simple-squiggle

simplify.js (37181B)

---

 import { isConstantNode, isParenthesisNode } from '../../utils/is.js';
 import { factory } from '../../utils/factory.js';
 import { createUtil } from './simplify/util.js';
 import { createSimplifyConstant } from './simplify/simplifyConstant.js';
 import { hasOwnProperty } from '../../utils/object.js';
 import { createEmptyMap, createMap } from '../../utils/map.js';
 var name = 'simplify';
 var dependencies = ['config', 'typed', 'parse', 'add', 'subtract', 'multiply', 'divide', 'pow', 'isZero', 'equal', 'resolve', 'simplifyCore', '?fraction', '?bignumber', 'mathWithTransform', 'matrix', 'AccessorNode', 'ArrayNode', 'ConstantNode', 'FunctionNode', 'IndexNode', 'ObjectNode', 'OperatorNode', 'ParenthesisNode', 'SymbolNode'];
 export var createSimplify = /* #__PURE__ */factory(name, dependencies, _ref => {
   var {
     config,
     typed,
     parse,
     add,
     subtract,
     multiply,
     divide,
     pow,
     isZero,
     equal,
     resolve,
     simplifyCore,
     fraction,
     bignumber,
     mathWithTransform,
     matrix,
     AccessorNode,
     ArrayNode,
     ConstantNode,
     FunctionNode,
     IndexNode,
     ObjectNode,
     OperatorNode,
     ParenthesisNode,
     SymbolNode
   } = _ref;
   var simplifyConstant = createSimplifyConstant({
     typed,
     config,
     mathWithTransform,
     matrix,
     fraction,
     bignumber,
     AccessorNode,
     ArrayNode,
     ConstantNode,
     FunctionNode,
     IndexNode,
     ObjectNode,
     OperatorNode,
     SymbolNode
   });
   var {
     hasProperty,
     isCommutative,
     isAssociative,
     mergeContext,
     flatten,
     unflattenr,
     unflattenl,
     createMakeNodeFunction,
     defaultContext,
     realContext,
     positiveContext
   } = createUtil({
     FunctionNode,
     OperatorNode,
     SymbolNode
   });
   /**
    * Simplify an expression tree.
    *
    * A list of rules are applied to an expression, repeating over the list until
    * no further changes are made.
    * It's possible to pass a custom set of rules to the function as second
    * argument. A rule can be specified as an object, string, or function:
    *
    *     const rules = [
    *       { l: 'n1*n3 + n2*n3', r: '(n1+n2)*n3' },
    *       'n1*n3 + n2*n3 -> (n1+n2)*n3',
    *       function (node) {
    *         // ... return a new node or return the node unchanged
    *         return node
    *       }
    *     ]
    *
    * String and object rules consist of a left and right pattern. The left is
    * used to match against the expression and the right determines what matches
    * are replaced with. The main difference between a pattern and a normal
    * expression is that variables starting with the following characters are
    * interpreted as wildcards:
    *
    * - 'n' - matches any Node
    * - 'c' - matches any ConstantNode
    * - 'v' - matches any Node that is not a ConstantNode
    *
    * The default list of rules is exposed on the function as `simplify.rules`
    * and can be used as a basis to built a set of custom rules.
    *
    * To specify a rule as a string, separate the left and right pattern by '->'
    * When specifying a rule as an object, the following keys are meaningful:
    * - l - the left pattern
    * - r - the right pattern
    * - s - in lieu of l and r, the string form that is broken at -> to give them
    * - repeat - whether to repeat this rule until the expression stabilizes
    * - assuming - gives a context object, as in the 'context' option to
    *     simplify. Every property in the context object must match the current
    *     context in order, or else the rule will not be applied.
    * - imposeContext - gives a context object, as in the 'context' option to
    *     simplify. Any settings specified will override the incoming context
    *     for all matches of this rule.
    *
    * For more details on the theory, see:
    *
    * - [Strategies for simplifying math expressions (Stackoverflow)](https://stackoverflow.com/questions/7540227/strategies-for-simplifying-math-expressions)
    * - [Symbolic computation - Simplification (Wikipedia)](https://en.wikipedia.org/wiki/Symbolic_computation#Simplification)
    *
    *  An optional `options` argument can be passed as last argument of `simplify`.
    *  Currently available options (defaults in parentheses):
    *  - `consoleDebug` (false): whether to write the expression being simplified
    *    and any changes to it, along with the rule responsible, to console
    *  - `context` (simplify.defaultContext): an object giving properties of
    *    each operator, which determine what simplifications are allowed. The
    *    currently meaningful properties are commutative, associative,
    *    total (whether the operation is defined for all arguments), and
    *    trivial (whether the operation applied to a single argument leaves
    *    that argument unchanged). The default context is very permissive and
    *    allows almost all simplifications. Only properties differing from
    *    the default need to be specified; the default context is used as a
    *    fallback. Additional contexts `simplify.realContext` and
    *    `simplify.positiveContext` are supplied to cause simplify to perform
    *    just simplifications guaranteed to preserve all values of the expression
    *    assuming all variables and subexpressions are real numbers or
    *    positive real numbers, respectively. (Note that these are in some cases
    *    more restrictive than the default context; for example, the default
    *    context will allow `x/x` to simplify to 1, whereas
    *    `simplify.realContext` will not, as `0/0` is not equal to 1.)
    *  - `exactFractions` (true): whether to try to convert all constants to
    *    exact rational numbers.
    *  - `fractionsLimit` (10000): when `exactFractions` is true, constants will
    *    be expressed as fractions only when both numerator and denominator
    *    are smaller than `fractionsLimit`.
    *
    * Syntax:
    *
    *     simplify(expr)
    *     simplify(expr, rules)
    *     simplify(expr, rules)
    *     simplify(expr, rules, scope)
    *     simplify(expr, rules, scope, options)
    *     simplify(expr, scope)
    *     simplify(expr, scope, options)
    *
    * Examples:
    *
    *     math.simplify('2 * 1 * x ^ (2 - 1)')      // Node "2 * x"
    *     math.simplify('2 * 3 * x', {x: 4})        // Node "24"
    *     const f = math.parse('2 * 1 * x ^ (2 - 1)')
    *     math.simplify(f)                          // Node "2 * x"
    *     math.simplify('0.4 * x', {}, {exactFractions: true})  // Node "x * 2 / 5"
    *     math.simplify('0.4 * x', {}, {exactFractions: false}) // Node "0.4 * x"
    *
    * See also:
    *
    *     simplifyCore, derivative, evaluate, parse, rationalize, resolve
    *
    * @param {Node | string} expr
    *            The expression to be simplified
    * @param {Array<{l:string, r: string} | string | function>} [rules]
    *            Optional list with custom rules
    * @return {Node} Returns the simplified form of `expr`
    */
 
