1. What are the contributions mentioned in the paper "Eliminating dependent pattern matching without k" ?
In this paper, the authors propose a new criterion for dependent pattern matching without K, and prove it correct by a translation to eliminators in the style of Goguen, McBride, and McKinna [ 2006 ].. Thus it brings the benefits of dependent pattern matching to contexts where the authors can not assume K, such as HoTT.
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2. What are the future works mentioned in the paper "Eliminating dependent pattern matching without k" ?
In the future the authors would like to adapt the work in this paper to this setting, so their criterion would become usable in languages based on cubical type theory as well.. Future research will have to show how much of the original pattern matching algorithm can be salvaged in this setting.. The authors will give an example to illustrate the problem.. The authors can give a proof 𝑝 of ( 𝑥 ∶ ℕ ) → 0+𝑥≡𝑥+0, so they have ext 𝑝 ∶ 𝜆𝑥.
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![Figure 8. The unification transitions represented as type-theoretic terms. Compared to the transitions given by Goguen et al. [2006], these work with the homogeneous equality and Φ has an additional dependence on the equality proof. While these unification transitions are the most general ones we can construct, they are not the ones that we use for case splitting in practice. Rather, injectivity, conflict, and cycle are replaced by their more specialized variants injectivity' (61), conflict' (62), and cycle' (63).](/figures/figure-8-the-unification-transitions-represented-as-type-2wgrku93.webp)
![Figure 3. The structural order ≺ is used to check termination [Goguen et al., 2006].](/figures/figure-3-the-structural-order-is-used-to-check-termination-2hp28quw.webp)
