| import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.OrbitData |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.ActiveCapacity |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.TargetRate |
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.PopulationCompatibility |
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.AmbientDegree |
|
|
| namespace OAI |
|
|
| /-! Group assignments, orbit counts and extraction capacities. -/ |
|
|
| noncomputable section |
|
|
| namespace MatrixMultiplication.AllFieldGroupDegrees |
|
|
| open AllFieldHistory AllFieldHistorySupport AllFieldHistoryChildLaws |
| open AllFieldGroupOrbitData JointPopulation JointCanonicalization |
| open MatrixMultiplication.Foundation |
| open scoped BigOperators |
| attribute [local instance] Classical.propDecidable Classical.decEq |
|
|
| variable {K tick : ℕ} |
|
|
| abbrev Letter (sigma : Placement) (h : ActiveOrder K tick sigma) := |
| Fin (activeHalfLength h.val) → Fin 7 |
|
|
| abbrev Class (sigma : Placement) := ActiveOrder K tick sigma × Fin 17 |
|
|
| abbrev Symbol (sigma : Placement) (c : Class (K := K) (tick := tick) sigma) := |
| Statistic c.1.val |
|
|
| def projectedStatistic (sigma : Placement) |
| (c : Class (K := K) (tick := tick) sigma) : Letter (K := K) (tick := tick) sigma c.1 → Symbol (K := K) (tick := tick) sigma c := |
| statistic c.1.val |
|
|
| def classDesignated (right : Bool) (side : Fin 3) (sigma : Placement) |
| (c : Class (K := K) (tick := tick) sigma) (u : JointPopulation.Shape) : Prop := |
| designated right side sigma c.1 u |
|
|
| def classLaw (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 := compatibilityLaw right side sigma c.1 u |
|
|
| theorem tripleSide_eq_sideWord (allocation : Allocation) (m : ℕ) (sigma : Placement) |
| (s : Fin 3) (t : JointCoarseHashing.Triple (Pos (K := K) (tick := tick) allocation m sigma)) |
| (h : ActiveOrder K tick sigma) (i : JointPopulation.Positions (Counts (K := K) (tick := tick) allocation m sigma) h) : |
| tripleSide s t ⟨h, i⟩ = sideWord (Counts (K := K) (tick := tick) allocation m sigma) s t h i := by |
| fin_cases s <;> rfl |
|
|
| theorem tripleSide_triple (allocation : Allocation) (m : ℕ) (sigma : Placement) |
| (s : Fin 3) (e : Targets (K := K) (tick := tick) allocation m sigma) |
| (h : ActiveOrder K tick sigma) (i : JointPopulation.Positions (Counts (K := K) (tick := tick) allocation m sigma) h) : |
| tripleSide s (triple (Counts (K := K) (tick := tick) allocation m sigma) e) ⟨h, i⟩ = |
| shapeSide s ((e h).val i) := by |
| fin_cases s <;> rfl |
|
|
| theorem childWidth_le {ε : ℝ} (hε : 0 ≤ ε) (h : Active K tick) : |
| AllFieldHistoryMasks.childWidth ε h ≤ ε := by |
| have hp : 0 ≤ AllFieldHistoryMasks.pairWidth ε h := by |
| unfold AllFieldHistoryMasks.pairWidth |
| positivity |
| calc |
| AllFieldHistoryMasks.childWidth ε h ≤ AllFieldHistoryMasks.pairWidth ε h := |
| div_le_self hp (by norm_num) |
| _ ≤ ε := div_le_self hε (one_le_pow₀ (by norm_num)) |
|
|
| theorem fineCompatible_uniform (allocation : Allocation) (m : ℕ) |
| {ε χ : ℝ} (hε : 0 ≤ ε) (hχ : ε ≤ χ) (sigma : Placement) (side : Fin 3) |
| (e : Targets (K := K) (tick := tick) allocation m sigma) |
| (w : Raw (K := K) (tick := tick) allocation m sigma) |
| (hw : fineCompatible allocation m ε 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) |
| (classLaw false side sigma) (classLaw true side sigma) χ (sigma side) e w := by |
| constructor |
| · intro h u hd a |
| exact (hw false h u hd a).trans |
| (mul_le_mul_of_nonneg_left ((childWidth_le hε h.val).trans hχ) (Nat.cast_nonneg _)) |
| · intro h u hd a |
| exact (hw true h u hd a).trans |
| (mul_le_mul_of_nonneg_left ((childWidth_le hε h.val).trans hχ) (Nat.cast_nonneg _)) |
|
|
| def uniformCompatible (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) : Prop := |
| JointPopulationCompatibility.compatibleWithCoarse (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 |
| (ownWord allocation m sigma) (sigma side) (triple (Counts (K := K) (tick := tick) allocation m sigma) e) w |
|
|
