openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Representation /RepresentationGroupCells.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Representation.RepresentationTieOuterCells | |
| import OAI.Combinatorics.LowSpaceSubsetSum.Representation.RepresentationObservationCells | |
| namespace OAI | |
| /-! # A concrete tie-group execution retaining its shared buffer | |
| The trace includes the incoming pair of readers and dictionary, the completed | |
| buffer, lookahead scratch, optional snapshot, and the actual outer replay. | |
| Only the result state survives after this call returns. | |
| -/ | |
| namespace LowSpaceSubsetSum.RepresentationGroupCells | |
| variable {n : ℕ} | |
| abbrev Left (I : Instance n) (F : IndexedMask n) (P : ℤ) | |
| (right : Array (LeafRecord n)) (d : ℕ) := CountedParentJoin.State I F P .direct right d | |
| abbrev Right (I : Instance n) (F : IndexedMask n) (P t : ℤ) | |
| (right : Array (LeafRecord n)) (d : ℕ) := CountedParentJoin.State I F P (.complementary t) right d | |
| def run (I : Instance n) (F : IndexedMask n) (P t : ℤ) | |
| (right₁ right₂ : Array (LeafRecord n)) {d₁ d₂ : ℕ} | |
| (C frontCharge fillCharge outerCharge charge copyCharge : ℕ) (K : ℤ) | |
| (B fuel₁ fuel₂ : ℕ) (s₁ : Left I F P right₁ d₁) (s₂ : Right I F P t right₂ d₂) | |
| (r : StoredEntry.Dictionary I C) (budget : ℕ) : | |
| (Option (TieGroup.Result (Left I F P right₁ d₁) (Right I F P t right₂ d₂) | |
| (StoredEntry.Dictionary I C) × ℕ) × ℕ) × ℕ := | |
| let retained := ParentLiveState.reader s₁ + ParentLiveState.reader s₂ + ParentLiveState.dictionary r + 32 | |
| if frontCharge ≤ budget then | |
| let filled := RepresentationFillCells.run I F P (.complementary t) right₂ fillCharge K B s₂ | |
| (budget - frontCharge) | |
| let peak := retained + filled.2 + 16 | |
| match filled.1.1 with | |
| | none => ((none, filled.1.2 + 1), peak) | |
| | some (f, left) => | |
| let bufferCells := (6 * n + 5) * f.buffer.size + 1 | |
| let long := RepresentationObservationCells.hasKey I F P (.complementary t) right₂ K f.cursor | |
| let present := RepresentationObservationCells.hasKey I F P .direct right₁ K s₁ | |
| let snap := RepresentationObservationCells.snapshot I F P (.complementary t) right₂ copyCharge | |
| (long.1.1 && present.1.1) f.cursor left | |
| let peak := max peak (retained + bufferCells + ParentLiveState.reader f.cursor + | |
| long.2 + present.2 + snap.2 + 32) | |
| match snap.1.1 with | |
| | none => ((none, filled.1.2 + long.1.2 + present.1.2 + snap.1.2 + 8), peak) | |
| | some (saved, remaining) => | |
| let out := RepresentationTieOuterCells.run I F P t right₁ right₂ C outerCharge charge copyCharge K | |
| f.buffer saved long.1.1 fuel₁ fuel₂ s₁ ⟨f.cursor, r, 0, 0, 0⟩ remaining | |
| ((out.1.1.map (fun (outer, rest) => | |
| (⟨outer, f.queries, f.buffer.size, long.1.1 && present.1.1⟩, rest)), | |
| filled.1.2 + long.1.2 + present.1.2 + snap.1.2 + out.1.2 + 8), | |
| max peak (retained + bufferCells + ParentLiveState.reader f.cursor + out.2 + 32)) | |
| else ((none, 1), retained) | |
| theorem run_value (I : Instance n) (F : IndexedMask n) (P t : ℤ) | |
| (right₁ right₂ : Array (LeafRecord n)) {d₁ d₂ : ℕ} | |
| (C frontCharge fillCharge outerCharge charge copyCharge : ℕ) (K : ℤ) | |
| (B fuel₁ fuel₂ : ℕ) (s₁ : Left I F P right₁ d₁) (s₂ : Right I F P t right₂ d₂) | |
| (r : StoredEntry.Dictionary I C) (budget : ℕ) : | |
