openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /JointExtraction /PopulationRates.lean
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.Population | |
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.TypeCounts | |
| import OAI.LinearAlgebra.MatrixMultiplication.Entropy.ComplexTypeEntropy | |
| import OAI.LinearAlgebra.MatrixMultiplication.Entropy.ComplexSeparationAsymptotics | |
| import Mathlib.Analysis.SpecificLimits.Basic | |
| namespace OAI | |
| /-! Joint tensor extraction, compatibility and entropy estimates. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.JointPopulationRates | |
| open MatrixMultiplication.Foundation Filter JointPopulation | |
| open scoped BigOperators Topology | |
| theorem tendsto_log_multinomial_div_of_total_div_zero | |
| {A : Type*} [Fintype A] [DecidableEq A] (counts : ℕ → A → ℕ) | |
| (hratio : Tendsto | |
| (fun n : ℕ => ((∑ a, counts n a : ℕ) : ℝ) / (n : ℝ)) | |
| atTop (𝓝 0)) : | |
| Tendsto (fun n : ℕ => | |
| Real.log (Nat.multinomial Finset.univ (counts n) : ℝ) / (n : ℝ)) | |
| atTop (𝓝 0) := by | |
| have hupperlim : Tendsto | |
| (fun n : ℕ => (((∑ a, counts n a : ℕ) : ℝ) / (n : ℝ)) * | |
| Real.log (Fintype.card A : ℝ)) atTop (𝓝 0) := by | |
| simpa using hratio.mul_const (Real.log (Fintype.card A : ℝ)) | |
| refine squeeze_zero ?_ ?_ hupperlim | |
| · intro n | |
| apply div_nonneg | |
| · apply Real.log_nonneg | |
| exact_mod_cast (Nat.succ_le_of_lt (Nat.multinomial_pos Finset.univ (counts n))) | |
| · exact Nat.cast_nonneg n | |
| · intro n | |
| have hcardNat : Nat.multinomial Finset.univ (counts n) ≤ | |
| (Fintype.card A) ^ (∑ a, counts n a) := by | |
| simpa only [exactWords_card] using exactWords_card_le (counts n) | |
| have hcard : (Nat.multinomial Finset.univ (counts n) : ℝ) ≤ | |
| (Fintype.card A : ℝ) ^ (∑ a, counts n a) := by | |
| exact_mod_cast hcardNat | |
| have hlog : Real.log (Nat.multinomial Finset.univ (counts n) : ℝ) ≤ | |
| ((∑ a, counts n a : ℕ) : ℝ) * Real.log (Fintype.card A : ℝ) := by | |
| simpa only [Real.log_pow] using | |
| Real.log_le_log (Nat.cast_pos.mpr (Nat.multinomial_pos Finset.univ (counts n))) hcard | |
| calc | |
| Real.log (Nat.multinomial Finset.univ (counts n) : ℝ) / (n : ℝ) ≤ | |
| (((∑ a, counts n a : ℕ) : ℝ) * Real.log (Fintype.card A : ℝ)) / (n : ℝ) := | |
| div_le_div_of_nonneg_right hlog (Nat.cast_nonneg n) | |
| _ = (((∑ a, counts n a : ℕ) : ℝ) / (n : ℝ)) * | |
| Real.log (Fintype.card A : ℝ) := by ring | |
| theorem tendsto_log_multinomial_div_of_mass | |
| {A : Type*} [Fintype A] [DecidableEq A] (counts : ℕ → A → ℕ) | |
| (mass : ℝ) (p : A → ℝ) (hmass : 0 ≤ mass) | |
| (htotal : Tendsto (fun n : ℕ => (∑ a, counts n a : ℕ) / (n : ℝ)) | |
| atTop (𝓝 mass)) | |
| (hempirical : 0 < mass → ∀ a, Tendsto | |
| (fun n : ℕ => (counts n a : ℝ) / (∑ a, counts n a : ℕ)) atTop (𝓝 (p a))) : | |
| Tendsto (fun n : ℕ => Real.log (Nat.multinomial Finset.univ (counts n) : ℝ) / | |
| (n : ℝ)) atTop (𝓝 (mass * finiteEntropy p)) := by | |
| by_cases hzero : mass = 0 | |
| · subst mass | |
| simpa only [zero_mul] using tendsto_log_multinomial_div_of_total_div_zero counts htotal | |
| have hpos : 0 < mass := lt_of_le_of_ne hmass (Ne.symm hzero) | |
