| import Mathlib |
| import OAI.LinearAlgebra.MatrixFields.Construction.JointMarginalSupport |
|
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| namespace OAI |
|
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| namespace MatrixAllFields |
|
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| open scoped BigOperators Topology Polynomial |
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| section |
| noncomputable section |
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| namespace MatrixMultiplication.JointAmbientRateLimit |
|
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| open MatrixMultiplication.Foundation JointPopulation JointAmbientDegree Filter |
| open scoped BigOperators Topology |
|
|
| attribute [local instance] Classical.propDecidable |
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| theorem tendsto_log_succ_div_nat_of_ratio {n : ℕ → ℕ} {mass : ℝ} |
| (hsize : Tendsto (fun N => (n N : ℝ) / (N : ℝ)) atTop (𝓝 mass)) : |
| Tendsto (fun N => Real.log (n N + 1 : ℕ) / (N : ℝ)) atTop (𝓝 0) := by |
| let C : ℝ := |mass| + 2 |
| have hC : 0 < C := by dsimp [C]; positivity |
| have hmC : mass < C := by dsimp [C]; linarith [le_abs_self mass] |
| have hlog : Tendsto (fun N : ℕ => Real.log (N : ℝ) / (N : ℝ)) atTop (𝓝 0) := by |
| simpa only [Function.comp_def, id_eq] using |
| Real.isLittleO_log_id_atTop.tendsto_div_nhds_zero.comp tendsto_natCast_atTop_atTop |
| have hc : Tendsto (fun N : ℕ => Real.log (C + 1) / (N : ℝ)) atTop (𝓝 0) := |
| tendsto_const_nhds.div_atTop tendsto_natCast_atTop_atTop |
| have hu : Tendsto (fun N : ℕ => |
| (Real.log (C + 1) + Real.log (N : ℝ)) / (N : ℝ)) atTop (𝓝 0) := by |
| simpa only [add_div, add_zero] using hc.add hlog |
| apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hu |
| · exact Eventually.of_forall fun N => |
| div_nonneg (Real.log_nonneg (by exact_mod_cast Nat.succ_le_succ (Nat.zero_le (n N)))) |
| (Nat.cast_nonneg _) |
| · filter_upwards [(tendsto_order.1 hsize).2 C hmC, |
| eventually_ge_atTop (1 : ℕ)] with N hn hN |
| have hNr : (1 : ℝ) ≤ N := by exact_mod_cast hN |
| have hNp : (0 : ℝ) < N := lt_of_lt_of_le zero_lt_one hNr |
| have hsizeN := (div_lt_iff₀ hNp).mp hn |
| have hbound : (n N + 1 : ℕ) ≤ (C + 1) * (N : ℝ) := by |
| push_cast |
| nlinarith |
| have hl := Real.log_le_log (by positivity : (0 : ℝ) < (n N + 1 : ℕ)) hbound |
| rw [Real.log_mul (by positivity : C + 1 ≠ 0) hNp.ne'] at hl |
| exact div_le_div_of_nonneg_right hl hNp.le |
|
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| variable {H : Type*} [Fintype H] [DecidableEq H] |
|
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| omit [Fintype H] [DecidableEq H] in |
| theorem tendsto_classExponent_div_nat |
| (counts : ℕ → H → Shape → ℕ) (s : Fin 3) (h : H) |
| (cap : ℕ → ℝ) (mass capRate marginalRate : ℝ) |
| (hsize : Tendsto (fun N => (∑ a, counts N h a : ℕ) / (N : ℝ)) |
| atTop (𝓝 mass)) |
| (hcap : Tendsto (fun N => ((∑ a, counts N h a : ℕ) / (N : ℝ)) * cap N) |
| atTop (𝓝 capRate)) |
| (hmarginal : Tendsto (fun N => |
| ((∑ a, counts N h a : ℕ) / (N : ℝ)) * |
| finiteEntropy (fun a => (wordPopulation |
| (sideWord (counts N) s (triple (counts N) (canonicalTarget (counts N))) h) a : ℝ) / |
| Fintype.card (Positions (counts N) h))) atTop (𝓝 marginalRate)) : |
| Tendsto (fun N => |
| classExponent (counts N) s (triple (counts N) (canonicalTarget (counts N))) h (cap N) / |
| (N : ℝ)) atTop (𝓝 (capRate - marginalRate)) := by |
| let A : ℝ := (Fintype.card Shape : ℝ) + (Fintype.card (Fin 17) : ℝ) + 2 |
| have hlog := tendsto_log_succ_div_nat_of_ratio hsize |
| have hinv : Tendsto (fun N : ℕ => (1 : ℝ) / N) atTop (𝓝 0) := |
| tendsto_const_nhds.div_atTop tendsto_natCast_atTop_atTop |
| have htotal := (hcap.sub hmarginal).add |
