openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Counting /AnnealingMultipliers.lean
| import Mathlib | |
| namespace OAI | |
| /-! | |
| # The companion's multiplier update from empirical type counts | |
| The old multiplier cancels from the update relative to the current ideal | |
| multiplier. Relative accuracy of two positive type means, followed by the | |
| proved adjacent-parameter factor two, gives the required factor four at | |
| the next parameter. This is deterministic algebra of the actual update. | |
| -/ | |
| namespace ContingencyTables.AnnealingMultipliers | |
| variable {eta p q ph qh w ideal nextIdeal : ℝ} | |
| theorem empirical_positive (hp : 0<p) (he : eta≤1/4) | |
| (hph : (1-eta)*p≤ph) : 0<ph := by | |
| have h : 0<(1-eta)*p := mul_pos (by linarith) hp | |
| exact h.trans_le hph | |
| /-- The normalized ratio of two empirically observed positive means. -/ | |
| theorem relative_ratio (hp : 0<p) (hq : 0<q) (he0 : 0≤eta) (he : eta≤1/4) | |
| (hph : (1-eta)*p≤ph ∧ ph≤(1+eta)*p) | |
| (hqh : (1-eta)*q≤qh ∧ qh≤(1+eta)*q) : | |
| 0<ph*q/(qh*p) ∧ | |
| 1/2≤ph*q/(qh*p) ∧ ph*q/(qh*p)≤2 ∧ | |
| |Real.log (ph*q/(qh*p))|≤4*eta := by | |
| have hp' := empirical_positive hp he hph.1 | |
| have hq' := empirical_positive hq he hqh.1 | |
| have hden : 0<qh*p := mul_pos hq' hp | |
| have hr : 0<ph*q/(qh*p) := div_pos (mul_pos hp' hq) hden | |
| have hlow : (1-eta)*(p*q)≤ph*q := by nlinarith [mul_le_mul_of_nonneg_right hph.1 hq.le] | |
| have hupp : ph*q≤(1+eta)*(p*q) := by nlinarith [mul_le_mul_of_nonneg_right hph.2 hq.le] | |
| have dlow : (1-eta)*(p*q)≤qh*p := by nlinarith [mul_le_mul_of_nonneg_right hqh.1 hp.le] | |
| have dupp : qh*p≤(1+eta)*(p*q) := by nlinarith [mul_le_mul_of_nonneg_right hqh.2 hp.le] | |
| have hpq : 0≤p*q := (mul_pos hp hq).le | |
| have htwice : (1+eta)*(p*q)≤2*((1-eta)*(p*q)) := by | |
| have := mul_nonneg (show 1-3*eta≥0 by linarith) hpq | |
| nlinarith | |
| have hlo : 1/2≤ph*q/(qh*p) := by | |
| rw [le_div_iff₀ hden] | |
| linarith | |
| have hup : ph*q/(qh*p)≤2 := by | |
| rw [div_le_iff₀ hden] | |
| linarith | |
| have hstretch : (1+eta)*(p*q)≤(1+4*eta)*((1-eta)*(p*q)) := by | |
| have hcoeff : 0≤2*eta-4*eta^2 := by nlinarith | |
| have := mul_nonneg hcoeff hpq | |
| nlinarith | |
| have hscale : 0≤1+4*eta := by linarith | |
| have hrup : ph*q/(qh*p)≤1+4*eta := by | |
| rw [div_le_iff₀ hden] | |
| have := mul_le_mul_of_nonneg_left dlow hscale | |
| nlinarith | |
| have hinvup : (ph*q/(qh*p))⁻¹≤1+4*eta := by | |
| rw [inv_div] | |
| rw [div_le_iff₀ (mul_pos hp' hq)] | |
| have := mul_le_mul_of_nonneg_left hlow hscale | |
| nlinarith | |
| have hlogup := Real.log_le_sub_one_of_pos hr | |
| have hloglo := Real.one_sub_inv_le_log_of_pos hr | |
| refine ⟨hr,hlo,hup,abs_le.mpr ⟨?_,?_⟩⟩ <;> linarith | |
| /-- The concrete update w*(observed plain count)/(observed defect count) | |
| is good for the next parameter. Earlier multiplier errors do not compound. -/ | |
| theorem update_good (hp : 0<p) (hq : 0<q) (hw : 0<w) | |
| (he0 : 0≤eta) (he : eta≤1/4) | |
| (hph : (1-eta)*p≤ph ∧ ph≤(1+eta)*p) | |
| (hqh : (1-eta)*q≤qh ∧ qh≤(1+eta)*q) | |
| (hideal : w*p/q=ideal) | |
| (hnext : ideal/2≤nextIdeal ∧ nextIdeal≤2*ideal) : | |
| nextIdeal/4≤w*ph/qh ∧ w*ph/qh≤4*nextIdeal := by | |
| have h := relative_ratio hp hq he0 he hph hqh | |
| have hq' := empirical_positive hq he hqh.1 | |
| have hi : 0 < ideal := by rw [← hideal]; positivity | |
| have hid : w*ph/qh=ideal*(ph*q/(qh*p)) := by | |
| rw [← hideal] | |
| field_simp [ne_of_gt hp,ne_of_gt hq,ne_of_gt hq'] | |
| rw [hid] | |
| have hlo := mul_le_mul_of_nonneg_left h.2.1 hi.le | |
| have hup := mul_le_mul_of_nonneg_left h.2.2.1 hi.le | |
| constructor <;> nlinarith [hnext.1,hnext.2] | |
| end ContingencyTables.AnnealingMultipliers | |
| end OAI | |