openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Completion /EntropyAlgebra.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Entropy.ComplexEntropyContinuity | |
| import Mathlib.Analysis.SpecialFunctions.Exp | |
| namespace OAI | |
| /-! Readable tensor completion and its finite arithmetic realization. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.CompletionEntropyAlgebra | |
| open MatrixMultiplication.Foundation | |
| def binaryEntropy (a : ℝ) : ℝ := entropyTerm a + entropyTerm (1 - a) | |
| def conditionalMixEntropy (a : ℝ) (m : ℕ) (Hc Ho : ℝ) : ℝ := | |
| ((m : ℝ) * (binaryEntropy a + a * Hc + (1 - a) * Ho) - | |
| binaryEntropy (a ^ m) - a ^ m * ((m : ℝ) * Hc)) / (1 - a ^ m) | |
| theorem conditionalMixEntropy_eq_split (a : ℝ) (m : ℕ) (Hc Ho : ℝ) | |
| (hden : 1 - a ^ m ≠ 0) : | |
| conditionalMixEntropy a m Hc Ho = | |
| ((m : ℝ) * (binaryEntropy a + a * Hc + (1 - a) * Ho) - | |
| entropyTerm (a ^ m) - a ^ m * ((m : ℝ) * Hc)) / (1 - a ^ m) + | |
| Real.log (1 - a ^ m) := by | |
| dsimp [conditionalMixEntropy, binaryEntropy, entropyTerm] | |
| field_simp | |
| ring | |
| private theorem conditionalMixEntropy_cancel | |
| (a C O L hc ho : ℝ) (m : ℕ) | |
| (hden : 1 - a ^ m ≠ 0) | |
| (hcLog : Real.log a = C - L) | |
| (hoLog : Real.log (1 - a) = O - L) : | |
| conditionalMixEntropy a m (C + hc) (O + ho) = | |
| m * L + Real.log (1 - a ^ m) + | |
| m * ((a - a ^ m) / (1 - a ^ m) * hc + | |
| (1 - a) / (1 - a ^ m) * ho) := by | |
| dsimp [conditionalMixEntropy, binaryEntropy, entropyTerm] | |
| rw [Real.log_pow, hcLog, hoLog] | |
| field_simp | |
| ring | |
| private theorem ratio_range (x y : ℝ) (hx : 0 < x) (hy : 0 < y) : | |
| 0 < x / (x + y) ∧ x / (x + y) < 1 := by | |
| have hs : 0 < x + y := add_pos hx hy | |
| exact ⟨div_pos hx hs, (div_lt_one hs).mpr (by linarith)⟩ | |
| private theorem ratio_complement (x y : ℝ) (h : x + y ≠ 0) : | |
| 1 - x / (x + y) = y / (x + y) := by | |
| field_simp | |
| exact add_sub_cancel_left x y | |
| theorem binaryEntropy_cardinal_mixture (np nq : ℝ) (hnp : 0 < np) (hnq : 0 < nq) : | |
| let a := np / (np + nq) | |
| binaryEntropy a + a * Real.log np + (1 - a) * Real.log nq = | |
| Real.log (np + nq) := by | |
| let a := np / (np + nq) | |
| have hs : 0 < np + nq := add_pos hnp hnq | |
| have hcLog : Real.log a = Real.log np - Real.log (np + nq) := | |
| Real.log_div hnp.ne' hs.ne' | |
| have hoLog : Real.log (1 - a) = Real.log nq - Real.log (np + nq) := by | |
| rw [show 1 - a = nq / (np + nq) from ratio_complement np nq hs.ne', | |
| Real.log_div hnq.ne' hs.ne'] | |
| change binaryEntropy a + a * Real.log np + (1 - a) * Real.log nq = _ | |
| dsimp [binaryEntropy, entropyTerm] | |
| rw [hcLog, hoLog] | |
| ring | |
| theorem conditionalMixEntropy_incident | |
| (np nq hc ho : ℝ) (m : ℕ) | |
| (hnp : 0 < np) (hnq : 0 < nq) (hm : 0 < m) : | |
