12 research outputs found
Quotient completion for the foundation of constructive mathematics
We apply some tools developed in categorical logic to give an abstract
description of constructions used to formalize constructive mathematics in
foundations based on intensional type theory. The key concept we employ is that
of a Lawvere hyperdoctrine for which we describe a notion of quotient
completion. That notion includes the exact completion on a category with weak
finite limits as an instance as well as examples from type theory that fall
apart from this.Comment: 32 page
On choice rules in dependent type theory
In a dependent type theory satisfying the propositions as
types correspondence together with the proofs-as-programs paradigm,
the validity of the unique choice rule or even more of the choice rule says
that the extraction of a computable witness from an existential statement
under hypothesis can be performed within the same theory.
Here we show that the unique choice rule, and hence the choice rule,
are not valid both in Coquand\u2019s Calculus of Constructions with indexed
sum types, list types and binary disjoint sums and in its predicative
version implemented in the intensional level of the Minimalist Founda-
tion. This means that in these theories the extraction of computational
witnesses from existential statements must be performed in a more ex-
pressive proofs-as-programs theory
W-types in setoids
W-types and their categorical analogue, initial algebras for polynomial
endofunctors, are an important tool in predicative systems to replace
transfinite recursion on well-orderings. Current arguments to obtain W-types in
quotient completions rely on assumptions, like Uniqueness of Identity Proofs,
or on constructions that involve recursion into a universe, that limit their
applicability to a specific setting. We present an argument, verified in Coq,
that instead uses dependent W-types in the underlying type theory to construct
W-types in the setoid model. The immediate advantage is to have a proof more
type-theoretic in flavour, which directly uses recursion on the underlying
W-type to prove initiality. Furthermore, taking place in intensional type
theory and not requiring any recursion into a universe, it may be generalised
to various categorical quotient completions, with the aim of finding a uniform
construction of extensional W-types.Comment: 17 pages, formalised in Coq; v2: added reference to formalisatio
A realizability semantics for inductive formal topologies, Church's Thesis and Axiom of Choice
We present a Kleene realizability semantics for the intensional level of the
Minimalist Foundation, for short mtt, extended with inductively generated
formal topologies, Church's thesis and axiom of choice. This semantics is an
extension of the one used to show consistency of the intensional level of the
Minimalist Foundation with the axiom of choice and formal Church's thesis in
previous work. A main novelty here is that such a semantics is formalized in a
constructive theory represented by Aczel's constructive set theory CZF extended
with the regular extension axiom
Inductive and Coinductive Topological Generation with Church's thesis and the Axiom of Choice
Here 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,
i.e. 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.Comment: arXiv admin note: text overlap with arXiv:1905.1196
A conservativity result for homotopy elementary types in dependent type theory
We prove a conservativity result for extensional type theories over
propositional ones, i.e. dependent type theories with propositional computation
rules, using insights from homotopy type theory. The argument exploits a notion
of canonical homotopy equivalence between contexts, and uses the notion of a
type-category to phrase the semantics of theories of dependent types.
Informally, our main result asserts that, for judgements essentially concerning
h-sets, reasoning with extensional or propositional type theories is
equivalent.Comment: 67 pages, comments welcom