openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Representation /RepresentationFillCells.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Representation.RepresentationReaderCells | |
| import OAI.Combinatorics.LowSpaceSubsetSum.Storage.BudgetedFillCells | |
| namespace OAI | |
| /-! # Concrete bounded-space filling of a representation tie buffer -/ | |
| namespace LowSpaceSubsetSum.RepresentationFillCells | |
| variable {n : ℕ} | |
| def run (I : Instance n) (F : IndexedMask n) (P : ℤ) (mode : FilteredStream.Mode) | |
| (right : Array (LeafRecord n)) {d : ℕ} (charge : ℕ) (K : ℤ) (B : ℕ) | |
| (s : CountedParentJoin.State I F P mode right d) (budget : ℕ) := | |
| BudgetedFillCells.run charge (RepresentationReaderCells.read I F P mode right) | |
| (RepresentationKeyCells.pairKey I F P mode) CountedParentJoin.cells ParentLiveState.reader | |
| K B s budget | |
| theorem run_value (I : Instance n) (F : IndexedMask n) (P : ℤ) (mode : FilteredStream.Mode) | |
| (right : Array (LeafRecord n)) {d : ℕ} (charge : ℕ) (K : ℤ) (B : ℕ) | |
| (s : CountedParentJoin.State I F P mode right d) (budget : ℕ) : | |
| (run I F P mode right charge K B s budget).1 = | |
| BudgetedFill.execute charge (RepresentationKeys.read I F P mode right) | |
| (RepresentationKeys.pairKey I F P mode) CountedParentJoin.cells K B s #[] 0 budget := by | |
| rw [run, BudgetedFillCells.run_value] | |
| have hr : (fun s : CountedParentJoin.State I F P mode right d => | |
| (RepresentationReaderCells.read I F P mode right s).1) = | |
| RepresentationKeys.read I F P mode right := | |
| funext (RepresentationReaderCells.read_value I F P mode right) | |
| have hk : (fun x => (RepresentationKeyCells.pairKey I F P mode x).1) = | |
| RepresentationKeys.pairKey I F P mode := | |
| funext (RepresentationKeyCells.pairKey_value I F P mode) | |
| rw [hr, hk] | |
| theorem run_cells (I : Instance n) (F : IndexedMask n) (P : ℤ) (mode : FilteredStream.Mode) | |
| (right : Array (LeafRecord n)) {d : ℕ} (charge : ℕ) (K : ℤ) (B C : ℕ) | |
| (s : CountedParentJoin.State I F P mode right d) (budget : ℕ) | |
| (hn : 10000 ≤ n) (hd : d ≤ n) (ht : 2 ^ (d + 1) ≤ 4 * (C + 1)) (hB : B ≤ C) : | |
| (run I F P mode right charge K B s budget).2 ≤ inputScale I ^ 5 * (C + 1) := by | |
| let u := inputScale I | |
| have hu : n + 1 ≤ u := by unfold u inputScale; omega | |
| have hc : 12 * n + 32 ≤ 44 * u := by omega | |
| have hsc := Nat.mul_le_mul hc ht | |
| have hs : ∀ st : CountedParentJoin.State I F P mode right d, | |
| ParentLiveState.reader st ≤ 176 * u * (C + 1) := by | |
| intro st | |
| rw [ParentLiveState.reader_exact] | |
| nlinarith only [hsc] | |
| have hl : ∀ x, (RepresentationKeyCells.pairKey I F P mode x).2 ≤ 44 * u := by | |
| intro x | |
| exact (RepresentationKeyCells.pairKey_cells I F P mode x).trans hc | |
| have h := BudgetedFillCells.run_cells charge (RepresentationReaderCells.read I F P mode right) | |
| (RepresentationKeyCells.pairKey I F P mode) CountedParentJoin.cells ParentLiveState.reader K | |
| (u ^ 4 * (C + 1)) (44 * u) (176 * u * (C + 1)) (6 * n + 4) | |
| (fun st => RepresentationReaderCells.read_scale I F P mode right st C hn hd ht) | |
| hl hs CountedParentJoin.cells_le B s budget | |
| have hbuf := Nat.mul_le_mul_left (6 * n + 5) hB | |
| have hsmall : (run I F P mode right charge K B s budget).2 ≤ | |
| (u ^ 4 + 500 * u) * (C + 1) := by | |
| change (run I F P mode right charge K B s budget).2 ≤ _ at h | |
| nlinarith only [h, hbuf, hu] | |
| have hu3 : 500 ≤ u ^ 3 := by | |
| calc | |
| 500 ≤ 8 ^ 3 := by norm_num | |
| _ ≤ u ^ 3 := Nat.pow_le_pow_left (by omega) 3 | |
| have hm := Nat.mul_le_mul_right u hu3 | |
| have hp : 500 * u ≤ u ^ 4 := by simpa only [pow_succ, Nat.mul_assoc] using hm | |
| have hfactor : 2 * u ^ 4 ≤ u ^ 5 := by | |
| calc | |
| _ ≤ u * u ^ 4 := Nat.mul_le_mul_right _ (by omega) | |
| _ = _ := (pow_succ' _ _).symm | |
| change (run I F P mode right charge K B s budget).2 ≤ u ^ 5 * (C + 1) | |
| exact hsmall.trans (Nat.mul_le_mul_right (C + 1) (by omega)) | |
| end LowSpaceSubsetSum.RepresentationFillCells | |
| end OAI | |