| have htop : Tendsto (fun n => ∑ a, counts n a) atTop atTop := | |
| (tendsto_natCast_atTop_iff (R := ℝ)).mp | |
| (Tendsto.num (tendsto_natCast_atTop_atTop (R := ℝ)) hpos htotal) | |
| have hlog := tendsto_log_multinomial_of_empirical_tendsto counts p htop (hempirical hpos) | |
| apply (htotal.mul hlog).congr' | |
| filter_upwards [htop.eventually (eventually_gt_atTop (0 : ℕ))] with n hn | |
| have hn' : ((∑ a, counts n a : ℕ) : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr (Nat.ne_of_gt hn) | |
| rw [div_mul_div_comm, | |
| mul_comm (n : ℝ) ((∑ a, counts n a : ℕ) : ℝ), mul_div_mul_left _ _ hn'] | |
| theorem tendsto_total_div_of_coordinate_rates | |
| {A : Type*} [Fintype A] (counts : ℕ → A → ℕ) | |
| (mass : ℝ) (p : A → ℝ) (hp : ∑ a, p a = 1) | |
| (hcounts : ∀ a, Tendsto (fun n : ℕ => (counts n a : ℝ) / (n : ℝ)) | |
| atTop (𝓝 (mass * p a))) : | |
| Tendsto (fun n : ℕ => (∑ a, counts n a : ℕ) / (n : ℝ)) atTop (𝓝 mass) := by | |
| have h := tendsto_finsetSum Finset.univ (fun a _ => hcounts a) | |
| simpa only [← Finset.sum_div, ← Nat.cast_sum, ← Finset.mul_sum, hp, mul_one] using h | |
| theorem tendsto_empirical_of_coordinate_rates | |
| {A : Type*} [Fintype A] (counts : ℕ → A → ℕ) | |
| (mass : ℝ) (p : A → ℝ) (hmass : 0 < mass) (hp : ∑ a, p a = 1) | |
| (hcounts : ∀ a, Tendsto (fun n : ℕ => (counts n a : ℝ) / (n : ℝ)) | |
| atTop (𝓝 (mass * p a))) (a : A) : | |
| Tendsto (fun n : ℕ => (counts n a : ℝ) / (∑ a, counts n a : ℕ)) | |
| atTop (𝓝 (p a)) := by | |
| have htotal := tendsto_total_div_of_coordinate_rates counts mass p hp hcounts | |
| have h := (hcounts a).div htotal hmass.ne' | |
| rw [mul_div_cancel_left₀ _ hmass.ne'] at h | |
| apply h.congr' | |
| filter_upwards [eventually_ne_atTop (0 : ℕ)] with n hn | |
| exact div_div_div_cancel_right₀ (Nat.cast_ne_zero.mpr hn) _ _ | |
| theorem finiteEntropy_scalar {A : Type*} [Fintype A] | |
| (p : A → ℝ) (mass : ℝ) (hp : ∑ a, p a = 1) : | |
| finiteEntropy (fun a => mass * p a) = entropyTerm mass + mass * finiteEntropy p := by | |
| simp only [finiteEntropy, entropyTerm_mul, Finset.sum_add_distrib, | |
| ← Finset.sum_mul, ← Finset.mul_sum, hp, one_mul] | |
| theorem weighted_empirical_entropy_eq {A : Type*} [Fintype A] | |
| (counts : A → ℕ) (scale : ℝ) : | |
| ((∑ a, counts a : ℕ) / scale) * | |
| finiteEntropy (fun a => (counts a : ℝ) / (∑ a, counts a : ℕ)) = | |
| finiteEntropy (fun a => (counts a : ℝ) / scale) - | |
| entropyTerm ((∑ a, counts a : ℕ) / scale) := by | |
| by_cases hzero : (∑ a, counts a) = 0 | |
| · have hc : ∀ a, counts a = 0 := by | |
| intro a | |
| have hle : counts a ≤ ∑ b, counts b := | |
| Finset.single_le_sum (fun b _ => Nat.zero_le (counts b)) (Finset.mem_univ a) | |
| rw [hzero] at hle | |
| exact Nat.eq_zero_of_le_zero hle | |
| simp [hc, finiteEntropy] | |
| have hD : ((∑ a, counts a : ℕ) : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr hzero | |
| have hsum : (∑ a, (counts a : ℝ) / (∑ a, counts a : ℕ)) = 1 := by | |
| rw [← Finset.sum_div, ← Nat.cast_sum, div_self hD] | |
| have hfac (a : A) : (counts a : ℝ) / scale = | |
| ((∑ a, counts a : ℕ) / scale) * ((counts a : ℝ) / (∑ a, counts a : ℕ)) := by | |
| rw [div_mul_div_comm, | |
| mul_comm ((∑ a, counts a : ℕ) : ℝ) (counts a : ℝ), mul_div_mul_right _ _ hD] | |
