Twelve Term Solution

twelveterm.py

#!/usr/bin/env python
# There are a few logical leaps here that should be documented in a writeup
# In the mean time this is a simple proof that needs some verification
from sympy import Eq, symbols, solve
from itertools import combinations
from numpy import choose

s11, s21, s12, s22 = symbols('s11 s21 s12 s22')
c11, c21, c12, c22 = symbols('c11 c21 c12 c22')

Edf = symbols('Edf')
Esf = symbols('Esf')
Erf = symbols('Erf')
Etf = symbols('Etf')
Elf = symbols('Elf')
Eif = symbols('Eif')

Edr = symbols('Edr')
Elr = symbols('Elr')
Err = symbols('Err')
Etr = symbols('Etr')
Esr = symbols('Esr')
Eir = symbols('Eir')

D = (1+(s11-Edf)/(Erf)*Esf)*(1+(s22-Edr)/(Err)*Esr) - \
    ((s21-Eif)/(Etf))*((s12-Eir)/(Etr))*Elf*Elr

base_eqs = [
    Eq((((s11-Edf)/(Erf))*(1+(s22-Edr)/(Err)*Esr) -
        Elf*((s21-Eif)/(Etf))*(s12-Eir)/(Etr)) / D, c11),
    Eq((((s22-Edr)/(Err))*(1+(s11-Edf)/(Erf)*Esf) -
        Elr*((s21-Eif)/(Etf))*(s12-Eir)/(Etr)) / D, c22),
    Eq((((s21 - Eif)/(Etf))*(1+((s22-Edr)/(Err))*(Esr-Elf)))/D, c21),
    Eq((((s12 - Eir)/(Etr))*(1+((s11-Edf)/(Erf))*(Esf-Elr)))/D, c12),
]

if __name__ == '__main__':
    no_elf = [solve(eq, Elf)[0] for eq in base_eqs]
    easy_eqs = [Eq(no_elf[a], no_elf[b]) for a, b in [(0, 3), (1, 2), (1, 3)]]
    simplified = [Eq(solve(eq, Elr)[0], Elr) for eq in easy_eqs]

    for term in solve(simplified, [Elr, Etf, Etr])[0]:
        print(term)