openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Storage /BackgroundPreparationCells.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Storage.IntervalDomainCells | |
| import OAI.Combinatorics.LowSpaceSubsetSum.Storage.AllocationCells | |
| import OAI.Combinatorics.LowSpaceSubsetSum.Storage.BackgroundBlockCells | |
| import OAI.Combinatorics.LowSpaceSubsetSum.Representation.RepresentationBackgroundPreparation | |
| namespace OAI | |
| /-! # Live storage for the allocated background construction | |
| The scanned outside list is shared by all eight row copies. The six integer | |
| width computations run before the size guard and are charged on both branches. | |
| -/ | |
| noncomputable section | |
| namespace LowSpaceSubsetSum.BackgroundPreparationCells | |
| open scoped BigOperators | |
| variable {n : ℕ} | |
| private def assemble (outside : List (Fin n) × ℕ) (counts : Vector ℕ 8 × ℕ) : | |
| Option (Vector (List (Fin n)) 8) × ℕ := | |
| if h : (∑ i : Fin 8, counts.1[i.val]) = outside.1.length then | |
| let xs : Vector (Fin n) outside.1.length := ⟨outside.1.toArray, by simp only [List.size_toArray]⟩ | |
| let rows := StoredBackgroundBlocks.run xs counts.1 h | |
| (some rows.1, outside.2+counts.2+rows.2+20) | |
| else (none, outside.2+counts.2+20) | |
| private def core (outside : (List (Fin n) × ℕ) × ℕ) | |
| (counts : (Vector ℕ 8 × ℕ) × ℕ) : | |
| (Option (Vector (List (Fin n)) 8) × ℕ) × ℕ := | |
| if h : (∑ i : Fin 8, counts.1.1[i.val]) = outside.1.1.length then | |
| let xs : Vector (Fin n) outside.1.1.length := ⟨outside.1.1.toArray, by simp only [List.size_toArray]⟩ | |
| let rows := BackgroundBlockCells.run xs counts.1.1 h | |
| ((some rows.1.1, outside.1.2+counts.1.2+rows.1.2+20), | |
| outside.2+counts.2+rows.2+12*n+128) | |
| else ((none, outside.1.2+counts.1.2+20), outside.2+counts.2+12*n+128) | |
| private theorem core_value (outside : (List (Fin n) × ℕ) × ℕ) | |
| (counts : (Vector ℕ 8 × ℕ) × ℕ) : | |
| (core outside counts).1 = assemble outside.1 counts.1 := by | |
| unfold core assemble | |
| dsimp only | |
| split <;> simp only [BackgroundBlockCells.value] | |
| def run (_I : Instance n) (v : Vector (Fin n) n) (guesses : Vector ℕ 3) : | |
| (Option (Vector (List (Fin n)) 8) × ℕ) × ℕ := | |
| core (IntervalDomainCells.run v (3*mixerSize n) (n-3*mixerSize n)) | |
| (AllocationCells.run n guesses) | |
| theorem value (I : Instance n) (v : Vector (Fin n) n) (guesses : Vector ℕ 3) : | |
| (run I v guesses).1 = RepresentationBackgroundPreparation.run I v guesses := by | |
| have ho : (IntervalDomainCells.run v (3*mixerSize n) (n-3*mixerSize n)).1 = | |
| CanonicalBackground.run v := IntervalDomainCells.run_value v _ _ | |
| calc | |
| (run I v guesses).1 = assemble | |
| (IntervalDomainCells.run v (3*mixerSize n) (n-3*mixerSize n)).1 | |
| (AllocationCells.run n guesses).1 := core_value _ _ | |
| _ = assemble (CanonicalBackground.run v) | |
| (RepresentationAllocationExecution.run n guesses) := | |
| congrArg₂ assemble ho (AllocationCells.value n guesses) | |
| _ = RepresentationBackgroundPreparation.run I v guesses := rfl | |
| theorem cells (I : Instance n) (v : Vector (Fin n) n) (guesses : Vector ℕ 3) : | |
| (run I v guesses).2 ≤ 2048*(n+1)^2 := by | |
| have ho := IntervalDomainCells.run_cells v (3*mixerSize n) (n-3*mixerSize n) | |
| have hc := AllocationCells.cells n guesses | |
| have hl : (IntervalDomainCells.run v (3*mixerSize n) (n-3*mixerSize n)).1.1.length ≤ n := by | |
| rw [IntervalDomainCells.run_value] | |
| exact CanonicalBackground.length_le v | |
| unfold run core | |
| dsimp only | |
| split | |
| next h => | |
| have hr := BackgroundBlockCells.cells | |
| (⟨(IntervalDomainCells.run v (3*mixerSize n) (n-3*mixerSize n)).1.1.toArray, | |
| by simp only [List.size_toArray]⟩) | |
| (AllocationCells.run n guesses).1.1 h | |
| change _+_+_+12*n+128 ≤ _ | |
| nlinarith only [ho, hc, hl, hr, Nat.zero_le (n^2)] | |
| · change _+_+12*n+128 ≤ _ | |
| nlinarith only [ho, hc, Nat.zero_le (n^2)] | |
| end LowSpaceSubsetSum.BackgroundPreparationCells | |
| end | |
| end OAI | |