   var simplify = typed('simplify', {
     string: function string(expr) {
       return this(parse(expr), this.rules, createEmptyMap(), {});
     },
     'string, Map | Object': function stringMapObject(expr, scope) {
       return this(parse(expr), this.rules, scope, {});
     },
     'string, Map | Object, Object': function stringMapObjectObject(expr, scope, options) {
       return this(parse(expr), this.rules, scope, options);
     },
     'string, Array': function stringArray(expr, rules) {
       return this(parse(expr), rules, createEmptyMap(), {});
     },
     'string, Array, Map | Object': function stringArrayMapObject(expr, rules, scope) {
       return this(parse(expr), rules, scope, {});
     },
     'string, Array, Map | Object, Object': function stringArrayMapObjectObject(expr, rules, scope, options) {
       return this(parse(expr), rules, scope, options);
     },
     'Node, Map | Object': function NodeMapObject(expr, scope) {
       return this(expr, this.rules, scope, {});
     },
     'Node, Map | Object, Object': function NodeMapObjectObject(expr, scope, options) {
       return this(expr, this.rules, scope, options);
     },
     Node: function Node(expr) {
       return this(expr, this.rules, createEmptyMap(), {});
     },
     'Node, Array': function NodeArray(expr, rules) {
       return this(expr, rules, createEmptyMap(), {});
     },
     'Node, Array, Map | Object': function NodeArrayMapObject(expr, rules, scope) {
       return this(expr, rules, scope, {});
     },
     'Node, Array, Object, Object': function NodeArrayObjectObject(expr, rules, scope, options) {
       return this(expr, rules, createMap(scope), options);
     },
     'Node, Array, Map, Object': function NodeArrayMapObject(expr, rules, scope, options) {
       var debug = options.consoleDebug;
       rules = _buildRules(rules, options.context);
       var res = resolve(expr, scope);
       res = removeParens(res);
       var visited = {};
       var str = res.toString({
         parenthesis: 'all'
       });
 
       while (!visited[str]) {
         visited[str] = true;
         _lastsym = 0; // counter for placeholder symbols
 
         var laststr = str;
         if (debug) console.log('Working on: ', str);
 
         for (var i = 0; i < rules.length; i++) {
           var rulestr = '';
 
           if (typeof rules[i] === 'function') {
             res = rules[i](res, options);
             if (debug) rulestr = rules[i].name;
           } else {
             flatten(res, options.context);
             res = applyRule(res, rules[i], options.context);
 
             if (debug) {
               rulestr = "".concat(rules[i].l.toString(), " -> ").concat(rules[i].r.toString());
             }
           }
 
           if (debug) {
             var newstr = res.toString({
               parenthesis: 'all'
             });
 
