| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.GroupCore |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.OrbitCount |
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.MaskedAssignment |
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.CoarseReorder |
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.PopulationMixedCompatibility |
|
|
| namespace OAI |
|
|
| /-! Group assignments, orbit counts and extraction capacities. -/ |
|
|
| noncomputable section |
|
|
| namespace MatrixMultiplication.AllFieldGroupOrbitData |
|
|
| open AllFieldHistory AllFieldHistorySupport AllFieldHistoryChildLaws |
| open AllFieldHistoryGroupMasks AllFieldHistoryGroupedRecovery |
| open AllFieldGroupOrbitCount JointPopulation JointCanonicalization |
| open JointCoarseHashing PermutationMatching |
| open scoped BigOperators |
| attribute [local instance] Classical.propDecidable |
|
|
| variable {K tick : ℕ} |
|
|
| abbrev Counts (allocation : Allocation) (m : ℕ) (sigma : Placement) := |
| groupCounts (K := K) (tick := tick) allocation m sigma |
| abbrev Targets (allocation : Allocation) (m : ℕ) (sigma : Placement) := |
| Target (Counts (K := K) (tick := tick) allocation m sigma) |
| abbrev Pos (allocation : Allocation) (m : ℕ) (sigma : Placement) := |
| Position (Counts (K := K) (tick := tick) allocation m sigma) |
| abbrev Raw (allocation : Allocation) (m : ℕ) (sigma : Placement) := |
| GroupRaw (K := K) (tick := tick) allocation m sigma |
| abbrev Orbit (allocation : Allocation) (m : ℕ) (sigma : Placement) := |
| Fin 3 × GroupOrbit (K := K) (tick := tick) allocation m sigma |
| abbrev Variable (allocation : Allocation) (m : ℕ) (sigma : Placement) := |
| Fin 3 × Raw (K := K) (tick := tick) allocation m sigma |
|
|
| def tripleSide {P : Type*} (s : Fin 3) (t : Triple P) : Word P := |
| ![t.1, t.2.1, t.2.2] s |
|
|
| def reorder {P : Type*} (sigma : Placement) (t : Triple P) : Triple P := |
| (tripleSide (sigma 0) t, tripleSide (sigma 1) t, tripleSide (sigma 2) t) |
|
|
| @[simp] theorem tripleSide_reorder {P : Type*} (sigma : Placement) |
| (t : Triple P) (s : Fin 3) : |
| tripleSide s (reorder sigma t) = tripleSide (sigma s) t := |
| JointCoarseReorder.tripleSide_reorder sigma t s |
|
|
| theorem reorder_injective {P : Type*} (sigma : Placement) : |
| Function.Injective (reorder (P := P) sigma) := |
| JointCoarseReorder.reorder_injective sigma |
|
|
| theorem reorder_support {P : Type*} (sigma : Placement) (S : P → ℕ) |
| (t : Triple P) (ht : HasSupportSum S t) : HasSupportSum S (reorder sigma t) := |
| JointCoarseReorder.reorder_support sigma S t ht |
|
|
| def coarse (allocation : Allocation) (m : ℕ) (sigma : Placement) |
| (e : Targets (K := K) (tick := tick) allocation m sigma) : Triple (Pos (K := K) (tick := tick) allocation m sigma) := |
| reorder sigma (triple (Counts (K := K) (tick := tick) allocation m sigma) e) |
|
|
| def ambient (allocation : Allocation) (m : ℕ) (sigma : Placement) : |
| Finset (Triple (Pos (K := K) (tick := tick) allocation m sigma)) := |
| (ambientSet (Counts (K := K) (tick := tick) allocation m sigma) (fun h => 2 * activeHalfLength h.val)).image |
