7 research outputs found
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
Inductive and Coinductive Topological Generation with Church's thesis and the Axiom of Choice
In this work we consider an extension MFcind of the Minimalist Foundation MF for predicative constructive mathematics with the addition of inductive and coinductive definitions sufficient to generate Sambin's Positive topologies, namely Martin-Löf-Sambin formal topologies equipped with a Positivity relation (used to describe pointfree formal closed subsets). In particular the intensional level of MFcind, called mTTcind, is defined by extending with coinductive definitions another theory mTTind extending the intensional level mTT of MF with the sole addition of inductive definitions. In previous work we have shown that mTTind is consistent with Formal Church's Thesis CT and the Axiom of Choice AC via an interpretation in Aczel's CZF+REA. Our aim is to show the expectation that the addition of coinductive definitions to mTTind does not increase its consistency strength by reducing the consistency of mTTcind+CT+AC to the consistency of CZF+REA through various interpretations. We actually reach our goal in two ways. One way consists in first interpreting mTTcind+CT+AC in the theory extending CZF with the Union Regular Extension Axiom, REA_U, a strengthening of REA, and the Axiom of Relativized Dependent Choice, RDC. The theory CZF+REA_U+RDC is then interpreted in MLS*, a version of Martin-Löf's type theory with Palmgren's superuniverse S. A last step consists in interpreting MLS* back into CZF+REA. The alternative way consists in first interpreting mTTcind+AC+CT directly in a version of Martin-Löf's type theory with Palmgren's superuniverse extended with CT, which is then interpreted back to CZF+REA. A key benefit of the first way is that the theory CZF+REA_U+RDC also supports the intended set-theoretic interpretation of the extensional level of MFcind. Finally, all the theories considered, except mTTcind+AC+CT, are shown to be of the same proof-theoretic strength
Inductive and Coinductive Topological Generation with Church's thesis and the Axiom of Choice
In this work we consider an extension MFcind of the Minimalist Foundation MF
for predicative constructive mathematics with the addition of inductive and
coinductive definitions sufficient to generate Sambin's Positive topologies,
namely Martin-L\"of-Sambin formal topologies equipped with a Positivity
relation (used to describe pointfree formal closed subsets). In particular the
intensional level of MFcind, called mTTcind, is defined by extending with
coinductive definitions another theory mTTind extending the intensional level
mTT of MF with the sole addition of inductive definitions. In previous work we
have shown that mTTind is consistent with Formal Church's Thesis CT and the
Axiom of Choice AC via an interpretation in Aczel's CZF+REA. Our aim is to show
the expectation that the addition of coinductive definitions to mTTind does not
increase its consistency strength by reducing the consistency of mTTcind+CT+AC
to the consistency of CZF+REA through various interpretations. We actually
reach our goal in two ways. One way consists in first interpreting
mTTcind+CT+AC in the theory extending CZF with the Union Regular Extension
Axiom, REA_U, a strengthening of REA, and the Axiom of Relativized Dependent
Choice, RDC. The theory CZF+REA_U+RDC is then interpreted in MLS*, a version of
Martin-L\"of's type theory with Palmgren's superuniverse S. A last step
consists in interpreting MLS* back into CZF+REA. The alternative way consists
in first interpreting mTTcind+AC+CT directly in a version of Martin-L\"of's
type theory with Palmgren's superuniverse extended with CT, which is then
interpreted back to CZF+REA. A key benefit of the first way is that the theory
CZF+REA_U+RDC also supports the intended set-theoretic interpretation of the
extensional level of MFcind. Finally, all the theories considered, except
mTTcind+AC+CT, are shown to be of the same proof-theoretic strength
The Multiverse: Logical Modularity for Proof Assistants
Proof assistants play a dual role as programming languages and logical systems. As programming languages, proof assistants offer standard modularity mechanisms such as first-class functions, type polymorphism and modules. As logical systems, however, modularity is lacking, and understandably so: incompatible reasoning principles-such as univalence and uniqueness of identity proofs-can indirectly lead to logical inconsistency when used in a given development, even when they appear to be confined to different modules. The lack of logical modularity in proof assistants also hinders the adoption of richer programming constructs, such as effects. We propose the multiverse, a general type-theoretic approach to endow proof assistants with logical modularity. The multiverse consists of multiple universe hierarchies that statically describe the reasoning principles and effects available to define a term at a given type. We identify sufficient conditions for this structuring to modularly ensure that incompatible principles do not interfere, and to locally restrict the power of dependent elimination when necessary. This extensible approach generalizes the ad-hoc treatment of the sort of propositions in the Coq proof assistant. We illustrate the power of the multiverse by describing the inclusion of Coq-style propositions, the strict propositions of Gilbert et al., the exceptional type theory of PĂ©drot and Tabareau, and general axiomatic extensions of the logic
The Multiverse: Logical Modularity for Proof Assistants
Proof assistants play a dual role as programming languages and logical systems. As programming languages, proof assistants offer standard modularity mechanisms such as first-class functions, type polymorphism and modules. As logical systems, however, modularity is lacking, and understandably so: incompatible reasoning principles-such as univalence and uniqueness of identity proofs-can indirectly lead to logical inconsistency when used in a given development, even when they appear to be confined to different modules. The lack of logical modularity in proof assistants also hinders the adoption of richer programming constructs, such as effects. We propose the multiverse, a general type-theoretic approach to endow proof assistants with logical modularity. The multiverse consists of multiple universe hierarchies that statically describe the reasoning principles and effects available to define a term at a given type. We identify sufficient conditions for this structuring to modularly ensure that incompatible principles do not interfere, and to locally restrict the power of dependent elimination when necessary. This extensible approach generalizes the ad-hoc treatment of the sort of propositions in the Coq proof assistant. We illustrate the power of the multiverse by describing the inclusion of Coq-style propositions, the strict propositions of Gilbert et al., the exceptional type theory of PĂ©drot and Tabareau, and general axiomatic extensions of the logic