openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Duality /WitnessArithmetic.lean
| import Mathlib.Analysis.SpecialFunctions.Log.Basic | |
| import Mathlib.Analysis.SpecialFunctions.Exp | |
| import Mathlib.Tactic.NormNum | |
| namespace OAI | |
| /-! Dual matrix multiplication exponents and finite rectangular constructions. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.DualWitness | |
| structure State where | |
| rho : ℝ | |
| factors : ℕ | |
| hB : ℝ | |
| hA : ℝ | |
| hC : ℝ | |
| def initial : State := ⟨1 / 2, 1, 0, 0, 0⟩ | |
| def minus (m : ℕ) (s : State) : State := | |
| let rho' := s.rho ^ m | |
| { rho := rho' | |
| factors := m * s.factors | |
| hB := m * s.hB | |
| hA := m * ((s.rho - rho') / (1 - rho') * s.hB + | |
| (1 - s.rho) / (1 - rho') * s.hA) | |
| hC := m * ((s.rho - rho') / (1 - rho') * s.hB + | |
| (1 - s.rho) / (1 - rho') * s.hC) } | |
| def plusA (m : ℕ) (s : State) : State := | |
| let rho' := 1 - (1 - s.rho) ^ m | |
| { rho := rho' | |
| factors := m * s.factors | |
| hB := m * (s.rho / rho' * s.hB + (rho' - s.rho) / rho' * s.hA) | |
| hA := m * s.hA | |
| hC := Real.log ((Real.exp s.hA + Real.exp s.hC) ^ m - Real.exp (m * s.hA)) } | |
| def plusC (m : ℕ) (s : State) : State := | |
| let rho' := 1 - (1 - s.rho) ^ m | |
| { rho := rho' | |
| factors := m * s.factors | |
| hB := m * (s.rho / rho' * s.hB + (rho' - s.rho) / rho' * s.hC) | |
| hA := Real.log ((Real.exp s.hC + Real.exp s.hA) ^ m - Real.exp (m * s.hC)) | |
| hC := m * s.hC } | |
| def state2 : State := minus 2 initial | |
| def state4 : State := plusA 2 state2 | |
| def state8 : State := plusC 2 state4 | |
| def state24 : State := minus 3 state8 | |
| def state48 : State := plusC 2 state24 | |
| def state96 : State := minus 2 state48 | |
| def state288 : State := plusA 3 state96 | |
| def normalizedHidden (s : State) : ℝ := | |
| (s.rho * s.hB + (1 - s.rho) * max s.hA s.hC) / s.factors | |
| def rho8 : ℝ := 1 - (3 / 4 : ℝ) ^ 4 | |
| def rho24 : ℝ := rho8 ^ 3 | |
| def delta24 : ℝ := 3 * (1 - rho8) / (1 - rho24) * Real.log (9 / 7 : ℝ) | |
| def z48 : ℝ := Real.log ((1 + Real.exp (-delta24)) ^ 2 - 1) | |
| def rho48 : ℝ := 1 - (1 - rho24) ^ 2 | |
| def rho96 : ℝ := rho48 ^ 2 | |
| def delta96 : ℝ := 2 * z48 / (1 + rho48) | |
| def rho288 : ℝ := 1 - (1 - rho96) ^ 3 | |
| def z288 : ℝ := Real.log ((1 + Real.exp (-delta96)) ^ 3 - 1) | |
| def hiddenRate : ℝ := 9 / 64 * Real.log 3 + | |
| (1 - rho48) / 48 * z48 + (1 - rho288) / 288 * z288 | |
| def aspect : ℝ := 2 * hiddenRate / Real.log 2 | |
| theorem state_factors : state288.factors = 288 := by | |
| norm_num [state288, state96, state48, state24, state8, state4, state2, | |
| initial, minus, plusA, plusC] | |
| theorem state8_rho : state8.rho = rho8 := by | |
| norm_num [state8, state4, state2, initial, minus, plusA, plusC, rho8] | |
| theorem state24_rho : state24.rho = rho24 := by | |
| simp only [state24, minus, rho24, state8_rho] | |
| theorem state48_rho : state48.rho = rho48 := by | |
| simp only [state48, plusC, rho48, state24_rho] | |
| theorem state96_rho : state96.rho = rho96 := by | |
| simp only [state96, minus, rho96, state48_rho] | |
| theorem state288_rho : state288.rho = rho288 := by | |
| simp only [state288, plusA, rho288, state96_rho] | |
| theorem rho8_range : 0 < rho8 ∧ rho8 < 1 := by norm_num [rho8] | |
| theorem rho24_range : 0 < rho24 ∧ rho24 < 1 := by norm_num [rho24, rho8] | |
| theorem rho48_range : 0 < rho48 ∧ rho48 < 1 := by norm_num [rho48, rho24, rho8] | |
| theorem rho96_range : 0 < rho96 ∧ rho96 < 1 := by norm_num [rho96, rho48, rho24, rho8] | |
| theorem rho288_range : 0 < rho288 ∧ rho288 < 1 := by | |
| norm_num [rho288, rho96, rho48, rho24, rho8] | |
| end MatrixMultiplication.DualWitness | |
| end | |
| end OAI | |