| (reorder sigma) |
|
|
| def support (allocation : Allocation) (m : ℕ) (sigma : Placement) |
| (p : Pos (K := K) (tick := tick) allocation m sigma) : ℕ := |
| 2 * activeHalfLength p.1.val |
|
|
| theorem counts_support (allocation : Allocation) (m : ℕ) (sigma : Placement) |
| (h : ActiveOrder K tick sigma) (u : JointPopulation.Shape) |
| (hu : 0 < Counts (K := K) (tick := tick) allocation m sigma h u) : |
| u.1.val + u.2.1.val + u.2.2.val = 2 * activeHalfLength h.val := |
| jointCounts_support_sum allocation m h.val.val u hu |
|
|
| theorem coarse_injective (allocation : Allocation) (m : ℕ) (sigma : Placement) : |
| Function.Injective (coarse (K := K) (tick := tick) allocation m sigma) := |
| (reorder_injective sigma).comp (triple_injective (Counts (K := K) (tick := tick) allocation m sigma)) |
|
|
| theorem ambient_support (allocation : Allocation) (m : ℕ) (sigma : Placement) |
| (f : Triple (Pos (K := K) (tick := tick) allocation m sigma)) |
| (hf : f ∈ ambient allocation m sigma) : HasSupportSum (support allocation m sigma) f := by |
| obtain ⟨t, ht, rfl⟩ := Finset.mem_image.mp hf |
| exact reorder_support sigma _ t (Finset.mem_filter.mp ht).2.1 |
|
|
| theorem coarse_mem_ambient (allocation : Allocation) (m : ℕ) (sigma : Placement) |
| (e : Targets (K := K) (tick := tick) allocation m sigma) : |
| coarse allocation m sigma e ∈ ambient allocation m sigma := by |
| apply Finset.mem_image.mpr |
| refine ⟨triple (Counts (K := K) (tick := tick) allocation m sigma) e, ?_, rfl⟩ |
| apply targetSet_subset_ambient (Counts (K := K) (tick := tick) allocation m sigma) |
| (fun h => 2 * activeHalfLength h.val) (counts_support allocation m sigma) |
| simp only [targetSet, Finset.mem_image, Finset.mem_univ, true_and] |
| exact ⟨e, rfl⟩ |
|
|
| def sharing {sigma : Placement} (h : ActiveOrder K tick sigma) : Bool := |
| match h.val.val.1 with |
| | .stageA _ => true |
| | .stageB _ => true |
| | .stageC _ => false |
|
|
| def workComplement (w : Work K) : w.Statistic ≃ w.Statistic := |
| match w with |
| | .stageA _ => CWCompleteStatistics.orderedComplementStatistic |
| | .stageB _ => CWCompleteStatistics.complementStatistic |
| | .stageC _ => Equiv.refl _ |
|
|
| def complement {sigma : Placement} (h : ActiveOrder K tick sigma) : Statistic h.val ≃ Statistic h.val := |
| workComplement h.val.val.1 |
|
|
| def compatibilityLaw (right : Bool) (side : Fin 3) (sigma : Placement) |
| (h : ActiveOrder K tick sigma) (u : JointPopulation.Shape) : Statistic h.val → ℝ := |
| let weight : Fin 3 → ℕ := fun i => |
| if right then activeParentShape h.val (sigma i) - (shapeSide (sigma i) u).val |
| else (shapeSide (sigma i) u).val |
| let source : Fin 3 := if side = 2 ∧ weight 1 ≠ 0 then sigma 1 else sigma 0 |
| (if right then rightLaw h.val u source else leftLaw h.val u source) ∘ |
| (complement h).symm |
|
|
| def designated (right : Bool) (side : Fin 3) (sigma : Placement) |
| (h : ActiveOrder K tick sigma) (u : JointPopulation.Shape) : Prop := |
| sharing h = true ∧ |
| let weight : Fin 3 → ℕ := fun i => |
