ) * (maxNativeDegree (K := K) (tick := tick) allocation sigma + δ))) := by | |
| obtain ⟨t₁, ht₁, ht₁Max, h₁⟩ := | |
| exists_threshold_eventually_competitors_le (K := K) (tick := tick) | |
| allocation sigma 1 hδ hwidthMax | |
| obtain ⟨t₂, ht₂, _, h₂⟩ := | |
| exists_threshold_eventually_competitors_le (K := K) (tick := tick) | |
| allocation sigma 2 hδ hwidthMax | |
| refine ⟨min t₁ t₂, lt_min ht₁ ht₂, (min_le_left _ _).trans ht₁Max, ?_⟩ | |
| intro width hw hwt | |
| filter_upwards [eventually_eligibility_card_le (K := K) (tick := tick) | |
| allocation sigma width hδ, | |
| h₁ width hw (hwt.trans (min_le_left _ _)), | |
| h₂ width hw (hwt.trans (min_le_right _ _))] with m hm₀ hm₁ hm₂ | |
| have hm (side : Fin 3) : | |
| Real.exp ((m : ℝ) * (orderNativeDegree (K := K) (tick := tick) allocation sigma side + δ)) ≤ | |
| Real.exp ((m : ℝ) * (maxNativeDegree (K := K) (tick := tick) allocation sigma + δ)) := | |
| Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left | |
| (add_le_add_left (orderNativeDegree_le_max (K := K) (tick := tick) allocation sigma side) δ) (Nat.cast_nonneg m)) | |
| refine ⟨fun e => (hm₀ e).trans (hm 0), ?_⟩ | |
| intro e o ho v hv | |
| rcases v with ⟨side, w⟩ | |
| fin_cases side | |
| · exact (Nat.cast_le.mpr (Finset.card_le_card | |
| (competitors_subset_eligibility_of_first allocation m width sigma e o (0, w) rfl))).trans | |
| ((hm₀ e).trans (hm 0)) | |
| · have hh := hm₁ e o ho w hv | |
| rw [nativeCompatibilityRate_eq (K := K) (tick := tick) allocation sigma 1 (by decide)] at hh | |
| exact hh.trans (hm 1) | |
| · have hh := hm₂ e o ho w hv | |
| rw [nativeCompatibilityRate_eq (K := K) (tick := tick) allocation sigma 2 (by decide)] at hh | |
| exact hh.trans (hm 2) | |
| end MatrixMultiplication.AllFieldGroupNativeDegreeBound | |
| end | |
| end OAI | |