openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Entropy /FloorExponentialRates.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Recovery.RecoveryGrowth | |
| import Mathlib.Analysis.SpecialFunctions.Exp | |
| import Mathlib.Data.Fintype.BigOperators | |
| namespace OAI | |
| /-! Finite entropy, rate estimates and ordered asymptotic limits. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.FloorExponentialRates | |
| open Filter | |
| open scoped BigOperators Topology Classical | |
| def count (rate : ℝ) (m : ℕ) : ℕ := ⌊Real.exp ((m : ℝ) * rate)⌋₊ | |
| theorem count_pos {rate : ℝ} (hr : 0 ≤ rate) (m : ℕ) : 0 < count rate m := by | |
| have he : (1 : ℝ) ≤ Real.exp ((m : ℝ) * rate) := | |
| Real.one_le_exp (mul_nonneg (Nat.cast_nonneg _) hr) | |
| exact lt_of_lt_of_le Nat.zero_lt_one (Nat.le_floor (by simpa only [Nat.cast_one] using he)) | |
| theorem log_count_bounds {rate : ℝ} (hr : 0 ≤ rate) (m : ℕ) : | |
| (m : ℝ) * rate - Real.log 2 ≤ Real.log (count rate m : ℝ) ∧ | |
| Real.log (count rate m : ℝ) ≤ (m : ℝ) * rate := by | |
| have hp : (0 : ℝ) < count rate m := Nat.cast_pos.mpr (count_pos hr m) | |
| have hone : (1 : ℝ) ≤ count rate m := by exact_mod_cast count_pos hr m | |
| have hround := Nat.lt_floor_add_one (Real.exp ((m : ℝ) * rate)) | |
| have hhalf : Real.exp ((m : ℝ) * rate) / 2 ≤ (count rate m : ℝ) := by | |
| change Real.exp ((m : ℝ) * rate) < (count rate m : ℝ) + 1 at hround | |
| linarith | |
| constructor | |
| · have h := Real.log_le_log (by positivity : 0 < Real.exp ((m : ℝ) * rate) / 2) hhalf | |
| simpa only [Real.log_div (Real.exp_ne_zero _) (by norm_num : (2 : ℝ) ≠ 0), | |
| Real.log_exp] using h | |
| · have h := Real.log_le_log hp (Nat.floor_le (Real.exp_pos _).le) | |
| simpa only [Real.log_exp] using h | |
| theorem tendsto_log_count_div_nat {rate : ℝ} (hr : 0 ≤ rate) : | |
| Tendsto (fun m : ℕ => Real.log (count rate m : ℝ) / (m : ℝ)) | |
| atTop (𝓝 rate) := by | |
| have hinv : Tendsto (fun m : ℕ => Real.log 2 / (m : ℝ)) atTop (𝓝 0) := | |
| tendsto_const_nhds.div_atTop tendsto_natCast_atTop_atTop | |
| have hlo : Tendsto (fun m : ℕ => rate - Real.log 2 / (m : ℝ)) atTop (𝓝 rate) := by | |
| simpa only [sub_zero] using tendsto_const_nhds.sub hinv | |
| apply tendsto_of_tendsto_of_tendsto_of_le_of_le' hlo tendsto_const_nhds | |
| · filter_upwards [eventually_gt_atTop (0 : ℕ)] with m hm | |
| have hm' : (0 : ℝ) < m := Nat.cast_pos.mpr hm | |
| have h := div_le_div_of_nonneg_right (log_count_bounds hr m).1 hm'.le | |
| simpa only [sub_div, mul_div_cancel_left₀ rate hm'.ne'] using h | |
| · filter_upwards [eventually_gt_atTop (0 : ℕ)] with m hm | |
| have hm' : (0 : ℝ) < m := Nat.cast_pos.mpr hm | |
| have h := div_le_div_of_nonneg_right (log_count_bounds hr m).2 hm'.le | |
| simpa only [mul_div_cancel_left₀ rate hm'.ne'] using h | |
| theorem tendsto_log_prod_count_div_nat {I : Type*} [Fintype I] (rate : I → ℝ) | |
| (hr : ∀ i, 0 ≤ rate i) : | |
| Tendsto (fun m : ℕ => Real.log ((∏ i, count (rate i) m : ℕ) : ℝ) / (m : ℝ)) | |
| atTop (𝓝 (∑ i, rate i)) := by | |
| have h := tendsto_finsetSum Finset.univ (fun i _ => tendsto_log_count_div_nat (hr i)) | |
| apply h.congr | |
| intro m | |
| rw [Nat.cast_prod, Real.log_prod (s := Finset.univ) | |
| (f := fun i => (count (rate i) m : ℝ)) | |
| (fun i _ => Nat.cast_ne_zero.mpr (Nat.ne_of_gt (count_pos (hr i) m)))] | |
| simp only [div_eq_mul_inv, Finset.sum_mul] | |
| abbrev Labels {I : Type*} (rate : I → ℝ) (m : ℕ) := ∀ i, Fin (count (rate i) m) | |
| theorem card_labels {I : Type*} [Fintype I] (rate : I → ℝ) (m : ℕ) : | |
| Fintype.card (Labels rate m) = ∏ i, count (rate i) m := by | |
| simp only [Labels, Fintype.card_pi, Fintype.card_fin] | |
| theorem card_labels_pos {I : Type*} [Fintype I] (rate : I → ℝ) | |
| (hr : ∀ i, 0 ≤ rate i) (m : ℕ) : 0 < Fintype.card (Labels rate m) := by | |
| rw [card_labels] | |
| exact Finset.prod_pos fun i _ => count_pos (hr i) m | |
| theorem tendsto_log_card_labels_div_nat {I : Type*} [Fintype I] (rate : I → ℝ) | |
| (hr : ∀ i, 0 ≤ rate i) : | |
| Tendsto (fun m : ℕ => Real.log (Fintype.card (Labels rate m) : ℝ) / (m : ℝ)) | |
| atTop (𝓝 (∑ i, rate i)) := by | |
| simpa only [card_labels] using tendsto_log_prod_count_div_nat rate hr | |
| end MatrixMultiplication.FloorExponentialRates | |
| end | |
| end OAI | |