Function rtAuxBody

src/compute-engine/rubi/rubi-utils.ts:1388–1516

(u: Expression, n: number)

}

const productOf = (ce: ComputeEngine, fs: Expression[]): Expression =>
  fs.length === 0
    ? ce.One
    : fs.length === 1
      ? fs[0]
      : fs.reduce((p, f) => p.mul(f));

// Rubi NegQ[u] over a term (negative form per the PosAux heuristics)
function negFormQ(u: Expression): boolean {
  return !posQ(u) && !zeroQ(u);
}

/** sum terms of a base (Add/Subtract), through odd literal powers
 * (Rubi SumBaseQ companions); null when not a sum base */
function sumBaseTerms(u: Expression): Expression[] | null {
  if (u.operator === 'Add' || u.operator === 'Subtract') return sumTermsX(u);
  if (u.operator === 'Power' && u.ops) {
    const k = realNum(u.ops[1]);
    if (k !== null && Number.isInteger(k) && Math.abs(k % 2) === 1)
      return sumBaseTerms(u.ops[0]);
  }
  return null;
}
const sumBaseQ = (u: Expression): boolean => sumBaseTerms(u) !== null;
const negSumBaseQ = (u: Expression): boolean => {
  const ts = sumBaseTerms(u);
  return ts !== null && negFormQ(mmaFirstTerm(ts));
};
const allNegTermQ = (u: Expression): boolean => {
  const ts = sumBaseTerms(u);
  if (ts !== null) return ts.every(negFormQ);
  return negFormQ(u);
};
const someNegTermQ = (u: Expression): boolean => {
  const ts = sumBaseTerms(u);
  if (ts !== null) return ts.some(negFormQ);
  return negFormQ(u);
};
const atomBaseQ = (u: Expression): boolean => {
  if (!u.ops) return true;
  if (u.operator === 'Power' && u.ops) {
    const k = realNum(u.ops[1]);
    if (k !== null && Number.isInteger(k) && Math.abs(k % 2) === 1)
      return atomBaseQ(u.ops[0]);
  }
  return false;
};

/** Rubi RtAux[u, n] (TrigSquare and the complex-integer basis case
 * omitted — not exercised by Chapter 1) */
let rtAuxDepth = 0;
function rtAux(u: Expression, n: number): Expression {
  // recursion backstop: the WL definition recurses through rewritten
  // forms whose CE canonicalizations can re-create each other — fall
  // back to the literal principal root rather than overflow
  if (rtAuxDepth > 40) {
    if (process.env.RUBI_DEBUG_RT)
      console.error(`rtAux depth cap: ${u.toString().slice(0, 120)}`);