| if right then activeParentShape h.val (sigma i) - (shapeSide (sigma i) u).val |
| else (shapeSide (sigma i) u).val |
| if side = 1 then weight 2 = 0 |
| else if side = 2 then weight 0 * weight 1 = 0 else False |
|
|
| def fineCompatible (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) : Prop := |
| ∀ (right : Bool) h u, designated right side sigma h u → ∀ a : Statistic h.val, |
| |(JointPopulationCompatibility.shapeStatisticCount (Counts (K := K) (tick := tick) allocation m sigma) e h u |
| (fun i => statistic h.val (if right then (w h i).2 else (w h i).1)) a : ℝ) - |
| (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) * AllFieldHistoryMasks.childWidth ε h.val |
|
|
| def ownWord (allocation : Allocation) (m : ℕ) (sigma : Placement) |
| (w : Raw (K := K) (tick := tick) allocation m sigma) : Word (Pos (K := K) (tick := tick) allocation m sigma) := |
| fun p => groupCoarse 0 p.1 (w p.1 p.2) |
|
|
| def Compatible (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) : Prop := |
| tripleSide side (coarse allocation m sigma e) = ownWord allocation m sigma w ∧ |
| fineCompatible allocation m ε sigma side e w |
|
|
| def Admissible (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) : Prop := |
| groupIdealSide allocation m ε (sigma side) sigma e w ∧ |
| ∀ h j, CWStrands.weight (w h j).1 = (shapeSide (sigma side) ((e h).val j)).val ∧ |
| CWStrands.weight (w h j).2 = activeParentShape h.val (sigma side) - |
| (shapeSide (sigma side) ((e h).val j)).val |
|
|
| def orbitOf (allocation : Allocation) (m : ℕ) (sigma : Placement) |
| (e : Targets (K := K) (tick := tick) allocation m sigma) |
| (w : Raw (K := K) (tick := tick) allocation m sigma) : GroupOrbit (K := K) (tick := tick) allocation m sigma := |
| Quotient.mk'' (groupCoordinates allocation m sigma e w) |
|
|
| def usedOrbits (allocation : Allocation) (m : ℕ) (ε : ℝ) (sigma : Placement) |
| (e : Targets (K := K) (tick := tick) allocation m sigma) : Finset (Orbit (K := K) (tick := tick) allocation m sigma) := |
| Finset.univ.filter fun o => ∃ w, Admissible allocation m ε sigma o.1 e w ∧ |
| orbitOf allocation m sigma e w = o.2 |
|
|
| def full (allocation : Allocation) (m : ℕ) (sigma : Placement) |
| (e : Targets (K := K) (tick := tick) allocation m sigma) (o : Orbit (K := K) (tick := tick) allocation m sigma) : |
| Finset (Variable (K := K) (tick := tick) allocation m sigma) := |
| Finset.univ.filter fun v => v.1 = o.1 ∧ orbitOf allocation m sigma e v.2 = o.2 |
|
|
| def passing (allocation : Allocation) (m : ℕ) (sigma : Placement) (ε : ℝ) |
| (e : Targets (K := K) (tick := tick) allocation m sigma) (o : Orbit (K := K) (tick := tick) allocation m sigma) : |
| Finset (Variable (K := K) (tick := tick) allocation m sigma) := |
| (full allocation m sigma e o).filter fun v => completeKeep allocation m ε (sigma v.1) sigma v.2 |
|
|
| def competitors (allocation : Allocation) (m : ℕ) (ε : 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) : |
| Finset (Triple (Pos (K := K) (tick := tick) allocation m sigma)) := |
| (Finset.univ.image (coarse allocation m sigma)).filter fun t => |
| t ≠ coarse allocation m sigma e ∧ |