| theorem Compatible_uniform (allocation : Allocation) (m : ℕ) |
| {ε χ : ℝ} (hε : 0 ≤ ε) (hχ : ε ≤ χ) (sigma : Placement) (side : Fin 3) |
| (e : Targets (K := K) (tick := tick) allocation m sigma) |
| (w : Raw (K := K) (tick := tick) allocation m sigma) |
| (hw : Compatible allocation m ε sigma side e w) : |
| uniformCompatible allocation m χ sigma side e w := by |
| apply (JointPopulationCompatibility.compatibleWithCoarse_target |
| (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 |
| (ownWord allocation m sigma) (sigma side) e w).2 |
| refine ⟨?_, fineCompatible_uniform allocation m hε hχ sigma side e w hw.2⟩ |
| intro h i |
| have hh := congrFun hw.1 ⟨h, i⟩ |
| simpa only [coarse, tripleSide_reorder, tripleSide_triple] using hh |
|
|
| def conditionalCount (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) |
| (classLaw false side sigma) (classLaw true side sigma) χ |
| (sigma side) (ownWord allocation m sigma w) w |
|
|
| theorem uniformCompatible_count (allocation : Allocation) (m : ℕ) (χ : ℝ) |
| (sigma : Placement) (side : Fin 3) (w : Raw (K := K) (tick := tick) allocation m sigma) : |
| Fintype.card {e : Targets (K := K) (tick := tick) allocation m sigma |
| uniformCompatible allocation m χ sigma side e w} = |
| conditionalCount allocation m χ sigma side w := |
| JointPopulationCompatibility.compatibleWithCoarse_target_count |
| (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 |
| (ownWord allocation m sigma) (sigma side) w |
|
|
| theorem competitors_card_le_conditionalCount (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 ≤ |
| conditionalCount allocation m χ sigma v.1 v.2 := by |
| classical |
| let candidates : Finset (Targets (K := K) (tick := tick) allocation m sigma) := |
| Finset.univ.filter fun f => uniformCompatible allocation m χ sigma v.1 f v.2 |
| have hsub : competitors allocation m ε sigma e o v ⊆ |
| candidates.image (coarse allocation m sigma) := by |
| intro t ht |
| obtain ⟨f, hf, hc⟩ := (Finset.mem_filter.mp ht).2.2.2 |
| exact Finset.mem_image.mpr ⟨f, Finset.mem_filter.mpr |
| ⟨Finset.mem_univ _, Compatible_uniform allocation m hε hχ sigma v.1 f v.2 hc⟩, hf⟩ |
| calc |
| (competitors allocation m ε sigma e o v).card ≤ |
| (candidates.image (coarse allocation m sigma)).card := Finset.card_le_card hsub |
| _ ≤ candidates.card := Finset.card_image_le |
| _ = Fintype.card {f : Targets (K := K) (tick := tick) allocation m sigma |
| uniformCompatible allocation m χ sigma v.1 f v.2} := by |
| simp only [candidates, Fintype.card_subtype] |
| _ = conditionalCount allocation m χ sigma v.1 v.2 := |
| uniformCompatible_count allocation m χ sigma v.1 v.2 |
|
|
| theorem eligibility_card_le_sharedAmbient (allocation : Allocation) (m : ℕ) |
| (ε : ℝ) (sigma : Placement) |
| (e : Targets (K := K) (tick := tick) allocation m sigma) : |
| ((data allocation m ε sigma).eligibilityCompetitors e).card ≤ |
| (JointAmbientDegree.sharedAmbient (Counts (K := K) (tick := tick) allocation m sigma) |
| (fun h => 2 * activeHalfLength h.val) (sigma 0) |
| (triple (Counts (K := K) (tick := tick) allocation m sigma) e)).card := by |
| let a := JointAmbientDegree.sharedAmbient (Counts (K := K) (tick := tick) allocation m sigma) |
| (fun h => 2 * activeHalfLength h.val) (sigma 0) |
| (triple (Counts (K := K) (tick := tick) allocation m sigma) e) |
| have hs : (data allocation m ε sigma).eligibilityCompetitors e ⊆ |
| a.image (reorder sigma) := by |
| intro f hf |
| obtain ⟨ha, _, hx⟩ := |
| ((data allocation m ε sigma).mem_eligibilityCompetitors e f).1 hf |
| obtain ⟨t, ht, rfl⟩ := Finset.mem_image.mp ha |
| apply Finset.mem_image.mpr |
| refine ⟨t, Finset.mem_filter.mpr ⟨ht, ?_⟩, rfl⟩ |
| intro h |
| funext i |
| have hh := congrFun hx ⟨h, i⟩ |
| change tripleSide (sigma 0) t ⟨h, i⟩ = |
| tripleSide (sigma 0) (triple (Counts (K := K) (tick := tick) allocation m sigma) e) ⟨h, i⟩ at hh |
| simpa only [tripleSide_eq_sideWord] using hh |
| exact (Finset.card_le_card hs).trans Finset.card_image_le |
|
|
| end MatrixMultiplication.AllFieldGroupDegrees |
|
|
| end |
|
|
| end OAI |
|
|
