• DOI:10.23638/LMCS-13(4:19)2017
• Corpus ID: 263738

Statman's Hierarchy Theorem

@article{Westerbaan2017StatmansHT,
  title={Statman's Hierarchy Theorem},
  author={Bram Westerbaan and Bas Westerbaan and Rutger Kuyper and Carst Tankink and Remy Viehoff and HENK P. Barendregt},
  journal={ArXiv},
  year={2017},
  volume={abs/1711.05497}
}

• Bram Westerbaan, B. Westerbaan, +3 authors H. Barendregt
• Published 15 November 2017
• Mathematics
• ArXiv

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.