openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldGroups /CompatibilityTransfer.lean
| import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.OrbitData | |
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.PopulationCWNonzero | |
| import OAI.LinearAlgebra.MatrixMultiplication.CoppersmithWinograd.CWShapePhysicalPermutation | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.StatisticComplement | |
| namespace OAI | |
| /-! Group assignments, orbit counts and extraction capacities. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.AllFieldGroupCompatibilityTransfer | |
| open MatrixMultiplication.Foundation AllFieldHistory AllFieldHistoryChildLaws | |
| open AllFieldHistoryGroupMasks AllFieldHistoryGroupedRecovery AllFieldGroupOrbitData | |
| open JointPopulation JointCanonicalization JointCanonicalCW CWStrands InheritedMasks | |
| open scoped BigOperators | |
| attribute [local instance] Classical.propDecidable | |
| variable {K tick : ℕ} {sigma : Placement} | |
| theorem physical_coarseMask (allocation : Allocation) (m : ℕ) (sigma : Placement) | |
| (e : Targets (K := K) (tick := tick) allocation m sigma) | |
| (a : Fin 3 → Raw (K := K) (tick := tick) allocation m sigma) | |
| (hcoarse : reorder sigma (ownWord allocation m sigma (a 0), | |
| ownWord allocation m sigma (a 1), ownWord allocation m sigma (a 2)) = | |
| AllFieldGroupOrbitData.coarse allocation m sigma e) | |
| (s : Fin 3) : | |
| coarseMask (Counts allocation m sigma) | |
| (fun h => Fin (activeHalfLength h.val) → Fin 7) | |
| (fun h => Fin (activeHalfLength h.val) → Fin 7) | |
| groupCoarse s e (a s) := by | |
| have ht := reorder_injective sigma hcoarse | |
| intro h i | |
| have he := congrArg (fun t => tripleSide s t ⟨h, i⟩) ht | |
| fin_cases s <;> exact he | |
| def halfShape (right : Bool) (h : ActiveOrder K tick sigma) | |
| (u : JointPopulation.Shape) : Fin 3 → ℕ := | |
| if right then activeParentShape h.val - shapeNat u else shapeNat u | |
| def halfLaw (right : Bool) (side : Fin 3) (h : ActiveOrder K tick sigma) | |
| (u : JointPopulation.Shape) : Statistic h.val → ℝ := | |
| if right then rightLaw h.val u side else leftLaw h.val u side | |
| theorem physical_half_product_nonzero (F : Type*) [Field F] | |
| (allocation : Allocation) (m : ℕ) (sigma : Placement) | |
| (e : Targets (K := K) (tick := tick) allocation m sigma) | |
| (a : Fin 3 → Raw (K := K) (tick := tick) allocation m sigma) | |
| (hraw : groupRawSource F allocation m sigma (a 0) (a 1) (a 2) ≠ 0) | |
| (hcoarse : reorder sigma (ownWord allocation m sigma (a 0), | |
| ownWord allocation m sigma (a 1), ownWord allocation m sigma (a 2)) = | |
| AllFieldGroupOrbitData.coarse allocation m sigma e) | |
| (right : Bool) (h : ActiveOrder K tick sigma) (u : JointPopulation.Shape) | |
| (hu : 0 < Counts allocation m sigma h u) : | |
| (∏ i : Class (Counts allocation m sigma) e h u, | |
| shapeTensor (F := F) (Fin (activeHalfLength h.val)) (halfShape right h u) | |
| (if right then (a 0 h i.val).2 else (a 0 h i.val).1) | |
| (if right then (a 1 h i.val).2 else (a 1 h i.val).1) | |
