Shallow Embedding of Type Theory is Morally Correct
| Autor: | Nicolai Kraus, András Kovács, Ambrus Kaposi |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
050101 languages & linguistics
Computer science Agda Object theory 05 social sciences Proof assistant 02 engineering and technology Mathematical proof TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES Type theory TheoryofComputation_LOGICSANDMEANINGSOFPROGRAMS Metatheory 0202 electrical engineering electronic engineering information engineering Calculus Embedding 020201 artificial intelligence & image processing 0501 psychology and cognitive sciences Parametricity computer computer.programming_language |
| Zdroj: | Lecture Notes in Computer Science ISBN: 9783030336356 MPC |
| DOI: | 10.1007/978-3-030-33636-3_12 |
| Popis: | There are multiple ways to formalise the metatheory of type theory. For some purposes, it is enough to consider specific models of a type theory, but sometimes it is necessary to refer to the syntax, for example in proofs of canonicity and normalisation. One option is to embed the syntax deeply, by using inductive definitions in a proof assistant. However, in this case the handling of definitional equalities becomes technically challenging. Alternatively, we can reuse conversion checking in the metatheory by shallowly embedding the object theory. In this paper, we consider the standard model of a type theoretic object theory in Agda. This model has the property that all of its equalities hold definitionally, and we can use it as a shallow embedding by building expressions from the components of this model. However, if we are to reason soundly about the syntax with this setup, we must ensure that distinguishable syntactic constructs do not become provably equal when shallowly embedded. First, we prove that shallow embedding is injective up to definitional equality, by modelling the embedding as a syntactic translation targeting the metatheory. Second, we use an implementation hiding trick to disallow illegal propositional equality proofs and constructions which do not come from the syntax. We showcase our technique with very short formalisations of canonicity and parametricity for Martin-Lof type theory. Our technique only requires features which are available in all major proof assistants based on dependent type theory. |
| Databáze: | OpenAIRE |
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