| (run I F P t right₁ right₂ C frontCharge fillCharge outerCharge charge copyCharge K | |
| B fuel₁ fuel₂ s₁ s₂ r budget).1 = | |
| BudgetedTieGroup.execute frontCharge fillCharge outerCharge charge copyCharge | |
| (RepresentationKeys.read I F P .direct right₁) (RepresentationKeys.pairKey I F P .direct) | |
| (RepresentationKeys.read I F P (.complementary t) right₂) | |
| (RepresentationKeys.pairKey I F P (.complementary t)) CountedParentJoin.cells K | |
| (StoredEntry.report I C (CountedRepresentationEmit.emit I F)) | |
| (CountedParentJoin.snapshot I F P (.complementary t) right₂) | |
| B fuel₁ fuel₂ s₁ s₂ r budget := by | |
| unfold run BudgetedTieGroup.execute | |
| dsimp only | |
| split | |
| · simp only [RepresentationFillCells.run_value] | |
| cases hf : (BudgetedFill.execute fillCharge (RepresentationKeys.read I F P (.complementary t) right₂) | |
| (RepresentationKeys.pairKey I F P (.complementary t)) CountedParentJoin.cells K | |
| B s₂ #[] 0 (budget - frontCharge)).1 with | |
| | none => rfl | |
| | some p => | |
| rcases p with ⟨f, left⟩ | |
| dsimp only | |
| simp only [RepresentationObservationCells.hasKey_value, RepresentationObservationCells.snapshot_value] | |
| cases hs : (BudgetedSnapshot.execute copyCharge | |
| (CountedParentJoin.snapshot I F P (.complementary t) right₂) | |
| ((CountedFill.hasKey (RepresentationKeys.read I F P (.complementary t) right₂) | |
| (RepresentationKeys.pairKey I F P (.complementary t)) K f.cursor).1 && | |
| (CountedFill.hasKey (RepresentationKeys.read I F P .direct right₁) | |
| (RepresentationKeys.pairKey I F P .direct) K s₁).1) f.cursor left).1 with | |
| | none => rfl | |
| | some p => | |
| rcases p with ⟨saved, remaining⟩ | |
| dsimp only | |
| simp only [RepresentationTieOuterCells.run_value] | |
| · rfl | |
| private theorem peak_scale (u C : ℕ) (hu : 16 ≤ u) : | |
| 16 * (u ^ 8 * (C + 1)) ≤ u ^ 9 * (C + 1) := by | |
| calc | |
| _ = (16 * u ^ 8) * (C + 1) := by ac_rfl | |
| _ ≤ (u * u ^ 8) * (C + 1) := | |
| Nat.mul_le_mul_right _ (Nat.mul_le_mul_right _ hu) | |
| _ = _ := by rw [← pow_succ'] | |
| theorem run_cells (I : Instance n) (F : IndexedMask n) (P t : ℤ) | |
| (right₁ right₂ : Array (LeafRecord n)) {d₁ d₂ : ℕ} | |
| (C frontCharge fillCharge outerCharge charge copyCharge : ℕ) (K : ℤ) | |
| (B fuel₁ fuel₂ : ℕ) (s₁ : Left I F P right₁ d₁) (s₂ : Right I F P t right₂ d₂) | |
| (r : StoredEntry.Dictionary I C) (budget : ℕ) (hn : 10000 ≤ n) | |
| (hd₁ : d₁ ≤ n) (hd₂ : d₂ ≤ n) | |
| (ht₁ : 2 ^ (d₁ + 1) ≤ 4 * (C + 1)) (ht₂ : 2 ^ (d₂ + 1) ≤ 4 * (C + 1)) (hB : B ≤ C) : | |
| (run I F P t right₁ right₂ C frontCharge fillCharge outerCharge charge copyCharge K | |
| B fuel₁ fuel₂ s₁ s₂ r budget).2 ≤ inputScale I ^ 9 * (C + 1) := by | |
| let u := inputScale I | |
| let H := u ^ 8 * (C + 1) | |
| have hu : n + 1 ≤ u := by unfold u inputScale; omega | |
| have hup : 1 ≤ u := by omega | |
| have hbase : 1000 ≤ u ^ 7 := by | |
| have hp : u ≤ u ^ 7 := by | |
| simpa only [pow_one] using Nat.pow_le_pow_right hup (by decide : 1 ≤ 7) | |
| omega | |
| have hscale : 1000 * u ≤ u ^ 8 := by | |
| simpa only [pow_succ] using Nat.mul_le_mul_right u hbase | |
| have hsMul : 1000 * u * (C + 1) ≤ H := Nat.mul_le_mul_right (C + 1) hscale | |
| have huC : 1 ≤ u * (C + 1) := Nat.mul_pos hup (by omega) | |
| have hH : 32 ≤ H := by nlinarith only [hsMul, huC] | |
| have h5 : u ^ 5 * (C + 1) ≤ H := | |
| Nat.mul_le_mul_right (C + 1) (Nat.pow_le_pow_right hup (by decide : 5 ≤ 8)) | |
