openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /AuxiliarySeparation /Polynomial /Laws.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Entropy.ComplexEntropyContinuity | |
| import Mathlib.Analysis.SpecialFunctions.BinaryEntropy | |
| import Mathlib.Analysis.SpecialFunctions.Pow.Real | |
| import Mathlib.Tactic.FinCases | |
| namespace OAI | |
| /-! | |
| # Probability laws for the polynomial tensor inequalities | |
| The determinant filtration uses an arbitrary binary law, while the three-sector | |
| filtration uses the uniform law on three branches. | |
| -/ | |
| noncomputable section | |
| open scoped BigOperators | |
| open MatrixMultiplication.Foundation | |
| namespace MatrixMultiplication.AuxiliarySeparation | |
| /-- The common binary probability law used for every permuted character. -/ | |
| def binaryLaw (q : ℝ) (hq₀ : 0 ≤ q) (hq₁ : q ≤ 1) : FiniteLaw (Fin 2) where | |
| mass i := if i = 0 then q else 1 - q | |
| nonneg i := by | |
| fin_cases i | |
| · simpa using hq₀ | |
| · simpa using sub_nonneg.mpr hq₁ | |
| total := by simp [Fin.sum_univ_two] | |
| @[simp] theorem binaryLaw_zero (q : ℝ) (hq₀ : 0 ≤ q) (hq₁ : q ≤ 1) : | |
| (binaryLaw q hq₀ hq₁).mass 0 = q := by simp [binaryLaw] | |
| @[simp] theorem binaryLaw_one (q : ℝ) (hq₀ : 0 ≤ q) (hq₁ : q ≤ 1) : | |
| (binaryLaw q hq₀ hq₁).mass 1 = 1 - q := by simp [binaryLaw] | |
| /-- The finite-law entropy agrees exactly with the binary entropy convention. -/ | |
| theorem binaryLaw_entropy (q : ℝ) (hq₀ : 0 ≤ q) (hq₁ : q ≤ 1) : | |
| finiteEntropy (binaryLaw q hq₀ hq₁).mass = Real.binEntropy q := by | |
| simp [finiteEntropy, Fin.sum_univ_two, entropyTerm, Real.binEntropy, Real.log_inv] | |
| ring | |
| theorem binaryLaw_product (q : ℝ) (hq₀ : 0 ≤ q) (hq₁ : q ≤ 1) (f : Fin 2 → ℝ) : | |
| (∏ i, f i ^ (binaryLaw q hq₀ hq₁).mass i) = f 0 ^ q * f 1 ^ (1 - q) := by | |
| simp [Fin.prod_univ_two] | |
| /-- The three-sector entropy factor is `3 ^ p`. -/ | |
| theorem uniformThree_entropy_factor (p : ℝ) : | |
| Real.exp (p * finiteEntropy (FiniteLaw.uniform (Fin 3)).mass) = (3 : ℝ) ^ p := by | |
| rw [FiniteLaw.uniform_entropy] | |
| norm_num only [Fintype.card_fin] | |
| rw [Real.rpow_def_of_pos (by norm_num)] | |
| congr 1 | |
| ring | |
| theorem uniformThree_product (f : Fin 3 → ℝ) : | |
| (∏ i, f i ^ (FiniteLaw.uniform (Fin 3)).mass i) = | |
| f 0 ^ (1 / 3 : ℝ) * f 1 ^ (1 / 3 : ℝ) * f 2 ^ (1 / 3 : ℝ) := by | |
| simp [Fin.prod_univ_three, FiniteLaw.uniform_mass, mul_assoc] | |
| end MatrixMultiplication.AuxiliarySeparation | |
| end | |
| end OAI | |