Termination checking is a classic static analysis, and, within this focus, there are type-based approaches that formalize termination analysis as type systems (i.e., so that all well-typed programs terminate). But there are situations where a stronger termination property (which we call strongly-bounded termination) must be determined and, accordingly, we explore this property via a variant of the simply-typed λ-calculus called the bounded-time λ-calculus (BTC). This paper presents the BTC and its semantics and metatheory through a Coq formalization. Important examples (e.g., hardware synthesis from functional languages and detection of covert timing channels) motivating strongly-bounded termination and BTC are described as well.

People writing proofs or programs in dependently typed languages can omit some function arguments in order to decrease the code size and improve readability. Type checking such a program involves filling in each of these implicit arguments in a type-correct way. This is typically done using some form of unification.

One approach to unification, taken by Agda, involves sometimes starting to unify terms before their types are known to be equal: in some cases one can make progress on unifying the terms, and then use information gleaned in this way to unify the types. This flexibility allows Agda to solve implicit arguments that are not found by several other systems. However, Agda's implementation is buggy: sometimes the solutions chosen are ill-typed, which can cause the type checker to crash.

With Gundry and McBride's twin variable technique one can also start to unify terms before their types are known to be equal, and furthermore this technique is accompanied by correctness proofs. However, so far this technique has not been tested in practice as part of a full type checker.

We have reformulated Gundry and McBride's technique without twin variables, using only twin types, with the aim of making the technique easier to implement in existing type checkers (in particular Agda). We have also introduced a type-agnostic syntactic equality rule that seems to be useful in practice. The reformulated technique has been tested in a type checker for a tiny variant of Agda. This type checker handles at least one example that Coq, Idris, Lean and Matita cannot handle, and does so in time and space comparable to that used by Agda. This suggests that the reformulated technique is usable in practice.