| have hl : ∀ st : Left I F P right₁ d₁, ParentLiveState.reader st ≤ H := by | |
| intro st | |
| have h := ParentLiveBounds.reader_le I F P .direct right₁ st C ht₁ | |
| change _ ≤ 176 * u * (C + 1) at h | |
| nlinarith only [h, hsMul, huC] | |
| have hr : ∀ st : Right I F P t right₂ d₂, ParentLiveState.reader st ≤ H := by | |
| intro st | |
| have h := ParentLiveBounds.reader_le I F P (.complementary t) right₂ st C ht₂ | |
| change _ ≤ 176 * u * (C + 1) at h | |
| nlinarith only [h, hsMul, huC] | |
| have hd : ∀ st : StoredEntry.Dictionary I C, ParentLiveState.dictionary st ≤ H := by | |
| intro st | |
| have h := ParentLiveState.dictionary_scale I C st | |
| change _ ≤ 18 * u * (C + 1) + 8 at h | |
| nlinarith only [h, hsMul, huC] | |
| have hret : ParentLiveState.reader s₁ + ParentLiveState.reader s₂ + ParentLiveState.dictionary r + 32 ≤ | |
| 4 * H := by have h₁ := hl s₁; have h₂ := hr s₂; have h₃ := hd r; omega | |
| have hfill := (RepresentationFillCells.run_cells I F P (.complementary t) right₂ fillCharge K B C | |
| s₂ (budget - frontCharge) hn hd₂ ht₂ hB).trans h5 | |
| have hpeak : | |
| (run I F P t right₁ right₂ C frontCharge fillCharge outerCharge charge copyCharge K | |
| B fuel₁ fuel₂ s₁ s₂ r budget).2 ≤ 16 * H := by | |
| unfold run | |
| dsimp only | |
| split | |
| · cases hf : (RepresentationFillCells.run I F P (.complementary t) right₂ fillCharge K B s₂ | |
| (budget - frontCharge)).1.1 with | |
| | none => dsimp only; omega | |
| | some p => | |
| rcases p with ⟨f, left⟩ | |
| have hcap : f.buffer.size ≤ C := | |
| (RepresentationObservationCells.fill_capacity I F P (.complementary t) right₂ fillCharge K B | |
| s₂ (budget - frontCharge) f left hf).trans hB | |
| have hbuf := Nat.mul_le_mul_left (6 * n + 5) hcap | |
| have hbufH : (6 * n + 5) * f.buffer.size + 1 ≤ H := by | |
| nlinarith only [hbuf, hsMul, huC, hu] | |
| have hcursor := hr f.cursor | |
| have hlong := (RepresentationObservationCells.hasKey_cells I F P (.complementary t) right₂ K | |
| f.cursor C hn hd₂ ht₂).trans h5 | |
| have hpresent := (RepresentationObservationCells.hasKey_cells I F P .direct right₁ K | |
| s₁ C hn hd₁ ht₁).trans h5 | |
| have hsnap := (RepresentationObservationCells.snapshot_cells I F P (.complementary t) right₂ | |
| copyCharge ((RepresentationObservationCells.hasKey I F P (.complementary t) right₂ K f.cursor).1.1 && | |
| (RepresentationObservationCells.hasKey I F P .direct right₁ K s₁).1.1) | |
| f.cursor left C hn hd₂ ht₂).trans h5 | |
| dsimp only | |
| cases hs : (RepresentationObservationCells.snapshot I F P (.complementary t) right₂ copyCharge | |
| ((RepresentationObservationCells.hasKey I F P (.complementary t) right₂ K f.cursor).1.1 && | |
| (RepresentationObservationCells.hasKey I F P .direct right₁ K s₁).1.1) f.cursor left).1.1 with | |
| | none => dsimp only; omega | |
| | some p => | |
| rcases p with ⟨saved, remaining⟩ | |
| have ho := RepresentationTieOuterCells.run_cells I F P t right₁ right₂ C outerCharge charge copyCharge K | |
| f.buffer saved (RepresentationObservationCells.hasKey I F P (.complementary t) right₂ K f.cursor).1.1 | |
| fuel₁ fuel₂ s₁ ⟨f.cursor, r, 0, 0, 0⟩ remaining hn hd₁ hd₂ ht₁ ht₂ hcap | |
| change _ ≤ H at ho | |
| dsimp only | |
| omega | |
| · dsimp only; omega | |
| exact hpeak.trans (peak_scale u C (by omega)) | |
| end LowSpaceSubsetSum.RepresentationGroupCells | |
| end OAI | |