6,807 research outputs found
Proof-irrelevant model of CC with predicative induction and judgmental equality
We present a set-theoretic, proof-irrelevant model for Calculus of
Constructions (CC) with predicative induction and judgmental equality in
Zermelo-Fraenkel set theory with an axiom for countably many inaccessible
cardinals. We use Aczel's trace encoding which is universally defined for any
function type, regardless of being impredicative. Direct and concrete
interpretations of simultaneous induction and mutually recursive functions are
also provided by extending Dybjer's interpretations on the basis of Aczel's
rule sets. Our model can be regarded as a higher-order generalization of the
truth-table methods. We provide a relatively simple consistency proof of type
theory, which can be used as the basis for a theorem prover
Signatures and Induction Principles for Higher Inductive-Inductive Types
Higher inductive-inductive types (HIITs) generalize inductive types of
dependent type theories in two ways. On the one hand they allow the
simultaneous definition of multiple sorts that can be indexed over each other.
On the other hand they support equality constructors, thus generalizing higher
inductive types of homotopy type theory. Examples that make use of both
features are the Cauchy real numbers and the well-typed syntax of type theory
where conversion rules are given as equality constructors. In this paper we
propose a general definition of HIITs using a small type theory, named the
theory of signatures. A context in this theory encodes a HIIT by listing the
constructors. We also compute notions of induction and recursion for HIITs, by
using variants of syntactic logical relation translations. Building full
categorical semantics and constructing initial algebras is left for future
work. The theory of HIIT signatures was formalised in Agda together with the
syntactic translations. We also provide a Haskell implementation, which takes
signatures as input and outputs translation results as valid Agda code
Coinductive Formal Reasoning in Exact Real Arithmetic
In this article we present a method for formally proving the correctness of
the lazy algorithms for computing homographic and quadratic transformations --
of which field operations are special cases-- on a representation of real
numbers by coinductive streams. The algorithms work on coinductive stream of
M\"{o}bius maps and form the basis of the Edalat--Potts exact real arithmetic.
We use the machinery of the Coq proof assistant for the coinductive types to
present the formalisation. The formalised algorithms are only partially
productive, i.e., they do not output provably infinite streams for all possible
inputs. We show how to deal with this partiality in the presence of syntactic
restrictions posed by the constructive type theory of Coq. Furthermore we show
that the type theoretic techniques that we develop are compatible with the
semantics of the algorithms as continuous maps on real numbers. The resulting
Coq formalisation is available for public download.Comment: 40 page
Three equivalent ordinal notation systems in cubical Agda
We present three ordinal notation systems representing ordinals below epsilon zero in type theory, using recent type-theoretical innovations such as mutual inductive-inductive definitions and higher inductive types. We show how ordinal arithmetic can be developed for these systems, and how they admit a transfinite induction principle. We prove that all three notation systems are equivalent, so that we can transport results between them using the univalence principle. All our constructions have been implemented in cubical Agda
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