openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldConstruction /ConstructionRates.lean
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.ConstructionLabels | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.RecoveryBudget | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.SourceRates | |
| namespace OAI | |
| /-! Tensor extraction over arbitrary fields and its asymptotic rate. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.AllFieldConstructionRates | |
| open MatrixMultiplication.Foundation AllFieldHistory Filter | |
| open scoped BigOperators Topology Classical | |
| variable {K : ℕ} | |
| abbrev Copies (allocation : Allocation) (ε : ℝ) (m : ℕ) := | |
| ∀ tick : Fin (K + 2), InverseLinearRecovery.MaskRectangles | |
| (AllFieldHistoryRecovery.shiftCount (K := K) (tick := tick.val) allocation ε m) | |
| theorem card_copies (allocation : Allocation) (ε : ℝ) (m : ℕ) : | |
| Fintype.card (Copies (K := K) allocation ε m) = | |
| ∏ tick : Fin (K + 2), RecoveryGrowth.replicationCount | |
| (AllFieldHistorySupport.supportRate (K := K) (tick := tick.val) allocation) | |
| (AllFieldHistoryMasks.lossConstant (K := K) (tick := tick.val) allocation ε + 6) m := by | |
| rw [Fintype.card_pi] | |
| apply Finset.prod_congr rfl | |
| intro tick _ | |
| exact InverseLinearRecovery.card_recovery_copies | |
| (AllFieldHistorySupport.supportRate (K := K) (tick := tick.val) allocation) | |
| (AllFieldHistoryMasks.lossConstant (K := K) (tick := tick.val) allocation ε + 6) m | |
| theorem card_copies_range (allocation : Allocation) (ε : ℝ) (m : ℕ) : | |
| Fintype.card (Copies (K := K) allocation ε m) = | |
| ∏ tick ∈ Finset.range (K + 2), Fintype.card | |
| (InverseLinearRecovery.MaskRectangles | |
| (AllFieldHistoryRecovery.shiftCount (K := K) (tick := tick) allocation ε m)) := by | |
| rw [Fintype.card_pi] | |
| exact Fin.prod_univ_eq_prod_range (fun tick : ℕ => Fintype.card | |
| (InverseLinearRecovery.MaskRectangles | |
| (AllFieldHistoryRecovery.shiftCount (K := K) (tick := tick) allocation ε m))) (K + 2) | |
| theorem card_copies_pos (allocation : Allocation) (ε : ℝ) (m : ℕ) : | |
| 0 < Fintype.card (Copies (K := K) allocation ε m) := Fintype.card_pos | |
| theorem copies_subexponential (allocation : Allocation) (ε : ℝ) : | |
| Tendsto (fun m : ℕ => Real.log (Fintype.card (Copies (K := K) allocation ε m) : ℝ) / | |
| (m : ℝ)) atTop (𝓝 0) := by | |
| have h := RecoveryGrowth.tendsto_log_prod_div_nat (Finset.univ : Finset (Fin (K + 2))) | |
| (r := fun tick m => Fintype.card (InverseLinearRecovery.MaskRectangles | |
| (AllFieldHistoryRecovery.shiftCount (K := K) (tick := tick.val) allocation ε m))) | |
| (fun _ _ _ => Fintype.card_pos) | |
| (fun tick _ => AllFieldHistoryRecovery.replication_subexponential | |
| (K := K) (tick := tick.val) allocation ε) | |
| simpa only [Copies, Fintype.card_pi] using h | |
| theorem copies_rate (hK : 0 < K) (allocation : Allocation) (ε : ℝ) : | |
| Tendsto (fun m : ℕ => Real.log (Fintype.card (Copies (K := K) allocation ε m) : ℝ) / | |
| ((K : ℝ) * populationLength (K := K) allocation m)) atTop (𝓝 0) := by | |
| have hden : (K : ℝ) * denominator (K := K) allocation ≠ 0 := | |
| mul_ne_zero (Nat.cast_ne_zero.mpr hK.ne') | |
| (Nat.cast_ne_zero.mpr (denominator_pos allocation).ne') | |
| have h := (copies_subexponential (K := K) allocation ε).div tendsto_const_nhds hden | |
| simp only [zero_div] at h | |
| apply h.congr | |
| intro m | |
| rw [Pi.div_apply, population_length, Nat.cast_mul] | |
| ring | |
| theorem source_rank_rate (hK : 0 < K) (allocation : Allocation) (ε : ℝ) : | |
| Tendsto (fun m : ℕ => Real.log | |
| (AllFieldSource.rankBudget (Copies (K := K) allocation ε m) K | |
| (populationLength (K := K) allocation m) : ℝ) / | |
| ((K : ℝ) * populationLength (K := K) allocation m)) | |
| atTop (𝓝 (8 * Real.log 7)) := by | |
| have hp := (AllFieldSourceRates.tendsto_log_polynomial_overhead K hK).comp | |
| (populationLength_tendsto (K := K) allocation) | |
| have he := (AllFieldSourceRates.tendsto_log_exponential_source K hK).comp | |
| (populationLength_tendsto (K := K) allocation) | |
| have h := ((copies_rate hK allocation ε).add hp).add he | |
| simp only [zero_add] at h | |
| apply h.congr | |
| intro m | |
| rw [AllFieldSourceRates.log_rankBudget _ _ _ (card_copies_pos allocation ε m), add_div, add_div] | |
| rfl | |
| theorem source_rank_terminal_rate (hK : 0 < K) (allocation : Allocation) (ε : ℝ) : | |
| Tendsto (fun m : ℕ => Real.log | |
| (AllFieldSource.rankBudget | |
| (Copies (K := K) allocation ε (AllFieldTerminalRates.terminalDilation K m)) K | |
| (AllFieldTerminalRates.terminalLength allocation K m) : ℝ) / | |
| ((K : ℝ) * AllFieldTerminalRates.terminalLength allocation K m)) | |
| atTop (𝓝 (8 * Real.log 7)) := | |
| (source_rank_rate hK allocation ε).comp (terminalDilation_tendsto K) | |
| theorem source_rank_terminal_rate_of_cardinality | |
| (ActualCopies : ℕ → Type*) [∀ m, Fintype (ActualCopies m)] | |
| (hK : 0 < K) (allocation : Allocation) (ε : ℝ) | |
| (cards : ∀ᶠ m : ℕ in atTop, Fintype.card (ActualCopies m) = Fintype.card | |
| (Copies (K := K) allocation ε (AllFieldTerminalRates.terminalDilation K m))) : | |
| Tendsto (fun m : ℕ => Real.log (AllFieldSource.rankBudget (ActualCopies m) K | |
| (AllFieldTerminalRates.terminalLength allocation K m) : ℝ) / | |
| ((K : ℝ) * AllFieldTerminalRates.terminalLength allocation K m)) | |
| atTop (𝓝 (8 * Real.log 7)) := by | |
| apply (source_rank_terminal_rate hK allocation ε).congr' | |
| filter_upwards [cards] with m hm | |
| simp only [AllFieldSource.rankBudget, hm] | |
| end MatrixMultiplication.AllFieldConstructionRates | |
| end | |
| end OAI | |