Constructive Membership and Indexes in Trees

Trees carrying information stored in their nodes are a fundamental abstract data type. Approaching trees in a formal constructive environment allows us to realize properties of trees, inherent in their structure. Specifically we will look at the evidence provided by the predicates which operate on these trees. This evidence, expressed in terms of logical and programming languages, is realizable only in a constructive context. In the constructive setting, membership predicates over recursive types are inhabited by terms indexing the elements that satisfy the criteria for membership. In this paper, we motivate and explore this idea in the concrete setting of lists and trees. We first provide a background in constructive type theory and show relavent properties of trees. We present and define the concept of inhabitants of a generic shape type that corresponds naturally and exactly to the inhabitants of a membership predicate. In this context, (λx.True) ∈ S is the set of all indexes into S, but we show that not all subsets of indexes are expressible by strictly local predicates. Accordingly, we extend our membership predicates to predicates that compute and hold the state “from above” as well as allow “looking below”. The modified predicates of this form are complete in the sense that they can express every subset of indexes in S. These ideas are motivated by experience programming in Nuprl’s constructive type theory and the theorems for lists and trees have been formalized and mechanically checked.