| (if right then (a 2 h i.val).2 else (a 2 h i.val).1)) ≠ 0 := by | |
| have hsupp : ∀ (h : ActiveOrder K tick sigma) (u : JointPopulation.Shape), | |
| 0 < Counts allocation m sigma h u → | |
| ∀ s, shapeNat u s ≤ activeParentShape h.val s := by | |
| intro h u hu s | |
| exact jointCounts_support_le allocation m h.val.val u hu s | |
| have hsource : sourceTensor (F := F) (Counts (K := K) (tick := tick) allocation m sigma) | |
| (Left (fun h => activeHalfLength h.val)) (Right (fun h => activeHalfLength h.val)) | |
| (parentTensor (F := F) (fun h : ActiveOrder K tick sigma => activeHalfLength h.val) | |
| (fun h : ActiveOrder K tick sigma => activeHalfLength h.val) | |
| (fun h : ActiveOrder K tick sigma => activeParentShape h.val)) | |
| (a 0) (a 1) (a 2) ≠ 0 := by | |
| exact hraw | |
| have hp := JointPopulationCWNonzero.class_half_products_nonzero | |
| (Counts allocation m sigma) (fun h => activeHalfLength h.val) | |
| (fun h => activeHalfLength h.val) (fun h => activeParentShape h.val) | |
| (fun h => AllFieldHistoryRecovery.halfLength_le_eight h.val) hsupp e | |
| (a 0) (a 1) (a 2) hsource | |
| (physical_coarseMask allocation m sigma e a hcoarse 0) | |
| (physical_coarseMask allocation m sigma e a hcoarse 1) | |
| (physical_coarseMask allocation m sigma e a hcoarse 2) h u hu | |
| cases right | |
| · exact hp.1 | |
| · exact hp.2 | |
| theorem priority_half_product_nonzero (F : Type*) [Field F] | |
| (allocation : Allocation) (m : ℕ) (sigma : Placement) | |
| (e : Targets (K := K) (tick := tick) allocation m sigma) | |
| (a : Fin 3 → Raw (K := K) (tick := tick) allocation m sigma) | |
| (hraw : groupRawSource F allocation m sigma (a 0) (a 1) (a 2) ≠ 0) | |
| (hcoarse : reorder sigma (ownWord allocation m sigma (a 0), | |
| ownWord allocation m sigma (a 1), ownWord allocation m sigma (a 2)) = | |
| AllFieldGroupOrbitData.coarse allocation m sigma e) | |
| (right : Bool) (h : ActiveOrder K tick sigma) (u : JointPopulation.Shape) | |
| (hu : 0 < Counts allocation m sigma h u) : | |
| (∏ i : Class (Counts allocation m sigma) e h u, | |
| shapeTensor (F := F) (Fin (activeHalfLength h.val)) (halfShape right h u ∘ sigma) | |
| (if right then (a (sigma 0) h i.val).2 else (a (sigma 0) h i.val).1) | |
| (if right then (a (sigma 1) h i.val).2 else (a (sigma 1) h i.val).1) | |
| (if right then (a (sigma 2) h i.val).2 else (a (sigma 2) h i.val).1)) ≠ 0 := by | |
| have he := fun i : Class (Counts allocation m sigma) e h u => | |
| CWShapePhysicalPermutation.shapeTensor_permute (F := F) sigma (halfShape right h u) | |
| (fun s => if right then (a s h i.val).2 else (a s h i.val).1) | |
| simp_rw [he] | |
| exact physical_half_product_nonzero F allocation m sigma e a hraw hcoarse right h u hu | |
| theorem useful_half_window (allocation : Allocation) (m : ℕ) (ε : ℝ) | |
| (sigma : Placement) (s : Fin 3) | |
| (e : Targets (K := K) (tick := tick) allocation m sigma) | |
| (w : Raw (K := K) (tick := tick) allocation m sigma) | |
| (hw : groupIdealSide allocation m ε s sigma e w) | |
| (right : Bool) (h : ActiveOrder K tick sigma) (u : JointPopulation.Shape) | |
