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The Implicit Calculus of Constructions
In this paper, we introduce a new type system, the Implicit Calculus of Constructions, which is a Curry-style variant of the Calculus of Constructions that we extend by adding an intersection typeExpand
The Implicit Calculus of Constructions Extending Pure Type Systems with an Intersection Type Binder and Subtyping
In this paper, we introduce a new type system, the Implicit Calculus of Constructions, which is a Curry-style variant of the Calculus of Constructions that we extend by adding an intersection typeExpand
Forcing as a Program Transformation
  • Alexandre Miquel
  • Mathematics, Computer Science
  • IEEE 26th Annual Symposium on Logic in Computer…
  • 21 June 2011
TLDR
It is shown how to avoid the cost of the transformation by introducing an extension of Krivine's abstract machine devoted to the execution of proofs constructed by forcing, which induces new classical realizability models and present the corresponding adequacy results. Expand
The Not So Simple Proof-Irrelevant Model of CC
It is well-known that the Calculus of Constructions (CC) bears a simple set-theoretical model in which proof-terms are mapped onto a single object--a property which is known as proof-irrelevance. InExpand
A model for impredicative type systems, universes, intersection types and subtyping
  • Alexandre Miquel
  • Mathematics, Computer Science
  • Proceedings Fifteenth Annual IEEE Symposium on…
  • 26 June 2000
TLDR
It is shown that uncountable types such as the type of real numbers or Zermelo-Frankel sets can safely be axiomatized on the impredicative level of, say, ECC, without harm for consistency. Expand
The λ-calculus with constructors: Syntax, confluence and separation
Abstract We present an extension of the λ(η)-calculus with a case construct that propagates through functions like a head linear substitution, and show that this construction permits to recover theExpand
Ordered combinatory algebras and realizability †
TLDR
It is shown that $\mathcal{^KOCA}$ 's are equivalent to Streicher's abstract Krivine structures for the purpose of modeling higher-order logic, in the precise sense that they give rise to the same class of triposes. Expand
Implicative algebras: a new foundation for realizability and forcing
  • Alexandre Miquel
  • Mathematics, Computer Science
  • Mathematical Structures in Computer Science
  • 2 February 2018
TLDR
It is shown that each implicative algebra induces a (Set-based) tripos, using a construction that is reminiscent from the construction of a realizability tripos from a partial combinatory algebra. Expand
Realizability in the Unitary Sphere
TLDR
This paper derives from the semantics a set of typing rules for a simply-typed linear algebraic lambda-calculus, and shows how it extends both to classical and quantumlambda-calculi. Expand
Existential witness extraction in classical realizability and via a negative translation
  • Alexandre Miquel
  • Computer Science, Mathematics
  • Log. Methods Comput. Sci.
  • 23 January 2011
TLDR
It is shown that in the Sigma^0_1-case, Krivine's witness extraction method reduces to Friedman's through a well-suited negative translation to intuitionistic second-order arithmetic. Expand
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