| have h := finiteEntropy_scalar | |
| (fun a => (counts a : ℝ) / (∑ a, counts a : ℕ)) | |
| ((∑ a, counts a : ℕ) / scale) hsum | |
| simp_rw [← hfac] at h | |
| linarith | |
| theorem tendsto_weighted_empirical_entropy_of_coordinate_rates | |
| {A : Type*} [Fintype A] (counts : ℕ → A → ℕ) | |
| (mass : ℝ) (p : A → ℝ) (hp : ∑ a, p a = 1) | |
| (hcounts : ∀ a, Tendsto (fun n : ℕ => (counts n a : ℝ) / (n : ℝ)) | |
| atTop (𝓝 (mass * p a))) : | |
| Tendsto (fun n : ℕ => ((∑ a, counts n a : ℕ) / (n : ℝ)) * | |
| finiteEntropy (fun a => (counts n a : ℝ) / (∑ a, counts n a : ℕ))) | |
| atTop (𝓝 (mass * finiteEntropy p)) := by | |
| have htotal := tendsto_total_div_of_coordinate_rates counts mass p hp hcounts | |
| have h := (tendsto_finiteEntropy_of_tendsto hcounts).sub (tendsto_entropyTerm htotal) | |
| rw [finiteEntropy_scalar p mass hp, add_sub_cancel_left] at h | |
| simpa only [weighted_empirical_entropy_eq] using h | |
| def sideCounts (counts : Shape → ℕ) (s : Fin 3) (a : Fin 17) : ℕ := | |
| ∑ u, if shapeSide s u = a then counts u else 0 | |
| def sideMass (p : Shape → ℝ) (s : Fin 3) (a : Fin 17) : ℝ := | |
| ∑ u, if shapeSide s u = a then p u else 0 | |
| theorem sideCounts_total (counts : Shape → ℕ) (s : Fin 3) : | |
| (∑ a, sideCounts counts s a) = ∑ u, counts u := by | |
| unfold sideCounts | |
| rw [Finset.sum_comm] | |
| simp | |
| theorem sideMass_total (p : Shape → ℝ) (s : Fin 3) : | |
| (∑ a, sideMass p s a) = ∑ u, p u := by | |
| unfold sideMass | |
| rw [Finset.sum_comm] | |
| simp | |
| theorem tendsto_sideCounts_div_of_coordinate_rates | |
| (counts : ℕ → Shape → ℕ) (mass : ℝ) (p : Shape → ℝ) | |
| (hcounts : ∀ u, Tendsto (fun n : ℕ => (counts n u : ℝ) / (n : ℝ)) | |
| atTop (𝓝 (mass * p u))) (s : Fin 3) (a : Fin 17) : | |
| Tendsto (fun n : ℕ => (sideCounts (counts n) s a : ℝ) / (n : ℝ)) | |
| atTop (𝓝 (mass * sideMass p s a)) := by | |
| have hfun (n : ℕ) : (sideCounts (counts n) s a : ℝ) / (n : ℝ) = | |
| ∑ u, if shapeSide s u = a then (counts n u : ℝ) / (n : ℝ) else 0 := by | |
| unfold sideCounts | |
| rw [Nat.cast_sum, Finset.sum_div] | |
| apply Finset.sum_congr rfl | |
| intro u _ | |
| by_cases h : shapeSide s u = a <;> simp [h] | |
| simp_rw [hfun] | |
| rw [sideMass, Finset.mul_sum] | |
| apply tendsto_finsetSum | |
| intro u _ | |
| by_cases h : shapeSide s u = a | |
| · simpa only [h, ite_true] using hcounts u | |
| · simp only [h, ite_false, mul_zero] | |
| exact tendsto_const_nhds | |
| variable {H : Type*} [Fintype H] [DecidableEq H] | |
| omit [Fintype H] [DecidableEq H] in | |
| theorem target_side_population (counts : H → Shape → ℕ) | |
| (e : Target counts) (h : H) (s : Fin 3) (a : Fin 17) : | |
| wordPopulation (sideWord counts s (triple counts e) h) a = sideCounts (counts h) s a := by | |
| let w : JointTypeCounts.FixedCoarseWords (counts h) (shapeSide s) | |
| (sideWord counts s (triple counts e) h) := | |
| ⟨(e h).val, (e h).property, fun _ => rfl⟩ | |
| exact ((conditionalWords_nonempty_iff | |
| (sideWord counts s (triple counts e) h) | |
| (fun b u => if shapeSide s u = b then counts h u else 0)).mp | |
| ⟨JointTypeCounts.fixedCoarseWordsEquiv _ _ _ w⟩) a | |
| omit [Fintype H] [DecidableEq H] in | |
