/**
 * Substitution: the first cross-equation rule for systems. When one equation is
 * in SOLVED form `v = expr` (one side a bare variable not occurring in the
 * other), every occurrence of v in another equation is replaced by expr — the
 * "isolate, then substitute" move. Sound because, under `v = expr`, the target
 * and the substituted target are equivalent, so keeping the source and rewriting
 * the target preserves the system's (intersection) solution set.
 *
 * The same machinery serves calculus u-substitution later. Pure (no DOM);
 * rebuilds through the smart constructors and preserves the ids of v-free
 * subtrees (only paths touching v are rebuilt).
 */
import { type Equation, type Expr } from "./expr.js";
import type { System } from "./system.js";
/** If `eqn` is `v = expr` (or `expr = v`) with v a bare variable absent from
 *  expr, the variable it's solved for and that value; else undefined. */
export declare function solvedVariable(eqn: Equation): {
    variable: string;
    value: Expr;
} | undefined;
/** Substitute the SOURCE's solved variable into TARGET. Returns the rewritten
 *  target (the source's relation is irrelevant to the target's), or undefined
 *  if source isn't in `v = expr` form. */
export declare function substitute(source: Equation, target: Equation): Equation | undefined;
/** Substitute equation `sourceIndex` (in `v = expr` form) into `targetIndex`,
 *  keeping the source, every other equation, and the assumptions unchanged.
 *  Undefined if the indices coincide or the source isn't solved for a variable. */
export declare function substituteInSystem(system: System, sourceIndex: number, targetIndex: number): System | undefined;
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