| tripleSide v.1 t = tripleSide v.1 (coarse allocation m sigma e) ∧ |
| ∃ f, coarse allocation m sigma f = t ∧ Compatible allocation m ε sigma v.1 f v.2 |
|
|
| theorem full_pos (allocation : Allocation) (m : ℕ) (ε : ℝ) (sigma : Placement) |
| (e : Targets (K := K) (tick := tick) allocation m sigma) (o : Orbit (K := K) (tick := tick) allocation m sigma) |
| (ho : o ∈ usedOrbits allocation m ε sigma e) : 0 < (full allocation m sigma e o).card := by |
| obtain ⟨w, _, hw⟩ := (Finset.mem_filter.mp ho).2 |
| apply Finset.card_pos.mpr |
| exact ⟨(o.1, w), Finset.mem_filter.mpr ⟨Finset.mem_univ _, rfl, hw⟩⟩ |
|
|
| theorem competitor_mem (allocation : Allocation) (m : ℕ) (ε : 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) (t : Triple (Pos (K := K) (tick := tick) allocation m sigma)) |
| (ht : t ∈ competitors allocation m ε sigma e o v) : t ∈ ambient allocation m sigma := by |
| obtain ⟨f, hf, _⟩ := (Finset.mem_filter.mp ht).2.2.2 |
| rw [← hf] |
| exact coarse_mem_ambient allocation m sigma f |
|
|
| theorem competitor_shares (allocation : Allocation) (m : ℕ) (ε : 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) (t : Triple (Pos (K := K) (tick := tick) allocation m sigma)) |
| (ht : t ∈ competitors allocation m ε sigma e o v) : SharesSide (coarse allocation m sigma e) t := by |
| have h := (Finset.mem_filter.mp ht).2.2.1 |
| rcases v with ⟨side, w⟩ |
| fin_cases side |
| · exact Or.inl h |
| · exact Or.inr (Or.inl h) |
| · exact Or.inr (Or.inr h) |
|
|
| def data (allocation : Allocation) (m : ℕ) (ε : ℝ) (sigma : Placement) : |
| JointMaskedSelection.OrbitData |
| (Pos (K := K) (tick := tick) allocation m sigma) (Targets (K := K) (tick := tick) allocation m sigma) |
| (Orbit (K := K) (tick := tick) allocation m sigma) (Variable (K := K) (tick := tick) allocation m sigma) where |
| ambient := ambient allocation m sigma |
| support := support allocation m sigma |
| coarse := coarse allocation m sigma |
| support_ambient := ambient_support allocation m sigma |
| target_mem := coarse_mem_ambient allocation m sigma |
| orbits := usedOrbits allocation m ε sigma |
| full := full allocation m sigma |
| passing := passing allocation m sigma ε |
| full_pos := full_pos allocation m ε sigma |
| passing_subset := fun _ _ _ => Finset.filter_subset _ _ |
| competitors := competitors allocation m ε sigma |
| competitor_mem := fun e o _ v _ t ht => competitor_mem allocation m ε sigma e o v t ht |
| competitor_ne := fun _ _ _ _ _ _ ht => (Finset.mem_filter.mp ht).2.1 |
| competitor_shares := fun e o _ v _ t ht => competitor_shares allocation m ε sigma e o v t ht |
|
|
| theorem admissible_covered (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) : |
| ∃ o ∈ (data allocation m ε sigma).orbits e, |
| (side, w) ∈ (data allocation m ε sigma).full e o := by |
| refine ⟨(side, orbitOf allocation m sigma e w), ?_, ?_⟩ |
| · exact Finset.mem_filter.mpr ⟨Finset.mem_univ _, w, hw, rfl⟩ |
| · exact Finset.mem_filter.mpr ⟨Finset.mem_univ _, rfl, rfl⟩ |
|
|
| theorem orbits_card_le (allocation : Allocation) (m : ℕ) (ε : ℝ) (sigma : Placement) |
