openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Compression /CompressionRowCells.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Compression.CompressionInitialization | |
| import OAI.Combinatorics.LowSpaceSubsetSum.Storage.MeasuredRowCells | |
| namespace OAI | |
| /-! # Constant live scratch for a compression row's three binary searches -/ | |
| namespace LowSpaceSubsetSum.CompressionRowCells | |
| def low (x : ℤ) : (ℤ × ℕ) × ℕ := ((x%1,3),8) | |
| def high (P x : ℤ) : (ℤ × ℕ) × ℕ := ((x%P,3),8) | |
| def run (right : Array ℤ) (P : ℤ) (hs : FilteredRow.PrimarySorted right id 1) | |
| (mode : FilteredStream.Mode) (x : ℤ) : (PairCursor.State ℤ right.size × ℕ) × ℕ := | |
| let k := FilteredStream.steps right.size | |
| let rr := MeasuredRowCells.range right low ((0-x)%1) k hs | |
| (FilteredStream.steps_sufficient right.size) | |
| let q := match mode with | |
| | .direct => MeasuredRowCells.start right rr.1.1 (high P) P (x%P) k false | |
| | .complementary t => MeasuredRowCells.start right rr.1.1 (high P) P ((t-x)%P) k true | |
| ((⟨x,q.1.1⟩,1+rr.1.2+q.1.2+4),rr.2+q.2+32) | |
| theorem value (right : Array ℤ) (P : ℤ) (hs : FilteredRow.PrimarySorted right id 1) | |
| (mode : FilteredStream.Mode) (x : ℤ) : | |
| (run right P hs mode x).1 = CompressionInitialization.row right P hs mode x := by | |
| have h := MeasuredRowCells.range_value right low ((0-x)%1) | |
| (FilteredStream.steps right.size) hs (FilteredStream.steps_sufficient right.size) | |
| have hv := congrArg Prod.fst h | |
| have hw := congrArg Prod.snd h | |
| cases mode <;> | |
| simp only [run, MeasuredRowCells.start_value, hv, hw, Bool.false_eq_true, ↓reduceIte] <;> rfl | |
| theorem cells (right : Array ℤ) (P : ℤ) (hs : FilteredRow.PrimarySorted right id 1) | |
| (mode : FilteredStream.Mode) (x : ℤ) : (run right P hs mode x).2 ≤ 200 := by | |
| let rr := MeasuredRowCells.range right low ((0-x)%1) (FilteredStream.steps right.size) hs | |
| (FilteredStream.steps_sufficient right.size) | |
| have hr := MeasuredRowCells.range_cells right low ((0-x)%1) | |
| (FilteredStream.steps right.size) 8 hs (FilteredStream.steps_sufficient right.size) | |
| (fun _ => le_rfl) | |
| change rr.2 ≤ 80 at hr | |
| cases mode with | |
| | direct => | |
| have hq := MeasuredRowCells.start_cells right rr.1.1 (high P) P (x%P) | |
| (FilteredStream.steps right.size) 8 false (fun _ => le_rfl) | |
| change rr.2+(MeasuredRowCells.start right rr.1.1 (high P) P (x%P) | |
| (FilteredStream.steps right.size) false).2+32 ≤ _ | |
| omega | |
| | complementary t => | |
| have hq := MeasuredRowCells.start_cells right rr.1.1 (high P) P ((t-x)%P) | |
| (FilteredStream.steps right.size) 8 true (fun _ => le_rfl) | |
| change rr.2+(MeasuredRowCells.start right rr.1.1 (high P) P ((t-x)%P) | |
| (FilteredStream.steps right.size) true).2+32 ≤ _ | |
| omega | |
| end LowSpaceSubsetSum.CompressionRowCells | |
| end OAI | |