import { type Equation, type Expr } from "../expr.js";
import { type BranchingRule, type Rule } from "../rule.js";
type NoParams = Record<string, never>;
/**
 * Undo a square: a² = b branches into a = √b and a = −√b. Sound because a
 * true a² = b forces b to be a perfect square of a rational, where √b is
 * exact; where b is negative the original is false and the branches are
 * undefined — nothing is claimed. Equalities only; gesture: tap the square.
 */
export declare const sqrtBothSides: BranchingRule<NoParams>;
/**
 * The zero-product property: a·b·… = 0 branches into a = 0, b = 0, …
 * (one branch per factor; rationals form an integral domain). Gesture: tap
 * the product when the other side is zero.
 */
export declare const zeroProduct: BranchingRule<NoParams>;
/** Tap a square root to simplify it: evaluate a perfect square (√9 → 3) or pull
 *  out its largest square factor (√8 → 2√2). Integer radicands only; a
 *  square-free radicand (√7) is already simplest and a negative one has no real
 *  value — neither offers the move. */
export declare const simplifySqrt: Rule<NoParams>;
/** Read `a·x² + b·x + c = 0` (one side literal 0, the other an expanded
 *  single-variable quadratic with INTEGER coefficients, a ≠ 0). Each term must
 *  be a monomial cx⁰/cx/cx² — a product (factored) or fractional coefficient is
 *  not recognized (zero-product / clearing denominators come first). */
export declare function readQuadratic(eqn: Equation): {
    side: Expr;
    v: string;
    a: bigint;
    b: bigint;
    c: bigint;
} | undefined;
/**
 * The quadratic formula: `a·x² + b·x + c = 0` branches into
 * `x = (−b + √D)/(2a)` and `x = (−b − √D)/(2a)`, `D = b² − 4ac`. An exact
 * equivalence over a field (a ≠ 0) — emits nothing. The √D is left UNSIMPLIFIED
 * (a literal radical) so `simplify-sqrt` is the natural follow-up; a perfect
 * square then collapses to rational roots (so this subsumes factoring), and a
 * negative D leaves √(negative) — an undefined point, i.e. "no real solution".
 * Gesture: tap an expanded quadratic that equals zero.
 */
export declare const quadraticFormula: BranchingRule<NoParams>;
export {};
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