A Design Structure for Higher Order Quotients

The quotient operation is a standard feature of set theory, where a set is partitioned into subsets by an equivalence relation. We reinterpret this idea for higher order logic, where types are divided by an equivalence relation to create new types, called quotient types. We present a design to mechanically construct quotient types as new types in the logic, and to support the automatic lifting of constants and theorems about the original types to corresponding constants and theorems about the quotient types. This design exceeds the functionality of Harrison’s package, creating quotients of multiple mutually recursive types simultaneously, and supporting the equivalence of aggregate types, such as lists and pairs. Most importantly, this design supports the creation of higher order quotients, which enable the automatic lifting of theorems with quantification over functions of any higher order.