| (((hinv.add hlog).const_mul A).add (hlog.const_mul (Fintype.card Shape : ℝ))) |
| simp only [add_zero, mul_zero] at htotal |
| convert htotal using 1 |
| ext N |
| dsimp only [classExponent, A, Positions] |
| simp only [Fintype.card_fin, div_eq_mul_inv] |
| ring |
|
|
| omit [DecidableEq H] in |
| theorem tendsto_sum_classExponent_div_nat |
| (counts : ℕ → H → Shape → ℕ) (s : Fin 3) |
| (cap : ℕ → H → ℝ) (mass capRate marginalRate : H → ℝ) |
| (hsize : ∀ h, Tendsto (fun N => (∑ a, counts N h a : ℕ) / (N : ℝ)) |
| atTop (𝓝 (mass h))) |
| (hcap : ∀ h, Tendsto (fun N => ((∑ a, counts N h a : ℕ) / (N : ℝ)) * cap N h) |
| atTop (𝓝 (capRate h))) |
| (hmarginal : ∀ h, Tendsto (fun N => |
| ((∑ a, counts N h a : ℕ) / (N : ℝ)) * |
| finiteEntropy (fun a => (wordPopulation |
| (sideWord (counts N) s (triple (counts N) (canonicalTarget (counts N))) h) a : ℝ) / |
| Fintype.card (Positions (counts N) h))) atTop (𝓝 (marginalRate h))) : |
| Tendsto (fun N => (∑ h, |
| classExponent (counts N) s (triple (counts N) (canonicalTarget (counts N))) h (cap N h)) / |
| (N : ℝ)) atTop (𝓝 (∑ h, (capRate h - marginalRate h))) := by |
| have hs := tendsto_finsetSum Finset.univ (fun h _ => |
| tendsto_classExponent_div_nat counts s h (fun N => cap N h) |
| (mass h) (capRate h) (marginalRate h) (hsize h) (hcap h) (hmarginal h)) |
| simpa only [Finset.sum_div] using hs |
|
|
| theorem eventually_all_target_degree_upper |
| (counts : ℕ → H → Shape → ℕ) (sum : H → ℕ) |
| (cap : ℕ → H → ℝ) (sideRate : Fin 3 → ℝ) (D : ℝ) (hD : 0 ≤ D) |
| (hentropy : ∀ᶠ N in atTop, ∀ h w, localAllowed (counts N) sum h w → |
| finiteEntropy (fun a => (wordPopulation w a : ℝ) / |
| Fintype.card (Positions (counts N) h)) ≤ cap N h) |
| (hrate : ∀ s, Tendsto (fun N => (∑ h, |
| classExponent (counts N) s (triple (counts N) (canonicalTarget (counts N))) h (cap N h)) / |
| (N : ℝ)) atTop (𝓝 (sideRate s))) |
| (hside : ∀ s, sideRate s ≤ D) (η : ℝ) (hη : 0 < η) : |
| ∀ᶠ N in atTop, ∀ e : Target (counts N), |
| Real.log ((neighbors (counts N) sum (triple (counts N) e)).card : ℝ) / (N : ℝ) ≤ |
| D + η := by |
| have hs : ∀ᶠ N in atTop, ∀ s : Fin 3, |
| (∑ h, classExponent (counts N) s |
| (triple (counts N) (canonicalTarget (counts N))) h (cap N h)) / (N : ℝ) ≤ |
| D + η / 2 := by |
| rw [Filter.eventually_all] |
| intro s |
| filter_upwards [(tendsto_order.1 (hrate s)).2 (sideRate s + η / 2) (by linarith)] |
| with N hN |
| linarith [hside s] |
| have hc : Tendsto (fun N : ℕ => Real.log 3 / (N : ℝ)) atTop (𝓝 0) := |
| tendsto_const_nhds.div_atTop tendsto_natCast_atTop_atTop |
| filter_upwards [hentropy, hs, (tendsto_order.1 hc).2 (η / 2) (by linarith), |
| eventually_ge_atTop (1 : ℕ)] with N he hs hc hN |
| intro e |
| by_cases hz : (neighbors (counts N) sum (triple (counts N) e)).card = 0 |
| · simp only [hz, Nat.cast_zero, Real.log_zero, zero_div] |
| linarith |
| have hNp : (0 : ℝ) < N := by exact_mod_cast hN |
| have hupper (s : Fin 3) : |
| (∑ h, classExponent (counts N) s (triple (counts N) e) h (cap N h)) ≤ |
| (N : ℝ) * (D + η / 2) := by |
| simp_rw [classExponent_target_eq] |
| simpa only [mul_comm] using (div_le_iff₀ hNp).mp (hs s) |
| have hb := div_le_div_of_nonneg_right |
| (log_neighbors_card_le (counts N) sum (triple (counts N) e) (cap N) he |
| ((N : ℝ) * (D + η / 2)) hupper (Nat.pos_of_ne_zero hz)) hNp.le |
| rw [add_div, mul_div_cancel_left₀ _ hNp.ne'] at hb |
| linarith |
|
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| end MatrixMultiplication.JointAmbientRateLimit |
|
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| end |
| end |
|
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| end MatrixAllFields |
|
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| end OAI |
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