             if (newstr !== laststr) {
               console.log('Applying', rulestr, 'produced', newstr);
               laststr = newstr;
             }
           }
           /* Use left-heavy binary tree internally,
            * since custom rule functions may expect it
            */
 
 
           unflattenl(res, options.context);
         }
 
         str = res.toString({
           parenthesis: 'all'
         });
       }
 
       return res;
     }
   });
   simplify.defaultContext = defaultContext;
   simplify.realContext = realContext;
   simplify.positiveContext = positiveContext;
 
   function removeParens(node) {
     return node.transform(function (node, path, parent) {
       return isParenthesisNode(node) ? removeParens(node.content) : node;
     });
   } // All constants that are allowed in rules
 
 
   var SUPPORTED_CONSTANTS = {
     true: true,
     false: true,
     e: true,
     i: true,
     Infinity: true,
     LN2: true,
     LN10: true,
     LOG2E: true,
     LOG10E: true,
     NaN: true,
     phi: true,
     pi: true,
     SQRT1_2: true,
     SQRT2: true,
     tau: true // null: false,
     // undefined: false,
     // version: false,
 
   }; // Array of strings, used to build the ruleSet.
   // Each l (left side) and r (right side) are parsed by
   // the expression parser into a node tree.
   // Left hand sides are matched to subtrees within the
   // expression to be parsed and replaced with the right
   // hand side.
   // TODO: Add support for constraints on constants (either in the form of a '=' expression or a callback [callback allows things like comparing symbols alphabetically])
   // To evaluate lhs constants for rhs constants, use: { l: 'c1+c2', r: 'c3', evaluate: 'c3 = c1 + c2' }. Multiple assignments are separated by ';' in block format.
   // It is possible to get into an infinite loop with conflicting rules
 
   simplify.rules = [simplifyCore, // { l: 'n+0', r: 'n' },     // simplifyCore
   // { l: 'n^0', r: '1' },     // simplifyCore
   // { l: '0*n', r: '0' },     // simplifyCore
   // { l: 'n/n', r: '1'},      // simplifyCore
   // { l: 'n^1', r: 'n' },     // simplifyCore
   // { l: '+n1', r:'n1' },     // simplifyCore
   // { l: 'n--n1', r:'n+n1' }, // simplifyCore
   {
     l: 'log(e)',
     r: '1'
   }, // temporary rules
   // Note initially we tend constants to the right because like-term
   // collection prefers the left, and we would rather collect nonconstants
   {
     s: 'n-n1 -> n+-n1',
     // temporarily replace 'subtract' so we can further flatten the 'add' operator
     assuming: {
       subtract: {
         total: true
       }
     }
   }, {
     s: 'n-n -> 0',
     // partial alternative when we can't always subtract
     assuming: {
       subtract: {
         total: false
       }
     }
   }, {
     s: '-(c*v) -> v * (-c)',
     // make non-constant terms positive
     assuming: {
       multiply: {
         commutative: true
       },
       subtract: {
         total: true
       }
     }
   }, {
     s: '-(c*v) -> (-c) * v',
     // non-commutative version, part 1
     assuming: {
       multiply: {
         commutative: false
       },
       subtract: {
         total: true
       }
     }
   }, {
     s: '-(v*c) -> v * (-c)',
     // non-commutative version, part 2
     assuming: {
       multiply: {
         commutative: false
       },
       subtract: {
         total: true
       }
     }
   }, {
     l: '-(n1/n2)',
     r: '-n1/n2'
   }, {
     l: '-v',
     r: 'v * (-1)'
   }, // finish making non-constant terms positive
   {
     l: '(n1 + n2)*(-1)',
     r: 'n1*(-1) + n2*(-1)',
     repeat: true
   }, // expand negations to achieve as much sign cancellation as possible
   {
     l: 'n/n1^n2',
     r: 'n*n1^-n2'
   }, // temporarily replace 'divide' so we can further flatten the 'multiply' operator
   {
     l: 'n/n1',
     r: 'n*n1^-1'
   }, {
     s: '(n1*n2)^n3 -> n1^n3 * n2^n3',
     assuming: {
       multiply: {
         commutative: true
       }
     }
   }, {
     s: '(n1*n2)^(-1) -> n2^(-1) * n1^(-1)',
     assuming: {
       multiply: {
         commutative: false
       }
     }
   }, // expand nested exponentiation
   {
     s: '(n ^ n1) ^ n2 -> n ^ (n1 * n2)',
     assuming: {
       divide: {
         total: true
       }
     } // 1/(1/n) = n needs 1/n to exist
 
   }, // collect like factors; into a sum, only do this for nonconstants
   {
     l: ' v   * ( v   * n1 + n2)',
     r: 'v^2       * n1 +  v   * n2'
   }, {
     s: ' v   * (v^n4 * n1 + n2)   ->  v^(1+n4)  * n1 +  v   * n2',
     assuming: {
       divide: {
         total: true
       }
     } // v*1/v = v^(1+-1) needs 1/v
 