| theorem tendsto_weighted_target_side_entropy | |
| (counts : ℕ → H → Shape → ℕ) (e : ∀ n, Target (counts n)) | |
| (mass : H → ℝ) (p : H → Shape → ℝ) (hp : ∀ h, ∑ u, p h u = 1) | |
| (hcounts : ∀ h u, Tendsto (fun n : ℕ => (counts n h u : ℝ) / (n : ℝ)) | |
| atTop (𝓝 (mass h * p h u))) (h : H) (s : Fin 3) : | |
| Tendsto (fun n : ℕ => ((∑ u, counts n h u : ℕ) / (n : ℝ)) * | |
| finiteEntropy (fun a => | |
| (wordPopulation (sideWord (counts n) s (triple (counts n) (e n)) h) a : ℝ) / | |
| (∑ u, counts n h u : ℕ))) | |
| atTop (𝓝 (mass h * finiteEntropy (sideMass (p h) s))) := by | |
| have hside := tendsto_weighted_empirical_entropy_of_coordinate_rates | |
| (fun n => sideCounts (counts n h) s) (mass h) (sideMass (p h) s) | |
| ((sideMass_total (p h) s).trans (hp h)) | |
| (tendsto_sideCounts_div_of_coordinate_rates (fun n => counts n h) | |
| (mass h) (p h) (hcounts h) s) | |
| simpa only [target_side_population, sideCounts_total] using hside | |
| theorem tendsto_log_target_card_div | |
| (counts : ℕ → H → Shape → ℕ) (mass : H → ℝ) (p : H → Shape → ℝ) | |
| (hmass : ∀ h, 0 ≤ mass h) | |
| (htotal : ∀ h, Tendsto (fun n : ℕ => (∑ a, counts n h a : ℕ) / (n : ℝ)) | |
| atTop (𝓝 (mass h))) | |
| (hempirical : ∀ h, 0 < mass h → ∀ a, Tendsto | |
| (fun n : ℕ => (counts n h a : ℝ) / (∑ a, counts n h a : ℕ)) | |
| atTop (𝓝 (p h a))) : | |
| Tendsto (fun n : ℕ => Real.log (Fintype.card (Target (counts n)) : ℝ) / (n : ℝ)) | |
| atTop (𝓝 (∑ h, mass h * finiteEntropy (p h))) := by | |
| simp_rw [target_card] | |
| exact Separation.tendsto_log_nat_prod_div_nat Finset.univ | |
| (fun h _ n => Nat.multinomial_pos _ _) | |
| (fun h _ => tendsto_log_multinomial_div_of_mass | |
| (fun n => counts n h) (mass h) (p h) (hmass h) (htotal h) (hempirical h)) | |
| theorem tendsto_log_target_card_div_of_coordinate_rates | |
| (counts : ℕ → H → Shape → ℕ) (mass : H → ℝ) (p : H → Shape → ℝ) | |
| (hmass : ∀ h, 0 ≤ mass h) (hp : ∀ h, ∑ a, p h a = 1) | |
| (hcounts : ∀ h a, Tendsto (fun n : ℕ => (counts n h a : ℝ) / (n : ℝ)) | |
| atTop (𝓝 (mass h * p h a))) : | |
| Tendsto (fun n : ℕ => Real.log (Fintype.card (Target (counts n)) : ℝ) / (n : ℝ)) | |
| atTop (𝓝 (∑ h, mass h * finiteEntropy (p h))) := by | |
| apply tendsto_log_target_card_div counts mass p hmass | |
| · intro h | |
| exact tendsto_total_div_of_coordinate_rates (fun n => counts n h) | |
| (mass h) (p h) (hp h) (hcounts h) | |
| · intro h hpos a | |
| exact tendsto_empirical_of_coordinate_rates (fun n => counts n h) | |
| (mass h) (p h) hpos (hp h) (hcounts h) a | |
| theorem tendsto_log_targetSet_card_div_of_coordinate_rates | |
| (counts : ℕ → H → Shape → ℕ) (mass : H → ℝ) (p : H → Shape → ℝ) | |
| (hmass : ∀ h, 0 ≤ mass h) (hp : ∀ h, ∑ a, p h a = 1) | |
| (hcounts : ∀ h a, Tendsto (fun n : ℕ => (counts n h a : ℝ) / (n : ℝ)) | |
| atTop (𝓝 (mass h * p h a))) : | |
| Tendsto (fun n : ℕ => Real.log ((targetSet (counts n)).card : ℝ) / (n : ℝ)) | |
| atTop (𝓝 (∑ h, mass h * finiteEntropy (p h))) := by | |
| simpa only [target_card, targetSet_card] using | |
| tendsto_log_target_card_div_of_coordinate_rates counts mass p hmass hp hcounts | |
| end MatrixMultiplication.JointPopulationRates | |
| end | |
| end OAI | |