Signature:

Type type_system::getOutputType(Type input, const Expression* expr);
Type type_system::narrowType(Type input, const Expression* expr, bool assumeTrue);

This should dispatch via a visitor similar to matcher::narrowType.

So that we can call getOutputType(ExpressionConstant, ...) or getOutputType(ExpressionFieldPath, ...) and get the output type of the expression.
And narrowType on ExpressionCompare etc.

Make $expr call narrowType when the output of the expression is boolean, to wire things together. We  should check the expression is boolean to match cases $expr: {$eq: [...]} but not $expr: {$add: [...]}} which is valid, but requires the type system to model the thruthy check (which we will not do).

Separate tickets will provide implementations.