openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Storage /ExtraPreparationCells.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Storage.ResiduePreparationCells | |
| import OAI.Combinatorics.LowSpaceSubsetSum.Representation.RepresentationExtraPreparation | |
| namespace OAI | |
| /-! # Full prepared-state memory through the bounded extra-prime draw -/ | |
| noncomputable section | |
| namespace LowSpaceSubsetSum.ExtraPreparationCells | |
| variable {n : ℕ} | |
| def run (I : Instance n) (hn : 10000 ≤ n) | |
| (hlog : Real.logb 2 ((mixerSize n : ℝ)+1)+1 ≤ (n : ℝ)/1000) | |
| (s : ((canonicalRepresentationData I).zeroSetup.extraHistoryCoupling (by omega)).Seed) : | |
| (Option (RepresentationExtraPreparation.Data n) × ℕ) × ℕ := | |
| let prepared := ResiduePreparationCells.run I hn hlog s.1 | |
| match h : prepared.1.1 with | |
| | none => ((none, prepared.1.2+1), prepared.2+9*n^3+512) | |
| | some data => | |
| let hs : (RepresentationResiduePreparation.run I hn hlog s.1).1 = some data := | |
| (congrArg Prod.fst (ResiduePreparationCells.value I hn hlog s.1)).symm.trans h | |
| let out := RepresentationScalarCells.extra (canonicalRepresentationData I).zeroSetup | |
| (((canonicalRepresentationData I).zeroSetup.trialHistoryCoupling.ideal s.1).1.1.1.1) | |
| (by omega) data.primes.stored.stored.layout.parameters.widths[3] | |
| (RepresentationExtraPreparation.width_ideal I hn hlog s.1 data hs) s.2 | |
| ((out.1.1.map (fun q => ⟨data,q⟩), prepared.1.2+out.1.2+1), | |
| prepared.2+out.2+9*n^3+512) | |
| theorem value (I : Instance n) (hn : 10000 ≤ n) | |
| (hlog : Real.logb 2 ((mixerSize n : ℝ)+1)+1 ≤ (n : ℝ)/1000) | |
| (s : ((canonicalRepresentationData I).zeroSetup.extraHistoryCoupling (by omega)).Seed) : | |
| (run I hn hlog s).1 = RepresentationExtraPreparation.run I hn hlog s := by | |
| have hp₁ := congrArg Prod.fst (ResiduePreparationCells.value I hn hlog s.1) | |
| have hp₂ := congrArg Prod.snd (ResiduePreparationCells.value I hn hlog s.1) | |
| simp only [run, hp₂, RepresentationExtraPreparation.run] | |
| split | |
| next h => | |
| split | |
| · rfl | |
| next data hs => | |
| have he := (hp₁.symm.trans h).symm.trans hs | |
| cases he | |
| next data h => | |
| simp only [RepresentationScalarCells.extra_value] | |
| split | |
| next hs => | |
| have he := (hp₁.symm.trans h).symm.trans hs | |
| cases he | |
| next data' hs => | |
| have he : data = data' := Option.some.inj ((hp₁.symm.trans h).symm.trans hs) | |
| subst data' | |
| rfl | |
| theorem cells_add (I : Instance n) (hn : 10000 ≤ n) | |
| (hlog : Real.logb 2 ((mixerSize n : ℝ)+1)+1 ≤ (n : ℝ)/1000) | |
| (s : ((canonicalRepresentationData I).zeroSetup.extraHistoryCoupling (by omega)).Seed) : | |
| (run I hn hlog s).2 ≤ (ResiduePreparationCells.run I hn hlog s.1).2+12*n^3+608 := by | |
| unfold run | |
| dsimp only | |
| split | |
| · omega | |
| next data h => | |
| have hs : (RepresentationResiduePreparation.run I hn hlog s.1).1 = some data := | |
| (congrArg Prod.fst (ResiduePreparationCells.value I hn hlog s.1)).symm.trans h | |
| have hx := RepresentationScalarCells.extra_cells (canonicalRepresentationData I).zeroSetup | |
| (((canonicalRepresentationData I).zeroSetup.trialHistoryCoupling.ideal s.1).1.1.1.1) | |
| (by omega) data.primes.stored.stored.layout.parameters.widths[3] | |
| (RepresentationExtraPreparation.width_ideal I hn hlog s.1 data hs) s.2 | |
| change _+_+9*n^3+512 ≤ _ | |
| omega | |
| theorem cells (I : Instance n) (hn : 10000 ≤ n) | |
| (hlog : Real.logb 2 ((mixerSize n : ℝ)+1)+1 ≤ (n : ℝ)/1000) | |
| (s : ((canonicalRepresentationData I).zeroSetup.extraHistoryCoupling (by omega)).Seed) : | |
| (run I hn hlog s).2 ≤ inputScale I^13*(compressionDictionaryCapacity n+1) := by | |
| have h1 := cells_add I hn hlog s | |
| have h2 := ResiduePreparationCells.cells_add I hn hlog s.1 | |
| have h3 := PrimePreparationCells.cells_add I hn hlog s.1.1 | |
| have hbase := RepresentationStoredPreparationCells.cells I hn s.1.1.1 | |
| have htotal : (run I hn hlog s).2 ≤ (RepresentationStoredPreparationCells.run I s.1.1.1).2+(48*n^3+2240) := by | |
| calc | |
| _ ≤ (ResiduePreparationCells.run I hn hlog s.1).2+12*n^3+608 := h1 | |
| _ ≤ ((PrimePreparationCells.run I hn hlog s.1.1).2+18*n^3+800)+12*n^3+608 := | |
| Nat.add_le_add_right (Nat.add_le_add_right h2 (12*n^3)) 608 | |
| _ ≤ (((RepresentationStoredPreparationCells.run I s.1.1.1).2+18*n^3+832)+18*n^3+800)+12*n^3+608 := | |
| Nat.add_le_add_right (Nat.add_le_add_right | |
| (Nat.add_le_add_right (Nat.add_le_add_right h3 (18*n^3)) 800) (12*n^3)) 608 | |
| _ = _ := by ring | |
| exact htotal.trans (ScalarStageCells.aggregate n (inputScale I) (compressionDictionaryCapacity n) | |
| (RepresentationStoredPreparationCells.run I s.1.1.1).2 (48*n^3+2240) | |
| (by unfold inputScale; omega) (by unfold inputScale; omega) hbase le_rfl) | |
| end LowSpaceSubsetSum.ExtraPreparationCells | |
| end | |
| end OAI | |