   }, {
     s: 'v^n3 * ( v   * n1 + n2)   ->  v^(n3+1)  * n1 + v^n3 * n2',
     assuming: {
       divide: {
         total: true
       }
     }
   }, {
     s: 'v^n3 * (v^n4 * n1 + n2)   ->  v^(n3+n4) * n1 + v^n3 * n2',
     assuming: {
       divide: {
         total: true
       }
     }
   }, {
     l: 'n*n',
     r: 'n^2'
   }, {
     s: 'n * n^n1 -> n^(n1+1)',
     assuming: {
       divide: {
         total: true
       }
     } // n*1/n = n^(-1+1) needs 1/n
 
   }, {
     s: 'n^n1 * n^n2 -> n^(n1+n2)',
     assuming: {
       divide: {
         total: true
       }
     } // ditto for n^2*1/n^2
 
   }, // Unfortunately, to deal with more complicated cancellations, it
   // becomes necessary to simplify constants twice per pass. It's not
   // terribly expensive compared to matching rules, so this should not
   // pose a performance problem.
   simplifyConstant, // First: before collecting like terms
   // collect like terms
   {
     s: 'n+n -> 2*n',
     assuming: {
       add: {
         total: true
       }
     } // 2 = 1 + 1 needs to exist
 
   }, {
     l: 'n+-n',
     r: '0'
   }, {
     l: 'v*n + v',
     r: 'v*(n+1)'
   }, // NOTE: leftmost position is special:
   {
     l: 'n3*n1 + n3*n2',
     r: 'n3*(n1+n2)'
   }, // All sub-monomials tried there.
   {
     l: 'n3^(-n4)*n1 +   n3  * n2',
     r: 'n3^(-n4)*(n1 + n3^(n4+1) *n2)'
   }, {
     l: 'n3^(-n4)*n1 + n3^n5 * n2',
     r: 'n3^(-n4)*(n1 + n3^(n4+n5)*n2)'
   }, {
     s: 'n*v + v -> (n+1)*v',
     // noncommutative additional cases
     assuming: {
       multiply: {
         commutative: false
       }
     }
   }, {
     s: 'n1*n3 + n2*n3 -> (n1+n2)*n3',
     assuming: {
       multiply: {
         commutative: false
       }
     }
   }, {
     s: 'n1*n3^(-n4) + n2 * n3    -> (n1 + n2*n3^(n4 +  1))*n3^(-n4)',
     assuming: {
       multiply: {
         commutative: false
       }
     }
   }, {
     s: 'n1*n3^(-n4) + n2 * n3^n5 -> (n1 + n2*n3^(n4 + n5))*n3^(-n4)',
     assuming: {
       multiply: {
         commutative: false
       }
     }
   }, {
     l: 'n*c + c',
     r: '(n+1)*c'
   }, {
     s: 'c*n + c -> c*(n+1)',
     assuming: {
       multiply: {
         commutative: false
       }
     }
   }, simplifyConstant, // Second: before returning expressions to "standard form"
   // make factors positive (and undo 'make non-constant terms positive')
   {
     s: '(-n)*n1 -> -(n*n1)',
     assuming: {
       subtract: {
         total: true
       }
     }
   }, {
     s: 'n1*(-n) -> -(n1*n)',
     // in case * non-commutative
     assuming: {
       subtract: {
         total: true
       },
       multiply: {
         commutative: false
       }
     }
   }, // final ordering of constants
   {
     s: 'c+v -> v+c',
     assuming: {
       add: {
         commutative: true
       }
     },
     imposeContext: {
       add: {
         commutative: false
       }
     }
   }, {
     s: 'v*c -> c*v',
     assuming: {
       multiply: {
         commutative: true
       }
     },
     imposeContext: {
       multiply: {
         commutative: false
       }
     }
   }, // undo temporary rules
   // { l: '(-1) * n', r: '-n' }, // #811 added test which proved this is redundant
   {
     l: 'n+-n1',
     r: 'n-n1'
   }, // undo replace 'subtract'
   {
     s: 'n*(n1^-1) -> n/n1',
     // undo replace 'divide'; for * commutative
     assuming: {
       multiply: {
         commutative: true
       }
     } // o.w. / not conventional
 
   }, {
     s: 'n*n1^-n2 -> n/n1^n2',
     assuming: {
       multiply: {
         commutative: true
       }
     } // o.w. / not conventional
 
   }, {
     s: 'n^-1 -> 1/n',
     assuming: {
       multiply: {
         commutative: true
       }
     } // o.w. / not conventional
 
