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cdec422 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 | 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
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