gcd.egi

--
-- Multivariate polynomial GCD reduction (design/cas-simplification.md G1).
--
-- Stage 2 of the Poly/Poly fraction reduction: cancel the multivariate
-- polynomial GCD (subresultant PRS over the rationals), treating every
-- atom -- symbols, parameters, symbolic applications such as 'cos --
-- uniformly as a variable. Fail-open outside the supported shape.
--

declare symbol x, y, a, b, c, r, G, M, X, Y, θ

-- Basic multivariate cancellations.
assertEqual "linear factor" ((x * y + y^2) / (x + y)) y
assertEqual "square over factor" ((x + y)^2 / (x + y)) (x + y)
assertEqual "difference of squares" ((x^2 - y^2) / (x - y)) (x + y)

-- Schwarzschild-style: a common polynomial factor containing parameters
-- cancels; the remaining X / Y folds into Laurent form.
assertEqual "parametric common factor"
  (((c^2 * r - 2 * G * M) * X) / ((c^2 * r - 2 * G * M) * Y))
  (X / Y)

-- T2-style: the common factor contains an application atom (cos θ).
assertEqual "application atom factor"
  ((b + a * cos θ)^2 / (b + a * cos θ))
  (b + a * cos θ)

-- Rational coefficients are cleared by a common scale (value preserved).
assertEqual "rational coefficients" (((x + y) / 2) / (x + y)) (1 / 2)

-- Coprime fractions stay untouched.
assertEqual "coprime untouched"
  (show ((x^2 - y) / (x - 1)))
  "(x^2 - y) / (x - 1)"

-- Value preservation.
assertEqual "value preserved 1" ((x * y + y^2) / (x + y) - y) 0
assertEqual "value preserved 2" ((x^2 - y^2) / (x - y) - (x + y)) 0