| (e : Targets (K := K) (tick := tick) allocation m sigma) : |
| (((data allocation m ε sigma).orbits e).card : ℝ) ≤ |
| (3 * groupOrbitPrefactor (K := K) (tick := tick) allocation sigma) * |
| ((m : ℝ) + 1) ^ groupDegree K tick sigma := |
| taggedUsedGroupOrbits_card_le_real allocation m sigma _ |
|
|
| def actualCompatible (allocation : Allocation) (m : ℕ) (ε : ℝ) (sigma : Placement) |
| (t : Triple (Pos (K := K) (tick := tick) allocation m sigma)) |
| (v : Variable (K := K) (tick := tick) allocation m sigma) : Prop := |
| ∃ f, coarse allocation m sigma f = t ∧ Compatible allocation m ε sigma v.1 f v.2 |
|
|
| def actualUseful (allocation : Allocation) (m : ℕ) (ε : ℝ) (sigma : Placement) |
| (t : Triple (Pos (K := K) (tick := tick) allocation m sigma)) |
| (v : Variable (K := K) (tick := tick) allocation m sigma) : Prop := |
| ∃ f, coarse allocation m sigma f = t ∧ |
| groupIdealSide allocation m ε (sigma v.1) sigma f v.2 |
|
|
| def actualKeep (allocation : Allocation) (m : ℕ) (ε : ℝ) (sigma : Placement) |
| (v : Variable (K := K) (tick := tick) allocation m sigma) : Prop := |
| completeKeep allocation m ε (sigma v.1) sigma v.2 |
|
|
| @[simp] theorem actualCompatible_coarse (allocation : Allocation) (m : ℕ) (ε : ℝ) |
| (sigma : Placement) (e : Targets (K := K) (tick := tick) allocation m sigma) |
| (v : Variable (K := K) (tick := tick) allocation m sigma) : |
| actualCompatible allocation m ε sigma (coarse allocation m sigma e) v ↔ |
| Compatible allocation m ε sigma v.1 e v.2 := by |
| constructor |
| · rintro ⟨f, hf, hv⟩ |
| exact (coarse_injective allocation m sigma hf) ▸ hv |
| · exact fun hv => ⟨e, rfl, hv⟩ |
|
|
| @[simp] theorem actualUseful_coarse (allocation : Allocation) (m : ℕ) (ε : ℝ) |
| (sigma : Placement) (e : Targets (K := K) (tick := tick) allocation m sigma) |
| (v : Variable (K := K) (tick := tick) allocation m sigma) : |
| actualUseful allocation m ε sigma (coarse allocation m sigma e) v ↔ |
| groupIdealSide allocation m ε (sigma v.1) sigma e v.2 := by |
| constructor |
| · rintro ⟨f, hf, hv⟩ |
| exact (coarse_injective allocation m sigma hf) ▸ hv |
| · exact fun hv => ⟨e, rfl, hv⟩ |
|
|
| theorem competitor_coverage (allocation : Allocation) (m : ℕ) (ε : 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) |
| (he : Compatible allocation m ε sigma v.1 e v.2) |
| (t : Triple (Pos (K := K) (tick := tick) allocation m sigma)) |
| (hne : t ≠ coarse allocation m sigma e) |
| (hc : actualCompatible allocation m ε sigma t v) : |
| t ∈ (data allocation m ε sigma).competitors e o v := by |
| obtain ⟨f, hf, hcomp⟩ := hc |
| apply Finset.mem_filter.mpr |
| refine ⟨?_, hne, ?_, f, hf, hcomp⟩ |
| · exact Finset.mem_image.mpr ⟨f, Finset.mem_univ _, hf⟩ |
| · rw [← hf] |
| exact hcomp.1.trans he.1.symm |
|
|
| theorem mem_passing (allocation : Allocation) (m : ℕ) (ε : 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) : |
| v ∈ (data allocation m ε sigma).passing e o ↔ |
| v ∈ (data allocation m ε sigma).full e o ∧ actualKeep allocation m ε sigma v := |
| Finset.mem_filter |
|
|
| end MatrixMultiplication.AllFieldGroupOrbitData |
|
|
| end |
|
|
| end OAI |
|
|
