Strictly Monotone Brouwer Trees for Well-founded Recursion Over Multiple Arguments

Abstract

Ordinals can help prove termination for dependently typed programs. Brouwer trees are a particular ordinal notation that make it very easy to assign sizes to higher order data structures. They extend natural numbers with a limit constructor, so a function's size can be the supremum of the sizes of values from its image. These can then be used to define well-founded recursion: any recursive calls are allowed so long as they are on values whose sizes are strictly smaller than the current size. Unfortunately, Brouwer trees are not algebraically well-behaved. They can be characterized equationally as a join-semilattice, where the join takes the maximum of two trees. However, it does not interact well with the successor constructor, so it does not interact properly with the strict ordering used in well-founded recursion. We present Strictly Monotone Brouwer trees (SMB-trees), a refinement of Brouwer trees that are algebraically well-behaved. SMB-trees are built using functions with the same signatures as Brouwer tree constructors, and they satisfy all Brouwer tree inequalities. However, their join operator distributes over the successor, making them suited for well-founded recursion or equational reasoning. We show how, using dependent pairs and careful definitions, an ill-behaved definition can be turned into a well-behaved one, with light axiomatic requirements. We implement a recursively-defined maximum operator for Brouwer trees that matches on successors and handles them specifically. Then, we define SMB-trees as the subset of Brouwer trees for which the recursive maximum computes a least upper bound. Finally, we show that every Brouwer tree can be transformed into a corresponding SMB-tree by joining it with itself an infinite number of times. All definitions and theorems are implemented in Agda.