Results 1  10
of
20
The type theoretic interpretation of Constructive Set Theory: inductive definitions
 Logic, Methodology and Philosophy of Science VII
, 1986
"... Abstract. We present a generalisation of the typetheoretic interpretation of constructive set theory into MartinLöf type theory. The original interpretation treated logic in MartinLöf type theory via the propositionsastypes interpretation. The generalisation involves replacing MartinLöf type ..."
Abstract

Cited by 148 (9 self)
 Add to MetaCart
(Show Context)
Abstract. We present a generalisation of the typetheoretic interpretation of constructive set theory into MartinLöf type theory. The original interpretation treated logic in MartinLöf type theory via the propositionsastypes interpretation. The generalisation involves replacing MartinLöf type theory with a new type theory in which logic is treated as primitive. The primitive treatment of logic in type theories allows us to study reinterpretations of logic, such as the doublenegation translation. Introduction. The typetheoretic interpretation of Constructive ZermeloFrankel set theory, or CZF for short, provides an explicit link between constructive set theory and MartinLöf type theory A crucial component of the original typetheoretic interpretation of CZF is the propositionsastypes interpretation of logic. Under this interpretation, arbitrary formulas of CZF are interpreted as types, and restricted formulas as small types. By a small type we mean here a type represented by an element of the type universe that is part of the type theory in which CZF is interpreted. The propositionsastypes representation of logic is used in proving the validity of three schemes of CZF, namely Restricted Separation, Strong Collection, and Subset Collection. Validity of Restricted Separation follows from the representation of restricted propositions as small types, while the validity of both Strong Collection and Subset Collection follows from the typetheoretic axiom of choice, that holds in the propositionsastypes interpretation of logic Our first aim here is to present a new typetheoretic interpretation of CZF. The novelty lies in replacing the pure type theory like ML 1 + W with a suitable logicenriched type theory. By a logicenriched intuitionistic type theory we mean an
Propositions as [Types]
, 2001
"... Image factorizations in regular categories are stable under pullbacks, so they model a natural modal operator in dependent type theory. This unary type constructor [A] has turned up previously in a syntactic form as a way of erasing computational content, and formalizing a notion of proof irrelevanc ..."
Abstract

Cited by 37 (3 self)
 Add to MetaCart
(Show Context)
Image factorizations in regular categories are stable under pullbacks, so they model a natural modal operator in dependent type theory. This unary type constructor [A] has turned up previously in a syntactic form as a way of erasing computational content, and formalizing a notion of proof irrelevance. Indeed, semantically, the notion of a support is sometimes used as surrogate proposition asserting inhabitation of an indexed family. We give rules for bracket types in dependent type theory and provide complete semantics using regular categories. We show that dependent type theory with the unit type, strong extensional equality types, strong dependent sums, and bracket types is the internal type theory of regular categories, in the same way that the usual dependent type theory with dependent sums and products is the internal type theory of locally cartesian closed categories. We also show how to interpret rstorder logic in type theory with brackets, and we make use of the translation to compare type theory with logic. Specically, we show that the propositionsastypes interpretation is complete with respect to a certain fragment of intuitionistic rstorder logic. As a consequence, a modied doublenegation translation into type theory (without bracket types) is complete for all of classical rstorder logic.
Weyl’s predicative classical mathematics as a logicenriched type theory
, 2006
"... Abstract. In Das Kontinuum, Weyl showed how a large body of classical mathematics could be developed on a purely predicative foundation. We present a logicenriched type theory that corresponds to Weyl’s foundational system. A large part of the mathematics in Weyl’s book — including Weyl’s definitio ..."
Abstract