| (hu : 0 < Counts allocation m sigma h u) : | |
| typeWindow (halfLaw right s h u) (AllFieldHistoryMasks.childWidth ε h.val) | |
| (fun i : Class (Counts allocation m sigma) e h u => | |
| statistic h.val (if right then (w h i.val).2 else (w h i.val).1)) := by | |
| cases right | |
| · exact (hw.2 h u hu).1 | |
| · exact (hw.2 h u hu).2 | |
| theorem own_coarse_guard (allocation : Allocation) (m : ℕ) (sigma : Placement) | |
| (e : Targets (K := K) (tick := tick) allocation m sigma) | |
| (a : Fin 3 → Raw (K := K) (tick := tick) allocation m sigma) | |
| (hcoarse : reorder sigma (ownWord allocation m sigma (a 0), | |
| ownWord allocation m sigma (a 1), ownWord allocation m sigma (a 2)) = | |
| AllFieldGroupOrbitData.coarse allocation m sigma e) | |
| (side : Fin 3) : | |
| tripleSide side (AllFieldGroupOrbitData.coarse allocation m sigma e) = | |
| ownWord allocation m sigma (a (sigma side)) := by | |
| rw [← hcoarse, tripleSide_reorder] | |
| generalize sigma side = s | |
| fin_cases s <;> rfl | |
| theorem transferY (F : Type*) [Field F] (allocation : Allocation) (m : ℕ) (ε : ℝ) | |
| (sigma : Placement) (e : Targets (K := K) (tick := tick) allocation m sigma) | |
| (a : Fin 3 → Raw (K := K) (tick := tick) allocation m sigma) | |
| (hraw : groupRawSource F allocation m sigma (a 0) (a 1) (a 2) ≠ 0) | |
| (hcoarse : reorder sigma (ownWord allocation m sigma (a 0), | |
| ownWord allocation m sigma (a 1), ownWord allocation m sigma (a 2)) = | |
| AllFieldGroupOrbitData.coarse allocation m sigma e) | |
| (huseX : groupIdealSide allocation m ε (sigma 0) sigma e (a (sigma 0))) : | |
| Compatible allocation m ε sigma 1 e (a (sigma 1)) := by | |
| refine ⟨own_coarse_guard allocation m sigma e a hcoarse 1, ?_⟩ | |
| intro right h u hd | |
| apply JointPopulationCompatibility.shape_count_window_of_typeWindow | |
| (Counts allocation m sigma) e h u _ _ _ | |
| intro hu | |
| have hz : halfShape right h u (sigma 2) = 0 := by | |
| cases right <;> simpa [designated, halfShape, shapeNat] using hd.2 | |
| have hw := CWStrandCompatibility.zeroZ_transfer | |
| (fun _ : Class (Counts allocation m sigma) e h u => halfShape right h u ∘ sigma) | |
| (fun i => if right then (a (sigma 0) h i.val).2 else (a (sigma 0) h i.val).1) | |
| (fun i => if right then (a (sigma 1) h i.val).2 else (a (sigma 1) h i.val).1) | |
| (fun i => if right then (a (sigma 2) h i.val).2 else (a (sigma 2) h i.val).1) | |
| (priority_half_product_nonzero F allocation m sigma e a hraw hcoarse right h u hu) | |
| (statistic h.val) (complement h) | |
| (AllFieldGroupStatisticComplement.statistic_wordComplement h) | |
| (halfLaw right (sigma 0) h u) (AllFieldHistoryMasks.childWidth ε h.val) | |
| (fun _ => hz) (useful_half_window allocation m ε sigma (sigma 0) e _ huseX right h u hu) | |
| simpa [compatibilityLaw, halfLaw] using hw | |
| theorem transferZ (F : Type*) [Field F] (allocation : Allocation) (m : ℕ) (ε : ℝ) | |
| (sigma : Placement) (e : Targets (K := K) (tick := tick) allocation m sigma) | |
| (a : Fin 3 → Raw (K := K) (tick := tick) allocation m sigma) | |
