openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Compression /CompressionJoinErasure.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Compression.BudgetedCompressionJoin | |
| import OAI.Combinatorics.LowSpaceSubsetSum.Compression.CompressionQueryBounds | |
| namespace OAI | |
| /-! # Compression dictionary erasure through the full stopped join | |
| The bounded dictionary carries its invariant and distinct-key count in the | |
| execution state. Erasure preserves all join cursors and counters, including | |
| prefixes which overflow the dictionary. | |
| -/ | |
| namespace LowSpaceSubsetSum.CompressionData | |
| variable {n : ℕ} (D : CompressionData n) | |
| def leftReference (B : ℕ) (e : Equiv.Perm (Fin n)) (q : CompressionSetup.PrimeSpace n) | |
| (z : Fin q.val) (fuel : ℕ) := | |
| BudgetedCompressionJoin.reference q.val z.val B | |
| (D.storedRightArray e q 0) (D.storedRightArray e q 1) | |
| (CompressionCallbacks.report D.input (compressionDictionaryCapacity n)) | |
| (D.streamLength e 0) (D.streamLength e 1) fuel | |
| (D.groupStart e q 0 .direct) (D.groupStart e q 1 (.complementary z.val)) | |
| (CompressionCallbacks.empty D.input (compressionDictionaryCapacity n)) | |
| def rightReference (B : ℕ) (e : Equiv.Perm (Fin n)) (q : CompressionSetup.PrimeSpace n) | |
| (z : Fin q.val) (tree : BinaryTrie.Tree) (fuel : ℕ) := | |
| let target := (D.input.target : ℤ) - z.val | |
| BudgetedCompressionJoin.reference q.val target B | |
| (D.storedRightArray e q 2) (D.storedRightArray e q 3) | |
| (CompressionCallbacks.probe D.input tree D.input.target) | |
| (D.streamLength e 2) (D.streamLength e 3) fuel | |
| (D.groupStart e q 2 .direct) (D.groupStart e q 3 (.complementary target)) false | |
| def eraseCompression (a : CompressionCallbacks.Dictionary D.input (compressionDictionaryCapacity n)) := | |
| a.val.keys | |
| theorem leftReference_value (B : ℕ) (e : Equiv.Perm (Fin n)) | |
| (q : CompressionSetup.PrimeSpace n) (z : Fin q.val) (fuel : ℕ) : | |
| StreamJoin.mapUser D.eraseCompression (D.leftReference B e q z fuel).1.state = | |
| D.leftJoin B e q z fuel := by | |
| let report := fun x y a => | |
| (CompressionCallbacks.report D.input (compressionDictionaryCapacity n) x y a).1 | |
| let report' := BinaryTrie.callback (compressionDictionaryCapacity n) | |
| (fun x y => WeightEncoding.weightKey D.input (x + y)) | |
| have hf : ∀ x y a, report' x y (D.eraseCompression a) = | |
| (D.eraseCompression (report x y a).1, (report x y a).2) := by | |
| intro x y a | |
| exact (CompressionCallbacks.report_value D.input (compressionDictionaryCapacity n) x y a).symm | |
| have h₀ : (fun s => (CompressionReader.read q.val .direct (D.storedRightArray e q 0) s).1) = | |
| D.groupRead e q 0 .direct := funext (CompressionReader.groupRead_value D e q 0 .direct) | |
| have h₁ : (fun s => (CompressionReader.read q.val (.complementary z.val) | |
| (D.storedRightArray e q 1) s).1) = D.groupRead e q 1 (.complementary z.val) := | |
| funext (CompressionReader.groupRead_value D e q 1 (.complementary z.val)) | |
| unfold leftReference | |
| rw [BudgetedCompressionJoin.reference_value, StreamJoin.stopped_eq] | |
| change StreamJoin.mapUser D.eraseCompression (StreamJoin.run _ _ _ _ report _ _ _ _ _ _ _) = _ | |
| rw [StreamJoin.run_mapUser _ _ _ _ report report' D.eraseCompression hf, h₀, h₁] | |
| rfl | |
| theorem rightReference_value (B : ℕ) (e : Equiv.Perm (Fin n)) | |
| (q : CompressionSetup.PrimeSpace n) (z : Fin q.val) (tree : BinaryTrie.Tree) (fuel : ℕ) : | |
| (D.rightReference B e q z tree fuel).1.state = D.rightJoin B e q z tree fuel := by | |
| let target := (D.input.target : ℤ) - z.val | |
| have h₂ : (fun s => (CompressionReader.read q.val .direct (D.storedRightArray e q 2) s).1) = | |
| D.groupRead e q 2 .direct := funext (CompressionReader.groupRead_value D e q 2 .direct) | |
| have h₃ : (fun s => (CompressionReader.read q.val (.complementary target) | |
| (D.storedRightArray e q 3) s).1) = D.groupRead e q 3 (.complementary target) := | |
| funext (CompressionReader.groupRead_value D e q 3 (.complementary target)) | |
| have hr : (fun x y a => (CompressionCallbacks.probe D.input tree D.input.target x y a).1) = | |
| StreamWitness.remember (fun x y => BinaryTrie.contains tree | |
| (WeightEncoding.weightKey D.input ((D.input.target : ℤ) - (x + y)))) := by | |
| funext x y a | |
| exact CompressionCallbacks.probe_value _ _ _ _ _ _ | |
| unfold rightReference | |
| dsimp only | |
| rw [BudgetedCompressionJoin.reference_value, StreamJoin.stopped_eq, h₂, h₃, hr] | |
| rfl | |
| end LowSpaceSubsetSum.CompressionData | |
| end OAI | |