There's an aside about how I've chosen precedence. The way I've defined operator precedence and binding, (\x.x) y is not a valid expression (parentheses only me

There's an aside about how I've chosen precedence. The way I've defined operator precedence and binding, (\x.x) y is not a valid expression (parentheses only mean function application), and (λx.x y) returns y. You can argue I defined things wrong, but it is all in how you define things.