# Représentation d’algèbres non libres en théorie des types

Abstract : This thesis has two parts: one is the presentation of an extension of the calculus of inductive constructions($\mathcal{CIC}$) of Christine Paulin and Thierry Coquand, allowing the definition of non free types, the second is the development of a formal proof, in the proof assistant \coq, of self-stabilization of an distributed algorithm. The first theme is developed in part II and the second in part III, the part I is a general presentation of $\mathcal{CIC}$, which is the central subject of this work since it is also the underlying formalism of \coq. Our extension of $\mathcal{CIC}$ is mainly based on a new notion: \emph{normalized types}. A normalized type is built with a type $A$ and a function \nf\ on this type. Intuitively the normalized, noted $\norm(A,\nf)$ represents the quotient of type $A$ by the relation $R_\nf$ defined by $(R_\nf \ x\ y) \Leftrightarrow (\nf\ x)=(\nf\ y)$. We show that this equality can be included in the conversion without losing properties of $\mathcal{CIC}$: logical consistency, decidability of type checking etc.
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Cited literature [268 references]

https://hal.archives-ouvertes.fr/tel-02346101
Contributor : Pierre Courtieu <>
Submitted on : Friday, January 31, 2020 - 2:01:14 PM
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• HAL Id : tel-02346101, version 1

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Pierre Courtieu. Représentation d’algèbres non libres en théorie des types. Logique en informatique [cs.LO]. Université Paris 11, 2001. Français. ⟨tel-02346101⟩

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