Propositional equality for gradual dependently typed programming
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
Gradual dependent types can help with the incremental adoption of dependently typed code by providing a principled semantics for imprecise types and proofs, where some parts have been omitted. Current theories of gradual dependent types, though, lack a central feature of type theory: propositional equality. Lennon-Bertrand et al. show that, when the reflexive proof refl is the only closed value of an equality type, a gradual extension of the Calculus of Inductive Constructions (CIC) with propositional equality violates static observational equivalences. Extensionally-equal functions should be indistinguishable at run time, but they can be distinguished using a combination of equality and type imprecision. This work presents a gradual dependently typed language that supports propositional equality. We avoid the above issues by devising an equality type of which refl is not the only closed inhabitant. Instead, each equality proof is accompanied by a term that is at least as precise as the equated terms, acting as a witness of their plausible equality. These witnesses track partial type information as a program runs, raising errors when that information shows that two equated terms are undeniably inconsistent. Composition of type information is internalized as a construct of the language, and is deferred for function bodies whose evaluation is blocked by variables. We thus ensure that extensionally-equal functions compose without error, thereby preventing contexts from distinguishing them. We describe the challenges of designing consistency and precision relations for this system, along with solutions to these challenges. Finally, we prove important metatheory: type safety, conservative embedding of CIC, weak canonicity, and the gradual guarantees of Siek et al., which ensure that reducing a program’s precision introduces no new static or dynamic errors.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.004 | 0.002 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it