Cited by 8 (7 self)
 Add to MetaCart
(Show Context)
Abstract. In Das Kontinuum, Weyl showed how a large body of classical mathematics could be developed on a purely predicative foundation. We present a logicenriched type theory that corresponds to Weyl’s foundational system. A large part of the mathematics in Weyl’s book — including Weyl’s definition of the cardinality of a set and several results from real analysis — has been formalised, using the proof assistant Plastic that implements a logical framework. This case study shows how type theory can be used to represent a nonconstructive foundation for mathematics. Key words: logicenriched type theory, predicativism, formalisation 1
A typetheoretic framework for formal reasoning with different logical foundations
 Proc of the 11th Annual Asian Computing Science Conference
, 2006
"... different logical foundations ..."
Implicit and noncomputational arguments using monads
, 2005
"... Abstract. We provide a monadic view on implicit and noncomputational arguments. This allows us to treat Berger’s noncomputational quantifiers in the Coqsystem. We use Tait’s normalization proof and the concatenation of vectors as case studies for the extraction of programs. With little effort one ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We provide a monadic view on implicit and noncomputational arguments. This allows us to treat Berger’s noncomputational quantifiers in the Coqsystem. We use Tait’s normalization proof and the concatenation of vectors as case studies for the extraction of programs. With little effort one can eliminate noncomputational arguments from extracted programs. One thus obtains extracted code that is not only closer to the intended one, but also decreases both the running time and the memory usage dramatically. We also study the connection between Harrop formulas, lax modal logic and the Coq type theory.
EM + Ext − + ACint is equivalent to ACext
, 2004
"... It is well known that the extensional axiom of choice (ACext) implies the law of excluded middle (EM). We here prove that the converse holds as well if we have the intensional (‘typetheoretical’) axiom of choice ACint, which is provable in MartinLöf’s type theory, and a weak extensionality princip ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
It is well known that the extensional axiom of choice (ACext) implies the law of excluded middle (EM). We here prove that the converse holds as well if we have the intensional (‘typetheoretical’) axiom of choice ACint, which is provable in MartinLöf’s type theory, and a weak extensionality principle (Ext−), which is provable in MartinLöf’s extensional type theory. In particular, EM ⇔ ACext holds in extensional type theory. The following is the principle ACint of intensional choice: if A, B are sets and R a relation such that (∀x: A)(∃y: B)R(x, y) is true, then there is a function f: A → B such that (∀x: A)R(x, f(x)) is true. It is provable in MartinLöf’s type theory [8, p. 50]. It follows from ACint that surjective functions have right inverses: If =B is an equivalence relation on B and f: A → B, we say that f is surjective if (∀y: B)(∃x: A)(y =B f(x)) is true. With R(y, x) def = (y =B f(x)), surjectivity
Categories and Subject Descriptors: F.4.1 [Mathematical Logic and Formal Languages]: Mathematical Logic—Lambda Calculus and Related Systems; Mechanical Theorem Proving General Terms: Theory
"... We construct a logicenriched type theory LTTw that corresponds closely to the predicative system of foundations presented by Hermann Weyl in Das Kontinuum. We formalise many results from that book in LTTw, including Weyl’s definition of the cardinality of a set and several results from real analysi ..."
Abstract
 Add to MetaCart
We construct a logicenriched type theory LTTw that corresponds closely to the predicative system of foundations presented by Hermann Weyl in Das Kontinuum. We formalise many results from that book in LTTw, including Weyl’s definition of the cardinality of a set and several results from real analysis, using the proof assistant Plastic that implements the logical framework LF. This case study shows how type theory can be used to represent a nonconstructive foundation for mathematics.
Vladimir Voevodsky. The following people were the official participants.
"... This book is freely available at ..."
A pluralist approach to the formalisation of
, 2010
"... We present a programme of research for pluralist formalisations, that is, formalisations that involve proving results in more than one foundation. A foundation consists of two parts: a logical part, which provides a notion of inference, and a nonlogical part, which provides the entities to be reaso ..."
Abstract
 Add to MetaCart
(Show Context)
We present a programme of research for pluralist formalisations, that is, formalisations that involve proving results in more than one foundation. A foundation consists of two parts: a logical part, which provides a notion of inference, and a nonlogical part, which provides the entities to be reasoned about. An LTT is a formal system composed of two such separate parts. We show how LTTs may be used as the basis for a pluralist formalisation. We show how different foundations may be formalised as LTTs, and also describe a new method for proof reuse. If we know that a translation Φ exists between two logicenriched type theories (LTTs) S and T, and we have formalised a proof of a theorem α in S, wemay wish to make use of the fact that Φ(α) isatheoremofT. We show how this is sometimes possible by writing a proof script MΦ. For any proof script Mα that proves a theorem α in S, if we change Mα so it first imports MΦ, the resulting proof script will still parse, and will be a proof of Φ(α) inT. In this paper, we focus on the logical part of an LTTframework and show how the above method of proof reuse is done for four cases of Φ: inclusion, the double negation translation, the Atranslation and the Russell–Prawitz modality. This work has been carried out using the proof assistant Plastic. 1.
Classical Predicative LogicEnriched Type Theories ✩
, 906
"... A logicenriched type theory (LTT) is a type theory extended with a primitive mechanism for forming and proving propositions. We construct two LTTs, named LTT0 and LTT ∗ 0, which we claim correspond closely to the classical predicative systems of second order arithmetic ACA0 and ACA. We justify this ..."
Abstract
 Add to MetaCart
A logicenriched type theory (LTT) is a type theory extended with a primitive mechanism for forming and proving propositions. We construct two LTTs, named LTT0 and LTT ∗ 0, which we claim correspond closely to the classical predicative systems of second order arithmetic ACA0 and ACA. We justify this claim by translating each secondorder system into the corresponding LTT, and proving that these translations are conservative. This is part of an ongoing research project to investigate how LTTs may be used to formalise different approaches to the foundations of mathematics. The two LTTs we construct are subsystems of the logicenriched type theory LTTW, which is intended to formalise the classical predicative foundation presented by Herman Weyl in his monograph Das Kontinuum. The system ACA0 has also been claimed to correspond to Weyl’s foundation. By casting ACA0 and ACA as LTTs, we are able to compare them with LTTW. It is a consequence of the work in this paper that LTTW is strictly stronger than ACA0. The conservativity proof makes use of a novel technique for proving one LTT conservative over another, involving defining an interpretation of the stronger system out of the expressions of the weaker. This technique should be applicable in a wide variety of different cases outside the present work. Key words: type theory, logicenriched type theory, predicativism, Hermann Weyl, second order arithmetic