openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldGroups /SourceSupport.lean
| import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.OrbitData | |
| namespace OAI | |
| /-! Group assignments, orbit counts and extraction capacities. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.AllFieldGroupSourceSupport | |
| open MatrixMultiplication.Foundation AllFieldHistory AllFieldHistorySupport | |
| open AllFieldHistoryChildLaws AllFieldHistoryGroupMasks AllFieldHistoryGroupedRecovery | |
| open AllFieldGroupOrbitData JointPopulation JointCanonicalization JointCanonicalCW | |
| open scoped BigOperators | |
| attribute [local instance] Classical.propDecidable | |
| variable {K tick : ℕ} | |
| theorem masks_of_prepared (F : Type*) [CommRing F] (allocation : Allocation) | |
| (m : ℕ) (ε : ℝ) (sigma : Placement) | |
| (x y z : GroupRaw (K := K) (tick := tick) allocation m sigma) | |
| (hT : groupPreparedSource F allocation m ε sigma x y z ≠ 0) : | |
| completeKeep allocation m ε 0 sigma x ∧ | |
| completeKeep allocation m ε 1 sigma y ∧ | |
| completeKeep allocation m ε 2 sigma z := by | |
| by_contra hn | |
| apply hT | |
| exact ite_eq_right hn | |
| theorem raw_nonzero (F : Type*) [CommRing F] (allocation : Allocation) | |
| (m : ℕ) (ε : ℝ) (sigma : Placement) | |
| (x y z : GroupRaw (K := K) (tick := tick) allocation m sigma) | |
| (hT : groupPreparedSource F allocation m ε sigma x y z ≠ 0) : | |
| groupRawSource F allocation m sigma x y z ≠ 0 := by | |
| have hm := masks_of_prepared F allocation m ε sigma x y z hT | |
| simpa only [groupPreparedSource, ExactRecovery.delete, ite_eq_left hm] using hT | |
| theorem marginal_sideMask (allocation : Allocation) (m : ℕ) (side : Fin 3) | |
| (sigma : Placement) (w : GroupRaw (K := K) (tick := tick) allocation m sigma) | |
| (hm : marginal allocation m side sigma w) : | |
| JointCanonicalMixed.sideMask (groupCounts allocation m sigma) | |
| (fun h => activeHalfLength h.val) (fun h => activeHalfLength h.val) | |
| (fun h => AllFieldHistoryRecovery.halfLength_le_eight h.val) side w := by | |
| intro h a | |
| exact (hm h a).trans | |
| (marginal_population_eq (groupCounts allocation m sigma) | |
| (projectTarget allocation m sigma (canonicalTarget (activeCounts allocation m))) | |
| (canonicalTarget (groupCounts allocation m sigma)) h side a) | |
| theorem marginal_source_nonzero (F : Type*) [CommRing F] (allocation : Allocation) | |
| (m : ℕ) (ε : ℝ) (sigma : Placement) | |
| (x y z : GroupRaw (K := K) (tick := tick) allocation m sigma) | |
| (hT : groupPreparedSource F allocation m ε sigma x y z ≠ 0) : | |
| JointCanonicalMixed.maskedSource (F := F) (groupCounts allocation m sigma) | |
| (fun h => activeHalfLength h.val) (fun h => activeHalfLength h.val) | |
| (fun h => activeParentShape h.val) (fun _ => true) | |
| (fun h => AllFieldHistoryRecovery.halfLength_le_eight h.val) x y z ≠ 0 := by | |
| have hm := masks_of_prepared F allocation m ε sigma x y z hT | |
| have hx := marginal_sideMask allocation m 0 sigma x hm.1.1 | |
| have hy := marginal_sideMask allocation m 1 sigma y hm.2.1.1 | |
| have hz := marginal_sideMask allocation m 2 sigma z hm.2.2.1 | |
| rw [JointCanonicalMixed.maskedSource, ExactRecovery.delete, | |
| ite_eq_left (And.intro hx (And.intro hy hz))] | |
| exact raw_nonzero F allocation m ε sigma x y z hT | |
| theorem prepared_source_support (F : Type*) [CommRing F] (allocation : Allocation) | |
| (m : ℕ) (ε : ℝ) (sigma : Placement) | |
| (x y z : GroupRaw (K := K) (tick := tick) allocation m sigma) | |
| (hT : groupPreparedSource F allocation m ε sigma x y z ≠ 0) : | |
| reorder sigma | |
| (ownWord allocation m sigma x, ownWord allocation m sigma y, | |
| ownWord allocation m sigma z) ∈ ambient allocation m sigma := by | |
| apply Finset.mem_image.mpr | |
| refine ⟨(ownWord allocation m sigma x, ownWord allocation m sigma y, | |
| ownWord allocation m sigma z), ?_, rfl⟩ | |
| exact JointCanonicalMixed.maskedSource_support (groupCounts allocation m sigma) | |
| (fun h => activeHalfLength h.val) (fun h => activeHalfLength h.val) | |
| (fun h => activeParentShape h.val) (fun _ => true) | |
| (fun h => AllFieldHistoryRecovery.halfLength_le_eight h.val) x y z | |
| (marginal_source_nonzero F allocation m ε sigma x y z hT) | |
| end MatrixMultiplication.AllFieldGroupSourceSupport | |
| end | |
| end OAI | |