Statman's Hierarchy Theorem

TL;DR

Statman's Hierarchy Theorem establishes that the reducibility relation between types in the simply typed lambda calculus forms a well-ordering of order type ω+4, revealing a precise structure of type reducibility.

Contribution

The paper provides a self-contained, syntactic, and simplified proof of the well-ordering of reducibility relations, introducing new relations and analyzing their order types.

Findings

The reducibility relation 0_{eta7} forms a well-ordering of order type 5+4.

Relations 0_h and 0_{h^+} have order types 5+5 and 8 respectively.

Five equivalence classes of 0_{h^+} correspond to canonical term models of Statman.

Abstract

In the Simply Typed $\lambda$-calculus Statman investigates the reducibility relation $\leq_{\beta\eta}$ between types: for $A,B \in \mathbb{T}^0$, types freely generated using $\rightarrow$ and a single ground type $0$, define $A \leq_{\beta\eta} B$ if there exists a $\lambda$-definable injection from the closed terms of type $A$ into those of type $B$. Unexpectedly, the induced partial order is the (linear) well-ordering (of order type) $\omega + 4$. In the proof a finer relation $\leq_{h}$ is used, where the above injection is required to be a B\"ohm transformation, and an (a posteriori) coarser relation $\leq_{h^+}$, requiring a finite family of B\"ohm transformations that is jointly injective. We present this result in a self-contained, syntactic, constructive and simplified manner. En route similar results for $\leq_h$ (order type $\omega + 5$) and $\leq_{h^+}$ (order type $8$) are obtained. Five of the equivalence classes of $\leq_{h^+}$ correspond to canonical term models of Statman, one to the trivial term model collapsing all elements of the same type, and one does not even form a model by the lack of closed terms of many types.