Eliminating reflection from type theory
Théo Winterhalter,Matthieu Sozeau,Nicolas Tabareau +2 more
- pp 91-103
Reads0
Chats0
TLDR
The first effective syntactical translation from ETT to ITT is provided with uniqueness of identity proofs and functional extensionality and is defined and proven correct in Coq and yields an executable plugin that translates a derivation in ETT into an actual Coq typing judgment.Abstract:
Type theories with equality reflection, such as extensional type theory (ETT), are convenient theories in which to formalise mathematics, as they make it possible to consider provably equal terms as convertible. Although type-checking is undecidable in this context, variants of ETT have been implemented, for example in NuPRL and more recently in Andromeda. The actual objects that can be checked are not proof-terms, but derivations of proof-terms. This suggests that any derivation of ETT can be translated into a typecheckable proof term of intensional type theory (ITT). However, this result, investigated categorically by Hofmann in 1995, and 10 years later more syntactically by Oury, has never given rise to an effective translation. In this paper, we provide the first effective syntactical translation from ETT to ITT with uniqueness of identity proofs and functional extensionality. This translation has been defined and proven correct in Coq and yields an executable plugin that translates a derivation in ETT into an actual Coq typing judgment. Additionally, we show how this result is extended in the context of homotopy type theory to a two-level type theory.read more
Citations
More filters
Journal ArticleDOI
The MetaCoq Project
Matthieu Sozeau,Abhishek Anand,Simon Boulier,Cyril Cohen,Yannick Forster,Fabian Kunze,Gregory Malecha,Nicolas Tabareau,Théo Winterhalter +8 more
TL;DR: This work generalizes Template-Coq to handle the entire polymorphic calculus of cumulative inductive constructions, as implemented by Coq, including the kernel’s declaration structures for definitions and inductives, and implement a monad for general manipulation of Coq ’s logical environment.
Proceedings ArticleDOI
Signatures and Induction Principles for Higher Inductive-Inductive Types
Ambrus Kaposi,András Kovács +1 more
TL;DR: In this article, the authors propose a general definition of higher inductive-inductive types (HIITs) using a small type theory, named the theory of signatures, which allows the simultaneous definition of multiple sorts that can be indexed over each other.
Posted Content
Gradualizing the Calculus of Inductive Constructions
TL;DR: A crucial trade-off is observed between graduality and the key properties of normalization and closure of universes under dependent product that CIC enjoys, which informs and paves the way towards the development of malleable proof assistants and dependently-typed programming languages.
Journal ArticleDOI
Gradualizing the Calculus of Inductive Constructions
TL;DR: In this paper , gradual variations on the Calculus of Inductive Construction (CIC) have been investigated for swifter prototyping with imprecise types and terms, and a parametrized presentation of Gradual CIC (GCIC) has been developed, which encompasses all three variations and develops their metatheory.
Book ChapterDOI
Shallow Embedding of Type Theory is Morally Correct
TL;DR: The authors consider the standard model of a type theoretic object theory in Agda and prove that shallow embedding is injective up to definitional equality, by modelling the embedding as a syntactic translation targeting the metatheory.
References
More filters
Book
Interactive Theorem Proving and Program Development
Yves Bertot,Pierre Castéran +1 more
TL;DR: The similarity between Fixpoint and fix makes it easier to understand the need for the various parts of this construct, and the construction of higher-order types and simple inductive types defined inside a section is helpful to understanding the form of the induction principle.
Dissertation
Towards a practical programming language based on dependent type theory
TL;DR: This thesis is concerned with bridging the gap between the theoretical presentations of type theory and the requirements on a practical programming language.
Proceedings ArticleDOI
A model of type theory in cubical sets
TL;DR: A model of type theory with dependent product, sum, and identity, in cubical sets is presented, and is a step towards a computational interpretation of Voevodsky's Univalence Axiom.
Dissertation
Dependently Typed Functional Programs and their Proofs
TL;DR: This thesis shows that the adoption of this uniqueness as axiomatic is sufficient to make pattern matching admissible, and develops technology for programming with dependent inductive families of datatypes and proving those programs correct.
Posted Content
The Simplicial Model of Univalent Foundations (after Voevodsky)
TL;DR: Voevodsky as discussed by the authors constructed a model of univalent type theory in the category of simplicial sets and showed that it is at least as consistent as ZFC with two inaccessible cardinals.
Related Papers (5)
Correct-by-construction model transformations from partially ordered specifications in Coq
Iman Poernomo,Jeffrey Terrell +1 more