   }, {
     l: 'n^1',
     r: 'n'
   }, // can be produced by power cancellation
   {
     s: 'n*(n1/n2) -> (n*n1)/n2',
     // '*' before '/'
     assuming: {
       multiply: {
         associative: true
       }
     }
   }, {
     s: 'n-(n1+n2) -> n-n1-n2',
     // '-' before '+'
     assuming: {
       addition: {
         associative: true,
         commutative: true
       }
     }
   }, // { l: '(n1/n2)/n3', r: 'n1/(n2*n3)' },
   // { l: '(n*n1)/(n*n2)', r: 'n1/n2' },
   // simplifyConstant can leave an extra factor of 1, which can always
   // be eliminated, since the identity always commutes
   {
     l: '1*n',
     r: 'n',
     imposeContext: {
       multiply: {
         commutative: true
       }
     }
   }, {
     s: 'n1/(n2/n3) -> (n1*n3)/n2',
     assuming: {
       multiply: {
         associative: true
       }
     }
   }, {
     l: 'n1/(-n2)',
     r: '-n1/n2'
   }];
   /**
    * Takes any rule object as allowed by the specification in simplify
    * and puts it in a standard form used by applyRule
    */
 
   function _canonicalizeRule(ruleObject, context) {
     var newRule = {};
 
     if (ruleObject.s) {
       var lr = ruleObject.s.split('->');
 
       if (lr.length === 2) {
         newRule.l = lr[0];
         newRule.r = lr[1];
       } else {
         throw SyntaxError('Could not parse rule: ' + ruleObject.s);
       }
     } else {
       newRule.l = ruleObject.l;
       newRule.r = ruleObject.r;
     }
 
     newRule.l = removeParens(parse(newRule.l));
     newRule.r = removeParens(parse(newRule.r));
 
     for (var prop of ['imposeContext', 'repeat', 'assuming']) {
       if (prop in ruleObject) {
         newRule[prop] = ruleObject[prop];
       }
     }
 
     if (ruleObject.evaluate) {
       newRule.evaluate = parse(ruleObject.evaluate);
     }
 
     if (isAssociative(newRule.l, context)) {
       var makeNode = createMakeNodeFunction(newRule.l);
 
       var expandsym = _getExpandPlaceholderSymbol();
 
       newRule.expanded = {};
       newRule.expanded.l = makeNode([newRule.l.clone(), expandsym]); // Push the expandsym into the deepest possible branch.
       // This helps to match the newRule against nodes returned from getSplits() later on.
 
       flatten(newRule.expanded.l, context);
       unflattenr(newRule.expanded.l, context);
       newRule.expanded.r = makeNode([newRule.r, expandsym]);
     }
 
     return newRule;
   }
   /**
    * Parse the string array of rules into nodes
    *
    * Example syntax for rules:
    *
    * Position constants to the left in a product:
    * { l: 'n1 * c1', r: 'c1 * n1' }
    * n1 is any Node, and c1 is a ConstantNode.
    *
    * Apply difference of squares formula:
    * { l: '(n1 - n2) * (n1 + n2)', r: 'n1^2 - n2^2' }
    * n1, n2 mean any Node.
    *
    * Short hand notation:
    * 'n1 * c1 -> c1 * n1'
    */
 
 
   function _buildRules(rules, context) {
     // Array of rules to be used to simplify expressions
     var ruleSet = [];
 
     for (var i = 0; i < rules.length; i++) {
       var rule = rules[i];
       var newRule = void 0;
       var ruleType = typeof rule;
 
       switch (ruleType) {
         case 'string':
           rule = {
             s: rule
           };
 
         /* falls through */
 
         case 'object':
           newRule = _canonicalizeRule(rule, context);
           break;
 
         case 'function':
           newRule = rule;
           break;
 
         default:
           throw TypeError('Unsupported type of rule: ' + ruleType);
       } // console.log('Adding rule: ' + rules[i])
       // console.log(newRule)
 
 
       ruleSet.push(newRule);
     }
 
     return ruleSet;
   }
 
   var _lastsym = 0;
 
   function _getExpandPlaceholderSymbol() {
     return new SymbolNode('_p' + _lastsym++);
   }
 
   function mapRule(nodes, rule, context) {
     var resNodes = nodes;
 
     if (nodes) {
       for (var i = 0; i < nodes.length; ++i) {
         var newNode = applyRule(nodes[i], rule, context);
 
         if (newNode !== nodes[i]) {
           if (resNodes === nodes) {
             resNodes = nodes.slice();
           }
 
           resNodes[i] = newNode;
         }
       }
     }
 
     return resNodes;
   }
   /**
    * Returns a simplfied form of node, or the original node if no simplification was possible.
    *
    * @param  {ConstantNode | SymbolNode | ParenthesisNode | FunctionNode | OperatorNode} node
    * @param  {Object | Function} rule
    * @param  {Object} context -- information about assumed properties of operators
    * @return {ConstantNode | SymbolNode | ParenthesisNode | FunctionNode | OperatorNode} The simplified form of `expr`, or the original node if no simplification was possible.
    */
 
