# The calculus of constructions and higher order logic

@inproceedings{Geuvers1995TheCO, title={The calculus of constructions and higher order logic}, author={J. H. Geuvers}, year={1995} }

The Calculus of Constructions (CC) ((Coquand 1985]) is a typed lambda calculus for higher order intuitionistic logic: proofs of the higher order logic are interpreted as lambda terms and formulas as types. It is also the union of Girard's system F ! ((Girard 1972]), a higher order typed lambda calculus, and a rst order dependent typed lambda calculus in the style of de Bruijn's Automath ((de Bruijn 1980]) or Martin-LL of's intuitionistic theory of types ((Martin-LL of 1984]). Using the… Expand

#### 18 Citations

Continuation-Passing Style and Strong Normalisation for Intuitionistic Sequent Calculi

- Computer Science, Mathematics
- Log. Methods Comput. Sci.
- 2009

The intuitionistic fragment of the call-by-name version of Curien and Herbelin's \lambda\_mu\_{\~mu}-calculus is isolated and proved strongly normalising by means of an embedding into the simply-typed lambda-Calculus, the first proof of strong normalisation through a reduction-preserving embedding. Expand

Extending Models of Second Order Predicate Logic to Models of Second Dependent Type Theory

- Mathematics, Computer Science
- CSL
- 1996

Here it is proved that for all formulas ϕ, ϕ is true in M if and only ifπ is inhabited in S(M), and this equivalence is proved for models M that are full models of classical second order predicate logic. Expand

Some Axioms for Mathematics

- Computer Science
- FSCD
- 2021

This paper presents a theory, the theory U, where the proofs of several logical systems can be expressed: Minimal, Constructive, and Ecumenical predicate logic, and it is proved that, when a proof in U uses only 19 symbols of a sub-theory, then it is aProof in that sub- theory. Expand

Continuation-Passing Style and Strong Normalisation for Intuitionistic Sequent Calculi

- Mathematics, Computer Science
- TLCA
- 2007

The intuitionistic fragment of the call-by-name version of Curien and Herbelin's ???µ???-calculus is isolated and proved strongly normalising by means of an embedding into the simply-typed λ-Calculus, the first proof of strong normalisation through a reduction-preserving embedding. Expand

Extending Models of Second Order Predicate Logic to Models of Second Order Dependent Type Theory

- 2007

1 I n t r o d u c t i o n We describe a method for constructing a model of second order dependent type theory (AP2) out of a model of classical second order predicate logic (CPRED2). We show tha t… Expand

Formally Proving the Correctness of Functional Programs

- Computer Science
- 2010

Methods for proving functional programs correct in the proof assistant Coq, a small core dependently typed functional programming language with a proof assistant for specifying properties of these programs and proving them, are investigated and compared. Expand

Expressing theories in the λΠ-calculus modulo theory and in the Dedukti system

- Mathematics
- 2016

Defining a theory, such as arithmetic, geometry, or set theory, in predicate logic just requires to chose function and predicate symbols and axioms, that express the meaning of these symbols. Using,… Expand

Dedukti : a Logical Framework based on the λ Π-Calculus Modulo Theory

- 2016

Dedukti is a Logical Framework based on the λΠ-Calculus Modulo Theory. We show that many theories can be expressed in Dedukti: constructive and classical predicate logic, Simple type theory,… Expand

Sequent Style Proof Terms for HOL

- 2005

In this work we present proof terms for a Gentzen sequent style presentation of HOL. Existing implementations of proof terms for HOL are natural deduction style systems. Sequent style proof terms… Expand

Enhancing the expressivity and automation of an interactive theorem prover in order to verify multicast protocols

- Computer Science
- 2006

This thesis was motivated by a case study involving the formalisation of arguments that simplify the verification of tree-oriented multicast protocols and developed a novel system of proof terms for the HOL logic that is described in this thesis. Expand

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