Results 1  10
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19
Explicit substitutions
, 1996
"... The λσcalculus is a refinement of the λcalculus where substitutions are manipulated explicitly. The λσcalculus provides a setting for studying the theory of substitutions, with pleasant mathematical properties. It is also a useful bridge between the classical λcalculus and concrete implementatio ..."
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Cited by 410 (11 self)
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The λσcalculus is a refinement of the λcalculus where substitutions are manipulated explicitly. The λσcalculus provides a setting for studying the theory of substitutions, with pleasant mathematical properties. It is also a useful bridge between the classical λcalculus and concrete implementations.
On the existence of StoneČech compactification
, 2009
"... Introduction. In 1937 E. Čech and M.H. Stone independently introduced the maximal compactification of a completely regular topological space, thereafter called StoneČech compactification [8, 18]. In the introduction of [8] the nonconstructive character of this result is so described: “it must be e ..."
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Cited by 2 (1 self)
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Introduction. In 1937 E. Čech and M.H. Stone independently introduced the maximal compactification of a completely regular topological space, thereafter called StoneČech compactification [8, 18]. In the introduction of [8] the nonconstructive character of this result is so described: “it must be emphasized that β(S) [the StoneČech compactification of S] may be
A minimalist twolevel foundation for constructive mathematics
, 811
"... We present a twolevel theory to formalize constructive mathematics as advocated in a previous paper with G. Sambin [MS05]. One level is given by an intensional type theory, called Minimal type theory. This theory extends the settheoretic version introduced in [MS05] with collections. The other lev ..."
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Cited by 2 (1 self)
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We present a twolevel theory to formalize constructive mathematics as advocated in a previous paper with G. Sambin [MS05]. One level is given by an intensional type theory, called Minimal type theory. This theory extends the settheoretic version introduced in [MS05] with collections. The other level is given by an extensional set theory that is interpreted in the first one by means of a quotient model. This twolevel theory has two main features: it is minimal among the most relevant foundations for constructive mathematics; it is constructive thanks to the way the extensional level is linked to the intensional one which fulfills the “proofsasprograms ” paradigm and acts as a programming language.
Decidability in Intuitionistic Type Theory is functionally decidable
 Mathematical Logic Quarterly 42
, 1996
"... In this paper we show that the usual intuitionistic characterization of the decidability of the propositional function B(x) prop [x: A], i.e. to require that the predicate (∀x ∈ A) B(x)∨¬B(x) is provable, is equivalent, when working within the framework of MartinLöf’s Intuitionistic Type Theory, to ..."
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Cited by 2 (1 self)
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In this paper we show that the usual intuitionistic characterization of the decidability of the propositional function B(x) prop [x: A], i.e. to require that the predicate (∀x ∈ A) B(x)∨¬B(x) is provable, is equivalent, when working within the framework of MartinLöf’s Intuitionistic Type Theory, to require that there exists a decision function φ: A → Boole such that (∀x ∈ A) (φ(x) =Boole true) ↔ B(x). Since we will also show that the proposition x =Boole true [x: Boole] is decidable, we can alternatively say that the main result of this paper is a proof that the decidability of the predicate B(x) prop [x: A] can be effectively reduced by a function φ ∈ A → Boole to the decidability of the predicate φ(x) =Boole true [x: A]. All the proofs are carried out within the Intuitionistic Type Theory and hence the decision function φ, together with a proof of its correctness, is effectively constructed as a function of the proof of (∀x ∈ A) B(x)∨¬B(x). 1 The basic lemmas
General recursion and formal topology
 In Proceedings Workshop on Partiality and Recursion in Interactive Theorem Provers, volume 43 of EPTCS
"... It is well known that general recursion cannot be expressed within MartinLöf’s type theory and various approaches have been proposed to overcome this problem still maintaining the termination of the computation of the typable terms. In this work we propose a new approach to this problem based on th ..."
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It is well known that general recursion cannot be expressed within MartinLöf’s type theory and various approaches have been proposed to overcome this problem still maintaining the termination of the computation of the typable terms. In this work we propose a new approach to this problem based on the use of inductively generated formal topologies. 1
Quotients over Minimal Type Theory
 In Computation and Logic in the Real World CiE 2007, Siena, volume 4497 of LNCS
, 2007
"... Abstract. We consider an extensional version, called qmTT, of the intensional Minimal Type Theory mTT, introduced in a previous paper with G. Sambin, enriched with proofirrelevance of propositions and effective quotient sets. Then, by using the construction of total setoid a ̀ la Bishop we build ..."
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Abstract. We consider an extensional version, called qmTT, of the intensional Minimal Type Theory mTT, introduced in a previous paper with G. Sambin, enriched with proofirrelevance of propositions and effective quotient sets. Then, by using the construction of total setoid a ̀ la Bishop we build a model of qmTT over mTT. The design of an extensional type theory with quotients and its interpretation in mTT is a key technical step in order to build a two level system to serve as a minimal foundation for constructive mathematics as advocated in the mentioned paper about mTT.