openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldGroups /AssignmentLoss.lean
| import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.AssignmentCore | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.AdmissibilityCore | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.Compatibility | |
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.MaskedGoodSelection | |
| namespace OAI | |
| /-! Group assignments, orbit counts and extraction capacities. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.AllFieldGroupAssignmentRetention | |
| open AllFieldHistory AllFieldHistoryGroupMasks AllFieldHistoryGroupedRecovery | |
| open AllFieldGroupOrbitData JointCoarseHashing | |
| attribute [local instance] Classical.propDecidable Classical.decEq | |
| variable {K tick : ℕ} | |
| theorem good_failure_fraction_le (allocation : Allocation) (m : ℕ) (ε : ℝ) | |
| (sigma : Placement) (k : ℕ) (U : Finset (ZMod (37 ^ k))) | |
| (hAP : ∀ x ∈ U, ∀ y ∈ U, ∀ z ∈ U, x + y = 2 * z → x = z ∧ y = z) | |
| (N : ℝ) (s : Sample (Pos (K := K) (tick := tick) allocation m sigma) k) | |
| (e : Targets (K := K) (tick := tick) allocation m sigma) | |
| (hg : (data (K := K) (tick := tick) allocation m ε sigma).Good k U N e s) | |
| (o : Orbit (K := K) (tick := tick) allocation m sigma) (ho : o ∈ (data (K := K) (tick := tick) allocation m ε sigma).orbits e) : | |
| ((((data (K := K) (tick := tick) allocation m ε sigma).passing e o).filter | |
| (fun v => assigned (K := K) (tick := tick) allocation m ε sigma k U s v ≠ some (coarse (K := K) (tick := tick) allocation m sigma e))).card : ℝ) / | |
| (((data (K := K) (tick := tick) allocation m ε sigma).full e o).card : ℝ) ≤ 1 / N := by | |
| have ha (v : Variable (K := K) (tick := tick) allocation m sigma) | |
| (hv : v ∈ (data (K := K) (tick := tick) allocation m ε sigma).passing e o) : | |
| Admissible (K := K) (tick := tick) allocation m ε sigma v.1 e v.2 := | |
| AllFieldGroupAdmissible.full_admissible (K := K) (tick := tick) allocation m ε sigma e o ho v | |
| ((mem_passing (K := K) (tick := tick) allocation m ε sigma e o v).mp hv).1 | |
| have hc (v : Variable (K := K) (tick := tick) allocation m sigma) | |
| (hv : v ∈ (data (K := K) (tick := tick) allocation m ε sigma).passing e o) : | |
| Compatible (K := K) (tick := tick) allocation m ε sigma v.1 e v.2 := | |
| AllFieldGroupCompatibility.admissible_compatible (K := K) (tick := tick) allocation m ε sigma v.1 e v.2 (ha v hv) | |
| have hh := (data (K := K) (tick := tick) allocation m ε sigma).good_passing_actual_failure_fraction_le k U hAP N e o ho s hg | |
| (actualCompatible (K := K) (tick := tick) allocation m ε sigma) (actualUseful (K := K) (tick := tick) allocation m ε sigma) | |
| (actualKeep (K := K) (tick := tick) allocation m ε sigma) | |
| (fun v hv => (actualCompatible_coarse (K := K) (tick := tick) allocation m ε sigma e v).mpr (hc v hv)) | |
| (fun v hv => (actualUseful_coarse (K := K) (tick := tick) allocation m ε sigma e v).mpr (ha v hv).1) | |
| (fun v hv => ((mem_passing (K := K) (tick := tick) allocation m ε sigma e o v).mp hv).2) | |
| (fun v hv f _ hn hcomp => competitor_coverage (K := K) (tick := tick) allocation m ε sigma e o v (hc v hv) f hn hcomp) | |
| rw [Finset.filter_congr_decidable] at hh | |
| exact hh | |
| end MatrixMultiplication.AllFieldGroupAssignmentRetention | |
| end | |
| end OAI | |