| (hraw : groupRawSource F allocation m sigma (a 0) (a 1) (a 2) ≠ 0) | |
| (hcoarse : reorder sigma (ownWord allocation m sigma (a 0), | |
| ownWord allocation m sigma (a 1), ownWord allocation m sigma (a 2)) = | |
| AllFieldGroupOrbitData.coarse allocation m sigma e) | |
| (huseX : groupIdealSide allocation m ε (sigma 0) sigma e (a (sigma 0))) | |
| (huseY : groupIdealSide allocation m ε (sigma 1) sigma e (a (sigma 1))) : | |
| Compatible allocation m ε sigma 2 e (a (sigma 2)) := by | |
| refine ⟨own_coarse_guard allocation m sigma e a hcoarse 2, ?_⟩ | |
| intro right h u hd | |
| apply JointPopulationCompatibility.shape_count_window_of_typeWindow | |
| (Counts allocation m sigma) e h u _ _ _ | |
| intro hu | |
| have hz : halfShape right h u (sigma 0) * halfShape right h u (sigma 1) = 0 := by | |
| cases right <;> simpa [designated, halfShape, shapeNat] using hd.2 | |
| have hp := priority_half_product_nonzero F allocation m sigma e a hraw hcoarse right h u hu | |
| by_cases hy : halfShape right h u (sigma 1) = 0 | |
| · have hw := CWStrandCompatibility.zeroY_transfer | |
| (fun _ : Class (Counts allocation m sigma) e h u => halfShape right h u ∘ sigma) | |
| (fun i => if right then (a (sigma 0) h i.val).2 else (a (sigma 0) h i.val).1) | |
| (fun i => if right then (a (sigma 1) h i.val).2 else (a (sigma 1) h i.val).1) | |
| (fun i => if right then (a (sigma 2) h i.val).2 else (a (sigma 2) h i.val).1) | |
| hp (statistic h.val) (complement h) | |
| (AllFieldGroupStatisticComplement.statistic_wordComplement h) | |
| (halfLaw right (sigma 0) h u) (AllFieldHistoryMasks.childWidth ε h.val) | |
| (fun _ => hy) (useful_half_window allocation m ε sigma (sigma 0) e _ huseX right h u hu) | |
| have hy' : (if right then activeParentShape h.val (sigma 1) - | |
| (shapeSide (sigma 1) u).val else (shapeSide (sigma 1) u).val) = 0 := by | |
| cases right <;> simpa [halfShape, shapeNat] using hy | |
| simpa [compatibilityLaw, halfLaw, hy'] using hw | |
| · have hx : halfShape right h u (sigma 0) = 0 := (Nat.mul_eq_zero.mp hz).resolve_right hy | |
| have hw := CWStrandCompatibility.zeroX_transfer | |
| (fun _ : Class (Counts allocation m sigma) e h u => halfShape right h u ∘ sigma) | |
| (fun i => if right then (a (sigma 0) h i.val).2 else (a (sigma 0) h i.val).1) | |
| (fun i => if right then (a (sigma 1) h i.val).2 else (a (sigma 1) h i.val).1) | |
| (fun i => if right then (a (sigma 2) h i.val).2 else (a (sigma 2) h i.val).1) | |
| hp (statistic h.val) (complement h) | |
| (AllFieldGroupStatisticComplement.statistic_wordComplement h) | |
| (halfLaw right (sigma 1) h u) (AllFieldHistoryMasks.childWidth ε h.val) | |
| (fun _ => hx) (useful_half_window allocation m ε sigma (sigma 1) e _ huseY right h u hu) | |
| have hy' : (if right then activeParentShape h.val (sigma 1) - | |
| (shapeSide (sigma 1) u).val else (shapeSide (sigma 1) u).val) ≠ 0 := by | |
| cases right <;> simpa [halfShape, shapeNat] using hy | |
| simpa [compatibilityLaw, halfLaw, hy'] using hw | |
| end MatrixMultiplication.AllFieldGroupCompatibilityTransfer | |
| end | |
| end OAI | |