 
   function applyRule(node, rule, context) {
     //    console.log('Entering applyRule("', rule.l.toString({parenthesis:'all'}), '->', rule.r.toString({parenthesis:'all'}), '",', node.toString({parenthesis:'all'}),')')
     // check that the assumptions for this rule are satisfied by the current
     // context:
     if (rule.assuming) {
       for (var symbol in rule.assuming) {
         for (var property in rule.assuming[symbol]) {
           if (hasProperty(symbol, property, context) !== rule.assuming[symbol][property]) {
             return node;
           }
         }
       }
     }
 
     var mergedContext = mergeContext(rule.imposeContext, context); // Do not clone node unless we find a match
 
     var res = node; // First replace our child nodes with their simplified versions
     // If a child could not be simplified, applying the rule to it
     // will have no effect since the node is returned unchanged
 
     if (res instanceof OperatorNode || res instanceof FunctionNode) {
       var newArgs = mapRule(res.args, rule, context);
 
       if (newArgs !== res.args) {
         res = res.clone();
         res.args = newArgs;
       }
     } else if (res instanceof ParenthesisNode) {
       if (res.content) {
         var newContent = applyRule(res.content, rule, context);
 
         if (newContent !== res.content) {
           res = new ParenthesisNode(newContent);
         }
       }
     } else if (res instanceof ArrayNode) {
       var newItems = mapRule(res.items, rule, context);
 
       if (newItems !== res.items) {
         res = new ArrayNode(newItems);
       }
     } else if (res instanceof AccessorNode) {
       var newObj = res.object;
 
       if (res.object) {
         newObj = applyRule(res.object, rule, context);
       }
 
       var newIndex = res.index;
 
       if (res.index) {
         newIndex = applyRule(res.index, rule, context);
       }
 
       if (newObj !== res.object || newIndex !== res.index) {
         res = new AccessorNode(newObj, newIndex);
       }
     } else if (res instanceof IndexNode) {
       var newDims = mapRule(res.dimensions, rule, context);
 
       if (newDims !== res.dimensions) {
         res = new IndexNode(newDims);
       }
     } else if (res instanceof ObjectNode) {
       var changed = false;
       var newProps = {};
 
       for (var prop in res.properties) {
         newProps[prop] = applyRule(res.properties[prop], rule, context);
 
         if (newProps[prop] !== res.properties[prop]) {
           changed = true;
         }
       }
 
       if (changed) {
         res = new ObjectNode(newProps);
       }
     } // Try to match a rule against this node
 
 
     var repl = rule.r;
 
     var matches = _ruleMatch(rule.l, res, mergedContext)[0]; // If the rule is associative operator, we can try matching it while allowing additional terms.
     // This allows us to match rules like 'n+n' to the expression '(1+x)+x' or even 'x+1+x' if the operator is commutative.
 
 
     if (!matches && rule.expanded) {
       repl = rule.expanded.r;
       matches = _ruleMatch(rule.expanded.l, res, mergedContext)[0];
     }
 
     if (matches) {
       // const before = res.toString({parenthesis: 'all'})
       // Create a new node by cloning the rhs of the matched rule
       // we keep any implicit multiplication state if relevant
       var implicit = res.implicit;
       res = repl.clone();
 
       if (implicit && 'implicit' in repl) {
         res.implicit = true;
       } // Replace placeholders with their respective nodes without traversing deeper into the replaced nodes
 
 
       res = res.transform(function (node) {
         if (node.isSymbolNode && hasOwnProperty(matches.placeholders, node.name)) {
           return matches.placeholders[node.name].clone();
         } else {
           return node;
         }
       }); // const after = res.toString({parenthesis: 'all'})
       // console.log('Simplified ' + before + ' to ' + after)
     }
 
     if (rule.repeat && res !== node) {
       res = applyRule(res, rule, context);
     }
 
     return res;
   }
   /**
    * Get (binary) combinations of a flattened binary node
    * e.g. +(node1, node2, node3) -> [
    *        +(node1,  +(node2, node3)),
    *        +(node2,  +(node1, node3)),
    *        +(node3,  +(node1, node2))]
    *
    */
 
 
   function getSplits(node, context) {
     var res = [];
     var right, rightArgs;
     var makeNode = createMakeNodeFunction(node);
 
     if (isCommutative(node, context)) {
       for (var i = 0; i < node.args.length; i++) {
         rightArgs = node.args.slice(0);
         rightArgs.splice(i, 1);
         right = rightArgs.length === 1 ? rightArgs[0] : makeNode(rightArgs);
         res.push(makeNode([node.args[i], right]));
       }
     } else {
       // Keep order, but try all parenthesizations
       for (var _i = 1; _i < node.args.length; _i++) {
         var left = node.args[0];
 
