openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Storage /LayoutPreparationCells.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Storage.LayoutSamplerCells | |
| import OAI.Combinatorics.LowSpaceSubsetSum.Storage.CanonicalMixerCells | |
| import OAI.Combinatorics.LowSpaceSubsetSum.Representation.RepresentationLayoutPreparation | |
| namespace OAI | |
| /-! # Executed canonical layout preparation and its retained arrays | |
| The sampled permutation and sign mask remain live while the three mixer | |
| counts and integer widths are prepared. The mixer preparation already | |
| accounts for its shared arrays and merge state. | |
| -/ | |
| noncomputable section | |
| namespace LowSpaceSubsetSum.LayoutPreparationCells | |
| variable {n : ℕ} | |
| def assemble | |
| (layout : (Option (Vector (Fin n) n × IndexedMask n) × ℕ) × ℕ) | |
| (prepare : Vector (Fin n) n → | |
| ((Vector (List (Fin n)) 3 × MixerParameterExecution.Parameters) × ℕ) × ℕ) : | |
| (Option (RepresentationLayoutPreparation.Data n) × ℕ) × ℕ := | |
| match layout.1.1 with | |
| | none => ((none, layout.1.2+1), layout.2+32) | |
| | some (v, signs) => | |
| let prepared := prepare v | |
| ((some ⟨v, signs, prepared.1.1.1, prepared.1.1.2⟩, layout.1.2+prepared.1.2+1), | |
| layout.2+prepared.2+8*n+64) | |
| def run (I : Instance n) | |
| (s : (canonicalRepresentationData I).zeroSetup.layoutCoupling.Seed) : | |
| (Option (RepresentationLayoutPreparation.Data n) × ℕ) × ℕ := | |
| assemble (LayoutSamplerCells.run (canonicalRepresentationData I).zeroSetup s) | |
| (CanonicalMixerCells.prepare I) | |
| theorem value (I : Instance n) | |
| (s : (canonicalRepresentationData I).zeroSetup.layoutCoupling.Seed) : | |
| (run I s).1 = RepresentationLayoutPreparation.run (canonicalRepresentationData I) s := by | |
| have h := LayoutSamplerCells.run_value (canonicalRepresentationData I).zeroSetup s | |
| have h₁ := congrArg Prod.fst h | |
| have h₂ := congrArg Prod.snd h | |
| simp only [run, assemble, RepresentationLayoutPreparation.run, h₁, h₂] | |
| split <;> simp_all only [CanonicalMixerCells.prepare_value] | |
| private theorem scalar_bound (n u C L P : ℕ) (hu : n+2 ≤ u) (hu0 : 10000 ≤ u) | |
| (hL : L ≤ u^6) (hP : P ≤ u^8*C) : L+P+8*n+64 ≤ u^10*(C+1) := by | |
| have h68 : u^6 ≤ u^8 := pow_le_pow_right' (by omega) (by decide) | |
| have hpoly : 8*n+64 ≤ u^8 := by | |
| have h2 : u^2 ≤ u^8 := pow_le_pow_right' (by omega) (by decide) | |
| nlinarith only [hu, hu0, h2] | |
| have hmajor : 3*u^8 ≤ u^10 := by | |
| have hcoef : 3 ≤ u^2 := by nlinarith only [hu0] | |
| calc | |
| _ ≤ u^2*u^8 := Nat.mul_le_mul_right _ hcoef | |
| _ = _ := by rw [← pow_add] | |
| have hL' := hL.trans (h68.trans (Nat.le_mul_of_pos_right _ (by omega : 0 < C+1))) | |
| have hp' := hP.trans (Nat.mul_le_mul_left (u^8) (show C ≤ C+1 by omega)) | |
| have hsmall := hpoly.trans (Nat.le_mul_of_pos_right _ (by omega : 0 < C+1)) | |
| have hfinal := Nat.mul_le_mul_right (C+1) hmajor | |
| simp only [Nat.mul_assoc] at hfinal | |
| omega | |
| private theorem assemble_cells (u C : ℕ) | |
| (layout : (Option (Vector (Fin n) n × IndexedMask n) × ℕ) × ℕ) | |
| (prepare : Vector (Fin n) n → | |
| ((Vector (List (Fin n)) 3 × MixerParameterExecution.Parameters) × ℕ) × ℕ) | |
| (hu : n+2 ≤ u) (hu0 : 10000 ≤ u) (hL : layout.2 ≤ u^6) | |
| (hP : ∀ v, (prepare v).2 ≤ u^8*C) : | |
| (assemble layout prepare).2 ≤ u^10*(C+1) := by | |
| unfold assemble | |
| cases h : layout.1.1 with | |
| | none => | |
| change layout.2+32 ≤ _ | |
| have hs := scalar_bound n u C layout.2 0 hu hu0 hL (Nat.zero_le _) | |
| omega | |
| | some pair => | |
| rcases pair with ⟨v, signs⟩ | |
| exact scalar_bound n u C layout.2 (prepare v).2 hu hu0 hL (hP v) | |
| private theorem raw_bound (n u : ℕ) (hu : n+1 ≤ u) (hu0 : 10000 ≤ u) : | |
| (n+1)*(3*n^3+128)+32*(n+1)^2 ≤ u^6 := by | |
| have hp3 := Nat.pow_le_pow_left (show n ≤ u by omega) 3 | |
| have hp2 := Nat.pow_le_pow_left hu 2 | |
| have hu3 : u ≤ u^3 := by | |
| simpa only [pow_one] using pow_le_pow_right' (by omega : 1 ≤ u) (by decide : 1 ≤ 3) | |
| have ht : 3*n^3+128 ≤ 4*u^3 := by nlinarith only [hp3, hu3, hu0] | |
| have hm := Nat.mul_le_mul hu ht | |
| have hm' : (n+1)*(3*n^3+128) ≤ 4*u^4 := by convert hm using 1; ring | |
| have h24 : u^2 ≤ u^4 := pow_le_pow_right' (by omega) (by decide) | |
| have h46 : 36*u^4 ≤ u^6 := by | |
| have hcoef : 36 ≤ u^2 := by nlinarith only [hu0] | |
| calc | |
| _ ≤ u^2*u^4 := Nat.mul_le_mul_right _ hcoef | |
| _ = _ := by rw [← pow_add] | |
| nlinarith only [hm', hp2, h24, h46] | |
| theorem cells (I : Instance n) (hn : 10000 ≤ n) | |
| (s : (canonicalRepresentationData I).zeroSetup.layoutCoupling.Seed) : | |
| (run I s).2 ≤ inputScale I^10*(compressionDictionaryCapacity n+1) := by | |
| apply assemble_cells (inputScale I) (compressionDictionaryCapacity n) | |
| · unfold inputScale; omega | |
| · unfold inputScale; omega | |
| · exact (LayoutSamplerCells.run_cells (canonicalRepresentationData I).zeroSetup s).trans | |
| (raw_bound n (inputScale I) (by unfold inputScale; omega) (by unfold inputScale; omega)) | |
| · intro v | |
| exact CanonicalMixerCells.prepare_cells_capacity I v hn | |
| end LowSpaceSubsetSum.LayoutPreparationCells | |
| end | |
| end OAI | |