| let a := np / (np + nq) | |
| conditionalMixEntropy a m (Real.log np + hc) (Real.log nq + ho) - | |
| Real.log ((np + nq) ^ m - np ^ m) = | |
| m * ((a - a ^ m) / (1 - a ^ m) * hc + | |
| (1 - a) / (1 - a ^ m) * ho) := by | |
| let a := np / (np + nq) | |
| change conditionalMixEntropy a m (Real.log np + hc) (Real.log nq + ho) - | |
| Real.log ((np + nq) ^ m - np ^ m) = _ | |
| have hs : 0 < np + nq := add_pos hnp hnq | |
| have ha : 0 < a ∧ a < 1 := ratio_range np nq hnp hnq | |
| have hden : 0 < 1 - a ^ m := | |
| sub_pos.mpr (pow_lt_one₀ ha.1.le ha.2 (Nat.ne_of_gt hm)) | |
| have hcLog : Real.log a = Real.log np - Real.log (np + nq) := | |
| Real.log_div hnp.ne' hs.ne' | |
| have hoLog : Real.log (1 - a) = Real.log nq - Real.log (np + nq) := by | |
| rw [show 1 - a = nq / (np + nq) from ratio_complement np nq hs.ne', | |
| Real.log_div hnq.ne' hs.ne'] | |
| have hfactor : (np + nq) ^ m - np ^ m = | |
| (np + nq) ^ m * (1 - a ^ m) := by | |
| dsimp [a] | |
| rw [div_pow] | |
| field_simp | |
| have hlog : Real.log ((np + nq) ^ m - np ^ m) = | |
| m * Real.log (np + nq) + Real.log (1 - a ^ m) := by | |
| rw [hfactor, Real.log_mul (pow_ne_zero _ hs.ne') hden.ne', Real.log_pow] | |
| rw [conditionalMixEntropy_cancel a (Real.log np) (Real.log nq) | |
| (Real.log (np + nq)) hc ho m hden.ne' hcLog hoLog, hlog] | |
| ring | |
| theorem conditionalMixEntropy_gibbs (q hc ho : ℝ) (m : ℕ) (hm : 0 < m) : | |
| conditionalMixEntropy (Real.exp hc / (Real.exp hc + Real.exp ho)) m | |
| (Real.log q + hc) (Real.log q + ho) - m * Real.log q = | |
| Real.log ((Real.exp hc + Real.exp ho) ^ m - Real.exp ((m : ℝ) * hc)) := by | |
| let a := Real.exp hc / (Real.exp hc + Real.exp ho) | |
| have ha : 0 < a ∧ a < 1 := ratio_range _ _ (Real.exp_pos hc) (Real.exp_pos ho) | |
| have hden : 1 - a ^ m ≠ 0 := | |
| (sub_pos.mpr (pow_lt_one₀ ha.1.le ha.2 (Nat.ne_of_gt hm))).ne' | |
| have hmean : m * ((a - a ^ m) / (1 - a ^ m) * Real.log q + | |
| (1 - a) / (1 - a ^ m) * Real.log q) = m * Real.log q := by | |
| field_simp | |
| ring | |
| have h := conditionalMixEntropy_incident (Real.exp hc) (Real.exp ho) | |
| (Real.log q) (Real.log q) m (Real.exp_pos hc) (Real.exp_pos ho) hm | |
| change conditionalMixEntropy a m (Real.log (Real.exp hc) + Real.log q) | |
| (Real.log (Real.exp ho) + Real.log q) - | |
| Real.log ((Real.exp hc + Real.exp ho) ^ m - Real.exp hc ^ m) = | |
| m * ((a - a ^ m) / (1 - a ^ m) * Real.log q + | |
| (1 - a) / (1 - a ^ m) * Real.log q) at h | |
| rw [Real.log_exp, Real.log_exp, add_comm hc (Real.log q), | |
| add_comm ho (Real.log q), hmean, ← Real.exp_nat_mul hc m] at h | |
| change conditionalMixEntropy a m (Real.log q + hc) (Real.log q + ho) - | |
| m * Real.log q = _ | |
| linarith | |
| end MatrixMultiplication.CompletionEntropyAlgebra | |
| end | |
| end OAI | |