         if (_i > 1) {
           left = makeNode(node.args.slice(0, _i));
         }
 
         rightArgs = node.args.slice(_i);
         right = rightArgs.length === 1 ? rightArgs[0] : makeNode(rightArgs);
         res.push(makeNode([left, right]));
       }
     }
 
     return res;
   }
   /**
    * Returns the set union of two match-placeholders or null if there is a conflict.
    */
 
 
   function mergeMatch(match1, match2) {
     var res = {
       placeholders: {}
     }; // Some matches may not have placeholders; this is OK
 
     if (!match1.placeholders && !match2.placeholders) {
       return res;
     } else if (!match1.placeholders) {
       return match2;
     } else if (!match2.placeholders) {
       return match1;
     } // Placeholders with the same key must match exactly
 
 
     for (var key in match1.placeholders) {
       if (hasOwnProperty(match1.placeholders, key)) {
         res.placeholders[key] = match1.placeholders[key];
 
         if (hasOwnProperty(match2.placeholders, key)) {
           if (!_exactMatch(match1.placeholders[key], match2.placeholders[key])) {
             return null;
           }
         }
       }
     }
 
     for (var _key in match2.placeholders) {
       if (hasOwnProperty(match2.placeholders, _key)) {
         res.placeholders[_key] = match2.placeholders[_key];
       }
     }
 
     return res;
   }
   /**
    * Combine two lists of matches by applying mergeMatch to the cartesian product of two lists of matches.
    * Each list represents matches found in one child of a node.
    */
 
 
   function combineChildMatches(list1, list2) {
     var res = [];
 
     if (list1.length === 0 || list2.length === 0) {
       return res;
     }
 
     var merged;
 
     for (var i1 = 0; i1 < list1.length; i1++) {
       for (var i2 = 0; i2 < list2.length; i2++) {
         merged = mergeMatch(list1[i1], list2[i2]);
 
         if (merged) {
           res.push(merged);
         }
       }
     }
 
     return res;
   }
   /**
    * Combine multiple lists of matches by applying mergeMatch to the cartesian product of two lists of matches.
    * Each list represents matches found in one child of a node.
    * Returns a list of unique matches.
    */
 
 
   function mergeChildMatches(childMatches) {
     if (childMatches.length === 0) {
       return childMatches;
     }
 
     var sets = childMatches.reduce(combineChildMatches);
     var uniqueSets = [];
     var unique = {};
 
     for (var i = 0; i < sets.length; i++) {
       var s = JSON.stringify(sets[i]);
 
       if (!unique[s]) {
         unique[s] = true;
         uniqueSets.push(sets[i]);
       }
     }
 
     return uniqueSets;
   }
   /**
    * Determines whether node matches rule.
    *
    * @param {ConstantNode | SymbolNode | ParenthesisNode | FunctionNode | OperatorNode} rule
    * @param {ConstantNode | SymbolNode | ParenthesisNode | FunctionNode | OperatorNode} node
    * @param {Object} context -- provides assumed properties of operators
    * @param {Boolean} isSplit -- whether we are in process of splitting an
    *                    n-ary operator node into possible binary combinations.
    *                    Defaults to false.
    * @return {Object} Information about the match, if it exists.
    */
 
 
   function _ruleMatch(rule, node, context, isSplit) {
     //    console.log('Entering _ruleMatch(' + JSON.stringify(rule) + ', ' + JSON.stringify(node) + ')')
     //    console.log('rule = ' + rule)
     //    console.log('node = ' + node)
     //    console.log('Entering _ruleMatch(', rule.toString({parenthesis:'all'}), ', ', node.toString({parenthesis:'all'}), ', ', context, ')')
     var res = [{
       placeholders: {}
     }];
 
     if (rule instanceof OperatorNode && node instanceof OperatorNode || rule instanceof FunctionNode && node instanceof FunctionNode) {
       // If the rule is an OperatorNode or a FunctionNode, then node must match exactly
       if (rule instanceof OperatorNode) {
         if (rule.op !== node.op || rule.fn !== node.fn) {
           return [];
         }
       } else if (rule instanceof FunctionNode) {
         if (rule.name !== node.name) {
           return [];
         }
       } // rule and node match. Search the children of rule and node.
 
