Understanding Can-Filling in Proof Theory and Type Theory

Can-filling is a technique used in proof theory and type theory to establish the existence of certain objects, such as functions or types, by constructing them directly from given assumptions. The name "can-filling" comes from the idea of filling in a "can" or a container with a specific content, where the content is determined by the assumptions made about the object being constructed.

In more detail, can-filling is a method for proving the existence of an object by showing that it can be constructed from existing objects, using a set of rules or axioms that define how the objects can be combined. The object being constructed is often called the "target" or "goal" object, and the existing objects are called "inputs" or "premises".

For example, in type theory, can-filling can be used to prove the existence of a function that takes one type as input and returns another type as output, by showing that it can be constructed from given types using the rules of type inference. Similarly, in proof theory, can-filling can be used to prove the validity of a statement by showing that it can be derived from a set of axioms and rules of inference.

Can-filling is a powerful technique for establishing the existence of objects in various contexts, and it has found applications in a wide range of fields, including mathematics, computer science, and philosophy.