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Formalising formulasastypesasobjects
 Types for Proofs and Programs
, 2000
"... Abstract. We describe a formalisation of the CurryHowardLawvere correspondence between the natural deduction system for minimal logic, the typed lambda calculus and Cartesian closed categories. We formalise the type of natural deduction proof trees as a family of sets Γ ⊢ A indexed by the current ..."
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Abstract. We describe a formalisation of the CurryHowardLawvere correspondence between the natural deduction system for minimal logic, the typed lambda calculus and Cartesian closed categories. We formalise the type of natural deduction proof trees as a family of sets Γ ⊢ A indexed by the current assumption list Γ and the conclusion A and organise numerous useful lemmas about proof trees categorically. We prove categorical properties about proof trees up to (syntactic) identity as well as up to βηconvertibility. We prove that our notion of proof trees is equivalent in an appropriate sense to more traditional representations of lambda terms. The formalisation is carried out in the proof assistant ALF for MartinLöf type theory. 1
Categorical models for explicit substitutions
 Proc. of FoSSaCS'99
, 1999
"... This paper concerns itself with the categorical semantics ofcalculi extended with explicit substitutions. For the simplytypedcalculus, indexed categories seem to provide the right categorical framework but because these structures are inherently nonlinear, alternate models are needed for linear ..."
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This paper concerns itself with the categorical semantics ofcalculi extended with explicit substitutions. For the simplytypedcalculus, indexed categories seem to provide the right categorical framework but because these structures are inherently nonlinear, alternate models are needed for linearcalculi extended with explicit substitutions. We propose to replace indexed categories by presheaves and obtain a semantics which can be specialised to both the linear and the intuitionistic case. The basic models of a calculi of linear (or intuitionistic) explicit substitutions are called linear (or cartesian) contexthandling categories. Then we add extra categorical structure to model the connectives of the logic, obtaining Lcategories as models of the (;I;,)fragment ofintuitionistic linear logic and Ecategories as models of the simply typedcalculus. The!type constructor is then modelled by a monoidal adjunction between an E and Lcategory. Finally, soundness and completeness of our categorical model is proven. 1
V Implementing FormulasasTypesasObjects Implementing FormulasasTypesasObjects
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Directed by Thierry Coquand
, 2004
"... 1 Introduction to categories and categorical models.................................... 1 2 Definition of some related (or not) theories.......................................... 2 2.1 Generalized algebraic theories................................................. 2 ..."
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1 Introduction to categories and categorical models.................................... 1 2 Definition of some related (or not) theories.......................................... 2 2.1 Generalized algebraic theories................................................. 2