 
       if (node.args.length === 1 && rule.args.length === 1 || !isAssociative(node, context) && node.args.length === rule.args.length || isSplit) {
         // Expect non-associative operators to match exactly,
         // except in any order if operator is commutative
         var childMatches = [];
 
         for (var i = 0; i < rule.args.length; i++) {
           var childMatch = _ruleMatch(rule.args[i], node.args[i], context);
 
           if (childMatch.length === 0) {
             // Child did not match, so stop searching immediately
             break;
           } // The child matched, so add the information returned from the child to our result
 
 
           childMatches.push(childMatch);
         }
 
         if (childMatches.length !== rule.args.length) {
           if (!isCommutative(node, context) || // exact match in order needed
           rule.args.length === 1) {
             // nothing to commute
             return [];
           }
 
           if (rule.args.length > 2) {
             /* Need to generate all permutations and try them.
              * It's a bit complicated, and unlikely to come up since there
              * are very few ternary or higher operators. So punt for now.
              */
             throw new Error('permuting >2 commutative non-associative rule arguments not yet implemented');
           }
           /* Exactly two arguments, try them reversed */
 
 
           var leftMatch = _ruleMatch(rule.args[0], node.args[1], context);
 
           if (leftMatch.length === 0) {
             return [];
           }
 
           var rightMatch = _ruleMatch(rule.args[1], node.args[0], context);
 
           if (rightMatch.length === 0) {
             return [];
           }
 
           childMatches = [leftMatch, rightMatch];
         }
 
         res = mergeChildMatches(childMatches);
       } else if (node.args.length >= 2 && rule.args.length === 2) {
         // node is flattened, rule is not
         // Associative operators/functions can be split in different ways so we check if the rule matches each
         // them and return their union.
         var splits = getSplits(node, context);
         var splitMatches = [];
 
         for (var _i2 = 0; _i2 < splits.length; _i2++) {
           var matchSet = _ruleMatch(rule, splits[_i2], context, true); // recursing at the same tree depth here
 
 
           splitMatches = splitMatches.concat(matchSet);
         }
 
         return splitMatches;
       } else if (rule.args.length > 2) {
         throw Error('Unexpected non-binary associative function: ' + rule.toString());
       } else {
         // Incorrect number of arguments in rule and node, so no match
         return [];
       }
     } else if (rule instanceof SymbolNode) {
       // If the rule is a SymbolNode, then it carries a special meaning
       // according to the first character of the symbol node name.
       // c.* matches a ConstantNode
       // n.* matches any node
       if (rule.name.length === 0) {
         throw new Error('Symbol in rule has 0 length...!?');
       }
 
       if (SUPPORTED_CONSTANTS[rule.name]) {
         // built-in constant must match exactly
         if (rule.name !== node.name) {
           return [];
         }
       } else if (rule.name[0] === 'n' || rule.name.substring(0, 2) === '_p') {
         // rule matches _anything_, so assign this node to the rule.name placeholder
         // Assign node to the rule.name placeholder.
         // Our parent will check for matches among placeholders.
         res[0].placeholders[rule.name] = node;
       } else if (rule.name[0] === 'v') {
         // rule matches any variable thing (not a ConstantNode)
         if (!isConstantNode(node)) {
           res[0].placeholders[rule.name] = node;
         } else {
           // Mis-match: rule was expecting something other than a ConstantNode
           return [];
         }
       } else if (rule.name[0] === 'c') {
         // rule matches any ConstantNode
         if (node instanceof ConstantNode) {
           res[0].placeholders[rule.name] = node;
         } else {
           // Mis-match: rule was expecting a ConstantNode
           return [];
         }
       } else {
         throw new Error('Invalid symbol in rule: ' + rule.name);
       }
     } else if (rule instanceof ConstantNode) {
       // Literal constant must match exactly
       if (!equal(rule.value, node.value)) {
         return [];
       }
     } else {
       // Some other node was encountered which we aren't prepared for, so no match
       return [];
     } // It's a match!
     // console.log('_ruleMatch(' + rule.toString() + ', ' + node.toString() + ') found a match')
 
 
     return res;
   }
   /**
    * Determines whether p and q (and all their children nodes) are identical.
    *
    * @param {ConstantNode | SymbolNode | ParenthesisNode | FunctionNode | OperatorNode} p
    * @param {ConstantNode | SymbolNode | ParenthesisNode | FunctionNode | OperatorNode} q
    * @return {Object} Information about the match, if it exists.
    */
 
 
   function _exactMatch(p, q) {
     if (p instanceof ConstantNode && q instanceof ConstantNode) {
       if (!equal(p.value, q.value)) {
         return false;
       }
     } else if (p instanceof SymbolNode && q instanceof SymbolNode) {
       if (p.name !== q.name) {
         return false;
       }
     } else if (p instanceof OperatorNode && q instanceof OperatorNode || p instanceof FunctionNode && q instanceof FunctionNode) {
       if (p instanceof OperatorNode) {
         if (p.op !== q.op || p.fn !== q.fn) {
           return false;
         }
       } else if (p instanceof FunctionNode) {
         if (p.name !== q.name) {
           return false;
         }
       }
 
       if (p.args.length !== q.args.length) {
         return false;
       }
 
       for (var i = 0; i < p.args.length; i++) {
         if (!_exactMatch(p.args[i], q.args[i])) {
           return false;
         }
       }
     } else {
       return false;
     }
 
     return true;
   }
 
   return simplify;
 });